Solid Bodies through Viscous Liquid. 705 



Of the four integrals in (31) , 



the first =if inYne int dn = ±V; 



Also the second and third together give 



l ' dn =J^2 Y ' 



9(1 + 

 4a 



^ v/ (2 ^ - f V n rt(P*dn, 



and this is the only part which could present any difficulty. 

 We have, however, already considered this integral in con- 

 nexion with the motion of a plane and its value is expressed 

 by (16). Thus 



1 ° ^ + 2a 2 V + 2a7ri ]_„ ^ -t)* J " l ° j 



2 d* 



The first term 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, farther, there has been no acceleration 

 for a Jong time, the third term also vanishes, and we obtain 

 the result appropriate to a uniform motion 



F = _ o = — birapv V = — biriia V , 



la 1 



as in (1). The general result (32) is that of Boussinesq and 

 Basset. 



As an example of (32), we may suppose (as formerly for 

 the plane) that Y(t) = from — cc to ; Y(t) =ht from to 

 T i ; Y(t) = hr x , when t > r x . Then if t < t 1? 



F= -* M [i + l? + -5Sj ! • • • (33) 



and when t > T l5 



When £ is very great (34) reduces to its first term. 



The more difficult problem of a sphere falling under the 

 influence of gravity has been solved by Boggio (loe. cit.). 

 In the case where the liquid and sphere are initially at rest, 

 the solution is comparatively simple : but the analytical form 



