PHILLIPS AND MOORE. — ALGEBEA OF PROJECTIVE GEOMETRY. 751 



And consequently 



A = (AC I DP) -(BC I DP). (4) 



It is to be observed that X is finite for all positions of A and B dis- 

 tinct from P, but becomes infinite when one of these points coincides 

 with P. One of the points A or B being fixed, there is an unique 

 position of the other which gives A a particular value. 



Distance. 



15. In this way we determine the relation between the magnitudes 

 of vectors on lines intersecting on f, but arrive at no relation between 

 vectors not so situated. To obtain a more general relation we define 

 distance as a scalar quantity, or number, determined by two points 

 not on f and such that distances AB * along any line are proportional 

 to the corresponding vectors B — A. In symbols 



distance AB = AB = K(B - A) 



when K is a constant for pairs of points A, B on a given line but may 

 (and usually does) change from line to line. 



It is not obvious that there exists a distance satisfying the above 

 definition. We shall first show (if it exists) what such a distance 

 must be. We shall find that it is not unique and then shall make a 

 further assumption of a function theoretic nature. Finally we prove 

 that the distance found has the properties required. 



From the definition we see that for points on a line, if 



B - A = \ (D - C), 



AB = XCD. 



From the latter equation follows the former provided AB is not zero. 

 In particular 



AB = - BA (5) 



showing that distance is directed. Putting in this equation B = A, 

 it follows that 



AA = 0. (6) 



Furthermore, since 



B-A + C-B = C-A, 



* The notation AB will be used to denote the distance from A to B. 



