MOORE. — MINIMUM GEOMETRY. 207 



Then the subgroup of the similarity group which preserves angle are 

 those for which b\ = b^. 



The transformations which preserve distance and reverse angle are 



x 1 = x + a, 



y 1 = a 2 x — y -f c 2 . 

 Those which preserve angle and reverse distance are, 



x 1 = — x +' Cjj 

 ij 1 = a 2 x + y + c 2 . 



Those which reverse both distance and angle are, 



X 1 = — X + Ci, 



y l = a-ix — y + c%. 



In this geometry instead of having the ordinary similarity trans- 

 formations we have two kinds, one which multiplies distance leaving 

 angle invariant and one which multiplies angle leaving distance 

 invariant. There is a three parameter group of motions leaving both 

 invariant. We have also the "umlegung" which reverses distance 

 and the dual which reverses angle. 



Since the distance between two points is the same as the distance 

 between any other two points, one on each line joining the two given 

 points to the point F, the question arises, what are the most general 

 analytic transformation of the plane into itself which will preserve 

 distance. In the first place / and F must be invariant. Let the 

 transformation be, 



x = f(x\ if), 

 y = g(x\ y x ). 

 If it preserves distance, 



ds = dx = ~ x dx 1 + -^ dy 1 = dx 1 . 

 Hence, 



dy 1 U ' 



dx 1 



Therefore f(x\ y l ) is independent of y 1 and must have z 1 with coeffi- 

 cient unity, that is, 



/ = a; 1 -f- a, 



and ^(a: 1 , y l ) is any arbitrary function. We will take g so that in a 



