54 Mr Basset, On the Motion of a Ring [Jan. 31, 
Let AB be the initial position of the projection of the ring 
on the paper, which is supposed to be perpendicular to its plane. 
The impulsive forces which must be applied to the ring and 
barrier in order to produce the initial motion, consist of a linear 
impulse €, and a couple G=Ao, about the diameter which is 
perpendicular to AB. Hence, if OC = Ao/¢,, the impulse of the 
whole motion consists of a linear impulse €, along OP. 
Since there are no impressed forces, it follows that if A’B’ be 
the position of the ring at any subsequent time, the motion must 
be such that it could be instantaneously produced by applying to 
the ring and barrier an impulse €, at P along OP. This impulse 
may be supposed to be applied by means of proper ag 
connecting the ring and barrier with P. 
If @ be the angle which the axis of the ring makes with OP, 
the conditions that the force constituent of the impulse should be 
equal to €, lead to equations (3). The condition that the couple 
constituent should vanish gives 
Aé — (Rw + EN IEG = (0) 
or A6=£, PC’ cos 6 = 62+ Ao. 
Differentiating with respect to ¢, substituting the value of 
é in terms of @ from (4), and then integrating, we shall ob- 
tain (5). 
Since the momentum due to the circulation alone is always 
perpendicular to the plane of the ring, it follows that if a ring 
initially at rest be set in motion by means of a couple about 
a diameter, the direction of this momentum will be changed; 
and the opposition which the liquid exerts against this action on 
the part of the ring will produce a couple tending to oppose the 
rotation of the ring. Also, since the impressed couple can produce 
no effect on the lear momentum of the system, it follows that 
the effect of changing the direction of the momentum due to the 
circulation, will be to cause the ring to move with a velocity of 
translation, which gives rise to a linear component of momentum 
of the whole system, such that the resultant of the latter and €, 
(whose direction has been changed) must be a momentum equal 
to ¢,, and whose direction coincides with the original direction 
ole. 
5. We shall now investigate the stability of the motion of 
a ring, which is moving parallel to its axis in the direction of 
the cyclic motion. 
