154 Mr Hardy, Sir Georr/e Stokes and the 



the condition B 1 but a condition first foi-mulated by Arzela*, 



VIZ. 



C: Quasi-uniform convergence by intervals {convergenza 

 uniforme a tratti). ^ The series is said to he quasi- uniformly con- 

 vergent by intervals if to every positive e and every N correspond a 

 division of (a, h) into a finite number v (e, N) of intervals 8,. (e, N), 

 and a corresponding number of numbers n,(e, Nj, all greater than A^, 

 and such that (A) is true for ?? = 7?,.(?- = 1, % ...,v) and all values 

 of X which belong to 8,.. 



The deduction of Arzela's criterion from B 3, in the manner 

 sketched above, was first made by Hobsonf. 



There is one further point which seems worth noticing here, 

 although it is not directly connected with Stokes's memoir. Dini J 

 proved that if u^ (x) ^ for all values of n and x, and s (x) is con- 

 tinuous throughout {a, b), then the series is uniformly convergent 

 throughout (a, b). This theorem is now almost intuitive. For it 

 is obvious that, for series of positive terms, quasi-uniform conver- 

 gence in any one of the senses B 1, B 2, or B 3 involves uniform 

 convergence in the corresponding sense A 1, A 2, or A 3. If then 

 s {x) is continuous throughout (a, b) it is continuous for every f of 

 (a, b) ; and therefore the series is quasi- uniformly convergent for 

 every f ; and therefore uniformly convergent for every |; and 

 therefore uniformly convergent throughout (a, b). 



7. Let us now consider Stokes's definitions and proofs in the 

 light of the preceding discussion. 



It is clear, in the first place, that Stokes has in his mind some 

 phenomenon characteristic of a small, hit fixed, neighbourhood of 

 a point. 



' Let u^ -{-U.+ ... (66)', he says§, ' be a convergent infinite series 

 havmg U for its sum. Let v, + v, ■]-... (Q7) be another infinite 

 series of which the general term v.,, is a function of the positive 

 variable h and becomes equal to Un when h vanishes. Suppose 

 that for a sufiiciently small value of h and all inferior values the 

 series (67) is convergent, and has V for its sum. It might at first 

 sight be supposed that the limit of V for A=0 was necessarily 

 equal to U. This however is not true.... 



' Theorem. The limit of V can never differ from U unless 

 the convergency of the series (67) becomes infinitely slow when h 

 vanishes. 



* ' Sulle serie di funzioni', Memorie dl Bologna, ser. 5, vol. 8, 1900 up 131-186 

 701-744. ' 



t L. c, pp. 380-382. 



J L.c. (German edition), pp. 148-149. See also Bromwich, Infinite series, p 125 

 (Ex. 6). ■ '^' 



§ p. 279. 



