For an **acnodal** cubic the six imaginery inflections disappear, and there remain three real inflections lying in a line.

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.

There is thus a complete division into the five kinds, the complex, simplex, crunodal, **acnodal** and cuspidal.

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.

And it then appears that there are two kinds of non-singular cubic cones, viz, the simplex, consisting of a single sheet, and the complex, consisting of a single sheet and a twin-pair sheet; and we thence obtain (as for cubic curves) the crunodal, the **acnodal** and the cuspidal kinds of cubic cones.

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.