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In §102 of a book titled, 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,

Gravitational Potential in the Thin Ring (TR) Approximation

<math>~\Phi_\mathrm{TR}(\varpi,z)</math> <math>~=</math> <math>~-\biggl[ \frac{2GM}{\pi } \biggr]\frac{K(k)}{\sqrt{(\varpi+a)^2 + z^2}}</math> <math>\mathrm{where:}~~~k \equiv \{4\varpi a/[ (\varpi+a)^2 + z^2]\}^{1 / 2}</math>

See also, §III.4, Exercise (4) in Kellogg (1929). As is reviewed in an accompanying chapter titled, 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.