108 MULTIPLE ALGEBRA. 



cases the particular algebra is little more than an application or 

 interpretation of the general. 



Grassmann observes that any kind of multiplication of 7i-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 = units will be 



ft 



required for the products of two factors, n for the 



-j . O 



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 x . It gives rise 



n I 

 to multiple quantities of higher orders, n = units being required 



to express the products of two factors, n- - Q - units for 



-j . O 



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 2 factors in terms of 



71 1 



n g units, etc. This kind of multiplication is associative, like the 



<H 



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 



* Crelle's Journ. f. Math., vol. xlix, p. 138. 



