48 Mr Basset, On the Motion of a Ring [Jan. 31, 
rise to sharp edges; and the ring will be supposed to be generated 
by the revolution of S about OZ Then O will be the centre of 
inertia of the ring, OZ its axis of unequal moment, which I shall 
call the aais of the ring; and I shall cal! the circle described by @ 
the circular axis of the ring. 
2. Let the ring be introduced into an infinite liquid which is 
at rest, and held fixed; let the circular aperture be closed up by 
means of a plane diaphragm, whose area is o;.and let cyclic 
irrotational motion be generated by applying to every point of this 
diaphragm a uniform impulsive pressure «p, where p is the density 
of the liquid, and then let the diaphragm be removed. 
The velocity potential of this cyclic motion will be 
b=«Q, 
where Q is a monocyclic function whose cyclic constant is unity, 
and « is the circulation, round any closed circuit, which embraces 
the ring once only. 
The resultant momentum of the cyclic motion will be parallel 
to the direction of the impulsive pressure in the diaphragm, and 
equal to €,; and the energy to Kx«*/2, where 
6, = cpa — Kp ffOndS, 
Le) ae 
where n is the z-direction cosine of the normal to the ring drawn 
outwards, and dS an element of its surface. 
If the ring be set in motion, the kinetic energy and momentum 
of the ring and liquid will be determined by the equations 
2T =P (uw +0") + Rw’ + A (o7 +0) + Co, + Ke’...... (1), 
E=Pu, y=Bv,, FC=Rwt+ a 
N=Ao,, p=Ao,, v=Co, 
Since the liquid is incapable of producing a couple about the 
axis of the ring, w, = const. = w throughout the motion. 
Hence, if the ring be let go after the cyclic motion has been 
generated, it will remain at rest ; for the only possible motion will 
be in the direction of its axis, and consequently 
2T = Rw’? + Cw? + Ki = its initial value, 
therefore w=0. 
