290 
MATHEMATICS: C. N. MOORE Proc. N. A. S. 
@, which property we shall denote by Sq. We require the class g to have 
the properties (L) and (P) , and the further property of being a sub-class of 
the class ^, which property we shall designate as 5^^. Hence is J also on g 
to g. 
We require the operation / to have the linear property (L) as defined by 
B. H. Moore, and the properties Mi and M2, defined as follows : 
(Ml) ai<yi<ayO ^ 72-73 = 7i72- 3 • ai (772) (a) ^ (Jys) (o-) S ^2(772) (o-) (0-), 
(Ms) flH<7i<a2-0 ^ 72-73 = 7i72-(r"3 <7'-D • ai[(/72) (o-")-(-/72) ((t')] 
^ (/73)(cr")-(/73)(<r') 
^a2[(/72) (<t")-(/72) (OL 
For the explanation of the other postulates we need certain auxiliary 
definitions. By the notation a' < a", g">g\ we shall mean that o-" con- 
tains all the elements of a' and at least one element not found in c'. For 
a given ^ on @ to a given a, and a given g' such that there exists 
a<a' , we shall write 
lim B(<j) = a 
<r|(r<ff' 
in the case that, corresponding to an arbitrary positive number e, there 
exists a (t^<g' such that for every a having the property (j<(j' ,\%{(t) 
— a I < ^, or in symbols 
This latter definition is based on the fundamental definition of limit in 
General Analysis given by B. H. Moore in the paper in these Proceedings 
referred to in the first foot-note. 
We now postulate for the class @ the following properties : 
(U)^ Bither(7'-D.H U [(r<(7']® or H co , (7'>(7o- D HU [o-<o-']®, 
(n)^ (T'.D-Hnk>(7']. 
In the typical instances in view in the formation of this general theory, of 
the two alternatives in (U) one holds and the other does not hold; how- 
ever it is not assumed that this disjunction between the two alternatives 
shall be presupposed. In order to avoid notational reference to these al- 
ternatives, it is convenient to introduce a property — of sets o- of @ ; if the 
first alternative holds, every cr of © has the property — , in notation d\ if 
the first alternative does not hold, the sets a- are the sets cr>a-u; further 
for brevity a property • (the negation of — ) is used ; thus every c is a o- or 
a 
We define 
(7Li^Uk<(r'] Q), 
cr\ k>(7'], o-;+i =a[<T><7'n] (cr|n = l,2,3---.), 
ip{(T)^(p{a_i) {(t), <p((t)=0 (a), 
