438 MR. J. MEKCER: FUNCTIONS OF POSITIVE AND NEGATIVE TYPE, 



On comparing these latter functions with h (s, t), it will be seen that we have 



h, (ft, t) = h (s, t) a = s == a 1( a < * =E a 1; 



= elsewhere ; 



and 



/t 2 ( s> t) = h (s, t) aj :< s < b, i < t ^ 6, 



= elsewhere. 

 It follows from this that we have 



at each point of the square in which these functions are defined. But it is easily seen 

 that, as K (s, t) is of positive type, the functions h r (s, t) satisfy the requirements of 

 the theorem enunciated in 17. Thus h(s, t) is merely the sum of two functions of 

 the same nature as K (x, t), and hence, as it is obvious d priori that the sum of any 

 number of functions of positive type is a function of positive type, it appears that we 

 do not in this way obtain any new species of these functions. 



The reader may convince himself in a similar manner that the same conclusion 

 holds in regard to the more general function considered in the preceding paragraph. 



23. Although the result of 21 proves to be so barren in this respect, it may be 

 applied to obtain an interesting property of the symmetrical minors of the deter- 

 minant of the integral equation 



t, ....... (7) 



when K (s, /) is of positive type. Adopting the notation and hypothesis of the 

 paragraph referred to, let A(X) be the determinant of the above integral equation 

 when h (.<?, t) replaces K (s, t). Then, since 



A (X) = 1 -X f h (.s- 1; Sl ) ds, + - f" [" h (*' * 8 ) ds, ds 2 - ... 



Ja Z:\JnJa \' s l) ''2/ 



A (X) = D ( X a " a2 ' " <M / K ('"" a2 ' "' a "\ (17) 



\ a 1} a 2 , .... aj/ \a 1; a 2) ..., a m j 



H a a S 1 ,S 3 ,...,S 



it is easily seen from (16) that 



where 



and is, therefore, a symmetrical m th minor of D (X), the determinant of (7), in accord- 

 ance with FREDHOLM'S definition. But, as we have shown that h (a, t) is of positive 



_ /!, Oj, ...,o m \ x f / a lt a,, ..., m \ , 



~ K ( ai , a 2 , ..., aj ~ X J a K U l5 a,, ..., oj dSl+ - 



