1887.] A. Mukhopadhyay — Memoir on Plane Analytic Geometry. 811 



that, when referred to the asymptotes, the equation must assume the 

 form 



Ax^ + 2B.xy^By^ + --l^^ = (69) 



Now, in this equation, the axis of x being an asymptote, one value of a: 

 must be infinite, and, therefore, in this equation, regarded as a quadratic 

 in X, we must have A = ; similarly, the axis of y being the other 

 asymptote, we must bave B = ; so that (69) reduces to 



2H^^+^^=<^ ^''^ 



To calculate H, we remark that, since the original axes are at right 

 angles, we have w = |, and, as also A = 0, B = 0, the invariant relation 



sin^o) "~ sin^ O 



reduces to 



-H2=(a6-/i2) sin2 0, (71) 



where O is the angle between the asymptotes, 



ax^-^2hxy + by^ = (72) 



But, a, /8 being the angles which the asymptotes make with the axes, 

 we have = a — /?, and, from equation (72), 



2\/h'^-ah 



tan 12 = 



sin O = 



a-\-h ' 



2\/h^-ah 



/ 

 V 



I (a-6)2 + 4/^2V 



so that (71) becomes 



4(a6-^2)2 



H2- 



"(a -6)2 + 4/^-2' 



and (70) gives for the required equation 



A x/{(«-^)H4/.2} 



which is the same result as that obtained before. It may be noted 

 that the value of H might have been obtaiaed with equal ease by using 

 the other invariant relation 



a-\-h -2h c os o> _ A + B - 2H cos fi 

 sin^ w "" sin^ O 



The geometrical meaning of this equation of the hyperbola is easily 

 seen, viz., taking p^^, p^ for the squares of the semi-axes of the conic, and 

 remembering that our original axes were rectangular, we have from (53), 



