Editing ThreeDimensionalConfigurations/JacobiEllipsoids

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=Jacobi Ellipsoids=
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As has been detailed in an [[ThreeDimensionalConfigurations/HomogeneousEllipsoids#Gravitational_Potential|accompanying chapter]], the gravitational potential anywhere inside or on the surface, <math>~(a_1,a_2,a_3) ~\leftrightarrow~(a,b,c)</math>, of an homogeneous ellipsoid may be given analytically in terms of the following three coefficient expressions:
<div align="center">
<table align="center" border=0 cellpadding="3">
<tr>
  <td align="right">
<math>
~A_1
</math>
  </td>
  <td align="center">
<math>
~=
</math>
  </td>
  <td align="left">
<math>~2\biggl(\frac{b}{a}\biggr)\biggl(\frac{c}{a}\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{b}{a}\biggr) \biggl[  \frac{(b/a) \sin\theta - (c/a)E(\theta,k)}{(1-k^2) \sin^3\theta} \biggr] \, ,
</math>
  </td>
</tr>

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

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

<tr>
  <td align="right">
<math>~\theta = \cos^{-1} \biggl(\frac{c}{a} \biggr)</math>
  </td>
  <td align="center">
&nbsp; &nbsp; &nbsp; and &nbsp; &nbsp; &nbsp; 
  </td>
  <td align="left">
<math>~k = \biggl[\frac{1 - (b/a)^2}{1 - (c/a)^2} \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 3, &sect;17, Eq. (32)</font></td></tr>
</table>
</div>

==Equilibrium Conditions for Jacobi Ellipsoids==
Pulling from Chapter 6 &#8212; specifically, &sect;39 &#8212; of [<b>[[Appendix/References#EFE|<font color="red">EFE</font>]]</b>], we understand that the semi-axis ratios, <math>~(\tfrac{b}{a},\tfrac{c}{a})</math> associated with Jacobi ellipsoids are given by the roots of the equation,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~a^2 b^2 A_{12}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~c^2 A_3 \, ,</math>
  </td>
</tr>
<tr><td align="center" colspan="3">[<b>[[Appendix/References#EFE|<font color="red">EFE</font>]]</b>], <font color="#00CC00">&sect;39, Eq. (4)</font> </td></tr>
</table>
</div>
and the associated value of the square of the equilibrium configuration's angular velocity is,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{\Omega^2}{\pi G \rho}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~2B_{12} \, ,</math>
  </td>
</tr>
<tr><td align="center" colspan="3">[<b>[[Appendix/References#EFE|<font color="red">EFE</font>]]</b>], <font color="#00CC00">&sect;39, Eq. (5)</font> </td></tr>
</table>
</div>
where,
<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>~A_{12}</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~-\frac{A_1-A_2}{(a^2 - b^2)} \, ,</math>
  </td>
</tr>
<tr><td align="center" colspan="3">[<b>[[Appendix/References#EFE|<font color="red">EFE</font>]]</b>], <font color="#00CC00">&sect;21, Eq. (107)</font></td></tr>

<tr>
  <td align="right">
<math>~B_{12}</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~A_2 - a^2A_{12} \, .</math>
  </td>
</tr>
<tr><td align="center" colspan="3">[<b>[[Appendix/References#EFE|<font color="red">EFE</font>]]</b>], <font color="#00CC00">&sect;21, Eq. (105)</font></td></tr>
</table>
</div>


Taken together, we see that, written in terms of the two primary coefficients, <math>~A_1</math> and <math>~A_3</math>, the pair of defining relations for Jacobi ellipsoids is:


<div align="center" id="JacobiConstraints">
<table border="1" align="center" cellpadding="8"><tr><td align="center">
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>~f_J</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~\biggl(\frac{b}{a}\biggr)^2 \biggl[ \frac{2(1-A_1)-A_3}{1 - (b/a)^2} \biggr]-\biggl(\frac{c}{a}\biggr)^2  A_3 =0 </math>
  </td>
</tr>
<tr><td align="center" colspan="3">and</td></tr>
<tr>
  <td align="right">
<math>~\frac{\Omega^2}{\pi G \rho}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~2\biggl\{2 - (A_1+A_3) - \biggl[ \frac{2(1-A_1)-A_3}{1 - (b/a)^2} \biggr] \biggr\}</math>
  </td>
</tr>
</table>
</td></tr>
</table>
</div>

==Roots of the Governing Relation==

===Constraint on Axis-Ratio Relationship===
To simplify notation, here we will set,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\chi \equiv \frac{b}{a}</math>
  </td>
  <td align="center">
&nbsp; &nbsp; &nbsp; and &nbsp; &nbsp; &nbsp; 
  </td>
  <td align="left">
<math>~\upsilon \equiv \frac{c}{a} \, ,</math>
  </td>
</tr>
</table>
</div>
in which case the governing relation is,
<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>~f_J</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{\chi^2}{1-\chi^2} \biggl[ 2(1-A_1)-A_3\biggr]-\upsilon^2  A_3 =0 \, .</math>
  </td>
</tr>
</table>
</div>

Our plan is to employ the [https://brilliant.org/wiki/newton-raphson-method/ Newton Raphson method] to find the root(s) of the <math>~f_J = 0</math> relation, typically holding <math>~\upsilon</math> fixed and using the Newton-Raphson technique to identify the corresponding "root" value of <math>~\chi</math>.  Using this approach, the [https://brilliant.org/wiki/newton-raphson-method/ Newton Raphson technique] requires specification of, not only the function, <math>~f_J</math>, but also its first derivative,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~f_J^'</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{df_J}{d\chi} \, .</math>
  </td>
</tr>
</table>
</div>

Let's determine the requisite expression, using a prime superscript to indicate differentiation with respect to <math>~\chi</math>.  
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~f_J^'</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[ 2(1-A_1)-A_3\biggr]\biggl[ \frac{2\chi}{(1-\chi^2)^2} \biggr]
-\frac{\chi^2}{1-\chi^2} \biggl[ 2A_1^'+A_3^'\biggr]
-\upsilon^2  A_3^' \, ,
</math>
  </td>
</tr>
</table>
</div>
where, given that <math>~\theta</math> does not depend on <math>~\chi</math>,
<div align="center">
<table align="center" border=0 cellpadding="3">
<tr>
  <td align="right">
<math>
~A_1^'
</math>
  </td>
  <td align="center">
<math>
~=
</math>
  </td>
  <td align="left">
<math>~\frac{2\upsilon}{\sin^3\theta}  \cdot \frac{d}{d\chi}\biggl\{ \frac{\chi}{k^2} \biggl[ F(\theta,k) - E(\theta,k) \biggr] \biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>
~=
</math>
  </td>
  <td align="left">
<math>~\frac{2\upsilon}{k^3 \sin^3\theta}  \cdot \biggl\{ [ F - E ] [k - 2\chi k^'  ] 
+\chi k [ F^' - E^' ]\biggr\} \, ,
</math>
  </td>
</tr>

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

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>
~=
</math>
  </td>
  <td align="left">
<math>
~\frac{2}{(1-k^2)^2\sin^3\theta}  \biggl\{ 
\biggl[  \chi \sin\theta - \upsilon E\biggr]\biggl[ (1-k^2) +2\chi kk^' \biggr] + \chi(1-k^2) \biggl[  \sin\theta - \upsilon E^'\biggr]
\biggr\}\, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~k^'</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{d}{d\chi}\biggl[\frac{1 - \chi^2}{1 - \upsilon^2} \biggr]^{1/2} = \frac{-\chi}{(1 - \chi^2)^{1/2}(1 - \upsilon^2)^{1/2}} \, , 
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~F^'</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{\partial F(\theta,k)}{\partial k} \cdot k^' \, , 
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~E^'</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{\partial E(\theta,k)}{\partial k} \cdot k^' \, . 
</math>
  </td>
</tr>
</table>
</div>

Now, according to [http://functions.wolfram.com/EllipticIntegrals/EllipticF/introductions/IncompleteEllipticIntegrals/ShowAll.html online WolframResearch documentation] &#8212; see, in particular, the subsection titled, "Representations of Derivatives" &#8212;
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{\partial F(z|m)}{\partial m}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{E(z|m)}{2(1-m)m} - \frac{F(z|m)}{2m} - \frac{\sin(2z)}{4(1-m)\sqrt{1-m\sin^2(z)}} \, ,
</math>
  </td>
</tr>
</table>
</div>
and,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{\partial E(z|m)}{\partial m}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{E(z|m) - F(z|m)}{2m} \, ,</math>
  </td>
</tr>
</table>
</div>
where, <math>~z~\leftrightarrow~\theta</math>, and,
<div align="center">
<math>~m \equiv k^2 ~~~~\Rightarrow~~~~\frac{dm}{dk} = 2k \ .</math>
</div>
Hence, we have,

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

<tr>
  <td align="right">
<math>~F^'</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[\frac{\partial F(z|m)}{\partial m} \cdot \frac{dm}{dk}\biggr] k^' 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[ \frac{E(\theta,k)}{2(1-k^2)k^2} - \frac{F(\theta,k)}{2k^2} - \frac{\sin(2\theta)}{4(1-k^2)\sqrt{1-k^2\sin^2\theta}} \biggr] 2kk^' \, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~E^'</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[ \frac{\partial E(z|m)}{\partial m} \cdot \frac{dm}{dk}\biggr] k^' 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[ E(\theta,k) - F(\theta,k) \biggr] \frac{k^'}{k} \, .
</math>
  </td>
</tr>
</table>
</div>
This, then, gives us all of the expressions necessary to specify the derivative, <math>~f_J^'</math> analytically.



<table border="1" cellpadding="5" align="center">
<tr>
  <th align="center" colspan="1">
<font size="+1">Table 1:&nbsp; Double-Precision Evaluations</font><p></p>
Related to Table IV in [<b>[[Appendix/References#EFE|<font color="red">EFE</font>]]</b>], <font color="#00CC00">Chapter 6, &sect;39 (p. 103)</font>
</th>
</tr>
<tr><td align="left">
<pre>
         b/a      c/a            omega2              angmom              5L/M                fJ              fJderiv

        1.00   0.582724     3.742297785D-01     3.037510987D-01     4.232965627D+00     0.000000000D+00     0.000000000D+00
        0.96   0.570801     3.739782202D-01     3.039551227D-01     4.235808832D+00     1.377942479D-06     1.636908401D-01
        0.92   0.558330     3.731876801D-01     3.046006837D-01     4.244805137D+00    -6.821687132D-07     1.676406830D-01
        0.88   0.545263     3.717835971D-01     3.057488283D-01     4.260805266D+00     8.533280272D-07     1.715558312D-01
        0.84   0.531574     3.696959199D-01     3.074667323D-01     4.284745355D+00    -4.622993727D-08     1.754024874D-01
        0.80   0.517216     3.668370069D-01     3.098368632D-01     4.317774645D+00     2.805300664D-08     1.791408327D-01
        0.76   0.502147     3.631138118D-01     3.129555079D-01     4.361234951D+00     3.221800126D-07     1.827219476D-01
        0.72   0.486322     3.584232032D-01     3.169377270D-01     4.416729718D+00     3.274773094D-08     1.860866255D-01
        0.68   0.469689     3.526490289D-01     3.219229588D-01     4.486202108D+00     1.202999164D-08     1.891636215D-01
        0.64   0.452194     3.456641138D-01     3.280805511D-01     4.572012092D+00     2.681560312D-07     1.918668912D-01
        0.60   0.433781     3.373298891D-01     3.356184007D-01     4.677056841D+00     1.037186290D-08     1.940927000D-01
        0.56   0.414386     3.274928085D-01     3.447962894D-01     4.804956583D+00     1.071021385D-07     1.957166395D-01
        0.52   0.393944     3.159887358D-01     3.559412795D-01     4.960269141D+00     8.098003093D-08     1.965890756D-01
        0.48   0.372384     3.026414267D-01     3.694732246D-01     5.148845443D+00     1.255768368D-07     1.965308751D-01
        0.44   0.349632     2.872670174D-01     3.859399647D-01     5.378319986D+00     1.329168636D-08     1.953277019D-01
        0.40   0.325609     2.696779847D-01     4.060726774D-01     5.658882201D+00    -9.783004411D-08     1.927241063D-01
        0.36   0.300232     2.496925963D-01     4.308722159D-01     6.004479614D+00     1.044268276D-07     1.884168286D-01
        0.32   0.273419     2.271530240D-01     4.617497270D-01     6.434777459D+00    -4.469279448D-08     1.820477545D-01
        0.28   0.245083     2.019461513D-01     5.007767426D-01     6.978643856D+00     7.996820889D-08     1.731984783D-01
        0.24   0.215143     1.740514751D-01     5.511400218D-01     7.680488329D+00     1.099319693D-07     1.613864645D-01
        0.20   0.183524     1.436093757D-01     6.180687545D-01     8.613182979D+00     5.068010978D-08     1.460685065D-01
        0.16   0.150166     1.110438660D-01     7.109267615D-01     9.907218635D+00    -2.170751250D-08     1.266576761D-01
        0.12   0.115038     7.728058393D-02     8.487699974D-01     1.182815219D+01     3.613784147D-09     1.025686850D-01
        0.08   0.078166     4.416740942D-02     1.079303624D+00     1.504078558D+01     3.319018649D-08     7.332782508D-02
        0.04   0.039688     1.541513490D-02     1.582762691D+00     2.205680933D+01    -6.674246644D-09     3.882477311D-02
</pre>
</td></tr>
</table>

<span id="Table2"><b>With regard to our Table 1 (immediately above):</b></span>   Given each pair of axis ratios, <math>~(\tfrac{b}{a},\tfrac{c}{a})</math> &#8212; copied from Table IV of [<b>[[Appendix/References#EFE|<font color="red">EFE</font>]]</b>] (see columns 1 and 2 of our Table 1) &#8212; and the corresponding coefficient values, <math>~A_1</math>, <math>~A_2</math>, and <math>~A_3</math>, as tabulated in [[ThreeDimensionalConfigurations/HomogeneousEllipsoids#Table2|Table 2 of our accompanying discussion]], we calculated corresponding values of <math>~\Omega^2</math> (column 3) and total angular momentum (column 4) in the units used in Table IV of [<b>[[Appendix/References#EFE|<font color="red">EFE</font>]]</b>], as well as  (column 5) the total angular momentum in units used by [http://adsabs.harvard.edu/abs/1995ApJ...446..472C Christodoulou, ''et al.'' (1995, ApJ, 446, 472)] &#8212; see [[ThreeDimensionalConfigurations/HomogeneousEllipsoids#Example_Evaluations|our related discussion of these physical quantities]].  We also have tabulated associated values of the function, <math>~f_J</math>, (column 6) and its first derivative, <math>~f_J^'</math>, (column 7) as defined immediately above.  Notice that <math>~f_J</math> is very nearly zero in all cases, which indicates that each axis-ratio pair indeed identifies a configuration that lies along the Jacobi sequence.

<table border="1" cellpadding="5" align="center">
<tr>
  <th align="center" colspan="1">
<font size="+1">Table 2:&nbsp; Jacobi Sequence</font>
</th>
</tr>
<tr><td align="left">
<pre>
   b/a       c/a        A1        A2        A3      omega2       a       5L/M

  0.990699  0.580000  0.512818  0.518962  0.968220  0.374217  1.868761  4.233113
  0.901558  0.552381  0.481786  0.549836  0.968378  0.372621  1.960046  4.251259
  0.820783  0.524762  0.450993  0.580215  0.968792  0.368424  2.057217  4.299402
  0.747135  0.497143  0.420459  0.610088  0.969452  0.361716  2.161309  4.377683
  0.679613  0.469524  0.390210  0.639442  0.970348  0.352587  2.273548  4.486951
  0.617393  0.441905  0.360273  0.668258  0.971469  0.341129  2.395412  4.628802
  0.559798  0.414286  0.330684  0.696516  0.972800  0.327439  2.528716  4.805667
  0.506257  0.386667  0.301483  0.724187  0.974329  0.311620  2.675723  5.020964
  0.456291  0.359048  0.272719  0.751241  0.976040  0.293786  2.839307  5.279337
  0.409492  0.331429  0.244450  0.777636  0.977914  0.274062  3.023190  5.587020
  0.365507  0.303810  0.216744  0.803324  0.979931  0.252593  3.232298  5.952388
  0.324034  0.276190  0.189686  0.828246  0.982067  0.229546  3.473314  6.386811
  0.284807  0.248571  0.163376  0.852329  0.984295  0.205118  3.755577  6.906010
  0.247591  0.220952  0.137939  0.875480  0.986581  0.179549  4.092599  7.532311
  0.212179  0.193333  0.113527  0.897587  0.988885  0.153130  4.504785  8.298565
  0.178382  0.165714  0.090333  0.918505  0.991162  0.126229  5.024664  9.255452
  0.146026  0.138095  0.068601  0.938044  0.993355  0.099316  5.707871 10.486253
  0.114948  0.110476  0.048654  0.955953  0.995393  0.073010  6.659169 12.140357
  0.084989  0.082857  0.030927  0.971879  0.997194  0.048162  8.105501 14.522397
  0.055982  0.055238  0.016051  0.985298  0.998651  0.026008 10.663879 18.396951
  0.027738  0.027619  0.005032  0.995331  0.999637  0.008539 16.979084 26.660547
</pre>
</td></tr>
</table>

<b>With regard to our Table 2 (immediately above):</b>   Here we specified twenty-one values of the axis ratio, <math>~\tfrac{c}{a}</math>, (column 2) and used our Newton-Raphson-based root finder to identify corresponding values of the companion axis ratio, <math>~\tfrac{b}{a}</math>, (column 1) that satisfies the governing relation, <math>~f_J = 0</math>.

===Angular Momentum Constraint===

<table border="1" align="center" cellpadding="8" width="80%"><tr><td align="left">
<div align="center"><b>Angular Momentum Determination</b></div>
In the above tables, the square of the angular momentum, <math>L^2</math>, for each equilibrium Jacobi ellipsoid has been determined in the following manner:
<table border="0" align="center" cellpadding="8">

<tr>
  <td align="right"><math>L^2</math></td>
  <td align="center"><math>=</math></td>
  <td align="left"><math>I^2\Omega^2</math></td>
</tr>

<tr>
  <td align="right">&nbsp;</td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>\frac{M^2\Omega^2}{5^2}(a^2 + b^2)^2</math>
  </td>
</tr>

<tr>
  <td align="right">&nbsp;</td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>\frac{M^2}{5^2}\biggl(\frac{\Omega^2}{\pi G \rho}\biggr)\biggl[\frac{3G M}{2^2abc} \biggr](a^2 + b^2)^2</math>
  </td>
</tr>

<tr>
  <td align="right"><math>\Rightarrow ~~~ \frac{L^2}{GM^3 \bar{a}}</math></td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>\frac{1}{5^2}\biggl(\frac{\Omega^2}{\pi G \rho}\biggr)\biggl[\frac{3}{2^2abc} \biggr]\frac{(a^2 + b^2)^2}{\bar{a}}</math>
  </td>
</tr>

<tr>
  <td align="right">&nbsp;</td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>\frac{3}{2^2\cdot 5^2}\biggl(\frac{\Omega^2}{\pi G \rho}\biggr)\frac{(a^2 + b^2)^2}{\bar{a}^4} \, .</math>
  </td>
</tr>
<tr><td align="center" colspan="3">[<b>[[Appendix/References#EFE|<font color="red">EFE</font>]]</b>], <font color="#00CC00">&sect;39, p. 103, Eq. (16)</font> </td></tr>
</table>

----

Normalizing in a different manner, we have:
<table border="0" align="center" cellpadding="8">

<tr>
  <td align="right"><math>L^2</math></td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>\frac{M^2}{5^2}\biggl(\frac{\Omega^2}{4\pi G \rho}\biggr)\biggl[4\pi G\rho \biggr](a^2 + b^2)^2</math>
  </td>
</tr>

<tr>
  <td align="right"><math>\Rightarrow ~~~j^2 \equiv \frac{L^2}{4\pi G M^{10/3}\rho^{-1 / 3}}</math></td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>\frac{M^{-4/3}}{5^2}\biggl(\frac{\Omega^2}{4\pi G \rho}\biggr)\biggl[\rho \biggr]^{4 / 3}(a^2 + b^2)^2</math>
  </td>
</tr>

<tr>
  <td align="right">&nbsp;</td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>\frac{M^{-4/3}}{5^2}\biggl(\frac{\Omega^2}{4\pi G \rho}\biggr)\biggl[\frac{3M}{4\pi abc} \biggr]^{4 / 3}(a^2 + b^2)^2</math>
  </td>
</tr>

<tr>
  <td align="right">&nbsp;</td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>\frac{1}{2^2\cdot 5^2}\biggl[\frac{3}{4\pi } \biggr]^{4 / 3}\biggl(\frac{\Omega^2}{\pi G \rho}\biggr)\frac{(a^2 + b^2)^2}{\bar{a}^4}</math>
  </td>
</tr>

<tr>
  <td align="right">&nbsp;</td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
\frac{1}{3}\biggl[\frac{3}{4\pi } \biggr]^{4 / 3} \biggl[\frac{L^2}{GM^{3}\bar{a}} \biggr] 
= 0.049365924 \biggl[\frac{L^2}{GM^{3}\bar{a}} \biggr] 
\, .
</math>
  </td>
</tr>
</table>

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


Alternatively, let's choose a value for the system's total angular momentum, <math>~L > 4.23296</math>, and solve for the axis-ratio pair that identifies that configuration's location along the Jacobi sequence.  We'll adopt the units used by [http://adsabs.harvard.edu/abs/1995ApJ...446..472C Christodoulou ''et al'' (1995)], that is, <math>~G = 1</math>, <math>~\pi \rho = 1</math> and <math>~M = 5</math>, hence,

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

<tr>
  <td align="right">
<math>~a^3</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{3Ma^2}{4\pi(bc)\rho} = \frac{15}{4}\biggl(\frac{b}{a}\biggr)^{-1}  \biggl(\frac{c}{a}\biggr)^{-1}\, .</math>
  </td>
</tr>
</table>
</div>

Given that the relationship between <math>~L</math> and <math>~\Omega</math> in equilibrium Jacobi ellipsoids is,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~L</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~a^2\biggl[1 + \biggl(\frac{b}{a}\biggr)^2\biggr]\Omega </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl[ \frac{15}{4}\biggl(\frac{b}{a}\biggr)^{-1}  \biggl(\frac{c}{a}\biggr)^{-1} \biggr]^{2/3}
\biggl[1 + \biggl(\frac{b}{a}\biggr)^2\biggr]\Omega </math>
  </td>
</tr>
</table>
</div>

the [[#JacobiConstraints|constraint on <math>~\Omega^2</math> given above]] implies that,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~L^2 \biggl[ \frac{4}{15}\biggl(\frac{b}{a}\biggr)  \biggl(\frac{c}{a}\biggr) \biggr]^{4/3}
\biggl[1 + \biggl(\frac{b}{a}\biggr)^2\biggr]^{-2}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~2\biggl\{2 - (A_1+A_3) - \biggl[ \frac{2(1-A_1)-A_3}{1 - (b/a)^2} \biggr] \biggr\} \, .</math>
  </td>
</tr>
</table>
</div>
Or, again adopting the shorthand notation,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\chi \equiv \frac{b}{a}</math>
  </td>
  <td align="center">
&nbsp; &nbsp; &nbsp; and &nbsp; &nbsp; &nbsp; 
  </td>
  <td align="left">
<math>~\upsilon \equiv \frac{c}{a} \, ,</math>
  </td>
</tr>
</table>
</div>

we seek roots of the function,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~f_L</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~L^2  - \biggl[ \frac{3^4\cdot 5^4}{2^5}  \biggr]^{1/3}\chi^{-4/3} \upsilon^{-4/3}(1 + \chi^2)^{2}
\biggl\{[2 - (A_1+A_3)] - \biggl[ 2(1-A_1)-A_3\biggr](1-\chi^2)^{-1} \biggr\} = 0 \, .</math>
  </td>
</tr>
</table>
</div>

As [[#Constraint_on_Axis-Ratio_Relationship|above]], we will hold <math>~\upsilon</math> fixed and use the Newton-Raphson technique to identify the corresponding "root" value of <math>~\chi</math>.  Hence, we need to specify, not only the function, <math>~f_L</math>, but also its first derivative,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~f_L^'</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~\frac{\partial f_L}{\partial \chi} \, .</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~- \biggl[ \frac{3^4\cdot 5^4}{2^5}  \biggr]^{1/3}\upsilon^{-4/3} \frac{\partial}{\partial \chi} \biggl\{
\chi^{-4/3} (1 + \chi^2)^{2}
[2 - (A_1+A_3)] 
- \chi^{-4/3} (1 + \chi^2)^{2}(1-\chi^2)^{-1}[ 2(1-A_1)-A_3] 
\biggr\}</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~- \biggl[ \frac{3^4\cdot 5^4}{2^5}  \biggr]^{1/3}\upsilon^{-4/3} \biggl\{
-\frac{4}{3}\chi^{-7/3} (1 + \chi^2)^{2}[2 - (A_1+A_3)] 
+4\chi^{-1/3} (1 + \chi^2)[2 - (A_1+A_3)] 
-\chi^{-4/3} (1 + \chi^2)^{2}(A_1^'+A_3^') 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+ \frac{4}{3} \chi^{-7/3} (1 + \chi^2)^{2}(1-\chi^2)^{-1}[ 2(1-A_1)-A_3] 
- 4\chi^{-1/3} (1 + \chi^2)(1-\chi^2)^{-1}[ 2(1-A_1)-A_3] 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
- 2 \chi^{-1/3} (1 + \chi^2)^{2}(1-\chi^2)^{-2}[ 2(1-A_1)-A_3] 
- \chi^{-4/3} (1 + \chi^2)^{2}(1-\chi^2)^{-1}[ -2A_1^'-A_3^'] 
\biggr\}</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~- \biggl[ \frac{3\cdot 5^4}{2^5} \biggr]^{1/3}\upsilon^{-4/3} \chi^{-7/3} (1 + \chi^2)\biggl\{
[12\chi^2-4(1 + \chi^2)][2 - (A_1+A_3)] 
-3\chi (1 + \chi^2)(A_1^'+A_3^') 
+ 3\chi (1 + \chi^2)(1-\chi^2)^{-1}[ 2A_1^' + A_3^'] 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+ (1-\chi^2)^{-2}\{ 4  (1 + \chi^2)(1-\chi^2)[ 2(1-A_1)-A_3] 
- 12\chi^{2} (1-\chi^2)[ 2(1-A_1)-A_3] 
- 6 \chi^{2} (1 + \chi^2)[ 2(1-A_1)-A_3] \}
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~- \biggl[ \frac{3\cdot 5^4}{2^5} \biggr]^{1/3}\upsilon^{-4/3} \chi^{-7/3} (1 + \chi^2)\biggl\{
[8\chi^2-4]A_1
+ 3\chi (1 + \chi^2)(1-\chi^2)^{-1} [ (1+\chi^2)A_1^' + \chi^2A_3^' ]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+ 2(1-\chi^2)^{-2} [ 2-2A_1-A_3] [ (4\chi^2-2)(1-\chi^2)^{2} +
2  (1 + \chi^2)(1-\chi^2)  - 6\chi^{2} (1-\chi^2) - 3 \chi^{2} (1 + \chi^2) 
] \biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~- \biggl[ \frac{3\cdot 5^4}{2^5} \biggr]^{1/3}\upsilon^{-4/3} \chi^{-7/3} (1 + \chi^2)\biggl\{
3\chi (1 + \chi^2)(1-\chi^2)^{-1} [ (1+\chi^2)A_1^' + \chi^2A_3^' ]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~+[8\chi^2-4]A_1
+ 2(1-\chi^2)^{-2} [ 2-2A_1-A_3] [ - \chi^2 - 9\chi^4    + 4\chi^6 ] \biggr\}
</math>
  </td>
</tr>
<!--
<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+ 4\chi^2 - 8\chi^4 + 4\chi^6 -2+4\chi^2 - 2\chi^4  + 2  - 2\chi^4  - 6\chi^{2}  + 6\chi^4 - 3 \chi^{2} - 3 \chi^4 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~-2 + 2
+ 4\chi^2 +4\chi^2  - 6\chi^{2}  - 3 \chi^{2} - 8\chi^4  - 2\chi^4   - 2\chi^4  + 6\chi^4 - 3 \chi^4 + 4\chi^6  
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~- \chi^2 - 9\chi^4    + 4\chi^6  
</math>
  </td>
</tr>
-->
</table>
</div>


What values of <math>~L</math> should we choose?  In association with our [[ThreeDimensionalConfigurations/EFE_Energies#Conserve_Only_L|discussion of warped free-energy surfaces]], we'd like to specify the eccentricity, <math>~e</math>, of a Maclaurin spheroid and adopt the angular momentum of ''that'' configuration.  According to our [[Apps/MaclaurinSpheroids#Maclaurin_Spheroids_.28axisymmetric_structure.29|accompanying discussion of the properties of Maclaurin spheroids]], 
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~L_\mathrm{Mac}^2</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~2^2a^4\Omega^2</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~2^3a^4 [ A_1 -A_3(1-e^2)]_\mathrm{Mac} </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~2^3 \biggl[\frac{3\cdot 5}{2^2}(1-e^2)^{-1/2}  \biggr]^{4/3} [ A_1 -A_3(1-e^2)]_\mathrm{Mac} </math>
  </td>
</tr>

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

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

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

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

Note, for example, that if <math>~e = 0.85</math>, the square-root of this expression gives, <math>~L_\mathrm{Mac} = 4.7148806</math>, which matches the angular momentum that was used by [http://adsabs.harvard.edu/abs/1995ApJ...446..472C Christodoulou ''et al'' (1995)] to generate their Figure 3.

==Sequence Plots==

===EFE Diagram===
<div align="center">
<table border="1" cellpadding="5" width="80%">
<tr>
<td align="left">Jacobi Sequence: (blue) Points defined by data in Table IV of [<b>[[Appendix/References#EFE|<font color="red">EFE</font>]]</b>], <font color="#00CC00">Chapter 6, &sect;39 (p. 103)</font>; (red) points generated here from [[#Roots_of_the_Governing_Relation|above-defined roots of the governing relation]].</td>
<td align="center">
Figure 2 extracted<sup>&dagger;</sup> from p. 902 of [http://adsabs.harvard.edu/abs/1965ApJ...142..890C S. Chandrasekhar (1965)]<p></p>
"''The Equilibrium and the Stability of the Riemann Ellipsoids.  I''"<p></p>
ApJ, vol. 142, pp. 890-921 &copy; [http://aas.org/ American Astronomical Society]
</td>
</tr>
<tr>
  <td align="center">
[[File:JacobiSequenceB.png|300px|Jacobi Sequence]]
  </td>
  <td align="center">
<!-- [[File:NormanWilson78D.png|650px|center|Norman &amp; Wilson (1978)]] -->
[[File:ChandrasekharFig2annotated.png|340px|Chandrasekhar Figure2]]
  </td>
</tr>
<tr><td align="left">&nbsp;</td>
<td align="left"><sup>&dagger;</sup>Original figure has been annotated (maroon-colored text and arrow added) for clarification.</td>
</tr>
<tr>
  <td align="center">
[[File:OverlapAttempt3.png|300px|Jacobi Sequence]]
  </td>
  <td align="center">
<!-- [[File:NormanWilson78D.png|650px|center|Norman &amp; Wilson (1978)]] -->
[[File:OverlapAttempt1.png|340px|Chandrasekhar Figure2]]
  </td>
</tr>
</table>
</div>

===Other Sequence Depictions===

<table border="1" cellpadding="5" align="center">
<tr>
<td align="center">[[File:JacobiOmega2vsJ2.png|250px|Jacobi Omega2 versus J2]]</td>
<td align="center">[[File:JacobiOmega2vsTau.png|250px|Jacobi Omega2 versus Tau]]</td>
<td align="center">[[File:JacobiTauVsJ2.png|250px|Jacobi Tau versus J2]]</td>
</tr>
</table>

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

<tr>
  <td align="right">
<math>j</math>
  </td>
  <td align="center">
<math>\equiv</math>
  </td>
  <td align="left">
<math>
 \biggl( \frac{3}{2^8 \pi^4}  \biggr)^{1/6}  \frac{L}{(GM^3\bar{a})^{1 / 2}} \, .
</math>
  </td>
</tr>
</table>

===Bifurcation from Maclaurin to Jacobi Sequence===
According to Table IV of [<b>[[Appendix/References#EFE|<font color="red">EFE</font>]]</b>], <font color="#00CC00">Chapter 6, &sect;39 (p. 103)</font>, the Jacobi Sequence bifurcates from the Maclaurin sequence at &hellip;
<ul>
<li><math>(b/a, c/a) = (1.000, 0.582724)</math></li>
<li><math>\Omega^2/(\pi G \rho) = 0.374230 ~~~~ \Rightarrow ~~~ \omega_0^2/(4\pi G \rho) = 0.093558</math></td></li>
<li><math>L/(G M^3 \bar{a})^{1 / 2} = 0.303751 ~~~~ \Rightarrow ~~~ j^2 = \biggl( \frac{3}{2^8 \pi^4}  \biggr)^{1/3}  \frac{L^2}{(GM^3\bar{a})} = 0.049365924 (0.303751)^2 = 0.004554731</math></td></li>
</ul>

According to Tables D.3 and D.4 (pp. 485 &amp; 486) of [<b>[[Appendix/References#T87|<font color="red">T87</font>]]</b>], the Jacobi Sequence bifurcates from the Maclaurin sequence at &hellip;
<ul>
<li><math>(a/\bar{a}, b/\bar{a}, c/\bar{a}) = (1.197234, 1.197234, 0.697657) ~~~~ \Rightarrow ~~~ (b/a, c/a) = (1.000000, 0.582724)</math></li>
<li><math>\Omega^2/(2\pi G \rho) = 0.1871148 ~~~~ \Rightarrow ~~~ \omega_0^2/(4\pi G \rho) = 0.0935574</math></td></li>
<li><math>J/(G M^3 \bar{a})^{1 / 2} = 0.303751 ~~~~\Rightarrow ~~~~ j^2 = 0.004555</math></td></li>
<li><math>\tau \equiv T/|W| = 0.1375 </math></td></li>
</ul>

In the paragraph on p. 467 of { {Hachisu86bfull }} that immediately follows Eq. (47), we find &hellip;
<ul>
  <li><math>\Omega^2/(4\pi G\rho) = 0.09356</math></li>
  <li><math>j^2 = 0.004555</math></li>
</ul>

===Bifurcation of Poincar&eacute;'s Sequence of Pear-Shaped Configurations from the Jacobian Sequence===
According to Eq. (28) of [<b>[[Appendix/References#EFE|<font color="red">EFE</font>]]</b>], <font color="#00CC00">Chapter 6, &sect;40 (p. 106)</font>, a pear-shaped configuration bifurcates from the Jacobi Sequence at &hellip;
<ul>
<li><math>(b/a, c/a) = (0.432232, 0.345069)</math></li>
<li><math>\Omega^2/(\pi G \rho) = 0.284030 ~~~~ \Rightarrow ~~~ \omega_0^2/(4\pi G \rho) = 0.0710075</math></li>
<li>From the [[#Angular_Momentum_Constraint|above expression]], <math>\frac{L}{(GM^3\bar{a})^{1 / 2}} = 0.389536</math> and <math>j^2 = \biggl( \frac{3}{2^8 \pi^4}  \biggr)^{1/3}  \frac{L^2}{(GM^3\bar{a})} = 0.0074907 </math></li>
</ul>
According to the first row of properties in Table I of {{ EHS82 }} &hellip;
<ul>
<li><math>\Omega^2/(4\pi G \rho) = 0.07101 </math></li>
<li><math>j^2 = \biggl( \frac{3}{2^8 \pi^4}  \biggr)^{1/3}  \frac{L^2}{(GM^3\bar{a})} = 0.007821 </math></li>
<li><math>\tau \equiv T/|W| = 0.1628 </math></li>
</ul>

===Bifurcation of Dumbbell sequence from Jacobian Sequence===
According the last pair of equations on p. 128 of [<b>[[Appendix/References#EFE|<font color="red">EFE</font>]]</b>], <font color="#00CC00">Chapter 6, &sect;45</font>, a dumbbell-shaped configuration (neutral point belonging to the fourth harmonic distortion) bifurcates from the Jacobi Sequence at &hellip;
<ul>
<li><math>(b/a) = (0.2972)</math> and <math>\cos^{-1}(c/a) = 75.081~\mathrm{degrees}~~~~\Rightarrow ~~~~ (c/a) = 0.2575</math>.</li>
</ul>
Chronologically, this result for <math>(b/a, c/a)</math> appears first in Eq. (93) on p. 635 of {{ Chandrasekhar67_XXXIIfull }}.  Then, in Eq. (66) on p. 302 of {{ Chandrasekhar68_XXXVfull }} &#8212; we find <math>\cos^{-1}(c/a) = 75.068~\mathrm{degrees}</math>, along with a footnote [5] which states, <font color="darkgreen">"The value <math>\cos^{-1}(c/a) = 75.081~\mathrm{degrees}</math> found earlier differs slightly; but the difference is not outside the limits of accuracy of the numerical evaluation."</font>

According to the first row of properties in Table II of {{ EHS82 }} &hellip;
<ul>
<li><math>\Omega^2/(4\pi G \rho) = 0.0532 </math></li>
<li><math>j^2 = \biggl( \frac{3}{2^8 \pi^4}  \biggr)^{1/3}  \frac{L^2}{(GM^3\bar{a})} = 0.01157 </math></li>
<li><math>\tau \equiv T/|W| = 0.1863 </math></li>
</ul>

I have not (yet) found the corresponding value of <math>\Omega^2</math> in any of Chandrasekhar's publications, but if we combine the value of <math>\Omega^2</math> obtained from {{ EHS82 }} with the values of <math>(b/a, c/a)</math> obtained from [<b>[[Appendix/References#EFE|<font color="red">EFE</font>]]</b>], we find &hellip;
<ul>
<li><math>\Omega^2/(\pi G \rho) = 0.2128</math></li>
<li>From the [[#Angular_Momentum_Constraint|above expression]], <math>\frac{L}{(GM^3\bar{a})^{1 / 2}} = 0.48242</math> &nbsp; and &nbsp; <math>j^2 = \biggl( \frac{3}{2^8 \pi^4}  \biggr)^{1/3}  \frac{L^2}{(GM^3\bar{a})} = 0.01149</math></li>
</ul>
This value of <math>j^2</math> is very close to the value obtained by {{ EHS82 }}.

In the paragraph at the top of the right-hand column of p. 467 of {{Hachisu86bfull }}, we find &hellip;
<ul>
  <li><math>\Omega^2/(4\pi G\rho) = 0.0535</math></li>
  <li><math>j^2 = 0.01157</math></li>
</ul>

{{ CKST95bfull }} grab parameter values from a variety of sources.  In subsection "B" (''Jacobi Ellipsoid to Binary'') of their Table 1 (p. 494) and in the first paragraph of their &sect;3.2 (p. 492), they state &hellip;
<ul>
  <li><math>(b/a, c/a) = (0.29720, 0.25746)</math></li>
  <li><math>\Omega^2/(4\pi G \rho) = 0.0532790</math></li>
  <li><math>j^2 = 0.0115082</math></li>
</ul>

=See Also=
* [[Apps/MaclaurinSpheroids#Maclaurin_Spheroids_.28axisymmetric_structure.29|Properties of Maclaurin Spheroids]]
* [[Apps/MaclaurinSpheroids/GoogleBooks#Excerpts_from_A_Treatise_of_Fluxions|Excerpts from Maclaurin's (1742) ''A Treatise of Fluxions'']]


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