420 MOORE AND PHILLIPS. 



dyadic does not, in general, set up a contact transformation, i. e., 

 intersecting lines do not, in general, go into intersecting lines. What 

 we have shown is that the contact of pairs of lines in the singular 

 pencils of g is preserved. 



The lines of a pencil j^ + ^q, in general, transform into a pencil of 

 complexes j/ + X^'. In this pencil are two special complexes that 

 are lines. These correspond to the two lines of the pencil p -}- \q 

 that belong to g. The lines of a plane transform into. a two parameter 

 linear system of complexes. The special complexes of this system are 

 one system of generators on a quadric. These correspond to the 

 lines of g which lie in the plane (and so envelope a conic). Similarly, 

 the lines through a point go into a two parameter linear system of 

 complexes whose quadric of singular elements corresponds to the cone 

 of lines passing through the point and belonging to g. 



Any two-two dyadic can be expressed as the sum of a self -con jugate 

 and an anti-self-conjugate dyadic. For 



rs = XriSi = |2(/-i.Si + SiVi) + ^2(?-i5i — SiVi) = ^{rs ^- sr) 



The first term is seen to be self-conjugate and the second term anti- 

 self-conjugate. 



A self-conjugate two-hoo dyadic is analogous to a polarity. For such 

 a dyadic rs we have the relation 



r{sp) = {j)r)s. 

 Hence, if 



{qr){sp) = 0, 

 then 



{pr){sq) = 0, 



that is, if p transforms into a complex (or line) in inAolution with q, 

 then q transforms into a complex (or line) in involution with p. 



Express rs in terms of six linearly independent complexes qu q^,. . .qe, 



rs = I,aikqiqk- 



If the dyadic is self-conjugate 



^aikqiqk = Saa^t^,. 

 Hence 



O-ik = 0-ki- 



Such a dyadic can always be reduced to the form 



