MR. F. W. DYSON ON THE POTENTIAL OF AN ANCHOR RING. 1057 
Substituting for cos a, c/r^, and for 
aP \ l 'pt 13,, cos n X + + /3« A*+i cos x) |, 
and retaining only for the present the terms of the second degree in . 
we find that at P 
-\T _ I •2„2„ i 1 -^ (I3n“ O O \ \ v*/5)i/9«+3 
V0 = 47r'«“C <i - - Y ( - o- A. A;+i j + ^ 
G,.- 
•> 
Therefore at Q 
V„ = .W2 (f - .ft,ft.) £ 
— 27ra®c Y(A.A:+i) £ I - 
(I r 
' 0 \/(r' + r — 2cr sin 9 cos 0) 
\_Su2yi'a, p. 59.] 
Expanding these integrals, we find that at a point R, x outside the ring, 
V„ 
:7rcr 
— o-A^A^+i) (log - + 
, , 8c 1 \ 
E ^ i; ) 
-^-r cos X 
•2 log - 1 
° E 
-\- cr % /3, 1 /3„^ 1 £ COS X + + 
Adding this to the values already obtained for the potential of a ring whose cross- 
section is an exact circle of radius a, and to that for a distribution of surface density 
a 1 /8„ cos nx on such a ring, we find that 
_^ 0 
2irci“ 
8c 
logjr + 
1 1 
‘“Sr-G 
t cos Y o” I c; 
4 
s \r“‘*x + 
cos (ft 4- 1) X 
11 
+ 
u cos {n -t 1) % 
Mj ft -f 1 
a cos (ft — 1) X 
E/ ft {n — 1) 
-f . . . 
+ s(%*-a-ftft,.)nog'^ + 
8o 
*"« K - Gi 
E. 
O 1 Sc 1 
2 log — — 1 
° E 
c 
. \ 
COSXy- 
A o-S/3«y8;,+i yp,cosx + -^ -1- 
G T 
(If' 
ilDCCCXCIlI. — A. 
