θ, 1)α(φ, 1)β = 0 between their parameters θ and φ; at a united point φ = θ, and the value of θ is given by an equation of the order α + β. The extension to curves of any given deficiency D was made in the memoir of Cayley, “On the correspondence of two points on a curve,”—Proc. Lond. Math. Soc. t. i. (1866; Collected Works, vol. vi. p. 9),—viz. taking P, P′ as the corresponding points in an (α, α′) correspondence on a curve of deficiency D, and supposing that when P is given the corresponding points P′ are found as the intersections of the curve by a curve Θ containing the co-ordinates of P as parameters, and having with the given curve k intersections at the point P, then the number of united points is a = α + α′ + 2kD; and more generally, if the curve Θ intersect the given curve in a set of points P′ each p times, a set of points Q′ each g times, &c., in such manner that the points (P, P′) the points (P, Q′) &c., are pairs of points corresponding to each other according to distinct laws; then if (P, P′) are points having an (α, α′) correspondence with a number = a of united points, (P, Q′) points having a (β, β′) correspondence with a number = b of united points, and so on, the theorem is that we have
p (a − α − α′) + q (b − β − β′) + ... = 2kD.
The principle of correspondence, or say rather the theorem of united points, is a most powerful instrument of investigation, which may be used in place of analysis for the determination of the number of solutions of almost every geometrical problem. We can by means of it investigate the class of a curve, number of inflections, &c.—in fact, Plücker’s equations; but it is necessary to take account of special solutions: thus, in one of the most simple instances, in finding the class of a curve, the cusps present themselves as special solutions.
Imagine a curve of order m, deficiency D, and let the corresponding points P, P′ be such that the line joining them passes through a given 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