ON DOUBLE POLYADICS — THE LINEAR MATRIX EQUATION. 363 



becomes half ((p*<p) or <p2S which, if ab and cd are any two terms of the 

 dyadic, is the sum of all terms of the form abed — a-dc-b, or 

 symbolically 



2ma = abed - a-dc-b. (30) 



The case of dyadics whose antecedents and consequents are com- 

 binatorial products of vectors in A-space, the sum of the classes of 

 antecedent and consequent being equal to the dimensionality of the 

 space, has also been treated; 3 and by applying multiple algebra to a 

 projective point space of three dimensions, various types of dyadics 

 occur, with corresponding Hamilton-Cayley equations. 4 



By analogy it is plain we shall expect m\ for the double polyadic to 

 be the sum of A : M, the K -tuple dot product of each antecedent into 

 its own consequent. Using the expansion (23) and noting that E; : M f 

 = ma we have 2 A : M = Zm« which is the same as mi by the usual 

 matrix theory. 5 If we agree to write, by analogy with Gibbs, 2 A : M = 

 ■<ps, called the scalar of <p, we shall have 



wi = SA : M = <p S . (31) 



By analogy with Gibbs' ((pj))s we may now define the following 

 scalar: 



((AM, BN)) S = A :MB :N - A :NB :M (32) 



where A, M, B, and N are any four polyadics whatever. If <p and 9 

 are any two double polyadics the scalar ((<p, 8))s is to be found by 

 multiplying the double polyadics term by term and adding the scalars 

 of all the terms. This scalar will be treated later on as a kind of 

 product of the two double polyadics. For the present it will suffice 

 to note that if 6 = <p the scalar becomes twice the coefficient of (p n ~ 2 in 

 the Hamilton-Cayley equation, or 



2ms = {{fp, <p)) a . (33) 



The proof is simple. Let <p be expanded in the form (23), and note 

 that by (24) we have 



Ei : Mi = m«. (34) 



Multiplying <p into itself we shall have terms of the form E,M,E,M, 

 whose scalar on developing by (32) vanishes, and terms of the form 

 E,M,E/,.M/.-, whose scalar is muvii-k — mucin m a two-row minor from 

 the determinant of the n 2 elements m pq of <p. Each of the latter 

 terms will occur twice, with all possible selections of pairs of subscripts. 



