The periodic solution of a rigid body in a central newtonian field

The equations of motion of a rigid body about a fixed point in a central Newtonian field is reduced to the equation of plane motion under the action of potential and gyroscopic forces, using the isothermal coordinates on the inertia ellipsoid.The construction of periodic solutions nearby equilibrium points, by using the Liapunov theorem of holomorphic integral are obtained and the necessary and sufficient conditions for the stability of the system are given.