Editing VE/RiemannEllipsoids (section)

=Steady-State 2<sup>nd</sup>-Order Tensor Virial Equations=
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<font size="-1">[[H_BookTiledMenu#Ellipsoidal_.26_Ellipsoidal-Like|<b>Steady-State<br />2<sup>nd</sup>-Order<br />Tensor Virial<br />Equations</b>]]</font>
|}
Drawing from our [[VE#Virial_Equations_.28Rotating_Frame.29|accompanying discussion of virial equations as viewed from a rotating frame of reference]], here we employ the 2<sup>nd</sup>-order tensor virial equation (TVE),
<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}_{ij} + \mathfrak{W}_{ij} + \delta_{ij}\Pi 
+ \Omega^2 I_{ij} - \Omega_i\Omega_k I_{kj} + 2\epsilon_{ilm}\Omega_m \int_V \rho u_lx_j dx \, ,
</math>
  </td>
</tr>
</table>
to determine the equilibrium conditions of uniform-density <math>~(\rho)</math> ellipsoids that have semi-axes, <math>~(a_1, a_2, a_3) \leftrightarrow (a, b, c),</math> and an internal velocity field, <math>~\vec{u}</math>  (as [[#Adopted_.28Internal.29_Velocity_Field|prescribed below]]), that preserves this specified ellipsoidal shape, as viewed from a frame of reference that is rotating with angular velocity, <math>~\vec\Omega</math>.  Because each of the indices, <math>~i</math> and <math>~j</math>, run from 1 to 3, inclusive, this TVE appears to provide nine equilibrium constraints; and once the values of the density and the three semi-axes are specified, there appear to be seven unknowns:  <math>~\Pi</math> and the three pairs of velocity-field components <math>~(\Omega_1, \zeta_1)</math>, <math>~(\Omega_2, \zeta_2)</math>, <math>~(\Omega_3, \zeta_3).</math>  In practice, however, only five constraints are relevant/independent because, as is encapsulated in &hellip;
<table border="0" width="60%" align="center" cellpadding="10"><tr><td align="left">
<div align="center"><font color="maroon">'''Riemann's Fundamental Theorem'''</font></div>
<font color="darkgreen">&hellip; non-trivial solutions are obtained only if no more than two of the three pairs of velocity-field components are different from zero.</font>
</td></tr></table>
<span id="SummaryTable">Following EFE</span>, we will set <math>~\Omega_1 = \zeta_1 = 0</math>, in which case the only applicable TVE constraint relations are the five identified in the following table of equations.


<table border="1" align="center" cellpadding="5">
<tr>
  <td align="center" colspan="2">Indices</td>
  <td align="center" rowspan="2">Each Associated 2<sup>nd</sup>-Order TVE Expression</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_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>

  </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
+ \biggl[ \frac{c^2}{c^2 + a^2}\biggr]^2 \zeta_2^2  a^2
+ \biggl\{
\Omega_2^2   + 2 \biggl[ \frac{a^2}{a^2+c^2}\biggr]\Omega_2 \zeta_2 
- (2\pi G \rho)A_3 
\biggr\}c^2 
</math>
  </td>
</tr>
</table>

  </td>
</tr>

<tr>
  <td align="center"><math>~2</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\{
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>
</tr>
</table>

  </td>
</tr>

<tr>
  <td align="center"><math>~3</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\{
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>
</table>

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

==General Coefficient Expressions==

In the context of our discussion of configurations that are triaxial ellipsoids, we begin by adopting the <math>~(\ell, m, s)</math> subscript notation to identify which semi-axis length is the (largest, medium-length, smallest).  As has been detailed in an [[ThreeDimensionalConfigurations/HomogeneousEllipsoids#Derivation_of_Expressions_for_Ai|accompanying chapter]], the gravitational potential anywhere inside or on the surface of an homogeneous ellipsoid may be given analytically in terms of the following three coefficient expressions:
<div align="center">
<table align="center" border=0 cellpadding="3">
<tr>
  <td align="right">
<math>
~\frac{A_\ell}{a_\ell a_m a_s}
</math>
  </td>
  <td align="center">
<math>
~=
</math>
  </td>
  <td align="left">
<math>~\frac{2}{a_\ell^3}
\biggl[  \frac{F(\theta,k) - E(\theta,k)}{k^2 \sin^3\theta} \biggr] \, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>
~\frac{A_s}{a_\ell a_m a_s}
</math>
  </td>
  <td align="center">
<math>
~=
</math>
  </td>
  <td align="left">
<math>
~\frac{2}{a_\ell^3} \biggl[  \frac{(a_m/a_s) \sin\theta - E(\theta,k)}{(1-k^2) \sin^3\theta} \biggr] \, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>
~\frac{A_m}{a_\ell a_m a_s} = \frac{2 - (A_\ell + A_s)}{a_\ell a_m a_s}
</math>
  </td>
  <td align="center">
<math>
~=
</math>
  </td>
  <td align="left">
<math>~
\frac{ 2}{a_\ell^3 }
\biggl[ \frac{
E(\theta, k) 
-~(1-k^2)
F(\theta, k)
-~(a_s/a_m)k^2\sin\theta}{k^2 (1-k^2)\sin^3\theta}
\biggr] \, ,
</math>
  </td>
</tr>

</table>
</div>
where, <math>~F(\theta,k)</math> and <math>~E(\theta,k)</math> are incomplete elliptic integrals of the first and second kind, respectively, with arguments,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\theta = \cos^{-1} \biggl(\frac{a_s}{a_\ell} \biggr)</math>
  </td>
  <td align="center">
&nbsp; &nbsp; &nbsp; and &nbsp; &nbsp; &nbsp; 
  </td>
  <td align="left">
<math>~k = \biggl[\frac{1 - (a_m/a_\ell)^2}{1 - (a_s/a_\ell)^2} \biggr]^{1/2} \, .</math>
  </td>
</tr>
</table>
</div>

===Specific Case of a<sub>1</sub> > a<sub>2</sub> > a<sub>3</sub>===

When we discuss configurations in which <math>~a_1 > a_2 > a_3 > 0</math>  &#8212; such as Jacobi, Dedekind, or ''most'' Riemann S-Type ellipsoids &#8212; we must adopt the associations, <math>~(A_1, a_1) \leftrightarrow (A_\ell, a_\ell)</math>, <math>~(A_2, a_2) \leftrightarrow (A_m, a_m)</math>, and <math>~(A_3, a_3) \leftrightarrow (A_s, a_s)</math>.  This means that the coefficients, <math>~A_1</math>, <math>~A_2</math>, and <math>~A_3</math> are defined by the expressions,
<div align="center">
<table align="center" border=0 cellpadding="3">
<tr>
  <td align="right">
<math>
~A_1
</math>
  </td>
  <td align="center">
<math>
~=
</math>
  </td>
  <td align="left">
<math>~2\biggl(\frac{a_2}{a_1}\biggr)\biggl(\frac{a_3}{a_1}\biggr)
\biggl[  \frac{F(\theta,k) - E(\theta,k)}{k^2 \sin^3\theta} \biggr] \, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>
~A_3
</math>
  </td>
  <td align="center">
<math>
~=
</math>
  </td>
  <td align="left">
<math>
~2\biggl(\frac{a_2}{a_1}\biggr) \biggl[  \frac{(a_2/a_1) \sin\theta - (a_3/a_1)E(\theta,k)}{(1-k^2) \sin^3\theta} \biggr] \, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>
~A_2
</math>
  </td>
  <td align="center">
<math>
~=
</math>
  </td>
  <td align="left">
<math>~2 - (A_1+A_3) \, ,</math>
  </td>
</tr>

</table>
</div>
where, the arguments of the incomplete elliptic integrals are,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\theta = \cos^{-1} \biggl(\frac{a_3}{a_1} \biggr)</math>
  </td>
  <td align="center">
&nbsp; &nbsp; &nbsp; and &nbsp; &nbsp; &nbsp; 
  </td>
  <td align="left">
<math>~k = \biggl[\frac{1 - (a_2/a_1)^2}{1 - (a_3/a_1)^2} \biggr]^{1/2} \, .</math>
  </td>
</tr>
<tr><td align="center" colspan="3">[ [[Appendix/References#EFE|EFE]], <font color="#00CC00">Chapter 3, &sect;17, Eq. (32)</font> ]</td></tr>
</table>
</div>

===Specific Case of a<sub>1</sub> > a<sub>3</sub> > a<sub>2</sub>===

When we discuss configurations in which <math>~a_1 > a_3 > a_2 > 0</math>  &#8212; these are usually referred to in [[Appendix/References#EFE|EFE]] as prolate S-Type Riemann ellipsoids  &#8212; we must instead adopt the associations, <math>~(A_1, a_1) \leftrightarrow (A_\ell, a_\ell)</math>, <math>~(A_2, a_2) \leftrightarrow (A_s, a_s)</math>, and <math>~(A_3, a_3) \leftrightarrow (A_m, a_m)</math>.  This means that the coefficients, <math>~A_1</math>, <math>~A_2</math>, and <math>~A_3</math> are defined by the expressions,
<div align="center">
<table align="center" border=0 cellpadding="3">
<tr>
  <td align="right">
<math>
~A_1
</math>
  </td>
  <td align="center">
<math>
~=
</math>
  </td>
  <td align="left">
<math>~2 \biggl( \frac{a_2}{a_1} \biggr)\biggl( \frac{a_3}{a_1} \biggr) 
\biggl[  \frac{F(\theta,k) - E(\theta,k)}{k^2 \sin^3\theta} \biggr] \, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>
~A_2
</math>
  </td>
  <td align="center">
<math>
~=
</math>
  </td>
  <td align="left">
<math>
~2 \biggl( \frac{a_3}{a_1} \biggr) \biggl[  \frac{(a_3/a_1) \sin\theta - (a_2/a_1)E(\theta,k)}{(1-k^2) \sin^3\theta} \biggr] \, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>
~A_3 = 2 - (A_1 + A_2)
</math>
  </td>
  <td align="center">
<math>
~=
</math>
  </td>
  <td align="left">
<math>~
\frac{2a_2 a_3}{a_1^2} 
\biggl[ \frac{
E(\theta, k) 
-~(1-k^2)
F(\theta, k)
-~(a_2/a_3)k^2\sin\theta}{k^2 (1-k^2)\sin^3\theta}
\biggr] \, ,
</math>
  </td>
</tr>

</table>
</div>
where, the arguments of the incomplete elliptic integrals of the first and second kind are,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\theta = \cos^{-1} \biggl(\frac{a_2}{a_1} \biggr)</math>
  </td>
  <td align="center">
&nbsp; &nbsp; &nbsp; and &nbsp; &nbsp; &nbsp; 
  </td>
  <td align="left">
<math>~k = \biggl[\frac{1 - (a_3/a_1)^2}{1 - (a_2/a_1)^2} \biggr]^{1/2} \, .</math>
  </td>
</tr>
<tr><td align="center" colspan="3">[ [[Appendix/References#EFE|EFE]], <font color="#00CC00">Chapter 7, &sect;48d, footnote to Table VII (p. 143)</font> ]</td></tr>
</table>
</div>
NOTE:  All ''irrotational'' ellipsoids belong to this category of configurations.

===Specific Case of a<sub>2</sub> > a<sub>1</sub> > a<sub>3</sub>===

When we discuss configurations in which <math>~a_2 > a_1 > a_3 > 0</math>  &#8212; for example, ''most'' Riemann ellipsoids of Types I, II, &amp; III &#8212; we must instead adopt the associations, <math>~(A_1, a_1) \leftrightarrow (A_m, a_m)</math>, <math>~(A_2, a_2) \leftrightarrow (A_\ell, a_\ell)</math>, and <math>~(A_3, a_3) \leftrightarrow (A_s, a_s)</math>.  This means that the coefficients, <math>~A_1</math>, <math>~A_2</math>, and <math>~A_3</math> are defined by the expressions,
<div align="center">
<table align="center" border=0 cellpadding="3">
<tr>
  <td align="right">
<math>
~A_2
</math>
  </td>
  <td align="center">
<math>
~=
</math>
  </td>
  <td align="left">
<math>~2 \biggl( \frac{a_1}{a_2} \biggr)\biggl( \frac{a_3}{a_2} \biggr)
\biggl[  \frac{F(\theta,k) - E(\theta,k)}{k^2 \sin^3\theta} \biggr] \, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>
~A_3
</math>
  </td>
  <td align="center">
<math>
~=
</math>
  </td>
  <td align="left">
<math>
~2\biggl( \frac{a_1}{a_2}\biggr) \biggl[  \frac{(a_1/a_2) \sin\theta - (a_3/a_2)E(\theta,k)}{(1-k^2) \sin^3\theta} \biggr]  \, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>
~A_1 = 2 - (A_2 + A_3)
</math>
  </td>
  <td align="center">
<math>
~=
</math>
  </td>
  <td align="left">
<math>~
\frac{ 2a_1 a_3}{a_2^2 }
\biggl[ \frac{
E(\theta, k) 
-~(1-k^2)
F(\theta, k)
-~(a_3/a_1)k^2\sin\theta}{k^2 (1-k^2)\sin^3\theta}
\biggr] \, ,
</math>
  </td>
</tr>

</table>
</div>
where, the arguments of the incomplete elliptic integrals are,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\theta = \cos^{-1} \biggl(\frac{a_3}{a_2} \biggr)</math>
  </td>
  <td align="center">
&nbsp; &nbsp; &nbsp; and &nbsp; &nbsp; &nbsp; 
  </td>
  <td align="left">
<math>~k = \biggl[\frac{1 - (a_1/a_2)^2}{1 - (a_3/a_2)^2} \biggr]^{1/2} \, .</math>
  </td>
</tr>
</table>
</div>

===Oblate Spheroids [a<sub>2</sub> = a<sub>1</sub> > a<sub>3</sub>]===

Starting with the case of <math>~a_2 > a_1 > a_3 > 0</math>  and setting <math>~a_2 = a_1</math>, we recognize, first, that <math>~k = 0</math>.  Hence, we have,

<table align="center" border=0 cellpadding="3">
<tr>
  <td align="right">
<math>
~A_3
</math>
  </td>
  <td align="center">
<math>
~=
</math>
  </td>
  <td align="left">
<math>
~2\biggl[  \frac{ \sin\theta - (a_3/a_1)E(\theta,0)}{\sin^3\theta} \biggr]  \, ,
</math>
  </td>
</tr>
</table>

==Adopted (Internal) Velocity Field==

EFE (p. 130) states that the &hellip; <font color="#007700">kinematical requirement, that the motion <math>~(\vec{u})</math>, associated with <math>~\vec{\zeta}</math>, preserves the ellipsoidal boundary, leads to the following expressions for its components:</font>
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~u_1</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~- \biggl[ \frac{a_1^2}{a_1^2 + a_2^2}\biggr] \zeta_3 x_2 + \biggl[ \frac{a_1^2}{a_1^2+a_3^2}\biggr] \zeta_2 x_3 \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~u_2</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~- \biggl[ \frac{a_2^2}{a_2^2 + a_3^2}\biggr] \zeta_1 x_3 + \biggl[ \frac{a_2^2}{a_2^2+a_1^2}\biggr] \zeta_3 x_1 \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~u_3</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~- \biggl[ \frac{a_3^2}{a_3^2 + a_1^2}\biggr] \zeta_2 x_1 + \biggl[ \frac{a_3^2}{a_3^2+a_2^2}\biggr] \zeta_1 x_2 \, .</math>
  </td>
</tr>
<tr><td align="center" colspan="3">[ [[Appendix/References#EFE|EFE]], <font color="#00CC00">Chapter 7, &sect;47, Eq. (1)</font> ]</td></tr>
</table>

==Equilibrium Expressions==
[<b>[[Appendix/References#EFE|<font color="red">EFE</font>]]</b> &sect;11(b), p. 22] <font color="#007700">Under conditions of a stationary state, [the tensor virial equation] gives,</font>
<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>~2 \mathfrak{T}_{ij} + \mathfrak{W}_{ij} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~- \delta_{ij}\Pi \, .</math>
  </td>
</tr>
</table>
</div>
<font color="#007700">[This] provides six integral relations which must obtain whenever the conditions are stationary</font>.

When viewing the (generally ellipsoidal) configuration from a rotating frame of reference, the 2<sup>nd</sup>-order TVE takes on the more general form:
<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}_{ij} + \mathfrak{W}_{ij} + \delta_{ij}\Pi 
+ \Omega^2 I_{ij} - \Omega_i\Omega_k I_{kj} + 2\epsilon_{ilm}\Omega_m \int_V \rho u_lx_j dx
\, .
</math>
  </td>
</tr>
<tr><td align="center" colspan="3">[ [[Appendix/References#EFE|EFE]], <font color="#00CC00">Chapter 2, &sect;12, Eq. (64)</font> ]</td></tr>
</table>

EFE (p. 57) also shows that &hellip; <font color="#007700">The potential energy tensor &hellip; for a homogeneous ellipsoid is given by</font>
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{\mathfrak{W}_{ij}}{\pi G\rho}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~-2A_i I_{ij} \, ,</math>
  </td>
</tr>
<tr><td align="center" colspan="3">[ [[Appendix/References#EFE|EFE]], <font color="#00CC00">Chapter 3, &sect;22, Eq. (128)</font> ]</td></tr>
</table>
<font color="#007700">where</font>
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~I_{ij}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\tfrac{1}{5} Ma_i^2 \delta_{ij} \, ,</math>
  </td>
</tr>
<tr><td align="center" colspan="3">[ [[Appendix/References#EFE|EFE]], <font color="#00CC00">Chapter 3, &sect;22, Eq. (129)</font> ]</td></tr>
</table>

<font color="#007700">is the moment of inertia tensor.</font>  Expressions for all nine components of the kinetic energy tensor, <math>~\mathfrak{T}_{ij}</math> are derived in [[#Appendix_E:_.C2.A0_Kinetic_Energy_Components|Appendix E]], below; and expressions for each of the six Coriolis components can be found in [[#Appendix_B:_.C2.A0Coriolis_Component_u1x2|Appendices B, C, &amp; D]].

===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.

===Off-Diagonal Elements===

Notice that the off-diagonal components of both <math>~I_{ij}</math> and <math>~\mathfrak{W}_{ij}</math> are zero.  Hence, the equilibrium expression that is dictated by each off-diagonal component of the 2<sup>nd</sup>-order TVE is,
 
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>~0</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
2 \mathfrak{T}_{ij} - \Omega_i\Omega_k I_{kj} + 2\epsilon_{ilm}\Omega_m \int_V \rho u_lx_j d^3x
\, .
</math>
  </td>
</tr>
</table>

For example &#8212; as is explicitly illustrated on p. 130 of EFE &#8212; for <math>~i=2</math> and <math>~j=3</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}_{23} - \Omega_2\Omega_3 I_{33} + 2\Omega_1 \cancelto{0}{\int_V \rho u_3x_3 d^3x}
- 2\Omega_3 \int_V \rho u_1x_3 d^3x \, ,
</math>
  </td>
</tr>
<tr><td align="center" colspan="4">[ [[Appendix/References#EFE|EFE]], <font color="#00CC00">Chapter 7, &sect;47, Eq. (3)</font> ]</td></tr>
</table>
whereas for <math>~i=3</math> and <math>~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}_{32} - \Omega_3 \Omega_2 I_{22} + 2\Omega_2 \int_V \rho u_1x_2 d^3x
- 2\Omega_1 \cancelto{0}{\int_V \rho u_2 x_2 d^3x}
\, .
</math>
  </td>
</tr>
<tr><td align="center" colspan="4">[ [[Appendix/References#EFE|EFE]], <font color="#00CC00">Chapter 7, &sect;47, Eq. (4)</font> ]</td></tr>
</table>

<table border="1" cellpadding="8" align="center" width="80%"><tr><td align="left">
Given our adoption of a uniform-density configuration whose surface has a precisely ellipsoidal shape and, along with it, our adoption of the above specific prescription for the internal velocity field, <math>~\vec{u}</math>, we recognize that,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\int_V \rho u_i x_j d^3x</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~0</math>
  </td>
  <td align="right">&nbsp; &nbsp; &nbsp; if  &nbsp; &nbsp;<math>~i = j \, .</math>
</tr>
<tr><td align="center" colspan="4">[ [[Appendix/References#EFE|EFE]], <font color="#00CC00">Chapter 7, &sect;47, Eq. (5)</font> ]</td></tr>
</table>
This has allowed us to set to zero one of the integrals in each of these last two expressions.  In what follows, we will benefit from recognizing, as well, that,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\mathfrak{T}_{32} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\mathfrak{T}_{23}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{1}{2} \int_V \rho v_2 v_3 d^3x \, .</math>
  </td>
</tr>
</table>
</td></tr></table>

Our first off-diagonal element is, then,

<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}_{23} - \Omega_2\Omega_3 I_{33} 
- 2\Omega_3 \rho \int_V u_1 z 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+a^2}\biggr]  \biggl[ \frac{c^2}{c^2 + a^2}\biggr] \zeta_2 \zeta_3 a^2
- \Omega_2\Omega_3 c^2 
- 2 \biggl[ \frac{a^2}{a^2+c^2}\biggr]\Omega_3 \zeta_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\Omega_3  
+ \biggl[ \frac{\zeta_2 a^2}{a^2 + c^2 }\biggr] \biggl[ 2\Omega_3 + \frac{\zeta_3 b^2}{b^2+a^2}\biggr]     
\biggr\} c^2 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </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 \, .
</math>
  </td>
</tr>
</table>

The second is,
 
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>~0</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
2 \mathfrak{T}_{32} - \Omega_3 \Omega_2 I_{22} + 2\Omega_2 \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+a^2}\biggr]  \biggl[ \frac{c^2}{c^2 + a^2}\biggr] \zeta_2 \zeta_3  a^2
- \Omega_3 \Omega_2 b^2 
- 2  \biggl[ \frac{a^2}{a^2 + b^2}\biggr]\Omega_2 \zeta_3 b^2
</math>
  </td>
</tr>

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

<tr>
  <td align="right">
&nbsp;
  </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>
</table>

===How Solution is Obtained ===
Adding this pair of governing expressions 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>~
\biggl[ 2 \mathfrak{T}_{23} - \Omega_2\Omega_3 I_{33} 
- 2\Omega_3 \int_V \rho u_1x_3 dx \biggr]
+
\biggl[2 \mathfrak{T}_{32} - \Omega_3 \Omega_2 I_{22} + 2\Omega_2 \int_V \rho u_1x_2 dx
\biggr]
</math>
  </td>
</tr>
<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~4 \mathfrak{T}_{23} - \Omega_2\Omega_3(I_{22}+ I_{33} )
+
2 \int_V \rho u_1 (\Omega_2 x_2 - \Omega_3 x_3) dx \, ;
</math>
  </td>
</tr>
<tr><td align="center" colspan="3">[ [[Appendix/References#EFE|EFE]], <font color="#00CC00">Chapter 7, &sect;47, Eq. (6)</font> ]</td></tr>
</table>
and subtracting the pair 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[ 2 \mathfrak{T}_{23} - \Omega_2\Omega_3 I_{33} 
- 2\Omega_3 \int_V \rho u_1x_3 dx \biggr]
-
\biggl[2 \mathfrak{T}_{32} - \Omega_3 \Omega_2 I_{22} + 2\Omega_2 \int_V \rho u_1x_2 dx
\biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\Omega_2\Omega_3 (I_{22} - I_{33} )
- 2 \int_V \rho u_1 ( \Omega_2 x_2 + \Omega_3 x_3) dx \, .
</math>
  </td>
</tr>
<tr><td align="center" colspan="3">[ [[Appendix/References#EFE|EFE]], <font color="#00CC00">Chapter 7, &sect;47, Eq. (7)</font> ]</td></tr>
</table>