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



and this is true independently of the number of points, s u s 2 , ...,s n and their situation 

 in the interval (a, b). 



Conversely, by an appeal to the theory of integral equations, we may prove that 

 any continuous symmetric function K (s, t) defined in Q, which satisfies this condition, 

 is of positive type. For it will be remembered that, according to FREDHOLM'S 

 theory,* the singular values of the equation 



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



J a 



are the zeros of the integral function 



D(X)= 1-xf *(*,) rf*i+ f l**^ 1 '*')*^,... 



J /i : J a . a \"1) J/ 



Applying our hypothesis that (G) holds for all values of ,<?,, ,s 2 , ..., $, it appears that 



C XV' 

 the coefficient of ' f- in the series on the right cannot be negative ; moreover, 



tL ; 



HILBKUT has proved that every continuous symmetric function has its singular values 

 all real. It follows, therefore, that, if X,. is any one of the zeros of D (X), we shall 

 have 



where the series in the square brackets on the left is not negative and that on the 

 right is positive ; and hence, that X r must be positive. Since we have seen that, for 

 K (,s, t) to be of positive type, it is sufficient that all the singular values of (7) should be 

 positive, we may now state the following theorem : 



In order that a continuous symmetric function K (s, t) defined in the square 

 a<s<6, a-^t^l may be of positive type, it is necessary and sufficient that the 

 functions 



should never take negative values when the variables s lt s 2 , ..., s n ... each range over 

 the closed interval (a, b). 



* Fide 'Acta Mathematical XXVII (1903). 



