iMK. F. W. DYSOI'T 01^ THE POTENTIAL OF AN ANCHOR RING. 
1089 
These equations give the velocity of the ring in steady motion, and the form of its 
cross-section. The cross-section is slightly elongated in the direction of the motion 
of the ring. The quantities /Sg and are very small ; for example, when cr = ’3, 
the case of a very thick ring, 
V = 2-96 X W- , A’ = - - 063 , As = - ‘ 006 , Ar = ' 005 . 
lire 
§ 37. Fluted oscillations of a Circular Vortex-Riny. 
Let the cross-section of the ring when disturbed be given by 
R = n {1 + S (a,, sin nx + A^ cos nf)] .(78) 
At an externa] point, not far from the ring, the stream-line function is given by 
m = 
me 
TT 
log|-2- 
86' 
lOG - - 1 
” R 
'5 a? cos X 
cll 
+ - 2 - (a„ sin nx + P>, cos nx) 
al’’ 
/ili 
(79). 
Let the central circle of the ring have moved a distance 2 q from the plane ot x, y. 
Then 
+ + = 2 [(««;, -f ctrx,) sin nx + (oA« + «A^) cos 
vz^ cc 
Now 
and 
Therefore 
+ a 2n (a„ cos ny — A. sin ny) ("y -f ^ 2 ^ -f c 
0R 
0y 
sill X 
0c 
cos y. 
dc 
~U = 
0rt 
cosy 
(h 
'‘~o 
— sin y. 
02„ ~ 
R 
1 
1 (Z'F 
R = 
^ R/y ’ 
~ m Rf/R 
R=7-- 
J)l 
1 8c . 
11 ^ 1 . 
17 (c — R cos x) 
m 
5 <? 1 . 
-smy-h — ^^.,siny 
C 
+ 
IT 
(c — It COS x) 
1 ((.“ 
7 . - TG ('■A/ cos ny — A: sin iiy) 
li 
6 z 
(80). 
( 81 ). 
MDCCCXCIII.— A. 
