352 PROCEEDINGS OF THE AMERICAN ACADEMY. 



d) k=.K, X<1< L. 



t=l 



1 —\ n \ 



(32) «^.,,,+i = -^- 2 / -^^ 



(logx) t=\'J ^ 



^ i^zm jzze 



n i—^ J=i 



'==^ J= 



< integrationg 



e) ^ = K, L <i I < e^. 



(33) 1 r ri ri ri 



c c c 



(I — L) integrations 





c c »=1 J=J 



< integrations 



III. Rr,< Rr^. 



Here we have the formula (29) again. 

 In § 4 we shall consider the n series : 



(34) ■ ^,,= 2^,,,^ r;Zl'o"""V 



which are such that if we multiply each by its proper factor x"" {\ogx)^''' , 

 where hj. j is given in (26), we obtain the n series (25). We thus 

 reduce the proof of convergence of (25) to the question of the con- 

 vergence of (34), and this last question will be settled by reference to 

 certain formulae to be established in the next section. 



§3. 



Lemmas Concerning Multiple Integrals. 



We now prove a number of lemmas, which will be useful in the proof 

 of convergence, and which also verify the statement that we have made 

 that the integrals of the last section, in which the lower limit is zero, 

 converge. 



Lemma I. Jf i is a function of x, continuous in the interval 

 <C X < c, and such that \b\\ log x \'~^, (t an integer > 1) is in- 



