RECENT ADVANCES IN SCIENCE 9 



G. H. Hardy and J. E. Littlewood (Rend. Circ. Mat. di 

 Palermo, 1916, 41, 1— 18) continue and complete a paper of theirs 

 published in 191 2 relating to the summability of series by Borel's 

 exponential method. 



G. H. Hardy's shortened paper (Compt. Rend. 19 16, 162, 

 463-5 ; Science Progress, 191 6, 10, 434) on the application of 

 Abel's method of summation to the theory of ordinary Dirich- 

 let's series was given in its English and fuller form in the Quart. 

 Journ. Math. (1916, 47, 176-92). 



W. H. Young (Proc. Lond. Math. Soc. 1916, 15, 354-9) gives 

 simple proofs of certain results he had communicated elsewhere 

 on functions of what he now calls " upper " and "lower " type 

 (remember Baire's upper and lower semi-continuous functions). 



Mrs. (Grace Chisholm) Young (ibid. 360-84) proves three 

 fundamental theorems, already known to be true for special 

 functions, on the derivates of a function which is restricted 

 merely to be measurable and finite everywhere. 



D. C. Gillespie (Annals of Math. 191 5, 17, 61-3) defines f(x) 

 to be " integrable in the Cauchy sense " if the general term 

 in the sum whose limit is the integral is f(x m _ 1 )(x m — x m _ 1 ), 

 and " integrable in the Riemann sense," if it is/(£ m ) (x m — x m _ l ), 

 where f m is any point in the interval x m _ 1 . . . x m . Since 

 the usual — and correct historically — condition of the continuity 

 of f(x) is not added to Cauchy 's criterion of integrability, 

 Gillespie arrives at the paradoxical-sounding conclusion of the 

 equivalence of integrability in Cauchy 's sense with that in Rie- 

 mann 's sense. 



The lectures on Lebesgue's integral given by C. de la Vallee 

 Poussin of Louvain at Harvard University in 191 5 have already 

 been mentioned in Science Progress (191 6, 10, 617). The 

 same author developed the subject in lectures at the College de 

 France in 191 5-16, and has now published these lectures as a 

 book : Integrates de Lebesgue, Fonctions d'ensemble, Classes de 

 Baire (Paris, 1916) in the collection of monographs on the theory 

 of functions of which Emile Borel is the general editor. The book 

 gives a very good picture of that branch of the theory of 

 functions which may be said to date from Borel's new defi- 

 nition (1898) of the measure of aggregates, Lebesgue's work on 

 integration and " additive functions of an aggregate," and the 

 researches of Baire and Lebesgue on the analytical representa- 

 bility of functions by infinite series of other functions. 



