746 
PROFESSOR W. M. HICKS ON THE THEORY OF VORTEX RINGS. 
also 
0= const — ^U 1 2 +Ac?.i// 1 
f nPPPiOl’P 
p = Ad^—xp^—^d. U 3 2 
=n—icHJ's 2 
along the boundary. Suppose for the moment that 
U 3 2 =Z+m cos r+w cos 2v 
U 3 /2 =/ / +w / cos v+n' cos 2v 
Then the equations of condition are 
Now 
Further 
IT— \d'l '=A d(xfj 2 —V'i) — 2 
d'm—dnn 
d'n —dn 
u 2 ^{WV» 2 
(C—c) 4 
« 4 S 2 
4) , +g) , l 
(C-c) 8 (t + 7c 2 - 
s ~~ 
1 + 5/c 2 — 4& cos v + 2/c 2 cos 2r 
2£(1 -F) 
and along the surface this becomes 
2k 
—{1 + 5V+^A 3 —^2 cos v +(i/ 3 i 3 —A+ 2 4 3 ) cos 2y} 
Also substituting in (20) the values of A' ]5 A' 3 determined from (15) 
^2 h |?=iA'oCi - (i*- i)¥i V+iA'A&-iA) 
+ { -iA' 0 (2U~5)- 4 ^ r +|A' 0 |}4 cos „ 
+ \ -A' 0 (L : -V)- l 57 +A »W -f (iA'»2L 3 -5 + 
4a 2 V 
^2 M ^ 3 cos2y 
Hence 
a 2 S \ a~k 2S /2 
6a 2 V 
gAo{l + (3L 3 4)^ 3 2 }+ ^ ^2 3 2 -^-oA^2 
+ -A' 0 (L 3 -4)^. 
4« 2 V 
' y/2 
h+h^'oPi r cos v 
+ -|A' 0 (4A 3 +A)+^' 0 (2L 3 - 
2 « 2 V 
-^(A'oLa-H-^jj cos 2v 
(34) 
