| No. of consts. | |
| 1. With a given cusp | 5 |
| 2. ” cusp on a given line | 6 |
| 3. ” cusp | 7 |
| 4. ” a given node | 6 |
| 5. ” node on given line | 7 |
| 6. ” node | 8 |
| 7. non-singular | 9 |
he determines in every case the characteristics (μ, ν) of the corresponding systems of cubics (4p), (3p, 1l), &c. The same problems, or most of them, and also the elementary problems in regard to quartics are solved by Zeuthen, who in the elaborate memoir “Almindelige Egenskaber, &c.,” Danish Academy, t. x. (1873), considers the problem in reference to curves of any order, and applies his results to cubic and quartic curves.
The methods of Maillard and Zeuthen are substantially identical; in each case the question considered is that of finding the characteristics (μ, ν) of a system of curves by consideration of the special or degenerate forms of the curves included in the system. The quantities which have to be considered are very numerous. Zeuthen in the case of curves of any given order establishes between the characteristics μ, ν, and 18 other quantities, in all 20 quantities, a set of 24 equations (equivalent to 23 independent equations), involving (besides the 20 quantities) other quantities relating to the various forms of the degenerate curves, which supplementary terms he determines, partially for curves of any order, but completely only for quartic curves. It is the discussion and complete enumeration of the special or degenerate forms of the curves, and of the supplementary terms to which they give rise, that the great difficulty of the question seems to consist; it would appear that the 24 equations are a complete system, and that (subject to a proper determination of the supplementary terms) they contain the solution of the general problem.
17. Degeneration of Curves.—The remarks which follow have reference to the analytical theory of the degenerate curves which present themselves in the foregoing problem of the curves which satisfy given conditions.
A curve represented by an equation in point-co-ordinates may break up: thus if P1, P2, ... be rational and integral functions of the co-ordinates (x, y, z) of the orders m1, m2 ... respectively, we have the curve P1α1P2α2 ... = 0, of the order m, = α1m1 + α2m2 + ..., composed of the curve P1 = 0 taken α1 times, the curve P2 = 0 taken α2 times, &c.
Instead of the equation P1α1P2α2 ... = 0, we may start with an equation u = 0, where u is a function of the order m containing a parameter θ, and for a particular value say θ = 0, of the parameter reducing itself to P1α1P2α2.... Supposing θ indefinitely small, we have what may be called the penultimate curve, and when θ = 0 the ultimate curve. Regarding the ultimate curve as derived from a given penultimate curve, we connect with the ultimate curve, and consider as belonging to it, certain points called “summits” on the component curves P1 = 0, P2 = 0 respectively; a summit Σ is a point such that, drawing from an arbitrary point O the tangents to the penultimate curve, we have OΣ as the limit of one of these tangents. The ultimate curve together with its summits may be regarded as a degenerate form of the curve u = 0. Observe that the positions of the summits depend on the penultimate curve u = 0, viz. on the values of the coefficients in the terms multiplied by θ, θ², ...; they are thus in some measure arbitrary points as regards the ultimate curve P1α1P2α2 ... = 0.
It may be added that we have summits only on the component curves P1 = 0, of a multiplicity α1 > 1; the number of summits on such a curve is in general = (α1² − α1)m1². Thus assuming that the penultimate curve is without nodes or cusps, the number of the tangents to it is = m² − m, = (α1m1 + α2m2 + ...)² − (α1m1 + α2m2 + ...). Taking P1 = 0 to have δ1 nodes and κ1 cusps, and therefore its class n1 to be = m1² − m1 − 2δ1 − 3κ1, &c., the expression for the number of tangents to the penultimate curve is
= (α1² − α1) m1² + (α2² − α2) m2² + ... + 2α1α2m1m2 +
+ α1 (n1 + 2δ1 + 3κ1) + α2 (n2 + 2δ2 + 3κ2) + ...
where a term 2α1α2m1m2 indicates tangents which are in the limit the lines drawn to the intersections of the curves P1 = 0, P2 = 0 each line 2α1α2 times; a term α1(n1 + 2δ1 + 3κ1) tangents which are in the limit the proper tangents to P1 = 0 each α1 times, the lines to its nodes each 2α1 times, and the lines to its cusps each 3α1, times; the remaining terms (α1² − α1)m1² + (α2² − α2)m2² + ... indicate tangents which are in the limit the lines drawn to the several summits, that is, we have (α1² − α1)m1² summits on the curve P1 = 0, &c.
There is, of course, a precisely similar theory as regards line-co-ordinates; taking Π1, Π2, &c., to be rational and integral functions of the co-ordinates (ξ, η, ζ) we connect with the ultimate curve Π1α1Π2α2 ... = 0, and consider as belonging to it, certain lines, which for the moment may be called “axes” tangents to the component curves Π1 = 01, Π2 = 0 respectively. Considering an equation in point-co-ordinates, we may have among the component curves right lines, and if in order to put these in evidence we take the equation to be L1γ1 .. P1α1 ... = 0, where L1 = 0 is a right line, P1 = 0 a curve of the second or any higher order, then the curve will contain as part of itself summits not exhibited in this equation, but the corresponding line-equation will be 1Λδ1 ... Π1α1 = 0, where Λ1 = 0,... are the equations of the summits in question, Π1 = 0, &c., are the line-equations corresponding to the several point-equations P1 = 0, &c.; and this curve will contain as part of itself axes not exhibited by this equation, but which are the lines L1 = 0,... of the equation in point-co-ordinates.