THE DYADICS IN THREE DIMENSIONS. 399 



If the sum of [AB] and [CD] is a line, AB and CD intersect. For, if 



[AB] + [CD] = [PQ], 

 then 



[ABP] + [CDP] = [PQP] = 0. 



This shows that [ABP] and [CDP] are the same plane and hence AB 

 and CD lie in a plane and so intersect. 

 The product of p by itself is 



[pp] = [{AB + CD){AB + CD)] = 2(AB CD). 



If 2? is a complex line [AB] and [CD] do not intersect and so (AB CD) 

 is not zero. If, however, p is a line [AB] 



(j}p) = (AB-AB) = 0. (19) 



Hence the necessary and sufficient condition that a matrix 



P = Ikifcll 

 represent a simple line is , 



Let 



be a complex line and 



(pp) = 0. 

 p = [AB] + [CD] 



I = [XY] 



be a line. The equation 



(pi) = (ABXY) + (CDXY) = 

 is a linear equation in the coordinates 



Xi Xjc 



of the line I. Hence the lines satisfying this equation constitute a 

 linear complex. This complex is the totality of lines I satisfying the 

 equation 



(p/) = 0. (20) 



This is a different thing from j^ which in a sense is the envelope of the 

 lines just as a point in a plane is different from the set of lines passing 

 through it. For this reason we call p a complex line to distinguish it 



