Page:Encyclopædia Britannica, Ninth Edition, v. 6.djvu/756

720 CURVE line obtain the node with two imaginary tangents (con jugate or isolated point, or acnode), nor again the real double tangent with two imaginary points of contact ; but this is of little consequence, since in the general theory the distinction between real and imaginary is not attended to. The singularities (1) and (3) have been termed proper singularities, and (2) and (4) improper ; in each of the first- mentioned cases there is a real singularity, or peculiarity in the motion; in the other two cases there is not; in (2) there is not when the point is first at the node, or when it is secondly at the node, any peculiarity in the motion ; the singularity consists in the point coming twice into the same position ; and so in (4) the singularity is in the line coming twice into the same position. Moreover (1) and (2) are, the former a proper singularity, and the latter an improper singularity, as regards the motion of the point ; and similarly (3) and (4) are, the former a proper singu larity, and the latter an improper singularity, as regards tJie motion of the line. But as regards the representation of a curve by an equa tion, the case is very different. First, if the equation be in point -coordinates, (3) and (4) are in a sense not singularities at all. The curve (*$#, y, z} m 0, or general curve of the order m, has double tangents and inflexions ; (2) presents itself as a singu larity, for the equations d x (*^c, y, z} m 0, d y (*c, y, z) m = 0, d,(*$x, y, z) = 0, implying (*$x, y, z} m = 0), are not in general satisfied by any values (a, b, c) whatever of (x, y, z}, but if such values exist, then the point (a, b, c) is a node or double point ; and (1) presents itself as a further singularity or sub-case of (2), a cusp being a double point for which the two tangents become coincident. In line-coordinates all is reversed : (1) and (2) are not singularities ; (3) presents itself as a sub-case of (4). The theory of compound singularities will be referred to further on. In regard to the ordinary singularities, we have m, the ordei class numl er of double points, cusps, double tangents, inflexions ; and this being so, Pliicker s " six equations " are (1) n~ m (m - I) - 25 - SK, (2) i = 3m (m-2)- 68 -8*t, (3) r = ii (TO - 2) (?w 2 - 9) - (m 2 - m - 6)(2S (4) (5) (6) = n (n- l)-2T- = 3tt (w-2)-6r- It is easy to derive the further forms- (7) L-K =B(n-m), (8) 2(r -8) = (n - m) (n + m -9), (9) m(m + 3) - 8 - 2ie = k n(n + 3) - T - 2, (10) |(TO-l)(m-2)-8-/c=4 (7t-l) (n-2)-r-t, (11, 12) 7?i 3 -2S-3/<: =?i 2 -2T-3z, =m + n, the whole system being equivalent to three equations only ; and it may be added that using a to denote the equal quantities 3m + 1 and 3n 4- K everything may be expressed in terms of m, n, a. We have K = a- 3??, i - a - 3m, 2r = ?i -n It is implied in Pliicker s theorem that, m, n, 8, K, r, i signifying as above in regard to any curve, then in regard to the reciprocal curve n, m, T, j, 8, K will have the same significations, viz., for the reciprocal curve these letters denote respectively the order, class, number of nodes, cusps, double tangents, and inflexions. The expression 7rt(??i + 3)-8- IK is that of the number of the disposable constants in a curve of the order m with 8 nodes and K cusps (in fact that there shall be a node is 1 condition, a cusp 2 conditions) and the equation (9) thus expresses that the curve and its reciprocal contain each of them the same number of disposable constants. For a curve of the order m, the expression m (m - I) - 8 - K is termed the "deficiency" (as to this more hereafter) ; the equation (10) expresses therefore that the curve and its reciprocal have each of them the same deficiency. The relations ?rt 2 - 28- 3/c= n 2 - 2r- 3t, = m + 7i,present themselves in the theory of envelopes, as will appear further on. With regard to the demonstration of Pliicker s equations it is to be remarked that we are not able to write down the equation in point-coordinates of a curve of the order m, having the given numbers 8 and K (f nodes and cusps. We can only use the general equation (*$, y, z) m = 0, say for shortness u = Q, of a curve of the mth order, which equation, so long as the coefficients remain arbitrary, represents a curve without nodes or cusps. Seeking then, for this curve, the values n, i, T of the class, number of inflexions, and number of double tangents, first, as regards the class, this is equal to the number of tangents which can be drawn to the curve from an arbitrary point, or what is the same thing it is equal to the number of the points of contact of these tangents. The points of contact are found as the intersections of the curve u by a curve depending on the position of the arbitrary point, and called the " first polar " of this point ; the order of the first polar is = m-l, and the number of intersections is thus =m(ml). But it can be laown, analytically or geometrically, that if the given curve has a node, the first polar passes through this node, which therefore counts as two intersections, and that if the curve has a cusp, the first polar passes through the cusp, touching the curve there, and hence the cusp counts as three intersections. But, as is evident, the node or cusp is not a point of contact of a proper tangent from the arbitrary point; we have, there fore, for a node a diminution 2, and for a cusp a diminu tion 3, in the number of the intersections ; and thus, for a curve with 8 nodes and K cusps, there is a diminution 28 + 3/c, and the value of n is n = m(m - 1) - 28 - 3*. Secondly, as to the inflexions, the process is a similar one ; it can be shown that the inflexions are the intersec tions of the curve by a derivative curve called (after Hesse who first considered it) the Hessian, defined geometrically as the locus of a point such that its conic polar in regard to the curve breaks up into a pair of lines, and which has an equation H = 0, where H is the determinant formed with the second differential coefficients of u in regard to the variables (x, y, z) ; H = is thus a curve of the order 3(m - 2), and the number of inflexions is = 3m(m 2). But if the given curve has a node, then not only the Hessian passes through the node, but it has there a node the two branches at which touch respectively the two branches of the curve ; and the node thus counts as six intersections ; so if the curve has a cusp, then the Hessian not only passes through the cusp, but it has there a cusp through which it again passes, that is, there is a cuspidal branch touching the cuspidal branch of the curve, and besides a simple branch passing through the cusp, and hence the cusp counts as eight intersections. The node or cusp is not an inflexion, and we have thus for a node a diminution 6, and for a cusp a diminution 8, in the number of the intersections ; hence for a curve with & nodes and K cusps, the diminution is =68 + 8*, and the number of inflexions is i = 3m(m - 2) 68 - 8*. Thirdly, for the double tangents ; the points of contact of these are obtained as the intersections of the curve by a curve II = 0, which has not as yet been geometrically defined, but which is found analytically to be of the order (m - 2)(m 2 - 9) ; the number of intersections is thus = m(m - 2)(m 2 - 9) ; but if the given curve has a node then there is a diminution = 4(? 2 - m 6), and if it has a cusp

then there is a diminution = 6(m 2 m 6), where, how-