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- 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.

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