PROFESSOR W. M. HICKS ON THE THEORY OF VORTEX RINGS. 
773 
and in general when p is very small =10 11 
2p 
z=f X 10 3 
or 
w/v=10* V /|=1'22X 10* 
Coming back to the general equation of condition, it is clear that, since vf'p 3 occurs 
on the left with a positive coefficient, whereas the highest on the right is vkp 2 , with a 
given value of x, it is possible to satisfy the condition of stability in the state of the 
ring defined by x, by taking w~p sufficiently large. If then w 2 p be very large, the 
ring would be stable up to a considerable value of x, but as the aperture increased, a 
point might be reached at last where it would become unstable. These considerations 
apply to any value of p. When x is very'small the equation for \/v becomes 
(writing y=\/v, w 2 p/v 2 =z 2 ) 
2 P , 1 
f-i^f-i^{ n ( n +H z *M n *-‘ 6n +*)}y+^ p {( n +iy+n-3}==' 
One root of this is 2 ; the others are given by 
y*+^ p y-^ p {(n+l)z 2 +n-S} = Q 
which gives the roots already found in Case II., as was to be expected. The solid 
ring has two periods, whereas just as a hollow begins to be formed it suddenly 
possesses two additional ones, given by X=0, and \=2. It will be interesting to see 
how these are introduced. To do this we must have recourse to equations (62a) and 
determine the amplitudes of the inner and outer vibrations respectively corresponding 
to any given value of X. 
Taking the second of the equations (62a) and putting therein u— 0, x— small, we 
get 
</LX(X — nv)~ M{(1 -\-p)\ 2 -\-2v\ — n(n-\- 1 )w 2 p — n(n —3)v 2 } = 0 
with the value q — 0. Hence M = 0 unless X is one of the roots (65). In other words, 
as the hollow forms' the outside vibrations are not immediately affected, but the 
internal vibrations introduce two periods peculiar to themselves, one standing vibra¬ 
tions (X=0) and another (\=2v) travelling in the direction of the cyclic motion with 
velocity 2v/n=2JJ/rn, where r is the radius of the outside surface. 
19. In the general case 
[pX 2 -f- AaX— n(np-\- \)u 2 -\-nkau}{(p-\-p)\ 2 —AaX —n{np—l)v 2 —nkav—n(n+1 )iv 2 p) 
_ ^(xs _ n 2 u 2 )(\ 2 - n 2 v 2 ) = 0 
5 G 
MDCCCLXXXV. 
