770. 
PROFESSOR W. M. HICKS ON THE THEORY OF VORTEX RINGS. 
The second condition is 
— 4c 3 +& 3 c 2 + ISbccl — Ab 3 cl>27d 2 .(67) 
Here every term is positive, except when iv 2 p<v 2 . Treating this as an equality, 
and writing w°p / 'v' 2 ‘=y, Ave get a cubic equation in y. Let the roots of this be a, 6, y 
in order of magnitude. The inequality may then be written 
(y- a ){y-P)(y-y)> ° 
Hence, either y>a or between /3 and y. 
When the central IioIIoav is very small, x is also very small. If x—0 
b=-^-v 
1 +p 
c = —l)w'p-\-(n 2 — 3n+ 4)t’ 2 } 
d= 
2 n 
1 + p 
{(n+ 1 )w~p-\- (n — 3)v z 
•2~> 
In this put for the moment n(n -b 1) w~p /T 3 -f n 2 —3n=2^ 
Then 
c = - r f^+ 2 )^ 
d=--£i? 
1 + P 
and the above condition becomes 
- 2 -(^_p2)3-pf 2 ^t 2 l 2 -pi^d^(£-p2)-27f- 
8p 3 
(1 +P) 
n £> o 
or 
whence 
— (8 + 4,,—2^jaj)f S +( 11 + 16/2+4/1 
(f-4-2^( f +- +1) l>0. 
•* .-YWsi±ie±^ >0 
1 + P/ ^ 2(1 + /,) 
In this case the two roots a and /3 of the cubic in y become equal, and the condition 
of stability is 
n(n-\- lW 3 p+n(n — - >0 
p +1 
This is always the case Avhen n> 3. 
