444 ME. J. MERCER: FUNCTIONS OF POSITIVE AND NEGATIVE TYPE 



Now, if we write s = t = a l in (19) and (24), it is easy to see that 



and hence that K A (a,, a^ constantly increases with X, so long as the latter is negative. 

 Consequently, when e is any positive quantity, and we take c to he 



it follows from the theorem of the preceding paragraph that 



K A (i, rtj) -/c(i, i)+ 



can only vanish for positive values of X. Thus, as D (X) has no negative roots, h (.s, ) 

 is of positive type,* and, therefore, in virtue of the remark at the end of 27, 



Lt H A (s, s) > ( < s < b). 

 \-i*- ^ 



Using the formula (2G), it will he seen that this becomes 



f( S) s ) - Lf&iL?ff_ > (<<&).. 



But, as e may he taken as small as we please, this is evidently impossible unless 

 / '( rf i> - s ) vanishes. It follows that, as a, and s may each have any assigned values 

 belonging to (a, b), we must have 



/(.s-, t,) = (a < .s- < /;, a < < b). 

 We have thus shown that, in the case of a function of positive type, the series 



has K (s, t) for its sum-function. It was shown in 27 that the convergence of this 

 series is absolute, and, by an application of DINI'S theorem, it may be shown that the 

 convergence is also uniform in the square a S s : b, a < t ^ b. Hence, if /i (s), 

 i/> 2 (6'), ..., i/i n (s), ... arc a complete system of normal functions relating to a function 

 (s,t) of positive type and X 1; X 2 , ..., X,,, ... are the corresponding singular values, 

 then the series 



=i X B 



converges both absolutely and uniformly, and its sum-function is K (s, t). 

 * Owing to the fact that A (X) has only positive roots. 



