742 
PROFESSOR W. M. HICKS ON THE THEORY OF VORTEX RINGS. 
therefore 
— 2a z \Jk= (1 + 5& 2 — 4 k cos v-\-2k 2 cos 2v)k ~^ 
o/c 
or along the surface 
— 2a 2 UXq(l -fa., cos 2v)=(l +5^ — 4^ cos v-{-2k 1 z cos 2v)k'~. 
Ohj 
Suppose for the moment 
7 
k-jj- = q 0 -f -q 1 cos v-\-q z cos 2v 
then 
therefore 
q 0 ( 1 -f 5/q 2 ) — 2 q 1 k l = — 2a 2 U 0 & 1 
?i—4#i=0 
q 3 — 2< Zi^i *f - ( lJn = ~ 2a 2 U 0 & 1 a 3 . 
^(1 — 3 Zq 2 ) = — 2 a 2 U 0 Zq 
?i-4&i? 0 =0 
Qq « 2 2 q= ^ ^ 
These give the equations (C) below. 
Before proceeding to introduce the last condition of equality of pressure on both 
sides of the outer boundary of the core, it will be convenient to collect here and 
partly discuss the equations already obtained. They are 
A (no cos v in i/q) 
iA 0 (L 1 -2)^ 1 +^+^A+f^^ 1 3 =0 
2 )^2+i t!~^4 ^2"^ 7/9 ^2°+2A(^oH—"TV^ h 2 ) — 0 
/2 
v/2 
1 
J 
(1) 
( 2 ) 
B (no cos 2v) 
't^o(Li—2)^ 1 2 +4.A 1 + ^-| + 1R e 3 Zq 2 +^—Zq 4 +^a 3 (A 0 -f /Q Zq 2 ) — 0 
16 
v^‘ 
-f A 0 (L i -2)^+iA 1 + ^|+| eA" +V+ift(A o+^f 
-4A {i a(a»- ff v)+iAo(L 2 -3)fe+i f -f «A-¥ }=o 
( 1 ) 
(2) 
