380 A. MukKopadliyay — Memoir on Plane Analytic Geometry. [No. 3, 



(x — ae)^ y^ 

 1 _i_ _ — 1 



and the first pedal, being the circle on the major axis as diameter, is 



(x — ae)2 -\- y^ = o,^, 



the coordinates of any point on which may be expressed by means of a 



single parameter, viz.j 



x = a (e -\- cos <^), 



y=:a sin ^, 



and hence the equation of any tangent may be thrown into the form 



(x — ae) cos 1> + y sin (p = a. 



A line at right-angles to this through the origin (which is now the 



focus) is 



X sin <l> — y cos ^ = 0, 



and, as the second pedal of the conic, or the first pedal of the circle, is 



the locus of the intersection of the two lines, we have, by solving for 



sin <l> and cos <^, 



. .ay . acB 

 sm </> = -T—- y , cos </) = '-o-r-2 T » 



where (^x., y) is, of course, a point on the pedal, viz., the actual equation 

 is 



aS (x'^-\-7/) = {x^-iry^ - aexy^ , 

 which quartic, therefore, is the second pedal of the given conic with 

 respect to a focus. To see that this is the inverse of a conic, we have 

 only to take its inverse, viz., substituting for x and y 



h^x h^y 



^2_|_^2' ^2_^^2 



respectively, the second-pedal-quartic is seen to be the inverse of 

 which is, of course, a conic, viz., this may be written 



a2 "*■ 62 ~ ^ ~ ~ah^ * 



which is equivalent to 





k^e\^ y^ k^ 



"^52-54 



It may be noted that any two conies having a common focus have two of 

 their common chords passing through the intersection of their directrices ; 

 in the present case, therefore, two of the chords of intersection of this 

 conic and the given conic are parallel to the directrices ; one of these 

 chords is found, by subtracting the equations of the conies, to be the line 



•_ A:2-fc2 



