Editing Apps/PapaloizouPringle84 (section)

====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>