180 ANNUAL REPORT SMITHSONIAN INSTITUTION, 1912. 



incommensurable; this function is indeed discontinuous, as much so 

 for the commensurable values as for the incommensurable values. 

 If we look a little closer, we perceive that the discontinuity is not of 

 the same nature in these points; we would observe, m fact, that the 

 commensurable numbers occupy infinitely less space on the axis of 

 the x's than do the incommensurable numbers. The ensemble of 

 these commensurable numbers is of dimension zero — that is to say, 

 it can be confined within intervals whose total extent is less than any 

 number given in advance. Speaking in more concrete terms, if we 

 choose a number at random, the probabOity that it be commensurable 

 is equal to zero.^ We therefore conclude that the function equal to 

 X for the incommensurable values of the variable is, on an average, 

 continuous for these incommensurable values whatever be its values 

 for the commensurable values — that is to say, that if we choose, in 

 the neighborhood of an incommensurable value, for which we study 

 the continuity, another value talcen at random, it is infinitely probable 

 that this value taken at random will also be incommensurable. It 

 is, then, infinitely probable that the variation of the function will be 

 infinitely small when the variation of the variable is small. 



This remark enables us to understand that it would not have been 

 possible to define analytically a function all of whose points should 

 be actually points of total discontinuity. It is only in pomts deter- 

 mined according to the dej&nition of the function, and playing a par- 

 ticular part in this definition, that the function is actually discontin- 

 uous on an average. 



The passage from the finite to the infinite, when we are concerned 

 with the discontinuity of functions is, then, not effected after the 

 manner which is the most usual in classical mathematical physics, 

 where the matter is supposed to be continuous, and where we replace 

 the finite by the continuous. We are led to conceive a different 

 process, which appears, besides, more m harmony with the molecular 

 conception and which consists in replacing the very great finite by the 

 denumerable infinite. 



This is the way in which the analytical generalization of such curves 

 as the curves H presents itself from this point of view: Let us con- 

 sider a number written in the form of an interminate decimal frac- 

 tion and let us imagine that the figures wliich follow the decimal 

 point are grouped in successive periods, each period containing many 

 more figures than the preceding period. To each period we shall 

 make correspond one term of a series, this term bemg equal to zero 

 if in the corresponding period the ratio of the number of even figures 

 to the number of odd figures is comprised between 0.4 and 0.6, while 

 if this ratio is not comprised between these limits, tlie term corre- 



V gco give one's self a random number, he can agree to choose at random the successive figures of the deci- 

 maHraction which is equal to it; the probability that this decimal fraction be finite or periodic is e^'idently 

 equal to zero. • 



