AND THEIR CONNECTION WITH THE THEORY OF INTEGRAL EQUATIONS. 443 



28. If e is any arbitrarily assigned positive quantity, it follows from the absolute 

 convergence of the series on the right of (24) that we can choose m great enough to 



ensure the inequality 



1.1. (*\.i. tt\\ 



(25) 



And, when this is done, it is easily seen that, since 



I ^ (s) ^n (0 | . |Xi/, n (*)*UO | (u > m) 



A. 

 we have 



X_(X n -X) 



Again, from (20), we see that a negative number L' can be chosen with so great an 

 absolute value that, when X < L', 



j K A (, -Il m (X ; ,, t) - \K (., - i ^ *' <' 



L =' A 



while, from (24) and (25), we deduce 



Adding the three inequalities just written, we obtain 



In other words, we have proved the theorem 



L< K, (*, t) = /'(*, ( s .,- < h, , < t < //). 



A^-- c 



29. It may be proved* that, if < is any constant and a^ any point of the interval 

 (a, />), then the solving function corresponding to the characteristic function 



is 



H (s t} - K (a t} - - c< i> 

 H * f '" 



K. 



(26) 



whilst the corresponding determinant is easily seen to be 



A (X) = I) (X) 



Of. BATEMAN, 'Messenger of Mathoraatics' (1908), p. 184. The result in question follows from 



equations (24), (25), and (26), by writing f(s) = * (tl ' *' , ;/ (t) = K (a lt t) and observing that 



c 

 3 L 2 



