80 
MR. F. W. DYSON ON THE POTENTIAL OF AN ANCHOR RING. 
Ths coefficient of kiv vanishes, as it necessarily should (Basset, ‘ Hydrodynamics,’ 
vol. 1 , art. 171), and the kinetic energy of the forward motion and the cyclic motion 
is given by 
2T=rMir3(l + -^-0-3 + 
/ 
I 0 , \ dX." + 8 X + 1 p 
+ pc K- (X-- 0-2 
2X3 + 9x2 + GX - 
64 
The first part of this result agrees, as far as it goes, with the result of § 11 . 
§ 15. As the cyclic motion through a ring is of considerable interest, it seems worth 
while to give other proofs of some of the above results. 
To find the circulation we may integrate the velocity along the axis of the ring 
from — 00 to 00 , and then along a semicircle from co to — co. 
t/; = {A|J;^rrr d- 
The velocity of the fluid at any point on the axis is the limit of 
indefinitely small. 
Consider the part arising from the term A^J^ct or 
1 d^lr 
nr dus 
where tit is 
d' 
nr COS (f) d(j) 
y/— 2nTC COS (j) + c~) 
It = limit of 
:• A r 
nr dcT 1 
Jo 
w COS d<p 
y/ (z~ + — 2nrC COS (p + c”) 
1* • ^ £• j Aj (.i.~ A c ctc cos (pij I 1 I 
= limit ot 1 - , - 3 -cos <p d(p 
{z^ + — 2vjc cos (p + c~y ^ ^ 
'0 
= limit of 
A, 
nr (z~ A c^y 
1 — 
me cos <p 
o , o 
z" A c" 
1 “b .,2 
me cos <p\ 
A C' 
cos (p d(p 
ttA^c 
(.2 AC^)?- 
We may notice that at the centre of the ring this 
ttA, 
c~ 
* As we are only considering the cyclic motion, only the parts of A^, An, &c., which involve v, are 
denoted by these letters now. 
