RECENT ADVANCES IN SCIENCE 195 



Arithmetic and Algebra. — H. W. Stager has published A 

 Sylow Factor Table of the First Twelve Thousand Numbers giving 

 the Possible Number of Sylow Sub-Groups of a Group of Given 

 Order between the Limits of o and 12,000 (Washington, 1916), 

 and there is a short account of it in Nature (191 7, 99, 164). 



G. A. Miller {Bull. Amer. Math. Soc. 191 7, 23, 283-7), con- 

 tinuing an article of his published in 19 10, considers the groups 

 generated by two operators of the same prime order such that 

 the conjugates of one under the powers of the other are com- 

 mutative. 



Olive C. Hazlett {Annals of Math. 1916, 18, 81-98) considers 

 certain rational, integral invariants of nilpotent algebras, and 

 proves theorems analogous to theorems about invariants of 

 algebraic forms, and in particular their finiteness. 



Analysis. — A review of W. B. Ford's Studies in Divergent 

 Series and Summability (New York, 191 6), which was mentioned 

 in the last number of Science Progress ( 1 91 7, 12, 12), was given 

 by C. N. Moore in the Bull. Amer. Math. Soc. (1917, 23, 308-14). 

 Like most of the reviews in this Bulletin, it is very full and well 

 done and contains criticisms and a list of misprints which must 

 be very useful to the author and his readers. 



S. A. Joffe {Quart. Journ. Math. 1916, 47, 103-26) verifies 

 the calculations of J. W. L. Glaisher for the first twenty-seven 

 Eulerian numbers and gives computations for five more numbers 

 of the series. Joffe 's calculations were made from central 

 differences of zero. 



We have been awaiting, since its announcement in the 

 Encyklopddie in 1898, the publication of A. Pringsheim's 

 Vorlesungen fiber Zahlen- und Funktionenlehre . The first two 

 parts of the first volume were published at Leipzig and Berlin 

 in 1 91 6, and an account of their contents is given in the Rev. 

 sent. (1917, 25 [1], 82). 



O. Szasz {Math. Ann. 1915-6, 77, 482-96) finds conditions 

 that a sequence of powers of a variable approximates to any 

 given continuous function of that variable. 



A. Denjoy {Journ. de Math. 191 5, 1, 105-240) has a long 

 memoir on the derivatives of continuous functions, and a note 

 {Compt. Rend. 1916, 162, 377-80) on derivation and its inverse 

 calculus. Mrs. (Grace Chisholm) Young {ibid. 380-2) writes on 

 the derivatives of a function. 



G. Vitali {Atti delta R. Accad. delle Scienze di Torino, 19*5-6, 



