  An analytical method is presented for treating the problem of a uniformly
rotating, self-gravitating ring without a central body in Newtonian gravity.
The method is based on an expansion about the thin ring limit, where the
cross-section of the ring tends to a circle. The iterative scheme developed
here is applied to homogeneous rings up to the 20th order and to polytropes
with the index n=1 up to the third order. For other polytropic indices no
analytic solutions are obtainable, but one can apply the method numerically.
However, it is possible to derive a simple formula relating mass to the
integrated pressure to leading order without specifying the equation of state.
Our results are compared with those generated by highly accurate numerical
methods to test their accuracy.
