The nature of the triple point depends upon the relation of the line at infinity to the cubic before inversion. Thus the line at infinity may cut the cubic in three distinct points all real, or one real and two imaginary, in one real and two coincident points (an ordinary tangent), or in three coincident points (an inflectional tangent). Hence the quartic may have at the triple point three distinct tangents all real, or one real and two imaginary, one real and two coincident, or all coincident. This quartic may be generated in a manner similar to that used for the curves already discussed. We showed in the section on nodal cubics that a system of conics through A, B, C, D, and a projective pencil of rays with its vertex at A generate by the intersection of corresponding elements a cubic with a node at A. Invert the whole figure from A and then project:—the pencil of rays remains a pencil; the system of conics becomes a system of unicursal cubics having a common node at A and passing through five other common points; the cubic inverts and projects into a quartic with a triple point at A, passing through the five other common points of the system of cubics. The three points of inflection of a nodal cubic lie on a right line. Inverting:—there are three points on a circular quartic with a triple point whose osculating circles pass through the triple point, and these three points lie on a circle through the triple point. Let these three points be designated by A, B, and C. The lines from the triple point O to the points A, B, C, and the common chord of the osculating circles at two of them form a harmonic pencil. Through one of these points, A, and the triple point draw a circle touching the quartic; the point of contact is on the common chord of the osculating circles at B and C.