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 + 3λt2. 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
3λt2 + 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, κ2-3λ = 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.
Types of Curves