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Version vom 2009-02-14
author Klaus Thoden <kthoden@mpiwg-berlin.mpg.de>
date Thu, 02 May 2013 11:08:12 +0200
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1 <?xml version="1.0" encoding="utf-8" ?>
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5 <html>
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7 <head><title></title>
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9 <meta name="generator" content="TeX4ht (http://www.cse.ohio-state.edu/~gurari/TeX4ht/mn.html)" />
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10 <meta name="originator" content="TeX4ht (http://www.cse.ohio-state.edu/~gurari/TeX4ht/mn.html)" />
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12 <meta name="src" content="Einst_Ueber_de_1910.tex" />
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13 <meta name="date" content="2005-03-10 17:36:00" />
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14 <link rel="stylesheet" type="text/css" href="Einst_Ueber_de_1910.css" />
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15 </head><body >
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17 <!--l. 12--><p class="noindent"><pb/></p>
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18 <div class="center" >
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20 <!--l. 13--><p class="noindent">
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21 </p><!--l. 14--><p class="noindent"><span
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22 class="cmr-12x-x-120">2</span><span
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23 class="cmbx-12x-x-120">. </span><span
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24 class="cmbxti-10x-x-144">Über einen Satz </span> <br/><span
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25 class="cmbxti-10x-x-144">der Wahrscheinlichkeitsrechnung</span>
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26 <span
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27 class="cmbxti-10x-x-144">und seine </span> <br/><span
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28 class="cmbxti-10x-x-144">Anwendung in der Strahlungstheorie; </span> <br/><span
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29 class="cmbxti-10x-x-144">von</span>
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30 <span
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31 class="cmbxti-10x-x-144">A. Einstein und L. Hopf.</span></p></div>
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32 <div class="center" >
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34 <!--l. 20--><p class="noindent">
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35 </p><!--l. 21--><p class="noindent">----------</p></div>
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36 <div class="center" >
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38 <!--l. 24--><p class="noindent">
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39 </p><!--l. 25--><p class="noindent"><span
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40 class="cmsy-10x-x-120">§ </span>1. Das physikalische Problem als Ausgangspunkt.</p></div>
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41 <!--l. 29--><p class="indent"> Will man in der Theorie der Temperaturstrahlung irgend <br/>eine Wirkung der Strahlung berechnen, etwa die auf einen <br/>Oszillator wirkende Kraft, so verwendet man dazu stets als <br/>analytischen Ausdruck für die elektrische oder magnetische
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42 <br/>Kraft Fouriersche Reihen der allgemeinen Gestalt
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43 </p>
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44 <center class="par-math-display" >
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45 <img
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46 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Ueber_de_1910/fulltext/img/Einst_Ueber_de_19100x.png" alt=" sum -t t- nA n sin 2 p nT + Bn cos 2 p n T . " class="par-math-display" /></center>
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47 <!--l. 40--><p class="nopar">
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48 </p><!--l. 43--><p class="noindent">Hierbei ist das Problem gleich auf einen bestimmten Raum-<br/>punkt spezialisiert, was für das Folgende ohne Bedeutung <br/>ist, <span
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49 class="cmmi-12">t </span>bedeutet die variable Zeit, <span
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50 class="cmmi-12">T </span>die sehr große Zeitdauer, <br/>für welche die Entwickelung gilt. Bei der Berechnung irgend-<br/>welcher Mittelwerte -- und nur solche kommen in der Strahlungs-<br/>theorie überhaupt vor -- nimmt man die einzelnen Koeffi-<br/>zienten <span
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51 class="cmmi-12">A</span><sub ><span
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52 class="cmmi-8">n</span></sub>, <span
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53 class="cmmi-12">B</span><sub ><span
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54 class="cmmi-8">n</span></sub> als unabhängig voneinander an, man setzt voraus, <br/>daß jeder Koeffizient unabhängig von den Zahlenwerten der <br/>anderen das Gauss sche Fehlergesetz befolge, so daß die
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55 <br/>Wahrscheinlichkeit<sup ><span
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56 class="cmr-8">1</span></sup>) <span
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57 class="cmmi-12">dW </span>einer Kombination von Werten <span
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58 class="cmmi-12">A</span><sub >
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59 <span
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60 class="cmmi-8">n</span></sub>, <span
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61 class="cmmi-12">B</span><sub ><span
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62 class="cmmi-8">n</span></sub> <br/>sich aus den Wahrscheinlichkeiten der einzelnen Koeffizienten <br/>einfach als Produkt darstellen müsse.</p>
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63 <table width="100%"
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64 class="equation"><tr><td><a
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65 id="x1-2r1"></a>
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66 <center class="math-display" >
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67
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68 <img
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69 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Ueber_de_1910/fulltext/img/Einst_Ueber_de_19101x.png" alt="d W = WA .WA ... WB .WB ... dA1 ... dB1 ... 1 2 1 2 " class="math-display" /></center></td><td width="5%">(1)</td></tr></table>
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70 <!--l. 58--><p class="nopar">
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71 </p><!--l. 62--><p class="indent"> Da bekanntlich die Strahlungslehre, so wie sie exakt aus <br/>den allgemein anerkannten Fundamenten der Elektrizitäts-<br/>----------
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72 </p><!--l. 67--><p class="indent"> 1) Unter ,,Wahrscheinlichkeit eines Koeffizienten&#8220; ist offenbar <br/>folgendes zu verstehen: Wir denken uns die elektrische Kraft in sehr <br/>vielen Zeitmomenten in Fourier sche Reihen entwickelt. Derjenige <br/>Bruchteil dieser Entwickelungen, bei welchem ein Koeffizient in einem <br/>bestimmten Wertbereich liegt, ist die Wahrscheinlichkeit dieses Wert-<br/>bereiches des betreffenden Koeffizienten.
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73 <pb/>
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74 </p><!--l. 78--><p class="indent">
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75
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76 </p><!--l. 79--><p class="noindent">theorie und der statistischen Mechanik folgt, in unlösbare Wider-<br/>sprüche mit der Erfahrung führt, liegt es nahe, dieser ein-<br/>fachen Annahme der Unabhängigkeit zu mißtrauen und ihr die <br/>Schuld an den Mißerfolgen der Strahlungstheorie zuzuschreiben.
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77 </p><!--l. 85--><p class="indent"> Im folgenden soll nun gezeigt werden, daß dieser Ausweg <br/>unmöglich ist, daß sich vielmehr das physikalische Problem <br/>auf ein rein mathematisches zurückführen läßt, das zum <br/>statistischen Gesetze (1) führt.
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78 </p><!--l. 91--><p class="indent"> Betrachten wir nämlich die aus einer bestimmten Rich-<br/>tung herkommende<sup ><span
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79 class="cmr-8">1</span></sup>) Strahlung, so hat diese gewiß einen höheren <br/>Grad von Ordnung, als die gesamte in einem Punkte wirkende <br/>Strahlung. Die Strahlung aus einer bestimmten Richtung
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80 <br/>können wir aber immer noch auffassen als von sehr vielen <br/>Emissionszentren herrührend, d. h. wir können die Fläche, <br/>welche die Strahlung aussendet, noch in sehr viele unabhängig <br/>voneinander ausstrahlende Flächenelemente zerlegen; denn der <br/>Entfernung dieser Fläche vom Aufpunkt sind ja keine Grenzen <br/>gesteckt, also auch nicht ihrer gesamten Ausdehnung. In <br/>diese von den einzelnen Flächenelementen herrührenden Strah-<br/>lungselemente führen wir wieder ein höheres Ordnungsprinzip <br/>ein, indem wir diese Strahlungselemente alle als von gleicher <br/>Form und nur durch eine zeitliche Phase verschieden auf-<br/>fassen; mathematisch gesprochen: die Koeffizienten der Fourier-<br/>schen Reihen, welche die Strahlung der einzelnen Flächenele-<br/>mente darstellen, seien für alle Flächenelemente dieselben, nur <br/>der Anfangspunkt der Zeit von Element zu Element verschie-<br/>den. Können wir Gleichung (1) unter Zugrundelegung dieser <br/>Ordnungsprinzipien beweisen, so gilt sie a fortiori für den <br/>Fall, daß man diese Ordnungsprinzipien fallen läßt. Be-<br/>zeichnet der Index <span
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81 class="cmmi-12">s </span>das einzelne Flächenelement, so erhält <br/>die dort ausgesandte Strahlung die Form:
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82 </p>
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83 <center class="par-math-display" >
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84 <img
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85 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Ueber_de_1910/fulltext/img/Einst_Ueber_de_19102x.png" alt=" sum t---ts- (n) an sin 2p n T . " class="par-math-display" /></center>
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86 <!--l. 124--><p class="nopar">
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87 </p><!--l. 127--><p class="noindent">Die gesamte von uns betrachtete Strahlung wird also dar-<br/>gestellt durch die Doppelsummen:
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88 </p>
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89 <table width="100%"
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90 class="equation"><tr><td><a
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91 id="x1-3r2"></a>
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92 <center class="math-display" >
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93 <img
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94 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Ueber_de_1910/fulltext/img/Einst_Ueber_de_19103x.png" alt=" ( ) sum s sum n a sin 2p n t-cos 2 pn ts - cos 2p n t-sin 2p n ts . n T T T T " class="math-display" /></center></td><td width="5%">(2)</td></tr></table>
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95 <!--l. 136--><p class="nopar">
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96 </p><!--l. 139--><p class="noindent">----------
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97 </p><!--l. 142--><p class="indent"> 1) genauer; ,,einem bestimmten Elementarwinkel <span
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98 class="cmmi-12">d</span><span
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99 class="cmbx-12">x</span> entsprechende&#8220; <br/>Annalen der Physik. IV. Folge. 38. <pb/>
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100 </p><!--l. 148--><p class="indent">
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101
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102 </p><!--l. 149--><p class="noindent">Vergleichung von (2) und (1) führt also zu den Ausdrücken:</p>
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103 <table width="100%"
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104 class="equation"><tr><td><a
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105 id="x1-4r3"></a>
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106 <center class="math-display" >
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107 <img
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108 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Ueber_de_1910/fulltext/img/Einst_Ueber_de_19104x.png" alt=" sum ts { An = an s cos 2 pn T , sum ts Bn = an n sin 2 pn -- , T " class="math-display" /></center></td><td width="5%">(3)</td></tr></table>
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109 <!--l. 160--><p class="nopar">
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110 </p><!--l. 164--><p class="noindent"><span
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111 class="cmmi-12">n </span>ist eine sehr große Zahl, <span
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112 class="cmmi-12">t</span><sub ><span
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113 class="cmmi-8">s</span></sub> kann jeden Wert zwischen 0 <br/>und <span
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114 class="cmmi-12">T </span>annehmen, die einzelnen Summanden
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115 </p>
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116 <center class="par-math-display" >
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117 <img
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118 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Ueber_de_1910/fulltext/img/Einst_Ueber_de_19105x.png" alt="cos 2p n ts bzw. sin 2 pn ts T T " class="par-math-display" /></center>
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119 <!--l. 174--><p class="nopar">
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120 </p><!--l. 178--><p class="noindent">liegen also regellos zwischen <span
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121 class="cmsy-10x-x-120">-</span>1 und +1 verteilt und sind <br/>gleich wahrscheinlich positiv wie negativ. Können wir für <br/>eine Kombination von Summen solcher Größen allgemein die <br/>Gültigkeit unserer Gleichung (1) nachweisen, so ist damit auch <br/>die Unmöglichkeit erwiesen, irgend ein Ordnungsprinzip in die <br/>im leeren Raum sich ausbreitende Strahlung einzuführen.
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122 </p>
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123 <div class="center" >
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124
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125 <!--l. 188--><p class="noindent">
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126 </p><!--l. 189--><p class="noindent"><span
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127 class="cmsy-10x-x-120">§ </span>2. Formulierung des allgemeinen mathematischen Problems.</p></div>
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128 <!--l. 193--><p class="indent"> Wir stellen uns also folgendes mathematische Problem: <br/>Gegeben ist eine sehr große Anzahl von Elementen, deren <br/>Zahlenwerte <span
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129 class="cmmi-12"><img
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130 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Ueber_de_1910/fulltext/img/cmmi12-b.png" alt="a" class="12x-x-b" /> </span>ein bekanntes statistisches Gesetz befolgen <br/>(entsprechend den <span
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131 class="cmmi-12">t</span><sub ><span
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132 class="cmmi-8">s</span></sub>). Von jedem dieser
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133 Zahlenwerte werden <br/>gewisse Funktionen <span
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134 class="cmmi-12">f</span><sub ><span
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135 class="cmr-8">1</span></sub> (<span
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136 class="cmmi-12"><img
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137 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Ueber_de_1910/fulltext/img/cmmi12-b.png" alt="a" class="12x-x-b" /></span>) <span
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138 class="cmmi-12">f</span><sub ><span
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139 class="cmr-8">2</span></sub> (<span
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140 class="cmmi-12"><img
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141 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Ueber_de_1910/fulltext/img/cmmi12-b.png" alt="a" class="12x-x-b" /></span>)<span
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142 class="cmmi-12">...</span> gebildet (entsprechend
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143 <br/><img
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144 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Ueber_de_1910/fulltext/img/Einst_Ueber_de_19106x.png" alt=" ) sin 2 p n ts-n .cos 2p n ts T T" class="left" align="middle" />. Diese Funktionen müssen wir <br/>noch einer Einschränkung unterwerfen: Es ergibt sich nämlich <br/>aus der Wahrscheinlichkeit, daß eine der Größen <span
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145 class="cmmi-12"><img
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146 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Ueber_de_1910/fulltext/img/cmmi12-b.png" alt="a" class="12x-x-b" /> </span>zwischen <br/><span
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147 class="cmmi-12"><img
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148 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Ueber_de_1910/fulltext/img/cmmi12-b.png" alt="a" class="12x-x-b" /> </span>+ <span
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149 class="cmmi-12">d<img
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150 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Ueber_de_1910/fulltext/img/cmmi12-b.png" alt="a" class="12x-x-b" /> </span>liegt, ein statistisches Gesetz für die <span
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151 class="cmmi-12">f</span>; die Wahr-<br/>scheinlichkeit <span
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152 class="cmmi-12"><img
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153 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Ueber_de_1910/fulltext/img/cmmi12-27.png" alt="f" class="cmmi-12x-x-27" align="middle" /></span><img
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154 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Ueber_de_1910/fulltext/img/Einst_Ueber_de_19107x.png" alt="(f)" class="left" align="middle" /> <span
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155 class="cmmi-12">df, </span>daß <span
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156 class="cmmi-12">f </span>einen Zahlenwert zwischen <span
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157 class="cmmi-12">f </span> <br/>und <span
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158 class="cmmi-12">f </span>+ <span
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159 class="cmmi-12">df</span>
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160 habe, sei nun stets eine solche Funktion, daß der <br/>Mittelwert
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161 </p>
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162 <center class="par-math-display" >
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163 <img
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164 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Ueber_de_1910/fulltext/img/Einst_Ueber_de_19108x.png" alt=" + integral oo
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165 f-= f f (f) df = 0.
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166 - oo
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167 " class="par-math-display" /></center>
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168 <!--l. 216--><p class="nopar">
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169 </p><!--l. 219--><p class="noindent">(Es ist leicht einzusehen, daß unsere Funktionen sin und cos <br/>wirklich diese Voraussetzung erfüllen; denn wenn jeder Wert <br/>von <span
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170 class="cmmi-12">t</span><sub ><span
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171 class="cmmi-8">s</span></sub> zwischen 0 und <span
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172 class="cmmi-12">T </span>gleich wahrscheinlich ist, verschwinden <br/>die Mittelwerte <span class="overline"> sin 2 <span
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173 class="cmmi-12"><img
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174 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Ueber_de_1910/fulltext/img/cmmi12-19.png" alt="p" class="12x-x-19" /> n</span><img
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175 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Ueber_de_1910/fulltext/img/Einst_Ueber_de_19109x.png" alt="ts -- T" class="frac" align="middle" /></span> und <img
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176 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Ueber_de_1910/fulltext/img/Einst_Ueber_de_191010x.png" alt="---------ts-) cos 2 p n-- . T" class="left" align="middle" />
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177 <pb/>
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178 </p><!--l. 229--><p class="indent">
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179
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180 </p><!--l. 230--><p class="indent"> Wir fassen nun eine (sehr große) Anzahl <span
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181 class="cmmi-12">Z </span>solcher Ele-<br/>mente <span
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182 class="cmmi-12"><img
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183 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Ueber_de_1910/fulltext/img/cmmi12-b.png" alt="a" class="12x-x-b" /> </span>zu einem System zusammen. Zu einem derartigen <br/>System gehören bestimmte Summen
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184 </p>
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185 <center class="par-math-display" >
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186 <img
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187 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Ueber_de_1910/fulltext/img/Einst_Ueber_de_191011x.png" alt=" sum sum
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188 (Z)f1(a), (Z)f2(a) ... " class="par-math-display" /></center>
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189 <!--l. 239--><p class="nopar">
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190 </p><!--l. 242--><p class="noindent">(entsprechend den Koeffizienten <span
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191 class="cmmi-12">A</span><sub ><sub ><span
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192 class="cmmi-6">n</span></sub></sub><img
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193 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Ueber_de_1910/fulltext/img/Einst_Ueber_de_191012x.png" alt="/" class="left" align="middle" /> <span
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194 class="cmmi-12">a</span><sub ><sub ><span
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195 class="cmmi-6">n</span></sub></sub><span
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196 class="cmmi-12">,</span> <span
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197 class="cmmi-12">B</span><sub ><sub ><span
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198 class="cmmi-6">n</span></sub></sub><img
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199 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Ueber_de_1910/fulltext/img/Einst_Ueber_de_191013x.png" alt="/" class="left" align="middle" /> <span
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200 class="cmmi-12">a</span><sub ><sub ><span
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201 class="cmmi-6">n</span></sub></sub>). Wir stellen <br/>uns die Aufgabe, das statistische Gesetz zu ermitteln, welches <br/>eine Kombination dieser Summen befolgt.
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202 </p><!--l. 250--><p class="indent"> Zunächst müssen wir über einen prinzipiellen Punkt Klar-<br/>heit schaffen:
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203 </p><!--l. 253--><p class="indent"> Das statistische Gesetz, das die Summen <span
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204 class="cmex-10x-x-120"><img
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205 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Ueber_de_1910/fulltext/img/cmex10-c-50.png" alt=" sum " class="10-120x-x-50" /></span>
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206 selbst be-<br/>folgen, wird gar nicht von der Anzahl <span
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207 class="cmmi-12">Z </span>der Elemente un-<br/>abhängig sein. Das können wir leicht an dem einfachen <br/>Spezialfall sehen, daß <span
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208 class="cmmi-12">f</span>(<span
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209 class="cmmi-12"><img
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210 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Ueber_de_1910/fulltext/img/cmmi12-b.png" alt="a" class="12x-x-b" /></span>) nur die Werte +1 und <span
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211 class="cmsy-10x-x-120">-</span>1 an-<br/>nehmen könne. Dann ist offenbar:
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212 </p>
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213 <center class="par-math-display" >
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214 <img
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215 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Ueber_de_1910/fulltext/img/Einst_Ueber_de_191014x.png" alt=" sum sum
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216 (Z+1) = (Z)± 1 " class="par-math-display" /></center>
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217 <!--l. 262--><p class="nopar">
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218 </p><!--l. 266--><p class="noindent">und
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219 </p>
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220 <center class="par-math-display" >
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221 <img
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222 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Ueber_de_1910/fulltext/img/Einst_Ueber_de_191015x.png" alt=" sum ------ sum ---- 2 = 2 + 1. (Z+1) (Z) " class="par-math-display" /></center>
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223 <!--l. 273--><p class="nopar">
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224 </p><!--l. 277--><p class="noindent">Der quadratische Mittelwert der Summe wächst also pro-<br/>portional mit der Anzahl der Elemente. Wollen wir also zu <br/>einem von <span
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225 class="cmmi-12">Z </span>unabhängigen statistischen Gesetze gelangen, so <br/>dürfen wir nicht die <span
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226 class="cmex-10x-x-120"><img
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227 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Ueber_de_1910/fulltext/img/cmex10-c-50.png" alt=" sum " class="10-120x-x-50" /></span>
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228 betrachten, sondern, da <span class="overline"> <span
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229 class="cmex-10x-x-120"><img
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230 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Ueber_de_1910/fulltext/img/cmex10-c-50.png" alt=" sum " class="10-120x-x-50" /></span>
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231 <sup ><span
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232 class="cmr-8">2</span></sup></span><img
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233 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Ueber_de_1910/fulltext/img/Einst_Ueber_de_191016x.png" alt="/" class="left" align="middle" /> <span
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234 class="cmmi-12">Z </span>kon-<br/>stant bleibt, die Größen
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235 </p>
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236 <center class="par-math-display" >
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237 <img
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238 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Ueber_de_1910/fulltext/img/Einst_Ueber_de_191017x.png" alt=" sum
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239 ---- S = V~ --. Z " class="par-math-display" /></center>
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240 <!--l. 289--><p class="nopar">
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241 </p>
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242 <div class="center" >
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243
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244
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245 <!--l. 293--><p class="noindent">
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246 </p><!--l. 294--><p class="noindent"><span
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247 class="cmsy-10x-x-120">§ </span>3. Statistisches Gesetz der einzelnen <span
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248 class="cmmi-12">S</span>.</p></div>
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249 <!--l. 298--><p class="indent"> Ehe wir nun eine Kombination aller Größen
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250 </p>
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251 <center class="par-math-display" >
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252 <img
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253 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Ueber_de_1910/fulltext/img/Einst_Ueber_de_191018x.png" alt=" sum
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254 S(n) = --(Z) V~ fn(a)- Z " class="par-math-display" /></center>
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255 <!--l. 303--><p class="nopar">
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256 </p><!--l. 306--><p class="noindent">untersuchen, wollen wir das Wahrscheinlichkeitsgesetz einer <br/>einzelnen solchen Größe aufstellen.
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257 </p><!--l. 309--><p class="indent"> Wir betrachten eine Vielheit von <span
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258 class="cmmi-12">N</span>-Systemen der oben <br/>definierten Art. Zu jedem System gehört ein Zahlenwert <span
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259 class="cmmi-12">S</span>. <br/>Diese Größen befolgen wegen der statistischen Verteilung der <span
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260 class="cmmi-12"><img
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261 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Ueber_de_1910/fulltext/img/cmmi12-b.png" alt="a" class="12x-x-b" /> </span> <br/>ein gewisses Wahrscheinlichkeitsgesetz, so daß die Anzahl der <br/>Systeme, deren Zahlenwert zwischen <span
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262 class="cmmi-12">S </span>und <span
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263 class="cmmi-12">S </span>+ <span
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264 class="cmmi-12">dS</span>
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265 liegt:</p>
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266 <table width="100%"
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267 class="equation"><tr><td><a
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268 id="x1-5r4"></a>
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269 <center class="math-display" >
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270 <img
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271 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Ueber_de_1910/fulltext/img/Einst_Ueber_de_191019x.png" alt="dN = F (S) d S . " class="math-display" /></center></td><td width="5%">(4)</td></tr></table>
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272 <!--l. 319--><p class="nopar">
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273 <pb/>
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274 </p><!--l. 326--><p class="indent">
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275
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276 </p><!--l. 327--><p class="noindent">Fügen wir nun zu den aus <span
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277 class="cmmi-12">Z</span>-Elementen bestehenden Systemen <br/>noch je ein weiteres Element, d. h. gehen wir von <span
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278 class="cmmi-12">S</span><sub ><span
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279 class="cmmi-8">Z</span></sub> zu <span
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280 class="cmmi-12">S</span><sub ><span
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281 class="cmmi-8">Z</span><span
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282 class="cmr-8">+1</span></sub> <br/>über, so werden die einzelnen Glieder unserer Vielheit ihren <br/>Zahlenwert ändern und in ein anderes Gebiet <span
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283 class="cmmi-12">dS </span>einrücken. <br/>Wenn es trotzdem möglich sein soll, zu einem von <span
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284 class="cmmi-12">Z</span>
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285 unab-<br/>hängigen statistischen Gesetz zu gelangen, so darf sich bei <br/>diesem Übergang die Anzahl <span
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286 class="cmmi-12">dN </span>nicht ändern. Es muß also <br/>in ein bestimmtes (in unserem einfachsten Fall eindimensionales) <br/>Gebiet <span
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287 class="cmmi-12">dS </span>die gleiche Anzahl von Systemen ein- wie austreten. <br/>Bezeichnet <img
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288 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Ueber_de_1910/fulltext/img/cmr12-8.png" alt="P" class="12x-x-8" /> die Zahl der Systeme, welche vom Übergang <br/>von <span
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289 class="cmmi-12">Z </span>zu <span
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290 class="cmmi-12">Z </span>+ 1 Elementen einen gewissen Zahlenwert <span
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291 class="cmmi-12">S</span><sub ><span
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292 class="cmr-8">0</span></sub>
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293 durch-<br/>schreiten und zwar sowohl der Größe wie der Richtung nach, <br/>so muß:</p>
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294 <table width="100%"
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295 class="equation"><tr><td><a
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296 id="x1-6r5"></a>
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297 <center class="math-display" >
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298 <img
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299 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Ueber_de_1910/fulltext/img/Einst_Ueber_de_191020x.png" alt="divP = 0, " class="math-display" /></center></td><td width="5%">(5)</td></tr></table>
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300 <!--l. 347--><p class="nopar">
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301 </p><!--l. 351--><p class="noindent">also
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302 </p>
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303 <center class="par-math-display" >
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304 <img
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305 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Ueber_de_1910/fulltext/img/Einst_Ueber_de_191021x.png" alt="d P ----= 0 d S " class="par-math-display" /></center>
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306 <!--l. 358--><p class="nopar">
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307 </p><!--l. 362--><p class="noindent">und, da ja <img
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308 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Ueber_de_1910/fulltext/img/cmr12-8.png" alt="P" class="12x-x-8" /> für <span
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309 class="cmmi-12">S </span>= <span
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310 class="cmsy-10x-x-120"><img
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311 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Ueber_de_1910/fulltext/img/cmsy10-c-31.png" alt=" oo " class="10-120x-x-31" /> </span>jedenfalls gleich 0 sein muß, auch</p>
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312 <table width="100%"
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313 class="equation"><tr><td><a
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314 id="x1-7r6"></a>
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315 <center class="math-display" >
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316 <img
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317 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Ueber_de_1910/fulltext/img/Einst_Ueber_de_191022x.png" alt="P = 0 .
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318 " class="math-display" /></center></td><td width="5%">(6)</td></tr></table>
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319 <!--l. 368--><p class="nopar">
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320 </p><!--l. 372--><p class="noindent">Nun ist:
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321 </p>
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322 <center class="par-math-display" >
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323 <img
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324 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Ueber_de_1910/fulltext/img/Einst_Ueber_de_191023x.png" alt=" sum V~ ------- (Z+1)f (a) Z f(a) S(Z+1) = -- V~ -------- = S(Z) ------+ V~ ------ , Z + 1 Z + 1 Z + 1 " class="par-math-display" /></center>
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325 <!--l. 381--><p class="nopar">
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326 </p><!--l. 385--><p class="noindent">oder, da <span
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327 class="cmmi-12">Z </span>eine sehr große Zahl sein soll:</p>
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328 <table width="100%"
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329 class="equation"><tr><td><a
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330 id="x1-8r7"></a>
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331 <center class="math-display" >
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332 <img
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333 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Ueber_de_1910/fulltext/img/Einst_Ueber_de_191024x.png" alt=" S(Z) f(a)- S(Z+1) = S(Z)- 2Z + V~ -- . Z " class="math-display" /></center></td><td width="5%">(7)</td></tr></table>
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334 <!--l. 393--><p class="nopar">
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335 </p><!--l. 396--><p class="noindent">Die Anzahl <img
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336 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Ueber_de_1910/fulltext/img/cmr12-8.png" alt="P" class="12x-x-8" /> setzt sich also aus zwei Teilen zusammen, <br/>einem <img
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337 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Ueber_de_1910/fulltext/img/cmr12-8.png" alt="P" class="12x-x-8" /><sub ><span
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338 class="cmr-8">1</span></sub>, der vom Summanden <span
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339 class="cmsy-10x-x-120">-</span><span
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340 class="cmmi-12">S</span><img
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341 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Ueber_de_1910/fulltext/img/Einst_Ueber_de_191025x.png" alt="/" class="left" align="middle" /> 2 <span
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342 class="cmmi-12">Z</span> und einem <img
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343 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Ueber_de_1910/fulltext/img/cmr12-8.png" alt="P" class="12x-x-8" /><sub ><span
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344 class="cmr-8">2</span></sub>, der <br/>von <span
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345 class="cmmi-12">f</span>(<span
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346 class="cmmi-12"><img
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347 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Ueber_de_1910/fulltext/img/cmmi12-b.png" alt="a" class="12x-x-b" /></span>)<img
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348 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Ueber_de_1910/fulltext/img/Einst_Ueber_de_191026x.png" alt="/" class="left" align="middle" /> <img
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349 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Ueber_de_1910/fulltext/img/Einst_Ueber_de_191027x.png" alt=" V~ -- Z" class="sqrt" /> herrührt.
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350 </p><!--l. 404--><p class="indent"> <img
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351 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Ueber_de_1910/fulltext/img/cmr12-8.png" alt="P" class="12x-x-8" /><sub ><span
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352 class="cmr-8">1</span></sub> enthält alle diejenigen <span
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353 class="cmmi-12">S</span>, welche in einem positiven <br/>Abstand <span
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354 class="msam-10x-x-120"><img
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355 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Ueber_de_1910/fulltext/img/msam10-c-35.png" alt="&lt;=" class="10-120x-x-35" /></span> <span
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356 class="cmmi-12">S</span><sub ><span
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357 class="cmr-8">0</span></sub><img
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358 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Ueber_de_1910/fulltext/img/Einst_Ueber_de_191028x.png" alt="/" class="left" align="middle" />2 <span
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359 class="cmmi-12">Z</span>
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360 vom Werte <span
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361 class="cmmi-12">S</span><sub ><span
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362 class="cmr-8">0</span></sub> gelegen waren; und zwar <br/>durchschreiten diese Glieder <span
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363 class="cmmi-12">S</span><sub ><span
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364 class="cmr-8">0</span></sub> in negativer Richtung. Ihre <br/>Anzahl ist, da <span
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365 class="cmmi-12">S</span><sub ><span
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366 class="cmr-8">0</span></sub><img
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367 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Ueber_de_1910/fulltext/img/Einst_Ueber_de_191029x.png" alt="/" class="left" align="middle" /> 2 <span
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368 class="cmmi-12">Z</span> eine sehr kleine Zahl ist, bis auf un-<br/>endlich kleine Größen höherer Ordnung:</p>
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369 <table width="100%"
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370 class="equation"><tr><td><a
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371 id="x1-9r8"></a>
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372 <center class="math-display" >
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373 <img
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374 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Ueber_de_1910/fulltext/img/Einst_Ueber_de_191030x.png" alt="P = - -S0-F (S ) . 1 2 Z 0 " class="math-display" /></center></td><td width="5%">(8)</td></tr></table>
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375 <!--l. 417--><p class="nopar">
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376 </p><!--l. 420--><p class="noindent">Zur Anzahl <img
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377 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Ueber_de_1910/fulltext/img/cmr12-8.png" alt="P" class="12x-x-8" /><sub ><span
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378 class="cmr-8">2</span></sub> kommt ein Beitrag aus jeder beliebigen posi-<br/>tiven und negativen Entfernung <img
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379 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Ueber_de_1910/fulltext/img/cmr12-1.png" alt="D" class="12x-x-1" /> von <span
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380 class="cmmi-12">S</span><sub ><span
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381 class="cmr-8">0</span></sub>, und zwar ein <br/><pb/>
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382 </p><!--l. 426--><p class="indent">
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383
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384 </p><!--l. 427--><p class="noindent">positiver oder negativer Beitrag, je nachdem <img
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385 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Ueber_de_1910/fulltext/img/cmr12-1.png" alt="D" class="12x-x-1" /> negativ oder <br/>positiv ist. In der Entfernung <img
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386 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Ueber_de_1910/fulltext/img/cmr12-1.png" alt="D" class="12x-x-1" /> ist die Anzahl <span
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387 class="cmmi-12">dN </span>gegeben <br/>durch</p>
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388 <center class="par-math-display" >
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389 <img
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390 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Ueber_de_1910/fulltext/img/Einst_Ueber_de_191031x.png" alt="F (S0 + D) dS = F (S0 + D) d D , " class="par-math-display" /></center>
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391 <!--l. 437--><p class="nopar"></p><!--l. 441--><p class="noindent">oder, da doch nur kleine Werte von <img
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392 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Ueber_de_1910/fulltext/img/cmr12-1.png" alt="D" class="12x-x-1" /> ins Gewicht fallen, durch <br/></p>
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393 <center class="par-math-display" >
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394 <img
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395 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Ueber_de_1910/fulltext/img/Einst_Ueber_de_191032x.png" alt="{ ( ) } dF-- F (S0) + D dD dD . S0 " class="par-math-display" /></center>
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396 <!--l. 449--><p class="nopar">
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397 </p><!--l. 452--><p class="noindent">Von dieser Anzahl durchqueren alle diejenigen den Wert <span
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398 class="cmmi-12">S</span><sub ><span
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399 class="cmr-8">0</span></sub> <br/>in positiver Richtung, die, von einem negativen <img
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400 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Ueber_de_1910/fulltext/img/cmr12-1.png" alt="D" class="12x-x-1" /> herkommend, <br/>ein so großes <span
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401 class="cmmi-12">f</span>(<span
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402 class="cmmi-12"><img
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403 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Ueber_de_1910/fulltext/img/cmmi12-b.png" alt="a" class="12x-x-b" /></span>) haben, daß
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404 </p>
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405 <center class="par-math-display" >
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406 <img
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407 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Ueber_de_1910/fulltext/img/Einst_Ueber_de_191033x.png" alt="f(a) V~ ---&gt;= |D | , Z " class="par-math-display" /></center>
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408 <!--l. 461--><p class="nopar">
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409 </p><!--l. 464--><p class="noindent">also die Anzahl
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410 </p>
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411 <center class="par-math-display" >
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412 <img
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413 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Ueber_de_1910/fulltext/img/Einst_Ueber_de_191034x.png" alt=" + integral o o
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414
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415 f (f )df . -D V~ Z- " class="par-math-display" /></center>
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416 <!--l. 471--><p class="nopar">
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417 </p><!--l. 474--><p class="noindent">In der negativen Richtung geht analog die Anzahl
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418 </p>
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419 <center class="par-math-display" >
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420 <img
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421 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Ueber_de_1910/fulltext/img/Einst_Ueber_de_191035x.png" alt=" V ~ - -D integral Z f (f) df .
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422 - oo
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423 " class="par-math-display" /></center>
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424 <!--l. 481--><p class="nopar">
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425 </p><!--l. 484--><p class="noindent">So wird:</p>
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426 <center class="par-math-display" >
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427
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428 <img
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429 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Ueber_de_1910/fulltext/img/Einst_Ueber_de_191036x.png" alt=" { } integral 0 (d F ) integral + oo
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430 P2 = dD F (S0) + D ---- f (f)d f d D S0 V~ - - oo - D V~ -Z integral oo { ( ) } - integral D Z d-F- - dD F (S0) + D d D f (f)d f . 0 S0 - oo
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431 " class="par-math-display" /></center>
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432 <!--l. 501--><p class="nopar">
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433 </p><!--l. 504--><p class="noindent">Durch partielle Integration geht dies über in:</p>
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434 <center class="par-math-display" >
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435 <img
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436 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Ueber_de_1910/fulltext/img/Einst_Ueber_de_191037x.png" alt=" integral 0 { ( ) } ( ) D2- dF-- V ~ -- V~ -- P2 = - d D D .F (S0) + 2 dD f - D Z . Z - oo S0 integral oo { 2( ) } ( ) - dD D .F (S ) + D-- d-F- f - D V~ Z- . V~ Z-. 0 2 d D S 0 0 " class="par-math-display" /></center>
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437 <!--l. 522--><p class="nopar"></p><!--l. 525--><p class="noindent">Da nun nach Voraussetzung
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438 </p>
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439 <center class="par-math-display" >
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440 <img
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441 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Ueber_de_1910/fulltext/img/Einst_Ueber_de_191038x.png" alt=" + integral oo
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442 f f(f )df = 0
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443 - oo
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444 " class="par-math-display" /></center>
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445 <!--l. 531--><p class="nopar"> <pb/>
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446 </p><!--l. 538--><p class="indent">
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447
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448 </p><!--l. 539--><p class="noindent">wird, wenn wir <img
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449 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Ueber_de_1910/fulltext/img/cmr12-1.png" alt="D" class="12x-x-1" /><img
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450 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Ueber_de_1910/fulltext/img/Einst_Ueber_de_191039x.png" alt=" V~ -- Z" class="sqrt" /> = <span
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451 class="cmmi-12">f </span>als Variable einführen:</p>
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452 <table width="100%"
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453 class="equation"><tr><td><a
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454 id="x1-10r9"></a>
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455 <center class="math-display" >
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456 <img
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457 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Ueber_de_1910/fulltext/img/Einst_Ueber_de_191040x.png" alt=" 1 (d F ) integral + oo
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458 P 2 = - ---- ---- f 2f (f)d f { 2Z dD S0 ( ) - oo
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459 -1-- dF-- --2 = - 2 Z dD .f . S0 " class="math-display" /></center></td><td width="5%">(9)</td></tr></table>
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460 <!--l. 553--><p class="nopar">
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461 </p><!--l. 556--><p class="noindent">(8) und (9) in (6) eingesetzt, ergeben die Differentialgleichung:
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462 </p>
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463 <center class="par-math-display" >
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464 <img
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465 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Ueber_de_1910/fulltext/img/Einst_Ueber_de_191041x.png" alt=" ---d F S F + f 2----= 0, d S " class="par-math-display" /></center>
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466 <!--l. 563--><p class="nopar">
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467 </p><!--l. 567--><p class="noindent">deren Lösung:</p>
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468 <table width="100%"
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469 class="equation"><tr><td><a
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470 id="x1-11r10"></a>
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471 <center class="math-display" >
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472 <img
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473 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Ueber_de_1910/fulltext/img/Einst_Ueber_de_191042x.png" alt=" - S22 F = const.e 2f , " class="math-display" /></center></td><td width="5%">(10)</td></tr></table>
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474 <!--l. 574--><p class="nopar">
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475
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476 </p><!--l. 578--><p class="noindent">das Gausssche Fehlergesetz ausspricht.
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477 </p>
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478 <div class="center" >
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479
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480 <!--l. 583--><p class="noindent">
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481 </p><!--l. 584--><p class="noindent"><span
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482 class="cmsy-10x-x-120">§ </span>4. Statistisches Gesetz einer Kombination aller <span
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483 class="cmmi-12">S</span><sup ><span
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484 class="cmr-8">(</span><span
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485 class="cmmi-8">n</span><span
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486 class="cmr-8">)</span></sup>.</p></div>
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487 <!--l. 588--><p class="indent"> Wir dehnen nun die Betrachtungen des vorigen Paragraphen <br/>vom eindimensionalen Fall auf den beliebig vieler Dimen-<br/>sionen aus. Wir haben diesmal eine Kombination von vielen <br/>Größen <span
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488 class="cmmi-12">S</span><sup ><span
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489 class="cmr-8">(</span><span
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490 class="cmmi-8">n</span><span
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491 class="cmr-8">)</span></sup> zu betrachten. Die Anzahl der in einem un-<br/>endlich kleinen Gebiete <span
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492 class="cmmi-12">dS</span><sup ><span
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493 class="cmr-8">(1)</span></sup> <span
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494 class="cmmi-12">dS</span><sup ><span
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495 class="cmr-8">(2)</span></sup><span
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496 class="cmmi-12">...</span> liegenden Systeme sei:</p>
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497 <table width="100%"
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498 class="equation"><tr><td><a
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499 id="x1-12r11"></a>
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500 <center class="math-display" >
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501 <img
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502 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Ueber_de_1910/fulltext/img/Einst_Ueber_de_191043x.png" alt=" ( (1) (2) ) (1) (2) d N = F S , S ... dS d S ... " class="math-display" /></center></td><td width="5%">(11)</td></tr></table>
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503 <!--l. 600--><p class="nopar">
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504 </p><!--l. 603--><p class="noindent">Wieder fordern wir, daß <span
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505 class="cmmi-12">dN </span>sich nicht ändern soll, wenn wir <br/>von <span
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506 class="cmmi-12">S</span><sub><img
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507 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Ueber_de_1910/fulltext/img/Einst_Ueber_de_191044x.png" alt="(Z)" class="left" align="middle" /></sub><sup> <span
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508 class="cmr-8">(</span><span
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509 class="cmmi-8">n</span><span
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510 class="cmr-8">)</span></sup>
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511 zu <span
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512 class="cmmi-12">S</span><sub><img
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513 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Ueber_de_1910/fulltext/img/Einst_Ueber_de_191045x.png" alt="(Z+1)" class="left" align="middle" /></sub><sup> <span
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514 class="cmr-8">(</span><span
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515 class="cmmi-8">n</span><span
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516 class="cmr-8">)</span></sup> übergehen, wieder führt dies zu der Diffe-<br/>rentialgleichung (5)
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517 </p>
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518 <center class="par-math-display" >
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519 <img
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520 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Ueber_de_1910/fulltext/img/Einst_Ueber_de_191046x.png" alt="div P = 0 . " class="par-math-display" /></center>
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521 <!--l. 611--><p class="nopar">
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522 </p><!--l. 614--><p class="noindent">Nur hat die Anzahl <img
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523 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Ueber_de_1910/fulltext/img/cmr12-8.png" alt="P" class="12x-x-8" /> in unserem jetzigen Fall Komponenten <br/>in jeder Richtung
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524 <span
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525 class="cmmi-12">S</span><sup ><span
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526 class="cmr-8">(1)</span></sup><span
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527 class="cmmi-12">, S</span><sup ><span
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528 class="cmr-8">(2)</span></sup><span
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529 class="cmmi-12">...</span><span
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530 class="cmmi-12">, </span>die wir mit <img
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531 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Ueber_de_1910/fulltext/img/cmr12-8.png" alt="P" class="12x-x-8" /><sup ><span
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532 class="cmr-8">(1)</span></sup>, <img
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533 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Ueber_de_1910/fulltext/img/cmr12-8.png" alt="P" class="12x-x-8" /><sup ><span
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534 class="cmr-8">(2)</span></sup><span
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535 class="cmmi-12">...</span> be-<br/>zeichnen wollen. (5) nimmt also die Gestalt an
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536 </p>
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537 <center class="par-math-display" >
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538 <img
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539 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Ueber_de_1910/fulltext/img/Einst_Ueber_de_191047x.png" alt=" (n) sum @-P--- n @ S(n) = 0.
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540 " class="par-math-display" /></center>
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541 <!--l. 624--><p class="nopar">
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542 </p><!--l. 627--><p class="noindent">Zwischen <span
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543 class="cmmi-12">S</span><sub><img
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544 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Ueber_de_1910/fulltext/img/Einst_Ueber_de_191048x.png" alt="(Z)" class="left" align="middle" /></sub><sup> <span
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545 class="cmr-8">(</span><span
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546 class="cmmi-8">n</span><span
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547 class="cmr-8">)</span></sup> und <span
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548 class="cmmi-12">S</span><sub><img
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549 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Ueber_de_1910/fulltext/img/Einst_Ueber_de_191049x.png" alt="(Z+1)" class="left" align="middle" /></sub><sup> <span
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550 class="cmr-8">(</span><span
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551 class="cmmi-8">n</span><span
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552 class="cmr-8">)</span></sup> besteht, wie früher Gleichung (7), <br/>daher bleiben die Betrachtungen des vorigen Paragraphen <br/>vollkommen gültig zur Berechnung der einzelnen <img
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553 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Ueber_de_1910/fulltext/img/cmr12-8.png" alt="P" class="12x-x-8" /><sup ><span
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554 class="cmr-8">(</span><span
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555 class="cmmi-8">n</span><span
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556 class="cmr-8">)</span></sup>. Es <br/>wird also
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557 </p>
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558 <center class="par-math-display" >
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559 <img
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560 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Ueber_de_1910/fulltext/img/Einst_Ueber_de_191050x.png" alt=" ---@ F P(n) = S(n)F + f2n------ @ S(n) " class="par-math-display" /></center>
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561 <!--l. 637--><p class="nopar"> <pb/>
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562 </p><!--l. 644--><p class="indent">
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563
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564 </p><!--l. 645--><p class="noindent">Wir können diesen Ausdruck noch vereinfachen, indem wir <br/>alle <span class="overline"><span
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565 class="cmmi-12">f</span><sub><span
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566 class="cmmi-8">n</span></sub><sup><span
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567 class="cmr-8">2</span></sup></span> als gleich annehmen. Dies kommt ersichtlich nur <br/>darauf hinaus, daß wir die einzelnen Funktionen <span
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568 class="cmmi-12">f</span><sub ><span
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569 class="cmmi-8">n</span></sub> mit passen-<br/>den Konstanten multipliziert denken. (Im speziellen Fall <br/>unserer sin und cos ist diese vereinfachende Annahme von <br/>selbst erfüllt.)
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570 </p><!--l. 653--><p class="indent"> So erhalten wir schließlich für die Funktion <span
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571 class="cmmi-12">F </span>die Diffe-<br/>rentialgleichung:</p>
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572 <table width="100%"
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573 class="equation"><tr><td><a
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574 id="x1-13r12"></a>
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575 <center class="math-display" >
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576 <img
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577 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Ueber_de_1910/fulltext/img/Einst_Ueber_de_191051x.png" alt=" sum @ ( --- @ F ) n ---(n) S(n)F + f2----(n) = 0. @ S @ S " class="math-display" /></center></td><td width="5%">(12)</td></tr></table>
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578 <!--l. 662--><p class="nopar">
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579 </p><!--l. 666--><p class="noindent">Zur Lösung dieser Differentialgleichung führt uns die Be-<br/>trachtung des über den ganzen Raum erstreckten Integrals:
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580 </p>
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581 <table width="100%"
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582 class="equation"><tr><td><a
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583 id="x1-14r13"></a>
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584 <center class="math-display" >
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585 <img
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586 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Ueber_de_1910/fulltext/img/Einst_Ueber_de_191052x.png" alt=" integral sum n1 { ( --- )2} -1 n S(n)F + f 2-@ F-- d S(1) ...dS(n1) F 0 @ S(n) { integral n1 { ( ) ( )} sum (n) -2-@-F-- (n) --2@-log-F-- (1) (n1) = n S F + f @ S(n) S + f @ S(n) d S ...dS . 0 " class="math-display" /></center></td><td width="5%">(13)</td></tr></table>
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587 <!--l. 685--><p class="nopar">
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588 </p><!--l. 688--><p class="noindent">Nun ist aber:
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589 </p>
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590 <center class="par-math-display" >
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591 <img
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592 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Ueber_de_1910/fulltext/img/Einst_Ueber_de_191053x.png" alt=" integral sum n1 { ( --- @ F ) } n S(n)F + f 2---(n) S(n) dS(1)...d S(n1) 0 @ S integral ( sum n1 sum n1 ) = F nS(n)2 + f2- n S(n)-@-F-- dS(1)...d S(n1), @ S(n) 0 0 " class="par-math-display" /></center>
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593 <!--l. 705--><p class="nopar">
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594 </p><!--l. 709--><p class="noindent">oder wenn wir den zweiten Summanden partiell integrieren <br/>und bedenken, daß im Unendlichen <span
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595 class="cmmi-12">F </span>= 0 sein muß,
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596 </p>
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597 <center class="par-math-display" >
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598 <img
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599 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Ueber_de_1910/fulltext/img/Einst_Ueber_de_191054x.png" alt=" ( ) integral sum n1 (n)2 --- (1) (n ) = F n S - f 2 .n1 dS ...d S 1 . 0 " class="par-math-display" /></center>
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600 <!--l. 719--><p class="nopar">
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601 </p><!--l. 722--><p class="noindent">Dieser Ausdruck verschwindet aber, weil
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602 </p>
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603 <center class="par-math-display" >
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604 <img
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605 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Ueber_de_1910/fulltext/img/Einst_Ueber_de_191055x.png" alt=" integral
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606 F S(n)2d S(1)...d S(n1) " class="par-math-display" /></center>
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607 <!--l. 729--><p class="nopar">
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608 </p><!--l. 733--><p class="noindent">nichts anderes ist, als der im letzten Paragraphen abgeleitete <br/>Mittelwert <span class="overline"><span
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609 class="cmmi-12">S</span><sup ><span
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610 class="cmr-8">(</span><span
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611 class="cmmi-8">n</span><span
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612 class="cmr-8">)2</span></sup></span><span
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613 class="cmmi-12">,</span>
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614 falls nur ein einziges <span
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615 class="cmmi-12">S </span>betrachtet wird; für <br/>diesen folgt aus Gleichung (10)
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616 </p>
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617 <center class="par-math-display" >
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618 <img
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619 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Ueber_de_1910/fulltext/img/Einst_Ueber_de_191056x.png" alt="--- --- S2 = f2 . " class="par-math-display" /></center>
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620 <!--l. 741--><p class="nopar"> <pb/>
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621 </p><!--l. 748--><p class="indent">
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622
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623 </p><!--l. 749--><p class="noindent">Andererseits wird durch partielle Integration:</p>
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624 <center class="par-math-display" >
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625 <img
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626 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Ueber_de_1910/fulltext/img/Einst_Ueber_de_191057x.png" alt=" { ( ) } integral sum (n) --- @ F ---@ log F (1) (n ) S F + f2 ---(n) f2 ----(n)-- dS ...d S 1 integral ( @ S( @ S )} --2 sum --@--- (n) -2--@-F-- (1) (n1) = f log F @ S(n) S F + f @ S(n) d S ...d S , " class="par-math-display" /></center>
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627 <!--l. 766--><p class="nopar">
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628 </p><!--l. 770--><p class="noindent">was nach Gleichung (12) ebenfalls verschwindet.
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629 </p><!--l. 773--><p class="indent"> Somit ist erwiesen, daß das Integral (13) verschwindet; <br/>dies ist aber wegen des quadratischen Charakters des Inte-<br/>granden nur möglich, wenn überall für jedes <span
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630 class="cmmi-12">n</span>
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631 gilt:</p>
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632 <table width="100%"
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633 class="equation"><tr><td><a
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634 id="x1-15r14"></a>
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635 <center class="math-display" >
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636 <img
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637 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Ueber_de_1910/fulltext/img/Einst_Ueber_de_191058x.png" alt=" --- S(n)F + f 2-@-F-- = 0. @ S(n) " class="math-display" /></center></td><td width="5%">(14)</td></tr></table>
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638 <!--l. 782--><p class="nopar">
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639 </p><!--l. 785--><p class="noindent">So gelangen wir also für <span
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640 class="cmmi-12">F </span>zu einem statistischen Gesetz, <br/>welches in bezug auf jedes <span
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641 class="cmmi-12">S</span><sup ><span
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642 class="cmr-8">(</span><span
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643 class="cmmi-8">n</span><span
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644 class="cmr-8">)</span></sup> mit dem Gaussschen Fehler-<br/>gesetz identisch ist:</p>
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645 <table width="100%"
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646 class="equation"><tr><td><a
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647 id="x1-16r15"></a>
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648 <center class="math-display" >
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649 <img
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650 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Ueber_de_1910/fulltext/img/Einst_Ueber_de_191059x.png" alt=" 2 2 - S(21f)2- - S(22f)2- F = const.e .e . " class="math-display" /></center></td><td width="5%">(15)</td></tr></table>
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651 <!--l. 795--><p class="nopar">
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652 </p><!--l. 798--><p class="noindent">Die Wahrscheinlichkeit einer Kombination von Werten <span
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653 class="cmmi-12">S</span><sup ><span
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654 class="cmr-8">(</span><span
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655 class="cmmi-8">n</span><span
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656 class="cmr-8">)</span></sup> <br/>setzt sich also einfach als Produkt aus den Wahrscheinlich-<br/>keiten der einzelnen <span
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657 class="cmmi-12">S</span><sup ><span
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658 class="cmr-8">(</span><span
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659 class="cmmi-8">n</span><span
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660 class="cmr-8">)</span></sup>
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661 zusammen.
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662 </p><!--l. 804--><p class="indent"> Es ist klar, daß, wenn für <span
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663 class="cmmi-12">S</span><sup ><img
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664 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Ueber_de_1910/fulltext/img/Einst_Ueber_de_191060x.png" alt="(1)" class="left" align="middle" /></sup><span
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665 class="cmmi-12">, S</span><sup ><img
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666 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Ueber_de_1910/fulltext/img/Einst_Ueber_de_191061x.png" alt="(2)" class="left" align="middle" /></sup><span
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667 class="cmmi-12">...</span> die Gleichung (15) <br/>gilt, dieselbe Gleichung für eine Kombination von Größen
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668 </p>
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669 <center class="par-math-display" >
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670 <img
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671 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Ueber_de_1910/fulltext/img/Einst_Ueber_de_191062x.png" alt="S(n)'= a S(n) n " class="par-math-display" /></center>
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672 <!--l. 812--><p class="nopar">
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673 </p><!--l. 816--><p class="noindent">erfüllt ist. In diesem Falle tritt statt <span class="overline"><span
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674 class="cmmi-12">f</span><sup ><span
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675 class="cmr-8">2</span></sup></span> die Größe <span
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676 class="cmmi-12"><img
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677 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Ueber_de_1910/fulltext/img/cmmi12-b.png" alt="a" class="12x-x-b" /></span><sub >
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678 <span
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679 class="cmmi-8"><img
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680 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Ueber_de_1910/fulltext/img/cmmi8-19.png" alt="p" class="8x-x-19" /></span></sub><sup ><span
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681 class="cmr-8">2</span></sup> <span class="overline"><span
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682 class="cmmi-12">f</span><sup ><span
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683 class="cmr-8">2</span></sup></span> <br/>in die Exponenten ein. Von der Art der <span
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684 class="cmmi-12">S</span><sup ><span
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685 class="cmr-8">(</span><span
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686 class="cmmi-8">n</span><span
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687 class="cmr-8">)</span><span
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688 class="cmsy-8">'</span></sup> sind aber die <br/>Koeffizienten <span
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689 class="cmmi-12">A</span><sub >
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690 <span
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691 class="cmmi-8">n</span></sub>, <span
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692 class="cmmi-12">B</span><sub ><span
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693 class="cmmi-8">n</span></sub> unseres physikalischen Problems; und <br/>zwar ist</p>
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694 <center class="par-math-display" >
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695 <img
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696 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Ueber_de_1910/fulltext/img/Einst_Ueber_de_191063x.png" alt=" (n) An S = --V ~ ---, an Z " class="par-math-display" /></center>
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697 <!--l. 826--><p class="nopar">
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698 </p><!--l. 830--><p class="noindent">also</p>
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699 <center class="par-math-display" >
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700 <img
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701 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Ueber_de_1910/fulltext/img/Einst_Ueber_de_191064x.png" alt="a = a V~ Z- n n " class="par-math-display" /></center>
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702 <!--l. 836--><p class="nopar">
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703 </p><!--l. 840--><p class="noindent">zu setzen.
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704 </p><!--l. 843--><p class="indent"> Somit ist auch die Gültigkeit der Gleichung (1) und die <br/>Unmöglichkeit erwiesen, eine wahrscheinlichkeits-theoretische <br/>Beziehung zwischen den Koeffizienten der die Temperatur-<br/>strahlung darstellenden Fourierreihe aufzustellen.
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705 </p>
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706 <div class="center" >
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707
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708 <!--l. 849--><p class="noindent">
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709 </p><!--l. 850--><p class="noindent">(Eingegangen 29. August 1910.)</p></div>
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710
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711 <div class="center" >
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712
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713 <!--l. 854--><p class="noindent">
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714 </p><!--l. 855--><p class="noindent">----------</p></div>
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715
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716 </body></html>
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717
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718
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719