144 ON THE MOTION OF SOLIDS THROUGH A LIQUID. [CHAP. V. 



ing in consequence of this motion are easily seen to be - v cos %, 

 and T sin ^, respectively, so that we must write for -, 



, d f 3a 8 \ , .a b . 2 



& . ii -| r ^ J cos % + V 4 ^ v sm X cos % + ^ c -&amp;gt; 

 ctt . c / c 



where terms which obviously contribute nothing to the integral 

 (50) have been omitted. Again 



= ... + 1 ^ 5 4 iwsin 2 %cos^ + &c (51), 



c 



similar omissions being made. Now 



I sin^cos 2 %cZ% = |, I sm 3 ^cos 2 x^ = T ^, 



JO J 



so that we have finally for the resultant fluid pressure on B in the 

 direction AB, 



d f 3a 3 \ 67rpa 3 6 3 ,_. 



- f + - U --T-UV (02). 



This result is correct to the order j- inclusive. Since 



dc , 



7t =-(+&amp;gt;, 



(52) may also be written 



We proceed to examine some particular cases, keeping only the 

 most important terms in each. 



(a) Let b a, v = u, so that the motion is symmetrical with 

 respect to the plane bisecting AB at right angles, and is the same 

 as if this plane formed a rigid boundary to the fluid on either 

 side of it. We have thus the solution of the case where a sphere 

 moves directly towards or away from a fixed plane wall. The force 

 repelling the sphere from the wall is* 



* Stokes, Cnmb. Trans. Vol. Tin. (1813). 



