226 Proceedingsqf Philosophical Societies. [Sept. 



quantity, which corresponds with the unknown fraction, and 

 which is to be looked upon as an arbitrary function of those 

 of the new variable quantities which ought to be eliminated. 



A Memoir upon the Vibration of Elastic Surfaces, by M. 

 Fourier. — " The application of mathematical analysis to the 

 study of natural phenomena is composed of two distinct parts. 

 The first consists in expressing all the physical conditions of the 

 question by means of the calculus ; the second consists in inte- 

 grating the differential equations obtained by this means, and in 

 deducing the complete knowledge of the phenomenon in question 

 from these integrals . This memoir belongs to the second branch 



of the application of analysis The general integrals of these 



equations have not as yet been obtained ; that is to say, of those 

 which contain, in limited terms, so many entirely arbitrary 

 functions, as the order and nature of the differential equations 

 allow. We principally endeavoured to discover these general 

 integrals under a form which would be proper to show clearly 



the progress and laws of the phenomena The differential 



equation of the movement of elastic surfaces was not known 

 some years ago, when the attention of mathematicians was 

 attracted to this question by the Institute. At that time this 

 equation was drawn up, which is of the fourth order, and differs 

 entirely from that of flexible surfaces. But it was necessary to 

 integrate this last equation, and also that of elastic plates. The 

 principal object of the memoir is to prove that the general inte- 

 grals of these equations are expressed by definite integrals, by 

 means of the theorems which we formerly gave in our Researches 

 concerning Heat. If it be considered that these very same 

 theorems serve to determine the laws of the propagation of heat 

 in solid matter, the oscillations of strings, and flexible or elastic 

 surfaces, and the movement of waves upon the surface of liquids, 

 the utility and extent of this new method of analysis will be 

 acknowledged." 



The author then gives the general integrals of vibrating sur- 

 faces, whose dimensions are infinite. The integral of the 

 equation of elastic plates, developed in a regular series according 

 to the powers of the variable quantities, may be summed up ; but 

 the expression to which this process leads cannot be used for 

 the resolution of the physical question, as it would present a 

 function which is very simple in itself under an extremely com- 

 plicated form. 



_ In rendering the agitation of sonorous bodies sensible to the 

 sight, or in measuring the duration of the vibrations by the 

 comparative value of the sounds thus produced, the results which 

 are observed always coincide with those which arise from the 

 particular values ; and these very relations are now confirmed by 

 the examination of the general integrals. 



If the two extremities of an elastic plate are supported upon 

 fixed obstacles, the movement is composed of a multitude of 



