X. On the sextactic points of a plane curve

It is, in my memoir “On the Conic of Five-pointic Contact at any point of a Plane Curve”, 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 orderm . The result arrived at is that the number of sextactic points is =m (12m —27), the points in question being the intersections of the curvem with a curve of the order 12m —27, the equation of which is (12m 2 —54m +57)H Jac. (U, H, ΩH¯ ) +(m —2) (12m —27)H Jac. (U, H, ΩU¯ ) +40(m —2)2 Jac. (U, H, Ψ)=0, where U=0 is the equation of the given curve of the orderm , 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.