THE DYADICS IN THREE DIMENSIONS. 401 



is what is called the pole of a with respect to the complex. All the 

 lines of the complex, lying in the plane a, pass through the point [ap]. 

 If / and /' are polar lines with respect to jj, 



p - X/ + txl'. 



If then X is a point on /, [XI] = and so 



[Xp] = m[A7'] 



which shows that the polar plane of X with respect to p contains the 

 line /'. 



If the product (pq) of two complexes p and q is zero, the complexes 

 are said to be in involution. In this case we shall say that the two 

 complexes intersect from analogy with the case of two lines which 

 intersect if their product is zero. 



If two lines h, h satisfy a linear relation 



Xi h + X2 h = 0, X], X2 5^ 



they coincide in position. If three lines satisfy a linear relation 



Xi/i + X2/2 + X3/3 = Xi,X2, or X3 F^ 



any line cutting two of them cuts the third and so they belong to a 

 plane pencil. If four lines satisfy a linear relation 



Xi/i + X2/2 + X3/3 + X4/4 = XiXo.Xs, or Xi 7^ 



any line cutting three cuts the fourth also. Hence they belong to the 

 same system of generators on a quadric. If five lines satisfy a linear 

 relation, the two lines cutting four of them will cut the fifth also and 

 so they belong to a linear congruence. If six lines satisfy a linear 

 relation they belong to a linear complex. 



Similarly if two complex lines satisfy a linear relation the linear 

 complexes represented by them are identical. 



If three complex lines satisfy a linear relation, the linear complexes 

 have a common congruence. If four complex lines satisfy a linear 

 relation, the complexes have one system of generators on a quadric 

 surface in common. If five satisfy a linear relation, the complexes 

 have two lines in common. If six complex lines satisfy a linear rela- 

 tion the complexes are in involution with a fLxed complex. 



