FOLIUM

folium, in mathematics, a curve invented and discussed by rene descartes. its cartesian equation is x^3 + y^3 = 3axy. the curve is symmetrical about the line x = y, and consists of two infinite branches asymptotic to the line x + y + a = o and a loop in the first quadrant. it may be traced by giving m various values in the equations x = 3am/(1 + m^3), y = 3am^2 (1 + m^3), since by eliminating m between these relations the equation to the curve is obtained. hence it is _unicursal_ (see curve). the area of the loop, which equals the area between the curve and its asymptote, is 3a/2. [illustration.]