777 



In order to determine the solution of (1) according to Volterra's 

 method we have to investig-ate again the cases a and h separately. 



a. Supposing that /. = — n and the given function f{x) satisfies 

 the conditions mentioned in tiieorem A, we have to consider the 

 equation of the second i<ind into wiiicii (16) passes foi' our case, viz. 



.(..) = ^;^ + yJ— ^^=-M5)^^5 . . (20) 

 Its solution is the absolutely and unitbrralj' convergent series : 



a a 



^ in — 1 



. . = ./'" + ' (5m) ^Sm f^^, f?5, . . (21) 



K §,„_!—§,„ 

 a 



which now moreover represents tiie only possible continuous solution 

 of (1) for ;. = — 71. 



b. Supposing that l=: — n-\- l„ (0 < R (A„) < 1) and ƒ(,*,•) satisfies 

 the conditions mentioned in theorem B, we arrive at equation (17), 

 in which the kernel, represented by (19), passes for onr case into 



— r(« + 1 — ;.„) r().„)j„ {s V'l 



so that (17) becomes: 



F (,<•) = r{n + 1 - ;.„) r(K„)jh {^^^^s,) « (^=) di . 



(22) 



This integral equation of the first kind has as kernel : 



A'(,f, g) = r(« + i-;,„) r(;.„) /„ {z VJ^,), 



so that A'„ (.1', .1') =1= and we consequently arrive at its solution by 

 determining the solution of (16), in ;vvhicli n = and /' has been 

 replaced by F; if we replace moreover /.„ by n-\-K, (16), consider- 

 ing (18), assumes the following form: 



^^ ^(^-;.)^(;.+;.)rf.^•j(.<— g)'-^"+') ^ ^2j k:,^;. ^-'^ ^ 



Its solution is the absolutely and uniformly comergent series: 



50* 



