Except under certain limited conditions to be determined later a curve with infinite ordinates can not be of great statistical value. The parabola, κ^2-3λ = 0, obtained by equating the discriminant of this quadratic to zero separates the points on the (λ, κ) plane which correspond to curves of _no_ infinite points from those corresponding to curves of _two_ infinite points. [Illustration: _Types of Curves_] 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. The area between the two ordinates corresponding to t = ±3 is 0.99998 of the total area under the curve, so that when neither of the minimum points corresponds to points closer than three units to the origin of the base curve the curve may be practically valuable. A moment’s consideration will show that the abscissas of the two minimum points must be practically the same as that of the corresponding infinite ordinates. The roots of the quadratic 3λt^2 + 2κt + 1 = 0 are numerically greater than 3 for all pairs of values of (λ, κ) lying above the line 27λ - 6κ + 1 = 0 As statistically promising within the discriminant of the cubic we then have the shaded area of the (λ, κ) plane.