150 Afr Hardy, Sir George Stokes and the 



I shall write the series in the form 



00 



S tin (^f^) ; 



1 

 and I shall suppose, for simplicity, that every term of the series is 

 continuous, and the series convergent, for every x of the interval 

 a^x ^b. I shall denote the sum of the series by s (x) ; and I shall 

 write 



Sn (OC) = II, (x) + Uo {x)+ ...■\- iln (.«), S (x) = Sn (x) + r,, (cc). 



The fundamental inequality in all my definitions will be of the t3^pe 



\rn(a!)'\^e (A), 



I shall refer to this inequality simply as (A). 



When we define uniform convergence, in one sense or another, 

 we have to choose various numbers in a definite logical order, those 

 which are chosen later being, in general, functions of those which 

 are chosen before. I shall write each number in a form in which 

 all the arguments of which it is a function appear explicitly : thus 

 no (^, e) is a function of ^ and e, Uo (e) one of e alone. 



It will sometimes happen that one of the later numbers depends 

 upon several earlier numbers already connected hy functional rela- 

 tions, so that it is really a function of a selection of these numbers 

 only. Thus h may have been determined as a function of e ; and 

 ??o niay have to be determined as a function of ^, e, and h, so that 

 it is in reality a function of ^ and e only. I shall express this by 

 writing 



«o = ^2o(^, e, S) = no(|^, e); 

 and I shall use a similar notation in other cases of the same kind. 

 3. The first three senses of uniform convergence are as follows. 

 A 1 : Uniform convergence throughout an interval. The 

 series is said to he uniformly convergent throughout the interval (a, b) 

 if to every positive e corresponds a.n no (e) such that (A) is true^for 

 n ^ Wo (e) and a^x^^b. 



This is the ordinary or ' classical ', and most important, sense, 

 the sense in which uniform convergence is defined in every treatise 

 on the theory of series. 



A 2 : Uniform convergence in the neighbourhood of a 

 point. The series is said to be uniformly convergent in the 

 neighbourhood of the point ^ of the interval (a, b) if an interval 

 (f — 8 (I), ^ -\-B (^))* can be found throughout luhich it is uniformly 

 convergent ; that is to say %f a positive 8{^) exists such that (A) 

 is true for every positive e, for n ^ n^ (^, S, e) = ??o (?> e), and for 



* A trivial change is of course required in the definition if t = « or ^ = b. The 

 same point naturally arises in the later definitions. 



