MOLECULAR THEORIES AND MATHEMATICS BOREL. l7l 



These diverse generalizations have rapidly acquired the rights 

 of a citizen in mathematical physics; the employment of integral 

 equations, of which the classical types are the equation of Volterra 

 and the equation of Fredliolm, has there become current. Although 

 these theories may be well laio\vn to all, it is perhaps useful to recall 

 briefly their origin by a particularly simple example. We shall thus 

 better understand their significance from the point of view where we 

 stand to-day. 



Let us consider a system composed of a finite number of material 

 points, each of which can deviate only a small amount from a certain 

 position of stable equilibrium. The" differential equations wliich 

 determine the variations of these deviations from then* positions 

 of equilibrium can, under certam hypotheses and to a first approxi- 

 mation, be regarded as linear in respect to these deviations. If, 

 moreover, we introduce the hypothesis that the system satisfy the 

 law of the conservation of energy, the differential equations take 

 a form very smiple and classic, from which one easily deduces the 

 fact that the motion may be considered as the superposition of a 

 certain number of periodic motions. The number of these elementary 

 periodic motions is equal to the number of degrees of freedom; it is 

 three times the number of the material points, if each of these points 

 can be arbitrarily displaced in the neighborhood of its position of 

 equilibrium. The periods of the simple periodic motions are the 

 specific constants of the system, which depend only on its configura- 

 tion and the hypotheses made concerning the forces put into opera- 

 tion by its deformation, but which does not depend upon the initial 

 conditions: positions and velocities. These initial conditions deter- 

 mine the arbitrary constants which figure in the general mtegral 

 and which are two in number for each period: the intensity and the 

 phase. 



Suppose now that the number of material points become very 

 great and let us identify each of them with a molecule of a solid body, 

 a bar of steel for example; if the hypotheses made continue to be 

 verified, and that is what one admits in the theory of elasticity, 

 their consequences will subsist also; we shall have, then, a very large 

 number of characteristic constants, each of these constants defining 

 a proper period of the system. Let us increase to mfinity the number 

 of the molecules. The system of differential equations, infuiitely 

 great in number, is then replaced ])y a finite number of partial differ- 

 ential equations, whose fundamental properties are obtamed by 

 passing to the limit. In particular, the proper periods can be deter- 

 mined and we establish the remarkable fact that these periods can be 

 calculated with precision and without ambiguity if we take care to 

 define them by commencmg with the longest period; there is only a 



