ay2 = x(x2 - 2bx + c), a > 0
In the third Case the Equation was yy = ax3 + 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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