ON DOUBLE POLYADICS THE LINEAR MATRIX EQUATION. 371 



which is the same as the difference of two determinants 



(01110222 — OU2^22l) — ("121^212 — "122021l) 



and this, again, agrees with the cubic determinant formed with the 

 element of the two dyadics as layers, namely 



biixy "121 



"112, "122 



0211j '^221 

 "212) "222 



A number of results follow immediately. We might have formed 

 the left side of (54) by taking the multiple dot product of Ei, E 2 , etc. 

 into each of the arbitrary double polyadics, as Ei: 01, E 2 : 01, etc. in the 

 first row, and similarly for the other rows. Instead of (51) we should 

 have had the determinant of the M;/ by developing after the manner 

 (23) instead of (27). The rest of our reasoning would have been un- 

 changed, however, hence the value of the expression would have re- 

 mained the same. The change would in fact be equivalent to chang- 

 ing rows into columns and columns into rows in all the non-signant 

 layers of the cubic determinant, which, since the second and third 

 indices are signant, leaves the result unchanged. This is illustrated 

 by the example just given, where the last index refers to the rows in- 

 stead of to the columns as in (41). If we let 0' stand for 2CMA when 

 <p stands for 2AM, and say that the conjugate 0' of a double polyadic 

 <p is formed from it by interchanging antecedent and consequent, it is 

 clear that E : <p = <p' : E. We therefore have the identity 



((01, 02, ■ ■ ■ , <Pn))s — ((<Pl> <P2 >" ' ', <Pn'))s 



(55) 



which may also be deduced directly from the definition of these scalars. 

 The well known fact that the Hamilton-Cayley equation for any 

 dyadic or matrix is identical with that of its conjugate appears as a 

 special case of this identity. 



Also if n — p out of the arbitrary double polyadics are allowed to 

 become equal to the idemfactor, the identitv (54) becomes in virtue 

 of (47), 



0i:Ej, c^i'.Eo, •••, ci :E, 



02 : Ei, 



02 '• E 2 , 



pi 



02 



E, 



Ei 



E, 



= in - p)!((<ph 02, • • • , <p P ))s (56) 



