748 
PROFESSOR W. M. HICKS ON THE THEORY OF VORTEX RINGS. 
Hence, applying the conditions (34) 
rV rl / 4. A n i 
n - AV=idU/+ M<fe-«-«3nK+ - *»* 
1 
v'a 
f A ' /T A' & 
2a^l Ao(L3 ^ + v/2 2 0 
IK 
= ss/ A «+K V) f-W 1 i IK 
v/- 
v/2\ ^ 
4Aa 4 , 
' \/2 k > 
Now (29a) to the lowest order 
and 
A , 4A « 4 7, 
A °+ ^2 
r\/2 ' v/2 
= ~(Mi+m') 
7T 
■v/2 
A ' o =-^ 2 ^ ( 18 ) 
U ‘=4^ fr ° m < 32 ) 
i/f 2 — .//, — A''. 4 (7; 2 s —/, y : ) — ~~A 0 ( L 3 — L,) from (15, 23) 
= -S-^2 A »(E-L I ) 
Substituting, the above equations become 
327T 2 «V' 9 
fic,ad' 
O 2 
ft 
The latter is 
—/r 2 a(L 2 — ^)4- 4a W+|/x 2 a 1 
= - W2A, (l-^-HrA* (l-{£) V 
A/ A 4Aftt 7 0 
2 k 2 V A ° v/2 - 
= -( ft +^ V 
Mr 
W_rW-R i V+ . ¥ ./ 
Av-^V 
M 
^ a \ 1 + V/ + 2 4 2 V 1 V~*l 
1 M 
