772 
PROFESSOR W. M. HICKS ON THE THEORY OF VORTEX RINGS. 
The condition of the reality of the roots of equation (70) is 
4 n s (n +1 ) 3 z 3 + n\n +1 ) 2 z 2 +18 nY(n-\- 1 ) | (w+1 )z — -|z+ 4 n h j (n +1 )z — = 
> 27n 4 \ (n+l)z— 
91 2 
or 
(n-j-l) s 2 8 — 2«(n-}-l) 2 z 2 -}-^(w-|- l)^n + —^ —>0 
For all values of n. This may be written 
nh(z — l) 2 +n 2 |3z 3 — 4z z -{- ^1 — 7 j + n| 32 3 — 2z 2 -b 1 ^z —j + 2 3 > 0 (71) 
Call this expression Y n . 
Then the condition for stability of steadiness is Y, > 0, or 
t(z — l) 2 +<|z(3z — 1) (s — l)-}- 1 ^— ^ j-j- j 3z 3 — 2z 2 +^z — 2 ~ [ +z 3 >0 
and 
Y n —Y l — (n — 1) (n 3 +?2d-1)2(2 —1) 3 +(^ +1)"j 2i(3z— 1 )(z— 1 )-j—2 — 
P P 
+ 32 8 -22 2 + 18 2-~ 
P P". 
= (n—1 ) (n°-\-n)z(z —!)*-{- n\ 2(32—l)(z— 1 )+—2—^ [• — 2 3 +Y T 
Here n~ or >2. Hence, if p>l 
Y„ — h i>(u—1)[ 112 2 +162+ 4+ YJ 
and Yj is positive. Hence Y„>Y 1 . It is therefore only heedful to determine the 
condition of stability when n=l, which is given above. It is 
When p= 10 1 this is 
for p= 10 -2 
/ I Q\ O 97 
8s 3_82 2 +2 1+ '■ z--- : T>0 . 
\ pj p p~ 
■ • . (72) 
( 2 - 5.2 . . .)(z 2 +4.2 . . . 2 -J- 6 .5 . . .)>0 
(z—28)(z 2 +272+1206) > 0 
