20 CONCRETE REPRESENTATIONS OF 

 serves angles and leaves the form of this equation unaltered 



az'+P - 



is 1 



yZ '+8 ' 



This is a conformal transformation since any transformation 

 between two complex variables has this property. 



To find the relations between the coefficients in order that 

 the fundamental circle may be unchanged, let its equation be 



x*+y 2 +k=Q or zz+fc=0. 



This becomes (az+j3)(^+)8)+&(yz+S)(yz+S)=0. 

 Hence aj8+fcy8=0 



and fc( 



aa 

 therefore aa=88, 



so that ^=i=-&=-i=l. 



8 * ft ky 



We have a=xS and a=xS, and also a=-r-8, 



A 



therefore 1*1 = ! 



The general transformation is therefore 2 



, where |X| = 1. 



By any such homographic transformation the cross-ratio 

 of four numbers remains unchanged, i.e. 

 (zfr, z 3 z 4 )=(z' 1 z' 2 , zV 4 ). 



1 The only other type of transformation possible is 



_ 



ff __ _ - ^ ^ _ __ --- ^ j 



yz' + d y/ + fl 

 but this only differs from the former by a reflexion in the axis of x, 2=2*, z=i*. 



2 When, as is often taken to be the case, the fundamental circle is the z-axia, the 

 conditions are simply that the coefficienta a, (3, y, 8 be all real numbers. 



