In this paper the stability of a new class of exact symmetrical solutions in the Newtonian gravitational (n+1)-body problem is studied. This class of solution follows
from a suitable geometric distribution of the (n+1)-bodies, and initial conditions, so that
the solution is represented geometrically by an oscillating regular polygon with n sides
rotating non-uniformly about its center. The body having a mass m_0 is at the center of the
polygon, while nbodies having the same mass m are at the vertices of the polygon and
move about the central body in identical elliptic orbits. It is proved that for n=2 and for
regular polygons 3<= n<= 6 each corresponding solution is unstable for any value of the
central mass m_0. For n=>7
the solution is linearly stable and the eccentricity of the particles’ orbits if both \mu = m_0 / m > 141.477 and the eccentricity of the particles' orbits e
is sufficiently smal
