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In &sect;102 of a book titled, [https://www.amazon.com/Theory-Potential-W-D-Macmillan/dp/0486604861/ref=sr_1_2?s=books&ie=UTF8&qid=1503444466&sr=1-2&keywords=the+theory+of+the+potential ''The Theory of the Potential'', MacMillan (1958; originally, 1930)] derives an analytic expression for the gravitational potential of a uniform, infinitesimally thin, circular "hoop" of radius, <math>~a</math>.  Throughout our related discussions, we generally will refer to this additional Key Equation from MacMillan as providing an expression for the,
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<td align="center" colspan="1"><font color="#770000">'''Gravitational Potential in the Thin Ring (TR) Approximation'''</font></td>
<td align="center" colspan="1" rowspan="2">[[File:FlatColorContoursCropped.png|220px|Contours for Thin Ring Approximation]]</td>
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{{ Math/EQ TRApproximation }}
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See also, &sect;III.4, Exercise (4) in [https://archive.org/details/foundationsofpot033485mbp Kellogg (1929)].  As is reviewed in an accompanying chapter titled, ''[[Apps/DysonWongTori#Thin_Ring_Approximation|Dyson-Wong Tori]],'' a number of research groups over the years have re-derived this "thin ring" approximation in the context of their search for effective and insightful ways to determine the gravitational potential of axisymmetric systems.