Page:Encyclopædia Britannica, Ninth Edition, v. 6.djvu/762

726 CURVE and in its equation writing x : y :d  : 6 : cj> we have </> a certain irrational function of 0, and the theorem is that the coordinates x,y,z of any point of the given curve can be expressed as proportional to rational and integral functions of 6, <f>, that is, of 6 and a certain irrational function of 8. In particular if D = 0, that is, if the given curve be unicursal, the transformed curve is a line, <f> is a mere linear function of 0, and the theorem is that the coordinates x,y,z of a point of the unicursal curve can be expressed as proportional to rational and integral functions of ; it is easy to see that for a given curve of the order m, these functions of 6 must be of the same order m. If D=l, then the transformed curve is a cubic; it can be shown that in a cubic, the axes of coordinates being properly chosen, fy can be expressed as the square root of a quartic function of ; and the theorem is that the coordinates x,y,z of a point of the bicursal curve can be expressed as proportional to rational and integral functions of 0, and of the square root of a quartic function of 0. And so if D = 2, then the transformed curve is a nodal quartic ; < can be expressed as the square root of a sextic function of 0, and the theorem is, that the coordinates x,y,z of a point of the tricursal curve can be expressed as proportional to rational and integral functions of 0, and of the square root of a sextic function of 0. But D = 3, we have no longer the like law, viz. $ is not expressible as the square root of an octic function of 6. Observe that the radical, square root of a quartic func tion, 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 coordinates x, y, z of a point of the bicursal curve, or in particular the coordinates of a point of the cubic, can be expressed as proportional to rational and integral functions of the elliptic functions sn, cnu, dnw ; in fact, taking the radical to be Jl - Q 1 . 1 - k~6 2 , and writing 6 = snu, the radical becomes cn. dim ; 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 (#, y, 2) 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 geome- triche 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 docs 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 coordinates x,y,z, and X ,Y ,Z rational and integral functions, all of the same order, of the coordinates a; ,?/,^, and there is thus a (1,1) correspondence given by these equations between the two points (x,y,z) and (a: ,?/,/). To explain this, observe that starting from the equations x :?/ :z = X : Y :Z, to a given point (x,y,z) there corresponds one point (x ,]/,^), but that if n be the order of the functions X,Y,Z, then to a given point a; ,7/ ,z there would, if the curves X = 0, Y = 0, Z = had no common intersections, cor respond n 2 points (x,y,z). If, however, the functions are such that the curves X = 0, Yr:0, Z = have k common intersections, then among the ?i 2 points are included these k points, which are fixed points independent of the point (a^j/ X) ; so that, disregarding these fixed points, the number of points (x,y,z) corresponding to the given point (x ,-!/,^) is=n*-k; and in particular if & = n a -l, then we have one corresponding point ; and hence the original system of equations a/ :/ :/=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 = have ?i 2 - 1 common intersections. The most simple example is in the two systems of equations xf : y : z = yz : zx : xy and x :y : z y z : z x : x y 1 ; where 2/2 = 0, zx = 0, xy = are conies (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 n, 2 -! 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 Q have a, +a^ + a. 3 ... +a n -i common intersec tions, where the a x points are ordinary points, the Oj points are double points, the a 3 points are triple points, &c., on each curve, we have the condition but to this must be joined the condition a 1 + 3a, + 6a 3 ... + i(ji -!)(- 2) a n - 1 <=1 t n(n + 3) - 2, (without which the transformation would be illusory) ; and the conclusion is that a lt a^, ...a. n - may be any numbers satisfying these two equations. It may be added that the two equations together give a. 3 + 3a 3 ...+i(n-l)(tt-2) 0,,-! = ^ (n -!)(- 2), which expresses that the curves X = 0, Y = 0, Z = are unicursal. The transformation maybe applied to any curve u = Q, which is thus rationally transformed into a curve w = 0, by a rational trans formation 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 conies which satisfy given conditions, and to other geometrical questions, contained in the Comptes Rendus, t. Iviii. et seq. (1864 to the present time). The theorem of united points in regard to points in a right lino 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 a double courbure dont les points peuvent se determiner individuellement applica tion du principe de correspondence dans la the orie de ces courbes." The theorem is as follows . if in a unicursal curve two points have an (a,/3) correspondence, then the number of united points (or points each corresponding to itself) is = o + /J. In fact in a unicursal curve the coordinates 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(a,/3) correspondence between the two points is given by an equation of the form (*0,l) a (<t>,I)P=*Q between their parameters and <t> ; at a united point <j> = 0, and the value of is given by an equation of the order a+. The extension to curves of any given de ficiency D was made in the memoir of Cayley, "On the correspondence of two points on a curve," Proc. Lond. Math. Soc., t. i.(1866), viz., taking P,P as the corresponding points in an (a, a ) 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 containing the coordinates of P as parameters, and having with the given curve k intersections at the point P, then the number of united points is == o + a + 2&D ; and more generally, if the curve intersect the given curve in a set of points P each p times, a set of points Q each q 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 (a,o ) correspondence with a number = of united points, (P,Q ) points having a (&,$ ) correspondence with a number =6 of united points, and so on, the theorem is that we have p (a - a - o ) + q(b - ft - ) + ... = 2&D. 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 in vestigate the class of a curve, number of inflexions, <kc., in fact, Pliicker s equations ; but it is necessary to tako 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 corre sponding points P,P be such that the line joining them passes through a given point ; this is an (m ,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 and (as special solutions) the cusps, and we have thus the relation n + K = 2m - 2 + 2D ; or, writing D = ^(m - l)(m - 2) - 5 - c, this is n = m(m - 1) - 25 - 3/c, 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 conies which satisfy given conditions, and on the number of solutions of very many other geo metrical problems. There is one application of the theory of the (a, a ) cor

respondence between two planes which it is proper to notice.