be a sufficient excuse for the employment of a notation which has hitherto been hardly admitted into Cambridge text-books. I have, however, confined myself rigorously to the demonstration of such elementary properties as are required in the course of this work. I should be glad if the very slight sketch contained in Chapter III. should be the means of inducing any of my readers to refer to the original memoirs on this and kindred subjects.

2.

3.

Identical relation between the Trilinear Co-ordinates of a Point ib.

Distance between two given Points

4—6.

Investigation of Equations of certain Straight Lines

6

7.

Every Straight Line may be represented by an Equation of the

First Degree

9

8.

9.

Every Equation of the First Degree represents a Straight Line

Point of Intersection of Two Straight Lines

II

12

10.

II.

Equation of a Straight Line passing through Two given Points ib.

Equation of a Straight Line passing through the Point of Inter-

section of Two Given Straight Lines

13

12.

13.

14.

Condition that Three Points may lie in the same Straight Line ib.

Condition that Three Straight Lines may intersect in a Point

Condition that Two Straight Lines may be parallel to one

another. Line at Infinity

Equation of a Straight Line, drawn through a given Point,

parallel to a given Straight Line

16.

Inclination of a Straight Line to a side of the Triangle of

Reference

The Anharmonic Ratio of a Pencil is equal to that of the range

ARTS.

22.

23.

24.

25.

26.

27-29.

The Bisectors of any Angle form, with the Lines containing it,

an Harmonic Pencil

Anharmonic Ratio of a given Pencil

Fourth Harmonic to Three given Straight Lines

Harmonic Relation of Points and Lines

ON INVOLUTION. Definitions

Anharmonic Properties of Points and Lines in Involution

CHAPTER II.

SPECIAL FORMS OF THE EQUATION OF THE SECOND DEGREE.

I.

Every Equation of the Second Degree represents a Conic

Section

5.

Condition of Tangency. Every Parabola touches the Line at

Equation of the Conic touching the Three Sides of the Triangle

Equations of the Four Circles which touch the Three Sides of

Equation of the Circle, with respect to which the Triangle of

Reference is self-conjugate

Equation of the Conic which touches two sides of the Triangle

21.

of Reference in the points where they meet the third

Any Chord of a Conic is divided harmonically by the Conic,

any Point, and its Polar

23.

24.

25.

26.

27.

Equation of a Line joining Two given Points.

Equation of the Tangent at a given Point

Pole of a given Straight Line.

Condition of Tangency. Condition for a Parabola
Co-ordinates of the Centre

8.

Sign changed by interchange of two Consecutive Lines or

Columns

9.

Multiplication of a Determinant by a given Quantity

II.

Condition that a Quadratic Function may be resolvable into

Two Factors

ib.

12, 13. Pascal's Theorem

EXAMPLES

68

70

CHAPTER IV.

ON THE CONIC REPRESENTED BY THE GENERAL EQUATION OF THE

2.

SECOND DEGREE.

To find the Point in which a Straight Line, drawn in a given

direction through a given Point of the Conic, meets the

Condition that a given Straight Line may touch the Conic

Condition that the Conic may break up into Two Straight

73

74

76

77

10.

Equation of the pair of Tangents, drawn to the Conic from a

given external Point

All Circles pass through the same two points at infinity

86

16.

All Conics, similar and similarly situated to each other, intersect

in the same two points in the line at infinity

ib.

ARTS.

17.

one another. Directrix of a Parabola

Locus of the intersection of two Tangents at right angles to

PAGE

87

18.

To find the magnitudes of the axes of the Conic

89

19.

To find the area of the Conic. Criterion to distinguish between

an Ellipse and an Hyperbola

EXAMPLES

CHAPTER V.

TRIANGULAR CO-ORDINATES.

I.

Definition of the Triangular Co-ordinates of a Point

94

95

96

The degree of a curve is the same as the class of its reciprocal,

and vice versa

100

6, 7.

The polar reciprocal of a conic is a conic

ΙΟΙ

8.

Equation of the Polar Reciprocal of one Conic with regard to

another

The anharmonic ratio of the Pencil, formed by four intersect-

ing straight lines, is the same as that of the range formed

by their poles

106

109

18.

19.

Polar reciprocal of one Circle with regard to another

Reciprocation with respect to a Point

IIO

112

20.

Instances of Transformation of Theorems by Reciprocation with

respect to a circle

ib.

21.

Corresponding Points and Lines. The angle between the radius

vector and tangent in any curve is equal to the correspond-

ing angle in the Reciprocal Curve