512 Mr. J. W. Peck on the Special 



point whose undisturbed distance from a fixed end is x. 

 V is the velocity of the elastic wave in the system. If the 

 arbitrary set of initial displacements be given by f(x), and of 

 initial velocities by <£(#), the value of y at time t is 



" . mirxV > mirVt I _ nnrVin , a . 



Sim T [A.«.- r +^ ? B„ S m-^-J; . (2) 



or more conveniently in terms of the fundamental period T, 



", . irnrx r A 2m7rt T -^ 2mTrt~\ /ON 



2 M - 7 -|A«ooB- T -+ 3is: B.Bm- T -J, • (3) 



wD ere 



2 C l 

 = lie 



. . . miTx 

 j(x) sin — j — ax, 



2 C l 

 B m = 7 <£(#) sin '^^dx. 



2 f ' , , , . ?>?7T.£ 



? * ( * )8m ~r 



The following deductions from these well-known results are, 

 of course, applicable to a variety of special cases, e. g. trans- 

 verse vibrations in thin strings, longitudinal vibrations in 

 wires, rods, &c, torsional vibrations in rods. 



It will be convenient to begin with two special cases : — 

 1. Suppose the system to start from rest so that (f>(x) = 

 for all values of x between and l. The rest epochs are 

 given by 



or 



«=?. (4) 



where k is any integer. This gives these epochs for any 

 harmonic constituent m ; and it is obvious that even in the 

 most general case both the sets of epochs occur always for 

 any harmonic constituent considered separately. But the 

 question is, whether among these epochs for any and every 

 constituent m are to be found those of the fundamental mode. 

 Only when this overlapping happens for every harmonic 

 mode, will the general rest epochs or undisturbed epochs 

 occur. In the present case it is seen from equation (4) above, 

 that Jc can always be chosen so as to be a multiple of m, what- 

 ever constituent is being dealt with, and therefore among the 

 rest epochs of any mode there will always be found those of 

 the fundamental. Hence in this case the general rest epochs 

 occur, and obviously at times T/2, T, 3T/2 ; &c. 



