Editing 2DStructure/AxisymmetricInstabilities (section)

==Poincar&eacute;-Wavre Theorem==
As [ <b>[[Appendix/References#T78|<font color="red">T78</font>]] </b>] points out &#8212; see his pp. 78 - 81 &#8212; Poincar&eacute; and Wavre were the first to, effectively, prove the following theorem:
<table border="0" align="center" width="75%" cellpadding="5">
<tr><td align="left">For rotating, self-gravitating configurations<font color="darkgreen"> "any of the following statements implies the three others: &nbsp; (i) the angular velocity is a constant over cylinders centered about the axis of rotation, (ii) the effective gravity can be derived from a potential, (iii) the effective gravity is normal to the isopycnic surfaces, (iv) the isobaric- and isopycnic-surfaces coincide."
</font></td></tr>
</table>
Among other things, this implies that for rotating barotropic configurations not only is the equation of state given by a function of the form, <math>~P = P(\rho)</math>, but it must also be true that,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{\partial \dot\varphi}{\partial z}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~0 \, .</math>
  </td>
</tr>
<tr>
  <td align="center" colspan="3">
[ <b>[[Appendix/References#T78|<font color="red">T78</font>]] </b>], &sect;4.3, Eq. (30)
  </td>
</tr>
</table>
</div>

NOTE:  We should investigate how this theorem comes into play in the context of our [[ThreeDimensionalConfigurations/RiemannTypeI#Riemann_Type_1_Ellipsoids|accompanying discussion of Type 1 Riemann ellipsoids]].  These are equilibrium triaxial, uniform-density configurations in which the system's internal vorticity vector does not align with the tumble-axis of the ellipsoid and, therefore apparently <math>~\dot\varphi</math> is not independent of <math>~z</math>.