| y²z² | y²z² | |||
| + | — | - | — | = 0 . |
| a² | b² |
The quantity which multiplies x² represents the two tangents at the double point (y, z); but this quantity is a perfect square and hence we have a cusp. In this way the point (x, z) may be shown to be a cusp. Lastly, when a parabola is inverted from the focus, we obtain a tricuspidal quartic.
The trinodal quartic can be generated in a manner analogous to that shown for the nodal cubic. Let two projective pencils of rays have their vertices at A and B, the locus of intersection of corresponding rays is a conic through A and B. Invert from any point O in the plane, and we obtain two systems of co-axial circles, O A being the axis of one and O B of the other. The locus of intersection of corresponding circles is a bicircular quartic having a node at O. Projecting the whole figure we have the following theorem:—two projective systems of conics through O P Q A and O P Q B generate by their corresponding intersections a trinodal quartic having its nodes at O, P, and Q, and passing through A and B.
It is evident that the quartic generated in this way may have three nodes, one node and two cusps, two nodes and one cusp, or three cusps, depending upon the nature of the conic inverted and the centre of inversion. Making this the basis of classification we thus distinguish four varieties of unicursal quartics. To these must be added a fifth variety, viz: the quartic with a triple point. Each of these varieties will be considered separately.
The method of treating unicursal quartics given in this and the next four sections is in some respects similar to that suggested by Cayley in Salmon’s Higher Plane Curves. But the method here sketched out is very different in its point of view and much wider in its application, yielding a multitude of new theorems not suggested by Cayley’s method.
TRINODAL QUARTICS.
The quartic with three double points is a curve of the sixth class having four double tangents and six cusps (Salmon’s H. P. C. Art. 243). Hence its reciprocal is of the sixth degree with four double points, six cusps, three double tangents, and no points of inflection.
The locus of intersection of tangents to a conic at right angles to one another is a circle. Inverting:—the locus of intersection of circles through the node and tangent a nodal, bicircular quartic and at right angles to one another is a circle. Projecting:—through the three nodes of a quartic draw two conics, each touching the quartic and intersecting so that the two tangents to the conics at their point of intersection, together with the lines from it to two of the nodes, form a harmonic pencil; the locus of all such intersections is a conic through these two nodes. Whenever the two tangents to the quartic from the third node, together with the lines from it to the other two nodes, form a harmonic pencil, this last conic breaks up into two right lines.
Any chord of a conic through O is cut harmonically by the conic and the polar of O. Inverting from O and projecting:—from one of the nodes of a trinodal quartic draw the two tangents to the quartic (not tangents at the node); draw the conic through these two points of contact and the three nodes; any line through the first mentioned node is cut harmonically by this conic, the quartic and the line joining the other two nodes.
If a triangle circumscribe a conic, the three lines from the angular points of the triangle to the points of contact of the opposite sides intersect in a point. Inverting and projecting:—through the three nodes of a quartic draw three conics touching the quartic; through the point of intersection of two of these conics, the point of contact of the third, and the three nodes draw a conic; three such conics can be drawn and they pass through a fixed point.