of Consonances of the Form h : 1. 499 



nate increases and diminutions of the maximum resultant dis- 

 placement, the duration of which can be arrived at by the 

 considerations employed by Smith in determining the duration 

 of beats. The duration of one such increase and diminution 



can be shown by the known formula to be r™ — at* 

 J AM— N 



Assume that the transmitting mechanism of the ear pos- 

 sesses such powers of transformation that any regular sequence 

 of increases and diminutions of maximum resultant displace- 

 ment is capable of giving rise, by transformation, to a subjec- 

 tive note having the same period as that of one increase and 

 diminution. This assumption only differs from that made 

 above in definiteness of form ; for the algebraic series which 

 is above proved to give rise to transformations of this de- 

 scription, is itself an assumption. 



It immediately follows, by considerations differing little 

 from those made use of in the ordinary investigation of the 

 motion of a particle under the action of a uniform force, that 

 the coefficient of the term in question will contain the square 



of the periodic time — that is to say, the coefficient //tit-m^ S 



and this is the essential point proved by the more complete 

 analytical investigation above given. 



76. Though perhaps defective as a complete demonstration 

 of the rationale of the origin of difference-tones, these consi- 

 derations render the general meaning of the coefficients of the 

 difference-tone terms in the above equations tolerably clear. 

 And we have thus sketched a method, in which the doctrine of 

 transformation arising out of the Smith's beats, as the resultant 

 forms pass through the transmitting mechanism of the ear, 

 forms the basis for the further explanation of the phenomena 

 of beats as we find them. 



The Resultant Wave-forms of Mistimed Consonances. 



11. I am principally acquainted with these forms as drawn 

 by means of Donkin's harmonograph. The curves (Plates 

 IV.-VII.) that accompany this paper exhibit all the points 

 on which it will be necessary to touch. 



78. It is hardly possible to be acquainted with these curves 

 without seeing that the figures formed by the vertices which 

 occur in the curves are in some way related to the phenomena 

 of the mistimed consonances. And as I had myself consider- 

 able difficulty in coming to definite conclusions as to the real 

 nature of this relation, and do not know of any published dis- 

 cussion of the subject, I add this article dealing with the rela- 



