1887.] A. Mukhopadhyay — Memoir on Plane Analytic Geometry. 349 



Substituting from (197), (198), and (199) in (196), the equation of tlio 

 locus sought is found to be 



\bH^-{-a^rj^' hH^'-Va^y)^)~ '^ ^ ^ 



We have, therefore, the 



Theorem. — If from any point P, tangents are drawn to the conic 



and P is constrained to move on any curve 



_ r(ar, 2/)=0, 

 the locus of the middle point of the polar chord of P with regard to S is 



"(ffs.rfs)-^ 



Similarly, if we consider the parabola 



2/2 = 4a2?, 

 any two points on the curve are 



{a tan2 6, 2a tan 6), (a tan^ </>, 2a tan ^), * 



so that the coordinates of the point of intersection of the tangents are 

 given by 



X = a tan tan <^ 

 Y=a (tan ^4- tan </>), 

 and the middle point of the polar chord is given by 



I=^(tan2(94-tan2<^), 



rj = a (tan 6-^ tan <p). 

 These give 



= — + 2 tan ^ tan ^, 



a2 a 



whence 



^'-^''{y=,. 



2a 

 Hence, substituting in F (x, y) = 0, we have the 



Theorem. — If from any point P tangents are drawn to the parabola 

 y^ = 4a^, 

 and P is constrained to move on the curve 



r(:«,2/) = o, 



the locus of the middle point of the polar chord of P with regard to the 

 parabola is 



»e-^. .)-<-■ 



We will here simply add that the result obtained above in equation 

 (200) is an immediate consequence of a new method which we propose 



to call the Method of Elliptic Inversion. 



26th October, 1887. 



45 



