470 MR. W. M. HICKS ON THE MOTION OF TWO SPHERES IN A FLUID. 
consider the case where one sphere is inside the other. An approximation to the 
value of the coefficients of vf and v 1 v. 2 is given in § 15. It is remarkable that in the 
case of two cylinders the coefficients of the terms in u 2 , v 2 are equal, while those of 
u l u 2 and v 1 v 2 are equal and opposite. But this is due to the fact that in a cylinder 
the image of a doublet (or a source) is a single doublet, whatever be the direction of 
the axis of the original doublet. 
9. If S S : be in a line through the centre the infinite trail of images of § 2 cuts 
out, and we are left with an image source and sink, and a line sink between them, 
supposing S to be outside S x . Let now S and Sj approach together and become a 
doublet whose strength is p. Then we shall get a single doublet as its image whose 
, aa T S'S' 
strength = — L. 
& b 
fi/r 
— - as in the former case. This we might have deduced at 
bbj 0 
once from the case of the external doublet in § 4, considered as the image of its 
image. 
Fisr. 5, 
If we proceed to find the kinetic energy, as in the previous case, we must clearly be 
led to the same form for the result, viz.: when A is moving with a velocity u from B 
2T = pi 1 M 2 jl + 3S:(^ 
where p., t ... . are the strengths of the doublets inside A alone. But in this case the 
relations between the p, p, cr are given by the equations (a, b being the radii of sphere) 
p«- 
c + p n - 
p«-l 
p»=. 
C + o-„ = 
b~ 
C + pn- 
whence 
a~ 
pn = 
¥ 
p,ip„~\ ■ 
c + p,i-1 
C" — 0" . « 
c 
Pi,~\ —7/U_i + u~ — 0 
