ON DOUBLE POLYADICS — THE LINEAR MATRIX EQUATION. 



369 



where the consequents Er • -E n are the fundamental A-ads and the 

 antecedents are any arbitrary A-adics. Let E y be any one of the 

 fundamental A-ads. We have 



tPi'-Ej — B;;. 



(50) 



The binary assemblage of n 2 arbitrary polyadics B tJ are now to be 

 formed into a determinant 



Bn, B12, 



B21, B22, 



. Bl n 



, Bo n 



B,,i, B n 2> 



B, 



(51) 



where the law of expansion is defined to be as follows: the leading 

 term is written [BnB 2 2- ■ -B„„] and signifies the determinant in the 

 ordinary sense whose elements are the scalar elements of the polyadics 

 which enter: namely, if we expand as 



B, :y = 6 iA Bi + bifA -\ \- 6«/nB f 



(52) 



the leading term in expanding (51) is the determinant of ordinary 

 character 



Olllj O112, 

 &221> "222> 



' , b-22n 



Until} Onnli " " "j ^nnn 



(53) 



the other terms in the expansion of (51) are of similar character, and 

 are formed by writing out the determinant (51) as if the elements were 

 scalars, enclosing each term in square brackets to signify the determi- 

 nant formed from it as in (53). The signs will occur as in ordinary 

 determinants if we adopt the rule that, in every term, the order of the 

 polyadics shall be the order of the rows in the original determinant. The 

 meaning of the determinant (51) is thus defined without ambiguity. 

 We note that by this rule the order of rows in (53) corresponds to the 

 order of rows in (51), in the sense that each row of (53) consists of the 

 elements of a single polyadic from the corresponding row of (51). 



It is then apparent that (51) is the sum of n! ordinary determinants, 

 hence the sum of in!) 2 terms on complete expansion. The order of the 

 rows is non-signant, in the sense that, if two rows in (51) be inter- 

 changed, the value of the expression is quite unchanged; for on ex- 



