296 DYNAMICAL THEOEY OF SOUND 



about which the oscillations take place. For the rest, we have 

 octaves of the primary tones, together with a difference- and 

 a summation-tone. If the approximation were continued we 

 should obtain combination-tones of higher order, as in the 

 former case. 



When, as in the case of the tympanic membrane, the 

 free period 2?r/\//A is relatively long, the most important 

 combination- tone is the difference-tone (n^ n 2 ), on account of 

 the relative smallness of the corresponding denominator in (7). 



The theory of combination-tones here reproduced has not 

 been accepted without question. The difference-tones, as 

 already mentioned, were known as a fact since the time of 

 Tartini, and a plausible explanation had been given by 

 Thomas Young (1800). According to this view the beats 

 between the two tones, as the interval increases, ultimately 

 blend, as if they were so many separate impulses, into 

 a continuous tone having the frequency of the beats. The 

 difficulty of this explanation is that the actual impulses 

 during a beat are as much positive as negative, so that 

 it does not appear how any appreciable residual effect in 

 either direction could be produced, if the vibrating system 

 be symmetrical. It is true that if we turn to the figure on 

 p. 23, it is apparently periodic, with the period of the in- 

 termittence ; but from the point of view of Fourier's theorem 

 the lower harmonics are all wanting, and the only two which are 

 present are precisely the two which are used in constructing the 

 figure. On the Helmholtz theory of audition the intermittent 

 excitation of a particular resonator m times a second is a wholly 

 different phenomenon from the excitation of an altogether 

 distinct resonator whose natural frequency is m. Young's 

 view appears indeed to be inadmissible on any dynamical 

 theory of audition, at least in the case of infinitely small 

 vibrations. On the other hand it is true, as we have seen, 

 that given a finite amplitude, and an unsymmetrical system, 

 a vibration of the type shewn in Fig. 10, p. 23, does actually 

 generate (among others) a vibration whose period corresponds 

 to the fluctuations there shewn. The distinction between the 

 two theories might therefore, from a merely practical point of 



