458 ME. W. M. HICKS ON THE MOTION OF TWO SPHERES IN A FLUID. 
which holds good for points where r<b, whence when r—a{<b) the flow into the 
sphere at any point (6) is 
^00 net' 1 - 1 
y,i+1 
Expand the potential due to the sources, &c., inside the spheres in a series of 
spherical harmonics 
V=S;^Y,(r>a) 
Hence the flow out of the sphere, for points just outside, is 
- 2 0 > + l)^Y„ 
and this must be equal to the other, whence 
Y„= 
■Mif 1 p ’ audY ° =0 
and 
V =-2 
n a 
2n+l 
1 oi+ 1 (br) n+l * 
a? 
v2n+l 1 f( 2;i+\ 
^ ( 6 > 0 - 
+i 
Consider 
X= 
yY 2 — 2Ar cos 6 + A 
n +1 (&r) i! 
_ /^oo A -p /A<Cc 
2—C —Ti ( r>a 
the potential for a source // at a point on 0 S inside the sphere at a distance X from 
the centre. Then 
CL 
Comparing this with the expression for Y, we see that if we make X=— and 
fj.' = -X source 
tt a ^ao A“ „ 1 C K 7 u! 
,+h 
d\ 
b yC 2 — 2 Ar cos 6 + A 2 a J ov r2 ~ 2Ar cos 6 + A 2 
?. c., Y is the potential of a source at the distance y from O whose magnitude is equal 
