438 THEOKY OF HEAT. [CHAP. IX. 



Elastic Surfaces, read at a sitting of the Academy of Sciences 1 , 

 Gth June, 1816 (Art. VI. 10 and 11, and Art. vii. 13 and 14). 

 They consist in the two formulae S and 8 , Art. 40G, and in the two 

 integrals expressed, one by the first equation of Art. 412, the other 

 by the last equation of the same Article. We then gave several 

 other proofs of the same results. This memoir contained also the 

 integral of equation (c), Art. 409, under the form referred to in 

 that Article. &quot;With regard to the integral (/3/3) of equation (a), 

 Art. 413, it is here published for the first time. 



423. The propositions expressed by equations (A) and (B ), 

 Arts. 418 and 417, may be considered under a more general point 

 of view. The construction indicated in Arts. 415 and 41 G applies 



Sill f ?)j ^-^ 77 7 1 ) 



not only to the trigonometrical function - - ; but suits 



oc oc 



all other functions, and supposes only that when the number p 

 becomes infinite, we find the value of the integral with respect to 

 a, by taking this integral between extremely near limits. Now 

 this condition belongs not only to trigonometrical functions, but is 

 applicable to an infinity of other functions. We thus arrive at 

 the expression of an arbitrary function f(x) under different very 

 remarkable forms ; but we make no use of these transformations 

 in the special investigations which occupy us. 



With respect to the proposition expressed by equation (A), 

 Art. 418, it is equally easy to make its truth evident by con 

 structions, and this was the theorem for which we employed them 

 at first. It will be sufficient to indicate the course of the proof. 



1 The date is inaccurate. The memoir was read on June 8th, 1818, as appears 

 from an abstract of it given in the Bulletin dcs Sciences par la Societe Philomatique, 

 September 1818, pp. 129 136, entitled, Note relative mix vibrations des surfaces 

 elastiques et au mouvement des ondes, par M. Fourier. The reading of the memoir 

 further appears from the Analyse des travaux de V Academic des Sciences pendant 

 Vannee 1818, p. xiv, and its not having been published except in abstract, from a 

 remark of Poissoii at pp. 150 1 of his memoir Sur les Equations aux differences 

 partielles, printed in the Memoires de VAcademie des Sciences, Tome in. (year 1818), 

 Paris, 1820. The title, Memoire sur les vibrations des surfaces glastiques, par 

 M. Fourier, is given in the Analyse, p. xiv. The object, &quot;to integrate several 

 partial differential equations and to deduce from the integrals the knowledge of the 

 physical phenomena to which these equations refer,&quot; is stated in the Bulletin, 

 p. ISO. LA. I 1 .] 



