92 



Mr Hardy, On the convergence 



Theorem 10. // a.m.,n *'* of bounded variation, and 2St/,„,,j, is 

 regularly convergent, then SSa,„,,„M,„,„ is regidarly convergent. 



In virtue of Theorem 8, it is enough to prove this when o.,n,n 

 is real. In virtue of Theorem 9, it is enough to prove it when 



am.,n > 0, A„,a,rt,,i ^ 0, Ana,„,„ > 0, ^m.,na.m,n > 0. 



In the first place, by Theorem 3, every row and column of the 

 series S2a,„^„iA,«,,i is convergent. 



In the second place, we have 



m n m n m n 



p-l («. q q-l P V pg 



+ 2 A^ a^, g 2 2 %i, j + , 2 A^ a^^ ^ 22 Uij + a^^ ^ 2 2 Uij * . 



m in n ii m n in n 



It follows that, if _p ^ m, q^n,we have 



p q 



<a 



m,n-'^'m,,n> 



2^2^ a^l y ^fji, V 



mn 



where Hm^n is the upper bound of 



1 '^ " i 

 '^'%Ui,j (/-t ^ m, n ^ v). 



\ mn I 



Now 



p q mn f P 1 p n m q^ 



22 a^, ^iV,"- 22 «M,^ '?''/",>'= 2 2 + 22+22 



1 1 ' 11 \in+l n+1 m+1 1 1 w+l 



II 



and so 



p q 



21 

 1 1 



1 1 ' ' 11 



where h^^n is the upper bound of 



22«t,j 



for all values of k, I, /x, and i/ such that fi'^k, v^l, and Tom 

 or l>n. 



* See pp. 124—125 of my paper quoted in § 1, where the general form of this 

 identity, for multiple series of any order, is given. Similar transformations of 

 double series were given independently by M. Krause, ' Uber Mittelwertsatze im 

 Gebiete der Doppelsummen und Doppelintegrale', Leipziger Berichte, vol. 55, 1903, 

 pp. 240 — 263. See also Bromwich, 'Various extensions of Abel's Lemma', Proc. 

 London Math. Soc, ser. 2, vol. 6, 1907, pp. 58—76, where further interesting 

 applications of the identity are made, 



