Page:Scientific Papers of Josiah Willard Gibbs - Volume 2.djvu/124

108 cases the particular algebra is little more than an application or interpretation of the general.

grassmann observes that any kind of multiplication of n-fold quantities is characterized by the relations which hold between the products of $$n$$ independent units. In certain kinds of multiplication these characteristic relations will hold true of the products of any of the quantities.

Thus if the value of a product is independent of the order of the factors when these belong to the system of units, it will always be independent of the order of the factors. The kind of multiplication characterized by this relation and no other between the products is called by Grassmann algebraic, because its rules coincide with those of ordinary algebra. It is to be observed, however, that it gives rise to multiple quantities of higher orders. If $$n$$ independent units are required to express the original quantities, $$n \frac{n + 1}{2}$$ units will be required for the products of two factors, $$n \frac{(n + 1)(n + 2)}{2. 3}$$ the products of three factors, etc.

Again, if the value of a product of factors belonging to a system of units is multiplied by —1 when two factors change places, the same will be true of the product of any factors obtained by addition of the units. The kind of multiplication characterized by this relation and no other is called by Grassmann external or combinatorial. For our present purpose we may denote it by the sign $$\times .$$ It gives rise to multiple quantities of higher orders, $$n \frac{n - 1}{2}$$ units being required to express the products of two factors, $$n \frac{(n - 1)(n - 2)}{2. 3}$$ units for products of three factors, etc. All products of more than $$n$$ factors are zero. The products of n factors may be expressed by a single unit, viz., the product of the $$n$$ original units taken in a specified order, which is generally set equal to 1. The products of $$n - 1$$ factors are expressed in terms of $$n$$ units, those of $$n - 1$$ factors in terms of $$n \frac{n - 1}{2}$$ units, etc. This kind of multiplication is associative, like the algebraic.

Grassmann observes, with respect to binary products, that these two kinds of multiplication are the only kinds characterized by laws which are the same for any factors as for particular units, except indeed that characterized by no special laws, and that for which all products are zero. The last we may evidently reject as nugatory. That for which there are no special laws, i.e., in which no equations