Editing Apps/PapaloizouPringle84 (section)

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

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<math>~0</math>
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<math>~=</math>
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<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>
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&nbsp;
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&nbsp;
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<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>
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&nbsp;
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<math>~=</math>
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<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>
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&nbsp;
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&nbsp;
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<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>
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