FUNCTIONS IN THE THEORY OF INTEGRAL EQUATIONS. 143 



13. Let us now suppose n > ; then, for values of t in (0, ], it is possible to 



find a positive number such that, for all values of p which are greater than it, 

 is less than unity. For these values of p and t we have . 



and so 

 Since 



for values of p and t which are not negative, we see from this that, for values of t in 



(0,), P (p, t) is less than a fixed positive number P independent of p. We 



\ p / 



p 

 therefore have 



Jo 



As the right-hand member converges to zero when p increases indefinitely, it 

 follows that the relation (25) is true ; and hence that 



(22) 



It may be shown in a similar way that 



* (27) 



where x(0 l8 anv function which has a limited second derivation in (0,^), and is 

 such that 



x() = o. x'() = i. x'.(0>o (osssi). 



It follows from the result obtained in 10 that 



Let us denote 



i {/[-* W] +/[+* ()]>. 



where s is a fixed point of the open interval (0, TT), by X (t) ; and let us suppose, for 

 the moment, that X( + 0) is finite. Then, if c is an arbitrarily assigned positive 

 number, we can choose a so small that 



