PROFESSOR W. M. HICKS ON THE THEORY OF VORTEX RINGS. 749 
therefore 
4:Trcid' p 2 V = — 2P'2 3 ^ / (^y 1 — 2L 2 -f-1 
1 
kd-\-k x , 
—i(pi~ fri Zjji ^F • 0*1+/) 
,X ft 
Aj, J 
This gives V, the velocity of translation. 
The first equation is 
1 
IT Z9.m&n%h 2 ^ /X 2 2 ^ (/ A l + / A )'^> + 90 o 27. 2 & 
32t r 2 (W 
32t r 3 a 2 7q 2 
1 
or 
1 fjL^dr 
^llW 3 I ^ i-2_Z. ft 1 /(^2 -^l) 
^7 W, 
327r 3 a 3 n = {/q> 2 d' — 0^+/) 2 ^} yyr+ yTV+ 2 1 ^ ~ 2 (/H ~ hTZ7T 2 / )(L 3 —L x ) j- —' 
7„ 2_7„ 2 r* }\ 
/c 2 K l / 
V 
fd 
k*-k* 
(3G) 
V-V 
li'd 
K~K 
, 2 
Another equation between a, & i5 & 2 is found from the fact that the volume of the 
core is constant, i.e., (7) 
87 f 3 a 3 (A: 2 2 —& 1 2 ) = volume =msay 
therefore 
HhII 
“= Wd’ - O,+/)¥}( 1 - gj ■-y ^+2 ^ 
h 2 
1 ——H 
Z- 2 
lt() 
Write 
7,2 
A/o 
*iW=® 
( \ 
7 , 2 
_o 
/h 
h 2 H- 
i-h / 
(Ln —L,) 
r P 
U 
Then 
4777.IT 
W*-(1 ~*) +7hM~ 1 ) + V<»IF-Gi- f“) log®[ (37) 
To find a 2 , /I 2 , A 2 , e 2 , we must take account of one order higher and the terms in 
cos 2v. This gives another equation, which, with the previous ones (33a), will be 
sufficient to determine them. As we do not require them for our present purposes 
we shall postpone their consideration to another occasion. 
It remains to discuss the results already obtained, viz., the three expressions, which 
give /3 X , V, and the relation between a and the Jc’s (or It and r x , r 2 ). 
The formulae are, writing x for k x /k^ 
(pi+tO f — 
/dq 
iMi {log ~+3(l -*) J —ifi' jt-fx-Pj-log : 
a 
(30) 
5 D 
MDCCCLXXXV. 
