110 MULTIPLE ALGEBRA. 



indeterminate product,* is a distributive function of a, ft, and y. It 

 is evidently not affected by changing the order of the letters. It is, 

 therefore, an algebraic product in the sense in which the term has 

 been defined. 



So, again, if we prefix S to an indeterminate product to denote 

 the sum of all terms obtained by giving the factors every possible 

 order, those terms being taken negatively which are obtained by an 

 odd number of simple permutations, 



for instance, will be a distributive function of a, /3, y, which is 

 multiplied by 1 when two of these letters change places. It will 

 therefore be a combinatorial product. 



It is a characteristic and very important property of an indeter- 

 minate product that every product of all its factors with any other 

 quantities is also a product of the indeterminate product and the 

 other quantities. We need not stop for a formal proof of this pro- 

 position, which indeed is an immediate consequence of the definitions 

 of the terms. 



These considerations bring us naturally to what Grassmann calls 

 regressive multiplication, which I will first illustrate by a very 

 simple example. If n, the degree of multiplicity of our original 

 quantities, is 4, the combinatorial product of aX/3xy and <5xe, viz., 



aX/3xyxSxe, 



is necessarily zero, since the number of factors exceeds four. But if 

 for Sxe we set its equivalent 



S\e-e\S, 



we may multiply the first factor in each of these indeterminate pro- 

 ducts combinatorially by aX/3xy, and prefix the result, which is a 

 numerical quantity, as coefficient to the second factor. This will give 



(aX/3XyX(5)e (aX/3xyXe)S. 



Now, the first term of this expression is a product of aX/3xy, 8, and 

 e, and therefore, by the principle just stated, a product of aX/3xy 

 and S\e. The second term is a similar product of aX/3xy and e\S. 

 Therefore the whole expression is a product of aX/3xy and S\e \S t 

 that is, of aX/3xy and <5xe. That is, except in sign, what Grass- 

 mann calls the regressive product of aX/3xy arid Sxe. 



To generalize this process, we first observe that an expression of 

 the form 



in which each term is an indeterminate product of two combinatorial 

 products, and in which S denotes the sum of all terms obtained by 



* This notation must not be confounded with Grassmann's use of the vertical line. 



