'Epochs in Vibrating Systems. 



provided 



. Ti , . To , . To 



arc tan T = ^ arc tan r ,f = A arc tan ~ == . 



515 



(13) 



the angles being taken as before. 



The times for the undisturbed epochs o£ a given harmonic 

 are obviously in quadrature with those for the rest epochs ; 

 but the set of conditions (10) which make the general rest 

 epochs possible are incompatible with those which make the 

 general undisturbed epochs possible, and vice versa. The 

 conditions (10) and (13) evidently agree with those of the 

 special cases 1. and II. above. In I. all the angles of the 

 condition (10) become zero, and of the condition (13) 90° 

 or multiples thereof. The reverse holds in II., and the 

 necessity for the absence of the even harmonics in the cases 

 remarked upon is shown by (10) and (13). 



These results are all very clearly illustrated by the usual 

 graphical representation for simple harmonic motion. 



Take a line XX' to represent the rest position, YY' the 

 undisturbed position. Represent each harmonic by a vector 



from rotating in the positive direction with a uniform 

 angular velocity m times that of the fundamental. The 

 general motion will then be represented by the group o\' 

 vectors (which are infinite in number though in any actual 

 system only the earlier ones in the harmonic series will be 

 appreciable) rotating simultaneously and starting all at once 

 from positions scattered round the circle. These initial posi- 

 tions will be determined by the initial conditions. A general 



