1887.] A. Mukhopadhyay — Memoir on Plane Analytic Geometry. 323 



hence we have 



ZOAB=ZCAE = ^-«/r 



Z OB A = »/'-</>. 

 Therefore ZOAB = Z.OBA, and 0A = OB, just as it should be, so that 

 the geometric meaning is the equality of two tangents to a circle drawn 

 from any external point. Lastly, if we draw any two tangents OA, OB 

 to any conic, and, if OA, OB, BA intersect a directrix at C, D, E, we 

 have as before 



ZOAB = ^-«/^, ZOBA = «/r-(|i. 

 Now draw through the centre two radii- vectores of the curve (pj, pg), 

 making angles 6, <p with the conjugate axis ; then, from the polar equation 

 to the curve, we have 



^^ "■ 1 - eS sin2^' ^2 ~ 1 - eS sinS<;»' 

 so that 



b h 



which furnish the geometrical meanings of the symbols \, fx in the 

 statement of the theorem. Substituting for X, ix. in our original equation, 

 we have 



Py cos V+P2 cos ^ 

 whence 



Pi _ sin (i// — <|>) _ OA 



^ "" sin {6 - x{/) " OB' 

 and this asserts that the tangents OA, OB arc proportional to the central 

 radii- vectores which are obviously parallel to them. In the case of the 

 circle, the indeterminateness in the position of the axes makes all the 

 radii- vectores equal, so that, as shewn before, 



OA=OB, il/-<p=e-if/. 

 It may be remarked that we might have started from the polar instead 

 of the Cartesian equations, as just shewn, and thus worked up to the 

 value of tan xj/ given above ; it is also useful to notice that, though the 

 theorem was obtained from a very particular form of the equation of a 

 central conic, it is perfectly true for the general conic, inasmuch as 

 the eccentricity only appears in the final result. 



§. 24. Similar Codes, 



§. 24. Generation of Similar Conies. Given any conic, any other 

 conic which is concentric with it, and similar and similarly situated, 

 may be generated as the locus of a point through whicli any two chords 

 of the conic being drawn at right angles to each other, the sum of the 



