182 ANNUAL KEPORT SMITHSONIAN INSTITUTION, 1912. 



We may also connect with these considerations the theory of 

 denumerable probabihties, that is to say, the study of probabihties, 

 in the case where either the infinitude of trials or the infinitude of 

 possible cases is denumerable, a, study lying between the study of 

 probabilities in the finite cases and the study of continuous proba- 

 bilities. 



VII. 



In spite of the interest of problems relating to functions of a real 

 variable, it is the theory of functions of a complex variable which, 

 since the immortal discoveries of Cauchy, is really the center of 

 analysis. The analogy between the theory of the functions which 

 Cauchy has called monogenic functions and that wliich we often 

 call analytical functions and the theory of Laplace's equation which 

 potentials satisfy, is certainly one of the most fruitful of the analo- 

 gies of analysis. We luiow all the advantage that Riemann has 

 reaped from the theory of potential and from the intuition of 

 physics in his profound researches upon the functions of a complex 

 variable. 



It is then natural to ask one's self what new ideas can the molecular 

 theories bring forward in the domain of complex variables. Here 

 again we shall be led to replace the very large finite number by the 

 denumerable infinit}^. It is easy to form series each term of wliich 

 presents a singular point, the ensemble of the terms of the series 

 thus possessing a denumerable infinitude of singular points. These 

 singular points may, for example, be so chosen as to coincide with 

 all such points among the points inside of a square whose two coordi- 

 nates are rational. The most simple series that we can thus form 

 presents itself under the form of the saim of a series of fractions 

 each of which admits a unique pole, wliich is a simple pole. The 

 physical interpretation in the domain of the real of such a series 

 leads one to consider the potential of a system composed of an 

 infinitude of isolated points, the mass concentrated in each of the 

 points being finite (which leads to the admission that the density 

 in each such point is infinite if the point be abstractly considered as 

 a simple geometrical point without dimensions). We suppose, as 

 is well understood, that the series v/hose terms denote the values 

 of the masses is convergent, which amounts to saying that the total 

 mass is finite, although concentrated in an infinitude of distmct 

 points, for example, in all the points whose two coordinates are 

 rational numbers. The potential with wiiich we are now concerned 

 is in the case of a plane what we call logarithmic potential. We 

 should be able to reason in an analogous manner in space of three 

 dimensions; we should then have the Newtonian potential ])roperly 

 speaking. 



