different mean values 75 



I propose to complete the researches of my above quoted paper 

 by proving the theorem: 



Theorem 2. Whenever Holder s {and therefore Gesaro's) Kth 

 mean exists and is finite, then lim s]f exists too, and both have the 



same value, ayid vice versa. 



The demonstration of both theorems is based upon relations 

 between the mean values which it is possible to calculate com- 

 pletely, as I have found, in a most simple manner. 



In §§ I to VI I determine the expression of s^^"^ by 



hf {X = l,% ...{n-K)), 



in § VII the expression of h\l^ by 



4'^ (X = l, 2, ...7i), 



in § VIII the expression of S^^^ by 



h^;:^ (\ = i, 2, ...7i), 



and finally in § IX I consider two more general mean values. 

 I. From the definitions follow 



" A=i "■A=] ri 



,f= 1 S' ^J A<'',= i W hf+ 1 (2Af ' - ;,f >) + . . . 

 ^\=2 X ^^ w |_2 ^ 3 ^ ^ 



+»^j(„_2)e,-(„-3)esi 



Adding a term which is zero, 



(2)_ (n-l)(n-2) (2) _ 1 (1 , (2) , 1 . (2) J i-'^ j^2) \ 



etc. Now I suppose that, by continuing in this manner, I have 

 arrived at the formula 



1 n—K , . 



''' A = l 



where c is a function of the indices n and k, and the coefiicients 



n, K 



<^a,k(X = 1, 2, ...) are, for each k, definite numbers which are the 

 values of a function of X for \ = 1, 2, — 



