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), κ^2 - 3λ = 0 The evident close geometrical connection between the two discriminants suggests arranging the discriminant of the cubic curve in the following form: (κ^2 - 3λ) (16κ^2 - 117λ^2 - 18λ - 1) - 27λ^3(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, 16κ^2 - 117λ^2 - 18λ - 1 = 0, and to the parabola, κ^2 - 3λ = 0, premit of the ready plotting of the curve with sufficient accuracy. The general shape of the curve is shown in Figure 1. It is to be noted that one branch of the curve is within the parabola, almost coinciding with it, while the other crosses it at λ = 1/24. From the original form of this equation it appears that the two branches of this discriminant meet just inside the parabola in the end points with approximate co-ordinates (0.043, ±0.360). The geometry of the cusp and end-points on the discriminant curve is suggestive of interesting development in detail. Values of λ and κ for points on the discriminant give curves with two modes coinciding. All points on one side of the discriminant have three real and distinct modes, and all on the other have one real and two imaginary modes. To determine on which side the points giving three real modes lie we examine a point inside the discriminant. When κ = 0 the modal equation becomes 3λt^3 + (1 + 6λ)t = 0. ______________ Hence the roots are t = 0 and t = ±√(-(1 + 6λ)/3λ). The quantity under the radical is positive for values of λ between 0 and-⅙. Therefore, all points within the discriminant curve yield tri-modal curves and all without uni-modal curves. [Illustration: _The plane of λ and κ_ _Fig. I_ (_The horizontal scale is twice the vertical scale_)] The infinite values of dy/dx arise from zero values of the quadratic, 1 + 2κt + 3λt^2. The greatest possible number of modes for any one curve is therefore five, three from the cubic and two from the quadratic. Since for infinite values of dy/dx the corresponding ordinates are infinite, it is advisable to study the location of the infinite points of the curve, rather to the neglect of the idea of maximum values at such points. =Infinite Ordinates.= The infinite points on a curve are given by the values of t satisfying the equation 3λt^2 + 2κt + 1 = 0.