124, 125.] TWO SPHERES. 143 



is therefore satisfied by making 



f~f,+A&amp;gt; 

 where 



. rb 3 . ita* /6 3 6 5 3 cos 2 - 1 



The condition at the surface of ^4, viz. 



d$ 



j = u cos 6, 

 ds 



is however no longer satisfied ; but it is plain from the course of 

 the above investigation that the error in the normal velocity there 



will be of the order ^ , and that if this be rectified by the addi- 



c 



tion of a properly chosen term &amp;lt;f&amp;gt; 3 to the above value of &amp;lt;, the 



etfect of this at the surface of B will be of the order ^-5-. In the 



c y 



particular cases examined below we shall suppose a and 6 both 

 small in comparison with c, and shall not take into account small 

 quantities of so high an order as that last written. We have then 

 at the surface of B 



- = v cos 

 ds 



j- = i (v + 3a ^j sin x + * f - J u 



sin % cos x + &c; 



The total effect of the fluid pressure on the sphere B evidently 

 reduces to a force in the direction AB, the amount of which is 



f* 



I p . 2-7T& sin x bdx cos ^ ............ (50), 



where ^ is to be found from (18). In calculating -^ we must re 



member, as in Art. 102, that the origin B of the polar co-ordi 

 nates s, % is itself in motion with velocity v in the direction BA. 

 The rates at which the values of s, % for a fixed point are increas- 



