JANUABT 23, 1914] 



SCIENCE 



119 



suit is in sharp contrast with the continu- 

 ity of the sum which he demonstrates for 

 the binomial and other real power series. 

 At the same time he establishes the circular 

 form of the region of convergence of the 

 binomial series. Here, then, appears the 

 initial cleavage between the theories of real 

 and of analytic functions. 



The difference between the trigonometric 

 and the power series in respect to contin- 

 uity is naturally to be sought in the charac- 

 ter of the convergence at the points of con- 

 tinuity and of discontinuity. This differ- 

 ence was pointed out by Stokes in 1847 and 

 bySeidel a year later. Both discovered the 

 infinitely increasing slowness of conver- 

 gence of the series on approaching a dis- 

 continuity of its sum. Consequently a dis- 

 continuity can not be enclosed in any inter- 

 val, however small, in which the conver- 

 gence is throughout "von gleichem Grade." 

 In more modern parlance, the convergence 

 is non-uniform. Seidel in his introduction 

 explicitly points out that the erroneousness 

 of Cauchy's conclusion (see above) is ob- 

 vious from the existence of discontinuities 

 in functions represented by Fourier's series, 

 and he is evidently incited thereby to seek 

 a cause for the discontinuity. The origin 

 of Stokes's study is sufficiently obvious 

 from its title : ' ' On the Critical Values of ' 

 the Sum of Periodic Series." His failure 

 to appreciate the importance of his own 

 convergence discussion is evident from the 

 fact that it is not even mentioned in the 

 opening analysis of his lengthy memoir. 



A third discoverer of uniform conver- 

 gence was Weierstrass, who is known to 

 have been in possession of the notion as 

 early as 1841. Through his followers ( Heine 

 and others) it gradually percolated into 

 the mathematical literature. Unlike Seidel 

 and Stokes, he thoroughly realized its im- 

 portance. As Osgood has well said in his 

 Functionentheorie, he developed uniform 



convergence into one of the most important 

 organs "(methods) of modem analysis." 

 The origin of the notion in the case of 

 Weierstrass I have been unable to ascertain. 

 A conjecture or surmise may therefore be 

 pardoned. As is well known, the work of 

 Weierstrass is rooted in that of Abel, the 

 central theme or core being the theory of 

 Abelian functions. It would not seem to 

 be altogether improbable that both Weier- 

 strass 's theory of the analytic function and 

 his concept of uniform convergence had as 

 their starting point Abel's memoir on the 

 binomial series. For here, on the one hand, 

 with the demonstration of the circular form 

 of the region of convergence of the binomial 

 series, we find a proof of the continuity of 

 the series which involves implicitly the 

 idea of uniform convergence; on the other 

 hand, we have in the footnote a series with 

 discontinuities due, in fact, to non-uniform 

 convergence. It would be a small matter 

 for the discriminating Weierstrass to see 

 that the continuity of the sum could not be 

 carried over from the binomial to the trigo- 

 nometric series, because there was not the 

 same kind of convergence in the latter case. 

 If this surmise is correct, the discovery of 

 uniform convergence in the case of the third 

 ■ discoverer also is closely connected with a 

 Fourier series. 



I have dwelt at some length on uniform 

 convergence because its discovery marks 

 both the culmination of the first and older 

 epoch in the treatment of functional series, 

 and the beginning of a new one. In uni- 

 form convergence and a study of the dis- 

 continuous we have sought for the chief 

 springs of the modern theory of functions 

 of a real variable. By so doing we are led 

 to assign as a fourth great service of Fou- 

 rier's series the genesis of this theory. It 

 is not to be forgotten, however, that other 

 sources have also copiously contributed. 

 The morphology of one member of a body 



