PROFESSOR W. M. HICKS ON THE THEORY OP VORTEX RINGS. 745 
Equations B (1) (2) and C (3) become when the values already obtained are 
substituted 
” fA 0 (Ll — + + 16 + V+2 a 2^0 + : ^T = 0 
-|A 0 (l 3 -i)V+!a„(V-V)+£+£ %¥+«72 
+lft(A„+^f v)+i|{iA 0 (V+ 3 V)+;|W- V) 
-f A„(2L, + 3)^-19^ V-K(a„-~ V 1 = 0 
•i/8A(3A 0 +^»n=0 
(33) 
Subtracting one-half the third from the first 
iI A 0 &i 3 +A 3 ^ 23 ^ —1^2A 4_ bi a 2^ 8 A°+ —0 
Eliminating e 2 from the first and second 
_s a It t [I l 'i 1 | a / 1 -1 \ - SAa 4 ,, 2 2\ | i^ fll 
_lj 37 A . 4Aa 4 . o\ i _/3 2 / , , 4Aa 4 
+ s^ 3 ^o+ ^2 «i J 2/^2 1^0+ 
_l&/l 
2 hl 2 
A„(l/+3lf)+™(9V-V)-iAl 2 (3A 0 +~VH = 0 
(33a) 
In order to determine V, the velocity of translation, and the quantities a 3 , /3. 2 , e 2 , A 3 , 
it will be necessary to consider the remaining condition, viz., of equality of pressure 
on both sides of the outer boundary of the core. We take the general case where 
the densities of the fluid in the outside and inside are different. In this case, even 
when there is no additional circulation in the outer fluid, the velocities on the two 
sides of the surface will be different. Let dashed letters refer to the outer fluid; also 
let p denote pressure, d density, and let II = pressure at an infinite distance. Then 
therefore 
const (vel) 2 
P=n-^ , .U /3 . 
If Uo denote the velocity along the surface inside the core 
p= const — -|<LU 2 2 -j- Ac/.i/q 
