MR. F. W. DYSON ON THE POTENTIAL OF AN ANCHOR RING. 
47 
are solutions of the equation 
d'yj/' 
^ 
1 d-^fr 
w dm 
= 0. 
For putting xfj = zucf), this equation becomes 
d^cf) _j_ 1 
dz^ dm^ ‘ m dm m’" 
^3$ = 0, 
that is, <I) cos ^ is a solution of Laplace’s equation. 
§ 2. Expansion of the Functions .—Let the plane through the axis of the ring and 
a point P, cut the ring in the two circles whose centres are C and C^. 
2 
Let 
CP = R 
and the angle 
Also, for convenience, put 
and 
CiP = Pi ; OC = c ; 
OOP - X- 
2 . 
When P is less than c the above integrals may be expanded in ascending powers 
of P/c or s. 
_ d^ _ 
0 v/{r^ + — 2cr sin 0 cos ^} 
. 1233 Pd /, 4 Ei 
-4--loP’ — 
^ 2242 up 
2 
1.2 
[Cayley, ‘Ell. Func.,’ p. 54.J 
