362 PROCEEDINGS OP THE AMERICAN ACADEMY. 



III. Here we have : 



t^l ' 



< x-^^'""-''^' 2 f-dx...C^dx Cx''^'<-''^^ B ^5.| log X \'<-^ dx 









t integrations 



X 



^^'' 2 L(r/-r,) ] 'J^l^^g"^''""'^^ (Lemma III.) 



X 



< 3^' fc B\\ogx f ^-^ dx < il/^'+\ 



It is obvious that (43) is true in this case for y = ^-j + 1. 



This completes the proof of the inequalities (48). 



From (48) v?e now infer at once that 4>k,!,g {l > 0) approaches zero at 

 the poi7it X = 0. 



The inequalities (48) furnish the sufficient condition of TVeierstrass 

 for the absolute and uniform convergence of the series (34) in the in- 

 terval < a; ^ c. Accordingly the series (25) converges absolutely 

 and uniformly in any sub-interval <^ e < x ^ c. 



It remains now to show that we have a solution in the system 

 of functions z^^. For this purpose let us select any sub-interval 

 < e < X ^ c. Since the point x = is excluded, the coefficients i^'' 

 are continuous in this interval ; and in it tlie system of differential 

 equations (IG) is satisfied by the functions ^^ ^. For if we multiply the 

 series (25) for z,-j by b'^f^ we obtain the absolutely and uniformly con- 

 vergent series : 



g=oo 



5=0 



Taking the sum of such series for all values of i and^ we have : 



0^'') 2 2 ^'^'^^-j^ 2 2 2 ^'^■^'^'.j.^ 



i—\ j-l 1=1 j—\ q=0 



5=:co ,_ i=7?l j- 



2 2 2 ^i'^'.>.^ • 



0=0 L i=i j-i -i 



