24 CONCRETE REPRESENTATIONS OF 



B, 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 \L must be infinite and the sphere becomes a 

 plane. 



Let 2.^/k. 



Then 



x z +y z 

 By the transformation r'=s 6' =6 



this becomes ds z =dr' z +r' z de' z =dx' z +dy' z . 

 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+pz fc=0, 



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



x'+g) z +(y'+f) z 



y+f (x+g) z +(y+f) z 



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

 Geometric, Vorlesungen. 



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

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

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



