306 A. Mtikhopadhyay — Memoir on Plane Analytic Geometry. [Ko. Z, 



side becomes 



ax^-\-2}ixy + ly'^-^2gx-\-2fy-\-c, 

 while, A and (ah — h^) beiug trauslation-invariants, the i-ight hand side 

 remains unaltered, and the equation sought is accordingly 



^=^-_ _ (^^) 



It follows from (G) that the point of intersection of the asymptotes in 

 (55) coincides with the centre of the conic, and that, accordingly, the 

 centre is the pole of the line at infinity. It is also clear that the asymp- 

 totes will be at right angles to each other and the conic will be a rec- 

 tangular hyperbola, if (a-\-b) = 2h cos w, in oblique coordinates, and 

 {a-\-b) =0 in rectangular coordinates. 



§. 13. Eccentricity. — The eccentricity may be calculated in 

 different ways according to the definition we employ. 



First method, 



where a, /3 are the semi-axes of the conic. We have 



a"* 

 which give 



(2_-e3)2 _ (a2 + ^ 



and this, by substitution from (39) and (4*0), becomes 

 (2 - e2)8 _ (A+ B)2 

 l-"^~ AB • 

 But, from the invariants (42) and (43), we have 



a-\-h — 2h cos <o 



A+J3 =: - 



(56) 



AB = 



sin*a> 

 ah - 7i2 



sm'^w 

 BO that equation (56) becomes 



(2 - e2)2 (a+ h-2h cos (o)^ 



(57) 



1 — e^ {ab — h^) sin^w 



"which is the familiar equation. It is clear from (57) that (1 — e^) and 

 {ah — h^) are simultaneously positive, zero, or negative j hence, we have 



e^ ^ = yl 

 according as 



;i2 / = 7 ah, 



