Epochs in Vibrating Systems. 513 



But the general undisturbed epochs will in this case not 

 always occur. The condition is 



cos ^— = 0, 

 or 



-^ «> 



giving the series o£ times for any harmonic m. But here the 

 cancelling of 2k + 1 by m can only occur when m is an odd 

 number. Hence among the undisturbed epochs of any odd har- 

 monic are always found those of the fundamental ; but this 

 coincidence will not occur for the even harmonics. Hence 

 for a system starting from rest, the general undisturbed epochs 

 will only occur if the initial displacements be such that the 

 even harmonics are absent. 



II. Suppose the system to start from its undisturbed con- 

 figuration, i. e.f(x)=0 for all values of x between and V. 

 This gives exactly the reverse of the previous case. The rest 

 epochs do not in general occur ; but will, if the initial set of 

 velocities be such that the even harmonics are absent ; while 

 the undisturbed epochs will always occur. This may be 

 shown by taking the special form of equation (3) for this case 

 and treating it as above. 



III. The general case in which both the displacements and 

 velocities are given arbitrarily. Taking equation (3) above, 

 the condition that the rest epoch should occur is 



cot'^^- 2 ^ (6) 



Observing that A m is a distance (or angle) and B m a velocity 

 (linear or angular), denote 27rA m /B TO by the time r m . Then 

 equation (6) may be written 



t_ __ arc cot (T ffl /T m ) 4- kir ,^. 



T 27T7JI 



where T m is the period for the mode m, and h is any integer. 

 This gives the times of recurrence of the rest epochs for any 

 harmonic m ; but it is clear that in general the times of the 

 epochs for the fundamental will not be found among those of 

 the constituent harmonics : for r m and T,„ are quite inde- 

 pendent, the first depending upon the initial conditions of 

 displacement and velocity, the other only upon the elastieiry 

 (or its equivalent), the inertia, and the extent of the system. 



