Agraïments: The authors would like to thank the two reviewers for their helpful comments concerning this work. GR was supported by a grant from the National Science Foundation (DMS-1211675).

For a class of potential functions including those used for the planar n-body and n-vortex problems, we investigate co-circular central configurations whose center of mass coincides with the center of the circle containing the bodies. Useful equations are derived that completely describe the problem. Using a topological approach, it is shown that for any choice of positive masses (or circulations), if such a central configuration exists, then it is unique. It quickly follows that if the masses are all equal, then the only solution is the regular n-gon. For the planar n-vortex problem and any choice of the vorticities, we show that the only possible cocircular central configuration with center of vorticity at the center of the circle is the regular n-gon with equal vorticities.