404 MOORE AND PHILLIPS. 



Since this is true for all elements X of the same kind, that is, of the 

 same dimension it is easy to show that 



/l - /2 = 0, 



which shows that /i and I-i are identical. 



If I is complementary to X, the idemf actor I in (24) satifies the 

 equation 



For, if we multiply each side of (24) by ^ we get 



(^Y) = (^7X) = {^I-X). 



Hence 



[(^ - ^I)X\ = 



for all elements X and so 



^-^1 = 0. 



8. Conjugate, self-conjugate and anti-self-conjugate dy- 



adics. The dyadic <l>c obtained by interchanging the antecedents and 

 consequents of $ is called the conjugate of $. Thus if 



$ = A,B, + A.Bo + ....+ AnBn, 

 <i>e = B,Ay + ^2^2 + ....+ BnAn. 



If X is complementary to the consequents, it is clear that 



^X = ± X<S>c. 



The sign is plus or minus according as \XBi] is equal to [fiiX] or to its 

 negative. Similarly if 7 is complementary to the antecedents 



7$ = ± $c7- 



A dyadic $ is called self-conjugate if 



$ = $c 



and anti-self-conjugate if 



In each case the antecedents and consequents must be quantities of 

 the same kind. Any dyadic whose antecedents and consequents are 

 quantities of the same kind can he expressed as the sum of a self-conjugate 

 and of an anti-self -con jugate dyadic. For. 



$ = 1 ($ + <J>^) + 1 ($ - $,) 



