DUNKEL. — LINEAR DIFFERENTIAL EQUATIONS. 



161 



t=il-L ,1 



(55) 



log a: I'' ^ J=i ^ ^ 



dx I 



. C-\dx\ fB3n^\\ogxf'^-^\dx 



t—lL 



c c c 



t integrations 



c 



< "S I lo? X Y-' \ . ^^'^' , { B\\ogx f--^ dx (Lemma VII.) 



^ if *^ 2 n^g ^ |'-i-('-^) j B\logx 1'""^ (f:r 

 c 



< J/^^i^^l^ r^ I log X r^^ c^o; < i/''^+^ ^^^ . 

 = log X J ' * ' - G 



Therefore : 





'L l-L 



C ' C 





The limit of the left side of (54) for x = is zero by Lemma VI., 

 while the same thing is true of the left side of (55) from the inequali- 

 ties. Therefore (43) is true in this case for q ^ q^ -\- 1. 

 From (33) we have for case (e) : 



(56) 



log 



t=l-L 



c c 



{I — L) integrations 



X X 



+ 2 Jl\d^\^'^J\\dx\JBM^^\\ogx\^^-'\dx\^^. 



c c c 



t integrations 



In the first part of the bracket we have used the inequality obtained 

 from (52) : 



I log ^^-"1 </>.,.., .+il^^''"§^^(-)- 



Now tlie only difference between the inequalities (53) and (56) is in 

 ilie powt'r of ] log a; I outside the brackets; and, since | log a: | " ^ 

 I log a: \'~'-, all the results that we have obtained from (53) will follow 

 also from (50). Then for all the sub-cases of IL, (43) and (48) are 

 true iov q = Qi -\- L 



