204 PROCEEDINGS OF THE AMERICAN ACADEMY. 



The coordinates of a line will be the dual of the coordinates of a 

 point and therefore will be the numbers corresponding to the points 

 in which the line cuts/ and OP. The angle between two lines will be the 

 difference between the numbers corresponding to the points in which 

 it cuts /. The number corresponding to the point in which the line, 



ux + vy = w, 

 cuts /is Hence the angle between the lines 



u 2 x 2 + v 2 y2 = w 2 , 

 is 



(2) 9 = ^ ~ ** 



V1V2 



Curvature. Using the ordinary definition of curvature we can now 

 d erive the expression for the curvature, in this geometry, for a curve 



V = fix). 

 The tangent at the point (xi, yi) is 



y - yi = f 0) - xi), 



and the tangent at the near by point (.1*1 + dx, y\ + dy) is, 



y — yi— dy = f (xi + dx) {x - x x - dx). 



The angle between the tangents is then 



/ / (.r 1 + ^)-/'(.r 2 ). 



Hence dividing this by the element of arc we have for the curvature, 



v / (x) + f" (x) dx + f" Or) dx* +••••-/' fo) „.. 

 K== : dx ~ j {X) - 



From this we see at once that the curves of zero curvature are straight 

 lines. The curves of constant curvature are defined by the differen- 

 tial equation, 



*y 1. 



dx* - k 

 That is the curves of constant curvature are 



(4) y = ^a- 2 + ci.r + c 2 . 



This curve is then the analog of the circle in the euclidean plane. It 



