PROFESSOR W. M. HICKS ON THE THEORY OF VORTEX RINGS. 779 
When the added circulation is in the same direction as that of the core—as has 
been supposed throughout, v > u. 
If there are no added circulations, u— 0, iv=v, and 
-±Aa(l= 
and the condition of stability is 
p> Id 
2 
1 — x 
2 log 1 jx 
(1 
The maximum value of the right hand member is when 
or 
1 + x 2 log x 
x(l—x) 2 ~' (l—x) s 
0 
1 —a? 3 + 2a? log x— 0 
whence x=l, which makes the condition become 
p> 0. 
Now x is always < 1, and since p is always positive, the condition is satisfied. 
In general 
w H 
w =— = ——— 
27 Tak 2 87m 3 & 2 3 
_ Pi + p' 
87 TCiPk^ 
_ jU-x 
W “87raV 
Also 
P — — 47tA(X 3 (^2 3 — ^ 1 s ) 
Hence condition of stability is that 
A 2 
k:■ 
7vc) 
h. 
/ log 1 / x 1 
^ (1— x) 2 \ 
When a? is small the right hand side is evidently negative and the condition is 
satisfied. When x— 1 the right hand member is 0. 
Consequently 
pxp > 0 
