779 



u (.«) = ,r (■^■) ^- ^ . , ^ " (bj di ■ ■ ■ ■ (25) 



^ iV Va; — i; 

 a 



has the only possible continnons solution : 



ui.v)^(';y " i(.v-i) -^ I-(t^,){iz\/w^yr^''){é)di , 0<R{!i)<s+l (26) 



if the conditions are satisfied : 



II. fp {x) is an analytical function, regular for a^x^b, which 

 has x = a as zero of order *■; and the series for '['"■^U"), which we 

 may draw up by means of our definition, is uniformly convergent 

 for (I £ X < b. 



So, not only (24) but also (.11) with — n<^R{X)<^ — ?i-)-l repre- 

 sents the only possible continuous solution of (23), if the conditions 

 mentioned under I hold good for /"(•''')• Here too we can introduce 



ƒ(")(.«) 



some simplification by writing ^ (/(.)•), so that ot the 



' •' *' r{l—k)r{n + }i) ^ ^ 



integral equation : 



a a I 



— w <] _R (/.) <^ — H -f 1 {n pos. integer or zero) 

 the only possit)le continuous solution may be written in the form: 



r(n + X)(^^J "j{x-è) -^"i-d- „)(L^|/x-t)./(."+'-" ')(£)</t . (^^^ 



< i^ (ft) < s -f 1 



under the conditions : 



ff'{a;) 

 III. — ^,-- is an analytical function, regular for <i<x<b, 



(,K-a)i-("+^) • *" =- = 



having x^a as zero of order .v ; and the series ensuing from it for 



(ƒ(/'+■—"-'•) (.r) is uniformly convergent for a^x^b. 



/•'"+i) x) 

 2. As (21) with ' — = ff{x) and (26) both represeut the ojdy 



possible continuous solution of (25), in the same way (24) with 



=: (f{x) and (28) both the only possible continuous 



