Editing Apps/PapaloizouPringle84 (section)

===Analyses of Configurations with Uniform Specific Angular Momentum===
====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>.

====Equivalent Dimensionless Expression====

Now, as should be clear from [[SR#Barotropic_Structure|our introductory description of barotropic structures]], because the initial, unperturbed [[Apps/PapaloizouPringleTori#Solution|Papaloizou-Pringle torus]] is a polytropic configuration, the functions <math>~P_0(\varpi,z)</math> and <math>~\rho_0(\varpi,z)</math> can both be expressed in terms of the [[Appendix/Ramblings/PPToriPt1A#Equilibrium_Configuration|dimensionless enthalpy distribution]],

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

<tr>
  <td align="right">
<math>~\Theta_H(\varpi,z) </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~1 - \frac{1}{\beta^2}\biggl[ \chi^{-2} 
- 2 ( \chi^2 + \zeta^2 )^{-1/2} + 1 \biggr] \, ,</math>
  </td>
</tr>
</table>
</div>
where,

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

<tr>
  <td align="right">
<math>~\chi \equiv \frac{\varpi}{\varpi_0} \, ,</math>
&nbsp; &nbsp; &nbsp; &nbsp; 
<math>~\zeta \equiv \frac{z}{\varpi_0} \, ,</math>
&nbsp; &nbsp; &nbsp; &nbsp;
<math>~\beta^2 \equiv \frac{2n}{\mathfrak{M}_0^2} \, ,</math>
  </td>
  <td align="center">
<math>~</math>
  </td>
  <td align="left">
<math>~</math>
  </td>
</tr>

</table>
</div>
and <math>~\mathfrak{M}_0</math> is the Mach number of the circular, azimuthal flow at the pressure and density maximum.  Specifically (see also equation 1.1 of [http://adsabs.harvard.edu/abs/1985MNRAS.216..553B Blaes85]),

<div align="center">
<math>~P_0 = P_\mathrm{max} \Theta_H^{n+1} </math> 
&nbsp; &nbsp; &nbsp; &nbsp; and &nbsp; &nbsp; &nbsp; &nbsp; 
<math>~\rho_0 = \rho_\mathrm{max} \Theta_H^{n} \, .</math> 
</div>

Making these state-variable substitutions in the PDE that we have just presented for comparison with Kojima's work, we have,

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

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

In an effort to make this entire expression dimensionless, let's define a dimensionless enthalpy perturbation via the relation,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\delta W</math>
  </td>
  <td align="center">
<math>~\equiv \biggl[ \frac{\Omega_0 \rho_\mathrm{max}}{P_\mathrm{max}} \biggr]W^' \, ,</math>
  </td>
  <td align="left">
<math>~</math>
  </td>
</tr>
</table>
</div>
and multiply our expression through by <math>~(\varpi_0^2 \Omega_0/P_\mathrm{max})</math>.  This gives,

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

<tr>
  <td align="right">
<math>~
0 
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{1}{\chi} \frac{\partial}{\partial\chi} \biggl[  \Theta_H^n \chi \cdot \frac{\partial (\delta W) }{\partial \chi} \biggr]
+ \frac{\partial}{\partial \zeta} \biggl[ \Theta_H^n \frac{\partial (\delta W) }{\partial \zeta} \biggr] 
+ \biggl[ 
\Theta_H^{n-1} \biggl( \frac{{\bar\sigma}}{\Omega_0} \biggr)^2
\frac{ n\Omega_0^2 \rho_\mathrm{max} \varpi_0^2}{(n+1)P_\mathrm{max}  }-  \frac{\Theta_H^n m^2 }{\chi^2} \biggr]\delta W 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\Theta_H^n \cdot \frac{\partial^2 (\delta W) }{\partial \chi^2} 
+\Theta_H^n \cdot  \frac{\partial^2 (\delta W) }{\partial \zeta^2} 
+ \biggl[ \Theta_H^{n-1} \biggl( \frac{{\bar\sigma}}{\Omega_0} \biggr)^2 \frac{2n}{\beta^2} -  \frac{\Theta_H^n m^2 }{\chi^2} \biggr]\delta W 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+\frac{1}{\chi} \frac{\partial (\delta W) }{\partial \chi} \biggl[\Theta_H^n + n\chi \Theta_H^{n-1} \frac{\partial \Theta_H}{\partial\chi} \biggr]
+ n \Theta_H^{n-1} ~\frac{\partial (\delta W) }{\partial \zeta} \frac{\partial \Theta_H}{\partial \zeta} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\Theta_H^{n-1} \biggl\{
\Theta_H \cdot \frac{\partial^2 (\delta W) }{\partial \chi^2} 
+\Theta_H \cdot  \frac{\partial^2 (\delta W) }{\partial \zeta^2} 
+ \biggl[\frac{\Theta_H}{ \chi } + n \frac{\partial \Theta_H}{\partial\chi} \biggr]\frac{\partial (\delta W) }{\partial \chi} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+ \biggl[ n \frac{\partial \Theta_H}{\partial \zeta} \biggr] \frac{\partial (\delta W) }{\partial \zeta} 
+ \biggl[ \frac{2n }{\beta^2}\biggl( \frac{{\bar\sigma}}{\Omega_0} \biggr)^2  -  \frac{m^2 \Theta_H}{\chi^2} \biggr]\delta W 
\biggr\} \, .
</math>
  </td>
</tr>
</table>
</div>

[It appears as though I'm on the right track because this expression is very similar to equation (3.2) of Blaes85!]

====Change to Off-Axis Polar-Coordinate System====
In his effort to derive an ''analytic'' solution to this eigenvalue problem, [http://adsabs.harvard.edu/abs/1985MNRAS.216..553B Blaes (1985)] adopted a different meridional-plane coordinate system.  As is illustrated in his Figure 1, Blaes shifted from the (dimensionless) rectilinear <math>~(\chi,\zeta)</math> system to a (dimensionless) polar-coordinate <math>~(x,\theta)</math> system whose origin sits at the pressure-maximum of the initial, unperturbed Papaloizou-Pringle torus.  Mapping between these two coordinate systems is accomplished via the relations (see equation 2.1 of [http://adsabs.harvard.edu/abs/1985MNRAS.216..553B Blaes85]),

<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>~x^2 = (1-\chi)^2 + \zeta^2</math>
  </td>
  <td align="center">
&nbsp; &nbsp; &nbsp; &nbsp; and &nbsp; &nbsp; &nbsp; &nbsp; 
  </td>
  <td align="left">
<math>~\theta = \tan^{-1}\biggl[\frac{\zeta}{1-\chi}\biggr] \, ;</math>
  </td>
</tr>
<tr><td align="center" colspan="3">&nbsp; &nbsp; or &nbsp; &nbsp; &nbsp; &nbsp;</td></tr>
<tr>
  <td align="right">
<math>~\chi = 1 - x\cos\theta</math>
  </td>
  <td align="center">
&nbsp; &nbsp; &nbsp; &nbsp; and &nbsp; &nbsp; &nbsp; &nbsp; 
  </td>
  <td align="left">
<math>~\zeta = x\sin\theta \, .</math>
  </td>
</tr>
</table>
</div>

Mapping of partial derivatives is accomplished via the relations,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{\partial}{\partial\chi}</math>
  </td>
  <td align="center">
&nbsp; &nbsp; &nbsp; <math>~\rightarrow</math>&nbsp; &nbsp; &nbsp; 
  </td>
  <td align="left">
<math>~
- \cos\theta \cdot \frac{\partial}{\partial x}
+\frac{\sin\theta}{x}\cdot \frac{\partial}{\partial\theta} \, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\frac{\partial}{\partial\zeta}</math>
  </td>
  <td align="center">
&nbsp; &nbsp; &nbsp; <math>~\rightarrow</math>&nbsp; &nbsp; &nbsp; 
  </td>
  <td align="left">
<math>~
\sin\theta \cdot \frac{\partial}{\partial x}
+\frac{\cos\theta}{x}\cdot \frac{\partial}{\partial\theta} \, .
</math>
  </td>
</tr>
</table>
</div>

This means, for example, that,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{\partial^2(\delta W)}{\partial \chi^2}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{\partial}{\partial\chi}\biggl[  
- \cos\theta \cdot \frac{\partial(\delta W)}{\partial x}
+\frac{\sin\theta}{x}\cdot \frac{\partial(\delta W)}{\partial\theta}
\biggr]</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
-\cos\theta \cdot\frac{\partial}{\partial x}\biggl[  
- \cos\theta \cdot \frac{\partial(\delta W)}{\partial x}
+\frac{\sin\theta}{x}\cdot \frac{\partial(\delta W)}{\partial\theta}
\biggr] +
\frac{\sin\theta}{x} \cdot \frac{\partial}{\partial\theta}\biggl[  
- \cos\theta \cdot \frac{\partial(\delta W)}{\partial x}
+\frac{\sin\theta}{x}\cdot \frac{\partial(\delta W)}{\partial\theta}
\biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
-\cos\theta \biggl[  
- \cos\theta \cdot \frac{\partial^2(\delta W)}{\partial x^2}
-\frac{\sin\theta}{x^2}\cdot \frac{\partial(\delta W)}{\partial\theta}
+\frac{\sin\theta}{x}\cdot \frac{\partial^2(\delta W)}{\partial x ~\partial\theta}
\biggr] 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+
\frac{\sin\theta}{x} \biggl[  
\sin\theta \cdot \frac{\partial(\delta W)}{\partial x}
- \cos\theta \cdot \frac{\partial^2(\delta W)}{\partial\theta~\partial x}
+\frac{\cos\theta}{x}\cdot \frac{\partial(\delta W)}{\partial\theta}
+\frac{\sin\theta}{x}\cdot \frac{\partial^2(\delta W)}{\partial\theta^2}
\biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[  
\cos^2\theta \cdot \frac{\partial^2(\delta W)}{\partial x^2}
+\frac{\sin\theta\cos\theta}{x^2}\cdot \frac{\partial(\delta W)}{\partial\theta}
-\frac{\sin\theta\cos\theta}{x}\cdot \frac{\partial^2(\delta W)}{\partial x ~\partial\theta}
\biggr] 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+\biggl[  
\frac{\sin^2\theta}{x} \cdot \frac{\partial(\delta W)}{\partial x}
- \frac{\sin\theta\cos\theta}{x} \cdot \frac{\partial^2(\delta W)}{\partial\theta~\partial x}
+\frac{\sin\theta\cos\theta}{x^2}\cdot \frac{\partial(\delta W)}{\partial\theta}
+\frac{\sin^2\theta}{x^2}\cdot \frac{\partial^2(\delta W)}{\partial\theta^2}
\biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\cos^2\theta \cdot \frac{\partial^2(\delta W)}{\partial x^2}
+\frac{\sin^2\theta}{x^2}\cdot \frac{\partial^2(\delta W)}{\partial\theta^2}
+ \frac{\sin^2\theta}{x} \cdot \frac{\partial(\delta W)}{\partial x}
+\frac{2\sin\theta\cos\theta}{x^2}\cdot \frac{\partial(\delta W)}{\partial\theta}
-\frac{2\sin\theta\cos\theta}{x}\cdot \frac{\partial^2(\delta W)}{\partial x ~\partial\theta} \, .
</math>
  </td>
</tr>
</table>
</div>

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

<tr>
  <td align="right">
<math>~\frac{\partial^2(\delta W)}{\partial \zeta^2}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{\partial}{\partial\zeta}\biggl[  
\sin\theta \cdot \frac{\partial(\delta W)}{\partial x}
+\frac{\cos\theta}{x}\cdot \frac{\partial(\delta W)}{\partial\theta}
\biggr]</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\sin\theta \cdot \frac{\partial}{\partial x}\biggl[  
\sin\theta \cdot \frac{\partial(\delta W)}{\partial x}
+\frac{\cos\theta}{x}\cdot \frac{\partial(\delta W)}{\partial\theta}
\biggr] +
\frac{\cos\theta}{x} \cdot \frac{\partial}{\partial\theta}\biggl[  
\sin\theta \cdot \frac{\partial(\delta W)}{\partial x}
+\frac{\cos\theta}{x}\cdot \frac{\partial(\delta W)}{\partial\theta}
\biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\sin\theta \biggl[  
\sin\theta \cdot \frac{\partial^2(\delta W)}{\partial x^2}
-\frac{\cos\theta}{x^2}\cdot \frac{\partial(\delta W)}{\partial\theta}
+\frac{\cos\theta}{x}\cdot \frac{\partial^2(\delta W)}{\partial x ~\partial\theta}
\biggr] 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+\frac{\cos\theta}{x} \biggl[  
\cos\theta \cdot \frac{\partial(\delta W)}{\partial x}
+\sin\theta \cdot \frac{\partial^2(\delta W)}{\partial\theta~\partial x}
-\frac{\sin\theta}{x}\cdot \frac{\partial(\delta W)}{\partial\theta}
+\frac{\cos\theta}{x}\cdot \frac{\partial^2(\delta W)}{\partial\theta^2}
\biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ 
\sin^2\theta \cdot \frac{\partial^2(\delta W)}{\partial x^2}
+\frac{\cos^2\theta}{x^2}\cdot \frac{\partial^2(\delta W)}{\partial\theta^2}
+\frac{\cos^2\theta}{x}  \cdot \frac{\partial(\delta W)}{\partial x}
-\frac{2\sin\theta \cos\theta}{x^2}\cdot \frac{\partial(\delta W)}{\partial\theta}
+\frac{2\sin\theta \cos\theta}{x}\cdot \frac{\partial^2(\delta W)}{\partial x ~\partial\theta} \, .
</math>
  </td>
</tr>
</table>
</div>

A significant amount of simplification occurs when these two expressions are added.  Specifically, we have,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~
\frac{\partial^2(\delta W)}{\partial \chi^2}
+\frac{\partial^2(\delta W)}{\partial \zeta^2}
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ 
\frac{\partial^2(\delta W)}{\partial x^2}
+\frac{1}{x^2}\cdot \frac{\partial^2(\delta W)}{\partial\theta^2}
+\frac{1}{x}  \cdot \frac{\partial(\delta W)}{\partial x} \, .
</math>
  </td>
</tr>
</table>
</div>

With this as a start, a coordinate mapping of our above-derived dimensionless "eigenvalue problem" expression gives,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~0</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\Theta_H \biggl[ \frac{\partial^2 (\delta W) }{\partial \chi^2} + \frac{\partial^2 (\delta W) }{\partial \zeta^2} \biggr]
+ \biggl[ \frac{2n }{\beta^2}\biggl( \frac{{\bar\sigma}}{\Omega_0} \biggr)^2  -  \frac{m^2 \Theta_H}{\chi^2} \biggr]\delta W 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+ \biggl[\frac{\Theta_H}{ \chi } + n \frac{\partial \Theta_H}{\partial\chi} \biggr]\frac{\partial (\delta W) }{\partial \chi} 
+ \biggl[ n \frac{\partial \Theta_H}{\partial \zeta} \biggr] \frac{\partial (\delta W) }{\partial \zeta} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\Theta_H \biggl[ \frac{\partial^2(\delta W)}{\partial x^2}
+\frac{1}{x^2}\cdot \frac{\partial^2(\delta W)}{\partial\theta^2}
+\frac{1}{x}  \cdot \frac{\partial(\delta W)}{\partial x}  \biggr]
+ \biggl[ \frac{2n }{\beta^2}\biggl( \frac{{\bar\sigma}}{\Omega_0} \biggr)^2  -  \frac{m^2 \Theta_H}{(1-x\cos\theta)^2} \biggr]\delta W 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+ \biggl\{\frac{\Theta_H}{ \chi } + n \biggl[ - \cos\theta \cdot \frac{\partial\Theta_H}{\partial x}
+\frac{\sin\theta}{x}\cdot \frac{\partial \Theta_H}{\partial\theta}  \biggr]\biggr\} 
\biggl[- \cos\theta \cdot \frac{\partial (\delta W)}{\partial x}
+\frac{\sin\theta}{x}\cdot \frac{\partial (\delta W)}{\partial\theta} \biggr]  
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+ n\biggl[ \sin\theta \cdot \frac{\partial\Theta_H}{\partial x}
+\frac{\cos\theta}{x}\cdot \frac{\partial\Theta_H}{\partial\theta} \biggr] \biggl[ \sin\theta \cdot \frac{\partial (\delta W)}{\partial x}
+\frac{\cos\theta}{x}\cdot \frac{\partial (\delta W)}{\partial\theta} \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\Theta_H \biggl[ \frac{\partial^2(\delta W)}{\partial x^2}
+\frac{1}{x^2}\cdot \frac{\partial^2(\delta W)}{\partial\theta^2}
+\frac{1}{x}  \cdot \frac{\partial(\delta W)}{\partial x}  \biggr]
+ \biggl[ \frac{2n }{\beta^2}\biggl( \frac{{\bar\sigma}}{\Omega_0} \biggr)^2  -  \frac{m^2 \Theta_H}{(1-x\cos\theta)^2} \biggr]\delta W 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+ \biggl\{- \frac{\cos\theta\cdot\Theta_H}{ (1-x\cos\theta)} + n \biggl[ \cos^2\theta \cdot \frac{\partial\Theta_H}{\partial x}
-\frac{\sin\theta \cos\theta}{x}\cdot \frac{\partial \Theta_H}{\partial\theta}  \biggr]\biggr\} 
\cdot \frac{\partial (\delta W)}{\partial x}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+ n\biggl[ \sin^2\theta \cdot \frac{\partial\Theta_H}{\partial x}
+\frac{\sin\theta\cos\theta}{x}\cdot \frac{\partial\Theta_H}{\partial\theta} \biggr] \cdot \frac{\partial (\delta W)}{\partial x}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+ \biggl\{\frac{\sin\theta\cdot\Theta_H}{ x(1-x\cos\theta) } + n \biggl[ - \frac{\sin\theta\cos\theta}{x} \cdot \frac{\partial\Theta_H}{\partial x}
+\frac{\sin^2\theta}{x^2}\cdot \frac{\partial \Theta_H}{\partial\theta}  \biggr]\biggr\} 
\cdot \frac{\partial (\delta W)}{\partial\theta} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+ n\biggl[ \frac{\sin\theta\cos\theta}{x} \cdot \frac{\partial\Theta_H}{\partial x}
+\frac{\cos^2\theta}{x^2}\cdot \frac{\partial\Theta_H}{\partial\theta} \biggr] \cdot \frac{\partial (\delta W)}{\partial\theta} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\Theta_H \biggl[ \frac{\partial^2(\delta W)}{\partial x^2}
+\frac{1}{x^2}\cdot \frac{\partial^2(\delta W)}{\partial\theta^2}\biggr]
+ \biggl[ \frac{2n }{\beta^2}\biggl( \frac{{\bar\sigma}}{\Omega_0} \biggr)^2  -  \frac{m^2 \Theta_H}{(1-x\cos\theta)^2} \biggr]\delta W 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+ \biggl\{\frac{\Theta_H}{x}- \frac{\cos\theta\cdot\Theta_H}{ (1-x\cos\theta)} + n \biggl[ \frac{\partial\Theta_H}{\partial x}
\biggr]\biggr\} 
\cdot \frac{\partial (\delta W)}{\partial x}
+ \biggl[\frac{\sin\theta\cdot\Theta_H}{ x(1-x\cos\theta) } + 
\frac{n}{x^2}\cdot \frac{\partial \Theta_H}{\partial\theta}  \biggr] 
\cdot \frac{\partial (\delta W)}{\partial\theta} \, .
</math>
  </td>
</tr>

</table>
</div>

Finally, after multiplying through by <math>~x^2</math> and rearranging terms, we obtain,

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

<tr>
  <td align="right">
<math>~0</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\Theta_H x^2\cdot \frac{\partial^2(\delta W)}{\partial x^2}
+\Theta_H \cdot \frac{\partial^2(\delta W)}{\partial\theta^2}
+ \biggl\{\Theta_H x \biggl[\frac{1-2x \cos\theta}{ 1-x\cos\theta}\biggr] + nx^2 \cdot \frac{\partial\Theta_H}{\partial x}
\biggr\} \cdot \frac{\partial (\delta W)}{\partial x} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+ \biggl[\frac{\Theta_Hx\sin\theta}{ (1-x\cos\theta) } + 
n\cdot \frac{\partial \Theta_H}{\partial\theta}  \biggr] 
\cdot \frac{\partial (\delta W)}{\partial\theta} 
+ \biggl[ \frac{2n x^2}{\beta^2}\biggl( \frac{{\bar\sigma}}{\Omega_0} \biggr)^2  -  \frac{m^2 x^2\Theta_H}{(1-x\cos\theta)^2} \biggr]\delta W \, .
</math>
  </td>
</tr>

</table>
</div>

Note that, because [http://adsabs.harvard.edu/abs/1985MNRAS.216..553B Blaes (1985)] is only considering tori with a uniform specific angular momentum distribution, <math>~q=2</math>, the frequency ratio, <math>~(\bar\sigma/\Omega_0)</math>, may be rewritten as,

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

<tr>
  <td align="right">
<math>~\frac{\bar\sigma}{\Omega_0}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{\sigma + m{\dot\varphi}_0(\varpi)}{\Omega_0} = \frac{\sigma}{\Omega_0} + m\chi^{-q}
= \frac{\sigma}{\Omega_0} + \frac{m}{(1-x\cos\theta)^2} \, .</math>
  </td>
</tr>
</table>
</div>

As can be confirmed by comparing it to equation (3.2) of [http://adsabs.harvard.edu/abs/1985MNRAS.216..553B Blaes (1985)] &#8212; which, to facilitate comparison, has been extracted and displayed in the following framed image &#8212; our just-derived PDE matches the 2<sup>nd</sup>-order, two-dimensional PDE that defines the ''dimensionless'' eigenvalue problem discussed by Blaes (1985).

<div align="center" id="EigenvalueBlaes85">
<table border="1" cellpadding="5" width="80%">
<tr><td align="center" bgcolor="orange">
Equation (3.2) extracted without modification from p. 558 of [http://adsabs.harvard.edu/abs/1985MNRAS.216..553B O. M. Blaes (1985)]<p></p>
"''Oscillations of Slender Tori''"<p></p>
Monthly Notices of the Royal Astronomical Society, vol. 216, Issue 3, October 1985, pp. 553-563<br />&copy; Royal Astronomical Society<br />
https://doi.org/10.1093/mnras/216.3.553
</td></tr>
<tr>
  <td align="center">
[[File:Blaes85Eq3.2.png|600px|center|Blaes (1985, MNRAS, 216, 553)]]
<!-- [[Image:AAAwaiting01.png|400px|center|Whitworth (1981) Eq. 5]] -->
  </td>
</tr>
<tr>
  <td align="left">
This digital image has been posted by permission of author, O.M. Blaes (via email dated 19 July 2020), and by permission of Oxford University Press on behalf of the Royal Astronomical Society (via email dated 31 July 2020). 
<div align="center">Original publication website:  https://academic.oup.com/mnras/article/216/3/553/1005987</div>
  </td>
</tr>
</table>
</div>

<span id="isolatingBlaes85">Isolating on the right-hand side terms that explicitly involve the dimensionless eigenfrequency, <math>~\nu \equiv (\sigma/\Omega_0)</math>, this governing PDE may also be written in the form (see equations 4.1 and 4.2 of Blaes85),</span>
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\hat{L} (\delta W)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~-2n(1-\Theta_H)(M\nu^2 + N\nu)(\delta W) \, ,</math>
  </td>
</tr>
</table>
</div>
where,

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

<tr>
  <td align="right">
<math>~\hat{L} (\delta W)</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~
\Theta_H x^2\cdot \frac{\partial^2(\delta W)}{\partial x^2}
+\Theta_H \cdot \frac{\partial^2(\delta W)}{\partial\theta^2}
+ \biggl\{\Theta_H x \biggl[\frac{1-2x \cos\theta}{ 1-x\cos\theta}\biggr] + nx^2 \cdot \frac{\partial\Theta_H}{\partial x}
\biggr\} \cdot \frac{\partial (\delta W)}{\partial x} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+ \biggl[\frac{\Theta_Hx\sin\theta}{ (1-x\cos\theta) } + 
n\cdot \frac{\partial \Theta_H}{\partial\theta}  \biggr] 
\cdot \frac{\partial (\delta W)}{\partial\theta} 
+ \biggl[ \frac{2n x^2 m^2}{\beta^2(1-x\cos\theta)^4}  -  \frac{m^2 x^2\Theta_H}{(1-x\cos\theta)^2} \biggr]\delta W \, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~M</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~\frac{x^2}{(1-\Theta_H)\beta^2} \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~N</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~\frac{2mx^2}{(1-\Theta_H)\beta^2(1-x\cos\theta)^2} \, .</math>
  </td>
</tr>

</table>
</div>

====Introduce Coordinate-Parameter &eta;====

Following the lead of [http://adsabs.harvard.edu/abs/1985MNRAS.216..553B Blaes (1985)], and without loss of generality, we can everywhere replace the dimensionless function representing the unperturbed equilibrium enthalpy distribution,  <math>~\Theta_H</math> &#8212; that varies from unity at the cross-sectional center of the torus to zero at the torus surface &#8212; with the parameter,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\eta^2</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~1 - \Theta_H \, ,</math>
  </td>
</tr>
</table>
</div>
that varies from zero at the (cross-sectional) center to unity at the surface.  Making this substitution, our governing PDE becomes,

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

<tr>
  <td align="right">
<math>~0</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
(1-\eta^2) x^2\cdot \frac{\partial^2(\delta W)}{\partial x^2}
+(1-\eta^2) \cdot \frac{\partial^2(\delta W)}{\partial\theta^2}
+ \biggl\{(1-\eta^2) x \biggl[\frac{1-2x \cos\theta}{ 1-x\cos\theta}\biggr] + nx^2 \cdot \frac{\partial(1-\eta^2) }{\partial x}
\biggr\} \cdot \frac{\partial (\delta W)}{\partial x} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+ \biggl[\frac{(1-\eta^2) x\sin\theta}{ (1-x\cos\theta) } + 
n\cdot \frac{\partial (1-\eta^2) }{\partial\theta}  \biggr] 
\cdot \frac{\partial (\delta W)}{\partial\theta} 
+ \biggl[ \frac{2n x^2}{\beta^2}\biggl( \frac{{\bar\sigma}}{\Omega_0} \biggr)^2  -  \frac{m^2 x^2(1-\eta^2) }{(1-x\cos\theta)^2} \biggr]\delta W 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
x^2 (1-\eta^2)\cdot \frac{\partial^2(\delta W)}{\partial x^2}
+ (1-\eta^2) \cdot \frac{\partial^2(\delta W)}{\partial\theta^2}
+ \biggl\{x (1-\eta^2) \biggl[\frac{1-2x \cos\theta}{ 1-x\cos\theta}\biggr] -2 nx^2 \eta \cdot \frac{\partial \eta }{\partial x}
\biggr\} \cdot \frac{\partial (\delta W)}{\partial x} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+ \biggl[\frac{(1-\eta^2) x\sin\theta}{ (1-x\cos\theta) } -2n\eta \cdot \frac{\partial \eta }{\partial\theta}  \biggr] 
\cdot \frac{\partial (\delta W)}{\partial\theta} 
+ \biggl[ \frac{2n x^2}{\beta^2}\biggl( \frac{{\bar\sigma}}{\Omega_0} \biggr)^2  -  \frac{m^2 x^2(1-\eta^2) }{(1-x\cos\theta)^2} \biggr]\delta W \, .
</math>
  </td>
</tr>
</table>
</div>