268 SCIENCE PROGRESS 



In connection with G. H. Hardy's researches on Dirichlet's 

 divisor problem (see Science Progress, 191 6, 11, 93), the 

 investigation by Hardy (Proc. Lond. Math. Soc. 1916, 15, 192) 

 of the average order of two arithmetical functions occurring 

 in the above problem is of interest. 



H. H. Mitchell (Trans. Amer. Math. Soc. 191 6, 17, 165) 

 writes on the generalised Jacobi-Kummer cyclotomic function. 



Algebra and Analysis. — J. Littlejohn, in a paper read to the 

 Royal Society of Edinburgh on July 3, 191 6, showed, in the 

 case of numerical equations, how certain operators — differentia- 

 tions and integrations with respect to the coefficients — could be 

 applied to evaluate the roots. 



It is well known that abstract definitions of groups were 

 gradually arrived at by Cayley in papers published in 1854, 

 1859, and 1878, that Hamilton (1856) first denned abstractly the 

 groups of the regular polyhedra, and that Kronecker (1870) 

 first gave a really abstract definition of a group in the case in 

 which the operators are commutative. Miss Josephine E. 

 Burns (Amer. Journ. Math. 191 5, 37, 191), starting from an 

 article by G. A. Miller which was published in 191 1, proves a 

 few general theorems relative to the groups generated by two 

 operators satisfying certain defining relations, and then gives 

 abstract definitions of the substitution-groups of degree 8 and 

 some applications of the above theorems. 



To the Royal Society of South Africa on April 19, 191 6, Sir 

 Thomas Muir read two papers on determinants : one gave the 

 discovery of the connection of Pfaffians with the difference- 

 product, and established a series of theorems bringing Pfaffians 

 into relation with permanents and other integral functions ; 

 and the other was a note on the so-called Vahlen relations 

 between the minors of a matrix. 



Prof. E. T. Whittaker, in a paper read to the Royal Society 

 of Edinburgh on June 14, 191 6, gave a general process for 

 expressing a continued fraction as a continuant, and showed 

 how to express the differential coefficient of a continued fraction 

 as the ratio of two determinants the constituents of which are 

 definite functions of the terms of the continued fraction. 



C. N. Haskins (Trans. Amer. Math. Soc. 191 6, 17, 181) 

 investigates the measurable bounds and the distribution of 

 functional values of summable functions, and Dunham Jackson 

 (ibid. 178) proves otherwise one of the theorems of Haskins. 



