Cambridge:

PRINTED BY C. J. CLAY, M.A.

AT THE UNIVERSITY PRESS.

PREFACE.

THE prominence which the modern geometrical methods have recently acquired in the studies of the University of Cambridge, appears to justify the publication of a treatise devoted exclusively to these branches of Mathematics. This remark applies more especially to the method of Trilinear Co-ordinates, which forms the subject of the greater part of the following work. My object in writing on this subject has mainly been to place it on a basis altogether independent of the ordinary Cartesian system, instead of regarding it as only a special form of Abridged Notation.

A desire not unduly to increase the size of the book has prevented me from proceeding beyond Curves of the Second Degree.

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.

1.

2.

3.

4-6.

7.

Investigation of Equations of certain Straight Lines

Every Straight Line may be represented by an Equation of the
First Degree.

DEFINITION of Trilinear Co-ordinates
Identical relation between the Trilinear Co-ordinates of a Point ib.
Distance between two given Points

1

4

9

8.

9.

Every Equation of the First Degree represents a Straight Line
Point of Intersection of Two Straight Lines

11

12

10.

11.

Equation of a Straight Line passing through Two given Points. ib. Equation of a Straight Line passing through the Point of Intersection of Two given Straight Lines

13

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

15. Equation of a Straight Line, drawn through a given Point, parallel to a given Straight Line

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

ANHARMONIC RATIO. Definitions

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

ARTS.

PAGE

25

22.

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

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

3333

34

35

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

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

39

40

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
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

Equation of a Line joining Two given Points

52

53

ib.

54

ib.

iò.

55

56

62