364 PROCEEDINGS OF THE AMERICAN ACADEMY. 



first derivatives with regard to x, and satisfy the canonical system of 

 equations at every point. The terms of the series for Zf.j are gicen by the 

 formulae (26), in ichich the functions 4>k,i,q ^'"^ continuous in x through- 

 out the whole of the interval S, and reduce to zero when a; = 0, except 

 when q := 0. Tlie functions 4>i.i,o ^^^ constants, zero in all cases but the 

 following : 



§5. 



Linear Independence of the Solutions of the Canonical 



System. 



"We have shown that corresponding to each elementary divisor 

 (r — r^f"^ there are e^ solutions of (16) obtained by giving A the values 

 1, 2, ... e^; and for the development of these solutions we have re- 

 quired that I l)l:'j I I log X V^" shall be integrable up to a: = 0, where 

 Cf: is determined for the root r^ by the condition (23). We have, then, 

 n solutions which may be written as follows : 



( when Rr^-^ R r^ 

 x^" cf)"" ^ -l or Rr,, — Rr^, k -^^ k and I ^ L, 



'k,l ~ '^k,l ) 



or k = K and I ^X, 



(62) z"] ^ = /« (log x)^ ^ <^J; \ Rr^ = Rr,, k ^ k and L < I < e^ , 

 z ' =x'' (\ogx) <^ ' X< l<e^. 



where the functions ^ ' are continuous in x and 



(63) 



{K=^\,2,...m) (A= 1, 2, . . . e,). 



It is worth while to note three facts in regard to the z's, which will be 

 useful later on : 



K A 



I. 2 ' does not involve log x explicitly whenever A ^ Z; 



k,l 

 (64) II. limit x~^'z1''' = when Rr, > Rr,; 



III. limit xr^" (log x)~ ^'^~ z"' =: Q in all cases except the 



one, k = K and / = e^ ; and then the limit is — -r . 



■^ ^^ All 



{e. - A) 



