1887.] A. Mukhopadhyay — Memoir on Plane Analytic Geometry. 319 



2(Area of triangle) 



A= 



p 



whicli, by the help of equation (77), reduces to 



2 \/(a S'D' -(«+&) AS'2} 

 ^ ~ (^h-h^') S'- A • 



Hence, we have the 



Theorem. — If from any point {x\ y') two tangents be drawn to a 

 conic given by the general equation, the length of the chord of contact 

 is 



2 >/{ AS'D'-(a+5) A S'^] 



(afe-z.^ys'^X ' ^^^) 



where S', D' are the point-functions of the conic and of its director- 

 circle, respectively. 



Various particular cases may be deduced from the general formula 

 in (80). Thus, if the tangents be drawn from any point on the director- 

 circle, that is, if the tangents be orthogonal, the length of the chord of 

 contact is 



2^'\/- (a + 5 ) A 

 {ab-h^) S'- A • 

 Again, if two tangents be drawn from the directrix of a parabola to the 

 curve, the length of the chord is 



2S' /I^ = 2S' ^^^ 



f\/a ^ g \/h 



If the curve is an equilateral hyperbola, the director- circle degenerates 

 into the centre of the conic, and the chord in question, beiag the line at 

 infinity, is of infinite length ; this also follows from (SO), for in this 

 case 



SO that the numerator becomes the square root of a zero-quantity, while 

 the denominator also vanishes, and, therefore, the limiting value of the 

 apparently indeterminate expression is really infinite. 



Again, we can easily find the area of the triangle formed by the 

 chord of contact with the lines joining the centre to the points of con- 

 tact. For the chord of contact, being the polar of {x\ y'), is 



{ax'-{-hy'^g)x+{hx'-^hy'^'j)y + gx-^fy''\-c = 0, ... (81) 

 and the centre being 



\ab-h^' ah-ny' 



the perpendicular from the centre on the line in (81) is given by 



