PROFESSOR W. M. HICKS ON THE THEORY OF VORTEX RINGS. 
775 
2/*'p 
(l-as)(l +pp) 
r 2 
c = — j (ft 2 + up + n{rq> +1)p) -fft 3 —ftp+ftp(ft+ l)p/x 2 
I o P + P / , 4/F 2 1 1 
"^^(l-*)^ I-®/ 4 + (l-x) 2 J 1+pp 
d= 
2 /x'n 
n JP ~ 1 ) ~ ( n P~\r 1 ) ( n + 1 Wp 
(l-x)(l+pp) 
e =Tpo [" ( n °~ P (»+ 1 X n P+ 1 )'^ f 
1 —X 
1 + 
p\ 
(73) 
-> /*! 
£C(1 — ®) 
{np+l)^-np +1 
—KiWp 
®(1 -a;) 2 
The general investigation of the question of stability would be very laborious, but 
it may be useful to put down the values of the coefficients when the aperture becomes 
very large, or x— 1. They are 
In which 
b— —nix' 
c— — n(?i+l)/x 3 3 —(^ s +^)/ A 1 
d=- L jr{(n-{- l)ix. 2 2 p-ix 1 (l-\-[x 1 )—2} 
r 
e = w 3 |ii|(w+ 1 Ws i! -y (1 + Pi) 
(74) 
P'l+P- 1 
20. Pulsations .—Let the pulsations be given by 
k=k l ( l + £) Jc=Jc 2 (l +>?) 
Then since the volume of the core remains unaltered 
Since the stream function is now many-valued, it will be advisable to use the 
potential function for the part of the motion due to this: denoting it by <f), it is of 
the form 
(f)=\/C— c{AP 0 -f BQ 0 } 
5 g 2 
