34 Transactions. 



Art. VII. — On the Rarmonic Conic of Two Given Conies. 



By Evelyn G. Hogg, M.x\., Christ's College, Christchurch. 

 [Read before the Philosophical Institute of Canterbury, 4th Jidy, 1908.] 



1. Let two conies be taken — say, 



Si ^ ^hyz + 2mi2a; + 2niXy = o 

 8.2 ^ '2l,yz + 2m.2zx + %h^y = o. 

 These conies are respectively the isogonal transformations of the lines 



Lj ^ l-i^x + m{y + n-^z = o 

 L2 ^ I2X + 7n.2y + '>h^ = 0. 

 The tangent to Sj at any point P will transform isogonally into a conic 

 touching Lj at the point which is the isogonal conjugate of P. 



Let a common tangent to Si and S2 touch these conies in Pi and P.^ 

 respectively, and let its equation be 



Xx -\- fxy -\- vz = 0. 

 This transforms isogonally into the conic 



\yz + fxzx + vxy = 0, 

 which touches Li and L.^ in the points which are the isogonal conjugates 

 of Pi and P2 respectively. Hence, since this conic has double contact 

 with Li and L.2, its equation is of the form 



L1L.2 - (px + qy + rzY = 0. 

 Comparing this with the form 



Xyz + fjizx + vxy = 

 we have p- = l^l^, q- = niim^, r"- = n^n^, and this shows that the four 

 conies which are the isogonal transformations of the common tangents of 

 Si and So touch Lj and L^ along the four Imes 



y'likx ± \' yn^nioy ± s' ihu^z = 0. 

 These four lines determine on Li and Lj eight points which are the 

 isogonal conjugates of the points of contact of the four common tangents 

 of Si and S2. The four points lying on Lj and the four on L, may be 

 regarded as the intersections of Lj and Lg with a quartic curve. 

 The equation of this quartic will be of the form 



LjLo [ax"" + bif^ + cz- + Ifyz + 2gzx + 2hxy) + CiC^c^Ci = 0, 

 where c^^ v l^L^x + V-,Nj^ni.,y -}- V 11^11.2 z 



C.2 ^ Vy^x — V7n{m.2y — Vn^n^z 

 Cs ^ — VI1I2X + Vviiiiuy — -/nii^z 

 C4 ^ — %'' 1-J..2X - VniiVi.yy + \ niu^z. 

 If we now let a = /1/2, h = miyuo, c = n{ii.2 



'2f = -{m{iu + ?//oUi), 2(j =^ -{nil, + "0/1), 2// = -{lyvu^-liiih) 

 the quartic reduces to 



{m{ii, - w.^ii^fy'^z^ + {nj, - nJifz-x'^ + (hni.2 - kni{)xY- 

 + 2 [{nJ, + ^2^1) {h'>^h + y'h) xh/z + {liVi., + Imi) {m^n^ + nir/ii^ y'^zx 

 + {m{i\, + m,n^ (n,], + nj^) zlry] = 0. 



