Motion in the plane of symmetry in the N-centres problem.

The stability of circular satellite orbits in the symmetry plane of the problem of N = 2n centers (an even number of centers) is examined in the case when all the gravitating fixed centers are of the same mass and separated from each other by the same distance. An analytical formula is obtained for the critical radius of the circular orbit. In addition, the periodicity of noncircular finite motions of a satellite in the symmetry plane is considered for small and large distances from the origin of the reference frame.