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author Klaus Thoden <kthoden@mpiwg-berlin.mpg.de>
date Thu, 02 May 2013 15:04:28 +0200
parents 22d6a63640c6
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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_Kinet_de_1902.tex" />
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13 <meta name="date" content="2005-02-25 10:08:00" />
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14 <link rel="stylesheet" type="text/css" href="Einst_Kinet_de_1902.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-12">6. </span><span
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23 class="cmbxti-10x-x-120">Kinetische Theorie des W</span><span
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24 class="cmbxti-10x-x-120">ärmegleichgewichtes </span> <br/><span
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25 class="cmbxti-10x-x-120">und des</span>
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26 <span
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27 class="cmbxti-10x-x-120">zweiten Hauptsatzes der Thermodynamik; </span> <br/><span
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28 class="cmbxti-10x-x-120">von A.</span>
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29 <span
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30 class="cmbxti-10x-x-120">Einstein.</span></p></div>
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31 <div class="center" >
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33 <!--l. 19--><p class="noindent">
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34 </p><!--l. 20--><p class="noindent">----------</p></div>
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35 <!--l. 23--><p class="indent"> So gross die Errungenschaften der kinetischen Theorie <br/>der Wärme auf dem Gebiete der Gastheorie gewesen sind, so <br/>ist doch bis jetzt die Mechanik nicht im stande gewesen, eine <br/>hinreichende Grundlage für die allgemeine Wärmetheorie zu
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36 <br/>liefern, weil es bis jetzt nicht gelungen ist, die Sätze über <br/>das Wärmegleichgewicht und den zweiten Hauptsatz unter <br/>alleiniger Benutzung der mechanischen Gleichungen und der <br/>Wahrscheinlichkeitsrechnung herzuleiten, obwohl Maxwell&#8217;s <br/>und Boltzmann&#8217;s Theorien diesem Ziele bereits nahe ge-<br/>kommen sind. Zweck der nachfolgenden Betrachtung ist es, <br/>diese Lücke auszufüllen. Dabei wird sich gleichzeitig eine <br/>Erweiterung des zweiten Hauptsatzes ergeben, welche für die
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37 <br/>Anwendung der Thermodynamik von Wichtigkeit ist. Ferner <br/>wird sich der mathematische Ausdruck für die Entropie vom <br/>mechanischen Standpunkt aus ergeben.
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38 </p>
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39 <div class="center" >
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41 <!--l. 41--><p class="noindent">
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42 </p><!--l. 42--><p class="noindent"><span
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43 class="cmsy-10">§ </span>1. Mechanisches Bild für ein physikalisches System.</p></div>
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44 <!--l. 45--><p class="indent"> Wir denken uns ein beliebiges physikalisches System dar-<br/>stellbar durch ein mechanisches System, dessen Zustand durch <br/>sehr viele Coordinaten <span
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45 class="cmmi-10">p</span><sub ><sub ><span
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46 class="cmr-5">1</span></sub></sub><span
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47 class="cmmi-10">,</span> <span
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48 class="cmmi-10">...</span> <span
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49 class="cmmi-10">p</span><sub ><span
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50 class="cmmi-7">n</span></sub> und die dazu gehörigen Ge-<br/>schwindigkeiten
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51 </p>
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52 <center class="par-math-display" >
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53 <img
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54 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Kinet_de_1902/fulltext/img/Einst_Kinet_de_19020x.png" alt="dp1- dpn- d t , ... dt " class="par-math-display" /></center>
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55 <!--l. 53--><p class="nopar">
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56 </p><!--l. 57--><p class="noindent">eindeutig bestimmt sei. Die Energie <span
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57 class="cmmi-10">E </span>derselben bestehe aus <br/>zwei Summanden, der potentiellen Energie <span
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58 class="cmmi-10">V </span>und der lebendigen <br/>Kraft <span
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59 class="cmmi-10">L. </span>Erstere sei eine Function der Coordinaten allein, <br/>letztere eine quadratische Function der
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60 </p>
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61 <center class="par-math-display" >
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62 <img
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63 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Kinet_de_1902/fulltext/img/Einst_Kinet_de_19021x.png" alt="d-pn- ' dt = pn, " class="par-math-display" /></center>
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64 <!--l. 66--><p class="nopar">
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65 </p><!--l. 70--><p class="noindent">deren Coefficienten beliebige Function der <span
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66 class="cmmi-10">p </span>sind. Auf die <br/>Massen des Systems sollen zweierlei äussere Kräfte wirken. <br/><pb/>
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67 </p><!--l. 76--><p class="indent">
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68
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69 </p><!--l. 77--><p class="noindent">Die einen seien von einem Potentiale <span
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70 class="cmmi-10">V</span> <sub ><span
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71 class="cmmi-7">a</span></sub> ableitbar und sollen <br/>die äusseren Bedingungen (Schwerkraft, Wirkung von festen <br/>Wänden ohne thermische Wirkung etc.) darstellen; ihr Potential <br/>kann die Zeit explicite enthalten, doch soll seine Ableitung <br/>nach derselben sehr klein sein. Die anderen Kräfte seien <br/>nicht von einem Potential ableitbar und seien schnell ver-<br/>änderlich. Sie sind als diejenigen Kräfte aufzufassen, welche <br/>die Wärmezufuhr bewirken. Wirken solche Kräfte nicht, ist <br/>aber
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72 <span
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73 class="cmmi-10">V</span> <sub ><span
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74 class="cmmi-7">a</span></sub> explicite von der Zeit abhängig, so haben wir einen <br/>adiabatischen Process vor uns.
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75 </p><!--l. 91--><p class="indent"> Wir werden auch statt der Geschwindigkeiten, lineare <br/>Functionen derselben, die Momente <span
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76 class="cmmi-10">q</span><sub ><span
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77 class="cmr-7">1</span></sub><span
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78 class="cmmi-10">,</span> <span
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79 class="cmmi-10">...</span> <span
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80 class="cmmi-10">q</span><sub ><span
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81 class="cmmi-7">n</span></sub> als Zustands-<br/>variable des System einführen, welche durch <span
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82 class="cmmi-10">n</span>
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83 Gleichungen <br/>von der Form
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84 </p>
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85 <center class="par-math-display" >
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86 <img
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87 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Kinet_de_1902/fulltext/img/Einst_Kinet_de_19022x.png" alt=" -@L- qn = @ p'n " class="par-math-display" /></center>
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88 <!--l. 99--><p class="nopar">
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89 </p><!--l. 103--><p class="noindent">definirt sind, wobei <span
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90 class="cmmi-10">L </span>als Function der <span
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91 class="cmmi-10">p</span><sub ><sub ><span
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92 class="cmr-5">1</span></sub></sub><span
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93 class="cmmi-10">,</span> <span
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94 class="cmmi-10">...</span> <span
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95 class="cmmi-10">p</span><sub ><span
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96 class="cmmi-7">n</span></sub> und <br/><span
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97 class="cmmi-10">p</span><sub ><span
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98 class="cmr-7">1</span></sub><sup ><span
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99 class="cmsy-7">'</span></sup><span
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100 class="cmmi-10">,</span> <span
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101 class="cmmi-10">...</span> <span
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102 class="cmmi-10">p</span><sub ><span
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103 class="cmmi-7">n</span></sub><sup ><span
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104 class="cmsy-7">'</span></sup> zu denken ist.
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105 </p>
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106 <div class="center" >
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107
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108 <!--l. 107--><p class="noindent">
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109 </p><!--l. 108--><p class="noindent"><span
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110 class="cmsy-10">§ </span>2. Ueber die Verteilung der möglichen Zustände unter <span
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111 class="cmmi-10">N </span> <br/>identischen adiabatischen stationären Systemen, bei nahezu <br/>gleichem Energieinhalt.</p></div>
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112 <!--l. 112--><p class="indent"> Seien unendlich viele (<span
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113 class="cmmi-10">N</span>) Systeme gleicher Art vorhanden, <br/>deren Energieinhalt zwischen den bestimmten sehr wenig ver-<br/>schiedenen Werten <span class="overline"><span
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114 class="cmmi-10">E</span></span> und <span class="overline"><span
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115 class="cmmi-10">E</span></span> + <span
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116 class="cmmi-10"><img
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117 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Kinet_de_1902/fulltext/img/cmmi10-e.png" alt="d" class="10x-x-e" /> E</span>
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118 continuirlich verteilt sind. <br/>Aeussere Kräfte, welche nicht von einem Potential ableitbar <br/>sind, sollen nicht vorhanden sein und <span
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119 class="cmmi-10">V</span> <sub ><span
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120 class="cmmi-7">a</span></sub> möge die Zeit nicht <br/>explicite enthalten, sodass das System ein conservatives System <br/>ist. Wir untersuchen die Zustandsverteilung, von welcher wir <br/>voraussetzen, dass sie stationär sei.
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121 </p><!--l. 123--><p class="indent"> Wir machen die Voraussetzung, dass ausser der Energie <br/><span
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122 class="cmmi-10">E </span>= <span
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123 class="cmmi-10">L </span>+ <span
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124 class="cmmi-10">V</span> <sub ><span
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125 class="cmmi-7">a</span></sub> + <span
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126 class="cmmi-10">V</span> <sub ><span
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127 class="cmmi-7">i</span></sub> oder einer Function dieser Grösse, für das <br/>einzelne System keine Function der Zustandsvariabeln <span
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128 class="cmmi-10">p </span>und <span
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129 class="cmmi-10">q </span> <br/>allein vorhanden sei, welche mit der Zeit sich nicht ändert; <br/>auch fernerhin seien nur Systeme betrachtet, welche diese <br/>Bedingung erfüllen. Unsere Voraussetzung ist gleichbedeutend <br/>mit der Annahme, dass die Zustandsverteilung unserer Systeme <br/>durch den Wert von <span
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130 class="cmmi-10">E </span>bestimmt sei, und sich aus jeden be-<br/>liebigen Anfangswerten der Zustandsvariabeln, welche nur
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131 <br/><pb/>
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132 </p><!--l. 136--><p class="indent">
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133
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134 </p><!--l. 137--><p class="noindent">unserer Bedingung für den Wert der Energie Genüge leisten, <br/>von selbst herstelle. Existirte nämlich für das System noch <br/>eine Bedingung von der Art
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135 <span
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136 class="cmmi-10"><img
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137 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Kinet_de_1902/fulltext/img/cmmi10-27.png" alt="f" class="cmmi-10x-x-27" align="middle" /></span>(<span
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138 class="cmmi-10">p</span><sub ><span
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139 class="cmr-7">1</span></sub><span
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140 class="cmmi-10">.</span> <span
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141 class="cmmi-10">...</span> <span
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142 class="cmmi-10">q</span><sub ><span
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143 class="cmmi-7">n</span></sub>) = const., welche <br/>nicht auf die Form <span
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144 class="cmmi-10"><img
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145 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Kinet_de_1902/fulltext/img/cmmi10-27.png" alt="f" class="cmmi-10x-x-27" align="middle" /> </span><img
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146 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Kinet_de_1902/fulltext/img/Einst_Kinet_de_19023x.png" alt="(E)" class="left" align="middle" /> = const. gebracht werden kann, so <br/>wäre offenbar durch geeignete Wahl der Anfangsbedingungen <br/>zu erzielen, dass für jedes der <span
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147 class="cmmi-10">N </span>Systeme <span
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148 class="cmmi-10"><img
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149 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Kinet_de_1902/fulltext/img/cmmi10-27.png" alt="f" class="cmmi-10x-x-27" align="middle" /> </span>einen beliebigen <br/>vorgeschriebenen Wert hätte. Da sich diese Werte aber mit <br/>der Zeit nicht ändern, so folgt z. B., dass der Grösse <span
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150 class="cmex-10"><img
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151 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Kinet_de_1902/fulltext/img/cmex10-50.png" alt=" sum " class="10x-x-50" /></span>
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152 <span
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153 class="cmmi-10"><img
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154 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Kinet_de_1902/fulltext/img/cmmi10-27.png" alt="f" class="cmmi-10x-x-27" align="middle" /></span>,
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155 <br/>erstreckt über alle Systeme, bei gegebenem Werte von <span
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156 class="cmmi-10">E</span>, <br/>durch geeignete Wahl der Anfangsbedingungen, jeder beliebige <br/>Wert erteilt werden könnte. <span
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157 class="cmex-10"><img
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158 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Kinet_de_1902/fulltext/img/cmex10-50.png" alt=" sum " class="10x-x-50" /></span>
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159 <span
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160 class="cmmi-10"><img
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161 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Kinet_de_1902/fulltext/img/cmmi10-27.png" alt="f" class="cmmi-10x-x-27" align="middle" /> </span>ist nun andererseits aus <br/>der Zustandsverteilung eindeutig berechenbar, sodass anderen
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162 <br/>Werten von <span
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163 class="cmex-10"><img
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164 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Kinet_de_1902/fulltext/img/cmex10-50.png" alt=" sum " class="10x-x-50" /></span>
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165 <span
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166 class="cmmi-10"><img
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167 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Kinet_de_1902/fulltext/img/cmmi10-27.png" alt="f" class="cmmi-10x-x-27" align="middle" /> </span>andere Zustandsverteilungen entsprechen. <br/>Man ersieht also, dass die Existenz eines zweiten solchen <br/>Integrals <span
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168 class="cmmi-10"><img
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169 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Kinet_de_1902/fulltext/img/cmmi10-27.png" alt="f" class="cmmi-10x-x-27" align="middle" /> </span>notwendig zur Folge hat, dass durch <span
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170 class="cmmi-10">E </span>allein die <br/>Zustandsverteilung noch nicht bestimmt wäre, sondern dass <br/>dieselbe notwendig vom Anfangszustande der Systeme abhängen
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171 <br/>müsste.
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172 </p><!--l. 161--><p class="indent"> Bezeichnet man mit <span
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173 class="cmmi-10">g </span>ein unendlich kleines Gebiet aller <br/>Zustandsvariabeln
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174 <span
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175 class="cmmi-10">p</span><sub ><sub ><span
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176 class="cmr-5">1</span></sub></sub><span
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177 class="cmmi-10">,</span> <span
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178 class="cmmi-10">...</span> <span
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179 class="cmmi-10">p</span><sub ><span
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180 class="cmmi-7">n</span></sub><span
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181 class="cmmi-10">, q</span><sub ><sub ><span
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182 class="cmr-5">1</span></sub></sub><span
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183 class="cmmi-10">,</span> <span
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184 class="cmmi-10">...</span> <span
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185 class="cmmi-10">q</span><sub ><span
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186 class="cmmi-7">n</span></sub><span
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187 class="cmmi-10">, </span>welches so gewählt <br/>sein soll, dass <span
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188 class="cmmi-10">E </span>(<span
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189 class="cmmi-10">p</span><sub ><sub ><span
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190 class="cmr-5">1</span></sub></sub><span
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191 class="cmmi-10">,</span> <span
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192 class="cmmi-10">...</span> <span
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193 class="cmmi-10">q</span><sub ><span
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194 class="cmmi-7">n</span></sub>) zwischen <span class="overline"><span
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195 class="cmmi-10">E</span></span> und
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196 <span class="overline"><span
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197 class="cmmi-10">E</span></span> + <span
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198 class="cmmi-10"><img
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199 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Kinet_de_1902/fulltext/img/cmmi10-e.png" alt="d" class="10x-x-e" /> E </span>liegt, <br/>wenn die Zustandsvariabeln dem Gebiete <span
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200 class="cmmi-10">g </span>angehören, so ist
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201 <br/>die Verteilung der Zustände durch eine Gleichung von folgender <br/>Form zu charakterisiren
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202 </p>
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203 <center class="par-math-display" >
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204 <img
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205 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Kinet_de_1902/fulltext/img/Einst_Kinet_de_19024x.png" alt=" integral
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206
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207 dN = y (p1, ... qn) d p1 ... dqn, g " class="par-math-display" /></center>
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208 <!--l. 175--><p class="nopar">
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209 </p><!--l. 178--><p class="noindent"><span
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210 class="cmmi-10">dN </span>bedeutet die Anzahl der Systeme, deren Zustandsvariable <br/>zu einer bestimmten Zeit dem Gebiete <span
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211 class="cmmi-10">g </span>zugehören. Die <br/>Gleichung sagt die Bedingung aus, dass die Verteilung <br/>stationär ist.
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212 </p><!--l. 184--><p class="indent"> Wir wählen nun ein solches unendlich kleines Gebiet <span
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213 class="cmmi-10">G. </span> <br/>Die Anzahl der Systeme, deren Zustandsvariable zu irgend <br/>einer bestimmten Zeit <span
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214 class="cmmi-10">t </span>= 0 dem Gebiete <span
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215 class="cmmi-10">G</span>
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216 angehören, ist dann <br/>
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217 </p>
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218 <center class="par-math-display" >
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219 <img
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220 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Kinet_de_1902/fulltext/img/Einst_Kinet_de_19025x.png" alt=" integral
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221 dN = y (P1, ... Qn) d P1 ... dQn, G " class="par-math-display" /></center>
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222 <!--l. 194--><p class="nopar">
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223 </p><!--l. 198--><p class="noindent">wobei die grossen Buchstaben die Zugehörigkeit der abhängigen <br/>Variabeln zur Zeit
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224 <span
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225 class="cmmi-10">t </span>= 0 andeuten sollen. <pb/>
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226 </p><!--l. 205--><p class="indent">
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227
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228 </p><!--l. 206--><p class="indent"> Wir lassen nun die beliebige Zeit <span
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229 class="cmmi-10">t </span>verstreichen. Besass <br/>ein System in <span
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230 class="cmmi-10">t </span>= 0 die bestimmten Zustandsvariabeln <span
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231 class="cmmi-10">P</span><sub ><span
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232 class="cmr-7">1</span></sub><span
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233 class="cmmi-10">,</span> <span
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234 class="cmmi-10">...</span> <span
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235 class="cmmi-10">Q</span><sub ><span
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236 class="cmmi-7">n</span></sub><span
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237 class="cmmi-10">, </span> <br/>so besitzt es zur Zeit <span
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238 class="cmmi-10">t </span>= <span
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239 class="cmmi-10">t </span>die bestimmten Zustandsvariabeln <br/><span
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240 class="cmmi-10">p</span><sub ><span
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241 class="cmr-7">1</span></sub><span
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242 class="cmmi-10">,</span> <span
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243 class="cmmi-10">...</span> <span
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244 class="cmmi-10">q</span><sub ><span
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245 class="cmmi-7">n</span></sub><span
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246 class="cmmi-10">. </span>Die Systeme, deren Zustandsvariabeln in
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247 <span
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248 class="cmmi-10">t </span>= 0 dem <br/>Gebiete <span
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249 class="cmmi-10">G </span>angehörten, und zwar nur diese, gehören zur Zeit <span
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250 class="cmmi-10">t </span>= <span
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251 class="cmmi-10">t </span> <br/>einem bestimmten Gebiete <span
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252 class="cmmi-10">g </span>an, sodass also die Gleichung gilt:
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253 </p>
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254 <center class="par-math-display" >
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255 <img
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256 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Kinet_de_1902/fulltext/img/Einst_Kinet_de_19026x.png" alt=" integral
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257
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258 d N = y (p1, ... qn) . g " class="par-math-display" /></center>
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259 <!--l. 219--><p class="nopar">
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260 </p><!--l. 222--><p class="noindent">Für jedes derartige System gilt aber der Satz von Liouville, <br/>welcher die Form hat:
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261 </p>
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262 <center class="par-math-display" >
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263 <img
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264 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Kinet_de_1902/fulltext/img/Einst_Kinet_de_19027x.png" alt=" integral integral
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265 dP1, ... dQn = d p1, ... dqn. " class="par-math-display" /></center>
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266 <!--l. 230--><p class="nopar">
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267 </p><!--l. 233--><p class="noindent">Aus den drei letzten Gleichungen folgt
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268 </p>
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269 <center class="par-math-display" >
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270 <img
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271 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Kinet_de_1902/fulltext/img/Einst_Kinet_de_19028x.png" alt="y (P1, ... Qn) = y (p1, ... qn).1) " class="par-math-display" /></center>
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272 <!--l. 240--><p class="nopar">
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273 </p><!--l. 243--><p class="noindent"><span
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274 class="cmmi-10"><img
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275 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Kinet_de_1902/fulltext/img/cmmi10-20.png" alt="y" class="10x-x-20" /> </span>ist also eine Invariante des Systems, welche nach dem <br/>obigen die Form haben muss
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276 <span
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277 class="cmmi-10"><img
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278 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Kinet_de_1902/fulltext/img/cmmi10-20.png" alt="y" class="10x-x-20" /> </span>(<span
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279 class="cmmi-10">p</span><sub ><span
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280 class="cmr-7">1</span></sub><span
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281 class="cmmi-10">,</span> <span
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282 class="cmmi-10">...</span> <span
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283 class="cmmi-10">q</span><sub ><span
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284 class="cmmi-7">n</span></sub>) = <span
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285 class="cmmi-10"><img
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286 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Kinet_de_1902/fulltext/img/cmmi10-20.png" alt="y" class="10x-x-20" /></span><sup ><span
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287 class="cmsy-7">*</span></sup>(<span
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288 class="cmmi-10">E</span>)<span
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289 class="cmmi-10">. </span>Für <br/>alle betrachteten Systeme ist aber <span
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290 class="cmmi-10"><img
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291 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Kinet_de_1902/fulltext/img/cmmi10-20.png" alt="y" class="10x-x-20" /></span><sup ><span
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292 class="cmsy-7">*</span></sup>(<span
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293 class="cmmi-10">E</span>) nur unendlich wenig <br/>verschieden von <span
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294 class="cmmi-10"><img
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295 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Kinet_de_1902/fulltext/img/cmmi10-20.png" alt="y" class="10x-x-20" /></span><sup ><span
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296 class="cmsy-7">*</span></sup>(<span class="overline"><span
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297 class="cmmi-10">E</span></span>) = const., und unsere Zustandsgleichung <br/>lautet einfach
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298 </p>
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299 <center class="par-math-display" >
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300 <img
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301 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Kinet_de_1902/fulltext/img/Einst_Kinet_de_19029x.png" alt=" integral
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302 dN = A dp1, ... dqn, g " class="par-math-display" /></center>
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303 <!--l. 254--><p class="nopar">
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304 </p><!--l. 258--><p class="noindent">wobei <span
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305 class="cmmi-10">A </span>eine von den <span
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306 class="cmmi-10">p </span>und <span
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307 class="cmmi-10">q </span>unabhängige Grösse bedeutet.
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308 </p>
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309 <div class="center" >
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310
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311 <!--l. 261--><p class="noindent">
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312 </p><!--l. 262--><p class="noindent"><span
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313 class="cmsy-10">§ </span>3. Ueber die (stationäre) Wahrscheinlichkeit der Zustände <br/>eines Systems <span
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314 class="cmmi-10">S</span>,
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315 das mit einem System <span
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316 class="cmex-10"><img
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317 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Kinet_de_1902/fulltext/img/cmex10-50.png" alt=" sum " class="10x-x-50" /></span>
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318 von relativ unendlich <br/>grosser Energie mechanisch verbunden ist.</p></div>
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319 <!--l. 266--><p class="indent"> Wir betrachten wieder unendlich viele (<span
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320 class="cmmi-10">N</span>) mechanische <br/>Systeme, deren Energie zwischen zwei unendlich wenig ver-<br/>schiedenen Grenzen <span class="overline"><span
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321 class="cmmi-10">E</span></span> und <span class="overline"><span
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322 class="cmmi-10">E</span></span> + <span
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323 class="cmmi-10"><img
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324 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Kinet_de_1902/fulltext/img/cmmi10-e.png" alt="d" class="10x-x-e" /> </span><span class="overline"><span
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325 class="cmmi-10">E</span></span> liege. Jedes solche mecha-<br/>nische System sei wieder eine mechanische Verbindung eines
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326 <br/>Systems <span
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327 class="cmmi-10">S </span>mit den Zustandsvariabeln <span
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328 class="cmmi-10">p</span><sub ><span
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329 class="cmr-7">1</span></sub><span
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330 class="cmmi-10">,</span> <span
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331 class="cmmi-10">...</span><span
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332 class="cmmi-10">q</span><sub ><span
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333 class="cmmi-7">n</span></sub> und eines <br/>Systems <span
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334 class="cmex-10"><img
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335 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Kinet_de_1902/fulltext/img/cmex10-50.png" alt=" sum " class="10x-x-50" /></span>
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336 mit den Zustandsvariabeln <span
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337 class="cmmi-10"><img
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338 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Kinet_de_1902/fulltext/img/cmmi10-19.png" alt="p" class="10x-x-19" /></span><sub ><span
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339 class="cmr-7">1</span></sub> <span
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340 class="cmmi-10">...</span> <span
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341 class="cmmi-10"><img
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342 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Kinet_de_1902/fulltext/img/cmmi10-1f.png" alt="x" class="10x-x-1f" /></span><sub ><span
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343 class="cmmi-7">n</span></sub><span
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344 class="cmmi-10">. </span>Der Aus-<br/>druck für die Gesamtenergie beider Systeme soll so beschaffen <br/>sein, dass jene Terme der Energie, welche durch Einwirkung <br/>der Massen eines Teilsystems auf die des anderen Teilsystems
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345 <br/>----------
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346 </p><!--l. 280--><p class="indent"> 1) Vgl. L. Boltzmann, Gastheorie, II. Teil. <span
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347 class="cmsy-10">§ </span>32 u. <span
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348 class="cmsy-10">§ </span>37. <pb/>
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349 </p><!--l. 285--><p class="indent">
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350
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351 </p><!--l. 286--><p class="noindent">hinzukommen, gegen die Energie <span
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352 class="cmmi-10">E </span>des Teilsystems <span
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353 class="cmmi-10">S </span>zu ver-<br/>nachlässigen seien. Ferner sei die Energie <span
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354 class="cmmi-10">H </span>des Teilsystems <span
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355 class="cmex-10"><img
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356 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Kinet_de_1902/fulltext/img/cmex10-50.png" alt=" sum " class="10x-x-50" /></span>
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357 <br/>unendlich gross gegen <span
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358 class="cmmi-10">E. </span>Bis auf unendlich Kleines höherer <br/>Ordnung lässt sich dann setzen:
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359 </p>
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360 <center class="par-math-display" >
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361 <img
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362 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Kinet_de_1902/fulltext/img/Einst_Kinet_de_190210x.png" alt="E = H + E. " class="par-math-display" /></center>
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363 <!--l. 294--><p class="nopar">
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364 </p><!--l. 296--><p class="noindent">Wir wählen nun ein in allen Zustandsvariabeln <span
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365 class="cmmi-10">p</span><sub ><span
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366 class="cmr-7">1</span></sub> <span
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367 class="cmmi-10">...</span> <span
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368 class="cmmi-10">q</span><sub ><span
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369 class="cmmi-7">n</span></sub>, <br/><span
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370 class="cmmi-10"><img
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371 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Kinet_de_1902/fulltext/img/cmmi10-19.png" alt="p" class="10x-x-19" /></span><sub ><span
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372 class="cmr-7">1</span></sub> <span
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373 class="cmmi-10">...</span> <span
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374 class="cmmi-10"><img
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375 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Kinet_de_1902/fulltext/img/cmmi10-1f.png" alt="x" class="10x-x-1f" /></span><sub ><span
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376 class="cmmi-7">n</span></sub> unendlich kleines Gebiet <span
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377 class="cmmi-10">g</span>, welches so beschaffen sei, <br/>dass E zwischen den constanten Werten <span class="overline"><span
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378 class="cmmi-10">E</span></span> und
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379 <span class="overline"><span
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380 class="cmmi-10">E</span></span> + <span
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381 class="cmmi-10"><img
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382 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Kinet_de_1902/fulltext/img/cmmi10-e.png" alt="d" class="10x-x-e" /> </span><span class="overline"><span
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383 class="cmmi-10">E</span></span> liege. <br/>Die Anzahl <span
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384 class="cmmi-10">dN </span>der Systeme, deren Zustandsvariabeln dem <br/>Gebiet <span
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385 class="cmmi-10">g</span>
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386 angehören, ist dann nach dem Resultate des vorigen <br/>Paragraphen:
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387 </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_Kinet_de_1902/fulltext/img/Einst_Kinet_de_190211x.png" alt=" integral
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391 dN = A d p1 ... d xn. g " class="par-math-display" /></center>
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392 <!--l. 309--><p class="nopar">
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393 </p><!--l. 312--><p class="noindent">Wir bemerken nun, dass es in unserem Belieben steht, statt <span
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394 class="cmmi-10">A </span> <br/>irgend eine stetige Function der Energie zu setzen, welche <br/>für <span
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395 class="cmmi-10">E </span>+ <span class="overline"><span
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396 class="cmmi-10">E</span></span> den Wert <span
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397 class="cmmi-10">A </span>annimmt. Dadurch ändert sich nämlich <br/>unser Resultat nur unendlich wenig. Als diese Function wählen
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398 <br/>wir <span
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399 class="cmmi-10">A</span><span
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400 class="cmsy-10">'</span><span
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401 class="cmmi-10">.e</span><sup ><span
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402 class="cmsy-7">-</span><span
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403 class="cmr-7">2</span><span
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404 class="cmmi-7">hE</span></sup><span
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405 class="cmmi-10">, </span>wobei <span
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406 class="cmmi-10">h </span>eine vorläufig beliebige Constante <br/>bedeutet, über welche wir bald verfügen werden. Wir <br/>schreiben also:
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407 </p>
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408 <center class="par-math-display" >
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409 <img
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410 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Kinet_de_1902/fulltext/img/Einst_Kinet_de_190212x.png" alt=" i ntegral
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411 d N = A' e-2hEdp ... dx . 1 n g " class="par-math-display" /></center>
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412 <!--l. 326--><p class="nopar">
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413 </p><!--l. 329--><p class="noindent">Wir fragen nun: Wie viele Systeme befinden sich in Zuständen, <br/>sodass <span
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414 class="cmmi-10">p</span><sub ><span
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415 class="cmr-7">1</span></sub> zwischen <span
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416 class="cmmi-10">p</span><sub ><span
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417 class="cmr-7">1</span></sub>
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418 und <span
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419 class="cmmi-10">p</span><sub ><span
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420 class="cmr-7">1</span></sub> + <span
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421 class="cmmi-10">dp</span><sub ><span
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422 class="cmr-7">1</span></sub><span
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423 class="cmmi-10">,p</span><sub ><span
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424 class="cmr-7">2</span></sub> bez. <span
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425 class="cmmi-10">p</span><sub ><span
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426 class="cmr-7">2</span></sub> und <span
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427 class="cmmi-10">p</span><sub ><span
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428 class="cmr-7">2</span></sub> + <span
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429 class="cmmi-10">dp</span><sub ><span
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430 class="cmr-7">2</span></sub> <span
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431 class="cmmi-10">...</span> <span
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432 class="cmmi-10">q</span><sub ><span
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433 class="cmmi-7">n</span></sub> <br/>zwischen <span
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434 class="cmmi-10">q</span><sub ><span
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435 class="cmmi-7">n</span></sub> und <span
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436 class="cmmi-10">q</span><sub ><span
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437 class="cmmi-7">n</span></sub> + <span
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438 class="cmmi-10">dq</span><sub ><span
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439 class="cmmi-7">n</span></sub><span
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440 class="cmmi-10">, <img
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441 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Kinet_de_1902/fulltext/img/cmmi10-19.png" alt="p" class="10x-x-19" /></span><sub ><span
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442 class="cmr-7">1</span></sub> <span
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443 class="cmmi-10">...</span> <span
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444 class="cmmi-10"><img
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445 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Kinet_de_1902/fulltext/img/cmmi10-1f.png" alt="x" class="10x-x-1f" /></span><sub ><span
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446 class="cmmi-7">n</span></sub>
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447 aber beliebige, mit den <br/>Bedingungen unserer Systeme verträgliche Werte besitzen?
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448 <br/>Nennt man diese Anzahl <span
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449 class="cmmi-10">dN</span><span
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450 class="cmsy-10">'</span><span
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451 class="cmmi-10">, </span>so erhält man:
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452 </p>
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453 <center class="par-math-display" >
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454 <img
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455 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Kinet_de_1902/fulltext/img/Einst_Kinet_de_190213x.png" alt=" ' '-2hE integral - 2hH dN = A e dp1 ... dqn e dp1 ... dxn. g " class="par-math-display" /></center>
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456 <!--l. 340--><p class="nopar">
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457 </p><!--l. 343--><p class="noindent">Die Integration erstreckt sich dabei auf jene Werte der Zu-<br/>standsvariabeln, für welche <span
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458 class="cmmi-10">H </span>zwischen <span class="overline"><span
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459 class="cmmi-10">E</span></span> <span
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460 class="cmsy-10">- </span><span
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461 class="cmmi-10">E </span>und <span class="overline"><span
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462 class="cmmi-10">E</span></span> <span
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463 class="cmsy-10">- </span><span
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464 class="cmmi-10">E </span>+ <span
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465 class="cmmi-10"><img
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466 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Kinet_de_1902/fulltext/img/cmmi10-e.png" alt="d" class="10x-x-e" /> </span><span class="overline"><span
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467 class="cmmi-10">E</span></span> <br/>liegt. Wir behaupten nun, der Wert
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468 von <span
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469 class="cmmi-10">h </span>sei auf eine und <br/>nur eine Weise so zu wählen, dass das in unserer Gleichung
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470 <br/>auftretende Integral von <span
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471 class="cmmi-10">E </span>unabhängig wird.
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472 </p><!--l. 352--><p class="indent"> Das Integral <span
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473 class="cmsy-10"><img
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474 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Kinet_de_1902/fulltext/img/cmsy10-73.png" alt=" integral " class="10x-x-73" /></span> <span
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475 class="cmmi-10">e</span><sup ><span
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476 class="cmsy-7">-</span><span
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477 class="cmr-7">2</span><span
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478 class="cmmi-7">hH</span></sup> <span
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479 class="cmmi-10">d<img
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480 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Kinet_de_1902/fulltext/img/cmmi10-19.png" alt="p" class="10x-x-19" /></span><sub ><span
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481 class="cmr-7">1</span></sub> <span
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482 class="cmmi-10">...</span> <span
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483 class="cmmi-10">d<img
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484 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Kinet_de_1902/fulltext/img/cmmi10-1f.png" alt="x" class="10x-x-1f" /></span><sub ><span
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485 class="cmmi-7">n</span></sub><span
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486 class="cmmi-10">, </span>wobei die Grenzen der <br/>Integration durch die Grenzen E und <span
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487 class="cmmi-10">E </span>+<span class="overline"> <span
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488 class="cmmi-10"><img
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489 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Kinet_de_1902/fulltext/img/cmmi10-e.png" alt="d" class="10x-x-e" /> E</span></span> bestimmt sein <br/>mögen, ist nämlich bei bestimmtem <span
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490 class="cmmi-10"><img
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491 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Kinet_de_1902/fulltext/img/cmmi10-e.png" alt="d" class="10x-x-e" /> </span><span class="overline"><span
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492 class="cmmi-10">E</span></span>
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493 offenbar lediglich <br/><pb/>
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494 </p><!--l. 360--><p class="indent">
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495
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496 </p><!--l. 361--><p class="noindent">Function von E allein; nennen wir dieselbe <span
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497 class="cmmi-10"><img
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498 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Kinet_de_1902/fulltext/img/cmmi10-1f.png" alt="x" class="10x-x-1f" /></span>(<span
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499 class="cmmi-10">E</span>)<span
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500 class="cmmi-10">. </span>Dass in <br/>dem Ausdruck für <span
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501 class="cmmi-10">dN</span><span
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502 class="cmsy-10">'</span>
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503 auftretende Integral lässt sich dann <br/>in der Form schreiben:
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504 </p>
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505 <center class="par-math-display" >
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506 <img
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507 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Kinet_de_1902/fulltext/img/Einst_Kinet_de_190214x.png" alt="x (E-- E). " class="par-math-display" /></center>
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508 <!--l. 369--><p class="nopar">
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509 </p><!--l. 372--><p class="noindent">Da nun <span
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510 class="cmmi-10">E </span>gegen <span class="overline"><span
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511 class="cmmi-10">E</span></span> unendlich klein ist, so lässt sich dies bis <br/>auf unendlich Kleines höherer Ordnung in der Form schreiben:
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512 </p>
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513 <center class="par-math-display" >
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514 <img
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515 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Kinet_de_1902/fulltext/img/Einst_Kinet_de_190215x.png" alt=" -- -- -- x (E - E) = x(E)- Ex'(E). " class="par-math-display" /></center>
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516 <!--l. 379--><p class="nopar">
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517 </p><!--l. 382--><p class="noindent">Die notwendige und hinreichende Bedingung dafür, dass jenes <br/>Integral von <span
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518 class="cmmi-10">E</span>
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519 unabhängig ist, lautet also
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520 </p>
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521 <center class="par-math-display" >
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522 <img
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523 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Kinet_de_1902/fulltext/img/Einst_Kinet_de_190216x.png" alt=" -- x'(E) = 0. " class="par-math-display" /></center>
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524 <!--l. 390--><p class="nopar">
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525 </p><!--l. 393--><p class="noindent">Nun lässt sich aber setzen
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526 </p>
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527 <center class="par-math-display" >
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528 <img
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529 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Kinet_de_1902/fulltext/img/Einst_Kinet_de_190217x.png" alt=" -2hE x (E) = e .w(E). " class="par-math-display" /></center>
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530 <!--l. 399--><p class="nopar">
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531 </p><!--l. 403--><p class="noindent">wobei <span
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532 class="cmmi-10"><img
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533 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Kinet_de_1902/fulltext/img/cmmi10-21.png" alt="w" class="10x-x-21" /> </span>(<span
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534 class="cmmi-10">E</span>) = <span
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535 class="cmsy-10"><img
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536 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Kinet_de_1902/fulltext/img/cmsy10-73.png" alt=" integral " class="10x-x-73" /></span> <span
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537 class="cmmi-10">d<img
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538 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Kinet_de_1902/fulltext/img/cmmi10-19.png" alt="p" class="10x-x-19" /></span><sub ><span
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539 class="cmr-7">1</span></sub> <span
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540 class="cmmi-10">...</span> <span
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541 class="cmmi-10">d<img
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542 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Kinet_de_1902/fulltext/img/cmmi10-1f.png" alt="x" class="10x-x-1f" /></span><sub ><span
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543 class="cmmi-7">n</span></sub><span
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544 class="cmmi-10">, </span>erstreckt über alle Werte der <br/>Variabeln, deren Energiefunction zwischen E und <span
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545 class="cmmi-10">E </span>+ <span
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546 class="cmmi-10"><img
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547 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Kinet_de_1902/fulltext/img/cmmi10-e.png" alt="d" class="10x-x-e" /> E </span>liegt.
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548 </p><!--l. 408--><p class="indent"> Die gefundene Bedingung für <span
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549 class="cmmi-10">h </span>nimmt also die Form an:
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550 </p>
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551 <center class="par-math-display" >
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552 <img
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553 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Kinet_de_1902/fulltext/img/Einst_Kinet_de_190218x.png" alt=" -- -- { w'(E) } e-2hE.w (E). -2h + ----- = 0, w (E) " class="par-math-display" /></center>
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554 <!--l. 415--><p class="nopar">
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555 </p><!--l. 419--><p class="noindent">oder
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556 </p>
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557 <center class="par-math-display" >
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558 <img
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559 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Kinet_de_1902/fulltext/img/Einst_Kinet_de_190219x.png" alt=" -- 1w'(E)- h = 2w (E) . " class="par-math-display" /></center>
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560 <!--l. 426--><p class="nopar">
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561 </p><!--l. 430--><p class="indent"> Es giebt also stets einen und nur einen Wert für <span
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562 class="cmmi-10">h</span>, <br/>welcher die gefundenen Bedingungen erfüllt. Da ferner, wie <br/>im nächsten Paragraphen gezeigt werden soll,
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563 <span
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564 class="cmmi-10"><img
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565 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Kinet_de_1902/fulltext/img/cmmi10-21.png" alt="w" class="10x-x-21" /> </span>(<span
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566 class="cmmi-10">E</span>) und <span
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567 class="cmmi-10"><img
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568 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Kinet_de_1902/fulltext/img/cmmi10-21.png" alt="w" class="10x-x-21" /></span><span
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569 class="cmsy-10">'</span>(<span
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570 class="cmmi-10">E</span>) <br/>stets positiv sind, ist auch <span
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571 class="cmmi-10">h </span>stets eine positive Grösse.
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572 </p><!--l. 437--><p class="indent"> Wählen wir <span
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573 class="cmmi-10">h </span>in dieser Weise, so reducirt sich das <br/>Integral auf eine von <span
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574 class="cmmi-10">E</span>
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575 unabhängige Grösse, sodass wir für <br/>die Zahl der Systeme, deren Variabeln <span
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576 class="cmmi-10">p</span><sub ><sub ><span
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577 class="cmr-5">1</span></sub></sub><span
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578 class="cmmi-10">,</span> <span
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579 class="cmmi-10">...</span> <span
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580 class="cmmi-10">q</span><sub ><span
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581 class="cmmi-7">n</span></sub>
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582 in den be-<br/>zeichneten Grenzen liegen, den Ausdruck erhalten
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583 </p>
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584 <center class="par-math-display" >
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585 <img
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586 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Kinet_de_1902/fulltext/img/Einst_Kinet_de_190220x.png" alt="dN '= A''e-2hE .dp1 ... dqn. " class="par-math-display" /></center>
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587 <!--l. 446--><p class="nopar">
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588 </p><!--l. 449--><p class="noindent">Dies ist also auch bei anderer Bedeutung von <span
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589 class="cmmi-10">A</span><span
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590 class="cmsy-10">''</span> der Aus-<br/>druck für die Wahrscheinlichkeit, dass die Zustandsvariabeln <br/>eines mit einem System von relativ unendlich grosser Energie <br/>mechanisch verbundenen Systems zwischen unendlich nahen <br/>Grenzen liegen, wenn der Zustand stationär geworden ist.
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591 <pb/>
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592 </p><!--l. 460--><p class="indent">
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593
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594 </p>
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595 <div class="center" >
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596
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597 <!--l. 461--><p class="noindent">
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598 </p><!--l. 462--><p class="noindent"><span
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599 class="cmsy-10">§ </span>4. Beweis dafür, dass die Grösse <span
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600 class="cmmi-10">h </span>positiv ist.</p></div>
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601 <!--l. 465--><p class="indent"> Sei <span
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602 class="cmmi-10"><img
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603 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Kinet_de_1902/fulltext/img/cmmi10-27.png" alt="f" class="cmmi-10x-x-27" align="middle" /></span>(<span
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604 class="cmmi-10">x</span>) eine homogene, quadratische Function der Variabeln <br/><span
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605 class="cmmi-10">x</span><sub ><span
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606 class="cmr-7">1</span></sub> <span
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607 class="cmmi-10">...</span> <span
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608 class="cmmi-10">x</span><sub ><span
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609 class="cmmi-7">n</span></sub><span
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610 class="cmmi-10">. </span>Wir betrachten die Grösse <span
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611 class="cmmi-10">z </span>= <span
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612 class="cmsy-10"><img
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613 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Kinet_de_1902/fulltext/img/cmsy10-73.png" alt=" integral " class="10x-x-73" /></span> <span
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614 class="cmmi-10">dx</span><sub ><span
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615 class="cmr-7">1</span></sub> <span
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616 class="cmmi-10">...</span> <span
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617 class="cmmi-10">x</span><sub ><span
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618 class="cmmi-7">n</span></sub><span
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619 class="cmmi-10">, </span>wobei <br/>die Integrationsgrenzen dadurch bestimmt sein mögen, dass <br/><span
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620 class="cmmi-10"><img
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621 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Kinet_de_1902/fulltext/img/cmmi10-27.png" alt="f" class="cmmi-10x-x-27" align="middle" /></span>(<span
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622 class="cmmi-10">x</span>) zwischen einem gewissen Wert <span
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623 class="cmmi-10">y </span>und <span
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624 class="cmmi-10">y </span>+ <img
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625 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Kinet_de_1902/fulltext/img/cmr10-1.png" alt="D" class="10x-x-1" /> liege, wobei <img
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626 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Kinet_de_1902/fulltext/img/cmr10-1.png" alt="D" class="10x-x-1" /> <br/>eine Constante sei. Wir behaupten, dass <span
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627 class="cmmi-10">z</span>, welches allein <br/>von <span
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628 class="cmmi-10">y </span>Function ist, stets mit wachsendem <span
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629 class="cmmi-10">y </span>zunimmt, wenn <br/><span
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630 class="cmmi-10">n &gt; </span>2.
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631 </p><!--l. 475--><p class="indent"> Führen wir die neuen Variabeln ein <span
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632 class="cmmi-10">x</span><sub ><span
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633 class="cmr-7">1</span></sub> = <span
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634 class="cmmi-10"><img
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635 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Kinet_de_1902/fulltext/img/cmmi10-b.png" alt="a" class="10x-x-b" />x</span><sub ><span
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636 class="cmr-7">1</span></sub><sup ><span
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637 class="cmsy-7">'</span></sup> <span
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638 class="cmmi-10">...</span> <span
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639 class="cmmi-10">x</span><sub ><span
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640 class="cmmi-7">n</span></sub> = <span
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641 class="cmmi-10"><img
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642 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Kinet_de_1902/fulltext/img/cmmi10-b.png" alt="a" class="10x-x-b" />x</span><sub ><span
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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643 class="cmmi-7">n</span></sub><sup ><span
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644 class="cmsy-7">'</span></sup><span
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645 class="cmmi-10">, </span> <br/>wobei <span
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646 class="cmmi-10"><img
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647 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Kinet_de_1902/fulltext/img/cmmi10-b.png" alt="a" class="10x-x-b" /> </span>= const., dann ist:
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648 </p>
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649 <center class="par-math-display" >
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650 <img
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651 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Kinet_de_1902/fulltext/img/Einst_Kinet_de_190221x.png" alt=" integral
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652 z = an dx1'... d xn'. " class="par-math-display" /></center>
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653 <!--l. 483--><p class="nopar">
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654 </p><!--l. 486--><p class="noindent">Ferner erhalten wir <span
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655 class="cmmi-10"><img
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656 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Kinet_de_1902/fulltext/img/cmmi10-27.png" alt="f" class="cmmi-10x-x-27" align="middle" /></span>(<span
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657 class="cmmi-10">x</span>) = <span
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658 class="cmmi-10"><img
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659 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Kinet_de_1902/fulltext/img/cmmi10-b.png" alt="a" class="10x-x-b" /></span><sup ><span
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660 class="cmr-7">2</span></sup> <span
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661 class="cmmi-10"><img
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662 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Kinet_de_1902/fulltext/img/cmmi10-27.png" alt="f" class="cmmi-10x-x-27" align="middle" /></span>(<span
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663 class="cmmi-10">x</span><span
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664 class="cmsy-10">'</span>).
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665 </p><!--l. 490--><p class="indent"> Die Integrationsgrenzen des gewonnenen Integrals lauten <br/>also für <span
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666 class="cmmi-10"><img
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667 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Kinet_de_1902/fulltext/img/cmmi10-27.png" alt="f" class="cmmi-10x-x-27" align="middle" /></span>(<span
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668 class="cmmi-10">x</span><span
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669 class="cmsy-10">'</span>)
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670 </p>
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671 <center class="par-math-display" >
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672 <img
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673 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Kinet_de_1902/fulltext/img/Einst_Kinet_de_190222x.png" alt=" y y D a2- und a2-+ a2-. " class="par-math-display" /></center>
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674 <!--l. 497--><p class="nopar">
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675 </p><!--l. 500--><p class="noindent">Ist ferner <img
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676 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Kinet_de_1902/fulltext/img/cmr10-1.png" alt="D" class="10x-x-1" /> unendlich klein, was wir annehmen, so erhalten wir
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677 </p>
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678 <center class="par-math-display" >
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679 <img
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680 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Kinet_de_1902/fulltext/img/Einst_Kinet_de_190223x.png" alt=" integral
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681 z = an-2 dx1' ... dxn'. " class="par-math-display" /></center>
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682 <!--l. 508--><p class="nopar">
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683 </p><!--l. 511--><p class="noindent">Hierbei ist <span
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684 class="cmmi-10">y</span><span
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685 class="cmsy-10">' </span>zwischen den Grenzen
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686 </p>
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687 <center class="par-math-display" >
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688 <img
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689 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Kinet_de_1902/fulltext/img/Einst_Kinet_de_190224x.png" alt="y-- und y--+ D. a2 a2 " class="par-math-display" /></center>
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690 <!--l. 518--><p class="nopar">
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691 </p><!--l. 521--><p class="noindent">Obige Gleichung lässt sich auch schreiben
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692 </p>
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693
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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_Kinet_de_1902/fulltext/img/Einst_Kinet_de_190225x.png" alt=" ( ) z(y) = an- 2z-y- . a2 " class="par-math-display" /></center>
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697 <!--l. 527--><p class="nopar">
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698 </p><!--l. 530--><p class="noindent">Wählt man <span
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699 class="cmmi-10"><img
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700 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Kinet_de_1902/fulltext/img/cmmi10-b.png" alt="a" class="10x-x-b" /> </span>positiv und <span
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701 class="cmmi-10">n &gt; </span>2<span
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702 class="cmmi-10">, </span>so ist also stets
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703 </p>
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704 <center class="par-math-display" >
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diff changeset
705 <img
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706 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Kinet_de_1902/fulltext/img/Einst_Kinet_de_190226x.png" alt=" z(y) -(-y-)&gt; 1, z a2 " class="par-math-display" /></center>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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707 <!--l. 538--><p class="nopar">
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708 </p><!--l. 542--><p class="noindent">was zu beweisen war.
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diff changeset
709 </p><!--l. 545--><p class="indent"> Dieses Resultat benutzen wir, um zu beweisen, dass <span
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710 class="cmmi-10">h </span> <br/>positiv ist.
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diff changeset
711 </p><!--l. 548--><p class="indent"> Wir fanden
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diff changeset
712 </p>
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diff changeset
713 <center class="par-math-display" >
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714 <img
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diff changeset
715 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Kinet_de_1902/fulltext/img/Einst_Kinet_de_190227x.png" alt=" w'(E) h = 12------, w (E) " class="par-math-display" /></center>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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716 <!--l. 554--><p class="nopar">
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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717 </p><!--l. 558--><p class="noindent">wobei
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diff changeset
718 </p>
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diff changeset
719 <center class="par-math-display" >
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720 <img
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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diff changeset
721 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Kinet_de_1902/fulltext/img/Einst_Kinet_de_190228x.png" alt=" integral
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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diff changeset
722 w (E) = dp1 ... d qn, " class="par-math-display" /></center>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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723 <!--l. 564--><p class="nopar"> <pb/>
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724 </p><!--l. 571--><p class="indent">
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725
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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diff changeset
726 </p><!--l. 572--><p class="noindent">und E zwischen E und <span
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diff changeset
727 class="cmmi-10">E </span>+ <span
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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728 class="cmmi-10"><img
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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diff changeset
729 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Kinet_de_1902/fulltext/img/cmmi10-e.png" alt="d" class="10x-x-e" /> </span><span class="overline"><span
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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diff changeset
730 class="cmmi-10">E</span></span><span
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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731 class="cmmi-10">. <img
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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732 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Kinet_de_1902/fulltext/img/cmmi10-21.png" alt="w" class="10x-x-21" /> </span>(<span
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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733 class="cmmi-10">E</span>) ist der Definition nach <br/>notwendig positiv, wir haben nur zu zeigen, dass auch <span
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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734 class="cmmi-10"><img
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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735 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Kinet_de_1902/fulltext/img/cmmi10-21.png" alt="w" class="10x-x-21" /></span><span
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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736 class="cmsy-10">'</span>(<span
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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737 class="cmmi-10">E</span>) <br/>stets positiv ist.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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738 </p><!--l. 577--><p class="indent"> Wir wählen <span
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739 class="cmmi-10">E</span><sub ><span
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740 class="cmr-7">1</span></sub> und <span
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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741 class="cmmi-10">E</span><sub ><span
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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742 class="cmr-7">2</span></sub>, sodass <span
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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743 class="cmmi-10">E</span><sub ><span
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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744 class="cmr-7">2</span></sub> <span
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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745 class="cmmi-10">&gt; E</span><sub ><span
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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746 class="cmr-7">1</span></sub>, und beweisen, <br/>dass <span
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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747 class="cmmi-10"><img
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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748 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Kinet_de_1902/fulltext/img/cmmi10-21.png" alt="w" class="10x-x-21" /> </span>(<span
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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749 class="cmmi-10">E</span><sub ><span
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750 class="cmr-7">2</span></sub>) <span
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751 class="cmmi-10">&gt; <img
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752 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Kinet_de_1902/fulltext/img/cmmi10-21.png" alt="w" class="10x-x-21" /> </span>(<span
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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753 class="cmmi-10">E</span><sub ><span
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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754 class="cmr-7">1</span></sub>) und zerlegen <span
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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755 class="cmmi-10"><img
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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756 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Kinet_de_1902/fulltext/img/cmmi10-21.png" alt="w" class="10x-x-21" /> </span>(<span
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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757 class="cmmi-10">E</span><sub ><span
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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758 class="cmr-7">1</span></sub>) in unendlich viele <br/>Summanden von der Form
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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759 </p>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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760 <center class="par-math-display" >
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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761 <img
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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diff changeset
762 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Kinet_de_1902/fulltext/img/Einst_Kinet_de_190229x.png" alt=" integral
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
763
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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764 d (w (E1)) = dp1 ... d pn d q1 ... dqn. " class="par-math-display" /></center>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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765 <!--l. 586--><p class="nopar">
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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766 </p><!--l. 589--><p class="noindent">Bei dem angedeuteten Integral besitzen die <span
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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767 class="cmmi-10">p </span>bestimmte und <br/>zwar solche Werte, dass
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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768 <span
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
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769 class="cmmi-10">V </span><span
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
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770 class="msam-10"><img
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
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771 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Kinet_de_1902/fulltext/img/msam10-35.png" alt="&lt;=" class="10x-x-35" /> </span><span
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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772 class="cmmi-10">E</span><sub ><span
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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773 class="cmr-7">1</span></sub><span
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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774 class="cmmi-10">. </span>Die Integrationsgrenzen des <br/>lntegrals sind so charakterisirt, dass <span
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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775 class="cmmi-10">L </span>zwischen
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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776 <span
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
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777 class="cmmi-10">E</span><sub ><span
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
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778 class="cmr-7">1</span></sub> <span
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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779 class="cmsy-10">- </span><span
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
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780 class="cmmi-10">V </span>und <br/><span
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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781 class="cmmi-10">E</span><sub ><span
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
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782 class="cmr-7">1</span></sub> + <span
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
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783 class="cmmi-10"><img
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
784 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Kinet_de_1902/fulltext/img/cmmi10-e.png" alt="d" class="10x-x-e" /></span><span class="overline"><span
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
785 class="cmmi-10">E</span></span> <span
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
786 class="cmsy-10">- </span><span
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
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787 class="cmmi-10">V </span>liegt.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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788 </p><!--l. 593--><p class="indent"> Jedem unendlich kleinen derartigen Summanden entspricht <br/>aus <span
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
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789 class="cmmi-10"><img
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
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790 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Kinet_de_1902/fulltext/img/cmmi10-21.png" alt="w" class="10x-x-21" /> </span>(<span
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
791 class="cmmi-10">E</span><sub ><span
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792 class="cmr-7">2</span></sub>) ein Term von der Grösse
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793 </p>
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794 <center class="par-math-display" >
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795 <img
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796 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Kinet_de_1902/fulltext/img/Einst_Kinet_de_190230x.png" alt=" integral
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797 d [w (E2)] = dp1 ... dpn dq1 ... d qn, " class="par-math-display" /></center>
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798 <!--l. 600--><p class="nopar">
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799 </p><!--l. 604--><p class="noindent">wobei die <span
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diff changeset
800 class="cmmi-10">p </span>und <span
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801 class="cmmi-10">dp </span>die nämlichen Werte haben wie in <span
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802 class="cmmi-10">d</span><img
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803 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Kinet_de_1902/fulltext/img/Einst_Kinet_de_190231x.png" alt="[w(E1)]" class="left" align="middle" /><span
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804 class="cmmi-10">, </span> <br/><span
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805 class="cmmi-10">L </span>aber zwischen den Grenzen <span
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806 class="cmmi-10">E</span><sub ><span
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807 class="cmr-7">2</span></sub> <span
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808 class="cmsy-10">- </span><span
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809 class="cmmi-10">V </span>und <span
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810 class="cmmi-10">E</span><sub ><span
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diff changeset
811 class="cmr-7">2</span></sub> <span
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812 class="cmsy-10">- </span><span
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813 class="cmmi-10">V </span>+ <span
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814 class="cmmi-10"><img
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diff changeset
815 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Kinet_de_1902/fulltext/img/cmmi10-e.png" alt="d" class="10x-x-e" /> </span><span class="overline"><span
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816 class="cmmi-10">E</span></span> liegt.
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817 </p><!--l. 609--><p class="indent"> Es ist also nach dem eben bewiesenen Satze
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818 </p>
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819 <center class="par-math-display" >
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820 <img
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821 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Kinet_de_1902/fulltext/img/Einst_Kinet_de_190232x.png" alt="d [w (E2)] &gt; d [w(E1)]. " class="par-math-display" /></center>
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822 <!--l. 615--><p class="nopar">
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823 </p><!--l. 618--><p class="noindent">Folglich
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824 </p>
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825 <center class="par-math-display" >
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826 <img
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827 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Kinet_de_1902/fulltext/img/Einst_Kinet_de_190233x.png" alt=" sum sum
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828 d [w (E2)] &gt; d [w(E1)], " class="par-math-display" /></center>
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829 <!--l. 625--><p class="nopar">
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830 </p><!--l. 629--><p class="noindent">wobei <span
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831 class="cmex-10"><img
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832 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Kinet_de_1902/fulltext/img/cmex10-50.png" alt=" sum " class="10x-x-50" /></span>
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833 über alle entsprechende Gebiete der <span
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834 class="cmmi-10">p </span>zu erstrecken ist. <br/>
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835 </p><!--l. 632--><p class="indent"> Es ist aber
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836 </p>
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837 <center class="par-math-display" >
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838 <img
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839 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Kinet_de_1902/fulltext/img/Einst_Kinet_de_190234x.png" alt=" sum
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840 d [w (E1)] &gt; w(E1), " class="par-math-display" /></center>
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841
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842 <!--l. 637--><p class="nopar">
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843 </p><!--l. 641--><p class="noindent">wenn das Summenzeichen über alle <span
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844 class="cmmi-10">p </span>erstreckt wird, sodass <br/>
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845 </p>
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846 <center class="par-math-display" >
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847 <img
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848 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Kinet_de_1902/fulltext/img/Einst_Kinet_de_190235x.png" alt="V &lt;= E1. " class="par-math-display" /></center>
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849 <!--l. 647--><p class="nopar">
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850 </p><!--l. 650--><p class="noindent">Ferner ist
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851 </p>
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852 <center class="par-math-display" >
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853 <img
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854 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Kinet_de_1902/fulltext/img/Einst_Kinet_de_190236x.png" alt=" sum
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855 d [w (E2)] &lt; w(E2), " class="par-math-display" /></center>
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856 <!--l. 656--><p class="nopar">
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857 </p><!--l. 660--><p class="noindent">weil das Gebiet der <span
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858 class="cmmi-10">p</span>, welches durch die Gleichung
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859 </p>
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860 <center class="par-math-display" >
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861 <img
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862 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Kinet_de_1902/fulltext/img/Einst_Kinet_de_190237x.png" alt="V &lt;= E2 " class="par-math-display" /></center>
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863 <!--l. 666--><p class="nopar">
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864 </p><!--l. 670--><p class="noindent">bestimmt wird, das durch die Gleichung
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865 </p>
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866 <center class="par-math-display" >
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867 <img
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868 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Kinet_de_1902/fulltext/img/Einst_Kinet_de_190238x.png" alt="V &lt;= E1 " class="par-math-display" /></center>
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869 <!--l. 676--><p class="nopar">
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870 </p><!--l. 680--><p class="noindent">definirte Gebiet vollständig in sich einschliesst. <pb/>
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871 </p><!--l. 686--><p class="indent">
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872
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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873 </p>
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874 <div class="center" >
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875
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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876 <!--l. 687--><p class="noindent">
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877 </p><!--l. 688--><p class="noindent"><span
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878 class="cmsy-10">§ </span>5. Ueber das Temperaturgleichgewicht.</p></div>
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879 <!--l. 691--><p class="indent"> Wir wählen nun ein System <span
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880 class="cmmi-10">S </span>von ganz bestimmter Be-<br/>schaffenheit und nennen es Thermometer. Es stehe mit dem <br/>System <span
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881 class="cmex-10"><img
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882 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Kinet_de_1902/fulltext/img/cmex10-50.png" alt=" sum " class="10x-x-50" /></span>
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883 von relativ unendlich grosser Energie in mecha-<br/>nischer Wechselwirkung. Ist der Zustand des Ganzen stationär, <br/>so ist der Zustand des Thermometers durch die Gleichung <br/>definirt
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884 </p>
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885 <center class="par-math-display" >
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886 <img
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887 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Kinet_de_1902/fulltext/img/Einst_Kinet_de_190239x.png" alt="d W = A e-2hEd p1 ... dqn, " class="par-math-display" /></center>
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888 <!--l. 701--><p class="nopar">
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889 </p><!--l. 705--><p class="noindent">wobei <span
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890 class="cmmi-10">d W </span>die Wahrscheinlichkeit dafür bedeutet, dass die <br/>Werte der Zustandsvariabeln des Thermometers innerhalb der <br/>angedeuteten Grenzen liegen. Dabei besteht zwischen den <br/>Constanten <span
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891 class="cmmi-10">A </span>und <span
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892 class="cmmi-10">h </span>die Gleichung
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893 </p>
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894 <center class="par-math-display" >
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895 <img
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896 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Kinet_de_1902/fulltext/img/Einst_Kinet_de_190240x.png" alt=" integral
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897 1 = A. e-2hE dp1 ... dqn, " class="par-math-display" /></center>
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898 <!--l. 712--><p class="nopar">
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899 </p><!--l. 716--><p class="noindent">wobei die Integration über alle möglichen Werte der Zustands-<br/>variabeln erstreckt ist. Die Grösse <span
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900 class="cmmi-10">h </span>bestimmt also den Zu-<br/>stand des Thermometers vollkommen. Wir nennen <span
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901 class="cmmi-10">h </span>die Tem-<br/>peraturfunction, indem wir bemerken, dass nach dem Gesagten <br/>jede an dem System <span
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902 class="cmmi-10">S </span>beobachtbare Grösse <span
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903 class="cmmi-10">H</span>
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904 Function von <span
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905 class="cmmi-10">h </span> <br/>allein sein muss, solange <span
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906 class="cmmi-10">V</span> <sub ><span
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907 class="cmmi-7">a</span></sub> unverändert bleibt, was wir an-<br/>genommen haben. Die Grösse <span
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908 class="cmmi-10">h </span>aber hängt lediglich vom <br/>Zustande des Systems <span
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909 class="cmex-10"><img
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910 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Kinet_de_1902/fulltext/img/cmex10-50.png" alt=" sum " class="10x-x-50" /></span>
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911 ab (<span
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912 class="cmsy-10">§ </span>3), ist also unabhängig davon, <br/>wie <span
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913 class="cmex-10"><img
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914 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Kinet_de_1902/fulltext/img/cmex10-50.png" alt=" sum " class="10x-x-50" /></span>
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915 mit <span
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916 class="cmmi-10">S </span>thermisch verbunden ist. Es folgt daraus un-<br/>mittelbar der Satz: Ist ein System <span
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917 class="cmex-10"><img
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918 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Kinet_de_1902/fulltext/img/cmex10-50.png" alt=" sum " class="10x-x-50" /></span>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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919 mit zwei unendlich <br/>kleinen Thermometern <span
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920 class="cmmi-10">S </span>und <span
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921 class="cmmi-10">S</span><span
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922 class="cmsy-10">' </span>verbunden, so kommt diesen <br/>beiden Thermometern dieselbe Grösse <span
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923 class="cmmi-10">h </span>zu. Sind <span
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924 class="cmmi-10">S </span>und <span
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925 class="cmmi-10">S</span><span
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926 class="cmsy-10">'</span> <br/>identische Systeme, so kommt ihnen auch noch derselbe Wert <br/>der beobachtbaren Grösse <span
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927 class="cmmi-10">H</span>
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928 zu.
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929 </p><!--l. 727--><p class="indent"> Wir führen nun nur identische Thermometer <span
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930 class="cmmi-10">S </span>ein und <br/>nennen <span
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931 class="cmmi-10">H </span>das beobachtbare Temperaturmaass. Wir erhalten <br/>also den Satz: Das an <span
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932 class="cmmi-10">S </span>beobachtbare Temperaturmaass <span
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933 class="cmmi-10">H </span> <br/>ist unabhängig von der Art, wie <span
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934 class="cmex-10"><img
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935 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Kinet_de_1902/fulltext/img/cmex10-50.png" alt=" sum " class="10x-x-50" /></span>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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936 mit <span
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937 class="cmmi-10">S </span>mechanisch ver-<br/>bunden ist; die Grösse <span
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938 class="cmmi-10">H </span>bestimmt <span
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939 class="cmmi-10">h</span>, dieses die Energie E <br/>des Systems <span
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940 class="cmex-10"><img
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941 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Kinet_de_1902/fulltext/img/cmex10-50.png" alt=" sum " class="10x-x-50" /></span>
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942 und diese dessen Zustand nach unserer Vor-<br/>aussetzung.
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943 </p><!--l. 736--><p class="indent"> Aus dem Bewiesenen folgt sofort, dass zwei Systeme <br/><span
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944 class="cmex-10"><img
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945 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Kinet_de_1902/fulltext/img/cmex10-50.png" alt=" sum " class="10x-x-50" /></span>
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946 <sub ><span
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947 class="cmr-7">1</span></sub> und <span
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948 class="cmex-10"><img
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949 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Kinet_de_1902/fulltext/img/cmex10-50.png" alt=" sum " class="10x-x-50" /></span>
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950 <sub ><span
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951 class="cmr-7">2</span></sub> im Falle mechanischer Verbindung kein im statio-<br/><pb/>
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952 </p><!--l. 742--><p class="indent">
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953
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954 </p><!--l. 743--><p class="noindent">nären Zustand befindliches System bilden können, wenn nicht <br/>zwei mit ihnen verbundene Thermometer <span
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955 class="cmmi-10">S </span>gleiches Tem-<br/>peraturmaass oder, was dasselbe bedeutet, sie selbst gleiche <br/>Temperaturfunction besitzen. Da der Zustand der Systeme <br/><span
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956 class="cmex-10"><img
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957 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Kinet_de_1902/fulltext/img/cmex10-50.png" alt=" sum " class="10x-x-50" /></span>
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958 <sub ><span
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959 class="cmr-7">1</span></sub> und
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960 <span
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961 class="cmex-10"><img
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962 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Kinet_de_1902/fulltext/img/cmex10-50.png" alt=" sum " class="10x-x-50" /></span>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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963 <sub ><span
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964 class="cmr-7">2</span></sub> durch die Grössen <span
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965 class="cmmi-10">h</span><sub ><span
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966 class="cmr-7">1</span></sub> und <span
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967 class="cmmi-10">h</span><sub ><span
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parents:
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968 class="cmr-7">2</span></sub> oder <span
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
969 class="cmmi-10">H</span><sub ><span
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
970 class="cmr-7">1</span></sub> und <span
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
971 class="cmmi-10">H</span><sub ><span
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
972 class="cmr-7">2</span></sub> <br/>vollständig definirt wird, so folgt, dass das Temperaturgleich-<br/>gewicht lediglich durch die Bedingungen <span
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
973 class="cmmi-10">h</span><sub ><span
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
974 class="cmr-7">1</span></sub> = <span
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
975 class="cmmi-10">h</span><sub ><span
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
976 class="cmr-7">2</span></sub> oder
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
977 <span
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
978 class="cmmi-10">H</span><sub ><span
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
979 class="cmr-7">1</span></sub> = <span
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
980 class="cmmi-10">H</span><sub ><span
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
981 class="cmr-7">2</span></sub> <br/>bestimmt sein kann.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
982 </p><!--l. 754--><p class="indent"> Es bleibt jetzt noch übrig, zu zeigen, dass zwei Systeme <br/>von gleicher Temperaturfunction <span
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
983 class="cmmi-10">h </span>(oder gleichem Temperatur-<br/>maass <span
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
984 class="cmmi-10">H</span>) mechanisch verbunden werden können zu einem <br/>einzigen System von gleicher Temperaturfunction.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
985 </p><!--l. 760--><p class="indent"> Seien zwei mechanische Systeme <span
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casties
parents:
diff changeset
986 class="cmex-10"><img
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
987 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Kinet_de_1902/fulltext/img/cmex10-50.png" alt=" sum " class="10x-x-50" /></span>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
988 <sub ><span
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
989 class="cmr-7">1</span></sub> und <span
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
990 class="cmex-10"><img
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
991 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Kinet_de_1902/fulltext/img/cmex10-50.png" alt=" sum " class="10x-x-50" /></span>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
992 <sub ><span
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
993 class="cmr-7">2</span></sub> mechanisch <br/>zu einem System verschmolzen, so jedoch, dass die Terme <br/>der Energie unendlich klein sind, welche Zustandsvariabeln <br/>beider Systeme enthalten. Sowohl <span
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
994 class="cmex-10"><img
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
995 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Kinet_de_1902/fulltext/img/cmex10-50.png" alt=" sum " class="10x-x-50" /></span>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
996 <sub ><span
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
997 class="cmr-7">1</span></sub> als <span
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
998 class="cmex-10"><img
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
999 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Kinet_de_1902/fulltext/img/cmex10-50.png" alt=" sum " class="10x-x-50" /></span>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
1000 <sub ><span
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
1001 class="cmr-7">2</span></sub> seien verknüpft <br/>mit einem unendlich kleinen Thermometer <span
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
1002 class="cmmi-10">S. </span>Die Angaben <br/><span
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
1003 class="cmmi-10">H</span><sub ><span
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
1004 class="cmr-7">1</span></sub> und <span
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
1005 class="cmmi-10">H</span><sub ><span
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
1006 class="cmr-7">2</span></sub> desselben sind bis auf unendlich Kleines jeden-<br/>falls dieselben, weil sie sich nur auf verschiedene Stellen, eines <br/>einzigen, im stationären Zustande befindlichen Systems be-<br/>ziehen. Ebenso natürlich die Grössen <span
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
1007 class="cmmi-10">h</span><sub ><span
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
1008 class="cmr-7">1</span></sub> und <span
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
1009 class="cmmi-10">h</span><sub ><span
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
1010 class="cmr-7">2</span></sub>. Wir denken
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
1011 <br/>uns nun unendlich langsam die beiden Systemen gemeinsame <br/>Terme der Energie gegen Null hin abnehmen. Hierbei ändern <br/>sich sowohl die Grössen <span
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
1012 class="cmmi-10">H</span>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
1013 und <span
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
1014 class="cmmi-10">h</span>, als auch die Zustands-<br/>verteilungen beider Systeme unendlich wenig, da diese allein <br/>durch die Energie bestimmt sind. Ist dann die vollständige
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
1015 <br/>mechanische Trennung von <span
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
1016 class="cmex-10"><img
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
1017 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Kinet_de_1902/fulltext/img/cmex10-50.png" alt=" sum " class="10x-x-50" /></span>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
1018 <sub ><span
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
1019 class="cmr-7">1</span></sub> und <span
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
1020 class="cmex-10"><img
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
1021 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Kinet_de_1902/fulltext/img/cmex10-50.png" alt=" sum " class="10x-x-50" /></span>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
1022 <sub ><span
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
1023 class="cmr-7">2</span></sub> ausgeführt, so bleiben <br/>gleichwohl die Beziehungen
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
1024 </p>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
1025 <center class="par-math-display" >
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
1026 <img
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
1027 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Kinet_de_1902/fulltext/img/Einst_Kinet_de_190241x.png" alt="H1 = H2, h1 = h2 " class="par-math-display" /></center>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
1028 <!--l. 783--><p class="nopar">
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
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1029 </p><!--l. 787--><p class="noindent">bestehen und die Zustandsverteilung ist unendlich wenig ver-<br/>ändert. <span
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
1030 class="cmmi-10">H</span><sub ><span
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
1031 class="cmr-7">1</span></sub> und <span
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
1032 class="cmmi-10">h</span><sub ><span
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
1033 class="cmr-7">1</span></sub>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
1034 beziehen sich aber nur mehr auf <span
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
1035 class="cmex-10"><img
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
1036 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Kinet_de_1902/fulltext/img/cmex10-50.png" alt=" sum " class="10x-x-50" /></span>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
1037 <sub ><span
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
1038 class="cmr-7">1</span></sub>, <br/><span
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
1039 class="cmmi-10">H</span><sub ><span
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
1040 class="cmr-7">2</span></sub> und <span
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
1041 class="cmmi-10">h</span><sub ><span
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
1042 class="cmr-7">2</span></sub> nur mehr auf <span
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
1043 class="cmex-10"><img
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
1044 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Kinet_de_1902/fulltext/img/cmex10-50.png" alt=" sum " class="10x-x-50" /></span>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
1045 <sub ><span
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
1046 class="cmr-7">2</span></sub>. Unser Process ist streng um-<br/>kehrbar, da er sich aus einer Aufeinanderfolge von stationären <br/>Zuständen zusammensetzt. Wir erhalten also den Satz:
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
1047 </p><!--l. 794--><p class="indent"> Zwei Systeme von der gleichen Temperaturfunction <span
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
1048 class="cmmi-10">h </span> <br/>lassen sich zu einem einzigen System von der Temperatur-<br/>function <span
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
1049 class="cmmi-10">h </span>verknüpfen, sodass sich deren Zustandsverteilung <br/>unendlich wenig ändert. <pb/>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
1050 </p><!--l. 803--><p class="indent">
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
1051
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
1052 </p><!--l. 804--><p class="indent"> Gleichheit der Grössen <span
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
1053 class="cmmi-10">h </span>ist also die notwendige und <br/>hinreichende Bedingung für die stationäre Verknüpfung (Wärme-<br/>gleichgewicht) zweier Systeme. Daraus folgt sofort: Sind die <br/>Systeme <span
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
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1054 class="cmex-10"><img
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
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1055 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Kinet_de_1902/fulltext/img/cmex10-50.png" alt=" sum " class="10x-x-50" /></span>
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1056 <sub ><span
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1057 class="cmr-7">1</span></sub> und <span
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1058 class="cmex-10"><img
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1059 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Kinet_de_1902/fulltext/img/cmex10-50.png" alt=" sum " class="10x-x-50" /></span>
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1060 <sub ><span
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1061 class="cmr-7">2</span></sub>, und die Systeme <span
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1062 class="cmex-10"><img
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1063 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Kinet_de_1902/fulltext/img/cmex10-50.png" alt=" sum " class="10x-x-50" /></span>
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1064 <sub ><span
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1065 class="cmr-7">1</span></sub> und <span
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1066 class="cmex-10"><img
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1067 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Kinet_de_1902/fulltext/img/cmex10-50.png" alt=" sum " class="10x-x-50" /></span>
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1068 <sub ><span
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1069 class="cmr-7">3</span></sub> statiouär
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1070 <br/>mechanisch verknüpfbar (im Wärmegleichgewichte), so sind <br/>es auch <span
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1071 class="cmex-10"><img
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1072 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Kinet_de_1902/fulltext/img/cmex10-50.png" alt=" sum " class="10x-x-50" /></span>
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1073 <sub ><span
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1074 class="cmr-7">2</span></sub> und
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1075 <span
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1076 class="cmex-10"><img
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1077 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Kinet_de_1902/fulltext/img/cmex10-50.png" alt=" sum " class="10x-x-50" /></span>
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1078 <sub ><span
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1079 class="cmr-7">3</span></sub>.
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1080 </p><!--l. 813--><p class="indent"> Ich will hier bemerken, dass wir bis jetzt von der Vor-<br/>aussetzung, dass unsere Systeme mechanische seien, nur inso-<br/>fern Gebrauch gemacht haben, als wir den Liouville&#8217;schen <br/>Satz und das Energieprincip verwendet haben. Wahrschein-<br/>lich lassen sich die Fundamente der Wärmetheorie für noch <br/>weit allgemeiner definirte Systeme entwickeln. Solches wollen <br/>wir hier jedoch nicht versuchen, sondern uns auf die mecha-<br/>nischen Gleichungen stützen. Die wichtige Frage, inwiefern <br/>sich der Gedankengang von dem benutzten Bilde loslösen und <br/>verallgemeinern lässt, werden wir hier nicht behandeln.
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1081 </p>
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1082 <div class="center" >
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1083
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1084 <!--l. 826--><p class="noindent">
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1085 </p><!--l. 827--><p class="noindent"><span
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1086 class="cmsy-10">§ </span>6. Ueber die mechanische Bedeutung der Grösse <span
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1087 class="cmmi-10">h.</span><sup ><span
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1088 class="cmr-7">1</span></sup>)</p></div>
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1089 <!--l. 830--><p class="indent"> Die lebendige Kraft <span
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1090 class="cmmi-10">L </span>eines Systems ist eine homogene <br/>quadratische Function der Grössen <span
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1091 class="cmmi-10">q. </span>Durch eine lineare <br/>Substitution lassen sich stets Variable <span
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1092 class="cmmi-10">r </span>einführen, sodass die <br/>lebendige Kraft in der Form erscheint
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1093 </p>
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1094 <center class="par-math-display" >
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1095 <img
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1096 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Kinet_de_1902/fulltext/img/Einst_Kinet_de_190242x.png" alt="L = 1(a r2+ a r2+ ...+ a r 2) 2 1 1 2 2 n n " class="par-math-display" /></center>
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1097 <!--l. 839--><p class="nopar">
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1098 </p><!--l. 843--><p class="noindent">und dass
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1099 </p>
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1100 <center class="par-math-display" >
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1101 <img
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1102 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Kinet_de_1902/fulltext/img/Einst_Kinet_de_190243x.png" alt=" integral integral
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1103
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1104 d q1 ... dqn = dr1...drn, " class="par-math-display" /></center>
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1105 <!--l. 850--><p class="nopar">
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1106 </p><!--l. 854--><p class="noindent">wenn man die Integrale über entsprechende unendlich kleine <br/>Gebiete ausdehnt. Die Grössen <span
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1107 class="cmmi-10">r </span>nennt Boltzmann Momen-<br/>toiden. Die mittlere lebendige Kraft, welche einer Momentoide <br/>entspricht, wenn das System mit einem anderen, von viel <br/>grösserer Energie, ein System bildet, nimmt die Form an:
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1108 </p>
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1109 <center class="par-math-display" >
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1110 <img
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1111 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Kinet_de_1902/fulltext/img/Einst_Kinet_de_190244x.png" alt=" integral '' -2h[V+a1r2+a2r2+...+anrn2] anr2n -A- integral e--------1----2--------.--2--.dp1-... d-pn.dr1-... d-rn= 1-. A''e-2h [V+a1 r21+a2r22+...+an rn2].dp1 ... dpn dr1 ... drn 4h " class="par-math-display" /></center>
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1112
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1113 <!--l. 868--><p class="nopar">
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1114 </p><!--l. 871--><p class="noindent">----------
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1115 </p><!--l. 874--><p class="indent"> 1) Vgl. L. Boltzmann, Gastheorie, II. Teil, <span
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1116 class="cmsy-10">§§ </span>33, 34, 42. <pb/>
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1117 </p><!--l. 879--><p class="indent">
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1118
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1119 </p><!--l. 880--><p class="indent"> Die mittlere lebendige Kraft aller Momentoiden eines <br/>Systems ist also dieselbe und gleich:
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1120 </p>
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1121 <center class="par-math-display" >
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1122 <img
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1123 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Kinet_de_1902/fulltext/img/Einst_Kinet_de_190245x.png" alt="1-- L- 4h = n , " class="par-math-display" /></center>
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1124 <!--l. 886--><p class="nopar">
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1125 </p><!--l. 890--><p class="noindent">wobei <span
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1126 class="cmmi-10">L </span>die lebendige Kraft des Systems bedeutet. </p>
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1127 <div class="center" >
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1128
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1129 <!--l. 891--><p class="noindent">
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1130 </p><!--l. 892--><p class="noindent"><span
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1131 class="cmsy-10">§ </span>7. Ideale Gase. Absolute Temperatur.</p></div>
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1132 <!--l. 895--><p class="indent"> Die entwickelte Theorie enthält als speciellen Fall die <br/>Maxwell&#8217;sche Zustandsverteilung der idealen Gase. Verstehen <br/>wir nämlich in <span
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1133 class="cmsy-10">§ </span>3 unter dem System <span
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1134 class="cmmi-10">S </span>ein Gasmolecül, unter <br/><span
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1135 class="cmex-10"><img
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1136 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Kinet_de_1902/fulltext/img/cmex10-50.png" alt=" sum " class="10x-x-50" /></span>
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1137 die Gesamtheit aller anderen, so folgt für die Wahrschein-<br/>lichkeit, dass die Werte der Variabeln <span
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1138 class="cmmi-10">p</span><sub ><span
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1139 class="cmr-7">1</span></sub><span
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1140 class="cmmi-10">...</span><span
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1141 class="cmmi-10">q</span><sub ><span
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1142 class="cmmi-7">n</span></sub> von <span
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1143 class="cmmi-10">S </span>in
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1144 <br/>einem in Bezug auf alle Variabeln unendlich kleinen Gebiet <span
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1145 class="cmmi-10">g </span> <br/>liegen, der Ausdruck
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1146 </p>
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1147 <center class="par-math-display" >
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1148 <img
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1149 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Kinet_de_1902/fulltext/img/Einst_Kinet_de_190246x.png" alt=" integral
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1150 dW = A e-2hE dp1 ... dqn. g " class="par-math-display" /></center>
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1151 <!--l. 905--><p class="nopar">
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1152 </p><!--l. 908--><p class="noindent">Auch erkennt man sogleich aus unserem, für die Grösse <span
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1153 class="cmmi-10">h </span>in <br/><span
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1154 class="cmsy-10">§ </span>3 gefundenen Ausdruck, dass die Grösse <span
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1155 class="cmmi-10">h </span>bis auf unend-<br/>lich Kleines die nämliche wäre für ein Gasmolecül anderer <br/>Art, welches in dem Systeme vorkommt, in dem die Systeme <span
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1156 class="cmex-10"><img
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1157 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Kinet_de_1902/fulltext/img/cmex10-50.png" alt=" sum " class="10x-x-50" /></span>
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1158 ,
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1159 <br/>welche <span
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1160 class="cmmi-10">h </span>bestimmen, für beide Molecüle bis auf unendlich <br/>Kleines identisch sind. Damit ist die verallgemeinerte Max-<br/>well&#8217;sche Zustandsverteilung für ideale Gase erwiesen. --
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1161 </p><!--l. 914--><p class="indent"> Ferner folgt sofort, dass die mittlere lebendige Kraft der <br/>Schwerpunktsbewegung eines Gasmolecüles, welches in einem <br/>System <span
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1162 class="cmmi-10">S </span>vorkommt, den Wert 3/4 <span
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1163 class="cmmi-10">h</span>
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1164 besitzt, weil dieselbe drei <br/>Momentoiden entspricht. Nun lehrt die kinetische Gastheorie, <br/>dass diese Grösse proportional dem vom Gase bei constanten
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1165 <br/>Volumen ausgeübten Druck ist. Setzt man diesen definitions-<br/>gemäss der absoluten Temperatur <span
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1166 class="cmmi-10">T </span>proportional, so hat man <br/>eine Beziehung von der Form
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1167 </p>
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1168 <center class="par-math-display" >
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1169 <img
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1170 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Kinet_de_1902/fulltext/img/Einst_Kinet_de_190247x.png" alt=" -- -1-- 1w-(E)-- 4h = z.T = 2w'(E), " class="par-math-display" /></center>
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1171 <!--l. 925--><p class="nopar">
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1172 </p><!--l. 929--><p class="noindent">wobei <span
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1173 class="cmmi-10">z </span>eine universelle Constante, <span
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1174 class="cmmi-10"><img
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1175 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Kinet_de_1902/fulltext/img/cmmi10-21.png" alt="w" class="10x-x-21" /> </span>die in <span
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1176 class="cmsy-10">§ </span>3 eingeführte <br/>Function bedeutet.
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1177 <pb/>
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1178 </p><!--l. 935--><p class="indent">
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1179
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1180 </p>
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1181 <div class="center" >
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1182
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1183 <!--l. 936--><p class="noindent">
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1184 </p><!--l. 937--><p class="noindent"><span
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1185 class="cmsy-10">§ </span>8. Der zweite Hauptsatz der Wärmetheorie als Folgerung der <br/>mechanischen Theorie.</p></div>
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1186 <!--l. 940--><p class="indent"> Wir betrachten ein gegebenes physikalisches System <span
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1187 class="cmmi-10">S </span> <br/>als mechanisches System mit den Coordinaten <span
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1188 class="cmmi-10">p</span><sub ><span
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1189 class="cmr-7">1</span></sub> <span
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1190 class="cmmi-10">...</span> <span
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1191 class="cmmi-10">p</span><sub ><span
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1192 class="cmmi-7">n</span></sub><span
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1193 class="cmmi-10">. </span>Als <br/>Zustandsvariable in demselben führen wir ferner die Grössen
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1194 </p>
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1195 <center class="par-math-display" >
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1196 <img
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1197 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Kinet_de_1902/fulltext/img/Einst_Kinet_de_190248x.png" alt="d p d p ---1 = p1'...---n = pn' dt dt " class="par-math-display" /></center>
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1198 <!--l. 949--><p class="nopar">
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1199 </p><!--l. 953--><p class="noindent">ein. <span
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1200 class="cmmi-10">P</span><sub ><span
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1201 class="cmr-7">1</span></sub> <span
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1202 class="cmmi-10">...</span> <span
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1203 class="cmmi-10">P</span><sub ><span
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1204 class="cmmi-7">n</span></sub> seien die äusseren Kräfte, welche die Coordi-<br/>naten des Systems zu vergrössern streben. <span
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1205 class="cmmi-10">V</span> <sub ><span
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1206 class="cmmi-7">i</span></sub> sei die poten-<br/>tielle Energie des Systems, <span
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1207 class="cmmi-10">L </span>dessen lebendige Kraft, welche <br/>eine homogene quadratische Function der <span
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1208 class="cmmi-10">p</span><sub ><span
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1209 class="cmmi-7"><img
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1210 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Kinet_de_1902/fulltext/img/cmmi7-17.png" alt="n" class="7x-x-17" /></span></sub><sup ><span
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1211 class="cmsy-7">'</span></sup> ist. Die Be-<br/>wegungsgleichungen von Lagrange nehmen für ein solches <br/>System die Form an
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1212 </p>
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1213 <center class="par-math-display" >
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1214 <img
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1215 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Kinet_de_1902/fulltext/img/Einst_Kinet_de_190249x.png" alt=" [ ] @-(Vi---L)+ -d- @-L- - P = 0, (n = 1,..n = n). @pn dt @pn' n " class="par-math-display" /></center>
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1216 <!--l. 964--><p class="nopar">
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1217 </p><!--l. 967--><p class="noindent">Die äusseren Kräfte setzen sich aus zweierlei Kräften zu-<br/>sammen. Die einen, <span
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1218 class="cmmi-10">P </span><sub><span
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1219 class="cmmi-7"><img
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1220 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Kinet_de_1902/fulltext/img/cmmi7-17.png" alt="n" class="7x-x-17" /></span></sub><sup><span
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1221 class="cmr-7">(1)</span></sup>, sind diejenigen Kräfte, welche die <br/>Bedingungen des Systems darstellen, und von einem Potential <br/>ableitbar sind, welches nur Function der <span
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1222 class="cmmi-10">p</span><sub ><span
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1223 class="cmr-7">1</span></sub> <span
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1224 class="cmmi-10">...</span> <span
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1225 class="cmmi-10">p</span><sub ><span
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1226 class="cmmi-7">n</span></sub> ist (adia-<br/>batische Wände, Schwerkraft etc.):
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diff changeset
1227 </p>
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diff changeset
1228 <center class="par-math-display" >
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diff changeset
1229 <img
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diff changeset
1230 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Kinet_de_1902/fulltext/img/Einst_Kinet_de_190250x.png" alt=" @V P (n1) = --a-. @pn " class="par-math-display" /></center>
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1231 <!--l. 978--><p class="nopar">
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diff changeset
1232 </p><!--l. 981--><p class="noindent">Da wir Processe zu betrachten haben, welche mit unendlicher <br/>Annäherung aus stationären Zuständen bestehen, haben wir <br/>anzunehmen, dass <span
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1233 class="cmmi-10">V</span> <sub ><span
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diff changeset
1234 class="cmmi-7">a</span></sub> die Zeit zwar explicite enthalte, dass <br/>aber die partiellen Ableitungen der Grössen <span
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1235 class="cmmi-10"><img
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diff changeset
1236 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Kinet_de_1902/fulltext/img/cmmi10-40.png" alt="@" class="10x-x-40" /> V</span> <sub ><span
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diff changeset
1237 class="cmmi-7">a</span></sub><span
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1238 class="cmmi-10">/<img
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diff changeset
1239 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Kinet_de_1902/fulltext/img/cmmi10-40.png" alt="@" class="10x-x-40" /> p</span><sub ><span
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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diff changeset
1240 class="cmmi-7"><img
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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diff changeset
1241 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Kinet_de_1902/fulltext/img/cmmi7-17.png" alt="n" class="7x-x-17" /></span></sub> nach
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1242 <br/>der Zeit unendlich klein seien.
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1243 </p><!--l. 989--><p class="indent"> Die anderen Kräfte, <span
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1244 class="cmmi-10">P</span><sub> <span
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diff changeset
1245 class="cmmi-7"><img
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1246 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Kinet_de_1902/fulltext/img/cmmi7-17.png" alt="n" class="7x-x-17" /></span></sub><sup><span
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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diff changeset
1247 class="cmr-7">(2)</span></sup> = <span
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diff changeset
1248 class="cmmi-10">II</span><sub ><span
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diff changeset
1249 class="cmmi-7"><img
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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diff changeset
1250 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Kinet_de_1902/fulltext/img/cmmi7-17.png" alt="n" class="7x-x-17" /></span></sub><span
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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1251 class="cmmi-10">, </span>seien nicht von einem <br/>Potential ableitbar, welches nur von den <span
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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diff changeset
1252 class="cmmi-10">p</span><sub ><span
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
1253 class="cmmi-7"><img
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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diff changeset
1254 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Kinet_de_1902/fulltext/img/cmmi7-17.png" alt="n" class="7x-x-17" /></span></sub> abhängt. Die <br/>Kräfte <span
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
1255 class="cmmi-10">II</span><sub ><span
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parents:
diff changeset
1256 class="cmmi-7"><img
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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diff changeset
1257 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Kinet_de_1902/fulltext/img/cmmi7-17.png" alt="n" class="7x-x-17" /></span></sub> stellen die Kräfte dar, welche die Wärmezufuhr <br/>vermitteln.
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1258 </p><!--l. 993--><p class="indent"> Setzt man <span
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1259 class="cmmi-10">V</span> <sub ><span
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diff changeset
1260 class="cmmi-7">a</span></sub> + <span
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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diff changeset
1261 class="cmmi-10">V</span> <sub ><span
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1262 class="cmmi-7">i</span></sub> = <span
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diff changeset
1263 class="cmmi-10">V, </span>so gehen die Gleichungen (1) <br/>über in
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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1264 </p>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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1265 <center class="par-math-display" >
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1266 <img
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1267 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Kinet_de_1902/fulltext/img/Einst_Kinet_de_190251x.png" alt=" { } @-(V--L)- d-- -@L- IIn = @pn + dt @ pn' . " class="par-math-display" /></center>
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1268 <!--l. 1001--><p class="nopar">
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1269 </p><!--l. 1004--><p class="noindent">Die Arbeit, welche durch die Kräfte <span
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1270 class="cmmi-10">II</span><sub ><span
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1271 class="cmmi-7"><img
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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1272 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Kinet_de_1902/fulltext/img/cmmi7-17.png" alt="n" class="7x-x-17" /></span></sub> in der Zeit <span
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1273 class="cmmi-10">dt </span>dem <br/>System zugeführt wird, ist dann die Darstellung der vom <br/><pb/>
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1274 </p><!--l. 1010--><p class="indent">
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1275
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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1276 </p><!--l. 1011--><p class="noindent">System <span
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1277 class="cmmi-10">S </span>während <span
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1278 class="cmmi-10">dt </span>aufgenommenen Wärmemenge <span
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1279 class="cmmi-10">dQ</span>, welche <br/>wir im mechanischen Maass messen wollen.
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1280 </p>
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1281 <center class="par-math-display" >
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1282 <img
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1283 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Kinet_de_1902/fulltext/img/Einst_Kinet_de_190252x.png" alt=" sum sum -@V- sum -@-L dQ = Hn dpn = @pn dpn- @pn dpn{ } sum -dpn -d- @-L- + d t d t @p'n dt. " class="par-math-display" /></center>
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1284 <!--l. 1027--><p class="nopar">
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1285 </p><!--l. 1030--><p class="noindent">Da aber
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1286 </p>
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1287 <center class="par-math-display" >
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1288 <img
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1289 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Kinet_de_1902/fulltext/img/Einst_Kinet_de_190253x.png" alt=" sum { } sum sum
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1290 pn'd- -@L-' d t = d pn'@-L' - -@L'd pn'. dt @ pn @ pn @ pn " class="par-math-display" /></center>
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1291 <!--l. 1040--><p class="nopar">
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1292 </p><!--l. 1044--><p class="noindent">ferner
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1293 </p>
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1294 <center class="par-math-display" >
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1295 <img
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1296 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Kinet_de_1902/fulltext/img/Einst_Kinet_de_190254x.png" alt=" sum @ L ' sum @ L ' sum @L ' @pn'pn = 2L, @pn-dpn + @-pn'dpn = dL, " class="par-math-display" /></center>
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1297 <!--l. 1052--><p class="nopar">
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1298 </p><!--l. 1056--><p class="noindent">so ist
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1299 </p>
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1300 <center class="par-math-display" >
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1301 <img
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1302 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Kinet_de_1902/fulltext/img/Einst_Kinet_de_190255x.png" alt="dQ = sum @L--dpn + dL. @pn " class="par-math-display" /></center>
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1303 <!--l. 1062--><p class="nopar">
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1304 </p><!--l. 1065--><p class="noindent">Da ferner
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1305 </p>
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1306 <center class="par-math-display" >
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1307 <img
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1308 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Kinet_de_1902/fulltext/img/Einst_Kinet_de_190256x.png" alt=" 1 L T = 4-zh-= -nz-, " class="par-math-display" /></center>
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1309 <!--l. 1071--><p class="nopar">
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1310 </p><!--l. 1075--><p class="noindent">so ist
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1311 </p>
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1312 <table width="100%"
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1313 class="equation"><tr><td><a
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1314 id="x1-2r1"></a>
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1315
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1316 <center class="math-display" >
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1317 <img
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1318 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Kinet_de_1902/fulltext/img/Einst_Kinet_de_190257x.png" alt=" dQ dL sum @ V ---- = nz----+ 4z h ----d pn. T L @ pn " class="math-display" /></center></td><td width="5%">(1)</td></tr></table>
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1319 <!--l. 1082--><p class="nopar">
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1320 </p><!--l. 1086--><p class="indent"> Wir beschäftigen uns nun mit dem Ausdruck
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1321 </p>
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1322 <center class="par-math-display" >
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1323 <img
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1324 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Kinet_de_1902/fulltext/img/Einst_Kinet_de_190258x.png" alt=" sum -@V- @ pn dpn. " class="par-math-display" /></center>
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1325 <!--l. 1091--><p class="nopar">
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1326 </p><!--l. 1094--><p class="noindent">Derselbe stellt die Zunahme des Systems an potentieller Energie <br/>dar, welche stattfinden würde während der Zeit <span
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1327 class="cmmi-10">dt</span>, wenn <span
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1328 class="cmmi-10">V </span> <br/>nicht explicite von der Zeit abhängig wäre. Das Zeitelement <span
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1329 class="cmmi-10">dt </span> <br/>sei so gross gewählt, dass an die Stelle jener Summe deren
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1330 <br/>Mittelwert für unendlich viele gleichtemperirte Systeme <span
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1331 class="cmmi-10">S </span>ge-<br/>setzt werden kann, aber doch so klein, dass die expliciten <br/>Aenderungen von <span
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1332 class="cmmi-10">h </span>und <span
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1333 class="cmmi-10">V </span>nach der Zeit unendlich klein seien.
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1334 </p><!--l. 1105--><p class="indent"> Unendlich viele Systeme <span
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1335 class="cmmi-10">S </span>im stationären Zustande, welche <br/>alle identische <span
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1336 class="cmmi-10">h</span>
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1337 und <span
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1338 class="cmmi-10">V</span> <sub ><span
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1339 class="cmmi-7">a</span></sub> besitzen, mögen übergehen in neue <br/>stationäre Zustände, welche durch die allen gemeinsamen Werte <br/><span
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1340 class="cmmi-10">h </span>+ <span
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1341 class="cmmi-10"><img
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1342 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Kinet_de_1902/fulltext/img/cmmi10-e.png" alt="d" class="10x-x-e" /> h, V </span>+ <span
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1343 class="cmmi-10"><img
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1344 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Kinet_de_1902/fulltext/img/cmmi10-e.png" alt="d" class="10x-x-e" /> V </span>charakterisirt sein mögen. ,,<span
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1345 class="cmmi-10"><img
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1346 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Kinet_de_1902/fulltext/img/cmmi10-e.png" alt="d" class="10x-x-e" /></span>&#8220; bezeichne <br/>allgemein die Aenderung einer Grösse beim Uebergang des <br/>Systems in den neuen Zustand; das Zeichen ,,<span
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1347 class="cmmi-10">d</span>&#8220; bezeichne <br/>nicht mehr die Aenderung mit der Zeit, sondern Differentiale <br/>bestimmter Integrale. --
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1348 <pb/>
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1349 </p><!--l. 1119--><p class="indent">
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1350
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1351 </p><!--l. 1120--><p class="indent"> Die Anzahl der Systeme, deren Zustandsvariable vor der <br/>Aenderung innerhalb des unendlich kleinen Gebietes <span
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1352 class="cmmi-10">g </span>sich <br/>befinden, ist durch die Formel gegeben
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1353 </p>
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1354 <center class="par-math-display" >
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1355 <img
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1356 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Kinet_de_1902/fulltext/img/Einst_Kinet_de_190259x.png" alt=" integral
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1357 - 2 h (V+L) dN = Ae dp1 ... dqn, " class="par-math-display" /></center>
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1358 <!--l. 1128--><p class="nopar">
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1359 </p><!--l. 1132--><p class="noindent">dabei steht es in unserer Willkür, für jedes gegebene <span
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1360 class="cmmi-10">h </span>und <span
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1361 class="cmmi-10">V</span> <sub ><span
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1362 class="cmmi-7">a</span></sub> <br/>die willkürliche Constante von <span
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1363 class="cmmi-10">V </span>so zu wählen, dass die Con-<br/>stante <span
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1364 class="cmmi-10">A </span>der Einheit gleich wird. Wir wollen dies thun, um <br/>die Rechnung einfacher zu gestalten, und die so genauer defi-<br/>nirte Function <span
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1365 class="cmmi-10">V</span> <sup ><span
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1366 class="cmsy-7">*</span></sup> nennen.
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1367 </p><!--l. 1139--><p class="indent"> Man sieht nun leicht, dass die von uns gesuchte Grösse <br/>den Wert erhält:
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1368 </p>
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1369 <table width="100%"
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1370 class="equation"><tr><td><a
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1371 id="x1-3r2"></a>
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1372 <center class="math-display" >
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1373 <img
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1374 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Kinet_de_1902/fulltext/img/Einst_Kinet_de_190260x.png" alt=" sum * integral
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1375 @-V-d pn = N1 d{e-2h(V +L)}.V*d p1 ... dqn, @ pn " class="math-display" /></center></td><td width="5%">(2)</td></tr></table>
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1376 <!--l. 1146--><p class="nopar">
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1377 </p><!--l. 1150--><p class="noindent">wobei die Integration über alle Werte der Variabeln zu er-<br/>strecken ist. Dieser Ausdruck stellt nämlich die Vermehrung <br/>der mittleren potentiellen Energie des Systems dar, welche <br/>einträte, wenn zwar die Zustandsverteilung sich gemäss <span
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1378 class="cmmi-10"><img
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1379 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Kinet_de_1902/fulltext/img/cmmi10-e.png" alt="d" class="10x-x-e" /> V</span> <sup ><span
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1380 class="cmsy-7">*</span></sup> <br/>und
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1381 <span
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1382 class="cmmi-10"><img
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1383 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Kinet_de_1902/fulltext/img/cmmi10-e.png" alt="d" class="10x-x-e" /> h </span>änderte, <span
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1384 class="cmmi-10">V </span>aber sich nicht explicite veränderte.
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1385 </p><!--l. 1158--><p class="indent"> Ferner erhalten wir:
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1386 </p>
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1387 <table width="100%"
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1388 class="equation"><tr><td><a
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1389 id="x1-4r3"></a>
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1390 <center class="math-display" >
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1391 <img
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1392 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Kinet_de_1902/fulltext/img/Einst_Kinet_de_190261x.png" alt=" integral
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1393 sum @-V- 1- -2h(V*+L) ' 4z h @pn dpv = 4 zN d{e integral }.h.V.dp1 ... d qn { = 4zd [hV-]- 4z- e-2h(V*+L)d[hV ] N d p1 ... dqn. " class="math-display" /></center></td><td width="5%">(3)</td></tr></table>
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1394 <!--l. 1172--><p class="nopar">
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1395 </p><!--l. 1175--><p class="noindent">Die Integrationen sind hier und im Folgenden über alle mög-<br/>lichen Werte der Variabeln zu erstrecken. Ferner hat man <br/>zu bedenken, dass die Anzahl der betrachteten Systeme sich <br/>nicht ändert. Dies liefert die Gleichung:
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1396 </p>
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1397 <center class="par-math-display" >
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1398 <img
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1399 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Kinet_de_1902/fulltext/img/Einst_Kinet_de_190262x.png" alt=" integral * d(e-2h[(V +L)])d p1 ... dqn = 0, " class="par-math-display" /></center>
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1400 <!--l. 1186--><p class="nopar">
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1401 </p><!--l. 1190--><p class="noindent">oder
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1402 </p>
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1403 <center class="par-math-display" >
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1404 <img
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1405 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Kinet_de_1902/fulltext/img/Einst_Kinet_de_190263x.png" alt=" integral integral
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1406 e-2 h (V *+L)d(h V)d p1 ... d qn + dh e-2h(V*+L)d(L)
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1407 dp1 ... d qn = 0, " class="par-math-display" /></center>
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1408 <!--l. 1200--><p class="nopar">
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1409 </p><!--l. 1204--><p class="noindent">oder
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1410 </p>
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1411 <table width="100%"
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1412 class="equation"><tr><td><a
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1413 id="x1-5r4"></a>
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1414 <center class="math-display" >
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1415 <img
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1416 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Kinet_de_1902/fulltext/img/Einst_Kinet_de_190264x.png" alt=" integral -- -4z e-2h(V*+L)d(hV )dp1 ... d qn + 4zL dh = 0. N
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1417 " class="math-display" /></center></td><td width="5%">(4)</td></tr></table>
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1418 <!--l. 1211--><p class="nopar">
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1419 <pb/>
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1420 </p><!--l. 1218--><p class="indent">
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1421
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1422 </p><!--l. 1219--><p class="indent"> <span class="overline"><span
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1423 class="cmmi-10">V</span> </span> und <span class="overline"><span
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1424 class="cmmi-10">L</span></span> bezeichnen die Mittelwerte der potentiellen Energie <br/>und der lebendigen Kraft der <span
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1425 class="cmmi-10">N </span>- Systeme. Durch Addition <br/>von (3) und (4) erhält man:
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1426 </p>
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1427 <center class="par-math-display" >
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1428 <img
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1429 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Kinet_de_1902/fulltext/img/Einst_Kinet_de_190265x.png" alt=" sum @-V* -- -- 4zh @pn dpn = 4 zd[hV ]+ 4zL .dh, " class="par-math-display" /></center>
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1430 <!--l. 1228--><p class="nopar">
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1431 </p><!--l. 1232--><p class="noindent">oder, weil
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1432 </p>
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1433 <center class="par-math-display" >
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1434 <img
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1435 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Kinet_de_1902/fulltext/img/Einst_Kinet_de_190266x.png" alt=" n n h = 4-L, dh = -4-L2.dL, " class="par-math-display" /></center>
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1436 <!--l. 1239--><p class="nopar">
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1437 </p>
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1438 <center class="par-math-display" >
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1439 <img
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1440 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Kinet_de_1902/fulltext/img/Einst_Kinet_de_190267x.png" alt=" sum
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1441 4 zh @-V-dpn = 4zd[h V]- n z @-L. @pn L " class="par-math-display" /></center>
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1442 <!--l. 1247--><p class="nopar">
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1443 </p><!--l. 1250--><p class="noindent">Setzt man diese Formel in (1) ein, so erhält man
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1444 </p>
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1445 <center class="par-math-display" >
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1446 <img
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1447 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Kinet_de_1902/fulltext/img/Einst_Kinet_de_190268x.png" alt=" [ -- ] [ -*] d-Q-= d 4 zhV * = d V-- . T T " class="par-math-display" /></center>
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1448 <!--l. 1257--><p class="nopar">
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1449 </p><!--l. 1260--><p class="noindent"><span
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1450 class="cmmi-10">dQ/T </span>ist also ein vollständiges Differential. Da
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1451 </p>
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1452 <center class="par-math-display" >
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1453 <img
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1454 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Kinet_de_1902/fulltext/img/Einst_Kinet_de_190269x.png" alt="-- ( ) L-= n z, also d L- = 0 T T " class="par-math-display" /></center>
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1455 <!--l. 1265--><p class="nopar">
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1456 </p><!--l. 1269--><p class="noindent">ist, so lässt sich auch setzen
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1457 </p>
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1458 <center class="par-math-display" >
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1459 <img
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1460 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Kinet_de_1902/fulltext/img/Einst_Kinet_de_190270x.png" alt=" ( ) -dQ- E*-- T = d T . " class="par-math-display" /></center>
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1461 <!--l. 1275--><p class="nopar">
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1462 </p><!--l. 1278--><p class="noindent"><span
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1463 class="cmmi-10">E</span><sup ><span
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1464 class="cmsy-7">*</span></sup><span
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1465 class="cmmi-10">/T </span>ist also bis auf eine willkürliche additive Constante der <br/>Ausdruck für die Entropie des Systems, wobei <span
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1466 class="cmmi-10">E</span><sup ><span
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1467 class="cmsy-7">*</span></sup> = <span
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1468 class="cmmi-10">V</span> <sup ><span
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1469 class="cmsy-7">*</span></sup> + <span
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1470 class="cmmi-10">L </span> <br/>gesetzt ist. Der zweite Hauptsatz erscheint also als not-<br/>wendige Folge des mechanischen Weltbildes.
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1471 </p>
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1472 <div class="center" >
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1473
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1474 <!--l. 1284--><p class="noindent">
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1475 </p><!--l. 1285--><p class="noindent"><span
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1476 class="cmsy-10">§ </span>9. Berechnung der Entropie.</p></div>
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1477 <!--l. 1288--><p class="indent"> Der für die Entropie <span
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1478 class="cmmi-10"><img
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1479 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Kinet_de_1902/fulltext/img/cmmi10-22.png" alt="e" class="10x-x-22" /> </span>gefundene Ausdruck <span
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1480 class="cmmi-10"><img
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1481 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Kinet_de_1902/fulltext/img/cmmi10-22.png" alt="e" class="10x-x-22" /> </span>= <span
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1482 class="cmmi-10">E</span><sup ><span
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1483 class="cmsy-7">*</span></sup><span
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1484 class="cmmi-10">/T </span> <br/>ist nur scheinbar so einfach, da <span
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1485 class="cmmi-10">E</span><sup ><span
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1486 class="cmsy-7">*</span></sup> aus den Bedingungen des <br/>mechanischen Systems erst berechnet werden muss. Es ist <br/>nämlich
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1487 </p>
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1488 <center class="par-math-display" >
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1489 <img
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1490 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Kinet_de_1902/fulltext/img/Einst_Kinet_de_190271x.png" alt=" * E = E + E0, " class="par-math-display" /></center>
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1491 <!--l. 1297--><p class="nopar">
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1492 </p><!--l. 1301--><p class="noindent">wobei <span
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1493 class="cmmi-10">E </span>unmittelbar gegeben, <span
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1494 class="cmmi-10">E</span><sub ><span
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1495 class="cmr-7">0</span></sub> aber durch die Bedingung
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1496 </p>
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1497 <center class="par-math-display" >
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1498 <img
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1499 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Kinet_de_1902/fulltext/img/Einst_Kinet_de_190272x.png" alt=" integral
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1500 - 2h(E+E0) e d p1 ... dqn = N " class="par-math-display" /></center>
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1501 <!--l. 1307--><p class="nopar">
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1502 </p><!--l. 1311--><p class="noindent">als Function von <span
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1503 class="cmmi-10">E </span>und <span
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1504 class="cmmi-10">h </span>zu bestimmen ist. Man erhält so:
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1505 </p>
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1506 <center class="par-math-display" >
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1507 <img
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1508 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Kinet_de_1902/fulltext/img/Einst_Kinet_de_190273x.png" alt=" * { integral } e =-E- = E--+ 2z log e-2hEd p1 ... dqn + const. T T " class="par-math-display" /></center>
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1509 <!--l. 1318--><p class="nopar"> <pb/>
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1510 </p><!--l. 1325--><p class="indent">
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1511
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1512 </p><!--l. 1326--><p class="noindent">In dem so gefundenen Ausdruck ist die der Grösse <span
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1513 class="cmmi-10">E </span>zuzu-<br/>fügende willkürliche Constante ohne Einfluss auf das Resultat, <br/>und das als ,,const&#8220; bezeichnete dritte Glied ist von <span
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1514 class="cmmi-10">V </span>und <span
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1515 class="cmmi-10">T </span> <br/>unabhängig.
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1516 </p><!--l. 1333--><p class="indent"> Der Ausdruck für die Entropie <span
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1517 class="cmmi-10"><img
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1518 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Kinet_de_1902/fulltext/img/cmmi10-22.png" alt="e" class="10x-x-22" /> </span>ist darum merkwürdig, <br/>weil er lediglich von <span
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1519 class="cmmi-10">E</span>
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1520 und <span
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1521 class="cmmi-10">T </span>abhängt, die specielle Form <br/>von <span
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1522 class="cmmi-10">E </span>als Summe potentieller Energie und lebendiger Kraft <br/>aber nicht mehr hervortreten lässt. Diese Thatsache lässt
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1523 <br/>vermuten, dass unsere Resultate allgemeiner sind als die be-<br/>nutzte mechanische Darstellung, zumal der in <span
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1524 class="cmsy-10">§ </span>3 für <span
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1525 class="cmmi-10">h </span>ge-<br/>fundene Ausdruck dieselbe Eigenschaft aufweist.
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1526 </p>
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1527 <div class="center" >
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1528
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1529 <!--l. 1339--><p class="noindent">
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1530 </p><!--l. 1340--><p class="noindent"><span
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1531 class="cmsy-10">§ </span>10. Erweiterung des zweiten Hauptsatzes.</p></div>
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1532 <!--l. 1343--><p class="indent"> Ueber die Natur der Kräfte, welche dem Potential <span
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1533 class="cmmi-10">V</span> <sub ><span
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1534 class="cmmi-7">a</span></sub> <br/>entsprechen, brauchte nichts vorausgesetzt zu werden, auch <br/>nicht, dass solche Kräfte in der Natur vorkommen. Die mecha-<br/>nische Theorie der Wärme verlangt also, dass wir zu rich-<br/>tigen Resultaten gelangen, wenn wir das Carnot&#8217;sche Princip <br/>auf ideale Processe anwenden, welche aus den beobachteten <br/>durch Einführung beliebiger <span
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1535 class="cmmi-10">V</span> <sub ><span
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1536 class="cmmi-7">a</span></sub> erzeugt werden können. Natür-<br/>lich haben die aus der theoretischen Betrachtung jener Processe <br/>gewonnenen Resultate nur dann reale Bedeutung, wenn in <br/>ihnen die idealen Hülfskräfte <span
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1537 class="cmmi-10">V</span> <sub ><span
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1538 class="cmmi-7">a</span></sub> nicht mehr vorkommen.
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1539 </p><!--l. 1356--><p class="indent"> Bern, Juni 1902.
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1540 </p>
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1541 <div class="center" >
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1542
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1543 <!--l. 1358--><p class="noindent">
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1544 </p><!--l. 1359--><p class="noindent">(Eingegangen 26. Juni 1902.)</p></div>
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1545 <!--l. 1362--><p class="indent"> </p>
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1546 <div class="center" >
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1547
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1548 <!--l. 1363--><p class="noindent">
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1549 </p><!--l. 1364--><p class="noindent">----------</p></div>
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1550
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1551 </body></html>
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1552
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1553
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1554