Prof. J. D. Everett on Resultant Tones. 1S9 



freezing- vessel ( — ~ , cf. page 683) an hundredfold. The 



accuracy for which Jones strives would have been attained 

 therefore, cceteris paribus, by using a vessel not of a 1 litre's 

 capacity but of that of a million litres ! " Loomis un- 

 doubtedly worked with precaution " 



Gottingen, November 1895. 



XXIV. On Resultant Tones. 

 By Professor J. D. Everett, F.R.S* 



1. ri^HE received theory of the generation of resultant tones 

 JL in the ear may be summed up with rough accuracy 

 as follows f: — 



The druinskin being pulled inwards by the end of the 

 handle of the " hammer/' which is attached to its centre, 

 offers unsymmetrical resistance to displacement in the inward 

 and outward, directions, so that the equation for the move- 

 ment of its centre in free vibration would be 



— x = co 2 x + ax 2 , 

 or to a closer approximation 



— x = co 2 x + ax 2 + (3x*, 

 co, a, ft being constants. 



The value of co when the second is the unit of time is less 

 than 60, hence the frequency of free vibration, being coj2ir, is 

 less than 10. 



When two harmonic forces of frequencies m and n act upon 

 the drumskin, they produce, in addition to their own tones, 

 certain " resultant tones," the one of largest amplitude being 

 the "first difference-tone," of frequency n—m. The next 



/ 71 — 17l\ 2 



largest amplitude is about I ) of this, and belongs to 



the "first summation- tone/' of frequency n + m. Neither 

 of these tones will be audible unless the excursion x is so large 

 that ax 2, is sensible in comparison with co 2 oc. There will also 

 be difference-tones of frequencies 2?n—n and 2n—m, but 

 neither of these will be audible unless fix' 6 is sensible com- 

 pared with co 2 x. 



2. This theory does not appear sufficient to account for the 

 loud resultant tones which are sometimes heard. When a 

 Helmholtz siren is driven rapidly, with the rows of holes 9, 



* Communicated by the Physical Society : read January 24, 1896. 

 t See Tonempfindungen, Appendix xii. ; Rayleigh on Sound, art. 68 ; 

 Bosanquet, Proc. Phys. Soc. vol.iv. p. 240, arts. 57-69. 



