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



moreover, it is easily proved that, in virtue of the symmetry of K (s, t), 



22 r 



2 2 x^Xf, K (s, t) (ds dt) 



a=l /3 = 1 Jj a 



can be written as 



T r f r, a K (s, + u s, + v) + 2x^x 2 K (s , + , s 2 + v) + X/K (s 2 + u,s 2 + v\] dudv. . ( 5 ) 



L i \ i i j / \ * / \ /-i \ / 



J 7J J V) 



From this and the equation (4) we finally obtain the inequality 



r* f* P 1 f" 1 \ ? 2 



, J /I J T) */ TJ 



where F 2 (w, v) is the integrand of (5). 



The function F 2 (u, v) is, of course, dependent on the real constants x l and x 2 ; let 

 us suppose it possible to choose them in such a way that 



F,(0, 0) = a-A^i, *0 + 2x 1 x 2 *(s l , s 2 ) + x 2 2 K (s 2 , s 2 ) 



takes a negative value, say ft. Owing to the fact that K (s, t) is continuous, it is 

 then clear that we can chose rj so small that 



for u S 17, | v ^ r). From this we deduce the inequality 



as in the corresponding place in 6 ; and hence, as this is impossible for sufficiently 

 small values of e, it follows that, when y 1 and s 2 lie in the open interval (a, 6), and x l 

 and x 2 are real, F 2 (0, 0) is not negative. Accordingly, since K (s, t) is continuous, it is 

 easily seen that every function K (s, t) which is of positive type in the square a < s ^ b, 

 a ^ t < b is such that, when x { and x 2 are any real numbers, 



~ Si ~ 



X\K (Si, $l) + ZX&K ($!, S 2 ) + X/K (S 2 , S 2 ) > ( a ~ Si ~ , ) . 



V 'A/ 03 U J 



9. The reader will now be prepared for a general theorem of which those already 

 considered are particular cases. After having been through the latter in detail it will 

 be sufficient to sketch the general proof. 



Take any n distinct point s l} s s ,...,s n in the open interval (a, b), and suppose that e 

 and 17 are so small that the intervals [s a (i? + e), s a + (^ + e)] (a = 1, 2,...,) form a 

 non-overlapping set contained within (a, b). Now let 



6(8)= $x a 8^(s;s a ), 



