MB. F. W. DYSON ON THE POTENTIAL OP AN ANCHOR RING. 
1077 
Let 
1 V 
loo’ —- — = K. 
^ a 
Then the potential of an elliptic ring at outside points not far from the ring is 
2TTah Ik — ie cos 2v + 
K - 1 
cosh w — 4- cosh — 4- cosh^ Un e '4 cos 
V 
+ [ — § cosh ue~^’‘ + Y2 cosh^ Uq e cos Sv 
.(48). 
At the surface of the ring u = Uq and K = Kg = log (IGc/d) — Uq, and 
d 
V = 2TTah 1 Kg — -ie cos 2v + ~ 
K, - 1 ' 1 + 2e-^o\ 
, cosh Uq cos V 
O j 
2 — e-2“o 
e"-"" cosh Ur. cos Sv 
24 ^ 
(49). 
The complexity of the expressions renders it useless to find the potential any 
further, in the general case : but for a very flat ring, i.e., where b is negligible 
compared with a, the expressions take much simpler forms. 
27. For a very flat ring 
Y = 4tt ah 
1 IH 
9 I 2-4 2 I 2-4.6 r ' 
111 8c , 
1 8c 
logg-l 
Rcos 
X 
+ . . . 
(50) 
/ (D) log I" = i (log ^ - \e cos 2v - , 
f {^) ^ ^ (®“'" CO® cos 'y) + ^ ’ 
8c 
U" (D) log i = : 
R 16 
log 
16c 
v — 
5 e cos 
4v 
Therefore, 
