50 Mr Basset, On the Motion of a Ring [Jan. 31, 
Substituting the values of w and w from (3) we obtain, 
AC 2 Agee (p+ +)t + 0 cos 8+ 63 (pr p) eos’? (5) 
=f (@) say, 
where o is the initial value of 6. 
The character of the motion depends upon the roots of the 
equation f (@) =0, which we shall now consider. 
The roots are 
Case I. Let R> P. 
Tn order that the roots may be real, we must have 
: R 
Cie Wevareest 
If this condition be satisfied, one root will be positive and < 1, 
and the other will be negative and less than —1. Hence 6 will 
vanish when 6 has some value 8 lying between 0 and 77/2, and the 
ring will oscillate between the angles 8 and — £. 
/ 
But if o>t,/aE 
both roots will be imaginary, and @ will never vanish. Hence the 
ring will make a complete revolution. 
Cage 10 Ibem JP ss Jt, 
In this case both roots are real, and one of them is positive and 
<1 provided be sufficiently small; but if be sufficiently large 
both roots will be negative and <—1. In order that one root 
should not be <— 1, it 1s necessary that 
ae 
JAR 
If this condition be satisfied, the ring will oscillate between the 
angles 8 and — 8, where 8 lies between 0 and 7; but if 
26, 
NENTS 
the ring will make a complete revolution. 
@> 
