78 
MR. F. W. DYSON ON THE POTENTIAL OF AN ANCHOR RING. 
Differentiating the equations of § 3 (E), 
^ If . I . /^ + 2 21-1 
- (JiCt) = - j - 1 _ - s cos X + ( -rrr- cos 
dli 
16 
2 x)^ 
+ 
9 / + 12 
64 
3^ _ o 
-3x)s' + 
3^ + 2 . 6Z + 7 
128 
+ 
96 
o 60^ - 47 
cos2x-^^7^cos4x 
(Z 1 [ s /20Z-3 , , . . , 
> (Jottr) = — i _ cos X - - ^ cos X + ti cos 3x )s~ 
clR 
/20Z + 37 3Z-1 \ . 
+ ( —-cos 2 x - A cos 4x j + 
64 
^ (Js^) = (- 2 cos 2 x - (I cos X - i cos 3x) s - ^sE . . 
f/li 
Also 
1 1 / 
= ;(i - 5cosx)"^ 
c — R cos X 
_ 1 J 1 _t_ I O ^ / 1 L L '■ 
= ;' + , + ■» + 2 oos X 5 1 + ^ + - 
*■> 5 /t -4.' 
06'' . 06^ 
4 ' 8 
,2 
4 
+ 2 cos 2 x ^(1 + 5 ®) + 2 cos 3x ^ + . 
Therefore 
2- d 
(Jl-) 
R dx 
2ir 
J 0 fZR c — R cos X c 
3s^ 
- - 41 + 2 + J + ...) 1 - 4 
I + 
6 '^ — 
3Z + 2 
128 
Similarly 
Therefore 
27r 
R fZx 
'2rr 
~ f, Ir 
Rcosx 
c — R cos X 
&c. 
= 3 
27r 
/f 
^TT 
C 
Ai + ^ + o -r + • ■ 
Stt 
= — {A] + A;jcr'" 4 - oAgcr'* + &c.]. 
