Editing ThreeDimensionalConfigurations/RiemannStype (section)

==Maclaurin Spheroid Limit==
===Basic Relations===

====Expressions Supplied by EFE====

In Chapter 7, &sect;48(b) (pp. 137 - 138) of [[Appendix/References#EFE|[<font color="red">EFE</font>] ]], Chandrasekhar shows that <font color="orange">"&hellip; a ''stable'' [[Apps/MaclaurinSpheroidSequence|Maclaurin spheroid]]</font> can be considered as <font color="orange">a limiting <math>(a_2/a_1 \rightarrow 1)</math> form of a [S-type] Riemann ellipsoid."</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 S-type Riemann 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[4 B_{11} - \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;48(b), p. 137, Eq. (60)</font> </td></tr>
</table>

<span id="zeta3">and,</span>
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\zeta_3^2</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>4(\Omega_\mathrm{Mc} - \Omega_3)^2 \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow ~~~ \zeta_3</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\Omega_\mathrm{Mc} \mp \biggl[ 4B_{11} - \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;48(b), p. 137, Eqs. (58) &amp; (61)</font> </td></tr>
</table>
<span id="B11">where, </span>
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>B_{11}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>A_1 - a_1^2 A_{11} \, .</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, Eq. (105)</font></td></tr>
</table>

====Evaluating A<sub>11</sub>====

As has been stated in [[ThreeDimensionalConfigurations/HomogeneousEllipsoids#Evaluating_Aℓℓ|a separate derivation]], quite generally,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\frac{A_{\ell\ell}}{a_\ell a_m a_s}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\int_0^\infty \biggl[ (a_\ell^2 + u)^5(a_m^2 + u)(a_s^2 + u) \biggr]^{-1 / 2} du \, .
</math>
  </td>
</tr>
</table>
Now, in the case of Maclaurin spheroids, <math>a_m = a_\ell</math>, so this integral expression becomes,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\frac{A_{\ell\ell}}{a_\ell^2 a_s}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\int_0^\infty \biggl[ (a_\ell^2 + u)^6 (a_s^2 + u) \biggr]^{-1 / 2} du
=
\int_0^\infty \frac{du}{(a_\ell^2 + u)^3 (a_s^2 + u)^{1 / 2} } 
=
\int_0^\infty \frac{du}{v^m \sqrt{w}} \, ,
</math>
  </td>
</tr>
</table>
where, <math>m = 3, v = (a_\ell^2 + u), w = (a_s^2 + u), b = d = 1, a = a_s^2, c = a_\ell^2, k = (a_s^2 - a_\ell^2).</math>  Before applying the limits, carrying out the integration gives,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\int \frac{du}{v^m \sqrt{w}}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
-~\frac{1}{(m-1)(a_s^2 - a_\ell^2)} \biggl[
\frac{\sqrt{w}}{v^{m-1}} + \biggl(m - \frac{3}{2}\biggr) \int \frac{du}{v^{m-1} \sqrt{w}}
\biggr]
</math>
  </td>
</tr>
<tr>
  <td align="center" colspan="3">
[<b>[[Appendix/References#CRC|<font color="red">CRC</font>]]</b>] chapter on INTEGRALS, p. 408, Eq. (154)<br />
[https://www.academia.edu/36550954/I_S_Gradshteyn_and_I_M_Ryzhik_Table_of_integrals_series_and_products_Academic_Press_2007_ GR7<sup>th</sup>], p. 91, Eq. (2.248.8)</td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{1}{2(a_\ell^2 - a_s^2)} \biggl[
\frac{\sqrt{w}}{v^{2}} + \biggl(\frac{3}{2}\biggr) \int \frac{du}{v^{2} \sqrt{w}}
\biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{1}{2(a_\ell^2 - a_s^2)} \biggl[
\frac{\sqrt{w}}{v^{2}}\biggr] 
+
\frac{3}{4(a_\ell^2 - a_s^2)} 
\biggl\{
\frac{1}{(a_\ell^2 - a_s^2)} \biggl[
\frac{\sqrt{w}}{v} + \biggl(\frac{1}{2}\biggr) \int \frac{du}{v \sqrt{w}}
\biggr]\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{1}{2(a_\ell^2 - a_s^2)} \biggl[\frac{\sqrt{w}}{v^{2}}\biggr] 
+
\frac{3}{4(a_\ell^2 - a_s^2)^2} \biggl[\frac{\sqrt{w}}{v}\biggr] 
+
\frac{3}{8(a_\ell^2 - a_s^2)^2} 
\biggl\{
\int \frac{du}{v \sqrt{w}}
\biggr\}
</math>
  </td>
</tr>
<tr>
  <td align="center" colspan="3">
[<b>[[Appendix/References#CRC|<font color="red">CRC</font>]]</b>] chapter on INTEGRALS, p. 407, Eq. (148)<br />
[https://www.academia.edu/36550954/I_S_Gradshteyn_and_I_M_Ryzhik_Table_of_integrals_series_and_products_Academic_Press_2007_ GR7<sup>th</sup>], p. 90, Eq. (2.246)
</td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{1}{2(a_\ell^2 - a_s^2)} \biggl[\frac{\sqrt{w}}{v^{2}}\biggr] 
+
\frac{3}{4(a_\ell^2 - a_s^2)^2} \biggl[\frac{\sqrt{w}}{v}\biggr] 
+
\frac{3}{8(a_\ell^2 - a_s^2)^2} 
\biggl\{
\frac{2}{(a_\ell^2 - a_s^2)^{1 / 2}} \tan^{-1}\biggl[ \frac{\sqrt{w}}{(a_\ell^2 - a_s^2)^{1 / 2}} \biggr]
\biggr\}\, .
</math>
  </td>
</tr>
</table>
Applying the limits then gives,

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

<tr>
  <td align="right">
<math>\int_0^\infty \frac{du}{v^m \sqrt{w}}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{3\pi}{8(a_\ell^2 - a_s^2)^{5 / 2}} 
-~\frac{a_s}{2(a_\ell^2 - a_s^2)a_\ell^4}  
-
\frac{3a_s}{4(a_\ell^2 - a_s^2)^2 a_\ell^2}  
-
\frac{3}{4(a_\ell^2 - a_s^2)^{5 / 2}} 
\biggl\{
\tan^{-1}\biggl[ \frac{a_s}{(a_\ell^2 - a_s^2)^{1 / 2}} \biggr]
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{3\pi}{8 a_\ell^5 e^5} 
-~\frac{(1-e^2)^{1 / 2}}{2a_\ell^5 e^2}  
-
\frac{3(1-e^2)^{1 / 2}}{4a_\ell^5 e^4 }  
-
\frac{3}{4a_\ell^5 e^5} 
\biggl\{
\sin^{-1}(1-e^2)^{1 / 2}
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
1.49957 - 0.16703 - 0.27593 - 0.29421 = 0.76239
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow ~~~ \frac{A_{\ell\ell} }{a_\ell^3 (1 - e^2)^{1 / 2} }</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{1}{4a_\ell^5 e^5}\biggl\{
\frac{3\pi}{2 } 
-3 \biggl[\sin^{-1}(1-e^2)^{1 / 2}\biggr]
-~2e^3(1-e^2)^{1 / 2}  
- 3e(1-e^2)^{1 / 2}
\biggr\}  
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow ~~~ a_\ell^2 A_{\ell\ell} </math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{3(1-e^2)^{1 / 2}}{4 e^5}
\underbrace{\biggl[ \frac{\pi}{2 } - \sin^{-1}(1-e^2)^{1 / 2}\biggr]}_{\sin^{-1}e}
-  
\frac{1}{4 e^4}( 2e^2   + 3)(1-e^2)  
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
0.36562 - 0.13436 = 0.23126
</math>
  </td>
</tr>
</table>

====Derivation Check====

At the top of p. 141 (&sect;48) of [<b>[[Appendix/References#EFE|<font color="red">EFE</font>]]</b>] &#8212; see also Table VI on p. 142 &#8212; Chandrasekhar emphasizes that the point at which the <math>x = +1</math> self-adjoint (S-type) Riemann ellipsoid intersects the Maclaurin spheroid has <math>f = +2, a_3 = 0.30333, e = 0.95289, \Omega_3^2 = 0.11005</math>. As Table I (p. 78) records, the Maclaurin spheroid having this eccentricity has <math>\Omega_\mathrm{Mc}^2 = 0.44022.</math>  At this point, therefore,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>B_{11}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\tfrac{1}{4}\Omega^2_\mathrm{Mc} = 0.110055 \, .</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;48(c), Eq. (141)</font></td></tr>
</table>

Note as well that, from the [[#zeta3|above expression for (&zeta;<sub>3</sub>)<sup>2</sup>]], we have,
<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) = 0.66034 \, ;</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow ~~~ f </math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\frac{\zeta_3}{\Omega_3} = 2.0 \, ,</math>
  </td>
</tr>
</table>
as it should be.

For this same value of the eccentricity, we recognize as well that, <math>A_{1} = 0.34132</math>.  Therefore, drawing from the [[#B11|above expression for B<sub>11</sub>]], we also deduce that,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>a_1^2 A_{11}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>A_1 - B_{11} = 0.23127 \, .</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, Eq. (105)</font></td></tr>
</table>
<font color="red"><b>Hooray!!</b></font> &nbsp; This matches the numerical value of <math>a_\ell^2 A_{\ell\ell} = 0.23126</math> determined above for <math>e = 0.95289</math>.  So it seems reasonable to conclude that the general expression, itself, is correct.

===Frequency Ratio===
====Determination and Plot====

Given that, for all Maclaurin spheroids,
<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>
we conclude that along the entire Maclaurin spheroid sequence, the index symbol,

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

<tr>
  <td align="right">
<math>B_{11} = A_1 - a_{11}^2 A_{11} </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} 
+
\frac{1}{4 e^4}( 2e^2   + 3)(1-e^2)  
-
\frac{3(1-e^2)^{1 / 2}}{4 e^5}\biggl[
\frac{\pi}{2 } - \sin^{-1}(1-e^2)^{1 / 2}\biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>  
\frac{3}{4e^5} \biggl[\frac{4}{3} \cdot e^2\sin^{-1}e \biggr](1-e^2)^{1/2} 
-
\frac{3}{4 e^5} \underbrace{\biggl[ \frac{\pi}{2 } - \sin^{-1}(1-e^2)^{1 / 2}\biggr]}_{\sin^{-1}e}(1-e^2)^{1 / 2}
-
\frac{(1-e^2)}{e^2}  
+
\frac{1}{4 e^4}( 2e^2   + 3)(1-e^2)  
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>  
\frac{3(1-e^2)^{1/2}}{4e^5} \biggl[\frac{4}{3} \cdot e^2\sin^{-1}e -\sin^{-1}e \biggr] 
+
\frac{(3-2e^2)(1-e^2)}{4e^4} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>  
\frac{1}{4e^4}\biggl\{
(1-e^2)^{1/2} (4 e^2 - 3) \frac{\sin^{-1}e}{e} 
+
(3-2e^2)(1-e^2)  
\biggr\}  \, .
</math>
  </td>
</tr>
</table>


We can therefore write,
<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 (\mathcal{H} - 1)^{1 / 2} \, ,</math>
  </td>
<td align="center">&nbsp; &nbsp; &nbsp; and, &nbsp; &nbsp; &nbsp; 
  <td align="right">
<math>\frac{\zeta_3}{\Omega_\mathrm{Mc}}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>1 \mp (\mathcal{H} - 1)^{1 / 2} \, ,</math>
  </td>
</tr>
</table>

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

<tr>
  <td align="right">
<math>\mathcal{H} \equiv \frac{4B_{11}}{\Omega^2_\mathrm{Mc}}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{1}{2e^2}\biggl\{
(1-e^2)^{1/2} (4 e^2 - 3) \frac{\sin^{-1}e}{e} 
+
(3-2e^2)(1-e^2)  
\biggr\}  
\biggl\{ (3-2e^2)(1 - e^2)^{1 / 2} \cdot \frac{\sin^{-1} e}{e} - 3(1-e^2) \biggr\}^{-1}
\, .
</math>
  </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 Riemann S-type ellipsoids (<i>i.e.</i>, here), 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 [[3Dconfigurations/DescriptionOfRiemannTypeI#Description_of_Riemann_Type_I_Ellipsoids|Type I Riemann ellipsoids]], 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.
 

<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_StypeB.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">
Each Riemann S-type ellipsoid sequence &#8212; uniquely identified by its frequency ratio, <math>f = \zeta_3/\Omega_3</math> &#8212; intersects the Maclaurin spheroid sequence at a specific value of the meridional-plane eccentricity, <math>e = (1 - a_3/a_1)^{1 / 2}.</math>  Analogous to Figure 12 from &sect;48, p. 139 of [<b>[[Appendix/References#EFE|<font color="red">EFE</font>]]</b>], this plot displays how <math>f_\mathrm{Mc}</math> varies with <math>e</math> along the Maclaurin spheroid sequence; for a given  <math>e</math>, the solid black curve is associated with the ''direct'' configuration while the dashed orange curve is associated with the ''adjoint'' configuration.  The circular, solid-purple marker identifies the point where the Jacobi sequence <math>(f_\mathrm{Mc} = 0)</math> bifurcates from the Maclaurin spheroid sequence; bifurcation to the Dedekind sequence is not displayed here because its frequency ratio <math>(f_\mathrm{Mc} = \pm \infty)</math> is off scale.  The square, solid-green markers identify the location of the pair of models (direct/adjoint) for which <math>\Omega^2_\mathrm{Mc}</math> is maximum (see the square, solid-green marker in the left-hand panel).  The circular, solid-orange marker identifies the (degenerate) pair of models <math>(f_\mathrm{Mc} = +2)</math> where the "<math>x = +1</math>" sequence bifurcates from the Maclaurin spheroid sequence.
  </td>
</tr>
</table>


<table border="1" align="center" width="80%" cellpadding="8"><tr><td align="left">
Drawing from [[Appendix/Ramblings/PowerSeriesExpressions#Maclaurin_Spheroid_Index_Symbols|one of our mathematics appendices]], we know that when <math>e \ll 1</math>,

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

<tr>
  <td align="right">
<math>\biggl[ 4e^4 B_{11}\biggr]_{e \ll 1}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>  
\biggl\{
(1-e^2)^{1/2} (4 e^2 - 3) \frac{\sin^{-1}e}{e} 
+
(3-2e^2)(1-e^2)  
\biggr\}_{e \ll 1} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>  
(3-2e^2)(1-e^2)  
+ (4 e^2 - 3)
\biggl\{ 1 - \biggl[ \frac{1}{2} \biggr]e^2 - \biggl[\frac{1}{2^3}\biggr]e^4
- \biggl[\frac{1}{2^4}\biggr]e^6
- \biggl[\frac{5}{2^7 }\biggr]e^8
- \cdots
\biggr\} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>  
\times
\biggl\{ 1 + \biggl[\frac{1}{2\cdot 3}\biggr]e^2 
+ \biggl[\frac{1\cdot 3 }{2\cdot 4\cdot 5}\biggr]e^4
+ \biggl[ \frac{1 \cdot 3\cdot 5}{2\cdot 4\cdot 6\cdot 7}\biggr]e^6 
+ \biggl[ \frac{1 \cdot 3\cdot 5 \cdot 7}{2\cdot 4\cdot 6\cdot 8\cdot 9}\biggr]e^8 
+ \cdots
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>  
(3-2e^2)(1-e^2)  
+ (4 e^2 - 3)\biggl\{ 1 - \biggl[ \frac{1}{2} \biggr]e^2 - \biggl[\frac{1}{2^3}\biggr]e^4
- \biggl[\frac{1}{2^4}\biggr]e^6
- \cdots
\biggr\} 
</math>
  </td>
</tr>

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

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>  
+ (4 e^2 - 3)\biggl\{ 1 - \biggl[ \frac{1}{2} \biggr]e^2 
- \cdots
\biggr\} 
\times \biggl[\frac{1\cdot 3 }{2\cdot 4\cdot 5}\biggr]e^4
+ (4 e^2 - 3)
\times \biggl[ \frac{1 \cdot 3\cdot 5}{2\cdot 4\cdot 6\cdot 7}\biggr]e^6 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>  
(3-2e^2)(1-e^2)  
+ (4 e^2 - 3) - \biggl[ \frac{1}{2} \biggr]e^2(4 e^2 - 3) - \biggl[\frac{1}{2^3}\biggr]e^4(4 e^2 - 3)
- \biggl[\frac{1}{2^4}\biggr]e^6(4 e^2 - 3)
</math>
  </td>
</tr>

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

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>  
+ \biggl\{ 1  \biggr\} 
\times \biggl[\frac{1\cdot 3 }{2\cdot 4\cdot 5}\biggr]e^4(4 e^2 - 3)
+ \biggl\{ - \biggl[ \frac{1}{2} \biggr]e^2 \biggr\} 
\times \biggl[\frac{1\cdot 3 }{2\cdot 4\cdot 5}\biggr]e^4(4 e^2 - 3)
+ (4 e^2 - 3)
\times \biggl[ \frac{1 \cdot 3\cdot 5}{2\cdot 4\cdot 6\cdot 7}\biggr]e^6 
+ \mathcal{O}(e^8)
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>  
3-5e^2 + 2e^4
+ (4 e^2 - 3) - \frac{1}{2} (4 e^4 - 3e^2) - \frac{1}{2^3}(4 e^6 - 3e^4)
+ \frac{3}{2^4}\cdot e^6
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>  
+ \frac{1}{2\cdot 3}(4 e^4 - 3e^2)
- \frac{1}{2^2 \cdot 3}(4 e^6 - 3e^4)
+ \frac{1}{2^4}(e^6)
+  \biggl[\frac{1\cdot 3 }{2\cdot 4\cdot 5}\biggr](4 e^6 - 3e^4)
+ \biggl[\frac{1\cdot 3^2 }{2^4\cdot 5}\biggr](e^6)

- \biggl[ \frac{1 \cdot 3\cdot 5}{2\cdot 4\cdot 6\cdot 7}\biggr](3e^6 )
+ \mathcal{O}(e^8)
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>  
-e^2 + 2e^4  - 2 e^4 + \tfrac{3}{2}e^2 - \tfrac{1}{2} e^6 + \tfrac{3}{8}e^4 + \frac{3}{2^4}\cdot e^6
+ \tfrac{2}{3} e^4 - \tfrac{1}{2}e^2
- \tfrac{1}{3} e^6 + \tfrac{1}{4}e^4
+ \tfrac{1}{16}(e^6)
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>  
+  \biggl[\frac{3 }{2^3\cdot  5}\biggr](4 e^6 - 3e^4)
+ \biggl[\frac{3^2 }{2^4\cdot 5}\biggr](e^6)
- \biggl[ \frac{5}{2^4\cdot 7}\biggr](3e^6 )
+ \mathcal{O}(e^8)
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>  
- \tfrac{1}{2} e^6 + \tfrac{3}{8}e^4 + \frac{3}{2^4}\cdot e^6
+ \tfrac{2}{3} e^4 
- \tfrac{1}{3} e^6 + \tfrac{1}{4}e^4
+ \tfrac{1}{16}(e^6)

+  \biggl[\frac{3 }{2^3\cdot  5}\biggr](4 e^6 - 3e^4)
+ \biggl[\frac{3^2 }{2^4\cdot 5}\biggr](e^6)
- \biggl[ \frac{5}{2^4\cdot 7}\biggr](3e^6 )
+ \mathcal{O}(e^8)
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>  
e^4 \biggl[\frac{3}{2^3} + \frac{2}{3} + \frac{1}{2^2} - \frac{3^2}{2^3\cdot 5}  \biggr]
+ \mathcal{O}(e^6)
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>  
e^4 \biggl[3^2 \cdot 5 + 2^4\cdot 5 + 2\cdot 3\cdot 5 - 3^3  \biggr] \frac{1}{2^3 \cdot 3\cdot 5}
+ \mathcal{O}(e^6)
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>  
e^4 \biggl[\frac{2^4}{3\cdot 5}\biggr] 
+ \mathcal{O}(e^6)
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow ~~~ B_{11}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>  
\biggl[\frac{2^2}{3\cdot 5}\biggr] 
+ \mathcal{O}(e^2)
</math>
  </td>
</tr>
</table>

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



<table border="1" align="center" cellpadding="8">
<tr>
  <td align="center" colspan="13"><b>Table: &nbsp; Key Values Along the Limiting Maclaurin Spheroid Sequence</b></td>
</tr>

<tr>
  <td align="center" rowspan="2">Model</td>
  <td align="center" rowspan="2">e</td>
  <td align="center" rowspan="2"><math>\Omega^2_\mathrm{Mc}</math></td>
  <td align="center" rowspan="2"><math>B_{11}</math></td>
  <td align="center" rowspan="2"><math>\mathcal{H} \equiv \frac{4B_{11}}{\Omega^2_\mathrm{Mc}}</math></td>
  <td align="center" colspan="4">Direct</td>
  <td align="center" colspan="4">Adjoint</td>
</tr>

<tr>
  <td align="center"><math>\Omega_3</math></td>
  <td align="center"><math>\zeta_3</math></td>
  <td align="center"><math>\frac{\Omega_3}{\Omega_\mathrm{Mc}}</math></td>
  <td align="center"><math>f = \frac{\zeta_3}{\Omega_3}</math></td>
  <td align="center"><math>\Omega_3</math></td>
  <td align="center"><math>\zeta_3</math></td>
  <td align="center"><math>\frac{\Omega_3}{\Omega_\mathrm{Mc}}</math></td>
  <td align="center"><math>f = \frac{\zeta_3}{\Omega_3}</math></td>
</tr>

<tr>
  <td align="center" rowspan="1">Nonrotating Sphere &amp; <br />"<math>x = -1</math>" Bifurcation</td>
  <td align="right">0.00000</td>
  <td align="right">0.00000</td>
  <td align="center"><math>\frac{4}{15}</math></td>
  <td align="center"><math>\infty</math></td>
  <td align="center"><math>-~\biggl(\frac{4}{15}\biggr)^{1 / 2}</math><br />Note: (a)</td>
  <td align="right"><math>+~\biggl(\frac{16}{15}\biggr)^{1 / 2}</math></td>
  <td align="center"><math>\infty</math></td>
  <td align="center"><math>-~2</math><br />Note: (a)</td>
  <td align="center"><math>+~\biggl(\frac{4}{15}\biggr)^{1 / 2}</math><br />Note: (a)</td>
  <td align="right"><math>-~\biggl(\frac{16}{15}\biggr)^{1 / 2}</math></td>
  <td align="center"><math>\infty</math></td>
  <td align="center"><math>-~2</math><br />Note: (a)</td>
</tr>

<tr>
  <td align="center" rowspan="1">Jacobi/Dedekind Bifurcation</td>
  <td align="center">0.81267<br />Note: (b)</td>
  <td align="center">0.37423<br />Note: (b)</td>
  <td align="center"><math>\frac{\Omega^2_\mathrm{Mc}}{2}</math><br />Note: (c)</td>
  <td align="center"><math>2</math></td>
  <td align="center"><math>\Omega_\mathrm{Mc}</math><br />Note: (c)</td>
  <td align="center"><math>0</math><br />Note: (c)</td>
  <td align="center"><math>1</math></td>
  <td align="center"><math>0</math></td>
  <td align="center"><math>0</math><br />Note: (c)</td>
  <td align="center"><math>2\Omega_\mathrm{Mc}</math><br />Note: (c)</td>
  <td align="center"><math>0</math><br />Note: (c)</td>
  <td align="center"><math>\pm \infty</math><br />Note: (c)</td>
</tr>

<tr>
  <td align="center" rowspan="1">"<math>x = +1</math>" Bifurcation</td>
  <td align="right">0.952886702</td>
  <td align="right">0.440219895</td>
  <td align="center"><math>\frac{\Omega^2_\mathrm{Mc}}{4}</math><br />Note: (d)</td>
  <td align="center"><math>1</math></td>
  <td align="center"><math>\frac{\Omega_\mathrm{Mc}}{2}</math><br />Note: (d)</td>
  <td align="center"><math>\Omega_\mathrm{Mc}</math><br />Note: (d)</td>
  <td align="center"><math>\frac{1}{2}</math></td>
  <td align="center"><math>+~2</math></td>
  <td align="center"><math>\frac{\Omega_\mathrm{Mc}}{2}</math><br />Note: (d)</td>
  <td align="center"><math>\Omega_\mathrm{Mc}</math><br />Note: (d)</td>
  <td align="center"><math>\frac{1}{2}</math></td>
  <td align="center"><math>+~2</math></td>
</tr>

<tr>
<td align="left" colspan="13">
Notes:
<ol type="a">
  <li>[<b>[[Appendix/References#EFE|<font color="red">EFE</font>]]</b>], p. 142, last line of Table VI includes &hellip; <math>\Omega^2 = 4/15 \approx 0.26667</math> and <math>f = -2</math>.</li>
  <li>[<b>[[Appendix/References#EFE|<font color="red">EFE</font>]]</b>], p. 78, Table I.</li>
  <li>[<b>[[Appendix/References#EFE|<font color="red">EFE</font>]]</b>], p. 138, &sect;48b(iv).</li>
  <li>[<b>[[Appendix/References#EFE|<font color="red">EFE</font>]]</b>], p. 138, &sect;48b(v).</li>
</ol>
</td>
</tr>
</table>