A form which presents itself is when two ovals, one inside the other, unite, so as to give rise to a crunode - in default of a better name this may be called, after the curve of that name, a limacon.
As mentioned with regard to a branch generally, an infinite branch of any kind may have cusps, or, by cutting itself or another branch, may have or give rise to a crunode, &c.
It will readily be understood how the like considerations apply to other cases, - for instance, if the line is a tangent at an inflection, passes through a crunode, or touches one of the branches of a crunode, &c.; thus, if the line S2 passes through a crunode we have pairs of hyperbolic legs belonging to two parallel asymptotes.