778 



" ^'''^ "" TiTSTfrin+X) dwj öiZIiy^JH^ '^^ + j 



/^(i— ;i) r(nH-A) 



• J i/g„,-i-.f,„ rfW (ë.«-.Wi)i ->+')- + -" ' ' 



which now represents moreover the only possible continnons solution 

 of (1) for — ?i < R W < — ?i + 1. 



It is moreover iin|)ortant to observe, that though (J) for c=0 

 passes into the equation of Abel (12), the substitution of r=:0 in 

 (2]) and (24) does not lead to its solution. As we saw, tiie kernel 

 K{.K, §) eontinues to satisfy the conditions mentioned in theorem A 

 or B if c =: 0, but, for ).^ — n and c =: 0, A'„+i {x, i) s^ 0, so 

 tliat (16) is cancelled. 



And if - n <^ R {).) <^ — ?; + 1 and ; = 0, (17) passes into 



.1 X 



from which the solution ensues directly : 



sir^A^ Ft;.) ^ r /i"){E) 



a 



under the conditions for f{:v) mentioned in theorem B. 



5. The expressions (21) and (24) may not be easily reduced, 

 even though we should make use of (7) and so we accept the 

 conditions mentioned under I (§ 2) for /(x). As in this case (11) 

 must represent the same u {.c) we arrive at the following conclusions : 



1. Not only (21) but also (11) with / =: — n represents 



the only possible continuous solution of (20). if the conditions 



mentioned under I hold good for /'(,(•)• If we introduce some 



f."+V{.v) 



simplification here by supposing" ;z= y (.t), we can say that 



n/ 



the integral equation of the second kind: 



