MULTIPLE ALGEBRA. 109 



subsist between the products of a system of independent units, is also 

 rejected by Grassmann, as not appearing to afford important applica- 

 tions. I shall, however, have occasion to speak of it, and shall call 

 it the indeterminate product. In this kind of multiplication, n z units 

 are required to express the products of two factors, and n z 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 = l, j-j = l> k.k = l, etc., 



V = 0, j.^ = 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 com- 

 binatorial by the following process. He defines the complement 

 (Erganzung) 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 com- 

 plement 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 afford 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 multi- 

 plying 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 2 to an indeterminate product to denote the sum of all the 

 terms obtained by taking the factors in every possible order. Then, 



Sal/Sly, 

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



