3λt^3 + 2κt^2 + (1 + 6λ)t + 2κ = 0
is
16κ^4 - κ^2(1 + 66λ + 117λ^2) + 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),
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.
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