AND THEIR CONNECTION WITH THE THEORY OF INTEGRAL EQUATIONS. 419 



The quantity inside the square brackets is evidently a particular value of a 

 quadratic form whose coefficients are K(a 1 ,a l ), /f(a a , 2 ), * (^, ), 2* (MI, 2 ), ... ; 

 and, when ranges through the class 0, the numbers (a,), 0(a 2 ), ..., #() will 

 assume all possible real values. 



It is thus suggested that we are to look upon the double integral(l), when Granges 

 through , as the limiting case of a quadratic form whose variables assume all possible 

 real values. The function K(S, t) clearly takes the place of the coefficients of the 

 form. Moreover, when K(.S, t) is of positive type, the double integral (I) corresponds 

 to a quadratic form which cannot take negative values for real values of the variable ; 

 and similarly in regard to the case when K (*, t) is of negative type. 



Now the question, whether a quadratic form does, or does not, take both signs, as 

 the variables assume all real values, has been shown to depend on the signs of certain 

 determinants whose elements are coefficients of the form.* The considerations we 

 have just indicated seem, therefore, to point to the existence of properties of the 

 function K (.s, t) which will decide its type, without directly considering the integral 

 (1). It is the object of the present section to show that this is actually the case. 



5. Let us, for the present, confine our attention to a function K (.s-, t) of positive 

 type, so that 



for all functions 6 belonging to 8. 



We shall, in the first place, define a particular class of the functions B. Let s l be 

 any point of the open interval (<i, 6), and suppose that e and -q are any two positive 

 numbers which are so small that the points A' 1 (^ + e) also belong to the interval. 

 Then the continuous function which is zero for a < a S- i rj e and SjH-ij + eSEs 2E 6, 

 which is equal to unity for KITJ -s* 2= *i + r}, and winch is a linear function of .s in the 

 intervals (s l T? e, s, 17), (.S^ + T?, Xi + rj + e), will be denoted by &,^(* ; *i). The values 

 of the function in these latter intervals will be given by 



respectively, and will evidently be positive numbers less than unity at interior points. 

 Consider now the values of the function 



at the various points of the square a^s^b, a^t^b oi the (s, t) plane. In the 

 accompanying figure this large square, which we shall denote by Q, is intersected by 



* See, for example, BROMWICH, ' Quadratic Forms and their Classification by means of Invariant 

 Factors' (1907), chap, ii., where necessary conditions are obtained. It is not difficult to obtain conditions 

 which are both necessary and sufficient. 



3 H 2 



