FUNCTIONS IN THE THEORY OF INTEGRAL EQUATIONS. 129 



stated in a variety of particular forms. Without exhausting all possibilities, we shall 

 mention two corollaries which will be of use in the sequel. 



It has already been pointed out that the four systems of normal functions defined 

 in G satisfy the condition (1) of the above theorem ; and it will be shown, in a paper 

 to be published shortly,* that the condition (2) is also satisfied. Further, a function 

 X (f), which has a Lebesgue integral in an interval (y, 8) (0 : y < 8 S IT), may be 

 regarded as a uniformly continuous function of a in any interval whatever, for values 

 of t belonging'to (y, 8). We have thus the first of the corollaries mentioned : 



Letf(s) be a function, defined for all values of s, which has a Lebesgue integral in 

 any finite interval ; and let x (0 be a limited function which has a Lebesgue integral 

 in an interval (y, 8) (0 ^ y < 8 : IT). Then, as the positive integer n increases 

 indefinitely, each of the eight integrals 



converges uniformly to zero in any finite interval, t 



A sufficient condition that \ (s, t) may be a uniformly continuous function of s 

 in (y', 8'), for values of t belonging to (y, 8) is that x (*'. should be a continuous 

 function of the two variables ,s and t in the rectangle y' ^ s s 8', y : t s 8. Hence we 

 have the second corollary : 



Let f(s) be a function, defined for all values of s, which has a Lebesgue integral in 

 any finite interval ; and let x ( s be a continuous function of the two variables 



* In this paper I shall prove the following theorem : 



Let f(s) be any function whose square has a Lebesgue integral in (0, *). Let \l>\ (x), ^2 (<), , f ('). k 

 the complete system of normal functions whidi, for suitable values of /i, satisfy the differential equation 



;gr +& + /) = - 



and an assigned pair of boundary conditions at the end points of (0, IT). Then the series 



[ fc (0 /(O * J + [j* ** (0 /(O *] 



is convergent and its sum is 



["[/(*>? 



Jo 



in all cases. 



We shall see below (IV., 9, 11, 12) that the four systems of normal functions each satisfy the require- 

 ments of this theorem. 



In the meantime the reader will be able to deduce the particular cases here required from a result 

 obtained by A. C. DIXON (" On a Property of Summable Functions," ' Proc. Camb. Phil. Soc., 1 vol. xv., 

 pp. 211-216). 



t The limitation that n should be integral may be removed without difficulty. HOBSOX has proved the 

 equivalent of this corollary for the case in which x (0 has limited total fluctuation in (y, S), ride " On the 

 Uniform Convergence of FOURIER'S Series," 'Proc. Lond. Math. Soc.,' ser. 2, vol. 5, pp. 277-281 ; 'The 

 Theory of Functions of a Real Variable ' (1907), pp. 683-687. 

 VOL. CCXI. A. 8 



* 



