302 PRICEEDINGS OF SECTION A. 



Similarly the axis of P2 — 



^. P ^ y _ 



touches the cubic at the joint Q^ whose co-ordinates are— 



[(^. - uy {. - \f (A - /Ol- 

 Now the equations of (X.Pi) and (X.P2) are both satisfied by the 

 point Q3, 



\y^^,-,f ^^[.-\f i/(A-;0^], 



which clearly is on the cubic curve. 



Hence the cubic may be reo;arded as the locus of the intersection of 

 the axes of the extremities of chords of 2 drawn through O. 



The axes of P3 and P4 will touch the cubic at — 

 Q3[A^(^, _ „)3 ,,3(,, _ xf v'{X - /t?] and Q,[_Q,' 0/ 0/] 

 respectively, and intersect each other on the curve at the point — 



[}? (;. - vY 0,^ /^^ {^ - ^f 0/ "^ (^ - /O^ e/]. 



We may show that, in general, four axes pass through any point on 

 the cubic, and find the equation of those axes as follows : — 



Take any point (0,^,71) on the cubic ; the conic — 



S, = '^ + §' + Zi = 

 a f3 y 



will meet 2 in four points. Suppose that a fS'y' are the co-ordinates of 

 a point of intersection of 2 and Si. 

 The axis of {a (i'y) is — 



a! ^ li' ^ y' 



also — "i J- ^ 4- Zl = 



a' (5 y 



1/^0"^ ^^0+ ^V.~ ' 



we have for the equation of the four axes — 



1 1 1 



v/a„(/3y, -/3,y) VfS' {ya, - y^a) Vy^ {a(3, — a,fi) ~ 



If now (a,/?,y,) be the point Q3, and if the above quartic be expanded, 

 we know that two of its factors are (X.P,) and (X.P2), and find that the 

 other factor is the square of (X.P3), multiplied by a constant quantity. 

 Hence we infer that the conic Si touches 2 at the point P,, a conclusion 

 which may be verified by finding the tangent at P3, viz. — 



+ -r.--J-^< + -^ : = 0. 



a^k{ix — J/) ^o'^C" — ^) 70" l'^ — A') 



