212 PROCEEDINGS OF THE AMERICAN ACADEMY. 



Similarly we find ^ = v. (It is to be observed that for the angle 

 between two lines, the one is always taken, which does not contain 

 the point F. The other angle is the sum of two infinite angles). 

 The angle between the lines, 



uix + V\y = 1, 

 uix + v 2 y = 1, 

 is 



P = U1V2 — u 2 vi. 



These formulas for distance and angle show the directed character of 

 both. For the angles of the triangle X Y Z we have, 



X= 1, Y= 1, Z= -1, 



corresponding to the lengths of the sides opposite, 



ZY =1, XZ = 1, YX = -1. 



This triangle will do for triangle of reference for both distance and 

 angle. The angle coordinates of a line are the angles which it makes 



Z(l,-1) 



f 



r(i,o) 



Figure 3. 



with YZ and XZ. If the equation of a line be written in the form 

 above it will be perfectly dual and the condition that the point (x, y) 

 be on the line (u, v) is precisely, 



ux + vy = 1. 



