RECENT ADVANCES IN SCIENCE 543 



and 191 $, to the theory of Riemann's Zeta function and the 

 theory of the distribution of prime numbers. " Our answers 

 to these questions are naturally tentative and fragmentary. 

 The importance and difficulty of the problems dealt with 

 should be a sufficient apology for the incompleteness and mis- 

 cellaneous character of the results." 



F. Hallberg (Journ. Indian Math. Soc. 191 7, 9, 174-86) 

 writes on infinite series and arithmetical functions, a subject 

 which is connected with the Zeta function. 



Algebra. — W. H. Metzler (Proc. Roy. Soc. Edinburgh, 191 7, 

 37, 324-6) extends so as to involve k instead of two determi- 

 nants a theorem given by Muir in 1888 on the equality of the 

 sums of two sets of determinants formed in a certain way out 

 of two given determinants. 



Olive C. Hazlett (Trans. Amer. Math. Soc. 191 7, 18, 167-76) 

 considers linear associative division algebras over a general 

 algebraic field F, which may be described as sets of numbers 

 satisfying all the conditions for a field except that multipli- 

 cation is not necessarily commutative. 



O. E. Glenn {ibid. 443-62) generalises the theory of asso- 

 ciated forms founded by Hermite in 1856 and developed by 

 Clebsch (1872), Sylvester, and many others. 



W. A. Manning (ibid. 463-79) gives a much wider generalisa- 

 tion of Bochert's theorem (1892, 1897) connecting degree and 

 class of a multiply transitive group. It may be noticed that, 

 since in the theory of groups of substitutions the two problems 

 of the order and of the class of groups of given degree are in 

 large measure interchangeable, Bochert's inequalities, which 

 limit the degree of a multiply transitive group in terms of its 

 class, are second in importance only to Sylow's theorem. 



Analysis. — In 1903 G. H. Hardy proved a theorem about 

 the convergence of certain multiple series ; and he now (Proc. 

 Camb. Phil. Soc. 191 7, 19, 86-95) states and proves the leading 

 theorems of a more general class of such series in a form more 

 systematic and general than has been given to them before. 

 In the first place, some known theorems concerning simply 

 infinite series are recapitulated with certain changes of form, 

 and then the corresponding theorems for double series are 

 obtained in a form as closely analogous as possible . The further 

 simple generalisation to multiple series in general is left to 

 the reader. 



