MR. F. W. DYSON ON THE POTENTIAL OP AN ANCHOR RING. 
93 
and 
t/" (<^) = A + A eos 4x. 
All terms which are of order hig’her than cr'^ have been neglected ; and throughout, 
it has been assumed that oj^c^dTra^ is of zero order in a, of the second order, ySg of 
the third, and so on—an assumption which will be justified by the result. 
The value of 
at the surface of the ring is 
+ 2 + cr^ + 
2\ + 1 ^ . 108A +27 ^ , 6X + 1 „ 
2048 
, . /2\ + 3 , 12X + 5 
+ <r+ 
cr^ + 
+ A. 
32 
+ IN 
/36X + 51 ,, 
V 32 
M + 
(O'^C" 
4^«;n' + \ + 
I [ \ 1 , 9X + 3 , X \ , A /i I ^2X + 11 c ^2 
+ cosy jAj cr d- o--^ + - o-yS^j + Ag^l d- ~ ~2 
+ A, 1 o- + (- 2o- - o-ft 
+ OO.S 2x I A, ('" cr-^ + 0-* - ft + i 
256 
+ ^2(^-1- — 2 ^ (1 + I 
0 0 / o 
co^C'' / a'^ 
d- A 4 5(7 + ^^^^2 (^2 + ~ 0-^8 
+ cos 3x {a, + "** <>•» + J trft - ft) + A, (A 0-5 - 1 ft) 
+ Aj J + 2A, - 0-ft 
+ cos 4x I A, .r* + ®A| ^ ,.=ft + ^ ^ft - ft + 
H" A 2 (itIf i /^a) “i" A 3 ( 3 % cT"' /32) 
+ A 4 ,- d- 6A- d- 
00 / o 
/ <7" 
“ ■ 47rftn2 
— o-N 
Equating the coefficients of cos ;)^, cos 2;)^;, cos 3;)^', cos zero, we obtain four 
equations to find 
0 o 
co’^r/ 
47rft' 
2 , N, N N 
