THE DYADICS IN THREE DIMENSIONS. 405 



and it is clear that |(<J> + $c) is self-conjugate and |(<l> — <J>c) anti-self- 

 conjugate. 



The one-three idemf actor is minus the conjugate of the three-one 

 and vice versa, because if $ is the three-one idemfactor a$ = a, or 

 a= — ^ctt, since [a.4i] = —[Aio]. The two-two idemfactor is 

 self-conjugate. 



9. Products of two dyadics. By the product ABCD of two 

 dyads AB and CD is meant the indeterminate product A[BC]D 

 obtained by taking the outer product (see ^4) of the adjacent factors 

 B, C. In case [BC] is not a number, the result is an indeterminate 

 product of three factors or a triad. If [BC] is a number, it is commuta- 

 tive with A and the result is the dyad (BC)AD. Similarly the product 

 of three dyads AB, CD, EF is defined as 



ABCDEF = A[BC][DE]F 



and the product of two dyads and an extensive quantity E is 



ABCDE^ A[BC][DE] 



etc. In each case it is clear that the product is associative so far as 

 dyads or dyads and extensive quantities are concerned. 



The product $^ of two dyadics is defined as the result of multi- 

 plying each dyad of $ by each dyad of ^ and adding the results. 

 Similarly the product of three or more dyadics is obtained by multi- 

 plying them distributively. Since the product of dyads is associ- 

 ative, the product of dyadics is associative. 



We have seen that the dyadic ^ as an operator transforms extensive 

 quantities X complementary to the consequents into extensive quan- 

 tities ^X of the same dimension as the antecedents. // the conse- 

 quents of $ are complementary to the antecedents of ^, ^^ is a dyadic 

 which as an operator is equivalent to the operator ^ folloived by the 

 operator <S>. For if 



Y = ^X, Z = $F, 



by the associative law 



In the same way, if X is complementary to the antecedents of $, Z*^ 

 is equivalent to the transformation Z<S> followed by ^ as postfactor. 



