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Referencing (equivalently, Version 1 of) the above-identified integral expression for the Gravitational Potential of an Axisymmetric Mass Distribution, Trova, Huré & Hersant (2012) offer the following assessment in §6 of their paper:

"The important question we have tried to clarify concerns the possibility of converting the remaining double integral … into a line integral … this question remains open."

We also have wondered whether there is a possibility of converting the double integral in this Key Equation into a single (line) integral. This is a particularly challenging task when, as is the case with Version 1 of the expression, the integrand is couched in terms of cylindrical coordinates because the modulus of the elliptic integral is explicitly a function of both ${\varpi }^{\text{'}}$ and $z^{\text{'}}$.

We have realized that if we focus, instead, on Version 2 of the expression and associate the meridional-plane coordinates of the anchor ring with the coordinates of the location where the potential is being evaluated — that is, if we set $({\varpi }_a,z_a)=(\varpi ,z)$ — the argument of the elliptic integral becomes,

$\mu =[\frac{2}{1+\coth {\eta }^{\text{'}}}{]}^{1/2},$

while the integral expression for the gravitational potential becomes,

$\mathrm{\Phi }(\varpi ,z){|}_{\mathrm{a}\mathrm{x}\mathrm{i}\mathrm{s}\mathrm{y}\mathrm{m}}$	$=$	$-2^{3/2}G{\varpi }^2\underset{\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{f}\mathrm{i}\mathrm{g}}{\iint }\left[\frac{\sinh {\eta }^{\text{'}}}{(\cosh {\eta }^{\text{'}}-\cos {\theta }^{\text{'}}{)}^5}{]}^{1/2}\right[\frac{1}{1+\coth {\eta }^{\text{'}}}{]}^{1/2}K(\mu )\rho ({\eta }^{\text{'}},{\theta }^{\text{'}})d{\eta }^{\text{'}}d{\theta }^{\text{'}}$
	$=$	$-2^{3/2}G{\varpi }^2\underset{{\eta }_{\mathrm{m}\mathrm{a}\mathrm{x}}}{\overset{{\eta }_{\mathrm{m}\mathrm{i}\mathrm{n}}}{\int }}\frac{K(\mu )\sinh {\eta }^{\text{'}}d{\eta }^{\text{'}}}{(\sinh {\eta }^{\text{'}}+\cosh {\eta }^{\text{'}}{)}^{1/2}}\underset{{\theta }_{\mathrm{m}\mathrm{i}\mathrm{n}}(\eta )}{\overset{{\theta }_{\mathrm{m}\mathrm{a}\mathrm{x}}(\eta )}{\int }}\rho ({\eta }^{\text{'}},{\theta }^{\text{'}})\left[\frac{d{\theta }^{\text{'}}}{(\cosh {\eta }^{\text{'}}-\cos {\theta }^{\text{'}}{)}^{5/2}}\right].$

Notice that, by adopting this strategy, the argument of the elliptical integral is a function only of one coordinate — the toroidal coordinate system's radial coordinate, ${\eta }^{\text{'}}$. As result, the integral over the angular coordinate, ${\theta }^{\text{'}}$, does not involve the elliptic integral function. Then — as is shown in an accompanying chapter titled, Attempt at Simplification — if the configuration's density is constant, the integral over the angular coordinate variable can be completed analytically. Hence, the task of evaluating the gravitational potential (both inside and outside) of a uniform-density, axisymmetric configuration having any surface shape has been reduced to a problem of carrying out a single, line integration. This provides an answer to the question posed by Trova, Huré & Hersant (2012).