► (155) 



WILSON AM) LEWIS. — RELATIVITY. 499 



and then applying the distributive hiw. The pro(hicts A*4> and «I>'A 

 are therefore themselves dyadies of the same type as f^. If in place of 

 4> we use the idemfactor I, then it is easily shown that 



I. A (= — A-I) 



is the anti-selfconjugate dyadic which is determined by the 2-vector A. 



fi = I«A = — Jiokiko — .4]3k]k3 — .Ii4kik4 



+ -ilikekl — ^423k2k3 — ^24k2k4 



+ -liskski + J2.sk.3k2 — ^34k3k4 



+ Jllk4ki + J2lk4k2 + J34k4k3 



If we denote by ^^ the 2-vector obtained by inserting the cross in the 

 dyads of 0, we have ^^ = (I • A)^ = — 2 A. 



It is this relation between 2-vectors and linear vector functions or 

 dyadies which enables Minkowski to replace a 2-vector by an anti- 

 selfconjugate (or alternating) matrix and vice versa. 



If Q and W are the two dyadies obtained from the two 2-vectors 

 A and A', we may form the product S]*12'. (This is the product//^ of 

 Minkowski). We can then write 



(r.A)-A' = (r-l^).fi' = r-iU-n'). (156) 



We employed (§57) the selfconjugate dyadic fi'fi = (I 'A) '(I 'A), 

 and another dyadic | (12*0 + fi* •!]*), where 9,* was defined as 

 v.* = I 'A*. This dyadic corresponds to the matrix S of Min- 

 kowski/^ and may be regarded as the dyadic representing stress in 

 four dimensional space. 



68 The expression (T'A)'A' may be transformed by (38). 



(r.Aj.A' = - r(A.A') + A«(rxA'). 



As A«(rxA') i.s a l-voctor, the complement of its complement is itself, by (26). 

 By rules (30) and (24 j 



IA.(rxAO]** = [Ax(rxA')*]* = [(r.A'*;xA]* = (r.A'*j«A*. 



Hence we obtain the important relation 



(r.A).A' = -r(A.A')+ (r.A'*).A*. 



By introducing dyadies and canceling the vector r, we have 



(I.A).(I.A') = - (A.A)I + (I. A'*). (LA*). 

 If we set 



■•I' = U(l'A.)'(I'A') + (I. A'*). (I. A*)], 

 we may write 



(I. A). (LA') = ^ - A(A.A')I, (LA'*). (LA*) = ^ + ^(A.A')!. 



The dyadic ^ is precisely the matrix .S of Minkowski. 



