400 MOORE AND PHILLIPS. 



from a linear complex. Where no ambiguity results we shall use the 

 word complex for either the complex line or the linear complex. 



A complex p can be represented as a linear function of two lines of 

 which one, /, can be any line not belonging to the linear complex, 



m = 0. 



For a number X can be found satisfying the equation. 



[ip - \l){p - \l)] = (pp) - 2\(jjI) + \\ll) = 0. (21) 



Since / is a line, by (19), (//) = 0. Also by assumption (pi) is not zero. 

 Hence if 



T, _ (PP) 



p — X/ is a line /' and so 



p = x/ + r. 



The two lines I and I' are said to be polar with respect to the complex. 

 Any line of the complex that intersects one of them will intersect the 

 other also. For, if 9 is a line of the complex cutting /, 



(pq) = 0, (Iq) = 0. 



Hence from the equation 



(pq) = Hlq) + (I'q) 



it is seen that (I'q) = and so q intersects /. 

 Let P be any point and 



p = [AB] + [CD] 

 a complex. Then 



[Pp] = [PAB] + [PCD] (22) 



is a plane. If Q is any point in this plane 



(PpQ) = {p-PQ) = 0. 



Hence [PQ] is a line of the complex. All lines passing through P 

 and lying in the plane [Pp] therefore belong to the complex. Hence 

 [Pp] is what is known as the polar plane 0/ P with respect to the complex. 

 Similarly, if a is any plane 



[ap] 



