of Consonances of the Form h : 1. 



493 



with the greatest intensity when the tones derived from terms 

 of lower orders are weak or evanescent. This fact has been 

 used by Konig as one of the most powerful objections to the 

 theory of combination-tones as hitherto expounded; and 

 indeed, the objectionable character of some of the hypothetical 

 derivations by combination given by Helmholtz* must have 

 struck many readers independently. 



60. I shall now examine Helmholtz's hypothesis of asym- 

 metry in a little more detail ; and I think it will appear that 

 it leads, by tolerably simple mathematical treatment, to the 

 development of the combination-tones of the higher orders 

 under the circumstances under which they actually exist and 

 independently of the combination-tones of the lower orders. 



61. Let represent the position of rest of a point free to 

 move along the line x between the points A and B, subject 

 to certain forces in that line. Suppose that these forces are of 

 the nature of springs tending to resist the departure from 0, 

 and that on arrival at the points A, B, at distances a, b, from 

 on either side, the springs ultimately go up against dead 

 walls, so that further displacement is resisted with an infinite 

 force. If we set off the forces as ordinates, they may be repre- 

 sented as in the figure ; and analytically they may be ex- 

 pressed by such an assumption as 



Jcx i 



V- 



('-9(»a 



Expanding this function in a series proceeding by powers of 



* Tonempfinthingen, 4th ed. p. 329; also p. 327, where the combina- 

 tions are supposed to be formed with the partials of the primaries as well. 



Phil Mag. S. 5. No. 71. Suppl Vol. 11. 2N 



