334 A. Mukhopadhyay— Memoir on Plane Analytic Geometry. [No. 3, 

 evolute of the conic 



a^'^h^- ' 

 with regard to the circle described on the line joining the foci as 

 diameter, is the curve 



fcf-(i)*}(e/6)':en9T=(r^)' <-' 



Here we may remark in passing that since the reciprocal polar of the 

 evolute of the reciprocal polar of any curve can be geometrically proved 

 to be the locus of the extremity of the polar subtangent, it is clear that 

 the curve in (127) is the locus of the extremity of the polar subtangent 

 of the evolute of the conic 



— I- — = 1. 



2 1^ 7 9* 



Hence, transforming to polar coordinates, we have the 



Theorem. — The locus of the extremity of the polar subtangent of 

 the curve 



3 2 



(ly _ /acos ey /hsmey 



which is, of course, the evolute of the conic, is the curve 



(!-!)'i.={c-=-'^m^ 



{o„..(=-y-^.(»lf)']* (1.8, 



which is, of course, the polar form of the equation (127). 



Again, by putting m= |, k'^=:abj we find that the reciprocal polar 

 of the evolute of the parabola 



with respect to a circle of radius \/ab, is the cubic curve 



^. -^+-. = 2 (129) 



X y a 



By the application of the same process to the parabola, a variety of 



new theorems may be obtained, viz., taking the parabola of the n^^ 



degree, 



2/ = ^^^ (132) 



the normal at any point {x, y) is 



XnJ^-^.Y ■\-X = x{\^\mjx^-^), (131) 



