252 Proceedings of the Royal Irish Academy. 



6. A useful property of this conic is the following : — 



" Tangents to the quartic, the corresponding points to which on the conic 

 are collinear with the pole of the bitangent, meet again on the conic." 



Fig. 2. 



Let P be 2A- - X + 2, 5X- + oX + 2, 2X- + 5X + 5, 

 and Q 2m' - ^f + 2, 5^^ + 5/i + 2, 2^- + Ofx + o, 



P, Q, and are collinear if 



X + ^ + 2 (1 + X;u] = ; - 

 the tangents are o: {1 + Xf - tj - zk^ = 0, 



and X {1 + jjif - y - zp,^ = 0, which meet in 



M^ - x^ M^ (1 + x)^ - x^i + ^j^ (1 + f^Y - (1 + ^f. 



Dividing by /u - X, and putting X/x = 0, and X + /i = - 2 (1 + 0), 

 the coordinates of -K are found to be 



:r. = W +70 + 4, 

 y = 0^ + + 4, 

 3 = 49^ + + 1. 



8=yj - ?/ - « = 27 (0 + 1)% 

 82/ - 3 - a; = 27, 

 8^ - a - 2/ = 270^ 



^W^^^z + v/Sz/ - 3 - A' + ysT^lT^ = 0, 



which proves the property stated. 



