336 Mr. R. F. Muirhead on 



C Qo' Qi' Q2' have a common point C. Hence P,/ Qo', P/ Q/ 

 and P2' Q2' are concurrent in a point which we shall call L'. 



Thus all old lines through L have their new lines through 

 L'. The point L' thus uniquely determined when L is given 

 we define to be the neiu point corresponding to the old 

 point L. 



It is clear that (subject to the exception previously men- 

 tioned) every old line in the plane has a new line, and every 

 old point has a new point, and that all the old points in an 

 old line have their new points in the new line corresponding" 

 to it. 



Take now the following construction : — 

 Let L and ^ be points collinear with C. 

 Draw L Pj Qi to meet C A in P^ C B in Qi. 

 „ ZP.Q, ,, CAinP,; 

 „ LP2Q2 „ CBinQ,. 

 „ / Pi ^1 „ C B in q^. 



Let L2 P2 Q2 and ^Pj q^ intersect in 0. Thus C ^P. Q, P. 

 is a complete quadrilateral. Hence L P2 Qo and 10 ^^q^ 

 are harmonic ranges. 



Now draw P, a P/, P^^P,', Q^ /3 Q^', Q,0Q/,qi^qi' to 

 meet C A' in P/, P^' and C B' in Q,', Q/, g/. 



Let Pi' Q/ and P/ Q2' intersect in L\ P2' Q/ and P/ 6^/ in 

 I', and L/ P/ Q2^ and T P/ q,' in 0^ 



Now 0^{CV Q/ Q2I =^ KV Q/ Q2I =/SK.Vi Qi Q.^ 



= L{Cq,Q,Q,}=L{lOV,q,} = -l. 



Thus C^i^ Q/ Q2^ is a harmonic range. 



Hence by the converse of the quadrilateral property, 

 L^ r C are collinear *. 



Thus we have proved that for the old points in C L, any 

 line through C\ the new points all lie in a certain line C L^ 

 This construction fails when C L coincides with C A or C B, 

 but it is obvious, by the former construction, that the new 

 point for P, any point in C A, is P^ in C' A^, where P a P^ 

 are collinear. 



We can now complete the definition of neiv lines in the 

 cases previousl}^ excepted^ by laying down the principle that 



* To prove tliis formally. Let C P/ F^' cut I' Q,' in p.,'. Then 

 and Qa'iPi'Ps'Ps' G}=1'{^,' '^2 P2' C}^l'\q,'Q,' Q,' C} = -1 , 



.-. Q;{p;iVCP/}=-i. 



Tkiis the harmonic pencils P/ (L' 0' P.' Q,') and Qo' (P^'^o' C P/) 

 have a common ray P/ Q^'. Hence the intersections of the other cor- 

 responding rays are collinear, i. e. L', l', and C are collinear. 



