of certain multi'ple series 91 



Theorem 9. The necessary and sufficient condition that a real 

 function a.m^n should he of bounded va,riation is that it should he of 

 the fo7'm Am,n — A\n,n, whcrc 



and A\n^n satisfies similar conditions. 



Suppose first that a-m^n is of tlie form indicated. It is plain 

 that the series 



^^7n-^m,n> ■^^7i-^7n,nt ■"-^^m, ?i-^?n., n> 

 m n 



and the corresponding series formed from ^',„„>i, are all convergent. 

 Further we have 



and similar inequalities for A„o,„_,i and \n,n(^m,n- Hence «,„,„ is 

 of bounded variation. 



Next suppose that A,n,n is of bounded variation, and let 



Pm,n = I ^m,n(^7n,n \ \^7n,7i^m>7i ^ ")> Pm, 7i =" " \^rn, 7i^7n,7i < ^)) 

 P 7)1,71 ~ I ^m,7iO-7n,7i I ( ^m, Ji C^m, ?i ?$ ^/i P m,7i ~ ^ \^m,n(^7n,7i > ^)- 



Suppose also that 



C» 00 00 00 



^-'711, Jl — -^ — JJlJL, VI ^^ 771, 71 — -^ — ' Z' M, I' • 



771 ji m rt 



Then it is plain that 



^7n-Drn,7i^ ^> ^n^in, 7i-^^i ^7n,7i-t^7n,n^^^ 



and that B'^^n satisfies similar conditions. 

 Also 



00 00 



J^7n, n — -O 7JJ , jj = -^ .i ^fi, V Ojh, V ^^ ^m, ?!. ^m ^7i "r '^j 

 m w 



(^771,71 ^^ -ttnijTi -O 7ji, n + ftjn ~r ttjj ft. 



But, by Theorems 7 and 2, we have 



where 0^., CJ, D,,, and D,/ are positive and steadily decreasing 

 functions. Thus 



^7n, >i ^^^ ■^7n,7i -^ on, 7i > 



where 



Am,7i — Bm^n "H ^m + Dn + Jif, A ,n,7i— B rn.^n + ^m + -L'oi + -^ > 



£^ and E' being suitably chosen constants; and it is clear that 

 Am,7i and A',n,7i will satisfy the conditions of the theorem if 

 E and E are sufficiently large. 



7—2 



