I 



94 Mr Hardy, On the convergence H 



11 

 where 6, (f), a, h, and c are real, and a, ac - ¥, and the real part of s 

 are positive, is regularly convergent except for certain special 

 values of 9 and 0. In either of these series, of course, the cosine \ 

 may be replaced by a sine. 



In order to prove the converse of Theorem 10 we require two 

 lemmas analogous to Lemmas a and yS. 



Lemma e. If SSc^.n is a divergent series of positive terms, 

 we catt find €„i^n so that (1) e^^^n decreases luhen ni or n increases, 

 (2) €m,n tends regularly to zero, and (3) the series Sl,e^n,nGm,n is 

 divergent. 



(1) Suppose first that at least one row or column of the 

 original series, say the vth row 2c„i, ^, is divergent. By Lemma a, 

 we can choose a steadily decreasing sequence 97^, with limit zero, 

 so that XvmCm,p is divergent. We take 



€vi, n = Vm (n ^ v), €,n, n = {u > v), 



and it is plain that the conditions of the lemma are satisfied 



(2) Suppose that every row and column is convergent; 

 and let 



(m) (w) 



Then 'S,j„^ is divergent. We choose a steadily decreasing sequence 

 r]m so that 'S^Tjmjvi is divergent. Then SSc'„,,,„, where 



is divergent; and so Xjn, where 



yn ^ ^ Vm Cm, n j 

 (m) 



is divergent. We now choose a steadily decreasing sequence ^„, 

 with limit zero, so that ^^nln is divergent. It is clear that, if 

 we write 



// y 



(^ m,n — Vmbn^^m.jn — ^?w, n C^/i,, ^ , 



all the conditions of the lemma will be satisfied. 



Lemma ^. If SSc^.w is a divergent series of positive terms, we 

 can choose a sequence of pairs of integers (nii, Ui), tending to infinity ! 

 luith i, so that the series X%c\n,n, where c'm,n = if 771 = nit, n^ni 

 or m^nii, n = ni, and c,n,n = c,,,,,,,, otherwise, is divergent. 



The modification to be made in the series is effected by 

 drawing perpendiculars on to the axes from the points {mi, n^), 

 and annulling all terms which correspond to points on these 

 perpendiculars. Let a^ denote the sum of the terms whose 



