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



The determinant of n rows and columns, whose elements are these minors, will 



therefore be 



T fa lt a 2 , ..., a m \~] n h As,, .s 2 , ..., s n \ 

 [ U,, a 2 , ..., aj] \8 l} .s- 2 , ..., sj 



But, by the theory of determinants, we also know that it is equal to* 

 i, 2, , a Y B ~ 1 A'i, s 2 , ..., s n , ..., ci m \ 



ota, 2, , / \n *2, > - s > a i> > 



Thus, equating these two values, we find 



, f*i, % .... \ _. K /i, ^ ., *., ., .. K 



" 



hi "a, -. 



Now, in virtue of our hypothesis that K (x, /) is of positive type, it follows from 10 

 that the quotient on'tlio right-hand side of this equation has a denominator which is 

 positive and a numerator which is not negative. Hence we have 



and thus, as this is true for all values of n, the theorem of 10 shows that h(t<,t) is 

 of positive type. 



22. In the light of this result, it appears that each function of positive type can 

 be used to generate an infinite series of such functions. We might, therefore, expect 

 to obtain other species of functions of positive type by taking K(S, t) to be of the kind 

 considered in 14-20. 



For simplicity, let us consider the function 



where 



(',) = (a,) <(,) * 0. 



Confining our attention to the triangle s^t, it will be. seen that the variables 

 s and t can be related to the constant a, by either of the inequalities : - 



(i) s ^ t ^ 

 (ii) s < ! == t, 

 (iii) a, < x < t. 



The reader may find it convenient to refer to the accompanying diagram, in which 

 * Fide SCOTT and MATHEWS, op. df., pp. 67, 68. 



