AND THEIR CONNECTION WITH THE THEOEY OF INTEGRAL EQUATIONS. 439 



type, the function A (X) has its zeros all real and positive. It follows, therefore, from 

 (17) that all the zeros qf the minor 



Dx !,..., 

 \ a,, a,, ...,a m 



are real and positive. Since the minor must be identically zero if 



, a a , ..., a m \ _ Q 

 ( 2 , ..., aj ~ 



(of. 11), we have thus proved the theorem : 



The zeros of all symmetrical minors of the determinant of an integral equation of 

 the second kind, whose characteristic function is of positive type, are all real and 

 positive. 



In particular, as K A (, t), the solving function of (7), is denned by 



K A (M) = D(X; s,t)/D(\), 

 where 



it appears that, when s = t, the solving function only vanishes for positive values of X. 



PART IV. THE EXPANSION OF FUNCTIONS OF POSITIVE AND NEGATIVE TYPE. 



24. It is to be remarked that HILBERT and SCHMIDT have been able to give very 

 little information about the expansion of a given symmetric characteristic function in 

 a series of products of normal functions. HILBERT* has indeed shown incidentally 

 that, if the number of singular values is finite, 



and SCHMIDT f in his dissertation has established that this equation remains valid 

 when the series on the right is uniformly convergent. The latter theorem is, of 

 course, much wider than the former as regards its generality ; but it has the defect 

 that the uniform convergence, which it postulates, is not connected with any other of 

 the properties of K (s, t). In the present section we shall attempt to remedy this in 

 some measure by proving that the equality (18) certainly holds when K (s, t) is of 

 positive or negative type. 



* 'Gott, Nachr.,' 1904, p. 73. 



t Printed with additions in ' Math. Ann.,' Band LXIII. The theorem referred to will be found on 

 'pp. 449, 450. From a remark made on p. 453 I gather that it is originally due to HILBERT. 



