These curves are instances of unicursal bicircular quartics.
Similarly a cubic through the two circular points is termed a circular cubic; a quartic through the two points is termed a circular quartic, and if it passes twice through each of them, that is, has each of them for a node, it is termed a bicircular quartic. Such a quartic is of course binodal (m = 4, 6= 2, K = o); it has not in general, but it may have, a third node or a cusp. Or again, we may have a quartic curve having a cusp at each of the circular points: such a curve is a " Cartesian," it being a complete definition of the Cartesian to say that it is a bicuspidal quartic curve (m= 4, 6 = o, K= 2), having a cusp at each of the circular points.
The circular cubic and the bicircular quartic, together with the Cartesian (being in one point of view a particular case thereof), are interesting curves which have been much studied, generally, and in reference to their focal properties.
There will be from each circular point X tangents (X, a number depending on the class of the curve and its relation to the line infinity and the circular points, 2 for the general conic, 1 for the parabola, 2 for a circular cubic, or bicircular quartic, &c.); the X tangents from the one circular point and those from the other circular point intersect in X real foci (viz.