336 A. Mukliopadliyay — Memoir on Plane Analytic Geometry . [No. 3, 



which represents a circnlar cubic, of which a? = a is an asymptote, and 

 the point at infinity a point of inflexion. 



Again, the reciprocal polar of the evolute of the evolute of the 

 parabola 



2/' = 4a (x+2a), 

 the origin now being the centre of curvature at the vertex, with respect 

 to a circle of radius a, is the quartic 



y^(dx^-^2y'') = a^x^ (141) 



Similarly, the reciprocal polar of the evolute of the parabola 

 2/2 = 4a (x-\-2a), 

 with respect to a circle of radius h, is the cubic 



ax^ = k'^y^. 

 It is useful to notice that if we are given any curve 



u=f(x,y) = 0, (142) 



the normal at any point {x, y) is 



,^^ ^ du ,^^ ^ du ., .^ 



(Y-,)-=(X-.)^, (143) 



while the line at right angles to this through the origin is 



At the common point of intersection of these two lines, we have 



m*(Sf}^'-t{'i-^ "«' 



ie)"+(5)>= £(»£-'!) <"« 



whence it follows that if (|^, rj) be the point inverse to (X, Y), the 

 coordinates are given by 



du 



^"X^ + Y^- ^'- du__^dAi ^^^^^ 



dx dy 

 du 



^^^^ 72 ^^ /n.ox 



''=x^+y^= ^--iH-^u (148) 



y -—-x—- 

 dx dy 



Therefore, the equation of the reciprocal polar of the evolute of the 



curve given by (142) is obtained by eliminating x and y from the three 



equations (142), (147), (148); and, the general theory being thus given, 



the question is reduced to one of elimination. 



It is interesting to note that if the coordinates of any point on the 



given curve can be expressed in terms of a single variable parameter <^, 



