178 AKNUAL EEPOET SMITHSONIAN INSTITUTION, 1912. 



equations and calculations, and whicli involve a relatively small 

 number of s^^'mbols and numerical constants. It is, then, not a priori 

 absurd to suppose that one might imagine a physical model containing 

 also a relatively small number of parameters and leading to the same 

 equations. However, so long as this model shall not be imagined, 

 and perhaps it never will be, the analytical or geometrical researches 

 on functions of a very large but finite number of variables will be able 

 to offer interest to the physicists. 



VI. 



We have already observed that it is an ordinary proceeding in mathe- 

 matics to replace a very large finite by an infinite. What may this 

 procedure give when we apply it to discontinuous physical phenomena 

 whose complexity seems tied up to a very great number of molecules? 

 Such are, for example, the phenomena of the Brownian movement 

 which we observe when very fine particles are in suspension in an ap- 

 parently quiet liquid. These phenomena enter the category of those 

 concerning which we were speaking a moment ago, for which only a 

 statistical foreknowledge is possible. 



Can we construct an analytical image of it ? Prof. Perrin has 

 already remarked^ that the observed trajectories in the Brownian 

 movement suggested the notion of continuous fimctions possessmg no 

 derivative or that of continuous curves possessing no tangent. If 

 we observe these trajectories with optical mstruments more and more 

 perfected, we see at each new magnification, new details, the curvi- 

 linear arc that we could have traced becomes displaced by a sort of 

 broken line whose sides form finite angles with each other; so it is up 

 to the limit of the magnifications capable of actual realization. If we 

 admit that the movement is produced by the impacts of the molecules 

 against the particle, we should conclude from this that we would 

 obtain, with a sufficient magnification, the exact form of the trajectory 

 which would present itself under the form of a broken line with 

 rounded off angles and which would not be sensibly modified by a 

 still further magnification. 



But the analyst is not forbidden to retreat mdefinitely into the 

 supposed obtainment of this ultimate state and to thus arrive at the 

 conception of a curve in which the sinuosities become finer and finer 

 in proportion as he cmplo3^s a higher magnification, without his ever 

 obtaining the ultimate smuosities. This is indeed the geometrical 

 image of a continuous function not admitting of a derivative. 



We obtam thus a curve of the same nature, but rather too special 

 to stop to consider, when we study the function which Boltzmann 



1 Jean rerrin: La discontinuity dc la matifere. Revue du vwis, mars 1906. See also Jean Perrin: Les 

 atomes (Alcan, 1913). 



