Template:LSU CT99CommonTheme2

Suppose we rewrite (Version 1 of) the above-highlighted Key integral expression such that the (primed) coordinate location of each mass element is mapped from cylindrical coordinates <math>~(\varpi^', z^')</math> to a toroidal-coordinate system <math>~(\eta^',\theta^')</math> whose anchor ring cuts through the meridional plane at the cylindrical-coordinate location, <math>~(\varpi_a,z_a)</math>. This desired mapping is handled via the pair of relations,

<math>~\varpi^' = \frac{\varpi_a \sinh\eta^'}{(\cosh\eta^' - \cos\theta^')} \, ,</math>

and

<math>~(z^' - z_a) = \frac{\varpi_a \sin\theta^'}{(\cosh\eta^' - \cos\theta^')} \, ,</math>

and the corresponding expression for each differential mass element is,

<math>~\delta M(\eta^',\theta^') = \biggl[\frac{2\pi \varpi_a^3 \sinh\eta^'}{(\cosh\eta^' - \cos\theta^')^3} \biggr] \rho(\eta^', \theta^') d\eta^' d\theta^'</math>.

This gives, what we will refer to as the,

Gravitational Potential of an Axisymmetric Mass Distribution (Version 2)
<math>~\Phi(\varpi,z)\biggr|_\mathrm{axisym}</math>	<math>~=</math>	<math>~ - \frac{G}{\pi} \iint\limits_\mathrm{config} \biggl[ \frac{\mu}{\varpi^{1 / 2}} \biggr] \biggl[ \frac{\varpi_a \sinh\eta^'}{(\cosh\eta^' - \cos\theta^')} \biggr]^{- 1 / 2}K(\mu) \biggl[\frac{2\pi \varpi_a^3 \sinh\eta^'}{(\cosh\eta^' - \cos\theta^')^3} \biggr] \rho(\eta^', \theta^') d\eta^' d\theta^' </math>
	<math>~=</math>	<math>~ - 2G \biggl( \frac{\varpi_a^5}{\varpi} \biggr)^{1 / 2} \iint\limits_\mathrm{config} \biggl[\frac{ \sinh\eta^'}{(\cosh\eta^' - \cos\theta^')^5} \biggr]^{1 / 2} \mu K(\mu) \rho(\eta^', \theta^') d\eta^' d\theta^' \, ,</math>

where the square of the argument of the elliptic integral is,

<math>~\mu^2</math>

<math>~=</math>

<math>~ \frac{ 4\varpi \varpi_a \sinh\eta^'}{(\cosh\eta^' - \cos\theta^')}\biggl\{ \biggl[ \varpi+ \frac{\varpi_a \sinh\eta^'}{(\cosh\eta^' - \cos\theta^')} \biggr]^2 + \biggl[z- z_a - \frac{\varpi_a \sin\theta^'}{(\cosh\eta^' - \cos\theta^')} \biggr]^2 \biggr\}^{-1} \, . </math>