1. The first problem relating to the motion of a solid body in a viscous liquid which was successfully attacked was that of a sphere, the solution of which was given by Professor Stokes in 1850, in his memoir “On the Effect of the Internal Friction of Fluids on Pendulums,” ‘Cambridge Phil. Soc. Trans.,’ vol. 9, in the following cases: (i.) when the sphere is performing small oscillations along a straight line; (ii.) when the sphere is constrained to move with uniform velocity in a straight line; (iii.) when the sphere is surrounded by an infinite liquid and constrained to rotate with uniform angular velocity about a fixed diameter: it being supposed, in the last two cases, that sufficient time has elapsed for the motion to have become steady. In the same memoir he also discusses the motion of a cylinder and a disc. The same class of problems has also been considered by Meyer and Oberbeck, the latter of whom has obtained the solution in the case of the steady motion of an ellipsoid, which moves parallel to any one of its principal axes with uniform velocity. The torsional oscillations about a fixed diameter, of a sphere which is either filled with liquid or is surrounded by an infinite liquid when slipping takes place at the surface of the sphere, forms the subject of a joint memoir by Helmholtz and Piotrowski. Very little appears to have been effected with regard to the solution of problems in which a viscous liquid is set in motion in any given manner and then left to itself. The solution, when the liquid is bounded by a plane which moves parallel to itself, is given by Professor Stokes at the end of his memoir referred to above; and the solutions of certain problems of two-dimensional motion have been given by Stearn. In the present paper I propose to obtain the solution for a sphere moving in a viscous liquid in the following cases:—(i.) when the sphere is moving in a straight line under the action of a constant force, such as gravity; (ii.) when the sphere is surrounded by viscous liquid and is set in rotation about a fixed diameter and then left to itself.