1. The hydrodynamical problem of finding the waves or oscillations on a gravitating mass of liquid which, when undisturbed, is rotating as if rigid with finite angular velocity, in the form of an ellipsoid or spheroid, was first successfully attacked by M. Poincaré in 1885. In his important memoir, “Sur l’Équilibre d’une Masse Fluide animée dun Mouvement de Rotation,” Poincare has (§ 13) obtained the differential equations for the oscillations of rotating liquid, and shown that, by a transformation of projection, the determination of the oscillations of any particular period is reducible to finding a suitable solution of Laplace’s equation. He then applies Lamé’s functions to the case of the ellipsoid, showing that the differential equations are satisfied by a series of Lamé’s functions referred to a certain auxiliary ellipsoid, the boundary-conditions, however, involving ellipsoidal harmonics, referred to both the auxiliary and actual fluid ellipsoid. At the same time, Poincare’s analysis does not appear to admit of any definite conclusions being formed as to the nature and frequencies of the various periodic free waves. The present paper contains an application of Poincaré’s methods to the simpler case when the fluid ellipsoid is one of revolution (Maclaurin’s spheroid). The solution is effected by the use of the ordinary tesseral or zonal harmonics applicable to the fluid spheroid and to the auxiliary spheroid required in solving the differential equation. The problem is thus freed from the difficulties attending the use of Lamé’s functions, and is further simplified by the fact that each independent solution contains harmonics of only one particular degree and rank.