DRS. GUY BARLOW AND H. B. KEENE ON THE ANALYSIS OF SOUND. 
163 
(c) The beating tones 
~[(h l) 2 n ± p, 
IT 
(i i) ± p, 
(l, y) 6n ± p, &c.] beating at 2 p, 
— [( 1 , 1) n ± 2 p, 
7T 
( 1 j ■ 5 ) + 2 p, 
(|, y) 5n + 2 p, &c.] beating at 4 p, 
^[(i,n-2»±3p, 
7r 
( 1 , 4) 4 n ± 3 p, 
(b i) bw ± 3 p, &c.] beating at Qp, 
&c., 
where the numbers in ( ) brackets indicate the relative amplitudes of the beating pairs. 
The sound in the telephone is, therefore, remarkably complex. It will be noticed 
that the constituents of the original note (a) have frequencies which may be written 
n 4 - p, 2w -f- 2p, 3 n + 3 p, &c. 
and these can be associated with components in (c) to give beating at frequency p. This 
appears to be the explanation of the curious fact that in an experiment with a current 
rich in harmonics the beating heard agrees in frequency with the galvanometer oscilla¬ 
tions p, although with a S.H. current the beating frequency has the double value 2 p. 
Examination of the above table shows that this abnormal beating at p depends essen¬ 
tially on the co-operation of consecutive harmonics of the original current. In all cases 
the ear appears to appreciate only the slowest beats which are present. 
The more general case in which the interruptions are of unequal intervals and unequally 
spaced may be treated in a similar way to the above. It is only necessary that the 
interruptions shall be strictly periodic so that they may be represented by a Fourier 
series. 
Simple Response .—If the galvanometer system has a natural undamped period — 
and is made exactly dead-beat, the equation of motion due to a S.H. force is 
x + 2 n 0 x + n 2 x —‘F cos pt. 
The solution of this for the steady state gives the amplitude of motion 
a — 
F 
<+p 2 
When the frequency p of the force becomes very small compared with that of the galvano- 
F 
meter n 0 , the amplitude has the maximum value a m = — corresponding with the centre 
n 0 
of the response. 
Hence 
2 
a 
n 0 
n 2 +p 2 
2 A 
VOL. CCXXII.-A. 
