728 
PROFESSOR W. M. HICKS ON THE THEORY OF YORTEX RINGS. 
These conditions of stability only reach so far as fluted vibrations and pulsations 
are concerned. The question of the stability for twisted and beaded vibrations is not 
considered. J. J. Thomson has proved that simply continuous rings of the same 
density as the rest of the fluid are stable for these kinds of vibrations. The general 
case yet remains to be investigated. 
Some of the simpler results are here collected for the sake of reference. 
(distance of from sectional centre)/?" 
-^(l-x+logl /x) 
(38) 
(distance of C 2 from sectional centre)/?’ 
r 
4 a 
x 
1 —x 
log l/x 
(38) 
a? ,3 ( 1 — x) = - 
m 
2tt 2 
r 2 2 _ 2 / , A2 , /V , o nl-X-xlogllx , „ , log l/aj 
—^zp=^p—{pi+nr +—+ 2 /*— (i-xf — 
\—x 
2x 
’( 1 + p)=(l-xV+(l+x)-^,l°g 1 /x. . . 
(43) 
(42) 
The general formula for the velocity of translation is given in eq. (44), the following 
are cases 
y =^T-i) . (45) 
No core 
cd 
a. 
II 
> 
47 ra 
Continuous core 
< 
II 
47ra 
Ordinary ring 
47ra 
El. 
t H )~'i>2ira /x. 2 p 
(46) 
8a 
For hollow with no added circulations 
where 
V= El. 
87 ra 
64a 2 
lo g-Y7 
TT 
( 48 ) 
jx = circulation due to vortex filaments 
fi l = added inner circulation 
jx. 2 = wdiole outer circulation 
p = ratio of density of outside fluid to density of core 
m = volume of core 
r — outer sectional radius 
