  We prove for generalisations of quasi-homogeneous $n$-body problems with
center of mass zero and $n$-body problems in spaces of negative constant
Gaussian curvature that if the masses and rotation are fixed, there exists, for
every order of the masses, at most one equivalence class of relative equilibria
for which the point masses lie on a circle, as well as that there exists, for
every order of the masses, at most one equivalence class of relative equilibria
for which all but one of the point masses lie on a circle and rotate around the
remaining point mass. The method of proof is a generalised version of a proof
by J.M. Cors, G.R. Hall and G.E. Roberts on the uniqueness of co-circular
central configurations for power-law potentials.
