THE DYADICS IN THREE DIMENSIONS. 419 



where the B's are any four independent points. The q's are not any 

 four complexes but four that have two given hues in common. This 

 dyadic transforms planes into lines or complexes. The planes which 

 are transformed into lines envelope a quadric whose equation is 



{qiqk){Bia){Bka) = 0. 



Associated with the two-one dyadic is a covariant plane whose property 

 is the dual of the covariant point of the two-three. 



14. Two-two dyadics. A two-two dyadic has the form 



rS = riSi + r2S-2 + . . . . VnSn, 



the r's and s's being complexes (or lines). Since any complex can be 

 expressed as a linear function of six that are linearly independent, 

 the dyadic can be reduced to the form 



rs = nsi -f r2S2 + r^Ss + ViS^ -\- r-^Sr, -\- r^s%, 



in which any six linearly independent complexes (or lines) can be taken 

 as antecedents or consequents. 



As an operator, the dyadic determines a transformation of complexes 

 -p into complexes 



V' = r(sp) = I,ri{sip). 



The lines that transform into lines belong to a quadratic complex g, 

 consisting of lines p satisfying the equation 



[r{sp)-risp)] = 0. 



Similarly, the complexes that transform into lines are the complexes 

 p satisfying this quadratic equation. 



The lines of the quadratic complex g transform into the lines of 

 another quadratic complex g'. If two lines p and q oi g intersect, the 

 pencil of lines p + X? will, in general, transform into the pencil of 

 complexes p' + Xg'. If, however, p -\-\q is the pencil of lines of g 

 which lie in a singular ^* plane, p' -f \q' will, for all values of X, 

 represent a line. Hence the lines in the singular planes of g transform 

 into the lines in the singular planes of g'. It is to be noticed that the 



14 In general the lines of a quadratic line complex which lie in a given plane 

 envelope a conic. There are however oo ^ planes in which these conies degen- 

 erate. These planes are called singular planes. See Jessop, page 89. 



