292 MATHEMATICS: C. N. MOORE Proc. N. A. S. 
(II) lim , - ^M] = 1 (a'n), 
(III) — ^— T (7"r;) (o-) as function of <j is of the class % (n), 
the symbol J^y] indicating that the operation / has been repeated n times 
(IV) <^o(a-„) as a function of <j is of the class g {n), 
(V) lim <^on(o-) = ^ {n) 
(VI) (D^o,) (<7) = _i(o-)>0 (,7,n>l), (D,p„0 (<r) = 1 (.r), 
(VII) (DtA„) (<7,) = - (<r,M>l). 
We have then as the foundation of our theory : 
2: = (5I;Sj3; (g; @on @ to 51, -LP . ^on @ to 51. I'P^G*^ . 
cvon @ to 51 ^P^TT A ?c. I II III IV V VI VII 
^on @ to on ^ to g. LMiMilj^Ij^ 
We can now readily prove the two following properties of the operations 
D and J as applied to the class g : 
(4) {D[^m]) M = <pM {D<p,) (a) + ^i(cr) {D<p2) (a) (cr). . 
(5) iJ[<P2iD<p,)]) (a) = ^M^M - U[UD<p-2)]) M 
We next formulate a further limit definition that is also based on the 
definition of E. H. Moore to which reference has already been made. 
We shall say that 7/(0-) approaches a limit a as to a if, corresponding to every 
positive e, we can find a such that for a>(Tg we have 1 77(0-) — a I < ^, or in 
symbols 
(6) lim77(o-) = a:- = D : Ho-g , o->(7g- D 77(0-) — a | 
(T 
We are now ready to define the two generalized limits with which we are 
concerned. Given any function r]{a), we set 
(7) (C-„,) (a) = [n!/vM]{J-v) (<r) - , . , , ^, U"v) (<t), 
(Po{(^) <Po{l) •••<pQ{0-n-l) 
(8) (Mv) M = [l/MT)KJv){<r), 
(9) {H„v)M = (M'riKa), 
where C and H are used, as is customary, to connote Cesaro and Holder. 
If for a fixed n lim (C^r?) (a) exists, we define this limit as the generalized 
limit of type (Cn) for rjia). If lim {Hnri){<j) exists, we define this limit as 
the generalized limit of type {Hn) for 77(0-). 
