Editing SSC/StabilityConjecture/Bipolytrope51 (section)

===Original Manipulations===
As has been shown in our [[SSC/Structure/BiPolytropes/Analytic51#Derivation|accompanying discussion]], the value of <math>\xi_i</math> at which the maximum-mass turning point resides along each sequence is given by a root of the analytic expression,

<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>
\biggl(\frac{\pi}{2} + \tan^{-1} \Lambda_i\biggr) (1+\ell_i^2) [ 3 + (1-m_3)^2(2-\ell_i^2)\ell_i^2] 
</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
m_3 \ell_i [(1-m_3)\ell_i^4 - (m_3^2 - m_3 +2)\ell_i^2 - 3] \, ,
</math> 
  </td>
</tr>
</table>
where,

<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>
\ell_i
</math>
  </td>
  <td align="center">
<math>\equiv</math>
  </td>
  <td align="left">
<math>
\frac{\xi_i}{\sqrt{3}} \, ,
</math> 
  </td>
</tr>
<tr>
  <td align="right">
<math>
m_3
</math>
  </td>
  <td align="center">
<math>\equiv</math>
  </td>
  <td align="left">
<math>
3\biggl( \frac{\mu_e}{\mu_c}\biggr) \, ,
</math> 
  </td>
</tr>
<tr>
  <td align="right">
<math>
\Lambda_i
</math>
  </td>
  <td align="center">
<math>\equiv</math>
  </td>
  <td align="left">
<math>
\frac{1}{m_3 \ell_i}\biggl[1 + (1 - m_3)\ell_i^2  \biggr] \, .
</math> 
  </td>
</tr>
</table>

In what follows, we start from scratch and re-derive an analytic expression from which the value of <math>\nu_\mathrm{max}(\mu_e/\mu_c)</math> can be obtained. At the conclusion of this "new" derivation, we present a [[#Example|table]] in which high-precision determinations of <math>\nu_\mathrm{max}</math> have been recorded for a range of values of <math>\mu_e/\mu_c = m_3/3</math>.  The last column of this [[#Example|table]] lists "earlier fractional errors" of our determinations via this earlier-derived analytic expression.  The tiny errors signal that our more recently derived expression (below) is identical to this earlier expression (immediately above).