Editing ParabolicDensity/Axisymmetric/Structure/Try8thru10 (section)

===9<sup>th</sup> Try===

====Starting Key Relations====

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

<tr>
  <td align="left"><font color="orange"><b>Density:</b></font></td>
  <td align="right">
<math>\frac{\rho(\varpi, z)}{\rho_c}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\biggl[1 - \chi^2 - \zeta^2(1-e^2)^{-1} \biggr] 
\, ,</math>
  </td>
</tr>

<tr>
  <td align="left"><font color="orange"><b>Gravitational Potential:</b></font></td>
  <td align="right">
<math>\frac{ \Phi_\mathrm{grav}(\varpi,z)}{(-\pi G\rho_c a_\ell^2)} </math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{1}{2} I_\mathrm{BT}  
- A_\ell \chi^2  - A_s \zeta^2  
 + \frac{1}{2}\biggl[(A_{s s} a_\ell^2) \zeta^4
+ 2(A_{\ell s}a_\ell^2 )\chi^2 \zeta^2
+ (A_{\ell \ell} a_\ell^2)  \chi^4 \biggr]
\, .
</math>
  </td>
</tr>

<tr>
  <td align="left"><font color="orange"><b>Vertical Pressure Gradient:</b></font></td>
  <td align="right"><math>\biggl[\frac{1}{(\pi G\rho_c^2 a_\ell^2)} \biggr] \frac{\partial P}{\partial \zeta}</math></td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
\frac{\rho}{\rho_c} \cdot  \biggl[
2A_{\ell s}a_\ell^2 \chi^2\zeta - 2A_s \zeta  
+  2A_{ss} a_\ell^2  \zeta^3   
\biggr]
</math>
  </td>
</tr>
</table>

====Play With Vertical Pressure Gradient====

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

<tr>
  <td align="right"><math>\biggl[\frac{1}{(\pi G\rho_c^2 a_\ell^2)} \biggr] \frac{\partial P}{\partial \zeta}</math></td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
\biggl[1 - \chi^2 - \zeta^2(1-e^2)^{-1} \biggr] \biggl[
2A_{\ell s}a_\ell^2 \chi^2\zeta - 2A_s \zeta  
+  2A_{ss} a_\ell^2  \zeta^3   
\biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">&nbsp;</td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
\biggl[ (2A_{\ell s}a_\ell^2 \chi^2 - 2A_s )\zeta  +  2A_{ss} a_\ell^2  \zeta^3   \biggr]
- \chi^2 \biggl[ (2A_{\ell s}a_\ell^2 \chi^2 - 2A_s )\zeta  +  2A_{ss} a_\ell^2  \zeta^3   \biggr]
- \zeta^2(1-e^2)^{-1}\biggl[ (2A_{\ell s}a_\ell^2 \chi^2 - 2A_s )\zeta  +  2A_{ss} a_\ell^2  \zeta^3   \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">&nbsp;</td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
(2A_{\ell s}a_\ell^2 \chi^2 - 2A_s )\zeta  +  2A_{ss} a_\ell^2  \zeta^3 
- (2A_{\ell s}a_\ell^2 \chi^4 - 2A_s \chi^2)\zeta  -  2A_{ss} a_\ell^2 \chi^2 \zeta^3 
- (1-e^2)^{-1}\biggl[ (2A_{\ell s}a_\ell^2 \chi^2 - 2A_s )\zeta^3  +  2A_{ss} a_\ell^2  \zeta^5   \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">&nbsp;</td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
\biggl[ (2A_{\ell s}a_\ell^2 \chi^2 - 2A_s ) - (2A_{\ell s}a_\ell^2 \chi^4 - 2A_s \chi^2)\biggr]\zeta  
+  \biggl[ 2A_{ss} a_\ell^2  -  2A_{ss} a_\ell^2 \chi^2 - (1-e^2)^{-1}(2A_{\ell s}a_\ell^2 \chi^2 - 2A_s )\biggr]\zeta^3 
+ \biggl[ - (1-e^2)^{-1}2A_{ss} a_\ell^2 \biggr] \zeta^5 
\, .
</math>
  </td>
</tr>
</table>
Integrate over <math>\zeta</math> gives &hellip;

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

<tr>
  <td align="right"><math>\biggl[\frac{1}{(\pi G\rho_c^2 a_\ell^2)} \biggr] \int \biggl[\frac{\partial P}{\partial \zeta}\biggr] d\zeta </math></td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
\biggl[ (A_{\ell s}a_\ell^2 \chi^2 - A_s ) - (A_{\ell s}a_\ell^2 \chi^4 - A_s \chi^2)\biggr]\zeta^2  
+  \frac{1}{2}\biggl[ A_{ss} a_\ell^2  -  A_{ss} a_\ell^2 \chi^2 - (1-e^2)^{-1}(A_{\ell s}a_\ell^2 \chi^2 - A_s )\biggr]\zeta^4 
+ \frac{1}{3}\biggl[ - (1-e^2)^{-1}A_{ss} a_\ell^2 \biggr] \zeta^6 + ~\mathrm{const}
</math>
  </td>
</tr>

<tr>
  <td align="right">&nbsp;</td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
\biggl[-A_s \zeta^2 + \frac{1}{2}A_{ss}a_\ell^2 \zeta^4 + \frac{1}{2}(1-e^2)^{-1}A_s\zeta^4 - \frac{1}{3}(1-e^2)^{-1}A_{ss} a_\ell^2  \zeta^6 \biggr]\chi^0
+ \biggl[ A_{\ell s}a_\ell^2 \zeta^2 + A_s\zeta^2
- \frac{1}{2}A_{ss}a_\ell^2 \zeta^4 - \frac{1}{2}(1-e^2)^{-1}(A_{\ell s}a_\ell^2 \zeta^4 )
\biggr]\chi^2  
+  \biggl[- A_{\ell s}a_\ell^2 \zeta^2 \biggr]\chi^4 + ~\mathrm{const.}
</math>
  </td>
</tr>
</table>

====Now Play With Radial Pressure Gradient====

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

<tr>
  <td align="right"><math>\biggl[\frac{1}{(-\pi G\rho_c a_\ell^2)} \biggr] \frac{\partial \Phi}{\partial \chi}</math></td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
\frac{\rho}{\rho_c} \cdot   \biggl\{
- 2A_\ell \chi + \frac{1}{2}\biggl[
4(A_{\ell s} a_\ell^2)\zeta^2\chi
+ 4(A_{\ell\ell} a_\ell^2)\chi^3
\biggl] \biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">&nbsp;</td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
2\biggl[1 - \chi^2 - \zeta^2(1-e^2)^{-1} \biggr]  \biggl[
(A_{\ell s} a_\ell^2 \zeta^2 - A_\ell )\chi 
+ A_{\ell\ell} a_\ell^2 \chi^3
\biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">&nbsp;</td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
2\biggl[ (A_{\ell s} a_\ell^2 \zeta^2 - A_\ell )\chi + A_{\ell\ell} a_\ell^2 \chi^3\biggr]
- 2\chi^2
\biggl[ (A_{\ell s} a_\ell^2 \zeta^2 - A_\ell )\chi + A_{\ell\ell} a_\ell^2 \chi^3\biggr]
- 2\zeta^2(1-e^2)^{-1}
\biggl[(A_{\ell s} a_\ell^2 \zeta^2 - A_\ell )\chi + A_{\ell\ell} a_\ell^2 \chi^3\biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">&nbsp;</td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
2(A_{\ell s} a_\ell^2 \zeta^2 - A_\ell )\chi 
+ 2\biggl[ A_{\ell\ell} a_\ell^2 
+
(A_\ell - A_{\ell s} a_\ell^2 \zeta^2 ) \biggr]\chi^3
- 2A_{\ell\ell} a_\ell^2 \chi^5
+ 2(1-e^2)^{-1}
\biggl[(A_\ell\zeta^2 - A_{\ell s} a_\ell^2 \zeta^4 )\chi - A_{\ell\ell} a_\ell^2 \zeta^2\chi^3\biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">&nbsp;</td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
2\biggl[ (A_{\ell s} a_\ell^2 \zeta^2 - A_\ell ) + (1-e^2)^{-1}(A_\ell\zeta^2 - A_{\ell s} a_\ell^2 \zeta^4 )\biggr]\chi 
+ 2\biggl[ A_{\ell\ell} a_\ell^2 + (A_\ell - A_{\ell s} a_\ell^2 \zeta^2 ) - (1-e^2)^{-1}A_{\ell\ell} a_\ell^2 \zeta^2\biggr]\chi^3
- 2A_{\ell\ell} a_\ell^2 \chi^5
 </math>
  </td>
</tr>
</table>
Add a term <math>j^2 \sim (j_4^2\chi^4 + j_6^2\chi^6)</math> to account for centrifugal acceleration &hellip;

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

<tr>
  <td align="right"><math>\biggl[\frac{1}{(\pi G\rho_c^2 a_\ell^2)} \biggr] \frac{\partial P}{\partial \chi}
=
\biggl[\frac{1}{(-\pi G\rho_c a_\ell^2)} \biggr] \frac{\partial \Phi}{\partial \chi} 
+ \frac{j^2}{\chi^3}\biggl[\frac{\rho}{\rho_c}\biggr]</math></td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
2\biggl[ (A_{\ell s} a_\ell^2 \zeta^2 - A_\ell ) + (1-e^2)^{-1}(A_\ell\zeta^2 - A_{\ell s} a_\ell^2 \zeta^4 )\biggr]\chi 
+ 2\biggl[ A_{\ell\ell} a_\ell^2 + (A_\ell - A_{\ell s} a_\ell^2 \zeta^2 ) - (1-e^2)^{-1}A_{\ell\ell} a_\ell^2 \zeta^2\biggr]\chi^3
- 2A_{\ell\ell} a_\ell^2 \chi^5
+ \frac{j^2}{\chi^3}\biggl[1 - \chi^2 - \zeta^2(1-e^2)^{-1} \biggr]
 </math>
  </td>
</tr>

<tr>
  <td align="right">&nbsp;</td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
2\biggl[ (A_{\ell s} a_\ell^2 \zeta^2 - A_\ell ) + (1-e^2)^{-1}(A_\ell\zeta^2 - A_{\ell s} a_\ell^2 \zeta^4 )\biggr]\chi 
+ 2\biggl[ A_{\ell\ell} a_\ell^2 + (A_\ell - A_{\ell s} a_\ell^2 \zeta^2 ) - (1-e^2)^{-1}A_{\ell\ell} a_\ell^2 \zeta^2\biggr]\chi^3
- 2A_{\ell\ell} a_\ell^2 \chi^5
 </math>
  </td>
</tr>

<tr>
  <td align="right">&nbsp;</td>
  <td align="center">&nbsp;</td>
  <td align="left">
<math>
+ \frac{(j_4^2\chi^4 + j_6^2\chi^6)}{\chi^3}
- \frac{(j_4^2\chi^4 + j_6^2\chi^6)}{\chi^3}\biggl[\chi^2 \biggr]
- \frac{(j_4^2\chi^4 + j_6^2\chi^6)}{\chi^3}\biggl[\zeta^2(1-e^2)^{-1} \biggr]
 </math>
  </td>
</tr>

<tr>
  <td align="right">&nbsp;</td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
2\biggl[ (A_{\ell s} a_\ell^2 \zeta^2 - A_\ell ) + (1-e^2)^{-1}(A_\ell\zeta^2 - A_{\ell s} a_\ell^2 \zeta^4 )\biggr]\chi 
+ 2\biggl[ A_{\ell\ell} a_\ell^2 + (A_\ell - A_{\ell s} a_\ell^2 \zeta^2 ) - (1-e^2)^{-1}A_{\ell\ell} a_\ell^2 \zeta^2\biggr]\chi^3
- 2A_{\ell\ell} a_\ell^2 \chi^5
 </math>
  </td>
</tr>

<tr>
  <td align="right">&nbsp;</td>
  <td align="center">&nbsp;</td>
  <td align="left">
<math>
+ (j_4^2\chi + j_6^2\chi^3)
- (j_4^2\chi + j_6^2\chi^3)\biggl[\zeta^2(1-e^2)^{-1} \biggr]
- (j_4^2\chi^3 + j_6^2\chi^5)
 </math>
  </td>
</tr>

<tr>
  <td align="right">&nbsp;</td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
2\biggl[ (A_{\ell s} a_\ell^2 \zeta^2 - A_\ell ) + (1-e^2)^{-1}(A_\ell\zeta^2 - A_{\ell s} a_\ell^2 \zeta^4 )\biggr]\chi 
+ 2\biggl[ A_{\ell\ell} a_\ell^2 + (A_\ell - A_{\ell s} a_\ell^2 \zeta^2 ) - (1-e^2)^{-1}A_{\ell\ell} a_\ell^2 \zeta^2\biggr]\chi^3
- 2A_{\ell\ell} a_\ell^2 \chi^5
 </math>
  </td>
</tr>

<tr>
  <td align="right">&nbsp;</td>
  <td align="center">&nbsp;</td>
  <td align="left">
<math>
- \biggl[j_4^2\zeta^2(1-e^2)^{-1} - j_4^2\biggr]\chi
- \biggl[j_4^2 + j_6^2\zeta^2(1-e^2)^{-1} - j_6^2  \biggr]\chi^3 
- \biggl[j_6^2\biggr]\chi^5
 </math>
  </td>
</tr>

<tr>
  <td align="right">&nbsp;</td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
\biggl[ 2(A_{\ell s} a_\ell^2 \zeta^2 - A_\ell ) + 2(1-e^2)^{-1}(A_\ell\zeta^2 - A_{\ell s} a_\ell^2 \zeta^4 ) - j_4^2\zeta^2(1-e^2)^{-1} + j_4^2\biggr]\chi 
 </math>
  </td>
</tr>

<tr>
  <td align="right">&nbsp;</td>
  <td align="center">&nbsp;</td>
  <td align="left">
<math>
+ \biggl[ 2A_{\ell\ell} a_\ell^2 + 2(A_\ell - A_{\ell s} a_\ell^2 \zeta^2 ) - 2(1-e^2)^{-1}A_{\ell\ell} a_\ell^2 \zeta^2
- j_4^2 - j_6^2\zeta^2(1-e^2)^{-1} + j_6^2  \biggr]\chi^3 
+ \biggl[-j_6^2 - 2A_{\ell\ell} a_\ell^2 \biggr]\chi^5
 </math>
  </td>
</tr>
</table>

Integrate over <math>\chi</math> gives &hellip;

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

<tr>
  <td align="right"><math>\biggl[\frac{1}{(\pi G\rho_c^2 a_\ell^2)} \biggr] \int \biggl[\frac{\partial P}{\partial \chi}\biggr] d\chi </math></td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
\biggl[ (A_{\ell s} a_\ell^2 \zeta^2 - A_\ell ) + (1-e^2)^{-1}(A_\ell\zeta^2 - A_{\ell s} a_\ell^2 \zeta^4 ) - \frac{1}{2}j_4^2\zeta^2(1-e^2)^{-1} + \frac{1}{2}j_4^2\biggr]\chi^2 
 </math>
  </td>
</tr>

<tr>
  <td align="right">&nbsp;</td>
  <td align="center">&nbsp;</td>
  <td align="left">
<math>
+ \biggl[ \frac{1}{2}A_{\ell\ell} a_\ell^2 + \frac{1}{2}(A_\ell - A_{\ell s} a_\ell^2 \zeta^2 ) - \frac{1}{2}(1-e^2)^{-1}A_{\ell\ell} a_\ell^2 \zeta^2
- \frac{1}{4}j_4^2 - \frac{1}{4}j_6^2\zeta^2(1-e^2)^{-1} + \frac{1}{4}j_6^2  \biggr]\chi^4 
- \biggl[\frac{1}{6}j_6^2 + \frac{1}{3}A_{\ell\ell} a_\ell^2 \biggr]\chi^6
 </math>
  </td>
</tr>
</table>

====Compare Pair of Integrations====

<table border="1" align="center" cellpadding="8">

<tr>
  <td align="center" width="6%">&nbsp;</td>
  <td align="center" width="47%">Integration over <math>\zeta</math></td>
  <td align="center">Integration over <math>\chi</math></td>
</tr>

<tr>
  <td align="center"><math>\chi^0</math></td>
  <td align="right"><math>-A_s \zeta^2 + \frac{1}{2}A_{ss}a_\ell^2 \zeta^4 + \frac{1}{2}(1-e^2)^{-1}A_s\zeta^4 - \frac{1}{3}(1-e^2)^{-1}A_{ss} a_\ell^2  \zeta^6 </math></td>
  <td align="left">none</td>
</tr>

<tr>
  <td align="center"><math>\chi^2</math></td>
  <td align="right">
<math>A_{\ell s}a_\ell^2 \zeta^2 + A_s\zeta^2 - \frac{1}{2}A_{ss}a_\ell^2 \zeta^4 - \frac{1}{2}(1-e^2)^{-1}(A_{\ell s}a_\ell^2 \zeta^4 )</math>
  </td>
  <td align="left">
<math>(A_{\ell s} a_\ell^2 \zeta^2 - A_\ell ) + (1-e^2)^{-1}(A_\ell\zeta^2 - A_{\ell s} a_\ell^2 \zeta^4 ) - \frac{1}{2}j_4^2\zeta^2(1-e^2)^{-1} + \frac{1}{2}j_4^2</math>
  </td>
</tr>

<tr>
  <td align="center"><math>\chi^4</math></td>
  <td align="right">
<math>- A_{\ell s}a_\ell^2 \zeta^2 </math>
  </td>
  <td align="left">
<math>\frac{1}{2}A_{\ell\ell} a_\ell^2 + \frac{1}{2}(A_\ell - A_{\ell s} a_\ell^2 \zeta^2 ) - \frac{1}{2}(1-e^2)^{-1}A_{\ell\ell} a_\ell^2 \zeta^2
- \frac{1}{4}j_4^2 - \frac{1}{4}j_6^2\zeta^2(1-e^2)^{-1} + \frac{1}{4}j_6^2  </math>
  </td>
</tr>

<tr>
  <td align="center"><math>\chi^6</math></td>
  <td align="right">
none
  </td>
  <td align="left">
<math>
- \frac{1}{6}j_6^2 - \frac{1}{3}A_{\ell\ell} a_\ell^2 
</math>
  </td>
</tr>
</table>

Try, <math>j_6^2 = [-2A_{\ell\ell}a_\ell^2]</math> and <math>\frac{1}{2}j_4^2 = [A_\ell + (A_{\ell s} a_\ell^2) \zeta^2 ]</math>.

<table border="1" align="center" cellpadding="8">

<tr>
  <td align="center" width="6%">&nbsp;</td>
  <td align="center" width="47%">Integration over <math>\zeta</math></td>
  <td align="center">Integration over <math>\chi</math></td>
</tr>

<tr>
  <td align="center"><math>\chi^0</math></td>
  <td align="right"><math>-A_s \zeta^2 + \frac{1}{2}A_{ss}a_\ell^2 \zeta^4 + \frac{1}{2}(1-e^2)^{-1}A_s\zeta^4 - \frac{1}{3}(1-e^2)^{-1}A_{ss} a_\ell^2  \zeta^6 </math></td>
  <td align="left">none</td>
</tr>

<tr>
  <td align="center"><math>\chi^2</math></td>
  <td align="right"><math>A_{\ell s}a_\ell^2 \zeta^2 + A_s\zeta^2
- \frac{1}{2}A_{ss}a_\ell^2 \zeta^4 - \frac{1}{2}(1-e^2)^{-1}(A_{\ell s}a_\ell^2 \zeta^4 )</math></td>
  <td align="left">
<math>
(A_{\ell s} a_\ell^2 \zeta^2 - A_\ell ) + (1-e^2)^{-1}(A_\ell\zeta^2 - A_{\ell s} a_\ell^2 \zeta^4 ) - \frac{1}{2}j_4^2\zeta^2(1-e^2)^{-1} + \frac{1}{2}j_4^2
</math>
<br /><math>=</math><br />
<math>
(A_{\ell s} a_\ell^2 \zeta^2 - A_\ell ) + (1-e^2)^{-1}(A_\ell\zeta^2 - A_{\ell s} a_\ell^2 \zeta^4 ) - [A_\ell + (A_{\ell s} a_\ell^2) \zeta^2 ]\zeta^2(1-e^2)^{-1} + [A_\ell + (A_{\ell s} a_\ell^2) \zeta^2 ]
</math>
<br /><math>=</math><br />
<math>
2(A_{\ell s} a_\ell^2) \zeta^2\biggl[1 - \zeta^2 (1-e^2)^{-1} \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="center"><math>\chi^4</math></td>
  <td align="right">
<math>- A_{\ell s}a_\ell^2 \zeta^2 </math>
  </td>
  <td align="left">
<math>
\frac{1}{2}A_{\ell\ell} a_\ell^2 + \frac{1}{2}(A_\ell - A_{\ell s} a_\ell^2 \zeta^2 ) - \frac{1}{2}(1-e^2)^{-1}A_{\ell\ell} a_\ell^2 \zeta^2
- \frac{1}{4}j_4^2 - \frac{1}{4}[-2A_{\ell\ell}a_\ell^2]\zeta^2(1-e^2)^{-1} + \frac{1}{4}[-2A_{\ell\ell}a_\ell^2]  
</math>
<br /><math>=</math><br />
<math>
\frac{1}{4}\biggl[2(A_\ell - A_{\ell s} a_\ell^2 \zeta^2 ) - 2[A_\ell + (A_{\ell s} a_\ell^2) \zeta^2 ] \biggr] = - A_{\ell s}a_\ell^2 \zeta^2
</math>
  </td>
</tr>

<tr>
  <td align="center"><math>\chi^6</math></td>
  <td align="right">
none
  </td>
  <td align="left">
<math>
0 
</math>
  </td>
</tr>
</table>

What expression for <math>j_4^2</math> is required in order to ensure that the <math>\chi^2</math> term is the same in both columns?

<table border="0" align="center" cellpadding="8">

<tr>
  <td align="right"><math>
\frac{1}{2}j_4^2 \biggl[ 1 - \zeta^2(1-e^2)^{-1}\biggr]</math></td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
\biggl[ A_{\ell s}a_\ell^2 \zeta^2 + A_s\zeta^2 - \frac{1}{2}A_{ss}a_\ell^2 \zeta^4 - \frac{1}{2}(1-e^2)^{-1}(A_{\ell s}a_\ell^2 \zeta^4 )\biggr]
-
\biggl[(A_{\ell s} a_\ell^2 \zeta^2 - A_\ell ) + (1-e^2)^{-1}(A_\ell\zeta^2 - A_{\ell s} a_\ell^2 \zeta^4 )  \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">&nbsp;</td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
\biggl[ A_s\zeta^2 - \frac{1}{2}A_{ss}a_\ell^2 \zeta^4 - \frac{1}{2}(1-e^2)^{-1}(A_{\ell s}a_\ell^2 \zeta^4 )\biggr]
+
\biggl[( A_\ell ) - (1-e^2)^{-1}(A_\ell\zeta^2 ) + (1-e^2)^{-1}( A_{\ell s} a_\ell^2 \zeta^4 ) \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">&nbsp;</td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
\biggl[ A_s\zeta^2 - \frac{1}{2}A_{ss}a_\ell^2 \zeta^4 + \frac{1}{2}(1-e^2)^{-1}(A_{\ell s}a_\ell^2) \zeta^4\biggr]
+
A_\ell\biggl[1 - (1-e^2)^{-1}\zeta^2 \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right"><math>
\Rightarrow ~~~ \frac{1}{2}j_4^2 \biggl[ 1 - \zeta^2(1-e^2)^{-1}\biggr]
-
A_\ell\biggl[1 - \zeta^2(1-e^2)^{-1} \biggr]
</math></td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
\frac{1}{2}(1-e^2)^{-1}(A_{\ell s}a_\ell^2) \zeta^4
+ \biggl[ A_s \biggr]\zeta^2 
- \frac{1}{2}\biggl[ A_{ss}a_\ell^2 \biggr] \zeta^4
</math>
  </td>
</tr>
</table>
Now, considering the following three relations &hellip;

<table border="0" align="center" cellpadding="8">

<tr>
  <td align="right">
<math>
\frac{3}{2}(A_{ss}a_\ell^2)
</math>
  </td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
(1-e^2)^{-1} - (A_{\ell s}a_\ell^2) \, ;
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>
A_s
</math>
  </td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
A_\ell + e^2(A_{\ell s}a_\ell^2) \, ;
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>
e^2(A_{\ell s}a_\ell^2)
</math>
  </td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
2 - 3 A_\ell \, ;
</math>
  </td>
</tr>
</table>
we can write,

<table border="0" align="center" cellpadding="8">

<tr>
  <td align="right"><math>
\frac{1}{2}j_4^2 \biggl[ 1 - \zeta^2(1-e^2)^{-1}\biggr]
-
A_\ell\biggl[1 - \zeta^2(1-e^2)^{-1} \biggr]
</math></td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
\frac{1}{2}(1-e^2)^{-1}(A_{\ell s}a_\ell^2) \zeta^4
+ \biggl[ A_\ell + e^2(A_{\ell s}a_\ell^2) \biggr]\zeta^2 
- \frac{1}{3}\biggl[ (1-e^2)^{-1} - (A_{\ell s}a_\ell^2)\biggr] \zeta^4
</math>
  </td>
</tr>

<tr>
  <td align="right"><math>\Rightarrow ~~~
3j_4^2 \biggl[ 1 - \zeta^2(1-e^2)^{-1}\biggr]
-
3A_\ell\biggl[2 - 2\zeta^2(1-e^2)^{-1} \biggr]
</math></td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
3(1-e^2)^{-1}(A_{\ell s}a_\ell^2) \zeta^4
+ 6\biggl[ A_\ell + e^2(A_{\ell s}a_\ell^2) \biggr]\zeta^2 
- 2\biggl[ (1-e^2)^{-1} - (A_{\ell s}a_\ell^2)\biggr] \zeta^4
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
(A_{\ell s}a_\ell^2)\biggl\{2\zeta^4 + 3\zeta^4(1-e^2)^{-1} + 6 e^2\zeta^2 \biggr\}
- 2\zeta^4 (1-e^2)^{-1} + 6A_\ell \zeta^2 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
- 2\zeta^4 (1-e^2)^{-1} + 6A_\ell \zeta^2 
+
\biggl[2 - 3A_\ell  \biggr]\biggl\{2\zeta^4 + 3\zeta^4(1-e^2)^{-1} + 6 e^2\zeta^2 \biggr\}\frac{1}{e^2}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
- 2\zeta^4 (1-e^2)^{-1}  
- 3A_\ell\biggl\{2\zeta^4 + 3\zeta^4(1-e^2)^{-1} + 4 e^2\zeta^2 \biggr\}\frac{1}{e^2}
+
\biggl\{4\zeta^4 + 6\zeta^4(1-e^2)^{-1} + 12 e^2\zeta^2 \biggr\}\frac{1}{e^2}
</math>
  </td>
</tr>
</table>


<table border="0" align="center" cellpadding="8">

<tr>
  <td align="right"><math>\Rightarrow ~~~
3j_4^2 \biggl[ 1 - \zeta^2(1-e^2)^{-1}\biggr]
</math></td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
- 2\zeta^4 (1-e^2)^{-1} 
+ \frac{3A_\ell(1-e^2)^{-1}}{e^2}\biggl\{
\biggl[2e^2(1-e^2) - 2e^2\zeta^2 \biggr]
- \biggl[2\zeta^4(1-e^2) + 3\zeta^4 + 4 e^2(1-e^2)\zeta^2 \biggr]
\biggr\}
+
\biggl\{4\zeta^4 + 6\zeta^4(1-e^2)^{-1} + 12 e^2\zeta^2 \biggr\}\frac{1}{e^2}
</math>
  </td>
</tr>
</table>