360 PROCEEDINGS OF THE AMERICAN ACADEMY. 



since the greatest value of e^ — A + < — 1 is 



e^ — X + L — I — e^ — X + (ej^. — e^ + \) — I = ej^ — 1. 



X 



(51) 1 <^,_ ^^ ^^^j I < ^P'lJb I log X r^-i dx = M^^ ~M{x) 







It is obvious that the above result holds also for II. (c) (31). From 

 (32) we shall obtain in the same way for II. {d) : 



X 



(52) 1 log X t' I <A^ ,_^^^, I < M^^ lJb 1 log X r--V.r = ^^' ^ M(x) 





 yi/9i 7" 



' ' -" log X C 



'» 



In all three sub-cases (o), (c), and (d), it is easily seen that (43) is 

 true for 5' = y^ + !• We have now left of case II. the sub-cases (6) 

 and (e). 



From (30) we have for case (6) : 



(^^) 1 r /• 1 r ] r 1 T 



\<f>i ! ^1 1 ^ n i7-7 -\dx\... I -\dx\ -If^' ^,M{x) \dx\ 



c c c 



(/ — Z) integrations 



t—l-L ^ 



+ 2 r^l<^-^l • •• / -l^^l /^^^'|loga;r'"^K-^lT 



c c 



< integrations 



In the first part of the bracket we have replaced | (^^. j^ . ^ | by 



M'^^-^M(x), using inequality (51). 



We will consider the two parts of (53) separately. 



(54) X XX 



,, \_^ TiK^I... f^-\dx\ f-3f'^^^M(x)\dx\<M^^^M(c) 

 \log x\' ^ J x J ^ J X (J C 



c c 



{I — L) integrations 



^ iH/^i+i L (Lemma VII.) 



