point O; this is an (m − 1, m − 1) correspondence, and the value of k is = 1, hence the number of united points is = 2m − 2 + 2D; the united points are the points of contact of the tangents from O and (as special solutions) the cusps, and we have thus the relation n + κ = 2m − 2 + 2D; or, writing D = ½(m − 1)(m − 2) − δ − κ, this is n = m(m − 1) − 2δ − 3κ, which is right.
The principle in its original form as applying to a right line was used throughout by Chasles in the investigations on the number of the conics which satisfy given conditions, and on the number of solutions of very many other geometrical problems.
There is one application of the theory of the (α, α′) correspondence between two planes which it is proper to notice.
Imagine a curve, real or imaginary, represented by an equation (involving, it may be, imaginary coefficients) between the Cartesian co-ordinates u, u′; then, writing u = x + iy, u′ = x′ + iy′, the equation determines real values of (x, y), and of (x′, y′), corresponding to any given real values of (x′, y′) and (x, y) respectively; that is, it establishes a real correspondence (not of course a rational one) between the points (x, y) and (x′, y′); for example in the imaginary circle u2 + u′2 = (a + bi)2, the correspondence is given by the two equations x2 − y2 + x′2 − y′2 = a2 − b2, xy + x′y′ = ab. We have thus a means of geometrical representation for the portions, as well imaginary as real, of any real or imaginary curve. Considerations such as these have been used for determining the series of values of the independent variable, and the irrational functions thereof in the theory of Abelian integrals, but the theory seems to be worthy of further investigation.
16. Systems of Curves satisfying Conditions.—The researches of Chasles (Comptes Rendus, t. lviii., 1864, et seq.) refer to the conics which satisfy given conditions. There is an earlier paper by J. P. E. Fauque de Jonquières, “Théorèmes généraux concernant les courbes géométriques planes d’un ordre quelconque,” Liouv. t. vi. (1861), which establishes the notion of a system of curves (of any order) of the index N, viz. considering the curves of the order n which satisfy ½n(n + 3) − 1 conditions, then the index N is the number of these curves which pass through a given arbitrary point. But Chasles in the first of his papers (February 1864), considering the conics which satisfy four conditions, establishes the notion of the two characteristics (μ, ν) of such a system of conics, viz. μ is the number of the conics which pass through a given arbitrary point, and ν is the number of the conics which touch a given arbitrary line. And he gives the theorem, a system of conics satisfying four conditions, and having the characteristics (μ, ν) contains 2ν − μ line-pairs (that is, conics, each of them a pair of lines), and 2μ − ν point-pairs (that is, conics, each of them a pair of points,—coniques infiniment aplaties), which is a fundamental one in the theory. The characteristics of the system can be determined when it is known how many there are of these two kinds of degenerate conics in the system, and how often each is to be counted. It was thus that Zeuthen (in the paper Nyt Bydrag, “Contribution to the Theory of Systems of Conics which satisfy four Conditions” (Copenhagen, 1865), translated with an addition in the Nouvelles Annales) solved the question of finding the characteristics of the systems of conics which satisfy four conditions of contact with a given curve or curves; and this led to the solution of the further problem of finding the number of the conics which satisfy five conditions of contact with a given curve or curves (Cayley, Comptes Rendus, t. lxiii., 1866; Collected Works, vol. v. p. 542), and “On the Curves which satisfy given Conditions” (Phil. Trans. t. clviii., 1868; Collected Works, vol. vi. p. 191).
It may be remarked that although, as a process of investigation, it is very convenient to seek for the characteristics of a system of conics satisfying 4 conditions, yet what is really determined is in every case the number of the conics which satisfy 5 conditions; the characteristics of the system (4p) of the conics which pass through 4p points are (5p), (4p, 1l), the number of the conics which pass through 5 points, and which pass through 4 points and touch 1 line: and so in other cases. Similarly as regards cubics, or curves of any other order: a cubic depends on 9 constants, and the elementary problems are to find the number of the cubics (9p), (8p, 1l), &c., which pass through 9 points, pass through 8 points and touch 1 line, &c.; but it is in the investigation convenient to seek for the characteristics of the systems of cubics (8p), &c., which satisfy 8 instead of 9 conditions.
The elementary problems in regard to cubics are solved very completely by S. Maillard in his Thèse, Recherche des caractéristiques des systèmes élémentaires des courbes planes du troisième ordre (Paris, 1871). Thus, considering the several cases of a cubic
| 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
+ 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
