# 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 $~(\varpi^', z^')$ to a toroidal-coordinate system $~(\eta^',\theta^')$ whose anchor ring cuts through the meridional plane at the cylindrical-coordinate location, $~(\varpi_a,z_a)$. This desired mapping is handled via the pair of relations,

 $~\varpi^' = \frac{\varpi_a \sinh\eta^'}{(\cosh\eta^' - \cos\theta^')} \, ,$ and $~(z^' - z_a) = \frac{\varpi_a \sin\theta^'}{(\cosh\eta^' - \cos\theta^')} \, ,$

and the corresponding expression for each differential mass element is,

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

This gives, what we will refer to as the,

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

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

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