773 



l-F- 



of (ll):(.r — I) - I^(i-y.){iz\/.r. — §) is of order and ft satis- 



{x—^y-!' 



fies R[n)>0. 



As a condition in the deducing of (Ij from (5) it arises tliat (8) 

 must be uniformly convergent for a ^ x <l b, whicii lias the conse- 

 quence : f[x) continuous as R(k) <^1. 



We may thetefore agree upon tlie following: 



I. It is an analytical function, regular for n<x<h. 



(x-af-' 



having x = a as zero of order s [so that development (7) obtains], 

 and if the series, which we may draw up according to our defini- 

 tion for /'("+!-") (,t) [cf. (10)], converges uniformly for a^.v^b, — 

 (11) will be a continuous solution of (1), provided /?(^.) <^ 1 and 

 0</^(fO<.- + l. 



3. For c = 0, (1) passes into Abel's integral equation: 



for which we now find the solution in a general form by substi- 

 tuting in (11) also :=zO, viz. 



u(x)= ~ ^dï, .... (13) 



^ ^ JT r(foJ (.«-§)i-." ^ ' 



available under the same conditions as mentioned under I (§ 2). 

 Abel solved (12) on the su[)position <[/?(/.)<[ 1, and found as 

 solution (13), in which ix =r ;.. 



LiouviLLE ') extended Abel's problem to the supposition n <^ 

 <] /2 (A) <^ — n -|- 1 (k positive integer or zero) and found as solution 

 (13), in which (i=zzn -\- A. 



4. Let us now take (1) as a special case of Volterka's integral 

 equation of the first kind, which has the general form : 



/ix)= JK(.v,^)uii)d^ (14) 



then (1) appears to ensue from this, if we take as kernel 



/i^(.i;,|) = ^(l-/,)^^Y(.^— 5)^"^ /_;(5|/.^3g). . . (15) 



1) Journ. de I'Ec. Polytechn , Gab. 21 (1832). 



