768 
PROFESSOR W. M. HICKS ON THE THEORY OF VORTEX RINGS. 
whence 
and 
y. ' 2 v ^ n{(n — 3 )v~ + (n+l)vrp} 
1+p 1+p 
—v + \/{v 2 -\-n( 1 + p)((n — 3 )r 2 -t- (?i + l)w 3 p)} 
(65) 
In order that the motion may be stable the expression under the root must always 
be positive, whatever value n has, = > 1. The least value is when n— 1. Hence the 
condition of stability is 
2 v?p( 1 + p) > (1 + 2p)-r 3 
or 
/A~ > 
/ a _l + 2p_ 
2(l + p)p 
( 66 ) 
In the ordinary theory where p=l and /x 2 =/h the ring is therefore always stable. 
When there is no additional circulation the condition becomes p> 1/^/2 or d<d'^/2. 
The ordinary simple ring will therefore be stable even when the density of its core is 
as large as y/2 times the surrounding fluid. In any case, whatever the density of 
the core may be, the ring will be stable provided it has an additional outside circula¬ 
tion given by (66). In general the two values of X will be of opposite sign, and will 
correspond to waves travelling with and against the cyclic motion. The positive root 
gives that in the same direction and the negative in the opposite. In the simplest 
cases of equal density and no added circulation the roots are —nv and (n— l)v. 
18. Case IIL—Ab internal added circulation, with a hollow. Here Pi = 0, u— 0, 
and the equation for X gives one root zero, and the cubic 
(l -Hpp)X 3 +AapX 3 — [n{n — p)v 2 - f- np(n +1) orp +? i A apv -f A 3 a 3 } X 
—nA a {(iip — 1 )v 3 -f- (n +1 )iv 2 p +A civ } = 0 
in which since 
/x'= —47 tA<x 3 (& 2 3 — h 2 ) 
1 —x 
Denote the cubic by 
X 3 -f- 6X 3 +cX+ d= 0 
We shall first investigate the signs and finiteness of the different coefficients, and 
o o 
then pass on to the question of the reality of the roots. 
2 rp 1 — x n 
X - 
1 — x 1 — x n + (1 + x n ) p 
1 +PP 
