ON 



DOUBLE POLYADICS — THE LINEAR MATRIX EQUATION. 377 



The fact that any scalar (36) is unaltered by changing all the double 

 dyadics into their conjugates, and is also unaltered by changing the 

 order of double dyadics, appears in the double symmetry of (64). A 

 further and quite distinct fact could not have been brought out so 

 long as our notation represented poly adics as formal vectors : a scalar 

 formed from double dyadics, like (64), is unaltered if all the dyadics 

 A, M, B, N are replaced by their conjugates. This is equivalent to 

 interchanging the two subscripts throughout the expression on the 

 right of (64); whereas changing the double polyadics into their con- 

 jugates is equivalent to everywhere interchanging a with m and at the 

 same time b with n; and changing the order of the polyadics is equiva- 

 lent to everywhere interchanging a with b and at the same time m 

 with ?;. 



More generally a polyadic which is a sum of terms of the form 

 aia 2 - • &k where the A' factors are vectors in space of N dimensions 

 may be regarded as one of a set of K! polyadics obtained from one 

 another by making like permutations of factors in every term. A 

 glance at the definition (10) of multiple dot product is sufficient to 

 show that all scalars (36) are unaltered when all the antecedents and 

 consequents of every double polyadic have the factors of all their 

 terms permuted in the same manner. If each polyadic consists of a 

 single polyad and we apply the definition of Art. 5, the truth of the 

 proposition is evident, hence by the distributive principle holds 

 universally. 



Examples like (64) or (69) may be generalized in three ways, ac- 

 cording as we increase N, the dimensions of the space, K, the order of 

 the polyadics, or p, the order of the scalars. 



If N increases we still have double dyadics and the form of (69) is 

 unaltered, but the summation is performed over a larger group of 

 number-pairs, namely 11, 12, 21, 13, 31,- • •, NN. 



If A' increases the number of subscripts increases but (69) still con- 

 sists of 4 scalar terms and the summation is over pairs of unlike 

 number-triplets, quadruplets, etc. 



If p increases the order of the determinant (68) increases, likewise 

 the number of terms in (69). Thus if we keep N = 2, K = 2, but 

 make p = 3 there are 4 cubic determinants whose sum is ((AL, BM, 

 CN))s or 72 scalar terms in all. 



If we have double dyadics in 3 dimensions with p = 2, ((AM, BN))s 

 will be the sum of 36 expressions of the same form as (69) or 144 scalar 

 terms. 



It is usually possible to neglect the dimensionality of the space and 



