What is "semicubical parabola"

Parabola \Pa*rab"o*la\, n.; pl. Parabolas. [NL., fr. Gr. ?; -- so called because its axis is parallel to the side of the cone. See Parable, and cf. Parabole.] (Geom.)

1. A kind of curve; one of the conic sections formed by the intersection of the surface of a cone with a plane parallel to one of its sides. It is a curve, any point of which is equally distant from a fixed point, called the focus, and a fixed straight line, called the directrix. See Focus.

2. One of a group of curves defined by the equation y = ax^ n where n is a positive whole number or a positive fraction. For the cubical parabola n = 3; for the semicubical parabola n = 3/2. See under Cubical, and Semicubical. The parabolas have infinite branches, but no rectilineal asymptotes.

In mathematics, a cuspidal cubic or semicubical parabola is an algebraic plane curve defined parametrically as

x = t 

y = at. 
Its implicit equation is

y − ax = 0, 
which can be solved in to yield the equation

$$y = \pm ax^{3 \over 2}.$$

This cubic curve has a singular point at the origin, which is a cusp.

If one sets , , and , one gets

X = u

Y = u.
This implies that, for any value of , the curve is homothetic to the curve for which , or, equivalently, that the curves corresponding to different values of differ only by the choice of the unit length.