11269 lines
1.6 MiB
Executable File
11269 lines
1.6 MiB
Executable File
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url: https://de-wikisource-org.translate.goog/wiki/Translatorische_Bewegung_des_Licht%C3%A4thers?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-US
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saved date: Thu May 30 2024 00:24:43 GMT-0400 (Eastern Daylight Time)
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--><meta charset=utf-8>
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<title>Translational movement of the light ether - Wikisource</title>
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||
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<h1 id=firstHeading class="firstHeading mw-first-heading"><span class=mw-page-title-main><font style=vertical-align:inherit><font style=vertical-align:inherit>Translational movement of the light ether</font></font></span></h1>
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<td width=40%><font style=vertical-align:inherit><font style=vertical-align:inherit>Author:</font></font></td>
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<td width=60%><b><span id=ws-author><a href="https://de-wikisource-org.translate.goog/wiki/Wilhelm_Wien?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-US" title="Wilhelm Vienna"><font style=vertical-align:inherit><font style=vertical-align:inherit>Wilhelm Vienna</font></font></a></span> </b></td>
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<td><font style=vertical-align:inherit><font style=vertical-align:inherit>Title:</font></font></td>
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<td><b><span id=ws-title><font style=vertical-align:inherit><font style=vertical-align:inherit>About the questions that concern the translational movement of the light ether</font></font></span></b></td>
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<td><font style=vertical-align:inherit><font style=vertical-align:inherit>out of:</font></font></td>
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<td><font style=vertical-align:inherit><font style=vertical-align:inherit>Annals of Physics 301 (supplement), 1898, pp. I-XVIII</font></font></td>
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<td><font style=vertical-align:inherit><font style=vertical-align:inherit>Editor:</font></font></td>
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<td><font style=vertical-align:inherit><font style=vertical-align:inherit>G. and E. Wiedemann</font></font></td>
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<td><font style=vertical-align:inherit><font style=vertical-align:inherit>Date of creation:</font></font></td>
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<td><font style=vertical-align:inherit><font style=vertical-align:inherit>1898</font></font></td>
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<td><font style=vertical-align:inherit><font style=vertical-align:inherit>Publisher:</font></font></td>
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<td><span id=ws-publisher><font style=vertical-align:inherit><font style=vertical-align:inherit>John Ambr. Bart</font></font></span></td>
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<td><font style=vertical-align:inherit><font style=vertical-align:inherit>Place of publication:</font></font></td>
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<td><span id=ws-place><font style=vertical-align:inherit><font style=vertical-align:inherit>Leipzig</font></font></span></td>
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<td><font style=vertical-align:inherit><font style=vertical-align:inherit>Source:</font></font></td>
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<td><span id=ws-scan><a rel=nofollow class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-US&u=http://gallica.bnf.fr/ark:/12148/bpt6k153068.image.f976"><font style=vertical-align:inherit><font style=vertical-align:inherit>Gallica</font></font></a><font style=vertical-align:inherit><font style=vertical-align:inherit> , </font></font><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-US&u=https://commons.wikimedia.org/wiki/File:Translatorische_Bewegung_des_Licht%25C3%25A4thers.djvu" class=extiw title="commons:File:Translatory movement of the light ether.djvu"><font style=vertical-align:inherit><font style=vertical-align:inherit>Commons</font></font></a></span></td>
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<span typeof=mw:File><a href="https://de-wikisource-org.translate.goog/wiki/Datei:Wikipedia-logo-v2.svg?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-US" class=mw-file-description><img src="data:image/png;base64,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" decoding=async width=20 height=18 class=mw-file-element srcset data-file-width=103 data-file-height=94 sizes></a></span> <a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-US&u=https://de.wikipedia.org/wiki/%25C3%2584ther_(Physik)" class=extiw title="w:ether (physics)"><font style=vertical-align:inherit><font style=vertical-align:inherit>Article in Wikipedia</font></font></a>
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<td colspan=2 align=center><span style=color:#669966><b><font style=vertical-align:inherit><font style=vertical-align:inherit>complete</font></font></b></span></td>
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<b><font style=vertical-align:inherit><font style=vertical-align:inherit>Complete!</font></font></b><font style=vertical-align:inherit><font style=vertical-align:inherit> This text has been </font></font><a href="https://de-wikisource-org.translate.goog/wiki/Hilfe:Korrekturlesen?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-US" title=Help:Proofreading><font style=vertical-align:inherit><font style=vertical-align:inherit>read twice on the basis of source correction</font></font></a><font style=vertical-align:inherit><font style=vertical-align:inherit> . The notation follows the original text.
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<th colspan=2 bgcolor=#f9f9f9 align=center><span id=textdaten_index><a href="https://de-wikisource-org.translate.goog/wiki/Index:Translatorische_Bewegung_des_Licht%C3%A4thers.djvu?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-US" title="Index: Translational movement of the light ether.djvu"><font style=vertical-align:inherit><font style=vertical-align:inherit>Index page</font></font></a></span></th>
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<div style="width:em;text-align:justify;margin:0px auto;background-color:#ffffff;font-family:sans-serif;line-height:150%;border:0px;padding-left:2em;text-indent:em">
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<div class=prp-pages-output lang=de><span><span class=pagenum id=i title="Page: Translational movement of the light ether.djvu/1"> <span class="pagenumber noprint" style=color:#666666;display:inline;margin:0px;padding:0px>[<b><a href=https://de.wikisource.org/wiki/Page%3A_Translational_movement_of_the_light_ether.djvu/1 title="Page: Translational movement of the light ether.djvu/1">i</a></b>]</span> </span></span>
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<div style=text-align:center;font-size:130%;line-height:20pt>
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<b><font style=vertical-align:inherit><font style=vertical-align:inherit>On the questions concerning the translational movement of the light ether; by W. Vienna.</font></font></b>
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<div style=text-align:center;font-size:100%;line-height:20pt><font style=vertical-align:inherit><font style=vertical-align:inherit>
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(Representation for the 70th meeting of German natural scientists and doctors in Düsseldorf, 1898; Section Physics.)
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</font></font></div>
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<hr style=width:5em;text-align:center;text-align:-moz-center;margin-right:auto;margin-left:auto;margin-top:0px;margin-bottom:0px>
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<p><font style=vertical-align:inherit><font style=vertical-align:inherit>The question of whether the light ether takes part in the movements of bodies or not, and whether mobility can be attributed to it at all, has occupied physicists for a long time and there are countless assumptions and conjectures that one has to make about the properties of the carrier of electromagnetic phenomena held. However, there can be no doubt that everything we know about the ether is contained in </font></font><span style=letter-spacing:0.2em;left:0.1em><font style=vertical-align:inherit><font style=vertical-align:inherit>Maxwell</font></font></span><font style=vertical-align:inherit><font style=vertical-align:inherit> 's theory of electromagnetism and everything else belongs to the realm of pure speculation. Accordingly, I have not set myself the task of providing a literary report on the innumerable theories that have the light ether as their subject, but have endeavored to highlight the questions that we have to answer on the basis of </font></font><span style=letter-spacing:0.2em;left:0.1em><font style=vertical-align:inherit><font style=vertical-align:inherit>Maxwell</font></font></span><font style=vertical-align:inherit><font style=vertical-align:inherit> 's theory regarding the mobility of the ether have to provide.</font></font></p>
|
||
<p><font style=vertical-align:inherit><font style=vertical-align:inherit>If we make the assumption that the ether has mobility, further questions immediately arise, namely whether this movement requires energy expenditure, i.e. whether the ether is to be attributed inert mass, and then whether the ether is also set in motion by the movement of solid bodies . The latter does not appear to be the case according to many experiments, especially after the extensive experiments of </font></font><span style=letter-spacing:0.2em;left:0.1em><font style=vertical-align:inherit><font style=vertical-align:inherit>Lodge</font></font></span><font style=vertical-align:inherit><font style=vertical-align:inherit> , which were carried out with rapidly rotating metal masses or in the vicinity of high-speed circular saws.</font></font></p>
|
||
<p><font style=vertical-align:inherit><font style=vertical-align:inherit>We will first compare the assumptions as to whether mobility can be attributed to the ether or not and then move on to the discussion of the empirical facts.</font></font></p><span style=display:none></span> <span><span class=pagenum id=ii title="Page: Translational movement of the light ether.djvu/2"> <span class="pagenumber noprint" style=color:#666666;display:inline;margin:0px;padding:0px>[<b><a href=https://de.wikisource.org/wiki/Page%3A_Translational_movement_of_the_light_ether.djvu/2 title="Page: Translational movement of the light ether.djvu/2">ii</a></b>]</span> </span></span>
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<div style=text-align:center>
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<h3><span class=mw-headline id=Die_Annahme_der_Beweglichkeit_des_Aethers.><font style=vertical-align:inherit><font style=vertical-align:inherit>The assumption of the mobility of the aether.</font></font></span></h3>
|
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</div>
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||
<p><font style=vertical-align:inherit><font style=vertical-align:inherit>The tendency to bring the properties of the ether into agreement with those of ponderable matter has led to the assumption that the ether can carry out movements in the manner of a liquid, although not a single experiment indicates the existence of such movements. But if one ascribes mobility to the ether, then, as </font></font><span style=letter-spacing:0.2em;left:0.1em><font style=vertical-align:inherit><font style=vertical-align:inherit>Hertz</font></font></span><font style=vertical-align:inherit><font style=vertical-align:inherit> first noted, it follows strictly from </font></font><span style=letter-spacing:0.2em;left:0.1em><font style=vertical-align:inherit><font style=vertical-align:inherit>Maxwell</font></font></span><font style=vertical-align:inherit><font style=vertical-align:inherit> 's theory that under the influence of the pressure forces generated by a variable electromagnetic system, it must carry out movements that can be calculated, if one makes certain assumptions about the inertia of the ether.</font></font></p>
|
||
<p><span style=letter-spacing:0.2em;left:0.1em><font style=vertical-align:inherit><font style=vertical-align:inherit>Helmholtz</font></font></span><font style=vertical-align:inherit><font style=vertical-align:inherit> gave the basic principles for the calculation of these flows under the assumption that the inertia and compressibility of the aether is zero. However, he did not give any specific examples that would allow this theory to be tested against experience, and I will therefore give here two examples from which some conclusions can be drawn as to the meaning of these assumptions.</font></font></p>
|
||
<p><font style=vertical-align:inherit><font style=vertical-align:inherit>Currents in the ether are only excited by electromagnetic tensions when the field is neither static nor stationary, i.e. when the conditions of time are still changeable.</font></font></p>
|
||
<p><font style=vertical-align:inherit><font style=vertical-align:inherit>As a first example I introduce an electrified colon, which carries equal quantities of positive and negative electricity at a very small distance from each other, which increase proportionally with time.</font></font></p>
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<p><font style=vertical-align:inherit><font style=vertical-align:inherit>If we designate </font><font style=vertical-align:inherit>the coordinates </font><font style=vertical-align:inherit>with x, y </font><i><font style=vertical-align:inherit>, </font></i></font><i><font style=vertical-align:inherit><font style=vertical-align:inherit>z, </font></font></i><font style=vertical-align:inherit><i><font style=vertical-align:inherit>the time with </font></i></font><i><font style=vertical-align:inherit><font style=vertical-align:inherit>t</font></font></i><font style=vertical-align:inherit><font style=vertical-align:inherit> , </font><i><font style=vertical-align:inherit>the</font></i><font style=vertical-align:inherit> components of the electrical </font><font style=vertical-align:inherit>forces with </font><font style=vertical-align:inherit>Maxwell </font><span style=letter-spacing:0.2em;left:0.1em><font style=vertical-align:inherit>'s</font></span><font style=vertical-align:inherit> differential equations</font></font><i><font style=vertical-align:inherit></font></i><font style=vertical-align:inherit></font><i><font style=vertical-align:inherit></font></i><font style=vertical-align:inherit></font><i><font style=vertical-align:inherit></font></i><font style=vertical-align:inherit></font><span style=letter-spacing:0.2em;left:0.1em><font style=vertical-align:inherit></font></span><font style=vertical-align:inherit></font></p>
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<annotation encoding=application/x-tex>
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{\displaystyle {\begin{array}{ll}A{\frac {dL}{dt}}={\frac {\partial Z}{\partial y}}-{\frac {\partial Y}{\partial z}}&A{\frac {dX}{dt}}={\frac {\partial M}{\partial z}}-{\frac {\partial N}{\partial y}}\\\\A{\frac {dM}{dt}}={\frac {\partial X}{\partial z}}-{\frac {\partial Z}{\partial x}}&A{\frac {dY}{dt}}={\frac {\partial N}{\partial x}}-{\frac {\partial L}{\partial z}}\\\\A{\frac {dN}{dt}}={\frac {\partial Y}{\partial x}}-{\frac {\partial X}{\partial y}}&A{\frac {dZ}{dt}}={\frac {\partial L}{\partial y}}-{\frac {\partial M}{\partial x}}\\\\{\frac {\partial L}{\partial x}}+{\frac {\partial M}{\partial y}}+{\frac {\partial N}{\partial z}}=0\ &{\frac {\partial X}{\partial x}}+{\frac {\partial Y}{\partial y}}+{\frac {\partial Z}{\partial z}}=0.\end{array}}}
|
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</annotation>
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</semantics>
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</math></span><img src="data:image/svg+xml;base64,<svg xmlns:xlink="http://www.w3.org/1999/xlink" width="42.941ex" height="27.176ex" style="vertical-align: -12.803ex; margin-bottom: -0.201ex;" viewBox="0 -6101.5 18488.6 11700.7" role="img" focusable="false" xmlns="http://www.w3.org/2000/svg" aria-labelledby="MathJax-SVG-1-Title">
<title id="MathJax-SVG-1-Title">{\displaystyle {\begin{array}{ll}A{\frac {dL}{dt}}={\frac {\partial Z}{\partial y}}-{\frac {\partial Y}{\partial z}}&amp;A{\frac {dX}{dt}}={\frac {\partial M}{\partial z}}-{\frac {\partial N}{\partial y}}\\\\A{\frac {dM}{dt}}={\frac {\partial X}{\partial z}}-{\frac {\partial Z}{\partial x}}&amp;A{\frac {dY}{dt}}={\frac {\partial N}{\partial x}}-{\frac {\partial L}{\partial z}}\\\\A{\frac {dN}{dt}}={\frac {\partial Y}{\partial x}}-{\frac {\partial X}{\partial y}}&amp;A{\frac {dZ}{dt}}={\frac {\partial L}{\partial y}}-{\frac {\partial M}{\partial x}}\\\\{\frac {\partial L}{\partial x}}+{\frac {\partial M}{\partial y}}+{\frac {\partial N}{\partial z}}=0\ &amp;{\frac {\partial X}{\partial x}}+{\frac {\partial Y}{\partial y}}+{\frac {\partial Z}{\partial z}}=0.\end{array}}}</title>
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</svg>" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden=true style=vertical-align:-12.803ex;margin-bottom:-0.201ex;width:42.941ex;height:27.176ex alt="{\displaystyle {\begin{array}{ll}A{\frac {dL}{dt}}={\frac {\partial Z}{\partial y}}-{\frac {\partial Y}{\partial z}}&A{\frac {dX}{dt}}={\frac {\partial M}{\partial z}}-{\frac {\partial N}{\partial y}}\\\\A{ \frac {dM}{dt}}={\frac {\partial X}{\partial z}}-{\frac {\partial Z}{\partial x}}&A{\frac {dY}{dt}} ={\frac {\partial N}{\partial x}}-{\frac {\partial L}{\partial z}}\\\\A{\frac {dN}{dt}}={\frac { \partial Y}{\partial x}}-{\frac {\partial X}{\partial y}}&A{\frac {dZ}{dt}}={\frac {\partial L}{\partial y} }-{\frac {\partial M}{\partial x}}\\\\{\frac {\partial L}{\partial x}}+{\frac {\partial M}{\partial y}}+ {\frac {\partial N}{\partial z}}=0\ &{\frac {\partial X}{\partial x}}+{\frac {\partial Y}{\partial y}}+{\ frac {\partial Z}{\partial z}}=0.\end{array}}}"></span></td>
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</table><span><span class=pagenum id=iii title="Page: Translational movement of the light ether.djvu/3"> <span class="pagenumber noprint" style=color:#666666;display:inline;margin:0px;padding:0px>[<b><a href=https://de.wikisource.org/wiki/Page%3A_Translational_movement_of_the_light_ether.djvu/3 title="Page: Translational movement of the light ether.djvu/3">iii</a></b>]</span> </span></span><font style=vertical-align:inherit><font style=vertical-align:inherit>We satisfy these equations using the following expressions:
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<mi><font style=vertical-align:inherit><font style=vertical-align:inherit>
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e.g
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</font></font></mi>
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<mrow class=MJX-TeXAtom-ORD>
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<mn><font style=vertical-align:inherit><font style=vertical-align:inherit>
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2
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|
||
</msup>
|
||
</mstyle>
|
||
</mrow>
|
||
<annotation encoding=application/x-tex>
|
||
{\displaystyle r^{2}=x^{2}+y^{2}+z^{2}}
|
||
</annotation>
|
||
</semantics>
|
||
</math></span><img src=data:image/svg+xml;base64,PHN2ZyB4bWxuczp4bGluaz0iaHR0cDovL3d3dy53My5vcmcvMTk5OS94bGluayIgd2lkdGg9IjE3LjYyNWV4IiBoZWlnaHQ9IjMuMDA5ZXgiIHN0eWxlPSJ2ZXJ0aWNhbC1hbGlnbjogLTAuNjcxZXg7IiB2aWV3Qm94PSIwIC0xMDA2LjYgNzU4OC43IDEyOTUuNyIgcm9sZT0iaW1nIiBmb2N1c2FibGU9ImZhbHNlIiB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIGFyaWEtbGFiZWxsZWRieT0iTWF0aEpheC1TVkctMS1UaXRsZSI+Cjx0aXRsZSBpZD0iTWF0aEpheC1TVkctMS1UaXRsZSI+e1xkaXNwbGF5c3R5bGUgcl57Mn09eF57Mn0reV57Mn0rel57Mn19PC90aXRsZT4KPGRlZnMgYXJpYS1oaWRkZW49InRydWUiPgo8cGF0aCBzdHJva2Utd2lkdGg9IjEiIGlkPSJFMS1NSk1BVEhJLTcyIiBkPSJNMjEgMjg3UTIyIDI5MCAyMyAyOTVUMjggMzE3VDM4IDM0OFQ1MyAzODFUNzMgNDExVDk5IDQzM1QxMzIgNDQyUTE2MSA0NDIgMTgzIDQzMFQyMTQgNDA4VDIyNSAzODhRMjI3IDM4MiAyMjggMzgyVDIzNiAzODlRMjg0IDQ0MSAzNDcgNDQxSDM1MFEzOTggNDQxIDQyMiA0MDBRNDMwIDM4MSA0MzAgMzYzUTQzMCAzMzMgNDE3IDMxNVQzOTEgMjkyVDM2NiAyODhRMzQ2IDI4OCAzMzQgMjk5VDMyMiAzMjhRMzIyIDM3NiAzNzggMzkyUTM1NiA0MDUgMzQyIDQwNVEyODYgNDA1IDIzOSAzMzFRMjI5IDMxNSAyMjQgMjk4VDE5MCAxNjVRMTU2IDI1IDE1MSAxNlExMzggLTExIDEwOCAtMTFROTUgLTExIDg3IC01VDc2IDdUNzQgMTdRNzQgMzAgMTE0IDE4OVQxNTQgMzY2UTE1NCA0MDUgMTI4IDQwNVExMDcgNDA1IDkyIDM3N1Q2OCAzMTZUNTcgMjgwUTU1IDI3OCA0MSAyNzhIMjdRMjEgMjg0IDIxIDI4N1oiPjwvcGF0aD4KPHBhdGggc3Ryb2tlLXdpZHRoPSIxIiBpZD0iRTEtTUpNQUlOLTMyIiBkPSJNMTA5IDQyOVE4MiA0MjkgNjYgNDQ3VDUwIDQ5MVE1MCA1NjIgMTAzIDYxNFQyMzUgNjY2UTMyNiA2NjYgMzg3IDYxMFQ0NDkgNDY1UTQ0OSA0MjIgNDI5IDM4M1QzODEgMzE1VDMwMSAyNDFRMjY1IDIxMCAyMDEgMTQ5TDE0MiA5M0wyMTggOTJRMzc1IDkyIDM4NSA5N1EzOTIgOTkgNDA5IDE4NlYxODlINDQ5VjE4NlE0NDggMTgzIDQzNiA5NVQ0MjEgM1YwSDUwVjE5VjMxUTUwIDM4IDU2IDQ2VDg2IDgxUTExNSAxMTMgMTM2IDEzN1ExNDUgMTQ3IDE3MCAxNzRUMjA0IDIxMVQyMzMgMjQ0VDI2MSAyNzhUMjg0IDMwOFQzMDUgMzQwVDMyMCAzNjlUMzMzIDQwMVQzNDAgNDMxVDM0MyA0NjRRMzQzIDUyNyAzMDkgNTczVDIxMiA2MTlRMTc5IDYxOSAxNTQgNjAyVDExOSA1NjlUMTA5IDU1MFExMDkgNTQ5IDExNCA1NDlRMTMyIDU0OSAxNTEgNTM1VDE3MCA0ODlRMTcwIDQ2NCAxNTQgNDQ3VDEwOSA0MjlaIj48L3BhdGg+CjxwYXRoIHN0cm9rZS13aWR0aD0iMSIgaWQ9IkUxLU1KTUFJTi0zRCIgZD0iTTU2IDM0N1E1NiAzNjAgNzAgMzY3SDcwN1E3MjIgMzU5IDcyMiAzNDdRNzIyIDMzNiA3MDggMzI4TDM5MCAzMjdINzJRNTYgMzMyIDU2IDM0N1pNNTYgMTUzUTU2IDE2OCA3MiAxNzNINzA4UTcyMiAxNjMgNzIyIDE1M1E3MjIgMTQwIDcwNyAxMzNINzBRNTYgMTQwIDU2IDE1M1oiPjwvcGF0aD4KPHBhdGggc3Ryb2tlLXdpZHRoPSIxIiBpZD0iRTEtTUpNQVRISS03OCIgZD0iTTUyIDI4OVE1OSAzMzEgMTA2IDM4NlQyMjIgNDQyUTI1NyA0NDIgMjg2IDQyNFQzMjkgMzc5UTM3MSA0NDIgNDMwIDQ0MlE0NjcgNDQyIDQ5NCA0MjBUNTIyIDM2MVE1MjIgMzMyIDUwOCAzMTRUNDgxIDI5MlQ0NTggMjg4UTQzOSAyODggNDI3IDI5OVQ0MTUgMzI4UTQxNSAzNzQgNDY1IDM5MVE0NTQgNDA0IDQyNSA0MDRRNDEyIDQwNCA0MDYgNDAyUTM2OCAzODYgMzUwIDMzNlEyOTAgMTE1IDI5MCA3OFEyOTAgNTAgMzA2IDM4VDM0MSAyNlEzNzggMjYgNDE0IDU5VDQ2MyAxNDBRNDY2IDE1MCA0NjkgMTUxVDQ4NSAxNTNINDg5UTUwNCAxNTMgNTA0IDE0NVE1MDQgMTQ0IDUwMiAxMzRRNDg2IDc3IDQ0MCAzM1QzMzMgLTExUTI2MyAtMTEgMjI3IDUyUTE4NiAtMTAgMTMzIC0xMEgxMjdRNzggLTEwIDU3IDE2VDM1IDcxUTM1IDEwMyA1NCAxMjNUOTkgMTQzUTE0MiAxNDMgMTQyIDEwMVExNDIgODEgMTMwIDY2VDEwNyA0NlQ5NCA0MUw5MSA0MFE5MSAzOSA5NyAzNlQxMTMgMjlUMTMyIDI2UTE2OCAyNiAxOTQgNzFRMjAzIDg3IDIxNyAxMzlUMjQ1IDI0N1QyNjEgMzEzUTI2NiAzNDAgMjY2IDM1MlEyNjYgMzgwIDI1MSAzOTJUMjE3IDQwNFExNzcgNDA0IDE0MiAzNzJUOTMgMjkwUTkxIDI4MSA4OCAyODBUNzIgMjc4SDU4UTUyIDI4NCA1MiAyODlaIj48L3BhdGg+CjxwYXRoIHN0cm9rZS13aWR0aD0iMSIgaWQ9IkUxLU1KTUFJTi0yQiIgZD0iTTU2IDIzN1Q1NiAyNTBUNzAgMjcwSDM2OVY0MjBMMzcwIDU3MFEzODAgNTgzIDM4OSA1ODNRNDAyIDU4MyA0MDkgNTY4VjI3MEg3MDdRNzIyIDI2MiA3MjIgMjUwVDcwNyAyMzBINDA5Vi02OFE0MDEgLTgyIDM5MSAtODJIMzg5SDM4N1EzNzUgLTgyIDM2OSAtNjhWMjMwSDcwUTU2IDIzNyA1NiAyNTBaIj48L3BhdGg+CjxwYXRoIHN0cm9rZS13aWR0aD0iMSIgaWQ9IkUxLU1KTUFUSEktNzkiIGQ9Ik0yMSAyODdRMjEgMzAxIDM2IDMzNVQ4NCA0MDZUMTU4IDQ0MlExOTkgNDQyIDIyNCA0MTlUMjUwIDM1NVEyNDggMzM2IDI0NyAzMzRRMjQ3IDMzMSAyMzEgMjg4VDE5OCAxOTFUMTgyIDEwNVExODIgNjIgMTk2IDQ1VDIzOCAyN1EyNjEgMjcgMjgxIDM4VDMxMiA2MVQzMzkgOTRRMzM5IDk1IDM0NCAxMTRUMzU4IDE3M1QzNzcgMjQ3UTQxNSAzOTcgNDE5IDQwNFE0MzIgNDMxIDQ2MiA0MzFRNDc1IDQzMSA0ODMgNDI0VDQ5NCA0MTJUNDk2IDQwM1E0OTYgMzkwIDQ0NyAxOTNUMzkxIC0yM1EzNjMgLTEwNiAyOTQgLTE1NVQxNTYgLTIwNVExMTEgLTIwNSA3NyAtMTgzVDQzIC0xMTdRNDMgLTk1IDUwIC04MFQ2OSAtNThUODkgLTQ4VDEwNiAtNDVRMTUwIC00NSAxNTAgLTg3UTE1MCAtMTA3IDEzOCAtMTIyVDExNSAtMTQyVDEwMiAtMTQ3TDk5IC0xNDhRMTAxIC0xNTMgMTE4IC0xNjBUMTUyIC0xNjdIMTYwUTE3NyAtMTY3IDE4NiAtMTY1UTIxOSAtMTU2IDI0NyAtMTI3VDI5MCAtNjVUMzEzIC05VDMyMSAyMUwzMTUgMTdRMzA5IDEzIDI5NiA2VDI3MCAtNlEyNTAgLTExIDIzMSAtMTFRMTg1IC0xMSAxNTAgMTFUMTA0IDgyUTEwMyA4OSAxMDMgMTEzUTEwMyAxNzAgMTM4IDI2MlQxNzMgMzc5UTE3MyAzODAgMTczIDM4MVExNzMgMzkwIDE3MyAzOTNUMTY5IDQwMFQxNTggNDA0SDE1NFExMzEgNDA0IDExMiAzODVUODIgMzQ0VDY1IDMwMlQ1NyAyODBRNTUgMjc4IDQxIDI3OEgyN1EyMSAyODQgMjEgMjg3WiI+PC9wYXRoPgo8cGF0aCBzdHJva2Utd2lkdGg9IjEiIGlkPSJFMS1NSk1BVEhJLTdBIiBkPSJNMzQ3IDMzOFEzMzcgMzM4IDI5NCAzNDlUMjMxIDM2MFEyMTEgMzYwIDE5NyAzNTZUMTc0IDM0NlQxNjIgMzM1VDE1NSAzMjRMMTUzIDMyMFExNTAgMzE3IDEzOCAzMTdRMTE3IDMxNyAxMTcgMzI1UTExNyAzMzAgMTIwIDMzOVExMzMgMzc4IDE2MyA0MDZUMjI5IDQ0MFEyNDEgNDQyIDI0NiA0NDJRMjcxIDQ0MiAyOTEgNDI1VDMyOSAzOTJUMzY3IDM3NVEzODkgMzc1IDQxMSA0MDhUNDM0IDQ0MVE0MzUgNDQyIDQ0OSA0NDJINDYyUTQ2OCA0MzYgNDY4IDQzNFE0NjggNDMwIDQ2MyA0MjBUNDQ5IDM5OVQ0MzIgMzc3VDQxOCAzNThMNDExIDM0OVEzNjggMjk4IDI3NSAyMTRUMTYwIDEwNkwxNDggOTRMMTYzIDkzUTE4NSA5MyAyMjcgODJUMjkwIDcxUTMyOC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class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden=true style=vertical-align:-0.671ex;width:17.625ex;height:3.009ex alt="{\displaystyle r^{2}=x^{2}+y^{2}+z^{2}}"></span><font style=vertical-align:inherit><font style=vertical-align:inherit>,</font></font><span class=mwe-math-element><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style=display:none>
|
||
<math xmlns=http://www.w3.org/1998/Math/MathML alttext="{\displaystyle \varrho =x^{2}+y^{2}}"><semantics>
|
||
<mrow class=MJX-TeXAtom-ORD>
|
||
<mstyle displaystyle=true scriptlevel=0>
|
||
<mi><font style=vertical-align:inherit><font style=vertical-align:inherit>
|
||
ϱ</font></font>
|
||
</mi>
|
||
<mo><font style=vertical-align:inherit><font style=vertical-align:inherit>
|
||
=
|
||
</font></font></mo>
|
||
<msup>
|
||
<mi><font style=vertical-align:inherit><font style=vertical-align:inherit>
|
||
x
|
||
</font></font></mi>
|
||
<mrow class=MJX-TeXAtom-ORD>
|
||
<mn><font style=vertical-align:inherit><font style=vertical-align:inherit>
|
||
2
|
||
</font></font></mn>
|
||
</mrow>
|
||
</msup>
|
||
<mo><font style=vertical-align:inherit><font style=vertical-align:inherit>
|
||
+
|
||
</font></font></mo>
|
||
<msup>
|
||
<mi><font style=vertical-align:inherit><font style=vertical-align:inherit>
|
||
y
|
||
</font></font></mi>
|
||
<mrow class=MJX-TeXAtom-ORD>
|
||
<mn><font style=vertical-align:inherit><font style=vertical-align:inherit>
|
||
2
|
||
</font></font></mn>
|
||
</mrow>
|
||
</msup>
|
||
</mstyle>
|
||
</mrow>
|
||
<annotation encoding=application/x-tex>
|
||
{\displaystyle \varrho =x^{2}+y^{2}}
|
||
</annotation>
|
||
</semantics>
|
||
</math></span><img 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class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden=true style=vertical-align:-0.671ex;width:11.739ex;height:3.009ex alt="{\displaystyle \varrho =x^{2}+y^{2}}"></span><font style=vertical-align:inherit><font style=vertical-align:inherit>,</font></font><span class=mwe-math-element><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style=display:none>
|
||
<math xmlns=http://www.w3.org/1998/Math/MathML alttext="{\displaystyle \varphi =at/r}"><semantics>
|
||
<mrow class=MJX-TeXAtom-ORD>
|
||
<mstyle displaystyle=true scriptlevel=0>
|
||
<mi><font style=vertical-align:inherit><font style=vertical-align:inherit>
|
||
φ</font></font>
|
||
</mi>
|
||
<mo><font style=vertical-align:inherit><font style=vertical-align:inherit>
|
||
=
|
||
</font></font></mo>
|
||
<mi><font style=vertical-align:inherit><font style=vertical-align:inherit>
|
||
a
|
||
</font></font></mi>
|
||
<mi><font style=vertical-align:inherit><font style=vertical-align:inherit>
|
||
t
|
||
</font></font></mi>
|
||
<mrow class=MJX-TeXAtom-ORD>
|
||
<mo><font style=vertical-align:inherit><font style=vertical-align:inherit>
|
||
/
|
||
</font></font></mo>
|
||
</mrow>
|
||
<mi><font style=vertical-align:inherit><font style=vertical-align:inherit>
|
||
r
|
||
</font></font></mi>
|
||
</mstyle>
|
||
</mrow>
|
||
<annotation encoding=application/x-tex>
|
||
{\displaystyle \varphi =at/r}
|
||
</annotation>
|
||
</semantics>
|
||
</math></span><img src="data:image/svg+xml;base64,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" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden=true style=vertical-align:-0.838ex;width:8.899ex;height:2.843ex alt="{\displaystyle \varphi =at/r}"></span><font style=vertical-align:inherit><font style=vertical-align:inherit>. The components of the electrical forces are then the partial derivatives of the function</font></font></p>
|
||
<center>
|
||
<span class=mwe-math-element><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style=display:none>
|
||
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a
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t
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<mi mathvariant=normal><font style=vertical-align:inherit><font style=vertical-align:inherit>
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∂</font></font>
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||
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|
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<mi mathvariant=normal><font style=vertical-align:inherit><font style=vertical-align:inherit>
|
||
∂</font></font>
|
||
</mi>
|
||
<mi><font style=vertical-align:inherit><font style=vertical-align:inherit>
|
||
e.g
|
||
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</mrow>
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<mrow>
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(
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1
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|
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<mi><font style=vertical-align:inherit><font style=vertical-align:inherit>
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||
r
|
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</font></font></mi>
|
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</mfrac>
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|
||
<mo><font style=vertical-align:inherit><font style=vertical-align:inherit>
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)
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</mrow>
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<mo><font style=vertical-align:inherit><font style=vertical-align:inherit>
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.
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{\displaystyle at{\frac {\partial }{\partial z}}\left({\frac {1}{r}}\right).}
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</center>
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<p><font style=vertical-align:inherit><font style=vertical-align:inherit>This is the potential of an electric colon in the point</font></font><span class=mwe-math-element><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style=display:none>
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r
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=
|
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0
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{\displaystyle r=0}
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a
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t
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/
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l
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{\displaystyle at/l}
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</semantics>
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</math></span><img src="data:image/svg+xml;base64,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" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden=true style=vertical-align:-0.838ex;width:3.925ex;height:2.843ex alt="{\displaystyle at/l}"></span><font style=vertical-align:inherit><font style=vertical-align:inherit>. The line connecting both charges</font></font><span class=mwe-math-element><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style=display:none>
|
||
<math xmlns=http://www.w3.org/1998/Math/MathML alttext="{\displaystyle l}"><semantics>
|
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|
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<mstyle displaystyle=true scriptlevel=0>
|
||
<mi><font style=vertical-align:inherit><font style=vertical-align:inherit>
|
||
l
|
||
</font></font></mi>
|
||
</mstyle>
|
||
</mrow>
|
||
<annotation encoding=application/x-tex>
|
||
{\displaystyle l}
|
||
</annotation>
|
||
</semantics>
|
||
</math></span><img src="data:image/svg+xml;base64,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" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden=true style=vertical-align:-0.338ex;width:0.693ex;height:2.176ex alt="{\displaystyle l}"></span><font style=vertical-align:inherit><font style=vertical-align:inherit>is parallel to the </font></font><i><font style=vertical-align:inherit><font style=vertical-align:inherit>z</font></font></i><font style=vertical-align:inherit><font style=vertical-align:inherit> -axis. The components of </font></font><span style=letter-spacing:0.2em;left:0.1em><font style=vertical-align:inherit><font style=vertical-align:inherit>Poynting</font></font></span><font style=vertical-align:inherit><font style=vertical-align:inherit> 's energy flow are proportional to the quantities</font></font></p>
|
||
<table width=100%>
|
||
<tbody>
|
||
<tr>
|
||
<td valign=center style=text-align:left width=5%></td>
|
||
<td align=center><span class=mwe-math-element><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style=display:none>
|
||
<math xmlns=http://www.w3.org/1998/Math/MathML alttext="{\displaystyle {\mathfrak {P}}=ZM-YN=Aa^{2}tx\left({\frac {1}{r^{6}}}-{\frac {3z^{2}}{r^{8}}}\right)}"><semantics>
|
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<mrow class=MJX-TeXAtom-ORD>
|
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<mstyle displaystyle=true scriptlevel=0>
|
||
<mrow class=MJX-TeXAtom-ORD>
|
||
<mrow class=MJX-TeXAtom-ORD>
|
||
<mi mathvariant=fraktur><font style=vertical-align:inherit><font style=vertical-align:inherit>
|
||
P
|
||
</font></font></mi>
|
||
</mrow>
|
||
</mrow>
|
||
<mo><font style=vertical-align:inherit><font style=vertical-align:inherit>
|
||
=
|
||
</font></font></mo>
|
||
<mi><font style=vertical-align:inherit><font style=vertical-align:inherit>
|
||
Z
|
||
</font></font></mi>
|
||
<mi><font style=vertical-align:inherit><font style=vertical-align:inherit>
|
||
M
|
||
</font></font></mi>
|
||
<mo><font style=vertical-align:inherit><font style=vertical-align:inherit>
|
||
−</font></font>
|
||
</mo>
|
||
<mi><font style=vertical-align:inherit><font style=vertical-align:inherit>
|
||
Y
|
||
</font></font></mi>
|
||
<mi><font style=vertical-align:inherit><font style=vertical-align:inherit>
|
||
N
|
||
</font></font></mi>
|
||
<mo><font style=vertical-align:inherit><font style=vertical-align:inherit>
|
||
=
|
||
</font></font></mo>
|
||
<mi><font style=vertical-align:inherit><font style=vertical-align:inherit>
|
||
A
|
||
</font></font></mi>
|
||
<msup>
|
||
<mi><font style=vertical-align:inherit><font style=vertical-align:inherit>
|
||
a
|
||
</font></font></mi>
|
||
<mrow class=MJX-TeXAtom-ORD>
|
||
<mn><font style=vertical-align:inherit><font style=vertical-align:inherit>
|
||
2
|
||
</font></font></mn>
|
||
</mrow>
|
||
</msup>
|
||
<mi><font style=vertical-align:inherit><font style=vertical-align:inherit>
|
||
t
|
||
</font></font></mi>
|
||
<mi><font style=vertical-align:inherit><font style=vertical-align:inherit>
|
||
x
|
||
</font></font></mi>
|
||
<mrow>
|
||
<mo><font style=vertical-align:inherit><font style=vertical-align:inherit>
|
||
(
|
||
</font></font></mo>
|
||
<mrow>
|
||
<mrow class=MJX-TeXAtom-ORD>
|
||
<mfrac>
|
||
<mn><font style=vertical-align:inherit><font style=vertical-align:inherit>
|
||
1
|
||
</font></font></mn>
|
||
<msup>
|
||
<mi><font style=vertical-align:inherit><font style=vertical-align:inherit>
|
||
r
|
||
</font></font></mi>
|
||
<mrow class=MJX-TeXAtom-ORD>
|
||
<mn><font style=vertical-align:inherit><font style=vertical-align:inherit>
|
||
6
|
||
</font></font></mn>
|
||
</mrow>
|
||
</msup>
|
||
</mfrac>
|
||
</mrow>
|
||
<mo><font style=vertical-align:inherit><font style=vertical-align:inherit>
|
||
−</font></font>
|
||
</mo>
|
||
<mrow class=MJX-TeXAtom-ORD>
|
||
<mfrac>
|
||
<mrow>
|
||
<mn><font style=vertical-align:inherit><font style=vertical-align:inherit>
|
||
3
|
||
</font></font></mn>
|
||
<msup>
|
||
<mi><font style=vertical-align:inherit><font style=vertical-align:inherit>
|
||
e.g
|
||
</font></font></mi>
|
||
<mrow class=MJX-TeXAtom-ORD>
|
||
<mn><font style=vertical-align:inherit><font style=vertical-align:inherit>
|
||
2
|
||
</font></font></mn>
|
||
</mrow>
|
||
</msup>
|
||
</mrow>
|
||
<msup>
|
||
<mi><font style=vertical-align:inherit><font style=vertical-align:inherit>
|
||
r
|
||
</font></font></mi>
|
||
<mrow class=MJX-TeXAtom-ORD>
|
||
<mn><font style=vertical-align:inherit><font style=vertical-align:inherit>
|
||
8th
|
||
</font></font></mn>
|
||
</mrow>
|
||
</msup>
|
||
</mfrac>
|
||
</mrow>
|
||
</mrow>
|
||
<mo><font style=vertical-align:inherit><font style=vertical-align:inherit>
|
||
)
|
||
</font></font></mo>
|
||
</mrow>
|
||
</mstyle>
|
||
</mrow>
|
||
<annotation encoding=application/x-tex>
|
||
{\displaystyle {\mathfrak {P}}=ZM-YN=Aa^{2}tx\left({\frac {1}{r^{6}}}-{\frac {3z^{2}}{r^{8}}}\right)}
|
||
</annotation>
|
||
</semantics>
|
||
</math></span><img src="data:image/svg+xml;base64,<svg xmlns:xlink="http://www.w3.org/1999/xlink" width="38.848ex" height="6.343ex" style="vertical-align: -2.505ex;" viewBox="0 -1652.5 16726.3 2730.8" role="img" focusable="false" xmlns="http://www.w3.org/2000/svg" aria-labelledby="MathJax-SVG-1-Title">
<title id="MathJax-SVG-1-Title">{\displaystyle {\mathfrak {P}}=ZM-YN=Aa^{2}tx\left({\frac {1}{r^{6}}}-{\frac {3z^{2}}{r^{8}}}\right)}</title>
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</svg>" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden=true style=vertical-align:-2.171ex;width:34.779ex;height:6.009ex alt="{\displaystyle {\mathfrak {R}}=YL-XM=3Aa^{2}tz{\frac {x^{2}+y^{2}}{r^{8}}}.}"></span></p></td>
|
||
</tr>
|
||
</tbody>
|
||
</table>
|
||
<p><font style=vertical-align:inherit><font style=vertical-align:inherit>Now let's set</font></font></p>
|
||
<table width=100%>
|
||
<tbody>
|
||
<tr>
|
||
<td valign=center style=text-align:left width=5%></td>
|
||
<td align=center><span class=mwe-math-element><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style=display:none>
|
||
<math xmlns=http://www.w3.org/1998/Math/MathML alttext="{\displaystyle {\begin{array}{lllll}x=\varrho \cos \vartheta &&y=\varrho \sin \vartheta &&{\frac {dx}{dt}}=\alpha ={\frac {d\varrho }{dt}}\cos \vartheta -\varrho \sin \vartheta {\frac {d\vartheta }{dt}}\\\\{\frac {d\vartheta }{dt}}=\eta &&{\frac {d\varrho }{dt}}=\zeta &&{\frac {dy}{dt}}=\beta ={\frac {d\varrho }{dt}}\sin \vartheta +\varrho \cos \vartheta {\frac {d\vartheta }{dt}}\\\\&&&&{\frac {dz}{dt}}=\gamma ,\end{array}}}"><semantics>
|
||
<mrow class=MJX-TeXAtom-ORD>
|
||
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|
||
<mrow class=MJX-TeXAtom-ORD>
|
||
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|
||
<mtr>
|
||
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|
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|
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|
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|
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|
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|
||
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|
||
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|
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|
||
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|
||
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|
||
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|
||
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|
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|
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|
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|
||
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|
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|
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|
||
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|
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|
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|
||
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|
||
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|
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<annotation encoding=application/x-tex>
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{\displaystyle {\begin{array}{lllll}x=\varrho \cos \vartheta &&y=\varrho \sin \vartheta &&{\frac {dx}{dt}}=\alpha ={\frac {d\varrho }{dt}}\cos \vartheta -\varrho \sin \vartheta {\frac {d\vartheta }{dt}}\\\\{\frac {d\vartheta }{dt}}=\eta &&{\frac {d\varrho }{dt}}=\zeta &&{\frac {dy}{dt}}=\beta ={\frac {d\varrho }{dt}}\sin \vartheta +\varrho \cos \vartheta {\frac {d\vartheta }{dt}}\\\\&&&&{\frac {dz}{dt}}=\gamma ,\end{array}}}
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</annotation>
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</semantics>
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</math></span><img src=data:image/svg+xml;base64,<svg xmlns:xlink="http://www.w3.org/1999/xlink" width="61.229ex" height="19.176ex" style="vertical-align: -8.749ex; margin-bottom: -0.256ex;" viewBox="0 -4379.3 26362.6 8256.3" role="img" focusable="false" xmlns="http://www.w3.org/2000/svg" aria-labelledby="MathJax-SVG-1-Title">
<title id="MathJax-SVG-1-Title">{\displaystyle {\begin{array}{lllll}x=\varrho \cos \vartheta &amp;&amp;y=\varrho \sin \vartheta &amp;&amp;{\frac {dx}{dt}}=\alpha ={\frac {d\varrho }{dt}}\cos \vartheta -\varrho \sin \vartheta {\frac {d\vartheta }{dt}}\\\\{\frac {d\vartheta }{dt}}=\eta &amp;&amp;{\frac {d\varrho }{dt}}=\zeta &amp;&amp;{\frac {dy}{dt}}=\beta ={\frac {d\varrho }{dt}}\sin \vartheta +\varrho \cos \vartheta {\frac {d\vartheta }{dt}}\\\\&amp;&amp;&amp;&amp;{\frac {dz}{dt}}=\gamma ,\end{array}}}</title>
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</tbody>
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<p><font style=vertical-align:inherit><font style=vertical-align:inherit>this is what the equation of incompressibility requires</font></font></p>
|
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<center>
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<span class=mwe-math-element><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style=display:none>
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<math xmlns=http://www.w3.org/1998/Math/MathML alttext="{\displaystyle {\frac {\partial \alpha }{\partial x}}+{\frac {\partial \beta }{\partial y}}+{\frac {\partial \gamma }{\partial z}}=0,}"><semantics>
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<mrow class=MJX-TeXAtom-ORD>
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<mstyle displaystyle=true scriptlevel=0>
|
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<mrow class=MJX-TeXAtom-ORD>
|
||
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<mi mathvariant=normal><font style=vertical-align:inherit><font style=vertical-align:inherit>
|
||
∂</font></font>
|
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</mi>
|
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|
||
α</font></font>
|
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</mi>
|
||
</mrow>
|
||
<mrow>
|
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<mi mathvariant=normal><font style=vertical-align:inherit><font style=vertical-align:inherit>
|
||
∂</font></font>
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</mi>
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<mi><font style=vertical-align:inherit><font style=vertical-align:inherit>
|
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|
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|
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|
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|
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<mo><font style=vertical-align:inherit><font style=vertical-align:inherit>
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+
|
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|
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<mrow class=MJX-TeXAtom-ORD>
|
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|
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|
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|
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|
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|
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|
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|
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|
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<mi><font style=vertical-align:inherit><font style=vertical-align:inherit>
|
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e.g
|
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|
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</mrow>
|
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|
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|
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|
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<mn><font style=vertical-align:inherit><font style=vertical-align:inherit>
|
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|
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|
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<mo><font style=vertical-align:inherit><font style=vertical-align:inherit>
|
||
,
|
||
</font></font></mo>
|
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</mstyle>
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</mrow>
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<annotation encoding=application/x-tex>
|
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{\displaystyle {\frac {\partial \alpha }{\partial x}}+{\frac {\partial \beta }{\partial y}}+{\frac {\partial \gamma }{\partial z}}=0,}
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</annotation>
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</semantics>
|
||
</math></span><img src=data:image/svg+xml;base64,<svg xmlns:xlink="http://www.w3.org/1999/xlink" width="21.133ex" height="6.176ex" style="vertical-align: -2.505ex;" viewBox="0 -1580.7 9098.9 2659.1" role="img" focusable="false" xmlns="http://www.w3.org/2000/svg" aria-labelledby="MathJax-SVG-1-Title">
<title id="MathJax-SVG-1-Title">{\displaystyle {\frac {\partial \alpha }{\partial x}}+{\frac {\partial \beta }{\partial y}}+{\frac {\partial \gamma }{\partial z}}=0,}</title>
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<p><font style=vertical-align:inherit><font style=vertical-align:inherit>is if we assume that because of the symmetry around the </font></font><i><font style=vertical-align:inherit><font style=vertical-align:inherit>z</font></font></i><font style=vertical-align:inherit><font style=vertical-align:inherit> -axis the sizes</font></font><span class=mwe-math-element><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style=display:none>
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,
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ζ</font></font>
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,
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{\displaystyle \vartheta }
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|
||
<p><font style=vertical-align:inherit><font style=vertical-align:inherit>The differential equations derived by </font></font><span style=letter-spacing:0.2em;left:0.1em><font style=vertical-align:inherit><font style=vertical-align:inherit>Helmholtz</font></font></span><font style=vertical-align:inherit><font style=vertical-align:inherit> , which express that the electromagnetic</font></font><span><span class=pagenum id=iv title="Page: Translational movement of the light ether.djvu/4"> <span class="pagenumber noprint" style=color:#666666;display:inline;margin:0px;padding:0px>[<b><a href=https://de.wikisource.org/wiki/Seite%3ATranslatorische_Bewegung_des_Licht%C3%A4thers.djvu/4 title="Seite:Translatorische Bewegung des Lichtäthers.djvu/4">iv</a></b>]</span> </span></span><font style=vertical-align:inherit><font style=vertical-align:inherit>Currents caused by tensions in turn produce electromagnetic forces that balance themselves with those acting from outside</font></font></p>
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<td valign=center style=text-align:left width=5%><font style=vertical-align:inherit><font style=vertical-align:inherit>(1)</font></font></td>
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0
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P
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x
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+
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|
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{\displaystyle {\begin{cases}0={\frac {\partial P}{\partial x}}+A\left[{\frac {\partial {\mathfrak {P}}}{\partial t}}+\beta \left({\frac {\partial {\mathfrak {P}}}{\partial y}}-{\frac {\partial {\mathfrak {Q}}}{\partial x}}\right)-\gamma \left({\frac {\partial {\mathfrak {R}}}{\partial x}}-{\frac {\partial {\mathfrak {P}}}{\partial z}}\right)\right]\\0={\frac {\partial P}{\partial y}}+A\left[{\frac {\partial {\mathfrak {Q}}}{\partial t}}+\gamma \left({\frac {\partial {\mathfrak {Q}}}{\partial z}}-{\frac {\partial {\mathfrak {R}}}{\partial y}}\right)-\alpha \left({\frac {\partial {\mathfrak {P}}}{\partial y}}-{\frac {\partial {\mathfrak {Q}}}{\partial x}}\right)\right]\\0={\frac {\partial P}{\partial z}}+A\left[{\frac {\partial {\mathfrak {R}}}{\partial t}}+\alpha \left({\frac {\partial {\mathfrak {R}}}{\partial x}}-{\frac {\partial {\mathfrak {P}}}{\partial z}}\right)-\beta \left({\frac {\partial {\mathfrak {Q}}}{\partial z}}-{\frac {\partial {\mathfrak {R}}}{\partial y}}\right)\right]\end{cases}}}
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<title id="MathJax-SVG-1-Title">{\displaystyle {\begin{cases}0={\frac {\partial P}{\partial x}}+A\left[{\frac {\partial {\mathfrak {P}}}{\partial t}}+\beta \left({\frac {\partial {\mathfrak {P}}}{\partial y}}-{\frac {\partial {\mathfrak {Q}}}{\partial x}}\right)-\gamma \left({\frac {\partial {\mathfrak {R}}}{\partial x}}-{\frac {\partial {\mathfrak {P}}}{\partial z}}\right)\right]\\0={\frac {\partial P}{\partial y}}+A\left[{\frac {\partial {\mathfrak {Q}}}{\partial t}}+\gamma \left({\frac {\partial {\mathfrak {Q}}}{\partial z}}-{\frac {\partial {\mathfrak {R}}}{\partial y}}\right)-\alpha \left({\frac {\partial {\mathfrak {P}}}{\partial y}}-{\frac {\partial {\mathfrak {Q}}}{\partial x}}\right)\right]\\0={\frac {\partial P}{\partial z}}+A\left[{\frac {\partial {\mathfrak {R}}}{\partial t}}+\alpha \left({\frac {\partial {\mathfrak {R}}}{\partial x}}-{\frac {\partial {\mathfrak {P}}}{\partial z}}\right)-\beta \left({\frac {\partial {\mathfrak {Q}}}{\partial z}}-{\frac {\partial {\mathfrak {R}}}{\partial y}}\right)\right]\end{cases}}}</title>
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</svg>" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden=true style=vertical-align:-6.338ex;width:53.395ex;height:13.843ex alt="{\displaystyle {\begin{cases}0={\frac {\partial P}{\partial x}}+A\left[{\frac {\partial {\mathfrak {P}}}{\partial t}} +\beta \left({\frac {\partial {\mathfrak {P}}}{\partial y}}-{\frac {\partial {\mathfrak {Q}}}{\partial x}}\right) -\gamma \left({\frac {\partial {\mathfrak {R}}}{\partial x}}-{\frac {\partial {\mathfrak {P}}}{\partial z}}\right) \right]\\0={\frac {\partial P}{\partial y}}+A\left[{\frac {\partial {\mathfrak {Q}}}{\partial t}}+\gamma \ left({\frac {\partial {\mathfrak {Q}}}{\partial z}}-{\frac {\partial {\mathfrak {R}}}{\partial y}}\right)-\alpha \ left({\frac {\partial {\mathfrak {P}}}{\partial y}}-{\frac {\partial {\mathfrak {Q}}}{\partial x}}\right)\right]\ \0={\frac {\partial P}{\partial z}}+A\left[{\frac {\partial {\mathfrak {R}}}{\partial t}}+\alpha \left({\ frac {\partial {\mathfrak {R}}}{\partial x}}-{\frac {\partial {\mathfrak {P}}}{\partial z}}\right)-\beta \left({\ frac {\partial {\mathfrak {Q}}}{\partial z}}-{\frac {\partial {\mathfrak {R}}}{\partial y}}\right)\right]\end{cases} }}"></span></td>
|
||
</tr>
|
||
</tbody>
|
||
</table>
|
||
<p><font style=vertical-align:inherit><font style=vertical-align:inherit>Here </font></font><i><font style=vertical-align:inherit><font style=vertical-align:inherit>P</font></font></i><font style=vertical-align:inherit><font style=vertical-align:inherit> means the hydrostatic pressure.</font></font></p>
|
||
<p><font style=vertical-align:inherit><font style=vertical-align:inherit>Let us put the above values of into these equations</font></font><span class=mwe-math-element><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style=display:none>
|
||
<math xmlns=http://www.w3.org/1998/Math/MathML alttext="{\displaystyle {\mathfrak {P,\ Q,\ R,}}\alpha ,\beta ,\gamma }"><semantics>
|
||
<mrow class=MJX-TeXAtom-ORD>
|
||
<mstyle displaystyle=true scriptlevel=0>
|
||
<mrow class=MJX-TeXAtom-ORD>
|
||
<mrow class=MJX-TeXAtom-ORD>
|
||
<mi mathvariant=fraktur><font style=vertical-align:inherit><font style=vertical-align:inherit>
|
||
P
|
||
</font></font></mi>
|
||
<mo mathvariant=fraktur><font style=vertical-align:inherit><font style=vertical-align:inherit>
|
||
,
|
||
</font></font></mo>
|
||
<mtext mathvariant=fraktur>
|
||
|
||
</mtext>
|
||
<mi mathvariant=fraktur><font style=vertical-align:inherit><font style=vertical-align:inherit>
|
||
Q
|
||
</font></font></mi>
|
||
<mo mathvariant=fraktur><font style=vertical-align:inherit><font style=vertical-align:inherit>
|
||
,
|
||
</font></font></mo>
|
||
<mtext mathvariant=fraktur>
|
||
|
||
</mtext>
|
||
<mi mathvariant=fraktur><font style=vertical-align:inherit><font style=vertical-align:inherit>
|
||
R
|
||
</font></font></mi>
|
||
<mo mathvariant=fraktur><font style=vertical-align:inherit><font style=vertical-align:inherit>
|
||
,
|
||
</font></font></mo>
|
||
</mrow>
|
||
</mrow>
|
||
<mi><font style=vertical-align:inherit><font style=vertical-align:inherit>
|
||
α</font></font>
|
||
</mi>
|
||
<mo><font style=vertical-align:inherit><font style=vertical-align:inherit>
|
||
,
|
||
</font></font></mo>
|
||
<mi><font style=vertical-align:inherit><font style=vertical-align:inherit>
|
||
β</font></font>
|
||
</mi>
|
||
<mo><font style=vertical-align:inherit><font style=vertical-align:inherit>
|
||
,
|
||
</font></font></mo>
|
||
<mi><font style=vertical-align:inherit><font style=vertical-align:inherit>
|
||
γ</font></font>
|
||
</mi>
|
||
</mstyle>
|
||
</mrow>
|
||
<annotation encoding=application/x-tex>
|
||
{\displaystyle {\mathfrak {P,\ Q,\ R,}}\alpha ,\beta ,\gamma }
|
||
</annotation>
|
||
</semantics>
|
||
</math></span><img src=data:image/svg+xml;base64,<svg xmlns:xlink="http://www.w3.org/1999/xlink" width="15.796ex" height="2.843ex" style="vertical-align: -0.838ex;" viewBox="0 -863.1 6801.2 1223.9" role="img" focusable="false" xmlns="http://www.w3.org/2000/svg" aria-labelledby="MathJax-SVG-1-Title">
<title id="MathJax-SVG-1-Title">{\displaystyle {\mathfrak {P,\ Q,\ R,}}\alpha ,\beta ,\gamma }</title>
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</svg> class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden=true style=vertical-align:-0.838ex;width:15.796ex;height:2.843ex alt="{\displaystyle {\mathfrak {P,\Q,\R,}}\alpha ,\beta ,\gamma }"></span><font style=vertical-align:inherit><font style=vertical-align:inherit>one, so we get</font></font></p>
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<annotation encoding=application/x-tex>
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{\displaystyle 0={\frac {\partial P}{\partial \varrho }}+\varrho A^{2}a^{2}\left({\frac {3\varrho ^{2}}{r^{8}}}-{\frac {2}{r^{6}}}-{\frac {6zt}{r^{8}}}{\frac {1}{\varrho }}{\frac {\partial \psi }{\partial \varrho }}\right),}
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<title id="MathJax-SVG-1-Title">{\displaystyle 0={\frac {\partial P}{\partial \varrho }}+\varrho A^{2}a^{2}\left({\frac {3\varrho ^{2}}{r^{8}}}-{\frac {2}{r^{6}}}-{\frac {6zt}{r^{8}}}{\frac {1}{\varrho }}{\frac {\partial \psi }{\partial \varrho }}\right),}</title>
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</svg>" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden=true style=vertical-align:-2.505ex;width:38.555ex;height:6.343ex alt="{\displaystyle 0={\frac {\partial P}{\partial z}}+zA^{2}a^{2}\left({\frac {3\varrho ^{2}}{r^{8 }}}-{\frac {6t}{r^{8}}}-{\frac {\partial \psi }{\partial z}}\right).}"></span></p></td>
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</tr>
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</tbody>
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</table>
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<p><font style=vertical-align:inherit><font style=vertical-align:inherit>The angular velocity</font></font><span class=mwe-math-element><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style=display:none>
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<math xmlns=http://www.w3.org/1998/Math/MathML alttext="{\displaystyle \eta }"><semantics>
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<mstyle displaystyle=true scriptlevel=0>
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<mi><font style=vertical-align:inherit><font style=vertical-align:inherit>
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η</font></font>
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</mi>
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</mstyle>
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<annotation encoding=application/x-tex>
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{\displaystyle \eta }
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</annotation>
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</semantics>
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</math></span><img src="data:image/svg+xml;base64,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" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden=true style=vertical-align:-0.838ex;width:1.169ex;height:2.176ex alt="{\displaystyle \eta }"></span><font style=vertical-align:inherit><font style=vertical-align:inherit>has completely fallen out, so it does not need to have a value other than zero.</font></font></p>
|
||
<p><font style=vertical-align:inherit><font style=vertical-align:inherit>If we eliminate </font></font><i><font style=vertical-align:inherit><font style=vertical-align:inherit>P</font></font></i><font style=vertical-align:inherit><font style=vertical-align:inherit> from this , we get</font></font></p>
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||
<table width=100%>
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<tbody>
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||
<tr>
|
||
<td valign=center style=text-align:left width=5%><font style=vertical-align:inherit><font style=vertical-align:inherit>(2)</font></font></td>
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||
<td align=center><span class=mwe-math-element><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style=display:none>
|
||
<math xmlns=http://www.w3.org/1998/Math/MathML alttext="{\displaystyle \varrho z-t{\frac {\partial \psi }{\partial \varrho }}+{\frac {8zt}{r^{2}}}\left(z{\frac {\partial \psi }{\partial \varrho }}-\varrho {\frac {\partial \psi }{\partial z}}\right)=0.}"><semantics>
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<mrow class=MJX-TeXAtom-ORD>
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<mstyle displaystyle=true scriptlevel=0>
|
||
<mi><font style=vertical-align:inherit><font style=vertical-align:inherit>
|
||
ϱ</font></font>
|
||
</mi>
|
||
<mi><font style=vertical-align:inherit><font style=vertical-align:inherit>
|
||
e.g
|
||
</font></font></mi>
|
||
<mo><font style=vertical-align:inherit><font style=vertical-align:inherit>
|
||
−</font></font>
|
||
</mo>
|
||
<mi><font style=vertical-align:inherit><font style=vertical-align:inherit>
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||
t
|
||
</font></font></mi>
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<mrow class=MJX-TeXAtom-ORD>
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||
<mfrac>
|
||
<mrow>
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<mi mathvariant=normal><font style=vertical-align:inherit><font style=vertical-align:inherit>
|
||
∂</font></font>
|
||
</mi>
|
||
<mi><font style=vertical-align:inherit><font style=vertical-align:inherit>
|
||
ψ</font></font>
|
||
</mi>
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||
</mrow>
|
||
<mrow>
|
||
<mi mathvariant=normal><font style=vertical-align:inherit><font style=vertical-align:inherit>
|
||
∂</font></font>
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||
</mi>
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||
<mi><font style=vertical-align:inherit><font style=vertical-align:inherit>
|
||
ϱ</font></font>
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</mi>
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</mrow>
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</mfrac>
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||
</mrow>
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||
<mo><font style=vertical-align:inherit><font style=vertical-align:inherit>
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+
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||
</font></font></mo>
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||
<mrow class=MJX-TeXAtom-ORD>
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||
<mfrac>
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||
<mrow>
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<mn><font style=vertical-align:inherit><font style=vertical-align:inherit>
|
||
8th
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||
</font></font></mn>
|
||
<mi><font style=vertical-align:inherit><font style=vertical-align:inherit>
|
||
e.g
|
||
</font></font></mi>
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{\displaystyle \varrho z-t{\frac {\partial \psi }{\partial \varrho }}+{\frac {8zt}{r^{2}}}\left(z{\frac {\partial \psi }{\partial \varrho }}-\varrho {\frac {\partial \psi }{\partial z}}\right)=0.}
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<title id="MathJax-SVG-1-Title">{\displaystyle \varrho z-t{\frac {\partial \psi }{\partial \varrho }}+{\frac {8zt}{r^{2}}}\left(z{\frac {\partial \psi }{\partial \varrho }}-\varrho {\frac {\partial \psi }{\partial z}}\right)=0.}</title>
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<p><font style=vertical-align:inherit><font style=vertical-align:inherit>One can see immediately from this equation that</font></font><span class=mwe-math-element><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style=display:none>
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{\displaystyle \psi }
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1
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t
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<annotation encoding=application/x-tex>
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{\displaystyle 1/t}
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|
||
</math></span><img src="data:image/svg+xml;base64,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" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden=true style=vertical-align:-0.838ex;width:3.165ex;height:2.843ex alt="{\displaystyle 1/t}"></span><font style=vertical-align:inherit><font style=vertical-align:inherit>must contain. For</font></font><span class=mwe-math-element><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style=display:none>
|
||
<math xmlns=http://www.w3.org/1998/Math/MathML alttext="{\displaystyle t=0}"><semantics>
|
||
<mrow class=MJX-TeXAtom-ORD>
|
||
<mstyle displaystyle=true scriptlevel=0>
|
||
<mi><font style=vertical-align:inherit><font style=vertical-align:inherit>
|
||
t
|
||
</font></font></mi>
|
||
<mo><font style=vertical-align:inherit><font style=vertical-align:inherit>
|
||
=
|
||
</font></font></mo>
|
||
<mn><font style=vertical-align:inherit><font style=vertical-align:inherit>
|
||
0
|
||
</font></font></mn>
|
||
</mstyle>
|
||
</mrow>
|
||
<annotation encoding=application/x-tex>
|
||
{\displaystyle t=0}
|
||
</annotation>
|
||
</semantics>
|
||
</math></span><img src=data:image/svg+xml;base64,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 class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden=true style=vertical-align:-0.338ex;width:5.101ex;height:2.176ex alt="{\displaystyle t=0}"></span><font style=vertical-align:inherit><font style=vertical-align:inherit>the charge of the electric colon is zero. So at the moment the charge begins, the currents in the ether would become infinite.</font></font></p>
|
||
<p><font style=vertical-align:inherit><font style=vertical-align:inherit>Since </font></font><span style=letter-spacing:0.2em;left:0.1em><font style=vertical-align:inherit><font style=vertical-align:inherit>Maxwell</font></font></span><font style=vertical-align:inherit><font style=vertical-align:inherit> 's differential equations are completely fulfilled, there is no reason to exclude such a charge that increases from zero in proportion to time.</font></font></p>
|
||
<p><font style=vertical-align:inherit><font style=vertical-align:inherit>A solution to the differential equation (2) is</font></font></p>
|
||
<center>
|
||
<span class=mwe-math-element><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style=display:none>
|
||
<math xmlns=http://www.w3.org/1998/Math/MathML alttext="{\displaystyle \psi ={\frac {r^{2}z}{10t}}}"><semantics>
|
||
<mrow class=MJX-TeXAtom-ORD>
|
||
<mstyle displaystyle=true scriptlevel=0>
|
||
<mi><font style=vertical-align:inherit><font style=vertical-align:inherit>
|
||
ψ</font></font>
|
||
</mi>
|
||
<mo><font style=vertical-align:inherit><font style=vertical-align:inherit>
|
||
=
|
||
</font></font></mo>
|
||
<mrow class=MJX-TeXAtom-ORD>
|
||
<mfrac>
|
||
<mrow>
|
||
<msup>
|
||
<mi><font style=vertical-align:inherit><font style=vertical-align:inherit>
|
||
r
|
||
</font></font></mi>
|
||
<mrow class=MJX-TeXAtom-ORD>
|
||
<mn><font style=vertical-align:inherit><font style=vertical-align:inherit>
|
||
2
|
||
</font></font></mn>
|
||
</mrow>
|
||
</msup>
|
||
<mi><font style=vertical-align:inherit><font style=vertical-align:inherit>
|
||
e.g
|
||
</font></font></mi>
|
||
</mrow>
|
||
<mrow>
|
||
<mn><font style=vertical-align:inherit><font style=vertical-align:inherit>
|
||
10
|
||
</font></font></mn>
|
||
<mi><font style=vertical-align:inherit><font style=vertical-align:inherit>
|
||
t
|
||
</font></font></mi>
|
||
</mrow>
|
||
</mfrac>
|
||
</mrow>
|
||
</mstyle>
|
||
</mrow>
|
||
<annotation encoding=application/x-tex>
|
||
{\displaystyle \psi ={\frac {r^{2}z}{10t}}}
|
||
</annotation>
|
||
</semantics>
|
||
</math></span><img src="data:image/svg+xml;base64,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class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden=true style=vertical-align:-1.838ex;width:8.639ex;height:5.676ex alt="{\displaystyle \psi ={\frac {r^{2}z}{10t}}}"></span>
|
||
</center>
|
||
<p><font style=vertical-align:inherit><font style=vertical-align:inherit>It follows</font></font></p>
|
||
<table width=100%>
|
||
<tbody>
|
||
<tr>
|
||
<td valign=center style=text-align:left width=5%></td>
|
||
<td align=center><span class=mwe-math-element><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style=display:none>
|
||
<math xmlns=http://www.w3.org/1998/Math/MathML alttext="{\displaystyle \zeta ={\frac {1}{\varrho }}{\frac {\partial \psi }{\partial z}}=\left({\frac {2z^{2}+r^{2}}{10t}}\right){\frac {1}{\varrho }},}"><semantics>
|
||
<mrow class=MJX-TeXAtom-ORD>
|
||
<mstyle displaystyle=true scriptlevel=0>
|
||
<mi><font style=vertical-align:inherit><font style=vertical-align:inherit>
|
||
ζ</font></font>
|
||
</mi>
|
||
<mo><font style=vertical-align:inherit><font style=vertical-align:inherit>
|
||
=
|
||
</font></font></mo>
|
||
<mrow class=MJX-TeXAtom-ORD>
|
||
<mfrac>
|
||
<mn><font style=vertical-align:inherit><font style=vertical-align:inherit>
|
||
1
|
||
</font></font></mn>
|
||
<mi><font style=vertical-align:inherit><font style=vertical-align:inherit>
|
||
ϱ</font></font>
|
||
</mi>
|
||
</mfrac>
|
||
</mrow>
|
||
<mrow class=MJX-TeXAtom-ORD>
|
||
<mfrac>
|
||
<mrow>
|
||
<mi mathvariant=normal><font style=vertical-align:inherit><font style=vertical-align:inherit>
|
||
∂</font></font>
|
||
</mi>
|
||
<mi><font style=vertical-align:inherit><font style=vertical-align:inherit>
|
||
ψ</font></font>
|
||
</mi>
|
||
</mrow>
|
||
<mrow>
|
||
<mi mathvariant=normal><font style=vertical-align:inherit><font style=vertical-align:inherit>
|
||
∂</font></font>
|
||
</mi>
|
||
<mi><font style=vertical-align:inherit><font style=vertical-align:inherit>
|
||
e.g
|
||
</font></font></mi>
|
||
</mrow>
|
||
</mfrac>
|
||
</mrow>
|
||
<mo><font style=vertical-align:inherit><font style=vertical-align:inherit>
|
||
=
|
||
</font></font></mo>
|
||
<mrow>
|
||
<mo><font style=vertical-align:inherit><font style=vertical-align:inherit>
|
||
(
|
||
</font></font></mo>
|
||
<mrow class=MJX-TeXAtom-ORD>
|
||
<mfrac>
|
||
<mrow>
|
||
<mn><font style=vertical-align:inherit><font style=vertical-align:inherit>
|
||
2
|
||
</font></font></mn>
|
||
<msup>
|
||
<mi><font style=vertical-align:inherit><font style=vertical-align:inherit>
|
||
e.g
|
||
</font></font></mi>
|
||
<mrow class=MJX-TeXAtom-ORD>
|
||
<mn><font style=vertical-align:inherit><font style=vertical-align:inherit>
|
||
2
|
||
</font></font></mn>
|
||
</mrow>
|
||
</msup>
|
||
<mo><font style=vertical-align:inherit><font style=vertical-align:inherit>
|
||
+
|
||
</font></font></mo>
|
||
<msup>
|
||
<mi><font style=vertical-align:inherit><font style=vertical-align:inherit>
|
||
r
|
||
</font></font></mi>
|
||
<mrow class=MJX-TeXAtom-ORD>
|
||
<mn><font style=vertical-align:inherit><font style=vertical-align:inherit>
|
||
2
|
||
</font></font></mn>
|
||
</mrow>
|
||
</msup>
|
||
</mrow>
|
||
<mrow>
|
||
<mn><font style=vertical-align:inherit><font style=vertical-align:inherit>
|
||
10
|
||
</font></font></mn>
|
||
<mi><font style=vertical-align:inherit><font style=vertical-align:inherit>
|
||
t
|
||
</font></font></mi>
|
||
</mrow>
|
||
</mfrac>
|
||
</mrow>
|
||
<mo><font style=vertical-align:inherit><font style=vertical-align:inherit>
|
||
)
|
||
</font></font></mo>
|
||
</mrow>
|
||
<mrow class=MJX-TeXAtom-ORD>
|
||
<mfrac>
|
||
<mn><font style=vertical-align:inherit><font style=vertical-align:inherit>
|
||
1
|
||
</font></font></mn>
|
||
<mi><font style=vertical-align:inherit><font style=vertical-align:inherit>
|
||
ϱ</font></font>
|
||
</mi>
|
||
</mfrac>
|
||
</mrow>
|
||
<mo><font style=vertical-align:inherit><font style=vertical-align:inherit>
|
||
,
|
||
</font></font></mo>
|
||
</mstyle>
|
||
</mrow>
|
||
<annotation encoding=application/x-tex>
|
||
{\displaystyle \zeta ={\frac {1}{\varrho }}{\frac {\partial \psi }{\partial z}}=\left({\frac {2z^{2}+r^{2}}{10t}}\right){\frac {1}{\varrho }},}
|
||
</annotation>
|
||
</semantics>
|
||
</math></span><img src="data:image/svg+xml;base64,<svg xmlns:xlink="http://www.w3.org/1999/xlink" width="28.577ex" height="6.343ex" style="vertical-align: -2.505ex;" viewBox="0 -1652.5 12304 2730.8" role="img" focusable="false" xmlns="http://www.w3.org/2000/svg" aria-labelledby="MathJax-SVG-1-Title">
<title id="MathJax-SVG-1-Title">{\displaystyle \zeta ={\frac {1}{\varrho }}{\frac {\partial \psi }{\partial z}}=\left({\frac {2z^{2}+r^{2}}{10t}}\right){\frac {1}{\varrho }},}</title>
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</svg> class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden=true style=vertical-align:-2.338ex;width:19.62ex;height:6.009ex alt="{\displaystyle -\gamma ={\frac {1}{\varrho }}{\frac {\partial \psi }{\partial \varrho }}={\frac {2z}{10t}}.}"></span></p></td>
|
||
</tr>
|
||
</tbody>
|
||
</table><span><span class=pagenum id=v title="Page: Translational movement of the light ether.djvu/5"> <span class="pagenumber noprint" style=color:#666666;display:inline;margin:0px;padding:0px>[<b><a href=https://de.wikisource.org/wiki/Seite%3ATranslatorische_Bewegung_des_Licht%C3%A4thers.djvu/5 title="Seite:Translatorische Bewegung des Lichtäthers.djvu/5">v</a></b>]</span> </span></span><font style=vertical-align:inherit><font style=vertical-align:inherit>So the ether would flow parallel to the streamlines,
|
||
</font></font><p><font style=vertical-align:inherit><font style=vertical-align:inherit>in which the planes laid by the </font></font><i><font style=vertical-align:inherit><font style=vertical-align:inherit>z</font></font></i><font style=vertical-align:inherit><font style=vertical-align:inherit> -axis are the surfaces</font></font><span class=mwe-math-element><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style=display:none>
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r
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|
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2
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|
||
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|
||
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|
||
<mi><font style=vertical-align:inherit><font style=vertical-align:inherit>
|
||
e.g
|
||
</font></font></mi>
|
||
<mo><font style=vertical-align:inherit><font style=vertical-align:inherit>
|
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=
|
||
</font></font></mo>
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<mrow class=MJX-TeXAtom-ORD>
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c
|
||
</font></font></mi>
|
||
<mi mathvariant=normal><font style=vertical-align:inherit><font style=vertical-align:inherit>
|
||
O
|
||
</font></font></mi>
|
||
<mi mathvariant=normal><font style=vertical-align:inherit><font style=vertical-align:inherit>
|
||
n
|
||
</font></font></mi>
|
||
<mi mathvariant=normal><font style=vertical-align:inherit><font style=vertical-align:inherit>
|
||
s
|
||
</font></font></mi>
|
||
<mi mathvariant=normal><font style=vertical-align:inherit><font style=vertical-align:inherit>
|
||
t
|
||
</font></font></mi>
|
||
<mo><font style=vertical-align:inherit><font style=vertical-align:inherit>
|
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.
|
||
</font></font></mo>
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|
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|
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{\displaystyle r^{2}z={\rm {const.}}}
|
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class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden=true style=vertical-align:-0.338ex;width:12.245ex;height:2.676ex alt="{\displaystyle r^{2}z={\rm {const.}}}"></span><font style=vertical-align:inherit><font style=vertical-align:inherit>cut. However, such a flow is hydrodynamically impossible because of the speed</font></font><span class=mwe-math-element><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style=display:none>
|
||
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γ</font></font>
|
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|
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|
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|
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|
||
{\displaystyle \gamma }
|
||
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|
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||
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|
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<math xmlns=http://www.w3.org/1998/Math/MathML alttext="{\displaystyle \varrho =0}"><semantics>
|
||
<mrow class=MJX-TeXAtom-ORD>
|
||
<mstyle displaystyle=true scriptlevel=0>
|
||
<mi><font style=vertical-align:inherit><font style=vertical-align:inherit>
|
||
ϱ</font></font>
|
||
</mi>
|
||
<mo><font style=vertical-align:inherit><font style=vertical-align:inherit>
|
||
=
|
||
</font></font></mo>
|
||
<mn><font style=vertical-align:inherit><font style=vertical-align:inherit>
|
||
0
|
||
</font></font></mn>
|
||
</mstyle>
|
||
</mrow>
|
||
<annotation encoding=application/x-tex>
|
||
{\displaystyle \varrho =0}
|
||
</annotation>
|
||
</semantics>
|
||
</math></span><img src="data:image/svg+xml;base64,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" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden=true style=vertical-align:-0.671ex;width:5.463ex;height:2.509ex alt="{\displaystyle \varrho =0}"></span><font style=vertical-align:inherit><font style=vertical-align:inherit>becomes infinite.</font></font></p>
|
||
<p><font style=vertical-align:inherit><font style=vertical-align:inherit>As a second case, we consider an electrified point with the charge </font></font><i><font style=vertical-align:inherit><font style=vertical-align:inherit>e</font></font></i><font style=vertical-align:inherit><font style=vertical-align:inherit> , which moves through space </font><font style=vertical-align:inherit>with the constant speed </font></font><i><font style=vertical-align:inherit><font style=vertical-align:inherit>v</font></font></i><font style=vertical-align:inherit><font style=vertical-align:inherit> . This case is completely treated by </font></font><span style=letter-spacing:0.2em;left:0.1em><font style=vertical-align:inherit><font style=vertical-align:inherit>Heaviside</font></font></span><font style=vertical-align:inherit><font style=vertical-align:inherit> and his solution gives the following values of the electric and magnetic forces, related to a coordinate system fixed in the electrified point in whose </font></font><i><font style=vertical-align:inherit><font style=vertical-align:inherit>x</font></font></i><font style=vertical-align:inherit><font style=vertical-align:inherit> -axis the movement takes place.</font></font></p>
|
||
<table width=100%>
|
||
<tbody>
|
||
<tr>
|
||
<td valign=center style=text-align:left width=5%></td>
|
||
<td align=center><span class=mwe-math-element><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style=display:none>
|
||
<math xmlns=http://www.w3.org/1998/Math/MathML alttext="{\displaystyle {\begin{array}{lclcc}X={\frac {1}{v}}{\frac {\partial U}{\partial x}}\left(1-A^{2}v^{2}\right),&&Y={\frac {1}{v}}{\frac {\partial U}{\partial y}},&&Z={\frac {1}{v}}{\frac {\partial U}{\partial z}},\\\\M=-A{\frac {\partial U}{\partial z}},&&N=A{\frac {\partial U}{\partial y}},&&L=0.\end{array}}}"><semantics>
|
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<mrow class=MJX-TeXAtom-ORD>
|
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<mstyle displaystyle=true scriptlevel=0>
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</mtable>
|
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</mrow>
|
||
</mstyle>
|
||
</mrow>
|
||
<annotation encoding=application/x-tex>
|
||
{\displaystyle {\begin{array}{lclcc}X={\frac {1}{v}}{\frac {\partial U}{\partial x}}\left(1-A^{2}v^{2}\right),&&Y={\frac {1}{v}}{\frac {\partial U}{\partial y}},&&Z={\frac {1}{v}}{\frac {\partial U}{\partial z}},\\\\M=-A{\frac {\partial U}{\partial z}},&&N=A{\frac {\partial U}{\partial y}},&&L=0.\end{array}}}
|
||
</annotation>
|
||
</semantics>
|
||
</math></span><img src="data:image/svg+xml;base64,<svg xmlns:xlink="http://www.w3.org/1999/xlink" width="53.033ex" height="11.843ex" style="vertical-align: -5.338ex;" viewBox="0 -2800.6 22833.6 5098.9" role="img" focusable="false" xmlns="http://www.w3.org/2000/svg" aria-labelledby="MathJax-SVG-1-Title">
<title id="MathJax-SVG-1-Title">{\displaystyle {\begin{array}{lclcc}X={\frac {1}{v}}{\frac {\partial U}{\partial x}}\left(1-A^{2}v^{2}\right),&amp;&amp;Y={\frac {1}{v}}{\frac {\partial U}{\partial y}},&amp;&amp;Z={\frac {1}{v}}{\frac {\partial U}{\partial z}},\\\\M=-A{\frac {\partial U}{\partial z}},&amp;&amp;N=A{\frac {\partial U}{\partial y}},&amp;&amp;L=0.\end{array}}}</title>
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||
</math></span><img src="data:image/svg+xml;base64,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class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden=true style=vertical-align:-0.838ex;width:9ex;height:2.843ex alt="{\displaystyle {\mathfrak {P,\Q,\R}}}"></span></p>
|
||
<table width=100%>
|
||
<tbody>
|
||
<tr>
|
||
<td valign=center style=text-align:left width=5%></td>
|
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<td align=center><span class=mwe-math-element><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style=display:none>
|
||
<math xmlns=http://www.w3.org/1998/Math/MathML alttext="{\displaystyle {\mathfrak {P}}={\frac {{\mathfrak {A}}\varrho ^{2}}{\left(r^{2}-A^{2}v^{2}\varrho ^{2}\right)^{3}}},\quad {\mathfrak {Q}}=-{\frac {{\mathfrak {A}}xy}{\left(r^{2}-A^{2}v^{2}\varrho ^{2}\right)^{3}}},}"><semantics>
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|
||
<mi mathvariant=fraktur><font style=vertical-align:inherit><font style=vertical-align:inherit>
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||
P
|
||
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|
||
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|
||
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||
=
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||
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<mfrac>
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||
<mrow>
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||
<mrow class=MJX-TeXAtom-ORD>
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<mrow class=MJX-TeXAtom-ORD>
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||
<mi mathvariant=fraktur><font style=vertical-align:inherit><font style=vertical-align:inherit>
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||
A
|
||
</font></font></mi>
|
||
</mrow>
|
||
</mrow>
|
||
<msup>
|
||
<mi><font style=vertical-align:inherit><font style=vertical-align:inherit>
|
||
ϱ</font></font>
|
||
</mi>
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||
<mrow class=MJX-TeXAtom-ORD>
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||
<mn><font style=vertical-align:inherit><font style=vertical-align:inherit>
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2
|
||
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||
</mrow>
|
||
</msup>
|
||
</mrow>
|
||
<msup>
|
||
<mrow>
|
||
<mo><font style=vertical-align:inherit><font style=vertical-align:inherit>
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||
(
|
||
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|
||
<mrow>
|
||
<msup>
|
||
<mi><font style=vertical-align:inherit><font style=vertical-align:inherit>
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||
r
|
||
</font></font></mi>
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||
<mrow class=MJX-TeXAtom-ORD>
|
||
<mn><font style=vertical-align:inherit><font style=vertical-align:inherit>
|
||
2
|
||
</font></font></mn>
|
||
</mrow>
|
||
</msup>
|
||
<mo><font style=vertical-align:inherit><font style=vertical-align:inherit>
|
||
−</font></font>
|
||
</mo>
|
||
<msup>
|
||
<mi><font style=vertical-align:inherit><font style=vertical-align:inherit>
|
||
A
|
||
</font></font></mi>
|
||
<mrow class=MJX-TeXAtom-ORD>
|
||
<mn><font style=vertical-align:inherit><font style=vertical-align:inherit>
|
||
2
|
||
</font></font></mn>
|
||
</mrow>
|
||
</msup>
|
||
<msup>
|
||
<mi><font style=vertical-align:inherit><font style=vertical-align:inherit>
|
||
v
|
||
</font></font></mi>
|
||
<mrow class=MJX-TeXAtom-ORD>
|
||
<mn><font style=vertical-align:inherit><font style=vertical-align:inherit>
|
||
2
|
||
</font></font></mn>
|
||
</mrow>
|
||
</msup>
|
||
<msup>
|
||
<mi><font style=vertical-align:inherit><font style=vertical-align:inherit>
|
||
ϱ</font></font>
|
||
</mi>
|
||
<mrow class=MJX-TeXAtom-ORD>
|
||
<mn><font style=vertical-align:inherit><font style=vertical-align:inherit>
|
||
2
|
||
</font></font></mn>
|
||
</mrow>
|
||
</msup>
|
||
</mrow>
|
||
<mo><font style=vertical-align:inherit><font style=vertical-align:inherit>
|
||
)
|
||
</font></font></mo>
|
||
</mrow>
|
||
<mrow class=MJX-TeXAtom-ORD>
|
||
<mn><font style=vertical-align:inherit><font style=vertical-align:inherit>
|
||
3
|
||
</font></font></mn>
|
||
</mrow>
|
||
</msup>
|
||
</mfrac>
|
||
</mrow>
|
||
<mo><font style=vertical-align:inherit><font style=vertical-align:inherit>
|
||
,
|
||
</font></font></mo>
|
||
<mspace width=1em></mspace>
|
||
<mrow class=MJX-TeXAtom-ORD>
|
||
<mrow class=MJX-TeXAtom-ORD>
|
||
<mi mathvariant=fraktur><font style=vertical-align:inherit><font style=vertical-align:inherit>
|
||
Q
|
||
</font></font></mi>
|
||
</mrow>
|
||
</mrow>
|
||
<mo><font style=vertical-align:inherit><font style=vertical-align:inherit>
|
||
=
|
||
</font></font></mo>
|
||
<mo><font style=vertical-align:inherit><font style=vertical-align:inherit>
|
||
−</font></font>
|
||
</mo>
|
||
<mrow class=MJX-TeXAtom-ORD>
|
||
<mfrac>
|
||
<mrow>
|
||
<mrow class=MJX-TeXAtom-ORD>
|
||
<mrow class=MJX-TeXAtom-ORD>
|
||
<mi mathvariant=fraktur><font style=vertical-align:inherit><font style=vertical-align:inherit>
|
||
A
|
||
</font></font></mi>
|
||
</mrow>
|
||
</mrow>
|
||
<mi><font style=vertical-align:inherit><font style=vertical-align:inherit>
|
||
x
|
||
</font></font></mi>
|
||
<mi><font style=vertical-align:inherit><font style=vertical-align:inherit>
|
||
y
|
||
</font></font></mi>
|
||
</mrow>
|
||
<msup>
|
||
<mrow>
|
||
<mo><font style=vertical-align:inherit><font style=vertical-align:inherit>
|
||
(
|
||
</font></font></mo>
|
||
<mrow>
|
||
<msup>
|
||
<mi><font style=vertical-align:inherit><font style=vertical-align:inherit>
|
||
r
|
||
</font></font></mi>
|
||
<mrow class=MJX-TeXAtom-ORD>
|
||
<mn><font style=vertical-align:inherit><font style=vertical-align:inherit>
|
||
2
|
||
</font></font></mn>
|
||
</mrow>
|
||
</msup>
|
||
<mo><font style=vertical-align:inherit><font style=vertical-align:inherit>
|
||
−</font></font>
|
||
</mo>
|
||
<msup>
|
||
<mi><font style=vertical-align:inherit><font style=vertical-align:inherit>
|
||
A
|
||
</font></font></mi>
|
||
<mrow class=MJX-TeXAtom-ORD>
|
||
<mn><font style=vertical-align:inherit><font style=vertical-align:inherit>
|
||
2
|
||
</font></font></mn>
|
||
</mrow>
|
||
</msup>
|
||
<msup>
|
||
<mi><font style=vertical-align:inherit><font style=vertical-align:inherit>
|
||
v
|
||
</font></font></mi>
|
||
<mrow class=MJX-TeXAtom-ORD>
|
||
<mn><font style=vertical-align:inherit><font style=vertical-align:inherit>
|
||
2
|
||
</font></font></mn>
|
||
</mrow>
|
||
</msup>
|
||
<msup>
|
||
<mi><font style=vertical-align:inherit><font style=vertical-align:inherit>
|
||
ϱ</font></font>
|
||
</mi>
|
||
<mrow class=MJX-TeXAtom-ORD>
|
||
<mn><font style=vertical-align:inherit><font style=vertical-align:inherit>
|
||
2
|
||
</font></font></mn>
|
||
</mrow>
|
||
</msup>
|
||
</mrow>
|
||
<mo><font style=vertical-align:inherit><font style=vertical-align:inherit>
|
||
)
|
||
</font></font></mo>
|
||
</mrow>
|
||
<mrow class=MJX-TeXAtom-ORD>
|
||
<mn><font style=vertical-align:inherit><font style=vertical-align:inherit>
|
||
3
|
||
</font></font></mn>
|
||
</mrow>
|
||
</msup>
|
||
</mfrac>
|
||
</mrow>
|
||
<mo><font style=vertical-align:inherit><font style=vertical-align:inherit>
|
||
,
|
||
</font></font></mo>
|
||
</mstyle>
|
||
</mrow>
|
||
<annotation encoding=application/x-tex>
|
||
{\displaystyle {\mathfrak {P}}={\frac {{\mathfrak {A}}\varrho ^{2}}{\left(r^{2}-A^{2}v^{2}\varrho ^{2}\right)^{3}}},\quad {\mathfrak {Q}}=-{\frac {{\mathfrak {A}}xy}{\left(r^{2}-A^{2}v^{2}\varrho ^{2}\right)^{3}}},}
|
||
</annotation>
|
||
</semantics>
|
||
</math></span><img src=data:image/svg+xml;base64,<svg xmlns:xlink="http://www.w3.org/1999/xlink" width="47.611ex" height="7.009ex" style="vertical-align: -3.171ex;" viewBox="0 -1652.5 20499.2 3017.9" role="img" focusable="false" xmlns="http://www.w3.org/2000/svg" aria-labelledby="MathJax-SVG-1-Title">
<title id="MathJax-SVG-1-Title">{\displaystyle {\mathfrak {P}}={\frac {{\mathfrak {A}}\varrho ^{2}}{\left(r^{2}-A^{2}v^{2}\varrho ^{2}\right)^{3}}},\quad {\mathfrak {Q}}=-{\frac {{\mathfrak {A}}xy}{\left(r^{2}-A^{2}v^{2}\varrho ^{2}\right)^{3}}},}</title>
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<p><font style=vertical-align:inherit><font style=vertical-align:inherit>Let's sit again</font></font></p>
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<annotation encoding=application/x-tex>
|
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{\displaystyle {\mathfrak {S}}={\frac {\mathfrak {A}}{\left(r^{2}-A^{2}v^{2}\varrho ^{2}\right)^{3}}}.}
|
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|
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</semantics>
|
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</math></span><img src="data:image/svg+xml;base64,<svg xmlns:xlink="http://www.w3.org/1999/xlink" width="21.55ex" height="6.676ex" style="vertical-align: -3.171ex;" viewBox="0 -1508.9 9278.5 2874.4" role="img" focusable="false" xmlns="http://www.w3.org/2000/svg" aria-labelledby="MathJax-SVG-1-Title">
<title id="MathJax-SVG-1-Title">{\displaystyle {\mathfrak {S}}={\frac {\mathfrak {A}}{\left(r^{2}-A^{2}v^{2}\varrho ^{2}\right)^{3}}}.}</title>
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</svg>" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden=true style=vertical-align:-3.171ex;width:21.55ex;height:6.676ex alt="{\displaystyle {\mathfrak {S}}={\frac {\mathfrak {A}}{\left(r^{2}-A^{2}v^{2}\varrho ^{2}\right) ^{3}}}.}"></span>
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</center>
|
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<p><font style=vertical-align:inherit><font style=vertical-align:inherit>so we get from equations (1)</font></font></p>
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<tbody>
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<td valign=center style=text-align:left width=5%></td>
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<td align=center><span class=mwe-math-element><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style=display:none>
|
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<math xmlns=http://www.w3.org/1998/Math/MathML alttext="{\displaystyle 0={\frac {\partial P}{\partial \varrho }}+A\left(-v{\frac {\partial {\mathfrak {S}}}{\partial x}}x\varrho -\varrho v{\mathfrak {S}}+{\frac {\partial \psi }{\partial \varrho }}\left[3{\mathfrak {S}}+x{\frac {\partial {\mathfrak {S}}}{\partial x}}+\varrho {\frac {\partial {\mathfrak {S}}}{\partial \varrho }}\right]\right),}"><semantics>
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" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden=true style=vertical-align:-2.338ex;width:20.11ex;height:5.843ex alt="{\displaystyle 3{\mathfrak {S}}+x{\frac {\partial {\mathfrak {S}}}{\partial x}}+\varrho {\frac {\partial {\mathfrak {S}}} {\partial \varrho }},}"></span>
|
||
</center>
|
||
<p><font style=vertical-align:inherit><font style=vertical-align:inherit>This results in the elimination of </font></font><i><font style=vertical-align:inherit><font style=vertical-align:inherit>P</font></font></i></p>
|
||
<table width=100%>
|
||
<tbody>
|
||
<tr>
|
||
<td valign=center style=text-align:left width=5%><font style=vertical-align:inherit><font style=vertical-align:inherit>(3)</font></font></td>
|
||
<td align=center><span class=mwe-math-element><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style=display:none>
|
||
<math xmlns=http://www.w3.org/1998/Math/MathML alttext="{\displaystyle 0=v\varrho {\frac {\partial {\mathfrak {U}}}{\partial x}}+{\frac {\partial \psi }{\partial x}}{\frac {\partial {\mathfrak {U}}}{\partial \varrho }}-{\frac {\partial \psi }{\partial \varrho }}{\frac {\partial {\mathfrak {U}}}{\partial x}}.}"><semantics>
|
||
<mrow class=MJX-TeXAtom-ORD>
|
||
<mstyle displaystyle=true scriptlevel=0>
|
||
<mn><font style=vertical-align:inherit><font style=vertical-align:inherit>
|
||
0
|
||
</font></font></mn>
|
||
<mo><font style=vertical-align:inherit><font style=vertical-align:inherit>
|
||
=
|
||
</font></font></mo>
|
||
<mi><font style=vertical-align:inherit><font style=vertical-align:inherit>
|
||
v
|
||
</font></font></mi>
|
||
<mi><font style=vertical-align:inherit><font style=vertical-align:inherit>
|
||
ϱ</font></font>
|
||
</mi>
|
||
<mrow class=MJX-TeXAtom-ORD>
|
||
<mfrac>
|
||
<mrow>
|
||
<mi mathvariant=normal><font style=vertical-align:inherit><font style=vertical-align:inherit>
|
||
∂</font></font>
|
||
</mi>
|
||
<mrow class=MJX-TeXAtom-ORD>
|
||
<mrow class=MJX-TeXAtom-ORD>
|
||
<mi mathvariant=fraktur><font style=vertical-align:inherit><font style=vertical-align:inherit>
|
||
U
|
||
</font></font></mi>
|
||
</mrow>
|
||
</mrow>
|
||
</mrow>
|
||
<mrow>
|
||
<mi mathvariant=normal><font style=vertical-align:inherit><font style=vertical-align:inherit>
|
||
∂</font></font>
|
||
</mi>
|
||
<mi><font style=vertical-align:inherit><font style=vertical-align:inherit>
|
||
x
|
||
</font></font></mi>
|
||
</mrow>
|
||
</mfrac>
|
||
</mrow>
|
||
<mo><font style=vertical-align:inherit><font style=vertical-align:inherit>
|
||
+
|
||
</font></font></mo>
|
||
<mrow class=MJX-TeXAtom-ORD>
|
||
<mfrac>
|
||
<mrow>
|
||
<mi mathvariant=normal><font style=vertical-align:inherit><font style=vertical-align:inherit>
|
||
∂</font></font>
|
||
</mi>
|
||
<mi><font style=vertical-align:inherit><font style=vertical-align:inherit>
|
||
ψ</font></font>
|
||
</mi>
|
||
</mrow>
|
||
<mrow>
|
||
<mi mathvariant=normal><font style=vertical-align:inherit><font style=vertical-align:inherit>
|
||
∂</font></font>
|
||
</mi>
|
||
<mi><font style=vertical-align:inherit><font style=vertical-align:inherit>
|
||
x
|
||
</font></font></mi>
|
||
</mrow>
|
||
</mfrac>
|
||
</mrow>
|
||
<mrow class=MJX-TeXAtom-ORD>
|
||
<mfrac>
|
||
<mrow>
|
||
<mi mathvariant=normal><font style=vertical-align:inherit><font style=vertical-align:inherit>
|
||
∂</font></font>
|
||
</mi>
|
||
<mrow class=MJX-TeXAtom-ORD>
|
||
<mrow class=MJX-TeXAtom-ORD>
|
||
<mi mathvariant=fraktur><font style=vertical-align:inherit><font style=vertical-align:inherit>
|
||
U
|
||
</font></font></mi>
|
||
</mrow>
|
||
</mrow>
|
||
</mrow>
|
||
<mrow>
|
||
<mi mathvariant=normal><font style=vertical-align:inherit><font style=vertical-align:inherit>
|
||
∂</font></font>
|
||
</mi>
|
||
<mi><font style=vertical-align:inherit><font style=vertical-align:inherit>
|
||
ϱ</font></font>
|
||
</mi>
|
||
</mrow>
|
||
</mfrac>
|
||
</mrow>
|
||
<mo><font style=vertical-align:inherit><font style=vertical-align:inherit>
|
||
−</font></font>
|
||
</mo>
|
||
<mrow class=MJX-TeXAtom-ORD>
|
||
<mfrac>
|
||
<mrow>
|
||
<mi mathvariant=normal><font style=vertical-align:inherit><font style=vertical-align:inherit>
|
||
∂</font></font>
|
||
</mi>
|
||
<mi><font style=vertical-align:inherit><font style=vertical-align:inherit>
|
||
ψ</font></font>
|
||
</mi>
|
||
</mrow>
|
||
<mrow>
|
||
<mi mathvariant=normal><font style=vertical-align:inherit><font style=vertical-align:inherit>
|
||
∂</font></font>
|
||
</mi>
|
||
<mi><font style=vertical-align:inherit><font style=vertical-align:inherit>
|
||
ϱ</font></font>
|
||
</mi>
|
||
</mrow>
|
||
</mfrac>
|
||
</mrow>
|
||
<mrow class=MJX-TeXAtom-ORD>
|
||
<mfrac>
|
||
<mrow>
|
||
<mi mathvariant=normal><font style=vertical-align:inherit><font style=vertical-align:inherit>
|
||
∂</font></font>
|
||
</mi>
|
||
<mrow class=MJX-TeXAtom-ORD>
|
||
<mrow class=MJX-TeXAtom-ORD>
|
||
<mi mathvariant=fraktur><font style=vertical-align:inherit><font style=vertical-align:inherit>
|
||
U
|
||
</font></font></mi>
|
||
</mrow>
|
||
</mrow>
|
||
</mrow>
|
||
<mrow>
|
||
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<p><font style=vertical-align:inherit><font style=vertical-align:inherit>If the speed in the ether is to remain finite everywhere, it must</font></font></p>
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||
</math></span><img src="data:image/svg+xml;base64,PHN2ZyB4bWxuczp4bGluaz0iaHR0cDovL3d3dy53My5vcmcvMTk5OS94bGluayIgd2lkdGg9IjcuOTI4ZXgiIGhlaWdodD0iNS42NzZleCIgc3R5bGU9InZlcnRpY2FsLWFsaWduOiAtMi4wMDVleDsiIHZpZXdCb3g9IjAgLTE1ODAuNyAzNDEzLjYgMjQ0My44IiByb2xlPSJpbWciIGZvY3VzYWJsZT0iZmFsc2UiIHhtbG5zPSJodHRwOi8vd3d3LnczLm9yZy8yMDAwL3N2ZyIgYXJpYS1sYWJlbGxlZGJ5PSJNYXRoSmF4LVNWRy0xLVRpdGxlIj4KPHRpdGxlIGlkPSJNYXRoSmF4LVNWRy0xLVRpdGxlIj57XGRpc3BsYXlzdHlsZSB7XGZyYWMge1xwYXJ0aWFsIFxwc2kgfXtccGFydGlhbCB4fX09MH08L3RpdGxlPgo8ZGVmcyBhcmlhLWhpZGRlbj0idHJ1ZSI+CjxwYXRoIHN0cm9rZS13aWR0aD0iMSIgaWQ9IkUxLU1KTUFJTi0yMjAyIiBkPSJNMjAyIDUwOFExNzkgNTA4IDE2OSA1MjBUMTU4IDU0N1ExNTggNTU3IDE2NCA1NzdUMTg1IDYyNFQyMzAgNjc1VDMwMSA3MTBMMzMzIDcxNUgzNDVRMzc4IDcxNSAzODQgNzE0UTQ0NyA3MDMgNDg5IDY2MVQ1NDkgNTY4VDU2NiA0NTdRNTY2IDM2MiA1MTkgMjQwVDQwMiA1M1EzMjEgLTIyIDIyMyAtMjJRMTIzIC0yMiA3MyA1NlE0MiAxMDIgNDIgMTQ4VjE1OVE0MiAyNzYgMTI5IDM3MFQzMjIgNDY1UTM4MyA0NjUgNDE0IDQzNFQ0NTUgMzY3TDQ1OCAzNzhRNDc4IDQ2MSA0NzggNTE1UTQ3OCA2MDMgNDM3IDYzOVQzNDQgNjc2UTI2NiA2NzYgMjIzIDYxMlEyNjQgNjA2IDI2NCA1NzJRMjY0IDU0NyAyNDYgNTI4VDIwMiA1MDhaTTQzMCAzMDZRNDMwIDM3MiA0MDEgNDAwVDMzMyA0MjhRMjcwIDQyOCAyMjIgMzgyUTE5NyAzNTQgMTgzIDMyM1QxNTAgMjIxUTEzMiAxNDkgMTMyIDExNlExMzIgMjEgMjMyIDIxUTI0NCAyMSAyNTAgMjJRMzI3IDM1IDM3NCAxMTJRMzg5IDEzNyA0MDkgMTk2VDQzMCAzMDZaIj48L3BhdGg+CjxwYXRoIHN0cm9rZS13aWR0aD0iMSIgaWQ9IkUxLU1KTUFUSEktM0M4IiBkPSJNMTYxIDQ0MVEyMDIgNDQxIDIyNiA0MTdUMjUwIDM1OFEyNTAgMzM4IDIxOCAyNTJUMTg3IDEyN1ExOTAgODUgMjE0IDYxUTIzNSA0MyAyNTcgMzdRMjc1IDI5IDI4OCAyOUgyODlMMzcxIDM2MFE0NTUgNjkxIDQ1NiA2OTJRNDU5IDY5NCA0NzIgNjk0UTQ5MiA2OTQgNDkyIDY4N1E0OTIgNjc4IDQxMSAzNTZRMzI5IDI4IDMyOSAyN1QzMzUgMjZRNDIxIDI2IDQ5OCAxMTRUNTc2IDI3OFE1NzYgMzAyIDU2OCAzMTlUNTUwIDM0M1Q1MzIgMzYxVDUyNCAzODRRNTI0IDQwNSA1NDEgNDI0VDU4MyA0NDNRNjAyIDQ0MyA2MTggNDI1VDYzNCAzNjZRNjM0IDMzNyA2MjMgMjg4VDYwNSAyMjBRNTczIDEyNSA0OTIgNTdUMzI5IC0xMUgzMTlMMjk2IC0xMDRRMjcyIC0xOTggMjcyIC0xOTlRMjcwIC0yMDUgMjUyIC0yMDVIMjM5UTIzMyAtMTk5IDIzMyAtMTk3UTIzMyAtMTkyIDI1NiAtMTAyVDI3OSAtOVEyNzIgLTggMjY1IC04UTEwNiAxNCAxMDYgMTM5UTEwNiAxNzQgMTM5IDI2NFQxNzMgMzc5UTE3MyAzODAgMTczIDM4MVExNzMgMzkwIDE3MyAzOTNUMTY5IDQwMFQxNTggNDA0SDE1NFExMzEgNDA0IDExMiAzODVUODIgMzQ0VDY1IDMwMlQ1NyAyODBRNTUgMjc4IDQxIDI3OEgyN1EyMSAyODQgMjEgMjg3UTIxIDI5OSAzNCAzMzNUODIgNDA0VDE2MSA0NDFaIj48L3BhdGg+CjxwYXRoIHN0cm9rZS13aWR0aD0iMSIgaWQ9IkUxLU1KTUFUSEktNzgiIGQ9Ik01MiAyODlRNTkgMzMxIDEwNiAzODZUMjIyIDQ0MlEyNTcgNDQyIDI4NiA0MjRUMzI5IDM3OVEzNzEgNDQyIDQzMCA0NDJRNDY3IDQ0MiA0OTQgNDIwVDUyMiAzNjFRNTIyIDMzMiA1MDggMzE0VDQ4MSAyOTJUNDU4IDI4OFE0MzkgMjg4IDQyNyAyOTlUNDE1IDMyOFE0MTUgMzc0IDQ2NSAzOTFRNDU0IDQwNCA0MjUgNDA0UTQxMiA0MDQgNDA2IDQwMlEzNjggMzg2IDM1MCAzMzZRMjkwIDExNSAyOTAgNzhRMjkwIDUwIDMwNiAzOFQzNDEgMjZRMzc4IDI2IDQxNCA1OVQ0NjMgMTQwUTQ2NiAxNTAgNDY5IDE1MVQ0ODUgMTUzSDQ4OVE1MDQgMTUzIDUwNCAxNDVRNTA0IDE0NCA1MDIgMTM0UTQ4NiA3NyA0NDAgMzNUMzMzIC0xMVEyNjMgLTExIDIyNyA1MlExODYgLTEwIDEzMyAtMTBIMTI3UTc4IC0xMCA1NyAxNlQzNSA3MVEzNSAxMDMgNTQgMTIzVDk5IDE0M1ExNDIgMTQzIDE0MiAxMDFRMTQyIDgxIDEzMCA2NlQxMDcgNDZUOTQgNDFMOTEgNDBROTEgMzkgOTcgMzZUMTEzIDI5VDEzMiAyNlExNjggMjYgMTk0IDcxUTIwMyA4NyAyMTcgMTM5VDI0NSAyNDdUMjYxIDMxM1EyNjYgMzQwIDI2NiAzNTJRMjY2IDM4MCAyNTEgMzkyVDIxNyA0MDRRMTc3IDQwNCAxNDIgMzcyVDkzIDI5MFE5MSAyODEgODggMjgwVDcyIDI3OEg1OFE1MiAyODQgNTIgMjg5WiI+PC9wYXRoPgo8cGF0aCBzdHJva2Utd2lkdGg9IjEiIGlkPSJFMS1NSk1BSU4tM0QiIGQ9Ik01NiAzNDdRNTYgMzYwIDcwIDM2N0g3MDdRNzIyIDM1OSA3MjIgMzQ3UTcyMiAzMzYgNzA4IDMyOEwzOTAgMzI3SDcyUTU2IDMzMiA1NiAzNDdaTTU2IDE1M1E1NiAxNjggNzIgMTczSDcwOFE3MjIgMTYzIDcyMiAxNTNRNzIyIDE0MCA3MDcgMTMzSDcwUTU2IDE0MCA1NiAxNTNaIj48L3BhdGg+CjxwYXRoIHN0cm9rZS13aWR0aD0iMSIgaWQ9IkUxLU1KTUFJTi0zMCIgZD0iTTk2IDU4NVExNTIgNjY2IDI0OSA2NjZRMjk3IDY2NiAzNDUgNjQwVDQyMyA1NDhRNDYwIDQ2NSA0NjAgMzIwUTQ2MCAxNjUgNDE3IDgzUTM5NyA0MSAzNjIgMTZUMzAxIC0xNVQyNTAgLTIyUTIyNCAtMjIgMTk4IC0xNlQxMzcgMTZUODIgODNRMzkgMTY1IDM5IDMyMFEzOSA0OTQgOTYgNTg1Wk0zMjEgNTk3UTI5MSA2MjkgMjUwIDYyOVEyMDggNjI5IDE3OCA1OTdRMTUzIDU3MSAxNDUgNTI1VDEzNyAzMzNRMTM3IDE3NSAxNDUgMTI1VDE4MSA0NlEyMDkgMTYgMjUwIDE2UTI5MCAxNiAzMTggNDZRMzQ3IDc2IDM1NCAxMzBUMzYyIDMzM1EzNjIgNDc4IDM1NCA1MjRUMzIxIDU5N1oiPjwvcGF0aD4KPC9kZWZzPgo8ZyBzdHJva2U9ImN1cnJlbnRDb2xvciIgZmlsbD0iY3VycmVudENvbG9yIiBzdHJva2Utd2lkdGg9IjAiIHRyYW5zZm9ybT0ibWF0cml4KDEgMCAwIC0xIDAgMCkiIGFyaWEtaGlkZGVuPSJ0cnVlIj4KPGcgdHJhbnNmb3JtPSJ0cmFuc2xhdGUoMTIwLDApIj4KPHJlY3Qgc3Ryb2tlPSJub25lIiB3aWR0aD0iMTMzOSIgaGVpZ2h0PSI2MCIgeD0iMCIgeT0iMjIwIj48L3JlY3Q+CjxnIHRyYW5zZm9ybT0idHJhbnNsYXRlKDYwLDcyNSkiPgogPHVzZSB4bGluazpocmVmPSIjRTEtTUpNQUlOLTIyMDIiIHg9IjAiIHk9IjAiPjwvdXNlPgogPHVzZSB4bGluazpocmVmPSIjRTEtTUpNQVRISS0zQzgiIHg9IjU2NyIgeT0iMCI+PC91c2U+CjwvZz4KPGcgdHJhbnNmb3JtPSJ0cmFuc2xhdGUoOTksLTczNikiPgogPHVzZSB4bGluazpocmVmPSIjRTEtTUpNQUlOLTIyMDIiIHg9IjAiIHk9IjAiPjwvdXNlPgogPHVzZSB4bGluazpocmVmPSIjRTEtTUpNQVRISS03OCIgeD0iNTY3IiB5PSIwIj48L3VzZT4KPC9nPgo8L2c+CiA8dXNlIHhsaW5rOmhyZWY9IiNFMS1NSk1BSU4tM0QiIHg9IjE4NTYiIHk9IjAiPjwvdXNlPgogPHVzZSB4bGluazpocmVmPSIjRTEtTUpNQUlOLTMwIiB4PSIyOTEzIiB5PSIwIj48L3VzZT4KPC9nPgo8L3N2Zz4=" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden=true style=vertical-align:-2.005ex;width:7.928ex;height:5.676ex alt="{\displaystyle {\frac {\partial \psi }{\partial x}}=0}"></span>
|
||
</center>
|
||
<p><font style=vertical-align:inherit><font style=vertical-align:inherit>be, then we have</font></font></p>
|
||
<center>
|
||
<span class=mwe-math-element><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style=display:none>
|
||
<math xmlns=http://www.w3.org/1998/Math/MathML alttext="{\displaystyle v\varrho ={\frac {\partial \psi }{\partial \varrho }},\quad v=-\alpha .}"><semantics>
|
||
<mrow class=MJX-TeXAtom-ORD>
|
||
<mstyle displaystyle=true scriptlevel=0>
|
||
<mi><font style=vertical-align:inherit><font style=vertical-align:inherit>
|
||
v
|
||
</font></font></mi>
|
||
<mi><font style=vertical-align:inherit><font style=vertical-align:inherit>
|
||
ϱ</font></font>
|
||
</mi>
|
||
<mo><font style=vertical-align:inherit><font style=vertical-align:inherit>
|
||
=
|
||
</font></font></mo>
|
||
<mrow class=MJX-TeXAtom-ORD>
|
||
<mfrac>
|
||
<mrow>
|
||
<mi mathvariant=normal><font style=vertical-align:inherit><font style=vertical-align:inherit>
|
||
∂</font></font>
|
||
</mi>
|
||
<mi><font style=vertical-align:inherit><font style=vertical-align:inherit>
|
||
ψ</font></font>
|
||
</mi>
|
||
</mrow>
|
||
<mrow>
|
||
<mi mathvariant=normal><font style=vertical-align:inherit><font style=vertical-align:inherit>
|
||
∂</font></font>
|
||
</mi>
|
||
<mi><font style=vertical-align:inherit><font style=vertical-align:inherit>
|
||
ϱ</font></font>
|
||
</mi>
|
||
</mrow>
|
||
</mfrac>
|
||
</mrow>
|
||
<mo><font style=vertical-align:inherit><font style=vertical-align:inherit>
|
||
,
|
||
</font></font></mo>
|
||
<mspace width=1em></mspace>
|
||
<mi><font style=vertical-align:inherit><font style=vertical-align:inherit>
|
||
v
|
||
</font></font></mi>
|
||
<mo><font style=vertical-align:inherit><font style=vertical-align:inherit>
|
||
=
|
||
</font></font></mo>
|
||
<mo><font style=vertical-align:inherit><font style=vertical-align:inherit>
|
||
−</font></font>
|
||
</mo>
|
||
<mi><font style=vertical-align:inherit><font style=vertical-align:inherit>
|
||
α</font></font>
|
||
</mi>
|
||
<mo><font style=vertical-align:inherit><font style=vertical-align:inherit>
|
||
.
|
||
</font></font></mo>
|
||
</mstyle>
|
||
</mrow>
|
||
<annotation encoding=application/x-tex>
|
||
{\displaystyle v\varrho ={\frac {\partial \psi }{\partial \varrho }},\quad v=-\alpha .}
|
||
</annotation>
|
||
</semantics>
|
||
</math></span><img 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class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden=true style=vertical-align:-2.338ex;width:20.621ex;height:6.009ex alt="{\displaystyle v\varrho ={\frac {\partial \psi }{\partial \varrho }},\quad v=-\alpha .}"></span>
|
||
</center>
|
||
<p><font style=vertical-align:inherit><font style=vertical-align:inherit>So the ether flows with respect to the coordinate system moving with the speed </font></font><i><font style=vertical-align:inherit><font style=vertical-align:inherit>v</font></font></i><font style=vertical-align:inherit><font style=vertical-align:inherit> in the direction </font></font><i><font style=vertical-align:inherit><font style=vertical-align:inherit>x</font></font></i><font style=vertical-align:inherit><font style=vertical-align:inherit> at the same time as the charge with the same speed in the opposite direction, i.e. it rests with respect to a resting coordinate system. This result is remarkable because it shows that the movement of electric quanta is no reason for a movement of the ether, as </font></font><span style=letter-spacing:0.2em;left:0.1em><font style=vertical-align:inherit><font style=vertical-align:inherit>Helmholtz</font></font></span><font style=vertical-align:inherit><font style=vertical-align:inherit> assumes.</font></font></p>
|
||
<p><font style=vertical-align:inherit><font style=vertical-align:inherit>On the other hand, movements can occur if the aether has an inertia other than zero. I give the calculation for this case because it gives an idea of the magnitude of the density that would have to be assigned to the aether in a given case. Then, in addition to the terms of equations (1), there are also the components of the accelerations</font></font></p>
|
||
<center>
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||
</math></span><img src="data:image/svg+xml;base64,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" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden=true style=vertical-align:-2.005ex;width:20.222ex;height:5.509ex alt="{\displaystyle s{\frac {d\alpha }{dt}},\quad s{\frac {d\beta }{dt}},\quad s{\frac {d\gamma }{dt}}}"></span>
|
||
</center>
|
||
<p><font style=vertical-align:inherit><font style=vertical-align:inherit>to add, where </font></font><i><font style=vertical-align:inherit><font style=vertical-align:inherit>s</font></font></i><font style=vertical-align:inherit><font style=vertical-align:inherit> denotes the density of the ether and</font></font></p>
|
||
<table width=100%>
|
||
<tbody>
|
||
<tr>
|
||
<td valign=center style=text-align:left width=5%></td>
|
||
<td align=center><span class=mwe-math-element><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style=display:none>
|
||
<math xmlns=http://www.w3.org/1998/Math/MathML alttext="{\displaystyle {\begin{array}{l}{\frac {d\alpha }{dt}}={\frac {\partial \alpha }{\partial t}}+\alpha {\frac {d\alpha }{dx}}+\beta {\frac {\partial \alpha }{\partial y}}+\gamma {\frac {\partial \alpha }{\partial z}},\\\\{\frac {d\beta }{dt}}={\frac {\partial \beta }{\partial t}}+\alpha {\frac {d\beta }{dx}}+\beta {\frac {\partial \beta }{\partial y}}+\gamma {\frac {\partial \beta }{\partial z}},\\\\{\frac {d\gamma }{dt}}={\frac {\partial \gamma }{\partial t}}+\alpha {\frac {d\gamma }{dx}}+\beta {\frac {\partial \gamma }{\partial y}}+\gamma {\frac {\partial \gamma }{\partial z}}\end{array}}}"><semantics>
|
||
<mrow class=MJX-TeXAtom-ORD>
|
||
<mstyle displaystyle=true scriptlevel=0>
|
||
<mrow class=MJX-TeXAtom-ORD>
|
||
<mtable columnalign=left rowspacing=4pt columnspacing=1em>
|
||
<mtr>
|
||
<mtd>
|
||
<mrow class=MJX-TeXAtom-ORD>
|
||
<mfrac>
|
||
<mrow>
|
||
<mi><font style=vertical-align:inherit><font style=vertical-align:inherit>
|
||
d
|
||
</font></font></mi>
|
||
<mi><font style=vertical-align:inherit><font style=vertical-align:inherit>
|
||
α</font></font>
|
||
</mi>
|
||
</mrow>
|
||
<mrow>
|
||
<mi><font style=vertical-align:inherit><font style=vertical-align:inherit>
|
||
d
|
||
</font></font></mi>
|
||
<mi><font style=vertical-align:inherit><font style=vertical-align:inherit>
|
||
t
|
||
</font></font></mi>
|
||
</mrow>
|
||
</mfrac>
|
||
</mrow>
|
||
<mo><font style=vertical-align:inherit><font style=vertical-align:inherit>
|
||
=
|
||
</font></font></mo>
|
||
<mrow class=MJX-TeXAtom-ORD>
|
||
<mfrac>
|
||
<mrow>
|
||
<mi mathvariant=normal><font style=vertical-align:inherit><font style=vertical-align:inherit>
|
||
∂</font></font>
|
||
</mi>
|
||
<mi><font style=vertical-align:inherit><font style=vertical-align:inherit>
|
||
α</font></font>
|
||
</mi>
|
||
</mrow>
|
||
<mrow>
|
||
<mi mathvariant=normal><font style=vertical-align:inherit><font style=vertical-align:inherit>
|
||
∂</font></font>
|
||
</mi>
|
||
<mi><font style=vertical-align:inherit><font style=vertical-align:inherit>
|
||
t
|
||
</font></font></mi>
|
||
</mrow>
|
||
</mfrac>
|
||
</mrow>
|
||
<mo><font style=vertical-align:inherit><font style=vertical-align:inherit>
|
||
+
|
||
</font></font></mo>
|
||
<mi><font style=vertical-align:inherit><font style=vertical-align:inherit>
|
||
α</font></font>
|
||
</mi>
|
||
<mrow class=MJX-TeXAtom-ORD>
|
||
<mfrac>
|
||
<mrow>
|
||
<mi><font style=vertical-align:inherit><font style=vertical-align:inherit>
|
||
d
|
||
</font></font></mi>
|
||
<mi><font style=vertical-align:inherit><font style=vertical-align:inherit>
|
||
α</font></font>
|
||
</mi>
|
||
</mrow>
|
||
<mrow>
|
||
<mi><font style=vertical-align:inherit><font style=vertical-align:inherit>
|
||
d
|
||
</font></font></mi>
|
||
<mi><font style=vertical-align:inherit><font style=vertical-align:inherit>
|
||
x
|
||
</font></font></mi>
|
||
</mrow>
|
||
</mfrac>
|
||
</mrow>
|
||
<mo><font style=vertical-align:inherit><font style=vertical-align:inherit>
|
||
+
|
||
</font></font></mo>
|
||
<mi><font style=vertical-align:inherit><font style=vertical-align:inherit>
|
||
β</font></font>
|
||
</mi>
|
||
<mrow class=MJX-TeXAtom-ORD>
|
||
<mfrac>
|
||
<mrow>
|
||
<mi mathvariant=normal><font style=vertical-align:inherit><font style=vertical-align:inherit>
|
||
∂</font></font>
|
||
</mi>
|
||
<mi><font style=vertical-align:inherit><font style=vertical-align:inherit>
|
||
α</font></font>
|
||
</mi>
|
||
</mrow>
|
||
<mrow>
|
||
<mi mathvariant=normal><font style=vertical-align:inherit><font style=vertical-align:inherit>
|
||
∂</font></font>
|
||
</mi>
|
||
<mi><font style=vertical-align:inherit><font style=vertical-align:inherit>
|
||
y
|
||
</font></font></mi>
|
||
</mrow>
|
||
</mfrac>
|
||
</mrow>
|
||
<mo><font style=vertical-align:inherit><font style=vertical-align:inherit>
|
||
+
|
||
</font></font></mo>
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<title id="MathJax-SVG-1-Title">{\displaystyle {\begin{array}{l}{\frac {d\alpha }{dt}}={\frac {\partial \alpha }{\partial t}}+\alpha {\frac {d\alpha }{dx}}+\beta {\frac {\partial \alpha }{\partial y}}+\gamma {\frac {\partial \alpha }{\partial z}},\\\\{\frac {d\beta }{dt}}={\frac {\partial \beta }{\partial t}}+\alpha {\frac {d\beta }{dx}}+\beta {\frac {\partial \beta }{\partial y}}+\gamma {\frac {\partial \beta }{\partial z}},\\\\{\frac {d\gamma }{dt}}={\frac {\partial \gamma }{\partial t}}+\alpha {\frac {d\gamma }{dx}}+\beta {\frac {\partial \gamma }{\partial y}}+\gamma {\frac {\partial \gamma }{\partial z}}\end{array}}}</title>
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<p><font style=vertical-align:inherit><font style=vertical-align:inherit>are.</font></font><span><span class=pagenum id=vii title="Page: Translational movement of the light ether.djvu/7"> <span class="pagenumber noprint" style=color:#666666;display:inline;margin:0px;padding:0px>[<b><a href=https://de.wikisource.org/wiki/Seite%3ATranslatorische_Bewegung_des_Licht%C3%A4thers.djvu/7 title="Seite:Translatorische Bewegung des Lichtäthers.djvu/7">vii</a></b>]</span> </span></span><font style=vertical-align:inherit><font style=vertical-align:inherit>In the case just considered, the system is stationary with respect to the moving coordinate system. So it is</font></font></p>
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src=data:image/svg+xml;base64,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 class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden=true style=vertical-align:-2.005ex;width:21.649ex;height:5.676ex alt="{\displaystyle {\frac {\partial \alpha }{\partial t}}={\frac {\partial \beta }{\partial t}}={\frac {\partial \gamma }{\partial t}} =0.}"></span>
|
||
</center>
|
||
<p><font style=vertical-align:inherit><font style=vertical-align:inherit>Let's set the values of</font></font><span class=mwe-math-element><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style=display:none>
|
||
<math xmlns=http://www.w3.org/1998/Math/MathML alttext="{\displaystyle \alpha ,\beta ,\gamma }"><semantics>
|
||
<mrow class=MJX-TeXAtom-ORD>
|
||
<mstyle displaystyle=true scriptlevel=0>
|
||
<mi><font style=vertical-align:inherit><font style=vertical-align:inherit>
|
||
α</font></font>
|
||
</mi>
|
||
<mo><font style=vertical-align:inherit><font style=vertical-align:inherit>
|
||
,
|
||
</font></font></mo>
|
||
<mi><font style=vertical-align:inherit><font style=vertical-align:inherit>
|
||
β</font></font>
|
||
</mi>
|
||
<mo><font style=vertical-align:inherit><font style=vertical-align:inherit>
|
||
,
|
||
</font></font></mo>
|
||
<mi><font style=vertical-align:inherit><font style=vertical-align:inherit>
|
||
γ</font></font>
|
||
</mi>
|
||
</mstyle>
|
||
</mrow>
|
||
<annotation encoding=application/x-tex>
|
||
{\displaystyle \alpha ,\beta ,\gamma }
|
||
</annotation>
|
||
</semantics>
|
||
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1
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A
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∂</font></font>
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<mi><font style=vertical-align:inherit><font style=vertical-align:inherit>
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ψ</font></font>
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<mrow>
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<mi mathvariant=normal><font style=vertical-align:inherit><font style=vertical-align:inherit>
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∂</font></font>
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<mi><font style=vertical-align:inherit><font style=vertical-align:inherit>
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x
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<mi mathvariant=normal><font style=vertical-align:inherit><font style=vertical-align:inherit>
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∂</font></font>
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</mi>
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<mi><font style=vertical-align:inherit><font style=vertical-align:inherit>
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ϱ</font></font>
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[
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(
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<mrow class=MJX-TeXAtom-ORD>
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<mrow>
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<mi mathvariant=normal><font style=vertical-align:inherit><font style=vertical-align:inherit>
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∂</font></font>
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2
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<mi><font style=vertical-align:inherit><font style=vertical-align:inherit>
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ψ</font></font>
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<mrow>
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<mi mathvariant=normal><font style=vertical-align:inherit><font style=vertical-align:inherit>
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∂</font></font>
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</mi>
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<msup>
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<mi><font style=vertical-align:inherit><font style=vertical-align:inherit>
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ϱ</font></font>
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</mi>
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<mrow class=MJX-TeXAtom-ORD>
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<mn><font style=vertical-align:inherit><font style=vertical-align:inherit>
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2
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</mrow>
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<mo><font style=vertical-align:inherit><font style=vertical-align:inherit>
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−</font></font>
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1
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</font></font></mn>
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<mi><font style=vertical-align:inherit><font style=vertical-align:inherit>
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ϱ</font></font>
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</mi>
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</mfrac>
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</mrow>
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<mrow class=MJX-TeXAtom-ORD>
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<mfrac>
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<mrow>
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<mi mathvariant=normal><font style=vertical-align:inherit><font style=vertical-align:inherit>
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∂</font></font>
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</mi>
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<mi><font style=vertical-align:inherit><font style=vertical-align:inherit>
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ψ</font></font>
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</mi>
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</mrow>
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<mrow>
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<mi mathvariant=normal><font style=vertical-align:inherit><font style=vertical-align:inherit>
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∂</font></font>
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</mi>
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||
<mi><font style=vertical-align:inherit><font style=vertical-align:inherit>
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||
ϱ</font></font>
|
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</mi>
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</mrow>
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<mo><font style=vertical-align:inherit><font style=vertical-align:inherit>
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+
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<mi><font style=vertical-align:inherit><font style=vertical-align:inherit>
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ψ</font></font>
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</mrow>
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<mrow>
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<mi mathvariant=normal><font style=vertical-align:inherit><font style=vertical-align:inherit>
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∂</font></font>
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<msup>
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<mi><font style=vertical-align:inherit><font style=vertical-align:inherit>
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x
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2
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)
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<annotation encoding=application/x-tex>
|
||
{\displaystyle {\frac {1}{A}}{\frac {\partial \psi }{\partial x}}{\frac {\partial }{\partial \varrho }}\left[{\frac {1}{\varrho ^{2}}}\left({\frac {\partial ^{2}\psi }{\partial \varrho ^{2}}}-{\frac {1}{\varrho }}{\frac {\partial \psi }{\partial \varrho }}+{\frac {\partial ^{2}\psi }{\partial x^{2}}}\right)\right]}
|
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</annotation>
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</semantics>
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<title id="MathJax-SVG-1-Title">{\displaystyle {\frac {1}{A}}{\frac {\partial \psi }{\partial x}}{\frac {\partial }{\partial \varrho }}\left[{\frac {1}{\varrho ^{2}}}\left({\frac {\partial ^{2}\psi }{\partial \varrho ^{2}}}-{\frac {1}{\varrho }}{\frac {\partial \psi }{\partial \varrho }}+{\frac {\partial ^{2}\psi }{\partial x^{2}}}\right)\right]}</title>
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{\displaystyle {\begin{array}{lr}{\frac {1}{\varrho }}{\frac {\partial \psi }{\partial \varrho }}=&v+{\frac {1}{\varrho }}{\frac {\partial \psi _{1}}{\partial \varrho }},\\\\{\frac {1}{\varrho }}{\frac {\partial \psi }{\partial x}}=&{\frac {1}{\varrho }}{\frac {\partial \psi _{1}}{\partial x}},\end{array}}}
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<p><font style=vertical-align:inherit><font style=vertical-align:inherit>is. It is</font></font></p>
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<p><font style=vertical-align:inherit><font style=vertical-align:inherit>To integrate the differential equation, we set</font></font></p>
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ϱ</font></font>
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<mi><font style=vertical-align:inherit><font style=vertical-align:inherit>
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φ</font></font>
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<mo><font style=vertical-align:inherit><font style=vertical-align:inherit>
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.
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{\displaystyle \psi _{1}=\varrho \varphi .}
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||
</center>
|
||
<p><font style=vertical-align:inherit><font style=vertical-align:inherit>Then it will be</font></font></p>
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<span class=mwe-math-element><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style=display:none>
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<math xmlns=http://www.w3.org/1998/Math/MathML alttext="{\displaystyle {\frac {\partial ^{2}\varphi }{\partial \varrho ^{2}}}+{\frac {1}{\varrho }}{\frac {\partial \varphi }{\partial \varrho }}-{\frac {1}{\varrho ^{2}}}\varphi +{\frac {\partial ^{2}\varphi }{\partial x^{2}}}=-{\frac {\varrho {\mathfrak {U}}A}{s}}}"><semantics>
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∂</font></font>
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2
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<mrow>
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<mi mathvariant=normal><font style=vertical-align:inherit><font style=vertical-align:inherit>
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<mi><font style=vertical-align:inherit><font style=vertical-align:inherit>
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<mi><font style=vertical-align:inherit><font style=vertical-align:inherit>
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{\displaystyle {\frac {\partial ^{2}\varphi }{\partial \varrho ^{2}}}+{\frac {1}{\varrho }}{\frac {\partial \varphi }{\partial \varrho }}-{\frac {1}{\varrho ^{2}}}\varphi +{\frac {\partial ^{2}\varphi }{\partial x^{2}}}=-{\frac {\varrho {\mathfrak {U}}A}{s}}}
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<title id="MathJax-SVG-1-Title">{\displaystyle {\frac {\partial ^{2}\varphi }{\partial \varrho ^{2}}}+{\frac {1}{\varrho }}{\frac {\partial \varphi }{\partial \varrho }}-{\frac {1}{\varrho ^{2}}}\varphi +{\frac {\partial ^{2}\varphi }{\partial x^{2}}}=-{\frac {\varrho {\mathfrak {U}}A}{s}}}</title>
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<p><font style=vertical-align:inherit><font style=vertical-align:inherit>We first consider the differential equation</font></font></p>
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{\displaystyle {\frac {\partial ^{2}\varphi _{1}}{\partial \varrho ^{2}}}+{\frac {1}{\varrho }}{\frac {\partial \varphi _{1}}{\partial \varrho }}+{\frac {1}{\varrho ^{2}}}{\frac {\partial ^{2}\varphi _{1}}{\partial \vartheta ^{2}}}+{\frac {\partial ^{2}\varphi _{1}}{\partial x^{2}}}=-{\frac {\varrho {\mathfrak {U}}A}{s}}\sin \vartheta .}
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<title id="MathJax-SVG-1-Title">{\displaystyle {\frac {\partial ^{2}\varphi _{1}}{\partial \varrho ^{2}}}+{\frac {1}{\varrho }}{\frac {\partial \varphi _{1}}{\partial \varrho }}+{\frac {1}{\varrho ^{2}}}{\frac {\partial ^{2}\varphi _{1}}{\partial \vartheta ^{2}}}+{\frac {\partial ^{2}\varphi _{1}}{\partial x^{2}}}=-{\frac {\varrho {\mathfrak {U}}A}{s}}\sin \vartheta .}</title>
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</center>
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<p><font style=vertical-align:inherit><font style=vertical-align:inherit>whose integral is</font></font></p>
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φ</font></font>
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<annotation encoding=application/x-tex>
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{\displaystyle \varphi _{1}={\frac {A}{4\pi s}}\int \int \int {\frac {d\varrho '\ d\vartheta '\ dx'\ \varrho '^{2}\ {\mathfrak {U}}'\sin \vartheta '}{\sqrt {\left(x-x'\right)^{2}+\varrho ^{2}+\varrho '^{2}-2\varrho \varrho '\cos(\vartheta -\vartheta ')}}},}
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<title id="MathJax-SVG-1-Title">{\displaystyle \varphi _{1}={\frac {A}{4\pi s}}\int \int \int {\frac {d\varrho '\ d\vartheta '\ dx'\ \varrho '^{2}\ {\mathfrak {U}}'\sin \vartheta '}{\sqrt {\left(x-x'\right)^{2}+\varrho ^{2}+\varrho '^{2}-2\varrho \varrho '\cos(\vartheta -\vartheta ')}}},}</title>
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{\displaystyle S={\frac {A}{4\pi s}}\int \int \varrho '^{2}d\varrho 'dx'{\mathfrak {U}}'R,}
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<title id="MathJax-SVG-1-Title">{\displaystyle S={\frac {A}{4\pi s}}\int \int \varrho '^{2}d\varrho 'dx'{\mathfrak {U}}'R,}</title>
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{\displaystyle \varkappa ^{2}={\frac {4\varrho '\varrho }{(z'-z)^{2}+(\varrho +\varrho ')^{2}}},\quad K=\int \limits _{0}^{\frac {\pi }{2}}{\frac {d\varphi }{\sqrt {1-\varkappa ^{2}\sin ^{2}\varphi }}},\quad E=\int \limits _{0}^{\frac {\pi }{2}}{\frac {d\varphi }{\sqrt {1-\varkappa ^{2}\sin ^{3}\varphi }}}.}
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<title id="MathJax-SVG-1-Title">{\displaystyle \varkappa ^{2}={\frac {4\varrho '\varrho }{(z'-z)^{2}+(\varrho +\varrho ')^{2}}},\quad K=\int \limits _{0}^{\frac {\pi }{2}}{\frac {d\varphi }{\sqrt {1-\varkappa ^{2}\sin ^{2}\varphi }}},\quad E=\int \limits _{0}^{\frac {\pi }{2}}{\frac {d\varphi }{\sqrt {1-\varkappa ^{2}\sin ^{3}\varphi }}}.}</title>
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</svg>" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden=true style=vertical-align:-4.671ex;width:82.428ex;height:11.009ex alt="{\displaystyle \varkappa ^{2}={\frac {4\varrho '\varrho }{(z'-z)^{2}+(\varrho +\varrho ')^{2}}},\quad K=\int \limits _{0}^{\frac {\pi }{2}}{\frac {d\varphi }{\sqrt {1-\varkappa ^{2}\sin ^{2}\varphi }}},\quad E=\int \limits _{0}^{\frac {\pi }{2}}{\frac {d\varphi }{\sqrt {1-\varkappa ^{2}\sin ^{3}\varphi }}}.}"></span>
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<p><font style=vertical-align:inherit><font style=vertical-align:inherit>Then </font></font><i><font style=vertical-align:inherit><font style=vertical-align:inherit>S</font></font></i><font style=vertical-align:inherit><font style=vertical-align:inherit> satisfies the differential equation</font></font></p>
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<mi><font style=vertical-align:inherit><font style=vertical-align:inherit>
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S
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<mi><font style=vertical-align:inherit><font style=vertical-align:inherit>
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<mi><font style=vertical-align:inherit><font style=vertical-align:inherit>
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<mo><font style=vertical-align:inherit><font style=vertical-align:inherit>
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−</font></font>
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<mi><font style=vertical-align:inherit><font style=vertical-align:inherit>
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S
|
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<mo><font style=vertical-align:inherit><font style=vertical-align:inherit>
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2
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<mi><font style=vertical-align:inherit><font style=vertical-align:inherit>
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S
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</font></font></mi>
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<p><font style=vertical-align:inherit><font style=vertical-align:inherit>and so it is</font></font><span class=mwe-math-element><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style=display:none>
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<p><font style=vertical-align:inherit><font style=vertical-align:inherit>These are the same expressions that give the velocities of the circular vortex rings in a liquid, where the </font></font><i><font style=vertical-align:inherit><font style=vertical-align:inherit>x</font></font></i><font style=vertical-align:inherit><font style=vertical-align:inherit> -axis is the axis of the vortex rings when the rotation speed of the liquid particles is about the circular rotation axis</font></font></p>
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<p><font style=vertical-align:inherit><font style=vertical-align:inherit>away. With constant </font></font><i><font style=vertical-align:inherit><font style=vertical-align:inherit>e</font></font></i><font style=vertical-align:inherit><font style=vertical-align:inherit> and </font></font><i><font style=vertical-align:inherit><font style=vertical-align:inherit>s</font></font></i><font style=vertical-align:inherit><font style=vertical-align:inherit> it has a maximum for</font></font></p>
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src="data:image/svg+xml;base64,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" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden=true style=vertical-align:-2.838ex;width:9.904ex;height:6.176ex alt="{\displaystyle v={\frac {1}{{\sqrt {3}}A}}}"></span>
|
||
</center>
|
||
<p><font style=vertical-align:inherit><font style=vertical-align:inherit>and is the same</font></font></p>
|
||
<center>
|
||
<span class=mwe-math-element><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style=display:none>
|
||
<math xmlns=http://www.w3.org/1998/Math/MathML alttext="{\displaystyle {\frac {e^{2}A}{{\sqrt {3}}s}}.}"><semantics>
|
||
<mrow class=MJX-TeXAtom-ORD>
|
||
<mstyle displaystyle=true scriptlevel=0>
|
||
<mrow class=MJX-TeXAtom-ORD>
|
||
<mfrac>
|
||
<mrow>
|
||
<msup>
|
||
<mi><font style=vertical-align:inherit><font style=vertical-align:inherit>
|
||
e
|
||
</font></font></mi>
|
||
<mrow class=MJX-TeXAtom-ORD>
|
||
<mn><font style=vertical-align:inherit><font style=vertical-align:inherit>
|
||
2
|
||
</font></font></mn>
|
||
</mrow>
|
||
</msup>
|
||
<mi><font style=vertical-align:inherit><font style=vertical-align:inherit>
|
||
A
|
||
</font></font></mi>
|
||
</mrow>
|
||
<mrow>
|
||
<mrow class=MJX-TeXAtom-ORD>
|
||
<msqrt>
|
||
<mn><font style=vertical-align:inherit><font style=vertical-align:inherit>
|
||
3
|
||
</font></font></mn>
|
||
</msqrt>
|
||
</mrow>
|
||
<mi><font style=vertical-align:inherit><font style=vertical-align:inherit>
|
||
s
|
||
</font></font></mi>
|
||
</mrow>
|
||
</mfrac>
|
||
</mrow>
|
||
<mo><font style=vertical-align:inherit><font style=vertical-align:inherit>
|
||
.
|
||
</font></font></mo>
|
||
</mstyle>
|
||
</mrow>
|
||
<annotation encoding=application/x-tex>
|
||
{\displaystyle {\frac {e^{2}A}{{\sqrt {3}}s}}.}
|
||
</annotation>
|
||
</semantics>
|
||
</math></span><img src="data:image/svg+xml;base64,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" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden=true style=vertical-align:-2.838ex;width:5.672ex;height:6.676ex alt="{\displaystyle {\frac {e^{2}A}{{\sqrt {3}}s}}.}"></span>
|
||
</center>
|
||
<p><font style=vertical-align:inherit><font style=vertical-align:inherit>In cathode rays we have electrical charges that fly through space at almost as high a speed.</font></font></p>
|
||
<p><font style=vertical-align:inherit><font style=vertical-align:inherit>Let's assume there would be</font></font><span class=mwe-math-element><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style=display:none>
|
||
<math xmlns=http://www.w3.org/1998/Math/MathML alttext="{\displaystyle 6\cdot 10^{4}}"><semantics>
|
||
<mrow class=MJX-TeXAtom-ORD>
|
||
<mstyle displaystyle=true scriptlevel=0>
|
||
<mn><font style=vertical-align:inherit><font style=vertical-align:inherit>
|
||
6
|
||
</font></font></mn>
|
||
<mo><font style=vertical-align:inherit><font style=vertical-align:inherit>
|
||
⋅</font></font>
|
||
</mo>
|
||
<msup>
|
||
<mn><font style=vertical-align:inherit><font style=vertical-align:inherit>
|
||
10
|
||
</font></font></mn>
|
||
<mrow class=MJX-TeXAtom-ORD>
|
||
<mn><font style=vertical-align:inherit><font style=vertical-align:inherit>
|
||
4
|
||
</font></font></mn>
|
||
</mrow>
|
||
</msup>
|
||
</mstyle>
|
||
</mrow>
|
||
<annotation encoding=application/x-tex>
|
||
{\displaystyle 6\cdot 10^{4}}
|
||
</annotation>
|
||
</semantics>
|
||
</math></span><img src="data:image/svg+xml;base64,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" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden=true style=vertical-align:-0.338ex;width:6.221ex;height:2.676ex alt="{\displaystyle 6\cdot 10^{4}}"></span><font style=vertical-align:inherit><font style=vertical-align:inherit>electrostatic units are transported per second and if we take the speed to be a third of the speed of light, then a tube 50 cm long would constantly have a charge of</font></font><span class=mwe-math-element><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style=display:none>
|
||
<math xmlns=http://www.w3.org/1998/Math/MathML alttext="{\displaystyle 3\cdot 10^{-4}}"><semantics>
|
||
<mrow class=MJX-TeXAtom-ORD>
|
||
<mstyle displaystyle=true scriptlevel=0>
|
||
<mn><font style=vertical-align:inherit><font style=vertical-align:inherit>
|
||
3
|
||
</font></font></mn>
|
||
<mo><font style=vertical-align:inherit><font style=vertical-align:inherit>
|
||
⋅</font></font>
|
||
</mo>
|
||
<msup>
|
||
<mn><font style=vertical-align:inherit><font style=vertical-align:inherit>
|
||
10
|
||
</font></font></mn>
|
||
<mrow class=MJX-TeXAtom-ORD>
|
||
<mo><font style=vertical-align:inherit><font style=vertical-align:inherit>
|
||
−</font></font>
|
||
</mo>
|
||
<mn><font style=vertical-align:inherit><font style=vertical-align:inherit>
|
||
4
|
||
</font></font></mn>
|
||
</mrow>
|
||
</msup>
|
||
</mstyle>
|
||
</mrow>
|
||
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|
||
{\displaystyle 3\cdot 10^{-4}}
|
||
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0
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{\displaystyle x=0}
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</math></span><img src="data:image/svg+xml;base64,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" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden=true style=vertical-align:-0.671ex;width:9.335ex;height:2.509ex alt="{\displaystyle \varrho =1{\rm {mm}}}"></span><font style=vertical-align:inherit><font style=vertical-align:inherit>nearly</font></font></p>
|
||
<center>
|
||
<span class=mwe-math-element><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style=display:none>
|
||
<math xmlns=http://www.w3.org/1998/Math/MathML alttext="{\displaystyle {\frac {1}{2}}10^{-13}{\frac {1}{s}}.}"><semantics>
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|
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<mfrac>
|
||
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|
||
1
|
||
</font></font></mn>
|
||
<mn><font style=vertical-align:inherit><font style=vertical-align:inherit>
|
||
2
|
||
</font></font></mn>
|
||
</mfrac>
|
||
</mrow>
|
||
<msup>
|
||
<mn><font style=vertical-align:inherit><font style=vertical-align:inherit>
|
||
10
|
||
</font></font></mn>
|
||
<mrow class=MJX-TeXAtom-ORD>
|
||
<mo><font style=vertical-align:inherit><font style=vertical-align:inherit>
|
||
−</font></font>
|
||
</mo>
|
||
<mn><font style=vertical-align:inherit><font style=vertical-align:inherit>
|
||
13
|
||
</font></font></mn>
|
||
</mrow>
|
||
</msup>
|
||
<mrow class=MJX-TeXAtom-ORD>
|
||
<mfrac>
|
||
<mn><font style=vertical-align:inherit><font style=vertical-align:inherit>
|
||
1
|
||
</font></font></mn>
|
||
<mi><font style=vertical-align:inherit><font style=vertical-align:inherit>
|
||
s
|
||
</font></font></mi>
|
||
</mfrac>
|
||
</mrow>
|
||
<mo><font style=vertical-align:inherit><font style=vertical-align:inherit>
|
||
.
|
||
</font></font></mo>
|
||
</mstyle>
|
||
</mrow>
|
||
<annotation encoding=application/x-tex>
|
||
{\displaystyle {\frac {1}{2}}10^{-13}{\frac {1}{s}}.}
|
||
</annotation>
|
||
</semantics>
|
||
</math></span><img src="data:image/svg+xml;base64,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|
||
</center>
|
||
<p><font style=vertical-align:inherit><font style=vertical-align:inherit>Outside the tube, noticeable movements would only occur if the ether density was extremely low.</font></font><span><span class=pagenum id=ix title="Page: Translational movement of the light ether.djvu/9"> <span class="pagenumber noprint" style=color:#666666;display:inline;margin:0px;padding:0px>[<b><a href=https://de.wikisource.org/wiki/Seite%3ATranslatorische_Bewegung_des_Licht%C3%A4thers.djvu/9 title="Seite:Translatorische Bewegung des Lichtäthers.djvu/9">ix</a></b>]</span> </span></span><font style=vertical-align:inherit><font style=vertical-align:inherit>Nothing definite can be said about what happened in the immediate vicinity of the cargo. </font></font><sup id=cite_ref-1 class=reference><a href=#cite_note-1><font style=vertical-align:inherit><font style=vertical-align:inherit>[1]</font></font></a></sup></p>
|
||
<div style=text-align:center>
|
||
<h3><span class=mw-headline id=Reflexion_an_bewegten_durchsichtigen_Medien.><font style=vertical-align:inherit><font style=vertical-align:inherit>Reflection on moving transparent media.</font></font></span></h3>
|
||
</div>
|
||
<p><font style=vertical-align:inherit><font style=vertical-align:inherit>An example where the tensions in the aether would cause movement is the reflection of electromagnetic plane waves at the boundary of moving insulators. Let us denote the angle of incidence by</font></font><span class=mwe-math-element><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style=display:none>
|
||
<math xmlns=http://www.w3.org/1998/Math/MathML alttext="{\displaystyle \varphi }"><semantics>
|
||
<mrow class=MJX-TeXAtom-ORD>
|
||
<mstyle displaystyle=true scriptlevel=0>
|
||
<mi><font style=vertical-align:inherit><font style=vertical-align:inherit>
|
||
φ</font></font>
|
||
</mi>
|
||
</mstyle>
|
||
</mrow>
|
||
<annotation encoding=application/x-tex>
|
||
{\displaystyle \varphi }
|
||
</annotation>
|
||
</semantics>
|
||
</math></span><img src=data:image/svg+xml;base64,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 class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden=true style=vertical-align:-0.838ex;width:1.52ex;height:2.176ex alt="{\displaystyle \varphi }"></span><font style=vertical-align:inherit><font style=vertical-align:inherit>, with the index </font></font><i><font style=vertical-align:inherit><font style=vertical-align:inherit>e</font></font></i><font style=vertical-align:inherit><font style=vertical-align:inherit> the incident components, with </font></font><i><font style=vertical-align:inherit><font style=vertical-align:inherit>r</font></font></i><font style=vertical-align:inherit><font style=vertical-align:inherit> the reflected components, is according to the known laws</font></font></p>
|
||
<table width=100%>
|
||
<tbody>
|
||
<tr>
|
||
<td valign=center style=text-align:left width=5%></td>
|
||
<td align=center><span class=mwe-math-element><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style=display:none>
|
||
<math xmlns=http://www.w3.org/1998/Math/MathML alttext="{\displaystyle Y_{e}=\sin \left({\frac {x\ \sin \varphi +z_{1}\cos \varphi }{\lambda }}-{\frac {t}{T}}\right)2\pi ,}"><semantics>
|
||
<mrow class=MJX-TeXAtom-ORD>
|
||
<mstyle displaystyle=true scriptlevel=0>
|
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<msub>
|
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<mi><font style=vertical-align:inherit><font style=vertical-align:inherit>
|
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Y
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</font></font></mi>
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</mrow>
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<mo><font style=vertical-align:inherit><font style=vertical-align:inherit>
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=
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</font></font></mo>
|
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<mi><font style=vertical-align:inherit><font style=vertical-align:inherit>
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sin
|
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</font></font></mi>
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<mrow>
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<mfrac>
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<mrow>
|
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<mi><font style=vertical-align:inherit><font style=vertical-align:inherit>
|
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x
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</font></font></mi>
|
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<mtext>
|
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</mtext>
|
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<mi><font style=vertical-align:inherit><font style=vertical-align:inherit>
|
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sin
|
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</font></font></mi>
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<mo><font style=vertical-align:inherit><font style=vertical-align:inherit>
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</mi>
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<mo><font style=vertical-align:inherit><font style=vertical-align:inherit>
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+
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|
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<msub>
|
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<mi><font style=vertical-align:inherit><font style=vertical-align:inherit>
|
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e.g
|
||
</font></font></mi>
|
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<mrow class=MJX-TeXAtom-ORD>
|
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<mn><font style=vertical-align:inherit><font style=vertical-align:inherit>
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1
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</font></font></mn>
|
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</mrow>
|
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</msub>
|
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<mi><font style=vertical-align:inherit><font style=vertical-align:inherit>
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cos
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</font></font></mi>
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<mo><font style=vertical-align:inherit><font style=vertical-align:inherit>
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<mi><font style=vertical-align:inherit><font style=vertical-align:inherit>
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</mi>
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</mrow>
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<mi><font style=vertical-align:inherit><font style=vertical-align:inherit>
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</mi>
|
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</mfrac>
|
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</mrow>
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<mo><font style=vertical-align:inherit><font style=vertical-align:inherit>
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−</font></font>
|
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</mo>
|
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<mrow class=MJX-TeXAtom-ORD>
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<mfrac>
|
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<mi><font style=vertical-align:inherit><font style=vertical-align:inherit>
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t
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</font></font></mi>
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</mi>
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,
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|
||
</mstyle>
|
||
</mrow>
|
||
<annotation encoding=application/x-tex>
|
||
{\displaystyle Y_{e}=\sin \left({\frac {x\ \sin \varphi +z_{1}\cos \varphi }{\lambda }}-{\frac {t}{T}}\right)2\pi ,}
|
||
</annotation>
|
||
</semantics>
|
||
</math></span><img src="data:image/svg+xml;base64,<svg xmlns:xlink="http://www.w3.org/1999/xlink" width="38.456ex" height="6.176ex" style="vertical-align: -2.505ex;" viewBox="0 -1580.7 16557.4 2659.1" role="img" focusable="false" xmlns="http://www.w3.org/2000/svg" aria-labelledby="MathJax-SVG-1-Title">
<title id="MathJax-SVG-1-Title">{\displaystyle Y_{e}=\sin \left({\frac {x\ \sin \varphi +z_{1}\cos \varphi }{\lambda }}-{\frac {t}{T}}\right)2\pi ,}</title>
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<p><font style=vertical-align:inherit><font style=vertical-align:inherit>If we move the plate with the speed </font></font><i><font style=vertical-align:inherit><font style=vertical-align:inherit>v</font></font></i><font style=vertical-align:inherit><font style=vertical-align:inherit> in the direction </font></font><i><font style=vertical-align:inherit><font style=vertical-align:inherit>z</font></font></i><font style=vertical-align:inherit><font style=vertical-align:inherit> , we have to use </font></font><span style=letter-spacing:0.2em;left:0.1em><font style=vertical-align:inherit><font style=vertical-align:inherit>Lorentz</font></font></span><font style=vertical-align:inherit><font style=vertical-align:inherit> for the reflected waves</font></font></p>
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<annotation encoding=application/x-tex>
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{\displaystyle Y_{r}=R\ \sin \left({\frac {x\ \sin \varphi -z_{1}\cos \varphi }{\lambda }}+{\frac {A^{2}vz_{1}}{T}}-{\frac {t}{T}}\right)2\pi ,}
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<title id="MathJax-SVG-1-Title">{\displaystyle Y_{r}=R\ \sin \left({\frac {x\ \sin \varphi -z_{1}\cos \varphi }{\lambda }}+{\frac {A^{2}vz_{1}}{T}}-{\frac {t}{T}}\right)2\pi ,}</title>
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<title id="MathJax-SVG-1-Title">{\displaystyle L_{r}=-R\ \sin \left({\frac {x\ \sin \varphi -z_{1}\cos \varphi }{\lambda }}+{\frac {A^{2}vz_{1}}{T}}-{\frac {t}{T}}\right)2\pi ,}</title>
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{\displaystyle N_{r}=-R\ \sin \left({\frac {x\ \sin \varphi -z_{1}\cos \varphi }{\lambda }}+{\frac {A^{2}vz_{1}}{T}}-{\frac {t}{T}}\right)2\pi }
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<title id="MathJax-SVG-1-Title">{\displaystyle N_{r}=-R\ \sin \left({\frac {x\ \sin \varphi -z_{1}\cos \varphi }{\lambda }}+{\frac {A^{2}vz_{1}}{T}}-{\frac {t}{T}}\right)2\pi }</title>
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<p><font style=vertical-align:inherit><font style=vertical-align:inherit>based on a coordinate system that moves with the plate. If we relate everything to a fixed coordinate system, we have</font></font><span class=mwe-math-element><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style=display:none>
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{\displaystyle z_{1}=z-vt}
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The factor </font></font><i><font style=vertical-align:inherit><font style=vertical-align:inherit>R</font></font></i><font style=vertical-align:inherit><font style=vertical-align:inherit> is not exactly the same as when the system is at rest. The electromagnetic pressure causes work to be done or consumed and this reduces or increases the energy of the radiation. However, the boundary conditions can only be fulfilled if one assumes that this change in energy is distributed between reflected and refracted rays as if the incoming wave already carried with it the energy increased or decreased in proportion to this work.</font></font><span><span class=pagenum id=x title="Page: Translational movement of the light ether.djvu/10"> <span class="pagenumber noprint" style=color:#666666;display:inline;margin:0px;padding:0px>[<b><a href=https://de.wikisource.org/wiki/Seite%3ATranslatorische_Bewegung_des_Licht%C3%A4thers.djvu/10 title="Seite:Translatorische Bewegung des Lichtäthers.djvu/10">x</a></b>]</span> </span></span><font style=vertical-align:inherit><font style=vertical-align:inherit>From the given values it follows if we </font><font style=vertical-align:inherit>neglect quantities of the order </font></font><i><font style=vertical-align:inherit><font style=vertical-align:inherit>Av</font></font></i><font style=vertical-align:inherit></font></p>
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<annotation encoding=application/x-tex>
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{\displaystyle {\mathfrak {T}}={\frac {\partial {\mathfrak {R}}}{\partial x}}-{\frac {\partial {\mathfrak {P}}}{\partial z}}={\frac {2\sin 2\varphi }{\lambda }}R\left\{{\frac {1}{\lambda }}\sin \left({\frac {2z\ \cos \varphi }{\lambda }}-{\frac {2vt\ \cos \varphi }{\lambda }}\right)2\pi \right\},}
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<title id="MathJax-SVG-1-Title">{\displaystyle {\mathfrak {T}}={\frac {\partial {\mathfrak {R}}}{\partial x}}-{\frac {\partial {\mathfrak {P}}}{\partial z}}={\frac {2\sin 2\varphi }{\lambda }}R\left\{{\frac {1}{\lambda }}\sin \left({\frac {2z\ \cos \varphi }{\lambda }}-{\frac {2vt\ \cos \varphi }{\lambda }}\right)2\pi \right\},}</title>
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 <use xlink:href="#E1-MJMATHI-3C6" x="3270" y="0"></use>
</g>
 <use xlink:href="#E1-MJMATHI-3BB" x="1730" y="-715"></use>
</g>
</g>
</g>
 <use xlink:href="#E1-MJSZ3-29" x="10150" y="-1"></use>
</g>
 <use xlink:href="#E1-MJMAIN-32" x="13226" y="0"></use>
 <use xlink:href="#E1-MJMATHI-3C0" x="13727" y="0"></use>
</g>
 <use xlink:href="#E1-MJSZ3-7D" x="15051" y="-1"></use>
</g>
 <use xlink:href="#E1-MJMAIN-2C" x="28552" y="0"></use>
</g>
</svg>" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden=true style=vertical-align:-2.505ex;width:66.963ex;height:6.176ex alt="{\displaystyle {\mathfrak {T}}={\frac {\partial {\mathfrak {R}}}{\partial x}}-{\frac {\partial {\mathfrak {P}}}{\partial z }}={\frac {2\sin 2\varphi }{\lambda }}R\left\{{\frac {1}{\lambda }}\sin \left({\frac {2z\ \cos \varphi }{\lambda }}-{\frac {2vt\ \cos \varphi }{\lambda }}\right)2\pi \right\},}"></span>
|
||
</center>
|
||
<p><font style=vertical-align:inherit><font style=vertical-align:inherit>Let's sit</font></font></p>
|
||
<center>
|
||
<span class=mwe-math-element><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style=display:none>
|
||
<math xmlns=http://www.w3.org/1998/Math/MathML alttext="{\displaystyle 0={\frac {\partial \alpha }{\partial x}}+{\frac {\partial \gamma }{\partial z}},}"><semantics>
|
||
<mrow class=MJX-TeXAtom-ORD>
|
||
<mstyle displaystyle=true scriptlevel=0>
|
||
<mn><font style=vertical-align:inherit><font style=vertical-align:inherit>
|
||
0
|
||
</font></font></mn>
|
||
<mo><font style=vertical-align:inherit><font style=vertical-align:inherit>
|
||
=
|
||
</font></font></mo>
|
||
<mrow class=MJX-TeXAtom-ORD>
|
||
<mfrac>
|
||
<mrow>
|
||
<mi mathvariant=normal><font style=vertical-align:inherit><font style=vertical-align:inherit>
|
||
∂</font></font>
|
||
</mi>
|
||
<mi><font style=vertical-align:inherit><font style=vertical-align:inherit>
|
||
α</font></font>
|
||
</mi>
|
||
</mrow>
|
||
<mrow>
|
||
<mi mathvariant=normal><font style=vertical-align:inherit><font style=vertical-align:inherit>
|
||
∂</font></font>
|
||
</mi>
|
||
<mi><font style=vertical-align:inherit><font style=vertical-align:inherit>
|
||
x
|
||
</font></font></mi>
|
||
</mrow>
|
||
</mfrac>
|
||
</mrow>
|
||
<mo><font style=vertical-align:inherit><font style=vertical-align:inherit>
|
||
+
|
||
</font></font></mo>
|
||
<mrow class=MJX-TeXAtom-ORD>
|
||
<mfrac>
|
||
<mrow>
|
||
<mi mathvariant=normal><font style=vertical-align:inherit><font style=vertical-align:inherit>
|
||
∂</font></font>
|
||
</mi>
|
||
<mi><font style=vertical-align:inherit><font style=vertical-align:inherit>
|
||
γ</font></font>
|
||
</mi>
|
||
</mrow>
|
||
<mrow>
|
||
<mi mathvariant=normal><font style=vertical-align:inherit><font style=vertical-align:inherit>
|
||
∂</font></font>
|
||
</mi>
|
||
<mi><font style=vertical-align:inherit><font style=vertical-align:inherit>
|
||
e.g
|
||
</font></font></mi>
|
||
</mrow>
|
||
</mfrac>
|
||
</mrow>
|
||
<mo><font style=vertical-align:inherit><font style=vertical-align:inherit>
|
||
,
|
||
</font></font></mo>
|
||
</mstyle>
|
||
</mrow>
|
||
<annotation encoding=application/x-tex>
|
||
{\displaystyle 0={\frac {\partial \alpha }{\partial x}}+{\frac {\partial \gamma }{\partial z}},}
|
||
</annotation>
|
||
</semantics>
|
||
</math></span><img src="data:image/svg+xml;base64,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" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden=true style=vertical-align:-2.005ex;width:14.807ex;height:5.676ex alt="{\displaystyle 0={\frac {\partial \alpha }{\partial x}}+{\frac {\partial \gamma }{\partial z}},}"></span>
|
||
</center>
|
||
<p><font style=vertical-align:inherit><font style=vertical-align:inherit>so</font></font></p>
|
||
<center>
|
||
<span class=mwe-math-element><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style=display:none>
|
||
<math xmlns=http://www.w3.org/1998/Math/MathML alttext="{\displaystyle \alpha ={\frac {\partial \psi }{\partial z}}\quad \gamma =-{\frac {\partial \psi }{\partial x}},}"><semantics>
|
||
<mrow class=MJX-TeXAtom-ORD>
|
||
<mstyle displaystyle=true scriptlevel=0>
|
||
<mi><font style=vertical-align:inherit><font style=vertical-align:inherit>
|
||
α</font></font>
|
||
</mi>
|
||
<mo><font style=vertical-align:inherit><font style=vertical-align:inherit>
|
||
=
|
||
</font></font></mo>
|
||
<mrow class=MJX-TeXAtom-ORD>
|
||
<mfrac>
|
||
<mrow>
|
||
<mi mathvariant=normal><font style=vertical-align:inherit><font style=vertical-align:inherit>
|
||
∂</font></font>
|
||
</mi>
|
||
<mi><font style=vertical-align:inherit><font style=vertical-align:inherit>
|
||
ψ</font></font>
|
||
</mi>
|
||
</mrow>
|
||
<mrow>
|
||
<mi mathvariant=normal><font style=vertical-align:inherit><font style=vertical-align:inherit>
|
||
∂</font></font>
|
||
</mi>
|
||
<mi><font style=vertical-align:inherit><font style=vertical-align:inherit>
|
||
e.g
|
||
</font></font></mi>
|
||
</mrow>
|
||
</mfrac>
|
||
</mrow>
|
||
<mspace width=1em></mspace>
|
||
<mi><font style=vertical-align:inherit><font style=vertical-align:inherit>
|
||
γ</font></font>
|
||
</mi>
|
||
<mo><font style=vertical-align:inherit><font style=vertical-align:inherit>
|
||
=
|
||
</font></font></mo>
|
||
<mo><font style=vertical-align:inherit><font style=vertical-align:inherit>
|
||
−</font></font>
|
||
</mo>
|
||
<mrow class=MJX-TeXAtom-ORD>
|
||
<mfrac>
|
||
<mrow>
|
||
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{\displaystyle 0={\frac {\partial {\mathfrak {T}}}{\partial t}}+{\frac {\partial \psi }{\partial z}}{\frac {\partial {\mathfrak {T}}}{\partial x}}-{\frac {\partial \psi }{\partial x}}{\frac {\partial {\mathfrak {T}}}{\partial z,}}}
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|
||
</center>
|
||
<p><font style=vertical-align:inherit><font style=vertical-align:inherit>The tension in the aether would only stop when it moves at the same speed as the moving plate. But this only applies to low speeds. For larger ones, rather complicated values would arise, depending on the period of oscillation.</font></font></p>
|
||
<p><font style=vertical-align:inherit><font style=vertical-align:inherit>The fact that the co-movement of the ether only eliminates the tensions in the ether to a first approximation is due to the fact that the movement causes aberration of the ray, which, as is well known, cannot be easily explained by the assumption of moving ether.</font></font></p>
|
||
<p><font style=vertical-align:inherit><font style=vertical-align:inherit>It does not seem entirely hopeless to carry out experiments to see whether the aether is carried along in the direction of movement when reflected from rapidly moving plates.</font></font></p>
|
||
<div style=text-align:center>
|
||
<h3><span class=mw-headline id=Die_Annahme_ruhenden_Aethers.><font style=vertical-align:inherit><font style=vertical-align:inherit>The assumption of dormant aether.</font></font></span></h3>
|
||
</div>
|
||
<p><font style=vertical-align:inherit><font style=vertical-align:inherit>From the foregoing, we cannot entirely deny the possibility that the aether is moving. But the difficulties of carrying out such an assumption should already be sufficiently apparent in the examples outlined. As soon as it is possible to do justice to all the facts observed so far, if one considers the ether to be at rest, this path will initially be recommended for its simplicity. However, we then violate a very general mechanical principle from the outset,</font></font><span><span class=pagenum id=xi title="Page: Translational movement of the light ether.djvu/11"> <span class="pagenumber noprint" style=color:#666666;display:inline;margin:0px;padding:0px>[<b><a href=https://de.wikisource.org/wiki/Seite%3ATranslatorische_Bewegung_des_Licht%C3%A4thers.djvu/11 title="Seite:Translatorische Bewegung des Lichtäthers.djvu/11">xi</a></b>]</span> </span></span><font style=vertical-align:inherit><font style=vertical-align:inherit>that the equality of effect and counter-effect, if we do not want to assume that the electromagnetic tensions that want to set the ether in motion are canceled out by a certain rigid structure. And in general, if we deny the ether mobility, it becomes a substrate with highly indeterminate properties that we actually only use to make the finite value of the speed of light more understandable.</font></font></p>
|
||
<p><font style=vertical-align:inherit><font style=vertical-align:inherit>But this path will be particularly recommended to anyone who is initially only interested in the most general presentation of the facts.</font></font></p>
|
||
<p><font style=vertical-align:inherit><font style=vertical-align:inherit>The assumption of a resting aether was actually that advocated by </font></font><span style=letter-spacing:0.2em;left:0.1em><font style=vertical-align:inherit><font style=vertical-align:inherit>Fresnel</font></font></span><font style=vertical-align:inherit><font style=vertical-align:inherit> , although there is still talk of a partial continuation of the aether. However, this continuation only takes place inside the weighable bodies as soon as they themselves are moved and can be completely replaced by the view that what is continued is not the ether itself, but the part of the electromagnetic energy that is present in ponderable bodies liable. This emerges very clearly in the calculation by </font></font><span style=letter-spacing:0.2em;left:0.1em><font style=vertical-align:inherit><font style=vertical-align:inherit>Reiff </font></font></span><sup id=cite_ref-2 class=reference><a href=#cite_note-2><font style=vertical-align:inherit><font style=vertical-align:inherit>[2]</font></font></a></sup><font style=vertical-align:inherit><font style=vertical-align:inherit> , which shows that the </font></font><span style=letter-spacing:0.2em;left:0.1em><font style=vertical-align:inherit><font style=vertical-align:inherit>Fresnel</font></font></span><font style=vertical-align:inherit><font style=vertical-align:inherit> coefficient of continuation for a light ray in a moving medium results when the ether itself is at rest, the electromagnetic energy partly in the ether, partly in of the ponderable substance is present.</font></font></p>
|
||
<p><font style=vertical-align:inherit><font style=vertical-align:inherit>A precise implementation of the theory based on the assumption of resting ether and unchangeably charged ions as well as a complete discussion of all essential observation results is contained in the work of HA </font></font><span style=letter-spacing:0.2em;left:0.1em><font style=vertical-align:inherit><font style=vertical-align:inherit>Lorentz </font></font></span><sup id=cite_ref-3 class=reference><a href=#cite_note-3><font style=vertical-align:inherit><font style=vertical-align:inherit>[3]</font></font></a></sup><font style=vertical-align:inherit><font style=vertical-align:inherit> . E. </font></font><span style=letter-spacing:0.2em;left:0.1em><font style=vertical-align:inherit><font style=vertical-align:inherit>Wiechert </font></font></span><sup id=cite_ref-4 class=reference><a href=#cite_note-4><font style=vertical-align:inherit><font style=vertical-align:inherit>[4]</font></font></a></sup><font style=vertical-align:inherit><font style=vertical-align:inherit> starts from very similar points of view </font><font style=vertical-align:inherit>.</font></font></p>
|
||
<p><span style=letter-spacing:0.2em;left:0.1em><font style=vertical-align:inherit><font style=vertical-align:inherit>From his assumption, Lorentz</font></font></span><font style=vertical-align:inherit><font style=vertical-align:inherit> directly obtains the </font></font><span style=letter-spacing:0.2em;left:0.1em><font style=vertical-align:inherit><font style=vertical-align:inherit>Fresnel</font></font></span><font style=vertical-align:inherit><font style=vertical-align:inherit> coefficient of the propagation of light through moving media, the aberration and the </font></font><span style=letter-spacing:0.2em;left:0.1em><font style=vertical-align:inherit><font style=vertical-align:inherit>Doppler</font></font></span><font style=vertical-align:inherit><font style=vertical-align:inherit> principle. All three are directly related and result from a general theorem according to which all are for bodies at rest</font></font><span><span class=pagenum id=xii title="Page: Translational movement of the light ether.djvu/12"> <span class="pagenumber noprint" style=color:#666666;display:inline;margin:0px;padding:0px>[<b><a href=https://de.wikisource.org/wiki/Seite%3ATranslatorische_Bewegung_des_Licht%C3%A4thers.djvu/12 title="Seite:Translatorische Bewegung des Lichtäthers.djvu/12">xii</a></b>]</span> </span></span><font style=vertical-align:inherit><font style=vertical-align:inherit>valid equations of small oscillations can be transferred to moving ones if instead of the time </font></font><i><font style=vertical-align:inherit><font style=vertical-align:inherit>t</font></font></i><font style=vertical-align:inherit><font style=vertical-align:inherit> the variable</font></font><span class=mwe-math-element><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style=display:none>
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</math></span><img src="data:image/svg+xml;base64,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" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden=true style=vertical-align:-0.671ex;width:1.894ex;height:2.343ex alt="{\displaystyle t_{1}}"></span><font style=vertical-align:inherit><font style=vertical-align:inherit>means the time that light takes to travel in the free ether from a fixed point to an arbitrarily viewed point, and </font></font><i><font style=vertical-align:inherit><font style=vertical-align:inherit>above all</font></font></i><font style=vertical-align:inherit><font style=vertical-align:inherit> is the ratio of the speed of the body to the speed of light.</font></font></p>
|
||
<p><font style=vertical-align:inherit><font style=vertical-align:inherit>There is a further correction element for the continuation coefficient, which is caused by the fact that the movement also causes a change in the oscillation period according to the </font></font><span style=letter-spacing:0.2em;left:0.1em><font style=vertical-align:inherit><font style=vertical-align:inherit>Doppler</font></font></span><font style=vertical-align:inherit><font style=vertical-align:inherit> principle. It also immediately follows that the influence of the earth's motion is </font></font><i><font style=vertical-align:inherit><font style=vertical-align:inherit>only</font></font></i><font style=vertical-align:inherit><font style=vertical-align:inherit> shown in the aberration and that the prismatic deflection and the observation of the wavelength are not influenced by gratings. It also follows that a stationary current does not have an inductive effect on another wire due to the movement of the earth, because the movement creates an electrostatic charge which compensates for the effect.</font></font></p>
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<p><font style=vertical-align:inherit><font style=vertical-align:inherit>With the induction effect, the influence of the earth's movement only occurs in proportion to the size</font></font><span class=mwe-math-element><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style=display:none>
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</math></span><img src="data:image/svg+xml;base64,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" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden=true style=vertical-align:-0.338ex;width:4.979ex;height:2.676ex alt="{\displaystyle v^{2}A^{2}}"></span><font style=vertical-align:inherit><font style=vertical-align:inherit>so there is no prospect of experimental confirmation.</font></font></p>
|
||
<p><font style=vertical-align:inherit></font><span style=letter-spacing:0.2em;left:0.1em><font style=vertical-align:inherit><font style=vertical-align:inherit>After Lorentz</font></font></span><font style=vertical-align:inherit><font style=vertical-align:inherit> 's theory, which has been worked out in detail, </font><font style=vertical-align:inherit>proves that the assumption of immobile ether is completely sufficient to interpret a number of the diverse and hitherto little-explained phenomena of the influence of movement on electromagnetic processes, we must now point out a difficulty of a fundamental nature with consistent implementation of this theory.</font></font></p>
|
||
<p><font style=vertical-align:inherit><font style=vertical-align:inherit>This difficulty is closely related to the fact that changing electromagnetic states give rise to forces that would set the aether in motion if it were mobile. Let us imagine a body in the free ether in the form of a thin plate, which has different radiating capacities for heat rays on both sides. Since, according to </font></font><span style=letter-spacing:0.2em;left:0.1em><font style=vertical-align:inherit><font style=vertical-align:inherit>Maxwell</font></font></span><font style=vertical-align:inherit><font style=vertical-align:inherit> 's theory, the emitted rays exert a pressure on the surface, this pressure would predominate on the side of the greater radiation and set the body in motion.</font></font><span><span class=pagenum id=xiii title="Page: Translational movement of the light ether.djvu/13"> <span class="pagenumber noprint" style=color:#666666;display:inline;margin:0px;padding:0px>[<b><a href=https://de.wikisource.org/wiki/Seite%3ATranslatorische_Bewegung_des_Licht%C3%A4thers.djvu/13 title="Seite:Translatorische Bewegung des Lichtäthers.djvu/13">xiii</a></b>]</span> </span></span><font style=vertical-align:inherit><font style=vertical-align:inherit>So here we have the case that a body sets its center of gravity in motion through its own internal energy. So if we assume the ether to be immobile, then there would be a violation of the general principle of the center of gravity. On the other hand, the assumption of mobile ether, which has inertia, would avoid this objection.</font></font></p>
|
||
<p><font style=vertical-align:inherit><font style=vertical-align:inherit>However, the principle of the center of gravity may possibly be of a special nature and be limited to certain groups of effects in which no moving forces occur in the ether, as is actually the case with the usually observed ponderomotive effects.</font></font></p>
|
||
<p><font style=vertical-align:inherit><font style=vertical-align:inherit>Under all circumstances, this point should be kept in mind for further theoretical training.</font></font></p>
|
||
<div style=text-align:center>
|
||
<h3><span class=mw-headline id=Die_Versuchsergebnisse.><font style=vertical-align:inherit><font style=vertical-align:inherit>The test results.</font></font></span></h3>
|
||
</div>
|
||
<p><font style=vertical-align:inherit><font style=vertical-align:inherit>After we have discussed the two theoretical setups that are to be separated from each other, let us take a look at the experiments that have been carried out so far.</font></font></p>
|
||
<p><font style=vertical-align:inherit><font style=vertical-align:inherit>The main experiments that relate to our question are as follows:</font></font></p>
|
||
<div style=text-align:center>
|
||
<h4><span class=mw-headline id=A._Versuche_mit_positivem_Ergebniss.><font style=vertical-align:inherit><font style=vertical-align:inherit>A. Experiments with positive results.</font></font></span></h4>
|
||
</div>
|
||
<p><font style=vertical-align:inherit><font style=vertical-align:inherit>1. The aberration of the light of the fixed stars. As is well known, the aberration found a simple explanation through the emission hypothesis of light. The difficulties in the undulation theory have only recently been eliminated by HA </font></font><span style=letter-spacing:0.2em;left:0.1em><font style=vertical-align:inherit><font style=vertical-align:inherit>Lorentz</font></font></span><font style=vertical-align:inherit><font style=vertical-align:inherit> by assuming a ether </font></font><i><font style=vertical-align:inherit><font style=vertical-align:inherit>at rest</font></font></i><font style=vertical-align:inherit><font style=vertical-align:inherit> .</font></font></p>
|
||
<p><font style=vertical-align:inherit><font style=vertical-align:inherit>2. The </font></font><span style=letter-spacing:0.2em;left:0.1em><font style=vertical-align:inherit><font style=vertical-align:inherit>Doppler</font></font></span><font style=vertical-align:inherit><font style=vertical-align:inherit> principle is of general kinematic importance in its nature, but must still be taken into account when considering the question of moving or resting aether.</font></font></p>
|
||
<p><font style=vertical-align:inherit></font><span style=letter-spacing:0.2em;left:0.1em><font style=vertical-align:inherit><font style=vertical-align:inherit>3. Fizeau</font></font></span><font style=vertical-align:inherit><font style=vertical-align:inherit> 's experiment </font><font style=vertical-align:inherit>and its repetition by </font></font><span style=letter-spacing:0.2em;left:0.1em><font style=vertical-align:inherit><font style=vertical-align:inherit>Michelson</font></font></span><font style=vertical-align:inherit><font style=vertical-align:inherit> and </font></font><span style=letter-spacing:0.2em;left:0.1em><font style=vertical-align:inherit><font style=vertical-align:inherit>Morley</font></font></span><font style=vertical-align:inherit><font style=vertical-align:inherit> . A ray of light passing through flowing water in the direction of movement experiences an acceleration of the passage in proportion</font></font><span class=mwe-math-element><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style=display:none>
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class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden=true style=vertical-align:-0.838ex;width:17.526ex;height:3.176ex alt="{\displaystyle 1+v(1-(1/n^{2}))}"></span><font style=vertical-align:inherit><font style=vertical-align:inherit>, where </font></font><i><font style=vertical-align:inherit><font style=vertical-align:inherit>v</font></font></i><font style=vertical-align:inherit><font style=vertical-align:inherit> denotes the speed, </font></font><i><font style=vertical-align:inherit><font style=vertical-align:inherit>n</font></font></i><font style=vertical-align:inherit><font style=vertical-align:inherit> denotes the refractive index of the water. This result finds its complete explanation in the assumption of resting aether.</font></font></p><span style=display:none></span> <span><span class=pagenum id=xiv title="Page: Translational movement of the light ether.djvu/14"> <span class="pagenumber noprint" style=color:#666666;display:inline;margin:0px;padding:0px>[<b><a href=https://de.wikisource.org/wiki/Seite%3ATranslatorische_Bewegung_des_Licht%C3%A4thers.djvu/14 title="Seite:Translatorische Bewegung des Lichtäthers.djvu/14">xiv</a></b>]</span> </span></span>
|
||
<div style=text-align:center>
|
||
<h4><span class=mw-headline id=B._Versuche_mit_negativem_Ergebniss.><font style=vertical-align:inherit><font style=vertical-align:inherit>B. Experiments with negative results.</font></font></span></h4>
|
||
</div>
|
||
<p><font style=vertical-align:inherit></font><span style=letter-spacing:0.2em;left:0.1em><font style=vertical-align:inherit><font style=vertical-align:inherit>1. Arago's</font></font></span><font style=vertical-align:inherit><font style=vertical-align:inherit> experiment </font><font style=vertical-align:inherit>as to whether the movement of the earth influences the refraction of the light coming from the fixed stars.</font></font></p>
|
||
<p><font style=vertical-align:inherit><font style=vertical-align:inherit>2. </font></font><span style=letter-spacing:0.2em;left:0.1em><font style=vertical-align:inherit><font style=vertical-align:inherit>Ketteler</font></font></span><font style=vertical-align:inherit><font style=vertical-align:inherit> ’s interference experiment. The two beams of an interferential refractor are sent through two tubes filled with water and inclined towards each other in such a way that one beam hits one tube after the first reflection (on one glass plate), the other beam hits the second tube after the second reflection (on the other glass plate), i.e. runs in the opposite direction. Although both tubes are carried along by the earth's movement, there is no change in the interference fringes, although one beam is accelerated and the other is delayed.</font></font></p>
|
||
<p><font style=vertical-align:inherit><font style=vertical-align:inherit>Both results follow directly from the assumption of resting aether.</font></font></p>
|
||
<p><font style=vertical-align:inherit></font><span style=letter-spacing:0.2em;left:0.1em><font style=vertical-align:inherit><font style=vertical-align:inherit>3. Klinkerfues</font></font></span><font style=vertical-align:inherit><font style=vertical-align:inherit> ' experiment </font><font style=vertical-align:inherit>to determine whether the absorption line of sodium vapor was influenced by the movement of the earth.</font></font></p>
|
||
<p><font style=vertical-align:inherit></font><span style=letter-spacing:0.2em;left:0.1em><font style=vertical-align:inherit><font style=vertical-align:inherit>Klinkerfues</font></font></span><font style=vertical-align:inherit><font style=vertical-align:inherit> ' positive result </font><font style=vertical-align:inherit>would be incompatible with the theory of resting aether. However, the shift found is so small that observation errors cannot be ruled out.</font></font></p>
|
||
<p><font style=vertical-align:inherit></font><span style=letter-spacing:0.2em;left:0.1em><font style=vertical-align:inherit><font style=vertical-align:inherit>4. Des Coudres</font></font></span><font style=vertical-align:inherit><font style=vertical-align:inherit> 's experiment </font><font style=vertical-align:inherit>as to whether the induction effect of two coils of wire on a third is influenced by the fact that the direction of the induction of each coil falls once in the direction of the earth's movement, then in the direction perpendicular to it.</font></font></p>
|
||
<p><font style=vertical-align:inherit><font style=vertical-align:inherit>HA </font></font><span style=letter-spacing:0.2em;left:0.1em><font style=vertical-align:inherit><font style=vertical-align:inherit>Lorentz</font></font></span><font style=vertical-align:inherit><font style=vertical-align:inherit> has proven that when the aether is at rest, this influence only depends on the square of the ratio of the speed of the earth to the speed of light, and is therefore not observable because the movement of the earth creates an electrostatic charge on the current conductors, which cancels out the first-order effect.</font></font></p>
|
||
<p><font style=vertical-align:inherit></font><span style=letter-spacing:0.2em;left:0.1em><font style=vertical-align:inherit><font style=vertical-align:inherit>5. Lodge</font></font></span><font style=vertical-align:inherit><font style=vertical-align:inherit> 's experiments </font><font style=vertical-align:inherit>to investigate to what extent the surrounding aether is carried along by the movement of heavy or magnetizable masses.</font></font></p>
|
||
<p><font style=vertical-align:inherit></font><span style=letter-spacing:0.2em;left:0.1em><font style=vertical-align:inherit><font style=vertical-align:inherit>6. Zehnder</font></font></span><font style=vertical-align:inherit><font style=vertical-align:inherit> 's experiments </font><font style=vertical-align:inherit>as to whether the ether through the</font></font><span><span class=pagenum id=xv title="Page: Translational movement of the light ether.djvu/15"> <span class="pagenumber noprint" style=color:#666666;display:inline;margin:0px;padding:0px>[<b><a href=https://de.wikisource.org/wiki/Seite%3ATranslatorische_Bewegung_des_Licht%C3%A4thers.djvu/15 title="Seite:Translatorische Bewegung des Lichtäthers.djvu/15">xv</a></b>]</span> </span></span><font style=vertical-align:inherit><font style=vertical-align:inherit>Movement of a piston is moved in an air-diluted space.</font></font></p>
|
||
<p><font style=vertical-align:inherit><font style=vertical-align:inherit>The experiments of both observers were carried out with sensitive interference methods and gave negative results, so they are in perfect agreement with the assumption of a resting aether.</font></font></p>
|
||
<p><font style=vertical-align:inherit></font><span style=letter-spacing:0.2em;left:0.1em><font style=vertical-align:inherit><font style=vertical-align:inherit>7. Mascart</font></font></span><font style=vertical-align:inherit><font style=vertical-align:inherit> 's experiments </font><font style=vertical-align:inherit>on the rotation of the plane of polarization in quartz. There was no change in rotation when the light rays were once in the direction of the earth's movement and then in the opposite direction.</font></font></p>
|
||
<p><font style=vertical-align:inherit><font style=vertical-align:inherit>HA </font></font><span style=letter-spacing:0.2em;left:0.1em><font style=vertical-align:inherit><font style=vertical-align:inherit>Lorentz</font></font></span><font style=vertical-align:inherit><font style=vertical-align:inherit> gave the theory of this phenomenon and found that, assuming the aether is at rest, the earth's movement changes the existing rotation and independently adds a second one.</font></font></p>
|
||
<p><font style=vertical-align:inherit><font style=vertical-align:inherit>The negative result of </font></font><span style=letter-spacing:0.2em;left:0.1em><font style=vertical-align:inherit><font style=vertical-align:inherit>Mascart</font></font></span><font style=vertical-align:inherit><font style=vertical-align:inherit> 's observations would show that in quartz these two rotations caused by the influence of the earth's movement just cancel each other out.</font></font></p>
|
||
<p><font style=vertical-align:inherit></font><span style=letter-spacing:0.2em;left:0.1em><font style=vertical-align:inherit><font style=vertical-align:inherit>8. Roentgen</font></font></span><font style=vertical-align:inherit><font style=vertical-align:inherit> 's experiment to determine </font><font style=vertical-align:inherit>whether magnetic forces are generated by the movement of the earth from a charged capacitor.</font></font></p>
|
||
<p><font style=vertical-align:inherit><font style=vertical-align:inherit>The negative result of this experiment is not compatible with the assumption of a resting aether.</font></font></p>
|
||
<p><font style=vertical-align:inherit><font style=vertical-align:inherit>Electric charges and magnets would also have to produce magnetic or electrical forces through the movement of the earth. The absence of these forces would also be incompatible with the requirement of a resting aether.</font></font></p>
|
||
<p><font style=vertical-align:inherit></font><span style=letter-spacing:0.2em;left:0.1em><font style=vertical-align:inherit><font style=vertical-align:inherit>9. Fizeau</font></font></span><font style=vertical-align:inherit><font style=vertical-align:inherit> 's </font><font style=vertical-align:inherit>experiment on the influence of the earth's movement on the rotation of the plane of polarization through glass columns. </font><font style=vertical-align:inherit>The positive result of this experiment has recently been questioned. It would </font><font style=vertical-align:inherit>not be compatible with the assumption of resting aether according to HA </font></font><span style=letter-spacing:0.2em;left:0.1em><font style=vertical-align:inherit><font style=vertical-align:inherit>Lorentz</font></font></span><font style=vertical-align:inherit><font style=vertical-align:inherit> 's investigations .</font></font></p>
|
||
<p><font style=vertical-align:inherit><font style=vertical-align:inherit>10. The </font></font><span style=letter-spacing:0.2em;left:0.1em><font style=vertical-align:inherit><font style=vertical-align:inherit>Michelson</font></font></span><font style=vertical-align:inherit><font style=vertical-align:inherit> and </font></font><span style=letter-spacing:0.2em;left:0.1em><font style=vertical-align:inherit><font style=vertical-align:inherit>Morley</font></font></span><font style=vertical-align:inherit><font style=vertical-align:inherit> experiment . If the aether is at rest, the time it takes for a ray of light to travel back and forth between two plates of glass must change as the plates move. The change depends on the size</font></font><span class=mwe-math-element><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style=display:none>
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</math></span><img src="data:image/svg+xml;base64,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" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden=true style=vertical-align:-0.338ex;width:4.979ex;height:2.676ex alt="{\displaystyle v^{2}A^{2}}"></span><font style=vertical-align:inherit><font style=vertical-align:inherit>but should be observable when interference is used.</font></font></p>
|
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<p><span style=display:none></span> <span><span class=pagenum id=xvi title="Page: Translational movement of the light ether.djvu/16"> <span class="pagenumber noprint" style=color:#666666;display:inline;margin:0px;padding:0px>[<b><a href=https://de.wikisource.org/wiki/Seite%3ATranslatorische_Bewegung_des_Licht%C3%A4thers.djvu/16 title="Seite:Translatorische Bewegung des Lichtäthers.djvu/16">xvi</a></b>]</span> </span></span><font style=vertical-align:inherit><font style=vertical-align:inherit>The negative result is incompatible with the assumption of resting aether. This assumption can only be maintained by the hypothesis that the length dimensions of solid bodies are changed in the same proportion by the movement through the resting ether in order to compensate for the lengthening of the path of the light ray.</font></font></p>
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||
<p><font style=vertical-align:inherit><font style=vertical-align:inherit>The assumption of moving aether would give rise to the possibility that the aether is carried along by the movement of the earth and rests relative to it. This would explain all negative test results. But then the explanation of the aberration would remain.</font></font></p>
|
||
<div style=text-align:center>
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<h3><span id=Gravitation_und_Tr.C3.A4gheit.></span><span class=mw-headline id=Gravitation_und_Trägheit.><font style=vertical-align:inherit><font style=vertical-align:inherit>Gravity and inertia.</font></font></span></h3>
|
||
</div>
|
||
<p><font style=vertical-align:inherit><font style=vertical-align:inherit>The fact that gravity occupies an exceptional position and has no noticeable relationship to the other natural phenomena has often been emphasized. Its attribution to pressure forces is made more difficult by the fact that the energy reserve of a gravitating system has its greatest value when the individual parts of the mass are at an infinite distance. However, it is not always emphasized clearly enough that the acceleration of heavy masses is most likely related to gravity, because two independent definitions of mass are obtained through acceleration and gravity, which, as far as the very precise observations here go, are perfect to match. If one demands a further explanation of gravity, it would also have to give an account of why work is required to accelerate heavy masses. The fact that the two definitions of mass agree would then have to be a consequence of this explanation. It cannot be said with certainty whether such a theory can also be based on the ether, but it is probable.</font></font></p>
|
||
<p><font style=vertical-align:inherit><font style=vertical-align:inherit>However, it must also be emphasized here that it is by no means certain whether all effects can be traced back to tensions in the ether, just as it remains doubtful whether the processes in the ether can be completely satisfactorily represented by the laws of mechanics.</font></font></p>
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||
<p><span style=display:none></span> <span><span class=pagenum id=xvii title="Page: Translational movement of the light ether.djvu/17"> <span class="pagenumber noprint" style=color:#666666;display:inline;margin:0px;padding:0px>[<b><a href=https://de.wikisource.org/wiki/Seite%3ATranslatorische_Bewegung_des_Licht%C3%A4thers.djvu/17 title="Seite:Translatorische Bewegung des Lichtäthers.djvu/17">xvii</a></b>]</span> </span></span><font style=vertical-align:inherit><font style=vertical-align:inherit>If we now summarize the results, the impression is that there are still a number of questions to be answered before we can decide on the path that science should take.</font></font></p>
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<p><font style=vertical-align:inherit><font style=vertical-align:inherit>As we have seen, the assumption of moving ether without inertia leads to improbable consequences.</font></font></p>
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<p><font style=vertical-align:inherit><font style=vertical-align:inherit>As an experiment that would be important for this assumption, we recommend trying to see whether the ether is set in motion by the movement of reflecting transparent media.</font></font></p>
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<p><font style=vertical-align:inherit><font style=vertical-align:inherit>But since the ether is not set in motion by the movement of solid bodies, as far as is known so far, a negative result is likely.</font></font></p>
|
||
<p><font style=vertical-align:inherit><font style=vertical-align:inherit>The following difficulties stand in the way of the assumption that the ether is completely at rest:</font></font></p>
|
||
<p><font style=vertical-align:inherit><font style=vertical-align:inherit>1. Violation of the principle of the center of gravity (regarding the equality of effect and counteraction).</font></font></p>
|
||
<p><font style=vertical-align:inherit><font style=vertical-align:inherit>2. The negative results of the experiments of </font></font><span style=letter-spacing:0.2em;left:0.1em><font style=vertical-align:inherit><font style=vertical-align:inherit>Michelson</font></font></span><font style=vertical-align:inherit><font style=vertical-align:inherit> and </font></font><span style=letter-spacing:0.2em;left:0.1em><font style=vertical-align:inherit><font style=vertical-align:inherit>Morley</font></font></span><font style=vertical-align:inherit><font style=vertical-align:inherit> , that of </font></font><span style=letter-spacing:0.2em;left:0.1em><font style=vertical-align:inherit><font style=vertical-align:inherit>Roentgen</font></font></span><font style=vertical-align:inherit><font style=vertical-align:inherit> and possibly the experiments of </font></font><span style=letter-spacing:0.2em;left:0.1em><font style=vertical-align:inherit><font style=vertical-align:inherit>Mascart</font></font></span><font style=vertical-align:inherit><font style=vertical-align:inherit> and </font></font><span style=letter-spacing:0.2em;left:0.1em><font style=vertical-align:inherit><font style=vertical-align:inherit>Fizeau</font></font></span><font style=vertical-align:inherit><font style=vertical-align:inherit> .</font></font></p>
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||
<p><font style=vertical-align:inherit><font style=vertical-align:inherit>It would therefore be urgently desirable to repeat the following experiments or to carry out new ones.</font></font></p>
|
||
<p><font style=vertical-align:inherit><font style=vertical-align:inherit>1. Does the earth's movement affect the rotation of the plane of polarization</font></font></p>
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||
<dl>
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||
<dd><font style=vertical-align:inherit><font style=vertical-align:inherit>
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a) naturally rotating substances,
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||
</font></font></dd>
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</dl>
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||
<dl>
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||
<dd><font style=vertical-align:inherit><font style=vertical-align:inherit>
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||
b) through glass columns.
|
||
</font></font></dd>
|
||
</dl>
|
||
<p><font style=vertical-align:inherit><font style=vertical-align:inherit>2. Does the movement of the earth cause the magnetic forces required by the theory through the movement of electrical charges and the corresponding electrical forces through the movement of magnets?</font></font></p>
|
||
<p><font style=vertical-align:inherit><font style=vertical-align:inherit>When the results of these experiments are completely clear, it will become clear whether the otherwise simple theory of resting aether should be retained or abandoned. Should it have to be abandoned, it seems to me that only the </font><font style=vertical-align:inherit>way out indicated by </font></font><span style=letter-spacing:0.2em;left:0.1em><font style=vertical-align:inherit><font style=vertical-align:inherit>Des Coudres</font></font></span><font style=vertical-align:inherit><font style=vertical-align:inherit> would remain; namely influence of gravity on the light ether. This assumption seems to me to be equivalent to the assumption of a low inertial mass of the light ether.</font></font></p>
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||
<p><span style=display:none></span> <span><span class=pagenum id=xviii title="Page: Translational movement of the light ether.djvu/18"> <span class="pagenumber noprint" style=color:#666666;display:inline;margin:0px;padding:0px>[<b><a href=https://de.wikisource.org/wiki/Seite%3ATranslatorische_Bewegung_des_Licht%C3%A4thers.djvu/18 title="Seite:Translatorische Bewegung des Lichtäthers.djvu/18">xviii</a></b>]</span> </span></span><font style=vertical-align:inherit><font style=vertical-align:inherit>It would then be explained that the earth pulls the ether with it due to its significant gravity, while the movement of small solid bodies on the earth has no influence. The negative result of the experiments mentioned would be easily explained.</font></font></p>
|
||
<p><font style=vertical-align:inherit></font><span style=letter-spacing:0.2em;left:0.1em><font style=vertical-align:inherit><font style=vertical-align:inherit>But then the difficulties in explaining the aberration to which HA Lorentz</font></font></span><font style=vertical-align:inherit><font style=vertical-align:inherit> drew attention </font><font style=vertical-align:inherit>would essentially remain . </font><font style=vertical-align:inherit>However, whether these cannot be overcome if the co-movement of the ether under the influence of gravity is taken into account requires a special investigation. For this purpose, the hydrodynamic problem would have to be solved: determining the movements of a liquid through which a point moves at a constant speed, which attracts the individual parts of the liquid according to </font></font><span style=letter-spacing:0.2em;left:0.1em><font style=vertical-align:inherit><font style=vertical-align:inherit>Newton</font></font></span><font style=vertical-align:inherit><font style=vertical-align:inherit> 's law.</font></font></p>
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||
<p><font style=vertical-align:inherit><font style=vertical-align:inherit>The </font></font><span style=letter-spacing:0.2em;left:0.1em><font style=vertical-align:inherit><font style=vertical-align:inherit>Maxwell</font></font></span><font style=vertical-align:inherit><font style=vertical-align:inherit> 's tensions that would set the ether in motion are always so small, because they appear multiplied by the reciprocal speed of light, that the movements generally become imperceptible even with a very small inertial mass.</font></font></p>
|
||
<p><font style=vertical-align:inherit><font style=vertical-align:inherit>The task of the theory would then be to look for examples where the movement of the ether could actually be observed.</font></font></p>
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</div>
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<hr style=width:25%>
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<div style=font-size:90%>
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<ol class=references>
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<li id=cite_note-1><span class=mw-cite-backlink><a href=#cite_ref-1 aria-label="Jump high" title="Jump high"><font style=vertical-align:inherit><font style=vertical-align:inherit>↑</font></font></a></span> <span class=reference-text><font style=vertical-align:inherit><font style=vertical-align:inherit> I used </font></font><span style=letter-spacing:0.2em;left:0.1em><font style=vertical-align:inherit><font style=vertical-align:inherit>Jamin</font></font></span><font style=vertical-align:inherit><font style=vertical-align:inherit> 's interferential refractor to test whether a light beam passing through a vacuum tube is accelerated by the cathode rays; but the result was definitely negative.</font></font></span></li>
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||
<li id=cite_note-2><span class=mw-cite-backlink><a href=#cite_ref-2 aria-label="Jump high" title="Jump high"><font style=vertical-align:inherit><font style=vertical-align:inherit>↑ </font></font></a></span> <span class=reference-text><span style=letter-spacing:0.2em;left:0.1em><font style=vertical-align:inherit><font style=vertical-align:inherit>Reiff</font></font></span><font style=vertical-align:inherit><font style=vertical-align:inherit> , Wied. Ann. 50. p. 367. 1893</font></font></span></li>
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<li id=cite_note-3><span class=mw-cite-backlink><a href=#cite_ref-3 aria-label="Jump high" title="Jump high"><font style=vertical-align:inherit><font style=vertical-align:inherit>↑ </font></font></a></span> <span class=reference-text><span style=letter-spacing:0.2em;left:0.1em><font style=vertical-align:inherit><font style=vertical-align:inherit>Lorentz</font></font></span><font style=vertical-align:inherit><font style=vertical-align:inherit> , </font></font><a href="https://de-wikisource-org.translate.goog/wiki/Versuch_einer_Theorie_der_electrischen_und_optischen_Erscheinungen_in_bewegten_K%C3%B6rpern?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-US" title="Attempt at a theory of electrical and optical phenomena in moving bodies"><font style=vertical-align:inherit><font style=vertical-align:inherit>attempt at a theory of electrical and optical phenomena in moving bodies</font></font></a><font style=vertical-align:inherit><font style=vertical-align:inherit> . Leyden 1895 [WS:Template:1893].</font></font></span></li>
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<li id=cite_note-4><span class=mw-cite-backlink><a href=#cite_ref-4 aria-label="Jump high" title="Jump high"><font style=vertical-align:inherit><font style=vertical-align:inherit>↑ </font></font></a></span> <span class=reference-text><span style=letter-spacing:0.2em;left:0.1em><font style=vertical-align:inherit><font style=vertical-align:inherit>Wiechert</font></font></span><font style=vertical-align:inherit><font style=vertical-align:inherit> , Theory of Electrodynamics. Konigsberg 1896.</font></font></span></li>
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