8 CONCRETE REPRESENTATIONS OF 



identity u= const. In trilinear coordinates, for example, if a, 6, 

 c are the sides of the triangle of reference, A, u=ax-\-by+cz=2&. 

 The circular points appear in line-coordinates as an equation 

 of the second degree, 6><>>'=0, while every ordinary line satisfies 

 the identity G>W'= const. In trilinear coordinates 



-2cos B-2^ cos C 



In rectangular cartesian coordinates, made homogeneous 

 by the introduction of a third variable z, the equation of the 

 line infinity is z=0, while for finite points z=l. The line- 

 coordinates of the line lx+my+nz=Q are I, ra, n, and in general 

 1 2 + m 2 = constant. When the equation is in the ' perpendicular ' 

 form, for example, the constant is unity. But for the line 

 infinity 1=0 and m=0 so that Z 2 +w 2 =0, and this is true also 

 for any line y=ix+b, i.e. for any line passing through one 

 or other of the points of intersection of the line z=0 with the 

 locus x 2 +y z =0. 



Now an equation of the second degree in point-coordinates 

 or in line-coordinates represents a conic. But the equation 

 Z 2 +w 2 =0 represents a degenerate conic consisting of two 

 (imaginary) pencils of lines, since Z 2 +m 2 decomposes into 

 linear factors. Similarly 2=0 as a point-equation, when 

 written z 2 =0, represents a degenerate conic consisting of two 

 coincident straight lines. These conies are just one conic 

 considered from the two different points of view of a locus and 

 of an envelope, for the reciprocal of the equation l 2 +m 2 =cn 2 

 is c(x 2 +y 2 )z 2 . When c=0 the point-equation represents a 

 circle of infinite radius z 2 =0, and the line-equation Z 2 +w 2 =0 

 represents the two pencils of lines passing through the two 

 points through which all circles pass. This degenerate conic 

 is called the Absolute. 



If we now replace the degenerate conic by a proper conic, 

 we get a more general form of geometry, which includes 

 ordinary Euclidean geometry as a special case. It also in- 

 cludes as special cases the geometries of Lobachevsky and 



