Configurations polygonales en équilibre relatif

We consider the $\text{N}$-body ($\text{N=p·n}$) problem (where the bodies are submitted to their mutual attractions derived from a potential of the type $\text{V(r)=Cte/r}{}^{\mathrm{2\alpha }}$ where $\text{0⩽α<∞}$). We prove the existence of relative equilibrium configurations (denoted C.E.R.) when the ponctual bodies are at the vertices of $\text{p}$ regular polygons centered around a given mass $\text{M}$, the masses being equal on the vertices of each polygon. In the Newtonian case ($\text{α=1/2}$) we enrich the last result for values of n lower or equal than $\text{472}$.