Editing Apps/DysonPotential (section)

===External Potential at Torus Surface===

====Initial Low Resolution====
Hence,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\biggl(\frac{a}{2}\biggr)V_2 </math>
  </td>
  <td align="center">
<math>~\approx</math>
  </td>
  <td align="left">
<math>~
\biggl[1 - \frac{1}{8}\biggl(\frac{a^2}{c^2}\biggr) \cos^2\biggl( \frac{\psi}{2}\biggr)\biggr]
\biggl(\frac{a}{c}\biggr) \biggl(\frac{R_1}{c}\biggr)^{-1}\biggl[1 + \frac{a}{c} \biggl(\frac{R_1}{c}\biggr)^{-1}\biggr]^{-1}\biggl\{ 2\ln \frac{4}{k^'}  + \frac{1}{2}\biggl( \ln\frac{4}{k^'} - 1 \biggr){k'}^2 \biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+ 
\biggl[\frac{1}{8}\biggl(\frac{a^2}{c^2}\biggr) \cos\psi \biggr] 
\frac{1}{2} \biggl[ 1 + \frac{a}{c}\biggl( \frac{R_1}{c}\biggr)^{-1} \biggr] 
\biggl\{ 1 + \frac{1}{2}\biggl( \ln \frac{4}{k'} - \frac{1}{2}\biggr)(k')^2  \biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~ cV_2 </math>
  </td>
  <td align="center">
<math>~\approx</math>
  </td>
  <td align="left">
<math>~
\biggl[1 - \cancelto{0}{\frac{1}{8}\biggl(\frac{a^2}{c^2}\biggr) \cos^2\biggl( \frac{\psi}{2}\biggr)}\biggr]
\biggl(\frac{R_1}{c}\biggr)^{-1}\biggl[1 + \frac{a}{c} \biggl(\frac{R_1}{c}\biggr)^{-1}\biggr]^{-1}\biggl\{ 4\ln \frac{4}{k^'}  + \biggl( \ln\frac{4}{k^'} - 1 \biggr){k'}^2 \biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+ 
\biggl[\frac{1}{8}\biggl(\frac{a}{c}\biggr) \cos\psi \biggr] 
\biggl[ 1 + \frac{a}{c}\biggl( \frac{R_1}{c}\biggr)^{-1} \biggr] 
\biggl\{ 1 + \frac{1}{2}\biggl( \ln \frac{4}{k'} - \frac{1}{2}\biggr)(k')^2  \biggr\} \, .
</math>
  </td>
</tr>
</table>
Hence,

<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\Rightarrow ~~~ cV_2 </math>
  </td>
  <td align="center">
<math>~\approx</math>
  </td>
  <td align="left">
<math>~
\biggl(\frac{R_1}{c}\biggr)^{-1}\biggl[1 + \frac{a}{c} \biggl(\frac{R_1}{c}\biggr)^{-1}\biggr]^{-1}\biggl\{ 4\ln \frac{4}{k^'}  + \biggl( \ln\frac{4}{k^'} - 1 \biggr){k'}^2 \biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+ 
\biggl[\frac{1}{8}\biggl(\frac{a}{c}\biggr) \cos\psi \biggr] 
\biggl[ 1 + \frac{a}{c}\biggl( \frac{R_1}{c}\biggr)^{-1} \biggr] 
\biggl\{ 1 + \frac{1}{2}\biggl( \ln \frac{4}{k'} - \frac{1}{2}\biggr)(k')^2  \biggr\} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~\approx</math>
  </td>
  <td align="left">
<math>~
\biggl\{ 1 + \frac{1}{2}  \biggl(\frac{a}{c}\biggr)\cos\chi 
+\biggl( \frac{a}{c}\biggr)^2  \biggl[
\frac{3}{8}\cos^2\chi -\frac{1}{8}\biggr] \biggr\}
\biggl[1 - \frac{1}{2}\biggl( \frac{a}{c}\biggr)  + \frac{1}{4} \biggl( \frac{a}{c}\biggr)^2 (1 - \cos\chi)\biggr]
\biggl\{ 2\ln \frac{4}{k^'}  + \frac{1}{2}\biggl( \ln\frac{4}{k^'} - 1 \biggr){k'}^2 \biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+ 
\frac{1}{8}\biggl(\frac{a}{c}\biggr) \cos\psi 
\biggl[ 1 + \frac{1}{2}\biggl( \frac{a}{c}\biggr) + \frac{1}{4}  \biggl(\frac{a}{c}\biggr)^2\cos\chi \biggr] 
\biggl\{ 1 + \frac{1}{2}\biggl( \ln \frac{4}{k'} - \frac{1}{2}\biggr)(k')^2  \biggr\} \, .
</math>
  </td>
</tr>
</table>
To order <math>~(a/c)^1</math>, this gives,

<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\Rightarrow ~~~ cV_2 </math>
  </td>
  <td align="center">
<math>~\approx</math>
  </td>
<td align="left">
<math>~
2\ln \frac{4}{k^'}  + \frac{1}{2}\biggl( \ln\frac{4}{k^'} - 1 \biggr){k'}^2 
+ \biggl(\frac{a}{c}\biggr)(\cos\chi -1) \ln \frac{4}{k^'}  
+ \frac{1}{8}\biggl(\frac{a}{c}\biggr) \cos\psi 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~\approx</math>
  </td>
<td align="left">
<math>~
2\ln \frac{4}{k^'}  
+ \biggl(\frac{a}{c}\biggr) \biggl\{ - 1  + \cos\chi  \ln \frac{4}{k^'}  + \frac{1}{8} \cos\psi 
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~\approx</math>
  </td>
<td align="left">
<math>~
2\ln \frac{4}{k^'}  
+ \biggl(\frac{a}{c}\biggr) \biggl\{ - 1  + \cos\chi \biggl[ \ln \frac{4}{k^'}  - \frac{1}{8} \biggr]
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~\approx</math>
  </td>
<td align="left">
<math>~
\ln\biggl( \frac{2^3c}{a}\biggr) 
+ \frac{a}{c}\biggl[ 1- \frac{1}{2} \cos\chi \biggr]
+ \biggl(\frac{a}{c}\biggr) \biggl\{  \frac{1}{2} \cos\chi \biggl[\ln\biggl( \frac{2^3c}{a}\biggr)  - \frac{1}{4} \biggr] -1
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~\approx</math>
  </td>
<td align="left">
<math>~
\ln\biggl( \frac{2^3c}{a}\biggr) 
+ \frac{a}{c} \biggl\{ 1- \frac{1}{2} \cos\chi +  \frac{1}{2} \cos\chi \biggl[\ln\biggl( \frac{2^3c}{a}\biggr)  - \frac{1}{4} \biggr] -1
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~\approx</math>
  </td>
<td align="left">
<math>~
\ln\biggl( \frac{2^3c}{a}\biggr) 
+  \frac{1}{2}\biggl(\frac{a}{c}\biggr) \biggl[\ln\biggl( \frac{2^3c}{a}\biggr)  - \frac{5}{4} \biggr] \cos\chi \, .
</math>
  </td>
</tr>
</table>



----

We are trying to match equation (6) in [http://adsabs.harvard.edu/abs/1893RSPTA.184.1041D Dyson's (1893b)] "Part II", that is,
<table border="0" cellpadding="5" align="center">

<tr>
  <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>

====High Resolution====


<table border="1" width="85%" align="center" cellpadding="8">
<tr><td align="left">
<div align="center">'''Summary'''</div>

<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~2\biggl(\frac{R_1}{c}\biggr)^{-1}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
1 + \frac{1}{2}\biggl(\frac{a}{c}\biggr)\cos\chi 
+ \frac{1}{2^3}\biggl(\frac{a}{c}\biggr)^2 \biggl[ 3\cos^2\chi - 1 \biggr] 
+  \frac{1}{2^4}\biggl(\frac{a}{c}\biggr)^3\biggl[ 5\cos^3\chi 
~-~  3\cos\chi \biggr]
~+~ \frac{1}{2^7} \biggl( \frac{a}{c}\biggr)^4 \biggl[ 3
~-~ 30 \cos^2\chi
~+~ 35 \cos^4\chi \biggr]
~+~ \mathcal{O}\biggl(\frac{a^5}{c^5}\biggr) \, .
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\biggl[1~+~\biggl( \frac{a}{c}\biggr)\biggl(\frac{R_1}{c}\biggr)^{-1}\biggr]^{-1} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
1 - \frac{1}{2}\biggl( \frac{a}{c}\biggr) ~+~ \frac{1}{2^2}\biggl(\frac{a}{c}\biggr)^2(1-\cos\chi ) 
~+~ \frac{1}{2^3}\biggl(\frac{a}{c}\biggr)^3 ( 2\cos\chi ~-~ 3\cos^2\chi  )
~+~ \frac{1}{2^5} \biggl(\frac{a}{c}\biggr)^4 (
-~9 \cos\chi 
~+~8\cos^2\chi  
~-~ 5\cos^3\chi )
~+~ \mathcal{O}\biggl(\frac{a^5}{c^5}\biggr) \, .
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~t_K</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
1 ~-~ \frac{1}{2^4}\biggl(\frac{a}{c}\biggr)^2 ( 1 -\cos\chi )
+ \frac{1}{2^7\cdot 3} \biggl(\frac{a}{c}\biggr)^3
( -13~+~12\cos\chi +2 \cos^2\chi  )
+ \frac{1}{2^8\cdot 3} \biggl(\frac{a}{c}\biggr)^4 (-17  + 21\cos\chi  - \cos^2\chi  ~-~ 2\cos^3\chi )
+ \mathcal{O}\biggl(\frac{a^5}{c^5}\biggr) \, .
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~1~+~\biggl( \frac{a}{c}\biggr)\biggl(\frac{R_1}{c}\biggr)^{-1} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
1 ~+~\frac{1}{2}\biggl( \frac{a}{c}\biggr) ~+~ \frac{1}{2^2}\biggl(\frac{a}{c}\biggr)^2\cos\chi 
~+~ \frac{1}{2^4}\biggl(\frac{a}{c}\biggr)^3 ( 3\cos^2\chi - 1 ) 
~+~  \frac{1}{2^5}\biggl(\frac{a}{c}\biggr)^4 ( 5\cos^3\chi ~-~  3\cos\chi )
~+~ \mathcal{O}\biggl(\frac{a^5}{c^5}\biggr) \, .
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~t_E</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
-\frac{1}{2^3} \biggl(\frac{a}{c}\biggr)^2\cos\chi   
~+~ \frac{1}{2^6\cdot 3} \biggl(\frac{a}{c}\biggr)^3 ( 13- 12\cos\chi   - 2\cos^2\chi  )
~+~ \frac{1}{2^7\cdot 3} \biggl(\frac{a}{c}\biggr)^4 ( 12  -21 \cos\chi   - 2\cos^2\chi - 2\cos^3\chi )
+ f_{E5}\biggl(\frac{a}{c}\biggr)^5
+ \mathcal{O}\biggl(\frac{a^6}{c^6}\biggr) \, .
</math>
  </td>
</tr>
</table>

</td></tr>
</table>


<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{\pi V_\mathrm{Dyson}}{GM/c} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
2K(\mu) \biggl[~2\biggl(\frac{R_1}{c}\biggr)^{-1}\biggr] \biggl[1 + \biggl(\frac{a}{c}\biggr) \biggl(\frac{R_1}{c}\biggr)^{-1} \biggr]^{-1}\biggl\{ t_K \biggr\}
+ 
E(\mu) \biggl[1  + \biggl(\frac{a}{c}\biggr) \biggl(\frac{R_1}{c}\biggr)^{-1}  \biggr] \biggl\{\biggl(\frac{c}{a}\biggr)  t_E \biggr\} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
2K(\mu) \biggl\{ ~2\biggl(\frac{R_1}{c}\biggr)^{-1}\biggr\} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
\times
\biggl\{ \biggl[1 + \biggl(\frac{a}{c}\biggr) \biggl(\frac{R_1}{c}\biggr)^{-1} \biggr]^{-1} \biggr\} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
\times
\biggl\{ t_K \biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+ 
E(\mu) \biggl\{ 1  + \biggl(\frac{a}{c}\biggr) \biggl(\frac{R_1}{c}\biggr)^{-1}   \biggr\} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
\times
\biggl\{\biggl(\frac{c}{a}\biggr)  t_E \biggr\} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
2K(\mu) \biggl\{ 
1 + \frac{1}{2}\biggl(\frac{a}{c}\biggr)\cos\chi 
+ \frac{1}{2^3}\biggl(\frac{a}{c}\biggr)^2 \biggl[ 3\cos^2\chi - 1 \biggr] 
+  \frac{1}{2^4}\biggl(\frac{a}{c}\biggr)^3\biggl[ 5\cos^3\chi 
~-~  3\cos\chi \biggr]
~+~ \frac{1}{2^7} \biggl( \frac{a}{c}\biggr)^4 \biggl[ 3
~-~ 30 \cos^2\chi
~+~ 35 \cos^4\chi \biggr]
\biggr\} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
\times
\biggl\{
1 - \frac{1}{2}\biggl( \frac{a}{c}\biggr) ~+~ \frac{1}{2^2}\biggl(\frac{a}{c}\biggr)^2(1-\cos\chi ) 
~+~ \frac{1}{2^3}\biggl(\frac{a}{c}\biggr)^3 ( 2\cos\chi ~-~ 3\cos^2\chi  )
~+~ \frac{1}{2^5} \biggl(\frac{a}{c}\biggr)^4 (
-~9 \cos\chi 
~+~8\cos^2\chi  
~-~ 5\cos^3\chi )
\biggr\} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
\times
\biggl\{ 
1 ~-~ \frac{1}{2^4}\biggl(\frac{a}{c}\biggr)^2 ( 1 -\cos\chi )
+ \frac{1}{2^7\cdot 3} \biggl(\frac{a}{c}\biggr)^3
( -13~+~12\cos\chi +2 \cos^2\chi  )
+ \frac{1}{2^8\cdot 3} \biggl(\frac{a}{c}\biggr)^4 (-17  + 21\cos\chi  - \cos^2\chi  ~-~ 2\cos^3\chi )
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+ 
E(\mu) \biggl\{ 
1 ~+~\frac{1}{2}\biggl( \frac{a}{c}\biggr) ~+~ \frac{1}{2^2}\biggl(\frac{a}{c}\biggr)^2\cos\chi 
~+~ \frac{1}{2^4}\biggl(\frac{a}{c}\biggr)^3 \biggl[ 3\cos^2\chi - 1 \biggr] 
~+~  \frac{1}{2^5}\biggl(\frac{a}{c}\biggr)^4\biggl[ 5\cos^3\chi 
~-~  3\cos\chi \biggr]
\biggr\} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
\times
\biggl\{-\frac{1}{2^3} \biggl(\frac{a}{c}\biggr)\cos\chi   
~+~ \frac{1}{2^6\cdot 3} \biggl(\frac{a}{c}\biggr)^2 ( 13- 12\cos\chi   - 2\cos^2\chi  )
~+~ \frac{1}{2^7\cdot 3} \biggl(\frac{a}{c}\biggr)^3 ( 12  -21 \cos\chi   - 2\cos^2\chi - 2\cos^3\chi )
+ f_{E5}\biggl(\frac{a}{c}\biggr)^4
\biggr\} ~+~ \mathcal{O}\biggl(\frac{a^5}{c^5}\biggr)
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
2K(\mu) \biggl\{ 
1 + \frac{1}{2}\biggl(\frac{a}{c}\biggr)\cos\chi 
+ \frac{1}{2^3}\biggl(\frac{a}{c}\biggr)^2 \biggl[ 3\cos^2\chi - 1 \biggr] 
+  \frac{1}{2^4}\biggl(\frac{a}{c}\biggr)^3\biggl[ 5\cos^3\chi 
~-~  3\cos\chi \biggr]
~+~ \frac{1}{2^7} \biggl( \frac{a}{c}\biggr)^4 \biggl[ 3
~-~ 30 \cos^2\chi
~+~ 35 \cos^4\chi \biggr]
\biggr\} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
\times
\biggl\{ 
1 ~-~ \frac{1}{2^4}\biggl(\frac{a}{c}\biggr)^2 ( 1 -\cos\chi )
+ \frac{1}{2^7\cdot 3} \biggl(\frac{a}{c}\biggr)^3
( -13~+~12\cos\chi +2 \cos^2\chi  )
+ \frac{1}{2^8\cdot 3} \biggl(\frac{a}{c}\biggr)^4 (-17  + 21\cos\chi  - \cos^2\chi  ~-~ 2\cos^3\chi )
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
- \frac{1}{2}\biggl( \frac{a}{c}\biggr) ~+~ \frac{1}{2^5}\biggl(\frac{a}{c}\biggr)^3 ( 1 -\cos\chi )
-  \frac{1}{2^8\cdot 3} \biggl(\frac{a}{c}\biggr)^4( -13~+~12\cos\chi +2 \cos^2\chi  )
+\frac{1}{2^2}\biggl(\frac{a}{c}\biggr)^2(1-\cos\chi ) - \frac{1}{2^6}\biggl(\frac{a}{c}\biggr)^4(1-\cos\chi )^2
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
~+~ \frac{1}{2^3}\biggl(\frac{a}{c}\biggr)^3 ( 2\cos\chi ~-~ 3\cos^2\chi  )
~+~ \frac{1}{2^5} \biggl(\frac{a}{c}\biggr)^4 (-~9 \cos\chi ~+~8\cos^2\chi  ~-~ 5\cos^3\chi )
\biggr\} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+ 
E(\mu) \biggl\{ -\frac{1}{2^3} \biggl(\frac{a}{c}\biggr)\cos\chi 
\biggl[ 1 ~+~\frac{1}{2}\biggl( \frac{a}{c}\biggr) ~+~ \frac{1}{2^2}\biggl(\frac{a}{c}\biggr)^2\cos\chi 
~+~ \frac{1}{2^4}\biggl(\frac{a}{c}\biggr)^3 ( 3\cos^2\chi - 1 ) 
\biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+ \frac{1}{2^6\cdot 3} \biggl(\frac{a}{c}\biggr)^2 ( 13- 12\cos\chi   - 2\cos^2\chi  )
\biggl[ 1 ~+~\frac{1}{2}\biggl( \frac{a}{c}\biggr) ~+~ \frac{1}{2^2}\biggl(\frac{a}{c}\biggr)^2\cos\chi 
\biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+ \frac{1}{2^7\cdot 3} \biggl(\frac{a}{c}\biggr)^3 ( 12  -21 \cos\chi   - 2\cos^2\chi - 2\cos^3\chi )
\biggl[ 1 ~+~\frac{1}{2}\biggl( \frac{a}{c}\biggr) \biggr]
+ f_{E5}\biggl(\frac{a}{c}\biggr)^4
\biggr\} 
~+~ \mathcal{O}\biggl(\frac{a^5}{c^5}\biggr)
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
2K(\mu) \biggl\{ 
1 + \frac{1}{2}\biggl(\frac{a}{c}\biggr)\cos\chi 
+ \frac{1}{2^3}\biggl(\frac{a}{c}\biggr)^2 \biggl[ 3\cos^2\chi - 1 \biggr] 
+  \frac{1}{2^4}\biggl(\frac{a}{c}\biggr)^3\biggl[ 5\cos^3\chi 
~-~  3\cos\chi \biggr]
~+~ \frac{1}{2^7} \biggl( \frac{a}{c}\biggr)^4 \biggl[ 3
~-~ 30 \cos^2\chi
~+~ 35 \cos^4\chi \biggr]
\biggr\} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
\times
\biggl\{ 
1 - \frac{1}{2}\biggl( \frac{a}{c}\biggr) 
+\frac{3}{2^4}\biggl(\frac{a}{c}\biggr)^2(1-\cos\chi ) 
+ \frac{1}{2^7\cdot 3} \biggl(\frac{a}{c}\biggr)^3
\biggl[ ( -13~+~12\cos\chi +2 \cos^2\chi  )
~+~ 12 ( 1 -\cos\chi )
~+~ 48 ( 2\cos\chi ~-~ 3\cos^2\chi  )
\biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+ \frac{1}{2^8\cdot 3} \biggl(\frac{a}{c}\biggr)^4 
\biggl[ (-17  + 21\cos\chi  - \cos^2\chi  ~-~ 2\cos^3\chi )
- ( -13~+~12\cos\chi +2 \cos^2\chi  )
- 12 (1-\cos\chi )^2
~+~ 24 (-~9 \cos\chi ~+~8\cos^2\chi  ~-~ 5\cos^3\chi )
\biggr]
\biggr\} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+ 
E(\mu) \biggl\{ 
~-~ \frac{1}{2^3} \biggl(\frac{a}{c}\biggr)\cos\chi 
~ -\frac{1}{2^4} \biggl(\frac{a}{c}\biggr)^2 \cos\chi  
~ -\frac{1}{2^5} \biggl(\frac{a}{c}\biggr)^3 \cos^2\chi 
~ -\frac{1}{2^7} \biggl(\frac{a}{c}\biggr)^4 ( 3\cos^3\chi - \cos\chi ) 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+ \frac{1}{2^6\cdot 3} \biggl(\frac{a}{c}\biggr)^2 ( 13- 12\cos\chi   - 2\cos^2\chi  ) 
~+~\frac{1}{2^7\cdot 3} \biggl(\frac{a}{c}\biggr)^3 ( 13- 12\cos\chi   - 2\cos^2\chi  ) 
~+~ \frac{1}{2^8\cdot 3} \biggl(\frac{a}{c}\biggr)^4 ( 13- 12\cos\chi   - 2\cos^2\chi  )\cos\chi 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+ \frac{1}{2^7\cdot 3} \biggl(\frac{a}{c}\biggr)^3 ( 12  -21 \cos\chi   - 2\cos^2\chi - 2\cos^3\chi ) 
~+~\frac{1}{2^8\cdot 3} \biggl(\frac{a}{c}\biggr)^4 ( 12  -21 \cos\chi   - 2\cos^2\chi - 2\cos^3\chi )
+ f_{E5}\biggl(\frac{a}{c}\biggr)^4
\biggr\} 
~+~ \mathcal{O}\biggl(\frac{a^5}{c^5}\biggr)
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
2K(\mu) \biggl\{ 
1 + \frac{1}{2}\biggl(\frac{a}{c}\biggr)\cos\chi 
+ \frac{1}{2^3}\biggl(\frac{a}{c}\biggr)^2 ( 3\cos^2\chi - 1 ) 
+  \frac{1}{2^4}\biggl(\frac{a}{c}\biggr)^3 ( 5\cos^3\chi ~-~  3\cos\chi )
~+~ \frac{1}{2^7} \biggl( \frac{a}{c}\biggr)^4 ( 3~-~ 30 \cos^2\chi~+~ 35 \cos^4\chi )
\biggr\} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
\times
\biggl\{ 
1 - \frac{1}{2}\biggl( \frac{a}{c}\biggr) 
+\frac{3}{2^4}\biggl(\frac{a}{c}\biggr)^2(1-\cos\chi ) 
+ \frac{1}{2^7\cdot 3} \biggl(\frac{a}{c}\biggr)^3
\biggl[ ( -13~+~12\cos\chi +2 \cos^2\chi  )
~+~ ( 12 -12\cos\chi )
~+~ ( 96\cos\chi ~-~ 144\cos^2\chi  )
\biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+ \frac{1}{2^8\cdot 3} \biggl(\frac{a}{c}\biggr)^4 
\biggl[ (-17  + 21\cos\chi  - \cos^2\chi  ~-~ 2\cos^3\chi )
+ ( 13~-~12\cos\chi -2 \cos^2\chi  )
+ (-12 + 24\cos\chi - 12\cos^2\chi)
~+~ (-~216 \cos\chi ~+~192\cos^2\chi  ~-~ 120\cos^3\chi )
\biggr]
\biggr\} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+ 
E(\mu) \biggl\{ 
~-~ \frac{1}{2^3} \biggl(\frac{a}{c}\biggr)\cos\chi 
+ \frac{1}{2^6\cdot 3} \biggl(\frac{a}{c}\biggr)^2\biggl[ ( 13- 12\cos\chi   - 2\cos^2\chi  ) ~ -12  \cos\chi  \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
~+~\frac{1}{2^7\cdot 3} \biggl(\frac{a}{c}\biggr)^3 \biggl[
( 13- 12\cos\chi   - 2\cos^2\chi  ) + ( 12  -21 \cos\chi   - 2\cos^2\chi - 2\cos^3\chi ) 
~ -~12  \cos^2\chi 
\biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
~+~ \frac{1}{2^8\cdot 3} \biggl(\frac{a}{c}\biggr)^4 \biggl[
( 13\cos\chi - 12\cos^2\chi   - 2\cos^3\chi  ) 
~+~( 12  -21 \cos\chi   - 2\cos^2\chi - 2\cos^3\chi )
~ -~6 ( 3\cos^3\chi - \cos\chi ) 
+ 2^8\cdot 3 f_{E5}
\biggr]
\biggr\} 
~+~ \mathcal{O}\biggl(\frac{a^5}{c^5}\biggr)
</math>
  </td>
</tr>
</table>

That is,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{\pi V_\mathrm{Dyson}}{GM/c} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
2K(\mu) \biggl\{ 
1 + \frac{1}{2}\biggl(\frac{a}{c}\biggr)\cos\chi 
+ \frac{1}{2^3}\biggl(\frac{a}{c}\biggr)^2 ( 3\cos^2\chi - 1 ) 
+  \frac{1}{2^4}\biggl(\frac{a}{c}\biggr)^3 ( 5\cos^3\chi ~-~  3\cos\chi )
~+~ \frac{1}{2^7} \biggl( \frac{a}{c}\biggr)^4 ( 3~-~ 30 \cos^2\chi~+~ 35 \cos^4\chi )
\biggr\} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
\times
\biggl\{ 
1 - \frac{1}{2}\biggl( \frac{a}{c}\biggr) 
+\frac{3}{2^4}\biggl(\frac{a}{c}\biggr)^2(1-\cos\chi ) 
+ \frac{1}{2^7\cdot 3} \biggl(\frac{a}{c}\biggr)^3
( -1  ~+~ 96\cos\chi~-~ 142\cos^2\chi  )
+ \frac{1}{2^8\cdot 3} \biggl(\frac{a}{c}\biggr)^4 
(-16  ~-~183\cos\chi  ~+~177 \cos^2\chi  ~-~ 122\cos^3\chi )
\biggr\} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+ 
E(\mu) \biggl\{ 
~-~ \frac{1}{2^3} \biggl(\frac{a}{c}\biggr)\cos\chi 
+ \frac{1}{2^6\cdot 3} \biggl(\frac{a}{c}\biggr)^2 ( 13- 24\cos\chi   - 2\cos^2\chi )
~+~\frac{1}{2^7\cdot 3} \biggl(\frac{a}{c}\biggr)^3 ( 25- 33\cos\chi   - 16\cos^2\chi  - 2\cos^3\chi )
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
~+~ \frac{1}{2^8\cdot 3} \biggl(\frac{a}{c}\biggr)^4 ( 12  -2 \cos\chi   - 14\cos^2\chi - 22\cos^3\chi + 2^8\cdot 3 f_{E5} )
\biggr\} 
~+~ \mathcal{O}\biggl(\frac{a^5}{c^5}\biggr)
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
2K(\mu) 
\biggl\{ 
1 - \frac{1}{2}\biggl( \frac{a}{c}\biggr) 
+\frac{3}{2^4}\biggl(\frac{a}{c}\biggr)^2(1-\cos\chi ) 
+ \frac{1}{2^7\cdot 3} \biggl(\frac{a}{c}\biggr)^3
( -1  ~+~ 96\cos\chi~-~ 142\cos^2\chi  )
+ \frac{1}{2^8\cdot 3} \biggl(\frac{a}{c}\biggr)^4 
(-16  ~-~183\cos\chi  ~+~177 \cos^2\chi  ~-~ 122\cos^3\chi )
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+  \frac{1}{2}\biggl(\frac{a}{c}\biggr)\cos\chi - \frac{1}{2^2}\biggl( \frac{a}{c}\biggr)^2\cos\chi 
+\frac{3}{2^5}\biggl(\frac{a}{c}\biggr)^3(\cos\chi-\cos^2\chi ) 
+ \frac{1}{2^8\cdot 3} \biggl(\frac{a}{c}\biggr)^4
( - \cos\chi  ~+~ 96\cos^2\chi~-~ 142\cos^3\chi  )
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+ \frac{1}{2^3}\biggl(\frac{a}{c}\biggr)^2 ( 3\cos^2\chi - 1 )
+ \frac{1}{2^4}\biggl(\frac{a}{c}\biggr)^3 (1- 3\cos^2\chi  )
~+~\frac{3}{2^7}\biggl(\frac{a}{c}\biggr)^4(1-\cos\chi )  ( 3\cos^2\chi - 1 )
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+ \frac{1}{2^4}\biggl(\frac{a}{c}\biggr)^3 ( 5\cos^3\chi ~-~  3\cos\chi )
~+~ \frac{1}{2^5}\biggl(\frac{a}{c}\biggr)^4 (3\cos\chi  ~-~5\cos^3\chi )
+ \frac{1}{2^7} \biggl( \frac{a}{c}\biggr)^4 ( 3~-~ 30 \cos^2\chi~+~ 35 \cos^4\chi )
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+ 
E(\mu) \biggl\{ 
~-~ \frac{1}{2^3} \biggl(\frac{a}{c}\biggr)\cos\chi 
+ \frac{1}{2^6\cdot 3} \biggl(\frac{a}{c}\biggr)^2 ( 13- 24\cos\chi   - 2\cos^2\chi )
~+~\frac{1}{2^7\cdot 3} \biggl(\frac{a}{c}\biggr)^3 ( 25- 33\cos\chi   - 16\cos^2\chi  - 2\cos^3\chi )
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
~+~ \frac{1}{2^8\cdot 3} \biggl(\frac{a}{c}\biggr)^4 ( 12  -2 \cos\chi   - 14\cos^2\chi - 22\cos^3\chi + 2^8\cdot 3 f_{E5} )
\biggr\} 
~+~ \mathcal{O}\biggl(\frac{a^5}{c^5}\biggr)
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
2K(\mu) 
\biggl\{ 
1  +  \frac{1}{2}\biggl(\frac{a}{c}\biggr)(\cos\chi - 1) 
+\frac{1}{2^4}\biggl(\frac{a}{c}\biggr)^2 \biggl[3(1-\cos\chi ) - 4 \cos\chi + 2 ( 3\cos^2\chi - 1 )\biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+ \frac{1}{2^7\cdot 3} \biggl(\frac{a}{c}\biggr)^3 \biggl[
36 (\cos\chi-\cos^2\chi ) 
+ 24 (1 ~-~  3\cos\chi - 3\cos^2\chi +5\cos^3\chi  )
+ ( -1  ~+~ 96\cos\chi~-~ 142\cos^2\chi  )
\biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+ \frac{1}{2^8\cdot 3} \biggl(\frac{a}{c}\biggr)^4 \biggl[ 
(-16  ~-~183\cos\chi  ~+~177 \cos^2\chi  ~-~ 122\cos^3\chi )
~+~18(1-\cos\chi )  ( 3\cos^2\chi - 1 )
+ ( - \cos\chi  ~+~ 96\cos^2\chi~-~ 142\cos^3\chi  )
~+~ 24 (3\cos\chi  ~-~5\cos^3\chi )
+ 6 ( 3~-~ 30 \cos^2\chi~+~ 35 \cos^4\chi )
\biggr]
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+ 
E(\mu) \biggl\{ 
~-~ \frac{1}{2^3} \biggl(\frac{a}{c}\biggr)\cos\chi 
+ \frac{1}{2^6\cdot 3} \biggl(\frac{a}{c}\biggr)^2 ( 13- 24\cos\chi   - 2\cos^2\chi )
~+~\frac{1}{2^7\cdot 3} \biggl(\frac{a}{c}\biggr)^3 ( 25- 33\cos\chi   - 16\cos^2\chi  - 2\cos^3\chi )
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
~+~ \frac{1}{2^8\cdot 3} \biggl(\frac{a}{c}\biggr)^4 ( 12  -2 \cos\chi   - 14\cos^2\chi - 22\cos^3\chi + 2^8\cdot 3 f_{E5} )
\biggr\} 
~+~ \mathcal{O}\biggl(\frac{a^5}{c^5}\biggr)
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
2K(\mu) 
\biggl\{ 
1  +  \frac{1}{2}\biggl(\frac{a}{c}\biggr)(\cos\chi - 1) 
+\frac{1}{2^4}\biggl(\frac{a}{c}\biggr)^2 (1-7\cos\chi +  6\cos^2\chi )
+ \frac{1}{2^7\cdot 3} \biggl(\frac{a}{c}\biggr)^3 (23 ~+~  60\cos\chi - 250\cos^2\chi +120\cos^3\chi  )
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+ \frac{1}{2^8\cdot 3} \biggl(\frac{a}{c}\biggr)^4 ( -16  ~-~94\cos\chi  ~+~147 \cos^2\chi  ~-~ 194\cos^3\chi + 210 \cos^4\chi )
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+ 
E(\mu) \biggl\{ 
~-~ \frac{1}{2^3} \biggl(\frac{a}{c}\biggr)\cos\chi 
+ \frac{1}{2^6\cdot 3} \biggl(\frac{a}{c}\biggr)^2 ( 13- 24\cos\chi   - 2\cos^2\chi )
~+~\frac{1}{2^7\cdot 3} \biggl(\frac{a}{c}\biggr)^3 ( 25- 33\cos\chi   - 16\cos^2\chi  - 2\cos^3\chi )
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
~+~ \frac{1}{2^8\cdot 3} \biggl(\frac{a}{c}\biggr)^4 ( 12  -2 \cos\chi   - 14\cos^2\chi - 22\cos^3\chi + 2^8\cdot 3 f_{E5} )
\biggr\} 
~+~ \mathcal{O}\biggl(\frac{a^5}{c^5}\biggr)
</math>
  </td>
</tr>
</table>

====Insert Expressions for K and E====


<table border="1" align="center" width="85%" cellpadding="8">
<tr><td align="left">
<div align="center">
'''Summary'''
</div>

<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~k'</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
2\biggl( \frac{a}{c}\biggr)^{1 / 2}\biggl( \frac{R_1}{c}\biggr)^{-1 / 2}
\biggl[1 + \frac{a}{c}\cdot  \biggl( \frac{R_1}{c} \biggr)^{-1} \biggr]^{-1} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~ \frac{4}{k'}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~2\biggl( \frac{a}{c}\biggr)^{- 1 / 2}\biggl( \frac{R_1}{c}\biggr)^{1 / 2}
\biggl[1 + \frac{a}{c}\cdot  \biggl( \frac{R_1}{c} \biggr)^{-1} \biggr] 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~2\biggl( \frac{c}{a}\biggr)^{1 / 2}
\biggl[  4 + \biggl( \frac{a}{c}\biggr)^2 - 4\biggl(\frac{a}{c}\biggr)\cos\chi  \biggr]^{1 / 4}
\biggl\{ 1 + \frac{a}{c}\cdot  \biggl[ 4 + \biggl( \frac{a}{c}\biggr)^2 - 4\biggl(\frac{a}{c}\biggr)\cos\chi  \biggr]^{-1 / 2} \biggr\} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl( \frac{2^3c}{a}\biggr)^{1 / 2}
\biggl[  1  - \biggl(\frac{a}{c}\biggr)\cos\chi  + \frac{1}{4}\biggl( \frac{a}{c}\biggr)^2 \biggr]^{1 / 4}
\biggl\{ 1 + \frac{a}{2c}\cdot  \biggl[ 1 - \biggl(\frac{a}{c}\biggr)\cos\chi  + \biggl( \frac{a}{2c}\biggr)^2 \biggr]^{-1 / 2} \biggr\} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~ \ln \frac{4}{k'}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{1}{2} \ln\biggl( \frac{2^3c}{a}\biggr)
+ \frac{1}{4}\ln\biggl[  1  - \biggl(\frac{a}{c}\biggr)\cos\chi  + \frac{1}{4}\biggl( \frac{a}{c}\biggr)^2 \biggr]
+ \ln\biggl\{ 1 + \frac{a}{2c}\cdot  \biggl[ 1 - \biggl(\frac{a}{c}\biggr)\cos\chi  + \biggl( \frac{a}{2c}\biggr)^2 \biggr]^{-1 / 2} \biggr\} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~\approx</math>
  </td>
  <td align="left">
<math>~
\frac{1}{2} \ln\biggl( \frac{2^3c}{a}\biggr)
+ \frac{1}{4}\ln\biggl[  1  - \biggl(\frac{a}{c}\biggr)\cos\chi  \biggr]
+ \ln\biggl[ 1 + \frac{a}{2c} \biggr] 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~\approx</math>
  </td>
  <td align="left">
<math>~
\frac{1}{2} \ln\biggl( \frac{2^3c}{a}\biggr)
+ \frac{a}{2c}\biggl[ 1
- \frac{1}{2} \cos\chi \biggr]
</math>
  </td>
</tr>
</table>

</td></tr>
</table>


Remember that ([[#Elliptic_Integral_Expressions|see above]]),
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~K(\mu)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
 <td align="left">
<math>~
\ln \frac{4}{k^'}  + \frac{1}{2^2}\biggl( \ln\frac{4}{k^'} - 1 \biggr){k'}^2
+ \frac{3^2}{2^6} \biggl( \ln\frac{4}{k^'} - \frac{7}{6} \biggr){k'}^4
+ \frac{5^2}{2^8} \biggl( \ln\frac{4}{k^'} - \frac{37}{30}  \biggr){k'}^6 + \cdots
</math>
  </td>
</tr>
</table>

And ([[#Elliptic_Integral_Expressions|see above]]), 
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~E(\mu)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
1 
~+~ \frac{1}{2}\biggl( \ln \frac{4}{k'} - \frac{1}{2}\biggr)(k')^2
~+~ \frac{3}{2^4}\biggl( \ln \frac{4}{k'} - 1 - \frac{1}{3\cdot 4}\biggr)(k')^4
~+~ \frac{3^2\cdot 5}{2^7\cdot 3}\biggl( \ln \frac{4}{k'} - 1 - \frac{1}{2\cdot 3} - \frac{1}{2\cdot 3\cdot 5}\biggr)(k')^6
~+~ \cdots
</math>
  </td>
</tr>
</table>

----

We are trying to match equation (6) in [http://adsabs.harvard.edu/abs/1893RSPTA.184.1041D Dyson's (1893b)] "Part II", that is,
<table border="0" cellpadding="5" align="center">

<tr>
  <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>

----

=====To First Order=====

<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~~(k')^{2}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl( \frac{2a}{c}\biggr) \biggl\{1 
~-~ \frac{1}{2}\biggl(\frac{a}{c}\biggr) (2~-~  \cos\chi )  
+ \frac{1}{2^3}\biggl( \frac{a}{c}\biggr)^2\biggl[ (3-\cos^2\chi) 
~+~2(2~-~ \cos\chi )^2  \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+~ \frac{1}{2^4}\biggl( \frac{a}{c}\biggr)^3\biggl[ (3\cos\chi - \cos^3\chi) 
~+~ (2~-~  \cos\chi )(3-\cos^2\chi) 
~-~  (2~-~ \cos\chi )^3   \biggr]
\biggr\}
~+~ \mathcal{O}\biggl(\frac{a^5}{c^5}\biggr) 
</math>
  </td>
</tr>
</table>

<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\ln \frac{4}{k'}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{1}{2} \ln\biggl( \frac{2^3c}{a}\biggr) + \frac{1}{2}\ln \Gamma 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{1}{2} \ln\biggl( \frac{2^3c}{a}\biggr) + \frac{1}{2}\biggl\{
\frac{1}{2}\biggl(\frac{a}{c}\biggr) (2~-~ \cos\chi )  
+ \frac{1}{2^3}\biggl( \frac{a}{c}\biggr)^2 (3-\cos^2\chi) 
~-~\frac{1}{2^3}\biggl(\frac{a}{c}\biggr)^2 (2~-~ \cos\chi )^2  
~+~ \mathcal{O}\biggl(\frac{a^3}{c^3}\biggr) 
\biggr\}
</math>
  </td>
</tr>
</table>

Hence,

<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{\pi V_\mathrm{Dyson}}{GM/c} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
2K(\mu) 
\biggl\{ 
1  +  \frac{1}{2}\biggl(\frac{a}{c}\biggr)(\cos\chi - 1) 
\biggr\}
+ E(\mu) \biggl\{ 
~-~ \frac{1}{2^3} \biggl(\frac{a}{c}\biggr)\cos\chi 
\biggr\} 
~+~ \mathcal{O}\biggl(\frac{a^2}{c^2}\biggr)
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~\approx</math>
  </td>
  <td align="left">
<math>~
\biggl[ 1  +  \frac{1}{2}\biggl(\frac{a}{c}\biggr)(\cos\chi - 1) \biggr] \biggl\{ 2K(\mu) \biggr\}
~-~ \frac{1}{2^3} \biggl(\frac{a}{c}\biggr)\cos\chi \biggl\{~E(\mu) \biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~\approx</math>
  </td>
  <td align="left">
<math>~
\biggl[ 1  +  \frac{1}{2}\biggl(\frac{a}{c}\biggr)(\cos\chi - 1) \biggr] \biggl\{ 2\ln \frac{4}{k'} + \frac{1}{2}\biggl[ \ln\frac{4}{k'} - 1 \biggr] k'^2 \biggr\}
~-~ \frac{1}{2^3} \biggl(\frac{a}{c}\biggr)\cos\chi \biggl\{~1 + \frac{1}{2}\cancelto{0}{\biggl[ \ln\frac{4}{k'} - \frac{1}{2} \biggr] k'^2}\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~\approx</math>
  </td>
  <td align="left">
<math>~
\biggl\{ 2\ln \frac{4}{k'} + \frac{1}{2}\biggl[ \ln\frac{4}{k'} - 1 \biggr] \biggl(\frac{2a}{c}\biggr) \biggr\}
+  \frac{1}{2}\biggl(\frac{a}{c}\biggr)(\cos\chi - 1) \biggl\{ 2\ln \frac{4}{k'} +\frac{1}{2} \cancelto{0}{\biggl[ \ln\frac{4}{k'} - 1 \biggr] k'^2} \biggr\}
~-~ \frac{1}{2^3} \biggl(\frac{a}{c}\biggr)\cos\chi \biggl\{~1 + \frac{1}{2}\cancelto{0}{\biggl[ \ln\frac{4}{k'} - \frac{1}{2} \biggr] k'^2}\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~\approx</math>
  </td>
  <td align="left">
<math>~
2\ln \frac{4}{k'} + \biggl(\frac{a}{c}\biggr)\biggl[ \ln\frac{4}{k'} - 1 \biggr]  
+  \biggl(\frac{a}{c}\biggr)(\cos\chi - 1) \biggl\{ \ln \frac{4}{k'} \biggr\}
~-~ \frac{1}{2^3} \biggl(\frac{a}{c}\biggr)\cos\chi 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~\approx</math>
  </td>
  <td align="left">
<math>~
2\ln \frac{4}{k'} + \biggl(\frac{a}{c}\biggr)\biggl[ \ln\frac{4}{k'} - 1   
+  (\cos\chi - 1) \biggl( \ln \frac{4}{k'} \biggr)
~-~ \frac{1}{2^3} \cos\chi \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~\approx</math>
  </td>
  <td align="left">
<math>~
2\ln \frac{4}{k'} + \biggl(\frac{a}{c}\biggr)\biggl[ - 1   
+  \cos\chi \biggl( \ln \frac{4}{k'} \biggr)
~-~ \frac{1}{2^3} \cos\chi \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~\approx</math>
  </td>
  <td align="left">
<math>~
\ln\biggl( \frac{2^3c}{a}\biggr) +
\frac{1}{2}\biggl(\frac{a}{c}\biggr) \biggl\{ (2~-~ \cos\chi ) - 2   
+  \cos\chi \biggl[  \ln\biggl(\frac{2^3c}{a}\biggr)\biggr]
~-~ \frac{1}{4} \cos\chi \biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~\approx</math>
  </td>
  <td align="left">
<math>~
\ln\biggl( \frac{2^3c}{a}\biggr) +
\frac{1}{2}\biggl(\frac{a}{c}\biggr) \biggl[  \ln\biggl(\frac{2^3c}{a}\biggr)
~-~ \frac{5}{4} \biggr]\cos\chi
</math>
  </td>
</tr>
</table>

=====To Second Order=====

<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{\pi V_\mathrm{Dyson}}{GM/c} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl\{ 
1  +  \frac{1}{2}\biggl(\frac{a}{c}\biggr)(\cos\chi - 1) 
+\frac{1}{2^4}\biggl(\frac{a}{c}\biggr)^2 (1-7\cos\chi +  6\cos^2\chi )
\biggr\}
\biggl\{
2K(\mu) 
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+ 
\biggl\{ 
~-~ \frac{1}{2^3} \biggl(\frac{a}{c}\biggr)\cos\chi 
+ \frac{1}{2^6\cdot 3} \biggl(\frac{a}{c}\biggr)^2 ( 13- 24\cos\chi   - 2\cos^2\chi )
\biggr\} 
\biggl\{
E(\mu) 
\biggr\}
~+~ \mathcal{O}\biggl(\frac{a^3}{c^3}\biggr)
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl\{ 
1  +  \frac{1}{2}\biggl(\frac{a}{c}\biggr)(\cos\chi - 1) 
+\frac{1}{2^4}\biggl(\frac{a}{c}\biggr)^2 (1-7\cos\chi +  6\cos^2\chi )
\biggr\}
\biggl\{
2\ln \frac{4}{k^'}  + \frac{1}{2}\biggl( \ln\frac{4}{k^'} - 1 \biggr){k'}^2
+ \frac{3^2}{2^5} \biggl( \ln\frac{4}{k^'} - \frac{7}{6} \biggr){k'}^4
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+ 
\biggl\{ 
~-~ \frac{1}{2^3} \biggl(\frac{a}{c}\biggr)\cos\chi 
+ \frac{1}{2^6\cdot 3} \biggl(\frac{a}{c}\biggr)^2 ( 13- 24\cos\chi   - 2\cos^2\chi )
\biggr\} 
\biggl\{
1 ~+~ \frac{1}{2}\biggl( \ln \frac{4}{k'} - \frac{1}{2}\biggr)(k')^2
\biggr\}
~+~ \mathcal{O}\biggl(\frac{a^3}{c^3}\biggr)
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl\{ 
1  +  \frac{1}{2}\biggl(\frac{a}{c}\biggr)(\cos\chi - 1) 
+\frac{1}{2^4}\biggl(\frac{a}{c}\biggr)^2 (1-7\cos\chi +  6\cos^2\chi )
\biggr\}
\biggl\{
2\biggl[\frac{1}{2} \ln\biggl( \frac{2^3c}{a}\biggr) + \frac{1}{2}\ln \Gamma\biggr]  + \frac{1}{2}\biggl[ \frac{1}{2} \ln\biggl( \frac{2^3c}{a}\biggr) + \frac{1}{2}\ln \Gamma - 1 \biggr]{k'}^2
+ \frac{3^2}{2^5} \biggl[ \frac{1}{2} \ln\biggl( \frac{2^3c}{a}\biggr) + \frac{1}{2}\ln \Gamma - \frac{7}{6} \biggr] {k'}^4
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+ 
\biggl\{ 
~-~ \frac{1}{2^3} \biggl(\frac{a}{c}\biggr)\cos\chi 
+ \frac{1}{2^6\cdot 3} \biggl(\frac{a}{c}\biggr)^2 ( 13- 24\cos\chi   - 2\cos^2\chi )
\biggr\} 
\biggl\{
1 ~+~ \frac{1}{2}\biggl[ \frac{1}{2} \ln\biggl( \frac{2^3c}{a}\biggr) + \frac{1}{2}\ln \Gamma - \frac{1}{2}\biggr](k')^2
\biggr\}
~+~ \mathcal{O}\biggl(\frac{a^3}{c^3}\biggr)
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl\{ 
1  +  \frac{1}{2}\biggl(\frac{a}{c}\biggr)(\cos\chi - 1) 
+\frac{1}{2^4}\biggl(\frac{a}{c}\biggr)^2 (1-7\cos\chi +  6\cos^2\chi )
\biggr\}
\biggl\{
\biggl[ \ln\biggl( \frac{2^3c}{a}\biggr) + \ln \Gamma\biggr]  + \frac{1}{2^2}\biggl[  \ln\biggl( \frac{2^3c}{a}\biggr) + \ln \Gamma - 2 \biggr]
\biggl( \frac{2a}{c}\biggr) \biggl[ 1 
~-~ \frac{1}{2}\biggl(\frac{a}{c}\biggr) (2~-~  \cos\chi )\biggr]
+ \frac{3}{2^6} \biggl[ 3 \ln\biggl( \frac{2^3c}{a}\biggr) + \cancelto{0}{3\ln \Gamma} - 7 \biggr] 4\biggl( \frac{a}{c}\biggr)^2
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+ 
\biggl\{ 
~-~ \frac{1}{2^3} \biggl(\frac{a}{c}\biggr)\cos\chi 
+ \frac{1}{2^6\cdot 3} \biggl(\frac{a}{c}\biggr)^2 ( 13- 24\cos\chi   - 2\cos^2\chi )
\biggr\} 
\biggl\{
1 ~+~ \frac{1}{2^2}\biggl[  \ln\biggl( \frac{2^3c}{a}\biggr) + \cancelto{0}{\ln \Gamma} - 1 \biggr]
\biggl( \frac{2a}{c}\biggr) \biggl[ 1 
~-~\cancelto{0}{ \frac{1}{2}\biggl(\frac{a}{c}\biggr)} (2~-~  \cos\chi )\biggr]
\biggr\}
~+~ \mathcal{O}\biggl(\frac{a^3}{c^3}\biggr) \, ,
</math>
  </td>
</tr>
</table>

where,

<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\ln\Gamma</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{1}{2}\biggl(\frac{a}{c}\biggr) (2~-~ \cos\chi )  + \frac{1}{2^3}\biggl( \frac{a}{c}\biggr)^2( -1 ~+~ 4\cos\chi -2\cos^2\chi )
~+~ \mathcal{O}\biggl(\frac{a^3}{c^3}\biggr) \, .
</math>
  </td>
</tr>
</table>

Hence,

<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{\pi V_\mathrm{Dyson}}{GM/c} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl\{ 
1  +  \frac{1}{2}\biggl(\frac{a}{c}\biggr)(\cos\chi - 1) 
+\frac{1}{2^4}\biggl(\frac{a}{c}\biggr)^2 (1-7\cos\chi +  6\cos^2\chi )
\biggr\}
\biggl\{
\biggl[ \ln\biggl( \frac{2^3c}{a}\biggr) + \frac{1}{2}\biggl(\frac{a}{c}\biggr) (2~-~ \cos\chi )  + \frac{1}{2^3}\biggl( \frac{a}{c}\biggr)^2( -1 ~+~ 4\cos\chi -2\cos^2\chi ) \biggr]  
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+ \frac{1}{2}\biggl( \frac{a}{c}\biggr) \biggl[  \ln\biggl( \frac{2^3c}{a}\biggr)  - 2 + \frac{1}{2}\biggl(\frac{a}{c}\biggr) (2~-~ \cos\chi )   \biggr]
\biggl[ 1 
~-~ \frac{1}{2}\biggl(\frac{a}{c}\biggr) (2~-~  \cos\chi )\biggr]
+ \frac{3}{2^4}\biggl( \frac{a}{c}\biggr)^2 \biggl[ 3 \ln\biggl( \frac{2^3c}{a}\biggr)  - 7 \biggr] 
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+ 
\biggl\{ 
~-~ \frac{1}{2^3} \biggl(\frac{a}{c}\biggr)\cos\chi 
+ \frac{1}{2^6\cdot 3} \biggl(\frac{a}{c}\biggr)^2 ( 13- 24\cos\chi   - 2\cos^2\chi )
\biggr\} 
\biggl\{
1 ~+~ \frac{1}{2}\biggl( \frac{a}{c}\biggr)\biggl[  \ln\biggl( \frac{2^3c}{a}\biggr) - 1 \biggr]
\biggr\}
~+~ \mathcal{O}\biggl(\frac{a^3}{c^3}\biggr) 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl\{ 
1  +  \frac{1}{2}\biggl(\frac{a}{c}\biggr)(\cos\chi - 1) 
+\frac{1}{2^4}\biggl(\frac{a}{c}\biggr)^2 (1-7\cos\chi +  6\cos^2\chi )
\biggr\}
\biggl\{
\biggl[ \ln\biggl( \frac{2^3c}{a}\biggr) + \frac{1}{2}\biggl(\frac{a}{c}\biggr) (2~-~ \cos\chi )  + \frac{1}{2^3}\biggl( \frac{a}{c}\biggr)^2( -1 ~+~ 4\cos\chi -2\cos^2\chi ) \biggr]  
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+ \frac{1}{2}\biggl( \frac{a}{c}\biggr) \biggl[  \ln\biggl( \frac{2^3c}{a}\biggr)  - 2 \biggr]
+ \frac{1}{2^2}\biggl( \frac{a}{c}\biggr)^2 (2~-~ \cos\chi )  
- \frac{1}{2^2}\biggl( \frac{a}{c}\biggr)^2 \biggl[  \ln\biggl( \frac{2^3c}{a}\biggr)  - 2    \biggr]
(2~-~  \cos\chi )
+ \frac{3}{2^4}\biggl( \frac{a}{c}\biggr)^2 \biggl[ 3 \ln\biggl( \frac{2^3c}{a}\biggr)  - 7 \biggr] 
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
~-~ \frac{1}{2^3} \biggl(\frac{a}{c}\biggr)\cos\chi 
~+~ \frac{1}{2^6\cdot 3} \biggl(\frac{a}{c}\biggr)^2 ( 13- 24\cos\chi   - 2\cos^2\chi )
~-~ \frac{1}{2^4} \biggl(\frac{a}{c}\biggr)^2\cos\chi \biggl[  \ln\biggl( \frac{2^3c}{a}\biggr) - 1 \biggr]
~+~ \mathcal{O}\biggl(\frac{a^3}{c^3}\biggr) 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\ln\biggl( \frac{2^3c}{a}\biggr) + \frac{1}{2}\biggl(\frac{a}{c}\biggr) (2~-~ \cos\chi )  
~-~ \frac{1}{2^3} \biggl(\frac{a}{c}\biggr)\cos\chi 
+ \frac{1}{2}\biggl( \frac{a}{c}\biggr) \biggl[  \ln\biggl( \frac{2^3c}{a}\biggr)  - 2 \biggr]
+  \frac{1}{2}\biggl(\frac{a}{c}\biggr)(\cos\chi - 1) \biggl[ \ln\biggl( \frac{2^3c}{a}\biggr) \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
~+~ \frac{1}{2^3}\biggl( \frac{a}{c}\biggr)^2( -1 ~+~ 4\cos\chi -2\cos^2\chi )   
+ \frac{1}{2^2}\biggl( \frac{a}{c}\biggr)^2 (2~-~ \cos\chi )  
- \frac{1}{2^2}\biggl( \frac{a}{c}\biggr)^2 \biggl[  \ln\biggl( \frac{2^3c}{a}\biggr)  - 2    \biggr]
(2~-~  \cos\chi )
+ \frac{3}{2^4}\biggl( \frac{a}{c}\biggr)^2 \biggl[ 3 \ln\biggl( \frac{2^3c}{a}\biggr)  - 7 \biggr] 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+  \frac{1}{2^2}\biggl(\frac{a}{c}\biggr)^2(\cos\chi - 1) (2~-~ \cos\chi ) 
+\frac{1}{2^4}\biggl(\frac{a}{c}\biggr)^2 (1-7\cos\chi +  6\cos^2\chi ) \ln\biggl( \frac{2^3c}{a}\biggr)
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
~+~ \frac{1}{2^6\cdot 3} \biggl(\frac{a}{c}\biggr)^2 ( 13- 24\cos\chi   - 2\cos^2\chi )
~-~ \frac{1}{2^4} \biggl(\frac{a}{c}\biggr)^2\cos\chi \biggl[  \ln\biggl( \frac{2^3c}{a}\biggr) - 1 \biggr]
~+~ \mathcal{O}\biggl(\frac{a^3}{c^3}\biggr) 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\ln\biggl( \frac{2^3c}{a}\biggr) 
+ \frac{1}{2^3}\biggl(\frac{a}{c}\biggr)\biggl\{ 4(2~-~ \cos\chi )  
~-~ \cos\chi 
+ \biggl[ 4 \ln\biggl( \frac{2^3c}{a}\biggr)  - 8 \biggr]
+ \biggl[ 4\ln\biggl( \frac{2^3c}{a}\biggr) \biggr] (\cos\chi - 1) 
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~~+~ \frac{1}{2^6\cdot 3} \biggl( \frac{a}{c}\biggr)^2\biggl\{
24( -1 ~+~ 4\cos\chi -2\cos^2\chi )   
+ 48 (2~-~ \cos\chi )  
- 48\biggl[  \ln\biggl( \frac{2^3c}{a}\biggr)  - 2    \biggr]
(2~-~  \cos\chi )
+ 36\biggl[ 3 \ln\biggl( \frac{2^3c}{a}\biggr)  - 7 \biggr] 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+  48(\cos\chi - 1) (2~-~ \cos\chi ) 
~+~12 (1-7\cos\chi +  6\cos^2\chi ) \ln\biggl( \frac{2^3c}{a}\biggr)
~+~  ( 13- 24\cos\chi   - 2\cos^2\chi )
~-~12\cos\chi \biggl[  \ln\biggl( \frac{2^3c}{a}\biggr) - 1 \biggr]
\biggr\}
~+~ \mathcal{O}\biggl(\frac{a^3}{c^3}\biggr) 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\ln\biggl( \frac{2^3c}{a}\biggr) 
+ \frac{1}{2^3}\biggl(\frac{a}{c}\biggr)\biggl\{ 4(2~-~ \cos\chi )  
~-~ \cos\chi 
+ \biggl[ 4 \ln\biggl( \frac{2^3c}{a}\biggr)  - 8 \biggr]
+ \biggl[ 4\ln\biggl( \frac{2^3c}{a}\biggr) \biggr] (\cos\chi - 1) 
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~~+~ \frac{1}{2^6\cdot 3} \biggl( \frac{a}{c}\biggr)^2\biggl\{ (-24 + 96\cos\chi-48\cos^2\chi)
+ (96~-~ 48\cos\chi )  
~+~ \biggl[2-  \ln\biggl( \frac{2^3c}{a}\biggr)  \biggr]
(96~-~ 48 \cos\chi )
+ \biggl[ 108 \ln\biggl( \frac{2^3c}{a}\biggr)  - 252 \biggr] 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+ 48(3\cos\chi - \cos^2\chi - 2 )
~+~ (12 - 84\cos\chi +  72\cos^2\chi ) \ln\biggl( \frac{2^3c}{a}\biggr)
~+~  ( 13- 24\cos\chi   - 2\cos^2\chi ) ~+~12\cos\chi
~-~12\cos\chi \biggl[  \ln\biggl( \frac{2^3c}{a}\biggr) - \biggr]
\biggr\}
~+~ \mathcal{O}\biggl(\frac{a^3}{c^3}\biggr) 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\ln\biggl( \frac{2^3c}{a}\biggr) 
+ \frac{1}{2}\biggl(\frac{a}{c}\biggr)\biggl[  \ln\biggl( \frac{2^3c}{a}\biggr) ~-~ \frac{5}{4}\biggr]\cos\chi 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~~+~ \frac{1}{2^6\cdot 3} \biggl( \frac{a}{c}\biggr)^2\biggl\{
-24 ~+~ 96\cos\chi -48\cos^2\chi    + 96~-~ 48\cos\chi   
+ 192  
- 252  
-96\cos\chi ~+~  ( 13- 12\cos\chi   - 2\cos^2\chi )
+ (144\cos\chi - 48\cos^2\chi -96 )
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+36 \biggl[ \ln\biggl( \frac{2^3c}{a}\biggr)\biggr]\cos\chi    
~+~(12 - 84\cos\chi +  72\cos^2\chi -96 + 108) \biggl[ \ln\biggl( \frac{2^3c}{a}\biggr)\biggr]
\biggr\}
~+~ \mathcal{O}\biggl(\frac{a^3}{c^3}\biggr) 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\ln\biggl( \frac{2^3c}{a}\biggr) 
+ \frac{1}{2}\biggl(\frac{a}{c}\biggr)\biggl[  \ln\biggl( \frac{2^3c}{a}\biggr) ~-~ \frac{5}{4}\biggr]\cos\chi 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~~+~ \frac{1}{2^6\cdot 3} \biggl( \frac{a}{c}\biggr)^2\biggl\{
-71 ~+~84\cos\chi -98\cos^2\chi    
~+~(24 - 48\cos\chi +  72\cos^2\chi ) \biggl[ \ln\biggl( \frac{2^3c}{a}\biggr)\biggr]
\biggr\}
~+~ \mathcal{O}\biggl(\frac{a^3}{c^3}\biggr) 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\ln\biggl( \frac{2^3c}{a}\biggr) 
+ \frac{1}{2}\biggl(\frac{a}{c}\biggr)\biggl[  \ln\biggl( \frac{2^3c}{a}\biggr) ~-~ \frac{5}{4}\biggr]\cos\chi 
~+~ \frac{1}{2^6\cdot 3} \biggl( \frac{a}{c}\biggr)^2\biggl\{
-71 ~+~84\cos\chi -98\cos^2\chi    
~+~24\ln\biggl( \frac{2^3c}{a}\biggr)(1 - 2\cos\chi +  3\cos^2\chi ) 
\biggr\}
~+~ \mathcal{O}\biggl(\frac{a^3}{c^3}\biggr) 
</math>
  </td>
</tr>
</table>

In an effort to compare this expression with equation (6) from Dyson's (1893b) "Part II", we should make the substitutions,
<div align="center">
<math>~\ln\biggl(\frac{2^3c}{a}\biggr) \rightarrow (\lambda +2)</math> &nbsp;  &nbsp;  &nbsp; and  &nbsp;  &nbsp;  &nbsp; <math>~2\cos^2\chi \rightarrow 1 + \cos2\chi \, .</math>
</div>

This means,

<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{\pi V_\mathrm{Dyson}}{GM/c}\biggr|_{\mathcal{O}(a^2/c^2)} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
~+~ \frac{1}{2^6\cdot 3} \biggl( \frac{a}{c}\biggr)^2\biggl\{
-71 ~+~84\cos\chi -49(1+\cos2\chi )   
~+~24(\lambda + 2)(1 - 2\cos\chi ) 
~+~36(\lambda+2)(1 + \cos2\chi ) 
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
~+~ \frac{1}{2^6\cdot 3} \biggl( \frac{a}{c}\biggr)^2\biggl\{
-71 ~+~84\cos\chi -49 -49 \cos2\chi    
~+~24(\lambda + 2 -2\lambda \cos\chi - 4\cos\chi) 
~+~36(\lambda+2 +\lambda\cos 2\chi + 2\cos 2\chi) 
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
~+~ \frac{1}{2^6\cdot 3} \biggl( \frac{a}{c}\biggr)^2\biggl\{
-71 ~+~84\cos\chi -49 -49 \cos2\chi   
~+~24\lambda + 48 -48\lambda \cos\chi - 96\cos\chi 
~+~36\lambda+72 +36\lambda\cos 2\chi + 72\cos 2\chi 
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
~+~ \frac{1}{2^6\cdot 3} \biggl( \frac{a}{c}\biggr)^2\biggl\{
60\lambda -48\lambda \cos\chi - 12\cos\chi +36\lambda\cos 2\chi + 23\cos 2\chi 
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
~+~ \biggl( \frac{a}{c}\biggr)^2\biggl\{
\frac{5\lambda}{16}  - \frac{(4\lambda + 1)}{16}~\cos\chi +\frac{3(\lambda+\tfrac{23}{36})}{16}\cos 2\chi  
\biggr\} \, .
</math>
  </td>
</tr>
</table>

This expression differs from the 2<sup>nd</sup>-order term in Dyson's equation (6) by the amount,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\Delta \biggr|_{\mathcal{O}(a^2/c^2)} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl( \frac{a}{c}\biggr)^2\biggl\{
\frac{5\lambda}{16}  - \frac{(4\lambda + 1)}{16}~\cos\chi +\frac{3(\lambda+\tfrac{23}{36})}{16}\cos 2\chi  
\biggr\} 
-
\biggl(\frac{a}{c}\biggr)^2 \biggl\{ \frac{\lambda - \frac{1}{2}}{16} + \frac{3(\lambda + \frac{17}{36})}{16}\cos2\chi \biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{1}{16\cdot 12}\biggl( \frac{a}{c}\biggr)^2\biggl\{
60\lambda  - (48\lambda + 12)~\cos\chi +(36\lambda+23)\cos 2\chi  
\biggr\} 
-
\frac{1}{16\cdot 12}\biggl(\frac{a}{c}\biggr)^2 \biggl\{ 12\lambda - 6 + (36\lambda + 17) \cos2\chi \biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{1}{16\cdot 12}\biggl( \frac{a}{c}\biggr)^2\biggl\{
48\lambda + 6  - (48\lambda + 12)~\cos\chi +(6)\cos 2\chi  
\biggr\} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{1}{2^5}\biggl( \frac{a}{c}\biggr)^2\biggl\{
(\cos 2\chi  -1)- (8\lambda + 2)~(1+\cos\chi) 
\biggr\} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{1}{2^4}\biggl( \frac{a}{c}\biggr)^2\biggl\{
(\cos\chi  - 1)- (4\lambda + 1)~ 
\biggr\} (1+\cos\chi)
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{1}{2^4}\biggl( \frac{a}{c}\biggr)^2 (\cos\chi  - 2- 4\lambda ) (1+\cos\chi)
</math>
  </td>
</tr>
</table>