Editing Appendix/Ramblings/ToroidalCoordinates (section)

==CCGF Expansion==
[http://adsabs.harvard.edu/abs/1999ApJ...527...86C Cohl &amp; Tohline (1999; hereafter CT99)] derive an expression for the Newtonian gravitational potential in terms of a ''Compact Cylindrical Green's Function'' expansion.  They show, for example, that when expressed in terms of cylindrical coordinates, the axisymmetric potential is,
<div align="center">
<math>
\Phi(R,z) = - \frac{2G}{R^{1/2}} q_0 ,
</math>
</div>
where,
<div align="center">
<math>
q_0 = \int\int (R')^{1/2} \rho(R',z') Q_{-1/2}(\Chi) dR' dz',
</math>
</div>
and the dimensionless argument (the modulus) of the special function, <math>~Q_{-1/2}</math>, is,
<div align="center">
<math>
\Chi \equiv \frac{R^2 + {R'}^2 + (z - z')^2}{2R R'} .
</math>
</div>
Note:  Here we are using <math>~\Chi</math> instead of <math>~\chi</math> (as used by CT99) to represent this dimensionless parameter in order to avoid confusion with our use of <math>~\chi</math>, above.  Next, following the lead of CT99, we note that according to the Abramowitz &amp; Stegun (1965),
<div align="center">
<math>Q_{-1/2}(\Chi) = \mu K(\mu) \, ,</math>
</div>
where, the function <math>~K(\mu)</math> is the complete elliptical integral of the first kind and, for our particular problem,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\mu^2</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~2(1+\Chi)^{-1}</math>
  </td>
</tr>

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

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[\frac{4R R'}{(R + {R'})^2 + (z - z')^2} \biggr] \, .
</math>
  </td>
</tr>
</table>
</div>

Hence, we can write,
<div align="center">
<math>
q_0 = \int\int (R')^{1/2} \rho(R',z') \mu K(\mu) dR' dz' \, .
</math>
</div>


===Confirmation Provided by Trova, Hur&eacute; and Hersant===

In their study of the potential of self-gravitating, axisymmetric discs, [http://adsabs.harvard.edu/abs/2012MNRAS.424.2635T A. Trova, J.-M. Hur&eacute; and F. Hersant (2012; MNRAS, 424, 2635)] write (see their equation 1),
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\Phi(R,Z)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
-2G \int\int \sqrt{\frac{\varpi}{R}} \rho(\varpi, z) k K(k) d\varpi dz \, ,
</math>
  </td>
</tr>
</table>
</div>
where, the modulus, <math>~k</math>, of the complete elliptical integral of the first kind is (see their equation 2),
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~k</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{2\sqrt{\varpi R}}{\sqrt{(\varpi + R)^2 + (Z-z)^2}} \, ,
</math>
  </td>
</tr>
</table>
</div>
and its relevant domain is, <math>~0 \leq k \leq 1</math>.  After associating <math>~z \leftrightarrow Z</math>, <math>~z^' \leftrightarrow z</math>,  and <math>~R^' \leftrightarrow \varpi</math>, we see that the modulus, <math>~k</math>, used by Trova et al. (2012), is precisely the same as the argument, <math>~\mu</math>, defined in CT99.  Hence, the two expressions for the axisymmetric potential, <math>~\Phi(R,Z)</math>, are identical.


===Recognition as Circle===
If we scale all of the lengths in [http://adsabs.harvard.edu/abs/1999ApJ...527...86C CT99's] expression for <math>~\Chi</math>, to <math>~a</math> and, along the lines of what was done above, define,
<div align="center">
<math>
~\chi \equiv \frac{R'}{a} ~~~~\mathrm{and}~~~~\zeta \equiv \frac{z-z'}{a} ,
</math>
</div>
we can rewrite the expression in the form,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~
\biggl(\frac{R}{a}\biggr)^2 + \chi^2 + \zeta^2 
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~2 \Chi \biggl( \frac{R}{a}\biggr) \chi \, .</math>
  </td>
</tr>
</table>
</div>
Now, because,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\biggl[ \chi - \Chi\biggl(\frac{R}{a}\biggr) \biggr]^2</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\chi^2 - 2\Chi\biggl(\frac{R}{a}\biggr)\chi + \Chi^2 \biggl(\frac{R}{a}\biggr)^2</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~~ 2\Chi\biggl(\frac{R}{a}\biggr)\chi </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\chi^2 - \biggl[ \chi - \Chi\biggl(\frac{R}{a}\biggr) \biggr]^2 + \Chi^2 \biggl(\frac{R}{a}\biggr)^2 \, ,</math>
  </td>
</tr>
</table>
</div>
we can further rewrite the expression as,

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

<tr>
  <td align="right">
<math>~
\biggl(\frac{R}{a}\biggr)^2 + \chi^2 + \zeta^2
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\chi^2 - \biggl[ \chi - \Chi\biggl(\frac{R}{a}\biggr) \biggr]^2 + \Chi^2 \biggl(\frac{R}{a}\biggr)^2</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~~
\biggl[ \chi - \Chi\biggl(\frac{R}{a}\biggr) \biggr]^2 +  \zeta^2
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl(\frac{R}{a}\biggr)^2(\Chi^2 -1) \, .</math>
  </td>
</tr>
</table>
</div>

Finally, if we adopt the ''specific'' scale factor, <math>~a = R</math>, we have,
<div align="center">
<math>
~(\chi - \Chi)^2 + \zeta^2 = (\Chi^2 - 1) .
</math>
</div>
So, a curve of constant <math>~\Chi</math> produces an off-center circle whose center is located at <math>~\Chi</math> and whose radius is <math>~\sqrt{\Chi^2 - 1}</math>.

===Relating CCGF Expansion to Toroidal Coordinates===
We see that curves of constant <math>\Chi</math> (as defined in CT99) are in every respect identical to curves of constant <math>\xi_1</math> (as defined in MF53).  The association is straightforward:
<table align="center" border="2" cellpadding="5" bgcolor="pink">
<tr>
  <th align="center">
<font color="red">EUREKA!</font>
  </th>
</tr>
<tr>
  <td align="center">
<math>
\Chi^2 - 1 = \frac{1}{\xi_1^2 - 1} .
</math>
  </td>
</tr>
</table>