62 Transactions of the Academy of Science of St. Louis 
is independent of the order of the variables x1, x2, - - - In 
the subsequent pages it will be assumed that every lee 
which enters the analysis is, symmetric, since most important 
applications may be reduced to this case. 
The Space E{n;. We have seen that the function d(x, 
where a@,,) is a constant element of E,,), is linear homogeneous in 
x. Let us now suppose that @,,) is no longer constant, but is de- 
pendent upon x in such a way that the following conditions are 
satisfied. (This dependence is indicated by the notation @,n)(x), 
and so to prevent confusion multiplication will be distinguished 
by a dot between factors). 
(1) @eny) (Ex) =a qn; (x) for all values of — and x 
(2) dey (ety): (&et+y) =aeny(X)- x +4 ny (y)-¥ ice all values of 
x and-y. 
It is quite evident that these functional relations are ad hoc 
sufficient to insure that the function @,)(x)-x be linear homo- 
geneous inx. This suggests the consideration of the aggregate 
of values of aq)(x), for every x in Eq;, asa single object. Asa 
further generalization it is possible to consider the totality of 
all such objects which are equivalent with respect to multiplica- 
tion, i.e., which differ by nullifying elements, as a single object 
itself, to be distinguished, say, by a subscript enclosed in brack- 
ets. hus @1:1; is the class of aggregates of elements equivalent 
to dq)(x), x ranging over E,), which satisfy conditions (1) and 
It is natural to suppose that we shall define similar objects 
a(n) for every positive integer m. In this more general case the 
elements @,,) are vectors whose components are fixed by 7 pa- 
rameters, i.€., @{nj is the class of all aggregates equivalent to 
Gn) (X1, Xz, °° * , X,), Satisfying conditions (1) and (2), and the 
additional requirement that @(n)(x1, x2, - - - , x,) be symmetric 
in every pair of its arguments. Letting aj; and 6,2; represent 
two such elements, we make the following definitions, which are 
applicable, of course, for any positive value of 7. 
a2} +6 2; =the class of all aggregates equivalent to a@)(x, y) 
+be(x,y) - 
£a (2; =the class of all aggregates equivalent to £a)(x, y) 
@(2}X =Ga)(x, y)-x, and thus is a function on Eq, to a set of 
vectors with one-parameter components. 
If, in addition, we adjoint the definitions of neighborhood 
and limit point that appear on p. 3, and assume that every ele- 
ment has a norm, we say that such elements a@,,, corresponding 
