46 
MR. F. W. DYSON ON THE POTENTIAL OF AN ANCHOR RING. 
Similarly 
is a solution. 
sm n 
Jo 
cos n(f) cl(p 
\/(z^ + c® — 2ccj cos ^ + Tjj^) 
Let V stand for either of these integrals. Then 
dP + i V 
dcP dzi 
is also a solution of Laplace’s equation. 
For different values of p and q the solutions are not all independent: as it is easily 
seen that 
p cos ncj) d(p 
\/(z^ + — 2cvy cos cf) + 
satisfies a linear partial differential equation of the second order in c and 2 . 
But two independent sets of solutions are obtained by giving different values to p 
. dP\ , I d \/^ /dV 
The only cases considered in this paper are when n =0 or n = 1. That is the 
solutions of the forms 
d y 
dc) Jo 
dfj) 
(z" A — 2ctjj cos (jy A 
d\P d 
d0 
dcj dz Jo \/(z" + c- — 2cm cos 0 + ’ 
and 
cos 
♦e)'f 
cos ^ d(f) 
y/(z^ + c" — '2rm cos (f> + c") 
cos (f>ld(l) 
's/+ 0 ^ — 2c-, cos (f) + uj”) 
as tlieee cover all the cases to whicli the functions are applied in this paper. 
It is easily seen that 
cos (f) d(j) 
■UT 
f 
J 0 
1 ./ (z^ + c” — 2cijj cos (j) + m~) 
d\p p' 
r_ 
I.. /I,? 
COS 0 dcf) 
dcj .'0 + c® — 2cc;r COS (f) + 
and, consequently, 
