RECENT ADVANCES IN SCIENCE 365 



number of results concerning expansions of polynomials, by 

 an application of the calculus of Residues ; though the results 

 could be built up by elementary methods, they are suggested 

 directly only by Cauchy's calculus. 



N. E. Norlund, Comptes Rendus, \6g (19 19), pp. 166-68, 

 221-223, investigates generalisations of Euler's polynomials ; 

 the ordinary Eulerian polynomials satisfy a difference equa- 

 tion resembling that satisfied by the Bernoullian polynomials, 

 and the extension effected by Norland consists in the discus- 

 sion of a polynomial in n variables, satisfying a system of 

 such difference equations ; in a subsequent paper (ibid., pp. 

 372-375) he studies a single difference equation which is a 

 generalisation of the Eulerian equation. 



S. A. Joffe, Quarterly Journal, xlviii (1919), pp. 193-271, 

 continues his researches (Quarterly Journal, xlvii) on Eulerian 

 numbers ; of these numbers he calculates eighteen more than 

 those previously calculated, making fifty in all. 



G. H. Hardy, Messenger, xlviii (191 8), pp. 90-100, gives an 

 account of some of the fundamental properties of ' Stieltje's 

 integrals.' 



S. Pollard, Messenger, xlviii (191 8), pp. 87-89, gives a new 

 and simplified proof of the fundamental exponential inequality 

 required to prove the equivalence of the definitions of the 

 Gamma function as an infinite integral and an infinite pro- 

 duct. 



L. Bairstow and Arthur Berry, Proc. Royal Soc, xcv (a), 

 (1919), pp. 457-475, investigate solutions of Poisson's and 

 Laplace's equations, with a view to their applications to 

 problems of Elasticity and Hydrodynamics ; the functions 

 discussed are of the nature of ' Green's functions.' 



M. Kuniyeda, Quarterly Journal, xlviii (1918), pp. 1 13-153, 

 investigates various special cases of oscillating Dirichlet's 

 integrals which have been untouched by G. H. Hardy in his 

 general investigations, Quarterly Journal, xliv (191 8). 



G. A. Larew, Trans. Amer. Math. Soc, xx (1919), pp. 1-22, 

 investigates Mayer's problems in the calculus of variations. 



V. Brun, Comptes Rendus, 168 (1919), pp. 544-546, attacks 

 the proof of Goldbach's theorem that every even number is the 

 sum of two primes, with the aid of the sieve of Eratosthenes. 

 He obtains some remarkable results, of which may be stated 



(1) For all sufficiently large numbers n, there exists a number 

 between n and n + >/n with not more than 1 1 prime factors. 



(2) All sufficiently large even numbers are expressible as the 

 sum of two numbers with not more than 9 prime factors. 



H. Cramer, Comptes Rendus, 168 ( 1919), pp. 539-541, enun- 



