1887. | un an Infinite Lrqud. a 53 
=) Let the roots be —p and —q, where g>p>1. 
1—Dsin’¢ 
“We must put cos 0 = Tae TOSRa 
where Des MGY seal ‘ 
(Oar li 
@+t) 7 +1) 
In order to obtain the path described by the centre of inertia 
O of the ring, we must substitute the value of @ in terms of ¢ in 
(4), and integrate the result. 
We can however ascertain the character of the motion of O 
without integrating (4). For differentiating (5) we obtain 
Ad=—* sin O- Ga (pj) sin 0 cos 8 
ERO? Th 
Therefore i= ale : 
bo 
and eo - (0 —o). 
Also the value of 2 may be written 
a= is (p- = cos’ 8 +s So cos a). 
The term in square brackets has its greatest value when 6 = 0, 
in which case 2=0; hence 2 can never become negative. The 
motion of O is such that O moves along the initial direction of the 
axis of the ring with a uniform velocity, superimposed upon which 
is a variable periodic velocity; and at the same time vibrates 
perpendicularly to this line. 
Z= 
[4. The equations of motion in the last article may be ob- 
tained in a different manner by means of Sir W. Thomson’s Theory 
of the Impulse. 
ie 
