155 



of reference, and the era of reckoning for the time, we have finally, 

 for an approximate solution of a suitable kind, 



=Acosp+Bcos at, 



r)= A sin pt + B sin at. 



The terms B, in this expression, represent a circular motion of 

 period , in the positive direction (that is, from the positive axis 



O 



of to the positive axis of ??), or in the same direction as that of the 

 rotation w ; and the terms A represent a circular motion, of period 



, in the contrary direction. Now, w being very great, p and a are 



p 



very nearly equal to one another ; but p is rather less than a, as the 



following approximate expressions derived from their exact values 

 expressed above, show : 



. 1 X 4 IX 4 t 1 X 4 1 X 4 



p=n + 8^~S^ : * + 8^ + 8^ 



Hence the form of solution simply expresses that circular vibrations 

 of the pendulum in the contrary directions have slightly different 



periods, the shorter, , when the motion of the pendulum follows 



0* 



that of the arm supporting it, and the longer, , when it is in the 



p 



contrary direction. The equivalent statement, that if the pendulum 

 be simply drawn aside from its position of equilibrium, and let go 

 without initial velocity, the vertical plane of its motion will rotate 



slowly at the angular rate (a p), is expressed most shortly by 



taking A=B, and reducing the preceding solution to the form 



= 2A cos ist cos n't, 



rj = 2 A sin rst cos n't, 

 where 



1 IX 4 



n'= (er + p)> or, approximately, n'ssn-f- j-^ 



and 



1 / s . i IX 4 

 Tz=.-(a p), or, approximately, -8*=- -5. 



2 o o> 



It is a curious part of the conclusion thus expressed, that the 

 faster the bearing arm is carried round, the slower does the plane of 

 a simple vibration of the pendulum follow it. When the bearing 

 arm is carried round infinitely fast, the plane of a vibration of the 



