774 
PROFESSOR W. M. HICKS ON THE THEORY OF VORTEX RINGS. 
When reduced this is the biquadratic equation 
\++bP+c\ 2 +d\+e= 0 
where 
b = Aap/(l+pp) 
c— — { (n 2 -|- np-\- n(np +1 )p)u 2 -\- (n 2 —np)v 2 -{-np(n-\-l)w i p 
-n(p-\- p)Aau-\-npAav-\-. A 3 cr }/(l -f- pp) 
d=Aa{ n(np +1) u 2 —n(np — l)v 2 —nAct( u-\-v) —n(n +1 )vrp }/(1 -\-pp) 
e—n z [(n 2 — 1 )u 2 v 2 + (n +1) (np -f-1 )u 2 w 2 p +A au {(np +1 )uv—{tip — 1 )v 2 
-(n-)rl)w 2 p} — A 2 a 2 uv\!(I-\-pp) 
Now 
therefore 
or 
therefore 
p — —47rAa 3 (& 3 2 — h 1 2 )= —47rAa 3 & 3 2 (l —r) 
Pi ~t p _ Pi 
87 rcPkcf 
-|Aa(l — x) 
u— 
Pi 
87 TCpk-p 
Pi 
87 ra°kpx 
v = xu —| Aa{ 1 — x) 
2 
A a=- - (xu — v) 
l—x v ’ 
A a 2 t fi' 
v 1 —x p l + p! 
U /X, IP /Xj 
v Pi + p P-i x(pi + p 
10 _ /X 2 
V pi + p 
If then \/v=y, the equation for y is 
y*+ W+ cy 2 +dy + e= 0 
where now (writing p'/(p 1 -\-p), Sec. =p, See.) 
