The centre is a conjugate point (or **acnode**) and the curve resembles fig.

It may be remarked that we cannot with a real point and line obtain the node with two imaginary tangents (conjugate or isolated point or **acnode**), nor again the real double tangent with two imaginary points of contact; but this is of little consequence, since in the general theory the distinction between real and imaginary is not attended to.

The branch, whether re-entrant or infinite, may have a cusp or cusps, or it may cut itself or another branch, thus having or giving rise to crunodes or double points with distinct real tangents; an **acnode**, or double point with imaginary tangents, is a branch by itself, - it may be considered as an indefinitely small re-entrant branch.

Secondly, if two of the intersections coincide, say if the line infinity meets the curve in a onefold point and a twofold point, both of them real, then there is always one asymptote: the line infinity may at the twofold point touch the curve, and we have the parabolic hyperbolas; or the twofold point may be a singular point, - viz., a crunode giving the hyperbolisms of the hyperbola; an **acnode**, giving the hyperbolisms of the ellipse; or a cusp, giving the hyperbolisms of the parabola.

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.