CONTENTS.
CHAPTER I.
##### SECTION I. - THE POINT.
RULE OF SIGNS,
Geometric sum or resultant,
Projections,
##### SECTION II. - CARTESIAN CO-ORDINATES.
Définitions,
4, 5
Distance between two points,
Condition that three points should be collinear,
Area of triangle or polygon in terms of the co-ordinates of vertices,
Co-ordinates of a point dividing in a given ratio the join of two given points,
Co-ordinates of harmonic and isotomic conjugates,
Theory of the mean centre,
14, 15
##### SECTION III.
Polar co-ordinates,
Transformation of co-ordinates,
19-22
##### SECTION IV. - COMPLEX VARIABLES.
Definition of, and mode of representation,
Sum or difference of two complex variables,
Product and quotient of ditto,
Examples on complex variables,
Miscellaneous Exercises,
CHAPTER II.
##### SECTION I. - CARTESIAN CO-ORDINATES.
To represent a right line by an equation,
30, 31
Standard form of equation,
Line parallel to one of the axes,
Equation of line defined,
Comparison of different forms of equation,
To find the angle between two lines,
36, 37
Power of point with respect to line or curve,
Length of perpendicular from a given point on a given line,
Equation of line through the intersection of two given lines,
Equation of line passing through two given points,
Co-ordinates of point of intersection of two given lines,
Equation of line through a given point and making a given angle with a given line,
Equation of line dividing the angle between two given lines into parts whose sines have a given ratio,
Condition that three lines should be concurrent,
Condition that general equation of the second degree should be the product of the equations of two lines,
If the general equation of the second degree represents two lines, to find the co-ordinates of their point of intersection,
Equation of bisectors of angles made by a given line-pair,
53, 54
Theory of anharmonic ratio,
55, 56
Harmonic system of points, harmonic conjugates,
Anharmonic ratio of four rays of a pencil,
Anharmonic ratio of four lines whose equations are given,
Harmonic pencil
##### SECTION II. - SYSTEMS OF THREE CO-ORDINATES.
Trilinear or Normal co-ordinates defined,
Isogonal conjugates; symmedian point and lines,
Brocard angle and Brocard points,
Barycentric or areal co-ordinates defined,
Relation between the normal and barycentric co-ordinates of a point,
Isotomic conjugates,
Trilinear equation of the join of two given points,
Trilinear poles and polars: Cotes' Theorem,
Relation (identical) between the equation of four lines, no three of which are concurrent,
Perspective; triangles in, axis and centre of,
Length of perpendicular from a given point on a given line,
Angle between two given lines,
73, 74
Line at infinity,
Cyclic points-Isotropic lines,
Conditions for parallelism and perpendicularity when general equation in trilinear co-ordinates represents two lines,
Condition that la+ mb+ ng= 0 may be antiparellel to g = 0,
Distance between two points,
Area of triangle the co-ordinates of whose vertices are given,
Area of triangle formed by three given lines,
80, 81
Complementary and anti-complementary points and figures,
Supplementary points,
Triangles in multiple perspective,
Isobaryc group of points,
Comparison of points and line co-ordinates, equation of a point,
The absolute co-ordinates of a line,
Equation of the cyclic points,
Miscellaneous Exercises,
CHAPTER II.
##### SECTION I. - CARTESIAN CO-ORDINATES.
To find the equation of a cirle,
Geometrical representation of power of point with respect to circle,
Equation of circle whose diameter is the intercept which a given circle makes on a given line,
Equation of tangent to a circle,
Equation of pair of tangents from given point to circle,
Equations of chords of contact of common tangents to two circles,
Pole and polar with respect to a circle,
Inverse points with respect to a circle,
Angle of intersection of two circles or curves
Mutual power of two circles,
Circle cutting three given circles at given angles,
Circle cutting three given circles orthogonally,
Circle touching three given circles,
Circle through three given points,
Frobenius's Theorem,
Condition that four circles should cut a fifth orthogonally or be tangential to it
Coaxal system circles,
Radical axis and limiting points of coaxal system,
Centre of similitude,
Circle of similitude,
##### SECTION II. - A SYSTEM OF TANGENTIAL CIRCLES.
Equations of circles in pairs touching three circles,
To investigate the condition that any number of circles should have one common tangential circle,
Tangent to nine-points circle at point of contact with incircle,
##### SECTION III. - TRILINEAR CO-ORDINATES.
Circumconic of triangle of reference,
Coates' Theorem,
Circumcircle of triangle of reference,
Circumcircle of polygon of any number of sides,
Tangents to circumconic at angular points,
Chord of, and tangent to, circumcircle,
Incircle of triangle of reference,
Sir Andrew Hart's method of finding equation of incircle,
Join of two points on incircle,
Tangent at any point on incircle,
Conditions that general equation should represent a circle,
Equation of circle thourgh three given points,
Equation of pedal circle of a given point,
Simson's line,
##### SECTION IV. - TANGENTIAL EQUATIONS.
Tangential equation of circumcircle of triangle,
Tangential equation of circumcircle of polygon,
Tangential equation of incircle of triangle,
Tangential equation of incircle n-sided polygon,
Tangential equation of circle whose centre and radius are given,
Nine-points circle, equation of,
Pascal's Theorem,
Miscellaneous Exercices on the circle,
CHAPTER IV.
##### CARTESIAN CO-ORDINATES.
Contracted form of the equation,
Change of origin,
Intersection of line and conic, discussion of equation,
Co-ordinates of centre of conic,
Distinction between hyperbola, parabola and ellipse,
Reduction of equation to centre as origin,
Line of centres,
Locus of middle points of system of parallel chords, diameters,
Conjugate diameters of central conics,
Axes; reduction of general equation to normal form,
Invariants of, and equation giving squares of semiaxes of conic represented by, the general equation,
Equation of new axes when referred to old,
Reduction of general equation to parabola,
Tangent to, and tangential equation of, general conic,
Ratio in which join of two given points in cut by conic,
Harmonic properties of polars,
Polar of a given point; tangent,
Condition that join of two points may be cut in given anharmonic ratio by the conic,
Equation of pair of tangents from an external point,
Orthoptic circle of conic,
Classification of conics,
Asymptotes defined and their equation found,
Asymptotes angle between,
Asymptotes hyperbola referred to as axes,
Newton's Theorem,
Locus of centre of conic through four points,
Exercices on general equation,
CHAPTER V.
##### THE PARABOLA.
The parabola; its axis, directrix, focus, vertex,
Latus Rectum,
Intrinsic angle of point on curve,
Equation of tangent to parabola,
Subtangent bisected at vertex,
Pedal of, with respect to focus,
Locus of middle points of system of parallel chords,
Diameter through intersection of two tangents bisects chord of contact,
Equation of, referred to any diameter and tangent at its extremity
Isoptic curve of parabola,
Normal defined and equation found,
Subnormal constant in parabola,
Joachimsthal's circle for,
Circle of curvature, radius and centre of,
Circle of curvature, equation of,
Locus of centre of curvature, evolute,
Poral equation of, the focus being pole,
Length of line drawn from a given point in a given direction to meet the parabola,
Relation between the perpendiculars from the angular points of a circumtriangle on any tangent,
Exercices on the parabola,
CHAPTER VI.
##### THE ELLIPSE.
Focus; directrix; eccentricity,
Standard from of equation; centre; latus-rectum,
Method of generating ellipse. Pohlke, Boscovich, Hamilton,
Eccentric angle; co-ordinates of a point on curve in terms of,
The auxiliary circle,
Ellipse, the orthogonal projection of a circle,
Locus of middle points of a system of parallel chords,
Equation of tangent,
Conjugate diameters; formulae of Chasles; theorem of Apollonius,
Given any two conjugate diameters, to find the axis; construction of Mannheim,
Equation of, referred to a pair of conjugate diameters,
Supplemental chords,
Schooten's method of describing ellipse,
Equation of normal,
Normals through a given point; Apollonian hyperbola,
Evolute of ellipse
Radius of curvature at any point of,
Equation of the four normals from given point,
Joachimsthal's circle for,
Laguerre's Theorem,
Purser's Parabola,
Length of perpendicular from focus on tangent,
Pedal of, with respect to either focus,
New method of drawing tangents to curve,
Chords passing through a focus,
Angle between two tangents,
Locus of intersection of rectangular tangents:
Angle between two tangents expressed in terms of focal vectors to point of intersection,
Property of three confocal ellipses,
Fregier's Theorem,
Locus of pole of tangent with respect to circle whose centre is one of the foci,
Reciprocal polar of a curve defined,
Reciprocal polar of a circle with respect to a circle,
Reciprocal polars of a system of confocal ellipses or coaxal circles,
Rectangle contained by segments of any chord passing through a fixed point is to square of parallel semidiameter in a constant ratio,
Any two tangents are proportional to parallel semidiameters,
To find the major axis of an ellipse confocal with a given one, and passing through a given point,
Elliptic co-ordinates defined,
Polar equation of, a focus being pole,
Sum of reciprocals of segments of focal chord,
Exercises on the ellipse,
CHAPTER VII.
##### THE HYPERBOLA.
Focus; directrix; eccentricity,
Standard form of equation; centre, transverse and conjugate axis,
Latus-rectum; equilateral hyperbola,
Co-ordinates of a point on, expressed in terms of a single variable,
Locus of middle points of a system of parallel chords,
Equation of tangent,
Conjugate hyperbola,
Conjugate hyperbola equation,
Equation of, referred to a pair of conjugate diameters,
Equation of normal,
Joachimsthal's circle for,
Lengths of perpendiculars from foci on tangent,
Pedal of, with respect to focus,
Reciprocal of, with respect to focus,
Polar equation of, the centre being pole,
Eccentricity given by secant of half angle between asymptotes,
Equation of, referred to asymptotes,
Co-ordinates of centre and radius of curvature at any point on curve,
Polar equation of, the focus being pole,
Area of equilateral hyperbola between an asymptote and two ordinates,
Hyperbolic functions defined,
Co-ordinates of a point on, expressed in terms of a single variable,
Locus of pole of chord subtending a right angle at a fixed point,
Exercices on hyperbola,
CHAPTER VIII.
##### SECTION I. - FIGURES INVERSELY SIMILAR.
Double point or centre of similitude; double lines,
Triangles inversely similar are orthologique,
##### SECTION II. - PENCILS INVERSELY EQUAL.
Double directions of two pencils inversely equal,
Generation of the equilateral hyperbola,
Locus of centre of circumequilateral hyperbola of a triangle,
Orthocentre lies on circumequilateral hyperbola of a triangle,
##### SECTION III. - TWIN POINTS.
Twin points defined,
To construct the twin of a given point,
Isogonal conjugates of inverse points with respect to circumcircle are twin points.
Twin points are at the extremities of a diameter of circumequilateral hyperbola,
Locus of middle point of join of twin points,
Barycentric co-ordinates of twin points,
##### SECTION IV. - TRIANGLES DERIVED FROM SAME TRIANGLE.
Pedal triangle of a point: sides of same,
Pedal triangle area of,
Antipedal triangles,
Harmonic transformation of a triangle,
Ditto, sides and area of,
Locus of points whose pedal triangles have a constant Brocard angle,
##### SECTION V. - TRIPOLAR CO-ORDINATES.
Tripolar co-ordinates defined; tripolar pair,
To construct a point, being given the mutual ratios of its tripolar co-ordinates,
Pedal triangles of a tripolar pair are inversely similar,
Isodynamic points (Neuberg),
Relation between the tripolar and normal co-ordinates of a point,
Lucas's Theorem,
Equation of join of two points; of circle through three points; area of triangle formed by three points; &c., &c., in tripolar co-ordinates,
CHAPTER IX.
##### SPECIAL RELATIONS OF CONIC SECTIONS.
S - kS' = O represents a curve passing through all the points of intersection of the curves S = 0 and S' = 0,
Special cases-
1°. General equation of a conic passing through four fixed points on a given conic,
2°. General equation of conics having double contact,
3°. General equation of conics touching two lines,
All circles pass through the cyclic points,
Every parabola touches the line at infinity,
Contact of different orders,
Osculating circle,
Parabola having contact of third order,
Focus defined as an infinitely small circle having imaginary double contact,
All confocal conics are inscribed in the same imaginary quadrilateral,
Antifoci,
Tangential equation of all conics confocal with a given one,
Construction of circle of curvature at any point of central conic,
Construction of chord of osculation at any point,
Four chords of osculation can be drawn through any point,
Through any point on a conic can be described three circles to osculate the conic elsewhere,
Steiner's Circle,
Six osculating circles of a given conic can be described to cut a given circle orthogonally, or to pass through a given point,
Malet's Theorem, viz. the centres of the six osculating circles which pass through a given point lie on a conic,
Four-pointic contact or hyperosculation,
Equation of a conic having double contact with two given conics,
Propeties of common chords of two conics having double contact with a third, or of three with a fourth,
Diagonals of quadrilaterals, inscribed in and circumscribed to a conic, form a harmonic pencil,
Brianchon's Theorem,
Twelve points of intersection of three conics, each of which has double contact with a fourth, lie six-by-six on four conics,
Through every point of a conic can be described a parabola to hyperosculate it at that point,
Extension of this theorem,
Equation representing foci of general conic,
Graves' Theorem,
Proprerties of arcs and tangents of confocal conics,
M Cullagh's, Chasles', and Fagnani's Theorem,
If a polygon circumscribe a conic, and if all the summits but one move on confocal conics, the locus of that summit will be a confocal conic,
Similar conics,
Homothetic figures defined,
Conditions that two conics should be homothetic,
Conditions of being similar, but not homothetic,
Properties of homothetic, and of similar conics,
Pascal's Theorem and line,
Steiner's Theorem,
M'Cay's Extension of Feuerbach's Theorem,
Equation giving squares of semiaxes of general conic,
Area of ellipse given by general equation,
Miscellaneous Exercises,
CHAPTER X.
##### THE GENERAL EQUATION - TRILINEAR CO-ORDINATES.
Aronhold's notation,
Some equations expressed in,
Condition that ax2 = 0 should represent a line-pair,
Co-ordinates of double points,
Co-ordinates of pole of line with respect to conic,
Tangential equation of conic,
Condition that pole of lx = 0 should lie on mx = 0,
Equation of point-pair in which a line intersects a conic,
Geometrical signification of the vanishing of a coefficient in the general trilinear equation,
Equation of conic having triangle of reference as autopolar triangle,
Equation of conic referred to focus and directrix,
Conditions that general trilinear equation should represent an ellipse, parabola, or hyperbola,
Orthoptic circle of a conic referred to its autopolar triangle,
Discussion of the equation ab = g2, and properties of curve it represents,
Anharmonic properties of four points on a conic,
Tangential equation of ab = g2,
Equation of circle or curvature at any point on ab = k2g 2,
Theory of envelopes,
Envelope of a system of confocal conics
Condition that join of points in which a given line cuts a given conic, subtends a right angle at origin,
CHAPTER XI.
##### THEORY OF PROJECTION.
Base line; infinite line; projection of a point,
Projection of lines,
Anharmonic ratio of pencil unaltered by projection; projective properties,
Parallel are projected into concurrent lines,
Curve of any degree projected into curve of same degree,
Projection of concentric circles,
Projection of coaxal circles,
Any conic can be projected into a circle whose centre is the projection of any point in plane of conic,
The pencil formed by the legs of a given angle, and the isotropic lines of its vertex has a given anharmonic ratio,
Projection of focal properties,
Projection of locus described by vertex of constant angle,
Orthogonal projection; axis and modulus of,
Figures orthogonally related,
Conic of any species projected into conic of same species,
Lhuilier's problem, viz., to project a given triangle into another similar to a third,
Two triangles orthogonally related are orthologique,
Steiner ellipse of triangle,
##### SECTIONS OF A CONE.
Cone of second degree; base edge, vertex, axis,
Cone right and oblique,
Antiparallel section,
Sections which are parabolas, ellipses, hyperbolas,
Foci of plane section of right cone,
Directrices of section,
CHAPTER XII.
##### THEORY OF HOMOGRAPHIC DIVISION.
Condition that four points form a harmonic system,
Condition that three point pairs have a common pair of harmonic conjugates,
Condition that two line pairs form a harmonic pencil,
Locus of point whence tangents to two given conics form a harmonic pencil,
Point and line harmonic conics of two given conics,
Projective rows defined,
1 to 1 correspondence defined,
Projective rows have a 1 to 1 correspondence and are homographic,
Points which correspond to infinity,
Similar rows,
Projective pencils defined,
In any two projective pencils the anharmonic ratio of any four rays is the same as that of their homologous rays,
Maclaurin's and Newton's methods of generating conics,
Double points defined,
Double points and homologous point pairs have constant anharmonic ratio,
Double points found geometrically,
Concentric or superposed projective pencils,
Involution defined; double points and central point of,
Involution hyperbolic and elliptic,
Involution symmetric, isogonal, orthogonal,
A system of conics through four fixed points cuts any transversal in involution,
CHAPTER XIII.
##### THEORY OF DUALITY AND RECIPROCAL POLARS.
Principle of duality,
Reciprocation defined,
Substitutions to be made in any theorem in order to get the reciprocal theorem,
Reciprocal of Pascal's theorem,
Properties proved by reciprocation,
Special results when reciprocating conic is a circle,
Equation of the reciprocal of any conic,
To find the centre of reciprocation so that the polar reciprocal of a given triangle may be in species,
Lionnet's triangle,
Tangential equation of conic given a focus and circumtriangle,
Equation of the Brocard ellipse,
CHAPTER XIV.
##### SECTION I. - ON A SYSTEM OF THREE FIGURES DIRECTLY SIMILAR.
Homothetic figures; double point or centre of similitude of,
To find the double point of two polygons directly similar,
Three directly similar figures; corresponding lengths, angles of rotation, double points, triangle and circle of similitude of
In ditto, triangle formed by three homologous lines is in perspective with triangle of similitude and locus of centre of perspective is the circle of similitude,
Invariable points and invariable triangle,
Invariable points form a system of three corresponding points,
Invariable triangle and triangle of similitude are in perspective,
Director point; adjoint points; annex triangles,
Annex triangles are directly similar to modular triangles,
Annex circles: Lionnet's circle and triangle,
Triangle formed by any three homologous points is orthologique with Lionnet's triangle,
Triangle formed by any three corresponding points is similar to pedal triangle of any of these points with respect to the corresponding annex triangle,
Some theorems deduced by last proposition,
Exercises by Neuberg,
##### SECTION II. - THEORY OF HARMONIC CHORDS.
Symmedian point defined,
Centres of inversion; Brocard ellipse and circle,
Brocard points and angle,
Harmonic polygons defined,
Polar of Symmedian point is the same with respect to circumcircle and harmonic polygon,
Tucker's circle of harmonic polygon,
If figures directly similar be described on the sides of a harmonic polygon, every system of homologous points lies on the first pedal of a conic,
Exercises,
##### SECTION III. - THE TRIANGLE.
Parallels to sides of a triangle, through its symmedian point, meet sides in six concyclic points,
Lemoine circle and hexagon; Tucker's circles,
Trilinear equation of the Brocard ellipse,
Equation and envelope of Tucker's circles,
First and second triangle of Brocard,
The invariable triangle is triply in perspective with original triangle,
Barycentric co-ordinates of centres of perspective,
##### SECTION IV. - VARIOUS CIRCLES AND CONICS.
Neuberg's circles,
Neuberg's circles equation of, in barycentric co-ordinates,
Neuberg's circles properties of,
First and second angle of Steiner,
Neuberg's circles-different modes of derivation,
M'Cay s circles,
M'Cay s circles, are special cases of annex circles,
Isogonal transformation, isogonal conjugates,
Isogonal transformation of any diameter of circumcircle is an equilateral hyperbola,
Isogonal transformation of a few lines,
Neuberg's hyperbolae,
Poles, with respect to triangle of reference, of Neuberg's hyperbolae, Kiepert's hyperbola, and circumcircle are collinear,
Fuhrmann's circles,
Point of contact of nine-points circle with incircle,
Three points on the common tangent of incircle, and nine-points circle,
Orthocentroidal circle defined and its equation found,
Orthocentroidal circle meets perpendiculars and medians in six points forming a harmonic hexagon,
The Brocard parabolae,
Artzt's parabolae (second group),
Kiepert's hyperbola,
Kiepert's triangle,
Kiepert's hyperbola, any two points on, whose parametric angles differ by a right angle are collinear with centre of nine-points circle,
Kiepert's hyperbola; properties of in connexion with Neuberg's circles,
Kiepert's hyperbola; parametric angles of some special points on,
If two Kiepert's triangles have their parametric angles complements, the orthologique centre of either and ABC is the centre of perspective of ABC and the other,
Jerabek's hyperbola,
Most general equation of circumequilateral hyperbola,
Steiner's ellipse, properties of,
Tarry's point is the centre of perspective of original triangle and that formed by centres of Neuberg's circles,
Steiner's axes are parallel to asymptotes of Kiepert's hyperbola,
The parametric angles of the first Brocard triangle and that formed by Neuberg's centres are complementary,
The foci of Steiner,
The foci of Steiner theorems concerning,
Envelope of sides of Kiepert's triangle,
Co-ordinates of focus of Kiepert's parabola,
Equation satisfied by the Brocard angle,
Equation whose roots are Steiner's angles,
Relation between the Brocard and Steiner angles,
Equation of the seventeen-point cubic and the points (special) it passes through,
CHAPTER XV.
##### INVARIANT THEORY OF CONICS.
Determinant of transformation,
Invariants, covariants, contravariants, mixed concomitants defined,
Pencil and net of conics, trilinear and tangential,
Polar reciprocal of one conic with respect to a second,
Three conics of the pencil S1 - kS2 = 0 represent line-pairs,
Equation of the three line-pairs,
Equation of tangent-pair to conic at points of intersection by a line,
Equation of asymptotes of conic given by general equation,
Lamé's equation,
Equation of bisectors of angles of a given line-pair in oblique co-ordinates,
Calculation of invariants for some particular conics,
Condition that a triangle may be inscribed in one conic, and circumscribed to another,
Three special relations which a triangle can have with respect to a conic,
Tact-Invariant of two conics,
Parallel to ellipse; equation of,
Condition that two conics should osculate,
Centres of six circles, which can be described through any point to osculate a conic, lie on a conic,
Invariant angles of two conics,
Tact-Invariant expressed as product of six anharmonic ratios,
Envelope of line cut harmonically by two conics,
Locus of point whence tangents to two conics form a harmonic pencil,
Anharmonic ratio of pencil from any point of S1 - kS2 = 0 to common points of S1 and S2,
Locus of centres of all conics of a given pencil,
Anharmonic ratio of four conics of a pencil defined,
Conics harmonically inscribed or circumscribed,
Orthoptic circle of conic given by general equation,
Locus of centre of conic harmonically inscribed in four conics,
Other properties of harmonic conics,
Harmonic envelope of two conics for which Q2 vanishes,
Conics for which Q1 and Q2 vanish,
Harmonic system of conics and their harmonic invariant,
Examples of conics which are harmonic,
Poncelet's Theorem,
Locus of third summit of a triangle circumscribed in a conic, two of whose summits move on another conic,
Equation of four common tangents to two conics,
Fourteen-point conic of two given conics,
Tangential equation of four points common to two conics,
Fourteen-line conic of two given conics,
Envelope of the eight common tangents of two conics at their points of intersection,
Autopolar triangle,
Autopolar triangle squares of sides of are covariants,
Mutual power of two conics,
Tact-Invariant of S - L12 = 0, and S - L22 = 0,
Orthogonal invariant, or the condition that two conics should cut orthogonally,
Frobenius's Theorem concerning two systems of five conics inscribed in the same conic,
Condition that four conics should cut a fifth orthogonally, or be tangential to it,
Equation of conic inscribed in a given conic and touching three given conics also inscribed in the same conic,
Orthogonal conics,
Equation of conic cutting orthogonally three given conics inscribed in the same conic,
Locus of double points of a given trilinear net of conics,
Locus of point whose polars with respect to three conics are concurrent,
Jacobian of three conics defined,
Ditto is the locus of the double points on lines cut in involution by the conics,
Ditto is the locus of the double point of all conics of the net l1S1 + l2S2 + l3S3 = 0,
Ditto various theorems concerning,
Envelope of line cutting three conics in involution,
Hermite envelope of net of conics,
Locus of point whence tangents to three conics form a pencil in involution,
Contravariants,
Conditions that general equation should represent an equilateral hyperbola or a parabola,
The covariant F of the cyclic points and any conic gives the orthoptic circle of that conic,
Orthoptic circle of different conics,
Foci, equation of; antifoci,
General equation of conic confocal with a given one,
Co-ordinates of foci,
The covariant F of two conics having double contact passes through their points of intersection,
Identical relations,
Fourteen-point conic of a quadrilateral expressed in terms of the equations of its four sides,
Any three conics are conjugate with respect to one infinite number of quadrilaterals,
Condition that three given conics should have a common point,
Number of independent invariants, &c., &c., of two conics,
The six summits of two triangles, autopolar with respect to a conic, lie on a conic,
Condition that two given lines should intersect on a given conic,
Identical relation connecting coefficients in the equations of six conics harmonically circumscribed to the same conic,
Miscellaneous Exercises,