414 MOORE AND PHILLIPS. 



For {^Bi) = —(Bi^), {^Ci) = — (Ctl) and since these products are 

 numbers they are commutative. Since {^B){C^) = [^■B{C^)], this 

 shows that B{C^) is a point in the plane ^. Every plane ^ therefore 

 passes by the correlation 



X = B{C^) 



into a point lying on ^. The correlation is therefore a null-system. 



A^ an operator on planes the anti-self -con jug ate dyadic gives the same 

 result as finding the poles of the planes with respect to the complex 



p = UiBiCr] + [B2C2] + [B3C3] + [B,C,]} 

 For 



[p^] = |2[5,C,^] = m{^Bi)Ci - {^Ci)Bi} = 



= h{i^B)C - {^C)B} 



= mSiCi - CiBi)^ = \{BC - CB)l 



This transformation therefore transforms each plane into its pole with 

 respect to the complex p. 



We have seen that any dyadic is the sum of two, one of which is 

 self-conjugate representing a polarity, the other anti-self-conjugate 

 representing a null system. The dyadic transforms any plane ^ into 

 a point 



B{m = hiSm + CiB^)} + UB{C^) - CiBk)} 



on the line joining the points into which it is transformed by the 

 polarity ^{BC + CB) and by the null-system UBC - CB). 

 With a one-one dyadic is associated a complex (or line) 



p = [BC] = [5iCi) + [B2C2] + [B,Cs] + [B,C,]. 



This is a covariant of BC as can be shown by the same argument 

 used to show that the scalar of the one-three dyadic is an invariant. 

 We shall call this the complex of the dyadic. 



If BC transforms ^ into a point of r] and 77 into a point on ^, the 

 intersection of ^ and r? is a line of the complex p. For, if 



(Bmv) = ^{Biv)(.Ci) = 0, (A) 



(BiC-v)^) = ^{Bi^){Cin) = 0, (B) 



then 



(p-^v) = ^{BiCi-^v) = S{(5,^)(C,r,) - (C,^){Biv)\ = 0. (C) 



Conversely, through each line of the complex pass pairs of planes 

 that are transformed in this way. For, if [^r/] is a line of the complex 



