the whole system being equivalent to three equations only; and
it may be added that using a to denote the equal quantities
3m + ι and 3n + κ everything may be expressed in terms of m, n, a.
We have
| κ = a − 3n, ι = a − 3m, 2δ = m2 − m + 8n − 3a. 2τ = n2 − n + 8m − 3a. |
It is implied in Plücker’s theorem that, m, n, δ, κ, τ, ι signifying as above in regard to any curve, then in regard to the reciprocal curve, n, m, τ, ι, δ, κ will have the same significations, viz. for the reciprocal curve these letters denote respectively the order, class, number of nodes, cusps, double tangent and inflections.
The expression ½m(m + 3) − δ − 2κ is that of the number of the disposable constants in a curve of the order m with δ nodes and κ cusps (in fact that there shall be a node is 1 condition, a cusp 2 conditions) and the equation (9) thus expresses that the curve and its reciprocal contain each of them the same number of disposable constants.
For a curve of the order m, the expression ½m(m − 1) − δ − κ is termed the “deficiency” (as to this more hereafter); the equation (10) expresses therefore that the curve and its reciprocal have each of them the same deficiency.
The relations m2 − 2δ − 3κ = n2 − 2τ − 3ι = m + n, present themselves in the theory of envelopes, as will appear farther on.
With regard to the demonstration of Plücker’s equations it is to be remarked that we are not able to write down the equation in point-co-ordinates of a curve of the order m, having the given numbers δ and κ of nodes and cusps. We can only use the general equation (*≬ x, y, z)m = 0, say for shortness u = 0, of a curve of the mth order, which equation, so long as the coefficients remain arbitrary, represents a curve without nodes or cusps. Seeking then, for this curve, the values, n, ι, τ of the class, number of inflections, and number of double tangents,—first, as regards the class, this is equal to the number of tangents which can be drawn to the curve from an arbitrary point, or what is the same thing, it is equal to the number of the points of contact of these tangents. The points of contact are found as the intersections of the curve u = 0 by a curve depending on the position of the arbitrary point, and called the “first polar” of this point; the order of the first polar is = m − 1, and the number of intersections is thus = m(m − 1). But it can be shown, analytically or geometrically, that if the given curve has a node, the first polar passes through this node, which therefore counts as two intersections, and that if the curve has a cusp, the first polar passes through the cusp, touching the curve there, and hence the cusp counts as three intersections. But, as is evident, the node or cusp is not a point of contact of a proper tangent from the arbitrary point; we have, therefore, for a node a diminution 2, and for a cusp a diminution 3, in the number of the intersections; and thus, for a curve with δ nodes and κ cusps, there is a diminution 2δ + 3κ, and the value of n is n = m(m − 1) − 2δ − 3κ.
Secondly, as to the inflections, the process is a similar one; it can be shown that the inflections are the intersections of the curve by a derivative curve called (after Ludwig Otto Hesse who first considered it) the Hessian, defined geometrically as the locus of a point such that its conic polar (§8 below) in regard to the curve breaks up into a pair of lines, and which has an equation H = 0, where H is the determinant formed with the second differential coefficients of u in regard to the variables (x, y, z); H = 0 is thus a curve of the order 3(m − 2), and the number of inflections is = 3m(m − 2). But if the given curve has a node, then not only the Hessian passes through the node, but it has there a node the two branches at which touch respectively the two branches of the curve; and the node thus counts as six intersections; so if the curve has a cusp, then the Hessian not only passes through the cusp, but it has there a cusp through which it again passes, that is, there is a cuspidal branch touching the cuspidal branch of the curve, and besides a simple branch passing through the cusp, and hence the cusp counts as eight intersections. The node or cusp is not an inflection, and we have thus for a node a diminution 6, and for a cusp a diminution 8, in the number of the intersections; hence for a curve with δ nodes and κ cusps, the diminution is = 6δ + 8κ, and the number of inflections is ι = 3m(m − 2) − 6δ − 8κ.
Thirdly, for the double tangents; the points of contact of these are obtained as the intersections of the curve by a curve Π = 0, which has not as yet been geometrically defined, but which is found analytically to be of the order (m − 2)(m2 − 9); the number of intersections is thus = m(m − 2)(m2 − 9); but if the given curve has a node then there is a diminution = 4(m2 − m − 6), and if it has a cusp then there is a diminution = 6(m2 − m − 6), where, however, it is to be noticed that the factor (m2 − m − 6) is in the case of a curve having only a node or only a cusp the number of the tangents which can be drawn from the node or cusp to the curve, and is used as denoting the number of these tangents, and ceases to be the correct expression if the number of nodes and cusps is greater than unity. Hence, in the case of a curve which has δ nodes and κ cusps, the apparent diminution 2(m2 − m − 6)(2δ + 3κ) is too great, and it has in fact to be diminished by 2{2δ(δ − 1) + 6δκ + 92κ(κ − 1)}, or the half thereof is 4 for each pair of nodes, 6 for each combination of a node and cusp, and 9 for each pair of cusps. We have thus finally an expression for 2τ, = m(m − 2)(m2 − 9) − &c.; or dividing the whole by 2, we have the expression for τ given by the third of Plücker’s equations.
It is obvious that we cannot by consideration of the equation u = 0 in point-co-ordinates obtain the remaining three of Plücker’s equations; they might be obtained in a precisely analogous manner by means of the equation v = 0 in line-co-ordinates, but they follow at once from the principle of duality, viz. they are obtained by the mere interchange of m, δ, κ, with n, τ, ι respectively.
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 |
