704 Lord Rayleigh on the Motion of 



vanishes with £, the integral may be written 



9< 



'** Jo V \Z' kJ 



=^■-1+4-^! «" at ^. • • • ( 25 ) 



in which t' = t . 4:p 2 v/a 2 . When t, or f, is great, 



|Jr^=^( i -.i+---) ; • • (26) 



so that 



Ultimately, when t is very great, 



v =?v/(i) w 



§ 5. The problem of the sphere moving with arbitrary 

 velocity through a viscous fluid is of course more difficult 

 than the corresponding problem of the plane lamina, but it 

 has been satisfactorily solved by Boussinesq * and by Basset 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=Y n e int , a the 

 radius, M' the mass of the liquid displaced by the sphere, 

 and s=^/(n/2v), v being as before the kinematic viscosity, 

 Stokes finds as the total force at time t 



F=-M'V„„{g + A)i + A( 1+ I)j,,„, . (29) 



Thus, if 



;f-v. 



Jo 



dn, (30) 



F= -M' f °° V, n**f(i + /-)i + /-(l+ -)]■ dn, 

 J L\2 Asa J Asa \ saj J 



.... (31) 



* C. R. t. 100. p. 935 (1885) ; Theorie Analytique de la Chaleur, t. ii. 

 Paris, 1903. 



f Phil. Trans. 1888 ; Hydrodynamics, 11. ch. xxii. 1888. 



