DUNKEL. — LINEAR DIFFERENTIAL EQUATIONS. 



357 



X Tt) X 



Gt,+i {x) = J^~^Gt, (x) dx = J^-Gt^ (x) dx + J^ Gt, {x) dx 



C C 7) 



\JlG,^(x)dx^ < fl\GtAx)\ \dx\ < i(^, + l)J^^^\dx\ 



" '' "> (0 < X <rj) 



£ , ..J Ooga^yi+l- (]og^)<i+l [ 



= |!>°-i'-['-(gl)"" 



Now choose X in the interval < .x < ?; so that : 



T. -7--r I - Gf, (x) dx 



<i 



= o H'^g-'^l'''^^' 



(0< x< X). 



Then 



-—J Gt^+i (x) 



(It 



1 ri 1 ri 



-i-n I -Gt, (x) dx + -T— r / -Gf, (x) dx 



< €, 



and : 



(0<x< X), 



x=o (log xyi+^ 



Therefore VI. is true for all values of t. 



Lemma VII. Jf 13 is a continuous function of x in the interval 

 < X' ^ c ^ 1, and such that it is not greater in absolute value than the 

 constant N, and if Gf is defined as in VI, then : 



(42) 

 For 



X X 



\GA^)\ = \ fldx... flf3dx\<N\logx\'. 



t integrations 



XX XX 



c c c c 



^^[/^.l^^l]'^^|log^r[l-|;^^-']<iV^|loga:l'. 



