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Referencing (equivalently, Version 1 of) the aboveidentified 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 and .
We have realized that if we focus, instead, on Version 2 of the expression and associate the meridionalplane coordinates of the anchor ring with the coordinates of the location where the potential is being evaluated — that is, if we set — the argument of the elliptic integral becomes,
while the integral expression for the gravitational potential becomes,






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, . As result, the integral over the angular coordinate, , 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 uniformdensity, 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).