The Kansas University Quarterly, Vol. I, No. 2, October 1892

by Various

NODAL CUBICS.

If an ellipse be inverted from one of its vertices, the inverse curve is symmetrical with respect to the axis; it has one point of inflection at infinity and the asymptote is an inflectional tangent. This asymptote is the inverse of the circle of curvature at the vertex. The cubic has two other points of inflection situated symmetrically with respect to the axis. Hence the three points of inflection lie on a right line, a projective theorem which is consequently true of all nodal cubics. The axis is evidently the harmonic polar of the point of inflection at infinity. Since the axis bisects the angle between the tangents at the node, it follows that the line joining a point of inflection to the node, the two tangents at the node, and the harmonic polar of the point of inflection, form a harmonic pencil. There are three such lines, one to each node, and three harmonic polars; these form a pencil in involution, the tangents at the node being the foci.

Since the asymptote is perpendicular to the axis, we have by projection the following theorem:—through a point of inflection I, draw any line cutting the cubic in B and C. Through P the point of intersection of the harmonic polar and inflectional tangent of I, draw two lines to B and C. The four lines meeting in P form a harmonic pencil. The point of contact of the tangent from I to the cubic is on the harmonic polar of I. Any two inflectional tangents meet on the harmonic polar of the third point of inflection.

The locus of the foot of the perpendicular from the focus of a conic on a tangent is the auxiliary circle. Inverting from the vertex, there are two points, A and B, on the axis of the curve, such that if a circle be drawn through one of them and the node, cutting at right angles a tangent circle through the node, their point of intersection will be on the tangent to the curve where it is cut by the axis. Projecting:—through a point of inflection I of a nodal cubic draw a line cutting the cubic in P and Q; there are two determinate points on the harmonic polar of I, which have the following property:—draw a conic through P, Q, and the node touching the cubic; draw another conic through one of these points, P, Q, and the node cutting the former, so that their tangents at their point of intersection, together with the lines from it to P and Q form a harmonic pencil; the locus of such a point of intersection is the tangent from I to the cubic.

If three conics circumscribe the same quadrilateral, the common tangent to any two is cut harmonically by the third. Inverting from one of the vertices of the quadrilateral: if three nodal, circular cubics have a common double point and pass through three other fixed points, the common tangent circle through the common node to any two of the cubics is cut harmonically by the third; i. e., so that the pencil from the node to the two points of intersection and the points of contact is harmonic. Projecting this:—given three nodal cubics having a common node and passing through five other fixed points; let a conic be passed through the common node and two of the fixed points, touching two of the cubics. The pencil from the common node to the points of contact and the point where the conic cuts the third cubic is harmonic.

The following theorem may be proved in similar manner:—given a system of cubics having a common node and passing through five other fixed points; let a conic be drawn through the common node and two of the fixed points; the lines drawn from the points where it cuts the cubics to the common node form a pencil in involution.

A variable chord drawn through a fixed point P to a conic subtends a pencil in involution at any point O on the conic. Inverting from O:—a system of circles through the double point of a nodal circular cubic and any other fixed point P, is cut by the cubic in pairs of points which determine at the node a pencil in involution. Projecting:—a system of conics through the node of a unicursal cubic, two fixed points on the curve, and any fourth fixed point, is cut by the cubic in pairs of points which determine at the node a pencil in involution.

We give another proof of the theorem that the three points of inflection of a nodal cubic lie on a right line. This is easily shown by inversion and is a beautiful example of the method.

There are three points on a conic whose osculating circles pass through a given point on the conic; these three points lie on a circle passing through the given point.[1] (Salmon’s Conics, Art. 244, Ex. 5.) By inverting from the given point and then projecting, we readily see that there are three points of inflection on a nodal cubic which lie on a right line. If the above conic be an ellipse, the three osculating circles are all real; but if it be a hyperbola, one only is real. Hence an acnodal cubic has three real points of inflection, while a crunodal one has one real and two imaginary.

The reciprocals of many of the theorems of this section are of interest and will be given under Quartics.

CUSPIDAL CUBICS.[2]

Inverting the parabola from its vertex we obtain the Cissoid of Diocles. The focus of the parabola inverts into a point on the cuspidal tangent which I shall call the focus of the cissoid. The circle of curvature at the vertex of the parabola inverts into the asymptote of the cissoid. This asymptote is also plainly the inflectional tangent, and the point at infinity is the point of inflection. The directrix of the parabola inverts into a circle through the cusp of the cissoid having the cuspidal tangent for a diameter. Hall calls this the directrix circle. The double ordinate of the parabola which is tangent to the circle of curvature of the vertex inverts into the circle usually called the base circle of the cissoid.[3]

The cissoid may fairly be called the simplest form of the cuspidal cubic. Its projection and polar reciprocal are both cuspidal cubics. I shall now deduce from the parabola a few simple propositions for the cissoid, and then extend them to all cuspidal cubics.

(1) It is known that the locus of the intersection of tangents to the parabola which are at right angles to one another, is the directrix. Inverting:—the locus of the intersection of tangent circles to the cissoid through the cusp and at right angles to each other is the directrix circle.

(2) For the parabola, two right lines O P and O Q, are drawn through the vertex of the parabola at right angles to one another, meeting the curve in P and Q; the line P Q cuts the axis at a fixed point, whose abscissa is equal to its ordinate. Inverting:—two right lines, O P and O Q, are drawn at right angles to one another through the cusp of the cissoid, meeting the curve in P and Q; the circle O P Q passes through the intersection of the axis and asymptote.

(3) If the normals at the points P, O, R, of a parabola meet at a point, the circle through P O R will pass through the vertex. Inverting:—through a fixed point and the cusp of a cissoid, three and only three circles can be passed, cutting the cissoid at right angles; these three points of intersection are collinear.

From the geometry of the cissoid we see that if any line be drawn parallel to the asymptote, cutting the curve in two points, B and C, the segment B C is bisected by the axis. Hence, projecting the curve we have the following theorem:—any line drawn through the point of inflection is cut harmonically by the point of inflection, the curve, and the cuspidal tangent. Thus the cuspidal tangent is the harmonic polar of the point of inflection. The polar reciprocal of this last theorem reads as follows:—if from any point on the cuspidal tangent the two other tangent lines be drawn to the curve, and a line to the point of inflection, these four lines form a harmonic pencil. These are fundamental propositions in the theory of cuspidal cubics.

(4) Projecting proposition (1) above, we have the generalized theorem:—through the point of inflection draw any line cutting the cubic in B and C; through B, C, and the cusp draw two conics tangent to the cubic, and intersecting in a fourth point such that the two tangents to the conics at their point of intersection, together with the two lines from it to B and C, form a harmonic pencil; the locus of all such intersections is a conic through B, C, and the cusp having the point of inflection and the cuspidal tangent for pole and polar.

(5) Reciprocating (4) we have:—through any point on the cuspidal tangent draw the two other tangents, B and C, to the cubic. Touching B, C, and the inflectional tangent draw two conics, such that the points of contact of their common tangent, together with the points where their common tangent cuts the tangents B and C, form a harmonic range; the envelope of such common tangents is a conic having the cuspidal tangent and the point of inflection for polar and pole.

(6) Projecting (2) we obtain the following:—through the point of inflection draw any line cutting the curve in B and C; take any other two points on the cubic such that the pencil from the cusp, O, O (B P C Q) is harmonic; the conic passing through O B P C Q will pass through the intersection of the cuspidal and inflectional tangents.

(7) Reciprocating (6):—from any point on the cuspidal tangent draw two other tangents, B and C, to the cubic; take any two other tangents, P and Q, such that the range cut from the inflectional tangent by B, C, P, Q, is harmonic; the conic touching B, C, P, Q, and the inflectional tangent will also touch the line joining the point of inflection and the cusp.

(8) Projecting (3):—through the point of inflection draw any line cutting the cubic in B and C; through the cusp O and the points B and C on the cubic and any other fixed point P, three, and only three, conics can be passed, such that the tangent to the conic and cubic at their remaining point of intersection, together with the lines from it to B and C, form a harmonic pencil; these three points of intersection are collinear.