In this Second Edition several new articles have been

added, especially in the latter part of the work, and the chapter on Reciprocal Polars considerably enlarged.

GONVILLE AND CAIUS COLLEGE,
August, 1866.

N. M. F.

In the Third Edition, I have rewritten some articles where the demonstrations were imperfect or obscure, and have added some examples, taken from various Cambridge Examination papers.

DECEMBER, 1875.

2.

3.

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

Distance between two given Points

4

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.

10.

11.

Every Equation of the First Degree represents a Straight Line 11

Point of Intersection of Two Straight Lines

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

12

12.

13.

14.

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

Condition that Three Straight Lines may intersect in a Point

Condition that Two Straight Lines may be parallel to one

another. Line at Infinity

ib.

14

ib.

15.

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

parallel to a given Straight Line

17

16.

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

The Anharmonic Ratio of a Pencil is equal to that of the range
in which it is cut by any Transversal

Definition of an Harmonic Pencil

23

ib.

25

ARTS.

22.

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

an Harmonic Pencil

PAGE

25

26

27

28

30

. 31

27-29. Anharmonic Properties of Points and Lines in Involution

CHAPTER II.

SPECIAL FORMS OF THE EQUATION OF THE SECOND DEGREE.

1. Every Equation of the Second Degree represents a Conic

Section .

2, 3. Equation of the Conic described about the Triangle of Refer-

Condition of Tangency. Every Parabola touches the Line at

Infinity.

Equation of the Conic touching the Three Sides of the Tri-

angle of Reference .

Equations of the Four Circles which touch the Three Sides of

the Triangle of Reference

11-15. Equation involving the Squares only of the Variables

16. Condition of Tangency

Equation of the Circle, with respect to which the Triangle

of Reference is self-conjugate.

51

20.

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

Pole of a given Straight Line

Condition of Tangency. Condition for a Parabola

Co-ordinates of the Centre.

ib.

ib.

55

EXAMPLES.

56

ON THE CONIC REPRESENTED BY THE GENERAL EQUATION OF THE

SECOND DEGREE.

2.

3.

To find the point in which a straight line, drawn in a given

direction through a given point of the conic, meets the conic

again

Equation of the Tangent at a given Point.

74

75

4, 5.

6.

Condition that a given Straight Line may touch the Conic

Condition for a Parabola .

76

78

7.

Condition that the Conic may break up into Two Straight

Lines

ib.

10.

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

given external Point

80

ARTS.

PAGE

15.

16.

17.

18.

All Circles pass through the same two points at infinity

All Conics, similar and similarly situated to each other, in-

tersect in the same two points in the line at infinity

Radical axis of two similar and similarly situated Conics.

Property of the nine-point Circle

. 88

19.

Equation of the nine-point Circle

92

20.

Locus of the intersection of two Tangents at right angles to

one another. Directrix of a Parabola

21.

To find the magnitudes of the axes of the Conic

22.

To find the area of the Conic. Criterion to distinguish be-

tween an Ellipse and an Hyperbola

1. Definition of the Triangular Co-ordinates of a Point

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

and vice versâ

106

6, 7.

The polar reciprocal of a conic is a conic

107

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

113

15.

Any straight line drawn through a given point A is divided

harmonically by any Conic Section and the polar of A with

respect to it

115