208 A. Mukhopadhyay — Memoir on Plane Analytic Geometry. [N'o. 3, 



is easy, viz., 



a (x-a^')(x -af)^2h(x - x')(y - i)-\-h(y - y'){y - y") = (30) 

 represents any conic through (x', y'), (x", y"), which may, it is useful 

 to notice, satisfy three other conditions : and the chord in question, 

 being the common chord of (29) and (30), must have for its equation 

 a(:,-:,'X:,-.^")^2h(x-x')(y-y")-^h(y-y')(y-y") 



= ax'''{-2kxy-^hy^-^2gx-\-2fy^c (31) 



I have not, however, been able to find if the conies (29) and (30) are 

 connected by any very special or peculiar relation : their centres are 

 not coincident ; the (;entre of (30) is not on the chord whose equation 

 is required ; their asymptotes, however, include equal angles, and their 

 axes are parallel ; in fact, they are similar and similarly situated, and, 

 therefore, necessarily equi-eccentric. 



The equation of the tangent at any point may be deduced, as usual, 

 from the equation of the chord ; or we may first obtain by Joachims- 

 thal's method the equation of the pair of tangents from an external 

 point, and thence obtain the equation of the tangent at any point of 

 the curve. The same equation, however, may be obtained by transform- 

 ation, if we know the equation of the tangents from the origin ; thus, 

 the conic being 



ax''^2hxy-\-hy'-{-2gX'\-2fy-[-c = 

 and (x', y') the external point, remove the origin to this point, so that 

 the conic becomes 



ax^ ■^2hxy-\- hy"" + 2g'x -\- 2fy + c' = 0, 

 where the values of /', g, c' are the same as in §. 4. If now y = mx be 

 any line through the new origin, it will touch the conic if the quadratic 

 in Xf 



(a + 2Am4-5«O^' + 2(/+/m)it; + c' = 0, 

 has equal roots, which condition gives 



c'(a+2/m+6m*) = (g'-^fmy, 

 and by substituting 



V 

 m = -, 



X 



we have for the equation of the tangents, referred to the new origin, 



cXax'-\-2hxy^hy') = (g'x-{-fy)\ 

 which may be written 



cXax'^-\-2hxy-\-by'''\-2g'x-{-2fy-\-c') = (g'x-^f'y-\-c'y. 



B/Cvorting to our old axes, we have at once the equation in the form 



(Conic) X (Point-function) = (Polar)^, 

 which is, of course, the same equation as that obtained by Joachims- 

 thal's method. 



