RECENT ADVANCES IN SCIENCE 617 



Charles de la Valine Poussin (Trans. Amer. Math. Soc. 

 191 5, 22, 435) gives a long and important memoir, which is 

 practically a whole original treatise, on Lebesgue's integral. 



Prof. M. B. Porter (Bull. Amer. Math. Soc. 1915, 22, 109) 

 proves an important theorem on a class of functions of limited 

 variation which Vitali in 1905 called " absolutely " continuous 

 functions. 



Prof. G. Mittag-LefTler (Sitzungsber . der Kgl. Bayer. Akad. 

 der Wiss. zu Munchen, Math.-phys. Klasse, i9 l S> I0 °) gives 

 an interesting account of his work, especially since 1898, on 

 the analytical representation of a one-valued branch of a 

 monogenic function, in which the existence of an analytic 

 function for any " star " of convergence was proved by actual 

 construction. This paper is particularly valuable for the 

 exposition it gives of the work of contemporary mathematicians 

 and the relation of Mittag-Leffler's ideas to theirs. An appen- 

 dix to this paper contains a proof of a theorem of Marcel Riesz 

 due to G. H. Hardy. A note to this paper is of great interest 

 for those who are interested in the early history of the theory 

 of functions : Weierstrass, who at the beginning of his work 

 did not know Cauchy's theorem on the radius of convergence 

 of a power-series, always began his lectures on the theory of 

 analytic functions by proving an equivalent theorem. 



Joseph Slepian (Trans. Amer. Math. Soc. 191 5, 16, 71) 

 studies the functions of a complex variable defined by an 

 ordinary differential equation of the first order and degree. 



Under the direction of Prof. E. T. Whittaker, the Mathe- 

 matical Department of the University of Edinburgh has shown 

 great activity of late. Besides the production of a series of 

 tracts, principally on subjects connected with the Mathematical 

 Laboratory, which are reviewed elsewhere in the present 

 number, a series of " research papers " has been issued by 

 this Department. These papers are separate copies of papers 

 lately published in various mathematical periodicals and pro- 

 vided with special blue paper covers. Of these papers the 

 eleventh is an introduction by L. R. Ford (Proc. Edinb. Math. 

 Soc. 1915, 33) of what he calls " successive oscillation func- 

 tions," which are derived from functions — in the most general 

 sense of the word — of a real variable. The oscillation function 

 of the fi r 5t order is well known from the work of Baire, but here 

 Ford considers an oscillation function of this one, and so on, 



