MULTIPLE ALGEBRA. Ill 



putting every different pair of the letters before the dividing line, the 

 negative sign being used for any terms which may be obtained by an 

 odd number of simple permutations of the letters, in other words, 

 the expression 



aX/3\yXS 



is a distributive function of a, /3, y, and <5, which is multiplied by 1 

 when two of these letters change places, and may, therefore, be regarded 

 as equivalent to the combinatorial product aX/3xyxS. Now, if 7i = 5, 

 the combinatorial product of 



pXa-Xr and aX/BXyXS 



is zero. But if we multiply the first member of each of the above 

 indeterminate products by pXarXr, and prefix the result as co- 

 efficient to the second member, we obtain 



which is what Grassmann calls the regressive product of pXvXr and 

 aXflXyxS. It is easy to see that the principle may be extended so 

 as to give a regressive product in any case in which the total number 

 of factors of two combinatorial products is greater than n. Also, that 

 we might form a regressive product by treating the first of the given 

 combinatorials as we have treated the second. It may easily be shown 

 that this would give the same result, except in some cases with a 

 difference of sign. To avoid this inconvenience, we may make the 

 rule, that whenever in the substitution of a sum of indeterminate 

 products for a combinatorial, both factors of the indeterminate products 

 are of odd degree, we change the sign of the whole expression. With 

 this understanding, the results which we obtain will be identical with 

 Grassmann's regressive product. The propriety of the name consists 

 in the fact that the product is of less degree than either of the factors. 

 For the contrary reason, the ordinary external or combinatorial 

 multiplication is sometimes called by Grassmann progressive. 



Regressive multiplication is associative and exhibits a very remark- 

 able analogy with the progressive. This analogy I have not time 

 here to develop, but will only remark that in this analogy lies in its 

 most general form that celebrated principle of duality, which appears 

 in various forms in geometry and certain branches of analysis. 



To fix our ideas, I may observe that in geometry the progressive 

 multiplication of points gives successively lines, planes and volumes; 

 the regressive multiplication of planes gives successively lines, points 

 and scalar quantities. 



The indeterminate product affords a natural key to the subject of 

 matrices. In fact, a sum of indeterminate products of the second 



