RECENT ADVANCES IN SCIENCE 349 



involves the existence of the other limit, and that when the 

 function s\ + k oscillates finitely the function h„^ oscillates 

 between the same limits. 



G. H. Hardy, Math. Zeiisckrift, 6, 1920, gives a simple proof 

 of Hilbert's Theorem that the series already proved by Hilbert, 

 Weyl, Weiner and Schur 



oe 



is convergent whenever %aj is convergent. 



Practically nothing has as yet been done in the application 

 of methods of summation of divergent series to double series. 



Lloyd L. Small, Ann. Math., 21, 1920, gives a general theorem 

 about a method of summation of double series analogous to his 

 general method of summation for simple series given in 191 8 

 {ibid. 20). 



C. A. Fischer, Bull. Amer. Math. Soc, 27, 1920, gives the ne- 

 cessary and sufficient conditions that a linear transformation 

 may be completely continuous. These results complete the 

 results of the same author published in 1919 {ibid., 25), and are 

 important in that there it is thus proved that a certain part of 

 the theory of integral equation due to Fredholm is applicable 

 to Stieltjes's integral equations of certain '^types. 



In a paper entitled, " Sur les d^veloppements en s^rie suivant 

 les inverses de polynomes donnas " {Bull. Soc. Math. France, 

 48, 1920), P. Appell collects the principal results given in his 

 papers of 191 3 {Comptes Rendus, 37, Bull. Set. Math., 37) and 

 adds new matter. 



R. D. Carmichael, Ann. Math., 22, 1920, discusses the expan- 

 sion of certain analytic functions in series. 



C. N. Moore, in a paper entitled, " On the Summability of 

 the Developments in Bessel's Functions " {Trans. Amer. Math, 

 Soc, 21, 1920), establishes sufficient conditions for the sum- 

 mability {Cesdro) at the origin and the uniform summability 

 near the origin of the development in Bessel functions of an 

 arbitrary function. This paper forms an important supple- 

 ment to the paper by W. H. Young {Proc. Lond. Math. Soc, 2, 18, 

 1 9 19), in which the convergence and the summability are con- 

 sidered, but not the behaviour of the series near the origin. 

 Many of the lemmas obtained incidentally in the process of 

 proving the main theorems have an interest of their own and 

 applications to Fourier series. 



Abanibhushan Dattor {Bull. Calcutta Math. Soc,' 11, 1920) 

 gives a generalisation of Neumann's Expansion in a series of 

 Bessel functions in the forms : 



fz = t bjn + ^{kz) : fz = t b„Jn + a.{kz)z 



— a 



