154 



Proceedings of the Uoyal Irish Academy. 



or substituting for P,, P\, Qi, Q\, &c. iu terms of Fi, F\, &c., we get 



r' = - ^ {%KK' + 8 (6^,6^3 + F,F, + H,H,) - 4 {G\a\ + F\F\ + H\H\) 



- 8 (PiP'j + 6^2^'o + H,H\) + (2X+ ^0(^' - 4X) ), 

 ^^ 



By considering the nature of the result of eliminating x, y, z between the 

 three annihilating conies 



Fiyz + Gizx + Hxxy - Kx^, F^yz + &c., F^yz + &c., 

 whose Jacobian is 



PV + G^V + ^V + (2(^3 - <5^'i) ft?'^ + &c., 



we see that if ^i be the invariant of the fourth degree in the coefficients 



(1) 64^V^' = 



F,, G„ H„ -K, 0, 



F„ G„ H„ 0, -K, 



^3, Gz, H„ 0, 0, -K 



K'-4.K, 2 {2H, - H\), 2 (26^3 - G\), SF\, 2F, - F\, 2F, - F' 



2 {2H, - H\), K' - AK, 2 (2^3 - F\), 2G, - G\, ^G\, 2G, - G', 



2 {2G, - G\), 2 {2F, - F\), K' - 4Z; 2B:^ - H\, 2H, - H\, ZH\ 



Again, as the three conies 



F\yz + G\zx + H\xy + JS'V, F\yz + &c., F\yz + &c., 



are the polar conies of the points 



Xi : yi : Zy ; x., : y^ : Zo] x^ : y^ : z^, 



or he - hzC^ : h-iCi - h^c : hyC^ - Ici ; Mt^i - ca^ : ac - CyCt^ : a^Cy - ac^ ; 



a^b-i - aJj : aj)i - ctbz : ah - a^by ; 



and the components of these various ratios are all minors of the determinant 



a bj Ci 



«2 b d 



ciz bs c 

 and further, since the Jacobian of these three conies is the Hessian multiplied 



by 



^1 2/1 si 



2/2 

 2/3 



