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Version vom 2009-02-14
author Klaus Thoden <kthoden@mpiwg-berlin.mpg.de>
date Thu, 02 May 2013 11:08:12 +0200
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1 <?xml version="1.0" encoding="utf-8" ?>
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4 <!--http://www.w3.org/TR/xhtml1/DTD/xhtml1-transitional.dtd-->
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5 <html>
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7 <head><title></title>
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8 <meta http-equiv="Content-Type" content="text/html; charset=iso-8859-1" />
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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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11 <!-- xhtml,html -->
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12 <meta name="src" content="Einst_Prinz_de_1906.tex" />
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13 <meta name="date" content="2005-03-10 16:56:00" />
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14 <link rel="stylesheet" type="text/css" href="Einst_Prinz_de_1906.css" />
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15 </head><body >
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17 <!--l. 17--><p class="noindent"><pb/></p>
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18 <div class="center" >
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20 <!--l. 18--><p class="noindent">
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21 </p><!--l. 19--><p class="noindent"><span
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22 class="cmr-12x-x-120">13. </span><span
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23 class="cmbxti-10x-x-144">Das Prinzip von der Erhaltung der Schwer-</span>
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24 <br/><span
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25 class="cmbxti-10x-x-144">punktsbewegung und die Tr</span><span
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26 class="cmbxti-10x-x-144">ägheit der Energie; </span> <br/><span
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27 class="cmbxti-10x-x-144">von A.</span>
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28 <span
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29 class="cmbxti-10x-x-144">Einstein.</span></p></div>
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30 <div class="center" >
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32 <!--l. 24--><p class="noindent">
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33 </p><!--l. 25--><p class="noindent">--------</p></div>
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34 <!--l. 28--><p class="indent"> In einer voriges Jahr publizierten Arbeit<sup ><span
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35 class="cmr-8">1</span></sup>) habe ich ge-<br/>zeigt, daß die Maxwellschen elektromagnetischen Gleichungen <br/>in Verbindung mit dem Relativitätsprinzip und Energieprinzip <br/>zu der Folgerung führen, daß die Masse eines Körpers bei <br/>Änderung von dessen Energieinhalt sich ändere, welcher Art <br/>auch jene Energieänderung sein möge. Es zeigte sich, daß <br/>einer Energieänderung von der Größe <img
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36 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Prinz_de_1906/fulltext/img/cmr12-1.png" alt="D" class="12x-x-1" /> <span
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37 class="cmmi-12">E</span> eine gleichsinnige <br/>Änderung der Masse von der Größe <img
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38 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Prinz_de_1906/fulltext/img/cmr12-1.png" alt="D" class="12x-x-1" /> <span
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39 class="cmmi-12">E/V</span> <sup ><span
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40 class="cmr-8">2</span></sup> entsprechen <br/>müsse, wobei <span
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41 class="cmmi-12">V </span>die Lichtgeschwindigkeit bedeutet.
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42 </p><!--l. 42--><p class="indent"> In dieser Arbeit will ich nun zeigen, daß jener Satz die <br/>notwendige und hinreichende Bedingung dafür ist, daß das <br/>Gesetz von der Erhaltung der Bewegung des Schwerpunktes <br/>(wenigstens in erster Annäherung) auch für Systeme gelte, in <br/>welchen außer mechanische auch elektromagnetische Prozesse <br/>vorkommen. Trotzdem die einfachen formalen Betrachtungen, <br/>die zum Nachweis dieser Behauptung durchgeführt werden <br/>müssen, in der Hauptsache bereits in einer Arbeit von <br/>H. Poincaré enthalten sind<sup ><span
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43 class="cmr-8">2</span></sup>), werde ich mich doch der Über-<br/>sichtlichkeit halber nicht auf jene Arbeit stützen.
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44 </p>
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45 <div class="center" >
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47 <!--l. 55--><p class="noindent">
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48 </p><!--l. 56--><p class="noindent"><span
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49 class="cmsy-10x-x-120">§ </span>1. <span
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50 class="cmbx-12">Ein Spezialfall.</span></p></div>
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51 <!--l. 59--><p class="indent"> <span
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52 class="cmmi-12">K </span>sei ein im Raume frei schwebender, ruhender starrer <br/>Hohlzylinder. In <span
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53 class="cmmi-12">A </span>sei eine Einrichtung, um eine bestimmte <br/>Menge <span
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54 class="cmmi-12">S </span>strahlender Energie durch den Hohlraum nach <span
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55 class="cmmi-12">B </span>zu <br/>senden. Während der Aussendung jener Strahlungsmenge
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56 <br/>wirkt ein Strahlungsdruck auf die linke Innenwand des Hohl-<br/>zylinders <span
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57 class="cmmi-12">K</span>, der letzterem eine gewisse nach links gerichtete <br/>Geschwindigkeit verleiht. Besitzt der Hohlzylinder die Masse <span
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58 class="cmti-12">M, </span> <br/>----------
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59 </p><!--l. 69--><p class="indent"> 1) A. Einstein, Ann. d. Phys. <span
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60 class="cmbx-12">18. </span>p. 639. 1905.
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61 </p><!--l. 71--><p class="indent"> 2) H. Poincaré, Lorentz-Festschrift p. 252. 1900. <pb/>
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62 </p><!--l. 75--><p class="indent">
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63
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64 so ist diese Geschwindigkeit, wie aus den Gesetzen des Strahlungs-<br/>druckes leicht zu beweisen, gleich <img
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65 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Prinz_de_1906/fulltext/img/Einst_Prinz_de_19061x.png" alt="V1" class="frac" align="middle" /><img
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66 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Prinz_de_1906/fulltext/img/Einst_Prinz_de_19062x.png" alt="MS" class="frac" align="middle" />, wobei <span
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67 class="cmmi-12">V </span>die Licht-<br/>geschwindigkeit bedeutet. Diese Geschwindigkeit behält <span
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68 class="cmmi-12">K </span>so <br/>lange, bis der Strahlenkomplex, dessen räumliche Ausdehnung <br/>im Verhältnis zu der des Hohl-<br/>raumes
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69 <img
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70 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Prinz_de_1906/fulltext/img/Einst_Prinz_de_19060x.png" alt="PIC" class="graphics" width="199.17122pt" height="113.8111pt" /><!--tex4ht:graphics
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71 name="img/Einst_Prinz_de_19060x.png" src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Prinz_de_1906/fulltext/Einst_Prinz_de_1906_001.eps"
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72 --> von <span
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73 class="cmmi-12">K </span>sehr klein sei, <br/>in <span
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74 class="cmmi-12">B </span>absorbiert ist. Die Dauer <br/>der Bewegung des Hohlzylin-<br/>ders ist (bis auf Glieder <br/>höherer Ordnung) gleich <span
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75 class="cmmi-12"><img
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76 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Prinz_de_1906/fulltext/img/cmmi12-b.png" alt="a" class="12x-x-b" /></span><img
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77 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Prinz_de_1906/fulltext/img/Einst_Prinz_de_19063x.png" alt="/" class="left" align="middle" /> <span
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78 class="cmmi-12">V </span>, <br/>wenn <span
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79 class="cmmi-12"><img
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80 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Prinz_de_1906/fulltext/img/cmmi12-b.png" alt="a" class="12x-x-b" /></span>; die Entfernung zwi-<br/>schen <span
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81 class="cmmi-12">A </span>und <span
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82 class="cmmi-12">B </span>bedeutet. Nach <br/>Absorption des Strahlenkomplexes in <span
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83 class="cmmi-12">B </span>ruht der Körper <span
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84 class="cmmi-12">K </span> <br/>wieder. Bei dem betrachteten Strahlungsvorgang hat sich <span
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85 class="cmmi-12">K </span> <br/>um die Strecke
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86 </p>
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87 <center class="par-math-display" >
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88 <img
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89 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Prinz_de_1906/fulltext/img/Einst_Prinz_de_19064x.png" alt=" 1 S a d = -----. -- V M V " class="par-math-display" /></center>
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90 <!--l. 99--><p class="nopar">
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91 </p><!--l. 103--><p class="noindent">nach links verschoben.
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92 </p><!--l. 106--><p class="indent"> Im Hohlraum von <span
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93 class="cmmi-12">K </span>sei ein der Einfachheit halber masse-<br/>los gedachter Körper <span
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94 class="cmmi-12">k </span>vorhanden nebst einem (ebenfalls masse-<br/>losen) Mechanismus, um den Körper <span
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95 class="cmmi-12">k</span>, der sich zunächst in <span
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96 class="cmmi-12">B </span> <br/>befinden möge, zwischen <span
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97 class="cmmi-12">B </span>und <span
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98 class="cmmi-12">A </span>hin und her zu bewegen. <br/>Nachdem die Strahlungsmenge <span
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99 class="cmmi-12">S </span>in <span
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100 class="cmmi-12">B </span>aufgenommen ist, werde <br/>diese Energiemenge auf <span
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101 class="cmmi-12">k </span>übertragen, und hierauf <span
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102 class="cmmi-12">k </span>nach <span
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103 class="cmmi-12">A </span> <br/>bewegt. Endlich werde die Energiemenge <span
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104 class="cmmi-12">S </span>in <span
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105 class="cmmi-12">A </span>wieder vom <br/>Hohlzylinder <span
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106 class="cmmi-12">K </span>aufgenommen und <span
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107 class="cmmi-12">k </span>wieder nach <span
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108 class="cmmi-12">B </span>zurück-<br/>bewegt. Das ganze System hat nun einen vollständigen Kreis-<br/>prozeß durchgemacht, den man sich beliebig oft wiederholt <br/>denken kann.
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109 </p><!--l. 120--><p class="indent"> Nimmt man an, daß der Transportkörper <span
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110 class="cmmi-12">k </span>auch dann <br/>masselos ist, wenn er die Ennergiemenge <span
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111 class="cmmi-12">S </span>aufgenommen hat, <br/>so muß man auch annehmen, daß der Rücktransport der <br/>Energiemenge <span
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112 class="cmmi-12">S </span>nicht mit einer Lagenänderung des Hohl-<br/>zylinders <span
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113 class="cmmi-12">K </span>verbunden sei. Der Erfolg des ganzen geschilderten <br/>Kreisprozesses besteht also einzig in einer Verschiebung <span
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114 class="cmmi-12"><img
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115 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Prinz_de_1906/fulltext/img/cmmi12-e.png" alt="d" class="12x-x-e" /> </span>des <br/>ganzen Systems nach links, welche Verschiebung durch Wieder-<br/>holung des Kreisprozesses beliebig groß gemacht werden kann. <br/>Wir erhalten also das Resultat, daß ein ursprünglich ruhendes <br/>System, ohne daß äußere Kräfte auf dasselbe wirken, die Lage
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116 <br/><pb/>
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117 </p><!--l. 135--><p class="indent">
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118
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119 </p><!--l. 136--><p class="noindent">seines Schwerpunktes beliebig viel verändern kann, und zwar <br/>ohne daß das System irgend eine dauernde Veränderung <br/>erlitte.
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120 </p><!--l. 140--><p class="indent"> Es ist klar, daß das erlangte Resultat keinen inneren <br/>Widerspruch enthält; wohl aber widerstreitet es den Grund-<br/>gesetzen der Mechanik, nach denen ein ursprünglich ruhender <br/>Körper, auf welchen andere Körper nicht einwirken, keine
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121 <br/>Translationsbewegung ausführen kann.
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122 </p><!--l. 147--><p class="indent"> Setzt man jedoch voraus, daß jeglicher Energie <span
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123 class="cmmi-12">E </span>die <br/>Trägheit <span
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124 class="cmmi-12">E</span><img
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125 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Prinz_de_1906/fulltext/img/Einst_Prinz_de_19065x.png" alt="/" class="left" align="middle" /> <span
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126 class="cmmi-12">V</span> <sup ><span
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127 class="cmr-8">2</span></sup>
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128 zukomme, so verschwindet der Widerspruch <br/>mit den Elementen der Mechanik. Nach dieser Annahme be-<br/>sitzt nämlich der Transportkörper, während er die Energie-<br/>menge <span
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129 class="cmmi-12">S </span>von <span
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130 class="cmmi-12">B </span>nach <span
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131 class="cmmi-12">A </span>transportiert, die Masse <span
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132 class="cmmi-12">S</span><img
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133 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Prinz_de_1906/fulltext/img/Einst_Prinz_de_19066x.png" alt="/" class="left" align="middle" /> <span
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134 class="cmmi-12">V</span> <sup ><span
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135 class="cmr-8">2</span></sup>; und <br/>da der Schwerpunkt <span
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136 class="cmti-12">des ganzen Systems </span>während dieses Vor-<br/>ganges nach dem Schwerpunktssatz ruhen muß, so erfährt <br/>der Hohlzylinder <span
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137 class="cmmi-12">K</span>
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138 während desselben im ganzen eine Ver-<br/>schiebung <span
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139 class="cmmi-12">S</span><span
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140 class="cmsy-10x-x-120">' </span>nach rechts von der Größe
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141 </p>
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142 <center class="par-math-display" >
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143 <img
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144 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Prinz_de_1906/fulltext/img/Einst_Prinz_de_19067x.png" alt="d'= a .S--.1--. V 2 M " class="par-math-display" /></center>
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145 <!--l. 164--><p class="nopar">
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146 </p><!--l. 168--><p class="indent"> Ein Vergleich mit dem oben gefundenen Resultat zeigt, <br/>daß (wenigstens in erster Annäherung) <span
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147 class="cmmi-12"><img
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148 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Prinz_de_1906/fulltext/img/cmmi12-e.png" alt="d" class="12x-x-e" /> </span>= <span
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149 class="cmmi-12"><img
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150 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Prinz_de_1906/fulltext/img/cmmi12-e.png" alt="d" class="12x-x-e" /></span><span
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151 class="cmsy-10x-x-120">' </span>ist, daß also die <br/>Lage des Systems vor und nach dem Kreisprozeß dieselbe ist. <br/>Damit ist der Widerspruch mit den Elementen der Mechanik <br/>beseitigt. </p>
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152 <div class="center" >
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153
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154 <!--l. 174--><p class="noindent">
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155 </p><!--l. 175--><p class="noindent"><span
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156 class="cmsy-10x-x-120">§ </span>2. <span
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157 class="cmbx-12">Über den Satz von der Erhaltung der Bewegung des</span>
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158 <br/><span
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159 class="cmbx-12">Schwerpunktes.</span></p></div>
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160 <!--l. 180--><p class="indent"> Wir betrachten ein System von <span
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161 class="cmmi-12">n </span>diskreten materiellen <br/>Punkten mit den Massen <span
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162 class="cmmi-12">m</span><sub ><span
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163 class="cmr-8">1</span></sub><span
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164 class="cmmi-12">, m</span><sub ><span
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165 class="cmr-8">2</span> </sub> <span
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166 class="cmmi-12">...</span> <span
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167 class="cmmi-12">m</span><sub ><span
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168 class="cmmi-8">n</span></sub> und den Schwerpunkts-<br/>koordinaten <span
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169 class="cmmi-12">x</span><sub ><span
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170 class="cmr-8">1</span></sub> <span
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171 class="cmmi-12">...</span><span
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172 class="cmmi-12">z</span><sub ><span
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173 class="cmmi-8">n</span></sub>. Diese materiellen Punkte seien in ther-<br/>mischer und elektrischer Beziehung nicht als Elementargebilde <br/>(Atome, Moleküle), sondern als Körper im gewöhnlichen Sinne <br/>von geringen Dimensionen aufzufassen, deren Energie durch die
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174 <br/>Schwerpunktsgeschwindigkeit nicht bestimmt sei. Diese Massen <br/>mögen sowohl durch elektromagnetische Vorgänge als auch durch <br/>konservative Kräfte (z. B. Schwerkraft, starre Verbindungen) <br/>aufeinander einwirken; wir wollen jedoch annehmen, daß sowohl <br/>die potenzielle Energie der konservativen Kräfte als auch die <br/><pb/>
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175 </p><!--l. 196--><p class="indent">
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176
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177 </p><!--l. 197--><p class="noindent">kinetische Energie der Schwerpunktsbewegung der Massen stets <br/>als unendlich klein relativ zu der ,,inneren&#8220; Energie der <br/>Massen <span
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178 class="cmmi-12">m</span><sub ><span
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179 class="cmr-8">1</span></sub> <span
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180 class="cmmi-12">...</span><span
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181 class="cmmi-12">m</span><sub ><span
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182 class="cmmi-8">n</span></sub> aufzufassen seien.
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183 </p><!--l. 202--><p class="indent"> Es mögen im ganzen Raume die Maxwell-Lorenzschen <br/>Gleichungen</p>
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184 <table width="100%"
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185 class="equation"><tr><td><a
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186 id="x1-2r1"></a>
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187 <center class="math-display" >
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188 <img
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189 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Prinz_de_1906/fulltext/img/Einst_Prinz_de_19068x.png" alt=" u 1 d X @ N @ M -- r + ------ = ----+ -----, V V dt @ y @ z v- 1- dY-- @-L- @ N- V r + V d t = @ z + @ x , w 1 dZ @ M @ L --r + -- ----= -----+ ----, { V V d t @ x @ y 1-d-L- @-Y- @-Z- V dt = @ z + @ y ,
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190 1-d-M- = @-Z- + @-X- , V d t @ x @ z 1- dN-- @-X- @-Y- V d t = @ y + @ x " class="math-display" /></center></td><td width="5%">(1)</td></tr></table>
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191 <!--l. 223--><p class="nopar">
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192 </p><!--l. 227--><p class="noindent">gelten, wobei
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193 </p>
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194 <center class="par-math-display" >
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195 <img
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196 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Prinz_de_1906/fulltext/img/Einst_Prinz_de_19069x.png" alt="r = @ X-+ @ Y-+ @-Z- @ x @ y @ z " class="par-math-display" /></center>
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197 <!--l. 233--><p class="nopar">
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198 </p><!--l. 237--><p class="noindent">die 4 <span
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199 class="cmmi-12"><img
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200 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Prinz_de_1906/fulltext/img/cmmi12-19.png" alt="p" class="12x-x-19" /></span>-fache Dichte der Elektrizität bedeutet.
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201 </p><!--l. 240--><p class="indent"> Addiert man die der Reihe nach mit
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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_Prinz_de_1906/fulltext/img/Einst_Prinz_de_190610x.png" alt=" V V V ---X x, --- Y x ...---N x 4 p 4 p 4 p " class="par-math-display" /></center>
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206 <!--l. 246--><p class="nopar">
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207 </p><!--l. 250--><p class="noindent">multiplizierten Gleichungen (1) und integriert man dieselben <br/>über den ganzen Raum, so erhält man nach einigen partiellen <br/>Integrationen die Gleichung</p>
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208 <table width="100%"
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209 class="equation"><tr><td><a
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210 id="x1-3r2"></a>
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211 <center class="math-display" >
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212 <img
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213 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Prinz_de_1906/fulltext/img/Einst_Prinz_de_190611x.png" alt=" integral r ---x (uX + v Y + w Z) d t {4p integral } { -d- -1- ( 2 2 2) + d t x .8 p X + Y ...+ N d t integral
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214 - V-- (Y N - Z M )d t = 0. 8p " class="math-display" /></center></td><td width="5%">(2)</td></tr></table>
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215 <!--l. 263--><p class="nopar">
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216 </p><!--l. 266--><p class="noindent">Das erste Glied dieser Gleichung stellt die von dem elektro-<br/>magnetischen Felde den Körpern <span
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217 class="cmmi-12">m</span><sub ><span
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218 class="cmr-8">1</span></sub> <span
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219 class="cmmi-12">...</span><span
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220 class="cmmi-12">m</span><sub ><span
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221 class="cmmi-8">n</span></sub> zugeführte Energie <br/>dar. Nach unserer Hypothese von der Abhängigkeit der <br/>Massen von der Energie hat man daher das erste Glied der
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222 <br/>Summe dem Ausdruck
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223 </p>
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224 <center class="par-math-display" >
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225 <img
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226 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Prinz_de_1906/fulltext/img/Einst_Prinz_de_190612x.png" alt=" sum
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227 V 2 x d-mn- n d t " class="par-math-display" /></center>
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228 <!--l. 277--><p class="nopar"> <pb/>
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229 </p><!--l. 284--><p class="indent">
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230
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231 </p><!--l. 285--><p class="noindent">gleichzusetzen, da wir nach dem Obigen annehmen, daß die <br/>einzelnen materiellen Punkte <span
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232 class="cmmi-12">m</span><sub ><span
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233 class="cmmi-8"><img
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234 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Prinz_de_1906/fulltext/img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sub> ihre Energie und daher auch <br/>ihre Masse <span
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235 class="cmti-12">nur </span>durch Aufnahme von elektromagnetischer Energie <br/>ändern.
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236 </p><!--l. 290--><p class="indent"> Schreiben wir ferner auch dem elektromagnetischen Felde <br/>eine Massendichte
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237 <img
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238 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Prinz_de_1906/fulltext/img/Einst_Prinz_de_190613x.png" alt="(re)" class="left" align="middle" /> zu, die sich von der Energiedichte durch <br/>den Faktor 1<img
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239 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Prinz_de_1906/fulltext/img/Einst_Prinz_de_190614x.png" alt="/" class="left" align="middle" /> <span
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240 class="cmmi-12">V</span> <sup ><span
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241 class="cmr-8">2</span></sup> unterscheidet, so nimmt das zweite Glied <br/>der Gleichung die Form an:
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242 </p>
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243 <center class="par-math-display" >
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244 <img
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245 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Prinz_de_1906/fulltext/img/Einst_Prinz_de_190615x.png" alt=" { integral } 2-d- V d t x red t . " class="par-math-display" /></center>
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246 <!--l. 300--><p class="nopar">
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247 </p><!--l. 304--><p class="indent"> Bezeichnet man mit <span
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248 class="cmmi-12">J </span>das im dritten Gliede der Gleichung (2) <br/>auftretende Integral, so geht letztere über in:
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249 </p>
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250 <table width="100%"
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251 class="equation"><tr><td><a
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252 id="x1-4r3"></a>
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253 <center class="math-display" >
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254 <img
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255 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Prinz_de_1906/fulltext/img/Einst_Prinz_de_190616x.png" alt=" sum ( d m ) d { integral } 1 xn----n + --- xre dt - ------J = 0. d t d t 4 p V " class="math-display" /></center></td><td width="5%">(2a)</td></tr></table>
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256 <!--l. 313--><p class="nopar">
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257 </p><!--l. 318--><p class="indent"> Wir haben nun die Bedeutung des Integrales <span
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258 class="cmmi-12">J </span>aufzu-<br/>suchen. Multipliziert man die zweite, dritte, fünfte und sechste <br/>der Gleichungen (1) der Reihe nach mit den Faktoren <span
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259 class="cmmi-12">N V </span>, <br/><span
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260 class="cmsy-10x-x-120">-</span><span
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261 class="cmmi-12">M V </span>, <span
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262 class="cmsy-10x-x-120">-</span><span
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263 class="cmmi-12">Z V </span>, <span
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264 class="cmmi-12">Y V </span>, addiert und integriert über den Raum, <br/>so erhält man nach einigen partiellen Integrationen
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265 </p>
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266 <table width="100%"
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267 class="equation"><tr><td><a
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268 id="x1-5r3"></a>
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269 <center class="math-display" >
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270 <img
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271 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Prinz_de_1906/fulltext/img/Einst_Prinz_de_190617x.png" alt=" integral ( ) dJ- = - 4p V r-- X + -v N - w-M d t = - 4V R , d t 4p V V x " class="math-display" /></center></td><td width="5%">(3)</td></tr></table>
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272 <!--l. 330--><p class="nopar">
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273 </p><!--l. 334--><p class="noindent">wobei <span
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274 class="cmmi-12">R</span><sub ><span
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275 class="cmmi-8">x</span></sub> die algebraische Summe der <span
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276 class="cmmi-12">X</span>-Komponenten aller <br/>vom elektromagnetischen Felde auf die Massen <span
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277 class="cmmi-12">m</span><sub ><span
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278 class="cmr-8">1</span></sub> <span
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279 class="cmmi-12">...</span><span
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280 class="cmmi-12">m</span><sub ><span
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281 class="cmmi-8">n</span></sub> aus-<br/>geübten Kräfte bedeutet. Da die entsprechende Summe aller <br/>von den konservativen Wechselwirkungen herrührenden Kräfte <br/>verschwindet, so ist <span
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282 class="cmmi-12">R</span><sub ><span
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283 class="cmmi-8">x</span></sub> gleichzeitig die Summe der <span
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284 class="cmmi-12">X</span>-Kom-<br/>ponenten <span
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285 class="cmti-12">aller </span>auf die Msssen <span
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286 class="cmmi-12">m</span><sub ><span
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287 class="cmmi-8"><img
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288 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Prinz_de_1906/fulltext/img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sub> ausgeübten Kräfte.
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289 </p><!--l. 343--><p class="indent"> Wir wollen uns nun zunächst mit Gleichung (3) befassen, <br/>welche von der Hypothese, daß die Masse von der Energie <br/>abhängig sei, unabhängig ist. Sehen wir zunächst von der <br/>Abhängigkeit der Massen von der Energie ab und bezeichnen <br/>wir mit æ<sub ><span
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290 class="cmmi-8"><img
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291 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Prinz_de_1906/fulltext/img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sub> die Resultierende aller <span
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292 class="cmmi-12">X</span>-Komponenten der auf <span
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293 class="cmmi-12">m</span><sub ><span
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294 class="cmmi-8"><img
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295 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Prinz_de_1906/fulltext/img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sub>
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296 <br/>wirkenden Kräfte, so haben wir für die Masse <span
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297 class="cmmi-12">m</span><sub ><span
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298 class="cmmi-8"><img
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299 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Prinz_de_1906/fulltext/img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sub> die Be-<br/>wegungsgleichung aufzustellen:
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300 </p>
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301 <table width="100%"
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302 class="equation"><tr><td><a
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303 id="x1-6r4"></a>
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304 <center class="math-display" >
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305 <img
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306 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Prinz_de_1906/fulltext/img/Einst_Prinz_de_190618x.png" alt=" 2 { } m d--xn = -d- m d-xn = æ , n d t2 d t n dt n " class="math-display" /></center></td><td width="5%">(4)</td></tr></table>
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307 <!--l. 355--><p class="nopar">
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308 <pb/>
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309 </p><!--l. 362--><p class="indent">
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310
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311 </p><!--l. 363--><p class="noindent">folglich erhalten wir auch:</p>
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312 <table width="100%"
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313 class="equation"><tr><td><a
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314 id="x1-7r5"></a>
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315 <center class="math-display" >
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316 <img
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317 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Prinz_de_1906/fulltext/img/Einst_Prinz_de_190619x.png" alt=" ( ) d- sum d-xn sum
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318 dt mn dt = æ n = Rx . " class="math-display" /></center></td><td width="5%">(5)</td></tr></table>
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319 <!--l. 370--><p class="nopar">
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320 </p><!--l. 373--><p class="noindent">Aus Gleichung (5) und Gleichung (3) erhält man</p>
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321 <table width="100%"
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322 class="equation"><tr><td><a
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323 id="x1-8r6"></a>
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324 <center class="math-display" >
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325 <img
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326 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Prinz_de_1906/fulltext/img/Einst_Prinz_de_190620x.png" alt="--J--- sum d-xn 4 pV + mn dt = knost. " class="math-display" /></center></td><td width="5%">(6)</td></tr></table>
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327 <!--l. 378--><p class="nopar">
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328 </p><!--l. 382--><p class="indent"> Führen wir nun die Hypothese wieder ein, daß die <br/>Größen <span
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329 class="cmmi-12">m</span><sub ><span
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330 class="cmmi-8"><img
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331 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Prinz_de_1906/fulltext/img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sub> von der Energie also auch von der Zeit abhängen, <br/>so stellt sich uns die Schwierigkeit entgegen, daß für diesen <br/>Fall die mechanischen Gleichungen nicht mehr bekannt sind; <br/>das erste Gleichheitszeichen der Gleichung (4) gilt nun nicht <br/>mehr. Es ist jedoch zu beachten, daß die Differenz</p>
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332 <center class="par-math-display" >
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333 <img
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334 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Prinz_de_1906/fulltext/img/Einst_Prinz_de_190621x.png" alt=" { } 2 -d- mn dxn- - mn d-xn-= dmn--d-xn dt dt d t2 dt d t 1 integral r d xn = -2- ------- (u X + vY + w Z) dt V 4p dt " class="par-math-display" /></center>
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335 <!--l. 402--><p class="nopar">
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336 </p><!--l. 406--><p class="noindent">in den Geschwindigkeiten vom zweiten Grade ist. Sind daher <br/>alle Geschwindigkeiten so klein, daß Glieder zweiten Grades <br/>vernachlässigt werden dürfen, so gilt auch bei Veränderlichkeit <br/>der Masse <span
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337 class="cmmi-12">m</span><sub ><span
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338 class="cmmi-8"><img
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339 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Prinz_de_1906/fulltext/img/cmmi8-17.png" alt="n" class="8x-x-17" /></span></sub> die Gleichung
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340 </p>
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341 <center class="par-math-display" >
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342 <img
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343 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Prinz_de_1906/fulltext/img/Einst_Prinz_de_190622x.png" alt=" ( ) d-- d-xn dt mn d t = æ n " class="par-math-display" /></center>
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344 <!--l. 416--><p class="nopar">
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345 </p><!--l. 420--><p class="noindent">sicher mit der in Betracht kommenden Genauigkeit. Es gelten <br/>dann auch die Gleichungen (5) und (6), und man erhält aus <br/>den Gleichungen (6) und (2a):</p>
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346 <table width="100%"
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347 class="equation"><tr><td><a
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348 id="x1-9r7"></a>
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349 <center class="math-display" >
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350 <img
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351 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Prinz_de_1906/fulltext/img/Einst_Prinz_de_190623x.png" alt=" [ integral ] d sum
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352 --- (mn xn) + x red t = knost. d t " class="math-display" /></center></td><td width="5%">(2b)</td></tr></table>
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353 <!--l. 428--><p class="nopar">
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354 </p><!--l. 433--><p class="indent"> Bezeichnet <span
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355 class="cmmi-12"><img
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356 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Prinz_de_1906/fulltext/img/cmmi12-18.png" alt="q" class="cmmi-12x-x-18" align="middle" /> </span>die <span
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357 class="cmmi-12">X</span>-Koordinate des Schwerpunktes der <br/>ponderabelen Massen und der Energiemasse des elektromagne-<br/>tischen Feldes, so ist
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358 </p>
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359 <center class="par-math-display" >
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360 <img
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361 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Prinz_de_1906/fulltext/img/Einst_Prinz_de_190624x.png" alt=" sum
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362 q = ---( sum mn-xn)-+- integral x-red-t-, mn + integral re dt " class="par-math-display" /></center>
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363 <!--l. 442--><p class="nopar">
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364 </p><!--l. 446--><p class="noindent">wobei nach dem Energieprinzip der Wert des Nenners der <br/><pb/>
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365 </p><!--l. 451--><p class="indent">
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366
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367 </p><!--l. 452--><p class="noindent">rechten Seite von der Zeit unabhängig ist.<sup ><span
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368 class="cmr-8">1</span></sup>) Wir können daher <br/>Gleichung (2b) auch in der Form schreiben:</p>
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369 <table width="100%"
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370 class="equation"><tr><td><a
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371 id="x1-10r7"></a>
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372 <center class="math-display" >
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373 <img
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374 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Prinz_de_1906/fulltext/img/Einst_Prinz_de_190625x.png" alt="d-q d t = konst. " class="math-display" /></center></td><td width="5%">(2c)</td></tr></table>
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375 <!--l. 458--><p class="nopar">
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376 </p><!--l. 462--><p class="indent"> Schreibt man also jeglicher Energie <span
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377 class="cmmi-12">E </span>die träge Masse <span
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378 class="cmmi-12">E</span><img
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379 src="http://foxridge.mpiwg-berlin.mpg.de/permanent/einstein/annalen/Einst_Prinz_de_1906/fulltext/img/Einst_Prinz_de_190626x.png" alt="/" class="left" align="middle" /> <span
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380 class="cmmi-12">V</span> <sup ><span
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381 class="cmr-8">2</span></sup> <br/>zu, so gilt -- wenigstens in erster Annäherung -- das Prinzip <br/>von der Erhaltung der Bewegung des Schwerpunktes auch für <br/>Systeme, in denen elektromagnetische Prozesse vorkommen.
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382 </p><!--l. 470--><p class="indent"> Aus der vorstehenden Untersuchung folgt, daß man ent-<br/>weder auf den Grundsatz der Mechanik, nach welchem ein <br/>ursprünglich ruhender, äußeren Kräften nicht unterworfener <br/>Körper keine Translationshewegung ausführen kann, verzichten <br/>oder annehmen muß, daß die Trägheit eines Körpers nach <br/>dem angegebenen Gesetze von dessen Energieinhalt abhänge. <br/>
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383 </p><!--l. 479--><p class="indent"> Bern, Mai 1906.
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384 </p><!--l. 481--><p class="noindent">----------
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385 </p><!--l. 484--><p class="indent"> 1) Nach der in dieser Arbeit entwickelten Auffassung ist der Satz <br/>von der Konstanz der Masse ein Spezialfall des Energieprinzipes.
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386 </p>
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387 <div class="center" >
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388
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389 <!--l. 488--><p class="noindent">
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390 </p><!--l. 489--><p class="noindent">(Eingegangen 17. Mai 1906.)</p></div>
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391 <div class="center" >
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392
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393 <!--l. 492--><p class="noindent">
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394 </p><!--l. 493--><p class="noindent">----------</p></div>
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395
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396 </body></html>
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397
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398
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399