﻿Motion of a Sphere in a Viscous Fluid. 543 



determine <£i(')> anc ^ ^ ne solution will contain four constants 

 -which may be adjusted so as to satisfy four boundary 

 conditions such as (5). 



If desired the process of approximation may be carried 

 still further so as to include terms o£ the third degree in the 

 velocity. To do this we must first substitute the value of 

 <f>i(r) sin 2 6 cos 6 in the operator 3 in the first term on the 

 right-hand side of (9). If now <j>i{r) sin 2 6 cos 6 be sub- 

 stituted in the last term SDi/^ which was previously neglected, 

 the resulting expression contains terms in sin 2 6, sin 2 6 cos 0, 

 and sin 4 6 cos 6, and hence by adding to the previous value 

 of ty a term of the form ^> 2 {r) [a sin 2 6+b sin 2 cos 6-\- 

 c* sin 4 6 cos 0~\ and suitably choosing the values of the nume- 

 rical constants a, b, c, an equation giving </> 2 (»') in terms of 

 <£ (r) and 4>±(r) may be obtained, and the approximation is 

 thus corrected so as to include the terms of the third power 

 in the velocity. 



In applying the above solutions to actual cases for com- 

 parison with experimental results, certain difficulties are met 

 with which are connected with the validity of Stokes's 

 solution and are perhaps best dealt with here. 



Reverting to the case of a sphere moving in an unbounded 

 fluid, Stokes's solution may be written 



3 „ . /„ 1 a 



t= J V «,(l-^)sin 2 0. 



At a great distance from the sphere this becomes 



yj/ = j Yar sin 2 0, 



and the corresponding velocities will be 



j, 1 "d^ 3 a a 



e. — I M *^W 



fsin a or 4 /• 



The velocity in the direction of motion of the sphere is 



3 Ya 



~ — (cos 2 <9-f-isin 2 0), 



1 V 



The total momentum' of the fluid in the direction of motion 

 of the sphere is obtained by integrating this over the whole 



