48 
MR. F. W. DYSON ON THE POTENTIAL OP AN ANCHOR RING. 
i:, 7 P 
cos (f) d(j> 
\/{^" + c- — 2cr sin 6 cos 
2 ri (2 cos^ 4> - I) cl4> 
_(i 
■'’iW' 
_ E^ 2 
COS^ (p 
/ E ^ \ 2E ^ 
_ 2 " 2 (^cos3<^ - - 1 + Ej2 _ E2 
Ri /f. E,2_IH 
v/jl -cos~</)^ 
dcp 
, , E,2 - E2 . „ ^ 
\/ U- o sill- 0 
p 2 
-‘M 
f!! , r E 2 - E2 . „ I 
- j“ v/|l j 
f 
^ n 
sin- (p f. d(p 
d(p 
. , . E,2 - E^ . „ , 
\/]l -— sill- (p 
» E ^ 22 Ej^ 
4Ei _ 2 
E “ 172 
-}-... 
, 1 E2 /. 4E. 1 \ , 12.3 E^ 
^ 2 Ei2 E - 1 . 2 j 22.4 Ej-i 
4E 
E^ 1.2 3 . 4 ; “*"■■■ 
2 r, 4E, 12 E2 /, 4Ei 2 \ , 1®. 32 Td 
E'i E 22 E/ \ E 1.2/ ■*" 22 .42 IV 
4Ei 
E 
2 
+ . . . 
1.2 3.4 
[Cayley, ‘Ell. Func.,’ p. 54.] 
2 M 
4Ej 
t- 1 4E. ^ 
jj, 6 logy -7 
2+p^^s ^ 
163 log ‘A _ 226 
I IP 
+ EH 1" 
1 
128 
J 
Noy”- 
Therefore 
_ 4^3 _ pQg ^ ]43 
= 4c-p--cosx + j^ 
= 4c" 1 — s cos X + j)- 
, 4E, , 8 s s2cos2y cos 3 y 
log y = log - - - cos X - j -y - f-- 
= 1 + 2 
cos X 
4 2 
$2 
— &c. 
8 
cos 2x — cos 3x — &c. 
E2 
41^2 
= + 16 cos X + ^ (1 + 2 cos 2x) + &c. 
64 
