The stability of a drop of incompressible fluid held together by the action of surface tension and made to rotate rigidly about an axis is determined, the effect of gravity being neglected. Two distinct problems are investigated. In the first is considered an isolated drop in the form of a surface of revolution and the manner in which its stability changes with angular speed is investigated. At zero angular speed, where the drop is spherical, infinitesimal disturbances are shown to be stable and beyond a certain critical angular speed a new linear series of equilibrium forms emerges, the original series becoming unstable.