26 CONCRETE REPRESENTATIONS OF 



the centre of rotation is real, upon the fundamental circle, or 

 imaginary. The first case is similar to ordinary rotation. 

 In the second the paths are all circles touching the fundamental 

 circle. In the third the paths all cut the fundamental circle ; 

 one of these paths is an orthogonal circle, the other paths are 

 the equidistant curves ; the motion is a translation along a 

 fixed line. 



22. It would appear that the representation by circles 

 is a sort of generalisation of the Cayley-Klein representation, 

 since a straight line is a circle whose centre is at infinity. 

 When the circles degenerate in this way, however, the fixed 

 circle becomes the line infinity, and the geometry degenerates 

 to Euclidean. 



It is of interest to deduce the general Cayley-Klein repre- 

 sentation from the circular one, but this cannot be done by a 

 conformal transformation. 



Abandoning the conformal representation, the transforma- 

 tion which changes circles orthogonal to x z +y z +k=Q into 

 straight lines is 



k 

 The points (r, 6), ( , 6) are both represented by the same 



point, so that this transformation gives a (1, 1) correspondence 

 between the pairs of real points which are inverse with respect 

 to the circle x z +y z +k=0 and the points which lie within the 



2 2 



circle x 2 +y 2 + =0, since for real values of r, r' z < -. Every 



A* /.' 



point upon the circle r 2 +fc=0 is thus to be considered double. 

 To a pair of imaginary points corresponds a point outside the 

 new fixed circle. Any circle, not orthogonal, is transformed 

 into a conic having contact with the circle kr z +p z =0 at the 

 two points which correspond to the intersections of the circle 

 with the fixed circle r 2 +k=0. 



In fact, any curve in the r'-plane which cuts the fixed circle 



