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
3λt3 + (1 + 6λ)t = 0.
Hence the roots are
t = 0 and
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.
The plane of λ and κ
