  In this paper we prove the infinitesimal uniqueness theorem for the Newton
potential of non simply connected bodies using the singularity theory approach.
  We consider the Newtonian potentials of the domains in ${\bf R}^n$ boundaries
of which are the vanishing cycles on the level hypersurface of a holomorphic
function with isolated singularity at 0. These domains don't have to be convex,
connected or simply connected, we also don't have any dimensional restrictions.
  We consider multiparametric families of such domains in the miniversal
deformation of the original function. We show that in the parameter space any
point has a neighborhood s.t. potentials of the domains corresponding to
parameters from this neighborhood are all different as functions of the
external parameter y.
