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



14. Near the end of chapter vii. of the Tonempfindungen 

 Helmholtz makes prominent mention of the slipping of the 

 hammer on the anvil as an important cause of resultant tones, 

 and appears to regard it as exemplifying his mathematical 

 formula for the restoring force as a function of the displace- 

 ment. But it is clear that if the hammer, which holds the 

 drumskin, is liable to shift in its supports, the restoring force 

 cannot be a mere function of the displacement, but must also 

 depend on the relative position and relative velocity of the 

 hammer and anvil at the moment considered. I accept all 

 the consequences which Helmholtz deduces in the passage 

 in question from this slipping, including its application to 

 explain first difference-tones ; but I regard these consequences 

 as lying outside the range of his general mathematical for- 

 mulae as given in Appendix xii. 



15. To sum up my objections to the received mathematical 

 theory of resultant tones : — 



First. It assumes that the reaction of the drumskin against 

 the air is a definite function of the displacement of the drum- 

 skin from a certain fixed position, whereas this reaction 

 depends also on the position and motion of the further end of 

 the hammer at the time. 



Secondly. Even if the vibrations of the drumskin were in 

 accordance w T ith the received formulae, there is plenty of scope 

 for the introduction of additional constituents on the road 

 from the drumskin to the liquid in the cochlea. The auditory 

 ossicles, with their ligamentous supports and attachments, 

 probably serve to protect the oval window of the cochlea 

 against shocks and jars, and to smooth down asperities in the 

 wave-form, thus mitigating the harshness of sounds and 

 rendering them more musical. The changes thus introduced 

 are very unlikely to fulfil the special conditions required for 

 the vanishing of the common fundamental. 



Thirdly. The received theory makes the common funda- 

 mental, when not coincident with the first difference-tone, 

 depend on a term involving the cube or some higher power 

 of the displacement. When the primaries are as 3:5, the 

 fundamental 1 comes in as 2m — n, and depends on the cube 

 of the displacement. When they are as 4 : 11, the tone 1, 

 which Koenig found to be the loudest resultant, is 3m-w, 

 and depends on the fourth power. When they are in the 

 ratio 4 : 15, as in Kcenig's experiment with the simple tones 

 ut 5 and si 6 , the common fundamental ut z , which was the 

 only resultant tone heard, is 4m— m, and depends on the fifth 

 power of the displacement ; the first difference-tone, which 

 depends on the second power and should in theory be the 



