77G 



= r(.. fi-J.) J , „, ^^", ~A'-i)-+'-. 



and consequently : 



(-1)" 



m=o '«• / (m +n—p + 1 — >.„) 



r(n^l-X„){^) '(x-§) 2 /„_,^_ (..l/^_§), 



SO that: 



K„ (.1-, §), /v, (.r, §) . . . /Cl- 1 (■!', §) are continuons for a ^ § ^ .r ^ 6, 



moreover K^ {x, aj) ^ O, K^ {.i\ :r) ^ O, . . . K„-\ {x, x) ^ O for a ^x^b, 



G{x,i) 

 wliile K„ {x, S.) = ^ - , iu wliieli ; 



so that: 



dG{u:,ï) 



ix (x,i) and — r are continuous for a<§<x<b, and 6- (x,x)=i^O 



o.v ^ ^ ^ 



for a^x ^ h. 



Consequently for — ?; <^ R{)) <[ — n -f- 1 t'ie suppositions mentioned 

 in theorem B are satisfied b^' K[x.i), also it' :^=0. 



Summarizing we can already state that (1) has only one continuous 

 solution, provided R{).)<^\ and certain conditions for f{x) are 

 satisfied. 



Tiiese necessary and sufficient conditions for f{x) mentioned in 

 A or B are doubtlessly satisfied if we impose on j\x) the conditions 

 mentioned under I (§ 2), so that we can complete I in this way 

 that (11) under the conditions mentioned there is the only possible 

 continuous solution of (1). [This holds good for (13) as a solution 

 of (12)]. As a conse()uence of this (11) is invariable by changing 

 f» within the limits indicated. 



