22 CONCRETE REPRESENTATIONS OF 



on the circle, i.e. the cross-ratio of the pencil 0(PQ, XY) 

 where is any point on the circle. 



17. The expression for the line-element can now be found 

 by making PQ infinitesimal. 



We have, by Ptolemy's Theorem, 



PX.QY=PQ.XY+PY.QX. 



Hence 



. PQ. 



Let OP (Fig. 1) cut the circle PXY again in E and the fixed 

 circle in A, B. Then R is a fixed point so that PR is constant. 



Also 



RX RY r> j , 

 == 7r = 7 =a nxed ratio =e, 



PX~PY 

 and PR. XY=PX . RY+PY . RX=2e.PX . PY. 





Therefore D 

 r 



position of P alone. 



r> v= ins an( l 

 . r I JrH 



therefore a function of the 



FIG. 1 



To find its value we may take any orthogonal circle through 

 P, say the straight line P.R. 



Then XY AB 



PX.PY PA.PB~k+x*+y*' 



Hence 



