1887.] A. Mukhopadhyay — Memoir on Plane Analytic Geometry. 335 



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



Y-Xn.^^-lX = 0, (132) 



so that, at the point of intersection of the lines given by (131) and (132), 

 we have 



1 1 + \2^2^2(^-l) I Y = \nx^-^. X. (1 + \nyoo^-% (133) 



I 1 + X2,i2^2(^-1) I X = ^ (1 + Xn^*-^-2), (134) 



and the inverse of the X, Y is given by 



I'^X ^_ 



^■\-^'^~ x{l-\\nyx' 

 h^ Y JcK Xux""-^ 



^-X2 + Y2-^(H.X^2/^— 2) ^ ^ 



r, . (136) 



^-X2 + Y2"^(l+Xn7/a;^-2) 



Tp-here i, rj are the coordinates of a point on the locus sought ; hence, 

 eliminating x, y between the equations (135) and (136), by virtue of the 

 relation in (130), we have, after replacing |, rjhj x, y respectively, the 



Theorem. — The reciprocal polar of the evolute of the parabola of 

 the n*^ degree 



y = Xx'^ 

 is the curve 



2/05^-2 (l + -. 'ly =\nh^i^-'^) (137) 



\ n x f 



where k is the constant of inversion. 



As before, by assigning particular values to X and oi in this equa- 

 tion, we may deduce various theorems. 



Thus, the reciprocal polar of the evolute of the parabola 

 2/2 =4:axy 

 with regard to a circle whose diameter is equal to the latus-rectum, is 

 the cubic curve 



r (cos2^4-cot2^) = 4acos(9, (138) 



of which x=:2a is an asymptote. 



Again, the reciprocal polar of the evolute of the parabola 

 ?/2 = 4iax, 

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



x^ = y^(a'-2x), (139) 



of which i» = - is an asymptote. 



Again, the reciprocal polar of the evolute of the parabola 

 y^ — 4a {x-} a), 

 the focus being now the origin, with regard to a circle whose diameter 

 is equal to the semi-latus-rectum, is the curve 



r cot ^ = a sill 0, (140) 



