UNICURSAL QUARTICS.

The inverse of a conic from any point not on the curve is a nodal bicircular quartic. This is shown by inverting the general equation of the conic

ax² + 2hxy + by² + 2gx + 2fy + c = 0 ;

	x		y
by substituting for x and y	———	and	——— ,
	x² + y²		x² + y²

we get the equation

ax² + 2hxy + by² + 2(gx + fy)(x² + y²) + c(x² + y²)² = 0.

The origin is evidently a double point on the curve, and is a crunode, acnode, or cusp according as the conic is a hyperbola, ellipse, or parabola. The factors of the terms of the fourth degree, viz: (x + iy) (x + iy) (x - iy) (x - iy), show that the two imaginary circular points at infinity are double points on the quartic, which is thus trinodal. Hence this nodal, bicircular quartic can be projected into the most general form of the trinodal quartic. Trinodal quartics are unicursal.

If the conic which we invert be a parabola, the quartic has two nodes and one cusp. If the conic be inverted from a focus, the quartic has the two circular points at infinity for cusps. This is best shown analytically as follows: let the equation of the conic, origin being at the focus, be written

x²		y²		2aex		b²
——	+	——	+	——	-	—	= 0.
a²		b²		a²		a²

Inverting this we have

x²		y²		2aex(x² + y²)		b²(x² + y²)²
——	+	——	+	——————	-	—————	= 0.
a²		b²		a²		a²

Now transform this equation so that the lines joining the origin to the circular points at infinity shall be the axes of reference. To do this

let x + iy = x₁ and x - iy = y₁;

	x₁ + y₁		x₁ - y₁
∴ x =	———	and y =	———
	2		2i

Making these substitutions and reducing we have (dropping the subscripts),

(x² + 2xy + y²)		(x² - 2xy + y²)		4aexy(x + y)		b²x²y²
——————	-	——————	-	——————	-	—————	= 0.
a²		b²		a²		a²

Making this equation homogeneous by means of z, we have

	(x² + 2xy + y²)		(x² - 2xy + y²)		4aexy(x + y)		b²x²y²
z²	——————	-	——————	-	——————	-	—————	= 0.
	a²		b²		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		4aexyz		b²y²
x²	—	-	—	-	———	-	——
	a²		b²		a²		a²

	yz²		yz²		2aexy² z
+ 2x	—	+	—	-	———
	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.