Editing VE/RiemannEllipsoids (section)

===The Three Diagonal Elements===
For <math>~i = j = 1</math>, 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>~
2 \mathfrak{T}_{11} + \mathfrak{W}_{11} + \Pi 
+ \Omega^2 I_{11} - \Omega_1\Omega_k I_{k1} + 2\epsilon_{1lm}\Omega_m \int_V \rho u_lx_1 d^3x
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
2 \mathfrak{T}_{11} + \mathfrak{W}_{11} + \Pi + \Omega^2 I_{11} 
- \Omega_1^2I_{11} 
+ 2 \Omega_3 \int_V \rho u_2x_1 ~d^3x
- 2\Omega_2 \int_V \rho u_3x_1 ~d^3x
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
2 \mathfrak{T}_{11} + \mathfrak{W}_{11} + \Pi 
+( \Omega_2^2 + \Omega_3^2) I_{11} 
+ 2 \Omega_3\rho \int_V u_2x ~d^3x
- 2\Omega_2\rho \int_V  u_3 x~ d^3x
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[ \frac{a^2}{a^2 + b^2}\biggr]^2 \zeta_3^2 I_{22}
+
\biggl[ \frac{a^2}{a^2+c^2}\biggr]^2 \zeta_2^2 I_{33}  
~-~(2\pi G\rho) A_1 I_{11} + \Pi 
+( \Omega_2^2 + \Omega_3^2) I_{11} 
+ 2  \biggl[ \frac{b^2}{b^2+a^2}\biggr] \Omega_3 \zeta_3 I_{11}
+ 2  \biggl[ \frac{c^2}{c^2 + a^2}\biggr]\Omega_2 \zeta_2 I_{11}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\Pi 
+ \biggl\{
( \Omega_2^2 + \Omega_3^2)  
+ 2  \biggl[ \frac{b^2}{b^2+a^2}\biggr] \Omega_3 \zeta_3 
+ 2  \biggl[ \frac{c^2}{c^2 + a^2}\biggr]\Omega_2 \zeta_2 
~-~(2\pi G\rho) A_1 
\biggr\} I_{11}
+
\biggl[ \frac{a^2}{a^2 + b^2}\biggr]^2 \zeta_3^2 I_{22}
+
\biggl[ \frac{a^2}{a^2+c^2}\biggr]^2 \zeta_2^2 I_{33}  
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow~~~ -\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 \zeta_3 
+ 2  \biggl[ \frac{c^2}{c^2 + a^2}\biggr]\Omega_2 \zeta_2 
~-~(2\pi G\rho) A_1 
\biggr\} a^2 
+ \biggl[ \frac{a^2}{a^2 + b^2}\biggr]^2 \zeta_3^2 b^2
+ \biggl[ \frac{a^2}{a^2+c^2}\biggr]^2 \zeta_2^2  c^2  \, .
</math>
  </td>
</tr>
</table>

Once we choose the values of the (semi) axis lengths <math>~(a, b, c)</math> of an ellipsoid &#8212; from which the value of <math>~A_1</math> can be immediately determined &#8212; along with a specification of <math>~\rho</math>, this equation has the following five unknowns:  <math>~\Pi, \Omega_2, \Omega_3,  \zeta_2, \zeta_3</math>.  Similarly, for <math>~i = j = 2</math>,

<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>~
2 \mathfrak{T}_{22} + \mathfrak{W}_{22} + \Pi 
+ \Omega^2 I_{22} - \Omega_2\Omega_k I_{k2} + 2\epsilon_{2lm}\Omega_m \int_V \rho u_lx_2 d^3x
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
2 \mathfrak{T}_{22} + \mathfrak{W}_{22} + \Pi 
+ (\Omega_1^2 + \Omega_3^2) I_{22} + 2\Omega_1 \rho \int_V u_3 y ~d^3x
- 2\Omega_3 \rho \int_V u_1 y ~d^3x
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[ \frac{b^2}{b^2 + c^2}\biggr]^2 \zeta_1^2  I_{33}
+
\biggl[ \frac{b^2}{b^2+a^2}\biggr]^2 \zeta_3^2 I_{11}
~-~( 2\pi G \rho) A_2 {I}_{22} 
+ \Pi 
+ (\Omega_1^2 + \Omega_3^2) I_{22} 
+ 2 \biggl[ \frac{c^2}{c^2+b^2}\biggr]\Omega_1 \zeta_1 I_{22} 
+ 2 \biggl[ \frac{a^2}{a^2 + b^2}\biggr] \Omega_3 \zeta_3  I_{22}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\Pi 
+
\biggl[ \frac{b^2}{b^2+a^2}\biggr]^2 \zeta_3^2 I_{11}
+ \biggl\{
(\Omega_1^2 + \Omega_3^2)  
+ 2 \biggl[ \frac{c^2}{c^2+b^2}\biggr]\Omega_1 \zeta_1  
+ 2 \biggl[ \frac{a^2}{a^2 + b^2}\biggr] \Omega_3 \zeta_3  
~-~( 2\pi G \rho) A_2 
\biggr\}{I}_{22} 
+ \biggl[ \frac{b^2}{b^2 + c^2}\biggr]^2 \zeta_1^2  I_{33}
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow~~~-\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 \zeta_3^2 a^2
+ \biggl\{
(\Omega_1^2 + \Omega_3^2)  
+ 2 \biggl[ \frac{c^2}{c^2+b^2}\biggr]\Omega_1 \zeta_1  
+ 2 \biggl[ \frac{a^2}{a^2 + b^2}\biggr] \Omega_3 \zeta_3  
~-~( 2\pi G \rho) A_2 
\biggr\}b^2 
+ \biggl[ \frac{b^2}{b^2 + c^2}\biggr]^2 \zeta_1^2  c^2 \, .
</math>
  </td>
</tr>
</table>

This gives us a second equation, but an additional pair of (for a total of seven) unknowns:  <math>~\Omega_1, \zeta_1</math>.  For the third diagonal element &#8212; that is, for <math>~i=j=3</math> &#8212; 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>~
2 \mathfrak{T}_{33} + \mathfrak{W}_{33} + \Pi 
+ \Omega^2 I_{33} - \Omega_3\Omega_k I_{k3} + 2\epsilon_{3lm}\Omega_m \int_V \rho u_lx_3 ~d^3x
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
2 \mathfrak{T}_{33} + \mathfrak{W}_{33} + \Pi 
+ (\Omega_1^2 + \Omega_2^2) I_{33}  + 2\Omega_2 \rho \int_V u_1 z ~d^3x
- 2\Omega_1 \rho  \int_V u_2 z ~d^3x
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl[ \frac{c^2}{c^2 + a^2}\biggr]^2 \zeta_2^2  I_{11}
+
\biggl[ \frac{c^2}{c^2+b^2}\biggr]^2 \zeta_1^2 I_{22}
- (2\pi G \rho)A_3 I_{33} + \Pi 
+ (\Omega_1^2 + \Omega_2^2) I_{33}  + 2 \biggl[ \frac{a^2}{a^2+c^2}\biggr]\Omega_2 \zeta_2 I_{33}
+ 2 \biggl[\frac{b^2}{b^2 + c^2}\biggr] \Omega_1 \zeta_1 I_{33} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\Pi
+
\biggl[ \frac{c^2}{c^2 + a^2}\biggr]^2 \zeta_2^2  I_{11}
+
\biggl[ \frac{c^2}{c^2+b^2}\biggr]^2 \zeta_1^2 I_{22}
+ \biggl\{
(\Omega_1^2 + \Omega_2^2)   + 2 \biggl[ \frac{a^2}{a^2+c^2}\biggr]\Omega_2 \zeta_2 
+ 2 \biggl[\frac{b^2}{b^2 + c^2}\biggr] \Omega_1 \zeta_1  
- (2\pi G \rho)A_3 
\biggr\}I_{33} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~ -\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 \zeta_2^2  a^2
+
\biggl[ \frac{c^2}{c^2+b^2}\biggr]^2 \zeta_1^2 b^2
+ \biggl\{
(\Omega_1^2 + \Omega_2^2)   + 2 \biggl[ \frac{a^2}{a^2+c^2}\biggr]\Omega_2 \zeta_2 
+ 2 \biggl[\frac{b^2}{b^2 + c^2}\biggr] \Omega_1 \zeta_1  
- (2\pi G \rho)A_3 
\biggr\}c^2 \, . 
</math>
  </td>
</tr>
</table>
This gives us three equations ''vs.'' seven unknowns.