320 A. Mukhopadliyay — Memoir on Plane Analytic Geometry. [No. 3, 



i(ax' + hy''^g){hf-hg)-^ilix' + hy'+f){hg-af) + (gx'^fy'-^c){ah--h^)} 



^ (ah-h^) [ (ax-Jfliy'-^g)^+(lix+ly'+f)'i] \ 



If, therefore, pi be tlie length of the perpendicular in question, this 



reduces to 



Pi = 



(82) 



(ah-h^) V'[(a+6)S'-D'| 

 Hence, as the length of the chord is given in (80), the area of the 

 triangle is written down to be 



It must be carefully noticed that the two triangles whose areas are given 

 in (77) and (83), being on opposite sides of the chord of contact, are 

 affected with opposite signs ; hence their algebraic sum establishes the 

 truth of a property enunciated by Prof. Nash, viz., we have the following 

 Theorem, — If two tangents are drawn from any point (a;', y') to 

 the conic 



ax'^-\-27ixy-^hy^-\-2gx + 2fy-\-c = 0, 

 the area of the quadrilateral formed by the two tangents and the two 

 lines joining the centre to the points of contact is 



,^. (»' 



where S' is the point- function of the conic. 



It is easy to remark that the geometrical meaning of the equation 

 of the conic is that, when the area of the quadrilateral vanishes, the 

 locus of the point must be the curve itself. Again, since we know from 

 geometry that the area of the quadrilateral is real or imaginary according 

 as the point is outside or inside the curve, we infer from (84) that any 

 given point is inside or outside the curve according as AS' is positive or 

 negative, which is equivalent to the statement that the point is inside 

 or outside according as the discriminant and the point-function have the 

 same or different signs, and the same result, of course, also follows from 

 the formula in (77). Here we may add that if from any point two 

 tangents be drawn to a conic, the angle between the two tangents will be 

 real, only if a certain relation holds amongst the coefficients in the equa- 

 tion of the conic ; thus, first taking the simple case when the tangents are 

 drawn from the origin, we have the tangents given by equation (74), viz. 



{ac-g^)x^-\-2(ch-fg)xy + (hc-P)y^'=0, 

 and clearly the angle between these two lines will be real, if 



{ch-fgfV {ac-g^){hc-P) 

 or A Z 0. 



