172 Prof. Barton and Miss Browning on 



impressed force. Finally, for p 2 = ?i 2 , S = tt/2, and this 

 corresponds with the case of maximum amplitude of 

 response. 



4. The smaller the damping of the responding system the 

 sharper is its resonance, the greater the damping the greater 

 is its range of resonance. That is to say, the smaller the 

 value of k the greater is the falling off of the response for a 

 given mistuning, and vice versa. For it is seen from the 

 first term on the right side of equation (4) that whenp 2 = n 2 

 the amplitude is a maximum, for n constant while p varies. 

 Further, when p 2 — n 2 is finite and of a given value it has a 

 less effect on the amplitude if the other term in the deno- 

 minator (2hif is large. 



By reference to the second term on the right side of 

 equation (4) we see that the ratio of successive amplitudes 

 of the free vibrations is e kjrlq = e kir/p nearly. But the 

 logarithmic increment X (per half wave) for this system is 

 the logarithm to the base e of this ratio. Hence we have 



_ kir p\ n\ , a 



\= — or k=— = — -, . . . . (b) 



p IT IT y 



where X = the log. dec. for the responding pendulum of the 

 same period as the forces. Thus by observations on the free 

 vibrations of a responding pendulum the value of k may be 

 found. 



It might be urged that in th** experimental arrangement 

 specified we have strictly speaking an instance of coupled 

 vibrations, and have not reached the ideal of forced vibrations. 

 That this is not the case may be ascertained as follows. 



On reference to '' Coupled Vibrations, II." (Phil. Mag. 

 Jan. 1918) we see that in coupled systems two superposed 

 vibrations occur, the ratio of their frequencies being 

 plq— \/(l + /3). Also by equation (24) p. 65 and (43a) 

 p. 68 of the same paper, we see that the ratio of the 

 amplitudes of these quick and slow vibrations for our re- 

 sponding systems is given by 



— M 



,J~ P '<r> e-<'- 1 WQ+P)= e -rr nearly for p large. . (7) 

 (l-f/o)/3 P 



In our experimental case p exceeds 2000 (being 700 gm. 

 /0"3 gm.), k is of the order one fifth, and /3 about one third. 

 Thus after 20 and 40 seconds, the ratio in question has fallen 

 to 1/20 and 1/1000 respectively. 



