MR. P. W. DYSON ON THE POTENTIAL OP AN ANCHOR RING. 
1049 
V = 277-a^ j \ + 2 + —a cos x + 
X-i , ?>(\ + U) 
16 
+ 
16 
cos 2x 
(T 
+ 
+ 
3(A--iV) 
5 (X. + 
-cosx +-cos 3x 
or 
9X , 7(X- M ^ , 35 (X + -rVV) . ‘ 
- + 71^ COS 2x + -- jo T 4 ■ cos 4x 
256 
128 
cr^ + &C. I . (6), 
where cr = ajc, and X = log (8c/a) — 2. 
Now, inside the ring, 
d^Y , IclY , , 
7? + 
Let 
then 
V = - 2772^ + V 
1> 
fll 4. _L = 0 
dz" du!^ w r?nr 
We may, therefore, assume that 
27r«2 
, . fE , (7E2 , 3cr2R3 , q/LV/.; ^ , q X . 1 
+ As cos X + j -o + -^ ;;i cos X + o-s (-g^ cos 2x + tw) + • • • |* 
+ Ag I ^ cos 2x + j -3 cos X + -4 (/s cos 2x + # 2 ) 
+ (^ 6 % cos 3x + cos x) 
+ ^ (toIa cos 4x + cos 2x + /ra) + •••(-, 
, , fE^ ^ , <7 R* ^ , 
+ A 4 cos 3x + I ^ cos 2 x + . 
+ Ag cos 2 x + ...|. 
Therefore, at the surface of the ring, 
^ 1 . A , A /o- . , , fSa^ , 25a^\ 
^ - - 2 + Ai + A2 (j + + A3(— + — 
27ra' 
4 ■ 128 
3cr- 
16 
512 /’ 
+ COS X {Ao (^1 + + Ag j, 
+ cos 2x + As — + Ag (^1 + — + + A 4 -j, 
+ cos 3x I Ag yyy -\- A^^ j, 
+ cos 4x \ Ag 
MDCCCXCIII.— a, 
768 
21o-^ 
10^ 
+ Ag 
(^)- 
6 s 
