**Cartesian equation: **

*ay*^{2} = *x*(*x*^{2} - 2*bx* + *c*), *a* > 0

Newton's classification of cubic curves appears in

*In the third Case the Equation was **yy* = *ax*^{3} + *bxx* + *cx* + *d** and defines a Parabola whose Legs diverge from one another, and run out infinitely contrary ways.*

The case divides into five species and Newton gives a typical graph for each species. The five types depend on the roots of the cubic in *x* on the right hand side of the equation.

(i) All the roots are real and unequal : *then the Figure is a diverging Parabola of the Form of a Bell, with an Oval at its vertex .*

This is the case for the graph drawn above.

(ii) Two of the roots are equal : *a Parabola will be formed, either Nodated by touching an Oval, or Punctate, by having the Oval infinitely small .*

(iii) The three roots are equal : *this is the Neilian Parabola, commonly called Semi-cubical .*

(iv) Only one real root : *If two of the roots are impossible, there will be a Pure Parabola of a Bell-like Form .*

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JOC/EFR/BS January 1997

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