Let

be the equation to an algebraical curve of the nth degree, the co-ordinates of any point on which in a system of linear co-ordinates are (x, y, z), u0, u1, u2 …. being homogeneous functions of x and y of degrees indicated by the attached suffixes; then

is the equation to its Hessian, which is a curve of the 3(n − 2)th degree.

Every one of the 3n(n − 2) points of intersection of H and U is a point of inflexion on U if it be not a multiple point on U. In this last case the intersection may or may not be a point of inflexion on some one of the branches of U; but in any case where H passes through a multiple point the total number 3n(n − 2) of inflexions suffers a reduction.