MR. F. W. DYSON ON THE POTENTIAL OF AN ANCHOR RING. 
73 
§ 12. Velocity Potential for a Ring moving at Right Angles to its The 
velocity potential $ is of the form 
{J j + A^a^J 2 + AgPi 3 • • •} cos (^. 
The constants are to be determined from the condition that the normal velocity at 
the surface of the ring is — w cos y; cos (f). 
Thus 
U COS V COS (b 
dR ^ ^ 
d(R 
= — Cll COS X COS <p. 
Now the formulae (C) of § 3 give on differentiation 
^ _ 1 
ds cs 
. . 2 
1 + s cos X + 
31 - 1 
. 6/ - 11 ^ \ 2 
+ “HT ^ cos 2x ) 
128 
4 ' 16 
COSX+ cos 3 X) S' 
, 2m - 83 ^ , 
+ 192 + 
, /m - 30 ,15^-28 „ \ c 
I -- Or\C! -V/ -J_ -- — ^ -vy 1 
, /135^ - 27 . 240Z - 83 ^ , 420Z - 809 , ^ 
+ (-- +-HLT- COs2x +-CCS 4x + 
1072 
(fh _L J _ 
ds [ 
S f 4:1 + n \ ^ 
COS X - 2 “ (- 32 “ cos X - ^2 cos 3x) s- 
+ 
121 - 1 12/ + 13 
64 
, /54/ - 15 
32 cos 2x + ti cos 4x) 
105/ + 61 
256 
cos 3x + -^4^ cos 5x + 
ds 
I - 2 cos 2x - cos X + i cos 3xj s - ^ .s~ 
+ 
/ 20/ + 107 i‘>i o L'i 1 
-cos X + m cos 3x + rf 8 cos dxjs^ + ...}■ 
df 
ds 
6 cos 3x — (10 cos 2x + cos 4x) s 
- (V cos X + fi cos 3x + -re cos 5x) + 
..}■ 
f/J. 1 . 
7 s’ - ^ - 24 cos 4x — cos 3x + f cos 5x) 5 - . . • j. 
7s" - (“ 120 cos 5x - . . 
ds c“s' 
MDCCCXCTir.—A. 
