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 z ka are given by the 

 formulae (26), in which the functions 4>*,i,q are continuous in x through- 

 out the whole of the interval b, and reduce to zero when x = 0, except 

 when q = 0. The functions 4> k ,i,o are 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 K solutions of (16) obtained by giving A the values 

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

 quired that [ b'£i | | log x r A " — shall be integrable up to x = 0, where 

 e K is determined for the root r K by the condition (23). We have, then, 

 n solutions which may be written as follows : 



( when Rr k ^. R r K 



z*' l = x K </>*' l 1 or Rr k --Rr K , k :j: k and I < L, 



(or 



k = k and / <J A, 



(62) *£ * = x r * (log x) l ~ L <f>* * Rr k = Rr K , k^ K and L <l<e k , 

 z^ = x « (log ar) </>^ A <;<<>„. 



„ A 



where the functions <£ ' are continuous in a: and 



l *> A A 1 I 



^k,i 'x=o " ( or k = k, and I < A, 



(63) kA j *■ 



<iLo=(/=A)T A</^«, 



(k = 1, 2, . . . hi) (A=l, 2, ...«.)■ 



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

 useful later on : 



K K 



I. z ' does not involve log x explicitly whenever A > I; 



(64) II. limit x~ ri z K ;) = when Rr K > Rr<; 



III. limit x K (log x) " ' s ' = in all cases except the 



one, k = k and / = e K ; and then the limit is 7 ; • 



(e K — A) ! 



