362 PROCEEDINGS OF THE AMERICAN ACADEMY. 



III. Here we have : 



t-i 



< 



2 f-dx...f-dx fx'^*-^ B M*\ log x \ e *- x dx 





 t integrations 



M* 2 r 1 n f B\\ogx \ e *- k dx (Lemma III.) 



X 



<M qi f0B\ log x | e * -1 dx < M qi+1 . 



It is obvious that (43) is true in this case for q — q x + 1. 



This completes the proof of the inequalities (48). 



From (48) we now infer at once that (f> k</>7 (q > 0) approaches zero at 

 the point x = 0. 



The inequalities (48) furnish the sufficient condition of Weierstrass 

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

 terval < x < 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 kl . For this purpose let us select any sub-interval 

 < e < x ^ c. Since the point x — is excluded, the coefficients b\' t 

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

 equations (1G) is satisfied by the functions z kJ . For if we multiply the 

 series (25) for z u by b' kJ we obtain the absolutely and uniformly con- 



vergent series 



5=0 



<jr=co 



i.J.9 



Taking the sum of such series for all values of i and 7' we have : 



i—m j—e i i=m j=e i q=<*> 



(57) 2 2 #*./= 2 2 2 #<*** 



i=l j=l i=l 3=1 1=0 



q—co — i=m j=e z - —1 



q-0 L j = l j-\ -J 



