I. The oscillations of a rotating ellipsoidal shell containing fluid

The paper contains an application of the analysis used by M. Poincaré, in his memoir “Sur l’équilibre d’une masse fluide animée d'un mouvement de rotation,” to the determination of the free oscillations of a system consisting of a fluid mass contained within a rigid ellipsoidal envelope, rotating about one of its principal axes. It is found that, when such a system is oscillating in one of its fundamental modes, the disturbances of the fluid are all expressible by means of Lamé functions, the functions involved being all of the same order; and a method of obtaining the frequencies of these oscillations, similar to that used by M. Poincaré for a fluid ellipsoid with a free surface, is given.