Hogg. — On the Harmonic Conic of Tivo Given Gomes. 37 



It is seen on inspection that 



ti+Vi = ^-2 + Ih = k-\- V-, = ii+Vi = 2L, 



and it is easily proved that 



^{t') = 2(/) = 4 [P + 2AAA] 

 t,t,y, = F' - 4AiA,SiS, 



= PilhVsPi — l^Sxyzh. 



It may also be noticed that the lines Li and L.^ are conjugate with 

 respect to all comes inscribed in the standard quadrilateral 



'^likx + Vm^m^y ± ^n-^n.^z = o. 

 Since t^ touches each of the conies Si and S.2 we have 



V^^iXq + V»hYo + x/n,Zo = 



with similar equations of condition for t.^, t^, and ti : hence the co-ordinates 

 of the four intersections of the conies S^ and P4 are 



(XoYoZo), (XoYiZ,), (X^YoZO, (XiY.Zo). 



The common tangents ^1, ^2, ^3, t^, are the axes of homology of the isogonal 

 conjugates of the intersections of S3 and S4. 



Let the equations of two rectangular hyperbolas S' S" referred to their 

 common self-conjugate triangle be 



S' ^E I'x'^ -f- m'y^ + n'z"^ = 

 S" = V'x' + m\f + h",s^ = 0. 

 The equations of their common tangents are 



X Vl'i:'{m'n"-vfn') ± y Vm'm"{ii'l!' - n"l') + z x/n'n"{l'm" ~L"m') = 0. 



Let one of these tangents touch S' and S" at ^'{x'y'z') and '^"{x"y"z") 

 respectively, then 



I'x' - k' \/l'L"{m'n" - m"n') 



l"x" = k" Vl'l"{m'?i" - m"n'), 

 and therefore x'x" = k'k" {m'n" — m"n') ; 



similarly, y'y" = k'k" {ji'l" - n"L') 



z'z" = k'k" (I'no" - i",u'). 

 Also, since V -f m' + n' = 0, I" + vi" -f 71" = 0, 



we have m'n" - m"ii' = n'l" - n"l' ^ I'm" - V'm' , 



and therefore x'x" = y'y" = z'z" 

 — that is to say, P' and P" are isogonal conjugates. 



Hence the points in which the harmonic conic of S' and S" cuts those 

 conies are isogonal conjugates in pairs with respect to the self-conjugate 

 triangle of the hyperbolas. 



