Editing Apps/DysonPotential (section)

===The Ratio R<sub>1</sub>/c===

Note that, via the law of cosines,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~R_1^2</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~(2c)^2 + R^2 - 4Rc\cos\chi</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~\biggl(\frac{R_1}{c}\biggr)^2</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~4 + \biggl( \frac{R}{c}\biggr)^2 - 4\biggl(\frac{R}{c}\biggr)\cos\chi</math>
  </td>
</tr>
</table>
At the surface of the torus, where <math>~R=a</math>, we therefore have,
<table border="0" cellpadding="5" align="center">

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

====Low Order====
Employing the [[Appendix/Ramblings/PowerSeriesExpressions#Binomial|binomial theorem]], we can write,
<table border="0" cellpadding="5" align="center">

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

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~\approx</math>
  </td>
  <td align="left">
<math>~\frac{1}{2}\biggl\{ 1 - \frac{1}{2} \biggl[ - \biggl(\frac{a}{c}\biggr)\cos\chi + \frac{1}{4}\biggl( \frac{a}{c}\biggr)^2 \biggr] 
+ \frac{3}{8}\biggl[ - \biggl(\frac{a}{c}\biggr)\cos\chi + \cancelto{0}{\frac{1}{4}\biggl( \frac{a}{c}\biggr)^2} \biggr]^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}\biggl\{ 1 + \frac{1}{2}  \biggl(\frac{a}{c}\biggr)\cos\chi - \frac{1}{8}\biggl( \frac{a}{c}\biggr)^2  
+ \frac{3}{8}\biggl(\frac{a}{c}\biggr)^2\cos^2\chi  \biggr\}</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~\approx</math>
  </td>
  <td align="left">
<math>~
\frac{1}{2}\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\}
</math>
  </td>
</tr>

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

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

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

====Higher Order====

Adopting the shorthand notation,
<div align="center">
<math>~\gamma \equiv \frac{1}{2}\biggl(\frac{R_1}{c}\biggr) \, ,</math>&nbsp; &nbsp; &nbsp; and &nbsp; &nbsp; &nbsp; <math>~b \equiv - \biggl(\frac{a}{c}\biggr)\cos\chi + \frac{1}{4}\biggl( \frac{a}{c}\biggr)^2 \, ,</math>
</div>
and employing the [[Appendix/Ramblings/PowerSeriesExpressions#Binomial|binomial theorem]], we can write,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\gamma = \biggl[ 1 + b \biggr]^{1 / 2}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ 
1 + \frac{1}{2}b - \frac{1}{2^3}b^2 + \frac{1}{2^4}b^3 - \frac{3\cdot 5}{2^7\cdot 3}b^4 + \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>~ 
1 + \frac{1}{2}\biggl[ - \biggl(\frac{a}{c}\biggr)\cos\chi + \frac{1}{4}\biggl( \frac{a}{c}\biggr)^2 \biggr] - \frac{1}{2^3}\biggl[ - \biggl(\frac{a}{c}\biggr)\cos\chi + \frac{1}{4}\biggl( \frac{a}{c}\biggr)^2 \biggr]^2 
</math>
  </td>
</tr>

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

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

<span id="gammaInverse">Also, we have,</span>

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

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

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

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

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

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
~+~ \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>
</table>
Hence,

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

<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 \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]
~+~ \mathcal{O}\biggl(\frac{a^5}{c^5}\biggr) \, .
</math>
  </td>
</tr>
</table>
And, adopting the shorthand notation,
<div align="center">
<math>~d \equiv 
\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]
 \, ,</math>
</div>
we have,
<table border="0" cellpadding="5" align="center">

<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 -d + d^2 - d^3 + d^4
~+~ \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>~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]
</math>
  </td>
</tr>

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

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

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

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

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

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>
~+~ \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>
</table>