DBS. GUY BARLOW AND H. B. KEENE ON THE ANALYSIS OF SOUND. 
161 
The tones present are therefore 
n x , ± ij 3n* F n x , o n F tti) cYc. 
The difference tones n — n x , 3 n — n x , &c., have zero value of frecpiency when 
with the amplitudes 
These represent the fundamental and “ subharmonic ” responses. 
We shall now suppose we are running through the fundamental response so that 
n — Hi nearly. Put n — n x — p where p is the small difference in frequency between 
interrupter and current. In going up, i.e. with increasing speed, through the response 
p changes from negative to positive. 
The general expression may now be written 
n 
n i> 3 w n jDffi tYc., 
A 
7T 
rA. 
7T 
l — , &c. 
7r 
y' = ~ sin n x t + — {cos pt — cos (2 n—p) t + } cos (2 n+p) t—-$ cos (4 n—p) t 
A IT 
+ -g cos (4 n+p)t — } cos (6n— p) t + &c.}. 
The frequencies of the components are 
p, n u 2 n + p, 4 n ± p, Gn + p, &c. 
The galvanometer responds to p. It is seen that there is a series of beating tones, 
2 n, 4 n, Gn, &c., and all these beat with the same frequency 2 p, i.e. twice the frequency 
of the oscillations of the galvanometer at the same instant (confirmed by experiment). 
The amplitudes of the beating tones are never equal, but they tend to equality for the 
higher harmonics of the series. This explains the beating which is heard in the tele¬ 
phone even when n x is well below the limits of audibility. 
For synchronism (p = 0) there is produced the single note of fundamental n x con¬ 
taining the even harmonics 2 n, 4w, &c. The original tone n x may be inaudible, and in 
any case it is weakened by interruption, but the addition of the harmonics will in general 
render audible the resulting note n v . 
AVhen n is near 2 n x put n — 2 n x = p. The component frequencies are then 
n u n x + p, 3 n x -j- p, 5 n x + 3 p, ln x + 3 p, &c. 
In this case there is no galvanometer response. There is only one beating pair, n x , 
n x + p, the amplitudes A/2, A/n of which are sufficiently near equality to give a marked 
beating, but, of course, this will not be heard unless n x is within the audible range. 
It will be noted that the beats have the frequency p in this case, instead of 2 p as above. 
Similarly, when n is near 3 n x it is easily shown that there is no response and no beating. 
[Confirmed by experiment.] 
