206 PROCEEDINGS OF THE AMERICAN ACADEMY. 



line / must be left invariant and X\ — x 2 must be transformed into 

 x\ — x\. Therefore the collineations which do this are, 



y x = a 2 x + b 2 y + c 2 . 



These transformations, as we shall see multiply angle by a constant, 

 therefore there is a four parameter group of transformations which 

 leave distance invariant and multiply angle by a constant. The five 

 parameter group of collineations, 



*> = to + J* 



y 1 = a 2 x + b 2 y + c 2 , 



is a magnification for both distance and angle. The ratio of magnifi- 

 cation for distance is at once seen to be 61. Since distance and angle 

 are dual conceptions, there is a four parameter group of collineations 

 which leave angle invariant and multiply distance by a constant. 

 We could have taken the equations in angle coordinates and then the 

 transformations leaving angle invariant would have had a form exactly 

 similar to (5). 



Similarity transformations do not exist in this plane in the same 

 sense that they exist in the euclidean plane for if the sides of a triangle 

 are left invariant the angles are not necessarily left invariant. If we 

 apply the transformation (5) to the two lines, 



uix + viy = 0, 

 u 2 x + v 2 y = 0, 



the angle between the transformed lines becomes, 



1 fu\ u 2 y 

 h \vi v 2t 



Hence by these transformations angle is multiplied by 7-. The 



02 



collineations which preserve both distance and angle are the trans- 

 formations of the three parameter group, 



x 1 = x + Ci, 



y l = a 2 x + y + c 2 . 



If we apply the transformation (6) to the above two lines, the angle 

 between the transformed lines becomes, 



