NON-EUCLIDEAN GEOMETRY 25 



or, using complex numbers, 



_(pp+k)(z'+p) 

 (z'+p)(z'+p) ' 



Whence z=*b^. 



z'+p 



A second inversion in the circle 



zz+qz+qz k=0 



gives 2 



This will not hold when the circle of inversion is a straight 

 line, 6=<f), Here inversion becomes reflexion and the formula is 



z==z ' e W(.-*> = 3/ e * e 



This combined with an inversion gives 



~z"+p 

 Now these transformations are always of the general form 



z^fJlM, w here X =1. 



/3z'+ a. 



In fact, this transformation is always of one or other of the 

 two forms z=z'e"* 



(when /S=0) or *= 



(by dividing above and below by /3). 



Hence the general displacement of a plane figure is equivalent 

 to a pair of inversions in two orthogonal circles. 



21. In the general transformation there are always two 

 points which are unaltered, for if 2' =2 we have the quadratic 

 equation 



/32 2 + (a- Xa)a+ &X/3=0. 



These form the centre of rotation, and the circles with these 

 points as limiting points are the paths of the moving points. 



There are three kinds of motions according as the roots of 

 this quadratic are real, equal, or imaginary, or according as 



