of certain multiple series 95 



representative points lie on the perpendiculars from (m, m) on 

 to the axes. Then So"„i is divergent. Applying Lemma to this 

 series we obtain the construction required, nii being in fact 

 always equal to Ui. 



Theorem 12. If S2a,„.^,i,i<„,,,j, is regularly convergent ivhen- 

 ever SSif.„i.,,i is regularly convergent, then a,n,n "^^ ^f hounded 

 variation. 



In the first place it follows from Theorem 4 that ■u,„,^n is, for 

 every value of n (or m), of bounded variation in m (or n). It 

 remains only to shew that SS | A,„^„a,„,_,i | is convergent. 



Suppose, on the contrary, that it is divergent. By Lemma e, 

 we can choose a sequence of positive numbers €,„,„, tending 

 regularly to zero, so that 2Sc,rt,n, where 



is divergent. We can then modify this series as in Lemma ^ 

 without destroying its divergence. 



Now let 



m n 

 U —^l.u 

 1 1 



and suppose that 



if m = mi, n ^ ni or m ^ mi, n = ni, and that otherwise 



^ in n, — 





this last formula being interpreted as meaning e^yi^n if 



These equations define u,n,^n uniquely for all values of m and n, 

 and it is plain that U'm,n tends regularly to zero, so that 2S^/,„,,t is 

 regularly convergent. On the other hand 



^ -^ '■hn,n ^'■m,n "i ■^ '^m,n ^i')n,n ^ m,n — -^ ■^ ^ w,n> 



11 11 11 



which tends to infinity with i. Thus ^'%a„r,nUm,n is not convergent. 

 This proves Theorem 12. Combining it with Theorem 10 we 

 obtain the analogue of Theorem 5, viz. 



Theorem 13. The necessary and sufficient condition that am^n 

 should be a convergence factor is tJiat it should be of bounded 

 variation. 



