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11.
Condition that a Quadratic Function may be resolvable into
Two Factors
ib.
12, 13. Pascal's Theorem
69
EXAMPLES
71
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 conic
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
8.
Equation of the Polar of a given Point
9.
Co-ordinates of the Pole of a given Straight Line
79
10.
Equation of the pair of Tangents drawn to the Conic from a
given external Point
ARTS.
PAGE
15.
16.
All Circles pass through the same two points at infinity
All Conics, similar and similarly situated to each other, in-
88
17.
18.
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
89
91
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
93
21.
To find the magnitudes of the axes of the Conic
94
22.
To find the area of the Conic. Criterion to distinguish be-
tween an Ellipse and an Hyperbola
96
98
CHAPTER V.
TRIANGULAR CO-ORDINATES.
1. Definition of the Triangular Co-ordinates of a Point
2. Formulæ relating to Straight Lines
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
105
106
107
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
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
23.
24.
25.
28.
29.
30.
31.
If four straight lines form an harmonic pencil, either pair
will be its own polar reciprocal with respect to the other 116
Condition that two pairs of straight lines may form an har-
monic pencil
Reciprocation with respect to a Circle
123
Reciprocation with respect to a Point
The three circles described on the diagonals of a complete
quadrilateral as diameters have a common radical axis
Foci of a quadrilateral
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 cor-
responding angle in the Reciprocal Curve
Definition of the Tangential Co-ordinates of a Straight Line. 131
Interpretation of the Negative Sign. Equations of certain
points .
4. Identical relation between the co-ordinates of any straight
133
An equation of the nth degree represents a curve of the nth
class
Equation of a Conic, touching the three sides of the triangle
of reference .
136
Equation of circumscribed Conic
138
Equation of the Pole of a given straight line, and of the
1-3.
4-7.
CHAPTER VIII.
146
ON THE INTERSECTION OF CONICS, AND ON PROJECTIONS.
Any two conics intersect in four points, real or imaginary.
Vertices of the quadrangle formed by these points
If the four points of intersection be all real, or all imaginary,
all the vertices are real. If two of the points of inter-
section be real, and two imaginary, one vertex only is real.
If the four points of intersection be all real, all the common
chords are real; if not, one pair only is real
Invariants of two Conics.
ON PROJECTIONS.
147
150
13.
14.
Definition of Projections
Projection to infinity
Any quadrilateral may be projected, in an infinite number of
ways, into a parallelogram, of which the angles are of any
magnitude
Any two conics may be projected into concentric curves
Also into similar and similarly situated curves
154
These projections may be effected in an infinite number of ways
Any two intersecting conics may be projected into hyperbolas
of any assigned eccentricity
Any two conics may be projected into conics of any eccentricity,
or into circles
155
Projection of the foci and directrices of a Conic
The anharmonic ratio of any pencil or range is unaltered by
projection
156
Any two lines, which make an angle A with each other, form
with the lines joining the circular points at infinity to their
point of intersection, a pencil of which the anharmonic ratio
is e(π-24)√=1
157
The anharmonic ratio of any four points on, or any four tan-
gents to, a conic, is constant
158
Any system of points in involution projects into a system in
involution, and the foci of one system project into the foci
of the other
26.
A system of Conics, passing through four given points, cut any
straight line in a system of points in involution.
159
ON THE DETERMINATION OF A CONIC FROM FIVE GIVEN GEOMETRICAL
CONDITIONS.
If five points be given, one conic only can be drawn
162
3.
If four points and one tangent be given, two conics can be
drawn
4.
If three points and two tangents be given, four conics can be
If two points and three tangents be given, four conics can be
If one point and four tangents be given, two conics can be
164
If five tangents be given, one conic only can be drawn
Reduction of certain other conditions to these
Conjugate triad
165
ON THE LOCUS OF THE CENTRE OF A SYSTEM OF CONICS WHICH