208 PKOCEEDINGS OF THE AMERICAN ACADEMY. 



given region the transformation will be (1,1), that is in the angle 

 between two lines through F a branch of a curve will go into a single 

 branch with ends on the lines through F into which the boundaries of 

 the angle transforms. 



By such a transformation the curves, 



u(x + a) + vg(x, y) = w, 



can be transformed into straight lines. So far then as length is con- 

 cerned these curves could be taken for straight lines and angle defined 

 accordingly. The whole geometry would then be the same. In this 

 character this geometry differs very much from ordinary geometry. 



Case II. 



The second kind of geometry discussed in A. P. G. had for funda- 

 mental system a line/ and a point F not on it. In this geometry angle 

 and distance are so related that any three parts of a triangle determine 

 it. A transformation then which preserves distance must also pre- 

 serve angle and there is no separation of these groups as in Case I. 



The general definitions of distance are the same here as before. 

 There is a metric example here, however, which will serve to make it 

 more concrete. Let the line / be the line at infinity. The distance 

 here defined then becomes equal to the area of the rectangles of which 

 BF is one diagonal and A one vertex. The properties of distance can 

 then be verified in this metrical case. 



Distance has the following properties : 



AB = AC if BC meets AF on /. That is the locus of points equi- 

 distant from a given point is a straight line. This line can be any line 

 of the plane, however, instead of a line through F as in Case I. 



The distance from a point A, not on/, to a point P of /is infinite if F 

 does not lie on AP. If F does lie on AP the distance is indeterminate. 



The distance from a point A to a point on AF, not on /, is zero. 



Distance is a- directed quantity, that is if a positive direction is 

 assigned for one line of the plane a positive direction is assigned for 

 each line of the plane. A construction was given for measuring on 

 any line beginning at an assigned point, a distance equal to a given 

 distance. 



From these most all properties of distance can be derived. 



Angle was defined as the dual of distance and with reference to the 

 same fundamental system. Angle then has the following properties: 



