1887.] A. Mukhopadhyay — Memoir on Plane Analytic Geometry. 307 



or according as the conic is an ellipse, a parabola, or an hyperbola. 

 In the equilateral hyperbola, we have 



a-i~b — 2h cos o> = 0, 



whence e=v2. 



8eco7id method, 



e = sec - , 



where <^ is the angle between the asymptotes. The equation of the 

 asymptotes from (55) being 



we have 



2 sin (o. V 7^2 — aZ> .^^. 



tan<;^ = — — . (5b) 



But sec2<^ 



a-\rh — 2h cos w 



(2cos^|_l)- 









2-sec4 

 z 





whence we have 



-1) 



Therefore, from equation (58), 



e^ — 1 (h^ — ah) sin^w 



(2 - 62)2 "^ (a-\-b- 2h cos (o)^' 

 which is the same equation as (57). 

 Third method. 



The eccentricity may be defined to be the ratio of the distance of 

 any point on the conic from a focus to its distance from the correspond- 

 ing directrix ; the calculation on the basis of this method will come in 

 most appropriately when we presently deal with Laplace's Linear Equa- 

 tion of a Conic (§§. 16 — 20; see, in particular, §, 20). 

 §. 14. Director-circle. — The director-circle of 

 S = ax^ + 2hxy + ht/^2gx-{-2fy + c = 

 being the locus of intersection of orthogonal tangents, its equation in 

 rectangular coordinates is known to be 



(ab-h^)(x^ + y^)^2(gh-fh)x + 2(fa-hg)y 



+ c(a-^h)-f-g^ = 0, (59) 



which may also be written in the form 



TJ^(a-\-h)S-iax-\-hy + gy-(hx-\-hy-\-fy = (60) 



