MOLECULAR THEORIES AND MATHEMATICS BOREL. 181 



spending to the period is equal to the term of the same order of a 

 certaiQ convergent series with positive terms. It is clear that, if the 

 lengths of the successive periods increase rapidly, it is infinitely 

 probable that a small number of periods only will furnish terms differ- 

 ent from zero. Consequently, the series which corresponds to the 

 decimal number will be termmate; this terminate series has a certain 

 sum, which remains the same so long as the decimal number varies so 

 little that the last one of the periods which furnished a term to the 

 series be not modified; at least, in the interval thus defined, it is ex- 

 tremely probable that the function corresponding to the decimal 

 number preserves this constant and well determined value — that is 

 to say, is represented by a horizontal line. However, there are in this 

 interval, as in every interval, particular decimal numbers for which 

 certain periods of liigh order, perhaps even an infinitude of such 

 periods, are irregular from the point of view of the distribution of the 

 even and odd figures. There are then mtervals which are extremely 

 small, and, on an average, extremely rare, but nevertheless every- 

 where dense, in which the curve runs up above the horizotital line 

 which in general represents it. In one of these points, which we may 

 call maxima of the curve, it is extremely probable that, if we take a 

 neighboring value of the variable at random, the function will dimin- 

 ish — that is to say, that this point has, on an average, the character 

 of a maximum in a point. 



In the preceding example the maxima are represented by intervals 

 more or less narrow but finite. One can in modifying shghtly the 

 definition obtain a curve which would coincide everywhere with the 

 axis of X, excepting in points not filling any interval. It suffices to 

 agree that, in the series which we have just defined, we replace by zero 

 every term which is followed by an infinitude of terms equal to zero. 

 The new series can then be different from zero only if the terms of the 

 fii'st series are all, starting from a certain order, different from zero. 



The study of the analytical models thus obtained lead one to 

 thoroughly examine the theory of functions of real variables and 

 even to conceive new notions, such as the notion of average derivative ^ 

 naturally suggested by the physical example of the function H. 

 Besides, it is necessary to observe that, in the study of these func- 

 tions, the notion of continuous ensemble is often combined with 

 the notion of denumerable ensemble; for example, it is easy to see 

 that the ensemble of decimal numbers whose figures are all odd 

 present certain characters of the ensemble of all the decimal num- 

 bers ; it has, as we say, the power of the continuum,^ but it is, however, 

 of zero dimension. 



•See Emile Borcl: Comptes rendus de rAcad6mio des Sciences, April 29, 1912. 



» If, in a decimal number all of whose figures are odd, we replace the respective figures 1 , 3, 5, 7, 9 by the 

 figures 0, 1 , 2, 3, ■!, we can consider the number as any number whatever written in the system whose base is 5 



85360°— SM 1912 13 



