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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)
$\mathrm{\Phi }(\varpi ,z){|}_{\mathrm{a}\mathrm{x}\mathrm{i}\mathrm{s}\mathrm{y}\mathrm{m}}$	$=$	$-2G{a}^2\left[\frac{(\cosh \eta -\cos \theta )}{\sinh \eta }{]}^{1/2}\underset{\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{f}\mathrm{i}\mathrm{g}}{\iint }\right[\frac{\sinh {\eta }^{\text{'}}}{(\cosh {\eta }^{\text{'}}-\cos {\theta }^{\text{'}}{)}^5}{]}^{1/2}\mu K(\mu )\rho ({\eta }^{\text{'}},{\theta }^{\text{'}})d{\eta }^{\text{'}}d{\theta }^{\text{'}},$
$\mathrm{w}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{e}\mathrm{:}{\mu }^2=\frac{2\sinh {\eta }^{\text{'}}\cdot \sinh \eta }{\sinh {\eta }^{\text{'}}\cdot \sinh \eta +\cosh {\eta }^{\text{'}}\cdot \cosh \eta -\cos ({\theta }^{\text{'}}-\theta )}.$