PHILLIPS AND MOORE. — ALGEBRA OF PROJECTIVE GEOMETRY. 787 



and A (BC + BD) = 2ABH 



= 2(AB + BH + HA) 



= 2AB + (BC + BD) + DA + CA 



= (AB + BC + CA) + (AB + BD + DA) 



= ABC + ABD 



= aC + aD. 



By duality the distributive law 



A (b + c) = Ab + Ac (39) 



is also true. 



46. We have defined the product of two lines and have expressed it 



in terms of three unit points when the segments representing the two 



lines have an end point in common. This product can also be ex- 

 pressed in terms of unit points when the segments representing the 

 lines do not have an end point in common. That is, we know that the 

 product AB • CD is the sector determined by the two lines and we 

 seek to express this sector in terms of the four points A, B^ C, D. Grass- 

 mann has called such a product a regressive product. Let the two 

 lines be represented by segments thus 



a = CD, b = AB 



and let X be the unit point at the intersection of a, b. 



Then we can express A in terms of B and X. 



(A + 1) A = X + AB. 



(40) 



