NON-EUCLIDEAN GEOMETRY 21 



To find the condition that this cross-ratio may be real, let V 

 be the amplitude, and r y the modulus of z f z jf then 



(ZZ ZZ ->- r ^2* *'-' M + '"-' ) 



IZjZjj, z 3 z 4 j - e 



r u r za 

 Hence we must have 



and the four points z x , z 2 , z 3 , z 4 are concyclic. 



16. Now to find the function of two points which is in- 

 variant during a motion ; the two points determine uniquely 

 an orthogonal circle, and if the transformation leaves this 

 circle unaltered it leaves unaltered the two points where it 

 cuts the fixed circle. Hence if these points are x, y, the cross- 

 ratio (zjZg, xy) for all points on this circle depends only on 

 Zj and z a . If the distance function is (PQ)=/)(z 1 22> x y)\ or > 

 as we may write it, /(z l5 z a ), then for three points P, Q, R, 

 (PQ)+(QR)=(PR), 01 



f(*v z 2 )+f(z 2 , z 3 )=f(z lt z 3 ). 



This is a functional equation by which the form of the function 

 is determined. Differentiating with respect to Zj, which may 

 for the moment be regarded simply as a parameter, we have 



f, (9 . QY d (PX\_ ft(9 . RY d (PX\ 



* (2 i' Za) QX ' dz- 1 (pYr / (z ZS) -RX' ^(PY) ' 



Hence 



f'fr, z 2 ) QX RY/PX RY\ ( PX (?r\_(z 1 z 3> xy) 

 f'(*i> **)~QY RX~\PY RXt \PY QX)-(zjZ 2 , xy) ' 



and (z^, xy)f'\(z l z z , xy)\=const.=fj.. 



Integrating, we have 



/(z 1 ,z a )=/tlog(z 1 z 2 , xy)+C. 



The constant of integration, G, is determined =0 by substitut- 

 ing in the original equation. Hence 



, XY), 

 (PQ, X Y) being the cross-ratio of the four points P, Q, X, Y 



