The name lemniscate is sometimes given to any **crunodal** quartic curve having only one real finite branch which is symmetric about the axis.

For a **crunodal** cubic the six inflections which disappear are two of them real, the other four imaginary, and there remain two imaginary inflections and 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.

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.