Mr. S. Taylor on Variations of Pitch in Beats. 61 



To find the corresponding number for the minimum position 

 (figs. 2 and 4), it is only necessary to change the sign of one of 

 the amplitudes ; the result therefore is 



Aj— Ag 



Aj being supposed >A2. 



The two principal cases are therefore the following : — 



I. Higher tone the louder, Aj^Ag, n^>nQ. 



For maximum intensity, number of vibrations per second 



_n^A^^]-n^^^_ A^{ni-~n^ _ A^{n^^n^) 



= /?Q-f- 



A. + As ''' A1 + A2 ' A^ + A 



For minimum intensity, number of vibrations per second 



_ n^A.-WgAg ^^ ^ k^[n^-n^) 

 Aj — A^ Aj A2 



II. Lower tone the louder, A2>A„ n^>n2. 



For maximum intensity, number of vibrations per second 



__ y^lA^ + ^^2A2 _^ ^^[n^-n^ _ k^{n^-n^ ^ 

 Ai+A^ -""^"^ A1 + A2 """^ A. + A^ 



For minimum intensity, number of vibrations per second 



■^2"*" 1 2""" 1 



These equations, which embody the conclusions already ob- 

 tained directly from the figures, are deduced by Helmholtz*, as 

 particular cases, from a more general analytical investigation. 



It will be observed that when Aj=A2 a discontinuity arises, 

 which indicates that the assumptions on which our equations 

 were obtained are here no longer applicable. The case has not 

 been considered by Helmholtz, but admits of easy direct treat- 

 ment. The displacement of the molecule depends on the ex- 

 pression 



sin ^irn-^t + sin ^irnjt, 

 or 



2 sin (?2j 4- n^TTt cos [n^—n^irt. 



The second factor, cos [n^ — n^irt, is a slowly varying function 

 of t which determines the number of beats per second. The first 

 factor vanishes whenever [n^-\-n^irt is a multiple of tt, i. e. 



whenever Hsa multiple of . This result is not, like the 



^ /I, ^-?^2 

 previous one, merely approximate, but rigorous. The successive 



* P. 622. 



