different mean values 81 



so that finally 



^n + K-l\ " ^ -^7 



K J A = l 



Analogous considerations lead to 



n 

 A<'')=1 ^!L_^ + fl_ M ^Zl ^ ...(10), 



\ K ) ,V'''\ K 



Formulae (9) and (10) prove theorem 1. 



IX. By similar considerations it is possible to arrive at a state- 

 ment about the equivalence of two means of the kind examined 

 in Part II of my above quoted paper. 



Let 6k , Ck denote the terms of two infinite sequences of positive 

 real numbers, which have, when we write 



n n 



1 1 



the properties (i) lim Bn= 'X^ , lim Cn = go , 



.... 1 ^ /c6, 1 « KC^ 



(n) - 2 ^- , - 2 rT 



tend to limits or oscillate between finite limits. Then putting as 

 before (o) ^ 



the means ^1'^^= w^K^T-i C^^)' 



t?=livt. (12), 



^n 2 



are connected by two analogous relations. From (11) and (12) follow 



b s^'\ = B s^'^-B J'\ (13), 



c s^'\=G f^-C J'\ (14). 



n n-l n n n-l n-l ^ ' 



VOL. XX. PART I. 6 



