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III. On the Motion of a Sphere in a Viscous Liquid. 



By A. B. BASSET, M.A. 

 Communicated by Lord RAYLEIGH, D.C.L., Sec. U.S. 



Received November 10, Read November 24, 1887. 



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,t 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 PiOTROWSM.J 



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 solu- 

 tions of certain problems of two-dimensional motion have been given by STEA.RN. 

 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. 



" ' Crelle, Jonrn. Math.,' vol. 73, p. 31. 

 t ' Crelle, Journ. Math.,' vol. 81, p. 62. 

 + ' Wissenschaftl. Abhandl.,' vol. 1, p. 17J. 

 ' Quart. Journ. Math., 1 vol. 17, p. 90. 



. -J 28.5.88 



