156 Mr Hardy, Sir George Stokes and uniform, convergence 



vanishes. For if Un, V^ represent the sums of the terms after 

 the nth in the series {QQ), (67) respectively, we have 



V^Vn + V,:, U=U^+Un'; 

 whence 



v,:=Y-u-{v,,-u,,)^-u,:. 



Now V-U, Yn- Un vanish with h, and Ua vanishes when n 

 becomes infinite. Hence for a sufficiently small value of h and 

 all inferior values, together with a value of n sufficiently large and 

 independent of h, the value of F,/ may be made numerically less 

 than ^ any given quantity e however small ; and therefore, by 

 definition, the convergency of the series (67) does not become in- 

 finitely sloiv when h vanishes.' 



Now this argument is, until we reach the last sentence, perfectly 

 accurate, and indeed, if we translate it into inequalities, substantially 

 identical with that given in § 5, under (2). Stokes proves, in fact, 

 that continuity at | involves quasi-uniform convergence at |. 

 Where he falls into error is simply in his final assertion that this 

 property is that which he has previously defined, the mistake being 

 due to a failure to observe that his intervals of values of h depend 

 upon a prior choice of e. In a word, he confuses, momentarily, 

 B 2 and B 3. The ordinary view that Stokes defined uniform 

 convergence in the same sense as Weierstrass compels us to suppose 

 that he confused B 3 with A 1 , or at any rate with A 2 : and this 

 is hardly credible. 



I add one final remark. If we could identify Stokes's idea with 

 B_3, instead of with B 2, we could acquit him of having made any 

 mistake at all, since B 3 really is a necessary and sufiicient con- 

 dition for continuity. We could then regard Stokes as having 

 anticipated Dini's theorem. This view, however, does not seem to 

 me to be tenable. 



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