AND THEIE CONNECTION WITH THE THEORY OF INTEGRAL EQUATIONS. 429 



positive and negative types to be mutually exclusive, save for the trivial case when 

 K (s, t) vanishes everywhere. For, if K (s, t) belongs to both classes, we must have 



for all points of the interval (a, b) ; and hence *(.<?,, s,) must be zero everywhere in 

 this interval. It follows, then, from a remark made in 11, that K(.S, t) is zero in the 

 whole square Q. 



PART III. CERTAIN FUNCTIONS OF POSITIVE TYPE. 



13. In the present section we propose to investigate certain species of functions 

 which are of positive type. The remark made at the end of the previous section 

 ( 12) will make it plain that there is no loss in thus limiting ourselves, since the 

 corresponding results for functions of negative type may be at once deduced by the 

 device there explained. 



Let us again consider the square Q of the (x, t) plane which is bounded by the lines 

 <t = a, s = b, t a, t = b ; and let us suppose that it is divided into two triangles by 

 the diagonal whose equation is s = t. The most direct method of denning a continuous 

 symmetric function in Q is, evidently, to define a continuous function in one of the 

 triangles, say that in which N < / ; and then to suppose this continued into the 

 remaining portion of the square by defining its value at a point for which .s- > t to be 

 that at its image by reflection in the diagonal. For example, if 9 (,s) is a continuous 

 function of s in the interval (ft, b), and we define K (.s, t) to be equal to 9 (s) in the 

 triangle .s ^~ t, then the continuation of this function into the triangle .s > t is 

 evidently 9 (t}. 



The theorem of 10 may be applied to the function we have just defined, and hence 

 the condition that it should be of positive type deduced. Instead of doing this, 

 however, we shall consider the more general function* 



(,<)-'()*(') (**t) 



*()*(*) (.a). 



where 9 (s) and 4> (>">') are both continuous in the interval (a, b). It will be remembered 

 that functions of this kind occur as GREEN'S functions of certain linear differential 

 equations of the second order, and that it is therefore of some interest to know when 

 they are of positive type. Accordingly we shall seek necessary and sufficient 

 conditions which will ensure that this is so. 



14. In the first place, let us suppose that 6 (s) and < (s) are any continuous 

 functions whatever ; and let 2 be the set of points belonging to (a, b) at which 

 neither of them vanish. This set will evidently be dense in itself in virtue of the 



* Of. BATEMAN, ' Messenger of Mathematics,' New Series, 1907, p. 93. 



