of Attractive and Repulsive Forces. 177 



dW 



— = Cxsin^^ + a x ) + C 2 sm(2q l z + « 2 ) + Cssin^^ -i- « 3 ) + &c, 



+ C / 1 sin(^4- a'i) + C / 2 sin(2^ + otf 2 ) + C / 3 sin(3^ + «' 3 ) + &c., 

 + Q f \ sin (q 3 z + a"x) + &c, 

 in which equation, it is to be observed, all the terms are periodic 



in such manner that the mean of all the values of -=— at any 



given instant is zero. Since the constants Oi, C 2 , C 3 , &c, 

 a \i a 2? a 3? & c v are arbitrary quantities, and not limited in num- 

 ber, we may by the application of Fourier's Theorem draw the 

 inference that the series of terms in the first line of the above 



d^t . 

 expression for -=— is capable of representing any arbitrary form 



of vibration satisfying the condition that at a given instant the 

 velocities at positions separated by the constant interval X x are 

 equal. From this conclusion it follows that the production in 

 any manner of a musical note whose wave-length is X x involves 

 the simultaneous production of musical notes whose wave- 

 lengths are ~, — ^ &c. Experience confirms this result, the 



ear being able to detect, together with the fundamental note 



corresponding to the wave-length X 1? harmonic notes, as they 



are called, corresponding to the other wave-lengths. If we 



take the first and second series of terms in the expression for 



dW 



—=— } and suppose the ratio of X x to \ 2 to be exactly equal to 



az 



that of two low numbers, we shall have the theoretical expres- 

 sion of musical concords. If X± be to \ 2 in a ratio expressed 

 by that of two high numbers nearly equal, the same two series 

 will express the phenomenon of beats. In short, by taking ad 

 libitum the number of the series and corresponding wave-lengths, 

 we can express every species of sound, from a pure musical 

 note accompanied by its harmonics to irregular and confused 

 noise ; and it may, I think, be regarded as evidence that sounds 

 have actually the composite character indicated by this theory, 

 that noises partake so far of the qualities of notes as to be re- 

 cognizable by the ear as being in some cases grave and in 

 others acute. 



9. It will now be proper to introduce certain considerations 

 which will point out the bearing of the foregoing argument 

 on the subject signified by the title of this communication. 

 It has been shown how it is possible, by the aid of Fourier's 

 theorem, to give a complete mathematical theory of aerial 

 sounds, whether musical or unmusical. But this theory alto- 



Phil. Mag. S. 5. Vol. 2. No. 10. Sept. 1876. N 



