The discriminant of the equation
3λt3 + 2κt2 + (1 + 6λ)t + 2κ = 0
is
16κ4 - κ2(1 + 66λ + 117λ2) + 3λ(1 + 6λ)3 = 0
This fourth degree curve crosses the horizontal or λ-axis at λ = 0 and at λ = -⅙ and when λ = 0 its equation reduces to 16κ^4 - κ^2 = 0 or κ = ±0, κ = ±¼. There is thus contact with the vertical or κ-axis at the origin and that axis is crossed at the points (0, ±¼). At the point (λ = -⅙, κ = 0) there is a cusp with the λ-axis for tangent. The other two intersections with the line λ = -⅙ are imaginary, indicating the presence of two branches to the curve.
The discriminant of the denominator of dy/dx is the parabola (in λ and κ),
κ2 - 3λ = 0
The evident close geometrical connection between the two discriminants suggests arranging the discriminant of the cubic curve in the following form:
(κ2 - 3λ) (16κ2 - 117λ2 - 18λ - 1) - 27λ3(1 - 24λ) = 0
From the equation in this, the well known uv + kws = 0 form, numerous elementary geometrical facts can be derived. The relations to the hyperbola, 16κ2 - 117λ2 - 18λ - 1 = 0, and to the parabola, κ2 - 3λ = 0, premit of the ready plotting of the curve with sufficient accuracy. The general shape of the curve is shown in Figure 1.