A SPHERE IN A VISCOUS LIQUID. 51 



and the value of the acceleration is 



8. It seems almost hopeless to attempt to determine the com pit te value of F from 

 the preceding equations, but, in the case of many liquids, v is a small quantity, and 

 (22) and (23) may then be solved by the method of successive approximation. For 

 a first approximation 



r=F )=/-", 

 whence 



fF'-T)rfr f , rfr . } 



-'-- 



The integral on the right hand side of (23) cannot be evaluated in finite terms, and 

 we shall denote it by < (t). Putting T = ty, we obtain 



where 



TI 1.3... (2- 1) 

 U "- 2"n! 



Now 



fi i _ , - 



;.-*^=L^ 



Therefore 



and therefore 



When t is very large we may replace (1 e~ x ')/X< by (M)~ l , and we shall obtain 



which shows that <^ (t) = where t = CD . 



Another expression for <j> (t) may be obtained in the form of a series, for 



n _' 



