1086 
MR. F. W. DYSON ON THE POTENTIAL OF AN ANCHOR RING. 
Writing I for log (8c/R) — 2, and s for Vijc, we have 
cos 2x) 
T_7i^ — 1 + — 4 
J=^?+-^5COSX+(-^^+ 
15Z - 23 
+ 
/33^ + 1 
I 64 
cos X + cos 3x) 
105Z - 176 
768 + 3072 
, /540^ + 27 , 240^ - 23 ^ . 
+ ( 2048 + 768 + 
cos 4x) 4- 
We obtain, by differentiation, 
1 fZJ 1 I 
~cTc~ ~ c^si ^ “ 
'21 + 1 cos 2x\ _ /20Z + 5 
4 
s — 
32 
[Supra, p. 53) 
cos X — A cos 3x 
/36/ + 3 . 24Z + 3 ^ \ . 
-d-— cos 2x — yf 8 cos 4x ) + 
V 128 
64 
1 dYr 1 . 
Tdc) ^ - cV + ■ ■ •} 
^ (70). 
+ (2 cos 2x + i cos 4x.) s + . . .} 
Multiplying by sr' or c (I — s cos x) 
J^ = l- 
I + 1 
s cos X + 
, /3Z + 5 
+ (-^cosx 
2Z + 5 
16 
3Z - 1 
j^cos 2x)s^ 
192 
cos 3x) S' 
. /12Z+11 , 12Z + 17 ^ 15Z-8 , , ^ , 
+ (^LLTo - —cos 2x- -^^cos4x)s4+... 
\ 2048 
768 
1^^_ }_ 
c dc c ch 
, /2Z + 3 , cos2y\ 
cos X + (“I-+-^r j ^ 
+ ■ cos X + ^2 cos 3x) 
41 + 7 . 4Z + 1 
+ - 
128 
e del c 
■ = S {cos 2x - 
cos% , cos 3x^ 
4 4 '® 
121 + 9 
32 
+ I cos 2x + ^2 cos 4x^ . . I 
1 <Z\3 _ _ 1 ^2cos3x + (cos2x - h cos4x)5 + . •.} 
C 0 o 
c dc 
1 rf\*T®' I 1 (O ,1 _L I 
c s 
d- cos 2x d- xis cos 4x) -j- . . . j 
[ • (71) 
J 
