DUNKEL. — LTNEAR DIFFERENTIAL EQUATIONS. 359 



, ^ ^ ra / k = 1,2, . . . m\ 



< 48 > 1**1 S** 0=1,2,..., J' 



For q = 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 : 



W l* M}ffl+ il ^ 



t—i x z * 



x mr k -r K ) ^ f 1 1 d x | . . . f 1 1 dx | f x-^-^ BM* | log X | e «- A | <fe | 



c c c 



< integrations 



after replacing 



respectively by the greater values 



M q \ [ log a: | e « _A , B 

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



l*M, Sl +ll ^ **2 [^^]'"/^[log xl^l^KLemmalV.) 



c 



x a; 



< Jf*» C ( B\\ogx | e « -A \dx\<M qi CcB\ log x |^' _1 1 rf* | 



c c 



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 K ~~ 

 is integrable up to x = (cf. page 358), and therefore B I log x 'f*^' 

 must also be integrable up to x = 0. So (43) is verified for q — q x + 1. 



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



, , X XX 



(50) I 4>,, ,, ft+1 1 ^ 2 / \** • • •/ 1 **f B Mqi I l0 S * I 6 *"* d * 



t — 1 o - 



t integrations 

 t=l p 



% M qi 2 / -B I log a- \ e -~ x+t ~ l dx (Lemma II.) 



t—\ *J 



T - 1 



t=i Z 



2.p 



t=i «/ 



: M q * ^ I 5 I log a? I 6 *' -1 rfa: 



