CURVE 727 Imagine a curve, real or imaginary, represented by an equation (involving, it may be, imaginary coefficients) between the Cartesian coordinates u,u ; then, writing u = x + iy, u = x + iy t , the equation determines real values of (x,y), and of (x^y 1 ), corresponding to any given real values of (x ,i/) and (x,y) respectively; that is, it esta blishes a real correspondence (not of course a rational one) between the points (x,y) and (x ,y ) ; for example in the imaginary circle v? + u z ~*(a + bi) 2 , the correspondence is given by the two equations 3?-y^ + x z -y 2 = a?-b 2 , 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 inde pendent variable, and the irrational functions thereof in the theory of Abelian integrals, but the theory seems to be worthy of further investigation. The researches of Chasles (Comptes Rendus, t. Iviii., 1864, et seq.) refer to the conies which satisfy given con ditions. There is an earlier paper by De Jonquieres, " Thcoremes gonoraux concernant les courbes gdomctriques planes d uii 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 },n(n + 3) 1 conditions, then the index N ia the number of these curves which pass through a given arbitrary point. But Chasles in the first of his papers (February 1864), considering the conies which satisfy four conditions, establishes the notion of the two characteristics (fjL, v) of such a system of conies, vi?., p. is the number of the conies which pass through a given arbitrary point, and v is the number of the conies which touch a given arbitrary line. And he gives the theorem, a system of conies satisfying four conditions, and having the characteristics (//., j/) contains 2v - p. line-pairs (that is, conies, each of them a pair of lines), and 2^ v point-pairs (that is, conies, 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 conies in the system, and how often each is to be counted. It was thus that Zeuthen (in the paper Nyt Bydrag, " Contribu tion to the Theory of Systems of Coriics which satisfy four Conditions," Copenhagen, 1865, translated with an addi tion in the Nouvelles Annales) solved the question of finding the characteristics of the systems of conies 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 conies which satisfy five con ditions of contact with a given curve or curves (Cayley, Comptes Rendus, t. Ixiii., 1866), and "On the Curves which satisfy given Conditions" (Phil. Trans., t. clviii., 1868). It may be remarked that although, as a process of in vestigation, it is very convenient to seek for the character istics of a system of conies satisfying 4 conditions, yet what is really determined is in every case the number of the conies which satisfy 5 conditions ; the characteristics of the system (4/>) of the conies which pass through 4p points are (5p), (ip, It), the number of the conies 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 (9/:>), (&p, I/), &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 charac teristics of the systems of cubics (8/>) fec., which satisfy 8 instead of 9 conditions. The elementary problems in regard to cubics are solved very completely by Maillard in his These, Recherche des charactcrisques des syxtemes elcmentaires des courbes planes du troisieme ordre (Paris, 1871). Thus, considering the several cases of a cubic No. of const* 1. With a given cusp 5 cusp on given line 6 6. cusp 7 a given node 6 node on given line 7 node ... s 7. non-singular 9 he determines in every case the characteristics (p., v) of the corresponding systems of cubics (4/>), (3p, II), &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 " Almiridelige Egenskaber, <fec.," 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 (p, v) 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 char acteristics /j., v, and 18 other quantities, in all 20 quantities, a set of 24 equations (equivalent to 23 indepen dent 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 in 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. 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-coordinates may break up : thus if P],P 2 , . . be rational and integral functions of the coordinates (x,y,z) of the orders m^m^. . respectively, we have the curve P 1 a iP 2 a 2. . =0, of the order m, = a^m^ + a 2 ra 2 + . . , composed of the curve Pj = taken Oj times, the curve P 2 = taken a 2 times, &c. Instead of the equation P 1 a )P !! a 2. .=0, we may start with an equation ti = 0, where u is a function of the order m containing a parameter 6, and for a particular value say =*= 0, of the parameter reducing itself to Pj a iP 2 a 2. . . Supposing 6 indefinitely small, we have what maybe called the penultimate curve, and when = 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 P 1 = 0, P 2 = 0, respectively ; a summit 2 is a point such that, drawing from an arbitrary point the tangents to the penultimate curve, we have 02 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 w=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 6, 2 ,. . ; they are thus in some measure arbitrary points as regards the ultimate curve P, a i P./2. . = 0. It may be added that we have summits only on the component curves Pj^O, of a multiplicity aj >1 ; the number of summits on such a curve is in general = (a^ - aj m^. Thus assuming that the penultimate curve is without nodes or cusps, the number of the tangents to it is = ??i 2 -m, = (o 1 7 1 + o a 7?z a + . . ) 2 - (a^n^ + a,w a + . .). taking Pj-0 to have 5 X nodes and K L cusps, and therefore its class TJj to be-=m 1 a -m 1 -25 1 -3ft 1 &c., the expression for the number of tangents to the penultimate curve is = (af - a) m^ + (a 2 2 - a a ) m a 3 + . . + 2a 1 a.,m 1 w !i + . + !(! + 25 X + 3/c x ) + o s (i a + 28,, + 3if a ) + . . where a term 2a 1 a 2 r/r 1 ?n 2 indicates tangents which are in the limit the lines drawn to the intersections of the curves P 1 =0,P 2 = each line 2a,a 2 times ; a term a, (n t + 1ti l + 3/tj) tangents which are in the limit the proper tangents to P, = each a x times, the lines to its
nodes each 2a t times, and the lines to its cusps each 3a x times ;