1044 
MR. F. W. DYSON ON THE POTENTIAL OP AN ANCHOR RING. 
Section I. 
§ 1. To find solutions of Laplace’s Equation aiyplicable to space inside an anclioi 
ring. 
z 
Let O be the centre of the ring, OZ its axis. 
Let any section through OZ cut the ring in the circle whose centre is C. 
Let Q be any point in that section. 
Let OC = c, CA = a, and let the coordinates of Q, referred to CA and CB as axes, 
be X and 2 ; ; also let the polar coordinate of Q, referred to C as origin, CA as initial 
line, be E, and y, so that CQ = R and A ACQ = y. 
In cylindrical coordinates, Laplace’s equation is 
, 1 r/V , il?T , 1 
-f--. — 4 -•- = 0. 
^ dttj ^ dz^ ^ 
Writing zs = c — x, this becomes 
(PY (fiV _ 1 dV 
cLv“ dz^ c — X dx 
d}Y 
(c — xy 
We shall find solutions of this equation in descending powers of c. 
First, consider the case where V is independent of (j). 
Then 
Let 
[d^Y , d^Y\ (d^Y , d-V\ , dY 
^ \dx^ dz^) \dx- dz") dx 
V - U„ + - U„4 2 + 3 U« + 2 + 
c c 
then 
