Therefore, all pairs of values of λ and κ within the parabola, with the exception of the very narrow region also within the first discriminant curve, give uni-modal curves without infinite ordinates.
Types of Curves. Without entering into detailed proofs we will now investigate the general shape of the curves corresponding to values of λ and κ in each of the distinct regions of the plane of λ and κ.
In the region beneath the parabola and to the right from the shaded area of [Fig. I] the curve is essentially of the shape shown in [Fig. II]. This type includes the most common skew curves and hence is of great importance in statistics.
As the point (λ, κ) moves from the λ-axis the crest rises until the parabola is reached when the infinite ordinates appear as two coincident lines, shown in [Fig. III].
After the parabola is passed, the infinite ordinates separate and the curve apparently separates into three branches as in [Fig. IV].
In crossing the κ-axis to the left one asymptote moves off to infinity giving a curve of the type shown in [Fig. V].
Then the asymptote reappears giving a curve of the type shown in [Fig. VI].
This general shape is preserved as the point moves toward the λ-axis and when the point reaches the discriminant curve the middle branch is flattened at the minimum point.
For points within the discriminant curve two minimum points appear and the central branch now shows a maximum with a minimum point on either side as in [Fig. VII].
The Tri-modal Curves. The curves corresponding to values of (λ, κ) within the discriminant, because of the requirement that an element of area under the translated curve must always be equivalent to the corresponding element under the base or generating curve, can be of statistical value only under the following conditions.