1819.] Royal Academy of Sciences. 227 



isochronous vibrations, which run on together without interfering 

 with one another; but the relations are not the same as for 

 flexible strings : the octave, the twelfth, the seventeenth, are not 

 heard. This resonance then is not a general fact serving as the 

 foundation of the laws of harmony. According to the supposi- 

 tions which may be made,- and the arbitrary functions which 

 may be introduced into the calculation, a great number of partial 

 sounds may be suppressed at the beginning. Thus in a case 

 where the author lays down all the circumstances, the subordi- 

 nate sound, the lesser sharp, will occur at the triple octave of the 

 second major of the principal sound; an interval which is looked 

 upon as dissonant: the superior sounds, could not be determined 

 in the least. 



If an extended flexible surface of a rectangular figure, the 

 extremities of which are fixed, is considered, the movement may 

 be resolved into a multitude of partial movements, each of which 

 is expressed by a particular integral : the coefficients of the 

 different terms are limited integrals easy to be obtained, the 

 series being convergent ; the subordinate sounds have not in 

 general any commensurable relation : these sounds are totally 

 different from those produced by an elastic surface and the 

 monochord. In the case of there being only one dimension, 

 the flexible body is sonorous, the harmony is pure and complete ; 

 but as soon as a second dimension is added, all harmony ceases, 

 and there is nothing but a confused mixture of sounds, only 

 slightly distant from each other, and of which it is impossible to 

 discern the relation. 



If the surface is elastic, the equation, instead of being one of 

 the second order, is of the fourth. The subordinate sounds are 

 to one another, and to the principal sound, as one number to 

 another ; and it is upon this account that elastic surfaces yield 

 harmonious sounds. 



If constant retarding forces are made to enter into the calcu- 

 lation, the tone remains the same, but the sound weakens, the 

 motion ceases, or rather it passes into the neighbouring bodies, 

 and is propagated in them. The action of these forces rapidly 

 destroys the accidental effect of the initial disposition, and leaves 

 only the effect of the proper elasticity and the figure of the sono- 

 rous body in action for some time. 



If the elastic surface, being of a very small thickness, has its 

 other dimensions unlimited, the movement is propagated rapidly 

 along the whole extent of the surface ; foldings and annular 

 furrows are formed, which recede from the origin of the motion. 

 The question will then be to express in a single formula all the 

 variable states of the surface, so that its figure at any instant of 

 time may be exactly determined. This equation contains, under 

 the double sign . of definite integration, two auxiliary variable 

 quantities along with the three principal variable quantities. The 



r2 



