1090 
MR. 1\ AV. DYSON ON THE POTENTIAL OE AN ANCHOR RING. 
and 
m 
1 
X 
TT (c — K co,s %) 1C 
( 82 ). 
Writing K = a, and neglecting a in comparison with c 
8c 
log v — i sin X — 2:0 sin X + c cos X + — t cos nx — A. sin nx) 
rrrrp \ xt / TTct 
• • • '))X, 
= a + a ^ (a,, sin nx +cos ?ix) H-iS a (a„ cos nx — A; sin?Jx) . (83). 
This equation gives 
rt = 0, c = 0 
or V = (log ~ - i 
live \ ^ a 
a. 
« - - J-) A. = 1 
17(1 * 
• 
^ - 1 )«./! = 0 
iva~ 
h 
Therefore 
'Dl 
a.„ 
+ — 1)' — 0 . 
iv-a 
Therefore the time of an oscillation of the type 
a.,, sin nx + A cos ux 
IS 
2. 
TV 
27r-cd 
'HI , mi'll — 1) 
“ ^1 
ITO, 
(84). 
(85). 
The steady motion and small fluted oscillations of a single vortex ring have been 
worked out by Mr. Hicks by means of toroidal functions. Only the simplest case, 
that of a vortex ring in a fluid of equal density, with no added circulations, is 
considered in this paper; the same methods might be applied to other cases. The 
steady motion given above agrees with Mr. Hicks’ result, in the velocity V, and the 
value of Ai to fbc first order. 
