Vol.. 6, 1920 MATHEMATICS: MOORE AND PHILLIPS 
155 
NOTE ON GEOMETRICAL PRODUCTS 
By C. Iv. B. Moore and H. B. Philups 
DEPARTMENT OP Mathematics, Massachusetts Institute of Technoi^ogy 
Communicated by E. H. Moore, January 28, 1920 
Geometrical products may be divided into two general classes: those 
definable in a space of any number of dimensions and those definable only 
in a space of a definite number of dimensions. To the first class belong the 
progressive (outer) product of Grassmann and the inner products of Grass- 
mann, Gibbs^ and Lewis. ^ To the second class belong the regressive prod- 
uct of Grassmann and the cross product of Gibbs (as defined in the Gibbs- 
Wilson Vector Analysis) and the quaternion multiplications. In this 
paper we show that there is a series of geometrical products independent 
of dimensions that may be in a sense considered intermediate between the 
progressive product and the inner product. 
In terms of the units^ the progressive product is expressed by identities 
of the form ^12^34 = ^1234, the product containing all the units multiplied 
together. If there are any common units as in ^12^14 = 0, the product 
is zero. The inner product of Lewis is expressed by identities of the form 
^3.^123 = k\2, the common units being cancelled (with certain conventions 
as to algebraic sign). Unless one factor is entirely contained in the other 
the result is zero. 
If TTi and 7r2 are products of units whose first m factors are the same 
and in the same order we define as the product [iriTr^lrn of index m the re- 
sult of cancelling those m common units and taking the outer product of 
the remaining units in the order written. Thus 
[^12^13 ]l = ^23, [^123^124 ]2 = ^34- 
If TTi and 7r2 have m units in common but these are not the first m units, 
TTi and 7r2 are rewritten (with change of sign if necessary as required by the 
outer multiplication) in such a way that the first m factors are the same, 
and in the same order, the product is then formed as above, negative 
signs introduced being included.^ Thus 
[^12^23]l = — k\z, [^234^134]2 = ^21 = — ^12- 
If there are either more or less than m units in common, the product is 
zero. Thus 
[^123^124]l = ^[l23^124]3 = 0. 
The inner product is obtained when the index is equal to the dimension 
of the lowest factor. Thus 
[klkii\i = ^1.^12 = —^2. 
The outer product occurs when the index is zero. Thus [^12^34 ]o = ^1234- 
To obtain the product of M and A^, two homogeneous linear functions of 
units, they are multiplied distributively, the product of units being found 
