442 MR. J. MERCER: FUNCTIONS OF POSITIVE AND NEGATIVE TYPE, 



Now, when X is negative, the terms of the series on the right must be either zero 

 or positive. We conclude, therefore, that for all functions of the class B 



f f K (s, *) (it) 6 (t) d* dt>0 (X < 0). 



ft * (7 



Iii other words, K x (x, t) is of positive type for these values of X. Applying, then, 

 the theorem proved above ( 6, 10), we see that 



K, (*,*)><> (is. ==/>), (X<0) (22) 



27. "Returning to the formula (20) and writing s = t, we obtain 



L* [K A (*, ) -K, (X : , *)] = * (*, -") - i [ ^^- 



A-*---* = 1 A r 



Accordingly, from (21) and (22). it follows that 



(,.)> ilMiH! (23) 



= l A,, 



This is true, of course', for all values of m which are sufficiently great; and, further, 

 when we increase ni- we only add positive terms to the right-hand side. By a well- 

 known theorem of the elementary theory of series, we thus see that 



v [*. (*)? 

 ,,-t X,, 



converges for each value of .s in the interval (a, />) ; and hence, since 



2i^(,^,,(/)|<[^(,ff + [ 

 that the series 



convergers absolutely for each pair of values of the variables satisfying the inequalities 

 ' S s S ft, a S f : &. From this last result it follows that the function 



./M^M-s^*-^ ...... (24) 



has a definite finite value when the variables are restricted in the manner just 

 mentioned. In the paragraphs which follow we shall consider the properties of 

 f(s, t), and eventually prove that it is everywhere zero. It may be remarked that 

 the inequality (23) proves the relation 



S/(, -s) < K (s, s). 



