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



of a very simple case in which the forces acting on the solids can 

 be readily calculated by the direct method. 



125. Let us suppose that we have two spheres in motion in 

 the line joining their centres A, B. Let u be the velocity of the 

 first in the direction AB, v that of the second in the direction BA. 

 Further, P being any point of the fluid, let 



r, PB = s, 



also let a, b be the radii of the spheres and c the distance AB of 

 their centres. If the sphere B were absent, and its place occupied 

 by fluid, the velocity-potential fa due to the sphere A alone would 

 be, by Art. 102, 



To find the value of $ 1 in the neighbourhood of B we have 

 r 2 = c 2 2c5 cos % -f s 2 , 



r cos 6 = c s cos y, 

 so that 



&amp;lt; ua* 



, rt 



1 --- cos y 1 2 - cos v + - a 

 A 



3 cos 2 v 1 * 

 -; -*- -+&O. 



This gives at the surface of B t 



dfa 1 ?/ 

 - 



The relation which actually holds at the surface of B, viz. 



dA 



-f = V COS V, 



ote 



* We recognise the coefficients of 2 -, 3 -: 2 , &c., within the brackets, as the 



C C 



zonal harmonics of orders 1, 2, &c&amp;lt; respectively. In fact, remembering that ^ 



is the potential at the point P due to a small magnet of unit moment placed at A 

 with its axis pointing in the direction A B, we readily find from the definition of 

 the aforesaid zonal harmonics Q lt Q%, &c., that 



