Editing Apps/DysonPotential (section)

====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>