MR. F. W. DYSON ON THE POTENTIAL OF AN ANCHOR RING, 
1067 
v3 + — 4- 
8 ^ 192 ^ 
c d(f> 
\/2ct)s' cos + z'“^ 
+ na^ t (otii sin 7i tf)' -I- /3,i cos [ \2 
• 0 L 
C COS 71 (j) (1^ 
\/ P — 2cm' COS + s'^) 
3 Y 
(*-n + 
c dcji 
’ J + c" — 2ct<j'cos 
, (28). 
§ 18. These integrals admit of simplification. 
Denotinsf 
cos 7lcf) d(j) 
\/{c^ + — 2c7)s' cos (f) +2^"} 
by I 
dn , 1_ f r — a 
dm'-^ 
m 
since I cos w ^ is a solution of Laplace’s equation. 
Synimetry shows that 
therefore 
and 
— 4-1— — 0 
dc~ c dc'^ dz’^- c3 ^ 
^ I l!L\ T _ _|_ T 
dc^ dz'V ~ de^ c 
11 4- -ll Y T — _ -1 / - — 
dc dz'^J ^ dc\c dc 
C2 dc 
Substituting, we obtain 
V„ = 2™^oU + |^ + 
2 \2 
YA 
cdc ^ ^ \cdcj 
d(j) 
1 0 \/(c^ + — 2cx)s' cos 0 + z'") 
+ 47700 2 («„ sin n([>' + /3„ cos ) .ji-f—+ — — 
0 ^ 71 ^ . aPP 
O 0\ 0 
rr7?~\ a'^ 
32cV ci/c 
cos ncj) d^ 
0 \/(^“ + — 2cm' cos (j) + z'") 
dtj) 
\/(c" + m'^ — 2cm' COS (f) + z'~) 
(29). 
6 IT 2 
