AND THEIR CONNECTION WITH THE THEORY OF INTEGRAL EQUATIONS. 435 



nator satisfying these conditions had the same value at two distinct points, it would 

 necessarily have that value at all points of its domain which lie between them ( 15). 

 Thus, since the condition (l) and the continuity of 0(s), <f>(s) assure us of an interval 

 of the domain which lies between these points, the condition (2) would be violated. 

 Hence a discriminator of this kind must be a steadily increasing function ; and, 

 conversely, a steadily increasing discriminator satisfies (2). We may, therefore, 

 combine the results of the two preceding paragraphs in the theorem : 



The necessary and sufficient condition, that a function K (a, t), satisfying the 

 requirements of the theorem of 17, should be definite, is that its discriminator 

 should be a steadily increasing function whose domain is dense everywhere in (a, b). 



As an application of this theorem we may consider the function* 



= (t-a)(b-s) (,>0- 



The discriminator has the open interval (a, b) for its domain, and its value at any 

 point is 



which steadily increases with s. It follows from 17 and the theorem just stated 

 that K(M, t) is a definite function of positive type. 



21. Leaving the particular class of functions with which we have been dealing, 

 let us now suppose that K (s, t) is any function of positive type defined in the square 

 a : s < b, a^t^b. Let a 1; 3 , . .., a m be any m points of the interval (a, b] which 

 are such that 



A/,, a a , ..., a m 



A. 1 I 7* v i 



Then the function 



, lt 2 , 



will evidently be symmetric and continuous in the square a : s < b, a < t < b. 

 Again, when the function 



(Si, s 2 , ..., s n> i, a 2 , ..., 



1 ~ y n ft n 



o. W'l . vto* . . . . vv}i 

 H) 1J *? > 771^ 



is expressed as a determinant, it is easy to see that the minor obtained by suppressing 

 all but the i* h of the first n rows and all but the /* of the first r> columns is 



(Sit a l> CT 2> > Mr. 



* This is the generalised form of HILBERT'S classical function, vide 'Gott. Nachr.,' p. 227 (1904). 



3 K 2 



