Editing Apps/PapaloizouPringle84 (section)

====Kojima's Setup====
Presumably PP85 rewrote the equation in the latest form presented above in order to help make it clear how the equation simplifies (specifically, the last term on the right-hand side vanishes) for configurations that initially have uniform specific angular momentum &#8212; that is, for configurations in which <math>~h^' = 0</math>.  But other simplifications arise as well because the epicyclic frequency, <math>~\kappa</math>, also goes to zero in configurations with uniform specific angular momentum.  This means that the frequency ratio, <math>~{\bar\sigma}^2/D</math>, that appears in two terms of our derived expression goes to unity, that is,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{{\bar\sigma}^2}{D}\biggr|_{j_0-\mathrm{constant}} = \biggl[ \frac{{\bar\sigma}^2}{{\bar\sigma}^2 - \kappa^2}\biggr]_{j_0-\mathrm{constant}}</math>
  </td>
  <td align="center">
&nbsp; &nbsp; &nbsp; <math>~~\rightarrow~~</math>&nbsp; &nbsp; &nbsp; 
  </td>
  <td align="left">
<math>~1 \, .</math>
  </td>
</tr>
</table>
</div>

Implementing both of these simplifications, the latest form of our "eigenvalue problem" equation becomes,

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

<tr>
  <td align="right">
<math>~
\frac{ {\bar\sigma}^2  \rho_0^2 W^'}{\gamma P_0  } 
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~-
\frac{1}{\varpi} \frac{\partial}{\partial\varpi} \biggl[ \cancelto{1}{\biggl(\frac{{\bar\sigma}^2}{D} \biggr)} \rho_0 \varpi \cdot \frac{\partial W^'}{\partial \varpi} \biggr]
+ \cancelto{1}{\biggl(\frac{{\bar\sigma}^2}{D} \biggr)} \frac{\rho_0 m^2  W^' }{\varpi^2} 
- \frac{\partial}{\partial z} \biggl(\rho_0 \frac{\partial W^'}{\partial z} \biggr) 
- \frac{m W^' \bar\sigma}{\varpi} \frac{\partial}{\partial\varpi} \biggl[ \frac{\rho_0}{\varpi D} \cancelto{0}{\biggl( \frac{dj_0 }{d\varpi } \biggr)} \biggr] 
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow~~~~0
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{1}{\varpi} \frac{\partial}{\partial\varpi} \biggl[  \rho_0 \varpi \cdot \frac{\partial W^'}{\partial \varpi} \biggr]
+ \frac{\partial}{\partial z} \biggl(\rho_0 \frac{\partial W^'}{\partial z} \biggr) 
-  \frac{\rho_0 m^2  W^' }{\varpi^2} 
+ \frac{ {\bar\sigma}^2  \rho_0^2 W^'}{\gamma P_0  } \, .
</math>
  </td>
</tr>
</table>
</div>

As can be confirmed by comparing it to equation (12) of [http://adsabs.harvard.edu/abs/1986PThPh..75..251K Kojima (1986)] &#8212; which, to facilitate comparison, has been extracted and displayed in the following framed image &#8212; this expression matches the 2<sup>nd</sup>-order, two-dimensional PDE that defines the eigenvalue problem discussed by Kojima.

<div align="center" id="EigenvalueKojima86">
<table border="1" cellpadding="5" width="80%">
<tr><td align="center" bgcolor="orange">
Equations (12) &amp; (13) extracted without modification from p. 254 of [http://adsabs.harvard.edu/abs/1986PThPh..75..251K Kojima (1986)]<p></p>
"''The Dynamical Stability of a Fat Disk with Constant Specific Angular Momentum''"<p></p>
Progress of Theoretical Physics, vol. 75, pp. 251-261 &copy; The Physical Society of Japan
</td></tr>
<tr>
  <td align="center">
[[File:Kojima86Eq12.png|500px|center|Kojima (1986, Progress of Theoretical Physics, 75, 251)]]
<!-- [[Image:AAAwaiting01.png|400px|center|Whitworth (1981) Eq. 5]] -->
  </td>
</tr>
</table>
</div>

This expression also serves as the starting point for the stability analysis presented by [http://adsabs.harvard.edu/abs/1985MNRAS.216..553B Blaes (1985)] &#8212; see his equation (3.1), but note that he has replaced the adiabatic exponent with the polytropic index via the relation, <math>~\gamma = (n+1)/n</math>.