AND THEIR CONNECTION WITH THE THEORY OF INTEGRAL EQUATIONS. 427 



It may be remarked that, as a corollary of this theorem, we have the notable fact 

 that, if any continuous symmetric function is such that the integrals 



r* c b r* / \ r* r* r* /e e <> \ 



K ( 8l , Sl )d8 1 ,\ *(*'*') <&,<&,,..., I ... K ( Si > s *-> s ")d Sl d S2 ...ds n ,... 



Jn JaJn \*li *2/ J a J a J a \ S l, 2> > S n/ 



are none of them negative, then the functions (8) have the same property. 



11. The properties of the determinants (8) may be used to obtain some idea of the 

 nature of functions of positive type. Let us suppose, in the first place, that there is 

 a point (a^ i) belonging to Q at which one of these functions K (s, t) vanishes. The 



determinant K('' M evidently reduces to \_K(S, i)] 2 ; hence, because it can never 



be negative, 



(*, a,) = K (a l , ft) = 0. 



In other words, if we draw the square Q and the diagonal s t, the existence of a 

 point (j, j) on this diagonal at which K (.s', f) vanishes involves the fact that K (*, t) 

 vanishes everywhere on the lines drawn through this point parallel to the axes of .s- 

 and t. In particular, we deduce from this that a function K (.s, t) which is of posit ire 

 type, and is not zero everywhere in Q, cannot vanish everywhere on the diagonal s = t. 



More generally, let us suppose that there are points a,, a l} ..., a n of the interval 

 (a, b) such that 



, a a , ..., a n \ 



\a t , a.,, ..., a 



By considering the determinant whose elements are the first minors of the four 

 elements belonging to the first two rows and columns of 



*, ,, a,, ..., a n 



, ,, ,, ..., n (1Q) 



\8, a 1; a 2 , ...,/' 



we obtain the equation* 



K s, i, a a , ..., a n K /a 2 , a 3 , ..., a n 

 \s, a,, a 2 , ..., aj \a 2 , a s , ..., a n 



= K /S, a,, ..., n \ K (a 1 , a.,, ..., a n \ _ f /, a , -., n \l 2 

 \s, a 2 , ..., aj \a 1} a 2 , ..., aj \ 1; a 2 , ..., aj] 



Recalling that the first term on the right vanishes in virtue of our hypothesis, and 

 that neither of the terms in the product on the left can be negative, it is clear that 

 we have 



s, a,, ..., a n \ 





* Vide SCOTT and MATHEWS, ' Theory of Determinants' (1904), p. 62. 



3 I 2 



