30 CONCRETE REPRESENTATIONS OF 

 We have then 



But 



PX_amXOP PY_amYOP 

 OP ~ sin OXP' OP ~ sin YP 



and ?*. QY_smXOP sin YOQ smOYP ainOXQ 

 PY'QX sin YOP * sin XOQ ' sin OXP ' sin OYQ 

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



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



Hence we have the true distance (P$) given by 



>'<?', X'Y^P'W). 



FIG. 4 



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

 to that of 17. 



We find as before that (PQ, 



XY 



but in this case 

 and 



PY.QX' 



PX . P Y=x 2 + y z + k' 

 dx*+dy*)+ (ydx-xdy) 



so that cfo=-V . V(dx*+dy*)+(ydx-xdyY 



(x*+y*+k') z 



Comparing this with the expression in 23 we find 



