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
α1 + 4α2 + 9α3 + ... (n − 1)² αn−1 = n² − 1;
but to this must be joined the condition
α1 + 3α2 + 6α3 ... + ½n(n − 1) αn−1 = ½n (n + 3) − 2
(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
α2 + 3α3 ... + ½ (n − 1) (n − 2) αn−1 = ½ (n − 1) (n − 2),
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 (*
