ON DOUBLE POLYADICS — THE LINEAR MATRIX EQUATION. 379 



In the same way it is easy to see that any scalar derived from a set 

 of p unit double polyads, namely 



((WftMb ■••,I,E.))a (74) 



where subscripts run from 1 to n, will vanish unless, by interchanges 

 among the consequents, it can be brought to the form 



((E a E a ,E 6 E 6 , ...,E a E,)) 8 (75) 



wherein each antecedent is equal to its own consequent; and moreover 

 vanishes unless all antecedents are unequal. The sign is plus or minus 

 according as the number of interchanges needed to give the form (75) 

 is even or odd. 



We may now regard the scalars (36) as products of double polya- 

 dics formed according to this law, and, if each double polyadic is 

 developed as in (25), can evidently calculate these scalars with no 

 more labor than would be needed to multiply algebraic polynomials. 



12. Star Products. 



It is especially in the case p = 2 that the concept of the last article 

 appears useful. In keeping with the idea that ({<p\, <fi))s, the calcula- 

 tion of which has been exemplified in some detail, is a kind of product 

 of the two double polyadics, we may introduce the notation 



((<Pi, */>2))s = <P*<P2 (76) 



and speak of this scalar as the star product of the two double polyadics. 

 It is essentially a kind of double product, but since there is nothing 

 analogous for mere vector factors it is not necessary to write two stars 

 to emphasize the distinction. It may properly be compared, or rather 

 contrasted, with the double dot product of Gibbs. To illustrate, take 

 again the simplest case, K = 1, N = 2, so that the factors are dyadics 

 in space of two dimensions. By Art. 11 the multiplication table for 

 unit dyads will be 



ii, ij, ji, jj 



ii 0, 0, 0, +1 



ij 0, 0,-1, (77) 



ji 0, -1, 0, 



jj +1, 0, 0, 



