350 SCIENCE PROGRESS 



set of linear content zero. Similar questions are dealt with 

 by S. Banach {C.R., 173, 192 1, 457). 



Denjoy's extension of the Lebesgue integral to functions 

 whose set of points of non-summability is not restricted to have 

 content zero, but can be any non-dense closed set, has already 

 got into the textbooks (see Hobson : Real Variable, 2nd edition, 

 vol. i. cap. 8). The Denjoy integral has been applied by R. L. 

 Borger {Bull. Amer. Math. Soc, 27, 1921, 325) to widen still 

 further the conditions under which the Cauchy-Goursat Theorem 

 is valid. Denjoy himself, in a series of papers {Proc. Amst. 

 Acad., 23, 1921, So, 220 ; C.R., 172, 1921, 653, 833, 903, 1218 ; 

 173, 1 92 1, 127) continues his researches on " totalisation " and 

 on trigonometrical series. 



G. H. Hardy {Proc. Camb. Phil. Soc, 20, 1921, 304) finds a 

 necessary and sufficient condition that a series should be sum- 

 mable (C, i). 



There are two general methods of transforming a series into 

 one converging more rapidly ; the first is associated with Euler 

 and Stirling, and was developed later by Markoif ; the second, 

 invented by Kummer, was extended by Leclert and Catalan. 

 H. B. A. Bockwinkel {Nieuw. Ar chief voor Wiskunde, 13, 1921, 

 383) now shows that they are essentially the same. 



G. H. Hardy {Mess. Math., 50, 1921, 165) continues his use- 

 ful notes on the Integral Calculus with a note on Mellin's In- 

 version Formula, and S. Pollard {ibid., 50, 1921, 1 5 1 ) evaluates 

 some definite integrals by means of Fourier's Integral Theorem. 



J. Wolff {Proc. Amst. Acad., 23, 1921, 585) discusses a 

 theorem of Picard on the behaviour of a uniform analytical 

 function in the neighbourhood of an isolated essential singularity. 



The Italian mathematicians are unrivalled for the elegance 

 of their geometrical treatment of matters connected with the 

 theory of functions ; so that we welcome a series of articles 

 by G. Castelnuovo {Rend. Lincei, 30 (i), 1921, 50, 99, 195, 355) 

 on Abelian Functions and another by F. Severi {ibid., 30 (i), 

 1921, 163, 204, 231, 276, 296, 328, 365) on simple integrals of 

 the first kind belonging to an algebraic surface. 



A. Buhl {Annates de Toulouse, 10, 192 1, 175) discusses the 

 circumstances which reduce the number of double integrals of 

 the second kind belonging to an algebraic surface. 



On the elhptic function transformation of the seventh and 

 higher orders there are papers by A. Berry {Mess. Math., 50, 

 1 92 1, 187) and J. H. McDonald {Bull. Amer. Math. Soc, 27, 

 192 1, 366). 



H. Villat {Annates de I'ecole normale superieure, 38, 1921, 

 183) generalises the Schwarz method for conformal representa- 

 tion to the case of doubly-connected areas which need not be 

 symmetrical. 



