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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 $({\varpi }^{\text{'}},z^{\text{'}})$ to a toroidal-coordinate system $({\eta }^{\text{'}},{\theta }^{\text{'}})$ 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,

${\varpi }^{\text{'}}=\frac{{\varpi }_a\sinh {\eta }^{\text{'}}}{(\cosh {\eta }^{\text{'}}-\cos {\theta }^{\text{'}})},$

and

$(z^{\text{'}}-z_a)=\frac{{\varpi }_a\sin {\theta }^{\text{'}}}{(\cosh {\eta }^{\text{'}}-\cos {\theta }^{\text{'}})},$

and the corresponding expression for each differential mass element is,

$\delta M({\eta }^{\text{'}},{\theta }^{\text{'}})=\left[\frac{2\pi {\varpi }_a^3\sinh {\eta }^{\text{'}}}{(\cosh {\eta }^{\text{'}}-\cos {\theta }^{\text{'}}{)}^3}\right]\rho ({\eta }^{\text{'}},{\theta }^{\text{'}})d{\eta }^{\text{'}}d{\theta }^{\text{'}}$.

This gives, what we will refer to as the,

Gravitational Potential of an Axisymmetric Mass Distribution (Version 2)
$\mathrm{\Phi }(\varpi ,z){|}_{\mathrm{a}\mathrm{x}\mathrm{i}\mathrm{s}\mathrm{y}\mathrm{m}}$	$=$	$-\frac{G}{\pi }\underset{\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{f}\mathrm{i}\mathrm{g}}{\iint }\left[\frac{\mu }{{\varpi }^{1/2}}\right][\frac{{\varpi }_a\sinh {\eta }^{\text{'}}}{(\cosh {\eta }^{\text{'}}-\cos {\theta }^{\text{'}})}{]}^{-1/2}K(\mu )[\frac{2\pi {\varpi }_a^3\sinh {\eta }^{\text{'}}}{(\cosh {\eta }^{\text{'}}-\cos {\theta }^{\text{'}}{)}^3}]\rho ({\eta }^{\text{'}},{\theta }^{\text{'}})d{\eta }^{\text{'}}d{\theta }^{\text{'}}$
	$=$	$-2G\left(\frac{{\varpi }_a^5}{\varpi }{)}^{1/2}\underset{\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{f}\mathrm{i}\mathrm{g}}{\iint }\right[\frac{\sinh {\eta }^{\text{'}}}{(\cosh {\eta }^{\text{'}}-\cos {\theta }^{\text{'}}{)}^5}{]}^{1/2}\mu K(\mu )\rho ({\eta }^{\text{'}},{\theta }^{\text{'}})d{\eta }^{\text{'}}d{\theta }^{\text{'}},$

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

${\mu }^2$

$=$

$\frac{4\varpi {\varpi }_a\sinh {\eta }^{\text{'}}}{(\cosh {\eta }^{\text{'}}-\cos {\theta }^{\text{'}})}\left\{\right[\varpi +\frac{{\varpi }_a\sinh {\eta }^{\text{'}}}{(\cosh {\eta }^{\text{'}}-\cos {\theta }^{\text{'}})}{]}^2+[z-z_a-\frac{{\varpi }_a\sin {\theta }^{\text{'}}}{(\cosh {\eta }^{\text{'}}-\cos {\theta }^{\text{'}})}{]}^2{\}}^{-1}.$