III. On the sextactic points of a plane curve

Abstract

It is, in my memoir "On the Conic of Five-pointic Contact at any Point of a Plane Curve” (Phil. Trans, vol. cxlix. (1859) pp. 371—400), remarked that as in a plane curve there are certain singular points, viz. the points of inflexion, where three consecutive points lie in a line, so there are singular points where six consecutive points of the curve lie in a conic; and such a singular point is there termed a “sextactic point.” The memoir in question (here cited as “former memoir”) contains the theory of the sextactic points of a cubic curve; but it is only recently that I have succeeded in establishing the theory for a curve of the order m . The result arrived at is that the number of sextactic points is = m (12 m —27), the points in question being the intersections of the curve m with a curve of the order 12 m —27, the equation of which is (12 m 2 —54 m + 57) H Jac. (U, H, Ω H ¯ ) + ( m —2) (12 m —27) H Jac. (U, H, Ω U ¯ ) + 40 ( m —2) 2 Jac. (U, H, Ψ ) = 0, where U = 0 is the equation of the given curve m , H is the Hessian or determinant formed with the second differential coefficients ( a, b, c, f, g, h ) of U, and, (A, B, C, F, G, H) being the inverse coefficients (A = bc — f 2, &c.), then Ω = (A, B, C, F, G, H≬∂ x , ∂ y , ∂ z ) 2 H, Ψ = (A, B, C, F, G, H≬∂ x H, ∂ y H, ∂ z H) 2 ; and Jac. denotes the Jacobian or functional determinant, viz.