Vol. 8, 1922 
MA THEM A TICS: C. N. MOORE 
289 
for summing divergent series. Our genreal theorem includes as special 
cases this theorem, the analogous theorem of Landau^ for divergent inte- 
grals, and a further new theorem with regard to the equivalence of general- 
ized derivatives of the Cesaro and Holder type. It can also be extended to 
the case of multiple limits, so that it includes new theorems, analogous to 
those mentioned, with regard to multiple series, multiple integrals, and 
partial derivatives. As this latter extension involves more complicated 
formulas, it will not be dealt with in the present communication. 
Following the terminology introduced by E- H. Moore, ^ we indicate the 
basis of our general theory as follows: 
• ^ • ® • ® to 9( . ^on ® to H . gon @ to H . . yOn @ to ^ . on § to g^^ 
where [a] denotes the class of all real numbers a, ^ = [p] denotes a 
class of elements and @ = [cr ] denotes a class of sets a of elements p of the 
range *!|3; ® = [7], ^ = and g = [<^] are three classes of functions 7, 
t], and (p, respectively, on © to 5t (we shall restrict ourselves throughout 
to the consideration of single-valued functions) ; (^0 is a special function (p 
of the class g; and 7 is a function on ® to ^ and on § to g, that is, a 
functional transformation turning a function of the class @ into a function 
of the class ^ or a function of the class ^ into a function of the class g, 
denoted by Jy or Jr]. 
In order to make clear the relationship of our general theorem to the 
two special theorems referred to above, we will indicate here what the gen- 
eral basis reduces to in the particular instances III and IV: 
^"^^ [alln = 1,2, ]; ®^ [(7,^(1,2,... n)\n]; 
® = ^ = g ^ [all 7, ry, on @ to ^]; 
<^o(o-J = n {n)\ (Jy) (o-n) = 7(0-1) + 7(0-1) + +7(<^») (w). 
= [all a>0]; ^ ^ [a ^ {all 0<x< a) {a>0)]; 
@ = [all functions that are finite and integrable 
(Lebesgue) on every finite interval (0, a) ] ; 
[allr, = fly\x>0)]; g - [all ^ = ifU\x>0)]; 
=a{a); (Jy) (O = firiya); (Jv) (O = fin(na). 
We next proceed to make certain postulates with regard to the nature of 
the elements in our basis, readily seen to be verified in the specific in- 
stances cited. Thus we require the class ® to have the linear property 
(L) as defined by K. H. Moore, ^ and the property (P) defined by 
(P) 7i • 72 • D • 7i72®, 
the arguments of the two 7's being the same. It will then follow that 
7p = 7i • 72 7„ is of the class @. We require the class ^ to have the 
properties (L) and (P), and the further property of being a sub-class of the 
