MOORE. — MINIMUM GEOMETRY. 215 



The area of any closed curve is its perimeter. The dual of this from 

 formula (7), is 



Angle area = jKds. 

 For curves of constant curvature then, 



Angle Area = Line area X K. 

 The area of a pseudo circle which does not cut / that is of the curve 



A.r 2 + 2Bxy + Cif = 1, where AC - B 2 > 



is 



'y 



Line area = / (xdy — ydx) = / x 2 d (-) = / 



d 



A+»g) + o(gr 



2tt, 



- v — where A = AC - B 2 . 



14 2 . 



but p = ^ = — > hence p = ,—• The line area of the pseudo circle is 



7r Vp. If the pseudo circle cuts the line/ the area is infinite. 



Collineations of the plane. The general collineation of the plane, 

 which leave F and /invariant is, 



x 1 = ax + by, 

 y l = ex + dy. 



If this transformation be applied to Xi y^ — x^ y\ we have 



x\y\ — x\y\ = M (xiy 2 — x 2 yi), where M = ad — be. 



This is then a magnification for distance. The angle between the 

 lines, 



uix + viy = 1, 



u 2 x + v 2 y = 1, 

 becomes 



u\v\ — u\v\ = ^rj (mi» 2 — UoV\). 



1YI 



The transformation then divides angle. If it is applied to a pseudo 

 circle we have for curvature, 



