Prof. A. M. Mayer's Researches in Acoustics. 269 



will always be periodic if the sensation corresponding to their 

 generating motions is that of sound. Now Fourier has shown, 

 and states in his theorem, that any periodic curve can always be 

 reproduced by compounding harmonic curves (often infinite in 

 number) having the same axis as the given curve and having the 

 lengths of their recurring periods as 1, J, 3, J, &c. of the given 

 curve ; and the only limitation to its irregularity is that its ordi- 

 nates must be finite, and that the projection on the axis of a 

 point moving in the curve must always progress in the same di- 

 rection. Fourier demonstrates that the given curve can only 

 be reproduced by one special combination, and shows that, by 

 means of definite integrals, one can assign the definite sinusoids 

 with their amplitudes and differences of phase. Now Helm- 

 holtz* has shown that differences of phase in the constituent 

 elementary sounds do not alter the character of the compo- 

 site sound, and, therefore, that although the forms of the curve 

 corresponding to one and the same composite sound may be 

 infinite in variety (by reason of differences in phase in the com- 

 ponent curves), yet the composite sound is always resolved 

 into the same elements. This experimental result of Helmholtz 

 also conforms to the theorem of Fourier in reference to the 

 curves projected by such motions ; for he has shown that only 

 one series of sinusoidal resolution is possible. 



Fourier's theorem can be expressed as follows :— The con- 

 stants C, C t , C 2 , &c, and a v a 2 , &c, can be determined so that 

 a period of the curve can be defined by the following equationf : — 



y = C + C x sin (-^- + «i ) +C 2 sinY 2— +aA + 



But Fourier's theorem is the statement of a mathematical 

 possibility; and it does not necessarily follow that it can be im- 

 mediately translated into the language of dynamics without 

 experimental confirmation ; for, as Helmholtz remarks, " That 

 mode of decomposition of vibratory forms, such as the theorem 

 of Fourier describes and renders possible, is it only a mathe- 

 matical fiction, admirable because it renders computation facile, 

 but not corresponding necessarily to any thing in reality ? Why 

 consider the pendulum-vibration as the irreducible element of 

 all vibratory motion ? We can imagine a whole divided in a 

 multitude of different ways ; in a calculation we may find it con- 

 venient to replace the number 12 by 8 + 4, in order to bring 8 



* Tonempjindungen, p. 190 et seq. 



t For other and more convenient forms of expression of this theorem, 

 as well as for a demonstration of it, see pp. 52 and 60 of Donkin's 'Aeons- 

 tics' — the most admirable work ever written on the mathematical theory of 

 sound. 



