MOLECULAR THEORIES AND MATHEMATICS BOREL. 183 



The hypothesis that the attracting masses are simple material 

 points without dimensions is difficult to accept from a physical 

 point of view. We are thus led to carry out the analytical opera- 

 tion of dispersing this mass in a small circle (or small sphere) having 

 the point for center, without changing the potential outside of this 

 circle (or sphere). We shall call this circle (or sphere) the "sphere 

 of action" of the point which coincides with its center. We shall 

 choose its radius to be proportional to the mass concentrated at its 

 center. In such a fashion that if the series formed by the masses 

 conA'erges sufficiently rapid, we can so arrange that the radii of the 

 s]3heres of action also form a rapidly converging series, and that, at the 

 same time, the maximum densit}'' of the attracting mass be finite. 

 It is also easy, if we admit that we dispose arbitraril}^ of the distri- 

 bution of the masses and densities, to arrange so that the distribu- 

 tion in each sphere of action reduces to zero, and so for its derivatives, 

 over the whole surface of the sphere. The distribution of the density 

 is thus not only finite, but continuous throughout space. 



The hypothesis that we have made concerning the convergence of 

 the series whose tenns are the radii of the spheres of action implies 

 the convergence of the series whose terms are the projections of these 

 spheres on any straight line whatever. If, then, in this series we 

 suppress a certain 7iumber of the first terms, the remamder of the 

 series can be made less than any number fixed in advance. From this 

 we conclude that, in an interval small as you please, taken on the 

 straight line on wliich we project the splieres, we can find an infinite 

 number of points which appertain at the most to a finite number of 

 such projections, namely, those belonging to the spheres S which 

 correspond to the first terms of the series and which we have sup- 

 pressed for rcndermg the remamder less than the interval considered. 

 If we consider a plane perpendicular to the right line and passing 

 through one of these points (tliis point being chosen, as is possible, 

 distinct from the projections of the centers of the spheres S, finite in 

 number, concerning which we have just spoken), this plane will at 

 most intersect a finite number of spheres S without going through 

 their centers, but will be exterior to all the other spheres of action. It 

 is possible to modify tlie distribution of the matter within the spheres 

 S wliich are finite hi number and intersected by tlie plane in such 

 a manner as to replace these spheres by smaller spheres wdiicli do 

 not intersect the plane, this operation not modifymg the potential 

 outside of the spheres and tlie density remainmg finite, since the 

 operation relates to only a Imiited number of spheres. To sum up, 

 it is possible to find a plane perpendicular to any right liue whatever, 

 cutting out of this Ihie any segment whatever given ii\ advance and 

 such that in all the points of this plane the density sliall bo zero. 

 vSince our jiotential function is defined by a density everywhere finite 



