On Vibrations of a Heavy Material Point in a Fixed Ellipsoidal Bowl

The periodic motions of a material point are studied on the assumption that, throughout the motion, the point remains on a fixed absolutely smooth surface (in an ellipsoidal bowl), which is part of the surface of a triaxial ellipsoid. The motion occurs in a uniform field of gravity, and the largest axis of the ellipsoid is directed along the vertical.
Cases are considered where the motion of the point occurs along one of the principal sections of the surface in the neighborhood of a stable equilibrium at the lowest point of the bowl. An analytical representation of the corresponding periodic motions is obtained up to terms of degree five inclusive with respect to the magnitude of perturbation of the point from the equilibrium. The stability of these periodic motions is investigated.