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



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

 locus sought is found to be 



\hH^-\-a^r)^' hH'^^-a^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 



^ F(a.,2/)=0, 

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



-^te-ifs)-- 



Similarly, if we consider the parabola 



y^ = 4a^, 

 any two points on the curve are 



{a tan^ 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 6 tan <t> 



Y = a (tan ^-|-tan <^), 

 and the middle point of the polar chord is given by 



^=^(tan2^4-tan2^), 



7)=: a (tan ^-|-tan <^). 

 These give 



i-=— +2tan6'tan</>. 



whence 



^='^' ^- 



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



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

 y^ = 4iaXj 

 and P is constrained to move on the curve 



F(^,^)-0, 

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

 parabola is 



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. 



