30 CONCRETE REPRESENTATIONS OF 

 We have then 



(PQ)=H\og (PQ, XY)=n\og (pf 



But 



PX sinXOP PY_smYOP 

 OP ~ sin OXP' OP sin YP 



, PZ QY_BmXOP sin YOQ sinQFP emOXQ 

 PY ' QX~ sin YOP ' sin XOQ ' sin OXP ' sin YQ 



i.e. (PQ, XY)=(P'Q r , X'Y')(QP, XY) 



therefore (PQ, XY)*=(P'Q f , X'Y'). 



Hence we have the true distance (PQ) given by 



>'Q', X'Y')=(P'Q'). 



FIG. 4 



Then the line-element can be obtained in a manner similar 

 to that of 17. 



We find as before that (PQ, XY)=1+ 



r JL . 



but in this case PX . PY=x*+y*+k' 



and X 7 2 = -4^'(dx z + dy z )+ (ydx-xdy)*\l(dx z 



so that ^._^.y(^+%V(yfe-^) s . 



Comparing this with the expression in 23 we find 



