But if the given curve has a node, then not only the Hessian passes through the node, but it has there a node the two branches at which touch respectively the two branches of the curve; and the node thus counts as six intersections; so if the curve has a cusp, then the Hessian not only passes through the cusp, but it has there a cusp through which it again passes, that is, there is a cuspidal branch touching the cuspidal branch of the curve, and besides a simple branch passing through the cusp, and hence the cusp counts as eight intersections.
This method of solution fails when the discriminant R vanishes, for then the Hessian has equal roots, as also the cubic f.
Gordan has also shown that the vanishing of the Hessian of the binary n ic is the necessary and sufficient condition to ensure the form being a perfect n th power.
The French army, similarly arrayed, but with a few battalions attached to the cavalry wings, was more heterogeneous than the German, being composed of French, Hessian, German mercenaries, and Liegeois.
Prussia, on the other hand, announced its determination to carry out the principles of the Union and to maintain the Hessian constitution.