PROFESSOR W, M. HICKS ON THE THEORY OF VORTEX RINGS. 741 
Substituting for k, and putting in the terms involving A u from (20), the value 
along the outer surface is 
1 i. Hi 
y/2 Bk 
=4A 0 {l-(L 2 -l)v] +ift { -P 0 (L 3 -4)fc+4|'| + j 
+f e A*»+^‘( 2 ^+W+W+W 3 
e x JcA 
+ 
+ 
COS V 
{-4A„(L 1! -3)Ji-4^+j«A+^(W+4M s ) 
"-|A 0 (2L i -5)V-^+4A|-4A 0 (L 3 -4)+i|Lji 3 
+(ft+iA 2 )y^+|W 2 +f^+^|(3iV+¥^A 8 ) 
The lowest term in —SxJjJSv is 
y2{-lA 0 (L-2)i+4|+^+f^| 1 F} S m« 
Along the outer surface this is 
/2 { -iA 0 (L a -2)i 2 +4|+^A+ 
By Eq. (A. 2) below this is 
cos 2v 
sm v 
+ terms of the fourth order. 
— sA^o+^fVjsin v 
(27) 
(28) 
and is therefore of the third order. 
8. Pressure conditions .—Along the inner surface the pressure is zero, and since it 
is a stream line the velocity must be constant. Denote it by U, then 
(C 
dud 
dn ) 
- 
Now along this surface a L = 0, and therefore SxjjjSv is of the fourth order, whilst 
JcSxfjJBk is of the second. Hence it may be neglected, and 
ott (C-AS.H'i 
a-U= - -g-^ 
5 C 
M DCCCLXXXV. 
