Page:Scientific Papers of Josiah Willard Gibbs - Volume 2.djvu/132

theory of eliminations and substitutions, including the theory of matrices and determinants, seems to afford the most simple application of multiple algebra. I have already indicated what seems to me the appropriate foundation for the theory of matrices. The method is essentially that which Grassmann has sketched in his first Ausdehnungslehre under the name of the open product and has developed at length in the second.

In the theory of quantics Orassmann's algebraic product finds an application, the quantic appearing as a sum of algebraic products in Grassmann's sense of the term. As it has been stated that these products are subject to the same laws as the ordinary products of algebra, it may seem that we have here a distinction without an important difference. If the quantics were to be subject to no farther multiplications, except the algebraic in Grassmann's sense, such an objection would be valid. But quantics regarded as sums of algebraic products, in Grassmann's sense, are multiple quantities and subject to a great variety of other multiplications than the algebraic, by which they were formed. Of these the most important are doubtless the combinatorial, the internal, and the indeterminate. The combinatorial and the internal may be applied, not only to the quantic as a whole or to the algebraic products of which it consists, but also to the individual factors in each term, in accordance with the general principle which has been stated with respect to the indeterminate product and which will apply also to the algebraic, since the algebraic may be regarded as a sum of indeterminate products.

In the differential and integral calculus it is often advantageous to regard as multiple quantities various sets of variables, especially the independent variables, or those which may be taken as such. It is often convenient to represent in the form of a single differential coefficient, as

${\frac {d\tau }{d\rho }},$

a block or matrix of ordinary differential coefficients. In this expression, $\rho$ may be a multiple quantity representing say $n$ independent variables, and $\tau$ another representing perhaps the same number of dependent variables. Then $d\rho$ represents the $n$ differentials of the former, and $d\tau$ the $n$ differentials of the latter. The whole expression represents an operator which turns $d\rho$ into $d\tau ,$ so that we may write identically

$d\tau ={\frac {d\tau }{d\rho }}d\rho .$

Here we see a matrix of $n^{2}$ differential coefficients represented by a quotient. This conception is due to Grassmann, as well as the representation of the matrix by a sum of products, which we have