416 MOORE AND PHILLIPS. 



transforms the planes of the tetrahedron C1C2C3C4 into the points 

 Bi, Bo, Bz, Bi. Now let ^ represent a polarity. Then the lines 

 [Bid] belong to a quadric. This shows that 5jCj have the required 

 property. 



The three-three dyadic is the dual of the one-one. It transforms 

 points into planes and can be expressed as a sum of four terms exactly 

 dual to the one-one. Associated with each three-three dyadic 



ajS = ai/Si + CL2^2 + asjSa -\- 04)84 

 is a covariant complex 



P = [ai/3i] -f [aoiSo] + [03/33) + [a,8i]. 



As in case of the one-one dyadic it can be shown that if aj3 transforms 

 a point A into a plane passing through B and the point B into a plane 

 passing through A, then the line joining A and 5 is a line of the 

 complex p. 



13. Two-three and two-one dyadics. A two-three dyadic 

 has the form 



where the /S's are planes and the ^'s are either lines or complexes. 

 Since the planes can be expressed as linear functions of four not 

 through a point, the dyadic can be written 



q^ = qij3i + gaft + qsl3s + qil34. 



The consequents can be any four planes not passing through a point. 

 The antecedents cannot, however, be assigned arbitrarily. For, the 

 four complexes qi in general have two lines h and k in common. We 

 call these the singular lines of the dyadic. It is clear that (hq)^ = 

 {kq)0 = O. 



As an operator on points the dyadic determines a transformation of 

 points X into lines or complexes 



p = g(/3Z) = S9,(|3,Z). 



If the transform of X is a line, 



(pp) = S(gigfc)(/3iZ)(/3,Z) = 0. 



