﻿546 Mr. W. Ellis Williams on the 



The last term is of the second order in the velocity, and when 



these are neglected -~- = when ~^- = 0, so that the motion 



is steady with respect to both fixed and moving origins ; it 

 is in fact immaterial which system we choose. When 

 these terms are not neglected this is no longer the case, but 

 if reference is made to fig. 2 it will be seen that in the 

 immediate neighbourhood of the sphere the stream-lines are 



all parallel to the axis of z, and therefore ^ is zero, and 



the second term is therefore nowhere very important, and 

 the results obtained by referring the motion to a fixed 

 origin may be expected to give a better representation of 

 the results than if the origin is taken to move along with 

 the sphere. This was in fact found to be the case, and the 

 calculations being similar for both cases only those referred 

 to a fixed origin are given below. 



The solution for a sphere in a concentric sphere gives 



<£ (r)=^+Br + Cr 2 + Dr 4 . 



Substituting in (10) we obtain 



a particular solution of this is 



and the complementary function is 



I 



-r 2 + mr 6 + nr°+p, 



where I, m, n, and p are constants to be determined by the 

 boundary conditions. 



Thus the complete solution of equation (9) is 



f = (^ + Br + (V + VA V sin 2 9+ T~(- — +BV 



+ BCr 2 -BDr 4 )+i +mr 3 + nr 5 +p]v 2 sin 2 cos 6. 

 The boundary conditions to determine /, m, n, and p are 



