PROFESSOR W. M. HICKS OK THE THEORY OF VORTEX RINGS. 
7G3 
Again, 
Therefore 
&= v(«) 
YL 
—|L) cos |M 
sm wv 
Now 
Therefore 
^-^r^l{(£-i L ) cos sin nv ) dv 
sin OT +(^-i M )(° os ™-(-m 
dL 
du 2 
ddx n 
L—AR„T BT„ 
dT„ 
— |L=A -T-*-iE, +B Y-*-iT, 
du 
du 
= A{(«*—i)SP,-py + B{-(n=—i)SQ„-iT4 
= AR 
2?iSP„ 
" i p _ p 
-A- n -|-i x ?*—i 
—BT„ 
2»SQ„ 
Qn-l Q» +1 
*} 
Now P„/P„ +1 and Q„/Q„~i are both of order k. 
Also, 
(2w+1 )P„ +1 - 4»CP„+(2n -1 )P*_! = 0 
Therefore 
9 Ji!L 2;t + l 2n — 1 
P a+1 “'2C + 2C P s+1 
or to the lowest order 
2 n ( 2n 4-1 )k 
P«+i 
So 
2« ^-=(2?i-l)* 
1 
and therefore to the same order 
and similarly 
Hence 
(f) l -\-(f). 2 = const. 
——iL=w(AR ;i —BT W ). ....... 
pi=M(CB»—DT„) 
fV® {_(AR„~BT„) sin «+(CR„-DT„) cos nv] 
(55) 
(56) 
The part of <f> x containing u disappearing with a corresponding term in </> 2 , as it 
ought consistently with the fact that this part of cf> is independent of the path of 
integration, 
