  In this paper, we describe an analytical method for treating uniformly
rotating homogeneous rings without a central body in Newtonian gravity. We
employ series expansions about the thin ring limit and use the fact that in
this limit the cross-section of the ring tends to a circle. The coefficients
can in principle be determined up to an arbitrary order. Results are presented
here to the 20th order and compared with numerical results.

Astrophysics;General Relativity and Quantum Cosmology