Part 1

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Vol. I. OCTOBER, 1892 No. 2.

THE

KANSAS UNIVERSITY

QUARTERLY

CONTENTS

UNICURSAL CURVES BY METHOD OF INVERSION, _H. B. Newson_ FOREIGN SETTLEMENTS IN KANSAS, _W. H. Carruth_ THE GREAT SPIRIT SPRING MOUND, _R. H. S. Bailey_ ON PASCAL’S LIMAÇON AND THE CARDIOID, _H. C. Riggs_ DIALECT WORD-LIST, _W. H. Carruth_

PUBLISHED BY THE UNIVERSITY LAWRENCE, KANSAS

_Price of this number, 50 cents_

Entered at the Post-office in Lawrence as Second-class matter.

COMMITTEE OF PUBLICATION

E. H. S. BAILEY F. W. BLACKMAR W. H. CARRUTH C. G. DUNLAP E. MILLER S. W. WILLISTON V. L. KELLOGG, MANAGING EDITOR

JOURNAL PUBLISHING HOUSE LAWRENCE, KANSAS 1892

KANSAS UNIVERSITY QUARTERLY.

VOL. I. OCTOBER, 1892. NO. 2.

Unicursal Curves by Method of Inversion.

BY HENRY BYRON NEWSON.

This paper contains a summary of the work done during the last school year by my class in Modern Geometry. Since many of the results were suggested or entirely wrought out by class-room discussion, it becomes practically impossible to assign to each member of the class his separate portion. Many of the results were contributed by Messrs. M. E. Rice, A. L. Candy, H. C. Riggs, and Miss Annie L. MacKinnon.

The reader who is not familiar with the method of Geometric Inversion should read Townsend’s Modern Geometry, chapters IX and XXIV; or a recent monograph entitled, “Das Princep der Reziproken Radien,” by C. Wolff, of Erlangen.

When a conic is inverted from a point on the curve, the inverse curve is a nodal, circular cubic.

This is shown analytically as follows: let the equation of the conic be written

ax² + 2hxy + by² + 2gx + 2fy = 0;

which shows that the origin is a point on the curve. Substituting for

x y x and y ————————— and ———————— , x² + y² (x² + y²)

we have as the equation of the inverse curve

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

The terms of the second degree show that the origin is a double point on the cubic; and is a crunode, acnode, or cusp, according as the conic is a hyperbola, ellipse, or parabola. The terms of the third degree break up into three linear factors, viz: gx + fy, x + iy, and x - iy, which are the equations of the three lines joining the origin to the three points where the line at infinity cuts the cubic; thus showing that the cubic passes through the imaginary circular points at infinity.

Since the above transformation is rational, it follows that there is a (1, 1) correspondence between the conic and the cubic. This fact is also evident from the nature of the method of inversion. The cubic has its maximum number of double points, viz: one; and hence is unicursal. This unicursal circular cubic may be projected into the most general form of unicursal cubic; the cuspidal variety, however, always remaining cuspidal.

By applying the method of inversion to many of the well known theorems of conics, new theorems are obtained for unicursal, circular cubics. If one of these new theorems states a projective property, it may at once by the method of projection be extended to all unicursal cubics. Examples will be given below.

The following method of generating a unicursal cubic is often useful. Given two projective pencils of rays having their vertices at A and B; the locus of the intersection of corresponding rays is a conic through A and B. Invert the whole system from A. The pencil through A remains as a whole unchanged, while the pencil through B inverts into a system of co-axial circles through A and B, and the generated conic becomes a circular cubic through A and B, having a node at A. Now project the whole figure and we have the following:—given a system of conics through four fixed points and a pencil of rays projective with it and having its vertex at one of the fixed points, the locus of the intersection of corresponding elements of the two systems is a unicursal cubic, having its node at the vertex of the pencil, and passing through the three other fixed points.

Unicursal cubics are divisible into two distinct varieties, nodal and cuspidal. The nodal variety is a curve of the fourth class and has three points of inflection, one of which is always real. The cuspidal variety is of the third class and has one point of inflection (Salmon, H. P. C., Art. 147). Each of these varieties forms a group projective within itself; that is to say, any nodal cubic may be projected into every other possible nodal cubic, and the same is true with regard to the cuspidal. But a nodal cubic can not be projected into a cuspidal and vice versa.

In applying this method of investigation to the various forms of unicursal cubics and quartics, only a limited number of theorems are given in each case. It will be at once evident that many more theorems might be added, but enough are given in each case to illustrate the method and show the range of its application. It is not necessary to work out all the details, as this paper is intended to be suggestive rather than exhaustive.

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.

SYSTEMS OF CUBICS THROUGH NINE POINTS.

Let U and V be the equations of two given cubics, then U + kV is the equation of a system of cubics through their nine points of intersection. Twelve cubics of this system are unicursal, and the twelve nodes are called the twelve critic centres of the system. (See Salmon’s H. P. C., Art. 190.)

Let the equation of the system be written briefly

a + ka₁ + (b + kb₁) x + (c + kc₁) y + u₂ + u₃ = 0;

one, and only one, value of k makes the absolute term vanish; hence one, and only one, curve of the system passes through the origin, which may be any point in the plane. Make the equation of the system homogeneous by means of z, and differentiate twice with respect to z; we obtain thus the equations of the polar conics and polar lines of the origin with respect to the system.

The polar conics of the origin are given by

3(a + ka₁) + 2 { (b + kb₁) x + (c + kc₁)y } + u₂ = 0;

thus showing that the polar conics of any point, with respect to the system of cubics, form a system through four points. The polar lines of the origin are given by

3(a + ka₁) + (b + kb₁) x + (c + kc₁)y = 0,

which represents a pencil of lines through a point.

Suppose now the origin to be at one of the critic centres; then for a particular value, k₁, all terms lower than the second degree must vanish, so that

║ a b c ║ ║ ║ = 0. ║ a₁ b₁ c₁ ║

The factors of the terms of u₂, which involves k₁, represent the tangents at the double point to the nodal cubic, and also the polar conic of the origin with respect to this nodal cubic. Hence a critic centre is at one of the vertices of the self-polar triangle of its system of polar conics. The opposite side of this triangle is the common polar line of the critic centre with respect to its system of polar conics, and hence it is also the common polar line of the critic centre with respect to the system of cubics. The four basal points of the system of polar conics lie two and two upon the tangents at the double point of the nodal cubic.

When the origin is taken at one of the nine basal points of the system of cubics, a and a₁ both vanish. Hence it is readily seen that a basal point of a system of cubics is also a basal point of its system of polar conics and the vertex of its pencil of polar lines.

Suppose two of the basal points of the system of cubics to coincide, then every cubic of the system, in order to pass through two coincident points, must touch a common tangent at a fixed point. The common tangent is the common polar of its point of contact, both with respect to the system of cubics and to its system of polar conics. Hence the union of two basal points gives rise to a critic centre. The self-polar triangle of its system of polar conics here reduces to a limited portion of the common tangent. This line is not a tangent to the nodal cubic, but only passes through its double point.

Suppose three of the basal points of a system of cubics to coincide, such a point will then be a point of inflection on each cubic of the system. For, in the last case, if a line be drawn from the point of contact of the common tangent to a third basal point of the system, such a line will be a common chord of the system of cubics. Suppose, now, this third basal point be moved along the curves until it coincides with the other two; then the common chord becomes a common tangent, which cuts every cubic of the system in three coincident points, and hence is a common inflectional tangent.

Since the polar conic of a point of inflection on a cubic consists of the inflectional tangent and the harmonic polar of the point, and since the polar conics of a fixed point with respect to a system of cubics pass through four fixed points, it follows that in a system of cubics having a common point of inflection and a common inflectional tangent the harmonic polars of the common point of inflection meet in a point.

Since the common inflectional tangent is the common polar line of the common point of inflection, it follows that such a point is a critic centre of the system of cubics. One cubic of the system then has a node at the common point of inflection of the system, and forms an exception. The line which is the common inflectional tangent to the other cubics of the system cuts this also in three points, but is one of the tangents at the double point; the other tangent at the double point goes through the vertex of the pencil of harmonic polars.

It is evident that the nine basal points of a system of conics may unite into three groups of three each. The cubics will then all have three common points of inflection, and at these points three common inflectional tangents. These three points all lie on a line.

When four basal points of the system of cubics coincide, such a point is a double point on every cubic of the system. This is easily shown as follows, using the method of inversion. Let a system of conics through four points be inverted from one of the four points. The system of conics inverts into a system of cubics, having a common node and passing through three other finite fixed points and the two circular points at infinity. Since the common node counts as four points of intersection, it follows that any two cubics of the system, and hence all of them, intersect in nine points. This system can be projected into a system having a common double point and passing through any five other fixed points.

A number of theorems concerning the system of cubics can easily be inferred from known theorems concerning the system of conics. Since two conics of the system are parabolas, it follows that two cubics of the system are cuspidal. Since three conics of the system break up into pairs of right lines, it follows that three cubics of the system break up into a right line and a conic. Each right line and its corresponding conic intersect in the common double point. The line at infinity cuts the system of conics in pairs of points in involution, the points of contact of the two parabolas of the system being the foci; it follows on inversion that the pairs of tangents to the cubics at their common node form a pencil in involution, the two cuspidal tangents being the foci.

If the four basal points of the system of conics lie on a circle, this circle inverts into a right line, and one cubic then consists of this right line and the lines joining the centre of inversion to the circular points at infinity. This theorem may be stated for the system of cubics as follows: if the conic determined by the five basal points of the system of cubics (not counting the common double points), break up into right lines, the line passing through three of the five points, together with the lines joining the other two points to the common node, constitute a cubic of the system.

If three of the four basal points of the system of conics lie on a line, the conics consist of this line and a pencil of lines through the fourth basal point. Inverting from this fourth point and then projecting, we have a system of cubics consisting of a pencil of lines and a conic through the vertex and the four other fixed points. Hence, when the five fixed points of such a system of cubics lie on a conic through the common node, this conic is a part of every cubic of the system. If we invert the above system of conics from one of the three points on the right line, and then project, we obtain a system of cubics which consists of a system of conics through four fixed points, and a fixed right line through one of these four points. Hence, if two of the five basal points of such a system of cubics be on a line through the common node, this line is a part of every cubic of the system.

If a system of conics having one basal point at infinity be inverted from one of the remaining basal points, this point at infinity inverts to the center of inversion, and we obtain a system of cubics having five coincident basal points and hence passing through only four others. The system of cubics is now so arranged that one tangent at their common double point is common to all. Only one cubic of the system is cuspidal. As before three cubics break up into a right line and conic.