# Template:LSU CT99CommonTheme3

Now, beginning with Version 2 of our expression for the Gravitational Potential of Axisymmetric Mass Distributions, let's also map the (unprimed) cylindrical coordinate pair, $~(\varpi, z)$, to the same (but, unprimed) toroidal coordinate system, $~(\eta,\theta)$, and place the toroidal coordinate system's anchor ring in the equatorial plane of the cylindrical-coordinate system such that, $~(\varpi_a,z_a) = (a,0)$. This gives, what we will refer to as the,
 Gravitational Potential of an Axisymmetric Mass Distribution (Version 3) $~\Phi(\varpi,z)\biggr|_\mathrm{axisym}$ $~=$ $~ - 2G a^2 \biggl[ \frac{(\cosh\eta - \cos\theta)}{\sinh\eta} \biggr]^{1 / 2} \iint\limits_\mathrm{config} \biggl[\frac{ \sinh\eta^'}{(\cosh\eta^' - \cos\theta^')^5} \biggr]^{1 / 2} \mu K(\mu) \rho(\eta^', \theta^') d\eta^' d\theta^' \, ,$  $\mathrm{where:}~~~\mu^2 = \frac{ 2 \sinh\eta^'\cdot \sinh\eta}{ \sinh\eta^'\cdot \sinh\eta + \cosh\eta^'\cdot\cosh\eta -\cos(\theta^' - \theta) } \, .$