Editing Apps/DysonPotential (section)

===Relationship Between Angles===

Drawing on the Law of Cosines, [[#The_Ratio_R1.2Fc|as above]], we can state that on the torus surface,
<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 + a^2 - 4ac\cos\chi</math>
  </td>
</tr>
</table>
Alternatively, applying the Law of Cosines to the angle, <math>~\psi</math>, we have,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~(2c)^2</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~R_1^2 + a^2 - 2aR_1\cos\psi</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~\cos\psi</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{R_1^2 + a^2 - 4c^2}{2aR_1} \, .</math>
  </td>
</tr>
</table>
Therefore, anywhere along the surface of the torus, we can switch from one of these angles to the other via the relation,
<table border="0" cellpadding="5" align="center">

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

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

====Cosine &psi; Expansion====
Employing the [[Appendix/Ramblings/PowerSeriesExpressions#Binomial|binomial theorem]], we therefore can write,

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

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

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

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

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~- \biggl[\cos\chi  ~-~ \frac{1}{2}\biggl( \frac{a}{c}\biggr) \biggr] \biggl\{ 1 
~+~ \frac{1}{2} \biggl(\frac{a}{c}\biggr) ~+~ \frac{1}{2^3} \biggl(\frac{a}{c}\biggr)^2\biggl[ 3  ~-~ \cos\chi \biggr]
+~ \frac{1}{2^5} \biggl(\frac{a}{c}\biggr)^3 \biggl[ 10
~-~ 3 \cos\chi \biggr]~+~ \frac{1}{2^7} \biggl(\frac{a}{c}\biggr)^4 \biggl[3 \cos^2\chi  ~+~ 5\cdot 7 \biggr]
~+~ \mathcal{O}\biggl(\frac{a^5}{c^5}\biggr) \biggr\} 
</math>
  </td>
</tr>
</table>

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

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

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

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

====Cosine-Squared Expansion====

Letting,
<div align="center">
<math>~b \equiv \biggl[- \biggl(\frac{a}{c}\biggr) + \frac{1}{4}\biggl(\frac{a}{c}\biggr)^2 \cos\chi \biggr] \, ,</math>
</div>
via the [[Appendix/Ramblings/PowerSeriesExpressions#Binomial|binomial theorem]] we have,
<table border="0" cellpadding="5" align="center">

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

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

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

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

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

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

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

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

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

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

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

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

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

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

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>
~+~ \biggl(\frac{a}{c}\biggr)^3\biggl[
\frac{1}{2^2}~-~ \cos\chi ~+~ \frac{5}{4} \cos^2\chi  ~-~ \frac{1}{2} \cos^3\chi    
\biggr]
~+~\biggl(\frac{a}{c}\biggr)^4 \biggl[ \frac{1}{2^2}  ~-~ \frac{17}{2^4} \cos\chi  
~+~  \frac{3}{2} \cos^2\chi  
~-~ \frac{3}{4} \cos^3\chi 
~+~\frac{1}{2^4} \cos^4\chi 
\biggr]
~+~ \mathcal{O}\biggl( \frac{a^5}{c^5} \biggr)
</math>
  </td>
</tr>
</table>

====Cosine-Cubed Expansion====

Again, letting,
<div align="center">
<math>~b \equiv \biggl[- \biggl(\frac{a}{c}\biggr) + \frac{1}{4}\biggl(\frac{a}{c}\biggr)^2 \cos\chi \biggr] \, ,</math>
</div>
via the [[Appendix/Ramblings/PowerSeriesExpressions#Binomial|binomial theorem]] we have,

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

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

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl[\frac{1}{2}\biggl( \frac{a}{c}\biggr) ~-~ \cos\chi  \biggr]^3 \biggl\{
1 -\frac{3}{2}\biggl[ b \biggr] + \frac{3\cdot 5}{2^3} \biggl[ b \biggr]^2 - \frac{3\cdot 5\cdot 7}{2^4\cdot 3}\biggl[ b \biggr]^3 + \frac{3\cdot 5\cdot 7\cdot 9}{2^7\cdot 3}\biggl[ b \biggr]^4 + \mathcal{O}(b^5)
\biggr\} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl[\frac{1}{2}\biggl( \frac{a}{c}\biggr) ~-~ \cos\chi  \biggr]^3 \biggl\{
1 -\frac{3}{2}\biggl[ - \biggl(\frac{a}{c}\biggr) + \frac{1}{4}\biggl(\frac{a}{c}\biggr)^2 \cos\chi  \biggr] + \frac{3\cdot 5}{2^3} \biggl[ - \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">
&nbsp;
  </td>
  <td align="left">
<math>~
- \frac{5\cdot 7}{2^4}\biggl[ - \biggl(\frac{a}{c}\biggr) + \frac{1}{4}\biggl(\frac{a}{c}\biggr)^2 \cos\chi  \biggr]^3 
+ \frac{5\cdot 7\cdot 9}{2^7}\biggl[ - \biggl(\frac{a}{c}\biggr) + \frac{1}{4}\biggl(\frac{a}{c}\biggr)^2 \cos\chi  \biggr]^4 + \mathcal{O}(b^5)
\biggr\} 
</math>
  </td>
</tr>

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

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

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+ \frac{5\cdot 7\cdot 9}{2^7}\biggl[ \biggl(\frac{a}{c}\biggr)^2 - \frac{1}{2} \biggl(\frac{a}{c}\biggr)^3 \cos\chi  + \frac{1}{2^4} \biggl(\frac{a}{c}\biggr)^4 \cos^2\chi \biggr]^2 + \mathcal{O}(b^5)
\biggr\} 
</math>
  </td>
</tr>

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