Hogg. — Ou the Harmonic Conic of Two Given Conies. 35 



The isogonal transformation of this — on which lie the eight points of 

 contact of the common tangents of Si and S.^ — is the conic 



{m-^n^ — uuni)~x~ + {nj.;^ — nj^)'^ jf + {liVi^ — L^m^ z^ 



+ 2 [(7?i/o+ wii) {linio + 4'?^]) y^ + (4^2 + ^Wi) {min., + niM^ zx 



-\-hn{ii.2 + m.^i^ {n^l.^ + yijt-^) xy] = o, 



which is the F conic of the two circumconics Si and S^. 



The form of the above quartic also shows that on the conic F He 

 eight points which are the isogonal conjugates of the points in which the 

 four lines 



VI1I2 X ± v^??JiW.2 y + ^n-{n^ z — 

 intersect the conic 



F^ ^ l-^L x~ + m^in., y- + ?ii"2 ^~ — ('«i"2 + m.^h) yz 



— ("i4 + ''-'^1) ■«'i' — (^iWa + Itni^) xy = 0. 



If ( ^, — , — ) and ( — , — , - ) be respectively the co-ordinates of two 

 V /j m^ nj \ U m, nj 



points Oi and 0.^, then the conic F' passes through the six points in which 

 the lines joining the vertices of the triangle of reference to 0^ and 0.2 meet 

 the opposite sides of that triangle ; but tlaese six points are the points in 

 which the conies 



53 = \' i-^x + Vm-^y + \/niZ = 



54 = s/lc^x -\- Vm.^j + \^n^z — 



touch the sides of the triangle of reference. 



Hence the conic F^ is the F conic of S;, and S4. 



To obtain the equations of the four common tangents of Sj and S.j let 

 us form the four conies 



Ti ^ LiLo ~ ^1^^ = 

 T2 ^ L1L2 ~ c.j'^ = 

 Tg ^ L1L.2 — C;/ = 



T4 ^ L1L.2 — Cf = 0, 



and write down their isogonal transformations, which will be the equa- 

 tions of the common tangents. 



Let Xo = ( ^m,7?.2 — Vm.^n^)'^, Xi = ( s^m{)h^ + '/m.My' 



Yo = {\/^. - W^,f, Yi = (sATJ, + V^,f 



Zo = i^T^i. - V'km.f, Zi = ( v'/^2 + VT^,)\ 



then the equations of the common tangents of the two conies Si and S2 

 are . 



^1 ^ Xoaj + Y^y -\- ZqZ = 



U = X,,x + Y1.7/ + Zi^ = 



t.^ = XjA- + Yo?/ -t- Zi^ = 



ti = Xi.c + Yi?/ + Zi^ = 0. 



These tangents constitute four of the group of eight lines represented by 

 the equation 



(\/mi«2 + \^m^niYx + ( V'n-J,^ + Vn^liYy + ( Vl^m^ ± \ l^nii)'^ z = 0. 



