1887.] A. Mukhopadhyay — Memoir on Plane Analytic Geometry. 309 



= [-(a+6) A +27icosa). a] -r (a6 - /i^) 

 — (a-\-h — 2h cos w) A 



" (ah - h^y 



which, by a glance at (53), is seen to represent, as before, the sum of 

 the squares of the semi-axes. From the vahie of the radius given above, 

 it is clear that, when the conic is an equilateral hybcrbola, the radius 

 vanishes, and the director-circle is a circle of infinitesimal radius, viz.y 

 it is the centre of the conic itself, and the asymptotes, therefore, are 

 the only tangents of the equilateral hyperbola which are at right angles 

 to each other. 



§. 15. Hyperbola referred to the asymptotes. — In this section, 



we purpose to investigate what form the general equation assumes when 

 the axes of coordinates are transformed to the asymptotes ; two methods 

 will be given, the first very direct and elementary, the second partly 

 geometrical and requiring a knowledge of the invariants given above. 



First method. 



Let the general equation of the second degree be 

 ax^ -f 2ha)y + hy^ + 2gx + 2/?/ + c = 0. 

 Transfer the coordinate axes to the centre of the conic, which is also 

 the point of intersection of the asymptotes ; the conic then becomes 



aa;H 2hxy-\-hy^ + ^i,\% = ^ (61) 



and the asymptotes are given by 



ax^-\-2hxy+hy^ = 0. (62) 



Now the equation of either asymptote may be taken tohe y = mx, so 

 that the two values of m are found, by substitution in (62), to be the 

 roots of the quadratic 



hm^^2hm + a = (63) 



Hence, if a, /3 be the angles which the two asymptotes make with the 

 axis of Xj both tan a and tan must satisfy (63), so that we have 

 h t&n^'a-\-2h tan a-l- a = 



or 6 sin'^a-f-2A sin a cos a+a cos2a = (64) 



and similarly, 



6 sin2^-f2/isin/3cos;8-i-acos2/3 = (65) 



Now, the angle between the original axes being w = -, the ordinary 



formulae for the transformation of coordinates (Salmon's Conies, §. 9, 

 Ed. 1879, p. 7) become in this ca^e 



y sin (j)=X. sin a-fY sin /?. 



X sin w = X cos a 4- Y cos (3. 

 Substituting these in (61), and arranging, we have for the equation of 

 the conic 



