498 Mr. R. H. M. Bosanquot on the Beats 



(oi ll+1 is ilio w+lth coefficient), 



Q + l )«»+i f e n f cos (n6— </>) <?/ n cos ((9 — tt<ft) \ 



besides other terms analogous to those shown above. 



72. In the neighbourhood of any consonance of the form 

 h : 1, the terms having the denominators (hp — q) 2 become 

 large ; this is Helmholtz's explanation of the origin of differ- 

 ence-tones, generalized. 



73. As the argument from the analytical expressions fails 

 to give perfect satisfaction unless the nature of the causes 

 involved be more directly demonstrated, I shall try to show 

 more simply how it is that this comes about. 



In periodic functions such as cos pt, cos (qt — e), the quan- 

 tities p, q are such that, if r, t' be the periodic times, 



, ■ 2tt 2tt 



pT = qT' = 27r, or p= — , tf=-^r 



If, then, 



Mt=1, 



Nt' = 1, 



M, N are the frequencies of the primaries, and 



^ = 2ttM, 2 = 2<7tN. 



In the case of a mistuned consonance of the form h: 1, the 

 denominator of the h difference-tone term in the above expres- 

 sion will be 47r 2 (AM~N)' 2 . And AM— N is the frequency of 

 the beat which gives rise to the transformation according to 

 all theories (putting h = 1 in the more general formula AM — kN). 



.'. t^ttf — tt is the time of duration of the beat of the result- 

 AM— N 



ant form, whether we call it the Smith's beat, or the bow of 



the pendulum curves. As the denominator diminishes, the 



time or duration of the beat increases. 



74. What happens, then, is that a force is developed, by 

 the influence of the higher terms in question, which acts for 

 a: time corresponding to the duration of the beat. And it is 

 matter of ordinary mechanical knowledge that, under these 

 circumstances, the space traversed is proportional to the square 

 of the time during which the action lasts ; so that when 

 the beat is lengthened the effect of the transformation is 

 strengthened. 



75. It is possible to found an independent treatment of the 

 subject on these considerations, the course of which would be 

 somewhat as follows. 



In mistuned consonances of the form h : 1 there are alter- 



