152 Mr Hardy, Sir George Stokes and the 



Young, in his paper already quoted, by means of the Heine-Borel 

 Theorem * ; and it plainly includes, as a particular case, Weierstrass's 

 theorem referred to above. 



5. It seems to me that the definition given by Stokes is not 

 any one of A 1 , A 2, A 3 ; and that, if we are to understand him 

 rightly, we must consider another parallel group of definitions. 

 These definitions differ from those given above in that (A) is 

 supposed to be satisfied, not for all sufficiently large values of n, 

 but only for an infinity o/ values. 



B 1 : Quasi-uniform convergence throughout an interval. 

 The series is said to he quasi-uniformly convergent tlvroughout (a, h) 

 if to every positive e and every N corresponds an n^ (e, N) greater than 

 N and such that (A) is true for n = n^ (e, N) and a^x^b. 



B 2 : Quasi-uniform convergence in the neighbourhood 

 of a point. The series is said to be quasi-uniformly convergent in 

 the neighbourhood of f if an interval (^ — 8(f), | + S(f)) can be 

 found throughout which it is quasi-uniformly convergent ; i.e., if a 

 positive 8(f) exists such that (A) is true for every positive e, every N, an 

 «o (f . 8, e, iV) = ?io (f , e. ^) greater than N, and f — 8 (f ) ^ .'c ^ f + 5 (f ). 



B3: Quasi-uniform convergence at a point, llie series 

 is said to be quasi-uniformly convergent for iV = ^ if to every positive 

 € and every N correspond a positive S (f, e, N) and an 



no{^,e,8,N) = n,(^,e,N), 



greater than N, such that (A) is true for n = ??o (!> f> N') and for 



Definition B 1 is to be attributed to Dini or to Darboux+. 

 Another form of it has been given by Hobson|. As Arzela and 

 Hobson§ have pointed out, a series is quasi-uniformly convergent 

 throughout an interval if, and only if, it can be made uniformly 

 convergent by an appropriate bracketing of its terms. 



Definition B 2 is for us at the moment of peculiar interest, 

 for (as I shall show in a moment) it is really this definition that 

 is given by Stokes. 



. Definition B 3 is also of great interest, both in itself and in 



* Choose € and determine 5 (^, e) and n^ (|, e), as in definition A3, for every f of 

 the interval. Every point of {a, b) is included in an interval {^-d, ^ + o). By the 

 Heine-Borel Theorem, every point of (a, b) is included in one or other of a finite 

 sub-set of these intervals. If N (e) is the largest of the Hq's corresponding to each of 

 the intervals of this finite sub-set, then (A) is true for n^N and a ^ .r ^ 6. 



This is the essence of the proof, though, like all proofs of the same character, it 

 requires a somewhat more careful statement if all apj^earance of dfpendence upon 

 Zermelo's AusicahUprinzip is to be avoided. 



t See Pringsheim, I. c. 



+ ' On modes of convergence of an infinite series of functions of a real variable', 

 Proc. London Math. Sac, ser. 2, vol. 1, 1903, pp. 373-387. Hobson (following Dini) 

 uses the expression ' simply uniformly'. 



§ L. c, p. 375. 



