422 MK. J. MERCER: FUNCTIONS Of POSITIVE AND NEGATIVE TYPE, 



for all values of e which are less than a certain positive number ( 5). But this is 

 evidently impossible, because, when e tends to zero, the right-hand side tends to zero, 

 and we arrive at the contradiction that a fixed positive quantity (viz., rfa) is less 

 than, or equal to, zero. 



We conclude that K (s l} i) cannot be negative when , lies in the open interval 

 (a, b) ; and hence, since K (s, s) is continuous in the same interval when regarded as 

 closed, we have the result that every function K (s, t) which is of positive type in the 

 square a < s ^ b, :<& satisfies the inequality 



K ( *,) > 0* (a == ! < b). 



7. This is a first condition which must be satisfied by these functions, and we may 

 obtain a second on similar lines. Let s, and tt a be any two distinct points of the open 

 interval (a, b), and, as before, let e and 17 be two positive numbers ; the latter will now 

 be supposed so small that the intervals [.s-, (r/ + e), *!+ (17 + e)], [$ 2 (r/ + e), s 2 + (>7 + e)] 

 are both contained within (a, b) and do not overlap. We now propose to consider the 

 values of the function 



[x.O^ (s ; Sl ) + x.A,, (* ; *)] OA, (t ; .s-,) + a- 2 0.., (t ; ,s 2 )] 



at points interior to Q, when x-i and x. 2 are any real constants. For this purpose we may 

 make use of a diagram (tig. 2) which is an obvious extension of the one employed in 

 the previous paragraph. The square Q is divided in this case not by eight, but by sixteen 

 lines, viz., those whose equations are * = s a 77, .s- = ,v a + (77 + e) ; t = s ft + rj, t = s/, (77 + e) 

 (a., ft = 1, 2). By giving a and /3 all possible values in the equations just written, it 

 will be seen that we obtain four sets of eight, for each of which we can distinguish a 

 square q^ bounded by the lines .s = ,s- a + T), t = .^ + ?? ; moreover, these squares will 

 evidently have borders d^ of width e. It is not difficult to see that, in those parts 

 of Q which are exterior to the borders d afi (a, ft = 1, 2), we have either 



or 



that in the square q afi we have 



* The reader may compare this with the fact that, when we have a quadratic form which only assumes 

 non-negative values, and we put all the variables save one (say x^ equal to zero, we deduce that the 

 coefficient of x-f must be > 0. 



t Both these pairs of equalities will hold in certain parts of the square, but we only require that at 

 least one of them should be true. 



