Editing 3Dconfigurations/DescriptionOfRiemannTypeI

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=Description of Riemann Type I Ellipsoids=
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! style="height: 125px; width: 125px; background-color:#ffeeee;" |[[H_BookTiledMenu#Three-Dimensional_Configurations|<b>Type I<br />Riemann<br />Ellipsoids</b>]]
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An earlier, less succinct version of this chapter is titled, [[ThreeDimensionalConfigurations/DescriptionOfRiemannTypeI|ThreeDimensionalConfigurations/DescriptionOfRiemannTypeI]].  Links to even earlier ramblings are [[#See_Also|provided below]].

==Analytic Determination of Equilibrium Model Parameters==
Drawing heavily from &sect;47 (pp. 129 - 132) of [[Appendix/References#EFE|[<font color="red">EFE</font>] ]], in a separate chapter we show how the steady-state 2<sup>nd</sup>-order tensor virial equations can be used to derive the equilibrium structure of [[VE/RiemannEllipsoids#Riemann_Ellipsoids_of_Types_I,_II,_&_III|Riemann Ellipsoids of Type I, II, &amp; III]].  By definition, for these types of Riemann Ellipsoids, 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 that chapter's [[VE/RiemannEllipsoids#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>.

<font color="red">'''STEP #1:'''</font>  Choose the trio of semi-axis lengths that will define the surface of a uniform-density ellipsoid.  By definition, for Riemann Type I ellipsoids the choice must be restricted to the domain,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>a_2 \ge a_1 \ge a_3</math>
  </td>
  <td align="center">
&nbsp; &nbsp; &nbsp; and &nbsp; &nbsp; &nbsp;
  </td>
  <td align="left">
<math>
2a_1 \ge (a_2 + a_3) \, .
</math>
  </td>
</tr>
<tr><td align="center" colspan="3">
[http://www.kendrickpress.com/Riemann.htm BCO2004], Chapter X, &sect;6, top of p. 186; English translation of Riemann (1861)<br />
[ [[Appendix/References#EFE|EFE]], <font color="#00CC00">Chapter 7, &sect;51, Eq. (186)</font> ]
</td></tr>
</table>


<font color="red">'''STEP #2:'''</font>  Evaluate the integral expressions for the index symbols, <math>A_1</math>, <math>A_2</math>, and <math>A_3</math>, which will be used to evaluate the gravitational potential on the surface of, and throughout the interior of, the chosen ellipsoid.  Because, for Type I ellipsoids, <math>a_2 \ge a_1 \ge a_3</math>, presumably we must [[VE/RiemannEllipsoids#Specific_Case_of_a2_%3E_a1_%3E_a3|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 three most relevant index symbols 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>
Note as well that,
<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 a_2^2 - A_3 a_3^2}{a_2^2 - a_3^2} \biggr] \, .</math>
  </td>
</tr>
<tr><td align="center" colspan="3">[ [[Appendix/References#EFE|EFE]], <font color="#00CC00">Chapter 3, &sect;21, Eqs. (105) &amp; (107)</font> ]</td></tr>
</table>


<font color="red">'''STEP #3:'''</font>  Switching notation from <math>(a_1, a_2, a_3) \rightarrow (a, b, c)</math>, evaluate the intermediary parameters, <math>\beta_\pm</math> and <math>\gamma_\pm</math>:
<table border="0" align="center" width="80%" cellpadding="5"><tr><td align="left">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\beta </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">[ [[Appendix/References#EFE|EFE]], <font color="#00CC00">Chapter 7, &sect;47, Eq. (16)</font> ]</td></tr>

<tr>
  <td align="right">
<math>
\gamma 
</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">[ [[Appendix/References#EFE|EFE]], <font color="#00CC00">Chapter 7, &sect;47, Eq. (17)</font> ]</td></tr>
</table>

As is emphasized in [[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>"
</td></tr></table>


<font color="red">'''STEP #4:'''</font>  Using the following set of five constraint equations, determine the values of the five structural parameters, <math>\Pi</math>, <math>\Omega_2</math>, <math>\zeta_2</math>, <math>\Omega_3</math>, and  <math>\zeta_3</math>.  The resulting parameter values will reflect the equilibrium properties of a "Direct" (Jacobi-like) configuration if <math>(\beta, \gamma) \rightarrow (\beta_+, \gamma_+)</math>, whereas they will reflect the equilibrium properties of the "Adjoint" (Dedekind-like) configuration if <math>(\beta, \gamma) \rightarrow (\beta_-, \gamma_-)</math>.

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

<tr>
  <td align="right">
<math>\biggl[ \frac{3\cdot 5}{2\pi (abc) \rho} \biggr]\Pi</math>
  </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>

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

<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>~
-~\Omega_2 \beta \biggl[ \frac{a^2 + c^2}{c^2} \biggr] \, ;
</math>
  </td>
</tr>
<tr><td align="center" colspan="3">[ [[Appendix/References#EFE|EFE]], <font color="#00CC00">Chapter 7, &sect;47, Eq. (12)</font> ]</td></tr>
</table>

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

<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>\zeta_3</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>~
-~\Omega_3 \gamma \biggl[ \frac{a^2 + b^2}{b^2} \biggr] \, .
</math>
  </td>
</tr>
<tr><td align="center" colspan="3">[ [[Appendix/References#EFE|EFE]], <font color="#00CC00">Chapter 7, &sect;47, Eq. (12)</font> ]</td></tr>
</table>

==Maclaurin Spheroid Limit==

===Basic Relations===

In Chapter 7, &sect;51(c) (pp. 165 - 166) of [[Appendix/References#EFE|[<font color="red">EFE</font>] ]], Chandrasekhar shows that <font color="orange">"&hellip; the ''entire'' [[Apps/MaclaurinSpheroidSequence|Maclaurin sequence]] can be considered as limiting <math>(a_2/a_1 \rightarrow 1)</math> forms of the Riemann Ellipsoids of type I."</font>  First, we [[Apps/MaclaurinSpheroidSequence#Equilibrium_Angular_Velocity|recall]] that, as viewed from the inertial frame, each Maclaurin spheroid of eccentricity, <math>e = (1 - a_3^2/a_1^2)^{1 / 2}</math>, rotates uniformly with angular velocity,
<table border="0" align="center" cellpadding="8">
<tr>
  <td align="right"><math>\Omega_\mathrm{Mc}^2 \equiv \frac{\omega_0^2}{\pi G \rho} = 2e^2 B_{13} </math></td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
2(3-2e^2)(1 - e^2)^{1 / 2} \cdot \frac{\sin^{-1} e}{e^3} - \frac{6(1-e^2)}{e^2} \, .</math>
  </td>
</tr>
<tr>
  <td align="center" colspan="3">
[<b>[[Appendix/References#EFE|<font color="red">EFE</font>]]</b>], &sect;32, pp. 77-78, Eqs. (4) &amp; (6)
</td>
</tr>
</table>

Given the specified value of the semi-axis ratio, <math>a_3/a_1</math>, the properties of the limiting Riemann Type I ellipsoid are given by the expressions,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>2\Omega_3</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\Omega_\mathrm{Mc} \pm \biggl[16 B_{13} + \Omega^2_\mathrm{Mc}\biggr]^{1 / 2} \, ,</math>
  </td>
</tr>
<tr><td align="center" colspan="3">[<b>[[Appendix/References#EFE|<font color="red">EFE</font>]]</b>], <font color="#00CC00">Chapter 7, &sect;51(c), p. 166, Eq. (215)</font> </td></tr>
</table>

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

<tr>
  <td align="right">
<math>\zeta_3</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>2(\Omega_\mathrm{Mc} - \Omega_3) \, ,</math>
  </td>
</tr>
<tr><td align="center" colspan="3">[<b>[[Appendix/References#EFE|<font color="red">EFE</font>]]</b>], <font color="#00CC00">Chapter 7, &sect;51(c), p. 165, Eq. (212)</font> </td></tr>
</table>
where,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>B_{13}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\biggl[ \frac{A_1 a_1^2 - A_3 a_3^2}{a_1^2 - a_3^2} \biggr] \, ;</math>
  </td>
</tr>
<tr><td align="center" colspan="3">[<b>[[Appendix/References#EFE|<font color="red">EFE</font>]]</b>], <font color="#00CC00">Chapter 3, &sect;21, Eqs. (105) &amp; (107)</font></td></tr>
</table>
and,

<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>
\frac{1}{e^2} \biggl[\frac{\sin^{-1}e}{e} - (1-e^2)^{1/2}  \biggr](1-e^2)^{1/2} \, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>
~A_3
</math>
  </td>
  <td align="center">
<math>
~=
</math>
  </td>
  <td align="left">
<math>
\frac{2}{e^2} \biggl[(1-e^2)^{-1/2} -\frac{\sin^{-1}e}{e} \biggr](1-e^2)^{1/2} \, .
</math>
  </td>
</tr>
<tr>
  <td align="center" colspan="3">
[<b>[[Appendix/References#EFE|<font color="red">EFE</font>]]</b>], &sect;17, p. 43, Eq. (36)
</td>
</tr>
</table>

<font color="red">REMINDER:</font>&nbsp; For a given choice of the eccentricity, there are two viable solutions &hellip; the ''direct'' configuration and its ''adjoint.''  In the context of [[ThreeDimensionalConfigurations/RiemannStype#Expressions_Supplied_by_EFE|Riemann S-type ellipsoids]], this pair of solutions arises from the choice of the sign <math>(\pm)</math> in the  expression for <math>\Omega_3</math>; in the context of Type I Riemann ellipsoids (<i>i.e.</i>, here) the pair arises from the choice of the sign <math>(\mp)</math> in the <font color="red">STEP #3</font> determination of <math>\beta</math> and <math>\gamma</math>.  In both physical contexts, the ''direct'' (Jacobi-like) solution results from selecting the ''inferior'' sign while the ''adjoint'' (Dedekind-like) solution results from selecting the ''superior'' sign.

===Frequency Ratio===

In the context of [[ThreeDimensionalConfigurations/RiemannStype#Equilibrium_Conditions_for_Riemann_S-type_Ellipsoids|Riemann S-type ellipsoids]], we have found it useful to examine model sequences along which the frequency ratio,
<table border="0" align="center" cellpadding="8">
</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>
is constant.  Below, we will examine how such sequences behave across the domain of Type I Riemann Ellipsoids.  In anticipation of this discussion, here we examine how <math>f</math> varies along the ''limiting'' Maclaurin spheroid sequence.  

Adopting the parameter,
<table border="0" align="center" cellpadding="8">
</tr>
  <td align="right"><math>\mathcal{H} \equiv \frac{16B_{13}}{\Omega^2_\mathrm{Mc}}</math></td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
\frac{8}{e^2}
</math>
  </td>
</tr>
</table>
we have the relation,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\frac{2\Omega_3}{\Omega_\mathrm{Mc}}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>1 \pm  (1 + \mathcal{H} )^{1 / 2} \, .</math>
  </td>
</tr>
</table>
But we also see that,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>f_\mathrm{Mc} = \biggl(\frac{\zeta_3}{\Omega_3}\biggr)_\mathrm{Mc}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>2\biggl[\frac{\Omega_\mathrm{Mc}}{\Omega_3} - 1 \biggr] \, .</math>
  </td>
</tr>
</table>
Combining these last two expressions gives,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\frac{4}{f_\mathrm{Mc} +2}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>1 \pm  (1 + \mathcal{H} )^{1 / 2}</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>1 \pm  \biggl[ 1 + \frac{8}{e^2} \biggr]^{1 / 2}</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow ~~~ \frac{4e}{f_\mathrm{Mc} +2}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>e \pm  ( 8 + e^2 )^{1 / 2}</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow ~~~ f_\mathrm{Mc}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>4e\biggl[ e \pm  ( 8 + e^2 )^{1 / 2} \biggr]^{-1} - 2 \, .</math>
  </td>
</tr>
</table>

In an [[Apps/MaclaurinSpheroidSequence#Maclaurin_Spheroid_Sequence|accompanying discussion of the Maclaurin spheroid sequence]], a number of different plots have been used to display how various physical parameters vary along the sequence.  The solid curve that appears in Figure 1 of that discussion has been redrawn as a black-dotted curve in the left-hand panel of Figure 1 of ''this'' chapter (immediately below); it shows how <math>\Omega^2_\mathrm{Mc}</math> varies with the spheroid's eccentricity, <math>e</math>.  The small solid-green square marker identifies the location along the sequence where the system with the maximum angular velocity resides:
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\biggl[ e, \frac{\omega_0^2}{\pi G \rho} \biggr]</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~\biggl[ 0.92995, 0.449331 \biggr] \, .</math>
  </td>
</tr>
<tr>
  <td align="center" colspan="3">[<b>[[Appendix/References#EFE|<font color="red">EFE</font>]]</b>], &sect;32, p. 80, Eqs. (9) &amp; (10)</td>
</tr>
</table>

 
<table border="1" align="center" cellpadding="8" width="80%">
<tr>
  <td align="center" colspan="2">
<b>Figure 1: &nbsp; Parameter Variations Along the Maclaurin Spheroid Sequence</b>
  </td>
</tr>
<tr>
<td align="center">
[[File:JacobiSequenceTooA.png|400px|center|JacobiSequenceToo]]
</td>
<td align="center">
[[File:f_McA.png|400px|center|FrequencyRatio]]
</td>
</tr>
<tr>
  <td align="left" width="50%">Analogous to Figure 5 from &sect;32, p. 79 of [<b>[[Appendix/References#EFE|<font color="red">EFE</font>]]</b>]; shows how the square of the normalized rotation frequency varies with eccentricity, <math>e = (1 - a_3/a_1)^{1 / 2},</math> along the (black-dotted) Maclaurin sequence and along the Jacobi sequence (series of purple circular markers).</td>
  <td align="left">&nbsp;</td>
</tr>
</table>


<table border="1" cellpadding="10" align="center" colspan="10">
<tr>
  <td align="center" rowspan="3">Note:</td>
  <td align="center" rowspan="3"><math>e</math></td>
  <td align="center" rowspan="3"><math>\Omega_\mathrm{Mc}^2</math></td>
<td align="center" rowspan="7" bgcolor="gray" width="1%">&nbsp;</td>
  <td align="center" colspan="4">Limiting Riemann S-type Ellipsoids</td>
<td align="center" rowspan="7" bgcolor="gray" width="1%">&nbsp;</td>
  <td align="center" colspan="4" bgcolor="pink">Limiting Type I Riemann Ellipsoids</td>
</tr>
<tr>
  <td align="center" colspan="2">Direct</td>
  <td align="center" colspan="2">Adjoint</td>
  <td align="center" colspan="2">Direct</td>
  <td align="center" colspan="2">Adjoint</td>
</tr>
<tr>
  <td align="center" colspan="1"><math>\Omega_3^2</math></td>
  <td align="center" colspan="1"><math>f = \frac{\zeta_3}{\Omega_3}</math></td>
  <td align="center" colspan="1"><math>(\Omega_3^\dagger)^2</math></td>
  <td align="center" colspan="1"><math>f^\dagger = \frac{\zeta_3^\dagger}{\Omega_3^\dagger}</math></td>
  <td align="center" colspan="1"><math>\Omega_3^2</math></td>
  <td align="center" colspan="1"><math>f = \frac{\zeta_3}{\Omega_3}</math></td>
  <td align="center" colspan="1"><math>(\Omega_3^\dagger)^2</math></td>
  <td align="center" colspan="1"><math>f^\dagger = \frac{\zeta_3^\dagger}{\Omega_3^\dagger}</math></td>
</tr>

<tr>
  <td align="center">(a)</td>
  <td align="right">0.00000</td>
  <td align="right">0.00000</td>
  <td align="right">000</td>
  <td align="right">000</td>
  <td align="right">000</td>
  <td align="right">000</td>
  <td align="right">000</td>
  <td align="right">000</td>
  <td align="right">000</td>
  <td align="right">000</td>
</tr>

<tr>
  <td align="center">(b)</td>
  <td align="right">0.81267</td>
  <td align="right">0.37423</td>
  <td align="right">000</td>
  <td align="right">000</td>
  <td align="right">000</td>
  <td align="right">000</td>
  <td align="right">000</td>
  <td align="right">000</td>
  <td align="right">000</td>
  <td align="right">000</td>
</tr>

<tr>
  <td align="center">(c)</td>
  <td align="right">0.92995</td>
  <td align="right">0.44933</td>
  <td align="right">000</td>
  <td align="right">000</td>
  <td align="right">000</td>
  <td align="right">000</td>
  <td align="right">000</td>
  <td align="right">000</td>
  <td align="right">000</td>
  <td align="right">000</td>
</tr>

<tr>
  <td align="center">(d)</td>
  <td align="right">0.95289</td>
  <td align="right">0.44022</td>
  <td align="right">000</td>
  <td align="right">000</td>
  <td align="right">000</td>
  <td align="right">000</td>
  <td align="right">000</td>
  <td align="right">000</td>
  <td align="right">000</td>
  <td align="right">000</td>
</tr>

<tr>
  <td align="left" colspan="13">
Notes:
<ol type="a">
<li>Nonrotating sphere, <math>c/a \rightarrow 1</math>; also, self-adjoint Riemann S-type ellipsoid</li>
<li>Bifurcation to Jacobi (''direct'') and Dedekind (''adjoint'') sequences</li>
<li>Configuration with maximum <math>\Omega^2_\mathrm{Mc}</math></li>
<li>Onset of dynamical instability; also, self-adjoint Riemann S-type ellipsoid
<li>Infinitesimally thin disk, <math>c/a \rightarrow 0</math>
</ol>
  </td>
</tr>
</table>

==Example Equilibrium Models==

<table border="1" align="center" cellpadding="5">
<tr>
  <td align="center" width="50%">[[ThreeDimensionalConfigurations/RiemannStype#Fig2|Riemann S-Type Ellipsoids]]</td>
  <td align="center">Type I Riemann Ellipsoids</td>
</tr>
<tr><td align="center" colspan="2">
[[File:EFEfig17firstSmall.png|800px|center|EFE Figure 17]]
</td></tr>
</table>

===Extracted from Table 4 of XXVIII===
Here are equilibrium model parameters drawn from Table 4 of {{ Chandrasekhar66_XXVIIIfull }}.

<table border="1" align="center" cellpadding="5">
<tr>
<td align="center" colspan="16" bgcolor="lightgreen">
'''Data Extracted from Table 4 (p. 858) of <br />
{{ Chandrasekhar66_XXVIIIfigure }}
</td>
<td align="center" colspan="3" rowspan="2">Our<br /> (reverse-engineered)<br /> Determination<br />of Index Symbols</td>
</tr>
<tr>
  <td align="center" colspan="16">''The Properties of a Few Riemann Ellipsoids of Type I''</td>
</tr>
<tr>
  <td align="center" rowspan="2"><math>\frac{a_2}{a_1}</math></td>
  <td align="center" rowspan="2"><math>\frac{a_3}{a_1}</math></td>
  <td align="center" rowspan="10" bgcolor="lightgrey" width="3%">&nbsp;</td>
  <td align="center" colspan="6">Direct</td>
  <td align="center" rowspan="10" bgcolor="lightgrey" width="3%">&nbsp;</td>
  <td align="center" colspan="6">Adjoint</td>
  <td align="center" rowspan="2" bgcolor="red"><font color="white">A<sub>1</sub></font></td>
  <td align="center" rowspan="2" bgcolor="red"><font color="white">A<sub>2</sub></font></td>
  <td align="center" rowspan="2" bgcolor="red"><font color="white">A<sub>3</sub></font></td>
</tr>
<tr>
  <td align="center"><math>\Omega_2</math></td>
  <td align="center"><math>\Omega_3</math></td>
  <td align="center"><math>\zeta_2</math></td>
  <td align="center"><math>\zeta_3</math></td>

  <td align="center"><math>(\zeta_2/\Omega_2)</math></td>
  <td align="center"><math>(\zeta_3/\Omega_3)</math></td>

  <td align="center"><math>\Omega_2^\dagger</math></td>
  <td align="center"><math>\Omega_3^\dagger</math></td>
  <td align="center"><math>\zeta_2^\dagger</math></td>
  <td align="center"><math>\zeta_3^\dagger</math></td>

  <td align="center"><math>(\zeta_2/\Omega_2)^\dagger</math></td>
  <td align="center"><math>(\zeta_3/\Omega_3)^\dagger</math></td>
</tr>
<tr>
  <td align="right">1.05263</td>
  <td align="right">0.41667</td>
  <td align="right">+0.14834</td>
  <td align="right">+0.73257</td>
  <td align="right">-1.41355</td>
  <td align="right">-2.61578</td>

  <td align="right">-9.52912</td>
  <td align="right">-3.57069</td>

  <td align="right">+0.50185</td>
  <td align="right">+1.30617</td>
  <td align="right">-0.41783</td>
  <td align="right">-1.46707</td>
  
  <td align="right">-0.83258</td>
  <td align="right">-1.12318</td>

  <td align="right">0.43008706</td>
  <td align="right">0.40190235</td>
  <td align="right">1.16801059</td>
</tr>
<tr>
  <td align="right">1.25000</td>
  <td align="right">0.50000</td>
  <td align="right">+0.39259</td>
  <td align="right">+0.66536</td>
  <td align="right">-2.19983</td>
  <td align="right">-1.93895</td>

  <td align="right" bgcolor="white">-5.60338</td>
  <td align="right" bgcolor="white">-2.91414</td>

  <td align="right">+0.87993</td>
  <td align="right">+0.94583</td>
  <td align="right">-0.98148</td>
  <td align="right">-1.36398</td>
  
  <td align="right" bgcolor="white">-1.11541</td>
  <td align="right" bgcolor="white">-1.44210</td>

  <td align="right">0.50823343</td>
  <td align="right">0.37944073</td>
  <td align="right">1.11232585</td>
</tr>
<tr>
  <td align="right">1.44065</td>
  <td align="right">0.49273</td>
  <td align="right">+0.57179</td>
  <td align="right">+0.59896</td>
  <td align="right">-2.24560</td>
  <td align="right">-1.49425</td>

  <td align="right">-3.92732</td>
  <td align="right">-2.49474</td>

  <td align="right">+0.89032</td>
  <td align="right">+0.69996</td>
  <td align="right">-1.44219</td>
  <td align="right">-1.27866</td>

  <td align="right">-1.61986</td>
  <td align="right">-1.82676</td>

  <td align="right">0.52403947</td>
  <td align="right">0.32351421</td>
  <td align="right">1.15244632</td>
</tr>
<tr>
  <td align="right">1.66667</td>
  <td align="right">0.33333</td>
  <td align="right">+0.71251</td>
  <td align="right">+0.52815</td>
  <td align="right">-2.37502</td>
  <td align="right">-1.19714</td>

  <td align="right">-3.33331</td>
  <td align="right">-2.26667</td>

  <td align="right">+0.71251</td>
  <td align="right">+0.52815</td>
  <td align="right">-2.37502</td>
  <td align="right">-1.19714</td>

  <td align="right">-3.33331</td>
  <td align="right">-2.26667</td>

  <td align="right">0.41805282</td>
  <td align="right">0.20718125</td>
  <td align="right">1.37476593</td>
</tr>
<tr>
  <td align="right">1.36444</td>
  <td align="right">0.09518</td>
  <td align="right">+0.05632</td>
  <td align="right">+0.40707</td>
  <td align="right">-6.68275</td>
  <td align="right">-1.24612</td>

  <td align="right">-118.657</td>
  <td align="right">-3.06119</td>

  <td align="right">+0.63035</td>
  <td align="right">+0.59414</td>
  <td align="right">-0.59714</td>
  <td align="right">-0.85376</td>

  <td align="right">-0.94731</td>
  <td align="right">-1.43697</td>

  <td align="right">0.14374587</td>
  <td align="right">0.09152713</td>
  <td align="right">1.76472699</td>
</tr>
<tr>
  <td align="right">1.69351</td>
  <td align="right">0.11813</td>
  <td align="right">+0.15764</td>
  <td align="right">+0.38504</td>
  <td align="right">-6.27092</td>
  <td align="right">-1.02536</td>

  <td align="right">-39.7800</td>
  <td align="right">-2.66300</td>

  <td align="right">+0.73061</td>
  <td align="right">+0.44893</td>
  <td align="right">-1.35309</td>
  <td align="right">-0.87944</td>

  <td align="right">-1.85200</td>
  <td align="right">-1.95897</td>

  <td align="right">0.18178501</td>
  <td align="right">0.08464699</td>
  <td align="right">1.73356799</td>
</tr>
<tr>
  <td align="right">1.52303</td>
  <td align="right">0.05315</td>
  <td align="right">+0.03311</td>
  <td align="right">+0.29600</td>
  <td align="right">-9.85239</td>
  <td align="right">-0.84580</td>

  <td align="right">-297.565</td>
  <td align="right">-2.85743</td>

  <td align="right">+0.52221</td>
  <td align="right">+0.38805</td>
  <td align="right">-0.62474</td>
  <td align="right">-0.64518</td>

  <td align="right">-1.19634</td>
  <td align="right">-1.66262</td>

  <td align="right">0.08593434</td>
  <td align="right">0.04618515</td>
  <td align="right">1.86788051</td>
</tr>
<tr>
  <td align="right">1.78590</td>
  <td align="right">0.06233</td>
  <td align="right">+0.08952</td>
  <td align="right">+0.28558</td>
  <td align="right">-9.19424</td>
  <td align="right">-0.74657</td>

  <td align="right">-102.706</td>
  <td align="right">-2.61422</td>

  <td align="right">+0.57083</td>
  <td align="right">+0.31825</td>
  <td align="right">-1.4418</td>
  <td align="right">-0.66992</td>

  <td align="right">-2.52580</td>
  <td align="right">-2.10501</td>

  <td align="right">0.10258739</td>
  <td align="right">0.04358267</td>
  <td align="right">1.85382994</td>
</tr>
<tr>
  <td align="left" colspan="19">NOTE: &nbsp; All frequencies are given in the unit of <math>(\pi G \rho)^{1 / 2}</math>.</td>
</tr>
</table>


Given the values of <math>\Omega_2</math> and <math>\Omega_3</math> from this table, [[ThreeDimensionalConfigurations/DescriptionOfRiemannTypeI#ReverseEngineered|we have reverse-engineered this problem]] and determined what numerical values of <math>A_1</math>, <math>A_2</math>, and <math>A_3</math> were used by {{ Chandrasekhar66_XXVIII }} for various models. These values have been recorded in the last three columns of the table.

===Extracted from Table 6a of XXVIII===

<table border="1" align="center" cellpadding="5" width="80%">
<tr>
<td align="center" colspan="12" bgcolor="lightgreen">
'''Data Extracted from Table 6a (p. 871) of <br />
{{ Chandrasekhar66_XXVIIIfigure }}
<br />&nbsp;<br />[&nbsp;Also appears as Table XIII(a) on p. 170 of [[Appendix/References#EFE|<font color="red">EFE</font>]] ]
</td>
<td align="center" colspan="3" rowspan="2">Our<br /> (reverse-engineered)<br /> Determination<br />of Index Symbols</td>
</tr>
<tr>
  <td align="center" colspan="12">''The Properties of Marginally Overstable Riemann Ellipsoids of Type I''</td>
</tr>
<tr>
  <td align="center" rowspan="2"><math>\frac{a_2}{a_1}</math></td>
  <td align="center" rowspan="2"><math>\frac{a_3}{a_1}</math></td>
  <td align="center" rowspan="11" bgcolor="lightgrey" width="3%">&nbsp;</td>
  <td align="center" colspan="4">Direct</td>
  <td align="center" rowspan="11" bgcolor="lightgrey" width="3%">&nbsp;</td>
  <td align="center" colspan="4">Adjoint</td>
  <td align="center" rowspan="2" bgcolor="red"><font color="white">A<sub>1</sub></font></td>
  <td align="center" rowspan="2" bgcolor="red"><font color="white">A<sub>2</sub></font></td>
  <td align="center" rowspan="2" bgcolor="red"><font color="white">A<sub>3</sub></font></td>
</tr>
<tr>
  <td align="center"><math>\Omega_2</math></td>
  <td align="center"><math>\Omega_3</math></td>
  <td align="center"><math>\zeta_2</math></td>
  <td align="center"><math>\zeta_3</math></td>

  <td align="center"><math>\Omega_2^\dagger</math></td>
  <td align="center"><math>\Omega_3^\dagger</math></td>
  <td align="center"><math>\zeta_2^\dagger</math></td>
  <td align="center"><math>\zeta_3^\dagger</math></td>
</tr>
<tr>
  <td align="right" bgcolor="lightblue">1.0000</td>
  <td align="right" bgcolor="lightblue">0.3033</td>
  <td align="right" bgcolor="lightblue">&nbsp;0.0000</td>
  <td align="right" bgcolor="lightblue">+0.7073</td>
  <td align="right" bgcolor="lightblue">&nbsp;0.0000</td>
  <td align="right" bgcolor="lightblue">-2.7417</td>

  <td align="right" bgcolor="lightblue">&nbsp;0.0000</td>
  <td align="right" bgcolor="lightblue">+1.3708</td>
  <td align="right" bgcolor="lightblue">&nbsp;0.0000</td>
  <td align="right" bgcolor="lightblue">-1.4147</td>

  <td align="right" bgcolor="lightblue">0.341295655</td>
  <td align="right" bgcolor="lightblue">0.341295655</td>
  <td align="right" bgcolor="lightblue">1.317408690</td>
</tr>
<tr>
  <td align="right">1.0526</td>
  <td align="right">0.3712</td>
  <td align="right">+0.1283</td>
  <td align="right">+0.7176</td>
  <td align="right">-1.5014</td>
  <td align="right">-2.5977</td>

  <td align="right">+0.4898</td>
  <td align="right">+1.2972</td>
  <td align="right">-0.3931</td>
  <td align="right">-1.4371</td>

  <td align="right">0.39892471</td>
  <td align="right">0.37240741</td>
  <td align="right">1.22866788</td>
</tr>
<tr>
  <td align="right">1.1111</td>
  <td align="right">0.4230</td>
  <td align="right">+0.2153</td>
  <td align="right">+0.7098</td>
  <td align="right">-1.8984</td>
  <td align="right">-2.3978</td>

  <td align="right">+0.6812</td>
  <td align="right">+1.1922</td>
  <td align="right">-0.6000</td>
  <td align="right">-1.4275</td>

  <td align="right">0.44194613</td>
  <td align="right">0.38437206</td>
  <td align="right">1.17368182</td>
</tr>
<tr>
  <td align="right">1.1765</td>
  <td align="right">0.4560</td>
  <td align="right">+0.2942</td>
  <td align="right">+0.6901</td>
  <td align="right">-2.1276</td>
  <td align="right">-2.1787</td>

  <td align="right">+0.8032</td>
  <td align="right">+1.0751</td>
  <td align="right">-0.7794</td>
  <td align="right">-1.3984</td>

  <td align="right">0.47156283</td>
  <td align="right">0.38075039</td>
  <td align="right">1.14768677</td>
</tr>
<tr>
  <td align="right" bgcolor="yellow">1.2500</td>
  <td align="right" bgcolor="yellow">0.4703</td>
  <td align="right" bgcolor="yellow">+0.3639</td>
  <td align="right" bgcolor="yellow">+0.6633</td>
  <td align="right" bgcolor="yellow">-2.2794</td>
  <td align="right" bgcolor="yellow">-1.9637</td>

  <td align="right" bgcolor="yellow">+0.8778</td>
  <td align="right" bgcolor="yellow">+0.9579</td>
  <td align="right" bgcolor="yellow">-0.9450</td>
  <td align="right" bgcolor="yellow">-1.3599</td>

  <td align="right" bgcolor="yellow">0.48950275</td>
  <td align="right" bgcolor="yellow">0.36484494</td>
  <td align="right" bgcolor="yellow">1.14565231</td>
</tr>
<tr>
  <td align="right">1.3333</td>
  <td align="right">0.4676</td>
  <td align="right">+0.4269</td>
  <td align="right">+0.6329</td>
  <td align="right">-2.3842</td>
  <td align="right">-1.7621</td>

  <td align="right">+0.9150</td>
  <td align="right">+0.8458</td>
  <td align="right">-1.1125</td>
  <td align="right">-1.3186</td>

  <td align="right">0.49697204</td>
  <td align="right">0.33963373</td>
  <td align="right">1.16339423</td>
</tr>
<tr>
  <td align="right">1.4286</td>
  <td align="right">0.4474</td>
  <td align="right">+0.4877</td>
  <td align="right">+0.5999</td>
  <td align="right">-2.4626</td>
  <td align="right">-1.5752</td>

  <td align="right">+0.9178</td>
  <td align="right">+0.7400</td>
  <td align="right">-1.3082</td>
  <td align="right">-1.2768</td>

  <td align="right">0.49205257</td>
  <td align="right">0.30602987</td>
  <td align="right">1.20191756</td>
</tr>
<tr>
  <td align="right">1.5385</td>
  <td align="right">0.4053</td>
  <td align="right">+0.5550</td>
  <td align="right">+0.5635</td>
  <td align="right">-2.5307</td>
  <td align="right">-1.3984</td>

  <td align="right">+0.8807</td>
  <td align="right">+0.6390</td>
  <td align="right">-1.5937</td>
  <td align="right">-1.2330</td>

  <td align="right">0.47004307</td>
  <td align="right">0.26291500</td>
  <td align="right">1.26704192</td>
</tr>
<tr>
  <td align="right">1.6722</td>
  <td align="right">0.3278</td>
  <td align="right" bgcolor="pink">+0.7107</td>
  <td align="right" bgcolor="pink">+0.5142</td>
  <td align="right" bgcolor="pink">-2.4011</td>
  <td align="right" bgcolor="pink">-1.1673</td>

  <td align="right" bgcolor="pink">+0.7107</td>
  <td align="right" bgcolor="pink">+0.5142</td>
  <td align="right" bgcolor="pink">-2.4011</td>
  <td align="right" bgcolor="pink">-1.1673</td>

  <td align="right">0.42132864</td>
  <td align="right">0.19910085</td>
  <td align="right">1.37957051</td>
</tr>
<tr>
  <td align="left" colspan="15">NOTE: &nbsp; All frequencies are given in the unit of <math>(\pi G \rho)^{1 / 2}</math>.  Also &hellip;
<ul><li>The model highlighted with a light-blue background is a Maclaurin spheroid; in this case we determined the values of the index symbols, <math>A_1</math>, <math>A_2</math>, and <math>A_3</math>, from the analytic expressions given in our [[Apps/MaclaurinSpheroids#Gravitational_Potential|accompanying discussion of the gravitational potential of oblate spheroids]].</li>  <li>The model highlighted with a yellow background is the example that we have used elsewhere in this MediaWiki chapter to illustrate the trajectories of Lagrangian fluid elements.</li> <li>The model whose frequencies have been highlighted with a pink background appears to be self-adjoint.</li></ul> </td>
</tr>
</table>

===Extracted from Table 6b of XXVIII===

<table border="1" align="center" cellpadding="5">
<tr>
<td align="center" colspan="12" bgcolor="lightgreen">
'''Data Extracted from Table 6b (p. 871) of <br />
{{ Chandrasekhar66_XXVIIIfigure }}
<br />&nbsp;<br />[&nbsp;Also appears as Table XIII(b) on p. 170 of [[Appendix/References#EFE|<font color="red">EFE</font>]] ]
</td>
<td align="center" colspan="3" rowspan="2">Our<br /> (reverse-engineered)<br /> Determination<br />of Index Symbols</td>
</tr>
<tr>
  <td align="center" colspan="12">''The Properties of Marginally Overstable Riemann Ellipsoids of Type I''</td>
</tr>
<tr>
  <td align="center" rowspan="2"><math>\frac{a_2}{a_1}</math></td>
  <td align="center" rowspan="2"><math>\frac{a_3}{a_1}</math></td>
  <td align="center" rowspan="9" bgcolor="lightgrey" width="3%">&nbsp;</td>
  <td align="center" colspan="4">Direct</td>
  <td align="center" rowspan="9" bgcolor="lightgrey" width="3%">&nbsp;</td>
  <td align="center" colspan="4">Adjoint</td>
  <td align="center" rowspan="2" bgcolor="red"><font color="white">A<sub>1</sub></font></td>
  <td align="center" rowspan="2" bgcolor="red"><font color="white">A<sub>2</sub></font></td>
  <td align="center" rowspan="2" bgcolor="red"><font color="white">A<sub>3</sub></font></td>
</tr>
<tr>
  <td align="center"><math>\Omega_2</math></td>
  <td align="center"><math>\Omega_3</math></td>
  <td align="center"><math>\zeta_2</math></td>
  <td align="center"><math>\zeta_3</math></td>

  <td align="center"><math>\Omega_2^\dagger</math></td>
  <td align="center"><math>\Omega_3^\dagger</math></td>
  <td align="center"><math>\zeta_2^\dagger</math></td>
  <td align="center"><math>\zeta_3^\dagger</math></td>
</tr>
<tr>
  <td align="right">1.1582</td>
  <td align="right">0.1411</td>
  <td align="right">+0.0618</td>
  <td align="right">+0.5209</td>
  <td align="right">-4.1927</td>
  <td align="right">-1.8047</td>

  <td align="right">+0.5802</td>
  <td align="right">+0.8927</td>
  <td align="right">-0.4469</td>
  <td align="right">-1.053</td>

  <td align="right">0.19506141</td>
  <td align="right">0.15833429</td>
  <td align="right">1.64660480</td>
</tr>
<tr>
  <td align="right">1.1846</td>
  <td align="right">0.1238</td>
  <td align="right">+0.0558</td>
  <td align="right">+0.4903</td>
  <td align="right">-4.7796</td>
  <td align="right">-1.6695</td>

  <td align="right">+0.5829</td>
  <td align="right">+0.8229</td>
  <td align="right">-0.4573</td>
  <td align="right">-0.9947</td>

  <td align="right">0.17573713</td>
  <td align="right">0.13759829</td>
  <td align="right">1.68666459</td>
</tr>
<tr>
  <td align="right">1.2124</td>
  <td align="right">0.1057</td>
  <td align="right">+0.0480</td>
  <td align="right">+0.4554</td>
  <td align="right">-5.4901</td>
  <td align="right">-1.5236</td>

  <td align="right">+0.5737</td>
  <td align="right">+0.7479</td>
  <td align="right">-0.4598</td>
  <td align="right">-0.9277</td>

  <td align="right">0.12951075</td>
  <td align="right">0.09436674</td>
  <td align="right">1.77612251</td>
</tr>
<tr>
  <td align="right">1.2418</td>
  <td align="right">0.0866</td>
  <td align="right">+0.0389</td>
  <td align="right">+0.4146</td>
  <td align="right">-6.4045</td>
  <td align="right">-1.3629</td>

  <td align="right">+0.5506</td>
  <td align="right">+0.6658</td>
  <td align="right">-0.4523</td>
  <td align="right">-0.8488</td>

  <td align="right">0.12951075</td>
  <td align="right">0.09436674</td>
  <td align="right">1.77612251</td>
</tr>
<tr>
  <td align="right">1.2727</td>
  <td align="right">0.0666</td>
  <td align="right">+0.0286</td>
  <td align="right">+0.3658</td>
  <td align="right">-7.6880</td>
  <td align="right">-1.1810</td>

  <td align="right">+0.5098</td>
  <td align="right">+0.5737</td>
  <td align="right">-0.4308</td>
  <td align="right">-0.7529</td>

  <td align="right">0.1025817</td>
  <td align="right">0.07183867</td>
  <td align="right">1.82557964</td>
</tr>
<tr>
  <td align="right">1.3050</td>
  <td align="right">0.0455</td>
  <td align="right">+0.0176</td>
  <td align="right">+0.3044</td>
  <td align="right">-9.7572</td>
  <td align="right">-0.9654</td>

  <td align="right">+0.4436</td>
  <td align="right">+0.4661</td>
  <td align="right">-0.3873</td>
  <td align="right">-0.6305</td>

  <td align="right">0.07218729</td>
  <td align="right">0.04860482</td>
  <td align="right">1.87920789</td>
</tr>
<tr>
  <td align="right">1.3707</td>
  <td align="center">&epsilon;</td>
  <td align="right">+2.1492 &epsilon;<sup>3/2</sup></td>
  <td align="right">+1.4485 &epsilon;<sup>1/2</sup></td>
  <td align="right">-2.2795 &epsilon;<sup>-1/2</sup></td>
  <td align="right">-4.4390 &epsilon;<sup>1/2</sup></td>

  <td align="right">+2.2795 &epsilon;<sup>1/2</sup></td>
  <td align="right">+2.1135 &epsilon;<sup>1/2</sup></td>
  <td align="right">-2.1492 &epsilon;<sup>1/2</sup></td>
  <td align="right">-3.0422 &epsilon;<sup>1/2</sup></td>

  <td align="right">---</td>
  <td align="right">---</td>
  <td align="right">---</td>
</tr>
<tr>
  <td align="left" colspan="15">NOTE: &nbsp; All frequencies are given in the unit of <math>(\pi G \rho)^{1 / 2}</math>.</td>
</tr>
</table>

===Extracted from Table 6c of XXVIII===

<table border="1" align="center" cellpadding="5">
<tr>
<td align="center" colspan="12" bgcolor="lightgreen">
'''Data Extracted from Table 6c (p. 872) of <br />
{{ Chandrasekhar66_XXVIIIfigure }}
<br />&nbsp;<br />[&nbsp;Also appears as Table XIII(c) on p. 170 of [[Appendix/References#EFE|<font color="red">EFE</font>]] ]
</td>
<td align="center" colspan="3" rowspan="2">Our<br /> (reverse-engineered)<br /> Determination<br />of Index Symbols</td>
</tr>
<tr>
  <td align="center" colspan="12">''The Properties of Marginally Overstable Riemann Ellipsoids of Type I''</td>
</tr>
<tr>
  <td align="center" rowspan="2"><math>\frac{a_2}{a_1}</math></td>
  <td align="center" rowspan="2"><math>\frac{a_3}{a_1}</math></td>
  <td align="center" rowspan="7" bgcolor="lightgrey" width="3%">&nbsp;</td>
  <td align="center" colspan="4">Direct</td>
  <td align="center" rowspan="7" bgcolor="lightgrey" width="3%">&nbsp;</td>
  <td align="center" colspan="4">Adjoint</td>
  <td align="center" rowspan="2" bgcolor="red"><font color="white">A<sub>1</sub></font></td>
  <td align="center" rowspan="2" bgcolor="red"><font color="white">A<sub>2</sub></font></td>
  <td align="center" rowspan="2" bgcolor="red"><font color="white">A<sub>3</sub></font></td>
</tr>
<tr>
  <td align="center"><math>\Omega_2</math></td>
  <td align="center"><math>\Omega_3</math></td>
  <td align="center"><math>\zeta_2</math></td>
  <td align="center"><math>\zeta_3</math></td>

  <td align="center"><math>\Omega_2^\dagger</math></td>
  <td align="center"><math>\Omega_3^\dagger</math></td>
  <td align="center"><math>\zeta_2^\dagger</math></td>
  <td align="center"><math>\zeta_3^\dagger</math></td>
</tr>
<tr>
  <td align="right">1.2907</td>
  <td align="right">0.1573</td>
  <td align="right">+0.0979</td>
  <td align="right">+0.5082</td>
  <td align="right">-4.6947</td>
  <td align="right">-1.6093</td>

  <td align="right">+0.7206</td>
  <td align="right">+0.7791</td>
  <td align="right">-0.6376</td>
  <td align="right">-1.0498</td>

  <td align="right">0.21984120</td>
  <td align="right">0.15276149</td>
  <td align="right">1.62739731</td>
</tr>
<tr>
  <td align="right">1.4954</td>
  <td align="right">0.1563</td>
  <td align="right">+0.1427</td>
  <td align="right">+0.4633</td>
  <td align="right">-5.1812</td>
  <td align="right">-1.3221</td>

  <td align="right">+0.7908</td>
  <td align="right">+0.6109</td>
  <td align="right">-0.9355</td>
  <td align="right">-1.0027</td>

  <td align="right">0.22570513</td>
  <td align="right">0.12700933</td>
  <td align="right">1.64728554</td>
</tr>
<tr>
  <td align="right">1.6417</td>
  <td align="right">0.1431</td>
  <td align="right">+0.1769</td>
  <td align="right">+0.4219</td>
  <td align="right">-5.5580</td>
  <td align="right">-1.1381</td>

  <td align="right">+0.7788</td>
  <td align="right">+0.5056</td>
  <td align="right">-1.2612</td>
  <td align="right">-0.9496</td>

  <td align="right">0.21358447</td>
  <td align="right">0.10455951</td>
  <td align="right">1.68185603</td>
</tr>
<tr>
  <td align="right">1.7679</td>
  <td align="right">0.1233</td>
  <td align="right">+0.2211</td>
  <td align="right">+0.3784</td>
  <td align="right">-6.0132</td>
  <td align="right">-0.9802</td>

  <td align="right">+0.7280</td>
  <td align="right">+0.4200</td>
  <td align="right">-1.8137</td>
  <td align="right">-0.8829</td>

  <td align="right">0.19209450</td>
  <td align="right">0.08326199</td>
  <td align="right">1.72464351</td>
</tr>
<tr>
  <td align="right">1.8651</td>
  <td align="right">0.0976</td>
  <td align="right">+0.2856</td>
  <td align="right">+0.3310</td>
  <td align="right">-6.7416</td>
  <td align="right">-0.8341</td>

  <td align="right">+0.6299</td>
  <td align="right">+0.3471</td>
  <td align="right">-2.8546</td>
  <td align="right">-0.7942</td>

  <td align="right">0.16820821</td>
  <td align="right">0.06283471</td>
  <td align="right">1.76895708</td>
</tr>
<tr>
  <td align="left" colspan="15">NOTE: &nbsp; All frequencies are given in the unit of <math>(\pi G \rho)^{1 / 2}</math>.</td>
</tr>
</table>

==Examination of Lagrangian Flow==

===EFE Rotating Cartesian Frame===
Concentric triaxial ellipsoids are defined by the expression,
<table border="0" align="center" cellpadding="5">
<tr>
  <td align="right"><math>P</math></td>
  <td align="center"><math>=</math></td>
  <td align="right"><math>\biggl(\frac{x}{a}\biggr)^2 + \biggl(\frac{y}{b}\biggr)^2 + \biggl(\frac{z}{c}\biggr)^2 \, ,</math></td>
</tr>
</table>
where <math>0 \le P \le 1</math> is a constant.  As viewed from the rotating reference frame, the velocity flow-field everywhere inside <math>(0 \le P < 1)</math>, and on the surface <math>(P = 1)</math> of the Type I Riemann ellipsoid is given by the expression &#8212; see, for example, an [[ThreeDimensionalConfigurations/ChallengesPt6#Riemann_Flow|accompanying discussion of the Riemann flow-field]],

<table border="0" align="center" cellpadding="5">
<tr>
  <td align="right"><math>\mathbf{u}_\mathrm{EFE}</math></td>
  <td align="center"><math>=</math></td>
  <td align="right"><math>
\boldsymbol{\hat\imath} \biggl\{ - \biggl[ \frac{a^2}{a^2 + b^2} \biggr] \zeta_3 y + \biggl[ \frac{a^2}{a^2 + c^2} \biggr] \zeta_2 z \biggr\}
+
\boldsymbol{\hat\jmath} \biggl\{ +\biggl[ \frac{b^2}{a^2 + b^2} \biggr] \zeta_3 x \biggr\}
+
\mathbf{\hat{k}} \biggl\{ - \biggl[ \frac{c^2}{a^2 + c^2} \biggr] \zeta_2 x  \biggr\}
\, .</math>
  </td>
</tr>
</table>
In an [[ThreeDimensionalConfigurations/ChallengesPt6#EFE_Rotating_Frame|accompanying discussion]], we have shown that,
<table border="0" align="center" cellpadding="5">
<tr>
  <td align="right"><math>\mathbf{u}_\mathrm{EFE} \cdot \nabla P</math></td>
  <td align="center"><math>=</math></td>
  <td align="right"><math>0 \, ,</math></td>
</tr>
</table>
which means that, at every location inside and on the surface of the configuration, the velocity vector is orthogonal to a vector that is normal to the ellipsoidal surface at that location.

===Tilted Coordinate System===

<table border="1" align="center" width="60%" cellpadding="8">
<tr>
  <td align="center" colspan="8">'''Figure 1: &nbsp; Tilted Reference Frame'''</td>
</tr>
<tr>
<td align="left">

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

<tr>
  <td align="right">
<math>~\boldsymbol{\hat\imath}</math>
  </td>
  <td align="center">
<math>~\rightarrow</math>
  </td>
  <td align="left">
<math>~\boldsymbol{\hat\imath'} \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\boldsymbol{\hat\jmath}</math>
  </td>
  <td align="center">
<math>~\rightarrow</math>
  </td>
  <td align="left">
<math>~\boldsymbol{\hat\jmath'}\cos\theta - \boldsymbol{\hat{k}'}\sin\theta \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\boldsymbol{\hat{k}}</math>
  </td>
  <td align="center">
<math>~\rightarrow</math>
  </td>
  <td align="left">
<math>~\boldsymbol{\hat\jmath'}\sin\theta + \boldsymbol{\hat{k}'}\cos\theta \, .</math>
  </td>
</tr>
</table>
</td>
<td align="center">[[File:PrimedCoordinates3.png|250px|Primed Coordinates]]</td>
<td align="left">

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

<tr>
  <td align="right">
<math>~x</math>
  </td>
  <td align="center">
<math>~\rightarrow</math>
  </td>
  <td align="left">
<math>~x' \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~y</math>
  </td>
  <td align="center">
<math>~\rightarrow</math>
  </td>
  <td align="left">
<math>~y' \cos\theta - z' \sin\theta \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~z - z_0</math>
  </td>
  <td align="center">
<math>~\rightarrow</math>
  </td>
  <td align="left">
<math>~y' \sin\theta + z'\cos\theta \, .</math>
  </td>
</tr>
</table>

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

As we have detailed in our [[ThreeDimensionalConfigurations/ChallengesPt6#For_Arbitrary_Tip_Angles|accompanying discussion]], as viewed from this "tipped"  frame, the concentric ellipsoidal surfaces of a Type I Riemann ellipsoid are defined by the expression,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>P'</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\biggl[\frac{y'\cos\theta - z'\sin\theta}{b}\biggr]^2 
+ \biggl[\frac{z_0 + z'\cos\theta + y'\sin\theta}{c}\biggr]^2
+\biggl(\frac{x'}{a}\biggr)^2 \, .
</math>
  </td>
</tr>
</table>

and the <span id="CompactFlowField">velocity flow-field</span> is given by the expression,

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

<tr>
  <td align="right">
<math>\boldsymbol{u'}_\mathrm{EFE}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\boldsymbol{\hat\imath'} \biggl\{ 
- \biggl[\frac{a^2}{a^2+b^2}\biggr] \zeta_3 (y'\cos\theta - z'\sin\theta) 
+ \biggl[ \frac{a^2}{a^2 + c^2}\biggr] \zeta_2 (z_0 + y'\sin\theta + z'\cos\theta)
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+
\biggl[\boldsymbol{\hat\jmath'} \cos\theta  - \mathbf{\hat{k}'} \sin\theta \biggr] \biggl\{ 
\biggl[ \frac{b^2}{a^2 + b^2}\biggr] \zeta_3 x'
\biggr\}
+
\biggl[\boldsymbol{\hat\jmath'} \sin\theta  + \mathbf{\hat{k}'} \cos\theta \biggr] \biggl\{ 
- 
\biggl[ \frac{c^2}{a^2 + c^2} \biggr] \zeta_2 x'
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\boldsymbol{\hat\imath'} \biggl\{ 
- \biggl[\frac{a^2}{a^2+b^2}\biggr] \zeta_3 (y'\cos\theta - z'\sin\theta) 
+ \biggl[ \frac{a^2}{a^2 + c^2}\biggr] \zeta_2 (z_0 + y'\sin\theta + z'\cos\theta)
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+
\boldsymbol{\hat\jmath'}  \biggl\{ 
\cos\theta \biggl[ \frac{b^2}{a^2 + b^2}\biggr] \zeta_3 
-
\sin\theta\biggl[ \frac{c^2}{a^2 + c^2} \biggr] \zeta_2 
\biggr\}x'
-
\mathbf{\hat{k}'} \biggl\{ 
\sin\theta \biggl[ \frac{b^2}{a^2 + b^2}\biggr] \zeta_3 
+
\cos\theta \biggl[ \frac{c^2}{a^2 + c^2} \biggr] \zeta_2 
\biggr\}x' \, .
</math>
  </td>
</tr>
</table>

We also have explicitly demonstrated that, for any arbitrarily chosen value of the tilt angle, <math>\theta</math>,
<table border="0" align="center" cellpadding="5">
<tr>
  <td align="right"><math>\mathbf{u'}_\mathrm{EFE} \cdot \nabla P'</math></td>
  <td align="center"><math>=</math></td>
  <td align="right"><math>0 \, .</math></td>
</tr>
</table>

===Preferred Tilt===

As we discuss [[ThreeDimensionalConfigurations/ChallengesPt6#For_Specific_Tip_Angle|elsewhere]], if we specifically choose,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\tan\theta</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
- \frac{\beta \Omega_2}{\gamma \Omega_3} = - \biggl[ \frac{c^2 (a^2 + b^2)}{b^2(a^2 + c^2)}\biggr]\frac{\zeta_2}{\zeta_3} \, ,
</math>
  </td>
</tr>
</table>
the component of the flow-field in the <math>\mathbf{\hat{k}'}</math> direction vanishes; that is, in this specific case, as viewed from the tilted reference frame, all of the fluid motion is confined to the x'-y' plane.  Notice that this plane is not parallel to any of the three principal planes of the Type I Riemann ellipsoid.  <font color="red">I have not seen this fluid-flow behavior previously described in the published literature.  Maybe Norman Lebovitz will know.</font>

The three panels of Figure 2, and the text description that follows, have been drawn from a [[ThreeDimensionalConfigurations/ChallengesPt2#COLLADA-Based_Representation|separate discussion]].


<div align="center">
<table border="1" align="center" cellpadding="8">
<tr>
  <th align="center">Figure 2a</th>
  <th align="center">Figure 2b</th>
</tr>
<tr>
  <td align="left" bgcolor="lightgrey">
[[File:B125c470B.cropped.png|500px|EFE Model b41c385]]
  </td>
  <td align="left" bgcolor="lightgrey">
[[File:B125c470A.cropped.png|500px|EFE Model b41c385]]
  </td>
</tr>
<tr>
  <td align="center" colspan="2" bgcolor="lightgrey">
[[File:DataFileButton02.png|75px|file = Dropbox/3Dviewers/AutoRiemann/TypeI/Lagrange/TL15.lagrange.dae]] <font size="+2">&#x21b2;</font>
  </td>
</tr>
<tr>
  <th align="center" colspan="2">Figure 2c</th>
</tr>
<tr>
  <td align="center" bgcolor="white" colspan="2">
[[File:ProjectedOrbitsFlipped2.png|600px|EFE Model b41c385]]<br />
  <div align="center">[[File:DataFileButton02.png|75px|file = Dropbox/3Dviewers/RiemannModels/RiemannCalculations.xlsx --- worksheet = TypeI_1b]] <font size="+2">&#x21b2;</font></div>
  </td>
</tr>
</table>
</div>

As has been described in an [[ThreeDimensionalConfigurations/RiemannTypeI#Figure3|accompanying discussion of Riemann Type 1 ellipsoids]], we have used COLLADA to construct  an animated and interactive 3D scene that displays in purple the surface of an example Type I ellipsoid; panels a and b of Figure 2 show what this ellipsoid looks like when viewed from two different perspectives.  (As a reminder &#8212; see the [[#explanation| explanation accompanying Figure 2 of that accompanying discussion]] &#8212; the ellipsoid is tilted about the x-coordinate axis at an angle of 61.25&deg; to the equilibrium spin axis, which is shown in green.)  Yellow markers also have been placed in this 3D scene at each of the coordinate locations specified in the [[#ExampleTrajectories|table that accompanies that discussion]].  From the perspective presented in Figure 2b, we can immediately identify three separate, nearly circular trajectories; the largest one corresponds to our choice of z<sub>0</sub> = -0.25, the smallest corresponds to our choice of z<sub>0</sub> = -0.60, and the one of intermediate size correspond to our choice of z<sub>0</sub> = -0.4310.  When viewed from the perspective presented in Figure 2a, we see that these three trajectories define three separate planes; each plane is tipped at an angle of &theta; = -19.02&deg; to the ''untilted'' equatorial, x-y plane of the purple ellipsoid.

===Vorticity===
Here we examine the expression for the vorticity from several different coordinate-system orientations.

====wrt Rotating Body Frame====

As [[#EFE_Rotating_Cartesian_Frame|provided above]], the steady-state velocity field as viewed from the rotating ellipsoid's body frame is,

<table border="0" align="center" cellpadding="5">
<tr>
  <td align="right"><math>\mathbf{u}_\mathrm{EFE}</math></td>
  <td align="center"><math>=</math></td>
  <td align="right"><math>
\boldsymbol{\hat\imath} \biggl\{ - \biggl[ \frac{a^2}{a^2 + b^2} \biggr] \zeta_3 y + \biggl[ \frac{a^2}{a^2 + c^2} \biggr] \zeta_2 z \biggr\}
+
\boldsymbol{\hat\jmath} \biggl\{ +\biggl[ \frac{b^2}{a^2 + b^2} \biggr] \zeta_3 x \biggr\}
+
\mathbf{\hat{k}} \biggl\{ - \biggl[ \frac{c^2}{a^2 + c^2} \biggr] \zeta_2 x  \biggr\}
\, .</math>
  </td>
</tr>
</table>

Hence, as viewed with respect to the body frame, we find,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\nabla\times \boldsymbol{u}_\mathrm{EFE}</math>
  </td>
  <td align="center">
<math>\equiv</math>
  </td>
  <td align="left">
<math>
\boldsymbol{\hat\imath} \biggl\{ \frac{\partial u_z}{\partial y} - \frac{\partial u_y}{\partial z}  \biggr\}
+
\boldsymbol{\hat\jmath} \biggl\{ \frac{\partial u_x}{\partial z} - \frac{\partial u_z}{\partial x} \biggr\}
+
\boldsymbol{\hat{k}}  \biggl\{ \frac{\partial u_y}{\partial x} - \frac{\partial u_x}{\partial y}  \biggr\}
</math>
  </td>
</tr>

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

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\boldsymbol{\hat\jmath} \zeta_2
+
\boldsymbol{\hat{k}}  \zeta_3 \, .
</math>
  </td>
</tr>
</table>

====Another Choice====

Now let's view the flow from a "tilted plane" in which the vorticity vector aligns with the <math>\boldsymbol{\hat{k}''}</math> axis.

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

<tr>
  <td align="right">
<math>\boldsymbol{u''}_\mathrm{EFE}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\boldsymbol{\hat\imath''} \biggl\{ 
- \biggl[\frac{a^2}{a^2+b^2}\biggr] \zeta_3 (y''\cos\chi - z''\sin\chi) 
+ \biggl[ \frac{a^2}{a^2 + c^2}\biggr] \zeta_2 (z_0 + y''\sin\chi + z''\cos\chi)
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+
\boldsymbol{\hat\jmath''}  \biggl\{ 
\cos\chi \biggl[ \frac{b^2}{a^2 + b^2}\biggr] \zeta_3 
-
\sin\chi\biggl[ \frac{c^2}{a^2 + c^2} \biggr] \zeta_2 
\biggr\}x''
-
\mathbf{\hat{k}''} \biggl\{ 
\sin\chi \biggl[ \frac{b^2}{a^2 + b^2}\biggr] \zeta_3 
+
\cos\chi \biggl[ \frac{c^2}{a^2 + c^2} \biggr] \zeta_2 
\biggr\}x'' \, .
</math>
  </td>
</tr>
</table>

The vorticity is,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\nabla\times \boldsymbol{u''}_\mathrm{EFE}</math>
  </td>
  <td align="center">
<math>\equiv</math>
  </td>
  <td align="left">
<math>
\boldsymbol{\hat\imath''} \biggl\{ \frac{\partial u_z''}{\partial y''} - \frac{\partial u_y''}{\partial z''}  \biggr\}
+
\boldsymbol{\hat\jmath''} \biggl\{ \frac{\partial u_x''}{\partial z''} - \frac{\partial u_z''}{\partial x''} \biggr\}
+
\boldsymbol{\hat{k}''}  \biggl\{ \frac{\partial u_y''}{\partial x''} - \frac{\partial u_x''}{\partial y''}  \biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\boldsymbol{\hat\jmath''} \biggl\{ \biggl[\frac{a^2}{a^2+b^2}\biggr] \zeta_3 (\sin\chi) 
+ \biggl[ \frac{a^2}{a^2 + c^2}\biggr] \zeta_2 (\cos\chi) \biggr\}
+
\boldsymbol{\hat\jmath''} \biggl\{ \sin\chi \biggl[ \frac{b^2}{a^2 + b^2}\biggr] \zeta_3 
+
\cos\chi \biggl[ \frac{c^2}{a^2 + c^2} \biggr] \zeta_2  \biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>
+
\boldsymbol{\hat{k}''}  \biggl\{ \cos\chi \biggl[ \frac{b^2}{a^2 + b^2}\biggr] \zeta_3 
-
\sin\chi\biggl[ \frac{c^2}{a^2 + c^2} \biggr] \zeta_2  \biggr\}
+
\boldsymbol{\hat{k}''}  \biggl\{ \biggl[\frac{a^2}{a^2+b^2}\biggr] \zeta_3 (\cos\chi ) 
- \biggl[ \frac{a^2}{a^2 + c^2}\biggr] \zeta_2 (\sin\chi )
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\boldsymbol{\hat\jmath''} \biggl\{ \biggl[\frac{a^2}{a^2+b^2} + \frac{b^2}{a^2 + b^2}\biggr] \zeta_3 (\sin\chi) 
+ \biggl[ \frac{a^2}{a^2 + c^2} + \frac{c^2}{a^2 + c^2}\biggr] \zeta_2 (\cos\chi) 
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>
+
\boldsymbol{\hat{k}''}  \biggl\{ 
\biggl[\frac{a^2}{a^2+b^2} + \frac{b^2}{a^2 + b^2}\biggr] \zeta_3 (\cos\chi ) 
- \biggl[ \frac{a^2}{a^2 + c^2} + \frac{c^2}{a^2 + c^2}\biggr] \zeta_2 (\sin\chi )
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\boldsymbol{\hat\jmath''} \biggl\{ \zeta_3 (\sin\chi) 
+ \zeta_2 (\cos\chi) 
\biggr\}
+
\boldsymbol{\hat{k}''}  \biggl\{ 
\zeta_3 (\cos\chi ) 
- \zeta_2 (\sin\chi )
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\boldsymbol{\hat\jmath''} \biggl\{ \tan\chi 
+ \frac{\zeta_2}{\zeta_3}  
\biggr\}\zeta_3\cos\chi
+
\boldsymbol{\hat{k}''}  \biggl\{ 
1 
- \frac{\zeta_2}{\zeta_3} \tan\chi 
\biggr\}\zeta_3 \cos\chi \, .
</math>
  </td>
</tr>
</table>
So, in order for the <math>\boldsymbol{\hat{\jmath}''}</math> component to be zero, we choose,

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

<tr>
  <td align="right">
<math>\tan\chi</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
-\frac{\zeta_2}{\zeta_3} \, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow ~~~ \cos\chi</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\biggl[1 + \frac{\zeta_2^2}{\zeta_3^2}\biggr]^{-1 / 2} \, ,
</math>
  </td>
</tr>
</table>
in which case we have,

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

<tr>
  <td align="right">
<math>\nabla\times \boldsymbol{u''}_\mathrm{EFE}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\boldsymbol{\hat{k}''}  \biggl\{ 
1 + \frac{\zeta_2^2}{\zeta_3^2}  
\biggr\}\zeta_3 \biggl[1 + \frac{\zeta_2^2}{\zeta_3^2}\biggr]^{-1 / 2}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\boldsymbol{\hat{k}''}  \biggl[ 
1 + \frac{\zeta_2^2}{\zeta_3^2}  
\biggr]^{1 / 2}\zeta_3 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\boldsymbol{\hat{k}''}  (\zeta_2^2 + \zeta_3^2)^{1 / 2} \, .
</math>
  </td>
</tr>
</table>
<font color="red"><b>This makes sense!</b></font>  

Now, can we retrieve the "rotating body frame" expression simply by transforming the coordinates?  Well &hellip;

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

<tr>
  <td align="right">
<math>\boldsymbol{\hat{k}''}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
- \boldsymbol{\hat{\jmath}} \sin\chi + \boldsymbol{\hat{k}} \cos\chi
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow ~~~ \nabla\times \boldsymbol{u''}_\mathrm{EFE} = \boldsymbol{\hat{k}''}  \zeta_3</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
(\zeta_2^2 + \zeta_3^2)^{1 / 2} \biggl[- \boldsymbol{\hat{\jmath}} \sin\chi + \boldsymbol{\hat{k}} \cos\chi \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
(\zeta_2^2 + \zeta_3^2)^{1 / 2} \biggl[- \boldsymbol{\hat{\jmath}} \tan\chi + \boldsymbol{\hat{k}} \biggr]\cos\chi
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
(\zeta_2^2 + \zeta_3^2)^{1 / 2} \biggl[\boldsymbol{\hat{\jmath}} \biggl( \frac{\zeta_2}{\zeta_3}\biggr) + \boldsymbol{\hat{k}} \biggr]
\biggl[1 + \frac{\zeta_2^2}{\zeta_3^2}\biggr]^{-1 / 2}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\boldsymbol{\hat{\jmath}} \zeta_2 + \boldsymbol{\hat{k}} \zeta_3 \, .
</math>
  </td>
</tr>

</table>
<font color="red"><b>Hooray!</b></font>

====wrt Lagrangian Orbital Planes====

As viewed from the "preferred tilted plane," the steady-state velocity field is,

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

<tr>
  <td align="right">
<math>\boldsymbol{u'}_\mathrm{EFE}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\boldsymbol{\hat\imath'} \biggl\{ 
- \biggl[\frac{a^2}{a^2+b^2}\biggr] \zeta_3 (y'\cos\theta - z'\sin\theta) 
+ \biggl[ \frac{a^2}{a^2 + c^2}\biggr] \zeta_2 (z_0 + y'\sin\theta + z'\cos\theta)
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+
\boldsymbol{\hat\jmath'}  \biggl\{ 
\cos\theta \biggl[ \frac{b^2}{a^2 + b^2}\biggr] \zeta_3 
-
\sin\theta\biggl[ \frac{c^2}{a^2 + c^2} \biggr] \zeta_2 
\biggr\}x'
-
\boldsymbol{\hat{k}'} \cancelto{0}{\biggl\{ 
\tan\theta \biggl[ \frac{b^2}{a^2 + b^2}\biggr] \zeta_3 
+
\biggl[ \frac{c^2}{a^2 + c^2} \biggr] \zeta_2 
\biggr\}} \cos\theta x'
</math>
  </td>
</tr>
</table>
where,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\tan\theta</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
- \frac{\beta \Omega_2}{\gamma \Omega_3} = - \biggl[ \frac{c^2 (a^2 + b^2)}{b^2(a^2 + c^2)}\biggr]\frac{\zeta_2}{\zeta_3} \, .
</math>
  </td>
</tr>
</table>

Hence, the vorticity is,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\nabla\times \boldsymbol{u'}_\mathrm{EFE}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
- \boldsymbol{\hat\imath'} \biggl\{ \frac{\partial u'_y}{\partial z'}  \biggr\}
+
\boldsymbol{\hat\jmath'} \biggl\{ \frac{\partial u'_x}{\partial z'}  \biggr\}
+
\boldsymbol{\hat{k}'}  \biggl\{ \frac{\partial u'_y}{\partial x'} - \frac{\partial u'_x}{\partial y'}  \biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\boldsymbol{\hat\jmath'} \frac{\partial }{\partial z'}\biggl\{ 
\biggl[\frac{a^2}{a^2+b^2}\biggr] \zeta_3 (z'\sin\theta) 
+ \biggl[ \frac{a^2}{a^2 + c^2}\biggr] \zeta_2 (z'\cos\theta)
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>
+~
\boldsymbol{\hat{k}'} \frac{\partial }{\partial x'}  \biggl\{ 
x'\cos\theta \biggl[ \frac{b^2}{a^2 + b^2}\biggr] \zeta_3 
-
x'\sin\theta\biggl[ \frac{c^2}{a^2 + c^2} \biggr] \zeta_2 
\biggr\}
+ ~
\boldsymbol{\hat{k}'} \frac{\partial }{\partial y'}  \biggl\{ 
\biggl[\frac{a^2}{a^2+b^2}\biggr] \zeta_3 (y'\cos\theta ) 
- \biggl[ \frac{a^2}{a^2 + c^2}\biggr] \zeta_2 (y'\sin\theta )
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\boldsymbol{\hat\jmath'} \biggl\{ 
\biggl[\frac{a^2}{a^2+b^2}\biggr] \zeta_3 \tan\theta 
+ \biggl[ \frac{a^2}{a^2 + c^2}\biggr] \zeta_2 
\biggr\}\cos\theta
+~
\boldsymbol{\hat{k}'} \biggl\{ 
\biggl[ \frac{b^2}{a^2 + b^2}\biggr] \zeta_3 
-
\tan\theta\biggl[ \frac{c^2}{a^2 + c^2} \biggr] \zeta_2 
\biggr\}\cos\theta
+ ~
\boldsymbol{\hat{k}'} \biggl\{ 
\biggl[\frac{a^2}{a^2+b^2}\biggr] \zeta_3  
- \biggl[ \frac{a^2}{a^2 + c^2}\biggr] \zeta_2 \tan\theta 
\biggr\}\cos\theta
</math>
  </td>
</tr>
</table>


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

<tr>
  <td align="right">
<math>\Rightarrow ~~~ \biggl(\frac{1}{\cos\theta}\biggr)\nabla\times \boldsymbol{u'}_\mathrm{EFE}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\boldsymbol{\hat\jmath'} \biggl\{ 
\biggl[\frac{a^2}{a^2+b^2}\biggr] \zeta_3 \tan\theta 
+ \biggl[ \frac{a^2}{a^2 + c^2}\biggr] \zeta_2 
\biggr\}
+~
\boldsymbol{\hat{k}'} \biggl\{ 
\biggl[ \frac{b^2}{a^2 + b^2}\biggr] \zeta_3 
+ ~
\biggl[\frac{a^2}{a^2+b^2}\biggr] \zeta_3  
-
\biggl[ \frac{c^2}{a^2 + c^2} \biggr] \zeta_2\tan\theta 
- \biggl[ \frac{a^2}{a^2 + c^2}\biggr] \zeta_2 \tan\theta 
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\boldsymbol{\hat\jmath'} \biggl[\frac{a^2}{a^2+b^2}\biggr] \biggl\{ 
\zeta_3 \tan\theta 
+ \frac{b^2}{c^2}\biggl[ \frac{c^2(a^2+b^2)}{b^2(a^2 + c^2)}\biggr] \zeta_2 
\biggr\}
+~
\boldsymbol{\hat{k}'} \biggl\{ 
\zeta_3  
-
\zeta_2\tan\theta \biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\boldsymbol{\hat\jmath'} \biggl[\frac{a^2}{a^2+b^2}\biggr] \biggl[ 
1 
- \frac{b^2}{c^2} 
\biggr]\zeta_3 \tan\theta
+~
\boldsymbol{\hat{k}'} \biggl\{ 
\zeta_3  
-
\zeta_2\tan\theta \biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\boldsymbol{\hat\jmath'} \biggl[ 
\frac{a^2(c^2 - b^2)}{c^2(a^2+b^2)}  
\biggr]\zeta_3\tan\theta
+~
\boldsymbol{\hat{k}'} \biggl\{ 
\zeta_3  
-
\zeta_2\tan\theta \biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
-~\boldsymbol{\hat\jmath'} \biggl[ 
\frac{a^2(c^2 - b^2)}{b^2(a^2 + c^2)}  
\biggr]\zeta_2
+~
\boldsymbol{\hat{k}'} \biggl\{ 
\zeta_3  
-
\zeta_2\tan\theta \biggr\} \, .
</math>
  </td>
</tr>
</table>

----

Can we obtain this result by starting from the original, rotating body frame coordinate expression, then transforming the coordinates ?

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

<tr>
  <td align="right">
<math>\nabla\times \boldsymbol{u}_\mathrm{EFE}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\boldsymbol{\hat\jmath} \zeta_2
+
\boldsymbol{\hat{k}}  \zeta_3 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\biggl[ \boldsymbol{\hat\jmath'}\cos\theta - \boldsymbol{\hat{k}'}\sin\theta \biggr] \zeta_2
+
\biggl[ \boldsymbol{\hat\jmath'}\sin\theta + \boldsymbol{\hat{k}'}\cos\theta \biggr] \zeta_3 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\boldsymbol{\hat\jmath'}\biggl[ \zeta_2 \cos\theta 
+
\zeta_3 \sin\theta \biggr] 
+
\boldsymbol{\hat{k}'} \biggl[ \zeta_3\cos\theta  
-
\zeta_2 \sin\theta \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow ~~~ \biggl( \frac{1}{\cos\theta} \biggr) \nabla\times \boldsymbol{u}_\mathrm{EFE}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\boldsymbol{\hat\jmath'}\biggl[ \zeta_2 
+
\zeta_3 \tan\theta \biggr] 
+
\boldsymbol{\hat{k}'} \biggl[ \zeta_3  
-
\zeta_2 \tan\theta \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\boldsymbol{\hat\jmath'}\biggl[ 1 
+
\biggl(\frac{\zeta_3}{\zeta_2}\biggr) \tan\theta \biggr] \zeta_2
+
\boldsymbol{\hat{k}'} \biggl[ \zeta_3  
-
\zeta_2 \tan\theta \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\boldsymbol{\hat\jmath'}\biggl\{ 1 
-
\biggl[ \frac{c^2 (a^2 + b^2)}{b^2(a^2 + c^2)}\biggr] 
\biggr\} \zeta_2
+
\boldsymbol{\hat{k}'} \biggl[ \zeta_3  
-
\zeta_2 \tan\theta \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\boldsymbol{\hat\jmath'}\biggl\{ 
\biggl[ \frac{b^2(a^2 + c^2) - c^2 (a^2 + b^2)}{b^2(a^2 + c^2)}\biggr] 
\biggr\} \zeta_2
+
\boldsymbol{\hat{k}'} \biggl[ \zeta_3  
-
\zeta_2 \tan\theta \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
-~\boldsymbol{\hat\jmath'} 
\biggl[ \frac{a^2 (c^2 - b^2 )}{b^2(a^2 + c^2)}\biggr] 
\zeta_2
+
\boldsymbol{\hat{k}'} \biggl[ \zeta_3  
-
\zeta_2 \tan\theta \biggr]\, .
</math>
  </td>
</tr>
</table>
<font color="red"><b>Q.E.D.</b></font>

----

Let's see if we can rewrite this expression in a more physically insightful way.

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

<tr>
  <td align="right">
<math>\biggl[ 1 + \tan^2\theta \biggr]^{1 / 2} \nabla\times \boldsymbol{u}_\mathrm{EFE}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\boldsymbol{\hat\jmath'}\biggl[ \zeta_2  
+
\zeta_3 \tan\theta \biggr] 
+
\boldsymbol{\hat{k}'} \biggl[ \zeta_3  
-
\zeta_2 \tan\theta \biggr] \, ,
</math>
  </td>
</tr>
</table>
where,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\tan\theta</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
- \biggl[ \frac{c^2 (a^2 + b^2)}{b^2(a^2 + c^2)}\biggr]\frac{\zeta_2}{\zeta_3} \, .
</math>
  </td>
</tr>
</table>

====Summary Vorticity Expressions====

<ol><li>
Written in terms of the (unprimed) body-frame coordinates,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\nabla\times \boldsymbol{u}_\mathrm{EFE}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\boldsymbol{\hat\jmath} \zeta_2
+
\boldsymbol{\hat{k}}  \zeta_3 \, .
</math>
  </td>
</tr>
</table>
</li>
<li>
If we view the fluid motion from a (double-primed) frame that is tilted with respect to the (unprimed) body frame by the angle, <math>\chi</math>, such that,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\tan\chi</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
-\frac{\zeta_2}{\zeta_3} \, ,
</math>
  </td>
</tr>
</table>
then <math>\boldsymbol{\hat{k}''}</math> will align with the vorticity vector and the vorticity vector will have only one component, namely,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\nabla\times \boldsymbol{u''}_\mathrm{EFE}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\boldsymbol{\hat{k}''}  (\zeta_2^2 + \zeta_3^2)^{1 / 2} \, .
</math>
  </td>
</tr>
</table>

</li>
<li>
If we view the fluid motion from a (single-primed) frame that is tilted with respect to the (unprimed) body frame by an angle, <math>\theta</math>, such that the motion of Lagrangian fluid elements is everywhere parallel to the x'-y' plane &#8212; that is, such that there is no Lagrangian fluid motion in the <math>\boldsymbol{\hat{k}'}</math> direction &#8212; we find,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\tan\theta</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
- \frac{\beta \Omega_2}{\gamma \Omega_3} = - \biggl[ \frac{c^2 (a^2 + b^2)}{b^2(a^2 + c^2)}\biggr]\frac{\zeta_2}{\zeta_3} \, ,
</math>
  </td>
</tr>
</table>
and,
<table border="0" cellpadding="5" align="center">

  <td align="right">
<math>\nabla\times \boldsymbol{u'}_\mathrm{EFE}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\boldsymbol{\hat\jmath'}\overbrace{\biggl[ \zeta_2\cos\theta 
+
\zeta_3 \sin\theta \biggr]}^{\mathrm{due~to~vertical~shear}} 
+
\boldsymbol{\hat{k}'} \underbrace{\biggl[ \zeta_3  \cos\theta
-
\zeta_2 \sin\theta \biggr]}_{\zeta_L} \, .
</math>
  </td>
</table>
</li>

<li>
[[#Vorticity_Implied_by_Lagrangian_Fluid_Motions|From below]], the contribution to the vorticity that is provided by the Lagrangian orbital-element-based description of the motion of the fluid is,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\zeta_\mathrm{L}</math>
  </td>
  <td align="center">
<math>\equiv</math>
  </td>
  <td align="left">
<math>
\biggl\{ \frac{\partial \dot{y}'}{\partial x'} - \frac{\partial \dot{x}'}{\partial y'}  \biggr\}
=
\biggl[ 
\biggl(\frac{x_\mathrm{max}}{y_\mathrm{max}}\biggr)
+
\biggl(\frac{y_\mathrm{max}}{x_\mathrm{max}}\biggr)\biggr] \dot\varphi \, .
</math>
  </td>
</tr>
</table>
<ol type="A">
  <li>
Adopting the parameter (Model001 evaluation in parentheses),
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\Lambda</math>
  </td>
  <td align="center">
<math>\equiv</math>
  </td>
  <td align="left">
<math>
\biggl[ \frac{a^2}{a^2 + b^2} \biggr] \zeta_3  \cos\theta    
-
\biggl[ \frac{a^2}{a^2 + c^2} \biggr] \zeta_2 \sin\theta 
= - 1.332892 \, ,
</math>
  </td>
</tr>
</table>
we have found that,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\frac{x_\mathrm{max}}{y_\mathrm{max}}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\biggl\{ \Lambda \biggl[ \frac{a^2 + b^2}{b^2} \biggr] \frac{\cos\theta}{\zeta_3} \biggr\}^{1 / 2} 
= 1.025854
\, ,
</math>
  </td>
  <td align="center">&nbsp; &nbsp; &nbsp; and, &nbsp; &nbsp; &nbsp; </td>
  <td align="right">
<math>\dot\varphi</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\biggl\{ \Lambda \biggl[ \frac{b^2}{a^2 + b^2} \biggr] \frac{\zeta_3 }{\cos\theta} \biggr\}^{1 / 2}
= 1.299300
\, ,
</math>
  </td>
</tr>
</table>
which implies,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\zeta_L\biggr|_\mathrm{Model001}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
2.599446 \, .
</math>
  </td>
</tr>
</table>

  </li>
  <li>
Alternatively, we have found that,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\frac{x_\mathrm{max}}{y_\mathrm{max}}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{a(c^2\cos^2\theta + b^2\sin^2\theta)^{1 / 2}}{bc}
= 1.025854
\, ,
</math>
  </td>
  <td align="center">&nbsp; &nbsp; &nbsp; and, &nbsp; &nbsp; &nbsp; </td>
  <td align="right">
<math>\dot\varphi</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{\zeta_3}{\cos\theta}
\biggl[\frac{ abc ( c^2 \cos^2\theta + b^2 \sin^2\theta )^{1 / 2}}{c^2(a^2 + b^2)}  \biggr] 
= -1.299300
\, ,
</math>
  </td>
</tr>
</table>
which implies,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\zeta_L\biggr|_\mathrm{Model001}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
-2.599446 \, .
</math>
  </td>
</tr>
</table>
  </li>
  <li>
In step #3, immediately above, we have determined that the <math>\boldsymbol{\hat{k}'}</math> component of the fluid vorticity is given by the expression,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\zeta_L</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
(\zeta_3\cos\theta - \zeta_2\sin\theta) 
= -2.599446 \, .
</math>
  </td>
</tr>
</table>

  </li>
</ol>
</li>
</ol>

<table border="1" width="80%" align="center" cellpadding="5"><tr><td align="left">
<font color="red">It appears as though we have separately derived three expressions for the quantity, <math>\zeta_L</math>.  It would be great if we could demonstrate analytically that the three expressions are, indeed, identical.</font>

Keep in mind that the definition of <math>\tan\theta</math> establishes the relation,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>b^2(a^2 + c^2)\zeta_3\sin\theta</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
- c^2 (a^2 + b^2) \zeta_2 \cos\theta
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow ~~~ \frac{\zeta_3}{c^2(a^2 + b^2) \cos\theta}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
- \frac{\zeta_2}{b^2(a^2 + c^2)  \sin\theta}
</math>
  </td>
</tr>
</table>

----

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

<tr>
  <td align="right">
<math>\Upsilon</math>
  </td>
  <td align="center">
<math>\equiv</math>
  </td>
  <td align="left">
<math>
(c^2\cos^2\theta + b^2\sin^2\theta)^{1 / 2} \, .
</math>
  </td>
</tr>
</table>

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

<tr>
  <td align="right">
<math>\zeta_\mathrm{L}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\biggl[ 
\biggl(\frac{x_\mathrm{max}}{y_\mathrm{max}}\biggr)
+
\biggl(\frac{y_\mathrm{max}}{x_\mathrm{max}}\biggr)\biggr] \dot\varphi 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\biggl[ 
\frac{a\Upsilon}{bc} + \frac{bc}{a\Upsilon}
\biggr] 
\frac{\zeta_3}{\cos\theta}
\biggl[\frac{ abc \Upsilon}{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>
\frac{\zeta_3}{\cos\theta}\biggl\{
\biggl[ \frac{a\Upsilon}{bc} \biggr] 
\biggl[\frac{ abc \Upsilon}{c^2(a^2 + b^2)}  \biggr] 
+
\biggl[ \frac{bc}{a\Upsilon} \biggr] 
\biggl[\frac{ abc \Upsilon}{c^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>
\frac{b^2\zeta_3}{(a^2 + b^2)\cos\theta}\biggl\{
\biggl[\frac{ a^2 \Upsilon^2}{b^2c^2}  \biggr] 
+
1 
\biggr\}
</math>
  </td>
</tr>

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

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{\zeta_3}{c^2(a^2 + b^2)\cos\theta}\biggl\{
a^2 c^2\cos^2\theta + a^2b^2\sin^2\theta 
+
b^2c^2(\sin^2\theta + \cos^2\theta) 
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{\zeta_3}{c^2(a^2 + b^2)\cos\theta}\biggl\{
c^2(a^2 + b^2)\cos^2\theta  
\biggr\}
+
\frac{\zeta_3}{c^2(a^2 + b^2)\cos\theta}\biggl\{
b^2(a^2 + c^2)\sin^2\theta 
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\zeta_3\cos\theta - \zeta_2\sin\theta \, .
</math>
  </td>
</tr>
</table>
<font color="red"><b>Q.E.D.</b></font>

----

Alternatively, pulling from the expressions that have been derived in terms of the parameter, <math>\Lambda</math>, we find,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\zeta_\mathrm{L}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\biggl[ 
\biggl(\frac{x_\mathrm{max}}{y_\mathrm{max}}\biggr)
+
\biggl(\frac{y_\mathrm{max}}{x_\mathrm{max}}\biggr)\biggr] \dot\varphi 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\biggl\{ \Lambda \biggl[ \frac{a^2 + b^2}{b^2} \biggr] \frac{\cos\theta}{\zeta_3} \biggr\}^{1 / 2}
\biggl\{ \Lambda \biggl[ \frac{b^2}{a^2 + b^2} \biggr] \frac{\zeta_3 }{\cos\theta} \biggr\}^{1 / 2}
+
\biggl\{ \Lambda^{-1} \biggl[ \frac{b^2}{a^2 + b^2} \biggr] \frac{\zeta_3}{\cos\theta} \biggr\}^{1 / 2}
\biggl\{ \Lambda \biggl[ \frac{b^2}{a^2 + b^2} \biggr] \frac{\zeta_3 }{\cos\theta} \biggr\}^{1 / 2}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\Lambda
+
\biggl[ \frac{b^2}{a^2 + b^2} \biggr] \frac{\zeta_3}{\cos\theta} 
</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] \zeta_3  \cos\theta    
-
\biggl[ \frac{a^2}{a^2 + c^2} \biggr] \zeta_2 \sin\theta 
+
\biggl[ \frac{b^2}{a^2 + b^2} \biggr] \frac{\zeta_3}{\cos\theta}\biggl\{\sin^2\theta + \cos^2\theta\biggr\} 
</math>
  </td>
</tr>

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

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

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\zeta_3  \cos\theta    
-
\zeta_2\sin\theta \, .
</math>
  </td>
</tr>
</table>
<font color="red"><b>Q.E.D.</b></font>
</td></tr></table>

===Lagrangian Fluid Trajectories===

====Off-Center Ellipse====

The yellow dots in Figures 2a and 2b trace three different, nearly circular, closed curves.  These curves each show what results from the intersection of the surface of the triaxial ellipsoid and a plane that is tilted with respect to the x = x' axis by the specially chosen angle, <math>\theta</math>; the different curves result from different choices of the intersection point, <math>z_0</math>.  Several additional such curves are displayed in Figure 2c.  Each of these curves necessarily also identifies the trajectory that is followed by a fluid element that sits on the surface of the ellipsoid.  

We have determined that the <math>y'(x')</math> function that defines each closed curve is describable analytically by the expression,

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

<tr>
  <td align="right">
<math>
\biggl[ \frac{y' - y'_0}{y'_\mathrm{max}} \biggr]^2
</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
1 - \biggl( \frac{x'}{x'_\mathrm{max}} \biggr)^2 \, ,
</math>
  </td>
</tr>
</table>
where (see independent derivations with identical results from [[ThreeDimensionalConfigurations/ChallengesPt2#OffCenter|ChallengesPt2]] and [[ThreeDimensionalConfigurations/ChallengesPt6#Are_Orbits_Exact_Circles|ChallengesPt6]]),
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math> x'_\mathrm{max}  </math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
a\biggl[ 1 -\frac{z_0^2 \cos^2\theta}{(c^2  \cos^2\theta + b^2 \sin^2\theta )} \biggr]^{1 / 2}  
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
a\biggl[ \frac{(c^2 - z_0^2) \cos^2\theta + b^2\sin^2\theta}{(c^2  \cos^2\theta + b^2 \sin^2\theta )} \biggr]^{1 / 2} \, ,  
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>y'_\mathrm{max} </math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
bc \biggl[ (c^2  \cos^2\theta   + b^2 \sin^2\theta )  - z_0^2 \cos^2\theta   \biggr]^{1 / 2} ( c^2 \cos^2\theta + b^2 \sin^2\theta )^{-1}
 </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
b c \biggl[ \frac{(c^2 - z_0^2) \cos^2\theta + b^2\sin^2\theta}{(c^2  \cos^2\theta + b^2 \sin^2\theta )^2} \biggr]^{1 / 2} \, ,  
 </math>
  </td>
</tr>

<tr>
  <td align="right">
<math>y'_0 </math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
-  \frac{z_0 b^2\sin\theta }{ ( c^2 \cos^2\theta + b^2 \sin^2\theta )} \, .
</math>
  </td>
</tr>

</table>

This is the equation that describes a closed ellipse with semi-axes, <math>(x'_\mathrm{max}, y'_\mathrm{max})</math>, that is offset from the z'-axis along the y'-axis by a distance, <math>y'_0</math>.  Notice that the degree of flattening,

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

<tr>
  <td align="right">
<math>\biggl[ \frac{x'}{y'} \biggr]_\mathrm{max} </math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{ a( c^2 \cos^2\theta + b^2 \sin^2\theta )^{1 / 2}}{bc} \, ,
</math>
  </td>
</tr>
</table>
is independent of <math>z_0</math>; that is to say, the degree of flattening of all of the elliptical trajectories is identical!  Notice, as well, that the y-offset, <math>y'_0</math>, is linearly proportional to <math>z_0</math>.

In a [[ThreeDimensionalConfigurations/ChallengesPt6#Plot_Off-Center,_Slightly_Flattened_Ellipse|separate discussion]], we have demonstrated that the [[#CompactFlowField|compact version of the ''tilted'' flow-field]] is everywhere orthogonal to the elliptical trajectory whose analytic definition is given by the off-set ellipse equation.

====Associated Lagrangian Velocities====
Let's presume that, as a function of time, the x'-y' coordinates and associated velocity components of each Lagrangian fluid element are given by the expressions,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~x'</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~x_\mathrm{max}\cos(\dot\varphi t)</math>
  </td>
<td align="center">&nbsp; &nbsp; &nbsp; and, &nbsp; &nbsp; &nbsp;
  <td align="right">
<math>~y' - y_0</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~y_\mathrm{max}\sin(\dot\varphi t) \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\dot{x}'</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~- x_\mathrm{max}~ \dot\varphi \cdot \sin(\dot\varphi t) = (y_0 - y') \biggl[ \frac{x_\mathrm{max}}{y_\mathrm{max}} \biggr] \dot\varphi </math>
  </td>
<td align="center">&nbsp; &nbsp; &nbsp; and, &nbsp; &nbsp; &nbsp;
  <td align="right">
<math>~\dot{y}' </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~y_\mathrm{max}~\dot\varphi \cdot \cos(\dot\varphi t) = x' \biggl[ \frac{y_\mathrm{max}}{x_\mathrm{max}}\biggr] \dot\varphi \, .</math>
  </td>
</tr>
</table>

If this is the correct description of the Lagrangian motion in a <math>z' = 0</math> plane of motion, then the velocity components, <math>\dot{x}'</math> and <math>\dot{y}'</math>, must match the [[ThreeDimensionalConfigurations/ChallengesPt6#SpecificTipAngle|respective components of the Riemann flow-field]], namely,

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

<tr>
  <td align="right">
<math>\boldsymbol{u'}_\mathrm{EFE}\biggr|_{z'=0}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\boldsymbol{\hat\imath'} \biggl\{\biggl[ \frac{a^2}{a^2 + c^2}\biggr] \zeta_2 z_0
\biggr\}
+
\boldsymbol{\hat\imath'} \biggl\{ 
\cancelto{0}{z' (c^2 - b^2 )\tan\theta}  
- y' [c^2  + b^2 \tan^2\theta ]
\biggr\}\frac{\zeta_3 \cos\theta}{c^2}\biggl[\frac{a^2}{a^2+b^2}\biggr] 
+
\boldsymbol{\hat\jmath'} \biggl[ \frac{b^2}{a^2 + b^2}\biggr] \frac{\zeta_3~x' }{\cos\theta} \, .
</math>
  </td>
</tr>
</table>
First, let's compare the <math>\boldsymbol{\hat\jmath'}</math> components.

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

<tr>
  <td align="right">
<math>x' \biggl[ \frac{y'}{x'}\biggr]_\mathrm{max} \dot\varphi</math>
  </td>
  <td align="center">
<math>\leftrightarrow</math>
  </td>
  <td align="left">
<math>
x' \biggl[ \frac{b^2}{a^2 + b^2}\biggr] \frac{\zeta_3}{\cos\theta}
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow ~~~ \dot\varphi</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\biggl[ \frac{b^2}{a^2 + b^2}\biggr] \frac{\zeta_3}{\cos\theta}\biggl[ \frac{x'}{y'}\biggr]_\mathrm{max} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\biggl[ \frac{b^2}{a^2 + b^2}\biggr] \frac{\zeta_3}{\cos\theta}
\biggl[\frac{ a( c^2 \cos^2\theta + b^2 \sin^2\theta )^{1 / 2}}{bc}  \biggr] 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{\zeta_3}{\cos\theta}
\biggl[\frac{ abc ( c^2 \cos^2\theta + b^2 \sin^2\theta )^{1 / 2}}{c^2(a^2 + b^2)}  \biggr] 
\, .
</math>
  </td>
</tr>
</table>

Now let's compare the <math>\boldsymbol{\hat\imath'}</math> components.

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

<tr>
  <td align="right">
<math>(y_0 - y') \biggl[ \frac{x'}{y'}\biggr]_\mathrm{max} \dot\varphi</math>
  </td>
  <td align="center">
<math>\leftrightarrow</math>
  </td>
  <td align="left">
<math>
\biggl[ \frac{a^2}{a^2 + c^2}\biggr] \zeta_2 z_0
-
y' [c^2  + b^2 \tan^2\theta ]
\frac{\zeta_3 \cos\theta}{c^2}\biggl[\frac{a^2}{a^2+b^2}\biggr] 
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow ~~~ (y_0 - y') \dot\varphi</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\biggl\{ \biggl[ \frac{a^2}{a^2 + c^2}\biggr] \zeta_2 z_0
-
y' [c^2  + b^2 \tan^2\theta ]
\frac{\zeta_3 \cos\theta}{c^2}\biggl[\frac{a^2}{a^2+b^2}\biggr] 
\biggr\} \biggl[ \frac{y'}{x'}\biggr]_\mathrm{max} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\biggl\{ \biggl[ \frac{a^2}{a^2 + c^2}\biggr] \zeta_2 z_0
-
y' \biggl[c^2  + b^2 \tan^2\theta \biggr]
\frac{\zeta_3 \cos\theta}{c^2}\biggl[\frac{a^2}{a^2+b^2}\biggr] 
\biggr\}  
\biggl[\frac{bc}{ a( c^2 \cos^2\theta + b^2 \sin^2\theta )^{1 / 2}}  \biggr] \, .
</math>
  </td>
</tr>
</table>
Inserting the just-derived expression for <math>\dot\varphi</math> into this last expression gives,

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

<tr>
  <td align="right">
<math>(y_0 - y') </math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\biggl\{ \biggl[ \frac{a^2}{a^2 + c^2}\biggr] \zeta_2 z_0
-
y' \biggl[c^2  + b^2 \tan^2\theta \biggr]
\frac{\zeta_3 \cos\theta}{c^2}\biggl[\frac{a^2}{a^2+b^2}\biggr] 
\biggr\}  
\biggl[\frac{bc}{ a( c^2 \cos^2\theta + b^2 \sin^2\theta )^{1 / 2}}  \biggr]
\biggl[ \frac{a^2 + b^2}{b^2}\biggr] \frac{\cos\theta}{\zeta_3}
\biggl[\frac{bc}{ a( c^2 \cos^2\theta + b^2 \sin^2\theta )^{1 / 2}}  \biggr] 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\biggl\{ \biggl[ \frac{a^2}{a^2 + c^2}\biggr] \zeta_2 z_0
-
y' \biggl[c^2  + b^2 \tan^2\theta \biggr]
\frac{\zeta_3 \cos\theta}{c^2}\biggl[\frac{a^2}{a^2+b^2}\biggr] 
\biggr\}  
\frac{\cos\theta}{\zeta_3}
\biggl[\frac{c^2 (a^2 + b^2)}{ a^2( c^2 \cos^2\theta + b^2 \sin^2\theta )}  \biggr] 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\biggl\{ \biggl[ \frac{a^2}{a^2 + c^2}\biggr] \zeta_2 z_0
\biggr\}  
\frac{\cos\theta}{\zeta_3}
\biggl[\frac{c^2 (a^2 + b^2)}{ a^2( c^2 \cos^2\theta + b^2 \sin^2\theta )}  \biggr] 
+
\biggl\{ 
-
y' \biggl[c^2  + b^2 \tan^2\theta \biggr]
\frac{\zeta_3 \cos\theta}{c^2}\biggl[\frac{a^2}{a^2+b^2}\biggr] 
\biggr\}  
\frac{\cos\theta}{\zeta_3}
\biggl[\frac{c^2 (a^2 + b^2)}{ a^2( c^2 \cos^2\theta + b^2 \sin^2\theta )}  \biggr] 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\biggl\{ \biggl[ \frac{c^2 (a^2 + b^2)}{b^2(a^2 + c^2)}\biggr] \frac{\zeta_2 }{\zeta_3}
\biggr\}  
\biggl[\frac{b^2 z_0 \cos\theta}{ ( c^2 \cos^2\theta + b^2 \sin^2\theta )}  \biggr] 
-
y' \, .
</math>
  </td>
</tr>
</table>

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

<tr>
  <td align="right">
<math>\tan\theta</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
- \biggl[ \frac{c^2 (a^2 + b^2)}{b^2(a^2 + c^2)}\biggr]\frac{\zeta_2}{\zeta_3} \, .
</math>
  </td>
</tr>
</table>
Hence,

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

<tr>
  <td align="right">
<math>(y_0 - y') </math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\biggl[\frac{- b^2 z_0 \sin\theta}{ ( c^2 \cos^2\theta + b^2 \sin^2\theta )}  \biggr] 
-
y' \, .
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow ~~~ y_0 </math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\biggl[\frac{- b^2 z_0 \sin\theta}{ ( c^2 \cos^2\theta + b^2 \sin^2\theta )}  \biggr] 
\, .
</math>
  </td>
</tr>
</table>

<font color="red">SUCCESS !!!</font>

====Vorticity Implied by Lagrangian Fluid Motions====

As we have stated [[#Associated_Lagrangian_Velocities|above]], the motion of fluid elements in the primed (preferred tilted-plane) coordinate system is given by the pair of expressions,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>u_x'= \dot{x}'</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>(y_0 - y') \biggl[ \frac{x_\mathrm{max}}{y_\mathrm{max}} \biggr] \dot\varphi </math>
  </td>
<td align="center">&nbsp; &nbsp; &nbsp; and, &nbsp; &nbsp; &nbsp;
  <td align="right">
<math>u_y' = \dot{y}' </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>x' \biggl[ \frac{y_\mathrm{max}}{x_\mathrm{max}}\biggr] \dot\varphi \, .</math>
  </td>
</tr>
</table>
So, the contribution to the local vorticity that is provided by orbital motion of individual Lagrangian fluid elements is,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\zeta_\mathrm{L}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\biggl\{ \frac{\partial u'_y}{\partial x'} - \frac{\partial u'_x}{\partial y'}  \biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\biggl[ \frac{y_\mathrm{max}}{x_\mathrm{max}}\biggr] \dot\varphi
+
\biggl[ \frac{x_\mathrm{max}}{y_\mathrm{max}} \biggr] \dot\varphi 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\biggl[ \frac{x_\mathrm{max}^2 + y_\mathrm{max}^2}{x_\mathrm{max}y_\mathrm{max}} \biggr] \dot\varphi \, .
</math>
  </td>
</tr>
</table>
That is,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\zeta_\mathrm{L}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\biggl\{ 
\frac{a(c^2\cos^2\theta + b^2\sin^2\theta)^{1 / 2}}{bc}
+
\frac{bc}{a(c^2\cos^2\theta + b^2\sin^2\theta)^{1 / 2}}
\biggr\}
\frac{\zeta_3}{\cos\theta}
\biggl[\frac{ abc ( c^2 \cos^2\theta + b^2 \sin^2\theta )^{1 / 2}}{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\{ 
\frac{a^2(c^2\cos^2\theta + b^2\sin^2\theta)+b^2c^2}{c^2(a^2 + b^2)}
\biggr\}
\frac{\zeta_3}{\cos\theta}
</math>
  </td>
</tr>

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

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

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

<tr>
  <td align="right">
<math>\tan\theta</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
- \biggl[ \frac{c^2 (a^2 + b^2)}{b^2(a^2 + c^2)}\biggr]\frac{\zeta_2}{\zeta_3}
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow ~~~ \biggl[ \frac{b^2(a^2 + c^2)}{c^2 (a^2 + b^2)}\biggr]\frac{\zeta_3}{\cos\theta}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
- \frac{\zeta_2}{\sin\theta} \, ,
</math>
  </td>
</tr>
</table>
in which case,

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

<tr>
  <td align="right">
<math>\zeta_\mathrm{L}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\biggl\{ 
\frac{a^2 (c^2-b^2) }{c^2(a^2 + b^2)}
\biggr\}
\zeta_3\cos\theta
-
\frac{\zeta_2}{\sin\theta}
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\zeta_\mathrm{L}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\biggl\{ 
\frac{a^2 (c^2-b^2) }{c^2(a^2 + b^2)}
\biggr\}
\zeta_3\cos\theta
-
\frac{\zeta_2}{\sin\theta}
</math>
  </td>
</tr>
</table>

===Summary &amp; Example===

====Model001====
This particular set of seven key parameters has been drawn from [[Appendix/References#EFE|[<font color="red">EFE</font>] ]] Chapter 7, Table XIII (p. 170). The tabular layout presented here, also appears in a [[ThreeDimensionalConfigurations/ChallengesPt2#Example_Equilibrium_Model|related discussion labeled, ''Challenges Pt. 2'']].

<table border="1" align="center" cellpadding="5">
<tr>
  <td align="center" colspan="6"><b>Model001:</b> &nbsp; 
<math>\biggl(\frac{a_2}{a_1}, \frac{a_3}{a_1}\biggr) = \biggl(1.2500, 0.4703 \biggr)</math></td>
</tr>

<tr>
  <td align="center" colspan="2"><math>A_1</math></td>
  <td align="center" colspan="2"><math>A_2</math></td>
  <td align="center" colspan="2"><math>A_3</math></td>
</tr>

<tr>
  <td align="right" colspan="2">0.48955940032702523984</td>
  <td align="right" colspan="2">0.36486593343389634429</td>
  <td align="right" colspan="2">1.1455746662390784159</td>
</tr>

<tr>
  <td align="center" colspan="6" bgcolor="lightgreen"><b>Direct</b></td>
</tr>

<tr>
  <td align="center"><math>\beta_+</math></td>
  <td align="center"><math>\gamma_+</math></td>
  <td align="center"><math>\Omega_2 = \boldsymbol{\hat{\jmath}} \cdot \vec{\Omega}_f</math></td>
  <td align="center"><math>\Omega_3 = \boldsymbol{\hat{k}} \cdot \vec{\Omega}_f</math></td>
  <td align="center"><math>\zeta_2 = \boldsymbol{\hat{\jmath}} \cdot \vec{\zeta}</math></td>
  <td align="center"><math>\zeta_3 = \boldsymbol{\hat{k}} \cdot \vec{\zeta}</math></td>
</tr>

<tr>
  <td align="right">1.1343563893093</td>
  <td align="right">1.8050153443093</td>
  <td align="right">0.3639465285418</td>
  <td align="right">0.6633461900921</td>
  <td align="right">-2.2793843997547</td>
  <td align="right">-1.9636540847967</td>
</tr>

<tr>
  <td align="center"><math>\tan\theta</math></td>
  <td align="center"><math>\theta</math> (deg.)</td>
  <td align="center"><math>|z_0|_\mathrm{max}</math></td>
  <td align="center"><math>\frac{y_0}{z_0}</math></td>
  <td align="center"><math>\biggl[ \frac{x'}{y'}\biggr]_\mathrm{max}</math></td>
  <td align="center"><math>\dot\varphi</math></td>
</tr>

<tr>
  <td align="right">-0.3447989745608</td>
  <td align="right">-19.02414</td>
  <td align="right">0.6379200460018</td>
  <td align="right">-1.4003818611184</td>
  <td align="right">1.0258604183520</td>
  <td align="right">-1.2992789284526</td>
</tr>

<tr>
  <td align="center" colspan="4" rowspan="2">&nbsp;</td>
  <td align="center"><math>\boldsymbol{\hat{\jmath}'} \cdot \vec{\zeta}</math></td>
  <td align="center"><math>\zeta_L = \boldsymbol{\hat{k}'} \cdot \vec{\zeta}</math></td>
</tr>

<tr>
  <td align="right">-1.5148019600561</td>
  <td align="right">-2.5994048604237</td>
</tr>

<tr>
  <td align="center" colspan="6" bgcolor="yellow"><b>Adjoint</b></td>
</tr>

<tr>
  <td align="center"><math>\beta_-</math></td>
  <td align="center"><math>\gamma_-</math></td>
  <td align="center"><math>\Omega_2^\dagger = \boldsymbol{\hat{\jmath}} \cdot \vec{\Omega}_f^\dagger</math></td>
  <td align="center"><math>\Omega_3^\dagger = \boldsymbol{\hat{k}} \cdot \vec{\Omega}_f^\dagger</math></td>
  <td align="center"><math>\zeta_2^\dagger = \boldsymbol{\hat{\jmath}} \cdot \vec{\zeta}^\dagger</math></td>
  <td align="center"><math>\zeta_3^\dagger = \boldsymbol{\hat{k}} \cdot \vec{\zeta}^\dagger</math></td>
</tr>

<tr>
  <td align="right">0.1949846556907</td>
  <td align="right">0.8656436106907</td>
  <td align="right">0.8778334467750</td>
  <td align="right">0.9578800413643</td>
  <td align="right">-0.9450244149966</td>
  <td align="right">-1.3598596896888</td>
</tr>

<tr>
  <td align="center"><math>\tan\theta</math></td>
  <td align="center"><math>\theta</math> (deg.)</td>
  <td align="center"><math>|z_0|_\mathrm{max}</math></td>
  <td align="center"><math>\frac{y_0}{z_0}</math></td>
  <td align="center"><math>\biggl[ \frac{x'}{y'}\biggr]_\mathrm{max}</math></td>
  <td align="center"><math>\dot\varphi</math></td>
</tr>

<tr>
  <td align="right">-0.2064250069478</td>
  <td align="right">-11.663458271</td>
  <td align="right">0.5364347308466</td>
  <td align="right">-1.1444841125518</td>
  <td align="right">0.8936566385511</td>
  <td align="right">-0.7566275461198</td>
</tr>

<tr>
  <td align="center" colspan="4" rowspan="2">&nbsp;</td>
  <td align="center"><math>\boldsymbol{\hat{\jmath}'} \cdot \vec{\zeta}^\dagger</math></td>
  <td align="center"><math>\zeta_L^\dagger = \boldsymbol{\hat{k}'} \cdot \vec{\zeta}^\dagger</math></td>
</tr>

<tr>
  <td align="right">-0.6505985480434</td>
  <td align="right">-1.5228299477827</td>
</tr>
</table>


As a consequence, the time-dependent x'-y' coordinate positions of individual Lagrangian fluid elements is precisely describe by the expressions,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>x'</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>x'_\mathrm{max}\cos(\dot\varphi t)</math>
  </td>
<td align="center">&nbsp; &nbsp; &nbsp; and, &nbsp; &nbsp; &nbsp;
  <td align="right">
<math>y' - y_0</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>y'_\mathrm{max}\sin(\dot\varphi t) \, ,</math>
  </td>
</tr>
</table>

and &#8212; see [[ThreeDimensionalConfigurations/DescriptionOfRiemannTypeI#Examination_of_Lagrangian_Flow_in_One_Specific_Model|an accompanying discussion]] (alternatively, [[ThreeDimensionalConfigurations/ChallengesPt6#Are_Orbits_Exact_Circles|ChallengesPt6]]) for details &#8212; the values of these additional key parameters are &hellip;

<table border="0" cellpadding="5" align="center">
<tr>
  <td align="center" colspan="4">&nbsp;</td>
  <td align="center" rowspan="7" colspan="1" bgcolor="lightgrey">&nbsp; </td>
  <td align="center" colspan="2">'''Example Values'''</td>
</tr>

<tr>
  <td align="right">
<math>\tan\theta </math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>- \frac{\zeta_2 }{ \zeta_3 } \biggl[ \frac{a^2 + b^2}{a^2 + c^2} \biggr]\frac{c^2}{b^2} = -0.344793</math>
  </td>
  <td align="center">&nbsp; &nbsp; &nbsp; &nbsp;</td>
  <td align="right">
<math>\theta =</math>
  </td>
  <td align="left">
<math>- 19.0238^\circ</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>
z_0^2
</math>
  </td>
  <td align="center">
<math>\le</math>
  </td>
  <td align="left">
<math>
c^2 + b^2\tan^2\theta
</math>
  </td>
  <td align="center">&nbsp; &nbsp; &nbsp; &nbsp;</td>
  <td align="right">
<math>z_0^2 \le</math>
  </td>
  <td align="left">
<math>0.40694</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>
\frac{y_0}{z_0}
</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
- \frac{b^2 \sin\theta}{(c^2\cos^2\theta + b^2\sin^2\theta)}
</math>
  </td>
  <td align="center">&nbsp; &nbsp; &nbsp; &nbsp;</td>
  <td align="right">
<math>\frac{y_0}{z_0} =</math>
  </td>
  <td align="left">
<math>+ 1.400377</math>
  </td>
</tr>

<tr>
  <td align="right">
For <math>1 \ge h > 0 </math>, &nbsp; &nbsp; &nbsp;<math>
\biggl| \frac{x'_\mathrm{max}}{a}  \biggr|
</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\biggl[ \frac{(c^2 - z_0^2) \cos^2\theta + b^2\sin^2\theta}{(c^2  \cos^2\theta + b^2 \sin^2\theta )} \biggr]^{1 / 2}\cdot h
</math>
  </td>
  <td align="center">&nbsp; &nbsp; &nbsp; &nbsp;</td>
  <td align="right">
&nbsp; &nbsp; <math>\biggl| \frac{x'_\mathrm{max}}{a}  \biggr|</math>
  </td>
  <td align="left">
varies with choice of <math>z_0^2</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>
\frac{x'_\mathrm{max}}{y'_\mathrm{max}}  
</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{a(c^2\cos^2\theta + b^2\sin^2\theta)^{1 / 2}}{bc}
</math>
  </td>
  <td align="center">&nbsp; &nbsp; &nbsp; &nbsp;</td>
  <td align="right">
&nbsp; &nbsp; <math>\frac{x'_\mathrm{max}}{y'_\mathrm{max}} =</math>
  </td>
  <td align="left">
<math>+ 1.025854</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>
\dot\varphi
</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{\zeta_3}{\cos\theta}
\biggl[\frac{ abc ( c^2 \cos^2\theta + b^2 \sin^2\theta )^{1 / 2}}{c^2(a^2 + b^2)}  \biggr] 
</math>
  </td>
  <td align="center">&nbsp; &nbsp; &nbsp; &nbsp;</td>
  <td align="right">
<math>\dot\varphi =</math>
  </td>
  <td align="left">
<math>-1.2993</math>
  </td>
</tr>
</table>

====Example Sequences====

Let's plot sequences in which &hellip;
<ul>
  <li><math>f \equiv (\zeta_2^2 + \zeta_3^2)^{1 / 2}/( \Omega_2^2 + \Omega_3^2)^{1 / 2}</math> is constant; this is the analog of <math>f = \zeta/\Omega_f</math> in [[ThreeDimensionalConfigurations/RiemannStype#Equilibrium_Conditions_for_Riemann_S-type_Ellipsoids|Riemann S-type Ellipsoids]].</li>
  <li><math>\zeta_L / (\zeta_2^2 + \zeta_3^2)^{1 / 2}</math> is constant.</li>
  <li><math>(x'/y')_\mathrm{max}</math> is constant.</li>
</ul>

=See Also=

* [[3Dconfigurations/DescriptionOfRiemannTypeI|Description of Riemann Type I Ellipsoids]] (best)
* [[ThreeDimensionalConfigurations/DescriptionOfRiemannTypeI|Description of Riemann Type I Ellipsoids]] (older introduction)
* [[ThreeDimensionalConfigurations/RiemannTypeI#Riemann_Type_1_Ellipsoids|Riemann Type 1 Ellipsoids]] (oldest introduction)
* [[ThreeDimensionalConfigurations/Challenges#Challenges_Constructing_Ellipsoidal-Like_Configurations|Construction Challenges (Pt. 1)]]
* [[ThreeDimensionalConfigurations/ChallengesPt2|Construction Challenges (Pt. 2)]]
* [[ThreeDimensionalConfigurations/ChallengesPt3|Construction Challenges (Pt. 3)]]
* [[ThreeDimensionalConfigurations/ChallengesPt4|Construction Challenges (Pt. 4)]]
* [[ThreeDimensionalConfigurations/ChallengesPt5|Construction Challenges (Pt. 5)]]
* [[ThreeDimensionalConfigurations/ChallengesPt6|Construction Challenges (Pt. 6)]]

* Related discussions of models viewed from a rotating reference frame:
** [[PGE/RotatingFrame#Rotating_Reference_Frame|PGE]]
** <font color="red"><b>NOTE to Eric Hirschmann &amp; David Neilsen...  </b></font>I have moved the earlier contents of this page to a new Wiki location called [[Apps/RiemannEllipsoids_Compressible|Compressible Riemann Ellipsoids]].

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