432 MOORE AND PHILLIPS. 



B'2, Bz, Bi determine a second point, etc. The four points thus 

 obtained he in a plane. 



The double product of the one-two dyadic B[CD] and the three-one 

 idem factor /la is the two-one dyadic 



$= [BAWCB-a] 



= [BA][{Ca)D - {Ba)C\ 

 = B-Aa-[DC - CD] 

 - [BI)]C - [BC\D. 



To interpret this let B[CD] transform the lines YZ, ZX, XY of a 

 plane ^ into the points X' , Y' , Z' respectively. Then $ will transform 

 the plane ^ into a complex 



[5Z)](C^) - \BC\{J)i) 



which is a linear function of the three lines A' A"', YY' , ZZ' . Hence 

 the complex contains the quadric of these lines. If we take four 

 points X, Y , Z, W of ^ four quadrics will thus be determined which 

 all belong to the same complex. In this case the lines of the four 

 point will transform into points of a four line in another plane. Since 

 the dyadic can be so determined that this four line is arbitrary, this 

 shows that if the corresponding triangles of a four point in one plane 

 and a four line in another plane are joined as above the four quadrics 

 determined will belong to a complex. 

 If 



\BCD\ = 0, 

 [CDKB^ + [DB]m + [BC]{D^) = 



for every plane ^. Hence 



[CD]B = [BD]C - [BC]D 



which shows that in that case the dyadic 4> is the conjugate of B[CD]. 

 The double product of B[CD] and a/1 is the plane of the dyadic 



{Ba)[CDA] = [BCD]. 



Let 2^1, P2, Pz be three lines intersecting in a point of this plane. If 

 ps transforms into a point of the plane pi ^2 and p2 into a point of 

 P3, pi, the general theorem shows that pi will transform into a point 

 of the plane p-zps. If [BCD] = this will be true of any three lines 

 intersecting in a point. 



