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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 Liue 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.
12.
13.
14.
15.
16.
Equation of a Straight Line passing through Two given Points
Equation of a Straight Line passing through the Point of Inter-
section of Two Given Straight Lines
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
Equation of a Straight Line, drawn through a given Point, paral-
lel to a given Straight Line
Inclination of a Straight Line to a side of the Triangle of Refer-
ib.
The Anharmonic Ratio of a Pencil is equal to that of the range
in which it is cut by any Transversal
ARTS.
22.
The Bisectors of any Angle form, with the Lines containing it,
an Harmonic Pencil.
Anharmonic Ratio of a given Pencil .
23.
24.
Fourth Harmonic to Three given Straight Lines
PAGE
25
26
27
29
30
27-29. 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 Sec-
tion
Equation of the Conic described about the Triangle of Refer-
Condition of Tangency. Every Parabola touches the Line at
Infinity.
35
Equation of the Circumscribing Circle
Equation of the Conic touching the Three Sides of the Triangle
of Reference
Equations of the Four Circles which touch the Three Sides of
the Triangle of Reference
40
42
Condition of Tangency
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
9880
59
бо
66
Sign changed by interchange of two Consecutive Lines or Co-
lumns
EXAMPLES
71
CHAPTER IV.
ON THE CONIC REPRESENTED BY THE GENERAL EQUATION OF THE
SECOND DEGREE.
2. 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
72
74
75
78
79
81
4, 5.
6.
Condition that a given Straight Line may touch the Conic Condition for a Parabola
Lines
JO.
given external Point
JI.
.
Condition that the Conic may break up into Two Straight
Equation of the Polar of a given Point
Co-ordinates of the Pole of a given Straight Line
Equation of the pair of Tangents, drawn to the Conic from a
Co-ordinates of the Centre
33
All Circles pass through the same two points at infinity
All Conics, similar and similarly situated to each other, intersect
in the same two points in the line at infinity.
87
17.
Radical axis of two similar and similarly situated Conics
88
90
91
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
To find the area of the Conic. Criterion to distinguish between
an Ellipse and an Hyperbola
92
93
CHAPTER V.
TRIANGULAR
CO-ORDINATES.
1.
Definition of the Triangular Co-ordinates of a Point
99
Definition of a Polar Reciprocal
The degree of a curve is the same as the class of its reciprocal,
and vice versâ
The polar reciprocal of a conic is a conic
Equation of the Polar Reciprocal of one Conic with regard to
another
104
105
тоб
The anharmonic ratio of the Pencil, formed by four intersecting
straight lines, is the same as that of the range formed by their
poles
Any straight line drawn through a given point A is divided har-
monically by any Conic Section and the polar of A with re-
spect to it
112
CONTENTS.
If four straight lines form an harmonic pencil, either pair will
be its own polar reciprocal with respect to the other
Condition that two pairs of straight lines may form an harmonic
pencil
xi
115
ΓΙ
118
The three circles described on the diagonals of a complete quad-
rilateral as diameters have a common radical axis
Foci of a quadrilateral
25.
28.
29.
30.
31.
Orthocentre of a triangle
The director circles of all conics which touch four given straight
lines have a common radical axis
Polar reciprocal of a Circle with regard to any point.
Instances of Transformation of Theorems by Reciprocation with
respect to a point .
Corresponding Points and Lines. The angle between the radius vector and tangent in any curve is equal to the corresponding angle in the Reciprocal Curve
Co-ordinates of the foci of a Conic
122
124
119
120
CHAPTER VII.
TANGENTIAL CO-ORDINATES.
Definition of the Tangential Co-ordinates of a Straight Line
2. Interpretation of the Negative Sign. Equations of certain points 131
3. General Equation of a Point
132
Identical relation between the co-ordinates of any straight Line. 133
6. An equation of the nth degree represents a curve of the nth
7. Equation of a Conic, touching the three sides of the triangle of
reference
8. Equation of circumscribed Conic
9. Equation of the Pole of a given straight line, and of the centre.