Xy 2 -4z 3 +g2x 2 y+g3x 3, and also the special form axz 2 -4by 3 of the cuspidal cubic. An investigation, by non-symbolic methods, is due to F.
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.
For a cuspidal cubic the six imaginary inflections and two of the real inflections disappear, and there remains one real inflection.
The oval may unite itself with the infinite branch, or it may dwindle into a point, and we have the crunodal and the acnodal forms respectively; or if simultaneously the oval dwindles into a point and unites itself to the infinite branch, we have the cuspidal form.
Crunodal or acnodal), or cuspidal; and we see further that there are two kinds of non-singular curves, the complex and the simplex.
Each singular kind presents itself as a limit separating two kinds of inferior singularity; the cuspidal separates the crunodal and the acnodal, and these last separate from each other the complex and the simplex.
The singular kinds arise as before; in the crunodal and the cuspidal kinds the whole curve is an odd circuit, but in an acnodal kind the acnode must be regarded as an even circuit.