NON-EUCLIDEAN GEOMETRY 17 



figures for angular and linear metric. He remarks that in 

 ordinary Euclidean geometry the conies which play the role 

 of absolute are also distinct, namely the one is a double line 

 and the other is a point-pair. We have seen above, how- 

 ever, that these are just different aspects of the same 

 degenerate conic. The double line is the locus, or assemblage 

 of point-elements, the point-pair or pair of imaginary pencils 

 is the envelope, or assemblage of line-elements. 



CONFORM REPRESENTATION BY CIRCLES 



12. We shall next consider a very useful representation 

 which has important applications in the theory of functions, 

 that in which straight lines are represented by circles. 1 Since 

 a circle requires three conditions to determine it, one condition 

 must be given. Hence if the circle 



represents a straight line, the constants, g, /, c, must be 

 connected by a linear relation, which may be written 



2gg'+2ff'=c+c'. 



But this relation expresses that the circle cuts orthogonally 

 the fixed circle 



x 2 +y 2 +2g'x+2f'y+c'=0. 



Hence the circles which represent the straight lines of a geometry 

 form a linear system cutting a fixed circle orthogonally. 



Similarly in three dimensions if planes are represented by 

 spheres they will cut a fixed sphere orthogonally. 



13. Thus we find at once that there are three forms of 

 geometry, according as the fundamental circle is real, vanish- 

 ing, or imaginary. 



A difficulty, however, presents itself. Two orthogonal 



1 An interesting account of this representation, from the point of view of elementary 

 geometry, is given by H. S. Carslaw, Proc. Edinburgh Math. Soc., 28 (1910). The 

 following account, which was suggested by Professor Carslaw's paper, appeared in the 

 same volume. 



C 



