RECENT ADVANCES IN SCIENCE 179 



credited (cf. Science Progress, 191 7, 12, 6, for reference to 

 an allied article by Burns). 



Arithmetic and Algebra. — L. J. Rogers (Proc. Lond. Math. 

 Soc. 191 7, 16, 315-36) gives two theorems of combinatory 

 analysis and some allied identities. Major P. A. MacMahon 

 (ibid. 191 8, 16, 352-4) makes a small contribution to com- 

 binatory analysis, and (ibid. 17, 25-41) shows that there is a 

 one-to-one correspondence between combinations which are 

 derived from m, identical sets of n, different letters and magic 

 squares of order n, which are such that the numbers, placed in 

 the cells, have for each row and column a sum m. He then 

 generalises this theory. The problem involved is the enumera- 

 tion of particular sets of partitions of certain multipartite 

 numbers. 



G. A. Miller (Annals of Math. 191 7, 19, 44-8) explains a 

 different method of solving E. H. Moore's (1896) problem of 

 possible arrangements of the players at card tournaments, 

 and emphasises the connection of this problem with the theory 

 of substitution groups. 



E. T. Bell (ibid. 191 8, 19, 210-16) finds many theorems 

 which express sums 2f(n) as functions of [pc], which denotes 

 the greatest positive integer contained in x, where x assumes 

 successively each of a given class of values, not necessarily 

 integral, and n runs through all members of a given set of 

 integers. 



S. Wigert (Acta Math. 191 7, 41, 197-218) shows that, for 

 values of z in the neighbourhood of the origin, the function 

 defined by Lambert's series, which is an analytic function 

 whose domain of existence is limited by the imaginary axis, 

 allows of a certain functional asymptotic equation ; and 

 then applies this result to obtain a result on an arithmetical 

 function studied by Voronoi and Landau. 



W. H. Metzler (Proc. Roy. Soc. Edinburgh, 191 8, 38, 57- 

 60) discusses a determinantal equation whose roots are the 

 products of the roots of given equations. 



Connected with O. E. Glenn's work of 191 7 (cf. Science 

 Progress, 191 8, 12, 543) are his researches (Annals of Math. 

 191 8, 19, 201-6) on the covariant expansion of a modular form. 



Analysis. — H. F. Price (Amer. Journ. Math. 191 8, 40, 108- 

 12) considers the " fundamental regions " of Klein for certain 

 finite groups in two complex variables, 



