TRANSACTIOXS OF SECTION A. 627 



The matter, however, is not so simple. All agree that notes corresponding- to 

 the difference tone are heard under some circumstances, but many deny that they 

 are produced as Von Helmholtz supposed, and would therefore deny that they are 

 combinatioa tones. Again, Von Helmholtz, who was the most prominent sup- 

 porter of the objective reality of these notes, was also the author of the theory 

 which explains their production within the ear itself. 



It is therefore better to begin with the second question. Do notes correspond- 

 ing in frequency with the combination tones accompany the two fundamental 

 notes as air-waves under any circumstances ? 



The physical evidence for and against an affirmative answer is as following : — 



Von Helmholtz stated that he had set membranes in motion by combination 

 tones produced by the siren, and air resonators in motion by combination tones 

 produced by the harmonium (Ellis, p. 157). I am not aware that the experiment 

 with membranes has been repeated in the same form, but 0. Lummer (' Verb, 

 Phys. Gesell.,' Berlin, 1886, No. 9, p. 66) claimed to have detected the tones by 

 means of the microphone. On the other hand, Konig (p. 130) denies that com- 

 bination tones are reinforced by resonators ; and Bosanquet satisfied himself that 

 the ordinary first difference tone is incapable of exciting a resonator. 



More recently, the writer and Mr. Edser, using as a resonator a tuning-fork 

 of frequency 64, the motions of which were detected by attaching to one of the 

 prongs a mirror which formed one of a system by which Michelson's interference 

 bands were produced, have obtained evidence of the objective character of the sum- 

 mation and difference tones produced by a siren. They have confirmed these 

 results in the case of the summation tone by a Eayleigh vane-resonator as modi- 

 fied by Boys ('Phil. Mag.,' April 1895, p. 341). 



The only objection which, as far as the writer knows, has been broug'ht against 

 these experiments is that the tones detected must be of very small intensity ; and 

 Mr. Bosanquet has stated (in a letter) that he does not wish to be understood as 

 denying the existence of very feeble combination tones. 



It is unnecessary to quote experiments made by various observers with tunino-- 

 forks, as the use of these instruments is in general opposed to the directions and 

 theories of Helmholtz. (Ellis, pp. 157, 158.) 



On the whole, then, the evidence appears to be in favour of the view that 

 objective notes of the same frequency as the combination tones do exist, at all 

 events in special cases. Their relative intensity to the fundamental notes has not 

 been determined, and is probably small. 



Turning next to questions of theory, three explanations have been given of 

 objective combination tones — viz., that they are due (1) to beats, (2) to finite dis- 

 placements of the vibrating particles, and (3) to intermittence. 



Among other objections to the first theory, Helmholtz pointed out that it 

 would not explain the summation tone. (Ellis, p. 156). 



Hence Konig suggested that the summation tone might be due to the beats of 

 partials (Konig, p. 126). This explanation requires that, if ;j and q are the fre- 

 quencies of the fundamentals, jo + rj = n (p - q) where n is an integer. The writer and 

 Mr. Edser have, however, obtained evidence of the objective existence of the summa- 

 tion tone when ;> / y = 16 / 9, so that n = 25 / 7 {loc. cit. p. 352). (See Ellis, p. 630.) 

 Appunn and Preyer have suggested that the summation tone is the beat tone 

 tetween the first partial of the higher note and the difference tone, for 

 ~P — (li -q)=l} + q- Konig (p. 127) strongly opposes the adequacy of this explana- 

 tion, which is contrary to his own observations on beats, and which fails to explain 

 why the difference tone should not produce equally permanent effects by beating 

 with the first partial of the lower note, thus giving 2q'^{p — q) = ?,q —p or p — Sq. 



The theory of finite displacements is due to Helmholtz, who has shown (Ellis, 

 p. 412) that if the elastic forces are not symmetrical about the position of equili- 

 brium, the fundamental tones will be accompanied by the second partials and the 

 difference and summation tones. 



He has, however, also proved (Ellis, p. 420) that if in an instrument such as 

 the siren the opening of one hole afl'ects the pressure under which the air is simul- 



k 



S S 2 



