1088 
MR. F. W. DYSON ON THE POTENTIAL OF AN ANCHOR RING. 
— ^ 11 + A + ^ A + COS X (— 2c7 - 0-^2) -f cos 2x (^ + - cr/3. 
.A 
- 0-^2 cos ox + cos 4x ( — 0-^83 + o-= 2 
IS constant. 
Ecj Liating the coefficients of cos Xj cos 2 x, cos 3 x, cos 4 x to zero, 
I X+l|3X + 5.., o \ \ / II ^ .1 I 182 
^0^ { -^ H-^ O" — 4 ^^2) + «1 ( — 1 + ^ 9. 
“ 4 Mft3 + o-A) — h 
32 
CCr 
Xa- , 12\ -f 7 . 
~l6 + 
768 
(T 
— ^'2~ 7\ + 
\ + 2 
+ “.(i + ^-^<^’-5A + iA) + «4i- 
Clr 
64 
M«2 \ 2 
Xo 
1 
-^2,^- (^ + ^"^^2 - 0-^3) = 0 
■V-~ . ySs 
3X — 1 q 'Xo' r, \ , / CT" , Ha 
+ (32 + i 
V 
— 
- «2 4 - 2 % + = 0 
^4 n ^ n 2X 1 ,Tp . 10 9 , 
“3072 ~ i ^^3-O’ P 2 + i P 2 
32 
+ ^'1 fils + §') + ^^3 ( “ 32 “ 
128 
CT 
V , . q/S2 
+ «3 2 + 27 ^ 
Now, to = ■§• Mot ; therefore, if m be the strength of the vortex, 
ni =11^^ = -|- M ||(c — E, cos x) E (^E c/x 
= iM7r«~c( I + ^ 
Substituting then for M a^, and solving the above equations : 
-rj in r4x + 7 12 X + 9 o'! 
‘;i~8-i^"'i 
TTC 
^2=- 
/33-- 
/34 = 
J2X + 
0-- + 
I2X + // 
3.211^ 
<T^ 
CT 
168X + 63 
^1024 
84X3 XI lx + 41 
512 
r 
)* 
( 74 ). 
(76). 
(77). 
