This paper is devoted chiefly to the consideration of the surface oscillations of water contained in a vessel in the shape of a circular cylinder with its axis vertical, when the motion is slightly disturbed from a uniform rotation about the axis of the vessel. The work was undertaken with the hope of finding some indication of the effect of the depth of the water in the vessel on the period of the surface waves, and for the purpose a vessel of circular cross-section was naturally chosen. It is shown that a slight change of shape does not affect the periods of the oscillations. The solution of the corresponding problem when the surface oscillations take the form of “long waves” or “tidal waves” is well known, and the present paper deals only with “short waves,” for which the horizontal velocity is not the same at all depths.