PROFESSOR W. M. HICKS ON THE THEORY OF VORTEX RINGS. 
771 
When n— 2 
or 
When n— 1 
or 
6iv 2 pz,[ 2 
P + 1 
2 ^ 2p +1 0 
vrp> —v l 
r 6(p+l; 
2 »v>( 2 -- + i)»* 
w z p > 
2p +1 9 
-ir 
2(P + 1) 
(08) 
(69) 
To n— 1 corresponds a vibration of the ring as a whole about its state of steady 
motion, whereby it describes a circle, without change in cross section, round its mean 
position. Instability therefore for n= 1 actually means instability in steadiness, 
whereas instability for n— 2 means instability of form. 
For a ring of the same density as the fluid, and just at the point of forming its 
hollow, the condition of stability of steadiness is w 2 > fv 3 , and for stability of form 
w 2 >{v 2 . 
For a ring with a very dense core, the corresponding conditions are w 2 p>\y l and 
urp > }v 2 . 
When the hollow is very large x is nearly equal to unity. In the case x—1 
b = — nv 
c= —n[n -\-1 )w 2 
v, 2 
d=—{(n-\-l)w % p — 2v 2 }v 
and the equation to find \/v=y becomes, writing iv 2 /v~=z 
y^—ny 2 — n{n-\-l)zy-\-n 2 \{n-\-l)z—■- j = 0.(70) 
When p is infinitely large (density of core infinitely small) the roots of this are n, 
and ±x/{n(n-{-!)%}. The last two agree with the result obtained for a hollow only, 
as might be expected. Hence, when the density of the core is small the ring is 
stable when its aperture is very large. 
When p= 1 and 2=1 (the ordinary case treated), the roots are —n and n^y/n. 
This is, therefore, also stable when the aperture is large. 
