758 



PROCEEDINGS OF THE AMERICAN ACADEMY. 



when one of the points is on f. The locus of points at a constant dis- 

 tance from a point A is a line b through the intersection of FA with f. 

 For fixed distance A and b are correspondents with respect to a definite 

 correlation in which F and f are corresponding elements and f, the 

 locus of points whose lines pass through them. On any line distances 

 AB are proportional to the vectors B — A. Equal vectors on different 

 lines have equal length if F is on f, but not otherwise. 



Angle. 



22. Point vectors. Just as A — B may be considered as a point of 

 zero magnitude on f, or as a vector associated with the segment AB not 

 crossing f, so a — b may be considered as a line of zero magnitude 

 passing through F or as a point vector associated with the angular seg- 

 ment (or vector) not containing F. 



Figure 11. 



Point vectors have properties dual to those of line vectors. Thus, if 



a — b = — d 



ac and bd are points of a line through F. The relation of the lines a, 

 b, c, d is shown in the diagram (Figure 11). 



If a-b = X(c-d) 



where c, d are distinct and a, b, c^ pass through a point, we shall say 

 the angle ab is X times the angle cd. In this ca.se, if p is the line join- 

 ing F to the common point, 



A=(ac|dp)~ (bcldp) (8) 



