Ou 
Double Products and Strains in Hyperspace. 
Obviously, if the numbers of the elements in the factors are 4 and /, the 
product belongs to class + 7— x. It is of fundamental importance to 
observe that the way in which the regressive product is defined by 
reference to the progressive product of elements, is sufficient to insure 
the distributivity of the regressive product relative to sums. 
To justify the double definition, it is necessary to show that the two 
rules lead to the same result. For this, it is convenient to consider all 
the primary elements as expressed linearly in terms of x independent 
primary unit elements, all elements of higher classes as expressed in terms 
of the unit elements of those classes. Then, inasmuch as the distributive 
law applies to the regressive product and the product of two sums of 
terms may be resolved into the sum of the products of each pair of 
terms, of which one is selected from the first factor and one from the 
seeond, it is sufficent to prove the equivalence of the two rules for factors 
made up of certain of the units. As the factors contain respectively & 
and / units, and as there are only ~ units in all, there must be # + / — x 
units common to the two factors; it being understood that if any unit 
is repeated in both factors, it may be repeated in the list of common 
units as many times as it occurs in the factor in which it occurs least 
frequently, unless so great a repetition is not required to make up a total 
of &£-+ 2—n common units, after all those which are common to both 
factors, but are not repeated in more than one of the factors have been 
counted. Let the product of the 4+ 2—~»x units common to the two 
factors be J/, and let the two factors be written as AMZ and JZ2. 
Consider the product 4d/ >< I/B, where it is clear that 4 contains 7 — / 
and 2 contains x — # units. According to the first rule it is necessary 
to select x — & elements from J/2 to form with the % elements in 4AM 
a Scalar coefficient for the remaining elements of J/2. If any of these 
n — are taken from J, the resulting scalar will surely have a repeated 
unit and will vanish. Hence all x —% should be taken from #4, and 
according to the first rule the product is 
(3) AM < MB = (AMB) M 
The same result is obtained by a similar application of the second rule. 
A further word on the geometrical interpretation will considerably 
facilitate the expression of some of the following remarks. If the primary 
elements be interpreted as vectors issuing from a fixed point in Euclidean 
space of x dimensions, the elements of the second class will be conceived 
as plane areas and in particular the product of two vectors will be the 
parallelogram included by them, the elements of the third class will be 
three-dimensional volumes, and so on until the scalars which are elements 
of the zth class will be -dimensional volumes, and in particular the 
product of 7 vectors will be the x-dimensional parallelepiped constructed 
upon them. It appears therefore that the necessary and sufficient con- 
dition that & vectors  < », be linearly dependent is that their (progres- 
sive) product be zero. The regressive product is the product of two 
factors, which, regarded as spaces, have a total dimensionality greater 
