Editing SSC/Stability/n1PolytropeLAWE/Pt3 (section)

===Fourth Attempt===

====XXXX====
If we assume that, <math>~\alpha_e = (3 - 4/2) = 1</math> and <math>~\sigma_c^2 = 0</math>, then the relevant envelope LAWE is,

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

<tr>
  <td align="right">
<math>~0</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{d^2x}{d\eta^2} + \biggl\{ 4 -2Q \biggr\}\frac{1}{\eta} \cdot \frac{dx}{d\eta} 
~-~  \biggl[ 2 Q \biggr]   \frac{x}{\eta^2} \, ,
</math>
  </td>
</tr>
</table>
where,

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

<tr>
  <td align="right">
<math>~Q \equiv - \frac{d \ln \phi}{ d\ln \eta}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[1- \eta\cot(\eta-B_0) \biggr] \, .
</math>
  </td>
</tr>
</table>


Let's work through the analytic derivatives again.  Keeping in mind that,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{d}{d\eta}\biggl[\cot(\eta - B) \biggr]</math>
  </td>
  <td align="center">
<math>~=</math>
  <td align="left">
<math>~
- \biggl[ 1 + \cot^2(\eta - B)\biggr] \, ;
</math>
  </td>
</tr>
</table>
and that the,

<table border="0" cellpadding="5" align="center">
<tr>
  <td align="center" colspan="3"><font color="maroon"><b>Precise Solution to the Polytropic LAWE</b></font></td>
</tr>

<tr>
  <td align="right">
<math>~x_P</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{b(n-1)}{2n}\biggl[1 + \biggl(\frac{n-3}{n-1}\biggr) \biggl( \frac{1}{\eta \phi^{n}}\biggr) \frac{d\phi}{d\eta}\biggr]</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~-b\biggl[ \biggl( \frac{1}{\eta \phi}\biggr) \frac{d\phi}{d\eta}\biggr]</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{b}{\eta^2}\biggl[ -\frac{d\ln \phi}{d\ln \eta}\biggr] </math>
  </td>
</tr>

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

<tr>
  <td align="right">
<math>~\Rightarrow ~~~ x_P</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{b}{\eta^2} \biggl[1- \eta\cot(\eta-B_0) \biggr] \, .
</math>
  </td>
</tr>
</table>

As we have [[#First_Attempt|already tried once, above]], let's try a more general form of this expression, namely,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~x_Q</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
A + \frac{C}{(\eta - F)^2} \biggl[1 - (\eta-D) \cot(\eta-B) \biggr] \, .
</math>
  </td>
</tr>
</table>

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

<tr>
  <td align="right">
<math>~\frac{dx_Q}{d\eta}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[1 - (\eta-D) \cot(\eta-B) \biggr] \frac{d}{d\eta}\biggl[ \frac{C}{(\eta - F)^2} \biggr]
- \frac{C}{(\eta - F)^2} \frac{d}{d\eta} \biggl[ (\eta-D) \cot(\eta-B) \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[1 - (\eta-D) \cot(\eta-B) \biggr]\biggl[ \frac{-2C}{(\eta - F)^3} \biggr]
- \frac{C}{(\eta - F)^2} \biggl[ \cot(\eta-B) \biggr]
+ \frac{C(\eta - D)}{(\eta - F)^2}\biggl[1 + \cot^2(\eta - B)\biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{C}{(\eta-F)^3} \biggl\{
- 2 + 2(\eta-D) \cot(\eta-B) 
- (\eta - F) \biggl[ \cot(\eta-B) \biggr]
+ (\eta - D)(\eta - F) \biggl[1 + \cot^2(\eta - B)\biggr]
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{C}{(\eta-F)^3} \biggl\{
[(\eta - D)(\eta - F) - 2] 
+ (\eta - 2D + F) \cot(\eta-B) 
+ (\eta - D)(\eta - F) \cot^2(\eta - B)
\biggr\} \, .
</math>
  </td>
</tr>
</table>

And,

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

<tr>
  <td align="right">
<math>~\frac{d^2x_Q}{d\eta^2}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl\{
[(\eta - D)(\eta - F) - 2] 
+ (\eta - 2D + F) \cot(\eta-B) 
+ (\eta - D)(\eta - F) \cot^2(\eta - B)
\biggr\} \frac{d}{d\eta}\biggl[\frac{C}{(\eta-F)^3} \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~+\frac{C}{(\eta-F)^3} \cdot \frac{d}{d\eta} \biggl\{
[(\eta - D)(\eta - F) - 2] 
+ (\eta - 2D + F) \cot(\eta-B) 
+ (\eta - D)(\eta - F) \cot^2(\eta - B)
\biggr\} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl\{
[(\eta - D)(\eta - F) - 2] 
+ (\eta - 2D + F) \cot(\eta-B) 
+ (\eta - D)(\eta - F) \cot^2(\eta - B)
\biggr\} \biggl[\frac{-3C}{(\eta-F)^4} \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~+\frac{C}{(\eta-F)^3}  \biggl\{
[2\eta - (D+F) ] + \cot(\eta-B)
- (\eta - 2D + F) \biggl[1 + \cot^2(\eta-B) \biggr]
+ [2\eta -(D+F) ] \cot^2(\eta - B)
- 2[\eta^2 -\eta(D+F) + DF]\cot(\eta - B)\biggl[1 +  \cot^2(\eta - B)\biggr]
\biggr\} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{C}{(\eta-F)^4} \biggl\{
-3[(\eta - D)(\eta - F) - 2] 
- 3(\eta - 2D + F) \cot(\eta-B) 
- 3 (\eta - D)(\eta - F) \cot^2(\eta - B)
\biggr\} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~+\frac{C}{(\eta-F)^3}  \biggl\{
[2\eta - (D+F) ] 
- (\eta - 2D + F) 
+ \cot(\eta-B)
- (\eta - 2D + F) \cot^2(\eta-B)
+ [2\eta -(D+F) ] \cot^2(\eta - B)
- 2[\eta^2 -\eta(D+F) + DF]\cot(\eta - B)
- 2[\eta^2 -\eta(D+F) + DF]\cot^3(\eta - B)
\biggr\} 
</math>
  </td>
</tr>
</table>

====YYYY====
And,

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

<tr>
  <td align="right">
<math>~\frac{d^2 x_Q}{d\eta^2}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl\{
[(\eta - D)(\eta - F) - 2] 
+ (\eta - 2D + F) \cot(\eta-B) 
+ (\eta - D)(\eta - F) \cot^2(\eta - B)
\biggr\} \frac{d}{d\eta}\biggl[\frac{C}{(\eta-F)^3} \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+\frac{C}{(\eta-F)^3} \cdot \frac{d}{d\eta}\biggl\{
[(\eta - D)(\eta - F) - 2] 
+ (\eta - 2D + F) \cot(\eta-B) 
+ (\eta - D)(\eta - F) \cot^2(\eta - B)
\biggr\} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl\{
[(\eta - D)(\eta - F) - 2] 
+ (\eta - 2D + F) \cot(\eta-B) 
+ (\eta - D)(\eta - F) \cot^2(\eta - B)
\biggr\}\biggl[\frac{-3C}{(\eta-F)^4} \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+\frac{C}{(\eta-F)^3} \cdot \biggl\{
\frac{d}{d\eta}\biggl[ (\eta - 2D + F) \cot(\eta-B) \biggr]
+\frac{d}{d\eta}\biggl[  (\eta - D)(\eta - F) \cot^2(\eta - B)\biggr]
\biggr\} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{C}{(\eta-F)^4} \biggl\{
-3[(\eta - D)(\eta - F) - 2] 
-3(\eta - 2D + F) \cot(\eta-B) 
-3(\eta - D)(\eta - F) \cot^2(\eta - B)
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+
(\eta-F)\frac{d}{d\eta}\biggl[ (\eta - 2D + F) \cot(\eta-B) \biggr]
+(\eta-F)\frac{d}{d\eta}\biggl[  (\eta - D)(\eta - F) \cot^2(\eta - B)\biggr]
\biggr\} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{C}{(\eta-F)^4} \biggl\{
-3[(\eta - D)(\eta - F) - 2] 
-3(\eta - 2D + F) \cot(\eta-B) 
-3(\eta - D)(\eta - F) \cot^2(\eta - B)
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+ (\eta-F) \cot(\eta-B)  \frac{d}{d\eta}\biggl[ (\eta - 2D + F) \biggr]
+ (\eta-F) (\eta - 2D + F)  \frac{d}{d\eta}\biggl[ \cot(\eta-B) \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+(\eta-F) \cot^2(\eta - B) \frac{d}{d\eta}\biggl[  \eta^2 -\eta(D+F) + DF \biggr]
+(\eta-F) (\eta - D)(\eta - F) \frac{d}{d\eta}\biggl[ \cot^2(\eta - B)\biggr]
\biggr\} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{C}{(\eta-F)^4} \biggl\{
-3[(\eta - D)(\eta - F) - 2] 
-3(\eta - 2D + F) \cot(\eta-B) 
-3(\eta - D)(\eta - F) \cot^2(\eta - B)
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+ (\eta-F) \cot(\eta-B) 
- (\eta-F) (\eta - 2D + F)  \biggl[ 1 + \cot^2(\eta - B)\biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+(\eta-F) \cot^2(\eta - B) \biggl[  2\eta - (D+F) \biggr]
-2 (\eta-F) (\eta - D)(\eta - F) \cot(\eta - B)\biggl[ 1 + \cot^2(\eta - B)\biggr]
\biggr\} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{C}{(\eta-F)^4} \biggl\{
-3[(\eta - D)(\eta - F) - 2] 
- (\eta-F) (\eta - 2D + F) 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+ \biggl[ (\eta-F)
-3(\eta - 2D + F) 
-2 (\eta-F) (\eta - D)(\eta - F)\biggr] \cot(\eta - B)
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+\biggl[ (\eta-F) [  2\eta - (D+F) ]
-3(\eta - D)(\eta - F) 
-2 (\eta-F) (\eta - D)(\eta - F) \cot(\eta - B)
- (\eta-F) (\eta - 2D + F)\biggr] \cot^2(\eta - B)
\biggr\} 
</math>
  </td>
</tr>
</table>

So the envelope LAWE becomes,

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

<tr>
  <td align="right">
<math>~\frac{(\eta-F)^4}{C} \cdot \mathrm{LAWE}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{(\eta-F)^4}{C} \cdot \frac{d^2x_Q}{d\eta^2} + \frac{(\eta-F)^4}{C} \biggl[ 1 + \eta\cot(\eta-B_0)  \biggr] \frac{2}{\eta} \cdot \frac{dx_Q}{d\eta} 
~-~ \frac{(\eta-F)^4}{C}  \biggl[ 1- \eta\cot(\eta-B_0)  \biggr]   \frac{2x_Q}{\eta^2} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ \biggl\{
-3[(\eta - D)(\eta - F) - 2] 
- (\eta-F) (\eta - 2D + F) 
+ \biggl[ (\eta-F)
-3(\eta - 2D + F) 
-2 (\eta-F) (\eta - D)(\eta - F)\biggr] \cot(\eta - B)
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+\biggl[ (\eta-F) [  2\eta - (D+F) ]
-3(\eta - D)(\eta - F) 
-2 (\eta-F) (\eta - D)(\eta - F) \cot(\eta - B)
- (\eta-F) (\eta - 2D + F)\biggr] \cot^2(\eta - B)
\biggr\} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+ (\eta-F) \biggl[ 1 + \eta\cot(\eta-B_0)  \biggr] \frac{2}{\eta} \biggl\{ 
[(\eta - D)(\eta - F) - 2] 
+ (\eta - 2D + F) \cot(\eta-B) 
+ (\eta - D)(\eta - F) \cot^2(\eta - B)
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
~-~ \frac{(\eta-F)^4}{C}  \biggl[ 1- \eta\cot(\eta-B_0)  \biggr]   \frac{2}{\eta^2} \biggl\{
A + \frac{C}{(\eta - F)^2} \biggl[1 - (\eta-D) \cot(\eta-B) \biggr]
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ 
-3[(\eta - D)(\eta - F) - 2] 
- (\eta-F) (\eta - 2D + F) 
+ \biggl[ (\eta-F)
-3(\eta - 2D + F) 
-2 (\eta-F) (\eta - D)(\eta - F)\biggr] \cot(\eta - B)
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+ (\eta-F) \biggl[ 1 + \eta\cot(\eta-B_0)  \biggr] \frac{2}{\eta} \biggl\{ 
[(\eta - D)(\eta - F) - 2] 
+ (\eta - 2D + F) \cot(\eta-B) 
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+\biggl[ (\eta-F) [  2\eta - (D+F) ]
-3(\eta - D)(\eta - F) 
-2 (\eta-F) (\eta - D)(\eta - F) \cot(\eta - B)
- (\eta-F) (\eta - 2D + F)\biggr] \cot^2(\eta - B)
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+ (\eta-F) \biggl[ 1 + \eta\cot(\eta-B_0)  \biggr] \frac{2}{\eta} \biggl[ 
(\eta - D)(\eta - F) \cot^2(\eta - B)
\biggr]
~-~ (\eta-F)^2  \biggl[ 1- \eta\cot(\eta-B_0)  \biggr]   \frac{2}{\eta^2} \biggl[1 - (\eta-D) \cot(\eta-B) \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
~-~ \frac{(\eta-F)^4}{C}  \biggl[ 1- \eta\cot(\eta-B_0)  \biggr]   \frac{2A}{\eta^2} \, .
</math>
  </td>
</tr>
</table>

What does this reduce to if <math>~A = D = F = 0</math>.

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

<tr>
  <td align="right">
<math>~\frac{\eta^4}{C} \cdot \mathrm{LAWE}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ 
-3[(\eta )(\eta ) - 2] 
- (\eta) (\eta ) 
+ \biggl[ (\eta)
-3(\eta ) 
-2 (\eta) (\eta )(\eta )\biggr] \cot(\eta - B)
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+ (\eta) \biggl[ 1 + \eta\cot(\eta-B_0)  \biggr] \frac{2}{\eta} \biggl\{ 
[(\eta )(\eta ) - 2] 
+ (\eta ) \cot(\eta-B) 
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+\biggl[ (\eta) [  2\eta ]
-3(\eta )(\eta ) 
-2 (\eta) (\eta )(\eta) \cot(\eta - B)
- (\eta-) (\eta )\biggr] \cot^2(\eta - B)
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+ (\eta) \biggl[ 1 + \eta\cot(\eta-B_0)  \biggr] \frac{2}{\eta} \biggl[ 
(\eta )(\eta ) \cot^2(\eta - B)
\biggr]
~-~ (\eta)^2  \biggl[ 1- \eta\cot(\eta-B_0)  \biggr]   \frac{2}{\eta^2} \biggl[1 - (\eta) \cot(\eta-B) \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
~-~ \frac{(\eta)^4}{C}  \biggl[ 1- \eta\cot(\eta-B_0)  \biggr]   \frac{2A}{\eta^2} \, .
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ 
6 - 4\eta^2
-2 (\eta + \eta^3 ) \cot(\eta - B)
+ 2 \biggl[ 1 + \eta\cot(\eta-B_0)  \biggr] \biggl[
\eta^2 - 2 + \eta  \cot(\eta-B) 
\biggr]
- 2\biggl[ \eta^2 + \eta^3 \cot(\eta - B) \biggr] \cot^2(\eta - B)
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+ 2 \biggl[ 1 + \eta\cot(\eta-B_0)  \biggr]\biggl[ 
\eta^2 \cot^2(\eta - B)
\biggr]
~-~2  \biggl[ 1- \eta\cot(\eta-B_0)  \biggr] \biggl[1 - \eta \cot(\eta-B) \biggr]
~-~ \frac{2A\eta^2}{C}  \biggl[ 1- \eta\cot(\eta-B_0)  \biggr]    
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ 
6 - 4\eta^2
-2 (\eta + \eta^3 ) \cot(\eta - B)
+ 2\eta^2 - 4 + 2\eta  \cot(\eta-B) 
+ 2 \eta^3 \cot(\eta-B_0)
~-~4 \eta\cot(\eta-B_0) 
+ 2 \eta^2\cot(\eta-B_0) \cot(\eta-B) 
- 2\eta^2  \cot^2(\eta - B)
- 2 \eta^3  \cot^3(\eta - B)
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+ 2 \eta^2 \cot^2(\eta - B)
+ 2 \eta^3 \cot(\eta-B_0)  \cot^2(\eta - B)
-2 + 4 \eta\cot(\eta-B_0) - 2\eta^2\cot^2(\eta-B_0)  
~-~ \frac{2A\eta^2}{C}  \biggl[ 1- \eta\cot(\eta-B_0)  \biggr]    
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
- 2\eta^2
-2 \eta^3 \biggl[ \cot(\eta - B) + \cot^3(\eta - B) \biggr]
+ \cot(\eta-B_0) \biggl[ 2 \eta^3 
+ 2 \eta^2 \cot(\eta-B) 
+ 2 \eta^3  \cot^2(\eta - B)
\biggr] - 2\eta^2\cot^2(\eta-B_0)  
~-~ \frac{2A\eta^2}{C}  \biggl[ 1- \eta\cot(\eta-B_0)  \biggr]    
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
- 2\eta^2
+ 2 \eta^3\biggl[ \cot(\eta-B_0) - \cot(\eta-B)\biggr] \biggl[ 1 +  \cot^2(\eta - B)\biggr] 
+ 2 \eta^2  \cot(\eta-B_0)\biggl[ \cot(\eta-B)  - \cot(\eta-B_0)  \biggr]
~-~ \frac{2A\eta^2}{C}  \biggl[ 1- \eta\cot(\eta-B_0)  \biggr]  \, .  
</math>
  </td>
</tr>
</table>