MOLECULAR THEORIES AND MATHEMATICS BOKEL. 185 



to say, of the existence of the Ta3^1oi' series, with radius of conver- 

 gence different from zero — or of monogeneity; that is to say, of the 

 existence of tlie unique derivative ? This question has no meaning 

 so long as we can confound analyticity %vith monogeneity. On the 

 other hand, it takes a very clear signification as soon as we have 

 succeeded in constructmg nonanalytical monogenic functions. 



To-day I can not enter into the detail of the deductions whereby 

 this problem has been resolved.^ Here is the result: It is, mdeed, 

 monogeneity which is the essential character to which the funda- 

 mental property of analytical functions is due. This fundamental 

 property subsists for the nonanalytical monogenic functions as soon 

 as we specify clearly the nature of the domams in which these func- 

 tions are considered. I have proposed to call the domains satisf3^ing 

 these distinct conditions domains of Caiichy. A domain of Caucliy 

 is obtamed by cutting off from a continuous domain domains of 

 exclusion analogous to the spheres of action just mentioned, domains 

 which may bo infinite hi number, but whose sum can be supposed to 

 be less than any given number (just as the spheres or circles of exclu- 

 sion just considered, whose radii once chosen wo can multiply by 

 any number less than vmity, and are free to increase the upper limit 

 of tlie density in tlie same time that wo decrease the radii of exclusion). 



The series formed by these excluded domains should, as is well 

 understood, be supposed to be convergent; moreover, we ought to 

 suppose that its convergence is more rapid than that of a deterini- 

 nate series which it is not necessary to write here. Under these 

 conditions, which refer only to the domain and not to the function, 

 every function wliicli in Cauchy's domain satisfies the fu.ndamental 

 equation of monogenity possesses the essential property of the ana- 

 lytical function. We can calculate it throughout its domain of 

 existence by the Iviiowledgo of its derivatives at one point (the exist- 

 ence of the first derivative involves the existence of all the derivatives, 

 at least in a certain domain which forms part of the Cauchy domain) 

 and tliis mode of calculation implies the consequence that, if the 

 monogenic function be zero on an arc however small, it is zero in 

 every point of the domain of Cauchy. Two functions can not, then, 

 coincide on an arc without coinciding throughout their domain of 

 existence, in the generahzed sense. 



I can not develop the consequences of these results from point of 

 view of the theory of functions; but I would, in closing, submit to 

 you some reflections wliich they suggest, upon the relations between 

 mathematical and physical continuity. 



>See Emlle Borel: Definition et doraaine d'exlstence des fonctions irionogfenes uniformes (Fifth Inter- 

 national Congress of Mathematicians, Cambridge (England), 1912). Les fonctions monog6nes nou analy- 

 tiques (Bulletin de la Socl6t6 math6matlque de France, 1912). 



