6 E. B. Wilson, 
than x. Formula (3) shows that in case the factors are made up of 
units, the regressive product is the space common to the two spaces of 
the factors, that is, it is the intersection of the factors, taken, of course, 
with a certain magnitude. An examination of the rules for expanding 
the regressive product, especially as illustrated by the examples there 
given, shows at once that the result is true in general, and that the 
regressive product of two spaces is always the intersection of the spaces, 
taken with a proper numerical value. It should be noted that if the 
spaces of two factors in a regressive product do not exhaust the dimen- 
sionality of all space, that is, if the spaces of both factors lie in a sub- 
space of the 2-dimensional space, then the scalar coefficients which occur 
in the expansion of the product will be products of x vectors lying in 
that subspace and will therefore all vanish. That is to say, if the factors 
lie in a subspace, the regressive product must be zero. 
This fact may serve as foundation for the proof of the associative law 
for the multiplication of three factors, which may be denoted by X, Y, 
Z. Vf X and Y lie in a subspace of the -dimensional space, the regres- 
sive product XY is zero and hence XY times Z is necessarily zero. But 
if X and Y lie in a subspace, so must Y and the product YZ, which is 
the space common to Y and Z Hence X times YZ is also zero; and 
the associative law holds in this case. To prove the law in general it 
is sufficient, owing to the applicability of the distributive law, to prove 
it for the case that X, Y, Z are products of the units. Furthermore, it may 
be assumed that X and Y, and also VY and Z, exhaust the x dimensions 
of space. Let JZ represent the product of the units common to X, Y, Z; 
and let 4 be the product of the units other than those in J7 which are 
common to VY and Z; and similarly 2 and C for the pairs Z, X and 
X, Y. Then as X and Y and also Y and Z must exhaust all z-dimen- 
sions of space, it is obvious that every unit wlich occurs in X must 
occur in Y or Z, and similarly for Y and Z. Hence the factors may 
be written as 
Ge as Vea Coie Zi — Aelia 
and the two groupings of the factors give 
[X VY] Z= [((BMC)(CMA) (AMB) = [((BMCA) CM] (AMB) 
— (BMC A)(CAMB) M, 
X[VZ] = (BMC) [(CMA)(AMB)] = (BMC) (CAMB) MUA] 
= (BMCA)(CAMB)M, 
which are equal; and the associative law is proved. Care should, how- 
ever, be exercised against applying the law to cases to which it cannot 
apply, such as 
m=5, [(aBy)de](ea) = 1 Vea), (@ By) [(e) (Cea)| = 0. 
Here the products are not regressive, but progressive. 
In addition to the combinatorial product « <8 of two elements 
there is the dyadic product «8. This corresponds to the simplest 
type of Grassmann’s Liickenausdriicke. It is, according to Gibbs's 
