Editing Appendix/Ramblings/ToroidalCoordinates (section)

===But Not Every Circle Will Do===
It is very important to appreciate that, although surfaces of constant <math>\Chi</math> (or, equivalently, surfaces of constant <math>\xi_1</math>) are always off-center circles, it is not the case that every off-center circle will prove to be a <math>\Chi= \mathrm{constant}</math> surface in the most relevant toroidal coordinate system.  To be more specific, suppose we want to evaluate the potential at some location <math>(R,0)</math> inside or outside of a uniform-density torus whose meridional cross-section is a circle of radius <math>r_c</math> and whose center is located on the <math>x</math>-axis at position <math>x_0</math>.  The equation describing the cross-sectional surface of this torus is,
<div align="center">
<math>
(R' - x_0)^2 - {z'}^2 = r_c^2 .
</math>
</div>
Dividing through by the square of a (as yet unspecified) scale length, <math>a</math>, gives,
<div align="center">
<math>
\biggl[ \chi^2 - \frac{x_0}{a} \biggr]^2 - \zeta^2 = \frac{r_c^2}{a^2} .
</math>
</div>
This dimensionless expression will only describe a <math>\Chi = \mathrm{constant}</math> surface in an MF53 toroidal coordinate system if, simultaneously,
<div align="center">
<math>
\Chi = \frac{x_0}{a} ~~~~~\mathrm{and}~~~~~ \Chi^2 - 1 = \frac{r_c^2}{a^2} .
</math>
</div>
That is, only if,
<div align="center">
<math>
a = (x_0^2 - r_c^2)^{1/2} .
</math>
</div>
But in the above discussion we were only able to associate the dimensionless argument of the special function in CT99's CCGF expansion with the "radial" coordinate of the MF53 toroidal coordinate system by setting <math>a = R</math>, that is, only by setting the scale length equal to the cylindrical coordinate value <math>R</math> ''at which the potential is to be evaluated''.  So the surface of our torus will only align with a <math>\xi_1 = \mathrm{constant}</math> surface in a toroidal coordinate system if,
<div align="center">
<math>
R = (x_0^2 - r_c^2)^{1/2} .
</math>
</div>
This is a very tight constraint that usually will not be satisfied.