200 Prof. J. D. Everett on Resultant Tones. 



12, 15, and 18 open, the resultant tone of frequency 3 on the 

 same scale is the most prominent tone in the whole volume of 

 sound. 



3. The view which I desire to put forward is closely con- 

 nected with the well-known theorem of Fourier, that eveiy 

 periodic variation can be resolved in one definite way into 

 harmonic constituents, whose periods must be included in the 

 list 1, J, J, J, &c, where 1 denotes the period of the given 

 variation itself. The corresponding frequencies will be as 

 1, 2, 3, 4, &c. 



In the majority of cases, when this analysis is carried out, 

 the fundamental constituent, represented by 1 in the above 

 lists, is the largest or among the largest ; but in the case of 

 a variation compounded of two simple tones with frequencies 

 in the ratio of two integers, neither being a multiple of the 

 other, the fundamental will be absent, and the Fourier series 

 will consist of only two terms, which in the language of 

 acoustics are harmonics of the fundamental. 



4. Clearness of thought is facilitated in these matters by 

 supposing a curve to be drawn, in which horizontal distance 

 represents time and vertical distance represents the quantity 

 whose variation is in question. Since the variation is periodic, 

 the curve will consist of repetitions of one and the same form, 

 in other words it will consist of a number of equal and similar 

 waves, and the wave-length stands for the complete period of 

 the variation. 



The point on which I wish to insist is, that if such a curve 

 representing the superposition of two harmonics of the funda- 

 mental is in the first instance very accurately drawn, and is 

 then inaccurately copied in such a way that all successive 

 waves are treated alike, the inaccuracy is morally certain to 

 introduce the fundamental. 



Let y denote any ordinate, and 6 the time (or abscissa) 

 expressed in such a unit that 27r is the numerical value of 

 the wave-length or period ; then the amplitude of the funda- 

 mental is the square root of A 2 -{- B 2 , where 



1 f 2 *- 1 f 2,r 



A=- ycosOdO, B=- y S m0d0. 



7tJ j ttJ j 



In the original curve both A and B vanish. Let y' be the 

 altered value of y in the new and inaccurate curve, and let 

 z denote y' —y, then we have, between the above limits, 



jV cos 0J0= jV cos Odd +jr cos 6d$=$z cos 6d0, 

 since (*?/ cos Odd is zero. 



