RECENT ADVANCES IN SCIENCE 351 



1919, in a paper entitled, "A New Solution of Waring's Problem," 

 give a short account of a solution of Waring's Problem of the 

 existence of the function g{k) [solved by Hilbert in 1909 {Gott. 

 Nachrichten, 1909, and Ann. Math., 67, 1909)] which they have 

 recently discovered, which is important in that it brings 

 Waring's Problem, which is one of many similar problems of 

 Combinatory Analysis, into relation with the transcendental 

 side of the Analytic Theory of Numbers. This method yields 

 a great deal more than can be obtained by more elementary 

 methods. The paper opens with a short account of the main 

 features of this method, which js applicable to almost any 

 problem concerning the decomposition of integers into parts 

 of a particular kind. 



L. Tschakaloff {Ann. Math., 80, 1919) investigates the arith- 

 metical properties of the infinite series 



^a * x". 



V =0 



In a paper entitled " An American Tournament treated 

 by the Calculus of Symmetric Functions " {Quart. Journ., 49, 

 1920) P. A. MacMahon analyses by means of the powerful cal- 

 culus of symmetric functions the events in a tournament of n 

 players, where each player plays every other player, and when 

 the players are or are not in an assigned order. 



H. W. Richmond {Proc. Camb. Phil. Soc, 22, 19, 1920) dis- 

 cusses the classical problem of the determination of the formulae 

 for sets of four integers, such that the sum of their cubes is zero. 



P. A. MacMahon, in a memoir entitled " Congruences with 

 respect to Composite Moduli," in the Trans. Camh. Phil. Soc, 

 22, 21, 1920, puts together certain results in the Theory of the 

 Residues of Powers with respect to composite moduli. 



G. A. Miller {Trans. Amer. Math. Soc, 21, 1920) investigates 

 the properties of the subgroup of an abelian prime power 

 group which are conjugate under its group of isomorphisms. 



Louis C. Mathewson, in a paper entitled " On the Groups of 

 Isomorphisms of a System of Abelian Groups of Order p"" and 

 Type(«ii,i, . . ., i), inthe Amer. Journ. Math. ,22, 1920, studies 

 the groups of isomorphisms of the system of abelian groups of 

 order ^^, type {nii . . . i), w > i, and shows that these groups 

 may be built upon the group of isomorphisms of an abelian group 

 which contains no operations of order greater than p. 



G. A. Miller, in a paper on AbeHan Groups {Mess. Math., 

 49, 1920), reduces the determination of the properties of the 

 group of isomorphisms of G for a general abelian prime power 



