Editing Appendix/Ramblings/ToroidalCoordinates (section)

===Advantageous Coordinate System===

According to [http://adsabs.harvard.edu/abs/1986PThPh..75..251K Kojima's (1986)] review of the [http://adsabs.harvard.edu/abs/1984MNRAS.208..721P PP84] discussion &#8212; see his equation (14) &#8212; surfaces of constant density can be defined by the coordinate, <math>~\chi_{PP}</math>, where,
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<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\tan\chi_{PP}</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~
\frac{z}{ \frac{z^2}{\sqrt{\varpi^2 + z^2}} + \sqrt{\varpi^2 + z^2} - 1}
</math>
  </td>
</tr>
</table>
</div>

Indeed, equation (6.6) of [http://adsabs.harvard.edu/abs/1984MNRAS.208..721P PP84] defines the coordinate, <math>~\chi_{PP}</math>, via the expression,
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<tr>
  <td align="right">
<math>~\tan\chi_{PP}</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~
\frac{\cos\theta}{ \cos^2\theta + 1 - \varpi_0/r} \, ,
</math>
  </td>
</tr>
</table>
</div>
where,
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  <td align="right">
<math>~z = r\cos\theta</math>
  </td>
  <td align="center">
&nbsp; &nbsp; &nbsp; &nbsp; and &nbsp; &nbsp; &nbsp; &nbsp; 
  </td>
  <td align="left">
<math>~\varpi = r\sin\theta \, .</math>
  </td>
</tr>
</table>
</div>

Let's see if these match.  Starting from the Kojima expression, we have,

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<tr>
  <td align="right">
<math>~\tan\chi_{PP}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{r\cos\theta}{ \frac{r^2\cos^2\theta}{r} + r - 1}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{\cos\theta}{ \cos^2\theta + 1 - 1/r} \, .
</math>
  </td>
</tr>
</table>
</div>
Hence, they are the same, as long as we appreciate that Kojima assumes all length scales are normalized to <math>~\varpi_0</math>.  Let's express this coordinate in terms of the <math>~(\xi_1, \xi_2)</math> toroidal coordinates as defined by MF53, namely,

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<tr>
  <td align="right">
<math>
~\frac{z}{a}
</math>
  </td>
  <td align="center">
<math>
~=
</math>
  </td>
  <td align="left">
<math>
~\frac{(1-\xi_2^2)^{1/2}}{\xi_1 - \xi_2}  \, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>
~\frac{\varpi}{a} \equiv \biggl[ \biggl(\frac{x}{a}\biggr)^2 + \biggl(\frac{y}{a}\biggr)^2 \biggr]^{1/2}
</math>
  </td>
  <td align="center">
<math>
~=
</math>
  </td>
  <td align="left">
<math>
~\frac{(\xi_1^2 - 1)^{1/2}}{\xi_1 - \xi_2} \, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>
~\frac{r}{a} \equiv \biggl[ \biggl(\frac{x}{a}\biggr)^2 + \biggl(\frac{y}{a}\biggr)^2 + \biggl(\frac{z}{a}\biggr)^2\biggr]^{1/2}
</math>
  </td>
  <td align="center">
<math>
~=
</math>
  </td>
  <td align="left">
<math>
~\biggl[ \frac{\xi_1 + \xi_2}{\xi_1 - \xi_2} \biggr]^{1/2} \, .
</math>
  </td>
</tr>
</table>

Kojima's expression becomes:
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<tr>
  <td align="right">
<math>~\tan\chi_{PP}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~z \biggl[ \frac{z^2}{\sqrt{\varpi^2 + z^2}} + \sqrt{\varpi^2 + z^2} - 1 \biggr]^{-1}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{(1-\xi_2^2)^{1/2}}{\xi_1 - \xi_2}  \biggl\{ \biggl[\frac{(1-\xi_2^2)}{ (\xi_1 - \xi_2)^2}  \biggr] \biggl[ \frac{\xi_1 - \xi_2}{\xi_1 + \xi_2} \biggr]^{1/2}
+ \biggl[ \frac{\xi_1 + \xi_2}{\xi_1 - \xi_2} \biggr]^{1/2} - 1 \biggr\}^{-1}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~(1-\xi_2^2)^{1/2}  \biggl\{ \frac{(1-\xi_2^2)}{ (\xi_1 - \xi_2)}   \biggl[ \frac{\xi_1 - \xi_2}{\xi_1 + \xi_2} \biggr]^{1/2}
+ (\xi_1 + \xi_2)^{1/2} (\xi_1 - \xi_2)^{1/2} - (  \xi_1 - \xi_2 ) \biggr\}^{-1}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~(1-\xi_2^2)^{1/2}  \biggl\{ \frac{(1-\xi_2^2)}{ (\xi_1 - \xi_2)^{1/2} (\xi_1 + \xi_2)^{1/2}}  
+ (\xi_1 + \xi_2)^{1/2} (\xi_1 - \xi_2)^{1/2} - (  \xi_1 - \xi_2 ) \biggr\}^{-1}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~(1-\xi_2^2)^{1/2} (\xi_1 - \xi_2)^{1/2} (\xi_1 + \xi_2)^{1/2} \biggl\{ (1-\xi_2^2) + (\xi_1 - \xi_2)(\xi_1 + \xi_2)
- (  \xi_1 - \xi_2 )(\xi_1 + \xi_2)^{1/2} (\xi_1 - \xi_2)^{1/2}  \biggr\}^{-1}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~(1-\xi_2^2)^{1/2} (\xi_1^2 - \xi_2^2)^{1/2} \biggl\{ (1-\xi_2^2) + (\xi_1^2 - \xi_2^2)
- (  \xi_1 - \xi_2 )(\xi_1^2 - \xi_2^2)^{1/2} \biggr\}^{-1} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl[ \frac{(1-\xi_2^2)}{ (\xi_1^2 - \xi_2^2) } \biggr]^{1/2} \biggl\{ \biggl[ \frac{(1-\xi_2^2)}{(\xi_1^2 - \xi_2^2)}\biggr] + 1
- \biggl[ \frac{ (  \xi_1 - \xi_2 )}{(\xi_1 + \xi_2)} \biggr]^{1/2}\biggr\}^{-1} \, .
</math>
  </td>
</tr>
</table>
</div>
This does not appear to be very useful or productive!