RECENT ADVANCES IN SCIENCE 7 



terval or intervals to which we restrict ourselves. The results 

 lead to a notable extension of our knowledge of the behaviour 

 of power series on their circle of convergence. Again, con- 

 nected with this are also Young's (ibid, x-xi) investigations on 

 the connection between Legendre's series and Fourier series, 

 and (ibid, xi-xiii) on series of Bessel functions (cf. Proc. Roy. 

 Soc. 191 8, A, 94, 292-5). 



G. H. Hardy and J. E. Littlewood (Proc. Lond. Math. Soc. 

 1 gig, 17, xiii-xv) indicate a proof of a theorem given without 

 proof in their paper on Abel's convergence-theorem and its 

 converse, because this theorem contains the complete solution 

 of the problem of Cesaro summability for the Fourier series of 

 a bounded function. 



T. A. Brown (Proc. Edinburgh Math. Soc. 191 5-16, 34, Re- 

 search Paper, 191 6, No. 1) shows the relation between Whit- 

 taker's (191 5) " cardinal function " of the functions cotabular 

 with f(x) (Science Progress, 1916, 10, 435) and Fourier's re- 

 peated integral, gives a new derivation of Fourier's integral 

 formula, and extends the notion of the Fourier integral to the 

 case in which the variables involved are complex. 



E. J. Wilczynski (Amer. Math. Monthly, 191 9, 26, 9-12), 

 starting from the well-known fact that the existence of a scale 

 of relation is a necessary and sufficient condition that a power 

 series defines a rational function, gives the corresponding con- 

 dition for the expansion in a power series of a general algebraic 

 function ; this condition seems to have hitherto escaped notice. 

 W. F. Osgood (Trans. Amer. Math. Soc. 191 8, 19, 251-74) dis- 

 cusses the nature of an analytic transformation in the neigh- 

 bourhood of a singular point. M. J. M. Hill (Proc. Lond. Math. 

 Soc. 1 9 19, 17, 320-33) applies the same method that he used 

 in 1902, for obtaining by ordinary algebraic expansion the 

 continuations of series which have one and two singular points, 

 to the case of the hypergeometric series — which has three 

 singular points. 



Representations of analytic functions by means of con- 

 tinued fractions have certain advantages over those by power 

 series, but the great impediment to the use of continued frac- 

 tions in the theory of functions and differential equations is 

 the want of algorithms for adding, multiplying and differ- 

 entiating them. Whittaker (Proc. Roy. Soc. Edinburgh, 19 16, 

 36, 243-55 ; Research Paper, 191 6, No. 6) supplies in some 



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