of certain multiple series ' 93 



Hence, by Lemma 8, we can choose co so that 



p q m n 



11 11 



if m and n are greater than eo. Thus the double series is con- 

 vergent, and, since its rows and cokimns are convergent, it is 

 regularly convergent. 



When a^n,^n and its various differences are positive, this theorem 

 is nearly the same as that referred to in § 1. It is related to the 

 latter theorem, in fact, as what Dr Bromwich calls 'Abel's test' for 

 ordinary convergence is related to 'Dirichlet's test'.* The more 

 direct generalisation is as follows. 



Theorem 11. If a,n^n is of bounded variation and tends 

 regularly to zero, and 



m n 



2^ 2^ U/j,^ „ 



1 1 



is bounded, then S'Zam,n%n,7i ^■^ regularly convergent. 



The proof is similar to that of Theorem 10, and I need hardly 

 write it out at length. The theorem shews, for example, that 

 the series 



2S 



cos (md + n^) 

 (a + mco + nw'f ' 



where 6 and <^ are real, w'jui is positive or complex, and the real 

 part of s is positive, is regularly convergent except for certain 

 special values of 6, </>, and a; or again that the series 



^^ cos (md + n(f)) 



(arn^ + 2bmn + cn'-y ' 



* Theorem 10 itself does uot seem to have been enunciated before, even in the 

 specialised form. The nearest theorem which I have been able to find is one 

 given by G. N. Moore, ' On convergence factors in double series and the double 

 Fourier's series', Tia)is. Amer. Math. >Soc., Vol. 14, 1913, pp. 73 — 101. Moore's 

 theorem (a particular case of a theorem concerning Cesaro summability) is as 

 follows : if 



(1) 23«„j, ,j is convergent as a double series in Priugsheim's sense, 



VI n 



(2) |2S«^^J<:7i, 



(3) ^m, » ^ 0, 



00 00 



(4) lim 2 |«„,,,J =0, lim S|a„^,„|=0, 

 m-^cc »=1 w-^-a)m = l 



(5) 22 I M^, „ a,„, „ I 

 is convergent, then 22a„j „ Um,n *® convergent. 



