Page:Scientific Papers of Josiah Willard Gibbs - Volume 2.djvu/125

subsist between the products of a system of independent units, is also rejected by Grassmann, as not appearing to afford important applications. I shall, however, have occasion to speak of it, and shall call it the indeterminate product. In this kind of multiplication, $n^{2}$ units are required to express the products of two factors, and $n^{2}$ units for products of three factors, etc. It evidently may be regarded as associative.

Another very important kind of multiplication is that called by Grassmann internal. In the form in which I shall give it, which is less general than Grassmann's, it is in one respect the most simple of all, since its only result is a numerical quantity. It is essentially binary and characterized by laws of the form

$i.i=1,$⁠	$j.j=1,$⁠	$k.k=1,$⁠	etc.,
$i.j=0,$	$j.i=0,$	etc.,

where $i,j,k,$ etc., represent a system of independent units. I use the dot as significant of this kind of multiplication.

Grassmann derives this kind of multiplication from the combinatorial by the following process. He defines the complement (Ergänzung) of a unit as the combinatorial product of all the other units, taken with such a sign that the combinatorial product of the unit and its complement shall be positive. The combinatorial product of a unit and its complement is therefore unity, and that of a unit and the complement of any other unit is zero. The internal product of two units is the combinatorial product of the first and the complement of the second.

It is important to observe that any scalar product of two factors of the same kind of multiple quantities, which is positive when the factors are identical, may be regarded as an internal product, i.e., we may always find such a system of units, that the characteristic equations of the product will reduce to the above form. The nature of the subject may afibrd a definition of the product independent of any reference to a system of units. Such a definition will then have obvious advantages. An important case of this kind occurs in geometry in that product of two vectors which is obtained by multiplying the products of their lengths by the cosine of the angle which they include. This is an internal product in Grassmann's sense.

Let us now return to the indeterminate product, which I am inclined to regard as the most important of all, since we may derive from it the algebraic and the combinatorial. For this end we will prefix $\textstyle \sum$ to an indeterminate product to denote the sum of all the terms obtained by taking the factors in every possible order. Then,

$\textstyle \sum \displaystyle \alpha \left\vert \beta \right\vert \gamma ,$

for instance, where the vertical line is used to denote the