Editing Appendix/Ramblings/Bordeaux (section)

====Setup====

From our [[Apps/Wong1973Potential#D0andCn|accompanying discussion of Wong's (1973) derivation]], the exterior potential is given by the expression,
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<math>~\biggl( \frac{a}{GM} \biggr) \Phi_\mathrm{W}(\eta,\theta)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
-D_0
(\cosh\eta - \cos\theta)^{1 / 2} ~\sum_{n=0}^{\mathrm{nmax}} \epsilon_n \cos(n\theta) C_n(\cosh\eta_0)P_{n-\frac{1}{2}}(\cosh\eta) \, ,
</math>
  </td>
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</table>
[http://adsabs.harvard.edu/abs/1973AnPhy..77..279W Wong (1973)], &sect;II.D, p. 294, Eqs. (2.59) &amp; (2.61)
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where,
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<math>~D_0 </math>
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<math>~\equiv</math>
  </td>
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<math>~
\frac{2^{3/2} }{3\pi^2} \biggl[ \frac{\sinh^3\eta_0}{\cosh\eta_0}\biggr] 
=
\frac{2^{3/2} }{3\pi^2} \biggl[\frac{(R^2 - d^2)^{3 / 2}}{d^2 R}  \biggr] 
\, ,</math>
  </td>
</tr>

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  <td align="right">
<math>~C_n(\cosh\eta_0)</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
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<math>~(n+\tfrac{1}{2})Q_{n+\frac{1}{2}}(\cosh \eta_0) Q_{n - \frac{1}{2}}^2(\cosh \eta_0) 
- (n - \tfrac{3}{2}) Q_{n - \frac{1}{2}}(\cosh \eta_0)~Q^2_{n + \frac{1}{2}}(\cosh \eta_0) \, 
</math>
  </td>
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</table>
[http://adsabs.harvard.edu/abs/1973AnPhy..77..279W Wong (1973)], &sect;II.D, p. 294, Eq. (2.63)
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and where, in terms of the major ( R ) and minor ( d ) radii of the torus &#8212; or their ratio, &epsilon; &equiv; d/R, 
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  <td align="right">
<math>~\cosh\eta_0</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{R}{d} = \frac{1}{\epsilon} \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\sinh\eta_0</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{a}{d} = \frac{1}{d}\biggl[ R^2 - d^2 \biggr]^{1 / 2} = \frac{1}{\epsilon} \biggl[1 - \epsilon^2 \biggr]^{1 / 2} \, .</math>
  </td>
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</table>

These expressions incorporate a number of [[2DStructure/ToroidalGreenFunction#Basic_Elements_of_a_Toroidal_Coordinate_System|basic elements of a toroidal coordinate system]].  In what follows, we will also make use of the following relations:

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Once the primary scale factor, <math>~a</math>, has been specified, the illustration shown at the bottom of this inset box &#8212; see also our [[Apps/DysonWongTori#Self-Gravitating.2C_Incompressible_.28Dyson-Wong.29_Tori|accompanying set of similar figures]] used by other researchers &#8212; helps in explaining how transformations can be made between any two of the referenced coordinate pairs:  <math>~(\varpi, z)</math>, <math>~(\eta, \theta)</math>, <math>~(r_1, r_2)</math>.

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  <td align="right">
<math>~\varpi</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{a\sinh\eta}{(\cosh\eta - \cos\theta)}</math>
  </td>
<td align="center">&nbsp; &nbsp; &nbsp;<math>~\Rightarrow ~</math>&nbsp; &nbsp; &nbsp;</td>
  <td align="right">
<math>~\cos\theta</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\cosh\eta - \frac{a\sinh\eta}{\varpi}</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~z - Z_0</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{a\sin\theta}{(\cosh\eta - \cos\theta)}</math>
  </td>
<td align="center">&nbsp; &nbsp; &nbsp;<math>~\Rightarrow ~</math>&nbsp; &nbsp; &nbsp;</td>
  <td align="right">
<math>~\sin\theta</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{(z - Z_0)}{\varpi} \cdot \sinh\eta </math>
  </td>
</tr>
</table>
Given that (sin<sup>2</sup>&theta; + cos<sup>2</sup>&theta;) = 1, we have,
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<math>~1</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[ \frac{(z - Z_0)}{\varpi} \cdot \sinh\eta \biggr]^2 + \biggl[\cosh\eta - \frac{a\sinh\eta}{\varpi}\biggr]^2
</math>
  </td>
</tr>

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  <td align="right">
<math>~\Rightarrow ~~~ \coth\eta</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
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<math>~
\frac{1}{2a\varpi}\biggl[\varpi^2 + a^2 + (z - Z_0)^2  \biggr] \, .
</math>
  </td>
</tr>
</table>
We deduce as well that,

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  <td align="right">
<math>~\frac{2}{\coth\eta + 1}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{4a\varpi}{(\varpi + a)^2 + (z - Z_0)^2} \, ,
</math>&nbsp; &nbsp; &nbsp; &nbsp; and,
  </td>
</tr>

<tr>
  <td align="right">
<math>~\sinh\eta + \cosh\eta</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{\varpi^2 + a^2 + (z - Z_0)^2}{(\varpi + a)^2 + (z - Z_0)^2} \, .
</math>
  </td>
</tr>
</table>

----
Given the definitions, 
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  <td align="right">
<math>~r_1^2</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~(\varpi + a)^2 + (z - Z_0)^2 \, ,</math>
  </td>
</tr>

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  <td align="right">
<math>~r_2^2</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~(\varpi - a)^2 + (z - Z_0)^2 \, ,</math>
  </td>
</tr>
</table>
we can use the transformations,

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<tr>
  <td align="right">
<math>~\varpi</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{(r_1^2 - r_2^2)}{4a}</math> &nbsp; &nbsp; and,
  </td>
</tr>

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  <td align="right">
<math>~(z - Z_0)^2</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~r_2^2 - \frac{1}{16a^2}\biggl[ r_1^2 - r_2^2 - 4a^2 \biggr]^2 \, ,</math> &nbsp; &nbsp; or,
  </td>
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  <td align="right">
<math>~(z - Z_0)^2</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~r_1^2 - \frac{1}{16a^2}\biggl[ r_1^2 - r_2^2 + 4a^2 \biggr]^2 \, .</math>
  </td>
</tr>
</table>
Or we can use the transformations,

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  <td align="right">
<math>~\eta</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\ln \biggl(\frac{r_1}{r_2}\biggr) \, ,</math>
  </td>
</tr>

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  <td align="right">
<math>~\cos\theta</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{r_1^2 + r_2^2 - 4a^2}{2r_1 r_2} \, .</math>
  </td>
</tr>
</table>

----

Additional potentially useful relations can be found in an [[2DStructure/ToroidalGreenFunction#Using_Toroidal_Coordinates_to_Determine_the_Gravitational_Potential|accompanying chapter wherein we present a variety of basic elements of a toroidal coordinate system]].

[[File:WongTorusIllustration02.png|400px|center|Wong diagram]]
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