the order m, the number δ + κ of nodes and cusps is at most
= ½(m − 1)(m − 2); for a given curve the deficiency of the actual
number of nodes and cusps below this maximum number, viz.
½(m − 1)(m − 2) − δ − κ, is the “Geschlecht” or “deficiency,”
of the curve, say this is = D. When D = 0, the curve is said to be
unicursal, when = 1, bicursal, and so on.
The general theorem is that two curves corresponding rationally to each other have the same deficiency. [In particular a curve and its reciprocal have this rational or (1, 1) correspondence, and it has been already seen that a curve and its reciprocal have the same deficiency.]
A curve of a given order can in general be rationally transformed into a curve of a lower order; thus a curve of any order for which D = 0, that is, a unicursal curve, can be transformed into a line; a curve of any order having the deficiency 1 or 2 can be rationally transformed into a curve of the order D + 2, deficiency D; and a curve of any order deficiency = or > 3 can be rationally transformed into a curve of the order D + 3, deficiency D.
Taking x′, y′, z′ as co-ordinates of a point of the transformed curve, and in its equation writing x′ : y′ : z′ = 1 : θ : φ we have φ a certain irrational function of θ, and the theorem is that the co-ordinates x, y, z of any point of the given curve can be expressed as proportional to rational and integral functions of θ, φ, that is, of θ and a certain irrational function of θ.
In particular if D = 0, that is, if the given curve be unicursal, the transformed curve is a line, φ is a mere linear function of θ, and the theorem is that the co-ordinates x, y, z of a point of the unicursal curve can be expressed as proportional to rational and integral functions of θ; it is easy to see that for a given curve of the order m, these functions of θ must be of the same order m.
If D = 1, then the transformed curve is a cubic; it can be shown that in a cubic, the axes of co-ordinates being properly chosen, φ can be expressed as the square root of a quartic function of θ; and the theorem is that the co-ordinates x, y, z of a point of the bicursal curve can be expressed as proportional to rational and integral functions of θ, and of the square root of a quartic function of θ.
And so if D = 2, then the transformed curve is a nodal quartic; φ can be expressed as the square root of a sextic function of θ and the theorem is, that the co-ordinates x, y, z of a point of the tricursal curve can be expressed as proportional to rational and integral functions of θ, and of the square root of a sextic function of θ. But D = 3, we have no longer the like law, viz. φ is not expressible as the square root of an octic function of θ.
Observe that the radical, square root of a quartic function, is connected with the theory of elliptic functions, and the radical, square root of a sextic function, with that of the first kind of Abelian functions, but that the next kind of Abelian functions does not depend on the radical, square root of an octic function.
It is a form of the theorem for the case D = 1, that the co-ordinates x, y, z of a point of the bicursal curve, or in particular the co-ordinates of a point of the cubic, can be expressed as proportional to rational and integral functions of the elliptic functions snu, cnu, dnu; in fact, taking the radical to be √1 − θ²·1 − k²θ², and writing θ = snu, the radical becomes = cnu, dnu; and we have expressions of the form in question.
It will be observed that the equations x′ : y′ : z′ = X : Y : Z before mentioned do not of themselves lead to the other system of equations x : y : z = X′ : Y′ : Z′, and thus that the theory does not in anywise establish a (1, 1) correspondence between the points (x, y, z) and (x′, y′, z′) of two planes or of the same plane; this is the correspondence of Cremona’s theory.
In this theory, given in the memoirs “Sulle trasformazioni geometriche delle figure piani,” Mem. di Bologna, t. ii. (1863) and t. v. (1865), we have a system of equations x′ : y′ : z′ = X : Y : Z which does lead to a system x : y : z = X′ : Y′ : Z′, where, as before, X, Y, Z denote rational and integral functions, all of the same order, of the co-ordinates x, y, z, and X′, Y′, Z′ rational and integral functions, all of the same order, of the co-ordinates x′, y′, z′, and there is thus a (1, 1) correspondence given by these equations between the two points (x, y, z) and (x′, y′, z′). To explain this, observe that starting from the equations of x′ : y′ : z′ = X : Y : Z, to a given point (x, y, z) there corresponds one point ( x′, y′, z′), but that if n be the order of the functions X, Y, Z, then to a given point x′, y′, z′ there would, if the curves X = 0, Y = 0, Z = 0 had no common intersections, correspond n2 points (x, y, z). If, however, the functions are such that the curves X = 0, Y = 0, Z = 0 have k common intersections, then among the n2 points are included these k points, which are fixed points independent of the point ( x′, y′, z′); so that, disregarding these fixed points, the number of points (x, y, z) corresponding to the given point ( x′, y′, z′) is = n2 − k; and in particular if k = n2 − 1, then we have one corresponding point; and hence the original system of equations x′ : y′ : z′ = X : Y : Z must lead to the equivalent system x : y : z = X′ : Y′ : Z′; and in this system by the like reasoning the functions must be such that the curves X′ = 0, Y′ = 0, Z′ = 0 have n′2 − 1 common intersections. The most simple example is in the two systems of equations x′ : y′ : z′ = yz : zx : xy and x : y : z = y′z′ : z′ x′ : x′y′; where yz = 0, zx = 0, xy = 0 are conics (pairs of lines) having three common intersections, and where obviously either system of equations leads to the other system. In the case where X, Y, Z are of an order exceeding 2 the required number n2 − 1 of common intersections can only occur by reason of common multiple points on the three curves; and assuming that the curves X = 0, Y = 0, Z = 0 have α1 + α2 + α3 ... + αn−1 common intersections, where the α1 points are ordinary points, the α2 points are double points, the α3 points are triple points, &c., on each curve, we have the condition
but to this must be joined the condition
(without which the transformation would be illusory); and the conclusion is that α1, α2, ... αn−1 may be any numbers satisfying these two equations. It may be added that the two equations together give
which expresses that the curves X = 0, Y = 0, Z = 0 are unicursal. The transformation may be applied to any curve u = 0, which is thus rationally transformed into a curve u′ = 0, by a rational transformation such as is considered in Riemann’s theory: hence the two curves have the same deficiency.
Coming next to Chasles, the principle of correspondence is established and used by him in a series of memoirs relating to the conics which satisfy given conditions, and to other geometrical questions, contained in the Comptes rendus, t. lviii. (1864) et seq. The theorem of united points in regard to points in a right line was given in a paper, June–July 1864, and it was extended to unicursal curves in a paper of the same series (March 1866), “Sur les courbes planes ou à double courbure dont les points peuvent se déterminer individuellement—application du principe de correspondance dans la théorie de ces courbes.”
The theorem is as follows: if in a unicursal curve two points have an (α, β) correspondence, then the number of united points (or points each corresponding to itself) is = α + β. In fact in a unicursal curve the co-ordinates of a point are given as proportional to rational and integral functions of a parameter, so that any point of the curve is determined uniquely by means of this parameter; that is, to each point of the curve corresponds one value of the parameter, and to each value of the parameter one point on the curve; and the (α, β) correspondence between the two points is given by an equation of the form (*≬ 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
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
