MOLECULAR THEOEIES AND MATHEMATICS BOEEL. 179 



designates by H and Gibbs by rj, and which represents, in the case of 

 a gas, the logarithm of the probabilit}^ of a deternihiate distribution 

 of the velocities of the molecules. Each collision ])etween two mole- 

 cules gives a sudden variation to this function, which is thus repre- 

 sented by a [serrated or] staircase curve, the horizontal projections 

 of the steps corresponding to ihc intervals of time which separate two 

 collisions; the number of the collisions undergone by a molecule being 

 some billions per second (that is to say, of the order of magnitude of 

 10®) and the number of molecules of the order of magnitude of 10^* 

 (if we consider a mass of some grams of gas), the total number of col- 

 lisions per second is of the order of magnitude of lO^'' power; such 

 is the mmiber of steps projected on a portion of the axis of the ab- 

 scissae equal to unity, if the second is taken for the unit of time.' 

 Wliat the physicists consider is the mean behavior of the curve. They 

 replace the serrated curve by a more regular curve, having the same 

 mean behavior m the time intervals which are very small in compari- 

 son to the second, but very great m comparison to 10"^^ of a second. 



These diverse considerations bring interesting suggestions to the 

 analyst, on which I would like to dwell for a moment. 



In the first place, referring to the subject of continuous curves 

 without derivatives of which the Brownian movement has given us 

 the image, should the passage from the finite to the infinite lead to a 

 curve all of whose points are points of discontinuity, or to a curve 

 which admits an infinitude of points of discontinuity but also an 

 infinitude of points of contmuity? For properly understandmg the 

 question, it is necessary to briefly recall the capital distinction between 

 the denumerable infinity and the continitous infinity. An infhiite 

 ensemble is said to be denumerable if its terms can be enumerated 

 by means of integers; such is the case for the ensemble composed of 

 terms of a simple or multiple series; we can also cite as a denumerable 

 ensemble the ensemble of the rational numbers. On the other hand, 

 the ensemble of all the numbers comprised between and 1, both 

 commensura])le and incommensurable, is not denumerable; we say 

 that this ensem])le has tlie same power as the continuum. If we define 

 a discontinuous function by a series each term of which admits a point 

 of discontinuity, the ensemble of these points of discontinuity is 

 denumerable, as are the terms themselves. Can we determine a 

 function which shall be totally discontinuous — that is to say, one whose 

 points of discontinuity shall be all the points of a continuous ensem- 

 ble, and not merely those of a denumerable ensemble? It would 

 seem to be easy to imagkie such a function. Such is the function 

 often studied which is equal to 1 if a; is commensurable and to x if x is 



1 This discontinuity supposes, as is well understood, that wc consider the duration of a collision as less 

 than tlie mean interval of two collisions (in the whole mass), a liypothesis admissible with difficulty. 

 The schevta to which this hypothesis leads, is not less interesting from the analytical point of view. 



