AND THEIR CONNECTION WITH THE THEORY OF INTEGRAL EQUATIONS. 425 



where x lt x 2 , ..., x, t are any real constants; and consider the values of the function 

 6 (,s) (f) in Q. It will be seen on consideration that in this general case Q must be 

 regarded as divided by 8n lines, and that there are n 2 squares q^, each having a 

 border d^ (a, /8 1,2,..., n). It will also be seen that 



0(*)0(t) = X& in ^ (, = 1, 2,..., n), 

 outside the borders d afi , 



and that in the border d ai we have 



afi 



Proceeding then as in the case n = 2, we obtain the inequality 



where 



j K (.s, t) 6 (.s) (t) ds cfc-J 1 |" F. (u, v) du do < 4e (2y? + e) ( S 



.'/' i 



M, 



F n (, r) = X?K (! + , Sj + r)+ .r/K (s, + u, .s- 2 + r) + . . . + .'/: (* + , ,v,, + /) 



and hence we establish that F n (0, 0) is always > 0. Eventually we obtain the 

 general theorem : 



Every function K (x, t) which is of positive type in the square a^.^^h, ct^f^b 

 must be such that, when s lt s 2 , ..., s n are any points of the closed interval (a, b), we 



have 



X?K (.s- 1; s 1 ) + .r.j*K (*,, -s- a ) +...+.</: (s n , s n ) + '2,A\.v., K (.s- 1; s,)+ ... > 0, 



for all real values of x l} x 2 , ..., :r H . 



1 0. In accordance with the notation employed by FREUHOLM, let 



K (, A'j) /C (*', 6-3) ... K (*, A',,) 



Then, by the theory of quadratic forms, it is known that, in virtue of the inequality 

 which has just been obtained, we must have* 



(6) 



* Vide BROMWICH, 'Quadratic Forms and their Classification by means of Invariant Factors' (1906), 

 pp. 19, 20. 



VOL. CCIX. A. 3 I 



