metadata:
dcterms:identifier ECHO:SH99QARK.xml
dcterms:creator (GND:122095553) Waring, Edward
dcterms:title (la) Meditationes analyticae
dcterms:date 1785
dcterms:language lat
text (la) free

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<h>MEDITATIONES ANALYTICÆ,
A B
EDVARDO WARING,
REGIÆ SOCIETATIS ET ACADEMIÆ BONONIENSIS INSTITUTI SOCIO,
ET MATHESEOS APUD CANTAB. PROFESSORE LUCASIANO.</h>
<h>PRIMA EDITIO IMPRIMEBATUR ANNIS 1773, 4 & 5: SECUNDA CUM NONNULLIS
ADDITIONIBUS ANNIS 1783, 4 & 5, ET EDITA JUN. 4, 1785.</h>
<h>CANTABRIGIÆ,
T<sc>YPIS</sc> A<sc>CADEMICIS EXCUDEBAT</sc> J. A<sc>RCHDEACON</sc>;
Veneunt apud J. N<sc>ICHOLSON</sc>, Cantabrigiæ J. R<sc>IVINGTON</sc> & Filios, Londiai; et
J. & J. F<sc>LETCHER</sc>, et D. P<sc>RINCE</sc> & Co. Oxon.</h>
<h>MDCCLXXXV.</h>

<pb>[0008]

<pb>[0009]
<h>TO
S<sc>IR</sc> JOSEPH BANKS, B<sc>ART</sc>. P<sc>RESIDENT OF THE</sc> R<sc>oyal</sc> S<sc>OCIETY</sc>;</h>
<p>Who by his Voyages has greatly contributed to the Increa$e of botanical and natural
Knowledge; and on all Occa$ions has proved him$elf a zealous Patron and active Promoter
of every Science.</p>
<h>TO THE
R<sc>EV</sc>. NEVIL MASKELYNE, D.D.F.R.S. <sc>AND</sc> A<sc>STRONOMER</sc> R<sc>OYAL</sc>;</h>
<p>Di$tingui$hed for his Knowledge in Optics and phy$ical A$tronomy; who after proving
by his Ob$ervations in two Voyages the Certainty of the lunar Method in finding the Lon-
gitude at Sea, recommended it to the Board of Longitude; and $trenuou$ly affi$ted them in
bringing it into Practice.</p>
<h>AND TO
JOHN WILSON, E<sc>sq</sc>. F.R.S. <sc>AND</sc> K<sc>ING'S</sc> C<sc>OUNCIL IN THE</sc> L<sc>AW</sc>;</h>
<p>Who was in early Life my Defender, and during a Period of twenty-five Years has been
my Friend and Coun$ellor; from whom in mathematical Enquiries I have received greater
Affi$tance than from any other Per$on, and than whom I know no one po$$e$$ed of more
Knowledge or Acutene$s in the$e Studies.</p>
<h>THE FOLLOWING WORK</h>
<h>IS WITH THE GREATEST ESTEEM INSCRIBED.</h>
<h>BY THEIR VERY HUMBLE SERVANT.</h>
<h>E. WARING.</h>

<pb>[0010]

<pb>[0011]
<h>PRÆFATIO</h>
<p>CUM langue$cere jam apud no$trates videantur $tudia mathema-
tica, & juventutem academicam ab ii$dem dehortari non ce$$ent
plerique, mirum $ane erit me tempus & pecuniam in con$cribendis
operibus mathematicis & imprimendis impendere, nec labori illau-
dato parcere; argumenta quibus me tuear $ollicite non quæro, hoc
con$olans, me officii causâ hæc $crip$i$$e. Munus quod nactus e$$em
profe$$orium erat ornandum, & in$tituti ratio po$tulabat, ut mathe-
$eos fines pro virili latius proferrem, eju$que partes, quantum in me
e$$et, defenderem: rem interea lectori mathematico non ingratam me
facturum e$$e arbitror; $i hìc præfationis loco, di$qui$itionum, quæ in
hoc libro contineantur, brevem $ubnectam hi$toriam ortûs & pro-
gre$sûs.</p>
<p>Parabolæ quadraturam & approximationes ad areas curvarum per
polygona curvis in$cripta & circum$cripta invenerunt antiqui, $ed
notatione algebraicâ egentes haud multa præclara in hâc mathema-
tum parte efficere potuerunt.</p>
<p>_Archimedes_ invenit limites inter quæcunque parallelogramma in-
$cripta & circum$cripta curvis, etiamque inter in$criptos & circum-
$criptos cylindros $olidis a rotatione curvarum circa axes $uos gene-
ratis: rationes horum limitum a _Walli$io_ & _Neutono_ dicuntur ultimæ
rationes.</p>
<p>_Keplerus_ fingit curvam ex infinitis punctis con$i$tere, & ejus ordi-
natam maximam vel minimam evadere, cum per gradus in$en$ibiles
varietur; _Cavallerius_ lineas in indivi$ibiles partes divi$it; & exinde
deduxit aream curvæ, cujus æquatio e$t <math>y = a x^n</math>, ubi <math>n</math> e$t integer
numerus.</p>
<p>_Carte$ius_ invenit methodum ducendi tangentes ad curvas, circulis
ductis, qui curvam tangerent.</p>
<p>_Huddenius_ dedit regulam, ex quâ dici po$$it, utrum duæ radices
datæ æquationis $int inter $e æquales, necne; & exinde deduxit me-
thodum de maximis & minimis inve$tigandis rationalium quantita-
tum algebraicarum.</p>

<pb ii>[0012]<rh>PRÆFATIO.</rh>
<p>_Walli$ius, Fermatius, Barrovius, Slu$ius,_ aliique invenere proportio-
nalia incrementa ab$ci$$æ & ordinatæ ex principiis quæ nunc u$itata
$unt in methodo incrementorum vel fluxionum; hæc vero principia
ad irrationales quantitates nunquam ab iis applicata fuere, $emper
enim ita reducebant æquationem, ut in eâ nulli irrationales termini
contineantur: _Walli$ius_ ex interpolatione invenit aream curvæ, cujus
æquatio $it <math>y = a x^{n / m}</math>, animadvertit etiam rectangulum ex ordinatâ in
fluxionem ab$ci$$æ fluxioni areæ curvæ e$$e æquale.</p>
<p>Inter _Joannem Bernoulli, Keilium, Leibnitzium & Neutonum_ agitata
fuit quæ$tio, utrum _Neutonus_ an _Leibnitzius_ primus invenerit fluxio-
nem, vel, quod idem e$t, incrementum irrationalis quantitatis ab$que
ejus irrationalitatis exterminatione. _Neutonus_ in epi$tolâ ad _Collin-_
_$ium_ datâ Dec. 10, 1672, primum quod $cio, profitetur $e huju$ce pro-
blematis $olutionem cognovi$$e; in epi$tolâ ad _Oldenburgum_ cum _Leib-_
_nitzio_ communicandâ idem profitetur, $ed haud con$tat illum ejus me-
thodum alicui communica$$e. Ex epi$tolis _Leibnitzii_ dilucide con$tat
illum ejus $olutionem prædictis temporibus haud detexi$$e; in epi$tolâ
ad _Oldenburgum_ 21 Junii, 1677, $olutionem prædicti problematis pri-
mum dedit _Leibnitzius._ Ex his manife$to con$tat _Neutonum_ primum
detexi$$e methodum inveniendi fluxionem irrationalis quantitatis;
confitendum e$t tamen nullum te$timonium extare, ex quo credibile
$it _Leibnitzium_ eandem proprio marte haud detexi$$e; nec aliquid in
hoc mirabile videtur; res quidem in eo erant, ut ulterior vix ullus in
hâc mathe$eos parte daretur progre$$us $ine huju$ce problematis re$o-
lutione; nec fuit inve$tigatio difficilis, quæ huju$modi e$t: $it irratio-
nalis quantitas, cujus incrementum requiratur, <math>(a + b x + c x^2 +
&c.)^m</math>; $cribatur <math>v</math> pro <math>a + b x + c x^2 + &c.</math>, & re$ultat <math>(a + b x +
c x^2 + &c.)^m = v^m</math>; $ed incrementum quantitatis <math>v^m</math> prius inventum
fuit <math>m v^m-1</math> ductum in incrementum quantitatis <math>v</math>, at <math>v = a + b x +
c x^2 + &c.</math> cujus incrementum prius datum fuit <math>b + 2 c x + &c.</math> in
incrementum quantitatis <math>x</math>; in quantitate <math>m v^m-1 ×</math> incre. quantitatis
<math>v</math> pro <math>v</math> & ejus incremento $cribantur eorundem prædicti valores, &
confit problema: nemo pote$t e$$e te$tis in $uas partes, is mihi $em-

<pb iii>[0013]<rh>PRÆFATIO.</rh>
per dicendus e$t inventor, qui primus evulgaverit, vel $altem cum
amicis communicaverit; vix enim inveniatur aliquis dignus mathe-
matici nomine, qui de $uo ingenio multa a prioribus $criptoribus
reperta ip$e haud detexerit.</p>
<p>Fluxiones $ecundi, &c. ordinis ex primis per eundem modum
ac primas ex earum fluentibus inveniri po$$e docuere _Neutonus,_ alii-
que; ille primus dedit fluentem fluxionis <math>(a + b x^n)^m x^θ-1 x^.</math>: deinde
_Cragius_ animadvertendo eandem irrationalitatem in fluente & flu-
xione contentam e$$e, ex a$$umptis generalibus terminis pro coeffi-
cientibus invenit fluentem fluxionis <math>(a + b x^n + c x^2n + &c.)^m x^θ-1 x^.</math>:
fluxicnem ad alteram formulam reducendam e$$e, $i modo plures
ejus factores $int inter $e æquales primus ob$ervabat _foannes Ber-_
_noulli_; deinde _Neutonus_ inve$tigavit fluentem fluxionis <math>(a + b x^n +
c x^2n + &c.)^l × (e + f x^n + g x^2n + &c.)^m x^θ x^.</math>; & animadvertit, $i plures
factores datæ fluxionis habeant inter $e communem divi$orem, redu-
cendam e$$e datam fluxionem ad diver$am formulam; quod$i ordi-
nata $it fractio rationalis irreducibilis cum denominatore ex duobus
vel pluribus terminis compo$ito, re$olvendum e$$e denominatorem ad
diver$os $uos omnes primos; & $i divi$or $it aliquis, cui nullus alius
e$t æqualis, tum fluentem exprimi nequire: in hoc libro ob$ervatur,
$i <math>P</math> $it algebraica functio quantitatis <math>x</math>, dimen$iones quantitatis <math>x</math> in
<math>P</math> majores e$$e per unitatem quam dimen$iones eju$dem quantitatis
in <math>π</math>, ubi <math>P = π x^.</math>; ni dimen$iones quantitatis <math>x</math> in <math>P</math> nihilo $int
æquales, in quo ca$u dimen$iones quantitatis <math>x</math> in <math>P</math> majores erunt
quam dimen$iones quantitatis <math>x</math> in <math>π</math> per quantitatem majorem
quam unitatem; hinc facile $equitur, $i dimen$iones quantitatis <math>x</math> in
numeratore <math>π</math> fluxionis <math>{π / ρ} x^.</math> $int minores per unitatem quam ejus di-
men$iones in denominatore, fluentem fluxionis <math>{π / ρ} x^.</math> in finitis termi-
nis non exprimi po$$e. _Neutonus_ dicit omnem ordinatam duobus
modis in $eriem re$olvi po$$e, nam index vel affirmativus e$$e pote$t
vel negativus, & tentandus e$t uterque ca$us, & $i $erierum alterutra
tandem abrumpitur, habebitur area curvæ in finitis terminis; hìc

<pb iv>[0014]<rh>PRÆFATIO.</rh>
animadvertitur, quod, $i una $eries abrumpatur, rebus recte di$po$itis
abrumpetur altera; etiamque docetur methodus inveniendi fluentes
fluxionum huju$ce formulæ <math>(a + b x^n + &c. + (e + f x^n + &c.)^m)^r ×
x^θ × (A + B x^n + &c. + (e + f x^n + &c.)^m-1 × (g + &c.)) x^.</math> ex prin-
cipiis, iis a _Cragio_ traditis haud multum di$$imilibus; & ob$ervatur, $i
fluxio quantitatis <math>a + b x^n + &c.</math> per <math>(e + f x^n + &c.)^m</math> divi$a æqua-
lis $it fluxioni quantitatis <math>(e + f x^n + &c.)^m</math> per <math>a + b x^n + &c.</math> di-
vi$æ, pro fluente a$$umendam e$$e <math>(a + b x^n + &c. + (e + f x^n +
&c.)^m)^n × &c. &c.</math>, & non <math>(a + b x^n + &c. + (e + f x^n + &c.)^m)^r+1 ×
&c.</math>; a$$eritur etiam in fluentibus detegendis ea$dem radices omnis
irrationalis quantitatis u$urpandas e$$e; & e $ub$titutione datur me-
thodus inve$tigandi plures fluentes eju$dem fluxionis inter diver$os
variabilis valores contentas, quæ inter $e $unt æquales.</p>
<p>_Cragius_ primum fluentes quarundam fluxionum fluentes involven-
tium methodum inveniendi docuit; nempe fluentem fluxionis <math>P^. $.
π x^.</math> e$$e <math>P $. π x^. - $. P π x^.</math>; eandem perfecit _foannes Bernoulli_ per $e-
riem <math>$. n z^. = n z - {z^2 n^. / 2z^.} + {z^3 n^.. / 2 · 3 z^. ^2} - {z^4 n^... / 2 · 3 · 4z^. ^3} + &c.</math> ex hâc $erie
eruit <math>$. {t^. / 1 + t^2} = t - {1 / 3}t^3 + {1 / 5}t^5 - &c. = {t / 1(1 + t^2)} + {1 · 2 t^3 / 1 · 3 (1 + t^2)^2} +{1 · 2 · 4t^5 / 1 · 3 · 5 (1 + t^2)^3} + &c.</math> & $ic <math>$. {x^. / √(1 - x^2)} = {x^. / 1(1 - x^2)^{1 / 2}} - {1 / 3}{x^3 /(1 - x^2)^{3 / 2}}+ {1 / 5}{x^5 /(1 - x^2){^}5/2}
- &c.</math> unde <math>$. {x^. / √ (1 - x^2)} $. {x^. / √ (1 - x^2)} = $. {x x^. / 1 - x^2} -{1 / 3} $. {x^3 x^. / (1 - x^2)^2} + {1 / 5}$. {x^5 x^. / (1 - x^2)^3} - &c.</math> po$tea vero _Neutonus_ tradidit
fluentem fluxionis <math>x^. $. x^. $. x^. $. x^. .. $. x^. $. x^. $. y x^.</math>, quæ facile e prædictis
methodis acquiri pote$t. _Eulerus_ dedit fluentem fluxionis <math>$. x^α x^.
$. x^β x^. $. x^γ x^. .. $. {x^. / 1 - x} &c.</math>; in hoc libro continentur nonnulla huju$ce
generis exempla. Deduxit _Neutonus_ ex fluentibus <math>(l - 1)</math> datarum
fluxionum huju$ce formulæ <math>x^θ±rn-1 x^. (e + f x^n + g x^2n + &c.)^λ±s-1</math>,
ubi <math>θ, λ, e, f, g, &c.</math> maneant $emper eædem invariabiles quantitates,
<math>r</math> vero & <math>s</math> $int quicunque integri numeri, & <math>e + f x^n + g x^2n + &c.</math>

<pb v>[0015]<rh>PRÆFATIO.</rh>
quantitates <math>(l)</math> nominum, fluentem cuju$cunque fluxionis prædictæ
formulæ; $ed hìc ob$ervatur quo$dam ca$us, in quibus <math>λ</math> vel <math>θ</math> $it integer
numerus, excipiendos e$$e; etiámque $i modo $it exponentialis quan-
titas <math>e^(a+bx^n +cx^2n +&c.)^m (a + b x^n + c x^2n + &c.)^m±r x^θ±πn x^.</math>, ubi <math>v & π</math>
$int integri numeri & <math>a + b x^n + c x^2n + &c.</math> $it quantitas <math>l</math> nominum,
tum e fluentibus <math>l - 1</math> fluxionum independentium prædictæ formulæ,
quæ habent <math>l - 1</math> diver$os valores quantitatis <math>v</math> erui po$$e fluentes
omnium fluxionum eju$dem formulæ: & con$imiles propo$itiones de
fluentibus huju$ce generis, quæ in diver$as irrationales quantitate<_>s</_>
ducuntur; de fluentialibus fluxionibus; de fluentibus fluxionum hu-
ju$ce formulæ <math>(a + b x^n + c x^2n + &c.)^λ±π × x^θ±αn±βm x^.</math> vel <math>(a + b x^n
+ c x^2n + &c.)^λ±π × x^θ±αn x^. $. (e + f x^n + &c.) ^±ρx^θ′ ±βn x^.</math>, ubi <math>π, ρ, α
& β</math> $unt integri numeri; in hoc libro traduntur.</p>
<p>In datâ algebraicâ æquatione relationem inter ab$ci$$am & ejus
corre$pondentes ordinatas de$ignante pro <math>y</math> $ub$titutit <math>v^s</math> _Neutonus_; &
ex hâc $ub$titutione in quibu$dam ca$ibus re$ultant æquationes ad
areas, quarum fluxiones arearum per functionem algebraicam <math>v</math> in <math>v</math>
denotantur; quod quidem evenit, cum æquatio inter ab$ci$$am <math>x</math> &
ejus corre$pondentes ordinatas <math>y</math> $it <math>y^m + a x^n y^r = b x^s</math>. In Actis Phi-
lo$oph. Lond. 1764. a me primum docetur <math>v^n + (a + b x + c x^2)v^n-1
+ (a′ + b′x .. x^4)v^n-2 + &c. = 0</math> e$$e æquationem ad aream (<math>v</math>)
curvæ, cujus æquatio relationem inter ab$ci$$am & ejus corre$pon-
dentes ordinatas $it <math>y^n + (α + β x) y^n-3 + &c. = 0</math>, $i modo ea in
finitis terminis exprimi po$$it.</p>
<p>_Jacobus Bernoulli_ invenit fluentem fluxionalis æquationis <math>y^. + P y x^.
= X x^.</math>, ubi <math>P</math> & <math>X</math> $unt fuctiones quantitatis <math>x</math>.</p>
<p>_Eulerus_ idem principium ad fluxionales æquationes $uperiorum
ordinum applicavit, nempe invenit fluentem & ejus (<math>n</math>) multiplica-
tores æquationis <math>y^. ^n + a y^. ^n-1 x^. + b y^. ^n-2 x^. ^2 + c y^. ^n-3 x^. ^3 ... + A y x^. ^n = 0</math>, &
_D'Alembert_ detexit fluentem fluxionalis æquationis <math>y^. ^n + a y^. ^n-1 x^. + b y^. ^n-1 x^. ^2
... A y x^. ^n = X x^. ^n</math>; in hoc libro traditur methodus inveniendi, annon
fluxionalis æquatio reddi pote$t integrabilis ex ejus multiplicatione
in functionem quantitatum <math>x</math> & <math>x^.</math>.</p>
<p>_Joannes Bernoulli_ primum invenit fluxionem exponentialis quanti-

<pb vi>[0016]<rh>PRÆFATIO.</rh>
tatis <math>x^x</math>; dedit regulam pro inveniendo fractionis <math>{P / Q}</math> valore, cum
ejus numerator & denominator $imul evane$cant, erit enim <math>{P^. / Q^.}</math> vel
<math>{P^.. / Q^..}, &c.;</math> quæ regula in quibu$dam ca$ibus fallit; in hoc libro tra-
ditur regula pro inveniendo prædicto valore, quæ vix aut ne
vix unquam fallit; nempe evane$cant & numerator & denomi-
nator cum <math>x = a</math>, pro <math>x</math> $cribatur <math>a + v,</math> & reducantur & nume-
rator & denominator ad $eriem $ecundum dimen$iones quanti-
tatis <math>v</math> progredientem, & facile con$tabit valor fractionis: eadem
principia etiam ad inveniendas differentias, &c. inter quantitates, quæ
evadunt infinitæ magnæ, hìc applicantur: invenit etiam curvam, quæ
<math>n - 1</math> habet quadrabiles areas; e. g. $it <math>y = z + (a x^n + b x^n-1 +
c x^n-2 + &c.)^{r / 1}</math> relationem inter ab$ci$$am <math>x</math> & ejus corre$pondentes
ordinatas <math>y</math> exprimens, ubi <math>{r / s}</math> $it fractio ad minimos terminos reducta,
& <math>z x^.</math> fluxio, cujus fluens inveniri pote$t; & radices vero <math>α, β, γ, δ, &c.</math>
æquationis <math>a x^n + b x^n-1 + &c. = 0</math> $int po$$ibiles, tum inveniri po$-
$unt <math>n - 1</math> quadrabiles areæ curvæ prædictæ inter valores <math>α & β, β & γ,
γ & δ, &c.</math> contentæ: inve$tigavit etiam fluentem fluxionis <math>x^x x^.</math> inter
valores <math>0 & 1</math> quantitatis <math>x</math> contentam <math>= 1 - {1 / 2^2} + {1 / 3^3} - {1 / 4^4} + &c.</math>
docuit methodum detegendi areas curvarum, quarum relationes inter
ab$ci$$as & earum corre$pondentes ordinatas per homogeneam vel
algebraicam vel fluxionalem æquationem primi ordinis exprimuntur:
reduxit fluxionalem æquationem <math>y^m y^.. = x^n y^. ^2-p x^. ^p</math> ad fluxionalem æqua-
tionem primi ordinis: invenit etiam curvam omnia puncta contrariæ
flexuræ ad fluxionalem æquationem de$ignantem in quibu$dam ca$i-
bus e$$e algebraicam.</p>
<p>_Cote$ius_ invenit radices æquationis <math>x^n = 1 = 0</math>, ex quibus detexit
fluentem fluxionis <math>{x^{λ / μ}n-1 x^./x^2n ± a x^n + 1}</math>.</p>

<pb vii>[0017]<rh>PRÆFATIO.</rh>
<p>_Manfredi_ dedit fluentem fluxionalis æquationis primi ordinis, primi
vero gradus.</p>
<p>_Riccati_ dedit ca$us, in quibus fluens fluxionis <math>y^. + a y^2 x^. = X x^.</math> in-
note$cit.</p>
<p>_Brook Taylor_ transformavit datam fluxionalem æquationem, in
quâ <math>x</math> fluit uniformitur, in alteram, in quâ <math>y</math> fluit uniformiter.</p>
<p>_Mac Laurinus_ invenit qua$dam fluxiones, quarum fluentes inveniri
po$$unt ope ellipticorum & hyperbolicorum arcuum, de quâ re po$tea
$crip$ere _Eulerus, Le Grange_, & alii; at minus utiles $unt hæ inve$ti-
gationes, quoniam nullæ dantur tabellæ, quæ exhibeant prædictos
arcus.</p>
<p>_Comes Fagnanus_ tradidit fluentes quarundam fluxionalium æquatio-
num primi ordinis, in quibus $imiliter involvuntur variabiles <math>x & y</math>,
& exinde detexit ellipticos arcus, quorum differentia e$t finita quan-
titas, hujus generis plures adjecit _Eulerus._</p>
<p>_Clairaut_ invenit fluxionem <math>p x^. + q y^.</math> fluentem recipere, $i modo
<math>({p^. / x^.}) = ({q^. / x^.})</math>; _Fontaine_ idem perfecit pro fluxionibus <math>p x^. + p y^. +
r z^.</math>, & invenit <math>A ({B^. / x^.}) - B ({A^. / x^.}) + ({A^. / z^.}) - ({B^. / y^.}) = 0</math>, cum æqua-
tio fluxionalis <math>x^. + A y^. + B z^. = 0</math> $it integrabilis; eadem principia
diver$is modis ad fluxionales æquationes primi ordinis applicavere
_D' Alembert & Eulerus_; con$imilia principia a _Marchione Condorcet_ &
in hoc libro extenduntur ad fluxionales quantitates & æquationes
plures variabiles quantitates involventes & majores ordines habentes.</p>
<p>_Eulerus_ invenit fluentes fluxionum <math>(a + b x^n)^m ± r x^pn±vn-1 x^.</math> inter
valores <math>o</math> vel infinitum <math>& - {a / b}</math> quantitatis <math>x^n</math>, ubi <math>r & v</math> $unt integri
numeri, ex fluente fluxionis <math>(a + b x^n)^m x^pn-1 x^.</math> inter eo$dem valores
variabilis <math>x</math> contentâ, quod quidem e prædictâ _Neutoni_ regulâ erui
pote$t, nam pro <math>x</math> in fluente per prædictam _Neutoni_ regulam deductâ
$cribantur prædicti valores, & quantitatum re$ultantium differentia
erit fluens quæ$ita: hinc, cum generalis fluens acquiri po$$it, non

<pb viii>[0018]<rh>PRÆFATIO.</rh>
difficile erit fluentem prædictam detegere; ex $ub$titutione enim $e-
quitur: inve$tigatio fluentis inter <math>o</math> vel infin. & <math>{a / b}</math> non magis utilis
e$t quam inter quo$cunque duos alios variabilis <math>x</math> valores; in non-
nullis ca$ibus con$tat ejus magis facilis re$olutio; e. g. _Eulerus_ inve-
nit fluentem fluxionis <math>{z^m-1 z^. / 1 + z^n}</math>, inter valores <math>o</math> & infinitum quantitatis
<math>z = {π / n × $in. {n / m} π}</math>, ubi <math>π</math> denotat <math>{1 / 2}</math> peripheriam circuli, cujus radius e$t
1, unde facile deduci pote$t fluens fluxionis <math>{z^n+s-1 z^. / z^2n + 2 l k^n z^n + k^2n}</math>: _Mac_
_Laurinus_ detegit, $i modo <math>Q, G & P</math> denotent fluentes fluxionum
<math>x^. x^rn-1 (E + F x^n)^l-1 × $. x^. x^sn-1 (e × f x^n)^k & x^. x^rn+sn-1 × (E + F x^n)^l-1 &
x^. x^sn-1 (e + f x^n)^k+l</math> inter valores <math>o & - {E / F} = d</math> quantitatis <math>x^n</math> conten-
tas, <math>e^l d^sn Q = G × P</math>.</p>
<p>_Simp$on_ dedit legem $eriei, quæ deducit <math>$. x^m±rn-1 x^. (e + f x^n)^λ ± s</math> ex
datâ <math>$. x^m-1 (e + f x^n)^λ x^.</math>; ubi <math>r & s</math> $unt integri numeri.</p>
<p>_Le Grange_ invenit variationem incrementi æqualem e$$e incre-
mento variationis, $i modo infinite$ima $int; & deduxit _Lexell, Eu-_
_lerus, Le Grange_ fluxionales æquationes, quarum fluentes inveniri
po$$unt: in hoc opere animadvertitur, $i incrementum fluxionis <math>x^.</math>
æqualis $it fluxioni incrementi, fluxionem incrementi functionis
quantitatis <math>x</math> æqualem e$$e incremento fluxionis.</p>
<p>_D'Alembert_ primus dedit exempla de æquationibus relationes ex-
primentibus inter quantitates huju$ce formulæ <math>({P^. / x^.}), ({P^. / y^.}) &c.</math>;
ubi hæ quantitates re$pective denotant fluxiones quantitatis <math>P</math>, cum
<math>x, y, &c.</math> $olummodo habeantur variabiles: de hâc re plurima $crip-
$ere _Eulerus, Le Grange, Condorcet, Le Place_, aliique; at de his per-
pauca in hoc libro traduntur: ex datâ re$olutione æquationis prædicti
generis <math>({m^. / y^.}) p + ({p^. / y^.}) m = ({m^. / x^.}) q + ({q^. / x^.}) m</math> $equitur multipli-

<pb ix>[0019]<rh>PRÆFATIO.</rh>
cator cuju$cunque fluxionalis æquationis primi ordinis <math>p x^. + q y^. = 0</math>;
& con$imilia principia etiam applicari po$$unt ad fluxionales æqua-
tiones $uperiorum ordinum. Ex quibu$dam datis formulis multi-
plicatorum dedit _Eulerus_ exempla fluxionalium æquationum, quæ
redduntur integrabiles per prædictos multiplicatores: ex datâ $ub-
$titutione, quæ ita transformat datam fluxionalem æquationem, ut
evadat integrabilis, invenit multiplicatorem, qui datam æquationem
reddet integrabilem: docuit methodum inve$tigandi in quibu$dam
ca$ibus, annon data æquatio $it fluens datæ fluxionalis æquationis:
ex datis particularibus fluxionalis æquationis <math>y^.. + P y^. x^.^2 + Q y^2 x^. ^2 =
X x^.^2</math> valoribus, ubi <math>P, Q & X</math> $unt functiones quantitatis <math>x</math> eruit gene-
ralem valorem prædictæ æquationis; invenit fluentem fluxionalium
æquationum <math>{n y^. (e + f y + g y^2 + &c.) / √ (a y^2 + b y + c)} + {m x^. (e + f x + g x^2 + &c.) / √ (a x^2 + b x + c)}= 0 & {x^. / √ (a + b x^2 + c x^4)} + {y^. / √ (a + b y^2 + c y^4} = 0</math>: dedit etiam
plura exempla fluxionalium æquationum, quarum fluentes detegun-
tur: & $ub$equentem propo$itionem; $i modo <math>P = a</math>, ubi <math>a</math> $it quæ-
cunque invariabilis quantitas, $it generalis fluens datæ fluxionalis
æquationis, tum erit quæcunque functio quantitatis <math>P = a</math> generalis
fluens: nonnulla adjecit de $ubtangente curvæ, cujus æquatio <math>1 · 2 ·
3 .. x = y ab</math> _Walli$io_ primum tradita fuit. Plura exempla fluentium
inter valores <math>0 & 1</math> variabilis <math>x</math> contentarum dedit.</p>
<p>In hoc opere 1. ob$ervatur, quod$i plures dimen$iones fluxionum <math>y^. ^n,
&c.</math> in datâ fluxionali æquatione contineantur, tum inve$tigatio fluentis
exigit re$olutionem æquationis algebraicæ, cujus dimen$iones haud
$unt minores quam prædictæ (<math>n</math>). 2. Adjicitur etiam methodus in-
veniendi fluxionalem æquationem <math>n</math> ordinis, ex quâ methodo con$tat
fluxionalem æquationem <math>n</math> ordinis <math>n</math> recipere diver$as generales fluen-
tes primi ordinis, etiamque <math>n</math> diver$os generales multiplicatores; <math>n.{n - 1 / 2}</math> generales fluentes $ecundi ordinis; & $ic deinceps: exhinc de-

<pb x>[0020]<rh>PRÆFATIO.</rh>
duci po$$unt plurimæ fluxionales æquationes, quæ fluentes recipiunt.
3. Animadvertitur, $i generales fluentes fluxionalis æquationis <math>n</math> or-
dinis $int <math>P = a, P′ = a′, P″ = a″, &c.</math>, ubi <math>a, a′, a″</math>, $unt invaria-
biles quantitates, generalem fluentem datæ fluxionalis æquationis
e$$e quemcunque functionem quantitatum <math>P, P′, P″, &c.</math> 4. O$ten-
ditur quantitatem <math>P = 0</math> non nece$$ario e$$e fluentem fluxionalis
æquationis <math>α P^. + β P q^. = 0</math>, cum $it fluens prioris partis æquationis
<math>α P^. = 0</math>, at non fluens po$terioris <math>β P q^. = 0</math>; quæ e$t quantitas
<math>β P = 0</math> ducta in <math>q^.</math>: hìc animadvertendum e$t fluxionalem æqua-
tionem primi ordinis <math>p x^. + q y^. = 0</math> $emper reduci po$$e ad fluxiona-
lem æquationem <math>a P^. + b P c^. = 0</math>, ubi <math>p, q, a, b & c</math> $unt functiones
quantitatum <math>x & y, & P</math> e$t quæcunque a$$umpta functio quantita-
tum <math>x & y</math>: etiamque, $i modo $it <math>y = A × B × C × D × &c. + a</math>, ubi
<math>a</math> e$t invariabilis quantitas ad libitum a$$umenda, generalis fluens
datæ fluxionalis æquationis; <math>y = A, y = B, y = C, &c.</math> e$$e particula-
res fluentes fluxionalis æquationis: vel magis generaliter, $i modo
<math>φ:(x & y) × φ′ (x & y) × φ″ (x & y) × &c. + a = 0</math> $it generalis fluens
datæ fluxionalis æquationis, <math>φ (x & y) = 0, φ′ (x & y) = 0, φ″ (x & y)
= 0</math> e$$e particulares fluentes datæ fluxionalis æquationis; hinc pro
particulari fluente quærendæ $unt prædictæ quantitates <math>φ (x & y)
= 0, &c.</math> vel <math>φ (x & y) × φ′: (x & y) × &c. = 0</math>. Et $ic de fluxio-
nalibus æquationibus $uperiorum ordinum. 5. A$$eritur fluentes
<math>x^-β $. x^β-α-1 x^. $. x^α P x^. = x^-2 $. x^α-β-1 x^. $. x^β P x^., &c.</math> vel generaliter
<math>x^-π $. x^π-β-1 x^. $.^β-s-1 x^. $. x^1-2-1; x^. .. $. x^τ P x^. = x^-γ $. x^γ-β-1 x^. $. x^β-π-1 x^.
$. x^π-ε<?>-1 x^. $. &c....$. x^α P x^.</math>; in hi$ce duabus iidem continentur in-
dices <math>π, α, β, σ, γ, ε, τ, &c.;</math> & $i præcedens terminus in utri$que $it
<math>x^b-λ-1 x^.</math>, tum ducitur in <math>$. x^λ-&c.-1 x^. $. &c.</math> 6. Animadvertitur, quod,
$i detur algebraica æquatio <math>a y^n + b y^n-1 + c y^n-2 + d y^n-3 + &c. = 0</math>,
ubi <math>a, b, c, &c.</math> fint functiones ip$ius <math>x</math>, & curva per prædictam æqua-
tionem de$ignata haud $it compo$ita e duabus vel pluribus curvis, &
fluens fluxionis <math>{b x^. / a}</math> haud inveniri po$$it finitis terminis, tum curva
haud quadrari pote$t: con$imilis propo$itio etiam de fluente cuju$-

<pb xi>[0021]<rh>PRÆFATIO.</rh>
cunque algebraicæ functionis quantitatum <math>y & x</math> in <math>x^.</math> ductæ affirmari
pote$t; cujus $olutio petenda e$t e meis Mi$cel. Analyt. anno 1762
editis, in quibus primum docetur methodus inveniendi $ummam e
$ingulis valoribus cuju$cunque algebraicæ functionis quantitatum <math>x
& y</math> & earum fluxionum; deinde Actis Philo$oph. Londin. 1764, ex
datâ algebraicâ æquatione relationem inter <math>x & y</math> exprimente a me
datur methodus, e quâ inve$tigari po$$it, utrum fluens cuju$cunque
fluxionis, quæ $it algebraica functio literarum in datâ algebraicâ
æquatione contentarum & earum fluxionum, inveniri pote$t, necne.
7. A$$eritur $ummam fluentium cuju$cunque algebraicæ functionis li-
teræ <math>x & y</math> in <math>x^.</math> ductæ ad quemcunque valorem ab$ci$$æ <math>x</math> pertinentium
exprimi po$$e per finitos terminos, circulares arcus & logarithmos
radicum datæ algebraicæ æquationis. 8. Docetur etiam methodus,
ex datâ algebraicâ, fluentiali, exponentiali, &c. quantitate, quæ ex-
primit $ummam $eriei $ecundum dimen$iones literæ <math>x</math> progredien-
tis, inveniendi $ummam alternorum $eriei terminorum vel denique
$eriei terminorum, quorum di$tantiæ a $emet ip$is $it <math>n</math>, ex ii$dem
principiis ac iis, quæ cum pluribus aliis ad Regiam Societatem anno
1757 communicata fuerunt. 9. Circa tres axes $ecum quo$cunque
angulos facientes gyretur quæcunque curva, cujus ab$ci$$æ & ordi-
natæ $int re$pective <math>x & y</math>; transformetur data curva in alteram $ub-
$tituendo pro <math>x, az + bv + c</math>; & pro <math>y, pz + qv + r</math>; gyretur
hæc curva circa axem $uum; & e datis contentis $olidorum circa tres
axes rotatione datæ curvæ generatorum $equitur $olidum generatum
a rotatione po$terioris curvæ circa axem $uum.</p>
<p>Vice-comes _Brounker_ eruit $ummas $erierum <math>{1 / 1.2} + {1 / 3.4} + {1 / 5.6}+ &c. {1 / 2.3} + {1 / 4.5} + &c. {1 / 2.3.4} + {1 / 4.5.6} + &c.</math> _Gregory St._
_Vincent_ primum dedit <math>{1 / 1-x} = 1 + x + x^2 + &c.</math> _Mercator_ ope-
rando in literis ad eundem modum, quo arithmetici in numeris de-
cimalibus dividunt, invenit <math>$.{x^. / 1-x} = x - {1 / 2}x^2 + {1 / 3}x^3 - &c.</math> idem

<pb xii>[0022]<rh>PRÆFATIO.</rh>
in extractione radicum perfecit _Neutonus:_ eorum antece$$ores in re
algebraicâ divi$iones & extractiones in infinitum haud promovebant,
nullum per$picientes u$um, cui per applicationem _Walli$iani_ theore-
matis in quadraturis detegendis in$ervire potuerit: _Neutonus_ ex bi-
nomiali theoremate reduxit binomiales algebraicas quantitates ad
infinitas $eries, & exinde deduxit areas curvarum: quantitates multo
magis complexæ ad infinitas $eries adhuc $olummodo per vulgares
methodos divi$ionis & radicum extractionis olim traditas reduci po$-
$unt: in harum regularum exemplis docuit $eries vulgares, quibus
ex arcu corre$pondens $inus vel co$inus detegitur, & e logarithmo
numerus: $eriem <math>x - {1 / 2}x^2 + {1 / 3}x^3 - {1 / 4}x^4 + &c.</math> pro logarithmo primus
invenit vice comes _Brounker_; $eriem vero <math>t - {1 / 3}t^3 + {1 / 5}t^5 - &c.</math> pro
arcu detexit _Jacobus Gregory_; $eries <math>A - {c / 2} × {1 A - x P / 2} - &c.</math> pro
arcu ellip$eos, &c. ex novis principiis in hoc opere primum traditur;
$ed animadvertendum e$t in quâcunque fluente detegendâ, conver-
gentes e$$e debere $eries pro duobus valoribus variabilis datæ $eriei
quantitatis.</p>
<p>_Leibnitzius_ a _Neutono_ quæ$ivit ca$us, in quibus $eries exortæ vel ex
_Mercatoris_ methodo dividendi, vel ex _Neutonianâ_ radicum extractione
in infinitum convergent: quod _Leibnitzius_ quæ$ivit, in hoc libro pri-
mum peragitur; nunquam convergent $eries e prædictis methodis
deductæ $ecundum dimen$iones quantitatis <math>x</math> a$cendentes, $i <math>x</math> major
$it quam minima radix æquationum re$ultantium ex denominatore
vel ex radicalibus in infimos terminos depre$$is nihilo æqualibus e$$e
a$$umptis; nec de$cendentes, $i <math>x</math> minor $it quam maxima radix præ-
dicta. Convergentia pendet ex ratione quam habent radices præ-
dictæ ad <math>x</math>; e. g. $eries <math>m + {1 / 2}m^2 + {1 / 3}m^3 + &c. = $.{m / 1-m}</math> $emper
converget, cum <math>m</math> minor $it quam 1; $eries autem <math>x - {1 / 2}x^2 + {1 / 3}x^3 -
&c. = $.{x^. / 1 + x}</math>, cum <math>x</math> non major $it quam 1: $i <math>m = {x / 1 + x}</math>; tum
erit <math>m + {1 / 2}m^2 + {1 / 3}m^3 + &c. = x - {1 / 2}x^2 + {1 / 3}x^3 - &c.</math>; hinc per $e-
riem <math>m + {1 / 2}m^2 + {1 / 3}m^3 + &c.</math> $emper acquiri pote$t log. quantitatis

<pb xiii>[0023]<rh>PRÆFATIO.</rh>
<math>1 + x</math>, utcunque magna $it quantitas <math>x</math>: $eries autem <math>x - {1 / 2}x^2 + &c.</math>
divergit cum <math>x</math> major $it quam 1, ergo ex eâ non deduci pote$t log.
quantitatis, quæ major e$t quam 2: quantitatum <math>{a / 10}, {a / 100}, {a / 1000}, &c.</math>
(dato logar. numeri 10) logarithmi facile detegi po$$unt ex loga-
rithmo quantitatis <math>a</math>, & hinc ex $erie <math>x - {1 / 2}x^2 + &c.</math> detegi pote$t
log. quantitatis, cum ea fru$tra $ine tale transformatione quæreretur;
& magis convergentes evadent utræque $eries: eadem etiam appli-
cari po$$unt; cum quantitas, cujus logar. innote$cit, non $it 10, $ed
quicunque alius numerus: in omni ca$u $eries <math>m + {1 / 2}m^2 + {1 / 3}m^3 +
&c.</math> magis celeriter converget quam $eries <math>x - {1 / 2}x^2 + {1 / 3}x^3 - &c.</math>: in
hoc libro ex transformatione fluxionis $cribendo in eâ <math>z + a & z^.</math> pro
<math>x & x^.</math> o$tenditur, quod fluens inter duas proximas radices prædicta-
rum æquationum contenta $emper acquiri pote$t; etiamque, $i
quæcunque radices prædictæ inter duos valores quantitatis <math>x</math>, inter
quos requiritur fluens, con$i$tant, & fluens haud $it infinita; tum
nece$$e e$t plures interpolare diver$os valores quantitatis <math>a</math>, a$$umen-
dos ita quidem ut quantitas <math>z</math> incipiat ad $ingulas prædictas radices,
vel in ii$dem terminet; $in aliter vero omnes $eries re$ultantes haud
erunt convergentes: $i vero plurimi interpolentur valores quantitatis
<math>a</math> inter duas proxime $ucce$$ivas radices prædictas ita ut maxime cele-
riter convergant $eries, tum erunt differentiæ inter valores prædictos in
geometricâ progre$$ione; unde ex _Mercatoris & Neutoni_ methodo fluens
$olummodo detegi pote$t cum utrique valores variabilis quantitatis
$int minores quam minima radix vel negativa vel affirmativa; $ed ex
hâc $implici transformatione $emper detegi po$$it, cum prædicti va-
lores inter qua$cunque duas proximas radices interponantur, vel magis
generaliter cum ea finita $it: utrum quæcunque duæ datæ quantita-
tes inter ea$dem duas radices interponantur, necne; plerumque ex
earum $ub$titutione in datis æquationibus e mutatione $ignorum da-
tarum & re$ultantium æquationum facile con$tabit.</p>
<p>_Taylorus_ dedit $eriem <math>S = y ± {ay^. / x^.} + {a^2 y^.. / 1.2x^.^2} ± &c.</math></p>
<p>_Eulsrus, Mac Laurinus,_ &c. cum approximationes inventæ lente

<pb xiv>[0024]<rh>PRÆFATIO.</rh>
convergant, interpolavit alios terminos ad perparvas di$tantias a $e
invicem po$itos. _Eulerus_ advertebat, quod, cum ordinatæ fiant infi-
nitæ, tum haud convergent approximationes $ic inventæ, & ad in-
ve$tigandam aream curvæ inter duas ordinatas po$itam interpolavit
plures ordinatas, & deinde ex $erie <math>y = {s^. / n^.} - {s^.. / 2n^.^2} + {s^... / 2.3n^.^3} - &c.</math>
inve$tigat ordinatam, & ex ordinatâ per $eriem <math>s = $.yn^. + {y / 2} + {y^. / 12n}+ &c.</math> inve$tigat aream, &c.</p>
<p>_Arabes, Lucas de Burgo, Vieta, Harriottus, Ougbtredus, Girard, De_
_Lagny_, aliique tractarunt de approximationibus ad radices æquatio-
num, docuere methodos approximationum ea$dem, quibus nunc uti-
mur, cum inveniatur quantitas a radice quæ$itâ haud longe di$tans;
_Neutonus, Vietæ_ aliorumque ve$tigiis in$i$tens, eadem magis elucide
perfecit, & animadvertebat, quod $i <math>e</math> $it perparva quantitas quæ$ita,
& data æquatio $it <math>A + Be + Ce^2 + &c. = 0</math>, tum ex tribus termi-
nis <math>A + Be + Ce^2 = 0</math> nihilo æqualibus e$$e $uppo$itis $equitur ap-
proximatio ad valorem quantitatis <math>e</math>, quæ duplum præbet numerum
figurarum approximationis ex æquatione <math>A + Be = 0</math> inventæ;
ultimo a me primum o$tenditur veram e$$e hanc propo$itionem, &
radices æquationis <math>A + Be + Ce^2 + De^3 = 0</math> triplum numerum
illarum figurarum præbere; & $ic deinceps.</p>
<p>_Eulerus_ in algebraicâ æquatione <math>A = 0</math> pro <math>x</math> $ub$tituit ejus valo-
rem <math>(a)</math> prope, & a$$eruit <math>{Ax^. / A^.}</math> e$$e propiorem valorem; ejus demon-
$trationem primum dedit _Courtivron._</p>
<p>_Simp$on_ invenit continuas approximationes ad radices duarum
æquationum duas incognitas quantitates habentium.</p>
<p>Hic animadvertendum e$t omnes has methodos $olummodo parti-
culares præbere ca$us regulæ vulgariter dictæ fal$i, quæ in hoc libro
magis generaliter redditur.</p>
<p>In hoc libro 1. primum o$tenditur approximationem ex hâc methodo
inventam $emper eo magis accuratam fore, quo propior fuit quanti-
tas pro radice $ub$tituta radici quæ$itæ quam reliquis. E. g. $i quan-

<pb xv>[0025]<rh>PRÆFATIO.</rh>
titas data $it ferè intermedia inter duas proxime $ucce$$ivas radices,
i. e. $it radix æquationis ad limites, tum nova approximatio inventa
erit infinita quantitas; minime refert, utrum permagnam habeat ratio-
nem ad radicem quæ$itam, necne. 2. Traditur methodus ex datis ap-
proximationibus ad <math>m</math> radices æquationis <math>n</math> dimen$ionum inveniendi
approximationes ad $ingulas prædictas radices magis convergentes.
3. Sit æquatio <math>x^n - p x^n-1 + qx^n-2 - &c. = 0</math>, & $i una radix multo
major $it quam <math>m</math>, multo vero minor quam <math>n - m - 1</math>, traditur nova
methodus inveniendi $eriem radicem prædictam exprimentem. 4. Da-
tur etiam theorema quod dico incrementiale theorema, ex quo $i modo
dentur incrementa e $ingulis variabilibus in datâ quantitate conten-
tis, facile erui po$$unt ejus diver$a incrementa; & ex hoc theoremate
quamplurimæ æquationes in algebraicas transformari po$$int, ex qui-
bus approximationes ad radices deduci po$$int; etiamque adjicitur
nova methodus ex a$$umptis <math>m</math> diver$is valoribus pro radice <math>x</math>, & ab
eâ haud longe di$tantibus, inveniendi æquationem <math>m</math> dimen$ionum,
cujus radix erit approximatio ad radicem <math>x</math>: omnia præcedentia prin-
cipia etiam applicari po$$unt ad plures æquationes plures incognitas
quantitates habentes, etiamque ex iis deduci pote$t una in terminis
reliquarum.</p>
<p>_Neutonus_ dedit <math>(α^2m + β^2m + γ^2m + &c.)^{1 / 2m}</math>, ubi <math>α, β, γ, &c.</math> $unt ra-
dices datæ æquationis, pro maximâ radice.</p>
<p>_Daniel Bernoulli_ invenit maximam radicem datæ æquationis <math>x^n -
px^n-1 + qx^n-2 - rx^n-3 + &c. = 0</math>, $i modo ea $it po$$ibilis; in hoc
libro ob$ervatur idem perfici po$$e pro omnibus æquationibus irratio-
nales vel plures variabiles quantitates involventibus; docetur etiam
con$imilis methodus ulterius promota inveniendi radicem, quæ minor
$it quam maxima, major vero quam omnes reliquæ; & eadem
principia ad detegendas omnes reliquas radices promovere liceat.
Omnia ea, quæ prius protuli de convergentiis $erierum ex divi$ione &
extractione radicum ortarum, æque ad ha$ce approximationes appli-
cari po$$unt; $ed omnes hæ regulæ vel $upponunt omnes radices e$$e
po$$ibiles, vel po$$ibiles e$$e majores quam impo$$ibiles; at in æqua-

<pb xvi>[0026]<rh>PRÆFATIO.</rh>
tionibus $uperiorum dimen$ionum plerumque numerus impo$$ibilium
radicum multo major erit quam numerus po$$ibilium, & con$equen-
ter probabile erit, ut maxima impo$$ibilis radix major $it quam ma-
xima po$$ibilis; ergo regulæ non applicari po$$unt, ni inventum $it
po$$ibilem radicem e$$e majorem quam $ingulas impo$$ibiles. Datur
methodus facile deducendi infinitas æquationes, quarum radices $unt
omnes po$$ibiles.</p>
<p>In hoc libro docetur etiam methodus reducendi quantitates præ-
dictas ad $eries $ecundum dimen$iones quantitatis <math>x</math> progredientes,
&c. ita ut earum integrales innote$cant.</p>
<p>_Barrovius_ in quibu$dam ca$ibus deduxit ex æquatione relationem
inter <math>x & y</math> exprimente approximationem ad quantitatem <math>y</math> in termi-
nis quantitatis <math>x</math>; hanc propo$itionem magis generalem reddidit _Neu-_
_tonus_, cujus particularem ca$um dedit in rever$ione $erierum; eadem
principia ad fluxionales æquationes applicavit, $ed in hi$ce æquatio-
nibus, $i modo re$olutio $it generalis, tot $emper occurrent quantitates
ad libitum a$$umendæ, quot $it ordo fluxionalis æquationis, in quibus
ca$ibus plerumque occurrent quædam difficultates, quarum nonnullæ
in hoc libro traduntur; e.g. in generali fluente detegendâ per $eriem
fluxionalis æquationis (<math>n</math>) ordinis $emper occurrent homogeneæ flu-
xionales æquationes primi $ecundi, &c. u$que ad <math>n</math> ordinem. _Leib-_
_nitzius_ primum a$$ump$it quantitatem <math>y = a x^n + b x^n+m + cx^n+2m + &c.</math>
generales coefficientes habentem. _Eulerus_ recte in$tituit approxima-
tiones ad $ingulas quantitates & earum fluxiones, & ex iis per me-
thodos prædictas approximationes deduxit. In hoc libro 1. docentur
methodi adjudicandi, utrum hæ $eries convergent necne, ferè ex ii$-
dem principiis ac prius tradita fuerunt pro algebraicis æquationibus,
&c. 2. Prima approximatio ex quâcunque datâ hypothe$i ita deduci
pote$t. Sit æquatio <math>y^n - a y^n-1 + b y^n-2 - &c. = 0,</math> ubi <math>a, b, c, &c.</math>
$int functiones ip$ius <math>x</math>; deinde rejectis omnibus terminis, qui haud
maximi re$ultant ex datâ hypothe$i in prædictis quantitatibus <math>a, b, &c.</math>
re$ultet æquatio <math>y^n - α y^n-1 + β y^n-2 - &c. = 0;</math> quæ erit æquatio, cujus
radices erunt primæ approximationes ad $ingulas datæ æquationis
radices; $i modo quædam ex iis longe di$tent a reliquis, tum ex ob-

<pb xvii>[0027]<rh>PRÆFATIO.</rh>
$ervatione adjunctâ facile acquiri po$$unt; aliter exoritur re$olutio
quadraticæ &c. æquationis: idem etiam perfici po$$it per eundem
modum, $i modo quæcunque irrationales quantitates in datâ æqua-
tione contineantur: eadem principia applicari po$$unt ad plures
æquationes plures incognitas quantitates habentes: _Le Grange_ inve-
nit legem, quam ob$ervat rever$io $eriei ex meâ $erie pro $ummis po-
te$tatum; eadem $eries paulo aliter in hoc libro deducitur; etiamque
deteguntur approximationes ad maximam vel minimam datæ alge-
braicæ æquationis radicem; & invenitur <math>y</math> in terminis $ecundum di-
men$iones quantitatis <math>x</math> progredientibus, $i modo prius ita transfor-
metur data æquatio, ut una radix evadat multo major vel minor
quam quæcunque alia datæ æquationis radix.</p>
<p>_Briggius_ vel _Vieta_ & _Pa$cal_, primum quod $cio invenit binomiale
theorema: id ad radices extrahendas primus applicavit _Neutonus._</p>
<p>No$tras _Briggius_ reddidit calculum pro logarithmis faciliorem, ex
eo quod interpolavit plures quantitates ea methodo quam invenit dif-
ferentiarum ultimo inter $e æqualium; & prout primæ, $ecundæ, ter-
tiæ, &c. differentiæ exoriantur æquales, diver$as ex methodus habet le-
ges: hanc rem ulterius pro$ecuti $unt _Regnaldus & Moutonus,_ & hi omnes
invenere <math>n</math> differentias pote$tatis <math>n</math> quantitatum in arithmeticâ progre$-
$ione e$$e æquales; deinde ex huju$modi quantitatibus datis, quarum
differentiæ $int ultimo æquales, inve$tigavere legem, ex quâ interpolari
po$$unt prædictæ quantitates, quæ con$tant ex quantitatibus huju$ce
formulæ <math>A x^n - a x^n-1 + b x^n-2 - &c.</math> _Neutonus_ idem problema ma-
gis generale reddidit, a$$ump$it enim quantitatem prædictæ formulæ
<math>A x^n - a x^n-1 + b x^n-2 - &c. = y</math>, & pro <math>x & y</math> $crip$it qua$cunque quan-
titates <math>p, q, r, s, &c. & A, B, C, &c.</math> & ex æquationibus re$ultantibus
deduxit quantitatem prædictam; ex hâc methodo duxit curvam alge-
braicam per quotcunque data puncta: ex a$$umptis quantitatibus
hujus formulæ <math>A + B (x - p) + C · (x - p) · (x - q) + &c.</math> re$o-
lutionem magis facilem reddidere _Nicbole, Sterling & Walzius:_ in hoc
libro 1. quædam nova traduntur de hâc methodo differentiarum, de
ejus convergentiâ, cum termini haud ultimo evadant inter $e æquales,

<pb xviii>[0028]<rh>PRÆFATIO.</rh>
i.e. haud terminetur $eries. 2. Adjungitur etiam problema ex datâ
lege, cui <math>m</math> a$$umptæ quantitates conveniant, <math>n</math> vero haud conveniant,
ita corrigere legem, ut <math>n</math> po$teriores etiam ei conveniant. 3. Datur
methodus corre$pondentium valorum, i. e. $it <math>y = {x - q · x - r · x - s. &c. / p - q · p - r. p - s. &c.}× A + {x - p · x - r · x - s · &c. / q - p · q - r · q - s · &c.} × B + {x - p · x - q · x - s · &c. / r - p · r - q · r - s · &c.}× C + &c. = y.</math> 4. Datur altera ncva methodus differen-
tiarum. 5. Methodus corre$pondentium valorum applicatur ad plu-
res ca$us in transformatione æquationum ex Medit. Algehr. de$ump-
tos. 6. Eadem etiam applicatur ad datos corre$pondentes valores
trium vel plurium incognitarum quantitatum, &c.</p>
<p>_Brook Taylor_ invenit incrementum cuju$cunque algehraicæ integra-
lis; idem perficitur pro incremento fluentialis integralis ah _Le Grange,_
_Eulero,_ &c. & in hoc lihro; con$imilia principia etiam applicari po$$unt
ad incrementa integralium & fluentialium quantitatum detegenda:
integralem incrementi <math>z · z + z<x>.</x> · z + 2z<x>.</x> .. z + (n - 1)z<x>.</x>,</math> etiamque
incrementi <math>{1 / z · z + z<x>.</x> .. z + nz<x>.</x>}</math> dedit _Taylor: Facohus Stirling_ reduxit
omnes quantitates huju$ce formulæ <math>A + B z + C z^2 + D z^3 + &c.</math>
ad formulam <math>a + b z + c z · (z + 1) + d z · (z + 1) · (z + 2) +
&c.</math> & quantitates huju$ce formulæ <math>A + B z^-1 + C z^-2 + D z^-3 +
&c.</math> ad formulam <math>a + {b / z} + {c / z · z + 1} + {d / z · z + 1 · z + 2} + &c.</math> $i
vero termini in genere $int <math>x^z+n</math> in <math>{a / z} + {b / z · z + 1} + &c.</math> ubi <math>z</math> $it
di$tantia a primo $eriei termino, invenit $ummam <math>{a / (1 - x) · z} +{b - A x / (1 - x) · z · (z + 1)} + &c.</math> & ex quihu$dam æquationihus inter $uc-
ce$$ivos terminos & di$tantiam a primo $eriei termino invenit parti-
culares valores per vulgarem methodum infinitarum $erierum.</p>

<pb xix>[0029]<rh>PRÆFATIO.</rh>
<p>_De Moivre_ & _Daniel Bernoulli_ inve$tigavere $ummas recurrentium
$erierum, & e $ummâ datâ deduxere relationem inter $ucce$$ivos ter-
minos.</p>
<p>_Monmort_ transformavit $eriem <math>A + {B / r} + {C / r^2} + &c.</math> ita con$titutam,
ut ultimæ coefficientium differentiæ <math>d, d′, d″, &c.</math> evadant tandem in-
ter $e æquales, in $eriem finitam <math>{rA / r - 1} - {rd / (r - 1)^2} + {rd′ / (r - 1)^3} + &c.</math>
_Nicholas Bernoulli_ invenit $eriem æqualem $ummæ <math>n</math> terminorum hu-
ju$ce $eriei; huju$ce prohlematis altera datur $olutio in hoc opere:
_Monmort_ etiam deduxit $ummam $eriei, cujus numeratores con$tituant
lineam quamlihet erectam in triangulo arithmetico, denominatores
vero lineam quamlihet tran$ver$am.</p>
<p>_Bernoulli, Mac Laurin, Le Grange, Eulerus,_ &c. multa tradidere de
i$operimetricis prohlematihus, de quihus haud in hoc opere tractatur.</p>
<p>_Walli$ius_ dedit ex interpolatione arearum curvarum inter valores <math>0
& 1</math> ah$ci$$æ <math>x</math> contentarum, quarum ordinatæ $int re$pective <math>(1 - x^2)^0,
(1 - x^2)^1, (1 - x^2)^2, &c.</math> terminum intermedium inter primum &
$ecundum, viz. aream curvæ, cujus ordinata $it <math>(1 - x^2)^{1 / 2}</math>, i. e. circuli
inter prædictos valores ah$ci$$æ <math>x</math> contentam <math>= {2 · 4 · 4 · 6 · 6 · 8 · 8 · &c. / 3 · 3 · 5 · 5 · 7 · 7 · 9 · &c.}</math>;
Vice-comes _Brounker_ dedit <math>{2 · 4 · 4 · 6 · 6 · &c. / 3 · 3 · 5 · 5 · 7 · &c.} = {1 / 1 + {1 / 2} + {9 / 2} + {25 / 2} + &c.}</math>
de his continuis fractionihus plura & elegantia $crip$ere _Eulerus_ &
_Le Grange,_ &c.; e. g. $i continuæ fractionis termini $emper recurrant,
tum ejus valor e quadraticâ æquatione facile erui pote$t, & $ic facile
transformari pote$t radix quadratica cuju$cunque quantitatis in con-
tinuam fractionem: nonnulla de hâc re elegantia dedit _Lamhert:_ in
hoc lihro in novâ $pecie continuarum fractionum, in quihus inclu-
duntur radices, exprimitur radix cuju$cunque æquationis huju$ce for-
mulæ <math>x^m + ax^m-1 + b x + c = 0.</math></p>

<pb xx>[0030]<rh>PRÆFATIO.</rh>
<p>_Bernoulli_ & _Eulerus_ invenere plura continua contenta inter $e
æqualia; & plura, & multo magis generalia ex novis principiis in
hoc lihro continentur. E. g. Datur methodus ex $uh$tituendo <math>α x, β x,
γ x, &c.</math> pro <math>x</math> in datâ quantitate, uhi <math>α, β, &c.</math> $unt radices æquationis
<math>x^n - 1 = 0</math> inveniendi continuum contentum datæ quantitati æquale;
ex eo quod inveniuntur quantitates, quæ in $eriem datæ quantitati
vel fluenti æqualem ductæ, eam reddunt magis convergentem. 2. Et
ex a$$umptis æquationihus inter terminos facile con$tant continua
contenta datis quantitatihus æqualia.</p>
<p>_Facohus Gregary_ ex datis aliquot ordinatis æquidi$tantihus invenit
a$ymptoton hyperholæ generis logarithmici, cujus prohlematis ope
qua$dam $eries $ummavit.</p>
<p>_Facohus Bernoulli_ invenit maximum terminum hinomii ad integram
pote$tatem evecti, punctum vero ejus inflexûs dedit _De Moivre._</p>
<p>_Facohus Bernoulli_ a$$umit quantitatem <math>A</math>, cujus termini ad infinitam
di$tantiam $unt infinite parvi; ex quantitate <math>A</math> $uhtrahit eandem
quantitatem per quo$dam terminos diminutam; & re$ultat $eries,
cujus $umma e$t prædictorum terminorum $umma; feriem re$ultan-
tem $emper convergere, primum quod $cio in hoc lihro demon$tra-
tur: hoc prohlema magis generaliter perfecere _Goldhatch_ & _De_
_Moivre_ ex multiplicatione in factores nihilo æquales; & in hoc lihro
idem perficitur per additionem $ine multiplicatione. 2. Etiamque hi
duo cafus redduntur magis generales per additionem e $ingulis duo-
hus, trihus, &c. $ucce$$ivis terminis. 3. A$$umuntur duæ vel plu-
res $eries, & ex iis deducuntur $eries, quarum $ummæ innote$-
cant, e. g. ex $eriehus <math>{1 / a} + {1 / a + h}x^b + {1 / a + 2h}x^2b + &c., {1 / b} + {1 / b + h}x^b + {1 / b + 2h}x^2b + &c., {1 / c} - {1 / c + h}x^b + {1 / ε + 2 h}x^2b - {1 / c + 3h} x^3b +
&c.</math> inveniri pote$t $umma omnis $eriei, cujus generalis terminus e$t
<math>{a′z^m + b′ z^m-1 + &c./a + h z · a + h z + h · a + h z + 2 h .. a + h z + n h × b + h z · b + h z + h
· b + h z + 2 h .. b + h z + n h × h z + c · h z + {1 / 2} h + c · h z + h + c · h z +

<pb xxi>[0031]<rh>PRÆFATIO.</rh>
1{1 / 2}h + c..h z + {n″ / 2}h + c × &c.}</math>, ubi <math>n, n′, n″, &c. & m</math> $unt integri
numeri, & <math>z</math> di$tantia a primo $eriei termino. 4. Ex datis $ummis
$erierum <math>{1 / a} ± {1 / a + h} x^b + {1 / a + 2h}x^2h ± &c., {1 / a^2} ± {1 / (a + h)^2}x^h + &c.,{1 / a^3} ± {x^h / (a + h)^3} + &c., ... {1 / a^l} ± {1 / (a + h)^l}x^b + &c.; {1 / b} ± {1 / b + h} x^h +{1 / b + 2h} x^2h ± &c., {1 / b^2} ± {1 / (b + h)^2} x^h + &c., ... {1 / b^l′} ± {1 / (b+1)^l′} x^h + &c.</math>;
acquiri po$$unt per $implices æquationes $ummæ omnium $erierum,
quarum generales termini $unt <math>{a′z^m + b′z^m-1 + &c. / (z + a)^l × (h z + a + h)^l × (h z + a +
2h)^l .. (h z + a + n h)^l × (b + h z)^l′ . (b + h + h z)^l′ .. (b + n′h + hz)^l′ &c.}</math>.
5. Datur facilis methodus generaliter inveniendi ca$us, in quibus
$umma $eriei, cujus generalis terminus e$t <math>{1 / a + h z · a + h z + h · a + h z
+ 2 h · &c. × b + h z · b + h z + h · b + h z + 2 h. &c. × c + h z · c +
h z + h. &c. × &c.}</math>, in finitis terminis exprimi pote$t. 6. Animadver-
titur, $i in denominatore contineatur factor <math>h z + d</math>, qui non habet $uc-
ce$$ivum <math>h z + d + n h</math>, ubi <math>n</math> e$t integer numerus; in finitis terminis non
exprimi po$$e $eriei $ummam: <math>Nichole</math> invenit generalem terminum
datæ $eriei <math>{a′ z^m + b′ z^m-1 + &c. / a + h z · a + h z + h · a + h z + 2 h .. a + h z + n h} ={e / a + h z + n h} + {f / a + h z + n h. a + h z + (n - 1) h} + &c.</math>, in hoc
libro prædictus terminus invenitur <math>= {e′ / a + h z · a + h z + h · a + h z
+ 2 h .. a + z h + n h} + {f′ / a + h z · a + h z + h .. a + h z + (n-1) h}

<pb xxii>[0032]<rh>PRÆFATIO.</rh>
+ &c.</math>, ubi <math>m</math> non major e$t quam <math>n</math>; etiamque terminus <math>{a z^m + b z^m-1 + &c. / a + hz. a + hz
+ h .. a + hz + nh × b + hz. b + hz + h. b + hz + n′h × c + hz. &c.}</math>,
re$olvitur in plures $implices, quarum $ummæ innote$cunt. _De Moivre_
_& Bernoulli_ invenit $ummas $erierum ex inveniendo fluentes utriu$que
æquationis partis: _Eulerus_ exinde deduxit $ummas $erierum, quarum
generales termini $unt <math>{1 / a + hz. b + hz. c + hz. &c.} × e^z</math>, ex fluen-
tibus fluxionum <math>{x^a-1 x^. / 1 ± x^h}, {x^b-1 x^. / 1 ± x^h}, &c</math>, cum quantitates <math>a, b, c, &c.</math> $int
inæquales, & exinde facile erui po$$unt $erierum $ummæ, cum gene-
rales termini $int <math>{a″ z^m + b″ z^m-1 + &c. / a + h z · b + h z. &c.} e^z</math>: in hoc libro ex datâ $ummâ
<math>W</math> $eriei, cujus generalis terminus e$t <math>V</math>, & cujus fluxio detegi pote$t, de-
ducuntur $ummæ $erierum, quarum generales termini $unt <math>V (a′ z^m +
b′ z^m-1 + &c.)</math>; & ex $ummis $erierum, quarum generales termini $unt
<math>V × {1 / α + hz}, V × {1 / β + hz}, V × {1 / γ + hz}, &c.</math> deducuntur $ummæ $erie-
rum, quarum generales termini $unt <math>V × {a′ z^m + b′ z^m-1 + &c. / α + h z · β + h z · γ + h z · &c.}</math>;
& $imiliter de pluribus factoribus inter $e inæqualibus in denomina-
toribus contentis. 2. Summæ prædictarum $erierum per fluentes flu-
xionum <math>W^., x^{a / h} W^., x^{β / h} W^., &c.</math> exprimuntur, &c. 3. Summæ $erierum,
quarum generales termini $unt <math>{a′ z^m + b′ z^m-1 + &c. / α + z · β + z · γ + z · δ + z · &c.} ×
l · {l - 1 / 2} .. {l - nz / nz + 1} × e^nz</math> $emper exprimi po$$unt per finitos terminos,
circulares arcus & logarithmos, $i modo <math>α, β, γ, &c.</math> vel $int integri
numeri, vel habeant 2 pro $uo denominatore. 4. Dantur $ummæ $e-
rierum, quarum generales termini $unt <math>{a′ z^m + b′ z^m-1 + &c./1 · 2 · 3 · n z × α + z · β + z

<pb xxiii>[0033]<rh>PRÆFATIO.</rh>
· γ + z · &c.} × e^z</math>, ex fluentibus fluxionum <math>z^α × x^.</math>, ubi <math>{x^. / 1 ± x} = z^.,
z^β x^., z^γ x^., &c.</math> & $ic de pluribus con$imilibus. 5. Ex dato generali
termino invenitur $eriei $umma, $i modo generaliter in finitis termi-
nis exprimi po$$it, rejiciendo omnes ultimos factores & a$$umendo
pro $ummâ omnes reliquos in rationalem & integralem functionem
quantitatis _z_ di$tantiæ a primo $eriei termino cum coefficientibus de-
ducendis ductos, & exinde deducendo coefficientes quæ quidem $em-
per præbent $ummam, $i modo in finitis terminis exprimi po$$it; $in
aliter $eriem in infinitum progredientem. 6. Datur methodus inve-
niendi $ummas $erierum, quæ continent irrationales quantitates: no-
$tras _Landen_ po$t primam editionem huju$ce operis editam transfor-
mavit $eriem <math>x - x^-1 - {x^2 - x^-2 / 2} + &c. = log. x</math>, & quæ $eries, ut a
me primum ob$ervatur, $emper divergit, cum <math>x</math> $it po$$ibilis quantitas,
in $eriem $ecundum $inus & co$inus progredientem; $eries enim e$t dif-
ferentia inter duas vulgares $eries <math>x - {x^2 / 2} + {x^3 / 3} - &c. & x^-1 - {x^-2 / 2} +{x^-3 / 3} - &c.</math>, quarum una divergit, cum altera convergat; hæc $eries
ea$dem propo$itiones præbet, utrum $it transformatio in $inus, nec-
ne; &c. prior methodus calculum magis facilem plerumque reddit,
Hìc datur fluens fluxionis <math>s^(q) ^m c^(q) ^n × {x^. / √ (1 - x^2)}</math>.</p>
<p>_Joannes Bernoulli_ invenit $ummas quarundam $erierum <math>1 ± {1 / 2^λ} +{1 / 3^λ} ± {1 / 4^λ} + &c.</math> ex radicibus æquationis <math>x - A + {A^3 / 2 · 3} - &c. = 0</math>, vel
æquationis <math>x - {A^2 / 1 · 2} + {A^4 / 2 · 3 · 4} - &c. = 0</math>; _Eulerus_ invenit $ummas
plurium $erierum con$imilis formulæ ex ii$dem principiis & ra-
dicibus æquationis <math>e^±x ± e^∓x = (1 ± {x / i})^i ± (1 ± {x / i})^i = &c. =
v^i ± w^i = 0</math>; a _Cote$io_ datis; & exinde ex $ub$titutione plures de-

<pb xxiv>[0034]<rh>PRÆFATIO.</rh>
duxit; quarundam $erierum e divi$ione rationalium functionum inte-
grorum vel primorum numerorum, & ex comparatione $erierum ex-
inde ortarum cum $eriebus e prædictis _Joannis Bernoulli_, &c. deductis
$ummas inve$tigavit; invenit etiam $ummam $eriei <math>1 + {x / 2^2} + {x^2 / 3^2} +
&c.</math>, cum <math>x = {1 / 2}</math>; _Landen_ eandem invenit, cum <math>x = {1 / 2}</math> & cum $int duo
alii valores <math>α & β</math> quantitatis <math>x</math>: in hoc libro exinde deteguntur $um-
mæ $erierum, quarum generales termini $unt <math>{(a′z^m + b′z^m-1 + &c.) x^z / (a + n z)^h (b + n z)^h″
(c + n z)^h″ . &c.}</math>, ubi <math>m, h, h′, h″</math>, &c. $unt integri numeri, & <math>x vel = 1
vel {1 / 2} vel α vel β</math>. Cum duo vel plures divi$ores denominatoris ge-
neralis termini $int inter $e æquales, tum non deduci pote$t $eriei $um-
ma ex $ummis $erierum, quæ non habent totidem divi$ores inter $e
æquales. _Landen_ dedit limites ad quos appropinquant ultimo quæ-
dam quantitates; in hoc libro adjiciuntur de his quædam nova princi-
pia. _Naudei_ problemata ab _Eulero_ in re$olutionem aliorum problema-
tum transformantur, at non re$olvuntur; con$imilia quam plurima
facile adjici po$$unt, quorum nonnulla in hoc libro traduntur. 2. Da-
tur nova methodus inveniendi approximationes ad fluentes fluxionum
<math>{x^z x^. / 1 ± x^h}</math>, ubi <math>h:α</math> non habet rationalem rationem. 3. Cum dentur
quantitates infinite magnæ, docetur methodus detegendi earum $um-
mam, differentiam, &c.</p>
<p>_Jacobus Bernoulli_ primus (quod $cio) edidit $ummam $eriei <math>^n √ (a +
<_>n</_>√ (a + √ (a + &c.))) = x = <_>n</_>√ (x + a)</math>; _Joannes Bernoulli_ a$$erit $e
primum inveni$$e; dedit etiam $eries <math>x = ^n √ (a ^n √ (a &c.)) = ^n √ (ax)</math>,
unde <math>x^n-1 = a</math>; ob$ervatur in hoc libro, quod in $ingulis hi$ce for-
mulis plures dentur radices quam ex hi$ce $olutionibus deducantur.
_Joannes Bernoulli_ dedit regulam pro inveniendâ $ummâ <math>1^m + 2^m +
3^m + 4^m + &c.</math> ubi <math>m</math> e$t integer numerus; legem, quam ob$ervat hæc
$eries, debemus _I$aaco Milner._</p>

<pb xxv>[0035]<rh>PRÆFATIO.</rh>
<p>_Eulerus_ invenit proportionem, quam medius terminus $eriei <math>1 +
n x + &c. = (1 + x + x^2)^n</math> habet ad $ummam $eriei, &c.</p>
<p>Hoc opus moliebatur mecum, & in partes di$tribuebat eruditi$$imus
vir, mihique amici$$imus _Joannes Wil$on_ armiger, & laboris quidem
totius particeps fui$$et; ni $tudia foren$ia, quibus $e dedit, vetui$$ent;
adjutore tali tantoque non $ine magno & meo & rei mathematicæ
damno carui; ab illius enim ingenio hæc $cientia incrementa e$$et cap-
tura ab aliis vix aut ne vix quidem $peranda.</p>
<p>Jam re$tat ea, quæ in hoc libro continentur, breviter recen$ere.</p>
<p>In primo capite datur regula, quam ob$ervat fluxio cuju$cunque
exponentialis. 2. Animadvertitur in quadraturis minime ad rem
conducere, quomodo partes generantur, e. g. utrum per motum, nec-
ne; $olummodo opus e$t partes corre$pondentibus quantitatis a$-
$umptæ partibus convenire.</p>
<p>In $ecundo capite ob$ervatur dimen$iones variabilis quantitatis <math>x</math>
in fluxione minores e$$e quam ejus dimen$iones in fluente per unita-
tem; ni dimen$iones fluentis $int nihilo æquales, in quo ca$u dimen-
$iones in fluxione minores erunt quam ejus dimen$iones in fluente
per quantitatem majorem quam unitatem; unde con$tat, $i dimen-
$iones quantitatis <math>x</math> in numeratore $int minores per unitatem quam
ejus dimen$iones in denominatore, ejus fluentem in finitis terminis
haud exprimi po$$e. 2. Animadvertitur; $i modo fluxio, quæ $it
functio quantitatis <math>x^n</math> in <math>x^.</math>, reducatur ad functionem quantitatis x<_>-n</_> in
x^., vel ad functionem cuju$cunque quantitatis, quæ algebraicam habeat
relationem ad x<_>n</_> in <math>x^.</math>; & utriu$que fluens per vulgares methodos detega-
tur, & $erierum alterutra terminetur, tum (rebus recte di$po$itis) ter-
minari alteram. 3. Datur methodus inveniendi fluentem fluxionis <math>(a +
b x^n + c x^2n + &c. + (d + e x^n + f x^2n + &c.) (p + q x^n + r x^2n + &c.)^λ)^ν ×
(A + B x^n + &c. + &c.) x^.</math>, $i modo in finitis terminis exprimi po$$it.
4. Sit fluxio <math>(A + B)^m × (a + b x^n + &c. + &c.) x^.</math>; ubi <math>{A^. / B} = {B^. / A}</math> vel
magis generaliter $it <math>(A + B + C)^m × &c. x^.</math>, ubi <math>{A^. / B} = {B^. / C} = {C^. / A}</math>; & $ic
deinceps; tum non a$$umenda e$t quantitas <math>(A + B + C &c.)^m+1 ×

<pb xxvi>[0036]<rh>PRÆFATIO.</rh>
&c.</math> $ed <math>(A + B + C &c.)^m x &c.</math> pro fluente. 5. Traditur methodus
inveniendi fluentes in finitis terminis ($i modo per eos exprimi po$$int)
per infinitas $eries. 6. Traduntur fluentes quarundam fluxionalium
quantitatum non prius traditarum. 7. Quædam nova adjiciuntur de
correctionibus fluentium. 8. Ob$ervatur in corrigendis fluentibus
ea$dem radices $emper u$urpandas e$$e, i. e. nunquam - √ pro
+ √ $ub$tituendam e$$e. 9. Ex $ub$titutione inveniuntur diver$æ
fluentes eju$dem fluxionis inter $e æquales. 10. Animadvertitur
qua$dam e$$e exponentiales quantitates, quæ perpetuo mutantur de
po$$ibili in impo$$ibilem, & vice versâ de impo$$ibili in po$$ibilem.
11. Docetur methodus inveniendi, annon datæ fluentes inter datos
valores variabilis contentæ $int finitæ. 12. Ex datâ fluxione, cujus
fluens exprimit $ummam $eriei $ecundum dimen$iones quantitatis <math>x</math>
progredientis, deducitur fluxio, cujus fluens exprimit $ummam ter-
minorum prædictæ $eriei ad (<math>n</math>) di$tantias a $e$e po$itorum. 13.
Nonnullæ fluxiones non prius traditæ, quæ involvunt irrationales
functiones variabilis quantitatis <math>x</math> reducuntur ad fluxiones, quæ nul-
lam involvunt irrationalem quantitatem variabilis re$ultantis fluxionis.
14. Traditur methodus in genere detegendi, annon fluens datæ fluxi-
onis exprimi po$$it per finitos terminos, circulares arcus & logarith-
mos. 15. Datâ æquatione $imiliter involvente <math>X</math> & <math>x</math>, unde <math>x = φ:X</math>;
& fluxione <math>P x^.</math>, ubi <math>P</math> e$t functio quantitatis <math>x</math>; & in fluxione <math>P x^.</math>
pro <math>x & x^.</math> $criptis φ: <math>X & φ: X</math>; re$ultet <math>Q X^. = 0</math>; in quâ pro <math>X &
X^.</math> $cribantur <math>x & x^.</math>, exinde re$ulter <math>Q′ x^.</math>: $int <math>L & l, M & m</math> corre$pon-
dentes valores quantitatum <math>X & x</math>; tum erit fluens fluxionis <math>(P +
Q) x^.</math> inter valores <math>L & M</math> quantitatis <math>x</math> contenta, eadem ac fluens
inter valores <math>l & m</math> eju$dem quantitatis. 16. Sit fluxio <math>(a + b x^n +
.. x^m)^m+λ × x^pn+σn-1 x^.</math>, ubi <math>λ & σ</math> $unt quicunque integri affirmativi nu-
meri; tum ex quibu$cunque <math>r-1</math> independentibus fluentibus huju$ce
formulæ detegi po$$unt omnes aliæ eju$dem formulæ. 17. Per $ub-
$titutionem erui po$$unt fluxiones, quarum fluentes per pauciores
eju$dem formulæ exprimi po$$unt. 18. Sit <math>x^θ+zn+βm x (a + b x^n +
c x^m)^λ+π x^.</math>, ubi literæ <math>α, β & π</math> re$pective denotant affirmativos integros
numeros; tum ex datis <math>(α + β + π + 1)</math> fluentibus inter $e indepen-

<pb xxvii>[0037]<rh>PRÆFATIO.</rh>
dentibus fluxionum huju$ce formulæ detegi po$$unt fluentes omnium
fluxionum eju$dem formulæ, &c. 19. Sit fluxio <math>X^. $. y x^.</math>, ubi <math>X^.</math> e$t
fluxio, cujus fluens inveniri pote$t; & $it numerus fluentium inter $e
independentium in formulis fluxionum <math>Y x^. & X y x^.</math> contentarum re-
$pective <math>m & r</math>; tum ex <math>(m + r)</math> fluentibus fluxionum <math>X^. $. y x^.</math> inter $e
independentibus detegi po$$unt fluentes omnium fluxionum eju$dem
formulæ, &c. 20. Sit exponentialis fluxio <math>e^a + bx^n ..hxμn × x^r ± vn</math>, ubi <math>μ</math>
& <math>v</math> $unt integri numeri, tum ex fluentibus <math>(μ)</math> fluxionum prædictæ
formulæ independentibus acquiri po$$unt fluentes omnium fluxionum
eju$dem formulæ, &c. 21. Traditur nova generalis methodus inve-
niendi valorem fractionis, cum ejus numerator & denominator $imul
evane$cant. 22. Adjiciuntur quædam de nonnullis fluxionibus in
alias reducendis.</p>
<p>In capite tertio 1. ex datâ algebraicâ æquatione datur methodus
inveniendi æquationem, cujus radix e$t data fluxionalis quantitas.
2. Si <math>(n)</math> fluxionales æquationes ordinum <math>m, r, s, &c.</math> reducantur ad
unam, tum ob$ervatur æquationem re$ultantem non majorem quam
<math>m + r + s + &c.</math> ordinem habere; etiamque nonnulla de novis fluxio-
nalibus, &c. æquationibus in æquationes re$ultantes per operationem
introductis. 3. Datæ fluxionales æquationes <math>α. β. γ. δ. &c. = 0 &
π. ρ. σ. τ. &c. = 0</math> di$tingui po$$unt in $ub$equentes <math>α = 0 & π = 0;
α = 0 & ρ = 0; &c.; β = 0 & π = 0; β = 0 & ρ = 0; &c.</math> 4. Ex
datis duabus vel pluribus fluxionalibus æquationibus deducuntur duæ
vel plures aliæ, quarum variabiles quantitates $unt eædem. 5. Data
æquatio fluentes involvens reducitur ad fluxionalem, in quâ nulla
continetur fluens. 6. Datâ algebraicâ æquatione relationem inter <math>x
& y</math> exprimente; inveniuntur quidam novi ca$us, in quibus <math>x & y</math> fa-
cile exprimi po$$unt in terminis tertiæ <math>z.</math> 7. Datâ algebraicâ æqua-
tione relationem inter <math>x & y</math> exprimente; invenitur, annon fluens <math>(v)</math>
fluxionis, quæ e$t algebraica functio quantitatum <math>x & y</math> & earum flu-
xionum, in finitis terminis exprimi pote$t, viz. ex a$$umendo æquati-
onem algebraicam, quæ nece$$ario exprimit relationem inter <math>v & x</math>
vel <math>y</math>; idem per infinitas $eries perficitur. 8. Hinc datur methodus
inveniendi quam plurimas æquationes ad curvas, quarum areæ, &c. in

<pb xxviii>[0038]<rh>PRÆFATIO.</rh>
finitis terminis exprimi po$$unt. 9. Traditur nota, ex quâ $æpe dici
pote$t aream curvæ in finitis terminis non exprimi po$$e. 10. Summa e
$ingulis valoribus areæ curvæ, cujus æquatio relationem inter ab$ci$$am
<math>x</math> & ordinatam <math>y</math> de$ignans $it algebraica, in finitis terminis exprimi po-
te$t. 11. Sit data algebraica æquatio, cujus fluxio ducatur in quanti-
tates æquales ex datâ æquatione deductas, tum fluens generalis æqua-
tionis re$ultantis non erit data æquatio; & vice versâ eadem æqua-
tio præbeat plures fluxionales æquationes. 12. Nonnulla adjiciuntur
de correctione fluentium fluxionalium æquationum. 13. Datur me-
thodus detegendi, annon data æquatio $it generalis fluens datæ fluxi-
onalis æquationis. 14. Traduntur quædam de methodis inveniendi,
annon fluxionales æquationes $int integrabiles. 15. Datis <math>n.{n - 1 / 2}</math>
diver$is, i. e. independentibus æquationibus huju$ce generis <math>a $. y x^. +
b $. y^2 x^. + c $. y x x^. + &c.</math>, exinde deduci pote$t valor cuju$cunque
quantitatis huju$ce generis <math>p $. v z^. + q $. v^2 z^. + r $. v z z^. + &c.</math>, ubi
<math>n</math> e$t maxima dimen$io ad quam a$cendunt variabiles quantitates; <math>a,
b, c, &c., p, q, r, &c.</math> $unt invariabiles quantitates, & <math>a′ x + b′ y + c′ = z
& p′ x + q′ y + r′ = v</math>. 16. Datis generaliter contentis $olidorum a
rotatione curvæ circa tres axes generatorum, dabuntur contenta $oli-
dorum a rotatione eju$dem curvæ circa quo$cunque alios axes gene-
ratorum, &c. 17. Sit data fluxio <math>(a y + b x + c) x^. + (A y + B x +
C) y^. = 0</math>, ex datâ fluente fluxionis <math>y x^.</math> (quæ plerumque acquiri pote$t
e datâ æquatione) erui pote$t fluens omnis fluxionis formulæ <math>(h y^m +
(k + l x)y^m-1 + &c.) x^.; &c.</math> 18. Datâ algebraicâ vel fluxionali æqua-
tione <math>α = 0</math> & fluxionali quantitate <math>π</math>, in quibu$dam ca$ibus inveni-
tur ejus fluens ope datarum. 19. Datâ fluxionali vel fluxionalibus
æquationibus, inveniuntur functiones variabilium in iis contentarum
& earum fluxionum, quarum fluentes innote$cunt. 20. Si <math>M</math> $it
particularis valor fluxionalis æquationis, tum erit quæcunque functio
quantitatis _M_ valor fluxionalis æquationis. 21. Sint <math>α = 0 & π = 0</math>
duæ fluxionales æquationes <math>m & r</math> ordinum, quorum <math>m</math> minor e$t
quam <math>r</math>; & $it <math>K<x>α</x> + L<x>π</x>=V^.</math>, ubi <math>V</math> e$t fluxionalis quantitas ordi-
nis <math>(r - 1)</math>; duæ datæ fluxionales reducuntur ad duas fluxionales

<pb xxix>[0039]<rh>PRÆFATIO.</rh>
<math>V = con$t. & α = 0</math>. 22. Datâ fluente <math>n</math> ordinis, docetur metho-
dus ejus fluxionalem æquationem inveniendi. 23. Sunt <math>n</math> diver$æ
fluentes <math>α = 0, β = 0, γ = 0, &c.</math> primi ordinis; <math>n. {n - 1 / 2}</math> fluentes
$ecundi ordinis; &c.; <math>(n)</math> vero diver$i multiplicatores, qui reddunt
fluxionalem æquationem <math>(n)</math> ordinis integrabilem. 24. Generalis
fluens prædictæ fluxionalis æquationis erit <math>φ: (α, β, γ, δ, &c.)</math> 25.
Datis duabus quantitatibus <math>p & q</math>, quæ $unt functiones vel algebraicæ
vel fluxionales quantitatum <math>x & y</math>, traditur methodus detegendi, an-
non <math>p</math> $it functio quantitatis <math>x</math>. 26. Cum dimen$iones fluxionis ordi-
nis maxime $uperioris in datâ æquatione $int majores quam 1; tum
reducendæ $unt per extractionem ad unam dimen$ionem. 27. Non-
nulla adjiciuntur de methodis inveniendi fluentes fluxionalium æqua-
tionum $uperiorum ordinum. 28. Per infinitas $eries inveniuntur
multiplicatores datarum fluxionalium æquationum. 29. Dantur ca$us,
in quibus fluxionales æquationes, qui habent terminos variabilium <math>(m
& m-1) vel (m, m-1, m-2), &c.</math> dimen$ionum, integrari po$$unt.
30. Fluxionalis æquatio <math>(n)</math> ordinis, quæ $it homogenea, reducitur
ad fluxionalem æquationem <math>(n - 1)</math> ordinis. 31. Sint duæ vel tres
vel <math>m</math> homogeneæ fluxionales æquationes ordinis <math>(n)</math>, tres vel quatuor
<math>(m + 1)</math> variabiles quantitates <math>(x, y, z, &c.)</math> involventes; eæ reduci
po$$unt ad fluxionalem æquationem, cujus ordo non major e$t quam
<math>n m - 1</math>. 32. In $ub$titutionibus minime pro <math>x & y</math> $ub$tituantur
homogeneæ functiones nullarum dimen$ionum quantitatum <math>z & v</math>,
nam hæ quantitates reduci po$$unt ad functiones eju$dem quantita-
tis. 33. Detur æquatio algebraica, in quâ $imiliter involvuntur <math>x &
y</math>; inveniatur ejus fluxio <math>φ x^. + φ′ y^. = 0</math>; in hâc æquatione pro <math>y</math> in <math>φ</math>
& pro <math>x</math> in <math>φ′</math> $cribantur earum valores ex datâ æquatione deducti; &
æquationis re$ultantis generalis fluens for$an non erit data æquatio.
34. Detur fluxionalis æquatio <math>p x^. + q y^., & $it p^. = α x^. + β y^. & q^. =
π x^. + ρ y^.</math>; tum, $i <math>{β - p / 2}</math> $it functio quantitatis <math>x</math>, data æquatio red-
ditur integrabilis per multiplicatorem, qui e$t functio quantitatis <math>x</math>;
ex con$imilibus principiis per fluxionalem æquationem inferioris or-

<pb xxx>[0040]<rh>PRÆFATIO.</rh>
dinis invenitur, annon fluxionalis æquatio $uperioris ordinis recipiat
multiplicatorem, qui e$t functio quantitatum <math>x & x^.</math>. 35. Eadem prin-
cipia extenduntur ad detegendam fluentem fluxionalis æquationis <math>p y^. ^n
+ q y^. ^n-1 ... t y^. ^n-m = 0</math>, ubi 2 <math>m</math> minor e$t quam <math>n</math>; & <math>p</math> e$t functio quan-
titatum <math>x, x^., y, y^., .. y^. ^n-2m-1, &c.</math> 36. Per methodum in cor. 4. prob.
53. traditam detegi pote$t; annon data fluxionalis æquatio, quæ e$t
functio quantitatum <math>x, x^., y, y^., y^.., &c.</math> evadere po$$it integrabilis per
multiplicatorem, in quâ dantur omnes functiones quantitatum <math>y, y^.,
y^.., &c.</math>; & in eâ continentur variabiles quantitates, quæ denotant
functiones quantitatum <math>x & x^.</math> deducendas. 37. Sint <math>m</math> æquationes
<math>(m + 1)</math> variabiles quantitates involventes, quarum formulæ $int <math>a x^. ^n
+ b y^. ^n + &c. + a′ x^. t^. ^n-1 + b′ y^. t^. ^n-1 + ..a′ x t^.^n ^n + &c. + T t^.^n = 0, &c.</math>, tum
earum fluentes deducuntur. 38. Traditur methodus inveniendi <math>(l)</math>
fluxionales æquationes <math>(l + 1)</math> variabiles quantitates habentes, qua-
rum fluentes innote$cunt. 39. Generalis fluens fluxionalis æquatio-
nis deductæ ex transformando datam fluxionalem æquationem, in
quâ <math>x</math> fluit uniformiter in alteram, in quâ <math>y</math> fluit uniformiter, eadem
erit ac generalis fluens datæ fluxionalis æquationis. 40. Per infini-
tas $eries datur methodus inveniendi, annon fluens fluxionalis æqua-
tionis in finitis terminis exprimi po$$it. 41. Ex a$$umptis pro <math>x</math> & <math>y</math>
functionibus vel algebraicis vel fluentialibus quantitatis <math>z</math> deduci po$-
$unt fluxionales æquationes, quarum nec <math>x</math> per <math>y</math>, nec <math>y</math> per <math>x</math>, quam-
vis utraque per tertiam <math>z</math>, exprimi pote$t. 42. Inveniuntur quædam
functiones duarum diver$arum fluentialium quantitatum ex datâ
fluxionali æquatione deductarum, quæ erunt inter $e æquales. 43.
Sint <math>n</math> fluxionales æquationes <math>a = 0, &c.</math>, quarum ordo $it <math>l, (n + 1)</math>
variabiles quantitates <math>(x, y, z, &c.)</math> involventes; tum $int <math>m, m′, &c.</math>
functiones quantitatum <math>(x, y, z, &c.)</math> & earum fluxionum, quarum
ordines minores $unt quam <math>l</math>; & ita a$$umi po$$unt multiplicatores
<math>m, m′</math>, &c., ut exoriantur <math>n</math> diver$æ & independentes fluxionales æqua-
tiones, quæ integrari po$$unt.</p>
<p>In libro $ecundo 1. traduntur nonnulla nova de inveniendis incre-

<pb xxxi>[0041]<rh>PRÆFATIO.</rh>
mentis fluentialium quantitatum. 2. Sit quantitas <math>P</math> functio quan-
tatis <math>x</math>, cujus incrementum $it <math>Q</math>; tum erit <math>Q^.</math> æqualis incremento
fluxionis <math>P^.</math>, $i modo fluxio (<math>0^.</math>) incrementi (<math>0</math>) quantitatis <math>x</math> æqualis
$it incremento fluxionis <math>x^.</math>. 3. Incrementum <math>{a x^n + b x^n-1 + &c. / x(x + x)...(x + (n - 1)x^.)}</math>
in primâ editione huju$ce operis reducitur ad incrementa huju$ce ge-
neris <math>α + {β / x} + {γ / x(x + x)} + &c.</math>; & ob$ervatur, $i <math>β = b - n · {n - 1 / 2}a x<x>.</x> non $it = 0</math>, integralem in finitis terminis non exprimi po$$e. 4. Sit
incrementum fractio <math>{a x^m + b x^m-1 + &c. / x(x + x<x>.</x>) · (x + (n - 1)x<x>.</x>) × (x + p) .. (x + p + (m - 1) x<x>.</x> × &c.}</math>; ea reducitur ad integrabiles formulas, cum formu-
lis <math>{β / x}, {β′ / x + p}, &c.</math> adjunctis. 5. Si numerus literarum <math>o, p, &c</math>. $it <math>λ</math>,
tum ex <math>λ</math> independentibus integralibus huju$ce formulæ deduci po$-
$unt omnes eju$dem formulæ: & $i dimen$iones quantitatis <math>x</math> in nume-
ratore $int minores quam ejus dimen$iones in denominatore per duas
vel plures, tum <math>β + β′ + &c. = 0, &c.</math> 6. Dimen$iones quantitatis
<math>x</math> in incremento $emper erunt minores quam dimen$iones eju$dem
quantitatis in integrali per unitatem; ni dimen$iones quantitatis in
numeratore integralis æquales $int ejus dimen$ionibus in denomina-
tore, in quo ca$u minores erunt per quantitatem majorem quam
unitatem: hinc, $i dimen$iones in integrali minores $int per unitatem
quam ejus dimen$iones in incremento, tum ejus integralis in finitis
terminis non exprimi pote$t. 7. Si quicunque divi$or in denomi-
natore contineatur, qui non habeat alterum a $e di$tantem per <math>r x</math>,
ubi <math>r</math> e$t integer numerus; tum integralis in finitis terminis non ex-
primi pote$t. 8. Traditur methodus in genere inveniendi integralem
dati incrementi, $i modo finitis terminis exprimi po$$it, rejiciendo
fingulos ultimos terminos in denominatore contentos, &c. 9. Datur
methodus inveniendi integrales incrementorum, quæ duas vel plures
variabiles quantitates & earum incrementa involvunt. 10. Si inve-

<pb xxxii>[0042]<rh>PRÆFATIO.</rh>
niatur incrementum quantitatis <math>V</math> ordinis <math>m</math> ex hypothe$i quod <math>x</math>
$olummodo $it variabilis, & re$ultet quantitas <math>W</math>, deinde inveniatur
incrementum quantitatis <math>W</math> ordinis <math>n</math> ex hypothe$i, quod <math>y</math> $olum-
modo $it variabilis; eadem re$ultabit quantitas ac $i modo inveniatur
primo incrementum quantitatis <math>V</math> ordinis <math>n</math> ex hypothe$i quod <math>y</math> $o-
lummodo $it variabilis, & re$ultet quantitas <math>u</math>, deinde inveniatur in-
crementum <math>m</math> ordinis quantitatis <math>u</math> ex hypothe$i quod <math>x</math> $olummodo
$it variabilis; &c. 11. Si <math>A</math> $it integralis, quæ e$t functio quantitatis
<math>z</math> & ejus incrementorum; & a$$umatur æquatio, in quâ <math>x & z</math> $imili-
ter involvuntur, deinde inveniantur quantitas <math>z</math> & ejus incrementa in
terminis quantitatis <math>x</math> & ejus incrementorum; $cribantur hæ quanti-
tates pro $uis valoribus in <math>A</math>, & re$ultet <math>B</math>; in <math>B</math> pro <math>x</math> $cribatur <math>z</math> &
re$ultet <math>C</math>: $int <math>π & ρ, α & β</math> corre$pondentes valores quantitatum <math>x
& z</math>, tum integralis inter valores <math>α & β</math> variabilis <math>z</math> quantitatis <math>A + C</math>
contenta eadem erit ac integralis eju$dem quantitatis <math>A + C</math> inter valo-
res <math>π & ρ</math> eju$dem quantitatis <math>z</math>. 12. Traditur methodus inveniendi, an-
non integralis logarith. <math>{A / B}</math> inveniri po$$it ope logarithmorum, ubi <math>A
& B</math> $unt functiones variabilis <math>x</math>. 13. Datâ æquatione algebraicâ
<math>y^n + (a + b x) y^n-1 + &c, = 0</math>; traditur methodus inveniendi æqua-
tionem, cujus radix e$t quæcunque algebraica functio quantitatum
<math>x & y</math> & earum incrementorum. 14. Adjicitur nota, e quâ $æpe dici
pote$t prædictam æquationem in finitis terminis non exprimi po$$e.
15. Sit integralis æquatio <math>a x^r + b x^s y^t = c y^n</math>; & exinde con$tant
quantitates <math>x & y</math> in terminis quantitatis <math>z</math>, & con$equenter quæcun-
que algebraica functio quantitatum <math>x & y</math> & earum incrementorum
in algebraicis terminis quantitatis <math>z</math> & ejus incrementorum; & exinde
annon ejus integralis inveniri po$$it. 16. Datur methodus extermi-
nandi integrales quantitates ex datâ æquatione. 17. Nonnulla dantur
de correctione integralium incrementorum. 18. Ob$ervatur <math>n</math> e$$e
integrales primi ordinis incrementialis æquationis (<math>n</math>) ordinis; <math>n .{n - 1 / 2}</math> diver$as integrales $ecundi ordinis; &c. 19. Prædicta æquatio
<math>n</math> habet multiplicatores. 20. Generalis integralis incrementialis æqua-

<pb xxxiii>[0043]<rh>PRÆFATIO</rh>
tionis (<math>n</math>) ordinis erit quæcunque functio e $ingulis (<math>n</math>) generalibus
integralibus prædictæ æquationis; quæcunque functio ex $ingulis
<math>(n · {n - 1 / 2})</math> generalibus integralibus $ecundi ordinis prædictæ fluxi-
onalis æquationis; & $ic deinceps. 21. Sit æquatio <math>y<x>n′</x> + a y<x>n′ - 1</x> + b y<x>n′ - 2</x> ..
+ d y<x>′</x> + f y = 0</math>, & erit ejus generalis integralis <math>A e^αx + B e^βx + &c.
= y</math>, ubi <math>e^αx<x>.</x> = π, e^βx<x>.</x> = ρ, &c., & π, ρ, σ, &c.</math> $unt radices æqua-
tionis <math>v^n + a v^n-1 + &c. = 0</math>. Plura in hoc libro adjiciuntur de in-
tegralibus & incrementis, & incrementialibus æquationibus con$imilia
iis, quæ in priori libro de fluxionibus & fluxionalibus æquationibus
traduntur.</p>
<p>In libro tertio 1. datur ratio, quam habeant inter $e vera & appa-
rens convergentia. 2. Datâ lege, quam ob$ervant termini $eriei in
infinitum progredientis; traditur methodus digno$cendi, annon $eries
fit finita. 3. Dantur $eries, quæ $emper convergant ad omnes valores
earum incognitarum quantitatum; aliæ $eries, quæ $emper divergant.
4. Sit infinita æquatio <math>A = a x^r + b x^r+1 + &c.</math>; & $i coefficientes
haud cre$cant in majori quam quâcunque geometricâ ratione & fint
omnes affirmativæ tum datur una affirmativa radix & non plures;
&c. 5. Traditur æquatio, quæ habet infinitas & incognitas radices,
&c. 6. Docetur methodus reducendi duas vel plures æquationes in-
finitas in unam, ita ut incognitæ quantitates exterminentur. 7. Tra-
ditur methodus detegendi, annon data quantitas $it radix datæ infi-
nitæ æquationis. 8. Æquatio <math>1 - x + x^2 - x^3 + &c. = 0</math> habet
radicem <math>x = 1. &c.</math> 9. Sit æquatio <math>0 = a - b x + c x^2 - d x^3 +
&c.</math>; ubi <math>{b / a}</math> multo major $it quam <math>{c / b}, & {c / b}</math> quam <math>{d / c}, {d / c}</math> quam <math>{f / e}, &c.</math>; tum
erunt omnes radices datæ æquationis po$$ibiles: & <math>{a / b}</math> erit approxima-
tio ad minimam radicem; <math>{b / c}</math> approximatio ad $ecundam; &c. 10. Sint
<math>m</math> radices multo majores vel minores quam reliquæ; tum ex <math>(m + 1)</math>

<pb xxxiv>[0044]<rh>PRÆFATIO</rh>
primis vel ultimis terminis deduci po$$unt approximationes ad illas
radices; $i autem una radix multo major $it quam <math>m</math> radices, multo
autem minor quam reliquæ, tum datur ejus approximatio; & $ic de
pluribus. 11. Docetur methodus inveniendi maximam radicem datæ
æquationis, $i modo omnes $int po$$ibiles; & exinde rever$ionem $e-
riei <math>y = b x + c x^2 + &c.</math> 12. Traditur methodus inveniendi ap-
proximationes ad radicem datæ æquationis quæ multo major $it quam
<math>m</math> radices & multo minor quam reliquæ. 13. Si una radix multo
major vel minor $it quam quæcunque alia & vera ad <math>r</math> figuras; tum
ex <math>m + 1</math> primis vel ultimis terminis erui pote$t approximatio vera
ad <math>m × r</math> figuras prope. 14. Animadvertitur in æquationibus $upe-
riorum ordinum (ut maxime probabile e$t) numerum impoffibilium
radicum majorem e$$e quam numerum po$$ibilium, & con$equenter
maximum impo$$ibilem radicem majorem e$$e quam maximum
po$$ibilem, & methodum inveniendi approximationes per extracti-
onem radicum pote$tatum e $ingulis radicibus, &c. perraro in
æquationibus $uperiorum dimen$ionum u$ui in$ervire. 15. Erit <math>n{n - 1 / 2} · {n - r / r + 1} - n · n · {n - 1 / 2} .. {n - r + 1 / r} + n · {n - 1 / 2} · n · {n - 1 / 2} ..{n - r + 2 / r - 1} - &c. = 0 vel ± n · {n - 1 / 2} · {n - {r - 1 / 2}/{r + 1 / 2}}.</math> 16. Sit æquatio
<math>1 - (α + β + γ + &c.) {1 / x} + (α β + αγ + βγ + &c,) {1 / x^2} - (α β γ +
α β δ + &c.) {1 / x^3} + &c. = A = 0</math>; datur lex $eriei, quæ $it <math>A^n</math>. 17. Ex
terminis $eriei re$ultantis per divi$ionem numeratoris 1, &c. per da-
tam æquationem deducuntur approximationes ad maximam radicem
$ecundam, &c.; $i modo radices $int omnes po$$ibiles. 18. Approxi-
matio ad radicem inventa pendet ex hoc, nempe quo propius quan-
titas a$$umpta pro radice $it ad unam radicem quam ad reliquas. 19.

<pb xxxv>[0045]<rh>PRÆFATIO</rh>
Approximationes $ic inventæ ultimo convergent in majori quam
quâvis geometricâ progre$$ione. 20. Docentur ca$us, in quibus ap-
proximationes inventæ convergunt. 21. Si terminorum datæ æqua-
tionis $igna $emel $olummodo vel progrediantur <math>de + in +, vel -
in -</math>; vel mutentur <math>de + in -, vel - in +</math>; tum a$$umatur
quæcunque negativa quantitas in uno ca$u, affirmativa vero in al-
tero, pro approximatione ad radicem prædictam, & $emper conver-
gent approximationes per prædictam methodum inventæ. 22. Datis
approximationibus ad <math>r</math> radices datæ æquationis, inveniuntur appro-
ximationes magis appropinquantes. 23. Detur æquatio <math>A = 0</math>, ubi
<math>A</math> $it functio quantitatis <math>x</math>; fingatur <math>x</math> variabilis, & $it <math>a</math> valor quan-
titatis <math>x</math> prope; pro <math>x</math> $cribatur <math>a - e</math> & re$ultet æquatio <math>P - Q e +{1 / 2} R e^2 - {1 / 6} S e^3 + &c. = 0</math>; & ex terminis huju$ce æquationis deduci
po$$unt continuæ approximationes ad radices datæ æquationis. 24.
Ex datâ æquatione <math>n</math> invariabiles & incognitas quantitates involvente
& datis ad eas approximationibus, per <math>n</math> $ub$titutiones erui po$$unt
quantitates ad earum valores magis approximantes. 25. Datis in
finitis æquationibus duas vel plures incognitas quantitates habenti-
bus, invenitur una in terminis reliquarum. 26. Sit <math>y</math> $umma $eriei
$ecundum dimen$iones quantitatis <math>x, v</math> vero $umma $eriei $ecundum
dimen$iones quantitatis <math>z</math>; & detur $eries <math>w</math> $ecundum dimen$iones
quantitatum <math>z & x</math> progrediens; traditur methodus inveniendi <math>w</math> in
terminis quantitatum <math>y & v</math>. 27. Invenitur formula $eriei <math>p + a q x +
&c. = A</math>, cujus pote$tates <math>A^n</math> vel radices <math>A^{n / m}</math> $emper eandem ob$er-
vant legem. 28. Docetur methodus inveniendi approximationes
ad fluentes fluxionum reducendo datam fluxionem ad $eriem $ecun-
dum dimen$iones perparvarum quantitatum, vel variabilium, vel
unius vel plurium progredientem. 29. Generalis fluens fluxionis <math>y^. ^2 =
y z^.^2</math> erit <math>y = E x + F x′</math>, ubi <math>x = z + {z^3 / 2 · 3} + {z^5 / 2 · 3 · 4 · 5} + &c.,
x′ = 1 + {z^2 / 1 · 2} + {z^4 / 1 · 2 · 3 · 4} + &c., & z = log. (x + √ (1 + x^2)) =

<pb xxxvi>[0046]<rh>PRÆFATIO.</rh>
log. (x′ - √ (x′<_>2</_> - 1)), & E & F</math> $unt invariabiles quantitates ad li-
bitum a$$umendæ. 30. Datur $eries pro inveniendâ <math>$. {√(1 - cx^2)x^. / √(1 - x^2)}</math>,
ubi <math>c</math> e$t perparva quantitas; i.e. peripheriâ ellip$eos, quæ haud multum
differt a circulo. 31. Traduntur ca$us, in quibus $eries <math>a^m ± m a^m-1 x
+ &c. = (a ± x)^m, &c.</math> convergit. 32. Docetur methodus inveniendi,
annon $eries exorta ex reducendo algebraicam functionem quantita-
tis <math>x</math> ad $eriem (<math>P</math>) $ecundum ejus dimen$iones progredientem con-
vergat; etiamque annon $eries, quæ exprimit <math>$. P x^.</math>, convergat. 33. In
fluentibus, &c. inve$tigandis; docetur methodus interpolandi, ita ut $e-
ries maxime celeriter convergant. 34. Si transformetur fluxio, cujus
variabilis e$t <math>x</math>, in alteram cujus variabilis e$t <math>z</math>, quæ datam habeat
relationem ad <math>x</math>; & reducantur duæ fluxiones ad $eries $ecundum
dimen$iones quantitatum <math>x & z</math> re$pective progredientes; dantur con-
vergentiæ, quas habent inter $e duæ $eries. 35. Transformantur
datæ algebraicæ quantitates in terminos $ecundum dimen$iones quan-
titatis <math>x</math> progredientes, quarum formulæ $unt integrabiles. 36. <math>{a / z - α}= {a / z} + {a α / z (z + 1)} + {a α (α + 1) / z · z + 1 · z + 2} + {a α (α + 1) (α + 2) / z · z + 1 · z + 2 · z + 3} +
&c.</math> 37. Vera & apparens convergentia geometricæ $eriei e$t unifor-
mis. 38. Contineantur tres incognitæ quantitates (<math>x, y & z</math>) in
datâ æquatione, datur methodus inveniendi unam in terminis dua-
rum reliquarum. 39. Datâ æquatione involvente duas incognitas
quantitates <math>x & y</math>, datur $ub$equens methodus inveniendi <math>y</math> in termi-
nis quantitatis <math>x</math>; a$$umatur <math>y = A x^m + B x^n+m + C x^2n+m + &c.</math>; de-
inde <math>1^mo</math>. inveniatur approximatio ad coefficientem <math>A</math>; & exinde ap-
proximatio ad coefficientem <math>B</math>, & propior ad coefficientem <math>A</math>; & $ic
deinceps; &c. 40. Sit fluxionalis æquatio (<math>n</math>) ordinis, duas varia-
biles quantitates <math>x, y</math> & earum fluxiones involvens; ex eâ deducatur
$eries, quæ exprimat <math>y</math> in terminis quantitatis <math>x</math>; & $i modo ab initio
$eriei incipiat, $emper occurrent in eâ (<math>n</math>) invariabiles quantitates ad
libitum a$$umendæ. 41. In prædictâ re$olutione for$an occurrent re-

<pb xxxvii>[0047]<rh>PRÆFATIO.</rh>
$olutiones homogenearum fluxionalium æquationum ordinem (<math>n</math>)
non $uperantium. 42. Datâ fluxionali æquatione relationem inter
ab$ci$$am, ejus corre$pondentes ordinatas & earum fluxiones expri-
mente, invenitur fluxionalis æquatio relationem inter ab$ci$$am &
ejus corre$pondentes ordinatas ad a$ymptotos de$ignans. 43. Ex da-
tis relationibus inter valores <math>x & z</math> incognitæ quantitatis <math>x</math>, & inter
$ummas duarum $erierum exinde re$ultantium; traditur methodus in-
veniendi coefficientes ip$ius $eriei. 44. Promovetur regula vulgariter
dicta fal$i: viz. cum datæ approximationes ad duas vel plures radices
maxime accedant, vel cum dentur plures $ub$titutiones pro incognitâ
quantitate in datâ æquatione: & eadem perficiuntur in pluribus
æquationibus plures incognitas quantitates habentibus. 45. Erit
<math>x = ^m √ (a + {b / ^m √ (a + &c.)})</math>, re$olutio æquationis <math>x^m+1 - a x = b; &c.;
x = ^n √ (a + ^n √ (a + &c.))</math> radix æquationis <math>(x^n - a)^n - x^n &c. = 0.
&c.</math></p>
<p>In libro quarto 1. ad particularem ca$um, ubi <math>z<x>.</x></math> incrementum
quantitatis <math>z</math> di$tantiæ a primo $eriei termino e$t 1, applicantur ea,
quæ prius traduntur de fluxionalibus & incrementialibus æquationi-
bus. 2. Datur methodus transformandi æquationem inter $ummas
& terminos in incrementialem æquationem. 3. Summa nullius $e-
riei exprimi pote$t per algebraicam æquationem inter prædictam
$ummam & quantitatem <math>x</math>, $ecundum cujus dimen$iones progrediatur
$eries, relationem de$ignantem; ni dimen$iones quantitatis <math>x</math> vel ea-
rum differentiæ denotari po$$int per arithmeticas $eries; & differentiæ
inter dimen$iones quantitatis <math>z</math> di$tantiæ a primo $eriei termino in
numeratore & denominatore eædem $int: & non exprimi pote$t per
fluxionalem æquationem; ni prædictæ differentiæ $int eædem, vel uni-
formiter cre$cant vel decre$cant. 4. Sit $eries <math>a + b x + c x^2 + &c.</math>
$ecundum dimen$iones quantitatis <math>x</math> progrediens, cujus coefficientes
terminorum proxime $ub$equentium ad infinitam di$tantiam <math>:: r:1</math>;

<pb xxxviii>[0048]<rh>PRÆFATIO.</rh>
ducatur hæc $eries in functionem ip$ius <math>x = 0</math>, cum <math>x = α</math>; tum $i <math>x</math>
major $it quam <math>r</math>, $eries diverget; $in minor, converget. 5. Tran$-
formatur data $eries in alteram, cujus termini $int $ummæ e $ingu-
lis (<math>n</math>) $ucce$$ivis terminis. 6. Nova methodus adjicitur dividendi al-
teram quantitatem per alteram; & exinde detegendi $eries, quarum
$ummæ innote$cant. 7. Sit $eries <math>{1 / α · β · γ · δ · &c.} + {1 / α + 1 · β + 1 ·
γ + 1 · δ + 1 · &c.} x^b + {1 / α + 2 · β + 2 · γ + 2. δ + 2 · &c.} x^2b + &c.</math>,
& $i omnes reliquæ quantitates <math>γ, δ, &c.</math> di$tent a duabus <math>α & β</math> per
integros numeros; tum $ummæ $erierum inveniri po$$unt a fluentibus
fluxionum <math>{x^bα-1 x^. / 1 ± x^b} & {x^bβ-1 x^. / 1 ± x^b}</math>. 8. Inveniuntur $eries, quæ $unt gene-
rales vel particulares fluentes fluxionalis æquationis. 9. Ex datis
convergentibus $eriebus inveniuntur aliæ. 10. Deteguntur $ummæ
$erierum, quarum $inguli termini dantur ex infinitis $eriebus. 11.
Conce$sâ methodo detegendi generaliter annon una data $eries vel
quantitas $it algebraica functio aliarum, traditur methodus inveniendi,
annon detur algebraica vel fluxionalis relatio inter $ummam & ter-
minos datæ $eriei. 12. Generalis fluens æquationis <math>y^. ^n = y x^. ^n</math> erit <math>a (1
+ {x^n / 1 · 2 .. n} + {x^2n / 1 · 2 .. 2n} + &c.) + b x (1 + {x^n / 2 · 3 .. n + 1} + &c.) +
&c.</math> 13. Invenitur quantitas, quæ in $eriem ducta, præbeat $eriem
magis convergentem. 14. Inveniuntur ca$us, in quibus differentiæ
$ucce$$ivæ celeriter convergant, &c. 15. In detegendâ $ummâ $eriei,
cujus generalis terminus $it <math>{a z^m + b z^m-1 + &c. / z^n + p z^n-1 + &c.}</math> vel quæcunque alia
determinata functio quantitatis <math>x</math>, primum invenienda e$t $umma ter-
minorum, u$que donec terminus <math>z</math> evadat major quam maxima radix
æquationum <math>a z^m + b z^m+1 + &c. = 0 & z^n + p z^n-1 + &c. = 0, &c.</math>
16. Transformatur æquatio relationem inter $ucce$$ivas $ummas &

<pb xxxix>[0049]<rh>PRÆFATIO.</rh>
terminos de$ignans in infinitam fluxionalem æquationem relatio-
nem inter $ummam <math>S</math> vel terminum <math>t</math>, ejus fluxiones, & <math>z</math> di$tantiam
a primo $eriei termino & <math>z^.</math> exprimentem; & nonnuliæ $eries tradun-
tur. 17. Ex $ummis $erierum primi ordinis deducuntur $ummæ om-
nium $erierum $uperiorum ordinum. 18. Inveniuntur $ummæ $erie-
rum, quarum termini $unt irrationales. 19. Datur convergentia inter-
polationum $erierum. 20. Traditur methodus interpolationis $erie-
rum, cum numerus factorum in terminis contentorum non $it idem.
21. Dantur $eries interpolabiles. 22. Etiamque $ummæ $erierum
<math>{1 / 1 · 2 .. m} ± {x^n / 1 · 2 .. n + m} + {x^2n / 1 · 2 .. 2n + m} ± &c.</math> 23. Ex datis
fluentibus fluxionum <math>$. {x^. / 1 + a x^n}, $. {x^. / x} $. {x^. / 1 + a x^n}, $. {x^. / x} $. {x^. / x} $. {x^. / 1 + a x^n},
&c.</math> ad <math>h - 1, &c.</math>; terminos datur $eries pro inveniendâ $ummâ $eriei, cu-
jus generalis terminus e$t <math>{± a^z / nz + l. (nz + 1)^h-1}&c.</math> 24. Traduntur nonnulla
problemata _Naudeanis_ haud multum di$$imilia. 25. Adjicitur methodus
inveniendi continuum contentum datæ quantitati æquale. 26. Dan-
tur diver$æ $eries pro inveniendis $inubus & co$inubus arcuum qui
inter $e $unt <math>:: n:m</math>; & methodus inveniendi approximationes ad ra-
dices æquationis per $inus & co$inus. 27. Invenitur aggregatum plu-
rium fractionum, quæ $int infinitæ. 28. Adjungitur nova methodus in-
veniendi approximationem ad fluentem fluxionis <math>{x^α x^. / 1 ± x^b}</math>, cum <math>b:α</math> non
habeat rationalem rationem; quæ ad omnes fluxiones applicari pote$t.
29. Datur facilis methodus detegendi; annon $umma $eriei, cujus ge-
neralis terminus e$t <math>{a′ / α + z · β + z · γ + z · &c.}</math>, in finitis terminis ex-
primi po$$it: methodus generaliter detegendi, annon $umma omnis
$eriei, cujus generalis terminus $it data functio quantitatis <math>x</math>, in finitis
terminis exprimi po$$it, in methodo incrementorum traditur. 30.
Per $implices æquationes deducitur $umma $eriei, cujus generalis ter-

<pb xl>[0050]<rh>PRÆFATIO.</rh>
minus e$t <math>{a′ z^m + b′ z^m-1 + &c. / (α + z)^b (β + z)^b′ (γ + z)^b″ &c.}</math>, ex $ummis $erierum qua-
rum generales termini re$pective $unt <math>{1 / α + z}, {1 / (α + z)^2}, {1 / (α + z)^3} ..{1 / (α + z)^b}; {1 / β + z}, {1 / (β + z)^2} .. {1 / (β + z)^b}, &c.</math> 31. Invenitur $umma
$eriei, cujus generalis terminus e$t <math>{a′ z^m + b′ z^m-1 + &c. / 1 · 2 · 3 .. n z × m′ + z · m″ + z · &c.}</math>,
ubi <math>m′ & m″, &c.</math> $unt inæquales quantitates, per circulares vel logarith-
mos, &c. 32. Ducantur $ucce$$ivi termini in qua$cunque <math>m</math> arith meticas
independentes $eries, & ex $ummis <math>(m + 1)</math> $erierum re$ultantium
erui po$$unt per $implices æquationes omnium huju$ce generis $um-
mæ. 33. Sit <math>P = A + B x^n + C x^2n + &c.</math>; tum ex datis <math>P, $. x^α P^.,
$. x^β P^., &c.</math> erui pote$t $umma $eriei, cujus generalis terminus e$t
<math>{φ (a′z^m + b′z^m-1 + &c.) / α + z · β + z · γ + z · &c.}</math>; $i modo φ $it generalis terminus datæ $e-
riei <math>{P^. / x^.}</math>. 34. Erit <math>{1 / α} · {1 / β - α} · {1 / γ - α} · &c. + {1 / β} · {1 / α - β} · {1 / γ - β} · &c. +{1 / γ} · {1 / α - γ} · {1 / β - γ}.&c. + &c. = {1 / α β γ &c.}</math>. 35. Sit <math>(a + b x^n + c x^2n
+ ..x′^n)^k x^θ = x (A + B x^n + C x^2n + &c.)</math>; tum ex <math>(l - 1)</math> fluentibus
independentibus fluxionum formulæ <math>(a + b x^n + c x^2n + &c.)^k ± i x^θ±hn x^.</math>,
ubi <math>b & i</math> $unt integri numeri, deduci po$$unt $ummæ omnium $erie-
rum, quarum generales termini $unt <math>{φ (a′ z^m + b′ z^m-1 + &c.) / θ ± (r + zn) · θ ± (s + zn) · θ ± (t + z n · &c.)}</math>,
ubi <math>r, s, t, &c.</math> $unt diver$i integri numeri, & <math>φ</math> e$t
generalis terminus $eriei <math>A + B x^n + &c.. &c.</math> 36. Sit <math>(a + b x^n)^m
= a^m + m a^m-1 b x^n + &c.</math>; tum per conicas $ectiones inveniri pote$t
$umma $eriei, cujus generalis terminus e$t <math>m · {m - 1 / 2}...{m - z + 1 / z}× {a′ z^m′ + b′ z^m′-1 + &c. / n z + r n · n z + s n · n z + t n · &c.}</math>; $i modo <math>r, s, t, &c.</math> $int in-

<pb xli>[0051]<rh>PRÆFATIO</rh>
æquales fractiones, quarum denominatores vel $unt 1 vel 2: inveniri
pote$t per arcus conicarum $ectionum, $i denominatores non $int
majores quam 1, 2, 3 vel 4: & $ic e fluentibus in capite 2. libri 1.
traditis, erui po$$unt plures huju$modi $eries. 37. Erunt <math>x^-β $. x^β-α-1 x^.
$. x^α-1 x^. P = x^-α $. x^^α-β-1 x^. $. x^β-1 x^. P, &c.</math> 38. Dantur methodi inve-
niendi $ummabiles $eries, quæ $unt inter $e æquales. 39. Traditur
fluens fluxionis <math>s^(q) ^m · c^(q) ^n × {q x^. / √(1 - x^2)},</math> ubi <math>s^(q) & c^(q)</math> $unt $inus & co$i-
nus arcûs <math>qA; & A</math> e$t arcus, cujus co$inus e$t <math>x</math>. 40. Datur nova me-
thodus differentiarum, in quâ ex terminis ad di$tantias <math>n-1, n-2,
n-3, &c.,</math> a primo, qui $int <math>S^n-1, S^n-2, S^n-3, &c.</math> acquiri pote$t termi-
nus <math>S^n+m</math> ad di$tantiam <math>n+m</math> a primo; erit enim <math>S^m+n = (m + n)S^n-1
-(m + n) · {m + n-1 / 2} S^n-2 + &c.</math>; datis etiam <math>S^n, S^-n, S^n-1, S^-n+1,
S^n-2, S^-n+2, &c.</math> traditur lex pro <math>S^m</math>. 41. Adjicitur methodus corre$pon-
dentium valorum, i. e, $int <math>S^α & α, S^β & β, S^γ & γ &c.</math>, corre$pon-
dentes valores quantitatum <math>x & y</math>, invenitur <math>y = {x-α · x-β · &c. / α-β · α-γ}S^α + {x-a · x-γ · &c. / β-α · β-γ · &c.} × S^β + &c.</math> in omnibus his methodis cor-
re$pondentium valorum, $i modo inventæ quantitates fingantur nihilo
æquales, re$ultant æquationes, quæ præbebunt approximationes per
regulas fal$i datas. 42. Si modo detur formula functionis quantita-
tum <math>x, y, z, &c</math>., quarum tot corre$pondentes valores dentur, quot
incognitæ quantitates in datâ functione contineantur; tum ex iis de-
duci pote$t functio quæ$ita. 43. Dentur corre$pondentes valores
trium vel plurium quantitatum <math>x, y, z, &c.</math> inveniuntur quantitates,
quæ habeant prædictos corre$pondentes valores. 44. Datur correctio
datæ quantitatis, quæ invenit veros corre$pondentes valores in <math>n</math> ca$i-
bus, fallit vero in <math>m</math>. 45. A$$eritur $ummam omnium fractionum
<math>{A / α-β · α-γ · α-δ · &c.} + {B / β-α · β-γ · β-δ} + &c</math>., ubi <math>A</math> e$t
eadem rationalis & integralis functio quantitatis <math>α</math> ac <math>β</math> e$t quantita-

<pb xlii>[0052]<rh>PRÆFATIO.</rh>
tis <math>β; &</math> in <math>A</math> $imiliter involvuntur quantitates <math>β, γ, δ, &c.</math>; & $imi-
liter etiam ac quantitates <math>α, γ, δ, &c</math>. in <math>B</math>; &c. e$$e rationalem & in-
tegralem functionem quantitatum <math>α, β, γ, &c.</math> in quâ $imiliter invol-
vuntur prædictæ quantitates <math>α, β, γ, &c.</math>, quæ habeat tot dimen$iones
prædictarum quantitatum, quot $it differentia inter dimen$iones quan-
titatum <math>A & α-β. α-γ · α-δ. &c.;</math> $i differentia $it 0, tum erit
data quantitas; $i negativa $it, tum erit prædicta $umma = 0. 46.
Ad finem adjungitur methodus deductionis & reductionis; cujus po-
tius mentionem feci, quam aliquid particulariter adjecerim, ut via in
hâc maxime generali $cientiâ aliis $ternatur; ad quod maxime con-
feret tractatus de generalibus functionibus.</p>
<p>Quædam etiam alia adjiciuntur nova non alibi petenda; queri li-
ceat opera mathematica exterarum gentium in hanc academiam per-
raro migra$$e; unde evenit me nunquam vidi$$e opera multorum præ-
clarorum mathematicorum.</p>
<p>Pag. 147. dele Fig. 1. p. 95; pag, 467. 1. 17. pro _n_-1, lege _n_; l. antepen. pro _n_, lege _n_ + 1.</p>
<fig/>

<pb>[0053]
<h>MEDITATIONES ANALYTICÆ</h>
<h>CAPUT I.
_De Fluxionibus fluentium inveniendis._</h>
<h>PROB. I.</h>
<p it>DATA metbodo inveniendi $ucce$$ivos terminos convergentis $eriei
<math>a + b + c + d + e + f + &c</math>. in infinitum progredientis, $ummam
ejus invenire.</p>
<p>Addatur primus terminus <math>(a)</math> ad $ecundum <math>(b)</math>, & eorum $umma
<math>(a + b)</math> ad tertium <math>(c)</math>, trium terminorum $umma <math>(a + b + c)</math> ad
quartum <math>d, &</math> $ic deinceps; & quoniam $eries e$t convergens, hæ $uc-
ce$$ivæ $ummæ ad $ummam totius $eriei vergunt, & ultimo propius
accedunt ad eam quam pro quâvis datâ differentiâ: quæret vero ali-
quis, & recte quidem, quomodo cogno$ci pote$t has $ummas ad $um-
mam totius $eriei perpetuo vergere, & ultimo propius ad $e invicem
accedere quam pro datâ quâvis differentiâ: cui re$pondendum e$t,
hanc e$$e methodum: continuo inveniantur duæ quantitates, quarum
una major e$t quam quæ$ita $umma, altera vero minor, i. e. limites
inter quos interponitur $umma quæ$ita, & $i hi limites continuo ad
$e invicem vergant, & denique $i hi limites ultimo probari po$$int
propius ad $e invicem accedere quam pro quâvis datâ differentiâ,
tum hæ $ummæ propius ad $e invicem convergunt, & ultimo pro-
pius accedunt quam pro datâ quâvis differentiâ.</p>
<p>F<sc>IG</sc>. 1. Ex. Sit curva <math>AbcdBA</math> rectâ lineâ <math>AB</math> & curvâ <math>AcB</math> com-
prehen$a; & ductis tangentibus ad puncta curvæ <math>A & B</math>, $int interni
anguli <math>BAC & ABC</math> $imul a$$umpti haud majores quam duo recti:
ducatur linea <math>f c i</math> parallela lineæ <math>A B</math>, & tangens curvam in puncto

<pb 2>[0054]<rh>DEFLUXIONIBUS</rh>
<math>c</math>, & agantur lineæ <math>A c, c B</math>; erit in$criptum triangulum vel æquale
vel majus quam dimidium trapezii <math>A B i f A</math>, & con$equenter majus
quam dimidium areæ curvilineæ: & $ic ducantur lineæ <math>e g & h k</math> cur-
vam tangentes in punctis <math>b</math> & <math>d</math>, parallelæ autem lineis <math>A c & B c,</math> &
erunt triangula <math>A b c & c d B</math> re$pective majora quam dimidia trape-
ziorum <math>A e g c</math> & <math>c h k B</math>, ergo majora erunt quam dimidia curvilinea-
rum arearum <math>A b c A & c d B c;</math> & $ic continuo in$cribantur triangula,
quorum ba$es re$pective parallelæ $unt lineis tangentibus curvam in
eorum verticibus, & eodem modo probari pote$t hæc triangula in-
$cripta majora e$$e quam dimidia reliquarum curvilinearum arearum,
& exinde e curvilineâ areâ continuo aufertur quantitas major quam
dimidium reliquæ, &c. & con$equenter ultimo re$iduum erit minus
quam quæcunque data quantitas. Sit $eries, cujus primus terminus
<math>a</math> $it triangulum <math>A c B</math>, $ecundus vero <math>b</math> æqualis $it $ummæ duorum
triangulorum <math>A b c</math> & <math>c d B</math>, tertius vero terminus æqualis $it $ummæ
quatuor triangulorum $imiliter in$criptorum, &c.; hæc $eries erit con-
vergens, $ummæ enim $ucce$$ivæ <math>(a, a + b, a + b + c, &c.)</math> perpetuo
ad $ummam $eriei, i. e. ad aream curvæ <math>A b c d B A</math> vergunt, & ultimo
propius ad eam accedunt quam pro datâ quâvis differentiâ.</p>
<p>Cor. 1. Omnis quantitas, quæ continuo inter limites convergentis
$eriei ponitur, æqualis erit datæ $eriei $ummæ. Si non $it æqualis
datæ $eriei $ummæ, $it differentia <math>d</math>, & inveniantur limites <math>a b</math> & <math>a c,{a b c / e f g}</math> inter quos ponitur $eriei $umma & prædicta
quantitas, & qui propius ad $e invicem accedunt, quam pro differen-
tiâ <math>d</math>. Sint <math>e f</math> & <math>e g</math> re$pective $eriei $umma & prædicta quantitas, &
<math>e f</math> & <math>e g</math> majores erunt quam <math>a b</math>, minores vero quam <math>a c</math>; per hypo-
the$in vero differentia <math>f g=d</math> major e$t quam <math>b c</math>, ergo $umma <math>ef +
fg=eg</math> major erit quam <math>a c</math>, quod hypothe$i contradicit; unde <math>e f</math>
haud major vel minor erit quam <math>e g</math>. Q.E.D.</p>
<p>Hoc modo demon$travit Archimedes aream circuli æqualem e$$e
triangulo, cujus ba$is e$t ejus peripheria & perpendiculum radius.</p>

<pb 3>[0055]<rh>FLUENTIUM INVENIENDIS.</rh>
<p>Cor. 2. Sint duæ convergentes $eries, & $i $ingulus terminus unius
$eriei $it in datâ ratione ad $ingulum alterius terminum, tum prioris
$eriei $umma erit ad $ummam po$terioris $eriei in eâdem ratione.</p>
<p>Hujus corollarii demon$tratio eadem erit ac ea in prob. 2. lib. xii.
Eucl. elem. quam con$ulas.</p>
<h>PROB. II.</h>
<h it>Datis $ucce$$ivis $eriei $ummis, invenire terminos $ucce$$ivos.</h>
<p>Sint <math>S, S′, S″, S′″, S^4</math>, &c. $ucce$$ivæ $ummæ, & $eries vel finita, vel
in infinitum progrediens <math>a + b + c + d + e + &c.</math> i. e. $it
<math>S = a + b + c + d + e + &c.</math>
<math>S′= # b + c + d + e + &c.</math>
<math>S″= # c + d + e + &c.</math>
<math>S′″= # d + e + &c.</math>
&c. # &c. # &c. # tum <math>a=S-S′, b=S^1 -S^2,
c=S^2 -S^3, d=S^3 -S^4</math>, & $ic deinceps.</p>
<p>Ex. Sit di$tantia <math>(x)</math> termini quæ$iti a primo, & $int duæ $ucce$$ivæ
$ummæ <math>x × <r>x - e</r> × <r>x - 2e</r> × <r>x - 3e</r>...<r>x-<ol>n-2</ol>e</r> × <r>x-<ol>n-1</ol>e</r></math>, &
<math><r>x + e</r> × x × <r>x-e</r> × <r>x - 2e</r>...<r>x-<ol>n-2</ol>e</r></math>. Subtrahatur prior $umma e
po$teriori, i. e. de $ummâ <math><r>x + e</r> × x × <r>x-e</r> × <r>x - 2e</r>....<r>x-<ol>n-2</ol>e</r></math>
auferatur $umma <math><r>x-<ol>n-1</ol>e</r> × x × <r>x-e</r> × <r>x-2e</r>......<r>x-<ol>n-2</ol>e</r></math>
& re$iduum erit <math>ne × x × <r>x-e</r> × <r>x-2e</r>.....x-<r>n-2</r>e</math> terminus
quæ$itus.</p>
<p>F<sc>IG</sc>.2. Cor.1. Sit figura <math>a b c d e I A a</math> rectis <math>A a</math> & <math>A I</math> & curvâ <math>a b c I</math>
comprehen$a; in$cribantur & circum$cribantur parallelogramma $ub
ba$ibus <math>A B, B C, C D, &c.</math> æqualibus, quæ dicantur <math>e</math>; & lateribus <math>A a,
B b, C c, D d, &c.</math> figuræ lateri <math>A a</math> parallelis, contenta; & complean-
tur circum$cripta & in$cripta parallelogramma; & $it <math>x = AI</math>, &

<pb 4>[0056]<rh>DEFLUXIONIBUS</rh>
$umma in$criptorum parallelogrammorum <math>A b + B c + C d + &c.</math>
continuo $it <math>x × <r>x - e</r> × <r>x - 2e</r> .. × <r>x - <ol>n - 2e</ol></r> × <r>x - <ol>n - 1e</ol></r></math>; tum
$umma circum$criptorum parallelogrammorum erit proxima $umma,
viz. <math><r>x + e</r> × x × <r>x - e</r> × <r>x - 2e</r> .... <r>x - <ol>n - 2</ol>e</r></math>: circum$criptorum &
in$criptorum parellalogrammorum differentia per exemplum erit
<math>n e × x × <r>x - e</r> × <r>x - 2e</r> ... <r>x - <ol>n - 2</ol>e</r></math>. Sed hæc differentia æqualis
e$t parallelogrammo <math>Ab</math>, cujus ba$is e$t <math>e</math>; & con$equenter latus <math>Aa</math>,
$i parallelogrammum $it rectangulum, erit æquale; $in aliter propor-
tionale contento <math>{n e × x × <r>x - e</r> × <r>x - 2e</r> .. <r>x - <ol>n - 2</ol>e</r> / e} = n x × <r>x - e</r> ×
<r>x - 2e</r> .. <r>x - <ol>n - 2</ol>e</r></math>. Area vero curvæ inter duas prædictas $ummas
<math>x × <r>x - e</r> .. <r>x - <ol>n - 1</ol>e</r> & <r>x + e</r> × x × <r>x - e</r> .. <r>x - <ol>n - 2</ol>e</r></math>, viz. $ummas
circum$criptorum & in$criptorum parallelogrammorum $emper poni-
tur: $it <math>e = 0</math>, & hæ duæ $ummæ <math>x × <r>x - e</r> × <r>x - 2</r>e .. <r>x - <ol>n - 1</ol>e</r> &
<r>x + e</r> × x × <r>x - e</r> × <r>x - 2e</r> ... <r>x - <ol>n - 2</ol>e</r></math> fient <math>x × x × x × &c. = x^n</math>
inter $e & quantitati <math>x^n</math> æquales: $ed curvæ area inter has duas quan-
titates continuo ponitur, ergo area curvæ in hoc ca$u erit <math>x^n</math>: ordi-
nata <math>Aa</math> fuit <math>n × x × <r>x - e</r> × <r>x - 2e</r> ... <r>x - <ol>n - 2</ol>e</r></math>; quod, cum <math>e = 0</math>,
fit <math>n × x × x × x × &c. = n x^n-1</math>; ergo, $i area curvæ continuo $it <math>x^n</math>,
ejus ordinata corre$pondens erit <math>n x^n-1</math>.</p>
<p>Cor. 2. Sint ordinatæ <math>n × x × <r>x - e</r> × <r>x - 2e</r> ... <r>x - <ol>n - 2</ol>e</r></math>, & $umma
prædictorum in$criptorum parallelogrammorum erit <math><r>x + e</r> × x × <r>x - e</r>
× <r>x - 2e</r> ... <r>x - <ol>n - 2</ol>e</r></math>; & con$equenter, $i ordinata curvæ $it <math>n x^n-1</math>,
ejus area erit <math>x^n</math>.</p>
<p>Et $ic ratiocinari liceat de pluribus huju$cemodi quantitatibus.</p>

<pb 5>[0057]<rh>FLUENTIUM INVENIENDIS.</rh>
<h>THEOR. I.</h>
<p>F<sc>IG</sc>. 3. Fluxio vel velocitas quantitatis (<math>x = A p</math>) per <math>x^.</math> $emper de-
$ignetur, tum fiuxio ejus pote$tatis (<math>x^n = A P</math>) erit <math>n x^n-1 x^.</math>; fluat enim
uniformiter <math>x</math>, & $int tres $ucce$$ivi valores quantitatis <math>x</math> re$pective,
<math>x - 0, x, x + 0</math>; tum tres $ucce$$ivi valores quantitatis <math>x^n</math> erunt re$pective
<math><r n>x - 0</r> = x^n - n x^n-1 0 + n · {n - 1 / 2} x^n-1 0^2 - &c.
x^n = x^n ·
<r n>x + 0</r> = x<_>n</_> + n x<_>n-1</_> 0 + n · {n - 1 / 2} x<_>n-z</_> 0<_>2</_> + &c.</math>
$ubtrahantur hi tres $ucce$$ivi valores quantitatis (<math>x^n</math>) a $e in-
vicem, & re$ultant duæ $ucce$$ivæ differentiæ <math>n x^n-1 0 - n · {n - 1 / 2} x^n-1
0^2 + &c. & n x^n-1 0 + n · {n - 1 / 2} x^n-2 0^2 + &c.</math> quæ continuo cre$cunt;
unde, $i <math>x</math> fluat uniformiter, quantitatis (<math>x^n</math>) motus erit acceleratus:
$i vero $it motus acceleratus, tum $patium <math>n x^n-1 0 - n. {n - 1 / 2} x^n-2 0^2 +
&c.</math> minus, & <math>n x^n-1 0 + n. {n - 1 / 2} x^n-2 0^2 + &c.</math> majus erit quam $pa-
tium, quod corpus velocitate ad punctum <math>P</math> eodem tempore de$cri-
beret; ergo velocitas quantitatis (<math>x</math>) erit ad velocitatem quantitatis
(<math>x^n</math>) in minori ratione quam <math>0: n x^n-1 0 - n. {n - 1 / 2} x^n-1 0^2 + &c</math>: in
majori autem ratione quam <math>0: n x^n-1 0 + n. {n - 1 / 2} x^n-1 0^2 + &c.</math> i. e.
in minore ratione quam <math>x^.: n x^n-1 x^. - n. {n - 1 / 2} x^n-2 0 x^. + &c.</math> & in
majori quam <math>x^.: n x^n-1 x^. + n. {n - 1 / 2} x^n-2 0 x^. + &c.</math> ergo, $i velocitas
quantitatis (<math>x</math>) per <math>x^.</math> de$ignetur, velocitas quantitatis <math>x^n</math> inter duas

<pb 6>[0058]<rh>DEFLUXIONIBUS</rh>
quantitates <math>n x^n-1 x^. - n · {n - 1 / 2} x^n-2 0 x^. + &c. & n x^n-1 x^. + n × {n - 1 / 2}x^n-2 0 x^. + &c.</math> $emper ponetur, quicunque $it valor quantitatis (<math>0</math>).
Supponatur <math>0</math> nihilo æqualis, & duæ prædictæ quantitates fiunt <math>n x^n-1 x^.</math>
& inter $e æquales; unde velocitas quantitatis <math>x^n</math>, quæ inter eas po-
nitur, erit etiam <math>n x^n-1 x^.</math>.</p>
<p>Cor. I. Sit quantitas <math>x^{n / m}</math>, cujus fluxio requiritur; fingatur <math>x^{n / m} = y</math>,
& con$equenter <math>x^n = y^m</math>, unde <math>n x^n-1 x^. = m y^m-1 y^.</math>, & per reductionem
<math>y^. = {n x^n-1 x^. / m y^m-1} = {n / m}x<_>{n / m}-1
</_> x^.</math>.</p>
<p>Cor. 2. Sit quantitas <math>a x - b y + c z + d v, &c.</math> & $int <math>x^., y^., z^., v^.,
&c.</math> re$pective fluxiones quantitatum (<math>x, y, z, v, &c.</math>) i. e. earum $pa-
tia dato tempore uniformi motu de$cripta; & erit fluxio datæ quan-
titatis <math>a x^. - b y^. + c z^. + d v^. &c.</math> Velocitates vel fluxiones enim erunt
in eâdem ratione ac $patia uni$ormi motu dato tempore de$cripta;
hæc vero $patia $unt <math>a x^. - b y^. + c z^. + d v^., &c.</math> unde con$tat cor.</p>
<p>Cor. 3. Fluxio rectanguli <math>x y</math> erit <math>x y^. + y x^.</math>. Fluxio enim quadrati
<math><r 2>x + y</r> = (x^2 + y^2 + 2 x y) = 2 × <r>x + y</r> × <r>x^. + y^.</r> = 2 x x^. + 2 y y^. + 2 y x^.
+ 2 x y^.</math>; $ed fluxio quantitatis <math>x^2 + y^2</math> e$t <math>2 x x^. + 2 y y^.</math>, quæ $ubtra-
hatur e priori, re$ultat fluxio quantitatis <math>(2 x y) = 2 x y^. + 2 y x^.</math>, unde
fluxio rectanguli <math>(x y) = x y^. + y x^.</math>.</p>
<p>Cor. 4. Sit fractio <math>{x / y}</math>, cujus fluxio requiritur; fingatur <math>{x / y} = z</math>, &
exinde <math>x = y z</math>, & con$equenter <math>x^. = y z^. + z y^.</math>; unde <math>z^. = {x^. - z y^. / y} ={x^. - {x / y} y^. /y} = {y x^. - x y^. / y^2}</math>.</p>
<p>Cor. 5. Fluxio contenti <math>x y z</math> erit <math>x y × z^. + z × {. / x y}</math>, $ed <math>{. / x y} = x y^.
+ y x^.</math>; ergo fluxio contenti <math>x y z</math> erit <math>x y z^. + x z y^. + z y x^.</math>; & $ic
fluxio contenti <math>x y z v w</math> invenietur <math>x y z v w^. + x y z w v^. + x y v w z^.

<pb 7>[0059]<rh>FLUENTIUM INVENIENDIS.</rh>
+ x v w z y^. + y v w z x^.</math>: & fluxio contenti <math>x^n y^m z^r v^s</math> erit <math>n x^n-1 y^m z^r v^s x^.
+ m y^m-1 x^n z^r v^s y^. + r z^r-1 x^n y^m v^s z^. + s v^s-1 x^n y^m z^r v^.</math>; & $ic deinceps.</p>
<p>Cor. 6. Sit quantitas data quæcunque algebraica functio fluentium
quantitatum; $cribendo <math>y, z, v, &c.</math> pro quantitatibus in his functio-
nibus contentis inveniri pote$t ejus fluxio.</p>
<p>Ex. 1. Sit fluens <math><r m>a + b x^n + c x^2n + d x^3n + &c.</r></math> cujus fluxio requi-
ritur: $cribatur <math>y = a + b x^n + c x^2n + d x^3n + &c.</math> & exinde <math>y^. = n b x^n-1 x^.
+ 2 n c x^2n-1 x^. + 3 n d x^3n-1 x^.; & y^m = <r m>a + b x^n + c x^2n + d x^3n + &c.</r></math>
$ed fluxio quantitatis <math>y^m</math> e$t <math>m y^m-1 y^. = m × <r m-1>a + b x^n + c x^2n + &c.</r>
× <r>n b x^n-1 x^. + 2 n c x^2n-1 x^. + &c.</r></math> $i modo pro <math>y & y^.</math> $cribantur earum
valores <math>a + b x^n + c x^2n + &c. & n b x^n-1 x^. + 2 n c x^2n-1 x^. + &c.</math></p>
<p>Ex. 2. Sit fluens <math><r>a + b x^n + c x^2n + &c.</r> × <r λ>e + f x^n + g x^2n + &c.</r>
× <r μ>k + l x^n + m x^2n + &c.</r> × <r ,>p + q x^n + r x^2n + &c.</r> × &c.</math> & ejus
fluxio e prædictis principiis invenietur <math>n x^. × <r λ-1>e + f x^n + g x^2n + &c.</r>
× <r μ-1>k + l x^n + m x^2n + &c.</r> × <r γ-1>p + q x^n + r x^2n + &c.</r> × <r>b e k p +
λ f a k p + μ l a e p + v q a e k × x^n-1 + 2 c e k p + 2 λ g a k p + 2 μ m a e p
+ 2 v r a e k + b × e k q + e p l + k p f + λ f × a k q + a p l + b k p +
μ l × a e q + a p f + p e b + v q × a e l + a k f + k e b × x^2n-1 + &c.</r></math></p>
<p>D<sc>EF</sc>. Logarithmi dicuntur quantitates, quæ $unt proportionales
exponentibus pote$tatum vel radicum datæ quantitatis; e. g. $it <math>(A)</math>
logarithmus quantitatis <math>x</math>, tum <math>{n / m} A</math> erit logarithmus quantitatis <math>x^{n / m}</math>.</p>
<p>Hæ autem exponentes $unt men$uræ rationum quantitatum, ergo
logarithmi etiam erunt men$uræ earum rationum.</p>
<p>Cor. I. Hinc facile colligitur, $i modo logarithmi quantitatum <math>a & b</math>
dicantur re$pective <math>A & B</math>, logarithmum producti <math>a × b</math> e$$e <math>A + B</math>.</p>
<p>Cor. Hinc, $i quantitas (<math>x</math>) cre$cat proportionaliter, ejus logarith-
mus fluit uniformiter, i. e. $i quantitas (<math>x</math>) fluat eâdem ratione ac
quantitas ip$a, tum ejus logarithmus fluit uniformiter; & con$equen-

<pb 8>[0060]<rh>DEFLUXIONIBUS</rh>
ter quantitas <math>x</math> erit ad datam quantitatem <math>M:: x^.</math> ad fluxionem ejus
logarithmi <math>{M x^. / x}</math>; hinc fluxio logarithmi quantitatis (<math>x</math>) erit propor-
tionalis fluxioni <math>{x^. / x}</math>. Si vero pro <math>x & x^.</math> $cribantur <math>x + a & x^.</math>, re$ultat
<math>({M x^. / x + a})</math> fluxio logarithmi quantitatis (<math>x + a</math>).</p>
<h>PROB. III.</h>
<p>1. Sit exponentialis quantitas <math>x^y</math>, invenire ejus fluxionem: fingatur
<math>x^y = X</math>, unde <math>y × log. x = log. X</math>; hujus æquationis inveniatur
fluxio, & re$ultat log. <math>x × y^. + y × {x^. / x} = {X^. / X}</math>, unde <math>X × log. x, × y^. + X y{x^. / x} = X^.</math>; pro <math>X</math> $cribatur <math>x^y</math>, re$ultat fluxio quæ$ita <math>X^. = x^y × log. x
× y^. + y x^y-1 x^.</math>.</p>
<p>2. Sit data exponentialis quantitas <math>x^y × v</math>, & per præcedentem
methodum inveniri pote$t ejus fluxio <math>x^y v^. + v y x^y-1 x^. + v × x^y × log.
x × y^.</math>.</p>
<p>3. Sit exponentialis quantitas <math>x^y^x^v^w^&c.</math>, & ejus fluxio erit <math>y^x^v^w^&c. × x^y^x^v^w^&c. ^-1
x^. + x^y^z^v^w^&c. × z^v^w^&c. × y^z^v^w^&c. ^-1 × log. x × y^. + x^y^z^v^w^&c × y^z^v^w^&c. × v^w^&c. × z^v^w^&c. ^-1 × log. x ×
log. y × z^. + x^y^z^v^w^&c. × y^z^v^w^&c × z^v^w^&c. × w^&c. × v^w^&c ^-1 × log. x
× log. y × log.
z × v^. + x^y^z^v^w^&c. × y^z^v^w^&c × z^v^w^&c × v^w^&c × &c. w^&c.-1 × log. x × log. y ×
log. z × log. v × w^. + &c.</math> unde facile con$tabit lex, quam ob$ervat
hæc $eries.</p>

<pb 9>[0061]<rh>FLUENTIUM INVENIENDIS.</rh>
<p>4. Fluxio exponentialis <math>x^y^z^v^w^&c. × V</math> erit <math>V</math> × fluxion. exponent. <math>(x^y^z^v^w^&c.)
+ x^y^z^v^w^&c. × V^.</math>; hæc vero fluxio $ic exprimi pote$t <math>x^y^z^v^w^&c. × V^. +{V × y^z^v^w^&c. × A / x V^.} x^. + {z^v^w^&c. × B × x × log. x / y x^.} y^. + {v^w^&c. × C × y × log. y / z. y^.} z^. +{w^&c. × D × z × log. z / v z^.} v^. + {&c. × E × v × log. v / w v^.} w^. + &c.</math> ubi literæ <math>A,
B, C</math>, & præcedentes terminos re$pective denotant.</p>
<p>Sit <math>V</math> exponentialis quantitas, & facile con$tabit fluxio quæ$ita.</p>
<p>Et $ic inveniri pote$t fluxio contenti $ub pluribus exponentialibus
quantitatibus.</p>
<p>Cor. Si vero <math>x, y, z, &c.</math> $int quæcunque functiones quarumlibet
incognitarum quantitatum; & pro <math>x, y, z, &c.</math> & earum fluxionibus
in $erie prius traditâ $cribantur hæ functiones & earum fluxiones re-
$pective; invenietur fluxio quæ$ita.</p>
<p>5. Secundæ fluxiones inveniuntur e primis, tertiæ e $ecundis, &c.
eodem modo, quo primæ inveniuntur e fluentibus; ergo datæ fluxio-
nes pro fluentibus habeantur, & facile inveniri po$$unt earum flu-
xiones.</p>
<p>Ex. Sint <math>x^., x^.., x^..., &c.</math> re$pective prima, $ecunda, tertia, &c. fluxio-
nes quantitatis <math>(x)</math>; & $ic <math>y^., y^.., y^..., &c.</math> prima, $ecunda, tertia, &c. flux-
iones quantitatis <math>(y), &c.</math> invenire $ecundam, tertiam, &c. fluxionem
quantitatis <math>x^m y^n</math>: prima vero fluxio datæ quantitatis invenietur
<math>m x^m-1 y^n x^. + n y^n-1 x^m y^.</math>; ejus vero fluxio, quæ erit $ecunda fluxio
quantitatis <math>(x^m y^n)</math>, erit <math>m × <r>m - 1</r> x^m-2 y^n x^.^2 + 2 m n x^m-1 y^n-1 y^. x^. +
m x^m-1 y^n x^.. + n × <r>n - 1</r> y^n-2 x^m y^. ^2 + n y^n-1 x^m y^..</math>; & eodem modo inveniri
pote$t ejus <math>(x^m y^n)</math> tertia fluxio, &c.</p>

<pb 10>[0062]<rh>DE FLUXIONIBUS, &c.</rh>
<p>Cor. Hinc fluxio <math>n</math> ordinis rectanguli <math>x y</math> erit <math>x^. ^n y + n x^. ^n-1 y^. + n ×{n - 1 / 2} x^. ^n-2 y^. ^2 + n · {n - 1 / 2} · {n - 2 / 3} x^. ^n-3 y^. ^3 .. n x^. y^. ^n-1 + x y^. ^n</math>.</p>
<h>SCHOLIUM.</h>
<p>Duæ $unt methodi inveniendi areas curvarum, vel quod ad idem
redit, inveniendi $ummas quantitatum, quarum datur infinitus nu-
merus; altera e repetitis additionibus continuo vergit ad $ummam
quæ$itam, & ultimo propius ad eam accedit quam pro datâ quâvis
differentiâ, quod cogno$ci pote$t e limitibus $ucce$$ive deductis inter
quos con$i$tit quæ$ita $umma: hæc fuit methodus veterum. Altera
vero a$$umit $ummam tanquam quantitatem generaliter cognitam, &
ejus partes exinde deducit; & quantitatibus, quarum datur infinitus
numerus, fiunt prædictæ partes vel æquales vel propiores ad rationem
æqualitatis quam pro datâ quâvis differentiâ; & exinde concludit $um-
mam quantitatum, quarum datur infinitus numerus, e$$e a$$umptam
quantitatem: hæc fuit methodus recentiorum: in hâc vero methodo
minime refert, quo modo generantur partes ip$æ; $olummodo re$ert,
utrum partes quantitatis a$$umptæ quantitatibus, quarum datur in-
finitus numerus, conveniant, necne: $i vero partes con$iderentur tan-
quam motu generatæ, tum dicitur methodus fluxionum vel incre-
mentorum; $in aliter dici pote$t methodus integrationis.</p>

<pb 11>[0063]
<h>CAP. II.</h>
<h it>De inveniendis Fluxionum fluentibus.</h>
<h>PROB. IV.</h>
<p it>DATA quâcunque fluxione, quæ $it algebraica functio literæ <math>x</math> in <math>x^.</math>
ducta; invenire utrum ejus fluens exprimi pote$t terminis incognitæ
quantitatis <math>x</math>, necne.</p>
<p>Ca$. 1. Sit fluxio <math>a x^m x^.</math>, & ejus fluens erit <math>{a x^m+1 / m + 1}</math>.</p>
<p>1. Ca$. 2. Sit fluxio <math>x^θ-1 × <r λ-1>e + f x^n + g x^2n + k x^3n + &c.</r> ×
<r>a + b x<_>n</_> + c x<_>2n</_> + d x<_>3n</_> + &c.</r> × x^.</math>; a$$umatur pro ejus fluente <math>x^θ ×
<r λ>e + f x<_>n</_> + g x<_>2n</_> + k x<_>3n</_> + &c.</r> × <r>A + B x<_>n</_> + C x<_>2n</_> + &c.</r></math> tum per cor.
6. theor. 1. ejus fluxio invenietur <math>θ e A + <r>θ + λ n</r> f A + <r>θ + n</r> e B x^n +
<r>θ + 2 λ n</r> g A + <r>θ + n + λ n</r> f B + <r>θ + 2 n</r> e C x<_>2n</_> + &c. × x<_>θ-1</_> ×
<r λ-1>e + f x<_>n</_> + g x<_>2n</_> + &c.</r> x^.</math>; & ex æquatis inter $e corre$pondentibus
terminis re$ultabunt <math>θ e A = a, <r>θ + λ n</r> f A + <r>θ + n e</r> B = b, <r>θ + 2 λ n</r>
× g A + <r>θ + n + λ n</r> × f B + <r>θ + 2 n e</r> C = c, &c.</math> & exinde <math>A = {a / θ e},
B = {b - <r>θ + λ n</r> f A / <r>θ + n</r> e}, C = {c - <r>θ + 2 λ n</r> g A - <r>θ + n + λ n</r> f B / (θ + 2 n) e}, &c.</math></p>
<p>Ca$. 3. Sit fluxio <math>x^θ-1 × <r λ-1>e + f x^n + g x^2n + h x^3n + &c.</r> ×
<r μ-1>k + l x<_>n</_> + m x<_>2 n</_> + &c.</r> × <r>a + b x<_>n</_> + c x<_>2n</_> + d x<_>3n</_> + &c.</r> × x^.</math>; & ad mo-
dum præcedentem pro ejus fluente a$$umatur quantitas <math>x^θ × (e + f x^n
+ g x^2n + &c.)^λ × (k + l x^n + m x^2n + &c.)^μ × (α + β x^n + γ x^2n + &c.)</math>;
cujus inveniatur fluxio, & ex æquatis corre$pondentibus datæ & re-<aa>

<pb 12>[0064]<rh>DE INVENIENDIS</rh>
$ultantis fluxionis terminis inveniri pote$t prædicta fluens <math>x^θ ×
<r λ>e + f x<_>n</_> + g x<_>2n</_> + &c.</r> × <r μ>k + l x<_>n</_> + m x<_>2n</_> + &c.</r> × <r>{a / θ e k} (A) +
b - <ol>θ + λ n</ol> f k + <ol>θ + μ n</ol> e l A/θ + n e k
× x<_>n</_> + &c.</r></math> Et $ic deinceps.</p>
<p>Ca$. 4. Pro quantitatibus <math>a + b x^n + c x^2n + &c., d + e x^n + f x^2n
+ &c., g + h x^n + k x^2n + &c., &c.</math> $cribantur re$pective <math>R, S, T, &c.</math> &
$it data fluxio <math>(A + B x^n + C x^2n + &c.) x^θ-1 × R^λ-1 S^μ-1 T^ν-1 × &c. x^.</math>,
$i vero communem habeant divi$orem nonnullæ $ub$equentes quan-
titates <math>R, S, T, &c. & A + B x^n + C x^2n + &c.</math> tum ita reducatur
data fluxio, ut quantitates prædictæ nullum habeant communem di-
vi$orem, i. e. $imul colligantur $ub eodem vinculo omnes iidem divi-
$ores, qui in datâ fluxione continentur; e. g. $it <math>α + β x^n</math> communis
divi$or quantitatum <math>a + b x^n + c x^2n + &c. = R, d + e x^n + f x^2n +
&c. = S, & g + h x^n + k x^2n + &c. = T; i. e. <r>α + β x^n</r> × <r>l + m x^n + o x^2n
+ &c.</r> = a + b x^n + c x^2n + &c. = R, <r>α + β x^n</r> × <r>p + q x^n + r x^2n + &c.</r>
= d + e x^n + f x^2n + &c. = S, & <r>α + β x^n</r> × <r>s + t x^n + &c.</r> = g +
h x^n + k x^2n + &c. = T</math>; & reducatur data fluxio in hanc formulam
<math>x^θ-1 × <r>A + B x^n + C x^2n + &c.</r> × <r λ-1>α + β x^n</r> × <r λ-1>l + m x^n + o x^2n + &c.</r>
(R^λ-1) × <r μ-1>α + β x^n</r> × <r μ-1>p + q x^n + r x^2n + &c.</r> (S^μ-1) × <r ν-1>α + β x^n</r>
× <r ν-1>s + t x^n + &c.</r> (T^ν-1) = <r λ+μ+ν-3>α + β x^n</r> × <r λ-1>l + m x^n + o x^2n + &c.</r>
× <r μ-1>p + q x^n + r x^2n + &c.</r> × <r ν-1>s + t x^n + &c.</r> × x^θ-1 × <r>A + B x^n + &c.</r></math> a$$umatur
pro ejus fluente quantitas <math>x^θ × <r λ+μ+ν-2>α + β x^n</r> × <r λ>l + m x^n + o x^2n + &c.</r>
× <r μ>p + q x^n + r x^2n + &c.</r> × <r ν>s + t x^n + &c.</r> × <r>π + ξ x^n + σ x^2n + &c.</r></math>
per caput præcedens inveniatur ejus fluxio, & $upponantur datæ &
re$ultantis fluxionis termini corre$pondentes inter $e æquales; & ex-
inde inve$tigari po$$unt coefficientes <math>π, ξ, σ, &c.</math> quæ$itæ.</p>
<p>Ca$. 5. Sit data fluxio eadem ac in præcedente ca$u, & quantitatis
$ub eodem vinculo contentæ $int plures divi$ores inter $e æquales,

<pb 13>[0065]<rh>FLUXIONUM FLUENTIBUS.</rh>
viz. divi$ores huju$ce formulæ <math><r 2>α + β x^n</r></math> vel <math><r m>α + β x^n</r></math>, ubi litera <math>m</math> vel
binarium vel majorem numerum denotat; & ita reducatur data fluxio,
ut quantitatum $ub vinculis nulli inveniantur communes divi$ores,
vel in eâdem quantitate nulli inter $e æquales; deinde ex methodo
in præcedente ca$u traditâ inve$tigari pote$t fluens quæ$ita.</p>
<p>Ex. 1. Sit <math><r ν>α + β x^n</r> × <r>l + m x^n + o x^2n + &c.</r> = a + b x^n + c x^2n + &c.</math>
a$$umatur pro fluente quantitas <math>x^θ × <r νλ-ν+1>α + β x^n</r> × <r λ>l + m x^n + o x^2n + &c.</r>
× S^μ × T^ν × (π + ξ x^n + σ x^2n + &c.)</math>; & inveniatur ejus fluxio, quæ
fiat æqualis datæ fluxioni, & exinde detegi pote$t fluens quæ$ita.</p>
<p>Ex. 2. Sint <math><r ξ>α + β x^n</r> × <r θ>γ + δ x^n</r> × <r>l + m x^n + o x^2n + &c.</r> = a + b x^n +
c x^2n + &c. = R, & <r χ>α + β x^n</r> × <r φ>ε + ξ x^n</r> × <r>p + q x^n + r x^2n + &c.</r> =
d + e x^n + f x^2n + &c. = s</math>, & pro fluente fluxionis prædictæ
<math><r>A + B x^n + C x^2n + &c.</r> × x^θ-1 × R^λ-1 × S^μ-1 × T^ν-1 × &c. × x^.</math> a$$umenda
e$t quantitas <math>x^θ × <r ξ(λ-1)+χ(μ-1)+1>α + β x^n</r> ×
<r θ(μ-1)+1>ε + ξ x<_>n</_></r> × <r θ(λ-1)+1>λ + δ x<_>n</_></r> ×
<r λ>l + m x<_>n</_> + o x<_>2n</_> + &c.</r> × <r μ>p + q x<_>n</_> + r x<_>2n</_> + &c.</r> × T<_>ν</_> × &c.</math> & exinde
per methodum prius traditam deduci pote$t fluens quæ$ita.</p>
<p><n *> Cor. 1. Si fluxio $it fractio rationalis irreducibilis cum denomina-
tore ex duobus vel pluribus terminis compo$ito, re$olvendus e$t de-
nominator in divi$ores $uos omnes primos: & $i divi$or $it aliquis,
cui nullus alius e$t æqualis, curva quadrari nequit: $in duo vel plures
<math>λ</math> $int divi$ores æquales, viz. <math><r λ>α + β x^n + γ x^2n + &c.</r> = R^λ</math>, & $i
adhuc alii duo vel plures <math>μ</math> $int $ibi mutuo æquales & prioribus in-
æquales viz. <math><r μ>π + ξ x^n + σ x^2n + &c.</r> = T^μ</math>, tum pro fluente quæ$itâ
fluxionis <math>x^θ-1 × R^-λ × T^-μ × (a + b x^n + &c.) × x^.</math> a$$umenda e$t quan-
titas <math>R^-λ+1 × T^-μ+1 × x^θ × &c. × (A + B x^n + C x^2n + &c.)</math> & ex me-
thodo prius traditâ deduci po$$unt valores incognitarum coefficien-
tium <math>A, B, C, &c.</math> in hoc ca$u <math>λ & μ</math> $unt integri numeri.</p>
<p>Cor. 2. Si fluxio $it fractio irreducibilis, & ejus denominator con-
<fn *>Newt, Quadr, Curv. p. 54.</fn>

<pb 14>[0066]<rh>DE INVENIENDIS</rh>
tentum $it $ub factore rationali Q & factore $urdo irreducibili <math>R^π</math>;
inveniendi $unt lateris _R_ divi$ores omnes primi, & $i duo vel plures λ
divi$ores $int inter $e æquales, viz. <math><r γ>α + βx^n + γx^2n + &c.</r></math> & $i adhuc
alii duo vel plures _μ_ $int $ibi mutuo æquales & prioribus inæquales,
viz. <math><r μ>ρ + σx^n + τx^2n + &c.</r>, &c.</math> & $i rationalis factor <math>Q $it =
<r a>α + βx<_>n</_> + γx<_>2π</_> + &c.</r> × <r b>ρ + σx<_>n</_> + τx<_>2n</_> + &c.</r> × <r c>δ + εx<_>n</_> + ζx<_>2n</_> + &c.</r>
× &c.</math> tum pro fluente quæ$itâ fluxionis <math>x^θ-1 × Q^-1 × R^-θ × (a + bx^n
+ cx^2n + &c.) × x^.</math> a$$umenda e$t quantitas <math><r 1-a-πγ>α + βx^n + γx^2n + &c.</r>
× <r 1-b-πμ>ρ + σx^n + τ x^2n + &c.</r> × <r 1-c>δ + εx^n + ζx^2n + &c.</r> × &c. × S^1-π ×
(A + Bx^n + Cx^2n + &c.), ubi R^π = S^π × <r λπ>α + β x^n + γ x^2n + &c.</r> ×
<r πμ>ρ + σx<_>n</_> + τx<_>2n</_> + &c.</r> × &c.</math> & per methodum in hoc problemate
traditam deduci po$$unt valores incognitarum coefficientium <math>A, B, C, &c.</math></p>
<p>Ca$. 6. Sit fluens quæcunque algebraica functio literæ <math>x</math>; tum, $i
inveniatur ejus fluxio, ea minores habebit dimen$iones quantitatis <math>x</math>
quam fluens per unitatem; ni dimen$iones quantitatis <math>x</math> in prædictâ
algebraicâ functione nihilo $int æquales, i.e. maximæ dimen$iones
quantitatis <math>x</math> in numeratore æquales $int ejus maximis dimen$ionibus
in denominatore; in quo ca$u maximæ dimen$iones quantitatis <math>x</math> in
fluente majores erunt quam maximæ dimen$iones eju$dem quantitatis
in fluxione per quantitatem majorem quam unitatem. Maximæ di-
men$iones quantitatis <math>(x)</math> in irrationalibus quantitatibus per metho-
dum, quæ docetur in prob. 26. no$t. meditat. algebr. acquirendæ $unt;
unde e contra, datâ fluxione, quæ e$t functio algebraica quantitatis <math>x</math>
in fluxionem <math>x^.</math> ducta; & con$equenter datâ maximâ dimen$ione quan-
titatis <math>x</math> in eâ contentâ, datur etiam maxima dimen$io quantitatis <math>x</math>
in fluente contenta; exinde facile con$tant dimen$iones quantitatis
<math>x</math>, ad quas & haud plures a$$urgent $eriei termini, $i modo haud in
infinitum progrediatur. e. g. Sit fluxio <math>(A + Bx^n + Cx^2n ... + Mx^xn)

<pb 15>[0067]<rh>FLUXIONUM FLUENTIBUS.</rh>
x^θ-1 × R^λ-1 × S^μ-1 × T^γ-1 × &c. × x^.</math>, ubi <math>R = a + bx^n + cx^2n ...x^πn,
S = d + e x^n + fx^2n ...x^pn, T = l + bx^n + kx^2n ... x^σn, &c.</math>; a$$uma-
tur pro fluente quæ$itâ quantitas <math>(α + β x^n + γx^2n + &c.) × x^θ × R^λ ×
S^μ × T′ × &c.</math> deinde per methodum prius traditam inveniatur quan-
titas <math>α + β x^n + γx^2n + ... λ^kn-(π+ρ+σ+&c.)n</math>; i. e. $eries <math>α + βx^n +
γx^2n + &c.</math> non ultra terminum <math>x^k (π+ρ+σ+&c.)n</math>, ni in infinitum pro-
grediatur; & con$equenter non habet plures quam <math>κ - π - ρ - σ -
&c. + 1</math> terminos.</p>
<p>Ca$. 7. Si requiratur fluens prædictæ fluxionis <math>(A + Bx^n ... Mx^κn)
× x^θ-1 × R^λ-1 × S^μ-1 × T^γ-1 × x^.</math> in $erie $ecundum dimen$iones quan-
titatis <math>x^n</math> de$cendente; quæ in pluribus ca$ibus converget, quam præ-
cedens $eries: a$$umatur <math>(αx^φn + βx^(φ-1)n + γx^(φ-2)n + &c.) × x^θ × R^λ
× S^μ × T′ × &c.</math> pro fluente, ubi <math>φ = κ - π - ρ - σ - &c.</math>; & deinde
inveniatur fluxio fluentis a$$umptæ, & æquentur corre$pondentes datæ
& re$ultantis fluxionis termini, & exinde deduci po$$unt coefficientes
<math>α, β, γ, &c.</math></p>
<p>Hæc $eries $emper terminat, cum $eries ex priori methodo deducta
terminat.</p>
<p>Hic excipienda $unt exempla nonnullarum fluxionum, quarum
fluentium termini progrediuntur, u$que donec dimen$iones quanti-
tatis <math>(x)</math> evadunt nihilo æquales.</p>
<p>Ca$. 8. Sit fluxio rationalis functio literæ <math>x</math> in fluxionem <math>x^.</math> ducta,
& maximæ dimen$iones quantitatis <math>x</math> in denominatore $uperent maxi-
mas dimen$iones quantitatis <math>x</math> in numeratore per unitatem, tum ejus
fluens haud inveniri pote$t in finitis algebraicis terminis literæ <math>x</math>.
e.g. Sit fluxio <math>{a + bx + <r {1 / 2}>c + dx^3</r>/e + fx + <r {1 / 2}>g + hx^5</r> × x^.}</math>; maximæ dimen$iones quan-
titatis <math>x</math> in numeratore datæ fluxionis $unt <math>3 × {1 / 2} = 1{2 / 1}</math>; maximæ au-
tem dimen$iones eju$dem quantitatis <math>(x)</math> in denominatore $unt <math>5 × {1 / 2} =
2{2 / 1}</math>; & exinde dimen$iones quantitatis <math>(x)</math> in denominatore majores
$unt quam dimen$iones eju$dem quantitatis in numeratore per uni-

<pb 16>[0068]<rh>DEINVENIENDIS</rh>
tatem; & con$equenter fluens fluxionis in finitis algebraicis terminis
non exprimi pote$t.</p>
<p>Ca$. 9. Omnis fluxio duobus modis in $eriem re$olvi pote$t; nam
index <math>n</math> quantitatis <math>x</math> vel affirmativus e$$e pote$t vel negativus. Pro-
ponatur fluxio <math>{<r>3k - lxx</r>x^. / x^2 √ (k x - l x^3 + m x^4)}</math>, in quâ <math>n = 1:</math> hæc vel $ic $cribi
pote$t <math>x^-{5 / 2} × <r>3^k - l x^2</r> × <r -{1 / 2}>k - l x^2 + m x^3</r> x^.</math>; vel $ic <math>x^-2 × <r>- l + 3 k x^-2</r> ×
<r -{1 / 2}>m - lx<_>-1</_> + k x<_>-3</_></r> x^.</math>. Tentandus e$t ca$us uterque per præcedentes
methodos, & $i $erierum alterutra ob terminos tandem deficientes
abrumpitur ac terminatur, habebitur area curvæ in finitis terminis.</p>
<p>Si vero una $eries terminet; tum nece$$ario, rebus recte di$po$itis,
terminat altera. e. g. Sit fluxio <math>{α x^(λ-1)n-1 x^. / (α + βx^n)^λ}</math>, cujus fluens e$t
<math>{a / n(λ - 1) α} × {x^(λ-1)n / (α + βx^n)^λ-1}</math>; transformetur hæe fluxio in formulam
<math>{a x^n-1 x^. / (αx^-n + β)^λ}</math>, cujus fluens e$t <math>{a / n × (λ - 1)α} × {1 / (αx^-n + β)^λ-1}</math>: hæ autem
duæ fluentes $unt eædem & eâdem facilitate fere per methodum hic
traditam deducuntur.</p>
<p>Omnia hæc etiam applicari po$$unt ad $ub$equentem ca$um.</p>
<p>Ca$. 10. Datâ fluxione <math>A + B x^n + C x^2n + &c. + D + E x^n + F x^2n + &c.
× <r λ-1>p + q x^n + r x^2n + &c.</r> (α) + <r>G + H x^n + I x^2n + &c.</r> × <r μ-1>s + t x^n + &c.</r>
(β) + K + L x^n + &c. × αβ × <r γ>a + b x^n + c x^2n + &c. + <ol>d + e x^n + f x^2n + &c.</ol> ×
<ol>p + q x<_>n</_> + r x<_>2n</_> + &c.</ol>^λ</r> × <r π>g + b x<_>n</_> + &c. + <ol>k + l x<_>n</_> + m x<_>2n</_> + &c.</ol> ×
<ol>s + t x<_>n</_> + &c.</ol>^μ</r></math>, invenire, utrum ejus fluens exprimi pote$t in finitis
algebraicis terminis quantitatis <math>x^n</math>, necne.</p>
<p>A$$umatur pro ejus fluente quantitas <math><ol>a + b x^n + c x^2n + &c. +</ol>

<pb 17>[0069]<rh>FLUXIONUM FLUENTIBUS.</rh>
<ol><ol>d + e x<_>n</_> + f x<_>2n</_> + &c.</ol> × <ol>p + q x<_>n</_> + r x<_>2n</_> + &c.</ol>^λ</ol>^γ+1 × <r π+1>g + b x<_>n</_> + &c.
+ <ol>k + l x<_>n</_> + m x<_>2n</_> + &c.</ol> × <ol>s + t x<_>n</_> + &c.</ol>^μ</r> × <r>α′ + β′ x<_>n</_> + γ′ x<_>2n</_> + &c.</r></math>
hujus quantitatis inveniatur fluxio, quæ fiat æqualis datæ fluxioni,
& exinde deduci po$$unt coefficientes <math>α′, β′, γ′, &c.</math> quæ$itæ.</p>
<p>Sit fluens <math>(a + b x^n + c x^2n + &c.) x^θ × R^λ S^μ T^γ × &c.</math> ubi in quanti-
tatibus <math>R, S, T, &c.</math> contineantur irrationales functiones quantitatis
<math>x^n</math>, viz. <math>α, β, γ, &c.</math> & fluentis fluxio $it <math>R^λ-1 S^μ-1 T^μ-1 × &c. × x^θ-1 × Q</math>;
tum in quantitate <math>Q</math> involvuntur irrationales quantitates <math>α, β, λ, &c.</math>;
& rectangula $ub quibu$que duabus <math>αβ, αγ, βγ, &c.</math>; contenta $ub
quibu$que tribus, quatuor, <math>&c.; αβγ, &c.; &c.</math></p>
<p>Ea, quæ dicta fuerunt in præcedentibus ca$ibus de fluxionibus
prius datis ad omnes algebraicas fluxiones applicari po$$unt. Quibus
etiam adjici pote$t $ub$equens.</p>
<p>Ca$. 11. Sit data fluxio <math><r π>A + B x^n + C x^2n + &c. + <ol>d + e x^n + f x^2n + &c.</ol>
× <ol>p + q x^n + r x^2n + &c.</ol>^λ</r> × <r ->a + b x^n + c x^2n + &c. + <ol>b + k x^n + &c.</ol>
× <ol>p + q x^n + r x^2n + &c.</ol>^λ-1</r> × x^.</math>; & $i fluxio quantitatis <math>A + B x^n +
C x^2n + &c. (P)</math> divi$a per quantitatem <math><r>d + e x^n + f x^2n + &c.</r> ×
<r λ>p + q x<_>n</_> + r x<_>2n</_> + &c.</r> (Q)</math> æqualis $it fluxioni <math>Q^.</math> per quantitatem
<math>(P)A + B x^n + C x^2n + &c.</math> divi$æ, i.e. $i fluxio prioris partis per po$te-
riorem divi$a æqualis $it fluxioni po$terioris partis per priorem divi$æ;
tum haud a$$umi debet quantitas <math><r π + 1>A + B x^n + C x^2n + &c. + <ol>d + e x^n +
f x^2n + &c.</ol> × <ol>p + q x^n + r x^2n + &c.</ol>^λ</r> × <r>α′ + β′ x^n + &c.</r></math> $ed quantitas
<math><r π>A + B x^n + C x^2n + &c. + <ol>d + ex^n + f x^2n + &c.</ol> × <ol>p + q x^n + r x^2n + &c.</ol>^λ</r>

<pb 18>[0070]<rh>DE INVENIENDIS</rh>
× <r>α′ + β′ x^n + &c.γ′ × <ol>d + e x^n + &c.</ol> × <ol>p + q x^n + &c.</ol>^λ</r></math></p>
<p>Ex. Sit data fluxio <math><r n>x + √ (x^2 + a^2)</r> x^.</math>, cujus fluens requiritur:
quantitatis <math>(x + √ (x^2 + a^2))</math> fluxio e$t <math>x^. + {x x^. / √ (x^2 + a^2)}:</math> fluxio <math>(x^.)</math>
vero quantitatis <math>x</math> per quantitatem <math>(√(x^2 + a^2))</math> divi$a æqualis e$t
fluxioni <math>({xx^. / √ (x^2 + a^2)})</math> quantitatis <math>√(x^2 + a^2)</math> per <math>x</math> divi$æ, viz.
<math>{x^. / √ (x^2 + a^2)} = {x x^. / √ (x^2 + a^2) × x}</math>; unde a$$umatur pro fluente quæ$itâ
quantitas <math><r n>{x + √ (x^2 + a^2)</r> × <r>α′√(x^2 + a^2) + β′x:</r></math> & ejus fluxio in-
venietur <math><r n>x + √(x^2 + a^2)</r> × (nα′ + {n β x / √(x^2 + a^2} + {α′ x / √(x^2 + a^2)} + β′) × x^.}</math>.</p>
<p>Fiat igitur <math>nα′ + {n β′ x / √ (x^2 + a^2)} + {α′ x / √ (x^2 + a^2)} + β′ = 1</math>; re$ultant
<math>nβ′ + α′ = 0, & nα′ + β′ = 1</math>; unde <math>β′ - n^2 β′ = 1</math>; & con$equenter
<math>β = {1 / 1 - n^2} & α = {n / n^2 - 1}</math>; & fluens quæ$ita <math>=(x + √(x^2 + a^2))^n
× ({- n / n^2 - 1} √ (x^2 + a^2) + {1 / n^2 - 1}).</math></p>
<p>Con$imilia etiam affirmari po$$unt de trinomialibus <math>P + Q + R,
&c.</math> quantitatibus.</p>
<p>Hinc e præcedentibus principiis inveniri pote$t fluens cuju$cunque
datæ algebraicæ fluxionis, $i modo exprimi po$$it in $initis algebraicis
terminis literæ <math>x</math>.</p>
<p>Sæpe vero facilius inve$tigari pote$t fluens prædicta, $i abjiciantur
irrationales quantitates $ub vinculis contentæ; & fluxionis re$ultantis
inveniatur fluens; e quâ etiam $æpe erui pote$t fluens quæ$ita.</p>
<p>Et $ic de pluribus huju$cemodi quantitatibus.</p>
<p>Ca$. 12. In re$olutione ca$uum prius traditorum nonnunquam

<pb 19>[0071]<rh>FLUXIONUM FLUENTIBUS.</rh>
denominator termini deducti nihilo evadat æqualis, in quo ca$u fluens
fluxionis quæ$itæ in finitis terminis non exprimi pote$t.</p>
<p>Ca$. 13. Fluentes quarumcunquc algebraicarum fluxionum etiam
per infinitas $eries terminorum $ecundum dimen$iones quantitatis <math>(x)</math>
progredientium deduci po$$unt.</p>
<p>1. Ex irrationalibus quantitatibus facile con$tat numerus valcrum,
quos habet quæ$ita fluens; per infinitas $eries inveniantur $inguli
valores <math>(P, Q, R, S, T, &c.)</math> quæ$itæ fluentis, & exinde $umma $in-
gulorum valorum quæ$itæ fluentis; & $ic $umma rectangulorum $ub
$ingulis duobus; contentorum $ub $ingulis tribus; &c.: per medit.
algebr. inveniri pote$t, utrum hæ $ummæ per rationalem $unctionem
incognitæ quantitatis <math>x</math>, & exinde utrum fluens quæ$ita per prædictam
functionem exprimi pote$t.</p>
<p>Hæc methodus invenit $luentes, $i modo nulli valores fluentium
correctionem exigant.</p>
<p>2. Si vero correctionem exigant, tum corrigendi $unt $inguli <math>(n)</math>
valores <math>(P, Q, R, S, &c.)</math> addendo ad $ingulos <math>P, Q, R, S, &c.</math> inva-
riabiles quantitates generaliter a$$umptas <math>A, B, C, D, &c.</math>; & deinde
inveniendo $ummam <math>(α)</math> e $ingulis valoribus re$ultantibus <math>P + A,
Q + B, R + C, &c.; & (β)</math> $ummam eorum rectangulorum $ub qui-
bu$que duobus, <math>(γ)</math> contentorum $ub quibu$que tribus, <math>(δ)</math> $ub qui-
bu$que quatuor; &c. deinde per meditat. algebr. i. e. per methodum
communes divi$ores detegendi inveniatur, annon ita a$$umi po$$unt
quantitates <math>A, B, C, &c.</math> ut prædictæ $ummæ evadant $initæ ratio-
nales functiones incognitæ quantitatis <math>x</math>; $i hoc fieri po$$it, tum in-
venietur æquatio <math>v^n - αv^n-1 + βv^n-2 - γv^n-3 + &c. = 0</math>, cujus <math>(n)</math>
radices erunt <math>(n)</math> diver$i valores fluentis quæ$itæ.</p>
<p>Ex. Sit fluxio <math>√(a^2 - x^2)xx^.</math>; con$tat duos e$$e valores irrationalis
quantitatis <math>√(a^2 - x^2)</math>, & con$equenter duos e$$e valores fluentis <math>(y)</math>;
reducatur hæc fluxio in infinitas $eries, & re$ultant duo valores
<math>a x x^. - {1 / 2}{x^3 x^. / a} - {1 / 8}{x^5 x^. / a^3} + &c. - a x x^. + {x^3 x^. / 2a} + {x^5 x^. / 8a^3} - &c.</math> quo-

<pb 20>[0072]<rh>DE INVENIENDIS</rh>
rum fluentes erunt re$pective <math>{ax^2 / 2} - {x^4 / 8a} - {x^6 / 48 a^3} + &c. & - {ax^2 / 2}+ {x^4 / 8a} + {x^6 / 48a^3} - &c.</math> corrigantur hæ fluentes addendo quamcunque
quantitatem <math>A - {a^3 / 3}</math> ad unum valorem, & <math>A + {a^3 / 3}</math> ad alterum; & re-
$ultant duo valores <math>A - {a^3 / 3} + {a x^2 / 2} - {x^4 / 8a} - {x^6 / 48a^3} + &c., & A + {a^3 / 3} -{ax^2 / 2} + {x^4 / 8a} + {x^6 / 48a^3} - &c.</math>, quorum $umma e$t <math>2A</math>, & rectangulum
$ub duobus valoribus erit <math>A^2 - {1 / 9}a^6 + {a^4 x^2 / 3} - {a^2 x^4 / 3} + {x^6 / 9}; &c.</math></p>
<p>Cor. 1. Si modo $int <math>(n)</math> valores prædicti, & corrigatur unus valor
addendo quantitatem <math>A</math>; tum ex quantitate <math>A</math> & datâ fluxione de-
duci po$$unt correctiones $ingulorum <math>(n - 1)</math> reliquorum.</p>
<p>Cor. 2. Si $it fluxio $ecundi vel <math>m</math> ordinis, cujus fluens requiritur;
tum in correctione prædictorum <math>(n)</math> valorum a$$umi po$$unt <math>(m)</math>
quantitates ad libitum.</p>
<h>PROB. V.</h>
<p>_Datâ fluxione, quæ in $e continet fluentem_ <math>(v)</math>, _quæ baud exprimi po-_
_te$t in finitis algebraicis terminis variabilis quantitatis_ <math>(x)</math>; _invenire utrum_
_fluens datæ fluxionis $it finita algebraica functio quantitatis_ <math>(x)</math> _& præ-_
_dictæ fluentis_ <math>(v)</math>.</p>
<p>1. Collocentur termini $ecundum dimen$iones fluentis <math>v</math>, ita ut illi
primum locum occupent, in quibus maxima invenitur dimen$io
fluentialis quantitatis <math>v</math>; & $ic deinceps.</p>
<p>Terminus, in quo maximæ inveniuntur dimen$iones fluentialis
quantitatis <math>(v)</math>, $it rectangulum quantitatis <math>(W)</math>, quæ $it functio li-
teræ <math>v</math>, in quantitatem <math>B x^.</math> ductæ, in quâ haud continetur litera <math>v</math>:

<pb 21>[0073]<rh>FLUXIONUM FLUENTIBUS.</rh>
inveniatur fluens fluxionis <math>B x^.</math>, quæ dicatur <math>X</math>; deinde inveniatur
fluxio rectanguli <math>X × W</math>, & erit <math>W X^.</math> (<math>W B x^.</math> prædictus terminus) <math>+
X W^.</math>: in hâc vero po$teriori parte <math>X W^.</math> $æpe quantitas <math>v</math> haud tam
multas habet dimen$iones quam prædictus terminus <math>W</math>; & $ic his
operationibus repetitis $æpe continuo deprimi po$$unt dimen$iones
quantitatis <math>v</math>, ita ut tandem evane$cet, & exinde inveniri pote$t
fluens fluxionis quæ$ita.</p>
<p>Ex. 1. Sit <math>v = fluen.</math> flux. <math>{x^. / 1 - x}</math>, invenire fluentem fluxionis <math>v ×
x^n x^.</math>; ubi <math>n</math> e$t integer numerus. Hìc <math>v</math> e$t unica dimen$io fluentialis
quantitatis <math>v</math>, quæ dicatur <math>W</math>; & erit <math>x^n x^. = Bx^.</math>, cujus fluens e$t
<math>{x^n+1 / n + 1} = X</math>; unde fluctio rectanguli <math>X × W = {v × x^n+1 / n + 1}</math> erit <math>v x^n x^.</math>
(data fluxio) <math>+ {x^n+1 / n + 1} × {x^. / 1 - x} (v^.)</math>; $ed fluens fluxionis <math>- {x^n+1 / (n + 1)} ×{x^. / (1 - x)} = {x^n+1 / (n + 1)^2} + {x^n / (n + 1) × n} + {x^n-1 / (n + 1) × (n - 1)} ... {x / n + 1} -{v / n + 1}</math>, & con$equenter fluens quæ$ita erit <math>{vx^n+1 / n + 1} + {x^n+1 / (n + 1)^2} + {x^n / (n + 1).n}+ {x^n-1 / (n + 1)(n - 1)} ... {x / n + 1} - {v / (n + 1)}</math>: $it <math>n</math> negativus numerus; &
erit fluens quæ$ita <math>{vx^n+1 / n + 1} - {1 / n + 1}({x^n+2 / n + 2} + {x^n+3 / n + 3} ... {x^-1 / -1} + $.{x^. / x} + v).</math></p>
<p>Ex. 2. Sit <math>z^. = {x^. / 1 + x^2}</math>, & erit fluens fluxionis <math>zx^n x^.</math> per præceden-
tem methodum inventa <math>{1 / n + 1} × <r>x^n+1 z - {x^n / n} + {x^n-2 / n - 2} - {x^n-4 / n - 4} +{x^n-6 / n - 6} &c. ∓ z</r></math>; $i $uerit <math>{n + 1 / 2}</math> par numerus, erit <math>- z</math>; $in impar,
+ z: etiamque 2<_>do</_>. $i <math>{n / 2}</math> fuerit par numerus, pro <math>∓ z</math> $cribatur <math>- {1 / 2}</math>
log. <math>(1 + x^2)</math>; $in impar, $cribatur <math>+ {1 / 2}</math> log. <math>(1 + x^2)</math>. Si <math>n</math> $it nega-

<pb 22>[0074]<rh>DE INVENIENDIS</rh>
tivus numerus, tum fluens fluxionis <math>zx^n x^. = {1 / n + 1} (x^n+1 z - {x^n+2 / n + 2} +
+ {x^n+4 / n + 4} - &c..{x^-1 / - 1} vel ({x^-2 / - 2}) ± z)</math>; <math>1^mo</math>. erit <math>+ z</math>, $i <math>{n + 1 / 2}</math> $it im-
par negativus numerus; $in par, <math>- z: 2^do</math>. $i autem <math>{n / 2}</math> $it par negati-
vus numerus, tum pro <math>± z</math> $cribatur <math>$.{x^. / x} - {1 / 2} log. (1 + x^2)</math>; $in impar,
<math>+ {1 / 2} log. (1 + x^2) - $.{x^. / x}(log. x)</math>.</p>
<p>Et $ic ratiocinari liceat de fluentibus fluxionis
<math>z x^n x^., ubi z^. = {x^. / 1-x^2}</math>.</p>
<p>Ex. 3. Sit <math>v^. = {x^. / x}</math>, & data fluxio <math>v^m x^n x^.</math>, cujus fluens requiritur;
tum <math>v^m = W & x^n x^. = Bx^.</math>, unde <math>X = {x^n+1 / n + 1}</math>, & <math>X × W = v^m × {x^n+1 / n + 1}</math>,
cujus fluxio erit <math>v^m x^n x^.</math> (data fluxio) <math>+ {m / n + 1} v^m-1 × x^n x^.</math>; a$$umatur
<math>{m / n + 1} v^m-1 x^n x^.</math> tanquam data fluxio, & repetatur operatio, & erit <math>{m / n + 1}v^m-1 = W′ & x^n x^. = B′x^.</math>, cujus fluens e$t <math>{x^n+1 / n + 1} = X′</math>, unde <math>X′ × W′ ={mv^m-1 × x^n+1 / (n + 1) × (n + 1)}</math>, cujus fluxio erit <math>mv^m-1 × {x^n x^. / n + 1}</math> (data fluxio) +
<math>{m / (n + 1)^2} × (m - 1) v^m-2 x^. × x^n</math>; & $ic deinceps e repetitis operationibus
invenietur fluens quæ$ita <math>v^m × {x^n+1 / n + 1} - m{v^m-1 × x^n+1 / (n + 1)^2} + m · (m - 1)
× v^m-2 × {x^n+1 / (n + 1)^3} - m · (m - 1) · (m - 2) × v^m-3 × {x^n+1 / (n + 1)^4} + &c.</math>
quæ terminat, cum <math>m</math> $it integer po$itivus numerus.</p>
<p>Ex. 4. Sit <math>z^. = {ay^. / √ (1 = y^2)}</math>; $cribatur <math>u = √ (1 = y^2)</math>; & erit fluens

<pb 23>[0075]<rh>FLUXIONUM FLUENTIBUS.</rh>
fluxionis <math>z^n y^. = yz^n ± nauz^n-1 ∓ n · (n - 1)a^2 yz^n-2 ∓ n · (n - 1) ×
(n - 2)a^3 uz^n-3</math>. Per problema enim a$$umatur pro fluente <math>y z^n</math>, cu-
jus fluxio e$t <math>z^n y^. + nz^n-1 × {ay^. / √ (1 ∓ y^2)} × y</math>; deinde pro fluente quan-
titatis <math>+ nz^n-1 × {ayy^. / √ (1 ∓ y^2)}</math> a$$umenda e$t quantitas <math>∓ naz^n-1
√ (1 ∓ y^2)</math>, cujus fluxio erit <math>naz^n-1 × {yy^. / √ (1 ∓ y^2)} ∓ n × (n - 1)az^n-2 ay^.</math>,
& $ic deinceps; unde erit $luens $luxionis <math>(z^n y^.) = yz^n ± nauz^n-1 ∓
n · (n - 1)a^2 z^n-2 y ∓ &c.</math></p>
<p>Ex. 5. Sint <math>z^. = {ay^. / √ (1 - y^2)} & u = √ (1 - y^2)</math>, invenire fluentem
fluxionis <math>z^n u^r y^m y^.</math>: per problema <math>W = z^n & Bx^. = u^r y^m y^.</math>, cujus fluens
dicatur <math>V</math>; & erit fluxio rectanguli <math>(VW) = WV^. (z^n u^r y^m y^.)</math> data fluxio)
<math>+ nz^n-1 × V × {ay^. / u}, &c.</math> Scribantur pro fluentibus fluxionum <math>u^r y^m y^.,{Ay^. / u}, {B′y^. / u}, {Cy^. / u}, &c.</math> re$pective <math>A, B′, C, D, &c.</math> Et fluens quæ$ita per
problema invenietur <math>Az^n - naB′z^n-1 + n · (n - 1)a^2 Cz^n-2 - n ·
(n - 1) · (n - 2)a^3 Dz^n-3 + &c.</math> Aliter, $i fluens fluxionis <math>Bx^.</math> non
detegi po$$it, a$$umatur pro fluente quæ$itâ <math>{1 / (n + 1)a} × z^n+1 u^r+1 y^m - x</math>,
cujus fluxio e$t <math>z^n u^r y^m y^.</math> (data fluxio) <math>- {1 / (n + 1)a}z^n+1 u^r-1 y^m-1 (<r>r + 1</r>y^2 -
mu^2)y^. - x^.</math>, unde <math>x^. = - {1 / (n + 1)a}z^n+1 u^r-1 y^m-1 (<r>r + 1</r>y^2 - mu^2)y^.
= - {1 / (n + 1)a}z^n+1 u^r-1 y^m-1 (<r>r + 1 + m</r>y^2 - m)y^.</math>; & $ic deinceps, &c.
hìc autem animadvertendum e$t in omnibus fluentibus inve$tigandis,
$i <math>$. By^., $. B′y^., &c.</math> inveniri po$$it, tum prior; $in non, tum po$terior
methodus adhibenda e$t.</p>

<pb 24>[0076]<rh>DE INVENIENDIS</rh>
<p>Ex. 6. Sit <math>z^. = <r m>b + c y^n</r> y^.</math>; tum erit fluens fluxionis <math>(a z^2 y^n-2 y^.)
= {a / n-1} z^2 y^n-1
- {2a / n-1} × {1 / (m + 1). nc} <r m+1>b + c y^n</r> z + {2a / n-1} × {1 / (m + 1). nc}× $. <r 2m+1>b + c y^n</r> y^.</math>.</p>
<p>Et $ic de plurimis huju$cemodi exemplis.</p>
<p>Ex. 7. Sint <math>x A^. = P, P A^. = Q^., Q A^. = R^., R A^. = S^., &c.</math> & fluens
fluxionis <math>A^n x^.</math> invenietur <math>x A^n - n A^n-1 P + n × (n - 1) A^n-2 Q - n ×
(n - 1) × (n - 2) A^n-3 R + &c.</math></p>
<p>Ex. 8. Datis fluentibus fluxionum <math>y x^., y x x^., y x^2 x^., y x^3 x^., .. y x^n-1 x^.</math>:
$int <math>y x^. = A^., A x^. = B^., B x^. = C^., C x^. = D^., &c. ... L x^. = M^., M x^.
= N^., N x^. = O^., O x^. = P^., P x^. = Q^., Q x^. = R^., R x^. = S^., S x^. = T^.</math>,
quarum æquationum numerus $it <math>n</math>; invenire fluentem fluxionis <math>S x^.
= T^.</math>; datâ fluxionali æquatione <math>S x^. = T^.</math>, a$$umenda e$t <math>S x - α = T</math>
pro fluente, cujus fluxio e$t <math>S x^. + x S^. - α^. = T^.</math>, unde <math>x S^. = R x x^.
= α^.</math>: in fluxionali æquatione <math>R x x^. = α^.</math> a$$umenda e$t æquatio
<math>{R x^2 / 2} - β = α</math> pro fluente, cujus fluxio <math>R x x^. + {x^2 / 2} R^. - β^. = α^.</math>,
unde <math>β^. = {x^2 / 2} R^. = {x^2 / 2} Q x^.</math>; & $ic continuo repetitis operationibus in-
venietur <math>T = S x - R × {x^2 / 2} + Q × {x^3 / 2 · 3} - P {x^4 / 2 · 3 · 4} + &c. ... ±{1 / 2 · 3 · 4 ... (n - 1)} × (x^n-1 A - $. x^n-1 y x^.)</math>.</p>
<p>Ex. 9. Datis fluentibus fluxionum <math>y x^., y^2 x^., y^3 x^., y^4 x^., &c.</math> quæ dican-
tur re$pective <math>A, B, C, D, &c.</math> $int <math>A y^. = K^., K y^. = L^., L y^. = M^., &c.</math> &
$ic <math>B y^. = k^., k y^. = l^., l y^. = m^., &c.</math> etiamque <math>C y^. = p^., p y^. = q^., q y^. = r^.,
&c. & D y^. = P^., P y^. = Q^., Q y^. = R^., &c. &c.</math> tum erit <math>K = A y
- $. y A^. (y^2 x^. = B^.): L = K y - $. y K^. (y A y^.)</math>; $ed fluens
fluxionis <math>(A y y^.) = {A y^2 / 2} - $. {y^2 / 2} A^. ({y^3 x^. / 2} = {C^. / 2})</math>; unde <math>L = K y -

<pb 25>[0077]<rh>FLUXIONUM FLUENTIBUS.</rh>
{A y^2 / 2} + {C^. / 2} = {A y^2 - 2 B y + C / 2}: & M = L y - $. y L^. (y K y^.)</math>;
$ed fluens <math>$. y K y^. = {y^2 / 2} × K - $. {y^2 / 2} × K^. ({y^2 / 2} × A y^.)</math>; & $luens <math>$.{y^2 / 2} A y^. = {y^3 A - D / 2 · 3}</math>; fluens igitur fluxionis <math>y K y^. = {y^2 / 2} K - {y^3 A - D / 2 · 3}</math>,
& fluens quæ$ita <math>M = L y - {y^2 / 2} K + {y^3 A - D / 2 · 3}</math>; $cribantur pro <math>K &
L</math> earum re$pectivi valores, & re$ultat <math>M = {A y^3 - 2 B y^2 + C y / 2}(L y) - {A y^3 - B y^2 / 2} ({y^2 K / 2}) + {y^3 A - D / 2 · 3} = {y^3 A - 3 y^2 B + 3 C y - D / 2 · 3}:
&c.</math> & in genere terminus <math>H</math>, cujus di$tantia a primo $it <math>r</math>, erit
<math>{y^r A - r y^r-1 B + r. {r - 1 / 2} y^r-2 C - &c./1 · 2 · 3 ... r} = H</math>.</p>
<p>Eodem modo a$$umenda e$t pro fluente fluxionis <math>(k^. = B y^.)</math> quan-
titas <math>B y - λ</math>, & exinde deduci pote$t <math>λ^. = y B^. = C^.</math>, unde <math>k = B y
- C</math>; & $ic <math>l = {y^2 B - 2 y C + D / 2}, m = {y^3 B - 3 y^2 C + 3 y D - E / 2 · 3},
&c.</math>: per eandem methodum detegi po$$unt <math>p = C y - D, q ={y^2 C - 2 y D + E / 2}, r = {y^3 C - 3 y^2 D + 3 y E - F / 2 · 3}, &c.</math>; etiamque
<math>P = D y - E, Q = {y^2 D - 2 y E + F / 2}, &c.</math>; & $ic deinceps.</p>
<p>2. Ii$dem literis ea$dem quantitates denotantibus; $it <math>z^. =
(a - y)^r × A^. = a^r A^. - r a^r-1 y A^. + r. {r-1 / 2} a^r-2 y^2 A^. - r. {r-1 / 2} · {r-2 / 3}a^r-3 y^3 A^. + &c.</math> unde <math>z = a^r A - r a^r-1 B + r. {r-1 / 2} a^r-2 C - &c.</math></p>

<pb 26>[0078]<rh>DE INVENIENDIS</rh>
<p>Cor. Ex præcedentibus duobus ca$ibus con$tat, quod$i <math>y = a,
H = {z / 1 · 2 · 3 ... r}</math>.</p>
<p>Ex. 10. Sint <math>T^. = S α^.. S^. = R β^., R^. = Q γ^., Q^. = P δ^., P^. = O ε^., &c.</math></p>
<p>Sint etiam <math>α^. = p^., p β^. = q^., q γ^. = r^., r δ^. = s^., s ε^. = t^., &c.</math> & erit per
prob. <math>T = S p - R q + Q r - P s + O t - &c.</math> ad finem termi-
norum.</p>
<p>Et $ic $int <math>β^. = p^. <x>′</x>, p<x>′</x> γ^. = q^. <x>′</x>, q<x>′</x> δ^. = r^. <x>′</x>, r<x>′</x> ε^. = s^. <x>′</x>, &c.</math> & erit <math>S = R p<x>′</x>
- Q q<x>′</x> + P r<x>′</x> - O s<x>′</x> + &c.</math></p>
<p>Etiamque $int <math>γ^. = p^. <x>″</x>, p<x>″</x> δ^. = q^. <x>″</x>, q<x>″</x> ε^. = r^. <x>″</x>, &c.</math> & erit <math>R = Q p<x>″</x> -
P q<x>″</x> + O r<x>″</x> - &c.</math> unde e datis quantitatibus <math>p, q, r, s, t, &c. p<x>′</x>, q<x>′</x>, r<x>′</x>, s<x>′</x>,
&c. p<x>″</x>, q<x>″</x>, r<x>″</x>, s<x>″</x>, &c.</math> facile deduci po$$unt fluentes <math>T, S, R, Q, P, &c.</math></p>
<p>2. Sint <math>A^. = y α^., B^. = A β^., C^. = B γ^., D^. = C δ^., E^. = D ε^., &c.</math>
etiamque <math>y β α^. = P^.</math>, erit <math>B = A β - P</math>; $int <math>γ β^. = p^., p y α^. = Q^.</math>,
& erit <math>C = B γ - p A + Q = A β γ - P γ - p A + Q</math>; $int etiam
<math>δ γ^. = p^. <x>′</x>, p<x>′</x> β^. = q^. <x>′</x>, y q<x>′</x> α^. = R^.</math>, & erit <math>D = C δ - p<x>′</x> B + q<x>′</x> A - R</math>;
$int etiam <math>ε δ^. = p^. <x>″</x>, p<x>″</x> γ^. = q^. <x>″</x>, q<x>″</x> β^. = r^. <x>″</x>, y r<x>″</x> α^. = S^.</math>, & erit <math>E = Dε -
C p<x>″</x> + B q<x>″</x> - A r<x>″</x> + S</math>; & $ic deinceps.</p>
<p>Cor. Sint <math>α = β = γ = δ = ε = &c.</math> & erunt <math>p^. = p^. <x>′</x> = p^. <x>″</x> = &c.
= α α^.</math>, unde <math>p = p<x>′</x> = p<x>″</x> = &c. = {α^2 / 2}</math>, & $ic inveniri pote$t <math>q = q<x>′</x> = q<x>″</x> =
&c. = {α^3 / 2 · 3}, r = r<x>′</x> = r<x>″</x> = &c. = {α^4 / 2 · 3 · 4}</math>, & $ic deinceps; & erunt
<math>A^. = y α^., P = y α α^., Q^. = y {α^2 / 2} α^., R^. = {y α^3 / 6} α^., S^. = {y α^4 / 2 · 3 · 4} α^., T^. ={y α^5 / 2 · 3 · 4 · 5} α^., &c. & B^. = A α^., C^. = B α^., D^. = C α^., &c.</math> & per exem-
plum erunt <math>B = A α - P, C = B α - {α^2 / 2} A + Q = {α^2 / 2} A - P α + Q,

<pb 27>[0079]<rh>FLUXIONUM FLUENTIBUS.</rh>
D = C α - {α^2 / 2} B + {α^3 / 1 · 2 · 3} A - R = {α^3 / 1 · 2 · 3} A - {α^2 / 1 · 2} P + α Q - R,
E = D α - {α^2 / 2} C + {α^3 / 1 · 2 · 3} B - {α^4 / 1 · 2 · 3 · 4} A + S = {α^4 A / 1 · 2 · 3 · 4} -{α^3 / 1 · 2 · 3} P + {α^2 / 1 · 2} Q - α R + S</math>, & $ic invenitur <math>F = {α^5 / 1 · 2 · 3 · 4 · 5}A - {α^4 / 1 · 2 · 3 · 4} P + {α^3 / 1 · 2 · 3} Q - {α^2 / 1 · 2} R + α S - T</math>; & $ic deinceps.</p>
<p>Ex. 11. Fluens fluxionis <math>({x^bβ-1 / x^bx} x^. $. {x^bα-1 / 1±x^b} x^.) = - {1 / b} ({1 / α-β}x^b(β-α) $. {x^bα-1 / 1±x^b} x^. + ({1 / β-α} $. {x^bβ-1 / 1±x^b} x^.)) = A</math>: fluens fluxionis <math>({x^bγ-1 / x^bβ}× A x^.) = {1 / b^2} ({1 / (α-β) × (α-γ)} x^b(γ-α) $. {x^bα-1 / 1±x^b} x^. + {1 / (β - α) × (β - γ)}x^b(γ-β) $. {x^bβ-1 / 1±x^b} x^. + ({1 / (γ-α) × (γ-β)} $. {x^bγ-1 / 1±x^b} x^.)) = B</math>; fluens flu-
xionis <math>({x^bδ-1 / x^bγ} x^. × B) = - {1 / b^3}({1 / (α-β) (α-γ) (α-δ)} x^b(δ-α) $. {x^bα-1 / 1∓x^b} x^.
+ {1 / (β-α) (β-γ) (β-δ)} x^b(δ-β) $. {{x^bβ-1 / 1±x^b} x^. + {1 / (γ-α) (γ-β) (γ-δ)}x^h(δ-γ) $. {x^bγ-1 / 1±x^b} + ({1 / (δ-α) × (δ-β) × (δ-γ)} $. {x^bδ-1 / 1±x^b} x^.)) = C}</math> &
in genere $int <math>$. {x^bβ-1 / x^bα} x^. $. {x^bα-1 x^. / 1±x^b} = A, $. {x^bγ-1 / x^bβ} x^. × A = B, $. {x^bδ-1 x^. / x^bγ}B = C, $. {x^bε-1 x^. / x^bδ} × C = D, $. {x^bζ-1 x^. / x^bε} × D = E</math>, & $ic deinceps, u$que
ad terminos <math>$. {x^bλ-1 x^. / x^bκ} × P = Q, $. {x^bμ-1 x^. / x^bλ} × Q = R</math>; etiamque <math>$.{x^bα-1 x^. / 1±x^b} = A′, $. {x^bβ-1 x^. / 1±x^b} = B′, $. {x^bγ-1 x^. / 1±x^b} = C′, $. {x^bδ-1 x^. / 1±x^b} = D′, &c.</math> u$-

<pb 28>[0080]<rh>DE INVENIENDIS</rh>
que ad <math>$. {x^bλ-1 x^. / 1±x^b} = L′, $. {x^bμ-1 x^. / 1±x^b} = M′</math>; & $it (<math>n</math>) numerus literarum
(<math>α, β, γ, δ, ε, &c. κ, λ & μ</math>); tum erit <math>R = ± {1 / b^n-1} ({x^b(μ-α) A′ / (α-β) (α-γ) (α-δ) .. (α-λ) (α-μ)}+ {x^l(μ-β) B′ / (β-α) (β-γ) (β-δ) ... (β-λ) (β-μ)} +{x^b(μ-γ) C′ / (γ-α) · (γ-β) · (γ-δ) ... (γ-μ)} + {x^b(μ-δ) D′ / (δ-α) × (δ-β) × (δ-γ) .. (δ-μ)}.... + {x^b(μ-λ) L′ / (λ-α) (λ-β) (λ-γ) .. (λ-μ)} + {M′ / (μ-α) (μ-β) (μ-γ) .. (μ-λ)})</math>.
Signum erit + vel -, prout <math>n</math> $it impar vel par numerus.</p>
<p>Hoc facile con$tat ex eo quod aggregatum $ingulorum contento-
rum, viz. <math>(β-γ) × (β-δ) × (γ-δ) × (β-ε) × (γ-ε) × (δ-ε) × (β-ζ)
× (γ-ζ) × (δ-ζ) × &c. - (α-γ) × (α-δ) × (γ-δ) × (α-ε) × (γ-ε) ×
(δ-ε) × (α-ζ) × (γ-ζ) × (δ-ζ) × (ε-ζ) × &c. + (α-β) × (α-δ) ×
(β-δ) × (α-ε) × (β-ε) × (δ-ε) × (α-ζ) × (β-ζ) × (δ-ζ) × &c. -
(α-β) × (α-γ) × (β-γ) × (α-ε) × (β-ε) × (γ-ε) × (α-ζ) × (β-ζ)
× (γ-ζ) &c. + &c.</math> nihilo erit æquale. In primo contento non con-
tinetur litera <math>α</math>, in $ecundo non continetur litera <math>β</math>, in tertio non
continetur litera <math>γ</math>; & $ic deinceps.</p>
<p>1.2. Eadem methodus etiam deteget fluentes fluxionum, quæ duas
vel plures diver$as fluentes involvunt. e.g. Sint <math>A, B, C, &c.</math> $luen-
tes datarum fluxionum; & data fluxio, cujus fluens requiritur, $it
<math>A B C &c. × v^.</math>; tum per prob. pro fluente quæ$itâ a$$umenda e$t quan-
titas <math>A B C &c. × v - α</math>, cujus fluxio e$t <math>A B C &c. × v^.</math> (data fluxio)
<math>+ v B C &c. A^. + v A C &c. B^. + v A B &c. C^. + &c. - α^.</math>: deinde per
eandem methodum inveniantur fluentes fluxionum <math>v B C &c. A^., v A C
&c. B^., v A B &c. C^., &c.</math> & tandem invenietur fluens datæ fluxionis
quæ$ita.</p>
<p>2. Sit fluxio <math>z^m v^.</math>, ubi <math>z</math> denotat fluentem datæ fluxionis; ejus
fluens nonnunquam etiam deduci pote$t e $ub$equentibus prin-
cipiis.</p>

<pb 29>[0081]<rh>FLUXIONUM FLUENTIBUS</rh>
<p>A$$umatur pro fluente quantitas <math>z^m A + z^m-1 B + z^m-2 C + z^m-3 D
+ &c.</math> Hujus quantitatis inveniatur fluxio, & datæ fluxioni fiat
æqualis, & exinde inveniri po$$unt quantitates a$$umptæ <math>A, B, C,
&c.</math>.</p>
<p>3. Sit data fluxio <math>W B x^.</math>, ubi <math>B</math> $it data functio quantitatis <math>x; & W</math>
functio quantitatis <math>v</math>, cujus <math>v</math> fluxio $it <math>C x^.</math>, ubi <math>C</math> denotat functionem
quantitatis <math>x</math>; & vel per præcedentem methodum a$$umatur quanti-
tas <math>W $. B x^. - α</math> pro fluente, & $ic per prædictam methodum pro-
gredi liceat: vel data fluxio <math>W B x^.</math> ita $cribi pote$t <math>W × {B / C} × C x^. = W
× {B / C} v^.</math>; & exinde $i modo pro fluente fluxionis (<math>W v^.</math>) $cribatur <math>r</math>, per
methodum in problemate contentam a$$umatur quantitas <math>r × {B / C} - α</math>
pro fluente quæ$itâ, cujus fluxio e$t <math>W × {B / C} v^.</math> (data fluxio) <math>+ r × {B^. / C}- α^.</math>, unde <math>α^. = r × {B^. / C} = r × {B^. C - C^. B / C^2} = r Z^. = r × {Z^. / C x^.} × C x^.
= r × {Z^. / C x^.} v^.</math>; & eodem modo pro fluente fluxionis (<math>r × {Z^. / C x^.} × v^.</math>)
a$$umatur <math>{Z^. / C x^.} $. (r v^.) - β</math>, cujus fluxio erit <math>r × {Z^. / C x^.} v^.</math> (data fluxio) <math>+{Z^. ^. / C x^.} × $. (r v^.) - β^.</math>, unde pro fluente <math>β</math> a$$umatur quantitas <math>{Z^. ^. / C x^.} ×{1 / C x^.} × $. v^. ($. (r v^.) - γ</math>; & $ic deinceps.</p>
<p>Ex. 1. Sit fluxio <math>{X x^. / (l x)^n}</math>, ubi <math>X</math> $it functio quantitatis <math>x, & l x = v</math>
hyper. log. quantitatis <math>x</math>; hæc vero quantitas hoc modo $cribi pote$t <math>X x{x^. / x (l x)^n} = X x {v^. / v^n}</math>; a$$umatur quantitas <math>- X x × {1 / (n - 1) v^n-1} - α</math> pro

<pb 30>[0082]<rh>DE INVENIENDIS</rh>
ejus fluente, cujus fluxio erit <math>X x {v^. / v^n}</math> (data fluxio) <math>- (X^. x) × {1 / (n - 1) v^n-1}- α^.</math>, unde <math>α^. = - (X^. x) × {1 / (n-1) v^n-1} = - (X^. x) × {x / x^.} × {x^. / (n-1) x v^n-1}= - (X^. x) {x / x^.} × {v^. / (n-1) v^n-1}</math>; deinde a$$umatur pro fluente fluxionis <math>α^.</math>
quantitas <math>(X^. x) {x / x^.} × {1 / (n - 1) (n - 2) v^n-2} - γ</math>, & $ic deinceps: unde
$i modo $cribantur <math>(X^. x) = P x^., (P^. x) = Q x^., (Q^. x) = R x^., &c. V x^.</math>,
erit <math>$. ({X x^. / (l x)^n}) = {- X x / (n - 1) (l x)^n-1} - {P x / (n - 1) (n - 2) (l x)^n-2} -{Q x / (n - 1) (n - 2) (n - 3) (l x)^n-3} - .... {1 / (n - 1) (n - 2) ... 1} $. ({V x^. / l x}).</math></p>
<p>E. g. Sit <math>X = x^m</math>, & erit <math>$. ({x^m x^. / (l x)^n}) = {- x^m+1 / (n - 1) (l x)^n-1} -{<r>m + 1</r> x^m+1 / (n - 1) (n - 2) (l x)^n-2} - {<r 2>m + 1</r> x^m+1 / (n - 1) (n - 2) (n - 3) (l x)^n-3} ...{<r n-1>m + 1</r> / (n - 1) · (n - 2) .. 1} × $. {x^m x^. / l x}, &c.</math></p>
<p>Ex. 2. Sit data fluxio <math>a^x X x^.</math>, cujus fluens <math>$. a^x X x^. = {1 / l a} × a^x X -{1 / l a} $. a^x X^.</math>: $cribatur <math>X^. = P x^., P^. = Q x^., Q^. = R x^., &c.</math>; tum e præce-
dente methodo deduci pote$t <math>$. a^x X x^. = m a^x X - m^2 a^x P + m^3 a^x Q -
&c.</math> ubi <math>m = {1 / la} & la = log. a</math>: aliter hoc problema re$olvi pote$t e
principiis prius traditis, $int enim <math>X x^. = P^., P x^. = Q^., Q x^. = R^., &c.</math>
& erit <math>$. a^x X x^. = a^x P - (l a) a^x Q + (l a)^2 a^x R - &c.</math></p>
<p>4. Sit <math>βx^. × φ $. W x^. × π $. V x^. × F $. y x^.</math>, vel <math>βx^. φ $. W x^. π $. V x^. F
$. y x^. &c. &c.</math>, ubi in priore fluxione per <math>φ $. W x^., π $. V x^., F $. y x^.,

<pb 31>[0083]<rh>FLUXIONUM FLUENTIBUS.</rh>
&c.</math> de$igno datas functiones fluentium <math>$. W x^., $. V x^., $. y x^., &c.</math> in
po$teriori ca$u per <math>F $. y x^.</math> intelligo datam functionem fluentis <math>$. y x^.</math>;
per <math>π $. V x^. F $. y x^.</math> intelligo datam functionem fluentis <math>$. V x^. F $. y x^.</math>;
& $ic deinceps. Si fluentes <math>$. W x^. & $. V x^. & $. y x^., &c., & $. V x^. F $. y x^.,
&c.</math> inveniri po$$int vel in finitis terminis quantitatis (<math>x</math>), vel in ter-
minis reliquarum fluentium, &c. cum finitis terminis; tum primo in-
veniantur prædictæ fluentes in terminis reliquarum fluentium, &c.; &
in datâ fluxione ex iis exprimantur: deinde fluentes prædictarum
fluxionum per methodos in hoc & præced. problem. traditas deduci
po$$unt; e. g. fluentiales plerumque per eundum modum ac irratio-
nales quantitates tractandæ $unt, i. e. $it <math>Q × R^γ-1 × S^μ-1 × &c. × x^θ-1 x^.</math>
data fluxio, ubi literæ <math>Q, R, S, &c.</math> denotant fluentes fluxionum, quæ
$unt functiones quantitatis <math>x</math> in <math>x^.</math> ductæ tum a$$umenda e$t pro flu-
ente datæ fluxionis quantitas <math>P × R^λ × S^μ × &c.</math>; deinde inveniatur
ejus fluxio, & fiant corre$pondentes termini datæ & re$ultantis æqua-
tionis inter $e æquales, & ex æquationibus re$ultantibus deduci pote$t
fluens quæ$ita, $i modo ea in prædictis terminis exprimi po$$it; &c.
Horum ca$uum facile infinita dari po$$unt exempla.</p>
<p>Ex ii$dem principiis deduci po$$unt fluentes omnium huju$ce gene-
ris fluxionum.</p>
<h>PROB. VI.</h>
<h it>Datâ exponentiali fluxione, invenire utrum ejus fluens exprimi pote$t in
finitis algebraicis & exponentialibus terminis, necne.</h>
<p>Ob$ervatâ lege fluxionis exponentialis in prob. 3. traditâ, facile
re$olvi pote$t hoc problema. Nullæ aliæ enim in fluente continentur
exponentiales quantitates præter eas, quæ in datâ fluxione dantur.
Exponentialibus vero quantitatibus datis & earum fluxionibus inven-
tis, facile per $ub$titutiones in præcedentibus problematibus tradi-
tas erui po$$unt algebraicæ quantitates: $i fluxio contineat expo-

<pb 32>[0084]<rh>DE INVENIENDIS</rh>
nentiales $uperiorum ordinum quantitates, tum ejus fluens, terminis
ip$is inter $e comparatis, facile deduci pote$t: $i vero $it exponentialis
primi ordinis, tum vel e terminis ip$is inter $e comparatis; vel $i hoc
non valeat in quibu$dam ca$ibus, ex continuis approximationibus
deduci pote$t fluens quæ$ita.</p>
<p>E principiis pro detegendis irrationalium & fluentialium fluxionum
fluentibus, datis; $emper inveniri pote$t fluens cuju$cunque fluxionis,
quæ continet irrationales, fluentiales & exponentiales quantitates.</p>
<p>Ex. 1. Sit data fluxio <math>(y + n - 1) × <r n-1>y - 1</r> y^y y^. + log. y × y^y ×
(y^n - n y^n-1 + n. {n-1 / 2} y^n-2 - n. {n-1 / 2} · {n-2 / 3} y^n-3 + &c.) × y^.</math>.</p>
<p>In hâc fluxione unica exponentialis invenitur, viz. <math>y^y</math>; ergo nulla
alia continetur in quæ$itâ fluente: $ed log. <math>y × y^y y^.</math> ducitur in <math>y^n -
n y^n-1 + n · {n-1 / 2} y^n-2 - &c. = <r n>y-1</r></math>; ergo per problema fluens, $i
modo exprimi po$$it finitis terminis, erit <math>y^y × <r n>y-1</r></math>.</p>
<p>Ex. 2. Sit data fluxio <math>Q^x x^n x^.</math>; fluxio vero quantitatis <math>Q^x</math> e$t log. <math>Q
(m) Q^x x^.</math>; ergo pro fluente quæ$itâ a$$umatur <math>N Q^x × x^n - α</math>, cujus
fluxio erit <math>N m Q^x x^n x^.</math> (data fluxio, $i modo <math>N = {1 / m}) + n N Q^x x^n-1 x^.
- α^.</math>, unde <math>α^. = n N Q^x x^n-1 x^.</math>; & $ic iteratâ operatione a$$umatur pro
<math>α</math> quantitas <math>{n / m^2} Q^x x^n-1 - β</math>; & $ic deinceps; & tandem re$ultabit fluens
quæ$ita <math>{Q^x / m} × (x^n - {n x^n-1 / m} + {n × (n-1) / m^2} x^n-2 - {n. (n-1). (n-2) / m^3} x^n-3 + &c.)</math>
quæ $eries terminat, $i <math>n</math> $it integer po$itivus numerus: aliter pro flu-
ente fluxionis <math>Q^x x^n x^.</math> a$$umatur $umma <math>Q^x {x^n+1 / n+1} - α</math>, cujus $ummæ
inveniatur fluxio, & re$ultat <math>Q^x x^n x^.</math> (data fluxio) <math>+ m Q^x {x^n+1 / n+1}x^. - α^.</math>,
unde <math>α^. = {m / n + 1} Q^x x^n+1 x^.</math>; & $ic iteratis operationibus invenietur $eries

<pb 33>[0085]<rh>FLUXIONUM FLUENTIBUS.</rh>
<math>Q^x × ({x^n+1 / n+1} - {m x^n+2 / (n + 1) · (n + 2)} + {m^2 x^n+3 / (n + 1) · (n + 2) · (n + 3)} - &c.)</math>
quæ $emper in infinitum progreditur.</p>
<p>Sub$equenti modo hoc exemplum aliter re$olvi pote$t: a$$umatur
pro quæ$itâ fluente quantitas <math>Q^x × (A x^n + B x^n-1 + C x^n-2 + &c.)</math> &
$i æquetur ejus fluxio datæ, exinde re$ultabunt coe$$icientes quæ$itæ
<math>A, B, C, &c.</math></p>
<p>Ex. 3. Sit fluxio <math><r>a + b x</r>^<r m>f+g x</r> × <r l+mx+nx^2 >c + d x + e x^2</r> × <r s>b + k x</r> ×
(m <r m-1>f + g x</r> × g × log. <r>a + b x</r> + <r m>f + g x</r> × {b / a + b x} + <r>m + 2 n x</r>
× log. <r>c + d x + e x^2</r> + <r>l + m x + n x^2</r> × {d + 2 e x / c + d x + e x^2} + {s k / h + k x}) x^.
+ &c.</math>: pro ejus fluente a$$umenda e$t quantitas <math>A<r>a + b x</r>^<r m>f+g x</r> ×
^1+mx+nx^2 <r>c + d x + e x<_>2</_></r> × <r s>b + k x</r> + &c.</math>; & ex æquatâ ejus fluxione
datæ, erui pote$t fluens fluxionis datæ.</p>
<p>Cor. Sit data fluxio <math>X x^.</math>, ubi <math>X</math> e$t functio quantitatis <math>x</math>; in eâ pro
<math>x</math> $cribatur quæcunque fluentialis & exponentialis quantitas, & ejus
fluxio pro <math>x^.</math>; & re$ultat fluxio, quæ reduci pote$t ad priorem <math>X x^.</math>.</p>
<h>THEOR. II.</h>
<p>1. Datâ quantitate <math>A</math>, in quâ continentur duæ variabiles quanti-
tates <math>x & y</math>; $it ejus fluxio <math>A^. = a x^. + b y^.</math>; inveniantur fluxiones
quantitatum <math>a & b</math>, quæ $int re$pective <math>a^. = αx^. + βy^., b^. = πx^. + ξy^.</math>,
ubi <math>a, b, α, β, π, ξ</math> $unt functiones literarum <math>x & y</math>; tum erit <math>π = β</math>,
i.e. <math>π</math> erit eadem quantitas ac <math>β</math>.</p>
<p>Cor. Hinc, datâ fluxione <math>(A^. = a x^. + b y^.)</math> duas variabiles quanti-
tates <math>x & y</math> involvente, inveniri pote$t; utrum ejus fluens exprimi
pote$t, necne.</p>
<p>Inveniantur enim fluxiones quantitatum <math>a & b, & $i a^. = αx^. + βy^.,

<pb 34>[0086]<rh>DE INVENIENDIS</rh>
& b^. = π x^. + ξ y^., & π = β</math>, tum exprimi pote$t fluens; $in aliter vero
non.</p>
<p>2. Sint tres vel plures variabiles quantitates <math>(x, y, z, &c.)</math> in datâ
quantitate <math>(A)</math> contentæ, & $it <math>A^. = a x^. + b y^. + c z^. + &c. & a^. = α x^.
+ β y^. + γ z^. + &c. b^. = λ x^. + μ y^. + v z^. + &c. c^. = π x^. + ξ y^. + σ z^.
+ &c.</math> tum erit per præcedentem ca$um <math>β = λ, γ = π, ν = ξ, &c;</math>.</p>
<p>Cor. Datâ fluxione <math>A^. = a x^. + b y^. + c z^. + &c.</math> tres, &c. variabiles
quantitates involvente, inveniantur fluxiones quantitatum <math>a, b, c, &c.</math>
quæ $int re$pective <math>a^. = α x^. + β y^. + γ z^. + &c. b^. = λ x^. + μ y^. +
ν z^. + &c. c^. = π x^. + ξ y^. + σ z^. + &c.</math> & $i <math>β = λ, γ = π, ν = ξ, &c.</math>
etiamque $i fluxiones quantitatis <math>A</math> inveniantur per hanc methodum,
viz. 1<_>mo</_>. una quantitas vel <math>x</math> vel <math>y</math> vel <math>z &c.</math> in quantitate <math>A</math> contenta
$olummodo $upponatur variabilis; & <math>A^.</math> evadat <math>B</math>; deinde in quantitate
<math>B</math> tantummodo $upponatur quæcunque reliqua vel <math>y</math> vel <math>z</math> vel <math>x &c.</math>
variabilis, & <math>B^.</math> evadat quantitas <math>C</math>; tertio quæcunque adhuc reliqua
quantitas, i. e. quæ non prius $uppo$ita fuit variabilis, & evadat <math>C^.</math>
quantitas <math>D</math>; & $ic deinceps: & $i ultima quantitas $emper evadat
eadem; annon hæc vel illa vel quæcunque alia 1<_>mo</_>. $upponitur e$$e
invariabilis, & $ic deinccps; i. e. $i non refert, quo ordine quantitates
$upponuntur variabiles; ultima enim quantitas $emper eadem re$ul-
tat; tum ejus fluens exprimi pote$t: $in aliter vero non.</p>
<p>3. 1. Sit fluxio <math>A^. ^n</math> ordinis <math>(n)</math> duas vel tres vel plures variabiles
quantitates <math>(x, y, z, &c.)</math> & earum fluxiones <math>n</math> ordinis continens, i. e. $it
<math>A^. ^n = a x^. ^n + b x^. x^. ^n-1 + c x^. ^2 x^. ^n-2 + d x^. ^3 x^. ^n-3 + &c.
+ B x^.. x^. ^n-2 + C x^.. x^. x^. ^n-3 + &c.
+ D x^... x^. ^n-3 + &c.
(P^. ^n) + p x^. ^n-1 y^. + q x^. ^n-2 y^. ^2 + &c.
&c.
+ α y^. ^n + β y^. y^. ^n-1 + γ y^. ^2 y^. ^n-2 + &c.
+ Γ y^.. y^.. ^n-2 + &c. (Q^. ^n) + π y^. ^n-1 x^. + ξ y^. ^n-2 x^. ^2 + &c. &c. &c. &c.</math></p>

<pb 35>[0087]<rh>FLUXIONUM FLUENTIBUS.</rh>
<p>Supponantur omnes quantitates præter <math>x</math> e$$e invariabiles & erit
<math>P^. ^n = fluxi. n - 1</math> ordinis quantitatis. <math>a x^.</math>; deinde $upponantur omnes
quantitates præter <math>y</math> e$$e invariabiles, & erit <math>Q^. ^n = flux. n - 1</math> ordinis
quantitatis <math>α y^.</math>; & $ic de reliquis variabilibus quantitatibus.</p>
<p>3. 2. Sit <math>a^. = δ x^. + ε y^. + &c. & α^. = λ x^. + μ y^. + &c.</math> & erit coeffi-
ciens <math>p</math> termini <math>(x^. ^n-1 y^.) = <r>n - 1</r> ε + λ</math>, & $ic invenitur <math>π</math> coefficiens
termini <math>(y^. ^n-1 x^.) = <r>n - 1</r> λ + ε</math>; $ed e theor. con$tat <math>λ = ε</math>, ergo <math>p = π
= n λ</math>; & $ic e methodo inveniendi fluxiones datarum fluentium fa-
cile con$tant reliqui termini; & vice versâ e datis terminis inveniri
pote$t, utrum fluens datæ fluxionis $it integrabilis, necne.</p>
<p>3. 3. Inveniatur fluxio <math>(n - 1)</math> ordinis fluxionis <math>a x^. + α y^. + &c.</math>;
& $i haud eadem evadat ac data fluxio <math>(n)</math> ordinis; tum fluens datæ
fluxionis non integrari pote$t: $i vero eadem $it, tum per præcedentem
ca$um inveniatur, annon fluens fluxionis <math>a x^. + α y^. + &c.</math> integrari
pote$t; $i integrari po$$it, tum fluens datæ fluxionis etiam integrari
pote$t; $in aliter non.</p>
<p>4. Datâ quantitate <math>(A)</math> continente duas variabiles quantitates <math>x
& y:1^mo</math>. a$$umatur <math>x</math> tanquam invariabilis quantitas, & inveniatur
<math>(α)</math> fluxio ordinis <math>m</math> quantitatis <math>(A)</math>; deinde in fluxione <math>(α)</math> ordinis <math>m</math>
re$ultante a$lumantur <math>y, y^., y^.., .... & y^. ^m</math> tanquam invariabiles quanti-
tates, & inveniatur fluxio ordinis <math>r</math> fluxionis <math>(α)</math>: fluxio re$ultans
eadem erit ac fluxio inventa e $ub$equenti methodo, viz. a$$umatur <math>y</math>
tanquam invariabilis quantitas, & inveniatur fluxio <math>(β)</math> ordinis <math>r</math>
quantitatis <math>A</math>; deinde in fluxione <math>(β)</math> ordinis <math>(r)</math> re$ultante a$$uman-
tur <math>x, x^., x^.., &c.</math> tanquam invariabiles quantitates, & inveniatur fluxio
ordinis <math>m</math> fluxionis <math>(β)</math>.</p>
<p>Idem etiam verum e$t, i. e. fluxiones re$ultantes erunt inter $e æqua-
les, $i modo fluxiones ordinis <math>(m + r)</math> quantitatis <math>(A)</math> inveniantur,
i. e. inveniantur re$pective <math>(m + r)</math> fluxiones <math>A^., A^.., A^..., ... A^. ^m+r</math>, quarum
quæcunque <math>r</math> fluxiones inveniantur ex hypothe$i quod <math>y, y^., y^.., &c.</math>

<pb 36>[0088]<rh>DE INVENIENDIS</rh>
$int invariabiles quantitates; <math>(m)</math> vero reliquæ, quod <math>x, x^., x^.., &c.</math> $int
invariabiles quantitates. Minime refert, quo ordine inveniuntur
fluxiones prædictæ.</p>
<p>5. Eadem etiam affirmari po$$unt de fluxionibus quantitatis <math>A</math> tres
vel plures variabiles quantitates habentis.</p>
<p>5. 2. Con$imilia etiam prædicari po$$unt de fluxionibus quantitatis
<math>A</math>, in quâ continentur fluxiones $uperiorum <math>(m)</math> ordinum quantita-
tum variabilium <math>(x, y, z, &c.)</math>. Minime enim refert ordo, in quo de-
teguntur fluxiones; ex hypothe$i quod <math>x</math> vel <math>y</math> vel <math>z &c.</math> $it invariabilis.</p>
<p>6. Invenire generaliter, annon data fluxio cuju$cunque ordinis <math>m</math>
$it integrabilis; 1<_>mo</_>. a$$umatur quantitas <math>(A)</math> tanquam generalis fun-
ctio quantitatum (<math>x, y, z, v, &c.)</math>; deinde inveniatur fluxio ordinis
<math>(m)</math> quantitatis <math>A</math>; & æquentur corre$pondentes datæ & inventæ flu-
xionis termini: $i hoc non fieri po$$it, tum fluxio data non erit in-
tegrabilis.</p>
<p>Hic animadvertendum e$t, quod$i <math>M × x^. ^π^b × x^. ^π′^b′ × &c. × y^. ^ρ^k × y^. ^ρ′^k′ × &c.
× z^. ^σ^ι × z^. ^σ′^ι′ × &c. = Γ & N × x^. ^e^ε × x^. ^e′^ε′ × &c. × y^. ^f^ζ × y^. ^f′^ζ′ × &c. × z^. ^g^θ × z^. ^g′^θ′ × &c.
= Δ</math> $int duo termini re$ultantis fluxionis ordinis <math>(m)</math>; & utri-
u$que <math>(Γ & Δ)</math> inveniantur fluxiones ex hypothe$i aliquando ut <math>x</math> $o-
lummodo $it variabilis, & aliquando ut <math>x^.</math> vel <math>x^..</math>, vel <math>x^...</math>, vel &c., vel <math>y</math>
vel <math>y^.</math> vel <math>y^..</math> vel &c., vel <math>z</math> vel <math>z^.</math> vel <math>z^..</math> vel &c. $olummodo $it variabilis,
ita ut duæ re$ultantes fluxiones contineant eundem fluxionalem ter-
minum <math>x^. ^λ^θ × x^. ^λ′^θ′ × &c. × y^. ^μ^χ × y^. ^μ′^χ′ × &c. × z^. ^ν^i × z^. ^ν′^i′ × &c. = H</math>, & $int flu-
xiones re$ultantes <math>R × H & S × H</math>, ubi <math>R & S</math> $unt algebraicæ functi-
ones quantitatum <math>(x, y, z, &c.)</math>; tum erit <math>R:S</math> in datâ ratione, quæ
ratio facile deduci pote$t: $i hanc $emper ob$ervent rationem con$i-
miles quantitates ex iis in datâ fluxione contentis deductæ; tum flu-
xio integrabilis erit; $in aliter vero non.</p>
<p>Si <math>x^. ^π^b, x^. ^ρ^ε, &c.; y^. ^ρ^k, &c.</math> habeant maximas dimen$iones prædictarum flu-
xionum <math>x^. ^π, x^. ^ρ, &c.; y^. ^ρ, &c.</math> in duobus terminis <math>Γ & Δ</math>; tum in fluxionali

<pb 37>[0089]<rh>FLUXIONUM FLUENTIBUS.</rh>
termino <math>H</math> haud nece$$e e$t, ut fluxiones <math>x^. ^π, x^. ^ρ, &c.; y^. ^ρ, &c.</math> ad majo-
res dimen$iones a$cendant.</p>
<h>PROB. VII.</h>
<h>1. _Datâ fluxione duas variabiles_ <math>(x & y)</math> _quantitates involvente, in-_
_venire ejus fluentem_.</h>
<p>A$$umatur <math>x</math> tanquam invariabilis quantitas, & inveniatur fluens
quantitatis re$ultantis; deinde a$$umatur <math>y</math> tanquam invariabilis quan-
titas, & inveniatur fluens quantitatis exinde re$ultantis; & $i hæ duæ
fluentes omnes functiones, in quibus continentur duæ incognitæ
quantitates <math>(x & y)</math>, ea$dem habent; tum fluens ejus invenitur; aliter
vero haud inveniri pote$t.</p>
<p>Ex. 1. Sit fluxio <math>(6 y^5 + 6 y √(a^2 + x^2)) × y^. + <r>3 x^2 + {3 y^2 / √(a^2 + x^2}</r>
× x^.</math>, invenire ejus fluentem.</p>
<p>1<_>mo</_>. Supponatur <math>x</math> e$$e invariabilis; tum transformatur data fluxio
in fluxionem <math><r>6 y^5 + 6 y √(a^2 + x^2)</r> y^.</math>, cujus fluens erit <math>y^6 + 3 y^2
√(a^2 + x^2): 2^do</math>. vero $upponatur <math>y</math> e$$e invariabilis, & fluxio re$ul-
tans erit <math><r>3 x^2 + {3 y^2 / √(a^2 + x^2)}</r> x^.</math>, cujus fluens invenitur <math>x^3 + 3 y^2
√ (a^2 + x^2)</math>; $ed functio <math>3 y^2 √(a^2 + x^2)</math>, in quâ continentur duæ
incognitæ quantitates <math>x & y</math> in utri$que fluentibus eadem e$t; ergo
fluens inveniri pote$t, & erit <math>3 y^2 √(a^2 + x^2) + y^6 + x^3</math>.</p>
<p>Ex. 2. Sit data fluxio <math>9 x^2 y^3 x^. + 16 x^3 y^3 y^.</math>; $upponatur <math>y</math> e$$e in-
variabilis, & re$ultat fluxio <math>9 x^2 y^3 x^.</math>, cujus fluens e$t <math>3 x^3 y^3</math>; deinde
$upponatur <math>x</math> e$$e invariabilis, & re$ultat fluxio <math>16 y^3 x^3 y^.</math>, cujus fluens
e$t <math>4 x^3 y^4</math>; $ed <math>3 x^3 y^3</math> e$t functio in quâ continentur literæ <math>x & y</math> in
unâ fluxione, & <math>4 x^3 y^4</math> in alterâ: hæ autem duæ functiones haud
$unt eædem, ergo fluens datæ quantitatis haud inveniri pote$t.</p>
<p>2. Datâ fluxione tres vel plures variabiles quantitates <math>(x, y, z, v,
&c.)</math> habente, cogno$ci pote$t e præcedente methodo, utrum ejus fluens

<pb 38>[0090]<rh>DE INVENIENDIS</rh>
inveniri pote$t, necne: a$$umantur omnes variabiles quantitates præ-
ter unam tanquam invariabiles, & quantitatis re$ultantis inveniatur
fluens, & $ic de reliquis; & $i hæ fluentes re$ultantes ea$dem habe-
ant functiones duarum vel plurium variabilium quantitatum <math>(x, y, z,
&c.)</math> tum ejus fluens inveniri pote$t, $in aliter vero non.</p>
<p>Ex. 1. Sit fluxio <math>2 x^4 y^3 z z^. + <r>3 x^4 z^2 y^2 + 2 a x^2 y + {2 y / √(x^2 + y^2)} + 2 b y</r>
y^. + <r>4 x^3 y^3 z^2 + 2 a y^2 x + {2 x / √(x^2 + y^2)} + 3 e x^2 + 2 f x</r> × x^.</math>.</p>
<p>Primo $upponantur omnes quantitates <math>(x, y, &c.)</math> præter <math>z</math> e$$e
invariabiles, & re$ultat fluxio <math>2 x^4 y^3 z z^.</math>, cujus fluens e$t <math>x^4 y^3 z^2</math>;
deinde $upponantur omnes præter <math>y</math> invariabiles e$$e, & re$ultat fluxio
<math><r>3 x^4 z^2 y^2 + 2 a x^2 y + {2 y / √(x^2 + y^2)} + 2 b y</r> y^.</math>, cujus fluens e$t <math>x^4 z^2 y^3 +
a x^2 y^2 + 2 √(x^2 + y^2) + b y^2</math>; tertio $upponatur <math>x</math> $olummodo e$$e va-
riabilis, & re$ultat fluxio <math><r>4 x^3 y^3 z^2 + 2 a y^2 x + {2 x / √(x^2 + y^2)} + 3 e x^2 + 2 f x</r> x^.</math>,
cujus fluens e$t <math>x^4 y^3 z^2 + a y^2 x^2 + 2 √(x^2 + y^2) + e x^3 + f x^2</math>: in his
tribus fluentibus terminus (in quo continentur omnes literæ <math>z, y, x</math>)
idem e$t, viz. <math>x^4 y^3 z^2</math>: in duabus po$terioribus fluentibus termini, in
quibus continentur duæ literæ <math>x & y</math>, iidem $unt; viz. <math>a y^2 x^2 &
2 √(x^2 + y^2)</math>; unde fluens datæ fluxionis invenitur <math>x^4 y^3 z^2 + a x^2 y^2 +
2√(x^2 + y^2) + b y^2 + e x^3 + f x^2 + A</math>, ubi litera <math>A</math> denotat quantita-
tem ad libitum a$$umendam.</p>
<p>Cor. Hæc methodus haud $olummodo deteget, annon fluentes
exprimi po$$unt; $ed etiam fluentes ip$as; exhinc etiam facilius in-
veniri po$$unt fluentes fluxionum $uperiorum ordinum, maxime vero
ex iis ob$ervatis, quæ in theor. 2. tradita fuere.</p>
<p>3. Si vero, cum a$$umantur omnes quantitates præter unam tan-
quam invariabiles, fluentes re$ultantium fluxionum haud per vulgares
methodos cogno$ci po$$int; $ingula re$ultans fluxio in in$initas $eries

<pb 39>[0091]<rh>FLUXIONUM FLUENTIBUS.</rh>
reducatur, quarum termini progrediuntur $ecundum dimen$iones
eju$dem literæ, & ex hoc problemate inveniantur fluentes terminorum
$erierum re$ultantium; & id, quod requiritur, peractum erit.</p>
<p>Ex. Sit <math>{x^. + y^. / x + y}</math>; a$$umatur <math>x</math> invariabilis, & re$ultat <math>{y^. / x + y} = {y^. / x} - {yy^. / x^2}+ {y^2 y^. / x^3} - &c.</math> cujus fluens erit <math>{y / x} - {y^2 / 2x^2} + {y^3 / 3x^3} - &c.</math> deinde a$$u-
matur <math>y</math> invariabilis, & re$ultat fluxio <math>{x^. / x + y} = {x^. / x} - {yx^. / x^2} + {y^2 x^. / x^3} - {y^3 x^. / x^4}+ &c.</math> cujus fluens e$t log. <math>ax + {y / x} - {y^2 / 2x^2} + {y^3 / 2x^3} - &c.</math> termini
vero utriu$que $eriei, in quibus continentur literæ <math>(x & y)</math>, iidem
$unt; ergo fluens datæ fluxionis inveniri pote$t.</p>
<p>Et $ic de fluxionibus tres vel plures variabiles quantitates invol-
ventibus. Hæc principia etiam ad detegendum, annon fluentes fluxi-
onum $uperiorum ordinum exprimi po$$unt, facile applicari po$$unt.</p>
<h>PROB. VIII.</h>
<h it>_I._ Datis corre$pondentibus valoribus datæ fluentis & $ingularum varia-
bilium quantitatum in eâ contentarum, eam corrigere.</h>
<p>Scribantur dati valores $ingularum variabilium quantitatum pro
$uis valoribus in datâ fluente; & quantitas re$ultans $it <math>B</math>; $it vero
corre$pondens valor fluentis <math>A</math>; & addatur differentia <math>A - B</math> ad da-
tam fluentem, & $umma erit correcta fluens.</p>
<p>Ex. 1. Sit fluxio <math>ax^n x^.</math>, cujus fluens e$t <math>{ax^n+1 / n + 1}</math>; cum vero <math>x</math> fiat ni-
hilo æqualis, fiat prædicta fluens <math>A</math>: $cribatur pro <math>x</math> ejus valor <math>o</math> in
datâ fluente <math>{ax^n+1 / n + 1}</math>, & re$ultat <math>B = 0</math>; & exinde fluens correcta erit
<math>{ax^n+1 / n + 1} + A</math>.</p>

<pb 40>[0092]<rh>DE INVENIENDIS</rh>
<p>Ex. 2. Sit fluxio <math>3√(x^2 + y^2)<r>xx^. + yy^.</r></math>, cujus fluens e$t <math><r 1{1 / 2}>x^2 + y^2</r></math>.
Sint corre$pondentes valores quantitatum <math>x & y</math>, & datæ fluentis re-
$pective <math>α, β & A</math>; $cribantur in datâ fluente pro incognitis quanti-
tatibus <math>x & y</math> re$pective <math>α & β</math>, & re$ultat <math><r 1{1 / 2}>α^2 + β^2</r> = B</math>; & con$e-
quenter fluens erit <math><r 1{1 / 2}>x^2 + y^2</r> + A - B</math>.</p>
<p>Ex. 3. Sit fluxio <math>sby^s-1 y^.. + s · <r>s - 1</r> by^s-2 y^. ^2 + r · <r>r - 1</r> ax^r-2 x^. ^2</math>,
ejus fluens erit <math>sby^s-1 y^. + rax^r-1 x^.</math>, $i modo fluat uniformiter <math>x</math>;
prædictâ methodo $cribantur pro <math>x, y, x^., y^.</math> earum corre$pondentes
valores, & quantitas re$ultans $it <math>B</math>; $it <math>A</math> corre$pondens valor datæ
fluentis, & erit correcta fluens <math>sby^s-1 y^. + rax^r-1 x^. + <r>A - B</r> (Cx^.)</math>
quoniam <math>x^.</math> e$t data quantitas. Fluxionis vero <math>sby^s-1 y^. + rax^r-1 x^. +
Cx^.</math> fluens erit <math>by^s + ax^r + Cx</math>, corrigatur hæc fluens; & erit correcta
fluens <math>by^s + ax^r + Cx + D</math>, ubi <math>D</math> invariabilem quantitatem præ-
dictâ methodo acqui$itam denotat. Et $ic in genere de correctionibus
fluentium huju$ce generis.</p>
<p>Cor. 1. Sit <math>m</math> minor quam <math>n</math>; & data fluxio <math>n</math> ordinis, cujus fluens
(<math>π</math>) ordinis (<math>m</math>) erit fluxio ordinis (<math>n - m</math>): $it <math>v</math> quantitas, quæ fluit
uniformiter; & literæ <math>A, B, C, D,.. P, Q</math> invariabiles re$pective de-
notent quantitates; & erit correcta fluens <math>(m)</math> ordinis <math>= π +
(Av^m-1 + Bv^m-2 + Cv^m-3 .. Pv + Q) × v^. ^n-m</math>.</p>
<p>Ex. Sit fluxio <math>n × (n - 1) × (n - 2)..(n - m + 1) Lx^n-m x^. ^m</math>, ubi <math>x</math>
fluit uniformiter, cujus fluens erit <math>x^n</math>; fluens vero correcta erit <math>x^n +
Ax^m-1 + Bx^m-2 + Cx^m-3 .. Px + Q</math>.</p>
<p>2. Erit generalis fluens fluxionis <math>(yz^.) = $.yz^. + A</math>; & generalis
fluens fluxionis <math>(x^. $.yz^.) = $.x^. $.yz^. + Ax + B</math>; erit generalis flu-
ens fluxionis <math>(w^. $.x^. $.yz^.) = $.w^. $.x^. $.yz^. + A$.xw^. + Bw + C</math>, &
fluens fluxionis <math>(v^. $.w^. $.x^. $.yz^.) = $.v^. $.w^. $.x^. $.yz^. + A$.v^. $.xw^. +
B$.wv^. + Cv + D</math>, & fluens fluxionis <math>(u^. $.v^. $.w^. $.x^. $.yz^.) = $.u^. $.v^.
$.w^. $.x^. $.yz^. + A$.u^. $.v^. $.xw^. + B$.u^. $.wv^. + C$.vu^. + Du + E</math>,

<pb 41>[0093]<rh>FLUXIONUM FLUENTIBUS.</rh>
& $ic deinceps: hìc literæ <math>A, B, C, D, E, &c.</math> re$pective denotant
invariabiles coefficientes ad libitum a$$umendas.</p>
<h>PROB. IX.</h>
<p it>Datis <math>m</math> valoribus datæ fluentis (<math>π</math>) & corre$pondentibus valoribus
$ingularum variabilium (<math>y, z, &c.</math>) quantitatum in eâ contentarum, inve-
nire ejus fluentis <math>m</math> ordinis correctionem.</p>
<p>Scribantur dati <math>m</math> corre$pondentes valores $ingularum variabilium
quantitatum in datâ fluente pro quantitatibus ip$is, & $int quanti-
tates re$ultantes re$pective <math>ξ, σ, τ, υ, &c.</math> Sint vero <math>m</math> dati valores
datæ fluentis re$pective <math>a, b, c, d, &c.</math> & a$$umatur pro correctâ fluente
($i modo <math>x</math> fluat uniformiter) quantitas <math>π + Ax^m-1 + Bx^m-2 + Cx^m-3
.. Px + Q</math>; ubi literæ <math>A, B, C, &c.</math> incognitas & invariabiles deno-
tant quantitates.</p>
<p>Sint <math>m</math> valores quantitatis <math>x</math>, qui corre$pondent (<math>m</math>) datis valoribus
(<math>a, b, c, &c.</math>) datæ fluentis (<math>π</math>) re$pective <math>α, β, γ, δ, &c.</math> & re$ultant (<math>m</math>
$implices æquationes <math>Aα^m-1 + Bα^m-2 + Cα^m-3 + &c. = a - ξ; Aβ^m-1
Bβ^m-2 + Cβ^m-3 + &c. = b - σ; Aγ^m-1 + Bγ^m-2 + Cγ^m-3 + &c. =
c - τ, &c.</math> totidem (<math>m</math>) incognitas quantitates (<math>A, B, C, &c.</math>) haben-
tes; quæ ita reduci po$$unt, ut inveniantur valores incognitarum
quantitatum (<math>A, B, C, &c.</math>); & conficitur problema.</p>
<h>THEOR. III.</h>
<p>In corrigendis fluentibus fluxionum <math>Xx^.</math> eædem radices $emper ad-
hibendæ $unt; e. g. <math>$i + √(A)</math> (ubi <math>A</math> e$t functio variabilis quan-
titatis <math>x</math>) in fluxione occurrat; tum in fluentibus correctis etiam adhi-
benda e$t, $ed in ejus locum minime $ub$tituenda e$t radix <math>- √ (A)</math>;
in calculo enim fluentium non datur $altus, nempe nulla datur mu-
tatio in radicibus.</p>
<p>In corrigendis fluentibus, i. e. cum $ub$tituantur <math>a & b, &c.</math> pro
variabili <math>x</math> in generali fluente, in quantitatibus re$ultantibus eædem
radices $emper corre$pondenter u$urpandæ $unt.</p>

<pb 42>[0094]<rh>DE INVENIENDIS</rh>
<p>Ex. Requiratur fluens datæ fluxionis <math>(3 + 3 x √(1 - x^2))(e + 3 x
- (1 - x^2)^1{1 / 2})^{1 / m} × x^.</math> inter duos valores (<math>a & b</math>) quantitatis <math>x</math> contenta:
ejus fluens generalis erit <math>{m / m + 1} × (e + 3x - (1 - x^2)^1{1 / 2})^{m+1 / m} (P) + E</math>,
ubi <math>E</math> $it invariabilis ad libitum a$$umenda; fluens correcta quæ$ita
erit <math>{m / m + 1}((e + 3 a - (1 - a^2)^1{1 / 2})^{m+1 / m} - (e + 3 b - (1 - b^2)^1{1 / 2})^{m+1 / m})</math>;
vel <math>{m / m + 1}((e + 3 a + (1 - a^2)^1{1 / 2})^{m+1 / m} - (e + 3 b + (1 - b^2)^1{1 / 2})^{m+1 / m})</math>;
vel <math>{m / m + 1}(α (e + 3 a + (1 - a^2)^1{1 / 2})^{m+1 / m} - α (e + 3 b + (1 - b^2)^1{1 / 2})^{m+1 / m})</math>;
&c.; minime autem <math>{m / m + 1}(e + 3 a + (1 - a^2)^1{1 / 2})^{m+1 / m} - (e + 3 b -
(1 - b^2)^1{1 / 2})^{m+1 / m})</math>; vel <math>{m / m + 1}(α (e + 3a + (1 - a^2)^1{1 / 2})^{m+1 / m} - β(e + 3 b
+ (1 - b^2)^1{1 / 2})^{m+1 / m})</math>; vel &c.; ubi <math>1, α, β, &c.</math> $unt diver$æ radices
æquationis <math>y^m - 1 = 0; & + 1 & - 1</math> duæ radices æquationis <math>z^2 -
1 = 0</math>; & in priori ca$u iidem valores $emper earundem radicum (<math>y &
z</math>), in po$teriori haud $emper iidem valores prædicti in corre$ponden-
tibus terminis eju$dem correctæ fluentis continentur.</p>
<p>Et $ic de fluxionibus, in quibus plures continentur variabiles quan-
titates & earum fluxiones.</p>
<h>THEOR. IV.</h>
<p>Datâ fluxione (<math>Ax^.</math>), quæ e$t functio quantitatis <math>x</math> in <math>x^.</math>; $cribatur
data functio quantitatis <math>z</math> pro <math>x</math>, & ejus fluxio pro <math>x^.</math>; & re$ultet fluxio
(<math>Bz^.</math>); inveniantur valores quantitatis <math>z</math>, cum <math>x</math> evadat <math>vel = a, vel
= b</math>, qui $int re$pective <math>α, β, γ, δ, &c.; & π, ξ, σ, τ, &c.</math>; inveniantur
etiam valores fluentis datæ fluxionis inter duos valores <math>a & b</math> quanti-
tatis (<math>x</math>) po$iti; ii$dem radicibus, ut in præcedente regulâ docetur, in

<pb 43>[0095]<rh>FLUXIONUM FLUENTIBUS.</rh>
utroque ca$u $emper adhibitis: $int <math>α & π</math> duo valores prædicti quan-
titatis <math>z</math>, in quibus eædem $emper adhibentur radices, eædemque
etiam ac eæ in prædictis duobus ca$ibus u$urpatæ, cum <math>x</math> evadat <math>vel
= a, vel = b</math>; tum erit fluens fluxionis (<math>Ax^.</math>) inter valores <math>a & b</math>
quantitatis <math>x</math> contenta, eadem ac fluens fluxionis (<math>Bz^.</math>) inter valores
<math>α & π</math> quantitatis <math>z^.</math> contenta.</p>
<p>Ex. Sit fluxio <math>(2x + 3x √(20^2 - x^2))x^. = Ax^.</math>, cujus fluens e$t
<math>x^2 - (20^2 - x^2)^1{1 / 2} + E</math> quantitas invariabilis; $it <math>z^2 - 5z = x</math>,
unde <math>(2z^2 - 10z + 3 × (z^2 - 5z)(20^2 - <r 2>z^2 - 5z</r>)^{1 / 2})(2z - 5) × z^.
= Bz^.</math>: $int <math>6 & 14</math> valores quantitatis <math>x</math>, inter quos fluens datæ
fluxionis ponitur; tum erit fluens datæ fluxionis inter valores <math>14 & 6</math>
quantitatis <math>x</math> re$pective <math>14^2 + (20^2 - 14^2)^1{1 / 2} - (6^2 + (20^2 - 6^2)^1{1 / 2})
= P, & 14^2 - (20^2 - 14^2)^1{1 / 2} - (6^2 - (20^2 - 6^2)^1{1 / 2}) = Q</math>: cum $ub-
$tituantur <math>14 & 6</math> pro <math>x</math> in quantitate <math>P</math> adhibentur <math>+ (20^2 - 14^2)^1{1 / 2}
& + (20^2 - 6^2)^1{1 / 2} pro - (20^2 - x^2)^1{1 / 2}</math>, & in po$teriori quantitate <math>Q</math>
u$urpantur <math>- (20^2 - 14^2)^1{1 / 2} & - (20^2 - 6^2)^1{1 / 2}</math> pro eâdem quantitate
<math>- (20^2 - x^2)^1{1 / 2}</math>; i. e. in priori ca$u $emper applicatur radix <math>√(1) =
- 1</math>; in po$teriori vero ca$u <math>√(1) = + 1</math>: deinde inveniantur valores
quantitatis <math>z</math>, cum evadat <math>x = 14 vel = 6</math>; i. e. radices æquationum
<math>z^2 - 5z = 14 & z^2 - 5z = 6</math>; radices prioris æquationis erunt
<math>± √({25 / 4} + 14) + {5 / 2} = 7</math>, cum adhibeatur $ignum <math>+; & = - 2</math>,
$i modo adhibeatur $ignum <math>-</math>: & $imiliter radices æquationis <math>z^2 -
5 z = 6</math> erunt <math>± √({25 / 4} + 6) + {5 / 2} = 6</math> cum adhibeatur $ignum <math>+;
& = - 1</math>, $i modo u$urpetur $ignum <math>-</math>: unde inter valores <math>7 & 6</math>
quantitatis <math>z</math> duo valores fluentis fluxionis (<math>B z^.</math>) viz. <math>P & Q</math>; etiam-
que inter valores <math>- 2 & - 1</math> quantitatis <math>z</math> continentur duo valores
prædictis (<math>P & Q</math>) æquales fluentis fluxionis (<math>B z^.</math>).</p>

<pb 44>[0096]<rh>DE INVENIENDIS</rh>
<p>Cor. Hinc, $i modo <math>A x^.</math> $it fluxio, cujus fluens vel in finitis termi-
nis exprimi pote$t, vel non; vel etiam $it fluentialis vel exponentialis,
&c. fluxio; & deducatur fluxio <math>B z^.</math>, cujus variabilis quantitas e$t <math>z</math>,
ubi <math>z</math> e$t functio quantitatis <math>x</math>; tum ex hoc theoremate deduci po$-
$unt valores quamplurimi fluentis fluxionis <math>(B z^.)</math> inter <math>α & π, β & ξ,
γ & σ, &c:</math> valores quantitatis <math>z</math> contenti, qui erunt inter $e æqua-
les, $i modo corre$pondentes radices in <math>α & π, β & ξ, γ & σ, &c.</math> ir-
rationalium quantitatum in fluente <math>$. B z^.</math> contentæ, re$pective, ut
prius docetur, $int eædem.</p>
<p>Cor. 2. Con$imilia etiam applicari po$$unt ad ca$us, in quibus <math>x</math>
e$t functio quantitatis <math>z</math> e datâ æquatione relationem inter <math>x & z</math> ex-
primente deducenda; etiamque ad ca$us in quibus plures variabiles
quantitates & earum fluxiones primi, $ecundi, &c. ordinum in datâ
fluxione continentur.</p>
<h>PROB. X.</h>
<p it>Invenire fluentem <math>(W)</math> datæ fluxionis (<math>X x^. = y x^.</math>), quæ $it functio
quantitatis <math>x</math> in <math>x^.</math> inter valores <math>a & b</math> quantitatis <math>x</math> contentam; cum
inter valores <math>a & b</math> contineatur valor vel valores <math>(α, β, γ, δ, &c.)</math> quanti-
tatis <math>x</math>, in quibus evadat <math>X = 0</math>; vel valores <math>(π, ξ, σ, τ, &c.)</math>, in quibus
<math>X</math> evadat infinita quantitas, & con$equenter <math>{1 / X} = 0</math>: ubi in <math>X = 0 &{1 / X} = 0</math> $emper eædem adhibentur radices (e. g. quadraticæ <math>+ √ vel
- √, &c.</math>), eædemque etiam ac eæ in quantitatibus ex $ub$titutione
quantitatum <math>a & b</math> pro <math>x</math> in prædictâ quantitate <math>X</math> exortis; ut in theor.
<math>3, & 4.</math> docetur.</p>
<p>Primo inveniantur radices æquationum <math>X = 0 & {1 / X} = 0</math>, quæ $int
re$pective <math>α, β, γ, δ, &c.; & π, ξ, σ, τ, &c.</math>; in quibus eædem prædictæ
radices adhibentur; & $int <math>γ, ξ, σ, δ, &c.</math> $ucce$$ivæ radices inter <math>a & b</math>
inventæ, & con$equenter <math>α, γ, ξ, σ, δ, &c. b</math> $ucce$$ivæ quantitates.</p>

<pb 45>[0097]<rh>FLUXIONUM FLUENTIBUS.</rh>
<p>1<_>mo</_>. 1. Si fluens vel quantitas <math>W</math> continua, i.e. quæ in eâ ea$dem
radices involvit, ex valore (<math>a</math>) quantitatis <math>x</math> ad valorem <math>b</math>, $emper $it
finita & po$$ibilis; $cribantur <math>α, γ, ξ, σ, &c., b</math> $ucce$$ive pro <math>x</math> in flu-
ente <math>W</math>, & re$ultent quantitates <math>A, P, R, S, T, &c., B</math>; inveniantur
differentiæ <math>A - P = λ, P - R = μ, R - S = ν, S - T = π, &c.</math>:
$umma omnium quantitatum <math>λ μ, ν, &c.</math> affirmative $umptarum $it
<math>Σ</math>; tum erit <math>Σ</math> fluens quæ$ita.</p>
<p>2<_>do</_>. Scribantur <math>v + γ, v + δ, &c.; v + ξ, v + σ, &c.</math>; pro <math>x</math> in datâ
quantitate <math>W</math>, & re$ultent quantitates <math>Γ, Δ, &c.; P, Σ, &c.</math>; reducan-
tur hæ quantitates in convergentes $eries $ecundum dimen$iones
quantitatis <math>v</math> progredientes, viz. <math>C v^{m / n} + &c., D v^{m′ / n′} + &c., + &c.; R v^{r / s}
+ &c., S v^{r′ / s′} + &c. + &c.; ubi {m / n}, {m′ / n′}, &c.; {r / s}, {r′ / s′}, &c.</math>. $unt fractiones
ad minimos terminos reductæ: in his $eriebus & quantitatibus ut in
theor. <math>3, & 4.</math> docetur, $emper adhibendæ $unt eædem radices: 1. $i
omnes numeratores <math>m, m′, m″, m′″, &c., r, r′, &c.</math> $int pares, & con-
$equenter denominatores impares; tum in fluente <math>W</math> pro <math>x</math> $criban-
tur <math>a & b</math>; & differentia inter quantitates re$ultantes <math>A & B</math> erit
fluens quæ$ita.</p>
<p>2. Si omnes numeratores <math>m′^b′ vel r′^l, m′^b+1 r′^l+1, m′^b+2, ... m′^b+s vel r′^l+τ</math>,
prædictarum fractionum inter <math>m′^b vel r′^l, & m′^b+s vel r′^l+τ</math> po$itarum,
$int pares; quorum corre$pondentes valores quantitatis <math>x</math> $int re$pec-
tive <math>e, f, g, ... k</math>: tum in fluente <math>W</math> pro <math>x</math> $cribantur <math>e & k</math>; & differ-
entia <math>E - K</math> inter quantitates re$ultantes <math>E & K</math> erit fluens quæ$ita.
Si denominator $it par, tum evadit fluens impo$$ibilis.</p>
<p>Si numerator & denominator $it impar; tum inveniendæ $unt flu-
entes utrinque, quæ incipiunt ab eodem puncto.</p>
<p>F<sc>IG</sc>. <math>α.</math> 3. Sit fractio <math>1 + {r / s} vel 1 + {r′ / s′}, &c. vel = 0</math>, vel affirmativa
quantitas; tum erit fluens inter <math>a & b</math> contenta, infinita: duæ autem
huju$modi fluentes $emel, vel bis vel ter, &c. occurrent, quarum dif-
erentia non erit in$inita. e. g. Sit curva <math>b a f g b l</math>, cujus crura in

<pb 46>[0098]<rh>DE INVENIENDIS</rh>
infinitum pergentia <math>b a f & g b l</math>, $int continua; & con$equenter in fun-
ctione ab$ci$$æ, quæ de$ignat ordinatas <math>a m & b n</math> ad prædicta crura,
eædem radices involvuntur; tum erit differentia inter duas areas
<math>f a m o k & g b n o k</math> $emper finita quantitas.</p>
<p>4. Si vero continuus valor quantitatis <math>X</math>; i. e. quæ ea$dem radices
$emper involvit, inter valores <math>a & b</math> variabilis quantitatis vel ab$ci$$æ
<math>x</math> evadat impo$$ibilis; tum inter valores <math>a & b</math> non continetur po$$i-
bilis fluens vel area.</p>
<p>F<sc>IG</sc>. β. 5. Quamvis curva <math>a π ξ b</math> continue progreditur a puncto <math>a</math>
ad punctum <math>b</math>; tamen nec ordinata <math>a m</math>; nec area <math>a m α μ, a m β v,
&c.</math> proprie dici pote$t continuari a linea <math>a m</math> ad lineam <math>b n</math>.</p>
<p>6. In inve$tigandis diver$is ordinatis <math>α μ, α μ′ & α μ″ vel βν, βν′, βν″,
&c.</math> ad eandem ab$ci$$am $emper u$urpantur diver$æ radices in functione
ab$ci$$æ, quæ exprimit ordinatam: ab ordinata <math>a m</math> ad ordinatam <math>π h′</math>
eædem radices $emper u$urpandæ $unt in functione ab$ci$$æ, quæ ex-
primit ordinatas; & in functione ab$ci$$æ, quæ exprimit areas curvæ:
in functionibus ab$ci$$æ, quæ exprimunt ordinatas & areas curvæ ab
ordinatâ <math>ξ h</math> ad ordinatam <math>π h′</math> eædem $emper continentur radices, at
non omnes eædem cum iis, quæ u$urpantur in ii$dem functionibus ab
ordinatâ <math>a m</math> ad ordinatam <math>π h′</math>: ordinatâ <math>π h′</math> evadet eadem, utrum hæ
vel illæ radices u$urpantur: ab ordinatâ <math>ξ′ h</math> ad ordinatam <math>b n</math> eædem
radices, $ed non omnes eædem cum iis, quæ in hoc vel illo præce-
dente ca$u, i. e. in prædictis functionibus u$urpantur.</p>
<p>Eadem, quæ a$$erimus de areis & ordinatis, æque ad fluxiones &
fluentes applicari po$$unt.</p>
<p>F<sc>IG</sc>. <math>γ</math>. Ad has res magis illu$trandas: $it curva continua, quæ tria
habet continua crura; nempe <math>m b c d e f g h k l p q r n & α β γ & π ξ σ</math>;
requiratur area inter ordinatam <math>a m</math> & ordinatam <math>b n</math> contenta. 1<_>mo</_>.
Inveniatur area <math>a m b a</math>, deinde <math>b c d b</math>, tum area <math>d e f d & f g h f</math>; de-
inde area inter a$ymptoton <math>o i</math> crus <math>h k</math> & ab$ci$$am <math>h o</math> contenta; etiam-
que area inter a$ymptoton eandem <math>o i</math>, crus <math>l p</math> & ab$ci$$am <math>o p</math>; $i hæ
duæ areæ $int infinitæ, tum $umma areæ inter <math>a m & b n</math> erit infinita:
differentia autem inter has duas areas <math>i o h k & l p o i</math> $emper erit finita;

<pb 47>[0099]<rh>FLUXIONUM FLUENTIBUS.</rh>
ultimo inveniantur areæ <math>p q r & r n b</math>; $umma harum arearum erit
$umma quæ$ita.</p>
<p>7. Si fluens inveniri po$$it aliarum fluentium & finitorum termino-
rum ope; & dentur cafus, in quibus finiti termini & quædam præ-
dictæ fluentes evane$cunt; tum in iis ca$ibus detegi pote$t fluens ope
reliquarum fluentium.</p>
<p>8. Si diver$æ radices eju$dem irrationalis quantitatis $imiliter invol-
vantur in fluente, tum $imiliter etiam involventur in fluxione; & vice
versâ $i $imiliter involvantur in fluxione, tum $imiliter etiam invol-
ventur in generali fluente.</p>
<p>9. Si unum crus vel valor fluxionis, quæ e$t functio quantitatis <math>x</math>
in <math>x^.</math>, habeat dimen$iones quantitatis <math>(x)</math> in denominatore majores per
unitatem quam dimen$iones eju$dem quantitatis <math>(x)</math> in numeratore,
tum area prædicti cruris vel fluens prædictæ fluxionis non exprimi
pote$t in finitis algebraicis terminis quantitatis <math>(x)</math>.</p>
<h>PROB. XI.</h>
<h it>Invenire, quando fluens datæ fluxionis <math>X x^.</math>; vel area curvæ, cujus ordinata
e$t <math>X</math>, evadat impo$$ibilis.</h>
<p>Inveniatur, quando ordinata <math>X</math> evadat impo$$ibilis; in quibus
ca$ibus fluxio evadit impo$$ibilis, in ii$dem ca$ibus fluens etiam eva-
dit impo$$ibilis; $i quantitas <math>X</math> (ii$dem radicibus $emper adhibitis)
perpetuo maneat po$$ibilis inter duos valores <math>a & b</math> quantitatis <math>x</math>;
tum fluens corre$pondens inter eo$dem valores <math>a & b</math> quantitatis <math>x</math>,
etiam $emper po$$ibilis evadet.</p>
<p>Eædem radices irrationalium quantitatum in ordinata <math>X</math> $emper
u$urpandæ $unt.</p>
<p>2. Nonnullæ exponentiales, &c. quantitates <math>(X)</math>, etiamque fluentes
earundem quantitatum <math>X</math> in <math>x^.</math> ductarum, &c. continuo evadunt po$$i-
biles & impo$$ibiles. e. g. $it <math>e^x = <r x> - a</r></math>, ubi <math>e</math> e$t negativa quantitas:
hæc quantitas continuo mutatur de po$$ibili in impo$$ibilem, & vice

<pb 48>[0100]<rh>DE INVENIENDIS</rh>
versâ de impo$$ibili in po$$ibilem quantitatem: fluxionis log. <math>(- a) ×
<r x> - a</r> x^.</math> fluens erit <math><r x> - a</r></math>, quæ etiam continuo mutatur de po$$ibili in
impo$$ibilem, & de impo$$ibili in po$$ibilem quantitatem.</p>
<h>PROB. XII.</h>
<p it>Invenire, utrum fluens fluxionis <math>(X x^.)</math>, ubi <math>X</math> e$t functio quantitatis
<math>x</math> inter datam & infinitam di$tantiam contenta; i. e. inter valores finitos
& in$inite magnos quantitatis variabilis <math>(x)</math>; $it finita, necne.</p>
<p>Si maximæ dimen$iones variabilis <math>(x)</math> in numeratore quantitatis <math>X</math>
$int minores (per quantitatem majorem quam unitatem) quam di-
men$iones eju$dem quantitatis <math>(x)</math> in denominatore; tum fluens præ-
dicta erit finita, $in aliter non.</p>
<p>2. Si corre$pondentes exponentiales vel quantitates cuju$cunque
generis ad infinitam di$tantiam habeant rationem, i. e. $ingula præce-
dens ad ejus $ub$equentem, quæ major e$t quam ratio $ucce$$ivarum
quantitatum ad ea$dem di$tantias, cum dimen$iones variabilis quan-
titatis <math>x</math> in numeratore $int minores per unitatem quam dimen$iones
eju$dem quantitatis in denominatore; tum area erit finita; $in aliter
vero non. e. g. Sit data fluxio <math>e^-x x^. = X x^.</math>, etiamque duo $ucce$$ivi
valores quantitatis <math>X</math> re$pective <math>e^-x & e^-x-1</math>: corre$pondentes autem
valores fractionum ad ea$dem di$tantias <math>x & x + 1</math>, cum dimen$iones
quantitatis <math>x</math> in numeratore $int minores per unitatem quam dimen-
$iones eju$dem quantitatis in denominatore erunt ad infinitam di$tan-
tiam <math>{1 / x} & {1 / x + 1}</math>; $ed <math>e^-x:e^-x-1:: e:1:</math> $i <math>e</math> major $it quam <math>1</math>; tum
<math>e^-x</math> habet majorem rationem ad <math>e^-x-1</math>, cum <math>x</math> $it infinita quantitas,
quam <math>::{1 / x}:{1 / x + 1}</math>; ergo fluens contenta inter finitos & infinite mag-
nos valores quantitatis <math>x</math> in fluxione <math>e^-x x^.</math> erit finita; cum <math>e</math> $it major
quam <math>1</math>, & <math>x</math> $it affirmativus numerus: etiamque ex ii$dem principiis

<pb 49>[0101]<rh>FLUXIONUM FLUENTIBUS.</rh>
con$tat prædictam fluentem e$$e finitam, cum <math>e</math> minor $it quam 1, &
<math>x</math> $it negativa quantitas, i. e. <math>X</math> $it <math>e^x</math>: $in aliter non.</p>
<p>3. Invenire, utrum area vel fluens datæ fluxionis <math>Xx^.</math> inter duas
finitas di$tantias vel valores (<math>a & b</math>) ab$ci$$æ (<math>x</math>) $it finita, necne.</p>
<p>Primo inveniatur, annon inter duos prædictos valores <math>a & b</math> infi-
nita evadat ordinata <math>X</math>; $i non evadat infinita inter prædictos valores,
tum area non erit infinita: $i vero evadat infinita, cum <math>x = α</math>, tum
<math>x - α</math> erit radix denominatoris fractionis <math>X</math>, quæ exprimit ordina-
tam; pro <math>x</math> in datâ fractione <math>X</math> $cribatur <math>z + α</math>, & $i minimæ dimen-
$iones quantitatis (<math>z</math>) in numeratore contentæ $int minores per quan-
titatem minorem quam unitatem quam minimæ dimen$iones eju$dem
quantitatis (<math>z</math>) in denominatore; tum area quoad hoc crus in in-
finitum pergens erit finita; $in aliter vero non.</p>
<p>Et $ic de pluribus cruribus in infinitum pergentibus.</p>
<p>4. Si exponentiales, &c. quantitates exprimentes ordinatas <math>X</math>, cum
evadant infinitæ, ad corre$pondentes di$tantias ab$ci$$æ (<math>z</math>) evadant
minores quam omnes quantitates <math>{d / z}</math>, ubi <math>d</math> e$t data quantitas & <math>z</math> fere
<math>= 0</math>; tum area vel fluens <math>$. Xx^.</math> e$t finita quoad hoc crus, $in aliter
vero non.</p>
<p>5. In omnibus his ca$ibus a valore <math>a</math> ad valorem <math>b</math> $emper u$ur-
pandæ $unt eædem radices irrationalium quantitatum in ordinatâ <math>X</math>
contentarum.</p>
<p>Ex. Sit fluxio <math>{x^n x^. / (ax^m + bx^r + (a^2 x^2m + dx^s)^{1 / 2})^{1 / 2}}</math>; hæc fluxio di$tingui
pote$t in quatuor fluxiones, quarum prima <math>{x^n x^. / (ax^m + bx^r + (a^2 x^2m + dx^s)^{1 / 2})^{1 / 2}}={x^n x^. / (ax^m + bx^r + (ax^m + {d / 2a}x^s-m + &c.))^{1 / 2}} = {x^n x^. / (2ax^m + bx^r + {1 / 2}{d / a}x^s-m) + &c.)^{1 / 2}}

<pb 50>[0102]<rh>DE INVENIENDIS</rh>
= {x^n x^. / √(2a)x^{m / 2} + {bx^r + {1 / 2}{d / a}x^s-m /2√(2a)x^{m / 2}} + &c.}</math>; in his fluxionibus (<math>m</math>) $uppo-
nitur major quam <math>r</math> vel <math>s</math>; tum, $i <math>n</math> minor $it quam <math>{m / 2}</math> per quan-
titatem majorem quam unitatem, fluens inter finitum & infini-
tum valorem quantitatis <math>x</math> contenta, erit finita; $in aliter non: <math>2^da</math>.
fluxio <math>{x^n x^. / (ax^m + bx^r - (a^2 x^2m + dx^s)^{1 / 2})^{1 / 2}} = {x^n x^. / (ax^m + bx^r - ax^m - {d / 2a}x^s-m + &c.)^{1 / 2}} ={x^n x^. / b^{1 / 2} x^{r / 2} ± &c., vel √(-{d / 2a})x^{s-m / 2} ± &c., vel √(b-{d / 2a})x^{s-m=r / 2} ± &c.}</math>:
$i <math>n</math> minor $it per quantitatem majorem quam unitatem quam
major quantitas vel <math>{r / 2}</math> vel <math>{s-m / 2}</math>; tum fluens inter prædictos va-
lores contenta erit finita, $in aliter non. Et $imiliter ratiocinari
liceat de duobus reliquis fluxionibus <math>{x^n x^. / -(ax^m + bx^r + (a^2 x^2m + dx^s)^{1 / 2})^{1 / 2}}&{x^n x^. / -(ax^m + bx^r - (a^2 x^2m + dx^s)^{1 / 2})^{1 / 2}}</math></p>
<h>THEOR. V.</h>
<p>Datâ fluxione, quæ e$t algebraica functio literæ <math>x</math> in <math>x^.</math> ducta, &
quæ nullos in $e habet tran$cendentales terminos; $umma ejus fluen-
tis valorum $emper exprimi pote$t per finitos terminos, circulares
arcus & logarithmos.</p>
<p>Summa enim e $ingulis valoribus cuju$cunque irrationalis quanti-
tatis <math>(P)</math> nihilo $emper erit æqualis, & exinde $umma e $ingulis va-
loribus fluentis <math>(Px^.)</math>: $ummam vero e $ingulis valoribus fluentis

<pb 51>[0103]<rh>FLUXIONUM FLUENTIBUS.</rh>
(<math>Qx^.</math>), quæ e$t rationalis functio quantitatis <math>x</math> in <math>x^.</math> ducta, $emper in-
veniri po$$e finitorum terminorum, circularium arcuum & logarith-
morum ope, e $ub$equentibus con$tabit; ergo con$tat theorema.</p>
<p>E capite primo medit. algebr. con$tat $ummam $ingulorum valo-
rum cuju$cunque algebraicæ quantitatis haud tran$cendentalis e$$e
rationalem quantitatem.</p>
<h>PROB. XIII.</h>
<p it>Datâ quantitate vel algebraicâ vel fluentiali, quœ exprimit $ummam
$eriei $ecundum dimen$iones literœ <math>x</math> progredientis; invenire quantitatem
alternis $eriei terminis æqualem; vel denique $eriei terminis, quorum di-
$tantia a $emetip$is $it <math>n</math>.</p>
<p>1. Si modo $eries requiratur alternis $eriei terminis æqualis; pro <math>x,
x^., x^.., &c.</math> in datâ quantitate $cribantur re$pective <math>x, x^., x^.., &c.</math> & <math>- x,
-x^., -x^.., &c.</math> emergent duæ quantitates, quarum $emi$umma & $e-
midifferentia erunt quantitates de$ideratæ.</p>
<p>2. Si requiratur $umma terminorum, quorum di$tantia a $e invicem
$it <math>n</math>. Sint <math>α, β, γ, δ, ε, &c.</math> re$pectivæ radices æquationis <math>π^η - 1 = 0</math>;
in datâ quantitate pro <math>x, x^., x^.., &c.</math> $cribantur re$pective <math>αx, αx^., αx^..,
&c. βx, βx^., βx^.., &c. γx, γx^., γx^.., &c. &c.</math> & re$ultant <math>n</math> quantitates:
$umma harum <math>n</math> quantitatum re$ultantium per <math>n</math> divi$a erit $umma
quæ$ita, i.e. erit fumma primi, <math>n + 1, 2n + 1, 3n + 1, &c.</math> termino-
rum. Et $ic e principiis in prob. 27. tertiæ edit. & in præfatione no$t.
medit. algebr. traditis inveniri po$$unt $umma $ecundi, <math>n + 2, 2n + 2,
3n + 2, &c.</math> terminorum; $umma tertii, <math>n + 3, 2n + 3, 3n + 3, &c.</math>
terminorum; & $ic deinceps: methodus enim, quæ invenit $ummam
prædictorum terminorum $eriei ab expan$ione irrationalium vel fra-
ctionalium quantitatum in terminos $ecundum dimen$iones quanti-
tatis <math>x</math> progredientes, etiam deteget $ummam con$imilium termino-
rum $eriei progredientis $ecundum dimen$iones quantitatis <math>x</math>, quæ e$t
fluens $eriei ex expan$ione irrationalium vel fractionalium quantita-
tum in terminos $ecundum dimen$iones quantitatis <math>x</math> progredientes
in <math>x^.</math> ductæ, &c.</p>

<pb 52>[0104]<rh>DE INVENIENDIS</rh>
<p>Con$imilia etiam applicari po$$unt ad $eries, quæ ex expan$ione
irrationalium, &c. quantitatum in terminos $ecundum dimen$iones
quantitatum <math>x, y, z, &c.</math> progredientes, exoriuntur; etiamque ad $e-
ries, quæ $unt fluentes præcedentium $erierum in <math>x^., y^., &c.</math> ductarum.</p>
<h>PROB. XIV.</h>
<p it>Datam fluxionem, quæ $it functio variabilium quantitatum <math>(x, y, z, &c.)</math>
& earum fluxionum, in alteram transformare; quœ $it functio variabilium
quantitatum <math>(v, w, u, &c.)</math> & earum flux’ionum, & cujus variabiles quan-
titates datam habeant relationem ad variabiles datœ fluxionis quantitates.</p>
<p>Supponatur data fluxio <math>= X^.</math>, & re$ultans æquatio & datæ æqua-
tiones inter variabiles quantitates datæ (<math>x, y, &c.</math>) & quæ$itæ (<math>v, w,
&c.</math>) æquationis relationem exprimentes ita transformentur ut re$ul-
tet æquatio relationem inter fluxionem <math>X^.</math> & variabiles quantitates <math>v,
w, &c.</math> & earum fluxiones exprimens; ex hâc æquatione, $i modo
fieri po$$it, inveniatur quantitas <math>X^.</math> terminis vero variabilium quanti-
tatum <math>v, w, &c.</math> & earum fluxionum, & confit problema.</p>
<p>Ex hâc methodo $æpe reduci pote$t data fluxio in magis $implicem.</p>
<p>Ex. 1. Sit data fluxio <math>{-x^-n x^. / a + 2bx + cx^2} = X^.; & x = {1 / v}</math>; transfor-
mabitur data fluxio in fluxionem <math>{v^n v^. / av^2 + 2bv + c} = X^.</math>.</p>
<p>Ex. 2. Sit data fluxio <math>{x^{m / r} x^. /e + fx^n} = X^., & v^r = x</math>, & transformari
pote$t data fluxio in fluxionem <math>{rv^m+r-1 v^. / e + fv^rn} = X^.</math>.</p>
<p>Hinc con$tat, $i modo numerus dimen$ionum variabilis quantitatis
in datâ fluxione (e.g. $it data fluxio <math>{Ax^<r>r+{l / s}</r>n-1 x^. /<r γ>e + fx^n + gx^2n + &c.</r> × P}</math>) $ine
vinculo $it aliquota pars vel partes (<math>{l / s}</math>) dimen$ionum (<math>n</math>) eju$dem

<pb 53>[0105]<rh>FLUXIONUM FLUENTIBUS.</rh>
quantitatis $ub vinculo, transformari po$$e fluxionem in meliorem
formulam, $i modo pro <math>x^{n / s}</math> $cribatur <math>v, &c.</math> Si hâc methodo haud
bene proce$$it transformatio, nonnunquam transformari pote$t data
fluxio in præ$tantiorem formulam, $cribendo <math>v = x^{1 / sn}</math>, & reducendo
datam fluxionem in alteram, cujus variabilis quantitas e$t <math>v</math>.</p>
<p>Ex. 3. Sit data fluxio <math><r m>a + bx^n</r> x^rn-1 x^.</math>, ubi litera <math>r</math> integrum deno-
tat numerum; $cribatur <math>a + bx^n = v</math>, & erit <math>x^n = {v - a / b}</math>; unde fluxio
data <math><r m>a + bx^n</r> x^rn-1 x^. = {1 / n}v^m × ({v - a / b})^r-1 × {v^. / b}</math>, & quoniam <math>r - 1</math> e$t
integer numerus, facile reduci pote$t <math>(v - a)^r-1</math> per binomiale theo-
rema in $implices terminos, & exinde fluens datæ fluxionis inveniri
pote$t.</p>
<p>Si <math>r</math> haud $it integer numerus, $ed <math>- r - m - 1</math>. $it integer nume-
rus; & data fluxio <math><r m>a + bx^n</r>x^rn-1 x^. = <r m>{a / x^n} + b</r> × x^rn+nm-1 x^.</math>; $cribatur
<math>{a / x^n} + b = v</math>, & re$ultat <math>{a / v - b} = x^n</math>; unde <math><r m>{a / x^n} + b</r> × x^nr+nm-1 x^. = - v^m
× {1 / n}(v - b)^-m-r-1 × a^r+m v^.</math>.</p>
<p>Cor. Facile exhinc fingi po$$unt fluxiones, quæ reduci po$$unt in
magis $implices: a$$umatur enim $implex fluxio, & pro variabilibus
quantitatibus & earum fluxionibus in eâ contentis $cribantur quan-
titates magis compo$itæ & earum fluxiones; re$ultat fluxio, quæ fa-
cile reduci pote$t in prædictam $implicem.</p>
<p>Ex. 1. Sit data fluxio <math>x^9 × <r m>a + bx^s</r> x^.</math>; a$$umatur <math>x = <r π>ev^v + fv^r+n + gv^r+2n + &c.</r></math>;
$ub$tituatur hic valor & ejus fluxio pro quantitate <math>x</math>
& ejus fluxione in datâ fluxione, & re$ultat fluxio magis compo$ita

<pb 54>[0106]<rh>DE INVENIENDIS</rh>
<math><r m>a + b <ol>e v^ν + f v^ν+n + g v^ν+2n + &c.</ol>^πs</r> × <r θπ + π - 1>e + f v^n + g v^2n + &c.</r> ×
π v^(θ+1)πν-1 × (ν e + (n + ν) f v^n + &c.) × v^.</math>, quæ facile reduci pote$t
ad prædictam $implicem fluxionem.</p>
<p>Ex. 2. Sit data eadem fluxio <math>x^θ × <r m>a + b x^5</r> × x^.</math>; a$$umatur <math><r π>e v^ν +
f v^ν+n + g v^ν+2n + &c.</r> × <r λ>h + k v^n + l v^2n + &c.</r> = x</math>, $cribatur hic
valor & ejus fluxio pro <math>x</math> & <math>x^.</math> in datâ fluxione, & re$ultat fluxio ma-
gis compo$ita.</p>
<p>Sit fluxio <math>X x^.</math>, & $int dimen$iones quantitatis <math>x</math> in numeratore
quantitatis <math>X</math> minores per unitatem quam ejus dimen$iones in deno-
minatore; transformetur hæc fluxio in alteram <math>V v^.</math>, ubi <math>v</math> e$t quæ-
cunque algebraica functio quantitatis (<math>x</math>), & <math>V</math> con$imilis functio
quantitatis <math>v</math>; tum erunt dimen$iones quantitatis <math>v</math> in numeratore fra-
ctionis <math>V</math> minores per unitatem quam ejus dimen$iones in denomina-
tore prædictæ fractionis.</p>
<h>PROB. XV.</h>
<p it>Datam fluxionem irrationales quantitates babentem in alteram $œpe
transformare; $i modo fieri po$$it, quœ nullas babet irrationales quanti-
tates.</p>
<p>Per caput 5. medit. algebr. inveniatur, annon $ingula irrationalis,
&c. quantitas in datâ fluxione contenta exprimi pote$t in rationalibus
functionibus eju$dem literæ <math>v</math>; $i vero huju$ce generis inveniantur
rationales functiones, $ub$tituantur hæ rationales functiones, &c. pro
$uis valoribus in datâ fluxione, & re$ultat fluxio quæ$ita.</p>
<p>Ex. 1. Sit prædicta fluxio rationalis functio quantitatum <math>x^n &
(a + bx^n)^{1 / 5}</math> (ubi litera <math>s</math> integrum denotat numerum) in fluxionem <math>x^n-1 x^.</math>
ducta; eam transformare in alteram, quæ nullas habet irrationales
quantitates. Scribatur <math>(a + b x^n)^{1 / 5} = v</math>, unde <math>{v^5 - a / b} = x^n</math>, quibus

<pb 55>[0107]<rh>FLUXIONUM FLUENTIBUS.</rh>
quantitatibus pro $uis valoribus in datâ fluxione re$pective $ub$titutis,
& <math>{s v^5-1 v^. / n b}</math> pro <math>x^n-1 x^.</math>; re$ultat fluxio quæ$ita.</p>
<p>Ex. 2. Sit prædicta fluxio rationalis functio quantitatum <math>x^n &
({a + b x^n / c + d x^n})^{1 / 5}</math>, in fluxionem <math>x^n-1 x^.</math> ducta; $cribatur <math>({a + b x^n / c + d x^n})^{1 / 5} = v</math>,
& exinde <math>v^5 = {a + b x^n / c + d x^n}</math>, etiamque <math>c v^5 - a/b - d v^5 = x^n</math>, & <math>x^n-1 x^. ={s c v^5-1 × (b - d v^5) + s d v^5-1 × (c v^5 - a) / n (b - d v^5)^2} × v^.</math>; quibus quantitatibus
pro $uis valoribus in datâ fluxione $ub$titutis, re$ultat fluxio quæ$ita.</p>
<p>Cor. Sit fluxio rationalis functio quantitatum <math>x^n & (a + b x^n)^{1 / 5} ×
(c + d x^n)^{5-1 / 5}</math>; pro <math>(a + b x^n)^{1 / 5} × (c + d x^n)^{5-1 / 5}</math> $cribatur ejus valor <math>(c + d x^n)
× ({a + b x / c + d x^n})^{1 / 5}</math>, & reducitur fluxio in fluxionem præcedentis formulæ.</p>
<p>Ex. 3. Sit data fluxio rationalis functio quantitatum <math>x^n</math> &
<math>^2 √(a^2 + b x^n + c x^2n)</math> in <math>x^n-1 x^.</math> ducta: $cribatur <math>(a^2 + b x^n + c x^2n)^{1 / 2}
= a + p x^n v</math>, & exinde <math>b x^n + c x^2n = 2 a p x^n v + p^2 x^2n v^2</math>, unde
<math>x^n = {b - 2 a p v / p^2 v^2 - c}</math>, & <math>(a^2 + b x^n + c x^2n)^{1 / 2} = a + p x^n v = {p b v - p^2 a v^2 - c a / p^2 v^2 - c}</math>;
quibus quantitatibus pro $uis valoribus in datâ fluxione $ub$titutis,
& fluxione <math>{2 p^3 a v^2 - 2 p^2 b v + 2 a p c / (p^2 v^2 - c)^2} v^.</math> pro <math>n x^n-1 x^.</math>; re$ultat fluxio
quæ$ita.</p>
<p>Sit data fluxio functio rationalis quantitatum <math>x^n & (a + b x^n + c x^2n)^{1 / 2}</math>
in <math>x^n-1 x^.</math>; $cribatur <math>(a + b x^n + c x^2n)^{1 / 2} = √ (c) x^n + v</math>, & re$ultat <math>c x^2n
+ 2 √ (c) v x^n + v^2 = a + b x^n + c x^2n</math>; unde <math>x^n = {v^2 - a / b - 2 √ (c) v} &

<pb 56>[0108]<rh>DE INVENIENDIS</rh>
(a + b x^n + c x^2n)^{1 / 2} = √ (c)x^n + v = {√ (c)bv - cv^2 - ca / b √ (c) - 2 c v}</math>; & fluxio <math>x^n-1 x^.
= {1 / n} × {2 b v - 2 a √ (c) - 2 √ (c) v^2 / (b - 2 √ (c) × v)^2} × v^.</math>; quibus quantitatibus pro
$uis valoribus in datâ fluxione $ub$titutis, re$ultat fluxio quæ$ita.</p>
<p>Et $imiliter $cribatur <math>(a + b x^n + c x^2n)^{1 / 2} = √ (a) + √ (c) x^n + v</math>,
vel <math>(a + b x^n + c x^2n)^{1 / 2} = c^{1 / 2} (x^n + α)^{1 / 2} (x^n + β)^{1 / 2} = v (x^n + α) = (b^2 x^2n +
l (x^n + γ) (x^n + δ))^{1 / 2} = bx^n + v (x^n + γ) = (k^2 + t (x^n + ε) (x^n + ζ))^{1 / 2} =
k + v (x^n + ε), &c.</math> & facile transformari pote$t data in fluxionem,
cujus variabilis quantitas e$t <math>v</math>.</p>
<p>Eædem $ub$titutiones etiam transformant multas alias formulas in
magis $implices.</p>
<p>Ex. 3. Sit data fluxio <math>x^rn-1 x^. (a + (e + f x^n)^{1 / m})^{1 / 5}</math> in rationalem fun-
ctionem quantitatum <math>x^n & (e + f x^n)^{1 / m}</math>; ubi <math>r & m</math> $unt integri nu-
meri; $cribatur <math>(e + f x^n)^{1 / m} = v</math>, & erit <math>x^n = {v^m - e / f}</math>, & data fluxio
transformatur in alteram <math>({v^m - e / f})^r-1 × {m / n f} v^m-1 v^. × ^5 √ (a + v) ×</math>
rationalem functionem quantitatum <math>(v) & v^m - e</math>; cujus fluens facile
inveniri pote$t.</p>
<p>Ex. 4. Sit fluxio <math>x^rn-1 x^. × P</math>, ubi <math>P</math> $it rationalis functio quantitatum
<math>(a + (e + f x^n)^{1 / m})^{1 / 5}, (e + f x^n)^{1 / m} & x^n; & r, m & s</math> integri $int numeri;
$cribantur <math>v^5, v^5 - a, & {(v^5 - a)^m - e / f}</math>: & fluxione <math>{m s v^5-1 × (v^5 - a)^m-1 / f}v^.</math>
pro <math>a + (e + f x^n)^{1 / m}, (e + f x^n)^{1 / m}, x^n, & n x^n-1 x^.</math> in datâ fluxione; &
re$ultat fluxio quæ$ita.</p>

<pb 57>[0109]<rh>FLUXIONUM FLUENTIBUS.</rh>
<p>Ex. 5. Sit data fluxio <math>x^rn-1 x^. × (a + b x^n + (e + f x^n)^{1 / m})^{1 / 5}</math>, ubi literæ
<math>m</math> & <math>r</math> integros denotant numeros; $cribatur <math>e + f x^n = v^m</math>, unde <math>x^n ={v^m - e / f}</math>; & transformari pote$t data fluxio in fluxionem <math>({v^m - e / f})^r-1
× {m v^m-1 v^. / n f} × (a + b × {v^m - e / f} + v)^{1 / 5}</math>.</p>
<p>Ex. 6. Sint literæ <math>m & r</math> integri numeri, & fluxio data <math>n x^rn-1 x^.
(a + ({e + f x^n / g + b x^n})^{1 / m})^{1 / 5} ×</math> in rationalem functionem quantitatum <math>x^n &
({e + f x^n / g + b x^n})^{1 / m}</math>; $cribatur <math>{e + f x^n / g + b x^n} = v^m</math>, unde <math>{g v^m - e / f - b v^m} = x^n</math>, & tran$-
formari pote$t data fluxio in fluxionem <math>({g v^m - e / f - b v^m})^r-1 × (a + v)^{1 / 5} ×</math>
in flux. quantitatis <math>({g v^m - e / f - b v^m}) ×</math> functionem quantitatum <math>v &{g v^m - e / f - b v^m}</math>.</p>
<p>Ex. 7. Sit data fluxio <math>(a + (e^2 + f x^n + g x^2n)^{1 / 2})^{1 / 2} x^rn-1 x^. ×</math> quamcun-
que rationalem functionem quantitatum <math>x^n & (e^2 + f x^n + g x^2n)^{1 / 2}</math>, ubi
<math>r</math> $it integer numerus; $cribatur <math>e^2 + 2 e v x^n + v^2 x^2n = e^2 + f x^n +
g x^2n</math>, unde <math>x^2 = {2 e v - f / g - v^2}</math>; & data fluxio facile transformatur in
alteram <math>(a + e + {2 e v^2 - f v / g - v^2})^{1 / 5} × {1 / n} × ({2 e v - f / g - v^2})^r-1 ×</math> in flux. quan-

<pb 58>[0110]<rh>DE INVENIENDIS</rh>
titatis <math>{2 e v - f / g - v^2} ×</math> rationalem functionem quantitatum <math>v & e +{2 e v^2 - f v / g - v^2}</math>.</p>
<p>Ex. 8. Si vero $it data fluxio <math>x^. x^rn-1 × (e + f z + g z^2 + &c.)^{1 / 5} × P</math>,
ubi <math>z</math> vel = <math>(e + f x^n)^{1 / 5}</math> vel <math>({e + f x^n / b + k x^n})^{1 / 5} &c.</math> & <math>P</math> $it rationalis functio
quantitatis <math>x^n</math>; tum facile transformari pote$t data fluxio in alteram
magis $implicem $cribendo pro <math>(e + f x^n)^{1 / 5}</math> vel <math>({e + f x^n / b + k x^n})^{1 / 5}</math> ejus valo-
rem <math>(v)</math>; & pro <math>x^n</math> in priori ca$u <math>{v^5 - e / f}</math>, in po$teriori vero <math>{b v^5 - e / f - k v^5}</math>;
& earum fluxiones re$pective pro <math>n x^n-1 x^.</math>, in datâ fluxione; &c.</p>
<p>Ex. 9. Sit data fluxio <math>x^n-1 x^. ×</math> in rationalem functionem quan-
titatum <math>^m √ (a + b ^r √ (c + d ^5 √ (e + &c. + ({g + f x^n / b + k x^n})^{1 / t}))) = P,
<_>r</_>√ (c + d <_>5</_>√ (e + &c. + ({g + f x<_>n</_> / b + k x<_>n</_>})<_>{1 / t}</_>)) = Q, <_>5</_>√ (e + &c. + ({g + f x<_>n</_> / b + k x<_>n</_>})<_>{1 / t}</_>) = R,
&c., ({g + f x<_>n</_> / b + k x<_>n</_>})<_>{1 / t}</_> = U, & x<_>n</_></math>; a$$umantur <math>v = P, {v^m - a / b} = Q,{({v^m - a / b})^r - c/d} = R, &c., ({((v^m - a / b})^r - c)^5 /d
- e, &c.)^t = U, {b U^t - g / f - k U^t} = x^n</math>, & ejus fluxio pro <math>n x^n-1 x^.</math>; $cribantur hæ quantitates
pro $uis valoribus; & transformatur data fluxio in alteram, quæ erit
rationalis functio quantitatis <math>v</math> in <math>v^.</math>.</p>
<p>Ex. 10. Sit data fluxio rationalis functio quantitatum <math>(x^n +

<pb 59>[0111]<rh>FLUXIONUM FLUENTIBUS.</rh>
√ (1 + x<_>2n</_>))<_>{1 / m}</_> & √(1 + x<_>2n</_>) & x<_>n</_></math> in <math>x^n-1 x^.</math>, ubi <math>m</math> $it integer numerus;
$cribatur <math>x^n + √ (1 + x^2n) = y^m</math>, & re$ultat <math>x^n = {y^2m - 1 / 2y^m} & √(1 + x^2n)
= y^m - x^n</math>; unde facile reduci pote$t data fluxio in rationalem fun-
ctionem quantitatis <math>y</math> in <math>y^.</math>.</p>
<p>Et $ic de quamplurimis huju$cemodi exemplis; facile enim deduci
po$$unt infinitæ con$imiles $ub$titutiones, quæ transformant fluxio-
nem irrationales quantitates involventem in fluxionem rationales vel
minus irrationales quantitates $olummodo habentem; e contra inveniri
po$$unt infinitæ huju$modi irrationales fluxiones, quæ in rationales
transformari po$$unt; a$$umatur enim quæcunque rationalis fluxio, &
pro <math>x^n</math> vel quâcunque aliâ quantitate $cribatur irrationalis quantitas,
& pro ejus fluxione fluxio prædictæ quantitatis, &c. & exterminetur
quantitas <math>x</math> & ejus fluxio; frequenter re$ultabit irrationalis fluxio,
quæ facile reduci pote$t in a$$umptam rationalem.</p>
<p>Con$tat nullam a$$ignari po$$e fluxionem, cujus fluens finitis ter-
minis algebraicis inveniri pote$t, vel finitis $olummodo terminis loga-
rithmorum exprimi pote$t, &c. quæ haud transformari pote$t per al-
gebraicarum quantitatum $ub$titutiones in rationalem fluxionem; $ed
dantur fluxiones, quarum fluentes exprimuntur algebraicis & loga-
rithmicis terminis finitis conjunctim, quæ haud reduci po$$unt per
prædictas $ub$titutiones in rationales fluxiones: $ed de his $atis.</p>
<h>LEMMA.</h>
<p>1. Sit <math>n</math> integer numerus, & æquatio <math>x^n - p x^n-1 + q x^n-2 - rx^n-3 +
&c. = 0, &c.</math> $int etiam <math>a, b, c, d, &c.</math> producta differentiarum uni-
cuique radicum & radicibus reliquis interjacentium, hoc e$t, $it <math>a =
(α - β) × (α - γ) × (α - δ) × &c. = n α^n-1 - (n - 1) pα^n-2 + (n - 2)
qα^n-3 - &c. & b = (β - α) × (β - γ) × (β - δ) × &c. = nβ^n-1 - (n - 1)
pβ^n-2 + (n - 2)qβ^n-3 - &c. & c = (γ - α) × (γ - β) × (γ - δ) × &c.

<pb 60>[0112]<rh>DE INVENIENDIS</rh>
= nγ^n-1 - (n - 1)pγ^n-2 + (n - 2) qγ^n-3 - &c. & d = (δ - α) × (δ - β)
× (δ - γ) × &c. = n δ^n-1 - (n - 1) pδ^n-2 + (n - 2) q δ^n-3 - &c.&c. &</math>
erit fractio <math>{1 / x^n - p x^n-1 + q x^n-2 - &c.} = {{1 / a}/x - α} + {{1 / b}/x - β} + {{1 / c}/x - γ} +{{1 / d}/x - δ} + &c.</math></p>
<p>2. Sit <math>{x^s / x^n - px^n-1 + qx^n-2 - &c.} = {a / x - α} + {b / x - β} + {c / x - γ} + {d / x - δ}+ &c.</math> ubi exponens <math>s</math> denotat integrum po$itivum numerum, mino-
rem vero quam <math>n</math>; tum erit <math>a = {a^s / (α - β) × (α - γ) × (α - δ) × &c.} ={a^s / nα^n-1 - (n - 1)pα^n-2 + (n - 2)qα^n-3 - &c.}, b = {β^s / (β - α) × (β - γ) × (β - δ) × &c.},
c = {γ^s / (γ - α) × (γ - β) × (γ - δ) × &c.}, &c.</math></p>
<p>Cor. Sint <math>α & β</math> radices duæ quadraticæ ex eâdem quadraticâ ortæ,
quas ideo radices cognatas appellare licet, addanturque in unam
fractiones <math>{a / x - α} & {b / x - β};</math> nunc quicquid imaginarii e$t in utrâque
fractione $eor$im $umptâ $emper ex earum $ummâ <math>{(a + b)x - bα - αβ / (x - α) × (x - β)}</math>
evane$cet.</p>
<p>3. Sit <math>{1 / (x + α)^m × (x + β)^n} = {a / (x + α)^m} + {b / (x + α)^m-1} + {c / (x + α)^m-2}.....{A / (x + β)^n} + {B / (x + β)^n-1} + {C / (x + β)^n-2} + &c.</math> & erit <math>a = {1 / (β - α)^n}, b ={- n / (β - α)^n+1}, c = {n × {n + 1 / 2}/(β - α)^n+2}, &c. A = {1 / (α - β)^m}, B = {- m / (α - β)^m-2}, C =

<pb 61>[0113]<rh>FLUXIONUM FLUENTIBUS.</rh>
{m × {m + 1 / 2}/(α - β)^m-2}, &c.</math> eodem modo $upponatur <math>{1 / (x + α)^n × (x + β)^m × (x + γ)^r}= {a / (x + α)^n} + {b / (x + α)^n-1} + {c / (x + α)^n-2}....{A / (x + β)^m} + {B / (x + β)^m-1} +{C / (x + β)^m-2}....{P / (x + γ)^r} + {Q / (x + γ)^r-1} + {R / (x + γ)^r-2} + &c.</math> & exinde
deduci po$$unt coefficientes <math>a, b, c, &c. A, B, C, &c. P, Q, R, &c.</math> ex
reductione harum fractionum in communem denominatorem.</p>
<p>4. Literæ <math>α, β, γ, &c. a, b, c, &c.</math> ea$dem quantitates ac in ca$u 1. de-
notent: & erit quantitas <math>{Q / (x^n - px^n-1 + &c.) × R} = {Q / a × (x - α) × R} +{Q / b × (x - β) × R} + {Q / c × (x - γ) × R} + &c.</math> ubi literæ <math>Q & R</math> qua$cun-
que quantitates variabiles vel invariabiles de$ignare po$$unt.</p>
<p>5. Datâ fluxione, cujus denominator continetur $ub divi$oribus
<math>a, b, c, d, &c.</math> i. e. $it data fluxio <math>{Ax^. / P}</math>, ubi <math>P = a × b × c × d × &c.</math>
ea in alias transformari pote$t huju$ce formulæ <math>{lx^. / a} + {mx^. / b} + {nx^. / c}+ &c. = {Ax^. / P};</math> reducantur hæ fractiones in communem denomina-
torem, & re$ultat <math>{lbc &c. / abc &c.} + {mac &c. / abc &c.} + {nab &c. / abc &c.} + &c. = {A / P}:</math>
ita vero $æpe a$$umi po$$unt coefficientes <math>l, m, n, &c.</math> ut evane$cant
quidam irrationales termini.</p>
<p>6. Sit data fluxio <math>{ax^m + bx^m±1 + cx^m±2 + &c. / px^n + qx^n±1 + rx^n±2 + &c.} × x^.</math>, ubi <math>m</math> major
e$t quam <math>n</math>; tum reduci pote$t in alteram, ubi maximus index quan-
titatis <math>x</math> in numeratore minor e$t quam ejus maximus index <math>(n)</math> in

<pb 62>[0114]<rh>DE INVENIENDIS</rh>
denominatore. Dividatur numerator per denominatorem, viz. <math>px^n +
qx^n±1 + rx^n±2 + &c. (ax^m + bx^m±1 + cx^m±2 + &c.) {a / p}x^m-n +{bp - aq / p^2}x^m-n±1 + &c. = Q</math>; u$que donec maximus index re$idui
<math>(R)</math> minor $it quam index <math>n</math>, & id, quod requiritur, perficietur; erit
enim data fluxio <math>= Qx^. + {Rx^. / px^n + qx^n±1 + &c.}.</math></p>
<p>7. Erit fluxio <math>{ax<_>n-1</_> + bx<_>n-2</_> + cx<_>n-3</_> + &c. / px<_>n</_> + qx<_>n-1</_> + rx<_>n-2</_> + &c.} × x^. = {a / np} ×{npx<_>n-1</_> + (n - 1)qx<_>n-2</_> + (n - 2) r x<_>n-3</_> + &c. / px<_>n</_> + qx<_>n-1</_> + rx<_>n-2</_> + &c.} x^.</math> (fluxio, cujus
fluens e$t <math>{a / np} × log. (px^n + qx^n-1 + rx^n-2 + &c.) +{b - (n-1){aq / np})x^n-2 + (c-(n-2){ar / np})x^n-3 + &c./px^n + qx^n-1 + rx^n-2 + &c.}x^..</math></p>
<h>LEMMA.</h>
<p>1. Sit æquatio <math>x^2n - 2px^n + 1 = 0, & p</math> minor quam 1.</p>
<p>F<sc>IG</sc>. Sit circulus <math>ABEF &c. P</math>, cujus centrum $it <math>C, &</math> radius unitas.
E tabulis $inuum inveniatur arcus <math>A G</math>, cujus $inus $it <math>p.</math> E prædictis
tabulis inveniantur $inus arcuum <math>{(n - 1) × 90° + AG / n}, {(n - 5) × 90° + AG / n},{(n - 9) × 90° + AG / n}, {(n - 13) × 90° + AG / n}.... {- 3 × (n - 1)90° + AG / n}</math>;
qui $int re$pective <math>α, β, γ, δ, ε, &c.</math> tum <math>x^2 - 2αx + 1, x^2 - 2βx + 1,
x^2 - 2γx + 1, x^2 - 2δx + 1, &c.</math> erunt quadratici divi$ores quanti-
tatis <math>x^2n - 2px^n + 1.</math></p>

<pb 63>[0115]<rh>FLUXIONUM FLUENTIBUS.</rh>
<p>2. Sit data quantitas <math>x^n - 1</math>, cujus quadratum <math>x^2n - 2x^n + 1</math> erit
quantitas præcedentis formulæ. In hoc ca$u <math>p = 1</math> $inus arcûs <math>(AG
= 90°)</math>: inveniantur igitur $inus <math>α, β, γ, &c.</math> arcuum 90°, <math>{(n - 4) × 90° / n},{(n - 8) × 90° / n}, {(n - 12) × 90° / n}.... - {(3n - 4)90° / n}</math>; tum <math>x^2 - 2αx + 1,
x^2 - 2βx + 1, x^2 - 2γx + 1, &c.</math> erunt quadratici divi$ores qua-
drati <math>x^2n - 2x^n + 1; & x - 1</math> $implex divi$or datæ quantitatis
<math>x^n - 1</math>, cum <math>n</math> $it impar numerus; etiamque <math>x - 1 & x + 1</math> erunt
$implices divi$ores datæ quantitatis <math>x^n - 1</math>, cum <math>n</math> $it par numerus:
$i vero <math>n = 4m</math> $it numerus pariter par, tum data quantitas recipiet
$implices divi$ores <math>x + 1 & x - 1 &</math> quadraticum <math>x^2 + 1.</math></p>
<p>3. Sit data quantitas <math>x^n + 1 = 0</math>, cujus quadratum <math>x^2n + 2x^n + 1</math>
erit quantitas præcedentis formulæ; in hoc ca$u <math>p = - 1</math> $inus
arcûs <math>(AG = - 90°)</math>; inveniantur igitur $inus arcuum <math>{(n - 2) × 90° / n},{(n - 6) × 90° / n}, {(n - 10) × 90° / n}....{- (3n - 2) × 90° / n}</math>: $i vero <math>n = 4m + 2</math>
$it impariter par, tum unus $inus arcûs erit <math>0</math>: $int $inus inventi, cum
<math>n</math> vel <math>= 2m + 1</math>, vel <math>= 4m</math>, vel <math>4m + 2</math> re$pective, in primo ca$u
<math>- 1, α, β, γ, δ, &c.</math> in $ecundo <math>α, β, γ, δ, &c.</math> in tertio <math>0, α, β, γ, δ,
&c.</math> tum data quantitas con$tat in primo ca$u e quadraticis divi$ori-
bus <math>x^2 - 2αx + 1, x^2 - 2βx + 1, x^2 - 2γx + 1, &c. & e</math> $implici
divi$ore <math>x + 1</math>; etiamque e prædictis & quadratico divi$ore <math>x^2 + 1</math> in
po$tremo ca$u; &c.</p>
<p>4. Sit <math>n</math> impar numerus, <math>& x^n - 1 = (x - 1) × (x^2 - 2αx + 1) ×
(x^2 - 2βx + 1) × &c. &</math> erit <math>{1 / x^n - 1} = {{1 / n}/x - 1} + {{2 / n}αx - {2 / n}/x^2 - 2αx + 1} +{{2 / n}βx - {2 / n}/x^2 - 2βx + 1} + &c.</math></p>
<p>Sit <math>n</math> vero par numerus, <math>& x^n - 1 = (x - 1) × (x + 1) × (x^2 - 2αx + 1)

<pb 64>[0116]<rh>DE INVENIENDIS</rh>
× (x<_>2</_> - 2βx + 1) × &c.</math> & erit <math>{{1 / n}/x - 1} - {{1 / n}/x + 1} + {{2 / n}αx - {2 / n}/x^2 - 2αx + 1} +
&c. = {1 / x^n - 1} ·</math></p>
<p>Sit <math>n</math> pariter par numerus, & erit <math>x^n - 1 = (x - 1) × (x + 1) ×
(x^2 + 1) × (x^2 - 2αx + 1) × &c. & {1 / x^n - 1} = {{1 / n}/x - 1} - {{1 / n}/x + 1} -{{2 / n}/x^2 + 1} + {{2 / n}αx - {2 / n}/x^2 - 2αx + 1} + &c.</math></p>
<p>Sit <math>n</math> impar numerus & <math>x^n + 1 = (x + 1) × (x^2 - 2αx + 1) ×
(x^2 - 2βx + 1) × &c.</math> & erit <math>{- 1 / x^n + 1} = {- {1 / n}/x + 1} + {{2 / n}αx - {2 / n}/x^2 - 2αx + 1) × &c.}</math></p>
<p>Sit <math>n</math> par numerus, & <math>x^n + 1 = (x^2 + 1) × (x^2 - 2αx + 1) ×
(x^2 - 2 β x + 1) × &c.</math> & erit <math>{- 1 / x^n + 1} = {- {2 / n}/x^2 + 1} + {{2 / n}αx - {2 / n}/x^2 - 2 α x + 1} +{{2 / n}β x - {2 / n}/x^n - 2 β x + 1} + &c.</math> Hæc facile deduci po$$unte lemmate præced.</p>
<p>5. Sit <math>x^2n ± 2 p x^n + 1</math>, ubi <math>p</math> minor $it quam 1; & <math>x^2n ± 2 p x^n
+ 1 = (x^2 - 2 α x + 1) × (x^2 - 2 β x + 1) × (x^2 - 2 γ x + 1) ×
&c.</math> i. e. $int <math>α, β, γ, &c.</math> co$inus arcuum <math>{A / n}, {360° - A / n}, {360° + A / n},{2 × 360° - A / n}, {2 × 360° + A / n}, {3 × 360° - A / n}, &c.</math> ubi <math>A</math> $it arcus,
cujus co$inus $it <math>± P</math>, itemque $int <math>e, f, g, h, &c.</math> co$inus arcuum,
quorum unu$qui$que ad unumquemque arcum antecedentem eo-
dem ordine $umptum eam rationem habeat, quæ e$t numeri <math>n - 1</math>
ad 1. Et erit <math>{1 / x^2n ± 2 p x^n + 1} = - {{α ± p e / n - n p^2}x - {1 / n}/x^2 - αx + 1} - {{β ± pf / n - p^2 n}x - {1 / n}/x^2 - βx + 1} -{{γ ± pg / n - np^2}x - {1 / n}/x^2 - γx + 1} - &c.</math> ubi affirmativum $ignum affixum e$t quantitati
<math>p e, &c.</math>; $i modo $it <math>+ p</math>, $in aliter negativum.</p>

<pb 65>[0117]<rh>FLUXIONUM FLUENTIBUS.</rh>
<p>6. Sit <math>(x^n - α) × (x^n - β) = x^2n ± 2 p x^n + 1</math>; ubi <math>p</math> major e$t
quam 1; $cribatur <math>z = x^n √(α) & y = x^n √(β)</math>, & erit <math>{1 / x^2n - 2px^n + 1}= {α / β - α} × {1 / z^n - 1} + {β / α - β} × {1 / y^n - 1}; & {1 / x^2n + 2px^n + 1} = {α / α - β} × {1 / 1 + z^n}+ {β / β - α} × {1 / 1 + y^n}</math>.</p>
<p>7. Sit <math>{ex^. / e + fx^n + gx^2n}</math>; $cribatur <math>x^n = v^n √({e / g}), & f = 2 p √ (ge);</math>
& erit <math>{ex^. / e + f x^n + g x^2n} = {^2n √({e / g})v^./v^2n + 2pv^n + 1}</math>.</p>
<p>8. Sit <math>{z^m-1 z^. / (1 + z)^n ± (1 - z)^n}</math>; pro <math>z</math> $cribatur <math>{x - a / x + a}</math>, & re$ultat
<math>{z^m-1 z^. / (1 + z)^n ± (1 - z)^n} = {2ax^. × (x - a)^m-1 / (x + a)^m+1} × {{1 / (2x)^n ± (2a)^n}/(x + a)^n} ={2ax^. × (x - a)^m-1 / (x + a)^m-n+1 × ((2x)^n ± (2a)^n)}</math>.</p>
<p>Plura autem de divi$oribus inveniendis adjiciuntur, quam ratio
no$tri in$tituti exigit.</p>
<h>PROB. XVI.</h>
<h it>Invenire quantitatem, quæ in datam irrationalem quantitatem ducta,
rationale productum facit.</h>
<p>Datæ irrationalis quantitatis inveniantur $inguli diver$i valores,
quibus in $e$e continuo ductis, contentum erit rationalis quantitas;
dividatur hoc contentum per datam quantitatem, & quotiens erit
quantitas quæ$ita. Vid. pag. 152. Medit. Algebr.</p>
<p>Cor. Hinc transferri po$$unt e numeratore in denominatorem, &
vice versâ e denominatore in numeratorem, quæcunque irrationales

<pb 66>[0118]<rh>DE INVENIENDIS</rh>
quantitates <math>(A, &c.)</math>; inveniatur rationalis quantitas <math>B</math>, cujus data
irrationalis <math>A</math> e$t divi$or; $it vero <math>C = {B / A}</math>, pro <math>A</math> $ub$tituatur <math>{B / C}</math> &
confit corollarium.</p>
<p>Ex. 1. Sit fluxio <math>{Q x^. / + R^2 √(a) + S^2 √(b)}</math>; ubi literæ <math>Q, R, S, a, b</math> re-
$pective denotent qua$cunque rationales functiones literæ <math>x</math>; ducan-
tur & numerator & denominator in quantitatem <math>+ R√(a) - S√(b)</math>;
& re$ultat fluxio <math>{+ R Q√(a) - S Q√(b) / R^2 a - S^2 b}x^.</math>.</p>
<p>Ex. 2. Sit fluxio <math>{Q v^. / P + R √ (a) + S √ (b)}</math>; ducantur & numerator &
denominator in quantitatem <math>(P + R √ (a) - S √ (b)) × (P - R√(a)
+ S√(b)) × (P - R √ (a) - S √ (b))</math>; & re$ultat quantitas quæ$ita.</p>
<p>Fluens fluxionis <math>a^2 x^./a^2 + x^2 = z^.</math> erit circularis arcus, cujus radius e$t <math>α</math>
& tangens <math>x</math>; hæc vero fluens erit rectangulum impo$$ibilis quantitatis
<math>{√(- a^2) / 2} ×</math> in hyperb. logarith. quantitatis impo$$ibilis <math>{x + √(-a^2) / x - √(-a^2)}</math>,
i. e. erit logarith. impo$$ibilis quantitatis <math>({x + √(-a^2) / x - √(-a^2)})^{√(-a^2) / 2} =
({γ + √(^2 √- a^2) / γ - √(γ^2 - a^2)})^{√(-a^2) / 2}</math>, ubi <math>y</math> e$t $inus eju$dem arcûs.</p>
<p>Ex hac æquatione deduci pote$t <math>y = a × {e{z√(- 1) / a} + e{-z√(- 1) / a}/2} &
√ (a^2 - y ^2) = a × {e{z√(- 1) / a} - e{-z√(- 1) / a}/2√(- 1)}</math>.</p>

<pb 67>[0119]<rh>FLUXIONUM FLUENTIBUS.</rh>
<p>Logarithmus autem quantitatis <math>({x + √(-a^2) / x - √(-a^2)} = {(x + √(-a^2))^2 / x^2 + a^2})
= 2</math> logar. <math>(x + √(- a^2)) - log. (x^2 + a^2) = {2 / √(-a^2)} ×</math> in cir-
cularem arcum prædictum; unde log. quantitatis impo$$ibilis <math>x +
√(- a^2) = {1 / √(- a^2) ×</math> circularem arcum prædictum <math>+ {1 / 2}}</math> log. quan-
titatis <math>(x^2 + a^2)</math>. Logarithmus autem quantitatis <math>x - √(- a^2) ={1 / 2}</math> log. quantitatis <math>(x^2 + a^2) - {1 / √(-a^2)} ×</math> circularem arcum præ-
dictum.</p>
<h>PROB. XVII.</h>
<p it>Invenire fluentem cuju$cunque fluxionis, quæ e$t algebraica functio literæ
<math>x</math> in fluxionem <math>x</math> ducta; $i modo ea fluens per finitos terminos literæ <math>x</math>, ejus
circulares arcus & logarithmos exprimi po$$it.</p>
<p>1. Sit fluxio <math>{x^s x^. / x^n - p x^n-1 + &c.}</math>; ubi literæ <math>p, q, r, &c.</math> cognitas &
invariabiles coefficientes re$pective denotant; per lemma con$tat
<math>{x^s x^. / x^n - px^n-1 + &c.} = {ax^.x^s / x - α} + {bx^.x^s / x - β} + {cx^.x^s / x - γ} + &c.</math> ubi literæ <math>α,
β, γ, &c.</math> $unt radices datæ æquationis <math>x^n - px^n-1 + qx^n-2 - &c. = 0</math>;
quantitas vero <math>{ax^s x^. / x - α}</math> reduci pote$t in $ub$equentem <math>ax^s-1 x^. + aαx^s-2 x^.
aα^2 x^s-3 x^.... + aα^s-1 x^. + {aα^s x^. / x - α}</math>; fluentes autem harum fluxionum
erunt <math>{ax^s / s} + {aαx^s-1 / s - 1} + {aα^2 x^s-2 / s - 2} ... + α a^s-1 x + aα^s ×</math> log. <math>(x - α)</math>.
Et $ic de reliquis terminis <math>{bx^s x^. / x - β}, {cx^s x^. / x - γ}, &c.</math> Si vero aliquid imagi-

<pb 68>[0120]<rh>DE INVENIENDIS</rh>
narii in duabus corre$pondentibus fractionibus <math>{bx^s x^. / x - α} & {cx^s x^. / x - β}</math> $eor$im
$umptis contineatur, addantur in unam hæ duæ fractiones, & re$ultat
fractio <math>{(b + c)x^s+1 - (bβ + cα)x^s / (x - α) × (x - β)}x^.</math> ex impo$$ibilitate libera: divida-
tur numerator per denominatorem u$que donec $olummodo re$tat
<math>(Ax + B)x^.</math>; quotientis terminorum inveniantur fluentes; fluens
vero re$idui <math>{(Ax + B) × x^. / (x - α) × (x - β)}</math> erit <math>{1 / 2}A × log. (x - α) × (x - β) -{B + (α + β) × {1 / 2}A/{1 / 4}(α - β)^2} ×</math> arc. circul. cujus radius $it <math>{1 / 2} × (α - β) × √(-1)</math>,
& cujus tangens $it <math>x - {1 / 2}α - {1 / 2}β</math>; ubi <math>{1 / 2}(α - β)√(-1)</math> e$t affirma-
tiva po$$ibilis quantitas.</p>
<p>Cor. 1. Sit data fluxio <math>{a x^s + b x^s-1 + &c. / x^n - p x^n-1 + q x^n-2 - &c.}x^.</math>, e prædictis
con$tat fluentem $ingularium fluxionum <math>{a x^s x^. / x^n - p x^n-1 + &c.}</math> inveniri
po$$e circularium arcuum, logarithmorum & finitorum terminorum
ope; ergo fluens ex omnibus conjunctim; acqui$itis prædictis radici-
bus <math>α, β, γ, &c.</math> facile acquiri pote$t fluens quæ$ita.</p>
<p>Hinc fluens omnis fluxionis, quæ e$t rationalis algebraica functio
literæ <math>x</math> in fluxionem <math>x^.</math> ducta, inveniri pote$t circularium arcuum,
logarithmorum & finitorum terminorum ope.</p>
<p>2. Sit data fluxio quæcunque rationalis functio quantitatum <math>x^n &
(a + b x^n)^{1 / m}</math>, ducta in <math>x^n-1 x^.</math>; tum ejus fluens $emper inveniri pote$t
ope finitorum terminorum, circularium arcuum & logarithmorum.</p>
<p>Data fluxio enim transformari pote$t in rationalem fluxionem per
methodum in ex. 1. prob. 15. traditam; unde per prob. ejus fluens
prædictâ methodo inveniri pote$t.</p>

<pb 69>[0121]<rh>FLUXIONUM FLUENTIBUS.</rh>
<p>Ex. Sit data fluxio <math>{ax^n + bx^2n + &c. + A(α + βx^n)^{1 / m} + B × (α + βx^n)^{2 / m} + &c./p + qx^n + rx^2n + &c + P × (α + βx^n)^{1 / m} + Q(α + βx^n)^{2 / m} + &c}x^n-1 x^.</math>: per prædictum exemplum $cribatur <math>(α + βx^n)^{1 / m} = v</math>, & erit
<math>{v^m - α / β} = x^n</math>, & exinde <math>x^n-1 x^. = {mv^m-1 v^. / nβ}</math>; unde data fluxio erit ra-
tionalis, viz. <math>{a × {v^m - α / β} + b × {(v^m - α)^2 / β^2} + A × v + B v^2 + &c./p + q × {v^m - α / β} + r × {(v^m - α)^2 / β^2} + Pv + Qv^2 + &c.} ×{m v^m-1 v^. / nβ}</math>; & exinde per prob. ejus fluens inveniri pote$t ope finitorum
terminorum, circularium arcuum & logarithmorum.</p>
<p>3. Sit data fluxio quæcunque rationalis functio quantitatis <math>x^n</math> &
irrationalis quantitatis <math>({a + bx^n / c + dx^n})^{1 / m} = v in x^n-1 x^.</math> ducta; & per ex-
emp. 2. prob.15. transformari pote$t in rationalem functionem literæ
<math>v</math> in fluxionem <math>v^.</math> ductam; unde con$tat ejus fluentem inveniri po$$e
ope finitorum terminorum, circularium arcuum & logarithmorum.</p>
<p>4. Sit data fluxio rationalis functio quantitatum <math>x^n & (a + bx^n)^{1 / s} ×
(c + dx^n)^{s-1 / s}</math> in <math>x^n-1 x^.</math> ducta. Pro <math>(a + bx^n)^{1 / s} × (c + dx^n)^{s-1 / s}</math> $cribatur
ejus valor <math>(c + dx^n) × ({a + bx^n / c + d x^n})^{1 / s}</math>, & reducitur data fluxio in fluxio-
nem præcedentis formulæ; unde ejus fluens ope prædictarum quan-
titatum inveniri pote$t.</p>
<p>5. 1. Sit data fluxio rationalis functio quantitatum <math>x^n & (e + f x^n
+ g x^2n)^{1 / 2}</math> in <math>x^n-1 x^.</math> ducta; & <math>g x^2n + f x^n + e = g × (x^n - α) × (x^n - β)</math>;
unde <math>g x^2n + f x^n + e = g × {(x^n - α) × (x^n - β)^2 / x^n - β}</math>, & exinde <math>(g x^2n +

<pb 70>[0122]<rh>DE INVENIENDIS</rh>
f x^n + e)^{1 / 2} = g{1 / 2} × ({x^n - α / x^n - β})^{1 / 2} × (x^n - β).</math> Supponatur <math>({x^n - α / x^n - β})^{1 / 2}
= v</math>, & exinde <math>{βv^2 - α / v^2 - 1} = x^n</math>; quo valore pro <math>x^n</math> & ejus fluxione pro
<math>n x^n-1 x^.</math> in datâ fluxione $ub$tituto; re$ultabit fluxio, quæ erit ratio-
nalis functio literæ <math>v</math> in fluxionem <math>v^.</math> ducta.</p>
<p>2. Si vero radices æquationis <math>x^2n + {f / g} x^n + {e / g} = 0</math> $int impo$$ibiles,
& con$equenter prædictæ quantitates $ub$titutæ impo$$ibiles; tum
impo$$ibiles quantitates evitandi gratiâ ad $ub$titutiones $ub$equentes
recurrere præ$tat. Supponatur <math>(g x^2n + fx^n + e(b^2))^{1 / 2} = b + px^n v</math>,
& exinde <math>gx^2n + fx^n + h^2 = h^2 + 2pbx^n v + p^2 v^2 x^2n</math>; unde <math>x^n ={f - 2bpv / p^2 v^2 - g}, & (h^2 + fx^n + g x^2n)^{1 / 2} = b + px^n v = {pfv - hp^2 v^2 - gb / p^2 v^2 - g}</math>;
quibus quantitatibus pro $uis valoribus in datâ fluxione $ub$titutis,
& fluxione <math>{2p^3 bv^2 - 2p^2 fv + 2bpg / n(p^2 v^2 - g)^2}v^.</math> pro <math>x^n-1 x^.</math>; transformatur
data fluxio in rationalem fluxionem.</p>
<p>6. Sit data fluxio rationalis functio quantitatum <math>x^n, & ^m √(a + b
<_>r</_>√(c + d<_>s</_>√(e + &c. + (f + gx<_>n</_>)<_>{1 / t}</_>))) = P, <_>r</_>√(c + d<_>s</_>√(e + &c. +
(f + gx<_>n</_>)<_>{1 / t}</_>)) = Q, <_>s</_>√(e + &c. + (f + gx<_>n</_>)<_>{1 / t}</_>) = R, (f + gx<_>n</_>)<_>{1 / t}</_> = V</math>;
ubi literæ <math>r, s, &c.; t</math> $unt integri numeri; pro <math>P, Q, R, &c. &c. &
V & gx^n</math> $cribantur re$pective <math>v = P, {v^m - a / b} = Q, {({v^m - a / b})^r - c/d} = R,
&c.; ({({v^m - a / b})^r - c/d})^s - e, &c. = (f + gx^n)^{1 / t}, & (({({v^m - a / b})^r - c/d})^s &c.)^t
- f = gx^n; &</math> pro <math>x^n-1 x^.</math> ejus valor exinde deductus; unde facile

<pb 71>[0123]<rh>FLUXIONUM FLUENTIBUS.</rh>
quæcunque rationalis functio quantitatum <math>P, Q, R, &c., V & x^.^n</math> in
<math>x^n-1 x^.</math> ducta transformari pote$t in rationalem functionem quantita-
tum <math>v^m & v</math> in <math>v^.</math>.</p>
<p>7. Sit data fluxio rationalis functio quantitatum <math>x^n, ^m √(a + b
<_>r</_>√(c + d<_>s</_>√(e + &c. + ({f + gx<_>n</_> / b + kx<_>n</_>})<_>{1 / t}</_>))) = P, <_>r</_>√(c + d<_>s</_>√(e + &c.
+ ({f + gx<_>n</_> / h + kx<_>n</_>})<_>1/t</_>)) = Q, <_>s</_>√(e + &c. + ({f + gx<_>n</_> / h + kx<_>n</_>})<_>{1 / t}</_>) = R, &c.,
({f + gx<_>n</_> / h + kx<_>n</_>})<_>{1 / t}</_> = V</math> in <math>x^n-1 x^.</math> multiplicata; ubi <math>r, s, &c., t</math> $unt integri nu-
meri; pro <math>P, Q, R, &c., V, & x^n, & x^n-1 x^.</math> $cribantur re$pective <math>v = P,{v^m - a / b} = Q, {({v^m - a / b})^r - c/d} = R, &c., ({({v^m - a / b})^r &c./d &c.}&c.) = V,
& {f - bV^t / kV^t - g} = x^n, & {thg - tkf / n(kV^t - g)^2}V^t-1 V^.</math>; & exinde facile transformari
pote$t rationalis functio quantitatum <math>P, Q, R, &c., V & x^n</math>, in <math>x^n-1 x^.</math>
ducta, in rationalem functionem quantitatum <math>v^m & v</math> in <math>v^.</math>; & con$e-
quenter e methodis prius traditis con$tat ejus fluentem inveniri po$$e
ope finitorum terminorum, circularium arcuum & logarithmorum.</p>
<p>8. Sit data fluxio rationalis functio quantitatum <math>x^n & (a + bx^n +
(e + fx^n)^{1 / 2})^{1 / 2}, & (e + fx^n)^{1 / 2}</math> in <math>x^n-1 x^.</math> ducta: $cribatur <math>e + fx^n = v^2</math>
& exinde <math>(a + bx^n + (e + fx^n)^{1 / 2})^{1 / 2} = (a + b × {v^2 - e / f} + v)^{1 / 2}, & {v^2 - e / f}= x^n</math>; quibus quantitatibus pro $uis valoribus $ub$titutis, & <math>{2vv^. / nf}</math> pro
<math>x^n-1 x^.</math> in datâ fluxione; re$ultat fluxio, quæ erit rationalis functio

<pb 72>[0124]<rh>DE INVENIENDIS</rh>
quantitatum <math>v & (a + b × {v^2 - e / f} + v)^{1 / 2}</math>, cujus fluens etiam inve-
niri pote$t ope finitorum terminorum, circularium arcuum & loga-
rithmorum.</p>
<p>9. Sit data fluxio rationalis functio quantitatum <math>x^n & ({e + fx^n / h + kx^n})^{1 / 2}
= v, & (p + qv + rv^2)^{1 / 2}</math> in <math>x^n-1 x^.</math> ducta: $cribantur in datâ fluxione
pro <math>({e + fx^n / h + kx^n})^{1 / 2} & (p + qv + rv^2)^{1 / 2}</math>, & pro <math>x^n & x^n-1 x^.</math>; re$pective
<math>v, (p+qv + rv^2)^{1 / 2}, {hv^2 - e / f - kv^2} & {2bvv^. / n(f - kv^2)} + {2kvv^. × (bv^2 - e) / n(f - kv^2)^2}</math>;
& re$ultat fluxio, quæ e$t rationalis functio quantitatum <math>v & (p +
q v + rv^2)^{1 / 2}</math> in <math>v^.</math>, cujus fluens e methodis prius traditis innote$cit.</p>
<p>10. Sit data fluxio rationalis functio quantitatum <math>x^n &{(a + l({e + fx^n / h + kx^n})^{1 / s})^{1 / m} /(b + c({e + fx^n / h + kx^n})^{1 / s})^{1 / m}}(P) & ({e + fx^n / h + kx^n})^{1 / s}</math> in <math>x^n-1 x^.</math>, ubi <math>s</math> e$t integer
numerus: $cribatur <math>P = v</math>, & erit <math>z = ({bv^m - a / l - cv^m})^s = {e + fx^n / h + kx^n}</math>,
unde <math>{bz - e / f - kz} = x^n</math>; quibus quantitatibus pro <math>P, ({e + fx^n / h + kx^n}), x^n</math>, &
ejus <math>({bz - e / f - kz})</math> fluxione pro <math>n x^n-1 x^.</math>, in datâ fluxione $ub$titutis;
re$ultat fluxio, quæ erit rationalis functio quantitatis <math>v^m & v</math> in <math>v^.</math>,
cujus fluens e prædictis methodis innote$cet.</p>
<p>Et $ic de fluentibus plurium huju$ce generis fluxionum invenien-

<pb 73>[0125]<rh>FLUXIONUM FLUENTIBUS.</rh>
dis: e. g. cum data fluxio $it rationalis functio quantitatum <math>x^n & x^n
+ √(1 + x^2n) = v & √ (1 + x^2n)</math> in <math>x^n-1 x^.; &c.</math></p>
<p>10. Reducere datas fluxiones in alias, quarum fluentes innote$cunt
e multiplicatione earum numeratorum & denominatorum in a$$ump-
tas quantitates.</p>
<p>Ca$. 1. Datâ fluxione <math>{P x^n-1 x^. / Q^2 √(e + fx^n + gx^2n) + R^2 √(h + kx^n + lx^2n)</math>;
ubi literæ <math>P, Q, & R</math> re$pective denotant rationales functiones quan-
titatis <math>x^n</math>; ducantur & numerator & denominator huju$ce fluxionis in
quantitatem <math>Q √(e + fx^n + gx^2n) - R√(b + kx^n + lx^2n)</math>, & re$ultat
fluxio <math>{PQ√(e + fx^n + gx^2n) - PR √(h + kx^n + lx^2n) / Q^2 × (e + fx^n + gx^2n) - R^2 × (h + kx^n + lx^2n)} × x^n-1 x^.}</math>, cujus
fluens e regulis prius traditis innote$cit.</p>
<p>2. Sit data fluxio <math>{Px^n-1 x^. / Q × (R^m-1 (a + bx^n)^{m-1 / m} + R^m-2 S(a + bx^n)^{m-2 / m} × (d + ex^n)^{1 / m}
+ R^m-2 S^2 (a + bx^n)^{m-3 / m} × (d + ex^n)^{2 / m} ...S^m-1 × (d + ex^n)^{m-1 / m})}</math>; ubi <math>P, Q, R,
& S</math> denotant rationales functiones quantitatis <math>x^n, & m</math> e$t integer
numerus; ducatur hæc fluxio in <math>R × (a + bx^n)^{1 / m} - S(d + ex^n)^{1 / m}</math>; &
re$ultat fluxio <math>{P x^n-1 x^. / Q × R^m (a + bx^n) = S^m (d + ex^n)} × (R(a + bx^n)^{1 / m} - S
× (d + ex^n)^{1 / m})</math>; cujus fluens in formulis prius traditis continetur.</p>
<p>3. Datâ fluxione <math>{Px^n-1 x^. / Q√(e + fx^n) + R√(g + bx^n) + S√(l + mx^n)}</math>
ubi literæ <math>P, Q, R & S</math> qua$cunque rationales functiones quantitatis <math>x^n</math>
re$pective denotant: ducantur & numerator & denominator datæ flu-
xionis in quantitatem <math>(Q√(e + fx^n) - R√(g + hx^n) + S√(l + mx^n))
× (Q√(e + fx^n) - R√(g + hx^n) - S√(l + mx^n)) × (Q√(e + fx^n)
+ R√(g + hx^n) - S√(l + mx^n))</math>; & re$ultat fluxio, cujus fluens in-

<pb 74>[0126]<rh>DE INVENIENDIS</rh>
veniri pote$t, $i modo fluens fluxionis <math>{A x^n-1 x^. / B} × (e + f x^n)^{1 / 2} × (g + h x^n)^{1 / 2}
× (l + m x^n)^{1 / 2}</math> detegi po$$it, ubi literæ <math>A & B</math> denotant rationales
functiones quantitatis <math>x^n</math>.</p>
<p>4. Sit data fluxio <math>A^.</math>, cujus numerator $it <math>x^.</math> in quamcunque rationa-
lem functionem irrationalium algebraicarum functionum <math>α, β, γ, δ,
&c.</math> quantitatis <math>x</math> ducta, quæ functiones <math>α, β, γ, δ, &c.</math> haud in $e$e
multiplicantur; & cujus denominator $it quæcunque rationalis fun-
ctio irrationalium functionum algebraicarum <math>π, ξ, σ, τ, &c.</math> quantita-
tis (<math>x</math>): tum e datis fluentibus fluxionum, quæ exoriuntur ex quan-
titatibus <math>π α, π β, π γ, &c.; ξ α, ξ β, ξ γ, &c.; σ α, σ β, σ γ, &c.; &c.;
π ξ α, π ξ β, π ξ γ, &c.; π σ α, π σ β, &c.; ξ σ α, &c.; &c.; π ξ σ α, &c.; &c.;</math>
(quæ $unt quantitates <math>π, ξ, σ, &c.</math> in quantitates <math>α, β, γ, δ, &c.</math> re-
$pective ductæ; rectangula $ub quibu$que duabus prædictis (<math>π ξ, π σ,
ξ σ, &c.</math>) in quantitates <math>α, β, γ, δ, &c.</math> re$pective ductis; etiamque con-
tenta $ub quibu$que tribus, quatuor, &c. quantitatibus (<math>π, ξ, σ, τ, &c.</math>)
re$pective ductis in prædictas quantitates (<math>α, β, γ, δ, &c.</math>)) ductis
in qua$cunque vel ea$dem vel diver$as rationales functiones quanti-
tatis <math>x</math> in <math>x^.</math>; $emper detegi pote$t fluens fluxionis <math>(A^.)</math>.</p>
<p>Con$tat ex ducendo numeratorem & denominatorem datæ fluxio-
nis in quantitatem acqui$itam per methodum in prob. 26. medit. al-
gebr. traditam, ita ut denominator evadat rationalis quantitas.</p>
<p>4. Datâ fluxione <math>{P x^n-1 x^. / Q × α}</math>, ubi literæ <math>P, Q, & A, B, C, D, &c.</math> ra-
tionales functiones quantitatis <math>x^n</math> re$pective denotant, & litera <math>α</math> de-
notat quantitatem, quæ in <math>A √ (e + f x^n + g x^2n) + B √ (h + k x^n + l x^2n)
+ C √ (m + n′x^n + 0 x^2n) + D √ (p + q x^n + r x^2n) + &c. (T)</math> ducta,
creat rationalem functionem quantitatis <math>x^n</math>; ducantur numerator &
denominator datæ fluxionis in quantitatem <math>T</math>; & re$ultat fluxion, cu-
jus fluens e prædictis methodis inveniri pote$t; &c.</p>

<pb 75>[0127]<rh>FLUXIONUM FLUENTIBUS.</rh>
<h>THEOR. VI.</h>
<p>Rationalis functio cuju$cunque literæ <math>x</math> $emper dividi pote$t in $im-
plices divi$ores <math>x + a, &c.</math>: hinc etiam multæ fluxiones, quæ apparent
tanquam irrationales, transformari po$$unt in rationales fluxiones.</p>
<p>Ex. 1. Sit data fluxio quæcunque algebraica & rationalis functio
quantitatum <math>^m √ (a + b x^n) & x^n</math> in <math>x^n-1 x^.</math>; & facile con$tat methodus
transformandi datam fluxionem in rationalem functionem quantitatis
<math>^m √ (a + b x^n), &c.</math>; & exinde con$tat denominatorem dividi po$$e in $uos
$implices divi$ores huju$ce generis <math>^m √ (a + b x^n) + p, ^m √ (a + b x^n) + q,
&c.</math> ubi literæ <math>p & q</math> invariabiles quantitates re$pective denotant.</p>
<p>Ex. 2. Sit data fluxio quæcunque algebraica functio & rationalis
quantitatum <math>{^m √ (a + b x^n) / ^m √ (d + e x^n)} & x^n</math> in <math>x^n-1 x^.</math> ducta; & erit data fluxio
rationalis functio quantitatis <math>{^m √ (a + b x^n) / ^m √ (c + d x^n)}</math>; & con$equenter ejus de-
nominator dividi pote$t in $implices divi$ores huju$ce formulæ
<math>^m √ ({a + b x^n / c + d x^n}) + p, &c.</math> & $ic de reliquis exemplis prius traditis; &c.</p>
<h>PROB. XVIII.</h>
<p it>Datâ fluxione, quæ $it algebraica functio quantitatis <math>x^n</math>; invenire, an-
non ejus fluens inve$tigari pote$t logarithmorum po$$ibilium vel impo$$ibilium
quantitatum, i. e. circularium arcuum & finitorum terminorum ope.</p>
<p>1. Inveniantur omnes divi$ores denominatoris datæ fluxionis ma-
xime $implices, qui haud minores recipiunt dimen$iones incognitæ
quantitatis <math>x</math>; quam eas, quas habet data fluxio; & $int hi divi$ores

<pb 76>[0128]<rh>DE INVENIENDIS</rh>
re$pective <math>A, B, C, D, E, &c.</math> i. e. $it <math>A × B × C × D × E × &c. =</math> datæ
fluxionis denominatori, ubi nullus datur divi$or huju$ce formulæ
<math>(a + b x^n + c x^2n + &c.)^{r / s}</math>, & litera <math>s</math> haud divi$or e$t literæ <math>r</math>; tum
fluens datæ fluxionis for$an erit aggregatum <math>α × log. A + β × log.
B + γ × log. C + δ × log. D + &c.</math> ubi <math>α, β, γ, δ, &c.</math> invariabiles
quantitates re$pective denotant.</p>
<p>2. Sint vero duo divi$ores <math>A & B</math> impo$$ibiles & corre$pondentes,
i. e. $int <math>p + √ (- q^2) & p - √ (- q^2)</math>; tum <math>α log. A + β log. β</math> de-
notat aream circuli, ut prius docetur.</p>
<p>3. Denominator, &c. finitorum terminorum deducendus e$t e de-
nominatore dato, &c. per methodum in prob. 4. traditam; & con$e-
quenter $ummæ fluxionum finitorum terminorum per prædictam
methodum deductorum, & fluxionum logarithmorum & circularium
arcuum prius inventorum æquentur datæ fluxioni, $i modo po$$ibile
$it; & exinde deduci pote$t fluens quæ$ita.</p>
<p>Ex. Sit data fluxio <math>{b x^n-1 + {2 / 5} × (e + f x^n)^-{3 / 5} × f x^n-1 + (a + b x^n +
(e + f x^n)^{2 / 5} + 1)({4 / 3} d x^n-1 × (g + h x^n) + h x^n-1 × (c + d x^n)) × (c + d x^n)^{1 / 3} /(a + b x^n + (e + f x^n)^{2 / 5} + 1)}× x^.</math>. Primo animadvertendum e$t denominatorem habere $implicem
divi$orem <math>(a + b x^n + (e + f x^n)^{2 / 5} + 1) = H</math>, qui in $impliciores divi-
$ores haud re$olvi pote$t; & con$equenter $i fluens datæ fluxionis
inveniri po$$it finitorum terminorum, logarithmorum, & circula-
rium arcuum ope; tum quædam pars fluentis erit <math>α × log. H + β
× log. (c + d x^n) + γ log. (e + f x^n)</math>; ubi literæ <math>α, β, & γ</math> con$tantes
quantitates inveniendas denotant; <math>& (c + d x^n)^{1 / 3} & (e + f x^n)^{1 / 5}, &c.</math>
$unt irrationales quantitates in prædicto $implici divi$ore <math>(H)</math> con-
tentæ: & $ic de pluribus $implicibus divi$oribus in denominatore datæ

<pb 77>[0129]<rh>FLUXIONUM FLUENTIBUS.</rh>
fluxionis contentis argumentari liceat: <math>2^do</math>. pro $ingulis irrationali-
bus quantitatibus <math>(k + l x^n)^{1 / 5}</math> in datâ fluxione $ub$tituere liceat <math>a × log.
(A + B (k + l x^n)^{1 / 5} + C (k + l x^n)^{2 / 5} + D (k + l x^n)^{3 / 5} ... + (k + l x^n)^{5 / 5-1})</math>,
ubi litera <math>s</math> integrum denotat numerum, & <math>A, B, C, D; &c.</math> $unt
rationales functiones quantitatis <math>x^n</math>; pro parte fluentis quæ$itâ: 3<_>tio</_>.
$i vero $it <math>H^t</math>, ubi <math>t</math> $it integer numerus, eædem quantitates pro loga-
lithmicis partibus fluentis a$$umendæ $unt.</p>
<p>Hactenus de logarithmicis partibus fluentis; quarum, $i aliquis
evadat impo$$ibilis, ea ad circulares arcus reducenda e$t.</p>
<p>Nunc quoad partes fluentis, quæ finitos terminos recipiunt; <math>1^mo</math>.
ob$ervandum e$t in iis haud contineri $implicem divi$orem datæ flu-
xionis denominatoris, qui non habet alterum $ibi ip$i æqualem;
ergo, $i talis divi$or in denominatore contineatur, numerator partis
fluxionis, quæ per finitos terminos detegi pote$t, per eum divi$orem
dividi queat: quoad reliquos denominatoris terminos, & irrationales
datæ fluxionis numeratoris terminos, fluentes a$$umendæ $unt per
methodos in prob. 4. traditas; e. g. $int quantitates in denominatore
<math>(a′ + b′ x^n + &c.)^π, &c.</math> & in numeratore <math>(c′ + d′ x^n + &c.)^ρ, &c.</math> tum
pro parte fluentis quæ$itâ ad eas refertâ, & per finitos terminos inve-
$tigandâ, a$$umatur <math>(a′ + b′ x^n + &c.)^1-π × (c′ + d′ x^n + &c.)^1+ρ × (λ +
μ x^n + ν x^2n + &c.)</math>; ubi <math>λ, μ, ν, &c.</math> $unt con$tantes quantitates inve-
$tigandæ: & $ic de pluribus quantitatibus huju$ce generis.</p>
<p>Inveniantur fluxiones $ingularum prædictarum fluentium, deinde
fiat earum $lumma datæ fluxioni æqualis; & ex æquatis corre$pon-
dentibus datæ fluxionis & prædictæ $ummæ terminis deduci pote$t
ejus fluens; (quæ in hoc ex. erit <math>{1 / n}</math> log. <math>(a + b x^n + (e + f x^n)^{2 / 5} + 1)
+ {1 / n} (c + d x^n)^{4 / 3} × (g + h x^n))</math>; $i modo per finitos terminos quantitatis
<math>x^n</math>, logarithmos & circulares arcus algebraicarum functionum præ-
dictæ quantitatis <math>x^n</math> exprimi po$$it.</p>

<pb 78>[0130]<rh>DE INVENIENDIS</rh>
<p>In pleri$que ca$ibus præ$tat, ut negativi indices ex denominatore
& numeratore exterminentur.</p>
<p>Sit <math>{P x^. / Q}</math> fluxio logarithmi fractionis <math>{α / β}</math>; ubi <math>P & Q, α & β</math> $unt al-
gebraicæ functiones quantitatis <math>x</math>; tum erunt dimen$iones quantita-
tis <math>x</math> in denominatore <math>Q</math> majores quam ejus dimen$iones in numera-
tore <math>P</math> per unitatem; ni dimen$iones quantitatis <math>x</math> in numeratore <math>α</math>
æquales $int ejus dimen$ionibus in <math>β</math>, in quo ca$u dimen$iones deno-
minatoris <math>Q</math> $uperant dimen$iones numeratoris per quantitatem ma-
jorem quam unitatem.</p>
<h>THEOR. VII.</h>
<p>1. Sit <math>{x^. / x} = {y^. / y}</math>; erit <math>x = a y</math>, ubi <math>a</math> $it invariabilis coefficiens ad li-
bitum a$$umenda.</p>
<p>Cor. Hinc fluxio (<math>x^.</math>) nullius quantitatis per $uam quantitatem (<math>x</math>)
divi$a eandem habet quotientem, ac fluxio (<math>y^.</math>) alterius quantitatis per
$uam quantitatem <math>y</math> divi$a; ni <math>x = a y</math>.</p>
<p>2. Sint duæ prædictæ variabiles quantitates <math>x & y, & $it {x^. / y} = {y^. / x}</math>; &
erit <math>x x^. = y y^.</math>, & exinde <math>x^2 = y^2 + a</math>.</p>
<p>Ex. Sit <math>{y^. / √ (y^2 + a)}</math>, ejus fluens inveniri pote$t ope logarithmorum:
a$$umatur enim <math>x^2 = y^2 + a</math>, & erit <math>{x^. / y} = {y^. / x}</math> i. e. <math>{x^. / y} = {y y^. / y √ (y^2 + a)} ={y^. / √ (y^2 + a)}</math>; ergo, $i a$$umatur pro fluente <math>y + x = y + √ (y^2 + a)</math>, erit
<math>{y^. + x^. / x + y} = {y^. / x} = {x^. / y} = {y^. + {y y^. / √ (y^2 + a)}/√ (y^2 + a) + y} = {y^. / √ (y^2 + a)}</math>.</p>
<p>Cor. Hinc fluxio (<math>x^.</math>) nullius quantitatis (<math>x</math>) per alteram quantita-

<pb 79>[0131]<rh>FLUXIONUM FLUENTIBUS.</rh>
tem (<math>y</math>) divi$a eandem dabit quotientem, ac fluxio (<math>y^.</math>) alterius quan-
titatis per priorem quantitatem (<math>x</math>) divi$a; ni <math>x^2 = y^2 + a</math>.</p>
<p>3. Sint tres variabiles quantitates <math>x, y & z</math>; & $it <math>{x^. / z} = {y^. / x} = {z^. / y}</math>; &
per reductionem, viz. e primâ æquatione <math>{x^. / z} = {y^. / x}</math>, re$ultat <math>x x^. = z y^.</math>:
inveniatur ejus fluxio <math>x x^.. + x^.^2 = z^.y^.</math>, & con$equenter <math>z^. = {x x^.. + x^.^2 / y^.}</math>,
& exinde ob <math>({y^. / x} = {z^. / y})</math> invenitur fluxionalis æquatio <math>y y^.^2 = x^2 x^.. +
x x^.^2</math>, $i modo <math>y</math> fluat uniformiter.</p>
<p>4. Sit <math>{x^. / v} = {y^. / x} = {z^. / y} = {v^. / z}</math>; & per reductionem invenitur fluxionalis
æquatio <math>x^3 x^... + 4 x^2 x^. x^.. + x x^.^3 = y y^.^3</math>; & in genere $int <math>n</math> variabiles
quantitates, & <math>n - 1</math> æquationes prædicti generis, & $it fluxionalis
æquatio relationem inter <math>x & y</math> exprimens prædicti generis <math>A = y y^.^n-1</math>;
tum $int <math>n + 1</math> variabiles quantitates & <math>n</math> æquationes prædicti generis,
& erit æquatio inter <math>x & y</math> & earum fluxiones re$ultans <math>x A^. = y y^.^n</math>,
$i modo <math>y</math> fluat uniformiter.</p>
<p>Ex re$olutione fluxionalium æquationum quærendæ $unt fluentes
prædictarum æquationum.</p>
<p>5. E divi$oribus denominatoris femper detegi pote$t, annon fluens
datæ fluxionis inveniri pote$t ope finitorum terminorum, circula-
rium arcuum, & logarithmorum; con$tat enim e prob. 4. $ub$titutio,
quæ nece$$ario continet finitos terminos; & e divi$oribus denomina-
toris, &c. $emper con$tabunt termini, qui involvunt logarithmos &
circulares arcus, ni in ca$ibus hoc theorem. datis, i. e. ubi <math>α + β</math> $it
divi$or quæ$itus & <math>{α^. / β} = {β^. / α}</math>, & con$equenter <math>β = √ (α^2 + a)</math>, ubi <math>a</math>
denotat invariabilem quantitatem; & $ic ubi <math>α + β + γ</math> $it divi$or
quæ$itus & <math>{α^. / β} = {β^. / γ} = {γ^. / α}</math>; & $ic deinceps.</p>

<pb 80>[0132]<rh>DE INVENIENDIS</rh>
<p>In hoc ca$u facile con$tat ($i modo <math>α</math> $it rationalis algebraica fun-
ctio, & <math>β & γ</math> finitæ algebraicæ functiones literæ <math>x</math>) <math>β & γ</math> e$$e re-
$pective <math>p P {1 / 3} & q P {2 / 3}</math>; ubi literæ <math>p, q & P</math> denotant re$pective rationa-
les functiones literæ <math>x</math>: &c.</p>
<p>In detegendis prædictis divi$oribus denominatoris: $æpe e denomi-
natore in numeratorem; & vice versâ e numeratore in denominato-
rem, transformari debent irrationales quantitates.</p>
<p>Ni igitur re$olutioni ob$tet vulgaris algebra; $emper inveniri pote$t,
utrum fluens cuju$cunque datæ fluxionis inve$tigari pote$t ope finito-
rum terminorum, circularium arcuum & logarithmorum; necne.</p>
<h>PROB. XIX.</h>
<p it>_1._ Invenire, annon data fluxio, quæ e$t functio quantitatis <math>x</math> in <math>x^.</math>; $it
rationalis functio quantitatis (<math>z</math>), quæ e$t data functio quantitatis <math>x</math>, in <math>z</math>.</p>
<p>A$$umatur rationalis functio <math>{a + b z + c z^2 + d z^3 + &c. / A + B z + C z^2 + D z^3 + &c.} × z^.</math>: in
hâc functione pro <math>z & z^.</math> $cribatur data functio quantitatis <math>x</math> & ejus
fluxio; & ex æquatis corre$pondentibus terminis datæ & re$ultantis
fluxionis detegi pote$t, annon data fluxio $it rationalis functio quan-
titatis <math>z</math> in <math>z^.</math>.</p>
<p it>_2._ Invenire, utrum prædicta fluxio, quæ e$t functio quantitatis <math>x</math> in <math>x^.</math>;
$it rationalis functio quantitatum <math>z, v, w, &c.</math>; in fluxione rationalis
functionis quantitatum <math>x, v, w, &c.</math>; ubi literæ <math>z, v, w, &c.</math> re$pective de-
notant datas functiones quantitatis <math>x</math>; necne.</p>
<p>A$$umantur duæ rationales functiones <math>{a + b z + c v + d w + &c. +
e z^2 + f v^2 + &c. + g z v + h z w + &c. + k z^3 + &c. / A + B z + C v + D w
+ &c. + E z^2 + F v^2 + G z v + &c.} = P</math>, &
<math>{a′ + b′ z + c′ v + d′ w + &c. + e′ z^2 + &c. / A′ + B′ z + C′ v + D′ w + &c. + E′ z^2 + &c.} = Q</math>; in fluxione <math>P × Q^.</math>
pro <math>z, v, w, &c.</math>; & earum fluxionibus $cribantur re$pective earum

<pb 81>[0133]<rh>FLUXIONUM FLUENTIBUS.</rh>
valores in datis functionibus quantitatis <math>x</math> & ejus fluxionis; & ex
æquatis corre$pondentibus terminis datæ & re$ultantis fluxionis de-
tegi pote$t, annon data fluxio $it rationalis functio prædicta.</p>
<h>PROB. XX.</h>
<h it>Datis fluentibus fluxionum datæ formulæ; invenire alias fluxiones, qua-
rum fluentes e datis deduci po$$unt.</h>
<p>In fluxionibus, quarum fluentes dantur, pro variabili & ejus flu-
xione $cribantur quæcunque functio novæ variabilis & ejus fluxio;
tum $equitur fluxio, cujus fluens e datis innote$cit.</p>
<p>Ex. 1. Sit data fluxio <math>(A) y^r y^. × y^-{1 / 2} × (c y^2 + b y + a)^-{1 / 2}</math>, ubi <math>r</math> e$t
integer numerus po$itivus vel negativus; tum ex duabus independen-
tibus fluentibus huju$ce formulæ deduci po$$unt fluentes omnium
fluxionum eju$dem formulæ, i. e. utcunque variatur litera <math>r</math>, $i modo
invariabiles maneant coefficientes <math>a, b, & c</math>; pro <math>y & y^.</math> $cribantur in
datâ fluxione <math>x + p & x^.</math>; & ita a$$umantur coefficientes <math>a, b, c & p</math>
ut evadat fluxio <math>(B) {b′ (x^b + &c.) x^. / √ (x^3 + a′)}</math>, ubi litera <math>b</math> denotat integrum nu-
merum: unde, $i fluentes omnium fluxionum formulæ <math>y^r y^. × y^-{1 / 2} x
(c y^2 + b y + a)^-{1 / 2}</math> detegi po$$int, fluentes omnium fluxionum for-
mulæ <math>{x^b x^. / √(x^3 + a′)}</math> detegi po$$unt: 2. in fluxione <math>{x^b x^. / √(x^3 + a′)}</math> pro
<math>√ (x^3 + a′)</math> $cribatur <math>z, &c.</math> & re$ultat fluxio formulæ <math>(z^2 - a′)^{b+1 / 3} × z^.</math>;
unde <math>(C) z^1 × (z^2 - a)^± {1 / 3} vel ± {2 / 3} z^.</math>, ubi <math>l</math> e$t integer numerus: deinde
pro <math>y, x & z</math> in prædictis fluxionibus <math>A, B, & C</math> $cribatur <math>{e + f v / l′ + m v}</math>; &
re$ultant fluxiones formulæ <math>v^b × (l + m v)^±{1 / 2} × (p + q v + r′ v^2 +
s v^3)^±{1 / 2} v^., &c.</math>: & $imiliter ex $cribendo in his fluxionibus pro <math>v</math> quan-
titatem <math>{e′ + f′ w / l″ + m′ w}</math>, vel pro quâcunque irrationali quantitate in præ-

<pb 82>[0134]<rh>DE INVENIENDIS</rh>
dictis fluxionibus contentâ $cribendo con$imilem quantitatem <math>{e′ + f′ w / l′ + m′ w}</math>;
re$ultabunt formulæ fluxionum, quarum fluentes e prædictis deduci
po$$unt; & $ic deinceps.</p>
<p>Ex. 2. Sit fluxio <math>{y^. × A / √ (f ± √ (g + y^2))}</math>, ubi <math>A</math> e$t rationalis functio
quantitatis <math>√ (g + y^2)</math>; pro <math>√ (g + y^2)</math> $cribatur <math>x</math>, & re$ultat
æquatio $ormulæ <math>{x x^. × A′ / √ (x^2 - g) × √ (f ± x)}</math>, ubi <math>A′</math> e$t functio quan-
titatis <math>x</math>.</p>
<h>PROB. XXI.</h>
<h it>Invenire, annon fluens datæ fluxionis ex fluentibus quarundam datarum
deduci pote$t.</h>
<p>1. Ex formulâ irrationalitatis datæ fluxionis $æpe deduci po$$unt
formulæ quantitatum, quæ transformabunt prædictas fluxiones in
alias eju$dem irrationalitatis, quam habet data fluxio; deinde tran$-
formentur generaliter hæ fluxiones in alias, quæ eandem habent ir-
rationalitatem ac data fluxio, tum ducantur fluxiones re$ultantes in
invariabiles coefficientes, & $imul addantur; deinde fiat aggregatum
re$ultans datæ fluxioni æquale, $i modo po$$ibile $it; & perficitur pro-
blema.</p>
<p>Ex. 1. Transformare quantitatem irrationalitatem <math>^3 √ (a + z)</math>
habentem, ita ut eandem irrationalitatem habeat ac quantitas
<math>^2 √ (c + d x^2)</math>. Supponatur <math>^3 √ (a + z) = ^2 √ (c + d x^2)</math>, & exinde
<math>(a + z)^2 = (c + d x^2)^3</math>, unde <math>z = (c + d x^2)^{3 / 2} - a</math>; $cribatur hæc
quantitas pro <math>z</math> in quantitate <math>^3 √ (a + z)</math>, & ea transformabitur in
quantitatem <math>^2 √ (c + d x^2)</math>. Aliter magis generaliter $upponatur

<pb 83>[0135]<rh>FLUXIONUM FLUENTIBUS.</rh>
<math>{e + f z + g z^2 + &c. / e′ + f′ z + g′ z^2 + &c.} + {h + k z + l z^2 + &c. / h′ + k′ z + l′ z^2 + &c.} × ^3 √ (a + z) + &c. ={α + β x + γ x^2 + &c. / α′ + β′ x + γ′ x^2 + &c.} + {λ + μ x + v x^2 + &c. / λ′ + μ′ x + v′ x^2 + &c.} × √ (c + d x^2)</math>; deind
inveniatur <math>z</math> in terminis quantitatis <math>x</math>, & $ub$tituatur hæc quantitas
pro <math>z</math> in datâ fluxione; & perficietur exemplum.</p>
<p>Ex. 2. Transformare quantitates <math>x & √ (a + b x + c x^2)</math> in ratio-
nales functiones quantitatis <math>z</math>: a$$umatur rationalis functio <math>{d + e z / D + E z}</math>
quantitatis <math>z</math> pro <math>x</math>; $cribatur hæc functio pro <math>x</math> in quantitate
<math>√ (a + b x + c x^2)</math>; & re$ultat <math>a + b × {d + e z / D + E z} + c ({d + e z / D + E z})^2 ={H z^2 + I z + K / (D + E z)^2}</math>; fiat <math>4 H K = I^2</math>, & exinde deduci po$$unt diver$i
valores quantitatum <math>D, E, d, &c.</math> & con$equenter functiones quæ-
$itæ.</p>
<p>Ex. 3. Invenire; annon fluens datæ fluxionis, quæ e$t functio quan-
titatis <math>V</math> in <math>V^.</math> algebraice inveniri pote$t ope fluxionum <math>x^., {y^. / y}, {z^. / a^2 + z^2},
({a^2 - b^2 v^2 / t^2 - c^2 v^2})^{1 / 2} v^., ({a^2 + b^2 w^2 / t^2 + c^2 w^2})^{1 / 2} w^., &c.</math> i.e. finitorum terminorum,
logarithmorum, circularium, ellipticorum & hyperbolicorum arcuum,
&c. pro <math>x, y, z, v, w, &c.</math> in prædictis fluxionibus <math>x^., {y^. / y}, &c.</math> $ub$titu-
antur tales functiones quantitatis <math>V</math>; & pro <math>x^., y^., z^., v^., w^., &c.</math> barum
functionum fluxiones; quales eandem præbebunt irrationalitatem in
re$ultantibus ac in datâ fluxione: tum ex his fluxionibus generaliter
a$$umptis & $imul adjunctis detegi pote$t; annon ita a$$umi po$$unt
coefficientes fluxionum a$$umptarum, ut $umma evadat data fluxio.</p>
<p>Facile con$tat arcum hyperbolæ de$ignari po$$e per arcum impo$$i-
bilem ellip$eos; $cribatur enim <math>√ (-1) w</math> pro <math>v</math> in fluxione

<pb 84>[0136]<rh>DEINVENIENDIS</rh>
<math>({a^2 - b^2 v^2 / t^2 - c^2 v^2})^{1 / 2} v^.</math>, & re$ultat <math>({a^2 + b^2 w^2 / t^2 + c^2 w^2})^{1 / 2} × √ (- 1) w^. = √ (- 1)</math>
x fluxionem arcus prædicti hyperbolici.</p>
<p>Ex. 4. Sint tres fluxiones <math>({(a - b)^2 - x^2 / (a + b)^2 - x^2})^{1 / 2} x^. = P^.</math>, <math>({(a′ + b′)^2 - z^2 / (a′ - b′)^2 - z^2})^{1 / 2} z^.
= Q^., & ({e^2 - c y^2 / e^2 - y^2})^{1 / 2} y^. = R^.</math>; invenire, annon ita a$$umi po$$unt co-
efficientes, &c., ut fluens fluxionis <math>m P^. + n Q^. + r R^.</math>; ubi <math>m, n & r</math> $unt
invariabiles quantitates, in finitis terminis exprimi po$$it; $i modo
a$$umatur <math>a = a′, b = b′, & z = x</math>; & <math>R^.′ = 2 x^. + ({(a - b)^2 - x^2 / (a + b)^2 - x^2})^{1 / 2} x^.
+ ({(a + b)^2 - x^2 / (a - b)^2 - x^2})^{1 / 2} x^. = 2 x^. + {2 (a^2 + b^2) x^. - 2x^2 x^. / √ ((a - b)^2 - x^2) × √ ((a + b)^2 - x^2)}= T^.</math>: in hâc fluxione <math>(T^.)</math> pro <math>x</math> $cribatur <math>{a^2 - b^2 / a^2} × y ×{(a^2 - y^2)^{1 / 2} /(a^2 - {(a^2 - b^2) / a^2}y^2)^{1 / 2}}</math> & re$ultat fluxio <math>T^. = 4 × ({a^2 - {a^2 -
b^2 / a^2}y^2 /a^2 - y^2})^{1 / 2} y^.</math>
formulam datam habens.</p>
<h>THEOR. VIII.</h>
<p>Sit data fluxio <math>X x^.</math>; ita transformetur data fluxio, ut evadet <math>Z z^.</math>, ubi
<math>X & Z</math> $unt functiones quantitatum <math>x & z</math>; cum vero <math>x</math> evadat <math>α & β,
& z</math> re$pective <math>π & ξ</math>; tum fluens <math>$. X x^.</math> inter valores <math>α & β</math> quantita-
tis <math>x</math> contenta æqualis erit fluenti <math>$. Z z^.</math> inter valores <math>π & ξ</math> quantita-
tis <math>z</math>.</p>
<p>Ex. I. Sit <math>{x^m-1 x^. / (1 - x^n)^{n-k / n}}</math>; & $tatuatur <math>I - x^n = z^n</math>, ut prodeat <math>- $.{z^k-1 z^. / (1-z^n)^{n-m / n}}</math>, & erit <math>$. {x^m-1 x^. / (1 - x^n)^{n-k / n}} = $. {- z^k-1 z^. / (1 - z^n)^{n-m / n}}</math>, quarum utræque

<pb 85>[0137]<rh>FLUXIONUM FLUENTIBUS.</rh>
inter valores <math>0 & 1</math> quantitatum <math>x & z</math> continentur; cum enim <math>x</math> fiat
<math>0, z</math> evadet <math>I</math>; & vice versâ cum <math>x</math> fiat <math>I, z</math> evadet <math>0</math>.</p>
<p>Ex. 2. Sit <math>{x^k-1 x^. / 1 + x^n}</math>; $tatuatur <math>z = {1 / x}</math>, & evadet <math>{- z^n-k-1 z^. / 1 + z^n}</math>; & erit <math>$.{x^k-1 x^. / 1 + x^n} = - $. {z^n-k-1 z^. / 1 + z^n}</math>, quarum utræque fluentes inter valores <math>0</math>
& infinitum quantitatum <math>x & z</math> continentur: cum enim <math>x</math> fiat infi-
nita, <math>z</math> evadet <math>0</math>; &c.</p>
<p>Ex. 3. 1. Sit <math>x^m-1 x^. (1 - x^n)^k</math>; $cribatur <math>x = {y / (1 + y^n)^{1 / n}}</math>, & erit <math>1 - x^n ={1 / 1 + y^n}, & x^m = {y^m / (1 + y^n)^{m / n}}</math>; & exinde <math>x^m-1 x^. (1 - x^n)^k = {y^m-1 y^. / (1 + y^n)^k+1+{m / n}}</math>;
cujus fluens <math>$. x^m-1 x^. (1 - x^n)^k</math> inter valores <math>0 & 1</math> quantitatis <math>x</math> contenta
æqualis erit fluenti <math>$. {y^m-1 y^. / (1 + y^m)^k+1+{m / n}}</math> inter valores <math>0</math> & infinitum quan-
titatis <math>y</math> po$itæ.</p>
<p>2. Ponatur <math>x = (1 - z^n)^{1 / n}</math>, & exinde re$ultat <math>$. x^m-1 x^. (1 - x^n)^k = -
$. (1 - z^n)^{m-n / n} × z^kn+n-1 z^.</math>; unde hæ fluentes inter valores <math>0 & 1</math> quan-
titatum <math>x & z</math> po$itæ, evadunt æquales.</p>
<p>Pro <math>x</math> vel <math>y, &c.</math> in his & $ub$equentibus fluxionibus $cribatur <math>v</math>;
& erui po$$unt fluxiones, quarum fluentes inter prædictos valores in-
note$cunt.</p>
<h>THEOR. IX.</h>
<p>1. Sint fluxiones <math>A^., B^., C^., &c.</math>; quarum variabilis quantitas $it <math>x</math>; in
iis pro <math>x</math> $ub$tituantur quæcunque functiones literæ (<math>x</math>), & re$ultent
fluxiones <math>A^.′, A^.″, &c.</math>; <math>B^.′, B^.″, &c.</math>; <math>C^.′, &c.</math>; &c.: ducantur hæ fluxi-
ones in invariabiles quantitates <math>m, m′, &c.; n, n′, &c.; r, r′, &c.; &c.</math>;

<pb 86>[0138]<rh>DE INVENIENDIS</rh>
& $i earum $ummæ <math>m A^.′ + m′ A″ + &c. + n B^.′ + n′ B^.″ + &c. + rC′ +
&c.</math> fluens detegi po$$it per qua$cunque datas fluentes; tum ex iis
fluentibus detegi pote$t fluens functionis fluxionum <math>A^., B^., C^., &c.</math></p>
<p>In his functionibus acquirendis $æpe præ$tat; ut tales functiones
literæ <math>x</math> pro <math>x</math> in diver$is fluxionibus a$$umantur, quales prebent
fluxiones re$ultantes con$imiles irrationales quantitates involventes.</p>
<p>2. Sit <math>X</math> talis functio quantitatis <math>x</math>; qualis, $i modo in eâ <math>(X)</math> pro
<math>x</math> $cribatur etiam prædicta functio <math>X</math>, præbet quantitatem <math>(x)</math>: tum
in data fluxione <math>P x^.</math>, ubi <math>P</math> e$t functio quantitatis <math>x</math>, pro <math>x</math> $ub$titua-
tur <math>X</math>, & pro <math>x^.</math> $cribatur <math>X^.</math>; & re$ultet fluxio <math>Q x^.</math>; tum, $i in fluxione
<math>P x^. + Q x^.</math> pro <math>x</math> & ejus fluxione vel $cribatur <math>x</math> & ejus fluxio, vel <math>X</math> &
ejus fluxio; in utroque ca$u eadem re$ultabit fluxio: & con$equenter,
$i duo valores quantitatis <math>x</math> $int <math>l & m</math>, & iis corre$pondentes valores
quantitatis <math>X</math> $int re$pective <math>L & M</math>; tum (corre$pondentibus radici-
bus adhibitis) fluens fluxionis <math>P x^. + Q x^.</math> inter valores <math>l & m</math> quanti-
tatis <math>x</math>, eadem erit ac fluens eju$dem fluxionis <math>P x^. + Q x^.</math> inter va-
lores <math>L & M</math> eju$dem quantitatis <math>(x)</math> contenta.</p>
<p>Hæ quantitates <math>x & X</math> hoc modo $emper reciprocant; $i modo in
æquatione relationem inter quantitates <math>X & x</math> de$ignante, $imiliter
involvantur prædictæ quantitates <math>X & x.</math></p>
<p>Ex. I. Sit <math>a (x + X) + b X x - a b = 0</math> æquatio relationem
inter <math>X & x</math> exprimens, unde <math>x = {(b - X) a / a + b X}</math>; & fluxio <math>P x^. ={b{1 / 2} x^. x{1 / 2}/(a + {b^2 - a / b} x - x^2)^{1 / 2}}</math>; in eâ pro <math>x</math> $cribatur <math>a × {b-x / a+bx}(ob (b - x)
× ({a / b} + x) = a + {b^2 - a / b} x - x^2)</math>, & re$ultat fluxio <math>Q x^. = -{b a × (b - x)^{1 / 2} /x^{1 / 2} (a + bx)^1{1 / 2}} x^.</math> ea$dem irrationales quantitates <math>(b - x)^1 & (x + {a / b})^{1 / 2}</math>

<pb 87>[0139]<rh>FLUXIONUM FLUENTIBUS.</rh>
tantummodo continens: fluens $ummæ fluxionum <math>{1 / 2} P x^. + {1 / 2} Q x^. ={{1 / 2} b^{1 / 2} x^{1 / 2} x^. /(a + {b^2 - a / b} x - x^2)^{1 / 2}} - {^{1 / 2} b a (b - x)^{1 / 2} x^./x^{1 / 2} (a + bx)^1{1 / 2}} erit - b × {(b - x)^{1 / 2} × x^{1 / 2} /(a + b x)^{1 / 2}}= L</math>.</p>
<p>Fluens fluxionis <math>(P + Q) x^.</math> inter valores <math>α & β</math> quantitatis <math>x</math>
eadem erit ac fluens eju$dem fluxionis inter valores <math>a × {b - α / a + b α} &
a × {b - β / a + bβ}</math> quantitatis <math>x</math> contenta.</p>
<p>Cor. Si in fluxione <math>{x - ^{1 / 2} x^. /(a + {b^2 - a / b} x - x^2)^{1 / 2}} = R^.</math> pro <math>x</math> $cribatur <math>a ×{b - x / a + b x} = z</math>, re$ultabit <math>- R^.</math>; deinde $umma <math>P x^. + {a / b^{1 / 2}} R^. = +{b{1 / 2} x{1 / 2} x^. /(a + {b^2 - a / b}x - x^2)^{1 / 2}} + {a b-^{1 / 2} x - ^{1 / 2} x^. /(a + ({b^2 - a / b}) x - x^2)^{1 / 2}} = {(b x + a) × x - ^{1 / 2} x^./b^{1 / 2} × (a + {b^2 - a / b}x
- x^2)^{1 / 2}}= b^{1 / 2} x - ^{1 / 2} x^. × ({x + {a / b}/b - x})^{1 / 2}</math>; tum in $ummâ <math>P x^. + {a / b^{1 / 2}} R^. = b^{1 / 2} x - ^{1 / 2} x^. ×
({x + {a / b}/b - x})^{1 / 2}</math> pro <math>x</math> $cribatur <math>a × {b - x / a + b x} = z</math>, & re$ultabit quantitas
<math>Q x^. - {a / b^{1 / 2}} R^. = b^{1 / 2} z - ^{1 / 2} z^. × ({z + {a / b}/b-z})^{1 / 2}</math>; unde <math>b^{1 / 2} x - ^{1 / 2} x^. × ({x + {a / b}/b-x})^{1 / 2} +
b^{1 / 2} z - ^{1 / 2} z^. × ({z + {a / b}/b - z})^{1 / 2} = P x^. + {a / b^{1 / 2}} R^. + Q x^. - {a / b^{1 / 2}} R^. = P x^. + Q x^. =</math>
fluxioni, cujus fluens e$t 2<math>L</math>.</p>

<pb 88>[0140]<rh>DE INVENIENDIS</rh>
<p>Ex. 2. Sit fluxio <math>{x^m-2 x^. / (a - x^m)^{1 / m}}</math>; a$$umatur æquatio <math>x^m + z^m = a</math>, in quâ
literæ <math>x & z</math> $imiliter involvuntur; unde re$ultabit con$imilis fluxio
<math>{- z^m-2 z^. / (a - z^m)^{1 / m}}</math>; & con$equenter fluens prædictæ fluxionis inter duos va-
lores <math>k & l</math> quantitatis <math>x</math>, æqualis erit fluenti eju$dem fluxionis inter
valores <math>(a - k^m)^{1 / m} & (a - l^m)^{1 / m}</math> quantitatis <math>(x)</math> contenta.</p>
<h>PROB. XXII.</h>
<h it>Datis fluxionibus, quarum fluentes cogno$cuntur; invenire utrum datœ
fluxionis fluens earum ope inveniri pote$t, necne.</h>
<p>Ducantur $ingulæ datæ fluxiones in incognitas & invariabiles co-
efficientes, & fluxiones re$ultantes addantur vel detrahantur de datâ
fluxione; re$ultantis fluxionis inveniatur per prob. 4. fluens, i.e. per
prob. 4. con$tant dimen$iones ultimi quæ$itæ fluentis termini; a$$u-
matur igitur pro fluente quæ$itâ quantitas, quæ nece$$ario continet
fluentem re$ultantis fluxionis, $i modo ea in finitis algebraicis termi-
nis exprimi po$$it; a$$umptæ quantitatis inveniatur fluxio, quæ fiat
æqualis prædictæ re$ultanti fluxioni, $i modo fieri po$$it; & confit
problema.</p>
<p>Ex. 1. Datâ fluente fluxionis <math>(a + b x^n)^m x^.</math>; invenire fluentem flu-
xionis <math>(a + b x^n)^m x^n x^.</math>: ducatur data fluxio <math>(a + b x^n)^m x^.</math> in coefficien-
tem incognitam & invariabilem <math>A</math>, & re$ultat <math>A × (a + b x^n)^m x^.</math>; ad-
datur hæc re$ultans fluxio ad fluxionem, cujus fluens requiritur, &
fit <math>(A + x^n) × (a + b x^n)^m x^.</math>; $ed per prob. 4. fluens huju$ce fluxionis
hanc habet formulam <math>B × x × (a + b x^n)^m+1</math>; a$$umatur igitur hæc
quantitas pro quæ$itâ fluente, & ejus fluxio e$t <math>(B a + (b B +
(m + 1) n B b) x^n) × (a + b x^n)^m x^.</math>; fiat hæc fluxio eadem ac prædicta
fluxio, i.e. <math>B a = A, & (b B + (m + 1)n B b) x^n = x^n</math>; unde <math>B =

<pb 89>[0141]<rh>FLUXIONUM FLUENTIBUS.</rh>
{1 / b + (m + 1) n b}, & A = {a / b + (m + 1) n b}</math>; & fluens quæ$ita erit <math>B x ×
(a + b x^n)^m+1 - A ×</math> fluen. flux. <math>(a + b x^n)^m x^.</math>. Cum <math>1 + (m + 1) ×
n = 0</math>; tum ex datâ fluente haud deduci pote$t fluens quæ$ita.</p>
<p>Ex. 2. Sit fluxio <math>(a + b x^n)^m x^pn-1 x^.</math>, cujus fluens <math>V</math> datur; & fluens
fluxionis <math>(a + b x^n)^m × x^p n+n-1 x^.</math> invenietur <math>{(a + b x^n)^m+1 × x^pn / (m + p + 1) × n b} -{p a V / (m + p + 1) × b}</math>. Cum <math>m + p + 1 = 0</math>, tum etiam ex datâ fluente
haud acquiri pote$t quæ$ita.</p>
<p>In hoc & $ub$equentibus exemplis litera (<math>p</math>) denotat vel integrum
numerum vel fractionem vel denique quamcunque invariabilem
quantitatem.</p>
<p>Ex. 3. Datâ fluente fluxionis <math>(a + b x^n)^m × x^pn-1 x^.</math>; invenire fluentem
fluxionis <math>(a + b x^n)^m × x^pn+vn-1 x^.</math>; per prob. ducatur fluxio <math>(a + b x^n)^m
x^pn-1 x^.</math> in incognitam coefficientem <math>h</math>, & $ubducatur quantitas re$ul-
tans de datâ fluxione <math>(a + b x^n)^m x^pn+vn-1 x^.</math>, cujus fluens requiritur; &
exinde inveniatur fluens differentiæ <math>(a + b x^n)^m × (x^(p+v)n-1 - h x^pn-1)
× x^.</math>, ubi <math>h</math> e$t quantitas a$$umenda, ita ut $eries terminet: pro fluente
quæ$itâ a$$umatur quantitas <math>(a + b x^n)^m+1 × (A × x^(p+v-1)n + B x^(p+v-z)n
+ C x^(p+v-3)n + &c. L x^pn)</math>, cujus fluxio invenietur <math>(a + b x^n)^m × ((p +
v + m) × n b A x^(p+v)n-1 + ((p + v + m - 1) × n B b + (p + v - 1)
n A a) x^(p+v-1)n-1 + ((p + v + m - 2) n C b + (p + v - 2) n B a)
x^(p+v-2)n-1 + &c.) × x^.</math>; fiat hæc fluxio æqualis fluxioni <math>(a + b x^n)^m ×
(x^(p+v)n-1 - h x^pn-1) x^.</math>: i. e. <math>(p + v + m) × n b A = 1, & (p + v + m
- 1) × n B b + (p + v - 1) n A a = 0, &c.</math> unde <math>A = {1 / (p + v + m) × n b}</math>,
<math>B = - {(p + v - 1) × n a A / (p + v + m - 1) × n b} = {-(p + v - 1) a / (p + v + m) × (p + v + m - 1) n b^2}</math>,
& $ic deinceps: lex progre$$ionis hujus $eriei, $i modo $cribantur <math>p +
v = q & q + m = r</math>, erit <math>(a + b x^n)^m+1 × ({1 / r × n b} x^(q-1)n - {(q - 1) a A / (r - 1) b}

<pb 90>[0142]<rh>DE INVENIENDIS</rh>
x^(q-2)n + {(q - 2) a B / (r - 2) b} x^(q-3)n - {(q - 3) a C / (r - 3) b} x^(q-4)n + &c.</math> ad <math>v</math> terminos)
<math>± {p / p + m + 1} × {p + 1 / p + m + 2} × {p + 2 / p + m + 3}...{p + v - 1 / p + m + v} × {a^v / b^v} ×</math> fluen.
flux. <math>(a + b x^n)^m x^pn-1 x^.</math>: $ignum affixum erit -, $i <math>v</math> $it impar; $in
aliter +. Cum <math>p + m + l = 0</math>, ubi <math>l</math> e$t quicunque integer numerus
haud major quam <math>v</math>; tum ex datâ fluente non acquiri pote$t quæ-
$ita.</p>
<p>Ex. 4. Datâ prædictâ fluxione <math>(a + b x^n)^m x^pn-1 x^.</math>, invenire fluentem
fluxionis <math>(a + b x^n)^m × x^pn-vn-1 x^.</math>: a$$umatur <math>(a + b x^n)^m+1 × (A x^(p-v)n
+ B x^(p-v+1)n + C x^(p-v+2)n + &c.)</math> pro fluente quæ$itâ; $cribantur
<math>q = p - v - 1, s = m + q, t = p + m + 1</math>; fluens quæ$ita per me-
thodum prius traditam invenietur <math>(a + b x^n)^m+1 × ({1 / (q + 1) × n a} x^(p-v)n
- {(s + 2) b A / (q + 2) a} × x^(p-v+1)n + {(s + 3) b B / (q + 3) a} x^(p-v+2)n - {(s + 4) b C / (q + 4) a} x^(p-v+3)n
+ &c. ad <math>v</math> terminos) = {t - 1 / p - 1} × {t - 2 / p - 2} × {t - 3 / p - 3}..{t - v / p -v} × {b^v / a^v} ×
fluen. flux. (a + b x^n)^m x^pn-1 x^.</math>: $ignum affixum erit +, $i <math>v</math> $it par, $in
aliter -. Cum <math>p - v + l = 0</math>, ubi <math>l</math> e$t quicunque integer nume-
rus haud major quam <math>v - 1</math>; tum ex datâ fluente haud detegi pote$t
quæ$ita.</p>
<p>Ex. 5. Datâ fluente fluxionis <math>(a + b x^n)^m × x^pn-1 x^.</math>; invenire fluentem
fluxionis <math>(a + b x^n)^m+r × x^pn-1 x^.</math>, ubi <math>r</math> e$t integer po$itivus numerus:
$cribatur <math>a + b x^n = Q</math>; fluens quæ$ita erit <math>Q^m+1 × x^pn × ({Q^r-1 / (p + m + r) × n}+ {(m + r) a Q^r-2 / (p + m + r) × (p + m + r - 1) n} + {(m + r) × (m + r - 1) a^2 Q^r-3 / (p + m + r) × (p + m + r - 1)
× (p + m + r - 2)n} + &c. ad r terminos) + {m + 1 / p + m + 1} × {m + 2 / p + m + 2} ×{m + 3 / p + m + 3} ... {m + r / p + m + r} a^r × fluent</math>. fluxionis <math>(a + b x^n)^m x^pn-1 x^.</math>.</p>

<pb 91>[0143]<rh>FLUXIONUM FLUENTIBUS.</rh>
<p>Ex. 6. Ex ii$dem principiis e prædictâ fluente datâ inveniri pote$t
fluens fluxionis <math>(a + b x^n)^m-r × x^pn-1 x^.</math> e$$e <math>Q^m+1 × x^pn × ({-Q^-r / (m + r + 1) × n a}- {(p + m - r + 1) Q^1-r / (m - r + 1) × (m - r + 2) n a^2} - {(p + m - r + 1) × (p + m - r + 2) Q^2-r / (m - r + 1) × (m - r + 2) × (m - r + 3) n a^3}- &c.</math> ad <math>r</math> terminos) <math>+ {p + m / m} × {p + m - 1 / m - 1} × {p + m - 2 / m - 3} ... {p + m - r + 1 / m - r + 1}× {1 / a^r} × fluent. flux. (a + b x^n)^m × x^pn-1 x^..</math> Cum <math>m</math> $it integer numerus,
& minor quam <math>r</math>; tum ex datâ fluente non detegi pote$t quæ$ita.</p>
<p>Et in genere in omnibus $ub$equentibus ca$ibus ex datâ fluente non
detegi pote$t quæ$ita, cum ullus factor in denominatore nihilo evadat
æqualis.</p>
<p>Ex hi$ce $eriebus conjunctim deducendis con$equitur fluentem
fluxionis <math>(a + b x^n)^m+r × x^pn+vn-1 x^.</math> e$$e <math>{Q^m+r × x^(p+v)n / (p + m + v + r) × n} +{(m + r) × a A / (p + m + v + r - 1) × Q} + {(m + r - 1) × a B / (p + m + v + r - 2) × Q} + {(m + r - 2) a C / (p + m + v + r - 3)Q}</math>
ad <math>r</math> terminos <math>+ {m + 1 / m + p + v} × {a H / b x^n} - {p + v - 1 / m + p + v - 1} × {a I / b x^n} - {p + v - 2 / m + p + v - 2}× {a K / b x^n}</math> ad <math>v</math> terminos <math>± {(m + 1) × (m + 2) × (m + 3) .. (m + r)
× p × (p + 1) × (p + 2) .. (p + v - 1) / (p + v + m + 1) · (p + v + m + 2) ... (p + v + m + r)
× (p + m + 1) · (p + m + 2) .. (p + m + v)} × {a^v+r / b^v} × fluen</math>. flux. <math>(a + bx^n)^m
x^pn-1 x^.</math>: ubi literæ <math>A, B, C, &c. H, I, K, &c.</math> re$pective denotant præ-
cedentes terminos.</p>
<p>2. Eodem modo deduci pote$t fluens fluxionis <math>(a + bx^n)^m-r × x^pn+vn-1 x^.
= -{Q^m-r+1 x^(p+v)n / (m - r + 1) na} + {(p + m + v - r + 1) × Q A / (m - r + 2) × a} + {(p + m + v - r + 2) × Q B / (m - r + 3) × a
ad r terminos - {QH / bx^n} - {p + v - 1 / m + p + v - 1} × {a I / bx^n} - {p + v - 2 / m + p + v - 2} × {a K / b x^n}</math> ad

<pb 92>[0144]<rh>DE INVENIENDIS</rh>
<math>v</math> terminos <math>+ {m + p + v / m} × {m + p + v - 1 / m - 1} ... {m + p + v - r + 1 / m - r + 1} ×{p / p + m + 1} × {p + 1 / p + m + 2} × {p + 2 / p + m + 3} .. {p + v - 1 / p + m + v} × {a^v-r / (-b)^v} × fluen.</math> flux.
<math>(a + b x^n)^m x^pn-1 x^.</math>, ubi literæ <math>A, B, C, D, &c. H, I, K, &c.</math> etiam præ-
cedentes terminos re$pective denotant.</p>
<p>3. Fluens fluxionis <math>(a + bx^n)^m+r × x^(p-v)xn-1 x^.</math> invenietur <math>Q^m+r+1 × x^(p-v)n /(p - v) n a}- {m + p - v + r + 1 / (p - v + 1)} × {A b x^n / a} - {m + r + p - v + 2 / p - v + 2} × {B b x^n / a}</math> ad <math>v</math>
terminos <math>- {H b x^n / Q} + {m + r / p + m + r - 1} × {aI / Q} + {m + r - 1 / p + m + r - 2} × {a K / Q} +
&c. ad <math>r</math> terminos + {m + r + p - v + 1 / p - v} × {m + r + p - v + 2 / p - v + 1} ...{m + r + p / p - 1} × {m + 1 / m + p + 1} × {m + 2 / m + p + 2} ... {m + r / m + p + r} × {(- b)^v / a^v-r} × fluent</math>.
fluxionis <math>(a + b x^n)^m x^pn-1 x^.</math>: literis <math>A, B, C, &c. H, I, K, &c.</math> præce-
dentes terminos re$pective denotantibus.</p>
<p>4. Fluens fluxionis <math>(a + b x^n)^m-r × x^(p-v)n-1 x^.</math> invenietur <math>= -{Q^m-r+1 x^(p-v)n / (m - r + 1) n a} + {m + p - r - v + 1 / m - r + 2} × {Q A / a} + {m + p - r - v + 2 / m - r + 3} ×{Q B / a}</math> ad <math>r</math> terminos <math>- {m + p - v / p - v} × {Q H / a} - {m + p - v + 1 / p - v + 1} × {I b x^n / a}- {m + p - v + 2 / p - v + 2} × {K b x^n / a}</math> ad <math>v</math> terminos <math>+ {(p + m).(p + m - 1) / m × (m - 1) × (m - 2)..}{(p + m - 2)..(p + m - r - v + 1) / (m - r + 1) × (p - 1) × (p - 2)..(p - v)} × {(-b)^v / a^r+v} × fluent. flux. (a + b x^n)^m
x^pn-1 x^.</math>, literis <math>A, B, C, &c. H, I, K, &c.</math> præcedentes terminos re$pe-
ctive denotantibus.</p>
<p>Et $ic ad trinomiales, &c. terminos progredi liceat.</p>
<p>Ca$. 2. Datis fluentibus fluxionum <math>(a + b x^n + c x^2n + d x^3n + &c.)^m

<pb 93>[0145]<rh>FLUXIONUM FLUENTIBUS.</rh>
× x^pn-1 x^., (a + b x^n + c x^2n + d x^3n + &c.)^m × x^pn+n-1 x^., (a + b x^n + c x^2n
+ d x^3n + &c.)^m × x^pn+2n-1 x^.</math>, quarum numerus e$t <math>r</math>, $i modo <math>r</math> $it
numerus terminorum in quantitate <math>a + b x^n + c x^2n + &c.</math> conten-
torum: $cribantur literæ <math>A, B, C, D, &c.</math> pro prædictis fluentibus, &
erit <math>(a + b x^n + c x^2n + d x^3n + &c.)^m+1 × x^pn = p n a A + (p + m + 1)
n b B + (p + 2m + 2)ncC + (p + 3m + 3) n d D + &c.</math> Hoc fa-
cile demon$trari pote$t, ex inveniendo utriu$que æquationis partis
fluxiones; hæ fluxiones enim evadunt eædem.</p>
<p>Si igitur dentur quæcunque <math>r - 1</math> prædictæ fluentes, ex hâc æqua-
tione $equitur reliqua.</p>
<p>Sit terminus in prædictâ $erie <math>(p + l m + l) × F</math>, ubi <math>l</math> e$t integer
numerus, & <math>p + l m + l = 0</math>; cujus fluens <math>F</math> e$t reliqua; tum ex præ-
dictis <math>(r - 1)</math> fluentibus non deduci pote$t reliqua <math>(F)</math>.</p>
<p>Ca$. 2. Ii$dem datis; $cribatur <math>Q</math> pro fluente fluxionis <math>(a + b x^n
+ c x^2n + d x^3n + &c.)^m+1 x^pn-1 x^.</math>, & erit <math>(a + bx^n + c x^2n + d x^3n + &c.)^m+1
x^pn = (m + 1) × (n b B + 2 n c C + 3 n d D + &c.) + p n Q</math>; unde $i
modo dentur <math>r - 1</math> fluentes <math>B, C, D, &c.</math> dabitur etiam fluens <math>Q</math>.</p>
<p>Cor. Si fluxio con$tet ex <math>r</math> nominibus vel terminis in vinculo
radicali, & dentur <math>r - 1</math> fluentes prædicti generis; dabuntur etiam
omnes fluentes fluxionum, quæ addendo vel $ubducendo quantitates
<math>n, 2 n, 3 n, 4 n</math>, &c. de indice <math>p n - 1</math>; vel unitatem, duo, tres, qua-
tuor, &c.; de indice <math>m</math>; generari po$$unt.</p>
<p>Excipiendi $unt nonnulli ca$us, ut e $ub$equentibus con$tabit.</p>
<p>Ca$. 3. Si pro <math>e + f x^n + g x^2n + &c. · x^αn, & b + k x^n + l x^2n + &c..
x^βn</math>, <math>q + r x^n + s x^2n + &c..x^2n, &c.</math> $cribantur <math>R, S, T, &c.</math> & in flu-
xione <math>x^p n±σ n-1 x^. R^m±λ S^0±μ T^t±ν × &c.</math> maneant quantitates datæ
<math>p n - 1, n, m, o, t, e, f, g, h, k, l, q, r, s, &c.</math> & pro <math>σ, λ, μ, ν, &c.</math> $cri-
bantur in datâ fluxione $ucce$$ive quicunque integri numeri; & $i
dentur <math>α + β + γ + &c.</math> fluentes independentes re$ultantium fluxio-
num, ubi literæ <math>α + 1, β + 1, γ + 1, &c.</math> re$pective denotant nume-
rum terminorum in quantitatibus <math>R, S, T, &c.</math> contentorum; tum
dabuntur fluentes omnium prædictarum fluxionum re$ultantium.</p>
<p>Con$tat ex principiis in priori exemplo traditis:</p>

<pb 94>[0146]<rh>DE INVENIENDIS</rh>
<p>Ca$. 4. Datis $ingulis fluentibus fluxionum <math><r l>a + b x^n + c x^2n + &c. +
<ol>e + f x<_>n</_> + g x<_>2n</_> + &c.</ol>^ m</r> × x<_>θ±rn-1</_> x^.</math>, ubi <math>r</math> integrum denotat nume-
rum; datis etiam fluentibus <math>(s - 1)</math> fluxionum huju$ce formulæ
<math><r l>a + b x^n + c x^2n + &c. + <ol>e + f x^n + g x^2n + &c.</ol>^m</r> × <r m ± t>e + f x^n + g x^2n + &c.</r>
× x^θ±vn-1 x^.</math>, ubi <math>s</math> $it numerus terminorum in quantitate <math>e + f x^n +
g x^2n + &c.</math> contentorum, <math>& t & v</math> integros denotent numeros; exinde
inveniri po$$unt fluentes $ingularum fluxionum huju$ce formulæ</p>
<p>Eadem etiam principia facile applicari po$$unt ad omnes algebrai-
cas fluxiones utcunque compo$itas.</p>
<p>5. 1. Sint <math>a + b x^n + c x^2n + &c. = R, d + e x^n + f x^2n + &c. = S,
i + b x^n + k x^2n + &c. = T, &c.</math> tum per methodos in prob. 4. tra-
ditas ita reducatur data fluxio <math>(A + B x^n + C x^2n + &c.) × R^λ-1 ×
S^μ-1 × T^γ-1 × &c.</math>, ut in eâ quantitates prædictæ (<math>R, S, T, &c.)</math> nullum
habeant communem divi$orem, & in nullâ quantitate contineantur
plures divi$ores inter $e æquales; aliter generalis fluens prædictæ flu-
xionis ex paucioribus fluentibus independentium fluxionum, quam
ex numero hâc regulâ dato, detegi pote$t. e. g. Sit <math>R = a + b x^n +
c x^2n + ... x^pn = (α + β x^n)^l × (a′ + b′ x^n + &c.); S = d + e x^n + f x^2n
+ &c. ... x^qn = (α + β x^n)^b × (γ + δ x^n)^k × (d′ + e′ x^n + &c.)</math>; &c.:
& data fluxio <math>R^λ × S^μ × x^θ x^. = (α + β x^n)^λl+μb × (γ + δ x^n)^μk × (a′ + b′ x^n
.. x^(p-ηn)^λ × (d′ + e′ x^n .. x^(q-b-k)n)^μ × x^θ x^.</math>; per regulam datam ex <math>(p + q)</math>
fluentibus formulæ <math>R^λ±π × S^μ±ρ × x^θ±σ × x^.</math> inter $e independentibus
ubi literæ <math>π, ς & σ</math> quo$cunque integros denotant numeros, deduci
po$$unt omnes aliæ eju$dem formulæ at quoniam <math>R^λ ± π × S^μ±ρ × x^ς±σ
× x^. = (α + β x^n)^(λ±π)l+(μ±ρ)h × (γ + δ x^n)^(μ±ρ)k × (a′ + b′ x^n .. x^(p-1)n)^λ±π ×
(d′ + e′ x^n .. x^(q-b-k)n)^μ±ρ × x^θ±σ x^.</math>, exinde con$tat omnes fluentes præ-
dictæ formulæ deduci po$$e ex <math>(p + q - l - b - k + 2)</math> fluentibus
eju$dem formulæ inter $e independentibus.</p>
<p>2. Sit <math>R^m × S^σ × T^t × x^p n-1</math> quantitas ita reducta, ut quantitates <math>R =
e + f x^n .. + x^αn, S = b + k x^n .. + x^βn, T = q + r x^n + s x^2n .. + x^λn,

<pb 95>[0147]<rh>FLUXIONUM FLUENTIBUS.</rh>
&c., & x^p n-1</math> nullum habeant inter $e communem divi$orem; etiam-
que $ingulæ nullum habeant divi$orem, qui e$t quadratus <math>(x + a)^2</math>,
vel cubus <math>(x + a)^3</math>, &c.: tum raro ex paucioribus quam <math>(α + β + γ
+ &c.)</math> fluentibus fluxionum inter $e independentibus, formulæ
<math>R^m±λ × S^ο±μ × T^t±γ × x p^n±σ n-1 × x^. × (a + b x^n + c x^2n + &c.)</math>, ubi
literæ <math>λ, μ, ν, &c., & σ</math> integros denotant numeros, $emper deduci
po$$unt fluentes omnium fluxionum eju$dem $ormulæ.</p>
<p>Huc etiam applicari po$$unt con$imilia iis, quæ in theor. præce-
dentibus traduntur.</p>
<p>6. Sint <math>(a + b x^n + c x^2n + &c .. x^rn)^m+λ × x^p n±σ n-1 x^.</math>, ubi <math>σ</math> e$t qui-
cunque integer, <math>& m & λ</math> quicunque integri affirmativi numeri; tum
ejus fluens $emper vel inveniri pote$t, vel $olummodo exigit fluentem
fluxionis <math>{x^. / x}</math>.</p>
<p>Con$tat ex reducendo fluxionem datam in $eriem $ecundum di-
men$iones quantitatis <math>x</math> progredientem.</p>
<p>7. Si vero <math>m</math> affirmativa quantitas haud $it integer numerus; at <math>p,
λ & σ</math> $int integri affirmativi numeri; & tum ex <math>(r - 1)</math> fluentibus
fluxionum formulæ <math>(a + b x^n + c x^2n + .. + x^rn)^m+λ × x^p n+σ n-1 x^.</math> datis,
deduci
dictis <math>(r - 1)</math> fluentibus non deduci po$$unt omnes fluentes formulæ
<math>(a + b x^n + c x^2n + .. + x^rn)^m±λ × x^p n±σ n-1 x^.</math>; $ed ex <math>(r)</math> fluentibus in-
ter $e independentibus, i. e. quæ nec in finitis terminis, nec a $e in-
vicem exprimi po$$unt, formulæ <math>(a + b x^n + c x^2n + .. + λ^rn)^m±λ ×
x^pn±σ n-1 x^.</math> deduci po$$unt omnes fluentes eju$dem formulæ: etiam-
que ex omnibus, i.e. infinitis fluentibus fluxionum formulæ <math>(a + b x^n
+ c x^2n ... + x^rn)^m+λ × x^pn+σ n-1 x^.</math> non deduci po$$unt fluentes omnium
fluxionum formulæ <math>(a + b x^n + c x^2 n ... + x^r n)^m±r × x^p n±σ n-1 x^.</math>.</p>
<p>Literis <math>R, S, T, &c.</math> ea$dem quantitates ac in ca$. 5. 2. denotanti-
bus; $int <math>m, &c.</math> integri affirmativi numeri; & <math>o, t, &c.</math>, non integri
numeri; tum fluens omnis fluxionis formulæ <math>x^pn±σ n-1 x^. × R^m+λ ×
S^σ±μ × T^t±r × &c.</math>, ubi <math>λ</math> e$t affirmativus numerus, colligi pote$t ex

<pb 96>[0148]<rh>DE INVENIENDIS</rh>
<math>(β + γ + &c.)</math> diver$is fluentibus prædictæ formulæ inter $e indepen-
dentibus: 2. etiamque ex $ingulis, viz. infinitis fluentibus formulæ
<math>R^m+λ × S^ο±μ × T^t±r × &c. × x^pn±σ n-1 x^.</math> non colligi pote$t fluens omnis
fluxionis formulæ <math>R^m-λ × S^ο±μ × T^t±r × &c. × x^pn±σ n-1 x^.</math>.</p>
<p>8. Sit <math>p</math> integer affirmativus numerus, tum ex <math>(α + β + γ + &c.
- 1)</math> fluentibus inter $e independentibus formulæ <math>x^pn+σ n-1 × R^m+λ ×
S^ο+μ × T^t+r × &c. × x^.</math>; ubi literæ <math>σ, λ, μ, ν, &c.</math> $unt affirmativi numeri,
<math>& m, o, t, &c.</math> affirmativæ quantitates, deduci po$$unt fluentes omnium
fluxionum eju$dem formulæ; at ex fluentibus omnium fluxionum
huju$ce formulæ non deduci po$$unt fluentes omnium fluxionum for-
mularum, in quibus prædicti numeri <math>σ, λ, μ, ν, &c.</math> $unt negativi.</p>
<h>THEOR. X.</h>
<p>Sæpe per $ub$titutionem acquiri po$$unt particulares formulæ ge-
neralis fluxionis, quarum fluentes e paucioribus inter $e independen-
tibus quam per regulam a$$ignatis, deduci po$$unt; e. g. $it fluxio
<math>(a + b x^n)^λ±α x^(θ±β)n x^n-1 x^.</math>, quæ ex unâ datâ fluente huju$ce formulæ
acquiri pote$t, ubi literæ <math>α & β</math> integros denotant numeros: in eâ pro
<math>x^n</math> $cribatur <math>c + d z^n + e z^2n</math>, & re$ultat fluxio <math>(a + b c + b d z^n +
b e z^2n)^λ±α × (c + d z^n + e z^2n)^θ±β x^. (d z^n-1 + 2 e z^2n-1)z^.</math>: unde fluentes
omnium fluxionum huju$ce po$terioris formulæ erui po$$unt ex
fluente unius datæ fluxionis eju$dem formulæ</p>
<h>THEOR. XI</h>
<p>1. Sit <math>x^θ+αn+βm × (a + b x^n + c x^m)^λ+π x^.</math>; ubi literæ <math>α, β & π</math> re$pe-
ctive denotant affirmativos integros numeros: tum ex datis fluentibus
<math>(N + 1)</math> inter $e independentibus fluxionum huju$ce formulæ, ubi
$umma <math>α + β + π</math> nunquam major e$t quam <math>N</math>; detegi po$$unt flu-
entes omnium fluxionum prædictæ formulæ.</p>
<p>Hoc theorema ex ii$dem principiis; ac ea, quæ in prob. 22.
traduntur; deduci pote$t: e. g. $int <math>x^θ (a + b x^n + c x^m)^λ x^. = x^θ A^λ x^.

<pb 97>[0149]<rh>FLUXIONUM FLUENTIBUS.</rh>
= P^., x^θ+n × A^λ x^. = Q^.</math>; tum erit <math>$. x^θ+m A^λ x^. = {1 / (θ + 1) c + (λ + 1) mc}(x^θ+1 × A^λ+1 - (θ + 1) a P - ((λ + 1)n b + (θ + 1)b)Q)</math>; & $ic de-
inceps.</p>
<p>2. Sint <math>n & m</math> integri numeri, & <math>N</math> eorum maximus communis di-
vi$or; etiamque $it <math>v</math> major numerus <math>{n / N}</math> vel <math>{m / N}</math>; tum ex <math>v</math> independen-
tibus fluentibus fluxionum formulæ <math>x^θ±α n±β m (a + b x^n + c x^m)^λ±π x^.</math>
datis $emper acquiri po$$unt fluentes omnium fluxionum eju$dem for-
mulæ viz. <math>x^θ±α n±β m (a + b x^n + c x^m)^λ±π x^.</math>.</p>
<p>2. 2. Si vero <math>n & m</math> $int fractiones, &c.; reducendæ $unt ad mini-
mum communem denominatorem <math>(σ)</math>, & evadant <math>{μ / σ} & {μ′ / σ}</math>; inveniatur
<math>(N)</math> maximus communis divi$or numeratorum <math>μ & μ′</math>, & $it <math>v</math> major
numerus <math>{μ / N}</math> vel <math>{μ′ / N}</math>; tum ex <math>v</math> independentibus fluentibus fluxionum
prædictæ formulæ $emper deduci po$$unt fluentes omnium fluxionum
eju$dem formulæ</p>
<p>3. Sit fluxio <math>x^θ+α m+β n+λ r+δ s+&c. × (a + b x^m + c x^n + d x^r + e x^s +
&c.)^λ+π x^.</math>, in quâ <math>α, β, γ, δ, &c. & π</math> reperiuntur integri affirmativi nu-
meri: $it <math>α + β + λ + &c. + π</math> numerus non major quam <math>b</math>; tum
fluentes omnium fluxionum huju$ce formulæ inveniri po$$unt ex <math>s ×{s + 1 / 2} × {s + 2 / 3} × {s + 3 / 4} .. {s + b - 1 / b}</math> inter $e independentibus fluenti-
bus fluxionum eju$dem formulæ: ubi <math>s</math> e$t numerus indicum <math>m, n,
r, s, &c.</math></p>
<p>Con$imiles propo$itiones de fluentibus fluxionum formulæ
<math>x^θ+α n+β m+&c. (a + b x^n + c x^2n + .. + e x^δ n + k x^m + l x^2m + .. + x^rm +
p x^n+m + q x^n+2m + r x^m+2n + &c.)^λ+π x^.</math>, ubi literæ <math>α, β, &c., δ, ν, &c.; &
π</math> re$pective integros denotant affirmativos numeros; facile deduci
po$$unt.</p>

<pb 98>[0150]<rh>DE INVENIENDIS</rh>
<h>THEOR. XII.</h>
<p>1. Sit fluxio <math>X $. γ x^.</math>, ubi <math>X^.</math> e$t fluxio, cujus fluens $emper inveniri
pote$t; & $it numerus fluentium inter $e independentium in formulis
fluxionum <math>γ x^. & X γ x^.</math> contentarum, re$pective <math>m & r</math>; e quibus detegi
po$$unt omnes reliquæ earundem for mularum; tum ex <math>(m + r)</math> flu-
entibus fluxionum <math>X^. $. γ x^.</math> inter $e independentibus detegi po$$unt
omnes fluentes eju$dem formulæ</p>
<p>Con$tat ex eo, quod fluens datæ fluxionis <math>= X $. γ x^. - $. X y x^.</math>; &
$i modo dentur <math>m</math> independentes fluentes fluxionis <math>γ x^.</math>, per hypothe$in
dabuntur omnes; & $imiliter $i modo dentur (<math>r</math>) fluentes indepen-
dentes fluxionis <math>X y x^.</math>, dabuntur omnes ejus formulæ</p>
<p>Ex. Sit fluxio <math>(A + B x^n + C x^2n + &c.) × x^θ±σ n-1 x^. × $. R^m±λ ×
S^ο±μ × T^t±τ × &c. × x^x±τ π-1 x^.</math>, ubi literæ <math>σ, λ, μ, ν, &c. & τ</math> quo$cun-
que integros denotant numeros; & $umma <math>θ ± σ n</math>, vel <math>θ ± σ n + n</math>,
vel <math>θ ± σ n + 2 n</math>, vel <math>θ ± σ n + 3 n</math>, &c., non evadat nihilo æqualis;
etiamque <math>R = a + b x^n + c x^2n + .. x^αn, S = b + k x^n + l x^2n + .. x^β n,
T = q + r x^n + s x^2n + .. + x^γ n</math>, &c.; tum erit <math>X = {1 / θ ± σ n} A x^θ±σ n
+ {1 / θ ± σ n + n} B x^θ±σ n+n + {1 / θ ± σ n + 2n} C x^θ±σ n+2n + &c.; & γ =
R^m±λ × S^ο±μ × T^t±r × &c. × x^x±τ n-1</math>: numeri independentium fluen-
tium in formulis fluxionum <math>γ x^. & X γ x^.</math> contentarum erunt re$pective
<math>α + β + γ + &c., & α + β + γ + &c.</math>; ergo numerus fluentium inter
$e independentium, quarum fluxiones datam habent formulam,
non pote$t e$$e major quam <math>2(α + β + γ + &c.)</math>.</p>
<p>Si <math>x = θ ± l n</math>, ubi <math>l</math> e$t integer numerus; tum prædictus nume-
rus non pote$t e$$e major quam <math>α + β + γ + &c.</math></p>
<p>2. Sit fluxio <math>r x^. = s^.</math>, ubi <math>s = (a + b x^n + c x^2n ... b x^αn)^± π × (p + q x^n
+ r x^2n ... t x^β n)^m±p × x^k±a x^.</math>; a$$umatur <math>y = (A + B x^n + C x^2n ...
H x^δ n)^L±τ × (P + Q x^n + R x^2 n ... T x^ε n)^m′±v × x^k′±ξ</math>; ubi <math>π, ς, σ, τ, ν, ξ</math>
$unt quicunque integri numeri; tum ex datis fluentibus <math>δ + ε + &c.</math>
fluxionum formulæ <math>y x^.</math> haud a $e invicem pendentibus, in quibus

<pb 99>[0151]<rh>FLUXIONUM FLUENTIBUS.</rh>
<math>τ, υ, &c. & Ξ</math> haud eo$dem habent valores, erui po$$unt fluentes $in-
gularum fluxionum formulæ prædictæ <math>y x^.</math>; deinde ex $ingulis fluen-
tibus fluxionis <math>y x^.</math> datis & ex fluentibus <math>α + β + &c. + δ + ε + &c.</math>
fluxionum formulæ <math>r x^. $. y x^.</math> haud a $e invicem pendentibus, in qui-
bus <math>π, ξ, &c. & σ; τ, υ, &c.</math> haud eo$dem habent valores, erui po$$unt
fluentes $ingularum fluxionum formulæ <math>r x^. $. y x^.</math>.</p>
<p>3. Con$imiles propo$itiones etiam deduci po$$unt de fluentibus $u-
periorum ordinum; $æpe enim reduci po$$unt in fluentes primi or-
dinis. e. g. Sit fluxio <math>X^. × $. R x^. × $. Z x^.; & X^., X R x^. & X Z x^.</math> flu-
xiones, quarum fluentes detegi po$$unt, i. e. $int <math>$. X^. = X, $. X R x^. =
α, $. X Z x^. = β</math>; tum erit <math>$. X^. × $. R x^. × $. Z x^. = X × $. R x^. × $. Z x^. -
(α × $. Z x^. + β × $. R x^.) + ($. α Z x^. + $. β R x^.)</math>. Inveniantur numeri
fluentium inter $e independentium, ex quibus detegi po$$unt omnes
valores fluentium <math>$. R x^., $. Z x^., $. α Z x^. & $. β R x^.</math>; qui $int re$pective
<math>π, ξ, σ & τ</math>; tum ad plurimum ex <math>(π + ξ + σ + τ)</math> fluentibus inter
$e independentibus datæ fluxionis detegi po$$unt omnes fluentes eju$-
dem formulæ.</p>
<p>4. Sit data fluxio <math>X^. $. R x^. $. Z x^.</math>; cujus fluens <math>X $. r x^. $. Z x^. - $. X r x^.
$. Z x^.</math>: $int <math>X, r x^. & X r x^.</math> fluxiones, quarum fluentes detegi po$$unt,
i. e. $int re$pective <math>X, α & β</math>. Inveniatur numerus fluentium inter $e
independentium, ex quibus detegi po$$unt omnes valores fluentium
<math>Z x^., α Z x^. & β Z x^.</math>; qui fint re$pective <math>π, ξ & σ</math>; tum ex (<math>π + ξ + σ</math>)
ad plurimum fluentibus datæ fluxionis inter $e independentibus $em-
per detegi po$$unt fluentes omnium fluxionum eju$dem formulæ.</p>
<p>Cor. 1. Sit fluxio <math>A x^. $. B x^. $. C x^. $. D x^. .... $. P x^. $. Q x^. $. R x^. $. R x^.
= p x^.</math> in quâ $it <math>A x^. $. B x^. $. C x^. $. D x^. .... $. P x^. $. Q x^. $. R x^. = X^.</math>;
ubi per <math>X & R</math> denotentur eædem ac prius quantitates; tum quod
prius traditum fuit de fluentibus fluxionum <math>X^. $. R x^.</math>, etiam ad fluen-
tes fluxionis <math>p x^.</math> applicari pote$t.</p>
<p>Facile con$tant ca$us horum theorematum, in quibus e minore
numero independentium fluentium quam per has regulas a$$ignato
deduci po$$unt omnes fluentes datam formulam habentes.</p>

<pb 100>[0152]<rh>DE INVENIENDIS</rh>
<h>THEOR. XIII.</h>
<p>1. Pro <math>a + b x^n + c x^2n + d x^3n ... b x^(μ-1)n + k x^μn</math> $cribatur <math>p</math>, & pro
fluentibus fluxionum <math>e^p x^2-1 x^., e^p x^α+n-1 x^., e^p x^α+2n-1 x^., e^p x^α+3n-1 x^., ....
e^p x^α+(μ-1)n-1 x^., e^p x^α+μn-1 x^.</math> $ub$tituantur re$pective <math>A, B, C, D, E, ...
H, K</math>; tum, quoniam fluxio quantitatis <math>e^p x^α</math> erit <math>(α x^α-1 + n b x^α+n-1 l
+ 2 n c x^α+2n-1 l + 3 n d x^α+3n-1 l ... (μ - 1) n h x^α+(μ-1)n-1 l + μ n k x^α+μn-1 l) x^.
× e^p</math>, con$equitur fluens <math>(K)</math> fluxionis <math>e^p x^α+μn-1 x^. = {1 / μ n k l}(x^α - α A
- n b B l - 2 n c C l - 3 n d D l ... - (μ - 1) n b H l) e^p</math>.</p>
<p>Hìc logarithmus hyper. quantitatis <math>e</math> $upponitur <math>l</math>.</p>
<p>2. Sit exponentialis fluxio <math>e^a + b x^n + c x^2n ... b x^μn × x^r±νn x^.</math>, ubi <math>μ & υ</math>
funt integri numeri; tum ex fluentibus (<math>μ</math>) fluxionum prædictæ for-
mulæ independentibus, i. e. quæ habent <math>μ</math> diver$os valores quan-
titatis <math>ν</math>, datis; acquiri po$$unt fluentes omnium fluxionum eju$dem
formulæ.</p>
<p>3. Sit exponentialis fluxio <math>x^. e^a + b x^n + c x^2n ... b x^μn (α + β x^n + γ x^2n ...
ε x^ρn)^k=σ × (A + B x^n + C x^2n ... E x^Ξn)^l=τ × x^r±υn</math>; ubi <math>σ & υ</math> $unt qui-
cunque integri numeri; tum e fluentibus <math>μ + ξ + Ξ</math> fluxionum $u-
pradictæ formulæ independentibus erui po$$unt fluentes omnium
fluxionum eju$dem formulæ.</p>
<p>4. Sit exponentialis fluxio <math>e^(a + b x^n + c x^2n ... h x^πn)^m × (α + βx^n + γx^2n
... ε x^ρn)^k±σ × (A + B x^n + C x^2n ... E x^Ξn)^l±τ × x^r∓υn x^. = P x^.</math>; ubi <math>σ,
τ, υ, &c.</math> $unt quicunque integri numeri; tum ex omnibus fluentibus
formulæ <math>P x^.</math> datis; & ex <math>(π + ξ + Ξ)</math> independentibus fluentibus flu-
xionum formulæ <math>(a + b x^n + c x^2n .. + b x^πn)^m±r × P x^.</math>, ubi <math>r</math> eft etiam
quicunque integer numerus; acquiri po$$unt fluentes omnium fluxi-
onum eju$dem formulæ.</p>
<p>Facile con$tant hæ propo$itiones.</p>
<p>Cor. 1. Sit exponentialis fluxio <math>e^x x^a x^.</math>, cujus fluens <math>(P)</math> detur; tum
fluens fluxionis <math>e^x x^a+π x^.</math> erit <math>{1 / l} e^x (x^a+π - {1 / l} (a + π) x^a+π-1 + {1 / l2} (a + π)

<pb 101>[0153]<rh>FLUXIONUM FLUENTIBUS.</rh>
· (a + π - 1) x^a+π-2 - {1 / l3} (a + π). (a + π - 1). (a + π - 2) x^a+π-3 .....{1 / l^π-1} (a + π) · (a + π - 1) · (a + π - 2) ... (a + 2) · x^a+1) ± {1 / l^π} (a + π).
(a + π - 1) · (a + π - 2) ... (a + 1) × P</math>, $i <math>π</math> $it integer affirmativus
numerus: $ignum ± erit +, $i <math>π</math> $it par; $in aliter -.</p>
<p>2. Sit <math>π</math> negativus numerus, & erit <math>$. x^a-π e^x x^. = ({1 / a - π + 1} ×
x^a-π+1 - {l / a - π + 1} × {1 / a - π + 2} x^a-π+2 + {l / a - π + 1} · {l / a - π + 2} ·{1 / a - π + 3}x^a-π+3 .... {l / a - π + 1} · {l / a - π + 2} · {l / a - π + 3} ... {1 / a}x^a) e^x ={l / a - π + 1} · {l / a - π + 2} · {l / a - π + 3} ... {l / a} × P</math>: $ignum ± in hoc ca$u
etiam erit +, $i <math>π</math> $it par; $in aliter -.</p>
<p>Ea, quæ prius tradita fuere de algebraicis fluxionibus, æque ad
fluentiales applicari po$$unt.</p>
<h>PROB. XXIII.</h>
<p>1. Sit fluxio <math>(a + b x^n)^m × x^pn+υn-1 x^.</math>, invenire fluentem inter duos
valores (<math>0 & - {a / b}</math>) quantitatis <math>x^n</math> contentam.</p>
<p>In fluente ex. 3. prob. 22. inventâ pro <math>x^n</math> $cribantur re$pective <math>0 &
- {a / b}</math>; & erit re$ultantium differentia <math>{p / p + m + 1} · {p + 1 / p + m + 2} · {p + 2 / p + m + 3}... {p + v - 1 / p + m + v} × {a^v / b^v}</math> × fluen. flux. <math>(a + b x^n)^m × x^pn-1 x^.</math> inter duos valores
<math>0 & - {a / b}</math> quantitatis <math>x^n</math> contentam, quæ erit fluens quæ$ita.</p>
<p>2. Sit data fluxio <math>(a + b x^n)^m × x^pn-vn-1 x^.</math>; & erit ejus fluens inter
duos valores <math>0 & - {a / b}</math> quantitatis <math>x^n</math> contenta <math> = {p + m / p - 1} × {p + m - 1 / p - 2}

<pb 102>[0154]<rh>DE INVENIENDIS</rh>
× .. {p + m - v + 1 / p - v} × {b^v / a^v} ×</math> fluen. flux. <math>(a + b x^n)^m × x^pn-1 x^. (A)</math> inter
duos prædictos valores quantitatis <math>x^n</math> contentam. Hoc con$tat ab
ex. 4. prob. 22.</p>
<p>3. Sit data fluxio <math>(a + b x^n)^m+r x^pn-1 x^.</math>; & ejus fluens inter duos va-
lores <math>0 & - {a / b}</math> quantitatis <math>x^n</math> contenta, erit <math> = {m + 1 / p + m + 1} × {m + 2 / p + m + 2}× {m + 3 / p + m + 3} ... {m + r / p + m + r} × a^r × A</math> fluent. prædictam.</p>
<p>4. Sit data fluxio <math>(a + b x^n)^m-r × x^pn-1 x^.</math>; & ejus fluens inter duos
valores <math>0 & - {a / b}</math> quantitatis <math>x</math> contenta, erit <math> = {p + m / m} × {p + m - 1 / m - 1} ×{p + m - 2 / m - 2} ... {p + m - r + 1 / m - r + 1} × {A / a^r}</math>.</p>
<p>5. Sit data fluxio <math>(a + b x^n)^m+r × x^pn+vn-1 x^.</math>; & ejus fluens inter duos
valores <math>0 & - {a / b}</math> quantitatis <math>x^n</math> contenta, erit <math> = {(m + 1) × (m + 2)
× (m + 3) .. (m + r) × p × (p + 1) × (p + 2) .. (p + v - 1) / (p + m + 1) · (p + m + 2)
· (p + m + 3) .. (p + m + v) · (p + m + v + 1) ... (p + v + m + r)} × {a^v+r / b^v}× fluent. flux. (a + b x^n)^m × x^pn-1 x^. (A)</math>.</p>
<p>6. Fluens fluxionis <math>(a + b x^n)^m-r x^pn+vn-1 x^.</math>, inter duos prædictos
valores <math>0 & - {a / b}</math> quantitatis <math>x^n</math> contenta, erit <math> = {m + p + v / m} ×{m + p + v - 1 / m - 1} × ... {m + p + v - r + 1 / m - r + 1} × {p / p + m + 1} × {p + 1 / p + m + 2} × {p + 2 / p + m + 3}... {p + v - 1 / p + m + v} × {a^v-r / (-b)^v} × fluent. flux. (a + b x^n)^m x^pn-1 x^. (A)</math>.</p>
<p>7. Fluens fluxionis <math>(a + b x^n)^m+r x^pn-vn-1 x^.</math> inter valores <math>0 & - {a / b}</math>
quantitatis <math>x^n</math> contenta, erit <math> = {m + r + p - v + 1 / p - v} × {m + r + p - v + 2 / p - v + 1}

<pb 103>[0155]<rh>FLUXIONUM FLUENTIBUS.</rh>
... {m + r + p / p - 1} × {m + 1 / m + p + 1} × {m + 2 / m + p + 2} .. {m + r / m + p + r} × {(- b)^v / a^v-r}</math> × fluentem
prædictam <math>A</math>.</p>
<p>8. Fluens fluxionis <math>(a + b x^n)^m-r x^pn-vn-1 x^.</math>, inter valores <math>0 & - {b / a}</math>
quantitatis <math>x^n</math> contenta, erit <math> = {(p + m) × (p + m - 1) ×(p + m - 2)
.. (p + m - r - v + 1) / m × (m - 1) × (m - 2) ... (m - r + 1) ×
(p - 1) × (p - 2) ... (p - v)} × {-b^v / a^r+v} ×</math> fluentem <math>A</math>.</p>
<p>Hæc facile con$tant e prob. 22. & con$imilia de trinomialibus <math>(a +
b x^n + c x^2n)^m × x^pn-1 x^., &c.</math> fluxionibus facile detegi po$$unt. e. g.
fluens fluxionis <math>(a + b x^n + c x^2n)^m±u × x^pn=vn-1 x^. erit P × (a + b x^n +
c x^2n)^m=u × x^pn±v′n-1 - Q × $. (a + b x^n + c x^2n)^m=a × x^pn±βn-1 x^. - R ×
$ (a + b x^n + c x^2n)^m±α′ × x^pn±β′n-1 x^.</math>; ubi <math>α, β; α′ & β′, u & v</math> $unt
dati integri numeri; & fluentes duarum fluxionum <math>(a + b x^n + c x^2n)^m=a
× x^pn±βn-1 x^. & (a + b x^n + c x^2n)^m±a′ × x^pn±β′n-1 x^.</math> dantur, attamen
nec exprimi in finitis terminis, nec terminis finitis a $e invicem de-
duci po$$unt; & <math>P, Q & R</math> $unt rationales functiones quantita-
tis <math>x^n</math>.</p>
<p>9. 1. Si plures $int huju$ce generis <math>(a + b x^n)^m=r x^(p±v)n-1</math> fluxiones,
quarum aggregatum fluentium inter valores <math>0 & - {a / b}</math> quantitatis <math>(x)</math>
contentarum requiritur: inveniatur per problema $ingula fluens, &
exinde deduci pote$t earum $umma.</p>
<p>2. Si vero plures prædictæ quantitates per quantitates irrationales
denotentur; transformandæ $unt irrationales quantitates in terminos-
$ecundum dimen$iones quantitatis <math>x</math> progredientes; vel in tales ter-
minos, quorum fluentes per methodos prius traditas deduci po$$unt;
deinde inveniantur fluentes $ingularum fluxionum re$ultantium, &
exinde aggregatum quæ$itum.</p>
<p>Ex. 1. Invenire fluentem fluxionis <math>{(a + b x^n)^m / (b + k x^n)^β} x^pn-1 x^.</math> inter duos va-

<pb 104>[0156]<rh>DE INVENIENDIS</rh>
lores <math>0 & - {a / b}</math> quantitatis <math>x^n</math> contentam, ubi <math>A</math> $it fluens fluxio-
nis <math>(a + b x^n)^m × x^pn-1 x^.</math> inter duos prædictos valores quantitatis <math>x^n</math>
contenta.</p>
<p>Per binomiale theorema <math>(h + k x^n)^-β = h^-β × (1 - {β k / h} x^n +{β × (β + 1) × k^2 / 1 · 2 h^2} x^2n - {β × (β + 1) × (β + 2) × k^3 / 1 · 2 · 3^h3} x^3n + &c.)</math>; ducatur
hæc quantitas in <math>(a + b x^n)^m x^pn-1 x^.</math>, & fluxionum re$ultantium inter
valores <math>0 & - {a / b}</math> quantitatis <math>x</math> inveniantur fluentes; & erunt re$pe-
ctive <math>{A / h^β} × (1 + {p / p + m + 1} × {a / b} × {β k / h} + {p × (p + 1) / (p + m + 1) × (p + m + 2)}× {a^2 / b^2} × {β × (β + 1) k^2 / 2h^2} + {p × (p + 1) × (p + 2) / (p + m + 1) × (p + m + 2) × (p + m + 3)} ×{a^3 / b^3} × {β × (β + 1) × (β + 2) × k^3 / 1 · 2 · 3^h3} + &c.)</math></p>
<p>Ex. 2. Erit fluens fluxionis <math>x^. x^pn-1 × (a + b x^n)^m-1 $. (h + k x^n)^β ×
x^qn-1 x^. = $. {1 / h^m d^qn} × x^. × x^(p+q)n-1 × (a + b x^n)^m-1 × $. x^qn-1 × (h + k x^n)^m+β x^.</math>
inter duos valores <math>(0 & - {a / b} = d)</math> quantitatis <math>x^n</math> contenta, cum <math>p +
q + m + β = 0</math>.</p>
<p>Per binomiale theorema erit <math>y^. = (h + k x^n)^β × x^qn-1 x^. = x^. x^qn-1 (h^β
+ βh^β-1 k x^n + β · {β - 1 / 2} h^β-2 k^2 x^2n + &c.)</math>, cujus fluens erit <math>x^qn
({h^β / qn} + {β h^β-1 / (q + 1) × n} k x^n + β × {(β - 1) / 2} × {h^β-2 / (q + 2)n} k^2 x^2n + & c.)</math>; du-
catur hæc fluens in <math>z^. = x^. x^pn-1 × (a + b x^n)^m-1</math>, re$ultat fluxio <math>x^. ×
(a + b x^n)^m-1 × x^(p+q)^n-1 ({h^β / qn} + {β h^β-1 / (q + 1) n} k x^n + β × {β - 1 / 2} × {h^β-2 / (q + 2)n}

<pb 105>[0157]<rh>FLUXIONUM FLUENTIBUS.</rh>
k^2 x^2n + &c.)</math>; per methodos prius traditas fluens huju$ce fluxionis
inter valores <math>0 & - {a / b}</math> quantitatis <math>x^n</math> contenta, <math>invenitur = {h^β / qn} × G ×
(1 + {p + q / p + q + m} × {q / q + 1} × {β k / h} × d^n + {p + q / p + q + m} × {p + q + 1 / p + q + m + 1} × {q / q + 2} ×
β × {β - 1 / 2} × {k^2 / h^2} × d^2n + &c.)</math>, ubi <math>G = fluen</math>. flux. <math>x^(p+q)n-1 x^. × (a +
b x^n)^m-1</math> inter valores <math>0 & - {a / b}</math> quantitatis <math>x</math> contentæ. In hâc $erie
pro <math>p + q + m</math> $cribatur ejus valor <math>- β</math>, & pro <math>p + q</math> ejus valor <math>- β
- m</math>; & <math>evadet = {h^β / qn} × G × (1 + {β + m / 1} × {q / q + 1} × {k / h} d^n + (β + m) ×{β + m - 1 / 2} × {q / q + 2} × {k^2 / h^2} d^2n + &c.)</math>: per prædictum theorema erit
<math>x^qn-1 × (h + k x^n)^m+β × x^. = h^m+β × x^qn-1 x^. + (m + β) h^m+β-1 x^(q+1)n-1 k x^.
+ (m + β) × {m + β - 1 / 2} × h^m+β-2 x^(q+2)n-2 k^2 x^. + &c.</math>, cujus fluens erit <math>(P){h^m+β / q n} x^qn + {(β + m) / q + 1} × {h^m+β-1 / n} k x^(q+1)n + (β + m) × {β + m - 1 / 2 n × (q + 2)} h^β+m-2
× k^2 × x^q+2n + &c.</math>; ducatur hæc quantitas in <math>{q n / h^β+m x^qn}</math>, & pro <math>x^n</math> $criba-
tur <math>d^n</math>, & re$ultabit $eries prius inventa <math>{q n / h^β+m d^qn} × P = 1 + {β + m / 1} ×{q / q + 1} × {k / h} d^n + (β + m) × {β + m - 1 / 2} × {q / q + 2} × {k^2 / h^2} d^2n + &c.</math>; ducatur
hæc quantitas <math>{q n / h ^β+m d^qn} × P</math> in <math>{h^β / qn} × G</math>; & re$ultat quantitas prius tradita
<math>{1 / h^m d^qn} × P × G = {h^β / qn} × G × (1 + {β + m / 1} × {q / q + 1} × {k / h} d^n + &c.) = fluen</math>.
fluxionis <math>x^pn-1 x^. × (a + b x^n)^m-1 × $. (h + k x^n)^β x^qn-1 x^. = $. y z^.</math>. Q. E. D.</p>
<p>Cor. 1. Si vero <math>q + p + m + β</math> haud nihilo $it æqualis, $ed <math>p</math> augeatur

<pb 106>[0158]<rh>DE INVENIENDIS</rh>
vel diminuatur per quemcunque integrum numerum, ex hoc exemplo
detegi pote$t fluens fluxionis <math>y z^.</math>, inter duos valores <math>0 & - {a / b}</math> quanti-
tatis <math>x^n</math> contenta; e. g. $it <math>Q^. = z^. × fluen.</math> flux. <math>y^. x^n</math>, $ed <math>y^. = (h + k x^n)^β
× x^qn-1 x^.</math>, & exinde <math>x^n y^.</math> <math>erit = (h + k x^n)^β x^(q+1)n-1 x^.</math>; ergo fluens flux.
<math>x^n y^. erit = {(h + k x^n)^β+1 × x^qn / n k × (β + 1 + q)} - fluent</math>. flux. <math>{(h + k x^n)^β × x^qn-1 x^. × q h / k × (β + 1 + q)}{(q h y^. / k × (β + 1 + q)})</math>, unde <math>Q^. = {(h + k x^n)^β+1 x^qn / n k × (β + 1 + q)} z^. - {q h y z^. / k × (β + 1 + q)}</math>.</p>
<p>Cor. 2. Ii$dem literis ea$dem quantitates denotantibus, $int <math>Y, Z
& V</math> re$pective fluentes fluxionum <math>y^., z^. & z y^.</math> cum <math>h + k x^n = 0</math>, i. e.
inter valores <math>0 & - {h / k} (D)</math> quantitatis <math>x^n</math> contentæ; $int etiam <math>γ & π</math>
$imultanea incrementa fluentium fluxionum <math>x^(p+q)n-1 x^. × (h + k x^n)^β x^.</math>,
& <math>x^. x^pn-1 × (a + b x^n)^m+β, & p + q + m + β = 0</math>; tum ex hoc exemplo
con$tat <math>V = {γπ / a^β+1 D^pn}</math>, & con$equenter fluens fluxionis <math>y z^. = Y Z - V
({γ π / a^β+1 D^pn})</math>.</p>
<p>Cor. 3. Sit <math>z^. = - x^. x^n-r-2 × (a + b x^n)^m-1, & y^. = - x^. x^r × (h + k x^n)^β</math>;
pro <math>x</math> in hi$ce duabus fluxionibus $cribatur <math>{1 / v}</math>, & re$ultant fluxiones
<math>z^. = {v^. / v^2} × {1 / v^mn-r-2} × (a v^n + b)^m-1 = (a v^n + b)^m-1 × v^r-mn v^., & y^. = {v^. / v^2} ×{1 / v^r+βn} × (h v^n + k)^β = (h v^n + k)^β × v^-r-nβ-2 v^.</math>; unde, $i <math>{r - m n + 1 / n} -{r + β n + 1 / n} + β + m = 0</math>, fluens fluxionis <math>y z^.</math> inter duos valores <math>0 &
- {b / a} (d)</math> quantitatis <math>v^n</math> contenta erit per exemplum <math>{G P / k^m d^-(r+βn+1)}</math>, ubi
literæ <math>P & G</math> re$pective denotant fluentes (inter duos prædictos va-

<pb 107>[0159]<rh>FLUXIONUM FLUENTIBUS.</rh>
lores <math>0 & - {b / a}</math> quantitatis <math>v^n</math> contentas) fluxionum <math>v^. v^-r-βn-2 ×
(h v^n + k)^β+m & v^. × v^-(β+m)n-1 × (a v^n + b)^m-1</math>; cum autem fluens
fluxionis <math>v z^.</math> inter duos valores <math>(0 & - {b / a})</math> quantitatis <math>v^n</math> contineatur,
fluens fluxionis <math>y z^.</math> inter duos valores <math>({1 / 0} & - {a / b})</math> quantitatis <math>x^n</math> con-
tinebitur, quoniam <math>x = {1 / v}</math>; ergo fluens fluxionis <math>y z^.</math> inter <math>- {a / b}</math> & in-
finitum erit <math>{G P / k^m d^-(r+βn+1)}</math>.</p>
<p>Et $ic e præcedente coroll. invenietur fluens fluxionis <math>y z^.</math> inter duos
valores <math>(- {h / k} (D)</math> & infinitum) quantitatis <math>x^n</math> contenta <math> = Y Z - γ π
× {D^r+1-nm / b^β+1}</math>, ubi literæ <math>Y, Z, γ, π</math> re$pective denotant $imultaneas flu-
entes fluxionum <math>- x^. x^r (kx^n + h)^β, - x^. x^n-r-2 × (bx^n + a)^m-1, v^. v^-(m+β)n-1
× (hv^n + k)^β & v^. v^r-mn × (b + a v^n)^m+β.</math></p>
<p>Ex ii$dem principiis con$imilia proferre liceat de trinomialibus, &c.
quantitatibus; etiamque ex aliis transformationibus; $ed de his $atis.</p>
<h>PROB. XXIV.</h>
<p>_In inve$tigandis fluxionum fluentibus nonnunquam occurrunt ca$us, in_
_quibus numerator & denominator datæ fractionis_ <math>({P / Q})</math> _nihilo evadunt re-_
_$pective æquales; invenire ejus valorem._</p>
<p>1. Reducatur data fractio ad minimos terminos; i. e. dividatur &
numerator & denominator per maximum eorum communem divi$o-
rem, & $it quotiens re$ultans <math>{p / q}</math>; tum erit <math>{p / q}</math> valor fractionis quæ$itæ:
$i numerator & denominator <math>p & q</math> $imul nihilo evadant æquales,

<pb 108>[0160]<rh>DE INVENIENDIS</rh>
tum <math>{p^. / q^.}</math> erit valor fractionis quæ$itæ: $i <math>p^. & q^.</math> etiam nihilo evadant
æquales, tum erit prædictus valor <math>= {p^... / q^...}</math>: $i vero <math>p^.. & q^..</math> nihilo fiant
æquales, tum prædictus valor <math>= {p^... / q...;}</math>; & $ic deinceps: quod$i nihilo
$emper evadant æquales & numeratores & denominatores ex hâc me-
thodo inventi; tum reducenda e$t data fractio in alteram, ita ut om-
nes æquales divi$ores in unam $ummam colligantur, eodem modo,
qui prius docetur in prob. 4. & exinde per hanc methodum inveniri
pote$t valor fractionis <math>{P / Q}</math> quæ$itus.</p>
<p>Sub$equentia in nonnullis ca$ibus u$ui in$ervire poterunt.</p>
<p>2. Sint <math>P & Q</math> algebraicæ quantitates; & nihilo evadant æquales,
cum <math>x = a</math>; etiamque prædictæ quantitates <math>P & Q</math> habeant divi$ores
<math>(x - a)^m & (x - a)^n</math>, ubi literæ <math>m & n</math> denotant maximas pote$tates
quantitatis <math>x - a</math>, per quas dividi po$$unt quantitates <math>P & Q</math> re$pe-
ctive; tum, $i <math>m</math> major $it quam <math>n</math>, erit fractio <math>{P / Q} = 0</math>; $i <math>m</math> minor $it
quam <math>n</math>, erit <math>{P / Q}</math> infinita quantitas: $i <math>m = n</math>, erit <math>{P / Q}</math> finita quantitas.</p>
<p>3. Sit fractio <math>{Ax^b + Bx^b′ + Cx^b″ + &c. / A′x^k + B′x^k′ + C″x^k″ + &c.} = {P / Q}</math> quæcunque rationa-
lis functio quantitatis <math>x</math>, quæ nihilo evadat æqualis, cum <math>x = a</math>; $it
<math>(x - a)^m</math> communis divi$or quantitatum <math>P & Q</math>, etiamque maxima
pote$tas quantitatis <math>x - a</math>, per quam dividi pote$t vel <math>P</math> vel <math>Q</math>: $it <math>{P / Q}= {(x - a)^m / (x - a)^m} × {α / β}</math>; tum erit <math>{P / Q} = {α / β}</math>.</p>
<p>Cor. 1. Per hanc methodum $emper detegi pote$t valor fractionis <math>{P / Q}</math>;

<pb 109>[0161]<rh>FLUXIONUM FLUENTIBUS.</rh>
cum <math>P & Q</math> $int rationales functiones quantitatis <math>x</math>; idem etiam per-
fici pote$t per methodum prius traditam inveniendi fluxionem (<math>m</math> or-
dinis) quantitatum <math>P & Q</math>; i. e. <math>{P^. ^m / Q^. ^m}</math>; evadet enim <math>{P^. ^m / Q^. ^m} = {(x - a)^m-m × α / (x - a)^m-m × β}= {α / β}</math>; & $imiliter erit <math>{P^. ^r / Q^. ^r} = {(x - a)^m-r / (x - a)^m-r} × {α / β} = {α / β}</math>, $i modo <math>r</math> minor $it
quam <math>m; & x = a</math>.</p>
<p>Cor. 2. Con$tat methodum prius traditam inveniendi <math>{P^. ^m / Q^. ^m}</math> detegere
valorem fractionis <math>{P / Q}</math>, cum <math>m</math> $it integer numerus; $in aliter non.</p>
<p>4. Sit <math>{P / Q} = {p / q} × {α / β}</math>; ubi <math>P = o & Q = 0</math>, & <math>α & β</math> $int finitæ quan-
titates; tum inveniatur valor fractionis <math>{p / q} = r</math>, & erit <math>{P / Q} = {p / q} × {α / β}= γ × {α / β}</math>.</p>
<p>5. Si <math>(x - a)^m</math> $it divi$or quantitatis <math>A</math>; tum <math>(x - a)^m-1</math> erit divi$or
quantitatis <math>{A^. / x^.}</math>; & $ic deinceps.</p>
<p>6. Hoc problema generaliter re$olvi pote$t algebraice ex $ub$e-
quente propo$itione.</p>
<p>Pro <math>x</math> in datâ quantitate (<math>P</math>) $cribatur <math>v + a</math>, & reducatur quan-
titas (<math>P</math>) in alteram (<math>B</math>), quæ e$t functio quantitatis (<math>v</math>); deinde
reducatur quantitas (<math>B</math>) in $eriem <math>A v^l + B v^l′ + &c.</math> $ecundum di-
men$iones quantitatis <math>v</math> a$cendentem; tum erunt (<math>l</math>), viz. dimen$io-
nes quantitatis (<math>v</math>) in primo termino, numerus divi$orum, qui $unt
<math>(x - a)</math>, in quantitate <math>P</math> contentorum.</p>
<p>Et $imiliter pro <math>x</math> $cribatur <math>v + a</math> in quantitate <math>Q</math>, & reducatur ea

<pb 110>[0162]<rh>DEINVENIENDIS</rh>
in $eriem $ecundum dimen$iones quantitatis <math>v</math> a$cendentem <math>A′ v^λ +
B′ v^λ′ + &c.</math>: tum primo, $i <math>l</math> major $it quam <math>λ</math>, erit <math>{P / Q} = 0</math>; 2<_>do</_>. $i
<math>l = λ</math>, tum erit <math>{P / Q} = {A / A′}</math>; & denique, $i <math>l</math> minor $it quam <math>γ</math>, erit <math>{P / Q}</math>
infinita quantitas.</p>
<p>Ex. Sit fractio <math>{x - 1 - ^3 √(3x^2 - 3x - 9) / 5x - ^2 √(25x^2 + 7x - 28)}</math>, cujus numerator &
denominator nihilo evadunt re$pective æquales, cum <math>x = 4</math>; invenire
valorem fractionis, cum <math>x = 4</math>.</p>
<p>Pro <math>x</math> in numeratore & denominatore datæ fractionis $cribatur re-
$pective <math>v + 4</math>; & re$ultant <math>v + 3 - ^3 √(3v^2 + 21 v + 27) = P, &
5 v + 20 - √(25 v^2 + 207 v + 400) = Q</math>; reducantur irrationales
quantitates in $eriem $ecundum dimen$iones quantitatis <math>x</math> a$cenden-
tem, i.e. erunt <math>α = (27 + 21 v + 3 v^2)^{1 / 3} = 3 + {1 / 3} × {21 v / 9}({7 / 9}v) + &c.</math>,
& <math>β = (400 + 207 v + 25 v^2)^{1 / 2} = 20 + {207 / 40}v + &c.</math>; & con$equen-
ter <math>P = v + 3 - α = v + 3 - 3 - {7 / 9}v - &c. = {2 / 9}v - &c.</math>, &c <math>Q =
5 v + 20 - 20 - {207 / 40}v - &c. = {- 7 / 40}v - &c.</math>; unde <math>{{2 / 9}v/-{7 / 40}v} = {-80 / 63}</math>.</p>
<p>7. Sint <math>A & B</math> quæcunque functiones quantitatis <math>x</math>, tum <math>{A - A / B -B}</math> non
cuicunque quantitati dici pote$t æqualis.</p>
<p>Eadem principia etiam ad exponentiales, integrales, <math>&c</math>. quantitates
applicari po$$unt; nam reduci po$$unt omnes hæ quantitates in $eries
$ecundum dimen$iones quantitatis <math>v</math> progredientes.</p>

<pb 111>[0163]<rh>FLUXIONUM FLUENTIBUS.</rh>
<h>SCHOLIUM.</h>
<p>Datis fluentibus <math>n</math> datarum fluxionum; invenire, utrum fluens cu-
ju$cunque datæ fluxionis, earum & finitorum terminorum ope inve-
$tigari pote$t, necne.</p>
<p>1<_>mo</_>. Quærendæ $unt $ub$titutiones, quæ nece$$ario continent $in-
gulas fluxiones, quarum fluentes dantur; quod plerumque e doctrinâ
irrationalium quantitatum prius datâ, facile innote$cit; etiamque in-
note$cit $ub$titutio, quæ nece$$ario continet finitos terminos; quibus
quantitatibus pro $uis valoribus $ub$titutis, & quantitatum re$ultan-
tium aggregato datæ fluxioni æquale e$$e $uppo$ito, $i modo po$$ibile
$it; exinde deduci pote$t problematis re$olutio.</p>
<p>2<_>do</_>. Sæpe vero in detegendâ datæ fluxionis fluente, $i transformetur
data fluxio in alteram, cujus variabilis quantitas quandam habet re-
lationem ad variabilem datæ fluxionis quantitatem; re$ultabit fluxio,
cujus fluens per notas regulas inve$tigari pote$t.</p>
<p>3<_>tio</_>. Si transformentur etiam irrationales quantitates e denomina-
tore fluxionis in numeratorem, & vice versâ e numeratore in denomi-
natorem; $æpe re$ultabunt fluxiones, quarum fluentes innote$cunt.</p>
<p>E. g. Literæ <math>π, ξ σ τ, &c.</math> integros denotent numeros vel affirmativos
vel negativos, & <math>a, b, c, d, e, &c. b; p, q, r, s, t, &c. A, B, C, D, E, &c.
P, Q, R, S, T, &c. α, β, γ, δ, η, &c. κ, λ, μ, ν, &c.</math> qua$cunque invaria-
biles quantitates; <math>& x, y, z, v, &c.</math> variabiles quantitates; & dentur flu-
entes omnium fluxionum huju$ce formulæ <math>{x^π x^. / √(a + bx + cx^2 + dx^3 + ex^4)}'</math>
ubi <math>π vel = 0</math> vel $it affirmativus vel negativus integer nume-
rus; pro <math>x</math> $cribatur <math>z - α</math>, & con$tat inveniri po$$e fluentes omnium
fluxionum formulæ <math>{(z - α)^π z^. / √(p + q z + r z^2 + s z^3 + t z^4)}</math>; $it <math>π = - 1</math>
& fit fluxio formulæ <math>{z^. / (z - a) × √(p + q z + r z^2 + s z^3 + t z^4)}</math>;
etiamque e pluribus fluxionibus prædictæ formulæ in unam $um-

<pb 112>[0164]<rh>DEINVENIENDIS</rh>
mam collectis con$tabunt fluentes omnium fluxionum formulæ <math>(A){z^. / (bz - α) × √(p + q z + r z^2 + s z^3 + t z^4)}</math>; & exinde deduci po$$unt
fluentes omnium fluxionum formulæ <math>{x^π x^. / (x^n + p x^n-1 + q x^n-2 + &c.) × √(a + b x + c x^2 + d x^3 + e x^4)} (B)</math>: $it enim <math>x^n + p x^n-1 + &c. =
(x - α) × (x - β) × (x - γ) × &c</math>. per lem. 1. dividi pote$t hæc fluxio
in plures fluxiones formulæ <math>A</math>; $it <math>x^n + p x^n-1 + q x^n-2 + &c. = (x - α)^π
× (x - β)^ρ × (x - γ)^σ × &c.</math> & per idem lem. reduci pote$t hæc fluxio
in plures prædictæ formulæ pro <math>x</math> in fluxione (<math>B</math>) $cribatur <math>z^m</math>, &
pro <math>x^., m z^m-1 z^.</math>; & re$ultat fluxio formulæ <math>{z^(π+1)m-1 z^. / (z^nm + p z^(n-1)m + &c.) √(a + b z^m + c z^2m + d z^3m + e z^4m)}</math>,
cujus fluens per datas formulas inno-
te$cet.</p>
<p>2. In fluxione <math>{(P x^πm + Q x^(π-1)m + R x^(π-2)m + &c.) x^m-1 x^. / (x^nm + p x^nm-m + q x^nm-2m + &c.) × √(a + b x^m + c x^2m + d x^3m + ex^4m)} (C)</math>,
cujus fluens per prædictas fluentes acquiri pote$t,
$it <math>a + b x^m + c x^2m + d x^3m + e x^4m = (e x^2m - αx^m + β) × (x^2m - γ x^m
+ δ) = (e x^m - ε) × (x^3m - ι x^2m + η x^m - θ)</math>; & in fluxione <math>C</math> pro
<math>√(a + b x^m + c x^2m + d x^3m + e x^4m)</math> $cribatur ejus valor, vel
<math>{a + b x^m + c x^2m + d x^3m + e x^4m / √(a + b x^m + c x^2m + d x^3m + e x^4m)}</math>, vel <math>{(e x^2m - α x^m + β) × √(x^2m - γ x^m + δ) / √(e x^2m -
αx^m + β)}</math>,
vel <math>{e x^m - ε) × √(x^3m - ι x^2m + η x^m - θ) / √(e x^m - ε)}</math>, vel <math>{(x^3m - ι x^2m + ηx^m - θ) × √(e x^m - ε) / √(x^3m - ι x^2m + η x^m - θ)}</math> & re$ultant fluxiones $ub$equentium formularum
<math>{(P x^πm + Q x^(π-1)m + R x^(π-2)m + &c.) / (x^nm + p x^nm-m + q x^nm-2m + &c.)} × x^m-1 x^. × √(a + b x^m + c x^2m +

<pb 113>[0165]<rh>FLUXIONUM FLUENTIBUS.</rh>
d x^3m + e x^4m</math>); vel <math>{P x^πm + Q x^(π-1)m + R x^(π-2)m + &c.) / x^nm + p x^(n-1)m + q x^(n-2)m + &c.} × (x^2m - γ x^m + δ) ×
√({e x^2m - α x^m + β / x^2m - γ x^m + δ})x^m-1 x^.</math>; vel <math>{(P x^πm + Q x^(π-1)m + R x^(π-2)m + &c.) / (x^nm + p x^(n-1)m + q x^(n-2)m + &c.)}× (x^3m - ι x^2m + nx^m - θ) × √{(ex^m - ε) × x^. x^m-1 / x^3m - ι x^2m + n x^m - θ}</math>; vel <math>{P x^πm + Q x^(π-1)m + R x^(π-2)m
+ &c.) / x^nm + px^(n-1)m + q x^(n-2)m + &c.} × (ε x^m - ε) × √({x^3m - ι x^2m + n x^m - θ / e x^m - ε}) × x^m-1 x^.</math>;
unde omnes fluentes harum formularum per prædictas fluentes inve-
niri po$$unt.</p>
<p>3. Sit data fluxio hujus formulæ <math>{P x^πm + Q x^(π-1)m + &c.) × x^m-1 x^. / x^nm + p x^(n-1)m + q x^(n-2)m + &c.} ×
^4 √(a x^2m + b x^m + c)</math>; $cribatur <math>√(a x^2m + b x^m + c) = v x^m + √(c)
(d)</math>; & exinde <math>a x^2m + b x^m + c = v^2 x^2m + 2 d v x^m + c</math>, & con$equenter
<math>x^m = {2 d v - b / a - v^2}</math>, & <math>^4 √(a x^2m + b x^m + c) = √(v x^m + √(c)) =
√({2 d v^2 - b v / a - v^2} + d)</math>; quibus valoribus pro <math>x^m & ^4 √(a x^2m + b x^m + c)</math>
in datâ fluxione $ub$titutis, & fluxione quantitatis <math>{2 d v - b / a - v^2}</math> pro
<math>mx^m-1 x^.</math>; re$ultat fluxio formulæ prius traditæ: $i vero <math>a & c</math> $int negativæ
quantitates, & $int duæ radices æquationis <math>a w^2 + b w + c = 0</math> po$$i-
biles; viz. <math>√({b^2 - 4 a c / 4 a^2}) (λ)</math> $it po$$ibilis quantitas, & <math>μ</math> quæcun-
que invariabilis quantitas inter duas radices prædictas <math>{- b / 2 a} + λ</math> &
<math>{- b / 2 a} - λ</math> po$ita; $cribatur <math>z + μ</math> pro <math>x</math> in datâ fluxione, & <math>z^.</math> pro <math>x^.</math>;
& re$ultat fluxio huju$ce formulæ <math>Γ ^4 √(a z^2m + k z^m + l^2) z^m-1 z^.</math>, ubi
Γ $it rationalis functio quantitatis <math>z^m</math>, quæ fluxio reduci pote$t
per præcedentem $ub$titutionem: $i vero <math>c</math> $it negativa quantitas

<pb 114>[0166]<rh>DEINVENIENDIS</rh>
& radices æquationis <math>a w^2 + b w + c = 0</math> impo$$ibiles; tum fluxio
$emper erit impo$$ibilis quantitas, & con$equenter ejus fluens impo$$i-
bilis quantitas. Sit data fluxio <math>Γ × {x^m-1 x^. / ^4 √(a x^2m + b x^m + c)}</math>, ubi <math>Γ</math> in
hâc fluxione & $ub$equentibus e$t rationalis functio quantitatis <math>x^m</math>;
eadem $ub$titutio hanc fluxionem in fluxionem formularum prius
traditarum $emper transformabit: $it fluxio <math>Γ × ^4 √({a + b x^m / c + d x^m}) × x^m-1 x^.</math>,
$cribatur <math>^4 √({a + b x^m / c + d x^m}) = v</math>; & transformari pote$t hæc fluxio in fluxi-
onem, cujus variabilis quantitas $it <math>v</math>, præcedentis formulæ. Sint for-
mulæ <math>Γ × (a + b x^m)^{1 / 4} × (c + d x^m)^{3 / 4} x^m-1 x^., Γ × (a + b x^m)^{3 / 4} × (c + d x^m)^{3 / 4}x^m-1 x^., Γ × {x^m-1 x^. / (a + b x^m)^{3 / 4} × (c + d x^m)^{1 / 4}}, Γ × {x^m-1 x^. / (a + b x^m)^{3 / 4} × (c + d x^m)^{3 / 4}},
Γ × {x^m-1 x^. × (a + b x^m)^{3 / 4} /(c + d x^m)^{3 / 4}}, Γ × {x^m-1 × x^. × (a + b x^m)^{1 / 4} /(c + d x^m)^{3 / 4}}, Γ × {x^m-1 x^. × (a + b x^m)^{3 / 4} /(c + d x^m)^{1 / 4}}</math>.
In his formulis pro <math>(a + b x^m)^{3 / 4} & (c + d x^m)^{3 / 4}</math> $cribantur re$pective
<math>{a + b x^m / (a + b x^m)^{1 / 4}} & {c + d x^m / (c + d x^m)^{1 / 4}}</math>, & evadunt fluxiones formularum prius de-
ductarum.</p>
<p>4. Sit data fluxio <math>Γ × ^4 √(a + b x^m) × ^2 √(c + d x^m) × x^m-1 x^.</math>, $cribatur
<math>(a + b x^m)^{1 / 4} = v</math>, & exinde <math>{v^4 - a / b} = x^m, & ^2 √(c + d x^m) = √(c + {dv^4 - da / b})</math>
& transformari pote$t data fluxio in fluxionem <math>Δ v^. × ^2 √({dv^4 + bc - da / b})</math>,
ubi <math>Δ</math> denotat rationalem functionem literæ <math>v</math>: $int <math>Γ × {(a + b x^m)^{1 / 4} /(c + d x^m)^{1 / 2}}× x^m-1 x^.</math>, vel <math>Γ × {(c + d x^m)^{1 / 2} /(a + b x^m)^{1 / 4}} × x^m-1 x^.</math>, vel <math>Γ × {x^m-1 x^. / (a + b x^m)^{1 / 4} × (c + d x^m)^{1 / 2}}</math>,
& eodem modo pro <math>(a + b x^m)^{1 / 4}</math> $cribatur <math>v</math>, & transformentur hæ

<pb 115>[0167]<rh>FLUXIONUM FLUENTIBUS.</rh>
fluxiones in fluxiones, quarum variabilis quantitas e$t <math>v</math>, & re$ultant
fluxiones formularum prius traditarum. Sint fluxiones <math>Γ × (a + b x^m)^{3 / 4}× (c + d x^m)^{1 / 2} x^m-1 x^.</math>, vel <math>Γ × {(a + b x^m)^{3 / 4} /(c + d x^m)^{1 / 2}} × x^m-1 x^.</math>, vel <math>Γ × {(c + d x^m)^{1 / 2} /(a + b x^m)^{3 / 4}} ×
x^m-1 x^.</math>, vel <math>{Γx^m-1 x^. / (a + b x^m)^{3 / 4} × (c + d x^m)^{1 / 2}}</math>; in his fluxionibus pro <math>(a + b x^m)^{3 / 4}</math>
$cribatur <math>{a + b x^m / (a + b x^m)^{1 / 4}}</math>, & transformantur hæ fluxiones in fluxiones
præcedentium formularum.</p>
<p>5. Sit data fluxio <math>Γ × ^3 √(a + b x^m) × ^2 √(c + d x^m) × x^m-1 x^.</math>; $criba-
tur <math>v = ^3 √(a + b x^m)</math>, & re$ultat <math>x^m = {v^3 - a / b}</math>, & exinde <math>√(c + d x^m)
^2 √(c + d × {v^3 - a / b})</math>, & data fluxio transformatur in fluxionem <math>Δ ×
^4 √({d v^3 / b} + {b c - d a / b}) × v^.</math>, (ubi Δ erit rationalis functio literæ <math>v</math>),
quæ e$t fluxio formulæ prius traditæ.</p>
<p>Sint fluxiones <math>Γ × {^3 √(a + b x^m) / √(c + d x^m)} × x^m-1 x^., & Γ × {√(c + d x^m) × x^m-1 x^. / ^3 √(a + b x^m)}</math>
vel <math>Γ × {x^m-1 x^. / ^3 √(a + b x^m) × √(c + d x^m)}</math>, & per $ub$titutionem prius tra-
ditam hæ fluxiones reduci po$$unt in fluxiones formularum præce-
dentium.</p>
<p>6. Sit fluxio <math>{Δ / Θ} × √(a x^2m + b) × √(c x^2m + d) × √(e x^2m + f) × x^m-1 x^.</math>,
ubi <math>Δ</math> $it quæcunque rationalis functio quantitatis <math>x^m</math> $ine denomina-
tore, & <math>Θ</math> quæcunque rationalis functio ejus quadrati <math>x^2m</math>. Scriba-
tur in datâ fluxione pro <math>x^2m, v</math>; & transformetur data fluxio in alte-
ram, cujus variabilis quantitas e$t <math>v</math>; & re$ultat fluxio <math>Φ × v^. × ^2 √(av+b)

<pb 116>[0168]<rh>DE INVENIENDIS</rh>
× ^2 √(c v + d) × ^2 √(e v + f) + x v^. √(v) × √(a v + b) × √(c v + d) ×
√(ev + f)</math>, ubi <math>Φ & Χ</math> re$pective denotant rationales functiones quantita-
tis <math>v</math>; re$ultans fluxio habet formulam prius traditam: $int fluxiones <math>{Δ / Θ}× √(a x^2m + b) × {√(c x^2m + d) / √(e x^2m + f)} × x^m-1 x^.</math>, vel <math>{Δ / 9} × {√(a x^2m + b) × x^m-1 x^. / √(c x^2m + d) × √(e x^2m + f)}</math>,
vel <math>{Δ / Θ} × {x^m-1 x^. / √(a x^2m + b) × √(c x^2m + b) × √(e x^2m + f)}</math>; & per eandem
$ub$titutionem viz. <math>v = x^2m</math>, reduci pote$t data fluxio in formulas
prius traditas.</p>
<p>7. Sit fluxio <math>{Γ × x^m-1 x^. / Θ + Δ √(a x^m + b) + Φ √(e x^m + f) + Χ √(g x^m + b)
+ Ψ √(k x^m + l)}</math>,
ubi <math>Γ, Θ, Δ, Φ, Χ, Ψ</math> re$pective denotant rationales
functiones quantitatis <math>x^m</math>; inveniatur per prob. 16. quantitas, quæ in
denominatorem ducta rationale productum facit, ducatur ea in datam
fluxionem, & re$ultant fluxiones prædictarum formularum.</p>
<p>Sint fluxiones <math>{Γ × x^m-1 x^. / Θ + Δ√(a x^m + b) + Φ√(e x^m + d) + Χ√(e x^2m + f x^m + g)},{Γ x^m-1 x^. / Θ + Δ √(a x^m + b) + Φ √(c x^3m + d x^2m + e x^m + f)},{Γ x^m-1 x^. / Θ + Δ √(a x^2m + b x^m + c) + Φ √(d x^2m + e x^m + f)},{Γ x^m-1 x^. × √(a x^m + b) / Θ + Φ √(c x^m + d) + Χ √(e x^m + f) + Ψ √(g x^m + k)},{Γ x^m-1 x^. √(a + b x^m) / Θ + Φ√(c x^2m + d x^m + e) + Ψ√(g x^m + k)}, {Γ x^m-1 x^.√(a + b x^m) / Θ + Φ√(c x^3m + d x^2m + e x^m + f)},{Γ × x^m-1 x^. √(a x^2m + b x^m + c) / Θ + Δ √(d x^m + e) + Φ √(g x^m + k)}, {Γ × x^m-1 x^.√(a x^2m + b x^m + c) / Θ + Δ √(d x^2m + e x^m + f)},{Γ x^m-1 x^.√(a x^3m + b x^2m + c x^m + d) / Θ + Δ √(e x^m + f)}, &c. vel {Γ x^m-1 x^. / Θ + Δ^3vel{3 / 2} √(a x^2m + b) + √(d x^m + e)},

<pb 117>[0169]<rh>FLUXIONUM FLUENTIBUS.</rh>
{Γ x^m-1 x^. × (a x^m + b)^{1 / 3} vel {2 / 3} /Θ + √(d x^m + e)}, {Γ x^m-1 x^. ^2 √(a x^m + b) / Θ + ^3 √(d x^m + e)}; {Γ x^m-1 x^. √(a x^m + b) / Θ + ^3 √((d x^m + e)^2)},
&c. vel {Γ × x^m-1 x^. / Θ + Δ ^4 √(a x^m + b) + Φ ^4 √(c x^m + d)}, {Γ x^m-1 x^. ^4 √(a x^m + b) / Θ + Φ ^4 √(c x^m + d)},{Γ x^m-1 x^. / Θ + Δ ^2 √(a x^m + b) + Φ ^4 √(c x^m + d)}, {Γ x^m-1 x^. ^2 √(a x^m + b) / Θ + Φ ^4 √(a x^m + d)},{Γ x^m-1 x^. ^4 √(a x^m + b) / Θ + Φ ^2 √(c x^m + d)}, &c.</math> ducantur hæ fluxiones in irrationalem quan-
titatem deductam, ita ut denominator fiat rationalis quantitas; & re-
$ultant fluxiones formularum prædictarum.</p>
<p>8. Sit data fluxio <math>{Γ x^m-1 x^. / Θ + Φ ^2 √(a + ^2 √(b x^m + c)) + X√(d + √(b x^m + c))
+ Ψ √(e + √(b x^m + c)) + Δ √(f + √(b x^m + c))}</math>, ducatur denomi-
nator in quantitatem acqui$itam, quæ creat denominatorem rationa-
lem; & re$ultat fluxio, quæ ($cribendo <math>√(b x^m + c) = v</math>) erit formulæ
prius traditæ. Et $ic quamplurimæ huju$ce generis inveniri po$$unt
fluxiones, quæ transformari po$$unt in fluxiones formularum præce-
dentium. 1<_>mo</_>. A$$umatur fluxio præcedentis formulæ, in cujus de-
nominatore nulli contineantur irrationales termini; deinde per prob.
16. inveniatur quantitas <math>(H)</math> quæ ducta in numeratorem (<math>N</math>) creat
rationale productum (<math>M</math>). In datâ fluxione pro <math>N</math> $cribatur <math>{M / H}</math>; &
invenitur fluxio, quæ facile reduci pote$t in alteram præcedentis for-
mulæ: nonnullæ vero fluxiones in has formulas reduci po$$unt e $ub-
$titutionibus in prob. 17, &c. prolatis.</p>
<p>In datâ formulâ <math>{x^πm-1 x^. / √(a x^4m + b x^3m + c x^2m + d x^m + e)}</math> $cribatur <math>v^2 =

<pb 118>[0170]<rh>DE INVENIENDIS, &c.</rh>
a x^4m + b x^3m + c x^2m + d x^m + e</math>, & transformetur data fluxio in fluxi-
onem, cujus variabilis quantitas e$t <math>v</math>; & invenitur nova formula,
quæ reduci pote$t in datam. Et $ic deinceps. Si vero omnes radi-
ces æquationis <math>a x^4m + b x^3m + c x^2m + d x^m + e = 0</math> $int impo$$ibiles,
& <math>a</math> $it negativa quantitas; tum haud datur po$$ibilis fluens.</p>
<p>Methodis haud diffimilibus inveniri po$$unt quamplurimæ fluxiones,
quæ reduci po$$unt in fluxiones formularum <math>{Γ x^m-1 x^. / √(a x^5m + b x^4m + c x^3m + d x^2m
+ e x^m + f)} & {Γ x^m-1 x^. / √(a x^6m + b x^5m + c x^4m + d x^3m + e x^2m + f x^m + g)}, &c.</math></p>

<pb 119>[0171]
<h>CAP. III.
_De Fluxionalibus œquationibus._</h>
<h>PROB. XXV.</h>
<p>_DATA algebraicâ œquatione relationem inter ab$ci$$am_ (<math>x</math>) _& ejus_
_ordinatas_ (<math>y</math>) _exprimente, & datâ fluxionali quantitate quœ e$t alge-_
_braica functio ab$ci$$œ_ (<math>x</math>) _& ordinatœ_ (<math>y</math>) _& earum fluxionum; invenire_
_$ummam $ingulorum bujus functionis valorum; etiamque œquationem, cujus_
_radix e$t data fluxionalis quantitas._</p>
<p>E datâ æquatione inveniantur prima fluxio ordinatæ _(y)_, & $ic
$ecunda, tertia, &c. fluxiones ordinatæ quibus valoribus in datâ
fluxionali quantitate $ub$titutis, re$ultat algebraica quantitas, cujus
$umma $ingulorum valorum erui pote$t e primo capite medit. alge-
braic. Et $ic e prædicto capite inveniri po$$unt aggregata rectangu-
lorum e quibu$que duobus valoribus datæ functionis, contentorum e
quibu$que tribus, quatuor, &c. valoribus datæ functionis, i. e. inve-
niri po$$unt diver$æ quæ$itæ æquationis coefficientes, & con$equenter
æquatio ip$a.</p>
<p>Ex. 1. Datâ æquatione <math>y^2 - Ay + B = 0</math>, in quâ <math>A & B</math> $unt
quæcunquæ ab$ci$$æ (_x_) functiones; invenire æquationem, cujus ra-
dices $unt <math>{y^. / x^.}</math>. Sint duæ ordinatæ _(y)_ radices re$pective <math>α & β</math>; &
$umma quæ$itæ æquationis radicum erit <math>{α^. + β^. / x^.} = {A^. / x^.}</math>, quoniam <math>A =
α + β: & ob 2 yy^. - Ay^. - A^. y + B^. = 0</math>, & exinde <math>y^. = {A^. y - B^. / 2y - A}</math>, &

<pb 120>[0172]<rh>DE FLUXIONALIBUS</rh>
con$equenter duo valores fluxionis <math>y^.</math> e $cribendo <math>α & β</math> re$pective pro <math>y,
& α^. & β^.</math> re$pective pro <math>y^.</math>, evadent <math>{Aα - B^. / 2α - A}&{A^. β - B^. / 2β - A}</math>; unde rectangu-
lum $ub binis valoribus quæ$itæ æquationis radicum erit <math>{α^. β^. / x^. ^2} = {A^. α - B^. / 2α - A} ×{A^. β - B^. / (2β - A)x^. ^2}</math>, cujus denominator e$t <math>(4 α β - 2A(α + β) + A^2)x^. ^2
= (4B - A^2)x^. ^2</math>; numerator vero <math>A^. ^2 α β - A^. B^. (α + β) + B^. ^2 =
B A^. ^2 - A A^. B^. + B^. ^2</math>: & con$equenter, $i modo <math>v</math> ejus radicem de-
notet, æquatio quæ$ita erit <math>v^2 - {A^. / x^.}v + {BA^. ^2 - AA^. B^. + B^. ^2 / (4B - A^2)x^. ^2}</math>.</p>
<p>Ex. 2. Datâ æquatione <math>y^3 + By - C = 0</math>, in quâ <math>B & C</math> $unt
quæcunque functiones ab$cif$æ _x_; invenire æquationem, cujus radices
$unt <math>{y^. / x^.}</math>.</p>
<p>Sint tres radices ordinatæ (<math>α, β, γ</math>); tum $umma earum fluxionum
per <math>x^.</math> divi$arum erit <math>{α^. + β^. + γ^. / x^.} = 0</math>, quoniam <math>α + β + γ = 0: &{α^. β′ + α^. γ^. + β^. γ^. / x^. ^2} = {C^.- B^.α / (3α^2 + B)x^.} × {C^.- B^.β / (3β^2 + B)x^.} + {C^.- B^.α / (3α^2 + B)x^.} × {C^.- B^.γ / (3γ^2 + B)x^.}+ {C^.- B^.β / (3β^2 + B)x^.} × {C^. -B^.γ / (3γ^2 + B)x^.}</math>; reducantur hæ fractiones in commu-
nem denominatorem, & re$ultat denominator <math>(3α^2 + B)(3β^2 + B)
(3γ^2 + B)x^. ^3 = (27C^2 + 4B^3)x^. ^3</math>, numerator vero <math>(3 γ^2 + B) (C^.- B^.α)
(C^. - B^.β) + (3β^2 + B)(C^.- B^.α)(C^.- B^.γ) + (3α^2 + B)(C^.- B^. β)(C^.- B^. γ);</math>
ergo $umma rectangulorum e quibu$que binis in $e$e ductis erit
<math>{B^. ^2 B^2 - 3C^. ^2 B + 9CB^. C^. / (27C^2 + 4B^3)x^. ^2}</math>; & contentum $ub tribus radicibus in $e$e
ductis (<math>{α^. β^. γ^. / x^. ^3}</math>) erit <math>{C^. ^3 - B^. ^3 C + B^. ^2 C^. B / (27C^2 + 4B^3)x^. ^3}</math>; unde $i modo radix quæ$itæ

<pb 121>[0173]<rh>ÆQUATIONIBUS.</rh>
æquationis $it <math>v</math>, æquatio quæ$ita erit <math>v^3 + {B^2 B^. ^2 - 3C^. ^2 B + 9CB^. G^. / (27G^2 + 4B^3)x^. ^2}v - {C^. ^3 - B^. ^3 C + B^. ^2 C^. B / (27C^2 + 4B^3)x^. ^3} = 0</math>.</p>
<p>Ex. 3. Datâ eâdem æquatione <math>y^3 + By - C = 0</math>, ac in præcedenti
exemplo; invenire $ummam $ingularum quantitatum huju$ce generis
<math>{y x^. / y^.}</math>. Sint <math>α, β, γ</math> tres incognitæ quantitatis <math>y</math> radices; & tres quantita-
tes, quarum $umma quæritur, erunt re$pective <math>{3α^3 + Bα / C^. -B^.α}x^., {3β^3 + Bβ / C^. -B^. β}x^.,{3γ^3 + Bγ / C^. -B^. γ}x^.</math>; quibus fractionibus ad communem denominatorem re-
ductis, fit denominator <math>- B^. ^3 × αβγ + B^. ^2 C^.(αβ + αγ + βγ) - B^. C^. ^2
(α + β + γ) + C^. ^3 = C^. ^3 - B^. ^3 C + B^. ^2 BC^.</math>; & eodem modo invenitur
numerator <math>(4B^2 C^. B^. - 3BCB^. ^2 + 9CC^. ^2)x^.</math>; unde $umma quæ$ita <math>{αx^. / a^.}+ {βx^. / β^.} + {γx^. / γ^.} = {4B^. C^. B^2 + 9C^. ^2 C-3B^. ^2 CB / C^. ^3 -B^. ^3 C + B^. ^2 C^. B}x^.</math>.</p>
<p>Eodem fere modo inveniri po$$unt æ quationes, quarum radices $unt
quæcunque fluxionales functiones datæ algebraicæ vel datarum alge-
braicarum æquationum radicum.</p>
<h>PROB. XXVI.</h>
<h>_Transformare duas œquationes in unam; ita ut variabilis quantitas_ <math>y</math>, _&_
_ejus fluxiones exterminentur._</h>
<p>1. Sit altera æquatio algebraica <math>y^n - py^n-1 + qy^n-2 - ry^n-3 + &c. = 0</math>;
in quâ <math>p, q, r, &c.</math> re$pective denotant functiones variabilium
quantitatum (<math>z, x, &c.</math>) altera vero quæcunque fluxionalis æquatio
<math>A y^π y^. ^ρ y^.. ^σ y^... ^τ + &c. = 0</math>, in quâ continentur variabiles quantitates
(<math>x, y, z, &c.)</math>, &c.; eas in unam fluxionalem æquationem ita transfor-
mare, ut variabilis quantitas <math>y</math> & ejus fluxiones exterminentur.</p>

<pb 122>[0174]<rh>DE FLUXIONALIBUS</rh>
<p>Supponantur <math>α, β, γ, δ, &c.</math> diver$i valores variabilis quantitatis <math>y</math>
in datâ algebraicâ æquatione; quibus $ub$titutis in fluxionali, &
eorum fluxionibus pro incognitâ quantitate <math>y</math> & ejus fluxionibus;
re$ultant quantitates <math>A α^π α^. ^ρ α^.. ^σ &c. + &c. A β^π β^. ^ρ β″^σ &c. + &c.
A γ^π γ^. ^ρ γ^.. ^σ &c. + &c. &c.</math> ducantur hæ quantitates continuo in $e$e,
& contenti prob. 25. ope inveniatur aggregatum, quod nihilo fiat
æquale, & re$ultat æquatio quæ$ita.</p>
<p>2. 1. Duas fluxionales æquationes <math>A = 0 & B = 0</math> in unam tran$-
formare, ita ut variabilis quantitas <math>y</math> & ejus fluxiones exterminentur.</p>
<p>Sit <math>y^. ^π</math> fluxio variabilis quantitatis <math>y</math>, cujus ordo <math>π</math> e$t maximus eorum,
qui in datis æquationibus continentur; & $int duæ æquationes <math>Ay^. ^π ^m +
By^. ^π ^m-1 + Cy^. ^π ^m-2 + &c. = 0, & ay^. ^π ^n + by^. ^π ^n-1 + &c. = 0</math>, ubi <math>A, B, C, &c.
a, b, c, &c.</math> $unt quæcunque functiones variabilium (<math>x, z, &c.</math>) & earum
fluxionum, & quantitatum <math>y, y^, y^.., &c.</math> u$que ad <math>y^. ^π-1</math>; per methodum
in algebraicis æquationibus traditam ita reducantur hæ duæ æqua-
tiones in unam, ut exterminetur fluxio <math>y^. ^π</math>; & re$ultat æquatio fluxio-
nem <math>y^. ^π</math> haud involvens; deinde inveniatur fluxio æquationis re$ultan-
tis, quæ contineat in $e fluxionem <math>y^. ^π</math>; hujus æquationis ope & alte-
rius, quæ continet in $e $implicem pote$tatem fluxionis <math>y^. ^π</math>, exterminetur
fluxio <math>y^. ^π</math>; & re$ultant duæ æquationes folummodo involventes fluxio-
nes <math>y^., y^.., &c. y^. ^π - 1, &c.</math> in quibus haud continetur fluxio <math>y^. ^π</math>: deinde eo-
dem modo exterminentur fluxiones (<math>y^. ^π - 1, y^. ^π - 2, &c.</math>) & tandem extermi-
nabuntur quantitas <math>y</math> & ejus fluxiones.</p>
<p>2. 2. Sint <math>y^. ^π & y^. ^ρ</math> fluxiones maximi ordinis quantitatis <math>y</math> prædicta-
rum æquationum <math>A = 0 & B = 0</math> re$pective, & $it <math>ξ</math> minor quam <math>π</math>;
inveniatur fluxio ordinis <math>π - ξ</math> æquationis <math>B = 0</math>; & re$ultet fluxio-

<pb 123>[0175]<rh>ÆQUATIONIBUS.</rh>
nalis æquatio <math>C = 0</math> in quâ continetur prædicta fluxio <math>y^. ^π</math> ordinis <math>π</math>;
reducantur duæ æquationes <math>A = 0 & C = 0</math> in unam, ita ut exter-
minetur fluxio <math>y^. ^π</math>; & re$ultat æquatio <math>D = 0</math> haud involvens <math>y^. ^π</math>; cu-
jus $it fluxio <math>y^. ^π′</math> maxime $uperioris ordinis: $i <math>π′</math> major $it quam <math>ξ</math>,
tum inveniatur fluxio ordinis <math>π′ - ξ</math> æquationis <math>B = 0</math>, & re$ultet
æquatio <math>C′ = 0</math>; deinde reducantur duæ æquationes <math>D = 0 & C′ = 0</math>
in unam, ita ut exterminetur fluxio <math>y^. ^π′</math> & $ic deinceps: unde per hanc
& præcedentem methodum tandem exterminabuntur variabilis quan-
titas <math>y</math> & ejus fluxiones.</p>
<p>Cor. Sint duæ fluxionales æquationes <math>A = 0 & B = 0</math>, in quibus
fluxiones maximi ordinis quantitatis <math>y</math> $unt re$pective <math>y^. ^π & y^. ^ρ</math>; deinde
reducantur hæ duæ æquationes in unam, ita ut exterminentur quan-
titas <math>y</math> & ejus fluxiones, & re$ultet æquatio <math>C = 0</math>; tum in æquatione
<math>C = 0</math> haud continentur fluxiones $uperioris ordinis quam fluxio (<math>π</math>)
ordinis quantitatum & fluxionum in æquatione <math>B = 0</math> contentarum,
& fluxio (ξ) ordinis quantitatum & fluxionum in æquatione <math>A = 0</math>
contentarum.</p>
<p>2. Con$imilia etiam prædicari po$$unt de ordinibus fluxionum in
(<math>n - 1</math>) æquationibus re$ultantibus ex reductione (<math>n</math>) æquationum,
ita ut exterminentur variabilis quantitas <math>y</math> & ejus fluxiones.</p>
<p>3. Et $ic de <math>n</math> æquationibus in unam transformandis, ita ut
(<math>n - 1</math>) variabiles quantitates & earum fluxiones exterminentur.</p>
<p>4. Sint <math>n</math> fluxionales æquationes <math>a = 0, b = 0, c = 0, &c.; m, r, s,
&c.</math> ordinum re$pective; reducantur hæ fluxionales æquationes in
unam, ita ut exterminentur (<math>n - 1</math>) variabiles quantitates & earum
fluxiones, & non nece$$ario re$ultabit fluxionalis æquatio $uperioris
quam <math>m + r + s + &c.</math> ordinis.</p>
<p>Cor. Ex hâc methodo transformandi æquationes, ita ut variabiles
quantitates exterminentur, irrepunt in transformatas æquationes va-
lores, qui in datis æquationibus haud inveniuntur; eodem modo, quo

<pb 124>[0176]<rh>DE FLUXIONALIBUS</rh>
in algebraicas æquationes per con$imilem methodum transfor-
matas irrepunt valores, qui in datis haud inveniuntur, ut con$tat e
prob. 31. medit. algebraic. e. g. Sint duæ fluxionales æquationes
<math>(x^m + z^r) y^. + z^. = 0, & (x^b + y^k) × z^. + x^. = 0</math>; ita transformare has
duas fluxionales æquationes in unam, ut exterminetur <math>z & z^.</math>: in unâ
æquatione <math>z^.</math> ducitur in 1, in alterâ in <math>x^b + y^k</math>; ducatur re$pective
$ingula æquatio in coefficientem fluxionis <math>z^.</math>, quæ invenitur in alterâ;
& re$ultant <math>(x^m + z^r) × (x^b + y^k) y^. + (x^b + y^k) × z^. = 0, & (x^b + y^k) ×
z^. + x^. = 0</math>, $ubducatur po$terior æquatio de priori, & re$iduum erit
<math>(x^m + z^r) × (x^b + y^k) × y^. - x^. = 0</math>, in quibus duabus æquationibus
<math>(x^m + z^r) × (x^b + y^k) × y^. - x^. = 0, & (x^b + y^k) × z^. + x^. = 0</math> inveniun-
tur valores, qui in datis haud continentur, viz. in his re$ultantibus
æquationibus inveniuntur omnes valores, qui in datis continentur;
etiamque valores, qui in æquationibus <math>x^b + y^k = 0 & (x^b + y^k) × z^. + x^.
= 0</math> inveniuntur. Et $ic inveniatur fluxio æquationis <math>(x^m + z^r) × (x^b
+ y^k) × y^. - x^. = 0</math>, & re$ultat <math>(x^m + z^r) × (x^b + y^k) × y^.. + (m x^m-1 x^. + r z^r-1 z^.)
× (x^b + y^k) y^. + (h x^b-1 x^. + k y^k-1 y^.) (x^m + z^r) y^. - x^.. = 0</math>; hæc vero & datæ
æquationes <math>(x^b + y^k) z^. + x^. = 0</math> habent omnes valores, quos habent æqua-
tiones <math>(x^m + z^r) × (x^b + y^k) y^. - x^. + A v^. = 0 & (x^b + y^k) z^. + x^. = 0</math>,
ubi litera <math>A</math> denotat invariabilem quantitatem ad libitum a$$umen-
dam; & <math>v</math> quantitatem, quæ fluit uniformiter: e re$ultante & datâ
æquatione inveniatur æquatio, in quâ haud continetur fluxio <math>z^.</math>; & e
præcedente methodo inveniri po$$unt valores in duabus re$ultantibus
æquationibus, in quibus haud invenitur <math>z^.</math>, qui in datis haud inveni-
untur; & $ic repetitis operationibus tandem ita transformabuntur
æquationes, ut exterminetur <math>z</math>; & eâdem methodo argumentandi
continuo repetitâ, tandem deduci po$$unt valores, qui in re$ultante
æquatione inveniuntur, haud vero in datis æquationibus.</p>
<p>Et $ic de pluribus æquationibus in unam transformandis, ita ut
exterminentur duæ vel plures incognitæ quantitates.</p>
<p>Hoc problema aliter $æpe re$olvi pote$t e multiplicatione datarum
æquationum in a$$umptas quantitates, quæ reddunt quantitates eva-
ne$centes, quas exigat problema. Sed de his nimis.</p>

<pb 125>[0177]<rh>ÆQUATIONIBUS.</rh>
<h>THEOR. XIV.</h>
<p>1. Datâ fluxionali æquatione <math>α × β × γ × δ × &c. = 0</math>; ubi <math>α, β, γ, δ,
&c.</math> qua$cunque fluxionales quantitates re$pective denotant; ea de-
primi pote$t in diver$as æquationes <math>α = 0, β = 0, γ = 0, δ = 0</math>.
Facile con$tat.</p>
<p>2. Datis duabus fluxionalibus æquationibus vel algebraicâ & flu-
xionali æquatione <math>α × β × γ × δ × &c. = 0 & π × ξ × σ × &c. = 0</math>, quæ
recipiunt divi$ores <math>α, β, γ, &c. & π, ξ, σ, &c.</math> hæ æquationes di$tingui
po$$unt in $ub$equentes <math>α = 0 & π = 0; α = 0 & ξ = 0; α = 0 &
σ = 0; &c. β = 0 & π = 0; β = 0 & ξ = 0; β = 0 & σ = 0; &c.
γ = 0 & π = 0; γ = 0 & ξ = 0; &c.</math></p>
<p>Et $ic de pluribus æquationibus plures divi$ores habentibus.</p>
<h>PROB. XXVII.</h>
<h it>Datâ fluxionali œquatione; invenire utrum ea reduci pote$t in duas
vel plures alias, necne.</h>
<p>Inveniatur per meditat. algebr. annon duos vel plures divi$ores re-
cipiat data æquatio, & con$tat problema.</p>
<p>Ex. Sit data æquatio <math>(x y^.. + x^.^2 + y^. x^.) × (y x^. + {1 / n} x y^. + a x^m x^.) = 0</math>;
ubi <math>x</math> fluit uniformiter. Hæc fluxionalis æquatio continet duos divi-
$ores <math>x y^.. + x^.^2 + y^. x^., & y x^. + {1 / n} x y^. + a x^m x^.</math>; unde data æquatio dividi
pote$t in duas alias <math>x y^.. + x^.^2 + y^. x^. = 0 & y x^. + {1 / n} x y^. + a x^m x^. = 0</math>:
fluens prioris fluxionalis æquationis erit <math>x y^. + x x^. = 0</math>, quæ correcta
fit <math>x y^. + x x^. + b x^. = 0</math>, ubi litera <math>b</math> quamlibet invariabilem quanti-
tem denotat: fluens vero hujus æquationis erit <math>y + x + b × log.
x = 0</math>; quæ correcta $it <math>y + x + b × log. x + c = 0</math>, ubi <math>c</math> denotat
quamlibet invariabilem quantitatem: fluens vero po$terioris æqua-
tionis <math>(y x^. + {1 / n} x y^. + a x^m x^. = 0)</math> e$t <math>{1 / n} y x^n + {a x^m+n / m + n} + A = 0</math>, ubi <math>A</math>
quamcunque invariabilem quantitatem denotat; unde data fluxiona-

<pb 126>[0178]<rh>DE FLUXIONALIBUS</rh>
lis æquatio duas independentes recipit $olutiones, viz. <math>y + x + b ×
log. x + c = 0</math>, vel <math>{1 / n} y x^n + {a x^m+n / m + n} + A = 0</math>: hæc fluxionalis æquatio
ad re$olutionem duorum diver$orum problematum re$picit: in mul-
tis ca$ibus, $i fluxionales æquationes diver$os habeant ordines, fluxio-
nales æquationes $uperioris ordinis $olummodo pro re$olutione pro-
blematis habendæ $unt.</p>
<p>Si fluxio <math>y^.</math> maximi ordinis in datâ fluxionali æquatione plures (<math>n</math>)
habeat radices: inveniantur $inguli (<math>n</math>) valores prædictæ fluxionis <math>y^.,</math>
& re$ultant (<math>n</math>) diver$æ fluxionales æquationes, quarum fluentes re-
quirantur.</p>
<p>Cor. 1. Hinc facile deduci po$$unt infinitæ fluxionales æquationes,
quæ dividi po$$unt in alias: a$$umantur enim fluxionales æquationes,
quibus in $e$e ductis, exorietur fluxionalis æquatio, quæ facile reduci
pote$t in a$$umptas.</p>
<p>Cor. 2. Datâ fluxionali æquatione involvente variabiles quantitates
<math>x & y</math> & earum fluxiones; datâ etiam aliâ fluxionali æquatione ha-
bente duas variabiles quantitates <math>z & v</math> & earum fluxiones; invenire
annon <math>z</math> eadem e$t functio vel algebraica vel fluxionalis quantitatis
<math>v</math>, ac litera <math>x</math> e$t quantitatis <math>y</math>: pro <math>z & v</math> & earum fluxionibus $cri-
bantur <math>x & y</math> & earum fluxiones re$pective; deinde inveniatur, utrum
hæ æquationes re$ultantes communem habeant divi$orem, necne; &
confit coroll.</p>
<h>THEOR. XV.</h>
<p>Datis duabus fluxionalibus æquationibus relationem inter <math>z, y, x</math>
& earum fluxiones exprimentibus; ex iis deduci po$$unt duæ aliæ in-
finitis modis, quarum variabiles quantitates eædem $unt.</p>
<p>Ducantur datæ æquationes in qua$cunque quantitates, deinde
addantur vel $ubducantur a $e invicem, & re$ultabunt æquationes
quæ$itæ.</p>

<pb 127>[0179]<rh>ÆQUATIONIBUS.</rh>
<h>PROB. XXVIII.</h>
<h it>Datam æquationem fluentem involventem in fluxionalem reducere, in
quâ nulla continetur fluens.</h>
<p>1. Si fluens ducatur in variabilem quantitatem, dividatur æquatio
per illam quantitatem, & inveniatur fluxio quantitatis re$ultantis, &
exterminabitur ea fluens.</p>
<p>1. 2. Si autem majores $int dimen$iones (<math>n</math>) fluentialis quantitatis
exterminandæ, vel altior $it ordo (<math>m</math>) ejus fluentis; tum opus e$t toties
(<math>n</math>) vel (<math>m</math>) ordinis inveniendi fluxionem datæ æquationis.</p>
<p>Ex. 1. Sit data æquatio <math>x x^. + y y^. + (x + a)x^.</math> flu. <math>√(a^2 + x^2)x^. = 0</math>;
requiratur hanc æquationem ita reducere, ut exterminetur fluens <math>$.
√(x^2 + a^2) · x^.</math>.</p>
<p>Hæc fluens ducitur in variabilem quantitatem <math>(x + a) · x^.</math>, divida-
tur data æquatio per hanc quantitatem, & re$ultat æquatio <math>{x x^. + y y^. / (x + a) · x^.}+ fl. √(a^2 + x^2)x^. = 0</math>. Inveniatur fluxio æquationis re$ultantis, &
exterminabitur fluens prædicta.</p>
<p>Ex. 2. Sit data æquatio <math>x x^. + y y^. + (x + a) x^. $. <r n>√ (a^2 + x^2) · x^.^n</r></math>,
dividatur hæc æquatio per <math>(x + a) × x^.</math>, & re$ultat æquatio <math>{x x^. + y y^. / (x + a) · x^.}+ $. <r n>√ (x^2 + x^2) · x^.^n</r>
</math>; hujus æquationis inveniatur fluxio (<math>n</math>) ordinis,
& re$ultat æquatio a fluentibus libera.</p>
<p>Ex. 3. Sit data æquatio <math>x^2 + y^2 + (c + d x) × $. √(a^2 + x^2) x^. +
(fluen. x^. √(a^2 + x^2))^2 = 0</math>; maximæ dimen$iones fluentis <math>√(a^2 + x^2)x^.</math>
inveniuntur duæ, viz <math>(fluen. x^. √(a^2 + x^2))^2</math>, $ed hæ maximæ dimen-
$iones in nullam variabilem quantitatem ducuntur, ergo inveniatur
fluxio datæ æquationis, quæ erit <math>2 x x^. + 2 y y^. + (c + d x) × √(a^2 +
x^2) x^. + d x^.</math> fluen. <math>√(a^2 + x^2) x^. + 2</math> fluen, <math>x^. √(a^2 + x^2) × (a^2 + x^2) x^.

<pb 128>[0180]<rh>DE FLUXIONALIBUS</rh>
= 0</math>. In hâc æquatione <math>$. √(a^2 + x^2) x^.</math> ducitur in variabilem quan-
titatem <math>(2 a^2 + 2 x^2 + d) × x^.</math>; dividatur inventa æquatio per hanc
quantitatem, & re$ultat <math>{2 x x^. + 2 y y^. + (c + d x) √(a^2 + x^2) x^. / (2 a^2 + 2 x^2 + d) x^.} + $. x^.
√(a^2 + x^2) = 0</math>; hujus æquationis inveniatur fluxio, & re$ultat
fluxionalis æquatio a fluentibus libera.</p>
<p>Cor. 1. Et $ic iteratis operationibus de pluribus fluentibus exter-
minandis.</p>
<p>Cor. 2. Facile con$tabit ordo, ad quem a$$urget æquatio a fluen-
tialibus quantitatibus libera.</p>
<p>Cor. 3. Si quantitates irrationales utcunque ex his fluentibus com-
po$itæ in datâ æquatione contineantur, tum per meditationes alge-
braicas ab irrationalitate $uâ liberari pote$t æquatio re$ultans.</p>
<p>2. Hæc methodus autem transformandi æquationem, in quâ plures
inveniuntur dimen$iones (<math>n</math>) fluentialis quantitatis exterminandæ,
invenit fluxionalem æquationem $uperioris ordinis, quam nece$$ario
exigat problema; $ub$equenti igitur methodo uti oportet; $it fluens
fluxionis <math>Q</math>, quæ dicatur <math>V</math>, quantitas exterminanda; & $it data
fluentialis æquatio <math>A V^n + B V^n-1 + C V^n-2 + &c. = P</math>, cujus fluxio
e$t <math>A V^n + n V^n-1 A V^. + (n - 1) B V^n-2 V^. + V^n-1 B^. + &c. = P^.</math>; per
methodum in medit. algeb. traditam ita reducantur hæ duæ æquati-
ones <math>A V^n + B V^n-1 + C V^n-2 + &c. = P & A^. V^n + n V^n-1 A V^. +
(n - 1) B V^n-2 V^. + &c. = P^.</math> in unam, ut exterminetur fluentialis
quantitas <math>V</math>, & con$tat problema; i. e. generaliter $it æquatio <math>q = 0</math>
reducantur duæ æquationes <math>q = 0 & q^. = 0</math>, ita ut exterminetur fluens
<math>V</math>, & confit prob. Eâdem operatione $æpius repetitâ tolli po$$unt duæ
vel plures fluentiales quantitates e datâ æquatione; etiamque fluen-
tiales quantitates $uperiorum ordinum.</p>
<p>Si vero irrationales functiones fluentis <math>V</math> contineantur in datâ
æquatione <math>A = 0</math>, tum e methodo in med. algeb. traditâ ita reducan-
tur duæ æquationes <math>A = 0 & A^. = 0</math>, ut ex eâ exterminetur incog-
nita quantitas <math>V</math>.</p>

<pb 129>[0181]<rh>ÆQUATIONIBUS.</rh>
<p>Et $ic tolli po$$unt fluentiales quantitates e duabus vel pluribus
æquationibus plures variabiles quantitates habentibus.</p>
<h>PROB. XXIX.</h>
<p it>Datâ fluxionali œquatione duas incognitas quantitates <math>x & y</math> & earum
fluxiones involvente, in quâ <math>x</math> fluit uniformiter; eam ita transformare, ut
ex eâ exterminentur incognita quantitas <math>x</math> & ejus fluxio.</p>
<p>Inveniatur fluxio datæ fluxionalis æquationis, & æquatio data &
re$ultans ita reducantur, ut exterminetur fluxio <math>x^.</math>; æquationis re$ul-
tantis, in quâ $olummodo continetur quantitas <math>x</math> $ine ejus fluxioni-
bus, & quantitas <math>y</math> & ejus fluxiones, inveniatur fluxio; & data & po-
$terior re$ultans æquatio ita reducantur in unam, ut exterminetur
fluxio <math>x^.</math>; unde duæ inveniuntur æquationes, quæ $olummodo conti-
nent algebraicas functiones quantitatis <math>x</math> & nullas ejus fluxiones; ita
reducantur hæ duæ æquationes in unam, ut exterminetur incognita
quantitas <math>x</math>; & re$ultat æquatio, in quâ $olummodo continentur quan-
titas <math>y</math> & ejus fluxiones.</p>
<p>Cor. Datâ fluxionali æquatione <math>m + 2</math> ordinis unam $olummodo
variabilem quantitatem (<math>y</math>) & ejus fluxiones involvente; eam tran$-
formare in alteram, terminos variabilis quantitatis <math>y</math> & alterius <math>x</math>, quæ
fluit uniformiter; vel cujus fluxio <math>r</math> ordinis fluit uniformiter, ubi <math>r</math>
minor $it quam <math>m + 2</math>; involventem.</p>
<p>Inveniatur fluentialis æquatio datæ fluxionalis; fluentiali æqua-
tioni adjiciatur quantitas vel fluxio generalis, cujus fluens fluit uni-
formiter, i. e. quæ erit functio fluxionalis quantitatis, cujus fluens
fluit uniformiter; & perficitur corollarium.</p>
<h>PROB. XXX.</h>
<p it>Datâ œquatione relationem inter duas quantitates <math>x & y</math> & earum
fluxiones exprimente; invenire œquationem relationem inter duas alias
quantitates <math>z & v</math> exprimentem, ubi literœ <math>x & y</math> exprimi po$$unt in ter-
minis quantitatum <math>z & v</math>.</p>

<pb 130>[0182]<rh>DE FLUXIONALIBUS</rh>
<p>1. Sub$tituantur pro <math>x & y, x^. & y^., &c.</math> in datâ æquatione quæcun-
que algebraicæ functiones literarum <math>z & v</math>, & earum fluxionum; &
re$ultant æquationes quæ$itæ.</p>
<p>Ex. 1. Datâ æquatione relationem inter prædictas quantitates <math>x &
y</math> & earum fluxiones exprimente, pro literis <math>x & y</math> in datâ æquati-
one $cribantur re$pective <math>a z + b v + c & p z + q v + r</math>; & pro
fluxionibus <math>x^. & y^.</math> re$pective <math>a z^. + b v^. & p z^. + q v^.</math>; & $ic deinceps;
& re$ultat æquatio relationem inter <math>z & v</math> exprimens, quæ algebraicè
e data relatione inter <math>x & y</math> petenda e$t.</p>
<p>1. 2. Si relationes inter quantitates <math>v & z, x & y</math> exprimantur
per algebraicas æquationes; tum e reductione plurium æquationum
in unam, ita ut exterminentur omnes incognitæ quantitates præter <math>z
& v</math>, petenda e$t relatio inter quantitates <math>z & v</math>.</p>
<h>PROB. XXXI.</h>
<p it>Datis algebraicis & fluxionalibus œquationibus; invenire œquationem,
cujus radices quamcunque habeant a$$ignabilem relationem ad radices data-
rum œquationum.</p>
<p>Si relatio vel exprimatur per algebraicam vel fluxionalem quanti-
tatem, quæ $upponatur (<math>w</math>); vel per algebraicas vel fluxionales æqua-
tiones; tum perfici pote$t problema e reductione plurium (<math>n</math>) æqua-
tionum in pauciores (<math>n - m</math>) ita ut exterminentur (<math>m</math>) incognitæ
quantitates & earum fluxiones.</p>
<p>2. Si vero ver$etur relatio inter diver$os eju$dem incognitæ quan-
titatis (<math>v</math>) valores & eorum fluxiones.</p>
<p>Inveniantur æquationes, quarum radices $unt re$pective <math>v, v^2, v^3
.. v^n</math> ($i modo <math>n</math> $it maxima dimen$io ad quam a$cendat <math>v</math> in datâ re-
latione); etiamque æquationes, quarum radices $unt <math>v^., v^.^2, v^.^3, v^.^4 .. v^.^m</math>
($i modo <math>m</math> $it maxima dimen$io, ad quam a$cendat <math>v^.</math> in datâ rela-
tione); & $ic de fluxionibus $uperiorum ordinum: e re$ultantibus
æquationibus inveniri po$$unt (per medit. algebr.) æquationes, qua-

<pb 131>[0183]<rh>ÆQUATIONIBUS.</rh>
rum radices algebraicam vel fluxionalem relationem quæ$itam ha-
bent.</p>
<p>Cor. Con$tat æquationes hâc methodo derivatas multo plures ha-
bere radices, quam nece$$ario exigat problema: $æpe autem reduci
po$$unt per methodum divi$ores inveniendi.</p>
<p>3. Hoc problema $æpe recipere pote$t re$olutionem e $ub$titutione
datæ vel datarum æquationum; etiamque $ingulos ex deducendo va-
lores per infinitas $eries, & exinde quæ$itas radices inve$tigando.</p>
<h>PROB. XXXII.</h>
<p it>Datâ œquatione algebraicâ relationem inter ab$ci$$am <math>x</math> & ejus ordina-
tas <math>y</math> exprimente; invenire quo$dam ca$us, in quibus literœ <math>x & y</math> per fun-
ctionem tertiœ <math>z</math> exprimi po$$unt.</p>
<p>1. 1. Sit data æquat io <math>A y^n + (a + b x) y^n-1 + (d x + e x^2) y^n-2 +
&c. = 0</math>, in quâ continentur $olummodo termini (<math>n & n - 1</math>) dimen-
$ionum quantitatum (<math>x & y</math>): $cribatur pro <math>x</math> rectangulum <math>z y</math>, &
re$ultat æquatio <math>(A + b z + e z^2 + &c.)y^n + (a + d z + &c.)y^n-1 = 0</math>,
unde <math>y = - {a + d z + &c. / A + b z + e z^2 + &c.}</math>.</p>
<p>Cor. Area curvæ, cujus æquatio $it prædictæ formulæ; vel $olidum
ex ejus rotatione circa axim $uum tanquam ba$im generatum; vel
fluens fluxionis, quæ e$t rationalis functio quantitatum <math>x & y</math> in <math>x^.</math>
vel <math>y^., &c.</math> ducta; exprimi po$$unt ope finitorum terminorum ab$ci$$æ
& ordinatæ, & earum circularium arcuum & logarithmorum.</p>
<p>Et $ic progredi liceat ad æquationes, in quibus $olummodo conti-
nentur termini (<math>n, n - 1 & n - 2</math>) dimen$ionum; & $ic deinceps.</p>
<p>1. 2. Sit data æquatio <math>a y^n + b y^n-1 x + c y^n-2 x^2 + &c. + a y^m +
β y^m-1 x + γ y^m^.-2 x^2 + &c. = 0</math>, in quâ $olummodo continentur ter-
mini (<math>n & m</math>) dimen$ionum. Scribatur <math>y = x z</math>, quâ quantitate
pro ejus valore <math>y</math> in datâ æquatione $ub$titutâ, re$ultat <math>a z^n + b z^n-1

<pb 132>[0184]<rh>DE FLUXIONALIBUS</rh>
+ c z^n-2 + &c.) x^n + (α z^m + β z^m-1 + &c.) x^m = 0</math>; unde <math>x = (-{α z^m + β z^m-1 + &c. / a z^n + b z^n-1 + &c.})^{1 / n-m}, & y = x z</math>.</p>
<p>Cor. Si in æquatione relationem inter variabiles <math>x & y</math> exprimente
tres dimen$iones (<math>n, n - m, n - 2 m</math>) occurrant, quarum $umma tan-
tum $uperat mediam, quantum hæc media infimam; tum ope qua-
draticæ æquationis re$olutionis variabiles <math>y & x</math> per novam <math>z</math> exprimi
poterunt. Et $ic deinceps.</p>
<p>2. 1. Sit <math>Δ (x, y) = φ (x, y)</math>; ubi <math>Δ (x, y) & φ (x, y)</math> re$pective de-
notant homogeneas functiones literarum <math>x & y</math>: dimen$iones vero (<math>n</math>)
functionis <math>φ (x, y)</math> $uperent dimen$iones <math>(n - 1)</math> functionis <math>Δ (x, y)</math>
per unitatem: & $i in prædictis functionibus duæ $olummodo con-
tineantur radicales, viz. <math>√(α x + β y) & √(ε x + δ y)</math>: vel una $o-
lummodo <math>√(α x^2 + β x y + γ y^2)</math> vel <math>(α x + β y)^{1 / m}</math>; vel <math>({α x + β y / ε x + δ x})^{p / m}
= P</math>; vel <math>^m √(a + b ^r √(c + &c. ^s √({e + f x / h + k x}))) = Q, &c.</math>; tum
per prob. 15. fluens cuju$cunque rationalis functionis literarum <math>x & y</math>
in <math>x^.</math> vel <math>y^.</math> ductæ reduci pote$t ad fluentem fluxionis, quæ e$t rationa-
lis fractio variabilis quantitatis (<math>u</math>) in fluxionem <math>u^.</math> ductæ: $cribatur
enim <math>z y</math> pro <math>x</math> in datâ æquatione, & inveniatur valor quantitatis (<math>y</math>)
in terminis quantitatis (<math>z</math>); in hoc valore $olummodo continentur
radicales quantitates <math>√(α z + β) & √(ε z + δ)</math>; vel <math>√(α z^2 + β z + γ),
&c.</math>; & nullæ aliæ; unde con$tat fluxiones <math>x^. & y^.</math> $olummodo invol-
vere ea$dem radicales quantitates; & exinde rationalem functionem
literarum <math>x</math> vel <math>y</math> in <math>x^.</math> vel <math>y^.</math> ductam ea$dem $olummodo involvere ir-
rationales quantitates, & con$equenter eam per prob. 15. reduci
po$$e ad rationalem functionem variabilis quantitatis (<math>u</math>) in ejus
fluxionem (<math>u^.</math>) ductam.</p>
<p>2. 2. Sit <math>Δ (x, y) = φ (x, y)</math>; ubi <math>Δ (x, y) & φ(x, y)</math> re$pective de-
notant homogeneas functiones quantitatum (<math>x & y</math>) ab$que denomi-

<pb 133>[0185]<rh>ÆQUATIOXIBUS.</rh>
natore, quarum dimen$iones re$pective $unt <math>n - 1 & n</math>; functio au-
tem <math>Δ (x, y)</math> nullas in $e involvat radicales præter eas, quæ formulam
<math>√(α x + β y)</math> habeant; quarum etiam haud duæ vel plures in $e$e
ducantur: functio vero <math>φ (x, y)</math> contineat nullas radicales quantita-
tes; tum fluens fluxionis <math>y x^.</math> vel <math>x y^.</math> detegi pote$t ex fluentibus fluxi-
onum, quæ con$tant ex rationalibus functionibus quantitatis <math>u</math> in <math>u^.</math>,
& quantitatis <math>v</math> in <math>v^., &c.</math></p>
<p>Et $ic æquationes facile deduci po$$unt, quarum fluentes inveniri
po$$unt ope fluentium in $cholio præcedentis capitis traditarum.</p>
<p>Omnia hæc facile con$tant e $ub$titutione rectanguli <math>z y</math> pro <math>x</math> in
datâ æquatione.</p>
<p>3. 1. Si vero dimen$iones functionis <math>φ (x, y)</math> in datâ æquatione
<math>φ (x, y) = Δ (x, y)</math> majores $int quam dimen$iones functionis <math>Δ (x, y)</math>
per duas; tum fluens fluxionis <math>y x^.</math> in æquationibus, quæ ea$dem in-
volvunt radicales quantitates, ac eæ, quæ in tribus præcedentibus ca-
$ibus traduntur, inveniri pote$t ope rationalis fractionis: $cribatur
enim <math>z x</math> pro <math>y</math> in datâ æquatione <math>Δ (x, y) = φ (x, y)</math> & re$ultat <math>A x^2
= B</math>; ubi literæ <math>A & B</math> denotant functiones quantitatis <math>z</math>, in quibus
nullæ aliæ inveniuntur radicales quantitates præter eas, quæ radicali-
bus in datâ æquatione contentis corre$pondent. e. g. $it radicalis in
datâ æquatione contenta <math>√(α x + β y)</math>, & ejus corre$pondens radi-
calis in re$ultante æquatione erit <math>√(α + β z)</math>; $it etiam radicalis
quantitas in datâ æquatione <math>√(α x^2 + β x y + γ y^2)</math>, & ejus corre$pon-
dens radicalis in re$ultante æquatione erit <math>√(α + β z + γ z^2), &c.</math>
$ed ex æquatione prius tradita <math>A x^2 = B</math> $equitur <math>x = {B{1 / 2}/A{1 / 2}}</math>, cujus flu-
xio <math>x^. = {1 / 2}{B - {1 / 2}/A - {1 / 2}} × {A B^. - B A^. / A^2}</math>; $ed per $ub$titutionem prius traditam
<math>y = z x = {z B{1 / 2}/A{1 / 2}}</math>, & con$equenter <math>y x^. = {1 / 2} z × {A B^. - B A^. / A^2}</math>, quæ con-
tinebit nullas radicales quantitates, præter eas, quæ corre$pondent
irrationalibus datâ æquatione contentis: vel magis generaliter, $it <math>V</math>

<pb 134>[0186]<rh>DE FLUXIONALIBUS</rh>
homogenea & rationalis functio quantitatum <math>x & y</math>, cujus numerus
dimen$ionum $it impar, tum fluxio <math>V x^.</math> vel <math>V y^.</math> etiam continebit nul-
las radicales quantitates præter eas, quæ corre$pondent irrationalibus
datâ æquatione contentis.</p>
<p>3. 2. In genere $int <math>φ (x, y) & Δ (x, y)</math> homogeneæ functiones <math>n &
m</math> dimen$ionum, quarum <math>n</math> major $it quam <math>m</math>; & fluens omnis fluxi-
onis <math>A x^. + B y^.</math> ubi literæ <math>A & B</math> denotant homogeneas functiones
literarum <math>x & y</math> dimen$iones (<math>r</math>) habentes de$ignari pote$t per flu-
entem functionis variabilis quantitatis (<math>z</math>) ductæ in fluxionem <math>z^.</math>
quæ continebit nullas radicales quantitates præter eas, quæ corre-
$pondent irrationalibus datâ æquatione contentis: $i modo literæ <math>A</math>
& <matyh>B</math> nullas involvant radicales quantitates præter eas, quæ in datâ
æquatione continentur, & <math>{r + 1 / n - m}</math> $it integer numerus vel affirmativus
vel negativus.</p>
<p>Con$tat e $ub$titutione prædictâ <math>z y</math> pro <math>x</math>.</p>
<p>Eadem principia etiam applicari po$$unt ad fluentes fluxionalium
quantitatum $uperiorum ordinum. e. g. Sit <math>s</math> ordo fluxionalium
quantitatum & <math>r</math> dimen$io homogenearum functionum (<math>P, Q, R, &c.</math>)
quantitatum (<math>x & y</math>), quæ in fluxiones re$pective ducantur; deinde
pro <math>y</math> $cribatur <math>z x</math>, & pro <math>y^.</math> ejus fluxio; & $ic deinceps; tum in flu-
xione re$ultante nulla involvetur radicalis quantitas; quæ non cor-
re$pondet iis, quæ vel in datâ æquatione vel in quantitatibus (<math>P, Q,
R, &c.</math>) inveniuntur; $i modo <math>{r + s / n - m}</math> $it integer numerus vel affirma-
tivus vel negativus.</p>
<p>Et $ic facile deduci po$$unt con$imiles æquationes, quarum quædam
fluxiones a$$ignari po$$unt, & quarum fluentes exprimi po$$unt ope
fluentium in $cholio præcedentis capitis traditarum.</p>
<p>Hæc principia etiam ad plures æquationes plures incognitas quan-
titates habentes applicari po$$unt.</p>

<pb 135>[0187]<rh>ÆQUATIONIBUS.</rh>
<p>4. Sit æquatio <math>x^α (e + f x^n y^p + g x^2n y^2p + &c.)^λ + H x^β (l + m x^n y^p
+ o x^2n y^2p + &c.)^μ = K x^π (q + r x^n y^p + s x^2n y^2p + &c.)^γ + &c.</math> $cribatur
<math>v^n = x^n y^p</math>, & re$ultat æquatio <math>v^2 y^-{pα / n} (e + f v^n + g v^2n + &c.)^λ +
H v^β y^-{pβ / n} (l + m v^n + o v^2n + &c.)^μ = K v^π y^-{pπ / n} (q + r v^n + s v^2n +
&c.)^γ + &c.</math></p>
<p>Ex. Sit æquatio <math>x^α + a x^n+α y^p = b y^β,</math> i.e. <math>x^α (1 + α x^n y^p) = b y^β</math>;
$cribatur <math>v^n = x^n y^p</math>, & re$ultat æquatio <math>v^α y^-{pα / n} (1 + a v^n) = b y^β</math>,
unde <math>b y^{βn+pα / n} = v^α (1 + a v^n) & y = b{-n / βn + pα} ((v^α) (1 + α v^n)){n / βn + pα}</math>, &
exinde <math>x = v × b^{p / βn+pα} ((v^α) (1 + a v^n))^{-p / βn+pα}, &c.</math></p>
<p>5. Ducatur data æquatio in a$$umptas quantitates, & nonnunquam
deduci pote$t æquatio, cujus incognitæ quantitates <math>x & y</math> exprimi
po$$unt ope tertiæ <math>z</math>.</p>
<p>Ex. 1. Sit data æquatio <math>y^n + a x^m y^r = b x^s</math>, ducatur data æquatio
in <math>x^α</math>, & re$ultat <math>y^n x^α + a y^r x^m+z = b x^s+α</math>; $it <math>n:α::r:m + α</math>; unde
<math>α = {n m / r - n}</math>; $cribatur <math>y^n x^α = v^n</math>, & re$ultat æquatio <math>v^n + a v{r / n} = b x^s+α,
& y^n = {v^n / x^α}</math>.</p>
<p>Cor. Si <math>n</math> vel <math>m</math> vel <math>s = o</math>; vel <math>r = n</math>, vel <math>m = s</math>; tum ad invenien-
dos valores quantitatum <math>x & y</math> per novam quantitatem <math>v</math> haud nece$-
$ario ducenda e$t data æquatio in quantitatem <math>x^α</math>.</p>
<p>Ex. 2. Sit data æquatio <math>y^n + a x^m y^r + b x^p y^s = c x^q</math> ducatur hæc
æquatio in <math>x^α</math>, & re$ultat <math>y^n x^α + a x^m+α y^r + b x^p+α y^s = c x^q+α</math>.</p>
<p>Si omnes pote$tates quantitatum <math>y</math> vel <math>x</math> in datâ æquatione con-
tentæ $int inter $e æquales; vel duæ $int diver$æ pote$tates, quarum
una dimidium e$t alterius, &c. vel $i in æquationem horum generum
reduci po$$it data æquatio ex ejus multiplicatione in <math>x^α</math> vel <math>y^α</math>, ubi <math>α</math>
$it index ad libitum a$$umendus; vel $i modo ducatur data æquatio
in <math>x^α</math>; & vel pro <math>y^n x^α</math>, vel pro <math>y^r x^α+m</math>, vel pro <math>y^s x^p+α</math> $cribatur <math>v</math>, &
æquatio data reducatur in terminos quantitatum <math>v & x</math>; & $i æqua-

<pb 136>[0188]<rh>DE FLUXIONALIBUS</rh>
tiones re$ultantes $int prædictorum generum, &c. tum facile exprimi
po$$unt quantitates <math>x & y</math> per novam quantitatem <math>v</math>.</p>
<p>Et $ic de transformandis pluribus æquationibus, ita ut earum in-
cognitæ quantitates exprimantur per novam quantitatem <math>v</math>.</p>
<p>6. 1. Nonnunquam ita transformari po$$unt e $ub$titutione æqua-
tiones, ut earum incognitæ quantitates (<math>x, y, &c.</math>) exprimi po$$int in
terminis novæ a$$umptæ <math>v</math>; e.g. datâ algebraicâ æquatione duas in-
cognitas quantitates <math>x & y</math> involvente; invenire utrum $ingula quan-
titas ex iis $it rationalis functio tertiæ <math>z</math>, necne; a$$umantur rationales
functiones <math>a + b v + c v^2 + &c. = x</math>, vel <math>{a + b v + c v^2 + &c. / p + q v + r v^2 + &c.} = x
& a′ + b′v + c′v^2 + &c. = y</math> vel <math>{a′ + b′v + c′v^2 + &c. / p′ + q′v + r′v^2 + &c.} = y</math>; $ub$ti-
tuantur hæ quantitates pro $uis valoribus in datâ æquatione, & $up-
ponantur corre$pondentes termini re$ultantis æquationis nihilo æqua-
les; & ex æquationibus re$ultantibus deduci po$$unt incognitarum
quantitatum valores, $i modo ulli dentur.</p>
<p>2. Et $ic inveniri pote$t, utrum $ingula variabilis quantitas (<math>x, y,
&c.</math>) in datâ vel datis æquationibus contenta $it rationalis functio vel
irrationalis datæ formulæ variabilis <math>v</math>, necne; a$$umantur in genere
irrationales functiones datæ formulæ pro incognitis quantitatibus (<math>x,
y, &c.</math>); quibus pro $uis valoribus in datâ vel datis æquationibus $ub-
$titutis, & terminis re$ultantium æquationum corre$pondentibus ni-
hilo æqualibus e$$e $uppo$itis; ex æquationibus re$ultantibus erui
po$$unt incognitarum quantitatum valores, $i modo tales recipiant.</p>
<p>3. E prob. 54. medit. algebr. deduci po$$unt æquationes, qua-
rum incognitæ quantitates <math>x & y</math> exprimi po$$unt ope tertiæ <math>v</math> a$$u-
mendæ: 1. a$$umatur quæcunque functio literæ <math>v</math> pro <math>y</math>, deinde a$$u-
matur quæcunque functio literarum <math>v & y</math> pro <math>x</math>; ita transformentur hæ
duæ æquationes in unam, ut exterminetur litera <math>v</math>; & re$ultat æqua-
tio relationem inter literas <math>x & y</math> exprimens, ubi literæ <math>x & y</math> $unt

<pb 137>[0189]<rh>ÆQUATIONIBUS.</rh>
datæ functiones tertiæ <math>v.</math> Ex. g. a$$umatur <math>y = {a + b v + c v^2 + &c. / p + q v + r v^2 + &c.}& x y = v vel {x / y} = v</math>, quibus quantitatibus pro $uo valore <math>v</math> in primâ
æquatione $ub$titutis, re$ultant æquationes quæ$itæ.</p>
<p>4. A$$umantur quæcunque functiones quantitatis <math>v</math> pro <math>x & y</math>,
deinde a$$umatur æquatio relationem inter <math>x & y</math> exprimens formulæ,
quæ recipere pote$t prædictas functiones quantitatis <math>v</math> pro valoribus
corre$pondentibus quantitatum <math>x & y</math>; $ub$tituantur a$$umptæ fun-
ctiones quantitatis <math>v</math> pro $uis valoribus <math>x & y</math> in a$$umptâ æquatione;
& fiant corre$pondentes termini æquationis re$ultantis inter $e æqua-
les; & ex æquationibus re$ultantibus detegi pote$t æquatio quæ$ita.</p>
<p>Omnia hæc etiam ad fluxionales æquationes applicari po$$unt.</p>
<p>7. Sit æquatio <math>a(y^n + z^n) + b(y^n-1 z + z^n-1 y) + c(y^n-2 z^2 + z^n-2 y^2)
+ &c. + p(y^n-1 + z^n-1) + q(y^n-1 z + z^n-1 y) + &c. = 0</math>, in quâ $i-
militer involvuntur quantitates <math>z & y</math>; pro <math>y + z</math> $cribatur <math>w</math>, & pro
<math>z y</math> $cribatur <math>v</math>; & æquatio re$ultans erit <math>a w^n + (b - na) w^n-2 v +
(c - (n - 2)b + n. {n-3 / 2} a)w^n-4 v^2 + (d - (n-4) c + (n-2) ×{n-5 / 2}b - n.{n-4 / 2}.{n-5 / 3} a)w^n-6 v^3 + (e - (n-6)d + (n-4) × {n-7 / 2}c - (n-2) × {n-6 / 2} × {n-7 / 3}b + n × {n-5 / 2} × {n-6 / 3} × {n-7 / 4}a)w^n-8 v^4 +
&c. + pw^n-1 + (q - (n-1)p)w^n-3 v + (r - (n-3)q + (n-1) × {n-4 / 2}p)
w^n-5 v^2 + &c. = 0</math>. Lex coefficientium huju$ce $eriei facile con$tabit
ex ob$ervat. leg. $eriei in theor. 38. medit. algeb. contentæ.</p>
<p>Quantitas <math>v</math> nunquam a$$urgit ad majores quam <math>{n / 2}</math> dimen$iones, $i
<math>n</math> $it par numerus; vel majores quam <math>{n-1 / 2}</math>, $i <math>n</math> $it impar: $i vero
detur quantitas <math>v</math> e terminis quantitatis <math>w</math>, tum re$olutione quadra-
ticæ æquationis dantur <math>z & y</math> in terminis quantitatis <math>w</math>.</p>

<pb 138>[0190]<rh>DE FLUXIONALIBUS</rh>
<p>Cor. Sit <math>n = 2 vel 3, & v</math> haud a$$urgit ad majores quam unam;
$it <math>n = 4 vel 5, & v</math> haud a$$urgit ad majores quam 2; $it <math>n = 8</math> vel
9, & haud a$$urget <math>v</math> ad majores quam 4 dimen$iones; & quoniam
extractio biquadraticæ æquationis cogno$citur; ergo fluens cuju$cun-
que fluxionis, quæ e$t functio quantitatum <math>x & y</math> in <math>x^.</math> vel <math>y^.</math> ducta, per
methodos prius traditas deduci pote$t; $i modo dimen$iones datæ
æquationis relationem inter variabiles <math>x & y</math> exprimentis, in quâ <math>x &
y</math> $imiliter involvuntur, haud $uperent novem dimen$iones.</p>
<p>7. 2. Sit æquatio relationem inter variabiles <math>z & w</math> exprimens eju$-
modi, ut quantitas (<math>z</math>) $it cognita functio quantitatis <math>w</math>; vel utræque
(<math>z & w</math>) $int cognitæ functiones novæ quantitatis (<math>v</math>); deinde a$$u-
mantur duæ æquationes relationes inter <math>z, w, x & y</math> exprimentes, ita
ut ex datis valoribus quantitatum <math>z & w</math> $emper acquiri po$$unt cor-
re$pondentes valores incognitarum quantitatum <math>x & y</math>; deinde redu-
cantur hæ æquationes in unam, ita ut exterminentur incognitæ
quantitates <math>z & w</math>, & re$ultat æquatio relationem inter <math>x & y</math> expri-
mens, cujus utræque <math>x & y</math> facile de$ignari po$$unt ope tertiæ quanti-
tatis a$$umptæ: & con$equenter fluxio, quæ e$t functio quantitatum
<math>x & y</math> & earum fluxionum, ope prædictæ tertiæ quantitatis & ejus
fluxionum exprimi pote$t.</p>
<p>Con$imilia etiam ad tres vel plures æquationes applicari po$$unt.</p>
<p>Si vero plures dentur æquationes plures variabiles quantitates ha-
bentes, nonnunquam detegi po$$unt re$olutiones ex his vel ex aliis
principiis in medit. algeb. traditis.</p>
<p>8. Datis methodis inveniendi continuas approximationes ad flu-
entes datarum fluxionum, ex iis haud nunquam con$tant fluentes
ip$æ, i.e. haud nunquam terminatur $eries ip$a.</p>
<p>Con$tant ex infinitis $eriebus infinitæ formulæ harum æquationum.</p>
<p>Ex. Sit data æquatio <math>y^m + a x^n = b y^e x^r</math>; $eries, quæ exprimit flu-
entem fluxionis <math>y x^.</math>, pote$t e$$e huju$ce formulæ <math>A x y + B y^e+1 x^c+1 +
C y^2e+1 x^2c+1 + D y^3e+1 x^3c+1 + &c.</math> ubi <math>c = r - n</math>: inveniatur fluxio hu-
ju$ce $eriei & re$ultat æquatio <math>(P) y x^. = (A x + (e+1) B y^e x^c+1 +
(2e + 1) Cy^2e x^2c+1 + &c.)y^. + (A y + (c+1) B y^e+1 x^c + (2c+1)

<pb 139>[0191]<rh>ÆQUATIONIBUS.</rh>
C y^2e+1 x^2c + &c.)x^.:</math> fluxio autem datæ æquationis erit <math>(m y^m-1 -
e b y^e-1 x^r) y^. = (r b y^e x^r-1 - n a x^n-1)x^.</math>, in æquatione (<math>P</math>) pro <math>y^.</math> $cri-
batur ejus valor <math>{r b y^e x^r-1 - n a x^n-1 / m y^m-1 - e b y^e-1 x^r}x^.</math> e po$teriori æquatione flu-
xionali deductus, & re$ultat <math>y x^. = (Ax + (e+1) B y^e x^c+1 + (2e
+ 1) Cy^2e x^2c+1 + &c.) ({r b y^e x^r-1 - n a x^n-1 / m y^m-1 - e b y^e-1 x^r})x^. + (A y + (c+1)
B y^e+1 x^e + (2c+1) C y^2e+1 x^2c + &c.)x^.</math>; quâ reductâ, fit (<math>Q</math>)
<math>n a A x^n \\ - (mA - m) y^m # - e b y^e x^r \\ + e b A y^e x^r \\ - r b A y^e x^r \\ + n a (e + 1) B y^e x^c+n(r) \\ - m (c + 1) B y^e+m x^c # + &c. = 0.</math></p>
<p>In terminis <math>(mA - m) y^m & m (c + 1) B y^e+m x^c</math> pro <math>y^m</math> $cribatur
ejus valor <math>- a x^n + b y^e x^r e</math> datâ æquatione deductus, & re$ul-
tant re$pective <math>- (m A - m) a x^n + (m A - m) b y^e x^r & - m (c
+ 1) B a y^e x^n+c(r) + m (c+1) B y^2e x^r+c(n+2c)</math>; quibus quantitatibus
pro $uis valoribus in æquatione <math>Q</math> $ub$titutis, re$ultat <math>(Q)=
n A a \\ + m a(A - 1)
 # x^n -e b \\ x^n + e b A \\ - r b A \\ + n a (e + 1) B \\ + m (c + 1)a B \\ - (m A - m) b # y^e x^r + &c. = 0</math></p>
<p>Fiant corre$pondentes termini re$ultantis æquationis nihilo re$pe-
ctive æquales, & habebimus <math>nA + mA - m = 0</math>, (unde <math>A = {m / m + n}</math>);
& <math>(n (e + 1) + m (c + 1)) aB - (e - e A + r A + m A - m) b = 0</math>;
pro <math>A & r</math> in hâc æquatione $cribantur earum valores <math>{m / m + n} & n + c</math>
re$pective, & invenietur <math>B = {n e + m c / (n (e + 1) + m(c + 1))(m + n)}×{b / a}</math>, &

<pb 140>[0192]<rh>DE FLUXIONALIBUS</rh>
$ic deinceps: $eries, quæ exprimit <math>$. y x^. erit = {m / m + n} x y +{m c + n e / (m (c+1) + n(e+1))(m+n)}×{b / a} y^e+1 x^c+1 + {(m-e)(c+1) + r(e+1) / m×(2c+1) + n×(2e+1)}× {b B / a} y^2e+1 x^2c+1 + {(m-e)(2c+1) + r (2e+1) / m(3c+1) + n (3e+1)} × {bC / a} y^3e+1 x^3c+1 +{(m-e)(3c+1) + r (3e+1) / m (4c+1) + n (4e+1)} × {bD / a} y^4e+1 x^4c+1 + &c.</math></p>
<p>Cor. 1. Si <math>(m-e)(lc+1) + r(le+1) = 0</math>, cum <math>l</math> $it integer af-
firmativus numerus, tum terminatur $eries; aliter vero non.</p>
<p>Cor. 2. Si <math>e = 0</math>, tum hæc $eries facile transformari pote$t in bi-
nomiale theorema.</p>
<p>Ex. 2. Sit data æquatio <math>y^m = a x^n + b y^e x^c + n + d y^2e x^2c+n + f y^3e x^3c+n
+ &c.</math> & $eries, quæ exprimit aream pote$t e$$e huju$ce formulæ <math>$. y x^.
= A x y + B y^e+1 x^c+1 + C y^2e+1 x^2c+1 + D y^3e+1 x^3c+1 + &c.</math> Ex calculo
facile deduci po$$unt coefficientes <math>A, B, C, &c.</math></p>
<p>Ex. 3. Sit fluxionalis æquatio <math>y^m y^. = a x^n-1 x^. + (b y^e-1 x^c+n + d y^2e-1
x^2c+n + f y^3e-1 x^3c+n + &c.) y^. + (p y^e x^c+n-1 + q y^2e x^2c+n-1 + r y^3e x^3c+n-1 +
&c.) x^.</math>; & $eries quæ exprimit <math>$. y x^.</math> pote$t e$$e huju$ce formulæ <math>A x y
B y^e+1 x^c+1 + C y^2e+1 x^2c+1 + &c.</math></p>
<p>Facile infinitæ formulæ $erierum fingi po$$unt, quæ exprimunt
fluentes quarumcunque fluxionum, quæ $unt datæ functiones varia-
bilium quantitatum in datis algebraicis vel fluxionalibus æquationi-
bus contentarum; $ed de his po$tea in capite de infinitis $eriebus.</p>
<h>PROB. XXXIII.</h>
<p>1. _Datâ algebraicâ æquatione_ <math>n</math> _dimen$ionum_ <math>y^n + (a + b x) y^n-1 +
(c + d x + e x^2) y^n-2 + &c. = 0</math> _relationem inter ab$ci$$am_ <math>x</math> _& ejus_
_corre$pondentes ordinatas_ <math>y</math> _exprimente; invenire æquationem relationem_
_inter ab$ci$$am_ <math>x</math> _& ejus aream_ <math>v</math> _de$ignantem, $i modo curva generaliter_
_quadrari po$$it._</p>

<pb 141>[0193]<rh>ÆQUATIONIBUS.</rh>
<p>Datâ ab$ci$sâ <math>x, n</math> $unt diver$i valores ordinatæ <math>y</math>, qui datæ abfci$$æ
corre$pondent, & con$equenter <math>n</math> $unt diver$i areæ valores ei corre-
$pondentes; æquatio relationem inter ab$ci$$am <math>x</math> & aream <math>v</math> erit
huju$ce formulæ <math>v^n + (A + B x + C x^2) v^n-1 + (D + E x + F x^2
+ G x^3 + H x^4) v^n-2 + (K + L x + M x^2 + .. P x^6) v^n-3 &c. = 0</math>;
a$$umatur hæc æquatio pro æquatione quæ$itâ, & per problema
25. inveniatur æquatio, cujus radix e$t ordinata <math>y={v^. / x^.}</math>; $ingulis
re$ultantis & datæ æquationis terminis re$pective inter $e æquatis,
re$ultant æquationes, quarum incognitæ quantitates erunt re$pective
<math>A, B, C, D, &c.</math> & quarum valores per methodum communes divi$o-
res inveniendi facile erui po$$unt; $i vero curva haud quadraturam
admittat, tum nulli dantur prædicti communes divi$ores.</p>
<p>Ex. 1. Sit data æquatio <math>y^2 + x^2 - 1 = 0</math>, & $upponatur æquatio
ad aream <math>v^2 + (A + B x + C x^2) v + D + E x + F x^2 + G x^3 +
H x^4 = 0</math>; per prædictum problema inveniatur æquatio, cujus radix
e$t <math>{v^. / x^. = y}</math>; & $i modo evane$cat diver$orum areæ valorum $umma, cum
evane$cat ab$ci$$a <math>x</math>, i.e. $it <math>A = 0</math>; erunt <math>B & C</math> etiam nihilo æquales;
& per problema 25. <math>{(E + 2 F x + 3 G x^2 + 4 H x^3)^2 / 4(D + E x + F x^2 + G x^3 + H x^4)} = x^2 - 1</math>, &
con$equenter <math>4 H = 16 H^2, 4 G = 24 H G, 4 F - 4 H = 16 H F +
9 G^2, 4 E - 4 G = 8 H E + 12 F G, 4 D - 4 F = 6 G E + 4 F^2, -
4 E = 4 E F, - 4 D = E^2</math>; $ed e methodo communes divi$ores inve-
niendi, &c. con$tat has æquationes inter $e contradictorias e$$e, &
con$equenter curvam haud generaliter e$$e quadrabilem.</p>
<p>Ex. 2. 1. Sit æquatio <math>y^2 + (4 + 2 x)y + x^2 + 3 x + 1 = 0</math>; a$$umatur
æquatio <math>v^2 + (A + B x + C x^2)v + D x^4 + E x^3 + F x^2 + G x + H = 0</math>;
tum erunt <math>B = 4, C = 1 & A = 0</math>, $i modo evane$cat arearum
$umma, cum evane$cat ab$ci$$a; per exemplum primum problema-
tis 25. invenietur <math>{(D x^4 + E x^3 + F x^2 + G x + H)(2 x + 4)^2 - (x^2 + 4 x)(2 x + 4)(4 D x^3 +3 E x^2 + 2 F x + G) + (4 D x^3 + 3 E x^2 + 2 F x + G)^2 /
4Dx^4 + 4Ex^3 +

<pb 142>[0194]<rh>DE FLUXIONALIBUS</rh>
4Fx^2 + 4Gx + 4H - (4x + x^2)^2} =
x^2 + 3 x + 1</math>, reducatur hæc æquatio, & corre$pondentes termini in-
ter $e æquales e$$e $upponantur; re$ultat <math>16 D^2 - 8 D = -1</math>, unde
<math>D = {1 / 4}</math> quo valore in reliquis re$ultantibus terminis $ub$tituto, exo-
riuntur æquationes <math>0 = 0; & 9 E^2 - 32 E + 28 = 0</math>, unde <math>E</math> erit vel <math>{14 / 9}</math>
vel 2; $ub$tituatur <math>{14 / 9}</math> pro <math>E</math> in reliquis æquationibus, & inveniuntur
<math>F = 0, G = - 12, & H = - 12</math>; & con$equenter æquatio quæ$ita
erit <math>v^2 + (4 x + x^2)v + {1 / 4} x^4 + {14 / 9} x^3 - 12 x - 12 = 0</math>.</p>
<p>2. Sit <math>A</math> quæcunque quantitas, & con$equenter non evane$cat are-
arum $umma, cum evane$cat ab$ci$$a; tum erit æquatio ad aream
<math>v^2 + (A + 4 x + x^2) v + {1 / 4} x^4 + {14 / 9} x^3 + {1 / 2} A x^2 + (2 A - 12) x +{1 / 4} A^2 - 12 = 0.</math></p>
<p>Hæc æquatio facile deduci pote$t e priore, $cribendo pro <math>v</math> ejus valo-
rem <math>v + {A / 2}</math>: nam per hanc $ub$titutionem transformatur prior æqua-
tio ad aream, i.e. cujus diver$orum valorum $umma evane$cit, cum
ab$ci$$a evadat nihilo æqualis; in æquationem ad aream, cujus $umma
valorum erit <math>A</math>, cum <math>ab$ci$$a = 0</math>.</p>
<p>2. 2. Eodem modo $it æquatio ad aream <math>(v) v^n + (B x + Cx^2) v^n-1 +
(D x^4 + E x^3 + &c.)v^n-2 + &c. = 0</math>, cujus valorum $umma evane$cit,
cum <math>ab$ci$$a = 0</math>: in hâc æquatione pro <math>v</math> $cribatur <math>v + {A / n}</math>, & æquatio
re$ultans erit æquatio ad aream eju$dem curvæ, cujus $umma valo-
rum erit <math>A</math>, cum evane$cat ab$ci$$a <math>x</math>.</p>
<p>3. Paulo aliter progredi licet in hoc & multis aliis ca$ibus; e.g.
eâdem æquatione datâ, & ii$dem literis ea$dem quantitates denotan-
tibus; a$$umatur etiam æquatio ad aream <math>v^n + (A + B x + C x^2)

<pb 143>[0195]<rh>ÆQUATIONIBUS.</rh>
v^n-1 + (D + E x + F x^2 + G x^3 + H x^4) v^n-2 + &c. = 0</math>, & con$e-
quenter pro <math>{v^. / x^.}</math> $cribendo <math>y</math>, erit <math>n y v^n-1 + (n - 1)(A + B x + C x^2)
v^n-2 y + (n - 2)(D + E x + F x^2 + &c.)v^n-3 y + &c. + (B + 2 C x)
v^n-1 + (E + 2F x + 3 G x^2 + 4 H x^3)v^n-3 + &c. = 0;</math> e quibus æqua-
tionibus, $i methodis notis exterminetur <math>v</math>, habebimus æquationem,
quæ exprimit relationem inter <math>x & y</math>; hujus autem æquationis coef-
ficientes æquari debent coefficientibus datæ æquationis <math>y^n + (a + b x)
y^n-1 + (c + d x + e x^2) y^n-2 + &c. = 0</math>. & $i quantitates <math>A, B, C, &c.</math>
exinde determinari po$$int, curva quadratur; aliter autem quadrari
non pote$t.</p>
<p>4. Datâ præcedente algebraicâ æquatione <math>y^n + (a + b x) y^n-1 +
(c + d x + e x^2)y^n-2 + &c. = 0</math>; invenire utrum fluens <math>v</math> fluxionis
<math>((α + βx + γx^2 + &c. + x^3)y^m + (p + q x + r x^2 + &c. + x^s-1)y^m+1
+ (&c.))x^. = Δx^.</math> inveniri pote$t, necne; ubi literæ <math>s, m, &c.</math> integros
denotant numeros. Per medit. algeb. inveniatur æquatio, cujus ra-
dix $it <math>Δ</math>: a$$umatur æquatio ad fluentem <math>v^n + (A + B x + C x^2 ...
P x^m+s+1)v^n-1 + (D + E x^. + F x^2 + &c. + Q x^2m+2s+2)v^n-2 + &c. = 0</math>,
deinde inveniatur æquatio cujus radix $it <math>{v^. / x^.}</math>; re$ultantis & inventæ
æquationis, cujus radix $it <math>Δ</math>, fiant termini inter $e æquales, & e re-
$ultantibus exinde æquationibus per methodum communes divi$ores
inveniendi erui po$$unt valores quantitatum <math>A, B, C, &c. P; D, E, F,
&c. Q; &c.</math> $i vero nulli dentur corre$pondentes valores harum quan-
titatum, i.e. æquationes re$ultantes inter $e contradictoriæ $int; tum
finitis terminis haud exprimi pote$t fluens.</p>
<p>Cor. 1. Si pro <math>x^.</math> $cribatur <math>x^. ^r</math>, & requiratur fluens <math>(r)</math> ordinis: pro
<math>x^m+s+1</math> $cribe <math>x^m+s+r</math>; pro <math>x^2m+2s+2</math> $cribe <math>x^2m+2s+2r</math>, &c. & eodem modo
quo prius deduci pote$t re$olutio.</p>
<p>5. Datâ algebraicâ æquatione <math>A y^n + B y^n-1 + C y^n-2 + &c. = 0</math>,
(ubi literæ <math>A, B, C, &c.</math> qua$cunque rationales functiones literæ <math>x</math> re-
$pective denotant) relationem inter duas incognitas quantitates <math>x & y</math>

<pb 144>[0196]<rh>DE FLUXIONALIBUS</rh>
exprimente; datâ etiam quantitate, quæ e$t fluxionalis functio (<math>v^.</math>)
incognitarum datæ æquationis quantitatum (<math>x & y</math>); invenire utrum
fluens (<math>v</math>) prædictæ fluxionalis functionis inveniri pote$t, necne; i. e.
relatio inter incognitas quantitates <math>x & y & v</math> per algebraicam æqua-
tionem exprimi pote$t, necne.</p>
<p>Ex datâ algebraicâ æquatione inveniatur æquatio, cujus radix e$t
data fluxionalis functio variabilium datæ æquationis quantitatum
(<math>x & y</math>).</p>
<p>A$$umatur æquatio, cujus radix e$t fluens quæ$ita (<math>v</math>); & formula,
quæ nece$$ario continet radices quæ$itæ fluentis <math>v</math>, $i modo illæ radi-
ces finitis algebraicis terminis quantitatum <math>x & y</math> exprimi po$$int, viz.
<math>v^m + (A + B x + &c.) v^m-1 + (D + E x + &c.)v^m-2 + &c. = 0, vel
(α + β x + &c.)v^m + (A + B x + &c.)v^m-1 + &c. = 0</math>.</p>
<p>Hujus æquationis formula plerumque e datâ æquatione & fluxio-
nali quantitate facile erui pote$t: e numero enim radicum, quas
præbent data æquatio & fluxionalis quantitas, con$tat numerus di-
men$ionum quantitatis <math>v</math> in quæ$itâ æquatione contentarum; & $ic
de numero dimen$ionum, &c. quantitatum <math>x vel y</math> in quæ$itâ æquati-
one contentarum.</p>
<p>Ex a$$umptâ æquatione inveniatur altera, cujus radix e$t <math>v^.</math>; re$ul-
tantis æquationis, etiamque æquationis, cujus radix e$t fluxionalis
quantitas, terminis corre$pondentibus inter $e æqualibus e$$e $uppo-
$itis; re$ultant æquationes, e quibus methodo communes divi$ores
inveniendi erui po$$unt incognitæ coefficientes a$$umptæ æquationis;
$i vero hæ æquationes inter $e contradictoriæ $unt, tum fluens quæ-
$ita haud inveniri pote$t.</p>
<p>Et $ic de duabus vel pluribus æquationibus tres vel plures varia-
biles quantitates habentibus.</p>
<p>Cor. 1. Hinc inveniri po$$unt quamplurimæ algebraicæ curvæ,
quarum relationes inter earum areas & ab$ci$$as (<math>x</math>) exprimi po$$unt
per algebraicas æquationes: a$$umantur enim quæcunque datæ æqua-
tiones <math>Av^n + Bv^n-1 + Cv^n-2 + &c. = 0</math>, ubi <math>A, B, C, &c.</math> qua$cun-

<pb 145>[0197]<rh>Æ QUATIONIBUS.</rh>
que functiones ab$ci$$æ <math>x</math> re$pective denotant; & per prob. 25. inve-
niatur æquatio, cujus radix $it <math>{v^. / x^.} = y</math>; & invenitur æquatio relatio-
nem inter ab$ci$$am <math>x</math> & ejus ordinatas <math>y</math> exprimens; & cujus æqua-
tio, quæ exprimit relationem inter aream <math>v</math> prædictæ curvæ & ejus
ab$ci$$am <math>x</math>, e$t <math>A v^n + B v^n-1 + &c. = 0</math>.</p>
<p>Facile con$tat, $i modo æquatio <math>A v^n + B v^n-1 + &c. = 0</math> dividi
po$$it in duas alias, æquationem relationem inter ab$ci$$am <math>x</math> & ejus
ordinatas <math>y</math> exprimentem etiam dividi po$$e in duas alias; $in aliter
vero non.</p>
<p>Cor. 2. Hinc eâdem methodo etiam inveniri po$$unt quamplurimæ
algebraicæ curvæ, quarum relationes inter ab$ci$$as <math>x</math> & fluentem <math>v</math> cu-
ju$cunque algebraicæ functionis (<math>P</math>) ab$ci$$æ <math>x</math> & ordinatæ <math>y</math> in <math>x^.</math> ductæ
per algebraicam æquationem de$ignari po$$unt: a$$umatur enim æquatio
relationem inter <math>v & x</math> exprimens, cujus radix <math>v</math> e$t fluens fluxionalis
functionis; deinde inveniatur æquatio, cujus radix e$t <math>{v^. / x^.} = P</math>, pro <math>P</math>
$cribatur prædicta functio quantitatum <math>x & y</math>; & re$ultat æquatio
quæ$ita relationem inter <math>x & y</math> exprimens.</p>
<p>Et $ic de con$imilibus ad fluentes $uperiorum ordinum prædicta-
rum fluxionalium functionum quantitatum <math>x & y</math> & earum fluxio-
num applicandis.</p>
<p>6. Hi quinque ca$us etiam re$olutionem accipere po$$unt e methodo
convergentes $eries inveniendi: e. g. datâ algebraicâ æquatione <math>y^n +
(a + b x) y^n-1 + &c. = 0</math>; invenire, annon ejus area exprimi pote$t
in algebraicis terminis ab$ci$$æ <math>x</math>.</p>
<p>Per convergentes $eries inveniantur <math>n</math> valores ordinatæ <math>y</math>, qui $int
re$pective <math>y, y′, y″, &c.</math> & re$ultabunt <math>y = α x + β + {γ / x} + {δ / x^2} + &c.
y′ = α′ x + β′ + {γ′ / x} + {δ′ / x^2} + &c. y″ = α″ x + β″ + &c.</math> ubi <math>γ, γ′, γ″, &c.</math>
nihilo debent e$$e æquales, aliter area haud inveniri pote$t. Sit <math>v</math> area,

<pb 146>[0198]<rh>DE FLUXIONALIBUS</rh>
& ejus diver$i valores <math>v, v′, v″, &c.</math> erunt <math>v = {α x^2 / 2} + β x + A - {δ / x}- &c. v′ = {α′x^2 / 2} + β′x + A′ - {δ′ / x} - &c. v″ = α″x^2 + β″x + A″ -{δ″ / x} - &c. &c. = &c.</math></p>
<p>Ducantur hæ æquationes in $e$e, & re$ultat æquatio <math>v^n + (π x^2 +
ξ x + σ) v^n-1 + &c.</math> quæ erit æquatio ad aream, $i modo ea in alge-
braicis terminis inveniri po$$it; quod con$tabit e terminis re$ultantis
æquationis haud in infinitum progredientibus, i. e. haud in negati-
vas dimen$iones quantitatis <math>v</math> de$cendentibus.</p>
<p>2. Datâ algebraicâ æquatione <math>A y^n + B y^n-1 + &c. = 0</math>, ubi <math>A, B,
C, &c.</math> $unt algebraicæ functiones quantitatis <math>x</math>, & fluxionali quanti-
tate; per cap.1. medit. algebr. inveniri pote$t, utrum ejus fluens ex-
primi pote$t finitis algebraicis terminis ab$ci$$æ <math>x</math>, necne.</p>
<h>THEOR. XVI.</h>
<p>Si fluens cuju$cunque fluxionis e datâ æquatione deductæ genera-
liter exprimi po$$it finitis terminis ejus variabilium quantitatum <math>x &
y</math>; tum $umma e $ingulis valoribus prædictæ fluxionis exprimi pote$t
finitis terminis prædictarum variabilium quantitatum <math>x & y</math>; $i igitur
ita transformentur variabiles quantitates <math>x & y</math>, ut con$tet $ummam
prædictam haud exprimi po$$e generaliter finitis terminis prædictarum
quantitatum <math>x & y</math>, concludere licet fluentem ip$am haud exprimi
po$$e finitis terminis quantitatum <math>x & y</math>.</p>
<p>Ex. Datâ algebraicâ æquatione haud divi$orem recipiente <math>Ay^n +
(a + b x) y^n-1 + (c + d x + e x^2) y^n-2 + &c. = 0</math>, & relationem inter
ab$ci$$am <math>x</math> & ejus corre$pondentes ordinatas <math>y</math> exprimente. In datâ
æquatione pro literis <math>x & y</math> $cribantur re$pective <math>l v + m z + n′ &
p v + q z + r</math>, & ($i modo po$$ibile $it) re$ultet æquatio <math>(Pz + Q)
v^n-1 + (R z^2 + S z + T) v^n-2 + &c. = 0</math>, cujus primus terminus <math>v^n</math>
dee$t; & $i <math>P z + Q</math> haud $it divi$or quantitatis <math>R z^2 + S z + T</math>, tum
curva quadraturam haud recipiet finitis terminis.</p>

<pb 147>[0199]<rh>Æ QUATIONIBUS.</rh>
<p>Sit vero æquatio re$ultans in genere <math>(P z^m + Q z^m-1 + &c.) v^n-m +
(R z^m+1 + S z^m + T z^m-1 + &c.) v^n-m-1 + &c. = 0</math>, & $i fluens fluxionis
<math>{R z^m+1 + S z^m + T z^m-1 + &c. / P z^m + Q z^m-1 + &c.} z^.</math> finitis terminis haud exprimi po$$it,
tum curva haud finitis terminis quadraturam recipiet.</p>
<p>Eadem principia etiam applicari po$$unt ad contenta $olidorum,
ad fluentes quarumcunque fluxionum, &c.</p>
<h>THEOR. XVII.</h>
<p>1. Datâ algebraicâ æquatione <math>A y^n + (a + b x) y^n-1 + &c. = 0</math>,
relationem inter ab$ci$$am <math>x</math> & ejus corre$pondentes ordinatas <math>y</math> expri-
mente; $umma e $ingulis valoribus cuju$cunque algebraicæ functionis
literarum <math>x & y</math> & earum fluxionum $emper exprimi pote$t in finitis
terminis ab$ci$$æ & ordinatæ, earum circularium arcuum & logarith-
morum: $umma enim e $ingulis valoribus irrationalis partis nihilo
erit æqualis, & $umma e $ingulis valoribus rationalis functionis $em-
per inveniri pote$t ope finitorum terminorum, circularium arcuum &
logarithmorum.</p>
<p>2. Datâ fluxionali æquatione, & fluxionali quantitate continente
rationalem & irrationalem algebraicam quantitatem, cujus rationalis
quantitatis fluens inveniri pote$t; tum $umma e $ingulis valoribus
fluxionalis quantitatis inveniri pote$t; $umma enim e $ingulis valori-
bus irrationalis quantitatis erit nihilo æqualis.</p>
<h>PROB. XXXIV.</h>
<p>1. _Ita transformare œquationem fluxionalem, in quâ $olummodo conti-_
_nentur duœ variabiles quantitates_ (<math>x & y</math>), _quarum_ <math>x</math> _fluit uniformiter, ut_
_fluat uniformiter_ <math>y</math>.</p>
<p>F<sc>IG</sc>. 1. pag. 95. Sit quæcunque curva (_MRS_), cujus ab$ci$$a (<math>x =
A P</math>) fluit uniformiter; $it ejus fluxio <math>(P p)</math>, & fluxio ordinatæ <math>(y = P M)
M b</math>; $ecunda vero fluxio <math>2 m f</math>; nunc fluat ordinata (<math>PM</math>) unifor-

<pb 148>[0200]<rh>DE FLUXIONALIBUS</rh>
miter, $it ejus fluxio <math>M h</math>, tum prima ab$ci$$æ (<math>A P</math>) fluxio erit <math>b e</math>, &
$ecunda ejus fluxio <math>2 m e</math>: $ed duo triangula <math>M b e, f m e</math> $unt $imilia,
ergo <math>e h (x^.):h M (y^.)::2 m e (x^..)</math> $ecunda ab$ci$$æ fluxio, cum fluat
uniformiter <math>y:2 m f (-y^..)</math> $ecunda ordinatæ fluxio, cum fluat uni-
formiter <math>x</math>; & con$equenter <math>y^.. = - {y^. x^.. / x^.}</math>, quo valore pro <math>y^..</math> in datâ flu-
xionali æquatione $ub$tituto, & $ic deinceps; & ita transformabitur
æquatio, ut fluat uniformiter <math>y</math>. Aliter.</p>
<p>2. Sit <math>y = A x^n + B x^n-r + C x^n-2r + &c.</math> & erit <math>y^. = n A x^n-1 x^. +
(n - r) B x^n-r-1 x^. + (n - 2 r) C x^n-2r-1 x^. + &c. &
y^.. = n × (n - 1) A x^n-2 x^.^2 + (n - r)(n - r - 1) B x^n-r-2 x^.^2 + (n - 2 r)(n - 2 r - 1) C x^n-2r-2 x^.^2
+ nAx^n-1 x^.. + (n-r) Bx^n-r-1 x^.. + (n-2r) Cx^n-2r-1 x^.. + &c.
+ &c. = Qx^.^2 + Px^..</math>; fluat uniformiter <math>x</math>, & con$equenter eva-
ne$cet <math>x^..</math>, & erit <math>y^.. = Q x^.^2</math>; deinde fluat uniformiter <math>y</math>, & erit <math>y^.. = 0
= Q x^.^2 (y^.. cum x fluat uniformiter) + P x^.. {y^. x^.. / x^.}</math>, quoniam <math>P ={y^. / x^.}</math>; utræque enim hæ quantitates <math>P & {y^. / x^.}</math> $unt <math>n A x^n-1 + (n - r)
B x^n-r-1 + &c.</math> unde <math>Q x^.^2 + {y^. x^.. / x^.} = y^.. + {y^. x^.. / x^.} = 0</math>, & con$equenter <math>y^..
= - {y^. x^.. / x^.}, &c.</math></p>
<p>3. Sit <math>y^. = P x^., & 1^mo</math>. fluat uniformiter <math>x^.</math>; tum erunt <math>y^.. = P^. x^., y^.. =
P^.. x^., y^.... = P^... x^., y^.. ^... = P^.... x^., &c.</math>; unde <math>P = {y^. / x^.}, P^. = {y^.. / x^.}, P^.. = {y^... / x^.}, P^... = {y^.... / x^.},
&c.</math>: deinde 2<_>do</_>. fluat uniformiter <math>y</math>, tum erunt <math>P = {y^. / x^.}, P^. = - {y^. x^.. / x^.^2},
P^.. = - {y^. x^. x^... - 2 x^.. ^2 y^. / x^.^3}, P^... = - {y^. x^.^2 x^.... - 6y^. x^. x^.. x^... + 6 y^. x^.. ^3 / x^.^4}, &c.</math>: & ex æquatis
duobus diver$is valoribus quantitatum <math>P^., P^.., P^..., &c.</math>; re$ultabunt <math>P^.

<pb 149>[0201]<rh>ÆQUATIONIBUS.</rh>
= {y^.. / x^.} = - {y^. x^.. / x^. <_>2</_>}</math>, unde <math>y^.. = - {y^. x^.. / x^.}</math>; deinde <math>P^.. = {y^... / x^.} = - {y^. x^. x^... - 2 x^.. ^2 y^. / x^. ^3}</math>,
& exinde <math>y^... = - {y^. x^. x^... - 2 x^.. ^2 y^. / x^. ^2}</math>; tum <math>P^... = {y^.... / x^.} = - {y^. x^. ^2 x^.... - 6 y^. x^. x^.. x^... + 6 y^. x^.. ^3 / x^. ^4}</math>;
&c.: $cribantur valores exinde deducti <math>- {y^. x^.. / x^.}, - {y^. x^. x^... - 2 x^.. ^2 y^. / x^. ^2},
- {y^. x^. ^2 x^.... - 6 y^. x^. x^.. x^... + 6 y^. x^.. ^3 / x^. ^3}, &c.</math>, pro <math>y^.., y^..., y^...., &c.</math> in datâ fluxionali æqua-
tione; & re$ultat fluxionalis æquatio quæfita.</p>
<p>2. Datam æquationem involventem <math>x & y</math> & earum fluxiones, in
quâ <math>x</math> fluat uniformiter, ita transformare, ut nec <math>x</math> nec <math>y</math> fluat uni-
formiter: affumatur <math>y^. = p x^.</math>, & exinde <math>y^.. = p^. x^. + p x^.. = p^. x^.</math> cum <math>x</math>
fluat uniformiter; $ed <math>p = {y^. / x^.} & p^. = {y^.. x^. - x^.. y^. / x^. ^2}</math>, $cribatur hæc quan-
titas pro fuo valore <math>p^.</math> in æquatione <math>y^.. = p^. x^.</math>, & re$ultat æquatio <math>y^.. ={x^. y^.. - y^. x^.. / x^.}</math>; eodem modo <math>y^... = p^.. x^. + 2 p^. x^.. + p x^... = p^.. x^.</math> cum <math>x</math> fluat uni-
formiter; in hâc expre$$ione fcribantur <math>{y^. / x^.} & {y^.. x^. - x^.. y^. / x^. ^2}</math> pro <math>p & p^.</math>; &
re$ultat <math>y^... = p^.. x^. + 2({y^.. x^. - x^.. y^. / x^. ^2}) x^.. + {y^. / x^.} × x^...</math>, & con$equenter <math>p^.. ={y^... x^. ^2 - 2 y^.. x^. x^.. + 2 x^.. ^2 y^. - y^. x^. x^... / x^. ^3} = {y^... / x^.}; &c.</math>: in datâ fluxionali æquatione
$cribendi $unt igitur valores <math>{x^. y^.. - y^. x^.. / x^.} & {y^... x^. ^2 - 2 y^.. x^. x^.. + 2 x^.. ^2 y^. - y^. x^. x^... / x^. ^2}</math>
pro <math>y^.. & y^...</math>; & $ic deinceps: & re$ultat æquatio quæfita.</p>
<p>3. In genere $it <math>y^. = p m x^., & m x^.</math> con$tans quantitas; & erit <math>y^.. =
m p^. x^.</math>, $ed <math>p = {y^. / m x^.}</math> & exinde <math>p^. = {y^.. m x^. - m^. x^. y^. - m x^.. y^. / m^2 x^. ^2}</math>; $ub$tituatur
hic valor pro <math>p^.</math> in æquatione <math>y^.. = m x^. p^.</math>, & habebimus æquationem
<math>y^.. = {m x^. y^.. - m^. x^. y^. - m y^. x^.. / m x^.}</math>; per eundem modum inveniri po$$unt

<pb 150>[0202]<rh>DE FLUXIONALIBUS</rh>
<math>y^..., y^...., &c.</math>; $cribantur hi valores pro <math>y^.., y^..., y^...., &c.</math> in datâ æquatione, &
re$ultat quæfita.</p>
<p>Si vero requiratur, ut <math>q x^. + r y^.</math> fluat uniformiter, eodem pror$us
modo progrediendum e$t.</p>
<p>4. Datam fluxionalem æquationem <math>a = 0</math> relationem inter <math>x & y</math> &
earum fluxiones exprimentem, in quâ <math>p = φ (x & y)</math> data functio
quantitatum <math>x & y</math> fluit uniformiter, ita transformare, ut re$ultet
æquatio relationem inter <math>x & p</math> & earum fluxiones exprimens.</p>
<p>Reducantur duæ æquationes <math>α = 0 & p = φ (x & y, &c.)</math>, ita ut ex-
terminetur quantitas <math>y</math> & ejus fluxiones, & re$ultat æquatio quæfita.</p>
<h>PROB. XXXV.</h>
<p>_Duas vel plures_ <math>(n)</math> _fluxionales æquationes_ <math>(n + 1)</math> _vel plures variabiles_
_quantitates habentes, in quibus una_ <math>x</math> _fluit uniformiter, ita transformare,_
_ut fluat uniformiter_ <math>y</math>.</p>
<p>Pro <math>y^..</math> $cribatur in datis æquationibus ejus valor <math>- {y^. x^.. / x^.}</math>, & pro <math>y^...</math>
fluxio prius inventa, & $ic deinceps; & ita transformabuntur æqua-
tiones ut fluat uniformiter <math>y</math>.</p>
<p>Et $ic transformari po$$unt duæ vel plures <math>(n)</math> æquationes <math>(n + 1)</math>
variabiles quantitates habentes; in quibus fluit uniformiter <math>m x^.</math>, ita
ut fluat uniformiter <math>p x^. vel p x^. + q y^., &c.</math></p>
<h>THEOR. XVIII.</h>
<p>1. Sit algebraica æquatio <math>P = Q</math>, e quâ deducatur fluxionalis
<math>P^. = Q^.</math>: fi modo utri$que huju$ce æquationis partibus addantur vel
$ubducantur eædem quantitates; vel utræque partes fluxionalis
æquationis in ea$dem quantitates ductæ tum eadem manet
fluxionalis æquatio, (for$an vero etiam adjiciantur novæ), cujus
fluens erit <math>P = Q + a</math>, ubi <math>a</math> denotat invariabilem quantitatem ad

<pb 151>[0203]<rh>ÆQUATIONIBUS.</rh>
libitum a$$umendam: $i vero hæ duæ partes æquationis haud ducan-
tur in ea$dem $ed in æquales quantitates utcunque e datâ algebraicâ
<math>(P = Q)</math> æquatione deductas, tum particularis fluentialis æquatio
deducta e fluxionali æquatione re$ultante etiam erit <math>P = Q</math>; quod$i
generaliter corrigatur, i. e. in genere inveniatur æquatio fluentialis
ad fluxionalem æquationem re$ultantem, haud erit <math>P = Q + a</math>. e. g.
ducatur utraque pars fluxionalis æquationis re$ultantis in <math>P & Q</math>, &
re$ultat <math>P P^. = Q Q^.</math>, cujus fluentialis æquatio erit in genere <math>P^2 = Q^2
+ a</math>, haud <math>P = Q + a</math>; dividatur autem utraque pars fluxionalis
æquationis <math>(P^. = Q^.)</math> per <math>P & Q</math> re$pective, & re$ultat <math>{P^. / P} = {Q^. / Q}</math>, cujus
fluentialis æquatio in genere erit <math>P = a Q</math>, & haud <math>P = Q + a</math>.</p>
<p>Cor. 1. Hinc, $i modo detur æquatio, cujus inveniatur fluxio; & in
fluxionali æquatione re$ultante $ub$tituantur quæcunque quantitates
e datâ æquatione deductæ pro fuis valoribus; re$ultabunt diver$æ flu-
xionales æquationes, quarum eædem dantur particulares fluentiales,
quamvis for$an hæ generaliter correctæ haud erunt eædem.</p>
<p>Cor. 2. Eadem etiam affirmari poffunt de fluxionalibus æquationi-
bus $uperiorum ordinum, & de pluribus æquationibus plures varia-
biles quantitates habentibus.</p>
<p>2. E contra æquationes fluxionales diver$æ generaliter ea$dem præ-
beant fluentiales æquationes: a$$umatur enim fluentialis æquatio,
in quâ unus terminus $it invariabilis & ad libitum a$$umendus: ita
transformetur hæc æquatio, ut fiat alter terminus invariabilis & ad
libitum a$$umendus: harum duarum æquationum inveniantur fluxi-
ones, & duarum fluxionalium æquationum re$ultantium fluentes in-
venientur eædem æquationes.</p>
<p>Ex. Sit æquatio algebraica a$$umpta <math>a y^n + b x^r y^s + c x^t + d = 0</math>,
ubi termini <math>d & a</math> $unt invariabiles quantitates ad libitum a$$umendæ:
dividatur data æquatio per <math>y^n</math>, & re$ultat <math>a + b x^r y^s-n + c x^t y^-n +
d y^- n = 0</math>; inveniantur fluxiones ex utri$que his æquationibus, & re-
$ultant duæ fluxionales æquationes diver$æ, viz. <math>n a y^n-1 y^. + s b x^r y^s-1 y^.

<pb 152>[0204]<rh>DE FLUXIONALIBUS</rh>
+ r b y^s y^s x^r-1 x^. + t e x^t-1 x^. = 0 & r b x^r-1 y^s-n x^. + (s - n) b x y^s-n-1 y^. +
t c x^t-1 y^-n x^. - n c x^t y^-n-1 y^. - n d y^- n-1 y^. = 0</math>; quæ duæ fluxionales æqua-
tiones ea$dem præbent fluentiales.</p>
<p>In hâc æquatione duæ vel plures $unt invariabiles quantitates ad
libitum a$$umendæ.</p>
<p>Et $ic de fluxionalibus æquationibus $uperiorum ordinum, & plu-
ribus æquationibus.</p>
<h>PROB. XXXVI.</h>
<h>_Corrigere fluentes in genere._</h>
<p>1. Datis quibu$cunque fluxionalibus æquationibus; & quantitate,
quæ e$t fluxionalis functio literarum in datis fluxionalibus æquatio-
nibus contentarum; datam quantitatem generaliter corrigere.</p>
<p>Sit data quantitas fluxio <math>(n)</math> ordinis; & <math>v^. ^n-1</math> fluxio <math>n - 1</math> ordinis, quæ
fluit uniformiter; tum correctio primæ fluentis erit <math>a × v^. ^n-1</math>, ubi litera
<math>a</math> invariabilem quantitatem denotat; fi vero <math>V</math> $it prima fluens datæ
quantitatis, tum erit prima fluens correcta <math>V + a v^. ^n-1</math>: fit <math>w^. ^n-2</math> fluxio <math>n - 2</math>
ordinis, quæ fluit uniformiter, & correctio $ecundæ fluentis erit <math>f. a v^. ^n-1
+ b w^. ^n-2</math>, ubi <math>b</math> denotat invariabilem coefficientem; $it <math>W</math> fluens flu-
xionis <math>(V + a v^. ^n-1)</math>; & $ecunda fluens correcta erit <math>W + b w^. ^n-2</math>; & $ic
deinceps.</p>
<p>2. Fluentialem æquationem in genere corrigere; <math>1^mo</math>. ducatur utra-
que datæ æquationis pars in eandem quantitatem; inveniatur fluens
æquationis re$ultantis ($i modo fieri po$$it), quæ fit fluxionalis æqua-
tio <math>(n - 1)</math> ordinis; alteri parti æquationis fluentis inventæ addatur
vel $ubtrahatur quantitas <math>a v^. ^n-1</math>, ubi <math>a</math> denotat invariabilem quantita-
tem, & <math>v^. ^n-1</math> denotat fluxionem <math>n - 1</math> ordinis, quæ fluit uniformiter; &
vere corrigitur prædicta æquatio, & vera etiam e$t fluentialis æquatio,

<pb 153>[0205]<rh>ÆQUATIONIBUS.</rh>
quam de$ignat data fluxionalis: eodem modo inveniri pote$t correctio
æquationis re$ultantis; & $ic deinceps.</p>
<p>Ex. 1. Sit data fluxio <math>2 y y^. = 3 x^2 x^.</math>, cujus fluens e$t <math>y^2 = x^3</math>; al-
teri æquationis parti addatur vel $ubtrahatur invariabilis quantitas
<math>a</math> ad libitum a$$umenda, & erit fluens correcta <math>y^2 = x^3 + a</math>, quæ e$t
fluentialis æquatio prædictæ fluxionalis.</p>
<p>Cave vero, ne ita ratiocineris, quoniam <math>y^2 = x^3</math>, erit <math>y = x^{3 / 2}</math>, &
exinde correcta fluens <math>y = x^{3 / 2} + a</math>: hæc æquatio minime erit gene-
raliter fluentialis æquatio datæ fluxionalis <math>2 y y^. = 3 x^2 x^.</math>; $ub$ti-
tuatur enim pro <math>y</math> in datâ fluxionali æquatione ejus valor a$$ump-
tus <math>x^{3 / 2} + a</math>, & haud re$ultant utræque æquationis partes inter $e
æquales.</p>
<p>Ex. 2. Sit data fluxionalis æquatio <math>{x^. / x} = {2 y^. / y}</math>; & erit æquatio, cujus
fluxio e$t prædicta æquatio <math>log. x = log. y^2</math>; addatur invariabilis
quantitas ad alteram huju$ce æquationis partem, & re$ultat æquatio
vere correcta <math>log. x = 2 log. y + a</math>, unde <math>x = a y^2</math> e$t æquatio vere
correcta, $ed non <math>x = y^2 + a</math>.</p>
<p>Cor. 1. Hinc cavendum e$t in re$olutionibus problematum vel
arithmeticorum vel geometricorum, ne in corrigendis fluentibus erro-
res in $olutiones irrepant, & re$ultans fluentialis æquatio haud fit
æquatio, quam de$ignat problema: data quantitas ad libitum a$$u-
menda fluenti datæ fluxionalis æquationis femper adjungenda e$t, vel
fluenti æquationis e datâ (ducendo utra$que æquationis partes in
eandem & haud diver$as quantitates) deductæ; & haud quantitati e
prædictâ fluente quocunque modo deductæ.</p>
<p>3. Si data fluxionalis æquatio <math>P = 0</math> plures recipiat divi$ores <math>A, B,
C, &c.</math> i. e. <math>P = A × B × C × &c. = 0</math>; & inveniantur quantitates <math>α, β,
γ, &c.</math> quæ in datas <math>A, B, C, &c.</math> ductæ præbent quantitates, quarum
fluentes detegi poffunt, & quæ fint re$pective <math>W, X, Y, &c.</math> tum erunt
diver$i valores datæ fluxionalis æquationis <math>W + a = 0, X + b = 0,
Y + c = 0</math>, &c. ubi literæ <math>a, b, c, &c.</math> denotant invariabiles quanti-
tates ad libitum a$$umendas.</p>

<pb 154>[0206]<rh>DE FLUXIONALIBUS</rh>
<p>Cor. 3. Eadem etiam applicari po$$unt ad plures fluxionales æqua-
tiones plures variabiles quantitates involventes; $ingula enim fluxio-
nalis æquatio corrigenda e$t prædictâ methodo. Et $ic de æquatio-
nibus, in quibus continentur fluxiones $uperiorum ordinum.</p>
<p>4. Cum nulla inveniatur quantitas; quæ in datam fluxionalem
æquationem ducta, creat fluxionalem, cujus fluens inveniri pote$t,
tum e datâ fluxionali æquatione exprimente relationem inter quanti-
tates <math>x & y</math> & earum fluxiones inveniatur particularis valor (<math>Y</math>) quan-
titatis <math>y</math> terminis vero variabilis quantitatis <math>x</math>; ut corrigatur valor
quantitatis <math>y</math> inventus, $cribatur pro <math>y</math> ejus valor <math>Y + X</math> (correc. quæ-
$it.); & e datâ & re$ultante æquationibus inveniatur <math>X</math> terminis quan-
titatis <math>x</math>, & erit correctio quæ$ita.</p>
<p>Cor. In generali correctione fluxionalis æquationis primi ordinis
continetur invariabilis quantitas ad libitum a$$umenda, $ecundi vero
ordinis duæ continentur invariabiles quantitates ad libitum a$$umen-
dæ, & $ic deinceps.</p>
<p>Eadem etiam applicari po$$unt ad plures (<math>n</math>) æquationes fluxiona-
les plures (<math>n + 1</math>) variabiles quantitates quorumcunque ordinum in-
volventes.</p>
<p>5. Si vero generalis fluens haud inveniri po$$it finitis terminis, con-
fugiendum e$t ad infinitas $eries, e. g. datâ fluxionali æquatione <math>n</math>
ordinis duas involvente variabiles quantitates <math>x & y</math> & earum fluxio-
nes; e methodo inveniendi $eries progredientes $ecundum dimen$io-
nes cuju$cunque literæ <math>x</math> inveniatur $eries, in quâ continentur (<math>n</math>) li-
teræ invariabiles ad libitum a$$umendæ, & erit $eries vere correcta.</p>
<h>THEOR. XIX.</h>
<p>1. Datâ generali fluente fluxionalis æquationis (<math>m</math>) ordinis, quæ
continet (<math>m</math>) invariabiles quantitates (<math>A, B, C, D, &c.</math>) ad libitum
a$$umendas, etiamque (<math>r</math>) variabiles quantitates (<math>x, y, z, &c.</math>): datis
etiam (<math>m</math>) diver$is valoribus e $ingulis (<math>r</math>) variabilibus quantitatibus
(<math>x, y, z, &c.</math>); qui $int re$pective <math>a, a′, a″, &c.; b, b′, b″, &c.; c, c′, c″, &c.;

<pb 155>[0207]<rh>Æ QUATIONIBUS.</rh>
&c.</math>; $cribantur hi valores <math>a, b, c, &c.; a′, b′, c′, &c.; a″, b″, c″, &c.; &c.</math>;
re$pective pro $uis valoribus in datâ generali fluente, & re$ultabunt
(<math>m</math>) æquationes totidem (<math>m</math>) incognitas quantitates habentes (<math>A, B,
C, D, &c</math>) e quibus deduci po$$unt quantitates <math>A, B, C, D, &c.</math>; quæ
præbent correctiones datæ fluentis datis (<math>m</math>) valoribus e $ingulis va-
riabilibus re$pondentes.</p>
<p>2. Sint duæ vel plures (<math>n</math>) fluxionales æquationes (<math>m, r, s, &c.</math>) or-
dinum re$pective, tres vel plures (<math>n + 1</math>) variabiles quantitates <math>x, y,
w, &c.</math> & earum fluxiones habentes, in quibus <math>x^.</math> e$t con$tans; inve-
niantur generales fluentes variabilium <math>y, w, &c.</math>; quæ $int <math>y = φ:(x),
w = φ′:(x), &c.</math> vel quantitates quâcunque aliâ methodo de$ignatæ:
& $i modo $int <math>m + r + s + &c.</math> invariabiles quantitates ad libitum
a$$umendæ in hi$ce functionibus <math>φ, φ′, &c.</math> tum ex <math>m + r + s + &c</math>. in-
variabilibus quantitatibus ad libitum a$$umendis in unâ functione <math>φ</math>,
deduci po$$unt <math>m + r + s + &c.</math> invariabiles quantitates ad libitum
a$$umendæ in $ingulis reliquis; vel quod idem e$t, prædictæ invaria-
biles quantitates in $ingulis diver$is functionibus erunt eædem.</p>
<h>PROB. XXXVII.</h>
<p it>Datâ æquatione vel algebraicâ vel fluxionali, cujus radix <math>y</math> exprimit
$ummam $eriei <math>a x^h + b x^h±m + c x^h±1 + &c.</math> $ecundum dimen$iones literæ
<math>x</math> progredientis; invenire quantitatem alternis $eriei terminis æqualem;
& denιque $eriei terminis, quorum di$tantia a $emetip$is $it <math>n + 1</math>.</p>
<p>Si modo $eries requiratur alternis $eriei terminis æqualis; pro <math>x, x^.,
x^.., &c.</math> in datâ æquatione $cribantur re$pective <math>x, x^., x^.., &c. & - x,
- x^., - x^.., &c.</math> & exoriuntur duæ æquationes, quarum radices $int
<math>y & y′</math>; & erit $umma quæ$ita <math>{y + y′ / 2}</math>: $i vero requiratur $umma $e-
cundi, quarti, $exti, &c. terminorum, erit <math>{y - y′ / 2}</math> $umma quæ$ita. Si
requiratur $umma terminorum, quorum di$tantia a $e invicem $it <math>n + 1</math>;
$int <math>α, β, γ, δ, &c.</math> re$pective radices æquationis <math>x^n+1 - 1 = 0</math>, & in

<pb 156>[0208]<rh>DE FLUXIONALIBUS</rh>
datâ æquatione pro <math>x, x^., x^.., &c.</math> $cribatur re$pective <math>αx, αx^., αx^.., &c.
βx, βx^., βx^.., &c. γx, γx^., γx^.., &c. &c.</math> & re$ultant <math>n + 1</math> æquationes,
quarum corre$pondentes radices $int <math>y, y′, y″, &c.</math> & erit $umma quæ-
$ita<math>{y + y′ + y″ + &c. / n + 1}</math>: & $ic e medit. algebr. inveniri po$$unt $umma
e $ecundo, <math>n + 3, 2 n + 4, 3 n + 5</math>, &c. terminis; $umma e tertio,
<math>n + 4, 2 n + 5, 3 n + 6, &c.</math> terminis; & $ic deinceps.</p>
<p>Cor. Si requiratur æquatio, cujus radix (<math>z</math>) e$t $umma alternorum
$eriei terminorum, cujus $umma e$t <math>y</math>: per prob 25. inveniantur æqua-
tiones, quarum radices $unt <math>y & y′</math>: ergo e duabus re$ultantibus æqua-
tionibus & tertiâ <math>{y + y′ / 2} = z</math> inveniatur æquatio, cujus radix e$t <math>z</math>; &
con$it coroll.</p>
<p>Et $ic inveniri pote$t æquatio, cujus radix e$t <math>{y + y′ + y″ + &c. / n + 1}</math>.</p>
<p>Cor. Eodem modo ratiocinari liceat de pluribus æquationibus
plures incognitas quantitates habentibus: $ed haud tam facilis vel
elegans dici pote$t huju$ce problematis $olutio in æquationibus ac in
quantitatibus datis: in æquationibus enim plurimæ dantur radices,
quæ minime ad problema re$olvendum attinent.</p>
<h>PROB. XXXVIII.</h>
<p it>Invenire, annon data æquatio <math>W = 0</math> $it generalis fluens datæ fluxio-
nalis æquationis (<math>α = 0</math>) primi ordinis.</p>
<p>1. Sit <math>a</math> quantitas invariabilis ad libitum a$$umenda, quæ in datâ
fluxionali æquatione haud invenitur, tum ex æquatione <math>W = 0</math> inve-
niatur <math>a = V</math>, ubi <math>V</math> e$t quantitas in quâ non continetur litera (<math>a</math>);
tum, $i <math>{α / V^.}</math> ad minimos terminos reducta $it quantitas, quæ nullas
continet fluxiones, erit <math>W = 0</math> fluens generalis datæ fluxionalis æqua-
tionis; $in aliter non.</p>

<pb 157>[0209]<rh>Æ QUATIONIBUS.</rh>
<p>2. Si non deduci po$$it <math>a = V</math>; reducantur duæ æquationes <math>W = 0
& W^. = 0</math> in unam, ita ut exterminetur quantitas <math>a</math>, & re$ultet æqua-
tio fluxionalis (<math>β = 0</math>); tum, $i quotiens <math>({α / β})</math> ad minimos terminos
reducta nullas contineat fluxiones, <math>W = 0</math> erit fluens prædicta.</p>
<p>2. 2. Si quotiens <math>{α / β}</math> vel <math>{α / V^.}</math> contineat fluxiones, reducantur utræque
fluxionales æquationes <math>(α = 0 & β = 0)</math>, ita ut fluxiones <math>(x^., y^., &c.)</math>
in iis contentæ $olummodo habeant unam dimen$ionem, & re$ultent
quantitates <math>π = 0 & ξ=0, &c.</math>; tum $i quotiens <math>{(π / ξ)}</math> ad minimos terminos
reducta nullas contineat fluxiones, fluens quæ$ita erit <math>W = 0</math>. Si au-
tem non ita reduci po$$int prædictæ fluxionales æquationes <math>α = 0 &
β = 0</math>, ut $olummodo habeant unam dimen$ionem fluxiones contentæ;
tum per medit. algebr. exterminentur irrationales functiones fluxio-
num (<math>x^., y^., &c.</math>) ex prædictis æquationibus <math>α = 0 & β = 0</math>, & re$ult-
ent æquationes <math>γ = 0 & δ = 0</math>; deinde inve$tigetur, annon æquationes
<math>γ = 0 & δ = 0</math> communem habeant divi$orem, qui involvit fluxiones
<math>x^. & y^.</math>; $i communem habeant divi$orem, tum <math>W = 0</math> erit generalis
fluens fluxionalis æquationis <math>α = 0</math>; vel in æquatione <math>W = 0</math> conti-
netur divifor, qui erit fluens fluxionalis æquationis <math>α = 0</math>.</p>
<p>3. Si <math>W = 0</math> $it generalis fluens fluxionalis æquationis (<math>α = 0</math>) or-
dinis (<math>n</math>), & in eâ (<math>W = 0</math>) contineantur (<math>n</math>) invariabiles quantitates
(<math>a, b, c, &c.</math>) ad libitum a$$umendæ tum ex æquatione <math>W = 0</math> inve-
niatur æquatio <math>a = V</math>, cujus fluxio <math>V^. = 0</math> haud involvat quantitatem
<math>a</math>, deinde ex æquatione <math>V^. = 0</math> inveniatur <math>b = V′</math>, cujus fluxio <math>V^.′</math> haud
continet vel quantitatem <math>a</math> vel <math>b</math>, & $ic deinceps; u$que donec exter-
minantur omnes prædictæ (<math>n</math>) invariabiles quantitates ad libitum a$-
$umendæ, quæ in datâ fluxionali æquatione non continentur; &
re$ultet fluxionalis æquatio (<math>ξ = 0</math>) ordinis (<math>n</math>); tum, $i fractio <math>{α / ξ}</math>
ad minimos terminos reducta, nullas contineat fluxiones ordinis (<math>n</math>),
erit <math>W = 0</math> generalis fluens fluxionalis æquationis <math>α = 0</math>.</p>

<pb 158>[0210]<rh>DE FLUXIONALIBUS</rh>
<p>4. Si non deduci po$$it <math>a = V, &c.</math>; tum reducantur duæ æquatio-
nes <math>W = 0 & W^. = 0</math> in unam, ita ut exterminetur incognita quan-
itas <math>a</math>, & $ic de reliquis <math>b = V′</math>, &c.; & po$t $ingulas has reductiones4
re$ulter fluxionalis æquatio <math>ξ′ = 0</math> ordinis <math>(n)</math>, in quâ nec continetur
litera <math>a</math>, nec <math>b</math>, nec <math>c</math>, &c.: tum, $i fractio <math>{α / ξ′}</math> ad minimos terminos
reducta nullas contineat fluxiones ordinis <math>(n)</math>, erit <math>W = 0</math> generalis
fluens fluxionalis æquationis <math>α = 0</math>.</p>
<p>4. 2. Si quotientes <math>{α / ξ}</math> vel <math>{α / ξ′}</math> contineant fluxiones ordinis <math>(n)</math>, redu-
cantur utræque fluxionales æquationes <math>α = 0 & ξ′ = 0</math>, ita ut in iis
fluxio ordinis <math>(n)</math> variabilis quantitatis unam $olummodo habeat di-
men$ionem, & re$ultent æquationes <math>σ = 0 & γ = 0</math>; tum, $i fractiones <math>{σ / ξ}</math>
vel <math>{σ / γ}</math> ad minimos terminos reductæ contineant nullas fluxiones or-
dinis <math>(n)</math>, erit <math>W = 0</math> generalis fluens datæ fluxionalis æquationis
<math>α = 0</math>, $in aliter non.</p>
<p>4. 3. Si vero non ita reduci po$$int prædictæ æquationes, ut fluxio or-
dinis (<math>n</math>) variabilis quantitatis $olummodo habeat unam dimen$ionem;
tum per medit. algeb. ita reducantur duæ æquationes <math>α = 0 & ξ′ = 0</math>, ut
exterminentur irrationales functiones fluxionum in iis contentarum,
& re$ultent æquationes <math>β′ = 0 & σ′ = 0</math>; tum, $i <math>β′ = 0 & σ′ = 0</math> ha-
beant communem divi$orem, qui in $e continet fluxionem ordinis
(<math>n</math>) prædictæ variabilis quantitatis, fluentialis æquatio <math>W = 0</math> conti-
net in $e generalem fluentem fluxionalis æquationis in datâ fluxio-
nali æquatione <math>α = 0</math> contentæ.</p>
<p>5. Si vero dentur plures æquationes fluxionales <math>α = 0, β = 0,
γ = 0, &c.</math>; & fluentiales plures variabiles quantitates habentes; tum
per eandem methodum ita reducantur fluentiales æquationes, ut ex-
terminentur (<math>n</math>) invariabiles quantitates generaliter a$$umptæ, quæ in
datis fluxionalibus æquationibus (<math>α = 0, β = 0, &c.</math>) non continen-

<pb 159>[0211]<rh>ÆQUATIONIBUS.</rh>
tur, & refultent fluxionales æquationes <math>λ = 0, μ = 0, ν = 0, &c.;</math>
deinde inveniatur, annon datæ fluxionales æquationes <math>α = 0, β = 0,
γ = 0, &c.</math> & refultantes <math>λ = 0, μ = 0, ν = 0, &c.</math> habeant commu-
nem divi$orem, qui continet fluxionem (<math>n</math>) ordinis, quod per medit.
algebr. perfici pote$t ex reducendo fluxionales æquationes, ita ut ex-
terminentur irrationales functiones fluxionum maximi ordinis in
fingulis æquationibus contentæ, & exinde per methodum communes
divi$ores inveniendi divi$ores quæfiti deduci po$$unt, $i modo qui-
dam fint.</p>
<p>5. Hoc problema nonnunquam re$olutionem recipere pote$t ex
$ub$titutione valorum nonnullarum incognitarum quantitatum & ea-
rum fluxionum ex datis fluentialibus æquationibus deductarum pro
fuis valoribus in fluxionalibus æquationibus, & $i nihilo evadant
æquales prædictæ fluxionales æquationes; tum fluentiales erunt refo-
lutiones fluxionalium æquationum, fin aliter non.</p>
<p>Quamvis quantitates ex datâ fluentiali æquatione acquifitæ, & in
datâ fluxionali $ubftitutæ, eam reddent nihilo æqualem; attamen non-
nunquam data fluentialis æquatio non vere dici pote$t fluens datæ
fluxionalis, ex eo quod in generali fluente non continetur; & quam-
vis nonnunquam etiam in generali fluente continetur, attamen varia-
biles quantitates ex affumptis evadant invariabiles, tum non proprie
dici pote$t fluens datæ fluxionalis æquationis, nam omnino contradicit
hypothe$i fluxionalis æquationis; hoc femper evadet, ex eo quod ali-
qua quantitas in re$olutione affumitur invariabilis vel nihilo æqualis
vel penitus ex æquatione ejicitur, quæ in æquatione variabilis fuppo-
nitur. e.g. Sit data fluxionalis æquatio <math>α P^. + β P q^. = 0;</math> tum <math>P = 0</math>
erit fluens fluxionalis æquationis <math>α P^. = 0,</math> & non fluens fluxionalis
æquationis <math>β P q^. = 0,</math> ni <math>P</math> fit functio quantitatis <math>q</math>, & confequenter
non fluens datæ fluxionalis æquationis; attamen, $i modo $cribantur
<math>P = 0 & P^. = 0</math> pro earum valoribus in datâ fluxionali æquatione
<math>α P^. + β P q^. = 0,</math> tum proprie evane$cet terminus <math>α P^. = 0</math>, ex eo quod
<math>P^. = 0</math>; etiamque evane$cet terminus <math>β P q^.</math>, ex eo quod <math>P = 0,</math> at
non proprie; nam in datâ fluxionali æquatione <math>q</math> fupponitur variabilis;

<pb 160>[0212]<rh>DE FLUXIONALIBUS</rh>
fed in affumptâ æquatione pro particulari valore penitus ex æquati-
one ejicitur <math>q.</math> e. 2. Sit <math>x x^. + y y^. = y^. √(x^2 + y^2 - a^2)</math>; a$$umatur
<math>√(x^2 + y^2 - a^2) = 0</math> pro particulari fluente datæ fluxionalis æqua-
tionis; $i <math>x^2 + y^2 - a^2 = 0</math>; tum <math>x x^. + y y^. = 0</math>, etiamque <math>y^. √(x^2
+ y^2 - a^2) = 0</math> & con$equenter <math>x x^. + y y^. = y^. √(x^2 + y^2 - a^2)</math>;
fed non recte <math>y^. √(x^2 + y^2 - a^2) = 0</math>, nam <math>x^2 + y^2 - a^2</math> non e$t
functio fluentis (<math>y</math>) fluxionis (<math>y^.</math>), & con$equenter non recte <math>x x^. + y y^.
= y^. √(x^2 + y^2 - a^2)</math>: etiamque hæc particularis fluens in generali
non continetur, nam generalis fluens fluxionalis æquationis <math>x x^. + y y^.
= y^. √(x^2 + y^2 - a^2)(unde y^. = {x x^. + y y^. / √(x^2 + y^2 - a^2)})</math> erit <math>y = √(x^2
+ y^2 - a^2) + A</math>, ubi <math>A</math> e$t quæcunque invariabilis quantitas; &
non ita affumi pote$t <math>A</math>, ut præcedens <math>x^2 + y^2 - a^2 = 0</math> in hâc <math>y =
√(x^2 + y^2 - a^2) + A</math> contineatur: hæc autem fluens <math>x^2 + y^2 - a^2
= 0</math> potius dici pote$t re$olutio fluxionalis æquationis <math>x x^. + y y^. = 0</math> &
algebraicæ <math>√(x^2 + y^2 - a^2) = 0</math>.</p>
<p>Cor. Hinc etiam in fluxionalibus æquationibus $uperiorum ordinum,
$i quæcunque æquatio $it vere fluens prioris partis fluxionalis æqua-
tionis; fed non vere fluens reliquæ æquationis partis, attamen divifor
vel fluens fluxionalis quantitatis inferioris ordinis (<math>m</math>), quæ e$t divi-
for reliquæ æquationis partis, cujus partis fluxionum in eâ contenta-
rum ordo e$t major quam <math>m</math>; tum æquatio prædicta non vere erit
fluens datæ fluxionalis æquationis.</p>
<p>6. Datâ fluxionali æquatione <math>α = 0</math> vel <math>α P^. + β P q^. = 0</math>; invenire,
annon data æquatio <math>π = 0</math> vel <math>P = 0</math> fit fluens datæ fluxionalis æqua-
tionis: in datis æquationibus <math>α = 0 & π = 0</math> pro variabili <math>y</math> in iis
contentâ, ejus fluxione <math>y^., &c.</math> $cribantur <math>v - 0, v^. - 0^., &c.</math> ubi <math>0</math> e$t
quam minima quantitas; tum ita reducantur duæ re$ultantes æqua-
tiones, ut exterminentur variabilis <math>v</math> & ejus fluxiones, & refultet
æquatio <math>ξ = 0</math> relationem inter <math>x & 0</math> & earum fluxiones exprimens:
& fit <math>0^λ</math> minima pote$tas quantitatis <math>0</math> in æquatione refultante <math>ξ = 0,</math>
unde, $i modo fit fluxionalis æquatio primi ordinis, erit <math>{0^. / 0^λ} = $. X x^.</math>,

<pb 161>[0213]<rh>ÆQUATIONIBUS.</rh>
ubi <math>X</math> e$t functio quantitatis <math>x</math>, & exinde <math>{- 1 / (λ - 1)0^λ-1} = $. X x^. + A</math>,
ubi <math>A</math> e$t quæcunque invariabilis quantitas; unde <math>(1 - λ) × 0^λ-1 ={1 / $. X x^. + A}</math>; quod$i <math>y</math> in prædictâ æquatione <math>α = 0</math> $it vere fluens,
tum $altem <math>y ± 0,</math> ubi <math>o</math> e$t quam minima quantitas, erit etiam fluens;
ut vero <math>{1 / $. X x^. + A}</math> fiat nihilo æqualis, nece$$e e$t ut <math>A</math> evadat infinita
quantitas, & con$equenter <math>λ - 1</math> fit affirmativa quantitas.</p>
<p>Si <math>λ = 1</math>, tum erit log. <math>0 = $. X x^. + log. A</math>, unde <math>0 = A × e^$. X x^.</math>,
& $i <math>A</math>, tum etiam o evadet nihil.</p>
<h>PROB. XXXIX.</h>
<h it>Invenire, annon data algebraica æquatio $it particularis vel generalis
fluens datæ fluxionalis æquationis n ordinis.</h>
<p>I<_>mo</_>. Reducantur data algebraica & fluxionalis æquatio ad minimos
terminos; i. e. auferatur fingula quantitas, quæ ducitur in totam
æquationem vel fluxionalem vel algebraicam; etiamque reducantur
omnes fractiones ad minimos terminos ex dividendo earum numera-
tores & denominatores per maximos eorum communes divi$ores.</p>
<p>Datæ algebraicæ æquationis inveniantur fluxiones primi, fecundi,
<math>&c. n</math> ordinis; & exinde prima, fecunda, tertia, &c. fluxiones alte-
rius variabilis quantitatis <math>y</math> in datâ fluxionali & algebraicâ æquatione
contentæ quibus fluxionibus pro fuis valoribus in datâ fluxionali
æquatione $ub$titutis; æquatio re$ultans vel nihilo æqualis erit, vel
re$ultans & data algebraica æquatio communem habent divi$orem,
$i modo $it fluens: aliter; reducantur datæ algebraicæ & fluxionales
æquationes in unam, ita ut exterminentur omnes præter unam varia-
biles quantitates & earum fluxiones; & ex æquatione re$ultante &
principiis prius traditis facile con$tabit; annon data algebraica æqua-
tio fit fluens datæ fluxionalis.</p>

<pb 162>[0214]<rh>DE FLUXIONALIBUS</rh>
<p>Ex. 1. Sit <math>α P^. = 0</math> data fluxionalis æquatio, & <math>P × Q = 0</math> data al-
gebraica; fint <math>P^. = p x^. + p′ y^. & Q^. = q x^. + q′ y^.</math>; tum fluxio algebraicæ
æquationis <math>(P Q = 0)</math> erit <math>Q p x^. + Q p′ y^. + P q x^. + P q′ y^. = (Q p +
P q) x^. + (Q p′ + P q′) y^. = 0</math>; in datâ fluxionali æquatione <math>α P^. = α p x^.
+ α p′ y^. = 0</math> pro <math>x^.</math> $cribatur ejus valor <math>-{(Q p′ + P q′) / Q p + P q} × y^.</math> ex præcedente
æquatione deductus, & re$ultat <math>α p × {Q p′ + P q′ / Q p + P q} - αp′ = P × a ×{p q′ - q p′ / Q p + P q} = 0</math>; fed <math>P</math> erit communis divi$or æquationum <math>Q × P = 0</math>
& re$ultantis æquationis <math>P × α × {p q′ - q p′ / Q p + P q} = 0</math>, & con$equenter <math>P = 0</math>
erit fluens.</p>
<p>Ex. 2. Invenire, annon data æquatio <math>(A) y^m + x^n + y^r x^s = 0</math> fit
fluens fluxionalis æquationis <math>(B)m × (m - 1)y^m-2 y^.^2 + m y^m-1 y^.. +
n × (n - 1)x^n-2 x^.^2 + γ × (r - 1)y^r-2 x^s y^.^2 + r y^r-1 x^s y^.. + 2 r s y^r-1 x^s-1 y^. x^.
+ s(s - 1)y^r x^s-2 x^.^2 + m a y^m y^. x^. + n a x^n-1 y x^.^2 + r a y^r x^s y^. x^. + s a y^r+1
x^s-1 x^.^2 + b(y^m + x^n + y^r x^s)y^.^2 = 0</math>; datæ æquationis <math>A</math> inveniatur fe-
cunda fluxio, &c. & pro <math>y^..</math> in æquatione <math>B</math> $cribatur ejus valor e $e-
cundâ fluxione æquationis <math>A</math> deductus; in æquatione exinde reful-
tante $cribatur pro <math>y^.</math> ejus valor e primâ fluxione prædictæ æquationis
<math>A</math> deductus, & data æquatio erit divifor æquationis re$ultantis; e
principiis prius traditis con$tat datam algebraicam æquationem non
proprie pro fluente fluxionalis æquationis habendam effe.</p>
<p>2. Ex ii$dem principiis etiam inveniri poteft, utrum data flu-
xionalis æquatio fit fluens alterius datæ fluxionalis æquationis,
necne.</p>
<p>Si in datâ æquatione <math>π = 0</math>, quæ e$t fluens (<math>r</math>) ordinis datæ fluxi-
onalis æquationis <math>α = 0</math>, contineantur (<math>r</math>) invariabiles quantitates <math>a,
b, c, &c.</math> ad libitum affumendæ, quæ in datâ fluxionali æquatione haud
continentur; & quæ in diver$as functiones variabilium quantitatum
in datâ æquatione contentarum ducuntur; i. e. quæ inter $e $unt inde-

<pb 163>[0215]<rh>ÆQUATIONIBUS.</rh>
pendentes: tum, $i data æquatio <math>π = 0</math> $it fluens datæ fluxionalis
æquationis, i.e. $i in æquatione <math>α = 0</math> $cribantur quæcunque quan-
titates pro $uis valoribus ex æquatione <math>π = 0</math> deductis; & evane$cat
æquatio re$ultans, erit <math>π = 0</math> generalis fluens (<math>r</math>) ordinis fluxionalis
æquationis <math>α = 0</math>: $in autem non contineantur (<math>r</math>) prædicti generis
invariabiles quantitates, exceptis excipiendis erit particularis fluens:
particularis $emper continetur in fluente generali.</p>
<p>3. Sint <math>α = 0, β = 0, γ = 0, &c.</math> fluxionales æquationes ordinum
(<math>n, m, r, &c.</math>) re$pective; & <math>λ = 0, μ = 0, ν = 0, &c.</math> fluentiales æqua-
tiones, in quibus continentur (<math>n + m + r + &c.</math>) invariabiles & in-
dependentes ad libitum a$$umendæ quantitates, quæ in datis æqua-
tionibus non inveniuntur: in æquationibus <math>α = 0, β = 0, γ = 0, &c.</math>
$cribantur quæcunque quantitates pro $uis valoribus ex æquationibus
<math>λ = 0, μ = 0, ν = 0, &c.</math> deductis; & $i evane$cant æquationes re$ul-
tantes; tum erunt <math>λ = 0, μ = 0, ν = 0, &c.</math> generalis fluens datarum
fluxionalium æquationum <math>α = 0, β = 0, γ = 0, &c.</math></p>
<p>Cor. Sit <math>π = 0</math> æquatio, quæ continet (<math>m</math>) invariabiles quantitates
<math>a, b, c, d, &c.</math> ad libitum a$$umendas & inter $e independentes, quæ
in æquatione <math>ξ = 0</math> non inveniuntur: in æquatione <math>ξ = 0</math> $cribantur
quæcunque quantitates ex æquationibus <math>π = 0, π^. = 0, π^.. = 0, π^... = 0,
... π^. ^r = 0</math> utcunque deductæ pro $uis valoribus; & in æquatione re-
$ultante non plures quam (<math>r</math>) e prædictis invariabilibus quantitatibus
(<math>a, b, c, d, &c.</math>) evane$cent.</p>
<h>THEOR. XX.</h>
<p>Sit fluxionalis æquatio <math>r</math> ordinis; & $i modo ejus fluens (<math>r</math> ordinis)
inveniri po$$it, tum a fortiori ejus fluentes <math>r - 1, r - 2, &c.</math> ordinum
detegi po$$unt. Inveniatur enim ejus fluens (<math>r</math> ordinis), cujus prima
fluxio erit fluxionalis æquatio primi ordinis, & fluens <math>r - 1</math> ordinis
datæ fluxionalis æquationis; & fic deinceps; unde con$tat theor.</p>

<pb 164>[0216]<rh>DE FLUXIONALIBUS</rh>
<h>PROB. XL.</h>
<p it>Datâ fluλionali æquatione tres (<math>x, y, z,</math>) variabiles quantitates & ea-
rum fluxiones habente; invenire utrum una (<math>x</math>) ex iis exprimi pote$t in ter-
minis duarum reliquarum (<math>y, z</math>), necne.</p>
<p>1. A$$umatur <math>x</math> tanquam invariabilis quantitas, deinde inveniatur
quantitas (<math>p</math>), quæ in re$ultantem æquationem ducta, creat æquatio-
nem, cujus fluens (<math>P</math>) inveniri pote$t; deinde a$$umatur <math>y</math> tanquam
invariabilis, & inveniatur corre$pondens quantitas <math>q</math>, quæ in æqua-
tionem exinde re$ultantem ducta, creat fluxionem, cujus fluens (<math>Q</math>)
inveniri pote$t; 3<_>tio</_>. a$$umatur <math>z</math> invariabilis, & inveniatur corre$pon-
dens quantitas <math>r</math>, quæ in æquationem re$ultantem ducta præbet fluxi-
onem, cujus fluens (<math>R</math>) inveniri pote$t.</p>
<p>Si vero quantitates <math>p & q</math> habeant omnes eo$dem factores, in quibus
invenitur <math>z</math>; & quantitates <math>p & r</math> eo$dem involvant factores, in quibus
continetur <math>y</math>; & tertio quantitates <math>q & r</math> eo$dem habeant factores, in
quibus invenitur <math>x</math>; i. e. $int corre$pondentes; & $i omnes termini, in
quibus continentur literæ (<math>x & y & z</math>) iidem $int in tribus fluentibus
<math>P, Q & R</math>; & omnes termini, in quibus inveniuntur <math>z, y, x</math> $int iidem
in duabus fluentibus <math>P & Q, P & R, Q & R</math> re$pective; tum inveniri
pote$t fluens datæ fluxionalis æquationis, $in aliter vero non.</p>
<p>Ex. 1. Datâ fluxionali æquatione <math>((x^2 y + c)z^. + (x^2 z + d)y^. +
2 x z y x^.) √(z^2 + c x^2) + z z^. + c x x^. = 0</math>, invenire utrum ejus fluens
exprimi pote$t in algebraicis terminis quantitatum (<math>x, y, z</math>), necne.
Supponatur <math>x</math> e$$e invariabilis quantitas, & re$ultat <math>(x^2 z + d)
√(z^2 + c x^2)y^. + ((x^2 y + c)(√(z^2 + c x^2) + z)z^. = 0</math>; ducatur
hæc æquatio in <math>{1 / √(z^2 + c x^2)}</math>, & re$ultat <math>(x^2 z + d)y^. + (x^2 y + c +{z / √(z^2 + c x^2)})z^. = 0</math>, cujus fluens (<math>P</math>) erit <math>x^2 z y + d y + c z +
√(z^2 + c x^2) + A</math>: ubi <math>A</math> e$t functio quantitatis <math>x</math>: eodem modo
$upponatur <math>z</math> invariabilis, & ducatur æquatio re$ultans in prædictam

<pb 165>[0217]<rh>ÆQUATIONIBUS.</rh>
quantitatem <math>1/√(z^2 + c x^2)</math>, & invenietur fluens (<math>R</math>) re$ultantis fluxio-
nis <math>x^2 y z + d y + √(z^2 + c x^2) + B</math>, ubi <math>B</math> e$t functio quantitatis <math>z</math>:
& ultimo $upponatur <math>y</math> invariabilis, & ducatur æquatio re$ultans in
<math>{1 / √(z^2 + c x^2)}</math>, & fluxionis re$ultantis invenietur fluens <math>(Q) x^2 y z +
c z + √(z^2 + c x^2) + C</math>, ubi <math>C</math> e$t functio quantitatis <math>y</math>: terminus
vero <math>x^2 y z</math>, in quo continentur tres variabiles quantitates <math>x, y, z,</math> idem
e$t in tribus fluentibus <math>P, Q & R; &</math> termini, in quibus continetur <math>z</math>,
iidem $unt in fluentibus <math>P & Q</math>; termini, in quibus habetur <math>y</math>, iidem
$unt in fluentibus <math>P & R, &c.</math>; unde fluens datæ fluxionis invenitur
<math>x^2 y z + d y + c z + √(z^2 + c x^2) + A′ = 0</math>, ubi <math>A′</math> denotat inva-
riabilem quantitatem ad libitum a$$umendam.</p>
<p>2. 1. Sæpe vero ex hâc methodo haud con$tat re$olutio problema-
tis, quoniam haud cogno$cuntur methodi, e quibus deduci po$$unt
prædicti cognati multiplicatores; in his ca$ibus confugiendum e$t ad
infinitas $eries, i. e. inveniatur variabilis quantitas (<math>x</math>) in terminis
progredientibus $ecundum dimen$iones reliquarum (<math>y, z</math>), $i modo fieri
po$$it, & confit problema.</p>
<p>2. 2. For$an vero in fluxionalibus æquationibus $uperiorum (<math>n</math>)
ordinum haud inveniri pote$t variabilis quantitas (<math>x</math>) in terminis re-
liquarum (<math>y, z</math>); $ed for$an inveniri pote$t fluxio (<math>x^. ^m</math>) ordinis <math>m</math>, ubi
<math>m</math> minor e$t quam <math>n</math>, in terminis $ecundum dimen$iones reliquarum
(<math>y, z</math>) & earum fluxiones ordinum haud majorum quam <math>n - m</math> pro-
gredientibus, i. e. reduci pote$t data fluxionalis æquatio in alteram
minoris ordinis quam <math>n</math>.</p>
<p>2. 3. Eadem principia etiam applicari po$$unt ad æquationem qua-
tuor vel plures variabiles quantitates & earum fluxiones habentem;
& $ic ad duas vel plures æquationes tres vel plures variabiles quanti-
titates & earum fluxiones involventes.</p>
<p>3. Sit æquatio <math>P x^. + Q y^. + R z^. = 0; & M</math> quantitas, quæ in da-
tam æquationem ducta, præbet fluxionem, cujus fluens inveniri po-

<pb 166>[0218]<rh>DE FLUXIONALIBUS</rh>
te$t, i. e. $it quantitas <math>M P x^. + M Q y^. + M R z^.</math> integrabilis; & $int
<math>M^. = m x^. + m′ y^. + m″ z^., P^. = p x^. + p′ y^. + p″ z^., Q = q x^. + q′ y^. + q″ z^.
& R = r x^. + r′ y^. + r″ z^.</math>; tum ex theor. 2. erit <math>M p′ + Pm′ = M q
+ Q m, M p″ + P m″ = M r + R m & M q′ + Q m″ = M r′ + R m′</math>;
ducantur æquationes exinde re$ultantes <math>M p′ + P m′ - M q - Q m
= 0, M p″ + P m″ - M r - R m = 0 & M q″ + Q m″ - M r′ - R m′
= 0</math> re$pective in <math>R, Q & P</math>; & $umma trium quantitatum re$ultan-
tium erit <math>M(R × (p′ - q) - Q(p″ - r) + P(q″ - r′) = 0</math>; unde
æquatio erit integrabilis, $i modo <math>R(p′ - q) - Q(p″ - r) + P(q″ - r)
= 0</math>; $in aliter non.</p>
<p>3. 2. Sit æquatio <math>P x^. + Q y^. + R z^. + S v^. + T w^. + &c. = 0: 1^mo .</math>
a$$umantur omnes quantitates præter qua$cunque tres (<math>x, y & z</math>)
tanquam invariabiles, & re$ultat æquatio <math>P x^. + Q y^. + R z^. = 0</math>; per
præcedentem methodum inveniatur, annon hæc æquatio $it integrabi-
lis; $i modo $it integrabilis, tum a$$umantur omnes quantitates præter
tres alias (<math>x, y & v</math>) tanquam invariabiles; & inveniatur, annon
æquatio ex hâc hypothe$i re$ultans $it integrabilis; $i modo $it inte-
grabilis, tum a$$umantur omnes quantitates præter qua$cunque tres
alias tanquam invariabiles, & $i omnes æquationes ex huju$modi a$-
$umptis re$ultantes, $int integrabiles; tum integrari pote$t re$ultans
æquatio; $in aliter vero non.</p>
<p>Fluxiones <math>({M^.. P / y^. z^.}) = ({M^.. Q / x^. z^.}) = ({M^.. R / x^. y^.}); & ({M^.. P / y^. v^.}) = ({M^.. Q / x^. v^.})
= ({M^.. S / y^. x^.}), &c.; ({M^... P / y^. z^. v^.}) = ({M^... Q / x^. z^. v^.}) = ({M^... R / x^. y^. v^.}) = ({M^... S / x^. y^. z^.})</math>, &c.,
ubi quantitas <math>({V^... / x^. y^. z^.}) = Z</math> denotat quantitatem ex hac methodo in-
ventam; nempe ex deducendo fluxionem <math>Wx^.</math> quantitatis <math>V</math> ex hypo-
the$i quod <math>x</math> $olummodo $it variabilis; deinde fluxionem <math>Y y^.</math> quanti-
tatis <math>W</math>, ex hypothe$i quod <math>y</math> $olummodo $it variabilis; & tertio fluxi-
onem <math>Z z^.</math> quantitatis <math>Y</math> ex hypothe$i quod <math>z</math> $olummodo $it variabilis.</p>

<pb 167>[0219]<rh>Æ QUATIONIBUS.</rh>
<p>Cor. Sit æquatio fluxionalis (<math>n</math>) variabiles quantitates & earum
fluxiones habens; tum re$ultant <math>n × {n - 1 / 2} × {n - 2 / 3}</math> æquationes, e qui-
bus dici pote$t, annon data fluxionalis æquatio $it integrabilis.</p>
<p>3. 3. Si modo fluxiones variabilium $uperiorum (<math>n, n - 1, n - 2,
&c.</math>) ordinum in datâ æquatione (<math>α = 0</math>) contineantur: ducatur data
æquatio in <math>M</math>, functionem variabilium (<math>x, y, z, &c.</math>) in datâ æquati-
one contentarum, & earum fluxionum (<math>n - 1, n - 2, n - 3,. 3, 2, 1</math>)
ordinum; a$$umatur <math>N</math> functio incognita prædictarum (<math>x, y, z, &c.</math>)
variabilium; deinde inveniatur fluxio (<math>n</math>) ordinis functionis <math>N</math>; &
ex æquatis corre$pondentibus terminis re$ultantis & quantitatis <math>M α</math>
fluxionis <math>n</math> ordinis deduci pote$t, annon data æquatio $it integrabilis.</p>
<h>PROB. XLI.</h>
<p it>Datâ æquatione relationem inter quantitates <math>x & y</math> & earum fluxiones
de$ignante; & datâ quantitate, quæ e$t fluxionalis functio (<math>P</math>) literarum
<math>x & y</math>; invenire æquationem duas involventem incognitas quantitates <math>z
& v</math>, ita ut fluens quantitatis _(_quæ e$t eadem fluxionalis functio (<math>P</math>) lite-
rarum <math>z & v</math>, ac prædicta fuit functio literarum <math>x & y</math>_)_ algebraicè e
fluente datæ quantitatis, quæ fuit data fluxionalis functio literarum <math>x & y</math>,
deduci pote$t.</p>
<p>A$$umantur duæ æquationes, quæ exprimunt algebraicas relatio-
nes inter <math>x, y & z, v</math>; ita ut fluens functionis <math>P</math> literarum <math>z & v</math> de-
duci pote$t algebraicè e fluente functionis <math>P</math>, i. e. e datâ fluxionali
functione literarum <math>x & y</math>; ita reducantur æquationes re$ultantes,
ut exterminentur incognitæ quantitates <math>x & y</math>, & re$ultat quæ$ita
æquatio relationem inter <math>z & v</math> exprimens.</p>
<p>Ex. Datâ æquatione relationem inter <math>x & y</math> & earum fluxiones ex-
primente; $it data quantitas functio (<math>P = y x^.</math>) quantitatum <math>x & y</math>:
a$$umantur duæ æquationes <math>x = a z + b v + c & y = p z + q v + r</math>;

<pb 168>[0220]<rh>DE FLUXIONALIBUS</rh>
& con$equenter <math>a′ x + b′ y + c′ = z & p′ x + q′ y + r′ = v</math>; fluens
functionis <math>P = v z^.</math> inveniri pote$t, $i modo fluens fluxionis <math>y x^.</math> inve-
niri po$$it; nam <math>y x^. = (p z + q v + r) × (a z^. + b v^.)</math>, cujus fluens
<math>$. y x^. = {1 / 2}a p z^2 + r a z + {1 / 2} b q v^2 + r b v + b p z v + $. (a q - b p)v z^.</math>;
unde e fluente <math>$. y x^.</math> deduci pote$t <math>$. v z^.</math>; $cribantur hæ quantitates in
datâ æquatione pro $uis valoribus, & re$ultat æquatio quæ$ita relatio-
nem inter <math>z & v</math> exprimens.</p>
<p>Cor. 1. Datâ fluente fluxionis <math>y x^.</math>, ex eâ $emper inveniri pote$t
fluens fluxionis <math>vz^.</math> vel <math>zv^.</math>: datis fluentibus trium fluxionum <math>y x^.</math>,
<math>y^2 x^., y x x^.</math>; & ex iis $emper inveniri po$$unt fluentes fluxionum <math>v z^.,
v^2 z^., v z z^.; z v^., z^2 v^., z v v^.</math>: datis fluentibus $ex fluxionum <math>y x^., y^2 x^.,
y x x^., y^3 x^., y^2 x x^., y x^2 x^.,</math> & $emper ex iis algebraicè exprimi po$$unt
fluentes fluxionum <math>u z^., v^2 z^., v z z^.; v^3 z^., v z^2 z^., &c. z v^., z^2 v^., &c.</math>
Datis vero fluentibus (<math>n · {n - 1 / 2}</math>) inter $e independentibus fluxionum
huju$ce formulæ <math>y^n-1 x^., y^n-2 x x^., y^n-3 x^2 x^., &c. y^n-2 x^., y^n-3 x x^., y^n-4 x^2 x^.,
&c. y^n-3 x^., y^n-4 x x^., &c. ... y^2 x^., y x x^.; y x^.</math>; & ex iis $emper deduci po$-
$unt fluentes omnium con$imilium quantitatum <math>v^n-1 z^., v^n-2 z z^.,
v^n-3 z^2 z^., &c. v^n-2 z^., v^n-3 z z^., &c. ... v^2 z^., v z z^.; v z^.; z^2 v^., z v v^.,
&c...; z^n-1 v^., z^n-2 v v^., &c.</math></p>
<p>Cor. 2. Datis <math>n · {n - 1 / 2}</math> diver$is, i. e. inter $e independentibus æqua-
tionibus huju$ce generis <math>a $. y x^. + b $. y^2 x^. + c $. y x x^. + d $. y^3 x^. + &c.
... k $. y^n-1 x^. = l</math>; ubi literæ <math>a, b, c, d, &c. k</math> denotant datas invaria-
biles quantitates; tum $acile con$tabit omnis valor cuju$cunque
quantitatis huju$ce generis <math>p $. v z^. + q $. v^2 z^. + r $. v z z^. + &c. t $.
v^n-1 z^. + p′ $. z v^. + q′ $. z^2 v^. + &c.</math>; ubi <math>p, q, r, &c.; t, p′, q′, r′, &c.</math> datas
etiam denotant invariabiles quantitates.</p>
<p>Et $ic de datis con$imilibus quantitatibus.</p>

<pb 169>[0221]<rh>ÆQUATIONIBUS.</rh>
<h>THEOR. XXI.</h>
<p>1. Si area inter quo$cunque duos valores ab$ci$$æ (<math>x</math>) curvæ $emper
detegi po$$it, tum area inter quo$cunque datos valores cuju$cunque
ab$ci$$æ (<math>z</math>) eju$dem curvæ $emper detegi pote$t.</p>
<p>Transformetur enim data æquatio relationem inter ab$ci$$am <math>x</math> &
ejus ordinatam <math>y</math> exprimens in alteram relationem inter novam ab$ci$-
$am <math>z</math> & ejus corre$pondentes ordinatas (<math>v</math>) de$ignantem; per tri-
gonometriam cognitum e$t <math>a x + b y + c = z & px + qy + r = v</math>;
ubi <math>a, b, c; p, q, & r</math> invariabiles denotant coefficientes inveniendas,
unde fluxio datæ curvæ ad novam ab$ci$$am erit ut <math>vz^. = (p x + q y + r)
(ax^. + by^.)</math>; & con$equenter <math>$. vz^. = $. (px + qy + r)(ax^. + by^.) =
pa {x^2 / 2} + {qby^2 / 2} + rax + rby + qa$.yx^. + pbxy - pb$.yx^.</math>; & exinde
e fluente <math>$.yx^.</math> datâ $emper acquiri pote$t <math>$.vz^.</math></p>
<p>2. Datis generaliter centris gravitatis ad duas ab$ci$$as, tum ple-
rumque dabuntur centra gravitatis ad omnes ab$ci$$as: detur enim
generaliter <math>{$.yxx^. / $.yx^.}</math>, dabuntur plerumque <math>$. y x x^. & $. y x^.</math>: detur etiam
centrum gravitatis generaliter ad alteram ab$ci$$am, quod e præce-
dente non pendet; tum dabitur etiam generaliter <math>$.y^2 x^.</math>; $ed datis
generaliter <math>$. y x x^., $. y x^. & $.y^2 x^.</math> per præcedentem propo$itionem $em-
per acquiri po$$unt generaliter <math>$. v^2 z^., $. v z z^. & $.vz^.</math>, & con$equenter
generaliter centra gravitatis ad omnes ab$ci$$as.</p>
<p>Ex ii$dem principiis, $i dentur generaliter centra percu$$ionum,
viz. <math>{$.yx^2 x^. / $.yxx^.}</math> ad quinque ab$ci$$as; tum plerumque ea dabuntur ad om-
nes ab$ci$$as.</p>
<p>3. E datis generaliter contentis $olidorum a rotatione curvæ circa
tres diver$os axes generatorum, quorum unus $it <math>$.3, 1456 &c. ×
y^2 x^.</math>; dabuntur contenta $olidorum a rotatione eju$dem curvæ circa
quo$cunque alios axes generatorum: viz. e tribus independentibus
valoribus generalibus fluentium fluxionum <math>a y^2 x^. + βy x x^. + λyx^.</math>
erui pote$t generaliter <math>$.v^2 z^.</math>.</p>

<pb 170>[0222]<rh>DE FLUXIONALIBUS</rh>
<p>4. Circa tres axes $ecum quo$cunque datos angulos facientes gy-
retur quæcunque curva, cujus ab$ci$$æ & ordinatæ $unt re$pective <math>x
& y</math>; transformetur data curva in alteram $ub$tituendo pro <math>x = a′w
+ b′u + c′</math> & pro <math>y = p′ w + q′ u + r′</math>; ubi <math>w & u</math> re$pective denotant
ab$ci$$am & ejus corre$pondentes ordinatas novæ curvæ & <math>a′, b′, c′,
&c.; p′, q′ & r′</math> invariabiles coefficientes: gyretur hæc curva re$ultans
circa $uum axem, & e datis contentis $olidorum datæ curvæ circa tres
axes rotatione generatorum inter $e independentibus deduci pote$t
$olidum generatum e rotatione po$terioris curvæ circa $uum axem.</p>
<p>Con$tat e principiis prius traditis.</p>
<p>Ea, quæ prius tradita fuere de areis, centris gravitatis & percu$-
$ionum, & contentis $olidorum a rotatione datæ curvæ circa ejus axes;
mutatis mutandis, æque ad areas, prædicta centra & $olida cuju$cun-
que curvæ ex hâc $ub$titutione exortæ applicari po$$unt.</p>
<p>Cor. 4. Si pro <math>x & y</math> in datâ æquatione $cribantur re$pective <math>av^2 +
bzv + cz^2 + dz + ev + A & fv^2 + gvz + bz^2 + lz + mv + B</math>
vel quæcunque rationales vel irrationales quantitates; facile deduci
po$$unt huju$modi propo$itiones.</p>
<p>Et $ic de fluxionalibus functionibus quantitatum <math>z & v</math> pro datis
fluxionalibus functionibus quantitatum <math>x & y</math> in datâ æquatione vel
datis æquationibus $criptis; exinde enim con$imilia deduci po$$unt.</p>
<h>THEOR. XXII.</h>
<p>In fluxione <math>yx^.</math> pro <math>x & y</math> $cribantur re$pective <math>az^m + (b + cv)
z^m-1 + (d + ev + fv^2)z^m-2 + &c. & Az^r + (B + Cv)z^r-1 + (D +
Ev + Fv^2)z^r-2 + &c.</math> ubi <math>a, b, c, d, e, f, &c. A, B, C, D, E, F, &c.</math>
$unt quæcunque invariabiles coefficientes, <math>m</math> vero & <math>r</math> integri nu-
meri; & ex fluentibus independentibus <math>(m + r - 1).{m + r / 2}</math> fluxio-
num ex prædictis $ub$titutionibus re$ultantium datis, per $implices
æquationes erui po$$unt fluentes omnium fluxionum exinde re$ultan-
tium; etiamque e prædictis fluentibus datis erui po$$unt fluentes

<pb 171>[0223]<rh>ÆQUATIONIBUS.</rh>
quarumcunque fluxionum huju$ce formulæ <math>z^z v^β z^.</math>, $i modo <math>α + β</math>
haud major $it quam <math>m + r - 1, & α & β</math> $int integri numeri.</p>
<p>Totidem <math>(m + r - 1) × {m + r / 2}</math> enim in datâ fluxione <math>y x^.</math> continen-
tur diver$æ fluxiones, quarum fluentes $unt independentes, viz. <math>$. v z^.,
$. v^2 z^., $.vzz^., $.v^2 z^., $.v^2 zz^.,$.vz^2 z^., &c.</math></p>
<p>Et $ic ratiocinari liceat de aliis $ub$titutionibus in plurimis aliis
fluxionibus; viz. exponentialibus, fluentialibus, &c.</p>
<h>THEOR. XXIII.</h>
<p>1. Sit data æquatio <math>(a y + b x + c)x^. + (A y + B x + C)y^. = 0</math>;
& ex datâ fluente fluxionis <math>y x^.</math> (quæ plerumque acquiri pote$t e datâ
æquatione) erui pote$t fluens cuju$cunque formulæ <math>(by^m + (k + lx)
y^m-1 + &c.)x^. + (Hy^m + (K + Lx)y^m-1 + &c.)y^.</math>; ubi literæ <math>b, k, l,
&c. H, K, L, &c.</math> qua$cunque invariabiles coefficientes re$pective de-
notant; & <math>m</math> e$t integer numerus.</p>
<p>Si enim detur fluens fluxionis <math>y x^. = α</math>, tum dabitur fluens omnium
fluxionum formularum <math>y y^., y x^., x y^., xx^., x^., y^.;</math> ducatur data æquatio
in <math>x & y</math> re$pective, & re$ultant <math>(ayx + bx^2 + cx)x^. + (Ayx + Bx^2
+ Cx)y^. = π = 0 & (ay^2 + bxy + cy)x^. + (Ay^2 + Bxy + Cy)y^.
= ζ = 0</math>; $ed per hypothe$in dantur fluentes omnium fluxionum
hi$ce æquationibus contentarum præter duas fluentes (<math>P & Q</math>) flu-
xionum <math>y x x^. & yxy^.</math>; vel fluxionum <math>x^2 y^. & y^2 x^.</math>, quæ ex iis pendent;
erunt enim <math>x^2 y - 2P = $.x^2 y^. & y^2 x - 2Q = $.y^2 x^.</math>; $cribantur hæ
duæ quantitates pro $uis valoribus in æquationibus <math>π = 0 & ζ = 0</math> &
re$ultant <math>Bx^2 y + (a - 2B)P + {bx^3 / 3} + {cx^2 / 2} + AQ + C(xy - α) + F
= 0 & ay^2 x + (B - 2a)Q + bP + cα + {Ay^3 / 3} + {Cy^2 / 2} + G = 0</math>,
(ubi <math>F & G</math> $unt invariabiles quantitates) quæ $unt fluentes duarum
re$ultantium æquationum <math>π = 0 & ζ = 0</math>; i. e. exoriuntur duæ æqua-
tiones, quarum $ingulæ fluentes præter <math>P & Q</math> deduci po$$unt, ergo

<pb 172>[0224]<rh>DE FLUXIONALIBUS</rh>
facile acquiri po$$unt fluentes <math>P & Q</math>; & exinde dabitur fluens omnis
fluxionis huju$ce formulæ <math>(α′y^2 + βyx + γx^2 + δy + εx + τ)x^. +
(π′y^2 + ζ′yx + σx^2 + τy + λx + μ)y^.</math>, ubi <math>α′, β, γ, &c. π′, ζ′, σ, &c.</math>
$unt invariabiles quantitates; & $ic deinceps.</p>
<p>2. Sit data æquatio <math>(ay^n + (b + cx)y^n-1 + (d + ex + fx^2)y^n-2 +
&c.)x^. + (Ay^n + (B + Cx)y^n-1 + &c.)y^. = 0,</math> & dentur fluentes
$ingularum quantitatum huju$ce generis <math>(V)(by^m + (k + lx)y^m-1 +
&c.)x^. + (Hy^m + (K + Lx)y^m-1 + &c.)y^.</math>, ubi literæ <math>b, k, l, &c. H,
K, L, &c.</math> qua$cunque invariabiles quantitates re$pective denotant;
etiamque dentur (<math>n - 1</math>) fluentes fluxionum huju$ce generis <math>(W)
(αy^m+1 + (β + γx)y^m + &c.)x^. + (πy^m+1 + ζy^m x + &c.)y^.</math>, (ubi <math> n &
m</math> integri $unt numeri affirmativi, & <math>m</math> major quam <math>n; & α, β, γ, &c.
π, ζ, &c.</math> $unt invariabiles quantitates) quæ nec a fluentibus quibu$-
cunque in <math>V</math>, nec a $e invicem detegi po$$unt; tum ex omnibus flu-
entibus generis (<math>V</math>) & prædictis <math>n - 1</math> fluentibus detegi pote$t quæ-
cunque fluens generis <math>W</math>.</p>
<p>Ducatur data æquatio re$pective in (<math>m - n + 2</math>) quantitates <math>y^m-k+1,
y^m-n x, y^m-n-1 x^2, y^m-n-2 x^3, y^m-n-3 x^4, &c.; &</math> re$ultant <math>m - n + 2</math> di-
ver$æ æquationes: $ed in æquatione <math>W</math> continentur (<math>m + 1</math>) fluxio-
nes, viz. <math>y^m+1 x^., y^m x x^., y^m-1 x^2 x^., &c...yx^m x^.</math>; quarum fluentes for$an
nec a $e invicem, nec a fluentibus fluxionum in <math>V</math> contentarum de-
duci po$$unt: unde ex <math>(m + 1) - (m - n + 2) = n - 1</math> indepen-
dentibus fluentibus fluxionum formularum <math>y^m+1 x^., y^m xx^., y^m-1 x^2 x^.,
y^m-2 x^3 x^., &c.</math> & ex omnibus fluentibus generis <math>V</math> acquiri po$$unt
fluentes omnium fluxionum generis <math>W</math>.</p>
<p>In quamplurimis ca$ibus fluens <math>W</math> e paucioribus fluentibus quam
<math>n - 1</math> prædictis deduci pote$t.</p>
<p>In nonnullis ca$ibus denominator nihilo evadat æqualis, in quo
ca$u fluens quæ$ita e prædictis non detegi pote$t.</p>
<p>Eadem etiam ad algebraicas, fluxionales & incrementiales æqua-
tiones; & fluxionales, &c. quantitates applicari po$$unt; in pleri$que
ca$ibus facile deduci pote$t numerus datarum $luentium vel inte-
gralium, quas exigit data æquatio.</p>

<pb 173>[0225]<rh>ÆQUATIONIBUS.</rh>
<h>PROB. XLII.</h>
<p it>Datâ algebraicâ vel fluxionali œquatione <math>α = 0</math>, & fluxionali quanti-
tate <math>π</math>; in quibu$dam ca$ibus invenire, annon fluens quantitatis <math>π</math> detegi po-
te$t ope datarum fluentium.</p>
<p>Inveniatur, annon data quantitas <math>π</math> $it $umma quarumcunque di-
rectarum functionum quantitatis <math>α</math> in finitas quantitates vel variabiles
vel invariabiles ductarum; & fluxionum, quarum fluentes detegi po$-
$unt ope datarum fluentium.</p>
<p>Hoc vero $æpe perfici pote$t, <math>1^mo</math>. ex auferendo vel pro nihilo ha-
bendo qua$cunque quantitates in quantitate <math>π</math> contentas, quæ e datis
fluentibus detegi po$$unt: <math>2^do</math>. ex ob$ervando & inter $e compa-
rando irrationales quantitates, & dimen$iones & rationalium & ir-
rationalium quantitatum in <math>α & π</math> contentarum; exinde enim con-
$tabunt $unctiones quantitatum in <math>α</math> contentarum; etiamque quanti-
tates, in quas ducendæ $unt prædictæ functiones, ita ut præbeant
quantitates in <math>π</math> contentas, quæ haud detegi po$$unt ope datarum
fluentium: <math>3^tio .</math> e reducendo per methodum prius traditam datam
fluxionalem quantitatem in fluxionalem quantitatem, cujus ordo $it
minor quam ordo datæ fluxionalis æquationis; & deinde ex inveni-
endo, utrum fluens fluxionalis quantitatis re$ultantis ope datarum
fluentium deduci pote$t, necne.</p>
<p>Per hanc methodum, viz. reductionem fluxionalis quantitatis ita ut
ejus ordo minor $it quam datæ fluxionalis æquationis ordo, $æpe ex
re$ultante fluxione detegi pote$t, utrum fluens datæ fluxionalis quan-
titatis $ic exprimi pote$t, necne; cum in eâ plures involvantur varia-
biles quantitates & earum fluxiones.</p>
<h>PROB. XLIII.</h>
<p it>Data fluxionali œquatione duas variabiles quantitates <math>x & y,</math> & earum
fluxiones involvente; invenire functiones prœdictarum quantitatum <math>x & y</math>
& earum fluxionum, quarum fluentes dantur.</p>

<pb 174>[0226]<rh>DE FLUXIONALIBUS</rh>
<p>1. Dividatur data æquatio in duas partes $ibi invicem æquales, ita
ut fluens unius partis inveniri pote$t; tum etiam erit fluens alterius
partis.</p>
<p>2. Ducatur data æquatio in quamcunque quantitatem, & ita di-
vidatur fluxionalis æquatio re$ultans in duas inter $e æquales partes,
ut fluens unius partis inveniatur; & erit etiam fluens alterius.</p>
<p>Ex. 1. Sit fluxionalis æquatio <math>yx^. = nax^n-1 y^m x^. + may^m-1 x^m y^. +
bx^r x^. + cy^s y^., &</math> erit fluens fluxionis <math>yx^.,</math> i. e. area curvæ, cujus relatio
inter ab$ci$$am & ejus corre$pondentes ordinatas per datam fluxiona-
lem æquationem exprimitur <math>= a x^n y^m + {bx^r+1 / r + 1} + {cy^s+1 / s + 1} + A</math>, ubi <math>A</math>
$it invariabilis quantitas.</p>
<p>Ex. 2. Sit <math>y^2 x = P Q^. + Q P^. + A x^. + B y^.</math>, ubi <math>A & B</math> $unt
quæcunque functiones incognitarum quantitatum <math>x & y</math> re$pective;
tum erit fluens fluxionis 3, 1416 <math>y^2 x^.</math> (i.e. contentum $olidi generati
a rotatione curvæ, cujus relatio inter ab$ci$$as & ejus corre$ponden-
tes ordinatas de$ignatur per datam æquationem) <math>= 3, 1416 (QP +
$. Ax^. + $.By^. + a)</math>, ubi <math>a</math> e$t invariabilis quantitas.</p>
<p>Ex. 3. Sit æquatio fluxionalis <math>y x^. + n x y^. = by^. √(x^2n + cx^n + d)</math>,
ducatur data æquatio in quamcunque quantitatem, e. g. <math>y^n-1,</math> & re-
$ultat æquatio <math>y^n x^. + n y^n-1 x y^. = by^n-1 y^. √(x^2n + cx^n + d)</math>, unde <math>y^n x
= b$.y^n-1 y^. √(x^2n + cx^n + d) + a.</math></p>
<p>3. Si modo e fluentibus prædictâ methodo inve$tigatis deduci po$-
fint fluentes plurium fluxionum, tum harum fluxionum con$equentur
fluentes.</p>
<p>Ex. Sit fluxionalis æquatio <math>y^2 x^. = a x^r x^. + by^s y^.</math>; & $i modo detur
fluens fluxionis <math>y^2 x^. (A) = {a x^r+1 / r+1} + {by^s+1 / s + 1} + c</math>, dabitur etiam fluens
fluxionis <math>y x y^. (B^.)</math>; nam <math>{y^2 x - A / Q} = B = {y^2 x / Q} - {ax^r+1 / Q(r + 1)} -{bx^s+1 / Q(s + 1)} - {c / 2}</math>.</p>

<pb 175>[0227]<rh>ÆQUATIONIBUS.</rh>
<p>Et $ic de pluribus fluxionalibus æquationibus datis.</p>
<p>4. Sit æquatio vel algebraica vel fluxionalis <math>P = 0</math>, dentur fluen-
tes fluxionum <math>L, M, &c.</math> & $int etiam <math>R, S, T, &c.</math> fluxiones, quarum
fluentes e fluentibus datis fluxionum <math>L, M, &c.</math> vel $eparatim vel con-
junctim deduci po$$unt: tum fluens cuju$cunque fluxionis <math>αφ: (P)
+ aR + bS + cT + &c. + σ</math> (ubi <math>a, b, c, &c.</math> $unt invariabiles
quantitates; & <math>α</math> quæcunque fluxio vel quantitas, quæ pro ejus deno-
minatore haud habet functionem quantitatis <math>P</math> vel ejus divi$orem;
<math>& φ: (P)</math> $it quæcunque functio quantitatis <math>P = 0</math>, in quâ nulla
continetur pote$tas vel nihilo æqualis vel negativa) e prædictis flu-
entibus <math>L, M, &c.</math> deduci pote$t.</p>
<p>Cor. Datâ quâcunque fluxione (<math>Q</math>), facile deduci po$$unt infinitæ
fluxionales æquationes, quarum fluens fluxionis (<math>Q</math>) inveniri pote$t.
Dividatur fluxio <math>Q</math> in qua$cunque duas partes <math>α & β,</math> i.e. <math>Q = α + β</math>,
quarum una (<math>α</math>) inveniri pote$t ope datarum fluentium; tum vel
affumatur <math>β = 0</math>, vel quæcunque functio (<math>π</math>) quantitatis <math>β</math> nihilo
æqualis, quæ talis $it, ut functio (<math>π</math>) quantitatis <math>β</math> haud in $e po-
te$tatem contineat nihilo æqualem vel negativam; & deducitur æqua-
tio quæ$ita.</p>
<h>PROB. XLIV.</h>
<p it>Datâ fluxionali æquatione <math>P x^. + Q y^. = 0</math>, ubi <math>P & Q</math> $unt functiones
quantitatum <math>x & y;</math> invenire fluentem fluxionis <math>W x^. + Ry^.,</math> ubi <math>W & R</math>
$unt functiones quantitatum <math>x & y.</math>.</p>
<p>1<_>mo</_>. Si <math>W^. = αx^. + β y^., & R^. = πx^. + ζy^., & β = π;</math> tum integrari
pote$t data fluxio.</p>
<p>2. Sit <math>(A P + W)x^. + (AQ + R)y^.</math>, ubi <math>A</math> e$t functio quantita-
tum <math>x & y</math>, fluxio integrabilis; tum inveniatur ejus fluens, quæ etiam
erit fluens fluxionis <math>W x^. + Ry^.</math>.</p>
<p>3. Si modo fluens (<math>α = 0</math>) fluxionalis æquationis <math>P x^. + Q y^. = 0</math>
detegi po$$it, tum in datâ fluxione (<math>W x^. + Ry^.</math>) pro <math>y^.</math> $cribatur <math>- {Px^. / Q},</math>

<pb 176>[0228]<rh>DE FLUXIONALIBUS</rh>
& re$ultat <math>{W Q - RP / Q}x^.;</math> in hâc fluxione pro <math>y</math> $cribatur ejus valor,
qui erit functio quantitatis <math>x</math> ex datâ æquation <math>α = 0</math> deducta, unde
re$ultat data fluxio <math> = x^. φ: (x)</math>, cujus fluens (<math>M</math>) inveniri pote$t.</p>
<p>Sit <math>H =</math> con$t. fluens fluxionalis æquationis <math>P x^. + Q y^. = 0</math>, tum
erit <math>H -</math> con$t. <math>+ M</math> fluens datæ fluxionis <math>W x^. + Ry^.</math>.</p>
<p>4. Affumatur <math>W x^. + R y^. = V^.</math>; tum ita reducantur duæ æquatio-
nes <math>W x^. + R y^. = V^. & Px^. + Qy^. = 0</math> in unam, ut exterminentur vel
<math>y & y^.</math>; vel <math>x & x^.</math>, & refultat fluxionalis æquatio, cujus fluens e$t
quæ$ita.</p>
<p>Et $ic de pluribus æquationibus plures variabiles quantitates &
fluxiones $uperiorum ordinum habentibus.</p>
<p>5. Datâ fluxionali æquatione <math>r</math> ordinis, cujus variabiles quantitates
$int <math>x & y, & x^.</math> $it con$tans; datâ etiam fluxionali quantitate, quæ
$it functio variabilium <math>x, y</math> & fluxionum <math>x^., y^., y^.., ... y^. ^r + m</math> u$que ad or-
dinem <math>r + m</math>; invenire quantitatem fluxionalem, quæ haud implicat
fluxiones quantitatis <math>y</math>, quarum ordo $uperior $it ordini <math>r - 1</math>, i. e.
$uperior ordini fluxionis <math>y^. ^r - 1,</math> datæ fluxionali quantitati æqualem.</p>
<p>Inveniatur fluxio datæ æquationis ordinis <math>m</math>; ex æquatione re$ul-
tante deducatur valor fluxionis <math>y^. ^r+m;</math> $cribatur valor inventus pro <math>y^. ^r+m</math> in
datâ fluxionali quantitate, & exterminabitur <math>y^. ^r+m;</math> ex eodem proce$$u
repetito exterminari poffunt fluxiones <math>y^. ^r+m-1, y^. ^r+m-2, &c.</math> ad <math>y^. ^r-1; &</math> tandem
re$ultabit quantitas quæ$ita.</p>
<p>Eadem principia etiam applicari po$$unt ad plures fluxionales
æquationes & fluxionalem quantitatem datam plures variabiles quan-
titates & earum fluxiones habentes.</p>
<h>THEOR. XXIV.</h>
<p>Datis fluxionalibus æquationibus involventibus <math>x & y, &c. &</math> ea-
rum fluxiones, & datâ functione <math>P</math> quantitatum <math>x & y, &c. &</math> earum

<pb 177>[0229]<rh>ÆQUATIONIBUS.</rh>
fluxionum. Scribantur pro <math>x & y, &c. &</math> earum fluxionibus in datis
æquationibus & datâ functione <math>P</math> quæcunque functiones literarum
<math>z & v, &c. &</math> earum fluxiones; & re$ultabunt æquationes relationem
inter <math>z & v, &c. &</math> earum fluxiones exprimentes, & functio <math>Q</math> quan-
titatum <math>z & v, &c. &</math> earum fluxionum, cujus fluens e fluente fun-
ctionis <math>P</math> con$equitur $cribendo in eâ pro <math>x & y, &c. &</math> earum fluxio-
nibus prædictas functiones quantitatum <math>z, v, &c.</math> & earum fluxiones.</p>
<h>PROB. XLV.</h>
<p it>1. Invenire, quando duo valores quantitatis y in fluxionali æquatione
duas variabiles quantitates <math>x & y &</math> earum fluxiones <math>x^. & y^.</math> babente $int
inter $e æquales, i. e. in æquatione <math>αx^. = βy^.</math>, ubi <math>α & β</math> $unt quæcunque
algebraicæ functiones literarum <math>x & y.</math></p>
<p>Inveniatur fluens huju$ce æquationis, quæ $it <math>P = Q</math>, ubi <math>P & Q</math>
$unt etiam quæcunque functiones literarum <math>x & y</math>, & æquatio vere
correcta erit <math>P = Q + a</math>, ubi <math>a</math> e$t invariabilis quantitas ad libitum
a$$umenda; $ed duo valores quantitatis <math>y</math> fiunt inter $e æquales, cum
<math>βy^. = 0</math> & exinde <math>α x^. = 0, & β = 0;</math> in quâ æquatione exprimitur
relatio inter <math>x & y, &</math> in æquatione correctâ <math>P = Q + a</math> exprimitur
relatio inter tres incognitas quantitates <math>x, y & a</math>; unde duæ re$ultant
æquationes tres incognitas quantitates <math>x, y & a</math> habentes, & con$e-
quenter haud detegi pote$t, quando duæ radices quantitatis <math>y</math> fiant
inter $e æquales, ni detur quantitas corrigens <math>a.</math></p>
<p>Dato autem uno valore (<math>π</math>) quantitatum <math>x</math> vel <math>y</math>, quando fiant
duo valores quantitatis <math>y</math> inter $e æquales, exinde deduci po$$unt reli-
qui: dato enim valore quantitatis <math>x</math> vel <math>y</math> in æquatione <math>β = 0</math>, exinde
$equitur ejus corre$pondens valore quantitatis (<math>y</math> vel <math>x</math>), quibus valo-
ribus pro <math>y & x</math> in æquatione <math>P = Q + a</math> $ub$titutis, re$ultat valor
quantitatis <math>a,</math> & exinde duæ æquationes <math>β = 0 & P = Q + a</math>, quæ duas
incognitas quantitates <math>x & y</math> habent, e quibus deduci po$$unt reliqui
valores quantitatum <math>x & y</math>, quando duo valores quantitatis <math>y</math> fiant in-
ter $e æquales.</p>

<pb 178>[0230]<rh>DE FLUXIONALIBUS</rh>
<p>Si vero æquatio <math>P = Q</math> haud exprimi po$$it in finitis terminis, tum
confugiendum e$t ad infinitas $eries.</p>
<p>2. Datâ æquatione fluxionali <math>n</math> ordinis, i.e. in quâ continetur
fluxio <math>y^. ^n, & x</math> fluit uniformiter; invenire generaliter quando duo va-
lores quantitatum (<math>x</math> vel <math>y</math>) fiant inter $e æquales. Nece$$e e$t in hoc
ca$u, ut dentur <math>n</math> corre$pondentes valores $ingularum fluxionum in
datâ æquatione contentarum, e. g. <math>y^. ^n, y^. ^n-1, &c. x^.</math>, etiamque quantita-
tum <math>x & y</math>, quando duo valores quantitatum <math>x</math> vel <math>y</math> fiant inter $e
æquales; & his datis generaliter deduci pote$t, quando duo valores
prædictarum quantitatum fiant inter $e æquales.</p>
<p>Hoc con$tat ex eo, quod <math>n</math> incognitas & invariabiles coefficientes
habet æquatio vere correcta.</p>
<p>Eadem etiam applicari po$$unt ad duas vel plures æquationes tres
vel plures variabiles quantitates habentes.</p>
<p>Hæc ratiocinandi methodus haud $olummodo ad problemata re$ol-
venda, in quibus duo valores quantitatum <math>x</math> vel <math>y</math> inter $e æquales e$$e
$upponuntur, $ed etiam in quamplurima alia extendi pote$t.</p>
<p>3. Datâ. fluxionali æquatione duas variabiles quantitates & earum
fluxiones habente, invenire coefficientium con$titutionem.</p>
<p>Habeat æquatio <math>n</math> terminos, & con$equenter horum terminorum
$unt <math>n</math> coefficientes: $cribantur pro <math>n</math> corre$pondentibus valoribus
incognitarum quantitatum <math>(x & y) α, β, γ, δ, &c. π, ζ, σ, τ, &c.</math> pro
corre$pondentibus valoribus primarum fluxionum re$pective <math>α^., β^., γ^.,
δ^., &c. π^., δ^., σ^., τ^., &c.</math> & $ic de de$ignandis $ecundis, tertiis, &c. flu-
xionibus incognitarum quantitatum <math>(x & y):</math> $ub$tituantur hæ literæ
pro $uis quantitatibus in datâ æquatione, & re$ultant <math>n</math> æquationes
totidem incognitas coefficientes habentes, e quibus inveniri poffunt
coefficientium conftitutiones.</p>
<p>Multa de impo$$ibilibus radicibus, &c. deduci po$$unt fere ex eâ-
dem ratiocinandi methodo in fluxionalibus ac in algebraicis æquati-
onibus u$itatâ.</p>

<pb 179>[0231]<rh>ÆQUATIONIBUS.</rh>
<h>PROB. XLVI.</h>
<p it>Datis formulis fluxionum; invenire fluxiones prœdictarum formularum,
quarum fluentes exprimi po$$unt in finitis terminis, vel in terminis fluen-
tium, quarum dantur fluxiones.</p>
<p>J<_>mo</_>. Ex datis formulis fluxionum inveniantur formulæ quantita-
tum in prædictis formulis contentarum, ita ut fluentes fluxionum
re$ultantium datarum formularum re$pective exprimi po$$unt vel per
finitos terminos vel per fluentes prædictas, quarum fluxiones dantur:
deinde ex comparandis quantitatibus prædictarum formularum inter
$e $æpe inveniri pote$t formula quantitatis, quæ $atisfacit omnibus
conditionibus quæ$tionis prædictæ.</p>
<p>Ex. 1. Sit data fluxio <math>= P x^.</math>, cujus fluens acquiri pote$t, ubi <math>P</math> e$t
functio variabilium <math>x & y</math>; a$$umatur quæcunque functio (<math>Q</math>) quan-
titatis <math>x</math> pro fluente; tum erit <math>{Q^. / x^.} = P</math>, quæ e$t algebraica æquatio
exprimens relationem inter <math>x & y</math>, unde ex variabili <math>x</math> detegi pote$t
variabilis (<math>y</math>), ita ut fluens datæ fluxionis inveniri pote$t.</p>
<p>Ex. 2. Sint duæ formulæ <math>y x^. & y^2 x^.</math>, quarum fluentes requiruntur
in finitis terminis: a$$umatur pro fluente fluxionis <math>y x^.</math> quantitas for-
mulæ <math>{(a + b x + c x^2 + d x^3 + &c.)^1{1 / 2} × (A + Bx + Cx^2 + D x^3 + &c.)/(h + k x)^{λ / μ}}</math>;
cujus fluxio $it <math>P x^.</math>; tum fiat <math>P = y, & P^2 x^. = y^2 x^.</math> (exceptis excipi-
endis) etiam erit fluxio, cujus fluens in finitis terminis exprimi po-
te$t: & $ic detegi po$$unt plures huju$modi formulæ.</p>
<p>Ex. 3. Sint duæ fluxiones <math>y x^. & √ (x^.^2 + y^.^2)</math>; pro <math>y^.</math> $cribatur <math>z x^.</math>, &
evadent <math>$. y x^. = y x - $. x y^. = y x - $. z x x^. & √(1 + z^2) x^. = √ (x^.^2
+ y^.^2)</math>; invenire (<math>z</math>) functionem quantitatis <math>x</math>, quæ reddit fluxiones
<math>z x x^. & √ (1 + z^2) x^.</math> formularum, quarum fluentes exprimi po$$unt
in finitis terminis quantitatis (<math>x</math>): 1<_>mo</_>. a$$umatur <math>x = z^λ</math>, & evadent

<pb 180>[0232]<rh>DE FLUXIONALIBUS</rh>
duæ fluxiones <math>z x x^. = λ z^2λ z^. & √ (1 + z^2) x^. = λ (1 + z^2)^{1 / 2} × z^λ-1 z^.</math>;
unde, $i <math>λ</math> $it par numerus, fluentes duarum prædictarum fluxionum
$emper inveniri po$$unt.</p>
<p>Ex. 4. A$$umatur <math>x = {(1 - R^2)^{3 / 2} P^./- R^.}, y = {P R^. - R P^. (1 - R^2) / R^.}</math>, ubi
<math>R</math> fluat uniformiter; tum erit <math>√ (x^.^2 + y^.^2) = √ ({(1 - R^2)^3 P^.. ^2 - 6 R (1 - R^2)<x>2</x>
R R^. P^. P^.. + 9 R^2 R^.^2 P^.^2 × (1 - R^2) / R^.^2} + {R^2 (1 - R^2)^2 P^.. ^2 - 6 R^3 (1 - R^2)
R^. P^. P^.. + 9 R^4 R^.^2 P^.^2 / R^.^2}) = {(1 - R^2) P^.. - 3 R R^. P^. / R^.}</math>, cujus fluens e$t <math>- $.
R P^. + {(1 - R^2) P^. / R^.}</math>.</p>
<p>Cor. Hæc methodus in detegendis curvis algebraicis, quæ rectifi-
cari po$$unt, haud multum u$ui in$ervit.</p>
<h>THEOR. XXV.</h>
<p>Sit quæcunque fluxionalis æquatio duas involvens incognitas
quantitates <math>x & y</math> & earum fluxiones, in quâ <math>x</math> fluit uniformiter;
$emper erit quantitas, quæ in datam æquationem ducta, creat fluxio-
nem, cujus fluens inveniri pote$t.</p>
<p>Ex. g. Sit data fluxio <math>A^. = a x^. + b y^. = 0</math>, $int etiam <math>a^. = α x^. + β y^.,
b^. = π x^. + ξ y^., & π = β</math>; tum ip$ius æquationis fluens per theor. 2.
inveniri pote$t.</p>
<p>Si vero fluens haud inveniri po$$it; $emper datur quantitas <math>L</math>, quæ
in datam æquationem <math>a x^. + b y^. = 0</math> ducta, creat fluxionem <math>L a x^. +
L b y^.</math>, cujus fluens inveniri pote$t; $it <math>L^. = r x^. + s y^.</math>, & per prædictum
theorema erit <math>a s + L β = L π + b r</math>.</p>
<p>Con$imiles propo$itiones etiam de fluxionalibus æquationibus $u-
periorum ordinum e theor. 2. facile deduci po$$unt.</p>

<pb 181>[0233]<rh>ÆQUATIONIBUS.</rh>
<p>Cor. 1. Datâ unâ quantitate <math>L</math>, quæ in datam fluxionalem æquatio-
nem ducta, præbet fluxionem, cujus fluens inveniri pote$t; ex eâ <math>L</math>
facile deduci po$$unt infinitæ aliæ: $it enim data æquatio <math>a x^. + b y^.
= 0</math>, quæ ducta in <math>L</math> dat fluxionem <math>L a x^. + L b y^.</math>, cujus fluens in-
veniri pote$t, quæ $it <math>z</math>: i. e. $it <math>z^. = L a x^. + L b y^.</math>: $it <math>Z</math> quæcunque
functio quantitatis <math>z</math>, & fluens fluxionis <math>Z z^. = Z L a x^. + Z L b y^.</math> $em-
per deduci pote$t; i. e. fluxio e$t integrabilis: & con$equenter plures
deducuntur quantitates <math>Z L</math>, quæ in datam fluxionem <math>a x^. + b y^.</math> ductæ,
præbent fluxiones integrabiles.</p>
<p>Cor. 2. Inveniantur duæ quantitates <math>P & R</math>, quæ in <math>a x^. + b y^.</math> ductæ
dabunt fluxiones integrabiles <math>P a x^. + P b y^. = V^.; R a x^. + R b y^. = W^.</math>,
tum <math>V & {P / R}</math> erunt functiones quantitatis <math>W</math>.</p>
<p>Cor. 3. Sit <math>M</math> particularis valor fluxionalis æquationis vel fluxiona-
lium æquationum; tum erit quæcunque functio quantitatis <math>M</math>, etiam
particularis valor prædictæ fluxionalis æquationis, &c. nam particu-
laris $emper in generali continetur.</p>
<p>Cor. 4. Sit æquatio <math>P x^. + Q y^. = 0</math>; ducatur hæc æquatio in <math>M =
H × L</math> & re$ultat <math>M (P x^. + Q y^.) = 0</math>, quæ $it integrabilis æquatio,
i. e. $it <math>M P x^. + M Q y^.</math> fluxio, cujus fluens inveniri pote$t: $i vero
& <math>P & Q</math> nullum habeat divi$orem, qui reciprocat cum quantitate <math>H</math>;
tum per prob. 4. & 5, &c. <math>H</math> erit divi$or particularis fluentis fluxionis
<math>M (P x^. + Q y^.)</math>, & con$equenter <math>H = 0</math> erit quodam modo particula-
ris fluens fluxionalis æquationis <math>P x^. + Q y^. = 0</math>.</p>
<p>Ducatur data æquatio in <math>{1 / M} = {1 / H × L}</math>, & re$ultat <math>{1 / M} (P x^. + Q y^.)</math>;
$i vero & <math>P & Q</math> nullum habeat divi$orem <math>H</math>, tum per prædicta pro-
blemata <math>H</math> erit divi$or vel numeratoris vel denominatoris fluentis
fluxionis <math>M (P x^. + Q y^.)</math>, & con$equenter <math>H = 0</math> vel <math>{1 / H} = 0</math> dici po-
te$t quodammodo particularis valor fluxionalis æquationis <math>P x^. + Q y^.
= 0</math>.</p>

<pb 182>[0234]<rh>DE FLUXIONALIBUS</rh>
<p>Et $ic de fluxionalibus æquationibus $uperiorum ordinum: multi-
plicator autem $uperioris ordinis pote$t e$$e quæcunque fluxio cuju$-
cunque in$erioris ordinis.</p>
<h>THEOR. XXVI.</h>
<p>1. Sit quæcunque fluxionalis æquatio <math>P x^. + Q y^. = 0</math>, ducatur hæc
æquatio in <math>q</math>, & $it fluens fluxionis re$ultantis generalis <math>= S</math>; i. e.
<math>S^. = q P x^. + q Q y^.</math>; tum fluens datæ æquationis erit univer$aliter
quæcunque functio fluentis <math>S</math>.</p>
<p>Sit enim quæcunque functio quantitatis <math>(S) = φ: (S) = 0</math> fluens
quæ$ita, cujus fluxio erit <math>Δ: (S) × S^. = 0</math>, ubi <math>Δ: (S)</math> de$ignat fun-
ctionem quantitatis <math>S</math>; dividatur hæc æquatio per <math>Δ: S vel Δ: (S) × q</math>,
& re$ultat <math>S^. = 0</math> vel <math>{S^. / q} = P x^. + Q y^. = 0</math>, quæ e$t data æquatio; ergo
quæcunque functio quantitatis <math>S</math> erit re$olutio datæ æquationis.</p>
<p>Hic animadvertendum e$t, quamvis <math>φ: S = 0</math> e$t fluens datæ flu-
xionalis æquationis; attamen <math>V × φ: S = 0</math>, ubi <math>V</math> non e$t functio
quantitatis <math>S</math>, non proprie dici pote$t fluens prædictæ fluxionalis
æquationis, quamvis in eâ fluentem continet.</p>
<p>2. Sit æquatio <math>A x^. + B y^. = 0</math>, ducatur hæc æquatio in <math>q</math>, ita ut
fluens quantitatis <math>q A x^. + q B y^.</math> inveniri po$$it, quæ $it <math>P</math>; pro <math>P</math> vel
quâvis functione quantitatis <math>P</math> $cribatur <math>z</math>, & erit <math>z^. = 0</math>.</p>
<p>Et $ic de æquationibus fluxionalibus $uperiorum ordinum.</p>
<p>3. Sit æquatio <math>Q z^. = 0</math>, & erit ejus fluens <math>z = a</math> vel quæcunque
functio quantitatis <math>(z) = 0</math>; minime vero <math>Q = 0</math> proprie dici pote$t
re$olutio datæ æquationis; quamvis in ca$ibus pro hâc re de$ignatis
ejus fluens erit valor datæ æquationis.</p>
<h>THEOR. XXVII.</h>
<p>1. Sint <math>α = 0 & π = 0</math> duæ fluxionales æquationes <math>m & r</math> ordinum
re$pective, quorum <math>m</math> minor e$t quam <math>r, & K & L</math> duo multiplicatores

<pb 183>[0235]<rh>ÆQUATIONIBUS.</rh>
ita con$tituti ut <math>K α + L π = V^.</math>, ubi <math>V</math> e$t fluxionalis quantitas or-
dinis (<math>r - 1</math>); duæ datæ fluxionales æquationes reducuntur in duas
fluxionales <math>V = con$t. & α = 0</math>: & $imiliter deprimi po$$unt æqua-
tiones reductæ, &c.</p>
<p>Et $ic de pluribus fluxionalibus æquationibus plures variabiles
quantitates habentibus.</p>
<p>2. Sint datæ fluxionales æquationes <math>a = 0, b = 0, c = 0, &c.</math>; &
$int <math>α = 0, β = 0, &c.</math> generales fluentes quorumcunque ordinum
fluxionalium æquationum <math>a = 0, b = 0, &c.</math>; tum erit generalis fluens
fluxionalium æquationum <math>α = 0, β = 0, &c., c = 0, &c.</math> eadem ac
generalis fluens fluxionalium æquationum datarum <math>a = 0, b = 0,
c = 0, &c.</math></p>
<h>THEOR. XXVIII.</h>
<p>Si detur fluens generalis (<math>W = 0</math>) fluxionalis æquationis (<math>P = 0</math>),
facile exinde deduci pote$t multiplicator <math>M</math>, qui in datam æquatio-
nem ductus, reddit eam integrabilem.</p>
<p>In datâ generali fluente $it <math>C</math> quæcunque invariabilis quantitas ge-
neralis in datâ fluxionali æquatione haud contenta, & ita reducatur
per vulgarem algebram æquatio <math>W</math>, ut fiat <math>C × V = Z</math>, ubi <math>V & Z</math>
$unt quantitates in quibus haud continetur <math>C</math>; erit <math>{Z^. V - ZV^. / V^2 P} = M</math>
multiplicator quæ$itus.</p>
<p>Et $ic de fluxionalibus æquationibus $uperiorum ordinum.</p>
<p>E. g. Detur generalis fluens <math>W = 0</math> fluxionalis æquationis, quæ
continet duas <math>C & D</math> invariabiles coefficientes ad libitum a$$umendas;
ex æquatione <math>W = 0</math> inveniatur vel <math>C = α</math> vel <math>D = β</math>, in quibus (<math>α
& β</math>) haud continentur re$pective <math>C & D</math>, quarum fluxiones erunt
<math>α^. = 0 & β^. = 0</math>; deinde ex æquationibus <math>α^. = 0 & β^. = 0</math> re$pective de-
ducantur <math>D = π & C = ξ</math>, & evadunt duæ æquationes <math>π^. = 0 & ξ^. = 0</math>

<pb 184>[0236]<rh>DE FLUXIONALIBUS</rh>
eadem æquatio, cujus generalis fluens $ecundi ordinis e$t data æqua-
tio; duo vero inveniuntur ejus primi multiplicatores vel <math>{α^. / π^..}</math> vel <math>{β^. / ξ^.}</math>.</p>
<p>Et $ic deinceps.</p>
<h>THEOR. XXIX.</h>
<p>1. Sit quæcunque fluxionalis æquatio <math>n</math> ordinis <math>P = 0</math>, cujus ge-
neralis fluens $it <math>W = 0</math>, quæ continet qua$cunque <math>n</math> diver$as invari-
abiles quantitates <math>a, b, c, d, e, &c.</math> ad libitum a$$umendas & in datâ
fluxionali æquatione non contentas; ex æquatione <math>W = 0</math> inveniantur
<math>a = α, b = β, c = γ, d = δ, &c.</math> tum erunt <math>α^. = 0, β^. = 0, γ^. = 0,
δ^. = 0, &c. (n)</math> diver$æ æquationes, quæ $unt fluentes (<math>n - 1</math>) ordinis
datæ æquationis <math>P = 0</math>, in quibus continentur $olummodo <math>n - 1</math> in-
variabiles quantitates prædicti generis; deinde ex æquatione <math>α^. = 0</math>
inveniatur vel <math>b = β′</math> vel <math>c = γ′, &c.</math> & $ic ex æquationibus <math>β^. = 0,
γ^. = 0, δ^. = 0, &c.</math> inveniantur <math>a = α″, c = γ″, &c. a = α′″, b = β′″,
&c. &c.</math> re$pective, tum erunt <math>β^.′ = 0, γ^.′ = 0, &c. α^.″ = 0, γ^.″ = 0,
&c. α^.′″ = 0, &c. &c. (n × {n - 1 / 2})</math> diver$æ fluxionales æquationes, quæ
$unt fluentes (<math>n - 2</math>) ordinis datæ æquationis <math>P = 0</math>; & $ic inveniri
po$$unt <math>n · {n - 1 / 2} · {n - 2 / 3} ... {n - m + 1 / m}</math> diver$æ fluxionales æquatio-
nes, quæ $unt fluentes <math>m</math> ordinis fluxionalis æquationis <math>P = 0</math>; &
quæ continent (<math>m</math>) diver$as invariabiles quantitates ad libitum a$$u-
mendas, quæ in datâ æquatione haud continentur; $i <math>m = n</math>, tum ex
$ingulis hi$ce $ub$titutionibus eadem re$ultabit æquatio.</p>
<p>Cor. Sit quæcunque algebraica (<math>π = 0</math>) æquatio, cujus variabiles
$int <math>x, y, z, &c.; & n</math> invariabiles (<math>a, b, c, d, e, &c.</math>) tum ex eâ inve-
niantur (<math>n</math>) fluxionales æquationes <math>α = 0, β = 0, γ = 0, &c.</math>; ex qui-
bus exterminatur una invariabilis quantitas, viz. <math>a, b, c, &c.</math> re$pective;
deinde detegantur <math>n · {n - 1 / 2}</math> fluxionales æquationes $ecundi ordinis,

<pb 185>[0237]<rh>ÆQUATIONIBUS.</rh>
viz. <math>α′ = 0, β′ = 0, γ′ = 0, &c.</math> ex quibus $ingulis exterminantur quæ-
cunque duæ invariabiles <math>a & b, a & c, b & c, &c.</math> re$pective; & $ic
inveniantur <math>n · {n - 1 / 2} · {n - 2 / 3}</math> fluxionales æquationes tertii ordinis, ex
quibus exterminantur quæcunque tres invariabiles quantitates re-
$pective; & tandem invenientur fluxionales æquationes <math>n</math> ordinis, ex
quibus exterminantur (<math>n</math>) invariabiles quantitates; & omnes hæ flu-
xionales æquationes re$ultantes erunt eædem.</p>
<p>2. Si vero haud ita reduci po$$int prædictæ æquationes, ut inve-
niantur <math>a = α, b = β, c = γ, &c.</math>; tum ita reducantur duæ æquati-
ones <math>W = 0 & W = 0</math> in unam; ut exterminetur vel <math>a</math> vel <math>b</math> vel <math>c,
&c.</math>: & re$ultabunt prædictæ æquationes <math>b α^. = 0, k β^. = 0, l γ^. = 0,
&c.</math> ubi <math>b, k & l</math> $unt quæcunque algebraicæ quantitates; deinde
eodem modo reducantur duæ æquationes <math>b α^. = 0 & b α^.. + b^. α^. = 0;
k β^. = 0 & k β^.. + k β^. = 0; l γ^. = 0 & l γ^.. + l^. γ^. = 0; &c.</math> re$pective in
unam, ita ut exterminetur $ecunda invariabilis quantitas vel <math>b</math> vel <math>c,
&c.</math>; & re$ultant <math>n · {n - 1 / 2}</math> diver$æ fluxionales æquationes <math>b′ β^.′ = 0,
k′ γ^.′ = 0, &c., b″ α^.″ = 0, &c.</math>; ubi <math>b′, k′, &c.; b″, &c.</math> re$pective deno-
tant functiones algebraicarum quantitatum & fluxionum primi ordi-
nis; & $ic deinceps.</p>
<p>Cor. 3. Ex datâ generali fluente datæ æquationis methodo prius
traditâ deduci pote$t multiplicator, qui eam reddit integrabilem; $ed
con$tat e problemate, quod in æquatione fluxionali <math>n</math> ordinis (<math>n</math>)
dantur diver$æ fluxionales æquationes (<math>n - 1</math>) ordinis, quæ $unt flu-
entes generales datæ æquationis; & con$equenter (<math>n</math>) $unt proprii
multiplicatores diver$orum generum, qui ducti in datam æquationem
præbent fluxiones, quarum fluentes primi ordinis inveniri po$$unt;
& $ic de fluxionalibus æquationibus <math>n - 2, n - 3, n - 4, &c.</math> ordinum
inveniendis.</p>
<p>Hic autem ob$ervandum e$t, quod $i æquatio fluxionalis $it <math>P = 0
& M</math>. $it multiplicator eam reddens integrabilem, i. e. <math>M P</math> $it fluxio,

<pb 186>[0238]<rh>DE FLUXIONALIBUS</rh>
cujus fluens inveniri pote$t; per <math>M φ: ($. M P)</math> volo multiplicatorem
eju$dem haud vero diver$i generis.</p>
<h>THEOR. XXX.</h>
<p>Sint <math>A = 0, B = 0, C = 0, D = 0, &c. (n)</math> generales fluentes primi
ordinis fluxionalis æquationis <math>α = 0, &c.</math>; $it <math>α = 0</math> fluxionalis æqua-
tio <math>n</math> ordinis, variabiles quantitates <math>x, y</math>, & earum fluxiones <math>x^., y^., y^.., y^...,
... y^. ^n</math> involvens, & <math>x^.</math> $it con$tans; & in <math>A = 0</math> continetur <math>a</math> invaria-
bilis quantitas ad libitum a$$umenda, quæ in fluxionali æquatione
<math>α = 0</math> non continetur; & $imiliter in <math>B = 0</math> continetur invariabilis <math>b</math>,
quæ in prædictâ æquatione <math>α = 0</math> non invenitur; deinde in <math>C = 0,
D = 0, &c.</math> continentur re$pective invariabiles quantitates <math>c & d,
&c.</math> independentes & ad libitum a$$umendæ, quæ in <math>α = 0</math> non
inveniuntur, & in $ingulis his æquationibus <math>A = 0, B = 0, &c.</math> haud
continetur <math>y^. ^n</math>; tum ita reducantur æquationes <math>A = 0, B = 0, C = 0,
D = 0, &c.</math> ut exterminetur fluxio <math>y^. ^n-1</math> & re$ultant <math>n · {n - 1 / 2}</math> æquatio-
nes, quæ erunt fluentes $ecundi ordinis datæ fluxionalis æquationis
<math>α = 0</math>; & $ic deinceps; ita reducantur hæ æquationes <math>A = 0, B = 0,
C = 0, &c.</math> in unam, ut exterminentur fluxiones <math>y^. ^n-1, y^. ^n-2, y^. ^n-3, &c.</math> & tan-
dem re$ultabit æquatio, quæ nullas implicat fluxiones, & quæ gene-
ralis erit fluens datæ fluxionalis æquationis.</p>
<p>Cor. 1. Fluens generalis <math>n - 1</math> ordinis e $ingulis (<math>n</math>) æquationibus
<math>A = 0, B = 0, C = 0, D = 0, &c.</math> eadem evadet.</p>
<h>PROB. XLVII.</h>
<p it>Sint <math>x, y, w</math> tres variabiles quantitates, & $int <math>φ = 0 & φ′ = 0</math> duæ fun-
ctiones quantitatum <math>x, y & z</math>, in quibus functionibus contincantur <math>m + r</math>
invariabiles quantitates (<math>a, b, c, d, e, &c.</math>) ad libitum a$$umendæ; inve-

<pb 187>[0239]<rh>ÆQUATIONIBUS.</rh>
nire duas æquationes fluxionales <math>m & r</math> ordinum, quarum generales fluentes
$unt <math>φ = 0 & φ′ = 0</math>.</p>
<p>1. Ex utri$que æquationibus <math>φ = 0 & φ′ = 0</math> inveniantur <math>a = π,
& a = β</math> re$pective, unde <math>β = π, π^. = 0, & β^. = 0</math> tres æquatio-
nes, quarum una e$t algebraica, duæ vero reliquæ fluxionales primi
ordinis, in quibus haud continetur <math>a</math>; deinde ex æquatione <math>π = β</math>
inveniatur <math>b = π′</math>, etiamque ex æquationibus <math>π^. = 0, & β^. = 0</math>, inve-
niantur re$pective <math>b = ξ & b = γ</math>, unde re$ultant duæ fluxionales
primi ordinis æquationes <math>π^.′ = 0, & ξ = γ</math>, in quibus nec contine-
tur <math>a</math> nec <math>b</math>; tum ex æquationibus <math>π^.′ = 0, & ξ = γ</math>, inveniantur
<math>c = σ, & c = δ</math>, & exinde re$ultant duæ fluxionales æquationes <math>σ = δ,
& σ^. = 0</math>, quarum prior erit fluxionalis æquatio primi ordinis, po$te-
rior vero $ecundi ordinis, in quibus continetur nec <math>a</math> nec <math>b</math> nec <math>c</math>:
& $ic ex hâc methodo ratiocinandi continuo repetitâ, erui po$$unt duæ
æquationes quæ$itæ.</p>
<p>2. Si vero ex datâ æquatione <math>λ = 0</math> non inveniri po$$it <math>a = λ′</math>, & con-
$equenter <math>λ^.′ = 0</math> æquatio, quæ non involvit quantitatem <math>a</math>; tum re-
ducantur duæ æquationes <math>λ = 0 & λ^. = 0</math> in unam, ita ut extermine-
tur quantitas (<math>a</math>), & re$ultabit æquatio prædicta <math>λ^.′ = 0; &c.</math> aliter:
reducantur duæ æquationes <math>φ = 0 & φ′ = 0</math> in unam, ita ut exter-
minetur (<math>a</math>), & re$ultet æquatio <math>α = 0</math>; deinde inveniantur duæ
æquationes <math>φ^. = 0 & φ^.′ = 0</math>, & ita reducantur hæ duæ æquationes
in unam, ut exterminetur <math>a</math>; & re$ultet æquatio <math>π = 0</math>; unde re$ul-
tant duæ æquationes, una algebraica <math>α = 0</math>, altera vero fluxionalis
primi ordinis <math>π = 0</math>, in quibus non continetur invariabilis (<math>a</math>;) de-
inde ita reducantur tres æquationes <math>α = 0, α^. = 0 & π = 0</math> in duas
fluxionales æquationes (<math>μ = 0 & ν = 0</math>) primi ordinis, ita ut ex-
terminentur invariabiles quantitates (<math>a & b</math>): tum ita reducantur
æquationes <math>φ = 0 & φ′ = 0</math>, ut exterminetur invariabilis quantitas <math>b</math>,
& re$ultet æquatio <math>α′ = 0</math>; & ita reducantur æquationes <math>φ^. = 0 &
φ^.′ = 0</math> in unam, ut exterminetur quantitas <math>b</math>, & re$ultet æquatio

<pb 188>[0240]<rh>DE FLUXIONALIBUS</rh>
<math>π′ = 0</math>; unde re$ultant duæ æquationes, una algebraica <math>α′ = 0</math>, altera
vero fluxionalis primi ordinis <math>π′ = 0</math>, in quibus non continetur inva-
riabilis <math>b</math>; deinde ita reducantur tres æquationes <math>α′ = 0, α^.′ = 0 & π′ = 0</math>
in duas fluxionales æquationes (<math>μ′ = 0 & ν′ = 0</math>) primi ordinis, ita ut
exterminentur invariabiles quantitates <math>a & b</math>; tum inveniantur
fluxionales æquationes $ecundi ordinis <math>μ^. = 0, ν^. = 0; μ^.′ = 0, & ν^.′ = 0</math>,
in quibus non inveniuntur variabiles quantitates <math>a & b</math>: ita reducan-
tur prædictæ æquationes, ut exterminentur fluxiones <math>x^., y^., z^., &c.; a &
b</math>; & re$ultabit algebraica æquatio, in quâ non inveniuntur invaria-
biles <math>a & b</math>; unde re$ultant duæ æquationes una algebraica, & altera
fluxionalis $ecundi ordinis, in quibus non inveniuntur <math>a & b</math>; & $ic
deinceps ad exterminationem quantitatum (<math>a, b, &c. &c.</math>) progredi
liceat.</p>
<p>Con$imilia etiam applicari po$$unt ad tres vel plures datas æqua-
tiones quatuor vel plures variabiles quantitates & <math>b</math> invariabiles in-
volventes.</p>
<p>Ea, quæ in hoc prob. tradita fuere, $imiliter applicari po$$unt ad
plures (<math>n</math>) fluxionales æquationes <math>m, r, r, s, &c.</math> ordinum detegendas,
quarum variabilium quantitatum dantur <math>n</math> algebraicæ æquationes
generales relationes exprimentes.</p>
<h>THEOR. XXXI.</h>
<p>1. Sit fluxionalis æquatio <math>λ = 0</math> ordinis (<math>n</math>), cujus (<math>n</math>) generales
fluentes $int <math>α = 0, β = 0, γ = 0, δ = 0, &c.</math>; & cujus (<math>n</math>) generales
multiplicatores $int <math>π = 0, ξ = 0, σ = 0, τ = 0, &c.</math>; erunt <math>α^. = π λ,
β^. = ξ λ, γ^. = σ λ, &c.; & {α^. / β^.} = {π / ξ}, {α^. / γ^.} = {π / σ}, {β^. / γ^.} = {ξ / σ}, &c.</math>: hic <math>α, β, γ,
δ, &c.; π, ξ, σ, τ, &c.</math> $unt algebraicæ vel fluxionales quantitates mi-
norum ordinum quam (<math>n</math>).</p>
<p>2. Generalis fluens fluxionalis æquationis erit quæcunque functio
fluentium <math>α, β, γ, δ, &c.</math> i. e. erit <math>φ: (α, β, γ, δ, &c.)</math>: inveniatur enim
ejus fluxio, & re$ultat <math>φ′: (α, β, γ, δ, &c.) × α^. + φ″:(α, β, γ, δ, &c.) β^.

<pb 189>[0241]<rh>ÆQUATIONIBUS.</rh>
+ φ′″:(α, β, γ, δ, &c.) × γ^. + &c. = (φ′:(α, β, γ, δ, &c.) + φ″:(α, β,
γ, δ, &c.) {ξ / π} + φ′″:(α, β, γ, δ, &c.) {σ / π} + &c.) × α^. = (φ′:(α, β, γ, δ, &c.)
π + φ″:(α, β, γ, δ, &c.) ς + φ′″:(α, β, γ, δ, &c.) × σ + &c.) × λ = 0.</math></p>
<h>THEOR. XXXII.</h>
<p>Datâ fluxionali æquatione <math>P - p = 0</math>, ubi <math>P</math> $it quæcunque fun-
ctio literarum <math>x & y</math> & earum fluxionum, <math>p</math> functio literæ <math>x</math> & ejus
primarum fluxionum, formulæ vero ita ut multiplicator (<math>M</math>) $it fun-
ctio quantitatis <math>x</math>, qui ductus in <math>P</math> præbet fluxionem, cujus fluens,
i. e. $. <math>M P</math> inveniri pote$t; tum etiam erit $. <math>M (P - p)</math> fluxio, cu-
jus fluens innote$cet; $ed datâ generali fluente æquationis <math>P = 0</math> in-
veniri po$$unt omnes ejus multiplicatores, & con$equenter inveniri
po$$unt omnes multiplicatores æquationis <math>P - p = 0</math>, iidem enim
invenientur; & exinde deduci pote$t fluens æquationis prædictæ
<math>P - p = 0</math>.</p>
<p>E. g. Sit æquatio <math>A y^. ^n + B y^. ^n-1 ... P y x^.^n + Q x^.^n = 0</math>, ubi <math>A, B .. P, Q</math>
$unt functiones quantitatis <math>x</math>; tum, $i modo detur generalis valor
quantitatis <math>y</math> in æquatione <math>A y^. ^n + B y^. ^n-1 .. P y x^.^n = 0</math>, in quâ deficit ul-
timus terminus, & con$equenter omnes multiplicatores eam reddentes
integrabilem, qui vero multiplicatores erunt functiones quantitatis
<math>x</math>; con$equenter dabuntur etiam multiplicatores, qui reddunt æqua-
tionem <math>A y^. ^n + B y^. ^n-1 .. P y x^.^n + Q x^.^n = 0</math> integrabilem.</p>
<p>Ex. 2. Fluens fluxionalis æquationis <math>x^2 y^.. + (m + n + 1) x x^. y^. +
m n y x^.^2 = 0</math> erit vel <math>x^m+1 y^. + n y x^m x^. = a x^.</math>, vel <math>x^n+1 y^. + m y x^n x^. = a′x^.</math>;
ubi <math>x</math> fluit uniformiter:fluentes autem harum duarum fluxionalium
æquationum erunt <math>x^n y = {a x^n-m / n-m} + b</math> vel <math>x^m y = {a′ x^m-n / m-n} + b′</math>; ubi <math>a, b,
a′ & b′</math> $unt invariabiles quantitates ad libitum a$$umendæ.</p>

<pb 190>[0242]<rh>DE FLUXIONALIBUS</rh>
<p>Si vero a$$umantur <math>{a / n-m} = b′ & b = {-a′ / n-m}</math>; tum hæ duæ fluen-
tes erunt eædem.</p>
<h>PROB. XLVIII.</h>
<p it>Invenire fluxionalem æquationem, cujus particularis valor $it data æqua-
tio, viz. <math>y = φ:(x)</math>, vel quæcunque æquatio (<math>π = 0</math>) relationem inter <math>x</math>
& <math>y</math> exprimens.</p>
<p>A$$umatur quæcunque fluxionalis quantitas <math>P x^. + Q y^.</math>, ubi <math>P & Q</math>
$unt $unctiones quantitatum <math>x</math> & <math>y</math>: in hâc quantitate nonnunquam
pro <math>y</math> $cribatur <math>φ:x</math> vel pro quibu$dam quantitatibus $cribantur ea-
rum valores ex æquatione <math>π = 0</math> deducti; & pro <math>y^.</math> $cribatur ejus va-
lor ex æquatione <math>y = φ:(x)</math> vel ex æquatione <math>π = 0</math> deductus; &
re$ultabit fluxionalis æquatio, cujus particularis fluens erit <math>y = φ:
(x) vel π =0</math>.</p>
<p>2. Invenire æquationem fluxionalem, cujus particulares valores $unt
quæcunque datæ quantitates <math>α = 0, β = 0, γ = 0</math>, &c.</p>
<p>Inveniatur fluxionalis æquatio, cujus partiularis fluens erit <math>φ:(α)
× φ:(β) × φ:(γ) × &c. = 0</math>:vel inveniatur æquatio, cujus particu-
laris fluens erit <math>p + a π + b ς + c σ + &c. = 0</math>, ubi <math>p + a π = α,
p + b ς = β, p + c σ = γ, &c. & a, b, c, &c.</math> $unt coefficientes ad li-
bitum a$$umendæ; & perficitur problema.</p>
<p>Infinitis modis facile deduci pote$t huju$ce problematis re$olutio.</p>
<h>PROB. XLIX.</h>
<p it>Datis duabus quantitatibus <math>p & q</math>, quæ $unt functiones vel algebraicæ
vel fluxionales quantitatum <math>x & y</math>; invenire utrum <math>p</math> $it functio quanti-
tatis <math>q</math>, necne.</p>
<p>Sint <math>P & Q</math> functiones quantitatum <math>x & y</math>, quibus æquales funt <math>p &
q</math> re$pective, i. e. $int <math>P = p, Q = q</math>; ita reducantur hæ duæ æquatio-

<pb 191>[0243]<rh>ÆQUATIONIBUS.</rh>
nes in unam, ut exterminentur <math>x</math> & ejus fluxiones, & $i etiam eva-
ne$cat quantitas <math>y</math> & ejus fluxiones, tum <math>p</math> erit functio quantitatis <math>q</math>.</p>
<p>Et $ic de pluribus æquationibus plures variabiles quantitates ha-
bentibus.</p>
<p>Cor. Datâ fluente (<math>p</math>) fluxionalis æquationis, & <math>p & q</math> functionibus
quantitatum (<math>x & y</math>): per hoc problema detegi pote$t, annon <math>q</math> e$t
etiam fluens datæ æquationis.</p>
<h>PROB. L.</h>
<p it>_1._ Invenire fluxionales æquationes, quarum fluentiales innote$cunt. A$$u-
matur quæcunque æquatio pro fluentiali, inveniatur ejus fluxio, & divida-
tur per quamcunque quantitatem <math>N</math>; tum erit re$ultans æquatio fluxionalis,
cujus fluentialis innote$cit.</p>
<p>2. _Ex ii$dem principiis deduci po$$unt duæ vel tres vel plures_ (<math>n</math>) _flu-_
_xionales æquationes, tres vel quatuor & denique quemlibet numerum varia-_
_bilium & earum fluxionum involventes, quarum fluentiales innote$cunt._</p>
<p>A$$umantur quæcunque (<math>n</math>) æquationes involventes (<math>n + 1</math> vel plu-
res) variabiles quantitates & earum fluxiones, &c. pro fluentialibus;
inveniantur fluxiones cuju$cunque ordinis prædictarum (<math>n</math>) a$$umpta-
rum æquationum, & ducantur æquationes re$ultantes in qua$cunque
invariabiles quantitates; addantur $imul hæc producta, ita ut evadant
(<math>n</math>) diver$æ independentes æquationes; tum dividantur hæ (<math>n</math>) æqua-
tiones per qua$libet quantitates; & re$ultant (<math>n</math>) fluxionales æqua-
tiones, quarum (<math>n</math>) fluentiales innote$cunt.</p>
<p>A$$umendæ $unt æquationes pro fluentialibus huju$ce formulæ
<math>P + α = 0, &c.</math> ubi <math>P</math> e$t quælibet functio variabilium & earum flu-
xionum, & <math>α</math> e$t quæcunque invariabilis quantitas ad libitum a$$u-
menda.</p>

<pb 192>[0244]<rh>DE FLUXIONALIBUS</rh>
<h>THEOR. XXXIII.</h>
<p>De methodis inveniendi fluentes fluxionalium æquationum.</p>
<p>1. Sit æquatio, in quâ continentur variabiles quantitates <math>x</math> & <math>y</math> &
earum fluxiones <math>r</math> ordinum, & in quâ nulla quantitas fluit unifor-
miter; & $i $upponantur omnes termini, in quibus haud continentur
<math>x^. ^r vel y^. ^r</math> nihilo æquales, & $it æquatio re$ultans <math>P x^. ^r + Q y^. ^r = 0</math>, & a$$u-
matur æquatio <math>P x^. + Q y^. = 0</math>, & ducatur ea in quantitatem <math>q</math>, ita ut
fiat integrabilis, & inveniatur fluens æquationis re$ultantis <math>q P x^. + q Q y^.
= 0</math>, quæ dicatur <math>t</math>; fluens datæ æquationis erit functio quantitatis
<math>t</math>, quâ inventâ plerumque e principiis in theor. 2. datis deduci pote$t
fluens quæfita.</p>
<p>Demon$trari pote$t hoc theorema e methodo inveniendi fluxiones
fluentium.</p>
<p>Et $ic inveniri pote$t fluens <math>m</math> ordinis datæ æquationis, $i modo
fluentem recipiat data æquatio.</p>
<p>2. Si <math>y^. ^r^s</math> $it maxima dimen$io fluxionis <math>y^. ^r</math>, cujus ordo e$t maximus
datâ æquatione contentus; tum 1<_>mo</_>. ita reducenda e$t data æquatio,
ut haud plures quam una dimen$io quantitatis <math>y^. ^r</math> in eâ contineantur: &
re$ultabunt plures fluxionales æquationes <math>P = 0, Q = 0, R = 0, &c.</math>,
quarum diver$æ $unt refolutiones, diver$ique etiam multiplicatores;
e. g. $it fluxionalis æquatio <math>y^. ^r^2 - (A + B) y^. ^r + A B = 0</math>, ubi literæ <math>A</math>
& <math>B</math> re$pective denotant functiones quantitatum <math>y, y^., y^.., y^..., .. y^. ^r-1</math>; reli-
quarum variabilium & earum fluxionum; tum reduci pote$t data
fluxionalis æquatio ad duas alias <math>y^.^r - A = 0 & y^.^r - B = 0</math>, quarum
fluentes & multiplicatores po$$unt e$$e diver$i.</p>
<p>In re$olvendis fluxionalibus æquationibus, in quibus involvuntur
$olummodo <math>x & y, x^. & y^.: 1^mo</math>. ita reducendæ $unt datæ æquationes,
ut haud plures quam una dimen$io fluxionum <math>x^. & y^.</math> in datis æquatio-
nibus contineantur.</p>

<pb 193>[0245]<rh>ÆQUATIONIBUS.</rh>
<p>Ex. g. Sit æquatio fluxionalis <math>x^2 x^.^2 + x y x^. y^. = a^2 y^.^2</math>, erit <math>x^2 x^.^2 +
x y x^. y^. + {1 / 4}y^2 y^.^2 = (a^2 + {1 / 4} y^2) y^.^2</math>, unde <math>x x^. + {1 / 2}y y^. = ± √ (a^2 + {1 / 4}y^2) y^.</math></p>
<p>3. Transformetur data fluxionalis æquatio in eandem æquationis
partem, i. e. nihilo fiat æqualis; & quantitatis re$ultantis inveniatur
per methodos in præcedente capite traditas fluens, $i modo fieri po$-
$it, & invenitur fluens quæ$ita.</p>
<p>4. Si fluens haud præcedente methodo deduci po$$it; inve$tiganda
e$t quantitas, quæ in datam æquationem ducta, creat æquationem,
cujus fluens inveniri pote$t.</p>
<p>In hi$ce quantitatibus deducendis, quæ in datam æquationem
ductæ, præbent æquationes, quarum fluentes inveniri po$$unt; haud
inutiles for$an erunt $ub$equentes ob$ervationes.</p>
<p>1<_>ma</_>. Nullam compo$itam irrationalem quantitatem, i. e. irrationa-
lem quantitatem duobus vel pluribus terminis con$tantem involvi
nece$$e e$t plerumque in quæ$itâ re$ultante æquatione, quæ in datâ
haud invenitur, ni exponentiales in quæ$itâ fluente contineantur
quantitates; e. g. $it data fluxionalis æquatio <math>x x^. + 2 a (x x^. + y y^.)
(a^2 + x^2)^{1 / 2} (x^2 + y^2)^{1 / 2} = y y^. (a^2 + x^2)^{1 / 2}</math>; in hâc æquatione duæ conti-
nentur irrationales compo$itæ quantitates, viz. <math>(a^2 + x^2)^{1 / 2} & (x^2 + y^2)^{1 / 2}</math>,
ergo haud improbabile erit finitam æquationem, quæ exprimit fluen-
tem <math>(a^2 + x^2)^{1 / 2} + {2a / 3} (x^2 + y^2)^1{1 / 2} = {1 / 2} y^2 + b</math>, nullas alias compo$itas
irrationales quantitates involvere, ni in eâ contineantur exponentiales
vel fluentiales quantitates: in genere igitur a$$umatur quantitas præ-
dicti generis, quæ e$t generalis rationalis functio quantitatum <math>x^n</math> &
irrationalium quantitatum in datâ æquatione contentarum, ad pote-
$tatem vel radicem (<math>m</math>) evecta; (hìc for$an haud indignum e$t ob$er-
vatu generalem rationalem functionem quantitatum <math>x^n & v^m</math> e$$e fra-
ctionem <math>{(a x^r n + b x^(r+s)n + c x^(r+2s)n + &c.) v^em + (a′ x^r′ n + b x^(r′+s′)n + &c.) v^(e+b)m + &c. /
(A x^α n + B x^(α+β)n + C x^(α+2β)n + &c.) v^rm + (A′ x^α′ n + B′ x^(α′+β′)n
+ &c.) v^(r′+h′)m + &c.)}</math>; & exinde deduci pote$t fluens, ni in eâ con-
tineantur exponentiales vel fluentiales quantitates.</p>

<pb 194>[0246]<rh>DE FLUXIONALIBUS</rh>
<p>2<_>da</_>. In datis diver$is terminis datæ æquationis contineantur diver$æ
irrationales quantitates; $upponantur omnes termini, in quibus
continentur eædem irrationales quantitates, vel eædem duæ vel tres
diver$æ irrationales vel rationales quantitates nihilo æquales; tum
ex æquatione re$ultante $æpe detegi pote$t quantitas, quæ in da-
tam æquationem ducta, facit æquationem, cujus fluens inveniri
pote$t.</p>
<p>3<_>tia</_>. Dimen$iones quantitatis <math>y</math> in fluxionem <math>x^.</math> ductæ debent e$$e
majores per unitatem, quam dimen$iones eju$dem quantitatis in <math>y</math>
ductæ, in termino, in quo involvuntur $imul literæ <math>x & y</math>; & $ic de
líterâ <math>x</math>.</p>
<p>Ex. Sit æquatio <math>p y^. + q y x^. = r x^.</math>, ubi literæ <math>p, q & r</math> re$pective
denotant qua$cunque functiones literæ <math>x</math>; ducatur data æquatio in <math>P</math>
functionem quantitatis <math>x</math>, & re$ultat <math>P p y^. + P q y x^. = P r x^.</math>; $uppo-
natur <math>+ P p y</math> fluens fluxionis <math>P p y^. + P q y x^.</math>, & con$equenter <math>P p y^.
+ y (P p^. + P^. p) = P p y^. + P q y x^.</math>, unde <math>P p^. + p P^. = P q x^.</math>, & <math>P
(q x^. - p^.) = p P^.</math>, & exinde <math>{q x^. - p^. / p} = {P^. / P}, & $.{q x^. / p} = log. P p, &
P = e^$. {q x^. - p^. / p}, & P p y = e^$. {q x^. / p} y = e $.{q x^. - p^. / p} × r x^.</math>.</p>
<p>Cor. 1. Si <math>$. {q x^. / q}</math> $it logarithmus quantitatis Q, tum erit <math>P = {Q / p}</math>.</p>
<p>Cor. 2. Sit æquatio <math>p v^n-1 v^. + q v^n x^. = r x^.</math>; in eâ pro <math>v^n</math> & <math>v^n-1 v^.</math>
$cribantur re$pective <math>y & {y^. / n}</math>, re$ultat æquatio præcedentis formulæ
<math>{p / n} y^. + q y x^. = r x^.</math>.</p>
<p>Cor. 3. Sit <math>p v^. + v x^. = r v^- n+1 x^.</math>; ducatur hæc æquatio in <math>v^n-1</math>, &
re$ultat æquatio <math>p v^n-1 v^. + q v^n x^. = r x^.</math> præcedentis formulæ.</p>
<p>5. Sit fluxio <math>A x^. + B y^. + C x^. + D y^. = 0</math>, (ubi literæ <math>A & B</math> de-
notant functiones quantitatum <math>x & y</math> re$pective, & <math>C & D</math> funt quæ-
cunque functiones quantitatum <math>x & y</math> conjunctim) cujus inveniri
pote$t fluens: $int vero dimen$iones quantitatis <math>y</math> in functione <math>B</math>

<pb 195>[0247]<rh>Æ QUATIONIBUS.</rh>
majores quam dimen$iones eju$dem quantitatis <math>y</math> in functione <math>D</math>;
dividatur æquatio <math>A x^. + B y^. + C x^. + D y^. = 0</math> per functionem <math>P</math>
quantitatis <math>x</math>, & re$ultat æquatio in quâ dimen$iones quantitatis <math>y</math> in
fluxionem <math>x^.</math> ductæ minores erunt quam dimen$iones eju$dem quan-
titatis <math>y</math> in termino <math>{B / P}</math> (in quo continentur literæ <math>x & y</math> conjunctim)
in <math>y^.</math> ductæ per quantitatem majorem quam unitatem; $ed facile re-
duci pote$t æquatio <math>{A + C / P}x^. + {B + D / P}y^. = 0</math> ad æquationem integra-
bilem <math>A x^. + B y^. + C x^. + D y^. = 0</math> ex ejus multiplicatione in <math>P</math>: omnes
igitur datæ fluxionales æquationes, (ubi dimen$iones quantitatis <math>y</math> in
fluxionem <math>x^.</math> ductæ haud $uperant dimen$iones eju$dem quantitatis <math>y</math>
in termino, in quo involvuntur quantitates <math>x & y</math> conjunctim in <math>y^.</math>
ductæ per unitatem) reducendæ $unt ad fluxionales æquationes for-
mulæ prædictæ; quod $æpe perficietur, $i modo auferantur quanti-
tates in quibus continetur <math>x</math> e terminis, in quibus dimen$iones quan-
titatis <math>y</math> $unt nimis magnæ.</p>
<p>Et $ic de reducendis terminis, in quibus dimen$iones quantitatis <math>x</math>
$unt nimis magnæ.</p>
<p>Ex. Sit fluxionalis æquatio <math>8 y^7 x^-6 y^. + 7 a y^5 x^. + 5 a x y^4 y^. +
6 b x^-1 x^. = 0</math>: in hac æquatione dimen$iones (5) quantitatis <math>y</math> in <math>x^.</math>
ductæ haud $uperant prædictas dimen$iones (7) eju$dem quantitatis <math>y</math>
in <math>y^.</math> ductas per unitatem; ergo auferatur quantitas <math>(x^-6)</math>, in quâ con-
tinetur <math>x</math>, e termino prædicto <math>(8 y 7 x^-6 y^.)</math>, i. e. ducatur data æquatio in
<math>x^6</math>; & re$ultat æquatio quæ$ita <math>8 y^7 y^. + 7 a y^5 x^6 x^. + 5 a x^7 y^4 y^. +
6 b x^5 x^. = 0</math>, cujus fluens erit <math>y^8 + a y^5 x^7 + b x^6 = con$t.</math></p>
<p>6. Si vero multiplicator con$tet e pluribus terminis; a$$umatur pro
multiplicatore primus terminus e prædicta methodo inventus <math>+ p,
& e</math> methodo prius traditâ inveniatur proximus valor quantitatis <math>p</math>,
qui erit $ecundus terminus quæ$itus; & $ic deinceps. Hìc vero
animadvertendum e$t, multiplicatorem con$tare po$$e e terminis &
in numeratore & denominatore contentis; etiamque, quod$i nulla
exponentialis & fluentialis quantitas in datâ æquatione contineatur,

<pb 196>[0248]<rh>DE FLUXIONALIBUS</rh>
nullam exponentialem ni$i formulæ (<math>e^v</math>) vel in multiplicatore vel in
fluente nece$$ario contineri; & generaliter $i modo $int <math>π, ξ, σ, & c.</math>
exponentiales quantitates in datâ fluxionali æquatione contentæ,
nullas e$$e exponentiales quantitates in ejus multiplicatore vel
fluente nece$$ario contentas, ni$i formularum <math>π, ξ, σ, & c. e^v, e^π, e^p, e^σ,
& c.</math> exponentiales.</p>
<p>7. Eadem etiam principia applicari po$$unt ad detegendas fluentes
fluxionum, in quibus continentur fluxiones $uperiorum ordinum; $ed
a$$umi debent indices, & c. in terminis magis generalibus quam in
fluentibus fluxionum primi ordinis detegendis.</p>
<p>In hoc loco animadvertendum e$t.</p>
<p>1. Sit data æquatio, in quâ continentur quæcunque functiones
quantitatum <math>x & y</math> in <math>x^. & y^.</math> ductarum; æquationis multiplicator,
qui eam integrabilem reddit, quantitas erit algebraica, itemque fun-
ctio quantitatum <math>x & y</math>.</p>
<p>2. Si in datâ æquatione contineantur functiones quantitatum <math>x, y,
x^., y^., x^.. & y^..</math>; tum pote$t e$$e ejus multiplicator quæcunque functio
quantitatum <math>x, y, x^. & y^.</math>.</p>
<p>3. Si in datâ æquatione contineantur functiones quantitatum <math>x, y,
x^., y^., x^.., y^.., x^... & y^...</math>; tum pote$t e$$e ejus multiplicator quæcunque functio
quantitatum <math>x, y, x^., y^., x^.. & y^..</math>: & $ic deinceps.</p>
<p>Ex. 1. Sit <math>2 a y y^.. - 4 a y^.^2 - y^n+5 x^.^2 (1 + x^2)^{n-1 / 2} = 0</math>; ducatur hæc
æquatio in <math>{x x^. / y^4} + {(1 + x^2)y^. / y^5}</math>, & re$ultat <math>{2 a x x^. y^.. / y^3} + {2 a (1 + x^2)y^.. y^.. / y^4} -{4 a x x^. y^.^2 / y^4} - {4 a (1 + x^2) y^. ^3 / y^5} - (y^n+1 x^.^3 x + (1 + x^2) y^n x^.^2 y^.) (1 + x^2)^{n-2 / 2}
= 0</math>, cujus fluens reperietur <math>{a x^.^2 / y^2} + {2 a x x^. y^. / y^3} + {a (1 + x^2) y^.^2 / y^4} ={1 / n + 1}y^n+1 x^.^2 (1 + x^2)^{n+1 / 2} + C x^.^2</math>, ubi <math>C & x^.</math> $unt invariabiles quan-
titates.</p>

<pb 197>[0249]<rh>Æ QUATIONIBUS.</rh>
<p>Ex. 2. Sit <math>b y x^. y^. + b s x y^.^2 + b x y y^.. + {c y^.^t-1 y^.. / y^s-1 (a x^. + b x y^s y^.)^t-1} = 0</math>,
ducatur hæc æquatio in multiplicatorem <math>t y^s-1 (a x^. + b x y^s y^.)^t-1</math>; & re-
$ultat æquatio, cujus fluens erit <math>(a x^. + b x y^s y^.)^t + c y^.^t = con$t</math>.</p>
<p>Ex. 3. Sit <math>z^m+1 x^.^m x^.. + y^.^m+1 {z^. / z} = y^.^m y^..</math>; dividatur hæc æquatio per
<math>z^m+1</math>, & re$ultat <math>x^.^m x^.. + y^.^m+1 {z^. / z^m+2} = {y^.^m y^.. / z^m+1}</math>, cujus fluens erit <math>x^.^m+1 ={y^.^m+1 / z^m+1} + C v^.^m+1</math>, ubi <math>v</math> e$t quantitas, quæ fluit uniformiter.</p>
<p>Ex. 4. Sit <math>y^.. = 4 x^3 y^.. + 6 x^2 x^. y^. - 2 x y x^.^2</math>; ducatur hæc æquatio in
<math>2 x y^. - y x^.</math>, & æquationis re$ultantis fluens erit <math>x y^.^2 - y x^. y^. = 4 x^4 y^.^2
- 4 x^3 y y^. x^. + x^2 y^2 x^.^2 + C x^.^2</math>, ubi <math>C = con$t</math>.</p>
<p>8. Sæpe e quibu$dam datæ fluxionalis æquationis terminis nihilo
æqualibus e$$e $uppo$itis detegi pote$t multiplicator, qui in datam
fluxionalem æquationem multiplicatus, præbet fluxionem integrabi-
lem. Inveniantur enim multiplicatores, qui in hos & reliquos ter-
minos multiplicati, præbent fluxiones integrabiles; ex his vero $æpe
con$tabit multiplicator quæ$itus.</p>
<p>1. Sit fluxionalis æquatio <math>A + B = 0</math>; inveniatur quantitas (<math>a</math>),
quæ in <math>A</math> ducta, reddit fluxionem <math>a A</math> integrabilem, cujus fluens $it
<math>A′</math>; etiamque inveniatur quantitas <math>b</math>, quæ in <math>B</math> ducta, præbet fluxio-
nem <math>b B</math> integrabilem, cujus fluens $it <math>B′</math>; inveniantur tales functiones
quantitatum <math>A′ & B′</math>, viz. <math>φ A′ & Δ B′</math>, ut $it <math>φ A′:Δ B′::b:a</math>; & quan-
titas <math>a φ A′</math> in datam æquationem <math>A + B = 0</math> ducta, eam reddit in-
tegrabilem.</p>
<p>2. Sit æquatio <math>A + B + C + D + & c. = 0</math>, & $int fluentes fluxio-
num <math>a A, b B, c C, d D, &c</math>. re$pective <math>A′, B′, C′, D′, &c.:</math> deinde $it
<math>a φ:A′ = b φ′:B′ = c φ″:C′ = d φ″′:D</math>; & ex his æquationibus in-
veniri pote$t valor quantitatis <math>a φ A</math>, qui ductus in datam fluxionalem
æquationem, eam reddit integrabilem.</p>

<pb 198>[0250]<rh>DE FLUXIONALIBUS</rh>
<p>Cor. Hinc, $i detegi po$$it generaliter re$olutio prædictæ algebraicæ
æquationis <math>a φ:A′ = b φ′:B′</math>, detegi po$$unt generaliter multiplica-
tores; nam, $i generaliter detegi po$$int re$olutiones æquationum
<math>a φ:A′ = b φ′:B′</math>, quæ $it <math>Δ:α</math>; detegi pote$t generaliter re$olutio
æquationis <math>Δ:α = c φ″:C′</math>; & $ic deinceps.</p>
<p>Cor. 2. Hinc inveniri po$$unt fluxionales æquationes, quarum flu-
entes dantur:inveniantur enim duæ diver$æ fluxiones <math>A & B</math>, quæ
in datam quantitatem vel fluxionem ductæ, præbent fluxiones, qua-
rum fluentes cogno$cuntur; & erit <math>A + B = 0</math> fluxionalis æquatio,
cujus fluens inveniri pote$t.</p>
<p>Ex. 1. <math>y^.. - {m y^.^2 / y}, & y^n x^.^2 (α + 2 β x + γ x^2)^{n-4m+3 / 2m-2}</math> $unt fluxiones,
quæ in (<math>P</math>) factorem <math>{(β + γ x)x^. / (m - 1) y^2m-1} + {(α + 2 β x + γ x^2)y^. / y^2m}</math> ductæ,
præbent fluxiones, quarum fluentes $unt re$pective <math>{(β + γ x)x^. y^. / (m - 1)y^2m-1} +{(α + 2 β x + γ x^2)y^.^2 / 2y^2m} + {γ x^.^2 / 2(m - 1)^2 y^2m-23} & {1 / n - 2m + 1}y^n-2m+1 x^.^2
(α + 2 β x + γ x^2)^{n-2m+1 / 2m-2}</math>; & con$equenter fluens fluxionalis æquationis
<math>y^.. = {my^.^2 / y} + y^n x^.^2 (α + 2 β x + γ x^n)^{n-4m+3 / 2m-2} = 0</math> acquiri pote$t.</p>
<p>Cor. 1. In æquatione <math>y^.. - {m y^.^2 / y} + y^n x^.^2 (α + 2β x + γ x^2)^{n-4m+3 / 2m-2}= 0</math> $cribatur pro <math>y</math> ejus valor a$$umptus <math>z^{1 / 1-m}</math>, & re$ultat <math>{z^.. / 1 - m} +
z{n - m / ^1-m} x^.^2 (α + 2β x + γ x^2)^{n-4m+3 / 2m-2} = 0</math>.</p>
<p>Ex. 2. Sint <math>y^.. & {a y^n+1 x^.^2 / (α + 2 β x + γ x^2 + c y^2)^{n+4 / 2}}</math> quantitates, quæ in
eandem fluxionem <math>y^. (α + 2β x + γ x^2) - y x^.(β + γ x)</math> ductæ, præ-
bent fluxiones integrabiles, quarum fluentes $unt <math>{1 / 2}y^. ^2 (α + 2 β x + γ x^2)

<pb 199>[0251]<rh>ÆQUATIONIBUS.</rh>
<math>- y x^. y^. (β + γ x) + {1 / 2} γ x^. ^2 y^2</math>, & <math>{a y^n+2 x^. ^2 / (n + 2) (α + 2 β x + γ x^2 + c y^2)^{n+23 / 2}}</math>
unde fluens fluxionalis æquationis <math>y^.. + {a y^n+1 x^. ^2 / (α + 2 β x + γ x^2 + c y^2)^{n+4 / 2}}= 0</math> detegi pote$t.</p>
<p>Cor. Sit æquatio <math>y^.. + {a y^n+1 x^. ^2 / (x^2 + y^2)^{n+4 / 2}} = 0</math>, cujus fluens erit <math>{1 / 2}(y x^. -
x y^.)^2 + {a y^n+2 x^. ^2 / (n + 2) (x^2 + y^2)^{n+2 / 2}} = C x^.</math>: in hâc æquatione $cribatur <math>u x</math>
pro <math>y</math>, & erit <math>- x^2 u^. = y x^. - x y^.</math>. & fluens <math>{1 / 2}(y x^. - x y^.)^2 +{a y^n+2 x^. ^2 / (n + 2) (x^2 + y^2)^{n+2 / 2}} = {1 / 2} x^4 u^. ^2 + {a u^n+2 x^. ^2 / (n + 2) (1 + u^2)^{n+2 / 2}} = C x^.</math>, cujus
fluxionis fluens inveniri pote$t.</p>
<p>Ex. 3. Propo$itâ æquatione fluxionali <math>α y x^. + β x y^. = x^m y^n (γ y x^.
+ δ x y^.)</math>; omnes vero multiplicatores, qui ducti in <math>α y x^. + β x y^.</math>
præbent fluxiones, quarum fluentes detegi po$$unt, erunt functiones
quantitatis <math>x^α y^β</math> in fractionem <math>{1 / x y}</math> ductæ: eodem modo omnes mul-
tiplicatores, qui ducti in <math>x^m y^n (γ y x^. + δ x y^.)</math> præbent fluxiones, qua-
rum fluentes detegi po$$unt, erunt functiones quantitatis <math>x^γ y^δ</math> in fra-
ctionem <math>{1 / x^m+1 y^n+1}</math> ductæ; $ed hi multiplicatores po$$unt e$$e huju$modi
<math>x^μα-1 y^μβ-1</math> & <math>x^υγ-m-1 y^υδ-n-1</math>; unde, $i modo <math>μ α - 1 = υ γ - m - 1</math>,
& <math>μ β - 1 = υ δ - n - 1</math>, tum multiplicator inventus in propo$itam
æquationem ductus eam reddit integrabilem; $ed in hoc ca$u <math>μ ={γ n - δ m / α δ - β γ} & υ = {α n - βm / α δ - β γ}</math>.</p>
<p>Cor. Si <math>α δ - β γ = 0</math>, tum propo$ita æquatio evadet <math>α y x^. + β x y^.

<pb 200>[0252]<rh>DE FLUXIONALIBUS</rh>
= {γ / α} x^m y^n (α y x^. + β x y^.)</math>, unde erit <math>(α y x^. + β x y^.) (1 - {γ / α} x^m y^n) = 0</math>,
& exinde <math>α y x^. + β x y^. = 0</math>.</p>
<p>Multiplicator nonnunquam e$t algebraica functio variabilium
quantitatum & earum fluxionum in datâ æquatione contentarum;
nonnunquam e$t etiam algebraica, fluentialis & exponentialis functio
prædictarum quantitatum; & nonnunquam per infinitas $eries præ-
dictarum quantitatum $olummodo exprimi pote$t.</p>
<h>PROB. LI.</h>
<p it>Datâ formulâ æquationis fluxionalis, & quantitatis in prædictam æqua-
tionem ducendæ; invenire ca$us, in quibus quantitas re$ultans evadit inte-
grabilis.</p>
<p>A$$umantur generaliter datæ formulæ fluxionalis æquatio & ejus
multiplicator; ducantur hæ duæ quantitates in $e$e, & inveniantur
per methodos in præcedente capite traditas ca$us, in quibus datur
fluens æquationis re$ultantis; & perficitur problema.</p>
<p>Si innote$cant formulæ multiplicatorum, tum etiam innote$cent
formulæ fluentium, & con$equenter cum detur fluens ex a$$umptis
multiplicatoribus, tum etiam dabitur fluens ex a$$umptis fluentibus:
a$$umatur enim generaliter fluens datæ formulæ, cujus inveniatur
fluxio, dividatur hæc fluxio per datam fluxionalem æquationem, & $i
quotiens $it quantitas vel algebraica vel fluxionalis minoris quam
ordinis datæ fluxionalis æquationis; tum a$$umpta erit fluens datæ
fluxionalis, $in aliter non.</p>
<p>Hic for$an haud indignum e$t ob$ervatu; cum in datâ fluxio-
nali æquatione dimen$iones quantitatis <math>y</math> in <math>x</math> non $uperent ejus di-
men$iones in <math>y^.</math> ductas per unitatem; tum pro ejus multiplicatore
vel fluente a$$umenda e$t quantitas, quæ continet pote$tatem
vel radicem quantitatis <math>y</math>, quæ in nullam functionem quantitatis <math>x</math>
ducitur. e. g. Sit quantitas <math>y y^. + P y x^. + Q x^. = 0</math>, ubi <math>P & Q</math> $unt

<pb 201>[0253]<rh>ÆQUATIONIBUS.</rh>
functiones ip$ius <math>x</math>; tum multiplicatores po$$unt e$$e <math>(y + M)^n ×
(y + N)^m × &c.</math>: vel etiam ex datis fluxionibus re$ultantibus a datâ
fluxione in multiplicatorem ductâ erui pote$t formula fluentis; vel
facile a$$umi pote$t formula fluentis, & exinde erui po$$unt ca$us, in
quibus fluens datæ fluxionis habet formulam a$$umptam.</p>
<p>Ex. Sit data æquatio <math>y y^. + P y x^. + Q x^. = 0</math>, ducatur hæc æqua-
tio in <math>(y + M)^n</math>, ubi <math>P, Q & M</math> $unt functiones ip$ius <math>x</math>, & re$ultat
<math>(y + M)^n y y^. + P (y + M)^n y x^. + Q (y + M)^n x^. = 0</math>; $ed per præce-
dens caput <math>$. (y + M)^n y y^. = {(y + M)^n+2 / n + 2} - {M (y + M)^n+1 / n + 1}</math>, $i modo
<math>M</math> $it invariabilis; nunc autem $it <math>M</math> variabilis, & fluxio quantitatis
<math>{(y + M)^n+2 / n + 2} - {M (y + M)^n+1 / n + 1}</math> erit <math>(y + M)^n+1 (y^. + M^.) - M (y + M)^n
(y^. + M^.) - {M^. (y + M)^n+1 / n + 1}</math>; unde <math>(y + M)^n+1 (y^. + M^.) - M (y + M)^n
(y^. + M^.) - {M^. (y + M)^n+1 / n + 1} = (y + M)^n y y^. + P (y + M)^n y x^. + Q
(y + M)^n x^.</math>, & exinde <math>({n M^. / n + 1} - P x^.) y - ({M M^. / n + 1} + Q x^.) = 0</math>; fin-
gantur <math>{n M^. / n + 1} - P x^. = 0</math> & <math>{M M^. / n + 1} = - Q x^.</math>, & con$equenter erunt
<math>{n M^. / (n + 1) x^.} = P, & - {M M^. / (n + 1) x^.} = Q</math>, & data æquatio <math>y y^. + {n M^. / (n + 1)}y - {M M^. / (n + 1)} = 0</math>.</p>
<p>2. Hoc problema etiam e principiis in theor. 2. traditis re$olvi
pote$t. E. g. Sit fluxio <math>α x^. + β y^.</math>, cujus fluens inveniri pote$t; tum
erit <math>({α^. / y^.}) = ({β^. / x^.})</math>.</p>
<p>Ex. 1. Sit æquatio <math>P y x^. + (y + Q) y^. = 0</math> & multiplicator
<math>{1 / y^3 + M y^2 + N y}</math>; ubi <math>P, Q, M & N</math> $unt functiones quantitatis <math>x</math>:

<pb 202>[0254]<rh>DE FLUXIONALIBUS</rh>
ducatur æquatio in multiplicatorem, & re$ulta <math>{P y x^. / y^3 + M y^2 + N y} +{(y + Q)y^. / y^3 + M y^2 + N y} = 0</math>; ut vero fluens huju$ce fluxionis inveniatur,
nece$$e e$t fluxionem factoris <math>{P y / y^3 + M y^2 + N y}</math>, in quo <math>y</math> $olummodo
$umitur variabilis, per <math>y^.</math> divi$am, æqualem e$$e fluxioni quantitatis
<math>{y + Q / y^3 + M y^2 + N y}</math>, in quâ $olummodo $umitur <math>x</math> variabilis, per <math>x^.</math> divi$æ;
i.e. <math>- 2 P y^3 - P M y^2 = (y^3 + M y^2 + N y) {Q^. / x^.} - (y + 2){(y^2 M^. + y N^.) / x^.}</math>,
quæ reducta ad terminos $ecundum pote$tates quantitatis <math>y</math> progredi-
entes, evadet <math>0 = (2 P x^. + Q^. - M^.)y^3 + (P M x^. + M Q^. - N^. - Q M^.)
y^2 + (N Q^. - Q N^.) y</math>; unde $ingulis pote$tatibus $eor$im ad nihilum
productis, exorietur <math>N Q^. - Q N^. = 0</math>, & con$equenter <math>N = a Q</math>, ubi
<math>a</math> erit con$tans quantitas; quo valore pro <math>N</math> in duabus reliquis quan-
titatibus $ub$tituto, re$ultant <math>2 P x^. + Q^. - M^. = 0 & P M x^. + M Q^. -
a Q^. - Q M^. = 0</math>; ita reducantur hæ æquationes, ut exterminetur <math>x^.</math>, &
re$ultat æquatio <math>- M Q^. - M M^. + 2 a Q^. + 2 Q M^. = 0</math>, quâ multi-
plicatâ in <math>{1 / (2 a - M)^3}</math>; fluxionis re$ultantis fluens inventa dat
<math>{Q / (2 a - M)^2} = $. {M M^. / (2 a - M)^3} = {- 1 / 2 a - M} + {a / (2 a - M)^2} + b</math>. Sed
erit <math>2 P x^. = M^. - Q^.</math>, unde pro <math>M</math> a$$umatur quæcunque functio
quantitatis variabilis <math>x</math>, & exinde deduci po$$unt reliquæ quantitates
<math>Q, P & N</math>.</p>
<p>Ex. 2. Sit præcedens æquatio <math>P y x^. + (y + Q) y^. = 0</math>, & multiplica-
tor <math>{y^n-1 / y^2 + M y + N}</math>: ducatur æquatio in prædictum multiplicatorem
& re$ultat <math>{P y^n x^. + y^n-1 (y + Q) y^. / y^2 + M y + N} = 0</math>; ut vero fluens huju$ce fluxio-

<pb 203>[0255]<rh>ÆQUATIONIBUS.</rh>
nis inveniatur, nece$$e e$t fluxionem factoris <math>{P y^n / y^2 + M y + N}</math>, in quo <math>y</math>
$olummodo $umitur variabilis, per <math>y^.</math> divi$am, æqualem e$$e fluxioni
quantitatis <math>{(y + Q) y^n-1 / y^2 + M y + N}</math>, in quâ $olummodo $umitur <math>x</math> variabilis,
per <math>x^.</math> divi$æ; unde re$ultans æquatio ad terminos $ecundum pote$ta-
tes quantitatis <math>y</math> progredientes reducta, evadet <math>((n - 2) P x^. - Q^. +
M^.) y^n+1 + ((n - 1) P M x^. - M Q^. + N^. + Q M^.)y^n + (n P N x^. - N Q^.
+ Q N^.)y^n-1 = 0</math>; & e $ingulis pote$tatibus $eor$im nihil evadere $up-
po$itis, re$ultabunt æquationes <math>(n - 2) P x^. = Q^. - M^.; (n - 1) M P x^.
= M Q^. - Q M^. - N^.; & n N P x^. = N Q^. - Q N^.</math>; ex his æquationibus
inveniri po$$unt in quibu$dam ca$ibus corre$pondentes valores quan-
titatum <math>P, Q, M & N</math>.</p>
<p>Ex. 3. Sit data æquatio <math>y P x^. + (Q y + R) y^. = 0</math>, in quâ <math>P, Q & R</math>
$int functiones ip$ius <math>x</math>, & multiplicator <math>{y^m / (1 + S y)^n}</math>, ubi <math>S</math> e$t etiam
$unctio ip$ius <math>x</math>; ex hâc methodo inveniri po$$unt corre$pondentes
valores quantitatum <math>P, Q, R & S</math>, quando data æquatio in multipli-
catorem ducta, fiat integrabilis; etiamque corre$pondentes valores
earundem quantitatum, cum multiplicator $it <math>{y^m / (1 + S y + T y^2)^n}</math>; &c.
& $ic inveniri po$$unt ca$us, in quibus æquatio <math>(P y + Q) x^. + y y^. = 0</math>
redditur integrabilis per multiplicatorem <math>(y + R)^m × (y + S)^n</math>, ubi
<math>P, Q, R & S</math> $unt etiam functiones quantitatis <math>x</math>; & $imiliter detegi
po$$unt ca$us, in quibus æquatio <math>y^. + y^2 x^. + X x^. = 0</math> redditur inte-
grabilis per multiplicatorem <math>{1 / P y^2 + Q y + R}</math>, &c.</p>
<p>3. 1. Sit æquatio <math>(P x^m + Q x^m-1 + R x^m-2 + ... +T x^m-b) v^. +
(p x^m-1 + q x^m-2 + r x^m-3 + ... + t x^m-b-1) x^. = 0</math>; ducatur data
æquatio in quantitatem <math>a x^n + b x^n-1 + c x^n-2 + ... + e x^n-k</math>, & re-
$ultat <math>(a P x^m+n + (a Q + b P)x^m+n-1 + (a R + b Q + c P) x^m+n-2 + &c.)</math>

<pb 204>[0256]<rh>DE FLUXIONALIBUS</rh>
<math>v^. + (a p x^m+n-1 + (a q + b p) x^m+n-2 + (ar + bq + cp) x^m+n-3 + &c.)
x^. = 0</math>; & $i modo re$ultans quantitas $it integrabilis, tum fluxio
quantitatis <math>a p</math>, quæ e$t <math>a p^. + p a^. = (m + n) a P v^.</math>, & $imiliter fluxio
quantitatis <math>a q + b p</math>, quæ erit <math>a q^. + q a^. + b p^. + pb^. = (m + n - 1)
(a2 + bP) v^.</math>, etiamque <math>ar^. + ra^. + bq^. + qb^. + cp^. + pc^. = (m
+ n - 2) (aR + b2 + cP) v^.</math>; & $ic deinceps: unde re$ultant <math>(k +
b + 1)</math> æquationes involventes (<math>k + 2b + 3</math>) variabiles quantitates;
deinde ita a$$umantur hæ (<math>k + 2b + 3</math>) variabiles, ut $atisfaciant
prædictis (<math>k + b + 1</math>) fluxionalibus æquationibus, & con$equenter
a$$umi po$$unt ad libitum <math>b + 1</math> variabiles quantitates; at in pleri$-
que ca$ibus relationes inter variabiles quantitates a$$umere præ$tat,
ita ut $olutio magis facilis evadat; & $equitur fluens deductæ fluxio-
nalis æquationis.</p>
<p>2. Et $imiliter ducatur data æquatio in fractionalem quantitatem
<math>{a x^n + b x^n-1 + cx^n-2 + &c. / a′x^n′ + b′ x^n′-1 + c′ x^n′-2 + &c.}</math>; vel in <math>{(a x^n + b x^n-1 + cx^n-2 + &c.)^π / (a′x^n′ + b′ x^n-1 + c′ x^n-2 + &c.)^ρ}</math>;
vel in <math>{(a x^n + bx^n±{1 / l} + c x^n±{2 / l} + d x^n±{3 / l} ...x^n-3 + d x^n-1±{7 / l} + &c.)^π ×
(A x^i + B x^i±k′ + &c.)/(e x^n′ + f x^n′±1/l′ + g x^n′±2/l′ + &c.)^ρ ×
(A′ x^i′ + B′ x^i′±k″ + &c.)}</math>; vel denique in quamcunque functionem
quantitatis <math>x</math> & aliarum quantitatum, quæ $unt functiones variabilis
<math>v</math>, ita vero a$$umendarum, ut quantitas re$ultans evadat integrabilis.</p>
<p>Quantitates <math>P, Q, R, &c.; p, q, r, &c.; a, b, c, &c.; a′, b′, c′, &c.;
e, f, g, &c.; A, B, &c.; A′, B′, &c.</math>; denotant functiones quantitatis
<math>(v); & m, n, n′, i, i′, k′ & k″, π & ξ</math> $unt invariabiles; & <math>b, k, l & l′</math> $unt
integri numeri.</p>
<p>4. Sit æquatio <math>(a x^m + b x^m+b + cx^m+2b .... + M x^n-b-1 + P x^n-1
+ Qx^n+b-1 + R x^n+2b-1 + &c.) x^. + (px^n + qx^n+b + r x^n+2b + &c.) v^.
= 0</math>; ducatur ea in quantitatem <math>A x^l + B x^l+b + C x^l+2b ... D x^l+n-m-b-1
+ Ex^l+n-m-1 + Fx^l+n-m+b-1 + &c.</math>; & exorietur <math>(a A x^m+l + (aB + bA)

<pb 205>[0257]<rh>ÆQUATIONIBUS.</rh>
x^m+l+b + .... + (AM + ... + aD) x^l+n-b-1)x^. + (A p x^l+n + &c.) v^.
= 0</math>; tum a$$umantur <math>a A = α, a B + bA = β, &c.</math>, u$que ad æqua-
tionem <math>a D + ... + AM = θ</math>; deinde hoc modo a$$umantur reliquæ
<math>A P^. + P A^. + .... + E a^. + a E^. = (l + n) A p v^., A Q^. + 2A^. + ....
+ aF^. + Fa^. = (l + n + b)(Aq + Bp) v^.</math>, & $ic deinceps. In hoc ca$u
literæ <math>a, b, c, &c., M, P, 2, R, &c.; p, q, r, &c.; A, B, C, &c., E, F, &c.</math>
functiones literæ <math>v</math> denotent, & α, β, &c. θ $unt invariabiles quanti-
tates; deinde ita deducendæ $unt prædictæ quantitates, ut $atisfaciant
æquationibus a$$umptis, & inve$tigantur fluxionales æquationes, qua-
rum fluentes innote$cunt.</p>
<p>2. Et $imiliter ducatur data æquatio in quantitatem <math>{(Ax^l + B x^l+b +
&c.)^π × (H x^σ + 1 x^σ+h + &c.) / (A′ x^l′ + B′ x^l′+b + &c.)^ρ × (H′ x^σ′ + 1′ x^σ+h + &c.)}</math>; ubi <math>A, B, &c.; A′, B′, &c.; H, I, &c.;
H′, I′, &c.</math>; $unt functiones quantitatis <math>v, & π, ξ σ, &c.</math> $unt invaria-
biles quantitates: vel in quamcunque functionem literarum <math>x & v</math>, ita
a$$umendam, ut quantitas re$ultans evadat integrabilis.</p>
<p>3. Sit æquatio <math>(ax^m + bx^m+b + cx^m+2b + .... + k x^n-b + p x^n +
q x^n+b + &c.) v^. + (P x^n-1 + Q x^n+b-1 + &c.)x^. = 0;</math> ducatur data
æquatio in <math>Ax^-m + B x^-m+b + C x^-m+2b + &c. ... Dx^n-2m-b + Ex^n-2m +
Fx^n-2m+b + &c.</math>; & a$$umantur <math>Ab + Ba = 0, Ac + Bb + Ca = 0,
&c.</math> u$que ad terminum <math>A k + &c. = 0</math>, deinde hoc modo invenian-
tur reliqui <math>A P^. + P A^. = (n - m) × (Ap + ... + a E) v^., &c.</math>: vel
ejus multiplicator pote$t effe functio variabilis <math>x</math> & quantitatum, quæ
funt functiones variabilis <math>v</math>; $i modo $upponantur evane$centes
termini æquationis re$ultantis, in quibus dimen$iones quantitatis <math>x</math>
in <math>v^.</math> ductæ majores $int quam dimen$iones quorumcunque termi-
norum quantitatis <math>x</math> in <math>x^.</math> ductorum per quantitatem majorem quam
unitatem: hìc <math>a, b, &c.; P, Q, &c.; A, B, &c.</math> $unt functiones quan-
titatis <math>v^.</math>.</p>
<p>Ex his methodis inveniri pote$t infinita $eries, quæ reddit datam
æquationem integrabilem.</p>

<pb 206>[0258]<rh>DE FLUXIONALIBUS</rh>
<p>Eadem principia etiam ad omnes fluxionales æquationes quorum-
cunque ordinum applicari po$$unt.</p>
<h>PROB. LII.</h>
<h it>Datâ fluxionali æquatione $uperioris ordinis; invenire quantitatem, qua
in datam æquationem ducta, eandem baud raro reddet integrabilem.</h>
<p>I. A$$umatur multiplicator (<math>P</math>) primi generis, qui ducatur in da-
tam æquationem, & in multis ca$ibus per notas regulas fluentes in-
veniendi ita a$$umi pote$t <math>P</math>, ut inveniatur fluens æquationis re$ul-
tantis: $æpe quoque e præcedentibus principiis con$tabit $ormula
multiplicatoris <math>P</math>, & exinde hoc problema in præcedens tran$ibit.</p>
<p>2. Si vero in fluxionalibus æquationibus $uperiorum ordinum
haud inveniatur multiplicator primi generis <math>P</math>, affumatur multipli-
cator $ecundi generis <math>P x^. + 2 y^., &c.</math> & per prædictas regulas in
multis ca$ibus inveniri pote$t fluens fluxionalis æquationis; & fic
deinceps.</p>
<p>3. Invenire fluentem fluxionis <math>P y^. ^r + Q y^. ^r-1 + R y^. ^r-2 + &c. + T</math>; ubi
<math>P</math> e$t functio quantitatum <math>y^. ^r-1, y^. ^r-2, y^. ^r-3, .. y^. ^2 y^., & x^., y & x; Q</math> vero functio
quantitatum <math>y^. ^r-2, y^. ^r-3, ... y^. ^2, y^. & x^., y & x; R</math> autem functio quantitatum
<math>y^. ^r-3, y^. ^r-4, ... y^. ^2, y^. & x^., y & x, &c.; T</math> functio vel rationalis vel irrationalis
nonnullarum prædictarum fluxionum & quantitatum: in fluxione <math>P y^. ^r</math>,
a$$umantur omnes quantitates præter <math>y^. ^r-1</math> tanquam invariabiles, &
ex hâc hypothe$i inveniatur fluens fluxionis <math>P y^. ^r</math>, quæ $it π; deinde
inveniatur <math>π^.</math> ex hypothe$i quod <math>y^. ^r-1, y^. ^r-2, y^. ^r-3, ... y^.., y^., y, x^. & x</math> $int varia-
biles, quæ $it <math>P y^. ^r + H y^. ^r-1 + by^. ^r-2 + &c.</math>: tum inveniatur fluens fluxio-
nis <math>(Q - H)y^. ^r-1</math>, ex hypothe$i quod omnes quantitates præter <math>y^. ^r-2</math> $int

<pb 207>[0259]<rh>ÆQUATIONIBUS.</rh>
invariabiles; quæ $it ξ; deinde inveniatur <math>ξ^..</math> ex hypothe$i quod <math>y^. ^r-2,
y^. ^r-3,...y^.., y^., y & x</math> $int variabiles, quæ $it <math>(Q - H) y^. ^r-1 + Iy^. ^r-2 + &c.</math>;
tum inveniatur fluens fluxionis <math>(R - b - I)y^. ^r-2</math> ex hypothe$i quod om-
nes quantitates præter <math>y^. ^r-3</math> $int variabiles; & $ic deinceps: & ultimo
perveniendum e$t ad functionem quantitatis <math>x</math> in <math>x^. ^n</math> ductam, cujus
inveniatur fluens, & generaliter corrigatur; & fluens datæ fluxionis,
$i modo $it integrabilis, reperietur.</p>
<p>3. 2. Sit fluxionalis quantitas <math>(A) P y^. ^n + 2z^. ^n + R v^. ^n + &c. + F + &c.</math>,
in quâ plures continentur variabiles quantitates (y, z, v, &c.) & ea-
rum fluxiones; tum primo a$$umantur una (<math>y</math>) & ejus fluxiones $o-
lummodo variabiles, & ex hâc hypothe$i inveniatur fluens (π), cujus
(π) inveniatur fluxio ex hypothe$i quod omnes prædictæ quantitates
(<math>y, z, v, &c.</math>) & earum fluxiones $int variabiles, quæ $upponatur α;
$ubtrahatur hæc fluxio de datâ, & re$ultabit fluxio <math>A - α = B</math>, in
quâ nec continetur variabilis y, nec ejus fluxiones, $i modo fluens
datæ fluxionis exprimi po$$it: $i evane$cant prædictæ quantitates,
tum per eundem modum in quantitate <math>B</math> a$$umantur (<math>z</math>) & ejus flu-
xiones $olummodo variabiles, & ex hâc hypothe$i inveniatur fluens,
quæ $it ξ, cujus (ξ) inveniatur fluxio ex hypothe$i quod prædictæ
quantitates (<math>z, v, &c.</math>) & earum fluxiones $int variabiles, quæ $up-
ponatur β; $ubtrahatur hæc fluxio de <math>B</math>, & re$ultabit fluxio B - β,
in quâ nec continetur variabilis <math>z</math>, nec ejus fluxiones, $i modo fluens
datæ fluxionis exprimi po$$it: & $ic de reliquis variabilibus quanti-
tatibus.</p>
<p>Ex. 1. Sit <math>y^.. = y^n X x^. ^2</math>, ubi <math>X</math> $it functio quantitatis <math>x</math>; a$$umatur
<math>P</math> multiplicator primi generis, ducatur data æquatio in <math>P</math>, & re$ultat
<math>P y^.. = P y^n X x^. ^2</math>; per regulam hìc traditam pro fluente affumenda e$t
<math>P y^. - A^. = $. (P y^.. - P y^n X x^. ^2)</math>, $ed huju$ce æquationis fluxio erit <math>P y^..
+ y^. P^. - A^.. = Py^.. - P y^n X x^.</math>, unde <math>y^. P^. - A^.. = - P y^n X x^.</math>, cujus pro
fluente per eandem regulam a$$umenda e$t <math>y P^. + B^. = $. (y^. P^. - A^.. +

<pb 208>[0260]<rh>DE FLUXIONALIBUS</rh>
P y^n X x^. ^2)</math>, cujus fluxio erit <math>y^. P^. + y P^.. + B^.. = y^. P^. - A^.. + Py^n X x^.^2</math>,
unde <math>y P^.. + B^.. = - A^.. + P y^n X x^. ^2</math>; $ed ut corre$pondentes termini
huju$ce æquationis inter $e conveniant, nece$$e e$t, ut <math>B^.. = - A^.. & y P^..
= P y^n X x^. ^2,</math> & exinde <math>y^n = y & n = 1 & {P^.. / Px^. ^2} = X</math>.</p>
<p>Cor. Sit data æquatio <math>X x^. ^2 = m v^μ v^. ^2 + n v^μ+1 v^..</math>, $cribatur <math>v = y^{n / m+n}</math>
& re$ultant <math>v^. = {n / m + n} y^{-m / m+n} y^. & v^.. = -{mn / (m + n)^2} y^{-2m-n / m+n} y^. ^2 + {n / m + n}y^{-m / m+n} y^..</math>; & exinde <math>m v^μ v^. ^2 + n v^μ+1 v^.. = {n^2 / m + n} y {μn-m+n / m+n} y^.. = X x^. ^2</math>, hæc au-
tem æquatio e$t eju$dem formulæ ac æquatio <math>y^n X x^. ^2 = y^..</math>.</p>
<p>Ex. 2. Sit <math>y^.. + {A y^n x^. ^2 / (α + 2 βx + γ x^2)^{n+3 / 2}} = 0, ducatur hæc æquatio in
px^. + qy^. & re$ultat <math>p x^. y^.. + q y^. y^.. + {p Ay^n x^. ^3 + q A y^n y^. x^. ^2 / (α + 2βx + γ x^2)^{n+3 / 2}} = 0</math>; per
prædictam regulam pro fluente fluxionis p x^. y^.. + q y^. y^..(π^.) a$$umenda
e$t <math>p x^. y^. + {q y^. ^2 / 2} + Lx^. ^2</math>, cujus fluxio erit <math>p x^. y^.. + q y^. y^.. (π^.) + p^. x^. y^. +{q^. y^. ^2 / 2} + L^.x^. ^2</math>; $it igitur <math>p = P y & erit p^. = P^. y + Py^.</math>, unde p^. x^. y^. =
P x^. y^. ^2 + y P^. x^. y^.</math>; fingatur <math>P x^. y^. ^2 = - {q^.y^. ^2 / 2}</math>, & exinde <math>P x^. = -{q^. / 2}</math>, &
multiplicator <math>p x^. + q y^. erit -{q^. / 2} y + qy^.</math>; ducatur data æquatio <math>y^.. +{A y^n x^. ^2 / (α + 2β x + γ x^2)^{n+3 / 2}} = 0</math> in <math>- q^. y + 2 q y^.</math> & re$ultat æquatio <math>- q^. y y^..
+ 2 q y^. y^.. - {A y^n+1 q^. x^. ^2 - 2 q A y^n y^. x^. ^2 / (α + 2 β x + γ x^2)^{n+3 / 2}}</math>, cujus fluens erit <math>- y q^. y^. + q y^.^2
+ $.y y^. q^.. - $.{A y^n+1 q^. x^. ^2 - 2 q A y^n y^. x^. ^2 /(α + 2β x + γ x^2)^{n+3 / 2}} (H)</math>; $i vero fluens <math>H</math> inveniri

<pb 209>[0261]<rh>Æ QUATIONIBUS.</rh>
po$$it, erit <math>{- 2q A y^{n+1} x^. ^2 /(n + 1)(α + 2βx + γx^2)^{n+3 / 2}}</math>, cujus fluxio erit <math>{- 2qAy^n x^. ^2 y^. / (α + 2βx + γx^2)^{n+3 / 2}}+ {(- 2q^. (α + 2β x + γ x^2) + 2 (n + 3)q(β + γ x)x^.)Ay^n+1 x^. ^2 /(n + 1)(α + 2βx + γx^2)^{n+5 / 2}}</math>, unde
<math>A y^n+1 x^. ^2 q^. = - Ay^n+1 x^. ^2 {(2 q^. (α + 2β x + γ x^2) - 2(n + 3)q(β + γ x)x^.) / (n + 1)(α + 2β x + γ x^2)}</math>,
& exinde <math>(n + 1) (α + 2 β x + γ x^2)q^. = - 2(α + 2 β x + γ x^2)q^. + 2 x^.
(n + 3) q (β + γ x)</math>, & con$equenter <math>(n + 3) (α + 2 β x + γ x^2)q^. =
2 (n + 3) q (β + γ x)x^.</math>, unde <math>{q^. / q} = {2 (β + γ x)x^. / α + 2 β x + γ x^2}, & q = α + 2 β x
+ γ x^2</math>: multiplicator igitur quæ$itus erit <math>- 2 (β + γ x) y x^. +
2 y^. (α + 2 β x + γ x^2)</math>, qui ductus in datam æquationem <math>y^.. +{A y^n x^. ^2 / (α + 2 β x + γ x^2)^{n+3 / 2}} = 0</math>, præbet fluxionem, cujus fluens erit <math>- 2 y
x^. y^. (β + γ x) + y^. ^2 (α + 2 β x + γ x^2) + {2 / n + 1}{A y^n+1 x^. ^2 / (α + 2 β x + γ x^2)^{n+1 / 2}}+ γ y^2 x^. ^2 = C x^. ^2</math>, ubi <math>C</math> $it invariabilis quantitas ad libitum a$$u-
menda.</p>
<p>Ex. 3. Sit <math>y^.. + y X x^. ^2 = 0</math>, ubi <math>X</math> $it functio quantitatis <math>x</math>: ejus
multiplicator erit <math>- y q^.. + 2 q y^.</math>.</p>
<p>Ex. 4. Sit <math>y^.. + {A y^2 x^. ^2 /x{15 / 7}} = 0</math>, & ejus fluens invenietur <math>{24 x^. y^. / 243 A} - {6 y x^. y^. / 7 x{1 / 7}}+ x{6 / 7}y^. ^2 + {2 A y^3 x^. ^2 / 3^x{9 / 7}} - {3 y^2 x^. ^2 / 49^x{8 / 7}} = C x^.^2</math>.</p>
<p>Ex. 5. Sit <math>y^.. + y^- {5 / 3} x^. ^2 (α + 2 β x + γ x^2) = 0</math>, multiplicetur in quan-
titatem <math>36 y^{1 / 3} (β + γ x) x^. ^3 - 12 y^-{2 / 3} (α + 2 β x + γ x^2) x^. ^2 y^. + 4 y^.^3</math>
formulæ <math>P x^. ^3 + 2 Q x^. ^2 y^. + 3 R x^. y^. ^2 + 4 S y^. ^3</math>; & re$ultat æquatio, cujus
fluens erit <math>36 y^{1 / 3} (β + γ x) x^. ^3 y^. - 6 y^-{2 / 3} (α + 2 β x + γ x^2) x^. ^2 y^. ^2 + y^.^4
+ 9 y^-{4 / 3} (α + 2 β x + γ x^2)^2 x^. ^4 - 27 γ y^{4 / 3} x^. ^4 = Cx^.^4</math>.</p>

<pb 210>[0262]<rh>DE FLUXIONALIBUS</rh>
<p>Cor. In his æquationibus, quarum multiplicatores inveniuntur,
pro <math>y</math> & ejus fluxionibus <math>y^., y^.., &c.</math> $cribantur <math>f z^n</math>, vel quæcunque alia
quantitas, & ejus fluxiones; & re$ultant æquationes, quarum dabun-
tur etiam multiplicatores.</p>
<p>Ex. 6. Sit <math>y^2 y^.. + y y^. ^2 + A x x^. ^2 = 0</math>, ducatur ea in multiplicatorem
formulæ <math>3 L y^. ^2 + 2 M x^. y^. + N x^. ^2</math>, ubi <math>A</math> $it con$tans, & <math>L, M & N</math>
functiones quantitatis <math>x</math>; & æquationis re$ultantis inveniatur fluens
per notas regulas; invenietur multiplicator <math>3 y y^. ^2 + 3 A x x^. ^2</math>, qui
ductus in datam æquationem, dat fluxionem, cujus fluens erit <math>y^3 y^. ^3 +
3 A x y^2 x^. ^2 y^. - A y^3 x^. ^3 + A^2 x^3 x^. ^3 = C x^.^3</math>.</p>
<p>Cor. Sit <math>y = u x & y^. = p x^.</math>, unde <math>{x^. / x} = {u^. / p - u}</math>; ita reducatur
data æquatio, ut exprimat relationem inter <math>p & u</math>, quæ evadit <math>Au^. +
u p^2 u^. + p u^2 p^. - u^3 p^. = 0</math>; pro <math>p & p^.</math> $cribantur <math>{q / u} & {q^. u - u^. q / u^2}</math>, & re-
$ultat <math>A u^. + q q^. + q u q^. - u^2 q^. = 0</math>.</p>
<p>7. Sit æquatio <math>y^2 y^.. - {1 / 2} y y^. ^2 = a x x^. ^2</math> vel <math> y^.. - {1 / 2}{y^. ^2 / y} = {a x x^. ^2 / y^2}</math>, ducatur
hæc æquatio in <math>- 2 x^. ^2 + {2 x x^. y^. / y}</math>, & fluens quæ$ita erit <math>- 2 x^. ^2 y^. +{x x^. y^. ^2 / y} + {a x^2 x^. ^3 / y^2} = C x^.^3</math>.</p>
<p>8. Sit <math>y^.. - {2 y^. ^2 / 5 y} - {a x x^. ^2 / y^2} = 0</math>, ducatur hæc æquatio in <math>- {10 / 3}x y^{1 / 5} x^.^2
+ 2 x^2 y - {4 / 5} x^. y^.</math>, & fluxionis re$ultantis fluens erit <math>- {10 / 3}x y^{1 / 5} x^. ^2 y^. +
x^2 y^- {4 / 5} x^. y^. ^2 + {10 / 9} a x^3 y^-{9 / 5} x^. ^3 + {25 / 9} y^{6 / 5} x^. ^3 = C x^.^3</math>.</p>
<p>9. Sit <math>y^.. + {2 y^. ^2 / y} - {a x x^. ^2 / y^2} = 0</math>, ducatur ea in <math>y^2 x^. ^2</math>, & fluens fluxio-
nis re$ultantis erit <math>y^2 x^. ^2 y^. - {1 / 2} a x^2 x^. ^3 = C x^.^3</math>.</p>
<p>10. Sit æquatio <math>y^.. + {y^. ^2 / y} - {a x x^. ^2 / y^2} = 0</math>; multiplicetur ea per <math>3 y^3 y^.^2

<pb 211>[0263]<rh>ÆQUATIONIBUS.</rh>
- 3 a y^2 x x^. ^2</math>, & re$ultat fluxio, cujus fluens erit <math>y^3 y^. ^3 - 3 a x y^2 x^. ^2 y^. +
a y^3 x^. ^3 + a^2 x^3 x^. ^3 = C x^.^3</math>.</p>
<p>11. Sit <math>y^.. + y^2 X x^. ^2 = 0</math>, ubi <math>X = - {24β^3 /343 (α + βx)^{20 / 7}}</math>; & invenietur
ejus multiplicator <math>(α + β x - {8 / 7}β y (α + β x)^{1 / 7}) x^. + 2 (α + β x)^{8 / 7} y^.</math>,
qui ductus in datam æquationem præbet fluxionem, cujus fluens inve-
nietur <math>(α + β x - {8 / 7}β y (α + βx)^{1 / 7})x^.y^. + (α + βx)^{8 / 7} y^. ^2 - {16 β^3 y^3 x^. ^2 / 343 (α + β x)^{12 / 7}}- β y x^. ^2 + {4 β^2 y^2 x^. ^2 / 49 (α + β x)^{6 / 7}} = C x^.^2</math>.</p>
<p>12. Sit <math>y^.. + {A y^2 x^. ^2 /x{15 / 7}} = 0</math>, ducatur hæc æquatio in <math>{24 x^. / 343 A} - {6 y x^. / 7x^{1 / 7}}+ 2 x{6 / 7}y^.</math>, & re$ultat fluxio, cujus fluens erit <math>{24 x^. y^. / 343 A} - {6 y x^. y^. / 7 x^{1 / 7}} +
x{6 / 7} y^. ^2 + {2 A y^3 x^. ^2 /3 x^{9 / 7}} - {3 y^2 x^. ^2 /49 x^{8 / 7}} = C x^.^2</math>.</p>
<p>13. Sit <math>2 y^3 y^.. + y^2 y^. ^2 + X x^. ^2 = 0</math>, ubi <math>X = α + β x + γ x^2</math>; ducatur
ea in <math>{2 y^. ^3 / y} - {2 (α + β x + γ x^2) x^. ^2 y^. / y^3} + {2 x^. ^3 (β + 2 γ x) / y^2}</math>, & re$ultat
fluxio, cujus fluens erit <math>(y^2 y^. ^2 - (α + β x + γ x^2) x^. ^2)^2 + 4 y^3 x^. ^3 y^. (β +
2 γ x) - 4 γ y^4 x^. ^4 = C y^2 x^.^4</math>.</p>
<p>14. Si in æquatione <math>y^2 y^.. + y y^. ^2 + X x^. ^2 = 0</math> $cribatur <math>y y^. = {1 / 2} z^.</math>
$eu <math>y^2 = z</math>; re$ultat <math>{1 / 2}z^.. √(z) + X x^. ^2 = 0</math>, quæ ope multiplicatoris
<math>{3 z^. ^2 / 4√(z)} + 3 X x^. ^2</math> redditur integrabilis, $i modo <math>X = Ax</math> vel <math>α + β x</math>:
eodem modo $it æquatio <math>y^2 (2 y y^.. + y^. ^2) + X x^. ^2 = 0</math>, ubi <math>X = α
+ β x + γ x^2</math>; ponatur <math>y = z{2 / 3}</math> & prodit æquatio <math>{4 / 3}z{5 / 3}z^.. + X x^. ^2 = 0</math>,
per eundem modum integrabilis.</p>
<p>15. Sit <math>y^.. + {A y x^. ^2 / (B y^2 + C + 2 D x + E x^2)^2} = 0</math>; ubi <math>A, B, C, D, E</math>

<pb 212>[0264]<rh>DE FLUXIONALIBUS</rh>
$unt con$tantes, & ejus multiplicator invenietur <math>2 y^. (C + 2 D x +
E x^2) - 2 y x^. (D + Ex)</math>, cujus producti fluens erit <math>{y^.^2 / x^. ^2} (C + 2 D x +
E x^2) - {2 y y^. / x^.} (D + Ex) + {A y^2 / B y^2 + C + 2 D x + E x^2} + E y^2 = con$t</math>.</p>
<p>16. Sit æquatio <math>y^.. + {X^. / 2 X} y^. + {Y / X}x^. ^2 = 0</math>; ubi <math>Y & X</math> $unt functio-
nes quantitatum <math>y & x</math> re$pective; ducatur hæc æquatio in <math>X y^.</math>, &
re$ultat <math>X y^. y^.. + {1 / 2} Xy^. ^2 + Y y^. x^. ^2 = 0</math>, cujus fluens erit <math>{1 / 2} X y^. ^2 = - x^.^2
$. Y y^.</math>.</p>
<h>PROB. LIII.</h>
<p it>_1._ Sit <math>p x^. + q y^. = 0</math>, ducatur hæc æquatio in <math>P</math>, ubi <math>P</math>, $it functio
quantitatis <math>x</math>; invenire, annon ita a$$umi pote$t <math>P</math>, ut fluens fluxionis <math>P p x
+ P q y^.</math> inveniri pote$t.</p>
<p>Sint <math>p^. = α x^. + β q^. = π x^. + ξy^.</math>, tum erit <math>P β = P π + {P^. / x^.} × q</math>;
unde <math>{β - π / q} × x^. = {P^. / P}</math>; & exinde, $i differentia <math>β - π</math> per <math>q</math> divi$a $it
functio quantitatis <math>x</math>, ita a$$umi pote$t <math>P = e^$. {β-π / q} x^.</math>, ut fluens flu-
xionis <math>P p x^. + P q x^.</math> inveniri pote$t; $in aliter vero non.</p>
<p>1.2. Fluens terminorum datæ æquationis, in quibus continetur <math>y</math>, erit
<math>P $. q y^.</math>; ubi per <math>$. q y^.</math> intelligo fluentem (<math>V</math>) fluxionis <math>q y^.</math>, cum <math>x</math> fin-
gatur invariabilis; & fluens fluxionalis æquationis erit <math>P V + $. Q x^.
= A = 0</math>, ubi <math>Q x^. = (P p - W) x^., & W x^.</math> e$t fluxio rectanguli <math>P × V</math>
ex hypothe$i quod <math>x</math> $olummodo $it variabilis.</p>
<p>Aliter: inveniatur fluens fluxionis (<math>q y^.</math>) ex hypothe$i quod <math>x</math> $it in-
variabilis quantitas, quæ $it <math>V</math>; tum fiat <math>V P + P V = P Q′x^.</math>, ubi <math>Q′</math>
e$t ea pars quantitatis <math>p</math>, in quâ continetur <math>y</math> vel ejus functio & non
$olummodo quantitas <math>x</math> & ejus functiones; tum ex hypothe$i quod in

<pb 213>[0265]<rh>ÆQUATIONIBUS.</rh>
quantitatibus (<math>V & Q′</math>) $olummodo habetur variabilis <math>x</math>, quæratur
multiplicator <math>P</math> ex fluxionali æquatione <math>P V^. + V P = P Q′x^.</math>, unde
<math>{P^. / P} = {Q′x^. - V^. / V}</math>; ergo, $i <math>{Q x^. - V^. / V}</math> $it functio unica quantitatis <math>x</math> in <math>x^.</math>,
<math>P</math> erit etiam functio quantitatis <math>x</math>.</p>
<p>2. Sit æquatio <math>P y^. ^n - Q y^. ^n-1 x^. + R y^. ^n-2 x^. ^2 - S y^. ^n-3 x^. ^3 + &c. + π y^... x^. ^n-3 +
ξ y^. x^. ^n-2 + σ^.y^.x^. ^n-1 + τ y x^. ^n = X x^. ^n</math>, ubi <math>x</math> fluit uniformiter; & <math>P, Q,
R, S, &c. X</math> $unt functiones quantitatis <math>x</math>; tum ejus (<math>n</math>) multiplica-
tores (<math>M</math>) erunt functiones quantitatis <math>x</math>, quæ erunt (<math>n</math>) fluentes fluxio-
nalis æquationis <math>(M^.^n P) + (M^.^n-1 Q)x^. + (M^.^n-2 R)x^. ^2 + (M^.^n-3 S)x^. ^3 + ... ±
τ x^. ^n = M^. ^n P + (n P^. + Q x^.) M^. ^n-1 + (n · {n - 1 / 2} P^.. + (n - 1) Q^.x^. + R x^. ^2) M^. ^n-2 +
(n · {n - 1 / 2} · {n - 2 / 3} P^... + (n - 1). {n - 2 / 2} Q^.. x^. + (n - 2) R^.x^. ^2 + S x^. ^3) ×
M^. ^n-3 + &c. = 0</math>.</p>
<p>Ejus fluens erit <math>M P y^. ^n-1 - (M^. P + MP^. + MQx^.) y^. ^n-2 + (M^.. P + (2P^.
+ Qx^.) M^. + (P^.. + Q^.x^. + Rx^. ^2) M) y^. ^n-3 + (M^... P + (3 P^. + Qx^.) M^.. +
(3 P^.. + 2 Q^.x^. + R x^. ^2) M^. + (P^... + Q^..x^. + R^.x^. ^2 + S x^. ^3) M) y^. ^n-4 &c. =
α = x^. ^n-1 $. M X x^., &c.</math></p>
<p>Cor. Sit data prædicta æquatio <math>Py^. ^n - Qy^. ^n-1 x^. + R y^. ^n-2 x^. ^2 ... + π y∴ x^. ^n-3 +
ξ y^.. x^. ^n-2 + σ y^. x^. ^n-1 + τ y x^. ^n = X x^. ^n</math>; & $it <math>τ x^. ^n = {1 / M}((σM^. + Mσ^.)x^. ^n-1 - (ξ^.. M
+ 2 ξ^. M^. + ξ M^..)x^. ^n-2 + &c.)</math>; tum $emper detegi pote$t fluens; vel, $i <math>σx^. ^n-1 ={1 / M} $. (τMx^. ^n + (ξ^.. M + 2 ξ^. M^. + ξ M^..) x^. ^n-2 - (π^... M + 3 π^.. M^. + 3 π^.M^.. + π M^...) x^.^n-3
+ &c.)</math>; vel <math>ξx^. ^n-2 = {- 1 / M} $. $. (τ M^. x^. ^n - (σ M^. + M σ^.) x^. ^n-1 - (π^... M + 3 π^.. M^.

<pb 214>[0266]<rh>DE FLUXIONALIBUS</rh>
+ 3 π^. M^.. + πM^...)x^.<_>n-3</_> + &c.)</math>, &c.; tum fluens datæ fluxionalis æqua-
tionis $emper detegi pote$t.</p>
<p>Cor. Sit fluens datæ fluxionis <math>(P y^. ^n - Qy^. ^n-1 + &c.) = Γ = 0</math>; tum,
ut e præcedentibus con$tat, erit multiplicator Mφ:Γ.</p>
<p>3. Sit fluxionalis æquatio ordinis (<math>n</math>), viz. <math>P y^. ^n + Qy^. ^n-1x^. + Ry^. ^n-2x^.^2
+ &c. = 0</math>; ubi <math>P, Q, R, &c.</math> $unt functiones quantitatum <math>x & y</math> &
earum fluxionum ordinum haud majorum quam <math>n - 1, n - 2, n - 3</math>,
&c.; & <math>x</math> fluit uniformiter: $it multiplicator (<math>M</math>), qui reddit æqua-
tionem integrabilem, functio quantitatis <math>x</math>; tum inveniatur fluens
quantitatis <math>Py^. ^n</math> ex hypothe$i, quod <math>y^. ^n-1</math> $olummodo $it variabilis; quæ
$it <math>α</math>; & exinde inveniatur fluxio quantitatis <math>α M</math>, viz. <math>αM^. + Mα^.</math>, in
quâ $olummodo $upponitur (<math>x^.</math>) invariabilis; deinde inveniatur fluens
fluxionis <math>MQy^. ^n-1 x^. - αM^. - Mα^. + MPy^. ^n</math> ex hypothe$i, quod <math>y^. ^n-2</math> $o-
lummodo $it variabilis, quæ $it <math>-M^. $. α + M $. (Qy^. ^n-1 x^. - α^. + Py^. ^n)
= - M^.β′ + Mβ</math>: tum inveniatur fluxio quantitatis <math>β M - β′M^.</math>
ex hypothe$i quod <math>x^.</math> $olummodo $it invariabilis, quæ $it <math>Mγ +
M^. γ′ + M^.. γ″</math>: deinde inveniatur fluens fluxionis <math>MRy^. ^n-2x^.^2 - Mγ -
M^.γ′ - M^.. γ″ - Mα^. - αM^. + MPy^. ^n + MQy^. ^n-1 x^.</math> ex hypothe$i quod
<math>y^. ^n-3</math> $olummodo $it variabilis; quæ $it <math>Mδ + M^. δ′ + M^.. δ″</math>; cujus in-
veniatur fluxio ex hypothe$i quod <math>x^.</math> $olummodo $it invariabilis; &
$ic deinceps: & $i ultimo evane$cant quantitas <math>y</math> & ejus fluxiones;
tum integrari pote$t data æquatio per multiplicatorem, quæ e$t fun-
ctio quantitatis <math>x</math>; $in aliter vero non.</p>
<p>Cor. Si ex affumptis valoribus prædictarum quantitatum ultimo
evane$cant quantitas <math>y</math> & ejus fluxiones, tum dabuntur fluxionales
æquationes, quarum fluentes innote$cunt.</p>
<p>4. Datâ fluxionali æquatione <math>n</math> ordinis (<math>A = 0</math>), quæ $it data fun-
ctio quantitatis <math>y</math>, ejus fluxionum; <math>x & x^.</math>; ubi <math>x^.</math> e$t invariabilis; &

<pb 215>[0267]<rh>ÆQUATIONIBUS.</rh>
multiplicator eandem habeat formulam ac data æquatio, i. e. in eo in-
volvantur datæ functiones quantitatis <math>y</math> & ejus fluxionum <math>y^., y^.., y^..., &c.</math>
ad <math>y^. ^n-1</math>; per quamcunque datam methodum cum incognitis & cognitis
functionibus quantitatum <math>x & x^.</math>; invenire ejus fluentem, $i modo data
æquatio $it integrabilis.</p>
<p>1<_>mo</_>. Reducenda e$t data fluxionalis æquatio, ita ut fluxio maximi
ordinis in eâ contenta unam $olummodo habeat dimen$ionem, i. e. $it
<math>Py^. ^n + &c. = A = 0</math>; deinde ducatur data æquatio in multiplicatorem
<math>M,</math> & inveniatur fluens fluxionis <math>MPy^. ^n</math> ex hypothe$i quod <math>y^. ^n-1</math> $olum-
modo $it variabilis, quæ $it <math>α</math>; & exinde inveniatur fluxio quantitatis
<math>α</math> ex hypothe$i, quod quantitates <math>y, y^., y^.., ... y^. ^n-1 & x</math>; $int variabiles,
& re$ultet fluxionalis quantitas <math>B</math>; tum inveniatur fluens fluxionalis
quantitatis <math>MA - B</math> ex hypothe$i quod <math>y^. ^n-2</math> $olummodo $it variabilis,
$i <math>y^. ^n-1</math> in quantitate <math>MA-B</math> unam $olummodo habeat dimen$ionem,
quæ $it <math>γ</math>; $i autem in quibu$dam terminis quantitatis <math>MA - B</math> in-
veniantur aliæ dimen$iones præter unam fluxionis <math>y^. ^n-1</math>; tum ita a$$u-
mantur functiones <math>F, G, H, &c.,</math> ut evane$cant illæ dimen$iones flu-
xionis <math>y^. ^n-1</math>; deinde inveniatur fluxio quantitatis <math>γ</math> ex hypothe$i quod
<math>y, y^., y^.., ..., y^. ^n-2 & x</math> & prædictæ incognitæ functiones (<math>F, G, &c.</math>)
& earum fluxiones $int variabiles, & re$ultet Fluxionalis quantitas <math>C</math>:
tum inveniatur fluens fluxionalis quantitatis <math>MA - B - C</math> ex hypo-
the$i quod <math>y^. ^n-3</math> $olummodo $it variabilis, $i <math>y^. ^n-2</math> $olummodo habeat unam
dimen$ionem in quantitate <math>MA - B - C</math>, quæ $it <math>δ</math>; $i autem in qui-
bu$dam terminis quantitatis prædictæ <math>MA - B - C</math> inveniantur aliæ
dimen$iones præter unam fluxionis <math>y^. ^n-2</math>, tum ita a$$umantur quanti-
tates & earum fluxiones ut evane$cant illæ dimen$iones fluxionis <math>y^. ^n-2</math>:

<pb 216>[0268]<rh>DE FLUXIONALIBUS</rh>
deinde inveniatur fluxio quantitatis <math>δ</math> ex hypothe$i quod <math>y, y^., y^.., ...,
y^. ^n-3 & x</math> & prædictæ quantitates (<math>F, G, &c. &c.</math>) & earum fluxiones $int
variabiles, & re$ultet quantitas <math>D</math>; tum inveniatur fluens quantitatis
<math>MA - B - C - D</math> ex hypothe$i quod <math>y^. ^n-4</math> $olummodo $it variabilis,
& $ic deinceps; u$que donec evane$cant quantitas <math>y</math> & ejus fluxio-
nes ex fluxione deductâ: & $i ex hâc reductione re$ultent (<math>l</math>) inde-
pendentes æquationes totidem incognitas functiones quantitatum <math>x
& x^. &c.</math> involventes; tum ex fluentibus prædictarum fluxionalium
æquationum detegi pote$t data fluxionalis æquatio & ejus multipli-
cator: $i plures (<math>m</math>) $int incognitæ functiones prædictæ quam æqua-
tiones, tum for$an pro <math>m-l</math> functionibus a$$umi po$$unt quæcun-
que functiones quantitatis <math>x</math> & ejus fluxionum, &c.</p>
<p>Cor. Datâ æquatione <math>py^. ^n + qy^. ^n-1 + ry^. ^n-2 + sy^. ^n-3 ... ty^. ^n-m = 0</math>; ubi <math>2 m</math>
minor e$t quam <math>n, & p</math> e$t data functio quantitatum <math>x, x^., y, y^., y^..,</math>
u$que ad <math>y^. ^n-2m-1; & q</math> data functio quantitatum <math>x, x^., y, y^., y^.., &c. u$que
ad <math>y^. ^n-2m</math>; & r</math> e$t functio quantitatum <math>x, x^., y, y^., y^.., ... y^. ^n-2m+1; & s</math> fun-
ctio quantitatum <math>x, x^., y, y^.,... y^. ^n-2m+1</math>; & $ic deinceps: tum erit fluens
primi ordinis datæ æquationis <math>Ppy^. ^n-1 + (Pq - Pp^. - pP^.)y^. ^n-2 + (P.r
- Pq^. - qP^. + Pp^.. + 2P^. p^. + pP^..)y^. ^n-3 + (Pv - {. / Pr} + {.. / Pq} - {... / Pp})y^. ^n-4 +
&c.. y^. ^n-m+1 = 0(A) = Cx^.^n-1</math>, ubi <math>C</math> e$t invariabilis; $i modo <math>{m<x>.</x> / Pp} -{m-1<x>.</x> / Pq} + {m-2<x>.</x> / Pr} - {m-3<x>.</x> / Ps} + &c. = 0</math>, tum fluens datæ fluxionis (<math>n</math> ordinis)
detegi pote$t e fluente fluxionalis æquationis <math>(B) {m<x>.</x> / Pp} - {m-1<x>.</x> / Pq} + {m-2<x>.</x> / Pr} -
&c. = 0</math>: $i <math>P</math> $it fluxionalis quantitas <math>π</math> ordinis; & <math>p, q, r, s, &c.</math>
fluxionales quantitates <math>α, β, γ, δ, ε, &c.</math> ordinum re$pective; tum erit
prædicta (<math>B</math>) fluxionalis æquatio ordinis, qui inventus fuerit maxi-

<pb 217>[0269]<rh>ÆQUATIONIBUS.</rh>
mus inter numeros <math>α + m, β + m - 1, γ + m - 2, δ + m - 3, &c.</math></p>
<p>Æquatio re$ultans per con$imilem methodum reduci pote$t, $i 2 <math>m</math>
minor $it quam <math>n - 1</math>; & $ic deinceps.</p>
<h>THEOR. XXXIV.</h>
<p>1. Sit fluxionalis æquatio <math>α = 0</math>, quæ continet fluxionalem quantita-
tem <math>y^. ^n</math> ordinis <math>n</math>; & $i ducatur æquatio <math>α = 0</math> in <math>P</math> fluxionalem
quantitatem <math>y^. ^m</math> ordinis <math>m</math>, ita ut æquatio re$ultans evadat integrabilis;
tum plures erunt fluxionales æquationes ordinis (<math>m</math>), quæ $unt fluen-
tiales æquationes ordinis (<math>n - m</math>) datæ æquationis <math>α = 0</math>, i. e. in
fluentiali æquatione ordinis (<math>n - m</math>) datæ æquationis fluxio <math>y^. ^m</math> a$cen-
det ad majores quam unam dimen$iones.</p>
<p>2. Si vero dentur (<math>n</math>) algebraici multiplicatores, viz. <math>a, b, c, d, &c;</math> tum
una $olummodo datur generalis fluens, viz. <math>$. a$. b. $. c $. d &c. × α = 0</math>;
&, $i una $olummodo detur generalis fluens, tum dantur (<math>n</math>) alge-
braici multiplicatores; etiamque (<math>n</math>) diver$i algebraici multiplica-
tores, qui reddent datam æquationem integrabilem.</p>
<p>3. Sit fluxio <math>Mx^.^n + Nx^.^n-1 y^. + Px^.^n-2 y^.. + Qx^.^n-3 y^... + Rx^.^n-4 y^.... +
&c.; & π^. = Mx^. + M′y^.</math>; tum erit ejus fluens, ($i modo exprimi
po$$it), <math>= πx^.^n-1 + ζx^.^n-2 y^. + σx^.^n-3 y^.. + τx^.^n-4 y^... + &c. = A</math>; ubi <math>ζ, σ, τ,
&c.</math> $unt functiones quantitatis <math>x</math>; & exinde <math>N = M′ + φ:(x);
& P, Q, R, &c</math>. $unt etiam functiones quantitatis <math>x</math>.</p>
<p>Fluens fluxionis <math>A</math>, (i. e. fluens $ecundi ordinis datæ fluxionis), $i
modo integrabilis $it, erit <math>π′x^.^n-2 + ζ′x^.^n-3 y^. + σ′x^.^n-4 y^.. + τ′x^.^n-5 y^... + &c.</math>;
ubi <math>π′</math> e$t functio quantitatis <math>x</math> in <math>ay + b; & ζ′, σ′, τ′, &c.</math> $unt fun-
ctiones quantitatis <math>x:</math> & fluens omnis $uperioris ordinis (<math>m + 1</math>) datæ
fluxionis erit <math>π′^m x^.^n-m + ζ′^m x^.^n-m-1 y^. + σ′^m x^.^n-m-2 y^.. + τ′^m x^.^n-m-3 y^... + &c.</math>; ubi
<math>π′^m</math> fuerit functio quantitatis <math>x</math> in <math>a′y + b′</math>; & <math>ζ′^m, σ′^m, τ′^m, &c.</math> functiones
quantitatis <math>x</math>.</p>

<pb 218>[0270]<rh>DE FLUXIONALIBUS</rh>
<p>4. Sit fluxio <math>Mx^.^n + Nx^.^n-1 y^. + Px^.^n-2 y^.. + Qx^.^n-3 y^... + Rx^.^n-4 y^.... + + P′x^.^n-2 y^.^2
&c.</math>; ejus fluens erit ($i modo integrari po$$it,) <math>πx^.^n-1 + ζx^.^n-2 y^. +
σx^.^n-3 y^.. + τx^.^n-4 y^... + &c.</math>; ubi <math>M, N, P & P′, π & ζ</math> $unt functiones
quantitatum <math>x & y; & Q, R, &c. σ, τ, &c.</math> $unt functiones quantitatis
<math>x</math>; hæc fluxio eandem habet formulam ac fluxio in priori ca$u data.</p>
<p>5. Sit fluxio <math>Mx^.^n + Nx^.^n-1 y^. + Px^.^n-2 y^.. + Qx^.^n-3 y^... + Rx^.^n-4 y^.... + &c.
+ P′x^.^n-2 y^.^2 + Q′x^.^n-3 y^.y^..
+ Q′x^.^n-3 y^.^3</math>
ejus fluens erit ($i modo integrari po$$it) <math>πx^.^n-1 + ζx^.^n-2 y^. + σx^.^n-3 y^.. +
+ σ′x^.^n-3 y^.^2 τx^.^n-4 y^... + &c.</math>;
ubi literæ <math>M, N, P, P′, Q, Q′, Q″; π, ζ, σ, & σ′</math> $unt
functiones quantitatum <math>x & y; & R, &c., τ, &c.</math> $unt functiones
quantitatis <math>x</math>; hæc fluxio eandem habet formulam ac data fluxio in
præcedente ca$u; & $ic deinceps: unde, $i modo datæ fluxiones hu-
ju$ce generis integrari po$$int, continuo ex inveniendis earum fluen-
tibus deprimentur datæ fluxiones in alias, quarum formula e$t ma-
gis $implex.</p>
<h>PROB. LIV.</h>
<h it>Per infinitas $eries invenire multiplicatores datarum fluxionalium æqua-
tionum.</h>
<p>1. Ex datis fluxionalibus æquationibus (<math>α = 0</math>) inveniatur $eries, quæ
exprimit valorem unius variabilis in terminis reliquarum & earum
fluxionum; & $i data fluxionalis æquatio $it primi ordinis, tum in $e-
rie generali continetur una $olummodo invariabilis quantitas (<math>A</math>) ad
libitum a$$umenda; ex hâc $erie per methodum infinitarum $erierum
po$tea traditam inveniatur quantitas <math>A = P</math>; tum <math>A = P</math> erit ge-
neralis fluens æquationis <math>α = 0 & {P^. / α}</math> multiplicator quæ$itus.</p>
<p>2. Si data $it fluxionalis æquatio $uperioris (<math>m</math>) ordinis; tum in $erie
generali prædictâ inventâ continebuntur (<math>m</math>) invariabiles & inter $e

<pb 219>[0271]<rh>ÆQUATIONIBUS.</rh>
non dependentes (<math>A, B, C, D, &c.</math>): ex hâc $erie inveniatur quæcun-
que prædicta invariabilis quantitas, e. g. <math>A = P</math>; tum erit <math>A = P</math>
generalis fluens & <math>{P^. / α}</math> multiplicator quæ$itus.</p>
<h>THEOR. XXXV.</h>
<p>1. Sit æquatio fluxionalis <math>α = 0</math>, quæ ducta in <math>P</math> fiat fluxio, cujus
fluens <math>n</math> ordinis inveniri pote$t: $it <math>β</math> fluens <math>n</math> ordinis fluxionalis
æquationis <math>α = 0</math>, tum <math>{β^. ^n-1 / α}</math> erit quantitas, quæ ducta in <math>α</math> creat fluxio-
nem, cujus fluens (<math>n - 1</math>) ordinis inveniri pote$t; & $ic <math>{β^.^n-2 / α}, {β^.^n-3 / α}, &c.</math>
erunt multiplicatores, qui ducti in datam fluxionalem æquationem,
creant fluxiones, quarum fluentes <math>n - 2, n - 3, &c.</math> ordinum inveniri
po$$unt.</p>
<p>2. Et $ic $it <math>α = 0</math> data æquatio, & <math>P α = β^. & Q β = γ^., R γ = δ^.,
S δ = ε^., &c.</math> tum erunt <math>Q $. P α = γ^., R $. Q $. P α = δ^.</math>, & $ic <math>S $. R
$. Q $. P α = ε^., &c.</math> & multiplicatores, qui ducti in datam æquatio-
nem <math>α = 0</math> præbent æquationes fluxionales, quarum fluentes primi,
$ecundi, tertii, &c. ordinis inveniri po$$unt, erunt re$pective <math>P, {Q $. P α / α},{R $. Q $. P α / α}, {S $. R $. Q $. P α / α}, &c.</math></p>
<h>THEOR. XXXVI.</h>
<p>1. Sæpe e $ub$titutione $eparari po$$unt variabiles quantitates: e
methodis prius traditis deleri po$$unt quædam irrationales quantita-
tes, e quibus nonnunquam re$ultabit $eparatio; vel $æpe $ub$titutio
ex æquatione datâ $atis manife$ta erit, quæ $eparationem variabilium
quantitatum inducet; $æpe etiam di$tingui pote$t data æquatio in

<pb 220>[0272]<rh>DE FLUXIONALIBUS</rh>
diver$as partes, quarum $ingula e$t functio fluentis in ejus fluxionem,
quod e $ub$equentibus exemplis con$tabit.</p>
<p>Ex. 1. Sit <math>(x y^. + y x^.) √(a^4 - x^2 y^2) = {x x^. + yy ^./(x^2 + y^2)^{3 / 4}}</math>; facilis e$t ob$er-
vatio, ut <math>$. (x y^. + y x^.) = x y</math>, cujus quadratum <math>x^2 y^2</math> $olummodo con-
tinetur in quantitate <math>a^4 - x^2 y^2</math>; fluens vero fluxionis <math>x x^. + y y^.</math> erit
<math>{1 / 2}(x^2 + y^2)</math>; $ed quantitas <math>x^2 + y^2</math> $olummodo continetur in quan-
titate <math>(x^2 + y^2)^{3 / 4}</math>: $cribantur igitur <math>x y = z & x^2 + y^2 = v</math>; & exinde
re$ultabit æquatio <math>z^. √(a^4 - z^2) = {v^. / 2 v{3 / 4}}</math>, in quâ variabiles <math>z & v</math> $e-
parantur.</p>
<p>Ex. 2. Sit <math>a x^. = (x + y)^n × y^.</math>, $cribatur <math>x + y = z</math>, & erit <math>x^. + y^.
= z^.</math>, unde exorietur ($i modo exterminentur <math>x & x^.</math>) æquatio <math>a z^. -
a y^. = z^n y^.</math>, & con$equenter <math>y^. = {a z^. / a + z^n}</math>.</p>
<p>Ex. 3. Sit fiuxio <math>{2x^2 x^. + x y y^. + y^2 x^. / x^4 + x^2 y^2 + a^4} = {x x^. + y y^. / √(x^2 + y^2)}</math>, nunc $up-
ponatur irrationalis quantitas <math>√(x^2 + y^2) = z</math>, quâ quantitate &
ejus fluxionibus pro $uis valoribus in datâ æquatione $ub$titutis re-
$ultat <math>{z / x^2 z^2 + a^4} × (x z^. + z x^.) = z^.</math>; $cribatur <math>x z = p</math>, & ex hoc a$-
$umpto valore deduci pote$t <math>{p^. / p^2 + a^4} = {z^. / z}</math>.</p>
<p>Ex. 4. Sit æquatio <math>y^m (x x^. + y y^.) = x^n (y x^. - x y^.)</math>, quæ hoc modo
$cribi pote$t <math>{y^m-2 / x^n} (x x^. + y y^.) = {y x^. - x y^. / y^2}</math>; a$$umatur <math>{y x^. - x y^. / y^2} = p^.
& x x^. + y y^. = q q̈</math>, unde <math>{x / y} = p, & x^2 + y^2 = q^2</math>; his po$itis, deduci
pote$t æquatio <math>q^m-n-1 × q̈ = p^n × (p^2 + 1)^{m-n-z / 2} p^.</math>.</p>
<p>Ex. 5. Sit æquatio <math>{x^mt-1 y x^. / (b x^t + a y^n x^r)^m} = y^.</math>, $upponatur <math>(b x^t + a y^n x^r)^m

<pb 221>[0273]<rh>ÆQUATIONIBUS.</rh>
= z x^mt</math>; e datâ æquatione extermincntur <math>y</math> & ejus fluxio, & re$ultat
<math>{z^{1 / m} z^./m n z^{1 / m} + (r - t) mz^{1 / m}+1 - (m r - m t)(b z) - m n b} = {x^. / x}</math>.</p>
<p>Ex. 6. Sit <math>{y^u x^. / (b x^t + a y^n x^r)^m} = c x^{ut-mnt-t+r+n-ur / n} y^.</math>, & per præcedentem
methodum $eparari po$$unt indeterminatæ. Et $ic ex eâdem $ub-
$titutione $eparari po$$unt variabiles quantitates in æquationibus
<math>{y^u y^. / (b x^t + a y^n x^r)^m} = c x^{tu-n-tmn-ru+t-r / n} x^. & {(b x^t + f y^n x^r)^v y^u y^. / (b x^t + a y^n x^r)^m} =
c x^{ut-n-tmn-ru+t-r+ntv / n} x^.</math>.</p>
<p>Ex. 7. Sit æquatio <math>{y^n-1 y^. / (a x^t + b x^r + c y^n x^r)^m} = f x^t-r-1-mt x^.</math>; vel
<math>{(g x^t + h x^r + k y^n x^r)^e y^n-1 y^. / (a x^t + b x^r + c y^n x^r)^m} = x^t-r-1+te-mt x^.</math>, cum <math>c h = b k</math>; ex $ub$ti-
tutione <math>a x^t + b x^r + c y^n x^r = x^t z^{1 / m}</math> facile con$tat re$olutio, &c.</p>
<p>Ex. 8. Sit æquatio <math>{a y^. / b + (c y^n + f x)^u} = g y^1-n x^. & {a y^n-1 y^. / b + (c y^n + f x^m)^u}= g x^m-1 x^.</math>; $cribatur pro priori <math>(c y^n + f x)^u = z</math>, & pro $ecundâ <math>(cy^n
+ fx^m)^u = x</math>, & re$ultabunt duæ æquationes <math>{a z^{1-u / u} z^./n u b c g + n u c g z + a u f}= x^., & x^m-1 x^. = {a z^{1-u / u} z^./b c g n u + c g n u z + m a f u}</math>.</p>
<p>Ex. 9. Sit <math>{a y^n-1 y^. / b + (c y^n + p)^u} = g q x^.</math>, ubi <math>p & q</math> $unt algebraicæ
functiones literæ <math>x</math>; $i <math>q = {p^. / x^.}</math>, tum $eparari po$$unt indeterminatæ a
$cribendo <math>(c y^n + p)^u = z</math>, re$ultat enim æquatio <math>{a z^{1-u / u} z^./n b c g u + n c g u z + a u}= q x^. = p^.</math>.</p>
<p>Ex. 10. Sit æquatio <math>a^2 y^. - x^2 y^. + x y x^. = a x^. √ (x^2 + y^2 - a^2)</math>

<pb 222>[0274]<rh>DE FLUXIONALIBUS</rh>
$cribe <math>y = u √ (a^2 - x^2)</math>, & re$ultat <math>{u^. / √(u^2 - 1)} = {a x^. / a^2 - x^2}</math>, unde <math>u +
√(u^2 - 1) = n√({a + x / a - x})</math>; ubi <math>n</math> $it quæcunque con$tans quan-
titas.</p>
<p>11. Sit <math>x^m x^. + (b y + y^2){c x^. / x} = y y^.</math>, quæ ita $cribi pote$t <math>x^m x^. +{b c y x^. / x} = y^2 × ({y^. / y} - {c x^. / x^.})</math>; fingatur <math>{y^. / y} - {c x^. / x} = {p^. / p}</math>, unde <math>y = p x^c</math>, quâ
quantitate pro $uo valore <math>y</math> in datâ æquatione $ub$titutâ re$ultat <math>x^m x^.
+ b c p x^c-1 x^. = p p^. x^2c</math>; dividatur ea per <math>x^2c</math>, & erit re$ultans æquatio
<math>x^m-2c x^. + b c p x^-c-1 x^. = p p^.</math>, fiat <math>m - 2c = - c - 1</math>, & $eparantur
invariabiles.</p>
<p>Hæc methodus etiam nonnunquam applicari pote$t ad $ub$equentes
æquationes <math>x^m x^. + {a y^n x^. / x} + {b y^s x^. / x} = y^t y^.</math>; po$itis enim <math>n - 1</math> vel <math>s - 1
= t</math>; abbreviationem recipiet formula.</p>
<p>2. Sit æquatio <math>a′<x>′</x> x^m x^. + b y^q x^. = y^.</math>; $cribatur <math>y = z^{1 / 1-q}</math>, & re$ultat æqua-
tio <math>a x^m x^. + b z^{q / 1-q} x^. = {1 / 1 - q}z^{q / 1-q} z^.</math>; unde, $i vel <math>m = 0</math> vel <math>= {q / 1 - q}</math>, re-
$ultantis æquationis fluens inveniri pote$t.</p>
<p>3. Sit <math>x y^. + g y x^. + a x^n x^. + b y^n y^. = 0</math>; $cribatur <math>x = y^r u</math>, & exinde
deduci pote$t æquatio <math>y^r u y^. + g y(r y^r-1 u y^. + y^r u^.) + a y^rn u^n (r y^r-1 u y^.
+ y^r u^.) + b y^n y^. = 0</math>, cujus fluens inveniri pote$t, $i modo <math>g r + 1 = 0
& r n + r - 1 = n</math>.</p>
<p>Ex. 12. Sit <math>α y x^. + β x y^. + x^m y^n (γ y x^. + δ x y^.) = 0</math>, dividatur hæc
æquatio per <math>xy</math>, & re$ultat <math>{α x^. / x} + {β y^. / y} + x^m y^n ({γ x^. / x} + {δ y^. / y}) = 0</math>; fin-
gantur <math>x^α y^β = t & x^γ y^δ = u</math>, & inveniatur fluxionalis æquatio, cu-
jus variabiles quantitates $unt <math>t & u</math>, & evadit <math>t^{γ^n-δm / αδ-βγ}-1 t^. + u^{αn-βm / αδ-βγ}-1
u^. = 0</math>.</p>

<pb 223>[0275]<rh>ÆQUATIONIBUS.</rh>
<p>Ex. 13. Sit <math>y^. y + y^. (a + b x + n x^2) = y x^.(c + n x)</math>; $ub$tituatur
<math>{y(c + n x) / y + a + b x + n x^2} = u</math>; & inveniatur fluxionalis æquatio, cujus
variabiles quantitates $unt <math>x & u</math>, quæ erit <math>{x^. / (a + b x + n x^2)(c + n x)}= {u^. / u(n a + c^2 - b c + (b - 2 c)u + u^2)}</math>.</p>
<p>Ex. 14. Sit <math>(y - x)y^. = {n x^. (1 + y^2)^{3 / 2} /√(1 + x^2)}</math>, $cribatur <math>y = {x - u / 1 + x u}</math>, de-
inde inveniatur fluxionalis æquatio, cujus variabiles quantitates $unt
<math>x & u</math>, & re$ultabit <math>{x^. / 1 + x^2} = {u u^. / (1 + u^2)(n√(1 + u^2) + u)}</math>.</p>
<p>Ex. 15. Sit <math>(x^n x^. + a y^{-n-1-c / c} y^.)p = (x y^. + b y x^.)q</math>, $ub$tituantur
<math>y = t^s z^{r / n+1} & x = t^-{s / c} (a ± a b z^-{r / c})^{1 / n+1}</math>, ubi literæ <math>s & r</math> quantitates
invariabiles re$pective denotant; <math>& p & q</math> $unt algebraicæ functiones
literarum <math>x & y</math>, tales vero ut in $ingulis terminis in functione <math>p</math> vel <math>q</math>
contentis exponens quantitatis <math>y</math> in <math>c</math> ducta $emper $uperet exponen-
tem quantitatis <math>x</math> per datam quantitatem; e. g. $it terminus <math>e x^m y^n</math> in
functione <math>p</math>, & nece$$e e$t ut <math>c n - m</math> $emper in omnibus terminis
functione <math>p</math> contentis eandem præbeat $ummam, & re$ultat æquatio
<math>-{s / c} t^{-su-c-cs / c} t^. = {r z^{r-n-1 / n+1} z^. × {q / p}/(n + 1)(a ± a b z^-{r / c})^{n / n+1}}</math>; in quâ $eparantur varia-
biles <math>t & z</math>.</p>
<p>Ex. 16. Sit æquatio <math>(x^n x^. = a y^{-nf-c-f / c} y^.)p = (f x y^. + c y x^.)q</math>: $cri-
bantur <math>y = t^{s / f} z^{r / f(n+1)} & x = t^-{s / c} (a ± {a c z^-{r / c} /f})^{2 / n+1}</math>, ubi exponens
quantitatis <math>y</math> ducta in <math>c</math> $uperat exponentem quantitatis <math>x</math> ductam in
<math>f</math> in $ingulis terminis quantitatum <math>p</math> vel <math>q</math> per eandem quantitatem,

<pb 224>[0276]<rh>DE FLUXIONALIBUS</rh>
& exorietur <math>{s / c} t {- f c - f s n - s c / c f}t^. = {{r / n + 1}z^{r-n f-f / f n+f} × z^. × {q / p}/(a ± {a c / f}z^{-r / c}))^{n / n+c}}</math>, in quâ
$eparantur variabiles <math>t & z</math>.</p>
<p>Ex. 17. Sit <math>y^. + P y x^. = Q x^.</math>, $cribatur <math>y = X u,</math> & oritur <math>X u^. +
u X^. = Q x^. - P X u x^.</math>; ut fiat æquatio, cujus fluens detegi pote$t,
fingatur <math>X^. + P X x^. = 0</math>, unde <math>{X^. / X} = - P x^. & log. X = - $. P x^.
& X = e^- $. P x^.</math>; æquatio vero transformata erit <math>X u^. = Q x^. & u =
$.{Q x^. / X} = $.e^$. P x^. Q x^. = {y / X}</math>, unde <math>y = e^-$ · P x $. e^$. P x^. Q x^.</math>; & quanti-
tates <math>y & x</math> $eparabiles $unt, $i modo <math>P & Q</math> $int functiones quanti-
tatis <math>x</math>.</p>
<p>Ex. 18. Sit <math>x = {y x^. / y^.} + {a y^2 x^.^2 / y^.^2} + {b y^3 x^.^3 / y^.^3} + &c.</math>, $cribatur <math>t y^. = x^.</math>, &
æquatio re$ultans erit <math>x = y t + a y^2 t^2 + b y^3 t^3 + &c.</math>; in hâc æqua-
tione pro <math>t y</math> $cribatur <math>z</math>, & transformetur re$ultans æquatio in alteram
exprimentem relationem inter <math>x & z</math>, & evadet <math>x = z + a z^2 + b z^3
+ &c.</math>, unde <math>x^. = t y^. = {z y^. / y} = z^. + 2 a z z^. + 3 b z^2 z^. + &c.</math>; & ex-
inde log. <math>y = log. z + 2 a z + {3 b z^2 / 2} + &c. + con$t.</math></p>
<p>Cor. 1. Hinc con$tat in nonnullis ca$ibus quantitates <math>y & x</math> in datâ
fluxionali æquatione contentas po$$e exprimi per tertiam a$$umptam
<math>z</math>, at minime per $e ip$as.</p>
<p>2. Sit <math>y^.^n</math> maxima dimen$io fluxionis (<math>y^.</math>) in datâ æquatione contenta,
tum ea reduci pote$t in (<math>n</math>) diver$as fluxionales æquationes, quarum
$ingulis fluentibus corre$pondent prædicti valores quantitatum
<math>x & y</math>.</p>
<p>Ex. 19. Datâ fluxionali æquatione <math>P x^. = Q y^.</math>, quæ $it homogenea,
i.e. ubi <math>P & Q</math> $int homogeneæ functiones eju$dem dimen$ionum nu-
meri ip$arum <math>x & y</math>, invenire ejus fluentem.</p>

<pb 225>[0277]<rh>ÆQUATIONIBUS.</rh>
<p>In æquatione <math>{P x^. / Q} = y^.</math> pro <math>y</math> $cribatur <math>u x</math>, & quoniam <math>P & Q</math>
$unt quantitates quæ eundem habet numerum dimen$ionum, erit <math>{P / Q}</math>
functio quantitatis <math>u</math>, quæ dicatur <math>U</math>; ergo <math>U x^. = y^. = u x^. + x u^.</math>, unde
<math>{x^. / x} = {u^. / U - u} & log. x = $.{u^. / U - u}</math>.</p>
<p>Cor. 1. Quod $i <math>$.{u^. / U - u}</math> per logarithmos exprimi po$$it, habebitur
algebraica æquatio inter <math>x & u</math>, & exinde inter <math>u & y</math>, etiamque inter
<math>x & y</math>.</p>
<p>Cor. 2. Si in æquatione <math>P x^. = Q y^.</math>, ubi in literis <math>P & Q</math> haud $o-
lummodo continentur homogeneæ functiones eju$dem dimen$ionum
numeri, $ed in$int etiam quantitates huju$modi formulæ; log. <math>{√(x^2 +y^2) / x};
e^y:x</math>; ang. $in. <math>{x / √(x^2 + y^2)}</math>; co$. <math>{n x / y}</math>; &c. in quibus tran$cendentes
functiones afficiant functiones nullius dimen$ionis ip$arum (<math>x & y</math>);
pro <math>y</math> $cribatur <math>ux</math>, & re$ultabit æquatio huju$ce formulæ <math>U x^. = y^.</math>,
ubi <math>U</math> erit functio quantitatis <math>u</math>, unde <math>U x^. = u x^. + x u^.&{x^. / x} = {u^. / U - u}</math>
& log. <math>x = $.{u^. / U - u}</math>.</p>
<p>Cor. 3. Sit <math>p x^. + q y^. = 0</math>, ubi literæ <math>p & q</math> re$pective denotant
homogeneas functiones literarum <math>x & y</math>, quarum dimen$iones $unt
<math>m</math>; $it quantitas <math>L</math>, quæ in datam æquationem ducta, creat fluxio-
nem, cujus fluens inveniri pote$t; & erit <math>L p x^. + L q y^. = P x^. + Q y^.
= V^.</math>, ubi <math>P & Q</math> $unt homogeneæ <math>n</math> dimen$ionum functiones litera-
rum <math>x & y, & nV = P x + Q y</math>: $cribatur enim <math>y = x z</math>, & erit
<math>V = x^n Z</math>, ubi <math>Z</math> functio e$t quantitatis <math>z</math>, unde <math>V^. = n x^n-1 Z x^. +
x^n Z^.</math>; at quoniam <math>V^. = P x^. + Q y^. & y^. = x z^. + z x^.</math>, erit <math>V^. = (P +
Q z)x^. + Q x z^. = n x^n-1 Z x^. + x^n Z^.</math>, & exinde <math>P + Q z = n x^n-1 Z</math>; du-

<pb 226>[0278]<rh>DE FLUXIONALIBUS</rh>
catur hæc æquatio in <math>x</math>, & re$ultat <math>P x + Q z x = P x + Q y = n x^n Z
= n V</math>.</p>
<p>Hinc etiam erit <math>({V^. / x^.}) = P = {(P x^. + Q y^.)y - (P x + Q y)y^. / y x^. - x y^.} ={y V^. - n V y^. / y x^. - x y^.}</math>, $i modo <math>x</math> $olummodo fluat in quantitate (<math>{V^. / x^.}</math>); &c.</p>
<p>Eodem modo quoniam <math>p & q</math> $unt homogeneæ functiones quanti-
tatum <math>x & y</math>, erunt etiam <math>({p^. / y^.}) = {n p x^. - x p^. / y x^. - x y^.} & ({q^. / x^.}) = {y q^. - n q y^. / y x^. - x y^.};
&c.</math></p>
<p>Cor. 4. Sit <math>L^. = a x^. + b y^.</math>; e præcedentibus principiis invenientur
<math>a = {y L^. - m L y^. / y x^. - x y^.}, & b = {m L x^. - x L^. / y x^. - x y^.}; L</math> enim erit homogenea fun-
ctio quantitatum <math>x & y</math>.</p>
<p>Quoniam <math>V^. = L p x^. + L q y^., & V = n(L p x + L q y)</math>, erit <math>{V^. / V}= {p x^. + q y^. / n(p x + q y)}</math>, unde fluens <math>V</math> datæ æquationis <math>p x^. + q y^. = 0</math>,
facile deduci pote$t; & con$tat unum valorem quantitatis <math>L</math> e$$e
<math>{1 / p x + q y}</math>.</p>
<p>Aliter: $it <math>p x^. + q y^. = 0</math> æquatio, in quâ literæ <math>p & q</math> denotant
homogeneas <math>n</math> dimen$ionum functiones literarum <math>x & y</math>; $cribatur
<math>y = u x</math>, & evadent <math>p = x^n U & q = x^n W</math>, ubi literæ <math>U & W</math> deno-
tant functiones literæ <math>u</math>; & con$equenter <math>x^n U x^. + x^n W y^. = p x^. +
q y^. = 0</math>; dividatur hæc æquatio per <math>x^n</math>, & re$ultat <math>U x^. + W y^. = 0</math>;
pro <math>y^.</math> $cribatur ejus valor <math>u x^. + x u^.</math>, & re$ultat æquatio <math>(U + W u)x^.
+ W x u^. = 0</math>; dividatur hæc æquatio per <math>x (U + W u)</math>, & fit <math>{x^. / x} +{W u^. / U + W u} = 0</math> integrabilis æquatio.</p>
<p>Ex. 20. Sit <math>a x^n x^. + b y^n y^. + (x y^. - y x^.)({p / q} + {π / w}) = 0</math>, ubi <math>p & q</math>

<pb 227>[0279]<rh>ÆQUATIONIBUS.</rh>
$unt homogeneæ functiones quantitatum <math>x & y, π & w</math> $unt homoge-
neæ functiones earundem quantitatum, dimen$ionum, quarum dif-
ferentia e$t <math>n - 1</math>; $cribatur <math>x = yz</math> & deduci pote$t æquatio <math>a y^n z^n
(y z^. + z y^.) + b y^n y^. - y^2 z^. (y^k φ: (z) + y^n-1 φ: (z)) = 0</math>.</p>
<p>Ex. 21. Sit fluxionalis æquatio <math>a x^. + b y^. = 0</math>, cujus $inguli termini
quantitatum <math>a & b</math> habeant <math>m</math> vel <math>m - 1</math> dimen$iones; in hâc æqua-
tione pro <math>y & y^.</math> $cribantur <math>x v & x v^. + v x^</math>. re$pective, & re$ultat
æquatio formulæ <math>(P x^2 + Q x)v^. + (R x + S)x^. = 0</math>; ubi <math>P, Q, R
& S</math> $unt functiones quantitatis <math>v</math>; unde, $i modo <math>{Q v^. - S^. / S} = {2 P v^. - R^. / R}</math>,
tum $emper fluens æquationis re$ultantis detegi pote$t: ducatur enim
re$ultans æquatio in <math>e^$.{(Q v^.-s^.) / S}</math>, & exorietur æquatio, cujus fluens erit
<math>e^$.({Q v^.-s^.) / S} ({R x^2 + 2S x / 2}) = con$t</math>.</p>
<p>2. Scribantur in æquatione <math>(P x^2 + Q x)v^. + (R x + S)x^. = 0,
z^-n & - n z^-n-1 z^. pro x & x^.</math>, & re$ultat æquatio <math>(P z + Q z^n+1) v^. -
(n R + n S z^n)z^. =0</math>; & exinde con$tabit fluentem prædictæ æqua-
tionis inveniri po$$e, $i <math>{P v^. + n R^. / R} = {(n+1) Q v^. + n S^. / S}</math>; erit enim
<math>e^$. -{P v^. + n R^. / n R}(n R z + {n S / n + 1} z^n+1

) = con$t</math>.</p>
<p>3. Si vero termini $inguli quantitatum <math>a & b</math> habeant <math>n, n - 1 &
n - 2</math> dimen$iones, tum e $ub$titutione prius traditâ re$ultabit æquatio
formulæ <math>(P x^3 + Q x^2 + R x)v^. + (p x^2 + q x + r)x^. = 0</math>; & $ic de-
inceps; ubi <math>P, Q, R, p, q & r</math> $unt functiones quantitatis <math>v</math>; cujus
fluens inveniri pote$t, $i vel<math>{3 P v^. - p^. / p} = {2 Q v^. - q^. / q} = {R v^. - r^. / r}</math>; vel
<math>{P v^. + n p^. / p} = {(n + 1)Q v^. + n q^. / q} = {(2 n + 1)R v^. + n r^. / r}</math>; fluens enim
ejus in priori ca$u erit <math>e^$.{R v^.-r^. / r} ({p x^3 / 3} + {q x^2 / 2} + r x) = con$t.</math>; in po-

<pb 228>[0280]<rh>DE FLUXIONALIBUS</rh>
$teriori vero invenietur <math>e^$.-{P v^.+n p^. / np} (n p z + {n q / n + 1} z^n+1 + {n r / 2n + 1}z^2n+1)
= con$t</math>. $i <math>z = x^-n</math>.</p>
<p>4. Si vero terminos <math>n, n - 1, n - 2 ... n - m</math> dimen$ionum habeant
quantitates <math>a & b</math> in fluxionali æquatione <math>a x^. + b y^. = 0</math>, $cribantur
<math>x v & x v^. + v x^.</math> pro <math>y & y^</math>. in datâ æquatione, & re$ultat æquatio for-
mulæ <math>(P x^m+1 + Q x^m + R x^m-1 + S x^m-2 + &c.)v^. + (p x^m + q x^m-1
+ r x^m-2 + &c.)x^. = 0</math>, ubi <math>P, Q, R, &c. p, q, r, &c.</math> $unt functiones
ip$ius <math>v</math>; cujus fluens inveniri pote$t, $i modo vel<math>{(m + 1) Pv^. - p^. / p} ={m Q v^. - q^. / q} = {(m-1) R v^. - r^. / r} = {(m-2) S v^. - s^. / s} = &c. vel {P v^. + n p^. / p}= {(n + 1) Q v^. + n q^. / q}={(2n + 1) R v^. + n r^. / r} = {(3n + 1) S v^. + n s^. / s} =
&c.</math> fluens enim in priori ca$u erit <math>e^$.{(m+1) P v^.-p^. / p} ({p x^m+1 / m + 1} + {q x^m / m} +{r x^m-1 / m - 1} + &c.) = con$t.</math>; in po$teriori vero erit <math>e^$.-{P v^.+n p^. / n p} (n p z+{n q z^n+1 / n + 1}+ {n r z^2n+1 / 2n + 1} +&c. = con$t. $i z = x^-n</math>.</p>
<p>Ex. 22. Sit data algebraica æquatio homogenea relationem inter <math>x,
y & z</math> exprimens; <math>ubi z^. = P x^. + Q y^.</math>, in quâ literæ <math>P & Q</math> re$pective
denotant homogeneas functiones quantitatum <math>x & y</math>.</p>
<p>In datâ algebraicâ æquatione pro <math>y & z</math> $cribantur re$pective <math>v x &
w x</math>; & re$ultat algebraica æquatio relationem inter <math>v & w</math> exprimens;
in po$teriori æquatione pro <math>y & z</math> $ub$tituantur prædicti valores, &
re$ultat <math>w x^. + x w^. = x^m × (P′ x^. + Q′ (v x^. + x v^.))</math>, ubi <math>P′ & Q′</math> $unt
functiones quantitatum <math>v & w</math>: ex priori æquatione deduci pote$t
quantitas <math>v</math> in terminis quantitatis <math>w</math>, i. e. <math>v = φ:(w)</math>: $ub$tituatur
<math>φ:(w)</math> & ejus fluxio pro <math>v & v^.</math> in $ecunda æquatione, & re$ultat
æquatio <math>w x^. + x w^. = x^m (P″ x^. + Q″ (φ:(w) × x^. + x × φ^.:(w))</math>.</p>

<pb 229>[0281]<rh>ÆQUATIONIBUS.</rh>
<p>Huju$ce æquationis fluens facile detegi pote$t, cum <math>m = 0</math>; & in
multis aliis ca$ibus.</p>
<p>Et $ic progredi liceat ad fluxionales æquationes $uperiorum ordi-
num; etiamque ad inve$tigandos plures ca$us prædictarum fluxiona-
lium æquationum, quarum fluentes inveniri po$$unt.</p>
<p>Hinc facile deduci po$$unt infinitæ æquationes formularum prædi-
ctarum, quarum fluentes inveniri po$$unt.</p>
<p>Cor. Deducere quam plurimas fluxionales æquationes, quæ reduci
po$$unt in alias, quarum fluentes deduci po$$unt, vel quarum varia-
biles $eparantur.</p>
<p>A$$umantur æquationes fluxionales, in quibus $eparantur variabi-
les, vel quarum fluentes innote$cunt; pro variabilibus (<math>x & y</math>) & ea-
rum fluxionibus in his æquationibus $cribantur functiones quantita-
tum (<math>z & v</math>) & earum fluxionum; & re$ultant æquationes, quæ re-
duci po$$unt ad a$$umptas.</p>
<h>THEOR. XXXVII.</h>
<p>Fluxionalis æquatio, cujus termini haud videntur ea$dem habere
dimen$iones, $æpe transformari pote$t in alteram homogeneam, i. e.
cujus termini ea$dem habent dimen$iones. Hoc plerumque perfici
pote$t, $i modo ita transformetur æquatio, ut ii termini, qui haud ea$-
dem habent dimen$iones in datâ ea$dem præbeant in re$ultanti æqua-
tione: vel ut diruat transformatio omnes re$ultantis æquationis ter-
minos, in quibus haud eædem inveniuntur dimen$iones.</p>
<p>Ex. 1. Sit æquatio <math>x^. √(a x^2 + b z^3) = z^2 z^.</math>; $upponantur quanti-
tates <math>z^3 & x^2</math> ea$dem habere dimen$iones, i. e. $it <math>z^3 = y^2</math> quantitati
a$$umptæ, quæ ea$dem habet dimen$iones ac <math>x^2</math>; & exinde <math>z^2 z^. = {2 / 3} yy^.</math>;
$ub$tituantur pro <math>z^3 & z^2 z^.</math> in datâ æquatione <math>x^. √(a x^2 + b z^3) =
z^2 z^.</math> earum valores <math>y^2 & {2 / 3} y y^.</math>, & re$ultabit æquatio <math>x^. √(a x^2 + b y^2)
= {2 / 3} y y^.</math>, cujus termini manife$to ea$dem habent dimen$iones.</p>
<p>Ex. 2. Sit æquatio <math>x^3 x^. + {x^2 y^. / √(a + y)} = y^.</math>; ut liberetur data æqua-

<pb 230>[0282]<rh>DE FLUXIONALIBUS</rh>
tio a termino, cujus dimen$iones haud $unt æquales, fingatur <math>√(a +
y) = v</math>, & re$ultat <math>v^2 - a = y</math>, & exinde <math>2 v v^. = y^.</math>, unde deducitur
æquatio <math>x^3 x^. + 2 x^2 v^. = 2 v v^.</math>; nunc $upponantur <math>x^3 x^. & 2 x^2 v^.</math> ea$-
dem habere dimen$iones, & con$equenter <math>v</math> duplas habere dimen$io-
nes quantitatis <math>x</math>; a$$umatur igitur pro <math>v</math> quantitas <math>u^2</math> duarum di-
men$ionum, quâ $ub$titutâ pro <math>v</math> in datâ æquatione, re$ultabit homo-
genea æquatio <math>x^3 x^. + 4 x^2 u u^. = 8 u^3 u^.</math>.</p>
<p>Ex. 3. Sit æquatio <math>a y^n x^m x^. + b y^q x^p x^. + e x^r y^s y^. = 0</math>, $i haud $int
<math>n + m = q + p = r + s</math>; $cribatur <math>y = z^t</math>, unde re$ultabit <math>a z^nt x^m x^.
+ b z^q t x^p x^. + t e z^(s+1)t-1 x^r z^. = 0</math>; fingatur, $i modo po$$ibile $it, <math>n t +
m = q t + p = (s + 1) t + r - 1</math>, & re$ultabit homogenea æquatio.</p>
<p>Ex. 4. Sit data æquatio <math>a y y^. + b y x^. + c x y^. + d x x^. + e y^. + f x^.
= 0</math>; $i vero <math>b = c</math>, tum fluens datæ æquationis erit <math>{a y^2 / 2} + c y x +{d x^2 / 2} + e y + f x = A</math> con$t. quant. Si vero <math>b</math> haud $it <math>= c</math>; $criban-
tur <math>v + h & z + k</math> re$pective pro <math>y & x</math>, & earum fluxiones <math>v^. & z^.</math>
pro <math>y^. & x^.</math> in datâ æquatione; & re$ultat <math>a v v^. + b v z^. + c z v^. + d z z^.
+ (a b + c k + e) v^. + (b b + d k + f) z^. = 0</math>; fiant coefficientes <math>a b
+ c k + e & b b + d k + f</math> terminorum (<math>v^. & z^.</math>) nihilo re$pective
æquales, & re$ultant <math>a b + c k + e = 0 & b h + d k + f = 0</math>, unde
<math>b = {d e - c f / c b - d a} & k = {f a - b e / c b - d a}</math>, & æquatio re$ultans <math>a v v^. + b v z^. +
c z v^. + d z z^. = 0</math> erit homogenea.</p>
<p>Si vero <math>c b - d a = 0</math>, tum per hanc methodum haud reduci pote$t
data æquatio in homogeneam; in hoc ca$u erit <math>f x^. + (b y + d x) x^. +
n (b y + d x) y^. + e y^. = 0</math>, ponatur <math>b y + d x = z</math>, & erunt <math>{y^. / x^.} = -{f + z / e + n z} & y^. = {z^. - d x^. / b}</math>, unde <math>x^. = {z^. (e + n z) / - b f + e d + (- b + n d) z}</math>.</p>
<p>Ex. 5. Sit <math>(a + b x y + c x^2 y^2 + d x^3 y^3 + &c.) x^. + x^2 (l + m x y
+ n x^2 y^2 + &c.)y^. = 0</math>, vel <math>(a y + b x y^2 + c x^2 y^3 + &c.)x^. + (l x +

<pb 231>[0283]<rh>ÆQUATIONIBUS.</rh>
m x^2 y + n x^3 y^2 + &c.)y^. = 0</math>; $cribatur <math>z^-1</math> in his æquationibus pro
<math>y</math>, & exponentes omnium terminorum evadent eædem.</p>
<p>Si vero generaliter <math>(a x^π + b x^π-1 y^τ + c x^π-2 y^2τ + &c.) x^. + (l′ x^π y^τ-1
+ mx^π-1 y^2τ-1 + n x^π-2 y^3τ-1 + &c.) y^. = 0</math>; $cribatur <math>z</math> pro <math>y^τ</math>, & re-
$ultat æquatio homogenea.</p>
<p>Ex. 6. Sit <math>y^. + y^2 x^. = {a x^. / x^2}</math>, $cribatur <math>{1 / z}</math> pro <math>y</math> in data æquatione, &
re$ultat æquatio homogenea <math>x^2 z^. = x^. (x z - a z^2)</math>.</p>
<p>Ex. 7. Sit æquatio <math>(a x - b c y + b y^2) x^. + (d c x - 2 d x y +
c^2 y - 3 c y^2 + 2 y^3) y^. = 0</math>; $ub$tituatur pro <math>y</math> ejus valor a$$umptus
<math>{c ± √(c^2 + a z) / 2}</math>, & re$ultabit æquatio homogenea huju$ce formulæ
<math>α x x^. + β z x^. + γ x z^. + δ z z^. = 0</math>.</p>
<p>Et $imiliter progredi liceat in reductione datarum æquationum in
alias, quarum $inguli termini vel $int quantitates <math>n, n-1, n-2, &c.</math>
dimen$ionum.</p>
<p>Si tres vel plures (<math>x, y, z, &c.</math>) variabiles quantitates in datâ ho-
mogeneâ æquatione contineantur; pro literis <math>y, z, &c.</math> $cribantur re-
$pective <math>p x, q x, &c.</math> deinde per methodum, quæ de æquationibus duas
variabiles quantitates habentibus fuit tradita, progrediendum e$t.</p>
<h>PROB. LV.</h>
<h it>Invenire æquationes, quas reducere liceat ad homogeneas æquationes.</h>
<p>A$$umantur æquationes homogeneæ, $ub$tituantur quæcunque
functiones novarum quantitatum pro variabilibus quantitatibus in
a$$umptis æquationibus contentis; & re$ultant æquationes, quæ reduci
po$$unt ad homogeneas æquationes.</p>
<p>Cor. Datâ homogeneâ æquatione duas variabiles quantitates <math>x & y</math>
& earum fluxiones habente; pro <math>x</math> vel <math>y</math> $cribatur <math>{r / z^s}</math> in datâ æqua-
tione, & re$ultat æquatio; quæ facile reduci pote$t in priorem ex

<pb 232>[0284]<rh>DE FLUXIONALIBUS</rh>
æquatis terminis re$ultantis æquationis, quorum dimen$iones haud
videntur e$$e eædem.</p>
<p>Et $ic de infinitis aliis huju$cemodi æquationibus deducendis, quæ
vel in homogeneas æquationes vel in alias cuju$cunque datæ formulæ
reduci po$$unt.</p>
<p>2. Sit data fluxionalis homogenea æquatio (<math>n</math>) ordinis, quantitates
<math>x & y</math>, & earum fluxiones <math>x^., y^., ÿ, y^..., ..., y^. ^n-1 & y^. ^n</math> involvens, ubi <math>x</math> fluit
uniformiter: in eâ pro <math>y, y^., ÿ, ..., y^. ^n-1 & y^. ^n</math> $cribantur re$pective <math>v x,
v′ x^., {v″ x^.^2 / x}, {v′″ x^.^3 / x^2}, {v″″ x^.^4 / x^3}, ..., {v′^n-1 x^.^n-1 / x^n-2} & {v′^n x^.^n / x^n-1}</math>, & re$ultat æquatio algebraica
(<math>A = 0</math>) relationem inter quantitates <math>v, v′, v″, v′″, ... v′^n-1 & v′^n</math> ex-
primens: deinde, ob <math>y = v x</math>, erit <math>v x^. + x v^. = y^. = v′ x^.</math>, & exinde <math>{x^. / x}= {v^. / v′ - v}</math>; & $imiliter ob <math>y^. = v′ x^.</math>, erit <math>y^..; = v^.′ x^. = {v″ x^.^2 / x}</math>, unde <math>{x^. / x} ={v^.′ / v″}</math>; & denique <math>{x^. / x} = {v^. / v′ - v} = {v^.′ / v″} = {v^.″ / v′″ + v″} = {v^.′″ / v″″ + 2 v′″} = {v^..″″ / v″′″ + 3 v″″}= &c. = {v^.′^n-1 / v′^n + (n-2)v′^n-1}</math>: hæ autem $unt (<math>n-1</math>) diver$æ æquationes, viz.
<math>{v^. / v′ - v} = {v^.′ / v″} = &c.</math>, in quibus nec continetur <math>x</math> nec <math>x^.</math>: continentur
autem $olummodo <math>v, v′, v″, v′″, ..., v′^n-1 & v′^n</math>; & fluxiones <math>v^., v^.′, v^.″,
... v^.′^n-2 & v^.′^n-1</math>; deinde ex datâ algebraicâ æquatione <math>A = 0</math> inveniatur <math>v′^n</math>
in terminis reliquarum <math>v, v′, v″, v′″, v′^n-2 & v′^n-1</math>; & $ub$tituatur hæc quan-
titas in ultimâ æquatione <math>{v^.′^n-2 / v′^n-1 + (n-3)v′^n-2} = {v^.′^n-1 / v′^n + (n-2)v′^n-2}</math> pro ejus
valore <math>v′^n</math>; & re$ultant (<math>n-1</math>) fluxionales æquationes primi ordinis

<pb 233>[0285]<rh>ÆQUATIONIBUS.</rh>
involventes (<math>n</math>) quantitates <math>v, v′, v″, ..., v′^n-1</math> & earum fluxiones primi
ordinis: & exinde, $i modo reducantur hæ (<math>n-1</math>) æquationes in
unam, ita ut exterminentur omnes præter qua$cunque duas variabiles
& earum fluxiones, per prob. 26. re$ultabit fluxionalis æquatio invol-
vens duas prædictas variabiles quantitates & earum fluxiones ordinis,
qui non major e$t quam (<math>n-1</math>).</p>
<p>3. Sint duæ homogeneæ fluxionales æquationes primi ordinis invol-
ventes variabiles (<math>x, y&z</math>) & earum primas fluxiones: in his duabus æqua-
tionibus pro <math>y & z</math> $cribantur re$pective <math>v x & w x</math>, & pro <math>y^. & z^.</math> $ub$titu-
antur <math>v′x^. & w′x^.</math>, & re$ultabunt duæ algebraicæ æquationes <math>A = 0 & B = 0</math>
quatuor variabiles <math>v & w, v′ & w′</math> involventes: ex præcedente methodo
deduci pote$t <math>{x^. / x} = {v^. / v′ - v} = {w^. / w′ - w}</math>; deinde in æquatione <math>{v^. / v′ - v} ={w^. / w′ - w}</math> pro <math>v′ & w′</math> $cribantur re$pective functiones quantitatum <math>v &
w</math> ex æquationibus <math>A = 0 & B = 0</math> petitæ; & re$ultabit fluxionalis
æquatio primi ordinis relationem inter <math>v & w</math> & earum fluxiones ex-
primens.</p>
<p>4. Sint duæ, tres vel <math>m</math> homogeneæ fluxionales æquationes ordinis
(<math>n</math>), tres vel quatuor vel (<math>m+1</math>) variabiles quantitates (<math>x, y, z, &c.</math>)
involventes; pro <math>y, y^., ÿ, ..., y^.^n-1 & y^.^n</math> $cribantur <math>v x, v′ x^., {v″ x^.^2 / x}, ... {v′^n-1 x^.^n-1 / x^n-2}& {v′^n x^.^n / x^n-1}</math>; & pro <math>z, z^., z^.., ..., z^.^n</math> $ub$tituantur <math>w x, w′ x^., {w″ x^.^2 / x}, ... {w′^n x^.^n / x^n-1};
&c. &c.</math> & re$ultabunt (<math>m</math>) algebraicæ æquationes <math>(m) (n+1)</math> va-
riabiles quantitates habentes; & per methodum prius traditam de-
duei po$$unt æquationes <math>{x^. / x} = {v^. / v′ - v} = {v^.′ / v″} = {v^.″ / v′″ + v″} = &c. =

<pb 234>[0286]<rh>DE FLUXIONALIBUS</rh>
{v^.′<_>n-1</_> / v′<_>n</_> + (n - 2)v′<_>n-1</_>} = {w^. / w′ - w} = {w^.′ / w″} = {w^. ″ / w′″ + w″} = &c. = {w^. ′<_>n-1</_> / w′<_>n</_> + (n - 2)w′<_>n-1</_>}
= &c.</math>: in his æquationibus pro <math>v′^n, w′^n, &c.</math> $cribantur earum valo-
res; & re$ultabunt <math>m × n - 1</math> diver$æ fluxionales æquationes primi or-
dinis involventes <math>m × n</math> variabiles quantitates; $i hæ æquationes in
unam reducantur, ut exterminentur omnes præter duas variabiles
quantitates & earum fluxiones; tum re$ultabit fluxionalis æquatio,
cujus ordo non major e$t quam <math>m n - 1</math>.</p>
<p>5. Ex ii$dem principiis & reductione æquationum facile con$tat
($i modo dentur (<math>m</math>) homogeneæ fluxionales æquationes ordinum (<math>r,
s, t, &c.</math>); & reducantur per methodum prius traditam æquationes
deductæ in unam, ita ut exterminentur (<math>m - 1</math>) variabiles quantitates
& earum fluxiones) æquationem re$ultantem habere ordinem, qui
non major e$t quam <math>r + s + t + &c. - 1</math>.</p>
<h>THEOR. XXXVIII.</h>
<p>Defin. Æquatio relationem inter <math>x & y</math> & earum fluxiones expri-
mens dicitur homogenea, $i non $olum variabilibus <math>x & y</math>, $ed etiam
$ingulis earum fluxionibus <math>x^. & y^.</math>, itemque <math>y^.., y^... &c.</math> unam dimen$ionem
obtinentibus, omnes æquationis termini eundem dimen$ionum nu-
merum contineant.</p>
<p>Hinc; $i <math>y^. = p x^. & {p^. / x^.} = q</math>, litera <math>p</math> nullam habet dimen$ionem,
litera <math>q</math> unam habet dimen$ionem negativam, &c.</p>
<p>Ex ii$dem principiis, viz. $ub$titutionibus, fluxionales æquationes
$uperiorum ordinum reduci po$$unt ad fluxionales inferiorum ordi-
num æquationes.</p>
<p>Ex. 1. Datam æquationem fluxionalem $ecundi ordinis, quæ $it
homogenea, ad æquationem fluxionalem primi ordinis reducere.</p>
<p>Sint <math>y^. = p x^. & p^. = q x^.</math>, & habeatur æquatio inter quatuor quan-

<pb 235>[0287]<rh>ÆQUATIONIBUS.</rh>
titates <math>x, y, p & q</math> relationem exprimens: fingatur <math>y = u x & q = {v / x}</math>,
quibus valoribus pro <math>y & q</math> in datâ æquatione $ub$titutis, re$ultat
æquatio relationem inter <math>u, v & p</math> exprimens, ex quâ unam per duas
reliquas definire liceat: cum autem $it <math>y^. = p x^.</math>, erit <math>u x^. + x u^. = p x^.
& {x^. / x} = {u^. / p - u}</math>; deinde <math>p^. = q x^. = {v x^. / x}</math>, erit <math>{x^. / x} = {p^. / v}</math>, unde <math>{p^. / v} = {u^. / p - u}& p p^. - u p^. = v u^.</math>: $ed e præcedente æquatione inveniatur <math>v</math> in ter-
minis quantitatum <math>p & u</math>, $ub$tituatur valor inventus pro <math>v</math> in præ-
dictâ æquatione <math>p p^. - u p^. = v u^.</math>, & re$ultabit æquatio relationem
inter variabiles <math>p & u</math> & earum primas fluxiones exprimens.</p>
<p>Ex. 2. Si æquatio fluxionalis evadat homogenea, cum alteri varia-
bili <math>y</math> tribuantur <math>n</math> dimen$iones, tum ejus re$olutio eadem erit ac æqua-
tionis in ex. præced. contentæ; $cribatur enim <math>w^n</math> pro <math>y</math> & re$ultabit
æquatio homogenea; vel quod ad idem redit, fingantur <math>y = x^n u, y^. =
x^n-1 t x^. & y^.. = x^n-2 v x^.^2</math>; & ob <math>y^. = p x^. & p^. = q x^.</math>, erit <math>x u^. + n u x^. = t x^.,
& x t^. + (n - 1) t x^. = v x^.</math>, unde <math>{x^. / x} = {u^. / t - n u} = {t^. / v - (n - 1) t}</math>: $ub-
$tituantur etiam in datâ æquatione pro <math>y, p & q</math> prædicti valores, &
re$ultabit æquatio relationem exprimens inter <math>u, t & v</math>; ex quâ inve-
niri pote$t <math>v</math> in terminis quantitatum <math>t & u</math>; $cribatur hic valor in-
ventus pro <math>v</math> in æquatione <math>{u^. / t - n u} = {t^. / v - (n - 1) t}</math>, & re$ultabit æ-
quatio quæ$ita.</p>
<p>Sit <math>n = 0</math>, i. e. $i quantitati <math>y</math> & ejus fluxionibus nullæ tribuantur
dimen$iones, data æquatio evadet homogenea; hoc $cilicet ca$u vari-
abilis <math>x</math> cum $uis fluxionibus in $ingulis terminis ea$dem con$tituunt
dimen$iones.</p>
<p>Sit <math>n</math> infinitus, & dimen$iones e variabili <math>y</math> & ejus fluxionibus in $in-
gulis terminis evadent eædem.</p>
<p>Ex. gr. Sit <math>y^.. = c x^α y^β x^.^2-γ y^.^γ</math>, ejus fluentem invenire.</p>
<p>Quoniam <math>y^. = p x^., & y^.. = q x^.^2</math>, erit <math>y^.. = c x^α y^β p^γ x^.^2</math>; fingantur <math>y = x^n u,

<pb 236>[0288]<rh>DE FLUXIONALIBUS</rh>
y^. = x^n-1 t x^. & y^.. = x^n-2 v x^.^2</math>, & evadet <math>x^n-2 v = c x^α+nβ+γn-γ u^β t^γ</math>, unde
<math>n = {γ - α - 2 / β + γ - 1}</math>, & exinde <math>v = c u^β t^γ; & {u^. / t - n u} = {t^. / c u^β t^γ - (n - 1) t}</math>
æquat. quæ$it. primi ordinis.</p>
<p>Si <math>n</math> $it infinitus, $cribatur <math>y^. = p x^. & y^.. = q x^.^2</math>, & æquatio ita erit
comparata, ut in eâ tres variabiles <math>y, p & q</math> ubique eundem dimen-
$ionum numerum obtineant: $tatuantur <math>p = u y & q = v y</math>, quibus
quantitatibus pro $uis valoribus in datâ æquatione $ub$titutis, re$ul-
tabit æquatio relationem inter variabiles <math>x, u & v</math> exprimens, ex quâ
inveniatur <math>v</math> in terminis duarum reliquarum <math>u & x</math>: $ed quoniam
<math>p = u y</math> erit <math>y^. = p x^. = u y x^.</math>, & quoniam <math>y^.. = q x^.^2</math>, erit <math>y^.. = v y x^.^2 =
x^.(u y^. + y u^.)</math>; ex his æquationibus $equitur <math>{y^. / y} = u x^. = {v x^. - u^. / u}</math>, unde
<math>u^2 x^. + u^. = v x^.</math>, $ed invenitur valor quantitatis <math>v</math> in terminis quanti-
tatum <math>u & x</math>, $cribatur hic valor pro <math>v</math> in æquatione <math>u^2 x^. + u^. = v x^.</math>,
& re$ultat æquatio quæ$ita.</p>
<p>Si in æquatione <math>u^2 x^. + u^. = v x^.</math> $ub$tituantur <math>{y^. / y x^.} & {y^.. / y x^.} - {y^.^2 / y^2 x^.}</math> (ubi
<math>x^.</math> $it con$tans) pro $uis valoribus <math>u & u^.</math>, re$ultabit æquatio <math>{y^.. / y x^.} = v x^.</math>.</p>
<p>Facile con$tant ca$us, in quibus fluens fluxionalis æquationis inve-
niri pote$t ex æquatione re$ultante, &c.</p>
<p>Ex. 3. Si in datâ æquatione relationem inter <math>x & y</math> & earum flu-
xiones de$ignante, in quâ fluit uniformiter <math>x</math>, de$it altera incognita
quantitas <math>y</math>; $cribatur in datâ æquatione pro <math>y^.</math> ejus valor a$$umptus
<math>p x^.</math>, & pro fluxione <math>y^.. = p^. x^.</math>, pro <math>y^... = p^.. x^.</math> & $ic deinceps; & re$ultabit
æquatio, in quâ ordo fluxionum minor unitate quam in datâ, i. e. $i
data contineat $ecundas fluxiones, tum re$ultans continebit $olum-
modo primas; &c.</p>
<p>In genere $i <math>A</math> $it quantitas, quæ fluit uniformiter, $upponatur <math>x^. =
p A^.</math>, unde <math>x^.. = p^. A^., x^... = p^.. A^.</math> & $ic deinceps; quibus quantitatibus pro

<pb 237>[0289]<rh>ÆQUATIONIBUS.</rh>
$uis valoribus in datâ æquatione $ub$titutis re$ultabit æquatio, in quâ
ordo fluxionis quantitatis <math>p</math> minor erit unitate quam ordo quantita-
tis <math>x</math> in datâ æquatione inventus. Si nullus terminus in datâ æqua-
tione a$$umatur tanquam con$tans, tum a$$umi pote$t quæcunque
quantitas, cujus fluxio $it con$tans; $ed præ$tat a$$umere talem quan-
titatem fluentem uniformiter, quâ $cilicet a$$umptâ deleantur plurimæ
quantitates e datâ æquatione; vel ita transformentur quantitates, ut
exinde ejus fluens inve$tigari po$$it; vel reducatur æquatio in formu-
las prædictas, quarum re$olutiones docentur.</p>
<p>Æquationes, quæ haud apparent $ub hâc formulâ, e $ub$titutione
nonnunquam in eam reduci po$$unt, e.g. $it æquatio <math>x^m x^.. = y y^.. + y^. ^2 +
y^. ^2 y^2</math>, $cribatur <math>y y^. = z^.</math>, & re$ultat <math>x^m x^.. = z^.. + z^. ^2</math>.</p>
<p>Ex. 4. Si duæ variabiles (<math>x & y</math>) quantitates in datâ æquatione
contentæ $imul cum earum fluxionibus $emper in $ingulis terminis
ea$dem conficiant dimen$iones; & <math>x^.</math> $it con$tans quantitas; a$$uman-
tur <math>x = e^u & y = e^u t</math>, ubi <math>e = num</math>. cujus log. e$t 1; tum <math>x^. = e^u u^.
& x^.. = e^u (u^. ^2 + u^..) = 0</math>, quoniam <math>x^.</math> e$t con$tans, erit <math>u^.. = - u^. ^2;
y^. = e^u (t u^. + t^.),</math> & <math>y^.. = e^u (t^.. + 2 u^. t^. + t u^.. + t u^. ^2) = e^u (t^... + 2 u^. t^.)</math>,
quibus quantitatibus pro $uis valoribus in datâ æquatione $ub$titutis
re$ultabit æquatio, in quâ haud invenitur <math>u</math>.</p>
<p>Hoc in loco ob$ervandum e$t, nec quantitatem <math>t</math> nec <math>u</math> in æquatione
re$ultante fluere uniformiter, $ed <math>e^u</math>.</p>
<p>Ex. 5. Si variabilis <math>y</math> cum $uis fluxionibus <math>y^. & y^..</math> ubique eundem di-
men$ionum numerum adimpleat in datâ fluxionali æquatione, eam
in fluxionalem æquationem primi ordinis reducere.</p>
<p>Sint <math>y^. = p x^. & p^. = q x^.</math>, unde variabiles <math>y, p & q</math> ubique eundem
dimen$ionum numerum obtineant: $tatuantur <math>p = u y & q = v y &</math>
$ub$tituantur hæ quantitates in datâ æquatione pro $uis valoribus, &
re$ultabit æquatio relationem inter <math>x, u & v</math> exprimens, e quâ invenia-
tur <math>v</math> functio quantitatum <math>x & u</math>, ob <math>p = u y</math>, erit <math>y^. = u y x^.</math>, & exinde
<math>u x^. = {y^. / y} = {v x^. - u^. / u}</math>, unde <math>u^. + u^2 x^. = v x^.</math> æquatio quæ$ita.</p>

<pb 238>[0290]<rh>DE FLUXIONALIBUS</rh>
<p>Cor. · Ex formulis integralium æquationum prius traditis con$tant
formulæ huju$ce generis integrabiles.</p>
<p>Ex. 6. Si variabilis <math>y</math> cum $uis fluxionibus <math>y^., y^.., &c. y^. ^n</math> ubique eundem
dimen$ionum numerum adimpleat in datâ æquatione, eam in fluxio-
nalem æquationem (<math>n - 1</math>) ordinis reducere. Sub$tituantur pro, <math>y,
y^., y^.., y^..., &c.</math> in datâ æquatione re$pective <math>e^$. u x^., u x^. e^$. u x^., (u^2 x^. ^2 + u^. x^.)
e^$. u x^., (u^3 x^.^3 + 3 u u^. x^.^2 + u^.. x^.) e^$. u x^.</math>, &c. ubi <math>x^.</math> e$t con$tans, & re$ulta-
bit fluxionalis æquatio (<math>n - 1</math>) ordinis. Hoc etiam perfici pote$t e
præcedentibus principiis, quæ omnino eadem $unt ac ea hoc in loco
data.</p>
<p>E. g. Sit æquatio <math>y^.. + P x^. y^. + Q y x^. ^2 = 0</math>, $cribatur <math>y = e^$. u x^.</math>, unde
<math>y^. = e^$. u x^. u x^. & y^.. = e^$. u x^. (u^. x^. + u^2 x^. ^2)</math>; $ub$tituantur hæ quantitates
pro $uis valoribus in datâ æquatione, & re$ultabit <math>u^. + u^2 x^. + P u x^. +
Q x^. = 0</math>.</p>
<p>Cor. 1. Scribatur in æquatione (<math>u^. + u^2 x^. + P u x^. + Q x^. = 0</math>) M <math>z</math>
pro <math>u</math> & re$ultabit æquatio <math>M z^. + z (M^. + P M x^.) + M^2 z^2 x^. + Q x^.
= 0</math>, a$$umatur <math>M^. + P M x^. = 0</math>, & exinde <math>M = a e^-$. P x^.</math>, ubi <math>a</math> e$t
con$tans quantitas; & æquatio re$ultans <math>a e^- $. P x^. z^. + a^2 e^- 2 $. P x^. z^2 x^.
+ Q x^. = 0</math>.</p>
<p>Eodem modo $cribantur pro <math>u</math>; vel <math>{M / z}</math>, vel <math>K + M z</math>, vel <math>{K + M z / L + N z}</math>,
&e. in datâ æquatione, & re$ultantis æquationis fiant quicunque ter-
mini nihilo æquales, ita vero ut æquatio exinde re$ultans evadat ma-
xime $imples; &c.</p>
<p>Paulo aliter in quibu$dam ca$ibus progredi licet: e. g. $it æquatio
<math>y^.. - {n y^. x^. / x} + A x^n y^. x^. + B x^2n y x^.^2 = 0</math>; ponatur <math>y^. = x^n y u x^.</math>, $cribatur
<math>e^$. x^n u x^.</math> pro <math>y</math>, & ejus fluxio pro <math>y^.</math>, & $ic deinceps in datâ æquatione;
& re$ultabit <math>u^. + x^n u^2 x^. + A x^n u x^. + B x^n x^. = 0</math>, unde <math>x^n x^. = {- u^. / u^2 + A u + B}</math>;
$int radices æquationis <math>u^2 + A u + B = 0</math> re$pective <math>α & β</math>, tum erit

<pb 239>[0291]<rh>ÆQUATIONIBUS.</rh>
<math>y = a e^α $. x^n x^. + b e^β $. x^n x^.</math>, ubi <math>a & b</math> $unt quæcunque invariabiles
quantitates. Con$tat e $ub$titutione.</p>
<p>Facile deduci po$$unt formulæ æquationum, quarum ex his $ub$ti-
tutionibus <math>y = e^$. u x^.</math>, vel <math>y = e^u t & x = e^r u, &c.</math> fluentes deduci po$-
$unt: a$$umantur enim formulæ æquationum, quarum fluentes den-
tur, & exinde inve$tigentur æquationes, quæ prædictis $ub$titutio-
nibus præbent æquationes formularum a$$umptarum: e. g. fluentes
fluxionalium æquationum primi ordinis, in quibus dee$t una variabilis
quantitas, vel quæ in hanc formulam reducantur, inveniri po$$unt; &c.
& e priori $ub$titutione patet omnem æquationem, cujus $inguli ter-
mini habeant formulam <math>A y^n y^. ^m y^.. ^p x^. ^q</math> (ubi in $ingulis <math>A, n, m, & p</math> $unt
invariabiles & <math>n + m + p</math> eadem quantitas) tran$ire in æquationem
fluxionalem primi ordinis, in quâ dee$t <math>x</math>: deinde e $ub$titutione po-
$teriori con$tat omnem æquationem, cujus primus terminus $it <math>y^..</math>, cæ-
teri vero habeant formulam <math>A y^n y^. ^m x^. ^2-m</math>, in quâ <math>A, n & m</math> $unt inva-
riabiles, & <math>n + m + r (2 - m) = 1</math>, (ubi <math>r & 2 - m</math> in $ingulis terminis
eadem manet), tran$ire in fluxionalem æquationem $ecundi ordinis, in
cujus $ingulis terminis variabilis <math>t</math> cum ejus fluxionibus ea$dem con-
ficit dimen$iones, & dee$t variabilis <math>u</math>.</p>
<p>Idem etiam affirmari pote$t, $i modo pro <math>x^.</math> in $ingulis terminis
prædictis $cribatur <math>{z^. / z}</math>, vel fluxio cuju$cunque functionis quantitatis <math>x</math>.
Et $ic de inveniendis æquationibus, quæ $ub$titutionibus prædictis
reduci po$$int ad homogeneas æquationes primi ordinis, cætera$ve
æquationes, quarum fluentes dentur.</p>
<p>Ex. 7. Sit <math>a x^m x^. ^p = y^n y^. ^p-2 y^..</math>, ubi <math>x^.</math> e$t invariabilis quantitas; finga-
tur <math>x = e^bu & y = e^u t</math>, ubi <math>e</math> $it numerus, cujus logarith. <math>= 1</math>; tum
<math>x^. = h e^bu u^., y^. = e^u t^. + e^u t u^., y^.. = e^u (t^.. + 2 u^. t^. + t u^. ^2 + t u^..)</math>; quibus
valoribus pro <math>x, x^., y^. & y^..</math> in datâ æquatione $ub$titutis, re$ultat <math>e^bu(m+p) h^p u^. ^p
= e^u(n+p-1) t^n (t^. + t u^.)^p-2 (t^.. + 2 t^. u^. + t u^.. + t u^. ^2)</math>; ut vero de$truatur
exponentialis <math>e^u</math>, $cribatur <math>h u (m + p) = u (n + p - 1)</math>, unde <math>b =

<pb 240>[0292]<rh>DE FLUXIONALIBUS</rh>
{n + p - 1 / m + p}</math>; quo valore $ub$tituto re$ultat æquatio, in quâ nec conti-
netur quantitas <math>u</math>, nec ejus functiones, viz. <math>h^p u^. ^p = t^n (t^. + t u^.)^p-2 (t^.. +
2 t^. u^. + t u^.. + t u^. ^2)</math>: $ed quoniam per hypothe$in <math>x^.. = 0</math>, erit <math>u^.. = -
h u^. ^2 = - {n + p - 1 / m + p} u^. ^2</math>, quâ quantitate pro $uo valore $ub$titutâ,
re$ultat <math>({n + p - 1 / m + p})^p u^. ^p = t^n (t^. + t u^.)^p-2 (t^.. + 2 t^. u^. + {m - n + 1 / m + p}t u^. ^2) = 0</math>.</p>
<p>E. g. Sit æquatio <math>a x^-1 x^. = y^. ^-1 y^..</math>, ejus fluens haud inveniri pote$t e
præcedente exemplo, nam <math>m = - 1 & p = 1</math>, unde <math>m + p = 0, &
({n + p - 1 / m + p})^p</math> hâc methodo haud innote$cit.</p>
<p>In hoc exemplo animadvertendum e$t nec <math>t</math> nec <math>u</math> fluere uniformi-
ter, $ed <math>e^bu</math>; unde re$ultans æquatio hâc methodo haud generalem
re$olutionem recipiet.</p>
<p>Ex. 8. Sit æquatio <math>P y^.. + Q y^. x^. + R x^. ^2 = 0</math>; ubi <math>P, Q & R</math> $unt
homogeneæ functiones <math>n, n - 1 & n - 1</math> dimen$ionum literarum <math>x
& y</math> re$pective; $cribatur <math>y = x z</math>, & re$ultabunt <math>y^. = x z^. + z x^. & y^.. =
x z^.. + 2 z^. x^.</math>, $i modo <math>x</math> fluat uniformiter; quibus quantitatibus pro
$uis valoribus in datâ æquatione $ub$titutis, re$ultat æquatio <math>p x^2 z^..
+ q x z^. x^. + r x^. ^2 = 0</math>, ubi literæ <math>p, q & r</math> re$pective denotant functio-
nes literæ <math>z</math>.</p>
<p>9. Sit æquatio <math>P y^. ^n + Q y^. ^n-1 + R y^. ^n-2 + &c. = 0</math>, ubi literæ <math>P, Q, R,
&c.</math> $unt homogeneæ functiones <math>m, m - 1, m - 2, .... m - n + 1</math>,
dimen$ionum literarum <math>x & y</math> re$pective; fingatur <math>y = x z</math>, & exinde
<math>y^. = x z^. + z x^. & y^.. = x z^.. + 2 z^. x^., y^... = x z^... + 3 x^. z^..</math>, & $ic deinceps;
quibus quantitatibus pro $uis valoribus in datâ æquatione $ub$titutis,
re$ultat æquatio <math>p x^n z^. ^n + q x^n-1 z^. ^n-1 x^. + r x^n-2 z^. ^n-2 x^. ^2 + &c. = 0</math>, ubi li-
teræ <math>p, q, r, &c.</math> re$pective $unt functiones literæ <math>z, &c.</math></p>
<p>Sint <math>x & y</math> variabiles quantitates in datâ æquatione contentæ;

<pb 241>[0293]<rh>ÆQUATIONIBUS.</rh>
cavendum e$t, ne in $ub$titutionibus pro <math>x & y</math> $cribantur tales $un-
ctiones quantitatum a$$umptarum <math>z & v</math>, quales reddant quantitatem
<math>x</math> exinde deductam, pendentem e quantitate <math>y</math>; aliter in errores ut
nos inducamur probabile e$t: e. g. a$$umantur pro <math>x & y</math> quæcun-
que homogeneæ diver$æ functiones nullius dimen$ionis quantitatum
a$$umptarum <math>z & v</math>, $cribantur hæ functiones pro $uis valoribus in
datâ æquatione, & re$ultat homogenea æquatio, unde reducitur omnis
æquatio ad homogeneam; $ed in hoc ob$ervandum e$t, quod a$$ump-
tio duarum diver$arum homogenearum functionum nullius dimen-
$ionis præ$upponit quandam relationem inter quantitates <math>x & y</math> ex-
i$tentem.</p>
<h>PROB. LVI.</h>
<p it>Propo$itâ æquatione fluxionali, cujus fluens ex introductione novarum
variabilium pro iis in datâ æquatione contentis detegi pote$t; invenire
multiplicatorem, qui ductus in propo$itam æquationem, præbeat fluxionem,
cujus fluens inveniri pote$t.</p>
<p>Sit <math>P x^. + Q y^. = 0</math> propo$ita æquatio, quæ per $ub$titutionem re-
ducatur ad æquationem <math>P x^. + Q y^. = R t^. + S u^.</math>, ubi duæ variabiles
<math>t & u</math> introducuntur loco ip$arum <math>x & y</math>; dividatur æquatio <math>R t^. + S u^.
= 0</math> per <math>V</math>, cum igitur fluens fluxionis <math>{R t^. + S u^. / V}</math> inveniri po$$it, tum
etiam fluens fluxionis <math>{P x^. + Q y^. / V} = {R t^. + S u^. / V}</math> inveniri pote$t, unde
<math>{1 / V}</math> erit multiplicator, qui in datam æquationem ductus præbet fluxio-
nem, cujus fluens inveniri pote$t: in multiplicatore <math>{1 / V}</math> pro variabili-
bus <math>u & t</math> re$tituantur variabiles <math>x & y</math>, & re$ultat multiplicator quæ-
$itus.</p>
<p>Cor. Hinc multæ æquationes inveniri po$$unt, quarum multipli-

<pb 242>[0294]<rh>DE FLUXIONALIBUS</rh>
catores innote$cunt; in æquationibus, quarum multiplicatores dantur,
pro variabilibus, &c. $cribantur quæcunque functiones aliarum varia-
bilium quantitatum & earum fluxiones re$pective; & exinde deduci
po$$unt fluxionales æquationes, quarum multiplicatores detegi
po$$unt.</p>
<h>THEOR. XXXIX.</h>
<p>A$$umptâ generali relatione inter variabiles datarum æquationum
quantitates & novas; & his novis, pro earum valoribus in datis æqua-
tionibus generaliter expre$$is, fub$titutis; $æpe ita a$$umi po$$unt co-
efficientes vel indices in generali relatione vel æquationibus, ut eva-
ne$cant quidam termini, & exinde re$ultet æquatio, cujus integratio
datur.</p>
<p>Ex. 1. Sit <math>x^. = a x^m y^n y^. + b x^e y^b y^. + c x^g y^l y^. + &c.</math> fingatur <math>u =
y^z x^β</math>; ubi <math>α & β</math> $unt indeterminatæ quantitates; & erit integrabilis
data æquatio, cum <math>- {α / β} - 1 = {- m α / β} + n = {- e α / β} + h = {- g α / β}+ l, &c. vel {n + 1 / m - 1} = {h + 1 / e - 1} = {l + 1 / g - 1}, &c. &c.</math></p>
<p>Ex. 2. Sit æquatio <math>a y^l x^β x^. + b y^n x^γ x^. = y^.</math>; $cribatur <math>y = e x^π</math> &
re$ultat <math>a e^l x^π 1+β x^. + b e^n x^π n+γ x^. = π e x^π-1 x^.</math>; ut hi termini $e$e mu-
tuo de$truant, nece$$e e$t $upponere <math>π l + β = π n + γ = π - 1,</math> e
quibus æquationibus con$tabit <math>π = {γ - β / l - n} = {γ + 1 / 1 - n}</math>, & exinde <math>β ={l γ + l - n - γ / n - 1}; & a e^l-1 + b e^n-1 = {γ - β / l - n}</math>; & $i igitur <math>β ={l γ + l - n - γ / n - 1}, & a e^l-1 + b e^n-1 = {γ - β / l - n}</math> in datâ æquatione, erit
etiam <math>y = e x^{γ+1 / 1-n}</math>, $ed non erit ejus <math>y</math> generalis valor.</p>
<p>Ex. 3. Æquationem <math>a x^r x^. + b x^s y^2 x^. = y^.</math> in æquationem formulæ

<pb 243>[0295]<rh>ÆQUATIONIBUS.</rh>
<math>a x^m x^. + b y^2 x^. = y^.</math> reducere; $cribatur <math>x^s+1 = z</math>, & exinde <math>x = 1/z^{1 / s+1}</math>;
qui valor in datâ æquatione $ub$tituatur, & re$ultat <math>{a / s + 1} z^{r+1 / s+1}-1 z^. +{b / s + 1} y^2 z^. = y^.</math> æquatio formulæ quæ$itæ.</p>
<p>Si <math>s + 1 = 0</math>, fallit hæc methodus; in quo ca$u pro <math>x</math> $cribatur
<math>e^z</math>, & re$ultat æquatio prædictæ formulæ.</p>
<p>Ex. 4. Sit data æquatio <math>(P), (a + b x^n) x^2 z^. + (c + f x^n) x z x^. +
(a + b x^n) x^2 z^2 x^. + (g + h x^n) x^. = 0</math>; $cribantur <math>T y + S & T y^. + y T^. + S^.</math>
pro <math>z & z^.</math> re$pective in datâ æquatione, & re$ultat <math>(a + b x^n) x^2 T y^. +
(a + b x^n) x^2 y T^. + (a + b x^n) x^2 S^. + (c + f x^n) x T y x^. + (c + f x^n) x S x^.
+ (a + b x^n) (T y + S)^2 x^2 x^. + (g + h x^n) z^. = 0, (R)</math>; $upponatur
omnium terminorum $umma, in quibus continetur <math>y</math>, nihilo æqualis,
i. e. <math>(a + b x^n) x T^. + 2 (a + b x^n) x T S x^. + (c + f x^n) T x^. = 0</math>, unde
<math>{T^. / T} + 2 S x^. + {(c + f x^n) x^. / (a + b x^n) x} = 0</math>: fingatur <math>T = x^p</math>, & exinde $equitur
<math>{p x^. / x} + 2 S x^. + {(c + f x^n) x^. / (a + b x^n) x} = 0</math>; ex hâc æquatione con$tat <math>S ={- p a - p b x^n - c - f x^n / 2 x (a + b x^n)}</math>; $cribantur hæc quantitas & ejus fluxio
pro $uis valoribus <math>S & S^.</math> in æquatione <math>R</math>, & re$ultat fluxionalis æqua-
tio, cujus con$tant ca$us, qui integrationem admittunt, e ca$ibus æqua-
tionis <math>P</math>, qui integrationem recipiunt.</p>
<p>Cor. Ex his generalibus fluxionalibus æquationibus facile deduci
po$$unt æquationes magis particulares, quæ majorem habeant concin-
nitatem.</p>
<p>E. g. Sit æquatio <math>(P), (a + b x^n) x^2 z^. + (a + b x^n) x^2 z^2 x^. + (c +
f x^n) x z x^. + (g + h x^n) x^. = 0</math>, in eâ pro <math>z & z^.</math> $cribantur re$pective
<math>T y & T y^. + y T^.</math>, ubi <math>T</math> $it incognita functio quantitatis <math>x</math>; & re$ultat
æquatio <math>(Q) (a + b x^n) x^2 T y^. + (a + b x^n) y x^2 T^. + (a + b x^n) x^2 T^2 y^2 x^.
+ (c + f x^n) x T y x^. + (g + h x^n) x^. = 0</math>; $upponantur termini, in

<pb 244>[0296]<rh>DE FLUXIONALIBUS</rh>
quibus invenitur <math>y</math>, nihilo æquales, i.e. <math>(a + b x^n) y x^2 T + (c + f x^n)
x T y x^. = 0</math>, & con$equenter <math>{T^. / T} = - {c + f x^n / (a + b x^n) x}x^.</math>, cujus fluens erit
log. <math>T + {c / a} log. x + {f a - b c / n b a} log. (a + b x^n) = A</math> invariabili
quantitati ad libitum a$$umendæ, unde <math>T = {(a + b x^n)^{b c-a f / n b a} /x^{c / a}} × A</math>;
$cribatur <math>{(a + b x^n)^{b c-a f / n b a} /x^{c / a}}</math> pro <math>T</math> in re$ultante æquatione, & transfor-
matur re$ultans æquatio in $ub$equentem <math>y^. + {(a + b x^n)^{b c-a f / n b a} /x^{c / a}} Ay^2 x^. +{(g + b x^n) x^{c / a}-2 x^./A(a + b x^n)^{b c-a f / n b a}+1} = 0, (Q)</math>.</p>
<p>Cor. · Hujus æquationis facile con$tant ca$us, in quibus detegi
pote$t ejus fluens: inveniantur enim ca$us in quibus integrari pote$t
altera æquatio vel <math>P</math> vel <math>Q</math>, & exinde con$tant ca$us alterius æquatio-
nis <math>Q</math> vel <math>P</math>, qui integrationem admittunt.</p>
<p>Ex. 1. Sit <math>b c = a f, & A = 1</math>, & re$ultans æquatio erit <math>y^. + x^-{c / a} y^2 x^. +{(g + b x^n)x^{c / a}-2 x^./a + b x^n} = 0</math>; pro <math>x</math> & ejus fluxione $cribatur <math>t^{a / a-c} & {a / a - c}t^{c / a-c} t^.</math>, & exorietur <math>y^. + {a y^2 t^. / a - c} + {a (g + b t^{n a / a-c}) × t^-2 t^./(a - c)(a + b t^{n a / a-c})} = 0</math>.</p>

<pb 245>[0297]<rh>ÆQUATIONIBUS.</rh>
<h>PROB. LVII.</h>
<h it>Invenire fluentem datæ fluxionalis æquationis ex a$$umptâ æquationis
formulâ, quæ continet flucntem ip$am.</h>
<p>Inveniatur fluxio æquationis pro fluente a$$umptâ, &c. & ita re-
ducatur æquatio re$ultans, ut evadant termini datæ & re$ultantis
æquationis inter $e æquales; & id quod requiritur, fit.</p>
<p>Ex. 1. A$$umantur quantitates <math>{a x^n y^r + b x^m y^s / (a′ x^n′ y^r′ + b′ x^m′ y^s′)^p} = A</math>, vel
<math>{a x^n y^r + b x^m y^s + c y^t + d y^l + &c. / (a′x^ n′ y^r′ + b′ x^m′ y^s′ + c′ y^t′ + d′ y^l′ + &c.)^b} = A, &c.</math> pro fluente; & in-
veniatur ejus fluxio; rejiciantur earum denominatores; & re$ultant
fluxionales æquationes, quarum fluentes innote$cunt.</p>
<p>Ex. 2. Sit æquatio <math>(a x^2m-1 + b x^2m-2 y + c x^2m-3 y^2 + &c. + d x^2m-2
+ e x^2m-3 y + &c. + f x^2m-3 + g x^2m-4 y + &c. + &c. k x + l y + m′)
y^. = (A x^2m-1 + B x^2m-2 y + &c. + D x^2m-2 + E x^2m-3 y + &c.)x^.</math>.</p>
<p>Fingatur <math>(P x^m + Q x^m-1 y + R x^m-2 y^2 + &c. + S x^m-1 + T x^m-2 y
+ &c.)^α-β = M (p x^m + q x^m-1 y + r x^m-2 y^2 + &c. + s λ^m-1 + t x^m-2 y
+ &c.)^α+β</math>, cujus fluxio per fluentem ip$am dividatur, & æquationis
re$ultantis terminis corre$pondentibus inter $e æqualibus e$$e $uppo-
$itis, nonnunquam $equitur fluens particularis datæ æquationis.</p>
<p>Ex. 2. Æquatio <math>(a x + b + c y) y^. = (f x + g x + b y) x^.</math> $emper
reduci pote$t in homogeneam, cum <math>ab</math> haud æquat <math>f c</math>, $cribendo in
datâ æquatione pro <math>x & y</math> earum a$$umptos valores <math>z + α & v + β</math>;
& in æquatione re$ultante <math>(a z + (a α + b + c β) + c v)v^. = (f z +
(f α + g + b β) + b v) z^.</math> $upponendo terminos <math>a α + b + c β = 0 &
f α + g + b β = 0</math>, unde <math>β = {b f - a g / a b - c f} & α = {c g - h b / a b - c f}</math>, & data
æquatio transformatur in homogeneam <math>(a z + c v) v^. = (f z +
b v) z^.</math>.</p>

<pb 246>[0298]<rh>DE FLUXIONALIBUS</rh>
<p>Sit <math>a b = f c</math>, & con$equenter <math>(b + (a x + c y)) y^. = (g + l (a x
+ c y)) x^.</math>; $cribe <math>y^. = {z^. - a x^. / c}</math>, unde <math>(b + z) × {z^. - a x^. / c} = (g + l z)x^.</math>,
& con$equenter <math>{(b + z)z^. / g c + b a + a z + c l z} = x^.</math>, cujus fluens innote$cit.</p>
<p>Ex. 3. Sit æquatio <math>a x^m x^. + b y^2 x^. = y^.: 1^mo</math>. fingatur <math>y = c x^n +
d x^p v</math>, $cribatur hæc quantitas pro ejus valore in datâ æquatione, &
re$ultat <math>(a x^m + b c^2 x^2n + 2 b c d x^n+p v + b d^2 x^2p v^2)x^. = n c x^n-1 x^. +
p d x^p-1 v x^. + d x^p v^.</math>: fiant termini huju$ce æquationis <math>2 b c d x^n+p v x^. =
p d x^p-1 v x^., & b c^2 x^2n x^. = n c x^n-1 x^.</math>: colligi po$$unt e priori re$ultante
æquatione <math>n + p = p - 1</math>, unde <math>n = - 1; & 2 b c d = p d</math> & exinde
<math>2 b c = p:e</math> po$teriori vero <math>b c^2 = n c</math>, unde <math>b c = - 1 & c = -{1 / b},
& p = 2 b c = - 2</math>; & exinde <math>y = - {1 / b} x^-1 + x^-2 v</math>: æquatio vero
ex hâc $ub$titutione re$ultans erit <math>(a x^m + d^2 b x^-4 v^2) x^. = d x^-2 v^.</math>, &
con$equenter <math>{a / d}x^m+2 x^. + d b x^-2 v^2 x^. = v^.</math>; pro <math>x</math> $cribatur <math>- {1 / z}</math> in hâc
æquatione & re$ultat <math>- {a / d}z^- m-4 z^. - d b v^2 z^. = v^.</math>.</p>
<p>Sint <math>m = - 2, & 2 b c = p</math>; & fingatur <math>a + b c^2 = n c = - c</math>, &
con$equenter <math>c = {√(1 - 4 a b) - 1 / 2 b}</math>, & re$ultabit æquatio transfor-
mata <math>d^2 bx^4bc v^2 x^. = dx^2bc v^.</math>.</p>
<p>2<_>do</_>. Fingatur <math>y = c x^n + d x^p v^-1</math>, $cribatur hæc quantitas pro ejus
valore in æquatione datâ <math>a x^m + b y^2 x^. = y^.</math>, & re$ultat <math>(a x^m + b c^2 x^2n
+ 2 b c d x^p+n v^-1 + b d^2 v^2p v^-2)x^. = n c x^n-1 x^. + p d x^p-1 v^-1 x^. - d x^p
v^-2 v^.</math>: fiant <math>2 b c d x^p+n v^-1 x^. = p d x^p-1 v^-1 x^. & n c x^n-1 = b c^2 x^2n</math>, unde
<math>p + n = p - 1 & n = - 1, 2 b c d = p d & 2 b c = p, n c = - c =
b c^2 & b c = - 1</math>; & exinde <math>p = 2 b c = - 2</math>: & æquatio re$ultans
erit <math>a x^m x^. + b d^2 x^-4 v^-2 x^. = - d x^-2 v^-2 v^.</math>, & con$equenter <math>a x^m+2 v^2 x^.

<pb 247>[0299]<rh>ÆQUATIONIBUS.</rh>
+ b d^2 x^-2 x^. = - d v^.</math>; $cribatur <math>{x^m+3 / m + 3} = z</math> in hâc æquatione, & in-
venietur <math>a v^2 z^. + b d^2 (m + 3)^-{m+4 / m+3} z^-{m+4 / m+3} z^. = - d v^.</math>; & con$equenter
$i indeterminatæ quantitates <math>x & y</math> in datâ æquatione <math>a x^m x^. + b y^2 x^.
= y^.</math> $eparationem admittant, tum etiam indeterminatæ in æquatione
huju$ce formulæ <math>α z^{-m-4 / m+3} x^. + β v^2 z^. = v^.</math> $emper $eparari po$$unt.</p>
<p>Cor. 1. Sit <math>m = 0</math>, & data æquatio fit <math>a x^. + b y^2 x^. = y^.</math>, unde <math>x^. ={y^. / a + b y^2}</math>, in quâ $eparantur variabiles quantitates; & con$equenter
indeterminatæ in æquatione <math>α x^{-m-4 / m+3} = {-4 / 3} + β y^2 x^. = y</math> $eparari po$$unt,
reduci enim pote$t hæc æquatio in præcedentem: $it <math>m = {-4 / 3}</math> & con-
$equenter <math>{- m - 4 / m + 3} = {- 8 / 5}</math>, unde æquatio <math>α x^-{8 / 5} + β y^2 x^. = y^.</math> $emper
reduci pote$t in æquationem datam; & exinde in genere $i <math>m ={- 4 n / 2 n + 1}</math>, ubi <math>n</math> $it integer numerus, $emper reduci pote$t æquatio
<math>α x^m + β y^2 x^. = y^.</math> in æquationem datæ formulæ <math>a x^. + b y^2 x^. = y^.</math>.</p>
<p>Cor. 2. Fluens vero huju$ce æquationis <math>a x^. + b y^2 x^. = y^.</math>, i. e. <math>x^. ={y^. / a + b y^2}</math> generaliter inveniri pote$t circularium arcuum & logarith-
morum ope, ergo fluens æquationis <math>α x^{-4n / 2n+1} x^. + β y^2 x^. = y^.</math> $emper de-
tegi pote$t ope finitorum terminorum, circularium arcuum & loga-
rithmorum.</p>
<p>Aliter: Sit data æquatio <math>(P) y^. + a y^2 x^. = b x^m x^.</math>; a$$umatur <math>y =
c x^n-1 + {z^. / a z x^.}</math>, ubi <math>x^.</math> $it con$tans quantitas; tum erit <math>y^. = (n - 1)
c x^n-2 x^. + {z^.. / a z x^.} - {z^. ^2 / a x^. z^2}</math>, quibus quantitatibus pro $uis valoribus in

<pb 248>[0300]<rh>DE FLUXIONALIBUS</rh>
datâ æquatione <math>(P)</math> $ub$titutis, re$ultat <math>(P′) {z^.. / a z x^.} + (n - 1) c x^n-2 x^.
+ a c^2 x^2n-2 x^. + {2 c x^n-1 z^. / z} = b x^m x^.</math>: $cribantur <math>m = 2 n - 2 & b =
a c^2</math>, & fit re$ultans æquatio <math>(Q) z^.. + (n - 1) a c x^n-2 z x^. ^2 + 2 a c x^n-1 x^.
z^. = 0</math>; fingatur <math>z = A x^{-n+1 / 2} + B x^{-3n+1 / 2} + C x^{-5n+1 / 2} + &c.</math> unde <math>{z^. / x^.} =
- ({n - 1 / 2}) A x^{-n-1 / 2} - ({3 n - 1 / 2}) B x^{-3n-1 / 2} - ({5 n - 1 / 2}) C x^{-5n-1 / 2}
- &c. & {z^.. / x^. ^2} = ({n - 1 / 2} × {n + 1 / 2}) A x^{-n-3 / 2} + ({3 n - 1 / 2} × {3 n + 1 / 2})
B x^{-3n-3 / 2} + ({5 n - 1 / 2} × {5 n + 1 / 2}) C x^{-5n-3 / 2} + &c. = {n^2 - 1 / 4} A x^{-n-3 / 2}
+ {9 n^2 - 1 / 4} B x^{-3n-3 / 2} + &c.</math> quibus quantitatibus pro $uis valoribus
in æquatione <math>z^.. + (n - 1) a c x^n-2 z x^. ^2 + 2 a c x^n-1 x^. z^. = 0</math> $ub$ti-
tutis, re$ultat <math>z^.. = + ({n^2 - 1 / 4} A x^{-n-3 / 2} + {9 n^2 - 1 / 4} B x^{-3n-3 / 2} + &c.)x^. ^2
+ (n-1)acx^n-2 zx^. ^2 = + ((n-1)acAx^{n-3 / 2} + (n-1)acBx^{-n-3 / 2} + (n-1)acCx^{-3n-3 / 2} + &c.)x^. ^2
+ 2acx^n-1 x^. z^. = (-ac(n-1)Ax^{n-3 / 2} -(3n-1)acBx^{-n-3 / 2} -(5n-1)acCx^{-3n-3 / 2} -&c.)x^. ^2
0 = 0</math>.</p>
<p>Fiant corre$pondentes termini re$ultantis æquationis nihilo re-
$pective æquales, & re$ultant <math>z = A x^{-n+1 / 2} + {n^2 - 1 / 8 n a c} A x^{-3n+1 / 2} +{n^2 - 1 / 8} × {9 n^2 - 1 / 16n^2 a^2 c^2} A x^{-5n+1 / 2} + {n^2 - 1 / 4 × 2} × {9 n^2 - 1 / 4 × 4} × {25 n^2 - 1 / 4 × 6} × {A / n^3 a^3 c^3}x^{-7n+1 / 2} + &c. & {z^. / x^.} = {1 - n / 2} A x^{-n-1 / 2} + ({n^2 - 1 / 8 n a c})({1 - 3^n / 2}) A x^{-3n-1 / 2}
+ &c.</math> & exinde <math>y = c x^n-1 + {z^. / a z x^.} =

<pb 249>[0301]<rh>ÆQUATIONIBUS.</rh>
c x<_>n-1</_> - {1 / a} # {{n - 1 / 2} A x<_>{-n-1 / 2}</_> + ({3 n-1 / 2})({n<_>2</_> - 1 / 8}){A / n a c} x<_>{-3n-1 / 2}</_> + &c./Ax<_>{-n+1 / 2}</_> + {n<_>2</_> - 1 / 8}{A / n a c} x<_>{-3n+1 / 2}</_> + &c.} =
c x<_>n-1</_> - {1 / a x} # {{n - 1 / 2} + (3 n - 1)(n<_>2</_> - 1){x<_>-n</_> / 16 n a c} + &c./1 + {n<_>2</_> - 1 / 8}{x<_>-n</_> / n a c} + {n<_>2</_> - 1 / 8} × {9n<_>2</_> - 1 / 16}{x<_>-2n</_> / n<_>2</_> a<_>2</_> c<_>2</_>} + &c.}</math>.</p>
<p>Cor.. Hæc expre$$io erit finita, $i <math>(2 i + 1)^2 n^2 - 1 = 0</math>, ubi <math>i</math> $it
integer numerus, & <math>m = 2 n - 2 = {- 4 i - 2 ∓ 2 / 2 i + 1}</math>: data æquatio
in hoc ca$u erit <math>y^. + a y^2 x^. = a c^2 x^{-4i-2±2 / 2i+1} x^.</math>.</p>
<p>Sit <math>m = {- 4 i / 2 i + 1}</math>, & con$equenter <math>y^. + a y^2 x^. = a c^2 x^{-4i / 2i+1} x^.</math>; & erit
<math>a y x = a c x^n - {{n - 1 / 2} + ({3 n - 1 / 2})({n^2 - 1 / 8}) {x^-n / n a c} + &c./1 + {n^2 - 1 / 8}{x^-n / n a c} + {n^2 - 1 / 8} × {9 n^2 - 1 / 16}{x^-2n / n^2 a^2 c^2} + &c.} =
a c x^n + {{i / 2 i + 1} - {i (i^2 - 1) x^-{-1 / 2i+1} /2 (2i + 1)^2 a c} + &c./1 - {i (i + 1) / 2 (2 i + 1)}{x^{-1 / 2i+1} /a c} + &c.} + &c.</math></p>
<p>Cor. 2. Hæc re$olutio fluxionalis æquationis <math>a x^m x^. + b y^2 x^. = y^.</math>,
non erit generalis, nam in eâ haud continetur invariabilis quantitas,
quæ in fluxionali æquatione non invenitur.</p>
<p>3. Sit æquatio <math>y^. + (a x^b + b x^b-m + c x^b-2m + &c.) y^b x^. = (A x^i +
B x^i′) x^.</math>; & a$$umatur <math>y = {a x^n + b x^n-m + c x^n-2m + &c. / A x^r + B x^r-m + C x^r-2m + &c.}</math>, $ub$tituatur

<pb 250>[0302]<rh>DE FLUXIONALIBUS</rh>
hæc quantitas pro <math>y</math> & ejus fluxio pro <math>y^.</math> in datâ æquatione; & ex
æquatis corre$pondentibus terminis re$ultantis æquationis erui po$-
$unt ca$us, in quibus terminat $eries, i. e. fractio a$$umpta.</p>
<p>Eadem principia, i. e. methodus inveniendi valorem quantitatis <math>y</math>
in fractionibus, quarum numeratorum & denominatorum termini
$ecundum dimen$iones quantitatis (<math>x</math>) progrediuntur, ad quampluri-
mos ca$us applicari po$$unt.</p>
<p>Cor. Si modo pro variabilibus quantitatibus in quâcunque datâ
æquatione contentis $cribantur quæcunque functiones novarum va-
riabilium a$$umptarum, re$ultabunt æquationes, quæ facile reduci
po$$unt ad datam.</p>
<p>Ex. 1. Sit æquatio <math>a x^r x^. + b y x^-1 x^. + c y^2 x^. = y^.</math>, $cribantur in datâ
fluxionali æquatione <math>z^{b / b+1} u & z^{b / b+1}</math> pro <math>y & x, &</math> nonnunquam re$ultat
æquatio formulæ <math>α x^m x^. + β y^2 x^. = y^.</math>.</p>
<p>Ex. 2. Sit æquatio <math>a x^m x^. + b y x^p x^. + c y^2 x^. = y^.</math>, $cribatur pro <math>y</math>
ejus valor a$$umptus <math>{a / p} x^p + u</math>, ubi <math>p = m + 1</math>, & re$ultat æquatio
<math>{(a b p + a^2 c) / p^2} x^2p x^. + {(2 a c + b p) x^p / p} u x^. + c u^2 x^. = u^.</math>, quæ facile re-
duci pote$t in priorem.</p>
<p>Ex. 3. Sit <math>a x^m x^. + b y x^p x^. + c y^2 x^. = y^.</math>, $cribatur in eâ pro <math>y, u^-1</math>; &
pro <math>x, z^{1 / m+1}</math>; & re$ultat <math>{-c / m + 1} z^{-m / m+1} z^. - {b u / m + 1} z^{p-m / m+1} z^. - {a / m + 1} u^2 z^.
= u^.</math>.</p>
<p>Et $ic in genere $cribantur in datâ æquatione <math>a x^m x^. + b y^2 x^. = y^.</math>
pro <math>x & y</math> re$pective <math>α z^e u^b + β z^f u^k + γ z^g u^l + &c. & π z^n u^s +
ξ x^o u^t + &c.</math> & re$ultant æquationes, quæ facile reduci po$$unt ad
datam.</p>
<p>4. Si <math>u</math> fluat uniformiter, pro <math>u^.</math> $cribatur in quibu$dam locis datæ
æquationis data quantitas, & exinde e principiis prius traditis $æpe
deduci pote$t ejus fluens.</p>

<pb 251>[0303]<rh>ÆQUATIONIBUS.</rh>
<p>Ex. 1. Sit <math>√(x^.^2 + y^.^2) = u^.</math> con$tans quantitas, & data æquatio
<math>b y^m = {2 a y x^.. + a x^. y^. / u^. y^.}</math>; ducatur hæc æquatio in <math>{u^. y^. / 2√(y)}</math> & re$ultat
<math>{b y^m y^. u^. / 2√(y)} = {2 a y x^.. + a x^. y^. / 2√(y)}</math>, cujus fluens e$t <math>{b y^m+{1 / 2} /2 (m + {1 / 2})} = a x^.√(y) +
C √(x^.^2 + y^.^2)</math>, ubi <math>C</math> e$t quantitas invariabilis.</p>
<p>2. Sit æquatio <math>f = {x^.^2 -y y^.. / y^3 x^.^2}</math>, ubi <math>y x^.</math> e$t con$tans; ducatur hæc
æquatio in <math>2 y^.</math>, & fiet <math>2 f y^. = {2 x^.^2 y^. - 2 y y^. y^.. / y^3 x^.^2} = {2 y^. / y^3} - {2 y^. y^.. / x^.^2 y^2}</math>, $ed quo-
niam <math>y x^.</math> e$t con$tans, erit fluens <math>$. 2 f y^. = -{1 / y^2} - {y^.^2 / y^2 x^.^2} + C.</math></p>
<h>_PROB. LVIII._</h>
<h it>Invenire fluxionales æquationes, quarum dantur quædam particulares
re$olutiones.</h>
<p>A$$umatur quæcunque functio variabilium quantitatum <math>x & y</math> &
earum fluxionum pro uno datæ æquationis latere: $cribatur etiam
quæcunque functio variabilis quantitatis <math>x</math> pro variabili quantitate <math>y</math>,
& ejus fluxio pro <math>y^.</math>, & $ic deinceps, in functione a$$umptâ; & fiat
quantitas re$ultans, alterum æquationis latus; & re$ultat æquatio, cu-
jus re$olutio datur.</p>
<p>Ex. 1. A$$umatur functio <math>(y^n+1 + a y^n-1 + b y^n-3 ...P y + Q) y^.</math>; pro
quantitatibus <math>y & y^.</math> $cribantur <math>(A + √(p^n))^{1 / n} + (A - √(p^n))^{1 / n}</math> & ejus
fluxio re$pective, & re$ultare pote$t <math>(y^n+1 + a y^n-1 + b y^n-3 + &c.) y^. -
(2 A y + (n(A^2 - p^n)^{1 / n} + a) y^n-1 - (n · {n - 3 / 2}(A^2 - p^n)^{2 / n} - b) y^n-3 + (n ·

<pb 252>[0304]<rh>DE FLUXIONALIBUS</rh>
{n - 4 / 2} · {n - 5 / 3} (A^2 - p^n)^{3 / n} + c) y^n-5 + &c.) ({{1 / n} A^. + {1 / 2} p^{1 / 2} n-1 p^./(A + √(A^2 - p^n))^{n-1 / n}} +{{1 / n} A^. - {1 / 2} p^{1 / 2}n-1 p^./(A - √(A^2 - p^n))^{n-1 / n}})</math>, ubi literæ <math>a, b, c, &c. A & p</math> $unt quæcun-
que functiones quantitatis <math>x</math>, tum erit particularis valor quantitatis
<math>y = (A + √(p^n))^{1 / n} + (A - √(p^n))^{1 / n}</math>.</p>
<p>Aliter: Invenire fluxionales æquationes, quarum fluentes particu-
lares dantur.</p>
<p>A$$umatur fluentialis æquatio, & exinde deducatur fluxionalis; vel
inveniatur fluxio a$$umptæ æquationis, & in æquatione fluxionali
re$ultante pro quibu$cunque quantitatibus $ub$tituantur earum va-
lores e datâ æquatione acqui$iti, e. g. $it <math>V = 0</math> fluentialis æquatio
a$$umpta, inveniatur <math>V^. = p x^. + q y^. = 0</math>; e datâ æquatione inveni-
antur <math>y = w</math> functio quantitatis <math>x, & x = u</math> functio quantitatis <math>y</math>;
quibus quantitatibus pro $uis valoribus in quantitatibus <math>q & p</math> $ub-
$titutis, re$ultat æquatio <math>Y x^. + X y^. = 0</math>, ubi literæ <math>Y & X</math> denotant
functiones quantitatum <math>y & x</math> re$pective; & con$equenter relatio
particularis inter <math>x & y</math> variabiles quantitates æquationis <math>{x^. / X} + {y^. / Y} = 0</math>
detegi pote$t. Ex hoc principio in $ub$equentibus exemplis detegi
po$$unt generales fluentes; plures enim continentur invariabiles
quantitates ad libitum a$$umendæ in fluentiali quam in fluxionali
æquatione.</p>
<p>Cor.. Sit <math>V</math> functio quantitatum <math>x & y</math>, in quibus $imiliter invol-
vuntur variabiles quantitates <math>x & y</math>, & erit <math>X</math> eadem functio quanti-
tatis <math>x</math> ac <math>Y</math> quantitatis <math>y</math>.</p>

<pb 253>[0305]<rh>ÆQUATIONIBUS.</rh>
<h>THEOR. XL.</h>
<p>1. Sit fluxionalis æquatio <math>α = 0</math>, cujus fluens e$t <math>a</math> log. <math>A + b</math> log.
<math>B + c</math> log. <math>C + &c. = con$t.</math>; tum erit fluens etiam <math>A^a × B^b × C^c ×
&c. = con$t</math>.</p>
<p>Ex. Sit fluxionalis æquatio <math>{(a′ y^r + b′ y^r-1 + c′ y^r-2 + &c.) y^. / a y^n + b y^n-1 + c y^n-2 + &c.} +{(A′ x^s + B′ x^s-1 + C′ x^s-2 + &c.) x^. / A x^m + B x^m-1 + C x^m-2 + &c.} = 0</math>, ubi <math>r & s</math> $unt re$pective mi-
nores quam <math>n & m</math>, & $int <math>{(a′ y^r + b′ y^r-1 + &c.) / a y^n-1 + b y^n-1 + &c.} = {a′ / a}({α′ / y - α} + {β′ / y - β}{γ′ / y - γ} + &c.) & {(A′ x^s + B′ x^s-1 + &c.) / A x^m + B x^m-1 + &c.} = {A′ / A}({π′ / x - π} + {ξ′ / x - ξ} +{σ′ / x - σ} + &c.)</math>; tum erit æquatio relationem inter <math>x & y</math> exprimens
<math>((y - α)^α′ × (y - β)^β′ × (y - γ)^γ′ × &c.))^{a′ / a} × ((x - π)^π′ × (x - ξ)^ρ′ ×
(x - σ)^σ′ × &c.)^{A′ / A} = E</math> invariabili quantitati.</p>
<p>Sit fluxionalis æquatio, in quâ $imiliter involvuntur quantitates
<math>x, y</math> & earum fluxiones <math>x^., y^., &c.</math> tum in ejus generali fluente & mul-
tiplicatore $imiliter etiam involventur quantitates <math>x, y, &c.</math></p>
<p>2. Idem etiam affirmari pote$t de pluribus fluxionalibus æquationi-
bus plures variabiles quantitates involventibus.</p>
<p>Ex. 1. Sit data æquatio <math>a + b (x + y) + c x y = 0</math>, cujus fluxio
erit <math>(b + c x) y^. + (b + c y) x^. = 0</math>; in hâc fluxione pro <math>x & y</math> $criban-
tur re$pective <math>- {a + b y / b + c y} = x & - {a + b x / b + c x} = y</math> earum valores ex datâ
æquatione deducti; tum re$ultabit fluxionalis æquatio <math>{b^2 - c a / b + c y} y^. +

<pb 254>[0306]<rh>DE FLUXIONALIBUS</rh>
{b<_>2</_> - c a / b + c x} x^. = 0</math>, cujus generalis fluens erit <math>{b^2 - c a / c} × (log. (b + c y)
+ log. (b + c x)) = {b^2 - a / c} log. ((b + c y) × (b + c x)) = A</math>, ubi <math>A</math>
e$t quantitas ad libitum a$$umenda; $ed <math>(b + c y) × (b + c x) =
b^2 + c b (y + x) + c^2 y x</math>, & exinde <math>b (y + x) + c y x</math> equat invaria-
bilem quantitatem.</p>
<p>Ex. 2. Sit æquatio <math>a + 2 b (x + y) + c (x^2 + y^2) + 2 d x y = 0</math>,
cujus fluxio e$t <math>(2 b + 2 c x + 2 d y) x^. + (2 b + 2 c y + 2 d x) y^. = 0</math>,
in hâc fluxione pro <math>x & y</math> $cribantur <math>- {b + d y ∓ √(b^2 - a c + (2 b d
- 2 b c) y + (d^2 - c^2)y^2) / c} (P) & - {b + d x ∓ √(b^2 - a c + (2 b d -
2 b c) x + (d^2 - c^2) x^2) / c} (Q)</math>; & re$ultant duæ fluxionales æquationes
<math>{y^. / (b^2 - a c + (2 b d - 2 b c) y + (d^2 - c^2)y^2)^{1 / 2}} + {x^. / (b^2 - a c + (2 b d -
2 b c) x + (d^2 - c^2)x^2)^{1 / 2}} = 0, & (b + c x + + d Q) x^. + (b + c y +
d P)y^. = 0</math>.</p>
<p>Sit <math>b = 0 & {a c / c^2 - d^2} = n</math>; tum re$ultat æquatio <math>{x^. / √(x^2 + n)} +{y^. / √(y^2 + n)} = 0</math>.</p>
<p>3. Sit æquatio <math>{x^. / √(x^2 + n)} + {y^. / √(y^2 + n)} = 0</math>, cujus fluens e$t log.
<math>(x + √(x^2 + n)) + log. (y + √(y^2 + n)) = log. ((x + √(x^2 + n))
× (y + √(y^2 + n)) =</math> invariabili quantitati; unde æquatio <math>(x +
√(x^2 + n)) × (y + √(y^2 + n)) = x y + x √(y^2 + n) + y √(x^2 + n)
+ √(x^2 + n) × √(y^2 + n) = E</math> variabili quantitati; $cribatur enim

<pb 255>[0307]<rh>ÆQUATIONIBUS.</rh>
ob <math>(x + √(x^2 + n)) (y + √(y^2 + n)) = E</math>, pro <math>{x^. / y^.}</math> ejus valor <math>- {1 / E}(x + √(x^2 + n)) × (y + √(y^2 + n)) {√(x^2 + n) / √(y^2 + n)} = - {E / E} × {√(x^2 + n) / √(y^2 + n)}</math>,
unde <math>{x^. / √(x^2 + n)} = - {y^. / √(y^2 + n)}</math>.</p>
<p>Reducatur æquatio <math>(x + √(x^2 + n)) × (y + √(y^2 + n)) = E</math>,
ita ut exterminentur irrationales quantitates; per no$tr. medit. algeb.
erit <math>((x + √(x^2 + n)) (y + √(y^2 + n)) - E) × ((x - √(x^2 + n))
(y - √(y^2 + n)) - E) = n^2 + E^2 - 2 E x y - 2 E √(x^2 + n) ×
√(y^2 + n)</math>; ducatur hæc quantitas in <math>n^2 + E^2 - 2 E x y + 2 E √(x^2
+ n) × √(y^2 + n)</math>, & re$ultat æquatio quæ$ita <math>(n^2 + E^2)^2 - 4 E
(n^2 + E^2) x y - 4 E^2 ((x^2 + y^2) + n) n = L = 0</math> ab irrationalibus
libera.</p>
<p>In æquatione <math>{y^. / √(y^2 + n)} + {x^. / √(x^2 + n)} = 0</math> pro <math>y</math> $cribatur <math>√(A)
v + {B / 2 A^{1 / 2}}, & √(A) z + {B / 2 A^{1 / 2}}</math> pro <math>x</math>; & $i modo $it <math>{B^2 / 4A} (f^2) + n
= C</math>, tum evadet æquatio <math>{v^. / √(A v^2 + B v + C)} + {z^. / √(A z^2 + B z
+ C)} = 0</math>: deinde in æquatione <math>L = 0</math>, pro <math>x, y & n</math> $criban-
tur re$pective <math>√(A) z + {B / 2 A^{1 / 2}}, √(A) v + {B / 2 A^{1 / 2}} & C - f^2</math>; & re-
$ultat æquatio, quæ e$t generalis fluens æquationis <math>{v^. / √(A v^2 + B v + C)}+ {z^. / √(A v^2 + B v + C)} = 0</math>: aliter generalis fluens fluxionalis æqua-
tionis <math>{v^. / √(A v^2 + B v + C)} + {z^. / √(A v^2 + B v + C)} = {1 / √(A)}({v^. / √(v^2 + {B / A} v + {C / A})} + {z^. / √(z^2 + {B / A} z + {C / A})}) = 0</math> erit <math>{1 / √(A)}</math> (log. <math>(v

<pb 256>[0308]<rh>DE FLUXIONALIBUS</rh>
+ {1 / 2}{B / A} + √(v^2 + {B / A}v + {C / A})) + log. (z + {1 / 2}{B / A} + √(z^2 + {B / A}z + {C / A})))
= {1 / √(A)} (log. (v + {1 / 2}{B / A} + √(v^2 + {B / A}v + {C / A})) (z + {1 / 2}{B / A} + √(z^2 +{B / A}z + {C / A})) = A′</math> cuicunque invariabili quantitati; con$equenter <math>(v
+ {1 / 2}{B / A} + √(v^2 + {B / A}v + {C / A})) (z + {1 / 2}{B / A} + √(z^2 + {B / A}z + {C / A}))
= E</math> invariabili quantitati erit etiam fluens æquationis prædictæ.</p>
<p>Per eundem modum inveniri pote$t <math>(v + {2 / 1}{B / A} + √(v^2 + {B / A}v
+ {C / A}))^m × (z + {1 / 2}{B / A} + √(z^2 + {B / A}z + {C / A}))^r = E</math> generalis fluens
fluxionalis æquationis <math>{m v^. / √(v^2 + {B / A}v + {C / A})} + {r z^. / √(z^2 + {B / A}z + {C / A})}= 0</math>.</p>
<p>Ex. 2. Sit æquatio <math>a + 2 b (x + y) + c (x^2 + y^2) + 2 d x y + 2 e x y
(x + y) + f x^2 y^2 = A = 0</math>, cujus fluxio <math>(b + c x + d y + 2 e y x +
e y^2 + f y^2 x) x^. + (b + c y + d x + 2 e y x + e x^2 + f x^2 y) y^. = 0</math>, in
quantitate <math>b + c x + d y + 2 e y x + e y^2 + f y^2 x</math> pro <math>x</math> $cribatur <math>-{b + d y + e y^2 / c + 2 e y + f y^2} + √({(b + d y + e y^2)^2 - (c + 2 e y + f y^2) (a + 2 b y + c y^2) / (c + 2 e y + f y^2)^2})</math>;
& in quantitate <math>b + c y + d x + 2 e y x + e x^2 + f x^2 y</math> pro <math>y</math> $cribatur <math>-{b + d x + e x^2 / c + 2 e x + f x^2} + √({(b + d x + e x^2)^2 - (c + 2 e x + f x^2) (a + 2 b x + c x^2) / (c + 2 e x + f x^2)^2})</math>;
& re$ultabunt re$pective <math>√((e^2 - c f) y^4 + (2 e d - 2 c e - 2 b f) y^3
+ (2 e b + d^2 - c^2 - a f - 4 b e) y^2 + (2 b d - 2 e a - 2 b c) y + b^2
- c a) = √(A y^4 + B y^3 + C y^2 + D y + E), & √((e^2 - c f) x^4 +
(2 e d - 2 c e - 2 b f) x^3 + (2 e b + d^2 - c^2 - a f - 4 b e) x^2 + (2 b d
- 2 e a - 2 b c) x + b^2 - c a) = √(A x^4 + B x^3 + C x^2 + D x + E)</math>,
& exinde re$ultat fluxionalis æquatio <math>{x^. / √(A x^4 + B x^3 + C x^2 + D x + E)}

<pb 257>[0309]<rh>ÆQUATIONIBUS.</rh>
+ {y^. / √(A y^4 + B y^3 + C y^2 + D y + E)} = 0</math>, cujus data æquatio <math>A′ = 0</math>
erit fluens.</p>
<p>Ex datis coefficientibus <math>A, B, C, D & E</math>, & $uppo$itis quantitati-
bus <math>e^2 - c f = A m, 2 e d - 2 c e - 2 b f = B m, 2 e b + d^2 - c^2 - a f
- 4 b e = C m, 2 b d - 2 e a - 2 b c = D m & b^2 - c a = E m</math>; re$ul-
tant quinque æquationes, e quibus deduci po$$unt incognitæ quanti-
tates <math>a, b, c, d, e, f & m</math>; & con$equenter æquatio, quæ e$t fluens præ-
dictæ fluxionalis æquationis, <math>a′ + 2 b′ (x + y) + c′ (x^2 + y^2) + 2 d′ x y
+ 2 e′ x y (x + y) + f′ x^2 y^2 = 0</math>.</p>
<p>2. Sint <math>b = 0 & e = 0</math> & re$ultat æquatio <math>a + c (x^2 + y^2) + 2 d x y
+ f x^2 y^2 = 0</math>, unde fluxionalis æquatio <math>{x^. / √(- c f x^4 + (d^2 - c^2 - a f) x^2 - c a)}+ {y^. / √(- c f y^4 + (d^2 - c^2 - a f) y^2 - c a)} = {x^. / √(A x^4 + C x^2 + E)}+ {y^. / √(A y^4 + C y^2 + E)} = 0</math>, cujus fluens erit <math>- A + l (x^2 + y^2) +
2 x y √(l^2 + l C + A E) - E x^2 y^2 = 0</math>, ubi <math>l</math> e$t invariabilis quanti-
tas ad libitum a$$umenda.</p>
<p>Ex. 3. Sit æquatio <math>± {q̈ / √(a + bp^2 + cp^4 + dp^6)} ± {q̈ / √(a + bq^2 + cq^4 + dq^6)}</math>;
fingatur <math>x = p^2 & y = q^2</math>, & exinde deduci pote$t æquatio <math>±{x^. / √(a x + b x^2 + c x^3 + d x^4} ± {y^. / √(a y + b y^2 + c y^3 + d y^4)} = 0</math>.</p>
<p>In fluxione <math>{x^. / √(a + b x + c x^2 + d x^3 + e x^4)}</math> pro <math>x & x^.</math> $cribantur
<math>{α + β z / γ + δ z}</math> & ejus fluxio; & ita a$$umi po$$unt coefficientes, ut re$ultet
fluxio formulæ <math>{z^. / √(A + B z^2 + C z^4)}</math>.</p>

<pb 258>[0310]<rh>DE FLUXIONALIBUS</rh>
<h>THEOR. XLI.</h>
<p>Sit <math>X</math> eadem functio quantitatis <math>x</math>, ac <math>Y</math> $it quantitatis <math>y, & X x^. +
Y y^. = 0</math>; & detur generalis fluens <math>α = 0</math> huju$ce æquationis; ex æqua-
tione <math>X x^. + Y y^. = 0</math> con$tat <math>φ:(x) + φ:(y) = con$t. = φ:(c)</math>, unde
<math>c</math> erit con$imilis functio quantitatum <math>x & y</math>; i. e. <math>c = π:(x & y)</math>: in
æquationibus <math>π:(x & y) = c & φ:(x) + φ:(y) = φ:c</math>, pro <math>x & y</math>
$cribantur <math>a & b</math> & re$ultabit <math>c = π:(a & b) & φ:(a) + φ:(b) = φ:
(c)</math>; deinde in duabus æquationibus <math>φ:(a) + φ:(b) = φ:(c) & c = π

:a & b)</math> pro <math>b</math> $cribatur <math>a</math>, & re$ultabit <math>φ:(a) + φ:(b) = 2 φ:(a)
= φ:(c), & c = π:(a & b) = π′:(a)</math>: nunc a$$umatur æquatio
<math>2 X x^. + Y y^. = 0</math>, unde <math>2 φ:(x) + φ:(y) = con$t. = φ:(d)</math>: in hâc
æquatione <math>d = π:(x & y)</math> pro <math>x</math> $cribatur <math>π′:(x)</math>, & re$ultat æquatio
<math>d = π:(π′:(x) & y)</math>, quæ erit generalis fluens æquationis fluxionalis
<math>2 X x^. + Y y^. = 0</math>. In æquationibus <math>2 φ:(x) + φ:(y) = φ:(d) &
d = π:(π′ (x) & y)</math> pro <math>x & y</math> $cribantur <math>a & b</math> & re$ultant æquationes
<math>2 φ:(a) + φ:(b) = φ:(d) & d = π:(π′:(a) & a) = π″:(a):3^tia</math>.
a$$umatur æquatio <math>3 X x^. + Y y^. = 0</math>, unde <math>3 φ:(x) + φ:(y) = φ:(e)</math>:
in æquatione <math>π:(x & y) = e</math> pro <math>x</math> $cribatur <math>π″:(x)</math>, & re$ultat <math>e =
π:(π″:(x) & y)</math>; quæ erit generalis fluens prædictæ fluxionis <math>3 X x^.
+ Y y^. = 0</math>: nunc fingatur <math>x = a & y = b</math>, & denique <math>a = b</math>, tum
re$ultabunt <math>2:φ:(a) + φ:(b) = 3 φ:(a) = φ:(e) & e = π:(π″

:(a) & a) = π″′:(a)</math>, & $ic deinceps: ex hâc methodo deduci pote$t
fluens generalis fluxionalis æquationis <math>n X x^. + m Y y^. = 0</math>, ubi literæ
<math>n & m</math> integros denotant numeros.</p>
<h>THEOR. XLII.</h>
<p>Sit æquatio <math>A y^. ^n + B y^. ^n-1 x^. + C y^. ^n-2 x^. ^2 + D y^. ^n-3 x^. ^3 .. + L y^.. x^. ^n-2 + M y^. x^. ^n-3
+ N y x^. ^n = X x^. ^n</math>, ubi <math>A, B, C, .. M & N</math> invariabiles denotant coeffi-
cientes, & <math>X</math> e$t functio quantitatis <math>x</math>.</p>
<p>Ducatur hæc æquatio in <math>e^λx</math>, & re$ultat æquatio, cujus generalis

<pb 259>[0311]<rh>ÆQUATIONIBUS.</rh>
fluens e$t <math>e^λ x (A y^. ^n-1 + (B - λA) y^. ^n-2 x^. + (C - λ B + λ^2 A) y^. ^n-3 x^. ^2 + (D
- λ C + λ^2 B - λ^3 A) y^. ^n-4 x^. ^3 + ... + (M - λ L + λ^2 K - ... ± λ^n-3 C
∓ λ^n-2 B ± λ^n-1 A) y = x^. ^n-1 $. e^λ x X x^. + a x^. ^n-1</math>, $i modo <math>N - λ M +
λ^2 L - λ^3 K + ... ± λ^n-1 B ± λ^n A = 0</math>, & con$equenter <math>A λ^n - B λ^n-1
+ C λ^n-2 - D λ^n-3 + &c. = 0</math>; at <math>(n)</math> $unt diver$i valores quantitatis
<math>λ</math> in æquatione <math>A λ^n - B λ^n-1 + C λ^n-2 - &c. = 0</math>, qui $int re$pective
<math>λ, μ, ν, ξ, &c.</math>; unde ex <math>(n)</math> valoribus quantitatis <math>λ</math> exoriuntur totidem
<math>(n)</math> multiplicatores, viz. <math>e^λ x, e^μ x, e^γ x, &c.</math></p>
<p>Ducatur prædicta generalis fluens in <math>e^(μ-λ)x</math>, & detegi pote$t fluens
re$ultantis æquationis <math>e^μ x (A y^. ^n-2 + (B - (λ + μ) A) y^. ^n-3 x^. + (C - (λ
+ μ) B + (λ^2 + μ λ + μ^2) A) y^. ^n-4 x^. ^2 + (D - (λ + μ) C + (λ^2 + λ μ
+ μ^2) B - (λ^3 + λ^2 μ + λ μ^2 + λ^3) A) y^. ^n-5 x^. ^3 + &c. = x^. ^n-2 $. e^(μ-λ)x x^.
$. e^λx X x^. + {a / μ - λ} e^(μ-λ)x x^. ^n-2 + b x^. ^n-2 = -x^. ^n-2 ({1 / λ - μ} e^(μ-λ)x $. e^λx X x^.
+ {1 / μ - λ} $. e^μx X x^. - {a / μ - λ} e^(μ-λ)x - b) = x^. ^n-2 ({1 / μ - λ} e^(μ-λ)x $. e^λx X x^.
+ {1 / λ - μ} $. e^μx X x^. + a′ e^(μ-λ)x + b)</math>.</p>
<p>Hìc animadvertendum e$t <math>M - (λ + μ) L + (λ^2 + λ μ + μ^2) K
- (λ^3 + λ^2 μ + λ μ^2 + μ^3) 1 - .... ± (λ^n-1 + λ^n-2 μ + λ^n-3 μ^2 +
λ^n-4 μ^3 + .... μ^n-1) A = 0</math>.</p>
<p>Ducatur fluens re$ultantis æquationis in <math>e^(ν-μ)x</math>, & fluens æquatio-
nis exinde re$ultantis erit <math>e^νx (A y^. ^n-3 + (B - (λ + μ + ν) A) y^. ^n-4 x^. ^2 +
(C - (λ + μ + ν) B + (λ^2 + μ^2 + ν^2 + λ μ + λ ν + μ ν) A) y^. ^n-5 x^. ^2 +
(D - (λ + μ + ν) C + (λ^2 + μ^2 + ν^2 + λ μ + λ ν + μ ν) B + (λ^3
+ μ^3 + ν^3 + λ μ^2 + λ ν^2 + μ λ^2 + μ ν^2 + ν λ^2 + ν μ^2 + λ μ ν) A) y^. ^n-6 x^. ^3
+ &c. = x^. ^n-3 ({1 / λ - μ} × {1 / λ - ν} e^(ν-λ)x $. e^λ x X x^. + {1 / μ - λ} × {1 / μ - ν} e^(ν-μ)x

<pb 260>[0312]<rh>DE FLUXIONALIBUS</rh>
$. e^μ x X x^. + {1 / ν - μ} × {1 / ν - λ} $. e^ν x X x^. + (a″ e^(ν-λ) x + b′ e^(ν-μ) x + c) = 0</math>, & $ic
deinceps; ultimo erit <math>y = ± (({1 / λ - μ} × {1 / λ - ν} × {1 / λ - ξ} × {1 / λ - 0} × &c.)
× e^-λx $. e^λ x X x^. + {1 / μ - λ} × {1 / μ - ν} × {1 / μ - ξ} × {1 / μ - 0} × &c. × e^-μ x $. e^μ x X x^.
+ {1 / ν - λ} × {1 / ν - μ} × {1 / ν - ξ} × {1 / ν - 0} × &c. × e^-ν x $. e^νx X x^. + &c. + A′e^-λx
+ B′e^-μx + C′e^-νx + D′e^-ξx + &c.</math>, ubi literæ <math>A′, B′, C′, D′, &c.</math> re$pe-
ctive denotant invariabiles coefficientes ad libitum a$$umendas; & $ic
deinceps.</p>
<p>Erit <math>(λ - μ) × (λ - ν) × (λ - ξ) × (λ - 0) × &c. = n A λ^n-1 +
(n - 1) B λ^n-2 + (n - 2) C λ^n-3 + (n - 3) D λ^n-4 + &c.</math> & $ic de re-
liquis radicibus <math>μ, ν, ξ, &c.</math></p>
<p>2. Hoc theorema ex $ub$equentibus con$tabit: $it æquatio <math>N +
M x + L x^2 + K x^3 + I x^4 + H x^5 + ... + D x^n-3 + C x^n-2 + B x^n-1
+ A x^n = 0</math>, cujus radices $int <math>λ, μ, ν, ξ 0′, &c.</math>; tum erit <math>N + M λ +
L λ^2 + K λ^3 + I λ^4 + &c. = 0, & M + {λ^2 - μ^2 / λ - μ} L + {λ^3 - μ^3 / λ - μ} K +{λ^4 - μ^4 / λ - μ} I + &c. = M + (λ + μ) L + (λ^2 + μ^2 + μ λ) K + (λ^3 + μ^3
+ λ^2 μ + μ^2 λ) I + &c. = 0</math>, & $imiliter <math>L + (λ + μ + ν) K + (λ^2 +
μ^2 + ν^2 + λ μ + λ ν + μ ν) I + (λ^3 + μ^3 + ν^3 + λ^2 μ + λ^2 ν + μ^2 λ +
μ^2 ν + ν^2 λ + ν^2 μ + λ μ ν) H + &c. = 0, & K + (λ + μ + ν + ξ) +
K + (λ^2 + μ^2 + ν^2 + ξ^2 + λ μ + λ ν + μ ν + λ ξ + μ ξ + ν ξ) I +
(λ^3 + μ^3 + ν^3 + ξ^3 + λ^2 μ + μ^2 λ + λ^2 ν + μ^2 ν + ν^2 λ + &c.) H +
&c. = 0; &c.</math>: in primâ æquatione deductâ una $olummodo conti-
netur radix <math>λ</math> vel <math>μ</math> vel <math>ν, &c,;</math> in $ecundâ duæ; in tertiâ tres; & $ic
deinceps: prima prædicta æquatio incipit ab primâ datæ æquationis
coefficiente, $ecunda a $ecundâ, tertia a tertiâ, & $ic deinceps.</p>
<p>Coefficiens primi termini prædictarum æquationum erit 1; $e-
cundi termini erit $umma omnium radicum in iis re$pective conten-

<pb 261>[0313]<rh>ÆQUATIONIBUS.</rh>
tarum; tertii termini erit $umma omnium rationalium & non fractio-
nalium functionum radicum, quarum dimen$iones $unt duæ, i. e.
erit $umma quadratorum e $ingulis radicibus & rectangulorum $ub
quibu$que duabus in iis contentis; quarti termini coefficiens erit
$umma omnium con$imilium functionum radicum, quarum dimen-
$iones $unt tres, i. e. erit $umma cuborum e $ingulis radicibus, & $in-
gulorum valorum ex quadrato alterius radicis in alteram ducto, viz.
cujus $ormula eadem e$t ac <math>α^2 β</math>; & contentorum $ub quibu$que tribus
radicibus; & $ic deinceps: coefficientes omnium quantitatum in datis
æquationibus contentæ erunt 1.</p>
<p>Cor. 1. Po$itis <math>y^. = p x^., p^. = q x^., q^. = r x^., &c.</math> $cribantur hæ quan-
titates pro $uis valoribus in datâ fluxionali æquatione, & re$ultat <math>N y
+ M p + L q + K r + &c. = X</math>.</p>
<p>Cor. 2. In datâ prædictâ fluxionali æquatione pro <math>y, y^., y^..</math>, & $cri-
bantur <math>e ^$. u x^., u x^. e^$. u x^., (u^. x^. + u^2 x^. ^2) e ^$. u x^., &c.</math>; & re$ultat fluxionalis
æquatio <math>(n - 1)</math> ordinis, cujus generalis fluens facile innote$cit; nam
erit <math>{y^. / y x^.} = u</math>, quoniam <math>e ^$. u x^. = y</math>; & exinde e generali valore quanti-
tatis <math>y</math> prius dato facile con$tat generalis valor quantitatis <math>u</math>.</p>
<p>In re$ultante æquatione pro <math>x^.</math> $cribatur <math>{u^. / t}</math>, deinde pro <math>t + u^2</math> $cri-
batur <math>z</math>; & $imiliter ex in$initis $ub$titutionibus facile deduci po$$unt
fluxionales æquationes, quarum fluentes innote$cunt.</p>
<p>3. Sit fluxionalis æquatio <math>H y^. ^n + G y^. ^n-2 x^. ^2 + F y^. ^n-4 x^. ^4 + ... + C y^.... x^. ^n-4
+ B y^.. x^. ^n-2 + A y x^. ^n = 0</math>; ubi <math>n</math> e$t par numerus; ducatur hæc æqua-
tio in <math>y^.</math>, & re$ultat æquatio, cujus fluens e$t <math>H (y^. ^n-1 y^. - y^. ^n-2 y^.. + y^. ^n-3 y^... ....{1 / 2} y^. ^2 ^{1 / 2}n + G (y^. ^n-3 y^. - y^. ^n-4 y^.. + y^. ^n-5 y^... ... {1 / 2} y^. ^2 ^{1 / 2}n-1) x^. ^2 + &c. + C (y^... y^. - {1 / 2} y^.. ^2) x^. ^n-4 +
B ({1 / 2} y^. ^2) x^. ^n-2 + {1 / 2} A y^2 x^. ^n + A′ x^. ^n = 0</math>, ubi <math>A′</math> e$t invariabilis quantitas ad
libitum a$$umenda.</p>
<p>Si in hâc æquatione pro <math>y^.</math> $cribatur <math>e $. v x^., &c.</math>, vel <math>y^. = u x^., &c.</math>;

<pb 262>[0314]<rh>DE FLUXIONALIBUS</rh>
& exinde deduci po$$unt æquationes fluxionales, quarum fluentes in-
note$cunt.</p>
<p>4. Sit fluxionalis æquatio <math>a y^. ^n + b y^. ^n-1 x^. + c y^. ^n-2 x^. ^2 + d y^. ^n-3 x^. ^3 + ... f y^. ^n-m x^. ^m
... + b y^... x^. ^n-3 + i y^.. x^. ^n-2 + k y^. x^. ^n-1 + l y x^. ^n = X x^. ^n</math>; ducatur hæc æqua-
tio in datam quantitatem <math>p</math>, & $cribantur <math>a p = A, b p = B, c p = C,
d p = D, &c., f p = F, &c., b p = H, i p = I, k p = K & l p = L</math>;
ubi literæ <math>a, b, c, d, &c, h, i, k, l & p</math> functiones quantitatis <math>x</math> deno-
tant; tum integrari pote$t fluxionalis quantitas <math>p a y^. ^n + p b y^. ^n-1 x^. +
p c y^. ^n-2 x^. ^2 + &c.</math>, $i modo <math>A^. ^n - B^. ^n-1 + C^. ^n-2 - D^. ^n-3 + ... ± H^... ∓ I^.. ± K^. =
L = 0</math>: $i evane$cant termini <math>l, k, i, b, &c.</math> u$que ad <math>f</math>; tum erit <math>A^. ^n-m
- B^. ^n-m-1 + C^. ^n-m-2 - &c. = 0</math>, cum quantitas prædicta integrari po$$it;
fluens enim erit <math>A y^. ^n-1 + (B - A^.) y^. ^n-2 x^. + (C - B^. + A^..) y^. ^n-3 x^. ^2 + &c. =
x^. ^n-1 $. p X x^. =</math> con$t.</p>
<p>Cor. Si dentur omnes præter unam vel <math>l</math> vel <math>k</math> vel <math>i</math> vel <math>b, &c.</math> quanti-
tates prædictæ <math>l, k, i, b, &c ... c, b, a & p</math>; tum facile ex æquatione fluxio-
nali <math>A^. ^n - B^. ^n-1 + C^. ^n-2 - &c. = 0</math> acquiri pote$t quantitas vel <math>L</math> vel <math>K</math> vel
<math>I</math> vel <math>H, &c.</math> ei corre$pondens, & exinde vel <math>l = {L / p}</math>, vel <math>b = {H / p}</math>, vel <math>i ={I / p}, &c.</math></p>
<p>Cor. 2. Sint <math>a = a′ x^n, b = b′ x^n-1, c = c′ x^n-2, &c., b = b′ x^3, i =
i′ x^2, k = k′ x</math>; ubi literæ <math>a′, b′, c′, d′, &c., b′, i′, k′ & l′</math> invariabiles
denotant quantitates; ducatur data æquatio <math>a′ x^n y^. ^n + b′ x^n-1 y^. ^n-1 x^. +
c′ x^n-2 y^. ^n-2 x^. ^2 + &c. + b′ x^3 y^... x^. ^n-3 + i′ x^2 y^.. x^. ^n-2 + k′ x y^. x^. ^n-1 + l′ y x^. ^n = X x^. ^n</math>
in <math>x^λ</math>; & re$ultat æquatio, cujus fluens e$t <math>a′ x^n+λ y^. ^n-1 + (b′ - (n + λ) a′)
x^n+λ-1 y^. ^n-2 x^. + (c′ - (n + λ - 1) b′ + (n + λ) × (n + λ - 1) a′) x^n+λ-2

<pb 263>[0315]<rh>Æ QUATIONIBUS.</rh>
y^. ^n-3 x^. ^2 + (d′ - (n + λ - 2) c′ + (n + λ - 1) × (n + λ - 2) b′ -
(n + λ) × (n + λ - 1) × (n + λ - 2) a′) x^n+λ-3 y^. ^n-4 x^. ^3 + &c.</math> (cujus ul-
timi termini coefficiens <math>l′ - (λ + 1) k′ + (λ + 1) × (λ + 2) i′ - (λ + 1)
(λ + 2) (λ + 3)b′ + .... ± (λ + 1) × (λ + 2) × (λ + 3) × .... ×
(λ + n) a′ = 0) = x^. ^n-1 $. x^λ X x^. + con$t. x^. ^n-1</math>.</p>
<p>Eadem fluens $ic exprimi pote$t, viz. <math>x^. ^n-1 $. x^λ X x^. + con$t. x^. ^n-1 = {l′ / λ + 1}x^λ+1 y x^. ^n-1 + ({k′ - π / λ + 2}) x^λ+2 y^. x^. ^n-2 + ({i′ - ξ / λ + 3}) x^λ+3 y^.. x^. ^n-3 + ({b′ - σ / λ + 4})
x^λ+4 y^... x^. ^n-4 + &c.,</math> ubi literæ <math>π, ξ, σ, &c.</math> præcedentes coefficientes re-
$pective denotant.</p>
<p>Cor. 3. Sit <math>l′ - (λ + 1) k′ + (λ + 1) (λ + 2) i′ - (λ + 1) (λ + 2)
(λ + 3) h′ + .... ± (λ + 1) (λ + 2) (λ + 3) .... (λ + n) a′ = a′
(λ - α) (λ - β) (λ - γ) (λ - δ) × &c.</math>; unde <math>λ</math> vel <math>= α</math>, vel <math>= β</math>, vel
<math>= γ, &c.</math>: & exinde, $i modo ducatur data æquatio in <math>x^α, &c.</math> re$ul-
tabit æquatio, cujus fluens e$t datæ $imilis, ordinis vero inferioris
<math>(n - 1)</math>; & $ic deprimi pote$t re$ultans æquatio in fluxionalem æqua-
tionem <math>(n - 2)</math> ordinis, $i modo ea ducatur in <math>x^β; &</math> $ic eâdem
operatione <math>(n)</math> vicibus repetitâ, i. e. ductis re$ultantibus æquationibus
in <math>x^γ, x^δ, &c.</math> $ucce$$ive, & inventis fluxionum re$ultantium fluenti-
bus, tandem invenietur fluens quæ$ita.</p>
<p>5. Sit <math>y = a^n × e^-ax $. x^. $. x^. $. x^.</math> ad <math>n - 1</math> numerum <math>× $. e^a x X x^.</math>; re-
ducatur hæc æquatio, ita ut exterminentur fluentiales quantitates, &
re$ultat <math>y + {n y^. / a x^.} + {n (n - 1) y^.. / 2 a^2 x^. ^2} + {n (n - 1) (n - 2) y^... / 2. 3 a^3 x^. ^3} + {n (n - 1) (n - 2) (n - 3) y^.... / 2 · 3 · 4 a^4 x^. ^4}+ &c. = X.</math></p>
<p>Hoc $equitur etiam ex problemate; nulli in hoc ca$u requiruntur
multiplicatores, ni <math>x^.</math>.</p>
<p>6. Sit æquatio <math>X = a^n y^. + b^n-1 y^. x^. + c^n-2 y^. x^. ^2 + &c.</math>, $ub$tituatur <math>y = v
+ Z</math>, ubi <math>Z</math> e$t functio quantitatis <math>x</math>, quæ e$t particularis valor

<pb 264>[0316]<rh>DE FLUXIONALIBUS</rh>
quantitatis <math>y</math>, & re$ultabit fluxionalis æquatio <math>A v x^. ^n + B v^. x^. ^n-1 +
C v^.. x^. ^n-2 + &c. = 0</math>, ubi <math>a, b, c, &c.</math>, <math>A, B, C, &c.</math> $unt functiones
quantitatis <math>x</math>.</p>
<p>7. Sit æquatio <math>X = a y + {b y^. / x^.} + {c y^.. / x^. ^2} + {d y^... / x^. ^3} + &c.</math>; $tatuatur <math>x = log. v</math>,
unde <math>x^. = {v^. / v},</math> & erit <math>X = φ: (v)</math>; deinde inveniatur æquatio relatio-
nem inter <math>v & y</math> & earum fluxiones exprimens, quæ habet formulam
<math>V = a y + {b v y^. / v^.} + {c v^2 y^.. / v^. ^2} + {d v^3 y^... / v^. ^3} + &c.</math>: & $imiliter $it æquatio <math>V =
a y + {b v y^. / v^.} + {c v^2 y^.. / v^. ^2} + &c.;</math> $tatuatur <math>v = e^x</math>, & reducitur data æqua-
tio ad æquationem formulæ <math>X = a y + {b y^. / x^.} + {c y^.. / x^. ^2} + &c.</math></p>
<p>8. Sint <math>X = A y + B{ν^. / x^.} + C{y^.. / x^. ^2} ... {N^m+n y^. / x^. ^m+n} & P = A + B z + C z^2
... N z^m+n = Q R,</math> ubi <math>Q = a + b z + c z^2 ... l z^m & R = α + β z
+ γ z^2 .. ν z^n</math>: fingatur <math>X = a v + b{v^. / x^.} + c{v^.. / x^. ^2} ... l {v^. ^m / x^. ^m},</math> & erit <math>v = α y
+ β{y^. / x^.} + γ{y^.. / x^. ^2} ... + ν {y^. ^n / x^. ^n}</math>.</p>
<p>9. Sint <math>n</math> diver$i valores variabilis <math>y</math> in fluxionale æquatione <math>y^. ^n + p y^. ^n-1 x^.
+ q y^. ^n-2 x^. ^2 .. + S y x^. ^n = 0</math> (ubi <math>p, q, &c. S</math> datas functiones quantitatis
<math>x</math> denotant) re$pective <math>π, ξ, σ, τ, &c.</math> & erit generalis valor ip$ius <math>y</math> in
prædicta æquatione <math>= a π + b ξ + c σ + d τ + &c.</math> ubi <math>a, b, c, &c.</math>
invariabiles quantitates ad libitum a$$umendas re$pective de$ignant.</p>
<p>Cor. 4. Sit ζ valor variabilis <math>y</math> in fluxionale æquatione <math>(A) y^. ^n +
p y^. ^n-1 x^. + q y^. ^n-2 x^. ^2 ... S y x^. ^n + X x^. ^n = 0, (n)</math> vero diver$i valores quanti-
tatis <math>y</math> in æquatione <math>y^. ^n + p y^. ^n-1 x^. + q y^. ^n-2 x^. ^2 ... S y x^. ^n = 0</math> re$pective <math>π, ζ,

<pb 265>[0317]<rh>Æ QUATIONIBUS.</rh>
σ, τ, &c.</math> tum erit generalis valor ip$ius <math>y</math> in fluxionali æquatione <math>(A)
= a π + b ρ + c σ + d τ + &c. + Θ,,</math> ubi <math>a, b, c, d, &c.</math> $unt quæcun-
que invariabiles quantitates ad libitum a$$umendæ.</p>
<p>Cor. 2. Sint <math>h</math> valores variabilis <math>y</math> fluxionalis æquationis <math>(B) y^. ^n + p y^. ^n-1 x^.
+ q y^. ^n-2 x^. ^2 ... S y x^. ^n = X x^. ^n</math> (ubi <math>p, q ... S & X</math> $unt functiones ip$ius <math>x</math>)
<math>α, β, γ δ, &c.</math> tum <math>h - 1</math> diver$i valores variabilis <math>y</math> in æquatione <math>(A)</math>,
cum <math>X = 0;</math> erunt re$pective <math>α - β, α - γ, α - δ, &c.</math> $int <math>l</math> valores
variabilis <math>y</math> in fluxionali æquatione <math>y^. ^n + p y^. ^n-1 x^. + q y^. ^n-2 x^. ^2 ... S y x^. ^n = 7 x^. ^n,</math>
ubi τ etiam e$t functio ip$ius <math>x</math>, haud vero eadem ac <math>X</math>, re$pective <math>π,
ρ, σ, τ, &c.</math> & erunt <math>π - ρ, π - σ, π - τ, &c.</math> re$pective <math>l - 1</math> diver$i
valores ip$ius <math>y</math> in æquatione <math>(A)</math>, ubi <math>τ = 0:</math> & ex <math>n</math> diver$is valori-
bus ip$ius <math>y</math> in æquatione <math>(A)</math>, & uno valore variabilis <math>y</math> in æquatione
<math>(B)</math>; facile erui pote$t generalis valor ip$ius <math>y</math> in fluxionali æquatione
B: &c. Omnia hæc con$tant e $cribendo pro <math>y</math> & ejus fluxionibus in
datâ fluxionali æquatione earum valores prius a$$ignatos.</p>
<p>10. Sit fluxionalis æquatio <math>n</math> ordinis <math>X x^. ^n = P y x^. ^n + Q y^.x^. ^n-1 + Ry^..x^. ^n-2
+ S y^... x^. ^n-3 + T y^.... x^. ^n-4 + V y^. ^5 x^. ^n-5 + ... A y^. ^n, & $i P x^. ^m - m Q^.x^. ^m-1 + m.{m + 1 / 2} R^.. x^. ^m-2 - m · {m + 1 / 2}.{m + 2 / 3} S^... x^. ^m-3 + m · {m + 1 / 2} · {m + 2 / 3} · {m + 3 / 4}T^.... x^. ^m-4 - &c. = 0,</math> cum <math>m</math> vel 1 vel 2 vel 3, &c. ad numerum <math>m</math>; tum
reduci pote$t data æquatio ad fluxionalem æquationem <math>n - m</math> ordinis
ex <math>m</math> repetitis ii$dem multiplicatoribus (x^.).</p>
<p>11. Sit fluxionalis æquatio <math>p y^. ^n + q y^. ^n-1 x^. + r y^. ^n-2 x^. ^2 + s y^. ^n-3 x^. ^3 + t y^. ^n-4 x^. ^4 ...
+ P y^. ^n-m x^. ^m = X x^. ^n</math>, ubi omnes termini po$t terminum <math>P y^. ^n-m</math> ad termi-
num <math>X x^. ^n</math> de$unt; ejus fluens deduci pote$t ex <math>m</math> repetitis multipli-
catoribus, quorum primus inveniri pote$t e fluxionali æquatione

<pb 266>[0318]<rh>DE FLUXIONALIBUS</rh>
<math>p M^. ^m + m p^.. M^. ^m-1 + m · {m - 1 / 2}p^.. M^. ^m-2 + m.{m - 1 / 2} × {m - 2 / 3}p^... M^. ^m-3 + m.{m - 1 / 2}.{m - 2 / 3}.{m - 3 / 4}p^.... M^. ^m-4 + &c.
- q - (m - 1)q^. - (m - 1) × {m - 2 / 2}q^.. - (m - 1) × {m - 2 / 2} × {m - 3 / 3} q^... -&c.
+ # r # + (m - 2)r^. # + (m - 2) × {m - 3 / 2}r^..
- # s - # (m - 3)
+
(m)</math> ordinis; $ecundus, tertius, &c. e fluxionali æquatione <math>(m - 1,
m - 2, &c.)</math> ordinis deduci pote$t.</p>
<h>THEOR. XLIII.</h>
<p>Sit <math>π = 0</math> fluxionalis æquatio, in quâ continentur variabiles <math>x & y</math>
& earum fluxiones <math>x, y^., y^.., y^..., ...y^. ^n</math>; reducatur data æquatio, ita ut in eâ
contineatur tantummodo una dimen$io fluxionis maximi ordinis <math>y^. ^n</math> &
re$ultet <math>α = 0</math>, cujus multiplicator $it <math>M</math>; i.e. $it <math>M α = ρ^. = 0</math>; deinde
ita reducatur <math>ρ = 0</math>, ut $olummodo $implex dimen$io fluxionis <math>y^. ^n-1</math> in
eâ contineatur; & $ic deinceps redintegratâ operatione, u$que donec
inveniatur fluens quæ$ita. Hìc animadvertendum e$t, quod $ingula
extractio radicum tot diver$as fluentes præbet, quot diver$æ radices
exinde re$ultant.</p>
<p>Ex. 1. Sit fluxionalis æquatio <math>y^.. + ({α / x + a} + {β / x + b})y^.x^. + ({α β / (x + a)
(x + b)} - {α / (x + a)^2})y x^. ^2 = X x^. ^2;</math> tum erit <math>y = {1 / (x + a)^α} $. {(x + a)^α / (x + b)^β} x^.
$.(x + b)^β X x^..</math></p>
<p>2. Sit fluxionalis æquatio <math>y^... + ({α / x + a} + {β / x + b} + {γ / x + c})y^.. x^. +

<pb 267>[0319]<rh>ÆQUATIONIBUS.</rh>
({αβ / (x + α)(x + β)} + {αγ / (x + a)(x + c)} + {βγ / (x + b)(x + c)} - {2α / (x + a)^2} -{β / (x + b)^2})y^. x^. + ({αβγ / (x + a)(x + b)(x + c)} - {αβ / (x + a)^2 (x + b)} -{αβ / (x + a)(x + b)^2} - {αγ / (x + a)^2 × (x + c)} + {1 · 2 α / (x + a)^3})y x^. ^3 = X x^. ^3</math>; tum
erit <math>y = {1 / (x + a)^α} × $. {(x + a)^α / (x + b)^β} x^. $. {(x + b)^β / (x + c)^γ} x^. $. (x + c)^γ X x^.</math>.</p>
<p>3. Sit fluxionalis æquatio <math>y^. ^n + ({α / x + a} + {β / x + b} + {γ / x + γ} + {δ / x + δ}+ &c.)y^. ^n-1 x^. + ({αβ / (x + a)(x + b)} + {αγ / (x + a)(x + c)} + {αδ / (x + a)(x + d)}+ {βγ / (x + b)(x + c)} + {βδ / (x + b)(x + d)} + {γδ / (x + c)(x + d)} + &c. -{(n - 1)α / (x + a)^2} - {(n - 2)β / (x + b)^2} - {(n - 3)γ / (x + c)^2} - &c.)y^. ^n-2 x^. ^2 + ({αβγ / (x + a) (x + b) (x + c)}+ {αβδ / (x + a)(x + b)(x + d)} + {αγδ / (x + a)(x + c)(x + d)} +{βγδ / (x + b)(x + c)(x + d)} + &c. - {(n - 2)αβ / (x + a)^2 × (x + b)} - {(n - 2)αβ / (x + a)(x + b)^2}- {(n - 2)αγ / (x + a)^2 (x + c)} - {(n - 3)αγ / (x + a)(x + c)^2} - {(n - 3)βγ / (x + b)^2 (x + c)} -{(n - 3)βγ / (x + b)(x + c)^2} - {(n - 3)αδ / (x + a)^2 (x + d)} - {(n - 3)αδ / (x + a) (x + d)^2} - &c. +{(n - 1) · (n - 2)α / (x + a)^3} + {(n - 2).(n - 3)β / (x + b)^3} + {(n - 3) · (n - 4)γ / (x + c)^3} + &c.) y^. ^n-3
x^. ^3 &c. = X x^. ^n;</math> tum erit <math>y = {1 / (x + a)^z} × $. {(x + a)^α / (x + b)^β} x^. $. {(x + b)^β / (x + c)^γ} x^.
$. {(x + c)^γ / (x + d)^δ} x^. × $. &c. × X x^.</math>.</p>

<pb 268>[0320]<rh>DE FLUXIONALIBUS</rh>
<h>THEOR. XLIV.</h>
<p>Sit quæcunque fluxionalis æquatio <math>a y^. ^n + b = 0,</math> ubi <math>a & b</math>
funt functiones quantitatis <math>x</math>, ejus incrementi <math>x^.; y, y^., y^.., &c. ad
y^. ^n-1</math>, in quâ nullæ continentur fluentes; ducatur hæc æquatio in fun-
ctionem <math>p</math> quantitatum <math>x & x^.</math>, & $i re$ultans æquatio <math>p a y^. ^n + p b = 0</math>
$it integrabilis, tum reduci pote$t data fluxionalis æquatio ad fluxio-
nalem æquationem <math>n - 1</math> ordinis ope multiplicatoris <math>p</math>, qui ctiam de-
duci pote$t ope fluxionalis æquationis <math>n - 1</math> ordinis.</p>
<p>Inveniatur enim fluens fluxionis <math>p a y^. ^n</math> ex hypothe$i quod omnes
quantitates <math>x, x^., y^., ... y^. ^n-2</math> præter <math>y^. ^n-1</math> $int con$tantes, quæ $it <math>A</math>; deinde
fupponantur omnes quantitates <math>x, y, y^., ... y^. ^n-2</math> præter <math>y^. ^n-1 & x^.</math> in quan-
titate <math>A</math> variabiles, & ex hâc hypothe$i inveniatur fluxio <math>B</math> quantitatis
<math>A</math>, quæ auferatur de <math>p b</math>; & deinde ex eodem proce$$u repetito, i.e. ex
$upponendo omnes quantitates x, x^., y, y^., ... y^. ^n-3</math> præter <math>y^. ^n-2</math> con$tantes,
inveniatur fluens quantitatis <math>p b - B</math>, & $ic deinceps u$que donec $o-
lummodo contineantur in inventâ fluente <math>A′</math> variabiles <math>y, x & x^.</math>: in-
veniatur ejus fluxio <math>H</math> ex hypothe$i quod <math>x</math> $olummodo $it variabilis;
fiant corre$pondentes termini, i.e. termini, in quibus eædem inveni-
untur dimen$iones quantitatis <math>y, &c.</math> inter $e æquales, &c. & re$ultant
fluxionales æquationes haud majoris quam <math>n</math> ordinis, quarum varia-
biles quantitates $unt <math>p & x</math>; pro <math>p</math> in his æquationibus $cribatur
<math>e^$. v x^., &c.</math> & æquatio re$ultans relationem inter <math>x, x^., v</math> & ejus fluxio-
nes de$ignans haud majoris erit quam <math>n - 1</math> ordinis.</p>
<p>Ex. · Sit fluxionalis æquatio <math>α y^. + β x^. = 0</math>, ducatur hæc æquatio
in <math>p</math>, quæ $it functio quantitatis <math>x</math>, & re$ultat <math>p α y^. + p β y^. = 0</math>; in-
veniatur fluens fluxionis <math>α y^.</math> ex hypothe$i quod <math>y</math> $olummodo $it va-
riabilis, quæ $it π; deinde inveniatur fluxio rectanguli <math>p × π</math> ex hy-

<pb 269>[0321]<rh>ÆQUATIONIBUS.</rh>
pothe$i, quod <math>x</math> $olummodo $it variabilis, quæ erit <math>p x^.({π^. / x^.}) + π p^.
= p β x^.</math>; in hâc æquatione pro <math>p</math> $cribatur <math>e^$.v x^.</math> & re$ultat <math>({π^. / x^.}) + π v
= β</math>, in quâ nullæ continentur fluxiones, unde con$equitur valor
quantitatis <math>v</math>.</p>
<p>Cor. · Si duæ vel plures re$ultent æquationes ex corre$pondenti-
bus terminis inter $e æqualibus e$$e $uppo$itis, tum plerumque e me-
thodo prius traditâ reducendi plures æquationes in unam, ita ut va-
riabiles & earum fluxiones exterminentur, vel erui pote$t quantitas
<math>p</math>, vel faltem erui pote$t fluxionalis æquatio inferioris ordinis.</p>
<h>THEOR. XLV.</h>
<p>1. Si in datâ æquatione algebraicâ vel fluxionali contineatur ge-
neralis functio quantitatis (<math>π</math>), quæ $it functio quantitatum <math>x & y</math>, &
earum fluxionum; tum in multis ca$ibus, $i modo $cribatur <math>v = π</math>, &
ita reducantur duæ æquationes, ut exterminentur altera variabilis <math>x
vel y</math> & ejus fluxiones; re$ultabit æquatio, cujus fluens innote$cit.</p>
<p>2. Si vero fluens æquationis re$ultantis vel ex hâc $ub$titutione
vel ex quâcunque aliâ haud generaliter inveniri po$$it; in multis ca-
$ibus ita a$$umi po$$unt coefficientes, exponentes, &c. ut con$tabunt
ca$us particulares, in quibus fit re$ultans æquatio formulæ, cujus
fluens innote$cit.</p>
<p>3. 1. Dividatur data fluxionalis æquatio in duas partes, & $i fluens
(<math>P</math>) ex unâ parte exprimi po$$it, ducatur data æquatio in functionem
fluentis <math>P</math>, & inveniantur ca$us, in quibus fluens ex alterâ parte
etiam exprimi pote$t; tum con$equuntur æquationes, quarum fluentes
generales deduci po$$unt.</p>
<p>2<_>do</_>. Ducatur prior pars (<math>P^.</math>) datæ æquationis in functionem quan-
titatis <math>P</math>; $ecunda vero in quamcunque quantitatem quantitati <math>P</math>
æqualem $ed haud eandem, & inveniantur ca$us in quibus fluens

<pb 270>[0322]<rh>DE FLUXIONALIBUS</rh>
producti inveniri pote$t; & con$equuntur æquationes, quarum parti-
culares fluentes deduci po$$unt.</p>
<h it>_PROB. LIX._
Datâ æquatione fluxionali; invenire alias, quarum fluentes e datâ flu-
xionali æquatione deduci po$$unt.</h>
<p>A$$umantur æquationes relationem inter variabiles datæ & quæ$itæ
æquationis exprimentes, ita quidem ut relatio inter variabiles quæ$itæ
e relatione inter variabiles datæ æquationis a$$ignari pote$t; & exinde
deducatur æquatio relationem inter variabiles quæ$itæ æquationis
quantitates de$ignans, re$ultabit æquatio quæ$ita.</p>
<p>Ex. Sit data æquatio <math>y^. + a y^2 x^. = b x^m x^.</math>, in hâc æquatione $cri-
batur <math>x^n z = y</math>; tum exorietur æquatio <math>z^. + {n z x^. / x} + a x^n z^2 x^. = b x^m x^.</math>:
& facile patebit, $i modo detur fluens æquationis <math>y^. + a y^2 x^. = b x^m x^.</math>,
exinde deduci po$$e fluentem æquationis <math>z^. + {n z x^. / x} + a x^n z^2 x^. = b x^m x^.</math>;
etiamque $i modo cogno$cantur ca$us in quibus deduci pote$t fluens
prioris, facile etiam con$tari ca$us, in quibus deduci pote$t fluens po-
$terioris æquationis.</p>
<h>PROB. LX.</h>
<p it>Dato valore <math>M</math> variabilis quantitatis <math>y</math> in fluxionali æquatione contentæ,
i. e. $it <math>M</math> data functio alterius variabilis quantitatis <math>x</math>; in nonnullis ca-
$ibus per $ub$equentes methodos deducere generalem datæ æquationis fluentis
valorem, i. e. in genere corrigere datam æquationem.</p>
<p>Scribatur pro <math>y</math> ejus valor a$$umptus <math>M + z vel M z, &c.</math> in datâ
fluxionali æquatione; & re$ultantis æquationis nonnunquam detegi
pote$t generalis fluens.</p>

<pb 271>[0323]<rh>ÆQUATIONIBUS.</rh>
<p>Ex. 1. Sit <math>y^. + P y x^. + Q y^2 x^. + R x^. = 0</math>, ubi literæ <math>P, Q & R</math>
$int functiones quantitatis <math>x</math>; $ed per hypothe$in <math>M</math> e$t valor quanti-
tatis <math>y</math>, & con$equenter <math>M^. + P M x^. + Q M^2 x^. + R x^. = 0</math>; $cribatur
<math>M + z</math> pro <math>y</math> in datâ æquatione, & re$ultat <math>z^. + P z x^. + 2 Q M z x^. +
Q z^2 x^. = 0</math>; deinde $cribatur <math>{1 / v}</math> pro <math>z</math>, & proveniet <math>- {v^. / v^2} + {P x^. / v} +{2 Q M x^. / v} + {Q x^. / v^2} = 0</math>, quâ ductâ in <math>v^2</math>, re$ultat <math>- v^. + (P + 2 Q M)
v x^. + Q x^. = 0</math>, quæ e$t æquatio formulæ prius traditæ; ducatur
hæc æquatio in <math>S = e^-$.(P+2QM)x^.</math>, & exorietur æquatio, cujus fluens
erit <math>ve^-$.(P+2QM)x^. - $. Q S x^. = con$t.</math> $ed <math>v = {1 / y - M}</math>; & con$equenter
ducatur data æquatio in <math>v^2 e^-$.(P+2QM)x^. = {1 / (y - M)^2} e^-$.(P+2QM)x^. ={S / (y - M)^2}</math>; & re$ultat æquatio integralis <math>S v - $. Q S x^. = con$t</math>.</p>
<p>Cor. 1. Omnes quantitates quæ in datam æquationem ductæ, eam
reddunt integrabilem, in hâc formulâ <math>{S / (y - M)^2} ×</math> in functionem quam-
cunque quantitatis <math>({S / y - M} - $. Q S x^.)</math>, ubi <math>M</math> e$t cognita functio
quantitatis <math>x, & S = e^-$.(P+2QM)x^.</math>, continentur.</p>
<p>Cor. 2. Exhinc deduci pote$t generalis fluens æquationis <math>z^.. + S z^. x^.
+ T z x^.^2 = {E x^.^2 / z^3} e^-2 $. S x^.</math>; inveniatur fluens <math>v</math> æquationis <math>v^. + v^2 x^.
+ S v x^. + T x^. = 0</math>, fingatur <math>V = $. e^-2 $. v x^.-$. S x^. x^.</math>, eritque <math>z = e^$. v x^.
√(A + B V + C V^2)</math>, ubi literæ <math>A, B, C, &c.</math> $unt invariabiles quan-
titates, & <math>A C - {1 / 4} B^2 = E</math>.</p>
<p>Cor. 3. Sit æquatio fluxionalis <math>P = q</math>, ubi <math>P</math> $it functio variabilium
quantitatum <math>y & x</math> & earum fluxionum, <math>q</math> vero functio quantitatum
<math>x & x^.</math>; $cribantur in datâ æquatione <math>P = q</math> pro <math>y, y^., y^.., &c.</math> re$pective
<math>z + E, z^. + E^., z^.. + E^.., &c.</math> ubi <math>E</math> $it particularis valor quantitatis <math>y</math>,

<pb 272>[0324]<rh>DE FLUXIONALIBUS</rh>
tum re$ultabit æquatio, in quâ deficit terminus, qui fuit functio
$olummodo quantitatum <math>x & x^.</math>.</p>
<p>Ex. g. Sit <math>P y^.. + Q x^. y^. + R y x^.^2 = 0</math>, ubi literæ <math>P, Q & R</math> re$pec-
tive denotant functiones quantitatis <math>x</math>: $it <math>X</math> functio quantitatis <math>x</math>
quæ $it particularis valor quantitatis <math>y</math> in datâ æquatione integrali;
pro <math>y^., y^. & y^..</math> in datâ æquatione $cribantur re$pective <math>X z, X z^. + z X^.,
z X^.. + 2 z^. X^. + X z^..</math>, & re$ultat <math>(P z X^.. + Q z X^. x^. + R z X x^.^2) +
2 P X^. z^. + Q X x^. z^. + P X z^.. = 0</math>, $ed quoniam <math>X</math> e$t valor quantitatis
<math>y</math> in datâ æquatione, erit <math>P z X^.. + Q z X^. x^. + R z X x^.^2 = 0</math>, unde
<math>2 P X^. z^. + 2 X x^. z^. + P X z^.. = 0</math>.</p>
<h>PROB. LXI.</h>
<p it>Sint duæ vel plures <math>n</math> æquationes tres vel plures (<math>n + 1</math>) variabiles
quantitates & earum fluxiones habentes; ita reducere has æquationes in
unam, ut exterminentur omnes præter duas variabiles quantitates & earum
fluxiones, & haud nunquam vel re$ultent æquationes, quarum fluentes inno-
te$cunt; vel ita a$$umi po$$int coefficientes, ut evadant æquationes, quarum
fluentes dantur: exemplis docebitur prob. re$olutio.</p>
<p>1. Sint <math>x^. + (C x + D y) t^. = 0 & y^. + (K x + L y)t^. = 0</math>, unde
<math>- t^. = {x^. / C x + D y} = {y^. / K x + L y}</math> homogenea æquatio, cujus fluens
innote$cit.</p>
<p>2. Sub$tituantur pro quibu$dam functionibus quantitatum in datis
æquationibus contentarum quædam functiones novarum variabilium
a$$umptarum; & exinde nonnunquam deduci po$$unt æquationes,
quarum fluentes dantur.</p>
<p>Ex. 1. Sint æquationes <math>T′ x^. + T″ t^.(C x + D y) + θ t^. = 0 & T′ y^. +
T″ t^.(K x + L y) + θ′ t^. = 0</math>; ubi <math>T′, T″, θ, θ′</math> $unt functiones quantita-
tis <math>t</math>; ducatur po$terior æquatio in <math>v</math>, & ad productum re$ultans adda-
tur prior, & exorietur <math>T′ x^. + v T′ y^. + t^. (T″ (C x + D y) + v T″ (K x
+ L y) + θ + v θ′) = 0</math>: $it <math>(C + K v) x + (D + L v) y = n (x +

<pb 273>[0325]<rh>ÆQUATIONIBUS.</rh>
v y)</math>, unde <math>C + K v:D + L v::1:v</math>, & exinde acquiri pote$t quan-
titas <math>v</math>; pro <math>x + v y</math> in æquatione exortâ $cribatur <math>u</math>, & re$ultat <math>T′ u^.
+ T″ n u t^. + (θ + v θ′) t^. = 0</math>, cujus fluens innote$cit.</p>
<p>Ex. 2. Sint <math>x^. + (a x + b y + c z) t^. = 0, y^. + (e x + f y + g z) t^. = 0
& z^. + (h x + m y + n z) t^. = 0</math>; ducantur $ecunda & tertia æquatio-
nes in coefficientes invariabiles deducendas <math>v & μ</math>, & ad $ummam pro-
ductorum re$ultantium addatur prima æquatio; deinde $upponatur
<math>a + e ν + h μ = {b + f ν + m μ / ν} = {c + g ν + n μ / μ}</math>, & fingatur <math>x + ν y
+ μ z = u</math>; & re$ultabit æquatio <math>u^. + (a + e ν + h μ) u t^. = 0</math> cujus
fluens inveniri pote$t. Hic <math>ν & μ</math> tres habent valores, in præcedente
exemplo <math>v</math> duos habet valores.</p>
<p>Ex. 3. Sint <math>n</math> fluxionales æquationes <math>T (a x^. + b y^. + c z^. + d v^. + &c.)
+ T′ (e x + f y + g z + b v + &c.)t^. + θ t^. = 0, T(a′ x^. + b′ y^. + c′ z^. + d′ v^.
+ &c.) + T′(e′ x + f′ y + g′ z + h′ v + &c.) t^. + θ′ t^. = 0, T (a″ x^. + b″ y^.
+ c″ z^. + d″ v^. + &c.) + T′ (e″ x + f″ y + g″ z + h″ v + &c.) t^. + θ ″t^.
= 0, &c.</math>; ducantur hæ æquationes re$pective in <math>1, λ, μ, ν, &c.</math>; &
pro <math>a + λ a′ + μ a″ + ν a′″ + &c., b + λ b′ + μ b″ + ν b′″ + &c., c +
λ c′ + μ c″ + ν c′″ + &c., &c.</math> $cribantur re$pective <math>α, β, γ, δ, &c.</math>; deinde
a$$umatur <math>{e + λ e′ + μ e″ + ν e′″ + &c. / α} = {f + λ f′ + μ f″ + ν f′″ + &c. / β} ={g + λ g′ + μ g″ + ν g′″ + &c. / γ} = &c.</math>: ex hi$ce æquationibus inve-
niantur valores quantitatum <math>λ, μ, ν, &c.</math>; & exinde quantitates <math>α, β, γ,
&c.</math>; quibus inventis, re$ultat æquatio <math>T u^. + {1 / α} (e + λ e′ + μ e″ + ν e′″ +
&c.) T′ u t^. + (θ + λ θ′ + μ θ″ + &c.) t^. = 0</math>, cujus fluens innote$cit, ubi
<math>u = (a + λ a′ + μ a″ + &c.) x + (b + λ b′ + μ b″ + &c.) y + (c + λc ′
+ μc ″ + &c.) z + &c.</math></p>
<p>Per quantitates <math>T′, T′, θ, θ′, θ″, &c.</math> de$ignantur datæ functiones
quantitatis <math>t.</math></p>
<p>Ex. 4. Sint <math>n</math> fluxionales æquationes $uperiorum (<math>m</math>) ordinum, viz.
<math>T k v^. ^m + T′ l v^.^m-1 + T″ n v^.^m-2 + ... + T′^m (e x + f y + g z + &c.) t^. +

<pb 274>[0326]<rh>DE FLUXIONALIBUS</rh>
θ t^. = 0, T k′ v^. ^m + T′ l′ v^. ^m-1 + T″ n′ v^. ^m-2 + &c. + ... + T′<_>m</_> (e′ x + f′ y +
g′ z + &c.) t^. + θ t^. = 0, T k″ v^. ^m + T′ l″ v^. ^m-1 + T″ n″ v^. ^m-2 + ... + T′<_>m</_> (e″ x
+ f″ y + g″ z + &c.) t^. + θ″ t^. = 0</math>, ubi <math>v = a x + b y + c z + d v +
&c.</math>; ducantur hæ æquationes re$pective in <math>1, λ, μ, ν, &c.</math>; & a$$u-
matur <math>{e + λ e′ + μ e″ + &c. / a} = {f + λ f′ + μ f″ + &c. / b} = {g + λ g′ + μ g″ + &c. / c}= &c.</math> & exinde per $implices æquationes inveniantur quantitates <math>λ,
μ, ν, &c.</math>; quibus inventis re$ultat fluxionalis æquatio <math>T (k + λ k′ +
μ k″ + &c.) v^. ^m + T′ (l + λ l′ + μ l″ + &c.) v^. ^m-1 + T″ (n + λ n′ + μ n″ +
&c.) v^. ^m-2 + &c. + T′^m × ({e + λ e′ + μ e″ + &c. / a}) v t^. + (θ + λ θ′ + μ θ″ +
&c.) t^. = 0</math>.</p>
<p>Ex. 5. Sint <math>m</math> æquationes (<math>m + 1</math>) variabiles quantitates (<math>x, y, z, &c.
& t</math>) involventes, & $i in $ingulis terminis datæ æquationis $olum-
modo contineatur vel una dimen$io quantitatum (<math>x, y, z, v, &c.</math>), vel
earum fluxionum quorumcunque ordinum; vel nulla; & contineatur
unus terminus in $ingulis æquationibus, qui e$t functio quantitatis <math>t</math>
in <math>t^.^n</math>, ubi <math>t</math> e$t quantitas, quæ fluit uniformiter; i.e. $int æquationes
huju$ce formulæ <math>T x^. ^n + t y^. ^n + τ z^. ^n + &c. + T′ x^. ^n-1 t^. + t′ y^. ^n-1 t^. + τ′ z^. ^n-1 t^. +
&c. + &c. + .... + T′^n-1 x^. t^.^n-1 + t′^n-1 y^. t^.^n-1 + τ′^n-1 z^. t^.^n-1 + &c. + T′^n x t^.^n
+ t′^n y t^.^n + τ′^n z t^.^n + &c. + θ t^.^n = 0</math>, ubi <math>T, T′, T″, &c., t, t′, t″, &c.,
τ, τ′, τ″, &c. θ</math> $unt functiones quantitatis <math>t</math>; reducantur hæ æquatio-
nes per prob. 26. in unam, ita ut exterminentur omnes quantitates <math>y,
z, v, &c.</math>, & earum fluxiones præter unam <math>x</math> & ejus fluxiones <math>& t & t^.</math>,
tum re$ultabit æquatio huju$ce formulæ <math>P x^. ^α + Q x^. ^α-1 t^. + R x^. ^α-2 t^.^2 +
S x^. ^α-3 t^.^3 + &c. = L t^.^α</math>, ubi <math>P, Q, R, S, &c. & L</math> denotant functiones
quantitatis <math>t</math>, quæ facile deduci po$$unt.</p>
<p>Et $imiliter erui po$$unt formulæ æquationum re$ultantium ex
datis æquationibus, quarum formulæ haud $unt multo magis com-
po$itæ quam præcedentes.</p>

<pb 275>[0327]<rh>ÆQUATIONIBUS.</rh>
<p>Cor. Sint <math>m</math> æquationes (<math>m + 1</math>) variabiles quantitates (<math>x, y, z, v,
&c., & t</math>) involventes, quarum formulæ $int <math>a x^. ^n + b y^. ^n + c z^. ^n + &c. +
a′ x^. ^n-1 t^. + b′ y^. ^n-1 t^. + c′ y^. ^n-1 t^. + &c. + a″ x^. ^n-2 t^.^2 + b″ y^. ^n-2 t^.^2 + c″ y^. ^n-2 t^.^2 + &c. +
... + a′^n-1 x^. t^.^n-1 + b′^n-1 y^. t^.^n-1 + c′^n-1 z^. t^.^n-1 + &c. + a′^n x t^.^n + b′^n y t^.^n +
c′^n z t^.^n + &c. + T t^.^n = 0</math>, ubi literæ <math>a, a′, a″, ... a′^n; b, b′, b″, &c., ...
b′^n; c, c′, c″, &c., ... c′^n</math>; qua$cunque invariabiles quantitates re$pe-
ctive denotant, & <math>T</math> e$t quæcunque functio quantitatis <math>t</math>; reducantur
hæ <math>m</math> æquationes per prob. 26. in unam, ita ut exterminentur omnes
variabiles & earum fluxiones præter duas, viz. <math>x</math> & ejus fluxiones, & <math>t
& t^.</math>; tum re$ultabit æquatio formulæ <math>A x^. ^α + B x^. ^α-1 t^. + C x^. ^α-2 t^.^2 + D x^. ^α-3 t^.^3
+ ... + H x^. t^.^α-1 + I x t^.^α = T′ t^.^α</math>; literis <math>A, B, C, D, &c., H & I</math> in-
variabiles quantitates denotantibus, & <math>T</math> denotante functionem quan-
titatis <math>t</math>: huju$ce æquationis fluens in theor. 42. tradita fuit.</p>
<p>3. Re$olvi etiam po$$unt duæ vel plures <math>n</math> æquationes tres vel plu-
res (<math>n + 1</math>) variabiles quantitates habentes, a$$umendo generales
quantitates, quæ in $e continent re$olutiones quantitatum quæ$itas;
& eas pro valoribus e $ingulis variabilibus quantitatibus in datis
æquationibus contentis $cribendo, & corre$pondentes terminos re$ul-
tantes æquando.</p>
<p>Ex. 1. Sint <math>x^. + (a x + b y + c z) t^. = 0, y^. + (e′ x + f y + g z) t^.
= 0, z^. + (h x + m y + n z) t^. = 0</math>; a$$umantur <math>x = A e^f t + B e^g t +
H e^b t; y = A′ e^f t + B′ e^g t + C′ e^b t; & z = A″ e^f t + B″ e^g t + C″ e^b t</math>, ubi
<math>A, B, C, &c.; A′, B′, C′, &c.; A″, B″, &c.</math> $unt invariabiles quanti-
tates; $eribantur hæ quantitates pro $uis valoribus in datis æquationi-
bus, & e re$ultantibus æquationibus inve$tigari pote$t fluens quæ$ita.</p>
<p>Ex. 2. Sint <math>x^. + a y^. + T (c x + a y) t^. + θ t^. = 0, & y^. + b x^. +
T (f x + g y) t^. + ε t^. = 0</math>; $upponatur <math>x = A e^f $. T t^. + B e^g $. T t^. +
e^f $. T i $. e^-$ $. T i (E θ + H ε) t^. + e^g $. T t^. $. e^-g $. T t^. (L θ + M ε) t^.; y =
A′ e^f f. T t^. + B′ e^g $. T t^. + e^f f. T t^. $. e^- f $. T t^. (E′ θ + H′ ε) t^. + e^g $. T t^. $. e^-g $. T t^.
(L′ θ + M′ ε) t^.</math>, ubi <math>A, A′, B, B′, E, H, E′, H′, L, M, L′, M′</math>, $unt inva-
riabiles; & <math>T, θ & ε</math> $unt functiones quantitatis <math>t</math>; & deinde $cribantur

<pb 276>[0328]<rh>DE FLUXIONALIBUS</rh>
hæ quantitates, &c. pro $uis valoribus <math>x, y, x^., y^.</math>, in datis æquationi-
bus, & exinde inveniri pote$t fluens quæ$ita.</p>
<p>Facile deduci po$$unt infinitæ æquationes huju$modi, quarum
fluentes innote$cunt; a$$umantur enim æquationes pro fluentibus
quæ$itis, quarum inveniantur fluxiones, & methodis prius traditis
con$equuntur infinitæ æquationes fluxionales, quarum fluentes ac-
quiri po$$unt. Aliter: a$$umantur quæcunque æquationes relationem
inter variabiles quantitates, &c. exprimentes pro fluentibus ip$is; deinde
a$$umantur quæcunque fluxionales quantitates <math>l, m, n, &c.</math> & ex
a$$umptis æquationibus & earum fluxionibus, &c. inveniantur quan-
titates (<math>P, Q, R, &c.</math>) quæ æquant a$$umptas quantitates <math>l, m, n, &c.</math>
re$pective, tum <math>l = P, m = Q, n = R, &c.</math> erunt æquationes, qua-
rum fluens particularis erit a$$umptæ æquationes prædictæ.</p>
<h>THEOR. XLVI.</h>
<p>Sit æquatio fluxionalis <math>A = 0</math> ordinis (<math>n</math>), in quâ fluit uniformiter
<math>x</math>, cujus fluens $it <math>B = 0</math>; deinde transformentur utræque æquationes
<math>A = 0 & B = 0</math> re$pective in duas alias <math>C = 0 & D = 0</math>, $cribendo
pro <math>y^.., - {y^.. x^.. / x^.}</math>; & $ic deinceps, ut docetur in prob. 34; tum <math>D = 0</math> erit
fluens fluxionalis æquationis <math>C = 0</math>.</p>
<p>Ex. Sit æquatio <math>y^. x^. - x y^.. - a y^.. - {x y^.^2 / b} = 0</math>, in quâ fluit unifor-
miter <math>x</math>; $cribatur pro <math>y^..</math> ejus valor e prob. 34. deductus - <math>{y^. x^.. / x^.}</math>, &
re$ultat <math>x^. y^. + {x y^. x^.. / x^.} + {a y^. x^.. / x^.} - {x y^.^2 / b} = 0</math>; ducatur hæc æquatio in <math>x^.</math>
& re$ultat <math>x^.^2 y^. + x y^. x^.. + a y^. x^.. - {x x^. y^.^2 / b} = 0</math>, cujus fluens e$t <math>x x^. y^. +
a y^. x^. - {x^2 y^.^2 / 2 b} = 0</math>; quæ $i modo generaliter corrigatur, evadet <math>x x^. y^. +
a y^. x^. - {x^2 y^.^2 / 2 b} + A y^.^2 = 0</math>, ubi <math>A</math> denotat quantitatem invariabilem

<pb 277>[0329]<rh>ÆQUATIONIBUS.</rh>
ad libitum a$$umendam: aliter ducatur data æquatio in <math>{x^. / y^.^2}</math>, & re$ul-
tat <math>{x^.^2 y^. - x x^. y^.. / y^.^2} - {a x^. y^.. / y^.^2} - {x x^. / b} = 0</math>, cujus fluens generalis erit <math>{x x^. / y^.}+ {a x^. / y^.} - {x^2 / 2 b} = A</math> con$tant. quant. quæ æquatio eadem e$t ac præ-
cedens.</p>
<p>Cor.. Sit æquatio <math>p x^. + q y^. = 0</math>, inveniatur ejus fluxio ex hypo-
the$i quod <math>x^.</math> $it con$tans, & re$ultat <math>p^. x^. + q^. y^. + q y^.. = 0</math>, cujus
generalis fluens erit <math>p x^. + q y^. = C x^.</math>; deinde inveniatur ejus fluxio
ex hypothe$i quod <math>y^.</math> $it con$tans & re$ultat <math>p^. x^. + q^. y^. + p^. x^.. = 0</math>,
cujus fluens erit <math>p x^. + q y^. = C y^.</math>; quæ fluentes minime eædem $unt:
in æquatione <math>p^. x^. + q^. y^. + q y^.. = 0</math> pro <math>y^..</math> $cribatur ejus valor præ-
dictus <math>- {x^.. y^. / x^.}</math>, & re$ultat <math>p^. x^. + q^. y^. - {q x^.. y^. / x^.} = 0</math>, ducatur hæc æquatio
in <math>{1 / x^.}</math>, & re$ultat fluxio, cujus fluens e$t <math>p + {q y^. / x^.} = C</math>; vel quod idem
e$t <math>p x^. + q y^. = C x^.</math>; quæ eadem e$t ac fluens æquationis <math>p x^. + q y^. +
q y^.. = 0</math>.</p>
<p>Idem etiam affirmari pote$t de fluentibus quarumcunque fluxionum
hoc modo transformatarum: & e principiis prius traditis & datis
multiplicatoribus datarum fluxionalium æquationum facile erui po$-
$unt multiplicatores ex con$imili $ub$titutione fluxionalium æquatio-
num hoc modo transformatarum.</p>
<h>PROB. LXII.</h>
<p it>Datâ fluxionali æquatione duas variabiles quantitates <math>x & y</math> & earum
fluxiones habente, invenire utrum altera (<math>y</math>) exprimi pote$t in algebraicis
& haud exponentialibus terminis alterius (<math>x</math>), necne.</p>
<p>Inveniantur $inguli valores (<math>A, B, C, &c.</math>) quantitatis <math>y</math> (eâdem cor-
rectione adhibitâ) in terminis progredientibus $ecundum dimen$iones

<pb 278>[0330]<rh>DE FLUXIONALIBUS</rh>
alterius <math>x</math>; deinde per Med. Algeb. inveniatur annon $umma e $in-
gulis hi$ce valoribus $it algebraica & rationalis functio literæ <math>x</math>.</p>
<p>Ducantur quique duo e $ingulis prædictis valoribus in $e$e, deinde
inveniatur utrum aggregatum e $ingulis prædictis rectangulis $it al-
gebraica & rationalis functio prædictæ literæ <math>x</math>, necne.</p>
<p>Et $ic ducantur quique tres, quatuor, &c. prædicti valores in $e$e,
& inveniatur, utrum aggregata e $ingulis hi$ce contentis $int ratio-
nales & algebraicæ functiones literæ <math>x</math>, necne; $i vero detegantur
prædictæ rationales functiones, tum datur æquatio algebraica, quæ
exprimit relationem inter <math>x & y</math>; $in haud exprimi po$$int prædicta
aggregata per illas rationales functiones, tum nulla a$$ignari pote$t
æquatio algebraica, quæ exprimit relationem inter <math>x & y</math>.</p>
<p>Et $ic de inveniendis algebraicis æquationibus exprimentibus rela-
tiones inter qua$cunque functiones literarum <math>x & y</math>.</p>
<p>Omnia hæc etiam ad plures fluxionales æquationes applicari
po$$unt.</p>
<h it>_PROB. LXIII._
Datâ fluxionali æquatione exponentiales quantitates involvente; invenire
fluentem datæ æquationis, $i modo finitis terminis exprimi po$$it.</h>
<p>Per prob. 3. inveniri pote$t quantitas, quæ in datam exponentialem
datæ æquationis quantitatem ducta, exponentialem fluxionem dat,
cujus fluens inveniri pote$t; ducatur data æquatio in quantitatem
inventam, & re$ultantis æquationis inveniatur fluens; & confit
problema.</p>
<p>Sæpe vero exigit problema, ut prius transferantur exponentiales
quantitates e terminis datæ æquationis, in quibus continentur, in
reliquos; deinde per præcedentem methodum progrediendum e$t.</p>
<p>Ex. 1. Sit exponentialis æquatio fluxionalis <math>y^2 x^. + y x log. x × y^. + x y^.
= {p x^. + q y^. / x^y-1}</math>, ubi <math>p & q</math> $unt functiones literarum <math>x & y</math>, & $it fluxio
<math>p x^. + q y^.</math> integrabilis; ducatur data æquatio in <math>x^y-1</math>, & re$ultat æquatio,

<pb 279>[0331]<rh>ÆQUATIONIBUS.</rh>
cujus fluens invenietur <math>x^y × y = $. (p x^. + q y^.) + a</math>, ubi <math>a</math> denotat
invariabilem quantitatem ad libitum a$$umendam.</p>
<h it>_PROB. LXIV._
Datâ fluxionali æquatione, in quâ fluentes continentur; invenire ejus
fluentialem æquationem.</h>
<p>Hoc $æpe perfici pote$t ex ii$dem principiis, ac ea; quæ in præ-
cedente problemate de exponentialibus & fluxionalibus æquationibus
tradita fuere.</p>
<p>Ex. · Sit æquatio fluxionalis <math>r x^. $. x^m y^. + x^m+1 y^. = p x^. × x^-r+1</math>;
ducatur hæc æquatio in <math>x^r-1</math> & re$ultat <math>r x^r-1 x^. $. x^m y^. + x^m+r y^. = p x^.</math>,
cujus fluens e$t <math>x^r $. x^m y^. = $. p x^. + a</math>.</p>
<p>Æquationes in quibus fluentiales quantitates involvuntur, facile
reduci po$$unt, ita ut fluentiales quantitates exterminentur.</p>
<p>Eadem etiam applicari po$$unt ad plures fluxionales æquationes
fluentiales & exponentiales quantitates involventes.</p>
<h>PROB. LXV.</h>
<p it>Datâ fluxionali æquatione duas variabiles quantitates <math>x & y</math> & earum
fluxiones involvente; invenire utrum <math>x & y</math> $int algebraicæ, exponentiales
vel fluentiales functiones novæ a$$umptæ quantitatis <math>z</math>.</p>
<p>A$$umantur pro variabilibus <math>x & y</math> functiones quantitatis <math>z</math> gene-
ralibus terminis expre$$æ, quæ nece$$ario exprimunt valores quantita-
tum <math>x & y</math>; $cribantur hæ functiones pro $uis valoribus <math>x & y</math> in datâ
æquatione, & ex æquatis terminis re$ultantis æquationis corre$pon-
dentibus inveniantur functiones quantitatis <math>z</math> quæ$itæ, $i modo tales
recipiat data æquatio; & perficitur problema.</p>
<p>Cor. · Hinc inveniri po$$unt infinitæ æquationes fluxionales rela-
tionem inter duas variabiles quantitates <math>x, y</math> & earum fluxiones ex-
primentes, quarum variabiles (<math>x & y</math>) majori facilitate exprimi po$$unt

<pb 280>[0332]<rh>DE FLUXIONALIBUS</rh>
in terminis a$$umptæ variabilis <math>z</math>; quam variabilis <math>x</math> exprimi pote$t
in terminis quantitatis <math>y</math>, vel <math>y</math> in terminis quantitatis <math>x</math>.</p>
<p>A$$umantur valores variabilium <math>x & y</math> in terminis novæ a$$umptæ
quantitatis <math>z</math> inter $e re$pective re$pondentes; reducantur hæ duæ
a$$umptæ æquationes in unam, ita ut exterminetur variabilis quan-
titas <math>z</math>, & re$ultat quæ$ita æquatio relationem inter <math>x & y</math> & earum
fluxiones exprimens, cujus fluens datur in terminis a$$umptæ quan-
titatis <math>z</math>.</p>
<p>Ex. 1. A$$umantur <math>log. z + a z = y, & log. z + f z = x</math>, unde
<math>(a - f)z = y - x, & z = {y - x / a - f}</math>; $ed e primâ æquatione $equitur
<math>{z^. / z} + a z^. = y^.</math>; $cribantur igitur <math>{y - x / a - f} & {y^. - x^. / a - f}</math> pro <math>z & z^.</math> in hâc
æquatione & re$ultat æquatio quæ$ita <math>{y^. - x^. / y - x} + {a y^. - a x^. / a - f} = y^.</math>.</p>
<p>Ex. 2. A$$umantur <math>(log. z)^2 + a z = y, & log. z + f z = x</math>; unde
<math>2 f z log. z + f^2 z^2 - a z = x^2 - y</math>, & exinde <math>f^2 z^2 + a z = - x^2 +
2 f x z + y</math>, & con$equenter <math>z = {2 f x - a / 2 f^2} ∓ √({(2 f x - a)^2 / 4 f^4} - x^2 + y)</math>;
$cribatur hæc quantitas pro <math>z</math>, & ejus fluxio pro <math>z^.</math> in æquatione <math>{z^. / z} +
f z^. = x^.</math>; & re$ultat æquatio quæ$ita.</p>
<p>Ex. 3. A$$umantur æquationes <math>$. (e + f z^n)^m z^. + a x^n = y & $. {z^. / z} +
f z = x</math>, unde <math>{z^. / z} + f z^. = x^. & z^. = {z x^. / 1 + f z}</math>; $cribatur valor <math>{(1 + f z) z^. / z},
&c.</math> pro <math>x^., &c.</math> in æquatione <math>(e + f z^n)^m z^. + n a x^n-1 x^. = y^.</math>; deinde per
prob.26. ita reducantur æquationes re$ultantes, ut exterminetur <math>z</math>; &
re$ultat æquatio quæ$ita.</p>
<p>Et $ic inveniri po$$unt plures (<math>n</math>) æquationes (<math>n + 1</math>) variabiles
quantitates & earum fluxiones involventes, quarum variabiles quan-

<pb 281>[0333]<rh>ÆQUATIONIBUS.</rh>
titates exprimuntur per datas functiones a$$umptarum variabilium
quantitatum.</p>
<p>Hic animadvertendum e$t has re$olutiones non e$$e generales re$o-
lutiones æquationis re$ultantis: $i vero in a$$umptis re$olutionibus,
i. e. duabus æquationibus involventibus quantitates <math>x, y, & z</math> conti-
neatur quantitas ad libitum a$$umenda; & ea quantitas ex re$ultante
æquatione evane$cat, cum evane$cat quantitas <math>z</math>, tum a$$umptæ erunt
generales re$olutiones re$ultantis æquationis relationem inter <math>x & y</math>
exprimentis; $i modo $it fluxionalis æquatio primi ordinis: $i autem
duæ vel tres, &c. independentes & invariabiles quantitates in a$$umptis
re$olutionibus contineantur, tum $i modo reducantur a$$umptæ æqua-
tiones relationes inter <math>x, y, z, &c.</math> exprimentes in unam, ita ut ex-
terminentur omnes præter <math>x & y, &c.</math>; & $i modo evane$cant (<math>m</math>) in-
variabiles quantitates ex fluxionali æquatione (<math>m</math>) ordinis, tum erunt
a$$umptæ æquationes generales re$olutiones æquationis re$ultantis
(<math>m</math>) ordinis.</p>
<p>Sint duæ datæ æquationes (<math>A = 0 & B = 0</math>) exprimentes relationes
inter variabiles <math>x, y & z</math>; in quibus continetur invariabilis quantitas
ad libitum a$$umenda (<math>a</math>); tum ita reducantur tres æquationes <math>A = 0,
B = 0 & A^. = 0</math>, ut exterminentur duæ quantitates (<math>z & a</math>), & re$ul-
tat æquatio relationem inter <math>x & y</math> exprimens, cujus duæ datæ æqua-
tiones erunt generalis re$olutio.</p>
<p>Sint tres vel quatuor vel plures (<math>n</math>) æquationes (<math>A = 0, B = 0,
C = 0, &c.</math>) exprimentes relationes inter (<math>h + 1</math>) variabiles (<math>x, y, u, v,
&c.</math>), & (<math>n - h</math>) variabiles (<math>z, w, Z, &c.</math>), & duas vel tres vel plures
(<math>n - h</math>) invariabiles quantitates (<math>a, b, c, &c.</math>) ad libitum a$$umendas:
ita reducantur æquationes <math>A = 0, A^. = 0, B = 0, B^. = 0, C = 0, C^. = 0,
&c.</math> in (<math>b</math>) æquationes, ut exterminentur quantitates variabiles (<math>z, w,
Z, &c.</math>) & invariabiles <math>a, b, c, &c.</math>; tum re$ultant (<math>h</math>) æquationes,
quarum datæ erunt re$olutio.</p>
<p>Et $ic de æquationibus a$$umptis, in quibus variabiles habent flu-
xiones $uperiorum ordinum.</p>

<pb 282>[0334]<rh>DE FLUXIONALIBUS</rh>
<h>THEOR. XLVII.</h>
<p>In detegendâ fluxione <math>n</math> ordinis contenti <math>a × b × c × d × e × &c. f ×
g × h × &c.</math> coefficientes cuju$cunque contenti <math>a^. ^m b^. ^r c^. ^s d^. ^t, &c. f g h, &c.</math>
erit æqualis fractioni, cujus numerator e$t <math>n (n - 1) (n - 2) (n - 3)
..... (n - m - r - s - t - &c. + 1)</math>, denominator vero <math>1 · 2 · 3 ..
m × 1 · 2 · 3 ... r × 1 · 2 · 3 ... s × &c.</math> ubi <math>n = m + r + s + t + &c.</math></p>
<p>Cor. · Hinc $i fluxio reduci po$$it in prædictam formulam <math>= 0</math>,
ejus fluens erit <math>a × b × c × d × &c. = A x^n+1 + B x^n + C x^n-1 + &c.</math>
ubi <math>A, B, C, &c.</math> $unt quæcunque invariabiles quantitates, & <math>x</math> quan-
titas quæ fluit uniformiter.</p>
<h>THEOR. XLVIII.</h>
<p>Sit fluxionalis æquatio, <math>&c^.. (d^. × (c^. × (b^. a^.))) = a^. ^n b c d &c. +
((n - 1) b^. + (n - 2) c^. + (n - 3) d^. + &c.) a^.^n-1 + &c. = 0</math>, & erit
ejus generalis fluens <math>A + $ · {B / b} + $ · {$ · {C / c}/b} + $ · {$ · {D / d} $ · { / c}/b} + &c. = 0</math>, ubi <math>A,
B, C, D, &c.</math> $unt con$tantes quantitates ad libitum a$$umendæ nul-
lius, primi, $ecundi, tertii, &c. fluxionalis ordinis.</p>
<p>Cor. · Si vero <math>b, c, d, e, &c.</math> $int functiones quantitatis, quæ fluit
uniformiter, tum ejus fluens facile inveniri pote$t.</p>
<h>THEOR. XLIX.</h>
<p>1. Sint <math>x & y</math> re$pective quæcunque algebraicæ functiones quanti-
tatum <math>t & u</math> & earum fluxionum, & $i modo detur relatio algebraica
inter quantitates <math>t & u</math>; tum dabitur algebraica relatio inter quanti-
tates <math>x & y</math>.</p>
<p>Facile con$tat.</p>

<pb 283>[0335]<rh>ÆQUATIONIBUS.</rh>
<p>Cor. · Hinc facile deduci po$$unt quantitates <math>x & y</math>, quæ erunt
algebraicæ functiones ex $e ip$is, $i modo duæ aliæ (<math>t & u</math>) etiam
$int algebraicæ functiones ex $e ip$is: a$$umantur enim functiones
quantitatum <math>t, u</math>, & earum fluxionum, quæ $int <math>x, & y</math>; deinde in-
veniantur <math>t, & u</math> in terminis quantitatum <math>x & y</math>, & re$olvi pote$t
problema.</p>
<p>Ex. · Sint <math>y = {u √(t^.^2 - u^.^2) / t^.}, & x = t - {u u^. / t^.}</math>; ita reducantur hæ
æquationes ut inveniantur (<math>u & t</math>) termini quantitatum <math>x & y</math>,
& re$ultant <math>u = {y √(y^.^2 + x^.^2) / x^.} & t = x + {y y^. / x^.}</math>; & con$tat $i <math>t</math> $it alge-
braica functio quantitatis <math>u</math>, tum erit <math>x</math> algebraica functio quan-
titatis <math>y</math>.</p>
<p>2. Sit æquatio <math>{y^.. / φ:(x^. & y^.)} = {1 / a x + b y}</math>; ubi per <math>φ:(x^. & y^.)</math> denotatur
functio fluxionum <math>x^. & y^.</math> duarum dimen$ionum, & <math>x</math> fluit uniformi-
ter; in eâ pro <math>y^.</math> $cribatur <math>z x^.</math>, & re$ultat <math>(a x + b $. (z x^.)) × z^. = φ′:
(z) × x^.</math>, & con$equenter <math>b $. z x^. = {φ′:(z) x^. / z^.} - a x</math>, & exinde <math>b z = φ″:
(z) - {φ′:(z) z^.. / z^.^2} - a, & {z^.. / z^.} = (φ″:(z) - b z - a) z^.</math>, unde <math>log. {z^. / x^.} =
$. (φ″:(z) - b z - a) z^. + A = V + A</math>, & con$equenter <math>z^. = e^V + ^.^A x^.</math>,
unde <math>z^. e^-V-A = x^.</math>, cujus fluens <math>x = $. z^. e^-V-A & y = $. z x^. =
$. z z^. e^-V-A</math>.</p>
<p>Ex. 1. Sit æquatio <math>{- y^.. / x^.^2 + y^.^2} = {1 / y}</math>, pro <math>y^.</math> $cribatur <math>z x^.</math>, & re$ultat
æquatio <math>- y y^.. = z^. x^. × - $. z^. x^. = (1 + z^2) x^.^2</math>, unde <math>- $. z^. x^. ={(1 + z^2) x^. / z^.}</math>, & exinde <math>- z x^. = 2 z x^. - {(1 + z^2) z^.. / z^.^2} x^., & {z^.. / z^.} = {3 z z^. / 1 + z^2},
& log. {z^. / x^.} = {3 / 2} log. (1 + z^2)</math>, unde <math>z^. = a (1 + z^2)^{3 / 2} x^.</math>; & exinde <math>a x^.

<pb 284>[0336]<rh>DE FLUXIONALIBUS</rh>
= {z^. / (1 + z<_>2</_>)<_>{3 / 2}</_>}, & y^. = z x^. = {z z^. / a (1 + z<_>2</_>)<_>{3 / 2}</_>}</math>, & con$equenter <math>x = {a z / √(1 + z^2)}+ c & y = - {1 / a} (1 + z^2)^- {1 / 2} + b.</math> Aliter: ob <math>y y^.. + y^.^2 = - x^.^2 erit
y y^. = - x x^. + a x^.</math>, & con$equenter <math>y^2 = - x^2 + 2 a x + b</math>.</p>
<p>Ex. 2. Sit <math>{y^.. / x^.^2 + y^.^2} = {1 / y}</math>, pro <math>y^.</math> $cribatur <math>z x^.</math>, & re$ultat æquatio <math>y y^..
= z^. x^. $. z x^. = (1 + z^2) x^.^2</math>, unde <math>$. z x^. = {(1 + z^2) x^. / z^.}</math>, & exinde <math>{z^.. / z^.} ={z z^. / 1 + z^2}, & log. {z^. / a x^.} = {1 / 2} log. (1 + z^2)</math>, unde <math>{z^. / a (1 + z^2)^{1 / 2}} = x^., & a x ={1 / 2} × log. b × (z + √(1 + z^2)), & y^. = z x^. = {1 / a}{z z^. / (1 + z^2)^{1 / 2}}</math>, cujus fluens
e$t <math>y = {1 / a} × (1 + z^2)^{1 / 2} + c.</math> Eadem principia ulterius promovere
liceat.</p>
<p>3. Sit fluxionalis æquatio <math>A = 0</math> ordinis (<math>m</math>), in quâ continentur
(<math>r</math>) diver$æ invariabiles quantitates ad libitum a$$umendæ, quæ in datâ
<math>B = 0</math> æquatione (<math>m + r</math>) ordinis ea$dem variabiles quantitates &
earum fluxiones involvente non inveniuntur: $cribantur quantitates
& earum fluxiones ex æquatione <math>A = 0</math> acqui$itæ pro $uis valoribus
in æquatione <math>B = 0</math>, & $i ex nonnullis radicibus æquationis <math>A = 0</math>
pro $uis valoribus in æquatione <math>B = 0</math> $ub$titutis evane$cant prædictæ
(<math>r</math>) invariabiles quantitates, ex nonnullis vero non; tum æquatio
<math>A = 0</math> re$olvi pote$t in plures, quarum nonnullæ $unt generales
fluentes datæ fluxionalis æquationis <math>B = 0</math>, nonnullæ vero non.</p>
<p>4. Sit <math>V = 0</math> fluens fluxionalis æquationis <math>W = 0</math>; tum, ut prius
probatur, omnis functio quantitatis <math>V</math> erit etiam fluens æquationis
<math>B = 0</math>; unde fluens cuju$cunque fluxionalis æquationis contineat in
$e quemlibet numerum invariabilium & quodammodo variabilium
quantitatum ad libitum a$$umendarum. Si æquationis <math>V = 0</math> divi$or
(<math>P = 0</math>) maxime $implex $it fluens (<math>m</math>) ordinis fluxionalis æquationis

<pb 285>[0337]<rh>ÆQUATIONIBUS.</rh>
<math>W = 0</math>, & in $e contineat (<math>m</math>) invariabiles quantitates ad libitum a$-
$umendas, quæ in æquatione <math>W = 0</math> haud inveniuntur, tum <math>P = 0</math>
erit generalis fluens prædictæ æquationis <math>W = 0</math>, $in aliter particu-
laris.</p>
<h>THEOR. L.</h>
<p>1. Sit <math>V^. = p x^. + q y^. = ({V^. / x^.}) x^. + ({V^. / y^.}) y^. = D</math>; & detur æquatio
relationem inter <math>p, q, x & y</math> exprimens; tum ex eâ non datur valor
quantitatis (<math>V</math>).</p>
<p>2. Sint duæ æquationes relationes inter prædictas quantitates ex-
primentes; tum ex iis deduci po$$unt valores quantitatum <math>p & q</math> re-
$pective in terminis quantitatum <math>x & y</math>; & con$equenter facile con-
$tabit, annon fluxio <math>p x^. + q y^.</math> $it integrabilis, necne.</p>
<p>Sint <math>p & q</math> datæ functiones quantitatum <math>x & y</math>; ita reducantur hæ
functiones, ut exterminetur una incognita quantitas vel <math>p vel q</math> vel
<math>x vel y</math>; & re$ultabit æquatio relationem inter tres reliquas expri-
mens, unde inveniri pote$t quantitas <math>p</math> in terminis quantitatum <math>x & y</math>;
& exinde inveniri pote$t quantitas <math>q</math> in terminis quantitatum <math>x & y</math>,
& tum ex re$ultante æquatione a$$ignari pote$t valor quantitatis (<math>V</math>).</p>
<p>Si detur relatio inter <math>p & x vel y vel q</math>, tum ex eâ non deduci pote$t
valor quantitatis <math>V</math>.</p>
<p>3. Et $imiliter cum dentur relationes inter has quantitates <math>({V^. / x^.}) = p,
({V^. / y^.}) = q, ({V^. / z^.}) = r, ({V^.. / x^.^2}) = α, ({V^.. / x^. y^.}) = β, &c.</math>; tum $i duæ
$olummodo $int variabiles quantitates <math>x & y</math> & deduci poffint duæ ex
his quantitatibus in terminis quantitatum <math>x & y</math> & earum fluxionum,
exprimi pote$t <math>V</math> in terminis quantitatum <math>x & y</math> & earum fluxio-
num: $i vero tres invariabiles quantitates in datis æquationibus con-
tineantur; tum, $i tres prædictæ quantitates exprimi po$$int in ter-
minis quantitatum <math>x, y & z; V</math> exprimi pote$t in terminis quan-

<pb 286>[0338]<rh>DE FLUXIONALIBUS</rh>
titatum <math>x, y & z</math> & earum fluxionum, $in aliter vero non: & $ic
deinceps.</p>
<p>Omnes æquationes huju$ce generis reduci po$$unt ad fluxionales
æquationes; $cribantur enim in iis earum valores e fluxionalibus
æquationibus deducti, & re$ultant fluxionales æquationes quæ$itæ.</p>
<h>PROB. LXVI.</h>
<p it>Datis quibu$dam fluxionibus (<math>V^.</math>), in quibus continentur duæ vel plures
variabiles quantitates; invenire ca$us, in quibus earum fluentes (<math>V</math>) ex-
primi po$$unt: $i data fluxio dividi po$$it in fluxiones integrabiles huju$ce
generis <math>P x^.</math>, tum erit <math>P</math> functio quantitatis <math>x^.</math>; & $i vero reduci po$$it in
formulam <math>p x^. + q y^. + r z^. + s v^. + &c.</math> tum erunt <math>p, q, r, s, &c.</math> functi-
ones quantitatum <math>x, y, z, v, &c.</math> Con$imiles etiam propo$itiones etiam ad
fluxionales æquationes $uperiorum ordinum applicari po$$unt.</p>
<p>1. Sit <math>z^. = P x^.</math>, & erit <math>P</math> functio quantitatis <math>x</math>; & $ic $ic <math>z^. = p
(x^. + a y^.), & erit p</math> functio quantitatis <math>x + a y</math>; vel $imiliter $it <math>z^. =
p x ({x^. / x} + {y^. / a}) & erit p x = φ: (log. x + {y / a})</math>.</p>
<p>2. Sit <math>z^. = w^. + P x^., & erit P</math> functio quantitatis <math>x</math>; unde $it <math>z^. =
p x^. + {γ + α p / β} y^. = {γ / β} y^. + p (x^. + {α / β} y^.), & erit p = φ: (x + {α / β} y)</math>.</p>
<p>3. Sit <math>z^. = p x^. - p y^. + {z y^. / a}, & erit {p / z} (x^. - y^.) = {z^. / z} - {y^. / a} & con$e-
quenter {p / z} = φ: (x - y)</math>.</p>
<p>4. Sit <math>z^. = p x^. + q y^.</math>, tum fingatur <math>y</math> con$tans, & inveniatur <math>$. p x^.
= α; & erit z = α + φ: (y)</math>. Et $ic de pluribus variabilibus quan-
titatibus in data æquatione contentis.</p>
<p>5. Sit <math>z^. = p x^. + q y^.; & erit z = p x + q y - $. (x p^. + y q^.)</math>.</p>
<p>6. Sit <math>p x^. + q y^. = r v^. + s z^. + t w^., ubi p & q</math> $unt functiones
quantitatum <math>x & y; & r, s & t</math> $unt functiones quantitatum <math>v, z & w</math>:

<pb 287>[0339]<rh>ÆQUATIONIBUS.</rh>
$int <math>S^. = {M / h} × (p x^. + q y^.) & R^. = {L / k} (r v^. + s z^. + t w^.)</math>; tum erit <math>R^. ={h L S^. / k M}</math>, & con$equenter <math>R & {h L / k M}</math> erunt functiones quantitatis <math>S</math>: &
exinde <math>S & R</math> erunt functiones quantitatis <math>{k L / h M}</math>.</p>
<p>7. Sit <math>z^. = p x^. + q y^.</math>, ubi <math>p</math> e$t functio quantitatis <math>q</math>; tum erit <math>p^. =
Q q^.</math>, ubi <math>Q</math> e$t functio quantitatis <math>q</math>; unde <math>z = p x + q y - $. (x p^. +
y q^. = x Q q^. + y q^. = (x Q + y) q^.)</math>; & con$equenter <math>x Q + y & x</math> erunt
functiones quantitatis <math>q</math>.</p>
<p>Sit <math>z^. = p x^. + {p x / a} y^. = p x ({x^. / x} + {y^. / a}) = p x^. + x p^. - x p ({p^. / p} - {y^. / a})</math>;
unde <math>x p & z</math> erunt functiones quantitatum <math>{y / a} + log. x & log. p - {y / a}</math>,
cum <math>q = {p x / a}</math>.</p>
<p>8. Sæpe e $ub$titutione deduci po$$unt huju$ce generis problema-
tum re$olutiones. E. g. $it <math>z^. = p x^. + (p X + T) y^.</math>, ubi <math>X & T</math> $unt
functiones quantitatis <math>x</math>; $tatuatur <math>p = r - {T / X}</math>, & prodit <math>z^. = r x^. -{T / X}x^. + r X y^. = {- T x^. / X} + r X ({x^. / X} + y^.)</math>, unde <math>r X</math> erit functio quan-
titatis <math>$. {x^. / X} + y</math>: E. 2. Sit <math>V^. = p x^. + P x y^. + Π y^.</math>, ubi <math>P & Π</math> $unt
functiones ip$ius <math>p</math>, unde <math>V = p x + $. (P x y^. + Π y^. - x p^.)</math>; $tatuatur
<math>P x + Π = v</math>, & re$ultat <math>V = p x + $. (v y^. - {v p^. / P} + {Π p^. / P}) = p x +
$. {Π p^. / P} + $. v (y^. - {p^. / P})</math>, & con$equenter <math>v = φ: (y - $. {p^. / P})</math>.</p>
<p>9. Nonnunquam in his re$olutionibus e dato particulari valore
erui pote$t ejus generalis valor. E. g. Sit <math>z^. = p x^. + q y^.</math>, ubi <math>z =
M p + N q</math> (exi$tentibus <math>M & N</math> functionibus quibu$ve variabilium

<pb 288>[0340]<rh>DE FLUXIONALIBUS</rh>
<math>x & y</math>), & $it <math>V</math> particularis valor quantitatis (<math>z</math>); $tatuantur <math>z = V v,
& v^. = r x^. + s y^.</math>, & erunt <math>p = P v + V r & q = Q v + V s</math>, ubi <math>V^. =
p x^. + q y^. =</math> in hoc ca$u <math>P x^. + Q y^.</math>, ideoque <math>z = M p + N q =
(M P + N Q) v + V (M r + N s) = V v; at V = M P + N Q</math>, ergo
<math>M r + N s = 0</math>, unde <math>v^. = r (x^. - {M y^. / N}) = {r / N} (N x^. - M y^.)</math>; invenia-
tur multiplicator <math>R</math>, ita ut <math>R (N x^. - M y^.) = T^., & erit v^. = {r T^. / N R}</math>, &
exinde <math>{r / N R} = φ: (T)</math>.</p>
<p>10. Nonnunquam e notis methodis deduci pote$t formula particu-
laris valoris. E. g. Sit <math>z^. = p x^. + q y^. & z = p (α x + β y) + q (γ x
+ δ y)</math>, $tatuatur <math>u = {y / x}</math> functio nullius dimen$ionis, & <math>z = φ: (u)</math>,
unde <math>z^. = u^. φ′: u = ({y^. / x} - {y x^. / x^2} = {y^. / x} - {u x^. / x}) φ′: (u)</math>; & exinde <math>p = -{u / x} φ′: (u) = - {u z^. / x u^.} & q = {1 / x} φ′: (u) = {z^. / x u^.}</math>; $cribantur hæ quantitates
pro $uis valoribus in æquatione <math>z = p (a x + β y) + q (γ x + δ y)</math>, &
re$ultabit <math>{z^. / z} = {u^. / γ + (δ - α) u - β u^2} = flux. log. V</math>; eritque <math>V</math> parti-
cularis valor quantitatis <math>z</math>.</p>
<p>Hoc particulare valore dato, generalis facile per præcedentem me-
thodum inveniri pote$t.</p>
<p>11. Sit <math>z^. = p x^. + q y^.</math>, ubi <math>p & q</math> denotant functiones quantitatum
<math>x, y & z</math>; deinde ex æquatione <math>({p^. / y^.}) = ({q^. / x^.})</math> inveniatur <math>z</math> in terminis
quantitatum <math>x & y</math>; & exinde <math>p & q</math> in terminis earundem quantita-
tum <math>x & y</math>, quæ $int <math>P & Q</math>: tum detegatur multiplicator <math>M</math>, qui in
<math>P x^. + Q x^.</math> ductus, eam reddit integrabilem; unde <math>M P x^. + M Q y^. =
M z^. = S^.</math>; & con$equenter <math>M</math> erit functio quantitatis <math>z</math>.</p>
<p>Infinitæ con$imiles deduci po$$unt propo$itiones: primo enim a$$u-

<pb 289>[0341]<rh>ÆQUATIONIBUS.</rh>
mantur æquationes, quarum re$olutiones huju$ce formulæ dantur;
deinde ex $ub$titutione, &c. reducantur hæ æquationes ad alias, qua-
rum re$olutiones non apparent; tum facile reduci po$$unt po$teriores
æquationes ad priores.</p>
<p>Eadem principia etiam ad fluxionales æquationes $uperiorum or-
dinum extendi po$$unt.</p>
<h>THEOR. LI.</h>
<p>Si <math>S</math> $it functio quantitatum <math>x, y, z, v, w, &c.</math>; quæ evane$cat; cum
<math>z = a, v = b, w = c, &c.</math>; tum inveniatur fluxio <math>S^.</math> ex hypothe$i
quod <math>z, v, w, &c.</math> $int invariabiles & <math>z = a, v = b, w = c, &c.</math> &
reliquæ <math>x, y, &c.</math> variabiles; & erit <math>S^. = 0</math>; $i enim fluens <math>S = 0</math>, tum
ejus fluxio a fortiori <math>S^.</math> nihilo erit æqualis.</p>
<h>PROB. LXVII.</h>
<h it>Zua$cunque fluxionales quantitates per quantitates diver$œ formulœ
denotare.</h>
<p>Sit <math>V</math> functio quantitatum <math>x, y, v, &c.</math> tum erit <math>V^. = p x^. + q y^. + r z^.
+ &c. = x^. ({V^. / x^.}) + y^. ({V^. / y^.}) + z^. ({V^. / z^.}) + &c. & V^.. = p x^.. + q y^.. +
rz^.. + p^.x^. + q^.y^. + r^.z^. + &c.</math> Sint vero <math>p^. = αx^. + βy^. + γz^. + &c.
q^. = δ x^. + ε y^. + 7 z^. + &c. r^. = λ x^. + μ y^. + v z^. + &c.</math> unde per
theor. 2. erunt <math>β = δ, γ = λ, 7 = μ, &c.</math> & con$equenter <math>V^.. = p x^.. +
q y^.. + r z^.. + p^. x^. + q^. y^. + r^. z^. + &c. = p x^.. + q y^.. + r z^.. + α x^. ^2 + ε y^. ^2
+ v z^. ^2 + 2 β y^. x^. + 2 γ z^. x^. + 2 μ z^. y^. + &c. =</math> (per præcedent. no-
tat.) <math>x^.. ({V^. / x^.}) + y^.. ({V^. / y^.}) + z^.. ({V^. / z^.}) + x^. ^2 ({V^.. / x^. ^2}) + y^. ^2 ({V^.. / y^. ^2}) + z^. ^2 ({V^.. / z^. ^2})
+ 2y^. x^. ({V^.. / y^. x^.}) + 2 z^. x^. ({V^.. / x^. z^.}) + 2 y^. z^. ({V^.. / y^. z^.}) + &c.</math> ubi per <math>({V^.. / x^. y^.})</math>
denotatur $ecunda fluxio quantitatis <math>V</math> per <math>x^. y^.</math> divi$a, in quâ inve-
niendâ primum $olum modo habetur <math>x</math> variabilis, deinde $olummodo

<pb 290>[0342]<rh>DE FLUXIONALIBUS</rh>
habetur <math>y</math> variabilis: & $ic de omnibus quantitatibus hoc modo
denotatis: eâ dem vero methodo inveniri po$$unt valores fluxionum
<math>V^..., V^...., &c.</math> $ed hic animadvertendum e$t, $i modo $it quantitas <math>π x^. ^n x^.. ^m x^...^r
× &c. × y^. ^s y^.. ^t × &c. × z^. ^k z^.. ^l × &c.</math> tum per præcedentem notationem
$ic denotari debet, viz. <math>x^. ^n x^.. ^m x^...^r × &c. × y^. ^s × y^.. ^t × &c. × z^. ^k × z^.. ^l × &c. ×
({V^. ^n + m + r + &c. + s + t + &c. + k + l + &c. / x^. ^n+m+r+&c. y^. ^s+t+&c. z^. ^k+l+&c. &c.})</math>.</p>
<p>Hinc transformari pote$t quæcunque quantitas <math>V</math> & ejus fluxionum
functio in functiones eju$dem quantitatis <math>V</math> & ejus fluxionum ex
hypothe$i quod, 1<_>mo</_>. una quantitas fluat uniformiter, deinde altera;
& $ic deinceps.</p>
<h>PROB. LXVIII.</h>
<h>_Si vero detur quantitas, quœ $it fluens_ <math>$. P x^.</math> _fluxionis_ <math>P x^.</math>, _duas vel plu-_
_res quantitates_ <math>(x,y,z,&c.)</math> _involvens, ex hypotbe$i quod una_ <math>x</math> _$olummodo_
_fit variabilis; invenire fluxionem fluentis_ <math>$. P x^.</math> _ex bypotbe$i, quod_ <math>x, y, z,
&c.</math> _$int variabiles._</h>
<p>Inveniatur <math>P^. = Q x^. + R y^. + S z^. + &c.</math> tum erit fluxio quæ$ita
<math> = P x^. + y^. $. R x^. + z^. $. S x^. + &c.</math> ubi <math>$. R x^. & $. S x^., &c.</math> inveniri
debent ex hypothe$i, ut <math>x</math> $olummodo $it variabilis, quod facile
con$tat e theor. 2. $it enim <math>V = $. P x^.</math>, & exinde <math>V = P x^. + q y^. + &c.</math>
unde <math>P^. = α x^. + β y^. + &c. & q^. = β x^. + γ y^. + &c.</math> & con$e-
quenter fluxio quantitatis <math>P</math> ex hypothe$i, quod <math>y</math> $olummodo $it vari-
abilis, erit <math>β y^.</math>; ergo <math>R y^. = β y^.</math>; $ed <math>q^. = β x^. = R x^.</math>, $i modo <math>x</math> tantum
$it variabilis; & con$equenter <math>q = $. R x^., & V^. = P x^. + y^. $. R x^. + &c.
= ({V^. / x^.}) x^. + ({V^. / y^.}) y^. + &c.</math></p>
<p>Cor. 1. Sit quantitas <math>V = H $. P x^.</math>, & erit <math>V^. = H $. P x^. + H P^. x^. +
H y^. $. R x^. + H z^. $. S x^. + &c.</math> ubi <math>P^. = Q x^. + R y^. + S z^. + &c.</math></p>
<p>Cor. 2. Sit <math>V</math> functio quantitatum <math>x & u, & y = $. V x^.</math>; tum erit
<math>y^. = u^. $. x^.({V^. / u^.})</math>, $i modo <math>x</math> habeatur invariabilis: $upponatur enim

<pb 291>[0343]<rh>ÆQUATIONIBUS.</rh>
<math>y^. = p x^. + q u^.</math>, tum <math>p^. = α x^. + β u^. = V^.</math>, unde <math>({V^. / u^.}) = β</math>, $ed <math>q^. =
β x^. + γ u^.</math>; $i vero <math>x</math> $olummodo habeatur variabilis, erit <math>q^. = β x^. =
({V^. / u^.})x^.</math>, & exinde <math>q = $. x^.({V^. / u^.})</math>; unde <math>q u^. = u^. $. x^.({V^. / u^.}) = y^.</math> $i modo
<math>x</math> $it invariabilis; & $ic inveniri pote$t <math>y^.. = u^. ^2 $. x^.({V^.. / u^. ^2})</math>.</p>
<p>Cor. 3. Sit quantitas <math>V = $. π x^. $. P x^.</math>, ubi <math>x</math> $olummodo habetur
variabilis; a$$umantur <math>π^. = α x^. + β y^. + γ z^. + &c. & P^. = λ x^. + μ y^.
+ v z^. + &c.</math> tum erit fluxio quantitatis <math>(π $. P x^.) = (α x^. + β y^. + γ z^.
+ &c.)$. P x^. + π P x^. + π y^. $. μ x^. + π z^. $. v x^. + &c.</math> exinde <math>V^. = π x^.
$. P x^. + y^. $. x^. (β $. P x^. + π $. μ x^.) + z^. $. x^. (γ $. P x^. + π $. v x^.) + &c.</math>
ubi in fluxionibus <math>x^.(β $. P x^. + π $. μ x^.) & x^.(γ $. P x^. + π $. v x^.), &c.</math>
$olummodo habetur <math>x</math> variabilis.</p>
<p>Et $ic inveniri po$$unt fluxiones cuju$cunque ordinis quantitatis <math>V</math>,
quæ continet in $e fluentem cuju$cunque ordinis.</p>
<h>THEOR. LII.</h>
<p>Sit <math>V</math> functio quantitatum <math>x, y, &c.</math>; tum erit <math>V^. = p x^. + q y^. + &c.</math>;
$int etiam <math>y, x, &c.</math> functiones quantitatum <math>t, u, &c.</math> unde <math>x^. = a t^. +
b u^. + &c., & y^. = a′ t^. + b′ u^. + &c.;</math> pro <math>x^. & y^.</math> in prædictâ æquatione
<math>V^. = p x^. + q y^. + &c.</math> $cribantur earum valores, & re$ultat <math>V^. = (p a
+ q a′ + &c.)t^. + (p b + q b′ + &c.)u^. + &c.</math>: in hâc æquatione pro <math>p, q, &c.,
a′, b′, &c.</math> $cribantur earum valores <math>({V^. / x^.}), ({V^. / y^.}), &c.; ({y^. / t^.}), ({y^. / u^.}), &c.</math>;
& transformabitur data æquatio in aliam <math>V^. = t^. ({V^. / x^.})({x^. / t^.}) + t^. ({V^. / y^.})
({y^. / t^.}) + u^. ({V^. / x^.})({x^. / u^.}) + u^. ({V^. / y^.})({y^. / u^.}) + &c.</math></p>
<p>Et $ic facile transformari pote$t hæc æquatio in aliam, in quâ in-
volvuntur quantitates <math>t′, u′, &c.</math> ubi <math>t, u, &c.</math> $unt functiones quanti-

<pb 292>[0344]<rh>DE FLUXIONALIBUS</rh>
tatum <math>t′, u′, &c.; &</math> $ic deinceps: vel transformari pote$t data æquatio
<math>V^.. = p x^.. + q y^.. + r x^. y^. + &c.</math> in aliam præcedentis formulæ, in quâ
etiam involvuntur <math>t, u, &c.</math>; vel etiam <math>t′, u′, &c.</math>; & earum fluxiones:
$it enim <math>x^. = a t^. + b u^. + &c., & y^. = a′ t^. + b′u^. + &c.</math>, unde <math>x^.. = a t^..
+ t^. a^. + b u^.. + b^. u^. + &c., &c.</math> $int <math>a^. = c t^. + d u^. + &c., & b^. = c′ t^.
+ d′ u^. + &c.</math>: $cribantur hi valores pro <math>a^. & b^.</math> in præcedente æqua-
tione & re$ultat <math>x^.. = a t^.. + b u^.. + c t^. ^2 + (d + c′)t^. u^. + d′ u^. ^2, &c.</math>; & $i-
militer detegi pote$t <math>y^.. = a′ t^.. + b′ u^.. + e t^. ^2 + (f + e′)t^. u^. + f′ u^. ^2</math>, ubi <math>y^..
= a′ t^.. + t^. a^.′ + b′ u^.. + b′ u^. + &c., & a^.′ = e t^. + f u^. + &c., & b^. ′ = e′ t^.
+ f′ u^. + &c.</math>: $ub$tituantur hi valores, & <math>a t^.. + b u^. + &c., & a′ t^. +
b′ u^. + &c.</math> pro $uis valoribus <math>x^.., y^.., x^. & y^.</math>, in datâ æquatione; & re$ul-
tat æquatio in quâ $olummodo continentur fluxiones quantitatum
<math>t & u</math>.</p>
<p>Hæc fluxio facile transformari pote$t in terminos præcedentis for-
mulæ $eribendo pro <math>p, q, r, &c.</math> re$pective <math>({V^. / x^.}), ({V^. / y^.}), ({V^.. / x^. y^.}), &c</math>;
& pro <math>a, b, &c.; a′,b′, &c.</math>; $ub$tituendo <math>({x^. / t^.}), ({x^. / u^.}), &c., ({y^. / t^.}), ({y^. / u^.}),
&c.</math>; deinde pro <math>c, d, &c.; c′, d′, &c.; e, f, &c.; e′, f′, &c.</math>; $cribendo
<math>({x^.. / t^. ^2}) = ({a^. / t^.})</math> ob <math>a = ({x^. / t^.}), ({x^.. / t^. u^.}) = ({a^. / u^.}), &c.; ({x^.. / u^. t^.}) = ({b^. / t^.})</math> ob
<math>b = ({x^. / u^.}), ({x^.. / u^. ^2}) = ({b^. / u^.}), &c.; ({y^.. / t^. ^2}) = ({a^.′ / t^.})</math> ob <math>a^.′ = ({y^. / t^.}), ({y^.. / t^. u^.}) = ({a^.′ / u^.}),
&c.; ({y^.. / u^. t^.}) = ({b^.′ / t^.})</math> ob <math>b = ({y^. / u^.}), ({y^.. / u^. ^2}) = ({b^.′ / u^.}), &c.;
&c.</math>; in re$ultante æquatione.</p>
<p>Con$imiles etiam æquationes ex ii$dem principiis deduci po$$unt,
cum <math>t, u, &c.</math> $int functiones quantitatum <math>x, y, & c.</math> i. e. <math>t^. = b x^. + k y^., &
u^. = b′ x^. + k′ y^., &c.</math>; facile enim deduci po$$unt fluxiones <math>x^., y^.,
&c.</math> ex fluxionibus <math>t^., u^., &c.</math>; quibus pro $uis valoribus in datâ æqua-
tione $ub$titutis, re$ultat æquatio, quæ per methodum prius traditam
reduci pote$t ad alteram præcedentis formulæ.</p>

<pb 293>[0345]<rh>ÆQUATIONIBUS.</rh>
<h>PROB. LXIX.</h>
<p>1. Sit æquatio relationem inter quantitates <math>({V^. / x^.}), ({V^. / y^.}), ({V^. / z^.}),
&c. ({V^.. / y^. ^2}), ({V^.. / x^. z^.}), &c. ({V^... / y^. ^3}), &c.</math> de$ignans, i. e. $int
<math>V^. = p x^. + q y^. + r z^. + &c. & p^. = α x^. + β y^. + γ z^. + &c. q^. = δ x^. + ε y^. + 7z^. +
&c. r^. = λ x^. + μ y^. + σ z^. + &c.</math> & etiamque <math>α^. = a x^. + b y^. + &c.
β^. = d x^. + e y^. + &c. &c. δ^. = f x^. + g y^. + &c. &c. &c.</math> & $it data
æquatio relationem inter qua$cunque quantitates <math>p, q, r, &c. α, β, γ;
δ, ε, 7; λ, μ, ν; a, b; d, e; f, g, &c.</math> de$ignans; invenire utrum ea
transformari pote$t in æquationem relationem inter <math>V, x, y, z, &c.</math> &
earum fluxiones exprimentem: a$$umantur omnes quantitates præter
<math>V</math> & alteram <math>x</math> tanquam invariabiles, & æquationis re$ultantis abji-
ciantur parenthe$es, & fluxionalis æquationis exinde re$ultantis inve-
niatur fluens; & $ic de $ingulis reliquis (<math>y, z, &c.</math>) variabilibus quan-
titatibus; & ex fluxionibus fluentium $ic inventarum ad prædictas
formulas reductarum con$tabit, utrum data æquatio reduci pote$t in
fluxionalem, necne. E. g.</p>
<p>Si data æquatio relationem inter quantitates huju$ce generis <math>({V^. / x^.}),
({V^. / y^.}), ({V^.. / x^. ^2}), ({V^.. / x^. y^.}), ({V^... / y^. ^2}), ({V^... / x^. ^3}), &c. V, x, y, &c.</math> & earum fluxiones
de$ignet: a$$umantur omnes quantitates præter duas (<math>V & x vel y, &c.</math>)
tanquam con$tantes, & in æquatione re$ultante pro <math>({V^. / x^.</math>}) $cribatur
<math>{V^. / x^.}</math>, & pro <math>({V^.. / x^. ^2})</math> $cribatur <math>{V^.. / x^. ^2}</math>, & $ic deinceps; fluxionalis æquationis
re$ultantis inveniatur fluens; vel ducatur ea in multiplicatorem <math>M</math>,
ita ut ejus fluens inveniri po$$it, quæ dicatur <math>V</math>; nunc $upponantur
omnes prædictæ quantitates iterum variabiles, & inveniatur <math>V^.</math>, $ub-

<pb 294>[0346]<rh>DE FLUXIONALIBUS</rh>
ducatur hæc fluxio de datâ æquatione in fluxionales terminos mutatâ
& in multiplicatorem <math>M</math> ductâ & $i modo re$ultet fluxio, cujus fluens
inveniri pote$t, quæ $it <math>W</math>; tum <math>V ± W = 0</math> exprimet relationem
quæ$itam, $i modo per fluxionales æquationes exprimi po$$it, quam
etiam de$ignat data æquatio.</p>
<p>2. Eadem principia, quæ prius edita fuerunt de fluxionalibus
æquationibus tres vel plures variabiles quantitates & earum fluxiones
habentibus, etiam applicari po$$unt ad inveniendas con$imiles propo-
$itiones de æquationibus huju$ce generis. E. g. Omnis æquatio,
quæ e$t generalis re$olutio datæ æquationis prædictæ formulæ habet
tot vel plures invariabiles quantitates ad libitum a$$umendas, quot $it
maximus ordo fluxionis in datâ æquatione contentus.</p>
<p>3. Duæ æquationes exprimentes relationes inter quantitates huju$ce
generis <math>({z^. / x^.}), ({z^. / y^.}), &c.</math> reduci po$$unt ad unam, ita ut una variabilis
quantitas & ejus fluxiones exterminentur ex eâdem methodo, &c.
quæ prius tradita fuit de duabus æquationibus fluxionalibus in unam
reducendis, ita ut exterminentur variabilis quantitas & ejus fluxio-
nes, &c.</p>
<p>4. Fere omnia etiam proferre liceat de his, quæ in priori hujus ca-
pitis parte de fluxionalibus æquationibus con$cripta fuere.</p>
<p>E. g. Datâ generali re$olutione detegi pote$t æquatio, cujus fluens
datur per methodum in fluxionalibus æquationibus traditam; &
per con$imiles methodos deduci po$$unt quotcunque æquationes hu-
ju$modi, quarum particulares valores fluentium dantur: e. 2. a$$u-
matur quæcunque functio (<math>π</math>) quantitatum <math>x, y, z, &c.</math> pro <math>V</math>; etiam-
que quantitas <math>W</math>, quæ e$t data functio quantitatum <math>y, x, z, & V</math>, &
earum fluxionum; vel quantitatum formularum prædictarum <math>({V^. / x^.}),
({V^. / y^.}), ({V^.. / x^. ^2}), &c.</math>: in hâc quantitate <math>W</math> pro <math>V, V^., V^.., &c.; ({V^. / x^.}),
({V^. / y^.}), &c.</math> $cribantur earum valores ex dato valore <math>π</math> quantitatis <math>V</math>

<pb 295>[0347]<rh>ÆQUATIONIBUS.</rh>
deducti; & re$ultet quantitas <math>H</math>, quæ e$t functio quantitatum <math>x & y</math>,
& earum fluxionum; tum plerumque erit <math>V = π</math> particularis valor
æquationis <math>W = H</math>.</p>
<p>5. Si hæc vel quæcunque alia æquatio, cujus fluens innote$cit, tran$-
formetur in aliam per datas $ub$titutiones novarum pro variabilibus
in datâ æquatione contentis; ex re$olutione datæ æquationis facile
erui pote$t re$olutio re$ultantis æquationis.</p>
<p>6. Ex a$$umptis multiplicatoribus vel $ub$titutionibus deduci po$$unt
æquationes datæ formulæ, quarum fluentes exprimi po$$unt;; & ge-
neraliter omnia quæ tradita fuere de fluxionalibus æquationibus mu-
tatis mutandis ad æquationes huju$ce formulæ applicari po$$unt.</p>
<p>7. Principia in fluxionalibus æquationibus re$olvendis u$itata, ut
prius a$$eritur, etiam applicari po$$unt ad fluentes quantitatum hu-
ju$ce generis inveniendas.</p>
<p>Ex. Sit <math>p({V^. / x^.}) + q({V^. / y^.}) = 0</math>, tum erit fluens huju$ce æquationis
eadem ac fluens æquationis <math>{1 / p}x^. - {1 / q}y^. = 0</math>.</p>
<p>2. Nonnunquam $ub$tituendo <math>V^.</math> pro <math>x^. ({V^. / x^.}) + y^.({V^. / y^.}) + z^.
({V^. / z^.}) + &c.</math> in data æquatione; $i modo <math>x, y, z, &c.</math> $int variabiles;
vel <math>V^..</math> pro <math>x^..({V^. / x^.}) + y^..({V^. / y^.}) + ax^.y^.({V^.. / x^.y^.}) + &c.</math>; transformari po-
te$t data æquatio in fluxionalem, vel in aliam $implicioris formulæ
e. g. $it data æquatio <math>Mx^.({V^. / x^.}) + My^.({V^. / y^.}) = Wx^. + Zy^.</math>, $cribatur
<math>V^.</math> pro <math>x^.({V^. / x^.}) + y^.({V^. / y^.})</math>, & re$ultat fluxionalis æquatio <math>MV^. = Wx^.
+ Zy^.</math>.</p>
<p>3. Si <math>V</math> $it functio unius $olummodo quantitatis <math>x</math>, & in datâ
æquatione tantummodo fluxiones quantitatum <math>x</math> & <math>V</math> contineantur;

<pb 296>[0348]<rh>DE FLUXIONALIBUS</rh>
tum rejiciendo parenthe$es reduci pote$t data ad fluxionalem æqua-
tionem.</p>
<p>4. Si vero fluxiones plurium quam duarum variabilium (<math>V, x, y,
z, &c.</math>) in datâ æquatione contineantur, & con$equenter plures quam
duæ in datâ æquatione habeantur variabiles; tum, $i $emper occur-
rant quæcunque quantitates huju$ce formulæ <math>M({V^. ^n + m + r + &c. / x^. ^n y^. ^m z^. ^r &c.})</math>; & nulla
alia; in quâ continentur <math>V</math>, vel ejus fluxiones; ubi <math>M</math> e$t functio
quantitatum (<math>x, y, z, &c.</math>) & earum fluxionum; & <math>n′, m′, r′, &c.</math> $unt
minimi indices fluxionum (<math>x^., y^., z^., &c.</math>) in denominatoribus fractio-
num <math>({V^. ^n+m+r+&c. / x^. ^n y^. ^m z^. ^r &c.})</math>; reduci pote$t data æquatio ad alteram, in quâ
ordo fluxionis quantitatis <math>V</math> minor e$t per <math>n′ + m′ + r′ + &c.</math> quam
ordo fluxionis eju$dem quantitatis <math>V</math> in æquatione; $ub$tituantur enim
<math>v = ({V^. / x^.}), v′ = ({v^. / x^.}), v″ = ({v^. ′ / x^.}), ..., v′^n-1 = ({v^. ′^n-2 / x^.})</math>; deinde <math>w
= ({v^. ′^n-1 / y^.}), w′ = ({w^. / y^.}), w″ = ({w^.′ / y^.}),..., w′^m-1 = ({w^. ′^m-2 / y^.})</math>; tum <math>u =
({w^. ′^m-1 / z^.}), u″ = ({u^. / z^.}), u″ = ({u^. ′ / z^.}),..., u′^r-1 = ({u^. ′^r-2 / z^.}), &c.</math>: æquatio
in quâ introducuntur variabilis <math>u′^r-1</math> & ejus fluxiones loco variabilis <math>V</math>
& ejus fluxionum non involvit fluxionem quantitatis <math>u′^r-1</math> majoris or-
dinis quam maximus ordo fluxionis quantitatis <math>V</math> in datâ æquatione
per <math>n′ + m′ + r′</math> diminutus.</p>
<p>Hoc facilius perfici pote$t, $i modo pro <math>({V^. ^n′ / x^. ^n′})</math> $cribatur <math>U</math>; deinde
in æquatione re$ultante pro <math>({U^. ^m′ / y^. ^m′})</math> $cribatur <math>W</math>; tum pro <math>({W^. ^r′ / z^. ^r})</math> $cri-
batur <math>T</math>; & $ic deinceps.</p>

<pb 297>[0349]<rh>ÆQUATIONIBUS.</rh>
<p>E. g. Sit <math>({V^... / x^. ^2 y^.}) = P</math>, & erit <math>({V^.. / x^. y^.}) = $. P x^. + φ:y</math>, ubi <math>y</math> a$$u-
mitur con$tans; deinde <math>({V^. / y^.}) = $.$.P x^. + xφ:y + φ′:y</math>, ubi <math>y</math> ite-
rum a$$umitur con$tans; ultimo <math>z = $.^3 (Px^.) + x$. y^.φ:y + $. y^. φ′:y
+ φ″:(x)</math> ubi <math>x</math> a$$umitur con$tans; & $ic de fluxionalibus æqua-
tionibus $uperiorum ordinum.</p>
<p>Ex. 2. Sit <math>({z^.. / x^. ^2}) = P({z^. / x^.}) + Q</math>; ponatur <math>({z^. / x^.}) = v</math>, unde <math>({z^.. / x^. ^2})
= ({v^. / x^.})</math> & data æquatio fit <math>v^. = Pvx^. + Qx^.</math>, ubi <math>x</math> $umitur con$tans;
eodem modo $it <math>({z^.. / x^.y^.}) = P({z^. / x^.}) + Q</math>, $cribatur <math>({z^. / x^.}) = v</math>, & data
æquatio fit <math>({v^. / y^.}) = P v + Q</math>, in quo ca$u habetur <math>x</math> tanquam inva-
riabilis, & re$ultat æquatio <math>v^. + P v y^. + Q y^. = 0</math>, cujus e methodo
prius traditâ inveniri pote$t fluens.</p>
<p>5. Nonnunquam facile con$tat formula quantitatis, in quâ conti-
netur fluens vel re$olutio datæ æquationis.</p>
<p>Ex. 1. Sit fluxionalis homogenea æquatio prædictæ formulæ <math>A({V^. ^n / x^. ^n})
+ B({V^. ^n / x^. ^n-1 y^.}) + C({V^. ^n / x^. ^n-2 y^. ^2}) + D({V^. ^n / x^. ^n-3 y^. ^3}) + &c. = 0</math>: ubi <math>A, B, C,
D, &c.</math> $unt invariabiles quantitates; haud difficilis e$t ob$ervatio ut
particularis fluens huju$ce æquationis contineatur in formulâ <math>φ:(ax
+ by) = V</math>; $ub$tituatur hæc functio pro <math>V</math>, & valores quantatitum
<math>({V^. / x^. ^r y^. ^s})</math>, &c. exinde deducti pro $uis valoribus in datâ æquatione; &
re$ultat æquatio <math>(Aa^n + Ba^n-1 b + Ca^n-2 b^2 + Da^n-3 b^3 + &c.) × φ′:
(ax + by) = 0</math>, unde <math>a^n A + a^n-1 b B + a^n-2 b^2 C + a^n-3 b^3 D + &c.
= 0</math>; $int <math>α, β, γ, δ, &c.</math> radices æquationis <math>Ax^n + Bx^n-1 + Cx^n-2 +

<pb 298>[0350]<rh>DE FLUXIONALIBUS</rh>
Dx^n-3 + &c. = 0</math>, tum erunt <math>φ: (αx - y), φ′: (βx - y), φ″: (γx - y),
&c.</math> re$olutiones datæ æquationis; & magis generaliter <math>V = φ: (αx
- y) + φ′: (βx - y) + φ″: (γx - y) + &c.</math> erit re$olutio datæ
æquationis.</p>
<p>Sub$equens e$t elegans re$olutio particularis ca$us huju$ce exempli,
quem primus re$olvere docuit Clar. D'Alembert: $it æquatio <math>({V^.. / y^. ^2})
= a^2 ({V^.. / x^. ^2})</math>, tum <math>V^. = px^. + qy^., p^. = rx^. + sy^., q^. = sx^. + ty^.</math>; & per
æquationem erit <math>t = a^2 r</math>; unde ex his æquationibus deduci pote$t
<math>a p^. + q^. = (a r + s)(x^. + a y^.)</math>, & exinde <math>ar + s = φ:(x + a y)</math>; &
quoniam <math>a</math> vel affirmative vel negative accipi pote$t, erit <math>q + a p =
2 a φ′: (x + a y), & q - a p = 2 a φ′: (x - a y)</math>; unde <math>q = a φ: (x
+ a y) + a φ′: (x - a y), & p = φ: (x + a y) - φ′: (x - a y)</math>, & ex-
inde <math>V = π: (x + a y) + π′: (x - a y)</math>.</p>
<p>Ex. 2. Sit homogenea æquatio <math>A({V^. ^n / x^. ^n}) + B({V^. ^n / x^. ^n-1 y^.}) + C({V^. ^n / x^. ^n-1 z^.})
+ D({V^. ^n / x^. ^n-1 v^.}) + E({V^. ^n / x^. ^n-2 y^. ^2}) + F({V^. ^n / x^. ^n-2 y^.z^.}) + &c. = 0</math>, in quâ plu-
res variabiles <math>x, y, z, v, &c.</math> continentur, tum a$$umatur <math>V = φ: (ax
+ b y + c z + d v + &c.)</math>, $cribatur hæc functio pro <math>V</math>, & pro ter-
minis <math>({V^. ^n / x^. ^r y^. ^s z^. ^t &c.}), &c.</math> eorum valores exinde deducti; re$ultans
æquatio <math>A a^n + B a^n-1 b + C a^n-1 c + D a^n-1 d + E a^n-2 b^2 + F a^n-2 b c +
&c. = H = 0: $i H = (f a + g b + b c + &c.) × (f′ a + g′ b + b′ c +
&c.) × (f″ a + g″ b + b″ c + & c.) × &c.</math>; tum ex hâc methodo con$tat datæ
æquationis re$olutio; vel $i <math>H = (f a + g b + b c + &c.) (f′ a^2 + g′ b^2
b′ c^2 + l′ a b + m′a c + n′ b c + &c.) (&c.)</math>; & $ic deinceps; i. e. $i modo
algebraica expre$$io <math>H</math> haud in $implices divi$ores re$olvi po$$it, $ed
in $implices, quadraticos, cubicos, &c. etiam con$tat re$olutio.</p>
<p>Si hæ algebraicæ expre$$iones e fluxionalibus æquationibus $uperio-

<pb 299>[0351]<rh>ÆQUATIONIBUS.</rh>
rum ordinum deductæ haud recipiant $implices divi$ores; $ed qua-
draticos, cubicos, &c.; tum $emper reduci pote$t data æquatio ad flu-
xionales $ecundi, tertii, &c. ordinum æquationes.</p>
<p>Ex. 3. Sit <math>({z^.. / x^. y^.}) = a z</math>, & $tatuatur <math>z = e^p q</math>, ut quantitas expo-
nentialis <math>e^p</math> e calculo evane$cat, quoniam quantitas <math>z</math> unam ubique
tenet dimen$ionem; ponatur <math>z = e^xx γ</math>, unde <math>({z^.. / x^. y^.}) = α e^αx {γ^. / y^.} = a e^αx γ′</math>,
& $i <math>γ′ = γ</math>, exinde <math>{α γ^. / γ} = a y^. & γ = e^{a y / α}</math>, unde particularis valor ip$ius
<math>z = A e^αx+{ay / α}</math>; & $ic generaliter <math>z = A e^αx+{a / α}y + B e^βx+{b / β}y + C e^γx+{c / γ}y +
&c.</math> ubi literæ <math>A, B, C, &c. α, β, γ, &c. &c.</math> omnes recipere po$$unt
po$$ibiles valores.</p>
<p>Ex. 4. Sit æquatio <math>P ({V^. ^n / x^. ^n}) + Q ({V^. ^n / x^. ^n-1 y^.}) + R ({V^. ^n / x^. ^n-2 y^. ^2}) + &c. +
P′ ({V^. ^n-1 / x^. ^n-1}) + Q′ ({V^. ^n-1 / x^. ^n-2 y^.}) + R ({V^. ^n-1 / x^. y^. ^n-2}) + &c. + P″ ({V^. / x^. ^n-2}) + &c. +
L V = 0</math>; ubi in $ingulis terminis <math>V</math> vel ejus fluxio unam $olummodo
habet dimen$ionem; in hâc æquatione pro <math>V</math> & ejus fluxionibus $cri-
bantur <math>e^v × u</math> & ejus corre$pondentes fluxiones, ubi <math>v & u</math> $unt functiones
quantitatum <math>x & y</math>: functiones <math>v & u</math> pendent e functionibus <math>P, Q, R,
&c., P′, Q′, &c., P″, &c.</math> quo magis $implices $unt priores functiones, eo
magis plerumque $implices erunt po$teriores: $i <math>P, Q, R, &c., P′, &c.</math>
$int invariabiles; tum a$$umatur <math>V = e^x (φ:(a x + b y) + x φ′:(y) +
φ″:y) = e^x H</math>; vel <math>e^a/x+b′y H</math>; vel prior vel po$terior evadat quantitas
magis compo$ita; $ub$tituantur hæ quantitates pro $uis valoribus in
datâ æquatione, & ex terminis inter $e & nihilo re$pective æquatis
for$an deduci pote$t æquatio, cujus particularis re$olutio innote$cit.</p>
<p>6. In pleri$que ca$ibus haud nece$$e e$t, ut introducatur exponentialis
quantitas: in quibu$dam ca$ibus $implex algebraica expre$$io parti-
cularem datæ æquationis relationem præbebit. E. g. Sit <math>(x + y)^2

<pb 300>[0352]<rh>DE FLUXIONALIBUS</rh>
({V^.. / x^. y^.}) + m (x + y) ({V^. / x^.}) + m (x + y) ({V^. / y^.}) + n z = 0</math>: huju$ce
æquationis re$olutio erit <math>V = A (x + y)^λ φ:(x) + B (x + y)^λ+1 φ′:
(x) + C (x + y)^λ+2 φ″:(x) + &c.</math>; $ub$tituatur enim hæc quantitas
pro ejus valore in datâ æquatione, & ex æquatis corre$pondentibus
terminis re$ultant <math>n + 2 m λ + λ^2 - λ = 0, (n + 2 m λ + 2 m + λ^2
+ λ) B + (m + λ) A = 0, &c.</math>, unde <math>B = - {(m + λ) / 2 (m + λ)} A, C = -{(m + λ + 1) / 2 (2 m + 2 λ + 1)} B, D = - {(m + λ + 2) / 3 (2 m + 2 λ + 2)} C, &c.</math>; hæc $eries ter-
minat, quoties <math>m + λ + i = 0</math>, ubi <math>i</math> $it integer po$itivus numerus.</p>
<p>Cor. E datis valoribus quantitatum <math>n & m</math> duo exoriuntur valores
quantitatis <math>λ</math>, & exinde duæ $eries, quæ ad eandem reduci po$$unt.</p>
<p>Con$imilis æquationis <math>(n)</math> ordinis re$olutio erit con$imilis.</p>
<p>7. Si in datâ æquatione fluxionali huju$ce formulæ quantitates <math>x
& y</math> $imiliter involvantur; tum in ejus fluente vel re$olutione $imiliter
etiam involventur.</p>
<p>8. Nonnunquam a$$umatur quantitas compo$ita ex exponentialibus,
algebraicis, &c. E. g. Sit æquatio <math>({V^.. / y^. ^2}) = x^2 ({V^.. / x^. ^2})</math>, fingatur <math>V =
x^λ e^μ y (m × log. x + n y)</math> & $ub$tituantur pro <math>({V^.. / y^. ^2}) & ({V^.. / x^. ^2})</math> earum
valores in datâ æquatione; & ex æquatis corre$pondentibus terminis
re$ultantis æquationis deduci pote$t fluens vel re$olutio quæ$ita.</p>
<p>9. Aliquando ex a$$umptâ relatione inter variabiles datæ & quæ$itæ
æquationum reduci pote$t data æquatio ad alteram, cujus re$olutio
innote$cit. E. g. Sit æquatio <math>({V^.. / y^. x^.}) + P ({V^. / x^.}) + Q ({V^. / y^.}) + R z +
S = 0</math>; ponatur <math>V = e^u v</math>, ubi <math>u = φ:(x & y)</math>; $ub$tituatur hæc quan-
titas pro ejus valore <math>V</math> in datâ æquatione, &c.; & ita a$$umantur ter-
mini re$ultantis æquationis, ut evadat <math>({v^.. / x^. y^.}) + T ({v^. / x^.}) + e^- u S = 0</math>,
cujus fluens innote$cit.</p>

<pb 301>[0353]<rh>ÆQUATIONIBUS.</rh>
<p>Ex. 2. Sit <math>({V^.. / y^. ^2}) = P^2 ({V^.. / x^. ^2})</math>: introducantur binæ variabiles <math>t & u</math>,
quæ $unt $unctiones quantitatum <math>x & u</math>; & æquationis re$ultantis ter-
mini inter $e ita æquentur, ut evadat æquatio, cujus fluens innote$cit.</p>
<p>Per eundem modum introducendo variabiles <math>t & u</math> & æquando
terminos re$ultantis æquationis inveniri po$$unt ca$us quantitatum
<math>P, Q, R, &c.</math> in æquatione <math>({V^.. / y^. ^2}) + P ({V^.. / x^. ^2}) + Q ({V^. / y^.}) + R ({V^. / x^.})
+ S V + T = 0</math> contentarum, in quibus fluens ope reductionis hìc
dictæ deduci pote$t.</p>
<p>10. Methodus hìc tradita in hoc con$i$tit, ut huju$modi æquationes
ope introductionis binarum novarum variabilium <math>t & u</math> ad hanc for-
mulam <math>({V^.. / t^. u^.}) + P ({V^. / t^.}) + Q ({V^. / u^.}) + R V + S = 0</math>, vel ad quam-
cunque aliam formulam, cujus re$olutio innote$cit, reducantur.</p>
<p>Ex con$imili methodo reduci pote$t data fluxionalis æquatio hu-
ju$ce formulæ ad alteram; æquationis, cujus ordo minor e$t quam
ordo datæ fluxionalis æquationis, ope. E. g. Data æquatio <math>({V^.. / y^. ^2}) +
P ({V^.. / x^. ^2}) + Q ({V^. / x^.}) + Rv = 0</math> in æquationem <math>({V^.. / y^. ^2}) + F ({v^.. / x^. ^2}) + G
({v^. / x^.}) + H v = 0</math> transformari pote$t, æquationis <math>V = M ({v^. / x^.}) + N v</math>
ope, ubi literæ <math>P, Q, R, F, G, H, M & N</math> $unt $unctiones quanti-
titatis <math>x</math>.</p>
<p>Hoc conficitur reducendo duas vel plures æquationes in unam,
ita ut quædam variabiles & earum fluxiones exterminentur; &
æquando terminos re$ultantis æquationis, ita ut evadat æquatio, cu-
jus re$olutio innote$cit.</p>
<p>In æquationibus huju$ce generis minime adhuc datur methodus
inveniendi; annon data æquatio $it generalis re$olutio datæ fluxio-
nalis æquationis.</p>

<pb 302>[0354]<rh>DE FLUXIONALIBUS</rh>
<p>Æquatio fluxionalis <math>a × b × c × &c. = 0</math> huju$ce formulæ dividi
pote$t in æquationes <math>a = 0, b = 0, c = 0, &c.</math></p>
<h>THEOR. LIII.</h>
<p>Sit <math>π</math> particularis fluens fluxionis <math>p x^.</math>, tum erit ejus generalis fluens
<math>= π + a</math>.</p>
<p>Sit <math>σ</math> particularis fluens fluxionis <math>p x^. $. q x^.</math>, tum erit ejus generalis
fluens <math>= ς + a′ π + b</math>.</p>
<p>Sit <math>σ</math> particularis fluens fluxionis <math>p x^. $. q x^. $. r x^.</math>, & erit ejus generalis
fluens <math>σ + a″ ς + b′ π + c</math>.</p>
<p>Sit <math>τ</math> particularis fluens fluxionis <math>p x^. $. q x^. $. r x^. $. s x^.</math>, & erit ejus
generalis fluens <math>τ + a′″ σ + b″ ς + c′ π + d</math>, & $ic deinceps; ubi literæ
<math>a, b, c, d, &c; a′, b′, c′, &c.; a″, b″, &c.; a′″, &c.</math> invariabiles quantitates
ad libitum a$$umendas re$pective denotant.</p>
<p>Et $ic de generalibus fluentibus $uperiorum ordinum deducendis.</p>
<h>PROB. LXX.</h>
<p it>Sit fluxionalis æquatio <math>{α^. / l} = 0</math>, cujus fluens $it <math>α = C</math> (con$t.), deducere
quamcunque œquationem, in quâ variabiles $eparantur: a$$umatur <math>Z + v
= α</math>, & fingantur <math>z & v</math> quœcunque algebraicæ functiones quantitatum <math>π
& ς</math> re$pective; e datâ æquatione <math>{α^. / l} = 0</math> inveniatur œquatio relationem
inter duas variabiles quantitates <math>π & ς</math> exprimens, & re$ultat æquatio, in
quâ $eparantur variabiles <math>π & ς</math> & earum fluxiones.</p>
<p>Et ex eâdem methodo progredi liceat ad $ub$titutiones pro redu-
cendis fluxionalibus æquationibus $uperiorum (<math>n</math>) ordinum ad fluxio-
nales æquationes inferiorum <math>m</math> ordinum, in quibus $eparantur varia-
biles: $ed plerumque reductio fluxionalis æquationis <math>n</math> ordinis ad
ordinem inferiorem <math>n - 1</math>, in quâ $eparantur variabiles, &c. exigit
unam $ub$titutionem, ad ordinem inferiorem <math>n - 2</math> exigit duplicem
$ub$titutionem, & $ic deinceps.</p>

<pb 303>[0355]<rh>ÆQUATIONIBUS.</rh>
<h>THEOR. LIV.</h>
<p>Sit algebraica æquatio <math>A = 0</math>, in quâ <math>x & z</math> $imiliter involvuntur;
& fluxionalis æquatio <math>p x^. + q y^. = 0</math>, ubi <math>p & q</math> $unt functiones quan-
titatum <math>x & y</math>; reducantur hæ duæ æquationes <math>A = 0 & p x^. + q y^. = 0</math>
in unam, ita ut exterminentur <math>x & x^.</math>, & re$ultet æquatio <math>P z^. + Q y^.
= 0</math>; in æquatione <math>A = 0</math> pro <math>x</math> $cribantur <math>α & β</math>, quarum corre$pon-
dentes valores quantitatis <math>(z)</math> $int <math>π & ς</math>: $it <math>y = φ:(x)</math>, etiamque
<math>y = φ′: (z)</math>: tum, $i modo $cribantur <math>α, β, π & ς</math> pro <math>x</math> in quanti-
tate <math>φ:(x) + φ′: (x)</math>, & $int quantitates re$ultantes <math>A, B, C & D</math>,
erit <math>A - B = C - D</math>; modo corre$pondentes fluentes fluxionalium
æquationum <math>p x^. + q y^. = 0 & P x^. + Q y^. = 0</math> a$$umantur.</p>
<p>A$$ignari po$$unt fluentiales æquationes, quæ $unt fluentes data-
rum fluxionalium; at revera nulli u$ui in$erviunt: e. g. $i in datâ
fluentiali æquatione contineantur fluentiales quantitates, quarum
fluxiones $unt functiones quantitatum <math>x, y</math> & earum fluxionum non
integrabiles, tum vix ulli u$ui in$ervit integratio huju$modi; e. g. $it
fluentialis æquatio <math>y = $. {x^. / y} $. {x^. / y}</math>, cujus fluxionalis erit <math>y^2 y^.. + y y^. ^2 =
x^. ^2</math>; 2. $it <math>y = $. {x^. / y} $. {x^. / y} $. {x^. / y}</math>, cujus fluxionalis erit <math>y^3 y^... + 4 y^2 y^. y^.. + y y^. ^3
= x^. ^3, &c.</math> hæ fluentiales æquationes non pro fluentibus datarum
fluxionalium æquationum de$iderandis habendæ $unt.</p>
<h>SCHOLIUM.</h>
<p>1. In detegendis fluentibus fluxionalium æquationum plerumque
generales functiones variabilium tanquam independentes variabiles
tractandæ $unt; & exinde per regulas prius traditas de iis ratiocinari
liceat.</p>
<p>2. Prius docetur methodus inveniendi fluentes omnium fluxiona-
nalium æquationum per a$$umptiones rationalium functionum, &

<pb 304>[0356]<rh>DE FLUXIONALIBUS</rh>
rationalium & irrationalium quantitatum, vel earum radicum in da-
tis æquationibus contentarum, ni fa$tigium calculi prohibeat.</p>
<p>Con$imilia principia etiam applicari po$$unt ad fluentes fluxiona-
lium æquationum detegendas algebraicarum quantitatum, circula-
rium arcuum & logarithmorum ope.</p>
<p>3. Facile demon$trari pote$t, quod fluentes fere omnium fluxionum
(comparative loquendo) nec exprimi po$$int per finitos terminos, lo-
garithmos, circulares, nec per ellipticos nec hyperbolicos arcus; &c.:
hinc, $i $eparari po$$int variabiles quantitates in datâ fluxionali æqua-
tione, plerumque non inveniri po$$unt per prædictas methodos flu-
entes: $ed per $ub$titutiones algebraicarum functionum, &c. nova-
rum variabilium pro variabilibus in datâ æquatione contentis per-
raro $eparari po$$unt variabiles quantitates; ergo in pleri$que ca$ibus
ad infinitas $eries confugiendum e$t.</p>
<p>4. In æquationibus quantitates formularum <math>({V^. ^n / x^. ^m y^. ^r &c.})</math> involven-
tibus non adhuc datur generalis nota; ex quâ dici pote$t, annon data
$it generalis datarum æquationum re$olutio.</p>
<p>5. In detegendis fluentibus fluxionum, cum variabilis <math>(x)</math> evadat
infinita: $i fluens exprimatur per algebraicam functionem variabilis
<math>(x)</math>; tum ex $ub$titutione infinitæ & finitæ <math>(a)</math> quantitatis pro <math>x</math> fa-
cile con$tat fluens inter prædictos valores <math>(a)</math> & infinitum po$ita; $i
$olummodo in $ingulis quantitatibus addendis vel $ubtrahendis reji-
ciantur omnes quantitates, quæ $unt infinite minores quam quæcunque
aliæ in eâdem $ummâ vel differentiâ contentæ; & $i modo numera-
tores & denominatores nonnullarum fractionum nihilo evadant
æquales, tum earum valores per methodum prius traditam inve$ti-
gandi $unt.</p>
<p>6. Si fluens exprimi po$$it per circulares arcus, &c. variabilis, vel $it
tangens vel $ecans, &c.; tum facile per circularem arcum acquiri po-
te$t fluens, cum <math>x</math> evadat infinita.</p>
<p>7. Si vero fluens exprimi po$$it per plures logarithmos; viz. <math>a × log.

<pb 305>[0357]<rh>ÆQUATIONIBUS.</rh>
α + b × log. β + c × log. γ + &c.</math>; cujus nonnulli; e. g. log. <math>α, log.
β, log. γ, &c.</math> evadunt infiniti; reducantur omnes hi logarithmi in lo-
garithmum unius $ummæ, viz. <math>α × log · α + b × log. β + c × log. γ + &c.
= log · α^a × β^b × γ^c &c. = z</math>; & $i <math>z</math> $it finita quantitas, tum log. prædictæ
$ummæ erit finita; $in aliter non: inveniatur valor algebraicæ quan-
titatis <math>z</math>, & exinde acquiri pote$t ejus log.: $i fluens $it quæcunque
algebraica functio quantitatis <math>x</math>, & circularium arcuum & logarith-
morum ejus functionum; tum per principia prius tradita erui pote$t
ejus valor, cum <math>x</math> evadat infinita.</p>
<p>Si autem fluens exprimatur per datas alias fluentes; tum ex datis
curvis, $olidis, &c., per quæ innote$cunt datæ fluentes, erui pote$t
ejus valor inter quo$cunque valores variabilis in eâ contentæ.</p>
<p>Et $ic rationari liceat de fluxionalibus, æquationibus, &c.</p>
<p>8. Sint <math>n</math> fluxionales æquationes <math>a = 0, a′ = 0, a″ = 0, &c.</math> qua-
rum ordo $it <math>l, (n + 1)</math> variabiles quantitates <math>(x, y, z, &c.)</math> invol-
ventes; tum $int <math>m, m′, m″, &c.</math> functiones quantitatum <math>(x, y, z, &c.)</math>
& earum fluxionum, quarum ordines minores $unt quam <math>l</math>; & ita
a$$umi po$$unt multiplicatores <math>m, m′, m″, &c.</math>, ut exoriantur <math>(n)</math> diver$æ
& independentes fluxionales æquationes <math>m a + m′ a′ + m″ a″ + &c.
= 0</math>, quæ integrari po$$unt.</p>
<p>9. Generalis fluens fluxionalis æquationis <math>(n)</math> ordinis partiales
fluxiones involvens <math>(n)</math> arbitrarias functiones detegendas continet.</p>

<pb 306>[0358]
<h>DE
METHODO INCREMENTORUM.</h>
<h>LIBER. I.</h>
<p>D<sc>FF. I</sc>. DENOTENT quantitates <math>x<x>.</x>, y<x>.</x>, z<x>.</x>, &c.</math> prima incrementa
quantitatum <math>x, y, &c.</math> re$pective; eodem modo de$ignent
quantitates ita $criptæ <math>x<x>..</x>, y<x>..</x>, z<x>..</x>, &c.</math> $ecunda incrementa quantitatum
<math>x, y, z, &c.</math> vel prima incrementa quantitatum <math>x<x>.</x>, y<x>.</x>, z<x>.</x>, &c.</math> & $ic de-
inceps.</p>
<p>D<sc>EF</sc>. 2. Denotent quantitates <math>x, y, &c.</math> integrales quantitatum <math>x, y,
&c.</math> & $ic <math>x<x>″</x>, y<x>″</x>, &c.</math> $ecundas integrales quantitatum <math>x, y, &c.</math> & primas
integrales quantitatum <math>x<x>′</x>, y<x>′</x>, &c.</math></p>
<h>PROB. I.</h>
<p>_Datâ quantitate_ <math>P</math> _vel algebraicâ vel fluxionali, quæ $it functio quan-_
_titatum_ <math>x, y, z, &c.</math> _quarum incrementa $int_ <math>x, y, z, &c.</math> _ejus incrementum_
_inve$tigare._ 1. _Scribantur_ <math>x + x, y + y, z + z, &c.</math> _pro literis_ <math>x, y, z,
&c.</math> _re$pective, &_ <math>x + x, y + y, &c.</math> _pro_ <math>x & y, &c.</math> _in datâ æquatione,_
_& $it quantitas re$ultans_ <math>Q</math>; _tum erit_ <math>Q - P</math> _incrementum quæ$itum, i. e._
_differentia inter duos proxime $ucce$$ivos valores integralis dicitur incre-_
_mentum._</p>
<p>Secunda incrementa e primis eodem modo ac prima ex integrali-
bus deduci po$$unt; & $ic deinceps.</p>
<p>Ex. 1. Incrementum rectanguli <math>x y</math> erit <math>(x + x<x>.</x>)(y + y<x>.</x>) - x y =
x y<x>.</x> + y x<x>.</x> + x<x>.</x> y<x>.</x></math>.</p>

<pb 307>[0359]<rh>DE METHODO, &c.</rh>
<p>Ex. 2. Incrementum logarithmi quantitatis <math>(x</math>), cujus fluxio e$t <math>{x^. / x}</math>,
erit log. <math>(x + x<x>.</x>) - log. (x) = log · {x + x<x>.</x> / x}</math>, cujus fluxio erit <math>- {x<x>.</x>x^. / x(x + x<x>.</x>)}= {x^. / x + x<x>.</x>} - {x^. / x}</math>, $i modo <math>x<x>.</x></math> $it con$tans.</p>
<p>Ex. 3. Incrementum circularis arcus <math>$. {x^. / 1 + x^2}</math>, cujus radius e$t <math>1</math>, &
tangens <math>x</math>, erit differentia inter duos circulares arcus, quorum radius
e$t <math>1</math>, & tangentes re$pective <math>x & x + x<x>.</x></math>, i. e. <math>$. ({x^. / 1 + (x + x^.)^2 -{x^. / 1 + x^2}) = $. - {2xx<x>.</x> + x<x>.</x>^2 / (1 + x^2)(1 + (x + x<x>.</x>)^2)} × x^.</math>.</p>
<p>Ex. 4. Sit integralis <math>a x<x>n</x><x>′</x> × x<x>n + 1</x><x>′</x> &c. × x<x>n + m</x><x>′</x></math> ejus vero incrementum
erit, $i modo <math>x</math> cre$cat uniformiter, <math>a × (x<x>n+1</x><x>′</x> × x<x>n + 2</x><x>′</x> & .. x<x>n + m + 1</x><x>′</x> -
x<x>n</x><x>′</x> × x<x>n + 1</x><x>′</x> &c. × x<x>n + m</x><x>′</x>) = (m + 1)ax<x>.</x> (x<x>n + 1</x><x>′</x> × x<x>n + 2</x><x>′</x> ... x<x>n + m</x><x>′</x>)</math>; & ex eâdem
methodo con$tat incrementum integralis <math>{a / x<x>n</x><x>′</x> × x<x>n + 1</x><x>′</x> × &c. .. x<x>n + m</x><x>′</x>}</math> e$$e
<math>{a / x<x>n + 1</x> × x<x>n + 2</x><x>′</x> × &c. x<x>n + m + 1</x><x>′</x>} - {a / x<x>n</x><x>′</x> × x<x>n + 1</x><x>′</x> .. x<x>n + m</x><x>′</x>} = {- (m + 1)a x<x>.</x> / x<x>n</x><x>′</x> × x<x>n + 1</x><x>′</x> ... x<x>n + m + 1</x><x>′</x>}</math>, $i modo <math>x</math>
fluat uniformiter.</p>
<p>Cor. 1. Sint <math>S, S′, S″, S′″, &c. (n)</math> $ucce$$ivi valores integralis, tum
erit incrementum <math>n</math> ordinis <math>= S - n S′ + n · {n - 1 / 2} S″ - n · {n - 1 / 2} ·{n - 2 / 3} S′″ + &c.</math> incrementa enim primi ordinis $unt re$pective
<math>S - S, S′ - S″, S″ - S′″, &c.</math> $ecundi vero <math>S - S′ - (S′ - S″) =
S - 2 S′ + S″, &c.</math></p>

<pb 308>[0360]<rh>DE METHODO</rh>
<p>Cor. 2. Hinc deduci pote$t integralis primi $ub$equentis ordinis
quantitatis <math>(z) = z + z<x>.</x></math>, $ecundi vero $ub$equentis ordinis <math>= z + z<x>.</x>
+ (z<x>.</x> + z<x>..</x>) = z + 2 z<x>.</x> + z<x>..</x></math>, & exinde integralis $ub$equentis <math>n</math> ordi-
nis erit <math>z + n z<x>.</x> + n · {n - 1 / 2}z<x>..</x> + n · {n - 1 / 2} · {n - 2 / 3}z<x>..</x> + &c.</math></p>
<p>Ex hoc corollario $equitur incrementum contenti <math>(z × z<x>′</x> × z<x>″</x> ... z<x>n</x>)
= z<x>′</x> × z<x>″</x> .. z<x>n + 1</x> - z × z<x>′</x> × z<x>″</x> .. z<x>n</x> = z<x>′</x> × z<x>″</x> ... z · (z + (n + 1)z<x>.</x> + (n + 1).{n / 2}z<x>..</x> + &c. - z)</math>; etiamque incrementum fractionis <math>({1 / z · z<x>′</x> · z<x>″</x> ... z<x>n</x>})</math>
e$$e <math>{1 / z<x>′</x> × z<x>″</x> .. z<x>n + 1</x>} - {1 / z × z′ ... z<x>n</x>} = {z - z<x>n + 1</x> / z × z<x>′</x> × z<x>″</x> .. z<x>n + 1</x>} = - {z + (n + 1) × z<x>.</x> + (n + 1) · {n / 2}z<x>..</x> + &c. - z/z × z<x>′</x> ... z<x>n + 1</x>}</math>.</p>
<p>Cor. 2. Et $ic inveniri pote$t <math>{1 / z<x>′</x><x>m</x>} = {1 / z} - {m z<x>.</x> / z z<x>′</x>} + {m × (m - 1)z<x>.</x>^2 / zz<x>′</x>z<x>″</x>} -{m × (m - 1) × (m - 2)z<x>.</x>^3 / zzz<x>′</x>z<x>″</x>} + &c.</math> $i modo <math>z</math> fluat uniformiter; $uppo-
natur enim <math>{1 / z<x>m - 1</x>} = {1 / z} - {(m - 1)z<x>.</x> / zz<x>′</x>} + {(m - 1) · (m - 2)z<x>.</x>^2 / zz<x>′</x>z<x>″</x>} - &c.}</math> &
proximum incrementum $uper$criptarum quantitatum erit <math>- {z<x>.</x> / z z<x>′</x>} +{2(m - 1)z<x>.</x>^2 / z z<x>′</x>z<x>″</x>} - &c.</math> harum $umma erit <math>{1 / z<x>m</x>} = {1 / z} - {m z<x>.</x> / z z<x>′</x>} + {m (m - 1)z<x>.</x>^2 / zz<x>′</x>z<x>″</x>}- &c.</math> $i igitur vere a$$ignetur inferior fluentialis quantitas <math>x<x>m - 1</x></math>, vere
etiam a$$ignatur $uperior <math>x<x>m</x></math>.</p>

<pb 309>[0361]<rh>INCREMENTORUM.</rh>
<p>Et generaliter erit <math>{1 / z<x>m</x>} = {1 / z} - {mz<x>.</x> / zz<x>′</x>} + m · {m - 1 / 2}{2z<x>.</x>^2 - zz<x>..</x> + z<x>.</x>z<x>..</x> / z · z<x>′</x> · z<x>″</x>} -
&c.</math></p>
<p>Ex. 3. Sit integralis <math>1 · 2 · 3 · 4 ... z</math>, & ejus incrementum erit
<math>1 · 2 · 3 · 4 .. z · (1 - z)^2 = 1 · 2 · 3 · 4 .. z · (z + 1) - 1 · 2 · 3 · 4 .. z</math>,
$i modo incrementum quantitatis <math>z</math> $it <math>1</math>.</p>
<p>Ex. 4. Sit integralis <math>{1 / 1 · 2 · 3 · 4 ... z}</math>, & ejus incrementum erit <math>-{1 / 1 · 2 · 3 .. z - 1 · z + 1} = {1 / 1 · 2 · 3 .. z + 1} - {1 / 1 · 2 · 3 .. z}</math>.</p>
<p>Et $ic ex incrementis inveniri po$$unt integrales.</p>
<p>2. Sit <math>α</math> functio quantitatum <math>x, y, z, &c.</math>; inveniantur <math>α^., α^.., α^∴, α^n,
&c.</math>; ex hypothe$i; quod <math>x, y, z, &c.</math> fluunt uniformiter; & con$e-
quenter earum $ecundæ, tertiæ, &c. fluxiones nihilo $unt æquales;
in his quantitatibus <math>(α^., α^.., α^∴, &c.)</math> pro <math>x^., y^., z^., &c.</math> $cribantur re-
$pective <math>x, y, z, &c.</math>; & re$ultent quantitates <math>α<x>.</x>, α<x>..</x>, α<x>∴</x>, &c.</math>; tum erit
primum incrementum quantitatis <math>(α) = α<x>.</x> + {1 / 1 · 2}α<x>..</x> + {1 / 1 · 2 · 3}α<x>∴</x> +{1 / 1 · 2 · 3 · 4} α<x>....</x> + &c.</math> Ex hoc primo incremento per eandem metho-
dum detegi pote$t $ecundum, $i modo $upponantur <math>x^.., y^.., z^.., &c.</math> inva-
riabiles, & pro iis $cribantur re$pective <math>x<x>..</x>, y<x>..</x>, z<x>..</x>, &c. &c.</math></p>
<h>PROB. II.</h>
<p>_Invenire incrementa fluentium, quarum fluxiones $unt algebraicæ fun-_
_ctions quantitatis_ <math>x</math> _in_ <math>x^.</math> _ductæ: inveniri po$$unt e reductione fluxionum_
_in $eriem, cujus termini $ecundum dimen$iones quantitatis_ <math>x</math> _progrediuntur:_
_buju$ce $eriei inveniatur fluens, & deinde buju$ce fluentis detegatur incre-_
_mentum quæ$itum._</p>
<p>Ex. · Invenire incrementum fluentis <math>$. {x^. / 1 - x}:</math> erit <math>$. {x^. / 1 - x} =

<pb 310>[0362]<rh>DE METHODO</rh>
x + {1 / 2} x^2 + {1 / 3} x^3 + {1 / 4} x^4 + &c.</math> cujus incrementum quæ$itum erit
<math>(1 + x + x^2 + &c.) x^. + {1 / 2}(1 + 2 x + 3 x^2 + &c.)x^. ^2 + {1 / 3}(1 +
3 x + 6 x^2 + &c.) x<x>.</x>^3 + {1 / 4}(1 + 4 x + &c.) x<x>.</x>^4 + &c. = {x<x>.</x> / 1 - x} +{{1 / 2} x<x>.</x>^2 /(1 - x)^2} + {{1 / 3} x<x>.</x>^3 /(1 - x)^3} + {{1 / 4} x<x>.</x>^4 /(1 - x)^4} + &c.</math> $i modo incrementum <math>x<x>.</x></math> $it
con$tans, fingatur <math>z = 1 - x</math> & re$ultat incrementum fluentis
<math>$. {z^. / z} = {z<x>.</x> / z} - {z<x>.</x>^2 / 2z^2} + {z<x>.</x>^3 / 3z^3} + &c.</math></p>
<p>Et $ic de incrementis deducendis duplicatarum, &c. fluentium, vel
quarumcunque aliarum quantitatum, quæ $int functio finita vel infi-
nita algebraicarum vel exponentialium, &c. quantitatum. Aliter:
Earum incrementa detegi po$$unt e methodo in ca$u $ecundo præce-
dentis problematis traditâ.</p>
<p>2. Si vero e dato incremento fluentis <math>(v)</math> requiratur incrementum
quantitatis <math>x</math>; inveniatur per infinitas $eries quantitas <math>x</math> in terminis
$ecundum dimen$iones quantitatis <math>v</math> progredientibus, huju$ce quan-
titatis incrementum erit quæ$itum.</p>
<p>E. g. Sit <math>v = $.{x^. / 1 + x}</math>, & erit <math>x = v + {v^2 / 1 · 2} + {v^3 / 1 · 2 · 3} + &c.</math>
cujus incrementum <math>x<x>.</x> = v<x>.</x>(1 + v + {1 / 2}v^2 + {1 / 1 · 2 · 3}v^3 + &c.) + {1 / 2}v<x>.</x>^2
(1 + v + {1 / 1 · 2}v^2 + {1 / 1 · 2 · 3}v^3 + &c.) + {1 / 1 · 2 · 3}v<x>.</x>^3 (1 + v + {1 / 2}v^2 +{1 / 2 · 3}v^3 + &c.) + &c. = (1 + x)(v<x>.</x> + {1 / 1 · 2}v<x>.</x>^2 + {1 / 1 · 2 · 3}v<x>.</x>^3 + {1 / 1 · 2 · 3 · 4}v<x>.</x>^4
+ &c.)</math>. $i modo <math>v<x>.</x></math> $it con$tans.</p>
<h>THEOR. I.</h>
<p>Sit quantitas <math>P</math> functio quantitatis <math>x</math>, cujus incrementum $it <math>Q</math>;
tum erit fluxio <math>Q^.</math> quantitatis <math>(Q)</math> æqualis incremento fluxionis <math>P^.</math>,
$i modo fluxio <math>(o^.)</math> incrementi <math>(o)</math> quantitatis <math>x</math> æqualis $it incremento
fluxionis <math>x^.</math>.</p>

<pb 311>[0363]<rh>INCREMENTORUM.</rh>
<p>Scribatur in quantitate <math>P</math> pro <math>x</math> valor a$$umptus <math>x + x^. + o + o^.</math>,
& collocentur termini quantitatis re$ultantis $ecundum dimen$iones
fluxionum <math>x^. & o^.</math>, quæ re$pective denotant fluxiones quantitatum
<math>x & o</math>, i. e. quantitas re$ultans $it huju$ce formulæ <math>l + m(x^. + o^.)
+ (px^. ^2 + qx^. o^. + po^. ^2) + &c.</math>; deinde in quantitate <math>P</math> pro <math>x</math> $criba-
tur <math>x + o</math> & con$equitur ejus incrementum <math>(α)</math> ipsâ quantitate <math>P</math> au-
ctum, hujus incrementi <math>α</math> inveniatur fluxio quæ erit <math>m(x^. + o^.) - P^.</math>, $icut
e $ub$titutione prædictâ facile con$tat; tertio inveniatur fluxio quan-
titatis <math>P</math>, vel quod ad idem redit in datâ quantitate <math>P</math> pro <math>x</math> $cribatur
<math>x + x^.</math>, & collocentur termimi quantitatis re$ultantis $ecundum di-
men$iones fluxionis <math>x^.</math>, i. e. $int <math>a + bx^. + cx^. ^2 + &c.</math> tum erit <math>bx^.</math>
fluxio quantitatis <math>P</math>; in hâc fluxione pro <math>x</math> $cribatur <math>x + o</math> & pro <math>x^.</math>
ejus valor <math>x^. + o^.</math>, & e $ub$titutione facile con$tat e$$e <math>m(x^. + o^.)</math>, &
exinde pote$t theor.</p>
<p>Idem vero mutatis mutandis affirmari pote$t, $i <math>P</math> $it functio
quarumcunque quantitatum <math>x, y, z, &c.</math> mutatis etiam mutandis erit
incrementum <math>n</math> ordinis fluxionis <math>m</math> ordinis quantitatis <math>P</math> æquale flu-
xioni <math>m</math> ordinis incrementi <math>n</math> ordinis eju$dem quantitatis <math>P.</math></p>
<h>PROB. III.</h>
<p>1. _Invenire incrementum cuju$cunque fluentialis qua titatis_ <math>$. W.</math></p>
<p>Sit <math>$. W = u</math>, & incrementum quantitatis <math>$. W = $.(W + W<x>.</x>) - $.
W = $.W<x>.</x> = u.</math></p>
<p>Sit <math>W</math> functio quantitatis <math>x</math> in <math>x^.</math>, & in eâ pro <math>x</math> $cribatur <math>x + x^.</math> &
re$ultet quantitas <math>W′</math>, tum erit incrementum quæ$itum <math>$.W′ - $.W.</math></p>
<p>Ex. · Sit fluentialis quantitas <math>$. Vx^.</math>, cujus incrementum requiritur;
inveniatur incrementum quantitatis <math>Vx^.</math>, quæ erit <math>x^.V<x>.</x> + V(x^. <x>.</x>) + V(x^. <x>.</x>)</math>,
& exinde incrementum fluentis <math>$. Vx^. = $. x^. V<x>.</x> + $.V(x^. <x>.</x>) + $.V(x^. <x>.</x>)</math>;
at <math>$.Vx^. <x>.</x> = Vx<x>.</x> - $.x<x>.</x>V</math>, unde <math>{$.Vx^. / .} = $.x^.V<x>.</x> + Vx<x>.</x> - $.x<x>.</x>V^. + $.V^.(x^. <x>.</x>)</math>.</p>
<p>2. Si requiratur incrementum duplicis fluentialis formulæ <math>$.$.W</math>,
tum erit <math>($.$.W) = $.($.W) = $.$.W</math>, & $ic deinceps.</p>

<pb 312>[0364]<rh>DE METHODO</rh>
<p>3. Sint incrementa evane$centia, & ponantur <math>y^. = px^., p^. = qx^.,
q^. = r x^., &c.</math>; cum fuerit <math>V</math> functio quæcunque quantitatum <math>x, y, p,
q, r. s, &c.</math>; tum erit <math>V^. = M x^. + N y^. + P p^. + Q q^. + &c.</math>, & $imili
modo erit <math>V<x>.</x> = M x<x>.</x> + N y<x>.</x> + P p<x>.</x> + Q q<x>.</x> + &c.</math></p>
<p>Ex inveniendis incrementis quantitatum <math>$. Vx^., &c.</math> detegi po$$unt
maxima & minima, $i modo in incrementis re$ultantibus nihilo
fiant æqualia incrementa, & exinde fluentiales quantitates etiam ni-
hilo evadant æquales; $ed hic non de maximis & minimis, nec de
evane$centibus incrementis agitur.</p>
<h>PROB. IV.</h>
<p>_Transformare datum incrementum in alterum, cujus variabiles quanti-_
_tates_ <math>(z, v, &c.)</math> _datam babeant relationem ad variabiles_ <math>(x, y, &c.)</math> _in_
_dato incremento contentas. Sub$tituantur valores quantitatum_ <math>(x, y, &c.)</math>
_in terminis quantitatum_ <math>(z & v, &c.)</math> _expre$$i pro quantitatibus ip$is_ <math>(x, y,
&c.)</math> _in dato incremento, & eorum incrementa pro re$pectivis incrementis_
_quantitatum_ <math>(x, y. &c.)</math> _& re$ultat incrementum quæ$itum._</p>
<h>PROB. V.</h>
<h it>Datum incrementum in alterum datæ formulæ, $i modo $ieri po$$it,
reducere.</h>
<p>A$$umantur generaliter incrementa datæ formulæ, $upponatur eo-
rum $umma dato incremento æqualis, & exinde erui po$$unt coef-
ficientes, &c. incrementorum a$$umptorum.</p>
<p>Ex. 1. Sit <math>x^.</math> con$tans; reducere quantitates <math>x^n</math> ad incrementa for-
mulæ <math>x × (x - x<x>.</x>) × (x - 2x^.) .. (x - mx<x>.</x>)</math>.</p>
<p>A$$umantur <math>x^n = x × (x - x<x>.</x>) × (x - 2x<x>.</x>) .. (x - (n - 1)x<x>.</x>) +
bx<x>.</x>x × (x - x<x>.</x>) · (x - 2x<x>.</x>) .. (x - (n - 2)x<x>.</x>) + cx<x>.</x>^2 x × (x - x<x>.</x>) ...
(x - (n - 3)x<x>.</x>) + &c.</math> e reductione facile con$tat <math>b = n · {n - 1 / 2},
c = {5n^3 - 42n^2 + 127n - 135 / 3}, &c.</math></p>

<pb 313>[0365]<rh>INCREMENTORUM</rh>
<p>Ex. 2. Sit <math>x<x>.</x></math> con$tans, & inveniri pote$t <math>{1 / x^2} = {1 / x × (x + x<x>.</x>)} +{x<x>.</x> / x × (x + x<x>.</x>) × (x + 2 x<x>.</x>)} + {2 x<x>.</x>^2 / x(x + x<x>.</x>)(x + 2 x<x>.</x>)(x + 3 x<x>.</x>)} + {6 x<x>.</x>^3 / x(x + x<x>.</x>)...(x + 4 x<x>.</x>)}+ {24&c. / &c.} · &{1 / x^n} = {1 / x(x + x<x>.</x>)...(x + (n - 1)x<x>.</x>)} +{n · {n - 1 / 2}x<x>.</x>/x(x + x<x>.</x>)...(x + nx<x>.</x>)} + &c.</math></p>
<h>PROB. VI.</h>
<h>1. _Sit_ <math>x<x>.</x></math> _incrementum quantitatis <math>x</math> con$tans, invenire integralem quan-_
_titatis_ <math>Ax × (x - x<x>.</x>) · (x - 2 x<x>.</x>) ... (x - n x<x>.</x>)</math>.</h>
<p>Integralis erit <math>{A / (n + 2)x<x>.</x>}(x + x<x>.</x>) × x × (x - x<x>.</x>) .. (x - nx<x>.</x>)</math>; inve-
niantur enim duæ $ucce$$ivæ integrales <math>{A / (n + 2)x<x>.</x>} × (x + x<x>.</x>) × x.
(x - x<x>.</x>) .. (x - nx<x>.</x>) & {A / (n + 2)x<x>.</x>}x · (x - x<x>.</x>) · (x - 2x<x>.</x>) .. (x - nx<x>.</x>)
× (n - nx<x>.</x> - x<x>.</x>)</math>, & earum differentia erit quantitas data.</p>
<p>Co. · Dato incremento <math>x<x>.</x></math>; omnis quantitas huju$ce formulæ <math>Ax^n
+ Bx^n-1 + Cx^n-2 + Dx^n-3 + &c.</math> ubi <math>n</math> e$t integer affirmativus
numerus, & <math>A, B, C, D, &c.</math> invariabiles denotant quantitates, tran$-
formari pote$t in quantitates prædictæ formulæ <math>ax × (x - x<x>.</x>) ×
(x - 2 x<x>.</x>) ... (x - (n - 1) x<x>.</x>) + bx × (x - x<x>.</x>) .. (x - (n - 2) x<x>.</x>) +
cx × (x - x<x>.</x>) .. (x - (n - 3) x<x>.</x>) + d × &c.</math> æquatis autem terminis
corre$pondentibus huju$ce & datæ æquationis, i. e. $int <math>a = A, (n - 1)
× {n / 2}ax<x>.</x> + b = B, &c.</math> inveniri po$$unt coefficientes <math>a, b, &c.</math> & con-
$equenter reduci pote$t data quantitas ad incrementa formularum,
quarum innote$cunt integrales.</p>

<pb 314>[0366]<rh>DE METHODO</rh>
<p>Aliter: Invenire integralem prædictæ incrementialis quantitatis
<math>A x^n + B x^n-1 + C x^n-2 + &c.</math> a$$umatur quantitas <math>a x^n+1 + b x^n +
c x^n-1 + d x^n-2 + &c.</math> pro integrali deducendâ, cujus inveniatur
proxime $ucce$$ivus valor, qui erit <math>a (x + x<x>.</x>)^n+1 + b (x + x<x>.</x>)^n +
c (x + x<x>.</x>)^n-1 + &c.</math> unde differentia inter duos $ucce$$ivos valores erit
<math>a ((x + x<x>.</x>)^n+1 - x^n+1) + b ((x + x<x>.</x>)^n - x^n) + c ((x + x<x>.</x>)^n-1 - x^n-1) + &c. =
(n + 1)a x^n x<x>.</x> + (n + 1) · {n / 2} a x<x>.</x>^2 \\ + n b x<x>.</x> # x^n-1 + (n + 1) · {n / 2} · {n - 1 / 3} a x<x>.</x>^3 \\ + n · {n - 1 / 2} b x<x>.</x>^2 \\ + (n - 1) c x<x>.</x> # x^n-2 + &c.</math>
æquatis autem terminis hujus & datæ æquationis, re$ultant (<math>n + 1</math>)
æquationes <math>a x<x>.</x> = A, (n + 1) · {n / 2} a x<x>.</x>^2 + n b x<x>.</x> = B, &c.</math> unde facile con-
$tant coe$$icientes <math>a, b, c, &c.</math> & exinde integralis datæ quantitatis.</p>
<p>Aliter: A$$umatur quantitas <math>a x^n+1 + p</math> pro integrali quæ$itâ, cujus
incrementum erit <math>(n + 1) a x<x>.</x>x^n + (n + 1) × {n / 2} a x<x>.</x>^2 x^n-1 + &c. + p =
A x^n + B x^n-1 + C x^n-2 + &c.</math> & exinde <math>(n + 1) a x<x>.</x> = A, & p<x>.</x> = (B -
(n + 1) · {n / 2} a x<x>.</x>^2)x^n-1 + &c.</math> & con$equenter <math>n x<x>.</x> p = (B - (n + 1) × {n / 2} a x<x>.</x>^2)
x^n</math> prope; $cribatur igitur in præcedente æquatione <math>(B - (n + 1) ×{n / 2} a x<x>.</x>^2) x^n + q</math> pro <math>p</math>, & ejus incrementum pro <math>p<x>.</x></math>, & exinde deduci
pote$t propinquus valor quantitatis <math>q</math>; & $ic deinceps u$que donec
inveniatur integralis quæ$ita.</p>
<p>2. Sit <math>x<x>.</x></math> incrementum quantitatis <math>x</math> con$tans, & incrementialis
quantitatis <math>{x<x>.</x> / x × (x - x<x>.</x>) × (x - 2 x<x>.</x>) ... (x - n x<x>.</x>)}</math> integralis invenietur
<math>{x<x>.</x> / n x<x>.</x> × (x - x<x>.</x>) × (x - 2 x<x>.</x>) ... (x - n x<x>.</x>)} = {1 / n × (x - x<x>.</x>) ... (x - n x<x>.</x>)}</math>.</p>

<pb 315>[0367]<rh>INCREMENTORUM.</rh>
<p>Facile con$tat hic ca$us ex differentiâ inter duos $ucce$$ivos valores
integralis prædictæ.</p>
<p>3. Sit incrementum <math>{a x^m + b x^m-1 + c x^m-2 + d x^m-3 + e x^m-4 + &c. / x × (x + x<x>.</x>) × (x + 2x<x>.</x>) × (x + 3x<x>.</x>) .... (x + (n - 1) x<x>.</x>)}</math>
rationalis functio quantitatum <math>x & x<x>.</x></math> ad minimos terminos reducta;
ubi <math>m</math> e$t integer numerus, & æqualis vel minor quam <math>n, & x<x>.</x></math> e$t
con$tans; invenire annon integralis incrementi prædicti exprimi po-
te$t in finitis terminis.</p>
<p>Reducatur datum incrementum ad incrementa huju$ce generis
<math>α + ({β / x}) + {γ / x · (x + x<x>.</x>)} + {δ / x · (x + x<x>.</x>)(x + 2 x<x>.</x>)} + {ε / x (x + x<x>.</x>)(x + 2x<x>.</x>)(x + 3x<x>.</x>)}+ .... {θ / x (x + x<x>.</x>) (x + 2 x<x>.</x>) ... (x + (n - 1) x<x>.</x>)}</math>; $i <math>m = n</math>, tum <math>α = a,
β = b - α A, γ = c - β A′ - α B, δ = d - γ A″ - β B′ - α C, ε = e
- δ A′″ - γ B″ - β C′ - α D, &c.</math>; ubi <math>x (x + x<x>.</x>) (x + 2 x<x>.</x>) ... (x +
(n - 1) x<x>.</x>) = x^n + A x^n-1 + B x^n-2 + C x^n-3 + D x^n-4 + &c.; (x + x<x>.</x>)
(x + 2 x<x>.</x>) .. (x + (n - 1) x<x>.</x>) = x^n-1 + A′ x^n-2 + B′ x^n-3 + C′ x^n-4 +
&c., (x + 2 x<x>.</x>) (x + 3 x<x>.</x>) ... (x + (n - 1) x<x>.</x>) = x^n-2 + A″ x^n-3 +
B″ x^n-4 + D″ x^n-5 + &c., (x + 3 x<x>.</x>) (x + 4 x<x>.</x>) ... (x + (n - 1) x<x>.</x>) =
x^n-3 + A′″ x^n-4 + B′″ x^n-5 + C′″ x^n-6 + &c., &c.</math></p>
<p>Si <math>β = b - n · {n - 1 / 2} a x<x>.</x></math> haud $it nihilo æqualis, tum integralis non
exprimi pote$t, ni detur integralis incrementi <math>{β / x}</math>; in reliquis ca$ibus per
hanc methodum $emper inveniri pote$t integralis; erit enim <math>α {x / x<x>.</x>} + {γ / x<x>.</x> x}+ {δ / 2 x<x>.</x> x (x + x<x>.</x>)} + {ε / 3 x<x>.</x> x (x + x<x>.</x>) (x + 2 x<x>.</x>)} + ... + {θ / (n - 1) x<x>.</x> x (x + x<x>.</x>)
(x + 2 x<x>.</x>) .. (x + (n - 2) x<x>.</x>)}</math>, ubi <math>α, β, γ, δ, &c.</math> $unt invariabiles
quantitates.</p>

<pb 316>[0368]<rh>DE METHODO</rh>
<p>Si <math>m = n - 1</math>; tum integralis non exprimi pote$t, ni detur inte-
gralis incrementi (<math>{β / x}</math>): $i autem <math>m</math> minor $it quam <math>n - 1</math>, tum $em-
per per hanc regulam inveniri pote$t integralis.</p>
<p>Si vero <math>m</math> major $it quam <math>n</math>, tum reducenda e$t data fractio ad pro-
priam, ita ut maxima pote$tas quantitatis <math>x</math> in denominatore major
$it quam maxima pote$tas in numeratore.</p>
<p>4. Sit incrementum fractio <math>{a x^m′ + b x^m′+1 + c x^m′+2 + &c. / x · (x + x<x>.</x>)(x + 2 x<x>.</x>) ... (x + (n - 1) x<x>.</x>)
× (x + p)(x + p + x<x>.</x>)(x + p + 2 x<x>.</x>) ... (x + p + (m - 1) x<x>.</x>) × (x +
q) × (x + q + x<x>.</x>)(x + q + 2 x<x>.</x>) ... (x + q + (r - 1) x<x>.</x>) × &c.}</math>.</p>
<p>Semper reduci pote$t hæc fractio ad terminos progredientes $ecun-
dum formulas $ub$equentes <math>α + ({β / x}) + {γ / x · (x + x<x>.</x>)} + {δ / x (x + x<x>.</x>)(x + 2 x<x>.</x>)}+ {ε / x (x + x<x>.</x>)(x + 2 x<x>.</x>)(x + 3 x<x>.</x>)} + ... + {θ / x (x + x<x>.</x>)(x + 2 x<x>.</x>) .. (x + (n - 1) x<x>.</x>)}+ ({β′ / x + p}) + {γ′ / (x + p)(x + p + x<x>.</x>)} + {δ′ / (x + p)(x + p + x<x>.</x>)(x + p + 2 x<x>.</x>)}+ ... + {θ′ / (x + p)(x + p + x<x>.</x>)(x + p + 2 x<x>.</x>) .... (x + p + (m - 1) x<x>.</x>)}+ ({β″ / x + q}) + {γ″ / (x + q)(x + q + x<x>.</x>)} + {δ″ / (x + q)(x + q + x<x>.</x>)(x + q + 2 x<x>.</x>)}+ ... + {θ″ / (x + q)(x + q + x<x>.</x>)(x + q + 2 x<x>.</x>) .... (x + q + (r - 1) x<x>.</x>)}+ &c.</math>, ubi <math>α, β, γ, .. θ; β′, γ′, &c.; β″, γ″, &c.; p, q, &c</math>; $unt in-
variabiles quantitates.</p>
<p>Con$tat ex eo quod continentur <math>n + m + r + &c. + 1</math> invariabiles,
independentes & incognitæ coefficientes <math>α, β, γ, δ, &c.; β′, γ′, &c.; β″,
γ″, &c.</math>; in quantitate a$$umptâ.</p>

<pb 317>[0369]<rh>INCREMENTORUM.</rh>
<p>Si <math>β, β′, β″, &c.</math> nihilo $int re$pective æquales, vel integralis incre-
menti <math>{β / x} + {β′ / x + p} + {β″ / x + q} + &c.</math> finitis terminis exprimi po$$it, tum
integralis dati incrementi $emper ii$dem exprimi pote$t; $in aliter
non.</p>
<p>Hinc, $i modo numerus literarum <math>o, p, q, &c.</math> $it <math>λ</math>; ex <math>λ</math> indepen-
dentibus integralibus huju$ce formulæ datis deduci po$$unt omnes
eju$dem formulæ.</p>
<p>Si <math>m′ = n + m + r + &c. - 1</math>, tum nunquam exprimi pote$t in-
tegralis; $i <math>m′</math> major $it quam <math>n + m + r + &c.</math>; tum reducenda e$t
data ad propriam fractionem: $i <math>m′ = n + m + r + &c. - 2</math> vel mi-
nor; tum $emper <math>β + β′ + β″ + &c. = 0</math>.</p>
<p>5. Sit incrementum <math>{a x^m + b x^m-1 + c x^m-2 + &c. / x^π (x + x<x>.</x>)^π (x + 2 x<x>.</x>)^π ... (x + (n - 1) x<x>.</x>)^π × (x^π′ (x +
p + x<x>.</x>)^π′ × (x + p + 2 x<x>.</x>)^π′ ... (x + p + (r - 1) x<x>.</x>)^π′ × &c.}</math>; hoc incre-
mentum reduci pote$t ad terminos progredientes $ecundum $ub$e-
quentes formulas <math>α + ({β / x} + {γ / x^2} + {δ / x^3} + ... + {λ / x^π}) + ({β′ / x + p} +{γ′ / (x + p)^2} + {δ′ / (x + p)^3} ... + {λ′ / (x + p)^π}) + ({ε / x (x + x<x>.</x>)} + {ε′ / x (x + x<x>.</x>)(x + 2 x<x>.</x>)}+ .... + {ε′^n-2 / x(x + x<x>.</x>) ... (x + (n - 1) x<x>.</x>)}) + ({ξ / (x + p)(x + p + x<x>.</x>)} +{ξ′ / (x + p)(x + p + x<x>.</x>)(x + p + 2 x<x>.</x>)} + ... + {ξ′^r-2 / x(x + p + x<x>.</x>) .. (x + p + (r - 1) x<x>.</x>)})
+ {a′ ((x + x<x>.</x>)^2 - x^2) / x^2 × (x + x<x>.</x>)^2} + {a″ ((x + x<x>.</x>)^3 - x^3) / x^3 (x + x<x>.</x>)^3} + .. + {a′^n-2 ((x + (n - 1) x<x>.</x>)^π - x^π) / x^π × (x + x<x>.</x>)^π}+ {b′ (x + p + x<x>.</x>)^2 - (x + p)^2 / (x + p)^2 (x + p + x<x>.</x>)^2} + {b″ (x + p + x<x>.</x>)^3 - (x + p)^3 / (x + p)^3 (x + p + x<x>.</x>)^3} + .... +

<pb 318>[0370]<rh>DE METHODO</rh>
{b′^n′-3 (x + p + x<x>.</x>)^π′ - (x + p)^π′ / (x + p)^π′ (x + p + x<x>.</x>)^π′} + &c. .... {d′ (x + 2x<x>.</x>)^2 - x^2 / x^2 × (x + x<x>.</x>)^2 × (x + 2 x<x>.</x>)^2} +{d″ (x + 2 x<x>.</x>)^3 - x^3 / x^3 (x + x<x>.</x>)^3 (x + 2 x<x>.</x>)^3} + .... + {d′^π-3 (x + 2 x<x>.</x>)^π - x^π / x^π × (x + x<x>.</x>)^π × (x + 2 x<x>.</x>)^π} +{e × (x + p + 2 x<x>.</x>)^2 - (x + p)^2 / (x + p)^2 × (x + p + x<x>.</x>)^2 × (x + p + 2 x<x>.</x>)^2} + {e′(x + p + 2 x<x>.</x>)^3 - (x + p)^3 / (x + p)^3 × (x + p + x<x>.</x>)^3 × (x + p + 2 x<x>.</x>)^3}+ .... + {e′^π-3 (x + p + 2 x<x>.</x>)^π′ - (x + p)^π′ / (x + p)^π × (x + p + x<x>.</x>)^π′ × (x + p + 2 x<x>.</x>)^π′} +{f (x + 3 x<x>.</x>)^2 - x^2 / x^2 (x + x<x>.</x>)^2 × (x + 2 x<x>.</x>)^2 × (x + 3 x<x>.</x>)^2}+ &c. + {g (x + p + 3 x<x>.</x>)^2 - (x + p)^2 / (x + p)^2 × (x + p + x<x>.</x>)^2 × (x + p +
2 x<x>.</x>)^2 × (x + p + 3 x<x>.</x>)^2} + &c. + {l (x + (n - 1) x<x>.</x>)^2 - x^2 / (x^2)(x + x<x>.</x>)^2 (x + 2 x<x>.</x>)^2 (x + 3 x<x>.</x>)^2 ...
(x + (n - 1) x<x>.</x>)^2} + .... + {l′^π (x + (n - 1) x<x>.</x>)^π - x^π / x^π (x + x<x>.</x>)^π (x + 2 x<x>.</x>)^π ... (x + (n - 1) x^.)^π}+ {l″^π (x + p + (r - 1) x<x>.</x>)^2 - x<x>.</x>^2 / (x + p)^2 (x + p + x<x>.</x>)^2 (x + p + 2 x<x>.</x>)^2 (x + p + 3 x<x>.</x>)^2 ... (x + p + (r - 1) x<x>.</x>)^2}+ .... + {l′^π′ (x + p + (r - 1) x<x>.</x>)^π′ - x^π′ / (x + p)^π′ (x + p + x<x>.</x>)^π′ ... (x + p + (r - 1) x<x>.</x>)^π′}, &c.</math>; quorum
$ingulorum integrales detegi po$$unt, $i integralis quantitatis <math>{β / x} + {γ / x^2}+ {δ / x^3} + &c. + {β′ / x + p} + {γ′ / (x + p)^2} + {δ / (x + p)^3} + &c.</math> deduci po$$it.</p>
<p>Si data fractio $it impropria, tum reducatur ea ad propriam fracti-
onem.</p>
<p>Hæc principia etiam ad inveniendas integrales omnium rationalium
functionum variabilis (<math>x</math>) applicari po$$unt.</p>
<p>Si vero in denominatore contineantur duo factores <math>x + l x<x>.</x> & x +

<pb 319>[0371]<rh>INCREMENTORUM.</rh>
(l + r)x<x>.</x></math>, vel quicunque factores inter hos po$iti, ubi <math>r</math> $it integer
numerus: tum reducenda e$t data fractio ad con$imiles terminos, i.e.
eo$dem denominatores habentes ac ii, in quibus habetur conten-
tum <math>(x + l x<x>.</x>) (x + (l + 1)x<x>.</x>) (x + (l + 2)x<x>.</x>) (x + (l + 3)x<x>.</x>) ... (x
+ (l + r)x<x>.</x>)</math>. Et $ic de pluribus contentis huju$ce generis & eorum
pote$tatilus.</p>
<p>7. Dimen$iones quantitatis <math>x</math> in incremento $emper erunt minores
quam dimen$iones eju$dem quantitatis in integrali per unitatem, ni
dimen$iones quantitatis <math>x</math> in numeratore integralis æquales $int ejus
dimen$ionibus in ejus denominatore, in quo ca$u dimen$iones quan-
titatis <math>x</math> in numeratore incrementi minores erunt per quantita-
tem majorem quam unitatem quam dimen$iones quantitatis <math>x</math> in de-
nominatore.</p>
<p>Cor. Si dimen$iones quantitatis <math>x</math> in numeratore incrementi $int
minores quam dimen$iones quantitatis <math>x</math> in denominatore per uni-
tatem, tum ejus integralis non deduci pote$t.</p>
<p>Et $ic de fractionibus reducendis, &c. in qua$cunque formulas,
quarum integrales innote$cunt.</p>
<p>8. Inveniantur $inguli divi$ores denominatoris fractionis <math>A x^n +
B x^n-1 + &c.</math> qui $int <math>α, β, γ, δ, &c.</math> & $i quicunque divi$or invenia-
tur in denominatore, qui non habet alterum a $e di$tantem per in-
tegrum numerum <math>r</math> in <math>x<x>.</x></math> ductum, tum integralis dati incrementi
finitis terminis haud exprimi pote$t. E.g. Sit denominator <math>(x + {1 / 2} x<x>.</x>)
× (x - 4x<x>.</x>) × (x - 5x<x>.</x>)</math>, nunc divi$or <math>x + {1 / 2}x<x>.</x></math> non habet alterum in
denominatore a fe di$tantem per integrum numerum in <math>x<x>.</x></math>, ergo in-
tegralis dati incrementi finitis terminis haud exprimi pote$t.</p>
<p>Eadem principia etiam ad irrationales quantitates applicari
po$$unt.</p>
<p>1. Contineantur $olummodo in dato incrementi <math>{a′ x^m + b′x^m-1 + &c. / A x^n + B x^n-1 + &c.}</math>
denominatore duo $implices divi$ores <math>x + l x<x>.</x> & x + (l + r) x<x>.</x></math>; vel hi
duo & quicunque alii quorum numerus $it <math>τ</math> inter eos po$iti, ubi <math>r</math>

<pb 320>[0372]<rh>DE METHODO</rh>
$it integer numerus; tum a$$umenda e$t quantitas <math>((x + l x<x>.</x>) × (x +
(l + 1) x<x>.</x>) × (x + (l + 2) x<x>.</x>) ... (x + (l + r - 1) x<x>.</x>))^-1 × (α x^m+r+1-n +
β x^m+r-n + &c.)</math> ubi <math>α, β, &c.</math> $unt invariabiles coefficientes inve$ti-
gandæ, pro integrali quæ$itâ; inveniatur ejus incrementum, quod
dato incremento fiat æquale, & ex æquatis eorum corre$pondentibus
terminis deduci po$$unt invariables coefficientes <math>α, β, γ, &c.</math></p>
<p>2. Sit denominator <math>P = (x + a x<x>.</x>)(x + (a + b) x<x>.</x>)(x + (a + b + c) x<x>.</x>)
(x + e x<x>.</x>)(x + (e + f)x<x>.</x>)(x + (e + f + g) x<x>.</x>)(x + k x<x>.</x>)(x + (k + l) x<x>.</x>)
&c.)</math> ubi <math>b, c; f, g; l, &c.</math> re$pective denotant integros numeros; tum
a$$umenda e$t quantitas <math>((x + a x<x>.</x>)(x + (a + 1) x<x>.</x>)(x + (a + 2) x<x>.</x>)
... (x + (a + b + c - 1) x<x>.</x>)(x + e x<x>.</x>)(x + (e + 1) x<x>.</x>)(x + (e + 2) x<x>.</x>)
... (x^. + e + f + g - 1) x<x>.</x>)(x + k x<x>.</x>)(x + (k + 1) x<x>.</x>) ... (x + (k + l -
1) x<x>.</x>) × &c.)^-1 × (α x^π + β x^π-1 + &c.)</math> ubi <math>π = m - n + 1 + b + c +
f + g + l + &c.</math> pro integrali quæ$ita, cujus inveniatur incremen-
tum, & ex æquatis inter $e incrementi dati & inventi corre$ponden-
tibus terminis, inve$tigari po$$unt coefficientes <math>α, β, γ, &c.</math></p>
<p>Cor. 2. Si modo numerus quantitatum <math>a, e, l, &c.</math> fit <math>π</math>, & dentur
<math>π - 1</math> diver$æ independentes integrales incrementorum; tum ex iis
deduci po$$unt integrales e quibu$cunque incrementis prædicti ge-
neris.</p>
<p>3. Sit quantitas <math>{α′ x^m + b′x^m-1 + &c. / (x + α x<x>.</x>)^n × (x + (α + r) x<x>.</x>)^n}</math>, ubi <math>m, n & r</math> $unt
integri numeri: a$$umatur <math>((x + α x<x>.</x>)(x + (α + 1) x<x>.</x>)(x + (α + 2) x<x>.</x>)
... (x + (α + r - 1) x<x>.</x>))^-n × (α x^m+(r-2)n+1 + β x^m+(r-2)n + &c.</math>) pro inte-
grali quæ$itâ, cujus inveniatur incrementum; & ex æquatis dati & in-
venti incrementi corre$pondentibus terminis deduci po$$unt coeffi-
cientes <math>α, β, &c.</math></p>
<p>Con$imilis erit $ub$titutio, $i modo inter <math>(x + a x<x>.</x>)^n & (x + (α + r) x<x>.</x>)^n</math>
interponantur quidam termini formulæ <math>(x + (α + s) x<x>.</x>)^n</math>, ubi <math>s</math> e$t
integer numerus minor vero quam <math>r</math>.</p>
<p>Si vero plures horum generum, &c. quantitates in denominatore
contineantur. E.g. Sint <math>(x + α x<x>.</x>)^n & (x + (α + r) x<x>.</x>)^n, (x + β x<x>.</x>)^p

<pb 321>[0373]<rh>INCREMENTORUM.</rh>
& (x + (β + s) x<x>.</x>)^p, &c.</math> in denominatore contentæ, tum a$$umatur
pro denominatore integralis <math>((x + α x<x>.</x>)(x + (α + 1) x<x>.</x>) .. (x + (α +
r - 1) x<x>.</x>))^-n × ((x + β x<x>.</x>)(x + (β + 1) x<x>.</x>) ... (x + (β + s - 1) x<x>.</x>))^-p
× &c.</math> & per principia prius tradita deduci pote$t integralis incre-
menti dati, $i modo in finitis terminis exprimi po$$it.</p>
<p>Cor. 3. Hinc facile con$tat numerus diver$arum independentium
incrementorum prædicti generis integralium, e quibus deduci po$$unt
integrales $ingulorum eju$dem generis incrementorum.</p>
<p>4. Dato irrationali incremento; primo inveniendum e$t, annon
quædam irrationales quantitates $int $ucce$$ivi valores aliarum, & ex-
inde con$tabit $ub$titutio, quam integralis dati incrementi exigit; i.e.
abjiciantur ultimi $ucce$$ivi valores $ingulorum factorum in denomi-
natore, etiamque $ingulorum irrationalium in numeratore contento-
rum: horum factorum deducatur primus; tum a$$umantur pro inte-
grali prædicti factores (ultimis abjectis) in $e$e & in factores deducen-
dos ducti; & exinde erui pote$t integralis quæ$ita.</p>
<p>Ex. Sit incrementum, cujus numerator fit <math>(e + 2f + 4g)^l (e + 3f
+ 9g)^l (e + 4f + 16 g)^l ... (e + (z + 1) f + (z + 1)^2 g)^l × (a × (z +
1) + β) - ((a + (z^2 + 1) b)^m (a + (z^2 + 2) b)^m (a + (z^2 + 3) b)^m
... (a + (z + 1)^2 b)^m × (e + f + g)^l ... (e + zf + z^2 g)^l × (a z + β)</math>
& denominator <math>a^m (a + b)^m (a + 2b)^m (a + 3b)^m .. (a + (z + 1)^2 b)^m</math>;
pro denominatore integralis a$$umenda e$t quantitas <math>a^m (a + b)^m (a +
2b)^m ... (a + z^2 b)^m</math>; ubi ultimi factores, qui ex denominatore eva-
ne$cunt, cum <math>z</math> evadat minor per ejus incrementum; viz. <math>(a + (z^2
+ 1) b)^m (a + (z^2 + 2) b)^m .. (a + (z + 1)^2 b)^m</math> rejiciuntur; & pro
numeratore <math>± (e + 2f + 4g)^l (e + 3f + 9g)^l ... (e + zf + z^2 g)^l ×</math>
in terminum præcedentem huju$ce $eriei, qui facile cernitur, viz. <math>(e +
f + g) × (k + h z)</math>, ubi <math>k</math> & <math>h</math> $unt invariabiles coefficientes inve$ti-
gandæ huju$ce integralis inveniatur incrementum, & fiat dato in-
cremento æquale, & invenientur <math>h = α & k = β</math>: ob$ervandum e$t, $i
$int quædam irrationales quantitates in denominatore dati incrementi
contentæ, quæ non habent etiam in eo earum $ucce$$ivos valores, tum
minime exprimi pote$t in finitis terminis integralis dati incrementi.</p>

<pb 322>[0374]<rh>DE METHODO</rh>
<p>Per fucce$$ivos valores quantitatis <math>A</math>, intelligo valores re$ultantes e
$ub$titutione quantitatum <math>x + x<x>.</x>, x + 2 x<x>.</x></math>, &c. pro <math>x</math> in quantitate <math>A</math>.
Sit incrementum <math>B - A</math>, & $i <math>B</math> fit $ucce$$ivus valor quantitatis <math>A</math>;
tum erit <math>A</math> integralis quæ$ita.</p>
<p>Hæc methodus, cum $eries non terminet, $emper dabit $eriem ter-
minorum $ecundum reciprocas dimen$iones quantitatis <math>x</math> progredi-
entium.</p>
<p>5. Hæ vero irrationales quantitates reduci po$$unt ad plures infinitas
$eries $ecundum dimen$iones quantitatis <math>x</math> progredientes vel in $eries
formulæ <math>a + b x + c × x × (x - x<x>.</x>) + d x × (x - x<x>.</x>) × (x - 2 x<x>.</x>) +
&c.</math> vel formulæ <math>a + {b / x} + {c / x × (x + x<x>.</x>)} + {d / x × (x + x<x>.</x>) × (x + 2x<x>.</x>)} +
&c. &c. vel a x<x>.</x> + 2 b x x<x>.</x> + (4 c + 2 d x<x>.</x>^2) x^3 x<x>.</x> + &c.</math> vel formulæ
<math>a + b x^s + c x^s × (x - x<x>.</x>)^s + d x^s × (x - x<x>.</x>)^s × (x - 2 x<x>.</x>)^s + &c.</math> vel
formulæ <math>a + {b / x^s} + {c / x^s × (x - x<x>.</x>)^s} + &c.</math> in quibus $ingulis eædem ad-
hibentur radices; tum inveniatur aggregatum (<math>P</math>) ex integralibus
$ingularum $erierum; ducantur $ingulæ duæ prædictæ infinitæ inte-
grales in $e$e, & inveniatur aggregatum e $ingulis productis; ducan-
tur quæque tres, quatuor, &c. prædictæ integrales in $e$e, & inve-
niantur aggregata $ingularum quantitatum re$ultantium, quæ dican-
tur re$pective <math>Q, R, S, T, &c.</math>; tum erit <math>1 - {P / v} + {Q / v^2} - {R / v^3} + &c. = 0</math>
æquatio, cujus radix <math>v</math> e$t integralis quæ$ita.</p>
<p>6. Si vero in incremento dato contineantur incrementa duorum vel
plurium ordinum quantitatis <math>x</math>; tum abjiciantur omnes termini, in
quibus inveniuntur incrementa $uperiorum ordinum; & e terminis,
in quibus $olummodo contineatur incrementum primi ordinis, inve-
niatur integralis; inter incrementum inventæ integralis & datum
$umatur differentia, cujus integralis e præcedente methodo detegatur,
& $ic deinceps; & tandem detegetur integralis dati incrementi.</p>

<pb 323>[0375]<rh>INCREMENTORUM.</rh>
<h>THEOR. II.</h>
<p>Sit incrementum <math>(V<x>.</x><x>n</x>) n</math> ordinis & <math>x<x>.</x></math> con$tans; tum erit ejus inte-
gralis generaliter correcta <math>V + a x^n-1 + b x^n-2 + c x^n-3 ... k x + l</math>,
ubi <math>a, b, c; k, l</math> denotant qua$cunque invariabiles coefficientes ad li-
bitum a$$umendas. Hinc ad integrales quantitates applicari po$$unt
ea, quæ edita fuere de fluentibus fluxionum in prob. 8, & 9. libri præ-
cedentis.</p>
<h>THEOR. III.</h>
<p>1. Sint duæ vel plures variabiles quantitates <math>(x, y, z, &c.)</math> in datâ
incrementiali quantitate contentæ, a$$umantur omnes variabiles
quantitates præter quamcunque unam tanquam invariables & inve-
niatur integralis quantitatis re$ultantis; & $ic de $ingulis reliquis va-
riabilibus quantitatibus; & $i omnes termini integralium inventarum
ex a$$umendo omnes variabiles præter unam tanquam invariabiles, in
quibus inveniuntur plures quam una variabiles quantitates (<math>x, y, z,
&c.</math>) haud $int iidem, tum haud admittet integralem datum incre-
mentum: $i $int iidem, tum inveniantur eorum incrementa, & inve-
nietur, utrum integralis datæ quantitatis inveniri pote$t, necne.</p>
<p>2. Si incrementum unius variabilis quantitatis $it invariabilis,
tum incrementa cæterarum haud po$$unt e$$e dati numeri, nec
exprimi po$$unt, ni$i per literas de$ignantes earum incrementa; $i
enim dentur relationes inter incrementa unius variabilis quantitatis
& alterius, tum etiam deduci pote$t relatio inter earum integrales.</p>
<p>Ex. 1. Sit incrementum <math>x y z<x>.</x> + x z y<x>.</x> + y z x<x>.</x> + y x<x>.</x> z<x>.</x> + x y<x>.</x> z<x>.</x> +
z x<x>.</x> y<x>.</x> + x<x>.</x> y<x>.</x> z<x>.</x> + a x<x>.</x> + b y<x>.</x> + c z<x>.</x>;</math> $upponantur omnes termini præter
eos, in quibus inveniuntur minimæ dimen$iones incrementorum
quantitatum <math>x, y, z, &c.</math> nihilo æquales, & re$ultat quantitas incre-
mentialis <math>x y z<x>.</x> + x z y<x>.</x> + z y x<x>.</x> + a x<x>.</x> + b y<x>.</x> + c z<x>.</x></math>: nunc $upponantur
omnes præter <math>x</math> invariabiles, & re$ultat quantitas <math>y z x<x>.</x> + a x<x>.</x></math>, cujus
integralis e$t <math>y z x + a x</math>; & $ic $upponantur omnes quantitates præ-

<pb 324>[0376]<rh>DE METHODO</rh>
ter <math>y</math> invariabiles, & re$ultat incrementum <math>z x y<x>.</x> + b y<x>.</x></math>, cujus inte-
gralis inveniatur <math>x z y + b y</math>, & $ic $upponantur omnes præter tertiam
<math>z</math> invariabiles, & re$ultat incrementum <math>x y z<x>.</x> + c z<x>.</x></math>, cujus integralis
erit <math>x y z + c z</math>: hinc a$$umenda e$t quantitas <math>x y z + a x + b y + c z</math>
pro integrali; cujus incrementum erit datum.</p>
<p>Ex. 2. Sit incrementum datum <math>x y<x>.</x> + y x<x>.</x> + 2 y<x>.</x> x<x>.</x></math>; a$$umantur <math>x & y</math>
invariabiles alternatim, & re$ultabit integralis <math>x y</math>, cujus incrementum
erit <math>x y<x>.</x> + y x<x>.</x> + x<x>.</x> y<x>.</x></math>; $ubtrahatur hoc incrementum de dato, & re$ul-
tat differentia <math>x<x>.</x> y<x>.</x></math>, cujus integralis haud deduci pote$t, ni vel <math>x<x>.</x></math> vel <math>y<x>.</x></math>
$it con$tans.</p>
<p>3. Si vero incrementum quantitatis variabilis <math>x</math> $it datus numerus, &
con$equenter haud exprimatur in dato incremento; tum $upponantur
omnes præter unam e reliquis variabilibus invariabiles, & incrementi
re$ultantis inveniatur integralis, cujus deinde inveniatur incremen-
tum ex hypothe$i quod omnes mox a$$umptæ invariabiles $int varia-
biles: $ubducatur hoc incrementum de dato, & con$imili methodo
inueniatur integralis differentiæ re$ultantis; & tandem inveniri po-
te$t, utrum integralis dati incrementi exprimi pote$t, necne.</p>
<p>4. Et $ic inveniri pote$t, utrum integralis cuju$cunque quantitatis
duas vel plures variabiles quantitates, & earum incrementa quorum-
cunque ordinum involventis exprimi pote$t, necne.</p>
<p>5. Si una $olummodo contineantur variabilis quantitas <math>x</math> & ejus in-
crementa in dato incremento, tum e principiis prius traditis detegi
pote$t, utrum ejus integralis exprimi pote$t, necne: e. g. $it datum
incrementum <math>P x<x>.</x> + Q x<x>.</x>^2 + R x<x>.</x>^3 + &c.</math> ubi <math>P, Q, R, &c.</math> $unt fun-
ctiones quantitatis <math>x</math>; nunc $upponantur omnes termini præter <math>P x<x>.</x></math>
nihilo æquales, & inveniatur fluens <math>π</math> fluxionis <math>P x^.</math>, quæ erit integralis
incrementi dati, $i in eo haud contineantur incrementa quantitatis <math>x<x>.</x></math>;
inveniatur incrementum quantitatis <math>π</math>, quæ erit <math>P x<x>.</x> + Q x<x>.</x>^2 + R x<x>.</x>^3 +
&c. = {π^. x<x>.</x> / x^.} + {1 / 2}{π^.. x<x>.</x>^2 / x^. ^2} + {1 / 6}{π^... x<x>.</x>^3 / x^. ^3} + &c.</math> $i modo <math>x^. & x<x>.</x></math> $int invariabiles.</p>
<p>Si vero contineantur incrementa $uperiorum ordinum quantitatis
<math>x</math> in dato incremento, ex præcedentibus principiis etiam deduci po-

<pb 325>[0377]<rh>INCREMENTORUM.</rh>
te$t ejus integralis, $i modo integrari po$$it. E principiis in prob. 4.
I. 1. mutatis mutandis, & iis quæ $upra tradita fuere, inve$tigari po-
te$t, annon integralis cuju$cunque dati incrementi exprimi pote$t.</p>
<p>Ea, quæ de fluxionibus in theor. 2. tradita fuere mutatis mutandis
ad incrementa applicari po$$unt, e. g. inveniatur incrementum quan-
titatis <math>V</math> ordinis <math>m</math> ex hypothe$i, quod <math>x</math> $olummodo $it variabilis, &
re$ultet quantitas <math>W</math>; deinde inveniatur incrementum quantitatis <math>W</math>
ordinis <math>n</math> ex hypothe$i quod <math>y</math> $olummodo $it variabilis; & eadem re-
$ultabit quantitas, ac $i modo inveniatur 1<_>mo</_>. incrementum quanti-
tatis <math>V</math> ordinis <math>n</math> ex hypothe$i quod <math>y</math> $olummodo $it variabilis, & re-
$ultet quantitas <math>u</math>, & deinde inveniatur incrementum <math>m</math> ordinis quan-
titatis <math>u</math> ex hypothe$i quod <math>x</math> $olummodo $it variabilis.</p>
<p>Hinc e principiis in prædicto theor. 2. deduci pote$t, annon datum
incrementum duas vel plures variabiles quantitates & earum incre-
menta involvens, integrari pote$t.</p>
<p>In inve$tigandis integralibus incrementorum eædem radices in in-
tegrali & incremento $emper adhibendæ $unt, ut prius docetur de
fluentibus.</p>
<p>Si variabilis quantitas in denominatore a$cendat ad dimen$iones
majores per quantitatem majorem quam unitatem quam ejus dimen-
$iones in numeratore; tum integralis inter finitam & infinitam di$tan-
tiam erit finita; $in aliter non; &c. ut de fluentibus prius docetur.</p>
<h>PROB. VII.</h>
<h it>Transformare data incrementa in alia, quorum variabiles datas habent
relationes ad variabiles datarum quantitatum.</h>
<p>Ex datis relationibus inter variabiles datarum & quæ$itarum quan-
titatum $ub$tituantur illarum valores in terminis variabilium quæ$ita-
rum quantitatum, & exinde illarum incrementa pro incrementis va-
riabilium datarum quantitatum, & re$ultant quæ$ita incrementa.</p>
<p>Ex pleri$que $ub$titutionibus in prob. 14. I. 1. datis transformari
po$$unt irrationalia incrementa in rationalia.</p>

<pb 326>[0378]<rh>DE METHODO</rh>
<p>Ex. Sit relatio inter variabiles <math>z & x</math> datæ & quæ$itæ quantitatìs
<math>x = √ (z^2 + a^2)</math>, unde <math>x<x>.</x> = √ ((z + z<x>.</x>)^2 + a^2) - √ (z^2 + a^2); &
z = √ (x^2 - a^2) & z<x>.</x> = √ ((x<x>.</x> + x)^2 - a^2) - √ (x^2 - a^2)</math>; & $ic
inveniri po$$unt incrementa <math>z<x>..</x>, z<x>...</x>, &c.</math>: in quantitate <math>A</math>, quæ e$t fun-
ctio quantitatis <math>z</math> & ejus incrementorum, pro <math>z</math> & ejus incrementis
$cribantur eorum prædicti valores, & transformatur data quantitas
in alteram <math>B</math>, quæ e$t functio quantitatis <math>x</math> & ejus incrementorum.</p>
<p>Cor. 1. Sint <math>α & π</math>, etiamque <math>β & ξ</math> corre$pondentes valores prædi-
ctarum variabilium; tum integralis quantitatis <math>A</math> inter <math>α & β</math> contenta
eadem erit ac corre$pondens integralis quantitatis <math>B</math> inter <math>π & ξ</math>.</p>
<p>Cor. 2. Sit <math>A</math> data integralis, quæ e$t functio quantitatis <math>z</math> & ejus
incrementorum, & a$$umatur æquatio in quâ <math>x & z</math> $imiliter invol-
vuntur; deinde inveniantur quantitas <math>z</math> & ejus incrementa in terminis
quantitatis <math>x</math> & ejus incrementorum, quæ $it <math>B</math>; in quantitate <math>B</math> pro
<math>x</math> $cribatur <math>z</math> & re$ultet <math>C</math>; $int <math>π & ξ</math> valores quantitatis <math>x</math>, qui cor-
re$pondent valoribus <math>α & β</math> quantitatis <math>z</math>; tum integralis inter valo-
res <math>α & β</math> variabilis <math>x</math> quantitatis <math>A + C</math> contenta eadem erit ac inte-
gralis eju$dem quantitatis <math>A + C</math> inter valores <math>π & ξ</math> eju$dem quan-
titatis <math>(x)</math> contenta.</p>
<p>Principia e quibus detegi pote$t, utrum fluens fluxionalis æquatio-
nis tres vel plures variabiles quantitates involventis, exprimi pote$t,
necne; etiam ad incrementiales æquationes applicari po$$unt.</p>
<h>THEOR. IV.</h>
<p>1. Sit incrementum <math>{P x^m + Q x^m-1 + R x^m-2 + &c. / A × (x + αx<x>.</x>) × (x + (α + 1)x<x>.</x>) .. (x + (α + a) x<x>.</x>}</math>
ubi <math>a & m</math> $int integri numeri, & ejus integralis inveniri pote$t vel e
finitis terminis, vel e finitis terminis & integrali formulæ <math>{1 / x + a x<x>.</x>}</math>, vel
formulæ <math>{1 / x}</math>, quæ revera eadem haberi pote$t.</p>
<p>2. Et $ic $it denominator <math>(x + αx<x>.</x>) × (x + (α + 1) x<x>.</x>) .. (x + (α + a) x<x>.</x>)

<pb 327>[0379]<rh>ÆQUATIONIBUS.</rh>
× (x + β x<x>.</x>) × (x + (β + 1) x<x>.</x>) .. (x + (β + b) x<x>.</x>) × (x + γ) × (x +
(γ + 1) x<x>.</x>) × (x + (γ + 2) x<x>.</x>) ... (x + (γ + c) x<x>.</x>) × &c.</math> ubi <math>a, b, c,
&c.</math> $unt integri numeri; tum integralis cuju$cunque rationalis incre-
menti denominatorem prædictum habentis $emper detegi pote$t vel e
finitis terminis, vel e finitis terminis & integralibus incrementorum
huju$ce formulæ <math>{1 / x}</math>.</p>
<p>3. Sit denominator <math>(A x^n + B x^n-1 + &c.) × (x + γ)^m × &c. (x + δ)^r ×
&c.</math> ubi <math>r</math> major e$t quam <math>m, &c.</math> & quantitates <math>n, m & r, &c.</math> integri $unt
numeri, tum integralis cuju$cunque rationalis incrementi denomina-
torem prædictum habentis $emper detegi pote$t vel e finitis terminis,
vel e finitis terminis & integralibus incrementorum harum formula-
rum <math>{1 / x}, {1 / x^2}, {1 / x^3}, {1 / x^4} ... {1 / x^r}</math>.</p>
<p>Et in genere integralis cuju$cunque rationalis incrementi prædicti
generis $emper detegi pote$t vel e finitis terminis, vel e finitis terminis
& integralibus incrementorum formularum <math>{1 / x^2 + a x + b}, {1 / x^2 + a x + b)^2},{1 / (x^2 + a x + b)^3} ... {1 / (x^2 + a x + b)^t}, & {1 / x}, {1 / x^2} ... {1 / x^s}</math>, ubi <math>s</math> $it maxi-
mus numerus æqualium divi$orum in denominatore contentorum, &
<math>t</math> vel <math>= {s / 2} vel {s - 1 / 2}</math>.</p>
<p>Omnia hæc facile e prob. 6. deduci po$$unt; facile enim e prædicto
problemate reduci pote$t quodcunque rationale incrementum ad ejus
magis $implices divi$ores.</p>
<p>Idem etiam vere affirmari pote$t de quocunque incremento, quod
ad prædictum rationale reduci pote$t.</p>
<h>PROB. VIII.</h>
<p it>Invenire, annon datæ incrementialis quantitatis integralis ope aliarum
<math>(L, M, N, &c.)</math> detegi pote$t: ducantur incrementiales <math>(L, M, N, &c.)</math>

<pb 328>[0380]<rh>DE METHODO</rh>
quantitates in invariabiles <math>(α, β, γ, &c.)</math> coefficientes generaliter $umptas,
& quantitatum re$ultantium $umma e datâ incrementiali $ubtrahatur; & in
quantitate re$ultante ita a$$umantur coefficientes <math>α, β, γ, &c.</math> ut inveniatur
ejus integralis, $i modo fieri po$$it; & id quod requiritur, confit.</p>
<p>Eadem principia, quæ generaliter inveniunt, annon fluens datæ
fluxionis inveniri pote$t ope datarum fluxionum fluentium; etiam
detegent utrum integralis cuju$cunque incrementi detegi pote$t ope
integralium datorum incrementorum, necne.</p>
<h>PROB. IX.</h>
<p it>Dato incremento, quæ in $e continet integralem <math>v</math>, quæ haud exprimi
pote$t in finitis algebraicis terminis variabilis <math>x, &c.</math> invenire utrum in-
tegralis dati incrementi $it finita algebraica functio quantitatis <math>x</math> & præ-
dictæ integralis & earum incrementorum, necne.</p>
<p>Sit incrementum <math>v × π<x>.</x></math> cujus integralis requiritur; a$$umatur pro
integrali <math>v π</math>, cujus incrementum erit <math>v π<x>.</x></math> (datum) <math>+ π v<x>.</x> + π<x>.</x> v<x>.</x></math>; ergo
<math>vπ</math> - integ. increm. <math>(π v<x>.</x> + π<x>.</x> v<x>.</x>)</math> erit integralis quæ$ita.</p>
<p>Si vero in dato incremento vel contineantur plures dimen$iones
integralis <math>v</math>, vel quæcunque aliæ integrales quantitates; tum per
methodum (mutatis mutandis) in prob. 5. libri primi traditam pro
fluentibus con$imilium fluxionum progrediendum e$t.</p>
<p>Et fere eadem erit methodus detegendi integrales exponentialium
incrementorum, ac ea quæ prius tradita fuit ad inve$tigandas inte-
grales incrementorum præcedentium. Ex. g. Sit incrementum <math>e^z</math>,
& ejus integralis erit <math>{e^z / e^z<x>.</x> - 1}</math>, $i modo <math>z<x>.</x></math> $it con$tans.</p>
<p>Ex. Sit <math>z = - x</math>, & erit incrementi <math>e^-x</math> integralis <math>{e^-x / e^-x<x>.</x> - 1}</math>.</p>
<p>Facile acquiri po$$unt infinita huju$modi incrementa, quorum in-
tegrales innote$cunt; a$$umantur enim integrales, & deducantur ea-
rum incrementa, quorum igitur integrales dantur.</p>

<pb 329>[0381]<rh>INCREMENTORUM.</rh>
<p>2. Datis incrementis, quæ in $e involvunt irrationales quantitates
in prob. 15, &c. libri præcedentis traditas, facile e $ub$titutionibus
in problemate prædicto contentis
reduci po$$unt ad incrementa, quæ
nullas irrationales quantitates continent.</p>
<p>E $ub$titutione duorum valorum (<math>α & β</math>) variabilis quantitatis <math>x</math>
pro ipsâ <math>x</math> in datâ integrali re$ultabit valor integralis inter duos præ-
dictos valores variabilis <math>x</math> contentæ.</p>
<p>In his integralibus æque ac in fluentibus deducendis eædem ra-
dices $emper u$urpandæ $unt.</p>
<h>PROB. X.</h>
<h it>Invenire integralem incrementi <math>v r^z z<x>.</x></math>, ubi <math>z<x>.</x></math> $it invariabilis quantitas.</h>
<p>A$$umatur pro integrali <math>{v r^z z<x>.</x> / r^z<x>.</x> - 1 (R)} - P</math>, cujus incrementum e$t
<math>v r^z z<x>.</x> + v<x>.</x> r^z z<x>.</x> + v<x>.</x> {r^z z<x>.</x> / R} - P<x>.</x></math>; unde <math>P<x>.</x> = v<x>.</x> {r^z z<x>.</x> / R} + v<x>.</x> r^z z<x>.</x> = v<x>.</x> {r^z r^z<x>.</x> z<x>.</x> / R}</math>,
cujus integralis $it <math>v<x>.</x> {r^z z<x>.</x> / R^2} - Q</math>, & ex ii$dem principiis invenietur <math>Q ={r^2 z<x>.</x> × r^z / R^3} - S</math>, & $ic deinceps; unde integralis quæ$ita e$t <math>{v r^z z<x>.</x> / R} +{r^z <x>.</x> × r^z z<x>.</x> / R^2} v<x>.</x> + {r^2 z<x>.</x> × r^z z<x>.</x> / R^3} v<x>..</x> + &c.</math></p>
<p>Eadem methodus, quæ generaliter detegit vel fluentem fluxionis
vel integralem incrementi in finitis terminis, cum ita exprimi po$$it,
$emper eam exprimet per infinitam $eriem, $i finitis terminis exprimi
nequeat.</p>
<p>2. Data incrementiali quantitate, cujus integralis per $eriem <math>A +
B x^m + C x^2m + &c.</math> exprimatur, & per methodum in Medit. Algebr.
traditam $cribantur pro <math>x</math> & ejus incrementis quantitates <math>α x, β x, γx,
&c.</math> & earum incrementa; ubi <math>α, β, γ, &c.</math> $int re$pective radices
æquationis <math>x^n - 1 = 0</math>, & e quantitatibus re$ultantibus per Medit.

<pb 330>[0382]<rh>DE METHODO</rh>
Algeb. facile deduci pote$t quantitas æqualis $ummæ e $ingulis termi-
nis prædictæ $eriei, quorum di$tantiæ a $e invicem $int <math>n, 2 n, 3 n, &c.</math></p>
<p>Eadem etiam applicari po$$unt ad fluentiales & integrales æquati-
ones.</p>
<p>Per methodum in fluxionibus vel algebraicis æquationibus prius
traditam irrationales ex numeratore in denominatorem & vice versâ
ex denominatore in numeratorem trans$ormari po$$unt: & exinde
nonnunquam facilius deduci po$$unt integrales.</p>
<h>PROB. XI.</h>
<h it>Invenire, utrum integralis logaritb. <math>{A / B}</math> detegi pote$t ope logaritbmorum,
necne; ubi <math>A & B</math> $unt functiones quantitatis <math>x</math>.</h>
<p>Inveniantur $implices divi$ores numeratoris <math>A &</math> denominatoris <math>B</math>;
qui $int re$pective <math>α, β, γ, δ, &c.; π, ς, σ, τ, &c.</math>; & $i in numeratore
vel denominatore inveniatur divi$or, qui non habet alterum ei cor-
re$pondentem in denominatore vel numeratore, tum non inveniri
pote$t integralis per logarithmos; $in aliter $emper exprimi pote$t.</p>
<p>Sint <math>a & b</math> corre$pondentes divi$ores; tum, $i in <math>a</math> pro <math>x</math> $cribatur
<math>x + r x<x>.</x></math>, re$ultabit <math>b</math>; ubi <math>r</math> e$t integer numerus.</p>
<p>Si in divi$oribus <math>α, β, &c.</math> numeratoris pro <math>x</math> $cribantur <math>x + r x<x>.</x>,
x + r′ x<x>.</x></math>, &c., re$ultant <math>π, ς, &c.</math> divi$ores denominatoris; & vice
versâ, $i in reliquis <math>σ, τ, &c.</math> divi$oribus denominatoris $cribantur
<math>x + s x<x>.</x>, x + s′ x<x>.</x></math>, &c. pro <math>x</math>, re$ultant reliqui divi$ores numeratoris;
tum integralis dati incrementi log. <math>{A / B}</math> erit log. <math>{σ × σ′ × σ″ × σ′″ ... σ′^s-1 × τ × τ′ × τ″ × ... τ′^3′-1 × &c. / α × α′ × α″ ... α′^r-3 × β × β′ × β″ × ... β′^r′-1 × &c.}</math>: ubi, $i modo pro <math>x</math> in terminis <math>σ, σ′, σ″,
σ′″, &c.</math> $cribatur <math>x + x<x>.</x></math>, re$ultabunt quantitates <math>σ′, σ″, σ′″, σ″″, &c.</math>; &
$ic de reliquis <math>τ, τ′, τ″, &c.</math>; <math>α, α′, α″, &c.</math>; <math>β, β′, β″, &c.</math>, &c.</p>

<pb 331>[0383]<rh>INCREMENTORUM.</rh>
<p>Ex. Sit incrementum log. <math>{x × (x + p + 3 x<x>.</x>) / (x + 4 x<x>.</x>) (x + p)}</math>; tum per hoc prob.
erit integralis log. <math>{(x + p) (x + p + x<x>.</x>) (x + p + 2 x<x>.</x>) / x (x + x<x>.</x>) (x<x>.</x> + 2 x<x>.</x>) (x + 3 x<x>.</x>)}</math>: erit enim ejus
incrementum log. <math>{(x + p + x<x>.</x>) (x + p + 2 x<x>.</x>) (x + p + 3 x<x>.</x>) / (x + x<x>.</x>) (x + 2 x<x>.</x>) (x + 3 x<x>.</x>) (x + 4 x<x>.</x>)} - log.{(x + p) (x + p + x<x>.</x>) (x + p + 2 x<x>.</x>) / x (x + x<x>.</x>) (x + 2 x<x>.</x>) (x + 3 x<x>.</x>)} = log. {x × (x + p + 3 x<x>.</x>) / (x + 4 x<x>.</x>) (x + p)}</math>.</p>
<p>Eadem principia etiam ad circulares arcus, qui ex impo$$ibilibus
logarithmis deduci po$$unt; ad ellipticos, hyperbolicos, &c. arcus ap-
plicari po$$unt.</p>
<h>PROB. XII.</h>
<p it>Datâ æquatione relationem inter <math>x & y</math> exprimente, & conce$sâ radicum
extractione; invenire aggregatum e $ingulis valoribus cuju$cunque alge-
braicæ functionis quantitatum <math>x & y</math>, earum incrementorum & fluxionum.</p>
<p>Inveniatur e prædictis conce$$is primum incrementum quantitatis
<math>y</math>, & $ic $ecundum, tertium, &c. etiamque ejus fluxiones; quibus va-
loribus in datâ incrementiali quantitate $ub$titutis, re$ultat algebraica
quantitas, cujus $umma e $ingulis valoribus erui pote$t per Medit. Al-
gebr. Et $ic aggregatum rectangulorum e quibu$que duobus valori-
bus datæ functionis; contenta e quibu$que tribus, quatuor, &c. valo-
ribus datæ functionis deduci po$$unt; unde detegetur æquatio ip$a,
cujus radix e$t prædicta functio.</p>
<p>2. Sit æquatio <math>A y^n + (a + b x) y^n-1 + (c + d x + e x^2) y^n-2 + &c. = 0</math>,
& fluat <math>x</math> uniformiter, cujus incrementum $it <math>x<x>.</x></math>; tum pro $ingulis
datis corre$pondentibus valoribus quantitatum <math>x, y & x<x>.</x></math>, re$ultare po$-
$unt <math>n</math> diver$i corre$pondentes valores incrementi <math>y<x>.</x></math>, quorum $umma
erit <math>- {a + b x + b x<x>.</x> / A} - n y</math>.</p>

<pb 332>[0384]<rh>DE METHODO</rh>
<p>3. Datâ quantitate <math>x</math> & ejus incremento <math>x<x>.</x></math>, & numerus valorum in-
crementorum <math>y<x>.</x></math> pote$t e$$e <math>n × n</math>, ut facile con$tat e Medit. Algeb.</p>
<p>4. Datâ quantitate <math>x</math> & ejus incremento <math>x<x>.</x></math>, & $umma <math>n</math> valorum incre-
menti <math>y x<x>.</x> = - {a + b x / A} x<x>.</x></math>, cujus integralis erit <math>- ({a / A} x + {b / A} × int. x x<x>.</x>)
= ({a / A} x + {b / A} x × {x - x<x>.</x> / 2} + {b / A} x + c)</math>.</p>
<p>Et $ic de $ummis integralium incrementialium <math>y^2 x<x>.</x>, y^3 x<x>.</x>, &c.</math></p>
<p>5. Sit æquatio <math>(A + B x) y^n-1 + (C + D x + E x^2) y^n-2 + &c. = 0</math>, &
$i <math>A + B x</math> haud $it divi$or quantitatis <math>C + D x + E x^2</math>, tum $umma
e $ingulis integralibus quantitatis incrementialis <math>y x<x>.</x></math> haud exprimi
pote$t per finitos terminos qauntitatum <math>x & y</math>.</p>
<h>PROB. XIII.</h>
<p it>Datâ æquatione algebraicâ <math>(n)</math> dimen$ionum <math>y^n + (a + b x) y^n-1 +
&c. = 0</math>, invenire æquationem cujus radix (<math>z</math>) $it integralis quantitatis
<math>y x<x>.</x></math>, $i modo ea inveniri po$$it.</p>
<p>A$$umatur æquatio, in quâ dimen$iones quantitatis <math>z</math> inveniuntur
<math>n</math>, quoniam omni valori quantitatis <math>x</math> corre$pondent <math>n</math> diver$i valores
quantitatis <math>z; viz. z^n + (a x + {b / 2} x^2 - {b / 2} x x<x>.</x> + c) z^n-1 + &c. = 0</math>,
ubi <math>c</math> e$t invariabilis quantitas ad libitum a$$umenda, & <math>x<x>.</x></math> con$tans;
exinde inveniatur æquatio cujus radix erit <math>{z<x>.</x> / x<x>.</x>}</math>, & fiant coefficientes re-
$ultantis & datæ æquationis inter $e æquales, & exinde deduci po$$unt
coefficientes, $i modo integralis exprimi po$$it in finitis terminis.</p>
<p>Aliter: reduci pote$t data æquatio, ita ut inveniantur <math>n</math> valores
quantitatis <math>y</math> progredientes $ecundum dimen$iones qnantitatis <math>x</math>; de-
inde inveniatur per prædictas methodos integrales e $ingulis hi$ce
$eriebus, quarum $umma erit $umma radicum quæ$itæ æquationis;
ducantur quæque duæ, tres, &c. ex his integralibus in $e$e & inveni-

<pb 333>[0385]<rh>INCREMENTORUM.</rh>
antur aggregata e $ingulis hi$ce re$ultantibus $eriebus, & erunt re-
$pective coefficientes tertii, quarti, &c. terminorum quæ$itæ æquatio-
nis, & $ic deinceps; & exinde con$tabit æquatio quæ$ita.</p>
<p>Cor. In omnibus hi$ce re$olutionibus irrepit in terminos æquatio-
num quantitas ad libitum a$$umenda, quæ quantitas denotat corre-
ctionem quantitatis vel æquationis quæ$itæ.</p>
<p>2. Eædem etiam methodi applicari po$$unt ad inveniendas æquatio-
nes, quarum radices $unt quæcunque algebraicæ functiones quantita-
tum <math>x & y</math> incognitarum in datâ æquatione contentarum, & earum
incrementorum.</p>
<p>3. Et $ic de pluribus <math>m</math> æquationibus plures <math>(m + 1)</math> incognitas
quantitates habentibus, $ed ob$ervandum e$t in omnibus hi$ce æqua-
tionibus ac in æquationibus exprimentibus fluentes fluxionum, $i
modo $olummodo $it fluxio vel incrementum primi ordinis in quan-
titate, cujus æquatio exprimens ejus fluentem vel integralem requiri-
tur; tum una $olummodo quantitas in æquatione re$ultanti a$$umi
pote$t ad libitum; $i vero fluxiones vel incrementa $ecundi, tertii, &c.
ordinis in prædictâ quantitate contineantur, tum duæ, tres, &c.
quantitates ad libitum a$$umi po$$unt.</p>
<h>THEOR. V.</h>
<p>1. Sit æquatio integralis <math>a x^r + b x^s × y^t = c y^n</math>, e prob. 32. 1. 1.
con$tant quantitates <math>x & y</math> in terminis tertiæ <math>z</math>, & exinde earum in-
crementa in terminis tertiæ <math>z</math> & ejus incrementorum; unde deduci
pote$t quæcunque functio quantitatum <math>x & y</math>, & earum incremento-
rum in terminis quantitatis <math>z</math> & ejus incrementorum.</p>
<p>2. Sit algebraica æquatio homogenea, i. e. ejus $inguli termini
ea$dem conficiant dimen$iones, & inveniri pote$t quæcunque functio
ejus variabilium <math>x & y</math>, & earum incrementorum in terminis novæ
variabilis <math>z</math>, & ejus incrementorum.</p>
<p>Omnia, quæ de algebraicis æquationibus & fluxionalibus quanti-
tatibus in prob. 32, & 33. l. 1. tradita fuere, ad algebraicas æquationes

<pb 334>[0386]<rh>DE METHODO</rh>
& incrementiales quantitates (mutatis mutandis) applicari po$$unt;
& eadem principia quæ tradita fuere in fluxionalibus æquationi-
bus de detegendis rationalibus functionibus tertiæ <math>z</math>, quæ re$pective
æquant variabiles <math>x & y</math>, ad incrementiales æque applicari po$$unt.</p>
<h>PROB. XIV.</h>
<h it>Datam æquationem, in quâ integralis continetur quantitas; in incremen-
tialem reducere, ita ut exterminetur integralis quantitas.</h>
<p>Si integralis ducatur in variabilem quantitatem; dividatur æquatio
per eam quantitatem, & inveniatur incrementum quantitatis re$ul-
tantis, & exterminabitur integralis.</p>
<p>Si autem majores $int dimen$iones (<math>r</math>) integralis quantitatis exter-
minandæ, vel altior $it ordo (<math>n</math>) ejus integralis; in po$teriori ca$u
nece$$e e$t invenire incrementum (<math>n</math>) ordinis datæ æquationis, in
priori vero ca$u con$tabit reductio e prob. 28. 1. I. i. e. integrales ex
æquationibus exterminabuntur omnino per eandem methodum ac
fluentes e datis æquationibus.</p>
<p>Ex.. Sit data æquatio (<math>x + y</math>) × integ. <math>{x x<x>.</x> + y x^2 y / √(x^2 + y^2)} = √(b^2 + y^2)
- c x^2</math>; dividatur æquatio per <math>x + y</math>, & re$ultat integ. <math>{x x<x>.</x> + y x^2 y / √(x^2 + y^2)}= {√(b^2 + y^2) - c x^2 / x + y}</math>; inveniatur incrementum re$ultantis æquatio-
nis, & re$ultat <math>{x x<x>.</x> + y x^2 y / √(x^2 + y^2)} = increm.{√(b^2 + y^2) - c x^2 / x + y}</math>.</p>
<h>PROB. XV.</h>
<h>_Reducere duas incrementiales æquationes_ <math>A = 0</math> _&_ <math>B = 0</math> _in unam, ita_
_ut exterminentur variabilis quantitas <math>z</math> & ejus incrementa._</h>
<p>Sit infima incrementialis quantitas literæ <math>z</math> in unâ æquatione <math>z<x>.</x><x>n</x></math>,
in alterâ vero <math>z<x>.</x><x>n+p</x></math>; inveniatur <math>p</math> incrementum e $ingulis terminis

<pb 335>[0387]<rh>INCREMENTORUM.</rh>
prioris æquationis, & re$ultat infima incrementialis quantitas literæ
<math>z</math> in æquatione re$ultante <math>z<x>.</x><x>n+p</x></math> deinde e (<math>p - 1</math>) incremento e $ingulis
terminis prioris æquationis exterminetur incrementialis quantitas
<math>x<x>.</x><x>n+p-1</x></math>; & $ic reduci pote$t po$terior æquatio ad incrementialem æqua-
tionem, in quâ infimum incrementum quantitatis <math>z</math> $it <math>z<x>.</x><x>n</x></math>; unde duæ
dantur æquationes, in quibus infima incrementialis quantitas literæ
<math>z</math> erit <math>z<x>.</x><x>n</x></math>; ex his duabus æquationibus deduci pote$t æquatio, cujus
infima incrementialis quantitas literæ <math>z</math> erit <math>z<x>.</x><x>n-1</x></math>, & ex hâc & priori
datâ æquatione deduci pote$t altera æquatio, cujus infima incremen-
tialis quantitas literæ <math>z</math> e$t <math>z<x>.</x><x>n-1</x></math>, & ex his duabus æquationibus con$e-
quitur incrementialis æquatio, in quâ infimum incrementum quan-
titatis <math>z</math> e$t <math>z<x>.</x>n-2</math>; & $ic deinceps; & tandem ita reducetur æquatio, ut
exterminentur quantitas <math>z</math>, & ejus incrementa.</p>
<p>Cor.. In reducenda infimâ incrementiali quantitate <math>z<x>.</x><x>r</x></math> in utri$que
æquationibus ad unam $uperiorem <math>z<x>.</x><x>F-1</x></math> $emper inveniendum e$t pri-
mum incrementum quantitatum in utri$que æquationibus contenta-
rum; unde $i in duabus prædictis æquationibus <math>A = 0 & B = 0</math>, in
quibus <math>z<x>.</x><x>n</x></math> & <math>z<x>.</x><x>n+p</x></math> re$pective $int infima incrementa quantitatis <math>z</math>, etiam-
que fint <math>y<x>.</x><x>α</x></math> & <math>y<x>.</x><x>β</x></math> infima incrementa quantitatis <math>y</math>; & infimum incremen-
tum quantitatis <math>y</math> in æquatione re$ultante, in quâ haud inveniuntur
quantitas <math>z</math> vel ejus incrementa, haud pote$t e$$e majus quam <math>α + n
+ p</math> vel <math>β + n</math>.</p>
<p>Sed e prob. 36.1.1. con$tat, $i æquatio re$ultans $it incrementialis
cuju$cunque ordinis <math>α + n + p</math> vel <math>β + n, &c.</math> tot a$$umi po$$e cor-
re$pondentes valores variabilium <math>x & y</math> in eâ contentarum, & incre-
mentorum variabilis <math>y</math> diverforum, quot $it ordo datæ æquationis, $i
<math>x<x>.</x></math> $it con$tans.</p>
<p>2. Et $ic de pluribus (<math>m</math>) incrementialibus vel fluxionalibus æqua-

<pb 336>[0388]<rh>DE METHODO</rh>
tionibus in unam reducendis; hinc enim fluxionalis vel incrementia-
lis æquationis re$ultantis deduci pote$t ordo.</p>
<p>3. Æquationes incrementiales vel fluxionales po$$unt e$$e inter $e con-
tradictoriæ; hoc autem $emper con$tabit ex earum reductione in unam,
ita ut variabiles, &c. quantitates exterminentur, & $i hæc una re$ultans
æquatio aliquid ab$urdi in $e contineat, vel for$an ejus divi$ores; tum
aliquid contradictorium continent in $e æquationes incrementiales.</p>
<p>4. Ex hâc methodo transformandi æquationes, ita ut variabiles quan-
titates & earum incrementa exterminentur, irrepunt in æquationes
re$ultantes radices, quæ in datis haud inveniuntur; quomodo hæ
radices irrepant in re$ultantes æquationes, & radices ip$æ, e princi-
piis in medit. algebr. traditis, erui po$$unt.</p>
<h>THEOR. VI.</h>
<p>1. Sit incrementialis æquatio <math>A × B × C × &c. = 0</math>, quæ admittat
divi$ores <math>A, B, C, &c.</math> hæc incrementialis æquatio dividi pote$t in
plures incrementiales æquationes <math>A = 0, B = 0, C = 0, &c.</math></p>
<p>2. Sint duæ incrementiales æquationes <math>A × B × C × &c. = 0, &
a × b × c × &c. = 0</math>; tum hæ duæ incrementiales æquationes reduci
po$$unt ad alias, viz. <math>A = 0 & a = 0; A = 0 & b = 0; A = 0 & c = 0,
&c. B = 0 & a = 0; B = 0 & b = 0, &c. &c.</math></p>
<h>THEOR. VII.</h>
<p>E duabus datis incrementialibus æquationibus facile formari po$-
$unt infinitæ aliæ, quarum variabiles quantitates eandem habent in-
ter $e rationem. Ducantur duæ æquationes in invariabiles quan-
titates, & addantur vel $ubducantur æquationes re$ultantes de $e ip-
$is; & re$ultant novæ æquationes quæ$itæ.</p>
<h>THEOR. VIII.</h>
<p>Datâ incrementiali æquatione relationem inter <math>x & y</math> & earum in-
crementa <math>n</math> ordinum exprimente, $i modo ex eâ inveniatur æquatio,

<pb 337>[0389]<rh>INCREMENTORUM.</rh>
quæ haud continet incrementa majorum quam <math>n - 1</math> ordinum, tum
a$$umi pote$t una quantitas invariabilis ad libitum a$$umenda, & $ic
deinceps; unde in æquatione ex incrementis liberâ erunt <math>n</math> invaria-
biles quantitates ad libitum a$$umendæ.</p>
<p>Sit quantitas ip$a, vel ea ducta in quamcunque incrementialem
quantitatem inferioris ordinis, perfectum incrementum, cujus inte-
gralis dicatur <math>V</math>; & erit <math>V + C = 0</math> æquatio vere correcta, ubi <math>C</math>
denotat invariabilem quantitatem: $i modo $it incrementialis æquatio
ordinis <math>n, & x<x>.</x><x>m</x></math> $it incrementum ordinis <math>m</math>, qui minor $it quam <math>n</math>;
etiamque $it <math>x<x>.</x><x>m</x></math> con$tans; tum erit æquatio vere correcta <math>V + A(x<x>.</x><x>m</x>)^{n-1 / m}</math>
ubi <math>A</math> e$t invariabilis quantitas.</p>
<p>Et $ic progredi liceat ad æquationes integrales $uperiorum ordi-
num.</p>
<p>Quot invariabiles quantitates ad libitum a$$umi po$$unt, tot cor-
re$pondentes valores e $ingulis variabilibus quantitatibus, & e diver$is
incrementis variabilis quantitatis, quæ haud cre$cit uniformiter, a$-
$umi po$$unt.</p>
<h>PROB. XVI.</h>
<p>_Datâ æquatione relationem inter_ <math>x</math> _&_ <math>y</math> _& earum incrementa exprimente;_
_invenire æquationem, cujus radix e$t correctio ip$a: $cribatur_ <math>v + z</math> _pro_ <math>y</math>
_in datâ æquatione, ubi_ <math>v</math> _denotat particularem valorem quantitatis_ <math>y</math>; _ex_
_æquationibus re$ultantibus con$tat æquatio, cujus radix e$t_ <math>z</math> _correctio quæ-_
_$ita: vel $i modo per infinitas $eries convergentes inveniatur generalis valor_
_quantitatis_ <math>y</math> _in terminis quantitatis_ <math>x</math>, _con$equetur correctio quæ$ita._</p>
<p>Omnis mothodus, quæ detegit generalem correctionem datæ æqua-
tionis, etiam deducet generalem ejus integralem.</p>
<p>Et $ic de pluribus (<math>m</math>) æquationibus plures (<math>m + 1</math>) variabiles
quantitates & earum incrementa habentibus.</p>

<pb 338>[0390]<rh>DE METHODO</rh>
<h>PROB. XVII.</h>
<h it>Invenire quamplurimas incrementiales æquationes, quarum integrales
cogno$cuntur.</h>
<p>1. A$$umantur integrales æquationes, & ex iis deducantur incre-
mentiales.</p>
<p>2. Deducantur æquationes, quarum radices quamcunque habeant
relationem ad radices a$$umptarum vel inventarum æquationum, &
e radicibus a$$umptarum deduci po$$unt radices deductarum æquati-
onum.</p>
<p>3. A$$umatur <math>a</math> pro <math>y</math>, quæ $it quæcunque $unctio quantitatis <math>x</math>,
deinde $cribantur hæc quantitas & ejus incrementa pro $uis valoribus
in quantitate π, quæ $it functio variabilium <math>x, y</math> & earum incremen-
torum, & $it quantitas exinde re$ultans <math>ξ</math>; tum erit <math>α</math> particularis valor
incrementialis æquationis <math>π = ξ</math>.</p>
<p>4. Datâ incrementiali æquatione, e quâ deduci pote$t <math>F</math> integralis
incrementi (<math>P</math>); pro variabilibus & earum incrementis in datâ æqua-
tione & datâ quantitate <math>P</math> $cribantur quæcunque functiones novarum
variabilium, & earum incrementa; & re$ultant æquatio inter has
novas variabiles & earum incrementa relationem exprimens, & nova
quantitas <math>π</math>, cujus integralis deduci pote$t ex integrali quantitatis <math>P</math>.</p>
<p>5. Sit quæcunque incrementialis æquatio, dividatur ea in duas
æquales partes, ita ut integralis alterius partis inveniri pote$t, ea
erit etiam integralis alterius.</p>
<h>PROB. XVIII.</h>
<h it>_1._ Datâ integrali (<math>P = 0</math>) generali primi ordinis incrementialis
æquationis, invenire æquationem ip$am.</h>
<p>Sit <math>a</math> quantitas invariabilis in datâ integrali <math>P = 0</math> ad libitum
a$$umenda, & ex eâ deducatur <math>α = π</math>, & erit <math>π = 0</math> incrementialis
æquatio quæ$ita.</p>

<pb 339>[0391]<rh>INCREMENTORUM.</rh>
<p>Ex. Sit integralis <math>y = {z - z<x>.</x> / z<x>.</x> + n}y<x>.</x> + a</math>, ubi <math>z<x>.</x></math> e$t con$tans & <math>a</math> inva-
riabilis quantitas ad libitum a$$umenda; hujus æquationis invenia-
tur incrementum, & evadet <math>y<x>.</x> = {z - z<x>.</x> / z<x>.</x> + n}y<x>.</x> + {z<x>.</x> / z<x>.</x> + n}(y<x>.</x> + y<x>..</x>)</math>, quod re-
ductum fit <math>{z<x>.</x> + n - z / z<x>.</x> + n}y<x>.</x> = {z / z<x>.</x> + n}y</math>, unde <math>(z<x>.</x> + n - z)y<x>.</x> = zy<x>..</x></math> æqua-
tio, cujus integralis generalis e$t data.</p>
<p>2. Datâ integrali <math>(P = 0)</math> generali <math>n</math> ordinis incrementialis æqua-
tionis (<math>a = 0</math>); ex æquatione <math>P = 0</math>, quæ continet <math>n</math> invariabiles quan-
titates (<math>a, b, c, &c.</math>) ad libitum a$$umendas, inveniantur <math>n</math> æquationes
<math>a = π, b = ξ, c = σ, &c.</math> tum erunt <math>π. = 0, ξ. = 0, σ. = 0, &c. (n)</math> di-
verfæ æquationes, quæ $unt integrales <math>n - 1</math> ordinis incrementialis
æquationis <math>a = 0</math>; ex his æquationibus inveniri po$$unt <math>n. {n - 1 / 2}</math>
aliæ, quæ $unt integrales <math>n - 2</math> ordinis æquationis <math>a = 0</math>; & $ic de-
inceps u$que ad <math>n</math> æquationes detegendas, quæ $unt integrales primi
ordinis æquationis <math>a = 0</math>.</p>
<p>Si non inveniri po$$int <math>a = π</math>, tum reducantur duæ æquationes
<math>P = 0</math> & <math>P^. = 0</math> in unam, ita ut exterminetur invariabilis quantitas;
& $ic deinceps.</p>
<p>Cor. Sunt <math>n</math> multiplicatores, qui ducti in datam æquationem
<math>a = 0</math> prebent incrementa, quorum integrales cogno$cuntur; & $ic
deinceps, ut con$tat e theor. 29. libri præcedentis.</p>
<p>Sint <math>m, m′, m″, &c.</math> re$pectivi multiplicatores, qui ducti in æquatio-
nem incrementialem (<math>a = 0</math>) prebent (<math>n</math>) æquationes, quarum inte-
grales re$pective $unt <math>λ, λ′, λ″, λ′″, &c.</math>; tum erit generalis integralis
quæcunque functio (<math>n</math>) quantitatum <math>λ, λ′, λ″, λ′″, &c.</math> Et $ic de par-
ticularibus integralibus.</p>
<p>Facile etiam deduci po$$unt con$imiles propo$itiones de hâc re, quæ
prius de fluentialibus æquationibus tradita fuere.</p>

<pb 340>[0392]<rh>DE METHODO</rh>
<p>Hæc etiam applicari po$$unt ad fluentes vel integrales æquationum
fluxiones, & incrementa $imul involventium.</p>
<p>Sint duæ incrementiales æquationes <math>π = 0 & ξ = 0</math>, quarum com-
munis $it multiplicator; tum ex integrali alterius æquationis datâ
facile acquiri pote$t integralis alterius.</p>
<h>THEOR. IX.</h>
<p>1. Sit incrementialis æquatio <math>P z<x>.</x> = 0</math>; quæ haud continet <math>z</math> vel
<math>z<x>.</x>, &c.</math> tum erit ejus integralis <math>z = con$t.</math> haud vero <math>P = 0</math>; $i vero
$it <math>y<x>.</x> = P x<x>.</x></math> & fit <math>P = 0</math> functio quantitatis <math>y = con$t.</math> tum erit <math>y =
con$t.</math> particularis valor incrementialis æquationis <math>y<x>.</x> = P x<x>.</x></math>, & $ic de
pluribus huju$cemodi æquationibus.</p>
<p>2. Sit incrementialis æquatio <math>π = 0; & M</math> quantitas, cujus in-
tegralis deduci pote$t; tum $emper detegi pote$t integralis quantitatis
<math>φ: (π) + M ubi φ: (π)</math> $it functio quantitatis <math>π</math>, in quâ haud in-
venitur terminus, cujus dimen$io quantitatis <math>π</math> vel nulla e$t vel ne-
gativa.</p>
<p>3. Eadem etiam applicari po$$unt ad duas vel plures incrementia-
les æquationes <math>π = 0, ξ = 0, &c.</math> $emper enim detegi pote$t integralis
quantitatis <math>φ: (π & ξ, &c.) + M, ubi φ: (π, ξ, &c.)</math> de$ignat functio-
nem quantitatum <math>π, ξ, &c.</math> in quâ haud invenitur terminus, in quo
nulla continetur dimen$io e $ingulis quantitatibus <math>π, ξ, &c.</math> vel ne-
gativa.</p>
<p>4. Cuju$cunque incrementialis æquationis (<math>π = 0</math>) <math>n</math> ordinis, in
quâ continentur duæ variabiles <math>x & y</math> & earum incrementa, ubi al-
terius (<math>x</math>) primum incrementum e$t con$tans, $emper datur (<math>L</math>) fun-
ctio variabilium <math>x, y</math>, & earum incrementorum primi, $ecundi, &c.
(<math>n - 1</math>) ordinis, quæ ducta in datam æquationem præbet quantita-
tem, cujus integralis (<math>α</math>) datur.</p>
<p>5. Sit <math>α</math> generalis integralis æquationis <math>π = 0</math>, tum quæcunque
functio quantitatis <math>α</math> etiam erit integralis datæ æquationis <math>π = 0</math>; &
idem etiam valet, cum <math>α</math> $it particularis integralis.</p>

<pb 341>[0393]<rh>ÆQUATIONIBUS.</rh>
<p>6. Sit quæcunque integrabilis æquatio <math>π = ξ</math>, cujus incrementum
$it <math>π = ξ</math>, huju$ce æquationis ducatur alterum latus in <math>π</math>, alterum
vero in <math>ξ</math>, & evadit <math>ππ = ξξ.</math>: generalis integralis huju$ce æquatio-
nis haud erit <math>π = ξ</math>; hoc & con$imilia con$equuntur e libro præced.</p>
<p>7. Sint <math>π, ξ, σ, &c.</math> incrementa, quæ ducta in eundem multiplica-
torem <math>L</math>, præbent incrementa, quorum integrales innote$cunt; tum
erit <math>L (aπ + bξ + cσ + &c.)</math> incrementum, cujus integralis etiam
inveniri pote$t, $i modo <math>a, b, c, &c.</math> $int quæcunque invariabiles quan-
titates.</p>
<p>8. Datis formulis æquationis fluxionalis & ejus multiplicatoris,
inve$tigari po$$unt ca$us, in quibus ejus integralis deduci pote$t.</p>
<p>Ducatur æquatio datæ formulæ in multiplicatorem prædictum, &
quantitatis re$ultantis per methodos prius traditas inveniatur inte-
gralis, $i modo integralem recipiat quantitas re$ultans; vel e notis
integrabilitatis con$tare pote$t, annon quantitas re$ultans integratio-
nem admittit; vel aliter ex datâ incrementiali æquatione & formulâ
ejus multiplicatoris datâ facile con$tat formula ejus integralis, cujus
inveniatur incrementum, dividatur hoc incrementum per datum; &
exinde deduci po$$unt incognitæ quantitates in a$$umptâ integrali, $i
modo ea recipere po$$it formulam prædictam.</p>
<p>9. E $ub$titutionibus $æpe reduci po$$unt datæ incrementiales æqua-
tiones ad alias, quarum integrales innote$cunt; unde ex a$$umptis
æquationibus incrementialibus, quarum innote$cunt integrales, facile
e $ub$titutionibus deduci po$$unt aliæ, quarum integrales e prædictis
deduci po$$unt.</p>
<p>10. Sit æquatio incrementialis <math>P + Q = 0</math>; e $ub$titutione redu-
catur hæc æquatio ad $ub$equentem <math>R + S = 0</math>, unde <math>P + Q = R
+ S</math>: hinc $i vero <math>{R + S / V}</math> $it incrementum, cujus integralis innote$-
cit, tum etiam <math>{P + Q / V}</math> erit incrementum cujus integralis etiam inno-

<pb 342>[0394]<rh>DEMETHODO</rh>
te$cit; & <math>{1 / V}</math> erit multiplicator, qui in datam æquationem ductus,
præbet quantitatem, quæ integrari pote$t.</p>
<p>11. Ex irrationalibus quantitatibus in datâ incrementiali æqua-
tione contentis con$tabunt irrationales quantitates in multiplicatore
<math>M</math>, qui ductus in datam æquationem, eam reddit integrabilem; hæ
irrationales quantitates enim deduci po$$unt ex ii$dem principiis, quæ
prius tradita fuere de inveniendis irrationalibus quantitatibus in
multiplicatore datæ fluxionalis æquationis contentis.</p>
<p>12. Ex formulis etiam fluxionalium æquationum, quarum fluentes
innote$cunt, con$tabunt formulæ incrementialium æquationum, qua-
rum integrales corre$pondent fluentibus prædictis.</p>
<p>13. Et $ic in quibu$dam ca$ibus deduci po$$unt integrales generales
incrementialium æquationum ex earum particularibus valoribus.</p>
<p>14. Si modo datæ quantitatis incrementum $it con$tans; $criban-
tur in datâ æquatione quæcunque quantitates pro $uis valoribus ex
con$tante incremento, & earum incrementis diver$orum ordinum de-
ductæ; & haud nunquam exinde con$equitur incrementialis æquatio,
cujus integralis innote$cit.</p>
<p>15. A$$umatur integralis æquatio duas variabiles <math>x & y</math> involvens;
quarum incrementa re$pective $int <math>x<x>.</x> & y</math>; datæ æquationis invenia-
tur incrementum, in hoc vero incremento $cribantur quæcunque
quantitates pro $uis valoribus ex a$$umptâ integrali æquatione de-
ductæ, & re$ultat æquatio, cujus particularis integralis erit a$$umpta
æquatio.</p>
<p>16. Si vero detur incrementialis æquatio, in quâ $imiliter involvun-
tur variabiles <math>x & y</math>, & earum incrementa; tum in generali integrale
$imiliter etiam involventur variabiles quantitates <math>x, y, &c.</math> & $i <math>α</math>
fit valor quantitatis <math>y</math>, cum <math>x = β</math>; tum erit <math>β</math> valor quantitatis <math>y</math>,
cum <math>x = α</math>.</p>
<p>17. In quibu$dam ca$ibus deduci po$$unt integrales duarum vel plu-
rium (<math>n</math>) æquationum incrementialium tres vel plures variabiles quan-

<pb 343>[0395]<rh>INCREMENTORUM.</rh>
titates, & earum incrementa involventium: vel e reductione harum
(<math>n</math>) æquationum in unam, &c. ita ut exterminentur variabiles quan-
titates & earum fluxiones: vel ex a$$umptis generalibus quantitatibus
pro corre$pondentibus valoribus e $ingulis variabilibus quantitatibus,
quibus pro $uis valoribus in datis æquationibus incrementialibus $ub-
$titutis; e re$ultantibus æquationibus deduci po$$unt valores genera-
lium quantitatum a$$umptarum.</p>
<p>2. Facile deduci po$$unt (<math>n</math>) incrementiales æquationes <math>n + 1</math> va-
riabiles quantitates & earum incrementa involventes; quarum par-
ticulares valores fingularum variabilium in prædictis æquationibus
contentarum, innote$cunt: a$$umantur enim quæcunque (<math>n</math>) functi-
ones unius variabilis quantitatis & ejus incrementorum pro <math>n</math> reli-
quis, $cribantur hæ quantitates & earum incrementa pro $uis valori-
bus in quibu$cunque <math>n</math> diver$is quantitatibus <math>a, b, c, d, &c.</math> quæ $int
functiones (<math>n + 1</math>) variabilium prædictarum & earum incremento-
rum, & $int quantitates re$ultantes <math>P, Q, R, S, &c.</math> re$pective; tum
erunt <math>a = P, b = Q, c = R, d = S, &c. n</math> diver$æ æquationes, qua-
rum particulares valores erunt quantitates a$$umptæ.</p>
<p>18. Datâ incrementiali æquatione duas variabiles <math>x & y</math> & earum
incrementa involvente, ex principiis prius traditis, i. e. ex infinitis
$eriebus deduci pote$t, annon una <math>x</math> exprimi pote$t in algebraicis ter-
minis alterius <math>y</math>.</p>
<p>19. Per infinitas $eries detegi pote$t ex datâ incrementiali æqua-
tione (<math>a = 0</math>) duas variabiles quantitates <math>x & y</math> & earum incrementa
involvente generalis valor quantitatis <math>y</math> in terminis quantitatis <math>x</math>, qui
continet invariabilem quantitatem <math>a</math> ad libitum a$$umendam; etiam-
que $i modo data incrementialis æquatio $it majoris ordinis (<math>n</math>), tum
in prædicto valore continebuntur (<math>n</math>) invariabiles quantitates <math>a, b, c,
&c.</math> ad libitum a$$umendæ; deinde ex infinitâ $erie, quæ exprimit valo-
rem quantitatis <math>y</math> in terminis quantitatis <math>x</math>, inveniantur <math>a = π, b = ξ,
c = γ, &c.</math>; & exinde facile deduci po$$unt <math>n</math> multiplicatores, qui
reddunt datam æquationem <math>α = 0</math> integrabilem.</p>
<p>20. Sit <math>V<x>.</x> = P x<x>.</x></math>, tum <math>P</math> erit functio quantitatum <math>x & x<x>.</x></math>.</p>

<pb 344>[0396]<rh>DE METHODO, &c.</rh>
<p>Et $ic, mutatis mutandis, applicari licet reliqua principia de fluxio-
nalibus æquationibus tradita ad æquationes incrementiales, vel con-
junctim fluxionales & incrementiales.</p>
<h>THEOR. X.</h>
<p>Sit æquatio <math>y<x>1</x><x>n</x> + a y<x>1</x><x>n-1</x> + b y<x>1</x><x>n-2</x> ... d y<x>′</x> + f y = 0</math>, ubi <math>y<x>′</x>, y<x>″</x>, &c.</math> $unt præ-
cedentes integrales quantitatis <math>y</math>; fingatur <math>y = A e^αx</math>, ubi <math>x<x>.</x></math> $it con$tans;
tum erit <math>A e^αx+nxx<x>.</x> = y<x>1</x><x>n</x>, & A e^αx+nαx<x>.</x> + a A e^αx+(n-1)αx<x>.</x> .... d A e^αx+αx +
f A e^αx = 0</math>, dividatur hæc æquatio per <math>A e^αx</math>, & in æquatione re$ul-
tante pro <math>e^αx<x>.</x></math> $cribatur <math>v</math> & re$ultat æquatio <math>v^n + a v^n-1 + b v^n-2 +
&c. = 0</math>, cujus radices $int <math>π, ξ, σ, τ, &c.</math> inveniantur <math>α, β, γ δ, &c.</math>,
ita ut <math>e^αx<x>.</x> = π, e^βx<x>.</x> = ξ, e^γx<x>.</x> = σ, e^δx<x>.</x> = τ, &c.</math> tum erit <math>A e^αx + B e^βx +
C e^γx + D e^δx + &c. = y</math>, (ubi <math>A, B, C, &c.</math> $unt quæcunque invaria-
biles quantitates ad libitum a$$umendæ) integralis datæ æquationis.</p>
<p>Et $ic de infinitis con$imilibus æquationibus.</p>
<p>Sit æquatio <math>a y<x>1</x><x>n</x> + b y<x>1</x><x>n-1</x> + ... + d y<x>1</x> + f y = X</math>, vel <math>a y<x>.</x><x>n</x> + b y<x>.</x><x>n-1</x> + ...
+ d y<x>.</x> + f = X</math>; ubi <math>a, b, &c., d, f & X</math> $unt functiones quantitatum
<math>x & x<x>.</x>; & x<x>.</x></math> e$t con$tans; tum multiplicatores harum æquationum
$unt functiones quantitatum <math>x & x<x>.</x></math>.</p>
<h>SCHOLIUM.</h>
<p>Facilies e$t ob$ervatu, ut haud po$$ibile $it ulterius promovere generale
problema, quam ejus particulare; methodum generalium incremento-
rum, quam methodum fluxionum: haud difficile etiam erit pleraque in
methodo fluxionum contenta reddere magis generalia & ad methodum
incrementorum applicare; quæ præcipue in hoc libro perficiuntur:
$æpe etiam in primis incrementi terminis $i mutetur incrementum
in fluxionem, & inveniatur fluens fluxionis re$ultantis, exinde deduci
pote$t integralis dati incrementi.</p>
<p>Omnia in hoc libro (mutatis mutandis) etiam applicari po$$unt ad
æquationes, in quibus incrementa & fluxiones $imul continentur.</p>

<pb 345>[0397]
<h>DE
INFINITIS SERIEBUS.</h>
<h>LIBER III.</h>
<p>Def. SIT data infinita $eries <math>a + b + c + d + e + f + g + b +
&c.</math> & $i $ucce$$ivæ $ummæ <math>a, a + b, a + b + c, a + b +
c + d, &c.</math> continuo ad $ummam $eriei vergant, & ultimo proprius
accedant, quam data quævis differentia; tum hæc $eries dici pote$t
convergens.</p>
<p>Cor. Invenire, utrum data $eries $it convergens, necne: continuo
inveniantur limites, inter quos con$i$tit quæ$itæ $eriei $umma; & $i hi
limites continuo ad $e vergant, & ultimo propius ad $e invicem
accedant, quam quævis data differentia; tum hæc $eries dici pote$t
convergens.</p>
<p>2. Convergentia termini (<math>d</math>) revera erit in eâdem ratione, quam
habet terminus ip$e _d_ ad $ummam termini ip$ius & reliquorum $eriei
terminorum <math>d + e + f + g + b + &c.</math></p>
<p>3. Sint <math>A & P</math> quantitates inventæ, inter quas con$i$tit $umma
terminorum <math>c + d + e + f + g + &c.</math> i.e. $it <math>A</math> major quam præ-
dicta $umma & <math>P</math> minor, & <math>B & Q</math>, quantitates inventæ inter quas
con$i$tit $umma <math>d + e + f + g + &c.</math> apparens convergentia ter-
mini (<math>d</math>) $emper erit in eâdem ratione, quam habet <math>(A - P) -
(B - Q):A - P</math>, i.e. <math>{(A - P) - (B - Q) / A - P}:1</math>.</p>

<pb 346>[0398]<rh>DE INFINITIS</rh>
<p>4. Convergentia totius $eriei proportionalis erit ultimæ termino-
rum convergentiæ, i.e. terminorum ad infinitam di$tantiam; ea enim
$eries magis celeriter convergere dici pote$t, cujus termini ultimo
celerius convergunt ad $ummam $eriei quæ$itam.</p>
<p>5. In $erierum convergentiâ dijudicandâ, nece$$e e$t ut continuo
inveniantur quantitates, inter quas ponitur $eriei $umma: in $erie-
bus convergentibus, quarum termini alternatim $unt negativi &
affirmativi, plerumque continuo con$equuntur quantitates, quæ al-
ternatim $unt majores & minores quam $umma quæ$ita; hæ vero $eries
facile mutari po$$unt in $eries affirmativas, $i modo continuo $uman-
tur differentiæ inter affirmativos terminos & negativos proxime $ub-
$equentes pro terminis novæ $eriei quæ$itis. E.g. Sit $eries <math>1 - {1 / 2} +{1 / 3} - {1 / 4} + {1 / 5} - {1 / 6} + &c.</math> mutari pote$t hæc $eries in $ub$equentem affir-
mativam (ex $umendo differentiam continuo inter duos $ucce$$ivos
terminos) viz. <math>{1 / 1 · 2} + {1 / 3 · 4} + {1 / 5 · 6} + &c.</math> Series vero affirmativæ
haud eâdem facilitate in $eries regulariter alternatim affirmativas &
negativas mutari po$$unt, i.e. ita quidem ut omnes $ucce$$ivi termini
ultimo continuo fiant minores, & denique minores evadant quam quæ-
vis datæ quantitates; hoc vero in multis ca$ibus facile perfici pote$t:
E. g. $it $eries <math>S = {1 / m × (m + n)} + {1 / (m + 1) × (m + n + 1)} +{1 / (m + 2) · (m + n + 2)} + &c</math>. facile $e$e dividet in $eries <math>{1 / n} × ({1 / m} -{1 / m + n} + {1 / m + 1} - {1 / m + n + 1} + {1 / m + 2} - {1 / m + n + 2} + &c.)</math>; unde
terminis $eriei recte di$po$itis re$ultat <math>S = {1 / n}({1 / m} + {1 / m + 1} + {1 / m + 2} .. +{1 / m + n} - {1 / m + n} + {1 / m + n + 1} - {1 / m + n + 1} + {1 / m + n + 2} - {1 / m + n + 2}+ &c.) = {1 / n}({1 / m} + {1 / m + 1} + {1 / m + 2} ... + {1 / m + n - 1})</math>, $i <math>n</math> $it inte-
ger affirmativus numerus.</p>

<pb 347>[0399]<rh>SERIEBUS.</rh>
<p>6. Seriem affirmativam <math>t + t′ + t″ + &c.</math> cujus terminus generalis
$it <math>T</math> quæcunque data functio quantitatis <math>z</math> di$tantiæ a primo $eriei
termino, in regularem alteram alternatim affirmativam & negativam
transformare.</p>
<p>Fingatur <math>T</math> incrementum functionis quantitatis <math>z</math>, cujus incremen-
tum $upponitur 1; a$$umatur <math>φ: (z) - φ:(z + 1) = T</math>, ubi <math>φ: (z)</math>
denotat eandem functionem quantitatis <math>z</math>, ac <math>φ: (z + 1)</math> e$t functio
quantitatis <math>z + 1</math>; ex hâc æquatione inveniatur <math>φ:(z)</math>, & perficitur
problema.</p>
<p>Et $ic interponere liceat <math>n</math> terminos inter duos $ucce$$ivos datæ $e-
riei terminos.</p>
<p>7. Sæpe vero ex aliis methodis inve$tigari po$$unt quantitates, quæ
ex$uperant datæ $eriei $ummam; $i enim $ummam $eriei vel $erierum
novimus, quarum termini ex$uperant terminos datæ $eriei, tum $um-
ma data ex$uperat $ummam $eriei datæ: & $ic $æpe inveniri po$$unt
limites inter quos con$i$tit $umma $eriei quæ$ita.</p>
<p>Ex.1. Sit $eries <math>{1 / 1 · 2 · 3} + {1 / 4 · 5 · 6} + {1 / 7 · 8 · 9} + {1 / 10 · 11 · 12} + &c.</math>
& $it <math>{1 / 1 · 2} + {1 / 4 · 5} + {1 / 7 · 8} + &c. = S</math>, tum erit <math>{1 / 3}S</math> major quam datæ
$eriei $umma, quod facile con$tat ex hoc, nempe $inguli ejus <math>({1 / 3} S)</math>
termini majores $unt quam datæ $eriei corre$pondentes termini: &
$ic $umma <math>{1 / 4 · 5} + {1 / 7 · 8} + &c.</math> ducta in <math>{1 / 6}</math> major erit quam $umma
<math>{1 / 4 · 5 · 6} + {1 / 7 · 8 · 9} + &c.</math> & $ic deinceps.</p>
<p>Ex. 2. Summa $eriei <math>1 + {1 / 2} + {1 / 1 · 2 · 3} + {1 / 1 · 2 · 3 · 4} + &c.</math> minor
erit quam <math>1 + {1 / 2} + {1 / 4} + {1 / 8} + &c.</math> & $umma <math>{1 / 1 · 2} + {1 / 1 · 2 · 3} +{1 / 1 · 2 · 3 · 4} + &c.</math> minor erit quam <math>{1 / 2}(1 + {1 / 3} + {1 / 9} + {1 / 27} + &c.); &c.</math></p>

<pb 348>[0400]<rh>DE INFINITIS</rh>
<h>PROB. 1.</h>
<h>F<sc>IG</sc>. 4. _Conce$$is curvilinearum quadraturis; invenire limites, inter_
_quos con$i$tit $eriei $umma._</h>
<p>E terminis generaliter expre$$is inveniatur curva, cujus in$cripta
& circum$cripta pologona $int re$pectivi datæ $eriei termini, & area
curvæ minor erit quam $umma totius $eriei, major vero quam $umma
totius $eriei a primo termino diminutæ: $i modo curva inter initium
& $inem ab$ci$$æ non habeat punctum contrariæ flexuræ vel ordi-
natam maximam vel minimam.</p>
<p>Ex. · Sit $eries data <math>{1 / 2^{3 / 2}} + {1 / 3^{3 / 2}} + {1 / 4^{3 / 2}} ... {1 / z^{3 / 2}}</math>, ubi <math>z - 2</math> $it di$tantia
termini <math>{1 / z^{3 / 2}}</math> a primo: a$$umatur curva _FAPQRS_, &c. $it <math>A L =
A B = B C = C D = D E = &c.</math> in infinitum <math>= 1, & A P = {1 / 2^{3 / 2}} ={1 / O A^{3 / 2}}, BQ = {1 / 3^{3 / 2}} = {1 / O B^{3 / 2}}, C R = {1 / 4^{3 / 2}} = {1 / OC^{3 / 2}}, &c.</math> ergo $umma datæ
$eriei <math>{1 / 2^{3 / 2}} + {1 / 3^{3 / 2}} + {1 / 4^{3 / 2}} + &c. = $ummæ</math> arearum circum$criptorum
parallelogrammorum <math>P B + C Q + D R + E S + &c. = $ummæ</math>
arearum in$criptorum parallelogrammorum <math>L P + A Q + B R + C S
+ &c.</math> $ed curvilinea area _LNPQRS_, &c. major e$t quam prædicta
$umma circum$criptorum vel in$criptorum parallelogrammorum, &
curvilinea area <math>A P Q R S</math>, &c. minor: hæ vero curvilineæ areæ erunt
re$pective <math>{2 / O L^{1 / 2}} & {2 / O A^{1 / 2}} (2 & {2 / 2^{1 / 2}})</math>; ergo $umma datæ $eriei <math>{1 / 2^1{1 / 2}} + {1 / 3^1{1 / 2}}+ &c.</math> in infinitum inter duas quantitates <math>2 & {1 / 2^{1 / 2}}</math> ponitur.</p>
<p>Et $ic $umma $eriei <math>{1 / 3^1{1 / 2}} + {1 / 4^1{1 / 2}} + &c.</math> quæ e$t data $eries ab primo

<pb 349>[0401]<rh>SERIEBUS.</rh>
termino diminuta inter quantitates <math>{2 / 2^{1 / 2}} & {2 / 3^{1 / 2}}</math> continetur; & in genere
$umma $eriei <math>{1 / z^1{1 / 2}} + {1 / (z + 1)^1{1 / 2}} + {1 / (z + 2)^1{1 / 2}} + &c.</math> in in$initum inter
quantitates <math>{2 / (z - 1)^{1 / 2}} & {2 / z^{1 / 2}}</math> continetur; & $ic $umma <math>(z - 2)</math> ter-
minorum <math>{1 / 2^1{1 / 2}} + {1 / 3^1{1 / 2}} + {1 / 4^1{1 / 2}} ... {1 / (z - 1)^1{1 / 2}}</math> inter quantitates <math>2 -{2 / (z - 1)^{1 / 2}} & {2 / 2^{1 / 2}} - {2 / z^{1 / 2}}</math> ponitur.</p>
<p>Et $ic inveniri po$$unt limites, inter quos cuju$cunque con$imilis
$eriei con$i$tit $umma; &c.</p>
<p>Cor. 1. Sit $eries <math>{1 / 2^n} + {1 / 3^n} + {1 / 4^n} + &c.</math> in infinitum; & curvilineæ
areæ, inter quas con$i$tit $umma hujus $eriei, erunt infinitæ magnæ,
necne; prout <math>n</math> vel $it unitas vel minor $it quam unitas, necne.</p>
<p>Cor. 2. Si hæ curvæ habeant ordinatas maximas vel minimas, vel
puncta contrariæ flexuræ; tum ad $ingulam maximam vel minimam,
&c. corirgenda e$t area inventa.</p>
<h>PROB. II.</h>
<h it>Datâ lege, quam ob$ervant termini $eriei in infinitum progredientis;
invenire utrum ea $it finita vel infinite magna.</h>
<p>E datâ lege con$tant termini ad infinitam di$tantiam; $i vero ter-
minus (<math>A</math>) ad infinitam di$tantiam con$titutus $it minor quam quæ-
cunque quantitas huju$ce formulæ <math>{1 / n z}</math>, ubi <math>n</math> $it finita quantitas, &
<math>z</math> di$tantia termini a primo $eriei termino; tum $umma $eriei erit
finita quantitas, $in aliter vero non.</p>
<p>Con$tat e cor. 1. prob. præced.</p>
<p>2. Sit <math>z</math> di$tantia cuju$cunque termini a primo $eriei termino, etiam-

<pb 350>[0402]<rh>DE INFINITIS</rh>
que <math>T & T′</math> duo $ucce$$ivi termini datæ $eriei; & $it data æquatio rela-
tionem inter <math>T & T′</math> exprimens, viz. <math>T′ = T × (a z^r + {b / z^s} + {c / z^t} + &c.)</math>
ubi <math>a z^r + {b / z^s} + {c / z^t} + &c.</math> $it $eries $ecundum dimen$iones quantitatis <math>z</math>
de$cendens; tum, $i <math>1: - z + a z^r+1 + {b / z^r-1} + &c.</math> ubi <math>z</math> $upponitur
infinita quantitas, habet affirmativam & minorem quam rationem
æqualitatis per finitam rationem, tum $umma $eriei prædictæ erit
finita quantitas; $in aliter non.</p>
<p>Nam per coroll. prædict: $eries erit finita, $i modo duo $ucce$$ivi
termini (<math>T & T′</math>) $int re$pective <math>{1 / z^n} & {1 / (z + 1)^n}</math>; unde <math>{T / z^n} = {T′ / (z + 1)^n}</math>;
& exinde <math>T′ = (1 + {n / z} + n · {n - 1 / 2 z^2} + &c.) T</math>; cum <math>n</math> major $it quam
1, & affirmativa; $in aliter infinita; ducatur quantitas <math>1 + {n / z} + n · {n - 1 / 2 z^2}+ &c.</math> in <math>z</math> & re$ultat <math>z + n + n · {n - 1 / 2 z} + &c.</math>; ad hanc quanti-
tatem addatur quantitas <math>- z</math>; & $umma erit <math>n + n · {n - 1 / 2 z} + &c.</math>;
at $eries erit finita, cum <math>n</math> major $it quam unitas, &c. ergo con$tat
propo$itio.</p>
<p>3. Sint <math>T & T^m</math> $ucce$$ivi termini, quorum di$tantia a $e invicem
$it <math>m, & T^m = T (a z^r + {b / z^s} + {c / z^t} + &c.)</math> quæ $it $eries $ecundum
dimen$iones quantitatis <math>z</math> de$cendens; tum, $i <math>1: - z + a z^r+1 +{b / z^s-1} + &c.</math> ubi <math>z</math> $upponitur infinita quantitas, habet affirmativam
& minorem rationem quam <math>1:m</math> per finitam rationem, $umma
$eriei prædictæ erit finita quantitas; $in aliter non.</p>
<p>4. Datâ æquatione inter $ucce$$ivos datæ $eriei terminos relationem

<pb 351>[0403]<rh>SERIEBUS.</rh>
exprimente, facile e terminis datæ æquationis qui maximi $unt ad
infinitam di$tantiam & principiis in hoc problemate traditis erui po-
te$t, annon $eriei $umma $it finita quantitas.</p>
<p>Inveniatur enim ratio, quam habent termini $ucce$$ivi ad infinitam
di$tantiam po$iti; i. e. cum <math>z</math> evadat infinita quantitas, & per metho-
dos prius traditas detegi pote$t, annon $eries $it finita.</p>
<p>5. Si ad infinitam di$tantiam <math>T:T′</math> habeat rationem majorem per
finitam quam æqualitatis, tum $eries erit finita: $i termini $eriei $int
alternatim affirmativi & negativi, reducendi $unt ad $eriem affirmati-
vorum terminorum ex inveniendo differentias inter affirmativos &
negativos terminos $ucce$$ive.</p>
<p>Ex. 1. Sit <math>t′ = a t^n</math>, ubi <math>t^n</math> e$t (<math>n</math>) pote$tas quantitatis <math>t & a</math> inva-
riabilis quantitas; tum, $i <math>a t^n-1</math> minor $it quam 1, $eries converget;
$in major non.</p>
<p>Ex. 2. Sit terminus ad infinitam di$tantiam (i. e. cum <math>z</math> evadat
infinitus numerus) <math>= {z^n / e^mz}</math>; ubi literæ <math>e, n & m</math> denotant invariabiles
quantitates, quarum una <math>m</math> $altem e$t affirmativa; tum $eries $emper
erit finita, $i <math>e</math> major $it quam 1; $in aliter non; nam ad infinitam
di$tantiam erit <math>{T′ / T} = {z^n / e^mz}^-1 × {(z + 1)^n / e^mz+m} = {1 / e^m} (1 + {n / z} + n · {n - 1 / 2 z^2} + n · {n - 1 / 2 z} · {n - 2 / 3 z^2} + &c.) = (ob z infinitum numerum) {1 / e^m}</math>, unde con$tat
exemp.</p>
<p>Hinc, $i <math>z</math> $it infinitus numerus, con$tat <math>e^mz</math> majorem e$$e quam <math>z^n</math>,
cum <math>e</math> major $it quam 1.</p>
<p>Ex. 3. Sit terminus ad infinitam di$tantiam <math>{z^{1 / z} /z^z}</math>, tum, ob <math>z^{1 / z}</math> (cum
<math>z</math> evadat infinitus numerus) <math>= 1</math>; duo $ucce$$ivi termini <math>T & T′</math> erunt
<math>{1 / z^z} & {1 / (z + 1)^z+1}</math> prope, unde <math>T′:T::1:(1 + 1 + {1 / 1 · 2} + {1 / 1 · 2 · 3} +

<pb 352>[0404]<rh>DE INFINITIS</rh>
{1 / 1 · 2 · 3 · 4} + &c.) z = (1 + n) z</math>, ubi <math>1 + n</math> e$t numerus, cujus hy-
per. log. e$t 1; & con$tat $eriem e$$e finitam.</p>
<p>In omnibus his excipiendus e$t ca$us, cum terminus ad finitam
di$tantiam evadat infinitus.</p>
<p>Idem etiam deduci pote$t ex æquatione datâ inter $ucce$$ivas $um-
mas relationem de$ignante; facile enim reduci pote$t hæc æquatio ad
æquationem relationem inter $ucce$$ivos terminos exprimentem: &
de æquatione relationem inter $ucce$$ivas $ummas & terminos expri-
mente.</p>
<h>THEOR. I.</h>
<p>1. Sit $eries <math>A + {b x^p / a^p} + {c x^2p / a^2p} + {d x^3p / a^3p} + &c. = y, & 1^mo</math> in ter-
minis (<math>A, b, c, d, &c.</math>) continuo augeantur differentiæ inter dimen-
$iones quantitatis <math>z</math> di$tantiæ a primo $eriei termino per quantitatem,
quæ in terminis ad infinitam di$tantiam evadit infinite magna; tum
$i dimen$iones numeratoris $uperent dimen$iones denominatoris, $e-
ries erit divergens, $in aliter erit convergens, e. g. $eries <math>1 + α x^p +
α · (α + 1)x^2p + α · (α + 1) · (α + 2)x^3p + &c.</math> $emper diverget; $eries
vero <math>1 + {1 / α}{x^p / a^p} + {1 / α · (α + 1)}{x^2p / a^2p} + {1 / α · (α + 1) · (α + 2)}{x^3p / a^3p} + &c.</math> $em-
per converget, quicunque $it valor quantitatis <math>{x^p / a^p}</math>.</p>
<p>2. Sit <math>p</math> affirmativa quantitas, & differentiæ inter dimen$iones
quantitatis <math>z</math> in denominatore quantitatum (<math>A, b, c, d, &c.</math>) & earum
numeratore contentas ad infinitam di$tantiam continuo eædem ma-
neant, i. e. $it finita; tum $eries $emper converget, $i <math>x</math> inter <math>+ a &
- a</math> ponatur; $in aliter diverget; ni <math>x = a</math>, in qno ca$u $eries con-
verget vel diverget prout dimen$iones quantitatis <math>z</math> ad infinitam di-
$tantiam in denominatore $uperant ejus dimen$iones in numeratore

<pb 353>[0405]<rh>SERIEBUS.</rh>
per quantitatem majorem quam unitatem, necne. Sit <math>p</math> negativa &
$eries $emper converget, cum <math>x</math> major $it quam <math>a</math> vel <math>- a; &c.</math></p>
<p>3. In fluentibus fluxionalium æquationum detegendis, nece$$e e$t
ut convergens $it $eries inter duos valores (<math>α & β</math>) quantitatis <math>x</math> con-
tenta, i. e. <math>A + b (α ± β) + c(α^2 ± β^2) + d(α^3 ± β^3) + &c.</math> $it
convergens $eries; quod in $eriebus præcedentibus haud fieri pote$t,
ni utræque $eries <math>A + b α + c α^2 + d α^3 + &c. & A + b β + c β^2 +
d β^3 + &c.</math> $int convergentes.</p>
<p>Facile ex principiis hic traditis, $i modo $cribatur <math>v + α</math> pro <math>x</math> in
datâ $erie <math>A + b x + c x^2 + d x^3 + &c.</math> erui pote$t, annon $eries ex-
inde re$ultans (<math>A + b α + c α^2 + &c.) + (b + 2 c α + 3 d α^2 + &c.)
v + (c + 3 d α + &c.)v^2 + &c.</math> $it convergens.</p>
<p>Et ex ii$dem principiis erui pote$t; annon $eries, quæ $it quæcun-
que functio datæ $eriei, $it convergens.</p>
<p>Cor. 1. Series, cujus termini generaliter exprimuntur per algebrai-
cam & determinatam functionem haud vero exponentialem literæ <math>z</math>
di$tantiæ a primo $eriei termino, nunquam tam celeriter convergit;
quam $eries, cujus termini exprimuntur etiam per algebraicam & de-
terminatam functionem quantitatis <math>z</math> in <math>x^nz</math>; ubi <math>x</math> $it data quantitas,
minor vero quam unitas, & <math>n</math> quæcunque affirmativa quantitas; vel
<math>x</math> major quam unitas & <math>n</math> negativa quantitas.</p>
<p>In priori enim $erie termini <math>T & T′</math> $ucce$$ivi ad infinitam di$tan-
tiam haud habent inter $e rationem majorem vel minorem quam
æqualitatis per finitam rationem; in po$teriori vero erit ultimo <math>T:T′::1:x^n</math>.</p>
<p>Cor. 2. Po$terior vero $eries nunquam converget tam celeriter, quam
$eries formulæ huju$ce generis <math>x + {x^2 / 1 · 2} + {x^3 / 1 · 2 · 3} + {x^4 / 1 · 2 · 3 · 4} + ...{x^z / 1 · 2 · 3 · 4 ... z}</math>. Erit enim in hâc $erie ad infinitam di$tantiam
<math>T:T′::z:x</math>; i. e. in eadem ratione quam habet infinita quantitas
ad finitam.</p>

<pb 354>[0406]<rh>DE INFINITIS</rh>
<p>Hæc $eries vel $eries <math>x ± {x^2 / 2^2} + {x^3 / 3^3} + {x^4 / 4^4} + &c.</math> $emper convergit,
quicunque $it valor quantitatis <math>x</math>; & $eries <math>x + 1 · 2 x^2 + 1 · 2 · 3 x^3
+ &c. vel x + 4 x^2 + 27 x^3 + 256 x^4 + &c.</math> $emper diverget, qui-
cunque $it valor quantitatis <math>x</math>.</p>
<p>Et $ic e convergentiâ $eriei terminorum ad in$initam di$tantiam
dijudicari pote$t convergentia ip$ius $eriei.</p>
<h>THEOR. II.</h>
<p>Datâ quantitate <math>A</math>, quæ e$t functio quantitatis <math>x</math>, cujus termini
expan$i per $eriem <math>a x^r + b x^r+s + c x^r+2s + &c.</math> progrediuntur, tum
erit <math>{A^. / x^.} = r a x^r-1 + (r + s) b x^r+s-1 + &c.</math> & $ic erit <math>$. A x^. = {1 / r + 1}a x^r+1 + {1 / r + s + 1} b x^r+s+1 + &c.</math> $eries vero <math>{1 / r + 1} a x^r+1 + {1 / r + s + 1}b x^r+s+1 + &c.</math> converget, $i $eries <math>a x^r + b x^r+s + &c.</math> convergat; ple-
rumque etiam $eries <math>r a x^r-1 + (r + s) b x^r+s-1 + &c.</math> converget, $i
convergens $it prædicta $eries, $ed non $emper; quæ facile con$tabunt
e principiis prius traditis; & $ic deinceps.</p>
<h>THEOR. III.</h>
<p>1. Quamvis in$inita $eries data (<math>A</math>) haud convergat, tamen $i detur
finita quantitas <math>p</math>, quæ in $eriem convergentem <math>B</math> ducta præbet datam
quantitatem (<math>A</math>), i. e. <math>p B = A</math>; tum $umma datæ $eriei <math>A</math> quodam-
modo dici pote$t æqualis producto <math>p × B</math>: vel magis generaliter, $i
data $eries (<math>A</math>) $it functio quantitatis <math>x</math> ad $eriem $ecundum ejus di-
men$iones progredientem reducta, tum $eries <math>A</math> quodammodo dici
pote$t æqualis prædictæ functioni quantitatis <math>x</math>.</p>
<p>2. In ca$ibus, cum $eries <math>A</math> nec convergat nec divergat, & <math>a</math> $it valor
quantitatis <math>x</math>; tum plerumque inveniri pote$t $umma $eriei (<math>A</math>),
cum valor ejus incognitæ quantitatis <math>x</math> quam minime differt a dato
valore (<math>α</math>), i. e. $eries erunt convergentes; & exinde deduci pote$t va-
lor $ummæ quæ$itæ, cum <math>x = α</math>.</p>

<pb 355>[0407]<rh>SERIEBUS.</rh>
<p>E. g. Series <math>1 - x + x^2 - x^3 + x^4 - x^5 + &c.</math> $i <math>x = 1</math>, haud
convergit; facile etiamque con$tat, ut <math>{1 / 1 + x} × 1</math> erit <math>1 - x + x^2 -
&c.</math> ergo quodammodo erit <math>1 - 1 + 1 - 1 + &c.</math> in infinitum <math>={1 / 1 + 1} = {1 / 1 + 1 + 1} = {1 / 1 + 1 + 1 + 1} = &c. = vel {1 / 2} vel {1 / 3} vel {1 / 4}, &c.</math></p>
<p>Erit <math>{1 / 2}</math>, $i e $erie <math>{1 / 1 + x} = 1 - x + x^2 - &c.</math> exoriatur; erit <math>{1 / 3}</math>, $i e $erie
<math>{1 / 1 + x + x^2} = 1 - x + x^3 - x^4 + x^6 - x^7 + x^9 - x^10 + x^12 - &c.</math>; & genera-
liter erit <math>{1 / n + 1}</math>, $i e $erie <math>1 - x + x^n+1 - x^n+2 + x^2n+2 - x^2n+3 + x^3n+3 - x^3n+4
+ x^4n+4 - &c.</math> exoriatur.</p>
<p>Cor.. Hinc $ummæ $erierum, quæ tanquamd ivergentes apparent,
$æpe deduci po$$unt: revera for$an hi$ce $eriebus haud a$$ignari po$$unt
$ummæ.</p>
<p>3. Ex divi$ione alterius quantitatis per alteram deduci pote$t quæ-
cunque $eries; at animadvertendum e$t, ut hæc $eries non reverâ dici
po$$it quotiens, ni re$iduum nihilo $it æquale; $i re$iduum augeatur,
tum $eries divergit; quamvis quotientes $unt eædem, attamen fracti-
ones non $int æquales, nam re$idua po$$unt e$$e diver$a: e. g. $int
<tb math>
1 + 1) 1 (1 - 1 + 1 - 1 + 1 - 1 + &c. = 1 + 1 + 1) 1 (1 - 1 + 1 - 1 + 1 - &c.
1 + 1 # 1 + 1 + 1
- 1 # - 1 - 1
- 1 - 1 # - 1 - 1 - 1
+ 1 # + 1
&c. # &c.
</tb>
<math>= 1 + 1 + 1</math> (<math>n</math> $it numerus) <math>(1) 1 - 1 + 1 - 1 + 1 - &c.</math> in primo ca$u
re$idua $unt <math>+ 1 & - 1</math>, & con$equenter quotientes inventæ di$tant
a verâ per quantitates <math>{1 / 2} & - {1 / 2}</math>: in ultimo ca$u re$idua $unt <math>- n + 1
& 1</math>, & con$equenter quotientes inventæ di$tant a verâ per quantita-
tes <math>- {n - 1 / n} & {1 / n}</math>. Si $eries $int reciprocæ, viz. <math>a + b + c + &c. -

<pb 356>[0408]<rh>DE INFINITIS</rh>
a - b - c - &c. + a + b + c + &c. - a - b - c - &c. + &c.</math>,
tum $eries non convergit; & quamvis quotientes $int eædem, attamen,
$i re$idua per divi$ores divi$a non $int eadem, valores non iidem erunt.</p>
<h>THEOR. IV.</h>
<p>1. Convergentes $eries, cujus termini $unt in geometricâ progre$-
$ione, uniformiter convergunt; omnis enim terminus eandem habet
rationem ad reliquam $eriei $ummam.</p>
<p>2. Sit $eries <math>a - b + c - d + &c.</math>; $i <math>a</math> minor $it quam <math>b, b</math> quam
<math>c, c</math> quam <math>d, &c</math>,; tum $eries quodammodo dici pote$t e$$e divergens;
u$que donec $ub$equens terminus minor $it quam præcedens; etiam-
que quodammodo at plerumque non revera convergentia termini
$eriei dici pote$t e$$e in ratione quam habet datus terminus ad ejus
$ub$equentem.</p>
<p>3. Sint <math>R, S, T</math> tres $ucce$$ivi termini datæ $eriei generaliter ex-
pre$$i; & mutetur $eries de prædictâ divergentiâ in convergentiam
cum <math>R = S, &c.</math> & in $tatum prædictæ celerrimæ convergentiæ cum
<math>{S / R} = {T / S}</math>, vel cum $luxio fractionis <math>{S / R}</math> $it nihilo æqualis, &c. Plures
vero po$$unt e$$e $tatus celerioris convergentiæ prædicti generis, &c.
facile con$tabit numerus punctorum celerioris convergentiæ, &c. e
prædictis æquationibus, &c.</p>
<h>PROB. III.</h>
<p it>Invenire convergentes $eries, quœ dividi po$$unt in alias convergentes.</p>
<p>1. A$$umantur duæ $eries convergentes, ducantur hæ $eries in $e$e;
& inveniatur lex, quam ob$ervant termini $eriei re$ultantis; & facile
reduci pote$t hæc $eries ad duas alias.</p>
<p>2. Et $ic inveniri po$$unt e datis $eriebus convergentibus quam plu-
rimæ æquationes convergentes; inveniantur enim $unctiones datarum
$erierum, quæ $int convergentes: e principiis prius traditis facile
con$equuntur in$initæ $unctiones convergentium $erierum, quæ con-
vergentes $eries præbent: exhinc deduci po$$unt quædam $eries, quæ
methodis prius traditis haud erui po$$unt.</p>

<pb 357>[0409]<rh>SERIEBUS.</rh>
<h>REGULA.</h>
<p>1. Sit <math>x</math> incognita quantitas, & detur algebraica functio quanti-
tatis <math>x</math>, reducenda e$t data functio ad terminos $ecundum dimen$iones
quantitatis <math>x</math> progredientes. Hoc fit operando in literis ad eundem
modum, quo arithmetici in numeris decimalibus dividunt, radices
extrahunt vel a$$ectas æquationes $olvunt.</p>
<p>Ex. 1. Sit functio data <math>(a + b x^r + c x^2r + &c.)^m</math>, & per multino-
miale theorema evadet <math>a^m + m a^m-1 b x^r + m a^m-1 c \\ + m. {m - 1 / 2} a^m-2 b^2 # x^2r + &c.</math> quanti-
tas quæ$ita.</p>
<p>Ex. 2. Sit quantitas <math>(a + b x^r + c x^2r + &c.)^n (h + k x^r + l x^2r +
&c.)^m</math>, per prædictum theorema inveniantur <math>(a + b x^r + c x^2r + &c.)^n
= a^n + n a^n-1 b x^r + &c. (b + k x^r + l x^2r + &c.)^m = h^m + m h^m-1 k x^r
+ &c.</math> ducantur hæ $eries in $e$e & re$ultat <math>h^m a^n + (n b h + m k a)
a^n-1 h^m-1 x^r + &c.</math> quantitas quæ$ita.</p>
<p>Ex. 3. Sit quantitas <math>(a + b x^r + c x^2r + &c. + ((A + B x^r + C x^2r
+ &c.) (α + β x^r + γ x^2r + &c.)^n)^m + (π + ξ x^r + σ x^2r + &c.)^b)^k</math>; re-
ducatur ea ad $implices terminos $ecundum dimen$iones quantitatis <math>x</math>
progredientes, tum evadet <math>((a + (A α^n)^m + π^b)^k (= P^k) + k P^k-1 (b + m
(A α^n)^m-1 (α^n B + n A α^n-1 β) + b π^b-1 ξ) x^r (k P^k-1 Q) + k × {k - 1 / 2} P^k-2 Q^2 + \\ + k^. P^k-1 R
+ &c.</math> Series facile con$tat ex reducendo quantitatem ad pote$tatem
<math>k</math> elevandam in in$initonomialem quantitatem $ecundum dimen$iones
quantitatis <math>x^r</math> progredientem, & deinde in$initonomialem quantitatem
ad <math>k</math> pote$tatem elevando; & $ic de omnibus algebraicis quantitatibus
ad $eries $ecundum dimen$iones quantitatis <math>x</math> progredientes redu-
cendis.</p>
<p>2. Facile etiam transformari po$$unt hæ quantitates, ut progre-
diantur $ecundum dimen$iones reciprocas quantitatis <math>x</math>. E. g. Sit

<pb 358>[0410]<rh>DE INFINITIS</rh>
quantitas <math>(b x^r + a)^m</math>, & erit <math>(b x^r + a)^m = b^m x^m + m b^m-1 a x^(r-1)m +
m. {m - 1 / 2} b^m-2 a^2 x^(r-2)m + &c.</math>; & $ic deinceps.</p>
<p>Sæpe $eries, quæ progrediuntur $ecundum dimen$iones directas
quantitatis <math>x</math> convergunt; $æpe vero $eries, quæ progrediuntur $ecun-
dum ejus reciprocas convergunt; $æpe autem nec hæ, nec illæ con-
vergunt, quæ $atis manife$to con$tabunt e propo$. po$tea traditis.</p>
<h>THEOR. V.</h>
<p>Sit data $eries <math>a + b x + c x^2 + d x^3 + &c.</math> quæ in$inita evadit,
cum <math>x = p, vel x = q, &c.</math> tum a$$umatur <math>{1 / (p - x)^m × (q - x)^r} × &c.
× (A + B x + C x^2 + &c.) = a + b x + c x^2 + &c.</math> & e reductione
quantitatum <math>(p - x)^-m & (q - x)^-r, &c.</math> in $implices terminos $ecundum
dimen$iones quantitatis <math>x</math> progredientes, & ex æquatis corre$ponden-
tibus terminis utriu$que æquationis partis con$equentur coefficientes
<math>A, B, C, &c.</math> & in quibu$dam ca$ibus ita a$$umi po$$unt indices <math>(m, r,
&c.)</math> ut æquatio deducta magis celeriter convergat; vel aliæ functio-
nes quantitatis <math>x</math>, quæ evadunt infinitæ, cum <math>x = p vel x = q, &c.</math></p>
<p>Eadem principia ad plurimas con$imiles $eries applicari po$$unt.</p>
<h>THEOR. VI.</h>
<p>1. Sit æquatio <math>o = p - q x + r x^2 - s x^3 + &c.</math> in in$initum, cujus
in$initæ radices $unt <math>α, β, γ, δ, &c.</math> tum erunt <math>{q / p} = {1 / α} + {1 / β} + {1 / γ} + {1 / δ}+ &c.</math> $umma e $ingulis reciprocis radicibus; <math>{r / p} = {1 / αβ} + {1 / αγ} + {1 / βγ}+ {1 / αδ} + &c.</math> $umma e quibu$que duabus reciprocis radicibus in $e$e
ductis; & $ic deinceps.</p>

<pb 359>[0411]<rh>SERIEBUS.</rh>
<p>Cor. Hinc e prob. 1. & 3. medit. algebraic. deduci pote$t aggrega-
tum e $ingulis valoribus quantitatum reciprocarum <math>{1 / α^m} + {1 / β^m} + {1 / γ^m}+ {1 / δ^m} + &c.</math> vel e $ingulis valoribus reciprocarum quantitatum hu-
ju$ce generis <math>{1 / α^m β^n γ^r &c.} + {1 / α^n β^m γ^r &c.} + {1 / α^r β^m γ^n &c.} + &c.</math> & $ic de-
inceps; ubi <math>m, n, r, &c.</math> $unt a$$irmativæ quantitates.</p>
<p>Et $ic transformari pote$t data æquatio in alteram, cujus radices
$unt <math>{1 / α^m β^n γ^r &c.} + &c.</math></p>
<p>2. Sit æquatio <math>o = p - {q / x} + {r / x^2} - {s / x^3} + &c.</math> in in$initum, cujus
in$initæ radices $int <math>α, β, γ, δ, &c.</math> tum erunt <math>{q / p} = α + β + γ + δ +
&c. {r / p} = α β + α γ + β γ + α δ + β δ + γ δ + &c. {s / p} = α β γ + α β δ
+ α γ δ + β γ δ + &c. &c.</math></p>
<p>Cor. · Hinc e prædict. prob. medit. algebr. deduci pote$t aggre-
gatum e $ingulis valoribus contentorum <math>α^m β^n γ^r &c. + α^n β^m γ^r &c. +
α^r β^m γ^n &c.</math> ubi <math>m, n, r, &c.</math> $unt affirmativæ quantitates.</p>
<p>Et $ic transformari pote$t data æquatio in alteram, cujus radices
$unt <math>α^m β^n γ^r &c. + &c.</math></p>
<p>3. Sint duæ in$initæ æquationes, unam $olummodo incognitam
quantitatem <math>x & y</math> re$pective habentes; & quarum radices ex unâ $unt
<math>α, β, γ, δ, &c.</math> ex alterâ vero <math>π, ξ, σ, τ, &c.</math> & $i modo ex unâ æquatione
deduci po$$it aggregatum <math>(P)</math> e $ingulis quantitatibus huju$ce for-
mulæ <math>α^m β^n γ^r &c.</math> ex alterâ vero æquatione aggregatum <math>(Q)</math> e $ingulis
quantitatibus huju$ce formulæ <math>π^a ξ^b σ^c &c.</math> ubi omnes literæ <math>m, n, r,
&c.</math> vel $imul denotant affirmativas quantitates vel negativas, & $ic
omnes literæ <math>a, b, c, &c.</math> $imul $unt affirmativæ vel $imul negativæ:

<pb 360>[0412]<rh>DE INFINITIS</rh>
tum erit aggregatum e $ingulis quantitatibus huju$ce formulæ <math>(α^m β^n γ^r
&c. π^a ξ^b σ^c &c.) = P × Q.</math></p>
<p>4. Ducatur vel æquatio <math>o = p - q x + r x^2 - s x^3 + t x^4 - &c.</math>
vel æquatio <math>o = p - {q / x} + {r / x^2} - {s / x^3} + &c.</math> in arithmeticam $eriem
<math>0, 1, 2, 3, 4, &c.</math> & re$ultant æquationes, quarum radices $unt limites
inter datarum æquationum radices, &c.</p>
<p>5. E principiis in medit. algebr. traditis $æpe con$equuntur notæ
radicum impo$$ibilium, a$$irmativarum & negativarum in datis æqua-
tionibus contentarum. E. g. Sub primo termino $cribatur +; $ub
$ecundo $cribatur + vel - prout <math>{1 / 2} q^2</math> major $it quam <math>p r</math> vel non; $ub
tertio $cribatur + vel - prout <math>{1 / 3} r^2</math> major $it quam <math>q s</math>, necne; & $ub
quarto $cribatur + vel - prout <math>{1 / 4} s^2</math> major $it quam <math>r t</math>; & $ic dein-
ceps; & tot $altem erunt impo$$ibiles radices, quot $unt mutationes
$ignorum de + in - & - in +.</p>
<p>6. E medit. algebr. con$tat, quod $i modo quædam <math>n</math> radices æqua-
tionis <math>p - q x + r x^2 - &c. = 0 vel p - {q / x} + {r / x^2} - &c. = 0</math> $int po$-
$ibiles, tum etiam reperientur <math>n - 1</math> radices po$$ibiles æquationum
re$ultantium e multiplicatione ejus terminorum $ucce$$ivorum in
prædictæ arithmeticæ $eriei terminos.</p>
<p>7. Hinc facile formari pote$t in$inita æquatio, cujus radices haud
dantur; & tamen affirmari pote$t, ut nullas habeat po$$ibiles ra-
dices.</p>
<p>A$$umantur enim coefficientes, ita ut per prædictas regulas omnes
radices notas habeant impo$$ibilitatis; & id, quod requiritur, fit.</p>
<p>8. Si modo in datâ infinitâ æquatione pro <math>x</math> $cribantur quæcunque
duæ quantitates <math>α & β</math>, & quantitates re$ultantes mutentur de + in
- vel - in +, tum impar numerus radicum datæ æquationis inter
quantitates <math>α & β</math> interponitur; $in aliter nulla radix vel par nume-
rus radicum inter <math>α & β</math> invenietur; ni e $cribendo <math>γ</math> valorem inter
<math>α & β</math> contentum pro <math>x</math> $eries evadat divergens.</p>

<pb 361>[0413]<rh>SERIEBUS.</rh>
<p>1. Sit infinita æquatio <math>A = a x^r + b x^r+s + c x^r+2s + &c... + P x^l
+ Q x^l+s + &c.</math>; & $i coefficientes haud cre$cant in majori quam
quâcunque geometricâ ratione a$$ignabili, & $int omnes affirmativæ;
tum $emper datur una affirmativa radix & non plures, quæ quidem
erit minor quam quæcunque negativa in datâ æquatione contenta.</p>
<p>2. In affirmativâ $erie <math>a x^r + b x^r+s + c x^r+2s + &c.</math> fingantur duo
$ucce$$ivi termini ad infinitam di$tantiam <math>(l)</math> po$iti inter $e æquales,
i. e. <math>P x^l = Qx^l+s</math>, & con$equenter <math>x = {P / Q}^{1 / s} = α</math>; & exinde omnis
quantitas minor quam <math>α</math> vel - <math>α</math> pro <math>x</math> in prædictâ $erie con$tituta
$emper præbebit finitam $ummam; $in major $it quam <math>α</math>, tum $emper
præbebit infinitam.</p>
<p>2. 2. Si $eries $it negativa, tum ex pluribus terminis ad infinitam
di$tantiam po$itis inter $e æqualibus e$$e $uppo$itis acquiri po$$unt li-
mites quantitatum, quæ in datâ $erie pro <math>x</math> $ub$titutæ $emper præ-
bent finitas vel infinitas quantitates.</p>
<h>PROB. IV.</h>
<h it>Invenire æquationem, quæ infinitas babet cognitas radices.</h>
<p>Sint cognitæ radices <math>α, β, γ, δ, ε, &c.</math> & ducantur factores <math>(x - α)
× (x - β) × (x - γ) × (x - δ) × &c.</math> in $e$e, & contentum nihilo $iat
æquale; erit æquatio de$iderata.</p>
<p>Ex. · Sint radices <math>α, α^2, α^3, α^4, &c.</math> in infinitum; æquatio quæ$ita
erit <math>1 - {α / 1 - α} × {1 / x} + {α^3 / (1 - α) · (1 - α^2)} × {1 / x^2} - {α^6 / (1 - α) · (1 - α^2) · (1 - α^3)}{1 / x^3} + {α^10 / (1 - α) · (1 - α^2) · (1 - α^3) · (1 - α^4)} × {1 / x^4} - {α^15 / (1 - α) · (1 - α^2)}.{ / (1 - α^3) · (1 - α^4) · (1 - α^5)} × {1 / x^5} + &c. = 0.</math> Summa radicum e$t

<pb 362>[0414]<rh>DE INFINITIS</rh>
<math>{α / 1 - α}</math>, $umma rectangulorum $ub $ingulis binis <math>{α^3 / (1 - α) × (1 - α^2)}</math>,
& $ic deinceps.</p>
<p>Cor. Inveniatur æquatio, cujus radices $int quæcunque algebraica
functio huju$ce æquationis radicum; re$ultat æquatio, quæ etiam
habet infinitas radices.</p>
<p>Ex. 1. Ducatur hæc æquatio <math>1 - {α / 1 - α} × {1 / x} + {α^3 / (1 - α) · (1 - α^2)}× {1 / x^2} - &c. = 0</math> in alteram æquationem <math>1 - {β / 1 - β} × {1 / x} + {β^3 / (1 - β) ×
(1 - β^2)} × {1 / x^2} - &c. = 0</math>, & re$ultat æquatio <math>1 - {α + β - 2 α β / (1 - α) × (1 - β)}× {1 / x} + &c. = 0</math>, cujus radices erunt <math>α, α^2, α^3, &c.</math> in infinitum & <math>β,
β^2, β^3, &c.</math> in infinitum; & $ic deinceps.</p>
<h>PROB. V.</h>
<p it>Datis duabus infinitis æquationibus duas vel plures incognitas quanti-
tates <math>(x & y)</math> babentibus; eas in unam reducere, ita ut incognita quantitas
<math>(y)</math> exterminetur.</p>
<p>1<_>mo</_>. Collocentur termini utriu$que æquationis, ita ut illi termini
primum locum occupent, qui maximi $unt; ii vero proximum locum,
qui proxime majores $unt, & $ic deinceps. i. e. Sint termini in unâ
æquatione re$ultantes re$pective <math>a + b + c + d + e + &c. = 0</math>, in
alterâ vero <math>p + q + r + s + &c. = 0.</math> 1<_>mo</_>. Supponantur <math>a = 0 &
p = 0</math>, & ita reducantur hæ duæ æquationes, ut exterminetur incog-
nita quantitas <math>(y)</math>; deinde $upponantur pro propriori valore æquatio-
nis quæ$ito <math>a + b = 0 & p + q = 0</math>, & reducantur hæ duæ æquatio-
nes, ita ut exterminetur incognita quantitas <math>y</math>; & $ic deinceps; & con-
tinuo ad æquationem quæ$itam propius accedamus.</p>
<p>2. Datam æquationem unam <math>(x)</math> $olummodo incognitam quanti-

<pb 363>[0415]<rh>SERIEBUS.</rh>
tatem habentem in duas vel plures alias habentes duas vel plures in-
cognitas quantitates dividere.</p>
<p>A$$umatur æquatio duas vel plures incognitas quantitates <math>(x, y, &c.)</math>
habens; deinde addatur, $ubtrahatur, &c. a$$umpta æquatio e datâ;
& exoriuntur duæ æquationes duas vel plures incognitas quantitates
habentes, quæ facile reduci po$$unt ad datam. Et $ic de pluribus
æquationibus.</p>
<h>THEOR. VIII.</h>
<p>Sint duæ vel plures infinitæ æquationes duas vel plures incognitas
quantitates habentes, quarum $it una integer numerus vel fractio;
tum altera etiam erit integer numerus vel fractio, ni duo vel tres vel
plures valores reliquarum incognitarum quantitatum $int inter $e
æquales.</p>
<p>Multarum infinitarum æquationum inveniri po$$unt radices inte-
gri numeri ex principiis, quæ tradita fuere pro inveniendis radicibus,
qui $unt integri numeri, in finitis algebraicis æquationibus.</p>
<h>PROB. VII.</h>
<p it>Sint duæ infinitæ $eries, quarum termini $ecundum dimen$iones quanti-
tatis <math>x</math> collocantur; invenire nonnunquam infinitas $eries, quæ $int earum
communes divi$ores.</p>
<p>Dividantur hæ æquationes juxta methodum, per quam inveniuntur
communes divi$ores in algebraicis finitis æquationibus, & nonnun-
quam re$ultant communes divi$ores. E. g. Sint duæ infinitæ $eries
<math>1 - {5 / 3 · 2} x + {5 / 3 · 4} x^2 - {5 / 3 · 8} x^3 + &c. & 1 - {3 / 4} x + {3 / 8} x^2 - {3 / 16} x^3 +
&c.</math> dividantur hæ quantitates $ecundum methodum inveniendi com-
munes divi$ores, & operatio erit huju$modi
<math>1 - {5 / 6}x + {5 / 12}x^2 - {5 / 24}x^3 + &c.) 1 - {3 / 4}x + {3 / 8}x^2 - {3 / 16}x^3 + &c.) 1</math>
 # <math>{1 - {5 / 6}x + {5 / 12}x^2 - {5 / 24}x^3 + &c. / {1 / 12}x - {1 / 24}x^2 + {1 / 48}x^3 - &c.}</math> ducatur

<pb 364>[0416]<rh>DE INFINITIS</rh>
hoc re$iduum, fractionem evitandi gratiâ, in <math>{12 / x}</math>, & re$ulatat $eries
<math>1 - {1 / 2}x + {1 / 4}x^2 - {1 / 8}x^3 + &c.</math> dividatur prior divi$or per re$iduum, &
operatio erit
<math>1 - {1 / 2}x + {1 / 4}x^2 - &c.) 1 - {5 / 6}x + {5 / 12}x^2 - {5 / 24}x^3 + &c.) 1 - {1 / 3}x \\ 1 - {1 / 2}x + {1 / 4}x^2 - {1 / 8}x^3 + &c. \\ <ol>- {1 / 3}x + {1 / 6}x^2 - {1 / 12}x^3 + {1 / 24}x^4 - &c.</ol> \\ - {1 / 3}x + {1 / 6}x^2 - {1 / 12}x^3 + {1 / 24}x^4 - &c. \\ <ol>.........</ol></math>
unde con$tat has duas infinitas $eries eandem præbere $eriem <math>1 - {1 / 2}x
+ {1 / 4}x^2 - &c.</math> ductam re$pective in <math>1 - {1 / 3}x & 1 - {1 / 4}x</math>.</p>
<h>THEOR. IX.</h>
<p>1. Sint <math>α & β</math> duæ radices æquationis infinitæ <math>0 = a + b x + c x^2
+ d x^3 + &c. = A</math>, $cribatur in hâc æquatione pro <math>x</math> ejus valor a$-
$umptus <math>π</math>, qui inter duas proximas radices <math>α & β</math> ponitur; tum ple-
rumque erit quantitas re$ultans finita quantitas; $i <math>A</math> evadat infinita
quantitas, cum <math>x = λ</math> inter valores <math>α & β</math> po$ita; tum evadet iterum
finita quantitas, cum <math>x = μ</math> quantitas etiam inter <math>α & β</math> po$ita: &
plures po$$unt e$$e progre$$us huju$ce generis de finito ad infinitum,
&c. inter valores <math>α & β</math> quantitatis <math>x</math>.</p>
<p>2. Si modo $int <math>α, β, γ, &c.</math> $ucce$$ivæ radices datæ infinitæ æqua-
tionis; & <math>π</math> quantitas inter <math>α & β, & ς</math> inter <math>β & γ</math>: $cribantur <math>π & ς</math>
pro <math>x</math> in datâ infinitâ æquantione; & quantitatum exinde re$ultantium
mutabuntur $igna de + in - & - in +, ni detur quantitas inter
<math>π & ς</math> contenta, quæ $ub$tituta pro <math>x</math> præbet divergentem $eriem.</p>
<p>3. Si duæ æquationes <math>A + b x + c x^2 + d x^3 + &c. = 0 & A′ +
b x + c x^2 + d x^3 + &c. = 0</math> habeant po$$ibilem radicem <math>(α & β)</math> re-
$pective; & <math>A″</math> $it quantitas inter <math>A & A′</math> contenta, tum æquatio <math>A″
+ b x + c x^2 + d x^3 + &c. = 0</math> habet po$$ibilem radicem; ni inter
valores <math>α & β</math> quantitatis <math>x</math> $eries evadat divergens.</p>
<p>4. Divergens $eries in quantitatem nihilo æqualem ducta aliquando

<pb 365>[0417]<rh>SERIEBUS.</rh>
fiet convergens; in finitam quantitatem ducta nunquam producet
$eriem, quæ proprie dici pote$t convergens.</p>
<p>5. Inveniantur termini $eriei ad infinitam di$tantiam po$iti, quæ
e$t productum divergentis $eriei in quantitatem nihilo æqualem ductæ;
& ex iis per principia prius tradita $emper detegi pote$t, annon $eries
re$ultans $it convergens.</p>
<p>6. In $eriebus cuju$cunque generis fluens convergit magis celeriter
quam ejus fluxio; & $ic de con$imilibus.</p>
<h>PROB. VII.</h>
<h it>Datâ re$olutione æquationis, quæ $it infinita $eries irrationalium quantita-
tum; invenire æquationem ip$am.</h>
<p>Ita reducatur per prob. 26. medit. algebr. data re$olutio, ut exter-
minentur irrationales quantitates, & re$ultat æquatio quæ$ita.</p>
<h>PROB. VIII.</h>
<p it>Invenire, utrum data quantitas $it radix datæ infinitæ æquationis, necne:
$cribatur data radix pro ejus valore in datâ æquatione, & $i e quibu$cunque
principiis con$tet quantitatem re$ultantem ultimo ad nibil vergere, tum
con$tat quantitatem datam e$$e radicem datæ infinitæ æquationis.</p>
<p>E. g. Sit æquatio <math>0 = 1 - x - x^2 - x^3 - &c.</math> invenire, annon <math>{1 / 2}</math> $it
radix prædictæ æquationis: $cribantur <math>{1 / 2}, {1 / 4}, {1 / 8}, &c.</math> pro <math>x, x^2, x^3; &c.</math> in
datâ æquatione, & re$ultant $ucce$$iva re$idua <math>{1 / 2}, {1 / 4}, {1 / 8}, {1 / 16}, {1 / 32}, &c.</math> in in-
finitum, unde ultimo vergit ad nihil, & con$equenter <math>x = {1 / 2}</math> erit radix
datæ æquationis. Dividatur data æquatio per <math>{1 / 2} - x</math>, & quotiens in-
venietur <math>0 = 2 + 2 x + 2 x^2 + 2 x^3 + &c.</math>, cujus omnes radices $unt
impo$$ibiles.</p>
<p>Cor. Series <math>1 - x - x^2 - x^3 - &c. = {1 - 2 x / 1 - x}</math>; huju$ce $eriei nu-
merator <math>1 - 2 x = 0</math>, cum <math>x = {1 / 2}</math>: etiamque $eries <math>1 + x + x^2 + x^3

<pb 366>[0418]<rh>DE INFINITIS</rh>
+ &c. = {1 / 1 - x}</math> erit convergens, cum <math>x = {1 / 2}</math> ergo; <math>x = {1 / 2}</math> erit radis<?>
datæ æquationis <math>{1 - 2 x / 1 - x} = 1 - x - x^2 - &c. = 0</math>. Et $ic genera-
liter ratiocinari liceat; i.e. $i <math>numerator = 0</math>, & denominator ad $eriem
$ecundum dimen$iones quantitatis <math>x</math> progredientem reductus præbeat
convergentem $eriem, cum <math>x = α</math>; tum <math>α</math> erit radix $eriei, quæ oritur
ab expan$ione datæ fractionis in terminos $ecundum dimen$iones
quantitatis <math>x</math> progredientes.</p>
<p>2. Æquatio <math>0 = 1 + x + x^2 + x^3 + &c. = {1 / 1 - x}</math> habet unam ra-
dicem <math>x = - 1</math>; & æquatio <math>0 = 1 - x + x^2 - x^3 + &c. = {1 / 1 + x}</math>
habet radicem <math>x = 1</math>; nam $i pro <math>x</math> in utri$que æquationibus $criba-
tur quantitas inter <math>1 & - 1</math> po$ita, tum $emper evadent finitæ quan-
titates ad <math>0</math> magis appropinquantes, quo minus di$tet <math>(x) ab - 1</math> in
priori & <math>ab + 1</math> in po$teriori æquatione.</p>
<h>THEOR. X.</h>
<p>Sit $eries <math>A = 0</math>, quæ $it finita functio <math>(P)</math> algebraicæ quantitatis
<math>x</math>; tum radices æquationis <math>A = p</math> erunt etiam radices æquationis
<math>P = p</math>; & vice versâ radices æquationis <math>P = p</math> erunt radices æqua-
tionis <math>A = p</math>.</p>
<p>Ex.. Sit $eries <math>1 + x + x^2 + x^3 + &c. = a + √(- b)</math>, $ed e$t <math>{1 / 1 - x}= 1 + x + x^2 + &c. = a + √(- b)</math>, ergo nullæ radices inveniun-
tur in æquatione <math>1 + x + x^2 + &c. = a + √(- b)</math>, quæ in æqua-
tione <math>{1 / 1 - x} = a + √(- b)</math> haud inveniuntur.</p>
<p>Scribatur enim in $erie <math>1 + x + x^2 + x^3 + &c.</math> pro <math>(x)</math> valor a$-
$umptus <math>c + √(- d)</math>, & re$ultat <math>1 + c + c^2 + c^3 + &c. - d (1 +

<pb 367>[0419]<rh>SERIEBUS.</rh>
3c + 6c^2 + &c.) + d^2 (1 + 5c + 15c^2 + &c.) - d^3 (1 + 7c + 28c^2
+ &c.) + &c. (= {1 / 1 - c} - {d / (1 - c)^3} + {d^2 / (1 - c)^5} - &c.) + √(- d)
(1 + 2c + 3c^2 + &c. - d(1 + 4c + 10c^2 + &c.) + d^2 (1 + 6c +
21c^2 + &c.) - &c.)(= √(- d)({1 / (1 - c)^2} - {d / (1 - c)^4} + {d^2 / (1 - c)^6}- &c.) + &c.)</math>; unde <math>{1 / 1 - c} - {d / (1 - c)^3} + {d^2 / (1 - c)^5} - &c. = a &
√(- d)({1 / (1 - c)^2} - {d / (1 - c)^4} + {d^2 / (1 - c)^6} - &c.) = √(- b)</math>, &
exinde <math>{1 - c / (1 - c)^2 + d} = a & {1 / (1 - c)^2 + d} = √({b / d})</math>; & ex his dua-
bus æquationibus deduci po$$unt iidem & nulli alii valores quantita-
tum <math>c & d</math>, præter eos, qui ex æquatione <math>{1 / 1 - x} = a + √(- b)</math>, i.e.
<math>x = {a - 1 + √(- b) / a + √(- b)}</math>, re$ultant.</p>
<h>THEOR. XI.</h>
<p>Sit infinita æquatio <math>0 = a - bx + cx^2 - dx^3 + ex^4 - &c.</math>; ubi
<math>{b / a}</math> multo major $it quam <math>{c / b}</math>; & <math>{c / b}</math> multo major quam <math>{d / c}</math>; & <math>{d / c}</math> multo
major quam <math>{e / d}</math>; & <math>{e / d}</math> multo major quam <math>{f / e}</math>, & $ic deinceps in infini-
tum: tum erunt omnes radices datæ æquationis po$$ibiles; & ap-
proximatio <math>(α)</math> ad minimam radicem erit <math>{a / b}</math>; & approximatio ad radi-
cem <math>(β)</math>, quæ e$t minor quam omnes præter <math>α</math> erit <math>{b / c}</math>; vel radices
æquationis quadraticæ <math>a - bx + cx^2 = 0</math> erunt propriores approxi-
mationes ad <math>α & β</math>; & $imiliter <math>{c / d}</math> erit approximatio ad tertiam radi-

<pb 368>[0420]<rh>DE INFINITIS</rh>
cem <math>(γ)</math> datæ æquationis, quæ e$t minor quam omnes radices datæ
æquationis præter <math>α & β</math>; vel radices cubicæ æquationis <math>a - bx +
cx^2 - dx^3 = 0</math> erunt propriores approximationes ad radices <math>α, β & γ</math>
datæ æquationis; & $imiliter approximatio ad quartam minimam ra-
dicem erit <math>{d / e}</math>, & quatuor radices datæ æquationis <math>a - bx + cx^2 -
dx^3 + ex^4 = 0</math> erunt propriores approximationes ad quatuor mini-
mas radices <math>(α, β, γ & δ)</math> datæ æquationis; & $ic deinceps.</p>
<h>THEOR. XII.</h>
<p>Sit æquatio <math>y^n + a y^n-1 + b y^n-2 + c y^n-3 ... f y^n-m + g y^n-m-1 +
b y^n-m-2 ... + k y^n-m-r + l y^n-m-r-1 ... + p y^s+1 + q y^s + &c. = 0</math>, cu-
jus <math>m</math> radices $int multo majores quam reliquæ, tum ex $ummâ <math>m + 1</math>
primorum terminorum nihilo æquali e$$e $uppo$itâ, viz. <math>y^m + a y^m-1
+ b y^m-2 + c y^m-3 ... + f = 0</math> re$ultat æquatio, cujus radices $unt præ-
dictæ <math>m</math> radices prope: $i vero <math>r</math> radices datæ æquationis $int multo
minores quam prædictæ <math>m</math> radices, multo vero majores quam reliquæ;
tum ex $ummâ <math>r + 1</math> terminorum, quorum primus $it ultimus æqua-
tionis prius traditæ terminus <math>f y^n-m</math>, cæteri vero <math>r</math> $ub$equentes, viz.
<math>f y^r + g y^r-1 + b y^r-2 ... + k = 0</math>, nihilo æquali e$$e $uppo$itâ re$ultat
æquatio, cujus radices $unt <math>r</math> prædictæ radices prope; & $ic deinceps;
& denique $i <math>s</math> radices datæ æquationis $int multo minores quam re-
liquæ, tum ex $ummâ <math>s + 1</math> ultimorum terminorum nihilo æquali
e$$e $uppo$itâ re$ultat æquatio, cujus radices $unt <math>s</math> radices prædictæ.</p>
<p>Cor.. Sint <math>α, β, γ, δ, &c.</math> radices datæ æquationis <math>y^n - p y^n-1 +
q y^n-2 - r y^n-3 + &c. = 0</math>, quarum <math>α</math> multo major $it quam <math>β, β</math> quam
<math>γ, γ</math> quam <math>δ</math>, & $ic deinceps; tum erit <math>α = p</math> prope, <math>β = {q / p}</math> prope,
<math>γ = {r / q}</math> prope, & $ic deinceps.</p>

<pb 369>[0421]<rh>SERIEBUS.</rh>
<h>THEOR. XIII.</h>
<p>1. Sit æquatio <math>y^n + a y^n-1 + b y^n-2 + c y^n-3 + &c. = 0</math>, ubi <math>a, b, c,
&c.</math> $unt functiones ip$ius <math>x</math>, nunc ex hypothe$i quod <math>x</math> $it perparva
vel permagna quantitas vel ex quâcunque aliâ hypothe$i $int <math>α, β, γ,
&c.</math> proximi valores quantitatum <math>a, b, c, d, &c.</math> tum a$$umatur æqua-
tio <math>ν^n + α ν^n-1 + β ν^n-2 + γ ν^n-3 + δ ν^n-4 + &c. = 0, & n</math> radices hu-
ju$ce æquationis erunt primæ approximationes ad <math>n</math> diver$as datæ
æquationis radices.</p>
<p>2. Si vero quæcunque una radix multo minor $it quam quædam <math>m</math>
radices in datâ æquatione <math>y^n + a y^n-1 + b y^n-2 + ... f y^n-m + g y^n-m-1 +
b y^n-m-2 + &c. = 0</math> contentæ, multo vero major quam $ingulæ reli-
quæ (<math>n - m - 1</math>); tum erit <math>-{g / f}</math> prima approximatio ad prædictam
radicem; $i vero duæ radices <math>ε & τ</math> $int multo minores quam <math>m</math> radi-
ces prædictæ, multo vero majores quam <math>n - m - 2</math> reliquæ radices,
erunt radices æquationis <math>f y^n-m + g y^n-m-1 + b y^n-m-2 = 0</math> primæ ap-
proximationes ad duas radices prædictas; & $ic de pluribus.</p>
<p>3. Datis primis approximationibus <math>π, ξ, σ, &c.</math> ad qua$cunque (<math>m</math>)
radices datæ æquationis; alias <math>π′, ξ′, σ′, &c.</math> adhuc magis appropin-
quantes detegere: $cribatur <math>(y - π - π′) · (y - ξ - ξ′) · (y - σ - σ′)
· &c. × (y^n-m + p y^n-m-1 + q y^n-m-2 + &c.) = y^n + a y^n-1 + b y^n-2 + &c.
= 0</math>, & ex æquatis corre$pondentibus terminis re$ultantis æquationis
exorientur <math>n</math> æquationes totidem <math>n</math> incognitas quantitates <math>π′, ξ′, σ′, &c.
p, q, r, &c.</math> habentes; nunc abjectis omnibus terminis tanquam prope
nihilo æqualibus, in quibus plures quam una dimen$io quantitatum
<math>π′, ξ′, σ′, &c.</math> inveniuntur, & re$ultantes <math>n</math> æquationes in <math>m</math> alias redu-
cantur, ita ut exterminentur incognitæ quantitates <math>p, q, r, s, &c.</math> &
exorientur <math>m</math> æquationes totidem incognitas quantitates <math>π′, ξ′, σ′, &c.</math>
habentes; quibus in unam reductis, ita ut exterminentur omnes præ-
ter unam incognitæ quantitates, re$ultant valores approximationum
quæ$itarum <math>π′, ξ′, σ′, &c.</math> quæ invenientur re$pective <math>π′ = -

<pb 370>[0422]<rh>DE INFINITIS</rh>
{π^n + a π^n-1 + b π^n-2 + c π^n-3 + &c. / n π^n-1 + (n - 1) a π^n-2 + (n - 2) b π^n-3 + &c.}</math> prope, <math>ξ′ = -{ξ^n + a ξ^n-1 + b ξ^n-2 + c ξ^n-3 + &c. / n ξ^n-1 + (n - 1) a ξ^n-2 + (n - 2) b ξ^n-3 + &c.}</math> prope, <math>σ′ = -{σ^n + a σ^n-1 + b σ^n-2 + &c. / n σ^n-1 + (n - 1) a σ^n-2 + (n - 2) b σ^n-3 + &c.}</math> prope, &c.</p>
<p>In denominatoribus & numeratoribus harum fractionum rejician-
tur omnes termini tanquam nihilo æquales, qui ex hypothe$i a$$umptâ
multo minores evadunt quam reliqui; & e comparandis re$ultantis
æquationis <math>ν^n + α ν^n-1 + β ν^n-2 + &c. = 0</math> terminis, $æpe inveniri
po$$unt ejus radices. E. g. E comparandis quibu$que duobus proxime
$ucce$$ivis erui pote$t radix prope, $i modo longe di$tet a reliquis; &
$ic detegi pote$t quadratica æquatio, quæ continet duas radices prope
a reliquis longe di$tantes; & $ic deinceps.</p>
<p>Eadem fere principia ad fluxionales & incrementiales æquationes
applicari po$$unt.</p>
<p>Si plures dentur æquationes plures habentes incognitas quantita-
tes; tum ex ii$dem principiis erui po$$unt plures æquationes, quarum
radices $unt primæ approximationes ad radices datarum æquatio-
num.</p>
<p>4. Eadem principia etiam applicari po$$unt ad æquationes, in
quibus quæcunque irrationales continentur quantitates; rejectis enim
omnibus quantitatibus in irrationalibus terminis contentis, qui haud
maximi evadant ex a$$umptâ hypothe$i, e reliquis erui po$$unt ap-
proximationes ad $ingulas datæ æquationis radices, & ex iis primæ ap-
proximationes ad quantitatem vel quantitates quæ$itas; & ex primis
approximationibus deduci po$$unt $ecundæ approximationes e princi-
piis prius traditis: vel rejectis omnibus, quæ haud maximæ vel pro-
ximæ evadant ex a$$umptâ hypothe$i, e terminis re$ultantibus inve-
niri po$$unt approximationes ad radices quæ$itas vergentes; & $ic de-
inceps. Facile etiam con$tat numerus terminorum, qui vere obtineri
po$$unt ex rejectis quibu$dam datæ æquationis terminis.</p>

<pb 371>[0423]<rh>SERIEBUS.</rh>
<p>Si vero duo vel plures (<math>n</math>) valores quantitatis quæ$itæ $int prope
<math>= e</math> & inter $e æquales; tum inveniendi $unt termini datarum æqua-
tionum, qui nullam, unam, duas, vel denique (<math>n</math>) habent dimen$iones;
qui $int re$pective <math>A, B e, C e^2, D e^3, ... H e^n</math>; tum radices æquatio-
num <math>A + B e + C e^2 = 0</math> vel <math>A + B e + C e^2 + D e^3 + .. H e^n = 0</math>
erunt approximationes ad duas vel denique <math>n</math> radices æquationum,
quæ $unt prope inter $e æquales.</p>
<h>THEOR. XIV.</h>
<p>1. Datâ æquatione algebraicâ <math>x^n - p x^n-1 + q x^n-2 - r x^n-3 + s x^n-4 
- &c. = 0</math>, cujus una radix <math>α</math> multo major $it quam quæcunque alia,
& erit ea radix <math>p - p^-1 q + p^-2 r - s \\ - q^2 # p^-3 + &c.</math> cujus $eriei lex
eadem erit ac ea, quæ traditur pro inveniendâ $ummâ <math>α^m + β^m + γ^m +
δ^m + &c.</math> ubi <math>α, β, γ, δ, &c.</math> $unt radices datæ æquationis vel in meis
mi$cell. analyt. vel po$tea in medit. algeb. $i modo pro <math>m</math> $cribatur 1,
& termini in infinitum progrediantur; i.e. coefficiens generalis ter-
mini <math>p^m-1 q^α r^β s^γ &c.</math> ubi <math>l = 2 α + 3 β + 4 γ + &c.</math> erit <math>= m.{m - l + 1. m - 1 + 2 · m - l + 3 · m - l + 4 ... m - l + α + β + γ + &c. -1 / 1 · 2 · 3 .. α × 1 · 2 · 3 .. β × 1 · 2 · 3 .. γ × & c.}</math>;
$ignum affixum erit + vel -, $i <math>l</math> $it impar numerus, prout <math>α + β +
γ + &c.</math> $it impar vel par numerus, &c.; $ed in hoc ca$u <math>m = 1</math>, ergo
erit prædicta coefficiens <math>± {(2 - l) · (3 - l) ... (α + β + γ + &c. - l) / 1 · 2 · 3 .. α × 1 · 2 · 3 .. β × 1 · 2 · 3 .. γ × &c.}</math></p>
<p>2. Sit æquatio <math>P - Q x + R x^2 - S x^3 + T x^4 - &c. = 0</math>, cujus
una radix <math>π</math> $it multo minor quam quæcunque alia: $cribantur <math>{1 / x}= v, {Q / P} = p, {R / P} = q, {S / P} = r, {T / P} = s, &c.</math> & re$ultat æquatio <math>ν^n -
p ν^n-1 + q ν^n-2 - k ν^n-3 + &c. = 0</math>, cujus maxima radix <math>{1 / π}</math> erit <math>p -

<pb 372>[0424]<rh>DE INFINITIS</rh>
p^-1 q + p^-2 r - p^-3 s \\ - q^2 # + &c.</math> ut antea; ejus vero reciproca, i.e. radix
quæ$ita (<math>π</math>) erit <math>p^-1 + p^-3 q - p^-4 r + s \\ + 2q^2 # p^-5 - &c.</math> $eries prius tra-
dita eadem evadet ac hæc, $i modo in eâ pro <math>m</math> $cribatur <math>- 1</math> &
termini in infinitum progrediantur: hic animadvertendum e$t ulti-
mo duos primos terminos recte o$tendere duplum figurarum nume-
rum, tres vero primos terminos triplum figurarum numerum præ-
bere, &c.</p>
<p>2. 1. Sit æquatio <math>x^n - p x^n-1 + q x^n-2 + &c. = 0</math>, cujus radix <math>α</math>
multo major $it quam quæcunque alia, tum erit <math>α^λ = p^λ - λ p^λ-2 q +
λ p^λ-3 r - λ p^λ-4 s \\ + λ. {λ - 3 / 2} p^λ-4 q^2 # + &c.</math> $i in prædictâ $erie pro <math>m</math> $cribatur <math>λ</math>, &
termini in infinitum progrediantur, tum eadem evadet ac $eries prius
tradita.</p>
<p>2. Sit æquatio <math>P - Q x + R x^2 - &c. = 0</math>, cujus radix <math>π</math> multo
minor $it quam quæcunque alia in eâ contenta, transformetur hæc
æquatio in alteram <math>ν^n - p ν^n-1 + q ν^n-2 - &c. = 0</math> cujus radix <math>ν = {1 / x}</math>;
& erit <math>{1 / π^λ} = p^λ - λ p^λ-2 + &c.</math> ut antea, & <math>π^λ = p^-λ + λ p^-λ-2 q -
λ p^-λ-3 r + λ p^-λ-4 s \\ - λ. - {λ + 3 / 2} p^-λ-4 q^2 # - &c.</math> quæ $eries eadem erit ac $eries re$ul-
tans e $cribendo <math>- λ</math> pro <math>m</math> in $erie generali prius traditâ.</p>
<p>Cor.1. Termini omnium $erierum prædictarum, in quibus $olum-
modo continentur coefficientes <math>p & q</math>, erunt æquales quantitati
<math>({p + √(p^2 - 4 q) / 2})^±λ</math>; termini vero in quibus continentur coeffi-
cientes <math>p, q & r</math> erunt æquales quantitati <math>H^±λ</math>, $i modo <math>H</math> $it ma-
xima radix cubicæ æquationis <math>x^3 - p x^2 + q x - r = 0</math>; & $ic de-
inceps.</p>

<pb 373>[0425]<rh>SERIEBUS.</rh>
<p>Cor.2. Sint <math>{p^n - p · p^n-1 + q p^n-2 - r p^n-3 + &c. / n p^n-1 - (n - 1) p · p^n-2 + (n - 2) q p^n-3 - &c.} = α,{π^n - n p π^n-1 + (n - 1) q π^n-2 - (n - 2) r π^n-3 + &c. / n π^n-1 - (n - 1) q π^n-2 + (n - 2) q π^n-3 - &c.}=β,{ξ^n - n p ξ^n-1 + (n - 1) q ξ^n-2 - (n - 2) r ξ^n-3 + &c. / n ξ^n-1 - (n - 1) p ξ^n-2 + (n - 2) q ξ^n-3 - &c.}=γ,{σ^n - n p σ^n-1 + (n - 1) q σ^n-2 - (n - 2) r σ^n-3 + &c. / n σ^n-1 - (n - 1) p σ^n-2 + (n - 2) q σ^n-3 - &c.}=δ,&c.</math>
in infinitum, ubi <math>p - α = π, p - α - β = ξ, p - α - β - γ = σ, &c.</math>
tum quantitas ultimo re$ultans, i.e. ad infinitam di$tantiam po$ita,
$i modo reducatur in terminos $ecundum dimen$iones quantitatum
<math>p, q, r, s, &c.</math> progredientes, evadet <math>p - p^-1 q + p^-2 r - p^-3 s \\ - q^2 # + &c.</math>
$eries prius tradita.</p>
<p>Ex principiis in medit. algebr. ulterius promotis invenire liceat
radicem, quæ multo minor $it quam maxima radix; at multo major
quam quæcunque alia; & $ic deinceps; $ed omnia hæc e principiis
$ub$equentibus per $e per$picuis re$olutionem recipiant.</p>
<h>PROB. IX.</h>
<p it>1. Sit œquatio <math>x^n - p x^n-1 + q x^n-2 - r x^n-3 ... ± g x^n-m+1 ∓ h x^n-m 
± i x^n-m-1 ∓ k x^n-m-2 ± &c. = 0</math>, cujus radix <math>α</math> $it multo major quam
quœcunque alia (<math>β, γ, δ, &c.</math>) in datâ œquatione contenta, invenire radi-
cem α.</p>
<p>Per vulgarem algebram innote$cunt <math>p = α + (β + γ + δ + &c.),
q = α(β + γ + δ + &c.) + (β γ + β δ + γ δ + &c.), r = α(β γ +
β δ + γ δ + &c.) + (β γ δ + β γ ε + δ β ε + &c.), s = α(β γ δ + &c.)
+ (β γ δ ε + &c.) &c.</math> unde erit <math>α = p - (β + γ + δ + &c.)</math>, $ed <math>β +
γ + δ + &c. = {q / p} + {(β γ + β δ + γ δ + &c.) + (β^2 + γ^2 + δ^2 + &c.) / p}</math>,
at <math>(β^2 + γ^2 + δ^2 + &c.) + 2 (β γ + β δ + γ δ + &c.) =<pb 374>[0426]<rh>DE INFINITIS</rh>
{q<_>2</_> + 2α(S<_>3</_> + 2S<_>2</_>S<_>1</_> + 3S<_>1</_>S<_>1</_>S<_>1</_>) + S<_>4</_> + 4S<_>3</_>S<_>1</_> + 5S<_>2</_>S<_>2</_> + 10S<_>2</_>S<_>1</_>S<_>1</_> + 18S<_>1</_>S<_>1</_>S<_>1</_> / p<_>2</_>}</math>,
(ubi de$igno <math>S^3 = β^3 + γ^3 + δ^3 + &c. S^4 = β^4 + γ^4 + δ^4 + &c.
S^2 S^1 = β^2 γ + β^2 δ + γ^2 δ + &c. S^1 S^1 S^1 = β γ δ + β γ ε + β δ ε + &c.
S^3 S^1 = β^3 γ + β^3 δ + γ^3 β γ^3 δ + &c. S^2 S^2 = β^2 γ^2 + β^2 δ^2 + γ^2 δ^2
+ &c. S^2 S^1 S^1 = β^2 γδ + γ^2 βδ + δ^2 βγ + &c.</math> quæ fere eadem e$t
notatio, quâ u$us fui in mi$cell. analyt. anno 1762 edit.) & <math>β γ +
β δ + γ δ + &c. = {r + S^2 S^1 + 2S^1 S^1 S^1 / p}</math>; & $ic deinceps; unde con-
$tat <math>α = p - {q / p} - {q^2 / p^3} + {r / p^2} - &c.</math> eædem $eriei, quæ prius inventa
fuit.</p>
<p>2. Sit <math>δ</math> radix prædictæ æquationis, quæ multo minor $it quam <math>m</math>
radices datæ æquationis, multo vero major quam reliquæ <math>n - m - 1</math>;
tum ex principiis hic traditis deduci pote$t <math>δ = {i / b} - ({k / i} - {g i^2 / b^3}) +
&c.</math> termini, in quibus $olummodo continentur coefficientes termino-
rum (<math>x^n-1, x^n-2, x^n-3, .... x^n-m-1</math>) ob$ervant legem in ca$. 1. prob.
præced. traditam, cæterique legem haud multum diver$am habent,
mutatis indicibus de affirmativis in negativos, & mutato denomina-
tore de <math>b</math> & ejus pote$tatibus in <math>i</math> & ejus pote$tates.</p>
<p>Ex ii$dem principiis etiam detegi po$$unt approximationes ad duas
vel plures radices datæ æquationis, quæ fere $unt inter $e æquales;
etiamque eadem applicari po$$unt principia ad æquationes duas in-
cognitas quantitates habentes, ita ut una exprimatur in terminis al-
terius; etiamque ad qua$cunque æquationes, quæ reduci po$$unt ad
infinitas algebraicas æquationes.</p>
<p>Exhinc etiam con$tabunt leges, quas ob$ervant infinitæ $eries radi-
ces æquationum algebraicarum exprimentes.</p>

<pb 375>[0427]<rh>SERIEBUS.</rh>
<h>PROB. X.</h>
<h>_Datâ algebraicâ æquatione_ <math>x^n - p x^n-1 + q x^n-2 - r x^n-3 + &c. = 0</math>,
_cujus radix e$t_ <math>x</math>, _invenire valorem quantitatis_ <math>x</math>.</h>
<p>1<_>mo</_>. Sint omnes radices <math>α, β, γ, δ, ε, &c.</math> datæ æquationis po$$ibiles;
$int vero <math>α</math> multo major quam <math>β, β</math> quam <math>γ, γ</math> quam <math>δ, &c.</math> & inve-
niatur per algebr. meditat. $umma <math>α^2m + β^2m + γ^2m + δ^2m + &c. =
A^2m</math>; & <math>α</math> erit ad <math>A</math> in æquali vel majore ratione quam habet <math>1:^2m √(n)</math>.
Facile con$tat ex $upponendo omnes radices inter $e e$$e æquales.</p>
<p>2<_>do</_>. Inveniatur $umma rectangulorum <math>α^2m β^2m + α^2m γ^2m + β^2m γ^2m +
α^2m δ^2m + β^2m δ^2m + γ^2m δ^2m + α^2m ε^2m + &c. = A′^2m</math> per prædictas medi-
tationes; & <math>αβ</math> erit ad <math>A′</math> in æquali vel majore ratione quam <math>1:
^2m √(n · {n - 1 / 2})</math>.</p>
<p>3<_>tio</_>. Ex quam plurimis directis functionibus $ingularum radicum
inveniri pote$t maxima radix (<math>α</math>); & ex iis & quam plurimis directis
functionibus productorum $ub quibu$que duabus radicibus in dire-
ctam functionem $ingularum radicum erui pote$t radix <math>β</math>, quæ major
e$t quam omnes radices præter <math>α</math>; & $ic progredi licet ad invenienda
contenta <math>α β γ, α β γ δ, &c.</math> etiamque qua$cunque algebraicas earundem
radicum functiones.</p>
<p>Cor. 1. Ii$dem po$itis, <math>^2n √(α^2n + β^2n + γ^2n + δ^2n + &c.)</math> nunquam
differt a maxima radice <math>α</math> per quintam ejus partem; ni $it <math>n = 3</math>, in
quo ca$u vix differt per majorem partem: & <math>^4n √(α^4n + β^4n + γ^4n + &c.)</math>
nunquam differt a maximâ radice <math>α</math> per decuplam ejus partem, &c.
hic haud $upponitur <math>n = 2</math>.</p>
<p>Cor. 2. Limites, inter quos con$i$tit vera radix, inveniri po$$unt
$upponendo 1<_>mo</_> omnes vel qua$dam radices e$$e æquales, & 2<_>do</_> om-
nes præter quemdam numerum radicum evane$cere.</p>
<p>Cor. 3. Sit æquatio <math>x^n - p x^n-1 + q x^n-2 - &c. = 0</math>, cujus omnes
radices <math>α, β, γ, δ, &c.</math> $int po$$ibiles, quarum <math>α</math> major $it quam <math>β, β</math>

<pb 376>[0428]<rh>DE INFINITIS</rh>
quam <math>γ, γ</math> quam <math>δ, &c.</math>; invenire quantitatem, quæ habet ad radicem
<math>α</math> minorem quam datam rationem, viz. <math>r:1</math>, ubi <math>r</math> major e$t quam 1:
inveniantur logarithmi quantitatum <math>n & r</math>, qui dicantur <math>ν & ξ</math>, & par
numerus major quam <math>{ν / ξ}</math> dicatur <math>σ</math>; tum erit <math>(α^σ + β^σ + γ^σ + δ^σ + &c.)^{1 / σ}</math>
quantitas quæ$ita. Hæc principia etiam applicari po$$unt ad inve-
niendas approximationes intra datos limites con$titutas ad radices
<math>β, γ, &c.</math>, vel ad minimam radicem.</p>
<p>Cor. 4. Hinc e prob. primo meditat. algebr. con$equitur lex,
quam ob$ervat rever$io $eriei <math>x = a y - b y^2 + c y^3 - d y^4 ... y^m</math>, i. e.
$eries exinde deducta <math>y = A x + B x^2 + C x^3 + D x^4 + &c.</math> Inve-
niatur enim æquatio, cujus radices $unt reciprocæ datæ æquationis
radicum, quæ erit <math>v^m - {a / x} v^m-1 + {b / x} v^m-2 - {c / x} v^m-3 + {d / x} v^m-4 - &c. = 0</math>
ubi <math>v = {1 / y}</math>; $int <math>α, β, γ, δ, ... π</math> radices huju$ce æquationis, ubi <math>α</math> $it
maxima radix & <math>π</math> minima, maximus igitur erit valor radicis <math>y ={1 / π}</math>; tum <math>^2n √(α^2n + β^2n + γ^2n + &c.)</math> erit maxima radix <math>α</math>, $i modo <math>n</math> $it
infinitus affirmativus numerus, etiamque <math>^2n √(α^-1n + β^-2n + γ^-2n + &c.)</math>
erit minima radix <math>π</math>, & con$equenter maxima radix quantitatis <math>y = {1 / π}</math>;
$ed quoniam radices, ut con$tat ex ob$ervatione ad problema prædi-
ctum annexâ, eandem pror$us ob$ervant legem ac earum pote$tates;
$eries, quæ exprimit radicem <math>π = ^2n √(α^-2n + β^-2n + γ^-2n + &c.)</math> ean-
dem pror$us ob$ervat legem ac $eries pro $umma (<math>α^-1 + β^-1 + γ^-1 +
&c.</math>) cum radix $it <math>{-2n / 2n} = -1</math>, & con$equenter $eries erit per
prob. prædict. <math>p^-1 + p^-1-2 q - p^-1-3 r + p^-1-4 s \\ + 2 p^-1-4 q^2 - &c.</math> $i modo
æquatio $it <math>z^m - p z^m-1 + q z^m-2 - &c. = 0</math>; hoc autem in loco erunt

<pb 377>[0429]<rh>SERIEBUS.</rh>
re$pective <math>p = {a / x}, q = {b / x}, r = {c / x}, &c.</math> in ca$u propo$ito requiratur, ut
termini progrediantur $ecundum dimen$iones quantitatis <math>x</math>, unde in
terminis <math>p^-1-μ q^α r^β s^γ × &c.</math> infra $e po$itis, qui in eandem pote$tatem
quantitatis <math>x</math> ducuntur, quantitas <math>- 1 - μ + α + β + γ + &c. =
- 1 - α - 2β - 3γ - &c.</math> eandem conficiet $ummam, quæ erit
$ucce$$ive <math>-1, -2, -3, -4, -5, &c.</math> ubi <math>μ = 2 α + 3 β + 4 γ
+ &c.</math> & con$equenter $eries per prædictum problema invenietur
<math>y = {x / a} + {b x^2 / a^3} + {2 b^2 - c a / a^5} x^3 + &c.</math> unde e prædicto problemate con-
$tat lex, quam ob$ervat hæc $eries; exinde enim con$tat coefficientem
termini <math>p^-1-μ q^α r^β s^γ, &c.</math> e$$e <math>± {μ × (μ - 1)(μ - 2)(μ - 3) ... (μ - α - β - γ - &c. + 2) / 1 · 2 · 3 .. α × 1 · 2 · 3 .. β × 1 · 2 · 3 .. γ × &c.}</math>:
$ignum affixum erit + vel - prout <math>2 α +
3 β + 4 γ + &c.</math> $it par vel impar numerus; $i <math>α + β + γ + &c. = 1</math>,
i. e. una litera <math>α</math> vel <math>β</math> vel <math>γ, &c.</math> $it 1, cæteræ vero <math>0</math>; tum coefficiens
prædicta erit 1.</p>
<p>Et $ic, $i modo deficiant quidam datæ $eriei termini vel <math>a y</math> vel <math>b y^2,
&c.</math> facile deduci pote$t lex, quam ob$ervat $eries valorem radicis <math>π</math>
exprimens.</p>
<p>3. Summa <math>α^4m + β^4m + γ^4m + &c.</math> per $ummam <math>α^2m + β^2m + γ^2m +
&c.</math> divi$a erit approximatio ad <math>α^2m</math>. Et $ic de $ummis quantitatum
rationalium & irrationalium; reducantur rationales vel irrationa-
les quantitates in $eries, quarum primi termini $int maximi, & ex-
inde $umma e $ingulis valoribus rationalium vel irrationalium quan-
titatum prope æqualis erit prædictis terminis; &c.</p>
<p>4. Habeat prædicta æquatio radices impo$$ibiles, & e præcedente
methodo haud generaliter inve$tigari pote$t maxima radix: pendet
enim ex hoc, utrum impo$$ibilis radix major $it quam reliquæ radi-
ces, necne; i. e. impo$$ibilis vel po$$ibilis pars radicis impo$$ibilis
multo major $it quam reliquæ radices, necne; quâ inventâ, i. e. datâ

<pb 378>[0430]<rh>DE INFINITIS</rh>
approximatione ad impo$$ibilem vel po$$ibilem partem radicis deduci
pote$t approximatio ad alteram partem.</p>
<p>In quibu$cunque datis æquationibus $uperiorum dimen$ionum, qua-
rum radices haud dantur, maxime probabile e$t multo majorem e$$e
numerum impo$$ibilium quam po$$ibilium radicum, & con$equenter
ca$us non detegendi po$$ibilem radicem per hanc regulam multo ma-
jorem e$$e quam ca$us detegendi. Transformetur data æquatio in al-
teram, cujus radices $int reciprocæ radicum datâ æquatione conten-
tarum, & minima radix datæ æquationis vel negativa vel affirmativa
præbet maximam re$ultantis æquationis radicem; de eâ con$imiliter
ratiocinari liceat.</p>
<p>5. Sæpe approximationes ad valores radicum inve$tigari po$$unt e
comparandis diver$is æquationis terminis inter $e, cum quidam ter-
mini æquationis multo majores e$$e $upponantur quam reliqui.</p>
<p>Et con$imili methodo inve$tigari po$$unt approximationes ad qua$-
cunque algebraicas functiones datæ æquationis radicum: per meditat.
algebr. enim inveniri pote$t aggregatum e $ingulis valoribus datæ
functionis, ex $ingulis eorum quadratis, cubis, quadratoquadratis,
quadratis ex uno quocunque valore in quadrata e $ingulis reliquis
ductis, &c. unde per hoc problema inveniri pote$t approximatio ad
valorem vel diver$os valores datæ functionis; vel quod idem e$t, $em-
per deduci pote$t approximatio prædicta ex transformatione datæ
æquationis in alteram, cujus radices $unt data functio datæ æquatio-
nis radicum.</p>
<h>PROB. XI.</h>
<h>_Conce$sâ methodo detegendi radices_ <math>^2n √(-1)</math>, _invenire radices_ <math>n</math> _pote$tatis_
_impo$$ibilis quantitatis_ <math>a + b √(-1)</math>.</h>
<p>1. Sit <math>a</math> multo major quam <math>b</math>, tum ex vulgari extractione radicis <math>n</math> po-
te$tatis $emper inveniri pote$t radix quæ$ita; $i vero <math>a</math> $it negativa quan-
titas, ducatur data quantitas <math>- a + b √(-1)</math> in <math>-1</math>, & re$ultat (<math>a - b
√(-1))</math>; per prædictam & conce$$am methodos inveniantur radices

<pb 379>[0431]<rh>SERIEBUS.</rh>
<math>^n √(a - b √(- 1)) & ^n √(- 1)</math>, & e dividendo has radices per $e$e
con$equitur radix quæ$ita.</p>
<p>2. Sit <math>b</math> multo major quam <math>a</math>, ducatur data quantitas in <math>-
√(-1)</math>, & re$ultat <math>(b - a √(- 1))</math>; $ed radices <math>^n √(b - a √(- 1))
& ^n √(- √(-1)) = ^2n √(-1)</math> e prædictis methodis erui po$$unt,
ergo radix quæ$ita.</p>
<p>3. Si vero <math>a & b</math> $int prope inter $e æquales, tum for$an ab extra-
ctione <math>2 n</math> radicis ejus quadrati <math>a^2 - b^2 + 2 a b√(-1)</math> in <math>-√(-1)
= 2ab + (b^2 - a^2) √(-1)</math>; vel ab extractione radicum ex aliis
pote$tatibus erui pote$t radix quæ$ita; aliter vero in hoc ca$u $inga-
tur <math>α × (1 ± √(-1))</math> radix quæ$ita, & con$equenter erit <math>(1 - n .{n - 1 / 2} + n · {n - 1 / 2} · {n - 2 / 3} · {n - 3 / 4} - n · {n - 1 / 2} · {n - 2 / 3} · {n - 3 / 4} · {n - 4 / 5}. {n - 5 / 6} + &c.) α^n = M α^n = a′</math>, & erit <math>α = ({a′ / M})^{1 / n} = β</math> prope; $cri-
batur in æquatione <math>x^n - a - b √(-1) = 0</math> pro radice quæ$itâ <math>β +
π + (β′ + δ) √(-1)</math>, ubi <math>β′</math> quantitas a$$umpta paululum differt a
quantitate <math>β</math>; & ex maximis re$ultantis æquationis terminis, in quibus
invenitur <math>√(-1)</math>, necne, $eor$im nihilo æqualibus e$$e $uppo$itis;
re$ultant duæ æquationes, e quibus deduci po$$unt <math>π & ξ</math>; & $ic de-
inceps.</p>
<p>Hoc in loco vi$um e$t $ub$equentia $ubjicere hanc rem de impo$$i-
bilibus radicibus leviter per$tringentia, viz. 1. erit <math>(a + √(-a^2))^4n
+ (a - √(-a^2))^4n = 2 × (-4)^n a^4n, (a + √(-a^2))^4n+2 + (a -
√(-a^2))^4n+2 = 0, & (a + √(-a^2))^2n+1 + (a - √(-a^2))^2n+1 =
± 2^n+1 a^2n+1</math>, ubi litera <math>n</math> integrum denotat numerum; & $ignum
quantitati <math>2^n+1 a^2n+1</math> affixum e$t affirmativum necne, prout <math>2n + 1</math>
vel æquat <math>8 m + 1</math>, vel <math>8 m + 7</math>; necne; ubi <math>m</math> e$t integer numerus.</p>
<p>2. Quantitas <math>n · {n - 1 / 2} ... {n - r / r + 1} - n · n · {n - 1 / 2} · {n - 2 / 3} ... {n - r + 1 / r}+ n · {n - 1 / 2} · n · {n - 1 / 2} · {n - r + 2 / r - 1} - .... ± n · {n - 1 / 2} · {n - 2 / 3} .. {n - r / r + 1}

<pb 380>[0432]<rh>DE INFINITIS</rh>
æquat ± n · {n - 1 / 2} · {n - 2 / 3} ... {n - {r - 1 / 2}/{r + 1 / 2}}</math>, vel o; <math>erit = 0</math>, cum <math>r</math> $it
par numerus: $ignum affirmativum erit affixum, cum <math>r = 4s + 3</math>; $in
aliter negativum; ubi <math>s</math> e$t integer numerus.</p>
<h>PROB. XII.</h>
<p it>Datâ algebraicâ æquatione relationem inter <math>x & y</math> exprimente; invenire
legem, quam ob$ervat $eries de$ignans quantitatem <math>y</math> in $implicibus terminis
quantitatis <math>x</math>; $i modo dentur leges, quam ob$ervant quantitates buju$ce ge-
neris ad qua$cunque pote$tates elevatæ vel in $e$e ductæ.</p>
<p>Ita reducatur data algebraica æquatio relationem inter <math>x & y</math> ex-
primens, ut <math>π</math> functio quantitatis <math>x</math> multo magis appropinquet
ad unam quam ad ullam aliam radicem vel valorem quantitatis <math>y</math>;
$cribatur <math>v - π</math> pro <math>y</math> in datâ æquatione & $it æquatio re$ultans <math>α -
β v + γ v^2 - d v^3 + &c. = 0</math>; deinde fingatur <math>w = {1 / v}</math>, & erit <math>α w^n -
β w^n-1 + γ w^n-2 - δ w^n-3 + &c. = 0</math>, tum erit <math>{β / α}</math> approximatio ad ra-
dicem <math>w</math>, nunc per prob. 1. meditationum algebraicarum infinita $e-
ries, quæ erit $umma pote$tatis vel potius radix <math>({1 / n})</math> in$initæ pote-
$tatis <math>(n)</math> e $ingulis radicibus, i.e. pro <math>m</math> $cribatur 1; & pro <math>{β / α}, {γ / α}, {δ / α}</math>,
&c. $cribantur re$pective <math>p, q, r, s, &c.</math> erit <math>p - p^1-2 q + p^1-3 γ
- p^1-4 s + &c.</math>; coefficiens termini <math>p^1-μ q^α γ^β s^γ, &c. erit ±{(μ - 2) × (μ - 3) × (μ - 4) .. (μ - α - β - γ - &c.) / 1 · 2 · 3 ... α × 1 · 2 .. β × 1 · 2 .. γ × &c.}</math>, $ignum affi-
xum erit + vel -, prout <math>μ + α + β + γ + &c.</math> $it par vel impar
numerus; & re$ultantis infinitæ $eriei collocentur termini $ecundum
dimen$iones quantitatis <math>x</math>, vel potius $ecundum dimen$iones per-
parvæ quantitatis, & erit $eries quæ exprimit unam radicem vel valo-
rem quantitatis <math>y</math>.</p>

<pb 381>[0433]<rh>SERIEBUS.</rh>
<p>Cor. 1. Ex primo, binis, ternis vel $altem ex <math>n</math>, qui $it numerus di-
men$ionum datæ æquationis, maximis terminis, pro approximatione
ad valorem quantitatis <math>y</math> inventis $emper acquiri pote$t quantitas <math>π</math>
prædicta.</p>
<p>Cor. 2. Si requiratur radix prædicta in terminis $ecundum legem
<math>ax^r × (x ± x<x>.</x>)^r + b x^r+1 × (x ± x<x>.</x>)^r+1 + &c. + a′x^r+s × (x ± x<x>.</x>)^r+s +
b′x^r+s+1 × (x ± x<x>.</x>)^r+s+1 + &c.</math> progredientibus; reducatur $eries, quæ
exprimit radicem in terminis eju$dem formulæ, & exinde facile de-
duci pote$t per hoc problema $eries quæ$ita; & $ic inveniri pote$t ra-
dix prædicta in terminis progredientibus $ecundum alias leges in hoc
libro traditas; aliter e terminis maximis continuo deduci po$$unt ap-
proximationes quæ$itæ.</p>
<h>THEOR. XV.</h>
<p>Sit æquatio <math>1 - (α + β + γ + δ + &c.){1 / x} + (αβ + αγ + βγ +
&c.){1 / x^2} - (αβγ + αβδ + αγδ + βγδ + &c.){1 / x^3} + (αβγδ + αβγε
+ &c.){1 / x^4} - &c. = A</math>, i. e. $it <math>A = 0</math> æquatio, cujus radices $int <math>α, β,
γ, δ, ε, &c.</math>; tum erit <math>A^n = 1 - n (α + β + γ + δ + &c.){1 / x} + (n .{n - 1 / 2} (α^2 β^2 + γ^2 + δ^2 + &c.) + n × n (α β + α γ + β γ + α δ
+ &c.)){1 / x^2} - ((n · {n - 1 / 2} · {n - 2 / 3} (α^3 + β^3 + γ^3 + δ^3 + &c.) + n · n ·
+ α^2 γ + β^2 α + β^2 γ + &c.) + n · n · n (α β γ + α β δ +{n - 1 / 2} (α^2 β + α^2 γ + β^2 α + β^2 γ + &c.) + n · n · n (α β γ + α β δ +
αγδ + βγδ + &c.)){1 / x^3} + &c.</math>: in genere $it <math>α^m β^r γ^s δ^t, &c.</math> quodcun-
que productum in coefficiente ad terminum <math>(x^-b)</math> huju$e $eriei annexâ,

<pb 382>[0434]<rh>DE INFINITIS</rh>
tum erit <math>m + r + s + t + &c. = b</math>; cuncta etiam producta, in qui-
bus <math>m + r + s + t + &c. = b</math> $emper prædicto termino <math>x^-n</math> $unt ad-
jungenda; coefficiens prædicti termini <math>α^m β^r γ^s δ^t, &c.</math> erit <math>n · {n - 1 / 2} .{n - 2 / 3} ... {n - m + 1 / m} × n · {n - 1 / 2} · {n - 2 / 3} ... {n - r + 1 / r} × n · {n - 1 / 2} .{n - s + 1 / s} × n · {n - 1 / 2} · {n - 2 / 3} · {n - t + 1 / t} × &c.</math></p>
<p>Cor. 1. Sit <math>n = 1</math>, & omnes prædictæ coefficientes erunt 1, i. e. di-
vidatur unitas per <math>A</math>, & coefficientes e $ingulis productis <math>(α^m β^r γ^s δ^t,
&c.)</math> in terminis re$ultantibus erunt 1.</p>
<p>Cor. 2. Sit <math>α</math> maxima radix, <math>β</math> multo major quam quæcunque alia,
& $ic de reliquis (<math>γ, δ, &c.</math>), tum ita reducantur hæ coefficientes, ut
exterminentur omnes termini, in quibus invenitur <math>α</math> $olummodo;
& re$ultent termini, e quibus deduci pote$t propinquus valor quan-
titatis <math>β</math>; deinde exterminentur omnes termini in quibus $olummodo
inveniuntur <math>α & β</math>, vel termini maximi in quibus inveniuntur <math>α & β</math>,
& erui pote$t propinquus valor quantitatis <math>γ</math>; & $ic deinceps: e com-
parandis terminis re$ultantibus quam plurimarum functionum datæ
æquationis erui po$$unt quam preximi valores radicum datæ æqua-
tionis.</p>
<p>Cor. 3. Eadem principia etiam applicari po$$unt ad æquationes ir-
rationales quantitates involventes.</p>
<h>THEOR. XVI.</h>
<p>1. Sit æquatio <math>x^n - p x^n-1 + q x^n-2 - r x^n-3 + &c. = 0</math>, $tatuatur
<math>{1 / x^n - p x^n-1 + &c.} = x^-n + p x^-n-1 ... a x^m + b x^m-1 + c x^m-2 + d x^m-3
+ e x^m-4 + f x^m-s + &c.</math> tum, $i modo omnes radices (<math>α, β, γ, δ, &c.</math>) datæ
æquationis $int po$$ibiles, & <math>m</math> $it permagna quantitas, erit <math>{b / a} = (α)</math>

<pb 383>[0435]<rh>SERIEBUS.</rh>
maximæ radici datæ æquationis quam proxime: maximam radicem
volo vel affirmativam vel negativam.</p>
<p>2. Et <math>{b d - c^2 / a d - b c}</math> erit quam proxime (<math>β</math>) maxima radix ex omnibus
reliquis præter prædictam <math>α</math>.</p>
<p>3. Si vero numerator haud $it unitas, $ed quantitas <math>b x^μ + k x^μ-1 +
l x^μ-2 + &c.</math> i.e. $tatuatur <math>{b x^μ + k x^μ-1 + &c. / x^n - p x^n-1 + &c.} = b x^μ-n + (k + b p)
x^μ-n-1 ... a x^m + b x^m-1 + c x^m-2 + &c.</math> & ex eâdem pror$us methodo
inveniri po$$unt proximi valores radicum <math>α & β, &c.</math></p>
<p>4. Si vero maxima affirmativa & negativa radix $int quam proxime
inter $e æquales; in quo ca$u $it <math>n - m</math> par numerus, & erit <math>√({c / a})</math>
vel <math>√({d / b})</math> maxima radix (<math>α</math>) datæ æquationis prope; & <math>({d / c})</math> ma-
xima ex omnibus reliquis prope.</p>
<p>5. Sit <math>(x^n - p x^n-1 + q x^n-2 - &c.)^-1 = x^-n + p x^-n-1 ... + a x^-n-m
+ b x^-n-m-1 ... + d x^-n-r + e x^-n-r-1 ... + g x^-n-s + b x^-n-s-1 .... +
k x^-n-t + l x^-n-t-1 .... + &c.</math>; tum erit <math>{d / a}^{1 / r-m} = α</math> maxima radix præ-
dicta datæ æquationis; $i modo <math>m & r</math> $int permagnæ quantitates, &
omnes radices $int po$$ibiles.</p>
<p>Et ex ii$dem principiis ulterius promotis quam plurimæ aliæ regulæ
pro his inveniendis, etiamque con$imiles regulæ pro reliquis radici-
bus inveniendis deduci po$$unt.</p>
<p>6. Sit fractio <math>{A / B × C}</math> ad minimos terminos reducta, ubi <math>B</math> $it ratio-
nalis $unctio quantitatis <math>x, C</math> vero irrationalis nullum habens ratio-
nalem factorem; reducatur fractio <math>{A / BC}</math> in $implices terminos $ecun-
dum dimen$iones quantitatis <math>x</math> progredientes, viz. <math>α x^π + β x^π+β +

<pb 384>[0436]<rh>DE INFINITIS</rh>
γ x^π+2ρ .... a x^τ + b x^τ+ρ + c x^τ+2ρ + &c.</math>, & erit <math>^ρ √ ({a / b})</math> vel <math>^ρ √ ({b / a})</math>
maxima radix æquationis <math>B × C = 0</math> prope: $ecunda radix, quæ mi-
nor erit quam prædicta, major vero quam $ingulæ reliquæ, erit
<math>^ρ √ ({b d - c^2 / a d - b c}), &c.</math> prope; vel per $ingulas regulas prius traditas de-
duci pote$t.</p>
<p>Et $ic de reliquis radicibus detegendis.</p>
<p>Hæc principia ulterius etiam promovere liceat.</p>
<p>Si vero impo$$ibiles radices <math>π + √ (- ξ^2)</math> in datâ æquatione con-
tineantur, hæ regulæ $emper detegent radicem quæ$itam <math>α</math> vel <math>β</math>, cum
radix quæ$ita major $it quam <math>π + ξ</math> vel <math>π - ξ</math>.</p>
<p>Ex hâc methodo $æpe con$tabit approximatio ad maximam impo$-
$ibilem radicem; & ex his regulis plerumque deduci po$$unt approxi-
mationes vel ad maximas po$$ibiles vel impo$$ibiles radices.</p>
<h>PROB. XIII.</h>
<h it>Datâ algebraicâ æquatione <math>x^n - p x^n-1 + q x^n-2 - r x^n-3 + &c. = 0</math>,
invenire ejus radicis <math>(α)</math>) valorem.</h>
<p>A$$umatur <math>a</math> quantitas ab radice <math>α</math> datæ æquationis parum di$cre-
pans; pro radice <math>x</math> datæ æquationis $cribatur <math>a + b</math>, & re$ultat æqua-
tio <math>x^n - p x^n-1 + q x^n-2 - &c. = 0 = (a^n - p a^n-1 + q a^n-2 - &c.) +
(n a^n-1 - (n - 1) p a^n-2 + (n - 2) q a^a-3 - &c.) b + (n. {n - 1 / 2} a^n-2 -
(n - 1) × {n - 2 / 2} p a^n-3 + (n - 2) × {n - 3 / 2} q a^n-4 - &c.) b^2 + &c.</math> fin-
gatur quantitas <math>b</math> multo minor quam quantitas <math>a</math>, & neglectis quan-
titatibus in quibus inveniuntur ullæ dimen$iones quantitatis <math>b</math>, ($im-
plice exceptâ) ob earum parvitatem re$ultat $implex æquatio <math>a^n - p a^n-1
+ q a^n-2 - &c. + (n a^n-1 - (n - 1) p a^n-2 + (n - 2) q a^n-3 - &c.) b = 0</math>,
unde <math>b</math> erit prope <math>= - {a^n - p a^n-1 + q a^n-2 - &c. / n a^n-1 - (n - 1) q a^n-2 + (n - 2) q a^n-3 - &c.

<pb 385>[0437]<rh>SERIEBUS.</rh>
= π}</math>; in re$ultante æquatione pro quantitate <math>b</math> $cribatur <math>π + c</math>, &
per eandem methodum repetitam inveniatur valor quantitatis <math>c</math>
prope; & $ic deinceps.</p>
<p>2. Eædem omnino erunt approximationes inventæ, $i modo
in datâ æquatione 1<_>mo</_> ut in præcedente ca$u $cribatur <math>a + b</math> pro <math>x</math>,
deinde per prædictam methodum inveniatur <math>π</math> valor quantitatis <math>b</math>
prope; tum in datâ æquatione pro <math>x</math> $cribatur <math>a + π + c</math>, & per
methodum prius traditam inveniatur <math>ρ</math> valor quantitatis <math>c</math> prope;
deinde $cribatur in datâ æquatione pro <math>x</math> quantitas <math>a + π + ρ + d</math>
& per prædictam methodum inveniatur <math>σ</math> valor quantitatis
<math>d</math> prope; & $ic deinceps. Hinc valor incognitæ quantitatis <math>x</math> per
hanc methodum inventus erit <math>a - {a^n - p a^n-1 + q a^n-2 - &c. / n a^n-1 - (n - 1) p a^n-2 + &c.} -
{(a + π)^n - p (a + π)^n-1 + q (a + π)^n-2 - &c. / n (a + π)^n-1 - (n - 1) p (a + π)^n-2 + (n - 2) q (a + π)^n-3 - &c.} -
{(a + π + ρ)^n - p (a + π + ρ)^n-1 + q (a + π + ρ)^n-2 - &c. / n (a + π + ρ)^n-1 - (n - 1) p (a + π + ρ)^n-2 + (n - 2) q (a + π + ρ)^n-3 - &c.}
+ &c.</math> ubi litera <math>a</math> denotat proximum valorem quantitatis <math>x</math> datum,
literæ vero <math>π, ρ, σ, &c.</math> re$pectivos valores præcedentium fractio-
num.</p>
<p>Valores quantitatum <math>π, ρ, σ, τ, &c.</math> per præcedentem methodum,
i. e. per continuas $ub$titutiones in re$ultantibus haud in datis æqua-
tionibus, inventi, erunt iidem ac ii ex hâc methodo deducti.</p>
<p>3. Datâ æquatione <math>x^n - p x^n-1 + q x^n-2 - r x^n-3 + &c. = 0</math>, $up-
ponatur <math>x = a + e</math>, ubi <math>e</math> parvam habet rationem ad <math>a</math>; & $cribatur
in datâ æquatione pro <math>x</math> ejus valor <math>a + e</math>, & re$ultat æquatio <math>a^n -
p a^n-1 + q a^n-2 - &c. + (n a^n-1 - (n - 1) p a^n-2 + (n - 2) q a^n-3 - &c.)
e + (n · {n - 1 / 2} a^n-2 - (n - 1) × {n - 2 / 2} p a^n-3 + &c.) e^2 + &c. = P +
Q e + R e^2 + S e^3 + &c. = 0</math>, unde $it <math>P + Q e</math> prope <math>= 0</math>, & exinde
<math>e</math> prope <math>= - {P / Q}</math>, $ed erit <math>e = - {P / Q + R e + S e^2 + &c.}</math> prope <math>= -

<pb 386>[0438]<rh>DE INFINITIS</rh>
{P / Q - {R P / Q} + {S P^2 / Q^2} - &c.} = - {P Q^n-1 / Q^n - Q^n-2 R P + Q^n-3 S P^2 - &c.}</math>
prope <math>= - {P Q / Q^2 - R P}</math>.</p>
<p>4. Sit re$ultans æquatio eadem ac in præcedente exemplo <math>P + Q e
+ R e^2 + S e^3 + T e^4 + &c. = 0</math>, abjiciantur omnes termini ob par-
vitatem eorum, in quibus inveniuntur plures quam duæ dimen$iones
quantitatis <math>e</math>, & re$ultat æquatio <math>P + Q e + R e^2 = 0</math>, unde <math>e =
√ ({Q^2 - 4 R P / 4 R^2}) - {Q / 2 R}</math> prope.</p>
<p>Et $imiliter abjiciantur omnes præter terminos, in quibus inveni-
untur plures quam tres vel quatuor dimen$iones, &c. ex æquatione re-
$ultante inveniatur valor quantitatis <math>e</math>, & invenitur plerumque pro-
pior approximatio ad valorem quantitatis <math>e</math>.</p>
<p>Cor.. Datâ algebraicâ æquatione <math>x^n - p x^n-1 + q x^n-2 - &c. = 0</math>,
cujus radices $int re$pective <math>α, β, γ, δ, &c.</math> $it <math>α</math> minor quam <math>β,
β</math> quam <math>γ, &c.</math> $int etiam radices æquationis <math>n x^n-1 - (n - 1) p x^n-2 +
(n - 2) q x^n-3 - &c. = 0</math> re$pective <math>π, ρ, σ, τ, &c.</math> tum cognitum e$t
<math>π, ρ, σ, τ, &c.</math> limites e$$e, inter quos con$i$tunt datæ æquationis radi-
ces <math>β, γ, δ, &c.</math></p>
<p>A$$umantur vel <math>π</math> vel <math>ρ</math> vel <math>σ</math> vel <math>τ, &c.</math> tanquam approximatio ad
datæ æquationis radicem <math>x</math>, $cribatur in datâ æquatione pro <math>x</math> ejus va-
lor a$$umptus <math>π + z</math>, & per methodum prius traditam invenietur ejus
approximatio <math>- {π^n - p π^n-1 + q π^n-2 - &c. / n π^n-1 - (n - 1) p π^n-2 + (n - 2) q π^n-2 - &c.}</math>; $ed
quoniam <math>π</math> e$t radix æquationis <math>n π^n-1 - (n - 1) p π^n-2 + &c. = 0</math>,
erit approximatio inventa quantitas infinita, & con$equenter $eries
ex hâc methodo re$ultans infinite divergit; ergo convergentia $eriei
prædictæ haud pendet e ratione, quam habet quantitas a$$umpta pro
radice <math>α</math> ad radicem ip$am; $ed ex hoc, quod quantitas a$$umpta pro
radice multo propior $it ad radicem quæ$itam, quam ad ullam aliam
datæ æquationis radicem.</p>

<pb 387>[0439]<rh>SERIEBUS.</rh>
<h>THEOR. XVII.</h>
<p>Sit æquatio <math>x^n - p x^n-1 + q x^n-2 - &c. = 0</math>, cujus radices $int
<math>α, β, γ, δ, &c.</math> a$$umatur quantitas <math>a</math> pro radice <math>α</math>, & transformetur
data æquatio in alteram, cujus radices $unt minores vel majores quam
radices datæ æquationis per quantitatem <math>a</math>, i. e. $int <math>α - a, β - a,
γ - a, &c.</math> $int <math>a:α::1:1 + b, a:β::1:1 + k, a:γ::1:1 + l,
a:δ::1:1 + m, &c.</math> & $i <math>a</math> multo propior $it ad radicem <math>α</math> quam ad
<math>β</math> vel <math>γ</math> vel <math>δ, &c.</math> tum <math>b</math> multo minor erit quam <math>k</math> vel <math>l</math> vel <math>m, &c.</math>
& radices æquationis transformatæ erunt re$pective <math>b a, k a, l a, m a,
&c.</math> ergo <math>b × k × l × m × n′ × &c. a^n = ± (a^n - p a^n-1 + q a^n-2 - &c.) = Q a^n</math>
contentum $ub $ingulis transformatæ æquationis radicibus; & <math>(k l m n′
&c. + b k l m &c. + b k m n′ &c. + b l m n′ &c. + &c.)a^n-1 = ± (n a^n-1 -
(n - 1)p a^n-2 + (n - 2)q a^n-3 - &c.) = P a^n-1</math> aggregatum e $ingulis con-
tentis $ub <math>n - 1</math> radicibus transformatæ æquationis; ergo approximatio
per vulgares regulas inventa <math>- {a^n - p a^n-1 + q a^n-2 - &c. / n a^n-1 - (n - 1) p a^n-2 + (n - 2) q a^n-3 - &c.}
= {(a - α) × (a - β) × (a - γ) × (a - δ) × &c. / (a - α) × (a - β) × &c. + (a - β) × (a - γ) × &c. + (a - γ) ×
(a - δ) × &c. + &c.}</math>: ad quantitatem quæ$itam <math>α - a = b a::k l m n′
&c. ({2 / b}):k l m n′ &c. + b k l m &c. + b l m n′ &c. + &c. (P)</math>.</p>
<p>2. Si vero quantitas a$$umpta multo propior $it ad duas radices <math>α &
β</math> datæ æquationis quam ad reliquas, i. e. $i duæ radices transformatæ
æquationis, quæ $it <math>v^n ... T v^4 - S v^3 + R v^2 - Q v + P = 0</math>, $int
multo minores quam reliquæ tum radices æquationis <math>v^2 - {Q / R} v +
{P / R} = 0</math> erunt quam proxime illæ radices quæ$itæ $i vero tres, tum
radices æquationis <math>v^3 - {R / S} v^2 + {Q / S} v - {P / S} = 0</math> erunt quam proxime

<pb 388>[0440]<rh>DE INFINITIS</rh>
radices quæ$itæ; & $ic deinceps. Et facile methodo con$imili, i. e.
pro coefficientibus <math>P, Q, R, S, &c.</math> $cribendo earum valores in termi-
nis radicum <math>α, β, γ, &c.</math> inveniri po$$unt rationes, quas habent radices
inventæ per has regulas ad veras radices quæ$itas.</p>
<p>Ab extractione radicum con$tare po$$unt limites quantitatis a$$u-
mendæ pro primâ approximatione, ita ut $eries convergat; $ed æque
has approximationes ac datæ æquationis radices inve$tigare diffi-
cile e$t.</p>
<p>F<sc>IG</sc>. 1. 3. Aliter: $it æquatio <math>x^n - p x^n-1 + q x^n-2 - &c. = 0</math>, a$$u-
matur <math>x^n - p x^n-1 + q x^n-2 - &c. = y</math>, ubi <math>x</math> $it ab$ci$$a datæ curvæ, <math>y</math>
vero ordinata: curvam vero $ecet ab$ci$$a <math>A p</math> in tot (<math>n</math>) punctis,
(<math>b, k, l, m, n, 0, &c.</math>) quot dimen$iones ab$ci$$æ <math>x</math> inveniuntur in datâ
æquatione, & $int <math>f p, e q, d r, c s, b t, &c.</math> ejus (<math>n</math>) ordinatæ, quæ
re$pective $int maximæ; tum <math>A b, A k, A l, A m, A n, A o, &c.</math> erunt
re$pectivæ radices datæ æquationis <math>x^n - p x^n-1 + q x^n-2 - &c. = 0;
& A t, A s, A r, A q, A p, &c.</math> erunt re$pectivæ radices æquationis
<math>n x^n-1 - (n - 1) p x^n-2 + (n - 2) q x^n-3 - &c. = 0</math>; $i vero requira-
tur radix <math>A m</math> datæ æquationis <math>x^n - p x^n-1 + q x^n-2 - &c. = 0</math>,
a$$umatur quantitas <math>A a = a</math> pro quantitate approximante ad
radicem <math>A m</math>, $cribatur <math>a + π</math> pro radice <math>x</math> in datâ æquatione, &
per regulam prius traditam invenietur approximatio <math>π</math> prope <math>= -
{a^n - p a^n-1 + q a^n-2 - &c. / n a^n-1 - (n - 1) p a^n-2 + (n - 2) q a^n-3 - &c.} = a m × {a b × a k × a l × a n × a o × &c. / n × a r × a s ×
a t × a q × a p × &c.}</math>; unde approximatio inventa erit ad radicem
quæ$itam <math>::± a b × a k × a l × &c.:n × a r × a s × a t × &c.</math></p>
<p>Cor.. Hinc approximatio per hanc methodum inventa erit ad
veram radicem quæ$itam ultimo in ratione <math>m b × m k × m l × m n × m o
× &c.:n × m t × m s × m r × m q × m p × &c. ...</math> i. e. in ratione com-
po$itâ e rationibus <math>m b:m t, m k:m s, m l:m r, &c. & 1:n</math>; $it
vero hæc ratio, quæ erit data <math>A:B, & α</math> radix quæ$ita; erit appro-
ximatio <math>{A α / B}</math>, & differentia inter eam & veram radicem erit <math>{B - A / B} α</math>;

<pb 389>[0441]<rh>SERIEBUS.</rh>
ergo ultimo diver$æ approximationes per hanc methodum inventæ,
videntur e$$e prope <math>{A α / B}, {A / B} × {B - A / B} α, {A / B} × {(B - A)^2 / B^2} α, {A / B} × {(B - A)^3 / B^3} α,
&c.</math> i. e. prope quantitates in geometricâ progre$$ione, quamvis non
revera $unt; nam prædicta ratio <math>A:B</math> ultimo erit ratio æqualitatis.</p>
<p>Sint vero duo $ucce$$ivi termini $eriei re$pective <math>r & s</math>, & erit $um-
ma $eriei geometricæ <math>{r^2 / r - s}</math>; hæc vero methodus haud in ultimâ ap-
proximatione tam celeriter converget, quam præcedens: $i a$$umatur
$eries terminorum, quorum denominatores $ecundum pote$tates
quantitatis <math>r - s</math> progrediuntur, re$ultabit $eries in quibu$dam ca$i-
bus celeriter convergens.</p>
<p>Si vero impo$$ibiles radices in datâ æquatione contineantur, tum
$æpe regula $upra tradita approximationes magis convergentes de-
ducet.</p>
<p>4. Datâ æquatione <math>x^n - p x^n-1 + q x^n-2 - &c. = 0</math>, cujus radices
$int re$pective <math>α, β, γ, δ, &c.</math> i. e. $int <math>α</math> major quam <math>β, β</math> quam <math>γ,
&c.</math> pro <math>x</math> $cribatur <math>a + π</math>, ubi <math>a</math> major e$t quam maxima radix <math>α</math>,
tum invenietur <math>π</math> prope <math>= - {a^n - p a^n-1 + q a^n-2 - &c. / n a^n-1 - (n - 1) p a^n-2 + &c.} = -
{(a - α) × (a - β) × (a - γ) × (a - δ) × &c. / (a - α) × (a - β) × (a - γ) × &c. + (a - β) × (a - γ) × (a - δ) × &c. + &c.}</math>,
quicunque autem $it valor quantitatis <math>a</math>, $emper approximabit valor
ad $ingulas datæ æquationis radices, i. e. <math>α + π</math> propior erit ad
$ingulas datæ æquationis radices quam <math>α</math>; & $ic deinceps; approxi-
matio inventa erit ad radicem quæ$itam vel in ratione <math>1:n</math> vel in
majore ratione. Eadem etiam affirmari po$$unt de quantitate <math>a</math>, quæ
minor e$t quam minima radix.</p>
<p>Si vero impo$$ibiles radices in datâ æquatione contineantur, tum
haud dici pote$t, annon ulla quantitas a$$umpta major quam maxima
vel minor quam minima radix, $emper ad radicem maximam vel mi-
nimam appropinquabit.</p>

<pb 390>[0442]<rh>DE INFINITIS</rh>
<p>1. Sit æquatio <math>(x - p)^n = 0</math>, cujus omnes radices $int æquales;
a$$umatur <math>a</math> pro primâ approximatione, $cribatur <math>a + π</math> pro <math>x</math> in datâ
æquatione, & erit prima approximatio <math>π</math> prope <math>= - {(a - p)^n / n (a - p)^n-1}
= - {a - p / n}</math>; ergo, $i modo <math>λ</math> $it vera di$tantia quantitatis <math>a</math> a$$umptæ
a radice quæ$itâ, erunt <math>{n - 1 / n} λ</math> di$tantia $ecundæ approximationis a
vera radice, <math>{(n - 1)^2 / n^2} λ</math> di$tantia tertiæ a vera radice, <math>{(n - 1)^3 / n^3} λ</math> di-
ftantia quartæ, & $ic deinceps.</p>
<p>2. In quâcunque æquatione algebraicâ <math>x^n - p x^n-1 + &c. = 0</math>
$int <math>m</math> radices prope inter $e & quantitati <math>a</math> æquales; & e $ub$ti-
tutione <math>a + π</math> pro incognitâ quantitate <math>x</math> in datâ algebraicâ æqua-
tione, ubi <math>a</math> $it <math>α</math> prope, re$ultabit æquatio <math>P + Q π + R π^2 + S π^3
+ &c. = 0</math>; $i vero a$$umatur <math>m + 1</math> primorum huju$ce æquationis
terminorum $umma nihilo æqualis, i. e. <math>P + Q π + R π^2 ... H π^m
= 0</math>, tum <math>m</math> radices æquationis erunt prope <math>m</math> radices æquationis
<math>P + Q π + R π^2 + &c. = 0</math>, quæ $upponuntur inter $e prope
æquales.</p>
<p>In hoc ca$u plerumque haud detegi pote$t approximatio e $uppo-
nendo $ummam duarum vel trium vel quatuor, &c. ... vel <math>m</math> termi-
norum nihilo æqualem, i. e. vel <math>P + Q π = 0</math> vel <math>P + Q π + R π^2
= 0, &c.</math> ad <math>m</math> terminos.</p>
<p>5. Sit æquatio <math>P - Q e + R e^2 - &c. = 0</math>, cujus radices $int <math>α, β, γ,
&c.</math> & cujus una radix (<math>α</math>) perparvam habeat rationem ad quamcun-
que aliam; tum erunt <math>P = α β γ δ &c. Q = α β γ &c. + α β δ &c. +
α γ δ &c. + &c. + β γ δ &c. R = α β &c. + α γ &c. + α δ &c. +
&c. + β γ &c. + β δ &c. + γ δ &c. + &c.</math> i. e. <math>Q</math> erit prope <math>= β γ δ
+ &c. R = β γ &c. + β δ &c. + γ δ &c. + &c. &c.</math> $i vero $tatuatur
<math>P - Q e</math> prope <math>= α × β γ δ × &c. - β γ δ &c. × e = 0</math>, erit <math>e = {P / Q}</math> prope

<pb 391>[0443]<rh>SERIEBUS.</rh>
<math>= {α β γ δ &c. / β γ δ &c.} + &c. = α</math> prope: vel adhuc propior erit ejus valor <math>α -
α^2 × ({1 / β} + {1 / γ} + {1 / δ} + &c.) + α^3 ({1 / β^2} + {1 / γ^2} + {1 / δ^2} + &c. + {1 / β γ} + {1 / β δ}
+ {1 / γ δ} + &c.) - &c.</math>; & $ic deinceps.</p>
<p>2. Nunc $tatuatur <math>P - Q e + R e^2</math> prope <math>= (β γ &c. + β δ &c. +
γ δ &c.) e^2 - β γ δ &c. e + α β γ δ &c. = 0</math>, tum ab re$olutione æqua-
tionis con$equetur radix <math>= α + α^3 ({1 / β γ} + {1 / β δ} + {1 / γ δ} + &c.) &c.</math>
& generaliter $tatuantur <math>m</math> termini prædictæ æquationis <math>P + Q e +
R e^2 + S e^3 + &c... e^m-1 = 0</math>, & inveniri pote$t ejus radix <math>= α ± α^m
({1 / β γ δ &c.} + {1 / β δ ε &c.} + {1 / γ δ ε &c.} + &c.)</math> ubi in $ingulis factoribus
<math>{1 / β γ δ &c.}, {1 / β δ ε &c.}, &c.</math> tot contineantur diver$æ literæ <math>β, γ, δ, &c.</math> quot
unitates in <math>m - 1</math>; & per <math>{1 / β γ δ &c.} + {1 / β δ ε &c.} + &c.</math> de$igno aggre-
gatum e $ingulis quantitatibus prædicti generis. Signum affixum ±
erit affirmativum vel negativum, prout <math>m - 1</math> e$t par vel impar nu-
merus.</p>
<p>6. Sit æquatio <math>0 = t - s z + r z^2 - q z^3 + p z^4 ... l z^m ± k z^m+1 ±
h z^m+2 ± &c.</math>, in quâ una radix <math>α</math> $it perparva re$pectu habito ad re-
liquas; & con$tat, quod$i æquatio <math>0 = t - s z,</math> i. e. <math>z = {t / s}</math> præbeat va-
lorem quantitatis <math>α</math> ad numerum <math>π</math> figurarum verum; tum ultimo
re$olutio æquationis <math>0 = t - s z + r z^2 - q z^3 ... l z^m</math> præbebit va-
lorem prædictæ radicis <math>α</math> u$que ad numerum <math>m × π</math> figurarum verum.</p>
<p>7. Sit æquatio <math>0 = t - s z + r z^2 - q z^3 ... l z^m - k z^m+1 +
&c.</math>, ducatur hæc æquatio in <math>0 = a - b z + c z^2 - d z^3 ... f z^m -
g z^m+1 + &c.</math>, ita ut 1<_>mo</_> evadat æquatio huju$ce formulæ <math>0 = T - S z
... K z^m+1 + &c.</math> in quâ deficiunt termini <math>z^2, z^3, ... z^m</math>; tum ple-

<pb 392>[0444]<rh>DE INFINITIS</rh>
rumque, $i <math>{t / s}</math> $it valor quantitatis <math>z</math> ad <math>π</math> figuras verus, ultimo erit <math>{T / S}</math>
valor prædictæ quantitatis <math>z</math> u$que ad <math>m × π</math> $iguras verus; 2<_>do</_> eva-
dat æquatio formulæ <math>0 = T - S z + R z^2 - Q z^3 ... L z^m - K z^m+1
+ &c.</math> ita vero con$tituta, ut innote$cant radices æquationis re$ul-
tantis ex hypothe$i quod (<math>m + 1</math>) primi termini prædictæ æquationis
nihilo fiant æquales, i. e. <math>0 = T - S z + R z^2 - Q z^3 ... L z^m</math>; tum,
$i <math>{t / s}</math> $it valor quantitatis <math>z</math> ad <math>π</math> figuras verus, erit ultimo radix hu-
ju$ce æquationis valor prædictæ radicis ad <math>m × π</math> figuras verus.</p>
<p>Eadem principia etiam ad æquationes, in quibus duæ vel plures
continentur incognitæ quantitates, applicari po$$unt.</p>
<p>7. Sit æquatio <math>y^n - p y^n-1 + q y^n-2 - r y^n-3 + &c. = 0</math>, cujus una ra-
dix <math>α</math> multo major $it quam quæcunque alia; & $i requiratur $eries,
quæ $ecundum quantitates <math>e, f, g, &c.</math> reciproce progrediatur; $criba-
tur <math>p + {1 / e}</math> vel <math>a + {1 / e}</math> pro <math>y</math> in datâ æquatione, ubi <math>a</math> e$t <math>α</math> prope, &
re$ultet æquatio <math>e^n - P e^n-1 + Q e^n-2 - &c. = 0</math>; in hac æquatione
pro <math>e</math> $cribatur <math>P + {1 / f}</math>, & re$ultat æquatio <math>f^n - P′ f^n-1 + &c. = 0</math>;
deinde in hâc æquatione pro <math>f</math> $cribatur <math>P′ + {1 / g}</math>, & $ic deinceps;
unde <math>y = p + {1 / e} = p + {f / P f + 1} = p + {P′ g + 1 / P P′ g + P + g} =
{b P″ P′ + P′ + b / b P″ P′ P + P P′ + b P + b P″ + 1} = &c.</math> Lex huju$ce $eriei e lege
$eriei in theor. 45. medit. algebr. traditâ facile con$tabit.</p>
<p>Cor. 1. Approximationes ex hâc methodo inventæ mutatis mutan-
dis eædem erunt ac approximationes prius deductæ.</p>
<p>Cor. 2. Si vero duæ radices $int re$pective multo majores quam
reliquæ, tum e tribus primis terminis nihilo æqualibus e$$e $uppo$i-
tis, viz, æquatione <math>y^2 - p y + q = 0</math> con$tabit approximatio ad utram-

<pb 393>[0445]<rh>SERIEBUS.</rh>
que radicem. Et $ic de continuis approximationibus ad plures ma-
jores radices inveniendis.</p>
<p>Ex approximationibus ad duas vel plures majores radices datis, &
princip. $ub$. facile deduci po$$unt aliæ ad prædictas radices magis
appropinquantes.</p>
<p>8. Hæc principia etiam ad æquationem duas vel plures variabiles
quantitates habentem, vel ad plures æquationes applicari po$$unt.</p>
<p>Sit æquatio <math>x^n - p x^n-1 + q x^n-2 - &c. = 0, & $i - {α^n - p α^n-1 + q α^n-2 - &c. / n α^n-1 - (n - 1) p α^n-2 + (n - 2) q α^n-3 - &c.}
= β, & {(α + β)^n - p (α + β)^n-1 + q (α + β)^n-2 - &c. / n (α + β)^n-1 - (n - 1) p (α + β)^n-2 + &c.}
= - β</math>; vel $i prior fractio $it <math>l</math>, & proximæ ap-
proximationes per eandem methodum inventæ $int re$pective <math>m, n, o,
&c.</math> & tandem inveniatur approximatio <math>= - l - m - n - o - &c.</math>
tum $eries con$tans e diver$is approximationibus nec convergens nec
divergens dici pote$t.</p>
<p>Et con$imilia etiam ad æquationes diver$i generis, & plures æqua-
tiones plures incognitas quantitates habentes applicari po$$unt.</p>
<p>9. Sit data æquatio <math>x^n - p x^n-1 + q x^n-2 ... t x^n-m + &c. = 0</math>, a$$uma-
tur <math>a</math> pro approximatione ad minimam radicem, $cribatur <math>a ∓ v</math> pro
<math>x</math> in datâ æquatione & re$ultet æquatio <math>v^n - P v^n-1 + Q v^n-2 .. T v^n-m
+ &c. = 0</math> ubi <math>T</math> diver$um habeat $ignum a quantitate <math>t, & n</math> haud
<math>= m</math>; tum plerumque quantitates inventæ haud accedent ad mini-
mam radicem.</p>
<p>10. Si terminorum datæ æquationis $igna $emel $olummodo vel pro-
grediantur de + in +, vel - in -; vel mutabuntur de + in -,
vel - in +; tum a$$umatur quæcunque negativa quantitas in uno
ca$u, affirmativa vero in altero, major vero quam negativa vel affir-
mativa radix, pro approximatione ad radicem prædictam, & $emper
convergent approximationes per prædictam methodum inventæ.</p><pb 394>[0446]<rh>DE INFINITIS</rh>
<h>THEOR. XVIII.</h>
<p>Sit quadratica æquatio <math>x^2 - (a + s) x + a s = 0</math>, cujus radices
$unt <math>a & s</math>, & prima approximatio per methodum hic traditam inventa
erit <math>{as / a + s} = s - {s^2 / a} + {s^3 / a^2} - {s^4 / a^3} + &c.</math> per eandem methodum in-
veniantur $ucce$$ivæ approximationes, & approximatio ad <math>n - 1</math> di-
$tantiam a primâ erit <math>{s^b / a^b-1} - {s^b+1 / a^b} + {s^3b / a^3b-1} - {s^3b+1 / a^3b} + {s^5b / s^3b+1} - {s^5b+1 / a^5b} +
{s^7b / a^7b-1} - {s^7b+1 / a^7b} + &c.</math> ubi <math>b = 2^n-1</math>.</p>
<p>Cor. 1. Hinc con$tat, quod hæ approximationes ultimo convergent
in rationibus fractionis (<math>{s^b / a^b-1}</math>) inter $e, & con$equenter approxima-
tiones $ic inventæ ultimo vergent in ratione majori quam quâvis
geometricâ progre$$ione; idem etiam de approximationibus æquatio-
num $uperiorum ordinum vel quorumcunque diver$orum generum
vel plurium æquationum, &c. $ic inventis affirmari pote$t.</p>
<p>Cor. 2. Sit æquatio <math>x^n - p x^n-1 + q x^n-2 - r x^n-3 + &c. = 0</math>, cujus
radices $int <math>α, β, γ, δ, &c. σ</math>; ubi <math>σ</math> e$t radix quæ$ita, fingantur <math>{1 / α} +
{1 / β} + {1 / γ} + {1 / δ} + &c. = P, {1 / α β} + {1 / α γ} + {1 / β γ} + {1 / α δ} + &c. = Q, {1 / α β γ}
+ {1 / α β δ} + &c. = R, {1 / α β γ δ} + &c. = S, &c.</math> in quibus quantitatibus
haud continetur radix <math>σ</math>; & prima approximatio per prædictam
methodum inventa erit <math>{σ / 1 + P σ} = σ - P σ^2 + P^2 σ^3 - &c.</math> $ecunda
vero erit fractio, cujus numerator erit <math>{P σ^2 / (1 + P σ)^2} + {P Q σ^4 / (1 + P σ)^3} -

<pb 395>[0447]<rh>SERIEBUS.</rh>
{P R σ^5 / (1 + P σ)^4} + {P S σ^6 / (1 + P σ)^5} - &c.</math> denominator vero <math>{1 + P^2 σ^2 / 1 + P σ} -
{2 P Q σ^3 - Q σ^2 / (1 + P σ)^2} + {3 P R σ^4 - R σ^3 / (1 + P σ)^3} - {4 P S σ^5 - S σ^4 / (1 + P σ)^4} + &c.</math></p>
<h>PROB. XIV.</h>
<p it>Sit data æquatio <math>x^n - m x^n-1 + 1 x^n-2 - k x^n-3 .... d x^3 - c x^2 +
b x - a = 0</math>, & datis propinquis valoribus ad unam vel duas vel plures
ejus radices, invenire valores ad prædictas radices magis appropinquantes.</p>
<p>1. Sit <math>β</math> quantitas ab unâ $olummodo radice haud longe di$tans, &
$cribatur <math>(x^n-1 - P x^n-2 + Q x^n-3 ... S x^3 - T x^2 + V x + W) (x -
β - e) = x^n - m x^n-1 + l x^n-2 + &c. .. + d x^3 - c x^2 + b x - a = 0</math>;
inde re$ultant æquationes <math>P = m - β - e, Q = 1 - (β + e) P = l
- m (β + e) + (β + e)^2</math>, & $ic <math>R = k - (β + e) l + (β + e)^2 m -
(β + e)^3</math>, & ultimo re$ultat æquatio <math>(β + e)^n - m (β + e)^n-1 + l (β
+ e)^n-2 - k (β + e)^n-3 + &c. = 0</math>, $ed per hypothe$in <math>e</math> e$t perparva
re$pectu habito ad <math>β</math>, ergo ultimo erit <math>β^n - m β^n-1 + l β^n-2 - &c. =
- (n β^n-1 - (n - 1) m β^n-2 + (n - 2) l β^n-3 - &c.) e</math>, & exinde con-
$tat approximatio <math>e</math>: aliter vero ab ultimis terminis incipiendum e$t,
& facile ad eandem conclu$ionem pervenire liceat.</p>
<p>2. Sint <math>a, b, c, d, &c.</math> approximationes ad <math>r</math> radices datæ æquatio-
nis <math>x^n - m x^n-1 + l x^n-2 - &c. = 0</math>, invenire alias ad <math>r</math> radices magis
appropinquantes. Supponantur <math>α, β, γ, δ, ε, &c.</math> perparvæ quanti-
tates, quæ $int approximationes quæ$itæ; a$$umatur æquatio <math>(x^n-r -
P x^n-r-1 + Q x^n-r-2 - R x^n-r-3 + &c.) ((x - a - α) × (x - b - β)
× (x - c - γ) × (x - d - δ) × &c.) = x^n - m x^n-1 + l x^n-2 - k x^n-3
+ &c; = o</math>, huju$ce æquationis fiant corre$pondentes termini inter $e
æquales (omnibus quantitatibus $emper neglectis, vel pro nihilo ha-
bitis, in quibus plures quam una dimen$io quantitatum <math>α, β, γ, δ, &c.</math>
continentur), & ita reducantur æquationes re$ultantes, ut extermi-
nentur incognitæ quantitates <math>P, Q, R, S, &c.</math> & exorientur <math>r</math> $implices

<pb 396>[0448]<rh>DE INFINITIS</rh>
æquationes totidem incognitas quantitates <math>α, β, γ, δ, &c.</math> habentes, e
quibus inveniri po$$unt prædictæ quantitates.</p>
<p>1. 2. Sit æquatio <math>x^n - p x^n-1 + q x^n-2 - r x^n-3 + s x^n-4 - t x^n-5 +
&c. = 0</math>, cujus $int <math>a, b, c, d, &c.</math> approximationes ad <math>n</math> radices <math>α, β,
γ, δ, &c.</math> nunc fingantur diver$æ radices re$pective <math>a + π, b + ρ, c + σ,
d + τ, &c.</math> ducantur hæ quantitates in $e$e, & erit contentum <math>(x - a
- π) × (x - b - ρ) × (x - c - σ) × &c. = x^n - ((a + b + c + &c.)
+ (π + ρ + σ + τ + &c.)) x^n-1 + ((a b + a c + b c + &c.) + (b +
c + d + &c.) π + (a + c + d + &c.) ρ + (a + b + d + &c.) σ +
&c.) x^n-2 - ((a b c + a b d + b c d + &c.) + (b c + b d + c d + &c.)
π + &c.) x^n-3 + &c. = 0</math>, dicantur $umma e $ingulis quantitatibus
<math>(a + b + c + d, &c.)</math> productorum e $ingulis binis in $e ductis <math>(a b
+ a c + b c + &c.)</math>, contentorum e $ingulis ternis, quatuor, &c. re-
$pective <math>P, Q, R, S, T, &c.</math> deinde $upponantur <math>π + ρ + σ + τ +
&c. = p - P = α, a π + b ρ + c σ + d τ + &c. = (p - P) × P -
q + Q = P α - q + Q = β, a^2 π + b^2 ρ + c^2 σ + &c. = P β - Q α
+ r - R = γ, a^3 π + b^3 ρ + c^3 σ + &c. = P γ - Q β + R α - s +
S = δ, a^4 π + b^4 ρ + c^4 σ + &c. = P δ - Q γ + R β - S α + t - T</math>,
& $ic deinceps (unde cuidam in$picienti facile con$tat lex, quam ob-
$ervant termini <math>α, β, γ, δ, &c.</math>); inveniantur <math>n</math> $implices æquationes
huju$ce generis, & ex quibu$cunque <math>n</math> earum $implicibus æquationi-
bus erui po$$unt approximationes ad $ingulas radices quæ$itæ <math>π, ρ, σ,
τ, &c.</math></p>
<p>Cor. 1. E principiis prius traditis con$tabit, ut quantitates hâc
methodo inventæ eo magis approximent, quo minus di$tant appro-
ximationes datæ ab unâ radice quam ab reliquis; $i vero una appro-
ximatio data minus di$tet a duabus radicibus quam a reliquis, tum ad
inve$tigandam quantitatem magis adhuc appropinquantem utendum
e$t quadraticâ æquatione; & $ic deinceps.</p>
<p>Cor. 2. Rationes, quibu$cum vergunt ad radices quæ$itas approxi-
mationes inventæ, facile deduci po$$unt $cribendo pro $ingulis quan-
titatibus in approximationibus prædictis contentis earum functiones
datæ æquationis radicum.</p>

<pb 397>[0449]<rh>SERIEBUS</rh>
<p>Omnia hæc etiam ad æquationes, quæ haud $unt algebraicæ, ap-
plicari po$$unt.</p>
<h>PROB. XIX.</h>
<p>Sit quæcunque æquatio <math>A = 0</math>, ubi litera <math>A</math> de$ignat quamcunque
functionem quantitatis <math>x</math>; $ingatur <math>x</math> variabilis, <math>x^.</math> vero con$tans, &
inveniatur <math>{A^. / x^.} = p, {A^.. / x^. ^2} = q, {A^... / x^. ^3} = r, &c.</math> detur propinquus valor
quantitatis <math>x</math> in datâ æquatione, qui dicatur <math>a</math>; $cribatur <math>a</math> pro <math>x</math> in
quantitatibus <math>A, p, q, r, s, &c.</math> & pro quantitatibus re$ultantibus
$cribantur <math>P, Q, R, S, T, &c.</math> re$pective; deinde a$$umatur æquatio
<math>P - Q e + {1 / 2} R e^2 - {1 / 2 · 3} S e^3 + {1 / 2 · 3 · 4} T e^4 - &c.</math> in infinitum <math>= 0</math>,
$cribatur etiam <math>a - e</math> pro <math>x</math> in datâ æquatione <math>A = 0</math>, & re$ultet
æquatio <math>A′ = 0</math>, cujus radix <math>e</math> eadem erit ac ea in æquatione <math>P - Q e
+ {1 / 2} R e^2 - &c. = 0</math> contenta; nunc fingatur <math>P - Q e = 0</math>, & $it <math>α</math>
radix æquationis <math>A′ = 0</math>, quæ $it perparva, & æquationis <math>P - Q e = 0</math>
radix exprimetur per $eriem $ub$equentis formulæ, i. e. <math>e = {P / Q} = α
+ m α^2 + &c.</math> nunc a$$umatur <math>P - Q e + {1 / 2} R e^2 = 0</math>, & $ic exprime-
tur valor ejus radicis <math>e = α + m α^3 + &c.</math> & univer$aliter a$$umatur
aggregatum <math>m</math> terminorum præcedentis æquationis <math>P - Q e + {1 / 2} R e^2
- {1 / 6} S e^3 + &c. ... e^m-1 = 0</math>, & per $eriem huju$ce formulæ <math>e = α +
m′ α^m + &c.</math> exprimetur radix æquationis prædictæ <math>P - Q e + {1 / 2} R e^2
- {1 / 6} S e^3 + &c... = 0</math>.</p>
<p>Ex. Sit quantitas exponentialis <math>x^x</math> & augeatur <math>x</math> per <math>e</math>, tum evadet
<math>(x^x) = (x + e)^x+e = x^x + x^x (log. x + 1) e + {1 / 2} x^x (log. x^2 + 2 log. x
+ x^x + x^x-1) e^2 + &c. x^x</math> nunquam evadit <math>= 0</math>.</p>
<p>Cor.. Si duæ datæ æquationis radices $int prope inter $e & quan-
titati <math>a</math> æquales, tum a$$umatur <math>P - Q e + {1 / 2} R e^2 = 0</math>, & inveniatur
propior valor quantitatis <math>e</math>, & $ic ex aggregato <math>m</math> terminorum <math>P -
Q e + {1 / 2} R e^2 - &c. = 0</math> nihilo æquali a$$umpto, deduci pote$t propior

<pb 398>[0450]<rh>DE INFINITIS</rh>
valor quantitatis <math>e</math>, cum <math>m</math> radices datæ æquationis $int prope inter
$e & datæ quantitati <math>a</math> æquales.</p>
<h>THEOR. XX.</h>
<p>Datis duabus æquationibus <math>H y^n + (a + b x) y^n-1 + &c. = 0</math> &
<math>K y^m + (A + B x) y^m-1 + &c. = 0</math> duas incognitas quantitates <math>x & y</math>
habentibus, $int corre$pondentes radices incognitarum quantitatum
(<math>x & y</math>) re$pective <math>α, β, γ, δ, &c. & π, ρ, σ, τ, &c.</math> invenire ex approxi-
matione corre$pondentes valores (<math>α & π</math>) incognitarum quantitatum
(<math>x & y</math>).</p>
<p>Nece$$e e$t, ut prius dentur <math>k & l</math> ad prædictos corre$pondentes
valores (<math>α & π</math>) multo magis appropinquantes, quam ad ullam aliam
e corre$pondentibus incognitarum quantitatum (<math>x & y</math>) radicibus (<math>β,
γ, δ, &c. & ρ, σ, τ, &c.</math>) In datis æquationibus pro quantitatibus (<math>x
& y</math>) $cribantur re$pective <math>z - k & v - l</math>, & $int æquationes re$ul-
tantes <math>Γ + (Δ z + E v) + (ξ z^2 + H z v + Θ v^2) + &c. = 0 & (Π +
P z + Σ v) + (T z^2 + Ψ z v + Φ v^2) + &c. = 0</math>; & e con$titutione
coefficientium colligi po$$unt <math>Γ + Δ z + E v</math> prope <math>= 0, & Π + P z
+ Σ v</math> prope <math>= 0</math>, & exinde approximatio ad <math>z</math> erit <math>{E Π - Σ Γ / Σ Δ - E P}</math>, &
approximatio ad <math>v</math> erit <math>{Δ Π - P Γ / P E - Δ Σ}</math>.</p>
<p>2. Si vero $umantur plures termini vel ex unâ vel $ingulis duabus
æquationibus, quorum aggregatum nihilo fiat æquale, i. e. <math>Γ + (Δ z
+ E v)</math> prope <math>= 0 & + Π + (P z + Σ v) + (T z^2 + Ψ z v + Φ v^2)</math> pro-
pe <math>= 0</math>; vel <math>Π + P z + Σ v = 0 & Γ + (Δ z + E v) + ζ z^2 + H z v
+ θ v^2 = 0</math>; vel <math>Γ + (Δ z + E v) + (Ζ z^2 + H z v + Θ v^2) = 0 &
Π + (P z + Σ v) + (T z^2 + Ψ z v + Φ v^2) = 0; &c.</math> ita vero ut
omnes termini vel nullius & unius; vel nullius, unius & duarum;
&c. dimen$ionum contineantur in $ingulis duobus aggregatis, quæ
nihilo fiant æqualia, & de convergentiâ approximationum ad quan-<pb 399>[0451]<rh>SERIEBUS.</rh>
titates <math>z & v</math> ex his æquationibus inventarum dijudicare liceat e
theor. præced. & medit. algebr.</p>
<p>3. Si vero <math>k & l</math> multo magis approximent ad duas corre$pondentes
radices incognitarum quantitatum <math>(x & y)</math> re$pective quam ad ullas
alias, & pro <math>x & y</math> $ub$tituantur re$pective <math>z - k & v - l</math> in datis
æquationibus, & e con$titutione coefficientium colligi po$$unt vel <math>Γ +
Δ z + E v prope = 0 & II + (P z + Σ v) + (T z^2 + Ψ z v + Φ v^2)
prope = 0; vel II + P z + Σ v prope = 0, & Γ + (Δ z + E v) + (ζ z^2
+ H z v + Θ v^2) prope = 0;</math> e quibus æquationibus inveniri po$$unt
approximationes ad <math>z & v:</math> & $ic progredi licet, cum <math>k & l</math> ad tres vel
plures corre$pondentes incognitarum quantitatum radices multo ma-
gis approximent quam ad reliquas.</p>
<p>4. Sint duæ æquationes duas incognitas quantitates <math>x & y</math> habentes;
pro <math>x & y</math> in datis æquationibus $cribantur re$pective <math>h + e & m + 0′,</math>
& re$ultent æquationes <math>A + B e + C 0′ + D = 0 & P + Q e + R 0′
+ S = 0,</math> ubi literæ <math>D & S</math> in $ingulis terminis plures habent dimen-
$iones perparvarum quantitatum <math>e & 0′</math> quam unam; <math>& P, Q, R; A,
B & C</math>, habent vero nullas; tum approximationes ad quantitates
<math>e & 0′</math> per methodum hic traditam inventæ erunt re$pective <math>e =
{RA - CP / CQ - RB}, & 0′ = {QA - BP / RB - QC};</math> $i vero <math>CQ = RB,</math> tum approxima-
tiones $ic inventæ erunt infinitæ.</p>
<p>Hinc $int duæ datæ æquationes <math>α = 0 & π = 0,</math> ubi <math>α & π</math> $unt
functiones quantitatum <math>x & y;</math> inveniantur earum fluxiones, viz. <math>α^. =
a x^. + b y^. & π^. = p x^. + q y^.</math>; a$$umantur duæ æquationes <math>a h + b m = 0
& p h + q m = 0</math>, & ex iis inveniantur quantitatum <math>x & y</math> valores <math>(h
& m)</math> qui erunt limites inter valores quantitatum <math>x & y</math> in datis
æquationibus contenti, $cribantur <math>h + e & m + 0′</math> re$pective pro <math>x & y</math>
in datis æquationibus, & approximationes per methodum prius tra-
ditam inventæ erunt valde magnæ; ni duo valores quantitatum <math>x
& y</math> in datis æquationibus contenti $int re$pective <math>h & m</math> prope.</p>
<p>Et $ic de limitibus inventis ex fingendo <math>a = 0 & p = 0, &c.</math></p>

<pb 400>[0452]<rh>DE INFINITIS</rh>
<h>THEOR. XXI.</h>
<p>Sint duæ vel plures <math>n</math> invariabiles quantitates <math>(x, y, z, &c.)</math> in datâ
æquatione contentæ, quarum $int <math>a, b, c, &c.</math> re$pectivi valores prope;
$cribantur <math>a + α, b + β, c + γ, &c.</math> pro earum re$pectivis valoribus
in datâ æquatione; & re$ultet æquatio <math>A + B α + C β + D γ + &c.
+ K = 0</math>, ubi <math>A, B, C, D, &c.</math> denotant quantitates in re$ultante
æquatione contentas, in quibus haud continetur <math>α</math> vel <math>β</math> vel <math>γ</math> vel <math>&c.
K, K′, K″, &c.</math> vero $int quantitates, in quarum $ingulis terminis con-
tinentur plures quam una dimen$iones quantitatum <math>α</math> vel <math>β</math> vel <math>γ &c.</math></p>
<p>Et $ic $cribantur <math>a + α′, b + β′, c + γ′, &c.</math> pro earum re$pectivis
valoribus <math>(x, y, z, &c.)</math> in datâ æquatione & re$ultet æquatio <math>A +
B α′ + C β′ + D γ′ + &c. + K′ = 0</math>; & $ic $cribantur <math>a + α″, b + β″,
c + γ″, &c.</math> pro earum re$pectivis valoribus <math>(x, y, z, &c.)</math> in datâ
æquatione, & re$ultet <math>A + B α″ + C β″ + D γ″ + &c. + K″ = 0</math>; &
$ic deinceps: unde ex <math>n</math> $ub$titutionibus prædicti generis $i modo re-
jiciantur quantitates <math>K, K′, K″, &c.</math> re$ultabunt <math>n</math> $implices æquatio-
nes <math>A + B α + C β + D γ + &c. = 0, A′ + B α′ + C β′ + D γ′ +
&c. = 0, A″ + B α″ + C β″ + D γ″ + &c. = 0</math> totidem incognitas
quantitates <math>B, C, D, &c.</math> habentes, e quibus detegi po$$unt quantitates
ip$æ <math>B, C, D, &c.</math></p>
<p>2. Si vero duo $int valores quantitatis <math>x</math> prope inter $e & valori <math>a</math>
æquales, qui corre$pondent uni valori <math>(b, c, d, &c.)</math> e $ingulis reliquis
variabilibus quantitatibus; tum hæc erit formula æquationis quæ-
fitæ, viz. <math>P π^2 + q π + r ρ + s σ + t τ + &c. + q′ π ρ + r′ π σ + &c.
+ A = 0′,</math> ubi <math>a + π, b + ρ, θ + σ, d + τ, &c.</math> $unt corre$pondentes
valores e $ingulis variabilibus <math>x, y, z, &c.</math> in datâ æquatione contentis;
<math>P, P′, P″, q, r, s, t, &c. q′, r′, &c. A</math> $unt coefficientes quæ$itæ: $i vero
duo $int valores quantitatum <math>x & y</math> prope inter $e & quantitatibus
<math>a & b</math> re$pective æquales, qui corre$pondent uni valori <math>(c, d, &c.)</math> e
fingulis reliquis <math>(z, v, &c.)</math> tum erit formula æquationis quæ$itæ
<math>P π^2 + P′ π ρ + P″ ρ^2 + q π + r ρ + s σ + t τ + &c. + s′ π σ + s″ ρ σ
+ &c.</math></p>

<pb 401>[0453]<rh>SERIEBUS.</rh>
<p>Exhinc con$tat formula æquationis, in quâ plures valores plurium
variabilium quantitatum $int prope inter $e æquales.</p>
<p>Ea, quæ hic tradita fuerunt, facile ad plures æquationes plures
variabiles quantitates habentes applicari po$$unt.</p>
<p>Convergentiæ harum æquationum ex principiis prius traditis diju-
dicandæ $unt.</p>
<h>PROB. XV.</h>
<h it>Datis infinitis æquationibus duas vel plures incognitas quantitates haben-
tibus, invenire unam ex iis in terminis reliquarum.</h>
<p>Primo ita transformetur data æquatio, ut incognitæ quantitates in
eâ contentæ evadant perparvæ; $i requiratur $eries $ecundum dimen-
$iones prædictarum quantitatum a$cendens: $in requiratur de$cendens
$ecundum dimen$iones quarundam quantitatum; tum ita transfor-
metur data æquatio, ut evadant quantitates permagnæ; & perfici
pote$t hoc problema ex ii$dem principiis, quæ tradita fuere de finitis
æquationibus.</p>
<p>Si prima approximatio incognitæ quantitatis <math>y,</math> cujus infiniti in
datâ æquatione continentur termini, involvat quantitatem, quæ haud
in $e continet ullas incognitas quantitates; tum plerumque re$olu-
tionem infinitæ æquationis vel $ummationem infinitæ $eriei exiget
$ingulus terminus quæ$itus. E. g. Sit æquatio <math>1 + a - {a^3 / 1 · 3} +
{a^5 / 1 · 3 · 1 · 5} - &c. = l + {l^2 / 1 · 2} + {l^3 / 1 · 2 · 3} + {l^4 / 1 · 2 · 1 · 4} + &c.</math> inve-
nire <math>l</math> in terminis quantitatis <math>a</math>; a$$umatur <math>l = e</math> prope & <math>a</math> perparva
quantitas, $cribatur <math>e + p</math> pro <math>l</math>, & re$ultat <math>1 = e + {e^2 / 2} + {e^3 / 1 · 3} +
{e^4 / 1 · 3 · 4} + &c.</math> & exinde <math>e = log. 2;</math> etiamque <math>(1 + e + {e^2 / 2} + {e^3 / 1 · 3}

<pb 402>[0454]<rh>DE INFINITIS</rh>
+ &c.) p = 2p = a</math> prope; $cribatur <math>e + {a / 2} + q</math> pro <math>l</math> in datâ æqua-
tione & invenietur valor quantitatis <math>q = - {a^2 / 8}</math> prope; & $ic deinceps,
unde valor quantitatis <math>l = log. 2 + {a / 2} - {a^2 / 8} - {a^3 / 24}, &c.</math></p>
<p>Cor.. Omnis infinita $eries exorta ex hâc re$olutione erit <math>e + {e^2 / 2}
+ {e^3 / 1 · 2 · 3} + &c.</math> & haud difficilis erit inve$tigatio ex datâ vel datis
infinitis æquationibus, an finitus vel infinitus numerus infinitarum
$erierum in re$olutione prædictâ exorietur.</p>
<h>PROB. XVI.</h>
<p it>1. Sit y $umma datæ $eriei $ecundum dimen$iones quantitatis <math>x</math> progre-
dientis, <math>v</math> vero $umma $eriei $ecundum dimen$iones quantitatis <math>z</math> progredien-
tis, & detur $eries $ecundum dimen$iones quantitatum <math>z & x</math> progrediens,
cujus $umma $it <math>w</math>; invenire quantitatem <math>w</math> in $erie $ecundum dimen$iones
quantitatum <math>y & v</math> progrediente. Primo ita reducantur omnes hæ quan-
titates per methodos prius traditas, ut evadant perparvæ quantitates, qua-
rum dimen$iones in $erie quæ$itâ a$cendunt: vel permagnæ, quarum dimen-
$iones in prædictâ $erie de$cendunt; tum pro <math>x & z &</math> earum functionibus
in $erie <math>w</math> $cribantur earum valores e functionibus <math>y & v</math> deducti, & per-
ficitur problema.</p>
<p>Ex. 1. Datis logarithmis <math>y & v</math> duarum quantitatum <math>1 + x &
1 + z,</math> invenire logarithmum quantitatis <math>1 + x + z:</math> erunt <math>y =
x - {1 / 2}x^2 + {1 / 3}x^3 - &c. v = z - {1 / 2}z^2 + {1 / 3}z^3 - &c. w = (x + z) -
{1 / 2}(x + z)^2 + {1 / 3}(x + z)^3 - &c.</math> reducantur hæ æquationes ita ut
inveniatur <math>w</math> in terminis $ecundum dimen$iones quantitatis <math>y & v</math>

<pb 403>[0455]<rh>SERIEBUS.</rh>
progredientibus, & re$ultabit <math>w = y + v - y v + y ({1 / 2}v^2 - {1 / 1 · 3}v^3 +
+ v({1 / 2}y^2 - {1 / 1 · 3}y^3 +
{1 / 1 · 3 · 4}v^4 - &c.) - {3 / 4}y^2 v^2 + y^2 ({7 / 12}v^3 - &c.) - &c.
{1 / 1 · 3 · 4}v^4 - &c.) + v^2 ({7 / 12}y^3 - &c.)</math></p>
<p>Ex. 2. Sit <math>y = t - {t^3 / 3} + {t^5 / 5} - &c., v = T - {T^3 / 3} + {T^5 / 5} - &c., &
w = T + t - {(T + t)^3 / 3} + {(T + t)^5 / 5} - &c.;</math> tum erit <math>w = y + v -
y v (y + v) + {1 / 3}v y (5 (v^3 + y^3) + 7(v y^2 + y v^2)) - &c.</math>, quæ $eries
facile deduci pote$t ex a$$umendo $eriem <math>w = a (y + v) + b (y^2 v + v^2 y)
+ (c (y^4 v + v^4 y) + d (y^3 v^2 + v^3 y^2)) + (e (y^6 v + v^6 y) + f (y^5 v^2
+ v^5 y^2) + g (v^4 y^3 + y^4 v^3)) + &c.;</math> & in hâc $erie $cribendo pro <math>v
& y</math> earum datos valores; deinde ex æquando re$ultantem $eriem
datæ $eriei <math>T + t - {(T + t)^3 / 3} + &c.</math> erui po$$unt coefficientes <math>a, b, c,
d, &c.</math></p>
<h>PROB. XVII.</h>
<p it>Sit <math>p</math> con$tans quantitas, invenire formulam $eriei <math>(A) p + a q x +
(b r + c q^2)x^2 + (d s + e q r)x^3 + &c.</math> in infinitum progredientis, ita
ut ejus $ingulæ pote$tates <math>m & n</math>, & con$equenter radices eandem pror$us
ob$ervent legem, i. e. $i in $erie, quæ exprimit <math>m</math> pote$tatem $eriei <math>A,</math> pro
m $cribatur <math>n</math>, re$ultet $eries, quæ exprimet <math>n</math> pote$tatem $eriei <math>A</math>.</p>
<p>A$$umantur duæ $eries <math>n & m</math> pote$tates exprimentes, quæ eandem
pror$us ob$ervant legem, & con$equenter erunt huju$ce formulæ
<math>p^m + m p^m-1 q x + m r p^m-2 \\ + (a′ m^2 + b m) q^2 p^m-2 # x^2 + m. s p^m-3 \\ + (a′ m^2 + b m) q r p^m-3 # x^3 + &c.&c. \\ + &c.</math><pb 404>[0456]<rh>DE INFINITIS</rh>
$cribatur <math>n</math> pro <math>m</math> in hâc $erie & re$ultat <math>p^n + n p^n-1 q x + n r p^n-2 \\ + (a′ n^2 + b n) q^2 p^n-2 # x^2
+ n s p^n-3 \\ + (a′ n^2 + b n) q r p^n-3 # x^3 + &c.</math> Ducantur hæ duæ $eries in $e$e & $eriem
re$ultare oportet <math>p^n+m + (n + m) p^n+m-1 q x + (n + m) r p^n+m-2 \\ + (a′ (n + m)^2 + b (n + m) q^2) p^n+m-2 # x^2
+ &c. \\ + &c.</math> ut vero hæc $eries eadem fiat ac re$ultans, coefficiens omnis
termini, in quo continentur duæ literæ <math>q & r, q & s, &c.</math> vel <math>r & s,
&c.</math> & generaliter <math>Q & R</math>, erit <math>((A m^2 + B m) + (A n^2 + B n) + 2 m n)
Q R = A (m + n)^2 + B (m + n)</math>, unde <math>A = 1, & B</math> quæcunque
quantitas ad libitum a$$umenda; $it <math>R</math> idem ac <math>Q</math>, tum erit <math>A = {1 / 2}</math>.</p>
<p>Coefficiens omnis termini, in quo continentur tres literæ <math>Q × R × S</math>,
erit (<math>A m^3 + H m^2 + C m</math>); unde e præcedenti multiplicatione re$ul-
tabit <math>A (m^3 + n^3) + H (m^2 + n^2) + C (m + n) + (m^2 + b m) n +
(n^2 + b n) m + (m^2 + b′m) n + (n^2 + b′n) m + (m^2 + b″ m) n + (n^2
+ b″ n) m = A (m + n)^3 + H (m + n)^2 + C (m + n)</math> unde <math>A = 1,
H = b + b′ + b″, C</math> vero quæcunque quantitas ad libitum a$$umen-
da: in hoc ca$u per <math>m^2 + b m, m^2 + b′m, & m^2 + b″m</math> volo coefficientes
quantitatum <math>Q R, Q S & R S</math> in pote$tate <math>m</math> re$pective; coefficiens
vero omnis termini, in quo continentur quatuor literæ <math>Q, R, S, T</math> erit
<math>(m^4 + K m^3 + L m^2 + M m)</math>, ubi <math>3 K = H + H′ + H″ + H′″ + b
+ c + d + e + f + g, L = b c + d e + f g + C + C′ + C″ + C′″</math>,
exi$tentibus <math>m^3 + H m^2 + C m, m^3 + H′ m^2 + C′ m; m^3 + H″ m^2 + C″ m,
m^3 + H′″ m^2 + C′″ m; & m^2 + b m, m^2 + c m, m^2 + d m, m^2 + e m, m^2
+ f m, m^2 + g m</math> re$pective coefficientibus quantitatum <math>Q R S, Q R T,
Q S T, R S T; & Q R, S T; Q S, R T; Q T, R S; M</math> vero erit quæcun-
que quantitas ad libitum a$$umenda: & $ic inveniri po$$unt coeffici-
entes quantitatum, in quibus plures continentur literæ; etiamque
inveniri po$$unt innumeræ $eries diver$i generis, quarum pote$tates
vel functiones vel eandem ob$ervant legem, vel quamcunque aliam
legem.</p>

<pb 405>[0457]<rh>SERIEBUS.</rh>
<p>In theoremate $equente per (<math>{P^. ^n + m + r + &c. / x^. ^n y^. ^m z^. ^r &c.})</math> eadem, quæ in prob. 67.
libri primi, de$ignetur quantitas.</p>
<h>THEOR. XXII.</h>
<p>1. Sit quantitas <math>P</math>, quæ duas variabiles quantitates
<math>x & y</math> continet, quarum incrementa $int <math>e & 0</math>: novus valor
quantitatis <math>(P) = P + (({P^. / x^.}) e + ({P^. / y^.}) 0) + ({1 / 2} ({P^.. / x^. ^2}) e^2 + {1 / 2}
({P^.. / y^. ^2}) 0^2 + ({P^.. / x^. y^.}) e o) + ({1 / 6} ({P^... / x^. ^3}) e^3 + {1 / 2} ({P^... / x^. ^2 y^.}) e^2 o + {1 / 2} ({P^... / x^. y^. ^2})
e o^2 + {1 / 6} ({P^... / y^. ^3}) o^3) + &c.</math> in infinitum, ubi coefficiens termini <math>{P^. ^n+m / x^. ^n y^. ^m})
e^n 0^m</math> $emper erit <math>{1 / 1 · 2 · 3 .. n} × {1 / 1 · 2 · 3 .. m}</math>.</p>
<p>2. Si vero dentur tres (<math>x, y, z</math>) variabiles quantitates in quantitate
<math>P</math>, quarum incrementa $int <math>e, 0 & i</math>; evadet novus valor quantitatis
<math>(P) = P + ({P^. / x^.}) e + ({P^. / y^.}) 0 + ({P^. / z^.}) i + &c.</math>; tum erit coefficiens
termini <math>(({P^. ^n+m+r / x^. ^n y^. ^m z^. ^r}) e^n o^m i^r) = {1 / 1 · 2 · 3 .. n} × {1 / 1 · 2 · 3 .. m} × {1 / 1 · 2 · 3 .. r}</math>.</p>
<p>3. Si dentur quatuor vel plures (<math>x, y, z, v, &c.</math>) variabiles quanti-
tates in quantitate, quarum incrementa $int <math>e, o, i, k, &c</math>, tum coeffi-
ciens termini <math>({P^. ^n + m + r + s + &c. / x^. ^n y^. ^m z^. ^r v^. ^s &c.}) e^n 0^m i^r k^s &c.</math> in novo valore erit <math>=
{1 / 1 · 2 · 3 .. n} × {1 / 1 · 2 · 3 .. m} × {1 / 1 · 2 · 3 .. r} × {1 / 1 · 2 · 3 .. s}, &c.</math>: in hoc
novo valore cunctæ in$uper adjungendæ $unt quantitates prædicti ge-
neris, ubi indices <math>n, m, r & s</math> integros re$pective denotant numeros.</p>
<p>Et $ic deinceps.</p>

<pb 406>[0458]<rh>DE INFINITIS</rh>
<p>Cor. 1. Sint duæ æquationes <math>A = 0 & B = 0</math>, quæ $int re$pective
functiones quantitatum <math>x & y</math>; inveniantur fluxiones quantitatum <math>A
& B</math> ex hypothe$i quod <math>x</math> $olummodo $it variabilis & <math>x^.</math> $it con$tans,
quæ $int re$pective <math>({A^. / x^.}) = p, ({A^.. / x^. ^2}) = p′, ({A^... / x^. ^3}) = p″, &c. ({B^. / x^.})
= q, ({B^.. / x^. ^2}) = q′, ({B^... / x^. ^3}) = q″, &c.</math> deinde inveniantur fluxiones quan-
titatum <math>A & B</math> ex hypothe$i, quod <math>y</math> $olummodo $it variabilis & <math>y^.</math>
con$tans, & $int re$pective <math>({A^. / y^.}) = r, ({A^.. / y^. ^2}) = r′, ({A^... / y^. ^3}) = r″, &c.
({B^. / y^.}) = s, ({B^.. / y^. ^2}) = s′, ({B^... / y^. ^3}) = s″, &c.</math> ultimo $upponantur quanti-
tates <math>x & y</math> variabiles, <math>x^.</math> vero & <math>y^.</math> con$tantes, & $int <math>({A^.. / x^. y^.}) = n, &c.
({B^.. / x^. y^.}) = m, &c.</math> a$$umantur <math>k & l</math> proximi valores quantitatum <math>x & y</math>
re$pective, $cribantur hi valores pro <math>x & y</math> re$pective in quantitatibus
<math>A, B; p, p′, p″, &c. q, q′, q″, &c. r, r′, r″, &c. s, s′, s″, &c. n, m, &c.</math>
& $int quantitates re$ultantes <math>a, b; P, P′, P″, &c. Q, Q′, Q″, &c.
R, R′, R″, &c. S, S′, S″, &c. N, M, &c.</math> tum a$$umantur æquationes
<math>o = a + p e + r o′ + {1 / 2} p′ e^2 + n o′ e + {1 / 2} r′ o′^2 + &c. & o = b + q e +
s o′ + {1 / 2} q′ e^2 + m o′ e + {1 / 2} s′ o′^2 + &c.</math> $cribantur <math>k + e & l + o′</math> re$pective
pro <math>x & y</math> in datis æquationibus <math>A = o & B = o</math>, & re$ultent æqua-
tiones <math>A′ = o & B′ = o</math>, quarum radices <math>e & o′</math> erunt eædem ac radices
<math>e & o′</math> ex a$$umptis æquationibus deductæ. Nunc $upponantur <math>o =
a + p e + r o′ & o = b + q e + s o′</math>, & ex his æquationibus con$e-
quuntur <math>e = {s a - r b / r q - s p} & o′ = {q a - p b / p s - q r}</math>: hinc con$tat, quod in om-
nibus ca$ibus approximationes ex hâc methodo deductæ pendent ex
hoc, utrum quantitates <math>k & l</math> a$$umptæ pro radicibus ip$is $int multo

<pb 407>[0459]<rh>SERIEBUS.</rh>
propiores ad radices quæ$itas, quam ad reliquas incognitarum (<math>x & y</math>)
radices, necne.</p>
<p>Cor. 2. Si vel duæ vel plures radices quæ$itæ datarum æquationum
$int prope inter $e æquales, tum plures termini ex æquationibus
a$$umptis retineantur: quod etiam, cum radices quæ$itæ haud $int
prope æquales, propiorem approximationem deducet.</p>
<p>Eadem principia etiam applicari po$$unt ad plures (<math>m</math>) æquationes
plures <math>m</math> incognitas quantitates habentes.</p>
<p>Cor. 3. Sint duæ prædictæ æquationes <math>α = 0 & π = 0</math>, pro <math>x & y</math>
in his æquationibus $cribantur re$pective <math>h + k √ (-1) + a √ (-1)
+ e & m + n √ (-1) + o + u √ (-1)</math>, ubi <math>h + k √ (-1) & m
+ n √ (-1)</math> $unt quantitates multo propiores ad unum valorem
quantitatum <math>x & y</math> quam ad reliquos; $cribantur hi valores pro <math>x & y</math>
in datis æquationibus, & re$ultent <math>A + B e + C o′ + D + √ (-1)
(L a + M u + N) prope = o, P + Q e + R o′ + S + √ (-1) (T a
+ H u + K)</math> prope <math>= o</math>, ubi <math>P, Q, R, S, T, H, K; A, B, C, D, L, M
& N</math> $unt cognitæ quantitates; <math>D</math> vero & <math>S</math> in $ingulis terminis plu-
res habent dimen$iones quantitatum <math>e, o, a & u</math> quam unam; reli-
quæ vero habeant nullas: fingantur <math>A + B e + C o′ = o, P + Q e
+ R o′ = o, L a + M u + N = o & T a + H u + K = o</math>, ex hi$ce
æquationibus inveniri po$$unt approximationes ad valores quantita-
tum perparvarum <math>a, e, o′ & u</math>.</p>
<p>Si duo vel plures valores re$ultantium quantitatum <math>a, e, o′, u, &c.</math>
$int inter $e æquales, tum e principiis prius traditis erui po$$unt novæ
approximationes.</p>
<p>Ea, quæ prius tradita fuere de convergentia po$$ibilium radicum,
ad convergentiam impo$$ibilium radicum æque applicari po$$unt.</p>
<p>Ex. 1. Ex datâ fluente fluxionis <math>(1 - a x^2)^-{1 / 2} x^.</math> invenire fluentem
fluxionis <math>(1 - (a + b) x^2)^-{1 / 2} x^.</math>, $i modo <math>b</math> $it perparva quantitas.
Inveniatur incrementum quantitatis <math>(1 - (a + b) x^2)^-{1 / 2}</math> ex hy-
pothe$i quod <math>a</math> $olummodo $it variabilis & <math>b</math> $it ejus incrementum,
quod erit <math>(1 - a x^2)^-{1 / 2} + {b x^2 / 2 (1 - a x^2)^{3 / 2}} + {1 · 3 b^2 x^4 / 2^2 · 2 (1 - a x^2)^{5 / 2}} +

<pb 408>[0460]<rh>DE INFINITIS</rh>
{1 · 3 · 5 b^3 x^6 / 2^3 · 2 · 3 · (1 - a x^2)^{7 / 2}} + &c.</math> ducatur hæc $eries in <math>x^.</math> & inveniantur
fluxionum re$ultantium fluentes, & perficitur exemplum.</p>
<p>Ex. 2. Sit fluxio <math>(a + b x^n + c x^2n ... b x^ln)^m x^.</math>, augeantur coeffici-
entes <math>a, b, c, d, &c.</math> per perparva incrementa <math>α, β, γ, δ, &c.</math> & re$ultat
fluxio <math>(a + b x^n + c x^2n ... b x^ln)^m x^. + m (α + β x^n + γ x^2n + &c.) ×
(a + b x^n + c x^2n + &c.)^m-1 x^. + {1 / 2}m · (m-1) (α + b x^n + γ x^2n +
&c.)^2 × (a + b x^n + c x^2n + &c.)^m-2 x^. + &c.</math> $ed ex datis independen-
tibus fluentibus <math>l</math> fluxionum formulæ <math>(a + b x^n + c x^2n ... b x^ln)^m±π x^ρn x^.</math>,
ubi <math>π & ρ</math> $unt integri numeri, acquiri po$$unt omnium eju$dem for-
mulæ fluxionum fluentes, & con$equenter fluentes e $ingulis terminis
prædictæ fluxionis.</p>
<p>Ex. 3. Sit fluxio <math>(a + b x + c x^2 + d x^3)^m x^. = (a + b x)^m x^. + m
(a + b x)^m-1 × (c x^2 + d x^3) x^. + m × {m - 1 / 2} (a + b x)^m-2 × (c x^2 +
d x^3)^2 x^. + &c.</math>, ubi <math>c & d</math> $unt parvæ quantitates; tum erit $eries exprimens
fluentem datæ fluxionis <math>{1 / (m + 1) b} (a + b x)^m+1 + ({m / b^3} × ({(a + b x)^m+2 / m + 2} -
{2 a (a + b x)^m+1 / m + 1} + {a^2 (a + b x)^m / m}) × c) + ({m / b^4} ({(a + b x)^m+3 / m + 3} -
{3 a(a + b x)^m+2 / m + 2} + {3 a^2 (a + b x)^m+1 / m + 1} + {a^3 (a + b x)^m / m}) × d) + (m ·
{m - 1 / 2} {1 / b^5} × ({(a + b x)^m+4 / m + 4} - {4 a (a + b x)^m+3 / m + 3} + {6 a^2 (a + b x)^m+2 / m + 2}
- {4 a^3 (a + b x)^m+1 / m + 1} + {a^4 (a + b x)^m / m}) c^2) + &c.</math></p>
<p>Ex. 4. Sit fluxio <math>(a + b x + c x^2 + d x^3 + e x^4)^m x^. = (a + b x +
c x^2)^m x^. + m (a + b x + c x^2)^m-1 (d x^3 + e x^4) x^. + m.{m-1 / 2} (a + b x
+ c x^2)^m-2 (d x^3 + e x^4)^2 x^. + &c.</math>; ubi <math>d & e</math> $unt parvæ quantitates:
inveniantur fluentes harum fluxionum $ucce$$ive, & re$ultat $eries
quæ$ita.</p><pb 409>[0461]<rh>SERIEBUS.</rh>
<p>Ex. 5. Sit fluxio (<math>a^3 + a′ + (3 a^2 b + b′) x + (3 a b^2 + c′) x^2 + (b^3
+ d′) x^3)^m x^.</math>; ubi <math>a′, b′, c′, &c.</math> $unt parvæ quantitates; tum erit fluxio
<math>(a + b x)^3m x^. + m (a + b x)^3x(m-1) (a′ + b′ x + c′ x^2 + d′ x^3) x^. + m · {m-1 / 2}
(a + b x)^3x(m-2) (a′ + b′ x + c′ x^2 + d′ x^3)^2 x^. + &c.</math>, cujus fluens erit
<math>{1 / (3 m + 1) b}(a + b x)^3m+1 + &c. &c.</math></p>
<p>Ex. 6. Erit <math>$. {x^. / x + e} = log. (x + e) = log. x + {e / x} - {e^2 / 2 x^2} + {e^3 / 3 x^3} -
&c., & $. {x^. / x - e} = log. (x - e) = log. x - {e / x} - {e^2 / 2 x^2} - {e^3 / 3 x^3} - {e^4 / 4 x^4} -
&c.</math>, & erit <math>$. {x^. / a^2 - (x + e)^2} = $. {x^. / a^2 - x^2} + {e / a^2 - x^2} + &c.</math></p>
<p>Ex. 7. Sit <math>N</math> numerus, cujus log. e$t <math>v</math>; tum erit numerus, cujus
log. e$t <math>(v ± e) = N ± Ne + {N e^2 / 2} ± {N e^3 / 2 · 3} + {N e^4 / 2 · 3 · 4} + {N e^5 / 2 · 3 · 4 · 5}
+ &c.</math></p>
<p>Ex. 8. Sit radius <math>= 1 & A</math> arcus, cujus $inus $it <math>S</math> & co$inus <math>C, &
A ± e</math> arcus qui multum haud differt ab arcu <math>A</math>, i. e. $it e per-
parva quantitas; tum erit $inus arcus <math>(A ± e) = S ± C e - {1 / 2} S e^2
∓ {1 / 2 · 3} C e^3 + {1 / 2 · 3 · 4} S e^4 ± &c.</math> in infinitum <math>= S (1 - {1 / 2} e^2 + {1 / 2 · 3 · 4}
e^4 - &c.) ± C (e - {1 / 2 · 3} e ^3 + {1 / 2 · 3 · 4 · 5} e 5 - &c.) = S ×</math> co$. arc. <math>e
± C ×</math> $in. arc. <math>e</math>: & co$inus prædicti arcus <math>(A ± e) = C ± Se -
{1 / 2 · 3} C e^2 ∓ {1 / 1 · 2 · 3 · 4 · 5} S e^3 + {1 / 2 · 3 · 4 · 5 · 6 · 7} C e^4 ± &c.</math> in infini-
tum <math>= C (1 - {1 / 1 · 2} e^2 + {1 / 2 · 3 · 4} e^4 - &c.) ± S (e - {1 / 2 · 3} e^3 +
{1 / 2 · 3 · 4 · 5} e^5 + &c.) = ± S ×</math> $in. arc. <math>e + C ×</math> co$. arc. <math>e</math>.</p>

<pb 410>[0462]<rh>DE INFINITIS</rh>
<p>Cor. 1. Sit æquatio <math>y = A - {A^3 / 2 · 3} + {A^5 / 2 · 3 · 4 · 5} - &c.</math> vel <math>y = 1
- {A^2 / 2} + {A^4 / 1 · 2 · 3 · 4} - &c.</math>, vel magis generaliter <math>y = E(A - {A^3 / 2 · 3}
+ &c.) + F(1 - {A^2 / 2} + {A^4 / 2 · 3 · 4} - &c.)</math>, ubi <math>E & F</math> $unt quantitates
ad libitum a$$umendæ; tum erit fluens generalis fluxionalis æquatio-
nis <math>y^.. + y z^. ^2 = 0</math>, ubi <math>z = A</math>; unde generalis fluens huju$ce fluxio-
nalis æquationis erit <math>y = E ×</math> $in. arc. <math>z + F ×</math> co$. eju$dem arc. <math>z</math>.</p>
<p>Cor. 2. Erit <math>2 S × C =</math> $in. dupli arcus <math>A</math>, ergo <math>2 S C = 2 (A -
{A^3 / 2 · 3} + {A^5 / 2 · 3 · 4 · 5} - &c.) × (1 - {A^2 / 2} + {A^4 / 1 · 2 · 3 · 4} - &c.) = 2A -
{8 A^3 / 2 · 3} + {32 A^5 / 2 · 3 · 4 · 5} - {128 A^7 / 2 · 3 · 4 · 5 · 6 · 7} + &c. - &c.</math> Et $ic ex vulgari
trigonometriâ detegi po$$unt multæ con$imiles propo$itiones.</p>
<p>1.2. Et $imiliter $it <math>z</math> log. quantitatis <math>x + √(1 + x^2)</math>, tum erit <math>x = z
+ {z^3 / 2 · 3} + {z^5 / 2 · 3 · 4 · 5} + &c.</math> $it <math>z</math> log. quantitatis <math>x′ + √ (x′^2 - 1)</math>,
tum erit <math>x′ = 1 + {z^2 / 2} + {z^4 / 2 · 3 · 4} + {z^6 / 2 · 3 · 4 · 5 · 6} + &c.</math></p>
<p>Cor. Augeatur <math>z</math> per <math>e</math>, tum evadet <math>v = z + e + {(z + e)^3 / 2 · 3} +
{(z + e)^5 / 2 · 3 · 4 · 5} + &c. = x + e x′ + {1 / 2} e^2 x + {1 / 2 · 3} e^3 x′ + {1 / 2 · 3 · 4} e^4 x + &c.
= x (1 + {1 / 1 · 2} e^2 + {1 / 2 · 3 · 4} e^4 + {1 / 2 · 3 · 4 · 5 · 6} e^6 + &c.) + x′ (e +
{1 / 2 · 3} e^3 + {1 / 2 · 3 · 4 · 5} e^5 + &c.) = x ×</math> valorem quantitatis <math>x′</math> cum <math>z =
e + x′ ×</math> valorem quantitatis <math>x</math> cum <math>z = e</math>.</p>
<p>Ad eundem modum inveniri pote$t <math>v′ = 1 + {(z + e)^2 / 1 · 2} + {(z + e)^4 / 1 · 2 · 3 · 4}

<pb 411>[0463]<rh>SERIEBUS.</rh>
+ {(z + e^6 / 1 · 2 · 3 · 4 · 5 · 6} + &c. = x′ + e × x + {e^2 / 1 · 2} x′ + {e^3 / 1 · 2 · 3} x + &c. =
x′ (1 + {e^2 / 2} + {e^4 / 2 · 3 · 4} + {e^6 / 2 · 3 · 4 · 5 · 6} + &c.) + x (e + {e^3 / 1 · 2 · 3} + {e^5 / 1 · 2 · 3 · 4 · 5}
+ &c.) = x′ ×</math> valorem quantitatis <math>x′</math> cum <math>z = e + x ×</math> valorem
quantitatis <math>x</math> cum <math>z = e</math>; &c.</p>
<p>Cor. 1. Hinc generalis fluens fluxionalis æquationis <math>y^.. = y z^. ^2</math> erit
<math>E x + F x′ = y</math>, ubi <math>z = log. x + √ (x^2 + 1) = log. x′ + √ (x′^2 - 1).</math></p>
<p>Cor. 2. Erit <math>2 (z + {z^3 / 2 · 3} + {z^5 / 2 · 3 · 4 · 5} + &c.) × (1 + {z^2 / 2} + {z^4 / 2 · 3 · 4}
+ &c.) = 2z + {2^3 z^3 / 2 · 3} + {25 z^5 / 2 · 3 · 4 · 5} + &c.</math> Et $ic de omnibus pro-
po$itionibus iis con$imilibus ex vulgari trigonometriâ deductis.</p>
<p>Ex. 9. Sit <math>x</math> tangens arcus <math>A</math>; tum erit arcus, cujus tangens e$t
<math>(x ± e) = A ± {e / 1 + x^2} ± &c.</math></p>
<p>Ex. 10. Sit <math>y</math> $inus, & erit corre$pondens arcus <math>= $. {y^. / √ (1 - y^2)}</math>; &
arcus, cujus $inus e$t <math>y + e</math>, erit <math>$. {y^. / √ (1 - (y + e)^2)} = $. {y^. / √ (1 - y^2)}
+ {e / √ (1 - y^2)} + &c.</math></p>
<p>Ex. 11. Fluxio hyperbolici arcus (<math>A</math>) pote$t e$$e <math>({a^2 + x^4 / x^4})^{1 / 2} x^.</math>, ubi
<math>x y = a, & x</math> e$t ab$ci$$a & <math>y</math> ordinata hyperbolæ; $it etiam <math>x y = a
+ e</math>, tum erit fluxio ejus hyperbolici arcus <math>= ({x^4 + (a + e)^2 / x^4})^{1 / 2} x^.
= ({x^4 + a^2 / x^4})^{1 / 2} x^. + {e × (x^4 + a^2) - {1 / 2} × a / x^2} × x^. + ({1 / 2} {(x^4 + a^2) - {1 / 2} / x^2} - {1 / 2}
{(x^4 + a^2) - 1 {1 / 2} / x^2} a^2) e^2 x^. + &c.</math>, cujus fluens erit <math>A + e × &c. + &c.</math></p>

<pb 412>[0464]<rh>DE INFINITIS</rh>
<p>Ex. 12. Sit fluxio arcus elliptici vel hyperbolici <math>= {√ (t^4 + (c^2 ± t^2) x^2) / t √ (t^2 ± x^2)}
x^. = A^.</math>, ubi <math>c & t</math> $unt conjugatæ diametri & <math>x</math> ab$ci$$a: augeantur
diametri <math>t & c</math> per perparvas quantitates <math>a & e</math>, & evadet fluxio <math>A^. =
{√ ((t + a)^4 + ((c + e)^2 ± (t + a)^2) x^2) / (t + a) √ ((t + a)^2 ± x^2)} × x^. = {√ (t^4 + (c^2 ± t^2) x^2) / t √ (t^2 ± x^2)}
x^. (A^.) + {(2 t^3 a + (c e ± t a) x^2) t × (t^2 ± x^2) - (2t^2 ± x^2) a + (t^4 + (c^2 ± t^3) x^2) / (t^4 + (c^2 ± t^2))^{1 / 2} × t^2 × (t^2 ± x^2)^1{1 / 2}}
+ &c.</math>, cujus fluens erit arcus hyperbolæ vel ellip$eos.</p>
<p>Ex. 13. Sit fluxio <math>{√ (1 - c x^2) x^. / √ (1 - x^2)}</math>, ubi <math>c</math> $it perparva quantitas; tum
erit data fluxio ($i modo pro <math>√ (1 - x^3)</math> $cribatur <math>(P)) = {x^. / P} - {c x^2 x^. / 2 P} -
{c^2 x^4 x^. / 8 P} - {c^3 x^6 x^. / 16 P} - &c.</math> cujus fluentes reperiantur <math>A - {c / 2} × {1 · A - x P / 2}
- {c^2 / 8} × {3 B - x^3 P / 4} - {c^3 / 16} × {5 C - x^5 P / 6} - {5 c^4 / 128} × {7 D - x^7 P / 8} - &c.</math>,
ubi <math>A = $ · {x^. / P}, B = {1 · A - x P / 2}, C = {3 B - x^3 P / 3}, &c.</math></p>
<p>Hinc facile deduci pote$t peripheria ellip$eos, quæ haud multum
differt a circulo.</p>
<p>Hinc e principiis prius traditis, $i duæ fluxiones haud multum vel
in variabilibus vel in invariabilibus quantitatibus iis contentis inter
$e differant, & fluente alterius fluxionis datâ, inveniri pote$t approxi-
matio ad alteram.</p>
<p>Si inter fluxiones, quarum fluentes quæruntur, nulli dentur ter-
mini, qui vel nihil vel infiniti evadant; tum plerumque $eries re$ul-
tantes convergent; $in aliter vero non.</p>
<p>Eadem principia cum aliis prius traditis applicari po$$unt ad æqua-

<pb 413>[0465]<rh>SERIEBUS.</rh>
tiones, in quibus quantitates vel variabiles vel invariabiles paululum
di$tent a quantitatibus corre$pondentibus in datis æquationibus;
datæ enim æquationes præbent primas approximationes; in æquatio-
nibus $upra dictis ex primis vero approximationibus inventis facile
per principia prius tradita erui po$$unt approximationes propiores;
& $ic deinceps.</p>
<h>THEOR. XXIII.</h>
<p>Datâ æquatione <math>0 = p - q x + r x^2 - s x^3 + &c.</math> ubi <math>p, q, r, s, &c.</math>
$unt invariabiles quantitates; $it <math>x = α</math> prope, $cribatur <math>α × π</math> pro <math>x</math>
in datâ æquatione & invenietur <math>π = {p - r a^2 + 2 s α^3 - 3 t α^4 + &c. / q α - 2 r α^2 + 3 s α^3 - 4 t α^4 + &c.}</math>
prope; & $ic deinceps: $int vero <math>α, β, γ, δ, ε, &c.</math> $ucce$$ivæ approxi-
mationes e prob. 13. deductæ, i. e. $it <math>y = α + β + γ + δ + ε + &c.</math>
tum per præcedentem methodum invenietur <math>y = α × {α + β / α} × {α + β + γ / α + β}
× {α + β + γ + δ / α + β + γ} × &c.</math> quæ re$olutio etiam ad algebraicas plures va-
riabiles quantitates habentes, ad fluxionales, incrementiales, &c.
æquationes applicari pote$t.</p>
<h>PROB. XVIII.</h>
<h it>Datâ quantitate π, in quâ continentur quæcunque invariabiles quantitates
<math>a, b, c, d, &c.</math>; invenire ejus valorem, cum <math>a</math> nibilo evadat æqualis.</h>
<p>Ita reducatur data quantitas <math>π</math>, ut nec denominator nec numerator
habeat denominatorem, & con$equenter in quantitate re$ultante <math>ρ</math>
haud contineatur negativa pote$tas vel radix: in quantitate <math>ρ</math> pro <math>a</math>
$cribatur <math>a + 0</math>; & reducantur & numerator & denominator in duas
$eries $ecundum dimen$iones quantitatis <math>x</math> progredientes, quæ $int
re$pective <math>f o^λ + k o^μ + &c. & f′ o^λ′ + k′ o^μ′ + &c.</math>; tum $i <math>λ</math> major $it
quam <math>λ′</math>, erit <math>π = o</math>; $i <math>λ = λ′</math>, erit <math>π = {f / f′}</math>; $i autem <math>λ</math> minor $it
quam <math>λ′</math>, tum erit <math>π</math> infinita quantitas. Aliter:</p><pb 414>[0466]<rh>DE INFINITIS</rh>
<p>Ex datâ æquatione <math>y = π</math>, ubi <math>y</math> e$t quantitats a$$umpta haud in
quantitate <math>π</math> contenta; inveniatur valor quantitatis <math>a = σ</math>; deinde ex
æquatione <math>σ = 0</math> inveniatur valor quantitatis <math>y = τ, & τ</math> erit quanti-
tas quæ$ita.</p>
<p>Et $ic inveniri pote$t valor quantitatis <math>π</math>, cum plures quantitates
<math>a, b, c, &c.</math> fiant nihilo æquales.</p>
<h>THEOR. XXIV.</h>
<p>1. Summa vel differentia quarumcunque duarum vel plurium in-
finitarum $erierum convergentium, vel earum functio rationalis & in-
tegralis, erit etiam convergens $eries.</p>
<p>2. Sint duæ diver$æ $eries reguraliter divergentes $ub$equentis vero
formulæ <math>a + a′ r + a″ r^2 + a′″ r^3 + a″″ r^4 + &c.</math> in infinitum, tum ple-
rumque erit earum $umma vel differentia, &c. $eries divergens; facile
con$tant e differentiis inter corre$pondentes terminos duarum $erie-
rum per principia prius tradita ca$us, in quibus $eries convergit.</p>
<p>3. Summa vel differentia duarum $erierum infinitarum, quarum
una e$t divergens, altera vero convergens, $emper erit divergens.</p>
<p>4. Ducantur duæ convergentes $eries in $e$e, & $emper re$ultabit
$eries convergens, ut prius a$$eritur.</p>
<p>5. Ducatur divergens $eries in convergentem, & æquatio re$ultans
pote$t e$$e vel divergens vel convergens; pote$t e$$e convergens, cum
$umma $eriei convergentis in divergentem ductæ nihilo $it æqualis.</p>
<p>6. Ducatur divergens $eries in divergentem, & æquatio re$ultans
pote$t e$$e vel divergens vel convergens.</p>
<p>7. Dividatur vel convergens vel divergens $eries per convergentem
vel divergentem, & $eries re$ultans pote$t e$$e vel divergens vel con-
vergens; dividatur convergens $eries per alteram nihilo æqualem, &
$eries re$ultans erit divergens; $in aliter non.</p>

<pb 415>[0467]<rh>SERIEBUS.</rh>
<h>THEOR. XXV.</h>
<p>1. Sit quantitas <math>(a ± x)^m</math>, ubi <math>m</math> haud $it affirmativus numerus,
& $eries <math>a^m ± m a^m-1 x + m · {m - 1 / 2} a^m-2 x^2 ± &c. = (a ± x)^m</math> con-
verget, cum <math>a</math> major $it quam <math>x</math>, diverget autem cum minor $it.</p>
<p>1.2. Sit quantitas <math>(x ± a)^m</math>, & $eries <math>x^m ± m x^m-1 a + m · {m - 1 / 2} x^m-2 a^2
± &c.</math> erit convergens, cum <math>x</math> major $it quam <math>a</math>, &c.</p>
<p>Cum <math>x</math> prope <math>= a</math>, tum nec hæc nec illa $eries celeriter converget.</p>
<p>1.3. Sit <math>x = a & m</math> affirmativa quantitas, tum $eries <math>(a - a)^m = a^m ×
(1 - m + {m - 1 / 2} A - {m - 2 / 3} B + {m - 3 / 4} C ... ± {m - k / k + 1} H ±
{k + 1 - m / k + 2} I (1 + {k + 2 - m / k + 3} + {k + 3 - m / k + 4} P + {k + 4 - m / k + 5} Q + &c.</math>
in in$initum)) <math>= 0</math>; ubi literæ <math>A, B, C, ... H & I</math>; etiamque <math>P, Q,
&c.</math> præcedentes terminos re$pective denotant, & <math>k + 2</math> major e$t
quam <math>m</math>; ultimo convergit.</p>
<p>Sit <math>m</math> negativa quantitas, & $eries ultimo diverget.</p>
<p>1.4. Series <math>(a + x)^m = a^m + m a x^m-1 x + &c.</math> converget, cum <math>m</math> $it
affirmativa vel negativa & minor quam <math>1 & x = a</math>.</p>
<p>2. Sit <math>m</math> vel quantitas affirmativa vel negativa & minor quam 2; tum
$eries <math>$. (a + x)^m x^. = a^m x + {m / 2} a^m-1 x^2 + {m. (m-1) / 1 · 3} a^m-2 x^3 +
{m (m - 1)(m - 2) / 1 · 3 · 4} a^m-3 x^4 + &c.</math>, cum <math>x = a</math>, $emper convergit: $in
aliter non.</p>
<p>3. Sit <math>$. (a + x)^m x^. = a^m x + {m / 2} a^m-1 x^2 + {m (m - 1) / 1 · 3} a^m-1 x^3 + &c.</math>;
hæc $eries $emper convergit, $i <math>x = - a, & m</math> $it affirmativa vel ne-
gativa & minor quam 1.</p>
<p>4. Sit $eries <math>$. (x ± a)^m x^. = {x^m+1 / m + 1} ± a x^m + {m / 2} a^2 x^m-1 ± &c.</math>; hæc
$eries converget, cum <math>x</math> major $it quam <math>a</math>.</p>

<pb 416>[0468]<rh>DE INFINITIS</rh>
<p>1.2. Sit <math>x = a</math>, & $eries <math>$. (x - a)^m x^. = {x^m+1 / m + 1} - a x^m + {m / 2} a^2 x^m-1 - &c.</math>
converget, cum <math>m</math> $it affirmativa quantitas, vel negativa & minor
quam - 1: $it <math>x = a</math>, & $eries <math>$. (x + a)^m x^. = {x^m+1 / m + 1} + a x^m +
{m / 2} a^2 x^m-1 + &c.</math> converget, cum <math>m</math> vel $it affirmativa vel negativa &
minor quam 2.</p>
<h>THEOR. XXVI.</h>
<p>Sit quæcunque rationalis fluxio <math>{x^. / x^n - p x^n-1 + q x^n-2 - &c.}</math>; inve-
niatur ejus fluens in $erie $ecundum dimen$iones quantitatis <math>x</math> pro-
grediente, quæ $it <math>A x + B x^2 + C x^3 + &c.</math> Sint vero <math>α, β, γ, δ,</math> &c.
$ucce$$ivæ radices æquationis <math>x^n - p x^n-1 + q x^n-2 - &c. = 0</math>, i. e.
<math>α</math> minor $it quam <math>β, β</math> quam <math>γ</math>, &c. per regulas prius traditas inve-
niatur <math>{x^. / x^n - p x^n-1 + &c.} = {a x^. / α - x} + {b x^. / β - x} + {c x^. / γ - x} + &c.</math> redu-
cantur hæ fluxiones in $eries $ecundum dimen$iones quantitatis <math>x</math>
progredientes, quarum inveniantur fluentes & con$tat e præcedenti-
bus, quod $eries exinde orta <math>A x + B x^2 + C x^3 + &c.</math> haud conver-
get, ni <math>x</math> inter <math>+ α & - α</math>, inter <math>β & - β</math>, inter <math>γ & - γ</math>, &c. in-
terponatur; vel <math>x = - α</math>, cum plures valores quantitatis <math>x</math> $int <math>± α</math>,
tum in quibu$dam ca$ibus prius traditis converget $eries; ergo nece$$e
e$t ut duo valores <math>π & ρ</math> quantitatis <math>x</math>, inter quos requiratur fluens,
inter <math>+ α & - α, + β & - β, + γ & - γ, &c.</math> interponerentur, vel
<math>x = - α</math> in quibu$dam ca$ibus; aliter $eries erit divergens.</p>
<p>Con$tat, quod convergentia $eriei <math>A x + B x^2 + &c.</math> dijudicanda
e$t e convergentiâ i$tius $eriei, quæ inter omnes con$imiles $eries
fluentes fluxionum <math>{a x^. / α - x}, {b x^. / β - x}, {c x^. / γ - x}, &c.</math> re$pective exprimentes,
minime convergit: i. e. ex $erie <math>$. {a x^. / α - x} = {a x / α} + {a x^2 / 2 α^2} + {a x^3 / 3 α^3} + &c.</math></p>

<pb 417>[0469]<rh>SERIEBUS.</rh>
<p>2. Nunc transformetur data fluxio, ita ut ejus fluens ab quocunque
alio valore <math>c</math> quantitatis <math>x</math> inter <math>γ & δ</math> duas proxime $ucce$$ivas radi-
ces po$ito incipiat, i. e. $cribatur <math>z + c</math> pro <math>x</math> in datâ fluxione; &
con$tat, quod $i unus valor <math>c</math> quantitatis <math>x</math> inter radices qua$cunque
<math>γ & δ</math> interponatur, & $it differentia <math>d</math> inter <math>γ & c</math> minor quam
differentia inter <math>c & δ</math>, & $i modo transformetur data fluxio
<math>{x^. / x^n - p x^n-1 + &c.}</math>, ita ut fluens incipiat a puncto <math>c</math>, i. e. in datâ fluxione
pro <math>x & x^.</math> $cribantur <math>z + c & z^.</math>, fluxionis <math>{z^. / z^n - P z^n-1 + Q z^n-2 - &c.}</math>
re$ultantis fluentem inter duos valores <math>f & g</math>, qui inter duos valores
<math>γ - c & c - γ</math> quantitatis <math>z</math> re$pective continentur per $eriem con-
vergentem <math>a z + b z^2 + c z^3 + &c.</math> exprimi po$$e; vel quod idem e$t
fluentem fluxionis <math>{x^. / x^n - p x^n-1 + &c.}</math>, quæ inter duos valores <math>c ± f &
c ± g</math> quantitatis <math>x</math> continetur, per differentiam inter duas conver-
gentes $eries <math>a f + b f^2 + c f^3 + &c. & a g + b g^2 + c g^3 + &c.</math> ex-
primi.</p>
<p>3. Si vero <math>c = {γ + δ / 2}</math>, tum quæcunque fluens fluxionis <math>{x^. / x^n - p x^n-1 + &c.}</math>,
inter duos valores <math>γ & δ</math> quantitatis <math>x</math> e prædictâ $erie <math>a z + b z^2 +
c z^3 + &c.</math> inve$tigari pote$t. Hinc ex eâdem $erie <math>a z + b z^2 + c z^3
+ &c.</math> per præcedentem methodum derivatâ haud deduci pote$t fluens
fluxionis <math>{x^. / x^n - p x^n-1 + &c.}</math> inter duos valores <math>π & ρ</math> quantitatis <math>x</math> con-
tenta, ni <math>π & ρ</math> inter duas $ibi ip$is proximas radices quantitatis <math>x</math>
con$tituantur.</p>
<p>4. Sit <math>{P / (α - x)(β - x)(γ - x)&c.} = {a′ / α - x} + {b′ / β - x} + {c′ / γ - x} + &c.
= A + Bx + Cx^2 + Dx^3 + &c.</math> & $eries <math>A + B x + C x^2 + D x^3
+ &c.</math> convergit; $i <math>α, β, γ, &c.</math> $int majores quam <math>x</math>: $int duo valores
<math>a & b</math> quantitatis <math>x</math> inter duos divi$ores vel radices <math>β & α</math> $ibi ip$is

<pb 418>[0470]<rh>DE INFINITIS</rh>
proximas po$iti; $oribatur <math>z - π</math> pro <math>x</math> in datâ æquatione
<math>{P / (α - x)(β - x)&c.}</math>, & re$ultet <math>{Q / (λ - z)(μ - z)(ν - z)} = H +
L z + M z^2 &c.</math> quæ converget, cum <math>z</math> minor $it quam <math>λ & μ & ν, &c.</math>
$in major $it quam <math>λ</math> vel <math>μ</math> vel <math>ν</math>, &c.; tum $emper diverget: nunc
fingatur <math>π</math> major quam <math>a</math>, minor vero quam <math>b</math>, & quantitates <math>π - a
& b - π</math> inter minimam affirmativam radicem <math>λ = α - π</math> & minimam
negativam <math>μ = π - β</math> po$itæ, tum hæc $eries <math>H + L z + M z^2 +
&c.</math> quæ invenit aream inter duos valores <math>a & b</math> quantitatis <math>x</math>, maxime
celeriter converget, cum <math>π = {a + b / 2}</math>, ut prius docetur; & exinde <math>z = π
- a & b - π</math> re$pective.</p>
<p>5. Cum vero $eries exprimens fluentem fluxionis <math>{P x^. / (α - x)(β - x)(γ - x) &c.}
= (H + L z + M z^2 + &c.) z^.</math> per hanc methodum de-
ducta haud $atis celeriter convergat, tum interpolandi $unt plures <math>n</math>
termini. Invenire autem tales valores quantitatis <math>z</math>, ut $eries per
hanc methodum deductæ maxime celeriter convergant, pro <math>x</math> $criban-
tur $ucce$$ive <math>n</math> quantitates <math>z - π, z - ρ, z - σ, z - τ, &c.</math> & dif-
ferentiæ quantitatum <math>π, ρ, σ, τ, &c.</math> & radicis <math>α</math> ($i modo hæ differ-
entiæ inter <math>α & π, α & ρ, &c.</math> $int re$pective minores quam re$pectivæ
differentiæ inter quantitates <math>π, ρ, &c. & β</math>) erunt in geometricâ pro-
gre$$ione; $i vero differentiæ inter reliquas quantitates <math>σ, τ, &c. & β</math>
$int minores quam differentiæ prædictæ inter <math>α & σ, τ, &c.</math> re$pective,
tum differentiæ prædictæ inter <math>β & σ, τ, &c.</math> erunt re$pective in eâdem
geometricâ progre$$ione.</p>
<p>6. Convergentiæ duarum $erierum <math>x + {x^2 / 2 a} + {x^3 / 3 a^2} + {x^4 / 4 a^3} + &c. =
$. {ax^. / a - x}, & z + {z^2 / 2 b} + {z^3 / 3 b^2} + &c. = $. {b z^. / b - z}</math>, & $imiliter duarum $e-<pb 419>[0471]<rh>SERIEBUS.</rh>
rierum <math>x - {x^2 / 2 a} + {x^3 / 3 a^2} - &c. = $. {a x^. / a + x}, & z - {z^2 / 2 b} + {z^3 / 3 b^2} - &c.
= $. {b z^. / b-z}</math>; erunt inter $e æquales, cum <math>{x / a} = {z / b}</math>.</p>
<p>F<sc>IG</sc>. <math>α</math>. 7. In prædictis $eriebus interpolandis, ut $eries evadant
maxime convergentes, nece$$e e$t $eries prædictas $ucce$$ivas æqualiter
convergere: $int tres $ucce$$ivæ radices <math>α, β & γ</math> re$pective <math>A α, A β &
A γ</math>; & requiratur fluens inter prædictos valores <math>a & b</math> contenta, qui
1<_>mo</_> inter puncta <math>α & β</math> interjaceant: transformetur ab$ci$$a <math>x</math>, quæ in-
cipit <math>a</math> puncto <math>A</math>, ita ut incipiat <math>a</math> punctis <math>π, ρ, σ, τ, &c.</math>, & invenian-
tur fluentes inter puncta <math>a & π, π & c; c & ρ, ρ & d; d & σ, σ & e; &c.</math>;
tum ut convergentiæ evadant inter $e æquales, cum puncta <math>a & b</math>
multo propius ad punctum <math>β</math> quam ad punctum <math>α</math> accedant, erunt
<math>{π a / π β} = {π c / π β} = {ρ c / ρ β} = {ρ d / ρ β} = {σ d / σ β} = {σ e / σ β} = {τ e / τ β} = &c.,</math> unde <math>a π = c π,
c ρ = ρ d, σ d = σ e, &c.</math>; etiamque <math>β π, β ρ, β σ, &c.</math> erunt quantitates
in geometricâ progre$$ione; & $ic deinceps.</p>
<p>Sit <math>h</math> ultimum punctum divi$ionis, i. e. punctum divi$ionis, cum
ab$ci$$a incipiat <math>a</math> puncto <math>β</math>; tum erit prædicta quotiens <math>{π a / π β} = &c.
= {β b / γ β} vel {β b / α β}</math>, prout <math>γ β</math> minor $it quam <math>α β</math>, necne: in hoc ca$u pun-
ctum <math>β</math> inter puncta <math>a & b</math> ponitur.</p>
<p>Si autem <math>π</math> propius $it ad punctum <math>α</math> quam ad punctum <math>β</math>, & <math>ρ</math>
propius ad punctum <math>β</math> quam ad punctum <math>α</math>; tum erit <math>{a π / α π} = {π c / α π} =
{ρ c / ρ β} = {ρ d / ρ β} = {σ d / σ β} = &c.;</math> cum $eries prædictæ maxime celeriter con-
vergant.</p>
<p>8. Si vero fractio <math>{P / (x - α)(x - β)(x - γ)&c.} = {a′ / x - α} + {b′ / x - β} +

<pb 420>[0472]<rh>DE INFINITIS</rh>
&c. = A x^-n + B x^-n-1 + &c.</math> reducatur ad terminos $ecundum reci-
procas dimen$iones quantitatis <math>x</math> progredientes, tum $eries erit con-
vergens, $i modo <math>x</math> $it major quam maxima radix vel affirmativa vel
negativa <math>(α, β, γ, δ, &c.)</math>; aliter non.</p>
<p>9. Si vero quæcunque radices $int impo$$ibiles, e. g. $it <math>α = c +
d √(- 1)</math>, tum $i in $erie cujus termini $ecundum reciprocas dimen-
$iones quantitatis <math>x</math> progrediuntur, <math>x</math> $emper major $it quam ulla
po$$ibilis radix, & quam <math>c + d & c - d</math>; vel magis generaliter $i <math>x</math><_>n</_> in-
finite major $it quam <math>(c + d √ (- 1))^n + (c - d √ (- 1))^n</math> (cum <math>n</math> $it
infinita quantitas), & <math>x</math> major quam quæcunque po$$ibilis radix, tum
$eries exinde ortæ erunt convergentes: $i vero in $eriebus, quarum
termini $ecundum dimen$iones quantitatis <math>x</math> a$cendunt, quantitas <math>x</math>
$it minor quam quæcunque po$$ibilis radix etiamque quam <math>c + d &
c - d & e + f & e - f</math>, ubi <math>c ± d √ (- 1) & e ± f √(-1)</math>, &c. $int
impo$$ibiles radices, $eries per methodum prius traditam deduci po$-
$unt convergentes; vel magis generaliter $i in a$cendente $erie <math>x</math><_>n</_>
$emper infinite minor $it quam <math>{(c^2 + d^2)^n / (c + d √(- 1))^n + (c - d √(- 1))^n}</math>;
cum <math>n</math> $it quantitas infinite magna, &c., & <math>x</math> minor quam quæcun-
que po$$ibilis radix, tum $eries re$ultans $emper converget.</p>
<p>10. Si vero quantitas reducenda ad terminos progredientes $ecundum
dimen$iones quantitatis <math>x</math> $it quæcunque algebraica irrationalis fra-
ctio; fingantur irrationales quantitates re$pective & denominator ni-
hilo æquales, & inveniantur radices æquationum exinde re$ultantium,
& $i valor quantitatis <math>x</math> major $it quam quæcunque po$$ibilis vel impo$-
$ibilis radix æquationum re$ultantium, i. e. quam divi$or cuju$cunque
irrationalis quantitatis, &c. tum de$cendens $eries $emper converget;
$i vero valores <math>a & b</math> quantitatis <math>x</math> inter duas proximas radices vel di-
vi$ores prædictos ponantur, tum per præcedentem transformationem
inveniri po$$unt duæ a$cendentes $eries, quæ inveniunt fluentem
datæ quantitatis in <math>x^.</math> ductæ inter prædictos valores <math>a & b</math> quantitatis
<math>x</math> contentam, & quæ $emper convergent.</p>

<pb 421>[0473]<rh>SERIEBUS.</rh>
<p>Et $ic per transformationem prius traditam interpolari po$$unt in-
ter duos valores <math>a & b</math> quantitatis <math>x</math> quicunque alii, qui præbeant $e-
ries magis celeriter convergentes, e quibus con$equitur fluens quæ$ita.</p>
<p>Infinitæ $unt alie transformationes, quæ $eries magis convergentes
præbeant.</p>
<p>Con$imilia iis, quæ in hoc loco & in prob. præced. traduntur, ad
$eries $ecundum legem in prob. præced. traditam, & infinitas alias
facile applicari po$$unt; earum convergentiæ enim ex ii$dem princi-
piis dijudicandæ $unt.</p>
<p>11. Series $ecundum reciprocas pote$tates quantitatis <math>x</math> magis fre-
quenter convergent, quam $eries $ecundum directas pote$tates progre-
dientes, e. g. reciprocæ $eries ex algebraicis quantitatibus prius de-
ductæ $emper convergunt, cum <math>x</math> $it major quam maxima affirmativa
& negativa radix, i. e. quam ullus divi$or datæ quantitatis, u$que ad
ejus infinitam magnitudinem.</p>
<p>12. Hinc ad detegendam finitam fluentem fluxionis <math>{x^. / x^n - p x^n-1 +
q x^n-2 - &c.}</math> inter quo$cunque duos valores <math>a</math> & <math>b</math> quantitatis <math>x</math>, inter
quos ponuntur diver$æ radices <math>α, β, γ, &c.</math> æquationis <math>x^n - p x^n-1 +
q x^n-2 - &c. = 0</math>, per præcedentem methodum $altem requiruntur
(<math>2 n</math>) diver$æ $eries, quæ e radicibus (<math>x</math>) æquationis <math>x^n - p x^n-1 + q x^n-2
- &c. = 0</math> datis facile acquiri po$$unt.</p>
<p>In hoc ca$u ita transformetur data fluxio, ut fluentes a prædictis
radicibus <math>α, β, γ, &c.</math> incipiant, vel ab iis terminentur.</p>
<p>13. Sit fluxio <math>{(A x^m + B x^m-1 + C x^m-2 + &c.)x^. / x^n - p x^n-1 + q x^n-2 - &c.}</math>, ubi <math>n</math> $it integer nu-
merus, cujus fluens $it $eries <math>a x^r + b x^s + c x^t + &c.</math> terminorum
$ecundum dimen$iones quantitatis <math>x</math> progredientium, & $i quantitates
<math>A x^m + B x^m-1 + C x^m-2 + &c. & x^n - p x^n-1 + q x^n-2 - &c.</math> nullum
habeant communem divi$orem; tum omnia, quæ prius de $erie,
fluentem fluxionis <math>{x^. / x^n - p x^n-1 + q x^n-2 - &c.}</math> de$ignante, dicta fue-

<pb 422>[0474]<rh>DE INFINITIS</rh>
rint, æque ad $eriem prædictam <math>a x^r + b x^s + c x^t + &c</math>. applicari
po$$unt.</p>
<p>14. Sit fluxio <math>(a + b x + c x^2 + d x3.. x^n)^m × (A + B x^r + C x^s + &c.)
× x^.</math>, ubi <math>m</math> haud $it integer affirmativus numerus, & quantitates <math>a +
b x + c x^2 + d x^3 .. x^n = 0, & A + B x^r + C x^s + &c. = 0</math> nullas ha-
beant radices inter $e æquales: $int radices datæ quantitatis <math>a + b x
+ c x^2 .. x^n = 0</math> re$pective <math>α, β, γ, δ &c.</math> fluens vero datæ fluxionis
per $eriem <math>P x^b + Q x^k + R x^l + &c.</math> $ecundum dimen$iones quanti-
tatis <math>x</math> progredientem exprimatur; tum hæc $eries $emper invenietur
convergens, cum <math>x</math> minor $it quam <math>α & β & γ, &c.</math> $i vero major $it,
quam <math>α</math> vel <math>β</math> vel <math>γ</math>, &c. tum divergit $eries.</p>
<p>15. Si vero $cribantur <math>z + e & z^.</math> pro <math>x</math> & <math>x^.</math> in datâ fluxione, & in-
veniatur $eries $ecundum dimen$iones quantitatis <math>z</math> progrediens, viz.
<math>P′ z^b′ + Q′z^k′ + &c.</math> quæ $it fluens re$ultantis fluxionis; in hâc $erie
pro <math>z</math> $ub$tituantur duo diver$i valores <math>π & ρ</math> quantitatis <math>z</math>, quibus
corre$pondent duo valores <math>π + e & ρ + e</math> quantitatis <math>x</math>, & $i hi duo
valores <math>π + e & p + e</math> etiamque <math>e</math> inter duas proxime $ucce$$ivas, e. g.
<math>γ & δ</math> prædictæ æquationis radices ponantur; & <math>π & ρ</math> minores $unt
quam <math>± (e - γ) vel ± (e - δ)</math>, tum convergentes erunt duæ $eries
re$ultantes.</p>
<p>16. Hinc etiam ad detegendam finitam fluentem fluxionis <math>(a + b x
+ c x^2 + &c.)^m × (A + B x^r + &c.)x^.</math> inter duos quo$cunque quan-
titatis <math>x</math> valores contentam per præcedentem methodum requiruntur
$altem <math>2 n</math> diver$æ $eries, quæ e radicibus æquationis <math>a + b x + c x^2
+ d x^3 + &c... x^n = 0</math> datis facile deduci po$$unt; <math>n + 2</math> enim requi-
runtur diver$i valores prædictæ quantitatis <math>e</math>.</p>
<p>17. Eodem modo $it fluxio <math>(a + b x + c x^2 + &c.)^m (d + e x + f x^2 +
&c.)^n × (h + k x + l x^2 + &c.)^p &c. × (A x^r + B x^s + &c.)x^.</math>, ubi quan-
titates <math>a + b x + c x^2 + &c. d + e x + f x^2 + &c. h + k x + &c. &c.</math>
nullum habeant divi$orem inter $e æqualem vel communem, & $int
<math>α, β, γ, &c. λ, μ, ν, &c. π, ρ, σ, &c. &c.</math> re$pective radices æquationum
<math>a + b x + c x^2 + &c. = 0, d + e x + f x^2 + &c. = 0, h + k x + l x^2
+ &c. = 0, &c.</math> & $ic $cribantur <math>z + e & z^. pro x & x^.</math> in datâ fluxione,

<pb 423>[0475]<rh>SERIEBUS.</rh>
& e fluxione re$ultante inveniatur $eries $ecundum dimen$iones quan-
titatis <math>z</math> progrediens, quæ $it <math>P z^Γ + Q z^Δ + &c.</math> in hâc $erie pro <math>z</math>
$cribantur duo valores <math>τ & v</math>, quibus corre$pondent duo valores <math>τ + e
& v + e</math> quantitatis <math>x</math>: $i duo valores <math>τ + e & v + e</math> con$tituantur
inter duos proxime $ucce$$ivos valores <math>λ & μ</math> radicum <math>α, β, γ, &c.
λ, μ, ν, &c. π, ρ, σ, &c. &c. & τ & v</math> $int minores quam <math>± (e - λ)</math> vel
<math>± (e - μ)</math>, tum convergent $eries re$ultantes ex hâc $ub$titutione.</p>
<p>18. Hinc $i numerus radicum <math>α, β, γ, &c. λ, μ, ν, &c. π, ρ, &c. &c.</math>
$it <math>ν</math>, tum ex <math>2 ν</math> diver$is $eriebus proprie in$titutis prædicti generis
datis, quæ incipiunt a re$pectivis radicibus <math>α, β, γ, &c.</math> vel ad eas ter-
minantur, acquiri pote$t fluens datæ fluxionis inter quo$cunque duos
<math>φ & χ</math> quantitatis <math>x</math> valores contenta, ni fluens prædicta $it infinite
magna. Cum <math>T + e vel v + e</math> æqualis $it valori radicum prædicta-
rum, tum e principiis prius traditis in omnibus præcedentibus ca$i-
bus erui pote$t, utrum $eries re$ultans $it convergens, necne.</p>
<p>Sed hìc animadvertendum e$t quod $i in omnibus præcedentibus
ca$ibus valor quantitatis <math>x</math> a$$umptus $uperet maximam radicem præ-
dictarum æquationum, tum recurrendum e$t ad $eriem $ecundum
reciprocas pote$tates quantitatis <math>x</math> progredientem, & $emper converget
$eries.</p>
<p>In pleri$que ca$ibus præ$tat per methodum prius traditam plures
$eries a$$umere, quæ corre$pondent pluribus valoribus quantitatis <math>x</math>
inter <math>φ & χ</math> po$itis.</p>
<p>19. Si vero impo$$ibiles radices in prædictis æquationibus continean-
tur, quæ $int re$pective <math>x + a + √(- b^2) & x + a - √(- b^2)</math>, &
$i modo dici po$$it cum converget $eries <math>A x^r + B x^s + C x^t + &c. =
{1 / x^2 + 2 a x + a^2 + b^2}</math>, quod $emper evadet ut prius dicitur in $eriebus
$ecundum dimen$iones quantitatis <math>x</math> progredientibus, cum <math>a - b</math> ma-
jor $it quam <math>x</math>; in $eriebus autem $ecundum reciprocas ejus dimen-
$iones progredientibus, cum <math>x</math> major $it quam <math>a + b</math>, tum etiam dici<pb 424>[0476]<rh>DE INFINITIS</rh>
pote$t, cum fluentes fiuxionum prius traditarum etiam convergent; &
magis generaliter, $i <math>{(a + √(- b^2))^n + (a - √(- b^2))^n / 2}</math>, ubi <math>n</math> $it
infinita quantitas, infinite minor $it quam <math>x^n</math>, tum converget reciproca
$eries prædicta; $i <math>x^n</math> infinite minor $it quam <math>{(a^2 + b^2)^n / (a + √(- b^2))^n + (a - √(- b^2))^n}</math>,
tum a$cendens $eries $emper converget; cum autem ma-
jores $int hæ quantitates, tum $emper divergunt $eries prædictæ.</p>
<p>20. Sit fluxio <math>P^-r × Q^-m × R x^.</math>, cujus fluens requiritur, ubi literæ <math>P, Q
& R</math> de$ignant qua$cunque algebraicas $unctiones literæ <math>x^n</math> nullos di-
vi$ores communes habentes, <math>r & m</math> fractiones qua$cunque; a$$umatur
<math>P^-r+1 × Q^-m+1 × (A + B x^n + C x^2n + D x^3n + &c.)</math> pro fluente quæ-
$itâ, & $i hæc $eries <math>A + B x^n + C x^2n + D x^3n + &c.</math> in infinitum
progrediatur, tum hæc $eries $emper converget, cum $eries ad $implices
terminos ex divi$ione, extractione radicum, &c. reducta convergat.</p>
<p>21. Sit irrationalis quantitas duplicis formulæ <math>(a + b x + c x^2 + &c.
+ (e + f x + g x^2 + &c.)^m)^r = P</math>; ubi <math>m & r</math> $unt fractionales vel
negativi indices; reducatur hæc $eries ad terminos $ecundum dimen-
$iones quantitatis <math>x</math> progredientes, viz. <math>A x^t + B x^t+s + &c.</math>: $int <math>α,
β, γ, δ, &c.</math> radices æquationis <math>e + f x + g x^2 + &c. = 0</math>, quarum <math>α</math>
minor $it quam <math>β, β</math> quam <math>γ, γ</math> quam <math>δ, &c.</math>, deinde inveniantur ra-
dices (<math>x</math>) æquationis <math>a + b x + c x^2 + &c. + (e + f x + g x^2 + &c.)^m
= 0</math> (ii$dem valoribus radicis <math>^{1 / m} √</math> u$rtatis, qui in datâ $erie u$urpan-
tur) quæ $int <math>π, ρ, σ, τ, &c.</math>, quarum <math>π</math> minor $it quam <math>ρ, ρ</math> quam <math>σ,
σ</math> quam <math>τ, &c.</math>; tum, $i <math>x</math> minor $it quam minima radix <math>α, β, γ, &c.</math>;
<math>π, ρ, σ, &c.</math>; $eries a$cendens $emper erit convergens: $i major $it quam
maxima <math>α, β, γ, &c.; π, ρ, σ, &c.</math>; $eries reciproca, i. e. de$cendens $em-
per erit convergens.</p>
<p>Eadem etiam applicari po$$unt ad qua$cunque algebraicas irratio-
nales quantitates.</p>

<pb 425>[0477]<rh>SERIEBUS.</rh>
<p>Con$imilia etiam vere affirmari po$$unt de convergentiâ harum
algebraicarum quantitatum iis, quæ prius tradita fuere de convergen-
tiâ $erierum prius traditarum.</p>
<p>Con$imilia etiam affirmari po$$unt (mutatis mutandis) de conver-
gentiâ $erierum $ecundum dimen$iones quantitatis <math>x</math> progredientium,
quæ exprimunt fluentes omnium algebraicarum quantitatum, e. g.
<math>$. P x^.</math>, iis, quæ prius tradita fuere de convergentiâ $erierum, quæ $unt
fluentes fluxionum prius traditarum.</p>
<p>1.2. Hinc facile deduci po$$unt infinitæ divergentes $eries, quæ quo-
dammodo dici po$$unt æquales finitis quantitatibus; $it fluens con-
tenta inter valores <math>a & b</math> quantitatis <math>x</math> finita quantitas; vel, $i in datâ
functione pro <math>x</math> $cribantur <math>a & b</math> re$pective, re$ultent functiones, quæ
$int finitæ quantitates; deinde in datâ fluente vel functione pro <math>x</math> $cri-
batur <math>z + e, &c.</math> & fluens vel functio re$ultet, quæ in $eriem $ecundum
dimen$iones quantitatis <math>z</math> expan$a præbeat divergentem $eriem, cum
<math>z = a - e</math> vel <math>b - e, &c.</math>; tum re$ultans divergens $eries, vel diver$æ
re$ultantes $eries, earum $umma, differentia, & rationalis & integralis
functio quodammodo dici pote$t æqualis datæ finitæ quantitati ex
datâ functione facile deducendæ.</p>
<p>Et $ic de pluribus huju$cemodi functionibus.</p>
<p>Et idem mutatis mutandis affirmari pote$t de incrementialibus
quantitatibus $imiliter reductis.</p>
<p>1. 3. Sit quæcunque fluxio <math>A x^.</math>, ubi <math>A</math> $it functio quantitatis <math>x</math>; in
quantitate <math>A</math> pro <math>x</math> $cribantur re$pective <math>a, a + {1 / n}b, a + {2 / n}b, a + {3 / n} b
... a + {n-1 / n} b, a + b</math>, ubi <math>n</math> e$t integer numerus, & $int quantitates
re$ultantes re$pective <math>A, B, C, D,....P, Q, R</math>; tum fluens fluxionis
datæ inter valores <math>a & b</math> quantitatis <math>x</math> contenta, inter duas quanti-
tates <math>(A + B + C...P + Q)({a - b / n}), & (B + C + D...P +

<pb 426>[0478]<rh>DE INFINITIS</rh>
Q + R)({a - b / n})</math> ver$atur, $i nullus valor quantitatis <math>x</math> in æquatione <math>{A / x^.}
= 0</math> inter duos prædictos valores <math>a & b</math> contineatur.</p>
<h>THEOR. XXVII.</h>
<p>1. Sit data quantitas <math>A</math> functio incognitæ quantitatis <math>x</math>, cujus mi-
nima radix in æquationibus per methodum in præcedente proble-
mate traditam deductis $it <math>α</math>; deinde transformetur hæc quantitas <math>A</math>
in alteram (<math>B</math>), cujus incognita quantitas $it <math>z</math> functio quantitatis <math>x</math>,
& inveniatur minima radix æquationum ex quantitate <math>B</math> per præ-
dictam methodum re$ultantium, quæ $it <math>β</math>; reducantur quantitates
<math>A & B</math> ad $eries $ecundum dimen$iones quantitatum <math>x & z</math> a$cendentes,
& $int <math>a & π</math> corre$pondentes valores quantitatum <math>x & z</math>, tum $erie-
rum convergentiæ erunt majores vel minores prout rationes <math>α: a &
β: π</math> $int majores vel minores: $int <math>α′ & β′</math> maximæ radices prædicta-
rum æquationum, & reducantur quantitates <math>A & B</math> ad $eries $ecun-
dum dimen$iones quantitatum <math>x & z</math> de$cendentes, tum convergentiæ
$erierum re$ultantium erunt minores vel majores prout rationes <math>α′: a
& β′: π</math> $unt majores vel minores.</p>
<p>2. Sit data fluxio <math>Ax^.</math>, ubi <math>A</math> e$t prædicta functio quantitatis <math>x</math>; &
reducatur quantitas ad $eriem $ecundum dimen$iones quantitatis <math>x</math> vel
a$cendentem vel de$cendentem, & inveniatur ejus fluens; transfor-
metur data fluxio <math>A x^.</math> in alteram <math>B z^.</math>, cujus variabilis <math>z</math> e$t quæcunque
functio variabilis <math>x</math>; reducantur duæ fluxiones <math>A x^. & B z^.</math> ad $eries $e-
cundum dimen$iones quantitatum <math>x & z</math> re$pective a$cendentes vel
de$cendentes, & $int <math>a & b</math> valores quantitatis <math>x</math>, inter quos continetur
fluens quæ$ita, etiamque <math>π & ρ</math> eorum corre$pondentes valores quan-
titatis <math>z</math>: tum convergentiæ $erierum re$ultantium erunt majores vel
minores prout rationes <math>α′</math>: ad minorem quantitatem <math>a</math> vel <math>b</math>, vel <math>β′</math> ad
minorem <math>π</math> vel <math>ρ</math>, in de$cendentibus $eriebus; vel major <math>a</math> vel <math>b: α</math>, vel
major <math>π</math> vel <math>ρ: β</math> in a$cendentibus $eriebus, $int minores vel majores.</p>
<p>Facile con$tant e præced. theor.</p>

<pb 427>[0479]<rh>SERIEBUS.</rh>
<p>Ex. 1. Fluens fluxionis <math>$. {x^. / 1 + x} = 1 - {1 / 2}x + {1 / 3}x^3 - {1 / 4}x^4 + &c. vel
= {x / 1 + x} - {x^2 / 2 (1 + x)^2} + {x^3 / 3 (1 + x)^3} - {x^4 / 4 (1 + x)^4} + {x^5 / 5 (1 + x)^5} -
&c.</math>; hæc po$terior $eries, quæ progreditur $ecundum dimen$iones
quantitatis <math>{x / 1 + x}</math> (modo <math>x</math> $it affirmativa quantitas) $emper celerius con-
verget quam prior; nam <math>{x / 1 + x} = v</math> minor erit quam <math>x</math>: $it <math>x = 1</math>,
tum $eries erunt <math>1 - {1 / 2} + {1 / 3} - {1 / 4} + {1 / 5} - &c. & {1 / 2} - {1 / 2 · 4} + {1 / 3 × 8} - {1 / 4 × 16}
+ {1 / 5 × 32} - {1 / 6 × 64} + &c.</math>; hæc $eries multo celerius converget quam
prior, nam in priori $erie $ingulus terminus erit ad ejus $ucce$$ivum
prope in ratione æqualitatis, in po$teriori in ratione 2: 1 prope: ali-
ter, $it <math>{x / 1 + x} = v</math> & exinde <math>{x^. / 1 + x} = {v^. / 1 - v}</math>; radix æquationis <math>1 +
x = 0</math> e$t <math>- 1</math>, & radix æquationis <math>1 - v = 0</math> e$t <math>1</math>; cum autem re-
quiratur fluens inter duos valores <math>0 & {1 / 2}</math> quantitatis <math>v</math>, qui valoribus
<math>0 & 1</math> quantitatis <math>x</math> corre$pondent: minima vel unica radix prioris
æquationis <math>1 + x = 0</math> erit ad quantitatem <math>x:: ± 1: 1</math>; at unica radix
æquationis <math>1 - v = 0</math> erit ad quantitatem <math>v</math> (in majore quam præce-
dente ratione):: <math>1: {1 / 2}</math>; ergo $eries <math>1 - {1 / 2 · 2^2} + {1 / 3 · 2^3} - &c.</math> magis
celeriter coverget, quam corre$pondens $eries <math>1 - {1 / 2} + {1 / 3} - {1 / 4} + &c.</math></p>
<p>Ex. 2. Sit $eries <math>$. {x^. / 1 + x} = x - {1 / 2}x^2 + {1 / 3}x^3 - &c.</math>; pro <math>x & x^.</math> $cri-
bantur <math>{a + b v / c + d v} & {b c - d a / (c + d v)^2} v^.</math> in fluxione <math>{x^. / 1 + x}</math>, & re$ultat <math>{(b c - d a)v^. / (c + d v) (c + a + (d + b)v)}</math>:
cujus fluens reducta ad $eriem evadet <math>(b c - d a)
({1 / c · (c + a)} v - {2 c d + c b + a d / c^2 (c + a)^2} × {v^2 / 2} + &c.)</math>; huju$ce $eriei qui-

<pb 428>[0480]<rh>DE INFINITIS</rh>
cunque terminus ad infinitam di$tantiam erit ad ejus $ucce$$ivum-
quam proxime in minore duarum $ub$equentium ratione, viz. <math>c: d v
& c + a: (d + b) v</math>: $i requiratur fluens fluxionis inter duos valores
<math>α & β</math> quantitatis <math>x</math> contenta, & con$equenter inter duos corre$pon-
dentes valores <math>{c α - a / b - d α} & {c β - a / b - d β}</math> quantitatis <math>v</math>; tum convergentia
$eriei <math>x - {1 / 2} x^2 + {1 / 3} x^3 - &c.</math> i. e. $erierum <math>α - {1 / 2} α^2 + {1 / 3} α^3 - &c. &
β - {1 / 2} β^2 + {1 / 3} β^3 - &c.</math> major vel minor erit, prout minor ratio <math>1: α</math>
vel <math>1: β</math> major vel minor $it; & convergentia $eriei <math>{1 / c (c + a)} v -
{2 c d + c b + a d / c^2 (c + a)^2} × {v^2 / 2} + &c.</math> major vel minor erit prout minor ratio
<math>c: {c α - a / b - d α} × d, c: {c β - a / b - d β} × d; a + c: (d + b) × {c α - a / b - d α}</math> vel <math>a + c: (d
+ b) × {c α - a / b - d α}</math> major vel minor $it: convergentia totius dijudicanda
e$t ex minori convergentiâ.</p>
<p>Hoc con$tat ex theoremate præcedente; nam radix denominatoris
<math>1 + x = 0</math> in priori ca$u e$t <math>- 1</math>; in po$teriori ca$u radices denomi-
natoris $unt <math>= - {c / d} & = - {d + b / c + a}</math>; & rationes radicum ad <math>v</math> in po-
$teriori ca$u & radicis <math>- 1</math> ad <math>x</math> in priori erunt præcedentes.</p>
<p>Exhinc facile deduci po$$unt valores quantitatum <math>a, b, c & d</math>; in
quibus prædictæ rationes evadunt maximæ, & con$equenter $eries
convergunt maxime.</p>
<h>THEOR. XXVIII.</h>
<p>Sit <math>P</math> functio quantitatis <math>x</math>; reducatur in duas $eries, quarum altera
<math>a x^r + b x^n+r + c x^2n+r + &c. = A</math> $ecundum dimen$iones quantitatis
<math>x^n</math> a$cendit, altera vero <math>a′ x^r′ + b x^r′-n + &c.</math> $ecundum dimen$iones
eju$dem quantitatis <math>x^n</math> de$cendit; & $i una $eries <math>A</math> covergat cum <math>x
= α</math>, tum altera <math>B</math> $emper divergit cum <math>x = α</math>: ni cum <math>x = β</math>, ubi<pb 429>[0481]<rh>SERIEBUS.</rh>
$eries (<math>A</math>) exinde re$ultans $it ultima $eries convergens, vel prima quæ
non convergit; etiamque $it $eries <math>B</math> prima convergens, vel ultima quæ
non convergit: & in utroque cafu eadem $it $eries.</p>
<p>Ex. 1. Sit <math>$. {x^. / 1 + x^2} = x - {x^3 / 3} + {x^5 / 5} - {x^7 / 7} + &c. = - {1 / x} + {1 / 3 x^3} -
{1 / 5 x^5} + &c.</math>; tum $i una $eries convergit, altera nece$$ario diverget;
ni <math>x = 1</math>, in quo ca$u utraque $eries eadem evadit, at una e$t alterius
negativa, viz. una e$t <math>1 - {1 / 3} + {1 / 5} - {1 / 7} + &c.</math>, altera vero <math>- 1 + {1 / 3} +
{1 / 5} + {1 / 7} - &c.</math></p>
<p>Cor. Series, quæ e$t $umma duarum prædictarum $erierum <math>A ± B</math>
$emper divergit, ni utraque $eries eadem evadat, <math>A = B</math>; i. e. in
exemplo $eries <math>x - x^-1 - ({x^3 - x^-3 / 3}) + ({x^5 - x^-5 / 5}) - &c.</math> $em-
per divergit, ni <math>x = 1</math>.</p>
<h>THEOR. XXIX.</h>
<p>Reducatur functio data quantitatis <math>x</math> ad minimos terminos, ita uæ
quantitates in numeratore & denominatore contentæ nullum habeant
denominatorem: fingatur denominator <math>Q = 0</math>, & omnis irrationalis
quantitas vel in numeratore vel in denominatore contenta nihilo
æqualis, & $it <math>α</math> minima radix affirmativa vel negativa, at non <math>= 0</math>
prædictarum re$ultantium æquationum; deinde reducatur data fun-
ctio ad $eriem $ecundum dimen$iones quantitatis <math>x</math> a$cendentem, tum
hæc $eries $emper converget, $i quantitas <math>x</math> contineatur inter <math>α & - α</math>,
quod$i <math>x</math> major $it quam <math>α</math>, tum prædicta $eries diverget.</p>
<p>Si hæc $eries in <math>x</math> ducatur, tum $eries denotans fluentem inter duos
valores <math>a & b</math> quantitatis (<math>x</math>) contentam $emper converget, cum <math>a & b</math>
inter <math>α & - α</math> interponantur.</p>
<p>2. Reducatur prædicta functio ad $eriem $ecundum reciprocas di-
men$iones quantitatis <math>x</math> de$cendentem, & $it <math>λ</math> maxima radix prædi-
ctarum re$ultantium æquationum, & requiratur fluens inter duos
valores <math>a & b</math> quantitatis <math>x</math> contenta; $eries $emper converget, cum

<pb 430>[0482]<rh>DE INFINITIS</rh>
utræque <math>a & b</math> $int majores quam <math>λ</math>; $i una $it minor quam <math>λ</math>, tum
$emper diverget $eries prædicta.</p>
<p>3. Cum <math>x = α</math> in priori ca$u, vel <math>= λ</math> in po$teriori; tum in qui-
bu$dam ca$ibus evadet convergens $eries, in quibu$dam non; qui
ca$us facile con$tant e principiis prius traditis.</p>
<p>4. Si quædam radices $int impo$$ibiles, e. g. $int <math>a + b √(- 1) &
a - b √ (- 1)</math>; tum a$cendens $eries $emper converget, cum <math>x^-n</math> infinite
major $it quam <math>{2(a^n - n · {n - 1 / 2} a^n-2 b^2 + n · {n - 1 / 2} · {n - 2 / 3} · {n - 3 / 4} a^n-4 b^4 - &c.) / (a^2 + b^2)^n}</math>,
ubi <math>n</math> e$t quantitas infinite magna; & de$cendens $eries
$emper converget, cum <math>x^n</math> infinite major $it quam <math>2(a^n - n · {n - 1 / 2}
a^n-2 b^2 + n · {n - 1 / 2} · {n - 2 / 3} · {n - 3 / 4}a^n-4 b^4 + &c.)</math>.</p>
<p>Ea, quæ prius tradita fuerunt de interpolatione $erierum, hìc etiam
applicari po$$unt.</p>
<p>Cor. Exhinc deduci po$$unt infinitæ convergentes $eries, quarum
$ummæ innote$cunt; nam a$$umatur quæcunque functio quantitatis
<math>x</math>, &c. pro $ummâ, & reducatur ea ad $eriem $ecundum dimen$iones
quantitatis <math>x</math>, &c. progredientem; per theorema hìc datum ita a$$u-
matur valor quantitatis <math>x</math>, ut $eries evadat convergens; & confit co-
roll.</p>
<h>PROB. XIX.</h>
<h it>Transformare datam algebraicam quantitatem in alteram, cujus termini
progrediuntur $ecundum quantitates formulœ <math>a x^r (x ± 1)^r</math>.</h>
<p>Erunt <math>± x^n = x^n × (x ± 1)^n - n x^n+1 × (x ± 1)^n+1 + n · {n + 3 / 2} x^n+2
× (x ± 1)^n+2 - n · {n + 4 / 2} · {n + 5 / 3}x^n+3 × (x ± 1)^n+3 + n · {n + 5 / 2} · {n + 6 / 3} ·

<pb 431>[0483]<rh>SERIEBUS.</rh>
{n + 7 / 4}x^n+4 × (x ± 1)^n+4 - &c. & x^-n = x - {1 / 2} n × (x ± 1) - {1 / 2} n ± {n / 2} ×
x^{-n-1 / 2} × (x ± 1)^{-n-1 / 2} + {n^2 / 8} x^{-n-2 / 2} × (x ± 1)^{-n-2 / 2} ± {n × (n + 1) × (n - 1) / 2 · 4 · 6}
x^{-n-3 / 2} × (x ± 1)^{-n-3 / 2} - &c.</math></p>
<p>Scribantur hæ quantitates pro $uis valoribus in datâ algebraicâ
quantitate, & perficitur problema.</p>
<p>Ex vulgaribus vel novis, &c. modis multiplicandi, dividendi & radi-
ces extrahendi reduci pote$t algebraica quantitas ad $eriem infinitam
$ecundum prædictam legem progredientem.</p>
<p>Ex methodis haud di$$imilibus transformari po$$unt datæ alge-
braicæ quantitates in terminos $ecundum dimen$iones quantitatum
huju$ce formulæ <math>x^r × (x ± x<x>.</x>)^r × (x ± 2x<x>.</x>)^r ... (x ± n x<x>.</x>)^r</math>, vel $ecun-
dum infinitas alias progredientes.</p>
<h>THEOR. XXX.</h>
<p>Incrementum geometricæ $eriei <math>a x + a^2 x^2 + a^3 x^3 + &c. erit =
{a (x + x) / 1 - a x - a x^.} - {a x / 1 - a x} = ({a x / (1 - a x)^2} + {a^2 x^2 / (1 - a x)^3} + {a^3 x^3 / (1 - a x)^4} +
&c.) = {a x / (1 - a x<x>.</x>)} + ({a / (1 - a x)^2} - a) x + ({a^2 / (1 - a x)^3} - a^2)x^2 +
({a^3 / (1 - a x<x>.</x>)^4} - a^3) x^3 + &c.</math></p>
<h>THEOR. XXXI.</h>
<p>1. Sit $eries <math>A + B x + C x^3 + D x^5 + E x^7 + &c.</math> in infinitum; pro
ejus integrali primi ordinis, cum <math>x</math> cre$cat uniformiter, a$$umi pote$t
feries <math>a x + b x × (x - x) + c x^2 (x - x)^2 + d x^3 (x - x)^3 + &c.</math> cu-
jus incrementum erit <math>a x<x>.</x> + b x (x + x<x>.</x> - (x - x<x>.</x>)) + c x^2 ((x + x<x>.</x>)^2
- (x - x)^2) + &c. = a x<x>.</x> + 2 b x x<x>.</x> + (4 c + 2 d x<x>.</x>^2)x^3 x<x>.</x> + (6 d +

<pb 432>[0484]<rh>DE INFINITIS</rh>
8 e x<x>.</x>^3) x^5 x<x>.</x> + &c.</math> unde <math>a = {A / x^.}, B = 2 b x<x>.</x></math>, &c. vel $eries <math>a x + b x
(x - x<x>.</x>) + c x^3 × (x - x<x>.</x>)^3 + d x^5 · (x - x<x>.</x>)^5 + &c.</math> cujus incremen-
tum erit <math>a x<x>.</x> + 2 b x<x>.</x> x + 2 c x<x>.</x>^3 x^3 + (2 d x<x>.</x>^4 + 6 c) x<x>.</x> x^5 + &c.</math> vel $e-
ries <math>a x + b x^2 + c x^2 (x - x<x>.</x>)^2 + d x^4 (x - x<x>.</x>)^4 + &c.</math> cujus incre-
mentum erit <math>a x<x>.</x> + b x<x>.</x>^2 + 2 b x<x>.</x> x + 4 c x<x>.</x> x^3 + 8 d x<x>.</x>x^5 + &c.</math> & ex
æquatis corre$pondentibus harum & datæ $eriei <math>(A + B x + C x^3 +
&c.)</math> terminis inve$tigari po$$unt coefficientes <math>a, b, c, d,</math> &c.</p>
<p>2. Sit $eries infinita <math>A + B x + C x^2 + D x^3 + &c.</math> a$$umatur pro
ejus integrali primi ordinis, cum <math>x</math> cre$cat uniformiter, $eries <math>a x + b x
× (x - x<x>.</x>) · (x - 2x<x>.</x>) + c x^2 × (x - x)^2 × (x - 2x<x>.</x>)^2 + d x^3 · (x - x<x>.</x>)^3
· (x - 2x<x>.</x>)^3 + &c.</math> cujus incrementum erit <math>a x<x>.</x> + 3 b x<x>.</x> × x × (x - x<x>.</x>)
- c(3 x<x>.</x>^2 - 6 x x<x>.</x>) x^2 × (x - x<x>.</x>)^2 + &c. = a x<x>.</x> - 3 b x<x>.</x>^2 × x + (3 b x<x>.</x>
- 3 c x<x>.</x>^4)x^2 + &c.</math> & ex æquatis huju$ce & datæ $eriei corre$ponden-
tibus coefficientibus erui po$$unt quantitates <math>a, b, c, d,</math> &c.</p>
<p>Et $ic infinitæ diver$æ $eries $ecundum dimen$iones quantitatis <math>x</math>
progredientes facile deduci po$$unt, quæ datam $eriem de$ignant.</p>
<p>1. 2. Si vero integralis <math>n</math> ordinis prædictæ $eriei <math>A + B x + C x^2 +
&c.</math> requiratur: pro integrali quæ$itâ a$$umatur $eries <math>a x + b x ×
(x ± x<x>.</x>) × (x ± 2x<x>.</x>) ... (x ± nx<x>.</x>) + c x^2 × (x ± x<x>.</x>)^2 × (x ± 2x<x>.</x>)^2
...(x ± n x<x>.</x>)^2 + d x^3 × (x ± x<x>.</x>)^3 × (x ± 2x<x>.</x>)^3</math> &c. cujus incrementum
<math>n</math> ordinis inveniatur in terminis $ecundum dimen$iones quantitatis <math>x</math>
progredientibus, & re$ultantis & datæ $eriei termini corre$pondentes
inter $e æquentur; & exinde deduci po$$unt coefficientes <math>a, b, c, d,</math> &c.</p>
<p>3. Seriem infinitam <math>A + B x^-n-1 + C x^-n-2 + D x^-n-3 + &c.</math> in
alteram transformare, cujus termini $ecundum reciprocas dimen$iones
quantitatis <math>x</math> progrediuntur, & cujus integralis cuju$cunque <math>n</math> ordi-
nis exprimi etiam pote$t in terminis $ecundum eandem legem progre-
dientibus.</p>
<p>A$$umatur $eries <math>a + {b / x} + {c / x × (x + x<x>.</x>)} + {d / x × (x + x<x>.</x>) × (x + 2 x<x>.</x>)}
+ {e / x × (x + x<x>.</x>) × (x + 2 x<x>.</x>) × (x + 3 x<x>.</x>)} + &c.</math> & inveniatur incremen-

<pb 433>[0485]<rh>SERIEBUS.</rh>
tum <math>n</math> ordinis, & deinde æquentur termini datæ & re$ultantis æqua-
tionis corre$pondentes, & ex æquationibus re$ultantibus erui po$$unt
coefficientes <math>a, b, c,</math> &c.</p>
<p>4. Sit $eries <math>A x^s + B x^s+t + C x^s+2t + D x^s+3t + &c.</math> cujus inte-
gralis requiritur; pro integrali primi ordinis a$$umatur <math>a x^s × (x ± x<x>.</x>)^s
+ b x^s+1 × (x ± x<x>.</x>)^s+1 + c x^s+2 × (x ± x<x>.</x>)^s+2 + &c. &c. + a′ x^s+t ×
(x ± x<x>.</x>)^s+t + b′x^s+t+1 × (x ± x<x>.</x>)^s+t+1 + c′ x^s+t+2 × (x ± x<x>.</x>)^s+t+2 + &c. +
a″ x^s+2t × (x ± x<x>.</x>)^s+2t + b″x^s+2t+1 × (x ± x<x>.</x>)^s+2t+1 + &c. + &c.</math> invenia-
tur ejus incrementum, cujus termini fiant re$pective æquales cor-
re$pondentibus datæ $eriei terminis, & exinde erui po$$unt cofficien-
tes <math>a, b, c, &c. a′, b′, c′, &c. a″, b″, c″, &c. &c.</math></p>
<p>Si requiratur integralis <math>n</math> ordinis prædictæ $eriei, a$$umatur pro
integrali quæ$itâ quantitas <math>a x^s × (x ± x<x>.</x>)^s × (x ± 2x<x>.</x>)^s .. (x ± n x<x>.</x>)^s +
b x^s+1 × (x ± x<x>.</x>)^s+1 × (x ± 2 x<x>.</x>)^s+1 .. (x ± n x<x>.</x>)^s+1 + c x^s+2 × (x ± x<x>.</x>)^s+2
× (x ± 2x<x>.</x>)^s+2 .. (x ± n x<x>.</x>)^s+2 + &c. + a′ x^s+t × (x ± x<x>.</x>)^s+t × (x ± 2x<x>.</x>)^s+t
.. (x ± n x<x>.</x>)^s+t + b′x^s+t+1 × (x ± x<x>.</x>)^s+t+1 × (x ± 2x<x>.</x>)^s+t+1 .. (x ± nx<x>.</x>)^s+t+k
+ &c. + a″x^s+2t × (x ± x<x>.</x>)^s+2t × (x ± 2x<x>.</x>)^s+2t ... (x ± nx<x>.</x>)^s+2t + &c.
+ &c.</math> inveniatur ejus incrementum <math>n</math> ordinis, deinde ejus & datæ
$eriei corre$pondentes termini fiant inter $e æquales, & exinde erui
po$$unt cofficientes <math>a, b, c, &c. a′, b′, c′, &c. a″, b″, c″, &c.</math></p>
<p>5. In a$$umptis quantitatibus huju$ce formulæ <math>a x^r × (x - x<x>.</x>)^r ×
(x - 2 x<x>.</x>)^r × ... (x - n x<x>.</x>)^r</math>, $i <math>r</math> $it fractio, cujus denominator $it par
numerus, & impo$$ibiles quantitates irrepant in $eriem quæ$itam;
tum impo$$ibiles quantitates evitandi gratiâ a$$umantur quantitates
huju$ce formulæ vel <math>(- x)^r × (x<x>.</x> - x)^r × (2 x<x>.</x> - x<x>.</x>)^r ... (nx<x>.</x> - x)^r</math> vel
<math>x^r × (x + x<x>.</x>)^r + (x + 2 x<x>.</x>)^r ... (x + n x<x>.</x>)^r</math>, cujus incrementum e$t af-
firmativum.</p>
<p>Hæc methodus u$ui in$ervit, cum duo valores quantitatis, inter
quos requiritur integralis, $int perparvæ quantitates.</p>
<p>6. Sit $eries <math>A x^-s + B x^-s-t + C x^-s-2t + &c.</math> cujus integralis re-
quiritur in terminis $ecundum reciprocas dimen$iones quantitatis <math>x</math>
progredientibus: a$$umatur pro integrali primi ordinis quantitas
<math>a x^-{1 / 2}s × (x ± x<x>.</x>)^-{1 / 2}s + b x^-{1 / 2}s-{1 / 2} × (x ± x<x>.</x>)^-{1 / 2}s-{1 / 2} + c x^-{1 / 2}s-1 ×<pb 434>[0486]<rh>DE INFINITIS</rh>
(x ± x<x>.</x>)^-{1 / 2}s-1 + &c. + a′x^-{1 / 2}s-{1 / 2}t × (x ± x<x>.</x>)^-{1 / 2}s-{1 / 2}t + b′x^-{1 / 2}s-{1 / 2}t-{1 / 2} ×
(x ± x<x>.</x>)^-{1 / 2}s-{1 / 2}t-{1 / 2} + c′x^-{1 / 2}s-{1 / 2}t-1 × (x ± x<x>.</x>)^-{1 / 2}s-{1 / 2}t-1 + &c. + a″x^-{1 / 2}s-t
× (x ± x<x>.</x>)^-{1 / 2}s-t + &c. + &c.</math> cujus inveniatur incrementum primi or-
dinis, & ex terminis datæ & re$ultantis $eriei corre$pondentibus inter
$e æqualibus e$$e $uppo$itis, erui po$$unt coefficientes <math>a, b, c, &c. a′, b′,
c′, &c. a″, b″, c″, &c. &c.</math></p>
<p>1. 2. Si modo requiratur integralis <math>n</math> ordinis $eriei prius traditæ
<math>A x^-s + B x^-s-t + C x^-s-2t + &c.</math> & cujus termini $ecundum dimen-
$iones quantitatis <math>x</math> progrediuntur: pro integrali quæ$itâ a$$umatur
quantitas <math>a x^-{s / n} × (x ± x<x>.</x>)^-{s / n} × (x ± 2 x<x>.</x>)^-{s / n} .. (x ± (n - 1) x<x>.</x>)^-{s / n} + bx^-{s-1 / n}
× (x ± x<x>.</x>)^-{s-1 / n} × (x ± 2 x<x>.</x>)^-{s-1 / n} ... (x ± (n - 1) x<x>.</x>)^-{s-1 / n} + c x^-{s-2 / n} ×
(x ± x<x>.</x>)^-{s-2 / n} ... (x ± (n - 1) x<x>.</x>)^-{s-2 / n} + &c. + a′ x^{-s-t / n} × (x ± x<x>.</x>)^{-s-t / n} ×
(x ± 2 x<x>.</x>)^{-s-t / n} ... (x ± (n - 1) x<x>.</x>)^{-s-t / n} + b′ x^{-s-t-1 / n} × (x ± x<x>.</x>)^{-s-t-1 / n} × ...
(x ± (n - 1) x<x>.</x>)^{-s-t-1 / n} + &c. + a″x^{-s-2t / n} × (x ± x<x>.</x>)^{-s-2t / n} × (x ± 2 x<x>.</x>)^{-s-2t / n}
... (x ± (n - 1) x<x>.</x>)^{-s-2t / n} + &c.</math> cujus inveniatur incrementum, quod
reducatur ad terminos $ecundum reciprocas dimen$iones quantitatis
<math>x</math> progredientes, deinde e corre$pondentibus datæ & re$ultantis $eriei
re$pective inter $e æquatis erui po$$unt coefficientes <math>a, b, c, &c. a′, b′,
c′, &c. a″, b″, c″, &c.</math></p>
<p>7. Sit incrementum <math>P^n × Q^m × &c. × R</math>, cujus integralis requiritur;
ubi <math>P, Q, R, &c.</math> $unt functiones quantitatis <math>x, n</math> vero & <math>m</math> quæcunque
fractiones; ita reducatur hoc incrementum, ut in eo nulli conti-
neantur divi$ores; qui $int $ucce$$ivæ integrales; i. e. ne $it <math>T × T′</math>
divi$or quantitatis <math>P^n × Q^m × &c.</math> ubi <math>T</math> e$t $ucce$$iva integralis quan-
titatis <math>T′</math>; reducatur datum incrementum ad hanc formulam <math>T × T′ ×
T″ × T′″ ... T′^m-1 × T′^m × A^n × B^m × C^s × &c.</math> pro integrali quæ$itâ ($i re-
quiratur $eries a$cendens) a$$umatur $eries <math>T × T′ × T″ × T′″ ... × T′′^m-1
× A^n × A′^n × B^m × B′^m × C^s × C′^s × &c. (a + b x^r × (x<x>.</x> ± x<x>.</x>)^r + c x^r+1
(x ± x<x>.</x>)^r+1 + &c. + e x^2r × (x ± x<x>.</x>)^2r + f x^2r+1 (x ± x<x>.</x>)^2r+1 + &c. +

<pb 435>[0487]<rh>SERIEBUS.</rh>
l x^3r × (x ± x<x>.</x>)^3r + &c.)</math> cujus inveniatur incrementum, quod fiat
æquale dato incremento, & exinde deduci po$$unt coefficientes <math>a, b, c,
&c. e, f, &c. l, &c.</math></p>
<p>Si vero requiratur $eries de$cendens, tum $eriei præcedentis <math>a +
b x^r × (x ± x<x>.</x>)^r + c x^r+1 × (x ± x<x>.</x>)^r+1 + &c. &c.</math> loco $ub$tituatur vel
<math>a + b x^-{r / 2} × (x ± x<x>.</x>)^-{r / 2} + c x^-{r / 2}-{1 / 2} × (x ± x<x>.</x>)^-{r / 2}-{1 / 2} + &c. + e x^-2r ×
(x ± x<x>.</x>)^-2r + f x^-2r-{1 / 2} × (x ± x<x>.</x>)^-2r-{1 / 2} + &c.</math> vel infinitæ con$imiles
$ub$titutiones.</p>
<p>Facile con$tat, quod infinitæ aliæ $eries huju$modi pro integrali
quæ$itâ a$$umi po$$unt.</p>
<p>Ex transformatione harum $erierum in alias, quarum variabiles
quantitates quandam habeant relationem ad variabiles quantitates
datarum $erierum, deduci po$$unt $eries $ecundum dimen$iones alia-
rum quantitatum progredientes.</p>
<p>Hæc principia promovere liceat etiam ad inveniendas $eries $ecun-
dum datam legem progredientes, & quarum infinitæ aliæ functiones
præter fluentes & integrales $ecundum eandem legem progrediuntur.</p>
<p>Ut $eries huju$ce generis evadant convergentes vel magis celeriter
convergentes, præ$tat interpolare plures inter duos datos valores
<math>(z + a z<x>.</x> & z + b z<x>.</x>)</math> quantitatis <math>x</math>, i. e. $cribere pro <math>x</math> diver$os valo-
res <math>z + e z<x>.</x>, z′ + e′ z<x>.</x>, &c.</math>, ubi <math>e, e′, &c.</math> $unt integri numeri inter
duos valores <math>a & b</math> po$iti.</p>
<p>Interpolatio huju$ce generis, & $erierum exinde re$ultantium con-
vergentia e principiis in prob. traditis dijudicari po$$unt.</p>
<p>Eadem principia etiam applicari po$$unt ad $eries ex incrementia-
libus æquationibus re$ultantes.</p>
<h>PROB. XX.</h>
<p it>Transformare datam quantitatem in terminos $ecundum datam legem
progredientes: bi termini reducantur ad $implices terminos $ecundum dimen-
$iones quantitatis <math>x</math> progredientes, reducatur etiam data quantitas ad ter-
minos $implices $ecundum dimenfiones quantitatis <math>x</math> progredientes, & fiant

<pb 436>[0488]<rh>DE INFINITIS</rh>
coefficientes $ingulorum terminorum huju$ce progre$$us re$pective œquales
coefficientibus corre$pondentium terminorum e priore reductione deductorum.</p>
<p>Ex. 1. Sit quantitas <math>(a + b x)^m</math>; invenire $eriem, cujus lex $it <math>(e + f x)^2
+ (g x + h x^2)^2 + (k x^2 + l x^3)^2 + &c. = (a + b x)^m</math>; reducantur data
quantitas & termini $eriei prædictæ ad $eries, quarum termini $ecun-
dum dimen$iones quantitatis <math>x</math> progrediuntur, & evadunt <math>a^m + m a^m-1
b x + m · {m-1 / 2} a^m-2 b^2 x^2 + &c. & e^2 + 2 e f x + (f^2 + g^2) x^2 + 2 g h x^3
+ &c.</math> & ex æquatis corre$pondentibus terminis duarum re$ultan-
tium $erierum, viz. <math>a^m = e^2, m a^m-1 b = 2 e f, m · {m - 1 / 2} a^m-2 b^2 =
f^2 + g^2, &c.</math> con$equentur <math>e = a^{m / 2}, f = {m a^m-1 b / 2 e}, g = √ (m · {m - 1 / 2}
a^m-2 b^2 - f^2), &c.</math></p>
<p>Ex. 2. Ex eodem modo deduci pote$t <math>{a / z - α} = {a / z} + {a α / z · (z + 1)} +
{a × α (α + 1) / z · (z + 1) · (z + 2)} + {a α × (α + 1) × (α + 2) / z · (z + 1) · (z + 2) · (z + 3)} + &c.</math></p>
<p>Cor. 1. Si <math>α</math> vel æqualis vel major $it quam <math>z</math>, tum hæc $eries haud
converget; $in aliter paululum magis celeriter hæc $eries converget
quam $eries <math>{a / z} + {a α / z^2} + {a α^2 / z^3} + &c.</math></p>
<p>Cor. 2. Si vero incrementum quantitatis <math>z</math> $it 1, tum integralis $e-
riei <math>({a / z} + {a α / z · (z + 1)} + {a α × (α + 1) / z · (z + 1) · (z + 2)} + &c.)</math> erit <math>{a / z} + {a α / z}
+ {a α × (α + 1) / 2 z · (z + 1)} + &c.</math> hæc vero $eries converget, $i <math>z</math> major $it
quam <math>α</math>.</p>
<p>Hinc facile con$tat omnia, quæ traduntur de convergentiâ fluen-
tium datarum fluxionum æque de convergentiâ integralium con$i-
milium incrementorum affirmari po$$e.</p>

<pb 437>[0489]<rh>SERIEBUS.</rh>
<p>Facile e præcedentibus propo$itionibus deduci po$$unt convergen-
tiæ $erierum in hoc problemate traditarum; reducantur enim hæ $e-
ries ad alias $ecundum dimen$iones quantitatis <math>x</math> progredientes, & ex
convergentiis $erierum re$ultantium plerumque facile erui po$$unt
convergentiæ $erierum prædictarum.</p>
<h>PROB. XXI.</h>
<h it>Datis æquationibus algebraicis, invenire proximos valores unius incog-
nitæ quantitatis <math>x</math> terminis vero alterius <math>y</math>.</h>
<p>E comparandis terminis datæ æquationis inter $e $æpe con$tat
proximus valor unius incognitæ quantitatis <math>y</math>.</p>
<p>Fere vero præ$tat terminos datarum æquationum, qui maximi $int,
ex hypothe$i datâ quod <math>x</math> vel $it perparva vel permagna quantitas,
inter $e æquales e$$e $upponere, neglectis omnibus reliquis terminis,
qui minores $int, & ex æquationibus re$ultantibus deducere proximos
valores incognitæ quantitatis quæ$itæ <math>y</math>.</p>
<p>In datâ æquatione algebraicâ $cribatur (<math>a + p</math>) quantitatis <math>y</math> pro-
ximus valor <math>a</math> per incognitam quantitatem <math>p</math> auctus pro <math>y</math>, & e ter-
minis re$ultantis æquationis, qui ex hypothe$i datâ maximi re$ultant,
inter $e æquales e$$e $uppo$itis, neglectis iis, qui minores $int, deduci
pote$t proximus valor (<math>b</math>) quantitatis <math>p</math>; deinde vel in datâ æquatione
pro <math>y</math> $cribatur <math>a + b + q</math>, vel quod idem erit in re$ultante æquatione
pro <math>p</math> $cribatur <math>p + q</math>, & con$imili methodo inveniatur proximus va-
lor quantitatis <math>q</math>; & $ic deinceps; e. g. $it æquatio <math>y^3 + a^2 y - 2 a^3 +
a x y - x^3 = 0</math> ubi <math>x</math> $it perparva quantitas; $upponatur <math>x = 0</math> &
termini igitur qui maximi inveniuntur, erunt <math>y^3 + a^2 y - 2 a^3 = 0</math>,
unde <math>y = a</math> prope; $cribatur <math>a + p</math> pro <math>y</math> in datâ æquatione <math>y^3 + a^2 y
- 2 a^3 + a x y - x^3 = 0</math>, & re$ultat æquatio <math>p^3 + 3 a p^2 + (4 a^2 +
a x) p + a^2 x - x^3 = 0</math>; termini autem, qui maximi $int in hâc æqua-
tione, erunt <math>4 a^2 p + a^2 x = 0</math>, neglectis omnibus, qui minores $int,
unde <math>p = - {x / 4}</math> prope; $cribatur igitur pro <math>p</math> in re$ultante æquatione

<pb 438>[0490]<rh>DE INFINITIS</rh>
<math>- {x / 4} + q</math>, & inveniri po$$it <math>q = {x^2 / 64 a}</math> prope; & $ic deinceps; unde <math>y =
a - {x / 4} + {x^2 / 64 a} - {131 x^3 / 512 a^2} + &c.</math></p>
<p>2. Sit vero æquatio <math>y^3 + a x y + x^2 y - a^3 - 2 x^3 = 0</math> data, <math>x</math> vero
permagna quantitas; termini maximi, qui ex hâc hypothe$i in eâ con-
tinentur, erunt <math>y^3 + x^2 y - 2 x^3 = 0</math>, unde <math>y = x</math> prope; $cribatur
igitur <math>x + p</math> pro <math>y</math> in datâ æquatione, & re$ultat æquatio <math>p^3 + 3 x p^2
+ (4 x^2 + a x) p + a x^2 - a^3 = 0</math>, in quâ termini, qui maximi e præ-
dictâ hypothe$i inveniuntur, erunt <math>4 x^2 p + a x^2 = 0</math>, unde <math>p = - {a / 4}</math>
prope; $cribatur igitur in re$ultante æquatione <math>- {a / 4} + q</math> pro <math>p</math>, & e
terminis maximis inveniri pote$t <math>q = {a^2 / 64 x}</math> prope; & $ic deinceps; unde
<math>y = x - {a / 4} + {a^2 / 64 x} + {131 a^3 / 512 x^2} + &c.</math></p>
<p>Cor. 1. Facile con$tat, quod e divi$ione termini, in quem haud in-
greditur <math>q</math> vel <math>r, &c.</math> per coefficientem termini, in quo continetur una
$olummodo pote$tas quantitatis <math>p</math> vel <math>q</math> vel <math>r, &c.</math> $ucce$$ive, duplo
plures termini, uno dempto, valoris quantitatis <math>y</math> in terminis quanti-
tatis <math>x</math> detegentur.</p>
<p>Cor. 2. Sit <math>b</math> proximus valor quantitatis <math>y</math> in æquatione <math>P = 0</math>,
$cribatur <math>b + p</math> pro <math>y</math> in datâ æquatione <math>P = 0</math>, & re$ultet æquatio
<math>a + b p + c p^2 + d p^3 + &c. = 0</math>, ubi <math>a, b, c, d, &c.</math> $unt functiones
ip$ius <math>x</math>, tum a$$umantur vel <math>a + b p = 0</math>, vel <math>a + b p + c p^2 = 0,
&c.</math> & e po$terioribus a$$umptionibus con$equentur propiores valores
quantitatis <math>y</math>, quam e prioribus, ut facile con$tat e theor. 17.</p>
<p>Cor. 3. Convergentia $eriei: 1<_>mo</_> pendet ex hoc, quanto propior $it
valor a$$umptus pro quantitate <math>y</math> ad unam radicem <math>y</math> quam ad reli-
quas; tum etiam, $i modo æquatio a$$umpta $it <math>a + b p = 0</math>, pendet

<pb 439>[0491]<rh>SERIEBUS.</rh>
e ratione, quam habent quantitates a$$umptæ pro <math>{b / a}</math> ad quantitatem
ip$am <math>{b / a}, &c.</math></p>
<p>Cor. 4. Si <math>n</math> $int radices quantitatis <math>y</math> prope inter $e & radici quæ-
$itæ æquales, tum a$$umendi $unt <math>(n + 1)</math> termini æquationis <math>0 = a
+ b p + c p^2 .. p^n</math>, e quibus detegi pote$t propior valor $ingularum <math>n</math>
radicum a quantitate a$$umptâ haud multum di$tantium.</p>
<p>1. 1. Si vero <math>x</math> haud longe di$tet <math>a</math> quantitate <math>b</math>, tum $cribatur
quantitas <math>b</math> pro <math>x</math> in datâ æquatione, & ex æquatione re$ultante <math>(P
= 0)</math> inveniatur quantitas <math>y</math>, quæ multo magis approximat ad unam
radicem quam ad reliquas, quæ $it <math>f</math>: in datâ æquatione pro <math>x</math> $criba-
tur <math>b + z</math>; & pro <math>y</math> $cribatur <math>p + f</math>: fiant termini re$ultantis æqua-
tionis, qui ex hâc hypothe$i maximi evadant, nihilo æquales; & ex-
inde $equitur valor quantitatis <math>p</math> prope; unde $equitur <math>p + f</math> valor
quantitatis <math>y</math> prope; & $ic redintegratâ operatione deduci pote$t
quantitas ad valorem quantitatis <math>y</math> magis appropinquans; & $ic de-
inceps: $i inveniatur radix æquationis <math>P = 0</math> accurate; tum hæc
methodus evadet eadem ac duo præcedentes: & hic (mutatis mutan-
dis) applicari po$$unt omnia, quæ ad præcedentes methodos appli-
cantur.</p>
<p>Ex. Sit æquatio <math>y^2 - 5 x y + 4 x^2 - x = 0</math>, ubi <math>x = 2</math> prope; pro <math>x</math>
$cribatur <math>2 + z</math> & re$ultat <math>y^2 - (10 + 5 z) y + 14 + 15 z + 4 z^2
= 0: $i z = 0</math>, tum re$ultat <math>y^2 - 10 y + 14 = 0</math>, unde <math>y = 8</math> prope:
& con$equenter 8 vel 2 $unt quantitates, quæ multo magis approxi-
mant ad unam radicem quantitatis <math>y</math> quam ad reliquas; e. g. a$$u-
matur 8 quantitas approximans ad radicem quæ$itam: in re$ultante
æquatione pro <math>y</math> $cribatur <math>8 + p</math>, & re$ultat <math>64 + 16 p + p^2 - 80 -
10 p - 40 z - 5 p z + 14 + 15 z + 4 z^2 = 0</math>; collocentur maximi
termini huju$ce æquationis primi, iis vero proximi $ecundi, & $ic de-
inceps; i. e. <math>(A = 0) (64 - 80 + 14 - 40 z + 15 z + 16 p - 10 p)
+ (p^2 - 5 p z + 4 z^2) = (- 2 - 25 z + 6 p) + (p^2 - 5 p z + 4 z^2)
= 0</math>: a$$umantur termini, qui $unt maximi nihilo æquales, i. e. <math>- 2

<pb 440>[0492]<rh>DE INFINITIS</rh>
- 25 z + 6 p = 0</math>, & exinde re$ultabit valor quantitatis <math>p = {1 / 3} +
4 {1 / 6} z</math> prope: in æquatione <math>A = 0</math> pro <math>p</math> $cribatur <math>{1 / 3} + 4 {1 / 6} z + q</math>; & ex
terminis æquationis re$ultantis (qui maximi $unt) inveniatur valor
quantitatis <math>q</math> prope, viz. <math>a′ + b′z + c′z^2</math>; & $ic deinceps: & exinde
re$ultabit $eries quæ exprimit valorem quantitatis <math>y</math>, viz. <math>y = (8 + {1 / 3}
+ a′ + &c.) + (4{1 / 6} + b′ + &c.) z + (c′ + &c.) z^2 + &c. = (8 + {1 / 3} +
a′ + &c.) + (4{1 / 6} + b′ + &c.) (x - a) + (c′ + &c.) (x - a)^2 + &c.</math></p>
<p>Eadem principia, &c. ad fluxionales, &c. æquationes applicari
po$$unt.</p>
<p>1. 2. Ex methodis haud di$$imilibus inveniri pote$t quantitas <math>y</math> per
$eriem quantitatum, quæ $unt functiones alterius quantitatis <math>x</math> in
datâ æquatione contentæ, $ecundum datam legem progredientium,
e. g. $it $eries quantitatum per legem <math>A + B x × (x - 1) + C x^2 ×
(x - 1)^2 + D x^3 × (x - 1)^3 + &c.</math> progredientium, & per methodos
haud di$$imiles deduci po$$unt coefficientes <math>A, B, C, &c.</math> hæ vero coef-
ficientes ex $erie, cujus termini $ecundum quamcunque aliam datam
legem progrediuntur, etiam inve$tigari po$$unt.</p>
<p>4. Sit æquatio algebraica <math>y^n - p y^n-1 + q y^n-2 - &c. = 0</math>, ubi <math>p, q,
&c.</math> $unt functiones quantitatis <math>x</math>, inveniatur <math>y</math> in ejus <math>x</math> terminis, i. e.
$it <math>y = a′ x^r + b′ x^r+s + c′ x^r+2s + d′ x^r+3s + &c.</math> & convergentia huju$ce
$eriei pendet ex hoc, nempe quanto minus di$tant approximationes
$ucce$$ivæ ab unâ radice quam ab reliquis, &c. ut prius docetur: in
fluentibus, integralibus, &c. exinde deducendis, nece$$e e$t ut dentur
duo, &c. valores <math>a & b</math> quantitatis <math>x</math>, inter quos continentur fluentes,
integrales, &c. ut vero hæ $eries convergant, quantitates <math>a & b</math> haud
longe di$tent ab eâdem radice vel ex eâdem vel oppo$itis radicis
partibus; $i vero una litera <math>a</math> haud multum di$tet ab corre$pondente
radice æquationis <math>n y^n-1 - (n - 1) p y^n-2 + (n - 2) q y^n-3 - &c. = 0</math>,
vel duæ literæ ex oppo$itis partibus eju$dem radicis ponantur, tum
$eries prædictæ haud convergent: in ca$ibus quando $eries prædicta
vel non omnino vel non $atis convergat, præ$tet ut transformetur
data æquatio in eâ pro <math>x</math> $cribendo <math>z - α, z - β, z - γ, z - δ, &c.</math>
re$pective, ubi <math>α, β, γ, δ, &c. λ</math> inter valores <math>a & b</math> ponantur, ita qui-

<pb 441>[0493]<rh>SERIEBUS.</rh>
dem ut $eries re$ultans e primâ æquatione, derivatâ a $ub$titutione
<math>z - α</math> pro <math>x</math> in datâ æquatione, cum <math>z = α - a</math> vel <math>α - c</math>, conver-
get; & $imiliter $int $eries re$ultantes e $ub$titutionibus <math>β - c & β - d</math>
pro <math>z</math> in $ecundâ, <math>γ - d & γ - e</math> in tertiâ, & $ic deinceps u$que ad
ultimum <math>λ - b</math>, $emper convergentes; tum ex iis erui pote$t fluens
vel integralis quæ$ita.</p>
<p>Hìc animadvertendum e$t, quod $i omnes hæ $eries evadant con-
vergentes, nece$$ario nonnullæ ab$ci$$æ (<math>z</math>) incipiant a $ingulis radi-
cibus (<math>x</math>) inter <math>a & b</math> po$itis, vel ad eas terminentur.</p>
<p>Si <math>a & b</math> ex oppo$itis partibus eju$dem radicis ponantur; tum haud
convergent duæ $eries re$ultantes e $cribendo <math>a & b</math> pro <math>x</math>, ni ab$ci$$a
incipiat a radice inter <math>a & b</math> po$itâ.</p>
<p>5. Datis unâ duabus vel pluribus (<math>n</math>) æquationibus (<math>n + 1</math>) incogni-
tas quantitates habentibus, inveniantur valores quantitatis <math>x</math>, cum <math>y</math>
evadat vel infinita <math>vel = 0</math>, quorum $it minimus <math>α</math> & maximus <math>λ</math>;
tum $eries a$cendens $emper converget, cum <math>x</math> minor $it quam <math>α</math>; &
$eries de$cendens converget, cum <math>x</math> major $it quam <math>λ, &c.</math></p>
<p>Eadem principia etiam ad fluxionales & incrementiales æquationes
applicari po$$unt.</p>
<p>6. Si vero quædam quantitates vel <math>p</math> vel <math>q</math> vel <math>r, &c.</math> $int functiones,
quæ habent functionem quantitatis <math>x</math> in denominatore; fiat prædicta
functio nihilo æqualis, & haud converget $eries a$cendens <math>y = a′ x^r +
b′ x^r+s + c′ x^r+2s + &c.</math> vel $eries exprimens fluentem fluxionis <math>y x^., &c.</math>
$i <math>x</math> major $it quam ulla radix prædictæ æquationis; $i vero $it $eries
de$cendens, tum haud converget, ni <math>x</math> major $it quam quæcunque
radix prædictæ æquationis.</p>
<p>Inveniantur igitur valores (<math>π, ρ, σ &c.</math>) quantitatis <math>x</math> inter <math>a & b</math>
po$iti, in quibus <math>y</math> vel evadat infinita vel nihilo æqualis; & prædictæ
ab$ci$$æ (<math>z</math>) vel ab valoribus (<math>π, ρ, σ, &c.</math>) incipiant, vel ad eos termi-
nentur; præ$tat ut ab iis incipiant.</p>
<p>7. Sit <math>P</math> prædicta functio, & in quibu$dam ca$ibus pro quantitate
quæ$itâ <math>y</math> a$$umatur quantitas formulæ <math>P × (f x^r + g x^r+s + h x^r+2s +

<pb 442>[0494]<rh>DE INFINITIS</rh>
&c.)</math> $ub$tituatur hæc quantitas reducta ad infinitam $eriem pro <math>y</math> in
datâ æquatione, & ex æquatis corre$pondentibus terminis inter $e
deduci po$$unt coefficientes <math>f, g, b, &c.</math> In quibu$dam etiam ca$i-
bus a$$umatur quantitas prædictæ formulæ cum alterâ $erie <math>a′ x^r +
b′ x^r+s + &c.</math> pro <math>y</math>, vel cum pluribus quantitatibus prædictarum for-
mularum & $erie <math>a′ x^r + b′ x^r+s + &c.</math> & exinde erui po$$unt $eries
magis celeriter convergentes. Irrationales compo$itæ quantitates vel
ex divi$ione, extractione radicum, &c. ad $implices terminos reducendæ
$unt, quotie$cunque id exigat pro approximatione inveniendâ opus.</p>
<p>8. Cum vero ratio, quam habeat data approximatio ad radicem quæ-
$itam haud permultum ex$uperet rationes, quas habet data approxi-
matio ad unam, duas vel plures alias radices; tum ope quadraticæ,
cubicæ, &c. æquationis quærendæ $unt novæ approximationes ad
duas, tres vel plures radices datæ æquationis; unde con$tat ratio,
quare in his ca$ibus $i differentiæ indicum & $implicium & compo$i-
tarum quantitatum ad $implices reductarum vel $int <math>s</math> vel habeant
maximum communem denominatorem <math>s</math>; pro <math>y</math> a$$umenda e$t feries
huju$ce formulæ <math>a x^r + b x^r+{1 / 2}s + c x^r+s + d x^r+1{1 / 2}s + e x^r+2s + &c.</math> vel
<math>a x^r + b x^r+{1 / 3}s + c x^r+{2 / 3}s + d x^r+s + e x^r+1{1 / 3}s + &c. &c.</math></p>
<p>9. Si vero dentur quantitates ad duas vel plures huju$ce formulæ
radices æquationis maxime appropinquantes; tum e principiis prius
traditis erui po$$unt novæ approximationes ad radices prædictas pro-
pius accedentes.</p>
<h>PROB. XXII.</h>
<h it>_1._ Contineantur tres <math>(y, x, z)</math> incognitæ quantitates in datâ æquatione,
invenire <math>y</math> in terminis quantitatum <math>x & z</math>.</h>
<p>Primo reducantur duæ quantitates <math>x & z</math> ad alias <math>x′ & z′</math>, ut vel
evadant perparvæ vel permagnæ, vel altera perparva & altera vero
permagna, prout requiratur $eries a$cendens vel de$cendens $ecundum
dimen$iones quantitatum <math>x′ & z′</math>; deinde fingatur altera <math>x′</math> prope in
ratione directâ vel inversâ cuju$dam pote$tatis <math>m</math> quantitatis <math>z′</math>; & ex

<pb 443>[0495]<rh>SERIEBUS.</rh>
æquatis terminis inter $e, qui maximi re$ultent ex hâc hypothe$i, in-
veniatur (<math>a</math>) valor quantitatis <math>y</math> prope; $cribatur quantitas <math>a + π</math> pro
<math>y</math> in datâ æquatione, & ex eâdem methodo erui pote$t valor quan-
titatis <math>π</math> prope; & $ic deinceps.</p>
<p>Hoc prob. etiam re$olvi pote$t ex aliis principiis in prob. præced.
datis.</p>
<p>2. Si vero contineantur plures quam tres _(_<math>y, x, z</math>) incognitæ quan-
titates in datâ æquatione, per proce$$um omnino eundem erui pote$t
<math>y</math> in $erie e terminis quantitatum <math>x′, z′, &c.</math> con$tante.</p>
<p>3. Si vero detur fluxionalis vel incrementialis, &c. æquatio; ex
principiis in hoc problemate, &c. traditis erui pote$t valor quantitatis
<math>y</math> in $erie $ecundum dimen$iones reliquarum progrediente, $i modo <math>y</math>
in terminis reliquarum exprimi po$$it.</p>
<p>Si dentur plures <math>n</math> vel algebraicæ vel fluxionales vel incrementiales
æquationes, <math>n + 2</math> vel plures variabiles quantitates habentes; tum
ex principiis, quæ in hoc problemate, &c. prius tradita fuere, erui
pote$t altera <math>y</math> in $erie $ecundum dimen$iones reliquarum progredi-
ente, $i modo <math>y</math> in terminis reliquarum exprimi po$$it.</p>
<p>Hoc problema etiam re$olutionem accipere pote$t ex a$$umptâ $erie
progrediente $ecundum dimen$iones reliquarum quantitatum, i. e. prin-
cipiis, iis in prob. 21. contentis, analogis; etiamque ex a$$umptis qui-
bu$dam reliquarum quantitatum dimen$ionibus inter $e æqualibus.</p>
<h>PROB. XXIII.</h>
<h it>_1._ Datis duabus vel pluribus <math>(n)</math> algebraicis vel fluxionalibus vel in-
crementialibus æquationibus <math>n + 1</math> variabiles quantitates <math>(y, x, z, &c.)</math>
babentibus; invenire unam <math>(y)</math> ex iis in terminis reliquarum <math>(x, z, &c.)</math></h>
<p>Ita reducantur hæ æquationes, $cribendo in iis pro reliquis <math>x, z,
&c.</math> re$pective <math>x′ + a, z′ + b, &c.</math> ut <math>x′ & z′ &c.</math> perparvæ vel per-
magnæ evadant, prout requiratur $eries a$cendens vel de$cendens $e-
cundum earum dimen$iones; & $imiliter de earum fluxionibus, incre-
mentis, &c.: deinde ex æquatis terminis $ingularum æquationum,<pb 444>[0496]<rh>DE INFINITIS</rh>
qui maximi $unt, re$pective inter $e, erui po$$unt <math>α, β, γ, &c.</math> prope va-
lores quantitatis <math>y</math> re$pectivi in terminis quantitatum <math>x′, z′, &c.</math> $cri-
bantur hi valores inventi per variabiles quantitates incognitas aucti,
i. e. <math>x″ + α, z″ + β, &c.</math> pro <math>y</math>, & pro <math>y^., y^.., &c. y<x>.</x>, y<x>..</x>, &c.</math> earum valores
<math>x^. ″ + α^., z^. ″ + β^., x^.. ″ + α^.., z^. ″ + β^.., &c. x<x>.</x>″ + α<x>.</x>, z<x>.</x>″ + β<x>.</x>, x<x>..</x>′″ + β<x>..</x>, &c.</math>
re$pective in datis æquationibus, & ex æquationibus re$ultantibus fa-
cile detegi po$$unt valores prope quantitatum <math>x″, z″, &c.</math> re$pective, &
$ic deinceps; vel aliter e principiis prius traditis erui po$$unt formulæ
$erierum, quæ exprimunt quantitatem <math>y</math> terminis vero quantitatum
<math>x′, z′, &c.</math> quæ $int <math>y = A′ z′^k + B′ z′^k+1 + C′ z′^k+2l + &c. &c.</math> vel pro
incrementis primi ordinis <math>y = A x′^r × (x + 1)′^r + &c. &c. &c.</math> $cri-
bantur hæ quantitates pro $uis valoribus in datis æquationibus, & ex
$ingulis terminis re$ultantis æquationis nihilo æqualibus e$$e $uppo$i-
tis erui po$$unt coefficientes <math>A, B, C, &c. A′, B′, C′, &c. &c.</math></p>
<p>Convergentiæ $erierum e pluribus æquationibus deductarum vel
plures variabiles quantitates habentium e principiis prius traditis de
convergentiâ $eriei ex unâ æquatione erutæ, facile dijudicari po$$unt.</p>
<p>2. Dentur æquationes involventes variabiles quantitates <math>x, y, &c.</math>
& earum fluxiones, incrementa, &c. $int etiam <math>z & v</math> datæ functiones
quantitatum <math>x, y, &c.</math> invenire <math>v</math> in terminis $ecundum dimen$iones
quantitates <math>z</math> progredientibus: hoc perficitur vel e reductione plu-
rium æquationum in pauciores, ita ut exterminentur omnes incog-
nitæ quantitates præter <math>z & v</math>; vel ex a$$umptis $eriebus pro $ingulis
incognitis quantitatibus $ecundum dimen$iones quantitatis <math>z</math> progre-
dientibus, & ex earum $ub$titutionibus in datis æquationibus pro
$uis valoribus, & terminorum corre$pondentium coefficientibus nihilo
æqualibus e$$e $uppo$itis, erui po$$unt indices & coefficientes $erierum
a$$umptarum quæ$itæ.</p>
<h>THEOR. XXXII.</h>
<p>Sit fluxionalis æquatio relationem inter variabiles <math>x & y</math> & earum
fluxiones exprimens, ubi <math>x^.</math> $it con$tans; pro <math>y</math> $cribatur <math>A x^n</math> & pro <math>y^.,

<pb 445>[0497]<rh>SERIEBUS.</rh>
y^.., &c.</math> $cribantur re$pective <math>n A x^n-1 x^., n · (n - 1) A x^n-2 x^. ^2, &c.</math> in
datâ æquatione, ubi <math>A & n</math> $unt quæcunque invariabiles quantitates
a$$umendæ; & e terminis re$ultantis æquationis, qui inveniuntur
maximi ex hypothe$i datâ, quod <math>x</math> $it perparva vel permagna quan-
titas, inter $e æquatis, deduci pote$t quantitas <math>A x^n</math> proximus valor
quantitatis <math>y</math>; tum in datâ æquatione pro <math>y</math> $cribatur <math>A x^n + p</math>, & pro
<math>y^., y^.., &c.</math> re$pective <math>n A x^n-1 x^. + p^., n · (n - 1) A x^n-2 x^. ^2 + p^.., &c.</math> &
e terminis maximis re$ultantis æquationis inveniatur proximus valor
quantitatis <math>p</math>, qui $it <math>B x^m</math>; in æquatione re$ultante pro <math>p</math> $cribatur
ejus valor <math>B x^m + q</math>, & pro <math>p^., p^.., &c.</math> re$pective <math>m B x^m-1 x^. + q^., m ·
(m - 1) B x^m-2 x^. ^2 + q^.., &c.</math> & con$imili methodo inveniatur proximus
valor quantitatis <math>q</math>, & $ic deinceps; unde tandem innote$cet <math>y = A x^n
+ B x^m + &c.</math> prope.</p>
<p>Si <math>x = a</math> prope, tum $cribatur in datâ æquatione <math>a + z</math> vel <math>a + {1 / z}</math> pro
<math>x</math>, & ex methodo hìc traditâ inveniatur <math>y</math> in terminis quantitatis <math>z</math>.</p>
<p>2. Sit <math>r</math> ordo fluxionalis æquationis, & in $erie quæ exprimit va-
lorem quantitatis <math>y</math> in terminis quantitatis <math>x</math>, i. e. <math>y = A x^n + B x^m +
&c.</math> $emper tot invenientur invariabiles quantitates ad libitum a$$u-
mendæ, quot $it ordo prædictus, $i modo $eries ab initio terminorum
incipiat, i. e. recte in$tituatur. e. g. Sit æquatio <math>y^. = x^. + y x^. - 3 x x^. + x^2 x^.
+ y x x^.</math>, invenire quantitatem <math>y</math> in $erie $ecundum dimen$iones quan-
titatis <math>x</math> progrediente; $i a$$umatur <math>y = x</math> prope pro proximo valore
quantitatis <math>y</math>, tum haud incipit ab initio recte in$tituto $eries; finga-
tur <math>y = A + b x + p</math>, $cribatur hæc quantitas pro <math>y & b x^. + p^.</math> pro <math>y^.</math>
in datâ æquatione, & re$ultat <math>b x^. + p^. = x^. - 3 x x^. + A x^. + b x x^. +
p x^. + x^2 x^. + A x x^. + b x^2 x^. + p x x^.</math>; termini autem, qui ex hypothe$i
a$cendentis $eriei maximi $int, inveniantur <math>b x^. = x^. + A x^.</math>, unde <math>b =
1 + A</math>; & æquatio re$ultans <math>p^. = (A + b - 3) x x^. + (x^2 + p + b x^2
+ p x) x^.</math>; hujus re$ultantis æquationis termini maximi erunt <math>p^. =
(A + b - 3) x x^.</math>, unde <math>p = {1 / 2} (A + b - 3) x^2</math> prope, & con$equenter
<math>y = A + (A + 1) x + (A - 1) x^2 + &c.</math> ubi <math>A</math> e$t quantitas ad libi-
tum a$$umenda.</p>
<p>3. Si vero prædicta quantitas in fluente inventâ haud contineatur,

<pb 446>[0498]<rh>DE INFINITIS</rh>
tum fluens inventa erit valor particularis, qui nonnunquam in gene-
rali fluente fluxionalis æquationis, & $ic integrali, &c. haud contine-
tur. Idem etiam verum e$t de algebraicis æquationibus ut con$tat e
$ub$equen. exemp.</p>
<p>Sit æquatio <math>y = √ (a^2 + b x + c x^2) = a + {b x / 2 a} + ({c / 2 a} - {b^2 / 8 a^3}) x^2
+ &c.</math> In hâc $erie $emper continetur quantitas generalis <math>a</math>; nunc
$it <math>a = 0</math>, & re$ultat æquatio particularis <math>y = √ (b x + c x^2) = b^{1 / 2} x^{1 / 2}
+ {c x^1{1 / 2} / 2 b^{1 / 2}} + &c.</math> quæ $eries particularis in priori generali haud conti-
netur; eveniant enim quidam particulares ca$us, in quibus generalis
re$olutio fallit; quibus quidem ca$ibus generalis re$olutio præbet di-
vergentem $eriem, vel ejus quidam numeratores nihilo evadunt
æquales.</p>
<p>II. Datâ incrementiali æquatione relationem inter variabiles <math>x, y</math>
& earum incrementa relationem exprimente; invenire ejus integrale,
i. e. variabilem <math>y</math> in terminis $ecundum dimen$iones quantitatis <math>x</math> pro-
gredientibus.</p>
<p>1<_>mo</_>. Sit <math>x</math> permagna quantitas re$pectu habito ad reliquos æqua-
tionis terminos; tum abjectis omnibus terminis, qui haud maximi
re$ultant ex hâc $uppo$itione, e terminis exinde re$ultantibus inve-
niatur per methodum in fluxionalibus, &c. æquationibus re$olvendis
traditam proximus valor quantitatis <math>y</math> terminis vero quantitatis <math>x</math>,
qui $it <math>A x^-n</math>, deinde $cribatur <math>{± A / x^{1 / 2}n × (x ± x<x>.</x>)^{1 / 2}n} + p</math> pro <math>y</math> in datâ
æquatione, & ex æquatione re$ultante abjectis omnibus terminis
qui haud maximi re$ultant e prædictâ hypothe$i inveniatur pro-
ximus valor quantitatis <math>p</math>, qui $it <math>A x^-m</math>; tum pro <math>p</math> $cribatur
<math>{± A / x^{1 / 2}m × (x ± x<x>.</x>)^{1 / 2}m} + q</math>, & $ic deinceps; unde tandem re$ultabit valor
quantitatis <math>y</math> quæ$itus.</p>
<p>2<_>do</_>. Si vero <math>x</math> perparva $it re$pectu habito ad quo$dam æquationis
terminos, tum abjectis omnibus terminis, qui haud maximi re$ultant

<pb 447>[0499]<rh>SERIEBUS.</rh>
ex hâc hypothe$i, e reliquis deducatur proximus valor quantitatis <math>y</math>,
qui $it <math>A x^n</math>; deinde $cribatur pro <math>y</math> in datâ æquatione <math>A x^n × (x ± x<x>.</x>)^n
+ p</math>, & ex æquatione re$ultante inveniatur per con$imilem metho-
dum proximus valor quantitatis <math>p</math>; & $ic repetitâ operatione tandem
re$ultabit integralis quæ$ita.</p>
<p>Et $ic e principiis prius traditis erui pote$t integralis ex a$$u-
mendo quantitatem formulæ <math>A z^n + &c. + a + {b / z} + {c / z · z + z<x>.</x>} +
{d / z · z + z<x>.</x> · z + 2 z<x>.</x>} + &c.</math> pro integrali, $i <math>n</math> $it integer numerus, &
$ub$tituendo hanc quantitatem pro ejus valore in datâ æquatione,
ex æquatis corre$pondentium terminorum coefficientibus erui po$$unt
quantitates <math>A, &c. a, b, c, &c.</math></p>
<p>III. Aliter: Datâ fluxionali æquatione relationem inter variabiles
<math>x, y</math>, & earum fluxiones exprimente, invenire quantitatem <math>y</math> in termi-
nis $eriei progredientis $ecundum dimen$iones quantitatis <math>z</math>.</p>
<p>A$$umatur <math>y = A x^n</math> pro primo $eriei termino, quâ quantitate pro
ejus valore <math>y</math> in datâ æquatione $ub$titutâ, & ejus fluxionibus <math>n A x^n-1 x^.,
n · (n - 1) A x^n-2 x^. ^2, &c.</math> pro <math>y^., y^.., &c.</math> fiant omnes termini qui habent
maximas vel minimas dimen$iones quantitatis <math>x</math>, prout $eries requiri-
tur a$cendens vel de$cendens, inter $e æquales; & ex æquatione re$ul-
tante con$tabit primus terminus $eriei quæ$itæ <math>A x^n</math>; & pro reliquis
indicibus $cribe quantitatem <math>A x^n</math> & ejus fluxiones pro $uis valoribus
in datâ æquatione, $ubtrahe minimum indicem re$ultantis æquatio-
nis terminorum in $erie a$cendente vel maximum in $erie de$cendente
de $ingulis reliquis, & omnia re$idua $ibi ip$is & omnibus aliis addan-
tur, & $ic de reliquis; & e quantitatibus re$ultantibus con$tabunt dif-
ferentiæ indicum, i. e. quantitates <math>r, s, t, &c.</math></p>
<p>Scribatur igitur $eries <math>A x^n + B x^n+r + C x^n+s + D x^n+t + &c.</math> in
datâ æquatione pro <math>y</math>, & ejus $ucce$$ivæ fluxiones re$pective pro <math>y^., y^..,
&c.</math> & ex æquatis corre$pondentibus terminis re$ultantis æquationis,

<pb 448>[0500]<rh>DE INFINITIS</rh>
in quibus eædem inveniuntur dimen$iones quantitatis <math>x</math>, nihilo re-
$pective; re$ultabunt coefficientes <math>B, C, D, &c.</math> quæ$itæ.</p>
<p>Si vero duo vel tres vel plures valores quantitates <math>A x^n</math> $int inter $e
æquales, dividantur prædictæ differentiæ per duo, tres, &c. & re$ul-
tabunt differentiæ quæ$itæ, &c. i. e. $eries erunt huju$modi <math>A x^n +
B x^n+{1 / 2}r + C x^n+r + D x^n+{1 / 2}s + &c.</math> vel <math>A x^n + B x^n+{1 / 3}r + C x^n+{2 / 3}r +
&c.</math></p>
<p>Si primus valor quantitatis <math>y</math> inventus $it imaginarius, tum for$an
e $cribendo <math>v + a & z + b</math> pro <math>y & x</math> in datâ æquatione re$pective,
ita a$$umi po$$unt invariabiles quantitates <math>a & b</math>, ut haud evadat pri-
mus valor quantitatis <math>y</math> imaginarius; & ejus approximationes e me-
thodis prius traditis deduci po$$unt.</p>
<p>Hic animadvertendum e$t hanc methodum re$olutionis $emper
exigere, ut quantitates <math>v & z</math>, vel <math>v</math> vel <math>z</math>, $int perparvæ vel permagnæ,
prout requiritur $eries a$cendens vel de$cendens.</p>
<p>Per perparvas vel permagnas quantitates intelligo quantitates $ic
deductas; $it $eries deducta a$cendens <math>y = a′ z^λ + b′ z^μ + c′ x^ν + &c.
= P</math>; & inveniatur (<math>π</math>) valor quantitatis <math>z</math>, cum <math>P</math> evadat divergens
$eries; tum nece$$e e$t, ut <math>z</math> non major $it quam <math>± π</math>, aliter $eries
diverget: $it <math>P</math> $eries de$cendens, & <math>ρ</math> minimus valor quantitatis <math>z</math>,
cum evadat divergens, tum nece$$e e$t, ut <math>z</math> non minor $it <math>± ρ</math>, aliter
$eries diverget: aliter per perparvas intelligo quantitates multo ma-
gis approximantes ad unam quam ad reliquas radices.</p>
<p>In fluente fluxionalis æquationis <math>n</math> ordinis <math>n</math> continentur invaria-
biles quantitates pro conditione problematis a$$umendæ; unde a$$u-
mi po$$unt (<math>n</math>) corre$pondentes valores $ingularum quantitatum <math>x, x^.,
y, y^., y^.., &c.</math> Et $ic de incrementialibus, &c. æquationibus.</p>
<h>PROB. XXIV.</h>
<p>_Datâ fluxionali æquatione_ <math>n</math> _ordinis relationem inter quantitates_ <math>x, y,
{y^. / x^.}, {y^.. / x^. ^2}, ... {y^. ^n-1 / x^. ^n-1}, {y^. ^n / x^. ^n}</math> _exprimente, quarum_ <math>a, a′, b, c, d, e, ... A, B, C, D,

<pb 449>[0501]<rh>SERIEBUS.</rh>
E, F</math> _$int corre$pondcntes valores, & a quæ$itis valoribus baud longe di-_
_$tantes; invenire quantitates ad quæ$itos valores magis appropinquantes._</p>
<p>1. Reducatur æquatio $cribendo in eâ pro <math>{y^. ^n / x^. ^n}</math> ejus valorem <math>F + a′
(x - a)^λ</math>, & exinde pro <math>{y^. ^n-1 / x^. ^n-1}</math>, ejus valorem vel fluentem <math>E + F (x - a)
+ {a′ / λ + 1} (x - a)^λ+1</math>, & pro <math>{y^. ^n-2 / x^. ^n-2}</math> ejus fluentem <math>D + E (x - a) +
{1 / 2} F (x - a)^2 + {a′ / (λ + 1) (λ + 2)} (x - a)^λ+2</math>, & $ic deinceps; & ex
æquatis terminis qui maximi re$ultant ex hypothe$i quod <math>x - a</math> $it
perparva quantitas deduci po$$unt valores quantitatum <math>a & λ</math>; & $ic
deinceps, u$que donec inveniatur formula æquationis variabilem <math>y</math> in
terminis variabilis <math>x</math> exprimentis; & deinde, vel per continuas huju$ce
generis approximationes, vel per $ub$titutionem $eriei formulæ præ-
dictæ pro <math>y</math>, & ejus fluxionum pro <math>y^., y^.., y^..., &c.</math> & ex corre$pondentibus
datæ & re$ultantis æquationis terminis inter $e & nihilo re$pective
æquatis, erui pote$t $eries quæ$ita.</p>
<p>Aliter: pro <math>x</math> $cribatur <math>z + a</math>, & pro <math>{y^. ^n / x^. ^n}, {y^. ^n-1 / x^. ^n-1}, {y^. ^n-2 / x^. ^n-2}, {y^. ^n-3 / x^. ^n-3}, &c.</math> $cribantur
re$pective <math>F + a′ z^λ, E + F z + {a′ / λ + 1} z^λ+1, D + E z + {1 / 2} F z^2 +
{a′ / (λ + 1) (λ + 2)} z^λ+2, &c.</math> & per præcedentem methodum progredien-
dum e$t.</p>
<p>Hæc methodus etiam ad incrementiales æquationes applicari pote$t.</p>
<p>Si vero integrales vel fluentiales quantitates in datis incrementiali-
bus vel fluentialibus æquationibus contineantur, tum ex principiis hìc
datis, & iis de fluentibus, &c. fluentialium fluxionum, &c. inve$tigan-
dis prius traditis, erui pote$t $eries valorem quantitatis <math>y</math> exprimens.</p>
<p>2. Summa vel differentia duarum $erierum (<math>A′ & B′</math>), quarum una
<math>A</math> exprimit <math>y</math> in terminis $ecundum dimen$iones quantitates <math>x</math> a$cen-

<pb 450>[0502]<rh>DE INFINITIS</rh>
dentibus; altera vero <math>y</math> in terminis $ecundum dimen$iones quantitatis
<math>x</math> de$cendentibus erit data quantitas; at $i de$cendens $eries conver-
gat, tum a$cendens pene in omnibus ca$ibus diverget; & vice versâ
$i a$cendens $eries convergat, tum de$cendens $eries in prædictis ca$i-
bus etiam diverget.</p>
<p>3. Series, quæ exprimunt vel re$olutiones fluxionalium vel incremen-
tialium, &c. æquationum, vel fluentes &c. fluxionum, &c. quæ progre-
diuntur $ecundum dimen$iones quantitatis variabilis <math>x</math>, debent e$$e con-
vergentes, haud quidem $olummodo cum <math>α</math> $it datus valor quantitatis
<math>x</math>, $ed etiam cum <math>x = β</math>; in utri$que autem his ca$ibus ni <math>α</math> prope <math>= β</math>
plerumque haud converget $eries, ut facile con$tari pote$t ex iis, quæ
prius data fuerunt: $i non convergant utræque $eries, $cribatur <math>z + e</math>
pro <math>x</math> in datâ æquatione, & ex re$ultante æquatione inveniatur $eries
$ecundum dimen$iones quantitatis <math>(z) x - e</math> progrediens, quæ con-
vergit cum <math>z vel = α - e vel = a</math>; in hâc re$ultante æquatione pro
<math>z</math> $cribatur <math>z′ + e′</math>, & ex novâ re$ultante æquatione inveniatur $eries,
quæ convergit, cum <math>z′ vel = a + e′ vel a′</math>; deinde pro <math>z′</math> in æquatione
ultimo derivatâ $cribatur <math>z″ + e″</math>, & ex æquatione re$ultante invenia-
tur $eries quæ convergit cum <math>z″ vel = a′ + e″ vel = a″</math>; & $ic dein-
ceps; u$que donec inveniatur $eries, quæ convergit, cum <math>z′^m vel =
a′^m + e′^m+1 vel β - α + e + e′ + e″ .. + e′^m+1</math>. Sit vero $umma dua-
rum $erierum, quæ prius re$ultabant ex hypothe$i quod <math>z = α - e
& a</math>, etiamque cum <math>z′ = a + e′ & a′</math>, tertio cum <math>z″ = a′ + e″ & a″</math>,
& $ic deinceps; re$pective <math>π, ρ, σ, τ, &c.</math> tum erit $umma $eriei inter
duos valores (<math>α & βz</math>) quantitatis <math>x</math> contenta <math>= π + ρ + σ + &c.</math></p>
<p>Cor.. Sit æquatio algebraica relationem inter variabiles <math>x & y</math> de-
$ignans, inveniatur <math>y</math> vel <math>$. y x^., &c.</math> per $eriem $ecundum dimen$iones
quantitatis <math>x</math> progredientem; $cribatur <math>z + e</math> pro <math>x</math> in data æquatione,
& ex æquatione re$ultante inveniatur <math>y</math> vel <math>$. y x^., &c.</math> in $erie $ecun-
dum dimen$iones quantitatis <math>z</math> progrediente, quæ $it convergens; &
$ic inveniantur aliæ $eries convergentes, quæ corre$pondeant diver$is
valoribus quantitatis <math>e</math>, qui $int re$pective <math>e, e′, e″, &c.</math> & a$$umi po$-
$unt valores <math>e, e′, e″, &c.</math> ita ut e $eriebus re$ultantibus numero finitis

<pb 451>[0503]<rh>SERIEBUS.</rh>
deduci pote$t $umma $erierum vel fluentis inter quo$cunque duos va-
lores <math>α & β</math> quantitatis <math>x</math> contentæ.</p>
<p>Hìc ob$ervandum e$t, ut quo plures recte a$$umantur diver$i valo-
res <math>e, e′, e″, &c.</math> inter duos datos valores <math>α & β</math> quantitatis <math>x</math>, eo magis
celeriter convergent $eries re$ultantes; etiamque animadvertendum
e$t, $i ullus valor quantitatis <math>y</math> inter valores <math>α & β</math> quantitatis <math>x</math> conten-
tus evadat vel infinite magnus <math>vel = 0</math>; utra$que $eries re$ultantes
ex $ub$titutione valorum <math>α & β</math> pro <math>x</math> non evadere convergentes: in
interpolandis $eriebus nece$$e e$t, ut valores quantitatis (<math>x</math>) vel inci-
piant vel terminentur in punctis, in quibus quantitas <math>y</math> evadat <math>vel
= 0</math> vel infinita quantitas.</p>
<p>Ea, quæ de approximationibus fluentium fluxionum in infinitum
progredientium, æque ad has fluxionales æquationes applicari po$$unt.</p>
<p>4. Priu$quam vero finis huic parti de infinitis $eriebus, quæ expri-
munt valores variabilium (<math>x & y</math>) in datis fluxionalibus, &c. æqua-
tionibus contentarum imponatur, quædam animadvertenda $unt de
quibu$dam difficultatibus, quibus premitur methodus prius tradita
prædictos valores inveniendi: e. g. in æquatione fluxionali primi
ordinis, $æpe $i modo <math>a</math> $it proximus valor quantitatis <math>y</math> & $cribatur
<math>a + p</math> pro primâ approximatione, in æquationes $ub$equentes vel
<math>{q̈ / p} = b x^r x^.</math> vel <math>p^r p^. = b {x^. / x}</math> incidamus: $i modo $cribatur <math>A x^n</math> pro <math>p &
n A x^n-1 x^.</math> pro <math>p^.</math> in datâ æquatione tanquam approximatio, tum ma-
nife$to apparet, quod fallet hæc methodus; ergo vel $cribendæ $unt
in datâ æquatione <math>α + v & v^.</math> pro <math>p & p^.</math> in priori ca$u, & <math>α′ + z & z^.</math>
pro <math>x & x^.</math> in po$teriori; & deinde per methodos prius traditas progre-
diendum e$t; vel in priori ca$u a$$umenda e$t <math>p = D e^b′ x^r+1</math> prope ap-
proximatio, ubi <math>D</math> e$t quantitas ad libitum a$$umenda: $ub$tituantur
hæc quantitas per alteram aucta & ejus fluxio pro <math>p & p^.</math> in datâ
æquatione, & exinde per methodos prius traditas progredi liceat; po-
$tea enim ullâ urgetur difficultate opus: in po$teriori inveniatur <math>x</math> in
terminis quantitatis <math>p, &c.</math>; aliter <math>p = ((r + 1) b × log. x + D)^{1 / r+1}</math> prope;
& $ic deinceps.</p>

<pb 452>[0504]<rh>DE INFINITIS</rh>
<p>5. Hinc con$tat in fluxionali æquatione primi ordinis incidere liceat
in fluxionalem homogeneam æquationem primi ordinis; in fluxio-
nali $ecundi, tertii, &c. ordinis æquatione in quamcunque homoge-
neam fluxionalem æquationem $ecundi, tertii, &c. ordinis.</p>
<p>1. In fluxionali æquatione $ecundi ordinis $int termini maximi
<math>a y^.. ^b y^. ^s y^t + &c. = b x^r x^. ^s+2b</math>; ex hâc æquatione inveniatur fluens, & exinde
prima approximatio: aliter $cribatur <math>A x^n</math> pro <math>y</math> in datâ æquatione, &
re$ultat <math>n^s+b (n - 1)^b a A^t+s+b × x^(t+s+b)n-s-2b x^. ^s+2b + &c. = b x^r x^. ^s+2b</math>, $i
vero vel <math>t + s + b = 0</math> vel <math>r + s + 2b = 0</math> vel <math>n = 1</math>, tum haud ex
prædictâ methodo deduci pote$t re$olutio; in priori enim ca$u <math>A^t+s+b=0
= 1</math>, quicunque $it valor quantitatis <math>A</math>, ergo ad libitum a$$umi po-
te$t <math>A</math>; $i autem non $int <math>n^s+b (n - 1)^b a = b & r + s + 2b = 0</math>; tum
non a$$umi pote$t <math>A x^n</math> pro approximatione: in $ecundo ca$u <math>n = 0</math>,
cum <math>r + s + t</math> haud nihilo $it æqualis, & con$equenter <math>n^s+b = 0</math>, ni
<math>s + b = 0</math>; & iterum non a$$umi pote$t <math>A x^n</math> pro approximatione:
idem etiam vero e$t, cum <math>n - 1 = 0</math>, nam in eo ca$u <math>(n - 1)^b = 0</math>:
in his ca$ibus vel $cribendæ $unt quantitates <math>α + v & β + z &</math> ea-
rum fluxiones <math>v^., v^.., &c. & z^.</math> pro <math>y, x, y^., y^.., &c. & x^.</math> in datâ æquatione, &
ex æquatione re$ultante deducenda e$t æquatio: vel $ub$titutio in ca$u
primo u$urpanda e$t, in quâ ita a$$umi po$$unt coefficientes <math>F, E, D,
&c., a′, &c.</math>, ut per hanc methodum $emper detegi pote$t $eries; nulla
enim po$tea occurrit quantitas, quæ exigit fluentem fluxionis, i. e.
$eries incipit a termino, po$t quem nulla quantitas ad libitum a$$umi
pote$t: vel in priori ca$u pro <math>y</math> a$$umatur <math>A e^x^n</math>, & exinde pro <math>y^. & y^..</math>
re$pcctive <math>n A x^n-1 x^. e^x^n, & n · (n - 1) · A x^n-2 x^. ^2 e^x^n + n^2 A x^2n-2 x^. ^2 e^x^n</math>; $i
vero <math>s = - 2 h</math>, fallet etiam hæc methodus. Cum vero <math>y</math> haud facile
erui po$$it e quantitate <math>x</math>, for$an præ$tet transformare datam æqua-
tionem, ita ut fluat uniformiter <math>y</math>; deinde invenire <math>x</math> in $erie $ecundum
dimen$iones quantitatis <math>y</math> progrediente, tum ex rever$ione $erierum
deduci pote$t <math>y</math> in $erie $ecundum dimen$iones quantitatis <math>x</math> progre-
diente.</p>
<p>2. Sint maximi termini <math>v^. ^m ^π · v^. ^m-1 ^α · v^. ^m-2 ^β · v^. ^m-3 ^γ × &c. = a x^r x^. ^mπ+(m-1)α+(m-2)β+&c.</math>;

<pb 453>[0505]<rh>SERIEBUS.</rh>
deinde inveniantur radices, i. e. quantitas <math>v</math> in terminis quantitatis <math>x</math>
æquationis re$ultantis a $upponendo omnes terminos, qui maximi
$unt, $imul adjunctos nihilo æquales e$$e pro primis approximationi-
bus; $i irrationales quantitates in datâ vel datis æquationibus conti-
neantur, tum $emper rejiciendi $unt termini, qui haud maximi $unt,
in prædictâ æquatione contenti: vel aliter; $cribatur <math>A x^n</math> pro <math>z</math> in
datâ æquatione, & $i vel <math>π + α + β + γ + &c. = 0 vel - r = (π
+ α + β + γ + &c.) (m) - α - 2β - 3γ - &c. vel n = 1, 2, .. m;
&c.</math> fallit hæc methodus: in multis hi$ce ca$ibus erui pote$t appro-
ximatio $cribendo <math>a e^x^n</math> pro <math>y</math>, & ejus fluxiones pro <math>y^., y^.., &c.</math> re$pective,
& ex æquatis terminis erui pote$t approximatio quæ$ita; &c.: vel hìc
applicari po$$unt omnia in præcedente exemplo tradita.</p>
<p>6. Si hæc po$terior methodus fallat, tum ita transformetur æquatio
ut fluat uniformiter <math>y</math>, & inveniatur <math>x</math> in $erie $ecundum dimen$iones
quantitatis <math>y</math> progrediente; & denique $i hæc methodus fallat, tum
per methodum prius traditum $cribantur <math>v + α & z + β</math> & earum
fluxiones pro <math>x & y</math> & earum fluxionibus re$pective in datâ æqua-
tione, ubi <math>v & z</math> $unt perparvæ quantitates; vel diver$is aliis modis
transformentur variabiles datæ æquationis quantitates in alias, quæ
a$$ignabilem habent relationem ad variabiles datæ æquationis quan-
titates, &c.</p>
<p>In quibu$dam ca$ibus, cum utræque prædictæ quantitates nihilo
evadant æquales, ex earum coefficientibus nihilo æqualibus e$$e $up-
po$itis re$ultabunt indices quæ$iti.</p>
<p>Omnia hæc æque ad incrementiales æquationes applicari po$$unt.</p>
<p>7. In huju$modi re$olutionibus omnium algebraicarum, fluxiona-
lium, incrementialium, &c. æquationum primo reducendæ $unt æqua-
tiones, ita ut earum variabiles quantitates evadant perparvæ vel per-
magnæ, prout in $erie quæ$itâ earum dimen$iones a$cendunt vel de-
$cendunt.</p>
<p>8. Pleraque ex iis, quæ hìc traduntur de $eriebus ex algebraicis, flu-
xionalibus & incrementialibus æquationibus deductis æque ad $eries
ex infinitis aliis æquationibus derivatas applicari po$$unt.</p>

<pb 454>[0506]<rh>DE INFINITIS</rh>
<p>9. Si vero incrementialis $it æquatio, tum formula $eriei a$$umendæ
$it <math>y = A x^s (x ± x<x>.</x>)^s + B x^t (x ± x<x>.</x>)^t + &c.</math> $i $it incrementialis æqua-
tio primi ordinis; vel <math>y = A x^s × (x ± x<x>.</x>)^s × (x ± 2 x<x>.</x>)^s + B x^t ×
(x ± x<x>.</x>)^t × (x ± 2 x<x>.</x>)^t + &c.</math> $i modo $it incrementialis æquatio $e-
cundi ordinis; & $ic deinceps: vel <math>y = A x^s × (x ± x<x>.</x>)^s × (x ± b) +
B x^t × (x ± x^.)^t × (x ± c)</math> pro integrali incrementialis æquationis
primi ordinis a$$umi pote$t, & $ic de infinitis aliis.</p>
<p>Hic autem animadvertendum e$t, quod tot erunt quantitates in-
variabiles ad libitum a$$umendæ in $erie re$ultante ab initio, quot $it
ordo incrementialis æquationis, ut con$tat e præcedentibus.</p>
<p>hæc methodus etiam extendi pote$t ad duas vel plures algebraicas,
fluxionales, incrementiales, &c. æquationes tres vel plures variabiles
quantitates involventes.</p>
<p>10. Hæ methodi etiam applicari po$$unt ad æquationes ex infinitis
terminis, & earum fluxionibus, &c. con$tantes. E. g. Sit æquatio al-
gebraica <math>x = a y + b y^2 + c y^3 + d y^4 + &c.</math> & inveniri pote$t <math>y = {x / a}
- {b / a^3} x^2 + {2 b^2 - a c / a^5} x^3 + &c.</math> & $ic $it <math>x^. = a y + b y^3 + c y^5 + &c.</math>
& inveniri pote$t <math>y = {x / a} - {b / a^4} x^3 + {3 b^2 - a c / a^7} x^5 + &c.</math></p>
<p>2. Sit <math>a y + b y^2 + c y^3 + &c. = α x + β x^2 + γ x^3 + &c.</math> a$$umatur
<math>y = {α / a} x + l x^2 + m x^3 + n x^4 + &c.</math> quæ erit $eries formulæ quæ$itæ,
$cribatur hæc quantitas pro <math>y</math> in datâ æquatione & re$ultat <math>α x + (a l
+ b {a^2 / α^2}) x^2 + ({c α^3 / a^3} + {2 l α b / a} + a m) x^3 + &c. = α x + β x^2 + γ x^3
+ &c.</math> unde <math>a l + {b α^2 / a^2} = β</math> & con$equenter <math>l = {β a^2 - b α^2 / a^3}, & {c α^3 / a^3}
+ {2 l α b / a} + a m = γ</math>, &c.</p>
<p>Si vero dentur æquationes relationes inter quantitates huju$ce for-
mulæ <math>1 × 2 × 3 .. z,</math> earum fluxiones. &c. in quibus numerus facto-

<pb 455>[0507]<rh>SERIEBUS.</rh>
rum continuo augetur, de$ignantes; tum e continuis $ub$titutioni-
bus, approximationibus, interpolationibus, &c. inveniri pote$t una
quantitas in terminis alterius.</p>
<p>Converti po$$unt algebraicæ, fluxionales, &c. æquationes in feries,
quarum termini $ecundum prædictas formulas progrediuntur.</p>
<h>THEOR. XXXIII.</h>
<p>Cum <math>y</math> $it infinita quantitas, tum ejus fluxiones _(_<math>y^., y^.., y^..., &c.</math>) quo-
rumcunque ordinum erunt infinitæ quantitates, i. e. $it <math>y</math> quæcunque
functio quantitatis <math>x</math>, & $i <math>y</math> fiat infinita quantitas, cum <math>x</math> maneat
finita, tum $emper <math>y: x; y^.: x^.; y^..: x^. ^2; y^...: x^. ^3; &c.</math> erunt rationes infi-
nite magnæ, quarum $ingula $ub$equens infinite major erit quam
præcedens.</p>
<p>Idem etiam de incrementis integralium affirmari pote$t.</p>
<h>PROB. XXV.</h>
<p it>Datâ fluxionali æquatione <math>P = 0</math> variabiles quantitates <math>x, y</math> & earum
fluxiones babente, ubi <math>x & y</math> $unt ab$ci$$æ & corre$pondentes ordinatæ ad
curvam; invenire fluxionalem æquationem ejus a$ymptotos generaliter de-
$ignantem.</p>
<p>1<_>mo</_>. Fingatur <math>y</math> infinita quantitas, & ex hâc hypothe$i $int <math>A</math> ter-
mini qui maximi re$ultant, <math>B</math> vero termini qui $int maximi ex om-
nibus reliquis terminis; 2<_>do</_>. fingatur <math>x</math> infinita quantitas, <math>y</math> vero
finita, deinde $int termini, qui ex hâc hypothe$i evadant maximi vel
proxime $uperiores, re$pective <math>C & D</math>; tum æquationes <math>A + B = 0
& C + D = 0</math> de$ignabunt curvas ea$dem habentes a$ymptotos, quas
habet data curva.</p>
<h>THEOR. XXXIV.</h>
<p>Datâ præcedente fluxionali æquatione <math>(P = 0) n</math> ordinis, & literis
<math>A, B, C & D</math> ea$dem quantitates ac in præcedente problemate deno-

<pb 456>[0508]<rh>DE INFINITIS</rh>
tantibus; æquationis <math>A + B = 0</math> inveniatur generalis fluens <math>K = 0</math>,
& $i modo invariabiles quantitates ad libitum a$$umendæ eædem $int
in æquatione <math>K = 0</math> ac in fluente fluxionalis æquationis <math>P = 0</math>, i. e.
$i modo <math>a, b, c, d, &c.</math> corre$pondentes dati valores quantitatum <math>x, y,
x^., &c.</math> in unâ, $int etiam corre$pondentes valores prædictarum quan-
titatum in alterâ <math>K = 0</math>; & exinde deducantur valores quantitatis <math>x</math>,
cum <math>y</math> evadat infinita, qui $int <math>α, β, γ, δ, &c.</math> & $i valor <math>π</math> vel <math>ρ</math> quan-
titatis <math>x</math>, inter quos requiritur fluens, $it major quam radix <math>α</math> vel <math>β</math> vel
<math>γ</math> vel <math>δ, &c.</math> tum plerumque haud converget $eries $ecundum dimen-
$iones quantitatis <math>x</math> a$cendens; $i vero minor $it <math>π</math> vel <math>ρ</math> quam ulla
radix <math>α</math> vel <math>β</math> vel <math>γ</math> vel <math>δ, &c.</math> tum plerumque haud converget $eries $e-
cundum dimen$iones quantitatis <math>x</math> de$cendens.</p>
<p>Si vero ita transformetur data æquatio, ut ab$ci$$a <math>z</math> ab alio puncto
incipiat, tum e principiis prius traditis pro algebr. &c. æquationibus
facile con$tat, ut haud convergent $eries $ecundum dimen$iones quan-
titatis <math>z</math> progredientes, ni <math>π & ρ</math> inter duas proximas radices <math>β & γ</math>
quantitatis <math>x</math> con$tituantur.</p>
<h>THEOR. XXXV.</h>
<p>1. Sit prædicta fluxionalis æquatio <math>P = 0</math>, & ex eâ inveniatur æqua-
tio, cujus radices $int limites inter radices, <math>&c</math> datæ æquationis, i. e.
fingatur <math>y</math> folummodo variabilis, & inveniatur fluxio datæ æquatio-
nis, & $i valor quantitatis <math>x</math> pro radice <math>y</math> a$$umptus $it radix inventæ
æquationis, tum infinite divergit approximatio deducta; ejus con-
vergentia pendet ex hoc, nempe quo magis approximatio inventa ad
unam radicem accedat quam ad reliquas.</p>
<p>2. Ut $eries ex æquationibus huju$ce generis re$ultantes celeriter con-
vergant, nece$$e e$t plerumque interpolare plures $eries per methodos
con$imiles iis pro algebraicis æquationibus prius traditis; $ed caven-
dum e$t, ne diver$is correctionibus utamur, i. e. ut corre$pondentes
valores variabilium & earum fluxionum $emper conveniant.</p>
<p>Omnia hæc etiam ad incrementiales æquationes applicari po$$unt.</p>

<pb 457>[0509]<rh>SERIEBUS.</rh>
<p>3. Haud raro in hi$ce æquationibus quantitas <math>y</math> evadet impo$$ibilis
quantitas, & in hoc ca$u $eries nunquam converget, ni pro primâ
approximatione a$$umatur impo$$ibilis quantitas, cujus po$$ibilis pars
ab po$$ibili, impo$$ibilis vero ab impo$$ibili quantitatis parte haud
longe di$tet: hoc in ca$u valor quantitatis <math>y</math> nunquam exprimi pote$t
per $eriem $ecundum dimen$iones quantitatis <math>x</math> progredientem, ni <math>x</math>
$it impo$$ibilis quantitas.</p>
<p>4. Sit <math>y = A</math>, ubi <math>A</math> $it $eries $ecundum dimen$iones quantitatis <math>x</math>
progrediens, deinde detegatur $eries <math>y = B</math>, quæ exprimit <math>y</math> in termi-
nis novæ quantitatis <math>z</math>, ubi <math>z</math> $it data functio quantitatis <math>x</math> vel functio
quantitatum <math>x & y</math>; tum in multis ca$ibus e datâ functione prædictâ
& ca$ibus, in quibus convergit $eries <math>A</math>, & principiis hoc in capite
traditis erui po$$unt ca$us, in quibus convergit $eries <math>y = B</math>.</p>
<h>PROB. XXVI.</h>
<p it>Datis algebraicis, fluxionalibus, vel incrementialibus, &c. æquationibus,
quantitates <math>x & y, &c.</math> involventibus; invenire valorem quantitatis <math>y</math> per
terminos quo$cunque irrationales quantitatis <math>x</math> $ecundum datam legem pro-
gredientes, $i modo is per tales terminos exprimi po$$it.</p>
<p>Primo e terminis, qui maximi $unt, datarum æquationum in-
veniatur primus terminus (<math>α</math>) datæ formulæ: $cribatur <math>α + p</math> pro <math>y</math>
in datis æquationibus, & e terminis re$ultantium æquationum, qui
maximi $unt, inveniatur propinquus valor quantitatis <math>p</math> datæ formulæ,
& $ic iteratis operationibus tandem invenietur $eries quæ$ita: vel a$-
$umatur $eries generaliter expre$$a in terminis $ecundum datam for-
mulam progredientibus, $cribatur hæc $eries pro <math>y</math> in datâ æquatione
& ejus fluxiones, &c. pro <math>y^., y^.., &c.</math> & e coefficientibus corre$ponden-
tium terminorum re$ultantis æquationis nihilo æqualibus e$$e $uppo-
$itis erui pote$t $eries quæ$ita.</p>
<p>Ex. 1. Sit æquatio <math>y^. = (a + b x^m)^μ-1 x^τ x^.</math>, invenire quantitatem <math>y</math>
per $eriem con$tantem e terminis formulæ <math>(a + b x^m)^μ z^t;</math> a$$u-

<pb 458>[0510]<rh>DE INFINITIS</rh>
matur <math>y = (a + b x^m)^μ (A x^s + B x^s+m + C x^s+2m + &c.)</math> $cribatur hic
valor pro <math>y</math> & ejus fluxiones pro <math>y^., y^.., &c.</math> in datâ æquatione, & fiant
corre$pondentes termini re$ultantis æquationis nihilo re$pective æqua-
les, & ex æquationibus exinde deductis con$equentur <math>s = π + 1, A
= {1 / a s}, B = - {μ m b A + s b A / (s + m) a}, &c.</math> vel a$$umatur <math>(a + b x^m)^μ ×
{1 / π + 1 · a} x^m+1 - p</math> pro fluente quæ$itâ, & erit <math>p^. = {μ m + s / π + 1 · a} b x^m+π (a +
b x^m)^μ-1 x^.</math>, deinde a$$umatur pro <math>p</math> quantitas <math>{μ m + s / (π + 1)(m + π + 1) a^2}
b x^m+π+1 (a + b x^m)^μ + q</math> & erit <math>q^. = {(μ m + s) × (μ m + m + s) b^2 / (π + 1)(m + π + 1)} × x^2m+π
(a + b x^m)^μ-1 x^.</math>, & $ic deinceps; unde facile con$tat lex $eriei re$ul-
tantis.</p>
<p>Ex. 2. Sit <math>Z = - log. z</math>, invenire integarlem incrementi <math>v = {z<x>.</x> / z}</math>;
nunc fingatur integralis incrementi <math>({z<x>.</x> / z})</math> e$$e <math>Z + {a / z} + {b z / z · z + z<x>.</x>} +
{c z^2 <x>.</x> / z · z + z<x>.</x> · z + 2 z<x>.</x>} + &c.</math> cujus incrementum invenietur <math>Z<x>.</x> + {a z<x>.</x> / z · z + z<x>.</x>}
+ {2 b z^2 <x>.</x> / z · z + z<x>.</x> · z + 2 z<x>.</x>} + {3 c z^3 <x>.</x> / z · z + z<x>.</x> · z + 2 z<x>.</x> · z + 3 z<x>.</x>} + &c. = {z<x>.</x> / z}</math>; $ed
<math>Z<x>.</x></math> erit per prob. 2. lib. <math>2^di . = {z<x>.</x> / z} - {z^2 <x>.</x> / 2 z^2} + {z^3 <x>.</x> / 3 z^3} - {z^4 <x>.</x> / 4 z^4} + &c. = {z<x>.</x> / z} -
{z^2 <x>.</x> / 2 z · z + z<x>.</x>} - {z^3 <x>.</x> / 6 z · z + z<x>.</x> · z + 2 z<x>.</x>} - {z^4 <x>.</x> / 4 z · z + z<x>.</x> · z + 2 z<x>.</x> · z + 3 z<x>.</x>}
- &c.</math> $cribatur hæc quantitas pro ejus valore in æquatione prædictâ
<math>Z<x>.</x> + {a z<x>.</x> / z · z + z<x>.</x>} + &c. = {z<x>.</x> / z}</math>, unde <math>a = {1 / 2}, 2 b = {1 / 6}, 3 c = {1 / 4}, &c.</math> & con-
$equenter integralis quæ$ita erit <math>v = Z + {1 / 2 z} + {z<x>.</x> / 12 z · z + z<x>.</x>} +<pb 459>[0511]<rh>SERIEBUS.</rh>
{z<x>.</x>^2 / 12 z · z + z<x>.</x> · z + 2 z<x>.</x>} + {19 z<x>.</x>^3 / 30 z · z + z<x>.</x> · z + 2 z<x>.</x> · z + 3 z<x>.</x>} - &c.</math> In
hâc integrali deducendâ $upponitur <math>z<x>.</x></math> invariabilis quantitas.</p>
<p>Cor. · In detegendis integralibus incrementorum $æpe occurret
incrementum <math>{z<x>.</x> / z}</math>, cujus integralis $upra datur.</p>
<p>2. Si vero requiratur quæcunque functio (<math>v</math>) quantitatum <math>y & x</math>, vel
fluens (<math>v</math>) cuju$cunque fluxionis e quantitatibus <math>y & x &</math> earum flu-
xionibus compo$itæ, &c. per $eriem cujus termini progrediuntur vel
$ecundum dimen$iones quantitatis <math>x</math>, vel $ecundum quamcunque
aliam legem, cujus termini ad $ummam continuo convergunt; inve-
niatur proximus valor quantitatis <math>v</math>, qui $it <math>a</math>, deinde pro <math>v</math> $cribatur
<math>a + p</math>, & ex æquationibus re$ultantibus inveniatur proximus valor
quantitatis <math>p</math>, & $ic deinceps u$que donec tandem invenietur valor
quantitatis <math>v</math> quæ$itus.</p>
<p>Hæ æquationes reduci po$$unt ad unam, quæ exprimit relationem
inter variabiles <math>x & v</math>.</p>
<p>3. Sint vero duæ vel plures æquationes tres vel plures variabiles quan-
titates (<math>x′, y′, z′, &c.</math>) habentes; 1<_>mo</_>. ita reducendæ $unt datæ æqua-
tiones per methodos prius traditas; ut evadant earum variabiles (<math>x, y,
z, &c.</math>) perparvæ vel permagnæ, prout requiritur $eries a$cendens vel
de$cendens; deinde fiant termini utrarumque æquationum, qui $int
ex hypothe$i, quod <math>x</math> vel $it perparva vel permagna quantitas, maximi,
inter $e æquales; iis, qui minores $int, abjectis, & exinde deducantur
proximi valores quantitatum <math>y, z, &c.</math> in terminis quantitatis <math>x</math>, qui
$int re$pective <math>a, α, &c.</math> $cribantur <math>a + p, α + π, &c.</math> in datis æqua-
tionibus re$pective pro <math>y, z, &c.</math> & e terminis, qui maximi $int in re-
$ultantibus æquationibus, inveniantur proximi valores quantitatum
<math>p, π, &c.</math> & $ic deinceps; unde tandem eruentur valores quantitatum
<math>y, z, &c.</math> corre$pondentes.</p>
<p>Aliter: Per præcedentem methodum inveniantur proximi valores
quantitatum variabilium <math>x′, y′, z′, &c.</math> in datis æquationibus conten-
tarum & transformentur datæ æquationes, ita ut earum variabiles <math>x,

<pb 460>[0512]<rh>DE INFINITIS</rh>
y, z, &c.</math> evadant perparvæ vel permagnæ, tum $int <math>y = A x^n</math> prope,
<math>z = a x^m</math> prope, &c.; $cribantur hi proximi valores <math>A x^n, a x^r, &c.</math> pro
<math>y, z, &c.</math> in datis æquationibus, & e differentiâ indicum re$ultantium ex
hâc $ub$titutione colligi po$$unt formulæ $erierum <math>y = A x^n + B x^n+m
+ C x^n+2m + &c. & z = ax^r + bx^r+s + cx^r+2s + &c. &c.</math> $cribantur
hæ quantitates pro $uis valoribus in datis æquationibus, & ex æqua-
tis corre$pondentibus terminis re$ultantium æquationum con$equun-
tur coefficientes <math>A, B, C, &c. a, b, c, &c.</math></p>
<p>Si vero duo proximi valores quantitatis $int <math>A x^n</math>, tum a$$umenda
e$t $eries formulæ <math>Ax^n + Bx^n+{1 / 2}m + Cx^n+m + &c. = y</math>; $i tres, tum
$eries formulæ <math>Ax^n + Bx^n+{1 / 3}m + Cx^n+{2 / 3}m + Dx^n+m + &c.</math> a$$umenda
e$t; & $ic deinceps.</p>
<p>4. Eadem methodus etiam applicari pote$t ad fluxionales, inte-
grales, &c. æquationes. e. g. Sint fluxionales æquationes (<math>l, r, s, &c.</math>)
ordinum datæ, & ita transformentur, ut evadant variabiles perparvæ
vel permagnæ in iis pro <math>z & y</math> $cribantur <math>αx^m & A x^n, &c.</math> re$pective,
& per prædictas methodos progrediendum e$t; tum plerumque in
$ingulis re$ultantibus $eriebus continebuntur (<math>l + r + s + &c.</math>) varia-
biles quantitates ad libitum a$$umendæ aliter; in prædictis æquatio-
nibus pro <math>{y^. ^l / x^. ^l}, {y^. ^l-1 / x^. ^l-1}, {y^. ^l-2 / x^. ^l-2}, &c.; {z^. ^l / x^. ^l}, {z^. ^l-1 / x^. ^l-1}, {z^. ^l-2 / x^. ^l-2}, &c. &c.</math>; ubi <math>l</math> major e$t quam
<math>r</math> vel <math>s, &c.;</math> $cribantur re$pective <math>a + b λ^λ, c + ax + {b / λ + 1} x^λ+1, d +
c x + {a / 2} x^2 + {b / (λ + 1) (λ + 2)} λ^λ+2, &c; a′ + b′x^μ, c′ + a′ x + {b′ / μ + 1}
x′^μ+1, d′ + c′x + {a′ / 2} x^2 + {b′ / (μ + 1) (μ + 2)}x^μ+2, &c. &c.</math> re$pective; &
exinde deduci po$$unt quantitates <math>y, z, &c.</math> in terminis quantitatis (<math>x</math>),
quæ fluit uniformiter: $ed hìc animadvertendum e$t, quod nunquam
plures quam <math>l + r + s + &c.</math> po$$unt e$$e invariabiles quantitates ad
libitum a$$umendæ.</p>

<pb 461>[0513]<rh>SERIEBUS.</rh>
<h>THEOR. XXXVI.</h>
<p>Exii$dem æquationibus vel ad $ummas $erierum vel ad fluentes fluxi-
onum deduci po$$unt diver$æ incrementiales vel fluxionales, &c. æqua-
tiones, ut e $equentibus con$tabit; quæ autem omnes diver$æ æqua-
tiones, $i generaliter corrigantur, $emper ea$dem præbent $eries.</p>
<p>Ex. 1. Sit æquatio <math>y^2 = a^2 + bx + {x4 / c^2}</math>, ex hâc æquatione erui po-
te$t valor quantitatis <math>y</math> in terminis $ecundum dimen$iones quantitatis
<math>x</math> progredientibus <math>y = a + {b x / 2a} - {b^2 x^2 / 8a^3} + {b^3 x^3 / 16a^3} - &c.</math> $it <math>a = 0</math> &
erit $eries inventa <math>y = b^{1 / 2} x^{1 / 2} + {1 / 2} {x^{7 / 2} / b^{1 / 2} c^2} - &c.</math></p>
<p>Inveniatur fluxio datæ æquationis, & erit <math>2 y y^. = b x^. + {4x^3 x^. / c^2}</math>, unde
<math>y = b^{1 / 2} x^{1 / 2} + {1 / 2}{x^{7 / 2} / b^{1 / 2} c^2}&c.</math> $ed inveniatur hujus æquationis generalis fluens &
a$$umenda e$t quantitas <math>y^2 = a^2</math> prope, pro <math>y</math> $cribatur <math>a + p</math> in æquatione
<math>2 y y^. = b x^. + {4 x^3 x^. / c^2}</math>, & re$ultat <math>2 (a + p) p^. = b x^. + {4 x^3 x^. / c^2}</math>, & e ter-
minis, qui maximi $int, inter $e æqualibus e$$e $uppo$itis invenietur
<math>2 a p^. = b x^.</math> prope, unde <math>p = {bx / 2a}</math> prope, & exinde re$ultat prior $eries
<math>y = a + {b x / 2 a} + &c.</math></p>
<p>Dividatur data æquatio <math>y^2 = a^2 + b x + {x^4 / c^2}</math> per <math>x</math>, & re$ultat <math>{y^2 / x} =
{a^2 / x} + b + {x^3 / c^2}</math>, cujus fluxio erit <math>{2 y y^. / x} - {y^2 x^. / x^2} = {3 x^2 x^. / c^2} - {a^2 x^. / x^2}</math>: termini
huju$ce æquationis, qui maximi inveniuntur, erunt <math>{y^2 x^. / x^2} = {a^2 x^. / x^2}</math>; unde
<math>y^2 = a^2</math> prope, & <math>y = ± a</math> prope, $cribatur <math>a + p</math> pro <math>y</math> in fluxione

<pb 462>[0514]<rh>DE INFINITIS</rh>
<math>{2 y y^. / x} - {y^2 x^. / x^2} = {3 x^2 x^. / c^2} - {a^2 x^. / x^2}</math>, & re$ultat <math>{2 × (a + p)p^. / x} - {(2 a p + p^2) x^. / x^2}
= {3 x^2 x^. / c^2}</math>; maximi termini hujus æquationis $unt <math>{2 a p^. / x} - {2 a p x^. / x^2}</math>, fiant
nihilo æquales, & re$ultat <math>{p^. / p} = {x^. / x}</math>, unde <math>p = {b x / 2 a}</math> prope, & æquatio fiet
eadem ac prior <math>y = a + {b x / 2 a} + &c.</math> Et $ic deinceps.</p>
<p>Cor. 1. A$$umatur fluens <math>A + B = 0</math>, ubi <math>A & B</math> $unt quæcunque
functiones literarum <math>x & y, & a</math> invariabilis quantitas generaliter
a$$umpta, inveniatur quantitas <math>y</math> ex hâc æquatione in terminis $ecun-
dum dimen$iones quantitatis <math>x</math> progredientibus, i. e. $it <math>y = A x^n +
B x^n±m + C x^n±2m .... E x^n±rm + &c.</math> ubi <math>E</math> $it terminus in quem
primum ingreditur litera <math>a</math>; inveniatur fluxio præcedentis fluentis
<math>π^. = 0</math> in quâ haud continetur <math>a</math>, & ex hâc æquatione per methodos
prius datas inveniatur <math>y</math> in terminis quantitatis <math>x</math>, quæ erit <math>y = A x^n
B x^n±m + C x^n±2m + &c.</math> u$que ad terminum <math>D x^n±(r-1)m + E x^n±rm
+ &c.</math> ubi per prædictam methodum $i <math>p</math> $it proxima approximatio
vel quod idem e$t $i in datâ æquatione <math>π^. = 0</math> $cribatur <math>A x^n + B x^n±m
+ C x^n±2m ..... D x^n±(r-1)m + p</math> pro <math>y</math>, re$ultabit æquatio <math>{p^. / p} =
{(n±r m) x^. / x}</math> prope, cujus fluens erit <math>p = l x^n±rm</math> prope, ubi <math>l</math> denotat
quamcunque quantitatem ad libitum a$$umendam; unde hæc $eries
$emper eadem evadet ac præcedens.</p>
<p>Hinc facile deduci po$$unt infinitæ æquationes, quarum $eries ge-
nerales, quæ exprimunt <math>y</math> in terminis quantitatis <math>x</math>, erunt eædem.</p>
<p>Hæc principia facile applicari po$$unt ad fluxionales æquationes
$uperiorum ordinum, vel ad fluxionales æquationes plures variabiles
quantitates & earum fluxiones habentes.</p>
<p>Cor. 2. Series, quæ exprimunt generalem valorem quantitatis <math>y</math> in
terminis vero $ecundum dimen$iones quantitatis <math>x</math> progredientibus,

<pb 463>[0515]<rh>SERIEBUS.</rh>
eædem inveniuntur e diver$is æquationibus <math>A^. ^n + B^. ^n = 0, A^. ^n-1 + B^. ^n-1 +
a x^. ^n-1 = 0, A^. ^n-2 + B^. ^n-2 + a x x^. ^n-2 + b x^. ^n-2 = 0, &c.</math> u$que ad <math>A^. + B^. +
a x^n-1 x^. + b x^n-2 x^. + c x^n-3 x^. + &c. .... l x^. = 0, & A + B + a x^n +
b x^n-1 + c x^n-2 ... l′ = 0</math>, ubi literæ <math>a, b, c, ... l & l′</math> qua$cunque inva-
riabiles denotant quantitates.</p>
<p>Duæ $eries, quæ exprimunt prædictum generalem valorem, eædem
inveniuntur e duabus æquationibus <math>π = 0 & L × π = 0</math>, ubi <math>π</math> e$t
fluxionalis quantitas $uperioris ordinis quam quantitas <math>L</math>.</p>
<h>PROB. XXVII.</h>
<h it>Ex datis relationibus inter valores <math>x & z</math> incognitæ quantitatis <math>x</math> & inter
$ummas duarum $erierum re$ultantium; invenire coefficientes ip$ius $eriei.</h>
<p>Sit $umma $eriei huju$ce formulæ <math>a x^m + b x^m+r + c x^m+2r + &c. = Q</math>;
inveniatur valor (<math>R</math>) eju$dem $eriei, cum <math>x</math> evadat (<math>z</math>) quæcunque
data functio quantitatis <math>x,</math> i. e. <math>a z^m + b z^m+r + c z^m+2r + &c. = R =
A x^m′ + B x^m+r′ + C x^m′+2r′ + &c. + A′ x^b + B′ x^b+s + C′ x^b+2s + &c.</math>;
tum ex quantitate <math>Q = a x^m + b x^m+r + &c.</math> & relatione inter $um-
mas inveniatur <math>R = a′ x^m′ + b′ x^m′+r′ + c′ x^m′+2r′ + &c.</math>; & ex æquatis
corre$pondentibus terminis re$ultant <math>A = a′, B = b′, C = c′, &c.</math>,
& exinde deduci po$$unt coefficientes <math>a, b, c, &c.</math> quæ$itæ.</p>
<p>Ex. Cum <math>x</math> evadat <math>2 x = z</math>, tum $eries <math>a x^m + b x^m+r + c x^m+2r +
&c.</math> evadat <math>(a x^m + b x^m+r + c x^m+2r + &c.)^2</math>; in datâ $erie pro <math>x</math> $cri-
batur <math>2 x</math>, & re$ultat <math>a 2^m x^m + b × 2^m+r x^m+r + c × 2^m+2r x^m+2r + &c. =
(a x^m + b x^m+r + &c.)^2 = a^2 x^2m + 2 a b x^2m+r + (2 a c + b^2) x^2m+2r + &c.</math>;
& ex æquatis corre$pondentibus terminis re$ultant <math>a 2^m × x^m = a^2 x^2m</math>,
& confequenter <math>m = 2 m</math>; & exinde <math>m = 0 & 2^m a = 2^0 a = a = a^2</math>;
unde <math>a = 1; & 2 a b x^2m+r = 2 a b x^r = b × 2^m+r x^r</math>, unde <math>2^r = 2</math> & ex-
inde <math>r = 1; & c × 2^2r × x^2r = c × 2^2 × x^2 = (2 a c + b^2) x^2r = (2 c +
b^2)x^2r</math>, unde <math>4 c = 2 c + b^2</math>, & exinde <math>c = {b^2 / 2}</math>; & $ic deinceps; &

<pb 464>[0516]<rh>DE INFINITIS</rh>
tandem re$ultat $eries <math>a x^m + b x^m+r + c x^m+2r + &c. = 1 + b x + {b^2 / 2}
x^2 + {b^3 / 2 · 3} x^3 + {b^4 / 2 · 3 · 4} x^5 + &c.</math>; quæ ex logarithmo invenit nu-
merum.</p>
<p>Ex. 2. Sit prædicta $eries <math>a + b x^r + c x^2r + d x^3r + &c.</math>; cum vero
<math>x</math> evadat <math>2 x = z</math>, tum $eries <math>a + b x^r + &c.</math> evadat <math>(a + b x^r + c x^2
+ &c.)^s = a^s + s a^s-1 b x^r + s · {s - 1 / 2} a^s-2 b^2 x^2r \\ + s # a^s-1 c x^2r # + &c. = a + b × 2^r x^r
+ c × 2^2r x^2r + d × 2^3r × x^3r + &c.</math>, unde <math>a = a^s</math> & con$equenter <math>a = 1;
& 2^r b = s b</math> & exinde <math>2^r = s & r = {log. s / log. 2}; & s · {s - 1 / 2} a^s-2 b^2 + s a^s-1 c
= 2^2r × c & s · {s - 1 / 2} b^2 = (2^2r - s) c = (s^2 - s) c</math>, unde <math>c = {1 / 2} b^2</math>: etiamque
<math>s · {s - 1 / 2} · {s - 2 / 3} b^3 + s · (s - 1) b c + s d = 2^3r d = s^3 d</math> & con$equen-
ter <math>s · {s - 1 / 2} · {s - 2 / 3} b^3 + s · (s - 1) · b · {b^2 / 2} = (s^3 - s) d & s · {s - 1 / 2}·
{s + 1 / 3} b^3 = (s^3 - s) d & d = {b^3 / 1 · 2 · 3}: & $ic e = {b^4 / 1 · 2 · 1 · 4}, &c.</math>; & $e-
ries <math>a + b x^r + &c. = 1 + {b / 1}x^r + {b^2 / 1 · 2}x^2r + {b^3 / 1 · 2 · 3} x^3r + {b^4 / 1 · 2 · 3 · 4}
x^4r + &c.</math>, ubi <math>r = {log. s / log. 2}</math>, quæ erit <math>1</math>, cum <math>s = 2; & - 1</math> cum <math>s = {1 / 2}</math>.</p>
<p>Ex. Sit quantitas <math>a x + b x^2 + c x^3 + &c. = l</math>, quæ $it logarith-
mus quantitatis <math>1 + x</math>, at, cum numerus <math>(1 + x)</math> evadat <math>z = (1 + x)^2</math>;
tum logar. <math>l</math> evadat duplus; pro <math>x</math> $cribatur igitur <math>2 x + x^2</math> in
prædictâ $erie <math>a x + b x^2 + c x^3 + &c.</math>, & re$ultabit <math>a (2 x + x^2) + b
(4 x^2 + 4 x^3 + x^4) + c (2 x + x^2)^3 + &c. = 2 a x + (a + 4 b)x^2 +
(4 b + 8 c) x^3 + (b + 12 c + 16 d)x^4 + &c. = 2 l = 2 a x + 2 b x^2
+ 2 c x^3 + 2 d x^4 + &c.</math>: ex æquatis corre$pondentibus terminis re-

<pb 465>[0517]<rh>SERIEBUS.</rh>
$ultant <math>2 a = 2 a; 2 b = a + 4 b</math>, unde <math>b = - {a / 2}; & 4 b + 8 c = 2 c</math>,
& con$equenter <math>c = - {b / 3}; &c.</math></p>
<p>Et $ic progredi liceat ad plura & magis generalia exempla; etiam-
que ex re$olutionibus inter plures $ucce$$ivos terminos ad $eriem de-
ducendam.</p>
<p>2. Sint <math>V & W</math> datæ functiones quantitatis <math>x</math>, i. e. <math>V = X & W = X′</math>
invenire <math>V</math> in terminis quantitatis <math>W</math>.</p>
<p>Reducantur duæ æquationes <math>V = X & W = X′</math> ad unam, ita ut
exterminetur <math>x</math>, & re$ultat æquatio quæ$ita.</p>
<p>Aliter: Sit <math>V = a x^m + b x^m+n + c x^m+2n + &c. & W = a′ x^r + b′x^r+s
+ c′x^r+2s + &c.</math>; tum erit <math>V = A W^{m / r} + B W^{m / r}+t + C W^{m / r}+2t + &c.</math>;
in hâc æquatione pro <math>V & W</math> $cribantur earum valores <math>a x^m + &c.,
a′x^r + &c.</math>; & æquentur corre$pondentes termini æquationis re$ul-
tantis, & exinde facile erui po$$unt coefficientes <math>a, b, c, &c., a′, b′, &c.</math></p>
<p>Si <math>r = 0</math>, tum pro <math>W - a′</math> $cribatur <math>W′</math>; & evadet <math>V = A′ W′^{m / s} +
&c.</math></p>
<p>Si exponentiales, fluentiales, &c. quantitates in prædictis æqua-
tionibus <math>V = a x^m + &c. & W = a x^r + &c.</math> contineantur; tum ple-
rumque ita reducendæ $unt exponentiales, fluentiales, &c. quanti-
tates, ut progrediantur $ecundum dimen$iones perparvæ quantitatis
<math>0</math>, & deinde per methodum hìc traditam acquiri pote$t problematis
re$olutio.</p>
<h it>_PROB. XXVIII._
Reducere datam $eriem ad factores.</h>
<p>Sit $eries <math>a x^m + b x^m+n + c x^m+2n + &c.</math>: a$$umantur factores, qui
ad $eries reducti præbent $eries eju$dem formulæ ac data $eries, & qui
in $e$e continuo ducti continent tot $altem incognitas & indepen-
dentes coefficientes, quot re$ultant termini de$truendi.</p>

<pb 466>[0518]<rh>DE INFINITIS</rh>
<p>Ex. Sit quantitas <math>a + b x + c x^2 + &c.</math>; & erit æqualis contento
e $ingulis factoribus <math>(e + f x)^m × (e′ + f′x)^m′ × (e″ + f″x)^m″ × &c.</math>, qui-
bus factoribus <math>(e + f x)^m, (e′ + f′ x)^m′, &c.</math> ad $eries <math>e^m + m e^m-1 f x + &c.,
e′^m′ + m′ e′^m′-1 f′x + &c.</math> eju$dem ac datæ $eriei formulæ reductis; plures
continentur incognitæ & independentes coefficientes <math>e, e′, e″, &c.; f,
f′, f″, &c., &c., m, m′, m″, &c.</math> quam termini datæ $eriei æquandi; i. e.
æquentur corre$pondentes termini & re$ultant æquationes <math>e^m × e′^m′ × e″^m″
× &c. = a, m e^m-1 e′^m e″^m &c. f + m′ e′^m′-1 e^m e″^m″ &c. f′ + &c. = b</math>, & $ic dein-
ceps: hìc plures involvuntur incognitæ & independentes coefficientes
quam re$ultantes æquationes, & con$equenter facile erui po$$unt co-
efficientes, quæ $olutionem problematis præbent.</p>
<p>Cor. Sint factores <math>(e x^r + f x^s + g x^t + &c.) (e′ x^r′ + f′ x^s′ + g′ x^t′ +
&c.) (e^n x^r″ + f″ x^s″ + g″ x^t″ + &c.) &c. = a x^m + b x^m+n + &c.</math>, ubi <math>r
+ r′ + r″ + &c. = m, & s - r, &c. = n; e × e′ × e″ × &c. = a, &c.</math></p>
<p>Eadem principia etiam applicari po$$unt ad quantitates in quibus
incognitæ quantitates continentur.</p>
<h>PROB. XXIX.</h>
<p>Datâ æquatione quantitatem <math>v</math> involvente; ubi <math>v</math> $it quæcunque
quantitas, quæ e datâ quantitate <math>x</math> deduci pote$t; invenire valorem
quantitatis <math>x</math>.</p>
<p>1. Primo inveniatur <math>a</math> proximus valor quantitatis <math>x</math>, $cribatur quan-
titas <math>a</math> pro <math>x</math> in datâ æquatione, & $it quantitas re$ultans <math>A</math>; deinde
$cribatur <math>a + π</math> (ubi <math>π</math> $it perparva quantitas) pro <math>x</math> in datâ æqua-
tione, & $it quantitas re$ultans <math>B</math>; tum erit <math>A - B:A::π:{A π / A - B}</math>,
unde <math>a + {A π / A - B}</math> erit propior approximatio ad quantitatem <math>x</math>; & $ic
redintegratâ operatione propior adhuc con$tabit valor quantitatis <math>x</math>.</p>
<p>2. Si vero duo valores quantitatis <math>x</math> $int prope quantitati <math>a</math> & in-

<pb 467>[0519]<rh>SERIEBUS.</rh>
ter $e æquales; tum $cribendæ $unt pro <math>x</math> in datâ æquatione re$pec-
tive <math>a, a + π, a + 2 π</math>; & $int quantitates re$ultantes re$pective <math>A,
B, C</math>; erunt duæ radices quadraticæ æquationis <math>{1 / 2}(A - 2 B + C) e^2
- {1 / 2} (C - 4 B + 3 A) π e + A π^2 = 0</math>, quæ $int <math>α & β</math>, propiores ap-
proximationes ad duos valores quantitatis <math>x</math> prædictos, i. e. <math>a + α &
a + β</math> erunt prope duo valores quantitatis <math>x</math>: $i autem <math>α</math> multo mi-
nor $it quam <math>β</math>, tum erit <math>α = {2 A π / C - 4 B + 3 A}</math> prope.</p>
<p>Cor. 1. Sit æquatio <math>x^n - p x^n-1 + q x^n-2 - &c. = 0</math>: in hâc æqua-
tione pro <math>x</math> $cribantur re$pective <math>a & a + π</math>, & re$ultant quantitates
<math>a^n - p a^n-1 + q a^n-2 - &c. = A, & A + (n a^n-1 - (n - 1) p a^n-2 +
(n - 2) q a^n-3 - &c.) π + &c. = B</math>, & con$equenter <math>B - A = (n a^n-1
- (n - 1) p a^n-2 + (n - 2) q a^n-3 - &c.) π</math> prope: <math>a^n - p a^n-1 +
q a^n-2 - &c. = A::π:{a^n - p a^n-1 + q a^n-2 - &c. / n a^n-1 - (n - 1) p a^n-1 + (n - 2) q a^n-2 - &c.}
= e</math>; & exinde <math>a - e</math> erit propior valor quantitatis <math>x</math>.</p>
<p>Cor. 2. Sint vero <math>n</math> valores quantitatis <math>x</math> prope inter $e & quanti-
tati <math>a</math> æquales, $cribantur pro <math>x</math> in datâ æquatione re$pective <math>a, a + π,
a + 2π, a + 3 π, ... a + (n - 1) π</math>, & $int quantitates re$ultantes
re$pective <math>A, B, C, D, E, &c. & d = A - B, d′ = A - 2 B + C,
d″ = A - 3 B + 3 C - D, d′″ = A - 4 B + 6 C - 4 D + E, ....
d′^m-1 = A - m B + m. {m - 1 / 2} C - m · {m - 1 / 2} · {m - 2 / 3} D + &c.</math> & $int <math>α,
β, γ, δ, &c. n</math> radices <math>(e)</math> æquationis <math>A π^n - d π^n-1 e + {1 / 2} d′ π^n-2 e × (e - π)
- {1 / 2 · 3} d″ π^n-3 e × (e - π) × (e - 2 π) + {1 / 2 · 3 · 4} d′″ π^n-4 e × (e - π)
× (e - 2π) × (e - 3 π) - &c. = 0</math>, tum erunt <math>a + α, a + β, a + γ,
a + δ, &c.</math> re$pective approximationes ad <math>n</math> valores quantitatis <math>x</math> præ-
dictos.</p>
<p>Si vero $cribantur pro <math>x</math> in datâ æquatione <math>(n)</math> quantitates <math>a,
a + π, a + ρ, a + σ, a + τ, &c.</math>; & $int <math>(n)</math> quantitates re$ultantes
<math>A, B, C, D, E, &c.</math>; & $int <math>(n)</math> radices æquationis <math>A - ({A / π} - {B / π})e

<pb 468>[0520]<rh>DE INFINITIS</rh>
+ ({A / π p} + {B / π (π - ρ)} + {C / ρ (ρ - π)}) e × (e - π) - ({A / π ρ σ} - {B / π (π - ρ)
(π - σ)} - {C / ρ (ρ - π) (ρ - σ)} - {D / σ (σ - π) (σ - ρ)}) e × (e - π) ×
(e - ρ) + ({A / π ρ σ τ} + {B / π (π - ρ)(π - σ)(π - τ)} + {C / ρ (ρ - π)(ρ - σ)(ρ - τ)}
+ {D / σ (σ - π) (σ - ρ) (σ - τ)} + {E / τ (τ - π) (τ - ρ) (τ - σ)}) e × (e - π)
× (e - ρ) × (e - σ) + &c. = 0</math> re$pective <math>α, β, γ, δ, &c., &c.</math>; tum erunt
<math>a + α, a + β, a + γ, a + δ, &c.</math> approximationes ad <math>n</math> radices datæ
æquationis, quæ $unt prope inter $e æquales. Aliter: $int <math>(n)</math> radi-
ces prædictæ <math>α, β, γ, δ, &c.</math> radices <math>(e)</math> æquationis <math>{(e - π) × (e - ρ) × (e - σ) × (e - τ) &c. / ± π ρ σ τ
&c.} × A + {e × (e - ρ) × (e - σ) × (e - τ) × &c. / π × (π - ρ) × (π - σ) × (π - τ) × &c.}
× B + {e × (e - π) × (e - σ) × (e - τ) × &c. / ρ × (ρ - π) × (ρ - σ) × (ρ - τ) × &c.} × C + {e× (e - π) × (e - σ) × (e - τ) × &c. / σ × (σ - π) ×(σ - ρ) × (σ - τ) × &c.}
× D + {e × (e - π) × (e - ρ) × (e - σ) × &c. / τ × (τ - π) × (τ - ρ) × (τ - σ) × &c.} × E + &c. = 0</math>, tum erunt
<math>a + α, a + β, a + γ, a + δ, &c.</math> approximationes ad <math>(n)</math> prædictas
radices.</p>
<p>Lex huju$ce $eriei facile con$tat ex ob$ervatis ip$ius terminis.</p>
<p>Si vero <math>a, a + π, a + ρ, a + σ, &c.</math> $int multo propiores ad unam
radicem datæ æquationis quam ad reliquas; tum <math>(n)</math> radices prædictæ
æquationis, cujus radix e$t <math>e</math>, erunt re$pective <math>f - π, f - ρ, f - σ,
f - τ, &c.</math> prope, & radix quæ$ita <math>a + f</math> prope.</p>
<p>Cor. Hinc <math>1 - {π / π} = 0, 1 - {ρ / π} + {ρ × (ρ - π) / ρ π} = 0, 1 - {σ / π} + {σ × (σ - π) / ρ π}
- {σ × (σ - π) × (σ - ρ) / ρ π σ} = 0, 1 - {τ / π} + {τ × (τ - π) / π ρ} - {τ × (τ - π) × (τ - ξ) / π ρ σ}
+ {τ · (τ - π) · (τ - ρ) · (τ - σ) / π ρ σ τ} = 0</math>, & $ic deinceps; etiamque <math>{ρ / π} +<pb 469>[0521]<rh>SERIEBUS.</rh>
{ρ × (ξ - π) / π (π - ρ)} = 0, {σ / π} + {σ × (σ - π) / π(π - ρ)} + {σ · (σ - π) × (σ - ρ) / π(π - ρ)(π - σ)} = 0, {τ / π} +
{τ(τ - π) / π (π - ρ)} + {τ(τ - π)(τ - ρ) / π(π - ρ)(π - σ)} + {τ · (τ - π) · (τ - ρ) · (τ - σ) / π · (π - ρ) · (π - σ) · (π - τ)} = 0, &c.
& {σ × (σ - π) / ρ (ρ - π)} + {σ × (σ - π)(σ - ρ) / ρ (ρ - π)(ρ - σ)} = 0, {τ × (τ - π) / ρ (ρ - π)} + {τ × (τ - π)(τ - ρ) / ρ(ρ - π)(ρ - σ)}
+ {τ (τ - π) (τ - ρ) (τ - σ) / ρ(ρ - π)(ρ - σ)(ρ - τ)} = 0, &c., & {τ(τ - π) (τ - ρ) / σ(σ - π)(σ - ρ)} +
{τ(τ - π)(τ - ρ)(τ - σ) / σ(σ - π)(σ - ρ)(σ - τ)} = 0, &c.</math></p>
<p>Principia hìc tradita applicari po$$unt ad inveniendam $eriem, quæ
exprimit quantitatem <math>v</math> terminis vero $ecundum dimen$iones quanti-
tatis <math>z</math> progredientibus ex datâ æquatione relationem inter <math>x & y</math> ex-
primente, ubi literæ <math>z & v</math> re$pective denotant qua$cunque functiones
vel algebraicas vel fluentiales, integrales, &c. literarum <math>x & y</math>.</p>
<p>2. Datis <math>m</math> æquationibus <math>m</math> incognitas quantitates <math>x, y, z, &c.</math> ha-
bentibus, $int vero <math>α, β, γ, &c.</math> proximi valores incognitarum quanti-
tatum <math>x, y, &c.</math> re$pective inter $e corre$pondentes; a$$umantur <math>n + 1</math>
diver$i valores quantitatis <math>x</math>, viz. <math>α, α + π, α + π′, α + π″, &c. &c.</math>
& $ic a$$umantur <math>n + 1</math> diver$i corre$pondentes valores quantitatis <math>y</math>
qui $int re$pective <math>β, β + ρ, β + ρ′, β + ρ″, &c.</math> & $ic de reliquis, ubi
<math>π, π′ π″, &c. ρ, ρ′, ρ″, &c.</math> $unt perparvæ quantitates; $cribantur <math>α, β,
&c. α + π, β + ρ, &c. α + π′, β + ρ′, &c. &c.</math> pro $uis valoribus <math>x,
y, &c.</math> in datis æquationibus, & re$ultent quantitates <math>A, B, C, &c.</math> re-
$pective in unâ æquatione; <math>P, Q, R, &c.</math> in $ecundâ &c. & re$ulta-
bunt e primâ æquatione <math>n</math> $implices æquationes <math>a π + bρ + &c. =
B - A, a π′ + b ρ′ + &c. = C - A, a π″ + b ρ″ + &c. = D - A,
&c.</math> e $ecundâ vero æquatione <math>n</math> $implices æquationes <math>b π + k ρ +
&c. = Q - P, b π′ + k ρ′ + &c. = R - P, b π′ + kρ″ + &c. = S
- P, &c.</math> ex his <math>2 n, &c.</math> æquationibus inve$tigari po$$unt coefficien-
tes <math>a, b, &c. b, k, &c.</math> ultimo a$$umantur <math>m</math> æquationes <math>A + a e + b i
+ &c. = 0, P + b e + k i + &c. = 0; &c.</math> e quibus deduci po$$unt

<pb 470>[0522]<rh>DE INFINITIS</rh>
valores quantitatum <math>e, i &c.</math> & erunt <math>α + e, β + i, &c.</math> propiores
valores quantitatum <math>x, y, &c.</math></p>
<p>E. g. Sit <math>m = 2</math>; e duabus $implicibus æquationibus <math>a π + b ρ + A
- B = 0 & a π′ + b ρ′ + A - C = 0</math> deduci po$$unt coefficientes <math>a & b</math>,
& ex datis etiam æquationibus <math>k π + k ρ + P - Q = 0, & b π′ +
k ρ′ + P - R = 0</math> erui po$$unt coefficientes <math>b & k</math>; quibus datis fin-
gantur <math>a e + b i + A = 0, b e + k i + P = 0, e</math> quibus eruantur
valores quantitatum <math>e & i</math>, & erunt <math>α + e & β + i</math> propiores valores
quantitatum <math>x & y</math>.</p>
<p>In his $ub$titutionibus cavendum e$t, ne æquationes re$ultantes a
$e invicem $int dependentes.</p>
<p>In his ca$ibus $æpe præ$tat a$$umere <math>α = 0, β = 0, γ = 0, &c.</math> cum
<math>x</math> evadat <math>α</math>, tum <math>Γ</math> evadat <math>A</math>; etiamque cum <math>x</math> evadat <math>α + π</math>, tum <math>Γ</math>
evadat <math>B</math>; exinde cum <math>x</math> evadat <math>α + ρ</math>, tum <math>Γ</math> evadet <math>A + {B - A / π} ρ</math>
prope, $i modo <math>π & ρ</math> $int perparvæ quantitates.</p>
<p>2. Cum <math>n</math> quantitatites <math>x, y, &c.</math> evadant re$pective <math>α, β, &c.</math>; tum
<math>Γ</math> evadat <math>A</math>; etiamque cum prædictæ quantitates <math>x, y, &c.</math> evadant
<math>α + π, β + ρ, &c.</math>, tum <math>Γ</math> evadat <math>B</math>; & $imiliter cum <math>x, y, &c.</math> eva-
dant <math>α + π′, α + π″, &c.; β + ρ′, β + ρ″, &c.</math>; tum <math>Γ</math> evadat re$pe-
ctive <math>C, D, E, &c.</math></p>
<p>A$$umantur <math>(n)</math> $implices æquationes <math>a π + b ρ + &c. = B - A,
a π′ + b ρ′ + &c. = C - A, aπ″ + b ρ″ + &c. = D - A, a π′″ +
b ρ′″ + &c. = E - A, &c.</math>; ex his æquationibus inveniantur <math>(n)</math> in-
cognitæ quantitates <math>a, b, c, d, &c.</math>; tum erit <math>Γ - A = a λ + b μ + &c.</math>;
cum <math>x, y, &c.</math> evadant re$pective <math>α + λ, β + μ, &c.</math></p>
<p>1.2. Si vero duo vel plures valores quantitatum <math>x, y, &c.</math> $int <math>α, β,
&c.</math> prope; tum e pluribus a$$umptis re$pondentibus valoribus quan-
titatum <math>x, y, &c.</math> pro $uis valoribus in datis æquationibus $ub$titu-
tis; & ex principiis ii$dem, quæ in hoc prob. & in meditat. algebr.
traduntur, facile con$equi po$$unt valores ad veros valores magis ap-
propinquantes.</p>

<pb 471>[0523]<rh>SERIEBUS.</rh>
<p>Et $ic de <math>n</math> æquationibus plures <math>n + m</math> incognitas quantitates ha-
bentibus.</p>
<p>Convergentiæ harum $erierum pendent omnino ex ii$dem princi-
piis ac convergentiæ $erierum prius traditarum.</p>
<p>Hìc ob$ervandum e$t, $i quantitates re$ultantes prædictæ haud
cre$cant vel decre$cant ultimo in eâdem ratione ac i$tæ minimæ dif-
ferentiæ quantitatum ip$arum a veris valoribus, tum ex hâc methodo
haud deduci pote$t re$olutio; ex hypothe$i ultimarum rationum inter
prædictas differentias & quantitates re$ultantes $equitur re$olutio.</p>
<h>THEOR. XXXVII.</h>
<p>Sit <math>y = $. X x^.</math>, ubi <math>X</math> e$t functio quantitatis <math>x; & y = b</math> cum <math>x - a</math>;
& $i valor quantitatis <math>x</math> parum ex$uperet <math>a</math>, tum per prob. 5. libri
primi erit <math>y = X x - {1 / 2} P x^2 + {1 / 6} Q^3 - {1 / 24} R^4 + &c.</math> ubi <math>P x^. = X^., Q x^.
= P^., R x^. = Q^., &c.</math> $ed pro <math>x</math> $cribatur <math>z - a</math>, & invenietur <math>y = b
+ X (z - a) - {1 / 2} P (z - a)^2 + {1 / 6}Q(z - a)^3 - &c.</math></p>
<p>Cor.. Præ$tabit per intervalla procedere tribuendo ip$i <math>x</math> $ucce$$ive
valores <math>a, a′, a″, a′″, &c.</math> & pro $ingulis valores quantitatum <math>X, P, Q,
&c.</math> convenientes computando, & exinde aggregatum e $ingulis valo-
ribus $eriei <math>X x - {1 / 2} P x^2 + {1 / 6}Q^3 - &c.</math> detegendo, quod erit <math>y</math> fluens
quæ$ita.</p>
<h>THEOR. XXXVIII.</h>
<p>Sint <math>a & b</math> $imul valores quantitatum <math>x & y</math> re$pective; & $it <math>y</math> fun-
ctio quantitatis <math>x</math>; & 1<_>mo</_>. cum <math>x</math> fiat <math>x - n x<x>.</x></math>; tum <math>y</math> evaderet <math>y -
n y^. + n · {n + 1 / 2} y^.. - {n · (n + 1) · (n + 2) / 2 · 3} y^... + &c.</math> Ponatur <math>x - n x^.
= a</math> & erit <math>n = {x - a / x^.}</math> pene infinitus numerus; cum vero <math>x = a</math> per
hypothe$in erit <math>y = b</math>, & con$equenter <math>y = b + (x - a){y^. / x^.} - {1 / 2}{y^.. / x^. ^2} (x -

<pb 472>[0524]<rh>DE INFINITIS</rh>
a)^2 + {1 / 6}{y^... / x^. ^3} (x - a)^3 - &c.</math> 2<_>do</_>. Fiat vero <math>x</math> jam <math>x + n x^. & y</math> evade-
ret <math>y + n y^. + n · {n - 1 / 2} y^.. + &c.</math> $cribatur <math>{x - a / x^.}</math> pro <math>n</math> & fit <math>y = b +
{y^. / x^.} (x - a) + {1 / 2}{y^.. / x^. ^2} (x - a)^2 + {1 / 6}{y^... / x^. ^3} (x - a)^3 + &c.</math></p>
<p>Cor. 1. Si progre$$us ab <math>a</math> ad <math>x</math> in intervalla æqualia $ecundum
differentiam <math>a</math> di$pertiantur, & termini in $ingulis $eriebus ultimo
præcedentibus notentur per <math>′X, ′P, ′Q, ′R, &c.</math> ubi <math>y^. = X x^., X^. = P x^.,
P^. = Qx^., &c. & A, A′, A″, ... ′X, X</math> $int re$pective $ucce$$ivi valores
quantitatis <math>X</math> ad $ingula intervalla; & <math>B, B′, B″, .. ′P, P</math> $int re$pective
$ucce$$ivi valores quantitatis <math>P</math> ad prædicta intervalla, & $ic deinceps;
tum ex $ecundo ca$u re$ultant <math>y = b + α (A + A′ + A″ ... + ′X)
+ {1 / 2} α^2 (B + B′ + B″ ... + ′P) + {1 / 6} α^3 (C + C′ + C″ ... + ′Q) +
{1 / 24} α^4 (D + D′ + D″ ... + ′R) + &c.</math> Si inter limites & valorem
priorem <math>y = b + α (A′ + A″ ... X) - {1 / 2} α^2 (B′ + B″ ... + P) +
{1 / 6} α^3 (C′ + C″ ... + Q) - &c.</math> $umatur medium arithmeticum prodibit
$eries adhuc magis convergens <math>y = b + α (A′ + A″ + A′″ ... + ′X)
+ {1 / 2}α (A + X) + {1 / 4}α^2 (B - P) + {1 / 6}α^3 (C′ + C″ ... + ′Q) + {1 / 12}α^3 (C
+ Q) + {1 / 48}α^4 (D - R) + &c.</math></p>
<p>Convergentiæ harum $erierum e principiis prius traditis dijudicari
po$$unt.</p>
<p>Ex principiis prius traditis progredi liceat ad convergentiam $erie-
rum harum formularum deducendam, viz. <math>^m √(a - ^m √(b - ^m √(c - &c.)))</math>
vel <math>^m √(a - ^n√(b - &c.))</math> vel <math>^m√(a + b \\ ^m√(c + d) \\ <ol>^m√(e + &c.)</ol> # )</math>; & $ic de in-
finitis diver$orum generum $eriebus & æquationibus.</p>

<pb 473>[0525]<rh>SERIEBUS.</rh>
<h>THEOR. XXXIX.</h>
<p>1. Sit <math>x = ^n √(a + ^n √(a + ^n √(a &c.)))</math>; erit <math>x</math> vel radix æquati-
onis <math>x^n - a - x = 0</math>, vel <math>(x^n - a)^n - a - x = 0</math>, vel <math>((x^n - a)^n - a)^n
- a - x = 0</math>, & $ic deinceps; omnes enim hæ præcedentes æqua-
tiones erunt divi$ores $ub$equentium.</p>
<p>2. Sit <math>x = ^n √(a ^n √(a ^n √(a &c.)))</math>; & erit <math>x</math> radix æquationis <math>x^n-1 -
a = 0</math> vel <math>x^n^2 -1 - a^n+1 = 0</math> vel <math>x^n^3 -1 - a^n^2 +n+1 = 0</math> & denique
<math>x^n^m -1 - a^n^m -1 + n^m-2 + &c. .. + 1 = 0</math>, ubi <math>m</math> e$t integer numerus; omnes
enim hæ præcedentes æquationes erunt etiam divi$ores $ub$equentium.</p>
<p>Eadem principia etiam applicari po$$unt ad con$imiles $eries, in
quibus duæ vel plures continentur literæ vel radices, quæ eodem
ordine recurrunt; etiamque ad $eries $ub$equentes.</p>
<h>THEOR. XL.</h>
<p>1. Sit <math>x = ^n √a + b \\ ^m √a + b \\ ^m √a + &c.</math> # , & erit re$olutio æquationis
<math>x^m+1 - ax = b, &c.</math></p>
<p>2. Sit <math>x = ^m √a + b \\ ^n √c + d \\ ^m √a + b \\ &c.</math> # , & erit re$olutio æquationis
<math>(x^m - a)^n × (c x + d) = b^n x; &c.</math></p>
<p>Cor.. Sit cubica æquatio <math>x^3 - p x^2 + q x - r = 0</math>, a$$umantur
<math>m = 2, n & c = 1</math>, & re$ultat æquatio <math>x^3 + d x^2 - (a + b) x - a d

<pb 474>[0526]<rh>DE INFINITIS</rh>
= 0</math>, unde a$$umptis <math>d = - p, a = {r / d} = {-r / p}, & b = - (a + q)</math>,
re$ultat re$olutio cubicæ æquationis.</p>
<p>3. Sit <math>x = ^n√{a / b} \\ + ^n √{a / b} \\ + ^n √{a / b} \\ + &c.</math> # , unde <math>x^n+1 + b x^n = a, &c.</math></p>
<p>4. Sit <math>x = ^n√a + b \\ c + d \\ ^n √a + b \\ c + d \\ ^n √a + &c.</math> # , & erit æquatio exinde
re$ultans <math>c x^n+1 + d x^n - a c x - b x - d a = 0, &c.</math> Infinitæ eju$-
modi $olutiones harum æquationum facile dari po$$unt.</p>
<p>Si modo dentur quæcunque quantitates, quæ iterum perpetuo oc-
currunt; tum ex iis facile deduci po$$unt æquationes, quarum re$o-
lutiones per has methodos dantur.</p>
<p>Huju$ce generis quantitates in $e$e multiplicari vel per $e dividi
po$$unt per vulgares methodos, $ed plerumque producta & quotientes
re$ultantes erunt quantitates maxime compo$itæ; $ed tædet de his
plura adjicere.</p>
<p>Sit <math>z</math> di$tantia a primo $eriei termino, & $int <math>a, b, c, d, &c.</math> functio-
nes quantitatis <math>z</math>, tum ex ratione $ucce$$ivorum terminorum ad infi-
nitam di$tantiam facile con$tat, annon $eries fit convergens.</p>
<h>SCHOLIUM.</h>
<p>Veteres ad areas curvilinearum $igurarum approximabant, in$cri-
bendo in illis vel de$cribendo circa eas rectilineas figuras.</p>
<p>In curvis haud quadrabilibus, ubi etiam ordinata $it data functio
ab$ciffæ, methodi recentiorum vix aut ne vix $uperiores cen$endæ

<pb 475>[0527]<rh>SERIEBUS.</rh>
$unt; ut vero $eries per has methodos inventæ reddantur celeriter
convergentes, plurimæ interpolandæ $unt plerumque intermediæ $e-
ries, æque ac in$cribendæ vel de$cribendæ rectilineæ figuræ in curvi-
lineis figuris; & in multis ca$ibus rectilineæ figuræ magis conver-
gent quam $eries interpolandæ: reductio quantitatum ad terminos
$ecundum dimen$iones quantitatis <math>x</math> progredientes ni in fluentibus
detegendis perraro u$ui in$ervit; nam quantitates ip$æ plerumque
$ine tali reductione majore facilitate deduci po$$unt; & reductio præ-
dicta ad integrales a$cendentes, &c. detegendas non applicari pote$t.
In fluentibus detegendis ut prædicitur plerumque plurimæ interpo-
landæ $unt $eries, aliter haud convergit $eries.</p>

<pb 476>[0528]<rh>DE SUMMATIONE</rh>
<h>LIBER IV.</h>
<h>DE
SUMMATIONE SERIERUM, &c.</h>
<h>THEOR. I.</h>
<p>1. SINT $ummæ, i. e. integrales $ucce$$ivæ <math>s, s′, s″, &c.</math> & erunt
earum incrementa, i. e. $ucce$$ivi termini <math>s - s′ = t, s′ - s″
= t′, &c.</math> ubi per <math>z</math> $emper de$ignetur di$tantia termini dicti a primo
feriei termino.</p>
<p>Ex. 1. Sint <math>s & s′</math> re$pective <math>z × (z - 1) · (z - 2) .. (z - n) &
(z + 1) · z · (z - 1) ... (z - n + 1)</math>; tum erit <math>t = (n + 1) · z ·
(z - 1) · (z - 2) ...(z - n + 1)</math>, & con$equenter $i terminus gene-
ralis $eriei $it <math>t = z · (z - 1). (z - 2) ... (z - n + 1)</math>, tum erit ejus
fumma <math>s = {1 / n + 1}z · (z - 1) ... (z - n) + A</math>, ubi <math>A</math> $it $umma
omnium terminorum ab initio $eriei u$que ad terminum <math>z = n</math>.</p>
<p>Cor. Con$tat e prob. 6. lib. 2<_>di</_>. omnem quantitatem huju$ce ge-
neris <math>A z^n + B z^n-1 + C z^n-2 + &c.</math> ubi <math>n</math> $it integer numerus, re$olvi
po$$e in quantitates $ub$equentis formulæ <math>az · (z - 1) · (z - 2)
.. (z - n + 1) + bz · (z - 1) · (z - 2)..(z - n + 2) + c z. (z - 1)
...(z - n + 3) + dz · (z - 1) ..(z - n + 4) + &c.</math> unde <math>A = a,
b = B + n · {n -1 / 2}a, &c.</math> & $umma $eriei, cujus termini $unt prædicti
generis, detegi pote$t.</p>
<p>Ex. 2. Sint s & s′ re$pective <math>{1 / z · (z + 1) · (z + 2) .. (z + n - 1)} &

<pb 477>[0529]<rh>SERIERUM, &c.</rh>
{1 / (z + 1) · (z + 2) .. (z + n)}</math>, tum erit <math>s - s′ = t = {n / z(z + 1) · (z + 2)
.. (z + n)}</math>, & con$equenter $i <math>t = {1 / z · (z + 1) · (z + 2) .. (z + n)}</math>,
tum erit <math>s = {1 / n z · z + 1 · z + 2 ... z + n - 1} + A</math>, ubi <math>A</math> $it $umma
quorumcunque præcedentium terminorum.</p>
<p>Cor. 1. Con$tat e prob. prædictâ omnem quantitatem, quæ expandi
pote$t in infinitas $eries huju$ce generis <math>A z^-n + B z^-n-m + C z^-n-2m
+ &c.</math> ubi <math>n & m</math> $unt integri numeri affirmativi, & <math>A, B, C, &c.</math> in-
variabiles coefficientes, etiam ad infinitam $eriem huju$modi <math>{a / z(z + 1)
(z + 2) .. (z + (n - 1))} + {b / z(z + 1)(z + 2)(z + 3) ... (z + n)} +
{c / z(z + 1)(z + 2)(z + 3)(z + 4) .. (z + n + 1)} + &c.</math> reduci po$$e.</p>
<p>E. g. Series <math>{1 / z} + {1 / z^2} + {1 / z^3} + {1 / z^4} + &c.</math> in infinitum invenietur <math>=
{1 / z} + {1 / z. z + 1} + {1 · 2 / z. z + 1. z + 2} + {1 · 2 · 3 / z · z + 1 · z + 2 · z + 3} +
{1 · 2 · 1 · 4 / z · z + 1 · z + 2 · z + 3 · z + 4} + &c.</math> in infinitum. Sit <math>z = n</math>, &
hæc $eries ultimo convergit prope in eâdem ratione, quam $eries <math>{1 / 1} +
{1 / 2^n} + {1 / 3^n} + {1 / 4^n} + &c.</math>; prior <math>{1 / z} + {1 / z^2} + {1 / z^3} + &c.</math> $emper ultimo
convergit in majore ratione quam po$terior; ni <math>z</math> haud major $it
quam 1, in quo ca$u earum $umma erit infinita.</p>
<p>Cor. 2. Sit terminus generalis $eriei <math>{1 / z + a · z + b · z + c · z + d · &c.}</math>,
ubi <math>a, b, c, d, &c.</math> $int integri numeri & inæ quales, tum hic terminus

<pb 478>[0530]<rh>DE SUMMATIONE</rh>
$emper re$olvi pote$t in plures, quorum integrales ex hoc exemplo
deduci po$$unt.</p>
<p>Et ex dato termino generali erui pote$t $eriei $umma, & e theor. 2.
lib. 2<_>di</_>. corrigi pote$t $umma acqui$ita.</p>
<p>2. Ad hunc particularem ca$um, ubi <math>z<x>.</x></math> incrementum di$tantiæ a
primo $eriei termino <math>= 1</math>, applicari po$$unt omnia, quæ prius tradita
fuerunt de incrementialibus quantitatibus & æquationibus.</p>
<p>Ex. Sit generalis terminus <math>{α z^m + β z^m-1 + γ z^m-2 + &c. / z^n + a z^n-1 + b z^n-2 + &c.}</math>, ubi <math>z^n +
a z^n-1 + b z^n-2 + &c. = z(z + 1)(z + 2)..(z + n - 1)</math>; & <math>m</math> nu-
merus non major quam <math>n</math>; ($i <math>m</math> major $it quam <math>n</math>, tum terminus
facile reduci pote$t e divi$ione numeratoris per denominatorem ad
integram quantitatem & propriam fractionem prædicti generis); in-
venire $ummam $eriei, cujus generalis terminus e$t prædictus.</p>
<p>Statuatur <math>{α z^m + β z^m-1 + γ z^m-2 + δ z^m-3 + &c. / z(z + 1)(z + 2)(z + 3) ... (z + n - 1)} = p + {q / z} +
{r / z. (z + 1)} + {s / z(z + 1)(z + 2)} + {t / z · z + 1 · z + 2 · z + 3} + ...
+ {l / z · z + 1 · z + 2 .... (z + n - 1)}</math>; $i <math>m = n</math>, tum <math>p = α, q = β
- ap, r = γ - p b - a q, s = δ - p c - q b - r a + r, t = ε - p d
- q c - a r(b - a + 1) - s(a - 3)</math>, &c.; ubi <math>z + 1. z + 2. z + 3
... z + n - 1 = z^n-1 + a z^n-2 + b z^n-3 + c z^n-4 + d z^n-5 + &c.</math>, cu-
jus $eriei $umma e$t <math>A + p z + integ. ({q / z}) + {r / z} + {s / 2 z · z + 1} +
{t / 3 z · z + 1 · z + 2} + &c.</math>; ubi <math>A</math> e$t quantitas invariabilis pro con-
ditione problematis a$$umenda.</p>
<p>Cor. 1. Summa huju$ce $eriei inter quo$cunque duos <math>a′ & b′</math> valores
quantitatis <math>z</math> contenta erit <math>p (a′ - b′) + integ. ({q / z}) + ({r / a′} - {r / b′}) +

<pb 479>[0531]<rh>SERIERUM, &c.</rh>
({s / 2 a′ · a′ + 1} - {s / 2 b′ · b′ + 1}) + &c.</math>; $ed integ. <math>({q / z}) = {q / a′} + {q / a′ + 1}
+ {q / a′ + 2} ... {q / b′}</math>: erit infinita, $i <math>b</math> $it infinitus numerus; $umma ter-
minorum prædictorum $emper erit finita quantitas, cum <math>a′</math> $it finita
quantitas & <math>m</math> minor quam <math>n - 1</math>; $in aliter non.</p>
<p>Cor. 2. Si <math>q</math> haud $it <math>= 0</math>, tum $umma prædictorum terminorum
in infinitum progredientium in finitis terminis non exprimi pote$t,
$in aliter $emper exprimi pote$t: hinc, $i <math>m</math> minor $it quam <math>n - 1</math>,
tum $eriei prædictæ $umma finitis terminis $emper exprimi pote$t;
i. e. $it <math>b′</math> infinitus numerus & <math>q = 0</math>, $umma $eriei prædictæ <math>=
{r / a′} + {s / 2 a′ · a′ + 1} + {t / 3 a′ · a′ + 1 · a′ + 2} + &c.</math>; quæ $emper ter-
minat.</p>
<p>2. Sit generalis terminus <math>{α z^m + β z^m-1 + &c. / (z + e) (z + e + 1) (z + e + 2) ...
(z + e + n - 1)}</math>; tum eadem affirmari po$$unt de $ummâ $eriei ex-
inde re$ultantis, quæ prius affirmabantur de $erie, cujus generalis ter-
minus e$t <math>{α z^m + β z^m-1 + &c. / z (z + 1) (z + 2) ... (z + n - 1)}</math>: terminus enim
<math>{α z^m + β z^m-1 + &c. / (z + e) (z + e + 1) (z + e + 2) ... (z + e + n - 1)} = p + {q / z + e}
{r / z + e · z + e + 1} + {s / z + e · z + e + 1 · z + e + 2} + &c.</math></p>
<p>3. Sit terminus generalis <math>{α z^m + β z^m-1 + γ z^m-2 + &c. / z + a · z + b · z + c · z + d · &c.}</math>, ubi
<math>a - b, a - c, a - d, &c.</math> $unt integri numeri, & <math>m</math> minor quam
<math>n</math> per numerum majorem quam unitatem, & <math>n</math> e$t numerus facto-
rum <math>z + a, z + b, z + c, &c.</math> tum $eries $emper integrari pote$t, &c.:
ducatur enim numerator & denominator dati termini <math>{α z^m + β z^m-1 + γ z^m-2 + &c. / z + a · z + b

<pb 480>[0532]<rh>DE SUMMATIONE</rh>
· z + c · z + d · &c.}</math> re$pective in <math>(z + a + 1) (z + a + 2) (z + a
+ 3) ... (z + b - 1) × (z + b + 1) (z + b + 2) (z + b + 3) ...
(z + c - 1) × (z + c + 1) (z + c + 2) (z + c + 3) ... (z + d - 1)
× (z + d + 1) (z + d + 2) &c.</math>: & terminus generalis in terminum
generalem formulæ prius traditæ tran$it.</p>
<p>Sit generalis terminus <math>{α z^m + β z^m-1 + γ z^m-2 + &c. / z · z + 1 · z + 2 ... (z + n - 1) × z + e ·
z + e + 1 · z + e + 2 .. z + e + n′ - 1} = a + ({b / z} + {b′ / z + e}) +
({c / z · z + 1} + {c′ / z + e · z + e + 1}) + ({d / z · z + 1 · z + 2} + {d′ / z + e ·
z + e + 1 · z + e + 2}) + &c.</math>, ubi <math>m</math> e$t integer numerus non ma-
jor quam <math>n + n′</math>: ejus integralis erit <math>A + a z +</math> integ. increm. <math>({b / z} +
{b′ / z + e}) + {c / z} + {c′ / z + e} + {d / 2 z · z + 1} + {d′ / 2 z + e · z + e + 1} + &c.</math>:
exhinc con$tat integralem vel $ummam $eriei cujus generalis ter-
minus prius traditur detegi po$$e, $i modo <math>b & b′</math> nihilo $int re$pective
æquales.</p>
<p>Ex ii$dem principiis con$tat integralem generalis termini, vel $um-
mam $eriei, cujus terminus e$t <math>{α z^m + β z^m-1 + &c. / z + e · z + e + 1 · z + e + 2 ... (z + e
+ n - 1) × z + e′ · z + e′ + 1 · z + e′ + 2 ... (z + e′ + n′ - 1) × z + e″ · z + e″
+ 1 · z + e″ + 2 ... (z + e″ + n″ - 1) × &c.} = a + ({b / z + e} + {b′ / z + e′}
+ {b″ / z + e″} + &c.) + ({c / z + e · z + e + 1} + {c′ / z + e′ · z + e′ + 1} +
{c″ / z + e″ · z + e″ + 1} + &c.) + ({d / z + e · z + e + 1 · z + e + 2} +

<pb 481>[0533]<rh>SERIERUM, &c.</rh>
&c.) + &c.</math>, ubi <math>m</math> e$t integer numerus <math>= n + n′ + n″ + &c.</math>
ejus integralis vel $umma erit <math>A + a z +</math> integ. increm. <math>({b / z + e} +
{b′ / z + e′} + {b″ / z + e″} + &c.) + ({c / z + e} + {c′ / z + e′} + {c″ / z + e″} + &c.) +
({d / 2 · z + e · z + e + 1} + &c.) + &c.</math>: exhinc con$tat integralem
dati generalis termini finitis terminis detegi po$$e, $i modo integralis
incrementi (<math>{b / z + e} + {b′ / z + e′} + {b″ / z + e″} + &c.</math>) a$$ignari po$$it; &
con$equenter $ummam $eriei, cujus generalis terminus prius tradi-
tur; & $ic deinceps.</p>
<p>Sit generalis terminus <math>{α z^m + β z^m-1 + γ z^m-2 + &c. / z + a · z + b · z + c · &c. × z + a′ · z + b′ ·
z + c′ · z + d′ · &c. × z + a″ · z + b″ · z + c″ · &c.}</math>, ubi <math>b - a, c - a,
d - a, &c.; b′ - a′, c′ - a′, d′ - a′, &c.; b″ - a″, c″ - a″, d″ - a″,
&c., &c.</math>; $unt integri numeri, at non <math>a′ - a, a″ - a, a′″ - a′, &c.,
& a</math> minor e$t quam <math>b, b</math> quam <math>c, c</math> quam <math>d, &c.; & a′</math> quam <math>b′,
b′</math> quam <math>c′, &c.; &c.</math>: ducantur & numerator & denominator dati ter-
mini in <math>z + a + 1 · z + a + 2 · z + a + 3 ... z + b - 1 × z + b
+ 1 · z + b + 2 · z + b + 3 ... z + c - 1 × z + c + 1 · z + c + 2
... z + d - 1 × z + d + 1 · z + d + 2 ... &c. × z + a′ + 1 · z +
a′ + 2 · z + a′ + 3 ... z + b′ - 1 × z + b′ + 1 · z + b′ + 2 · z + b′
+ 3 ... z + c′ - 1 × z + c′ + 1 · z + c′ + 2 ... z + d′ - 1 × z +
d′ + 1 ... &c. × z + a″ + 1 · z + a″ + 2 · z + a″ + 3 .. z + b″ - 1
× z + b″ + 1 · z + b″ + 2 ... z + c″ - 1 × z + c″ + 1 · z + c″ + 2
... z + d″ - 1 × z + d″ + 1 · z + d″ + 2 ... &c. × &c.</math> & terminus
datus in terminum formulæ prius traditæ tran$it.</p>
<p>Sit (<math>n</math>) numerus quantitatum (<math>a, a′, a″, a′″, &c.</math>); tum ex (<math>n</math>) in-
dependentibus integralibus vel $ummis, viz. <math>{l / z + a}, {l′ / z + a′}, {l″ / z + a″},

<pb 482>[0534]<rh>DE SUMMATIONE</rh>
&c.</math>, deduci po$$unt integrales vel $ummæ omnium $erierum prædi-
ctarum formularum.</p>
<p>4. Sit generalis terminus <math>{α z^m + β z^m-1 + γ z^m-2 + &c. / (z + e)^b (z + e + 1)^b (z + e + 2)^b ....
(z + e + n - 1)^b} = {l / z + e} + {l′ / (z + e)^2} + {l″ / (z + e)^3} .. + {l′^b-1 / (z + e)^b}
+ {k / z + e · z + e + 1} + {(2 z + 2 e + 1) k′ / (z + e)^2 (z + e + 1)^2} + {(3 z^2 + 3 z e + e^2 + 3 z + 3 e + 1) k″ / (z + e)^3 (z + e + 1)^3}
+ ... {((z + e + 1)^b - (z + e)^b) k′^b-1 / (z + e)^b (z + e + 1)^b} + {i / z + e · z + e + 1 · z + e + 2} +
{((z + e + 2)^2 - (z + e)^2) i′ / (z + e)^2 (z + e + 1)^2 (z + e + 2)^2} + {((z + e + 2)^3 - (z + e)^3) i″ / (z + e)^3 (z + e + 1)^3 (z + e + 2)^3}
+ ... {((z + e + 2)^b - (z + e)^b) i′^b-1 / (z + e)^b (z + e + 1)^b (z + e + 2)^b} + {f / z + e · z + e + 1 · z +
e + 2 · z + e + 3} + {((z + e + 3)^2 - (z + e)^2) f′ / (z + e)^2 (z + e + 1)^2 (z + e + 2)^2 (z + e + 3)^2}
+ &c.</math>; tum ex <math>b</math> independentibus integralibus vel $ummis $erierum
harum formularum, e. g. <math>{l / z + e}, {l′ / (z + e)^2}, {l″ / (z + e)^3}, .. {l′^b-1 / (z + e)^b}</math> erui
po$$unt $ummæ omnium aliorum incrementorum vel $erierum earun-
dem formularum: vel generaliter $it terminus <math>{α z^m + β z^m-1 + γ z^m-2 + &c. / (z + a)^b (z + b)^b′ (z + c)^b″
× &c. × (z + a′)^i (z + b′)^i′ (z + c′)^i″ × &c. × (z + a″)^k (z + b″)^k′ × &c. × &c.}</math>;
ubi quantitates <math>a, a′, a″, &c.</math> $unt re$pective minores quam <math>b, c, &c.;
b′, c′, &c.; b″, c″, &c.; &c.; & b - a, c - a, &c.; b′ - a′, c′ - a′, &c.;
b″ - a″, c″ - a″, &c.</math> $unt integri numeri; at non <math>a′ - a</math> vel <math>a″ -a</math>
vel <math>a″ - a′, &c.</math> $eries, cujus prædictus fuit generalis terminus, eadem
erit ac $eries, cujus generalis terminus e$t <math>{α′ z^m′ + β′ z^m′-1 + γ′ z^m′-2 + &c. / (z + a)^π (z + a + 1)^π (z +

<pb 483>[0535]<rh>SERIERUM, &c.</rh>
a + 2)^π .. (z + b)^π (z + b + 1)^π .. (z + c)^π &c. × (z + a′)^ρ × (z + a′ + 1)^ρ
.. (z + b′)^ρ .. (z + c′)^ρ × &c. × (z + a″)^σ (z + a″ + 1)^σ × &c.} = {λ / z + a}
+ {λ′ / (z + a)^2} + {λ″ / (z + a)^3} ... + {λ′^n-1 / (z + a)^π} + {x / z + a′} + {x′ / (z + a′)^2} +
{x″ / (z + a′)^3} ... + {x′^ρ-1 / (z + a′)^ρ} + {θ / z + a″} + {θ′ / (z + a″)^2} + &c. + {ζ / (z + a)
(z + a + 1)} + {((z + a + 1)^2 - (z + a)^2)ζ′ / (z + a)^2 (z + a + 1)^2} ... + {((z + a + 1)^π - (z + a)^π^π-1)ξ′ / (z + a)^π (z + a + 1)^π}
+ &c. + &c.</math>; ubi <math>π, ρ, σ, &c.</math> re$pective $unt maximi inter indices
<math>h, b′, h″, &c.; i, i′, i″, &c.; k, k′, k″, &c.; &c.</math>; qui omnes $unt integri
numeri: $i enim numerator & denominator dati generalis termini
ducantur re$pective in <math>(z + a)^π-h × (z + a + 1)^π × (z + a + 2)^π
... × (z + b - 1)^π × (z + b)^π-h′ × (z + b + 1)^π × (z + b + 2)^π ..
(z + c - 1)^π × (z + c)^π-h″ × (z + c + 1)^π × (z + c + 2)^π × &c. ×
(z + a′)^ρ-i × (z + a′ + 1)^ρ × (z + a′ + 2)^ρ ... (z + b′ - 1)^ρ × (z + b)^ρ-i′
× (z + b′ + 1)^ρ × (z + b′ + 2)^ρ .. (z + c′ - 1)^ρ × (z + c)^ρ-i″ × (z +
c + 1)^ρ × &c. × &c.</math>; tum re$ultat po$terior generalis terminus: hinc
ex <math>π + ρ + σ + &c.</math> integralibus vel $ummis inter $e independentibus
$erierum, quarum generales termini $unt prædicti, viz. ex integrali-
bus vel $ummis $erierum, quarum termini $unt <math>{λ / z + a}, {λ′ / (z + a)^2}, ..
{λ′^n-1 / (z + a)^π}, {x / z + a′}, {x′ / (z + a′)^2}, ... {x′^ρ-1 / (z + a′)^ρ}; {θ / z + a″}, {θ′ / (z + a″)^2}, ...
{θ′^σ-1 / (z + a″)^σ}, &c.</math> deduci po$$unt integrales vel $ummæ omnium $erie-
rum, quarum generales termini $unt prædicti.</p>
<p>Hæc magis generaliter per diver$as methodos prius tradita fuere
in methodo incrementorum libro $ecundo contentorum.</p><pb 484>[0536]<rh>DE SUMMATIONE</rh>
<p>Cor.. Hinc facile inveniri po$$unt innumeræ $eries, quarum ge-
neralis $umma innote$cit. A$$umatur enim quæcunque functio
quantitatis <math>z</math> pro $ummâ <math>s</math>, & differentia inter duos ejus $ucce$$ivos
valores, i. e. inter <math>s & s′</math>, ubi <math>s′</math> $it valor, qui re$ultat $cribendo in
datâ functione <math>z + 1</math> pro <math>z</math>, invenietur terminus quæ$itus, i. e. <math>s - s′ = t</math>.</p>
<p>2. Sit generalis terminus <math>x^rz+n</math> in <math>{a / z} + {b / z · z + 1} + {c / z · z + 1 · z + 2}
+ {d / z · z + 1 · z + 2 · z + 3} + &c.</math> erit $umma æqualis <math>x^rz+n</math> in
<math>{a / 1 - x^r · z} + {b - A x^r / 1 - x^r · z · z + 1} + {c - 2 B x^r / 1 - x^r · z · z + 1 · z + 2} +
{d - 3 C x^r / 1 - x^r · z · z + 1 · z + 2 · z + 3}</math>, ubi quantitates <math>A, B, C, D, &c.</math>
de$ignant coefficientes terminorum præcedentes eos, in quibus repe-
riuntur; $cilicet <math>A = {a / 1 - x^r}, B = {b - A x^r / 1 - x^r}, C = {c - 2 B x^r / 1 - x^r}, &c.</math>
Fingatur enim <math>S = x^rz+n in {A / z} + {B / z · z + 1} + {C / z · z + 1 · z + 2}
+ &c.</math> deinde $cribatur <math>S - T</math> pro <math>S & z + 1</math> pro <math>z</math>, & re$ultat <math>S - T
= x^rz+n+r</math> in <math>{A / z + 1} + {B / z + 1 · z + 2} + {C / z + 1 · z + 2 · z + 3} + &c.</math>
hoc e$t <math>S - T = x^rz+n</math> in <math>{A x^r / z + 1} + {B x^r / z + 1 · z + 2} + {C x^r / z + 1 · z + 2 · z + 3}
+ &c.</math> quæ reducta ad formam ip$ius <math>S</math>, evadit <math>S - T = x^rz+n</math> in
<math>{A x^r / z} + {B x^r - A x^r / z · z + 1} + {C x^r - 2 B x^r / z · z + 1 · z + 2} + {D x^r - 3 C x^r / z · z + 1 · z + 2 · z + 3}
+ &c.</math> $ubducito valorem ip$ius <math>S - T</math> a valore ip$ius <math>S</math>, & relin-
quetur terminus <math>T = x^rz+n</math> in <math>{A (1 - x^r) / z} + {B (1 - x^r) + A x^r / z. z + 1} + {C (1 - x^r) + 2 B x^r / z · z + 1 · z + 2} + &c.</math>
Hic denique valor ip$ius <math>T</math> collatus cum

<pb 485>[0537]<rh>SERIERUM, &c.</rh>
illo in propo$itione dat <math>A (1 - x^r) = a, B (1 - x^r) + A x^r = b, C (1
- x^r) + 2 B x^r = c, &c.</math></p>
<p>Sit <math>r</math> negativa quantitas, & eadem erit lex $eriei progre$$ionis, ac ea
cum <math>r</math> $it affirmativa quantitas.</p>
<p>Ex. 2. Erit integralis incrementi <math>({A / z^n}) = {a / z^n-1} + {b / z^n} + {c / z^n+1} +
{d / z^n+2} + &c.</math>, ubi <math>z</math> denotat di$tantiam a primo $eriei termino, <math>& a =
{A / n - 1}, b = {(n - 1) a / 2}, c = {n b / 2} - {n · n - 1 / 2 · 3} a, d = {n + 1 / 2} c -
{(n + 1) · n / 2 · 3} b + {n + 1 · n · n - 1 / 2 · 3 · 4} a, e = {n + 2 / 2} d - {n + 2 · n + 1 / 2} c +
{n + 2 · n + 1 · n / 2 · 3 · 4} b - {n + 2 · n + 1 · n · n - 1 / 2 · 3 · 4 · 5} a, &c.</math></p>
<p>Ex. 3. Sit generalis terminus <math>x^rz+m</math> in <math>{A / z^n}</math>, tum erit ejus integralis
<math>x^rz+m × ({a / z^n} + {b / z^n+1} + {c / z^n+2} + {d / z^n+3} + &c.)</math>, ubi <math>a = {A / 1 - x^r}, b =
- {n x^r / 1 - x^r} a, c = - {1 / 1 - x^r} ((n + 1) b - (n + 1) · {n / 2} a) x^r, d = -
{1 / 1 - x^r} ((n + 2) c - (n + 2) · {n + 1 / 2} b + (n + 2) · {n + 1 / 2} · {n / 3} a) x^r,
e = - {1 / 1 - x^r} ((n + 3) d - (n + 3) · {n + 2 / 2} c + (n + 3) · {n + 2 / 2}·
{n + 1 / 3} b - (n + 3). {n + 2 / 2}· {n + 1 / 3}· {n / 2} a) x^r, &c.</math></p>
<p>Cor. Sit generalis $eriei terminus <math>{A / z^n} + {B / z^n+1} + {C / z^n+2} + {D / z^n+3} + &c.</math>,
vel <math>x^rz+m</math> in <math>{A / z^n} + {B / z^n+1} + &c.</math>, tum e duobus præcedentibus exemplis
erui pote$t ejus integralis in $eriebus $ecundum dimen$iones quantitatis
<math>z</math> progredientibus.</p>

<pb 486>[0538]<rh>DE SUMMATIONE</rh>
<p>Hæc facile demon$trari po$$unt e $cribendo <math>z & z + 1</math> pro <math>z</math> in in-
tegrali a$$umptâ, & duarum quantitatum re$ultantium differentiam
inveniendo, quæ erit generalis terminus datus.</p>
<p>In omnibus hi$ce ca$ibus, $i aliquis factor contineatur in denomi-
natore, qui non habet alium ab eo per integrum numerum di$tantem,
tum non integrari pote$t data $eries.</p>
<p>Et $ic de terminis diver$as exponentiales quantitates involventibus.</p>
<h>PROB. I.</h>
<p it>_1._ Datâ æquatione relationem exprimente inter $ucce$$ivas $ummas <math>s, s′,
s″, &c.</math> datæ $eriei, ejus $ucce$$ivos terminos <math>t, t′, t″, &c.</math> & quantitatem
<math>z</math> di$tantiam a primo $eriei termino, invenire æquationem inter $ummas
$ucce$$ivas & <math>z</math>.</p>
<p>Pro terminis <math>t, t′, t″, &c.</math> in datâ æquatione $cribantur eorum va-
lores <math>s - s′, s′ - s″, s″ - s′″, &c.</math> & re$ultat æquatio quæ$ita.</p>
<p>2. Datâ prædictâ æquatione, invenire æquationem relationem in-
ter <math>t, t′, t″, &c. & z</math> exprimentem: in datâ æquatione pro $ummis <math>s′,
s″, s′″, &c.</math> $cribantur <math>s - t, s - t - t′, s - t - t′ - t″, &c.</math> & habe-
bitur æquatio <math>A = 0</math>, in quâ $olummodo continetur $umma <math>s</math>; inve-
niatur incrementum huju$ce æquationis <math>(A = 0)</math> quod $it æquatio
<math>B = 0</math>: aliter in æquatione <math>(A = 0)</math> pro <math>s</math> $cribatur <math>s - t</math>, & pro <math>t,
t′, &c.</math> $cribantur <math>t′, t″, &c.</math> & pro <math>z, z + 1, &c.</math>; & re$ultat æquatio
<math>(C = 0)</math>, in quâ etiam $olummodo continetur $umma <math>s</math>, reducantur
duæ æquationes <math>A = 0 & C = 0</math>, vel quod idem e$t <math>A = 0 & B = 0</math>
in unam, ita ut exterminetur <math>s</math>, & re$ultat æquatio quæ$ita.</p>
<p>Cor.. Hinc facile deduci po$$unt infinitæ $eries, quarum $ummæ
dantur; a$$umatur enim æquatio ad $ummas, & ex eâ deducatur æqua-
tio ad terminos, e quâ erui po$$unt termini, i. e. $eries quæ$ita.</p>
<p>Ex. 1. Sit æquatio <math>(z - n) s = (z - 1) s′ = (z - 1) × (s - t)</math>,
unde <math>(n - 1) s = (z - 1) t</math>; pro <math>s</math> $cribatur <math>s - t = s′, & t′</math> pro <math>t, &
z + 1</math> pro <math>z</math>, & re$ultat æquatio <math>(n - 1) s = (n - 1) t + z t′</math>; $ubduc

<pb 487>[0539]<rh>SERIERUM, &c.</rh>
hanc æquationem de æquatione prædictâ <math>(n - 1) s = (z - 1) t</math> & re-
$ultat æquatio ad terminos <math>(z - n) t = z t′</math>.</p>
<p>3. Datam prædictam æquationem <math>(A = 0)</math> in alteram transfor-
mare, in quâ deficit <math>z</math>: inveniatur incrementum <math>A′ = 0</math> datæ æquatio-
nis ex hypothe$i quod <math>z = 1</math>; reducantur hæ duæ æquationes <math>A = 0
& A′ = 0</math> in unam, ita ut exterminetur <math>z</math>, & evadet æquatio quæ$ita.</p>
<p>4. Ex datis relationibus inter $ucce$$ivos terminos, invenire rela-
tiones inter $ucce$$ivos valores functionis quantitatis <math>z</math> di$tantiæ a
primo $eriei termino, quæ prædictos terminos denotant.</p>
<p>In datis relationibus pro <math>t, t′, t″, &c.</math> $cribantur re$pective <math>φ: z,
φ:z + 1, φ: z + 2, &c.</math>, & re$ultant æquationes quæ$itæ.</p>
<h>THEOR. II.</h>
<p>Sint <math>s - s′ = s<x>.</x> = t, s′ - s″ = t′ &c. t - t′ = s<x>..</x>, &c.</math> tum erunt <math>s -
2 s′ + s″ = t - t′ = s<x>..</x>, s - 3 s′ + 3 s″ - s′″ = t - 2 t′ + t″ = s<x>^...;</x></math>, &
in genere <math>s - n s′ + n · {n - 1 / 2} s″ - n · {n - 1 / 2} · {n - 2 / 3} s′″ + &c. =
t - (n - 1)t′ + (n - 1) · {n - 2 / 2}t″ - &c. = s<x>.</x><x>n</x> = t<x>.</x><x>n-1</x></math>; etiamque <math>s - n s<x>.</x>
+ n. {n - 1 / 2} s<x>..</x> - n. {n - 1 / 2} · {n - 2 / 3}s<x>...</x> + &c. = integ. increm. (t) - n t + n.
{n - 1 / 2} t<x>.</x> - n · {n - 1 / 2} · {n - 2 / 3} t<x>..</x> + &c. = s′^n & t′^n = t - n t<x>.</x> + n · {n - 1 / 2} t<x>..</x>
- n · {n - 1 / 2} · {n - 2 / 3} t<x>...</x> + &c. = s - (n + 1) s<x>.</x> + (n + 1) · {n / 2} s<x>..</x> - &c.</math></p>
<p>Cor.. Datam æquationem relationem inter $ucce$$ivos terminos
& $ummas exprimentem in incrementialem transformare; pro <math>s, s′,
s″, &c. t, t′, t″, &c.</math> $cribantur re$pective earum valores, & transfor-
matur data æquatio in incrementialem, cujus integralis vel erit <math>s</math> vel
<math>t</math>; & vice versâ data incrementialis æquatio, cujus integralis e$t <math>s</math>,

<pb 488>[0540]<rh>DE SUMMATIONE</rh>
transformari pote$t in æquationem relationem inter $ucce$$ivas $um-
mas, terminos, &c. de$ignantem, $cribendo in datâ æquatione pro
<math>s, s<x>.</x>, s<x>..</x>, &c.</math> earum valores in hoc theoremate a$$ignatos.</p>
<h>PROB. II.</h>
<p it>Detur æquatio relationem inter $ucce$$ivas $eriei $ummas <math>S, S′, &c. & z</math>
di$tantiam a primo $eriei termino exprimens; invenire, utrum $umma <math>S</math> ad
infinitam di$tantiam $it finita, necne.</p>
<p>Inveniatur ex datâ æquatione $eries de$cendens, quæ exprimit
$ummam <math>S</math> terminis vero quantitatis <math>z</math>, i. e. $it $eries <math>a z^m + &c. + A
B z^-r + &c.</math> ubi <math>m</math> $it affirmativa quantitas, <math>- r</math> vero negativa quan-
titas, tum $eries ad infinitam di$tantiam erit infinite magna: $it
<math>m = 0</math>, & con$equenter $eries <math>A + B z^-r + C z^-s + &c.</math> ubi <math>- r &
- s, &c.</math> $unt negativæ quantitates, tum $umma ad infinitam di$tan-
tiam erit finita, viz. <math>= A</math>; $i vero $it prædicta $eries <math>B z^-r + C z^-s +
&c.</math> tum erit $umma ad infinitam di$tantiam infinite parva.</p>
<p>Et $ic e datâ æquatione inter $ucce$$ivos terminos relationem ex-
primente & principiis prius traditis detegi pote$t, utrum $umma $it
finita, necne.</p>
<h>THEOR. III.</h>
<p>Sit data æquatio relationem inter $ucce$$ivos $eriei terminos <math>(T, T′,
&c.)</math> & $ummas, &c. exprimens <math>α × β × γ × &c. = 0</math>; tum erunt <math>α = 0,
β = 0, γ = 0, &c.</math> re$pective diver$æ æquationes, quæ diver$as $eries
datæ æquationis denotant. E. g. Sit æquatio data <math>z T^2 + (z^2 + z
+ 1) T T′ + (z^2 + z) T′^2 = 0</math>, ubi <math>z</math> $it di$tantia a primo $eriei ter-
mino; inveniantur datæ æquationis divi$ores <math>z T + (z + 1) T′ = 0
& T + z T′ = 0</math>, qui diver$as $eries datæ æquationis denotant; & $ic
deinceps.</p><pb 489>[0541]<rh>SERIERUM, &c.</rh>
<h>THEOR. IV.</h>
<p>Sit terminus <math>T′^n</math> vel $umma <math>S′^n+1</math>, cujus ordo <math>n</math> $it maximus; tum in
$erie ejus $ummam <math>S</math> exprimente, a$$umi po$$unt quæcunque <math>n + 1</math>
invariabiles quantitates ad libitum a$$umendæ, &c.</p>
<p>Hìc ad $eries applicari po$$unt omnia, quæ prius tradita fuerint de
pluribus <math>(n)</math> fluxionalibus vel incrementialibus æquationibus plures
(<math>n + 1</math>) variabiles quantitates habentibus, & earum correctionibus,
&c.</p>
<h>THEOR. V.</h>
<p>1. Datâ æquatione (<math>A = 0</math>) relationem inter $ucce$$ivas $ummas,
(<math>S, S′, &c.</math>) terminos (<math>T, T′, &c.</math>) & quantitatem <math>z</math> de$ignante; in
eâ (<math>A = 0</math>) pro <math>S, S′, &c. T, T′, &c. & z</math> $cribantur re$pective
<math>S′, S″, &c. T′, T″, &c. & z + 1</math>, & re$ultet æquatio <math>B = 0</math>; tum gene-
ralis $eries, quæ $it radix æquationis <math>r A + B = 0</math>, ubi <math>r</math> $it quæcun-
que invariabilis quantitas, haud eadem invenietur ac $eries, quæ erit
radix æquationis <math>B = 0</math>.</p>
<p>2. Sit prædicta æquatio data <math>A = a</math>, & re$ultans <math>B = b</math>; tum haud
eadem erit radix generalis æquationis <math>BA = ba, ac</math> æquationis
<math>B = b</math>.</p>
<p>Con$tant <math>e</math> theor. 9. libri $ecundi.</p>
<h>PROB. III.</h>
<p it>Datâ æquatione <math>A = 0</math> relationem exprimente inter terminos (<math>T, T′,
T″, &c.</math>) datæ $eriei $ecundum dimen$iones quantitatis <math>x</math> progredientis, in-
venire æquationem relationem inter $ucce$$ivos ejus fluxionis terminos (<math>t, t′,
&c.</math>) de$ignantem.</p>
<p>1. In æquatione <math>A = 0</math> pro <math>z</math> di$tantiâ datæ $eriei termini a primo
$cribatur <math>z + 1</math>, & pro <math>T & T′</math> $cribantur re$pective <math>T′ & T″</math>; datæ æqua-
tionis <math>A = 0</math> & re$ultantis <math>B = 0</math> inveniantur fluxiones <math>α = 0 & π = 0</math>;

<pb 490>[0542]<rh>DE SUMMATIONE</rh>
in æquationibus <math>α = 0 & π = 0</math> pro <math>T^., T^. ′ & T^. ″</math> $cribantur re$pective
<math>t, t′ & t″</math>; reducantur hæ æquationes ad unam, ita ut exterminen-
tur quantitates <math>T & T′ & T″</math>, & re$ultat æquatio quæ$ita.</p>
<p>2. Si vero plures termini <math>T, T′, T″, ... T′^n-1</math> in datâ æquatione con-
tineantur, tum inveniantur <math>n - 1</math> $ucce$$ivi valores datæ æquationis
$cribendo in eâ <math>z + 1, z + 2 ... z + n - 1; T′, T″, ... T′^n; T″, T′″,
..T′^n+1; &c.</math> $ucce$$ive pro <math>z, T, T′, &c.</math> & inve$tigentur eorum fluxiones,
& exorientur <math>n</math> novæ æquationes; in his æquationibus pro <math>T^., T^. ′, T^. ″,
&c.</math> $cribantur re$pective <math>t, t′, t″, &c.</math> & reducantur hæ <math>2 n</math> æquationes
ad unam, ita ut exterminentur quantitates <math>T, T′, T″, &c.</math> & re$ultat
æquatio quæ$ita.</p>
<p>Et $ic de inveniendis æquationibus relationem inter $ucce$$ivos ter-
minos $erierum exprimentibus, quarum termini algebraicam vel flu-
xionalem vel incrementialem habeant relationem ad $ucce$$ivos ter-
minos, inter quos dantur æquationes relationes exprimentes.</p>
<h>PROB. IV.</h>
<p it>Datis æquationibus relationem de$ignantibus inter $ummas (<math>S, S′, & s, s′)</math>
in diver$is $eriebus, & æquationibus ad terminos (<math>T, T′, &c.</math>) in alterutrâ;
invenire æquationes ad terminos in alterâ.</p>
<p>Scribantur <math>S - T, S - T - T′, &c. s - t, s - t - t′, &c.</math> pro <math>S′,
S″, &c. s′, s″, &c.</math> re$pective in datis æquationibus; deinde in æqua-
tionibus re$ultantibus progrediatur e relatione variabilium præce-
dente ad $ub$equentem, i. e. $cribatur <math>S - T</math> pro <math>S & s - t</math> pro <math>s; &
T′, T″, &c. t′, t″, &c.</math> re$pective pro <math>T, T′, &c. t, t′, &c. &c.</math> & ita re-
ducantur per vulgarem algebram datæ & re$ultantes æquationes, ut
exterminentur <math>S & s</math>; deinde per methodos prius traditas ita reducan-
tur hæ æquationes re$ultantes & æquatio ad terminos in alterutrâ,
ut exterminentur <math>T, T′, T″, &c.</math> & re$ultabit æquatio relationem in-
ter <math>t, t′, &c.</math> de$ignans.</p>

<pb 491>[0543]<rh>SERIERUM, &c.</rh>
<p>Et ex ii$dem principiis & datâ vel datis æquationibus relationes
inter $ucce$$ivas $ummas & terminos exprimentibus erui po$$unt
æquationes relationes inter $ucce$$ivas $ummas vel $ucce$$ivos termi-
nos $olummodo exprimentes.</p>
<h>PROB. V.</h>
<p it>Datis duabus $eriebus, i. e. methodo inveniendi terminos re$pectivos e
datâ eorum di$tantiâ <math>z</math> a primo $eriei termino; invenire relationem inter
terminos corre$pondentes duarum $erierum.</p>
<p>Scribantur pro corre$pondentibus duarum $erierum terminis re-
$pective <math>T & t</math>; & dentur duæ æquationes relationes inter <math>z & T, &
z & t</math> re$pective exprimentes; reducantur hæ duæ æquationes ad
unam, ita ut exterminetur <math>z,</math> & re$ultat æquatio relationem inter <math>T
& t</math> exprimens.</p>
<p>Ex.. Sint duæ $eries re$pective <math>0 + 1 + {1 / 4} + {1 / 9} + {1 / 16} + {1 / 25} + &c. & 0 +
{1 / 1 · 2} + {1 / 2 · 3} + {1 / 3 · 4} + {1 / 4 · 5} + &c.</math> & con$equenter erunt <math>{1 / z^2} = T
& {1 / z × z + 1} = t</math>; reducantur hæ æquationes, ita ut exterminetur <math>z</math>
& re$ultat æquatio <math>t T{1 / 2} + Tt = 1</math>.</p>
<p>Cor. · Datâ æquatione relationem exprimente inter di$tantiam a
primo $eriei termino (<math>z</math>) & terminum ip$um (<math>T</math>), & datâ æquatione
relationem exprimente inter corre$pondentes duarum $erierum ter-
minos (<math>T & t</math>); invenire æquationem relationem inter <math>z & t</math> exprimen-
tem; reducantur duæ æquationes relationem inter <math>z & T, T & t</math> ex-
primentes ad unam, ita ut exterminetur <math>T</math>, & fit corollarium.</p>
<p>Et $ic de relationibus inter terminos, &c. plurium æquationum
deducendis.</p>
<h>PROB. VI.</h>
<h it>Datis duabus $eriebus; invenire, annon earum $ummæ datam habent
inter $e generalem relationem.</h>

<pb 492>[0544]<rh>DE SUMMATIONE</rh>
<p>In datâ æquatione exprimente relationem inter <math>S & s</math> duas $ummas
quæ$itas, $cribantur pro <math>S & s</math> re$pective earum $ucce$$ivæ corre$pon-
dentes $ummæ <math>S - T & s - t</math>, modo <math>T & t</math> $int $ucce$$ivi dati termini
utriu$que $eriei re$pective; $cribantur etiam in datis $eriebus pro <math>S,
S - T</math>, & pro <math>s, s - t, &c.</math>; & $i eadem adhuc re$ultet relatio inter
<math>S & s; &c.</math>; tum datur relatio inter prædictas $ummas; $in aliter
vero non.</p>
<h>PROB. VII.</h>
<h it>Invenire $eriem, cujus $ummæ $ucce$$ivæ (<math>s</math>) quamcunque habeant rela-
tionem ad $ummas $ucce$$ivas (<math>S</math>) datæ $eriei.</h>
<p>In datâ æquatione relationem exprimente pro <math>S</math> $cribatur ejus va-
lor $ucce$$ivus <math>S - T</math>, $i modo <math>T</math> $it proximus terminus datæ $eriei;
ex æquatione re$ultante inveniatur proximus valor $ummæ $eriei
quæ$itæ <math>s</math>, & differentia inter duas $ucce$$ivas $ummas erit terminus
$eriei quæ$itus.</p>
<h>THEOR. VI.</h>
<p>1. Summa (<math>S</math>) nullius $eriei exprimi pote$t per algebraicam æqua-
tionem inter prædictam $ummam <math>S</math> & quantitatem <math>x</math>, $ecundum cu-
jus dimen$iones progreditur $eries, relationem de$ignantem; ni di-
men$iones quantitatis <math>x</math> vel earum differentiæ denotari po$$int per
unam vel duas vel denique <math>n</math> finitum numerum arithmeticarum $e-
rierum; etiamque, $i modo <math>z</math> denotet di$tantiam a primo datæ $eriei
termino, ni terminorum ip$orum coefficientes exprimi po$$int in ra-
tionalibus terminis literis <math>z</math>; etiamque, ni differentia inter dimen$io-
nes quantitatis <math>z</math> in eorum numeratore & denominatore contentas,
eadem $it; $i vero detur æquatio inter $ucce$$ivos datæ $eriei terminos
relationem exprimens, ni differentia prædicta eadem manet.</p>
<p>2. Summa nullius $eriei exprimi pote$t per fluxionalem æquatio-
nem relationem inter prædictam $ummam <math>S</math> & quantitatem <math>x</math>, $ecun-
dum cujus dimen$iones progreditur $eries, de$ignantem; ni dimen$i-

<pb 493>[0545]<rh>SERIERUM, &c.</rh>
ones quantitatis <math>x</math> vel earum differentiæ exprimi po$$int per unum
vel duo vel denique <math>n</math> numerum arithmeticarum $erierum; etiamque
ni coefficientes exprimi po$$int in rationalibus terminis literæ <math>z</math>, &
differentia inter dimen$iones quantitatis <math>z</math> in denominatore & nume-
ratore contentas eadem maneat, vel uniformiter augeatur vel minua-
tur: eadem etiam affirmari po$$unt de $eriebus, quarum relationes
per algebraicas æquationes inter $ucce$$ivos terminos de$ignantur.</p>
<p>Hæc vero principia applicari po$$unt ad deducendam unius $eriei
fummam ex alterâ, &c.</p>
<h>THEOR. VII.</h>
<p>In $eriebus ex divi$ione ortis eadem erit relatio inter terminos <math>t, t′,
t″, &c.</math> ac inter $ummas <math>s, s′, s″, &c.</math> $ucce$$ivas.</p>
<p>Sit enim æquatio data inter $ummas $ucce$$ivas <math>ps + qs′ + rs″ +
&c. = 0</math>, & ex hac æquatione per problema 1. inveniri pote$t æquatio
inter terminos $ucce$$ivos, quæ erit <math>pt + qt′ + rt″ + &c. = 0</math>: enim
<math>s′ + t = s, s″ + t′ = s′, s″′ + t″ = s″</math>, & $ic deinceps; unde <math>ps + qs′
+ rs″ + &c. = p(s′ + t) + q(s″ + t′) + &c. = (ps′ + qs″ + rs″″
+ &c. = 0) + pt + qt′ + rt″ + &c. = 0</math>, & exinde <math>pt + qt′ +
rt″ + &c. = 0</math>.</p>
<h>THEOR. VIII.</h>
<p>Sit $eries <math>{A / p + qx + rx^2 + sx^3 + &c.} = a + bx + cx^2 + dx^3 · ..
T′^α-3 x^n-3 + T′^α-2 x^n-2 + T′^α-1 x^n-1 + T′^α x^n + &c.</math> tum erit $emper ultimo
<math>pT′^α + qT′^α-1 + rT′^α-2 + sT′^α-3 + &c. = 0</math>, $i modo <math>A = b + kx + 1x^2
+ &c.</math> $it quantitas haud in infinitum pergens; con$tat e ducendo
$eriem <math>a + bx + cx^2 + &c.</math> in <math>p + qx + rx^2 + &c.</math> quod produ-
ctum exinde ortum erit <math>pa + (pb + qa)x + (pc + qb + ra)x^2
+ &c. = A.</math></p>

<pb 494>[0546]<rh>DE SUMMATIONE</rh>
<h>THEOR. IX.</h>
<p it>Sit $eries <math>S = a + b x + c x^2 + d x^3 + &c.</math> in infinitum, cujus rela-
tio inter $ucce$$ivos terminos <math>t, t′, t″, &c</math>, de$ignetur per æquationem <math>p t +
q t′ + r t″ + σ t′″ + &c. = 0</math> ubi <math>p, q, r, σ, &c.</math> $unt invariabiles quan-
titates; tum erit $umma $eriei <math>S = {a × (q x^n-1 + r x^n-2 + s x^n-3 + τ x^n-4
+ &c.) + b × (r x^n-1 + s x^n-2 + τ x^n-3 + &c.) + c × (s x^n-1 + τ + &c.) + d × (τ x^n-1 + &c.) / p x^n + q x^n-1 +
+ r x^n-2 + s x^n-3 + &c.}</math>,
ubi <math>n</math> $it minor per unitatem quam numerus lite.
rarum <math>p, q, r, &c.</math></p>
<p>Cor. 1. Hinc datâ æquatione relationem inter <math>n</math> $ucce$$ivos termi-
nos de$ignante; pro (<math>n - 1</math>) terminis a$$umi po$$unt quæcunque datæ
quantitates; ergo in hoc ca$u ita a$$umi po$$unt <math>n - 1</math> primi termini
ut numerator $it divi$or denominatoris, & con$equenter $eries prædicta
<math>a + b x + c x^2 + &c. = {1 / α + x}</math>; & $ic a$$umi po$$unt (<math>n - 1</math>) primi
termini, ut $eries <math>a + b x + c x^2 + &c.</math> (re$ultans ex æquatione <math>p t +
q t′ + r t″ + &c. = 0) = {π + ρ x / α′ + β′x + γ′x^2}</math>, & $ic deinceps.</p>
<p>Cor. 2. Fingatur <math>p × (x + α) × (x + β) × (x + γ) × &c. = p x^n +
q x^n-1 + r x^n-2 + &c.</math> & erunt <math>{A / x + α}, {B / x + β}, {C / x + γ}, &c., {F x + G / (x + α) ×
(x + β)}, {H x + 1 / (x + α) × (x + β)}</math>, &c. particulares valores $eriei per æqua-
tionem <math>p t + q t′ + r t′ + &c. = 0</math> denotatæ.</p>
<p>Cor. 3. Ex ultimâ relatione terminorum, quæ $it con$tans, i. e.
prædicti generis, $equitur methodus $ummandi $eries quam proxime,
in quibus relatio terminorum e$t variabilis: $it æquatio <math>p t (z^2 + a z
+ b) + q t′ (z^2 + c z + d) = 0</math>, & ultima relatio terminorum erit

<pb 495>[0547]<rh>SERIERUM, &c.</rh>
<math>p t z^2 + q t′ z^2 = 0</math>, & exinde per prob. erit $umma quæ$ita prope
<math>= {q / q + p} t.</math></p>
<p>Et $ic $it ultima relatio, i. e. relatio ad infinitam di$tantiam <math>p t +
q t′ + r t″ = 0</math>, & erit $umma terminorum longe di$tantium <math>=
{(q + r) t + s t′ / p + q + r}</math>, i. e. ubi <math>z</math> di$tantia a primo $it permagnus numerus:
& ex ii$dem principiis ulterius promotis corrigere liceat approxima-
tionem inventam. Con$tat e $cribendo <math>t & t′</math> pro <math>a & b, & 1</math> pro <math>x</math> in
huju$ce problematis re$olutione.</p>
<h>PROB. VIII.</h>
<p it>Æquationem relationem inter <math>n</math> $ucce$$ivos terminos (<math>t, t′, t″, t′″, &c.</math>)
viz. <math>p t + q t′ + r t″ + s t′″ + &c. = 0</math> exprimentem, cujus $eries $it
<math>a + b x + c x^2 + d x^3 + &c.</math> in plures dividere.</p>
<p>Summa $eriei prædictæ erit <math>{a(qx^n-1 + rx^n-2 + sx^n-3 + &c.) + b(rx^n-1 + s x^n-2 + &c.) + c(s x^n-1 + τ x^n-2 + &c.) + &c. /
p x^n + q x^n-1 + r x^n-2 + s x^n-3 + &c.}</math>; $ed e lem. cap. 2. lib. 1. dividi
pote$t hæc fractio in <math>n</math> $ub$equentes <math>{a / x - α} + {b / x - β} + {c / x - y} + &c.</math>
ubi <math>p (x - α) × (x - β) × (x - γ) × &c. = px^n + q x^n-1 + &c.</math> &
con$equenter data dividitur æquatio in alias, viz. <math>T - α T′ = 0, T -
β T′ = 0, &c.</math> & $eries <math>a + b x + c x^2 + &c.</math> dividitur in <math>n</math> $eries for-
mulæ <math>({a / x - α}) = - {a / α} - {a x / α^2} - {a x^2 / α^3} - &c.</math></p>
<p>2. Sit ultima relatio terminorum prædicta <math>p t + q t′ + r t″ + &c.
= 0</math>; dividi pote$t per præcedentem methodum $eries ejus in <math>n</math> alias,
quarum terminorum longe di$tantium $umma æquat terminum $e-
riei, cujus relatio datur: vel ob <math>(a + b x)^m = a^m + m{a^m × b x / a + b x} + m.

<pb 496>[0548]<rh>DE SUMMATIONE</rh>
{m + 1 / 2}{a^m × b^2 x^2 / (a + bx)^2} + m · {m + 1 / 2} · {m + 2 / 3} · {a^m × b^3 x^3 / (a + b x)^3} + &c. = {a^m+1 / a + b x} +
{(m + 1) a^m+1 × b x / (a + b x)^2} + {(m + 1) · (m + 2) a^m+1 b^2 x^2 / 2 # (a + bx)^3} + {(m + 1) × m + 2) × (m + 3) a^m+1 × b^3 x^3 / # 2 · 3 # (a + b x)^4} + &c. = {a^m+n / (a + b x)^n} + {(m + n) a^m+n b x / (a + b x)^n+1} +
{(m + n) × (m + n + 1) × a^m+n b^2 x^2 / # 2 # (a + b x)^n+2} + {(m + n) × (m + n + 1) × (m + n + 2) a^m+n+1 b^3 x^3 / # 2 # . # 3 # (a + b x)^n+3} + &c.</math> & quoniam ad infinitam di$tantiam ea-
dem invenietur relatio inter $ucce$$ivos terminos $eriei <math>a + b x + c x^2
+ d x^3 + &c. = {A + B x + C x^2 + &c. / (px^n + qx^n-1 + rx^n-2 + &c.)^m}</math>, ubi numerator terminat,
quicunque $it valor indicis <math>m</math>; exinde dividi pote$t $umma $eriei
<math>{a(q x^n-1 + r x^n-2 + s x^n-3 + &c.) + b(r x^n-1 + s x^n-2 + &c.) + c(s x^n-1 + &c.) / p x^n + q x^n-1 + r x^n-2 + &c.}</math>
multis modis in plures alias, &c.</p>
<p>Et $ic de pluribus con$imilibus methodis.</p>
<p>3. Convergentiæ $erierum, quarum dantur æquationes relationes
inter $ucce$$ivas $ummas & terminos exprimentes, deduci po$$unt ex
principiis de convergentiis incrementialium æquationum traditis, vel
nonnunquam ex convergentiis terminorum ad infinitam di$tantiam
po$itorum.</p>
<h>THEOR. X.</h>
<p>Sit <math>T</math> generalis $eriei functio quantitatis <math>z</math> di$tantiæ a primo $eriei
termino, tum ejus $umma per eandem methodum omnino inve$ti-
ganda e$t ac integralis incrementi, quod e$t eadem functio quantita-
tis <math>z</math>, cujus incrementum e$t 1.</p>

<pb 497>[0549]<rh>SERIERUM, &c.</rh>
<h>THEOR. XI.</h>
<p>Sit æquatio <math>A = 0</math> ad $ummas, <math>n</math> habens invariabiles quantitates
(<math>a, b, c, d, &c.</math>) ad libitum a$$umendas; dantur <math>n</math> diver$æ æquationes
relationes inter <math>T & T′</math>, vel <math>T′ & T″, &c.</math> exprimentes; dantur etiam
<math>n. {n - 1 / 2}</math> æquationes diver$æ relationes inter <math>T, T′ & T″, &c.</math>; expri-
mentes; & $ic deinceps.</p>
<p>Inveniatur $ucce$$ivus valor (<math>B = 0</math>) æquationis (<math>A = 0</math>); $cri-
bendo in eâ pro <math>z, z + 1</math>; pro <math>S, S′</math>; & $ic deinceps; ita reducantur
duæ æquationes <math>A = 0 & B = 0</math> ad unam, ut exterminentur quanti-
tates <math>a, b, c, d, &c.</math> $ucce$$ive; & re$ultabunt <math>n</math> diver$æ æquationes
<math>n - 1</math> invariabiles quantitates ad libitum a$$umendas habentes; tum
ita reducantur hæ re$ultantes æquationes, ut exprimant relationes in-
ter $ucce$$ivos terminos <math>T & T′</math>, & invenientur <math>n</math> diver$æ æquationes
prædictæ. Et $ic de reliquis.</p>
<h>THEOR. XII.</h>
<p>Datâ æquatione relationem inter $ucce$$ivos terminos exprimente,
& e prob. 18. lib. 2<_>di</_>. con$tat $emper dari multiplicatorem, qui in
datam æquationem ductus producet æquationem, cujus $ummatio
innote$cit.</p>
<p>Cor. 1. Si detur æquatio <math>A =</math> con$t. quæ $it $ummatio generalis
æquationis <math>B = 0</math>.</p>
<p>Ex incremento æquationis <math>A = con$t.</math> inveniatur æquatio ad ter-
minos <math>C = 0</math>; tum erit <math>{C / B}</math> unus multiplicator, qui reddit æquationem
<math>B = 0</math> $ummabilem.</p>
<p>Cor. 2. Si generalis $ummatio æquationis <math>a T^n + b T^n-1 + c T^n-2
... f T^2 + g T′ + h T = 0</math> (ubi <math>T, T′, ... T^n-2, T^n-1, T^n</math> $unt termini
$ucce$$ivi, & <math>a, b, c, ... f, g, h</math> $unt functiones ip$ius <math>z</math> di$tantiæ a
primo vel quocunque alio $eriei termino) innote$cat; tum innote$-

<pb 498>[0550]<rh>DE SUMMATIONE</rh>
cent diver$i multiplicatores, qui eam reddent $ummabilem; $int <math>π,
ρ, σ, τ, &c.</math> functiones quantitatis <math>z</math>; tum <math>π, ρ, σ, τ, &c.</math> erunt etiam
multiplicatores, qui reddent æquationem <math>a T^n + b T^n-1 + c T^n-2 ...
f T^2 + g T′ + b T = Z</math> $ummabilem, ubi <math>Z</math> e$t quæcunque functio
ip$ius <math>z</math>.</p>
<p>Hìc applicari po$$unt omnia, quæ prius traduntur de transforma-
tione, &c. incrementorum incrementialium æquationum, &c. facile
enim mutari pote$t incrementialis æquatio in æquationem inter $uc-
ce$$ivos terminos, &c.</p>
<h>PROB. IX.</h>
<h it>Invenire $eries, quæ $ummationem recipiunt.</h>
<p>1. A$$umatur $eries quælibet unitate determinata (utrum $it accu-
rate $ummabilis necne, nihil refert, modo ejus termini ad nihilum
perpetuo convergant); e quâ $eriem eandem termino $uo primo, vel
terminis duobus primis, vel terminis tribus primis, &c. multatam
$ubtrahe; inde illud $it, ut quod relinquitur vel æquale $it termino
primo $eriei a$$umptæ, vel terminis ejus duobus primis, vel terminis
tribus primis, & $ic in in$initum.</p>
<p>Hanc autem operandi rationem circa $eries hâc $ubtractione com-
paratas iterare licet, unde novæ $eries in in$initum exurgent, quæ
omnes $ummabiles erunt.</p>
<p>Ex. 1. Sumatur $eries <math>1 + {1 / 2} + {1 / 3} + {1 / 4} + {1 / 5} + {1 / 6} + &c.</math> e quâ $ubtra-
hatur ip$amet demptis <math>n - 1</math> terminis $uis primis, hoc e$t $ubtrahatur
<math>{1 / n} + {1 / n + 1} + {1 / n + 2} + {1 / n + 3} + &c.</math> & relinquitur <math>{n - 1 / n × 1} +
{n - 1 / 2 · (n + 1)} + {n - 1 / 3 · n + 2} + {n - 1 / 4 · n + 3} + {n - 1 / 5 · n + 4} + &c. = 1 + {1 / 2} +
{1 / 3} + {1 / 4}...{1 / n - 1}</math>: rur$us a $erie quam modo comparavimus, $ubtraha-
tur ip$amet demptis <math>m - 1</math> terminis $uis primis, hoc e$t <math>{n - 1 / m × (n + m - 1)}<pb 499>[0551]<rh>SERIERUM, &c.</rh>
+ {n - 1 / (m + 1) · (n + m)} + {n - 1 / (m + 2) · (n + m + 1)} + &c.</math> & relinquitur
<math>{(n - 1) × (m(n + m - 1) - n) / n × m × (n + m - 1)} + {(n - 1) × ((m + 1) × (n + m) - 2 × (n + 1)) / (n + 1) × (m + 1) × (n + m)}
+ {(n - 1) × ((m + 2) · (n + m + 1) - 3 × (n + 2)) / (n + 2) × (m + 2) × (n + m + 1)} + &c. = {n - 1 / 1 · n}
+ {n - 1 / 2 (n + 1)} + {n - 1 / 3 × (n + 2)}...{n-1 / (m - 1) · (n + m - 2)}</math>; & $ic deinceps.</p>
<p>Cor. 1. Sit data $eries <math>t + t^1 + t^2 + t^3 + t^4 + &c.</math> ubi per <math>t, t^1, t^2,
t^3, &c.</math> de$ignentur $ucce$$ivi ejus termini, & erit terminus generalis
$eriei ex datâ $erie per hanc methodum deductæ <math>= a t^m + b t^n + c t^t
+ d t^s + &c.;</math> ubi per <math>t^m, t^n, t^r, t^s, &c.</math> de$ignantur termini ad di$tan-
tias <math>m, n, r, s, &c</math>. re$pective a primo vel quocunque alio datæ $eriei
termino, & <math>a + b + c + d + &c. = 0;</math> $it <math>m</math> minor quam <math>n, n</math> quam
<math>r, r</math> quam <math>s,</math> & <math>m</math> minimus & <math>l</math> maximus index; tum erit $eriei, cujus
generalis terminus e$t <math>(a t^m + b t^n + c t^r + d t^s + &c.)</math>, $umma <math>= a(t^m
+ t^m+1 + t^m+2 ... + t^l-1) + b(t^n + t^n+1 + t^n+2 ... + t^l-1) + c(t^r +
t^r+1 + t^r+2 ... + t^l-1) + d(t^s + t^s+1 + t^3+2 ... + t^l-1) + &c.</math>: e. g.
$it data $eries <math>1 + {1 / 2} + {1 / 3} + {1 / 4} + &c.,</math> tum $umma $eriei, cujus termi-
nus e$t <math>a(t^m - t^n) = {a / z + m} - {a / z + n} = {a n - a m / z + m · z + n}</math>, erit <math>= a
({1 / z + m} + {1 / z + m + 1} + {1 / z + m + 2}...{1 / z + n - 1})</math>.</p>
<p>Cor. 2. Hinc facile con$tat $emper eandem evadere $eriem, utrum
primum $ubtrahatur e datâ $erie eadem ab <math>m</math> terminis multata, &
deinde ex $erie re$ultante <math>B</math> $ubtrahatur eadem <math>B</math> ab <math>n</math> terminis mul-
tata; vel primum $ubtrahatur e datâ $erie eadem ab <math>n</math> terminis mul-
tata, & deinde ex $erie re$ultante $ubtrahatur eadem ab <math>m</math> terminis
multata. Et con$imilia etiam affirmari po$$unt de pluribus con$imi-
libus repetitis operationibus.</p>
<p>2. Altera vero methodus erit, ut a$$umptâ quâlibet $erie infinitâ,
cujus termini tum ad nihil perpetuo convergant, tum procedant per

<pb 500>[0552]<rh>DE SUMMATIONE</rh>
pote$tates indeterminatæ <math>x</math>, $i multiplicetur $eries a$$umpta per bino-
mium vel multinomium utcunque conflatum ex quantitatibus datis
& indeterminatâ <math>x</math>, deinde ponatur binomium vel multinomium il-
lud nihilo æquale, &c. hinc elicietur valor indeterminatæ <math>x</math>, ad quem
$i quantitas i$ta re$tringatur, tunc $eries ex hâc multiplicatione genita
evadet etiam nihilo æqualis, quapropter $i transferantur termini ejus
primi ad alteram æquationis partem, $eries re$idua æqualis erit ter-
minis tran$latis.</p>
<p>Ex.1. Sumatur $eries <math>1 + {1 / 2}x + {1 / 3}x^2 + {1 / 4}x^3 + {1 / 5}x^4 + &c.</math> quæ $i mul-
tiplicetur per binomium <math>1 - a x^m = 0</math> erit $eries genita <math>1 + {1 / 2} x + {1 / 3} x^2
... {1 / m}x^m-1 - {(m + 1) × a - 1 / m + 1}x^m - {(m + 2)a - 2 / (m + 2).2} x^m+1 - {(m + 3) × a - 3 / (m + 3) · 3}
× x^m+2 - &c.</math> unde <math>1 + {1 / 2} x + {1 / 3} x^2 ... {1 / m}x^m-1 = P = {(m + 1) · a - 1 / m + 1}
x^m + {(m + 2) a - 2 / (m + 2) · 2} x^m+1 + {(m + 3) a - 3 / (m + 3) · 3} x^m+2 + &c.</math> multiplicetur
hæc æquatio re$ultans in <math>1 - b x^n</math> & re$ultat <math>(1 - b x^n) × P =
{(m + 1) × a - 1 / m + 1} x^m + {(m + 2) · a - 2 / (m + 2) · 2} x^m+1 + {(m + 3) · a - 3 / (m + 3) · 3}x^m+2 ... +
{(m + n)a - n / (m + n) × n}x^m+n-1 - {((n + 1) × (m + 1) × (m + n + 1) a - (m + n + 1) × (n + 1))
b - (m + 1) × ((m + n + 1) a - (n + 1)) / (m + 1) × (n + 1) ×
(m + n + 1)} x^m+n + {2(m + n + 2) × (m + 2) a - 2
{(m + 2) × (n + 2) - ((m + 2) × a - 2) × (m + n + 2) × (n + 2)^b / 2 · m + 2 ·
n + 2 · (m + n + 2)} x^m+n+1 +
{3(m + n + 3) · (m + 3) a - 3 (m + 3) · (n + 3) - ((m + 3) × a - 3) × (m + n + 3) · (n + 3)^b / 3 · m + 3 · n + 3 ·
m + n + 3}× x^m+n+2 + &c.</math> unde <math>(1 - b x^n) × P -
{(m + 1) × a - 1 / m + 1}x^m - {(m + 2)a - 2 / (m + 2) · 2}x^m+1 - {(m + 3)a - 3 / (m + 3) · 3} x^m+2 ... -

<pb 501>[0553]<rh>SERIERUM, &c.</rh>
{(m + n)a - n / (m + n) · n} x^m+n-1 = {(m + 1) · (m + n + 1) a - (m + 1) · (n + 1)-((m + 1) a - 1) × (m + n + 1) × (n + 1)^b / (m + n + 1) × (n + 1)}
× (n + 1) × (m + 1)} + {2(m + 2) · (m + n + 2)
a - 2(m + 2) · (n + 2) - ((m + 2)a - 2) × (m + n + 2) × (n + 2)^b / m + n + 2
· n + 2 · m + 2 · 2}
+ &c.</math></p>
<p>Et $ic iteratâ operatione novæ exurgent $eries.</p>
<p>Hæ methodi fallunt, cum $eries inventa $it infinita quantitas.</p>
<p>Cor. 1. Secunda methodus eadem $it ac prior, $i modo pro <math>x</math> $cri-
batur unitas, & vice versâ prima methodus $emper continetur in $e-
cundâ.</p>
<p>Cor. 2. Sit terminus <math>{A / az + b. cz + d · ez + f. g z + b · &c.} =
{B / a z + b} + {B′ / cz + d} + {B″ / ez + f} + {B″′ / g z + b} + &c. = {C / a z + b. cz + d}
+ {C′ / ez + f} + {C″ / gz + b} + &c. = {c / az + b. cz + d} + {c′ / ez + f. gz + b}
+ &c. = {D / a z + b. c z + d. e z + f} + {D′ / gz + b} + &c. = {d / a z + b. c z + d}
+ {d′ / gz + b. ez + f} + &c. = &c.,</math> ubi in denominatoribus
harum fractionum contineantur omnes divi$ores denominatoris datæ
fractionis & nulli alii.</p>
<p>Cor 3. Datis $eriebus, ex his methodis $æpe deduci po$$unt earum
$ummæ</p>
<p>A$$umatur $eries, ita ut factores in denominatore contenti conti-
neantur in duobus vel pluribus diver$is terminis a$$umptæ $eriei,
quorum di$tantiæ a $e invicem $int <math>m, n, &c.</math> & deinde per præce-
dentes methodos inveniatur $eries, quæ fiat datæ $eriei æqualis, & $it
coroll.</p>

<pb 502>[0554]<rh>DE SUMMATIONE</rh>
<p>Ex. 1. Sit $eries <math>S = {4n + 5 / 2 n + 1 · 2 n + 2 · 2 n + 3 · 2 n + 4} x +
{4n + 9 / 2 n + 3 · 2 n + 4 · 2 n + 1 · 2 n + 6} x^2 + {4 n + 13 / 2 n + 5 · 2 n + 6 · 2 n + 7 · 2 n + 8}
x^3 + &c.</math> quâpropter a$$umptâ $erie <math>{1 / 2 n + 1 · 2 n + 2} + {1 / 2 n + 3 · 2 n + 4}
x + {1 / 2 n + 5 · 2 n + 6} x^2 + &c. = S</math>, quoniam <math>$i 2 n + 1 & 2 n + 2</math>
fint factores in denominatore unius termini, tum <math>2 n + 3 · 2 n + 4</math>
erit denominator $ub$equentis termini: multiplicetur $eries a$$ump-
ta per <math>x - 1</math>, & re$ultat <math>(x - 1) S′ = ({- 1 / 2 n + 1 · 2 n + 2}) +
{8n + 10 / 2 n + 1 · 2 n + 2 · 2 n + 3 · 2 n + 4} x + &c. = - {1 / 2 n + 1 · 2 n + 2} +
2S,</math> pone <math>x - 1 = 0</math> & evadet datæ $eriei $umma <math>S = {1 / 2 n + 1 · 2 n + 2}</math>.</p>
<p>Cor.. Pone in prædictâ $ummâ <math>S</math> pro <math>n</math> duos diver$os valores <math>l</math> &
<math>l + m</math>, ubi <math>m</math> $it integer numerus, & erit duarum quantitatum re$ul-
tantium differentia $umma <math>m</math> terminorum, e. g. in hoc exemplo $cribe
in $ummâ <math>{1 / 2 n + 1 · 2 n + 2}</math> valores <math>n</math> & <math>n + m</math> pro <math>n</math>, & evadet dif-
ferentia <math>{1 / 2 n + 1 · 2 n + 2} - {1 / 2 n + m + 1 · 2 n + m + 2}</math> $umma <math>m</math>
primorum terminorum.</p>
<p>Ex. 2. Sit $eries <math>B = {1 / 1 · 2 · 3 · 4 · 5} + {4 / 4 · 5 · 6 · 7 · 8} +
{9 / 7 · 8 · 9 · 10 · 11} + {16 / 10 · 11 · 12 · 13 · 14} + &c.</math></p>
<p>Sit <math>n</math> numerus de$ignans locum termini, tum terminus ip$e erit
<math>{{1 / 3}n / 3n - 2 · 3 n - 1 · 3 n + 1 · 3n + 2}</math>; & $eries po$t <math>n</math> terminos erit

<pb 503>[0555]<rh>SERIERUM, &c.</rh>
<math>{1 / 3} × {(n + 1) / (3n + 1) · (3n + 2) · (3n + 4) · (3n + 5)} + {{1 / 3}(n + 2) / (3n + 4) · (3n + 5) · (3n + 7) · (3n + 8)}
+ {{1 / 3}(n + 3) / (3n + 7) × (3n + 8) × (3n + 10) × (3n + 11)}
+ &c.</math> quoniam denominator quilibet hujus $eriei e $actoribus qua-
tuor con$tat, quorum bini quique in termino proxime $equente repe-
tuntur, $umatur idcirco $eries <math>{1 / 3 n + 1 · 3 n + 2} + {x / 3 n + 4 · 3 n + 5}
+ {x^2 / 3 n + 7 · 3 n + 8} + &c. = S,</math> cujus bini factores in $ingulis deno-
minatoribus $emel tantummodo occurrunt; deinde, quoniam in $erie
datâ denominatoris cuju$libet pars ea quæ iterum occurrit in termino
proxime $equente con$picitur: multiplicetur $eries data per <math>x - 1</math>
($in vero ita accideret ut terminus unus vel duo vel plures terminis
repetitis interjaceant, fiat congruens multiplicatio $eriei datæ per
<math>x^2 - 1</math> vel <math>x^3 - 1, &c.)</math> prodibit $eries <math>- {1 / (3n + 1) × (3n + 2)} +
{18 n + 18 / 3 n + 1 · 3 n + 2 · 3 n + 4 · 3 n + 5} x + {18 n + 36 / 3 n + 4 · 3 n + 5 · 3 n + 7 · 3 n + 8}
{18 n + 54 / 3 n + 7 · 3 n + 8 · 3 n + 10 · 3 n + 11} x^3 &c. = (x - 1) S</math>: ponatur
<math>x = 1</math>, deinde fiat tran$latio primi termini ad alteram æquationis
partem; tum dividantur omnia per 54, hinc exurget $umma de$iderata
<math>{1 / 3} × (n + 1{ / 3 n + 1 · 3 n + 2 · 3 n + 4 · 3 n + 5} + {{1 / 3}(n + 2) / 3n + 1 · 3n + 1 · 3n + 1 · 3n + 8}
+ &c. = {1 / 54 · 3 n + 1 · 3 n + 2}</math>.</p>
<p>Ex. 3. Proponatur $eries <math>{19 / 1 · 2 · 3} × {1 / 4} + {28 / 2 · 3 · 4} × {1 / 8} + {39 / 3 · 4 · 5} × {1 / 16} +
{52 / 4 · 5 · 6} × {1 / 32} + &c.</math> cujus $umma requiritur.</p><pb 504>[0556]<rh>DE SUMMATIONE</rh>
<p>A$$umatur $eries <math>1 + {1 / 2} x + {1 / 3} x^2 + {1 / 4} x^3 + {1 / 5} x^4 + &c. = S</math>, in
quâ inveniuntur omnes factores e quibus conflantur denominatores
$eriei propo$itæ: fingantur multiplicatores bini <math>2 x - 1 & a x - b</math>;
prior a$$umatur, quoniam $eries data procedit per pote$tates fractio-
nis <math>{1 / 2}</math>; præterea cum ex horum multiplicatorum mutuo ductu gene-
retur quantitas <math>2 a x^2 - 2 b x - a x + b</math> multiplicetur $eries a$$umpta
per productum i$tud, quo facto emerget $eries <math>b - {3 b + 2 a / 1 · 2} x +
{9 a - 4 b / 1 · 2 · 3} x^2 + {16 a - 10 b / 2 · 3 · 4} x^3 + {25 a - 18 b / 3 · 4 · 5} x^4 + &c. = (2 x - 1) ×
(a x - b) × S</math>, tran$latione factâ erit <math>{9a - 4 b / 1 · 2 · 3} x^2 + {16 a - 10 b / 2 · 3 · 4} x^3 +
{25 a - 18 b / 3 · 4 · 5} x^4 &c. = (2 x - 1) × (a x - b) × S - b + {3 b + 2 a / 1 · 2} x</math>.
Comparetur nunc primus terminus hujus æquationis cum termino
primo $erie datæ, itemque $ecundus cum $ecundo; hinc erit <math>9 a - 4 b
= 19, & 16 a - 10 b = 28</math>, unde invenietur <math>a = 3, b = 2</math>, & $umma
$eriei propo$itæ erit = 1.</p>
<p>Eâdem operâ invenietur ($i ponatur <math>a x - b</math> in <math>3 x - 2</math>, erit jam
<math>x = {2 / 3} {19 / 1 · 2 · 3} × {4 / 9} + {28 / 2 · 3 · 4} × {8 / 27} + {39 / 3 · 4 · 5} × {16 / 81} + {52 / 4 · 5 · 6} × {32 / 243}
&c. = 2</math>.</p>
<p>Ex. 4. Sit $eries <math>1 + {1 / 2} x + {1 / 3} x^2 + {1 / 4} x^3 + {1 / 5} x^4 + &c. = S</math>, multi-
plicetur $eries a$$umpta per trinomium <math>a + b x + c x^2</math>, unde erit $e-
ries genita <math>{2 a + 3 b + 6 c / 1 · 2 · 3} x ^2 + {6 a + 8 b + 12 c / 2 · 3 · 4} x ^3 + {12 a + 15 b + 20 c / 3 · 4 · 5}
x ^4 &c. = (a + b x + c x^2) × S - a - {1 / 2} a x - b x</math>; fiat comparatio
terminorum $eriei genitæ cum terminis $eriei propo$itæ, e. g. $it $e-
ries <math>{19 x^2 / 1 · 2 · 3} + {28 x^3 / 2 · 3 · 4} + {39 x^4 / 3 · 4 · 5} + &c.</math> hinc orientur tres æquationes
<math>2 a + 3 b + 6 c = 19, 6 a + 8 b + 12 c = 28, 12 a + 15 b + 20 c

<pb 505>[0557]<rh>SERIERUM, &c.</rh>
= 39</math>, quarum ope invenientur <math>a = 2, b = - 7, c = 6</math>; erit igitur
$umma de$iderata <math>= (2 - 7 x + 6 x^2) × S - a + 6 x</math>: jam $i requi-
ratur, ut irrationalitas tollatur, pone <math>2 - 7 x + 6 x^2 = 0</math>, hinc eli-
cietur valor duplex quantitatis <math>x</math> nempe <math>{1 / 2} & {2 / 3}</math>, quapropter $i <math>x</math> re$tin-
guatur ad valorem <math>{1 / 2}</math> erit $umma $eriei <math>= 1</math>; $in vero <math>x</math> re$tinguatur
ad valorem <math>{2 / 3}</math>, erit $umma $eriei <math>= 2</math>.</p>
<p>3. Exempla hujus multiplicationis etiam ex præcedente additione
ulterius promotâ facile erui po$$unt. E. g. Sit $eries <math>p + q x + r x^2
+ s x^3 + &c. = S</math>, multiplicetur hæc $eries in <math>a</math>, in <math>b x</math> & in <math>c x^2</math>; &
re$ultant <math>a p + a q x + a r x^2 + &c., b p x + b q x^2 + b r x^3 + &c.,
& c p x^2 + c q x^3 + &c.</math>; quibus ad unam $ummam adjunctis, ea re$ultat
<math>a p + (a q + b p) x + (a r + b q + c p) x ^2 + &c. = (a + b x + c x^2)
× S</math>: $i <math>a + b x + c x^2 = 0</math>, & termini ad in$initam di$tantiam nihilo
evadant æquales; tum $eries re$ultans $emper converget.</p>
<p>Sint enim termini ad infinitam di$tantiam <math>{1 / z^0}</math>, ubi 0 e$t quantitas
quam minima, at finita; tum differentia inter duos $ucce$$ivos ter-
minos <math>{1 / z^0} & {1 / (z + 1)^0}</math>, ubi <math>z</math> e$t infinita quantitas <math>= {(z + 1)^0 - z^0 / z^0 × (z + 1)^0}
= {- O / z^0+1}</math> prope; unde termini ad infinitam di$tantiam decre$cunt in
reciprocâ ratione <math>(0 + 1)</math> di$tantiæ a primo $eriei termino, quæ e$t
major quam $implex <math>(1)</math>; ergo $umma $eriei erit finita.</p>
<h>THEOR. XIII.</h>
<p>Sit $eries <math>a + b x + c x^2 + &c.</math> $ecundum dimen$iones quantitatis
<math>x</math> progrediens, cujus coefficientes terminorum proxime $ub$equentium
ad infinitam di$tantiam hanc habent inter $e rationem, viz. $ingulus
præcedens ad ejus $ub$equentem <math>:: r: 1</math>; ducatur hæc $eries in fun-
ctionem ip$ius <math>x</math> nihilo æqualem, cum <math>x = a</math>; tum, $i <math>a</math> major $it
quam <math>r</math>, $eries re$ultans $emper diverget; $in minor vero converget.</p>
<p>Facile con$tat huju$ce theorematis demon$tratio.</p>

<pb 506>[0558]<rh>DE SUMMATIONE</rh>
<h>THEOR. XIV.</h>
<p>Sit $eries <math>a + b + c + d + e + &c. = S</math>, cujus termini generaliter
exprimantur per datam functionem <math>(φ)</math> quantitatis <math>z</math> di$tantiæ a
primo $eriei termino: addantur $imul quique duo $ucce$$ivi termini,
e. g. <math>(a + b) + (c + d) + (e + f) + &c.</math> & re$ultat $eries <math>(A) = S</math>
a primo termino incipiens; vel $eries <math>(B) = (b + c) + (d + e) +
(f + g) + &c. = S - a</math> a $ecundo incipiens; in functione <math>φ</math> pro <math>z</math>
$cribatur <math>z + 1 &</math> re$ultet functio <math>φ′</math>; deinde in quantitate <math>φ + φ′</math> $i
termini datæ $eriei $int omnes affirmativi; vel in <math>φ - φ′</math> $i termini $int
alternatim affirmativi vel negativi, pro <math>z</math> $cribatur <math>2 z′</math>, ubi <math>z′</math> e$t di-
$tantia a primo $eriei termino in $erie <math>A</math>, & re$ultat generalis termi-
nus $eriei <math>A</math>: & $imiliter in functione <math>φ</math> pro <math>z</math> $cribantur <math>z + 1 &
z + 2</math>, & re$ultent functiones <math>φ′ & φ″</math>, tum in quantitate <math>φ′ + φ″</math> vel
<math>φ′ - φ″</math> pro <math>z</math> $cribatur <math>2 z′</math>, & re$ultat generalis terminus $eriei <math>B</math>, ubi
<math>z′</math> denotat di$tantiam a primo $eriei termino.</p>
<p>Ex. 1. Sit $eries <math>1 + {1 / 2} + {1 / 3} + &c.</math>, cujus generalis terminus e$t
<math>{1 / z + 1} = φ</math>; tum, $i pro <math>z</math> $cribatur <math>z + 1</math>, re$ultabit <math>φ′ = {1 / z + 2}</math>; in
functione <math>φ + φ′ = {1 / z + 1} + {1 / z + 2} = {2 z + 3 / z + 1 · z + 2}</math> pro <math>z</math> $cribatur
<math>2 z′</math>, & re$ultat <math>{4 z′ + 3 / 2 z′ + 1 · 2 z′ + 2}</math> generalis terminus $eriei <math>(1 + {1 / 2}) +
({1 / 3} + {1 / 4}) + ({1 / 5} + {1 / 6}) + &c. = {3 / 2} + {7 / 12} + {11 / 30} + &c.</math>, ubi <math>z′</math> e$t di$tantia a
primo $eriei termino.</p>
<p>Ex. 2. Sit $eries <math>1 - {1 / 2} + {1 / 3} - {1 / 4} + &c.</math>, cujus generalis terminus
e$t <math>{1 / z + 1}</math>; pro <math>z</math> in functione <math>{1 / z + 1} = φ</math> $cribatur <math>z + 1</math>, & re$ultat
<math>φ′ = {1 / z + 2}</math>; & quoniam termini $unt alternatim negativi & affirma-
tivi in functione <math>φ - φ′ = {1 / z + 1} - {1 / z + 2} = {1 / z + 1 · z + 2}</math> pro <math>z</math>

<pb 507>[0559]<rh>SERIERUM, &c.</rh>
fcribatur <math>2 z′</math> & re$ultat <math>{1 / 2 z′ + 1 · 2 z′ + 2}</math> generalis terminus $eriei
<math>(1 - {1 / 2}) + ({1 / 3} - {1 / 4}) + ({1 / 5} - {1 / 6}) + &c. = {1 / 1 · 2} + {1 / 3 · 4} + {1 / 5 · 6} + &c.</math>;
literâ <math>z′</math> eandem quantitatem ac prius denotante.</p>
<p>Ex. 3. Sit data $eries <math>1 + {1 / 2} + {1 / 3} + {1 / 4} + &c.</math>, & requiratur generalis
terminus $eriei <math>({1 / 2} + {1 / 3}) + ({1 / 4} + {1 / 5}) + ({1 / 6} + {1 / 7}) + &c. = {5 / 2 · 3} + {9 / 4 · 5} +
{13 / 6 · 7} + &c.</math>: in generali termino <math>{1 / z + 1}</math> datæ $eriei pro <math>z</math> $cribatur
<math>z + 1 & z + 2</math>; & re$ultant <math>{1 / z + 2} = φ′ & {1 / z + 3} = φ″</math>: in termino
<math>φ′ + φ″ = {2 z + 5 / z + 2 · z + 3}</math> pro <math>z</math> $cribatur <math>2 z′</math> & re$ultat <math>{2 z′ + 5 / 2 z′ + 2 · 2 z′ + 3}</math>
generalis terminus quæ$itus.</p>
<p>2. Si modo addantur $imul quique <math>(n)</math> $ucce$$ivi termini datæ $eriei,
cujus generalis terminus e$t <math>φ</math>, a primo, $ecundo, & denique <math>(m + 1)</math>
termino incipiens; & re$ultet $eries <math>A</math>, cujus generalis terminus quæri-
tur: in termino <math>φ</math> pro <math>z</math> $cribantur <math>(n)</math> quantitates <math>z + m, z + m + 1,
z + m + 2, ... z + m + n - 2, z + m + n - 1</math>, & re$ultent quanti-
tates <math>φ′, φ″, φ′″, &c.</math>; in $ummâ <math>φ + φ′ + φ″ + φ′″ + &c.</math> pro <math>z</math> $cri-
batur <math>n z</math>, & re$ultabit generalis terminus $eriei <math>A</math> quæ$itus.</p>
<p>Ex. Sit data $eries <math>1 + {1 / h + 1} + {1 / 2h + 1} + &c.</math>, cujus generalis
terminus e$t <math>{1 / z h + 1}</math>: invenire generalem terminum $eriei, cujus ter-
mini $int $umma quorumcunque <math>(3)</math> $ucce$$ivorum terminorum a $e-
cundo incipientium. Pro <math>z</math> in generali termino $cribatur <math>z + 1</math>,
quoniam terminus a $ecundo incipit & exorietur <math>{1 / (z + 1) h + 1}</math>; in
hoc termino pro <math>z</math> $cribantur <math>z, z + 1 & z + 2, &</math> re$ultant <math>{1 / (z + 1)

<pb 508>[0560]<rh>DE SUMMATIONE</rh>
h + 1}, {1 / (z + 2) h + 1} & {1 / (z + 3) h + 1}</math>, cujus $umma e$t <math>{h^2 (3 z^2 + 12 z + 11) + h (6
z + 12) + 3 / (z + 1) h + 1 · (z + 2) h + 1 · (z + 3) h + 1}</math>; in hâc $ummâ pro <math>z</math> $cribatur <math>3 z′</math>
& re$ultabit <math>{h^2 (27 z′^2 + 36 z′ + 11) + h (18 z′ + 12) + 3 / (3 z′ + 1) h + 1 · (3 z′ + 2) h + 1 · (3 z′ + 3) h + 1}</math> ge-
neralis terminus $eriei <math>({1 / h + 1} + {1 / 2 h + 2} + {1 / 3 h + 1}) + ({1 / 4 h + 1}
+ {1 / 5 h + 1} + {1 / 6 h + 1}) + &c.</math>, ubi <math>z′</math> denotat di$tantiam a primo $e-
riei termino.</p>
<p>3. Si datæ $eriei termini $int alternatim negativi & affirmativi, &
<math>n</math> numerus terminorum $imul adjunctorum $it impar; tum erit $eriei
re$ultantis termini alternatim negativi & affirmativi; $in <math>n</math> $it par
numerus, tum omnes termini idem habebunt $ignum.</p>
<p>Cor. Ex $eriebus per hanc methodum transformatis & principiis
in prob. præced. traditis facile acquiri po$$unt multæ $eries, quarum
$ummæ innote$cunt.</p>
<p>Ex. Sit data $eries <math>1 + {1 / h + 1} + {1 / 2 h + 1} + {1 / 3 h + 1} + &c.</math>, cujus
generalis terminus e$t <math>{1 / h z + 1}</math>; $imul addantur tres $ucce$$ivi termini
<math>{1 / h z + 1} + {1 / h z + h + 1} + {1 / h z + 2 h + 1} = {3 h^2 z^2 + 6 h^2 z + 6 h z + 2 h^2 + 6 h + 3 / h z + 1 ·
h z + h + 1 · h z + 2 h + 1} = A</math>; etiamque tres $ucce$$ivi termini <math>{1 / h z + h + 1}
+ {1 / h z + 2 h + 1} + {1 / h z + 3 h + 1} = {3 h^2 z^2 + 12 h^2 z + 6 h z + 11 h ^2 + 12 h + 3 / h z + h + 1 · h z + 2 h + 1
· h z + 3 h + 1} = B</math>: in quantitatibus <math>A & B</math> per theor. pro <math>z</math> $cri-
batur <math>3 z</math>, & re$ultant quantitates <math>{27 h^2 z^2 + 18 h^2 z + 18 h z + 2 h^2 + 6 h + 3
/ 3 h z + 1 · 3 h z + h + 1 · 3 h z + 3 h + 1}

<pb 509>[0561]<rh>SERIERUM, &c.</rh>
= β & {27 h^2 z^2 + (36 h^2 + 18 h)z + 11 h^2 + 12 h + 3 / 3 h z + h + 1 · 3 h z + 2 h + 1 · 3 h z + 3 h + 1} = γ</math>: inveniatur dif-
ferentia <math>β - γ</math> inter has duas quantitates, quæ erit <math>{27 h^3 z^2 + (27 h^3 + 18 h^2) z + 6 h^3 + 9 h^2 + 3 h / 3 h z + 1 · 3 h z + h + 1 · 3 h z + 2 h + 1 · 3 h z + 3 h + 1} = τ</math>;
& con$equenter $umma $eriei, cujus
termini <math>τ</math> progrediuntur a valore <math>0</math> quantitatis <math>z</math> u$que ad valorem in$i-
nitum eju$dem quantitatis, erit 1; i. e. $umma $eriei a valore <math>0</math> ad valorem
infinitum quantitatis <math>z</math>, cujus generalis terminus e$t <math>{9 h^2 z^2 + (9 h^2 + 6 h) z + 2 h^2 + 3 h / 3 h z + 1 · 3 h z + b + 1 · 3 h z + 2 h + 1 · 3 h z + 3 h + 1}</math>, erit <math>{1 / 3 h}</math>.</p>
<p>Ex. Sit eadem $eries <math>1 + {1 / h + 1} + {1 / 2 h + 1} + &c.</math>; addantur tres
$ucce$$ivi termini <math>{1 / h z + 2 h + 1} + {1 / h z + 3 h + 1} + {1 / h z + 4 h + 1}
= {3 h^2 z^2 + (18 h^2 + 6 h) z + 26 h^2 + 18 h + 3 / h z + h + 1 · h z + 2 h + 1 · h z + 3 h + 1} = C</math>: in quantitate
<math>C</math> pro <math>z</math> $cribatur <math>3 z</math>, & re$ultat <math>{27 h^2 z^2 + (54 h^2 + 18 h) z + 26 h^2 + 18 h + 3 / 3 h z + 2 h + 1 · 3 h z + 3 h + 1 · 3 h z + 4 h + 1} = δ</math>;
inveniatur differentia inter quantitates <math>γ & δ</math> & re-
$ultat <math>{27 h^3 z^2 + (45 h^3 + 18 h^2) z + 18 h^3 + 15 h^2 + 3 h / 3 h z + h + 1 · 3 h z + 2 h + 1 · 3 h z + 3 h + 1 · 3 h z + 4 h + 1} = σ</math>,
& con$equenter $umma $eriei, cujus termini <math>σ</math> progrediuntur a valore <math>0</math>
quantitatis <math>z</math> ad valorem in$initum eju$dem quantitatis erit <math>{1 / h + 1}</math>; unde
$umma $eriei, cujus generalis terminus e$t <math>{9 h^2 z^2 + (15 h^2 + 6 h) z + 6 h^2 + 5 h + 1 / 3 h z + h + 1 · 3 h z + 2 h + 1 · 3 h z + 3 h + 1 · 3 h z + 4 h + 1}</math>,
a valore (<math>0</math>) quantitatis <math>z</math> ad
infinitum erit <math>{1 / 3 h · h + 1}</math>.</p>

<pb 510>[0562]<rh>DE SUMMATIONE</rh>
<p>Eadem etiam applicari po$$unt ad alias $eries, quarum termini $unt
alternatim affirmativi & negativi: in hoc ca$u, cum numerus termi-
norum, qui $imul adjunguntur, $it impar; tum termini $eriei re$ul-
tantis etiam erunt alternatim affirmativi & negativi, $in aliter non.</p>
<p>Con$imilia etiam prædicari po$$unt de ca$ibus in quibus quicunque
<math>n</math> termini $int affirmativi & <math>m</math> negativi, vel quocunque alio modo ter-
mini $int affirmativi & negativi.</p>
<p>4. Sint <math>T^m, T^n, T^r, T^s, &c.</math> termini $eriei <math>A</math>, quorum $it <math>T^m</math> quicunque
terminus datæ $eriei; & <math>T^n, T^r, T^s, &c.</math> termini prædictæ $eriei, quo-
rum di$tantiæ a termino <math>T^m</math> $int re$pective <math>n - m, r - m, s - m, &c.</math>;
i. e. in generali termino $eriei <math>A</math>, pro <math>z</math> di$tantiâ a primo $eriei ter-
mino $cribantur re$pective <math>z + m, z + n, z + r, z + s, &c.</math>; & re-
$ultabunt termini <math>T^m, T^n, T^r, T^s, &c.</math></p>
<p>Sint <math>T^m′, T^n′, T^r′, T^s′, &c.</math> termini $eriei <math>B</math>, cujus generalis termi-
nus, qui e$t functio quantitatis <math>z</math> di$tantiæ a primo $eriei termino,
$it <math>φ</math>; & $i in <math>φ</math> pro <math>z</math> $cribantur <math>z + m′, z + n′, z + r′, z + s′, &c.</math>,
re$ultabunt termini <math>T^m′, T^n′, T^r′, T^s′, &c.</math></p>
<p>Sint <math>T^m″, T^n″, T^r″, T^s″, &c.</math> termini $eriei <math>C</math>, cujus generalis termi-
nus $it <math>φ′</math> functio quantitatis <math>z</math> di$tantiæ a primo $eriei termino, & $i
in termino <math>φ′</math> pro <math>z</math> $cribantur re$pective <math>z + m″, z + n″, z + r″,
z + s″, &c.</math>, re$ultabunt termini <math>T^m″, T^n″, T^r″, T^s″, &c.</math></p>
<p>A$$umatur quantitas <math>a T^m + b T^n + c T^r + d T^s + &c. + a′ T^m′ +
b′ T^n′ + c′ T^r′ + d′ T^s′ + &c. + a″ T^m″ + b″ T^n″ + c″ T^r″ + &c. + &c.</math>,
pro termino $eriei quæ$itæ; tum ex $ummis (<math>S, S′, S″, &c.</math>) datarum
$erierum <math>A, B, C, &c.</math> acquiri pote$t $umma $eriei re$ultantis quæ$itæ,
erit enim <math>(a + b + c + d + &c.) × S + (a′ + b′ + c′ + d′ + &c.)
× S′ + (a″ + b″ + c″ + d″ + &c.) × S″ + &c. - (a + b + c + d +
&c.) (T + T′ + T″ + ... T^m-1) - (b + c + d + &c.) (T^m + T^m-1
+ T^m-2 ... T^n-1) - (c + d + &c.) (T^n + T^n+1 + &c. ... + T^r-1)
- &c. - (a′ + b′ + c′ + d′ + &c.) (T^1′ + T^2′ + T^3′ ... T^m′-1) - (b′
+ c′ + d′ + &c.) (T^m′ + T^m′+1 ... T^n′-1) - &c. - &c.</math></p>
<p>Cor. Sit data $eries <math>{1 / a} + {1 / a + 1} x + {1 / a + 2} x^2 + {1 / a + 3} x^3 + &c.

<pb 511>[0563]<rh>SERIERUM, &c.</rh>
= S</math>; ex $ummâ <math>S</math> datâ $emper acquiri pote$t $umma cuju$cunque
$eriei, cujus generalis terminus e$t <math>{b z^m + c z^m-1 + &c. / a + z + α · a + z + β · a + z + γ · &c.}x^z</math>;
ubi <math>α, β, γ, δ, &c.</math> $unt integri numeri, & <math>m</math> e$t numerus minor quam
numerus factorum in denominatore: a$$umatur <math>{A / a + z + α} +
{B / a + z + β} + {C / a + z + γ} + &c. = {b z^m + c z^m-1 + &c. / a + z + α · a + z + β · a + z + γ · &c.}</math>
& exinde inveniri po$$unt coefficientes <math>A, B, C, &c.</math>; deinde $uman-
tur <math>A = a′ x^α, B = b′ x^β, C = c′x^γ, &c.</math>; unde con$tabunt quanti-
tates <math>a′, b′, c′, &c.</math> & exinde erui pote$t $umma $eriei quæ$ita <math>= (a′ +
b′ + c′ + &c.) × S - a′ ({1 / a} + {1 / a + 1} x + {1 / a + 2}x^2 ... + {1 / α + α - 1}
x^z-1) - b′ ({1 / a} + {1 / a + 1} x ... {1 / a + β - 1}x^β-1) - c′ ({1 / a} + {1 / a + 1} x ...
+ {1 / a + γ - 1} x^γ-1) + &c. = (a′ + b′ + c′ + &c.)(S - ({1 / a} + {1 / a + 1}
... + {1 / a + α - 1}) - (b′ + c′ + &c.)({1 / a + α} + {1 / a + α + 1} ...
{1 / a + β - 1}) - (c′ + &c.) ({1 / a + β} + {1 / a + β + 1} + {1 / a + γ - 1})
- &c.</math></p>
<p>2. Sint duæ datæ $eries <math>{1 / a} + {1 / a + 1} x + {1 / a + 2} x^2 + &c. = S; &
{1 / b} + {1 / b + 1} x + {1 / b + 2} x^2 + {1 / b + 3} x^3 + &c. = S′; & {1 / c} + {1 / c + 1} x +
{1 / c + 2} x^2 + &c. = S″; &c.</math>; tum facile ex his $ummis datis acquiri
pote$t $umma cuju$cunque $eriei <math>{A z^m + B z^m-1 + C z^m-2 + &c. / a +
z + α · a + z + β · a + z + γ · &c × b + z + α′ · b + z + β′ · b + z + γ′ · &c. × c + z + α″ · c + z + β″ · &c.. &c.}

<pb 512>[0564]<rh>DE SUMMATIONE</rh>
x^z+b</math>, ubi <math>m, α, β, γ, &c.; α′, β′, γ′, &c.; α″, β″, &c.</math> $unt integri nu-
meri.</p>
<p>Cor. 2. Sit data $eries <math>{1 / a} - {1 / a + 1} x + {1 / a + 2} x^2 - {1 / a + 3} x^3 + &c.</math>;
tum ex eâ acquiri pote$t $umma cuju$cunque $eriei, cujus generalis
terminus e$t <math>{a′ z^m + b z^m-1 + c z^m-2 + &c. / 2 z + a · 2 z + a + 1 · 2 z + a + 2 .. 2 z + a + l + 1} x^z</math>,
ubi <math>z</math> denotat di$tantiam a primo $eriei termino & <math>m & l</math> $unt integri
numeri.</p>
<p>Cor. 3. Ex $ummis $erierum <math>{1 / α} + {1 / α + 1} x + {1 / α + 2} x^2 + &c. = S,
{1 / β} + {1 / β + 1} x + {1 / β + 2} x^2 + &c. = S′, {1 / γ} + {1 / γ + 1} x + {1 / γ + 2} x^2 +
&c. = S″, &c.</math>, etiamque $ummis $erierum <math>{1 / π} - {1 / π + 1} x + {1 / π + 2} x^2
- &c. = Σ, {1 / ρ} - {1 / ρ + 1} x + {1 / ρ + 2} x^2 - &c. = Σ′, {1 / σ} - {1 / σ + 1} x +
{1 / σ + 2} x^2 - &c. = Σ″, &c.</math> deduci pote$t $umma cuju$cunque $eriei,
quæ habet generalem terminum, cujus numerator e$t <math>a z^m + b z^m-1 +
c z^m-2 + &c.</math> in <math>x^bz+a</math>, ubi <math>m</math> e$t integer numerus & <math>z</math> di$tantia a primo
$eriei termino, & cujus denominator e$t <math>z + α · z + α + 1 · z + a
+ 2 · &c. × z + β · z + β + 1 · z + β + 2 · &c. × z + γ · z + γ
+ 1 · z + γ + 2 · &c. × &c. × 2 z + π · 2 z + π + 1 · 2 z + π + 2
· 2 z + π + 3 · &c. × 2 z + ρ · 2 z + ρ + 1 · 2 z + ρ + 2 · 2 z + ρ
+ 3 · &c. × 2 z + σ · 2 z + σ + 1 · 2 z + σ + 2 · 2 z + σ + 3 · 2 z
+ σ + 4 · &c. × &c.</math></p>
<h>THEOR. XV.</h>
<p>Quam plurimæ $eries deduci po$$unt ex dividendo alteram quanti-
tatem per alteram & mutando terminos denominatoris continuo.</p>

<pb 513>[0565]<rh>SERIERUM, &c.</rh>
<p>Ex. 1. Sit fractio <math>{1 / 1 + 1}</math>; a$$umantur diver$i denominatores <math>1 + 1
= (1 + {1 / 2}) + {1 / 2} = (1 + {3 / 4}) + {1 / 4} = (1 + {7 / 8}) + {1 / 8} = (1 + {15 / 16}) + {1 / 16} = &c.</math>; &
operatio erit <math>1 + 1 1 1
1 + 1
(1 + {1 / 2}) + {1 / 2} - 1 - {2 / 3}
- 1 - {1 / 3}
(1 + {3 / 4}) + {1 / 4} + {1 / 3} {+4 / 3 · 7}
{1 / 3} + {1 / 3 · 7}
(1 + {7 / 8}) + {1 / 8} - {1 / 3 · 7}{- 8 / 3 · 7 · 15}
&c.</math>; unde re$ultat <math>{1 / 1 + 1} = {1 / 2} =
1 - {2 / 3} + {4 / 3 · 7} - {8 / 3 · 7 · 15} + {16 / 3 · 7 · 15 · 31} - {32 / 3 · 7 · 15 · 31 · 63} +
{64 / 3 · 7 · 15 · 31 · 63 · 127} - &c.</math></p>
<p>Ex. 2. Sit fractio <math>{m / m + n}</math>, tum <math>m + n {m / m + n} 1
(m + {r / s} n) + {s - r / s} n - n {- s n / s m + r n}
- n - {(s - r) × n^2 / s m + r n}
(m + ({r / s} + {r × (s - r) / s^2})<r>n) + {(s - r)^2 / s^2}n + {(s - r) × n^2 / s m + r n} \\ &c.</r> {+ (s - r) × s^2 n^2 / (s m r + r n)(s^2 m + 2 r s - r^2) n} \\ &c.</math>,
unde erit <math>{m / m + n} = 1 - {s n / s m + r n} + {(s - r) × s n A / (s^2 m + 2 r s - r^2) n} -
{(s - r) s n B / (s^3 m - (s - r)^3 + s^3)n} + {(s - r) s n C / (s^4 m - (s - r)^4 + s^4)n} ... {s × (s - r) n L / (s^b m - (s - r)^b + s^b)n}</math>,<pb 514>[0566]<rh>DE SUMMATIONE</rh>
ubi <math>A, B, C, .. L</math> denotant præcedentes terminos & <math>h</math> de$ignat di$tan-
tiam a primo $eriei termino.</p>
<p>Hìc continuo adjicitur ad primum terminum denominatoris pars <math>{r / s}</math>
re$idui: in primo exemplo <math>{r / s} = {1 / 2}</math>.</p>
<p>Hæc methodus generaliter reddi pote$t ex a$$umendo diver$os va-
lores pro quantitatibus <math>{r / s}</math>, vel diver$as quantitates ad primum termi-
num adjungendas ad $ingulas divi$iones: etiamque ex a$$umendo
plures terminos continuo & in numeratore & denominatore con-
tentos.</p>
<p>Eadem principia etiam ad extractiones radicum applicari po$$unt,
& ex hâc complexâ methodo dividendi & radices extrahendi quam-
plurimæ in$initæ $eries deduci po$$unt, quarum $ummæ innote$cunt.</p>
<h>PROB. X.</h>
<h it>Ex datis quibu$dam $eriebus alias invenire, quarum $ummæ innote$cunt.</h>
<p>1. E datis $eriebus inveniatur earum quæcunque functio, quæ $it
convergens $eries; & invenitur $eries, cujus $umma e datis $eriebus &
datâ functione facile deduci pote$t.</p>
<p>1. 2. E datis $eriebus, quæ haud $unt $ummabiles $æpe deduci
po$$unt $eries, quæ $unt $ummabiles: inveniatur enim talis functio e
$ingulis $eriebus, ut $int $ummabiles, i. e. evane$cant e $ummis $erie-
rum re$ultantium $ummæ $erierum, quæ haud $unt $ummabiles.</p>
<p>Ex. 1. Sit <math>v^. = {x^. / 1 - x}</math>, & con$equenter <math>v = x + {1 / 2} x^2 + {1 / 3} x^3 + {1 / 4} x^4
+ &c.</math> ducatur hæc æquatio in <math>x^n x^.</math>, & re$ultat æquatio <math>v x^n x^. = x^n+1 x^.
+ {1 / 2} x^n+2 x^. + {1 / 3} x^n+3 x^. + &c.</math> inveniatur fluens ex utrâque æquationis
parte, & re$ultat <math>{v x^n+1 / n + 1} + {x^n+1 / (n + 1) · n + 1} + {x^n / n + 1 · n} ... + {x / n + 1} -

<pb 515>[0567]<rh>SERIERUM, &c.</rh>
{v / n + 1} = {x^n+2 / n + 2} + {x^n+3 / 2 × (n + 3)} + {x^n+4 / 3 × (n + 4)} + &c.</math> Fiant ter-
mini <math>{v x^n+1 / n + 1} - {v / n + 1} = 0</math>, unde <math>x = 1</math>, & re$ultant $eries $umma-
biles <math>{1 / n + 1} × ({1 / n + 1} + {1 / n} + {1 / n - 1} + {1 / n - 2} + &c.)</math> (quæ ad tot
terminos continuantur quot $unt unitates in <math>n + 1) = {1 / n + 2} +
{1 / 2 · (n + 3)} + {1 / 3 · (n + 4)} + &c.</math></p>
<p>Sit <math>n</math> impar numerus, tunc ex æquatione <math>{v x^n+1 / n + 1} - {v / n + 1} = 0</math> po-
terit deduci <math>x = - 1</math>, adeoque $eries infinita <math>- {1 / n + 2} + {1 / 2 · (n + 3)} -
{1 / 3 × (n + 4)} + {1 / 4 × (n + 5)} - &c. = {1 / n + 1} × ({1 / n + 1} - {1 / n} + &c.)</math> ad
tot terminos continuatæ quot $unt unitates in <math>n + 1</math>; quapropter $i <math>n</math> $it
numerus impar e $ummâ præcedentium $erierum emerget nova $eries
infinita <math>{1 / 2 × (n + 3)} + {1 / 4 × (n + 5)} + {1 / 6 × (n + 7)} + {1 / 8 × (n + 9)} +
&c.</math> æqualis $eriei finitæ <math>{1 / n + 1} × ({1 / n + 1} + {1 / n - 1} + {1 / n - 3} +
{1 / n - 5} + &c.)</math> ad tot terminos continuatæ quot $unt unitates in <math>{n + 1 / 2}</math>:
ex earundem $erierum differentiâ emerget altera $eries infinita <math>{1 / n + 2}
+ {1 / 3 × (n + 4)} + {1 / 5 × (n + 6)} + {1 / 7 × (n + 8)} + {1 / 9 × (n + 10)} + &c.</math>
æqualis $eriei finitæ <math>{1 / n + 1} × ({1 / n} + {1 / n - 2} + {1 / n - 4} + {1 / n - 6} &c.)</math>
ad tot terminos continuatæ quot $unt unitates in <math>{n + 1 / 2}</math>.</p>

<pb 516>[0568]<rh>DE SUMMATIONE</rh>
<p>2. Ducatur prædicta æquatio <math>{v x^n+1 / n + 1} + {x^n+1 / n + 1 · n + 1} + {x^n / n + 1 · n}
... - {v / n + 1} = {x^n+2 / n + 2} + {x^n+3 / 2 × (n + 3)} + {x^n+4 / 3 × (n + 4)} + &c.</math> in <math>x^m x^.</math>
& re$ultat <math>{v x^m+n+1 x^. / n + 1} + {x^n+m+1 x^. / n + 1 · n + 1} + {x^n+m x^. / n + 1 · n} ... - {v x^m x^. / n + 1} =
{x^n+m+2 x^. / n + 2} + {x^n+m+3 x^. / 2 · n + 3} + {x^n+m+4 x^. / 3 · n + 4} + &c.</math> inveniantur fluentes ex
utrâque æquationis parte, & re$ultat <math>{v x^n+m+2 / n + 1 · n + m + 2} +
{x^n+m+2 / n + 1 · n + m + 2 · n + m + 2} + {x^n+m+1 / n + 1 · n + m + 2 · n + m + 1} +
{x^n+m / n + 1 · n + m + 2 · n + m} ... - {v / n + 1 · n + m + 2} + {x^n+m+2 / (n + 1) · n + 1 · n + m + 2}
+ {x^n+m+1 / n + 1 · n · n + m + 1} + {x^n+m / n + 1 · n - 1 · n + m} .. + {x^m+2 / n + 1 · m + 2}
- {v x^m+1 / n + 1 · m + 1} - {x^m+1 / n + 1 · m + 1 · m + 1} - {x^m / n + 1 · m + 1 · m} -
{x^m-1 / n + 1 · m + 1 · m - 1} ... + {v / n + 1 · m + 1} = {x^n+m+3 / n + 2 · n + m + 3} +
{x^n+m+4 / 2 · n + 3 · n + m + 4} + {x^n+m+5 / 3 · n + 4 · n + m + 5} + &c.</math> Ut hæc
$eries fiat finita nece$$e e$t nihilo evadere æquales tres terminos
<math>{v x^n+m+2 / n + 1 · n + m + 2} - {v x^m+1 / n + 1 · m + 1} + {v / m + 1 · n + m + 2} ({- v / n + 1 · n + m + 2}
+ {v / n + 1 · m + 1})</math> in quibus invenitur quantitas logarithmica <math>v</math>.</p>
<p>Et $ic ducatur hæc æquatio re$ultans in <math>x^r x^.</math>, tum inveniatur
fluens ex utrâque æquationis parte; & deinde ducatur æquatio re$ul-
tans in <math>x^s x^.</math>, & inveniatur fluens ex utrâque æquationis parte; & $ic
deinceps; & novæ continuo oriuntur $eries, quarum $ummæ inve-
niuntur.</p>

<pb 517>[0569]<rh>SERIERUM, &c.</rh>
<p>3. Sit <math>z^. = {x^. / 1 + x^2}</math>, & exinde <math>z = x - {1 / 3} x^3 + {1 / 5} x^5 - {1 / 7} x^7 + {1 / 9} x^9 -
&c.</math> ducatur hæc æquatio in <math>x^n x^.</math>, & inveniatur fluens ex utrâque
æquationis parte: 1<_>mo</_>. $it <math>{n + 1 / 2}</math> integer numerus, & erit $umma
$eriei finitæ <math>{1 / n + 1} × (x^n+1 z - {x^n / n} + {x^n-2 / n - 2} - {x^n-4 / n - 4} + {x^n-6 / n - 6} &c. = z)</math>
æqualis $ummæ infinitæ $eriei <math>{x^n+2 / n + 2} - {x^n+4 / 3 · n + 4} + {x^n+6 / 5 · n + 6} - &c.</math>
Jam $i fuerit <math>{n + 1 / 2}</math> numerus par, ultimus terminus <math>z</math> afficietur $igno
negativo; $in fuerit <math>{n + 1 / 2}</math> numerus impar, ultimus terminus $igno
affirmativo afficietur.</p>
<p>Cor. · Sit <math>{n + 1 / 2}</math> numerus par, & $i modo fiat <math>x^n+1 z - z = 0</math>,
i. e. termini, qui haud inveniri po$$unt, nihilo æquales; tum erit
<math>x = ± 1</math> & exinde deduci po$$unt $eries, quæ exprimuntur finitis
terminis.</p>
<p>1. 2<_>do</_>. Sit <math>n</math> par numerus, & a$$umantur fluentes, erit <math>{x^n+2 / n + 2} -
{x^n+4 / 3 × (n + 4)} + {x^n+6 / 5 × (n + 6)} - {x^n+8 / 7 × (n + 8)} + &c.</math> in infinitum æqua-
lis $eriei finitæ <math>{1 / n + 1} × (x^n+1 z - {x^n / n} + {x^n-2 / n - 2} - {x^n-4 / n - 4} &c. ± log.
√(1 + x^2))</math>, ubi primus & ultimus terminus $emper $ervatur, & tot
intermedii quot unitates in <math>{1 / 2} n</math>.</p>
<p>Eodem proce$$u repetito plurium detegentur $erierum $ummæ, &c.
Et $ic deinceps.</p>
<p>4. Sit $eries a$$umpta <math>x + {1 / 3} x^3 + {1 / 5} x^5 + {1 / 7} x^7 + &c. = v</math>, ubi <math>v^. =
{x^. / 1 - x^2}</math> ducatur utraque æquationis pars in <math>x^n x^.</math>, & æquationis re$ul-

<pb 518>[0570]<rh>DE SUMMATIONE</rh>
tantis inveniatur fluens, & exinde re$ultabit finitæ $eriei $umma
<math>{v x^n+1 / n + 1} + {x^n / n + 1 · n} + {x^n-2 / n + 1 · n - 2} + {x^n-4 / n + 1 · n - 4} ... + {x^2 / n + 1 · 2}
+ {1 / 2} log. × (1 - x^2)</math> æqualis $ummæ infinitæ $eriei <math>{x^n+2 / n + 2} + {x^n+4 / 3 · n + 4}
+ {x^n+6 / 5 · n + 6} + {x^n+8 / 7 · n + 8} + &c.</math> $i <math>n</math> $it par numerus.</p>
<p>Si vero <math>n</math> $it impar numerus, tum erit $umma finitæ $eriei <math>{v x^n+1 / n + 1}
+ {x^n / n + 1 · n} + {x^n-2 / n + 1 · n - 2} + {x^n-4 / n + 1 · n - 4} ... + {x / n + 1 · 1} -
{v / n + 1} = {x^n+2 / n + 2} + {x^n+4 / 3 · n + 4} + &c.</math></p>
<p>Eodem proce$$u repetito plures detegentur $eries.</p>
<p>5. In genere $it $eries a$$umpta <math>v = x ± {x^n+1 / n + 1} + {x^2n+1 / 2n + 1} ±
{x^3n+1 / 3n + 1} + {x^4n+1 / 4n + 1} ± &c.</math> ubi <math>v^. = {x^. / 1 ± x^n}</math>, ducatur utraque æqua-
tionis pars in <math>x^m x^.</math>, & quoniam <math>{x^. / 1 ± x^n}</math> dividi pote$t per notas regu-
las in quantitates huju$ce generis <math>{x^. × (α + π x) / x^2 - 2 a x + 1} + {x^. × (β + ρ x) / x^2 - 2 b x + 1}
+ {x^. × (γ + σ x) / x^2 - 2 c x + 1} + &c.</math> igitur e præcedentibus ca$ibus facile inveniri
pote$t fluens fluxionis <math>v x^m x^.</math> per finitos terminos, &c. ergo invenitur
$umma infinitæ $eriei <math>{x^m+2 / m + 2} ± {x^n+m+2 / n + 1 · n + m + 2} + {x^2n+m+2 / 2 n + 1 × 2 n + m + 2}
± {x^3n+m+2 / (3 n + 1) × (3 n + m + 2)} + &c.</math> per finitos terminos, &c.; & $ic
continuo repetitâ hâc operatione plures $equuntur $eries $ummabiles.</p>
<p>6. Sit fractio finita <math>{1 / x^n - p x^n-1 + q x^n-2 - &c.} = x^r + P x^r-1 +

<pb 519>[0571]<rh>SERIERUM, &c.</rh>
Q x^r-2 + R x^r-3 + &c.</math> in infinitum; $int <math>α, β, γ, δ, &c.</math> radices æqua-
tionis <math>x^n - p x^n-1 + q x^n-2 - &c. = 0</math>; & ita transformetur finita
fractio, ut fiat <math>{1 / x^n - p x^n-1 + q x^n-2 - &c.} = {a / x - α} + {b / x - β} +
{c / x - γ} + {d / x - δ} + &c.</math> ($i vero duæ fractiones <math>{a / x - α} + {b / x - β}</math> in-
volvant corre$pondentes impo$$ibiles radices, tum reducendæ $unt ad
unam fractionem <math>{(b + a) x - (β a + b α) / (x - α) · (x - β)}</math>, & evane$cent impo$$ibiles
quantitates); ducatur æquatio <math>{a / x - α} + {b / x - β} + {c / x - γ} + &c.
= x^r + P x^r-1 + &c.</math> in datam finitam quantitatem <math>(A x^m + B x^m-1
+ C x^m-2 + &c.) × x^.</math>, e duobus præcedentibus ca$ibus con$tat
fluentem $eriei finitæ <math>(A x^m + B x^m-1 + C x^m-2 + &c.) × ({a / x - α} +
{b / x - β} + {c / x - γ} + &c.) × x^.</math> inveniri po$$e in finitis terminis, circula-
ribus arcubus & logarithmis; in his duobus ca$ibus fiant partes, quæ
involvunt circulares arcus & logarithmos, nihilo re$pective æquales;
& exinde erui po$$unt ca$us, in quibus $eries, $i modo unquam, evadet
finita quantitas; ergo $umma inventa finitæ $eriei æqualis erit $ummæ
infinitæ $eriei, i. e. fluenti fluxionis <math>x^. × (A x^m + B x^m-1 + C x^m-2 +
&c.) × (x^r + P x^r-1 + Q x^r-2 + &c.)</math>; & $ic deinceps.</p>
<h>PROB. XI.</h>
<p it>Datâ quantitate <math>{A z^m + B z^m-1 + C z^m-2 + &c. / z + α · z + β · z + γ · z + δ · &c.} × x^hz+m</math>, quæ ex-
primit $ingulos datæ $eriei terminos, ubi <math>z</math> denotat di$tantiam termini a
primo, literæ vero <math>A, B, C, D, &c., α, β, γ, δ, &c. h</math>, datas quantitates, &
<math>m</math> integrum numerum, re$pective denotant; invenire $ummam datæ $eriei.</p>
<p>1. Sit <math>v = {1 / 1 ∓ x^b} = 1 ± x^b + x^2b ± x^3b + x^4b ± &c.</math> ducatur data

<pb 520>[0572]<rh>DE SUMMATIONE</rh>
æquatio in <math>x^ba-1 x^.</math> & fit <math>{x^ba-1 x^. / 1 ∓ x^b} = x^ba-1 x^. ± x^ba+b-1 x^. + x^ba+2b-1 x^. ±
x^ba+3b-1 x^. + &c.</math> Inveniantur fluentes fluxionum ex utrâque æqua-
tionis parte, viz. <math>$. {x^hα-1 x^. / 1 ∓ x^b} = W = {x^hα / b α} ± {x^hα+h / b × (α + 1)} + {x^hα+2h / b × (α + 2)}
± &c.</math> ducatur hæc æquatio in <math>{x^hβ-1 x^. / x^hα}</math>, & re$ultat æquatio <math>W x^hβ-hα-1 x^.
= {x^hβ-1 x^. / b α} ± {x^hβ+h-1 x^. / b (α + 1)} + {x^hβ+2h-1 x^. / b × (α + 2)} ± &c.</math> inveniantur fluentes
fluxionum ex utrâque æquationis parte <math>X</math>, & evadunt <math>$. W x^hβ-hα-1 x^.
= X = {x^hβ / b^2 × α β} ± {x^hβ+h / b^2 × (α + 1) × (β + 1)} + {x^hβ+2h / b^2 × (α + 2) × (β + 2)}
± &c.</math> ducatur hæc æquatio in <math>{x^hγ-1 / x^hβ} x^.</math>, & re$ultat <math>X x^hγ-hβ-1 x^. =
{x^hγ-1 x^. / b^2 α β} ± {x^hγ+h-1 / b^2 × (α + 1) × (β + 1)} x^. + {x^hγ+2h-1 / b^2 × (α + 2) × (β + 2)} ± &c.</math>
inveniantur fluentes fluxionum ex utrâque æquationis parte, & erunt
<math>$. X x^hγ-hβ-1 = Y = {x^hγ / h^3 × α β γ} ± {x^hγ+h / h^3 × (α + 1) × (β + 1) × (γ + 1)} +
{x^hγ+2h / b^3 × (α + 2) × (β + 2) × (γ + 2)} ± &c.</math> ducatur hæc æquatio in
<math>{x^hδ-1 / x^hγ} x^.</math>, & re$ultat æquatio <math>Y x^hδ-hγ-1 x^. = {x^hδ-1 x^. / b^3 × α β γ} ± {x^hδ+h-1 x^. / b^3 × (α + 1) × (β + 1) × (γ + 1)}
+ {x^hδ+2h-1 x^. / b^3 × (α + 2) × (β + 2) × (γ + 2)} ± &c.</math> cujus
fluens e$t <math>$. r x^hδ-hγ-1 x^. = Z = {x^hδ / h^4 α β γ δ} ± {x^hδ+h / h^4 × (α + 1) × (β + 1) × (γ + 1) × (δ + 1)}
+ {x^hδ+2h / b^4 × (α + 2) × (β + 2) × (γ + 2) × (δ + 2)} ±
&c.</math> & $ic continuo repetitâ operatione, u$que donec includuntur in
denominatore omnes divi$ores <math>(z + α) · (z + β) · (z + γ) · (z + δ)
&c.</math> quorum ultimo inventus $it <math>z + π</math> & numerus <math>n</math>, & fluens <math>V</math>;

<pb 521>[0573]<rh>SERIERUM, &c.</rh>
ergo $umma $eriei cujus termini $emper $int <math>A × {1 / z + α · z + β · z + γ · z + δ &c. z + π}
x^bz+m</math> erit <math>b^n A × {x^m / x^hπ} × V</math>.</p>
<p>Per ex. prob. erit <math>X = $. W x^hβ-hα-1 x^. = {1 / b(β - α)} x^h(β-α) $. {x^hα-1 x^. / 1 ± x^b}
+ {1 / b(α - β)} $. {x^hβ-1 x^. / 1 ± x^b}, & Y = $. X x^hγ-hβ-1 x^. = {1 / b^2 (γ - α)(β - α)}
x^h(γ-α) $. {x^hα-1 x^. / 1 ± x^b} + {1 / b (γ - β)(α - β)} × x^h(γ-β) $. {x^hβ-1 x^. / 1 ± x^b} + {1 / (γ - α)(γ - β)}
$. {x^hγ-1 x^. / 1 ± x^b}</math>; & $ic deinceps.</p>
<p>Cor. Si omnes differentiæ <math>α - β, α - γ, β - γ; α - δ, β - δ,
γ - δ, &c.</math> $int integri numeri, tum fluentes omnium fluxionum
<math>{x^hα-1 x^. / 1 ± x^b}, {x^hβ-1 x^. / 1 ± x^b}, {x^hγ-1 x^. / 1 ± x^b}, &c.</math> a $e invicem pendent & deduci po$$unt;
& con$equenter $ummæ omnium $erierum, quarum generalis termi-
nus e$t <math>{(A z^m + B z^m-1 + &c.) × x^bz+m′ / (z + α)(z + β)(z + γ)(z + δ) &c.}</math>, a fluente fluxionis <math>{x^hα-1 x^. / 1 ± x^b}</math>
per divi$ionem, &c. deduci po$$unt.</p>
<p>Cor. 2. Si vero omnes reliqui indices <math>γ, δ, &c.</math> a duobus <math>α & β</math> di-
ftent per integros numeros, tum $ummæ $erierum inveniri po$$unt a
fluentibus fluxionum <math>{x^hα-1 x^. / 1 ± x^b} & {x^hβ-1 x^. / 1 ± x^b}</math>; & $ic deinceps.</p>
<p>Exhinc facile deduci po$$unt ca$us particulares, in quibus prædictæ
$eries æquant finitas quantitates.</p>
<p>Cor. Hinc ex finitis terminis, circularibus arcubus & logarithmis
deduci po$$unt $ummæ omnium $erierum, quarum termini $int præ-
dicti, ni duo <math>α & β</math> vel tres vel plures valores incognitarum quantita-
tum <math>α, β, γ, &c.</math> fint inter $e æquales, nam in i$to ca$u <math>α - β = 0, &c.</math>,
& con$equenter ejus reciproca evadit infinita: minime refert, annon
po$$ibiles vel impo$$ibiles $int prædictæ radices <math>α, β, γ, δ, &c.</math></p>

<pb 522>[0574]<rh>DE SUMMATIONE</rh>
<p>2. Nunc data $eries vel dividi pote$t in plures, quorum termini $int
re$pective <math>{A z^m x^bz+m′ / z + α · z + β · z + γ &c.}, {B z^m-1 x^bz+m′ / z + α · z + β · z + γ &c.},
{C z^m-2 x^bz+m′ / z + α · z + β · z + γ &c.}</math>, vel quocunque alio modo ita ut $umma e
fingularum $erierum terminis æqualis $it datæ quantitati; deinde in-
veniatur per $ub$equentem methodum $umma e $ingulis hi$ce $eriebus,
quibus ad unam $ummam adjunctis, ea erit $umma $eriei datæ quæ$ita.</p>
<p>2. Sit <math>V =</math> $umma $eriei, cujus termini $int <math>{1 / z + α · z + β · z + γ &c.}
x^bz</math>, i. e. <math>{1 / α · β · γ · δ} ± {1 / α + 1 · β + 1 · γ + 1 · δ + 1 · &c.} x^b +
{1 / α + 2 · β + 2 · γ + 2 · δ + 2 · &c.} x^2b + &c.</math> Inveniatur fluxio utri-
u$que æquationis partis, & re$ultat <math>V^. = {± h / α + 1 · β + 1 · &c.} x^h-1 x^. +
{2b / α + 2 · β + 2 · &c.} x^2b-1 x^. ± {3b / α + 3 · β + 3 · &c.} x^3b-1 x^. + &c.</math> du-
catur hæc æquatio in <math>{x / x^.}</math>, & re$ultat <math>{V^. x / x^.} = {± h / α + 1 · β + 1 · γ + 1 · &c.}
x^b + {2h / α + 2 · β + 2 · γ + 2 · &c.} x^2b ± {3h / α + 3 · β + 3 · γ + 3 · &c.} x^3b +
&c.</math> cujus flux. $it flux. quan. <math>{V^. x / x^.} = W^. = {± b^2 / α + 1 · β + 1 · γ + 1 · &c.} x^b-1 x^.
+ {2 · 2 h^2 / α + 2 · β + 2. &c.} x^2b-1 x^. ± {3 · 3b^2 / α + 3 · β + 3 · &c.} x^3b-1 x^. + &c.</math>
ducatur hæc æquatio in <math>{x / x^.}</math>, & re$ultat <math>{x W^. / x^.} = L = {± h^2 / α + 1 · β + 1 · &c.}
x^b + {2 · 2h^2 / α + 2 · β + 2 · &c.} x^2b ± {3 · 3h^2 / α + 3 · β + 3 · &c.} x^3b + &c.</math> inve-
niatur fluxio hujus æquationis, & re$ultat <math>L^. = {± h^3 / α + 1 · β + 1 · &c.}

<pb 523>[0575]<rh>SERIERUM, &c.</rh>
x^b-1 x^. + {2 · 2 · 2 h^3 / α + 2 · β + 2 · &c.} x^2b x^. + {3 · 3 · 3 h^3 / α + 3 · β + 3 · &c.} x^3b x^. + &c.</math> &
$ic continuo repetitâ operatone tandem invenietur $umma $eriei; cu-
jus terminus e$t <math>{A z^m / z + α · z + β · &c.}</math>; & $ic de reliquis.</p>
<p>Aliter: $it <math>A · (z + π) · (z + ρ) · (z + σ) · (z + τ) · &c. = A z^m
+ B z^m-1 + C z^m-2 + &c.</math> & $it $umma $eriei, cujus termini in genere
$int <math>{x^bz / z + α · z + β · z + γ · &c.}, V</math>, i. e. <math>V = {1 / α · β · γ · δ &c.} +
{1 / α + 1 · β + 1 · &c.} x^b + {1 / α + 2 · β + 2 · &c.} x^2b + &c.</math> ducatur hæc
æquatio in <math>x^πh</math>, & re$ultat <math>V × x^πh = {x^πh / α β γ δ &c.} + {x^πh+h / α + 1 · β + 1 · γ + 1 · &c.}
+ {x / α + 2 · β + 2 · γ + 2 · &c.} + &c.</math> inveniatur fluxio hujus æquatio-
nis, & erit flux. quan. <math>(V × x^πh) = W^. = {h π x^πh-1 x^. / α β γ δ &c.} + {b × (1 + π) x^hπ+h-1 x^. / α + 1 · β + 1. γ + 1 · &c.}
+ {b × (2 + π) x^hπ+2h-1 x^. / α + 2 · β + 2 · γ + 2 · &c.}</math>; ducatur hæc æquatio in
<math>{x^ρh / x^πh-1 x^.}</math>, & re$ultat <math>{W^. x^ρh / x^πh-1 x^.} = U = {b π x^ρh / α β γ δ &c.} + {b × (1 + π) x^ρh+h / α + 1 · β + 1 · γ + 1 · &c.}
+ {b × (2 + π) x^ρh+2h / α + 2 · β + 2 · γ + 2 · &c.}</math>, inveniatur fluxio utriu$que partis æqua-
tionis, & re$ultat flux. quan. <math>{W^. x^ρh / x^πh-1 x^.} = U^. = {b^2 π ρ x^ρh-1 x^. / α β γ δ &c.} + {b^2 × (1 + π) × (1 + ρ) x^ρh+h-1 x^. / α + 1 · β + 1 · γ + 1 · &c.}
+ {b^2 · (2 + π) · (2 + ρ) · x^ρh+2h-1 x^. / α + 2 · β + 2 · γ + 2 · &c.} + &c.</math> ducatur
hæc æquatio in <math>{x^σh / x^ρh-1 x^.}</math> & re$ultat <math>{U^. × x^σh / x^ρh-1 x^.} = M = {b^2 π ρ x^σh / α β γ δ &c.} +
{b^2 × (π + 1) · (ρ + 1) x^σh+h / α + 1 · β + 1 · γ + 1 · &c.} + {b^2 · (π + 2) · (ρ + 2) · x^σh+2h / α + 2 · β + 2 · γ + 2 · δ + 2 · &c.}</math>;<pb 524>[0576]<rh>DE SUMMATIONE</rh>
hujus æquationis inveniatur fluxio, & re$ultat <math>M^. = {h^3 π ρ σ x^σh-1 x^. / α β γ δ &c.} +
{h^3 · (π + 1) · (ρ + 1) · (σ + 1) x^σh+h-1 x^. / α + 1 · β + 1 · γ + 1 · δ + 1 · &c.} + {h^3 · (π + 2) · (ρ + 2) · (σ + 2) x^σh+2h-1 x^. / α + 2 · β + 2 · γ + 2 · δ + 2 · &c.}</math>
unde facile deduci pote$t $umma $eriei, cujus termini
$unt prædictæ datæ quantitates.</p>
<p>Si <math>α + z</math> vel <math>β + z</math> vel <math>γ + z, &c.</math> unquam evadant nihilo æqua-
les, tum in detegendis fluentibus terminorum prædictorum nece$$e
erit invenire fluentes fluxionum huju$ce formulæ <math>x^π x^. $. ({x^. / x})^n</math>, ubi <math>n</math>
denotat integrum numerum; hæ fluentes in libri primi capite $e-
cundo dantur.</p>
<p>Cor. 1. Hinc facile iterum con$tat, $i modo <math>α, β, γ, δ, &c.</math> $int inæ-
quales quantitates, $ummam cuju$cunque $eriei, cujus termini re$pective
denotantur per quantitatem <math>{A z^m + B z^m-1 &c. / H · (z + α) · (z + β) · (z + γ) · &c.}</math>, ubi
<math>m</math> e$t integer numerus; inveniri po$$e ope finitorum terminorum cir-
cularium arcuum & logarithmorum.</p>
<p>Cor. 2. Facile etiam e præcedentibus principiis deduci pote$t $um-
ma <math>m</math> terminorum inter quo$cunque datos terminos contenta, & e
principiis in medit. algeb. traditis con$equetur $umma terminorum,
quorum di$tantiæ a primo $eriei termino $int re$pective <math>n, n + m,
n + 2 m, n + 3 m, n + 4 m, &c</math>.</p>
<p>Cor. 3. Sit data $eries <math>A x^e + B x^e+n + Cx^e+2n + Dx^e+3n + &c.</math>, cu-
jus $umma $it functio quantitatis <math>x</math>, cujus fluxio inveniri pote$t; $it
<math>z</math> di$tantia a primo $eriei termino & a$$umatur functio <math>a z^m + b z^m-1
+ c z^m-2 + d z^m-3 + &c. = P</math>, ubi <math>m</math> e$t integer numerus; in fun-
ctione <math>P</math> pro <math>z</math> $cribantur <math>0, 1, 2, 3, 4, &c.</math> & re$ultent quantitates
<math>α, β, γ, δ, &c.</math>: tum ex datâ $erie $emper acquiri pote$t $umma $eriei
<math>α A z^e + β B z^e+n + γ C z^e+2n + &c</math>.</p>
<p>Cor. 4. Ducantur termini prædictæ feriei <math>A x^e + B x^e+n + C x^e+2n +

<pb 525>[0577]<rh>SERIERUM, &c.</rh>
&c</math>. re$pective in terminos <math>{β / α}, {β × (β + n) / α · (α + n)}, {β × (β + n)(β + 2n) / α(α + n)(α + 2n)}, &c.</math>;
& re$ultat $eries <math>{β / α} A x^e + {β × (β + n) / α (α + n)} B x^e+n + {β (β + n) (β + 2n) / α (α + n) (α + 2n)}
C x^e+2n + &c</math>., cujus $umma e $ummâ datæ $eriei $emper deduci pote$t,
$i <math>β = α + n × r, & r</math> $it integer numerus; nam in hoc ca$u prædicti
termini <math>{β / α}, {β / α} × {β + n / α + n}, &c.</math> evadunt functio <math>{(α + r n + n z)(α + r n + n z - n)
(α + r n + n z - 2n) · (α + r n + n z - 3 n) .. (α + n z + n) / α × (α + n) × (α + 2 n)(α + 3 n) · (α + 4 n) ... (α + (r - 1)n)}</math>,
ubi <math>z</math> e$t di$tantia a primo $eriei termino.</p>
<p>Cor. 5. Si fluens fluxionis <math>x^. × φ: x</math> detegi po$$it, tum $umma $eriei,
quæ e$t fluens fluxionis <math>x^. φ: (x) × (a z^m + b z^m-1 + &c.)</math> etiam expri-
mi pote$t.</p>
<h>PROB. XII.</h>
<p it>Datâ æquatione relationem inter <math>T, T′, T″, T′^n-1</math> $ucce$$ivos $eriei ter-
minos, & <math>z</math> di$tantiam a primo $eriei termino, $eriem ejus invenire. A$$u-
mantur pro <math>n</math> primis $eriei terminis re$pective <math>A, B, C, D, &c.</math> & e datâ
æquatione continuo inveniantur $ucce$$ivi termini quæ$itæ $eriei, & exinde
invenietur $eries quæ$ita.</p>
<h>PROB. XIII.</h>
<p it>Datâ æquatione relationem inter $ucce$$ivos $eriei terminos exprimente;
invenire æquationem, $ive ea algebraica $ive fluxionalis $it, ($i modo huju$ce
generis detur æquatio), cujus radix erit $eries quæcunque data, quæ de$i-
nitur æquatione, in quâ termini $eriei $unt unius tantum dimen$ionis.</p>
<p>1. Invenire æquationem, cujus radix e$t $eries <math>y = A + B x^m +
C x^2m + &c</math>. in quâ relatio coefficientium e$t con$tans, $cilicet <math>t D +
s C + r B + q A + &c. = 0</math>, ubi <math>A, B, C, D, &c.</math> $ucce$$ivos terminos,
& <math>q, r, s, &c.</math> invariabiles quantitates re$pective denotant.</p>

<pb 526>[0578]<rh>DE SUMMATIONE</rh>
<p><math>Sit <math>y = A + B x^m + C x^2m + D x^3m + E x^4m + &c. tum erit
 # t y = t A + t B x^m + t C x^2m + t D x^3m + t E x^4m + &c. &
s x^m y = # s A x^m + s B x^2m + s C x^3m + s D x^4m + &c.
r x^2m y = # r A x^2m + r B x^3m + r C x^4m + &c.
q x^3m y = # q A x^3m + q B x^4m + &c.
 # &c. # &c.</math>
$ed ex hypothe$i <math>t D + s C + r B + q A = 0, t E + s D + r C + q B
= 0</math>, & $ic in in$initum; unde evane$cent omnes $ub$equentes termini,
adeoque re$ultabit <math>(t + s x^m + r x^2m + q x^3m) y = t A + (t B + s A)
x^m + (t C + s B + r A) x^2m</math>.</p>
<p>2. Invenire fluxionalem æquationem exprimentem $eriem (<math>y</math>), cu-
jus $eriei æquatio relationem inter $ucce$$ivos (<math>T, T′, T″, &c.</math>) datæ
æquationis terminos de$ignans $it (<math>a z^α + b z^α-1 + c z^α-2 + &c.)
T x^m + (e z^β + f z^β-1 + g z^β-2 + &c.) T′ x^p±1m + (h z^γ + l z^γ-1 +
&c.) T″ x^p±2m + &c. = 0</math>, ubi <math>z</math> denotat di$tantiam a primo $eriei ter-
mino, & <math>a, b, c, &c. e, f, g, &c. h, l, &c</math>. con$tantes quantitates; etiam-
que <math>α, β, γ, &c.</math> integros numeros re$pective denotant.</p>
<p>Fingatur <math>y = A x^n + B x^n+m + C x^n+2m ... T x^n+rm + T′ x^n+(r+1)m +
T″ x^n+(r+2)m + &c.</math> nunc primo per methodum in prob. præced. tradi-
tam ex a$$umptâ æquatione <math>y = A x^n + B x^n+m + &c.</math> inveniatur
æquatio fluxionalis <math>y^. ^α + &c. =</math> $eriei, cujus generalis terminus $it
<math>(a z^z + b z^z-1 + &c.) T</math>; & $ic deducantur fluxionales æquationes,
quarum generales termini $erierum $int <math>(e z^β + f z^β-1 + &c.) T′, &c.</math>
& $ic deinceps; tum ita reducantur hæ fluxionales æquationes, i. e.
ducantur in tales dimen$iones quantitatis <math>x</math>, ut eædem ejus invenian-
tur dimen$iones in corre$pondentibus terminis (<math>T, T′, &c.</math>); & exinde
con$equentur fluxionales æquationes quæ$itæ.</p>
<p>Ex.. Requiratur fluxionalis æquatio, cujus radix $it <math>1 - {1 / 4} x^2 -
{3 / 64} x^4 - {3 / 256}x^6 - &c</math>. i. e. cujus relatio inter $ucce$$ivos terminos $it
<math>(2 z - 1) × (2 z + 1) T x^2 - (2 z + 2)^2 T′ = 0</math>; a$$umatur æquatio
<math>y = A + B x^2 + C x^4 + D x^6 + &c</math>. $ed quoniam factor <math>T</math> ductus

<pb 527>[0579]<rh>SERIERUM, &c.</rh>
fuit in <math>x^2</math>, con$equitur <math>m = 2</math>; & con$equenter omnes factores in
quantitates <math>T, T′, T″, &c.</math> ducti, qui $int <math>l z - α</math>, multiplicentur in
<math>{2 / l}</math>, ita ut evadant factores huju$ce generis <math>2 z - β</math>; factores <math>(2 z - 1,
2 z + 1, 2 z + 2)</math> vero in datâ æquatione contenti hanc habent for-
mulam $ine ullâ reductione.</p>
<p>Nunc inveniatur talis fluxionalis functio quantitatum <math>y & x</math> &
earum fluxionum, ut æqualis $it $eriei, cujus generalis terminus $it
<math>(2 z - 1) × (2 z + 1) × T</math>, ubi <math>T</math> $it generalis terminus $eriei a$-
$umptæ: per prob. prædic. ducatur æquatio a$$umpta in <math>x</math>, & re$ul-
tat <math>y x = A x + B x^3 + C x^5 + &c</math>. cujus fluxio erit <math>y x^. + x y^. = A x^.
+ 3 B x^2 x^. + 5 C x^4 x^. + &c. (2 z + 1) T x^2z x^.</math>, dividatur hæc æquatio
per <math>x</math> & evadet <math>{y x^. / x} + y^. = {A x^. / x} + 3 B x x^. + 5 C x^3 x^. + &c.</math> cujus fluxio
erit <math>{y^. x^. / x} - {y x^.^2 / x^2} + y^.. = - {A x^.^2 / x^2} + 3 B x^.^2 + 3 · 5 Cx^2 x^.^2 ... (2 z + 1) × (2 z - 1)
T x^2z-2 x^.^2</math>; ducatur hæc æquatio in <math>x^2</math> & evadet <math>(P) x x^. y^. - y x^.^2 +
x^2 y^.. = - A x^.^2 + 3 B x^2 x^.^2 + 3 · 5 C x^4 x^.^2 + &c.</math> & per eandem me-
thodum inveniatur <math>(Q) x y^.. + x^. y^. = 4 B x x^.^2 + 16 C x^3 x^.^2 ... 2 z ×
2 z T x^2z-1 x^.^2</math>; ducatur æquatio (<math>P</math>) in (<math>x</math>) talem dimen$ionem quanti-
tatis <math>x</math>, ut dimen$iones quantitatis <math>x</math> in terminum <math>T</math> æquationis <math>P</math>
ductæ inveniantur eædem ac dimen$iones quantitatis <math>x</math> in terminum
4ucce$$ivum <math>T′</math> æquationis (<math>Q</math>) ductæ unde re$ultat <math>- (x^2 x^. y^. - x y x^.^2
+ x^3 y^.. - (x y^.. + x^. y^.)) = (A + 4 B) x x^.^2 + (16 C - 3 B)x^3 x^.^2 ...
((2 z - 1) × (2 z + 1) T - (2 z + 2)^2 T′) x^2z+1 = 0</math>, unde fluxionalis
æquatio quæ$ita erit <math>(x^3 - x) y^.. - (1 - x^2) y^. x^. - x y x^.^2 = 0</math>.</p>
<p>Cor.. Ordo fluxionalis æquationis quæ$itæ $emper æquat indicem
vel <math>α</math> vel <math>β</math> vel <math>γ, &c.</math> ubi <math>α, β, γ, &c.</math> integri $int numeri, qui inve-
nitur maximus, & data $eries erit ejus particularis re$olutio.</p>
<p>Et ex datâ particulari re$olutione deprimi pote$t quæ$ita æquatio
ad inferiorem.</p>
<p>3. Si vero numerus factorum functionis quantitatis <math>z</math> in datâ
æquatione relationem inter $ucce$$ivos terminos <math>T, T′, &c.</math> exprimente

<pb 528>[0580]<rh>DE SUMMATIONE</rh>
continuo augeatur, tum omnino ex ii$dem principiis petenda e$t
$olutio.</p>
<p>4. Si vero diver$æ pote$tates $ub$equentium terminorum in datâ
æquatione contineantur, tum a$$umendo $eriem pro $ummis & ex eâ
deducendo $ingulos datæ æquationis terminos; & corre$pondentes
re$ultantis æquationis terminos inter $e æquando deduci pote$t $eries
quæ$ita.</p>
<h>PROB. XIV.</h>
<h it>1. Datis duabus vel pluribus convergentibus $eriebus <math>S, S′, S″, &c.</math> ex
iis deducere $eries, quarum con$equentur $ummæ.</h>
<p>Sint <math>α, β, γ, δ, &c.</math> quicumque $ucce$$ivi termini ex unâ $erie; <math>π, ρ,
σ, τ, &c.</math> ex alterâ, &c. tum erit <math>a α + b β + c γ + d δ + &c. k π +
l ρ + m σ + n τ + &c.</math> terminus generalis $eriei, cujus $umma facile
erui pote$t e $ummis datarum convergentium $erierum, ubi <math>a, b, c, d,
&c. k. l, m, &c.</math> $unt quæcumque finitæ quantitates.</p>
<p>2. Si vero quædam e prædictis $eriebus <math>S, S′, &c.</math> haud $int con-
vergentes, $ed earum termini continuo ad nihil vergant, & ultimo
minores $int quam quævis a$$ignandæ quantitates; <math>S″, S′″, &c.</math> vero
$int convergentes $eries; tum erui pote$t $umma $eriei, cujus genera-
lis terminus e$t <math>a α + b β + c γ + &c. + k π + l ρ + m σ + &c. +
&c. + A p + B q + &c. + D r + E s + &c.</math> ubi <math>α, β, γ, &c. π, ρ, σ,
&c. &c.</math> $int termini $ucce$$ivi $erierum datarum haud convergen-
tium; $ed <math>p, q, &c. r, s, &c.</math> termini $ucce$$ivi $erierum convergen-
tium; etiamque $int <math>a, b, c, &c. k, l, m, &c. &c. A, B, &c. D, E, &c.</math>
quæcunque finitæ quantitates, ita vero ut in coefficientibus $erierum,
quæ haud convergunt, fingantur <math>a + b + c + &c. = 0, k + l + m
+ &c. = 0</math>.</p>
<p>Si modo $eries progrediatur $ecundum dimen$iones quantitatis <math>x</math>,
tum <math>a, b, c, &c. k, l, m, &c. A, B, C, &c. D, E, &c.</math> $int functiones
quæcunque ip$ius <math>x</math>.</p>
<p>Hæc etiam principia applicari po$$unt ad $ummas $erierum, quæ
$unt tales functiones datarum convergentium $erierum, quæ nullas in

<pb 529>[0581]<rh>SERIERUM, &c.</rh>
$e reciprocas pote$tates involvunt; vel convergentes $eries præbent,
Inveniantur enim functiones datarum $erierum & quantitatum, qui-
bus hæ $eries æquales $unt, & erunt $ummæ $erierum re$ultantium
æquales $ummæ quantitatum re$ultantium e prædictis quantitatibus.</p>
<p>Cor. 1. Ex primis terminis $erierum, quarum termini ad nihil ver-
gunt; & ex datis primis terminis, & $ummis convergentium $erie-
rum, quæ haud in quantitates $imul $umptas nihilo æquales ducun-
tur, facile innote$cet $umma $eriei quæ$itæ.</p>
<p>Cor. 2. Datis $ummis quarumcunque $erierum, facile plerumque
detegi pote$t, utrum $umma cuju$cunque $eriei ex iis per methodos
prius traditas deduci pote$t, necne: ex ob$ervatis factoribus in deno-
minatore prædictæ $eriei & $erierum, quarum dantur $ummæ, con-
$equitur $ub$titutio, e quâ erui pote$t in $ummis datarum $erierum
$umma $eriei quæ$itæ.</p>
<p>Cor. 3. Ex datâ $erie <math>y = x + {1 / 2} x^2 + {1 / 3} x^3 + {1 / 4} x^4 + &c.</math> erui pote$t
$umma cuju$cunque $eriei, cujus generalis terminus e$t <math>± {A z^b + B z^b-1 + &c. / z × (z + m) × (z + n) × (z + p) × &c.}x^z</math>;
ubi <math>b, m, n, p, &c.</math> $unt integri nu-
meri: erunt enim <math>{1 / z}, {1 / z + m}, {1 / z + n}, &c.</math> diver$i termini datæ $eriei;
& con$equenter <math>{a x^z / z} + {b x^m+z / z + m} + {c x^n+z / z + n} + {d x^p+z / z + p} = a α + b β
+ c γ + d δ + &c.</math> ubi <math>α, β, γ, δ, &c.</math> $int termini ad di$tantias a primo
<math>m - 1, n - 1, p - 1, &c.</math> fiat <math>{a x^z / z} + {b x^z+m / z + m} + {c x^z+n / z + n} + &c. = {A z^b + B z^b-1 + &c. / z · z + m
z + n · z + p. &c.} x^z</math>, & inveniantur coe$$icientes <math>a, b, c, &c.</math> & id,
quod requiritur, exinde facile erui pote$t.</p>
<h>THEOR. XVI.</h>
<p>1. Ex fluentibus fluxionum, quæ finitis terminis exprimi po$$unt,
acquiri po$$unt $eries quantitatum, quarum $ummæ innote$cunt.</p>

<pb 530>[0582]<rh>DE SUMMATIONE</rh>
<p>Reducantur datæ fluxiones ad infinitas $eries $ecundum terminos,
quorum fluentes finitis terminis exprimi po$$unt, progredientes; in-
veniantur fluentes terminorum re$ultantium, & re$ultant $eries, qua-
rum $ummæ innote$cunt.</p>
<p>Ex. Sit fluxio <math>(a + b x^n)^{b / m} x^rn-1 x^.</math>, cujus fluens (exceptis excipien-
dis) in finitis terminis exprimi pote$t, cum <math>r</math> $it integer numerus; re-
ducatur data fluxio <math>(a + b x^n)^{b / m} x^rn-1 x^.</math> ad $eriem <math>a^{b / m} (x^rn-1 x^. (1 + {b b / m a}
x^n + {b / m} × {b - m / 2 m} × {b^2 / a^2} x^2n + &c.)</math> $ecundum dimen$iones quantitatis <math>x^n</math>
progredientem, cujus fluens erit <math>a^{b / m} x^rn ({1 / r n} + {1 / (r + 1) n} × {b b / m a}x^n +
{1 / (r + 2)n} × {b / m} × {b - m / 2 m} × {b^2 / a^2} x^2n + {1 / (r + 3) n} × {b / m} × {b - m / 2 m} × {b - 2 m / 3 m} ×
{b^3 / a^3} x^3n + &c.)</math>; hæc fluens autem innote$cit, ergo $umma $eriei <math>{1 / r n} +
{1 / (r + 1) n} × {b / m} × {b / a} x^n + {1 / (r + 2) n} × {b / m} × {b - m / 2 m} × {b^2 / a^2}x^2n + {1 / (r + 3) n}
× {b / m} × {b - m / 2 m} × {b - 2 m / 3 m} × {b^3 / a^3} x^3n + &c.</math> $emper finitis terminis ex-
primi pote$t.</p>
<p>1. 2. Si nova $eries incipiat ab <math>(l + 1)</math> termino datæ $eriei, tum
evadet <math>{b^l / a^l} x^ln × {b / m} × {b - m / 2 m} · {b - 2 m / 3 m} ... {b - (l - 1) m / l m} ({1 / (r + l) n} +
{1 / (r + l + 1)n} × {b - l m / (l + 1) m} × {b / a} x^n + {1 / (r + l + 2) n} × {b - l m / (l + 1)m} · {b - (l + 1)m / (l + 2) m}
× {b^2 / a^2} x^2n + &c.)</math>, ergo $umma $eriei <math>{1 / (r + l)n} + {1 / (r + l + 1)n} × {b - l m / (l + 1)m}
× {b / a}x^n + {1 / (r + l + 2)n} × {b - l m / (l + 1) m} × {h - (l + 1) m / (l + 2) m} × {b^2 / a^2} x^2n + &c.</math> cum
<math>r & l</math> $int integri numeri, in $initis terminis exprimi pote$t.</p>
<p>1. 3. Per cap. 3. medit, algebr, erui pote$t in finitis $umma e termini

<pb 531>[0583]<rh>SERIERUM, &c.</rh>
prædictæ $eriei ad <math>k</math> di$tantiam a $e po$itis, e. g. <math>{1 / (r + l) n} + {1 / (r + l + k) n}
× {h - l m / (l + 1) m} × {h - (l + 1) m / (l + 2) m} · {b - (l + 2) m / (l + 3) m} ... {h - (l + k - 1) m / (l + k) m} ×
{b^k / a^k} x^kn + {1 / (r + l + 2k)n} × {b - l m / (l + 1) m} · {h - (l + 1) m / (l + 2) m} ... {h - (l + 2 k - 1) m / (l + 2 k) m}
× {b^2k / a^2k} x^2kn + &c.</math></p>
<p>1. 4. Si requiratur ut $eries progrediatur $ecundum dimen$iones
quantitatis <math>x^-n</math>, tum evadet <math>b^{b / m} x^(r+{b / m},n ({1 / (r + {h / m}) n} + {1 / (r + {h / m} - 1) n} × {b / m} ×
{a / b x^n} + {1 / (r + {h / m} - 2)n} × {h / m} · {h - m / 2 m} × {a^2 / b^2 x^2n} + &c.) = $. x^rn-1 x^. (b x^n + a)^{b / m}</math>:
& $ic detegi pote$t $umma $eriei, quæ ab <math>(l + 1)</math> termino huju$ce $e-
riei incipiat; vel quæ erit $umma $ingulorum ad <math>k</math> di$tantiam a $e in-
vicem po$itorum terminorum eju$dem $eriei.</p>
<p>In his $eriebus pro <math>h</math> $cribatur <math>-h</math>, & evadunt termini $eriei alter-
natim affirmativi & negativi, & <math>h - m, h - 2 m, &c.</math> fiunt <math>- h - m,
- h - 2 m, &c.</math></p>
<p>1. 5. <math>Si - {h / m} - r</math> $it integer affirmativus numerus, tum ob <math>(a +
b x^n)^{b / m} x^rn-1 x^. = (a x^-n + b)^{b / m} x^{b / m}n+rn-1 x^.</math> pro <math>r</math> in $eriebus prius traditis
$cribatur <math>+ {b / m} + r</math>, & pro <math>a</math> $cribatur <math>b</math> & pro <math>b, a; & pro x^n, x^-n</math>; &
re$ultant $eries eju$dem formulæ, quarum $ummæ innote$cunt.</p>
<p>Ex. 2. Fluens fluxionis <math>(a + b x^n)^{b / m} x^rn-1 x^.</math> per in$initam $eriem $ic
exprimi pote$t <math>{1 / r n} x^rn × (a + b x^n)^{b / m} × (1 - {h / m} × {1 / (r + 1)} × {b x^n / a + b x^n} +
{h / m} · {h - m / m} × {1 / r + 1} · {1 / r + 2} · {b^2 x^2n / (a + b x^n)^2} - {h / m} · {h - m / m} · {h - 2m / m} × {1 / r + 1}

<pb 532>[0584]<rh>DE SUMMATIONE</rh>
· {1 / r + 2} · {1 / r + 3} × {b^3 x^3n / (a + b x^n)^3} + &c.)</math>; unde $i <math>r</math> $it integer numerus
(prædictis exceptis) $umma huju$ce $eriei in finitis terminis ex-
primi pote$t.</p>
<p>Et per transformationes prius traditas plures $eries erui po$$unt.</p>
<p>2. Ex fluentibus fluxionum, quæ exprimi po$$unt ex finitis ter-
minis, circularibus, hyperbolicis & ellipticis arcubus, logarithmis &
fluentibus datarum fluxionum, inveniri po$$unt $eries, quarum $ummæ
per prædictas fluentes exprimi po$$unt.</p>
<p>Series inveniri po$$unt ex reducendo datas fluxiones ad infinitas
$eries & earum fluentes inveniendo, ut in priori ca$u docetur.</p>
<p>Ex. Sit fluxio <math>(a + b x^n)^±{1 / 2} × x^rn+{1 / 2}n-1 x^.</math>, ubi <math>r</math> e$t integer numerus
vel affirmativus vel negativus, cujus fluens per $eriem $ic exprimi
pote$t <math>x^rn+{1 / 2}n ({2 / (2 r + 1) n} a^±{1 / 2} ± {1 / 2} × {2 / (2 r + 3) n} a^±{1 / 2}-1 b x^n ± {1 / 2} ×
{± 1 - 2 / 2 · 2} × {2 / (2 r + 5) n} × a^±{1 / 2}-2 b^2 x^2n ± {1 / 2} × {± 1 - 2 / 2 · 2} × {± 1 - 4 / 2 · 3} ×
{2 / (2 r + 7) n} × a^±{1 / 2}-3 b^3 x^3n ± ... ± {1 / 2} × {± 1 - 2 / 2 × 2} × {± 1 - 4 / 2 · 3} × {± 1 - 6 / 2 · 4}
... {± 1 - 2 z / 2 · z + 1} × {2 / (2 r + 2 z + 3)n} × a^±{1 / 2}-z-1 b^z+1 x^(z+1)n + &c.)</math></p>
<p>Et $imiliter reducenda e$t fluens ad $eriem $ecundum reciprocas
dimen$iones quantitatis <math>x^n</math> progredientem.</p>
<p>Ex. Sit fluxio <math>(a + b x^n)^±{1 / 3} vel {2 / 3} vel {1 / 4} vel {3 / 4} × x^rn-{1 / 3}nvel{1 / 4}nvel{1 / 2}n&c. x^.</math>, re-
ducatur hæc quantitas ad terminos $ecundum dimen$iones quan-
titatis <math>x^n</math> vel ejus reciprocas progredientes; & re$ultabit $eries, cujus
$umma per ellipticos, hyperbolicos, arcus, &c. innote$cit.</p>
<p>Eadem principia etiam ad fluentes omnium fluxionum applicari
po$$unt.</p>
<p>Ex. A$$umatur fluxio <math>(a + b x^2)^±{1 / 2}+m × x^r x^.</math>, ubi literæ <math>m & r</math> in-
tegros denotant numeros vel affirmativos vel negativos, cujus fluens
per algebraicam functionem circularium arcuum, logarithmorum &

<pb 533>[0585]<rh>SERIERUM, &c.</rh>
finitorum terminorum exprimi pote$t; reducatur fluxio ad infinitam
$eriem, & inveniatur ejus fluens, quæ erit <math>a^m±{1 / 2} x^r+1 ({1 / r + 1} + (m ± {1 / 2})
× {r + 1 / r + 3} × A × {b / a} x^2 + {m ± {1 / 2} - 1 / 2} × {r + 3 / r + 5} × B × {b^2 / a^2} x^4 + {m ± {1 / 2} - 2 / 3}
× {r + 5 / r + 7} × C × {b^3 / a^3} x^6 + {m ± {1 / 2} - 3 / 4} × {r + 7 / r + 9} × D × {b^4 / a^4} x^8 + ... + P × {b^l / a^l}
× x^2l ({m ± {1 / 2} - l + 1 / l} × {r + 2 l - 1 / r + 2 l + 1} + {m ± {1 / 2} - l / l + 1} × {r + 2 l + 1 / r + 2 l + 3} × Q {b / a} x^2
+ {m ± {1 / 2} - l - 1 / l + 2} × {r + 2 l + 3 / r + 2 l + 5} × R × {b^2 / a^2} x^4 + &c.))</math>, ubi literæ <math>A, B, C,
&c.; P, Q, R, &c.</math> denotant coefficientes præcedentium terminorum
& <math>l</math> e$t integer affirmativus numerus.</p>
<p>Cor. Hinc innote$cit $umma $eriei <math>{m ± {1 / 2} - l + 1 / l} × {r + 2 l - 1 / r + 2 l + 1} +
{m ± {1 / 2} - l / l + 1} × {r + 2 l + 1 / r + 2 l + 3} × Q × {b / a} x^2 + {m ± {1 / 2} - l - 1 / l +2} × {r + 2 l + 3 / r + 2 l + 5} × R × {b^2 / a^2}
x^4 + &c.</math>, ubi <math>Q, R, &c.</math> prædecentes coefficientes denotant.</p>
<p>Ducantur termini huju$ce $eriei in functionem <math>e + $ z + g z^2 +
h z^3 + &c.</math> quantitatis <math>z</math> di$tantiæ a primo $eriei termino, i. e. ducan-
tur primus terminus in <math>e</math>, $ecundus in <math>e + f + g + h + &c.</math>, tertius in
<math>e + 2 f + 4 g + 8 h + &c.</math>, quartus in <math>e + 3 f + 9 g + 27 h + &c.</math>; & $ic de-
inceps; & re$ultat $eries, cujus $umma innote$cit: vel ducantur termi-
ni $ucce$$ivi datæ $eriei in terminos <math>{β / α}, {β / α} × {β + n / α + n}, {β / α} × {β + n / α + n} × {β+ 2 n / α + 2 n^3},
&c.</math>, & re$ultat $eries <math>{m ± {1 / 2} - l + 1 / l} × {r + 2 l - 1 / r + 2 l + 1} × {β / α} + {m ± {1 / 2} - l / l + 1} ×
{r + 2 l + 1 / r + 2 l + 3} × {β / α} × {β + n / α + n} × Q × {b / a} x^2 + &c.</math>, cujus $umma innote$-
cit; $i <math>β - α = m′ n</math>, ubi <math>m′</math> e$t integer affirmativus numerus: nam
$eries <math>{β / α}, {β / α} · {β + n / α + n}, ... {β · β + n · β + 2 n ... β + z n / α · α + n · α + 2n ... α + z n} =<pb 534>[0586]<rh>DE SUMMATIONE</rh>
{β + z n.β + (z - 1) n.β + (z - 2) n ... β + (z - m′ + 1)n / α · α + n · α + 2 n ... α + (m′ - 1)n}</math> præ-
dicta functio quantitatis <math>z</math>.</p>
<p>Ad eundem modum ex reducendo fluxionem <math>(b x^2 ±a)^m±{1 / 2} ×
x^r ^.</math> ad $eriem <math>b^m±{1 / 2} x^2m±1+r+1 ({1 / 2 m ± 1 + r + 1} + (m ± {1 / 2}) ×
{2 m ± 1 + r + 1 / 2 m ±1 + r - 1} × A × {a / b x^2} + &c.)</math>, ubi _A_, &c. præcedentes coeffi-
cientes denotat, & principiis prius traditis erui po$$unt aliæ $eries,
quarum $ummæ innote$cunt.</p>
<p>Cor. Sit $eries <math>a + bx^n + cx^2n + dx^3n + &c. = S</math>, cujus fluxio
deduci pote$t; tum ex eâ erui pote$t $umma cuju$cunque $eriei <math>a ×
{α / β} × {α′ / β′} × {α″ / β″} × {α″′ / β″′}
× &c. + b × {α + n / β + n} × {α′ + n / β′ + n} × {α″ + n / β″ + n}
× {α″′ + n / β″′ + n} × &c.
× Ax^n + c × {α + 2n / β + 2n} × {α′ + 2n / β′ + 2n} × {α″ + 2n / β″ + 2n} × {α″′ + 2n / β″′ + 2n} × &c. × Bx^2n +
d × {α + 3n / β + 3n} × {α′ + 3n / β′ + 3n} × {α″ + 3n / β″ + 3n} × {α″′ + 3n / β″′ + 3n} × &c. × Cx^3n + &c.</math>, ubi
<math>β - α,β′ - α′,β″ - α″,β″′ - α″′</math>, &c. per _n_ divi$æ integri $unt affir-
mativi numeri & literæ _A,B,C,D,_ &c. præcedentes coefficientes
re$pective denotant.</p>
<h>PROB. XV.</h>
<p>_Ex datâ convergente $erie, vel $erie cujus termini ultimo nibilo fiunt æqua-_
_les, & cujus termini $ecundum dimen$iones quantitatis <math>x</math> progrediuntur, viz._
<math>(a x^s + b x^r+s + c x^2r+s + d x^3r+s + &c.)</math> _ita ut ad infinitam di$tan-_
_tiam dimen$iones evadant infinite magnæ vel negativæ vel affirmativæ, &_
_cujus $umma_ (S) _datur, $i modo $eries $it convergens; infinitas alias con$i-_
_miles detegere._</p>
<p>Ducatur data $eries in quotcunque quantitates huju$ce generis
<math>x^z x^.</math>; fluxionum re$ultantium inveniantur fluentes, viz. <math>$. S x^α x^.</math>; &
$ic ducantur $eries re$ultantes in quantitates huju$ce formulæ <math>x^β x^.</math>,

<pb 535>[0587]<rh>SERIERUM, &c.</rh>
& inveniantur fluxionum re$ultantium <math>x^β x^. $. S x^z x^.</math> fluentes,
& $i $eries data $it convergens, tum a fortiori $eries re$ultans etiam
erit convergens; & $ic deinceps: deinde ducatur data $eries in
quantitates huju$ce formulæ <math>x^π</math>, & inveniantur fluxiones quantitatum
re$ultantium, dividantur hæ fluxiones per <math>x^.</math>, & re$ultant novæ $eries,
quarum $ummæ erunt <math>({πx^π-1 Sx^. + x^π S^. / x^.})</math>, $i modo hæ re$ultantes
$eries convergant. Hæ re$ultantes $eries in pleri$que ca$ibus conver-
gent, $i modo data $eries convergat. Et $ic de fluxionibus novarum
$erierum, &c. u$que donec hæ $eries evadant tales, ut termini ad in-
finitam di$tantiam haud nihilo fiant æquales, tum ex omnibus his
$eriebus per prob. præced. inveniri po$$unt $eries, quarum $ummæ in-
note$cunt.</p>
<p>Et $ic e datis quibu$cunque $eriebus prædicti generis, facile deduci
po$$unt infinitæ aliæ per methodos in prob. præc. traditas, quarum
$ummæ innote$cunt; vel ex multiplicando datas $eries in qua$cunque
quantitates, ita ut re$ultent $eries quarum fluxiones vel fluentes vel
integrales vel incrementa producant convergentes $eries.</p>
<p>Et vice versâ ex principiis prius traditis inve$tigari pote$t, utrum
$umma datæ $eriei e $ummis quarumque $erierum datis inve$tigari
po$$it, nence.</p>
<p>Ex. 1. Sit æquatic data <math>S = 1 + {1 / 1 · 2}x + {1 / 1 · 2 · 3}x^2 + {1 / 1 ·2 · 1 · 4}
x^3 + &c.</math> ducatur hæc $eries in <math>1 - x</math>, & re$ultat $eries <math>(1 - x) × S
= 1 - {1 / 1 · 2}x - {1 / 1 · 3}x^2 - {1 / 1 · 2 · 4}x^3 - {1 / 1 · 2 · 1 · 5}x^4 - &c.</math> in infi-
nitum; $it <math>x = 1</math>, & evadet <math>1 = {1 / 2} + {1 / 1 × 3} + {1 / 1 · 2 × 4} + {1 / 1 · 2 · 3 × 5}
+ {1 / 1 · 2 · 1 · 4 × 6} + &c.</math> & $ic de con$imilibus e præcedentibus prin-
cipiis deducendis.</p>
<p>Ducatur data æquatio in <math>a + bx + cx^2 + dx^3 + &c. = 0</math>, & re-
$ultat $eries <math>a + ({1 / 2}a + b)x + ({1 / 1 · 2 · 3}a + {1 / 1 · 2}b + c)x^2 + &c. = 0.</math></p>

<pb 536>[0588]<rh>DE SUMMATIONE</rh>
<p>Ex. 2. Sit $eries <math>(1 - x^2)^-m = 1 + m x^2 + m.{m + 1 / 2}x^4 + m.{m + 1 / 2} · {m
+ 2 / 3} x^6 + &c.</math> ducatur æquatio in <math>x^.</math>, & inveniatur ejus fluens, &
re$ultat <math>x + {mx^3 / 3} + {m. m + 1 × x^5 / 1 · 5} + {m. m + 1. m + 2. x^7 / 1 · 3 · 7} + &c.
= $.{x^. / (1 - x^2)^m};</math> $it <math>m = {1 / 2}</math>, & erit <math>$.{x^. / √(1 - x^2)} = x + {x^3 / 1 · 3} +
{1 · 3 · 3 · x^5 / 1 · 2 · 3 · 4 · 5} + {1 · 3 · 3 · 5 · 5 x^7 / 1 · 2 · 3 · 4 · 5 · 6 · 7}
+ {1 · 3 · 3 · 5 · 5 · 7 · 7 x^9 / 1 · 2 · 3 · 4 · 5 · 6 · 7 · 8 · 9} + &c.</math>
Ducantur $ucce$$ivi termini huju$ce $eriei re$pective in quamcunque
algebraicam & integralem functionem quantitatis <math>z</math> di$tantiæ a primo
$eriei termino. E. g. Ducantur termini re$pective in terminos <math>{β / α},
{β.(β + n) / α.(α + n)},{β.β + n.β + 2n / α.α + n.α + 2n}</math>, &c., $i <math>{β - α / n}</math> $it integer numerus;
tum re$ultabit $eries, cujus $umma e prædictâ fluente deduci pote$t:
$i ex termino ad di$tantiam (_l_) a primo incipiat $eries, cujus termini
facile exprimi po$$unt per functionem vel algebraicam vel fluxiona-
lem vel incrementialem di$tantiæ a primo $eriei termino; e medit. al-
gebr. deduci pote$t $umma e $ingulis alternis, & terminis $eriei ad (_k_)
di$tantiam a $e po$itis. Ex his facile deduci po$$unt infinitæ aliæ, &c.</p>
<h>PROB. XVI.</h>
<p it>Datis $eriebus formularum, quæ haud per algebraicas vel fluxionales vel
incrementiales æquationes exprimi po$$unt; invenire, annon earum $ummæ
finitis terminis exprimi po$$unt.</p>
<p>Quæratur, annon pote$t e$$e $eries continuo ad nihil vergens & ul-
timo ad id accedens, quæ datæ $eriei fiat æqualis; $i modo $eries in-
venta ablatis quibu$dam ejus terminis auferatur a $e, & $ic deinceps;
vel quod magis generale erit, quæ datæ $eriei per qua$dam quantita-

<pb 537>[0589]<rh>SERIERUM, &c.</rh>
tes auctæ vel diminutæ æqualis erit, $i modo $eries prædicti generis
ducatur in quantitatem inve$tigandam nihilo æqualem.</p>
<p>Ex. 1. Sit $eries, <math>{2 · 3 ... (n) - 1 / 1.2.3... n} + {3 · 4 ... (n + 1) - 1 / 1 · 2 · 3 ... (n + 1)} +
{4 · 5 ... (n + 2) - 1 / 1 · 2 · 3 ... (n + 2)} + &c.</math> cujus generalis terminus erit <math>{(z + 2). / 1 · 2}
{(z + 3) ... (z + n) - 1 / · 3 .. (z + n)}</math>.</p>
<p>Facile con$tat e denominatoribus, &c. datæ $eriei ut $eries quæ$ita
$it <math>1 + {1 / 1 · 2} + {1 / 1 · 2 · 3} + {1 / 1 · 2 · 3 · 4} + &c.</math> & con$equenter $umma
datæ $eriei quæ$ita erit <math>1 + {1 / 1 · 2} + {1 / 1 · 2 · 3} ... + {1 / 1 · 2 · 3 ..(n - 1)}.</math></p>
<p>Ex. 2. Sit $eries <math>{e^2n + n^2 - 1 / e^(1 + n)^2} + {e^4n + n^2 - 1 / e^(2 + n)^2} + {e^6n + n^2 - 1 / e^(3 + n)^2} + &c.</math> & $e-
ries quæ$ita erit <math>{1 / e} + {1 / e^4} + {1 / e^9} + {1 / e^16} + &c.</math> & con$equenter $umma
$eriei datæ quæ$ita erit <math>{1 / e} + {1 / e^4} + {1 / e^9} + {e / e^16}...{1 / e^n^2}</math>.</p>
<p>Ex ob$ervatâ lege quam habent termini datæ $eriei plerumque fa-
cile erui pote$t $eries quæ$ita.</p>
<h>PROB. XVII.</h>
<h it>1. Invenire, annon datæ $eriei $umma e pluribus aliis datis & finitis
$eriebus inve$tigari pote$t.</h>
<p>Inve$tigetur, annon data $eries $it quæcunque directa functio $erie-
rum datarum & finitarum quantitatum, & perficitur problema.</p>
<p>2. Datâ particulari fluente, infinitæ $unt diver$æ fluxionales æqua-
tiones, quarum ea e$t re$olutio, & quarum diver$æ $unt generales re-
$olutiones; etiamque hoc verum erit in $eriebus, quæ particulares

<pb 538>[0590]<rh>DE SUMMATIONE</rh>
fluentes re$pective denotant, quamvis for$an vulgaris methodus unam
$olummodo deducat.</p>
<p>Ex. 1. Sit <math>y = z + 1 · 2z^2 + 1 · 2 · 3z^3 + &c.</math> $eries, quæ $emper
divergit; tum erit <math>yz = z^2 + 1 · 2z^3 + &c.</math> & exinde <math>{(yz^. + zy^.) / z^.}x
z = 1 x 2 z^2 + 1 x 2 · 3z^3 + &c. = y - z</math>, unde <math>y z^. + z y^. = {y - z / z}z^.</math>
una æquatio, cujus particularis re$olutio e$t prædicta $eries; huju$ce
fluxionalis æquationis innote$cit generalis re$olutio: inveniatur igi-
tur generalis re$olutio, & ita corrigatur ut fiat prædicta particularis
re$olutio <math>(P = 0)</math>, deinde quantitati _P_ addatur quæcunque functio
<math>Z</math> quantitatum <math>y</math> & <math>z</math>, quæ nihilo evadat æqualis, cum <math>P = 0</math>, tum
erit <math>P + Z = 0</math> æquatio, cujus particularis re$olutio erit data $e-
ries; &c.</p>
<p>Aliter: a$$umatur data æquatio <math>y = z + 1 · 2z^2 + &c.</math> aucta vel
diminuta per qua$cunque functiones quantitatum <math>z</math> & <math>y</math> ductas in di-
rectas pote$tates novæ invariabilis quantitatis _A_ ad libitum a$$umendæ;
deinde inveniatur per methodum prius traditam fluxionalis æquatio,
cujus æquatio a$$umpta $it generalis fluens, & erit fluxionalis æqua-
tio quæ$ita.</p>
<h>PROB. XVIII.</h>
<h it>Invenire infinitas $eries, quæ $int generales re$olutiones quæ$itarum
fluxionalium æquationum.</h>
<p>A$$umantur quantitates tot invariabiles quantitates ad libitum a$-
$umendas involventes, quot $it ordo fluxionalis æquationis quæ$itæ,
quæ facile reduci po$$unt ad infinitas $eries; harum quantitatum in-
veniantur fluxionales æquationes, quarum generales $int fluentes, &
perficitur prob.</p><pb 539>[0591]<rh>SERIERUM, &c.</rh>
<h>PROB. XIX.</h>
<p it>Invenire, quando termini datœ $eriei fiant maximi. Inveniantur,
quando proximi termini datæ $eriei fiant æquales, & exinde con$tabit,
quando fiant maximi; unde fiant maximi, cum $eries nec convergat nec
divergat; vel quod idem e$t, cum fluxio termini nibilo $it æqualis.</p>
<p>2. Invenire punctum inflexûs in datâ $erie, cujus termini vel re-
lationes inter terminos dantur.</p>
<p>A$$umantur tres termini in genere $ucce$$ivi <math>Q, R & S</math>, $upponan-
tur <math>Q - R = R - S</math>, & exinde detegentur quantitates <math>Q, R & S</math>, &
invenitur punctum inflexûs quam proxime; & idem deduci pote$t e
principiis fluxionum prius traditis.</p>
<p>Et $ic de inveniendis punctis vel terminis, in quibus con$i$tat quæ-
cunque data relatio: a$$umantur enim in genere termini datam ha-
bentes inter $e relationem, & ex æquationibus re$ultantibus facile
con$tabit re$olutio.</p>
<h>PROB. XX.</h>
<h it>Invenire $ummabiles $eries.</h>
<p>A$$umatur quæcunque quantitas pro $ummâ, & reducatur ea ad
$eriem, cujus termini $ecundum quamcunque legem progrediuntur,
ita autem ut $eries re$ultans converget, & perficitur prob. vel quod ad
idem redit: $it quæcunque finita quantitas vel rationalis vel irratio-
nalis, cujus inveniatur proximus valor; deinde differentiæ inter hunc
valorem & datam quantitatem inveniatur valor prope, & $ic deinceps
in infinitum, & re$ultabit convergens $eries.</p>
<p>Ex. 1. Erit <math>{1 / a - b} = {1 / a} + {b / a × (a - b)} = {1 / a} + {b / a × (a + c)} +
{b × (b + c) / a × (a + c) × (a - b)} = {1 / a} + {b / a · (a + c)} + {b · (b + c) / a · (a + c) · (a + d)} +
{b · (b + c) · (b + d) / a · (a + c) · (a + d) · (a - b)} = {1 / a} + {b / a · (a + c)} + {b · (b + c) / a · (a + c) · (a + d)}

<pb 540>[0592]<rh>DE SUMMATIONE</rh>
+ {b · (b + c) · (b + d) / a · (a + c) · (a + d) · (a + e)} + {b · (b + c) · (b + d) · (b + e) / a · (a + c) · (a + d) · (a + e) × (a - b)}
= &c.</math></p>
<p>2. Erit etiam <math>{1 / a + b} = {1 / a} - {b / a × (a + b)} = {1 / a} - {b / a × (a - c)} +
{b × (b + c) / a · (a - c) × (a + b)} = {1 / a} - {b / a · (a - c)} + {b · (b + c) / a · (a - c) · (a - d)} -
{b · (b + c) · (b + d) / a · (a - c) · (a - d) × (a + b)} = &c.</math></p>
<p>Facile demon$trari po$$int hæc theoremata e reductione fractionum
ad communes denominatores.</p>
<p>Cor. · Ex a$$umptis diver$is $eriebus numerorum pro quantitati-
bus <math>a, b, c, d, e, f, &c.</math> facile con$equentur $eries, quarum innote$cunt
$ummæ.</p>
<p>Ex. 2. Sit $eries <math>1 + b + b c + b c d + b c d e + b c d e f + &c.</math> cu-
jus termini continuo decre$cunt, & ultimo nihilo æquales fiunt, &
erit <math>1 = 1 - b + b × (1 - c) + bc × (1 - d) + bcd × (1 - e) +
bcde × (1 - f) + &c.</math></p>
<p>Hæc $eries non convergit celerius quam data.</p>
<p>Cor. · Sint <math>b = {m / a}, c = {m + p / a + p}, d = {m + q / a + q}, e = {m + r / a + r}, &c.</math> &
erit <math>1 = 1 - b + b × (1 - c) + &c. = 1 - {m / a} + {m / a} · {a - m / a + p} + {m / a} ×
{m + p / a + p} × {a - m / a + q} + {m / a} × {m + p / a + p} · {m + q / a + q} · {a - m / a + r} + &c.</math> nunc $tatu-
antur <math>q = 2p, r = 3p, s = 4p, &c.</math> & evadet hæc $eries <math>{a - m / a} +
{m / a} · {a - m / a + p} + {m / a} · {m + p / a + p} · {a - m / a + q} + &c. = {β - p - m / β - p} (1 + {m / β} + {m / β}
· {m + p / β + p} + {m · m + p · m + 2p / β · β + p · β + 2p} + &c. ubi β = a + p.</math></p>

<pb 541>[0593]<rh>SERIERUM, &c.</rh>
<p>Ex. 3. Sint <math>b - π = β, c - ρ = γ, d - σ = δ, e - τ = ε, &c.</math> tum
erit <math>0 = {α / a} - {α / a} × {β / b} - {α / a} × {π / b} × {γ / c} - {α / a} × {π / b} × {ρ / c} × {δ / d} - &c.</math> $i modo
fractiones novæ ultimo $int minores quam unitas, i. e. <math>0 = 1 -
{β - β′ / β} - {β′ / β} × {γ - γ′ / γ} - {β′ / β} × {γ′ / γ} × {δ - δ′ / δ} - {β′ / β} × {γ′ / γ} ×
{δ′ / δ} × {ε - ε′ / ε} - &c.</math>,
ubi <math>β′, γ′, δ′, ε′, &c.</math> $unt quantitates minores quam <math>β, γ, δ, ε, &c.</math></p>
<p>Si modo addantur quantitates, quæ continuo decre$cant in uno
termino, & detrahantur in $ucce$$ivis, vel prope; tum re$ultabit $eries,
cujus $umma innote$cit; hæc principia continent methodum $eries
deducendi, quarum $ummæ innote$cunt.</p>
<p>Hæc principia etiam applicari po$$unt ad inveniendas $eries conver-
gentes, quarum $ummæ $int quæcunque irrationales quantitates: $ed
perraro u$ui in$erviunt con$imiles transformationes, 1<_>mo</_>. enim nece$$e
e$t, ut detur methodus inveniendi quantitates <math>a, b, c, d, &c; α, β, γ,
&c.</math> e datis $eriei terminis, aliter non datur $eriei transformatio; i. e.
non ex datâ $erie acquiritur $eries convergens.</p>
<p>2. Datâ $erie convergente, inveniatur ejus functio vel algebraica
vel fluxionalis vel fluentialis vel incrementialis vel integralis e prin-
cipiis prius traditis, quæ evadit convergens, & per$icitur prob.</p>
<h>PROB. XXI.</h>
<h it>Invenire $ummas ferierum, quarum $inguli termini dantur ex infinitis
$eriebus.</h>
<p>1. Si vero dentur termini ex infinitis $eriebus: per infinitas $eries
inveniantur vel $ummæ harum infinitarum $erierum, vel approxima-
tiones $atis propinquæ ad illas $ummas; quibus in terminis $eriei, cujus
$umma requiritur, pro $uis valoribus $ub$titutis, re$ultat $eries, cujus
$umma inveniatur prope, & erit $umma quæ$ita prope.</p>
<p>Tum continuo per methodos prius traditas corrigendæ $unt ap-
proximationes inventæ, & exinde corrigetur $umma $eriei quæ$ita; &
$ic deinceps.</p>

<pb 542>[0594]<rh>DE SUMMATIONE</rh>
<p>2. Si vero termini e datis æquationibus pendeant, tum e datis
æquationibus vel deducantur valores terminorum datæ $eriei vel
approximationes ad terminos datæ $eriei $atis appropinquantes, &
tum per methodos prædictas continuo corrigatur $umma inventa,
& continuo re$ultabunt propiores approximationes ad $ummam
quæ$itam.</p>
<p>Ex. Literâ <math>z - 1</math> denotante di$tantiam termini cuju$cunque e primo,
terminus prædictus exprimatur per reciprocam $eriei <math>1 + {1 / 1 · 2}z^-1
+ {1 / 1 · 2 · 3}z^-2 + {1 / 1 · 2 · 3 · 4}z^-3 + &c.</math> ductam in <math>{1 / 2^z}</math>, i. e. $eries $it
<math>{{1 / 2} / 1 + {1 / 1 · 2} + {1 / 2 · 3} + &c.} + {{1 / 4} / 1 + {2 / 1 · 2} + {2^2 / 2 · 3} + &c.} + {{1 / 8} / 1 + {3 / 1 · 2}
+ {3^2 / 1 · 2 · 3} + &c.} + &c.</math> quæ $eries vere convergit; & facile erit di-
judicare de numero terminorum in fractione <math>1 + {z / 1 · 2} + {z^2 / 1 · 3} +
&c.</math> ad $ingulas operationes a$$umendorum, &c.: $ingulorum enim
terminorum præ$tat, ut inveniantur approximationes, quæ a veris
$ummis per quantitates prope æquales di$tant, & $i requiratur $umma
$eriei vera ad quendam limitem, tum ita inveniantur approximatio-
nes, ut earum aggregatum non produceret errorem limitem prædi-
ctum ex$uperantem.</p>
<p>In his æque ac in omnibus aliis $eriebus ex relatione, quam habent
termini $ucce$$ivi ad infinitam di$tantiam dijudicari pote$t convergen-
tia $eriei.</p>
<h>PROB. XXII.</h>
<p>_Conce$sâ metbodo detegendi generaliter, annon una data quantitas $it al-_
_gebraica functio aliarum datarum quantitatum, & datâ $erie $ecundum di-_
_men$iones incognitæ quantitatis_ <math>x</math> _progrediente, cujus $umma $it y; invenire_

<pb 543>[0595]<rh>SERIERUM, &c.</rh>
_annon a$$ignari pote$t æquatio, $ive algebraica $ive fluxionalis $it, quæ de-_
_$ignat relationem inter quantitates_ <math>x & y</math>, _& earum fluxiones._</p>
<p>1. Sit data $eries <math>a x^n + b x^n+m + c x^n+2m + &c. = y</math>, & e conce$sâ
methodo $equitur, utrum data $eries <math>y</math> $it algebraica functio literæ <math>x</math>,
i. e. utrum detur algebraica æquatio, quæ exprimit relationem inter
quantitates <math>x & y</math>, necne.</p>
<p>2. Inveniantur primæ, $ecundæ, &c. fluxiones datæ $eriei, quæ
erunt re$pective <math>y^. = n a x^n-1 x^. + (n + m)b x^n+m-1 x^. + (n + 2m)c x^n+2m-1 x^.
+ &c.</math> &c. jam vero e conce$sâ methodo detegatur utrum data $eries
<math>a x^n + b x^n+m + &c. = y</math> $it algebraica functio datæ quantitatis x, &
datarum $erierum <math>n a x^n-1 + (n + m) b x^n+m-1 + &c. = {y^. / x^.}, n · (n - 1)
a x^n-2 + (n + m) × (n + m - 1)b x^n+m-2 + &c. = {y^.. / x^. ^2}, &c.</math> necne; &
confit problema.</p>
<p>Ex. · Sit data $eries <math>1 + {x^n / 1 · 2 · 3 .. n} + {x^2n / 1 · 2 · 3 .. 2 n} + {x^3n / 1 · 2 ·3 .. 3n}
+ &c. = y</math>, cujus fluxio n ordinis erit <math>y^. ^n = (1 + {x^n / 1 · 2 · 3 ... n} +
{x^2n / 1 · 2 · 3 .. 2 n} + &c.)x^. ^n</math>; unde <math>y^. ^n = y x^. ^n</math>.</p>
<p>Cor. Generalis fluens æquationis <math>y^. ^n = y x^. ^n erit y = a (1 +
{x^n / 1 · 2 · 3 .. n} + {x^2n / 1 · 2 · 3 .. 2 n} + {x^3n / 1 · 2 · 3 .. 3 n} + {x^4n / 1 · 2 · 3 .. 4 n} + &c. in infi-
nitum) + b x (1 + {x^n / 2 · 3 .. n + 1} + {x^2n / 2 · 3 .. 2n + 1} + {x^3n / 2 · 3 .. 3 n + 1} +
&c.) + {c x^2 / 1 · 2.} (1 + {x^n / 3 · 4 .. n + 2} + {x^2n / 1 · 4..2n + 2}.. + {x^3n / 3 · 4 .. 3 n + 2} +
&c.) + {d x^3 / 1 · 2 ·3} (1 + {x^n / 1 · 5 .. n + 3} + {x^2n / 1 · 5..2n + 3} + {x^3n / 4 · 5 .. 3n + 3}

<pb 544>[0596]<rh>DE SUMMATIONE</rh>
+ &c.) + {e x^4 / 1 · 2 · 3 · 4} (1 + {x^n / 5 · 6 .. n + 4} + &c.) + &c. ... +
{b x^n-1 / 1 · 2 · 3 .. n - 1} × (1 + {x^n / n · n + 1 .. 2n - 1} + {x^2n / n(n + 1)(n + 2) .. (3n - 1)}
+ &c.</math> in infinitum); ubi literæ <math>a, b, c, d, e, &c.</math> & <math>b</math> invariabiles co-
efficientes ad libitum a$$umendas re$pective denotant.</p>
<p>3. Si termini $eriei <math>y = P</math> generaliter exprimantur per fractionem
huju$ce generis <math>{a z^m + b z^m-1 + c z^m-2 + &c. / A z^n + B z^n-1 + C z^n-2 + &c.} × x^rz+rs</math>, ubi literæ <math>r, s,
a, b, c, &c. A, B, C, &c.</math> invariabiles denotant quantitates, & <math>m</math> & <math>n</math>
$unt integri numeri, & <math>z</math> di$tantia a primo $eriei termino: tum e
prob. præced. deduci pote$t fluxionalis æquatio, cujus radix particu-
laris invenitur <math>y = P</math>.</p>
<p>Cor. 1. Ex hinc facile colligi pote$t fluentem generalis fluxionalis
æquationis huju$ce formulæ <math>y^. ^n + {a^n-1 / x} y^. x^. + {b / x^2} y^. ^n-2 x^. ^2 + {c^n-3 / x^3} y^. x^. ^3 + &c.
= {a′ - b′^z+1 x^z+1 / x^n - b′ x^n+1}</math> e$$e <math>y = A x^-α + B x^-β + C x^-γ + D x^-δ + &c. +
{a′ / α · β · γ · δ · &c.} + {a′ b′ x / α + 1 · β + 1 · γ + 1 · δ + 1 · &c.} + ...
{a′ b′^z x^z / α + z · β + z · γ + z · δ + z · &c.}</math>, ubi <math>a, b, c, &c.</math> $unt invariabiles
coefficientes facile deducendæ ex (<math>n</math>) invariabilibus quantitatibus <math>α, β,
γ, δ, &c.</math> datis, & (<math>n</math>) quantitates <math>A, B, C, D, &c.</math> $unt quæcunque
invariabiles quantitates ad libitum a$$umendæ, <math>& a′, b′, &c.</math> $unt datæ
invariabiles quantitates; & <math>z</math> e$t integer numerus.</p>
<p>4. Sit æquatio <math>(a + a′ x + a″ x^2 + &c.)x^. ^n + (b + b′ x + &c.)y x^. ^n
+ (c + &c.) y^. x^. ^n-1 + (d + &c.) y^.. x^. ^n-2 ... + (g + &c.) y^. ^n-1 x^. + b y^. ^n
= 0</math>, tum $i modo ejus generalis fluens $it <math>y = A + B x + C x^2 ...
G x^n-1 + H x^n + I x^n+1 + K x^n+2 + &c.,</math> ubi (<math>n</math>) literæ <math>A, B, C, &c.</math>
funt invariabiles quantitates ad libitum a$$umendæ, cæteræ vero <math>H,
I, &c.</math> invariabiles, erit <math>a + b A + c B + 2 d C + ... + 1 · 1. 3

<pb 545>[0597]<rh>SERIERUM, &c.</rh>
:: (n - 1) g G = - 1 · 2 · 3:: n × b H, b B + 1 · 2 c C + 2 · 3 d D
+ ... 2 · 3 · 4 .. (n + 1) I b = 0, &c.</math></p>
<h>PROB. XXIII.</h>
<h>1. _Invenire quantitatem, quæ in $eriem inveniendam magis celeriter_
_convergentem ducta præbet datam $eriem._</h>
<p>Huju$ce problematis re$olutio $olummodo deduci pote$t e terminis
ad infinitam di$tantiam po$itis; inveniatur enim quantitas $i modo
talis deduci po$$it, quæ vel in unum terminum cuju$cunque formulæ,
vel in plures terminos proxime convergentes ducta, præbet datam
$eriem, & per$icitur problema.</p>
<p>Ex. 1. Sit $eries infinita <math>(A) 1 + b + b^2 + b^3 + &c.</math> & erit <math>(1 + b)
× (1 + b^2 + b^4 + b^6 + &c. = B), & (1 + b)(1 + b^2)(1 + b^4 +
b^8 + &c. = C)</math>, & $imiliter <math>(1 + b)(1 + b^2)(1 + b^4)(1 + b^8 + b^16
+ &c. = D)</math>, &c. ubi <math>b</math> $it vel negativa vel affirmativa quantitas, &
erit <math>A</math> minus convergens quam <math>B, B</math> quam <math>C, C</math> quam <math>D, &c.</math></p>
<p>Ex. 2. Sit $eries <math>(P) x - {1 / 2}x^2 + {1 / 3}x^3 - {1 / 4}x^4 + &c.</math> a$$umatur
<math>(1 + x)^-1 × (x + Ax^2 + Bx^3 + Cx^4 + &c.) = P</math>, & invenientur
<math>A = {1 / 1 · 2}, B = {- 1 / 2 · 3}, C = {1 / 3 · 4}, &c.</math> i. e. $eries erit <math>(Q) x + {1 / 2}x^2 -
{1 / 2 · 3} x^3 ... {1 / z · z + 1}x^z+1</math>; a$$umatur <math>2. (1 + x)^-1 × ({1 / 2}x + ax^2 +
b x^3 + &c.) = Q</math>; & invenientur <math>a = + {3 / 4}, b = + {1 / 1 · 2 · 3}, c = -
{1 / 2 · 3 · 4}, d = + {1 / 3 · 4 · 5}, &c.</math> i. e. $eries erit <math>(R) {1 / 2}x + {3 / 4}x^2 + {1 / 1 · 2 · 3}
x^3 - {1 / 2 · 3 · 4}x^4 ... {1 / z · z + 1 · z + 2}x^z+2</math>, & $ic a$$umatur <math>3 (1 + x)^-1
({1 / 6}x + αx^2 + βx^3 + γx^4 + &c.) = R</math>, & invenietur $eries <math>(S) {1 / 6}x +
{5 / 12}x^2 + {11 / 36}x^3 + {1 / 1 · 2 · 3 · 4}x^4 - {1 / 2 · 3 · 1 · 5}x^5 ... {1 / z · z + 1 · z + 2 · z + 3}

<pb 546>[0598]<rh>DE SUMMATIONE</rh>
x^z+3</math>, & $ic deinceps: unde <math>P = {Q / 1 + x} = {R / 2 · (1 + x) · (1 + x)} =
{S / 2 · 3 · (1 + x) · (1 + x)^2} = {T / 2 · 3 · 4 · (1 + x) · (1 + x)^3} = &c.</math></p>
<p>Cor. Sit <math>x</math> minor quam 1, & omnes $eries <math>P, Q, R, S,</math> & c. evadent
convergentes; $it <math>x</math> eadem quantitas in omnibus prædictis $eriebus, &
erunt termini $erierum <math>(P, Q, R, S, &c.)</math> ad infinitam di$tantiam re-
$pective inter $e ut <math>± {1 / z}, {1 / z · z - 1}, {1 / z · z - 1 · z - 2}, {1 / z · z - 1 · z - 2 · z - 3}</math>,
& $ic deinceps; ubi <math>z</math> denotat di$tantiam a primo $eriei termino; &
con$equenter ad infinitam di$tantiam minor erit convergentia $eriei
<math>P</math> quam <math>Q, Q</math> quam <math>R, R</math> quam <math>S</math>, &c. termini enim ad prædictam
di$tantiam $eriei <math>P</math> infinite majores $unt quam termini $eriei Q, & $ic
deinceps: in $eriebus enim per functiones algebraicas quantitatis <math>z</math>
de$ignatis, convergentia pendet ex differentiâ inter dimen$iones
quantitatis <math>z</math> in numeratore & deneminatore contentas.</p>
<p>2. Ducatur $eries quæcunque <math>(P)</math> in quantitatem <math>α = 0</math>, (e. g. $it
<math>1 - x = 0</math>, & con$equenter <math>α = 1 - x</math>); & $i $eries <math>(P)</math> $it conver-
gens, tum $eries re$ultans <math>erit = 0</math>; $i $eries <math>(P)</math> $it divergens, tum
$eries re$ultans pote$t e$$e convergens vel divergens.</p>
<p>3. Datâ functione quantitatis <math>z</math>, quæ denotat quemcunque termi-
num $eriei <math>(P)</math>, cujus di$tantia a primo $it <math>z</math>; reducatur data functio
ad $eriem de$cendentem $ecundum dimen$iones quantitatis <math>z</math>; deinde
inveniatur quantitas <math>A</math>, ita ut, $i modo reducatur $inita quantitas <math>A</math>
ad infinitam $eriem, termini ad infinitam di$tantiam quam proxime
iidem evadant ac termini datæ $eriei; inveniatur $eries Q, quæ in
quantitatem <math>A</math> ducta, præbet datam $eriem <math>(P)</math>, & fere feries Q ma-
gis celeriter converget quam feries <math>P</math>.</p>
<p>4. Si vero detur relatio inter $ucce$$ivos terminos $eriei <math>(P)</math> ad in-
finitam di$tantiam, inveniatur quantitas <math>A</math>, quæ reducta ad $eriem
infinitam, cujus termini $ecundum dimen$iones quantitatis <math>z</math> de$cen-
dentes, eandem prope habent relationem inter $ucce$$ivos terminos
ac data relatio; tum deducatur $eries Q, ita ut <math>A × Q = P</math>, & fere Q
erit $eries magis celeriter convergens.</p>

<pb 547>[0599]<rh>SERIERUM, &c.</rh>
<p>Facile inve$tigari po$$int infinitæ quantitates <math>A</math>, quarum termini
$ucce$$ivi ad infinitam diftantiam vel quam proxime $int iidem, vel
habeant inter $e relationem, quam habent datæ $eriei termini.</p>
<p>E convergentiâ totius $eriei, i. e. terminorum ad infinitam di$tan-
tiam minime dici pote$t convergentia omnis partis; $ed fere facile
inveniri po$$unt termini, cum tran$eat $eries e divergentiâ in conver-
gentiam, vel cum fiat maxime convergens; vel convergentiæ incre-
mentum evadat maximum, &c.</p>
<h>THEOR. XVII.</h>
<p>Sit <math>y = a - b r^z - cr^2z - d r^3z - e r^4z - &c.</math> ubi <math>r, a, b, c, d, e,
&c.</math> $unt quantitates invariabiles; & $int <math>A, B, C, D, E, &c.</math> diver$i
dati valores quantitatis <math>y</math>, re$pondentes diver$is valoribus quantitatis
<math>z</math>, qui $unt <math>0, 1, 2, 3, 4, &c.</math> unde <math>A = a - b - c - d - e - &c.
B = a - b r - c r^2 - d r^3 - &c. C = a - b r^2 - c r^4 - d r^6 - e r^8
- &c. D = a - b r^3 - c r^6 - d r^9 - e r^12 - &c. &c.</math>, ubi tot a$-
$umantur æquationes, quot $unt incognitæ quantitates <math>a, b, c, d, e,
&c.</math> & pro <math>r</math> a$$umi pote$t quicunque numerus exceptis <math>1 & -1</math>; ex
his $implicibus æquationibus inveniri po$$unt coefficientes <math>a, b, c, d,
&c.</math>; & con$equenter æquatio <math>y = a - b r^z - c r^2z - &c.</math></p>
<p>Cor. 1. Sit <math>z</math> infinita quantitas & <math>r</math> minor quam 1, tum erit
<math>y = a</math>.</p>
<p>Cor. 2. Fiat <math>z</math> negativa quantitas, & erit <math>y = a - {b / r^z} - {c / r^2z} - &c.</math>
& $it <math>z</math> infinita, & <math>r</math> major quam 1; tum erit <math>y = a</math>.</p>
<p>Cor. 3. Si <math>r = 1 vel - 1</math>, tum a$$umenda e$t $eries <math>y = a + {b / z} +
{c / z^2} + {d / z^3} + &c.</math> vel <math>a + bz + cz^2 + &c.</math></p>
<p>Differentiæ quarumcunque quantitatum ad perparvas di$tantias
<math>(o)</math> a $e invicem po$itarum erunt plerumque in ratione di$tantiarum
ip$arum; & con$equenter quantitas exprimi pote$t per quantitatem

<pb 548>[0600]<rh>DE SUMMATIONE</rh>
huju$ce formulæ <math>A + B o</math>, & magis generaliter per quantitatem <math>A +
B o + C o^2 + D o^3 + &c.</math></p>
<p>Hoc verum e$t haud $olummodo in algebraicis $ed etiam in expo-
nentialibus quantitatibus.</p>
<p>Hinc prope corrigi po$$unt errores, qui irrepunt in plera$que ope-
rationes ad perparva intervalla in$titutas.</p>
<h>THEOR. XVIII.</h>
<p>Sint <math>A, B, C, D, E, &c.</math> quantitates quæcunque invariabiles, <math>z</math> vero
variabilis, tum in expre$$ione <math>A + B z + C z · {z - n / 2} + D z · {z - n / 2}·
{z - 2 n / 3}· &c.</math> $cribe numeros quo$vis æquidi$tantes <math>o, n, 2n, 3n, &c.</math>
pro <math>z</math>, & quantitates provenientes dicantur re$pective <math>a, b, c, d, e, &c.</math>
$ubtrahatur qui$que terminus $eriei <math>a + b + c + d + e + &c.</math> de
proxime præcedente, vocenturque termini re$iduorum <math>(a - b, b - c,
c - d, &c.)</math> primæ differentiæ; deinde $ubtrahatur differentia quæ-
que prima a proxime præcedente, & re$idua re$ultantia vocentur
$ecundæ differentiæ; $ubtrahatur etiam quæque $ecunda a pro-
xime præcedente, & re$idua re$ultantia dicantur tertiæ differen-
tiæ; & $ic deinceps: $it <math>a</math> primus $eriei terminus, <math>d′</math> prima prima-
rum differentiarum, <math>d″</math> prima $ecundarum, & $ic deinceps; & erit
expre$$io <math>a - d′{z / n} + {z / n} · {z - n / 2n} d″ - {z / n} · {z - n / 2n} · {z - 2n / 3n} d^m + &c. =</math>
termino, cujus di$tantia a primo $it <math>z</math>.</p>
<p>Cor. 1. Hinc <math>d′ = a - b; d″ = (a - b) - (b - c) = a - 2 b + c</math>;
& $ic <math>d′″ = a - 3 b + 3 c - d</math>; & denique <math>d′ = a - n b + n · {n - 1 / 2}
c - n · {n - 1 / 2} · {n - 2 / 3} d + &c.</math></p>
<p>Cor. 1. 1. Sint <math>a, b, c, d, e, &c.</math> ordinatæ ad curvam, quarum com-
munis di$tantia a $e invicem $it <math>n</math>, i. e. quarum ab$ci$$æ $unt <math>o, n, 2n,

<pb 549>[0601]<rh>SERIERUM, &c.</rh>
&c.</math> & æquatio relationem inter ab$ci$$am <math>z</math> & ejus corre$pondentes
ordinatas <math>y</math> de$ignans $it <math>y = A - {B z / n} + {C z / n} · {z - n / 2 n} - D {z / n} · {z - n / 2 n}.
· {z - 2 n / 3 n} + &c</math>. tum erunt <math>A = a, B = d′ = a - b, C = d″ = a -
2 b + c, D = d′″ = a - 3 b + 3 c - d, &c.</math></p>
<p>2. Sint <math>A<x>.</x> n = d′, A<x>..</x> n^2 = d″, A<x>...</x> n^3 = d′″, &c.</math> tum erit <math>y = A -
A<x>.</x> z + A<x>..</x> z · {z - n / 2} - A<x>...</x> z · {z - n / 2} · {z - 2 n / 3} + &c.</math> fiat <math>n = 0</math> & eva-
dent <math>A<x>.</x>, A<x>..</x>, A<x>...</x>, &c.</math> primæ, $ecundæ, tertiæ, &c. fluxiones primæ or-
dinaiæ <math>A</math>, & exinde ordinata <math>y = A - {A^. / z^.}z + {1 / 2}{A^.. / z^. ^2} z^2 - {1 / 2 · 3} {A^... / z^. ^3} z^3 +
{1 / 2 · 3 · 4} {A^.... / z^. ^4} z^4 - &c.</math> & ejus area <math>= A z - {1 / 2}{A^. / z^.} z^2 + {1 / 6} {A^.. / z^. ^2} z^3 - {1 / 24} {A^... / z^.^3} z^4
+ &c.</math> pro <math>A, A^., A^.., &c.</math> $cribantur <math>y, y^., y^.., &c.</math> & fit area <math> = y z -
{1 / 2} {y^. / z^.} z^2 + {1 / 6} {y^.. / z^. ^2} z^3 - {1 / 24} {y^... / z^. ^3} z^4 + &c.</math> $int <math>y^., y^..., &c.</math> negativæ quantitates,
& fit <math>area = y z + {1 / 2} {y^. / z^.} z^2 + {1 / 6} {y^.. / z^. ^2} z^3 + &c.</math></p>
<p>In hoc ca$u ab$ci$$a $emper $upponitur incipere a primâ vel ultimâ
ordinatâ.</p>
<p>Cor.. Hinc literæ <math>a, b, c, &c.</math> inveniuntur e differentiis <math>a, d′, d″, d′″,
&c.</math> fere per eandem legem ac differentiæ <math>d′, d″, d′″, &c.</math> e literis <math>a, b,
c, &c.</math> i. e. <math>b = a - d′, c = a - 2d′ + d″, d = a - 3 d′ + 3 d″ -
d′″, e = a - 4 d′ + 6 d″ - 4d′″ + d″″, &c.</math></p>
<p>Ex. 1. Sint termini $ucce$$ivi <math>{1 / z}, {1 / z + 1}, {1 / z + 2}, {1 / z + 3}, &c.</math> &
invenientur prædictæ differentiæ <math>{1 / z} - {1 / z + 1} = {1 / z · z + 1} = d′, {1 / z} -
{2 / z + 1} + {1 / z + 2} = {1 × 2 / z · z + 1 · z + 2} = d″</math>; & $ic <math>{1 × 2 × 3 / z · z + 1 · z + 2 · z + 3}

<pb 550>[0602]<rh>DE SUMMATIONE</rh>
= d′″</math>, & $ic deinceps: & vice versâ $int termini re$pectivi <math>{1 / z}, {1 / z · z + 1},
{1 · 2 / z · z + 1 · z + 2}, {1 · 2 · 3 / z · z + 1 · z + 2 · z + 3}</math>, & erunt differentiæ præ-
dictæ <math>d′, d″, d′″, &c.</math> re$pective <math>{1 / z + 1}, {1 / z + 2}, {1 / z + 3}, {1 / z + 4}, &c.</math></p>
<p>Ex. 2. Sint termini <math>a, a b, a b^2, a b^3, &c.</math> in geometricâ progre$$ione,
& erunt differentiæ prædictæ <math>d′, d″, d′″, &c.</math> re$pective <math>a = d, a - a b
= a × (1 - b) = d′, a - 2 a b + a b^2 = a × (1 - b)^2 = d″, &c.</math> i. e.
differentiæ $ucce$$ivæ <math>a, a(1 - b), a × (1 - b)^2; a × (1 - b)^3, &c</math>.
quæ $int <math>a, d′, d″, &c.</math> erunt quantitates in geometricâ progre$$ione,
& vice versâ $int prædictæ differentiæ in geometricâ progre$$ione, &
erunt quantitates ip$æ in geometricâ progre$$ione.</p>
<p>Ex. 3. Sit $eries terminorum <math>a, b, a + b, a + 2 b, 2 a + 3 b, 3 a + 5 b,
5 a + 8 b, &c.</math> ubi quique $ucce$$ivus terminus e$t $umma duorum
præcedentium, tum termini differentiarum <math>n</math> ordinis erunt iidem ac
termini datæ $eriei (<math>n</math> primis exceptis); primus terminus differentia-
rum <math>n</math> ordinis erit <math>m a - b b</math>, $i <math>m b + b a</math> $it terminus in datâ $erie,
cujus di$tantia a primo $it <math>n + 1</math>.</p>
<p>Si termini $int $ummæ (<math>m</math>) præcedentium terminorum in qua$cun-
que datas coefficientes ductorum, vel eorum quæcunque functiones;
tum exinde erui po$$unt differentiæ.</p>
<h>PROB. XXIV.</h>
<p it>Datâ $erie <math>a + b + c + d + e + &c.</math> in infinitum pergente & e ter-
minis $ucce$$ivis <math>a, b, c, d, &c.</math> con$tante, & inventis differentiis <math>d′ = a
- b, d″ = a - 2 b + c, & c.</math> $it <math>z</math> di$tantia a primo $eriei termino, tum
terminus, cujus di$tantia a primo $it <math>z</math>, per theor. præc. erit <math>a - d′ z + z.
{z - 1 / 2} d″ - z × {z - 1 / 2} × {z - 2 / 3} d′″ + &c.</math> quæ $eries in infinitum excur-
ret, ni bæ differentiæ fiant ultimo nibilo æquales. Invenire igitur, utrum
bæc $eries in infinitum progrediens, converget; necne.</p>

<pb 551>[0603]<rh>SERIERUM, &c.</rh>
<p>1<_>mo</_>. Sit data $eries talis, ut terminus qui$que ultimo $it ad termi-
num $ucce$$ivum in permagnâ ratione fere ut infinita quantitas ad
finitam, e. g. $it $eries <math>1 + {1 / 1 · 2} + {1 / 1 · 2 · 3} + {1 / 1 · 2 · 3 · 4}</math>, & denique
<math>{1 / 1 · 2 · 3 .. z} + {1 / 1 · 2 · 3 .. z + 1}</math>; $it <math>z</math> infinita quantitas & <math>{1 / 1 · 2 · 3 .. z}</math>
erit ad <math>{1 / 1 · 2 · 3 .. z · (z + 1)}</math> ut infinita <math>z + 1 ad 1</math>. In hoc ca$u $e-
ries data maxime convergit, $ed $eries re$ultans <math>1 - d′ (1 - {1 / 2}) z +
d″(1 - 2 × {1 / 2} + {1 / 1 · 2 · 3}) z × {z - 1 / 2} - d′″ × z × {z - 1 / 2} · {z - 2 / 3} + &c.</math>
minime converget: hoc con$tare pote$t e $ub$equente ratiocinatione;
$it primus terminus $eriei <math>a</math>, omnes vero reliqui <math>vel = 0</math>, vel $inguli
$ucce$$ivi infinite $int minores præcedentibus; & erit $eries prope
<math>a + 0′ + 0″ + 0′″ + &c.</math> & omnes primæ differentiæ primarum, $e-
cundarum, &c. differentiarum erunt re$pective <math>a</math>, unde $eries erit
<math>a - a z + a z × {z - 1 / 2} - a z × {z - 1 / 2} × {z - 2 / 3} + &c.</math> quæ nunquam
celeriter converget, ni <math>z</math> $it integer numerus.</p>
<p>2<_>do</_>. Si ultimus datæ æquationis terminus habeat ad ejus $ucce$$i-
vum rationem majorem quam æqualitatis, i. e. ultimi termini datæ
$eriei $int in geometricâ progre$$ione, cujus communis ratio e$t <math>1: r</math>;
tum, $i omnes termini $eriei $int in geometricâ progre$$ione, erunt
differentiæ <math>d′, d″, d′″, &c.</math> in geometricâ progre$$ione, cujus commu-
nis ratio e$t <math>1, 1 - r</math>: unde con$tat, $i 1 major $it quam <math>1 - r</math>, tum
converget hæc $eries, $in aliter vero non. Eadem etiam affirmari
po$$unt de $erie, cujus termini ultimo $unt in geometricâ progre$$ione.</p>
<p>3<_>tio</_>. Si termini ad infinitam di$tantiam propiores accedant ad
rationem æqualitatis quam pro quâvis finitâ differentiâ; tum, utrum
$eries deducta $it convergens, necne; pendet e ratione, quam habent
ultimo $ucce$$ivæ differentiæ <math>d′, d″, d′″, &c.</math> i. e. ex æquatione <math>a - z d′
+ z × {z - 1 / 2} d″ - &c.</math> prius traditâ, i. e. ex ratione, quam habent

<pb 552>[0604]<rh>DE SUMMATIONE</rh>
inter $e ultimo $ucce$$ivi $eriei termini, con$tabit, utrum $eries re$ul-
tans $it convergens, necne.</p>
<p>Hinc con$tat, ut hæc $eries plerumque maxime, cum data $eries
minime convergat; $emper autem minime cum maxime convergat
data $eries: pendet enim e rationibus, quas habent inter $e $ucce$$ivæ
differentiæ <math>d, d′, d″, &c.</math></p>
<h>THEOR. XIX.</h>
<p>1. Sit convergens $eries <math>a + b + c + d + e + f + &c.</math> quorum
terminorum primæ differentiæ $int re$pective <math>a′, b′, c′, d′, e′, &c.</math> $e-
cundæ vero <math>a″, b″, c″, d″, &c.</math> & $ic deinceps: differentiæ vero <math>n</math> or-
dinis $int re$pective <math>α, β, γ, δ, ε, &c.</math> tum erit <math>a = α + n β + n.
{n + 1 / 2}γ + n · {n + 1 / 2} · {n + 2 / 3} δ + n ·{n + 1 / 2} · {n + 2 / 3} · {n + 3 / 4} ε + &c.</math> in
infinitum.</p>
<p>Cor. · Si modo $eries data $it convergens; tum omnes $eries,
quorum termini $unt differentiæ cuju$cunque ordinis erunt etiam
convergentes.</p>
<p>Facile con$tat, $i modo $eries $it affirmativa; $in termini eju$dem
$eriei, (i. e. cujus affirmativi & negativi termini eandem pror$us ob-
$ervant legem) $int alternatim affirmativi & negativi, tum erunt præ-
dictæ differentiæ alternatim affirmativæ & negativæ, & $eries erit
convergens.</p>
<p>2. Summa $eriei <math>a + a′ + a″ + a′″ + &c. a′^n = (n + 1) a - n · {n + 1 / 2}
b + (n - 1) · {n / 2} · {n + 1 / 3} c - (n - 2) · {n - 1 / 2} · {n / 3} · {n + 1 / 4} d + &c.</math></p>
<h>THEOR. XX.</h>
<p>1. Sint <math>d′, d″, d′″, d″″, &c.</math> ut antea $ucce$$ivæ primæ differentiæ
terminorum $eriei <math>a + b + c + d + e + &c.</math> i. e. cujus termini $int
$ucce$$ive <math>a, b, c, d, e, &c.</math> Sit etiam $eries, cujus termini $unt $uc-

<pb 553>[0605]<rh>SERIERUM, &c.</rh>
ce$$ive <math>E + a + b + c + d + e + &c., E + b + c + d + e + &c.,
E + c + d + e + &c., E + d + e + &c., E + e + &c.</math> & $int pri-
mæ differentiæ $ucce$$ivæ re$pective <math>D′, D″, D′″, &c.</math> tum erit <math>D′ = a,
D″ = d′, D′″ = d″, &c.</math> & $eries, quæ e$t $umma prædictorum termi-
norum per theor. præc. <math>A - D′ (z + 1) + D″ (z + 1) × {z / 2} - D′″ ×
(z + 1) × {z / 2} × {z - 1 / 3} + &c.</math> ubi <math>A</math> $it quantitas ad libitum a$$umenda.</p>
<p>2. Sint $ucce$$ivæ prædictæ $ummæ re$pective <math>S, S′, S″, S′″, &c.</math> ter-
mini $ucce$$ivi novæ $eriei; <math>δ′, δ″, δ′″, &c.</math> prædictæ differentiæ; tum
erit $umma novæ $eriei <math>B (z + 2) - δ′ × (z + 2) × {z + 1 / 2} + δ″ ×
(z + 2) × {z + 1 / 2} · {z / 3} + &c.</math> ubi quantitas <math>B</math> ad libitum a$$umenda e$t;
& $ic deinceps.</p>
<p>Cor. 1. Hinc tot quantitates in hâc $erie a$$umi ad libitum po$-
$unt, quot $ucce$$ivæ $ummæ requiruntur.</p>
<p>3. Sit $eries <math>a + b + c + d + e + &c.</math> addatur unus terminus <math>A</math>,
& erit <math>A + a + b + c + d + e + &c.</math> $int $ucce$$ivæ differentiæ
prioris $eriei <math>d′, d″, d′″, &c.</math> re$pective, & erunt $ucce$$ivæ differentiæ
po$terioris $eriei <math>A - a = D′, d′ = D′ - D″, d″ = D″ - D′″, d′″ =
D″′ - D″″ &c.</math> $it $umma prioris $eriei <math>a - z d′ + z × {z - 1 / 2} d″ - &c.</math>
& <math>e</math> præced. con$tat etiam $ummam po$terioris e$$e <math>A - (z + 1) D′ +
(z + 1) × {z / 2} D″ + &c.</math> $i vero <math>A = 0</math> tum $umma hujus $eriei eadem
erit ac præcedentis $umma.</p>
<p>Cor. 2. Et $ic $i modo ab alio puncto incipiamus: lex enim fere
eadem erit, ut con$tat <math>e</math> præced. Et tot a$$umi po$$unt ad libitum
incognitæ quantitates, quot $unt illæ differentiæ, quæ ortum $uum
ducunt ante primam differentiam $eriei prædictæ <math>a - b</math>.</p>
<p>3. Erit <math>a - z d′ + z · {z - 1 / 2} d″ - z · {z - 1 / 2} · {z - 3 / 3} d′″ + &c. = a<pb 554>[0606]<rh>DE SUMMATIONE</rh>
(1 - z + z · {z - 1 / 2} - z · {z - 1 / 2} · {z - 2 / 3} + z · {z - 1 / 2} · {z - 2 / 3} · {z - 3 / 4}
- &c.) + b z (1 - (z - 1) + (z - 1) · {z - 2 / 2} - (z - 1) · {z - 2 / 2} ·
{z - 3 / 3} + &c.) + c · z · {z - 1 / 2} (1 - (z - 2) + (z - 2) · {z - 3 / 2} -
(z - 2) · {z - 3 / 2} · {z - 4 / 3} + &c.) + d × z · {z - 1 / 2} · {z - 2 / 3} (1 - (z - 3)
+ (z - 3) · {z - 4 / 2} - (z - 3) · {z - 4 / 2} · {z - 5 / 3} + &c.) - e, &c.</math> $i
modo data $eries $it <math>a + b + c + d + &c.</math></p>
<h>THEOR. XXI.</h>
<p>1. Sint $eries quantitatum <math>a^m, (a + b)^m, (a + 2 b)^m, (a + 3 b)^m, &c.</math>
quarum radices $unt in arithmeticâ progre$$ione, ubi <math>a & b</math> $unt affir-
mativæ quantitates; tum præcedentes differentiæ $ingulorum ordi-
num, qui $unt minores quam <math>m</math>, erunt minores quam eju$dem ordi-
nis $ub$equentes; e contra erunt majores quam $ub$equentes in ordi-
nibus qui proxime majores $unt quam <math>m</math>; & præcedentes differentiæ
primi ordinis majores erunt quam $ub$equentes, $i <math>m</math> negativa $it
quantitas, vel minor quam unitas.</p>
<p>Sit <math>m</math> integer affirmativus numerus, tum erunt differentiæ ordinis
<math>m</math> inter $e æquales.</p>
<p>Hæc principia etiam applicari po$$unt ad plures compo$itas quan-
titates.</p>
<p>2. Invenire valorem quantitatis <math>n</math>, cum præcedentes primæ differentiæ
evadant minores quam $ub$equentes, i. e. <math>a^m - n (a + b)^m + n · {n - 1 / 2}
(a + 2 b)^m - n · {n - 1 / 2} · {n - 2 / 3} (a + 3 b)^m + &c.</math> fiat major vel minor
quam <math>a^m - (n + 1) (a + b)^m + (n + 1) · {n / 2} (a + 2 b)^m - &c.</math>; & con-
$imiliter invenire valorem prædictum, cum primæ differentiæ evadant

<pb 555>[0607]<rh>SERIERUM, &c.</rh>
majores vel minores quam $ecundæ, &c. i. e. <math>a^m - n(a + b)^m + n.
{n - 1 / 2} (a + 2 b)^m - &c.</math> evadat major vel minor quam <math>(a + b)^m -
n (a + 2 b)^m + (n - 1) · {n / 2}(a + 3 b)^m - &c.</math> inveniatur prope valor
quantitatis <math>n</math>, cum prædictæ quantitates evadant æquales; & id, quod
requiritur, conficitur: & $imiliter re$olvi pote$t hoc problema, cum
dentur quantitates <math>n & m</math>; vel <math>n</math> & ratio, quam habent quantitates <math>a
& b</math> inter $e, & requiritur prædicta ratio, vel quantitas <math>m</math>.</p>
<p>Cor. 1. Erit quantitas <math>(A) a^m - l · (a + b)^m + l · {l - 1 / 2} (a + 2 b)^m
- l · {l - 1 / 2} · {l - 2 / 3} (a + 3 b)^m + &c.</math> ad finem $eriei nihilo æqualis,
$i <math>m</math> minor $it quam <math>l, & m & l</math> $int integri affirmativi numeri; $i vero
<math>m = l</math>, erit prædicta quantitas <math>= ± 1 · 2 · 3 · 4 · 5 ... l × b^m</math>: $ignum
affixum erit + $i <math>m</math> $it par, $in aliter -.</p>
<p>Hæc enim quantitas erit prima differentia (<math>l</math>) ordinis prædictarum
quantitatum in arithmeticâ progre$$ione, quæ $emper erit nihilo
æqualis, cum <math>l</math> major $it quam <math>m, & l & m</math> $int integri numeri.</p>
<p>Con$imiles propo$itiones affirmari po$$unt de ca$ibus, in quibus vel
<math>a</math> vel <math>b</math> vel utræque $unt negativæ quantitates, in utri$que prioribus
ca$ibus quantitates cre$cant u$que ad infinitum, & deinde decre$cant.</p>
<h>PROB. XXV.</h>
<p it>Detur $eries ordinatarum intervallis quibu$cunque ad $e invicem di$tan-
tium, pergens vero ex unâ tantum parte in infinitum; invenire lineam pa-
rabolicam, quœ tran$it per extremitates omnium.</p>
<p>Sint <math>A, A 1, A 2, A 3, A 4, &c.</math> ordinatæ in$i$tentes ab$ci$$æ in
angulis rectis, $itque <math>R</math> punctum quodlibet in ab$ci$sâ: & ponatur
<math>a = R A, b = R A 1, c = R A 2, d = R A 3, &c.</math> Attamen de$ignet
<math>T</math> ordinatam quamvis in genere, cujus di$tantia a puncto <math>R</math> $it <math>z</math>;
ponatur etiam

<pb 556>[0608]<rh>DE SUMMATIONE</rh>
<math>B = {A 1 - A / b - a}, C = {B 1 - B / c - a}, D = {C 1 - C / d - a}, E = {D 1 - D / e - a}, &c.
B 1 = {A 2 - A 1 / c - b}, C 1 = {B 2 - B 1 / d - b}, D 1 = {C 2 - C 1 / e - b}, &c.
B 2 = {A 3 - A 2 / d - c} \\ &c. # , C 2 = {B 3 - B 2 / e - c} \\ &c. # , D 2 = {C 3 - C 2 / f - c} \\&c. # , &c.</math>
atque erit ordinata <math>T = A + B × (z - a) + C × (z - a) × (z - b)
+ D × (z - a) × (z - b) × (z - c) + E × (z - a) × (z - b) ×
(z - c) × (z - d) + &c.</math> facile con$tabit hoc prob. e $cribendo in
datâ æquatione pro <math>z</math> in ordinata <math>z = A + B (z - a) + &c.</math> ejus
$ucce$$ivos valores <math>a, b, c, &c.</math></p>
<p>Cor.. Hinc, $i modo $cribantur fluxiones radicum pro differentiis
ordinatarum; & pro earum intervallis fluxiones ab$ci$$æ, inve$tigari
pote$t radix fluxionalis æquationis.</p>
<h>PROB. XXVI.</h>
<h it>Detur $eries or dinatarum utrinque excurrens in in$initum, invenire li-
neam parabolicam, quœ tran$ibit per extremitates omnium.</h>
<p>1. De$ignet <math>a</math> ordinatam in medio omnium, $intque <math>a 2, a 4, a 6,
a 8, &c.</math> eæ ex unâ parte; & <math>2 a, 4 a, 6 a, 8 a, &c.</math> ex alterâ, pergente
progre$$ione utrinque in infinitum. Collige earum differentias pri-
mas <math>7 B, 5 B, 3 B, 1 B, B 1, B 3, B 5, B 7;</math> $ecundas <math>6 b, 4 b, 2 b, b,
b 2, b 4, b 6</math>; tertias <math>5 C, 3 C, 1 C, C 1, C 3, C 5</math>; quartas <math>4 c, 2 c, c, c 2,
c 4</math>; & $ic in reliquis, auferendo $emper antecedentes de con$equenti-
bus ut in prob. præc. Sint jam <math>a, b, c, d, e, &c.</math> ordinata media & mediæ
differentiæ in ordinibus alternis re$pective; $intque <math>1 B & B 1, 1 C &
C 1, 1 D & D 1, &c.</math> duæ mediæ differentiæ in reliquis ordinibus, &
ponantur <math>B = 1 B + B 1, C = 1 C + C 1, D = 1 D + D 1, E =
1 E + E 1</math>; $itque intervallum inter quamvis ordinatam <math>T</math> & me-
diam <math>a</math> ad intervallum commune æquidi$tantium ordinatarum ut
<math>z</math> ad unitatem; eritque ordinata <math>T = a + {B z + b z^2 / 1 · 2} + {2 C z + c z^2 / 1 · 2}

<pb 557>[0609]<rh>SERIERUM, &c.</rh>
× {z^2 - 1 / 3 · 4} + {3 D z^. + d z^2 / 1 · 2} × {z^2 - 1 / 3 · 4} × {z^2 - 4 / 5 · 6} + {4 E z + e z^2 / 1 · 2} × {z^2 - 1 / 3 · 4}
× {z^2 - 4 / 5 · 6} × {z^2 - 9 / 7 · 8} + &c.</math> ubi notandum e$t ab$ci$$am <math>z</math> e$$e negativam,
quando ordinata quæ$ita jacet ad contrarias partes ordinatæ mediæ.</p>
<p>2. Sint jam <math>1 A & A 1</math> ordinatæ duæ mediæ; <math>A 3, A 5, A 7, A 9,
&c.</math> eæ ex unâ parte; & <math>3 A, 5 A, 7 A, 9 A, &c.</math> ex alterâ. Collige
earum differentias primas <math>8 a, 6 a, 4 a, 2 a, a, a 2, a 4, a 6, a 8, &c.</math>
$ecundas <math>7 B, 3 B, 1 B, B 1, B 3, B 5, B 7</math>; tertias <math>6 b, 4 b, 2 b, b, b 2,
b 4, b 6, &c.</math> & $ic deinceps; $ubducendo ubique priores de po$terio-
ribus: excerpe nunc differentias medias <math>a, b, c, d, e, &c.</math> ut & duas
medias in aliis ordinibus, $cilicet <math>1 A & A 1, 1 B & B 1, 1 C & C 1, &c.</math>
& ponantur <math>A = 1 A + A 1, B = 1 B + B 1, C = 1 C + C 1, &c.</math> &
cuju$vis ordinatæ di$tantia <math>T</math> a medio puncto $it ad intervallum com-
mune æquidi$tantium ut <math>z</math> ad binarium; eritque <math>T = {A + a z / 2} +
{3 B + b z / 2} × {z^2 - 1 / 4 · 6} + {5 C + c z / 2} × {z^2 - 1 / 4 · 6} × {z^2 - 9 / 8 · 10} + {7 D + d z / 2} ×
{z^2 - 1 / 4 · 6} × {z^2 - 9 / 8 · 10} × {z^2 - 25 / 12 · 14} + &c.</math></p>
<p>Uterque ca$us facile demon$trari pote$t, $cribendo in priori ca$u
in ordinatâ <math>T = a + &c.</math> pro <math>z</math>, ejus $ucce$$ivos valores <math>0, 1 & - 1,
2 & - 2, 3 & - 3, &c.</math> & pro <math>a, b, c, d, &c.</math> earum re$pectivos valo-
res <math>a, 2 a - 2 × a + a 2, 4 a - 4 × 2 a + 6 × a - 4 × a 2 + a 4, 6 a
- 6 × 4 a + 15 × 2 a - 20 × a + 15 × a 2 - 6 × a 3 + a 6, &c.</math> & pro
<math>B, C, D, &c.</math> re$pective <math>2 a - a 2, 4 a - 2 × 2 a + 2 × a 2 - a 4, 6 a
- 4 × 4 a + 5 × 2 a - 5 × a 2 + 4 × a 4 - a 6, &c.</math> quæ erunt re$pe-
ctive $ummæ quantitatum <math>2 a - a & a - a 2; 4 a - 3 × 2 a + 3 ×
a - a 2 & 2 a - 3 × a + 3 × a 2 - a 4; 6 a - 5 × 4 a + 10 × 2 a
- 10 × a + 5 × a 2 - a 4 & 4 a - 5 × 2 a + 10 × a - 10 × a 2 +
5 × a 4 - a 6, &c.</math>: in po$teriori vero ca$u pro <math>a, b, c, d, e, &c.</math> $cri-
bendo re$pective <math>1 A - A 1, 3 A - 3 × 1 A + 3 × A 1 - A 3, 5 A
- 5 × 3 A + 10 × 1 A - 10 × A 1 + 5 × A 3 - A 5, &c.</math>; & pro <math>A,
B, C, D, E, &c.; 1 A + A 1, 3 A - 1 A - A 1 + A 3, &c.</math>; quæ

<pb 558>[0610]<rh>DE SUMMATIONE</rh>
erunt re$pective $ummæ quantitatum <math>1 A & A 1; 3 A - 2 × 1 A +
A 1 & 1 A - 2 × A 1 + A 3; 5 A - 4 × 3 A + 6 × 1 A - 4 × A 1
+ A 3 & 3 A - 4 × 1 A + 6 × A 1 - 4 × A 3 + A 5; 7 A - 6 × 5 A
+ 15 × 3 A - 20 × 1 A + 15 × A 1 - 6 × A 3 + A 5 & 5 A - 6 ×
3 A + 15 × 1 A - 20 × A 1 + 15 × A 3 - 6 × A 5 + A 7; &c.</math></p>
<p>Eodem plane modo deduci pote$t inve$tigatio, cum incipiat ab$ci$$a
<math>z</math> a quocunque alio puncto, quod inter qua$cunque alias ordinatas
jacet. Hi quidem omnes ca$us e prob. præc. facile erui po$$unt.</p>
<p>Hìc vero animadvertendum e$t $eries ex his methodis deductas
haud convergere, ni ordinata quæ$ita quam proxime exprimi po$$it
per $eriem in terminis ab$ci$$æ <math>x</math> huju$ce formulæ <math>A + B x + C x^2
+ D x^3 + &c.</math> de$ignatam.</p>
<p>Ex. Sit formula $eriei <math>Q × (a + b x + c x^2 + d x^3 + e x^4 + &c.) = y</math>,
ubi <math>Q</math> $it quæcunque data functio quantitatis <math>x, & n</math> $int incognitæ
coefficientes <math>a, b, c, d, e, &c.</math> tum facile e datis <math>n</math> corre$pondentibus
valoribus quantitatum <math>x & y</math> per regulas prius traditas, $i modo $cri-
batur <math>{y / Q} = v</math>, acquiri po$$unt (<math>n</math>) coefficientes <math>a, b, c, d, &c.</math> Et $ic
de infinitis aliis.</p>
<p>In ca$ibus prius traditis præ$tat, fi terminus vel ordinata quæ$ita
longe di$tet a terminis vel ordinatis datis, ita transformare datas or-
dinatas, ut inveniantur aliæ, quæ haud longe di$tant a quæ$itâ or-
dinatâ.</p>
<p>Ex. 2. Sit æquatio <math>a x^n + b x^n-1 + c x^n-2 + &c. ... f x + g + {b / x} +
{k / x^2} + {l / x^3} ... + {p / x^m} = y</math>; ducatur hæc æquatio in <math>x^m</math>, & re$ultat <math>a x^n+m
+ b x^n+m-1 + c x^n+m-2 + &c. ... + p = v x^m</math>, cujus coefficientes <math>a, b,
c, d, &c.</math> inveniri po$$unt e datis corre$pondentibus valoribus quanti-
tatum <math>x & y</math>, eâdem methodo ac coefficientes æquationis <math>a x^n+m +
b x^n+m-1 + c x^n+m-2 .... + p = v</math>; $cribatur enim in priori æquatione
pro <math>y x^m</math> ejus valor <math>v</math>, & re$ultat $ecunda.</p>
<p>Ex. 3. Sit æquatio <math>a x^n + b x^n-1 + c x^n-2 .... + f x + g + {b x^s / x + α}

<pb 559>[0611]<rh>SERIERUM, &c.</rh>
+ {k x^s+1 / (x + α)^2} + &c. + {l x^t / x + β} + &c. = y</math>, ducatur hæc æquatio in
<math>(x + α)^2 × (x + β) × &c</math>. & re$ultat æquatio, cujus re$olutio ($i modo
dentur <math>α, β, γ, &c</math>.) haud differt a præcedente.</p>
<p>Cor. 3. Hinc per <math>n</math> puncta duci pote$t parabolico hyperbolica
curva eâdem facilitate ac parabolica curva per eundem punctorum
numerum.</p>
<p>Ex. 4. Sit <math>y = {1 / a x^n + b x^n-1 + c x^n-2 + &c.}</math>, $i modo pro <math>y</math> $criba-
tur <math>{1 / v}</math>, re$ultabit præcedens æquatio <math>v = a x^n + b x^n-1 + c x^n-2 + &c.</math>
ad parabolicam curvam.</p>
<p>Ex. 5. Sit æquatio <math>a x^n + b x^n-1 + cx^n-2 + &c./A x^m + B x^m-1 + C x^m-2 + &c.} = y</math>, & ejus re-
$olutio aliquantulum differt a præcedente.</p>
<h>PROB. XXVII.</h>
<h it>Invenire $olidum algebraicum, quod per <math>m + n + I</math> data puncta tran$eat.</h>
<p>A$$umatur æquatio <math>a + b x + c x^2 + d x^3 ... + f x^n + B z + C z^2
... F z^m = y</math>, ubi <math>x & z</math> $unt duæ ab$ci$$æ & <math>y</math> earum corre$pondens
ordinata; ex datis <math>m + n + 1</math> corre$pondentibus valoribus duarum
ab$ci$$arum & ordinatæ erui po$$unt coefficientes <math>a, b, c, &c. B, C, &c.</math>
Hæc e$t maxime $implex formula, quam for$an recipere pote$t pro-
blema.</p>
<h>PROB. XXVIII.</h>
<p it>1. Sit <math>P</math> algebraica functio quantitatum <math>x, y, z, &c.</math> dati generis, quœ
<math>m</math> incognitas quantitates invariabiles babet, & dentur <math>m</math> corre$pondentes
functionis <math>P</math> & quantitatum <math>x, y, z, &c.</math> valores; invenire <math>m</math> incognitas
quantitates <math>a, b, c, d, &c.</math></p>
<p>Scribantur in datâ functione <math>P</math> pro literis <math>x, y, z, &c.</math> earum cor-
re$pondentes valores re$pective & pro quantitatibus re$ultantibus (<math>P</math>)

<pb 560>[0612]<rh>DE SUMMATIONE</rh>
earum re$pectivi valores, & exorientur <math>m</math> æquationes totidem incog-
nitas quantitates <math>a, b, c, d, &c.</math> habentes, e quibus inveniri po$$unt
valores prædictarum quantitatum <math>a, b, c, d, &c.</math></p>
<p>2. Si vero dentur duæ vel plures quantitates <math>P, Q &c.</math> quæ $unt
functiones quantitatum <math>x, y, z, &c.</math> & corre$pondentes valores quan-
titatum <math>P, Q, &c. x, y, z, &c.</math> pro $uis quantitatibus re$pective $ub-
$tituantur, & $i exoriantur tot æquationes quot incognitæ quantitates;
tum ex iis deduci po$$unt incognitæ quantitates quæ$itæ.</p>
<p>Cor. 2. Si æquationes exortæ $int $implices, tum e $implicibus
æquationibus facile con$tabunt incognitæ quantitates quæ$itæ.</p>
<h>PROB. XXIX.</h>
<p it>_1._ Invenire $ummam fractionum, quarum numeratores cre$cunt juxta
progre$$ionem $eriei numerorum differentiam con$tantem habentium, vel
quarum ultimœ differentiœ eju$dem ordinis fere $unt inter $e in ratione
œqualitatis, quarumque denominatores con$tituant progre$$ionem quamlibet
geometricam.</p>
<p>Sint 1: 1 - x communis denominatorum ratio, etiamque <math>d′, d″, d′″,
&c.</math> differentiæ primæ, $ecundæ, tertiæ, &c. numeratorum datæ $eriei
<math>A + B x + C x^2 + D x^3 + &c.</math> & erit per theor. præc. quique terminus,
cujus di$tantia e primo $it <math>(p) = (A - p d′ + p · {p-1 / 2} d″ - p · {p - 1 / 1}
· {p - 2 / 2} d′″ + &c.)x^p</math>; $ed erit <math>{A / 1 - x} = A(1 + x + x^2 ... x^p + &c.),
{d′ / (1 - x)^2} = d′ (1 + 2x + 3x^2 ... p x^p-1 + &c.), {d″ / (1 - x)^3} = d″ (1 +
3x + 6x^2 ... p · {p + 1 / 2}x^p-1 + &c.), {d′″ / (1 - x)^4} = d′″ (1 + 4x + 10 x^2
... + p.{p + 1 / 2}·{p + 2 / 3}x^p-1 + &c.)</math>, &c. unde $umma $eriei erit
<math>{A / 1 - x} - {d′x / (1 - x)^2} + {d″x^2 / (1 - x)^3} - {d′″x^3 / (1 - x)^4} + &c.</math></p>

<pb 561>[0613]<rh>SERIERUM, &c.</rh>
<p>In hâc $ummâ inventâ ponatur <math>x = {1 / z}</math>, & evadet $eries data <math>A +
{B / z} + {C / z^2} + {D / z^3} + &c.</math> & ejus $umma <math>{A z / z - 1} - {d′z / (z - 1)^2} + {d″z / (z - 1)^3}
- {d′″z / (z - 1)^4} + &c.</math></p>
<p>Hæc facile con$tant ex hâc propo$itione, nempe <math>(1 - x)^-p = 1 +
p x + p · {p + 1 / 2}x^2 + p. {p + 1 / 2} · {p + 2 / 3}x^3 + &c.</math></p>
<p>2. Si vero requiratur $umma <math>n</math> primorum terminorum $eriei
<math>A + Bx + Cx^2 + Dx^3 + &c.</math> inveniatur $umma $eriei in in-
finitum progredientis, viz. <math>{A / 1 - x} - {d′x / (1 - x)^2} + {d″x^2 / (1 - x)^3} - &c.</math> e
quâ $ubtrahatur $umma $eriei, cujus di$tantia a primo $it <math>n</math>, in infini-
tum progrediens, i. e. $it <math>Q x^n</math> terminus $eriei oriundus po$t <math>n</math> nume-
rum terminorum, $it præterea <math>D′</math> prima primarum differentiarum
exurgentium po$t terminum <math>Q x^n, D″</math> prima $ecundarum, <math>D′″</math> prima
tertiarum, &c. tum erit $eries incipiens a termino <math>Q x^n</math> atque in
infinitum pergens <math>= {Q x^n / 1 - x} - {D′x^n+1 / (1 - x)^2} + {D″x^n+2 / (1 - x)^3} - &c.</math> $ed
<math>Q = A - nd′ + n · {n - 1 / 2}d″ - n · {n - 1 / 2}·{n - 2 / 4}d′″ + &c. D′ =
d′ - nd″ + n · {n - 1 / 2}d′″ - &c. D″ = d″ - n d′″ + &c.</math> unde <math>{Qx^n / 1 - x}
-{D′ x^n+1 / (1 - x)^2} + {D″ x^n+2 / (1 - x)^3} - &c. =
x^n ({A / 1 - x} - {nd′ / 1 - x} + {n · {n - 1 / 2}d″ / 1 - x} - &c.
- {xd′ / (1 - x)^2} + {n x d″ / (1 - x)^2} - &c.
+ {x^2 d″ / (1 - x)^3} - &c.</math>, $cribatur <math>x = {1 / z}</math> & re$ultat

<pb 562>[0614]<rh>DE SUMMATIONE</rh>
<math>{1 / z^n-1}({A / z - 1} - {nd′ / z - 1} + {n · {n - 1 / 2}d″ / z - 1} - {n · {n - 1 / 2} · {n - 2 / 3}d′″ / z - 1} + &c.
- {d′ / (z - 1)^2} + {n d″ / (z - 1)^2} - {n.{n - 1 / 2}d′″ / (z - 1)^2} + &c.
+ {d″ / (z - 1)^3 - {nd′″ / (z - 1)^3} + &c.
- {d′″ / (z - 1)^4} + &c.})</math></p>
<h>THEOR. XXII.</h>
<p>Sit convergens $eries <math>a + b x + c x^2 + d x^3 + e x^4 + &c.</math> in infi-
nitum, cujus termini lente convergant; $int primæ primarum, $e-
cundarum, tertiarum, &c. differentiæ re$pective <math>D, D′, D″, D′″, &c.</math>
i. e. <math>a - b = D, a - 2 b + c = D′, a - 3 b + 3 c - d = D″, &c.</math>
& erit <math>a + b x + c x^2 + d x^3 + e x^4 + f x^5 + &c. = {a / 1 - x} - {Dx / (1 - x)^2}
+ {D′x^2 / (1 - x)^3} - {D″ / (1 - x)^4} + &c. = {a / 1 - x} - {(a - b)x / (1 - x)^2} + {(a - 2b + c)x^2 / (1 - x)^3}
{(a - 3b + 3c - d)x^3 / (1 - x)^4} + &c. = {a / 1 + x} + {(a + b)x / (1 + x)^2} + {(a + 2b + c)x^2 / (1 + x)^2}
+ {(a + 3b + 3c + 3d)x^3 / (1 + x)^4} + &c.</math> ubi <math>a, a + b, a + 2b + c, a + 3b
+ 3c + d, &c.</math> $unt primæ $ucce$$ivæ $ummæ re$pective.</p>
<p>2. Sit convergens $eries <math>a - b x + c x^2 - d x^3 + &c.</math> tum erit <math>a -
b x + c x^2 - d x^3 + &c. = {a / 1 + x} + {(a - b)x / (1 + x)^2} + {(a - 2b + c)x^2 / (1 + x)^3} +
{(a - 3b + 3c - d)x^3 / (1 + x)^4} + &c. = {a / 1 - x} - {(a + b)x / (1 - x)^2} + {(a + 2b + c)x^2 / (1 - x)^3}
- {(a + 3b + 3c + d)x^3 / (1 - x)^4} + &c.</math></p>

<pb 563>[0615]<rh>SERIERUM, &c.</rh>
<p>In quibu$dam ca$ibus $eries $ic transformatæ magis celeriter con-
vergant, in aliis vero minime. E. g. 1. Si data $eries con$tet e termi-
nis in geometricâ progre$$ione, viz. $it $eries <math>a + a^2 x + a^3 x^2 + a^4 x^3
+ &c.</math> tum erit $eries hoc modo derivata <math>= {a / 1 - x} - {a - a^2 / (1 - x)^2}x +
{a - 2a^2 + a^3 / (1 - x)^3}x^2 - &c. = {a / 1 - x} - {1 - a / 1 - x}P x + {1 - a / 1 - x}Q x - {1 - a / 1 - x}
Rx + &c.</math> ubi <math>P, Q, R, &c.</math> præcedentes terminos re$pective deno-
tant, unde con$tat $eriem derivatam etiam e$$e geometricam $eriem,
cujus communis ratio e$t <math>1: {1 - a / 1 - x}x</math>. Si hæc ratio major $it quam
<math>1: ax</math>, tum magis celeriter converget $eries re$ultans quam data;
$in aliter vero non. 2. Si termini ad infinitam di$tantiam con$tituti
magis celeriter convergant quam geometrica $eries, tum $eries derivata
nunquam converget in majore ratione quam $eries e terminis in geo-
metricâ progre$$ione con$tans; for$an vero in minore ratione, & for-
$an etiam diverget. E. g. Si <math>1 - x</math> major $it quam <math>x</math>, tum $emper
$eries prædicta <math>{a / 1 - x} - {Dx / (1 - x)^2}, &c.</math> evadet ultimo in geometrica pro-
gre$$ione; $i minor $it, $emper diverget; $i vero <math>1 - x = x</math>, tum vel
diverget vel converget in ratione minori quam quæcunque $eries con-
$tans e terminis in geometricâ progre$$ione. 3. Si termini ad infinitam
di$tantiam con$tituti evadant prope in ratione æqualitatis; tum, $i
<math>1 - x</math> major $it quam <math>x</math>, $eries vel converget in ratione $eriei, cujus
termini $int in geometricâ progre$$ione, vel in majore; $in aliter $eries
re$ultantes vel po$$unt e$$e convergentes in ratione prædictâ geome-
tricâ, vel in majore, vel in minore, vel divergentes; $ed plerumque $i
modo con$tent ex algebraicis functionibus di$tantiæ a primo $eriei ter-
mino, & data $eries $it convergens, re$ultans etiam erit convergens.</p>
<p>3. Erit <math>1 + bx + b^2 x + b^3 x^3 + b^4 x^4 + &c. = {1 / 1 - bx} = {1 / 1 - ax}
- {(a - b)x / (1 - ax)^2} + {(a^2 - 2ba + b^2)x^2 / (1 - ax)^3} - {(a^3 - 3ba + 3ba^2 - b^3)x^3 / (1 - ax)^4}
+ &c.</math></p>

<pb 564>[0616]<rh>DE SUMMATIONE</rh>
<p>4. Infinitæ $eries haud nunquam reduci po$$unt ad diver$as geome-
tricas $eries, ita ut termini $ucce$$ivi magis celeriter convergant.</p>
<p>E terminis $eriei datæ, qui maximi $int ad in$initam di$tantiam
facile erui pote$t prima geometrica $eries; $ubtrahatur hæc $eries de
datâ infinitâ $erie, & e terminis re$ultantis $eriei, qui maximi $int ad
infinitam di$tantiam, inveniatur $ecunda geometrica $eries; $ubtraha-
tur hæc $eries de $erie prius re$ultante, & per methodum præceden-
tem continuo repetitam progrediendum e$t.</p>
<p>Si vero $eries per hanc methodum inventæ haud celeriter conver-
gant, transformetur ea in $eriem quantitatum huju$ce formulæ
<math>{a x^r / x - α} + {b x^r ′ / x^2 - β x - γ} + {c x^r″ / x^3 - δ x^2 + ε x - ι} + &c.</math> vel huju$ce
<math>{a x^q + b x^r / c x^s + d x^s}</math>, vel infinitarum aliarum formularum; vel in $eriem e
diver$is $eriebus, vel quantitatibus, vel rationalibus, vel irrationalibus
utcunque in $e$e ductis, &c. ita vero ut $eries re$ultans maxime cele-
riter convergat. E. g.</p>
<p>Sit data $eries <math>A + B x + C x^2 + D x^3 + &c.</math> a$$umatur
<math>{a / 1 - p x} + {b x / (1 - p x)^2} + {c x^2 / (1 - p x)^3} + &c.</math> vel <math>{a / 1 - p x} + {b x / 1 - q x}
+ {c x^2 / 1 - r x} + &c.</math> vel quæcunque alia $eries con$tans e terminis
vel rationalibus vel irrationalibus; $i modo hæ $eries fiant æquales
datæ $eriei, quicunque $it valor quantitatis <math>x</math>, & convergant.</p>
<p>Sit formula <math>A x + B x^3 + C x^5 + &c.</math> & a$$umi pote$t $eries <math>{A x / 1 + x^2}
+ {a x^3 / (1 + x^2)^2} + {b x^5 / (1 + x^2)^3} + &c.</math></p>
<p>Eadem etiam principia applicari po$$unt ad datas $eries exprimen-
das cuju$cunque aliæ formulæ vel rationalis vel irrationalis vel a$cen-
dentis vel de$cendentis. E. g. Sit $eries data <math>A + {B / x} + {C / x^2} + {D / x^3} +

<pb 565>[0617]<rh>SERIERUM, &c.</rh>
&c.</math> a$$umatur $eries <math>{a x / x - p} + {b / (x - p)^2} + {c / (x - p)^3} + {d / (x - p)^4} +
&c.</math> reducatur hæc $eries ad alteram $ecundum dimen$iones recipro-
cas quantitatis <math>x</math> progredientem, & evadet <math>a + {p a / x} + {(a p^2 + b) / x^2} +
{(a p^3 + 2 b p + c) / x^3} + {(a p^4 + 3 b p^2 + 3 c p + d) / x^4} + &c.</math> & ex æqua-
tis corre$pondentibus hujus & datæ æquationis terminis re$ultant
<math>a = A, p = {B / A}, b = C - a p^2, c = D - a p^3 - 2 b p</math>, & $ic deinceps.</p>
<h>REGULA.</h>
<p>1. In $ummatione $erierum $imul addantur plures primi termini,
& ex ultimâ ratione terminorum ad infinitam di$tantiam inveniatur
prima approximatio (<math>A</math>) ad $ummam e reliquis terminis; deinde in-
veniatur terminus (<math>t</math>), cujus $umma e$t <math>A</math>, etiamque differentia (<math>D</math>)
inter hunc terminum & ejus corre$pondentem datæ $eriei terminum;
ex ultimâ ratione differentiæ <math>D</math> inveniatur prima approximatio (<math>B</math>)
ad $ummam, cujus terminus e$t <math>D</math>, & erit <math>A ± B</math> propior approxi-
matio ad $ummam datæ $eriei quæ$itam; & $ic ex operatione continuo
repetitâ re$ultabunt novæ approximationes ad $ummam datæ $eriei
quæ$itam magis magi$que appropinquantes.</p>
<p>1. 2. Inveniatur e quâcunque methodo quantitas a $ummâ haud
longe di$tans; deinde inveniatur differentia inter terminos huju$ce &
datæ $eriei; & $eriei, cujus terminus e$t differentia prædicta, inve-
niatur proxime $umma; huju$ce quantitatis terminorum & diffe-
rentiæ prædictæ inveniatur differentia; deinde inveniatur quantitas
haud longe di$tans e $ummâ $eriei, cujus terminus e$t ea differentia;
& $ic deinceps; & ultimo ad $ummam $eriei accedere liceat.</p>
<p>Ex. 1. Sit terminus generalis <math>{a z^m + b z^m-1 + c z^m-2 + &c. / z^n + p z^n-1 + q z^n-2 + &c.}</math>, ubi <math>z</math>
$it di$tantia a primo $eriei termino, <math>m</math> minor $it quam <math>n</math> per quanti-
tatem majorem quam 1; i. e. $eries $it convergens: addantur plurimi

<pb 566>[0618]<rh>DE SUMMATIONE</rh>
primi termini, ita quidem ut <math>z</math> multo major fiat quam quæcunque
radix vel po$$ibilis vel impo$$ibilis æquationum <math>z^n + p z^n-1 + q z^n-2 +
&c. = 0, & a z^m + b z^m-1 + c z^m-1 + &c. = 0</math>; deinde ultima ratio
generalis termini ad infinitam di$tantiam erit <math>{a z^m / z^n} = {a / z^n-m}</math>, & con$e-
quenter prima approximatio <math>A</math> ad $ummam datæ $eriei erit <math>$ · {a z^. / z^n-m} =
{- a / (n - m - 1) z^n-m-1}</math>; terminus vero <math>t</math> (cujus $umma e$t <math>A</math>) erit
<math>{- a / n-m-1} × ({1 / z^n-m-1} - {1 / (z + 1)^n-m-1}) = {-a / n - m - 1} ×
{(n - m - 1) z^n-m-1 + (n - m - 1)({n - m - 2 / 2})z^n-m-1 + &c. / z^2n-2m-2 + (n - m - 1)z^2n-2m-1 + &c.}</math>; diffe-
rentia vero (<math>D</math>) inter hunc terminum & terminum <math>{a z^m + b z^m-1 + &c. / z^n + p z^n-1 + &c.}</math>
erit <math>{-((n - m - 2)({n - m - 1 / 2})a - (n - m - 1)(b - ap))z^n-m-3 + &c. / (n - m - 1)z^2n-2m-2 + &c.}</math>,
& con$equenter erit <math>B = (- (n - m - 2) × {1 / 2}a + (b - a p))(α) $.
{z^. / z^n-m+1} = (α) × {-1 / (n - m) z^n-m}</math>, & exinde invenitur propicr approxima-
tio <math>A ± B</math>; & $ic deinceps.</p>
<p>Aliter: pro $umma $eriei, cujus generalis terminus e$t <math>{a / z^n-a}</math>, a$$uma-
tur approximatio <math>{a′ / z^{n-m-1 / r} × (z + 1)^{n-m-1 / r} × (z + 2)^{n-m-1 / r} .. (z + r - 1)^{n-m-1 / r}}</math>,
ubi <math>r</math> e$t quicunque integer numerus, & per methodum prius datam
erui po$$unt $ucce$$ivæ approximationes.</p>
<p>2. Et $ic detegi po$$unt approximationes ad $ummam datæ $eriei,
cujus lex exprimitur per quamcunque irrationalem functionem quan-

<pb 567>[0619]<rh>SERIERUM, &c.</rh>
titatis <math>z</math>; etiamque con$tat numerus terminorum, qui primo $imul
addendi $unt: e. g. $it generalis terminus <math>{π / ρ}</math>; tum $umma totidem pri-
morum terminorum inveniatur, ut <math>z</math> evadat major quam ulla radix
æquationis <math>π = 0 vel ρ = 0</math>; quo magis quantitas (<math>z</math>) ex$uperat om-
nes prædictas radices, eo magis converget $eries per hanc methodum
re$ultans, quamvis non in eâdem ratione: vel hæc $umma e notis
regulis inve$tigari pote$t, irrationalis functio reducatur ad $eriem ter-
minorum progredientium $ecundum legem, cujus deduci po$$unt ad
re$pectivos terminos continuæ approximationes; & $ic deinceps.</p>
<p>E. g. Datis $ummis $erierum <math>1 + {1 / 4} + {1 / 9} + {1 / 16} + &c. = A, 1 + {1 / 8} +
{1 / 27} + {1 / 64} + &c. = B, 1 + {1 / 16} + {1 / 81} + {1 / 256} + &c. = C, &c.</math> in infinitum,
etiamque quantitate, quæ e$t talis functio quantitatis <math>z</math> di$tantiæ a
primo datæ $eriei termino, ut reducatur ad $eriem $ub$equentis for-
mulæ <math>a z^-2 + b z^-3 + c z^-4 + &c.</math> tum $umma $eriei erit <math>a A + b B
+ c C + &c.</math></p>
<p>Ex principiis in hoc problemata traditis erui pote$t lex, quam ob-
fervat convergentia datæ $eriei ex hâc methodo derivatæ.</p>
<p>Ex. 2. Sit $eries, cujus termini ultimo $int prope in geometricâ
progre$$ione, tum inveniatur $umma i$tius geometricæ $eriei; deinde
eruatur terminus geometricæ $eriei, $ubtrahatur hic terminus ab ejus
corre$pondente datæ $eriei termino, & deinde per eandem methodum
inveniatur approximatio ad $ummam $eriei, cujus terminus e$t re$ul-
tans differentia; & $ic continuo repetitâ operatione tandem exorietur
$umma quæ$ita.</p>
<p>E. g. Sit infinita $eries <math>a + b + c + d + e + f + g + h + k +
&c.</math> $int <math>r^2 a - c = a′, r^4 a - r′^2 a′ + e = a″, r^6 a - r′^4 a′ + r″^2 a″ - g
= a′″, &c.</math> etiamque <math>r^2 b - d = b′, r^4 b - r′^2 b′ + f = b″, r^6 b - r′^4 b′
+ r″^2 b″ - b = b′″</math>, & $ic deinceps; ubi <math>r = {b / a}, r′ = {b′ / a′}, r″ = {b″ / a″}, &c.</math>
tum erit $umma $eriei <math>{a^2 / a - b} - {a′^2 / a′ - b′} + {a″^2 / a″ - b″} - &c. = a + b
+ c + d + &c.</math></p>

<pb 568>[0620]<rh>DE SUMMATIONE</rh>
<p>Sub$tituantur pro <math>a′, b′; a″, b″; &c.</math> earum re$pective valores, &
facile con$tat exemplum.</p>
<p>Ex a$$umptis diver$is denominatoribus facile diver$is modis dividi
pote$t data infinita $eries in infinitas $eries geometricas.</p>
<p>2. Sit $eries <math>a - b x + c x^2 - d x^3 + &c.</math> & erit $umma quæ$ita
<math>{a / 1 + x} + {d′ x / (1 + x)^2} + {d″ x^2 / (1 + x)^3} + &c. = (1 - x) (a + (a - b) x
+ (a - b + c) x^2 + (a - b + c - d) x^3 + &c.) = (1 - x) ({a / 1 - x} -
{b x / (1 - x)^2} + {b + c / (1 - x)^3} x^2 - {b + 2 c + d / (1 - x)^4} x^3 + {b + 3 c + 3 d + c / (1 - x)^5} x^4 - &c.)</math>,
ubi <math>d′, d″, d′″, &c.</math> re$pective $int <math>a - b, a - 2 b + c, a - 3 b + 3 c
- d, &c.</math> i. e. prima primarum, $ecundarum, tertiarum, &c. differen-
tiarum.</p>
<p>1. 2. Sit $eries <math>a + b x + c x^2 + d x^3 + &c.</math> & erit $umma <math>{a / 1 - x}
- {d′ x / (1 - x)^2} + {d″ x^2 / (1 - x)^3} - &c. = (1 + x) (a - (a - b) x + (a -
b + c) x^2 - (a - b + c - d) x^3 + &c.) = (1 + x) ({a / 1 + x} +
{b x / (1 + x)^2} + {(c + b) / (1 + x)^3} x^2 + {b + 2 c + d / (1 + x)^4} x^3 + {b + 3 c + 3 d + e / (1 + x)^5}
x^4 + {b + r c + 6 d + 4 e + f / (1 + x)^6} x^5 + &c.)</math> cujus $eriei lex cuicunque
eam in$picienti facile con$tabit.</p>
<p>Sit <math>x = - 1</math> in priori $erie <math>a - b x + c x^2 - &c.</math> vel <math>x = 1</math> in
po$teriori <math>a + b x + c x^2 + &c.</math> & ex his methodis nihil plerumque
deduci pote$t, quoniam in his ca$ibus $eries evadant in$inite magnæ.</p>
<p>Et $ic multiplicetur vel dividatur data $eries vel per <math>1 - x</math> vel <math>1 + x</math>
vel per quamcunque aliam quantitatem; & exorientur novæ $eries,
quarum $ummæ vel per hanc, vel per alias methodos traditas, for$an
innote$cent.</p>
<p>Ex. 4. Sit generalis terminus $eriei, cujus $umma requiritur, <math>=

<pb 569>[0621]<rh>SERIERUM, &c.</rh>
{1 / z<_>2</_>}r<_>z</_></math>, a$$umatur <math>{1 / z^2} × {r^z / 1 - r}</math> pro approximatione ad $ummam $eriei
quæ$itam; $i vero hæc $it $umma $eriei, tum ejus terminus erit <math>{1 / z^2} ×
{r^z / 1 - r} - {1 / (z + 1)^2} × {r^z+1 / 1 - r} = {(1 - r)z^2 + 2z + 1 / z^2 (z + 1)^2} × {r^z / 1 - r}</math>, $ubtra-
hatur hic terminus de termino <math>{1 / z^2}r^z</math>, & re$ultat <math>{((z + 1)^2 (1 - r) - (1 - r)z^2 - 2z - 1)r^z / z^2 (z + 1)^2} (1 - r) =
{- (2z + 1)r^z+1 / z^2 (z + 1)^2 (1 - r)}</math>; deinde pro $ub$e-
quente approximatione a$$umatur <math>{- 2 / z^3} × {r^z+1 / (1 - r)^2}</math>, unde propior ap-
proximatio ad $ummam $eriei erit <math>{1 / z^2} × {r^z / 1 - r} - {2 / z^3} × {r^z+1 / (1 - r)^2} &c.</math>;
& $ic deinceps.</p>
<p>Cor. · Hinc deduci pote$t $umma cuju$cunque $eriei huju$ce ge-
neris, i. e. cujus terminus generalis $it functio algebraica quantitatis
<math>z</math> in <math>r^z</math>; reduci enim pote$t algebraica functio ($i haud per alias notas
methodos detegi po$$it $umma quæ$ita) ad terminos $ecundum reci-
procas dimen$iones quantitatis <math>z</math> progredientes, & deinde per metho-
dum hic traditam deduci pote$t $umma $eriei quæ$ita.</p>
<p>Cor. · Approximatio huju$cemodi $eriei $ic inventa pendet e ra-
tione prius prolatâ, etiamque e ratione quam habet <math>1 - r:r</math>.</p>
<p>Et $ic inveniri pote$t aggregatum e pluribus huju$cemodi $eriebus,
i. e. pluribus $eriebus, quarum termini ultimo $int in geometricâ
progre$$ione.</p>
<p>Hæc principia etiam applicari po$$unt ad $eries, quarum termini
in volvunt functionem quantitatis <math>z</math> continuo cre$centem; $ed hæ $e-
ries, cum <math>z</math> $it permagna quantitas, $ine ullâ transformatione celer-
rime convergunt.</p>
<p>3. Summa $eriei, cujus generalis terminus datur, etiam detegi po-
te$t ex reducendo datum terminum ad $eriem $ecundum dimen$iones
quantitatis <math>z</math> di$tantiæ a primo termino progredientem; deinde ex

<pb 570>[0622]<rh>DE SUMMATIONE</rh>
a$$umendo $eriem formulæ prius traditæ pro ejus integrali, & ejus
incrementum deducendo, & $eriei generalem terminum exprimenti
æquale reddendo.</p>
<p>Ex. 1. Sit generalis terminus <math>z^n</math>; a$$umatur pro ejus integrali
<math>a z^n+1 + b z^n + c z^n-1 + d z^n-2 + e z^n-3 + &c.</math>, cujus inveniatur in-
crementum <math>a((z + 1)^n+1 - z^n+1) + b((z + 1)^n - z^n) + c((z + 1)^n-1
- z^n-1) + &c. = ((n + 1)a) z^n + (nb + (n + 1) · {n / 2}a) z^n-1 + ((n
- 1) c + n · {n - 1 / 2} b + (n + 1) · {n / 2} · {n - 1 / 3}a)z^n-1 + ((n - 2) d +
(n - 1) · {n - 2 / 2} c + n · {n - 1 / 2} · {n - 2 / 3} b + (n + 1) · {n / 2} · {n - 1 / 3} · {n - 2 / 4}
a)z^n-3 + &c. = z^n</math>; & exinde <math>a = {1 / n + 1}, b = -{1 / 2}, c = {1 / 12}n, d = 0
× n. {n - 1 / 2}, e = - {1 / 120} × n · {n - 1 / 2} · {n - 2 / 3}</math>, &c.</p>
<p>Si <math>n</math> $it negativus numerus, tum pro <math>n</math> $cribatur <math>- n</math> in præce-
dente $erie, & re$ultat $eries quæ$ita.</p>
<p>Ex. 2. Invenire $ummam logarithmorum, quorum numeri $unt <math>a</math>,
<math>a + n, a + 2n, ... z - n;</math> huju$ce $ummæ decrementum $it log. <math>z - n</math>, quod
inveniatur log. <math>z - {n / z} - {n^2 / 2z^2} - {n^3 / 3z^3} - {n^4 / 4z^4} - &c.</math>: pro integrali huju$ce
decrementi a$$umatur <math>Az</math> log. <math>z + bz + c -</math> integ. incr <math>{n / 2z} + {d / z} + {e / z^2} +
{f / z^3} + &c.</math>, cujus decrementum erit <math>A(z log. z - (z - n) log.(z - n)) +
bn - {n / 2z} + d({1 / z} - {1 / z-n}) + e({1 / z^2} - {1 / (z-n)^2}) + f({1 / z^3} - {1 / (z - n)^3})
+ &c. = A (n log. z + n - {n^2 / 2z} - {n^3 / 2 · 3z^2} - {n^4 / 3 · 4z^3} - {n^5 / 4 · 5 z^5} -
&c.) + bn - {n / 2z} - {dn / z. z - n} - {2nz - n^2 / z^2 (z - n)^2} × e - {3 z^2 n - 3 n^2 z + n^3 / z^3 (z - n)^3} × f

<pb 571>[0623]<rh>SERIERUM, &c.</rh>
- &c. = An log. z + An + bn - {n / 2z} - {n^2 / 2z} A - ({1 / 2 · 3} An^3 + dn)z^-2 -
({1 / 3 · 4} An^4 + dn^2 + 2ne)z^-3 - ({1 / 4 · 5} An^5 + dn^3 + 3 n^2 e + 3 n f)
z^-4 - &c.</math>; fiant corre$pondentes huju$ce & dati incrementi termini
inter $e æquales, & re$ultant <math>An = 1</math>, unde <math>A = {1 / n}, An = - bn</math>; &
{1 / 2 · 3} A n^3 + d n = ({1 / 2 · 3} n^2 + d n) = {n^2 / 2}</math>, unde <math>d = {n / 3}</math>; & <math>{1 / 3 · 4} An^4 +
d n^2 + 2 n e = {n^3 / 3}</math> & exinde <math>{1 / 3 · 4} n^3 + {n^3 / 3} + 2 n e = {n^3 / 3}, & e = {-n^2 / 1 · 12}</math>, &c.</math>;
& $eries quæ$ita <math>{1 / n} z log. z - {1 / n}z + c - integ. incr. {n / 2z} + {n / 3z} - {n^2 / 24z^2} - &c.</math></p>
<p>Cum duo valores <math>α & β</math> quantitatis <math>(z)</math>, inter quos requiritur inte-
gralis, $int permagni; & inveniatur integralis incrementi <math>{1 / z}</math> inter præ-
dictos valores <math>α & β</math> po$ita, converget data $eries: $i vero <math>α & β</math> vel
<math>α</math> vel <math>β</math> habeat perparvam rationem ad quantitatem <math>n</math>; tum vel inve-
niatur $eries $ecundum dimen$iones quantitatis <math>z</math> a$cendens, vel in-
terpolentur plures $eries inter valores <math>α & β</math> quantitatis <math>z</math> prædictos.</p>
<p>Cor. Interpolare quantitates inter $ucce$$ivos terminos contenti
<math>1 · 2 · 3 · 4 .. z</math>; inveniatur ejus log. qui erit <math>l · 1 + l · 2 + l · 3 ..
+ l · z</math>; huju$ce $eriei inveniatur $umma pro dato valore quantitatis
<math>z</math>, & invenitur log. quantitatis quæ$itæ; e dato logarithmo invenia-
tur numerus ei corre$pondens, & perficitur problema.</p>
<p>Hæc principia etiam applicari po$$unt ad quam plurima con$imilia
contenta.</p>
<p>4. Sit æquatio data relationem inter $ucce$$ivos $eriei datæ termi-
nos (<math>t</math> & <math>t′</math>) exprimens, & facile per methodos infinitarum $erierum
prius datas deduci pote$t $eries, quæ exprimit $ummam datæ $eriei,
i. e. rejiciantur ex datâ vel datis æquationibus omnes termini, qui
perparvi $int re$pectu habito ad reliquos, & ex æquatione vel æquatio-

<pb 572>[0624]<rh>DE SUMMATIONE</rh>
nibus re$ultantibus inveniatur proximus valor $ummæ quæ$itæ, & $ic
deinceps: vel a$$umi po$$unt $eries, quæ continent integrales quæ$itas
in terminis quantitatum <math>z</math> & quorundam præcedentium terminorum,
quibus in datis æquationibus pro $uis valoribus $ub$titutis, & corre-
$pondentibus terminis re$ultantium æquationum inter $e æquatis ex-
inde deduci po$$unt $eries quæ$itæ. E. g. Sit $eries <math>(z^θ + a z^θ-1 +
b z^θ-z + &c.) t = (z^θ + e z^θ-1 + f z^θ-2 + &c.) t′</math>, & exinde <math>t′ =
{z^θ + az^θ-1 + &c. / z^θ + e z^θ-1 + &c.} t = (1 + {a - e / z} + {b - a e + e^2 - f / z^2} + &c.) t</math>;
a$$umatur quantitas <math>b z^α</math> pro $ummâ quæ$itâ, & con$equenter duo
$ucce$$ivi termini <math>t</math> & <math>t′</math> erunt re$pective <math>b (z^x - (z + 1)^x) = - b
(α z^x-1 + α · {α - 1 / 2} z^x-2 + &c.)</math>, & <math>b ((z + 1)^x - (z + 2)^x) = - b
(α z^x-1 + 3 α · {α - 1 / 2} z^x-2 + &c.)</math>, unde <math>t′ = {-b(αz^α-1 + 3α · {α - 1 / 2} z^α-2 + &c.) / -b(α z^α-1 + α · {α - 1 / 2} z^α-2 + &c.)} t = (1 + {α · (α - 1) / z} +
&c.) t</math>, & <math>α · (α - 1) = a - e</math>,
& con$equenter <math>α = {1 / 2} ± √ (a - e + {1 / 4})</math>; & $ic ex continuo repetitâ
operatione, i. e. $cribendo <math>b z^x + k z^β + &c.</math> pro $ummâ quæ$itâ fa-
cile deduci pote$t approximatio propior ad veram $ummam; & $ic de-
inceps.</p>
<p>Et $ic de quâcunque datâ relatione huju$modi inter $ucce$$ivos ter-
minos <math>t, t′, t″, &c.</math></p>
<p>Ex. 2. Sit <math>t′ = ({r z^θ + a z^θ-1 + b z^θ-2 + &c. / z^θ + e z^θ-1 + f z^θ-2 + &c.}) t = (r + {a - r e / z}
+ {b - rf - a e + r e^2 / z^2} + &c.)t</math>; pro $ummâ quæ$itâ $cribatur <math>{z^α r^z-π / 1 - r}</math>;
& termini $ucce$$ivi <math>t</math> & <math>t′</math> erunt <math>({z^α / 1 - r} - {(z + 1)^α r / 1 - r})
r^z-π = (z^α - {α z^α-1 r / 1 - r} + &c.) r^z-π</math>, & <math>({(z + 1)^α r - (z + 2)^α r^2 / 1 - r})

<pb 573>[0625]<rh>SERIERUM, &c.</rh>
r^α-π = (r z^a - {(2 r^2 - r) / 1 - r} α z^α-1 + &c.) r^z-π</math>, unde <math>t′ =
({r z^α - {2 r^3 - r / 1 - r} α z^α-1 + &c. / z^α - {α z^α-1 r / 1 - r} + &c.}) t = (r + r · {α / z} + &c.) t</math>; deinde fiat
<math>r + {a - r e / z} = r + r · {α / z}</math>, & con$equenter <math>r α = (a - r e)</math>; & $ic de-
inceps.</p>
<p>Hinc etiam e principiis hìc traditis deduci po$$unt propiores appro-
ximationes.</p>
<p>Si vero indices datæ æquationis $int re$pective <math>θ, θ - σ, θ - 2 σ</math>,
&c. vel data æquatio $it quæcunque algebraica vel etiam fluxionalis
vel incrementialis functio $ucce$$ivorum $eriei quæ$itæ terminorum,
& quantitatis <math>z</math> di$tantiæ a primo $eriei termino; ex principiis hic
traditis deduci pote$t ejus $umma.</p>
<p>Convergentia $eriei e principiis prius traditis dijudicari pote$t.</p>
<p>Sit æquatio <math>(z^θ + a z^θ-1 + &c.) t = (r z^θ-π - e z^θ-π-σ + &c.) t′ +
(s z^θ-π′ + &c.) t″ + &c.</math>, ubi <math>π, π′, σ</math>, &c. $unt affirmativæ quantitates;
& ultimo erit $eries maxime convergens.</p>
<p>Hæ $eries transformari po$$unt per methodos prius traditas, ita ut
haud raro magis celeriter convergant.</p>
<p>Ex. 3. Sit æquatio <math>(z^θ + a z^θ-1 + b z^θ-2 + &c.) t = (r z^θ + s z^θ-1
+ t z^θ-2 + &c.)t′</math>; reducatur data æquatio ad æquationem <math>t = (r + {a′ / z}
+ {b′ / z^2} + &c.)t′</math>; a$$umatur $umma $eriei <math>S = α × {z + m / z + n} t</math> prope; tum
erit <math>S′ = α × {z + m + 1 / z + n + 1}t′</math> prope; & exinde <math>S - S′ = α × ({z + m / z + n} t
- {z + m + 1 / z + n + 1} t′) = t</math>, unde <math>t = {- α (z + n) · (z + m + 1) / (1 - α) z + n - α m · z + n + 1} t′</math>
prope; reducatur hæc æquatio ad æquationem <math>t = ({- α / 1 - α} + {β / z} +<pb 574>[0626]<rh>DE SUMMATIONE</rh>
{γ / z^2} + &c.)t′</math>; & ex æquatis corre$pondentibus huju$ce & datæ æqua-
tionis terminis re$ultant tres æquationes <math>{- α / 1 - α} = r, β = a′ & γ = b′</math>
e quibus erui po$$unt <math>α = {r / r - 1}, m & n</math>: deinde a$$umatur $umma
$eriei <math>S = α × {z + m / z + n} t + π × {z + m′ / z + n′}t′</math> prope; & exinde per metho-
dum prius traditam, i. e. inveniendo differentiam <math>S - S′ = α ×
({z + m / z + n}t - {z + m + 1 / z + n + 1} t′) + π ({z + m′ / z + n′} - {z + m′ + 1 / z + n′ + 1})t″ = t</math>; &
in hâc æquatione $cribendo pro <math>t″</math> ejus valorem <math>(r + {a′ / z + 1} +
{b′/(z + 1)^2 + &c. = r + {a′ / z} + {b′ - a′ / z^2} + &c.)t′}</math>, & æquando corre$pon-
dentes re$ultantis & datæ æquationis terminos deduci po$$unt coeffi-
cientes <math>π, m′ & n′</math> & $ic deinceps: vel a$$umatur $umma <math>S = α
× {z^b + m z^b-1 + n z^b-2 + &c. / z^s + m′ z^s-2 + n′z^s-2 + &c.} × t</math> vel <math>S = (α z^b + m z^b-1 + n z^b-2 +
&c.)t</math>; pro <math>z</math> in his æquationibus $cribatur <math>z + 1</math>, & re$ultabunt
<math>S′ = α × {(z + 1)^b + m × (z + 1)^b-1 + &c. / (z + 1)^s + m′ × (z + 1)^s-1 + &c.} × t′</math>; &c.; & ex æquatione
<math>S - S′ = t</math> & æquatione exinde re$ultante <math>t = (a z^k + b z^k-1 + c z^k-2
+ &c.)t′</math>, acquiri po$$unt coe$$icientes <math>α, m, n, &c., m′, n′, &c.</math></p>
<p>5. Datâ æquatione relationem inter diver$os terminos <math>t, t′, t″,
... t″</math> exprimente; invenire $ummam $eriei, quæ prædictam ha-
beat relationem: a$$umatur <math>t = a + {b / z} + {c / z^2} + {d / z^3} + &c.</math>, &
exinde <math>t′ = a + {b / z + 1} + {c / (z + 1)^2} + &c. = a + {b / z} + {c - b / z^2} + &c.,
t″ = a + {b / z + 2} + {c / (z + 2)^2} + &c. = a + {b / z} + {c - 2b / z^2} + &c., &c.</math>;

<pb 575>[0627]<rh>SERIERUM, &c.</rh>
$cribantur hæ quantitates pro $uis valoribus in datâ æquatione; & ex
æquatis inter $e vel nihilo corre$pondentibus æquationum re$ultan-
tium terminis erui po$$unt coe$$icientes <math>a, b, c, &c.</math>; & exinde $umma
<math>S = a z +</math> integ. increm. <math>({b / z}) + {c / z} + {d - c / 2z · z + 1} + &c.</math>: vel a$$u-
matur <math>t = a + {b / z} + {c / z · z + 1} + {d / z · z + 1 · z + 2} + &c.,</math> & exinde
<math>t′ = a + {b / z + 1} + {c / z + 1 · z + 2} + &c. = a + {b / z} + {c - b / z · z + 1} + &c.,
t″ = a + {b / z + 2} + {c / z + 2 · z + 3} + &c. = a + {b / z} + {c - 2b / z · z. + 1} + &c.,</math>
&c. $cribantur hæ quantitates pro $uis valoribus in datâ æquatione,
& ex æquatis corre$pondentibus re$ultantis æquationis terminis erui
po$$unt coe$$icientes <math>a, b, c, &c.;</math> unde $umma quæ$ita erit <math>a z +</math> integ.
increm. <math>({b / z}) + {c / z} + {d / 2 z · z + 1} + &c.</math></p>
<p>Si vero termini haud exprimi po$$int per præcedentes $eries, tum
a$$umatur $eries <math>{((z + 1)^s - z^s)a / z^s . (z + 1)^s} + {((z + 1)^s+1 - z^s+1) b / z^s+1 · (z + 1)^s+1} +
{((z + 1)^s+2 - z^s+2)c / z^s+2 (z + 1)^s+2} + &c.</math>, vel <math>{((z + 1)^s - z^s)a′ / z^s (z + 1)^s} + {((z + 1)^s+1 - z^s+1)b′ / z^s+1 (z + 1)^s+1}
+ {((z + 1)^s+2 - z^s+2) c′ / z^s+2 (z + 1)^s+2} + &c. + {((z + 1)^s+r - z^s+r)a″ / z^s+r (z + 1)^s+r} +
((z + 1)^s+r+1 - z^s+r+1)b″/z^s+r+1 (z + 1)^s+r+1} + &c. + {((z + 1)^s+2r - z^s+2r)a′″ / z^s+2r (z + 1)^s+2r} +
{((z + 1)^s+2r+1 - z^s+2r+1)b′″ / z^s+2r+1 (z + 1)^s+2r+1} + &c.</math> vel <math>{((z + 1)^s - z^s)a / z^s (z + 1)^s} + {((z + 2)^s - z^s)b / z^s (z + 1)^s (z + 2)^s}
+ {((z + 3)^s - z^s)c / z^s (z + 1)^s ... (z + 3)^s}</math> vel <math>{((z + 1)^s - z^s)a / z^s (z + 1)^s} + {((z + 1)^s+r - z^s+r)b / z^s+r (z + 1)^s+r}
+ {((z + 1)^s+2r - z^s+2r)c / z^s+2r × (z + 1)^s+2r} + &c., &c.</math>, vel a$$umatur $eries <math>a{((z + 1)^r - z^r e^s) / z^r (z + 1)^r}
e^<gap> + b {((z + 1)^t - z^t e^l) / z^t (z + 1)^t} × e^lz + &c., &c.</math>; harum $erierum termino-

<pb 576>[0628]<rh>DE SUMMATIONE</rh>
rum $ummæ innote$cunt; vel infinitæ $eries haud multum di$$imiles
a$$umi po$$unt; reducantur hæ quantitates ad $eries $ecundum dimen-
$iones quantitatis <math>z</math> progredientes, & ex æquatis corre$pondentibus
re$ultantis & datæ æquationis terminis erui po$$unt coefficientes quæ-
$itæ: vel magis generaliter a$$umatur quæcunque $eries $ecundum
reciprocas dimen$iones quantitatis <math>z</math> pro $ummâ <math>S</math>, & deinde inveni-
atur ejus $ucce$$iva $umma <math>S′</math>, & exinde earum differentia <math>S - S′</math>, cu-
jus termini fiant æquales eorum corre$pondentibus terminis in datâ
$erie; & exinde erui po$$unt coefficientes quæ$itæ.</p>
<p>6. Eodem modo pro termino quæ$ito a$$umi pote$t quæcunque $eries
$ecundum reciprocas dimen$iones quantitatis <math>z</math> progrediens, &c.: vel
a$$umi pote$t pro $ummâ <math>S = At + Bt′ + Ct″ + &c.</math>, ubi literæ <math>t,
t′, t″, &c.</math> $ucce$$ivos terminos; & <math>A, B, C, &c.</math> functiones quantitatis
<math>z</math> re$pective denotant: vel quæcunque aliæ functiones terminorum <math>t,
t′, t″, &c.</math>, & quantitatis <math>z</math>: vel pro <math>S</math> in quibu$dam ca$ibus a$$uman-
tur exponentiales quantitates <math>A e^ax^n + bx^n-m + cx^n-2m + &c. + &c.</math></p>
<p>In datis æquationibus relationem inter $ummas <math>S, S′, &c.</math> & termi-
nos <math>t, t′, &c.</math> exprimentibus rejiciantur omnes termini, qui ex datâ
hypothe$i evadant multo minores quam reliqui; & ex æquationibus
re$ultantibus inveniantur approximationes <math>(π, &c.)</math> ad $ummas vel
terminos quæ$itos, &c.; pro $ummis vel terminis prædictis $ub$ti-
tuantur <math>π, &c.</math> per incognitas quantitates auctæ, & ex æquationibus
re$ultantibus per methodum hìc traditam inveniantur propiores ap-
proximationes; & $ic deinceps.</p>
<p>Hæ re$olutiones $æpe $olummodo præbent particularem valorem
$eriei, quod quidem evenit, quoniam pro primâ approximatione $u-
mitur particularis valor; i. e. haud incipitur a primo $eriei termino:
invenire autem generalem valorem perraro u$ui in$erviunt; primo enim
præ$tat valores ad valorem quæ$itum maxime appropinquantes a$$u-
mere, quod re$tinguit re$olutionem ad particularem, quæ nullâ diffi-
cultate urgetur.</p>
<p>7. Si generalis valor per $eriem infinitam quæratur; tum in $erie ex
incrementiali deductâ tot erunt invariabiles quantitates ad libitum

<pb 577>[0629]<rh>SERIERUM, &c.</rh>
a$$umendæ, quot continentur in generali integrali datæ incrementia-
lis æquationis.</p>
<p>Methodus has generales $eries inveniendi ex incrementialibus æqua-
tionibus vel æquationibus relationes inter $ucce$$ivas $ummas & ter-
minos exprimentibus ii$dem difficultatibus urgetur ac methodus in-
veniendi $eries ex fluxionalibus æquationibus datis. Series incre-
mentiales in fluxionales infinitas $eries transformari po$$unt.</p>
<p>Ex. 1. Sit æquatio <math>t′ = {z^2 / z^2 + r}t</math>; a$$umatur <math>t = a + {b / z} + {c / z · z + 1}
+ {d / z · z + 1 · z + 2} + &c.</math> unde <math>t′ = a + {b / z + 1} + {c / z + 1 · z + 2} + &c.
= a + {b / z} + {c - b / z · z + 1} + &c.</math> $cribantur hi valores pro <math>t & t′</math> in datâ
æquatione, & re$ultat <math>t = a + {r A / z} + {r + 1 / 2 · z + 1}B + {r + 4 / 3 · z + 2}C +
{r + 9 / 4 · z + 3}D + &c.</math></p>
<p>Si vero requiratur generalis valor radicis <math>t</math>; a$$umatur <math>t = az^n +
b z^n-1 + &c.</math> unde <math>t′ = a(z + 1)^n + b(z + 1)^n-1 + &c.</math> $cribantur
hæ quantitates pro $uis valoribus in datâ æquatione, & ex æquatis
inter $e corre$pondentibus re$ultantis æquationis terminis erui po$$unt
coefficientes <math>a, b, &c.</math> quæ$itæ.</p>
<p>Hæc propo$itio haud nunquam magis $acilem recipiet $olutionem,
$i modo a$$umatur <math>s = y t</math>, vel <math>t = yt′</math>, vel <math>s = Pyt</math>, &c. ubi <math>P</math> $it fun-
ctio quantitatis <math>z</math>, & <math>s</math> $it $umma $eriei quæ$ita, <math>t</math> vero & <math>t′</math> duo $uc-
ce$$ivi termini, &c.; & deinde ita reducantur æquationes ut re$ultet
æquatio relationem inter <math>y & z</math> exprimens, e quâ inveniatur <math>y</math> in ter-
minis quantitatis <math>z</math>, qui progrediuntur $ecundum formulas in præce-
denti libro traditas.</p>
<h>PROB. XXX.</h>
<p it>Transformare œquationem inter $ucce$$ivas $ummas & terminos de$ignan-
tem in infinitam fluxionalem œquationem relationem inter $ummam <math>S</math> vel

<pb 578>[0630]<rh>DE SUMMATIONE</rh>
terminum <math>t</math>, ejus fluxiones, & <math>z</math> di$tantiam a primo $eriei termino & <math>z^.</math> ex-
primentem.</p>
<p>In datâ æquatione (cum $olummodo contineantur $ucce$$ivi termini
<math>t, t′, t″, t′″, &c., z & z<x>.</x></math>) pro <math>t′, t″, t′″, t″″, &c. t′^m</math>, $cribantur re$pective
<math>t′ = t + {t^. / z^.} × z<x>.</x> + {t^.. / 2z^. ^2} z<x>.</x>^2 + {t^... / 2 · 3 z^. ^3} × z<x>.</x>^3 + {t^.... / 2 · 3 · 4 z^. ^4} × z<x>.</x>^4 + &c.,
t″ = t + {2t^. / z^.} × z<x>.</x> + {4t^.. / 2z^. ^2} × z<x>.</x>^2 + {8t^... / 1 · 3z^ · ^3} × z<x>.</x>^3 + {t^.... / 2 · 3 · 4z^. ^4} × z<x>.</x>^4</math> vel de-
nique <math>t′^m = t + {mt^. / z^.} × z<x>.</x> + {m^2 t^.. / 2z^. ^2} × z<x>.</x>^2 + {m^3 t^... / 2 · 3 z^. ^3} × z<x>.</x>^3 + {m^4 t^.... / 2 · 3 · 4 z^. ^4} × z<x>.</x>^4
+ &c.</math> & re$ultat æquatio relationem exprimens inter <math>t</math> & ejus flu-
xiones, <math>z & z^.</math>; ubi <math>z<x>.</x> & z^.</math> $unt invariabiles quantitates.</p>
<p>Si vero <math>s, s′, s″, &c., s′^m; t, t′, t″, &c., t′^n</math> in datâ æquatione continean-
tur; pro <math>s′^m</math> $cribatur <math>s + {ms^. / z^.} × z<x>.</x> + {m^2 s^.. / 2z^. ^2} × z<x>.</x>^2 + {m^3 s^... / 2 · 3 z^. ^3} × z<x>.</x>^3 + {m^4 s^.... / 2 · 3 · 4 z^. ^4}
× z<x>.</x>^4 + &c.</math>; & pro <math>t′ = S′^r - S′^r+1 = - {s^. / z^.} × z<x>.</x> - {((r + 1)^2 - r^2)s^.. / 2z^. ^2} × z<x>.</x>^2
- {((r + 1)^3 - r^3)s^... / 2 · 3 z^. ^3} × z<x>.</x>^3 - {((r + 1)^4 - r^4)s^.... / 2 · 3 · 4 z^. ^4} × z<x>.</x>^4 - &c.</math></p>
<p>Si <math>y = t ± {t^. / z^.} × z<x>.</x> + {t^.. / 2z^. ^2} × z<x>.</x>^2 ± {t^... / 2 · 3 z^. ^3} × z<x>.</x>^3 + {t^.... / 2 · 3 · 4 z^. ^4} × z<x>.</x>^4 ±
&c.</math>; cum <math>y = t</math> prope, tum erit <math>t = y ∓ {y^. / z^.} × z<x>.</x> + {y^.. / 2z^. ^2} × z<x>.</x>^2 ∓ {y^... / 2 · 3 z^. ^3} × z<x>.</x>^3
+ &c.</math>; & $imiliter $i <math>y = t ± {mt^. / z^.} × z<x>.</x> + {m^2 t^.. / 2z^. ^2} z<x>.</x>^2 ± {m^3 t^... / 2 · 3 z^. ^3} × z<x>.</x>^3 + &c.</math>;
cum <math>y = t</math> prope, tum erit <math>t = y ∓ {my^. / z^.} × z<x>.</x> + {m^2 y^.. / 1 · 2 z^. ^2} × z<x>.</x>^2 ∓ &c.</math></p>
<p>Hæc con$tant ex rever$ione $erierum.</p>
<p>Erit <math>s<x>.</x><x>m</x> = t<x>.</x><x>m - 1</x> = {s^. ^m / z^. ^m} × z<x>.</x>^m + α {s^. ^m+1 / z^. ^m+1} × z<x>.</x>^m+1 + β {s^. ^m+2 / z^. ^m+2} × z<x>.</x>^m+2 + γ {s^. ^m+3 / z^. ^m+3} × z<x>.</x> ^m+3

<pb 579>[0631]<rh>SERIERUM, &c.</rh>
+ δ &c.</math>; ubi erit coefficiens primi termini <math>= 1 = {∓ 1 / 1 · 2 · 3 · 4 .. m}
× (m × 1^m - m · {m - 1 / 2} × 2^m + m · {m - 1 / 2} · {m - 2 / 3} × 3^m - m · {m - 1 / 2} ·
{m - 2 / 3} · {m - 3 / 4} × 4^m + m · {m - 1 / 2} · {m - 2 / 3} · {m - 3 / 4} · {m - 4 / 5} × 5^m - &c.)</math>,
unde <math>m × 1^m - m · {m - 1 / 2} × 2^m + m · {m - 1 / 2} · {m - 2 / 3} × 3^m - &c. =
± 1 · 2 · 3 ... m</math>; $ignum affixum erit <math>+</math>, $i <math>m</math> $it impar numerus;
$in aliter -: $i <math>m</math> $it negativus numerus vel fractio, tum prædicta
infinita $eries erit æqualis termino, cujus di$tantia a primo $it <math>m + 1</math>,
$eriei <math>1 - 1 · 2 + 1 · 2 · 3 - 1 · 2 · 3 · 4 + &c.</math></p>
<p><math>Erunt ± 1 · 2 · 3 · 4 .. (m + 1) α = m × 1^m+1 - m · {m - 1 / 2} × 2^m+1
+ m · {m - 1 / 2} · {m - 2 / 3} × 3^m+1 - m · {m - 1 / 2} · {m - 2 / 3} · {m - 3 / 4} ×4^m+1 + m
· {m - 1 / 2} · {m - 2 / 3} · {m - 3 / 4} · {m - 4 / 5} ×5^m+1 + &c.; 1 · 2 · 3 .. (m + 2) β
= m × 1^m+2 - m · {m - 1 / 2} × 2^m+2 + m · {m - 1 / 2} · {m - 2 / 3} × 3^m+2 - m · {m - 1 / 2} ·
{m - 2 / 3} · {m - 3 / 4} × 4^m+1 + &c.; 1 · 2 · 3 .. (m + 3) γ = m × 1^m+1 - m · {m - 1 / 2}
× 2^m+3 + m · {m - 1 / 2} · {m - 2 / 3} × 3^m+3 - m · {m - 1 / 2} · {m - 2 / 3} · {m - 3 / 4} × 4^m+3
+ &c.</math>; & $ic deinceps.</p>
<p>Erit <math>m × 1^r - m · {m - 1 / 2} 2^r + m · {m - 1 / 2} · {m - 2 / 3} 3^r - m · {m - 1 / 2} ·
{m - 2 / 3} 4^r + &c. = 0</math>, $i modo $it <math>r</math> numerus minor quam <math>m</math>.</p>
<p>Sub$tituantur hæ quantitates pro $uis valoribus in datâ æquatione,
& re$ultat æquatio quæ$ita.</p>
<p>Ex. Sit <math>t = αz^2 t′</math>; in hâc æquatione pro <math>t</math> & <math>t′</math> $cribantur re$pective

<pb 580>[0632]<rh>DE SUMMATIONE</rh>
<math>s - s′ = {s^. / z^.} × z<x>.</x> - {s^.. / 2z^. ^2} × z^. ^2 + &c., & s′ - s^.. = {s^. / z^.} × z^. - {3 s^.. / 2z^. ^2} × z^. ^2 +
&c.</math>, quibus $ub$titutis re$ultat fluxionalis æquatio <math>2 s^. z^. × z^. - s^.. z<x>.</x>^2 &c.
= a z^2 (2 s^. z^. z<x>.</x> - 3 s^.. z<x>.</x>^2 &c.)</math>, ex quâ inveniri pote$t prima approxi-
matio ad $ummam <math>s</math>; & $ic progredi liceat ad inveniendas reliquas.
approximationes.</p>
<h>THEOR. XXIII.</h>
<p>1. Datis <math>n</math> $ummis quantitatis <math>(P) {a y^. ^m / y^. ^γ x^. ^n} + {b y^. ^m-1 / y^. ^γ x^. ^n-1} + {c y^. ^m-2 / y^. ^γ x^. ^n-2} + &c.</math>
(ubi <math>a, b, c, &c.</math> $unt datæ coefficientes & <math>m = n + r)</math> ad <math>n</math> diver$os
valores ab$ci$$æ <math>x</math>, & ordinatæ <math>y</math>; pro relatione inter ab$ci$$am <math>x</math> &
ejus corre$pondentes ordinatas <math>y</math> a$$umatur æquatio <math>A + B x + C x^2
+ D x^3 ... x^n-1 = y</math>, ex hâc æquatione inveniatur valor quantitatis
<math>P</math>; in quantitate re$ultante pro <math>x & y</math> $cribantur earum <math>n</math> corre$pon-
dentes valores, & re$ultabunt <math>n</math> $implices æquationes totidem incog-
nitas quantitates <math>A, B, C, D, &c.</math> habentes, e quibus con$tabunt co-
efficientes <math>A, B, C, D, &c.</math> quæ$itæ.</p>
<p>2. Sint duæ vel tres vel denique <math>n</math> diver$æ quantitates (<math>P</math>) eju$dem
generis, i. e. $int datæ coefficientes <math>a, b, c, d, &c.</math> in diver$is $ummis
diver$æ tum ex <math>n</math> datis vel eju$dem vel diver$i prædicti generis quan-
titatibus ad <math>n</math> diver$os datos valores ab$ci$$æ <math>x</math> & ordinatæ <math>y</math>, per me-
thodum hìc traditam erui pote$t æquatio <math>A + B x + C x^2 ... x^n-1
= y</math>, cui corre$pondent <math>n</math> datæ $ummæ.</p>
<p>Cor. · Ex inventis æquationibus <math>A + B x + C x^2 + &c. = y</math> erui
po$$unt quæcunque quantitates, quæ ad variabiles <math>x & y</math> habent rela-
tionem facile inve$tigandam.</p>
<p>Si vero a$$umatur æquatio <math>a x^r + b x^γ+s + c x^γ+2s + d x^r+35 + &c.
= y</math>, $cribatur <math>z = x^3</math>, & re$ultat æquatio prædictæ formulæ <math>a + b z
+ c z^2 + d z^3 + &c. = {y / x^γ} = v</math>, cujus re$olutio prius datur.</p>
<p>Omnia hæc etiam ad æquationes curvas de$ignantes, quæ habent
a$ymptotos, facile applicari po$$unt.</p>

<pb 581>[0633]<rh>SERIERUM, &c.</rh>
<h>THEOR. XXIV.</h>
<p>1. Sit $eries <math>y = a + b x + c x^2 + d x^3 + &c.</math> $upponantur <math>E, E^., E^..,
E^..., &c.</math> re$pectivi valores quantitatum <math>y, y^., y^.., y^..., &c.</math> cum <math>x = 0</math>; tum
erit integralis <math>y = E + {E^... x / x^.} + {E^.. x^2 / 1 · 2 x^. ^2} + {E^... x^3 / 1 · 2 · 3 x^. ^3} + {E^.... x^4 / 1 · 2 · 3 · 4 x^. ^4}
+ &c.</math> erunt enim <math>E = a, {E^. / x^.} = b, {E^.. / 1 · 2 x^. ^2} = c, {E^... / 1 · 2 · 3 x^. ^3} = d</math>, &
$ic deinceps.</p>
<p>Fluens vero fluxionis <math>$ · y x^. = a x + {b x^2 / 2} + {c x^3 / 3} + {d x^4 / 4} + &c. =
Ex + {E^. x^. ^2 / 2 x} + {E^.. x^3 / 1 · 2 · 3 x^. ^2} + {E^... x^4 / 1 · 2 · 3 · 4 x^. ^4} + &c.</math></p>
<p>F<sc>IG</sc>. 2. Sit ba$is <math>AP = z & z^. = 1</math>, cujus ordinata <math>P M = y</math>; $it
prima ordinata <math>A F = a, A B = 1</math> & area <math>A B E F = A</math>; $int <math>A, B, C,
D, E, &c.</math> $ucce$$ivæ areæ per ordinatas <math>y, y^., y^.., y^..., &c.</math> re$pective gene-
ratæ, i. e. fluentes fluxionum <math>y z^., y^. z^., y^.. z^., y^... z, &c.</math> tum erit <math>A F = a
= A - {B / 2} + {C / 12} - {E / 720} + {F / 30240} - &c.</math> nam <math>A = a + {a^. / 2} + {a^.. / 2 · 3}
+ {a^... / 2 · 3 · 4} + &c.</math> & $ic <math>B = a^. + {a^.. / 2} + {a^... / 6} + &c. C = a^.. + {a^... / 2} + &c.
D = a^... + {a^.... / 2} + &c.</math> &c. reducantur hæ æquationes ita ut extermi-
nentur <math>a^., a^.., a^..., &c.</math> & re$ultat <math>a = A - {B / 2} + {C / 12} - &c.</math> i. e. ex in-
tegralibus fluentis & ejus $ucce$$ivarum fluxionum ivenitur primus
terminus: lex coefficientium ita dici pote$t; $int <math>k = {1 / 2}, l = {1 / 6}, m = {1 / 24},
n = {1 / 120}, &c.</math> etiamque $int <math>K = k = {1 / 2}, L = K k - l = {1 / 12}, M = k L
- l K + m = 0, N = k M - l L + m K - n = - {1 / 720}, &c.</math> & erit
<math>a = A - K B + L C - M D + &c.</math></p>
<p>Cor. 1. Quoniam areæ <math>A, B, C, D, &c.</math> generantur, dum tran$it

<pb 582>[0634]<rh>DE SUMMATIONE</rh>
ordinata <math>P M</math> ab <math>A F</math> ad <math>B E</math>, ob <math>z^. = 1</math>, & exinde area <math>B</math> = differ.
inter ordinatas <math>B E & A F, C</math> = differen. inter fluxiones prædicta-
rum ordinatarum, <math>D</math> = differen. inter earum tertias fluxiones, &c.
$cribantur <math>α, β, δ, ε, &c.</math> igitur pro <math>BE - AF, B^. E - A^. F, B^... E - A^... F</math>,
<math>B^..... E - A^..... F, &c.</math> & evadet <math>a = A - {α / 2} + {β / 12} - {δ / 720} + &c.</math></p>
<p>Dividatur ba$is <math>A a</math> in æquales $ucce$$ivas partes <math>AB, BC, CD, &c.
= z^.</math>, pro $ummâ æquidi$tantium ordinatarum <math>A F + B E + C K +
&c.</math> (excipiatur ultima ordinata <math>a f</math>) $ub$tituatur <math>S</math>, & pro areâ
<math>A F f a</math>, & differentiis <math>a f - A F, a^. f - A^. F, a^... f - A F^..., &c.</math> $cribantur
re$pective <math>A, α, β, δ, ι, &c.</math> tum erit <math>S = {A / z} - {α / 2} + {z β / 12 z^.} - {z 3 δ / 720 z^. ^3}
+ {z^5 ι / 30240 z^. ^5} - {z7 θ / 1209600z^. ^7} + &c.</math></p>
<p>2. Po$itis ordinatâ <math>A F = a</math>, & lineis <math>A R</math> & <math>A r</math> ad oppo$ita latera
puncti <math>A</math> inter $e æqualibus; lineis <math>R V & r v</math> ordinatis ad <math>R & r</math>,
quæ terminant aream <math>R V v r</math>; & quâcunque ordinatâ <math>P M = y</math>;
fluat uniformiter ba$is, & denotent literæ <math>A, C, E, &c.</math> areas $uper
ba$im <math>R r</math> a re$pectivis ordinatis <math>y, y^.., y^...., &c.</math> generatas: $upponatur
<math>A R = z</math>, & erit media ordinata <math>A F = a = {A / 2 z} - {z C / 12 z^. ^2} + {7 z^3 E / 720 z^. ^4} -
{31 z^5 G / 30240 z^. ^6} + &c.</math> per theor. enim <math>{R r v V / 2 z} = {A / 2 z} = a + {z^2 a^.. / 2 · 3 z^. ^2} +
{z^4 a^.... / 2 · 3 · 4 · 5 z^. ^4} + &c.</math> vel <math>a = {A / 2 z} - {z^2 a^.. / 6 z^. ^2} - {z^4 a^.... / 120 z^. ^4} - &c.</math> & $imiliter
<math>a^.. = {C / 2 z} - {z^2 a^.... / 6 z^. ^2} - &c. a^.... = {E / 2 z} - &c.</math> ita reducantur hæ æquationes,
ut exterminentur <math>a^.., a^...., &c.</math> & con$tabit. prop.</p>
<p>Lex huju$ce $eriei $ic enuntiari pote$t; $int <math>k = {z^2 / 2 · 3 z^. ^2}, l = {z^2 k / 4 · 5 z^. ^2},
m = {z^2 l / 6 · 7 z^. ^2}, n = {z^2 m / 8 · 9 z^. ^2}, &c.</math> etiamque <math>K = {k / 2 z} = {z / 12 z^. ^2} L = k K

<pb 583>[0635]<rh>SERIERUM, &c.</rh>
- {l / 2 z}, M = k L - l K + {m / 2 z}, N = k M - l L + m K - {n / 2 z}, &c.</math>
tum erit <math>a = {A / 2 z} - K C + L E - M G + N I - &c.</math> & quoniam
areæ <math>C, E, &c.</math> $unt re$pective fluentes fluxionum <math>y^.. z^., y^.... z^., &c.</math> $i re-
$pectivæ differentiæ fluxionum primi, tertii, quinti, &c. ordinis ordi-
natarum <math>r v & R V</math> exprimantur per <math>β, δ, ι, θ, &c.</math> tum erit <math>a = {A / 2 z}
- {z β / 12 z} + {7 z^3 δ / 720 z^. ^3} - {31 z^5 ι / 30240 z^. ^5} + {127 z^7 θ / 1209600 z^. ^7} - &c.</math></p>
<p>Cor. 1. Sint <math>A F, B E, C K, &c.</math> $eries ordinatarum æquidi$tantium
$uper ba$im <math>A a</math>, quarum <math>A F</math> $it prima & <math>a f</math> ultima; <math>A B</math> eorum com-
munis differentia $it 2 z, & linea <math>A R a b A</math> retro $umpta <math>= z = {1 / 2} A B</math>
& <math>a r</math> $umpta in contrariâ directione <math>{1 / 2} a b = z</math>, & $i area <math>R V r v</math> ab
ordinatis <math>R V & r v</math> terminata dicatur <math>A</math>, & differentiæ inter primas,
tertias, quintas, &c. fluxiones ordinatarum <math>r v & R V</math> exprimantur
per <math>β, δ, ι, θ, &c.</math> & $umma ordinatarum <math>A F + B E + C K ... +
a f = S</math>; erit <math>S = {A / 2 z} - {z β / 12 z} + {7 z^3 δ / 720 z^. ^3} - {31 z^5 ι / 30240 z^. ^5} + {127 z^7 θ / 1209600 z^. ^7} -
{511 z^9 x / 47900160 z^. ^9} + &c.</math> unde <math>A = 2 z S + {z^2 β / 6 z^.} - &c.</math></p>
<p>Cor. 1. Hinc inveniri pote$t <math>m^r + (m + e)^r + (m + 2 e)^r ... n^r =
{1 / (r + 1) · e} (n^r - m^r) + {n^r - m^r / 2} + {r e / 12} (n^r-1 - m^r-1) - {r · r - 1 · r - 2 / 720}
e^3 × (n^r-3 - m^r-3) + &c.</math></p>
<p>Et $ic <math>(m + e)^r + (m + 3 e)^r + (m + 5 e)^r + .. + (n - e)^r = {1 / (r + 1) · 2 e}
× (n^r+1 - m^r+1) - {r e / 12} × (n^r-1 - m^r-1) + {7 r · r - 1 · r - 2 · e^3 / 720} (n^r-3 - m^r-3) - &c.</math></p>
<p>Si <math>r = - s</math> & $umma <math>{1 / m^s} + {1 / (m + e)^s} + {1 / (m + 2 e)^s} + {1 / (m + 3 e)^s}...
{1 / 22^s} (0) = {1 / (s - 1) e m^s-1} + {1 / 2 m^s} + {s e / 12 m^s+1} - {s · s + 1 · s + 2 · e^3 / 720 m^s+3} + &c.</math>

<pb 584>[0636]<rh>DE SUMMATIONE</rh>
<math>{s.s + 1.s + 2.s + 3.s + 4.e5 / 30240m^s+s} + &c.</math> & fic colligi pote$t <math>{1 / (m + e)^s}
+ {1 / (m + 3e)^s} + {1 / (m + 5e)^s} + &c. = {1 / (s - 1)2 e m^s-1} - {se / 12m^s+1} +
{7s.s + 1.s + 2.e^3 / 720m^s+3} - &c.</math></p>
<p>Cor.. Si termini $eriei $int alternatim affirmativi & negativi, in
priori ca$u per theor. erit $umma affirmativorum <math>= {A / 2e} - {α / 2} +
{2eβ / 12} - {8e^3 δ / 720} + &c.</math> $umma vero negativorum per cor. <math>1.{A / 2e} - {eβ / 12}
+. {7e^3 δ / 720} - &c.</math> ubi <math>2 e</math> $it communis di$tantia inter ordinatas, & <math>α =
af - AF; & β, δ, &c.</math> ut antea; ergo differentia inter has duas inve-
nietur <math>{A F - a f / 2} + {eβ / 4} - {e^3 δ / 48} + {e^5 η / 480} - {17e^7 θ / 80640} - &c.</math></p>
<p>Ex. 2. Sint <math>E A = 2, E a = n, A F = log. 2, a f = log. n; & β, δ, η,
&c.</math> denotent fluxiones diver$as ip$ius <math>A F</math>, tum logar. ultimi valoris
contenti <math>{2 / 3} × {4 / 5} × {6 / 7} ×... × {n-2 / n-1} × 2√(n) = {l, 2 - l,n / 2} - {β / 4} + {δ / 48} -
{η / 480} + &c. + l.2 + {l,n / 2} = {3 / 2} × l.2 - {β / 4} + {δ / 48} - {η / 480} + &c. = (ob
{3 / 2} × l.2 = 1 + {β / 12} - {7δ / 720} + {31η / 30240} - &c.) 1 - {2β / 12} + {8δ / 720} - &c. =
(ob β = {1 / 2}, δ = {2 / 8}, η = {24 / 32}, &c.)1 - {1 / 12} + {1 / 360} - {1 / 1260} + {1 / 1680} -
&c.</math> $ed quoniam circumferentia circuli <math>(c)</math> cujus radius e$t unitas, $it
<math>c = 8.{8 / 9}·{24 / 25}·{48 / 49}·{80 / 81}·&c.</math> quæ $eries ita $eribi pote$t <math>c = 4 × {4 / 9} · {16 / 25}
{36 / 49} · {64 / 81}·. {(n - 2)^2 / (n - 1)^2} × n</math>, unde <math>√(c) = {2 / 3} × {4 / 5} × {6 / 7} × {8 / 9} × &c.</math> in infinitum <math>×
2√(n), &</math> ejus <math>log. = {l,c / 2} = 1 - {1 / 12} + {1 / 360} - {1 / 1260} + {1 / 1680} - &c.</math></p>

<pb 585>[0637]<rh>SERIERUM, &c.</rh>
<p>Hæc applicari poffunt ad inveniendas $ummas logarithmorum
quantitatum in arithmeticâ progreffione.</p>
<p>1. Sit <math>3 × {21 / 25} × {77 / 81} × {165 / 169} × {285 / 289} × &c.</math> cujus ultimus valor $it <math>p, &</math>
erit <math>√(p)</math> ultimus valor contenti <math>{3 / 5} × {7 / 9} × {11 / 13} × {15 / 17} .... {n - 4 / n - 2}√(n)</math>; $it
r<x>_</x> quicunque numerus in progreffione <math>3, 7, 11, 15, &c. & N</math> nume-
rus, cujus logarithmus e$t <math>{1 / 2r} - {1 / 3r^3} + {8 / 5r^5} - &c.</math> tum erit <math>√(p)
= {√ r / N} × {3 / 5} × {7 / 9} × {11 / 13} × {15 / 17}... {r - 4 / n - 2}</math>.</p>
<p>2. Et ex his principiis deduci pote$t unicam medii termini ad $um-
mam unicarum binomii e$$e ut::<math>1: {√(c × (n + 1)) / 2N}</math> prope, ubi <math>N</math> $it
numerus, cujus log. e$t <math>{1 / 4 ×(r + 1)} - {1 / 24 × (r + 1)^2} + {1 / 20 × (r + 1)^5}
- &c. & r</math> $it pote$tas binomii cum $it par; vel <math>r + 1</math> $it pote$tas,
cum $it impar.</p>
<p>3. Supponatur ba$is <math>A a</math> dividi in æquales partes, & $it area <math>A F f a
= Q</math>, $umma extremarum ordinatarum <math>A F + a f = A, &</math> $umma om-
nium intermediarum <math>B E + C K + &c. = B</math>, ba$is <math>Aa = R, &</math> eædem
quantitates ut antea per <math>β, δ, η, &c.</math> denotentur; tum erit area <math>A F f a
= Q = ({A / 2 n + 2} + {nB / n^2 - 1}) R - {R^4 δ / 720n^2} + {R^6 η / 30240} × {n^2 + 1 / n^4} - &c.</math>
nam $upponâtur <math>e = {R / n}, S + {af - af / 2} = B + {A / 2} = {nQ / R} + {Rβ / 12n} -
{R^3 δ / 720n^3} + &c.</math> deinde <math>{A F + a f / 2} = {A / 2} = {Q / R} + {Rβ / 12} - {R^3 δ / 720} + &c.</math> ex-
terminetur <math>β</math> ex his duabus æquationibus, & re$ultat area præ-
dicta.</p>
<p>Rejiciantur <math>δ, η, θ; &c. &</math> erit area <math>A F f a = {AR / 2n + 2} + {nBR / n^2 - 1}</math>;
$upponatur <math>n = 2</math>, unde tres $olummodo erunt ordinatæ <math>& B</math> media

<pb 586>[0638]<rh>DE SUMMATIONE</rh>
ordinata, & erit area <math>A F f a = {A + 4B / 6}R - {R^4 δ / 1.720} + {5R^6 η / 11.30240}
- &c.</math> $it <math>n = 3 &</math> quatuor $olummodo $unt ordinatæ, & erit <math>B =</math>
$ummæ primæ & $ecundæ ordinatarum; & area <math>A F f a = {A + 3 B / 8}
× R - {R^4 δ / 1.720} + {R^6 η / 81.3024} - &c.</math> $i vero rejiciantur <math>δ, η, &c.</math> tum
erit area <math>A F f a = {A + 3B / 8} × R</math>.</p>
<p>Sint quinque ordinatæ, <math>A</math> $umma primæ & ultimæ, <math>B</math> $umma
$ecundæ & quartæ, <math>C</math> vero media ordinata; erit area <math>A F f a =
{7A + 32B + 12C / 90} × R - {31R^6 η / 6 × 16 × 16 × 30240} + &c.</math> nam per
regulam pro tribus ordinatis erit <math>{Q / R} = {A + 4C / 6} × R - {R^3 δ / 4 × 720} +
{5R^5 η / 11.30240} - &c.</math> dividatur $ingula ba$is in duas æquales partes, &
inveniatur per eandem regulam area e $ingulâ parte, addantur hæ
duæ partes in unam & re$ultat <math>{2Q / R} = {A + 4B + 2C / 6} - {R^3 δ / 31.720} +
{5R^5 η / 2 × 16 × 16 × 30240} - &c.</math> ex his duabus æquationibus extermine-
tur <math>δ, &</math> con$tabit area quæ$ita.</p>
<p>3. Ex his principiis etiam interpolari poffunt $eries. Sint <math>F N z</math>
figura, cujus $ucceffivæ ordinatæ ad puncta <math>A, B, C, D,&c.</math> $emper
$int æquales $ucceffivis $ummis ordinatarum figuræ <math>FMf, i. e. A F =
A F, Bε = A F + B E, Cx = B ε + C K, D λ = C x + D L, &c.</math>
& requiratur quæcunque intermedia ordinata <math>PN</math> figuræ <math>F N z</math>;
occurrat ordinata <math>P N</math> curvæ <math>F M f</math> in <math>M</math>; $int <math>A F = a, P M = y,
&</math> communis diftantia ordinatarum <math>AB = e, &</math> area <math>A F M P
= Q, y^. - a^. = β, y^... - a^... = δ, &c. & P N = {Q / e} + {a + y / 2} + {eβ / 12} -
{e^3 δ / 720} + {e^5 η / 30240} - &c.</math> quod demon$trari pote$t $upponendo <math>PN</math> $uc-

<pb 587>[0639]<rh>SERIERUM, &c.</rh>
ceffive in locos ordinatarum figuræ <math>F N z</math> ad puncta <math>A, B, C, D, &c.</math>
movere, & exinde ejus $ucceffivos valores per theor. inveniri:
$i vero exterminanda e$t area <math>Q</math>, $upponatur <math>A P = m, &</math> quoniam
<math>{a + y / 2} = {Q / m} + {mβ / 12} - {m^3 δ / 720} + {m^5 η / 30240} - &c.</math> $equitur <math>PN = {a + y / 2}
× {e + m / e} + {e^2 - m^2 / 12 e} × β - {e^4 - m^4 / 720 e} × δ + &c.</math></p>
<p>Sumantur <math>A R & P r</math> in oppo$itis directionibus re$pective <math>= {1 / 2} AB,
R V & r v</math> occurrant <math>F M f</math> in <math>V & v</math>; $it <math>Q</math> area <math>R V v r; & β, δ, η, &c.</math>
differentiæ; quibus prima, tertia, quinta, &c. fluxiones ordinatæ <math>r v</math>
$uperant re$pectivas fluxiones quantitatis <math>R V, & AB = e</math> ut prius;
tum erit ordinata <math>P N = {Q / e} - {e β / 1.12} + {7e^3 δ / 1.720} - {31e^5 η / 31.30240} - &c.</math></p>
<p>Ordinatæ ad puncta <math>A, B, C, D, &c.</math> dicuntur primariæ ordinatæ
figuræ <math>F N z</math> vel <math>F M f.</math> Si <math>P p = A B, & p n</math> occurrat <math>F N z</math> in <math>n &
F M f</math> in <math>m</math>, tum <math>p n = P N + p m</math> vel <math>P N - p m</math>, prout <math>P p</math> in hâc vel
illâ directione ab <math>P</math> $umatur; & hinc ex quâcunque <math>(PN)</math> intermediâ
ordinatâ datâ omnes aliæ ordinatæ figuræ <math>F N z</math>, quarum di$tantiæ ab
eâ <math>(P N)</math> $it <math>n × AB</math>, ubi <math>n</math> e$t integer numerus, facile deduci poffunt
vel addendo vel $ubtrahendo intermedias ordinatas figuræ <math>F M f</math>.</p>
<p>Cor. 1. Sint <math>T′ X & T′ X′</math> primariæ ordinatæ figuræ <math>F M f</math> adjacentes
ad ordinatam intermediam <math>P N</math>; bi$eca <math>T T′</math> in <math>x, &</math> occurrat ordinata
<math>x y</math> curvæ <math>F M f</math> in <math>y</math>; area <math>x y v r = q</math>, ordinata <math>T η</math> ad <math>T</math> curvæ <math>F N z</math>
$it <math>s, x y = y, r v = u</math>; tum <math>P N = s + {q / e} - {e / 1.12} × (u^. - y^.) +
{7 e^3 / 1.720}(u^... - y^...) - &c.</math> nam $i <math>R v = a &</math> area <math>R V v r = Q</math>, tum
<math>R V y x = Q - q & P N = {Q / e} - {e / 1.12} × (u^. - a^.) + {7e^3 / 1.720} × (u^... - a^...)
- &c.</math> per præced. <math>& T η = s = {Q - q / e} - {e / 1.12} × (y^. - a^.) +
{7e^3 / 1.720} × (y^... - a^...) - &c.</math> unde <math>P N - s = {q / e} - {e / 24}(u^. - y^.) + {7e^3 / 1.720}
× (u^... - y^...), & P N = s + {q / e} - {e / 24} × (u^. - y^.) + &c.</math></p>

<pb 588>[0640]<rh>DE SUMMATIONE</rh>
<p>Cor. 2. Ii$dem figuris <math>F n z & F M f</math> manentibus, $it ba$is <math>F f</math>
a$ymptos po$terioris figuræ; & $it <math>A π</math> præcedens primum terminum
<math>A F</math>, minor quam <math>A B, &</math> $umantur <math>A π = b B = c C = D d = &c.</math>
<math>P N</math> ultimo in hoc ca$u erit <math>= T η</math>; i.e. $upponendo <math>(PT′ = π A), π n
+ b e + c k + d l + &c.</math> ultimo <math>= A F + B E + C K + &c.</math> unde
<math>π n = A F - b e + B E - c k + C K - d l + &c.</math></p>
<p>Ex. 1. Sint <math>A F = 1, B E = {1 / 2}, C K = {1 / 3}, D L = {1 / 4}, &c.</math> tum erunt
primariæ ordinatæ figuræ <math>F N z</math> re$pective <math>1, {3 / 2}, {11 / 6}, {25 / 12}, {147 / 60}, &c.</math> $it <math>A π =
{1 / 2} A B, &</math> quoniam intermediæ differentiæ <math>b e = {2 / 3}, c k = {2 / 5}, d l = {2 / 7}, &c.</math>
erit <math>π n = 1 - {2 / 3} + {1 / 2} - {2 / 5} + {1 / 3} - {2 / 7} + {1 / 5} - {2 / 9} &c. = 2 × (1 - log. 2).</math></p>
<p>Ex. 2. Sit <math>A F = 1, B E = {1 / 4}, C K = {1 / 9}, D L = {1 / 16}, &c. &</math> $it <math>A π =
- {1 / 2} A B</math>, tum erunt intermediæ differentiæ <math>b e, c k, d l, &c.</math> re$pective <math>{4 / 9},
{4 / 25}, {4 / 49}, &c. &</math> ordinatæ <math>A F, B ε, C x, D λ, &c.</math> erunt <math>1, 1 + {1 / 4}, 1 + {1 / 4} + {1 / 9},
&c. & π n = 1 - {4 / 9} + {1 / 4} - {4 / 25} + {1 / 9} - {4 / 49} + &c.</math></p>
<p>Ex. 3. Sit <math>A F = 1, B E = {1 / 5}, C K = {1 / 9}, D L = {1 / 13}, &c. & A π = {1 / 2} A B</math>,
tum erit <math>π n = 1 - {1 / 3} + {1 / 5} - {1 / 7} + {1 / 9} - &c. =</math> octuplæ parti circumfe-
rentiæ circuli, cujus radius e$t 1,</p>
<p>Cor.. Cum termini progrediantur $ine $ine, & eorum $ecundæ dif-
ferentiæ decre$cant, ita ut ultimo evane$cant; $it <math>K</math> ultimus valor pri-
marum differentiarum terminorum, & erit <math>π n = {A B - A π / A B} × K +
A F + B E + C K + &c. - b e - c k - d l - &c.</math> quoniam in hoc
ca$u fluxiones quantitatum <math>r v & x y</math> ultimo evane$cunt, <math>P N</math> erit ul-
timo<math> = s + {q / e}, q = K × r x = K × (A B - A π); &</math> con$equenter <math>π n =
{AB - Aπ / A B} × K + A F - b e + B E - c k + &c.</math></p>
<p>Con$imile etiam deduci pote$t theorema, cum $ecundæ differentiæ
terminorum ad limitem appropinquent.</p>
<p>Sit $eries <math>1, 1 × 1, 1 × 2, 1 × 2 × 3, 1 × 2 × 3 × 4, &c.</math> propo$ita,
invenire terminum, qui intermedius $it inter duos primos terminos:
logarithmus termini quæ$iti erit <math>{K / 2} + log. 1 - log. {3 / 2} + log. 2 -
log. {5 / 2} + log. 3 - log. {7 / 2} + log. 4 - &c. = logar.</math> ultimi valoris con-<pb 589>[0641]<rh>SERIERUM, &c.</rh>
tenti <math>{2 / 3} × {4 / 5} × {6 / 7} × {8 / 9} × ... {n / n - 1} √ (n + 1) = log. {√ (c) / 2}</math>, & con$equenter
terminus ip$e æqualis erit dimidio radicis, quæ pote$t quadratum
$emicircumferentiæ circuli, cujus radius e$t <math>1</math>.</p>
<p>Hic ultimo pauca adjicienda $unt de convergentiâ $erierum in hâc
methodo æque ac præcedentibus deductarum.</p>
<p>1. Si ordinata curvæ, cujus fluens requiritur, in infinitum progre-
diatur, for$an ejus area $it finita; i. e. fluens $it finita, quamvis ejus
fluxio $it infinita; tum nunquam $eries pro areâ, in quâ continetur
prædicta infinita ordinata, converget; ni area, i. e. fluens terminet ad
ordinatam vel fluxionem, quæ e$t infinita.</p>
<p>1. Si ordinata haud $it infinita, $ed ejus fluxio cuju$cunque ordinis
$it infinita; tum hæc $eries haud ultimo converget, ni ab ordinatâ,
cujus fluxio e$t infinita, terminetur.</p>
<p>3. Si fluxio areæ vel fluentis cuju$cunque ordinis evadat nihilo
æqualis; tum hæc $eries pro areâ, in quâ continetur prædicta fluxio
nihilo æqualis, nunquam converget; ni area terminetur ad ordina-
tam, in quâ invenitur prædicta fluxio nihilo æqualis. Con$tant
e præceden.</p>
<p>4. Eo magis ceteris paribus converget $eries re$ultans, quo magis
di$tant termini datæ $eriei a prædictis punctis; & quo plures proprie
interponantur termini, eo magis $altem in ratione numeri termino-
rum interpolandorum converget $eries.</p>
<p>5. Si valores quantitatis <math>x</math> longe di$tent ab omni radice æquatio-
num re$ultantium ex $upponendo generales terminos, <math>&c. = 0</math>; tum
$eries <math>y</math> proprie per $eriem huju$ce formulæ <math>a + b x + c x^2 + &c.</math> ex-
hiberi pote$t, $in aliter non: $unt ca$us, cum <math>x</math> $it perparva, in qui-
bus <math>y</math> per $eriem <math>a + b x^r + c x^2r + &c.</math> vel <math>a x^s + b x^s+r + c x^s+2r + &c.,
&c.</math>, de$ignari pote$t; cum autem <math>x</math> $it permagna quantitas, $unt ca-
$us, in quibus <math>y</math> per $eriem <math>a + b x^-1 + c x^-2 + d x^-3 + &c.</math>, vel per
<math>a + b x^-r + c x^-2r + &c.</math>, vel per <math>a x^-s + b x^-s-r + c x^-s-2r + &c.;
&c.</math>; de$ignari pote$t; & ad hos ca$us con$imilia iis, quæ in hoc pro-
blemate ad $eriem <math>a + b x + c x^2 + &c.</math> dantur, etiam applicari
po$$unt.</p>

<pb 590>[0642]<rh>DE SUMMATIONE</rh>
<h>THEOR. XXIV.</h>
<p>Sit <math>t = v</math>, & erit <math>t′ = t + t<x>.</x> = v + v^. + {1 / 2} v^.. + {1 / 6} v^... + {1 / 24} v^.... + &c.</math>
& exinde <math>t<x>.</x> = v^. + {1 / 2} v^.. + {1 / 6} v^... + {1 / 24} v^.... + &c.</math> unde <math>t<x>.</x> + t<x>..</x> = v^. + ({1 / 2} + 1)
v^.. + ({1 / 6} + {1 / 2} + {1 / 2}) v^... + &c.</math> & con$equenter <math>t<x>..</x> = v^.. + v^... + {7 / 12} v^.... + {1 / 4} v^..... +
&c.</math> & ex ii$dem operationibus repetitis erui po$$unt <math>t<x>...</x> = v^... + 1 {1 / 2}
v^.... + {5 / 4} v^..... + &c. t<x>....</x> = v^.... + 2 v^..... + &c. &c.</math> lex, quam ob$ervat hæc $eries,
$ic enuntiari pote$t, $it <math>t<x>.</x><x>n-1</x> = v^. ^n-1 + α v^. ^n + β v^. ^n+1 + γ v^. ^n+2 + &c.</math> & erit
<math>t<x>.</x><x>n</x> = v^. ^n + ({1 / 1 · 2} + α) v^. ^n+1 + ({1 / 1 · 2 · 3} + {1 / 1 · 2} α + β) v^. ^n+2 + ({1 / 1 · 2 · 3 · 4}
+ {1 / 1 · 2 · 3} α + {1 / 1 · 2} β + γ) v^. ^n+3 + &c.</math> unde facile con$tabit lex $eriei
quæ$ita.</p>
<p>Cor. · Hinc ex datâ incrementiali æquatione relationem inter <math>t,
t<x>.</x>, t<x>..</x>, &c. z & z^.</math> exprimente deduci pote$t infinita fluxionalis æquatio:
$cribantur enim pro <math>t, t<x>.</x>, t<x>..</x>, &c.</math> in datâ æquatione earum valores, &
re$ultat æquatio quæ$ita.</p>
<h>THEOR. XXVI.</h>
<p>Ex rever$ione prædictarum $erierum inveniri po$$unt <math>v^. = t<x>.</x> - {1 / 2} t<x>..</x>
+ {1 / 3} t<x>...</x> - {1 / 4} t<x>....</x> + {1 / 5} t<x>.....</x> - &c. v^.. = t<x>..</x> - t<x>...</x> + {11 / 12} t<x>....</x> - {10 / 12} t<x>......</x> + &c. v^... = t<x>...</x> - 1 {1 / 2} t
+ {7 / 4} t<x>......</x> - &c. v^.... = t<x>....</x> - 2 t<x>......</x> + &c. &c.</math></p>
<p>Cor. · Scribantur hæ quantitates in datâ fluxionali æquatione pro
$uis valoribus <math>v, v^.., v^..., &c.</math> & re$ultat infinita incrementialis æquatio.</p>
<p>Series di$tingui po$$unt in diver$os ordines; prout quantitates, quæ
exprimunt earum terminos, continent unam, duas, tres vel plures
independentes & variabiles quantitates <math>x, y, z, v, &c.</math> E. g. Ea $eries,
cujus termini exprimi po$$unt per quantitatem, in quâ $olummodo
continetur una variabilis quantitas, dici pote$t $eries primi ordinis;
ea, quæ continet duas variabiles quantitates, dici pote$t $eries $ecundi
ordinis; & $ic deinceps.</p>

<pb 591>[0643]<rh>SERIERUM, &c.</rh>
<h>PROB. XXXI.</h>
<p it>1. Conce$sâ omnium $erierum primi ordinis $ummas generaliter inve-
niendi methodo; invenire $ummam cuju$cunque $eriei (<math>A</math>) $ecundi, tertii,
&c. ordinis.</p>
<p>A$$umantur omnes variabiles quantitates (<math>y, z, v, &c.</math>) præter
unam <math>x</math> in datâ quantitate, quæ exprimit $eriei terminos tanquam in-
variabiles; & exinde e conce$sâ methodo inveniatur $eriei re$ultantis
$umma; deinde in $ummâ re$ultante a$$umantur quantitas <math>x</math> & om-
nes reliquæ præter <math>y</math> tanquam invariabiles, & e prædictâ conce$sâ
methodo inveniatur $eriei re$ultantis $umma; & $ic progrediendum
e$t u$que donec omnes invariabiles & independentes quantitates (<math>x,
y, z, &c.</math>) $ejunctim tanquam variabiles a$$umptæ fuerint, i. e. toties
inveniantur $ummæ re$ultantium $erierum, quot variabiles & inde-
pondentes quantitates (<math>x, y, z, v, &c.</math>) in datâ $erie <math>A</math> contineantur;
ultima $umma re$ultans vere correcta erit $umma quæ$ita.</p>
<p>Ex. 1. Sint <math>z & v</math> quicunque integri numeri <math>1, 2, 3, 4, 5, &c.</math> & $it
data quantitas <math>A</math>, in cujus formulâ continentur omnes termini $e-
riei, cujus $umma requiritur, <math>{1 / 2 z + v} × {1 / 2 z + v + 1} × {1 / 2 z + v + 2}</math>,
i. e. $it $eries data
<math>+ {1 / 3 · 4 · 5} + {1 / 4 · 5 · 6} + {1 / 5 · 6 · 7} + &c. ubi z = 1 & v = 1, 2, 3, &c.
+ {1 / 5 · 6 · 7} + {1 / 6 · 7 · 8} + {1 / 8 · 9 · 10} + &c. ubi z = 2 & v = 1, 2, 3, &c.
+ {1 / 7 · 8 · 9} + {1 / 8 · 9 · 10} + {1 / 9 · 10 · 11} + &c. ubi z = 3 & v = 1, 2, 3, &c.
&c. &c. &c.</math>
ubi <math>z & v</math> cre$cunt per unitatem.</p>
<p>A$$umo <math>z</math> tanquam invariabilem quantitatem, pro quâ $cribatur
<math>α</math>; & $eriei, cujus termini exprimuntur per quantitatem re$ultantem
<math>{1 / 2 α + v} × {1 / 2 α + v + 1} × {1 / 2 α + v + 2}</math> inveniatur $umma, quæ erit

<pb 592>[0644]<rh>DE SUMMATIONE</rh>
<math>(B) {1 / 2 α + v} × {1 / 2(2α + v + 1)}</math>; in hâc $ummâ pro <math>α</math> $cribatur ejus
prior valor <math>z</math>, & re$ultat $umma <math>{1 / 2 z + v} × {1 / 2 (2 z + v + 1)}</math>: nunc
a$$umo <math>v</math> invariabilem, pro quâ $cribatur <math>β z</math> vero variabilem, &
evadet $umma <math>(B) {1 / 2 z + β} × {1 / 2 (2 z + β + 1)}</math>; inveniatur $umma $eriei,
cujus termini exprimuntur per quantitatem <math>{1 / 2 (2 z + β) (2 z + β + 1)}</math>,
quæ erit <math>(C) {1 / 2 (2 z + β)}</math>, pro <math>β</math> in hâc $ummâ (<math>C</math>) $cribatur iterum <math>v</math>,
& re$ultat $umma quæ$ita <math>{1 / 2 (2 z + v)}</math>.</p>
<p>2. Ex hinc facile con$tant $ummæ quæ$itæ inter quo$cunque
valores, vel negativos vel affirmativos quantitatum <math>x, y, z, v, &c.</math>
contentæ.</p>
<p>A$$umantur omnes quantitates (<math>y, v, z, &c.</math>) præter unam (<math>x</math>) in
prædictâ quantitate (<math>P</math>) tanquam invariabiles; & exinde e conce$sâ
methodo inveniatur $eriei re$ultantis $umma, quæ $it <math>B</math>; in hâc $um-
mâ pro <math>x</math> $cribantur <math>α & a</math>; ubi <math>a & α</math> $int quantitates, inter quas
continentur valores quantitatis <math>x</math>; & $int quantitates re$ultantes <math>A
& A′</math>; & $umma $eriei inter valores <math>a & α</math> quantitatis <math>x</math> erit <math>A′ -
A = Q</math>: deinde in quantitate <math>Q</math> ita a$$umantur omnes quantitates <math>a,
α, v, z, &c.</math> præter unam <math>y</math> tanquam invariabiles, & exinde e conce$sâ
methodo inveniatur $umma $eriei, cujus termini $unt <math>Q</math>, in quâ va-
lores quantitatis <math>y</math> inter <math>b & β</math> ponuntur; & $ic de reliquis; ultima
$umma $ic inventa erit quæ$ita.</p>
<p>Ex. Invenire $ummam $eriei, cujus terminus e$t <math>{1 / 2 z + v} × {1 / 2 z + v + 1}
× {1 / 2 z + v + 2}</math>; in quo incrementum quantitatis <math>v</math> e$t <math>1</math> & quanti-
tatis <math>z</math> e$t <math>{1 / 2}</math>; & valores termini (<math>v</math>) inter <math>1 & 4</math> ponuntur; valores
autem termini (<math>z</math>) inter <math>1 & 3</math> ponuntur.</p>

<pb 593>[0645]<rh>SERIERUM, &c.</rh>
<p>Primo $uppono <math>z</math> e$$e invariabilem quantitatem, & invenio $um-
mam $eriei exinde re$ultantis, quæ erit <math>{1 / 2 z + v} × {1 / 2 (2 z + v + 1)} (B)</math>;
at quoniam valores quantitatis (<math>z</math>) $upponuntur inter <math>1 & 4</math> con-
$i$tere; per prob. in hâc quantitate (<math>B</math>) pro <math>v</math> $cribantur re$pective
<math>1 & 4 + 1 = 5</math>, unde re$ultat $umma <math>{1 / 2 z + 1} × {1 / 2 (2 z + 2)} & {1 / (2 z + 5)} ×
{1 / 2 (2 z + 6)}</math>, & $umma inter valores prædictos contenta erit <math>{1 / 2 z + 1}
× {1 / 2 (2 z + 2)} - {1 / 2 z + 5} × {1 / 2 × (2 z + 6)}: 2^do</math>. $umma $eriei, cujus
terminus e$t <math>{1 / 2 z + 1} × {1 / 2 (2 z + 2)} - {1 / 2 z + 5} × {1 / 2 (2 z + 6)}</math>, erit
<math>{{1 / 2} / 2 z + 1} - {{1 / 2} / 2 z + 5}</math>; at, quoniam valores quantitatis <math>z</math> inter <math>1 & 3</math>
continentur, in hâc $ummâ pro <math>z</math> $cribantur re$pective <math>1 & 3 + z, =
3 + {1 / 2} = {7 / 2}</math>, & re$ultant re$pective <math>{{1 / 2} / 2 + 1} - {{1 / 2} / 2 + 5} = {2 / 21} & {{1 / 2} / 7 + 1} -
{{1 / 2} / 7 + 5} = {1 / 48}</math>; unde $umma quæ$ita erit <math>{2 / 21} - {1 / 48} = {25 / 336}</math>.</p>
<p>Cor. In his ca$ibus haud refert, an prius a$$umatur (<math>z</math>) tanquam
invariabilis quantitas, an <math>v</math>; & $ic de reliquis variabilibus quantita-
tibus; nam ex $ingulis hi$ce diver$is modis eadem re$ultabit $umma
quæ$ita.</p>
<h>PROB. XXXII.</h>
<h it>Invenire generales terminos $erierum huju$ce generis, quarum $ummæ
innote$cunt.</h>
<p>A$$umatur quæcunque functio quantitatum (<math>x, y, z, v, &c.</math>) pro
$ummâ quæ$itâ deinde (datis incrementis quantitatum <math>x, y, z,
v, &c.)</math> inveniatur generalis terminus $eriei, cujus $umma e$t data
functio, ex hypothe$i quod (<math>x</math>) $olummodo e$t variabilis; & $it quan-
titas re$ultans <math>P</math>: deinde a$$umatur (<math>P</math>) pro $ummâ, & ex hypothe$i

<pb 594>[0646]<rh>DE SUMMATIONE</rh>
quod <math>y</math> $olummodo $it variabilis inveniatur generalis terminus $eriei,
cujus $umma e$t <math>P</math>; & $ic deinceps, de $ingulis reliquis incognitis
quantitatibus: terminus ultimo re$ultans erit quæ$itus.</p>
<p>Ex. Sit <math>{1 / 2z + v}</math> functio quantitatum <math>z & v</math> pro $ummâ a$$umptâ:
<math>z, & v</math>, incrementa quantitatum <math>z & v</math>: a$$umatur (<math>v</math>) tanquam inva-
riabilis quantitas, & incrementum datæ functionis erit <math>P = {1 / 2z + v}
- {1 / 2 z + v + 2z<x>′</x>} = {2 z<x>′</x> / (2 z + v) (2 z + v + 2 z<x>′</x>)}</math>; deinde in hoc primo
incremento a$$umantur <math>z & z<x>′</x></math> tanquam invariabiles quantitates, & in-
veniatur incrementum quantitatis <math>P</math>, quod erit <math>{2 z<x>′</x> / (2 z + v) (2 z + v + 2 z<x>′</x>)}
- {2 z<x>′</x>/(2 z + v + v<x>′</x>) (2 z + 2 z<x>′</x> + v + v<x>′</x>)} = Q</math>; ergo <math>Q</math> erit quantitas,
vel terminus, cujus $umma pote$t e$$e data functio.</p>
<p>Cor. Quamvis unica $olummodo datur formula termini, quæ cor-
re$pondet datæ $ummæ; attamen dantur infinitæ diver$æ $ummæ,
quæ corre$pondent eidem termino: e. g. $it <math>π</math> functio (<math>n</math>) quantita-
tum (<math>x, y, z, v, w, &c.</math>), cujus terminus $it <math>R</math>; tum erit generaliter
<math>π + φ (y, z, v, w, &c.) + φ′ (x, z, v, w, &c.) + φ″ (x, y, v, w, &c.) +
φ′″ (x, y, z, w, &c.) + &c.</math>; ubi <math>φ (y, z, v, w, &c.)</math> denotat functio-
nem quamcunque omnium quantitatum <math>y, z, v, w, &c.</math> præter unam
<math>x; & φ′ (x, z, v, w, &c.)</math> denotat quamcunque functionem quantita-
tum (<math>x, z, v, w, &c.</math>) præter unam <math>y</math>; & $ic deinceps; $umma, cujus
terminus e$t <math>R</math>.</p>
<p>Cor. 2. Sit <math>π</math> functio quantitatum (<math>x, y, z, v, w, &c.</math>), & $int <math>α, β,
γ, δ, ε, &c.</math> re$pective incrementa quantitatum <math>x, y, z, v, w, &c.</math>: in
functione <math>π</math> pro <math>x, y, z, v, w, &c.</math> $cribantur re$pective <math>x + α, y + β,
z + γ, v + δ, w + ε, &c.</math>, & $int quantitates re$ultantes re$pective
<math>A, B, C, D, E, &c.</math>; $imiliter in prædicta functione pro <math>x & y; x & z,
y & z, z & v, &c.</math> $cribantur re$pective <math>x + α & y + β, x + α & z + γ,
y + β & z + γ, x + α & v + δ, &c.</math> & $int quantitates re$ultantes re$pe-
ctive <math>A B, A C, B C, A D, &c.</math>; deinde in <math>π</math> pro <math>x & y & z, x & y & v,

<pb 595>[0647]<rh>SERIERUM, &c.</rh>
x & z & v, y & z & v, x & y & w, &c.</math> $cribantur re$pective <math>x + α &
y + β & z + γ, x + α & y + β & v + δ, x + α & z + γ & v + δ,
y + β & z + γ & v + δ, &c.</math> & $int quantitates re$ultantes <math>A B C,
A B D, A C D, B C D, &c.</math>: & $ic pro <math>x & y & z & v, &c.</math> $cribantur
in functione <math>π</math> re$pective <math>x + α & y + β & z + γ & v + δ, &c.</math>, &
re$ultet quantitas <math>A B C D, &c.</math>; & $ic deinceps; tum erit terminus
(cujus $umma e$t <math>π) = π - (A + B + C + D + &c.) + (A B +
A C + B C + A D + B D + C D + &c.) - (A B C + A B D +
A C D + A B E + &c.) + (A B C D + A B C E + A B D E +
A C D E + B C D E + &c.) - (A B C D E + &c.) + (A B C D E F
+ &c.) - &c.</math></p>
<h>PROB. XXXIII.</h>
<p it>Ii$dem conce$$is; invenire $ummam $eriei, cujus termini exprimuntur per
datam quantitatem, in quâ continentur variabiles quantitates <math>(x, y, z, v,
&c.)</math> ubi literæ <math>x, y, z, &c.</math> diver$os denotent numeros vel quantitates, i. e.
in dato termino $it <math>x</math> nec idem uumerus ac <math>y</math> vel <math>z, &c.</math> nec idem numerus
$it <math>y</math> ac <math>z, &c. &c.</math></p>
<p>Dicatur $umma per prædictam methodum inventa <math>A</math>.</p>
<p>Pro duabus literis <math>x & y</math>, quæ nece$$ario diver$æ $upponuntur, $cri-
batur eadem litera <math>α</math>, & inveniatur per prædictam methodum $umma
(<math>a</math>) $eriei, cujus termini exprimuntur per quantitatem re$ultantem;
$cribatur etiam in datâ quantitate pro <math>x & z</math> (quæ $upponuntur $em-
per diver$æ) eadem litera <math>β</math>; & $ic pro <math>y & z</math> $cribatur eadem litera
<math>γ, &c.</math> & per methodum præcedentem inveniantur $ummæ (<math>b, c; &c.</math>)
$erierum, quarum termini exprimuntur per re$ultantes quantitates:
aggregatum $ummarum (<math>a + b + c + &c.</math>) e $ingulis huju$modi $e-
riebus dicatur <math>B</math>.</p>
<p>Pro tribus literis <math>x, y & z</math> quæ $upponuntur inter $e diver$æ, in
datâ quantitate $cribatur eadem litera <math>ε</math>, & per præcedentem metho-
dum inveniatur $umma (<math>e</math>) $eriei, cujus termini exprimuntur per
$eriem re$ultantem; & $i pro $ingulis tribus literis diver$os numeros
e$$e $uppo$itis $cribantur in datâ quantitate eædem literæ re$pective,
& per prædictam methodum inveniantur $ummæ (<math>f, g, &c.</math>) $erierum,

<pb 596>[0648]<rh>DE SUMMATIONE</rh>
quarum termini exprimuntur per re$ultantes quantitates re$pective;
& aggregatum (<math>e + f + g + &c.</math>) e $ingulis $ummis dicatur <math>C</math>.</p>
<p>Pro quatuor literis (<math>x, y, z, v</math>), quæ $upponuntur e$$e diver$æ, $cri-
batur in datâ quantitate eadem litera, & inveniatur $umma (<math>l</math>) $eriei,
cujus termini exprimuntur per quantitatem re$ultantem; & $ic $cri-
bantur pro $ingulis quatuor literis diver$os numeros e$$e $uppo$itis
eædem literæ, & inveniantur $ummæ $erierum, quarum termini ex-
primuntur per literas re$ultantes; & aggregatum ex his $ummis
dicatur <math>D</math>.</p>
<p>Pro literis <math>x & y</math>, quæ diver$æ $upponuntur, $cribatur in datâ
quantitate eadem litera; etiamque pro <math>z & v</math>, quæ diver$æ $unt, $cri-
batur eadem litera, $ed e priori diver$a; & inveniatur $umma $eriei,
cujus termini exprimuntur per quantitatem re$ultantem; & $ic inve-
niantur $ummæ $erierum, quarum termini exprimuntur per quanti-
tates ex con$imilibus $ub$titutionibus deductas: aggregatum ex his
$ummis dicatur <math>B B; &c.</math></p>
<p>Summa quæ$ita erit <math>A - 1 · B + 1 · 2C - 1 · 2 · 3 D + &c.
+ 1 · 1 B B - &c.</math></p>
<p>Hujus $eriei lex e prob. 3. medit. algebraic. colligi pote$t.</p>
<p>Ex. 1. Sit quantitas, quæ denotat re$pectivos terminos datæ $eriei,
cujus $umma requiritur, <math>{1 / x^4 + y^3}</math>, ubi <math>x & y</math> integros quo$cunque nu-
meros <math>1, 2, 3, &c.</math> a $e invicem diver$os re$pective denotant.</p>
<p>A$$umatur <math>x</math> tanquam invariabilis quantitas (<math>a</math>), & inveniatur
$umma $eriei, cujus termini erunt <math>{1 / a^4 + y^3}</math>; deinde in $ummâ inventâ
pro <math>a</math> $cribatur <math>x</math>, & a$$umatur <math>y</math> tanquam invariabilis quantitas, &
inveniatur $umma $eriei re$ultantis, quæ dicatur <math>A</math>.</p>
<p>Deinde pro <math>x & y</math> in datâ quantitate <math>{1 / x^4 + y^3}</math> $cribatur eadem litera
<math>x</math>, & inveniatur $umma $eriei, cujus termini exprimuntur per quan-
titatem re$ultantem <math>{1 / x^4 + x^3}</math>, quæ dicatur <math>B</math>; tum $umma quæ$ita
erit <math>A - B</math>.</p>

<pb 597>[0649]<rh>SERIERUM, &c.</rh>
<p>Si vero in datâ quantitate contineantur quæcunque irrationales
quantitates, eadem erit methodus problema re$olvendi ac præcedens.</p>
<p>Cor. 1. Si vero inter datas quantitates contineantur valores quan-
titatum variabilium (<math>x, y, z, &c.</math>); tum $umma acqui$ita corrigi pote$t
e $ub$titutione prædictarum quantitatum pro $uis valoribus; &c.</p>
<p>Cor. 2. Haud nece$$ario $emper denotant integros numeros literæ
(<math>x, y, z, v, &c.</math>) $ed quantitates cuju$cunque generis, & ex ii$dem
principiis con$tabit problematis re$olutio.</p>
<p>Cor. 3. Si quantitas <math>A</math> inveniri po$$it, tum $emper inveniri po$$unt
quantitates <math>B, C, D, BB, E, BC, &c.</math> i. e. datâ quantitate, quæ ex-
primit $ingulos $eriei terminos duas vel plures variabiles quantitates
(<math>x, y, z, &c.</math>) involventis; $i modo $umma huju$ce $eriei inveniri po$-
$it; tum $umma omnis $eriei, cujus termini exprimuntur per quanti-
tatem re$ultantem e $ub$titutione invariabilium quantitatum pro qui-
bu$libet literis <math>x, y, z, &c.</math> vel <math>x</math> pro <math>y, &c.</math> $emper inveniri pote$t.</p>
<p>Cor. 4. A$$umatur quæcunque quantitas, quæ duas vel plures va-
riabiles quantitates (<math>x, y, z, v, &c.</math>) involvat pro $ummâ $eriei; & e
differentiâ inter hanc quantitatem a$$umptam & ejus $ucce$$ivam
$ummam, & $ic deinceps; deduci pote$t quantitas, quæ exprimit $in-
gulos prædictæ $eriei terminos.</p>
<p>Cor. 5. Si vero maximæ dimen$iones variabilium quantitatum
(<math>x, y, z, &c.</math>) in denominatore haud $uperent maximas dimen$iones
prædictarum quantitatum in numeratore contentas per quantitatem
majorem quam numerum diver$arum variabilium quantitatum (<math>x, y,
z, v, &c.</math>) in datâ quantitate contentarum; tum $eries in infinitum
progrediens nece$$ario evadet in$inita.</p>
<p>Summa vero huju$modi $eriei, $i haud in finitis terminis, $emper
per infinitas $eries exprimi pote$t; per hoc problema enim reduci
pote$t ad re$olutionem $erierum, in quibus una $olummodo conti-
netur variabilis quantitas; vel ex a$$umptâ $erie formulæ e formulâ
dati termini facile acquirendæ, cujus terminus deducatur, & ex æqua-
tis corre$pondentibus datæ & re$ultantis $eriei terminis erui pote$t
prædicta re$olutio.</p>
<p>Convergentiæ harum $erierum ex ii$dem principiis ac convergen-

<pb 598>[0650]<rh>DE SUMMATIONE</rh>
tiæ $erierum unam variabilem quantitatem $olummodo habentium
dijudicari po$$unt.</p>
<h>THEOR. XXVI.</h>
<p>Series (<math>n</math>) ordinis dividi po$$unt in infinitas $eries <math>(n - 1)</math> ordinis;
ita quidem $æpe ut $ummæ ex $ingulis prædictis infinitis $eriebus in-
note$cant; & hæ $ummæ præbeant convergentem $eriem.</p>
<p>Ex. Sit $eries $ecundi ordinis, cum terminus generalis e$t <math>{1 / 2z + 1 · 2v + 1}</math>,
ubi literæ <math>z & v</math> quo$cunque integros numeros denotant; $it vero ter-
minus negativus, cum <math>{v - z / 4}</math> $it integer numerus, $in aliter affirmati-
vus: dividatur hæc $eries $ecundi generis in infinitam $eriem $erierum
primi generis, quarum $ummæ detegi po$$unt, & exorientur novæ in-
finitæ $eries: ex. g. dividatur data $eries in alias <math>{1 / 1 · 3} + {1 / 3 · 5} + {1 / 5 · 7}
+ &c.</math> (ubi differentia inter <math>v & z</math> $emper e$t 2), <math>{1 / 1 · 5} + {1 / 3 · 7} +
{1 / 5 · 9} + &c.</math> (ubi prædicta differentia e$t 4), <math>{1 / 1 · 7} + {1 / 3 · 9} + {1 / 5 · 11}
+ &c. &c.</math> tum innote$cunt $ummæ harum $erierum re$ultantium,
viz. <math>{1 / 2} × 1, {1 / 4} (1 + {1 / 3}) = {1 / 3}, {1 / 6} (1 + {1 / 3} + {1 / 5}) = {23 / 90}, &c.</math> tum erit $umma datæ
$eriei quæ$ita <math>= {1 / 1 · 2} - {1 / 1 · 3} + {23 / 2 · 3 × 1 · 3 · 5} - {176 / 2 · 1 · 1 · 3 · 5 · 7}
+ {1689 / 2 · 5 · 1 · 3 · 5 · 7 · 9} - &c.</math></p>
<p>Et $ic de pluribus huju$cemodi $eriebus ad novam $eriem reducendis.</p>
<p>Si vero contineantur plures (<math>n</math>) variabiles quantitates in dato ter-
mino; tum reducenda e$t data $eries ad infinitas alias, quæ continent
<math>n - 1</math> variabiles quantitates, & quarum $ummæ innote$cunt; & ex-
inde $equuntur novæ infinitæ $eries, &c.</p><pb 599>[0651]<rh>SERIERUM, &c.</rh>
<h>PROB. XXXIV.</h>
<h it>Datâ $erie ex irrationalibus terminis con$tante, invenire ejus $ummam.</h>
<p>Primo dentur $olummodo <math>n</math> irrationales quantitates; &, $i modo
po$$ibile $it, ita dividatur $eries in plures, ut unica, &c. $olummodo
contineatur irrationalis quantitas in $ingulis, & per methodos præce-
dentes inveniatur $umma e $ingulis hi$ce $eriebus.</p>
<p><math>2^do</math>. Si modo infinitæ dentur irrationales quantitates in datâ $erie,
nonnunquam ita transformari po$$unt irrationales quantitates e nu-
meratore in denominatorem; & vice versâ, e denominatore in nu-
meratorem per medit. algebraic. &c. ut evane$cant e $erie omnes irra-
tionales quantitates.</p>
<p>Ex. 1. Sit data $eries <math>{1 / 2 . 3 . (√ ({1 / 2}) + √ ({1 / 3}))} + {1 / 3 . 4 . (√ ({1 / 3}) + √ ({1 / 4}))}
+ {1 / 4 . 5 . (√ ({1 / 4}) + √ ({1 / 5}))} + {1 / 5 . 6 . (√ ({1 / 5}) + √ ({1 / 6}))} + &c.</math> transformen-
tur irrationales quantitates e denominatore in numeratorem; & quoniam
<math>(√ ({1 / 2}) + √ ({1 / 3})) × (√ ({1 / 2}) - √ ({1 / 3})) = {1 / 2} - {1 / 3} = {1 / 6}</math>; erit <math>{1 / 2 . 3 . (√ ({1 / 2}) + √ ({1 / 3}))}
= √ ({1 / 2}) - √ ({1 / 3})</math>, & $ic <math>{1 / 3 . 4 . (√ ({1 / 3}) + √ ({1 / 4}))} = √ ({1 / 3}) - √ ({1 / 4})</math>, &
$ic <math>{1 / 4 . 5 . (√ ({1 / 4}) + √ ({1 / 5}))} = √ ({1 / 4}) - √ ({1 / 5})</math>; unde $eries erit in genere
<math>√ ({1 / 2}) - √ ({1 / 3}) + √ ({1 / 3}) - √ ({1 / 4}) + √ ({1 / 4}) - &c.</math> in infinitum; & $um-
ma erit <math>√ ({1 / 2})</math>.</p>
<p>Cor. 1. Series con$tantes ex irrationalibus terminis, quarum $um-
mæ dantur, facile deduci po$$unt ex principiis prius traditis, quæ
deducunt $eries $ummabiles, quæ haud involvunt irrationales quan-
titates.</p>
<p>Ex. 1. A$$umatur $eries, haud re$ert utrum $ummabilis $it, necne;
modo ejus termini ad nihil continuo vergant, & denique propiores
ad id accedant, quam ulla quævis differentia; & $ubtrahatur ip$amet
a $eipsâ demptis <math>r</math> primis terminis; deinde transformentur termini

<pb 600>[0652]<rh>DE SUMMATIONE</rh>
re$ultantes e numeratore in denominatorem per medit. algeb. vel in
alios diver$æ formulæ mutentur, & invenitur $eries quæ$ita. E. g.
Sit $eries a$$umpta <math>1 + ({1 / 2})^{1 / n} + ({1 / 3})^{1 / n} + ({1 / 4})^{1 / n} + ({1 / 5})^{1 / n} + &c.</math> $ub-
trahatur ip$amet a $e ipsâ demptis <math>r</math> primis terminis, i. e. <math>({1 / r})^{1 / n} +
({1 / r + 1})^{1 / n} + &c.</math> & re$iduum erit <math>1 - ({1 / r})^{1 / n} + ({1 / 2})^{1 / n} - ({1 / r + 1})^{1 / n}
+ ({1 / 3})^{1 / n} - ({1 / r + 2})^{1 / n} + &c. = 1 + ({1 / 2})^{1 / n} + ({1 / 3})^{1 / n} ... ({1 / r - 1})^{1 / n}</math>:
transformentur hæ irrationales quantitates e numeratore in denomi-
natorem, & re$ultabunt <math>1 - ({1 / r}^{1 / n} = {r - 1 / r × (1 + ({1 / r})^{1 / n} + ({1 / r})^{2 / n} ... ({1 / r})^{n-1 / n})}
& ({1 / 2})^{1 / n} - ({1 / r + 1})^{1 / n} = {r - 1 / 2 × (r + 1) × ({1 / 2}^{n-1 / n} + {1 / 2}^{n-2 / n} ({1 / r + 1})^{1 / n} + {1 / 2}^{n-3 / n}
({1 / r + 1})^{2 / n} + .. ({1 / r + 1})^{n-1 / n)}}</math>, & $ic deinceps; & erit $umma <math>1 + ({1 / 2})^{1 / n} +
({1 / 3})^{1 / n} + ({1 / 4})^{1 / n} ... ({1 / r - 1})^{1 / n} = {r - 1 / 1 × r × (1 + ({1 / r})^{1 / n} + ({1 / r})^{2 / n} + &c. ({1 / r})^{n-1 / n})}
+ {r - 1 / 2 × (r + 1) × ({1 / 2}^{n-1 / n} + &c.)} + &c.</math></p>
<p>Ex. 2. Sit $eries a$$umpta <math>√ ({1 / 1 . 2}) - √ ({1 / 2 . 3}) x + √ ({1 / 3 . 4})
x^2 - &c.</math> ducatur hæc $eries in quamcunque quantitatem. E. g.
<math>1 + x</math> & re$ultat quantitas <math>√ ({1 / 1 . 2}) + (√ ({1 / 1 . 2}) - √ ({1 / 2 . 3})
= ({2 / 1 . 2 . 3 . (√ ({1 / 1 . 2}) + √ ({1 / 2 . 3}))}) x - (√ ({1 / 2 . 3}) -

<pb 601>[0653]<rh>SERIERUM, &c.</rh>
√ ({1 / 3 . 4}) = {2 / 2 . 3 . 4 . (√ ({1 / 2 . 3}) + √ ({1 / 3 . 4}))} x^2 + &c.</math></p>
<p>3. Ducantur $eries huju$modi in qua$cunque alias quantitates, &
facile inveniri po$$unt $eries irrationales $ummabiles omnino per eun-
dem modum ac $eries rationales: & $ic de reliquis methodis inve-
niendi rationales $ummabiles $eries applicandis ad irrationales $eries
$ummabiles inveniendas; etiamque irrationales $eries ope aliarum
$erierum deducendas.</p>
<p>Ex. 3. Sit $eries a$$umpta <math>{1 / √ (2)} + {1 / √ (3)} x + {1 / √ (4)} x^2 + &c.</math> du-
catur hæc $eries in quantitatem <math>1 - 2x</math>, & re$ultat <math>{1 / √ (2)} + {1 / √ (3)} x
+ {1 / √ (4)} x^2 + &c. - {2 / √ (2)} x - {2 / √ (3)} x^2 - {2 / √ (4)} x^3 - &c. = {1 / √ (2)}
- {2√ (3) - √ (2) / √ (2 . 3)} x - {2√ (4) - √ (3) / √ (3 . 4)} x^2 - {2√ (5) - √ (4) / √ (4 . 5)} x^3 -
&c.</math> Fingatur <math>2 x - 1 = 0</math>, & re$ultat <math>x = {1 / 2}</math> & $umma $eriei
re$ultantis <math>{2√ (3) - √ (2) / √ (6}) × {1 / 2} + {2√ (4) - √ (3) / √ (12)} × {1 / 4} + &c. =
{1 / √ (2)}</math>.</p>
<p>Omnia, quæ prius de rationalibus $eriebus quoad earum conver-
gentiam, &c. tradita $unt, etiam ad has irrationales applicari po$$unt.</p>
<h>THEOR. XXVII.</h>
<p>In $eriebus interpolandis, juxta ac in iis $ummandis, nece$$e e$t ut
inveniantur limites, inter quos con$i$tant termini quæ$iti, aliter de
iis nihil affirmandum e$t: e quibu$cunque finitis datis ordinatis ad
qua$cunque datas di$tantias a $e invicem po$itis minime ullas alias
interponere po$$umus, ni detur lex, quam ob$ervant termini in genere,
haud enim $unt data, e quibus deduci po$$unt quæ$ita: $i vero detur
lex, tum exinde deduci pote$t $ub$titutio, ex quâ interpolari po$$unt.</p>

<pb 602>[0654]<rh>DE SUMMATIONE</rh>
<p>Facile con$tat, quod haud interpolari po$$unt $eries per methodum
differentiarum, ni ultimæ differentiæ eju$dem ordinis $int prope in
ratione æqualitatis.</p>
<h>THEOR. XXVIII.</h>
<p>1. Sit curva parabolica <math>Q a b β c γ, &c.</math> invenire ordinatam <math>P M (y)</math>
ad ab$ci$$am <math>A P</math>; a$$umatur alia parabolica curva <math>R π b ρ c σ &c.</math>
quæ habet ordinatas (<math>v</math>) præcedenti curvæ communes <math>b l, c m, d n, e o,
&c.</math> quarum una ab$ci$$a <math>A n</math> $it <math>z</math>; pro radicibus <math>A α, A β, A γ, A δ,
&c.</math> prioris curvæ $cribantur <math>α, β, γ, δ, &c.</math> re$pective; & pro radicibus
po$terioris curvæ, quæ dicatur approximans curva, $ub$tituantur <math>π, ρ,
σ, τ, &c.</math> & pro <math>A P, x</math>; tum erit approximatio inventa ex hâc methodo
interpolandi, i. e. corre$pondens ordinata <math>L P (v)</math> in curvâ a$$umptâ ad
ordinatam <math>P M (y)</math> quæ$itam::<math>{(x - π) × (x - ρ) × (x - σ) × &c. / (z - π) × (z - ρ) × (z - σ) × &c.}:
{(x - α) × (x - β) × (x - γ) × &c. / (z - α) × (z - β) × (z - γ) × &c.}</math> prope, ob <math>y = H × (x - α) . (x - β)
. (x - γ) . &c., & v = K × (x - π) . (x - ρ) . (x - σ) . &c.</math></p>
<p>2. Si vero curva data habeat crura ad finitas ab$ci$$as (<math>a, b, c, &c.</math>)
in infinitum pergentia; tum radices <math>x - a, x - b, x - c, &c. z - a,
z - b, z - c, &c.</math> quæ per prædictas ab$ci$$as denotantur, in deno-
minatore continentur; & $ic de ab$ci$$is ad a$ymptotos in curvâ a$-
$umptâ; & exinde $equitur ratio, quam habet ordinata in curvâ a$-
$umptâ ad ordinatam quæ$itam.</p>
<p>3. Si vero impo$$ibiles evadant quædam radices prædictæ, tum ex
quibu$que duabus corre$pondentibus impo$$ibilibus radicibus con$e-
quetur quadratica quantitas po$$ibilis, quæ & in curvâ datâ & a$$umptâ
pro multiplicatione duarum radicum in $e$e a$$umenda e$t.</p>
<p>Cor. . Hinc approximatio plerumque maxime pendet e radicibus
quæ finitimæ $unt ordinatæ quæ$itæ; & in hâc æque ac in omnibus
aliis curvis pendet e rationibus, quas habent differentiæ inter ab$ci$-
$am <math>A P</math> & eas, quæ habent communes ordinatas; ad di$tantias earum
a punctis, in quibus ordinatæ vel po$$ibiles vel impo$$ibiles evadant
nihil vel infinite magnæ.</p>

<pb 603>[0655]<rh>SERIERUM, &c.</rh>
<p>Eadem principia etiam applicari po$$unt ad inveniendam conver-
gentiam $erierum huju$ce generis <math>a x^s + b x^s+t + c x^s+2t + &c.</math> ubi per
<math>s & t</math> quæcunque fractiones denotentur, vel irrationales cuju$cunque
generis quantitates.</p>
<h>THEOR. XXIX.</h>
<p>Omnis $eries e$t interpolabilis, cujus termini $unt interpolabiles.
Interpolentur enim diver$i termini & interpolatur data $eries.</p>
<h>THEOR. XXX.</h>
<p>Erit <math>{r . r + 1 . r + 2 . r + 3 ... q / p . p + 1 . p + 2 . p + 3 ... p + q - r} = {r . r + 1 . r + 2 ... p - 1 / q + 1 . q + 2 ... p + q - r}</math>
reducantur enim hæ fractiones ad communem denominatorem, &
con$tat theor.</p>
<p>Ex. 1. Sit $eries <math>{2 × 4 × 6 × 8 ... &c / 1 × 3 × 5 × 7 ... &c.}</math>; $ed quoniam factorum incre-
mentum e$t binarius, divide numeratores & denominatores per bina-
rium, & evadet $eries <math>{1 × 2 × 3 × 4 × &c. / {1 / 2} × 1{1 / 2} × 2{1 / 2} × &c.} = {r × (r + 1) × (r + 2) × &c. / p × (p + 1) × (p + 2) × &c.}</math>
& con$equenter erunt <math>r = 1 & p = {1 / 2}</math>; nunc $it <math>m</math> di$tantia inter pri-
mum $eriei terminum & quemvis alium <math>q</math>, tum erit <math>q = m + 1</math>, &
$eries <math>{r . r + 1 . r + 2 .. p - 1 / q + 1 ... p + q - r}</math> evadet <math>= {1 . 2 . 3 .. p - 1 / m + 2 . m + 3 . . p + m}</math>;
ergo terminus $eriei <math>({1 × 2 × 3 .. m + 1 / {1 / 2} × 1{1 / 2} × 2{1 / 2} ... m + {1 / 2}})</math>, cujus di$tantia a primo
$it <math>m</math>, æquat terminum $eriei <math>({1 × 2 × 3 ... p - 1 / m + 2 . m + 3 ... m + {1 / 2}})</math>, cujus di-
$tantia a primo $it <math>p - 2 = {1 / 2} - 2 = - 1{1 / 2}</math>.</p>
<p>Cor. . Hinc $eries huju$ce generis $emper interpolari pote$t, viz.
<math>{r . r + 1 . r + 2 .. p - 1 / q + 1 . q + 2 . . p + q - r}</math>, ubi <math>q</math> $uperat <math>r</math> per integrum numerum;
requiratur enim terminus, cujus di$tantia a primo $it <math>m</math>: a$$umatur

<pb 604>[0656]<rh>DE SUMMATIONE</rh>
<math>p = r + m + 1</math>, & erit terminus quæ$itus <math>{r · r + 1 · r + 2 .. q / p · p + 1 · p + 2. · p + q - r}</math>.</p>
<h>THEOR. XXXI.</h>
<p>1. In interpolandis quantitatibus, quæ $unt functiones numeri <math>z</math>,
quarum continuo augetur $actorum numerus, ubi <math>z</math> denotat $ucce$$ive
numeros <math>1, 2, 3, 4, &c.</math> $æpe ex interpolatione indicum con$tabit
formula $eriei, quæ terminum quæ$itum exprimet, ex quâ per $ub-
$titutionem acquiri pote$t $eries ip$a, i. e. terminus quæ$itus.</p>
<p>Ex. 1. Sint quantitates huju$ce generis <math>1, 1.2, 1.2.3, 1.2.1.4</math>,
& in genere <math>1. 2. 3... z</math>, vel quod idem e$t <math>z. z - 1. z - 2 ... 1,
z + 1. z. z - 1. z - 2 ... 1</math>, invenire terminum intermedium in-
ter <math>(T′) z. z - 1. z - 2.. 1 & (T′) z + 1. z. z - 1. z - 2.. 1</math>:
dimen$iones quantitatis <math>z</math> in his duobus terminis datis contentæ per
unitatem differunt, ergo termini intermedii dimen$iones different a
dimen$ionibus quantitatis <math>z</math> in duobus prædictis terminis per <math>{1 / 2}</math>: a$$u-
matur igitur $eries <math>T (a z^{1 / 2} + b z^-{1 / 2} + c z^-1{1 / 2} + &c.)</math> pro termino in-
termedio quæ$ito, in $erie <math>(S) a z^{1 / 2} + b z^-{1 / 2} + c z^-1{1 / 2} + &c.</math> pro <math>z</math>
$cribatur <math>z + {1 / 2}</math>, & re$ultat <math>(S′) a z^{1 / 2} + (b + {1 / 4}a) z^-{1 / 2} + (c - {1 / 4} b - {1 / 8} ×
{1 / 4}a) z^-1{1 / 2} + &c.</math> ducantur hæ duæ $eries <math>S & S′</math> in $e$e; & productum
fiat æquale factori <math>z + 1</math>; & ex æquatis corre$pondentibus terminis
$equitur $eries <math>(S) z{1 / 2} + {3 / 8}z^-{1 / 2} - {7 / 128}z^-{3 / 2} + {9 / 1024}z^-{5 / 2} - &c.</math> & con$e-
quenter terminus quæ$itus erit <math>T × S</math>.</p>
<p>2. Ex ii$dem principiis ulterius promotis inveniri pote$t terminus
ad quamcunque di$tantiam a dato termino: $ed major erit calculi
labor.</p>
<p>3. Sint quantitates <math>x, x. x + n, x. x + n. x + n^2, x. x + n.
x + n^2 . x + n^3, &c.</math> ubi <math>n</math> multo minor e$t quam <math>x</math> & unitas, tum $e-
ries e multiplicatione deducta pro generali termino converget. E. g.
Series erit <math>x^z+1 + {n - n^z+1 / 1 - n} x^z + {1 / 2}(({n - n^z+1 / 1 - n})^2 - {n^2 - n^2z+2 / 1 - n^2}) x^z-1
+ &c.</math></p>

<pb 605>[0657]<rh>SERIERUM, &c.</rh>
<p>In hâc $erie $cribatur pro <math>z</math> ejus valor a$$ignatus, & con$tat termi-
nus, cujus di$tantia e primo $it <math>z</math>.</p>
<p>Lex huju$ce $eriei con$tat e no$tris medit. algebr.</p>
<p>4. Datâ quâcunque $erie, & æquatione relationem inter terminos
datæ & novæ $eriei exprimente, ita ut e datæ $eriei terminis con$tent
novæ æquationis corre$pondentes termini, & $i termini datæ $eriei,
tum etiam termini novæ $eriei, interpolari po$$unt.</p>
<p>Indices interpolari po$$unt per eandem methodum ac aliæ alge-
braicæ quantitates.</p>
<h>THEOR. XXXII.</h>
<p>1. Si $ucce$$ivi termini continuo exoriantur e majori numero fa-
ctorum, qui $unt functiones quantitatis <math>z</math>, in $e$e ductorum; tum
inveniantur logarithmi terminorum $ucce$$ivorum, & deduci pote$t e
differentiis logarithmorum, &c. $eries exprimens logarithmum $um-
mæ, quæ for$an progreditur $ecundum vulgares leges. Si vero nu-
merus factorum in $erie deductâ de$ignetur etiam per functionem
quantitatis <math>z</math> con$tantem continuo e majori numero factorum, tum
inveniantur logarithmi prædictorum logarithmorum, & $ic deinceps;
& tandem re$ultabit $eries exprimens logarithmum logarithmi, &c.
cujus termini prope etiam per vulgares leges exprimi po$$unt, ni nu-
merus factorum continuo exprimatur per functionem quantitatis <math>z</math>;
quæ continuo con$tat e majori numero factorum. E. g. Sit $eries 1,
<math>1 × 2 × 3, 1 × 2 × 3 × 4 × 5 × 6, &c.</math> ubi <math>n</math> numerus factorum conti-
nuo $it <math>= z. {z + 1 / 2}</math>; inveniantur logarithmi e $ingulis terminis, &
erunt eorum $ucce$$ivæ differentiæ <math>l 2 + l 3, l 4 + l 5 + l 6, l 7
+ l 8 + l 9 + l 10, &c.</math> & deinde inveniantur $ecundæ differen-
tiæ, &c.</p>
<p>Hæ vero differentiæ exprimi po$$unt per notas methodos, ergo $e-
ries ip$a.</p>
<p>2. Sit $eries <math>a + b + c + d + e + &c. = S</math>, tum continuo inve-
niantur <math>l a; l (a + b) - l a; l (a + b + c) - l (a + b), l (a + b +

<pb 606>[0658]<rh>DE SUMMATIONE</rh>
c + d) - l (a + b + c), &c.</math> & tandem invenietur $eries, cujus $um-
ma e$t <math>l S</math>.</p>
<p>3. Hæ transformationes $æpe u$ui in$ervire po$$unt in interpola-
tione $erierum huju$modi, quæ aliter interpolationem vix aut ne vix
recipiant: facile enim e$t interpolare $eriem re$ultantem, eâ vero in-
terpolatâ, inve$tigari pote$t interpolatio $eriei, ex quâ deducta fuit.</p>
<p>4. Si vero $eries exprimatur per terminos, quorum indices per
prædictas functiones progrediuntur; tum e præcedente methodo in-
veniantur logarithmi $eriei, & $æpe deduci pote$t logarithmus $eriei
$ummæ in terminis $ecundum vulgares leges progredientibus.</p>
<p>Cor. · Fluxiones, vel incrementa, &c. quantitatum <math>1. 2. 3... z</math>
vel <math>1. 2. 3.. z^z</math>, vel infinitarum aliarum, facile deduci po$$unt e prin-
cipiis prius traditis pro interpolationibus con$imilium quantitatum.</p>
<p>Numerus factorum vel dimen$iones quantitatum, quæ ex hâc me-
thodo interpolantur, deduci po$$unt.</p>
<h>PROB. XXXV.</h>
<p it>Invenire fluentes & exinde $eries per datam fluentem interpolabiles.</p>
<p>A$$umantur quæcunque quantitates in datâ fluente tanquam con-
$tantes; vel eædem in datâ & quæ$itâ quantitate variabiles, quæcunque
vero aliæ variabiles, i. e. diver$æ relationem a$$ignabilem inter $e ha-
bentes; reducantur quantitates re$ultantes ad infinitas $eries, & re-
$ultant $eries, quæ per datam fluentem interpolabiles erunt.</p>
<p>Ex. 1. Sit data fluens <math>$. (a^2 - x^2){1 / 2} x^.</math>, a$$umatur index <math>{1 / 2}</math> variabilis,
i. e. pro indice <math>{1 / 2}</math> $cribantur re$pective <math>0, 1, 2, 3, 4, 5, &c.</math> denique <math>l</math>;
quæ $eries per <math>{1 / 2}</math> interpolabilis e$t, & re$ultant <math>$. (a^2 - x^2)^0 x^., $. (a^2 -
x^2)^1 x^., $. (a^2 - x^2)^2 x^., $. (a^2 - x^2)^3 x^., &c.</math> & con$equenter $eries in-
termedia inter has fluentes inter eo$dem valores quantitatis <math>x</math> con-
tentas, erit <math>$. (a^2 - x^2){1 / 2} x^.</math>, quæ per aream circuli exprimi pote$t, unde
intermedius terminus inter primum & $ecundum $eriei <math>{1 / 1}, {1 / 1} - {1 / 3}, {1 / 1} - {2 / 3}
+ {1 / 5}, {1 / 1} - {3 / 3} + {3 / 5} - {1 / 7}, {1 / 1} - {4 / 3} + {6 / 5} - {4 / 7} + {1 / 9}, &c.</math> erit <math>{1 / 8}</math> peripheriæ circuli,
cujus radius e$t 1.</p>
<p>2. A$$umatur index 2 variabilis, & erit prædicta fluens intermedia
inter fluentes fluxionum <math>(a^2 - x^0){1 / 2} x^., (a^2 - x^4){1 / 2} x^., (a^2 - x^8){1 / 2} x^., &c.</math></p>

<pb 607>[0659]<rh>SERIERUM, &c.</rh>
<p>3. Et $ic a$$umi po$$unt indices <math>2 & {1 / 2}</math> invariabiles, & quantitas <math>a</math>
variabilis, ita vero ut $imul $int interpolabiles omnes prædictæ varia-
biles per datam fluentem. Inveniantur per infinitas $eries fluentes
prædictarum fluxionum, & re$ultant $eries, quæ per datam fluentem
interpolabiles erunt.</p>
<p>Ex. 2. Sit <math>$. x^. √ (x^2 + 1)</math> data fluens, & erunt fluentes fluxionum
<math>x^. × (x^2 + 1)^0, x^. (x^2 + 1)^1, x^. (x^2 + 2)^2, &c</math>, per eam interpolabiles.</p>
<p>Cor.. Fluentes fluxionum <math>x^. (1 ± x^2){1 / 3}, x^. × (1 ± x^2){1 / 4}, x^. (1 ± x^2){2 / 3},
x^. × (1 ± x^2){3 / 4}</math> (& con$equenter arcus elliptici vel hyperbolici), & gene-
raliter <math>x^. (1 ± x^2)^{n / m} & x^. (1 ± x^2)^-{n / m}</math> deduci po$$unt ex interpolatione
$erierum, quæ oriuntur e fluentibus fluxionum <math>(1 ± x^2)^0 x^., (1 ± x^2)^1 x^.,
(1 ± x^2)^2 x^., &c.</math></p>
<p>Ex. 3. Sit <math>$. {x^. / x^2} √(1 + x^4)</math> data fluens, quæ exprimit hyperbolicum
arcum, ergo hyperbolicus arcus erit intermedius terminus inter pri-
mum & $ecundum $eriei expre$$æ per fluentes fluxionum <math>{x^. / x^2} (1 + x^2)^0,
{x^. / x^2} × (1 + x^4)^1, {x^. / x^2} × (1 + x^4)^2, &c.</math> terminum.</p>
<p>Cor.. Arcus hyperbolicus exprimi pote$t per $eriem <math>- {1 / x} + {1 / 2 × 3}
x^3 - {1 / 2^2 . 2 × 7} x^7 + {1.3 / 2^3 × 1.3 × 11} x^11 - {1 · 3 · 5 / 21 · 2 · 1 · 4 × 15} x^15 +
{1 · 3 · 1 · 7 / 2^5 · 1 · 3 · 1 · 5 × 19} x^19 - &c.</math> ubi <math>x</math> denotat ab$ci$$am ad a$ymptoton.</p>
<p>Si vero requiratur de$cendens $eries, tum erit <math>x - {1 / 2 × 3} x^-3 +
{1 / 2^2 . 2 × 7} x^-7 - {1 · 3 / 2^3 · 1 · 3 · 11} x^-11 + &c.</math> quæ quoad coefficientes
attinet, pror$us eandem ob$ervat legem ac præcedens.</p>
<p>Ex. 4. Sin; fluentes fluxionum <math>{x^. / (1 + x)^0}, {x^. / (1 + x)^2}, {x^. / (1 + x)^4}, {x^. / (1 + x)^6},
&c.... {x^. / (1 + x)^2n}</math>, quæ erunt <math>x, {- 1 / 1 + x}, {- 1 / 3 (1 + x)^3}, {- 1 / 5 (1 + x)^5}, &c.</math>

<pb 608>[0660]<rh>DE SUMMATIONE</rh>
hæ vero fluentes inter valores <math>0 & 1</math> quantitatis <math>x</math> contentæ erunt re-
$pective <math>1, {1 / 2}, {7 / 24}, {31 / 5 × 32}, ... {2^2n-1 - 1 / (2n - 1). 2^2n-1}</math>; & medius terminus in-
ter duos primos erit hyp. log. 2.</p>
<p>Omnia hæc etiam ad incrementiales & integrales quantitates ap-
plicari po$$unt. Facile con$imili methodo inveniri po$$unt in$initæ
interpolationes incrementialium quantitatum.</p>
<h>THEOR. XXXIII.</h>
<p>Si numerus factorum in $ucce$$ivis terminis contentorum per arith-
meticam $eriem de$ignetur, tum for$an relatio inter $ucce$$ivos ter-
minos $it con$tans, i. e. numerus factorum in eâ contentus idem
$emper maneat. E. g. Sint $ucce$$ivi termini <math>1, 1 × 2, 1 × 2 × 3, 1 × 2
× 3 × 4, &c.</math> quorum numerus factorum $emper augetur per unita-
tem, i. e. $unt <math>1, 2, 3, 4, &c.</math> $int <math>t & t′</math> duo $ucce$$ivi $eriei termini, <math>z</math>
vero di$tantia termini <math>t</math> a primo $eriei termino, tum erit <math>(z + 2) × t
= t′</math> con$tans relatio. Cum vero relatio detur con$tans inter $ucce$-
$ivos terminos, tum ex quibu$dam datis primis terminis erui po$$unt
reliqui; & con$equenter $i interpoletur quantitas <math>(π)</math> inter primum &
$ecundum terminum, cujus di$tantia ex primo termino $it <math>{n / m}</math>, e termino
vero $ecundo $it <math>{m - n / m}</math>; tum ex prædictâ con$tante relatione erui pote$t
quicunque terminus, cujus di$tantia a termino <math>t^r</math> $it <math>{n / m}</math>, ubi <math>r</math> e$t in-
teger numerus; a termino vero proxime $ub$equente $it <math>{m - n / m}</math>; nam
erit <math>π × ({n / m} + 2) ({n / m} + 3) ({n / m} + 4).... ({n / m} + r + 1)</math>. E. 2. Sit
<math>φ: (z) × t = t′, & φ: (z + 1) × t′ = t″</math>, tum erit <math>φ: (z + r - 1)t′^r-1
= t′, φ: (z + r)t′^r = t^r+1, &c.</math>: & $ic ratiocinari liceat, cum detur
quæcunque relatio inter $ucce$$ivos terminos.</p>

<pb 609>[0661]<rh>SERIERUM, &c.</rh>
<p>Con$imilia etiam applicari po$$unt ad $eries, quarum primæ, $e-
cundæ, tertiæ, &c. differentiæ numerorum terminorum erunt in arith-
meticâ progreffione; hæ $eries, ut prius o$tenditur, interpolari po$-
$unt e $ucce$$ivis logarithmis eju$dem ordinis ac differentiæ numero-
rum, quæ con$tantes evadunt.</p>
<p>Si quantitates interpolandæ vel con$tent ex interpolabilibus facto-
ribus; vel $int quæcunque con$tantes functiones quantitatum, qua-
rum interpolationes dantur; tum dantur etiam earum interpolationes.</p>
<p>Cor. · Facile deduci po$$unt infinitæ quantitates, quæ ex interpo-
latione datarum quantitatum interpolari po$$unt.</p>
<p>Inveniatur quæcunque con$tans functio datarum quantitatum, <math>&</math>
perficitur coroll.</p>
<p>Et vice versâ ex animadvertendo $eriem datam e$$e prædictam fun-
ctionem quantitatum, quarum interpolationes innote$cunt, con$eque-
tur ejus interpolatio. E. g. Ex conce$sâ interpolatione terminorum <math>1,
1 × 2, 1 × 2 × 3, 1 × 2 × 3 × 4, &c.</math> termini binomialis theorematis <math>{n / m},
{{n / m} + 1 / 2} A, {{n / m} + 2 / 3} B, {{n / m} + 3 / 4} C, &c.</math> ubi <math>A, B, C, &c.</math> præcedentes ter-
minos re$pective denotant, interpolari po$$unt: terminus enim in datâ
$erie <math>1, 1 · 2, 1 · 2 · 3, 1 · 2 · 3 · 4, &c.</math> ad <math>({n / m} - 2)</math> di$tantiam a primo
$it <math>π</math>, & quoniam terminus ad di$tantiam <math>z</math> a primo in $erie <math>1, 1 · 2, 1 · 2
· 3, &c</math>. ductus in <math>z + 2</math> $ub$equentem præbet terminum; ergo terminus
<math>π</math> ad di$tantiam <math>(z = {n / m} - 2)</math> a primo in <math>z + 2 = {n / m}</math> ductus præbet
terminum ad di$tantiam <math>z + 1 = {n / m} - 1</math> a primo in eadem $erie; &
$imiliter termini ad di$tantias <math>{n / m} - 1, {n / m}, {n / m} + 1, {n / m} + 2, &c.</math>, etiam-
que ad di$tantias <math>{n / m} - 2, {n / m} - 3, {n / m} - 4, &c.</math> po$iti evadent <math>{n / m} × π,

<pb 610>[0662]<rh>DE SUMMATIONE</rh>
{n / m} × ({n / m} + 1) π, {n / m} ({n / m} + 1) ({n / m} + 2) π, {n / m} ({n / m} + 1) ({n / m} + 2) ({n / m}
+ 3) π, &c.</math>, etiamque <math>π, ({1 / {n / m} - 1}) π, ({1 / {n / m} - 1}) ({1 / {n / m} - 2}) π, &c.</math>
hinc terminus ad di$tantiam <math>r</math> a primo in $erie <math>1, 1 · 2, 1 · 2 · 3, &c.</math>
æqualis erit termino ad di$tantiam <math>r - {n / m} + 2</math> a primo in $erie <math>1, {n / m}, {n / m}
× ({n / m} + 1), &c.</math> in <math>π</math> ducto.</p>
<h>THEOR. XXXIV.</h>
<p>1. Sint termini <math>π, ρ, & σ</math>; quorum di$tantiæ a primo in $erie <math>1, 1 × 2,
1 × 2 × 3, 1 × 2 × 3 × 4, &c.</math> $int re$pective <math>{n / m} - 2, {n / m} + r - 1 & r;</math> tum
terminus in $erie <math>{n / m}, {n / m} × {{n / m} + 1 / 2}, {n / m} × {{n / m} + 1 / 2} × {{n / m} + 2 / 3}, &c.</math> cujus di-
$tantia e primo $it <math>r</math> erit <math>{ρ / π σ}.</math></p>
<p>2. Datis terminis $eriei <math>(P) 1, 1 × 2, 1 × 2 × 3, 1 × 2 × 3 × 4, &c.</math>
ad qua$libet di$tantias a primo po$itis, invenire terminum $eriei <math>(Q)
{m / r}, {m / r}. {m + n / r + s}, · .., {m / r} · {m + n / r + s} · {m + 2 n / r + 2 s} · .. {m + z n / r + z s} ad α</math> di$tantiam
a primo con$titutum. Sint termini ad <math>{m / n} - 2 & {r / s} - 2</math> di$tantias a
primo $eriei <math>(P)</math> termino re$pective <math>π & ρ</math>; termini vero ad di$tantias
<math>{n / m} + α - 1 & {r / s} + a - 1</math> a primo in eâdem $erie re$pective <math>σ & τ</math>; tum
erit terminus quæ$itus <math>{σ / π} × {ρ / τ} × {n^z+1 / s^z+1}</math>.</p>
<p>Hæ po$teriores $eries in priori <math>P</math> continentur.</p>

<pb 611>[0663]<rh>SERIERUM, &c.</rh>
<p>Cor. · Si detur $eries <math>(R)</math> terminorum, qui $int quæcunque alge-
braica finita functio prædictæ $eriei <math>Q</math>; facile ex ii$dem datis inveniri
pote$t terminus in prædictâ $erie <math>R</math> ad quamcunque di$tantiam a pri-
mo po$itus.</p>
<p>Eâdem methodo ratiocinari liceat ex interpolationibus $eriei <math>(P)</math>
vel plurium $erierum <math>(P), (Q), &c.</math> conce$$is ad interpolationes
$erierum, quæ $int quæcunque finitæ functiones quantitatum e præ-
dictis $eriebus deductarum.</p>
<p>Hic animadvertendum e$t, quod$i detur $eries huju$modi alterna-
tim affirmativa <math>&</math> negativa, i. e. $eries $it <math>- a + a b - a b c + a b c d
- &c.</math> quæ conficitur e factoribus <math>- a, - b, - c, &c.</math> in $e$e ductis,
minime ejus interpolatio eadem erit ac interpolatio factorum affirma-
tivorum <math>a, a × b, a × b × c, &c.</math> in priori ca$u enim inter $ingulos duos
fucce$$ivos terminos nihilo evadet terminus interpolandus, &c.</p>
<p>Series duas vel plures variabiles quantitates involventes vel $um-
mationem vel interpolationem recipiant ex eâdem methodo, ac $eries
unam $olummodo variabilem quantitatem habentes.</p>
<h>THEOR. XXXV.</h>
<p>1. Erit <math>e^±x = 1 ± x + {x^2 / 1 · 2} ± {x^3 / 1 · 2 · 3} + {x^4 / 1 · 2 · 3 · 4} ± &c. =
(1 ± {x / i})^i</math>, ubi <math>i</math> e$t in$inita quantitas; erit etiam <math>{e^+x +e^-x / 2} = 1 +
{x^2 / 1 · 2} + {x^4 / 1 · 2 · 3 · 4} + &c. = {(1 + {x / i})^i + (1 - {x / i})^i / 2}</math>; & $ic erunt
<math>{e^+x - e^-x / 2} = x + {x^3 / 1 · 2 · 3} + &c. & {e^x√(-1) - e^-x√(-1) / 2} × - √ (- 1) = x
- {x^3 / 1 · 2 · 3} + &c.</math> = $in. arcus <math>x, & {e^x√(-1) + e^-x√(-1) / 2} = 1 - {x^2 / 1 · 2}
+ {x^4 / 1 · 2 · 3 · 4} - &c.</math> = co$in. arcus <math>x</math>. Erit etiam <math>e^x √ (- 1) = 1 -

<pb 612>[0664]<rh>DE SUMMATIONE</rh>
{x^2 / 2} + {x^4 / 2 · 3 · 4} - {x^6 / 2 · 3 · 4 ·5 · 6} + &c. + (x - {x^3 / 1 · 2 · 3} + &c.) × √ (- 1)
= co$. x + √ (- 1)</math> $in. <math>x</math>; & $imiliter <math>e^-x√(-1) = 1 - {x^2 / 1 · 2} +
{x^4 / 1 · 2 · 3 · 4} - &c. - √ (- 1)(x - {x^3 / 1 · 2 ·3} + {x^5 / 1 · 2 · 3 · 4 ·5} - &c.) =
co$in. x - √ (- 1)</math> $in. <math>x</math>, hic radius e$t 1.</p>
<p>For$an haud indignum e$t ob$ervatu, quod <math>1 ± √ (- n) = 1</math>.</p>
<p>2. Sint <math>α, β, γ, δ, ε, &c. (n)</math> radices æquationis <math>x^n - 1 = 0</math>, & erit
<math>{e^zx + e^βx + e^γx + &c. / n} = 1 + {x^n / 1 · 2 · 3 .. n} + {x^2n / 1 · 2 · 3 .. 2n} + &c. &
{α^n-m e^αx + β^n-m e^βx + γ^n-m e^γx + &c. / n} = {x^m / 1 · 2 · 3 .. m} + {x^n+m / 1 · 2 · 3 .. n + m} +
{x^n+2m / 1 · 2 · 3 .. n + 2 m} + &c.</math> ubi <math>m</math> minor e$t quam <math>n</math>; & $ic $it quæcun-
que exponentialis quantitas, quæ reducatur in $eriem <math>(P)</math> $ecundum
dimen$iones quantitatis <math>x</math> progredientem, viz. <math>A x^a + B x^a+b + C x^a+20
+ &c.</math> & $i in datâ exponentiali $cribantur <math>α x, β x, γ x, &c.</math> re$pe-
ctive pro <math>x</math>, & $unt quantitates re$ultantes re$pective <math>A, B, C, D, &c.</math>
tum erit <math>{A + B + C + D + &c. / n} =</math> $ummæ $eriei, cujus termini $int
A x^a + σx^a+nb + &c.</math> $i vero requiratur ut primus terminus ad <math>m - 1</math>
di$tantiam a primo ponatur, viz. $it $eries <math>π x^a+mb + ρ x^a+(m+n)b + &c.</math>
tum erit ejus $umma <math>{α^n-m A + β^n-m B + γ^n-m C + &c. / n}</math>.</p>
<p>Idem etiam de $eriebus ab exponentialibus fluentibus exortis affir-
mari pote$t.</p>
<p>Cor. 1. Hinc inveniri po$$unt $ummæ $erierum prædictarum <math>a (t
+ t^n + t^2n + &c.) + b (t^′ + t^n+1 + t^2n+1 + &c.) + c (t^2 + t^n+2 +
t^2n+2 + &c.)</math>, ubi <math>t^n, &c.</math> denotat terminum prædictæ $eriei, cujus di-
$tantia a primo $it <math>n, &c. & a, b, c, &c.</math> de$ignant qua$cunque invaria-
biles quantitates.</p>

<pb 613>[0665]<rh>SERIERUM, &c.</rh>
<h>THEOR. XXXVI.</h>
<p>1. Erit <math>(1 ± {x / i})^i = 1 ± x + {x^2 / 1 · 2} ± {x^3 / 1 · 2 · 3} + &c.</math> & exinde
<math>(1 ± {x / i})^i - 1 = ± x + {x^2 / 1 · 2} ± {x^3 / 1 · 2 · 3} + &c.</math> quantitatis <math>(1 ± {x / i})^i
- 1 = v^i - 1</math> inveniantur quadratici divi$ores, qui erunt <math>(1 ± {x / i})^2
- 2 α (1 ± {x / i}) + 1</math>, ubi <math>α</math> denotat re$pective co$inus arcuum <math>0°,
{4 × 90° / i}, {8 × 90° / i}, {12 × 90° / i}</math>, &c. i. e. co$. arcuum <math>(0 π, {2 π / i}, {4 π / i}, {6 π / i},
{8 π / i}, &c.)</math>; vel $ub formulâ <math>(1 ± {x / i})^2 - 2 (1 ± {x / i})</math> co$in. <math>{2kπ / i} + 1</math>,
ubi <math>k</math> $ucceffive denotat integros numeros <math>0, 1, 2, 3, 4, &c.</math> continean-
tur. In hâc formulâ pro co$. <math>{2 k π / i}</math> $cribatur ejus valor <math>1 - {2 k^2 π^2 / i^2}</math>
prope, & re$ultat <math>{x^2 / i^2} + {4 k^2 π^2 / i^2} ± {4 k^2 π^2 / i^3} x = (1 ± {x / i})^2 - 2 (1 ± {x / i})</math>
co$. <math>{2 k · π / i} + 1</math>, ducatur hæc quantitas in <math>{i^2 / 4 k^2 π^2} &</math> re$ultat <math>{x^2 / 4 k^2 π^2}
± {x / i} + 1</math>; quare expre$$io <math>e^x - 1 = x (1 + {x^2 / 1 · 2} + {x^3 / 1 · 2 · 3} + &c.)</math>
habebit divi$ores <math>x, (1 + {x / i} + {x^2 / 4 π^2}), (1 + {x / i} + {x^2 / 16 π^2}), (1 + {x / i} +
{x^2 / 36 π^2}), &c.</math></p>
<p>2. Erit <math>e^x - e^-x = (1 + {x / i})^i - (1 - {x / i})^i = 2 (x + {x^3 / 1 · 2 · 3} +
{x^5 / 1 · 2 · 3 · 4 · 5} + &c.) = v^i - z^i</math>, $i modo pro <math>1 + {x / i} & 1 - {x / i}</math> $cri-<pb 614>[0666]<rh>DE SUMMATIONE</rh>
bantur re$pective <math>v & z</math>; $ed inveniri po$$unt ejus divi$ores <math>v^2 -
2 v z</math> co$. <math>{2 k π / i} + z^2 = {4 x^2 / i^2} + {4 k^2 π^2 / i^2} - {4 k^2 π^2 x^2 / i^4}</math> (ob co$in. <math>{2 k π / i}
= 1 - {2 k^2 π^2 / i^2}</math>); ducatur hæc quantitas in <math>{i^2 / 4 k^2 π^2}</math> & re$ultat for-
mula ejus divi$oris <math>{x^2 / k^2 π^2} - {x^2 / i^2} + 1 = {x^2 / k^2 π^2} + 1</math>, unde prædicta
æquatio habebit divi$ores <math>1 + {x^2 / π^2}, 1 + {x^2 / 4 π^2}, 1 + {x^2 / 9 π^2}, &c.</math></p>
<p>3. Erit <math>{e^x + e^-x / 2} = 1 + {x^2 / 2} + {x^4 / 1 · 2 · 1 · 4} + &c. = {(1 + {x / i})^i + (1 - {x / i})^i / 2}
{v^i + z^i / 2}</math>, $i modo pro <math>1 + {x / i} & 1 - {x / i}</math> $cribantur re$pective <math>v & z</math>;
tum inveniri po$$unt ejus divi$ores formulæ <math>1 + {4 x^2 / (2 k + 1)^2 π^2}</math> &
con$equenter erunt <math>1 + {4 x^2 / π^2}, 1 + {4 x^2 / 9 π^2}, 1 + {4 x^2 / 25 π^2}, &c.</math></p>
<p>Et $ic inveuiri po$$int <math>x, 1 - {x^2 / π^2}, 1 - {x^2 / 4 π^2}, 1 - {x^2 / 9 π^2}, &c.</math> divi$ores
$eriei <math>x - {x^3 / 1 · 2 · 3} + {x^5 / 1 · 2 · 1 · 4 · 5} - &c.</math> etiamque <math>1 - {4 x^2 / π^2}, 1 -
{4 x^2 / 9 π^2}, 1 - {4 x^2 / 25 π^2}, &c.</math> divi$ores $eriei <math>1 - {x^2 / 1 · 2} + {x^4 / 1 · 2 · 1 · 4} - &c.</math></p>
<p>4. Erit <math>e^x - 2 co$. g + e^-x = 2 (1 - co$. g + {x^2 / 1 · 2} + {x^4 / 1 · 2 · 1 · 4}
+ &c.) = (1 + {x / i})^i - 2 co$. g + (1 - {x / i})^i = v^i - 2 co$. g + z^i</math>,
unde con$tabit ejus divi$orum formula, viz. <math>v^2 - 2 v z co$.
{2 k π ± g / i} + z^2 = 2 + {2 x^2 / i^2} - 2 (1 - {x^2 / i^2}^i co$. {2 (2 k π ± g) / i} =

<pb 615>[0667]<rh>SERIERUM, &c.</rh>
{4 x^2 / i^2} + {4 (2 k π ± g)^2 / i^2}</math>; ob <math>i</math> infinitum, co$in. <math>{2 (2 k π ± g) / i} = 1
- {2 (2 k π ± g)^2 / i^2}</math>.</p>
<p>5. Erit <math>e^b+x ± e^c-x = (1 + {b + x / i})^i ± (1 + {c - x / i})^i = v^i ± z^i</math>,
cujus factores habent formulam <math>v^2 - 2 v z^. co$. {m π / i} + z^2</math> ubi <math>m</math>
denotat imparem numerum $i $ignum affixum $it +, $in vero $it -,
tum parem denotat numerum; $ed ob <math>i</math> numerum infinite magnum
erit co$. <math>{m π / i} = 1 - {m^2 π^2 / 2 i^2}</math>, unde formula factorum prædicta evadet
<math>(v - z)^2 + {m^2 π^2 / i^2} v z</math>, in hâc formulâ pro <math>v & z</math> $cribantur earum
valores <math>1 + {b + x / i} & 1 + {c - x / i}</math>, & ducatur quantitas exinde de-
ducta in <math>i^2</math>, & re$ultabit <math>(b - c)^2 + m^2 π^2 + 4 (b - c) x + 4 x^2</math>,
dividatur ea per <math>(b - c)^2 + m^2 π^2</math> & re$ultat formula quæ$ita <math>1 +
{4 (b - c) x + 4 x^2 / (b - c)^2 + m^2 π^2}</math>.</p>
<p>Cor. 2. Ut unitas evadat primus terminus $eriei; dividatur
quantitas <math>e^b+x ± e^c-x</math> per <math>e^b ± e^c</math>, & re$ultat <math>{e^b+x ± e^c-x / e^b ± e^c} = (1 +
{4 (b - c) x + 4 x^2 / π^2 + (b - c)^2}) (1 + {4 (b - c)x + 4 x^2 / 9 π^2 + (b - c)^2}) &c.</math> $i <math>m</math> $it impar, vel
<math>= (1 + {2 x / b - c}) (1 + {4 (b - c)x + 4 x^2 / 4 π^2 + (b - c)^2}) (1 + {4 (b - c)x + 4 x^2 / 16 π^2 + (b - c)^2}),
&c.</math> $i <math>m</math> $it impar.</p>
<p>Et $imiliter inveniri pote$t $ormula divi$orum quantita-
tis <math>e^a+bx ± e^c+dx = (1 + {a + b x / i})^i ± (1 + {c + d x / i})^i = v^i ± z^i,
&c.</math></p>

<pb 616>[0668]<rh>DE SUMMATIONE</rh>
<p>Plures huju$modi quantitates in $e$e ducantur, vel per $e dividan-
tur; & exinde oriri po$$unt $eries, quæ recipiant formulas diver$orum
generum pro earum divi$oribus.</p>
<p>Si vero <math>a</math> vel <math>b</math> vel <math>c</math> vel <math>d</math> $int impo$$ibiles quantitates, viz. <math>p √ (- 1)</math>
& exponentialium $umma vel differentia <math>e p √ (- 1) ± e - p√(- 1)</math> deno-
tat $inum arcus circuli, qui pro ejus valore in æquationibus prius
traditis $ub$titutus, æquationes $pecie diver$as reddent.</p>
<p>Cor. 4. Hinc inveniri po$$unt $ummæ plurium $erierum; e factori-
bus quidem datis innote$cunt radices ip$æ, & e terminis datæ æqua-
tionis innote$cunt $ummæ prædictarum radicum, earum quadrato-
rum, cuborum, &c. & denique aggregatum ex quâcunque algebraicâ
functione $ingularum radicum, cujus functionis $inguli termini con-
$tituant terminos $eriei datæ.</p>
<p>Cor. 4. Sit <math>z =</math> datæ $eriei <math>= α × β × γ × δ × &c.</math> tum erit <math>{z^. / z x^.} =
{α^. / α x^.} + {β^. / β x^.} + {γ^. / γ x^.} + &c.</math> $i modo <math>x</math> $it variabilis quantitas. E. g.
Sit <math>z = 1 + {x / 1 · 2} + {x^2 / 1 · 2 · 3} + &c. = (1 + {x / i} + {x^2 / 4 π^2}) (1 + {x / i} +
{x^2 / 16 π^2})</math>, &c. ubi <math>i</math> e$t infinita quantitas, tum erit <math>{z^. / z x^.} = {1 / 1 · 2} +
{x / 1 2} + &c. = {{1 / i} + {2 x / 4 π^2} / 1 + {x / i} + {x^2 / 4 π^2}} + {{1 / i} + {2 x / 16 π^2} / 1 + {x / i} + {x^2 / 16 π^2}} + &c.</math> & $ic de
reliquis.</p>
<h>THEOR. XXXVII.</h>
<p>1. Sit <math>x = A - {A^3 / 1 · 2 · 3} + {A^5 / 1 · 2 · 1 · 4 · 5} - {A^7 / 1 · 2 · 1 · 4 · 1 · 6 · 7} + &c.</math>
tum erit <math>x</math> $inus arcus (<math>A</math>) circuli, cujus radius e$t unitas, & cujus
peripheria e$t <math>2 π</math>.</p>
<p>Si vero detur $inus (<math>x</math>) arcus circuli, tum ejus corre$pondentes

<pb 617>[0669]<rh>SERIERUM, &c.</rh>
arcus circuli, quorum $inus e$t <math>x</math>, erunt re$pective <math>{m π / n}, {n - m / n} π,
{2 n + m / n} π, {3 n - m / n} π, &c. - {n + m / n} π, - {2 n - m / n} π, - {3 n + m / n} π,
&c.</math> quæ erunt radices (<math>A</math>) æquationis <math>x - A + {A^3 / 1 · 2 · 3} - {A^5 / 1 · 2 · 1 · 4 · 5}
+ &c. = 0</math>, & exinde reciprocæ radices erunt re$pective <math>{n / m π}, {n / (n - m) · π},
{n / (2 n + m) · π}, &c. - {n / (n + m) · π}, - {n / (2 n - m) · π}, &c.</math> & con$equen-
ter per vulgarem algebram erit <math>(1 - {n A / m π}) (1 + {nA / (n + m) π}) (1 -
{n A / (n - m) π}) (1 + {n A / (2 n - m) π}) &c. = 1 - {A / x} + {A^3 / 1 · 2 · 3 x} -
{A^5 / 1 · 2 · 1 · 4 · 5 x} + &c.</math> unde $umma e $ingulis quantitatibus <math>{1 / m}, -
{1 / n + m}, {1 / n - m}, {- 1 / 2 n - m}, {1 / 2 n + m}, &c. &c.</math> erit <math>{π / n x}</math>; $umma pro-
ductorum e $ingulis duabus erit o; $ummæ contentorum e $ingulis
tribus, quatuor, quinque, &c. erunt re$pective <math>{- π^3 / 1 · 2 · 3 x n^3}, 0,
{π^5 / 1 · 2 · 1 · 4 · 5 x n^5}, 0, {- π^7 / 1 · 2 · 1 · 4 · 1 · 6 · 7 x n^7}, 0, &c.</math></p>
<p>Pro <math>{n / m π}</math> $cribatur <math>a</math>, & erunt reciprocæ radices prædictæ <math>{1 / a}, {1 / π - a},
{1 / 2 π + a}, {1 / 3 π - a}, {1 / 4 π + a}, &c., - {1 / π + a}, - {1 / 2 π - a}, - {1 / 3 π + a},
- {1 / 4 π - a}, &c.</math> quarum $umma erit <math>{1 / a} + {2 a / π^2 - a^2} - {2 a / 4 π^2 - a^2} +
{2 a / 9 π^2 - a^2} - {2 a / 16 π^2 - a^2} + &c.</math></p>

<pb 618>[0670]<rh>DE SUMMATIONE</rh>
<p>2. Sit æquatio <math>y = 1 - {A^2 / 1 · 2} + {A^4 / 1 · 2 · 1 · 4} - {A^6 / 1 · 2 · 1 · 4 · 1 · 6}
+ &c.</math> & erit <math>y</math> co$inus arcus (<math>A</math>) circuli, cujus radius e$t 1, & peri-
pheria dicatur <math>2 π</math>.</p>
<p>Si vero detur <math>y</math> co$inus arcus <math>A = {r π / n}</math>, tum erunt corre$pon-
dentes arcus circuli, qui $int radices æquationis <math>y = 1 - {A^2 / 1 · 2} +
{A^4 / 1 · 2 · 1 · 4} - &c.</math> re$pective <math>{r π / n}, - {r π / n}, {(2 n - r) · π / n}, - {(2 n - r) · π / n},
{(4 n - r) π / n}, - {(4 n - r) · π / n}, {(6 n - r) · π / n}, &c.</math> cujus reciprocæ radices
erunt <math>{n / r π}, - {n / r π}, {n / (2 n - r) · π}, - {n / (2 n - r) π}, {n / (4 n - r) π}, -
{n / (4 n - r)} π, &c.</math> per vulgarem algebram erit <math>1 - {A^2 / 1 · 2 (1 - y)} +
{A^4 / 1 · 2 · 1 · 4 (1 - y)} - {A^6 / 1 · 2 · 1 · 4 · 1 · 6 (1 - y)} + &c. = (1 - {n A / r π})
(1 + {n A / r π}) (1 - {n A / (2 n - r) π}) (1 + {n A / (2 n - r) π}) (1 - {n A / (4 n - r) π})
(1 + {n A / (4 n - r) π}) &c.</math> & con$equenter $umma e $ingulis quantitati-
bus <math>{n / r π}, - {n / r π}, {n / (2 n - r) π}, {- n / (2 n - r) π}, {n / (2 n + r) π}, {- n / (2 n + r) π},
{n / (4 n + r) π}, &c.</math> nihilo æqualis erit; $umma vero productorum e
$ingulis duabus prædictis quantitatibus erit <math>{- 1 / 1 · 2 (1 - y)}</math>; aggregata
vero contentorum e $ingulis tribus, quatuor, quinque, $ex, &c. erunt
re$pective <math>0, {1 / 1 · 2 · 1 · 4 (1 - y)}, 0, {- 1 / 1 · 2 · 1 · 4 · 1 · 6 (1 - y)}, 0, &c.</math></p><pb 619>[0671]<rh>SERIERUM, &c.</rh>
<p>Sit <math>{γ π / n} = a</math>, tum erunt reciprocæ radices prædictæ <math>{1 / a}, {1 / 2 π - a},
{1 / 2 π + a}, {1 / 4 π - a}, {1 / 4 π + a}, &c., - {1 / a}, - {1 / 2 π - a}, - {1 / 2 π + a}, -
{1 / 4 π - a}, - {1 / 4 π + a}, &c.</math></p>
<p>Cor.. Hinc e capite 1<_>mo</_>. medit. algeb. facile colligi po$$unt aggre-
gata e quicu$cunque rationalibus, & haud reciprocis functionibus ra-
dicum vel æquationis infinitæ <math>(A) 1 - {A^2 / 1 · 2 (1 - y)} + {A^4 / 1 · 2 · 3 · 4 (1 - y)}
- &c. = 0</math>, vel etiam æquationis <math>(B) 1 - {A / x} + {A^3 / 1 · 2 · 3 x} -
{A^5 / 1 · 2 · 3 · 4 · 5 x} + &c. = 0</math>, vel earum in $e$e ductarum.</p>
<p>Ex.. In æquatione (<math>A</math>) detur <math>y</math> co$inus arcus nihilo æqualis, tum
erit <math>n:r::2:1</math>; $cribantur igitur <math>2 & 1</math> re$pective pro <math>n & r</math> in ejus
radicibus prius traditis, & re$ultant radices <math>{2 / π}, - {2 / π}, {2 / 3 π}, - {2 / 3 π}, {2 / 5 π},
- {2 / 5 π}, {2 / 7 π}, &c.</math> quarum quadrata erunt <math>{4 / π^2}, {4 / π^2}, {4 / 9 π^2}, {4 / 9 π^2}, {4 / 25 π^2},
{4 / 25 π^2}, &c.</math> $ed per algebram erit $umma quadratorum <math>1 = 8
({1 / π^2} + {1 / 9 π^2} + {1 / 25 π^2} + &c.)</math> & exinde <math>{π^2 / 8} = 1 + {1 / 9} + {1 / 25} + {1 / 49} +
&c. = S</math>; $i vero <math>s = 1 + {1 / 4} + {1 / 9} + {1 / 16} + {1 / 25} + {1 / 36} + &c.</math> tum erit <math>S +
{1 / 4} s = s</math>, unde <math>s = 1 + {1 / 4} + {1 / 9} + {1 / 16} + &c. = {4^S / 3} = {π^2 / 2 · 3}</math>.</p>
<p>Et $ic inveniatur $umma quadrato-quadratorum e $ingulis prædictis
quantitatibus, quæ evadet <math>32 × ({1 / π^4} + {1 / 81 π^4} + {1 / 625 π^4} + &c.) = {1 / 3}</math>,
& exinde <math>S = 1 + {1 / 3^4} + {1 / 5^4} + {1 / 7^4} + &c. = {π^4 / 32 · 3}</math>; $ed a$$umatur

<pb 620>[0672]<rh>DE SUMMATIONE</rh>
<math>s = 1 + {1 / 2^4} + {1 / 3^4} + {1 / 4^4} + &c.</math> dividatur hæc æquatio per 2<_>4</_>, & re$ul-
tat <math>{s / 2^4} = {1 / 2^4} + {1 / 4^4} + {1 / 6^4} + {1 / 8^4} + &c.</math> unde <math>S = s - {s / 2^4}</math>, & con$equen-
ter <math>{16 S / 15} = {π^4 / 3 · 2 · 3 · 5} = s = 1 + {1 / 2^4} + {1 / 3^4} + {1 / 4^4} + {1 / 5^4} + &c.</math></p>
<p>Et in genere $it <math>R</math> $umma <math>2 λ</math> pote$tatum e $ingulis radicibus præ-
dictis, per prob. 1. medit. algebr. invenienda; tum erit <math>1 + {1 / 3^2λ} + {1 / 5^2λ}
+ {1 / 7^2λ} + &c. = {R π^2 / 2^2λ+1} = S′, & s = {2^2λ / 2^2λ-1} S = 1 + {1 / 2^2λ} + {1 / 3^2λ} +
{1 / 5^2λ} + {1 / 7^2λ} + &c.</math></p>
<p>Ex. 2. Reciprocæ radices æquationis <math>(B) 1 - {A / x} + {A^3 / 1 · 2 · 3 · x} -
&c. = 0</math> inventæ fuerunt <math>{n / m π}; {n / (n - m) π}, - {n / (n + m) π}; {n / (2 n + m) π},
- {n / (2 n - m) π}; &c.</math> quarum $umma e$t <math>{1 / x}</math>; ergo erit <math>{1 / 2 m^2} + {1 / n^2 - m^2}
+ {1 / 4 n^2 - m^2} + {1 / 9 n^2 - m^2} + {1 / 16 n^2 - m^2} + &c. = {π / 2 m n x}.</math></p>
<p>Sit <math>n = 2 m</math>, unde $inus <math>x = 1</math>, & con$equenter radices prædictæ
fient <math>{2 / π}, {2 / π}, - {2 / 3 π}, - {2 / 3 π}, {2 / 5 π}, {2 / 5 π}, - {2 / 7 π}, - {2 / 7 π}, &c.</math></p>
<p>Et con$equenter erit <math>1 - {1 / 3} + {1 / 5} - {1 / 7} + {1 / 9} - &c. = {π / 4}</math>; & $ic <math>1 -
{1 / 3^3} + {1 / 5^3} - {1 / 7^3} + &c. = {1 / 32} π^3</math>, & generaliter erit <math>1 - {1 / 3^2λ+1} + {1 / 5^2λ+1} -
{1 / 7^2λ+1} + &c. = {N π^2λ+1 / 2^2λ+2}</math>, ubi <math>N</math> æqualis erit aggregato <math>2 λ + 1</math> pote$ta-
tis e $ingulis prædictis radicibus, quod facile e prob. 1. medit. algebr.
deduci pote$t.</p>

<pb 621>[0673]<rh>SERIERUM, &c.</rh>
<p>Sit <math>x = 0</math>, & con$equenter <math>m = 0</math>, & æquatio re$ultat <math>A (1 - {A^2 / 2 · 3}
+ {A^4 / 2 · 3 · 4 · 5} - &c.) = 0</math>, & exinde <math>1 - {A^2 / 2 · 3} + {A^4 / 2 · 3 · 4 · 5} - &c.
= (1 - {A / π}) (1 + {A / π}) (1 + {A / 2 π}) (1 - {A / 2 π}) (1 + {A / 3 π}) (1 - {A / 3 π})
(&c.) = (1 - {A^2 / π^2}) (1 - {A^2 / 4 π^2}) (1 - {A^2 / 9 π^2}) &c.</math></p>
<p>Sit <math>y = 0</math> & $imiliter colligi pote$t <math>(1 - {A^2 / 1 · 2} + {A^4 / 1 · 2 · 3 · 4} - &c.)
= (1 - {A^2 / π^2}) (1 - {A^2 / 9 π^2}) (1 - {A^2 / 25 π^2}) &c.</math></p>
<p>3. Sit æquatio <math>z = A - {A^3 / 1 · 2} + {A^5 / 1 · 2 · 3 · 4} - &c.</math> In hâc æqua-
tione pro <math>A</math> $cribatur <math>√(- 1) a</math>, & re$ultat æquatio <math>{z / √(- 1)} = a +
{a^3 / 2 · 3} + {a^5 / 2 · 3 · 4 · 5} + &c.</math> A$$umatur <math>z = 0</math>, & dividatur æquatio
re$ultans per <math>a</math>, & re$ultat æquatio <math>0 = 1 + {a^2 / 2 · 3} + {a^4 / 2 · 3 · 4 · 5} + &c.
= (1 + {a^2 / π^2}) (1 + {a^2 / 4 π^2}) (1 + {a^2 / 9 π^2}) &c.</math></p>
<p>4. Ob $in. <math>x = x (1 - {x^2 / π^2}) (1 - {x^2 / 4 π^2}) (1 - {x^2 / 9 π^2}) &c.</math> & co$. <math>x
= (1 - {4 x^2 / π^2}) (1 - {4 x^2 / 9 π^2}) &c.</math> & pro <math>x</math> $cribatur <math>{m π / n}</math> & duæ re$ultant
æquationes, viz. $in. <math>{m π / n} = {m π / n} ({n^2 - m^2 / n^2}) ({4 n^2 - m^2 / 4 n^2}) ({9 n^2 - m^2 / 9 n^2})
&c.</math> & co$. <math>{m π / n} = ({n^2 - m^2 / n^2}) ({9 n^2 - m^2 / 9 n^2}) ({25 n^2 - m^2 / 25 n^2}) &c.</math> e
priori æquatione $equitur <math>π = {n / m}</math> $in. <math>{m π / n} × ({n^2 / n^2 - m^2}) ({4 n^2 / 4 n^2 - m^2})

<pb 622>[0674]<rh>DE SUMMATIONE</rh>
({9 n^2 / 9 n^2 - m^2}) &c.</math> Sit <math>n = 2 m</math>, & re$ultat <math>{π / 2} = {2 · 2 / 1 · 3} · {4 · 4 / 3 · 5} · {6 · 6 / 5 · 7} ·
{8 · 8 / 7 · 9} · &c.</math> $it <math>n = 4 m</math> & erit $in. <math>{m π / n} = $in. 45° = √(2)</math>, & exinde
<math>{π / 4 √(2)} = {4 · 4 / 3 · 5} · {8 · 8 / 7 · 9} · {12 · 12 / 11 · 13} · &c.</math> dividatur prior re$ultans æqua-
tio per po$teriorem, & habebitur <math>2 √(2) = {2 · 2 / 1 · 3} · {6 · 6 / 5 · 7} · {10 · 10 / 9 · 11} · &c.</math></p>
<p>Et $ic de pluribus huju$cemodi æquationibus deducendis.</p>
<p>Cor. · Ob $inum & co$inum per $eries datos facile erui po$$unt
tangens, cotangens, $ecans, co$ecans, &c. prædicti arcus <math>{m π / n}</math>.</p>
<p>Cor. · Hinc etiam multiplicari, dividi, &c. po$$unt $inus, co$inus,
&c. duorum vel plurium diver$orum arcuum. E. g. <math>{$in. ar. {m π / n} / $in. ar. {k π / n}} =
{m / k} ({n^2 - m^2 / n^2 - k^2}) ({4 n^2 - m^2 / 4 n^2 - k^2}) ({9 n^2 - m^2 / 9 n^2 - k^2}) &c.</math></p>
<p>5. Datâ præcedente æquatione <math>1 - {A / x} + {A^3 / 2 · 3 x} - {A^5 / 2 · 3 · 4 · 5 x}
+ &c. = 0</math>, cujus reciprocæ radices $int <math>α, β, γ, δ, &c.</math> Fingatur <math>Z
= 1 - {A / x} + {A^3 / 2 · 3 x} - &c.</math> unde con$tat <math>{Z^. / Z A^.} = (α + β + γ + δ +
&c.) + (α^2 + β^2 + γ^2 + δ^2 + &c.) A + (α^3 + β^3 + γ^3 + δ^3 + &c.)
A^2 + &c. = a + b A + c A^2 + &c.</math> $ed <math>Z = 1 - {A / x} + {A^3 / 2 · 3 x} - &c.
= 1 - {1 / x}</math> $in. arc. <math>A</math>, & con$equenter <math>{Z^. / Z A^.} = {co$. arc. A / x - $in. arc. A}</math>.</p>
<p>Scribatur <math>{p π / n} P</math> pro <math>A</math> in datâ æquatione, & quoniam <math>{m π / n}, {n - m / n} π,
&c.</math> $unt radices datæ æquationis, erunt <math>{p / m}, {p / n - m}, - {p / n + m}, -

<pb 623>[0675]<rh>SERIERUM, &c.</rh>
{p / 2 n - m}, {p / 2 n + m}, &c.</math> reciprocæ radices æquationis re$ultantis <math>1 -
{p π P / n x} + {p^3 π^3 P^3 / 2 · 3 n^3 x} - &c. = 0</math>: $tatuantur
<math>K = {p / m} + {p / n - m} - {p / n + m} - {p / 2 n - m} + &c.</math>
<math>L = {p^2 / m^2} + {p^2 / (n - m)^2} + {p^2 / (n + m)^2} + &c.</math>
<math>M = {p^3 / m^3} + {p^3 / (n - m)^3} - {p^3 / (n + m)^3} - &c.</math>
&c. &c. &c. & exinde con$tat <math>K + L
+ M + &c. = {p / m - p} + {p / n - m - p} - {p / n + m + p} - {p / 2 n - m + p}
+ &c. = {p π co$. arc. {p π / n} / n x - n $in. arc. {p π / n}}</math>, ubi <math>x = $in. arc. {m π / n}</math>: in hâc $erie
$cribatur <math>m = {k + l / 2} & p = {l - k / 2}</math>, & re$ultat $eries <math>{1 / k} + {1 / n - l} -
{1 / n + 1} - {1 / 2 n - k} + {1 / 2 n + k} + &c.</math> $eu <math>{1 / k} + {2 l / n^2 - l^2} - {2 k / 4 n^2 - k^2} +
&c. = {π co$. arc. {l - k / 2 n} / n $in. arc. {k + l / 2 n} π - n $in. arc. {(l - k) π / 2 n}}</math>.</p>
<p>Inveniri po$$unt etiam e præcedentibus principiis <math>{1 / 1 + b} - {1 / 4 + b} +
{1 / 9 + b} - {1 / 16 + b} + &c. = {1 / 2 b} - {π √(b) / (e^π√(b) - e^-π√(b))b} & {1 / 1 + b} +
{1 / 4 + b} + {1 / 9 + b} + &c. = {(e^π√(b) + e^-π√(b)) π √(b) / 2 b (e^π√(b) - e^-π√(b))} - {1 / 2 b}</math>.</p><pb 624>[0676]<rh>DE SUMMATIONE</rh>
<p>Erit <math>{πlog · 2 / 2} = 1 + {1 / 2 · 3^2} + {1 · 3 / 2 · 4 · 5^2} + {1 · 3 · 5 / 2 · 4 · 6 · 7^2} + &c.</math></p>
<p>Et $ic e meditat. algebraicis colligi po$$unt plurimæ con$imiles $e-
ries, i. e. $umma cuju$cunque functionis, in quâ haud reciproca vel
nulla dimen$io prædictarum radicum continetur: progredi etiam liceat
ad radices æquationis quæ e$t fluxio datæ æquationis; & $ic deinceps.
For$an vero quærat aliquis, quare ratiocinatio, quæ prius tradita fuit
de $eriebus $inum circuli ex ejus arcu exprimentibus haud applicari
pote$t ad $eries quæ detegunt $inum, i. e. ordinatam vel ellip$eos vel
plurimarum etiam ovalium con$imilium ex earum arcubus: cui re-
$pondendum e$t, ut $eries prædictæ haud omnes for$an habent ra-
dices po$$ibiles, vel omnes radices datæ infinitæ æquationis non in
prædictâ $erie continentur.</p>
<p>Si modo igitur dentur termini infinitæ $eriei, cujus omnes radices
innote$cunt; tum facile erui po$$unt e præcedente methodo infinitæ
$eries, quarum $ummæ innote$cunt, i. e. e $ummis $ingulorum valo-
rum quarumcunque algebraicarum functionum prædictarum radicum
medit. algeb. capit. primo inve$tigatis.</p>
<p>Hæ $eries, quarum $ummæ innote$cunt, haud nunquam u$ui in$er-
vire po$$unt in datis $eriebus $ummandis; reducantur enim datæ $eries
ad plures, $i ita reduci po$$int, quarum re$ultantes termini qui len-
ti$$ime convergant, $int $eries, quarum $ummæ innote$cunt; pro his
terminis $cribantur prædictæ $ummæ, & re$ultant $eries magis con-
vergentes.</p>
<h>THEOR. XXXVIII.</h>
<p>Ex. Datis fluentibus fluxionum <math>{x^. / 1 + ax^n}, {x^. / x} $.{x^. / 1 + ax^n}, {x^. / x} $.{x^. / x} $.{x^. / 1 + ax^n},
{x^. / x} $.{x^. / x} $.{x^. / x} $.{1 / 1 + ax^n}, &c.</math> quæ erunt <math>x - {ax^n+1 / n + 1} + {a^2 x^τn+1 / 2n + 1} - {a^3 x^3n+1 / 3n + 1} +
{a^4 x^4n+1 / 4n + 1} - &c., x - {ax^n+1 / (n + 1)^2} + {a^2 x^2n+1 / (2n + 1)^2} - {a^3 x^3n+1 / (3n + 1)^2} + &c.,

<pb 625>[0677]<rh>SERIERUM, &c.</rh>
x - {ax^n+1 / (n + 1)^3} + {a^2 x^2n+1 / (2n + 1)^3} - {a^3 x^3n+1 / (3n + 1)^3} + &c.,..., x - {ax^n+1 / (n + 1)^4}
+ {a^2 x^2n+1 / (2n + 1)^4} - {a^3 x^3n+1 / (3n + 1)^4} + &c.&c.</math> & fluentibus fluxionum <math>{x^z x^. / 1 + ax^n}</math>,
erui po$$unt $ummæ quarumcunque $erierum, quarum generalis ter-
minus e$t <math>± {nz + m · nz + m′ · nz + m″ · nz + m′″. &c. × a^z × x^nz+1 / nz + l · nz + l′ · nz + l″. &c. × (zn + 1)^b-1}</math>;
ubi <math>l, l′, l″, &c.</math>, $int inter $e diver$æ quantitates, & <math>z</math> e$t di$tantia a
primo $eriei termino.</p>
<p>Pro <math>$.{x^. / 1 + ax^n}, $.{x^. / x} × $.{x^. / 1 + ax^n}, $.{x^. / x} $.{x^. / x} $.{x^. / 1 + ax^n}, $.{x^. / x} $.{x^. / x} $.{x^. / x}
$.{x^. / 1 + ax^n}, &c.</math> $cribantur re$pective <math>A, B, C, D,... P, Q, R & S & T</math>;
ducatur ultima fluens in <math>x^1-2 x^.</math>, & inveniatur fluens fluxionis re$ul-
tantis, quæ erit <math>V = {1 / l - 1} x^l-1 × (T - {1 / (l - 1)}S + {1 / (l - 1)^2} R -
{1 / (l - 1)^3}Q + ... ± {1 / (l - 1)^b-4} C ∓ {1 / (l - 1)^b-3} B ± {1 / (l - 1)^b-2}A) ∓
{1 / (l - 1)^b-1} × $.{x^l-1 x^. / 1 + a x^n}</math>, ubi <math>b</math> de$ignat numerum literarum <math>A,B,C,...
R, S & T</math> per unitatem auctum: hæc autem fluens proprie correcta
æquat $ummam $eriei, cujus generalis terminus e$t <math>{±1 / nz + l. (nz + 1)^b-1}
× a^z × x^nz+1</math>; ducatur prædicta fluens <math>V</math> in <math>x^l′-l-1 x^.</math> & inveniatur per
præcedentem methodum fluens fluxionis <math>x^l′-l-1 x^. × V</math>, & ea de$ignat
$ummam $eriei, cujus generalis terminus e$t <math>{± a^z / nz + l. nz + l′ × (zn + 1)^b-1}
× x^nz+l′</math>, & $imiliter introducantur reliqui factores in denominatorem:
$it <math>nz + X</math> ultimus factor qui in denominatorem introducitur; tum
ducatur fluens prius re$ultans <math>W</math> in <math>x^m-x</math>, & invenjatur fluxio <math>(m - x)
x^m-x-1 x^. W + x^m-x W^.</math>; & con$equenter quantitas <math>♈ = (m - X)
x^m-x-1 W + x^m-x × {W^. / x^.}</math>, quæ æquat $ummam $eriei, cujus generalis terminus

<pb 626>[0678]<rh>DE SUMMATIONE</rh>
e$t <math>{±(nz + m) · a^z × x^nz+m-1 / nz + l · nz + l′.. nz + X × (nz + 1)^b-1}</math>; tum ducatur quantitas
<math>♈</math> in <math>x^m′-m+1</math> & re$ultat quantitas cujus fluxio per <math>x^.</math> divi$a præbet $um-
mam $eriei, cujus generalis terminus e$t <math>± {nz + m · nz + m′ · a^z · x^nz+m′-1 / nz + l · nz + l′ · &c. × (nz + 1)^b-1}</math>;
& $ic introduci po$$unt quicunque factores in numeratorem; ergo
con$tat theorema.</p>
<p>Hæc methodus fallit vel cum <math>l - 1 = 0</math> vel <math>l′ - 1 = 0</math> vel <math>l″ - 1
= 0, &c.</math></p>
<p>Si unquam occurrat in denominatore vel denominatoribus termi-
norum $eriei, cujus $umma quæritur, factor <math>nz + l = 0</math> vel <math>nz +
l′ = 0</math> vel <math>nz + l″ = 0, &c.</math>; tum, ut prius a$$eritur $emper termi-
nus erui pote$t e fluente fluxionis <math>x^π x^.$.({x^. / x})^n</math>, ubi <math>n</math> e$t integer nu-
merus, cujus fluens prius traditur.</p>
<p>2. Sit $eries, cujus generalis terminus e$t <math>{± a^z x^nz / (nz + l′)^b′ × (nz + l)^b}</math>,
& dentur fluentes <math>A, B, C,... Q, R, S & T</math> fluxionum <math>{x^l-1 x^. / 1 + ax^n}, {x^. / x}$.
{x^l-1 x^. / 1 + ax^n}, {x^. / x} $.{x^. / x} $.{x^l-1 x^. / 1 + ax^n}, {x^. / x}$.{x^. / x}$.{x^. / x}$.{x^l-1 x^. / 1 + ax^n}, &c.</math> quarum numerus e$t
<math>h</math>; etiamque dentur fluentes <math>A′, B′, C′,... Q′, R′, S′, & T′</math> fluxionum
{x^l′-1 x^. / 1 + ax^n}, {x^. / x} $.{x^l′-1 x^. / 1 + ax^n}, {x^. / x} $.{x^. / x} $.{x^l′-1 x^. / 1 + ax^n}, &c.</math> quarum numerus e$t <math>b′</math>; tum
erit quantitas <math>{1 / (l′ - l)^b′}x^-1 (T - {b′ / l′ - l}S + {b′ · {b′ + 1 / 2} / (l′ - l)^2} × R -
{b′ ×{b′ + 1 / 2} × {b′ + 2 / 3} / (l′ - l)^3}Q... ± &c. × A) + {1 / (l - l′)^b} × x^-l′ (T′ - {1 / l-l′}
S′ + {b · {b + 1 / 2} / (l - l′)^2}R′ - {b · {b + 1 / 2} · {b + 2 / 3} / (l - l′)^3} Q′ + ... ± &c. A′)</math> $umma $e-
riei quæ$ita.</p>

<pb 627>[0679]<rh>SERIERUM, &c.</rh>
<p>Ad eundem modum $umma $eriei, cujus generalis terminus e$t
<math>{(az^m + bz^m-1 + cz^m-2 + &c.)a^z x^nz / (nz + l)^b × (nz + l′)^b′ × (nz + l″)^b″ × &c.}</math> ex <math>(b)</math> fluentibus fluxio-
num <math>{x^l-1 x^. / 1 + ax^n}, {x^. / x} $.{x^l-1 x^. / 1 + ax^n}, {x^. / x} $.{x^. / x} $.{x^l-1 x^. / 1 + ax^n}, &c.; & (b′)</math> flu-
entibus fluxionum <math>{x^l′-1 x^. / 1 + ax^n}, {x^. / x} $.{x^l′-1 x^. / 1 + ax^n}, {x^. / x} $.{x^. / x} $.{x^l′-1 x^. / 1 + ax^n}, &c.
& (b″)</math> fluentibus fluxionum <math>{x^l′-1 x^. / 1 + ax^n}, {x^. / x} $.{x^l′-1 x^. / 1 + ax^n}, {x^. / x} $.{x^. / x} $.{x^. / x}$.
{x^l′-1 x^. / 1 + ax^n}, &c.; &c.;</math> $i modo <math>b, b′, b″, &c. & m</math> $int integri numeri, ac-
quiri pote$t.</p>
<p>Hìc ob$ervandum e$t omnes fluentes prædictas inter valores <math>o & x</math>
quantitatis <math>x</math> contentas e$$e, $i $eries <math>Ax^r + Bx^s + &c.</math> a$cendat $e-
cundum dimen$iones quantitatis <math>x</math>; vel inter infinitum <math>& x</math> $i $e-
ries de$cendat $ecundum dimen$iones quantitatis <math>x</math>: $i vero $eries
<math>Ax^r + Bx^s + &c.</math> nonnullos affirmativos & nonnullos negativos in-
dices quantitatis <math>x</math> contineat; tum in $erie a$cendente pro <math>x</math> $cribatur
<math>0</math>, vel in $erie de$cendente pro <math>x</math> $cribatur infinita quantitas, & re$ul-
tet quantitas <math>A′</math>; & pro fluente prædictæ fluxionis a$$umatur fluens
inter valores <math>0 & x</math> quantitatis <math>x + A′</math>.</p>
<h>THEOR. XXXIX.</h>
<p>1. Erit <math>(1 ± {a / x}) (1 ± {b / x}) (1 ± {c / x}) (1 ± {d / x}) (1 ± {e / x}) &c. = 1
± (a + b + c + d + e + &c.) × {1 / x} + (ab + ac + bc + ad + bd
+ &c.) × {1 / x^2} ± (abc + abd + acd + bcd + &c.) × {1 / x^3} + (abcd
+ &c.) × {1 / x^4} + &c.</math></p>

<pb 628>[0680]<rh>DE SUMMATIONE</rh>
<p>Cor. 1. Sint <math>a = {1 / 2^n}, b = {1 / 3^n}, c = {1 / 5^n}, d = {1 / 7^n}, &c. & x = τ</math> tum
erit <math>{1 / M} = (1 - {1 / 2^n}) (1 - {1 / 3^n}) (1 - {1 / 5^n}) (1 - {1 / 7^n})&c. = 1 - {1 / 2^n} -
{1 / 3^n} - {1 / 5^n} + {1 / 6^n} - {1 / 7^n} + &c.</math></p>
<p>Cor. 2. Sit <math>a = {1 / 2^2n}, b = {1 / 3^2n}, c = {1 / 5^2n}, d = {1 / 7^2n}, &c., & x = 1</math>; tum
erit <math>{1 / N} = (1 - {1 / 2^2n})(1 - {1 / 3^2n})(1 - {1 / 5^2n})(1 - {1 / 7^2n})&c. = 1 - {1 / 2^2n}
- {1 / 3^2n} - {1 / 5^2n} + {1 / 6^2n} - {1 / 7^2n} + &c.</math></p>
<p>In his duobus ca$ibus $igna affixa erunt + vel -, prout par vel
impar $it numerus divi$orum primorum numerorum <math>2, 3, 5, 7, &c.</math>
pro denominatore in $e$e ductorum.</p>
<p>Cor. 3. Erit <math>(1 + {1 / 2^n}) (1 + {1 / 3^n}) (1 + {1 / 5^n}) &c. = 1 + {1 / 2^n} + {1 / 3^n}
+ {1 / 5^n} + {1 / 6^n} + {1 / 7^n} + {1 / 10^n} + &c.</math>; in quâ continentur omnes naturales
numeri, qui non habent quadratum numerum pro divi$ore. Et $i-
militer <math>(1 + {1 / 2^2n}) (1 + {1 / 3^2n}) (1 + {1 / 5^2n})&c. = 1 + {1 / 2^2n} + {1 / 3^2n} + {1 / 5^2n}
+ {1 / 6^2n} + &c.</math></p>
<p>Cor. 4. Sint <math>2, 3, 5, 7, 11, &c.</math> primi numeri, & erit contentum
<math>(1 + 2) (1 + 3) (1 + 5) (1 + 7) &c. = 1 + (2 + 3 + 5 + 7 +
&c.) + 2 (3 + 5 + 7 + &c.) + 3(5 + 7 + &c.) + &c.</math> quantitas,
in quâ occurrunt omnes naturales numeri præter eos, qui plures di-
vi$ores æquales accipiant.</p>
<p>2. Erit <math>{1 / (1 ± {a / x}) (1 ± {b / x}) (1 ± {c / x}) (1 ± {d / x}) &c.} = 1 = (a + b<pb 629>[0681]<rh>SERIERUM, &c.</rh>
+ c + d + &c.) × {1 / x} + (a^2 + b^2 + c^2 + &c. + ab + ac + bc +
ad + &c.) × {1 / x^2} = (a^3 + b^3 + c^3 + &c. + a^2 b + a^2 c + b^2 c + &c.
+ abc + abd + acd + &c.) × {1 / x^3} + (a^4 + b^4 + &c. + a^3 b + &c.
+ a^2 b^2 + &c. + a^2 bc + &c. + abcd + &c.) × {1 / x^4} = &c.</math></p>
<p>Cor. 1. Sint <math>a = {1 / 2^n}, b = {1 / 3^n}, c = {1 / 5^n}, &c., & x = 1</math>; tum erit <math>M =
{1 / (1 - {1 / 2^n}) (1 - {1 / 3^n}) (1 - {1 / 5^n}) (1 - {1 / 7^n}) &c.} = 1 + {1 / 2^n} + {1 / 3^n} + {1 / 4^n}
+ {1 / 5^n} + {1 / 6^n} + {1 / 7^n} + {1 / 8^n} + &c.</math></p>
<p>Cor. 2. Sint <math>a = {1 / 2^2n}, b = {1 / 3^2n}, c = {1 / 5^2n}, &c.</math>; tum erit <math>N =
{1 / (1 - {1 / 2^2n}) (1 - {1 / 3^2n}) (1 - {1 / 5^2n}) &c.} = 1 + {1 / 2^2n} + {1 / 3^2n} + {1 / 4^2n} + {1 / 5^2n}
+ {1 / 6^2n} + &c.</math></p>
<p>Cor. 3. Erit <math>{1 / (1 + {1 / 2^n}) (1 + {1 / 3^n}) (1 + {1 / 5^n}) &c.} = 1 - {1 / 2^n} - {1 / 3^n}
+ {1 / 4^n} - {1 / 5^n} + {1 / 6^n} - {1 / 7^n} - {1 / 8^n} + {1 / 9^n} - &c., &c.;</math> $igna affixa erunt af-
firmativa vel negativa, prout numerus divi$orum, qui $unt primi nu-
meri, $it par vel impar.</p>
<p>Cor. 4. Sit <math>n = 1</math>, & erit <math>N = 1 + {1 / 4} + {1 / 9} + {1 / 16} + {1 / 25} + &c. = {π^2 / 6},

<pb 630>[0682]<rh>DE SUMMATIONE</rh>
& {1 / M} = 1 - {1 / 2} - {1 / 3} - {1 / 5} + {1 / 6} - {1 / 7} - &c. = 0, & {1 / N} = 1 - {1 / 2^2} - {1 / 3^2} -
{1 / 5^2} + {1 / 6^2} &c. = {6 / π^2}</math>.</p>
<p>3. Erit <math>L = (1 ± {a / x})^±m × (1 ± {b / x})^±m × (1 ± {c / x})^±m × (1 ± {d / x})^±m
× &c. = 1 ± m (a + b + c + d + &c.) × {1 / x} ± m × {± m - 1 / 2} (a^2 +
b^2 + c^2 + &c.) {1 / x^2} + m^2 (ab + ac + bc + ad + &c.) {1 / x^2} ± m ×
{± m - 1 / 2} × {± m - 2 / 3} (a^3 + b^3 + &c.) {1 / x^3} ± m^3 (abc + abd + acd
+ bcd + &c.) {1 / x^3} ± &c.</math></p>
<p>Cor. 1. Sint <math>a = {1 / 2^n}, b = {1 / 3^n}, c = {1 / 5^n}, d = {1 / 7^n}, &c. & x = 1</math>, tum erit
<math>(1 ± {1 / 2^n})^m (1 ± {1 / 3^n})^m (1 ± {1 / 5^n})^m (1 ± {1 / 7^n})^m &c. = 1 ± m ({1 / 2^n} +
{1 / 3^n} + {1 / 5^n} + {1 / 7^n} + &c.) + m · {m - 1 / 2}({1 / 2^2n} + {1 / 3^2n} + &c.) + m^2 ({1 / 2^n · 3^n}
+ {1 / 2^n · 5^n} + {1 / 3^n · 5^n} + {1 / 2^n · 7^n} + {1 / 3^n · 7^n} + &c.) ± m · {m - 1 / 2} · {m - 2 / 3}
({1 / 2^3n} + {1 / 3^3n} + {1 / 5^3n} + &c.) ± m^3 ({1 / 2^n · 3^n · 5^n} + {1 / 2^n · 3^n · 7^n} + {1 / 3^n · 5^n · 7^n}
+ &c.) ± &c.</math>; etiamque erit <math>(1 ± {1 / 2^n})^-m × (1 ± {1 / 3^n})^-m × (1 ±
{1 / 5^n})^-m × (1 ± {1 / 7^n})^-m &c. = 1 ∓ m ({1 / 2^n} + {1 / 3^n} + {1 / 5^n} + {1 / 7^n} + &c.) - m ×
{- m - 1 / 2} × ({1 / 2^2n} + {1 / 3^2n} + &c.) + m^2 × ({1 / 2^n · 3^n} + {1 / 2^n · 5^n} + {1 / 3^n · 5^n} +
{1 / 2^n · 7^n} + {1 / 3^n · 7^n} + {1 / 5^n · 7^n} + &c.) ∓ m^3 · ({1 / 2^n · 3^n · 5^n} + {1 / 2^n · 3^n · 7^n} +

<pb 631>[0683]<rh>SERIERUM, &c.</rh>
{1 / 2^n · 5^n · 7^n} + {1 / 3^n · 5^n · 7^n} + {1 / 2^n · 3^n · 11^n} + &c.) - &c.</math>, in hoc ca$u omnes na-
turales numeri integri, qui primi $unt, reciproce in primâ $ummâ con-
tinentur & in <math>m</math> ducuntur; & in $ub$equentibus aggregata e $ingulis con-
tentis huju$ce generis <math>π^-rn × ρ^-sn × σ^-tn &c.</math>, ubi <math>π, ρ, σ, &c.</math> $unt quicun-
que diver$i primi, & <math>r, s, t, &c.</math> integri numeri, in $erie continentur; &
hæc aggregata ducuntur in coefficientes <math>m · {m - 1 / 2} .. {m - r + 1 / r} × m.
{m - 1 / 2} .. {m - s + 1 / s} × m · {m - 1 / 2} .. {m - t + 1 / t} × &c.</math></p>
<p>Et $ic deduci po$$unt infinitæ con$imiles propo$itiones ex a$$umptis
quibu$cunque factoribus, & reductis a$$umptis quantitatibus per me-
thodos divi$ionis, radicum extractionis, &c. ad infinitas $eries, quæ
evadunt convergentes.</p>
<p>5. Erit <math>M = {1 / (1 - {1 / 2^n}) (1 - {1 / 3^n}) (1 - {1 / 5^n}) (1 - {1 / 7^n}) &c.} = {2^n / 2^n - 1}
× {3^n / 3^n - 1} × {5^n / 5^n - 1} · &c. = 1 + {1 / 2^n} + {1 / 3^n} + {1 / 4^n} + &c.</math>; & $imiliter <math>N
= {1 / (1 - {1 / 2^2n}) (1 - {1 / 3^2n}) (1 - {1 / 5^2n}) (1 - {1 / 7^2n}) &c.} = {2^2n / 2^2n - 1} × {3^2n / 3^2n - 1}
× {5^2n / 5^2n - 1} × &c. = 1 + {1 / 2^2n} + {1 / 3^2n} + {1 / 4^2n} + {1 / 5^2n} + &c.</math>; unde <math>{M / N} = (1
+ {1 / 2^n}) (1 + {1 / 3^n}) (1 + {1 / 5^n}) (1 + {1 / 7^n}) &c., & {MM / N} = {2^n + 1 / 2^n - 1} × {3^n + 1 / 3^n - 1}
× {5^n + 1 / 5^n - 1} × {7^n + 1 / 7^n - 1} × &c.</math></p>
<p>6. Erit log. <math>M = log. (1 + {1 / 2^n} + {1 / 3^n} + &c.) = - log. (1 - {1 / 2^n}) - log.
(1 - {1 / 3^n}) - log. (1 - {1 / 5^n}) - &c. = 1 ({1 / 2^n} + {1 / 3^n} + {1 / 5^n} + &c.) +

<pb 632>[0684]<rh>DE SUMMATIONE</rh>
{1 / 2} ({1 / 2^2n} + {1 / 3^2n} + {1 / 5^2n} + &c.) + {1 / 3}({1 / 2^3n} + {1 / 3^3n} + {1 / 5^3n} + &c.) + {1 / 4}({1 / 2^4n}
+ &c.)</math></p>
<p>Cor. · Quoniam prius inve$tigatæ fuerunt $ummæ $erierum <math>1 +
{1 / 2^2} + {1 / 3^2} + {1 / 4^2} + {1 / 5^2} + &c. = {π^2 / 6}, & 1 + {1 / 2^2n} + {1 / 3^2n} + {1 / 4^2n} + &c. =
{1 / a}π^2n</math>; ubi <math>n</math> e$t integer numerus; & con$equenter invenitur valor fra-
ctionis <math>{1 / (1 - {1 / 2^2n})} (1 - {1 / 3^2n}) (1 - {1 / 5^2n}) &c.</math> huic $eriei æqualis; unde
con$tat <math>2 n log · π - log · a = - log · (1 - {1 / 2^2n}) - log. (1 - {1 / 3^2n}) - &c.
= 1 ({1 / 2^2n} + {1 / 3^2n} + {1 / 4^2n} + &c.) + {1 / 2}({1 / 2^4n} + {1 / 3^4n} + &c.) + {1 / 3}({1 / 2^6n} + {1 / 3^6n}
+ &c.) + &c.</math></p>
<p>Inveniantur valores e pluribus huju$modi $eriebus, i. e. cum <math>2 n</math>
æquat re$pective <math>α, β, γ, &c.</math> qui $int <math>{π^α / b}, {π^β / c}, {π^γ / d}, &c.</math> unde eorum
logarithmi erunt re$pective <math>α log. π - log. b; β log · π - log. c;
γ log. π - log. d; &c.</math> inveniantur $eries, quæ his logarithmis re$pe-
ctive $unt æquales. Ducantur hæ $eries re$pective in quantitates <math>k,
l, m, &c.</math> & erit $erierum re$ultantium aggregatum <math>= (k α + l β +
m γ + &c.) log. π - log. b^k c^l d^m, &c.</math></p>
<p>Fiat <math>k α + l β + m γ + &c. = 0</math>, & re$ultant $eries, quæ cognitæ
quantitatis logarithmo æquales $unt.</p>
<h>THEOR. XL.</h>
<p>1. Sit <math>A = 1 + {1 / 2^n} + {1 / 3^n} + {1 / 4^n} + &c.</math> ducatur hæc æquatio in
<math>1 - {1 / 2^n}</math> & re$ultat <math>(1 - {1 / 2^n}) A = B = 1 + {1 / 3^n} + {1 / 5^n} + {1 / 7^n} + &c.</math>

<pb 633>[0685]<rh>SERIERUM, &c.</rh>
ducatur hæc æquatio in <math>(1 - {1 / 3^n})</math> & re$ultat <math>(1 - {1 / 3^n}) B = C =
1 + {1 / 5^n} + {1 / 7^n} + {1 / 11^n} + &c.</math> ducatur hæc æquatio in <math>1 - {1 / 5^n}</math>, & re-
$ultat <math>(1 - {1 / 5^n}) C = D = 1 + {1 / 7^n} + {1 / 11^n} + &c. &c.</math> unde <math>A =
(1 - {1 / 2^n}) (1 - {1 / 3^n}) (1 - {1 / 5^n}) (1 - {1 / 7^n}) (1 - {1 / 11^n}) &c. = {2^n - 1 / 2^n}.
{3^n - 1 / 3^n} · {5^n - 1 / 5^n} · &c.</math></p>
<p>2. Sit <math>A′ = 1 - {1 / 3^n} + {1 / 5^n} - {1 / 7^n} + {1 / 9^n} - {1 / 11^n} + &c.</math> ducatur hæc
æquatio in <math>1 + {1 / 3^n}</math>, & re$ultat <math>(1 + {1 / 3^n}) A′ = B′ = 1 + {1 / 5^n} - {1 / 7^n} +
&c.</math> ducatur re$ultans æquatio in <math>1 - {1 / 5^n}</math>, & re$ultat <math>(1 - {1 / 5^n}) B′ = C′
= 1 - {1 / 7^n} - {1 / 11^n} + &c.</math> unde <math>A′^-1 = (1 + {1 / 3^n}) (1 - {1 / 5^n}) (1 + {1 / 7^n})
(1 + {1 / 11^n}) &c. = {3^n + 1 / 3^n} · {5^n - 1 / 5^n} · {7^n + 1 / 7^n} · &c.</math></p>
<p>Hìc <math>1, 3, 5, 7, &c.</math> omnes primos numeros denotant, & $ignum
affixum erit <math>+</math> vel <math>-</math>, prout primus numerus $it <math>4 m - 1</math> vel <math>4 m - 3</math>,
ubi <math>m</math> e$t integer numerus.</p>
<p>3. Ex principiis prius traditis inveniri pote$t valor quantitatis <math>A</math>, cum
<math>n</math> $it quicunque integer numerus; & exinde e Walli$ii formulâ <math>{π / 2} =
{2 · 2 · 4 · 4 · 6 · 6 · 8 · 8 · &c. / 1 · 3 · 3 · 5 · 5 · 7 · 7 · &c.}</math>, vel formulis prius datis deduci po$-
$unt per multiplicationem vel divi$ionem vel radicum extractionem
harum & prædictarum formularum plurimæ quantitates, quarum
contenta innote$cunt vel per pote$tates quantitatis <math>π</math>, vel per datas al-
gebraicas quantitates.</p><pb 634>[0686]<rh>DE SUMMATIONE</rh>
<p>Ex. g. Sit <math>n = 1,</math> & erit <math>A = {π / 4} = {3 / 4} · {5 / 4} · {7 / 8} · {11 / 12}· &c.</math> at primo
inventum fuit <math>{π^2 / 6} = {4 / 3} · {3^2 / 2 · 4} · {5^2 / 4 · 6} · {7^2 / 6 · 8}· &c.</math> dividatur $ecunda per
primam, & exorietur <math>{2π / 4} = {3 / 2} · {5 / 6} · {7 / 6} · {11 / 10} · {13 / 14} · {17 / 18} · &c.</math> denuo divi-
datur hæc æquatio per primam, & re$ultat <math>{4 / 2} · {4 / 6} · {8 / 6} · {12 / 10} · {12 / 14}· &c. =
2 = {2 / 1} · {2 / 3} · {4 / 3} · {6 / 5} · {6 / 7}· &c.</math> dividatur Walli$ii æquatio <math>{π / 2} = {2 · 2 · 4 · 4 · 6 · 1 · 8 / 1 · 3 · 3 · 5 · 5 · 7 · 7}
&c.</math> $eu <math>{4 / π} = {3 · 3 / 2 · 4} · {5 · 5 / 4 · 6} · {7 · 7 / 6 · 8}· &c.</math> per <math>{π^2 / 8} = {3 · 3 / 2 · 4} · {5, 5 / 4 · 6} · {7 · 7 / 6 · 8}· &c.</math>
& re$ultat <math>{3^2 / π^3} = {9 · 9 / 8 · 10} · {15 · 15 / 14 · 16} · {21 · 21 / 20 · 22}· &c</math>. ubi in numeratoribus
occurrunt omnes numeri impares non primi.</p>
<p>Et $ic $it <math>n = 3</math> & inveniri pote$t <math>A = {π^3 / 3^2} = {3^3 / 3^3 + 1} · {5^3 / 5^3 - 1} ·
{7^3 / 7^3 + 1} · {11^3 / 11^3 + 1} · {13^3 / 13^3 - 1}· &c.</math> & quoniam <math>{π^6 / 945} = 1 + {1 / 2^6} + {1 / 3^6} +
{1 / 4^6} + &c. = {2^6 / 2^6 - 1} · {3^6 / 3^6 - 1} · {5^6 / 5^6 - 1}· &c.</math> con$equitur <math>{π^3 / 30} = {3^3 / 3^3 - 1}
· {5^3 / 5^3 - 1} · {7^3 / 7^3 - 1} · &c.</math> ex divi$ione po$terioris æquationis per primam
habebitur <math>{16 / 15} = {3^3 + 1 / 3^3 - 1} · {5^3 - 1 / 5^3 + 1} · &c.</math></p>
<p>3. Sit <math>A = 1 - {1 / 2^n} + {1 / 4^n} - {1 / 5^n} + {1 / 7^n} - {1 / 8^n} + {1 / 10^n} - {1 / 11^n} + &c. & n</math>
impar numerus; & erit <math>(1 + {1 / 2^n}) A = B = 1 - {1 / 5^n} + {1 / 7^n} - {1 / 11^n} +
&c.</math> & $imiliter <math>(1 + {1 / 5^n}) B = C = 1 + {1 / 7^n} - {1 / 11^n} + {1 / 13^n} &c.</math> & $ic
deinceps: ex his tandem fiet <math>A(1 + {1 / 2^n})(1 + {1 / 5^n})(1 - {1 / 7^n}) &c. = 1,</math>

<pb 635>[0687]<rh>SERIERUM, &c.</rh>
ubi numeri primi unitate excedentes multipla $enarii habent $ignum
-; deficientes +: $it <math>n = 1</math> & erit <math>A = {π / 3√(3)} = {2 / 3} · {5 / 6} · {7 / 6}· &c.</math> ubi
denominatores $unt omnes præter primum per 6 divi$ibiles, per hanc
expre$$ionem dividatur æquatio <math>{π^2 / 6} = {4 / 3} · {3 · 3 / 2 · 4} · {5 · 5 / 4 · 6} · {7 · 7 / 6 · 8} · {11 · 11 / 10 · 12} ·
&c.</math> & re$ultat <math>{π√(3) / 2} = {9 / 4} · {5 / 4} · {7 / 8} · {11 / 10}· &c.</math> hæc vero per illam divi$a,
dabit <math>{4 / 3} = {6 / 4} · {6 / 8} · {12 / 14} · {18 / 16} · {18 / 20}· &c.</math> Et $ic ex ii$dem principiis deduci po$-
$unt aliæ con$imiles æquationes.</p>
<p>Sit <math>A = 1 + {1 / 3} - {1 / 5} - {1 / 7} + {1 / 11} - {1 / 13} - &c. = {π / 2√(2)},</math> & exinde (A),
i. e. <math>{π / 2√(2)}(1 - {1 / 3})(1 + {1 / 5})(1 + {1 / 7})(1 - {1 / 11})(1 + {1 / 13}) &c. = 1</math>, ubi
primorum numerorum formularum <math>8 m + 1</math> vel <math>8 m + 3</math> $igna erunt
-, formularum <math>8 m + 5</math> vel <math>8 m + 7</math> $igna erunt +; unde <math>{π / 2√(2)}
= {3 / 2} · {5 / 6} · {7 / 8} · {11 / 10} · {13 / 14} · {17 / 16} · {19 / 18} &c.</math> Et ex hâc & prius traditis æquationibus
per hanc methodum erui po$$unt plurimæ con$imiles æquationes.</p>
<p>E divi$ione factorum in numeratore per eorum corre$pondentes in
denominatore erui po$$unt diver$æ $eries, quæ in $e$e ductæ præbent
$eriem æqualem contento factorum in $e$e continuo ductorum: in
re$olutione huju$ce problematis plerumque u$ui in$ervire po$$int ea,
quæ in tradita fuere.</p>
<h>THEOR. XLI.</h>
<p>Cum detur valor contenti, quod erit fractio, cujus denominator
con$tat e pluribus factoribus; reducatur hæc fractio ad infinitam $e-
riem terminorum, & con$tat valor infinitæ $eriei; vel ducatur prædicta
fractio in qua$cunque cognitas quantitates, quæ vel diruat quo$dam
factores datæ fractionis vel novos adjiciat, & reducatur re$ultans fra-
ctio ad infinitam $eriem terminorum, cujus $umma etiam innote$cit.</p>

<pb 636>[0688]<rh>DE SUMMATIONE</rh>
<p>Ex.1. Detur <math>{π / 4} = {3 / 3 + 1} · {5 / 5 - 1} · {7 / 7 + 1} · &c. = {3 · 5 · 7 / 4 · 4 · 8} &c. =
{1 / (1 + {1 / 3}) (1 - {1 / 5}) (1 + {1 / 7}) &c.}</math>; unde <math>{π / 4} × {1 / 1 + {1 / 2}} = {π / 6} =
{1 / (1 + {1 / 2})(1 + {1 / 3})(1 - {1 / 5})(1 + {1 / 7})&c.}</math>; reducatur hæc fractio ad infini-
tam $eriem, & re$ultat <math>{π / 6} = 1 - {1 / 2} - {1 / 3} + {1 / 4} + {1 / 5} + {1 / 6} - {1 / 7} &c.</math> ubi
primi numeri <math>4m - 1</math> habent $ignum -, & <math>4 m + 1</math> $ignum +; $i
<math>m</math> $it integer numerus.</p>
<p>Fuit <math>{π / 2} = {1 / (1 - {1 / 3}) (1 + {1 / 5})(1 - {1 / 7})(1 - {1 / 11}) &c.} = 1 + {1 / 3} - {1 / 5} +
{1 / 7} + {1 / 9} + {1 / 11} - &c.</math> ubi impares numeri $olummodo occurrunt, & primi
numeri formulæ <math>4m + 1</math> habent $ignum -, formulæ vero <math>4m - 1</math>
habent $ignum +; & $igna compo$itorum numerorum e primis
eorum divi$oribus deduci po$$unt: ducatur hæc æquatio in <math>{1 / 1 - {1 / 2}}</math> &
re$ultat <math>π = {1 / (1 - {1 / 2})(1 - {1 / 3})(1 + {1 / 5}) &c.} = 1 + {1 / 2} + {1 / 3} + &c.</math> duca-
tur eadem æquatio in <math>{1 / 1 + {1 / 2}}</math> & re$ultat <math>{π / 3} = {1 / (1 + {1 / 2}) (1 - {1 / 3}) &c.} = 1
- {1 / 2} + {1 / 3} + {1 / 4} - &c.</math></p>
<p>Ducatur æquatio <math>{π / 2} = {1 / (1 - {1 / 2})(1 - {1 / 3})(1 - {1 / 5}) (1 - {1 / 7} &c.}</math> in <math>1 + {1 / 3}/1 - {1 / 3}
× 2</math>, & re$ultat <math>π = {1 / (1 - {1 / 2}) (1 - {1 / 3}) (1 - {1 / 5}) &c.}</math> & $ic deinceps; & e
lege quam ob$ervat $eries ex evolutione fractionis prioris æquationis
exorta con$equitur lex quam ob$ervat $eries ex pofteriore fractione in
$implices terminos per præcedentem methodum reductâ.</p>
<p>Sit <math>0 = {1 / (1 + {1 / 2})(1 + {1 / 3})(1 + {1 / 5})(1 + {1 / 7})(1 + {1 / 11}) &c.} = 1 - 2 - {1 / 3} -
{1 / 5} + {1 / 6} - {1 / 7}&c.</math> ducatur hæc æquatio in finitum numerum factorum
<math>1 + {1 / 2}, 1 + {1 / 3}, 1 + {1 / 5}, &c.</math> & æquatio re$ultans etiam erit nihilo æqua-

<pb 637>[0689]<rh>SERIERUM, &c.</rh>
lis; ducatur etiam æquatio exinde re$ultans in finitum numerum
factorum huju$ce generis <math>1 - {1 / 2}, 1 - {1 / 3}, 1 - {1 / 5}, &c.</math> & contentum etiam
erit nihilo æquale; in quo ca$u omnes primi numeri ni$i finitus eo-
rum numerus in $erie præcedente methodo deductâ $ignum habent
-: & $ic de infinitis huju$cemodi $eriebus, fractionibus & terminis
detegendis, vel rationalibus vel irrationalibus.</p>
<p>1. In genere a$$umantur quæcunque algebraicæ quantitates vel ra-
tionales fractiones vel irrationales quantitates; inveniantur quæcun-
que functiones harum quantitatum, reducantur hæ functiones ad $e-
ries, $i modo $int infinitæ, convergentes; tum inveniuntur $eries,
quarum $ummæ innote$cunt.</p>
<p>2. Inveniantur quæcunque functiones prædictarum $erierum &
datarum infinitarum, quarum $ummæ innote$cunt; & $i modo $eries
exinde re$ultantes $int convergentes, tum inveniuntur $eries, quarum
fummæ innote$cunt.</p>
<h>THEOR. XLII.</h>
<p>1. Sit <math>(x^α + x^β + x^γ + x^δ + &c.)(x^a + x^b + x^c + x^d + &c.) =
x^a+α .... + N x^n, & N</math> erit numerus modorum, quibus confici pote$t
<math>n</math> ex additione cuju$cunque quantitatis vel <math>a</math> vel <math>b</math> vel <math>c, &c.</math> ad unam
e quantitatibus <math>α, β, γ, δ, &c.</math> Si vero <math>b</math> $it negativa quantitas, tum
de quantitatibus <math>α, β, γ, δ, &c.</math> $ubtrahenda e$t <math>b</math>.</p>
<p>2. Et $ic de pluribus factoribus in $e$e ductis: e. g. $it <math>(1 + x^α z)
(1 + x^β Z)(1 + x^γ Z)(1 + x^δ Z) &c. = 1 + Pz + Qz^2 ... N x^n z^m</math>,
tum coefficiens <math>N</math> erit diver$orum modorum numerus, quibus nume-
rus <math>n</math> pote$t e$$e $umma <math>m</math> diver$orum terminorum <math>α, β, γ, δ, &c.</math> $it
<math>z = 1</math> & erit <math>(1 + x)(1 + x^2)(1 + x^3) &c. = 1 + x + x^2 + 2x^3
... Px^n, & P</math> numerus modorum, quibus numerus <math>n</math> ex additione
diver$orum terminorum <math>1, 2, 3, 4, &c.</math> produci pote$t.</p>
<p>3. Sit <math>{1 / (1 + x^α z)(1 + x^β z)(1 + x^γ z)&c.} = 1 + Pz + Qz^2 +
Rz^3 ... N x^n z^m</math>, & erit <math>N</math> numerus modorum, quibus numerus <math>n</math> ex

<pb 638>[0690]<rh>DE SUMMATIONE</rh>
additione <math>m</math> quantitatum <math>α, β γ, δ, &c. 2α, 2β, 2γ, &c. 3α, 3β, 3γ,
&c.</math> produci pote$t.</p>
<p>4. Sit <math>{1 / (1 - x)(1 - x^2)(1 - x^3)...(1 - x^m)} = 1 + x + 2x^2 +
... N x^n &c.</math> & erit <math>N</math> numerus modorum, quibus produci pote$t <math>n</math> ex
additione numerorum <math>1, 2, 3, 4, &c.</math> $upponatur etiam <math>{x / 1 - x} · {x^2 / 1 - x^2}
· {x^3 / 1 - x^3}...{x^m / 1 - x^m} = {x^m.{m+1 / 2} / (1 - x)(1 - x^2)(1 - x^3)...(1 - x^m)} =
x^m.{m+1 / 2} + ..... + N x^n+m.{m+1 / 2}, &c.</math> unde <math>N</math> erit numerus diver$orum
modorum quibus numerus <math>n + m · {m + 1 / 2}</math> dividi pote$t in <math>m</math> inæqua-
les partes, & con$equenter quot modis <math>n</math> produci pote$t ex additione
numerorum <math>1, 2, 3, ... m</math>, tot modis numerus <math>n + m · {m + 1 / 2}</math> dividi
pote$t in <math>m</math> inæquales partes.</p>
<p>5. Sint <math>(A){1 / (1 - x)(1 - x^2)(1 - x^3)...(1 - x^m)} = 1 + P x...
Nx^n, &c. & (B){x^m / (1 - x) (1 - x^2) (1 - x^3) ... (1 - x^m)} = x^m + P′x^m+1
... M x^n, &c.</math> & exinde <math>(A-B) = {1 - x^m / (1 - x)(1 - x^2)...(1 - x^m)} =
{1 / (1 - x)(1 - x^2)...(1 - x^m-1)} = 1 + Px ... (N - M)x^n, &c.</math> unde
differentia inter numerum modorum, quibus <math>n & n - m</math> numeri con-
fici po$$unt ex numeris <math>1, 2, 3, ... m</math> æqualis erit numero modorum,
quibus <math>n</math> confici pote$t e numeris <math>1, 2, 3, ... m - 1.</math></p>
<p>Eodem modo ducatur <math>A</math> vel in <math>(1 - x^m)(1 - x^m-1)</math> vel <math>B</math> in <math>(1 -
x^m-1); &c.</math> & con$imilia deduci po$$unt theoremata; & $ic deinceps.</p>
<p>6. E$t <math>{1 / (1 - x)(1 - x^3)(1 - x^5)&c.} = (1 + x)(1 + x^2)(1 + x^3)<pb 639>[0691]<rh>SERIERUM, &c.</rh>
&c.</math> in infinitum <math>= 1 + x + x^2 + 2 x^3 + ... N x^n</math>: & exinde quot
diver$is modis per additionem numerus <math>n</math> confici pote$t ex integris
inæqualibus numeris, totidem modis idem numerus <math>n</math> confici pote$t
per additionem numerorum modo $int impares, utrum $int æquales
vel inæquales.</p>
<p>7. Contentum <math>(x^-1 + 1 + x^1) (x^-3 + 1 + x^3) (x^-9 + 1 + x^9)
(x^-27 + 1 + x^27) &c.</math> præbet omnes exponentes quantitatis <math>x</math> vel ex
additione vel ex $ubtractione numerorum <math>1, 3, 9, 27, &c.</math> & $imilia
etiam de reliquis numeris prædicari po$$unt.</p>
<p>8. Sit <math>(1 - x)^r (1 - x)^s (1 - x)^t = (1 - x)^r+s+t = 1 - (r + s + t)
x + .. N x^m &c.</math>; tum, $i numerus modorum, quibus <math>α</math> quantitates in <math>r</math>
contineri po$$unt, multiplicetur in numerum modorum, quibus <math>β</math>
quantitates in <math>s</math> continentur; & productum re$ultans multiplicetur
in numerum modorum, quibus <math>m - α - β</math> quantitates in <math>t</math> continen-
tur; erit aggregatum e $ingulis contentis re$ultantibus <math>N</math>; ubi <math>α, β
& γ</math> re$pective denotant quo$cunque numeros <math>1, 2, 3, &c.</math>: & $imili-
ter numerus modorum, quibus <math>m</math> quantitates in <math>r + s + t</math> contineri
po$$unt, etiam erit <math>N</math>.</p>
<p>9. Sit <math>(1 + x^r + x^s)^n = 1 + n x^r ... + N x^m &c.: & α r + β s
= m</math>; & inveniatur numerus modorum, quibus <math>α</math> quantitates conti-
neri po$$unt in <math>n</math>; qui ducatur in numerum modorum, quibus <math>β</math>
quantitates contineri po$$unt in <math>n</math>; tum erit $umma e $ingulis his pro-
ductis = <math>N</math>.</p>
<p>Hìc etiam adjicere liceat quamplurimas huju$ce generis propo$itio-
nes; in genere $i modo data quantitas vel $eries quocunque modo ex
aliis conficiatur, animadvertentur etiam quomodo producentur ejus
coefficientes, exponentes, &c. & re$ultabunt huju$ce generis propo$i-
tiones: $i modo in diver$am formulam transformetur data quantitas,
ita ut eædem coefficientes, exponentes, &c. vel aliæ ad has affigna-
bilem relationem habentes e diver$is modis producentur; tum etiam
re$ultabunt novæ huju$ce generis propo$itiones: hinc vix ulla datur
$eries, e quâ non facile deduci po$$unt propo$itiones huju$modi.</p>
<p>10. In multitudine (<math>n</math>) literarum <math>a b c d e f g h, &c.</math> numeri modorum,
quibus detrahi po$$unt quæcunque duæ proxime $ucce$$ivæ, viz. vel

<pb 640>[0692]<rh>DE SUMMATIONE</rh>
<math>a & b</math>, vel <math>b & c</math>, vel <math>c & d, &c.</math>; $emel, bis, ter, quater, &c. erunt
<math>n - 1, (n - 2) · {n - 3 / 2}, (n - 3) · {n - 4 / 2} · {n - 5 / 3}, &c.</math></p>
<p>Cor. Sit continua fractio <math>{a / b + a} \\ b + a \\ b + &c.</math> in infin.; # tum erunt
fractiones $ucce$$ivæ ad datam vergentes <math>a × {1 / b}, a × {b / b^2 + a}, a × {b^2 + a / b^3 + 2 b a},
a × {b^3 + 2 b a / b^4 + 3 b^2 a + a^2}, a × {b^4 + 3 b^2 a + a^2 / b^5 + 4 b^3 a + 3 b a^2}, &c.</math> in genere erit fra-
ctio, cujus numerator e$t <math>a × (b^n-1 + (n - 2) b^n-3 a + (n - 3) · {n - 4 / 2}
b^n-5 a^2 + (n - 4) · {n - 5 / 2} · {n - 6 / 3} b^n-7 a^3 + (n - 5) · {n - 6 / 2} · {n - 7 / 3}·
{n - 8 / 4} b^n-9 a^7 + &c.)</math>, denominator vero <math>b^n + (n - 1) b^n-2 a + (n - 2)
· {n - 3 / 2} b^n-4 a^2 + (n - 3) · {n - 4 / 2} · {n - 5 / 3} b^n-6 a^3 + &c.</math> approximatio
ad valorem prædictæ continuæ fractionis.</p>
<p>Hæ fractiones etiam $ic exprimi po$$unt, viz. <math>a × {1 / b}, a × {A / b A + a P},
a × {B / b B + a Q}, a × {C / b C + a R}, a × {D / b D + a S}, &c.</math>; ubi literæ <math>A, B, C,
D, &c.</math> re$pective denotant denominatores; & <math>P, Q, R, S, &c.</math> nume-
ratores præcedentium fractionum.</p>
<p>Cor. Hæ fractiones erunt approximationes ad radicem quadraticæ
æquationis <math>x^2 + b x - a = 0</math>, nam erit <math>x = {a / b + a} \\ b + &c. # = {a / b + x}</math>,
& exinde <math>x^2 + b x = a</math>.</p>
<p>Cor. In datâ quadraticâ <math>x^2 + b x - a = 0</math> pro <math>x</math> $cribatur <math>z - b</math>,

<pb 641>[0693]<rh>SERIERUM, &c.</rh>
& re$ultat <math>z^2 - b z = a</math>; unde per præcedens coroll. <math>z =
{a / - b + a} \\ <ol>- b</ol> + &c.</math> # , & exinde <math>x = b + {a / - b} + a \\ <ol>- b</ol> + a \\ <ol>- b</ol> + &c.</math> # altera radix
prædictæ quadraticæ; continua fractio <math>{a / b + a} \\ b + a \\ b + &c.</math> # eadem erit ac continua
fractio <math>{a / - b + a} \\ - b + a \\ - b + &c.</math> # , at ejus negativa; & con$equenter ea-
rum approximationes eædem.</p>
<p>Cor. Hinc deduci po$$unt multæ continuæ fractiones, quarum
$ummæ differunt tantummodo per datas quantitates; transformetur
enim quæcunque quadratica æquatio <math>x^2 + b x = a</math> in alteram <math>z^2 +
B x = A e</math> $cribendo <math>z + p</math> pro <math>x</math>, & erit <math>a \\ b + a \\ b + a \\ b + &c. # =
A \\ B + A \\ B + A \\ B + &c. # + p.</math></p>
<p>2. Et $imiliter in cubicâ æquatione <math>x^3 + d x^2 - (a + b)x -
a d = 0</math> pro <math>x</math> $cribatur <math>z + e</math> & re$ultet æquatio <math>x^3 + D x^2 -
(A + B)x - A D = 0</math>, tum erit <math>√ a + b \\ 1 + d \\ √ a + b \\ 1 + d \\ √ &c. # =

<pb 642>[0694]<rh>DE SUMMATIONE</rh>
√ A + B \\ <ol>1 + D</ol> \\ <ol>√ A + B</ol> \\ <ol>1 + D</ol> \\ <ol>√ &c.</ol> # + e</math>.</p>
<p>Et $ic facile erui po$$unt infinitæ con$imiles propo$itiones ex a$$u-
mendo diver$as approximationes ad quotientes, radices, &c.</p>
<h>THEOR. XLIII.</h>
<p>Fluentes fluxionum <math>x^i x^. π′ & x^i+a x^. π′</math>, ubi <math>i</math> $it infinitus numerus, <math>a</math>
finitus, <math>π′</math> vero quæcunque algebraìca finita functio literæ <math>x</math>, inter va-
lores <math>0 & 1</math> quantitatis <math>x</math> po$itæ, erunt inter $e æquales.</p>
<p>Reducatur enim algebraica functio <math>π′</math> ad terminos a$cendentes
$ecundum dimen$iones quantitatis <math>x</math>, qui $int <math>a x^π-1 + b x^π+ρ-1
+ c x^π+2ρ-1 + &c.</math> & evadent duæ fluentes prædictæ re$pective
<math>{a / i + π} + {b / i + π + ρ} + {c / i + π + 2ρ} + &c. & {a / i + α + π} + {b / i + α + π + ρ}
+ &c.</math> $ed quoniam <math>i</math> e$t infinitus, hæ duæ $eries a$cendentes erunt
inter $e in ratione æqualitatis.</p>
<h>PROB. XXXVI.</h>
<p it>Qua$dam fluentes evolvere per producta infinita.</p>
<p>1. Fluens fluxionis <math>{x^. / √(1 - x^2)}</math> inter valores (<math>0 & 1</math>) quantitatis <math>x</math> per
ex. 1. prob. 23. l. 1. erit <math>{2 · 4 · 6 · 8 .. (2 i) / 1 · 3 · 5 · 7 .. (2 i - 1)} $. {x^2i x^. / √(1 - x^2)}</math>; & $ic per
prædict. exem. erit <math>$. {x x^. / √(1 - x^2)} = {3 · 5 · 7 ... (2 i + 1) / 2 · 4 · 6 .. (2 i)} $. {x^2i+1 x^. / √(1 - x^2)^2}</math>.
Sit <math>i</math> infinitus & erit <math>$. {x^2i x^. / √(1 - x^2)} = $. {x^2i+1 x^. / √ (1 - x^2)} = $. {x^2i+2 x^. / √(1 - x^2)}</math>,
& con$equenter <math>$. {x x^. / √(1 - x^2)} (1): $. {x^. / √ (1 - x^2} ({π / 2})::{1 · 3 · 5 · 7 · &c. / 2 · 4 · 6 · &c.}

<pb 643>[0695]<rh>SERIERUM, &c.</rh>
: {2 · 4 · 6 · &c. / 1 · 3 · 5 · 7 · &c.}</math>, unde <math>{π / 2} = {2 · 2 / 1 · 3} · {4 · 4 / 3 · 5} · {6 · 6 / 5 · 7}· &c.</math></p>
<p>Ex. 2. Per ex. 1. prob. 23. <math>$. x^m-1 x^. (1 - x^n)^{k-n / n} (P) & $. x^μ-1 x^. (1 - x^n)^{k-n / n}
(Q)</math> inter valores <math>0 & 1</math> quantitatis <math>x</math> po$itæ, erunt re$pective
<math>{(m + k) (m + k + n) (m + k + 2 n) ... (m + k + i n) / m(m + n) (m + 2 n) ... (m + i n)} $. x^m+in+n-1 x^.
(1 - x^n)^{k-n / n}, & {(μ + k) (μ + k + n) ... (μ + k + in) / μ (μ + n) ... (μ + i n)} $. x^μ+in+n-1 x^.
(1 - x^n)^{k-n / n}</math>; cum vero <math>i</math> $it in$initus, tum evadunt hæ duæ fluentes in-
ter $e æquales, & con$equenter erit <math>{P / Q} = {μ(m + k) / m(μ + k)} · {(μ + n)(m + k + n) / (m + n)(μ + k + n)}
· {(μ + 2 n)(m + k + 2 n) / (m + 2 n)(μ + k + 2 n)}. &c.</math></p>
<p>A$$umatur <math>μ = n</math>, vel <math>2 n</math> vel <math>3 n</math> vel <math>4 n, &c.</math>, ita quidem ut ejus
fluens deduci pote$t; e. g. $it <math>μ = n</math>, & erit <math>$. x^n-1 x^. (1 - x^n)^{k-n / n} =</math>
con$t. <math>(C) - {1 / k} (1 - x^n)^{k / n} (R)</math>; po$ito vero <math>x = 0</math>, evane$cere oportet
fluentem, unde <math>C = {1 / k}</math>; ergo erit <math>Q = {1 / k}</math>, & exinde con$tat <math>P = {1 / k}.
{μ(m + k) / m(μ + k)} · {(μ + n)(m + k + n) / (m + n)(μ + k + n)}. &c. = {1 / k} · {n(m + k) / m(n + k)} · {2 n (m + k + n) / (m + n)(2 n + k)}.
{3 n (m + k + 2n) / (m + 2n)(3 n + k)} · &c.</math></p>
<p>Sit <math>m = {n + α / 2} & k = {n - α / 2}</math>, & re$ultat <math>P = {4 · n / (n - α)^2} · {2 · 4 n^2 / 9 n^2 - α^2}·
{4 · 6 n^2 / 25 n^2 - α^2}· &c. = {2 / n - α} · {2 n · 2 n / (n + α) (3 n - α)} · {4 n · 4 n / (3 n + α) (5 n - α)}.
&c. = {π / n $in. {m π / n}} = {π / n co$. {α π / 2 n}}</math>. Sit <math>k = {μ n / v}</math>, & evadet <math>$. x^m-1 x^.

<pb 644>[0696]<rh>DE SUMMATIONE</rh>
(1 - x^n)^{μ / υ}-1 = {υ / m μ} · {2(μ υ + n μ) / (m + n)(μ + υ)} · {3(μ υ + n μ + n υ) / (m + 2 n)(μ + 2 υ)} ·
{4 (m υ + n μ + 2 n υ) / (m + 3 n)(μ + 3 υ)} · &c.</math></p>
<p>Cum <math>$. {x^m-1 x^. / ^n √ (1 - x^n)^n-k} = $.{x^k-1 / ^n √(1 - x^n)^n-m} = {1 / k} · {n (m + k) / m(k + n)}.
{2 n (m + k + n) / (m + n)(k + 2 n)}. &c.</math> inter duos valores <math>0 & 1</math> quantitatis <math>x</math> po$itæ;
& $imiliter erit <math>$. {x^p-1 x^. / ^n √ (1 - x^n)^n-q} = $. {x^q-1 x^. / ^n √ (1 - x^n)^n-p} = {p + q / p q}.
{n (p + q + n) / (p + n)(q + n)} · {2 n (p + q + 2n) / (p + 2n)(q + 2n)} · &c.</math> & con$equenter <math>{$. x^m-1 x^. (1 - x^n)^{k-n / n} / $. x^p-1 x^. (1 - x^n)^{q-n / n}}
= {p q (m + k) / m k (p + q)} · {(p + n)(q + n)(m + k + n) / (m + n)(k + n)(p + q + n)}.{(p + 2 n)(q + 2 n) / (m + 2 n)(k + 2 n)}
{(m + k + 2 n) / (p + q + 2 n)} · &c.</math></p>
<h>PROB. XXXVI.</h>
<h it>Innvenire diver$a producia ex binis huju$modi formulis, quæ inter $e
$unt æqualia.</h>
<p>Sint producta re$pective <math>{a + b / a b} · {n (a + b + n) / (n + a)(n + b)} · {2 n (a + b + 2 n) / (a + 2 n)(b + 2 n)}
· &c. {c + d / c d} · {n (c + d + n) / (n + c) (n + d)}. &c.</math> & $ic <math>{p + q / p q} · {n (p + q + n) / (n + p) (n + q)} ·
&c., & {r + s / r s} · {n (r + s + n) / (n + r)(n + s)}. &c.</math> ducantur duo priora producta in
$e$e, & evadit productum <math>{a + b / a b} × {c + d / c d} × {n (a + b + n) / (a + n)(b + n)} ×
{n (c + d + n) / (c + n)(d + n)}. &c.</math> & $ic ducantur in $e$e duo po$teriora, & re$ul-

<pb 645>[0697]<rh>SERIERUM, &c.</rh>
tat <math>{p + q / p q} × {r + s / r s}. &c.</math> nunc nece$$e e$t ut fiat <math>(a + b)(c + d)p q r s
= a b c d (p + q)(r + s)</math>, & con$equenter $inguli ex his $ex factori-
bus utrinque $int æquales: & exinde vel $int <math>s = d, r = b, q = c,
p = r + s = b + d, c + d = a, & a + b = p + q</math>; vel $int <math>s = d,
r = b, q = r + s = b + d, p = a, c + d = p + q & a + b = c</math>;
tertio vero $int <math>s = d, r = p + q, q = b, p = c, c + d = a & a + b
= r + s = p + q + d</math>. Hinc oriuntur tres formulæ infinitorum
productorum, quæ erunt inter $e æqualia; etiamque <math>$. {x^p-1 x^. / ^n √ (1 - x^n)^n-q}
· $. {x^p+q-1 x^. / ^n √ (1 - x^n)^n-r} = $.{x^q-1 x^. / ^n √ (1 - x^n)^n-r} · $. {x^q+r-1 x^. / ^n √(1 - x^n)^n-p} = $.{x^p-1 x^. / ^n √ (1 - x^n)^n-r}
· $. {x^p+r-1 x^. / ^n √ (1 - x^n)^n-q}</math>; quæ fluentes inter valores <math>0 & 1</math> quantitatis <math>x</math> con-
tinentur.</p>
<p>Cum vero <math>p + q = n</math>, erunt tria prædicta producta <math>= {π / n r $in.{p π / n}}</math>.</p>
<p>Et $ic progredi liceat ad deducenda diver$a producta e ternis qua-
tuor, &c. huju$modi formulis, quæ inter $e $unt æqualia.</p>
<p>Per prob. prius tradita e fluentibus datarum fluxionum acquiri
po$$unt infinita producta, quorum contenta e prædictis fluentibus
acquiri po$$unt.</p>
<p>Erit <math>$. {x^m+zn-1 x^. / (1 - x^n)^{n-k / n}} × $. {x^m+k+xn-1 x^. / (1 - x^n)^{k / n}} = {m / m + α n} $.{x^m-1 x^. / (1 - x^n)^{n-k / n}}. $.
{x^m+k-1 x^. / (1 - x^n)^{k / n}}</math>, ubi valores $ingularum prædictarum fluentium inter va-
lores <math>0 & 1</math> quantitatis <math>x</math> a$$umendi $unt: con$tat e prob. 23. l. 1.</p>
<p>Et $ic e datis in$initis productis haud difficilis erit inve$tigatio,
utrum prædicta producta $int prædicti generis, necne; $i vero $int

<pb 646>[0698]<rh>DE SUMMATIONE</rh>
producta prædicti generis, tum facile acquiri po$$unt fluentes, quibus
æqualia $unt producta jam prædicta.</p>
<p>Singuli factores facile transformari po$$unt in infinitas alias irra-
tionales, & exinde contenta in infinitis irrationalibus factoribus de-
duci po$$unt.</p>
<h>PROB. XXVII.</h>
<p it>Datâ $erie $ecundum continua contenta in $e$e ducta progrediente, inve-
nire $eriem relationem inter $ucce$$ivos terminos exprimentem: $it data $e-
ries <math>p (1 + α)(1 + β)(1 + γ) &c.... (P) (1 + π)(1 + ρ)(1 + σ)
(1 + τ) &c.</math> tum erunt $ucce$$ivi termini <math>T^m = Pπ & T^m+1 = P (1 +
π)ρ</math>, ergo erit relatio quæ$ita <math>(1 + π)ρ T^m = π T^m+1</math>.</p>
<p>Cor. 1. Ex datâ relatione inter $ucce$$ivos terminos per prob. 2. lib.
præc. erui pote$t, annon $umma $eriei $it in$inita; $ed $eries $ecundum
continua contenta progrediens facile transformari pote$t in æquationem
relationem inter $ucce$$ivos terminos de$ignantem; ergo facile deduci
pote$t, annon $eries $ecundum prædicta contenta progrediens $it finita.</p>
<p>Cor. 2. Facile ex principiis prius traditis innote$cunt ca$us, qui ex
datis relationibus inter $ucce$$ivos terminos $eriei præbent fluxionales
æquationes, in quibus variabiles quantitates $unt $ummæ $erierum &
quantitas $ecundum cujus dimen$iones termini a$cendunt vel de-
$cendunt.</p>
<p>Et vice versâ $it data prædicta fluxionalis æquatio; vel relatio inter
$ucce$$ivos terminos, quæ exinde deduci pote$t; deinde ita redu-
cantur æquationes inter $ucce$$ivos terminos & æquationem <math>(1 + π)
ρ T^m = π T^m+1</math> vel quæcunque aliæ datæ con$imiles ex continuâ $erie
deductæ, ita ut exterminentur termini; & re$ultat æquatio relationem
inter <math>π, ρ, σ, &c.</math> de$ignans, ubi <math>π, ρ, σ, &c.</math> $unt $ucce$$ivi valores
eju$dem quantitatis: ex hâc æquatione inveniantur valores <math>π, ρ, σ, &c.</math>;
& re$ultat $eries continua quæ$ita.</p>
<p>Aliter: A$$umatur functio quantitatis <math>z</math> di$tantiæ a primo $eriei
termino pro $ummâ; deinde inveniantur $ucce$$ivi termini <math>T & T′</math>, &
exinde ex a$$umpto primo termino contentorum, ita ut corre$pon-

<pb 647>[0699]<rh>SERIERUM, &c.</rh>
deat primo termino $eriei, cujus termini $unt <math>T & T′, &c.</math>; & tum ex
terminis <math>T^m & T^m+1</math> & præcedente factore <math>1 + π</math> datis per æquatio-
nem <math>(1 + π)ρT^m = π T^m+1</math> facile acquiri pote$t factor <math>1 + ρ</math>.</p>
<h>PROB. XXXVIII.</h>
<p>E prob.23. l.1. per factores huju$modi exprimere fluentem cuju$cun-
que fluxionis huju$ce formulæ <math>(a + bx^n)^m-1 x^rn-1 x^.</math> inter valores <math>o & {-a / b}</math>
quantitatis <math>x^n</math> contentam: fluens fluxionis <math>(a + b x^n)^m-1 x^n-1 x^.</math>, quæ in
genere erit <math>{1 / m n b} × (a + b x^n)^m</math>, inter prædictos valores quantitatis <math>x</math>
contenta erit <math>{a^m / m n b}</math>; & con$equenter erit <math>(P) $. (a + b x^n)^m-1 x^(1+1)n-1 x^.
= {1 · 2 · 3 ... i / m + 1 · m + 2 .. m + i} × {a^i / b^i} × {a^m / m n b}</math>, & $ic inveniri pote$t <math>(Q)
$. (a + b x^n)^m-1 x^(r+i)n-1 x^. = {r · r + 1 · r + 2 .. (r + i - 1) / m + r · m + r + 1 · m + r + 2 .. (m + r + i - 1)}
× {a^i / b^i} × $. (a + b x^n)^m-1 x^rn-1 x^.</math>; nunc fingatur <math>i</math> infinita quantitas, &
re$ultat <math>P = Q</math>, & con$equenter <math>{r · r + 1 · r + 2 · .. π + i - 1 / m + r · m + r + 1 · m + r + 2 ..
(m + r + i - 1)} × {a^i-r / b^i-r}$. (a + bx^n)^m-1 x^rn-1 x^. = {1 · 2 · 3 · .. i / m · m + 1 · m + 2
· .. m + i} × {a^i-1 / b^i-1} × {a^m / m n b}</math>; unde <math>$. (a + b x^n)^m-1 x^rn-1 x^. = {1 · 2 · 3 · ..(i) × m + r · m + r + 1 · m + r + 2 · .. m + r + i - 1 / r · r + 1 · r + 2 .. (r + i - 1) × m · m + 1. m + 2 .. m + i} × {a^r-1 / b^r-1}
× {a^m / m n b}</math>.</p>
<p>Fallet hæc methodus, cum quidam factores in prædictâ fractione
contenti evadant nihilo æquales; in his ca$ibus e transformatione datæ
fluxionis in alteram $æpe deduci pote$t re$olutio, & ex ii$dem prin-

<pb 648>[0700]<rh>DE SUMMATIONE</rh>
cipiis, viz. transformationibus facile erui po$$unt in$initæ fluxiones,
quarum particulares fluentes innote$cunt.</p>
<p>Sæpe vero fluentes prædictæ detegi po$$unt ex earum divi$ione in
alias, $i enim fluentes prædictæ e $ingulis his innote$cant, tum fluen-
tes e prædictis exinde inve$tigari po$$unt.</p>
<h>THEOR. XLIV.</h>
<p>Si data $eries <math>A</math>, cujus $umma innote$cit, quæ per datam quantita-
tem (<math>α</math>) divi$a præbeat $eriem magis convergentem; & $imiliter quo-
tiens per aliam datam quantitatem (<math>β</math>) divi$a præbeat $eriem adhuc
magis convergentem; & $ic deinceps in infinitum; & ultima quo-
tiens $it unitas; tum erit <math>A = α × β × γ × δ × &c.</math></p>
<p>Ex. 1. Series <math>1 ± b + b^2 ± b^3 + b^4 ± &c.</math> in infinitum per <math>1 ± b</math>
divi$a præbet; $eriem <math>1 + b^2 + b^4 + b^6 + &c.</math> hæc autem $eries per
<math>1 + b^2</math> divi$a dat <math>1 + b^4 + b^8 + b^12 + &c.</math> quæ per <math>1 + b^4</math> divi$a
præbet $eriem <math>1 + b^8 + b^16 + &c., &c.</math>; quarum prima magis con-
vergit quam $ecunda, $ecunda quam tertia, &c.; $i modo <math>b</math> minor $it
quam 1.</p>
<p>Et con$equenter erit <math>(1 ± b) (1 + b^2) (1 + b^4) (1 + b^8) (1 + b^16)
(1 + b^32) &c. = {1 / 1 ∓ b}</math>.</p>
<p>Cor. Sit <math>b = {m / n}</math>, ubi <math>m</math> minor e$t quam <math>n</math>; tum erit <math>(1 ± {m / n}) (1 +
{m^2 / n^2}) (1 + {m^4 / n^4}) (1 + {m^8 / n^8})&c. = ({n ± m / n}) ({n^2 + m^2 / n^2}) ({n^4 + m^4 / n^4})
({n^8 + m^8 / n^8}) ({n^16 + m^16 / n^16}) &c.</math> in infinitum <math>= {1 / 1 ∓ {m / n}} = {n / n ∓ m}</math>.</p>
<p>Ex. 2. Series <math>1 ± b + b^2 ± b^3 + b^4 ± b^5 + b^6 ± &c.</math> per <math>1 ± b
+ b^2 ± b^3 + b^4 ... ± b^b</math> divi$a præbet $eriem magis convergentem
<math>1 ± b^b+1 + b^2b+2 ± b^3b+3 + b^4b+4 ± b^5b+5 + &c.</math>; hæc autem $eries per
$eriem <math>1 ± b^b+1 + b^2b+2 ± b^3k+3 + b^4k+4 .... b^rm+r</math> divi$a præbet $eriem
<math>1 ± b^(r+1).(b+1) + b^2(r+1)(b+1) ± b^3(r+1)(b+1) + b^4(r+1)(b+1) ± .. &c.</math>; hæc au-

<pb 649>[0701]<rh>SERIERUM, &c.</rh>
tem $eries per $eriem <math>1 ± b^(r+1)(b+1) + b^2(r+1)(b+1) ± b^3(r+1)(b+1) ... b^6(r+1)(b+1)</math>
divi$a dat quotientem <math>1 ± b^(s+1)(r+1)(b+1) + b^2(s+1)(r+1)(b+1) ± b^3(s+1)(r+1)(b+1
+ &c.</math>; & $ic deinceps. Si omnes numeri <math>b + 1, r + 1, s + 1, &c.</math>
$int impares; tum $igna ± vel po$$unt e$$e + vel -; $i autem unus
e prædictis numeris <math>b + 1, r + 1, &c.</math> $it par, tum omnia $igna in
$ub$equentibus quotientibus erunt +.</p>
<p>Hinc, $i <math>b</math> minor $it quam unitas, continuum productum <math>(1 ± b
+ b^2 ± ... b^b)(1 ± b^b+1 + b^2b+2 ± b^3b+3 ... b^rb+1)(1 ± b^(r+1)(b+1) +
b^2(r+1)(b+1) ± b^3(r+1)(b+1) + .. b^s(r+1)(b+1))(1 ± b^(s+1)(r+1)(b+1) + b^2(s+1)(r+1)(b+1) ±
.. b^t(s+1)(r+1)(b+1))(1 ± b^(t+1)(s+1)(r+1)(b+1) + &c.) &c.</math> in infinitum <math>=
{1 / 1 ∓ b}</math>.</p>
<p>Pro <math>b</math> in hoc contento $cribatur <math>{m / n}</math>, & re$ultat prædictum conti-
nuum contentum <math>(1 ± {m / n} + {m^2 / n^2} ±.. {m^b / n^b})(1 ± {m^b+1 / n^b+1} + {m^2b+2 / n^2b+2} ± ...
{m^r(b+1) / n^r(b+1)})(1 ± {m^(r+1)(b+1) / n^(r+1)(b+1)} ± ... {m^s(r+1)(b+1) / n^s(r+1)(b+1)}) &c. = ({n^b+1 ± m^b+1 / (n ∓ m)n^b})
({n^(r+1)(b+1) ± m^(r+1)(b+1) / (n^b+1 ∓ m^b+1) × n^r×(b+1)})({n^(s+1)(r+1)(b+1) ± m^(s+1)(r+1)(b+1) / (n^(r+1)(b+1) ∓ m^(r+1)(b+1)) n^s(r+1)(b+1)}) &c.</math> in
in$initum <math>= {n / n ∓ m}</math>.</p>
<p>2. Sit $eries <math>1 ± b^-1 + b^-2 ± b^-3 + b^-4 ± &c.</math>; tum in præceden-
tibus continuis factoribus pro <math>b</math> $cribatur <math>b^-1</math>, & re$ultat contentum
e novis continuis factoribus datæ quantitati æquale.</p>
<p>Ex. 3. A$$umatur quæcunque algebraica functio quantitatis <math>x</math>, quæ
$it <math>π</math>; in eâ pro <math>x</math> $cribatur <math>- x</math> & re$ultet quantitas <math>A</math>; deinde in <math>π</math>
pro <math>x</math> $cribantur <math>√(-1)x & - √(-1)x</math>, & re$ultent quantitates <math>B & B′</math>, quarum productum
<math>B × B′ = A′</math>: tum in quantitate <math>π</math> pro <math>x</math> $cri-
bantur <math>αx, α′x, α″x & α′″x</math>; ubi <math>α, α′, α″ & α′″</math> $unt radices æquationis <math>z^4 +
1 = 0</math>; & re$ultent quantitates <math>C, C′, C″ & C′″</math> re$pective: ducantur hæ
quantitates in $e$e, & dicatur contentum re$ultans <math>C × C′ × C″ × C″′ = A″</math>;
tum pro <math>x</math> in datâ quantitate <math>π</math> $cribantur <math>βx, β′x, β″x, β′″x,

<pb 650>[0702]<rh>DE SUMMATIONE</rh>
&c.</math>; ubi <math>β, β′, β″, β′″, &c.</math> $unt radices æquationis <math>z^8 + 1 = 0</math>, & re-
$ultent quantitates <math>D, D′, D″, D′″, D″″, D′″″, D″″″, D′″″″</math>; ducantur
hæ quantitates in $e$e, & dicatur contentum re$ultans <math>D × D′ × D″ ×
D′″ × &c... D^7 = A″</math>; & in genere $int <math>ε, ε′, ε″, ε′″, &c.</math> radices æqua-
tionis <math>z^2α + 1 = 0</math>; pro <math>x</math> in datâ quantitate <math>π</math> $cribantur <math>εx, ε′x, ε″x,
ε′″x, &c.</math>, & re$ultent quantitates <math>P, P′, P″, P′″, &c.</math>; ducantur hæ
quantitates in $e$e & $it contentum re$ultans <math>P × P′ × P″ × P′″ × &c.
= A′^α</math>; tum erit continuum contentum <math>A × A′ × A″ × A′″ × &c.</math> in
infinitum <math>= {1 / π}</math>; $i modo <math>π = 1 + a x^σ + b x^2σ + c x^3σ + &c.</math> in in-
finitum, & <math>x</math> minor $it quam ulla radix æquationis <math>π = 0</math>, vel ejus
denominatoris vel cuju$cunque irrationalis quantitatis in prædictâ
(<math>π</math>) contentæ nihilo æqualis redditæ; & <math>σ</math> $it affirmativa quantitas.</p>
<p>Si vero <math>π = Γ + a x^σ + b x^2σ + c x^3σ + &c.</math>, tum <math>{Γ / π} = {A / Γ} × {A′ / Γ^2}
× {A″ / Γ^4} × {A′″ / Γ^8} × {A″″ / Γ^16} × &c.</math> in in$initum.</p>
<p>Sit <math>σ</math> negativa quantitas, & <math>x</math> major quam ulla radix prædictarum
æquationum, & erit etiam <math>{Γ / π} = {A / Γ} × {A′ / Γ^2} × {A″ / Γ^4} × &c.</math></p>
<p>Si modo quantitas <math>π</math> transformetur in alteram <math>ρ</math>, & incognita
quantitas <math>x</math> $it data algebraica functio incognitæ quantitatis (<math>z</math>) in
quantitate <math>ρ</math> contentæ; tum ex methodo hìc traditâ erui po$$unt con-
tinua contenta quantitatibus <math>{1 / π} & {1 / ρ}</math> re$pective & con$equenter inter
$e æqualia.</p>
<p>4. Si <math>π</math> $it quæcunque fluxio <math>λ x^.</math>, ubi <math>λ</math> e$t algebraica functio quan-
titatis <math>x</math>; in fluente <math>$. λx^. = V</math> pro <math>x</math> $cribatur <math>- x</math> & re$ultet quanti-
tas <math>A</math>; deinde in <math>V</math> pro <math>x</math> $cribantur <math>√(- 1)x & - √ (- 1) x</math> & re-
$ultent quantitates <math>B & B′</math>, quarum productum <math>B × B′ = A′</math>; & $ic
progrediendum e$t, ut in priori ca$u; & re$ultent quantitates <math>A″, A′″</math>,

<pb 651>[0703]<rh>SERIERUM, &c.</rh>
& tum erit <math>{Γ / V} = {A / Γ} × {A′ / Γ^2} × {A″ / Γ^4} × {A′″ / Γ^8} × &c.</math>, $i modo <math>V = Γ + ax^σ +
bx^2σ + &c.</math>, & <math>x</math> minor $it quam quæcunque radix æquationis <math>λ = 0</math>,
vel ejus denominatoris vel cuju$cunque irrationalis quantitatis in
quantitate <math>λ</math> contentæ & nihilo æqualis redditæ; & <math>σ</math> $it affirmativa
quantitas: at <math>x</math> major $it quam ulla prædicta radix, $i <math>σ</math> $it negativa
quantitas.</p>
<p>In hoc ca$u æqualia continua contenta deduci po$$unt ex con$imili-
bus transformationibus iis in prioribus ca$ibus traditis.</p>
<h>PROB. XXXIX.</h>
<p>1. _Invenire fluentem fluxionis_ <math>{z^m-1 Z^. / 1 + Z^n}</math> _inter valores_ <math>0</math> _& infinitum quan-_
_titatis_ <math>Z</math>.</p>
<p>1<_>mo</_>. Inveniatur fluens prædicta inter valores <math>0 & 1</math> quantitatis <math>z</math>
contenta per $eriem a$cendentem <math>{1 / m}z^m - {1 / m + n}z^m+n + {1 / m + 2n}z^m+2n
- {1 / m + 3n}z^m+3n + &c.</math> deinde inveniatur fluens fluxionis prædictæ
inter valores <math>1</math> & infinitum quantitatis <math>z</math> per $eriem de$cendentem
<math>{1 / m - n}z^m-n - {1 / m - 2n}z^m-2n + {1 / m - 3n}z^m-3n + &c.</math> Nunc animadver-
tendum e$t <math>m</math> minorem e$$e quam <math>n</math>, aliter fluentem quæ$itam e$$e in-
finitam, & con$equenter fluentem quæ$itam æqualem e$$e $ummæ dua-
rum $erierum <math>{1 / m} - {1 / m + n} + {1 / m + 2n} - &c. & - ({1 / m - n} - {1 / m - 2n} +
{1 / m - 3n} - &c.)</math> quæ $umma <math>{1 / m} + {1 / n^2 - m^2} - {1 / 4n^2 - m^2} + {1 / 9n^2 - m^2}
- {1 / 16n^2 - m^2} + &c. = {π / n $in. {m / n}π}</math>, quod e præcedentibus facile colligi
pote$t.</p>

<pb 652>[0704]<rh>DE SUMMATIONE</rh>
<p>Con$tat e præcedente capite $eriem a$cendentem e$$e convergen-
tem, cum <math>z</math> minor $it quam <math>1</math>; de$cendentem $eriem e$$e convergen-
tem, cum <math>z</math> major $it quam <math>1</math>.</p>
<p>Erit <math>$.{z^m-1 z^. / 1 + z^n} = {1 / m} × {z^m / 1 + z^n} + {1 / m} × {n / n + m} × {z^n+m / (1 + z^n)^2} + {1 / m} × {n / n + m}
× {2n / 2n + m} × {z^2n+m / (1 + z^n)^3} + &c.</math>; hæc $eries convergit, cum <math>{z^n / 1 + z^n}</math> minor
$it quam <math>± 1</math>; divergit autem, cum major $it.</p>
<p>2. Sit fluxio <math>{(a z^m-1 + b z^m-2 + &c.)z^. / z^tn - pz^tn-n + q^tn-2n - &c.}</math>, ubi <math>t n</math> major e$t quam <math>m</math>;
$int radices æquationis <math>z^t - p z^t-1 + q z^t-2 - &c. = 0</math>, re$pective <math>α, β,
γ, δ, &c.</math> & erit <math>{(az^m-1 + &c.)z^. / z^tn - pz^tn-n + &c.} = {(Az^s-1 + &c.)z^. / z^n - α} + {(Bz^s-1 + &c.)z^. / z^n - β}
+ &c.</math> unde ex hoc problemate inveniri po$$unt fluentes inter valo-
res <math>0</math> & infinitum quantitatis <math>z</math> contentæ.</p>
<p>3. Sit fluxio <math>{z^s-1 z^. / z^n + 1}</math>, in eâ pro <math>z</math> $cribatur <math>{x / a}</math>, & re$ultabit <math>{a^n-s x^s-1 x^. / x^n + a^n}</math>,
unde fluens fluxionis <math>{x^s-1 x^. / x^n + a^n}</math> inter valores <math>0</math> & infinitum quantitatis
<math>x</math> erit <math>a^s-n × {π / n × $in.{s / n}π}</math>. Sit fluxio <math>{α x^s-1 x^. / x^n + a^n} + {β x^s-1 x^. / x^n + b^n} = {x^n+s-1 x^. / x^n + a^n . x^n + b^n}</math>;
hinc <math>αb^n + βa^n = 0 & α + β = 1</math>; & fluens inter valores <math>0</math> & infi-
nitum quantitatis <math>x</math> erit <math>(α × a^s-n + β × b^s-n) × {π / n × $in. {s / n}π}</math>.</p>
<p>Sit <math>a^n = (l + √(l^2 - 1)) k^n, & b^n = (l - √(l^2 - 1)) k^n</math>; i. e. $it flu-
xio <math>{x^n+s-1 x^. / x^2n + 2 l k^n x^n + k^2n}</math>, & facile inveniri po$$unt quantitates <math>α & β</math>,
& exinde fluens quæ$ita <math>(α a^s-n + β b^s-n) × {π / n × $in. {s / n} π} =</math> (cum <math>l</math> mi-

<pb 653>[0705]<rh>SERIERUM, &c.</rh>
nor $it quam <math>±1</math>) <math>{k^s-n × π / √(1 - l^2)n $in. {sπ / n}} ×</math> $in. arcus <math>{s / n} ρ, &c.</math>, ubi <math>ρ</math> e$t
arcus cujus co$inus e$t <math>l</math>.</p>
<p>Huju$ce generis plura problemata deduci po$$unt; transformetur
enim data fluxio in alteram, cujus variabilis quantitas <math>y</math> datam habeat
relationem ad variabilem datæ fluxionis quantitatem <math>x</math>; & $i modo cum
<math>x = α</math> tum <math>y = π</math>, & cum <math>x = β</math> tum <math>y = ρ</math>; duæ fluentes inter va-
lores <math>α & β, π & ρ</math> duarum variabilium <math>x & y</math> contentæ erunt re-
$pective inter $e æquales. Et $ic de pluribus huju$ce generis tran$-
formationibus; & ex his $eriebus deduci po$$unt aliæ inter $e æqua-
les, &c.</p>
<h>THEOR. XLV.</h>
<p>1. Sint <math>x & X</math> $inus arcuum, qui $unt inter $e ut <math>n: 1</math>, & radius
(<math>r</math>); tum erit <math>X = n x + {1 - n^2 / 1 · 3 r^2} × A x^3 + {9 - n^2 / 1 · 5 r^2} × Bx^5 + {25 - n^2 / 1 · 7 r^2}
× Cx^7 + {49 - n^2 / 1 · 9 r^2} × D x^9 + &c.</math></p>
<p>2. Sint <math>x</math> & <math>X</math> $inus arcuum, qui $unt inter $e ut <math>n:m</math>; tum erit <math>X =
{n / m}x + {1 - {n^2 / m^2} / 1. 3 r^2} A x^3 + {9 - {n^2 / m^2} / 1. 5 r^2} Bx^5 + {25 - {n^2 / m^2} / 1. 7 r^2} + &c. = {n / m}x +
{m^2 - n^2 / 1 · 3 m^2 r^2} A x^3 + {9 m^2 - n^2 / 1 · 5 m^2 r^2} B x^5 + {25 m^2 - n^2 / 1 · 7 m^2 r^2} C x^7 + &c. = √({r^2 - x^2 / r^2})
× ({n / m} x + {4 - {n^2 / m^2} / 1. 3 r^2} × A x^3 + {16 - {n^2 / m^2} / 1. 5 r^2} × B x^5 + {36 - {n^2 / m^2} / 1. 7 r^2} C x^7 + &c.)
= √({r^2 - x^2 / r^2})({n / m}x + {4 m^2 - n^2 / 1 · 3 m^2 r^2} Ax^3 + {16 m^2 - n^2 / 1 · 5 m^2 r^2} B x^5 + {36 m^2 - n^2 / 1 · 7 m^2 r^2}
C x^7 + &c.)</math>: cum <math>{n / m}</math> $it impar numerus, tum abrumpitur $eries <math>{n / m} x<pb 654>[0706]<rh>DE SUMMATIONE</rh>
+ {m^2 - n^2 / 1. 3m^2 r^2} A x^3 + {9 m^2 - n^2 / 1 · 5 m^2 r^2} B x^5 + &c.</math>; cum autem <math>{n / m}</math> $it par nu-
merus, tum abrumpitur $eries <math>√({r^2 - x^2 / r^2})({n / m} x + {4 m^2 - n^2 / 1 · 3 m^2 r^2} A x^3 +
{16 m^2 - n^2 / 1 · 5 m^2 r^2} B x^5 + &c.)</math>.</p>
<p>3. Sit <math>x</math> $inus arcus <math>A</math>, tum co$inus arcus <math>{n / m} A</math> erit <math>r - {n^2 x^2 / 1 · 2 m^2 r}
- {4 - {n^2 / m^2} / 1. 4 r^2} × A x^4 - {16 - {n^2 / m^2} / 1. 6 r^2} × B x^6 - {36 - {n^2 / m^2} / 1. 8 r^2} C x^8 + &c. = r -
{n^2 / 1 · 2 m^2 r} x^2 - {4 m^2 - n^2 / 1 · 4 m^2 r^2} × A x^4 - {16 m^2 - n^2 / 1 · 6 m^2 r^2} × B x^6 - {36 m^2 - n^2 / 1 · 8 m^2 r^2}
C x^8 - &c. = √(r^2 - x^2) × (1 + {1 - {n^2 / m^2} / 2r} x^2 + {9 - {n^2 / m^2} / 1. 4 r^2} × A x^4 +
{25 - {n^2 / m^2} / 1. 6 r^2} × B x^6 + {49 - {n^2 / m^2} / 1. 8 r^2} × C x^8 + &c.) = √(r^2 - x^2) (1 +
{m^2 - n^2 / 2 m^2 r^2} x^2 + {9 m^2 - n^2 / 1 · 4 m^2 r^2} × A x^2 + {25 m^2 - n^2 / 1 · 6 m^2 r^2} B x^4 + &c.)</math>. Cum <math>{n / m}</math>
$it par numerus, tum $eries <math>r - {n^2 / 1 · 2 m^2 r} x^2 - {4 m^2 - n^2 / 1 · 4 m^2 r^2} × A x^4
&c.</math> abrumpitur; cum autem <math>{n / m}</math> $it impar numerus, tum po$terior $e-
ries <math>√(r^2 - x^2) (1 + {m^2 - n^2 / 2 m^2 r^2} x^2 + {9 m^2 - n^2 / 1 · 4 m^2 r^2} × A x^4 + {25 m^2 - n^2 / 1 · 6 m^2 r^2} ×
B x^6 + &c.</math> terminat.</p>
<p>In omnibus his $eriebus literæ <math>A, B, C, D, &c.</math> præcedentium ter-
minorum coeffiicentes re$pective denotant.</p>
<p>Omnes hæ $eries $emper convergent; nam $inus <math>x</math> non pote$t e$$e
major quam <math>+ r</math> vel <math>- r</math>.</p>
<p>Cor. Sit <math>x</math> $inus quantitas valde parva & con$equenter ejus arcui

<pb 655>[0707]<rh>SERIERUM, &c.</rh>
prope æqualis, & <math>n {x / r}</math> non major quam 1, tum hæ $eries ex arcu <math>n x</math>
dato inveniunt ejus $inum vel co$inum prope: $i vero propior requi-
ratur approximatio, ea facile deduci pote$t.</p>
<p>Facile con$imiles $eries per qua$libet duas $ub$equentes radium,
$inum, co$inum, tangentem, cotangentem, co$ecantem, &c. vel plu-
res deduci po$$unt.</p>
<p>4. Si prædictæ reducantur ad $eries $ecundum dimen$iones quanti-
tatis <math>x</math> de$cendentes; tum erit <math>X = {n / m} x + {m^2 - n^2 / 1 · 3 m^2 r^2} A x^3 + {9 m^2 - n^2 / 1 · 5 m^2 r^2}
B x^5 + &c. = {(-4)^{n-m / 2m} x^{n / m} / r^{n-m / m}} - {n. n - m. r^2 / 4. n - m. m} A x^{n-2m / m} - {n - 2 m. n - 3 m. r^2 / 8. n - 2 m. m}
B x^{n-4m / m} - {n-4 m. n - 5 m. r^2 / 12. n - 3 m. m} C x^{n-6m / m} - {n - 6 m. n - 7 m. r^2 / 16. n - 4 m. m}
D x^{n-8m / m} - &c. = √(r^2 - x^2) ({n / m}{x / r} + {4 m^2 - n^2 / 1 · 3 m^2 r^2} A x^3 + {16 m^2 - n^2 / 1 · 5 m^2 r^2}
B x^5 &c.) = √(r^2 - x^2) (2(-4)^{n-2m / 2m} × {x^{n-m / m} / r^{n-m / m}} + {(n - m) × (n - 2 m) / 4. (m - n). m}
× A ({x / r})^{n-3m / m} + {(n - 3 m). (n - 4 m) / 8 × (2 m - n) × m} × B ({x / r})^{n-5m / m} + {(n - 5 m). (n - 6 m) / 12 × (3 m - n). m}
× C ({x / r})^{n-7m / m} + &c.)</math>.</p>
<p>5. Co$inus <math>y = r - {n^2 / 1 · 2 m^2 r} x^2 - {4 m^2 - n^2 / 1 · 4 m^2 r^2} A x^4 - {16 m^2 - n^2 / 1 · 6 m^2 r^2} B x^6
- &c. = - 2 × (-4)^{n-2m / 2m} {x^{n / m} / r^{n-m / m}} + {n × (n - m) / 4 m × (m - n)} A {x^{n-2m / m} / r^{n-3m / m}} + {(n - 2 m) / 8 m.}
{(n - 3 m) / (2 m - n)} B {x^{n-4m / m} / r^{n-5m / m}} + {(n - 4 m) × (n - 5 m) / 12 m × (3 m - n)} C {x^{n-6m / m} / r^{n-7m / m}} + & c. = √(r^2 - x^2)

<pb 656>[0708]<rh>DE SUMMATIONE</rh>
(1 + {m^2 - n^2 / 2 m^2 r^2} x^2 + {9 m^2 - n^2 / 1 · 4 m^2 r^2} A x^4 + {25 m^2 - n^2 / 1 · 6 m^2 r^2} B x^6 + &c.) =
√(r^2 - x^2) ((-4)^{n-m / 2m} × ({x / r})^{n-m / m} + {n - m. n - 2 m / 4 m × (m - n)} A ({x / r})^{n-3m / m} +
{n - 3 m. n - 4 m / 8 m. 2 m - m} B ({x / r})^{n-5m / m} + {n - 5 m. n - 6 m / 12 m. 3 m - n} × C ({x / r})^{n-7m / m}
+ &c.)</math>.</p>
<p>Con$imiles $eries facile deduci po$$unt pro $inubus vel co$inubus,
tangentibus vel cotangentibus, $ecantibus vel co$ecantibus, ver$is $i-
nubus, &c. in terminis $ecundum dimen$iones co$inus, tangentis, &c.
progredientibus.</p>
<p>6. Exhinc datâ $erie $ecundum dimen$iones quantitatis <math>x</math> progrediente
a $cribendo in eâ pro <math>x, x^2, x^3, &c.</math> vel <math>x^{n / m}, x^{n / m}-1, x^{n / m}-2, &c.</math> earum valo-
res hìc traditos, & ducendo hos valores in incognitas coefficientes,
& æquando coefficientes corre$pondentium terminorum re$ultantis
æquâtionis, ita ut evane$cant quantitas <math>x</math> & ejus pote$tates vel radi-
dices; ex æquationibus hinc re$ultantibus erui po$$unt prædictæ co-
efficientes quæ$itæ.</p>
<p>7. Sint <math>s & c</math> $inus & co$inus arcus <math>A</math>, cujus radius $it 1; & per <math>s n,
c n; s (n - 2), c (n - 2); s (n - 4), c (n - 4); &c.</math> de$ignentur $inus
& co$inus arcuum <math>n A, (n - 2) A, (n - 4) × A, &c.</math> tum ex princi-
piis prius traditis erui po$$unt <math>s^n = ± {1 / 2^n-1} (s n - n s (n - 2) + n.
{n - 1 / 2} s (n - 4) - n. {n - 1 / 2}· {n - 2 / 3} s (n - 6) + n. {n - 1 / 2}· {n - 2 / 3};
{n - 3 / 4}s (n - 8) - &c.)</math> cum <math>n</math> $it impar numerus; $ignum erit +, $i
<math>n = 4 m + 1</math>, bui <math>m</math> e$t integer numerus; $in aliter -: $i vero <math>n</math> $it
par numerus, &c., tum erit <math>s^n = ± {1 / 2^n-1} (c n - n c (n - 2) + n. {n - 1 / 2}
c (n - 4) - n. {n - 1 / 2}· {n - 2 / 3} c (n - 6) + &c.)</math>. Signum erit +, $i

<pb 657>[0709]<rh>SERIERUM, &c.</rh>
<math>n = 4 m</math>; $in aliter -; ultimus autem terminus erit <math>{1 / 2} × n. {n - 1 / 2}·
{n - 2 / 3}·.. {{1 / 2}n + 1 / {1 / 2}n}</math>. Et $imiliter erit <math>c^n = {1 / 2^n-1} (c n + n c (n - 1) + n.
{n - 1 / 2} c (n - 2) + n. {n - 1 / 2}· {n - 2 / 3} c (n - 3) + &c.)</math>. Cum <math>n</math> $it par
numerus, ultimus terminus erit <math>{1 / 2} × n.{n - 1 / 2}· {n - 2 / 3}... {{1 / 2}n + 1 / {1 / 2}n}</math>.</p>
<p>Ex hinc datâ $erie $ecundum dimen$iones quantitatis <math>s</math> vel <math>x</math> pro-
grediente deduci pote$t $eries ei æqualis $ecundum quantitates <math>s n,
s (n - 2), s(n - 4), &c. & c n, c (n - 2), c (n - 4), &c.</math> progrediens,
cum <math>n</math> $it integer numerus.</p>
<p>Cor. Hinc data algebraica æquatio <math>x^n - p x^n-1 + q x^n-2 - &c. = 0</math>
transformari pote$t in æquationem <math>a s (n) + b c (n - 1) + a′ s (n - 2)
+ b′ c (n - 3) + a″ s (n - 4) + b″ c (n - 5) + &c. = 0</math>, vel <math>a c n +
b s (n - 1) + a′ c (n - 2) + b′ s (n - 3) + a″ s (n - 4) + b″ s
(n - 5) + &c. = 0</math>, prout <math>n</math> $it impar vel par numerus: tum datâ
approximatione <math>α</math> ad valorem <math>s</math>, & con$equenter ad <math>c & s 2 & c 2, s 3
& c 3, s 4 & c 4.. s n & c n</math>; quoniam fluxiones quantitatum <math>s n
& c n</math> $unt <math>n × c (n) × A^. & n × s n × A^.</math>; ubi <math>A</math> e$t arcus, cujus $inus
e$t <math>s</math>; & $i <math>e & f</math> $int approximationes ad <math>s n & c n, e′ & f′</math> ad <math>s (n - 1)
& c (n - 1), e″ & f″</math> ad <math>s (n - 2) & c (n - 2), &c.</math>; & $cribantur hæ
approximationes pro $uis valoribus in datis æquationibus, & $it
quantitas re$ultans <math>B</math>, erit <math>{-B / n a c n + (n - 1) b s (n - 1) + (n - 2) a′ c
(n - 2) + (n - 3) b′ s (n - 3) + &c.}</math>, vel <math>{-B / n a s n + (n - 1) b c (n - 1) +
(n - 2) a′ s (n - 2) + &c.}</math> propior approximatio: & $ic deinceps.</p>

<pb 658>[0710]<rh>DE SUMMATIONE</rh>
<h>PROB. XL.</h>
<p>_Sit pote$tas_ <math>(1 + x + x^2)^n = (1 + x (1 + x))^n = x^n (1 + x)^n +
n x^n-1 (1 + x)^n-1 + n. {n-1 / 2} x^n-2 . (1 + x)^n-2 + n. {n-1 / 2}· {n-2 / 3}
x^n-3 (1 + x)^n-3 + &c.</math> _coefficientem medii ejus termini_ (<math>x^n</math>) _definire._</p>
<p>Coefficiens erit <math>1 + {n. n - 1 / 1 · 1} + {n (n - 1) (n - 2) (n - 3) / 1 · 2 · 1 · 2} +
{n (n - 1) (n - 2) (n - 3) (n - 4) (n - 5) / 1 · 2 · 1 · 1 · 1 · 3} + &c.</math></p>
<p>Con$tat ex binomiali theoremate.</p>
<p>Cor.. Sint <math>P, Q & R</math> re$pective coefficientes mediorum termino-
rum ad pote$tates <math>n, n + 1 & n + 2</math>; tum e $ub$titutione con$tabit
<math>{n + 2 / n + 1} (R - Q) - (Q - P) = 4 P</math>.</p>
<p>Cor.. Quoniam <math>{n + 2 / n + 1} (R - Q) - (Q - P) = 4 P</math>, erit etiam
<math>{n + 3 / n + 1} (S - R) - (R - Q) = 4 Q</math>, ubi <math>P, Q, R & S</math> $unt coeffici-
entes mediorum terminorum re$pective; ita reducantur hæ duæ
æquationes ad unam, ut exterminetur <math>n</math>, & re$ultat æquatio relatio-
nem inter <math>S, R, Q & P</math> de$ignans.</p>
<p>Et $ic progredi liceat ad inveniendos terminos medios, <math>&c. (n)</math> po-
te$tatis aliarum. E. g. Quantitatis <math>(A) (a + b x + c x^2)^n</math> medius ter-
minus erit <math>b^n + {n. n - 1 / 1 · 1} b^n-2 a c + {n. (n - 1) (n - 2) (n - 3) / 1 · 2 · 1 · 2} b^n-4 a^2 c^2
+ {n. (n - 1) (n - 2) (n - 3) (n - 4) (n - 5) / 1 · 2 · 1 · 1 · 1 · 3} b^n-6 a^3 c^3 + &c.</math>; &
exinde <math>{n + 2 / n + 1} (R - b Q) - b (Q - b P) = 4 a c P</math>, $i modo <math>P, Q & R</math>
denotent coefficientes mediorum terminorum quantitatis <math>A</math> ad pote-
$tates <math>n, n + 1 & n + 2</math> elevatæ.</p><pb 659>[0711]<rh>SERIERUM, &c.</rh>
<h>PROB. XLI.</h>
<h it>Ex datis fluentibus vel integralibus, quœ a $e invicem deduci po$$int; in-
venire infinitas $eries, quœ etiam a $e invicem po$$unt inve$tigari.</h>
<p>Reducantur $ingulæ datæ fluentes vel integrales ad infinitas $eries
$ecundum dimen$iones variabilis <math>x</math> progredientes, & exorientur $eries,
quæ a $e invicem deduci po$$unt.</p>
<p>Ex. 1. Ex datâ fluente <math>$. (a + bx^n)^m × x^pn-1 x^.</math> erui pote$t fluens flu-
xionis <math>(a + b x^n)^m+r × x^pn+vn-1 x^.</math>, ubi <math>r & v</math> integri $unt numeri vel
affirmativi vel negativi, <math>m</math> vero & <math>p</math> haud integri affirmativi numeri;
reducantur hæ fluxiones ad infinitas $eries, & re$ultant <math>a^m x^pn-1 x^. +
m a^m-1 b x^(p+1)n-1 x^. + m. {m - 1 / 2} a^m-2 b^2 x^(p+2)n-1 x^. + &c. & a^m+r x^pn+vn-1 x^.
+ (m + r). a^m+r-1 bx^(p+v+1)n-1 x^. + (m + r). {m + r - 1 / 2} a^m+r-2 b^2 x^(p+v+2)n-1 x^.
+ &c</math>. quarum fluentes erunt re$pective <math>{a^m x^pn / pn} + {ma^m-1 bx^(p+1)n / (p + 1)n} + &c.
& {a^m+r x^pn+vn / (p + v) n} + {(m + r)a^m+r-1 bx^(p+v+1)n / (p + v + 1) n} + &c.</math> unde con$tat $ummam
feriei <math>{1 / p + v} + {m + r / p + v + 1} × x^n + {(m + r). (m + r - 1) / 2. (p + v + 2)} x^2n +
{(m + r) · (m + r - 1) · (m + r - 2) / 2 · 3 · (p + v + 3)} x^3n + &c.</math> ($i modo <math>v & r</math> $int
integri numeri vel affirmativi vel negativi, & <math>p & m</math> haud integri
affirmativi numeri) deduci po$$e e $ummâ $eriei <math>{1 / p} + {m / p + 1}x^n +
{m · (m - 1) / 2 · (p + 2)}x^2n + {m · (m - 1) · (m - 2) / 2 · 3 · (p + 3)} x^3n + &c.</math></p>
<p>Cor. Datâ $ummâ unius $eriei prædicti generis, ex eâ deduci po$-
$unt $ummæ omnium $erierum eju$dem generis, quicunque integri
numeri $int valores quantitatum <math>r & v</math>: ex fluente enim fluxionis

<pb 660>[0712]<rh>DE SUMMATIONE</rh>
<math>(a + b x^n)^m+r x^pn+vn-1 x^.</math> ad unum integrum valorem quantitatum <math>r &
v</math> re$pective erui po$$unt fluentes eju$dem fluxionis ad omnes integros
valores quantitatum <math>r & v</math>.</p>
<p>Cor. 2. Ex datâ fluente <math>$. (a + b x^n)^m-1 x^pn-1 x^.</math> inveniri po-
te$t fluens fluxionis <math>(a + b x^n)^m-1 x^pn+vn-1 x^. = (a + bx^n)^m × ({1 / r n b}
x^(q-1)n - {(q - 1)a / (r - 1)b} A x^(q-2)n + &c.</math> a d <math>v</math> terminos) <math>± {p / p + m} · {p + 1 / p + m + 1}
... {p + v - 1 / p + m + v - 1} · {a^v / b^v}$. (a + bx^n)^m-1 x^pn-1 x^.</math>, ubi <math>p + v = q, q + m
- 1 = r; & A, &c.</math> $int coefficientes præcedentium terminorum.</p>
<p>Reducantur $inguli termini ad infinitas $eries $ecundum reciprocas
dimen$iones quantitatis <math>x^n</math> progredientes, & ex æquatis corre$ponden-
tibus terminis in quibus eædem inveniuntur dimen$iones <math>(p + v +
m - w)</math> quantitatis <math>x^n</math>, re$ultabit æquatio <math>{(m - 1) · (m - 2) · (m - 3) / 1 · 2 · 3}
{... (m - w + 1) / ... w - 1} × {1 / p + v + m - w} = m · {m - 1 / 2} · {m - 2 / 3} ...
{m - w + 2 / w - 1} × {1 / p + m + v - 1} - m · {m - 1 / 2} · {m - 2 / 3} ... {m - w + 3 / w - 2} ×
{1 / p + m + v - 1} × {p + v - 1 / p + v + m - 2} + m · {m - 1 / 2} · {m - 2 / 3} ... {m - w + 4 / w - 3}
× {1 / p + m + v - 1} × {p + v - 1 / p + m + v - 2} × {p + v - 2 / p + m + v - 3} - .. ± m · {m - 1 / 2}
· {m - 2 / 3} ... {m - w + v + 1 / w - v} × {1 / p + m + v - 1} × {p + v - 2 / p + m + v - 2} ×
{p + v - 3 / p + m + v - 3} ... {p + 1 / p + m} ± &c. ± {p / p + m} · {p + 1 / p + m + 1} · {p + 2 / p + m + 2}
... {p + v - 1 / p + m + v - 1} × (m - 1) · {m - 2 / 2} · {m - 3 / 3} ... {m - w + v / w - v} ×
{1 / p + m + v - w};</math> $i modo <math>v & w</math> $int integri, <math>p</math> vero & <math>m</math> haud integri

<pb 661>[0713]<rh>SERIERUM, &c.</rh>
numeri: hic animadvertendum e$t quod omnes termini po$t <math>w</math> pri-
mos rejiciendi $unt; etiamque adjiciendum e$t quod coefficiens ter-
mini, cujus di$tantia a primo $it <math>w - 1</math>, invenietur <math>{1 / p + m + v - 1} ·
{p + v - 1 / p + m + v - 2} · {p + v - 2 / p + m + v - 3} ... {p + v - w + 1 / p + m + v - w}</math>.</p>
<p>2. Transformari po$$unt diver$is modis datæ fluxiones in alias,
quarum fluentes e fluentibus datarum fluxionum deduci po$$unt; re-
ducantur hæ fluxiones ad $eries $ecundum dimen$iones quarundam
quantitatum progredientes, & inveniantur fluentes; & re$ultant $e-
ries, quæ a $e invicem deduci po$$unt: e. g. $it fluxio data <math>(a + bx^n)^m+r
× x^(p+v)n-1 x^. = (b + ax^-n)^m+r × x^(m+r+p+v)n-1 x^.</math>; reducantur hæ fluxiones
ad terminos $ecundum dimen$iones quantitatis <math>x^n</math> vel <math>x^-n</math> vel <math>(a +
b x^n)^m</math> vel <math>b + a x^-n</math>, &c. progredientes, & inveniantur fluentes $erie-
rum re$ultantium; & facile con$tat $eries re$ultantes, modo conver-
gant & proprie corrigantur, a $e invicem erui po$$e.</p>
<p>Ex. 2. Sint $eries <math>a^m+r x^(p+v)n (({1 / (p + v) n} + (m + r) {b x^n / (p + v + 1) n a}
+ (m + r) · ({c / (p + v + 2) n a} + {m + r - 1 / 2} × {b^2 / a^2} × {1 / p + v + 2}) x^2n
+ &c.) = $. (a + bx^n + cx^2n)^m+r × x^(p+v)n-1 x^.</math>. Ex $ummis duarum
$erierum independentium huju$ce generis erui po$$unt $ummæ om-
nium $erierum eju$dem generis, quicunque $int integri valores quan-
titatum <math>r & v</math>; & exinde inveniri po$$unt $ummæ omnium $erierum
exprimentium fluentes fluxionum <math>(c + b x^-n + a x^-2n)^m+r × x^(2(m+r)+p+v)n-1 ^x.
= (a + bx^n + cx^2n)^m+r × x^(p+v)n-1 x^.</math>, quarum termini $ecundum di-
men$iones quantitatis <math>x^n</math> vel ejus reciprocas progrediuntur; vel $erie-
rum, quæ exprimunnt fluentes fluxionum ex prædictis per transfor-
mationes, &c. re$ultantium.</p>
<p>Et $ic in multis aliis ca$ibus.</p>
<p>Eadem etiam affirmari po$$unt, $i modo hæ $eries incipiant a ter-
minis, quorum di$tantiæ a primis $int re$pective <math>r, s, t,</math> &c.; vel $e-

<pb 662>[0714]<rh>DE SUMMATIONE</rh>
ries con$tent ex quibu$que terminis prædictarum $erierum, quorum
di$tantiæ a $e invicem $int re$pective <math>ρ, σ, τ, &c.</math> vel $ucce$$ivi termini
ducantur in integram functionem <math>a z^m + b z^m-1 + c z^m-2 + &c.</math> quan-
titatis <math>z</math> di$tantiæ a primo $eriei termino re$pective, ubi <math>m</math> e$t integer
affirmativus numerus.</p>
<h>PROB. XLII.</h>
<p>_Invenire fluentem fluxionis_ <math>{x^a x^. / 1 ± x^n}</math>; _ubi_ <math>α</math> _e$t irrationalis quantitas._</p>
<p>1<_>mo</_>. A$$umatur <math>π</math> fractio prope <math>= α</math>, & inveniatur <math>$. {x^π x^. / 1 ± x^n},</math> quæ
$it <math>V</math>; tum erit fluens quæ$ita = <math>V × x^α-π</math> prope.</p>
<p>2<_>do</_>. Ex eâdem methodo inveniatur <math>W = $. (α - π) × $. V x^α-π-1 x^.</math>
prope; tum erit <math>V - W</math> quantitas ad fluentem magis approximans;
& $ic deinceps.</p>
<p>Eadem principia etiam applicari po$$unt ad inveniendas approxi-
mationes ad fluentes omnium fluxionum.</p>
<h>THEOR. XLVI.</h>
<p>Summæ omnium $erierum <math>{1 / α. β. γ. δ. &c.} + {1 / α + 1. β + 1. γ + 1. δ + 1. &c.}
× x^b + {1 / α + 2. β + 2. γ + 2. δ + 2. &c.} x^2b + &c. = S</math>
deduci po$$unt e $ummis $erierum <math>{1 / α} + {1 / α + 1}x^b + {1 / α + 2}x^2b + &c.
= A, {1 / β} + {1 / β + 1}x^b + {1 / β + 2}x^2b + &c. = B, {1 / γ} + {1 / γ + 1}x^b + {1 / γ + 2}
x^2b + &c. = C, {1 / δ} + {1 / δ + 1}x^b + {1 / δ + 2}x^2b + &c. = D, &c.:</math> erunt
<math>{1 / α - β. α - γ. α - δ. &c.} × A + {1 / β - α. β - γ. β - δ. &c.}B +

<pb 663>[0715]<rh>SERIERUM, &c.</rh>
{1 / γ - α · γ - β · γ - δ · &c.} × C + {1 / δ - α · δ - β · δ - γ · &c.} D + &c.
= ± S</math>; erit <math>+</math>, cum numerus factorum <math>α, β, γ, &c.</math> $it par, $in ali-
ter -.</p>
<p>Ex his $eriebus igitur facile erui po$$unt $ummæ omnium $erierum,
quarum generalis terminus e$t <math>{az^m + bz^m-1 + &c. / pz^n + qz^n-1 + &c.} × e^bz</math>, ubi <math>z</math> e$t
di$tantia a primo datæ $eriei termino, ni duæ quantitates <math>α, β, γ, &c.</math>
$int inter $e æquales.</p>
<p>Cor. Hinc <math>{1 / α + z} × {1 / α - β · α - γ · α - δ · &c.} + {1 / β + z} × {1 / β - α ·
β - γ · β - δ · &c.} + {1 / γ + z} × {1 / γ - α · γ - β · γ - δ · &c.} + {1 / δ + z} ×
{1 / δ - α · δ - β · δ - γ · &c.} + &c. = {± 1 / α + z · β + z · γ + z · δ + z. &c.}</math>:
$ignum erit <math>+</math>, $i numerus factorum <math>α - β, α - γ, α - δ, &c.</math> $it par;
$in aliter -.</p>
<h>PROB. XLIII.</h>
<h it>Invenire $ummam quarumque fractionum, cum earum denominatores ni-
bilo œquales, & con$equenter fractiones evadant infinitœ.</h>
<p>Supponantur quantitates in factoribus contentæ, quæ evadant ni-
hilo æquales, variabiles; & augeantur per incrementa quam minima;
reducantur fractiones re$ultantes ad communem denominatorem, &
fimul addantur, &c.; tum fractio re$ultans reducatur ad minimos
terminos per divi$ionem ejus numeratoris & denominatoris per maxi-
mum communem divi$orem, & fractio re$ultans erit $umma quæ-
$ita.</p>
<p>Ex. 1. Invenire $ummam fractionum <math>{1 / α - β · α - γ} + {1 / β - α · β - γ}</math>
oum <math>α fiat = β:</math> augeantur quantitates <math>α & β</math> per <math>e & 0</math>; tum evadent

<pb 664>[0716]<rh>DE SUMMATIONE</rh>
fractiones <math>{1 / α - β + e - 0. α - γ + e} + {1 / β - α + 0 - e · β - γ + 0}
= {(β - γ + 0) - (α - γ + e) / α - β + e - 0 · α - γ + e · β - γ + 0} (ob β - α = 0) =
{0 - e / e - 0 · α - γ + e · β - γ + 0}</math> (ob <math>e</math> & <math>0</math> quam minima) <math>= -
{1 / α - γ · β - γ} = - {1 / (α - γ)^2}</math>.</p>
<p>Et $ic $umma fractionum <math>{1 / α - β · α - γ · α - δ} + {1 / β - α · β - γ · β - δ}
+ {1 / γ - α · γ - β · γ - δ} = {1 / (α - δ)^3}</math>, cum <math>α = β = γ</math>; &c. & $imiliter
$int (<math>n</math>) quantitates <math>α, β, γ, δ, &c.</math> inter $e æquales <math>& θ</math> non <math>= α</math>; tum erit
<math>{1 / α - β · α - γ · α - δ × &c. × α - θ} + {1 / β - α · β - γ · β - δ · &c.. β - θ}
+ {1 / γ - α · γ - β · γ - δ × &c. γ - θ} + &c. = {±1 / (α - θ)^n}</math>.</p>
<h>THEOR. XLVII.</h>
<p>1. Summa $eriei <math>{1 / αβγδ&c.} ± {1 / α + 1 · β + 1 · γ + 1 · δ + 1 &c.}x^b
+ {1 / α + 2 · β + 2 · γ + 2 · δ + 2 &c.}x^2b ± &c.</math> $emper deduci pote$t e
fluentibus fluxionum <math>{x^bα-1 x^. / 1 ± x^b}, {x^bβ-1 x^. / 1 ± x^b}, {x^bγ-1 x^. / 1 ±x^b}, {x^bδ-1 x^. / 1 ± x^b}, &c.: 1^mo</math>. $it <math>b</math>
affirmativa quantitas, <math>&</math> hæ $eries $emper convergent, cum <math>x</math> minor
$it quam <math>1</math>; nunquam autem, cum major $it.</p>
<p>2. Cum vero <math>x = 1</math> & denominator $it <math>1 + x^b</math>; vel <math>x = - 1</math>, cum <math>b</math>
$it impar numerus, vel fractio cujus denominator $it impar numerus,
<math>&</math> denominator $it <math>1 - x^b</math>; tum $eries $emper convergent: $i autem
<math>x = 1</math>, & denominator $it <math>1 - x^b;</math> vel <math>x = -1</math>, cum <math>b</math> $it impar nu-

<pb 665>[0717]<rh>SERIERUM, &c.</rh>
merus vel fractio ut antea <math>&</math> denominator <math>1 + x^b</math>; tum fluentes $in-
gularum fluxionum <math>{x^bx-1 x^. / 1 ± x^b}, {x^bβ-1 x^. / 1 ± x^b}, {x^bγ-1 x^. / 1 ± x^b}, &c</math>. erunt infinitæ: at hæ
fluentes ducuntur re$pective in fractiones <math>{1 / α - β · α - γ · α - δ · &c.}
= α′, {1 / β - α · β - γ · β - δ · &c.} = β′, {1 / γ - α · γ - β · γ - δ · &c.} = γ′,
{1 / δ - α · δ - β · δ - γ · &c.} = δ′, &c.</math>; quarum $umma nihilo æqualis
e$t; <math>&</math> ex fluentium in has fractiones re$pective ductarum $ummâ
evane$cat logarithmus infinitus vel quantitas infinita.</p>
<p>Si quantitates <math>α, β, γ, δ, &c.</math> $int affirmativæ, tum inveniantur flu-
entes prædictarum fluxionum inter valores <math>0 & 1</math> quantitatis <math>x</math>.</p>
<p>3. Si quæcunque quantitates <math>α, β, &c.</math> $int negativæ, i. e. <math>- α, - β,
&c.</math>; tum pro <math>- α, - β, &c.</math> $cribantur <math>l - α, l′ - β, &c.</math>, ubi <math>l, l′,
&c.</math> $unt integri numeri proxime majores quam <math>α, β, &c.</math> re$pective;
& inveniantur fluentes omnium fluxionum <math>{x^(l-α)b-1 x^. / 1 ± x^b}, {x^(l′-β)b-1 x^. / 1 ± x^b}, &c.</math>
inter valores <math>0</math> & <math>x</math> quantitatis <math>x</math> contentæ, quæ $int <math>A, B, C, &c.</math> & quæ
ductæ re$pective in <math>α′, β′, &c.</math> con$tituant $ummam <math>α′A + β′B + &c.</math>
finitam; deinde inveniatur $umma <math>- x^-αb ({1 / αb} ± {x^b / (α - 1)b} + {x^2b / (α - 2)b}
± {x^3b / (α - 3)b} .. {x^(l-1)b / (α - l + 1)b}) × α′ = P, - x^-βb ({1 / βb} ± {x^b / (β - 1)b} + {x^2b / (β - 2)b} ..
{x^(l′-1)b / (β - l′ + 1)b}) × β′ = Q, &c.</math>; i. e. inveniantur fluentes inter valores infini-
tum <math>& x</math> quantitatis <math>x</math> fluxionum <math>{(x^-αb-1 ⥊ x^(l - α)b -1) x^. / 1 ± x^b} = x^-αb-1 x^. ±
x^-(α-1)b-1 x^. + x^-(α-2)b-1 x^. ± &c. = P, {(x^-βb-1 ⥊ x^(l′-β)b-1)x^. / 1±x^b} = x^-βb-1 x^.
± &c. = Q, &c.</math>; ducantur hæ quantitates re$pective in <math>α′, β′, &c.</math>;
tum erit $umma $eriei quæ$ita <math> = α′(A ∓ P) + β′(B ∓ Q) + &c. =
α′A + β′B + γ′C + &c. ∓ α′P ∓ β′Q ∓ &c.</math></p>

<pb 666>[0718]<rh>DE SUMMATIONE</rh>
<p>4. Si vero requiratur $umma prædictæ $eriei inter duos valores <math>m</math>
& <math>m′</math> quantitatis <math>z</math> contenta: inveniantur fluentes fluxionum
<math>{x^(m+α)b-1 ⥊ x^(m′+α)b-1 / 1 ± x^b} x^. = A, {x^(m+β)b-1 ⥊ x^(m′+β)b-1 / 1 ± x^b} x^. = B, {x^(m+γ)b-1 ⥊ x^(m′+γ)b-1 / 1 ± x^b}
x^. = C, &c.</math> inter valores <math>0 & x</math> quantitatis <math>x</math>, cum <math>m + a,
m′ + α m + β, m′ + β; m + γ, m′ + γ; &c.</math> $int affirmativæ quan-
titates: $in vero quædam e prædictis $int negativæ quantitates; e. g.
$int <math>m + α & m′ + α</math> negativæ quantitates, tum inveniatur fluens <math>(P)</math>
fluxionis <math>{x^(m+α)b-1 ⥊ x^(m′+α)b-1 / 1 ± x^b} x^.</math> inter valores <math>x</math> & infinitum quantitatis
<math>x:</math> $in vero <math>m + α</math> $it negativa & <math>m′ + α</math> affirmativa quantitas; &
<math>α + l</math> affirmativa quantitas, ubi <math>l</math> e$t minimus integer numerus, qui
pote$t $ummam <math>α + l</math> reddere affirmativam; inveniatur fluens inter
valores <math>x</math> & infinitum quantitatis <math>x</math> fluxionis <math>{x^(m+α)b-1 ⥊ x^(α+l)b-1 / 1 ± x^b} × x^.</math>,
quæ $it <math>P</math>; deinde inveniatur fluens inter valores <math>0 & x</math> quantitatis <math>x</math>
contenta fluxionis <math>{x^(α+l)b-1 ⥊ x^(α+m′)b-1 / 1 ± x^b} x^.</math>, quæ $it <math>P′</math> tum fluens flu-
xionis <math>{x^(m+α)b-1 ⥊ x^(m+α)b-1 / 1 ± x^b} x^.</math> quæ$ita erit <math>P′ + P =A; &</math> $ic de $in-
gulis reliquis fluentibus <math>B, C, &c.</math> corrigendis; <math>&</math> exinde $umma quæ-
$ita erit <math>a′A + β′B + γ′C + &c.</math></p>
<p>5. Si <math>α</math> vel <math>β</math> vel <math>γ &c.</math> $it negativus numerus <math>&</math> inter <math>m & m′</math> con-
tentus; tum unus terminus prædictæ $eriei evadit in$initus, i. e. <math>{1 / 0}</math>;
$in autem plures termini evadant infiniti; <math>&</math> coefficientes, in quas
ducitur idem infinitus valor, $imul $umptæ nihilo $int æquales; tum
fumma quæ$ita pote$t e$$e finita; $in aliter vero non.</p>
<p>6. Si vero fluens $it <math>$. {z^ba-1 z^. / a ± bz^b} = {1 / a} × {1 / bα} z^bα ± {b / a^2} × {1 / b × (α + 1)}
z^b(α+1) + &c.</math>, tum facile reduci pote$t ad prædictam formulam <math>{1 / a}

<pb 667>[0719]<rh>SERIERUM, &c.</rh>
{a^α / b^α} × $. {x^bα-1 x^. / 1 ± x^b}</math> e $cribendo <math>{a^{1 / b} / b^{1 / b}}x</math> pro <math>z</math>. Et $imiliter fluxio <math>{x^bα-1 x^. / 1 ± x^b}</math>
reduci pote$t ad fluxionem <math>- {z^-(α-1)b z^. / z^b ± 1}</math> e $cribendo <math>{1 / z}</math> pro <math>x, &c.</math></p>
<p>7. Sit data $eries <math>{1 / α · α′. α″. α′″ · &c. × β · β′ · β″ · β′″ · &c. × γ · γ′ · γ″ · γ′″ · &c · × &c.}
+ {1 / α + 1 · α′ + 1 · α″ + 1 · α′″ + 1 · &c. × β + 1 · β′ + 1 · β″ + 1. &c.}
× γ · γ′ + 1 · γ″ + 1 · &c.. &c. + {1 / α + 2 · α′ + 2 · &c. × β + 2 · β′
+ 2. &c.. γ + 2 · γ′ + 2 · &c.. &c.}. ..{1 / α + z · α′ + z · α″ + z · &c.
× β · β′ + z · β″ + z · &c. × γ + z · γ′ + z · &c. × &c.}</math> in infini-
tum; ubi <math>z</math> e$t di$tantia a primo datæ $eriei termino; <math>& α - α′, α - α″,
α - α″, &c.; β - β′, β - β″, β - β′″, &c.; γ - γ′, γ - γ″, γ - γ′″,
&c., &c.</math>, $unt integri numeri; at <math>α - β, α - γ, β - γ, &c.</math> non $unt
integri numeri: tum, $i $umma fractionum <math>{1 / α - α′ · α - α″ · α - α′″.
&c. × α - β · α - β′ · α - β″ · α - β′″ · &c. × α - γ · α - γ′ · α - γ″ · &c. · &c.}
+ {1 / α′ - α · α′ - α″ · α′ - α′″ · &c..α′ - β · α′ - β′ · α′ - β″ · α′ - β′″. &c.. α′ - γ.
α′ - γ′ · α′ - γ″ · &c.} + {1 / α″ - α · α″ - α′ · α″ - α′″ · &c.. α″ - β · α″ - β′.
α″ - β″ · α″ - β′″ · &c. × α″ - γ · α″ - γ′ · α″ - γ″ · α″ - γ′″. &c.. &c.} +
{1 / α′″ - α · α′″ - α′ · α′″ - α″. &c. × α′″ - β · α′″ - β′ · α′″ - β′ · α′″ - β′″ · &c. ×
α′″ - γ · α′″ - γ′ · α′″ - γ″ · α′″ - γ′″ · &c. · &c.} + &c. = 0</math>, etiamque $umma

<pb 668>[0720]<rh>DE SUMMATIONE</rh>
fractionum <math>{1 / β - α · β - α′ · β - α″ · β - α′″ · &c. × β - β′ · β - β″ · β - β′″ · &c. ×
β - γ · β - γ′ · β - γ″ · β - γ′″ · &c.. &c.} + {1 / β′ - α · β′ - α′ · β′ - α″ · β′ - α′″ · &c.
× β′ - β · β′ - β″ · β′ - β′″ · &c. × β′ - γ · β′ - γ′ · β′ - γ″ · β′ - γ″ · &c.. &c.} +
{1 / β″ - α · β″ - α′ · β″ - α″ · β″ - α′″ · &c. × β″ - β · β″ - β′ · β″ - β′″ · &c.
× β″ - γ · β″ - γ′ · β″ - γ″ · β″ - γ′″. &c. × &c.} + {1 / β′″ - α · β′″ - α′ · β′″ - α″.
β′″ - α′″ · &c. × β′″ - β · β′″ - β′ · β′″ - β″ · &c. × β′″ - γ · β′″ - γ′.
β′″ - γ″ · β′″ - γ′″ · &c. × &c.} + &c. = 0; & {1 / γ - α · γ - α′ · γ - α″ · γ - α′″.
&c. × γ - β · γ - β′ · γ - β″ · γ - β′″ · &c. × γ - γ′ · γ - γ″ · γ - γ′″ · &c. × &c.}
+ {1 / γ′ - α · γ′ - α′ · γ′ - α″ · γ′ - α′″ · &c. × γ′ - β · γ′ - β′ · γ′ - β″ · γ′ - β′″. &c. ×
γ′ - γ · γ′ - γ″ · γ′ - γ′″ · &c. × &c.} + {1 / γ″ - α · γ″ - α′ · γ″ - α″ · γ″ - α′″ · &c. ×
γ″ - β · γ″ - β′ · γ″ - β″ · γ″ - β′″ · &c. × γ″ - γ · γ″ - γ′ · γ″ - γ′″ · &c. × &c.}
+ {1 / γ′″ - α · γ′″ - α′ · γ′″ - α″ · γ′″ - α′″ · &c. × γ′″ - β · γ′″ - β′ · γ′″ - β″ · γ′″ - β′″
· &c. × γ′″ - γ · γ′″ - γ′ · γ′″ - γ″ · &c. × &c.} + &c. = 0, &c.</math>; $umma datæ
$eriei in finitis terminis exprimi pote$t; $in aliter vero non: hoc fa-
cile con$tat per additionem, &c. prius traditam.</p>
<p>8. Sit <math>h</math> negativa quantitas, <math>&</math> $i requirantur fluentes fluxionum
<math>{x^bα-1 x^. / 1 ± x^b}, {x^bβ-1 x^. / 1 ± x^b}, {x^bγ-1 x^. / 1 ± x^b}, &c.</math> ad inveniendam $umman $eriei <math>{1 / α · β · γ · δ &c.}
+ {1 / α + 1 · β + 1 · γ + 1 · &c.}x^b + {1 / α + 2 · β + 2 · γ + 2 · &c.} × x^2b +<pb 669>[0721]<rh>SERIERUM, &c.</rh>
&c.</math> de$cendentis $ecundum dimen$iones quantitatis <math>x^b</math>; quæ $eries
$emper converget, cum <math>x</math> major $it quam 1; $in minor $it, diverget.</p>
<p>Si <math>x = 1</math>; haud multum diver$a erit $eries, cum <math>h</math> $it negativa, quam
cum <math>h</math> $it affirmativa quantitas; & de eâ fere eadem prædicari po$$unt.</p>
<p>In hoc ca$u pro inve$tigandis fluentibus fluxionum <math>{x^bα-1 x^. / 1 ± x^b},{x^bβ-1 x^. / 1 ± x^b},
{x^bγ-1 x^. / 1 ± x^b}, &c.</math> inter valores <math>m & m′</math> quantitatis <math>z</math> di$tantiæ a primo $eriei
termino, $i <math>α, β, γ, &c.</math> $int affirmativæ quantitates, inveniatur fluens
fluxionis <math>{x^(m+b)α-1 ⥊x^(ml+b)α-1 / 1 ± x^b}x^.</math> (inter valores infinitum & <math>x</math> quantita-
tis <math>x) = A:</math> $i autem quædam <math>α, β, γ, &c.</math> $int negativæ; e. g. $it <math>α</math> ne-
gativa quantitas, <math>& l + α</math> minima affirmativa, ubi <math>l</math> e$t integer affir-
mativus numerus inter numeros <math>m & m′</math> po$itus; inveniatur fluens
fluxionis <math>{x^(m+α)b-1 ⥊x^(l+α)b-1 / 1 ± x^b}x^.</math> inter valores <math>0 & x</math> quantitatis <math>x,</math> quæ
fit <math>P</math>; deinde inveniatur fluens (<math>A′</math>) fluxionis <math>{x^(l+α)b-1 ⥊x^(m′+α)b-1 / 1 ± x^b}x^.</math> inter
valores infinitum & <math>x</math> quantitatis <math>x,</math> tum erit fluens quæ$ita <math>P ± A′ =
A;</math> & $ic de reliquis; ergo $umma $eriei quæ$ita erit <math>α′A + β′B + &c.</math></p>
<p>Omnia de hoc ca$u, cum <math>h</math> $it negativa quantitas, affirmari po$-
$unt; quæ prius affirmantur de ca$u, cum <math>h</math> $it affirmativa quantitas.</p>
<p>Con$imilia etiam applicari po$$unt ad inveniendas $ummas $erie-
rum, quarum generalis terminus habeat formulam <math>{az^m + bz^m-1 + &c. / pz^n + qz^n-1 + &c.}</math>,
ubi <math>z</math> e$t di$tantia a primo $eriei termino.</p>
<p>9. Sint datæ $eries <math>{1 / α^2} + {1 / (α + 1)^2}x^b + {1 / (α + 2)^2}x^2b + {1 / (α + 3)^2}x^3b
+ &c. = A, {1 / α} + {1 / α + 1}x^b + {1 / α + 2}x^2b + &c. = A′, {1 / β} + {1 / β+1}x^b +
{1 / β + 2}x^2b + &c. = B, {1 / γ} + {1 / γ + 1}x^b + {1 / γ + 2}x^2b + &c. = C, &c.</math>:

<pb 670>[0722]<rh>DE SUMMATIONE</rh>
generales termini harum $erierum $unt re$pective <math>{1 / (α + z)^2}x^zb, {1 / α + z}x^zb,
{1 / β + z}x^zb, {1 / γ + z}, &c.</math>; ducantur hæ fractiones in coefficientes <math>a, b, c,
d, e, &c.</math> inve$tigandas, & re$ultant <math>{a / (α + z)^2}, {b / α + z}, {c / β + z}, {d / γ + z},
&c.</math>; reducantur hæ fractiones ad communem denominatorem & $i-
mul addantur, & re$ultat $umma <math>{a × (β + z)(γ + z)&c. + b(α + z)(β + z)(γ + z)&c. + c × (α + z)^2 (γ + z)&c. + d(α + z)^2 (β + z) &c. + &c. / (α + z)^2 × β + z.γ + z. &c.}</math>
fiat numerator <math>((b + c + d) + &c.)z^l-1 + (a + b(α + β + γ + &c.)
+ c(2α + γ + &c.) + d(2α + β + &c.) + &c.)z^l-2 + &c. = 1,</math>
quicunque $it valor quantitatis <math>z;</math> & con$equenter coefficientes e $in-
gulis pote$tatibus quantitatis <math>z</math> nihilo re$pective æquales, & coeffi-
ciens quæ non ducitur in pote$tatem quantitatis <math>z</math> erit 1, i. e. <math>b + c
+ d + &c. = 0, a + b(α + β + &c.) + c(2α + γ + &c.) + &c. = 0;
&c.; & a β γ &c. + bα β γ &c. + cα^2 γδ&c. + dα^2 βδ&c. + &c. =
P = 1;</math> & re$ultant <math>l - 1</math> $implices æquationes involventes <math>l</math> incognitas
quantitates <math>a, b, c, d, &c.</math>; e quibus deduci po$$unt rationes, quas ha-
bent inter $e incognitæ <math>a, b, c, d, &c.</math> & ex ultimâ æquatione <math>P = 1</math>
erui po$$unt quantitates ip$æ <math>a, b, c, d, &c.</math> & con$equenter <math>{a / (α + z)^2}
+ {b / α + z} + {c / β + z} + {d / γ + z} + &c. = {1 / (α + z)^2 .β + z.γ + z.δ + z.&c.}:</math>
e. g. $int <math>γ = 0, δ = 0, &c.,</math> tum erit <math>a = {1 / β - α}, b = - {1 / (β - α)^2} &
c = - b;</math> & con$equenter <math>{1 / β - α} × {1 / (α + z)^2} - {1 / (β - α)^2} × {1 / α + z} +
{1 / (β - α)^2} × {1 / β + z} = {1 / (α + z)^2 .β + z};</math> ergo erit $umma <math>{1 / β - α} A -

<pb 671>[0723]<rh>SERIERUM, &c.</rh>
{1 / (β - α)^2}A′ + {1 / (β - α)^2}B</math> æqualis $ummæ $eriei <math>{1 / α^2 β} + {1 / (α + 1)^2 .β + 1}
x^b + {1 / (α + 2)^2 .β + 2}x^2b + &c.</math></p>
<p>10. Sit $eries <math>{1 / α^m .β^n .γ^r .δ^s .&c.} ± {1 / (α + 1)^m .(β + 1)^n .(γ + 1)^r .(δ + 1)^s .&c.}
x^b + {1 / (α + 2)^m × (β + 2)^n × (γ + 2)^r × (δ + 2)^s}x^2b ± &c.;</math> tum ejus
$umma deduci pote$t ex datis $ummis $erierum <math>{1 / α} ± {1 / α + 1}x^b + {1 / α + 2}
x^2b ± &c. = A, {1 / α^2} ± {1 / (1 + α)^2}x^b + {1 / (α + 2)^2}x^2b ± &c. = A′, {1 / α^3} ±
{1 / (α + 1)^3}x^b + {1 / (α + 2)^3}x^2b ± {1 / (α + 3)^3}x^3b + &c. = A″,...{1 / α^m} ± {1 / (α + 1)^m}
+ {x^2b / (α + 2)^m} ± {x^3b / (α + 3)^m} + &c. = A′^m-1</math>; etiamque <math>{1 / β} ± {x^b / β + 1} +
{x^2b / β + 2} ± {x^3b / β + 3} + &c. = B, {1 / β^2} ± {x^b / (β + 1)^2} + {x^2b / (β + 2)^2} ± &c. = B′,
{1 / β^3} ± {x^b / (β + 1)^3} + {x^2b / (β + 2)^3} ± &c. = B″,...{1 / β^n} ± {x^b / (β + 1)^n} + {x^2b / (β + 2)^n}
± {x^3b / (β + 3)^n} ± &c. = B′^m-1; & {1 / γ} ± {x^b / γ + 1} + {x^2b / γ + 2} ± &c. = C, {1 / γ^2}
± {x^b / (γ + 1)^2} + {1 / (γ + 2)^2} ± &c. = C′,...{1 / γ^r} ± {x^b / (γ + 1)^r} + {x^2b / (γ + 2)^r}
± {x^3b / (γ + 3)^r} + &c. = C′^r-1, & {1 / δ} ± {x^b / δ + 1} + {x^2b / δ + 2} ± {x^3b / δ + 3} + &c. =
D,...{1 / δ^s} ± {x^b / (δ + 1)^s} + {x^2b / (δ + 2)^s} ± {x^3b / (δ + 3)^s} + &c. = D′^s-1, &c.;</math> i. e.
dentur $ummæ $erierum, quarum generales termini $unt <math>{± 1 / α + z},
{± 1 / (α + z)^2}, {± 1 / (α + z)^3}, {± 1 / (α + z)^4},...{± 1 / (α + z)^m}; {± 1 / β + z}, {± 1 / (β + z)^2}, {± 1 / (β + z)^3},

<pb 672>[0724]<rh>DE SUMMATIONE</rh>
...{± 1 / (β + z)^n}; {± 1 / γ + z}, {± 1 / (γ + z)^2}, {± 1 / (γ + z)^3},...{± 1 / (γ + z)^r}; ±{1 / δ + z},
{± 1 / (δ + z)^2},...{± 1 / (δ + z)^s}; &c.;</math> ducantur hi generales termini in coeffi-
cientes deducendas <math>a, a′, a″,.. a′^m-1; b, b′, b″,.. b′^n-1; c, c′, c″,.. c′^r-1; d,
d′, d″,.. d′^s-1, &c.</math>; re$pective; & deinde reducantur omnes fracti-
ones re$ultantes ad communem denominatorem, qui erit <math>(α + z)^m
(β + z)^n (γ + z)^r (δ + z)^s &c.;</math> & $imul addantur; tum fingan-
tur coefficientes $ingularum pote$tatum quantitatis <math>z</math> in nume-
ratore contentarum nihilo re$pective æquales, & re$ultant <math>m + n
+ r + s&c. - 1</math> æquationes habentes <math>(m + n + r + s + &c.)</math> plu-
res incognitas quantitates <math>a, a′,.. a′^m-1; b, b′, b″,.. b′^n-1; c, c′, c″,..
c′^r-1; d, d′, d″,.. d′^s-1; &c.</math> per unitatem quam numerus æquationum;
ex his <math>(m + n + r + s + &c. - 1)</math> $implicibus æquationibus erui
po$$unt rationes, quas habent inter $e $ingulæ incognitæ quantitates
<math>a, a′,.. a′^m-1; b, b′,.. b′^n-1; c, c′,.. c′^r-1; d, d′, .. d′^s-1, &c.:</math> etiamque
re$ultat $implex æquatio, ex fingendo terminum, in quo haud conti-
netur <math>z,</math> unitati æqualem; & exinde erui po$$unt quantitates <math>a, a′, a″,
.. a′^m-1; b, b′, b″, &c., b′^n-1; c,.. c′^r-1, d,.. d′^s-1, &c.</math></p>
<p>Summa $eriei quæ$itæ erit <math>= a A + a′ A′ + a″ A″ + ... a′^m-1 A′^m-1 +
bB + b′B′ + b″B″ + .. + b′^m-1B′^m-1 + cC + c′C′ + c″C″ + .. c′^r-1C′^r-1 +
dD + d′D′ + d″D″ + ... d′^s-1 D′^s-1 + &c.</math></p>
<p>Cor. Si <math>m = 1</math>, tum coefficiens <math>a</math> in $ummam $eriei <math>({1 / α} ± {x^b / α + 1} +
&c. = A)</math> ducenda <math>= {1 / (α - β)^n × (α - γ)^r × (α - δ)^s × &c.}:</math> $i vero <math>m</math>
$it numerus major quam 1, tum vel e prædictis $implicibus æquatio-
nibus, vel e prob. 43. deduci po$$unt coefficientes <math>a, a′, a″, &c.</math></p>
<p>Cor. Summa $eriei <math>{1 / α} ± {x^b / α + 1} + {x^2b / α + 2} ± &c.</math> inter quo$cunque

<pb 673>[0725]<rh>SERIERUM, &c.</rh>
duos valores, quorum di$tantiæ a primo dantur, deduci pote$t e flu-
ente fluxionis <math>$.{x^αb-1 x^. / 1 ± x^b}</math> proprie correctâ; etiamque $umma $eriei <math>{1 / α^2} ±
{x^b / (α + 1)^2} + {x^2b / (α + 2)^2} ± {x^3b / (α + 3)^2} + &c.</math> deduci pote$t generaliter e
fluente fluxionis <math>$. {x^. / x}$.{x^αb-1 x^. / 1 ± x^b}</math> vere correctâ; quæ dici pote$t fluens $e-
cundi ordinis: & $imiliter $ummæ $erierum <math>{1 / α^3} ± {x^b / (α + 1)^3} + {x^2b / (α + 2)^3}
± {x^3b / (α + 3)^4} + &c., {1 / α^4} ± {x^b / (α + 1)^4} + {x^2b / (α + 2)^4} ± {x^3b / (α + 3)^4} + &c.,
&c.</math>, deduci po$$unt e fluentibus fluxionum <math>$.{x^. / x}$.{x^. / x}$.{x^2b-1 x^. / 1 ± x^b}, $.{x^. / x}$.{x^. / x}
$.{x^. / x}$.{x^αb-1 x^. / 1 ± x^b}, &c.</math> vere correctis; quæ dici po$$unt fluentes tertii,
quarti, &c. ordinum: fluens $uperiori ordinis nunquam generaliter
exprimi pote$t per fluentem inferioris ordinis.</p>
<p>Et $ic de $ummis reliquarum $erierum prædictarum inve$tigandis.</p>
<p>Cor. Hinc generaliter detegi pote$t $umma $eriei inter quo$cun-
que terminos contenta; cujus generalis terminus $it quæcunque ra-
tionalis functio quantitatis <math>z</math> di$tantiæ a primo $eriei termino in <math>x^bz</math>
ducta, viz. <math>{a z^m + b z^m-1 + c z^m-2 + &c. / f z^n + g z^n-1 + h z^n-2 + &c.} × x^bz,</math> ubi <math>m & n</math> $unt integri
numeri; $i modo generaliter dentur fluentes omnium fluxionum
$ub$equentium formularum <math>{x^bπ-1 x^. / 1 ± x^b}, {x^. / x}$.{x^bπ-1 x^. / 1 ± x^b}, {x^. / x}$.{x^. / x}$.{x^bπ-1 x^. / 1 ± x^b}, {x^. / x}$.{x^. / x}
$.{x^. / x}$.{x^bπ-1 x^. / 1 ± x^b}, {x^. / x}$.{x^. / x}$.{x^. / x}$.{x^. / x}$.{x^bπ-1 x^. / 1 ± x^b}, &c.</math> ubi <math>h & π</math> $int quæcunque
quantitates po$$ibiles vel impo$$ibiles: i. e. $i detur generaliter fluens
fluxionis <math>{x^bπ-1 x^. / 1 ± x^b},</math> tum dabitur $umma omnis $eriei, cujus generalis
terminus habeat prædictam formulam, & in cujus denominatore

<pb 674>[0726]<rh>DE SUMMATIONE</rh>
<math>fz^n + gz^n-1 + &c.</math> haud contineantur duo vel plures divi$ores vel fa-
ctores inter $e æquales; i. e. denominator <math>fz^n + gz^n-1 + &c.</math> non
dividi pote$t per quantitatem formulæ <math>(σz + ρ)^2</math>; & $imiliter $umma
omnis $eriei, cujus generalis terminus prædictam habeat formulam,
& in cujus denominatore <math>fz^n + gz^n-1 + &c.</math> haud contineatur cubi-
cus divi$or vel factor <math>(σz + ρ)^3</math>, deduci pote$t e fluentibus fluxionum
formularum <math>{x^bπ-1 x^. / 1 ± x^b} & {x^. / x}$.{x^bπ-1 x^. / 1 ± x^b}</math>; & $ic deinceps.</p>
<p>Cor. Sit data $eries, cujus generalis terminus e$t <math>{± 1 / (α + z)^m (β + z)^n (γ + z)^r (δ + z)^s &c.}x^bz</math>,
& dentur $ummæ $erierum inter valores <math>H & K</math>
quantitatis <math>z</math> contentarum <math>{1 / α + 1} ± {1 / α + 2}x^b + {1 / α + 3}x^2b ± &c., {1 / α^2} ±
{1 / (α + 1)^2}x^b + {1 / (α + 2)^2}x^2b ± {1 / (α + 3)^2}x^3b + &c., {1 / α^3} ± {x^b / (α + 1)^3} +
{x^2b / (α + 2)^3} ± &c., ...{1 / α^m} ± {x^b / (α + 1)^m} + {x^2b / (α + 2)^m} ± &c.; & {1 / β} ± {x^b / β + 1}
+ {x^2b / β + 2} ± &c., {1 / β^2} ± {x^b / (β + 1)^2} + {x^2b / (β + 2)^2} ± &c., ... {1 / β^n} ± {x^b / (β + 1)^n}
+ {x^2b / (β + 2)^n} ± &c.; & {1 / γ} ± {x^b / γ + 1} + {x^2b / γ + 2} ± &c., {1 / γ^2} ± {x^b / (γ + 1)^2}
+ {x^2b / (γ + 2)^b} ± &c., ... {1 / γ^r} ± {x^b / (γ + 1)^r} + {x^2b / (γ + 2)^r} ± &c.; & {1 / δ} ±
{x^b / δ + 1} + {x^2b / δ + 2} ± &c., {1 / δ^2} ± {x^b / (δ + 1)^2} + {x^2b / (δ + 2)^2} + &c.,... {1 / δ^s} ±
{x^b / (δ + 1)^s} + {x^2b / (δ + 2)^s} ± &c., &c.</math>; tum ex iis deduci pote$t $umma datæ
$eriei inter prædictos valores <math>H & K</math> contenta: $ummæ autem præ-
dictarum $erierum inter valores <math>H & K</math> quantitatis <math>z</math> contentarum
ex fluentibus fluxionum <math>{x^bα-1 x^. / 1 ± x^b}, {x^. / x} $.{x^bα-1 x^. / 1 ± x^b}, {x^. / x} $.{x^. / x} $.{x^bα-1 x^. / 1 ± x^b}, {x^. / x} $. {x^. / x} $. {x^. / x}

<pb 675>[0727]<rh>SERIERUM, &c.</rh>
$. {x^bα-1 x^. / 1 ± x^b}</math> u$que ad <math>m</math> terminos; <math>{x^bβ-1 x^. / 1 ± x^b}, {x^. / x} $.{x^βb-1 x^. / 1 ± x^b}, {x^. / x} $.{x^. / x} $.{x^βb-1 x^. / 1 ± x^b}</math>, u$que
ad <math>n</math> terminos; <math>{x^γb-1 x^. / 1 ± x^b}, {x^. / x} $. {x^γb-1 x^. / 1 ± x^b}, &c.</math> u$que ad <math>r</math> terminos; & $ic de-
inceps; inter valores <math>0 & x</math> quantitatis <math>x</math> contentis exprimi po$$unt.</p>
<p>Sit <math>h</math> affirmativa quantitas, hæ $eries convergent, cum <math>x</math> inter <math>1 &
- 1</math> contineatur; & $i <math>x</math> major $it quam 1, tum $emper divergent: $i
vero <math>x = 1</math>, tum $emper converget, necne; prout numerus dimen-
$ionum quantitatis <math>z</math> in numeratore $eriei ad affirmativam reductæ
minor $it per quantitatem majorem quam unitatem quam ejus di-
men$iones in denominatore, necne.</p>
<p>11. Erit $eries <math>{1 / r} ± {m / (r + 1)} · {bx^n / a} + m · {m - 1 / 2} × {1 / (r + 2)} · {b^2 x^2n / a^2} ±
m · {m - 1 / 2} · {m - 2 / 3} × {1 / (r + 3)} · {b^3 x^3n / a^3} + m. {m - 1 / 2} · {m - 2 / 3}· {m - 3 / 4} ×
{1 / (r + 4)} × {b^4 x^4n / a^4} + &c. = {n / a^m x^rn} $. (a ± b x^n)^m x^rn-1 x^.</math> vere correctæ du-
catur hæc æquatio in <math>x^bn-1 x^.</math> & inveniatur fluens utriu$que partis
æquationis re$ultantis, & evadet <math>{1 / r h} ± {m / r + 1 · h + 1} × {b x^n / a} +
{m · (m - 1) / 2 · r + 2 · h + 2} × {b^2 x^2n / a^2} ± {m · (m - 1) · (m - 2) / 2 · 3 · (r + 3) · (h + 3)} × {b^3 x^3n / a^3} +
{m · (m - 1) · m - 2 · m - 3 / 2 · 3 · 4 · (r + 4) · (b + 4)} × {b^4 x^4n / a^4} ± &c. = {n^2 x^-bn / a^m} $. x^(b-r)n-1 x^. $. x^rn-1 x^.
(a ± bx^n)^m</math>; ducatur hæc æquatio in <math>x^kn-1 x^.</math>, & in veniatur fluens æqua-
tionis re$ultantis, quæ erit <math>{1 / r h k} ± {m / r + 1 · h + 1 · k + 1} × {b x^n / a} +
{m · m - 1 / 2 · r + 2 · b + 2 · k + 2} × {b^2 x^2n / a^2} ± &c. = {n^3 / a^m} x^-kn $. x^(k-h)n-1 x^. $. x^(b-r)n-1 x^.
$. x^rn-1 x^. (a ± b x^n)^m</math>; & $ic deinceps; & con$tat, $i modo <math>m, r, h, k, &c.,</math>
$int quantitates, quæ habent formulam <math>± l + {1 / 2} vel ± l</math>, ubi <math>l</math> e$t in-
teger numerus, prædictas fluentes inveniri po$$e finitorum termino-
rum, circularium arcuum & logarithmorum ope.</p>

<pb 676>[0728]<rh>DE SUMMATIONE</rh>
<p>Cor. Hinc $it generalis terminus <math>m · {m - 1 / 2} · {m - 2 / 3} .. {m - z + 1 / z}
× {1 / r + z} · {a z^M + b z^M-1 + c z^M-2 + &c. / f z^N + g z^N-1 + &c.} × e^z</math>, ubi <math>e = {b x^n / a} & M & N</math>
$unt integri numeri; & omnes divi$ores denominatoris <math>f z^N + g z^N-1
+ &c.</math> habeant formulam <math>z ± l ± {1 / 2}λ</math>, in quâ <math>l & λ</math> $unt integri nu-
meri vel <math>0, & z</math> e$t di$tantia a primo $eriei termino, <math>& m & r</math> quanti-
tates formulæ prius traditæ <math>± l ± {1 / 2} λ</math>; & haud duo vel plures divi-
$ores $unt inter $e æquales; tum ejus $umma detegi pote$t ope fini-
torum terminorum, circularium arcuum & logarithmorum.</p>
<p>Cor. Con$tat e $ummis $erierum, quarum generales termini re$pe-
ctive $unt <math>V × {x^z / (z ± l ± {1 / 2}λ)^h} = L, V × {x^z / (z ± l ± {1 / 2} λ)^b-1} = L′, V ×
{x^z / (z ± l ± {1 / 2} λ)^b-2} = L″, ... V × {x^z / (z ± l ± {1 / 2} λ)^2} = L′^b-2, V × {x^z / (z ± l ± {1 / 2} λ)}
= L′^b-1; & V × {x^z / (z ± l′ ± {1 / 2} λ′)^b′} = M, V × {x^z / (z ± l′ ± {1 / 2} λ′)^b′-1} = M′,
{x^z / (z ± l′ ± {1 / 2} λ′)^b′-1} = M″, ... V × {x^z / (z ± l′ ± {1 / 2} λ′)^2} = M′^b-2, V ×
{x^z / (z ± l′ ± {1 / 2} λ)} = M′^b-1</math>, etiamque <math>V × {x^z / (z ± l″ ± {1 / 2} λ″)^b″} = N, V ×
{x^z / (z ± l″ ± {1 / 2} λ″)^b″-1} = N′, ... V × {x^z / (z ± l″ ± {1 / 2}λ″)^2} = N′^b-2, V ×
{x^z / (z ± l″ ± {1 / 2} λ)} = N′^b-1; &c.</math> per prob. acquiri po$$e $ummam $eriei,
cujus generalis terminus e$t <math>V × {a z^M + b z^M-1 + &c. / (z ± l ± {1 / 2}λ)^b × (z ± l′ ± {1 / 2} λ′)^b′ × (z
± l″ ± {1 / 2} λ″)^b″ × &c.}</math>, $i modo <math>M, h, h′, b″, &c.</math> $int integri numeri;
con$tat etiam ex additione prius u$itatâ generalem terminum e$$e <math>a L
+ a′ L′ + a″ L″ + &c. + bM + b′M′ + &c.</math>; & coefficientes <math>a, a′, &c.,
b, b′, &c.</math> exinde inve$tigari po$$e.</p>

<pb 677>[0729]<rh>SERIERUM, &c.</rh>
<p>Sint <math>2l, 2l′, 2l″, &c.; 2λ, 2λ′, 2λ″, &c.</math> integri numeri & <math>V = m · {m - 1 / 2}
· {m - 2 / 3}...{m - z + 1 / z} × {1 / μ + z}</math>; tum $ummæ prædictarum $erierum
detegi po$$unt e fluentibus fluxionum formularum <math>$. x^μn-1 x^. (a +
b x^n)^m, $. {x^. / x} $. (a + b x^n)^m x^μn-1 x^., $. {x^. / x} $. {x^. / x} $. (a + b x^n)^m x^μn-1 x^., $. {x^. / x} $. {x^. / x}
$. {x^. / x} $. (a + b x^n)^m x^μn-1 x^., &c.</math>; ubi quantitas <math>μ</math> habet prædictam for-
mulam <math>± l ± {1 / 2} λ, l & λ</math> exi$tentibus integris numeris.</p>
<p>Ex his fluentibus generaliter deductis detegi po$$unt $ummæ $erie-
rum inter quo$cunque duos valores quantitatis <math>z</math> po$itæ etiamque,
$i modo dentur fluentes harum fluxionum inter quo$cunque duos
valores quantitatis <math>x</math>, erui po$$unt corre$pondentes prædictarum $erie-
rum $ummæ.</p>
<p>E principiis prius traditis dijudicari po$$unt harum $erierum con-
vergentiæ.</p>
<p>Si modo fluentes harum fluxionum per finitos terminos, circulares
arcus, & logarithmos exprimi po$$int; tum per finitos terminos, cir-
culares arcus & logarithmos etiam exprimi po$$unt fluentes, quæ ex-
primunt $ummam e $ingulis terminis prædictæ $eriei, quorum di$tan-
tiæ a $e invicem $it <math>H.</math> Si enim $eries $ecundum dimen$iones quan-
titatis <math>x^n</math> progrediantur; tum in fluente inventâ pro <math>x^n</math> $cribantur
<math>α x^n, β x^n, γ x^n, &c.,</math> ubi <math>α, β, γ, &c.</math> $unt radices æquationis <math>w^H - 1 = 0</math>;
& deinde per methodum in medit. algebr. datam progrediendum e$t.</p>
<p>Con$imiles etiam propo$itiones erui po$$unt de $eriebus, quarum
$ummæ exprimi po$$unt per ellipticos arcus & alias fluentes.</p>
<p>12. Sit <math>$. {x^. / 1 + x} = log. (1 + x) = a</math>, tum erit <math>x = a + {a^2 / 1 · 2} +
{a^3 / 1 · 2 · 3} + &c., & e^a = 1 + x</math>; hinc e medit. algebr. <math>{α^n-m e^αn + β^n-m e^ρa
+ γ^n-m e^βa + δ^n-m e^δa + &c. / n} (P) = {a^m / 1 · 2 · 3 .. m} + {a^n+m / 1 · 2 · 3 .. n + m} +

<pb 678>[0730]<rh>DE SUMMATIONE</rh>
{a^2n+m / 1 · 2 · 3 .. 2n + m} + {a^3n+m / 1 · 2 · 3 .. 3n + m} + &c.</math>; ubi <math>α, β, γ, δ, &c.</math> $unt
(<math>n</math>) radices æquationis <math>x^n - 1 = 0</math>. 2. Erit <math>e^-a = 1 - a + {a^2 / 1 · 2} -
{a^3 / 1 · 2 · 3} + &c.</math> & exinde <math>{α^n-m e^-az + β^n-m e^-βa + γ^n-m e^-γa + &c. / n}(P) =
± {a^m / 1 · 2 · 3 .. m} ± {a^n+m / 1 · 2 .. n + m} ± {a^2n+m / 1 · 2 .. 2n + m} ± &c.</math>: $igna af-
fixa erunt +, $i <math>n & m</math> $int pares numeri; erunt -, $i <math>n</math> $it par & <math>m</math>
impar; erunt alternatim <math>+ & -</math>, $i <math>n</math> $it impar; $ignum affixum
primo termino erit +, $i <math>m</math> $it par, $in aliter -; & exinde uterque
ca$us evadit idem, cum <math>n & m</math> $int pares numeri.</p>
<p>Sint <math>α & δ</math> corre$pondentes impo$$ibiles radices, quæ $int <math>e^α + e^δ =
e^a+b√(-1) + e^a-b√(-1) = e^a (e^b√(-1) + e^-b√(-1))</math>; ergo ex $ummis
omnium quantitatum huju$ce formulæ <math>e^b√(-1) + e^-b√(-1) = 2 (1 -
{b^2 / 1 · 2} + {b^4 / 1 · 2 · 3 · 4} - {b^6 / 1 · 2 · 3 · 4 · 5 · 6} + &c.) = 2</math> co$. arcus <math>b</math> cir-
culi, cujus radius e$t 1, erui po$$unt $ummæ omnium quantita-
tum formulæ (<math>P</math>), vel $erierune <math>1 ± {1 / m + 1 · m + 2 .. n + m} a^n +
{1 / m + 1 · m + 2 .. 2n + m}a^2n ± &c.</math> erit <math>{e^a√(-1) - e^-a√(-1) / 2 √ (-1)} = a -
{a^3 / 1 · 2 · 3} + {a^5 / 1 · 2 .. 5} - {a^7 / 1 · 2 .. 7} + &c. = x</math> $in. prædicti arcus (<math>a</math>).</p>
<p>2. Sint fluentes <math>$. {x^. / √ (1 - α x^2)} = a, $. {x^. ′ / √ (1 - βx′^2)} = a, $.
{x^. ″ / √ (1 - γ x″^2)} = a, &c.</math>; tum erunt <math>x = a - {αa^3 / 1 · 2 · 3} + {α^2 a^5 / 1 · 2 .. 5} -
&c., x′ = a - {β a^3 / 1 · 2 · 3} + &c., x″ = a - {γ a^3 / 1 · 2 · 3} + &c.</math>; $i modo <math>$.
{x^. / √ (1 - α x^2)} = a, $. {x^. ′ / √ (1 - β x′^2)} = a, &c.</math> inter valores <math>0 & x, 0 &

<pb 679>[0731]<rh>SERIERUM, &c.</rh>
x′, &c.</math> quantitatum <math>x, x′, &c.</math> contineantur; etiamque <math>{α^n-m x + β^n-m x′ + γ^n-m x″ + &c. / n}
= {± a^m / 1 · 2 · 3 .. 2m} ± {a^2n+m / 1 · 2 · 3 .. 2n + 2m} ± {a^4n+m / 1 · 2 · 3
.. 4n + m} ± &c.</math> $igna affixa leges prius traditas ob$ervant.</p>
<p>Hæ $eries evadunt eædem ac priores, & con$equenter ex quantita-
tibus <math>e^αa, e^βa, e^γa, &c.</math> deduci po$$unt quantitates <math>x, x′, x″, &c.: $.
{x^. / √ (1 - α x^2)} = a, $ {x^. / √ (1 - β x′^2)} = a, $. {x^. / √ (1 - γ x″^2)} = a, &c.</math>
inter valores <math>o & x, o & x′, o & x″, &c.</math> quantitatum <math>x, x′, x″, &c.</math> con-
tentæ erunt arcus circuli, cujus radius e$t 1 & $inus po$$ibiles vel im-
po$$ibiles quantitates re$pective <math>x √ (α), x′ √ (β), &c.</math> in <math>{1 / √ (α)}, {1 / √ (β)},
{1 / √ (γ)}, &c.</math> re$pective ductæ; i. e. erunt <math>x = a - {α a^3 / 1 · 2 · 3} + &c.,
x′ = a - {β a^3 / 1 · 2 · 3} + {β^2 a^5 / 1 · 2 .. 5} - &c., &c.: & $. {- x^. / √ (1 - α x^2)} = a,
{- x^. ′ / √ (1 - β x′^2)} = a, $. {- x^. ″ / √ (1 - γ x″^2)} = a, &c.</math> inter valores <math>1 & x, 1 &
x′, 1 & x″, &c.</math> quantitatum <math>x, x′, x″, &c.</math> contentæ erunt arcus cir-
culi, cujus radius e$t 1, & con$inus po$$ibiles vel impo$$ibiles quanti-
tates <math>x √ (α), x √ (β), x √ (γ), &c.</math> in <math>{1 / √ (α)}, {1 / √ (β)}, {1 / √ (γ)}, &c.</math> re-
$pective ductæ; i. e. erunt <math>x = 1 - {α a^2 / 1 · 2} + {α^2 a^4 / 1 · 2 · 3 · 4} - &c., x′ =
1 - {β a^2 / 1 · 2} + &c., &c.</math></p>
<p>3. Sit <math>a + {a^2 / 1 · 2} + {a^3 / 1 · 2 · 3} + {a^4 / 1 · 2 · 3 · 4} + &c. = x</math>, ubi <math>a^. = {x^. / 1 + x}</math>;
ducatur hæc æquatio in <math>a^m-1 a^.</math>, & inveniatur fluens æquationis re$ul-
tantis, quæ erit <math>{1 / m + 1} a^m+1 + {1 / m + 2} × {1 / 1 · 2} a^m+2 + {1 / m + 3} × {1 / 1 · 2 · 3}

<pb 680>[0732]<rh>DE SUMMATIONE</rh>
a^m+3 + &c. = $. a^m-1 x a^. = $. a^m-1 {x x^. / 1 + x} = a^m-1 (x - {1 / m} a) - (m - 1)
a^m-2 (x - {1 / m - 1} a) + (m - 1) · (m - 2) (x - {1 / m - 2} a) - (m - 1) ·
(m - 2) · (m - 3) (x - {1 / m - 3} a) + &c.</math></p>
<p>4. Ad eundem modum $it <math>a - {a^2 / 1 · 2} + {a^3 / 1 · 2 · 3} - &c. = x</math>; ubi
<math>a^. = {x^. / 1 - x}</math>; ducatur hæc æquatio in <math>a^m-1 a^.</math> & re$ultat æquatio <math>a^m a^.
- {a^m+1 / 1 · 2} a^. + &c. = a^m-1 a^. x</math>; cujus fluens erit <math>{a^m+1 / m + 1} - {a^m+2 / m + 2} + &c. =
a^m-1 (- x + {1 / m} a) - (m - 1) a^m-2 (- x + {1 / m - 1} a) + (m - 1) · (m
- 2) a^m-3 (- x + {1 / m - 2} a) - &c.</math></p>
<p>5. Et $imiliter $it æquatio <math>a - {a^3 / 2 · 3} + {a^5 / 2 · 3 · 4 · 5} - &c. = x</math>, ubi
<math>a^. = {x^. / √ (1 - x^2)}</math>; ducatur prior æquatio in <math>a^m-1 a^.</math> & inveniatur fluens
re$ultantis fluxionis, quæ erit <math>{a^m / m + 1} - {a^m+3 / 2 · 3 · (m + 3)} + {a^m+5 / 2 · 3 · 4 · 5 · (m + 5)}
- &c. = - a^m-1 × √ (1 - x^2) + (m - 1) a^m-2 x + (m - 1) · (m - 2)
a^m-3 × √ (1 - x^2) - (m - 1) · (m - 2) · (m - 3) a^m-4 x - &c. = -
(a^m-1 - (m - 1) · (m - 2) a^m-3 + (m - 1) · (m - 2) · (m - 3) · (m - 4)
a^m-5 - &c.) √ (1 - x^2) + ((m - 1) · a^m-2 - (m - 1) · (m - 2) ·
(m - 3) a^m-4 + &c.) x</math>.</p>
<p>6. Sit æquatio <math>1 - {a^2 / 1 · 2} + {a^4 / 1 · 2 · 3 · 4} - &c. = x</math>, ubi <math>a^. = {- x^. / √ (1 - x^2)}</math>;
ducatur hæc æquatio in <math>a^m-1 a^.</math> & inveniatur fluens fluxionis re$ultan-
tis, quæ erit <math>{a^m / m} - {a^m+2 / 1 · 2 · (m + 2)} + {a^m+4 / 1 · 2 · 3 · 4 · (m + 4)} - &c. =
(a^m-1 + (m - 1) · (m - 2) a^m-3 + &c.) √ (1 - x^2) + ((m - 1) a^m-2
+ (m - 1) · (m - 2) · (m - 3) a^m-4 + &c.) x</math>.</p>

<pb 681>[0733]<rh>SERIERUM, &c.</rh>
<p>Et con$imilia applicari po$$unt ad omnes ca$us prius traditos; in
omnibus hi$ce ca$ibus $eries (quæ denotat $ummam) terminat cum <math>m</math>
$it integer affirmativus numerus: deinde ducantur æquationes re$ul-
tantes in <math>a^m′-1 a^.</math>, & inveniantur fluentes æquationum re$ultantium, &
introducitur novus factor in $inguli termini datæ $eriei denominato-
rem; & $erierum $ummæ in finitis terminis exprimi pote$t, $i modo <math>m
& m′</math> $int integri numeri.</p>
<p>7. Hinc con$tat ex hâc methodo deduci po$$e in finitis terminis $um-
mam omnis $eriei, cujus generalis terminus e$t <math>{± a^lz+b / 1 · 2 · 3 · 4 .. n z ×
(z + m) · (z + m′) · (z + m″) · &c.}</math>, ubi <math>z</math> e$t di$tantia a primo $eriei
termino; & <math>m, m′, m″, &c.</math> $unt quicunque diver$i integri numeri.</p>
<p>8. Sit æquatio <math>a + {a^2 / 1 · 2} + {a^3 / 1 · 2 · 3} + &c. = x</math>, ducatur hæc æquatio
in <math>a^r</math> & re$ultat <math>a^r+1 + {a^r+2 / 1 · 2} + {a^r+3 / 1 · 2 · 3} + &c. = a^r x</math>; inveniatur ejus
fluxio, & re$ultat <math>(r + 1) a^r + {(r + 2) a^r+1 / 1 · 2} + {(r + 3) a^r+2 / 1 · 2 · 3} + &c. =
r a^r-1 x + a^r {x^. / a^.} = r a^r-1 x + a^r (1 + x)</math> ob <math>a^. = {x^. / (1 + x)}</math>; ducatur hæc
æquatio in <math>a^r′-r+1</math> & inveniatur fluxio æquationis re$ultantis, quæ erit
<math>((r + 1) · (r′ + 1) a^r′ + {(r + 2) · (r′ + 2) / 1 · 2} a^r′+1 + &c.) a^. = (r a^r′ x +
a^r′+1 (1 + x)) = r r′ a^r′-1 x a^. + r a^r′ x^. + (r′ + 1) a^r′ (1 + x) a^. + a^r+1 x^.
= (r r′ a^r′-1 x + (r + r′ + 1 + a′) a^r′ (1 + x) a^.</math>; dividatur hæc æqua-
tio per <math>a^.</math> & re$ultat æquatio quæ$ita.</p>
<p>Ducatur æquatio re$ultans in <math>a^rl′-r′+1</math> & inveniatur æquationis re$ul-
tantis fluxio, & introducitur novus factor in numeratorem; & $ic de-
inceps. Per eandem methodum introduci po$$unt con$imiles factores
in numeratores omnium $erierum hìc traditarum. Literæ <math>r, r′, r″,
&c.</math> denotent qua$cunque quantitates po$$ibiles vel impo$$ibiles; cor-

<pb 682>[0734]<rh>DE SUMMATIONE</rh>
re$pondentes impo$$ibiles radices $int <math>r = e + f √ (- 1) & r′ =
e - f √ (- 1)</math>; ducantur eædem functiones harum radicum in $e-
riem po$$ibiles quantitates $olummodo involventem, & po$t $ingulam
multiplicationem per methodum hìc traditam inveniantur fluxiones
vel fluentes, & $eries re$ultans erit po$$ibilis.</p>
<p>9. Hinc con$tat ex hâc methodo deduci po$$e in $initis terminis $um-
mam omnis $eriei, cujus generalis terminus e$t <math>{± a^lz+b × (A z^m
+ B z^m-1 + C z^m-2 + &c.) / 1 · 2 · 3 .. n z · (z + k) · (z + k′) · (z + k″) · (z + k′″) · &c.}</math>;
ubi <math>z</math> ut antea denotat di$tan-
tiam a primo datæ $eriei termino, & literæ <math>k, k′, k″, &c.</math> de$ignant
integros diver$os numeros.</p>
<p>Eadem etiam perfici po$$unt ex additione $erierum hìc traditarum
in invariabiles quantitates deducendas ductarum.</p>
<p>Si requirantur $eries, e quarum $ummis evane$cunt termini, qui
circulares arcus & logarithmos continent; inveniantur $ummæ per
methodos prius traditas; & fiant termini, qui involvunt independentes
circulares arcus & logarithmos nihilo re$pective æquales, & ex æqua-
tionibus re$ultantibus erui po$$unt $eries quæ$itæ.</p>
<p>Hæ $eries etiam inveniri po$$unt ex prædictâ additione termino-
rum.</p>
<p>10. Et $imiliter ex datis $ummis $erierum, quarum generales termini
$unt <math>{a^lz+b / 1 · 2 · 3 .. n z · (α + z)}, {a^lz+b / 1 · 2 · 3 .. n z · β + z}, {a^lz+b / 1 · 2 · 3 .. n z · γ + z},
{a^lz+b / 1 · 2 · 3 .. n z · δ + z}, &c.</math> & principiis hìc traditis erui po$$unt $um-
mæ omnium $erierum formulæ, cujus generalis terminus exprimitur
per fractionem, cujus numerator e$t <math>a^lz+b × (A z^m + B z^m-1 + C z^m-2
+ &c.)</math>, & denominator <math>1 · 2 · 3 · n z × (α + z) · (α + z + 1) · (α +
z + 2) .. (α + z + e) × (β + z) (β + z + 1) (β + z + 2) .. (β + z
+ e′) × (γ + z) (γ + z + 1) (γ + z + 2) .. (γ + z + e″) × (δ + z)
(δ + z + 1) (δ + z + 2) .. (δ + z + e″′) × &c.</math>; ni quilibet factores
in denominatore contenti $int inter $e æquales.</p>

<pb 683>[0735]<rh>SERIERUM, &c.</rh>
<p>Ex $ummâ $eriei, cujus generalis terminus e$t <math>{± a^lz+b / 1 · 2 · 3 .. z · (α + z)}</math>,
datâ per principia hic tradita erui pote$t $umma $eriei, cujus genera-
lis terminus e$t <math>{± a^lz+b / 1 · 2 · 3 .. n z · (α + z)}</math>.</p>
<p>11. Sit $eries <math>A x^n + B x^n+m + C x^n+2m + D x^n+3m + &c. = y</math>, cujus in-
dices variabilis quantitatis <math>x</math> $int in arithmeticâ progre$$ione; ducan-
tur $ucce$$ivi termini $eriei re$pective in terminos $ucce$$ivos trium
arithmeticarum $erierum <math>r z + s, r′ z + s′ & r″ z + s″</math>, ubi <math>z</math> e$t di-
$tantia a primo datæ $eriei termino, & re$ultant tres $eries <math>s A x^n +
(r + s) B x^n+m + (2 r + s) C x^n+2m + &c. = S, s′ A x^n + (r′ + s′) B x^n+m
+ (2 r′ + s′) C x^n+2m + &c. = S′, s″ A x^n + (r″ + s″) B x^n+m + (2 r″ +
s″) C x^n+2m + &c. = S″</math>: ex $ummis duarum priorum $erierum inter
$e independentium datis acquiri pote$t tertiæ $umma: a$$umantur
enim duæ æquationes <math>e r + e′ r′ = r″ & e s + e′ s′ = s″</math>, e quibus erui
po$$unt <math>e & e′</math>, & exinde <math>e S + e′ S′ = S″</math> $ummæ quæ$itæ.</p>
<p>Et $imiliter ducantur $ucce$$ivi termini $eriei re$ultantis in $ucce$$ivos
$ecundæ arithmeticæ $eriei terminos, & re$ultat $eries, cujus $umma de-
duci pote$t ex $ummis trium eju$dem generis $erierum: & $imiliter du-
catur data $eries in (<math>h</math>) arithmeticas $eries $ucce$$ive; tum $umma $eriei
re$ultantis erui pote$t e $ummis (<math>h + 1</math>) independentium $erierum eju$-
dem generis datis; vel quod idem e$t, ducantur $ucce$$ivi termini præ-
dictæ $eriei in $ucce$$ivos terminos quantitatis <math>′A z^m + ′B z^m-1 + ′C z^m-2
+ &c. = (r z + s)(t z + l)(t′ z + l′)(t″ z + l″) &c.</math>, ubi <math>z</math> di$tantia a
primo datæ $eriei termino $ucce$$ive denotat <math>0, 1, 2, 3, &c.</math> tum ex
$ummis (<math>m + 1</math>) $erierum re$ultantium inter $e independentium datis
erui po$$unt $ummæ omnium eju$dem generis $erierum: vel magis
generaliter ducantur $ucce$$ivi termini $eriei <math>A x^n + B x^n+m + C x^n+2m
+ &c.</math> in <math>a φ:z + b φ′:z + c:φ″:z + d:φ′″:z + &c. = P</math>, & $i-
militer in <math>a′ φ:z + b′ φ′:z + c′ φ″:z + &c. = Q</math>; etiamque in <math>a″ φ
: z + b″ φ′:z + c″ φ″:z + &c., &c. = R</math>, ubi <math>φ:z, φ′:z, φ″:z, &c.</math>
$unt diver$æ datæ functiones quantitatis (<math>z</math>) di$tantiæ a primo datæ
$eriei termino; & <math>P, Q, R, &c.</math> $unt (<math>n</math>) quantitates inter $e indepen-
dentes, & $int $ummæ $erierum re$ultantium re$pective <math>A′, A″, A′″, A″″,

<pb 684>[0736]<rh>DE SUMMATIONE</rh>
&c.</math>; tum ex $ummis (<math>n</math>) diver$arum re$ultantium huju$ce generis $e-
rierum erui po$$unt $ummæ omnium eju$dem generis $erierum; e. g.
$it $umma $eriei re$ultantis ex ducendo $ucce$$ivos terminos datæ $eriei
<math>Ax^n + Bx^n+m + &c.</math> in $ucce$$ivos & corre$pondentes terminos quan-
titatis <math>(αφ: z + βφ′: z + γφ″: z + δφ″: z + &c. = T) = B</math>.</p>
<p>Fiant <math>p a + qa′ + r a″ + &c. = a, pb + qb′ + rb″ + &c. = β,
pc + qc″ + rc′ + &c. = γ, pd + qd′ + rd″ + &c. = δ, &c.</math>; & re-
$ultant <math>n</math> æquationes totidem (<math>n</math>) incognitates <math>p, q, r, s, &c.</math> invol-
ventes, e quibus inveniantur quantitates <math>p, q, r, s, &c.</math>; tum erit <math>p A′
+ q A″ + r A′″ + sA″″ + &c. = B.</math></p>
<p>Cor. Quoniam <math>y = Ax^n + B x^n+m + Cx^n+2m + &c.</math>, & exinde <math>ax^x-1 y
+ x^x {y^. / x^.} = (a + n) Ax^2+n-1 + (a + n + m) Bx^n+m-1 + (a + n + 2m)
Cx^n+2m-1 + &c.</math>; & $imiliter multiplicetur hæc æquatio in <math>x^β</math>, & inve-
niatur fluxio æquationis re$ultantis, & $equitur $umma $eriei re$ul-
tantis ex multiplicatione datæ $eriei in duas arithmeticas $eries: & $ic
ex datâ $eriei $ummâ acquiri pote$t $umma $eriei re$ultantis ex mul-
tiplicatione terminorum datæ $eriei in quotlibet arithmeticas $eries.</p>
<p>12. Sit $eries <math>A + Bx^n + Cx^2n + Dx^3n + &c. = P</math>; tum erit <math>{1 / α} A +
{1 / α + n}Bx^n + {1 / α + 2n}Cx^2n + .. {1 / α + mz}x^mz + &c. = {1 / α}(P - {1 / x^α}$.x^2 P^.)</math>;
ducatur hæc æquatio in <math>x^β-1 x^.</math>, & inveniatur ejus fluens, quæ erit
<math>{1 / α · β}A + {1 / α + n · β + n}Bx^n + {1 / α + 2n · β + 2n}Cx^2n + &c. = {1 / αβ}
P - {1 / α} × {1 / β - α}x^-2 $. x^2 P^. - {1 / β} × {1 / α - β} x^-β $. x^β P^.</math>; & in genere erit
$umma $eriei <math>{A / αβγδ&c.} + {Bx^n / α + n · β + n · γ + n · δ + n · &c.} +
{Cx^2n / α + 2n · β + 2n · γ + 2n · δ + 2n} + {Dx^3n / α + 3n · β + 3n · γ + 3n · δ + 3n}
+ &c. = {1 / αβγδ&c.} × P - {1 / α} · {1 / β - α} · {1 / γ - α} · {1 / δ - α} · &c. × x^-x $. x^x P^.

<pb 685>[0737]<rh>SERIERUM, &c.</rh>
- {1 / β} · {1 / α - β} · {1 / γ - β} · {1 / δ - β} · &c. × x^-β $.x^β P^. - {1 / γ} · {1 / α - γ} · {1 / β - γ}.
{1 / δ - γ}. &c. × x^-γ $. x^γ P^. - {1 / δ} · {1 / α - δ} · {1 / β - δ} · {1 / γ - δ} · &c. × x^-δ $. x^δ P^.
- &c.</math></p>
<p>Hoc facile deduci pote$t ex hoc theoremate, viz. <math>{1 / α} · {1 / β - α} · {1 / γ - α}
· &c. + {1 / β} · {1 / α - β} · {1 / γ - β} · {1 / δ - β} · &c. + {1 / γ} · {1 / α - γ} · {1 / β - γ} · {1 / δ - γ}
· &c. + &c. + {1 / δ} · {1 / α - δ} · {1 / β - δ} · {1 / α - δ} · &c. + &c. = {1 / αβγδ&c.}</math>.</p>
<p>Cor. Hinc, $i $umma (<math>P</math>) $eriei, cujus generalis terminus $it <math>φ</math>, de-
tur; $umma $eriei, cujus generalis terminus e$t <math>{φ / α + z · β + z · γ + z ·
δ + z · &c.}</math>; vel magis generaliter <math>{φ × (az^m + b z^m-1 + cz^m-2 + &c.) / α + z · β + z · γ + z · δ + z · &c.}</math>
ubi <math>m</math> e$t integer numerus; pendet ex fluentibus fluxionum <math>x^α P^., x^β P^.,
x^γ P^., x^δ P^., &c.</math>; vel quod idem e$t ex $ummis $erierum <math>n x^α+n ({B / α + n}
+ {2C / α + 2n}x^n + {3D / α + 3n}x^2n + {4E / α + 4n}x^3n + &c.), nx^β+n ({1 / β + n} +
{2C / β + 2n}x^n + {3D / β + 3n}x^2n + &c.), nx^γ+n ({1 / γ + n} + {2C / γ + 2n} x^n +
{3D / γ + 3n}x^2n + &c.), nx^δ+n ({B / δ + n} + {2C / δ + 2n}x^n + {3D / γ + 3n}x^2n + &c.)</math>,
&c.; ni duæ vel plures e quantitatibus <math>α, β, γ, δ, &c.</math> $int inter $e
æquales; in quo ca$u e præcedentibus petenda e$t $umma.</p>
<p>13. Sit <math>(a + bx^n)^m = A + Bx^n + Cx^2n + &c. = P</math>; ducatur hæc
æquatio in <math>x^rn-1 x^.</math>, & inveniatur fluens æquationis re$ultantis <math>φ ×
x^rn = A′ + B′x^n + C′x^2n + &c.</math>; ducatur hæc æquatio in <math>x^sn-1 x^.</math> & in-
veniatur fluens æquationis re$ultantis; & $ic deinceps; & re$ultant

<pb 686>[0738]<rh>DE SUMMATIONE</rh>
$eries <math>{A / r · s · t · &c.} + {Bx^n / r + n · s + n · t + n · &c.} + {Cx^2n / r + 2n · s + 2n ·
t + 2n · &c.} + &c.</math>; cujus $umma exprimi pote$t in finitis terminis,
$i literæ <math>r, s, t, &c.</math> integros denotent numeros; per prædictos & cir-
circulares arcus & logarithmos, $i denotent affirmativas fractiones,
quarum denominatores non majores $unt quam 2; per prædictos &
ellipticos & hyperbolicos arcus, $i denominatores non majores $int
quam 3; &c.</p>
<p>14. Sit <math>(a + bx^n + cx^2n ... x^ln)^m = A + Bx^n + Cx^2n = P</math>, ducatur
hæc æquatio in <math>x^rn-1 x^., x^r′n-1 x^., x^r″n-1 x^., &c.</math> & inveniantur fluentes æqua-
tionum re$ultantium <math>φ = A′ + B′x^n + C′x^2n + &c.</math>; ducantur æquatio-
nes re$ultantes in <math>x^sn-1 x^., x^s′n-1 x^., x^s″n-1 x^., &c.</math>, & inveniantur fluentes
fluxionum re$ultantium, & $ic deinceps; tum, $i <math>r, r′, r″</math>, &c., <math>s, s′, s″,
&c.</math>, &c. $int integri numeri, ex <math>(l - 1)</math> fluentibus independentibus
prædictis erui po$$unt reliquæ: $i vero differentiæ inter prædictas
quantitates <math>r, r′, r″, &c.; s, s′, s″, &c.; &c.</math>; $int integri numeri, tum ex
<math>l</math> independentibus erui po$$unt reliquæ; & $ic deinceps.</p>
<p>Cor. Erunt <math>x^-β $. x^β-α-1 x^. $. x^x-1 x^. P = x^-α $. x^α-β-1 x^. $. x^β-1 x^. P, x^-γ $. x^γ-β-1 x^.
$. x^β-α-1 x^. $. x^α-1 x^. P = x^-γ $.x^γ-α-1 x^. $.x^α-β-1 x^. $. x^β-1 x^. P = x^-β $.x^β-γ-1 x^. $. x^γ-α-1 x^.
$. x^α-1 x^. = x^-β $. x^β-α-1 x^. $. x^α-γ-1 x^. $. x^γ-1 x^. = x^-α $. x^α-β-1 x^. $. x^β-γ-1 x^. $. x^γ-1 x^.
= x^-α $. x^α-γ-1 x^. $. x^γ-β-1 x^. $. x^β-1 x^.</math>, & $ic deinceps: $i modo omnes flu-
entes inter eo$dem valores quantitatis <math>x</math> contineantur; $i vero quar-
tus adjiciatur index <math>δ</math>, tum re$ultant <math>1 · 2 · 3 · 4 = 24</math> fluentes inter
$e æquales; & $ic deinceps.</p>
<h>THEOR. XLVIII.</h>
<p>Sint <math>T^m, T^n, T^r, T^s, &c.</math> termini $eriei <math>A</math>, quorum di$tantiæ a primo
vel quocunque alio $eriei termino $int re$pective <math>m, n, r, s, &c.</math> i. e.
in generali termino <math>A′</math>, pro <math>z</math> di$tantiâ a primo $eriei termino $cri-
bantur <math>z + m, z + n, z + r, z + s, &c.</math>, & re$ultent termini <math>T^m, T^n,
T^r, T^s, &c.</math> $int <math>T′^m′, T′^n′, T′^r′, T′^s′, &c.</math> termini $eriei <math>B</math>, quorum di-
$tantiæ a primo $int re$pective <math>m′, n′, r′, s′, &c.</math>; i. e. in generali

<pb 687>[0739]<rh>SERIERUM, &c.</rh>
termino <math>B′</math> pro <math>z</math> di$tantiâ a primo $eriei termino $cribantur <math>z + m′,
z + n′, z + r′, z + s′, &c.</math>, & re$ultent $eriei termini <math>T′^m′, T′^n′, T′^r′,
T′^s′, &c.</math> $int etiam <math>T″^m″, T″^n″, T″^r″, T″^s″, &c.</math> termini $eriei <math>C</math>, quo-
rum di$tantiæ a primo $int re$pective <math>m″, n″, r″, s″, &c.</math>; i. e. in
generali termino <math>C′</math> pro <math>z</math> di$tantiâ a primo $eriei termino $cribantur
<math>z + m″, z + n″, z + r″, z + s″, &c.</math>, & re$ultent termini <math>T″^m′, T″^n″,
T″^r″, T″^s″, &c.</math>; & $ic deinceps.</p>
<p>A$$umatur quantitas <math>aT^m + bT^n + cT^r + dT^s + &c. + a′T′^m′ +
b′T′^n′ + c′T′^r′ + d′T′^s′ + &c. + a″T″^m″ + b″T″^n″ + c″T″^r″ + d″T″^s″
+ &c.</math> pro termino $eriei quæ$itæ, tum ex $ummis <math>(S, S′, S″, &c.)</math>
datarum $erierum <math>A, B, C, &c.</math> acquiri pote$t $umma $eriei re$ul-
tantis quæ$itæ, erit enim <math>(a + b + c + d + &c.) × S + (a′ + b′ +
c′ + d′ + &c.) × S′ + (a″ + b″ + c″ + d″ + &c.) × S″ + &c. - a(T^0
+ T^1 + T^2 + .. T^m-1) - b(T^0 + T^1 + T^2 + .. T^n-1) - c(T^0 +
T^1 + T^2 + .. T^r-1) - d(T^0 + T^1 + T^2 + .. T^s-1) - &c. - a′(T′^0
+ T′^1 + T′^2 .. + T′^m′-1) - b′(T′^0 + T′^1 + T′^2 + T′^3 + .. + T′^n′-1)
- c′(T′^0 + T′^1 + T′^2 + .. + T′^n′-1) - d′(T′^0 + T′^1 + .. T′^s′-1)
+ &c. - a″(T″^0 + T″^1 + T″^2 + .. + T″^m″-1) - b″(T″^0 + T″^1 +
T″^2 + T″^3 + .. + T″^n″-1) - c″(T″^0 + T″^1 + T″^2 + .. + T″^r″-1)
- &c.</math></p>
<p>Sit <math>a + b + c + d + &c. = 0, & a′ + b′ + c′ + d′ + &c. = 0</math>, &
a″ + b″ + c″ + d″ + &c. = 0, &c.</math> tum in finitis prædictis terminis
exprimi pote$t $umma re$ultantis $eriei.</p>
<h>PROB. XLVI.</h>
<h it>Invenire infinitas $eries, quarum $ummœ innote$cant.</h>
<p>1. A$$umatur quæcunque quantitas pro $ummâ, reducatur ea per
divi$ionem, extractionem radicum, &c. ad $eriem convergentem, &
re$ultat $eries, cujus $umma innote$cit.</p>
<p>2. Reducatur a$$umpta quantitas ad $eriem $ecundum dimen$iones
quarumcunque perparvarum quantitatum (<math>x</math>) in eâ contentarum
progrediens, & exoriuntur $eries, quarum $ummæ innote$cunt: an-
non prædictæ $eries convergent, deduci pote$t e principiis prius tra-

<pb 688>[0740]<rh>DE SUMMATIONE</rh>
ditis; viz. ex $upponendo denominatorem & $ingulas irrationales
quantitates nihilo æquales, & deducendo minimam radicem (<math>a</math>)
quantitatis <math>x</math> in æquationibus re$ultantibus; & $i <math>x</math> minor $it quam <math>α</math>,
tum $eries a$cendens $emper converget; $i autem <math>x</math> major $it quam
maxima prædictarum radicum, tum $emper converget, $i modo $it
$eries $ecundum dimen$iones quantitatis <math>x</math> de$cendens.</p>
<p>3. A$$umantur quantitates pro $ummis, deinde reducantur quan-
titates a$$umptæ ad $eries $ecundum dimen$iones unius, duarum,
trium, &c. quantitatum, vel $ecundum qua$dam datas functiones
quarundam quantitatum progredientes; ita quidem ut $eries evadant
convergentes; & deducuntur $eries, quarum $ummæ innote$cunt.</p>
<p>4. Ducantur æquationes deductæ in fluxiones formularum, quæ
reddent fluentes re$ultantium æquationum facile integrabiles; & ex-
inde erui po$$unt $eries $ummabiles; & $ic deinceps.</p>
<p>5. Inveniantur diver$i valores quantitatum a$$umptarum per qua$-
dam functiones $ecundum dimen$iones quarundam perparvarum
quantitatum a$cendentes vel permagnarum de$cendentes; & exinde
$æpe $acilius erui po$$unt fluentes earum $ummarum, differentiarum,
&c.; $ed animadvertendum e$t, $i a$cendens $eries convergat, tum de-
$cendens diverget, ni omnes radices quantitatum a$$umptarum po$$i-
biles vel impo$$ibiles inter $e quodammodo habeantur æquales; in
quo ca$u nonnunquam convergent & nonnunquam divergent & a$-
cendens & de$cendens $eries, & in algebraicis quantitatibus utræque
plerumque evadent eædem vel una affirmativa & altera negativa &
æqualis: in fluentibus proprie correctis eadem affirmari po$$unt.</p>
<p>6. Ex theor. <math>P + ({P^. / x^.}) e + {1 / 2}({P^. / x^. ^2}) e^2 + &c.</math> & <math>P +
(({P^. / x}) e + ({P^. / y^.})0 + &c.) + &c.</math> erui po$$unt $ummæ multarum $e-
rierum.</p>
<p>Ex. 1. Erit <math>a^b+n = a^b (1 + nA + {n^2 A^2 / 1 · 2} + {n^3 A^3 / 1 · 2 · 3} + {n^4 A^4 / 1 · 2 · 3 · 4} +
&c.)</math>, ubi <math>A</math> e$t logarithmus quantitatis <math>a</math>.</p>

<pb 689>[0741]<rh>SERIERUM, &c.</rh>
<p>Ex. 2. Erit <math>√(b + a x + a n) = A^{1 / 2} + {a n / 2 A^{1 / 2}} - {1 / 4} × {a^2 n^2 / 2 A^{3 / 2}} + {1 / 4} × {3 / 2} ×
{a^3 n^3 / 2 · 3 A^{5 / 2}} - {1 / 4} × {3 / 2} × {5 / 2} × {a^5 n^5 / 2 · 3 · 4 A^{7 / 2}} + {1 / 4} × {3 / 2} · {5 / 2} · {7 / 2} × {a^7 n^7 / 2 · 3 · 4 · 5 A^{9 / 2}} - &c.,
ubi A = √(b + a x)</math>.</p>
<p>Ex. 3. Erit <math>(x + b + n)^x+n = (x + b)^x (1 + ({x / x + b} + log. (x + b))
n) + {1 / 2}n^2 (x · (x - 1) · (x + b)^x-2 + 2(x + b)^x-1 + 2 x × (x + b)^x-1 ×
log. (x + b) + (x + b)^x × log. (x + b)^2) + &c.</math></p>
<p>Cum plures variabiles in a$$umptâ quantitate contineantur, tum ex
theor. 21. 1. 3. erui po$$unt plures $eries, quarum $ummæ innote$cunt.</p>
<p>Convergentiæ harum $erierum ex principiis prius traditis dijudicari
po$$unt.</p>
<p>7. Et $ic ex a$$umptis quantitatibus per quamcunque methodum con-
tinuo inveniantur quantitates, quæ ad eas $emper propius accedunt,
& ultimo propius accedunt quam pro datâ quâvis differentiâ, tum
inveniuntur $eries, quarum $ummæ innote$cunt: hinc ex omnibus
methodis approximationes ad qua$cunque radices vel quantitates con-
tinuo deducendi prius traditis erui po$$unt $eries, quarum $ummæ
$int radices, &c. ip$æ. E. g. Sint quæcunque quantitates perparvæ, &
reducantur quæcunque a$$umptæ quantitates, ita ut progrediantur $e-
cundum dimen$iones perparvarum quantitatum, & re$ultant $eries
convergentes, quarum $ummæ $unt quantitates a$$umptæ.</p>
<p>8. Sit <math>P</math> quæcunque functio quantitatis <math>(x) = 0</math>, i. e. <math>P = 0</math>; pro
valore quantitatis <math>x</math> a$$umatur <math>a</math>, & $cribatur <math>a</math> pro <math>x</math> in functione <math>P</math>,
& re$ultet quantitas <math>A</math>; $cribatur etiam in quantitatibus <math>{P^. / x^.}, ({P^.. / x^. ^2}),
({P^... / x^. ^3}), ({P^.... / x^. ^4}), &c.; a</math> pro <math>x</math>, & re$ultent <math>π, π′, π″, π′″, &c.</math>; tum, $i
<math>π((a + z) - a) + {1 / 2} π′ ((a + z)^2 - a^2) + {1 / 2 · 3} π″ ((a + z)^3 - a^3) +
{1 / 2 · 3 · 4} ((a + z)^4 - a^4) + &c.</math> minor $it quam <math>A</math>, & inter <math>a & a + z</math>

<pb 690>[0742]<rh>DE SUMMATIONE</rh>
nulla contineatur po$$ibilis vel impo$$ibilis radix æquationis <math>({P^. / x^.})
= 0</math>, erit radix datæ æquationis major quam <math>a + z</math>.</p>
<p>Eadem principia etiam ad duas vel plures æquationes duas vel
plures incognitas quantitates habentes, etiamque ad $ucce$$ivas ap-
proximationes detegendas applicari po$$unt.</p>
<p>9. A$$umantur quæcunque quantitates vel algebraicæ vel fluxionales
vel incrementiales, reducantur eæ ad diver$as $eries $ecundum dimen-
$iones diver$arum perparvarum vel permagnarum quantitatum pro-
gredientes, & re$ultant diver$æ $eries inter $e æquales; vel reducan-
tur prædictæ quantitates in diver$as $eries, & inveniantur fluentes vel
integrales inter eo$dem valores earundem variabilium contentæ, &
re$ultant diver$æ $eries inter $e æquales.</p>
<p>10. Sint <math>P & Q</math> infinitæ $eries, quæ convergunt, cum <math>x</math> $it quæcun-
que quantitas minor quam <math>α</math>, tum etiam $eries progrediens $ecundum
dimen$iones quantitatis <math>(x) = $ · P x^m x^. $ · Q x^n x^. vel = $ · x^n′ x^. $ · x^m′ x^. &c.
× $ · P x^m x^. $ · Q x^n x^.</math> converget, cum <math>x</math> minor $it quam <math>α</math>.</p>
<p>2. Si prædictæ quantitates $emper convergant, cum <math>x</math> major $it
quam <math>α</math>, tum etiam $eries prædictæ <math>= $ · P x^n x^. × $ · Q x^n x^. &c.</math> conver-
gent, cum <math>x</math> major $it quam <math>α</math>.</p>
<p>Con$imilia etiam affirmari po$$unt de pluribus quantitatibus eju$-
dem generis <math>P, Q, R, S, &c.</math></p>
<h>THEOR. XLIX.</h>
<p>Sit data $eries $ecundum dimen$iones quantitatis <math>x</math> progrediens, &
<math>v</math> functio quantitatis <math>x</math> perparva, vel permagna; tum inveniatur
<math>x = φ: v</math>, $cribatur <math>φ: v</math> in datâ $erie pro <math>x</math>, & reducatur re$ultans
ad $eriem $ecundum dimen$iones quantitatis <math>v</math> progredientem, ita ut
exoriatur $eries convergens, & invenitur $eries cujus $umma innote$cit.</p>
<p>Sunt ca$us, in quibus $eries re$ultans nunquam converget; e. g. $it
$eries <math>x - {1 / 2} x^2 + {1 / 3} x^3 - {1 / 4} x^4 + &c., & √(1 - x^2) = v</math> perparva
quantitas, & exinde <math>x = √(1 - v^2)</math>, $cribatur hæc quantitas pro <math>x</math>
in datâ $erie, & re$ultat $eries <math>(1 - v^2)^{1 / 2} - {1 / 2}(1 - v^2)^{2 / 2} + {1 / 3}(1 - v^2)^{3 / 2}

<pb 691>[0743]<rh>SERIERUM, &c.</rh>
- &c. = (1 - {1 / 2} + {1 / 3} - {1 / 4} + &c.) - {1 / 2} (1 - 1 + 1 - 1 + 1 - &c.)
v^2 + {1 / 8} (1 × (1 - 2) - 1 × (2 - 2) + 1 × 1 - 1 × 2 + 1 × 3 - &c.)
v^4 + &c.</math>; quæ $eries nunquam converget, nam <math>x - {1 / 2} x^2 + {1 / 3} x^3 -
&c. = $ · {x^. / 1 + x} = {- v v^. / √(1 - v^2) × (1 + √(1 - v^2))}</math>; $ed radix æqua-
tionis <math>1 + √(1 - v^2) = 0</math> erit <math>v = 0</math>, ergo $eries $ecundum dimen$i-
ones quantitatis <math>v</math> progrediens nunquam converget, cum <math>v</math> $it finita
quantitas.</p>
<h>THEOR. L.</h>
<p>Erit <math>{u / u + 1} = {u^. / u} - {u^. / u^2} + {u^. / u^3} - {u^. / u^4} + &c.</math>, unde <math>log. (u + 1) - log.
u = {1 / u} - {1 / 2 u^2} + {1 / 3 u^3} - &c.; at log. (1 + u) = $ · {u^. / 1 + u} = u - {1 / 2} u^2
+ {1 / 3} u^3 - &c.</math>, unde <math>log. u = u - u^-1 - {u^2 - u^-2 / 2} + {u^3 - u^-3 / 3} - &c.</math>:
hæc $eries nunquam converget, cum <math>u</math> $it po$$ibilis quantitas: $i au-
tem <math>u</math> $it impo$$ibilis <math>a + b √(- 1), & a^2 + b^2 = 1</math>, tum $eries $em-
per converget.</p>
<h>THEOR. LI.</h>
<p>Sit data æquatio relationem inter <math>P, Q, R, &c.</math> exprimens, quarum
haud duæ in $e$e ducuntur, vel $imul involvuntur; ducatur data
æquatio in <math>ρ′</math>; ubi <math>ρ′</math> e$t fluxio functionis e $ingulis quantitatibus
<math>P, Q, R, &c.</math>, $i modo fluens requiratur; vel incrementum functionis
e prædictis quantitatibus, $i modo integralis requiratur; tum inve-
niri pote$t fluens vel integralis æquationis re$ultantis.</p>
<p>Ex. Sint quantitates, quæ $int quæcunque functiones quantitatis
<math>x ± x^-1</math>; vel quantitatis, in quâ <math>x & x^-1</math> $imiliter involvuntur; tum
ea facile reduci pote$t ad functionem $inuum & co$inuum arcuum
circuli.</p>
<p>Erunt <math>{(x + √(x^2 - 1))^{n / m} - (x - √(x^2 - 1))^{n / m} / 2 √(- 1)} & {(x + √(x^2 - 1))^{n / m}

<pb 692>[0744]<rh>DE SUMMATIONE</rh>
+ (x - √(x^2 - 1))^{n / m} /2}</math> $inus & co$inus arcus circuli, qui erit ad arcum
cujus $inus e$t <math>√(1 - x^2)</math> & co$inus <math>x:: n: m</math>. Si quantitates in præ-
dictâ functione $int <math>z^{n / m} - z^{-n / m} vel z^{n / m} + z^{-n / m}</math>, tum pro <math>z</math> $cribatur <math>x +
√(x^2 - 1)</math>, & pro <math>z^{n / m} - z^{-n / m} & z^{n / m} + z^{-n / m}</math> $cribantur <math>2 √(- 1) s^{n / m}</math> &
<math>2 c^{n / m}</math>, ubi <math>s^{n / m}</math> e$t $inus arcus <math>{n / m} A & c^{n / m}</math> co$inus eju$dem arcus, & co$inus
arcus <math>A</math> e$t <math>x</math> & radius 1.</p>
<p>Cor. Hinc, $i in æquatione <math>u - u^-1 - {u^2 - u^-2 / 2} + {u^3 - u^-3 / 3} - &c.
= $ · {u^. / u}</math> pro <math>u</math> $cribatur <math>x + √(x^2 - 1)</math>, re$ultat <math>s - {s′ / 2} + {s″ / 3} - &c. =
$ · {x^. / √(-1) · √(x^2 - 1)} = $ · {x^. / √(1 - x^2)} = z</math> arcui, cujus radius e$t <math>1</math>
& co$in. <math>= x</math>.</p>
<p>2. Denotentur $inus & co$inus arcus <math>q x</math> per <math>s^(q) & c^(q)</math> re$pective, tum
erunt <math>$ · {s^(q) x^. / √(1 - x^2)}</math> & <math>$ · {c^(q) x^. / √(1 - x^2)}</math> re$pective <math>{c^(q) / q} & {s^(q) / q}</math>; nam $it
$inus <math>s^(q) = s′</math>, tum erit <math>{s^.′ / √(1 - s′^2)} = {s^.′ / c^(q)} = {q x^. / √(1 - x^2)}</math> & exinde <math>{s′ / q} =
$ · {c^(q) x^. / √(1 - x^2)}</math>; & ad eundem modum inveniri pote$t <math>{c^(q) / q} = $ · {s^(q) x^. / √(1 - x^2)}</math>.</p>
<p>Hoc etiam verum e$t, cum <math>q</math> $it fractio.</p>
<p>3. Fluens fluxionis <math>(s^(q) ^m × c^(q) ^n × {q x^. / √(1 - x^2)}) = {1 / m + 1}s^(q) ^m+1 c^(q) ^n-1 - {1 / m + 1}
× {n - 1 / m + 3}s^(q) ^m+3 × c^(q) ^n-3 + {1 / m + 1} × {n - 1 / m + 3} × {n - 3 / m + 5}s^(q) ^m+5 c^(q) ^n-5 - {1 / m + 1} ×
{n - 1 / m + 3} × {n - 3 / m + 5} × {n - 5 / m + 7}s^(q) ^m+7 c^(q) ^n-7 + {1 / m + 1} × {n - 1 / m + 3} × {n - 3 / m + 5} × {n - 5 / m + 7}

<pb 693>[0745]<rh>SERIERUM, &c.</rh>
× {n - 7 / m + 9} × s^(q) ^m+9 × c^(q) ^n-9 - &c.</math>; hæc $eries $emper terminat, cum <math>n</math> $it
impar numerus: vel fluens prædicta erit <math>{1 / n + 1} c^(q) ^n+1 s^(q) ^m-1 - {1 / n + 1} ·
{m - 1 / n + 3} c^(q) ^n+3 × s^(q) ^m-3 + {1 / n + 1} × {m - 1 / n + 3} × {m - 3 / n + 5} × c^(q) ^n+5 s^m-5 - {1 / n + 1} ×
{m - 1 / n + 3} · {m - 3 / n + 5} · {m - 5 / n + 7} × c^(q) ^n+7 s^m-7 + &c.</math>, quæ $emper terminat cum
<math>m</math> $it impar numerus.</p>
<p>4. Ducatur prædicta æquatio <math>s - {s′ / 2} + {s″ / 3} - {s′″ / 4} + &c. = z</math> in
<math>{x^. / √(1 - x^2)}</math> & inveniatur fluens æquationis re$ultantis, deinde ducatur
æquatio re$ultans in <math>{x^. / √(1 - x^2)}</math> & inveniatur fluens æquationis re$ul-
tantis; & $ic deinceps; & re$ultant $eries, quarum $ummæ innote$cunt.</p>
<p>5. Fluens fluxionis <math>({u^m-1 u^. / (1 + u^n)^e}) = {u^m / m} - {e u^m+n / m + n} + e · {e + 1 / 2} · {u^m+2n / m + 2 n}
&c. - M′ vel = - {u^m-en / e n - m} + {e u^m-en-n / e n - m + n} - e × {e + 1 / 2} × {u^m-en-2n / e n + 2 n - m}
&c. + M″</math>, ubi <math>M′ = {1 / m} - {e / m + n} + e · {e + 1 / 2} × {1 / m + 2 n}</math> &c., & <math>M″
= {1 / e n - m} - {e / e n + n - m} + e · {e + 1 / 2} · {1 / e n + 2 n - m} &c.</math>; quæ erunt
fluentes fluxionum <math>{u^m-1 u^. / (1 + u^n)^e} & {u^en-m-1 u^. / (1 + u^n)^e}</math> inter valores <math>0 & 1</math> quantita-
tis <math>u</math> contentarum: ducatur hæc æquatio in <math>u^en-2m-1 u^.</math>, & invenientur flu-
entes flux. re$ult. <math>{u^en-m / m · e n - m} - {e u^en+n-m / m + n · e n + n - m} + e · {e + 1 / 2}
{u^en+2n-m / m + 2 n · e n + 2 n - m} &c. - {M′u^en-2m / e n - 2 m} + {M′ / e n - 2 m} = {u^-m / m · e n - m}
- {e u^-m-n / m + n · e n + n - m} + e · {e + 1 / 2} {u^-m-2n / m + 2 n · e n + 2 n - m} &c. +

<pb 694>[0746]<rh>DE SUMMATIONE</rh>
{M″ u^en-2m / e n - 2 m} - {M″ / e n - 2 m}</math>; ducatur hæc æquatio in <math>{u^m-{1 / 2}en / 2√(-1)}</math>, & re$ul-
tat <math>{u^{1 / 2}en - u^-{1 / 2}en / 2 √(- 1) m · e n - m} - e{u^{1 / 2}en+n - u^-{1 / 2}en-n / 2 √(- 1) · m + n · e n + n - m} +
&c. = 0</math>; & per principia prius tradita e $cribendo <math>x + √(x^2 - 1)</math>
pro <math>u</math> facile reduci pote$t data æquatio ad $eriem $inuum: ducatur
iterum iterumque reducta æquatio in <math>{x^. / √(1 - x^2)}</math>; & inveniantur
fluentes re$ultantium æquationum; & re$ultant novæ $eries, quarum
$ummæ innote$cunt.</p>
<p>6. Erit <math>(2 c^({n / 2}))^-e = ((x + √(x^2 - 1))^{n / 2} + (x - √(x^2 - 1))^{n / 2})^-e =
(x + √(x^2 - 1))^{1 / 2}en - e (x + √(x^2 - 1))^{1 / 2}en+n + &c. = (x - (√x^2
- 1))^{1 / 2}en - e (x - √(x^2 - 1))^{1 / 2}n+n + &c.</math>; & exinde <math>((x + √(x^2 - 1))^{1 / 2}en
- (x - √(x^2 - 1))^{1 / 2}en) - e ((x + √(x^2 - 1))^{1 / 2}en+n - (x - √(x^2 - 1))^{1 / 2}en+n)
+ &c. = s^{1 / 2}en - e s^{1 / 2}en+n + &c. = 0</math>; vel ducatur <math>2 (c^({n / 2}))^-e = (x +
√(x^2 - 1))^{1 / 2}en - e (x + √(x^2 - 1))^{1 / 2}en+n + &c. in (x + √(x^2 - 1))^m-{1 / 2}en
& (2 c^({n / 2}))^-e = (x - √(x^2 - 1))^{1 / 2}en - e (x - √(x^2 - 1))^{1 / 2}en+n + &c.</math>
in <math>(x - √(x^2 - 1))^m-{1 / 2}en</math>, & $ubducatur po$terior re$ultans æquatio de
priori, & dividatur per <math>2√(- 1)</math> & re$ultabit <math>{s^(m-{1 / 2}en) / (2 c^({n / 2}))^e} = s^(m) - e s^(m+n) + e
· {e + 1 / 2} s^(m+2n) - &c.</math>; ducatur hæc æquatio in <math>{x^. / √(1 - x^2)}</math>, & inve-
niatur fluens æquationis re$ultantis; deinde ducatur fluens prædicta
in <math>{x^. / √(1 - x^2)}</math> & inveniatur fluens æquationis re$ultantis; & $ic de-
inceps; & re$ultabunt $eries quarum $ummæ innote$cunt.</p>
<p>7. Erit <math>{1 / (x - √(x^2 - 1))^{1 / 2}n + y (x + √(x^2 - 1))^{1 / 2}n} = (x + √(x^2 - 1)){n / 2}

<pb 695>[0747]<rh>SERIERUM, &c.</rh>
- y (x + √(x^2 - 1))^{3n / 2} + &c., &c., {1 / (x + √(x^2 - 1))^{1 / 2}n + y (x -
√(x^2 - 1))^{1 / 2}n} = (x - √(x^2 - 1))^{1 / 2}n - y (x - √(x^2 - 1))^{3 / 2}n + &c.</math>;
$ubtrahatur alter æquatio de alterâ, & dividatur per <math>2√(- 1)</math>, re-
$ultat <math>(1 - y) × {s^({1 / 2}n) / 1 + y^2 + 2 y c^(n)} = s^({n / 2}) - y s ({3n / 2}) + &c.</math>: multiplicetur
prima æquatio per <math>(x + √(x^2 - 1))^{n / 2}</math> & $ecunda per <math>(x - √(x^2 - 1))^{n / 2}</math>
& $ubtrahatur alter de alterâ, & dividatur per <math>2 √(- 1)</math>, & re$ultat
<math>{s^(n) / 1 + y^2 + 2 y c^(n)} = s^(n) - s^(2n) y + &c.</math>: multiplicentur hæ æquationes per
<math>{x^. / √(1 - x^2)}</math> & inveniantur fluentes æquationum re$ultantium termino-
rum, & deducuntur $eries, quarum $ummæ innote$cunt: vel multipli-
cetur prima æquatio per <math>(x + √(x^2 - 1))^m-{n / 2}</math>; & $ecunda per <math>(x -
√(x^2 - 1))^m-{n / 2}</math>, & collocentur quantitates in ordine <math>(x + √(x^2 - 1))^b
± (x - √(x^2 - 1))^b = H</math>, quæ denotat <math>s^(b)</math> $inum vel co$inum arcus
<math>h z</math>, ubi <math>z</math> e$t arcus, cujus co$inus e$t <math>x</math>; vel $cribatur <math>y^q</math> pro <math>y</math>, & de-
inde multiplicentur æquationes per <math>y^p-1 y^.</math> & inveniantur fluentes, ex
hypothe$i quod <math>y</math> $olummodo $it variabilis; & con$equuntur $eries,
quarum $ummæ innote$cunt.</p>
<p>8. In $ummis $erierum prius traditarum inve$tigandis, haud nece$$a-
rio introducendi $unt $inus & co$inus arcuum <math>n z</math>, in eorum loco re-
tineantur quantitates algebraicæ <math>{(x + √(x^2 - 1))^n + (x - √(x^2 - 1))^n / 2}
= P & {(x + √(x^2 - 1))^n - (x - √(x^2 - 1))^n / 2 √(- 1)} = Q</math>.</p>
<p>A$$umatur <math>log. u = u - u^-1 - {u^2 - u^-2 / 2} + &c.</math>; ducatur hæc
æquatio in <math>{u^. / u}</math>, & re$ultat fluxio, cujus fluens e$t <math>u + u^-1 - {u^2 + u^-2 / 2 · 2}

<pb 696>[0748]<rh>DE SUMMATIONE</rh>
+ {u^3 + u^-3 / 3 · 3} - &c. = $ · log. u × {u^. / u} = {1 / 2} (log. u)^2</math>; hæc $eries ut prædi-
citur $emper divergit, ni <math>u</math> $it impo$$ibilis quantitas <math>x + √(x^2 - 1)</math>:
log. impo$$ibilis quantitatis denotat circularem arcum, &c.: & $imi-
liter erui po$$unt omnes $eries, quæ ex introductione $inuum & co-
$inuum per hanc methodum deduci po$$unt.</p>
<p>9. Sint duæ æquationes <math>A = 0 & B = 0</math>, in quibus <math>a</math> in unâ $imili-
ter involvitur ac <math>β</math> in alterâ; tum $ubtrahatur una æquatio de alterâ,
& pro <math>α & β</math> $cribantur <math>x + √(x^2 - 1) & x - √(x^2 - 1); & s^(b) & c^(b)</math> pro
<math>{(x + √(x^2 - 1))^b - (x - √(x^2 - 1))^b / 2 √(- 1)} & {(x + √(x^2 - 1))^b + (x + √(x^2 - 1))^b / 2}</math>,
& re$ultat æquatio, in quâ involvuntur $inus & co-
$inus arcuum <math>h z</math>; e. g. $int duæ æquationes <math>(a + b)^q = a^q +
q a^q-1 b + q · {q - 1 / 2} a^q-2 b^2 + q · {q - 1 / 2} · {q - 2 / 3} a^q-3 b^3 + &c. & (a + c)^q
= a^q + q a^q-1 c + q · {q - 1 / 2} a^q-2 c^2 + &c.</math>; $ubtrahatur po$terior æqua-
tio de priori; deinde pro <math>c</math> ponatur eadem functio quantitatis <math>x +
√(x^2 - 1)</math> ac <math>b</math> $it quantitatis <math>x - √(x^2 - 1)</math>; & facile transformari
pote$t æquatio re$ultans in æquationes $inus & co$inus arcuum <math>h z</math>
involventes.</p>
<p>Et $imiliter transformari po$$unt prædictæ æquationes, &c. in
æquationes, quæ $unt con$imiles functiones quantitatum <math>x + √(x^2 - 1)
& x - √(x^2 - 1)</math>; vel quarumcunque aliarum datarum quantitatum;
ducantur æquationes re$ultantes in fluxiones, quarum fluentes $unt
functiones e $ingulis prædictis quantitatibus; re$ultant $eries, quarum
$ummæ innote$cunt e fluentibus quantitatum in æquationibus con-
tentarum; & $ic deinceps.</p>

<pb 697>[0749]<rh>SERIERUM, &c.</rh>
<h>PROB. XLV.
_Invenire $ummam $eriei_ <math>x + {x^2 / 2^2} + {x^3 / 3^2} + &c. cum x = {1 / 2}</math>:</h>
<p>Erit log. <math>{1 / 1 - x} = x + {x^2 / 2} + {x^3 / 3} + &c.</math> in hâc æquatione pro <math>x</math>
$cribatur <math>{v / v - 1}</math> & re$ultat <math>log. {1 / 1 - {v / v - 1}} = log. (1 - v) = {v / v - 1} +
{v^2 / 2 (v - 1)^2} + {v^3 / 3 (v - 1)^3} + &c. = (1 - v^-1)^-1 + {(1 - v^-1)^-2 / 2} +
{(1 - v^-1)^-3 / 3} + &c.</math>; in hâc æquatione pro <math>1 - v</math> $cribatur <math>u</math> & re$ul-
tat <math>log. u = 1 - u^-1 + {(1 - u^-1)^2 / 2} + {(1 - u^-1)^3 / 3} + &c. = - (1 - u)
- {(1 - u)^2 / 2} - {(1 - u)^3 / 3} - &c.</math>, quæ prius inventa fuit <math>= u - u^-1
- {u^2 - u^-2 / 2} + {u^3 - u^-3 / 3} - &c.</math>; prima $eries converget, cum <math>1 - u^-1</math>
minor $it quam <math>± 1</math>; $ecunda cum <math>1 - u</math> $it etiam minor quam <math>± 1</math>:
$ed <math>$ · {x^. / x} log. {1 / 1 - x} = x + {x^2 / 2^2} + {x^3 / 3^2} + &c., & $ · {x^. / 1 - x} log. x = (1 - x)
+ {(1 - x)^2 / 2^2} + {(1 - x)^3 / 3^2} + &c. - (1 + {1 / 4} + {1 / 9} + {1 / 16} + &c. = A)</math>, &
exinde con$tat log. <math>x × log. {1 / 1 - x} = $ · {x^. / x} × log. {1 / 1 - x} + $ · {x^. / 1 - x}
× log. x = (x + {x^2 / 2^2} + {x^3 / 3^2} + &c.) + (1 - x + {(1 - x)^2 / 2^2} + {(1 - x)^3 / 3^2}
+ &c. - A)</math>; fingatur <math>x = {1 / 2}</math>, & re$ultat <math>log · {1 / 2} × log. 2 = 2 × ({1 / 2} +
{{1 / 4} / 2^2} + {{1 / 8} / 3^2} + {{1 / 16} / 4^2} + &c.) - (A = 1 + {1 / 4} + {1 / 9} + {1 / 16} + &c.</math>) & exinde
valor $eriei <math>x + {x^2 / 4} + {x^3 / 9} + &c. cum x = {1 / 2}</math>.</p>

<pb 698>[0750]<rh>DE SUMMATIONE</rh>
<p>Et ex con$imilibus principiis detegi pote$t $umma $eriei <math>x + {x^2 / 2^2} +
{x^3 / 3^2} + &c. = {2 a^2 / 3}, {a^2 / 3} - {1 / 2} (log. 2)^2, {4 a^2 / 15} - (log. {√(5) - 1 / 2})^2</math>, vel <math>{2 / 15} a^2
- (log. {√(5) - 1 / 2})^2</math>; prout <math>x</math> $it <math>= 1, {1 / 2}, {3 - √(5) / 2}</math> vel <math>{√(5) - 1 / 2}</math>.</p>
<h>THEOR. LII.</h>
<p>1. Erit <math>z + 1 · z + 2 · z + 3 .... n z = 1 · 2 · 3 .. (n - 1) ×
(n + 1) · n + 2 · n + 3 ... (2 n - 1) × (2 n + 1) · 2 n + 2 · 2 n + 3
.. (3 n - 1) × (3 n + 1) · 3 n + 2 · 3 n + 3 ... (4 n - 1) × (4 n + 1)
· 4 n + 2 · 4 n + 3 &c. ad (n z - 1) × n^z</math>; nam <math>{1 · 2 · 3 .. n z / 1 · 2 · 3 .. z} = z + 1
· z + 2 · z + 3 .. n z = {1 · 2 · 3 .. n z / n · 2 n · 3 n .. n z} × n^z = 1 · 2 · 3 · n - 1 · n + 1
· n + 2 .. 2 n - 1 · 2 n + 1 · 2 n + 2 .. 3 n - 1 · 3 n + 1 ... (n z - 1)
× n^z</math>; $it <math>n = 2</math> & evadet <math>z + 1 · z + 2 · z + 3 ... n z = 1 · 3 · 5 · 7
... (2 z - 1) × 2^z</math>.</p>
<p>2. Sit <math>z</math> infinita quantitas & erit <math>z^z: (z - n)^z: 1: 1 - n + {n^2 / 2} -
{n^3 / 2 · 3} + {n^4 / 2 · 3 · 4} - &c., & z^z: (z + n)^z:: 1: 1 + n + {n^2 / 2} + {n^3 / 1 · 2 · 3}
+ {n^4 / 1 · 2 · 3 · 4} + &c.</math> unde tres quantitates <math>(z - n)^z, z^z & (z + n)^n</math>
erunt ultimo in ratione quantitatum <math>α, 1 & β</math>; ubi <math>α & β</math> $unt nu-
meri, qui corre$pondeant logarithmis <math>n & - n</math>. Erunt <math>z^z: (z + n)^z
:: 1: 1 + n + {1 / 2} n^2 + {1 / 6} n^3 + &c.</math> etiamque <math>(z + n)^z: (z + n)^z+n:: 1:
(z + n)^n:: 1: z^n</math> & con$equenter <math>z^z: (z + n)^z+n: 1: (1 + n + {1 / 2} n^2 +
{1 / 6} n^3 + &c.) z^n</math>.</p>
<p>Et $ic facile deduci po$$unt innumeræ propo$itiones huju$modi,
quæ o$tendunt limites, ad quos continuo vergunt datæ algebraicæ vel
exponentiales quantitates, u$que donec <math>z</math> evadat infinita.</p>

<pb 699>[0751]<rh>SERIERUM, &c.</rh>
<p>3. In genere, $i dentur quæcunque irrationales quantitates, earum
rationes inter $e ad infinitam di$tantiam detegere.</p>
<p>Primo rejiciantur omnes quantitates in iis, quæ ad infinitam di-
$tantiam infinite minores $int quam reliquæ, i. e. nihil valent per
extractionem radicum, &c. deinde reducantur reliquæ in $eries con-
vergentes, & ex iis deduci po$$unt relationes ad infinitam di$tantiam
quæ$itæ.</p>
<p>1. 1. Datâ quantitate <math>A</math> involvente alias <math>z, &c.</math> infinite magnas; in-
venire annon <math>A</math> finita $it: $i in finito numero factorum contineatur
<math>A</math>; tum ex ejus reductione ad $eries $ecundum dimen$iones quanti-
tatis <math>z</math> progredientes con$tat, annon quantitas <math>A</math> $it finita.</p>
<p>2. Sint datæ æquationes, quæ vere con$tituuntur, cum <math>z</math> $it quæ-
cunque finita quantitas; tum etiam vere con$tituentur, cum <math>z</math> evadat
infinita: & $i in datis æquationibus pro <math>z</math> $cribatur <math>z + n vel n z, &c.</math>,
veræ erunt æquationes re$ultantes: e. g. a$$umatur <math>1^2 . 2^2 . 3^2 .. z^2
× {e^2z / z^2z+1} = 2 π</math>, & exinde <math>1^2 . 2^2 .. z^2 × (z + 1)^2 × {e^2z+2 / (z + 1)^2z+3} = 2 π</math>, unde
<math>2 π × (z + 1)^2 × {e^2 × z^2z+1 / (z + 1)^2z+3} = 2 π</math>, & <math>e^2 × {z^2z+1 / (z + 1)^2z+1} = 1</math>, & con-
$equenter <math>{(z + 1)^2z+1 / z^2z+1} = 1 + 2 + {4 / 1 · 2} + {8 / 2 · 3} + {16 / 2 · 3 · 4} + {32 / 2 · 3 · 4 · 5}
+ {64 / 2 · 3 · 4 · 5 · 6} + &c. = e^2</math>, aliter æquatio a$$umpta non pote$t e$$e
vera.</p>
<p>3. Augeatur infinita quantitas (<math>z</math>) per (<math>n</math>) finitas quantitates <math>a, b,
c, &c.</math>; & $int quantitates re$ultantes per earum $ub$titutiones pro <math>z</math>
in datâ quantitate <math>A, B, C, D, &c.</math>; tum erit <math>α A + β B + γ C + δ D
+ &c. = 0</math>, cum <math>α + β + γ + δ + &c. = 0, & A</math> $it finita: e. g. $it
præcedens æquatio <math>1^2 · 2^2 · 3^2 · 4^2 .. z^2 {e^2z / z^2z+1} = 2 π</math>, pro <math>z</math> $cribatur <math>z + 1</math>
& re$ultat <math>1^2 · 2^2 · 3^2 .. z^2 × {e^2z+2 / (z + 1)^2z+1} = 2 π</math>, & exinde <math>1^2 · 2^2 · 3^2 ..

<pb 700>[0752]<rh>DE SUMMATIONE</rh>
z^2 × e^2z × ({e^2 / (z + 1)^2z+1} - {1 / z^2z+1}) = - 1^2 · 2^2 · 3^2 .. z^2 × e^2z ((1 + 2 +
{4 / 1 · 2} + {8 / 1 · 2 · 3} + &c. - e^2) z^-2z-1 - &c.)</math>, at $it <math>(1 + 2 + {4 / 2} +
{8 / 1 · 2 · 3} + &c.) - e^2 = 0</math>, & $ub$equentes quantitates erunt infinite
parvæ, cum <math>z</math> evadat infinita: ergo con$tat exemplum.</p>
<p>4. Inveniantur $ucce$$ivi termini datæ quantitatis, & per principia
prius tradita dijudicari pote$t, utrum $eries $it finita necne.</p>
<p>5. Sit fluxio <math>(a + b x^n)^m x^rn-1 x^.</math>, inveniatur fluens (<math>A</math>) huju$ce flu-
xionis inter valores <math>0 & -{a / b}</math> quantitatis <math>x^n</math> po$ita; deinde per prob.
23. inveniatur fluens (<math>P</math>) fluxionis <math>(a + b x^n)^m±h x^(r±i)n-1 x^.</math> inter
prædictos valores, ubi <math>h & i</math> $unt infinitæ quantitates; vel altera
infinita, altera vero $inita vel nulla: per eandem methodum inve-
niatur fluens (<math>B</math>) fluxionis <math>(a + b x^n)^s x^tn-1 x^.</math> inter eo$dem valores
po$ita & exinde per prædictum prob. inveniatur fluens (<math>Q</math>) fluxionis
<math>(a + b x^n)^s±h x^(t±i)n-1 x^.</math>; & ex æquando fluentes <math>P & Q</math> con$equitur
relatio inter <math>A & B</math> quæ involvit quantitates infinite magnas: du-
cantur plures huju$modi quantitates in $e$e, vel inveniantur fluen-
tes prædictarum fluxionum ex diver$is modis; & ad eas applicentur
con$imilia principia; & re$ultant æquationes involventes quantitates
infinite magnas, vel potius limites ad quos quantitates appropin-
quant prius quam quædam quantitates in iis contentæ evadant in$i-
nite magnæ.</p>
<p>6. Erit <math>1^2 × 2^2 × 3^2 ... z^2 × {e^2z / z^2z+1} = 2 π</math> peripheria circuli, cujus ra-
dius e$t <math>1: & (1 - m^2) × (2^2 - m^2) × (3^2 - m^2) ... (z^2 - m^2) ×
{e^2z / z^2z+1} = {m / 2} × $in. arc. m S</math>; etiamque <math>(z + m) × (z + m + 1) × (z + m
+ 2) ... (2 z + m) × {e^z / 2^2z z^z} = 2^m-{1 / 2}</math>.</p>

<pb 701>[0753]<rh>SERIERUM, &c.</rh>
<h>PROB. XLVI.</h>
<p it>Datâ $erie <math>(P) a + b x^n + c x^2n + d x^3n + &c. .. D x^(m-3)n +
C x^(m-2)n + B x^(m-1)n + A x^mn + &c.</math> in infinitum progrediente, ubi <math>a, b,
c, d, &c. D, C, B, A, &c.</math> $int quæcunque datæ functiones quantitatis <math>z</math>
di$tantiæ a primo $eriei termino; invenire legem, quam ob$ervant cceffi-
cientes ejus <math>(P)</math> quadrati, &c.</p>
<p>A$$umatur <math>A</math> coefficiens generalis termini cuju$cunque <math>x^mn</math>, tum
erit <math>2 (a A + b B + c C + &c.)</math> coefficiens termini <math>x^mn</math> $eriei, quæ $it
quadratum $eriei <math>P</math>: $int vero <math>l & L</math> coefficientes quorumcunque
terminorum <math>x^rn & x^(m-r)n</math>; tum erit <math>L l</math> generalis terminus $eriei <math>a A +
b B + c C + &c.</math> Ex hoc vero generali termino dato deducatur
$umma $eriei in infinitum, quæ correcta præbet $eriei $ummam <math>a A +
b B + c C + &c.</math> i. e. generalem terminum $eriei <math>(P^2) = a^2 + 2 a b x^2
+ &c.</math></p>
<p>2. Ex ii$dem principiis etiam inveniri pote$t generalis terminus
$eriei, quæ $it <math>P^s = a^s + s a^s-1 b x^2 + &c.</math> $i modo $ummæ $erierum
exinde ortarum in finitis terminis exprimi po$$int: in his autem
$eriebus continentur una, duæ vel plures variabiles quantitates;
e. g. $it <math>s = 2</math>, & una $olummodo in $erie re$ultante continetur va-
riabilis quantitas; $it <math>s = 3</math>, & una vel duæ variabiles quantitates in
$eriebus re$ultantibus continentur; $it <math>s = 4</math>, & una vel duæ vel tres
variabiles quantitates in $eriebus re$ultantibus continentur; & $ic
deinceps. Facile con$tant generales termini ad has $eries, quæcun-
que $it pote$tas <math>s</math>. Et $ic de lege cuju$cunque functionis huju$ce $e-
riei vel plurium $erierum inve$tigandâ.</p>
<h>NOVA METHODUS DIFFERENTIARUM.
THEOR. LIII.</h>
<p>Sit data $eries in omni parte celeriter convergens <math>a x + b x^2 + c x^3
+ d x^4 + e x^5 + &c.</math> in eâ pro <math>x</math> $cribantur $ucce$$ive <math>p, 2 p, 3 p, 4 p,

<pb 702>[0754]<rh>DE SUMMATIONE</rh>
5 p, &c.</math>; quantitates re$ultantes erunt <math>a p + b p^2 + c p^3 + &c. = S^1,
2 a p + 4 b p^2 + 8 c p^3 + &c. = S^2, 3 a p + 9 b p^2 + 27 c p^3 + &c. =
S^3, 4 a p + 16 b p^2 + 64 c p^3 + &c. = S^4, &c.</math>; & dentur prædictæ
$ummæ <math>S^1, S^2, S^3, S^4 & S^5</math>; invenire $ummam <math>S^6</math> prope.</p>
<p>Fingatur <math>α S^1 + β S^2 + γ S^3 + δ S^4 + ε S^5 = α (a p + b p^2 + c p^3 +
&c.) + β (2 a p + 4 b p^2 + 8 c p^3 + &c.) + γ (3 a p + 9 b p^2 + 27 c p^3
+ &c.) + δ(4 a p + 16 b p^2 + 64 c p^3 + &c.) + &c. = (α + 2 β +
3 γ + 4 δ + &c.) a p + (α + 4 β + 9 γ + 16 δ + &c.) b p^2 + (α+
8 β + 27 γ + 64 δ + &c.) c p^3 + (α + 16 β + 81 γ + 256 δ + &c.)
d p^4 + &c. = S^6 = 6 a p + 36 b p^2 + 216 c p^3 + 1296 d p^4 + &c.</math></p>
<p>Fiant corre$pondentes termini huju$ce æquationis re$pective inter
$e æquales, & exinde re$ultant æquationes</p>
<tb>
α+2β+3γ+4δ+5ε=6 # Differentiæ # 6γ+24δ+60ε=120=μ-2λ=π
α+4β+9γ+16δ+25ε=36 # 2β+6γ+12δ+20ε=30=λ # 18γ+96δ+300ε=720=ν-2μ=ρ
α+8β+27γ+64δ+125ε=216 # 4β+18γ+48δ+100ε=180=μ # 54γ+384δ+1500ε=430=ξ-2ν=σ
α+16β+81γ+256δ+625ε=1296 # 8β+54γ+192δ+500ε=1080=ν # 24δ+120ε=360=ρ-3π=τ
α+32β+243γ+1024δ+3125ε=7776 # 16β+162γ+768δ+2500ε=6480=ξ # 96δ+600ε=2160=σ-3ρ=υ
 # # 120ι=720=υ-4τ
</tb>
<p>unde <math>ε = 6, δ = - 15, γ = 20, β = - 15, α = 6</math>; & con$equenter
<math>S^6 = 6 S^5 - 15 S^4 + 20 S^3 - 15 S^2 + 6 S^1</math> prope; & in genere erit <math>S^n =
n S^n-1 - n · {n - 1 / 2} S^n-2 + n · {n - 1 / 2} · {n - 2 / 3} S^n-3 - n · {n - 1 / 2} · {n - 2 / 3}
{n - 3 / 4} S^n-4 .... ± n. {n - 1 / 2} · {n - 2 / 3} S^3 ∓ n · {n - 1 / 2} S^2 ± n S^1</math> prope;
ubi coefficientes inveniuntur eædem ac in $erie <math>- e^n + n e^n-1 f - n ·
{n - 1 / 2} e^n-2 f^2 + &c. = - (e - f)^n</math>, $i modo primus & ultimus termi-
nus abjiciantur.</p>
<p>Cor. Sit <math>n = 5</math>, re$ultat <math>S^5 = 5 S^4 - 10 S^3 + 10 S^2 - 5 S^1 =
5 (S^4 - S^1) - 10 (S^3 - S^2)</math>; $int <math>S^1, S^2, S^3 & S^4</math> logarithmi rationum
<math>r: r + p, r: r + 2 p, r: r + 3 p, r: r + 4 p</math> re$pective; tum erit <math>S^5</math>
logar. rationis <math>r: r + 5 p</math> prope: in genere erit <math>S^n = n (S^n-1 ± S^1) -
n · {n - 1 / 2} (S^n-2 ± S^2) + n · {n - 1 / 2} · {n - 2 / 3} (S^n-3 ± S^3) - &c.</math></p>

<pb 703>[0755]<rh>SERIERUM, &c.</rh>
<p>1. 2. Ii$dem literis ea$dem quantitates denotantibus, erit <math>S^m+n =
{m + n · n + m - 1 · m + n - 2 ... m + 2 / 1 · 2 · 3 ·, · n - 1} × S^n-1 - {n - 1 / 1} A × {m + 1 / m + 2}
S^n-2 + {n - 2 / 2} × B × {m + 2 / m + 3} S^n-3 - {n - 3 / 3} × C × {m + 3 / m + 4} S^n-4 + {n - 4 / 4} ×
D × {m + 4 / m + 5} S^n-5 - {n - 5 / 5} × E × {m + 5 / m - 6} × S^n-6 + &c.</math> u$que ad termi-
num S<_>1</_> prope, ubi literæ <math>A, B, C, D, E, &c.</math> præcedentes coefficientes
re$pective denotant, & <math>m</math> e$t integer numerus.</p>
<h>THEOR. LIV.</h>
<p>1. Si in præcedente $erie $cribantur <math>+ p, - p, 2 p, - 2 p, 3 p, - 3 p
.... n p, - n p</math>; exorientur quantitates <math>a p + b p^2 + c p^3 + d p^4 + &c.
= S^1, - a p + b p^2 - c p^3 + d p^4 - &c. = S^-1, 2 a p + 4 b p + 8 c p^3 +
&c. = S^2, - 2 a p + 4 b p^2 - 8 c p^3 + &c. = S^-2, 3 a p + 9 b p^2 + 27 c p^3 +
&c. = S^3, - 3 a p + 9 b p^2 - 27 c p^3 + &c. = S^-3, &c.</math> tum erit $umma
<math>S^m = {m · (m^2 - 1) · (m^2 - 4) · (m^2 - 9) · (m^2 - 16) ... (m^2 - (n - 1)^2) · (m - n) / n · (n^2 - 1) · (n^2 - 4) · (n^2 - 9) ... (n^2 - (n - 1)^2) × 2 n = 1 · 2 · 3 · 4 ... 2 n}
S^-n + A × {m + n / m - n} S^+n - {2 n / 1} × B × {m - n / m + (n - 1)} S^-n+1 - C × {m + (n - 1) / m - (n - 1)}
S^n-1 + {2 n - 1 / 2} D × {m - (n - 1) / m + (n - 2)} S^-n+2 + {m + (n - 2) / m - (n - 2)} E S^n-2 - {2 n - 2 / 3}
F × {m - (n - 2) / m + (n - 3)} S^-n+3 - G × {m + (n - 3) / m - (n - 3)} S^n-3 + &c.</math> prope; ubi
literæ <math>A, B, C, D, &c.</math> præcedentes coefficientes re$pective denotant:
generaliter vero coefficientes terminorum <math>S^-n+s & S^n-s</math> erunt re$pec.
tive <math>{2 n - s + 1 / s} × L × {m - (n - s + 1 / m + (n - s)} & M × {m + (n - s) / m - (n - s)}</math>, ubi li-
teræ <math>L & M</math> earum præcedentes coefficientes re$pective denotant, quæ
coefficientes affirmativæ vel negativæ a$$umendæ $unt, prout <math>s</math> $it par
vel impar numerus.</p>

<pb 704>[0756]<rh>DE SUMMATIONE</rh>
<p>2. Si pro <math>x</math> in prædictâ $erie $cribantur <math>p, - p, 2 p, - 2 p, 3 p,
- 3 p ..... (n - 1) p, - (n - 1) p, n p</math> re$pective; tum erit <math>S^m =
{m · (m^2 - 1) · (m^2 - 4) · (m^2 - 9) ... (m^2 - (n - 1)^2) / 1 · 2 · 3 · 4 ... (2 n - 1)} S^n - A ×
{m - n / m + (n - 1)} S^-n+1 - {2 n - 1 / 1} × B × {m + (n - 1) / m - (n - 1)} S^n-1 + C × {m - (n - 1) / m + (n - 2)}
S^-n+2 + {2 n - 2 / 2} × D × {m + (n - 2) / m - (n - 2)} S^n-2 - &c.</math> prope, ubi <math>A, B, C, &c.</math>
præcedentes coefficientes ut antea denotant, <math>&</math> generaliter coefficien-
tes terminorum <math>S^-n+s & S^n-s</math> erunt re$pective <math>L × {m - (n - s + 1) / m + n - s} &
{2 n - s / s} × M × {m + (n - s) / m - (n - s)}; L & M</math> earum præcedentes coefficientes
re$pective denotantibus, quæ autem coefficientes affumendæ $unt af-
firmativæ vel negativæ, prout <math>s</math> $it par vel impar numerus.</p>
<p>Cor. 1. Si <math>m</math> $it negativa quantitas; tum in hi$ce $eriebus pro <math>m</math>
$eribenda e$t <math>- m</math>, & re$ultat quantitas quæ$ita.</p>
<p>Cor. 2. Cum numerus terminorum <math>S^1, S^2, S^3, &c.</math> detur, tum hæc
methodus plerumque magis converget quam ea in theor. præcedente
tradita; & con$imilis propo$itio etiam vera e$t de $eriebus in theor.
$ub$eq. contentis.</p>
<h>THEOR. LV.</h>
<p>In $erie prædictâ <math>a x + b x^2 + c x^3 + d x^4 + &c.</math> pro <math>x</math> $cribantur
re$pective <math>p, q, r, s, t, &c. & m</math>; & dicantur $ummæ $erierum re$ul-
tantium re$pective <math>S^p, S^q, S^r, S^s, &c., & S^m</math>; tum erit
<math>S^m = {m × (m - q) × (m - r) × (m - s) × (m - t) × &c. / p × (p - q) × (p - r) × (p - s) × (p - t) × &c.} S^p +
{m · (m - p) × (m - r) × (m - s) × (m - t) × &c. / q · (q - p) · (q - r) · (q - s) · (q - t) × &c.} × S^q +
{m · (m - p) · (m - q) · (m - s) · (m - t) × &c. / r · (r - p) · (r - q) · (r - s) · (r - t) × &c.}
× S^r + &c. = a m + b m^2 + c m^3 + d m^4 + &c.</math> prope; hìc vero anim-

<pb 705>[0757]<rh>SERIERUM, &c.</rh>
advertendum e$t coefficientem cuju$cunque termini <math>S^π</math> e$$e
<math>{m · (m - p) · (m - q) · (m - r) · (m - s) · &c. / π · (π - p) · (π - q) · (π - r) · (π - s) · &c.}</math>, in cujus numeratore haud conti-
netur factor <math>m - π</math>.</p>
<p>Cor. Hinc facile deduci po$$unt quædam arithmeticæ propo$itio-
nes, e. g. $int quæcunque quantitates <math>p, q, r, s, t, &c.; & m</math>; tum per
theorema <math>{m · (m - q) · (m - r) · &c. / p · (p - q) · (p - r) · &c.} × S^p + {m · (m - p) · (m - r) · &c. / q · (q - p) · (q - r) · &c.}
S^q + {m · (m - p) · (m - q) · &c. / r · (r - p) · (r - q) · &c.} S^r + &c. = a m + b m^2 + c m^3 +
&c.</math> prope; unde <math>{m · (m - q) · (m - r) · &c. / p · (p - q) · (p - r) · &c.} × (a p + b p^2 + c p^3 +
&c.) + {m · (m - p) · (m - r) · &c. / q · (q - p) · (q - r) · &c.} × (a q + b q^2 + c q^3 + &c.) +
{m · (m - p) · (m - q) · &c. / r · (r - p) · (r - q) · &c.} × (a r + b r^2 + c r^3 + &c.) + &c. = a m
+ b m^2 + c^3 + &c.</math>, & con$equenter ex æquatis inter $e corre$pon-
dentibus terminis <math>{m · (m - q) · (m - r) · (m - s) · &c. / p (p - q) · (p - r) · (p - s) · &c.} (A) p +
{m · (m - p) · (m - r) · (m - s) · &c. / q (q - p) · (q - r) · (q - s) · &c.} (B) q + {m · (m - p) · (m - q) · (m - s) · &c. / r (r - p) · (r - q) · (r - s) · &c.}
(C) r + &c. = p A + q B + r C + &c. = m</math>; & in ge-
nere <math>p^i A + q^i B + r^i C + &c. = m^i</math>; $i modo <math>i</math> $it integer <math>&</math> non ma-
jor quam numerus quantitatum <math>p, q, r, s, t, &c.</math></p>
<p>Summa fractionum <math>{p / α - β · α - γ · α - δ · α - ε · &c.} +
{Q / β - α · β - γ · β - δ · β - ε · &c.} + {R / γ - α · γ - β · γ - δ · γ - ε · &c.}
+ {S / δ - α · δ - β · δ - γ · δ - ε · &c.} + &c. = π</math>; ubi <math>P, Q, R, S, &c.</math>
$unt rationales <math>&</math> integrales functiones quantitatum <math>α, β, γ, δ, &c.</math>;

<pb 706>[0758]<rh>DE SUMMATIONE</rh>
i. e. $unctiones, in quibus nullus negativus vel fractionalis index
continetur: <math>&</math> in quantitate <math>P</math> $imiliter involvuntur omnes quanti-
titates <math>β, γ, δ, &c.</math> præter unam <math>α</math>; & $ic in quantitatibus <math>Q, R, S, &c.</math>
$imiliter involvuntur omnes prædictæ quantitates præter unam <math>β, γ,
δ, &c.</math> re$pective: <math>&</math> in quantitate <math>P</math> $imiliter involvuntur quantitates
<math>β, γ, δ, ε, &c.</math>; ac quantitates <math>α, γ, δ, ε, &c.</math> in <math>Q; & α, β, δ, ε, &c.</math> in
<math>R; α, β, γ, ε, &c.</math> in <math>S, &c.</math>: & ultimo functio quantitatis <math>α</math> in <math>P</math> ea-
dem e$t ac functio quantitatis <math>β</math> in <math>Q</math>, & utraque eadem ac functio
quantitatis <math>γ</math> in <math>C</math>, & $ic deinceps: tum, $i numerus literarum <math>α, β,
γ, δ, &c.</math> $it <math>n</math>; & dimen$iones literarum <math>α, β, γ, &c.</math> in quantitatibus
<math>P, Q, R, S, &c.</math> contentarum $int <math>m, m - 1, m - 2, &c.</math>; ubi <math>n & m</math>
$unt integri numeri; erit <math>π</math> functio rationalis <math>&</math> integralis quantita-
tum <math>α, β, γ, δ, &c.</math>, in quâ $imiliter involvuntur quantitates <math>α, β, γ,
δ, &c.</math>; cujus dimen$iones quantitatum in eâ contentarum haud ex-
$uperant <math>m - n</math>; $i <math>m</math> $it minor quam <math>n</math>, tum erit <math>π = 0</math>.</p>
<p>Facile ex hi$ce vel pluribus $implicibus æquationibus erui po$$unt
plures huju$modi propo$itiones.</p>
<h>THEOR. LVI.</h>
<p>Sit $eries celeriter convergens <math>a x^b + b x^b+k + c x^b+2k + d x^b+3k + &c.</math>,
in eâ pro <math>x</math> $cribantur re$pective <math>p, q, r, s, t, &c. π</math>, & <math>m</math>; & dicantur
$ummæ exinde re$ultantes re$pective <math>S^p, S^q, S^r, S^s, &c. & S^m</math>; tum erit
<math>S^m = {m^b × (m^k - q^k) × (m^k - r^k) × (m^k - s^k) × &c. / p^b (p^k - q^k) × (p^k - r^k) × (p^k - s^k) × &c.} ×
S^p + {m^b × (m^k - p^k) × (m^k - r^k) × (m^k - s^k) × &c. / q^b (q^k - p^k) × (q^k - r^k) × (q^k - s^k) × &c.}
× S^q + {m^b × (m^k - p^k) × (m^k - q^k) × (m^k - s^k) × &c. / r^b
(r^k - p^k) × (r^k - q^k) × (r^k - s^k) × &c.}
S^r + &c. = a m^b + b m^b+k + c m^b+2k + d m^b+3k + &c.</math>
prope; coefficiens cuju$cunque termini <math>S^π</math> erit
<math>{m^b × (m^k - p^k) × (m^k - q^k) × (m^k - r^k) × &c. / π^b (π^k - p^k) × (π^k - q^k) × (π^k - r^k) × &c.}</math>, in cujus numeratore haud continetur factor
<math>m^k - π^k</math>.</p>

<pb 707>[0759]<rh>SERIERUM, &c.</rh>
<p>Cor. 1. Hinc con$tat $ummam e $ingulis quantitatibus
<math>{(m^k - q^k) × (m^k - r^k) × (m^k - s^k) × &c. / (p^k - q^k) × (p^k - r^k) × (p^k - s^k) × &c.} (A) + {(m^k - p^k) × (m^k - r^k) × (m^k - s^k) × &c. / (q^k - p^k) × (q^k - r^k) × (q^k - s^k) × &c.}
(B) + {(m^k - p^k) × (m^k - q^k) × (m^k - s^k) × &c. / (r^k - p^k) × (r^k - q^k) × (r^k - s^k) × &c.} (C) + &c. = 1</math>; etiam-
que <math>p^fk A + q^fk B + r^fk C + s^fk D + &c. = m^fk</math>, $i modo <math>$</math> $it integer
<math>&</math> non major quam numerus quantitatum <math>p, q, r, s, &c.</math>.</p>
<p>Cor. 2. Ducantur factores e $ingulis numeratoribus in $e$e; e. g.
<math>(m^k - q^k) × (m^k - r^k) × (m^k - s^k) × (m^k - t^k) × &c. = ± q^k r^k s^k t^k &c. ∓
(q^k r^k s^k &c. + q^k s^k t^k &c. + r^k s^k t^k &c. + &c.) m^k ± (q^k r^k &c. + q^k s^k &c.
+ r^k s^k &c. + q^k t^k &c.) m^2k ∓ (q^k &c. + r^k &c. + s^k &c. + t^k &c.) m^3k ±
&c.</math>; & $ic de $ingulis reliquis numeratoribus: in his numeratoribus re-
$ultantibus rejiciantur omnes termini, in quibus haud inveniuntur
eædem <math>(l)</math> dimen$iones quantitatis <math>m^k</math>, & pro $ingulis numeratoribus
a$$umatur coefficiens quantitatis <math>m^lk</math>; tum erit $umma $ingularum
fractionum re$ultantium nihilo æqualis; ni <math>l = 0</math>, in quo ca$u erit
$umma prædicta <math>= 1</math>.</p>
<p>Omnia hæc facile con$tant e $cribendo pro <math>x</math> in a$$umptâ quanti-
tate ejus valores <math>p, q, r, s, &c.</math> $ucce$$ive.</p>
<p>Eadem etiam perfici po$$unt pro $eriebus diver$arum formularum.</p>
<p>E libro tertio con$tat, ut $eries hoc modo deductæ plerumque cæ-
teris paribus magis celeriter convergent, cum quantitates interpo-
landæ a$$umantur in quâdam geometricâ quam in quâvis aliâ ra-
tione.</p>
<p>In prob. huju$ce generis re$olvendis primum inveniendi $unt ter-
mini tabulæ a quæ$ito haud longe di$tantes; inter quos nulli evadunt
vel in$initi, vel nihilo æquales.</p>
<p>Hinc facile erui po$$unt quam proxime areæ vel centra gravitatis,
&c. curvarum <math>&</math> $olidorum a datis ordinatis <math>&</math> $e invicem haud longe
di$tantia, &c.</p>

<pb 708>[0760]<rh>DE SUMMATIONE</rh>
<h>THEOR. LVII.</h>
<p>1. In quantitate <math>y</math> quæ$itâ $int <math>a, b, c, d, e, &c.</math> $ucce$$ivi valores
quantitatis <math>x; & S^a, S^b, S^c, S^d, S^e, &c.</math> eorum corre$pondentes valores
quantitatis <math>y</math>; tum pote$t e$$e <math>y = {(x - b) (x - c) (x - d) (x - e) &c. / (a - b) (a - c) (a - d) (a - e) &c.}
× S^a + {(x - a) × (x - c) (x - d) (x - e) &c. / (b - a) (b - c) (b - d) (b - e) &c.} × S^b +
{(x - a) (x - b) × (x - d) (x - e) &c. / (c - a) (c - b) (c - d) (c - e) &c.}
S^c + {(x - a) (x - b) (x - c) × (x - e) &c. / (d - a) (d - b) (d - c) (d - e) &c.}
× S^d + {(x - a) (x - b) (x - c) (x - d) × &c. / (e - a) (e - b) (e - c) (e - d) &c.} × S^e + &c.</math></p>
<p>2. Aliter: Eadem quantitas <math>y</math> $ic exprimi pote$t <math>y = S^a + (x - a)
({1 / a - b} S^a + {1 / b - a} S^b) + (x - a) (x - b) ({1 / a - b} × {1 / a - c} × S^a +
{1 / b - a} × {1 / b - c} S^b + {1 / c - a} × {1 / c - b} S^c) + (x - a) (x - b) (x - c)
({1 / a - b} × {1 / a - c} × {1 / a - d} S^a + {1 / b - a} × {1 / b - c} × {1 / b - d} S^b + {1 / c - a} ×
{1 / c - b} × {1 / c - d} S^c + {1 / d - a} × {1 / d - b} × {1 / d - c} S^d) + (x - a) (x - b)
(x - c) (x - d) ({1 / a - b} × {1 / a - c} × {1 / a - d} × {1 / a - e} S^a + {1 / b - a} × {1 / b - c}
× {1 / b - d} × {1 / b - e} × S^b + {1 / c - a} × {1 / c - b} × {1 / c - d} × {1 / c - e} S^c + {1 / d - a}
× {1 / d - b} × {1 / d - c} × {1 / d - e} S^d + {1 / e - a} × {1 / e - b} × {1 / e - c} × {1 / e - d} S^e) + &c.</math></p>
<h>PROB. XLVII.</h>
<p>A$$umatur data quantitas <math>A + B x + C x^2 + D x^3 + ... + L x^n-1
= y</math>, & dentur <math>(n)</math> valores <math>α, β, γ, δ, ... x</math> quantitatis <math>z</math>, qui vere cor-
re$pondent <math>(n)</math> valoribus <math>π, ρ, σ, τ, ... φ</math> quantitatis <math>y</math>; &</math> e regulâ

<pb 709>[0761]<rh>SERIERUM, &c.</rh>
prius traditâ deduci po$$unt; i. e. $int <math>A + B α + C α^2 + D α^3 + ...
+ L α^n-1 = π, A + B β + C β^2 + D β^3 + ... + L β^n-1 = ρ, A +
B γ + C γ^2 + D γ^3 + ... L γ^n-1 = σ, &c., A + B χ + C χ^2 + D χ^3
... + L χ^n-1 = φ</math>; dentur etiam <math>(m)</math> valores <math>a, b, c, d, &c.</math> quanti-
tatis <math>(x)</math>, quibus corre$pondent <math>(m)</math> valores <math>P, Q, R, S, &c.</math> quanti-
tatis <math>y</math>; $ed cum regulâ prius traditâ, viz. <math>A + B x + C x^2 + D x^3
+ &c. = y</math> haud concordant; eorum errores $int re$pective <math>p, q, r, s,
&c.</math>; i. e. $int <math>A + B a + C a^2 + D a^3 + ... + L a^n-1 = P - p, A
+ B b + C b^2 + D b^3 ... + L b^n-1 = Q - q, A + B c + C c^2 + D c^3
... L c^n-1 = R - r, &c.</math>: corrigere a$$umptam quantitatem <math>A + B x
+ C x^2 + D x^3 + ... + L x^n-1 = y</math>.</p>
<p>Scribantur <math>{p / (a - α) × (a - β) × (a - γ) ... (a - χ)} = M,
{q / (b - α) × (b - β) × (b - γ) × (b - δ) ... × (b - χ)} = N, {r / (c - α) ×
(c - β) × (c - γ) ... (c - χ)} = O, {s / (d - α) × (d - β) × (d - γ) ... (d - χ)}
= P, {t / (e - α) × (e - β) × &c.} = Q, &c.</math></p>
<p>Quantitas pro correctione a$$umatur <math>(x - α) × (x - β) × (x - γ) ×
(x - δ) × ... × (x - χ) (M + (x - a) ({1 / a - b} M + {1 / b - a} N) +
(x - a) × (x - b) ({1 / (a - b) × (a - c)} M + {1 / (b - a) × (b - c)} N +
{1 / (c - a) × (c - b)} O) + (x - a) × (x - b) × (x - c) ({1 / (a - b) × (a - c) × (a - d)}
M + {1 / (b - a) × (b - c) × (b - d)} N + {1 / (c - a) × (c - b) × (c - d)} O +
{1 / (d - a) × (d - b) × (d - c)} P) + (x - a) × (x - b) × (x - c) × (x - d)
({1 / (a - b) × (a - c) × (a - d) × (a - e)} M + {1 / (b - a) × (b - c) ×

<pb 710>[0762]<rh>DE SUMMATIONE</rh>
(b - d) × (b - e)} N + {1 / (c - a) × (c - b) × (c - d) × (c - e)} O +
{1 / (d - a) × (d - b) × (d - c) × (d - e)} P + {1 / (e - a) × (e - b) × (e - c) × (e - d)}
Q) + &c.)</math> Lex, quam ob$ervant $ucceffive coefficientes prædictorum
terminorum, erit <math>{1 / (a - b) × (a - c) × (a - d) × (a - e) × &c.},
{1 / (b - a) × (b - c) × (b - d) × (b - e) × &c.}, {1 / (c - a) × (c - b) × (c - d)
× (c - e) × &c.}, {1 / (d - a) × (d - b) × (d - c) × (d - e) × &c.}, {1 / (e - a) ×
(e - b) × (e - c) × (e - d) × &c.}, &c.</math>: termini autem $ucce$$ivi $unt
<math>M, N, O, P, Q, R, &c.</math>; etiamque $ucce$$ive ducuntur re$ultantes
quantitates in <math>(x - a), (x - a) × (x - b), (x - a) (x - b) (x - c),
(x - a) (x - b) (x - c) (x - d), &c.</math></p>
<p>In multis ca$ibus præ$tat, ut ex repetitis operationibus per primam
correctionem in hâc regulâ traditam, viz. prima correctio <math>=
{(x - α) × (x - β) × (x - γ) .. (x - χ) / (a - α) × (a - β) × (a - γ) .. (a - χ)} × p</math>, $ecunda <math>= {(x - α) ×
(x - β) × (x - γ) .. (x - χ) × (x - a) / (b - α) × (b - β) × (b - γ) .. (b - χ) × (b - a)} × q, &c.</math> detegantur novæ
correctiones, u$que donec re$ultat correctio quæ$ita.</p>
<p>2. Sint <math>π, ρ, σ, τ, &c.</math> quantitates, quæ omnes erunt nihilo æqua-
les, cum <math>x = α vel β vel γ vel δ, &c.</math>; etiamque erunt re$pective <math>P,
R, S, T, &c.</math>, cum <math>x</math> $it re$pective <math>a, b, c, d, &c.</math>: $it <math>λ</math> quantitas
nihilo æqualis, cum <math>x = b vel c vel d, &c. at = L</math>, cum <math>x = a</math>: & $i-
militer $it <math>μ = 0</math>, cum <math>x = a, vel c vel d, &c.</math>, at <math>= m</math>, cum <math>x = b</math>;
& $ic $it <math>v = 0</math>, cum <math>x = a vel b vel d, &c.</math>, at <math>= n</math>, cum <math>x = c</math>; &
$ic deinceps; deinde a$$umi pote$t correctio quæ$ita <math>= {π / P} × {λ / L} × p
+ {ρ / R} × {μ / m} × q + {σ / S} × {v / n} × r + &c.</math></p>

<pb 711>[0763]<rh>SERIERUM, &c.</rh>
<p>Ex principiis hìc traditis, &c. & datis <math>n</math> valoribus datæ quantitatis
<math>y</math> corre$pondentibus, & erroribus <math>(m)</math> valorum prædictæ quantitatis;
cum duæ vel plures variabiles <math>(x, y, z, &c.)</math> quantitates in datâ
æquatione contineantur; tum ex a$$umptis <math>(m + n)</math> æquationibus
huju$ce generis <math>a + b x + c z + &c. + d x^2 + e z x + f z^2 + &c. +
g x^3 + h x^2 z + &c. + &c. = y</math>, quæ corre$pondent <math>(m + n)</math> diver$is
corre$pondentibus valoribus quantitatum <math>(x, y, z, &c.)</math> erui po$$unt
<math>(m + n)</math> coefficientes <math>a, b, c, &c.</math> quæ$itæ.</p>
<h>METHODUS CORRESPONDENDENTIUM VALORUM.</h>
<h>PROB. XLVIII.</h>
<p>1. Sint <math>a, b, c, d, &c.</math> dati valores quantitatis <math>x</math>, cujus $int <math>S^a, S^b, S^c,
S^d, &c.</math> corre$pondentes valores quantitatis <math>y</math>: invenire functionem
quantitatis <math>(x)</math>, quæ pro valoribus <math>(a, b, c, d, &c.)</math> quantitatis <math>(x)</math>
habet <math>S^a, S^b, S^c, S^d, &c.</math> corre$pondentes valores quantitatis <math>(y)</math>.</p>
<p>A$$umatur <math>y = {(x - b) × (x - c) × (x - d) × (x - e) × &c. / (a - b) × (a - c) × (a - d) × (a - e) × &c.} × S^a
+ {(x - a) × (x - c) × (x - d) × (x - e) × &c. / (b - a) × (b - c) × (b - d) × (b - e) × &c.} × S^b + {(x - a) × (x - b) × (x - d) × (x - e) × &c. / (c - a) × (c - b)
× (c - d) × (c - e) × &c.} × S^c + {(x - a) × (x - b) × (x - c) × (x - e) × &c. / (d - a) × (d - b) × (d - c) × (d - e) × &c.}
× S^d + &c.</math>; tum evadent valores quantitatis <math>(y)</math> re$pective <math>S^a, S^b, S^c,
S^d, &c.</math>: cum eorum corre$pondentes valores quantitatis <math>(x)</math> fiant <math>a,
b, c, d, &c.</math> re$pective.</p>
<p>2. Ii$dem po$itis; cum <math>x</math> evadat <math>a, b, c, d, &c.</math>; tum <math>A, B, C, D, &c.</math>
evadat <math>α, β, γ, δ, &c.</math> re$pective; a$$umatur <math>y = {A / α} × {(x - b) × (x - c) × (x - d) × &c. / (a - b) × (a - c) ×
(a - d) × &c.} × S^a + {B / β} × {(x - a) × (x - c) × (x - d) × &c. / (b - a) × (b - c) × (b - d) × &c.} × S^b +
+ {C / γ} × {(x - a) × (x - b) × (x - d) × &c. / (c - a) × (c - b) × (c - d) × &c.} × S^c + {D / δ} × {(x - a) × (x - b)

<pb 712>[0764]<rh>DE SUMMATIONE</rh>
× (x - c) × &c. / (d - a) × (d - b)
× (d - c) × &c.} × S^d + &c.</math>; in quo ca$u valores quantitatis <math>(y)</math> erunt
etiam re$pective <math>S^a, S^b, S^c, S^d, &c.</math>, cum valores quantitatis <math>(x)</math> $int
<math>a, b, c, d, &c.</math>: e. g. $int <math>A, B, C, D, &c.</math> re$pective <math>x</math>, tum erit <math>α = a,
β = b, γ = c, δ = d, &c.</math>; unde <math>y = {x × (x - b) × (x - c) × (x - d) × &c. / a × (a - b) × (a - c) × (a - d) × &c.}
× S^a + {x × (x - a) × (x - c) × (x - d) × &c. / b × (b - a) × (b - c) × (b - d) × &c.} × S^b + {x × (x - a) × (x - b) × (x - d) × &c. / c × (c - a) ×
(c - b) × (c - d) × &c.} × S^c + &c.</math>; quæ eadem erit ac quantitas e
methodo differentiarum deducta.</p>
<h>THEOR. LVIII.</h>
<p>Quantitates a$$umptæ <math>A, B, C, D, &c.</math> fiant <math>α, β, γ, δ, &c.</math>; cum <math>x</math>
evadat <math>a, b, c, d, &c.</math> re$pective: deinde a$$umantur quantitates <math>p, q,
r, s, &c.</math>; quæ nihilo $unt æquales, cum <math>x = a</math>; etiamque <math>p′, q′, r′,
s′, &c.</math> quæ nihilo $unt æquales, cum <math>x = b</math>; & $imiliter <math>p″, q″, r″, s″,
&c.</math> nihilo æquales, cum <math>x = c</math>; & $ic de quantitatibus <math>p′″, q′″, r′″, s′″,
&c.; p″′, q″″, &c.</math>; quæ nihilo evadunt æquales, cum <math>x = d, e, &c.</math> re-
$pective: $int <math>H, I, K, L, &c.</math> valores quantitatum <math>p′ × p″ × p′″ × p″″ ×
&c., q × q″ × q′″ × q″″ × &c., r × r′ × r′″ × r″″ × &c., s × s′ × s″ × s″″ ×
&c., &c.</math>, cum <math>x</math> evadat <math>a, b, c, d, &c.</math>, re$pective; tum a$$umatur <math>y =
{A / α} × {p′ × p″ × p′″ × &c. / H} × S^a + {B / β} × {q × q″ × q′″ × &c. / I} × S^b + {C / γ} × {r × r′ × r″ × &c. / K}
× S^c + &c.; & S^a, S^b, S^c, S^d, &c.</math> erunt valores quantitatis
<math>(y)</math>, qui corre$pondent valoribus <math>a, b, c, d, &c.</math> quantitatis <math>x</math> re-
$pective.</p>
<h>THEOR. LIX.</h>
<p>Sit æquatio, quæ $it functio quantitatum <math>(x, y, z, &c.)</math>, in quâ con-
tinentur <math>(n)</math> coefficientes ad libitum a$$umendæ; dentur etiam <math>(n)</math>

<pb 713>[0765]<rh>SERIERUM, &c.</rh>
corre$pondentes valores $ingularum incognitarum quantitatum <math>(x, y,
z, &c.)</math>; invenire coefficientes prædictas.</p>
<p>Pro incognitis quantitatibus <math>(x, y, z, &c.)</math> $cribantur earum <math>(n)</math>
corre$pondentes valores, re$ultant <math>(n)</math> æquationes totidem <math>(n)</math> incog-
nitas quantitates, i. e. <math>(n)</math> coefficientes ad libitum a$$umendas, ha-
bentes; e quibus detegi po$$unt coefficientes ip$æ.</p>
<p>2. Sint <math>(m)</math> æquationes <math>(n + m)</math> incognitas quantitates <math>(x. y, z, &c.)</math>
habentes, & in $ingulis hi$ce æquationibus contineantur <math>(r)</math> incog-
nitæ coefficientes ad libitum a$$umendæ; tum ex datis <math>(r)</math> corre$pon-
dentibus valoribus $ingularum incognitarum quantitatum <math>(x, y, z,
&c.)</math> per methodum prius traditam erui po$$unt coefficientes ip$æ.</p>
<h>PROB. XLIX.</h>
<p>I_nvenire quantitatem, quæ evadat re$pective_ <math>S, S′, S″, S′″, &c.</math>; _cum_
<math>x = α, y = β, z = γ, &c.; x = α′, y = β′, z = γ′, &c.; x = α″,
y = β″, z = γ″, &c.; &c.</math></p>
<p>A$$umantur quæcunque functiones <math>(H, K, L, &c.; A, B, C, D, &c.)</math>
quantitatum <math>(x, y, z, &c.)</math>; in functionibus <math>H, K, L, &c.</math> pro incog-
nitis quantitatibus <math>(x, y, z, &c.)</math> $cribantur re$pective <math>α, β, γ, &c.; α′, β′,
γ′, &c.; α″, β″, γ″, &c.; α′″, β′″, γ′″, &c.; &c.</math>; & re$ultent quantitates <math>h,
k, l, &c.</math> re$pective: in functione <math>(A)</math> pro incognitis quantitatibus <math>(x, y,
z, &c.)</math> $cribantur prædictæ quantitates <math>α, β, γ, &c.; α′, β′, γ′, &c.; α″,
β″, γ″, &c.; &c.</math>; re$ultent quantitates <math>a, a′, a″, a′″, &c.</math>: & $imiliter in
functionibus <math>B, C, D, &c.</math>; $cribantur pro incognitis <math>(x, y, z, &c.)</math> præ-
dictæ quantitates & re$ultent quantitates <math>b, b′, b″, &c.; c, c′, c″, &c.; d,
d′, d″, &c.</math> Quantitas quæ$ita pote$t e$$e <math>{H / h} × {(A - a′) × (A - a″) × (A - d′″) × &c. / (a - a′) × (a - a″) ×
(a - a′″) × &c.} × S + {K / k} × {(B - b) × (B - b″) × (B - b′″) × &c. / (b′ - b) × (b′ - b″) × (b′ - b′″) × &c.} × S′
+ {L / l} × {(C - c) × (C - c′) × (C - c′″) × &c. / (c″ - c) × (c″ - c′) × (c″ - c′″) × &c.} × S″ + &c.</math></p>
<p>Cor. Si requiratur, ut quantitas detegenda $it maxime $implex;
tum pro <math>H, K, L, &c.</math> a$$umatur invariabilis quantitas; & pro <math>A, B,

<pb 714>[0766]<rh>DE SUMMATIONE</rh>
C, &c.</math> a$$umantur functiones quantitatum <math>(x, y, z, &c.)</math> huju$ce for-
mulæ <math>e x + f y + g z + &c. + d</math>; ubi literæ <math>e, f, g, &c.</math> & d invaria-
biles quantitates re$pective denotant.</p>
<h>PROB. L.</h>
<h it>Datâ nonnullorum ca$uum re$olutione, & formulâ in quâ generalis con-
netur; eam detegere.</h>
<p>Primo a$$umatur ca$us, $i modo $ieri po$$it, in quo una $olum-
modo continetur incognita quantitas, & ex eo ca$u deduci pote$t in-
cognita quantitas: & $ic de $ingulis reliquis incognitis quantitatibus,
i. e. continuo inveniantur ca$us, in quibus una reliqua vel ea cum
quibu$dam prius inventis $olummodo contineatur, & exinde deduci
pote$t prædicta reliqua quantitas: $i vero non innote$cat methodus
inveniendi ca$um, in quo una $olummodo incognita quantitas con-
tineatur; inveniantur duo independentes ca$us, in quibus duæ $olum-
modo contineantur incognitæ quantitates: & $i haud dentur duo in-
dependentes ca$us, in quibus $olummodo contineantur quæcunque
duæ incognitæ quantitates cum prædictis inventis, inveniantur tres
ca$us, in quibus continentur tres incognitæ quantitates cum prius in-
ventis; & $ic deinceps: & exinde detegi po$$unt incognitæ quanti-
tates in re$olutione contentæ, & con$equenter re$olutio quæ$ita.</p>
<p>Ex. Sit formula <math>V = {x^2 z + b x y + 1 / f x + g z}</math>; & detur ca$us, cum <math>x & y</math>
nihilo $int re$pective æquales, viz. $it <math>z = 3 & V = 2</math>; tum re$ultat
<math>{1 / g z} = V</math>, & exinde <math>g = {1 / z × V} = {1 / 2 · 3}: 2^do</math>. detur ca$us cum <math>y = 0 &
x = 5</math>, viz. $it <math>z = 4 & V = 3</math>, tum erit <math>V = {x^2 z + 1 / f x + {1 / 6} z} = {25 z + 1 / f x + {4 / 6}}
= 3</math>, & exinde <math>f = {101 - 2 / 15} = 6{3 / 5}</math>; & $imiliter $int <math>x, y & z</math> re$pec-
tive <math>1, 2 & 3</math> cum <math>V = 6</math>, tum erit <math>6 = {3 + 2 h + 1 / 6{3 / 5} × 1 + {1 / 6} × 3}</math>; & exinde
<math>h = 19{3 / 10}</math>.</p>

<pb 715>[0767]<rh>SERIERUM, &c.</rh>
<p>Cor. 1. Hoc problema ad re$olutionem quam plurimorum proble-
matum applicari pote$t, nam in multis ca$ibus facile innote$cet for-
mula prædicta.</p>
<p>Ex. 1. Sit æquatio <math>x^n - p x^n-1 + q x^n-2 - r x^n-3 + s x^n-4 - &c.
= 0</math>, cujus radices $int <math>α, β, γ, δ, &c.</math>; invenire æquationem cujus ra-
dices (<math>z</math>) $unt <math>α + β, α + γ, α + δ, β + γ, β + δ, γ + δ, &c.</math>: for-
mula æquationis erit <math>z^n·{(n-1) / 2}=m + a p z^m-1 + (a′ p^2 + b q)z^m-2 + (a″ p^3
+ b′ p q + c r)z^m-3 + (a′″ p^4 + b″ p^2 q + c′ p r + d q^2 + e s)z^m-4 +
&c. = 0</math>; $inguli enim termini eidem pote$tati <math>(z^m-x)</math> quantitatis (<math>z</math>)
annectendi ea$dem <math>(α)</math> habent dimen$iones quantitatum <math>p, q, r, s, &c.</math>;
$i modo dimen$iones quantitatum <math>p, q, r, s, &c.</math> $int re$pective <math>1, 2,
3, 4, &c.</math> I<_>mo</_>. A$$umatur æquatio <math>x^n - x^n-1 = 0</math>, in quâ <math>q, r, s, &c.</math>
de$unt & <math>p = 1</math>, tum erunt valores quantitatis <math>(x)</math> re$pective <math>1, 0, 0,
0, &c.</math>, & con$equenter <math>(n-1)</math> valores quantitatis (<math>z</math>) erunt <math>1</math>, cæ-
teræ vero nihilo æquales; unde æquatio erit <math>z^m - (n - 1)z^m-1 +
(n - 1) · {n - 2 / 2} z^m-2 - (n - 1) · {n - 2 / 2} · {n - 3 / 3}z^m-3 + &c. = 0</math>; &
exinde <math>a = - (n - 1), a′ = (n - 1) · {n - 2 / 2}, a″ = - (n - 1) ·
{n - 2 / 2} · {n - 3 / 3}, &c.</math>: 2<_>do</_>. a$$umatur <math>x^n - x^n-2 = 0</math>, ubi <math>q = - 1 &
p, q, r, s, &c.</math> de$unt, cujus radices $unt <math>1 & - 1, & 0, 0, 0, &c.</math>;
unde <math>(n - 2)</math> valores quantitatis <math>z</math> erunt <math>1, & n - 2</math> etiam <math>- 1</math>; cæ-
teræ vero erunt nihilo æquales; & æquatio erit <math>z^m -- (n - 2) z^m-2
+ (n - 2) · {n - 3 / 2}z^m-4 - &c. = 0</math>; unde <math>b, d, &c.</math> coefficientes ter-
minorum <math>q, q^2, q^3, &c.</math> erunt re$pective <math>- (n - 2), (n - 2) · {n - 3 / 2},
- (n - 2) · {n - 3 / 2} · {n - 4 / 3}, &c.</math>: & $imiliter ex a$$umptâ æquatione
huju$ce formulæ <math>x^n - p x^n-1 + q x^n-2 = 0</math>, cujus duæ radices (<math>α & β</math>)
innote$cunt & cæteræ nihilo $unt æquales, detegi po$$unt coefficien-
tes terminorum <math>p q & p^2 q^2</math>; & ex duabus æquationibus prædictæ for-
mulæ <math>x^n - p x^n-1 + q x^n-2 = 0</math>, quarum <math>(n - 2)</math> radices in duabus re

<pb 716>[0768]<rh>DE SUMMATIONE</rh>
$pectivis æquationibus nihilo $unt æquales, duæ vero reliquæ inno-
te$cunt & non $unt eædem in utrâque æquatione, deduci po$$unt co-
efficientes terminorum <math>p^3 q & p q^2; p^4 q & p^2 q^2</math>: & $imiliter ex tribus
diver$is æquationibus prædictæ formulæ detegi po$$unt coefficientes
terminorum <math>p^5 q, p^3 q^2, p q^3; p^6 q, p^4 q^2, p^2 q^3</math>: & $ic deinceps: eadem
methodus etiam applicari pote$t ad terminos, in quibus continentur
literæ <math>p, q, r, &c.</math></p>
<h>THEOR. LX.</h>
<p>Nonnullæ coefficientes unius erui po$$unt e coefficientibus alterius
ca$us; quod plerumque fieri pote$t, cum unus $it particularis ca$us
alterius; vel quendam nexum habent inter $e duorum ca$uum quæ-
dam coefficientes.</p>
<p>Deducantur coefficientes terminorum in unâ quantitate, & e datâ
relatione deduci po$$unt coefficientes alterius quantitatis.</p>
<p>Ex. Sit quantitas <math>A</math> in quâ continentur literæ <math>a, b, c, d, &c.</math>; $it
etiam quantitas <math>B</math>, in quâ omnes literæ <math>a, b, c, d, &c.</math> non involven-
tes literas <math>p, q, r, &c.</math> $imiliter involvuntur ac in quantitate <math>A</math>; tum
deducatur eadem $unctio quantitatum <math>A & B</math>, quæ $it <math>E & F</math>; in his
quantitatibus omnes literæ, quæ non involvunt literas <math>p, q, r, s, &c.</math>;
$imiliter involventur.</p>
<p>Ex. 1. Sit æquatio data in$erioris ordinis <math>x^4 + q x^2 - r x + s = 0</math>,
cujus radices $int <math>α, β, γ, δ</math>; & æquatio, cujus radices $int <math>α + β, α + γ,
α + δ, β + γ, β + δ, γ + δ</math>, erit <math>v^6 + 2 q v^4 + (q^2 - 4^s) v^2 - r^2 = 0</math>.</p>
<p>Ducatur data æquatio <math>x^4 + q x^2 - r x + s = 0</math> in $implicem æqua-
tionem (<math>x = 0</math>), cujus radix e$t (<math>0</math>); & $it æquatio <math>x^5 + q x^3 - r x^2
+ s x = 0</math>, cujus radices erunt <math>α, β, γ, δ, 0</math>; & radices æquationis $imi-
liter exinde re$ultantis erunt <math>α + β, α + γ, β + γ, α + δ, β + δ, γ + δ,
α, β, γ, δ</math>; & æquatio ip$a <math>(v^4 + q v^2 - r v + s) × (v^6 + 2 q v^4 + (q^2
- 4 s) × v^2 - r^2) = 0</math>; cujus æquationis omnes termini iidem erunt,
ac termini æquationis $imiliter re$ultantis e datâ æquatione <math>x^5 + q x^3
- r x^2 + s x - t = 0</math>; in quibus continentur $olummodo literæ <math>q, r, s.</math></p>
<p>Per æquationem $imiliter re$ultantem in hoc ca$u de$ignatur æqua-
tio, cujus radices $unt <math>λ + μ, λ + ν, λ + π, λ + ρ, μ + ν, μ + π,

<pb 717>[0769]<rh>SERIERUM, &c.</rh>
μ + ρ, ν + π, ν + ρ, π + ρ</math>; $i modo <math>λ, μ, ν, π, ρ</math> $int re$pective radices
æquationis <math>x^5 + q x^3 - r x^2 + s x - t = 0</math>.</p>
<h>SCHOLIUM.</h>
<p>Finis huic operi non e$t imponendus, priu$quam paucula de me-
thodo re$olvendi hæc cum multis aliis problematibus adjiciantur,
quæ dici pote$t methodus deductionis & reductionis; omnia enim ferè
mathematica nihil aliud volunt, quam e datâ methodo deductionis
quantitates reducere: per deductionis methodum de$igno quamcun-
que notam operationem; E. g. vel quantitatum additionem, $ub-
ductionem, multiplicationem, divi$ionem, extractionem earum radi-
cum, &c. quoniam hæc operatio bene nota e$$e $upponitur, $emper
perfici pote$t exinde deductio: $i vero exigat problema, ut iterum
iterumque repetatur data operatio, in multis ca$ibus propter calculi
laborem ad reductionis methodum confugere haud inutile videtur;
E. g. Sint <math>a x + b x^2 + c x^3 + &c. = y, a z + b z^2 + c z^3 + &c. = x,
a v + b v^2 + c v^3 + &c. = z, a w + b w^2 + c w^3 + &c. = v, &c.</math> fa-
cile con$tat ut labor hujus calculi repetitis operationibus valde cre$-
cit; & idem affirmari pote$t de inveniendis primis, $ecundis, tertiis,
&c. fluxionibus quantitatis <math>(a + b x^n + c x^2n + &c.) × (e + f x^n +
g x^2n + &c.)^m</math>.</p>
<p>1.1. Ex datâ quantitate deductâ & deductionis methodo ejus redu-
ctionem invenire, e$t problema maximè generale; huju$ce problematis
particulares erunt $ub$equentes ca$us, viz. $it <math>x</math> quantitas invenienda,
& <math>a</math> quantitas deducta; & methodus deductionis talis $it, ut <math>A x^n +
B x^n-1 + C x^n-2 + &c. = a</math>; reductio vero hujus quantitatis (<math>a</math>) ad
quæ$itam <math>x</math> eadem e$t, ac re$olutio algebraicæ æquationis <math>A x^n +
B x^n-1 + &c. = a</math>. 2. Sit quantitas deducta quæcunque fluens, &
data deductio $it vulgaris methodus inveniendi fluxiones datarum
fluentium, & methodus reductionis erit methodus inveniendi fluen-
tem ex datâ fluxione: hoc vero problema plerumque re$olutionem
recipiet ex a$$umptâ quantitate generali (cujus irrationalitas vel for-
mula ferè deduci pote$t e datâ quantitate & datâ etiam deductionis
methodo) pro quantitate quæ$itâ; $cribatur hæc quantitas pro ejus

<pb 718>[0770]<rh>DE SUMMATIONE</rh>
valore, & e datâ deductionis methodo inveniatur altera quantitas,
quâ datæ quantitati æquali e$$e $uppo$itâ, exinde erui pote$t pro-
blematis re$olutio: e. g. $it data quantitas <math>(a^2 + 7 x + x^2) × (a^2 +
x^2)^{5 / 2} = W</math>, cujus quantitas reducta $it <math>A</math>, & methodus deductionis <math>A
+ {A^. / x^.}</math>, i. e. <math>A + {A^. / x^.} = W</math>; $ed ex hâc methodo con$tat quæ$itam quan-
titatem <math>A</math>, $i modo algebraice exprimi po$$it, eandem habere irratio-
nalitatem ac quantitatem <math>W</math>; a$$umatur igitur <math>(a^2 + x^2){7 / 2} × (p + q x
+ r x^2 + &c.) = A</math>, & e prædictâ methodo inveniri po$$unt <math>p = 1,
q = 0, r = 0, &c.</math> ergo quantitas quæ$ita erit <math>(a^2 + x^2){7 / 2}</math>; generalis
autem re$olutio erit <math>{$. e^x (a^2 + 7 x + x^2) × (a^2 + x^2){7 / 2} x^. / e^x} + b e^-x = A</math>,
ubi <math>b</math> e$t quantitas ad libitum a$$umenda; unde, $i generalis requira-
tur re$olutio quantitatis <math>A</math>, a$$umi debet re$olutio, quæ prædictam
generalem continet.</p>
<p>Cor. Hinc con$tat e præcedenti $ub$titutione deductum e$$e $olum-
modo particularem valorem quantitatis <math>A</math>, quoniam $ub$titutio $olum-
modo e$t particularis & non continet in $e omnes valores quantitatis <math>A</math>.</p>
<p>3. Con$imilia etiam principia applicari po$$unt ad reductionem
datæ quantitatis in plures, quarum methodi deductionum dantur; &
$ic de æquationibus inter qua$cunque quantitates: hinc ex datâ me-
thodo deductionis erui po$$unt quælibet quantitates, quas reducere
liceat; a$$umantur enim quæcunque quantitates & ex iis per datam
methodum deductionis inveniantur quantitates quæ$itæ.</p>
<p>4. Hìc vero animadvertendum e$t, ut talis $it deductionis methodus,
quæ ($i modo eadem data quantitas $ub diversâ $pecie lateat) diver-
$am haud $pecie $ed re præbeat quantitatem deductam: e. g. $it me-
thodus deductionis talis ut e datâ quantitate <math>a x^n y^m</math> deducatur quan-
titas <math>n^2 a x^n-1 y^m + m^2 a x^n y^m-1</math>; $it data quantitas <math>x^n+m</math>, & quantitas
per prædictam methodum inve$tigationis inventa erit <math>(n + m)^2 x^n+m-1</math>;
at <math>x^n+m = x^n × x^m</math>, & quantitas per eandem methodum inventa
<math>n^2 x^n-1 × x^m + m^2 x^n × x^m-1 = (n^2 + m^2) x^n+m-1</math>, $ed <math>n^2 + m^2</math> non e$t
<math>(n + m)^2</math>, ergo ex eâdem quantitate <math>x^n+m</math> $ub diversâ $pecie <math>(x^n × x^m)</math>
latente per hanc methodum deductionis diver$æ exorientur quantita-

<pb 719>[0771]<rh>SERIERUM, &c.</rh>
tes, & methodus deductionis habet relationem ad qualitatem æque ad
quantitatem, ni ex aliis datis ad finitum re$pon$ionum numerum re-
$tinguatur quæ$tio propo$ita: $i modo talis $it methodus deductionis,
ut eædem quantitates $ub diver$is formulis latentes $emper ea$dem præ-
beant quantitates; tum quantitas data utcunque in diver$am formulam
vel etiam in in$initas approximationes transformari pote$t, & exinde
omnes regulæ de approximationibus in hoc & præcedente libro tra-
ditæ mutatis mutandis ad hunc ca$um applicari po$$unt.</p>
<p>5. Sæpe ex repetitis <math>n</math> operationibus & animadver$ione methodo-
rum, e quibus formantur termini, con$tabit lex, quam ob$ervat $e-
ries, quæ exprimit quantitatem deductam per operationem <math>n</math> vicies
repetitam, quæ quantitas dicatur terminus ad <math>n</math> di$tantiam; pro <math>n</math>
$cribatur <math>{r / m}</math>, & invenietur $eries, quæ exprimit quantitatem in eadem
$calâ ad <math>{r / m}</math> di$tantiam. E. g. Ex operatione inveniendi fluxiones da-
tarum quantitatum $æpius repetitâ con$tabit fluxio rectanguli <math>x y</math> or-
dinis vero <math>(n) = y x^. ^n + n y^. x^. ^n-1 + n. {(n - 1) / 2} y^.. x^. ^n-2 + n · {n - 1 / 2} · {n - 2 / 3}
y^... ^n-3 x^. + &c.</math> In hâc fluxione pro <math>n</math> $cribatur <math>{r / m}</math>, & re$ultat quantitas
in eâdem $calâ ad <math>{r / m}</math> di$tantiam po$ita. Ex. 2. Fluxio <math>r</math> ordinis
quantitatis <math>x^n</math> invenietur <math>n · (n - 1) · (n - 2) ... (n - r + 1) x^n-r x^. ^r +
r · {r - 1 / 2} × n · (n - 1) ... (n - r + 2) x^n-r+1 x^. ^r-2 x^.. + r · {r - 1 / 2} · {r - 2 / 3}
× n · (n - 1) · (n - 2) ... (n - r + 3) x^n-r+2 x^. ^r-3 x^... + &c.</math> quæ etiam
ita $cribi pote$t <math>n x^n-1 x^. ^r + r × n · (n - 1) x^n-2 x^. x^. ^r-1 + r · {r - 1 / 2} × n \\ + r · {r - 1 / 2} · n
(n - 1) · (n - 2) x^n-3 x^. ^r-2 x^. ^2 + &c.</math> Sit terminus <math>x^. ^a ^α × x^. ^b ^β × x^. ^c ^γ × x^. ^d ^δ × \\ · (n - 1) x^n-2 x^. ^r-2 x^..

<pb 720>[0772]<rh>DE SUMMATIONE</rh>
&c.</math>; ubi <math>a α + b β + c γ + &c. = r</math>; tum ejus coefficiens erit <math>N × n
× (n - 1) · (n - 2) × (n - 3) ... (n - t + 1)</math>, ubi <math>t = α + β + γ +
δ + &c.; & N</math> numerus qui e$t functio quantitatis <math>r</math> & prædictorum
numerorum <math>a, α, b, β, c, γ, &c.</math> facile deducenda: in hâc $erie pro <math>r</math>
$cribatur <math>{k / l},</math> & con$tabit lex quam ob$ervat quantitas ad di$tantiam
<math>{k / l}</math> in $calâ fluxionum pote$tatis <math>x^n</math> po$ita.</p>
<p>Sit <math>r</math> negativus numerus & $eries prædicta denotet quantitatis <math>x^n</math>
fluentes; $ed hic animadvertendum e$t, quod$i numerus factorum $it
negativus, tum de$ignat factores in denominatore contentos, $i nu-
merus factorum affirmativus denotet factores in numeratore. E. g.
Requiratur fluens <math>r</math> ordinis quantitatis <math>x^n</math>, hìc <math>r</math> e$t negativus nume-
rus, & numerus factorum <math>(n · n - 1 ... (n - r + 1))</math> e$t <math>r</math>; ergo
numerus factorum in denominatore contentorum erit <math>r</math>, & ultimus
factor <math>(n - r + 1)</math> erit in hoc ca$u <math>(n + r)</math>, & $eries prædicta evadet
<math>{x^n+r / n + 1 · n + 2 .. (n + r) · x^. ^r} + r · {r + 1 / 2} × {x^n+r+1 x^.. / (n + 1) · (n + 2) ... (n + r + 1) · x^. ^r+2}
- r · {r + 1 / 2} · {r + 2 / 3} × {x^n+r+2 x^... / (n + 1) · (n + 2) ... (n + r + 2) × x^. ^r+3} + &c.</math>
huju$ce $eriei lex e præcedente con$equitur: $i vero requiratur termi-
nus in $calâ fluentium hìc traditarum quantitatis <math>x^n</math> ad di$tantiam -
<math>{k / l}</math> a primo po$itus; $i <math>{k / l}</math> $it integer affirmativus numerus, tum $olum-
modo requiruntur fluentes datæ quantitatis <math>x^n</math>; $i vero <math>{k / l}</math> $it fractio
ad minimos terminos reducta, tum inveniantur quantitates in $calâ
$ub$equente (ubi di$tantiæ a primo $int, <math>&c. - 2, - 1, 0, 1, 2, &c.</math>
& earum corre$pondentes termini, <math>&c., {1 / n + 1 · n + 2}, {1 / n + 1}, 1, n, n.
(n - 1), &c.)</math> quarum di$tantiæ a primo termino $int re$pective <math>- {k / l},
- {k - 1 / l}, - {k - 2 / l} ... - {k - l + 1 / l}</math>; pro his <math>l</math> quantitatibus inventis

<pb 721>[0773]<rh>SERIERUM, &c.</rh>
$cribantur <math>A, B, C, D, &c.</math> re$pective; tum erunt quantitates in eâ-
dem $calâ, quarum di$tantiæ a $ecundo $it <math>- {k / l}, - {k - 1 / l}, - {k - 2 / l},
&c.</math> re$pective <math>(n + {k / l}) A, (n + {k - 1 / l}) B, (n + {k - 2 / l}) C, &c.</math> & $ic
deinceps; & quantitas quæ$ita erit <math>A x^n+{k / l} x^. ^{-k / l},</math> $i <math>x^. ^{1 / l}</math> $it con$tans quan-
titas.</p>
<p>Facile etiam con$tat, $i modo $it <math>v</math> quantitas, cujus fluxio (<math>v^. ^{1 / l}</math>) or-
dinis <math>{1 / l}</math> vel quantitas in $calâ fluxionum ad di$tantiam <math>{1 / l}</math> a datâ fluente
po$ita $it con$tans, ubi <math>l</math> $it integer numerus; & $it <math>z^. ^r+{1 / l}</math> datâ fluxio; e$$e
fluentem <math>r + {1 / l} ordinis = (ubi r e$t integer numerus) z + A v^r +
B v^r-{1 / l} + C v^r-{2 / l} + &c.... F v^{2 / l} + G v^{1 / l} + H</math>, ubi <math>A, B, C ... F, G,
H</math> $unt invariabiles quantitates ad libitum a$$umendæ.</p>
<p>Ferè omnia etiam, quæ prius tradita fuere de fluxionibus, fluxio-
nalibus æquationibus & earum multiplicatoribus, &c. ad ha$ce inter-
polationes applicari po$$unt: hìc for$an haud indignum e$t ob$ervatu,
quod $eries, cujus termini hanc habeant formulam <math>A x^ez+f</math>, ubi <math>z</math> e$t
di$tantia a primo $eriei termino, & <math>e & f</math> $unt invariabiles quantitates,
& <math>A</math> algebraica functio vel rationalis vel irrationalis literæ <math>z</math>, $emper
exprimi pote$t vel per fluxionalem æquationem, vel per interpolatio-
nem $eriei, quæ e$t re$olutio fluxionalis æquationis & con$equenter
e$t re$olutio fluxionalis æquationis ordinis vero fractionalis: $i enim
<math>A</math> $it rationalis functio quantitatis <math>z</math>, tum erit re$olutio fluxionalis
æquationis; $i vero radices rationalis functionis contineat quantitas
<math>A</math>, tum plerumque recurrendum e$t ad interpolationem $erierum ex
fluxionalibus æquationibus exortarum.</p>
<p>Dimen$iones cuju$cunque incognitæ quantitatis in $ucce$$ivis ter-
minis datæ $eriei per inæquales differentias augeantur vel diminuan-

<pb 722>[0774]<rh>DE SUMMATIONE, &c.</rh>
tur, tum $umma $eriei nec per finitam algebraicam nec fluxionalem
æquationem exprimi pote$t.</p>
<p>Sit $eries, cujui termini per hanc formulam <math>A x^α</math> exprimuntur, ubi
<math>α</math> e$t quæcunque functio quantitatis <math>z</math> di$tantiæ a primo $eriei termino
vel algebraica vel fluxionalis, vel incrementialis vel formulæ, cujus
differentia inter numerum factorum in numeratore & denominatore
contentorum haud eadem manet; e prædictis principiis, i. e. terminis
ad in$initam di$tantiam con$titutis, inveniri pote$t, utrum convergit,
necne; etiamque annon exprimi pote$t per algebraicam vel fluxiona-
lem, &c. æquationem.</p>
<p>De hâc deductionum & reductionum doctrinâ, i. e. $ub$titutionibus
mentionem fieri hìc cen$ui con$ilio, ut via in hâc maxime generali
$cientiâ ad multo majora $ternatur.</p>
<fig/>

<pb>[0775]
<h>CORRIGENDA.</h>
<p>SÆ P E pro in lege duct. vel multiplicat. in, & po$t verba reduct. & re$olv. pro in lege ad, &
pro incrementialis, lege incrementalis, & pro integralis, lege integrale, & nonnunquam pro
fiunt vel fiant, lege evadunt vel evadant, &c.; pag. 18. lin. 11. pro <math>β & α, lege β′ & α′</math>; l. 12.
pro <math>{1 / }</math>, lege <math>{x / }</math>; pag. 19. l. 13. pro pote$t, lege pote$t, necne; pag. 20. l. 17. pro (<math>v</math>),
lege (<math>v</math>), necne; p. 26. l. antepenult. pro <math>P</math>, lege <math>P^.</math>; pag. 45. l. 27. pro affirmativa, lege nega-
tiva; pag. 53. l. 10. pro <math>b x</math>, lege <math>b x^n</math>; pag. 51. 1. 8. pro <math>v,</math> lege <math>v, l, t, &c.</math>; l. antepenult. pro
fluxione, lege fluxio; pag. 58. l. 13. dele)<_>t</_>, & pro <math>U</math>, lege <math>U′</math>; pag. 60. l. ultim. pro <math>n + 2</math>, lege
<math>n + 1, & pro m - 1, lege m + 1</math>; pag. 61. l. 1. pro <math>n - 2</math>, lege <math>n + 1</math>; pag. 64. l. 8. pro par, lege
impariter par; l. 15. pro <math>± p</math>, lege <math>∓ p</math>; l. antepenult. & penult. pro <math>α x, β x & γ x, lege 2 α x,
2 β x & 2 γ x</math>; pag. 68. l. 11. pro $ingularium, lege $ingularum; pag. 71. l. 1. pro <math>x^. ^n</math>, lege x^n</math>;
pag. 72. l. 1. pro, cujus, lege in <math>v^.</math> ducta, cujus; pag. 73. l. 16. pro <math>Q ×, lege Q × ()</math>; pag.
74. l. 15. po$t <math>(π, ρ, σ, τ, &c.)</math>, lege etiamque <math>(π^2, ρ^2, σ^2, &c., π^3, ρ^3, σ^3, &c., &c.)</math>; pag. 75. l.
8. pro $uos, lege ejus; pag. 77. l. 3. pro ... +, lege ... + <math>E</math>; pag. 81. 1. 20. pro $luxione,
lege fluxionem; pag. 81. 1. 20. pro <math>z^.</math>, lege <math>z^-1 z^.</math>; pag. 81. 1. 6. pro functio, lege rationalis $un-
ctio; pag. 81. 1. 8. dele <math>V</math>; l. antepenult. pro $umma, lege earum $umma; pag. 86. l. 5. pro x
$emel, lege <math>z</math>; pag. 91. 1. 27. pro <math>m ± r, lege m ± λ</math>; pag. 91. 1. 9. pro reducendæ, lege redu-
cantur; pag. 91. 1. 26. pro <math>s</math>, lege <math>s^.</math>; pag. 99. l. 19. pro <math>X</math>, lege <math>X^.</math>; pag. 101. 1. 16. pro <math>σ & v</math>,
lege <math>σ, τ & v</math>; pag. 101. 1. 6. pro <math>y^.</math>, lege <math>y</math>; pag. 111. 1. 15. pro pro unde, lege unde<math>{P / Q}</math>; pag.
121. 1. 18. <math>pro = 0, lege = 0, &c.</math>; pag. 131. 1. 4. pro exigat, lege exigere; pag. 131. 1. 13. pro
<math>δ x, lege δ y</math>; pag. 141. 1. 7. pro coefficientibus, lege coefficientibus corre$pondentium termino-
rum; l. 17. pro <math>x^.</math>, lege <math>x</math>; pag. 141. 1. 10. pro $uo, lege ejus; pag. 150. l. 23. pro utri$que par-
tibus, lege ad utra$que partes; pag. 152. l. 24. & pag. 151. 1. 4. pro alteri parti, lege ad alteram
partem; 1. 22. pro fluenti, lege ad fluentem; pag. 151. 1. 23. pro <math>V^.</math>, lege <math>V^. = 0</math>; pag. 160.
1. ultim. pro <math>$. X x^.</math>, lege <math>X x^.</math>; pag. 161. 1. 23. pro <math>n - m</math>, lege <math>m</math>; pag. 161. 1. 7. pro <math>Q</math>, lege
<math>- Q; 1. 7. pro - r) = 0, lege - r)) = 0</math>; pag. 171. 1. 21. pro fluxionum, lege fluentes fluxio-
num; pag. 171. 1. 17. pro <math>m - 1 + 1, lege m - n + 1</math>; 1. 19. pro æquatione, lege quantitate;
pag. 171. 1. 17. pro prius, lege po$tea; pag. 181. 1. 5. pro) 2, lege)<_>2</_>; 1. 6. pro <math>R R^. P^. P^..</math>, lege
<math>R^. P^. P^..</math>; pag. 181. 1. 2. pro <math>π^..</math>, lege <math>π^.</math>; pag. 191. 1. 24. pro <math>x^n</math>, lege <math>x^n & y^n</math>; pag. 191. 1. 19.
<math>{q x^. / q}</math>, lege <math>{q x^. / p}</math>; pag. 201. 1. 25. pro <math>x</math>, lege <math>x^.</math>; pag. 201. 1. 1. pro <math>x^l+n-b-1, lege x^l+n-b-1 +
&c.</math>; 1. 20. pro vel, lege aliter; pag. 201. 1. antepenult. dele <math>x^.</math>; pag. 201. 1. penult. pro <math>X x^.</math>,
lege <math>X x^. ^2</math>; pag. 211. 1. 8. pro notas. lege datas, & pro inveniatur, lege tum invenietur; pag.
211. 1. 6. pro <math>X y^. ^2</math>, lege <math>X^. y^. ^2</math>; 1. 10. pro <math>P p x</math>, lege <math>P p x^.</math>; 1. 15. pro <math>P q x^.</math>, lege <math>P q y^.</math>; pag. 213.
1. 6. pro <math>p y^.</math>, lege <math>p y^..</math>, & pro <math>σ</math>, lege <math>σ</math>; l. ultim. pro <math>τ M^.</math>, lege <math>τ M</math>; pag. 225. l. 2. pro habet,
lege habent; pag. 231. 1. 7. pro <math>x z, lege x^2</math>; pag. 231. 1. 7. pro transformantur, lege transfor-
mare; 1. 8. pro reducatur æquatio in, lege reducere æquationem ad; pag. 231. 1. 6. pro præ-
dictis, lege e prædictis; pag. 251. 1. 2. pro <math>b^2 - a</math>, lege <math>b^2 - c a</math>; 1. 4. pro equat, lege æquat;
pag. 261. 1. 18. pro <math>+ K</math>, lege × <math>I</math>; 1. 19. pro <math>I</math>, lege <math>H</math>; 1. 20. pro <math>H</math>, lege <math>G</math>; pag. 261. 1.
penult. collocentur <math>n, n - 1, n - 2</math> $upra <math>y^., y^., y^.</math>; pag. 261. 1. 6. pro e fluxionali æquatione (<math>m</math>)
ordinis, lege e fluxionalibus æquationibus (<math>m - 1, m - 2, m - 3, &c.</math>) ordinum; & pro deduci po-
te$t, lege deduci pote$t; in his duobus ca$ibus <math>A, B, C, &c, N, M, L, &c.; p, q, r, &c.</math> denotant
datas functiones quantitatis <math>x</math>; pag. 261. 1. 11 & 14. pro <math>x, y, &c.</math>, lege <math>p, x, y, &c.</math>; 1. 16. pro
<math>y, x & x^.</math>, lege <math>p, y, x & x^.</math>; 1. 17. pro <math>x</math>, lege <math>p & x</math>, & pro $it variabilis, lege $int variabiles;
1. antepenult. pro <math>β y^.</math>, lege <math>p β x^.</math>; pag. 271. 1. 6. pro <math>y^.</math> $emel, lege <math>y</math>; pag. 271. 1. 3. pro <math>y^. ^n-1 & y^. ^n-2</math>
$emel lege <math>z^. ^n-1 & z^. ^n-2</math>; pag. 281. 1. 8. pro <math>n + 1, n & n - 1, lege n, n - 1 & n - 2</math>; 1. 13. pro

<pb>[0776]<rh>CORRIGENDA.</rh>
<math>x^. ^n-1</math>, lege <math>x^. ^n-1</math>; 1. 14. <math>pro = 0, lege = E</math>; pag. 281. 1. 8. pro termini, lege in terminis; 1. 9.
dele & re$ultant <math>u = {y √(y^. ^2 + x^. ^2) / x^.} & t = x + {y y^. / x^.}</math>; pag. 281. 1. 23. pro <math>B = 0</math>, lege <math>W = 0</math>; pag.
291. 1. 25. pro <math>P^.</math>, lege <math>P</math>; pag. 291. 1. 19. pro <math>a x^. y^.</math>, lege <math>2 x^. y^.</math>; pag. 291. 1. 11. pro <math>R</math>, lege <math>R′</math>;
pag. 301. 1. 11. pro <math>P x^.</math>, lege <math>P z^.</math>; 1. 9 & 10. pro <math>A</math>, lege <math>A</math>; pag. 301. 1. 7. pro <math>x + x^.</math>, lege
<math>x + x<x>.</x></math>; pag. 301. 1. penult. pro fluxionales, lege integralis; pag. 301. 1. 4. pro <math>z(1 - z)^2</math>, lege
<math>z^2</math>; pag. 311. 1. 6. pro <math>$. {z^. / z}</math>, lege <math>- $. {z^. / z}</math>; pag. 311. 1. 11. pro <math>n - n x<x>.</x></math>, lege <math>x - n x<x>.</x></math>; 1. 18. pro
<math>(n - 1)</math>, lege <math>- (n - 1)</math>; pag. 311. 1. 11. pro <math>ax</math>, lege <math>(n + 1)a x</math>; pag. 311. 1. penult. pro <math>a′^n-2</math>,
lege <math>a′^n-1</math>; 1. ult. pro <math>b′, b″, &c.</math>, lege <math>b′(), b″(), &c.</math>; pag. 318. pro <math>d′, d″, &c., e, e″, &c., f, g,
&c. l</math>, lege <math>d′(), d″(), &c., e(), e′(), f(), g(), &c.</math>; 1. 7. pro <math>l′^π</math>, lege <math>l′^π-2; 1. 8 & 9.</math>
pro <math>l″^π & l′^π-1</math>, lege <math>l″^π () & l′^π-1 ()</math>; pag. 321. 1. 12. pro <math>x^. + e</math>, lege <math>x +</math> (<math>e</math>; 1. 17. pro <math>l</math>, lege
<math>k</math>, & pro <math>π</math>, lege <math>π′</math>, & pro <math>π - 1</math>, lege <math>π′ - 1</math>; 1. 20. pro. lege modo <math>n</math> major $it quam <math>m</math> per duo
vel plures, $in aliter for$an exigat <math>(π′)</math> prædicti generis integralia; pag. 321. 1. 29. pro illarum,
lege harum; pag. 321. 1. 14. pro quæ $it <math>B</math>, lege $cribantur hæ quantitates pro $uis valoribus in
integrali <math>A</math> & re$ultet <math>B</math>; 1. 17 & 19. pro <math>x</math>, lege <math>z</math>; pag. 321. 1. 14. pro <math>{r^z z<x>.</x> / }</math>, lege <math>{r^x r^x<x>.</x> z<x>.</x> / R^2}</math>, & pro
<math>Q =</math>, lege <math>Q = v^..</math>; pag. 331. 1. 5. pro =, lege = -, & pro <math>{b / A}x</math>, lege <math>{b / A}x x^.</math>; pag. 331. 1. 15.
pro <math>P</math>, lege <math>P<x>.</x></math>; pag. 341. 1. 1. pro applicari, lege applicare; 1. 20. pro facilies, lege facile; pag.
351. 1. 8. pro de, lege ex; pag. 351. 1. 1. pro <math>x^m</math>, lege <math>x^mr</math>, & pro <math>(r - 1) m & (r - 2) m</math>, lege
<math>(m - 1)r & (m - 2)r</math>; pag. 371. 1. 2. pro <math>(n - 1)q.</math> lege <math>(n - 1)p</math>; pag. 371. 1. 1. pro radicum,
lege radicum ductis; pag. 371. 1. 17. pro 3. lege 3. Sit <math>α</math> æqualis vel major quam <math>β, β</math> quam <math>γ,
γ</math> quam <math>δ &c.</math>; tum; pag. 381. 1. 16. pro <math>(a - α)</math>, lege <math>- (α - a)</math>; pag. 391. 1. 5. pro <math>α</math>, lege <math>a</math>;
pag. 391. 1. 24. pro <math>x^x + x^x-1</math>, lege <math>1 + {1 / x}</math>; pag. 401. 1. 9. pro vel <math>γ, &c</math>, lege <math>γ, &c</math>. vel <math>α′ vel β′, &c.</math>
vel <math>α″, &c.</math>; pag. 401. 1. 7. dele <math>Q R</math>; pag. 401. 1. 6. pro <math>b</math> $emel lege <math>β</math>; 1. 17. pro + $emel
lege -; pag. 401. 1. 10. pro <math>+ N e^5</math>, lege <math>± N e^5</math>; pag. 411. 1. ult. pro <math>a x</math> $emel lege <math>a x^3</math>;
pag. 411. 1. 1. pro æquatione, lege quantitate; pag. 421. 1. 1. pro <math>1 - {1 / 2}x</math>, lege <math>x - {1 / 2} x^2</math>; 1. 2.
pro - $emper lege +; 1. 16. pro <math>1 - {1 / 2 · 2^2}</math>, lege <math>1 + {1 / 2 · 2^2}</math>, & pro -, lege +; pag. 421. 1. 11.
pro <math>{cα - a / b - dα}</math>, lege <math>{cβ - a / b - aβ}</math>; pag. 421. 1. 3. pro eadem, lege for$an eadem; 1. 7. pro <math>+ {1 / 5}</math>, lege <math>- {1 / 5}</math>; pag.
431. 1. 10. pro <math>x - x</math>, lege <math>x - x<x>.</x></math>; pag. 441. 1. 7. pro <math>x - a & (x - a)^2</math>, lege <math>x - 2 & (x - 2)^2</math>; pag. 444.
1. 2. pro quantitates, lege quantitatis; pag. 451. 1. 17. pro <math>a + e′</math>, lege <math>e′ - a</math>; 1. 19. pro <math>a′ + e″</math>, lege
<math>e″ - a</math>; 1. 21. pro <math>a′^m + e′^m+1</math>, lege <math>e′^m+1 - a′^m</math>; 1. 23. pro <math>a + e′ & a′ + e″</math>, lege <math>e′ - a & e″ - a′</math>; pag. 452.
1. 13. pro approximatione, lege approximatione; cum a$$umi po$$it, tum <math>(n)</math> ad libitum a$$umi
pote$t; 1. 14. pro cum <math>r + s + t</math>, lege cum <math>t + s + b</math> vel <math>r + s + 2b</math>; pag. 451. 1. 4. pro approxima-
tionibus, lege approximationibus inveniendis, & pro <math>z</math>, lege <math>v</math>; pag. 451. 1. 22. pro <math>{a^2 / α^2}</math>, lege
<math>{α^2 / a^2}</math>; pag. 451. 1. ult. pro <math>z^t</math>, lege <math>x^t</math>; pag. 461. 1. 3 & 4. pro <math>n - 1, n - 2, n - 3 & n</math>, lege <math>n - 2,
n - 3 & n - 1</math>; pag. 461. 1. pro <math>x^5</math>, lege <math>x^4</math>; 1.5. pro <math>x^2</math>, lege <math>x^2r</math>; pag. 461. 1. 15. pro <math>W′^{m / s},</math>

<pb>[0777]<rh>CORRIGENDA.</rh>
lege <math>W′^{ml / s}</math>; 1. 18. pro <math>a x^r</math>, lege <math>a′ x^r</math>; pag. 461. 1. 13. pro <math>p a^n-1 & q a^n-2</math>, lege <math>p a^n-2 & q a^n-3</math>;
pag. 471. 1. 12. dele in his ca$ibus $æpe præ$tat affumere <math>α = 0, β = 0, γ = 0, &c.</math>; pag. 481. 1.
5. pro <math>3 z e + e^2</math>, lege <math>6 z e + 3 e^2</math>; pag. 481. 1. 10. pro præcedentes, lege præcedentium; pag.
491. 1. 6. pro i. e. lege viz.; 1. 14. pro 0, lege infin.; 1. 17. pro <math>T t^{1 / 2} + T t = 1</math>, lege <math>T = t T^{1 / 2}
+ t</math>;
pag. 491. 1. 6. pro <math>τ</math>, lege <math>τ x^n-2</math>; pag. 491. 1. 5. pro <math>s t′</math>, lege <math>r t′</math>; pag. 491. 1. 2. pro
<math>(n + 1)</math>, lege <math>2(n + 1)</math>; 1. 3. pro <math>(n + 2)</math>, lege <math>3(n + 2)</math>; pag. 501. 1. 2. pro <math>× (n + 1) × (m + 1)</math>,
lege <math>× (m + 1)</math>; pag. 501. 1. 9. pro <math>{1 / }</math>, lege <math>{{1 / 2} / }</math>; 1. 13 & 14. pro <math>{1 / }</math>, lege <math>{{1 / 2} / }</math>; pag.
501. 1. 10. pro vel, lege &; pag. 501. 1. 8. pro <math>2 z′ + 5</math>, lege <math>4 z′ + 5</math>; pag. 501. 1. 7. pro <math>2 h^2 +
3 h</math>, lege <math>2 h^2 + 3 h + 1</math>; 1. 10. pro <math>h z + h + 1</math>, lege <math>h z + 4 h + 1</math>; pag. 511. 1. 9. pro <math>α + α</math>, lege
<math>α + a</math>; pag. 511. 1. 9. pro <math>{n + 1 / 2}</math>, lege <math>n + 1</math>; pag. 521. 1. 5. pro <math>{1 / h &c.}</math>, lege <math>{1 / h^2 &c.}</math>, & pro
<math>{1 / (γ - α)}</math>, lege <math>{1 / b^2 (γ - α)}</math>; pag. 521. 1. 9. pro <math>x</math>, lege <math>x^πb+2b</math>; pag. 521. 1. 6. pro re$ultantis, lege
re$ultantis & datæ; 1. 23. pro $ed, lege &; 1. 26. pro fingantur, lege evadant; 1. 27. pro 0, lege
0, &c.; pag. 521. 1. 3. pro $ummæ, lege $umma, & pro æquales, lege æqualis; 1. 20. pro termini
ad di$tantias a primo, lege termini datæ $eriei, qui incipiunt ad di$tantias ab ejus primo 0, &c.;
pag. 531. 1. ult. pro termini, lege terminis; pag. 531. 1. 14. pro <math>A, B, C, D, &c.</math>, lege <math>{A / a}, {B / b},
{C / c}, {D / d}, &c.</math>; pag. 541. 1. 8 & 9. pro <math>z^-1, z^-2 & z^-3</math>, lege <math>z, z^2 & z^3</math>; pag. 541. 1. 1. pro <math>b B +
1. 2 c C</math>, lege <math>a′ + b B + b′ A + 2 C c + &c</math>; pag. 571. 1. 12. pro tum, lege tum pro termino
<math>(t)</math>; pag. 571. 1. 6. pro <math>t^....</math>, lege 16 <math>t^....</math>; 1. 12. pro <math>t′</math>, lege <math>t′^r</math>; pag. 571. 1. 7. pro <math>m + 1</math>, lege <math>m - 1</math>;
1. 11 & 13. pro 1. 2 &, lege <math>± 1 · 2, &c.</math>; pag. 591. 1. 4. pro $it, lege a$$umatur; pag. 601. 1.
4. pro <math>r</math>, lege <math>r - 1</math>; pag. 611. 1. 13. pro $unt, lege $int; pag. 611. 1. 12. pro <math>x^2 & x^3</math>, lege <math>x &
x^2</math>; pag. 621. 1. 6. pro <math>π^2</math>, lege <math>π^2λ</math>; pag. 621. 1. 13. dele 4 bis; pag. 621. 1. 16. pro valores 0,
lege valores 0 vel infinitum; pag. 631. 1. 3. pro <math>A</math>, lege <math>A^-1</math>; pag. 631. 1. 1. pro <math>A</math>, lege <math>A′</math>;
pag. 641. 1. 2. pro <math>b</math>, lege <math>- b</math>; pag. 651. 1. 14. dele hinc; pag. 671. 1. 2 & 3. pro <math>2 m, 2n + 2m
& 4n + m</math>, lege <math>2m + 1, 2n + 2m + 1 & 4n + 2m + 1</math>, & pro <math>a^m, a^2n+m & a^4m+m</math>, lege <math>a^2m+1</math>,
<math>a^2n+2m+1 & a^4n+2m+1</math>; 1. 9. dele re$pective; 1. 10 & 16. pro ductæ, lege ducti; pag. 681. 1. ·
pro eædem, lege eædem po$$ibiles; pag. 681. 1. antepen. pro $unt, lege $unt <math>(n′)</math>; pag. 681. 1. 1
& 8. pro <math>(n)</math>, lege <math>(n′)</math>; 1. 7. pro <math>c″ & c′</math>, lege <math>c′ & c″</math>; pag. 681. 1. · pro <math>α + z, β + z, γ + z,
δ + z</math>, lege <math>α + 2n, β + 2n, γ + 2n, δ + 2 n</math>, re$pective; pag. 681. 1. 2. pro cujus, lege quarum;
pag. 691. 1. 3. & 6. pro alter, lege altera; pag. 711. 1. pen. pro <math>q^2</math>, lege <math>q</math>.</p>

<pb>[0778]

<pb>[0779]

<pb>[0780]

<pb>[0781]

<pb>[0781a]
<fig>
<cap it>Fig. 1.</cap>
<var>C _i k d_ B _f g c h e b_ A</var>
<cap it>Fig. 2.</cap>
<var>_e d c h a b_ I E D C B A</var>
<cap it>Fig. 3.</cap>
<var>A _p_ A P</var>
<cap it>Pag. 581, 2, 3.</cap>
<var>V R F A M P E B _v R_ K C L D _f a_</var>
<cap it>Pag. 586.</cap>
<var>η π V R F A _e b_ ε E B _k c_ κ K C _l d_ λ L D _m n p_ T N M P _y x v r_ Z T′ _f_</var>
<cap it>Pag. 602.</cap>
<var>A α Q π R _b l_ β ξ _m c_ γ σ _d n_ M L P δ _o e_</var>
<cap it>Fig. 4.</cap>
<var>O N L P A Q B R C S D T E F</var>
<cap it>Fig. γ</cap>
<var>_a m b_ π α _c d e g f h_ β ξ _k o i p l q r b n_ γ σ</var>
<cap it>Pag. 584.</cap>
<var>E F A E B M P K C _f a_</var>
<var it>h a m f o k k n b g l</var>
<cap it>Pag. 147.</cap>
<var>R A S _e f m p_ M _h_ P</var>
<cap it>Fig β</cap>
<var>_a m_ ξ _h_ μ″ μ′ μ α ν″ ν′ ν β π _h′ b n_</var>
<var it>h a m f k o g b n _A_ l</var>
<cap it>Fig. α</cap>
<var it>h l a m f n b g k o k _A_</var>
</fig>

<pb>[0782]

<pb>[0783]

<pb>[0784]