542 SCIENCE PROGRESS 



the roots are all real and separate. This theorem is proved 

 by a simple method based on the theory of integral equations, 

 and is a remarkable instance of the utility of such equations 

 in the solution of specific problems. 



E. W. Hobson, Proc. Lond. Math. Soc. (2) xviii. (19 19), 

 pp. 249-266, discusses Hellinger's integrals, which occurred, 

 in the first instance, in the theory of quadratic forms containing 

 an infinite number of variables. Hellinger regarded the 

 integrals as a new species of limit, but Hahn showed that 

 they were reducible to Lebesgue integrals. The present 

 author gives a much simpler method of carrying out this 

 reduction, and the results are extended to a wider class of 

 cases of which those treated by Hellinger and Hahn are only 

 particular cases. 



H. B. Philips, Amer. Journ. of Math., xli. (1919), pp. 266- 

 279, writes on functions of matrices, and studies the functions 

 represented by polynomials or convergent series in a matrix 

 or finite number of matrices. As a preliminary, the main 

 facts concerning the roots of matrices are developed. 



W. J. Johnston, Proc. Royal Soc, 678, A Vol. 96, pp. 331— 

 333, lays down a linear associative algebra suitable for the 

 discussion of electromagnetic relations and the theory of rela- 

 tivity. It is based on four fundamental units *, /, k, o, and, 

 with the products, itself has sixteen units. The scalar unit 

 is commutative with the others, which can receive any inter- 

 pretation consistent with the distributive and associative 

 principles. The vectors i, j, k are ultimately taken as rec- 

 tangular unit vectors in Euclidean space, and o is perpendicular 

 to the other three. The products i, j, i, k, j, etc., can remain 

 without interpretation. Sir Joseph Larmor, in the same 

 journal, has applied this new algebra to a discussion from a 

 new standpoint of the more extended principle of relativity 

 as given by Einstein. 



The following is a further selection from recent papers : — 



Matsusaburo Fujiwara, Science Reports of the Tohoko 

 Imperial University, viii. (1) (191 9), pp. 43~5 I > generalises 

 the Tauberian theorem to cover the case of double series. 



Friedrich Reisz, Potenzreihen mit vorgeschriebenen Augangs- 

 liedem, Acta Mathematica, xlii. (2) (191 9). 



O. E. Glenn, on a new treatment of theorems of finiteness, 

 Transactions of the American Mathematical Society, xx. (3) ( 1919)* 



E. W. Chittenden, on the theory of developments of an 

 abstract class in relation to the calcul functionnel, Transactions 

 of the American Mathematical Society, xx. (3) (1919). 



Pierre Humbert, Sur deux Polynomes associes aux poly- 

 nomes de Legendre, Bulletin de la Societe Mathematique de 

 France, xlvi. (191 9). 



