CURVE 721 ever, it is to be noticed that the factor (m? - m 6) is in the case of a curve having only a node or only a cusp the number of the tangents which can be drawn from the node or cusp to the curve, and is used as denoting the number of these tangents, and ceases to be the correct expression if the number of nodes and cusps is greater than unity. Hence, in the case of a curve which has 8 nodes and K cusps, the apparent diminution 2(m 2 - m - 6)(28 + SK) is too great, and it has in fact to be diminished by 2 {28(8 - 1) + 68*c + f *(* - 1)} , orthehalf thereof is4 for each pair of nodes, 6 for each combination of a node and cusp, and 9 for each pair of cusps. We have thus finally an ex pression for 2r, = m(m - 2)(m 2 - 9) - &c. ; or dividing the whole by 2, we have the expression for T given by the third of Pliicker s equations. It is obvious that we cannot by consideration of the equation ?< = 0in point-coordinates obtain the remaining three of Pliicker s equations ; they might be obtained in a precisely analogous manner by means of the equation v = in line-coordinates, but they follow at once from the^ principle of duality, viz., they are obtained by the mere interchange of m, 8, K with n, T, i respectively. To complete Pliicker s theory it is necessary to take account of compound singularities ; it might be possible, but it is at any rate, difficult to effect this by considering the curve as in course of description by the point moving along the rotating line ; and it seems easier to consider the compound singularity as arising from the variation of an actually described curve with ordinary singularities. The most simple case is when three double points come into coincidence,, thereby giving rise to a triple point ; and a somewhat more complicated one is when we have a cusp of the second kind, or node-cusp arising from the coincidence of a node, a cusp, an inflexion, and a double tangent, as shown in the annexed figure, which represents the singularities as on the point of coalescing. The general conclusion (see Cayley, Quart. Math. Jour., t. vii., 1866, " On the higher singularities of plane curves ") is that every singularity whatever may be considered as compounded of ordinary singularities, say we have a singularity = 8 nodes, K cusps, T double tangents, and i inflexions. So that, in fact, Pliicker s equations properly understood apply to a curve with any singularities whatever, By means of Pliicker s equations we may form a table m n S K T t
1 _ _
1
_
2 2
3
6
9 4 1
3
y
3
1
1 4 12
28
24
10
1
16 18 9
1 10 16 8 2
8 12 7 1 1 4 10 6
2 1 8 6 3
4 6 5 2 1 2 4 4 1 2 1 2
3
3 1
The table is arranged according to the value of m ; and we have m = 0, n = l, the point; w=l,w = 0, the line; m = 2, n = 2, the conic ; of m = 3, the cubic, there are three cases, the class being 6, 4, or 3, according as the curve is without singularities, or as it has 1 node, or 1 cusp; and so of m = 4, the quartic, there are nine cases, where observe that in two of them the class is =6, the re duction of class arising from two cusps or else from three nodes. The nine cases may be also grouped together into four, according as the number of nodes and cusps (8 + K) is = 0, 1, 2, or 3. The cases may be divided into sub-cases, by the con sideration of compound singularities ; thus when m = 4, n = 6, =3, the three nodes may be all distinct, which is the general case, or two of them may unite together into the singularity called a tacnode, or all three may unite together into a triple point, or else into an oscnode. We may further consider the inflexions and double tangents, as well in general as in regard to cubic and quartic curves. The expression for the number of inflexions 3m(m - 2) for a curve of the order m was obtained analytically by Pliicker, but the theory was first given in a complete form by Hesse inthe two papers " Ueber die Elimination, u.s.w.," and " Ueber die Wendepuncte der Curveu dritter Ordnung " (Crelle, t. xxviii., 1844) ; in the latter of these the points of inflexion are obtained as the intersections of the curve u Q with the Hessian, or curve A = 0, where A is the determinant formed with the second derived functions of u. We have in the Hessian the first instance of a co- variant of a ternary form. The whole theory of the in flexions of a cubic curve is discussed in a very interesting manner by means of the canonical form of the equation x 3 + y s + z 3 + 6lxy2 = ; and in particular a proof is given of Pliicker s theorem that the nine points of inflexion of a cubic curve lie by threes in twelve lines. It may be noticed that the nine inflexions of a cubic curve are three real, six imaginary ; the three real inflex ions lie in a line, as was known to Newton and Maclaurin. For an acnodal cubic the six imaginary inflexions disappear, and there remain three real inflexions lying in a line. For a crunodal cubic, the six inflexions which disappear are two of them real, the other four imaginary, and there remain two imaginary inflexions and one real inflexion. For a cuspidal cubic the six imaginary inflexions and two of the real inflexions disappear, and there remains one real inflexion. A quartic curve has 24 inflexions ; it was conjectured by Salmon, and has been verified recently by Zeuthen, that at most 8 of these are real. The expression |m(m - 2)(m 2 - 9) for the number of double tangents of a curve of the order m was obtained by Pliicker only as a consequence of his first, second, fourth, and fifth equations. An investigation by means of the curve 11 = 0, which by its intersections with the given curve determines the points of contact of the double tangents, is indicated by Cayley, " Recherches sur I elimina tion et la the"orie des courbes" (Crelle, t. xxxiv., 1847), and in part carried out by Hesse in the memoir "Ueber Cnrven dritter Ordnung " (Crelle, t. xxxvi., 1848). A better process was indicated by Salmon in the " Note on the double tangents to plane curves," Phil. Mag., 1858; consider ing the m - 2 points in which any tangent to the curve again meets the curve, he showed how to form the equation of a curve of the order (m - 2), giving by its intersection with the tangent the points in question; making the tangent touch this curve of the order (m - 2), it will be a double tangent of the original curve. See Cayley, "On the Double Tangents of a Plane Curve" (Phil. Trans., i. cxlviii., 1858), and Dersch (Math. Ann., t. vii., 1874). The solution.
VI. 91
