340 Mr. R. F. Muirhead on 



The equation oi: PQ is 



/Ai- + 7y + r = (6) 



The equation of* P'Q^ is 



+ z'ul3[a-h)=^0, . (7) 



Now suppose (6) to pass through a fixed point(A'o, ^/o? ^o) 

 so that 



jM'o + ^72/0 + 5-0 = (8) 



Eliminating p from (7) and (8) we have 



-\-x,'Pz,{aoc--l)''y,'h^z,{aa-l) +z,'a:,a^{a-h) = 0. (9) 



This shows that i£ the line P Q passes through a fixed 

 point Xq ?/o ~o ^^^ P^ Q^ passes through another whose co- 

 ordinates ^/, yj y Zq will be got by equating to zero separately 

 the coefficient of g in (9) and the part independent of q. 

 Solving these, we get 



a^o lyo •• ^o=h^{au-l)y^ + aa(h^—\)xo : ci{ij/3—l).v^ 



+ ^iau-l)y^:-Zo(au-l)(h^-l). 

 This we may write in the form 



~o ^fi yo 



Now the condition that a line through C corresponds to 

 itself is 



= — , , or mx' 



+ {7i — k)xy — ly'^ = 0. 



(11) 



This is a qu adratic, giving two values oi x:y which will 

 be real if n—k'^ + hnl is positive, i. e. if 



{/3(aa-~l)-aa(Z^/3-l)}2 + 4«(5^-l)5;8(a«-l)>0. 



Let us now take a new triangle formed by A B and the 

 two douhle-lines as new fundamental triangle, usino- f 77, f 

 as area! coordinates referred to it. In what follows we shall 

 use A, B, C for the three double-points. 



If 



then A^ = 



^=aiX + h^y-\-CiZ "^ 

 7) = a2X + h2y + CiZ > 



5' = a3.27 + % + C32 J 



• 02) 



h 

 h 



cz 



where A" 



o^ bo 



