RECENT ADVANCES IN SCIENCE 119 



due to Hill, of infinite determinants. In the same volume 

 Pierre Humbert shows how a well-known transformation of 

 Laplace, which is of great help in finding the solutions of linear 

 differential equations, gives also interesting results in the theory 

 of integral equations. 



In vol. xliv. (1914-15) of the Messenger of Mathematics there 

 are, as usual, a number of short papers, most of which are 

 decidedly interesting. Dr. J. W. L. Glaisher gives some new 

 relations between sums of the reciprocals of powers of integers ; 

 S. Ramanujan and G. H. Hardy supply calculations of some 

 definite integrals and work on other questions in the integral 

 calculus ; and there are various notes on pure and analytic 

 geometry, the theory of potential, elimination, the theory of 

 differential equations, and so on. Dr. Thomas Muir gives a 

 collection of properties of a pair of orthogonants ; and F. Jackson 

 gives a simple method for proving that Laplace's integrals for 

 the functions called P n (pc) and Q n (x) satisfy Legendre's equation. 

 In the Quarterly Journal of Mathematics (vol. xlvi. 1914), there 

 are two papers on substitution groups by G. A. Miller, two 

 papers connected with the generalised potential by C. E. Weather- 

 burn, a paper by W. Woolsey Johnson on the history, computa- 

 tion, and tabulation of Cotesian numbers, a paper by S. A. Joffe 

 on sums of like powers of natural numbers, which is closely 

 connected with Dr. Glaisher's paper of 1899, a paper by Dr. 

 J. R. Wilton on a point in the theory of partial differential 

 equations, and a paper by Tomlinson Fort on periodic solutions 

 of linear difference equations of the second order. Dr. J. R. Wil- 

 ton has a paper on Darboux's method of solution of partial 

 differential equations of the second order in vol. xiv. (1914-15) 

 of the Proceedings oj the London Mathematical Society ; and the 

 same volume also contains two important papers by Prof. 

 E. W. Hobson on (1) the representation of the symmetrical 

 nucleus of a linear integral equation, and (2) theorems relating 

 to functions defined implicitly, with applications to the calculus 

 of variations. The paper by Prof. W. H. and Mrs. Young is 

 of great interest in connection with the " Heine-Borel " and 

 analogous theorems. Prof. A. E. H. Love, in his Presidential 

 Address printed here, considers certain qualities by which 

 valuable mathematical research is characterised. The papers 

 in these Proceedings are often so technical as not to admit of 

 abstraction ; they are practically all of the greatest value — the 



