Editing VE/RiemannEllipsoids (section)

=Various Degrees of Simplification=

==Riemann Ellipsoids of Types I, II, &amp; III==
In this, most general, case, the two vectors <math>~\vec{\Omega}</math> and <math>~\vec\zeta</math> are not parallel to any of the principal axes of the ellipsoid, and they are not aligned with each other, but they both lie in the <math>~y-z</math>-plane &#8212; that is to say, <math>~(\Omega_1, \zeta_1) = (0, 0)</math>.  For a given specified density <math>~(\rho)</math> and choice of the three semi-axes <math>~(a_1, a_2, a_3) \leftrightarrow (a, b, c)</math>, all five of the expressions displayed in our above [[#SummaryTable|''Summary Table'']] must be used in order to determine the equilibrium configuration's associated values of the five unknowns:  <math>~\Pi, (\Omega_2, \zeta_2), (\Omega_3, \zeta_3)</math>.  Here we show how these five unknowns can be derived from the five constraint equations, closely following the analysis that is presented in &sect;47 (pp. 129 - 132) of [ [[User:Tohline/Appendix/References#EFE|EFE]] ].

===Constraints Due to Off-Diagonal Elements===
We begin by subtracting the constraint equation provided by the first off-diagonal element <math>~(i, j) = (2, 3)</math> from the constraint equation provided by the second off-diagonal element <math>~(i, j) = (3, 2) </math>.  This gives,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\biggl\{
1  
+ \frac{\zeta_2}{\Omega_2}\biggl[ \frac{a^2}{a^2 + c^2 }\biggr] \biggl[ 2 + \frac{\zeta_3}{\Omega_3}\biggl( \frac{b^2}{b^2+a^2}\biggr) \biggr]     
\biggr\} \Omega_2\Omega_3c^2 
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl\{
1  
+ \frac{\zeta_3}{\Omega_3}\biggl[ \frac{a^2}{a^2+b^2}\biggr]  \biggl[2 +  \frac{\zeta_2}{\Omega_2} \biggl( \frac{c^2}{c^2 + a^2} \biggr) \biggr]   
\biggr\} \Omega_2 \Omega_3b^2 
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~ 
c^2  
+ \frac{\zeta_2}{\Omega_2}\biggl[ \frac{2 a^2 c^2}{a^2 + c^2 }\biggr] \biggl[ 1 + \frac{\zeta_3}{2 \Omega_3}\biggl( \frac{b^2}{b^2+a^2}\biggr) \biggr]      
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
b^2 
+ \frac{\zeta_3}{\Omega_3}\biggl[ \frac{2 a^2 b^2}{a^2+b^2}\biggr]  \biggl[1 +  \frac{\zeta_2}{2\Omega_2} \biggl( \frac{c^2}{c^2 + a^2} \biggr) \biggr]   
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~ 
c^2  
+ \frac{\zeta_2}{\Omega_2}\biggl[ \frac{2 a^2 c^2}{a^2 + c^2 }\biggr]      
+ \frac{\zeta_2}{\Omega_2} \cdot \frac{\zeta_3}{\Omega_3} \biggl[ \frac{a^2 c^2}{a^2 + c^2 }\biggr] \biggl[ \frac{b^2}{b^2+a^2} \biggr]      
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
b^2 
+ \frac{\zeta_3}{\Omega_3}\biggl[ \frac{2 a^2 b^2}{a^2+b^2}\biggr]    
+ \frac{\zeta_3}{\Omega_3} \cdot \frac{\zeta_2}{\Omega_2} \biggl[ \frac{a^2 b^2}{a^2+b^2}\biggr]  \biggl[ \frac{c^2}{c^2 + a^2} \biggr]   
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~ 
c^2  + \frac{\zeta_2}{\Omega_2}\biggl[ \frac{2 a^2 c^2}{a^2 + c^2 }\biggr]      
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
b^2 
+ \frac{\zeta_3}{\Omega_3}\biggl[ \frac{2 a^2 b^2}{a^2+b^2}\biggr]    \, .
</math>
  </td>
</tr>
<tr><td align="center" colspan="3">[ [[User:Tohline/Appendix/References#EFE|EFE]], <font color="#00CC00">Chapter 7, &sect;47, Eq. (11)</font> ]</td></tr>
</table>

Adding the two instead gives,
<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>~
\biggl\{
1  
+ \frac{\zeta_2}{\Omega_2}\biggl[ \frac{a^2}{a^2 + c^2 }\biggr] \biggl[ 2 + \frac{\zeta_3}{\Omega_3}\biggl( \frac{b^2}{b^2+a^2}\biggr) \biggr]     
\biggr\} \Omega_2\Omega_3c^2 
+
\biggl\{
1  
+ \frac{\zeta_3}{\Omega_3}\biggl[ \frac{a^2}{a^2+b^2}\biggr]  \biggl[2 +  \frac{\zeta_2}{\Omega_2} \biggl( \frac{c^2}{c^2 + a^2} \biggr) \biggr]   
\biggr\} \Omega_2 \Omega_3b^2 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
b^2 + c^2 
+ \frac{\zeta_2}{\Omega_2}\biggl[ \frac{a^2 c^2}{a^2 + c^2 }\biggr] \biggl[ 2 + \frac{\zeta_3}{\Omega_3}\biggl( \frac{b^2}{b^2+a^2}\biggr) \biggr]     
+ \frac{\zeta_3}{\Omega_3}\biggl[ \frac{a^2 b^2}{a^2+b^2}\biggr]  \biggl[2 +  \frac{\zeta_2}{\Omega_2} \biggl( \frac{c^2}{c^2 + a^2} \biggr) \biggr]   
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
b^2 + c^2 
+ \frac{\zeta_2}{\Omega_2}\biggl[ \frac{2a^2 c^2}{a^2 + c^2 }\biggr]    
+ \frac{\zeta_3}{\Omega_3}\biggl[ \frac{2a^2 b^2}{a^2+b^2}\biggr]     
+ \frac{\zeta_2}{\Omega_2} \cdot  \frac{\zeta_3}{\Omega_3} \biggl[ \frac{2a^2 b^2 c^2}{(a^2 + c^2)( b^2+a^2 ) }\biggr]      \, .
</math>
  </td>
</tr>
<tr><td align="center" colspan="3">[ [[User:Tohline/Appendix/References#EFE|EFE]], <font color="#00CC00">Chapter 7, &sect;47, Eq. (10)</font> ]</td></tr>
</table>

The first of these relations cleanly gives an expression for the frequency ratio, <math>~\zeta_3/\Omega_3</math>, in terms of the ''other'' frequency ratio, <math>~\zeta_2/\Omega_2</math>.  This allows us to rewrite the second relation in terms of the ratio, <math>~\zeta_2/\Omega_2</math>, alone.  We obtain,

<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>~
b^2 + c^2 
+ \frac{\zeta_2}{\Omega_2}\biggl[ \frac{2a^2 c^2}{a^2 + c^2 }\biggr]    
+ \biggl\{ \frac{\zeta_3}{\Omega_3}\biggl[ \frac{2a^2 b^2}{a^2+b^2}\biggr] \biggr\}
+ \frac{\zeta_2}{\Omega_2} \biggl[\frac{c^2 }{(a^2 + c^2) }  \biggr]  \cdot  \biggl\{ \frac{\zeta_3}{\Omega_3} \biggl[ \frac{2a^2 b^2 }{( b^2+a^2 ) }\biggr] \biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
b^2 + c^2 
+ \frac{\zeta_2}{\Omega_2}\biggl[ \frac{2a^2 c^2}{a^2 + c^2 }\biggr]    
+ \biggl\{ c^2 - b^2  + \frac{\zeta_2}{\Omega_2}\biggl[ \frac{2 a^2 c^2}{a^2 + c^2 }\biggr] \biggr\}
+ \frac{\zeta_2}{\Omega_2} \biggl[\frac{c^2 }{(a^2 + c^2) }  \biggr]  \cdot  \biggl\{ c^2 - b^2  + \frac{\zeta_2}{\Omega_2}\biggl[ \frac{2 a^2 c^2}{a^2 + c^2 }\biggr] \biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
2c^2 
+ \frac{\zeta_2}{\Omega_2} \biggl[\frac{c^2 }{a^2 + c^2 }  \biggr] (4a^2 + c^2 - b^2  )
+  \biggl\{ \frac{\zeta_2}{\Omega_2}\biggl[ \frac{c^2}{a^2 + c^2 }\biggr] \biggr\}^22a^2
</math>
  </td>
</tr>
</table>

<table border="1" align="center" cellpadding="10" width="80%"><tr><td align="left">
ASIDE:  &nbsp; Alternatively, given that,

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

<tr>
  <td align="right">
<math>~ 
\frac{\zeta_2}{\Omega_2}\biggl[ \frac{c^2}{a^2 + c^2 }\biggr]   
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{1}{2a^2}\biggl[ b^2 - c^2+ \frac{\zeta_3}{\Omega_3}\biggl( \frac{2 a^2 b^2}{a^2+b^2}\biggr) \biggr]
</math>
  </td>
</tr>
</table>
the quadratic equation that governs the value of the frequency ratio, <math>~\zeta_3/\Omega_3</math> is &hellip;
<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>~
4 a^2 c^2 
+ \biggl[ b^2 - c^2+ \frac{\zeta_3}{\Omega_3}\biggl( \frac{2 a^2 b^2}{a^2+b^2}\biggr) \biggr]  (4a^2 + c^2 - b^2  )
+  \biggl[ b^2 - c^2+ \frac{\zeta_3}{\Omega_3}\biggl( \frac{2 a^2 b^2}{a^2+b^2}\biggr) \biggr]^2
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
4 a^2 c^2 
+ \biggl[ \frac{\zeta_3}{\Omega_3}\biggl( \frac{2 a^2 b^2}{a^2+b^2}\biggr) \biggr]  (4a^2 + c^2 - b^2  )
+ ( b^2 - c^2)  (4a^2 + c^2 - b^2  )
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+  (b^2 - c^2)^2 + 2(b^2 - c^2) \biggl[\frac{\zeta_3}{\Omega_3}\biggl( \frac{2 a^2 b^2}{a^2+b^2}\biggr)\biggr]
+ \biggl[\frac{\zeta_3}{\Omega_3}\biggl( \frac{2 a^2 b^2}{a^2+b^2}\biggr)\biggr]^2
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
4 a^2 c^2 
+ ( b^2 - c^2)  (4a^2 + c^2 - b^2  )
+  (b^2 - c^2)^2 
+ \biggl[ \frac{\zeta_3}{\Omega_3}\biggl( \frac{2 a^2 b^2}{a^2+b^2}\biggr) \biggr] 
\biggl[ (4a^2 + c^2 - b^2  )+ 2(b^2 - c^2) \biggr]
+ \biggl[\frac{\zeta_3}{\Omega_3}\biggl( \frac{2 a^2 b^2}{a^2+b^2}\biggr)\biggr]^2
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[\frac{\zeta_3}{\Omega_3}\biggl( \frac{2 a^2 b^2}{a^2+b^2}\biggr)\biggr]^2
+ \biggl[ \frac{\zeta_3}{\Omega_3}\biggl( \frac{2 a^2 b^2}{a^2+b^2}\biggr) \biggr] 
(4a^2 + b^2 - c^2 ) 
+ 4 a^2 b^2 
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow~~~ 0</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[ \biggl(\frac{\zeta_3}{\Omega_3} \biggr)^2 \frac{a^2 b^2}{(a^2+b^2)^2}\biggr]
+ \frac{1}{2}\biggl[ \frac{\zeta_3}{\Omega_3}\biggl( \frac{1}{a^2+b^2}\biggr) \biggr] 
(4a^2 + b^2 - c^2 ) 
+ 1 \, .
</math>
  </td>
</tr>
</table>

Now, in our discussion of Riemann S-Type ellipsoids, there is also a quadratic equation that governs the equilibrium frequency ratio, <math>~f \equiv \zeta_3/\Omega_3</math>.  It is, specifically,
<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>~
\biggl[ \frac{a^2 b^2}{(a^2 + b^2)^2} \biggr] f^2 
+ \biggl[ \frac{2a^2 b^2 B_{12}}{c^2 A_3 - a^2 b^2 A_{12}} \biggr]\frac{f}{a^2 + b^2} + 1 \, .
</math>
  </td>
</tr>
<tr><td align="center" colspan="3">[ [[User:Tohline/Appendix/References#EFE|EFE]], <font color="#00CC00">&sect;48, Eq. (35)</font> ]</td></tr>
</table>
Notice that the first and third terms of this quadratic equation exactly match the first and third terms of the quadratic equation, which we have just derived, that governs the same frequency ratio in Riemann ellipsoids of Types I, II &amp; III.  Does the second term match?  That is, is the coefficient of the linear term the same in both quadratic relations?  Well, &hellip;
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~ \frac{2a^2 b^2 B_{12}}{c^2 A_3 - a^2 b^2 A_{12}} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~2a^2 b^2 \biggl[ c^2 A_3 + a^2 b^2 \biggl( \frac{A_1 - A_2}{a^2-b^2} \biggr)\biggr]^{-1} \biggl[A_2 + a^2\biggl( \frac{A_1 - A_2}{a^2-b^2} \biggr)\biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~2a^2 b^2 \biggl[ c^2 A_3(a^2 - b^2) + a^2 b^2 (A_1 - A_2) \biggr]^{-1} \biggl[a^2 A_1  - b^2 A_2\biggr] \, .
</math>
  </td>
</tr>
</table>

Even appreciating that we can make the substitution, <math>~A_3 = (2 - A_1 - A_2)</math>, I don't see any way that this coefficient expression can be manipulated to match the associated coefficient in the other expression, namely, <math>~(4a^2 + b^2 - c^2)/2</math>.

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


This is a quadratic equation whose solution gives,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~4a^2 \cdot \frac{\zeta_2}{\Omega_2} \biggl[\frac{c^2 }{a^2 + c^2 }  \biggr] </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
- (4a^2 + c^2 - b^2  ) \pm \biggl[ (4a^2 + c^2 - b^2  )^2 - 16a^2 c^2 \biggr]^{1 / 2} \, .
</math>
  </td>
</tr>
</table>

For the other frequency ratio we therefore find,

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

<tr>
  <td align="right">
<math>~ 
2\biggl\{ b^2 -c^2
+ \frac{\zeta_3}{\Omega_3}\biggl[ \frac{2 a^2 b^2}{a^2+b^2}\biggr]   \biggr\}
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
2 \cdot \frac{\zeta_2}{\Omega_2}\biggl[ \frac{2 a^2 c^2}{a^2 + c^2 }\biggr]      
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
- (4a^2 + c^2 - b^2  ) \pm \biggl[ (4a^2 + c^2 - b^2  )^2 - 16a^2 c^2 \biggr]^{1 / 2}
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~ \Rightarrow ~~~
4a^2 \cdot \frac{\zeta_3}{\Omega_3}\biggl[ \frac{b^2}{a^2+b^2}\biggr] 
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
- (4a^2  + b^2 -c^2)  \pm \biggl[ (4a^2 + c^2 - b^2  )^2 - 16a^2 c^2 \biggr]^{1 / 2}
</math>
  </td>
</tr>
</table>
<span id="OffDiagonal">&nbsp;</span>
<table border="1" width="80%" align="center" cellpadding="10"><tr><td align="left">
<div align="center">'''SUMMARY: &nbsp; Riemann Ellipsoids of Types I, II, &amp; III'''</div>
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\beta \equiv~ -~\frac{\zeta_2}{\Omega_2} \biggl[\frac{c^2 }{a^2 + c^2 }  \biggr] </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{1}{4a^2}\biggl\{ (4a^2 -b^2 + c^2  ) \mp \biggl[ (4a^2 + c^2 - b^2  )^2 - 16a^2 c^2 \biggr]^{1 / 2} \biggr\} \, ;
</math>
  </td>
</tr>
<tr><td align="center" colspan="3">[ [[User:Tohline/Appendix/References#EFE|EFE]], <font color="#00CC00">Chapter 7, &sect;47, Eq. (16)</font> ]</td></tr>

<tr>
  <td align="right">
<math>~
\gamma \equiv~-~\frac{\zeta_3}{\Omega_3}\biggl[ \frac{b^2}{a^2+b^2}\biggr] 
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{1}{4a^2}\biggl\{ (4a^2  + b^2 -c^2)  \mp \biggl[ (4a^2 + c^2 - b^2  )^2 - 16a^2 c^2 \biggr]^{1 / 2} \biggr\} \, .
</math>
  </td>
</tr>
<tr><td align="center" colspan="3">[ [[User:Tohline/Appendix/References#EFE|EFE]], <font color="#00CC00">Chapter 7, &sect;47, Eq. (17)</font> ]</td></tr>
</table>

As is emphasized in [[User:Tohline/Appendix/References#EFE|EFE]] (Chapter 7, &sect;47, p. 131) "<font color="darkgreen">&hellip; the signs in front of the radicals, in the two expressions, go together.</font>  Furthermore, "<font color="darkgreen">the two roots &hellip; correspond to the fact that, consistent with Dedekind's theorem, two states of internal motions are compatible with the same external figure.</font>"

----

As has also been pointed out in [[User:Tohline/Appendix/References#EFE|EFE]] (Chapter 7, &sect;51, p. 158), from the steps that have led to the development and solution of the above pair of quadratic equations we can demonstrate that the following relations also hold:

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

<tr>
  <td align="right">
<math>~\beta^2 - 2\beta + \frac{c^2}{a^2} = \biggl[ \frac{c^2 - b^2}{2a^2}\biggr]\beta \, ,</math>
  </td>
  <td align="center">
&nbsp; &nbsp; &nbsp; &nbsp;
  </td>
  <td align="left">
<math>~\gamma^2 -2\gamma + \frac{b^2}{a^2} = \biggl[ \frac{b^2 - c^2}{2a^2} \biggr]\gamma</math>
  </td>
</tr>

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

<tr>
  <td align="right">
<math>~1 - 2\gamma + \biggl(\frac{a^2}{b^2}\biggr)\gamma^2 </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl[ \frac{4a^2 - c^2 - 3b^2}{2b^2}\biggr]\gamma \, .</math>
  </td>
</tr>
<tr><td align="center" colspan="3">[ [[User:Tohline/Appendix/References#EFE|EFE]], <font color="#00CC00">Chapter 7, &sect;51, Eqs. (161) - (163)</font> ]</td></tr>
</table>

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

===Constraints Due to Diagonal Elements===

Next, to simplify manipulations, let's replace the frequency ratios by these newly defined &#8212; and ''known'' &#8212; parameters, <math>~\beta</math> and <math>~\gamma</math>, in the three diagonal-element expressions that are written out in our above [[#SummaryTable|Summary Table]].


<table border="1" align="center" cellpadding="5">
<tr>
  <td align="center" colspan="2">Indices</td>
  <td align="center" rowspan="2">Rewritten Diagonal-Element Expressions</td>
</tr>
<tr>
  <td align="center" width="5%"><math>~i</math></td>
  <td align="center" width="5%"><math>~j</math></td>
</tr>
<tr>
  <td align="center"><math>~1</math></td>
  <td align="center"><math>~1</math></td>
  <td align="left">

<table align="left" border=0 cellpadding="3">
<tr>
  <td align="right">
<math>~\biggl[ \frac{3\cdot 5}{2^2\pi a b c\rho} \biggr] \Pi</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
-~\biggl\{ 
( \Omega_2^2 + \Omega_3^2)  
+ 2  \biggl[ \frac{b^2}{b^2+a^2}\biggr] \Omega_3 \biggl[ - \frac{\Omega_3\gamma (a^2 + b^2)}{b^2} \biggr] 
+ 2  \biggl[ \frac{c^2}{c^2 + a^2}\biggr]\Omega_2 \biggl[ -  \frac{\Omega_2 \beta( a^2 + c^2)}{c^2} \biggr] 
~-~(2\pi G\rho) A_1 
\biggr\} a^2 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
- \biggl[ \frac{a^2}{a^2 + b^2}\biggr]^2 \biggl[ - \frac{\Omega_3\gamma (a^2 + b^2)}{b^2} \biggr]^2 b^2
- \biggl[ \frac{a^2}{a^2+c^2}\biggr]^2 \biggl[ -  \frac{\Omega_2 \beta( a^2 + c^2)}{c^2} \biggr]^2  c^2  
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl\{ 
- \Omega_2^2 - \Omega_3^2  
+ 2 \Omega_3^2 \gamma 
+ 2  \Omega_2^2 \beta 
~+~(2\pi G\rho) A_1 
\biggr\} a^2 
- \biggl( \frac{a^4}{b^2}\biggr) \Omega_3^2\gamma^2  
- \biggl( \frac{a^4}{c^2}\biggr) \Omega_2^2 \beta^2   
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl\{ \Omega_2^2 \biggl[2  \beta - 1 - \biggl( \frac{a^2}{c^2}\biggr)  \beta^2 \biggr]
+ \Omega_3^2 \biggl[ 2  \gamma - 1  - \biggl( \frac{a^2}{b^2}\biggr)\gamma^2  \biggr]
~+~(2\pi G\rho) A_1 
\biggr\}a^2   
</math>
  </td>
</tr>
<tr><td align="center" colspan="3">[ [[User:Tohline/Appendix/References#EFE|EFE]], <font color="#00CC00">Chapter 7, &sect;51, Eq. (158)</font> ]</td></tr>
</table>

  </td>
</tr>

<tr>
  <td align="center"><math>~2</math></td>
  <td align="center"><math>~2</math></td>
  <td align="left">

<table align="left" border=0 cellpadding="3">
<tr>
  <td align="right">
<math>~\biggl[ \frac{3\cdot 5}{2^2\pi a b c \rho} \biggr]\Pi</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
-~ \biggl[ \frac{b^2}{b^2+a^2}\biggr]^2 \biggl[ - \frac{\Omega_3\gamma (a^2 + b^2)}{b^2} \biggr]^2 a^2
- \biggl\{
\Omega_3^2  
+ 2 \biggl[ \frac{a^2}{a^2 + b^2}\biggr] \Omega_3 \biggl[ - \frac{\Omega_3\gamma (a^2 + b^2)}{b^2} \biggr]  
~-~( 2\pi G \rho) A_2 
\biggr\}b^2 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
-~\Omega_3^2 \gamma^2 a^2
- \Omega_3^2 b^2  
+ 2 a^2 \Omega_3^2 \gamma  
~+~( 2\pi G \rho) b^2A_2 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
-~a^2 \Omega_3^2 \biggl[\gamma^2 - 2\gamma  
+ \biggl( \frac{b^2}{a^2}\biggr) \biggr]  
~+~( 2\pi G \rho) b^2A_2 
</math>
  </td>
</tr>
<tr><td align="center" colspan="3">[ [[User:Tohline/Appendix/References#EFE|EFE]], <font color="#00CC00">Chapter 7, &sect;51, Eq. (159)</font> ]</td></tr>
</table>

  </td>
</tr>

<tr>
  <td align="center"><math>~3</math></td>
  <td align="center"><math>~3</math></td>
  <td align="left">

<table align="left" border=0 cellpadding="3">
<tr>
  <td align="right">
<math>~\biggl[ \frac{3\cdot 5}{2^2\pi abc\rho} \biggr]\Pi</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
- \biggl[ \frac{c^2}{c^2 + a^2}\biggr]^2 \biggl[ -  \frac{\Omega_2 \beta( a^2 + c^2)}{c^2} \biggr]^2  a^2
- \biggl\{
\Omega_2^2   + 2 \biggl[ \frac{a^2}{a^2+c^2}\biggr]\Omega_2 \biggl[ -  \frac{\Omega_2 \beta( a^2 + c^2)}{c^2} \biggr] 
- (2\pi G \rho)A_3 
\biggr\}c^2 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
-  \Omega_2^2 \beta^2 a^2 
-\Omega_2^2c^2   + 2 a^2\Omega_2^2 \beta  
+ (2\pi G \rho)c^2 A_3  
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~-~a^2\Omega_2^2\biggl[
\beta^2 - 2 \beta 
+ \biggl( \frac{c^2}{a^2}\biggr)    \biggr]
+ (2\pi G \rho)c^2 A_3  
</math>
  </td>
</tr>
<tr><td align="center" colspan="3">[ [[User:Tohline/Appendix/References#EFE|EFE]], <font color="#00CC00">Chapter 7, &sect;51, Eq. (160)</font> ]</td></tr>
</table>

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


Using the <math>~(i, j) = (3, 3)</math> element to preplace <math>~\Pi</math> in the other two expressions, we obtain,

<table align="center" border=0 cellpadding="3">
<tr>
  <td align="right">
<math>~0
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\Omega_2^2\biggl[
\beta^2 - 2 \beta 
+ \biggl( \frac{c^2}{a^2}\biggr)    \biggr]
+
\Omega_2^2 \biggl[2  \beta - 1 - \biggl( \frac{a^2}{c^2}\biggr)  \beta^2 \biggr]
+ \Omega_3^2 \biggl[ 2  \gamma - 1  - \biggl( \frac{a^2}{b^2}\biggr)\gamma^2  \biggr]
~+~2\pi G\rho \biggl[ A_1 -  \biggl(\frac{c^2}{a^2}\biggr)A_3 \biggr] \, ;
</math>
  </td>
</tr>
</table>

and,

<table align="center" border=0 cellpadding="3">
<tr>
  <td align="right">
<math>~\Omega_2^2\biggl[
\beta^2 - 2 \beta 
+ \biggl( \frac{c^2}{a^2}\biggr)    \biggr]
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\Omega_3^2 \biggl[\gamma^2 - 2\gamma  
+ \biggl( \frac{b^2}{a^2}\biggr) \biggr]  
~+~2\pi G \rho \biggl[\biggl( \frac{c^2}{a^2}\biggr) A_3 -  \biggl( \frac{b^2}{a^2} \biggr)A_2 \biggr] \, .
</math>
  </td>
</tr>
</table>

Inserting the [[#OffDiagonal|various relations highlighted above]], these two expressions may be rewritten as,

<table align="center" border=0 cellpadding="3">
<tr>
  <td align="right">
<math>~0
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\Omega_2^2
\biggl[ \frac{c^2 - b^2}{2a^2}\biggr]\beta 
~-~\Omega_2^2 \biggl[ \frac{4a^2 - b^2 - 3c^2}{2c^2}\biggr]\beta
~-~\Omega_3^2 \biggl[ \frac{4a^2 - c^2 - 3b^2}{2b^2}\biggr]\gamma
~+~2\pi G\rho \biggl[ A_1 -  \biggl(\frac{c^2}{a^2}\biggr)A_3 \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\Omega_2^2\beta
\biggl[ \frac{c^2(c^2 - b^2) + a^2 (b^2 + 3c^2 - 4a^2)}{2a^2c^2}
\biggr]
~-~\Omega_3^2 \gamma \biggl[ \frac{4a^2 - c^2 - 3b^2}{2b^2}\biggr]
~+~2\pi G\rho \biggl[ A_1 -  \biggl(\frac{c^2}{a^2}\biggr)A_3 \biggr] \, ;
</math>
  </td>
</tr>
</table>

<span id="Temporary">and,</span>

<table align="center" border=0 cellpadding="3">
<tr>
  <td align="right">
<math>~\Omega_2^2
\biggl[ \frac{c^2 - b^2}{2a^2}\biggr]\beta</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\Omega_3^2 
\biggl[ \frac{b^2 - c^2}{2a^2} \biggr]\gamma~+~2\pi G \rho \biggl[\biggl( \frac{c^2}{a^2}\biggr) A_3 -  \biggl( \frac{b^2}{a^2} \biggr)A_2 \biggr] 
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~ \Omega_2^2 \beta
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
 ~-~\Omega_3^2 \gamma
~+~4\pi G \rho \biggl[ \frac{ c^2 A_3 -  b^2 A_2}{c^2 - b^2} \biggr] 
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~ -~\Omega_3^2 \gamma
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\Omega_2^2 \beta~-~4\pi G \rho \biggl[ \frac{ c^2 A_3 -  b^2 A_2}{c^2 - b^2} \biggr] \, .
</math>
  </td>
</tr>
</table>
Together, then,

<table align="center" border=0 cellpadding="3">
<tr>
  <td align="right">
<math>~0
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\Omega_2^2\beta
\biggl[ \frac{c^2(c^2 - b^2) + a^2 (b^2 + 3c^2 - 4a^2)}{2a^2c^2}
\biggr]
~+~2\pi G\rho \biggl[ A_1 -  \biggl(\frac{c^2}{a^2}\biggr)A_3 \biggr] 
~+~\biggl[ \frac{4a^2 - c^2 - 3b^2}{2b^2}\biggr]\biggl\{
\Omega_2^2 \beta~-~4\pi G \rho \biggl[ \frac{ c^2 A_3 -  b^2 A_2}{c^2 - b^2} \biggr] 
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\Omega_2^2\beta
\biggl[ \frac{c^2(c^2 - b^2) + a^2 (b^2 + 3c^2 - 4a^2)}{2a^2c^2}
~+~\frac{4a^2 - c^2 - 3b^2}{2b^2}\biggr]
~+~2\pi G\rho \biggl[ \frac{a^2A_1 - c^2 A_3}{a^2} \biggr]
~-~2\pi G \rho \biggl[ \frac{4a^2 - c^2 - 3b^2}{b^2}\biggr] 
\biggl[ \frac{ c^2 A_3 -  b^2 A_2}{c^2 - b^2} \biggr] 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\Omega_2^2\beta
\biggl\{ \frac{b^2[ c^2(c^2 - b^2) + a^2 (b^2 + 3c^2 - 4a^2)] + (4a^2 - c^2 - 3b^2)a^2 c^2}{2a^2 b^2 c^2}
\biggr\}
~+~2\pi G\rho \biggl\{
\biggl[ \frac{a^2A_1 - c^2 A_3}{a^2} \biggr]
~-~\biggl[ \frac{4a^2 - c^2 - 3b^2}{b^2}\biggr] 
\biggl[ \frac{ c^2 A_3 -  b^2 A_2}{c^2 - b^2} \biggr] 
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\Omega_2^2\beta
\biggl\{ \frac{b^2 c^2(c^2 - b^2) + a^2 b^2(c^2 - b^2 ) + ( c^2 - b^2)a^2 c^2 + a^2 (4a^2 -2b^2 - 2c^2 )(c^2 - b^2 ) }{2a^2 b^2 c^2}
\biggr\}
~+~2\pi G\rho 
\biggl[ \frac{b^2(a^2A_1 - c^2 A_3) + a^2(3b^2-4a^2 + c^2)B_{23} }{a^2b^2} \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\Omega_2^2\beta \biggl[ \frac{c^2 - b^2}{c^2} \biggr]
\biggl[ \frac{ 4a^4 - a^2 (b^2 + c^2) + b^2 c^2  }{2a^2 b^2 }
\biggr]
~+~2\pi G\rho 
\biggl[ \frac{a^2(3b^2-4a^2 + c^2)B_{23} + b^2(a^2A_1 - c^2 A_3)  }{a^2b^2} \biggr] \, ,
</math>
  </td>
</tr>
<tr><td align="center" colspan="3">[ [[User:Tohline/Appendix/References#EFE|EFE]], <font color="#00CC00">Chapter 7, &sect;51, Eq. (170)</font> ]</td></tr>
</table>
where,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~B_{23}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl[ \frac{A_2 b^2 - A_3 c^2}{b^2 - c^2} \biggr] \, .</math>
  </td>
</tr>
<tr><td align="center" colspan="3">[ [[User:Tohline/Appendix/References#EFE|EFE]], <font color="#00CC00">Chapter 3, &sect;21, Eqs. (105) &amp; (107)</font> ]</td></tr>
</table>

Similarly, given that ([[#Temporary|see just above]]),

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

<tr>
  <td align="right">
<math>~\Omega_2^2 \beta \biggl[ \frac{c^2 - b^2}{c^2} \biggr]
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
-~\Omega_3^2 \gamma \biggl[ \frac{c^2 - b^2}{c^2} \biggr]
~+~4\pi G \rho \biggl[ \frac{ c^2 A_3 -  b^2 A_2}{c^2 } \biggr] 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
-~\Omega_3^2 \gamma \biggl[ \frac{c^2 - b^2}{c^2} \biggr]
~+~4\pi G \rho \biggl[ \frac{ (c^2 -  b^2)B_{23} }{c^2 } \biggr] \, ,
</math>
  </td>
</tr>
</table>
we have,

<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>~
\biggl\{
-~\Omega_3^2 \gamma \biggl[ \frac{c^2 - b^2}{c^2} \biggr]
~+~4\pi G \rho \biggl[ \frac{ (c^2 -  b^2)B_{23} }{c^2 } \biggr] 
\biggr\}
\biggl[ \frac{ 4a^4 - a^2 (b^2 + c^2) + b^2 c^2  }{2a^2 b^2 }
\biggr]
~+~2\pi G\rho 
\biggl[ \frac{a^2(3b^2-4a^2 + c^2)B_{23} + b^2(a^2A_1 - c^2 A_3)  }{a^2b^2} \biggr] 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
-~\Omega_3^2 \gamma \biggl[ \frac{c^2 - b^2}{b^2} \biggr]
\biggl[ \frac{ 4a^4 - a^2 (b^2 + c^2) + b^2 c^2  }{2a^2 c^2 }
\biggr]
~+~2\pi G \rho \biggl\{
\biggl[ \frac{ [4a^4 - a^2 (b^2 + c^2) + b^2 c^2 ](c^2 -  b^2)B_{23}  }{a^2 b^2 c^2 }
\biggr]
~+~ 
\biggl[ \frac{a^2c^2 (3b^2-4a^2 + c^2)B_{23} + b^2c^2(a^2A_1 - c^2 A_3)  }{a^2b^2 c^2} \biggr] 
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
-~\Omega_3^2 \gamma \biggl[ \frac{c^2 - b^2}{b^2} \biggr]
\biggl[ \frac{ 4a^4 - a^2 (b^2 + c^2) + b^2 c^2  }{2a^2 c^2 }
\biggr]
~+~2\pi G \rho 
\biggl[ \frac{ a^2( b^2 + 3c^2 - 4a^2 ) B_{23} +c^2(a^2A_1 - b^2 A_2)  }{a^2 c^2 }
\biggr] \, .
</math>
  </td>
</tr>
<tr><td align="center" colspan="3">[ [[User:Tohline/Appendix/References#EFE|EFE]], <font color="#00CC00">Chapter 7, &sect;51, Eq. (171)</font> ]</td></tr>
</table>

Finally, looking back at the <math>~(i, j) = (3, 3)</math> constraint and recognizing that,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~-~
\Omega_2^2\beta (c^2 - b^2)
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
4\pi G\rho c^2
\biggl[ \frac{a^2(3b^2-4a^2 + c^2)B_{23} + b^2(a^2A_1 - c^2 A_3)  }{ 4a^4 - a^2 (b^2 + c^2) + b^2 c^2  } \biggr] \, ,
</math>
  </td>
</tr>
</table>

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

<tr>
  <td align="right">
<math>~2\biggl[ \frac{3\cdot 5}{2^2\pi abc\rho} \biggr]\Pi</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~-~2a^2\Omega_2^2\biggl[
\beta^2 - 2 \beta 
+ \biggl( \frac{c^2}{a^2}\biggr)    \biggr]
+ (4\pi G \rho)c^2 A_3  
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
(4\pi G \rho)c^2 A_3  
-~
(c^2 - b^2 )\Omega_2^2 \beta
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
(4\pi G \rho)c^2 A_3  
+~
4\pi G\rho c^2
\biggl[ \frac{a^2(3b^2-4a^2 + c^2)B_{23} + b^2(a^2A_1 - c^2 A_3)  }{ 4a^4 - a^2 (b^2 + c^2) + b^2 c^2  } \biggr] 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
4\pi G \rho c^2 \biggl\{ A_3  
+~
\biggl[ \frac{a^2(3b^2-4a^2 + c^2)B_{23} + b^2(a^2A_1 - c^2 A_3)  }{ 4a^4 - a^2 (b^2 + c^2) + b^2 c^2  } \biggr] 
\biggr\} \, .
</math>
  </td>
</tr>
</table>

==Riemann S-Type Ellipsoids==
In this case, we assume that <math>~\vec{\Omega}</math> and <math>~\vec\zeta</math> are aligned with each other and, as well, are aligned with the <math>~z</math>-axis; that is to say, in addition to setting <math>~(\Omega_1, \zeta_1) = (0, 0)</math> we also set <math>~(\Omega_2, \zeta_2) = (0, 0)</math>.  So, there are only three unknowns &#8212; <math>~\Pi, (\Omega_3, \zeta_3)</math> &#8212; and they can be determined by ignoring off-axis expressions and simultaneously solving the ''diagonal element'' expressions  displayed in our above [[#SummaryTable|''Summary Table'']].  Furthermore, two of the three diagonal-element expressions can be simplified because we are setting <math>~(\Omega_2, \zeta_2) = (0, 0)</math>.  The three relevant equilibrium constraints are:


<table border="1" align="center" cellpadding="5">
<tr>
  <td align="center" colspan="2">Indices</td>
  <td align="center" rowspan="2">2<sup>nd</sup>-Order TVE Expressions that are Relevant to Riemann S-Type Ellipsoids</td>
</tr>
<tr>
  <td align="center" width="5%"><math>~i</math></td>
  <td align="center" width="5%"><math>~j</math></td>
</tr>
<tr>
  <td align="center"><math>~1</math></td>
  <td align="center"><math>~1</math></td>
  <td align="left">

<table align="left" border=0 cellpadding="3">
<tr>
  <td align="right">
<math>~0</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[ \frac{3\cdot 5}{2^2\pi a b c\rho} \biggr] \Pi
+\biggl\{ 
\Omega_3^2
+ 2  \biggl[ \frac{b^2}{b^2+a^2}\biggr] \Omega_3 \zeta_3 
~-~(2\pi G\rho) A_1 
\biggr\} a^2 
+ \biggl[ \frac{a^2}{a^2 + b^2}\biggr]^2 \zeta_3^2 b^2
</math>
  </td>
</tr>
</table>

  </td>
</tr>

<tr>
  <td align="center"><math>~2</math></td>
  <td align="center"><math>~2</math></td>
  <td align="left">

<table align="left" border=0 cellpadding="3">
<tr>
  <td align="right">
<math>~0</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[ \frac{3\cdot 5}{2^2\pi a b c \rho} \biggr]\Pi
+ \biggl[ \frac{b^2}{b^2+a^2}\biggr]^2 \zeta_3^2 a^2
+ \biggl\{
\Omega_3^2  
+ 2 \biggl[ \frac{a^2}{a^2 + b^2}\biggr] \Omega_3 \zeta_3  
~-~( 2\pi G \rho) A_2 
\biggr\}b^2 
</math>
  </td>
</tr>
</table>

  </td>
</tr>

<tr>
  <td align="center"><math>~3</math></td>
  <td align="center"><math>~3</math></td>
  <td align="left">

<table align="left" border=0 cellpadding="3">
<tr>
  <td align="right">
<math>~0</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[ \frac{3\cdot 5}{2^2\pi abc\rho} \biggr]\Pi
- (2\pi G \rho)A_3 c^2 
</math>
  </td>
</tr>
</table>

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


The <math>~(i, j) = (3, 3)</math> component expression immediately identifies the value of one of the unknowns, namely,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\Pi</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl( \frac{2^3\pi^2}{3\cdot 5} \biggr) G \rho^2A_3 a b c^3 \, .
</math>
  </td>
</tr>
</table>

From the remaining pair of diagonal-element expressions, we therefore have,

<table align="center" border=0 cellpadding="3">
<tr>
  <td align="right">
<math>~
0 
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
a^2 \Omega_3^2
+ 2  \biggl[ \frac{b^2a^2}{b^2+a^2}\biggr] \Omega_3 \zeta_3 
+ \biggl[ \frac{a^2}{a^2 + b^2}\biggr]^2 \zeta_3^2 b^2
~+~(2\pi G\rho)(A_3 c^2 - A_1  a^2 ) \, ,
</math>
  </td>
</tr>
</table>

and,
<table align="center" border=0 cellpadding="3">
<tr>
  <td align="right">
<math>~
0
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[ \frac{b^2}{b^2+a^2}\biggr]^2 \zeta_3^2 a^2
+ 
b^2 \Omega_3^2  
+ 2 \biggl[ \frac{a^2b^2}{a^2 + b^2}\biggr] \Omega_3 \zeta_3  
~+~( 2\pi G \rho)(A_3 c^2 - A_2 b^2) \, .
</math>
  </td>
</tr>
</table>
Multiplying the first of these two expressions through by <math>~b^2</math> and the second through by <math>~a^2</math>, then subtracting the second from the first gives,

<table align="center" border=0 cellpadding="3">
<tr>
  <td align="right">
<math>~0</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
b^2\biggl\{ 
2  \biggl[ \frac{b^2a^2}{b^2+a^2}\biggr] \Omega_3 \zeta_3 
+ \biggl[ \frac{a^2}{a^2 + b^2}\biggr]^2 \zeta_3^2 b^2
~+~(2\pi G\rho)(A_3 c^2 - A_1  a^2 ) \biggr\}
</math>
  </td>
</tr>
<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
-~
a^2\biggl\{
\biggl[ \frac{b^2}{b^2+a^2}\biggr]^2 \zeta_3^2 a^2
+ 2 \biggl[ \frac{a^2b^2}{a^2 + b^2}\biggr] \Omega_3 \zeta_3  
~+~( 2\pi G \rho)(A_3 c^2 - A_2 b^2) 
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl\{ 
2  \biggl[ \frac{b^4 a^2}{b^2+a^2}\biggr] \Omega_3 \zeta_3 
~+~(2\pi G\rho)(A_3 c^2 - A_1  a^2 )b^2 \biggr\}
~-~
\biggl\{
2 \biggl[ \frac{a^4 b^2}{a^2 + b^2}\biggr] \Omega_3 \zeta_3  
~+~( 2\pi G \rho)(A_3 c^2 - A_2 b^2) a^2
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~ \biggl[ \frac{b^2 a^2}{b^2+a^2}\biggr] \Omega_3 \zeta_3  
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\pi G\rho \biggl[ \frac{(A_3 c^2 - A_2 b^2) a^2 ~-~(A_3 c^2 - A_1  a^2 )b^2}{ b^2 - a^2} \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\pi G\rho \biggl[ \frac{(A_1 - A_2)a^2b^2}{ b^2 - a^2} - A_3 c^2\biggr] \, .
</math>
  </td>
</tr>
<tr><td align="center" colspan="3">[ [[User:Tohline/Appendix/References#EFE|EFE]], <font color="#00CC00">Chapter 7, &sect;48, Eq. (30)</font> ]</td></tr>
</table>
Note that &#8212; as EFE has done and as we have recorded in a [[ThreeDimensionalConfigurations/JacobiEllipsoids#Equilibrium_Conditions_for_Jacobi_Ellipsoids|related discussion]] &#8212; the first term on the right-hand-side of this last expression can be expressed more compactly in terms of the coefficient, <math>~A_{12}</math>.

Alternatively, dividing the first expression through by <math>~a^2</math> and the second by <math>~b^2</math>, then adding the pair of expressions gives,

<table align="center" border=0 cellpadding="3">
<tr>
  <td align="right">
<math>~
0 
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\Omega_3^2
+ 2  \biggl[ \frac{b^2}{b^2+a^2}\biggr] \Omega_3 \zeta_3 
+ \biggl[ \frac{a^2b^2}{(a^2 + b^2)^2}\biggr] \zeta_3^2 
~+~(2\pi G\rho)(A_3 c^2 - A_1  a^2 )\frac{1}{a^2} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~+~
\biggl[ \frac{a^2 b^2}{(b^2+a^2)^2}\biggr] \zeta_3^2 
+ 
\Omega_3^2  
+ 2 \biggl[ \frac{a^2}{a^2 + b^2}\biggr] \Omega_3 \zeta_3  
~+~( 2\pi G \rho)(A_3 c^2 - A_2 b^2) \frac{1}{b^2}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
2\Omega_3^2 + 2   \Omega_3 \zeta_3 
+ 2\biggl[ \frac{a^2b^2}{(a^2 + b^2)^2}\biggr] \zeta_3^2 
~+~2\pi G\rho \biggl[ \frac{A_3 c^2 - A_1  a^2 }{a^2} + \frac{A_3c^2 - A_2 b^2}{b^2}\biggr] \, .
</math>
  </td>
</tr>
</table>

If we divide through by 2, then replace the product, <math>~\Omega_3\zeta_3</math>, in this expression by the relation derived immediately above, we have,

<table align="center" border=0 cellpadding="3">
<tr>
  <td align="right">
<math>~
\Omega_3^2  
+ \biggl[ \frac{a^2b^2}{(a^2 + b^2)^2}\biggr] \zeta_3^2 
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
~-~\pi G\rho \biggl[ \frac{b^2 (A_3 c^2 - A_1  a^2) + a^2(A_3c^2 - A_2 b^2 ) }{a^2b^2} \biggr] 
~-~   
\pi G\rho \biggl[ \frac{(A_1 - A_2)a^2b^2 - A_3 c^2(b^2 - a^2)}{ b^2 - a^2} \biggr]\biggl[ \frac{b^2+a^2}{b^2 a^2}\biggr] 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{\pi G\rho}{ a^2b^2(a^2-b^2) } 
\biggl\{ [ b^2 (A_3 c^2 - A_1  a^2) + a^2(A_3c^2 - A_2 b^2 )](b^2-a^2) 
~+~   
[ (A_1 - A_2)a^2b^2 - A_3 c^2(b^2 - a^2) ](b^2+a^2) 
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{\pi G\rho}{ a^2b^2(a^2-b^2) } 
\biggl\{ [  - A_1  a^2 b^2 - A_2 a^2 b^2 ](b^2-a^2) 
~+~   
(A_1 - A_2)a^2b^2 (b^2+a^2) 
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{2\pi G\rho}{ (a^2-b^2) } 
\biggl[ 
A_1   a^2 
- A_2  b^2 
\biggr] \, .
</math>
  </td>
</tr>
<tr><td align="center" colspan="3">[ [[User:Tohline/Appendix/References#EFE|EFE]], <font color="#00CC00">Chapter 7, &sect;48, Eq. (29)</font> ]</td></tr>
</table>

<span id="fDefined">It has become customary to characterize each Riemann S-Type ellipsoid by the value of its equilibrium frequency ratio, </span>
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~f</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~\frac{\zeta_3}{\Omega_3} \, ,</math>
  </td>
</tr>
</table>
in which case the relevant pair of constraint equations becomes,

<table border="1" align="center" cellpadding="10" width="80%"><tr><td align="center">

<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>~\biggl[ \frac{b^2 a^2}{b^2+a^2}\biggr] f \Omega_3^2   
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\pi G\rho \biggl[ \frac{(A_1 - A_2)a^2b^2}{ b^2 - a^2} - A_3 c^2\biggr] \, ;
</math>
  </td>
</tr>
<tr><td align="center" colspan="3">[ [[User:Tohline/Appendix/References#EFE|EFE]], <font color="#00CC00">Chapter 7, &sect;48, Eq. (34)</font> ]</td></tr>
</table>
and,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~
\Omega_3^2 \biggl\{1
+ \biggl[ \frac{a^2b^2}{(a^2 + b^2)^2}\biggr] f^2 \biggr\}
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{2\pi G\rho}{ (a^2-b^2) } 
\biggl[ 
A_1   a^2 
- A_2  b^2 
\biggr] \, .
</math>
  </td>
</tr>
<tr><td align="center" colspan="3">[ [[User:Tohline/Appendix/References#EFE|EFE]], <font color="#00CC00">Chapter 7, &sect;48, Eq. (33)</font> ]</td></tr>
</table>

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

These two equations can straightforwardly be combined to generate a quadratic equation for the frequency ratio, <math>~f</math>.  Then, once the value of <math>~f</math> has been determined, either expression can be used to determine the corresponding equilibrium value for <math>~\Omega_3</math> in the unit of <math>~(\pi G \rho)^{1 / 2}</math>.  The fact that the value of <math>~f</math> is determined from the solution of  a quadratic equation underscores the realization that, for a given specification of the ellipsoidal geometry <math>~(a, b, c)</math>, if an equilibrium exists &#8212; ''i.e.,'' if the solution for <math>~f</math> is real rather than imaginary &#8212; then two equally valid, and usually different (''i.e.,'' non-degenerate), values of <math>~f</math> will be realized.  This means that two different underlying flows &#8212; one ''direct'' and the other ''adjoint'' &#8212; will sustain the shape of the ellipsoidal configuration, as viewed from a frame that is rotating about the <math>~z</math>-axis with frequency, <math>~\Omega_3</math>.

==Jacobi and Dedekind Ellipsoids==
Describe &hellip;

==Maclaurin Spheroids==

Describe &hellip;