376 Proceedings of the Royal Irish Aeadem?/. 



3. Example. ^Poles of chords of a conic. 



I pass on now to consider the vector expressions for a conic, the 

 the pole of a chord, and the centre of the curve. 



By the first article, the general equation of a conic is 



Oya:^ + 2aixy + a^y'^'' 



and, by the second article, the tangent at x-^y^, is 



_ X {a^^ + aiyi) 4 y {a^x^ + a^y^) 

 X {a^i + a,y,) + y {a^x^ + a.jjy) 



Given both x^yx and Xj^o, the point 



a^xXn + ai {x\y^ + x.y^) 4 aiyiy2 

 a^x^Xi + ay {x,yn + x.y{) + a^y^y^i 



is situated on the tangent at x^yu and on that at x^yz- This point is 

 consequently the pole of the chord joining the two given points on the 

 curve. 



Two points on the curve may be considered as given by the quadratic 

 equation 



h^-" + 2hxxy + hnf- = 0, 



the vectors to these points being determined by substituting the roots 

 of this qiiadratic in the vector expression for the conic. The pole of 

 the line joining these two points is 



since ari?/2 + ^2yi =- 25], and 2:1^:2 = ^2. if 3/1^2 = ^o- 



The points at infinity on the conic are determined by the quadratic 



a^pc^ + la^xy + awy"- = 0, 



since, when this vanishes, the vectors to the points determined are 

 infinitely long. The pole of the chord joining these points, or the 

 centi"e of the conic is 



_ Oyflj - 2aiai + a^a^ 

 P'^ 2{a,a,-a,') 



