348 A. Mukhopadliyay — Memoir on Plane Analytic Geometry. [No. 3, 



P is constrained to move on any curve 



'F{x,y)=0 (193) 



Let 6, </> be the eccentric angles at the points of contact of the tangents ; 

 then the tang^ents are 



X 



- COS 



a 



'-\ 



sin ^ = 1, 



X 



- cos 

 a 



,.\ 



sin </> = !, 



nates 



5 0f P, 



we have 





COS 



2 



X = 



a. 



cos 





2 



Y = 



sin 

 h. 



cos 



2 

 2 



If, further, ^ ^ be the coordinates of the middle point of the chord of 

 contact the locus of which is sought, we have 



1=.^ (cos 6* + cos 0) (194) 



T/n- (sin ^-fsin </)) (195) 



The locus is obtained by eliminating ^, </> between these and 



cos -— sm -— \ 



\ cos ; — COS • — :: — ) 



^ 2 2 ^ 



From (194) and (L95), we have 



i e+(l> 6-<p 



- = cos — — - COS — — - 



a 2 2 



V . o^i> e-<p 



- - sm — — - cos -—- . 



2 2 



whence, squaring and adding. 



e-<^ ^ 



3 yf 



^^'^'— =^+P (1^^) 



Also, by division, from (194) and (195), 



'^"^ 2 ~h^ 

 whence 



. 6^<p ay) e + <p bi 



sin — — = — /— ^ , cos — -— = — . - 



2 Vh'e^a^ 2 v^6?'^2-faV 



(198), (199) 



