concept of uniform convergence • 149 



explanation. In the first place it must be remembered that Stokes 

 was primarily a mathematical physicist. He was also a most acute 

 pure mathematician ; but he approached pure mathematics in the 

 spirit in which a physicist approaches natural phenomena, not 

 looking for difficulties, but trying to explain those which forced 

 themselves upon his attention. The difficulties connected with 

 continuity and discontinuity are of this character. The theorem 

 that a convergent series of continuous functions has necessarily 

 a continuous sum is one whose falsity is open and aggressive : 

 examples to the contrary obtrude themselves on analyst and 

 physicist alike. The falsity of this theorem Stokes therefore 

 observed and corrected. The falsity of the corresponding theorem 

 concerning integration lies somewhat deeper. It is easy enough, 

 when one's attention has been called to it, to see that the proof 

 of Cauchy and Moigno is invalid. But there are no particularly 

 obvious examples to the contrary : simple and natural examples 

 are indeed somewhat difficult to construct*. And Stokes, his 

 suspicions never having been excited, seems to have accepted the 

 false theorem without examination or reflection. 



This is half the explanation. The second half, I think, lies in 

 the distinctions between different modes of uniform convergence 

 which I shall consider in a moment. 



Stokes's second mistake is more obvious and striking. He 

 proves, quite accurately, that uniform convergence implies con- 

 tinuity f. He then enunciates and otfers a proof ;J; of the converse 

 theorem, which is false. The error is not one merely of haste or 

 inattention. The argument is as explicit and as clearly stated in 

 one case as in the other ; and, up to the last sentence, it is perfectly 

 correct. He proves that continuity involves something, and then 

 states, without further argument, that this something is what he 

 has just defined as uniform convergence. It is merely this last 

 statement that is false. 



Stokes's mistake seems at first sight so palpable that I was for 

 some time quite at a loss to imagine how he could have made it. 

 A closer examination of his memoir, and a comparison of his work 

 with other work of a very much later date, has made the lapse a 

 good deal more intelligible to me ; and my attempts to understand 

 it have led me to a number of remarks which, although they 

 contain very little that is really novel, are, I think, of some 

 historical and intrinsic interest. 



2. There are no less than seven different senses, all important, 

 in which a series may be said to be uniformly convergent. 



* See Bromwich, Infinite sfrlea, pp. 110-118; Hardy, ' Notes on some points in 

 the integral calculus', XL, Messentjer of Matlieniatica, vol. 44, 1915, pp. 145-149. 

 t p. 282. I use ' uniform ' instead of Stokes's ' not infinitely slow '. 

 X p. 283. 



