DUNKEL. — LINEAR DIFFERENTIAL EQUATIONS. 353 



tegrable up to x ■= 0, then h can he integrated t times from x = as 

 follows : 



X XX 



(35) f {x) = I -dx . . . I -dx I hdx. 







t integrations 



Lemma II. If the conditions of Lemma I. hold, then: 



X XX X 



(36) \f{x)\ = \C-dx...l~dxibdx\<i\h\\\ogx \-'dx. 







t integrations 



(0<a; < 1). 



We will prove these lemmas by mathematical induction. They are 

 true when t =^ 1. Let us assume that they are true for a particular 

 value of t, say t = ti. 



Let X be any particular value of x in the interval < a; < 1, and 

 choose € at pleasure such that < € < X Then we have : 



X X X X 



\flftA^)dx^<J'l\f^(x)\dx<JldxJ\b\\logxf^-'dx 



e c e 



X t 



= log X l'\b\\ log X 1''-^ dx-log€ f\b\\ log X f^-^ dx 







X 



— /log a:- 1 ^ I I log X f'~^ dx 



e 

 c X 



= Iloge| r|6||logarl''-^</x+ r|6||loga:l''rfx 



6 « 



X 



-\\ogX\^ \b\\\ogxi^-^ dx 





 « X 



< \\og.\l^\b\\\ogxf^-^dx +j]b\\ log x|'' dx 



6 « 



€ X X 



< f\b\\\ogx I'' dx+ l'\b\\\ogx\''dx= I \b\\ log X f' dx. 







VOL. XXXVIII. — 23 



