MR. F. W. DYSON ON THE POTENTIAL OF AN ANCHOR RING. 83 
Therefore the momentum of the cyclic motion is 
= '27T^pc [A I — ^2 0 "^ — -^3 — 1 • 3. cr® — 1.3.5 . Ag cr® + . . .} 
= TT-Kpc I 1 — (X + 1 ) 0-- — ^ — &C. I . 
§ 16. The cyclic motion might have been started by applying an impulsive 
pressure Kp at all points of the aperture of the ring, this would produce an 
impulsive pressure at all points of the ring. To determine this, the velocity 
potential of the cyclic must be found at these points. It is easy to find an 
expression for the velocity potential at points not far from the surface of the ring. 
For 
Airp 
IT 
A^c — Ao — 
" c 
. ft® 
- 1-3.A^.- 
cl^ _ 1 fZ-v/r 
Tss f?E 
~ ~~ w JgCT + . . . } 
Ai 
cR (1 — s cos 
1 + ^ s cos X — 
9/ + 13 
1 + 2 21-1 
4 ~ 16 
31-2 
cos I ^ 
64 
cos X — 
64 
cos 3 y ) s® 
3Z + 2 . 61 + 7 
128 
+ 
96 
cos 2 x — 
X 
60^ - 47 
3072 
cos 4x) . 
+ 
Ajft^ 
(1 — s cos x) 
/.3i *a 
+ 
Agft* 
cHl® (1 — s cos x) 
A, ft® 
c^R^ (1 — s cos x) 
r , s , /201 - 3 . , „ \ , 
jcos X + 2 + ( 32 X + 3 ^ cos 3 x) 
/20Z + 37 3^-1 ^ ^ \ s . 
-^ cos 2 x- 1 ,^ 4 -cos 4xy 6 '^ &c. 
[2 cos 2 x + (I cos X — i cos 3x) 3 ' - ] 
(6 cos 3x + (8 cos 2 x — cos 4x) s] + &c 
Expanding ^-in ascending powers of cr, 
expressed in cosines of multiples of x 
The term independent of x is k/ 27 t 
O n integration, we obtain 
and multiplying, d<t>/(./x is easily 
M 2 
