Editing Apps/PapaloizouPringleTori (section)

===Comparison===
The solution derived by PP84 (presented above) is normalized such that the pressure maximum is always in the equatorial plane at radial coordinate <math>~\chi = 1</math>, and tori of different thicknesses are constructed by specifying different values of the dimensionless Bernoulli constant, <math>~C_\mathrm{B}^'</math>.  Using the HSCF approach, the position of the outer edge of the torus is fixed and tori of different thicknesses are selected by moving the inner edge, <math>~x_{in}</math>, around.  In order to examine whether or not the two solutions are, in fact, the same, we will re-write the expression given above for <math>~\chi_{-}/\chi_{+}</math> in terms of <math>~x_{in}</math>.  In order to do this, we need to convert <math>~C_\mathrm{HSCF}</math> to <math>~C_\mathrm{B}^'</math>.  Based on their definitions, the ratio
<div align="center">
<math>
\frac{C_\mathrm{B}^'}{C_\mathrm{HSCF} } = \frac{\varpi_0}{\varpi_{+}} = x_0 .
</math>
</div>
Hence,
<div align="center">
<math>
C_\mathrm{B}^' = x_0~C_\mathrm{HSCF} =  \biggl( \frac{2x_{in}}{1+x_{in}} \biggr) \biggl( \frac{1}{1+x_{in}} \biggr) = \frac{2x_{in}}{(1+x_{in})^2}.
</math>
</div>
Therefore,
<div align="center">
<math>
1 - 2C_\mathrm{B}^' = 1 - \frac{4x_{in}}{(1+x_{in})^2} = \frac{ (1+x_{in})^2 - 4x_{in} }{(1+x_{in})^2} = \frac{ (1-x_{in})^2 }{(1+x_{in})^2} \, ,
</math>
</div>
and,
<div align="center">
<math>
\frac{\chi_{-}}{\chi_{+}} =  \frac{ 1 - \sqrt{1-2C_\mathrm{B}^'} }{1 + \sqrt{1-2C_\mathrm{B}^'} }  =  \frac{ 1 - \frac{1-x_{in}}{1+x_{in}} }{1 + \frac{1-x_{in}}{1+x_{in}}}  =  \frac{  (1+x_{in}) - (1-x_{in})  }{  (1+x_{in}) + (1-x_{in}) } = x_{in}.
</math>
</div>
Hence, both the PP84 derivation and the HSCF derivation produce the same result.