Double Products and Strains in Hyperspace. 3 
Moreover, the system will be supposed, as usual, to contain all the 
elements which may be linearly derived from any given elements; 
and it may be assumed that the coefficients in this derivation 
are any real or complex scalars. It will not be necessary to go 
into further details as regards these matters which are the same in 
all linear associative or Grassmannian algebras. So much in regard 
to addition, subtraction, and linear dependence or independence. 
The primary elements a, ~, y, ... may be interpreted either as 
vectors issuing from a fixed point in Euclidean space of 2 dimensions 
or as points lying in a Euclidean space (supposed flat, of course) 
of w—1 dimensions. It is the former interpretation which will be 
most used in what follows. It should be noted however, that the 
algebraic system is independent of any geometric interpretation. 
If proofs are given by means of either of the said interpretations, 
it is merely because the geometric language facilitates expression. 
As a matter of fact in a Grassmannian algebra where the com- 
binatory products lead to elements of different types from the ele- 
ments which constitute the factors, the geometric language and 
conception are far more fruitful and convenient than in those 
algebras in which the product is always of the same type as the 
factors; and hence it will be used constantly in what follows. 
Two primary elements may be multiplied according to the com- 
binatory law 
(2) axp=—Bpx<ea 
to form a product which is an element of another type and may 
be called a secondary element or element of the second class. The 
use of the cross < for combinatory multiplication it in accord with 
Gibbs’s usage in his address on multiple algebra. In lke manner 
k elements, k =n, may be combined to form an element of the 
kth class. Such multiplication is called progressive ; it is associative 
and it is distributive relative to expressions such as (1). If an ele- 
ment of class & be multiplied into an element of class 4 k+/=n, 
the multiplication remains progressive; if k4-/ n, the product is 
of classmand is ascalar. The properties of progressive multiplication 
as contained in (2) and in the associative and distributive laws are 
simple and are treated in a variety of places.’ If the sum of the 
classes, 8 + J, of two elements is greater than v, the rules of pro- 
gressive multiplication give a zero value for the product and it 

1 For instance, in either of Grassmann’s Ausdehnungslehren, 1844 and 
1862. or in Whitehead’s Universal Algebra, volume 1, Cambridge Uni- 
versity Press. 
