1911] ON THE MOTION OF SOLID BODIES THEOUGH VISCOUS LIQUID 35 § 5. The problem of the sphere moving with arbitrary velocity through a viscous fluid is of course more difificult than the corresponding problem of the plane, lamina, but 'it has been satisfactorily solved by Boussinesq* and by Basse b f. The easiest road to the result is by the application of Fourier's theorem to the periodic solution investigated by Stokes. If the velocity of the .sphere at time t be V- Vneint, a the radius, M' the mass of the liquid displaced by the sphere, and s = vW2l/)> v being as before the kinematic viscosity, Stokes finds as the total force at time t (29) sa ^ ' r°° hus, if V=\ Vnein*dn, ........................... (30) J o dn ....... (31) ^ ' o (\ sa/ sa sa Of the four integrals in (31), f°° the first = \ in Vn eint dn = i V ; ' .o the fourth = ^ f Vneint dn = _9z/, V. Ait ,' d JjOj" Also t-he Hccoud and third together give 4>a and fclii.s is fcho only part which could present any difficulty. We have, however, already conaiderecl this integral in connexion with the motion of a piano antl its value ia expressed by (10). Thus ' (32) ( ^ Tho firafc tt»nu depends upon the inertia of the fluid, and is the same as would be obtained by ordinary hydrodynamics when v = 0. If there is no acceleration at the moment, this term vanishes. If, further, there has been no acceleration for a long time, the third term also vanishes, and we obtain the result appropriate to a uniform motion. an in (i). The general result (32) is that of Bouseinesq and Basset. * 6'. R. t. c. p. 985 (1883) ; Theorie Analytiftue de la Chaleur, t. n. Paris, 1903. t Phil, Tram. 1888 ; Hydrodynamics', Vol. rt. chap. xxn. 1888. 3—2d with Boggio's work.