Editing ParabolicDensity/Axisymmetric/Structure/Try8thru10 (section)

===7<sup>th</sup> Try===

====Introduction====
<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(\chi, \zeta)}{\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}(\chi,\zeta)}{(-\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="purple"><b>Specific Angular Momentum:</b></font></td>
  <td align="right">
<math>
\frac{j^2 }{(\pi G \rho_c a_\ell^4)} \cdot \frac{1}{\chi^3}
</math>
  </td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
2j_1 \chi  - 2 j_3 \chi^3 
\, .
</math>
  </td>
</tr>

<tr>
  <td align="left"><font color="purple"><b>Centrifugal Potential:</b></font></td>
  <td align="right">
<math>
\frac{\Psi }{(\pi G \rho_c a_\ell^2)}
</math>
  </td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
\frac{1}{2}\biggl[j_3 \chi^4   -2j_1 \chi^2  \biggr]\, .
</math>
  </td>
</tr>
</table>

<table border="1" align="center" width="80%" cellpadding="8"><tr><td align="left">
[[#Index_Symbol_Expressions|From above]], we recall the following relations:

<table align="center" border=0 cellpadding="3">

<tr>
  <td align="right">
<math>
4e^4(A_{\ell \ell}a_\ell^2 )
</math>
  </td>
  <td align="center">
<math>
=
</math>
  </td>
  <td align="left">
<math>
- (3 + 2e^2) (1-e^2) + \Upsilon
\, ;
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\frac{3}{2} e^4(A_{ss}a_\ell^2 ) </math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{( 4e^2 - 3 )}{(1-e^2)} 
+
\Upsilon
\, ;
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>
e^4(A_{\ell s}a_\ell^2 )
</math>
  </td>
  <td align="center">
<math>
=
</math>
  </td>
  <td align="left">
<math>
(3-e^2) 
-
\Upsilon
\, .
</math>
  </td>
</tr>
</table>
where,

<table align="center" border=0 cellpadding="3">

<tr>
  <td align="right">
<math>
\Upsilon
</math>
  </td>
  <td align="center">
<math>
\equiv
</math>
  </td>
  <td align="left">
<math>
3 (1 - e^2)^{1 / 2} \biggl[\frac{\sin^{-1}e}{e}\biggr] 
\, .
</math>
  </td>
</tr>
</table>
<font color="red">Crosscheck</font> &hellip; Given that,

<table align="center" border=0 cellpadding="3">

<tr>
  <td align="right">
<math>
\Upsilon
</math>
  </td>
  <td align="center">
<math>
=
</math>
  </td>
  <td align="left">
<math>
(3-e^2) - e^4(A_{\ell s}a_\ell^2 )
\, .
</math>
  </td>
</tr>
</table>
we obtain the pair of relations,

<table align="center" border=0 cellpadding="3">

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

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

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

<tr>
  <td align="right">
<math>
\Rightarrow ~~~ (A_{\ell \ell}a_\ell^2 )
</math>
  </td>
  <td align="center">
<math>
=
</math>
  </td>
  <td align="left">
<math>
\frac{1}{2}  - \frac{1}{4}(A_{\ell s}a_\ell^2 )
\, ;
</math>
  </td>
</tr>

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

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

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

<tr>
  <td align="right">
<math>\Rightarrow ~~~ (A_{ss}a_\ell^2 ) </math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{2}{3}\biggl[ \frac{1}{(1-e^2)} - (A_{\ell s}a_\ell^2 )\biggr]
\, .
</math>
  </td>
</tr>
</table>

</td></tr></table>

====RHS Square Brackets (TERM1)====
Let's rewrite the term inside square brackets on the RHS of the expression for the gravitational potential.

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

<tr>
  <td align="right">
<math>\biggl[ ~~ \biggr]_\mathrm{RHS}</math>
  </td>
  <td align="center">
<math>\equiv</math>
  </td>
  <td align="left">
<math>
\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="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
e^{-4} \biggl\{
\frac{2}{3}\biggl[ \frac{( 4e^2 - 3 )}{(1-e^2)} + \Upsilon\biggr] \zeta^4
+ 2\biggl[ (3-e^2) - \Upsilon \biggr]\chi^2 \zeta^2
+ \frac{1}{4}\biggl[ - (3 + 2e^2) (1-e^2) + \Upsilon \biggr]  \chi^4 
\biggr\}
</math>
  </td>
</tr>

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

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

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>
+~
e^{-4}\biggl\{ \frac{2}{3}\biggl[ (\zeta^2 - \chi^2)(\zeta^2-2\chi^2) - \frac{13}{8}\chi^4  \biggr]\Upsilon
\biggr\}
</math>
  </td>
</tr>

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

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>
+~
e^{-4}\biggl\{ \frac{2}{3}\biggl[ (\zeta^2 - \chi^2)(\zeta^2-2\chi^2) - \frac{13}{8}\chi^4  \biggr]\Upsilon
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
- ~ \frac{2e^{-4}}{(1-e^2)}\biggl\{
\zeta^4
- 3 (1-e^2)\chi^2 \zeta^2
+ \frac{3}{8} (1-e^2)^2 \chi^4 
\biggr\}
+ ~ \frac{8e^{-2}}{3(1-e^2)}\biggl\{
\zeta^4
- \frac{3}{4} (1-e^2)\chi^2 \zeta^2
- \frac{3}{16} (1-e^2)^2 \chi^4 
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>
+~
\frac{2e^{-4}}{3}\biggl[ (\zeta^2 - \chi^2)(\zeta^2-2\chi^2) - \frac{13}{8}\chi^4  \biggr]\Upsilon
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
- ~ \frac{2e^{-4}}{(1-e^2)}\biggl\{ \underbrace{
\biggl[\zeta^2 - (1-e^2)\chi^2\biggr]\biggl[ \zeta^2 - 2(1-e^2)\chi^2\biggr]
- \frac{13}{8}(1-e^2)^2\chi^4}_{-0.038855}
\biggr\}

+ ~ \frac{8e^{-2}}{3(1-e^2)}\biggl\{ \overbrace{
\biggl[\zeta^2 - (1-e^2)\chi^2\biggr]\biggl[ \zeta^2 + \frac{1}{4}(1-e^2)\chi^2\biggr]
+ \frac{1}{16}(1-e^2)^2\chi^4}^{-0.010124}
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>
+~
\frac{2e^{-4}}{3}\biggl[\underbrace{ (\zeta^2 - \chi^2)(\zeta^2-2\chi^2) - \frac{13}{8}\chi^4 }_{-0.061608} \biggr]\Upsilon
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
0.212119014
</math>
&nbsp; &nbsp; &nbsp; ([[#Example_Evaluation|example #1]], below) .
  </td>
</tr>
</table>

Check #1:
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>
(\zeta^2 - \chi^2)(\zeta^2-2\chi^2) - \frac{13}{8}\chi^4  
</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\zeta^4 -3\chi^2\zeta^2 +2\chi^4 - \frac{13}{8}\chi^4
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\zeta^4 -3\chi^2\zeta^2 + \frac{3}{8}\chi^4 \, .
</math>
  </td>
</tr>
</table>

Check #2:
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>
(\zeta^2 - \chi^2)(\zeta^2 + \frac{1}{4}\chi^2) + \frac{1}{16}\chi^4  
</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\zeta^4 - \frac{3}{4}\chi^2\zeta^2 - \frac{1}{4}\chi^4 + \frac{1}{16}\chi^4
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\zeta^4 - \frac{3}{4}\chi^2\zeta^2 - \frac{3}{16}\chi^4
</math>
  </td>
</tr>
</table>

====RHS Quadratic Terms (TERM2)====

The quadratic terms on the RHS can be rewritten as,
<table border="0" cellpadding="5" align="center">

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

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

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

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

<tr>
  <td align="right">&nbsp;</td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
\frac{(\Upsilon - 3)}{3e^2} (\chi + \sqrt{2}\zeta)(\chi - \sqrt{2} \zeta)
+ \chi^2 
</math>
  </td>
</tr>

<tr>
  <td align="right"><math>\mathrm{TERM2}</math></td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
0.401150 ~~~ 
</math>
([[#Example_Evaluation|example #1]], below) .
  </td>
</tr>
</table>

where, again,
<table align="center" border=0 cellpadding="3">

<tr>
  <td align="right">
<math>
\Upsilon
</math>
  </td>
  <td align="center">
<math>
\equiv
</math>
  </td>
  <td align="left">
<math>
3 (1 - e^2)^{1 / 2} \biggl[\frac{\sin^{-1}e}{e}\biggr] = 2.040835
\, .
</math>
  </td>
</tr>
</table>

====Gravitational Potential Rewritten====

In summary, then,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\frac{ \Phi_\mathrm{grav}(\chi,\zeta)}{(-\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="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{1}{3} \Upsilon 
-
\frac{(\Upsilon - 3)}{3e^2} (\chi + \sqrt{2}\zeta)(\chi - \sqrt{2} \zeta)
- \chi^2
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>
- ~ \frac{e^{-4}}{(1-e^2)}\biggl\{
\biggl[\zeta^2 - (1-e^2)\chi^2\biggr]\biggl[ \zeta^2 - 2(1-e^2)\chi^2\biggr]
- \frac{13}{8}(1-e^2)^2\chi^4
\biggr\}

+ ~ \frac{4e^{-2}}{3(1-e^2)}\biggl\{
\biggl[\zeta^2 - (1-e^2)\chi^2\biggr]\biggl[ \zeta^2 + \frac{1}{4}(1-e^2)\chi^2\biggr]
+ \frac{1}{16}(1-e^2)^2\chi^4
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>
+~
\frac{e^{-4}}{3}\biggl[ (\zeta^2 - \chi^2)(\zeta^2-2\chi^2) - \frac{13}{8}\chi^4  \biggr]\Upsilon
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{1}{3} \Upsilon 
-
\frac{(\Upsilon - 3)}{3e^2} (\chi + \sqrt{2}\zeta)(\chi - \sqrt{2} \zeta)
- \chi^2
+ ~ \frac{4}{3e^{2}(1-e^2)}\biggl\{
\biggl[\zeta^2 - (1-e^2)\chi^2\biggr]\biggl[ \zeta^2 + \frac{1}{4}(1-e^2)\chi^2\biggr]
+ \frac{1}{16}(1-e^2)^2\chi^4
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>
- ~ \frac{1}{e^4(1-e^2)}\biggl\{
\biggl[\zeta^2 - (1-e^2)\chi^2\biggr]\biggl[ \zeta^2 - 2(1-e^2)\chi^2\biggr]
- \frac{13}{8}(1-e^2)^2\chi^4
\biggr\}
+~
\frac{1}{3e^4}\biggl[ (\zeta^2 - \chi^2)(\zeta^2-2\chi^2) - \frac{13}{8}\chi^4  \biggr]\Upsilon
</math>
  </td>
</tr>

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

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

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>
- \chi^2
+ ~ \frac{4}{3e^{2}(1-e^2)}\biggl\{ \frac{1}{16}(1-e^2)^2\chi^4 \biggr\}
+ \frac{1}{e^4(1-e^2)}\biggl\{ \frac{13}{8}(1-e^2)^2\chi^4  \biggr\}
- \frac{\Upsilon}{3e^4}\biggl\{ \frac{13}{8}\chi^4 \biggr\}
</math>
  </td>
</tr>

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

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

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>
- ~\chi^2
+ ~ \biggl\{
\frac{(1-e^2)}{12e^{2}}
+ \frac{13(1-e^2)}{8e^4}
- \frac{13\Upsilon}{24e^4} \biggr\}\chi^4
\, .
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
0.767874 (row 1) + 0.5678833 (row 2) - 0.950574 (row 3)
=
0.3851876 &nbsp;.
  </td>
</tr>
</table>

====Example Evaluation====

Let's evaluate these expressions, borrowing from the [[#QuantitativeExample|quantitative example specified above]].  Specifically, we choose,

<table border="0" align="center" width="80%">
<tr>
  <td align="center"><math>\frac{a_s}{a_\ell} = 0.582724 \, ,</math></td>
  <td align="center"><math>e = 0.81267 \, ,</math></td>
  <td align="center">&nbsp;</td>
</tr>
<tr>
  <td align="center"><math>A_\ell = A_m = 0.51589042 \, ,</math></td>
  <td align="center"><math>A_s = 0.96821916 \, ,</math></td>
  <td align="center"><math>I_\mathrm{BT} = \frac{2}{3}\Upsilon = 1.360556 \, ,</math></td>
</tr>
<tr>
  <td align="center"><math>a_\ell^2 A_{\ell \ell} = 0.3287756 \, ,</math></td>
  <td align="center"><math>a_\ell^2 A_{s s} = 1.5066848 \, ,</math></td>
  <td align="center"><math>a_\ell^2 A_{\ell s} = 0.6848975 \, .</math></td>
</tr>
</table>
Also, let's set <math>\rho/\rho_c = 0.1</math> and <math>\chi = \chi_1 = 0.75 ~~\Rightarrow ~~ \chi_1^2 = 0.5625</math>.  This means that,

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

<tr>
  <td align="right">
<math>
\zeta_1^2
</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
(1-e^2)\biggl[1 - \chi^2 - \frac{\rho(\chi, \zeta)}{\rho_c} \biggr]
=
\biggl[1 - (0.81267)^2)\biggr]\biggl[1 - 0.5625 - 0.1\biggr] 
=
0.11460
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>
\Rightarrow ~~~ \zeta_1
</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
0.33853 \, .
</math>
  </td>
</tr>
</table>

So, let's evaluate the gravitational potential &hellip;
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\frac{ \Phi_\mathrm{grav}(\chi_1,\zeta_1)}{(-\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}  
- \biggl[\overbrace{A_\ell \chi^2  + A_s \zeta^2}^{\mathrm{TERM2}}  \biggr]
+ \frac{1}{2}\biggl[
\underbrace{(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 }_{\mathrm{TERM1}}
\biggr]
=
0.385187372
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\mathrm{TERM1} </math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
0.019788921 + 0.088303509 + 0.104026655 = 0.212119085
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\mathrm{TERM2} </math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
0.290188361 + 0.110961809
=
0.401150171 \, .
</math>
  </td>
</tr>
</table>

====Replace &zeta; With Normalized Density====

First, let's readjust the last, 3-row expression for the gravitational potential so that <math>\zeta^2</math> can be readily replaced with the normalized density.
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\frac{ \Phi_\mathrm{grav}(\chi,\zeta)}{(-\pi G\rho_c a_\ell^2)} </math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{1}{3} \Upsilon 
-
\frac{(\Upsilon - 3)}{3e^2} (\chi^2 - 2\zeta^2)
+ ~ \frac{4(1-e^2)}{3e^{2}}\biggl\{
\biggl[(1-e^2)^{-1}\zeta^2 - \chi^2\biggr]\biggl[(1-e^2)^{-1} \zeta^2 + \frac{1}{4}\chi^2\biggr]
\biggr\}
</math>
  </td>
</tr>

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

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>
- ~\chi^2
+ ~ \frac{1}{24e^4}\biggl\{
2 e^2(1-e^2)
+ 39(1-e^2)
- 13\Upsilon \biggr\}\chi^4
\, .
</math>
  </td>
</tr>
</table>
Now make the substitution,

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

<tr>
  <td align="right">
<math>\zeta^2</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
(1-e^2)\biggl[1 - \chi^2 - \rho^*\biggr] 
\, ,</math>
  </td>
</tr>
</table>
where,

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

<tr>
  <td align="right">
<math>\rho^*</math>
  </td>
  <td align="center">
<math>\equiv</math>
  </td>
  <td align="left">
<math>
\frac{\rho(\chi, \zeta)}{\rho_c} 
\, .</math>
  </td>
</tr>
</table>
We have,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\frac{ \Phi_\mathrm{grav}(\chi,\zeta)}{(-\pi G\rho_c a_\ell^2)} </math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{1}{3} \Upsilon 
-
\frac{(\Upsilon - 3)}{3e^2} \biggl\{ \chi^2 - 2(1-e^2)\biggl[1 - \chi^2 - \rho^*\biggr] \biggr\}
+ ~ \frac{4(1-e^2)}{3e^{2}}
\biggl\{\biggl[1 - \chi^2 - \rho^*\biggr] - \chi^2\biggr\}\biggl\{\biggl[1 - \chi^2 - \rho^*\biggr] + \frac{1}{4}\chi^2\biggr\}
</math>
  </td>
</tr>

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

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>
- ~\chi^2
+ ~ \frac{1}{24e^4}\biggl\{
2 e^2(1-e^2)
+ 39(1-e^2)
- 13\Upsilon \biggr\}\chi^4
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{1}{3} \Upsilon 
-
\frac{(\Upsilon - 3)}{3e^2} \biggl\{ -2+2e^2 + (3-2e^2)\chi^2 + (2-2e^2)\rho^* \biggr\}
+ ~ \frac{4(1-e^2)}{3e^{2}}
\biggl\{1 - 2\chi^2 - \rho^*\biggr\}\biggl\{1 - \frac{3}{4}\chi^2 - \rho^*\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>
- ~ \frac{(1-e^2)}{e^4}
\biggl\{1 - 2\chi^2 - \rho^*\biggr\}\biggl\{ 1 - 3\chi^2 - \rho^* \biggr\}
+~
\frac{\Upsilon}{3e^4}\biggl\{ (1-e^2) - (2-e^2)\chi^2 - (1-e^2)\rho^* \biggr\} 
\biggl\{(1-e^2) - (3-e^2)\chi^2 - (1-e^2)\rho^*  \biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>
- ~\chi^2
+ ~ \frac{1}{24e^4}\biggl\{
39 - 37e^2
- 2e^4
- 13\Upsilon \biggr\}\chi^4
</math>
  </td>
</tr>

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

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

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>
- ~\chi^2
+ ~ \frac{1}{24e^4}\biggl\{
39 - 37e^2
- 2e^4
- 13\Upsilon \biggr\}\chi^4
\, .
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
0.767874 (row 1) + 0.5678833 (row 2) - 0.950574 (row 3)
=
0.3851876 &nbsp;.
  </td>
</tr>
</table>

Now, let's group together like terms and examine, in particular, whether the coefficient of the cross-product, <math>\chi^2 \rho^*)</math>, goes to zero.

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

<tr>
  <td align="right">
<math>\frac{ \Phi_\mathrm{grav}(\chi,\zeta)}{(-\pi G\rho_c a_\ell^2)} </math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{1}{3} \Upsilon 
-
\frac{(\Upsilon - 3)}{3e^2} \biggl\{2e^2 -2 + (2 - 2e^2)\rho^* \biggr\}
</math>
  </td>
</tr>

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

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

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>
- ~\chi^2
+ ~ \frac{1}{24e^4}\biggl\{
39 - 37e^2
- 2e^4
- 13\Upsilon \biggr\}\chi^4
-
\frac{(\Upsilon - 3)}{3e^2} \biggl\{ 3\chi^2 - 2e^2\chi^2 \biggr\}

</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{1}{3} \Upsilon 
+
\frac{(\Upsilon - 3)}{3e^2} \biggl\{2(1 - e^2)(1 - \rho^*) \biggr\}
</math>
  </td>
</tr>

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

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

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>
- ~\chi^2
+ ~ \frac{1}{24e^4}\biggl\{
39 - 37e^2
- 2e^4
- 13\Upsilon \biggr\}\chi^4
-
\frac{(\Upsilon - 3)}{3e^2} \biggl\{ 3\chi^2 - 2e^2\chi^2 \biggr\}
-~ \biggl\{ \frac{\Upsilon}{3e^2} \biggr\}\biggl[(1-e^2) - (3-e^2)\chi^2  \biggr]\chi^2 

</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{1}{3} \Upsilon 
+
\frac{(\Upsilon - 3)}{3e^2} \biggl\{2(1 - e^2)(1 - \rho^*) \biggr\}
</math>
  </td>
</tr>

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

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>
+ ~ 
\biggl[(1 - \rho^*) \biggr]
\frac{(1-e^2)}{3e^{4}}\biggl\{ 
\biggl[- 3e^2 +9 - (3-e^2)\Upsilon \biggr]\chi^2  
\biggr\}
+ ~ 
\biggl[- 2\chi^2\biggr]
\frac{(1-e^2)}{3e^{4}}\biggl\{ 
\biggl[- 3e^2 +9 - (3-e^2)\Upsilon \biggr]\chi^2  
\biggr\}
</math>
  </td>
</tr>

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

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>
- ~\chi^2
+ ~ \frac{1}{24e^4}\biggl\{
39 - 37e^2
- 2e^4
- 13\Upsilon \biggr\}\chi^4
-
\frac{(\Upsilon - 3)}{3e^2} \biggl\{ 3\chi^2 - 2e^2\chi^2 \biggr\}
-~ \biggl\{ \frac{\Upsilon}{3e^2} \biggr\}\biggl[(1-e^2) - (3-e^2)\chi^2  \biggr]\chi^2 

</math>
  </td>
</tr>
</table>