PROFESSOR W. M. HICKS ON THE THEORY OF VORTEX RINGS. 767 
Substituting these values, and writing p for d'/d, the four equations of condition 
reduce to the two 
L {pX 2 +A aX— n (np +1 )w 2 + nkau } — Mq(X — nu)(\ + nv) = 0 
qh(\-\-nu)(\—nv) — Al{(p-\-p)\ 2 —Aa\—n(np—l)v 2 ‘—nAav—n(n-\-l)w 2 p} = Q 
Whence the equation to determine X is 
I pX 2 +A«X— n(np-\- l)w 2 +??A«w , q(\ — nu){\.-\-nv) 
| q(\-\-mi)(\—nv) , (p-\~ p)X 3 —AaX— n(np — l)-y 2 — nAav-n(n-\- l)iv z p 
( 63 ) 
17. Case I.—No rotational core. Here the periods are given by 
X 2 — n{n- j-l)w 2 =0. 
The time of vibration is therefore 
2-7T (47Tf/A.' 2 ) 3 ''j 
n(n + 1) /i 2V 7 n(n + 1) f 
or by (39) .(64) 
= m i 
2T\.y/n(n + l) J 
and is therefore the same whatever the size of the ring. Moreover, when n is large, 
the time varies inversely as n. The ring is always stable for vibrations of this mode. 
The expression for the time does not agree with that found in the former paper 
[I. § 13]. The reason of this is stated in a note at the end of the present paper. 
Case II. Continuous case .—Here /x L = 0 , 7^=0, a=y=0, p— 1, q= 0, and the 
equations of motion reduce to two 
— (1 + pj/3 — { n(n — 1 )v 2 A~n(n -\-l)LV 2 p-\-nAav}/3-\-Aa$=0 
—Aafi — (1 +p)S — {n(n— l)iv°p-\~nAav}8=0 
Here putting 
bk =L sin 
(1 + p)X 2 — AaX — { n(n — 1 )v 2 -f-n(w+1) uAp + nAciv } = 0. 
Since the core is continuous 
therefore 
p. ~ — ^TtAtdk 2 
