148 Mr Hardy, Sir George Stokes and the 



Sir George Stokes and the concept of uniform, convergence. Bv 

 G. H. Hardy, M.A., Trinity College. 



[Received 1 Jan. 1918. Read 4 Feb. 1918.] 



1. The discovery of the notion of uniform convergence is 

 generally and rightly attributed to Weierstrass, Stokes, and Seidel. 

 The idea is present implicitly in Abel's proof of his celebrated 

 theorem on the continuity of power series ; but the three mathe- 

 maticians mentioned were the first to recognise it explicitly and 

 formulate it in general terms*. Their work was quite independent, 

 and it would be generally agreed that the debt which mathematics 

 owes to each of them is in no way diminished by any anticipation 

 on the part of the others. Each, as it happens, has some special 

 claim to recognition. Weierstrass's discovery was the earliest, and 

 he alone fully realised its far-reaching importance as one of the 

 fundamental ideas of analysis. Stokes has the actual priority of 

 publication ; and Seidel's work is but a year later and, while 

 narrower in its scope than that of Stokes, is even sharper and 

 clearer. 



My object in writing this note is to call attention to and, so 

 far as I can, explain tw^o puzzling features in the justly famous 

 memoir-f- in which Stokes announces his discovery. The memoir 

 is remarkable in many respects, containing a general discussion of 

 the possible modes of convergence, both of series and of integi'als, 

 far in advance of the current ideas of the time. It contains also 

 two serious mistakes, mistakes which seem at first sight almost 

 inexplicable on the part of a mathematician of so much originality 

 and penetration. 



The first mistake is one of omission. It does not seem to have 

 occurred to Stokes that his discovery had any bearing whatever on 

 the question of term by term integration of an infinite series. The 

 same criticism, it is true, may be made of Seidel's paper. But 

 Seidel is merely silent on the subject. Stokes, on the other hand, 

 quotes the false theorem that a convergent series may always be 

 integrated term by term, and refers, apparently with approval, to 

 the erroneous proof offered by Cauchy and Moignoj. 



Of this there is, I think, a fairly simple and indeed a double 



* The idea was rediscovered by Cauchy, five or six years after tlie publication of 

 the work of Stokes and Seidel. See Pringsheim, ' Grundlageu der allgemeineu 

 Funktionenlehre ', Encyld. der Math. Wiss., II A 1, §17, p. 35. 



t ' On the critical values of the sums of periodic series', Trans. Canib. Phil. Soc, 

 vol. 8, 1847, pp. 533-583 (Mathematical and physical papers, vol. 1, pp. 236-313). 



X See p. 2-42 of Stokes's memoir (as printed in the collected papers). 



