Page:Encyclopædia Britannica, Ninth Edition, v. 6.djvu/758

722 C U B, Y E is btill in so far incomplete that we have no properties of the cui-ve II = 0, to distinguish one such curve from the several other curves which pass through the points of con tact of the double tangents. A quartic curve has 28 double tangents, their points^ of contact determined as the intersections of the curve by a curve 11 = of the order 14, the equation of which in a very elegant form was first obtained by Hesse (1849). Investigations in regard to them are given by Pliicker in the Theorie der Algebraischen Curven, and in two memoirs by Hesse and Steiner (Crelle, t. xlv., 1855), in respect to the triads of double tangents which have their points of contact on a conic, and other like relations. It was assumed by Pliicker that the number of real double tangents might be 28, 16, 8, 4, or 0, but Zeuthen has recently found that the last case does not exist. The Hessian A has just been spoken of as a covariant of the form u ; the notion of invariants and covariants belongs rather to the fotm u than to the curve u = represented by means of this form ; and the theory may be very briefly referred to. A curve u = may have some iuvariantive property, viz., a property independent of the particular axes of coordinates used in the representation of the curve by its equation ; for instance, the curve may have a node, and in order to this, a relation, say A = 0, must exist between the coefficients of the equation; supposing the axes of coordi nates altered, so that the equation becomes u = 0, and writ ing A = for the relation between the new co-efficients, then the relations A = 0, A = 0, as two different expressions of the same geometrical property, must each of them imply the other ; this can only be the case when A, A are func tions differing only by a constant factor, or say, when A is an invariant of u. If, however, the geometrical property requires two or more relations between the coefficients, say A = 0, B = 0, &c., then we must have between the new coefficients the like relations, A = 0, B -= 0, &c., and the two systems of equations must each of them imply the other; when this is so, the system of equations, A 0, B = 0, &c., is said to be invariantive, but it does not follow that A, B, &c., are of necessity invariants of u. Similarly, if we have a curve U = derived from the curve u = in a manner independent of the particular axes of co-ordinates, then from the transformed equation u deriving in like manner the curve U = 0, the two equations U = 0, U = must each of them imply the other ; and when this is so, TJ will be a covariant of u. The case is less frequent, but it may arise, that there are covariant systems U = 0, V = &c., and U = 0, V = 0, &c., each implying the other, but where the functions U, V, &c., are not of necessity covariants of u. The theory of the invariants and covariants of a ternary cubic function u has been studied in detail, and brought into connection with the cubic curve u = ; but the theory of the invariants and covariants for the next succeeding case, the ternary quartic function, is still very incomplete. In further illustration of the Pliickerian dual generation of a curve, we may consider the question of the envelope of a variable curve. The notion is very probably older, but it is at any rate to be found in Lagrange s Theorie des Fonctions analytiques (1798) ; it is there remarked that the equation obtained by the elimination of the para meter a from an equation / (x, y, a) = and the deri ved equation in respect to a is a curve, the envelope of the series of curves represented by the equation/ (x, y, a) = in question. To develop the theory, consider the curve corre sponding to any particular value of the parameter ; this has with the consecutive curve (or curve belonging to the con secutive value of the parameter) a certain number of inter sections, and of common tangents, which may be considered as the tangents at the intersections ; and the so-called envelope is the curve which is at the same time gene rated by the points of intersection and enveloped by the common tangents ; we have thus a dual generation. But the question needs to be further examined. Suppose that in general the variable curve is of the order m with 8 nodes and K cusps, and therefore of the class n with T double tangents and t inflexions, m, n, 8, K, T, i being connected by the Pliickerian equatiocs, the number of nodes or cusps may be greater for particular values of the parameter, but this is a speciality which may be here disregarded. Con sidering the variable curve corresponding to a given value of the parameter, or say simply the variable curve, the consecutive curve has then also 8 and K nodes and cusps, consecutive to those of the variable curve ; and it is easy to see that among the intersections of the two curves we have tho nodes each counting twice, and the cusps each counting three times ; the number of the remaining inter sections is = m 2 - 28 - 3*. Similarly among the common tangents of the two curves we have the double tangents each counting twice, and the stationary tangents each counting three times, and the number of the remaining common tangents is = n z - 2r - 3t ( = m 2 - 28 - 3/c, inas much as each of these numbers is as was seen = m + n). At any one of the m 2 - 28 - 3/< points the variable curve and the consecutive curve have tangents distinct from yet infinitesimally near to each other, and each of these two tangents is also infinitesimally near to one of the ?4 2 2r 3i common tangents of the two curves ; whence, attending only to the variable curve, and considering the consecutive curve as coming into actual coincidence with it, the n 2 - 2r 3t common tangents are the tangents to the variable curve at the m 2 - 28 3/< points respectively, and the envelope is at the same time generated by the m 2 -2S-3/c points, and enveloped by the n 2 2r-3i tangents ; we have thus a dual generation of the en velope, which only differs from PKicker s dual generation, in that in place of a single point and tangent we have the group of m. 2 - 28 - 3/c points and ?i 2 2r - 3i tangents. The parameter which determines the variable curve may be given as a point upon a given curve, or say as a para metric point ; that is, to the different positions of the par- metric point on the given curve correspond the different variable curves, and the nature of the envelope will thus depend on that of the given curve ; we have thus the envelope as a derivative curve of the given curve. Many well-known derivative curves present themselves in thia manner ; thus the variable curve may be the normal (or line at right angles to the tangent) at any point of the given curve ; the intersection of the consecutive normals is the centre of curvature ; and we have the evolute as at once the locus of the centre of curvature and the envelope of the normal. It may be added that the given curve is one of a series of curves, each cutting the several normals at right angles. Any one of these is a "parallel" of the given curve ; and it can be obtained as the envelope of a circle of constant radius having its centre on the given curve. We have in like manner, as derivatives of a given curve, the caustic, catacaustic, or diacaustic, as the case may be, and the secondary caustic, or curve cutting at right angles the reflected or refracted rays. We have in much that precedes disregarded, or at least been indifferent to, reality ; it is only thus that the concep tion of a curve of the m-ih order, as one which is met by every right line in m points, is arrived at ; and the curve itself, and the line which cuts it, although both are tacitly assumed to be real, may perfectly well be imaginary. For real figures we have the general theorem that imaginary intersections, &c., present themselves in conjugate pairs ; hence, in particular, that a curve of an even order is met by a line in an even number (which may be = 0) of points :

a curve of an odd order in an odd number of points, hence