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



and trie effect is greatly enhanced if the key of C 64 is held 

 down. In the latter case its note continues to be heard for 

 a long time after the keys which were struck are released. 

 From these experiments it appears probable that the sounding- 

 board of a piano possesses the same property which we have 

 proved to exist in the violin. 



12. I now come to the explanation of the experiments of 

 Professor Eiicker and Mr. Edser (Proc. Phys. Soc. vol. xiii. 

 p. 412, Phil. Mag. 1895, xxxix. p. 341). They were made 

 with a Helmholtz siren, and in each instance the two primaries 

 were produced in the same box, sometimes the upper and 

 sometimes the lower box. The following explanation is a 

 development of suggestions contained in Appendix xvi. of the 

 Tonempfindungen. 



The rate of escape of air from the box containing the two 

 rows of holes which are employed may as a first approxima- 

 tion be assumed to be jointly proportional, at each instant, 

 to the aperture for escape and the differential pressure 

 which produces the escape. Again, this differential pressure 

 may be regarded as the algebraic sum of two terms, one of 

 them constant, and representing its average value, while the 

 other represents the difference from the average due to the 

 varying amount of the aperture from instant to instant. As 

 a first approximation, equal increments of aperture must be 

 regarded as producing equal decrements of pressure, so that 

 the variable term will be proportional (with reversed sign) to 

 the excess of the aperture above its mean value. This excess 

 (defect being counted negative) will be a periodic function of 

 the time, and if the ratio of the two primaries in lowest terms 

 be m : n, the frequency for the complete period will be repre- 

 sented on the same scale by 1. In other words it will be the 

 period of their common fundamental. 



Let the aperture at time t be expressed in a Fourier series, 

 6 being put for 27rt/T, where T is the complete period ; and let 

 the variable part of the expression be denoted by/(0), while 

 a denotes the mean aperture, so that the aperture at time t is 

 «o+/(0)- We shall have 

 /(0) = Asin0+ ... +a 1 sm(md + € ] )+ . .. + \ sin (nO + e 2 ) + 



The largest amplitudes will be a v and b x corresponding to the 

 two primaries ; but A, which corresponds to the fundamental, 

 is likely to be sensible. 



The pressure at time t is proportional to 



0-/W, 



C being a constant ; and the aperture is 



«o +/(0). 



