AND THEIR CONNECTION WITH THE THEORY OF INTEGRAL EQUATIONS. 433 



an interval (c, d), lying within (a, b), such that at each of its points the function 

 B(f<i) <f>(*i) is zero. We shall, therefore, have (11) 



K (.<?, t) = (c < s : d, a^t^. b) 

 = (c^t^d, a < .s> < 6) ; 



in particular, K (s, t) will vanish everywhere in the square c ^ s ^ d, c ^ t < d. 

 Now, if x () is an y continuous function of ,s- defined in the interval (a, b), which is 

 zero in the intervals a ^ s < c, (/ < .s- S fr, hut does not vanish everywhere in (c, d), 

 we shall have 



* (, x (*) x (0 <i* * = I* * (*, x (*) x (0 ** dt 



by the properties of ^(.s) and K(.S-, i). It follows from this that, if K(S, t) is definite, 

 the domain of its discriminator must be dense everywhere in (a, b). 



Again, let us suppose that the discriminator of (*, t) lias a constant value p 

 throughout a certain interval (c, d). It will then be seen that within the square 



and hence, if x (*) is defined as before, that 



f f * (*, X (*) X (0 da dt = !> \\ d + () x (,} c/.sT. 



J a Ja l_Jc J 



It may be proved without difficulty that there exists a function x (.s) which is not 

 everywhere zero, and is such that 



WX*-0 .......... (13) 



For, let Xi ( s ) and X ( <s> ) ^* e an y ^ wo functions which are not mere multiples of one 

 another, and which satisfy the conditions imposed on x (*) Then, if either of the 

 integrals 



f <f>(s) Xl (s)ds, 



Jc 



is zero, we shall have an obvious solution of (13). On the other hand, if their 

 respective values /x,, /j. 2 be different from zero, it is easily seen that 



satisfies (13) ; and, in virtue of our hypothesis, x( s ) is not zero everywhere in (a, b). 

 We conclude, therefore, that we can always find a function x (*') which is such that 



VOL. CCIX. A. 3 K 



