MR. F. W. DYSON ON THE POTENTIAL OP AN ANCHOR RING. 
1075 
§ 26. Let the external point at which the potential is required be near the ring; 
let its coordinates referred to CA and CB be B cos x B sin x- 
The integral 
c d(f) 
+ C“ — 2ctij' cos (f) -t z'~) 
may be expanded in the form 
1 I 
logjr + 
^ 8c 
‘“8 r 
- 1 
E cos X 
+ 
f 
2 loo’ ^ _ 
° R 
IG 
8c 
3 3 loa ^ - 4 
° R 
+ 
Pd 
16 
cos 2;)^ y — + &C. (45). 
Now 
i + 2.4 log 
E 
= ilog8c-ni$m~-&c. 
_ 1 loo-B 4- . HH3 ^ 
2 n 2.4 0 P^2 ^ 2.4.6 A T>i ^ 
Let 
Rdx 
Pd 
■ {46)- 
(fd — &-) 
A^ = ?/ 
the second series is the real part of 
— I log \/i log y + Iri + • • • 
Calling this series S, 
— = — Ji-J-Pdi; t 1-1-3 1 _|_ 1 
dy |2 y -r 2.4 ^ + 2.4.6 ^5 • • *1 
= Vy‘^ —I —y. 
Therefore 
S = ^ _ 1 log (y + _ 1) _ ^ + const. 
Now, when y is very large 
S = — ^ log >^/a^ — Ir- — \ log y. 
Therefore 
*3 3 
^7 — 5 - i log 2y — ^ + const. = - i log - i log y- 
6x2 
