CURVE 717 NP) =x, and MP (or ON) = y, from these two axes respec tively ; where x is regarded as positive or negative according as it is in the sense Ox or Ox from O; and similarly y as positive or negative according as it is in the sense Oy or O// from O ; or what is the same thing x y In quadrant xy 1 , or N.E., we have + + xfy N.W + xi/ S.E. +
- V s.w.
Any relation whatever between (x, y] determines a curve, and conversely every curve whatever is determined by a relation between (x, y). Observe that the distinctive feature is in the exclusive use of such determination of a curve by means of its equa- tijn. The Greek geometers were perfectly familiar with ths property of an ellipse which in the Cartesian notation is + r 2 = 1, the equation of the curve ; but it was as one Ct of a number of properties, and in no wise selected out of ths others for the characteristic property of the curve. 1 We obtain from the equation the notion of an algebraical or geometrical as opposed to a transcendental curve, viz., an algebraical or geometrical curve is a curve having an equation }?(x, y) = 0, where F(x, y) is a rational and integral function of the coordinates (x, y) ; and in what follows we attend throughout (unless the contrary is stated) only to such curves. The equation is sometimes given, and may conveniently be used, in an irrational form, but we always imagine it reduced to the foregoing rational aul integral form, and regard this as the equation of the curve. And we have hence the notion of a curve of a given order, viz., the order of the curve is equal to that of the term or terms of highest order in the coordinates (x, y] conjointly in the equation of the curve ; for instance, xij - 1 = is a curve of the second order. It is to be noticed here that the axes of coordinates may be any two lines at right angles to each other whatever ; and that the equation of a curve will be different according to the selection of the axes of coordinates ; but the order is independent of the axes, and has a determinate value for any given curve. We hence divide curves according to their order, viz., a curve is of the first order, second order, third order, &c., ac- cjrding as it is represented by an equation of the first order, ax + by + c = 0, or say (*$#, y, 1 ) = ; or by an equation of the second order, ax" 1 + 2, ixy + Ijy 1 + 2fy + 2yx + c = 0, say (*$x, y, I) 2 = ; or by an equation of the third order, &c ; or what is the sam.9 thing, according as the equation is linear, quadric, cubic, &c. 1 There is no exercise more profitable for a student than that of tracing a curve from its equation, or say rather that of so tracing a considerable number of curves. And he should make the equations for himself. The equation should be in the first Instance a purely numeri cal one, where y is given or can be found as an explicit function of x; here, by giving different numerical values to x, the corresponding values of y may be found ; and a sufficient number of points being thus deter mined, the curve is traced by drawing a continuous line through these points. The next step should be to consider an equation involving literal coefficients ; thus, after such curves as y=x*, y=x(x-l)(x- 2), y = (x-l)V x-2, &c., he should proceed to trace such curves as y=(x-a)(x-b)(x-c), y (x-a)*Jx-b, &c., and endeavour to as certain for what different relations of equality or inequality between the coefficients the curve will assume essentially or notably distinct forms. The purely numerical equations will present instances" of nodes, cusps, inflexions, double tangents, asymptotes, &c., specialities which he should be familiar with before he has to consider their general theory. And he may then consider an equation such that neither coordinate can be expressed as an explicit function of the other of them (practically, an equation such as x 3 + y 3 - 3xy 0, which requires the solution of a cubic equation, belongs to this class); the problem of tracing the curve here frequently requires special methods, and it may easily be such as to require and serve as an exercise for the powers of an advanced alge braist. A curve of the first order is a right line ; and conversely every right line is a curve of the first order. A curve of the second order is a conic, or as it is also called a quadric ; and conversely every conic, or quadric, is a curve of the second order. A curve of the third order is called a cubic ; one of the fourth order a quartic ; and so on. A curve of the order m has for its equation (*$#, y, 1) = 0; and when the coefficients of the function are arbitrary, the curve is said to be the general curve of the order m. The number of coefficients is }>(m+ l)(m + 2) ; but there is no loss of generality if the equation be divided by one coefficient so as to reduce the coefficient of the corre sponding term to unity, hence the number of coefficients may be reckoned as (m + l)(m + 2) - 1, that is, fyn(m + 3) ; and a curve of the order m may be made to satisfy this number of conditions ; for example, to pass through %m(m + 3) points. It is to be remarked that an equation may break up ; thus a quadric equation may be (ax + by + c)(a x + b y + c) = 0, breaking up into the two equations ax + by + c Q, a x + b y + c = Q, viz., the original equation is satisfied if either of these is satisfied. Each of these last equations represents a curve of the first order, or right line ; and the original equation represents this pair of lines, viz., the pair of lines is considered as a quadric curve. But it is an improper quadric curve ; and in speaking of curves of the second or any other given order, we frequently imply that the curve is a proper curve represented by an equa tion which does not break up. The intersections of two curves are obtained by combining their equations ; viz., the elimination from the two equa tions of y (or x) gives for x (or y} an equation of a certain order, say the resultant equation ; and then to each value of x (or y) satisfying this equation there corresponds in general a single value of y (or x), and consequently a single point of intersection ; the number of intersections is thus equal to the order of the resultant equation in x (or y). Supposing that the two curves are of the orders m, n, respectively, then the order of the resultant equation is in general and at most = mn ; in particular, if the curve of the order n is an arbitrary line (n 1 ), then the order of the resultant equation is = ra/ and the curve of the order m meets therefore the line in m points. But the resultant equation may have all or any of its roots imaginary, and it is thus not always that there are m real intersections. The notion of imaginary intersections, thus presenting itself, through algebra, in geometry, must be accepted in geometry and it in fact plays an all-important parkin modern geometry. As in algebra we say that an equation of the m-th order has m roots, viz., we state this generally without in the first instance, or it may be without ever, distinguishing whether these are real or imaginary ; so in geometry we say that a curve of the m-th order is met by an arbitrary line in m points, or rather we thus, through algebra, obtain the proper geometrical definition of a curve of the TO-th order, as a curve which is met by an arbitrary line in m points (that is, of course, in m, and not more than m, points). The theorem of the m intersections has been stated in regard to an arbitrary line; in fact, for particular lines the resultant equation may be or appear to be of an order less than m ; for instance, taking m = 2, if the hyperbola xy - 1 = be cut by the line y = ft, the resultant equation in x is fix - 1 = 0, and there is apparently only the intersection - , = /3); but the theorem is, in fact, true for every line whatever : a curve of the order m meets every line whatever in precisely m points. We have, in the case just
referred to, to take account of a point at infinity on the
