24 CONCRETE REPRESENTATIONS OF 



R, and the representation is by taking the stereographic 

 projection. 



When k is negative the sphere has an imaginary radius, but 

 such an imaginary sphere can be conformly represented (by 

 an imaginary transformation) upon a real surface of constant 

 negative curvature, such as the surface of revolution of the 

 tractrix about its asymptote (the pseudosphere). 1 



When k is zero /x must be infinite and the sphere becomes a 

 plane. 



Let 2fjL+/k=p. 



Then ds= -^ 9 */dx*+~dy*= 

 22 



By the transformation r'=^, Q'=Q 



this becomes ds z =dr' 2 +r'*d0' 2 =dx' 2 +dy'*. 

 Hence when k is zero the geometry is the same as that upon a 

 plane, i.e. Euclidean geometry, and the representation is by 

 inversion, or reciprocal radii. 



20. Let us now return to the consideration of motions and 

 investigate the nature of the general displacement of a rigid 

 plane figure. 2 In ordinary space the general displacement of 

 a rigid plane figure is equivalent to a rotation about a definite 

 point, and this again is equivalent to two successive reflexions 

 in two straight lines through the point. Now the operation 

 which corresponds to reflexion in a straight line is inversion 

 in an orthogonal circle. The formulae for inversion in the 

 circle 



zz+pz+pzk=Q, 

 which is any circle cutting zz+/c=0 orthogonally, are 



1 Cf. Darboux, Theorie des surfaces, vii., chap. xi. Also Klein, Nichteuklidische 

 Geometric, Vorlesungen. 



2 Cf. Weber u. Wellstein, Encyklopadie der Elementar-Mathematik (2. Aufl. Leipzig, 

 1907), Bd. 2, Abschn. 2. Also, Klein u. Fricke, Vorlesungen iiber die Theorie der auto- 

 morphen Functionen (Leipzig, 1897), Bd. 1. 



