Relative equilibrium of polygon type

We consider the N-body (N = p . n) problem (where the bodies are submitted to their mutual attractions derived from a potential of the type V(r) = Cte/r(2 alpha) where 0 less than or equal to alpha < infinity). We prove the existence of relative equilibrium configurations (denoted C.E.R.) when the ponctual bodies are at the vertices of p regular polygons centered around a given mass ill, the masses bring equal on the vertices of each polygon. In the Newtonian case (alpha = 1/2) we enrich the last result for values of n lower or equal than 472. (C) 2001 Academie des sciences/Editions scientifiques et medicales Elsevier SAS.