M. Duhamel on the Multiple Sounds of Bodies. 419 



those two movements, according to the general principle of 

 superposition of small movements. Whence it follows that, 

 in all the points of the nodal line of any one of these two move- 

 ments, the other movement alone will be produced. But it is 

 easy to ascertain, by calculation, that the neighbouring points of 

 one of these lines will make an equal number of vibrations ; 

 and in passing, for example, from that in which the number of 

 vibrations is greatest towards the other, there will be a certain 

 number of vibrations the amplitude of which will successively 

 diminish, and will become null when we pass the line of sepa- 

 ration of the two regions related to each of the sounds : the 

 number of the vibrations will then remain the same until we 

 reach the limit of the region entered. 



This reasoning shows clearly that the surface will divide 

 generally into two or more parts, in each of which the num- 

 ber of vibrations will correspond to one of the two sounds ; 

 and it was very natural to conclude from thence that these 

 two sounds would be heard as proceeding from different sur- 

 faces. 



This proposition appeared entirely new when I stated it in 

 1840. And, in fact, in the works published up to that time, 

 none but vague suppositions are to be found, such as are met 

 with in the work of Father Mersenne, and to which he did 

 not himself adhere ; or again, some accidental result of calcu- 

 lation offered in particular examples, and from which more- 

 over no induction was drawn similar to mine. I may even 

 say that I did not remark these remote relations until a long 

 time after I had demonstrated my general proposition, and 

 upon a minute investigation of all that might have any analogy 

 with it. 1 shall add lastly, that it was so little expected as to 

 be at first disputed, principally by M. Savart, when I an- 

 nounced it as a theoretical consequence of the laws of motion ; 

 he even predicted to me that the experiments which I proposed 

 to make to confirm it would not yield the result I expected. 

 For my part, I had no doubt on the point ; and I shall in a 

 few words describe these experiments, confining myself to 

 those which relate to plates or to bells, as the most easily 

 performed, and more conclusive than those which relate to 

 strings. I first took a square plate about two decimetres 

 wide by four millimetres thick, fixed at the centre and free at 

 all its other points. On moving the bow perpendicularly to 

 the plane of the plate and at any one of its corners, the deepest 

 sound was obtained, and the nodal lines were the two parallel 

 to the sides, drawn through the centre. On the contrary, 

 when the bow was placed in the middle of one of the sides, 

 the plate quitting its state of rest, a sound was obtained rather 



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