Editing ThreeDimensionalConfigurations/JacobiEllipsoids (section)

===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>