of certain multiple seines 89 



and the double limit 



lim a^n., n= CL, 



all exist. In this case 0^ a.nd «„ tend to a when m and n tend to 

 infinity. 



Lemma 7. // SSum^n is regularly convergent, to the sum s, and 



m n 

 1 1 



then, given any positive number e, we can find co so that 



I *m,M S I < 6 



if either m or n is greater than co. 



We may suppose s = without loss of generality. Since the 

 double limit exists, we can choose w^ so that | 6',„,^„ \< e if ni and n 

 are both greater than ^i. When &)i is fixed we can choose o)., and 

 «»3 so that the inequality is satisfied for 1 ^m ^(o^, n> Wn and for 

 m > &)3, 1 •^ n ^Wj. We can then take to to be the greatest of <wi, 

 (On, and 0)3. , . 



Lemma S. In the same circumstances, we can choose w so that 



pq 

 m n 



< e 



if p^m, q^ n, and either m or n is greater than w. 

 This follows at once from Lemma 7 and the identity 

 1^ _ , 



m n 



4. I shall say that a^n^n is of bounded variation in (m, 71) if 



(1) a^n^n is, for every fixed value of m or n, of bounded 

 variation in ?i or m, 



(2) the series 



is convergent. And I shall say that ar,i^n is a convergence factor if 

 SSa„i_,j,Mm_,i is regularly convergent whenever SSw,„,,„ is regularly 

 convergent. My main object is to prove the analogue of Theorem 5 

 for double series, i.e. to establish the equivalence of these two 

 notions. 



VOL. XIX. PARTS II., III. 7 



