4aexyz(x + y) b²x²y² - ————————————— - ————————— = 0, a² a² which is the equation of the quartic referred to the triangle formed by the three nodes. We are now able to determine the nature of the node at the vertex (y, z). Factor x² out of all the terms which contain it; and arrange thus: ┌ ┐ ┌ ┐ │ z² z² 4aeyz b²y² │ │yz² yz² 2aey²z │ x² │——— - ——— - ————— - ——————│ + 2x │———— + ———— - ———————│ │ a² b² a² a² │ │ a² b² a² │ └ ┘ └ ┘ 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.