PROFESSOR W. M. HICKS ON THE THEORY OF VORTEX RINGS. 
743 
C (constant pressure inside) 
i A„ {1 — (L t — 1 )*>} +1«AH S ^X»+¥ % h *=~ ^ (1 + 3^) 
*1 
-fA„(2L 1 +3)i 1 s - 2 ^-1« 3 (a 0 -^ v)+f e*V+19 ^ *i‘= 0 
\/2 
4Aa 4 
( 1 ) 
( 2 ) 
( 3 ) 
Values of A 0 , A Y —Subtract C (2) from A (1) and divide by Jc L 
therefore 
Now (10) 
-A 4- —— “Z-’ 3 —0 
2 0 + /n 3 y/% 1 
A, = -|AAH^| V ■ 
A 0 +A 1 +A 2 =-f^- 4 ^|ViH« 1 ‘) 
Therefore to the fourth order 
(1—-p^v 
vrv/2 
47 Aa 4 7 9/ h 7 7. 9A 
■ 7 r x /2“v / 2^ l(4 + 1 ^ l) 
or 
A 0 = 
■^(i+WI-^-W+W) 
4A« 4 
or to the lowest orders respectively 
A _ 47 4A« y g 
A °“~7r v /2 _ 72 ^ 
1 
Ar 
,3 _47_ x. 2 , TAa 4 7 4 
2 7x72^ . + ^2 J 
(29) 
(29a) 
Fa/we of f3 v —In equations A divide (l) by k l} (2) by Jc 2 , and subtract 
iAo(L 2 —L x ) +^Aj ^ f (a 0 +^V)=0 
v/2 
v / 2 
Therefore 
r b / z. 
11 72 i 3 /1 m 
,A« 4 /7 0 7 . 0 \/« *i 2 \ 
