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



hence, by dividing through both the r th row and the r th column of this determinant 

 by < (s r ) (r = 1, 2, ..., n), its value is seen to be 



>i)^(* 8 )--^(*.) 



/(*,),/(,), ../(*) 



/(.s-,), /(,), ...,/(*) 



The determinant just written can be evaluated without difficulty, and thus we find 

 that (12) is 



Now, according to our hypothesis, /'(*,) is positive and each of the factors 

 [/'(*) /'(- s '-i)] is positive or zero, It follows, then, that (12) cannot take negative 

 values when the variables are each restricted to the set 2. Taking this in conjunction 

 with what was said in the previous paragraph, we see that the functions 



/ \ / 01, 03 i / oj, 2, ..., o n 



K (* i '^'' t UM/" t "U! ,!"',*". 



can never take negative values, when the variables ,S'j, s. 2 , ..., s n , ... each range over 

 the interval (ft, b), and hence, by the theorem of 10, that K(S, t) is of positive type. 

 We may, therefore, state our results in the following theorem : 



If 0(x) and (/>(*) are each continuous functions defined in the interval (a, b), the 

 necessary and sufficient conditions that the function 



should be of 'positive type are (l) /w <Ae discriminator of the function should be 

 positive and non-decreasing in its domain S, and (2) /ja, ^y a and ft are the lower 

 and upper limits of S, 6 (s) should be zero in the interval (a, a), < (s) zero in the 

 interval (/S, b), and both B(s) and <f>(s) zero at points of the open interval (a, /3) which 

 do not belong to S. 



As a corollary of this, by supposing that (f> (s) = 1 (a < s < b), the reader may 

 deduce the corresponding conditions for the function defined in 13. 



1 8. Let us now investigate under what circumstances a function K (s, t), which 

 satisfies the conditions stated in the enunciation of the theorem of 17, is definite. 

 If the domain of its discriminator is not dense everywhere, it will be possible to find 



