418 M. Duhamel on the Multiple Sounds of Bodies. 



defines the state of a string which makes at the same time 

 vibrations of the first and the third order, and which conse- 

 quently gives at once the fundamental note and its twelfth ? 

 Why should the string rather give the fundamental sound 

 than its twelfth; since on making a = only the said twelfth 

 will absolutely be heard, and making §=0, only the funda- 

 mental note will be heard ?" 



It must be acknowledged that this reasoning is little con- 

 clusive ; for might it not be that a single sound should be 

 heard, and be determined by the relation of a. to §? and if 

 several were heard, why should they be only those obtained 

 when a or § are null ? 



The obscurity of all this reasoning was inevitable; it par- 

 takes of the faulty direction which this great philosopher follow- 

 ed : he tried to demonstrate directly that two sensations must 

 be experienced at once, instead of seeking simply to refer the 

 phsenomenon to another. This reduction is the object which 

 ought always to be proposed ; and even when one class of 

 phaenomena is referred to another class as yet little known, 

 science has always made progress, new relations being disco- 

 vered, and instead of two difficulties, but one remaining. In 

 the present case, the object cannot be, as we have already said, 

 to explain the double sensation ; we must only attempt to 

 bring the phaenomenon investigated into another class, in 

 which this double sensation is admitted. 



Struck by the want of solidity in the reasons alleged in 

 favour of these various theories upon a question in itself so 

 interesting, I endeavoured, as Lagrange required, to ascer- 

 tain the absolute movement of the diflf'erent points of the vi- 

 brating body. Reasonings applicable to every kind of body, 

 followed by precise calculations relative to the simple case of 

 strings, led me to a proposition which may be stated thus : — 



When a body is made to vibrate by several causes which would 

 produce separately the simple sounds which it can give, its sur- 

 face generally divides itself into a certainjinite number of parts, 

 in each of which the vibrations have unequal durations. These 

 different durations have relation to sounds corresponding to the 

 different causes; and we are in the same position as if we had 

 several separate surfaces, each having a peculiar movement of 

 vibration. 



In order to ascertain the truth of this proposition in the 

 simple case of two coexistent sounds, I must first observe that 

 if each of them existed singly, it would correspond to a differ-, 

 ent system of nodal lines. Now if the two causes which would 

 produce each of these movements are made to act at the same 

 time, or successively, a movement would result composed of 



