310 A. Mukhopadhyay — Memoir on Plane Analytic Geometry. [No. 3, 



{a cos2a + 27i cos a sin a-\-h sm2a)X2 

 -f (a cos2^ + 27i cos /5 sin ^-f Z> sin2yg)Y2 



+ 2 fa cos a cos /3 + A.sin (a + y8)+6 sin a sin ^IXY 



But, by (64) and (65), the coefficients of X^ and Y2 vanish, and the 

 equation becomes 



2H«/ + ,-j-^, = 0, ^Q&) 



where H is the quantity to be calculated. For this purpose, we note 

 that, if m^, m^ be the two roots of the quadratic in m given by (63), we 

 have 



2h a 



Now, we see that 



H= cos a cos /? j a-\-h (tan a-f-tan j8) + 6 tan a tan /8 [ 



2(ah-h^) n 



= — ^^ — cos a COS p, 



6 



where 



cos^a cos^/S = ] (1-f mi2)(l + m2^) f "" •.• m^ = tan a, m2 = tan /3. 



62 



- (a -6)2 + MS* 



Therefore, H=±^-^-"-^. 



v/ [(a-6)H4A2] 

 and, finally, the equation (66) becomes 



*^=*r — ^(^,;^— ' (C7) 



which is, accordingly, the equation of the hyperbola referred to its 

 asymptotes, which was sought. 



Second method. 



The same result may also be obtained as follows. The equation of 

 the conic, referred to its centre, being, as before, 



a^2 + 2A^7/ + 6^2 + __A_,:,0, (68) 



and remembering that the absolute term is a rotation- invariant, we see 



