Editing Apps/MaclaurinSpheroidSequence (section)

===Uniform Rotation===

{{ HE83 }} sought to find bifurcation points along the Maclaurin spheroid sequence where the associated, deformed equilibrium configuration is uniformly rotating.  From the set of familiar [[AxisymmetricConfigurations/SolutionStrategies#SRPtable|simple rotation profiles]], therefore, they set
<div align="center">
<math>
\Psi = - \frac{1}{2} \varpi^2 \omega_0^2 \, ,
</math>
</div>
in which case, in their investigation, the condition (along the surface) for hydrostatic balance is,

<div align="center">
<math>\Phi - \frac{1}{2} \varpi^2 \omega_0^2 = C_0</math> .
</div>
Replacing <math>\varpi^2</math> with its equivalent expression in terms of oblate spheroidal coordinates gives,
<table border="0" align="center" cellpadding="5">
<tr>
  <td align="right">
<math>- ~\Phi + \frac{\omega_0^2}{2} a_0^2 (1 + \xi^2)(1 - \eta^2) </math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>- ~C_0 \, ,</math>
  </td>
</tr>
<tr>
  <td align="center" colspan="3">
{{ HE83 }}, &sect;A.1 (p. 587), Eq. (6)
  </td>
</tr>
</table>
which is the same as their Appendix (&sect;A.1) Eq. (6), except they chose a different sign when defining the constant, <math>C_0</math>.