FUNCTIONS IN THE THEORY OF INTEGRAL EQUATIONS. 121 



that is to say, less than 



When any positive number c is assigned, we can clearly choose a negative number 

 A, whose numerical value is so great that the first term is less than for values of 

 X :s A, and the second term is clearly less than h, for all values of X. Observing that 

 our choice of A is quite independent of m, we deduce the inequality 



s FT *<*) f *- (0/(0 * =s A+e, (x s A,), 



n = 1 A M n. J a 



which, in virtue of (4), may be written 



- X f K A (s, t) f(t] dt^h + f , (X A,). 



J a 



In a similar way, it may be shown that, if 



o-. (s) > & 

 for all values of n ^. N 2 , there exists a negative number A 2 such that 



for all values of X ^ A 2 . 



4. Let U(s) and L(s) be the upper and lower limits of indeterminacy of the 

 series (5), it being supposed that, if necessary, these may have either of the improper 

 values oo . In the first place, let us assume that U (s) is a finite number. 

 Corresponding to any positive number e, we can then find a positive integer N, such 



that 



<r n (s) < U(*) + e 



for all values of n ^ N\. It follows from what was said in the preceding paragraph 

 that we can choose a negative number AI in such a way that 



-X f K x (, t)f(t) dt^U (a) + 2, (X ^ A,). 



J a 



Thus we have 



IIn7-xf K,(M)/(O^^U(s) ........ (7) 



A-*- OB J a 



Again, if U(.s) = +00, this inequality is obviously true, provided that we interpret 

 " x < oo " as " x is a finite number, or has the improper value oo." Further, if 

 U (s) = oo, it is easily shown, by an argument similar to that just employed, that 



lim -X f* K A (, t)f(t) dt = - oo. 



.- J 



VOL. CCXI. A. 



