348 SCIENCE PROGRESS 



representation analytique d'une branche uniforme d'une fonc- 

 tion monogene," of which the fifth was pubHshed in 1904. 



R. D. Carmichael, in a paper " On the Convergence of Certain 

 Classes of Series of Functions " in Amer. Journ. Math., 42, 1920, 

 attacks the central convergence problem for series S{x) of the 

 form 



S{x) = -tc^e^nix) 



and also for series T{n) of the form 



T{x) = c, -f Ic, Pi{x)P,{x) - P,{x) 



where c ci . . . are constants and Vo{x), Vi{x) . . ., Pi{x), Pzix) 

 . . . are a given sequence of functions. The functions ?;;.(x) are 

 defined in such a way that a considerable variety of important 

 classes of series are included under the general form S{x). We 

 may cite, as examples, an ascending or descending power series 

 of the generalised Direchlet form 



where X^ X,i X2 • • • is an increasing sequence of real numbers 



tending to infinity. 



Alfred Kienast {Proc. Camb. Phil. Soc, 20, 1920) proves 



certain results with respect to the relation between the nature 



of Holder's mean of a sequence of complex numbers ai a^ . . . 



hm ; ,,. 

 a„ . . ., VIZ. hJk), 



where /f„'°' = «i + . . . + «n 





and that of ^^"^ s„*, 



where s„<°^ = «i + . . . + «« 





proving among other results that the existence of either limit 



