Editing Apps/DysonPotential (section)

===Objective===

As has been [[#His_Derived_Expression|reprinted above]], on p. 62 of Dyson's [http://adsabs.harvard.edu/abs/1893RSPTA.184...43D ''Part I''] we find his power-series expression for the external potential, namely,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{\pi V_\mathrm{Dyson}}{GM} \biggr|_{\mathcal{O}(a^4/c^4)}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{4K(\mu)}{R+R_1}\biggl\{ 1 ~-~ \frac{1}{8}\biggl(\frac{a^2}{c^2}\biggr) \cos^2\biggl( \frac{\psi}{2}\biggr)
- \frac{1}{768}\biggl(\frac{a}{c}\biggr)^4 \biggl[ 5 ~+~ 8\cos\psi ~-~ \cos^2\psi ~-~ 4\cos^3\psi ~-~ \frac{4c^2}{RR_1} \cos2\psi \biggr] \biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+ 
\frac{(R + R_1)E(\mu)}{RR_1}\biggl\{
\frac{1}{8}\biggl(\frac{a}{c}\biggr)^2 \cos\psi ~-~\frac{1}{192} \biggl(\frac{a}{c}\biggr)^4 \biggl[
2\cos^2\psi ~-~4\cos\psi ~+~ \frac{2c^2}{RR_1}\cos2\psi
\biggr] \biggr\} \, ,
</math>
  </td>
</tr>
</table>
where &#8212; as in the context of toroidal coordinates &#8212; we occasionally will make the substitution, <math>~e^\eta = R_1/R</math>,  and therefore,
<table border="0" cellpadding="5" align="center">

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  <td align="right">
<math>~\mu</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~\frac{R_1 - R}{R_1+R}
= \frac{e^\eta - 1}{e^\eta + 1} \, .
</math>
  </td>
</tr>
</table>

In order to facilitate matching boundary conditions at the surface of the torus, between the exterior and interior expressions for the gravitational potential, Dyson rewrites this ''Part I'' expression for the external potential and &#8212; explicitly evaluating it on the torus surface &#8212; sets, <math>~R = a</math>.  Specifically, on p. 1049 of Dyson's [http://adsabs.harvard.edu/abs/1893RSPTA.184.1041D ''Part II''] we find equation (6), which reads,
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  <td align="right">
<math>~\frac{V}{2\pi a^2}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\ln\biggl(\frac{8c}{a}\biggr) + \frac{1}{2}\biggl(\frac{a}{c}\biggr) \biggl[\ln\biggl(\frac{8c}{a}\biggr) - \frac{5}{4}\biggr] \cos\chi
+  \biggl\{ \frac{1}{16} \biggl[ \ln\biggl(\frac{8c}{a}\biggr) - \frac{5}{2} \biggr] + \frac{3}{16} \biggl[\ln\biggl(\frac{8c}{a}\biggr) +\frac{17}{36} - \frac{72}{36}\biggr]\cos2\chi\biggr\}\biggl(\frac{a^2}{c^2}\biggr)
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+  \biggl\{ \frac{3}{32}\biggl[ \ln\biggl(\frac{8c}{a}\biggr) - \frac{25}{12}\biggr]\cos\chi + \frac{5}{64}\biggl[ \ln\biggl(\frac{8c}{a}\biggr)+\frac{7}{24} - \frac{48}{24}\biggr]\cos3\chi  \biggr\} \biggl(\frac{a^3}{c^3}\biggr)
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+  \biggl\{  \frac{9}{256}\biggl[ \ln\biggl(\frac{8c}{a}\biggr) - 2\biggr] + \frac{7}{128}\biggl[ \ln\biggl(\frac{8c}{a}\biggr) - \frac{19}{168} - 2\biggr]\cos2\chi 
+ \frac{35}{1024} \biggl[ \ln\biggl(\frac{8c}{a}\biggr) - 2 + \frac{19}{120}\biggr]\cos4\chi  \biggr\} \biggl(\frac{a^4}{c^4}\biggr) ~+~\cdots
</math>
  </td>
</tr>
</table>

In order to obtain this alternate power-series expression, Dyson &hellip;
* Expresses angular variations in terms of the angle, <math>~\chi</math>, instead of the angle, <math>~\psi</math>; these two angles are identified in the [[#His_Derived_Expression|above schematic]].
* Employs power-series expansions of both elliptic integral functions, <math>~K(\mu)</math> and <math>~E(\mu)</math>.
* Uses the binomial theorem to develop a number of other power-series expressions.

In what follows we will attempt to demonstrate that this second (''Part II'', equation 6) expression is identical to the first.