216 PROCEEDINGS OF THE AMERICAN ACADEMY. 



If the collineation is a motion 



ad — be = 1. 



These transformations are the area preserving eollineations of the 

 plane in euclidean geometry. 

 If 



ad = be = —1, 



the transformation is an "umlegung," that is a transformation which 

 reverses the sign of distance. For both motion and umlegung the 

 curvature of a curve is an absolute invariant. 



In this geometry rotation about a point A is equivalent to a trans- 

 lation along the series of lines passing through the point where the 

 line AF meets /. If the rotation is to be about a finite point that 

 point must remain invariant. But as / and F are left invariant no 

 other finite point can be and hence there are no rotations except 

 identity. 



Parallel curves. Parallel curves are defined in ordinary geometry 

 as the envelope of circles of constant radius having their centers on a 

 fixed curve. In this geometry then a parallel to a given curve will be 

 the envelope of the lines which are the loci of points at a constant 

 distance from the points of the curve. The lines corresponding to the 

 same point of the given curve all pass through the same point of/ and 

 hence are parallel. That is in a set of parallels to a curve the tangents 

 at corresponding points are parallel. The parallels to the curve 



/(*, y) = o, 



will have for equation the eliminant of 



f(x h Vi) — 0, xyi — yxi = a. 



This however is exactly the process for finding the tangential equation 

 of the curve f(x, y) = O in euclidean geometry. Hence the parallels 

 to a given curve have the order equal to the class of the original curve 

 and vica versa. 



The parallels to the pseudo circle is the eliminant of Xi, yi between, 



Kx\+2Bx iyi +Cy\ = 1, 



xy\ — yxi = a, 

 which is 



Ax 2 + 2Bxy + Cy = a 2 (AC - B 2 ). 



This is again a pseudo circle having the same pair of tangents from F. 



