ON DOUBLE POLYADICS — THE LINEAR MATRIX EQUATION. 387 



and nil = ab': a'b 



= aa'bb' 



= A S B S (116) 



where As and Bs are respectively the first coefficients in the Hamilton- 

 Cayley equations for A and B; or, what is the same thing, they are 

 the sums of elements along the main diagonal of these matrices. 

 Stated in complete form the result is 



mi = A IS B 1S + A 2S B 2S + ->-+ A hS B hS (117) 



As remarked earlier, the symbolic process is valid by virtue of the 

 distributive character of all the steps involved. Thus for m% we may 

 put, in abbreviated language, 



2ma= ({#, <p)) s = ((ab' | a'b, cd' | c'd)) s 

 = ab': a'bcd': c'd - ab': c'dcd': a'b 

 = aa'bb'c c'dd'- ac'b'dca'd'b (118) 



Now the first term on the right is the product of the scalars of the four 

 dyadics or matrices which enter; that is, it is symbolically AsB$CsDs 

 or A isB isA jsB jg- The second term may be further transformed 



ac'b'dca'd'b = [ac'a'c] [bd'b'd] 



= [aa'cc'] s [bb'dd'] s 



= [AC] S [BD] S 



= [A i A,] 8 [B i B ] ] 8 (119) 



where [A iA 7 ]$ is the first coefficient in the Hamilton-Cayley equation 

 for the matrix AiAj, that is, for the product of Ai and Aj, it is the sum 

 of elements on the main diagonal of this product; and similarly for 

 [BiBj]s. Stated in complete form the result is 



2m 2 = ?[A iS B iS A jS B jS - {A { A ,) S (B ,-fl ,•) s ], [i,j = 1, 2,- •, h] (120) 



the summation being now with respect to all possible pairs of terms 

 selected from the left side of the original equation (111). It is to be 

 particularly noted that the summation includes the cases i = j; in 

 other words, by " pairs of terms " is meant not merely different terms 

 but also the same term taken twice. The reason is not far to seek. 

 By the transformation (112) any term AxB was changed into a sum of 

 terms of the form MN: X, but not, in general, into a single such term. 

 It follows that, although ((MN, MN))g, which is the star product of a 

 single term MN into itself, must vanish, yet a scalar A 2 $ B 2 g — 

 (A 2 ) s (B 2 )s symbolically derived from it does not in general vanish. 



