I'm not sure what you mean by a ring of wires. If you mean the rings are made up of wires perpendicular to the disk (like my rods) then you can avoid divisions by zero like I did, by passing the test mass directly beteen 2 wires. If you mean the rings are zero-thickness wires as continuous circles, you will run into an infinity (probably with elliptic integrals) and have to decide how to dodge it. Why not send me an email with your equations and I'll see if I can help.

In the meantime you can look at arxiv:0803.0556 v1 by Feng and Gallo. They have an approach using elliptic integrals, and they show how to easily compute them, and (not so easily) fix the singularities. I prefer a discrete number of rods for each ring. I've gone as high as 720/ring, but usually I only use 180 and with symmetry that's one per degree, and get great accuracy, without double precision. Also I only need about 40 to 80 rings. I don't need any fancy math methods, just stick to sines and cosines, and the new computers handle it all just fine.