A SPHERE IN A VISCOUS LIQUID. 59 



where F (0) = fi. Putting (t) for the last two terms, and integrating, we obtain 



) 0()dtt, ........ (51) 



- x <'-">0(u)<ft< ....... (52) 



For a first approximation we have 



a. = fie-", = - \flf~" = F' (). 



Whence, if $, x , and / have the same meanings as in 8, a second approximation 

 gives 



(53) 



x <'-><(u)du ........ (54) 



V' * * 



And a third approximation gives 



. N ^i/ rf [i f 



(0 + ., <** f**~**/rdr 



aay/TT t/<J J 



- X(i - 1 *()<fc. (56) 



Now we have shown in 8 that 



Also 



f. -'#(-) *- *()( + i)-f 



= J' rfr I* -*-* 



And the value of the last integral in (56) is given by (38) ; whence 



I 2 



