116 MULTIPLE ALGEBKA. 



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 Ausdehn- 

 ungslehre under the name of the open product and has developed 

 at length in the second. 



In the theory of quantics Grassmann'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 indi- 

 vidual 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 



dr 

 dp' 



a block or matrix of ordinary differential coefficients. In this 

 expression, p may be a multiple quantity representing say n inde- 

 pendent variables, and T another representing perhaps the same 

 number of dependent variables. Then dp represents the n differ- 

 entials of the former, and dr the n differentials of the latter. The 

 whole expression represents an operator which turns dp into dr, 

 so that we may write identically 



7 dr -, 

 cir = -y- dp. 

 dp 



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 



