DUNKEL. — LINEAR DIFFERENTIAL EQUATIONS. 359 



For (7 = they are obviously true. Assuming them true for a special 

 value of q, q =^ qi, we will consider the cases outlined in (24) in turn. 

 I. From (27) we have : 



a:'^('-/i-'-<> 



. 1 Z X X 



2 f-\dx\...f-\dx\C x-^^'^-''^^ B M'^' 1 log X \'<-^\ dx 



c c 



t integrations 



after replacing 



respectively by the greater values 



J/'?', |loga:l'''-^ B 

 (Cf. (28), (30), (32)). 



l<^..^,.+il ^ ^^'^2 [^^;r^]'"/^|log xr^-^KxI(LemmaIV.) 



X a: 



< M'^'C j B\logx \'<-' I rf;r 1 ^' J/*^' foB\\ogx \'^-'^ \ dx \ 



c c 



< M"' M{x) ^ J/'''+\ 



From Lemma V. it follows that the limit of the right side of (49) is 



zero, when x approaches zero ; for we have assumed that B | log x f^'''' 



is integrable up to x = (cf. page 358), and therefore ^jloga;|^*~' 

 must also be integrable up to .'■ = 0. So (43) is verified for q ^^ qi -}- 1. 

 II. From (29) we have for case (a) : 



(50) I \ , ,.+1 l^^J'ldx.. .f I dxj B J/'- 1 log X r«-^ dx 







t integrations 

 t=l ^ 



S M"^' 2 f^ I '«g •^' r*"^+'~^ dx (Lemma I L) 



< M''' 2 I ^ I log a; i"^-^ dx 

 t=i •/ 



