A SPHERE IN A VISCOUS LIQUID. 47 



and, therefore, 



The conditions that F (0) = F (0) = require that C = D = ; whence 



,\ 



' 



Va/3 J , 



= ~ + 



Also the preceding value of F() satisfies the conditions that F()e~*', and F'()e~*' 

 should each vanish when a = oo ; whence all the conditions are satisfied, and we 

 finally obtain 



r/3 8 a \ / \ . 



\.(T + 3rta + 2 ) ex p- (- ) d " 



3Va8ingf'/a \ f (r - a + )' 



" 



The first integral can be evaluated ; in the second put r a + a = 2u^/(vt) and 

 we obtain 



3Va sin' g r f 



V 77 " 



_ r + a | -^dM. (13) 



4. When t = the second integral vanishes, whence the initial value of /> is 



Vo 8 sin 3 

 *--' 



which is the known value of t/ in the case of a frictionless liquid, as ought to be the 

 case. 



When t is very large, we may put t = oo in the lower limit of the second integral, 

 which then 



whence 



