Editing Appendix/Ramblings/Bordeaux (section)

====Summary====

Once the major ( R ) and minor ( d ) radii of the torus have been specified, two key model parameters can be immediately determined, namely,
<div align="center">
<math>~a^2 \equiv R^2 - d^2\, ,</math> &nbsp; &nbsp; &nbsp; and, &nbsp; &nbsp; &nbsp; <math>~\cosh\eta_0 \equiv \frac{R}{d} \, ,</math> 
</div>
in which case also, <math>~\sinh\eta_0 = a/d \, .</math>  Once the mass-density ( &rho;<sub>0</sub> ) of the torus has been specified, the torus mass is given by the expression,
<div align="center">
<math>~M = 2\pi^2 \rho_0 d^2 R \, .</math>
</div>
In addition to the principal pair of meridional-plane coordinates, <math>~(\varpi, z)</math>, it is useful to define the pair of distances,
<table border="0" cellpadding="5" align="center">

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

<tr>
  <td align="right">
<math>~r_2^2</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~(\varpi - a)^2 + (z - Z_0)^2 \, ,</math>
  </td>
</tr>
</table>
where, the equatorial plane of the torus is located at <math>~z = Z_0</math>.  As we have shown above, the leading (n = 0) term in Wong's (1973) series-expression for the gravitational potential anywhere outside the torus is,

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

<tr>
  <td align="right">
<math>~\biggl( \frac{a}{GM} \biggr) \Phi_\mathrm{W0}(\varpi,z)\biggr|_\mathrm{exterior}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
- \frac{2^{3} }{3\pi^3} 
\biggl[ \frac{\sinh\eta_0}{\cosh\eta_0}\biggr] 
\frac{a}{ r_1 } \cdot  \boldsymbol{K}(k) 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
\times \biggl\{
K(k_0)\cdot K(k_0) [ \cosh\eta_0(1 - \cosh\eta_0) ] 
+ 2K(k_0)\cdot E(k_0) [ \cosh^2\eta_0  + 1 ] 
- E(k_0)\cdot E(k_0) [ \cosh\eta_0(1 + \cosh\eta_0)  ]
\biggr\}  \, .
</math>
  </td>
</tr>
</table>

where, the two distinctly different arguments &#8212; one with, and one without a zero subscript &#8212; of the complete elliptic-integral functions are,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~k</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~
\biggl[ \frac{2}{\coth\eta + 1} \biggr]^{1 / 2}  
=
\biggl[ \frac{4a\varpi}{(\varpi + a)^2 + (z - Z_0)^2} \biggr]^{1 / 2} 
= \biggl[ \frac{4a\varpi}{r_1^2} \biggr]^{1 / 2}
= \biggl[ \frac{r_1^2 - r_2^2}{r_1^2} \biggr]^{1 / 2} \, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~k_0</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~
\biggl[ \frac{2}{\cosh\eta_0 + 1} \biggr]^{1 / 2}  \, .
</math>
  </td>
</tr>
</table>

As we also have shown above, the second (n = 1) term in Wong's (1973) series-expression for the exterior gravitational potential is,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\biggl( \frac{a}{GM} \biggr) \Phi_\mathrm{W1}(\varpi,z)\biggr|_\mathrm{exterior}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
- \frac{2^{3} }{3\pi^3} \biggl[ \frac{\sinh\eta_0}{\cosh\eta_0}\biggr]  \biggl[\frac{ r_1^2 + r_2^2 - 4a^2}{2r_1 r_2} \biggr] \frac{a}{r_2} \cdot
\boldsymbol{E}(k) 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~\times
\biggl\{
K(k_0)\cdot K(k_0) [\cosh\eta_0(1 - \cosh\eta_0)] 
+~2K(k_0)\cdot E(k_0) [(3\cosh^2\eta_0 - 1)] 
-~5 E(k_0)\cdot E(k_0) [\cosh\eta_0(1+\cosh\eta_0)] 
\biggr\}
\, .
</math>
  </td>
</tr>
</table>
Note that a transformation from the <math>~(r_1, r_2)</math> coordinate pair to the toroidal-coordinate pair <math>~(\eta, \theta)</math> includes the expression,
<table border="0" cellpadding="5" align="center">

<tr>
  <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>
So this (n = 1) term's explicit dependence on "cos(n&theta;)" is clear.  Finally,  the third (n = 2) term in Wong's (1973) series-expression for the exterior gravitational potential is,

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

<tr>
  <td align="right">
<math>~\biggl( \frac{a}{GM} \biggr) \Phi_\mathrm{W2}(\eta,\theta)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
-\frac{2^{3} }{3\pi^3} \biggl[ \frac{\sinh\eta_0}{\cosh\eta_0}\biggr]
\times \cos(2\theta)
\biggl\{ 
\biggl[ \frac{r_1^2 + r_2^2}{2r_1 r_2} \biggr] \frac{a}{r_2} \cdot  \boldsymbol{E}(k)  
- 
\frac{a}{r_1}  \cdot \boldsymbol{K}(k) 
\biggr\} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
\times \frac{2^{3 / 2}}{3^2}\biggl\{
K ( k_0 ) \cdot K(k_0) \cdot \cosh\eta_0(1 - \cosh\eta_0) [ 16\cosh^2\eta_0 - 13]
+  2  K ( k_0 ) \cdot E(k_0)  [ 16\cosh^4\eta_0 -13\cosh^2\eta_0 + 3 ]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
-~E(k_0) \cdot E(k_0) \cdot \cosh\eta_0 (1 + \cosh\eta_0) [ 3 +16\cosh^2\eta_0 ] 
\biggr\} \, .
</math>
  </td>
</tr>
</table>