PROFESSOR W. M. HICKS OK THE THEORY OF VORTEX RINGS. 
765 
(CO) 
( 01 ) 
whence the part of the pressure depending on v is 
|= _^-u| -(- + 2A</) bi+ 5 -^(AR,-BT„) cos nv 
+ ;j-~(CR, - »T.) sin «■} + A( x - 2a z TJbk) 
Along the inside surface this vanishes. Therefore 
l jp&(2k,)-qW(2h )} + U 1 (U,+ 2 ka%)y-^ (pfVWJ-lv'A*h)} 
r 4 a s k 2 • I 
-A{^«+4a% 1 U, r } = 0 
-\{pW^h)-qiV^h))+v 1 {v l +2ka%)a-:^{p^^h)-<i-nA^)} 
+K^y-ia%V^=0 
Substituting for £ rj, &c., these become 
_4A^V a+ 4a^ {Ui(Ui _2Aa»i 1 ) + »U I ! p} y 
+ (2ahqV i -2a^qV i y-^qS-nV l Uj^=0 
5 F 
Now 
Hence outside 
and inside 
t T __ JL H 
2a 2 6k 
a 2 U=- K ' 
2 V2k 
dU/ 
dk 
w 
■ k 
a 2 U: 
2/2 h 
-A a*h 
dso 
d\] U . ~ 
wr =—7 — 2 Aa 2 . 
ClrC n> 
h X 
da 
= \/(2£) { cos ) sin nv 
MDCCCLNXXV. 
