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 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
sa                       ^    '
r hus, if                           V=\    Vnein*dn,   ........................... (30)
J o
dn ....... (31)
^    '
o               (\        sa/         sa         sa
Of the four integrals in (31),
the first     = \      in Vn eint dn = i V ; '
the fourth = ^ f Vneint dn = _9z/, V. Ait ,' d                            JjOj"
Also t-he Hccoud and third together give
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 ttnu 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.
32d with Boggio's work.