538 SCIENCE PROGRESS 



any individual term satisfies a certain differential equation, 

 the function represented by the sum of an infinite number of 

 < such terms will also satisfy it. Prof. Young proves that such 

 a conclusion is valid if the function f(x) has, in the interval 

 in which such an equation is to be satisfied, a differential 

 coefficient of suitable order, such that the Fourier series of a 

 function equal to this differential coefficient in this interval, 

 and having any convenient values elsewhere, converges in 

 the interval in question. This occurs in particular if the 

 differential coefficient has bounded variation. 



Considerable limitations still hold, however, as the author 

 points out, in directions, and more especially in other regions 

 where the nature of the convergence must be elucidated — 

 which are to form the subject of further investigation, and 

 on which it is not at first sight clear that the same method is 

 readily applicable. But this paper marks a very notable 

 advance in the pure theory and the logical basis of many of 

 the expansions which are used regularly by the applied mathe- 

 matician. 



The second paper to which we have referred above and 

 which we have already mentioned in Recent Advances (for 

 January 1920) is by G. H. Hardy and J. E. Littlewood, on 

 Abel's theorem and its converse. This paper is intimately 

 related to many others by the same authors. The main 

 object of the authors is to obtain as far-reaching a generalisation 

 as possible of Tauber's theorem — the converse of Abel's 

 well-known theorem relating to power-series, which, with 

 Stolz's generalisation, is described, for instance, in Bromwich's 

 treatise on Infinite Series. Let the power series be 2 a n n n or S, 

 and let its sum be f(x). Let X a H be called A, the point x=i 

 on the circle of convergence being in question. In describing 

 the conclusions of the authors, certain of their abbreviations 

 are necessary. Thus (K) may stand for the proposition that 

 A is convergent, (L) for the proposition that f(x) tends to A, 

 and (0) and (o) for the properties of the coefficients usually 

 written as 



n n 



Then Abel's theorem states that (K) implies (L) along a 

 path C which is a radius (o, 1) of the circle of convergence, 

 whereas Stolz's theorem is the same for a so-called Stolz path. 

 The authors have already proved several associated results, 

 for example that the theorem is not true for any regular path. 

 Tauber's theorem states that (L) and (o) imply (K) when the 

 path (C) is the radius (0,1), and this theorem has been ex- 

 tended to any Stolz path, whereas it is also known that (L) 



