1887.] A. Muldiopadhyay — Memoir on Plane Analytic Geo7netry. 305 



will have the same sign or opposite signs, according as (ah — h^) "7 or 

 ZO, or according as the curve is an ellipse or hyperbola. This com- 

 pletes the classification of conies. (§. 9). 



§. 12. Asymptotes. — In the ordinary text-books (cf. Smith's 

 Conies, §. 174, Ed. 1882, p. 187), the method of finding the equation of 

 the asymptotes of the general conic is given as follows : it is first proved 

 that the asymptotes of the conic in the particular case 



w 



h^ 



are given by 



and thence it is inferred that, in the general case, the equations of the 

 conic and asymptotes must differ only by a constant ; the logic of this 

 reasoning is, to say the least, hardly satisfactory ; the following method 

 is both easy and rigorously logical. 



The asymptotes being tangents to the conic at infinity, they may 

 be regarded as a pair of lines passing through the points of intersection 

 of the conic and the line at infinity. Now, the equation of the conic 

 being 



S = aa-?' + 27?.i;i/ + &I/2 -f 2(7^ + 2/?/ 4- c = 0, 

 and that of the line at infinity having been shewn (§. 3) to be 



X = 0, 

 any conic through their common points is 



S + X = 0; 

 and, in order that this may be a pair of lines, its discriminant must 

 vanish, whence, as usual, 



A 

 ah — h^" 

 and the asymptotes are given by 



S = Q, 

 which shews that the asymptotic constant in (41) is a constant which 

 must be equated to S, to furnish the equation of the asymptotes. 



The above process may be represented in a modified form as fol- 

 lows ; the conic 



aa-2 + 2hxy 4- hy^ + 2gx -f 2/?/ + c = 

 being transformed to the centre, becomes 



a^M- 2hxy + hy^ + -^A_ ^ Q, 



whence it at once follows that the quantity to be added to the right 

 hand side of this equation to give the asymptotes is the asymptotic 

 constant. Now, if we transform back to our old axes, the left hand 

 '39 



