DUNKEL. LINEAR DIFFERENTIAL EQUATIONS. 365 



"We will now show that the n solutions (62) are linearly independent. 

 Suppose they were not and that there were n constants Ck.,x uot all zero 

 such that : 



K-\ A=l \ ; J A./ 



It will be convenient to suppose that our notation is such that: 



(66) Rn ^Rr.S . ..<Rr„,. 



Consider first those equations of (65) for which Rr,^ ^= R r^. We 

 have: 



(67) li-jt 2 2 ^. . -"'' <•: ' = Rr, = Rr, 



Now let Z = 1 in (67), and consider the limit of each terra for any 

 given value, within the range indicated, for k. For each terra in which 

 R 7\ y R r,^^ Rri the limit is zero by II. We have left, now, only 

 the terms : 



(68) C ^ x"'^- z"^ ^ Rr, = Rr, = Rr,. 



According to I. no logarithms appear explicitly in (68), and we can 

 write : 



Now by (63) the limit of all such terms is zero except in the one case 

 K — i" and X = 1, and for this term we have : 



(70) limit C,.i X-'' 4:1 = 15°iit C,,r ^[l = C,,, . 



So in the case of Z = 1 the limit (67) turns out to be C^i when 

 we evaluate the limit term by term. Now this is impossible unless 

 C^. 1 = 0; and so we must have, writing now k in place of k : 



(71) a,i = Rr, = Rr,. 



Now consider in the same way the cases of (67) in which 1=2; and 

 choose any one of the values of k indicated. Here again by II. the limit 

 of each term is zero when R i\ > A' r^. ; and we have left the terms : 



(72) (^.,.-"'\^ Rr. = Rr, = Rr, 



A > 2. 



