In some recent papers the author has drawn attention to certain general properties of rotating fluids, especially to the differences which may be expected between two- and three-dimensional motion. Unfortunately, mathematical difficulties have so far prevented the solution of any three-dimensional problem in a rotationally moving fluid from being obtained, except in one case, when the motion is very slow. In this case, Prof. Proudman has shown how it is possible to approximate to the solution of the problem of the slow motion of a sphere in a rotating fluid. Even in this case the analysis is very complicated. There seems little prospect of obtaining a more general solution of the problem when the inertia terms which Proudman neglected are taken into account. On the other hand, it is shown in the following pages that a solution can be obtained in the case when the sphere moves steadily along the axis of rotation of the fluid. The limitation imposed by considering only a steady motion necessarily excludes the case considered by Proudman, for all slow steady motions of a rotating fluid are two-dimensional.