THE DYADICS IN THREE DIMENSIONS. 431 



As an operator this determines a point plane correlation, 



I = ^X = \Cp] [DqX] = C-pq-DX = [CDX]. 



This is the correlation Avhich transforms each point X into its polar 

 plane with respect to the complex [CD]. If CD represents a polarity 

 [CD] is zero and so the double product of CD and pq is zero. Hence 

 the condition that a correlation represent .a polarity is that it be apolar 

 with the two-two idemfactor. 



Let rs be any two-two dyadic. Symbolically we may write this 



rs = [CD] [EF]. 



The double product of rs and the idemfactor ^a is 



[rA][sa] = [CDA][EF-a] 



= [CDA] [{Ea)F - {Fa)E] 

 = CD-Aa-{FE- EF] 

 = CD-{FE- EF} 

 = [CDF]E - [CDE]F. 



This result has the same form as the product [CD • EF] where C, D, E, 

 F, are points in a plane. The dyadic determines a coUineation which 

 transforms a plane ^ into the plane 



V = [rA]isa^) = - [r-s^]. 



To interpret this coUineation geometrically let rs transform the sides 

 [^2^43], [.43.4i], [^1.42] of a triangle in ^ into the complexes pi, p2, pz 

 respectively. Let 



ai = [Aipi] 



be the polar plane of A, with respect to pi and similarly let 



ao = [A2P2], as = [.43^3]- 



The general theorem states that ai, a2, as, will intersect on rj. The 

 fact that this is true whatever points Ai, Ai, A3, An are taken on ^ 

 proves a geometrical theorem. Suppose for example rs represents 

 a coUineation. By such a coUineation four non coUinear points Ai 

 of one plane could be transformed into any four non-coUinear points 

 Bi of another plane. Join A\ to B2 B3, A^ to ^3 Bi, and A3 to Bi B^. 

 The three planes intersect in a point. SimUarly A2, A3, A^ and 



