434 ME. J. MERCEE: FUNCTIONS OF POSITIVE AND NEGATIVE TYPE, 



It thus appears that the discriminator of a definite function of positive type cannot 

 be constant throughout any interval. 



1 9. Conversely, we may show that every function of positive type, whose 

 discriminator (1) has a domain which is dense everywhere in (a, b), and (2) has not a 

 constant value in the whole of any interval, is definite. For, if this were not so, we 

 would be able to find a continuous function ty (s) other than zero, such that 



\" K (s,t)^(t)dt = (<. 9 ==&).* 



J a 



Supplying in the value of K (s, t), this equation may be written 



dt=*0 (<**==&).. . . (14) 



Now, as i// (*) is continuous, and is not zero everywhere, we can find an interval 

 (c, cl) of (a, I) within which it does not vanish ; also, as the domain of the discrimi- 

 nator is dense everywhere, it will be possible to find a point, and, therefore, a whole 

 interval (y, S), belonging both to (c, d) and the domain. The interval (y, 8) will thus 

 be such that in it the functions \]> (), 6 (s), (f) (s) do not vanish. It follows that in 

 this interval the function of a 



"dt .......... (15) 



has a derivative which does not vanish ; and hence, by a well-known theorem of the 

 differential calculus, that this function cannot be zero more than once in (y, S). It is, 

 therefore, evident that by contracting (y, S) sufficiently we can ensure for it the 

 additional property that (15) vanishes at no point belonging to it. 



Returning now to the equation (14), and supposing that s is confined to the 

 interval (y, S), we see that 



/(*) = -\'8(t)*(t)dt/f<l>(t)1,(t)dt. 



la I Js 



Hence, since both the numerator and the denominator on the right are 

 differentiable, and the latter does not vanish in (y, S), the function /(*) is 

 differentiable in this interval. In fact, by applying the ordinary rules, we obtain 



But this is impossible, because by our hypothesis f(s) cannot be constant in any 

 interval. We conclude, therefore, that K (s, t) is a definite function. 



20. It may be remarked that the conditions (l) and (2) of the preceding- 

 paragraph may be stated in another and more convenient form. For, if a discrimi- 



* Fide 3. 



