1887. ] in an Infimte Liquid. 57 
0) 
Ri Fu =0, 
As, +h y=0, 
poms 
From the first and third equations we obtain 
pee WH 
w=w' sin ata), 
Le Gt i a) 
U= —a> COS ( A é 
The fifth equation gives 
o, = - sin (F a “) + const. 
The second and fourth give 
v = const., 
Wes 
Urs Te Ae 
These equations show that the motion is stable for all displace- 
ments which do not tend to remove the centre of inertia from the 
plane of its motion; but the motion is unstable for all displace- 
ments which tend to produce this effect. If the disturbance is 
such that v=0, the disturbed motion will still be stable, but 
the axis of rotation will be shifted through a certain angle. 
vt + const. 
7. A third kind of steady motion, which is helicoidal, is ob- 
tained by first communicating to the ring an arbitrary angular 
velocity © about its axis; secondly by applying an impulsive 
couple G about an axis inclined at an arbitrary angle a to the axis 
of the ring; and thirdly by applying a determinate impulse in the 
plane of the axes of the ring and couple. 
In order that steady motion may be possible, it is necessary 
that v and therefore 7 should be zero throughout the motion. This 
condition may be secured by means of an impulsive force whose 
components are X = — (sina, Z=F, 
