440 ME. J. MEECER: FUNCTIONS OF POSITIVE AND NEGATIVE TYPE, 



25. In the paper referred to above, SCHMIDT* has proved that, if K (s, t) is any 

 continuous symmetric function, the solution of 



is given by 



\ib (s) ['' 

 X n X Jo 



provided that X is not one of the singular values X 1; X 2 , ... , X n , ... ;t moreover, the 

 convergence of the series on the right is both absolute and uniform. Now, when we 

 take 



it is known that, in virtue of one of the characteristic relations, 



< (s) = K, (s, t). 



It follows, therefore, from the above expansion and the homogeneous equations 



ft 



^j n () = \ n \jj n (.r) K (x, t) dx (n = 1 , 2, . . .) , 



J a 



that 



It should be remarked that SCHMIDT'S theorem only allows us to assume that the 

 series on the right of (19) is uniformly convergent with respect to s (a < s < b), for 

 each assigned value of t ; and hence, by symmetry, that it is uniformly convergent 

 with respect to t (a :S t < b), for each assigned value of s. When K (s, t) is of positive 

 type, we may establish the uniform convergence of the series in the whole of the square 

 a < s < 6, a ^ t ^b, as follows. If we write t = s in (19), it is clear that the terms 

 of the series on the right become functions of s, which, with the possible exception of a 

 finite number, are all of the same sign as X ; accordingly, by DINI'S theorem,;]; this series 

 is uniformly convergent in the interval < .s < b. But, in virtue of the inequality 



the terms of the series on the right of (19) are never greater in absolute value than 

 those of 



i v 



.' (Q1 



MA..-X) 



* Pp. 453, 454. 



t We shall always suppose this to be the case in what follows. 



| DINI, " Fondamenti per la teoria delle funzioni di variabili reali " (Pisa, 1878), 99. See also YOUNO, 

 "On Monotone Sequences of Continuous Functions," 'Proc. Camb. Phil. Soc.,' vol. XIV., pp. 520-3. 



