Part 2

If two of three basal points of the system of conics be at infinity, the system of cubics obtained by projection and inversion has six coincident basal points and hence only three others. This system has both tangents at the common node common to all cubics of the system. If the two basal points at infinity in the system of conics be coincident, all the conics are parabolas, and hence all the cubics of the system are cuspidal and have a common cuspidal tangent.

If three of the basal points of the system of conics be at infinity, the conics consist of the line at infinity and a pencil of lines through the finite basal point. Inverting from the latter, we obtain a system of cubics with seven coincident basal points. This system is made up of a pencil of lines meeting in the seven coincident basal points together with the two lines joining this to the other two basal points of the system. These two lines are part of every cubic of the system.

If one of the remaining basal points be moved up to join the seven coincident ones, one of these fixed lines becomes indeterminate, and the system of cubics through eight coincident points consists of a fixed line through the eight coincident points and the ninth fixed point together with any two lines of the pencil through the eight points. If the nine basal points coincide, any three lines through it form a cubic of the system.

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) │ 4aexyz(x + y) b²x²y² z² │ ———————————————— - ———————————————— │ - ————————————— - ————————— └ a² b² ┘ a² a²

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.

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.

The eight points of contact of two conics with their four common tangents lie on a conic, which is the locus of a point, the pairs of tangents from which to the two given conics form a harmonic pencil. Inverting and projecting:—two connodal trinodal quartics have four common tangent conics through the three nodes; their eight points of contact lie on another connodal trinodal quartic; if from any point on the last quartic four conics be drawn through the nodes and tangent in pairs to the first quartics, any line through a node is cut harmonically by these four conics.

The eight common tangents to two conics at their common points all touch a conic. Inverting and projecting:—two connodal trinodal quartics intersect in four other points; eight conics can be drawn through the three nodes tangent to the quartics at these points of intersection; these eight conics all touch another connodal trinodal quartic.

A series of conics through four fixed points is cut by any transversal in a range of points in involution. Inverting and projecting:—a series of connodal trinodal quartics can be passed through four other fixed points; any conic through the three nodes cuts the series of quartics in pairs of points which determine at a node a pencil in involution. The conic touches two of the quartics and the lines to the points of contact are the foci of the pencil.

If the sides of two triangles touch a given conic, their six angular points will lie on another conic. Inverting and projecting:—if two groups of three conics each be passed through three nodes and tangent to the quartic, their six points of intersection (three of each group) lie on another connodal trinodal quartic.

If the two triangles are inscribed in a conic, their six sides touch another conic. Inverting and projecting:—if two groups of three conics each be passed through the three nodes of a quartic so that the three points of intersection of each group lie on the quartic, these six conics all touch another connodal trinodal quartic.

A triangle is circumscribed about one conic, and two of its angular points are on a second conic; the locus of its third angular point is a conic.—Inverting and projecting:—if three conics be drawn through the three nodes of two connodal trinodal quartics so that they all touch one of the quartics and two of their points of intersection are on the other quartic, the locus of their third point of intersection is a connodal trinodal quartic.

A triangle is inscribed in one conic and two of its sides touch a second conic; the envelope of its third side is a conic. Inverting and projecting:—if three conics be drawn through the three nodes of two connodal trinodal quartics so that their three points of intersection lie on one of the quartics and two of them touch the other quartic, the envelope of the third conic is another connodal trinodal quartic.

The theorems of this section are stated in the most general terms and are still true when one or more of the nodes are changed into cusps. It is therefore not necessary to give separate theorems for the case of one cusp and two nodes.

NODAL BICUSPIDAL QUARTICS.

A quartic with one node and two cusps is a curve of the fourth class, having one double tangent and two points of inflection (see Salmon). Hence its reciprocal is also a nodal bicuspidal quartic, a fact of which frequent note will be made in this section.

The inverse of a conic with respect to a focus is a curve called Pascal’s Limaçon. From the polar equation of a conic, the focus being the pole, it is evident that the polar equation of the limaçon may be written in the form:

e 1 r = —— cos_x_ + —— ; p p

where e and p are constants, being respectively the eccentricity and semi-latus rectum of the conic.

From the above equation it is readily seen that the curve may be traced by drawing from a fixed point O on a circle any number of chords and laying off a constant length on each of these lines, measured from the circumference of the circle. The point O is the node of the limaçon; and the fixed circle, which I shall call the base circle, is the inverse of the directrix of the conic. This is readily shown as follows:—the polar equation of the directrix is r = p/(e cos_x_). Hence the equation of its inverse is r = (e cos_x_)/p, which is the equation of the base circle of the limaçon.

The envelope of circles on the focal radii of a conic as diameters is the auxiliary circle. Inverting:—the envelope of perpendiculars at the extremities of the nodal radii of a limaçon is a circle with its centre on the axis and having double contact with the limaçon. Projecting:—from any point on a nodal bicuspidal quartic draw lines to the three nodes and a fourth line forming with them a harmonic pencil; the envelope of all such lines is a conic through the two cusps and having double contact with the quartic; the chord of contact passes through the node and cuts the line joining the cusps so that this point of intersection, the two cusps, and intersection of the double tangent with the cuspidal line form a harmonic range. Reciprocating:—on any tangent to a nodal bicircular quartic take the three points where it cuts the two inflectional tangents and the double tangent, and a fourth point forming with these a harmonic range; the locus of all such points is a conic touching the two inflectional tangents and having double contact with the quartic; the pole of the chord of contact is on the double tangent; join this last point to the intersection of the inflectional tangents and join the node with the same intersection; these four lines form a harmonious pencil.

If the tangent at any point P of a conic meet the directrix in Q, the line P Q will subtend a right angle at the focus O; the circle P O Q has P Q for a diameter and hence cuts the conic at P at right angles. Inverting:—from any point P on the limaçon draw O P to the node O; draw O Q perpendicular to O P meeting the base circle in Q; P Q is normal to the limaçon at P. Projecting:—from any point P on a nodal bicuspidal quartic draw lines to the three nodes and a fourth harmonic to these three; from O draw lines to the two cusps and a fourth harmonic to these two and the line O P; the locus of the intersection of the fourth line of each pencil is a conic through the three nodes. Call this the basal conic of the quartic. Reciprocating:—on any given tangent to a nodal bicuspidal quartic take its points of intersection with the double tangent and the inflectional tangents, and a fourth point harmonic with these; on the double tangent take its points of intersection with the given tangent and the inflectional tangents, and a fourth point harmonic with these; the envelope of the lines joining the fourth point of these two ranges is a conic touching the double and inflectional tangents.

The locus of the foot of the perpendicular from the focus on the tangent to a conic is the auxiliary circle. Inverting:—draw a circle through the node tangent to a limaçon; draw the diameter O P of this circle; the locus of P is a circle having double contact with the limaçon, the axis being the chord of contact. Cor.; the locus of the centre of the tangent circle is also a circle. Projecting:—through the three nodes of a nodal bicuspidal quartic draw any conic touching the quartic; the locus of the pole with respect to this conic of the line joining the two cusps is a conic; draw the chord O P of the first conic through the node O and the pole of the line joining the two cusps; the locus of P is a conic through the cusps, having double contact with the quartic.

If chords of a conic subtend a constant angle at the focus, the tangents at the ends of the chords will meet on a fixed conic, and the chords will envelope another fixed conic; both these conics will have the same focus and directrix as the given conic. Inverting:—draw two nodal radii of a limaçon O P and O Q, making a given angle at O; the envelope of the circle P O Q is another limaçon; the locus of the intersection of circles through O tangent to the limaçon at P and Q is another limaçon. These two limaçons have the same node and base circle as the given one. Projecting:—through the node O of a nodal bicuspidal quartic draw a pencil of radii in involution; let O P and O Q be a conjugate pair of these nodal radii; the envelope of the conic through P, Q, and the three nodes, is another quartic of the same kind: also draw conics through the three nodes tangent to the quartic at P and Q; the locus of their point of intersection is another quartic of the same kind. These three quartics all have the same node, cusps, and base conic.

Every focal chord of a conic is cut harmonically by the curve, the focus, and directrix. Inverting:—every nodal chord of a limaçon is bisected by the base circle. Projecting:—every nodal chord of a nodal bicuspidal quartic is cut harmonically by the quartic, the base conic, and the line joining the two cusps. Reciprocating:—from any point on the double tangent of a nodal bicuspidal quartic draw the other two tangents to the quartic and a line to the intersection of the inflectional tangents; the fourth harmonic to these lines envelopes a conic.

Since the limaçon is symmetrical with respect to the axis, it follows that the two points of inflection are situated symmetrically with respect to the axis. Hence the line joining the two points of inflection is parallel to the double tangent. Therefore by projection we infer the following general theorem for the nodal bicuspidal quartic: the line joining the two cusps, the line joining the two points of inflection, and the double tangent meet in a point. Also the fourth harmonic points on each of these lines lie on a line through the node. Reciprocating:—the point of intersection of the cuspidal tangents, the point of intersection of inflectional tangents, and the node all lie on a right line. From the node draw a fourth harmonic to this right line and the tangents at the node; draw a fourth line harmonic to this right line and the inflectional tangents; draw a fourth harmonic to the cuspidal tangents and this right line; these three lines all meet in a point on the double tangent.

TRICUSPIDAL QUARTICS.

A tricuspidal quartic is a curve of the third class with one double tangent and no inflection. Its reciprocal is therefore a nodal cubic.

We shall begin by reciprocating some of the simpler properties of nodal cubics. Since the three points of inflection of a nodal cubic lie on a right line, it follows that the three cuspidal tangents of a tricuspidal quartic meet in a point. The reciprocal of the harmonic polar of a point of inflection is a point on the double tangent, found by drawing through the point of intersection of the three cuspidal tangents a line forming with them a harmonic pencil. Three such lines can be drawn and it is not difficult to distinguish them. All six lines form a pencil in involution, the lines to the points of contact of the double tangent being the foci. I shall call such a point on the double tangent the harmonic point of the cuspidal tangent. Since any two inflectional tangents of a nodal cubic meet on the harmonic polar of the third point of inflection, it follows that any two cusps of a trinodal quartic and the harmonic point of the third cuspidal tangent lie on a right line. Since the point of contact of the tangents from a point of inflection of a nodal cubic is on the harmonic polar of the point, it follows that the tangent to the tricuspidal quartic at the point where it is cut by a cuspidal tangent passes through the harmonic point of that cuspidal tangent.

The inverse of the parabola from a focus is the cardioid; and the inverse of the corresponding directrix is the base circle of the cardioid. The cardioid projects into a tricuspidal quartic and its base circle projects into a conic through the three cusps which has the same general properties as the base conic of the nodal bicuspidal quartic.

The circle circumscribing the triangle formed by the three tangents to a parabola passes through the focus. Inverting:—three circles through the cusp, and tangent to a cardioid, intersect in three collinear points. Projecting:—three conics through the three cusps of a tricuspidal quartic and touching the quartic intersect in three collinear points. Reciprocating:—if three conics touch the three inflectional tangents of a nodal cubic and the cubic itself, their three other common tangents intersect in a point.

Circles described on the focal radii of a parabola as diameters touch the tangent through the vertex. Inverting and projecting:—from a point on a tricuspidal quartic lines are drawn to the three cusps and a fourth line forming a harmonic pencil; the envelope of this fourth line is a conic through the three cusps and touching the quartic at the point where the latter is cut by one of the cuspidal tangents. There are three such conics, one corresponding to each cusp. At any cusp the tangent to its corresponding base conic, the cuspidal tangent, and the lines to the other two cusps form a harmonic pencil. Reciprocating:—on any tangent to a nodal cubic take the three points of intersection with the inflectional tangents and a fourth point forming with these a harmonic range; the locus of this fourth point is a conic touching the three inflectional tangents and the cubic. The tangent to the cubic where it is touched by the conic goes through a point of inflection. On any inflectional tangent the point of contact of this conic, the point of inflection, and the points of intersection of the other two inflectional tangents form a harmonic range.

The circle described on any focal chord of a parabola as diameter will touch the directrix. Inverting:—the circle described on any cuspidal chord of a cardioid will touch the base circle. Projecting:—through a cusp C draw any chord of a tricuspidal quartic meeting the quartic in P and O; draw a conic through P, O, and the other two cusps so that the pencil at P formed by the tangent to the conic and the lines to the cusps is harmonic; all such conics will touch the base conic of the cusp C. Reciprocating:—from O, on any inflectional tangent of a nodal cubic, draw two tangents P and Q to the cubic; draw a conic touching the tangents P and Q and the other two inflectional tangents so that the range on one of these tangents formed by the point of contact of the conic and the intersection of the three inflectional tangent is harmonic; the envelope of all such conics is a conic touching the three inflectional tangents.

The directrix of a parabola is the locus of the intersection of tangents at right angles to one another. Inverting and projecting:—through any point P on the base conic of a cusp C of the tricuspidal quartic, two conics can be drawn through the three cusps and touching the quartic; their two tangents at P and the lines to the other two cusps form a harmonic pencil; their two points of contact lie on a line through C. Reciprocating:—from any point on one of the inflectional tangents to a nodal cubic draw the two tangents P and Q; draw two conics each touching the cubic and the three inflectional tangents, one touching P and the other Q; the envelope of their other common tangent is a conic touching the three inflectional tangents; the two points of contact of any one of these common tangents and the points where it cuts the other two inflectional tangents form a harmonic range.