w 



èf 



X 



dx J 



' r co 1' • ^^^^ 



J o 



o I 



When tlie integral equation 



2A(.^;)=ƒl|>G?)A'(.^.-^),/;i 



is given, we find by (18) 



ip(.^) = AJ,{x) + ^,A(..) + AJ,{x) 4- . . . . 



00 



Therefore 2 {A^,-{-Ai,-\-^2)Tilx) be expressed in function of tf>. 







For 



2dtb r IAx-8) 







and, differentiating again 



d.c- J X- 







tlius 



=^w 





:> 



x-ii 



5. We now proceed to give some applications of formula (18). 

 First let the integral equation be 



X 



f{x)=L{^)I^{x-^)d^ 



Ü 



then 



«0 = 1, aj = rt, = . . . r= 

 hence 



^„ = 1, A, = 0, ^, = 1, ^ = ^, = ... = 

 til us 



2^f {x) = 2 f- + è f/(l'^) [2^0 O^- -|5) + 2/, 0^-/3)] ^ii 







or 



