To complete Plücker’s theory it is necessary to take account of compound singularities; it might be possible, but it is at any rate difficult, to effect this by considering the curve as in course of description by the point moving along the rotating line; and it seems easier to consider the compound singularity as arising from the variation of an actually described curve with ordinary singularities. The most simple case is when three double points come into coincidence, thereby giving rise to a triple point; and a somewhat more complicated one is when we have a cusp of the second kind, or node-cusp arising from the coincidence of a node, a cusp, an inflection, and a double tangent, as shown in the annexed figure, which represents the singularities as on the point of coalescing. The general conclusion (see Cayley, Quart. Math. Jour. t. vii., 1866, “On the higher singularities of plane curves”; Collected Works, v. 520) is that every singularity whatever may be considered as compounded of ordinary singularities, say we have a singularity = δ′ nodes, κ′ cusps, τ′ double tangents and ι′ inflections. So that, in fact, Plücker’s equations properly understood apply to a curve with any singularities whatever.
By means of Plücker’s equations we may form a table—
| m | n | δ | κ | τ | ι |
| 0 | 1 | − | − | 0 | 0 |
| 1 | 0 | 0 | 0 | − | − |
| 2 | 2 | 0 | 0 | 0 | 0 |
| 3 | 6 | 0 | 0 | 0 | 9 |
| ” | 4 | 1 | 0 | 0 | 3 |
| ” | 3 | 0 | 1 | 0 | 1 |
| 4 | 12 | 0 | 0 | 28 | 24 |
| ” | 10 | 1 | 0 | 16 | 18 |
| ” | 9 | 0 | 1 | 10 | 16 |
| ” | 8 | 2 | 0 | 8 | 12 |
| ” | 7 | 1 | 1 | 4 | 10 |
| ” | 6 | 0 | 2 | 1 | 8 |
| ” | 6 | 3 | 0 | 4 | 6 |
| ” | 5 | 2 | 1 | 2 | 4 |
| ” | 4 | 1 | 2 | 1 | 2 |
| ” | 3 | 0 | 3 | 1 | 0 |
The table is arranged according to the value of m; and we have m = 0, n = 1, the point; m = 1, n = 0, the line; m = 2, n = 2, the conic; of m = 3, the cubic, there are three cases, the class being 6, 4 or 3, according as the curve is without singularities, or as it has 1 node or 1 cusp; and so of m = 4, the quartic, there are ten cases, where observe that in two of them the class is = 6,—the reduction of class arising from two cusps or else from three nodes. The ten cases may be also grouped together into four, according as the number of nodes and cusps (δ + κ) is = 0, 1, 2 or 3.
The cases may be divided into sub-cases, by the consideration of compound singularities; thus when m = 4, n = 6, δ = 3, the three nodes may be all distinct, which is the general case, or two of them may unite together into the singularity called a tacnode, or all three may unite together into a triple point or else into an oscnode.
We may further consider the inflections and double tangents, as well in general as in regard to cubic and quartic curves.
The expression for the number of inflections 3m(m − 2) for a curve of the order m was obtained analytically by Plücker, but the theory was first given in a complete form by Hesse in the two papers “Über die Elimination, u.s.w.,” and “Über die Wendepuncte der Curven dritter Ordnung” (Crelle, t. xxviii., 1844); in the latter of these the points of inflection are obtained as the intersections of the curve u = 0 with the Hessian, or curve Δ = 0, where Δ is the determinant formed with the second derived functions of u. We have in the Hessian the first instance of a covariant of a ternary form. The whole theory of the inflections of a cubic curve is discussed in a very interesting manner by means of the canonical form of the equation x³ + y³ + z³ + 6lxyz = 0; and in particular a proof is given of Plücker’s theorem that the nine points of inflection of a cubic curve lie by threes in twelve lines.
It may be noticed that the nine inflections of a cubic curve represented by an equation with real coefficients are three real, six imaginary; the three real inflections lie in a line, as was known to Newton and Maclaurin. For an acnodal cubic the six imaginary inflections disappear, and there remain three real inflections lying in a line. For a crunodal cubic the six inflections which disappear are two of them real, the other four imaginary, and there remain two imaginary inflections and one real inflection. For a cuspidal cubic the six imaginary inflections and two of the real inflections disappear, and there remains one real inflection.