442 Lord Rayleigh on Dynamical Problems in 



law of independence fails during the development of the 

 vibrations from a state of rest under a vigorous bombardment. 

 The investigation of this matter is accordingly more com- 

 plicated than in the case of the free masses, and I do not 

 propose here to enter upon it. 



In a modification of the original problem of some interest 

 even the stationary distribution is not entirely independent 

 of phase. I refer to the case where the bombardment is 

 from one side only, or (more generally) is less vigorous on 

 one side than on the other. It is easy to see that a one-sided 

 bombardment would of necessity disturb a uniform distribu- 

 tion of phase, even if it were already established. The per- 

 manent state is accordingly one of unequal phase-distribution, 

 and is not, as for the symmetrical bombardment, independent 

 of the vigour with which the bombardment is conducted. 



But in one important particular case the simplicity of the 

 symmetrical bombardment is recovered. For if the number 

 of projectiles striking in a given time be sufficiently reduced, 

 the stationary condition must ultimately become one of uniform 

 phase-distribution. 



Under this limitation it is easy to see what the stationary 

 state must be. Since the ultimate distribution is uniform 

 with respect to phase, it must be the same from whichever 

 side the bombardment comes. Under these circumstances it 

 could not be altered if the bombardment proceeded indiffer- 

 ently from both sides, which is the case already investigated. 

 We conclude that, provided the bombardment be very feeble, 

 there is a definite stationary condition, independent both of 

 the amount of the bombardment and of its distribution between 

 the two directions. It is of course understood that from 

 whichever side a projectile be fired, the moment of firing is 

 absolutely without relation to the phase of the vibrator which 

 it is to strike. 



The problem of the one-sided bombardment may also be 

 attacked by a direct calculation of the distribution of ampli- 

 tude in the stationary condition. The first step is to estimate 

 the effect upon the amplitude of a given collision. From (6), 

 if v! be the velocity before collision, and u after, 



2q 



u = u'+ ., ^ (v— u f ). 



1 + g 



The fraction 2q/(l-\-q) occurs as a whole, and we might 

 retain it throughout. But inasmuch as in the final result 



