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Removing DESpecs directory which deserted to git
author Klaus Thoden <kthoden@mpiwg-berlin.mpg.de>
date Wed, 29 Nov 2017 16:55:37 +0100
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1 <?xml version="1.0" encoding="utf-8" ?>
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9 <meta name="generator" content="TeX4ht (http://www.cse.ohio-state.edu/~gurari/TeX4ht/mn.html)" />
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12 <meta name="src" content="Einst_Ueber_de_1906.tex" />
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13 <meta name="date" content="2005-03-10 16:57:00" />
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14 <link rel="stylesheet" type="text/css" href="Einst_Ueber_de_1906.css" />
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15 </head><body >
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17 <!--l. 13--><p class="noindent"><pb/></p>
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18 <div class="center" >
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20 <!--l. 14--><p class="noindent">
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21 </p><!--l. 15--><p class="noindent"><span
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22 class="cmr-12x-x-120">9. </span><span
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23 class="cmbxti-10x-x-144">Über eine Methode zur Bestimmung des Ver-</span>
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24 <br/><span
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25 class="cmbxti-10x-x-144">h</span><span
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26 class="cmbxti-10x-x-144">ältnisses der transversalen und longitudinalen</span>
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27 <br/><span
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28 class="cmbxti-10x-x-144">Masse des Elektrons; von A. Einstein.</span></p></div>
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29 <div class="center" >
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31 <!--l. 21--><p class="noindent">
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32 </p><!--l. 22--><p class="noindent">----------</p></div>
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33 <!--l. 25--><p class="indent"> Drei die Kathodenstrahlen betreffende Größen gibt es, <br/>welche einer präzisen Beobachtung zugänglich sind, nämlich <br/>die Spannung, welche den Strahlen ihre Geschwindigkeit ver-<br/>leiht (Erzeugungsspannung), die elektrostatische Ablenkbarkeit <br/>und die magnetische Ablenkbarkeit. Zwischen diesen drei <br/>Größen gibt es zwei voneinander unabhängige Beziehungen, <br/>deren Kenntnis für bedeutende Strahlengeschwindigkeiten von <br/>hervorragendem theoretischen Interesse ist. Eine dieser Be-<br/>ziehungen wurde für <span
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34 class="cmmi-12"><img
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35 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Ueber_de_1906/fulltext/img/cmmi12-c.png" alt="b" class="cmmi-12x-x-c" align="middle" /></span>-Strahlen von Hrn. Kaufmann unter-<br/>sucht, nämlich der Zusammenhang zwischen magnetischer und
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36 <br/>elektrostatischer Ablenkbarkeit.
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37 </p><!--l. 39--><p class="indent"> Im folgenden soll darauf aufmerksam gemacht werden, <br/>daß eine zweite Beziehung zwischen diesen Größen mit hin-<br/>reichender Genauigkeit bestimmt werden kann, nämlich die <br/>Beziehung zwischen Erzeugungsspannung und elektrostatischer <br/>Ablenkbarkeit der Kathodenstrahlen oder -- was dasselbe be-<br/>deutet -- das Verhältnis der transversalen zur longitudinalen <br/>Masse des Elektrons in Funktion der Erzeugungsspannung.
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38 </p><!--l. 49--><p class="indent"> Wenn das Quadrat der Geschwindigkeit der Elektronen <br/>sehr klein ist gegenüber dem Quadrat der Lichtgeschwindig-<br/>keit, so gelten für die Bewegung des Elektrons die Gleichungen
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39 </p>
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40 <center class="par-math-display" >
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41 <img
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42 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Ueber_de_1906/fulltext/img/Einst_Ueber_de_19060x.png" alt=" 2 d-x-= - e-X etc., d t2 m0 " class="par-math-display" /></center>
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43 <!--l. 57--><p class="nopar">
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45 </p><!--l. 61--><p class="noindent">wobei <span
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46 class="cmmi-12"><img
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47 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Ueber_de_1906/fulltext/img/cmmi12-22.png" alt="e" class="12x-x-22" /></span>/<span
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49 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Ueber_de_1906/fulltext/img/cmmi12-16.png" alt="m" class="cmmi-12x-x-16" align="middle" /></span><sub ><span
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50 class="cmr-8">0</span></sub> das Verhältnis der Ladung zur Masse des Elektrons, <br/><span
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51 class="cmmi-12">x, y, z </span>die Koordinaten des Elektrons und <span
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52 class="cmmi-12">X, Y, Z </span>die Kom-<br/>ponenten der elektrischen Kraft des Feldes bedeuten, falls <br/>andere Kräfte als elektrostatische nicht auf das Elektron <br/>wirken. Wir nehmen an, die Elektronen bewegen sich mit <br/>der Anfangsgeschwindigkeit Null von einem gewissen Punkte <br/><span
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53 class="cmmi-12">x</span><sub ><span
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54 class="cmr-8">0</span></sub><span
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55 class="cmmi-12">, y</span><sub ><span
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56 class="cmr-8">0</span></sub><span
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57 class="cmmi-12">, z</span><sub ><span
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58 class="cmr-8">0</span></sub> (Kathode) aus. Die Bewegung ist dann eindeutig <br/><pb/>
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59 </p><!--l. 74--><p class="indent">
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61 </p><!--l. 75--><p class="noindent">bestimmt durch obige Gleichungen; sie sei gegeben durch die <br/>Gleichungen</p>
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62 <center class="par-math-display" >
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63 <img
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64 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Ueber_de_1906/fulltext/img/Einst_Ueber_de_19061x.png" alt="x = f1(t) , y = f2(t) , z = f (t) . 3 " class="par-math-display" /></center>
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65 <!--l. 85--><p class="nopar">
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66 </p><!--l. 89--><p class="indent"> Denkt man sich alle elektrostatischen Kraftkomponenten <br/>überall mit
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67 <span
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68 class="cmmi-12">n</span><sup ><span
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69 class="cmr-8">2</span></sup> multipliziert, so bewegt sich nunmehr -- wie <br/>leicht aus den obigen Bewegungsgleichungen zu ersehen ist --<br/>das Elektron gemäß den Gleichungen
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70 </p>
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71 <center class="par-math-display" >
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72 <img
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73 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Ueber_de_1906/fulltext/img/Einst_Ueber_de_19062x.png" alt="x = f1 (n t) , y = f2 (n t) , z = f (n t) . 3 " class="par-math-display" /></center>
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74 <!--l. 102--><p class="nopar">
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75 </p><!--l. 106--><p class="noindent">Hieraus folgt, daß bei Proportionaländerung des Feldes wohl <br/>die Geschwindigkeit, nicht aber die Bahn der Elektronen sich <br/>ändert. Eine Änderung der Bahn tritt bei Proportionaländerung <br/>des Feldes offenbar erst bei solchen Elektrongeschwindigkeiten <br/>ein, bei welchen das Verhältnis von transversaler und longi-<br/>tudinaler Masse merklich von der Einheit abweicht. Wählt <br/>man das elektrostatische Feld derart, daß die Kathodenstrahlen <br/>eine stark gekrümmte Bahn durchlaufen, so werden bereits
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76 <br/>geringe Verschiedenheiten der transversalen und longitudinalen <br/>Masse einen beobachtbaren Einfluß <br/>auf die Bahnkurve haben. Neben-<br/>stehende schematische
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77 <img
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78 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Ueber_de_1906/fulltext/img/Einst_Ueber_de_19063x.png" alt="PIC" class="graphics" width="146.92769pt" height="158.99377pt" /><!--tex4ht:graphics
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79 name="img/Einst_Ueber_de_19063x.png" src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Ueber_de_1906/fulltext/Einst_Ueber_de_1906_001.eps"
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80 --> Skizze zeigt <br/>eine Anordnung, mittels welcher man <br/>das Verhältnis der transversalen zur <br/>longitudinalen Masse des Elektrons <br/>nach dem angedeuteten Prinzip be-<br/>stimmen könnte. Die Kathodenstrahlen <br/>erlangen zwischen der geerdeten Ka-<br/>thode <span
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81 class="cmmi-12">K </span>und der an die positive <br/>Klemme der Stromquelle <span
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82 class="cmmi-12">M </span>ange-<br/>schlossenen, zugleich als Blende dienenden Anode <span
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83 class="cmmi-12">A </span>ihre Ge-<br/>schwindigkeit, werden hierauf durch das mit <span
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84 class="cmmi-12">A </span>verbundene <br/>Röhrchen <span
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85 class="cmmi-12">t </span>in den Raum zwischen den Metallzylindern <span
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86 class="cmmi-12">R</span><sub ><span
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87 class="cmr-8">1</span></sub> <br/>und <span
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88 class="cmmi-12">R</span><sub ><span
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89 class="cmr-8">2</span></sub> eingeführt. <span
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90 class="cmmi-12">R</span><sub ><span
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91 class="cmr-8">1</span></sub> ist geerdet, <span
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92 class="cmmi-12">R</span><sub ><span
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93 class="cmr-8">2</span></sub> mit <span
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94 class="cmmi-12">t</span>, also mit dem
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95 </p><!--l. 134--><p class="indent"> <pb/>
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96 </p><!--l. 138--><p class="indent">
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97
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98 </p><!--l. 139--><p class="noindent">positiven Pol der Stromquelle leitend verbunden, deren nega-<br/>tiver Pol geerdet ist. Die Dimensionen seien so gewählt, daß <br/>sich langsame Kathodenstrahlen annähernd in einem Kreise <br/>bewegen, und zwar in geringer Entfernung von <span
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99 class="cmmi-12">R</span><sub ><span
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100 class="cmr-8">2</span></sub>. Die Strahlen <br/>gelangen hierauf in die mit <span
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101 class="cmmi-12">R</span><sub ><span
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102 class="cmr-8">2</span></sub> metallisch verbundene, etwas <br/>konische Metallröhre <span
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103 class="cmmi-12">t</span><span
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104 class="cmsy-10x-x-120">'</span>, in welcher sich der phosphoreszierende <br/>Schirm <span
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105 class="cmmi-12">S </span>befindet. Auf letzteren falle der Schatten des am <br/>inneren Ende von <span
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106 class="cmmi-12">t</span><span
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107 class="cmsy-10x-x-120">' </span>angeordneten vertikalen Drahtes <span
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108 class="cmmi-12">D</span>.
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109 </p><!--l. 150--><p class="indent"> Bei Anwendung langsamer Kathodenstrahlen erhält der <br/>Schatten von <span
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110 class="cmmi-12">D </span>auf <span
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111 class="cmmi-12">S</span>
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112 eine ganz bestimmte Lage (Nullage). <br/>Erhöht man die Erzeugungsspannung der Strahlen, so wandert <br/>der Drahtschatten. Durch Einschalten einer Batterie <span
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113 class="cmmi-12">B </span>in <br/>die Erdungsleitung von <span
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114 class="cmmi-12">R</span><sub ><span
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115 class="cmr-8">1</span></sub> werde jedoch der Schatten wieder <br/>in die Nullage zurückgeführt.
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116 </p><!--l. 158--><p class="indent"> Bezeichnet man mit <img
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117 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Ueber_de_1906/fulltext/img/cmr12-5.png" alt="TT" class="12x-x-5" /> das Potential, bei welchem die <br/>Ablenkung der schattenbildenden Strahlen erfolgt, so ist <img
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118 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Ueber_de_1906/fulltext/img/cmr12-5.png" alt="TT" class="12x-x-5" /> <br/>auch diejenige Spannung, welcher die in Ablenkung begriffenen <br/>Strahlen ihre kinetische Energie verdanken. Bezeichnet ferner <span
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119 class="cmmi-12"><img
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120 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Ueber_de_1906/fulltext/img/cmmi12-25.png" alt="r" class="cmmi-12x-x-25" align="middle" /> </span> <br/>den Krümmungsradius der schattenbildenden Strahlen, so ist
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121 </p>
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122 <center class="par-math-display" >
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123 <img
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124 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Ueber_de_1906/fulltext/img/Einst_Ueber_de_19064x.png" alt="mt- r-X-- ml = 2 TT . " class="par-math-display" /></center>
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125 <!--l. 168--><p class="nopar">
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126 </p><!--l. 171--><p class="noindent">Hierbei bedeutet <span
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127 class="cmmi-12"><img
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128 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Ueber_de_1906/fulltext/img/cmmi12-16.png" alt="m" class="cmmi-12x-x-16" align="middle" /></span><sub ><span
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129 class="cmmi-8">t</span></sub> die ,,transversale Masse&#8220; des Elektrons, <br/><span
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130 class="cmmi-12"><img
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131 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Ueber_de_1906/fulltext/img/cmmi12-16.png" alt="m" class="cmmi-12x-x-16" align="middle" /></span><sub ><span
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132 class="cmmi-8">l</span></sub> diejenige longitudinale Masse, welche durch die Gleichung
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133 </p>
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134 <center class="par-math-display" >
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135 <img
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136 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Ueber_de_1906/fulltext/img/Einst_Ueber_de_19065x.png" alt=" 2 Kinetische Energie = m v-- l2 " class="par-math-display" /></center>
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137 <!--l. 179--><p class="nopar">
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138 </p><!--l. 183--><p class="noindent">definiert ist und <span
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139 class="cmmi-12">X </span>die ablenkende elektrische Kraft.
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140 </p><!--l. 186--><p class="indent"> Nennt man <span
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141 class="cmmi-12">P </span>das Potential von <span
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142 class="cmmi-12">R</span><sub ><span
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143 class="cmr-8">2</span></sub> (Potential des posi-<br/>tiven Poles der Stromquelle <span
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144 class="cmmi-12">M</span>), <span
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145 class="cmmi-12">p </span>das Potential von <span
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146 class="cmmi-12">R</span><sub ><span
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147 class="cmr-8">1</span></sub>, bei <br/>welchem sich der Schatten in der Nullage befindet, so ist
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148 </p>
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149 <center class="par-math-display" >
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150 <img
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151 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Ueber_de_1906/fulltext/img/Einst_Ueber_de_19066x.png" alt="TT = P - a(P - p), " class="par-math-display" /></center>
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152 <!--l. 194--><p class="nopar">
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153 </p><!--l. 198--><p class="noindent">wobei <span
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154 class="cmmi-12"><img
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155 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Ueber_de_1906/fulltext/img/cmmi12-b.png" alt="a" class="12x-x-b" /> </span>eine von den Apparatdimensionen abhängige, gegen 1 <br/>kleine Konstante bedeutet. Ferner ist die Größe <span
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156 class="cmmi-12">X </span>der Span-<br/>nung <span
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157 class="cmmi-12">P </span><span
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158 class="cmsy-10x-x-120">- </span><span
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159 class="cmmi-12">p </span>proportional. Man erhält also aus obiger Gleichung
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160 </p>
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161 <center class="par-math-display" >
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162 <img
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163 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Ueber_de_1906/fulltext/img/Einst_Ueber_de_19067x.png" alt="mt-= konst.----P----p---- , ml P - a (P - p) " class="par-math-display" /></center>
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164 <!--l. 208--><p class="nopar">
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165 </p><!--l. 212--><p class="noindent">oder (mit einigen erlaubten Vernachlässigungen)
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166 </p>
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167 <center class="par-math-display" >
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168 <img
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169 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Ueber_de_1906/fulltext/img/Einst_Ueber_de_19068x.png" alt=" [ ] mt-= konst. 1- (1 + a) p . ml P " class="par-math-display" /></center>
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170 <!--l. 219--><p class="nopar">
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171 </p><!--l. 221--><p class="indent"> <pb/>
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172 </p><!--l. 223--><p class="indent">
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173
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174 </p><!--l. 224--><p class="noindent">Da <span
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175 class="cmmi-12"><img
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176 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Ueber_de_1906/fulltext/img/cmmi12-b.png" alt="a" class="12x-x-b" /> </span>offenbar mit genügender Genauigkeit ermittelt werden <br/>kann und <span
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177 class="cmmi-12">P </span>und <span
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178 class="cmmi-12">p </span>bis auf wenige Prozent genau meßbar sind, <br/>so ist die Genauigkeit, mit welcher die Abweichung der Größe <br/><span
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179 class="cmmi-12"><img
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180 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Ueber_de_1906/fulltext/img/cmmi12-16.png" alt="m" class="cmmi-12x-x-16" align="middle" /></span><sub ><span
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181 class="cmmi-8">t</span></sub>/<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_1906/fulltext/img/cmmi12-16.png" alt="m" class="cmmi-12x-x-16" align="middle" /></span><sub ><span
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184 class="cmmi-8">l</span></sub> von der Einheit ermittelt werden kann, im wesentlichen <br/>bestimmt durch die Genauigkeit, mit welcher auf die Nullage <br/>des Drahtschattens eingestellt werden kann. Man überzeugt <br/>sich leicht, daß letztere Genauigkeit so groß gemacht werden <br/>kann, daß eine Abweichung der Größe <span
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185 class="cmmi-12"><img
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186 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Ueber_de_1906/fulltext/img/cmmi12-16.png" alt="m" class="cmmi-12x-x-16" align="middle" /></span><sub ><span
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187 class="cmmi-8">t</span></sub>/<span
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188 class="cmmi-12"><img
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189 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Ueber_de_1906/fulltext/img/cmmi12-16.png" alt="m" class="cmmi-12x-x-16" align="middle" /></span><sub ><span
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190 class="cmmi-8">l</span></sub> von der Einheit <br/>um 0,3 Proz. (entsprechend einer Schattenverschiebung von <br/>ca. 1 mm, wenn <span class="overline"><span
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191 class="cmmi-12">DS</span></span> = 10 cm) noch bemerkt werden kann. Zu <br/>erwähnen ist insbesondere, daß die unvermeidlichen Schwan-<br/>kungen, denen beim Experiment das Potential <span
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192 class="cmmi-12">P </span>unterworfen <br/>ist, nur von unbedeutendem Einfluß auf die Genauigkeit der <br/>Messung sein können.
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193 </p><!--l. 246--><p class="indent"> Wir wollen noch die Beziehung zwischen <span
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194 class="cmmi-12"><img
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195 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Ueber_de_1906/fulltext/img/cmmi12-16.png" alt="m" class="cmmi-12x-x-16" align="middle" /></span><sub ><span
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196 class="cmmi-8">t</span></sub>/<span
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197 class="cmmi-12"><img
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198 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Ueber_de_1906/fulltext/img/cmmi12-16.png" alt="m" class="cmmi-12x-x-16" align="middle" /></span><sub ><span
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199 class="cmmi-8">l</span></sub> und <img
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200 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Ueber_de_1906/fulltext/img/cmr12-5.png" alt="TT" class="12x-x-5" /> in <br/>erster Annäherung angeben, wie sie sich aus den verschiedenen <br/>Theorien ergibt. Wird <img
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201 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Ueber_de_1906/fulltext/img/cmr12-5.png" alt="TT" class="12x-x-5" /> in Volt ausgedrückt, so gilt
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202 </p><!--l. 252--><p class="indent"> nach der Theorie von Bucherer:
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203 </p>
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204 <center class="par-math-display" >
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205 <img
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206 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Ueber_de_1906/fulltext/img/Einst_Ueber_de_19069x.png" alt="mt- --TT--- m = 1 - 0,0070 .10000 , l " class="par-math-display" /></center>
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207 <!--l. 257--><p class="nopar">
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208 </p><!--l. 261--><p class="indent"> nach der Theorie von Abraham:
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209 </p>
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210 <center class="par-math-display" >
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211 <img
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212 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Ueber_de_1906/fulltext/img/Einst_Ueber_de_190610x.png" alt="mt-= 1 - 0,0084 .--TT--- , ml 10000 " class="par-math-display" /></center>
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213 <!--l. 266--><p class="nopar">
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214 </p><!--l. 269--><p class="indent"> nach der Theorie von Lorentz und Einstein:
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215 </p>
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216 <center class="par-math-display" >
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217 <img
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218 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Ueber_de_1906/fulltext/img/Einst_Ueber_de_190611x.png" alt="m TT -t-= 1 - 0,0104 .------ . ml 10000 " class="par-math-display" /></center>
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219 <!--l. 274--><p class="nopar">
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220 </p>
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221
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222 <div class="center" >
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223
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224 <!--l. 278--><p class="noindent">
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225 </p><!--l. 279--><p class="noindent">----------</p></div>
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226 <!--l. 283--><p class="indent"> Da ich nicht in der Lage bin, selbst experimentell zu <br/>arbeiten, würde es mich freuen, wenn sich ein Physiker für <br/>die dargelegte Methode interessierte.
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227 </p><!--l. 287--><p class="indent"> Bern, August 1906. </p>
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228 <div class="center" >
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229
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230 <!--l. 288--><p class="noindent">
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231 </p><!--l. 289--><p class="noindent">(Eingegangen 4. August 1906.)</p></div>
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232 <div class="center" >
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233
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234 <!--l. 293--><p class="noindent">
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235 </p><!--l. 294--><p class="noindent">----------</p></div>
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236
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237 </body></html>
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238
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239
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240