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 .