We consider the perturbation of the velocity of an inviscid incompressible fluid rotating about an axis with uniform angular velocity Ω, due to the slow uniform motion, after an impulsive start, of a sphere of radius a along the axis with velocity − V. This problem was first considered by Taylor (4), who obtained a family of solutions for the case of steady motion of a sphere along the axis of rotation of the fluid. All the solutions satisfied the boundary condition at the surface of the sphere and also the condition that the relative velocity should vanish at infinity. It was thought possible that the indeterminacy lay in the manner in which the motion was started. However, an experiment carried out in connexion with the problem showed that if V/aΩ was less than about 0·16 a column of liquid of the same diameter as the sphere was apparently pushed along in front of it. This agreed with Proudman's observation (3) that if the perturbation was small and steady it would be two-dimensional, while in Taylor's solutions the relative motion of the fluid was not small. Taylor (5) suggested that three possibilities presented themselves. Either the motion never becomes steady, or it becomes steady but not small, or it becomes steady and two-dimensional. Of these he preferred the latter, which seems to occur in practice.