36 Transactions. 



The remaining set of four lines may be written 



l\ = Xja: + Yi?/ + Zi^ = 

 jj., = Xjo; + Yo'</ + Zo^ = 



_P3 = Xo« + Yi?/ + Zo^: = 

 _?;, = Xo.x + YqT/ + Zi^ = 0. 

 Let now the equations of the following conies 



F' - Ci^ = 0, FqI - c./ = 0, Fi - c^ = 0, F' - c;- = o 

 he formed. They are found to be respectively — 



Pi = Xi?/5 + Y^zx + Z,xy = 

 Pa ^ Xi?/^; + YqZx + ZgXi/ = 

 P;^ = XqI/z + Yi2;a7 + Z^xt/ = o 

 P4 ^ Xgt/^; + Yo^^a; + Zixy = 



— that is to say, these conies are the isogonal transformations of the 

 four lines p^ jh, Ih, Ih- 



Hence the sixteen points found on F may be regarded as the isogonal 

 conjugates of the sixteen points in which each of the lines Ci, c.,, c^, c^ is 

 met respectively by the isogonal transformations of the pairs ^1, jh ! k, P-z ', 



''St Pi > fit Pi- 



The isogonal conjugates of the four points in which Ci is met by the 

 conies Ti and Pj will lie on the conic which is the isogonal transforma- 

 tion of Cj, hence the sixteen points on F lie four by four on the four conies 



V liliyz + Viihm.iZX + S^iiii^xy = 0. 

 It may be easily shown that 



F ^ tipi + 4 V'AjAg ( V'/iia yz + Vmjm^ zx + Vnin.^ xy) 

 ^ hPi + 4 v^AiAa ( A//1/2 yz — VmiVi^, zx — Vn^n^ xy) 

 ^ t-ip^, + 4 ^/AiA.2 (— \//i/.2 yz + V 7)1(111=1 z-"^ — Vnitin, xy) 

 ^ tiPi + 4 V A^Aa (— \/ 1^1.2 yz — VviiM^ zx + V7?i«2 ^y) 

 where Aj and A., are the discriminants of Sj and S.j. 

 It also follows that 



F = i {fiPi + Lp, + t,p, + ^,:p,) = i 5 (^j;) . 

 The conic F^ passes through the intersection of the conies 



lil^x^ + m{m.,y~ + n-^v.^z'^ = 

 (miWa + ?ii2^ii) ?/^ + {ihl^ -\- njj) zx + (/i?^i.2 + /.2'7ii) «?/ = 0. 

 The former of these is the fourteen-point conic of the system of lines 



y IJ-i X + Vm-^m.^ y + Vn-^iu z = 0, 

 while the latter is the isogonal transformation of the line 



L ^^ {^)ii{ii., + niM^ X + (?ii/2 + «oZi) y + (/i"i.2 + 'a^"!) 5r = 0. 

 It may be at once shown that 



F = L'- - AjA^So 

 where S,, is the conic x'^ y- 



iyl^ m{in^ n{ii.i 



= 0. 



