RECENT ADVANCES IN SCIENCE 347 



Frederic Riesz contributes a paper on Lebesgue's integral 

 to Acta Math., 42, 1920. His object is to make the notion of an 

 integral independent of the theory of measure. His point of 

 view, therefore, differs in an important respect from that of 

 Borel, who begins by introducing his own notion of measure, 

 and so is limited to less general sets than those which can be 

 dealt with by the methods of this paper. 



P. J. Daniell {Ann. Math., 21, 1920) gives further properties 

 of the general integral defined in 191 8 {ibid., 19, 191 8). This 

 definition covers the Lebesgue integral and the Radon-Young 

 integral as particular cases. 



R. L. MiOOTQ {Trans. Amer. Math. Soc, 21, 1920) gives a defini- 

 tion of a simple continuous arc joining A to B, which does not 

 stipulate that the set of points M should be bounded, nor that 

 it should contain no proper connected subset, containing both 

 A and B. He shows that in a Euclidean space of two dimensions 

 every point-set that satisfies this definition is a Jordan curve. 

 The advantages of the proposed definition are explained, and 

 other definitions introduced. 



J. W. Alexander {Ann. Math., 21, 1920) gives a proof of 

 Jordan's Theorem, that a simple closed curve divides the 

 plane into two regions, which was first proved by O. Veblen 

 in 1906 {Trans. Amer. Math. Soc, 66). The argument is based 

 on elementary combinatorial properties of chains or systems 

 of polygons. 



M. Jaiuna {Tohuku Math. Journ., 17, 1920), gives an elemen- 

 tary proof of the theorem about Bertrand curves proved by 

 Briosche in Bull. Soc. Math. France, 17. 



In a paper entitled " On the Term by Term Integration of an 

 Infinite Series over an Infinite Range and the Inversion of the 

 Order of Integration in Repeated Infinite Integrals " {Proc.Camb. 

 Phil. Soc, 20, 1920), S. Pollard investigates first under what 

 conditions on the Lebesgue Theory of integration term by term 

 integration of an infinite series is permissible. He obtains 

 much wider conditions than can be obtained on the Riemann 

 Theory, when the condition of uniform convergence is almost 

 always involved. The problem for infinite integrals is dealt 

 with on similar lines. 



In a paper entitled " Sur les fonctions de lignes implicites " 

 {Bull. Soc France, 48, 1920), Paul L6vy considers the problem 

 of the correspondence between two functions u{s) and v{s} both 

 defined for a certain continuous series of values of s. He studies 

 the general characteristics that the relation between v and v 

 must have, if v is uniquely determined by u, making use of 

 Hadamard's results {Bull. Soc France, 34, 1908). 



G. Mittag-Leffier continues, in Acta Math., 42, 1920, in a 

 sixth note his series of important papers entitled " Sur la 



