G

A TREATISE

ON

CONIC SECTIONS:

CONTAINING

AN ACCOUNT OF SOME OF THE MOST IMPORTANT MODERN
ALGEBRAIC AND GEOMETRIC METHODS.

BY

GEORGE SALMON, D.D., D.C.L., LL.D.. F.R.S.,

REGIUS PROFESSOR OF DIVINITY IN THE UNIVERSITY OF DUBLIN.

SIXTH EDITION.

London:
LONGMANS, GREEN, AND CO.

1879.

Length of Perpendicular from a Point on a Line
Equations of Bisectors of Angles between two given Right Lines
Area of Triangle in terms of Coordinates of its Vertices.
Area of any Polygon

Condition that three Lines may meet in a Point (see also p. 34)
Area of Triangle formed by three given Lines

Equation of Line through the Intersection of two given Lines

Test that three Equations may represent Right Lines meeting in a Point
Connexion between Ratios in which Sides of a Triangle are cut by any Transversal

by Lines through the Vertices which meet in a Point

Polar Equation of a Right Line

of Loci leading to Equations of Higher Degree

Problems where it is proved that a Moveable Line always passes through a

Centre of Mean Position of a series of Points

Right Line passes through a Fixed Point if Constants in its Equation be
connected by a Linear Relation

Loci solved by Polar Coordinates

.

CHAPTER IV.

THE RIGHT LINE.-ABRIDGED NOTATION,

Meaning of Constant k in Equation a = kB

Bisectors of Angles, Bisectors of Sides, &c. of a Triangle meet in a Point

Equations of a pair of Lines equally inclined to a, ß

Theorem of Anharmonic Section proved

Algebraic Expression for Anharmonic Ratio of a Pencil

Homographic Systems of Lines

Expression of Equation of any Right Line in terms of three given ones

Harmonic Properties of a Quadrilateral proved (see also p. 317)

Homologous Triangles: Centre and Axis of Homology

Condition that two Lines should be mutually Perpendicular

Length of Perpendicular on a Line

Proof that middle Points of Diagonals of Quadrilateral lie in a Right Line
Intersections of Perpendiculars, of Bisectors of Sides, and of Perpendiculars at

pp. 149, 153, 155, 266

Condition that Equation of second Degree should represent Right Lines (see also

Number of conditions that higher Equations may represent Right Lines
Number of terms in Equation of nth Degree

Conditions that general Equation may represent a circle

Coordinates of Centre and Radius

Condition that two Circles may be concentric

that a Curve shall pass through the origin

Coordinates of Points where a given Line meets a given Circle

Imaginary Points

General definition of Tangents

Condition that Circle should touch either Axis

Equation of Tangent to a Circle at a given Point

Condition that a Line should touch a Circle

Equation of Polar of a Point with regard to a Circle or Conic
Length of Tangent to a Circle

Line cut harmonically by a Circle, Point, and its Polar

Equation of pair of Tangents from a given Point to a Circle .

Circle through three Points (see also p. 130)

Condition that four Points should lie on a Circle, and its Geometrical meaning

Polar Equation of a Circle .

CHAPTER VII.

EXAMPLES ON CIRCLE.

Circular Loci

Condition that intercept by Circle on a given Line may subtend a right Angle at

a given Point

If a Point A lie on the polar of B, B lies on the polar of A

Conjugate and self-conjugate Triangles

Conjugate Triangles Homologous

If two Chords meet in a Point, Lines joining their extremities transversely meet

on its Polar

Distances of two Points from the centre, proportional to the distance of each

from Polar of other

98

Expression of Coordinates of Point on Circle by auxiliary Angle

Problems where a variable Line always touches a Circle

Examples on Circle solved by Polar Coordinates