[Junior readers will find all essential parts of the theory of Analytical Geometry included in Chapters i., n., v., vi., x., XL, xn., omitting the articles marked with asterisks.]
CHAPTER
I.
THE POINT.
... ...... ..... ...
Des Cartes' Method of Coordinates Distinction of Signs
.
Distance between two Points Its sign
.
PACK 1
.
2
.
3
.
Coordinates of Point cutting that Distance in a given Ratio Transformation of Coordinates
does not change Degree of an Equation Polar Coordinates
.
.
4
.6 6
9
.
10
.
CHAPTER
II.
THE RIGHT LINE.
Two
Equations represent Points represents a Locus
A single Equation
.
... .
.11
,
.....16 ....
Geometric representation of Equations Equation of a Right Line parallel to an Axis .
.
.
12
.13
.
14
.15
through the Origin in any Position Meaning of the Constants
.
.
.
.
in Equation of a
. Right Line Equation of a Right Line in terms of its Intercepts on the Axes in terms of the Perpendicular on it from Origin, and the Angles with Axes Expression for the Angles a Line makes with Axes
.17
.
.
.
.
.
.
Angle between two Lines Equation of Line joining two given Points Condition that three Points shall be on one Right Line Coordinates of Intersection of two Right Lines .
.
it
.
.
Middle Points of Diagonals of a Quadrilateral are in a Right Line Equation of Perpendicular on a given Line .
of Perpendiculars of Triangle . of Perpendiculars at Middle Points of Sideb
makes
... ... .
.
18
.
.19 20
.21 23 .
25
(see also p. G2)
.
making a given Angle with a given Line
.
.
26
.26
.
.
of Line
24
.
27 27
27
CONTENTS.
IT
... ........
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
.
MOT 28 .
29
80
.81 32
.
.32
.
.
88
.
. 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
34
Polar Equation of a Right Line
86
.
.
CHAPTER EXAMPLES ON
Till,
HI. it
I; IT I
LINE.
.39
. . Investigation of Rectilinear Loci of Loci leading to Equations of Higher Degree . Problems where it is proved that a Moveable Line always passes through a .
.
.
Fixed Point Centre of
Mean
.
.
.
.
85
.36
.
Position of a series of Points
.
.47
.
50
.
.
47
..>.*!
Right Line passes through a Fixed Point connected by a Linear Relation Loci solved by Polar Coordinates
if
Constants in
THE RIGHT LINE. Meaning
of Constant
k
in Equation
a
=
Equation be
60
.
.
CHAPTER
its
IV.
ABRIDGED NOTATION. .
kfi
.
.63
.
.... .... .67
Bisectors of Angles, Bisectors of Sides, Ac. of a Triangle meet in a Point . Equations of a pair of Lines equally inclined to a, ft
84, 54
,
.55
.
Theorem of Anharmonic Section proved Algebraic Expression for Anharmonic Ratio Homographic Systems of Lines
55
of a Pencil
.
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 Perpendiculars at middle Points of Sides meet in a Point Angle between two Lines :
.
.
.59
.
.
.
.
Trilinear Equation of Parallel to a given Line of Line joining two Points
.
.
.
Proof that middle Points of Diagonals of Quadrilateral
lie in
.
.
.
59
.
.
.
.
a Right Line
.60 84, 60
.60 61
.61 62 .
Intersections of Perpendiculars, of Bisectors of Sides, and of Perpendiculars at . middle Points of Sides, lie in a Right Line .
...
Equation of Line at infinity Cartesian Equations a case of Trilinear Tangential Coordinates Reciprocal Theorems
.
62 63
.64 (
.
.
57 67
.
...... ..... .
Trilinear Coordinates
.
.56
.
. '
.
'
>5
6f
,
.
61
CHAPTER
V.
RIGHT LINES. Meaning of a
of
an Equation resolvable into Factors
.
.67
.
of the nth Degree
Homogeneous Equation
... .
.
. . . . Imaginary Right Lines Angle between two Lines given by a single Equation . Equation of Bisectors of Angles between these Lines Condition that Equation of second Degree should represent Right Lines .
68
,
.69 70
.71
.
Number Number
(see also
.72
.... ....
pp. 149, 153, 155, 266 of conditions that higher Equations of terms in Eqxiation of nth Degree
.
.
may
.
represent Right Lines
74
.
74
CHAPTER VL THE
CIRCLE.
Equation of Circle Conditions that general Equation may represent a circle Coordinates of Centre and Radius . . Condition that two Circles
may
be concentric
.
.
.
.
.76
.
.
that a Curve shall pass through the origin . Coordinates of Points where a given Line meets a given Circle . . . . . Imaginary Points
75
77
.
.77
.
.
.....
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
.
77
,
.
.77
,
.79
78 80, 81
.
.
.81
.
.
...
.
.
.
lie
on a
Circle,
CHAPTER
and
its
85
.85
.
.
82, 83
.84
.
.... ......
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) Polar Equation of a Circle
.
.
Equation of Polar of a Point with regard to a Circle or Conic . . Length of Tangent to a Circle .
Condition that four Points should
75
.
.
86
Geometrical meaning
.
86 87
VII.
EXAMPLES ON CIRCLE. Circular Loci
.
...... .... ...... ..... .
.
Condition that intercept by Circle on a given Line a given Point If a Point
A
lie
Conjugate and
on the polar of B,
self -con jugate
B
lies
.
.
may
.
.88
subtend a right Angle at
on the polar of
A
.
.
30
.91 91
Triangles
.92
. . Conjugate Triangles Homologous If two Chords meet in a Point, Lines joining their extremities transversely meet on its Polar
92
Distances of two Points from the centre, proportional to the distance of each from Polar of other
93
.
.
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
... .
.
.
.
94
95
96
CONTENTS.
vi
CHAPTER VIIL PROPERTIES OP
TWO OR MORE
CIRCLE*.
. . Equation of radical Axis of two Circles Locus of Points whence Tangents to two Circles have a given Ratio Radical Centre of three Circles . . . .
Properties of system of Circles having limiting Points of the system
The
Properties of Circles cutting
two
common
radical
to
two
Circles
Centres of Similitude
.
.
.101
.
102
Angles
. .
.
.
100
. .
(see pp. 130, 361)
.
.
.99
.
. .
99
.
Circles at right Angles, or at constant
Equation of Circle cutting three at right Angles
Common Tangent
Axis
.
.
.98
.
.
.
.103 .105 .708
Axis of Similitude Locus of centre of Circle cutting three given Circles at equal Angles All Circles cutting three Circles at the same Angle have a common Axis of .
.
.
.
.
.
Similitude
.
.
.
.
.
.
.
To
describe a Circle touching three given Circles (see also pp. 115, 135, 291) . Prof. Casey's Solution of this Problem . .
by Inversion
.
CHAPTER THE CIRCLE
.
.
110
.
113
.118 114
,
,
108
109, 131
Relation connecting common Tangents of four Circles touched by same fifth . . . Method of Inversion of Curves , . Quantities unchanged
102
.
J14
IX.
ABRIDGED NOTATION.
. 116 , , Equation of Circle circumscribing a Quadrilateral 1 18 . . . Equation of Circle circumscribing Triangle a, /3, -y 1 18 . . . Geometrical meaning of the Equation Locus of Point such that Area of Triangle formed by feet of Perpendiculars .119 . from it on sides of Triangle may be given .
Equation of Tangent to circumscribing Circle at any vertex Equation of Circle circumscribing a Quadrilateral . . Tangential Equation of circumscribing Circle . Conditions that general Equation should represent a Circle . Radical Axis of two Circles in Trilinear Coordinates . Equation of Circle inscribed in a Triangle
Its Tangential Equation
.
119
.
.
.119
.
.
.
.
.
.121 .121 .122
...... ...... ..... .
.
.123
.
184
1X5 . . Equation of inscribed Circle derived from that of circumscribing Feuerbach's theorem, that the four Circles which touch the sides of a Triangle . are touched by the same Circle . 127, 313, 359 .
Length of Tangent to a
Circle in Trilinear Coordinates
Tangential Equation to Circle whose Centre and Radius is given Distance between two Points expressed in Trilinear Coordinates
BIO-RUMINANT NOTATION
.
.
.128 128
.
.
.128
.
129
130 Determinant Expressions for Area of Triangle formed by three Lines 180 for Equations of Circles through three Points, or cutting three at right Angles .131 Condition that four Circles may have a common orthogonal Circle 134 Relation connecting mutual distances of four Points in a Plane . 185 i Proof of Prof. Casey's theorems .
.
.
.
CHAPTER
X.
GENERAL EQUATION OF SECOND DEORBE. Number
of conditions which determine a Conic . Transformation to Parallel Axes of Equation of second Degree
136
.
.
.
187
CONTENTS.
Vll
.
. Equations of Diameters Diameters of Parabola meet Curve at infinity . . Conjugate Diameters
-.
.
.
.
.
140 143
144
.
.
.
.146
145 146
.
.
.
Harmonic Property of Polars
139
.
......
.
Class of a Curve, defined
.
and Parabola
Distinction of Ellipse, Hyperbola, Coordinates of centre of Conic
Equation of a Tangent Equation of a Polar
PAGE 188
.... ..... ....
Discussion of Quadratic which determines Points where Line meets a Conic , Equation of Lines which meet Conic at infinity
.
.
.
(see also p. 296)
.
.
Polar properties of inscribed Quadrilateral (see also p. 319)
.
.147
.
.148
.
147
148
.
Equation of pair of Tangents from given Point to a Conic (see also p. 269) Bectangles under segments of parallel Chords in constant ratio to each other
149
.
150 151
. . Case where one of the Lines meets the Curve at infinity . Condition that a given Line should touch a Conic (see also pp. 267, 340) Locus of centre of Conic through four Points (see also pp. 254, 267, 271, 302, 820)
CHAPTER
152 163
XI.
CENTRAL EQUATIONS.
....
Transformation of general Equation to the centre Condition that it should represent right Lines
.
Centre, the Pole of the Line at infinity (see also p. 296) . . . Asymptotes of Curve
.
.
.
.155
155
.155
.
.
154
.
.156
. . . Equation of the Axes, how found Functions of the Coefficients which are unaltered by transformation Sum of Squares of Reciprocals of Semi-diameters at right Angles is constant . Sum of Squares of conjugate Semi-diameters is constant .
157
.
159
.
..... ....... ......
159
.
Polar Equation of Ellipse, centre being Pole
.
.
Figure of Ellipse investigated . Geometrical construction for the Axes (see also p. 173) Ordinates of Ellipse in given ratio to those of concentric Circle
Figure of Hyperbola
Conjugate Hyperbola
Asymptotes defined
.
.
.160
,
.161
.
.
.
Eccentricity of a Conic given by general Equation Equations for Tangents and Polars
.
.
.
CONJUGATE DIAMETERS:
.
.
.
.
166
.166 167
.168
...... ......
Angle between conjugate Diameters Locus of intersection of Tangents which cut at righ Angles .
269, 352)
164 165, 1GG
.
.
.
.
164
155, 164
.
.
their Properties (see also p. 159)
Equilateral Hyperbola : its Properties Length of central Perpendicular on Tangent
163
.
Expression for Angle between two Tangents to a Conic (see also p. 189)
Locus of intersection of Tangents at fixed Angle
162
.
.
.... ....
.
161
.
.
.
(see also pp. 166,
Supplemental Chords To construct a pair of Conjugate Diameters inclined at a given Angle . Relation between intercepts made by variable Tangent on two parallel Tangents (see also pp. 287, 299)
Or on two Conjugate Diameters Given two Conjugate Diameters
.
.
.
to find the
.
.
Axes
.
.
.
.
.
171 171 171
.172
.
.
169
.169
.
172 173, 176
CONTENTS.
viii
NORMAL
:
its
Properties
...... ......
AM 178
Point (see also p. 835) . .174 . Chord subtending a right Angle at any Point on Conic passes through a fixed Point on Normal (see also pp. 270, 286) . 175 .
To draw a Normal through a given
Coordinates of intersection of two Normals
.
.
,
.175
Properties of Foci Sum or difference of Focal Radii constant
.
.
.
.177
....
. Property of Focus and Directrix Rectangle under Focal Perpendiculars on Tangent Focal Radii equally inclined to Tangent .
179
.
.
is
177
constant
.
180
.
180
. Confocal Conies cut at Right Angles . . .181 . Tangents at any Point equally inclined to Tangent to Confocal Conic through the Point . 18!
...
..... ..... .....
Locus of foot of Focal Perpendicular on Tangent . . . .182 Angle subtended at the Focus by a Chord, bisected by Line joining Focus to its Pole (see also pp. 255, 284) 188 Line joining Focus to Pole of a Focal Chord is perpendicular to that Chord (see also p. 321)
.
.
.
.
Polar Equation, Focus being Pole Segments of Focal Chord hare constant Harmonic
Mean
Origin of names Parabola, Hyperbola, and Ellipse (see also
CONFOCAL CONICS ASYMPTOTES how found
.
.
:
.
.
.
.188
.
.
.
.
Intercepts on Chord between Curve and Asymptotes are equal Lines joining two fixed to variable Point make constant Intercept on Constant area cut off by Tangent . . .
.
.186
.
.191
190
Asymptote
192
.192
.
Mechanical method of constructing Ellipse and Hyperbola
CHAPTER
186
,
328)
p.
184 186
178, 194, 218
XII.
THE PARABOLA. Transformation of the Equation to the form
y^px
.
.
.
. Expression for Parameter of Parabola given by general Equation . ditto, given lengths of two Tangents and contained Angle (see also
p. 214)
.
.
.
.
.
Intercept on Axis Poles
by two .
197
.199
.
Parabola the limit of the Ellipse when one Focus passes to infinity
196
.
..;..,...
200
Lines, equal to projection of distance between their .
.
.
.
201
Subnormal Constant Locus of foot of Perpendicular from Focus on Tangent . . Locus of intersection of Tangents which cut at right Angles (see also pp. 285, 352) Angle between two Tangents half that between corresponding Focal Radii Circle circumscribing Triangle formed by three Tangents passes through Focus
202
(see also pp. 214, 274, 285, 320) Polar Equation of Parabola .
207
.
.
. .
CHAPTER
.
.
CONICS.
,
.
205
XIII.
Focal Properties . . . . . Locus of Pole with respect to a series of Confocal Conies . If a Chord of a Conic pass through a fixed Point 0, then tan ^PFO. tan constant (see also p. 331)
206
.207
.
........ EXAMPLES ON
Lod
.
.
.
204
208
.209
... .
P'FO
209
is
210
CONfENfS.
1*
Locus of intersection of Normals at extremities of a Focal Chord
211
(see also p. 335)
Expression for angle between tangents to ellipse from any point (see also pp. 166
......
. . . 189, 391) Radii Vectores through Foci hare equal difference of Reciprocals .
.
.
Examples on Parabola Three Perpendiculars of Triangle formed by three Tangents Directrix (see also pp. 247, 275, 290, 342)
.
.211
.
.
intersect
.212
.
.
213 also
....
.
.
.
.213
.
Locus of foot of Perpendicular from Focus on Normal . Coordinates of intersection of two Normals Locus of intersection of Normals at the extremities of Chords passing through a given Point (see also p. 338) . .
.
.
.
.
.
given three Tangents and Locus of Foci, given four Points ditto,
sum
of squares of
.
.
215
216
.
.
215
.215
.
(see also pp. 254, 268, 339)
213
214
.214
.
Given three Points on Equilateral Hyperbola, a fourth is given (see also p. 290) Circle circumscribing any self-conjugate Triangle with respect to an Equilateral . Hyperbola passes through centre (see also p. 322) Locus of intersection of Tangents which make a given Intercept on a given . . Tangent Locus of Centre, given four Tangents
212
on
.212
.
Area of Triangle formed by three Tangents . . Radius of Circle circumscribing an inscribed or circumscribing Triangle Locus of intersection of Tangents which cut at a given Angle (see pp. 256, 285)
Intersections of perpendiculars of four Triangles formed by four Lines lie on a right Line perpendicular to Line joining middle Points of Diagonals (see also p. 246)
ECCENTRIC ANGLE
Construction for Conjugate Diameters
Radius of Circle circumscribing an inscribed Triangle (see also Area of Triangle formed by three Tangents or three Normals
SIMILAR CONIC SECTIONS
p. 333)
Condition that Conies should be similar, and similarly placed Properties of similar Conies
Condition that Conies should be similar, but not similarly placed
CONTACT OP CONICS
Contact having double contact
.
.
.
.
.
.
.
.
220
.
.
222
.
.
224
.
.
.
222
223 225
.226 227
.228
a Conic, chords of intersectioa nre equally inclined to
the Axes (see also p. 234)
. . on a Circle Relation between three points whose osculating Circles meet tonic again
Condition that four Points of a Conic should in the
219
220
.
Osculating Circle defined Expressions and construction for Radius of Curvature (see also pp. 234, 242, . 374) If a Circle intersect
217
217
same Point
lie
Coordinates of centre of Curvature
.
Evolutes of Conies (see also p. 338)
.
.
229 229 229
230 231
CHAPTER XIV. ABRIDGED NOTATION. Meaning of the Equation S = kS' Three values of k for which it represents right Lines .
.232
.
.
.
.233
*
CONTENTS.
Equation of Conic passing through five given Points Equation of osculating Circle Equations of Conies having double contact with each other Every Parabola has a Tangent at infinity (see also p. 329) Similar and similarly placed Conies have
common
.
.
.
.
.
.
Points at infinity
.
.
Method
. pp. 320, 353) of finding Coordinates of Foci of
.
.
.
238
.
238
.
(see also
.239
.
. given Conic (see also p. 863) Relation between Perpendiculars from any Point of Conic on Bides of inscribed
Quadrilateral
.
.
.
.
Anharmonic Property of Conies proved (see also pp. Extension of Property of Focus and Directrix
240
.
.241
.
.
239
.239
.
252, 288, 318)
.
236
.237
. . . . concentric, touch at infinity All Circles have imaginary common Points at infinity (see also p. 325) . Form of Equation referred to a self-conjugate Triangle (see also p. 253)
common Tangents
234
235
.
.
if
Conies having same Focus have two imaginary
2:*3
'234
.
Result of substituting the Coordinates of any Point in the Equation of a Conic 241 Diameter of Circle circumscribing Triangle formed by two Tangents and their Chord . . . . .241 Property of Chords of Intersection of two Conies, each having double contact with a third . . .
.
.
.242
.
Diagonals of inscribed and circumscribed Quadrilateral pass through the same Point . . .242 . .
.... .
.
Conies have each
If three
double contact with a fourth,
Intersection intersect in threes
Brianchon's If
Theorem
(see also pp. 280, 316)
three Conies have a
Pascal's
Theorem
Steiner's
common
.
their
Chords of 243
.244
.
.
Chord, their other Chords intersect in a Point
(see also pp. 280, 301, 3L6, 319, 379)
Supplement to Pascal's Theorem
244
.245
.
.
(see also p. 379)
246
.
.
Circles circumscribing the Triangles formed by four Lines meet in a Point When five Lines are given, the five Points so found lie on a Circle
.
Given five Tangents, to find their Points of Contact . . . MacLaurin's Method of generating Conies (see also p. 299) 247, Given five Points on a Conic to construct it, find its centre, and draw Tangent at any of the Points . . Equation referred to two Tangents and their Chord Corresponding Chords of two Conies intersect on one of their Chords of Intersec-
...... ..... ..... .
.
.
tion (see also pp. 243, 245)
.
.
247 248 247
248
.249
.
.
246 247
.
Locus of Vertex of Triangle whose sides touch a given Conic, and base Angles move on fixed Lines (see also pp. 319, 349) .
.250
.
To
inscribe in
a Conic a Triangle whose
also pp. 273, 281, 301)
318)
Anharmonic
.
;
or
on a Conic
(see .
if
[abed]
=
{a'b'c'd'}, if
260 251
.262
.
.
.
.
.
ratio of four Points
meet in a Point with the given one
through fixed Points I
Method of generating Conies (see also p. 300) Points and Tangents of a Conic (see also pp. 240, 288,
Generalizations of MacLaurin's
Anharmonic Properties of
sides pass
the Lines aa'
they touch a Conic having double contact
252
Envelope of Chord joining corresponding Points of two homographic systems on a Conic -(see also p. 802) .253 . 253 . . . Equation referred to sides of a self -con jugate Triangle .
.
.
.... .
Locus of Pole of a given Line with regard to a Conic passing through four fixed Points (see also pp. 153, 268, 271, 302) or touching four right Lines (see also pp. 267, 277, 281, 321, 339)
254
254
CONTENTS.
*t
..... ..... ..... ...... ... ...
Focal properties of Conies (see also pp. 267, 277, 281, 321, 339)
.
.
Locus of Intersections of Tangents to a Parabola which cut at a given Angle also pp. 213, 285) Self -con jugate Triangle
common
to
two Conies
(see also pp. 348, 361)
.
.
when real, when imaginary Locus of Vertex of a Triangle inscribed in one Conic, and whose sides touch one another (see also
p. 349)
ENVELOPES, how found Examples of Envelopes
....
Formation of Trilinear Equation of a Conic from Tangential, and Criterion whether a Point be within or without a Conic
vice versa
.
256 257
257 259 260 262
Chord of
its
..... ...... ....... ..... .
.
Equation of a Conic having double contact with two given Conies
.
.
touching four Lines Locus of a Point whence sum or difference of Tangents to two Circles
256 256
.261
.
Discriminant of Tangential Equation Given two points of a Conic having double contact with a third, contact passes through one or two fixed Points
256
(see
.
262 262
262
constant
263
problem Tangent and Polar of a Point with regard to a Conic given by the general Equation DISCRIMINANTS defined ; discriminant of a Conic found (see also pp. 72, 149,
263
Malfatti's
. . . . . 153,155) Coordinates of Pole of a given Line Condition that a Line should touch a Conic (see also pp. 152, 340) Condition that two Lines should be conjugate .
is
.266
.
.
.
....... ...... .
265
.
266 266 267
Hearn's method of finding Locus of Centre of a Conic, four conditions being
given Equation of pair of Tangents through a given Point (see also p. 149) Property of Angles of a circumscribing Hexagon (see also p. 289) Test whether three pairs of Lines touch the same Conic .
.
.
.
'
.
.
Equations of Lines joining to a given Point intersections of two Curves Chord which subtends a right Angle at a fixed Point on Conic passes through a .
fixed Point
Locus of the
latter Point
.....
when Point on Curve
267 269 270
270 270
270
270 Envelope of Chord subtending constant Angle, or subtending right Angle at Point not on Curve 270 Given four Points, Polar of a fixed Point passes through fixed Point .271 Locus of intersection of corresponding Lines of two homographic pencils . 271 Envelope of Pole of a given Point with regard to a Conic having double contact varies
.
.
.
.
with two given ones Anharmonic Ratio of four Points the same .
.
.
.
.271
.
as that of their Polars
271
.
Equation of Asymptotes of a Conic given by general Equation (see also p, 3 10) Given three Points on Conic, and Point on one Asymptote, Envelope of other Locus of Vertex of a Triangle whose sides pass through fixed Points, and base Angles move along Conies .
To
inscribe in
..... ..... .....
a Conic a Triangle whose
also pp. 250, 281, 301)
sides pass
272 272 272
through fixed Points (see
Equation of Conic touching five Lines Coordinates of Focus of a Conic given three Tangents (see also pp. 239, 353) Directrix of Parabola passes through intersection of Perpendiculars of circum-
273 274
275
'
scribing Triangle (see also pp. 212, 247, 290, 342) (see also p. 277)
Locus of Focus given four Tangents
275
.
.
.
.
.
275
CONTENT'S.
iii
CHAPTER XV. RECIPBOCAL POLAIU8. . Principle of Duality Locus of Centre of Conic touching four Lines Locus of Focus of Conic touching four Lines
.... ....
MUM .276
.
277
277
277 common radical Axis having for Diameters Diagonals of complete Quadrilateral have common .277 . radical Axis Locus of Point where Tangent meeting two fixed Tangents is cut in a given ratio 277 279 Degree of Polar Reciprocal in general 280 . . Pascal's Theorem and Brianchon's mutually reciprocal
Director Circles of Conies touching four Lines have a Circles
.....
.
.
.
.
....... .... ...... .... .
Radical Axes and Centres of Similitude of Conies having double contact with a given one Polar of one Circle with regard to another . Reciprocation of Theorems concerning Angles at Focus
.
.
.
.
.
Envelope of Asymptotes of Hyperbolas having same Focus and Directrix . Reciprocals of equal Circles have same Parameter Relation between Perpendiculars on Tangent from Vertices of circumscribing Quadrilateral
..... ..... ..... .....
Tangential Equation of Reciprocal Conic Trilinear Equation given Focus
and either three Points or three Tangents
.
Reciprocation of Anharmonic Properties . Carnot's Theorem respecting Triangle cut by Conic (see also p. 319) Reciprocal, when Ellipse, Hyperbola, or Parabola ; when Equilateral Hyperbola of Reciprocal, how found Reciprocal of Properties of Confocal Conies
Axes
To
describe a Circle touching three given Circles to form Equation of Reciprocal
How
Reciprocal transformed from one origin to another Reciprocals with regard to a Parabola
288
284 285
286 287
287 288 288
289 290 291
.291
.
.
282
.
,
.
291
.
.
.
292
292
293
CHAPTER XVI. HARMONIC AND ANHARMONIC PROPERTIES. Anharmonic Ratio, when one Point
infinitely distant
.
.
.
...... ....
Lines from two fixed Points to a variable Point,
how
.
.
of Newton's
mode
of Generation
Theorems a Conic a Polygon whose
Chasles's extension of these
To inscribe in To describe a Conic touching Conic
.
.... .
.
.
.
.
.
. sides pass through fixed Points three Lines and having double contact with a given
(see also p. 359)
.
.
... .
297
298
299 299 299
800 801
.801
Anharmonic proof of Pascal's Theorem . . of Locus of Centre, when four Points are given . . . Envelope of Line joining corrftsponding Points of two Homographic Systems Criterion whether two Systems of Points be Homographic (see also p. 383) .
296
cut any Parallel to
Asymptote Asymptotes through any Point on Curve, how cut any Diameter Anharmonic Property of Tangents to Parabola How any Tangent cuts two Parallel Tangents . Proof, by Anharmonic Properties, of MacLaurin's Method of Generating Conies, Parallels to
295
.296
Centre the Pole of the Line at infinity . . . Asymptotes together with two Conjugate Diameters form Harmonic Pencil
.
801
302 .
302 304
CONTENTS.
Xiii
. Analytic condition that four Points should form a Harmonic System Locus of Point whence Tangents to two Conies form a Harmonic Pencil
also p. 345)
...
....... ...... .
Condition that Line should be cut Harmonically by two Conies
INVOLUTION Property of Centre of Foci
305
(see
.
.
.
.
.
807
808
.809
,
how found when two Pairs of corresponding Points are given Condition that six Points or Lines should form a system in Involution System of Conies through four Points cut any Transversal in Involution Foci,
System of Conies touching four Lines, when cut a Transversal
806 806
.
.
810
.
810
.
811
.
in Involution
813
.
Proof by Involution of Feuerbach's Theorem concerning the Circle through . middle Points of Sides of Triangle .
.313
.
CHAPTER
XVII.
THE METHOD OP PROJECTION.
.... .... .... ...
All Points at Infinity may be regarded as lying in a right Line Projective Properties of a Quadrilateral Any two Conies may be projected into Circles Projective proof of Carnot's of Pascal's Theorem
Theorem
(see also p. 289)
.
.
818 819
.819
.
820
on the same Conic,
lie
(see also p. 343) . Projections of Properties concerning Eight Angles Locus of Pole of a Line with regard to a system of Confocal Conies
.820 821
.
.
.
.
Triangles He on same Conic (see also p. 341) Chord of a Conic passes through a fixed Point, if the Angle it subtends at a fixed Point in Curve, has fixed Bisector
.The six Vertices of
two
self -con jug ate
..... ....
Every Section
is Ellipse,
Origin of these
Names
.
........ .
.
.
Method
....
of deducing properties of Plane
827 328
.
Orthogonal Projection Radius of Circle circumscribing inscribed Triangle
CHAPTER
.329
while a given
.
.
may be
Curves from Spherical
324
.
.....
Line passes to Infinity Determination of Focus of Section of a right Cone Locus of Vertices of right Cones from which a given Conic
324
826
.
circle,
323
.
Hyperbola, or Parabola
Every Parabola has a Tangent at an infinite Distance Proof that any Conic may be projected so as to become a
822
two
Analytic basis of Method of Projection
SECTIONS OP A CONE
822
.823
.
.
Projections of Theorems concerning Angles in general Locus of Point cutting in given ratio intercept of variable Tangent between fixed Tangents
816 317
.
.
Projection of Properties concerning Foci The six Vertices of two Triangles circumscribing a Conic,
.
.
.
.
.
cut
330
.331 331
,
,
.
.
.
331
332 .
333
XVIII.
INVARIANTS AND COVARIANTS. . . . Equation of Chords of Intersection of two Conies Locus of Intersection of Normals to a Conic at the extremities of Chords passing through a given Point Condition that two Conies should touch .
.... .
.
,
334
335
836
CONTENTS.
XIV
Criterion whether Conies intersect in
..... ..... two
and two iinaginaiy Points or not
real
Equation of Curve parallel to a Conic Equation of Evolute of a Conic Meaning of the Invariants when one Conic is a pair of Lines Criterion whether six lines touch the same conic
.
.
.
.
.
Equation of pair of Tangents whose Chord is a given Line . Equation of Asymptotes of Conic given by Trilinear Equation Condition that a Triangle self -con jugate with regard to one Conic should be inscribed or circumscribed about another .
.
.
.... .
Six vertices of two self -conjugate Triangles
on a Conic
lie
.
.
...... ..... .... ..... .... ...... ....... ....
PAGB 337
337 838
838 339 340 340
340 341
Circle circumscribing self-conjugate Triangle cuts the director circle orthogonally Centre of Circle inscribed in self -conjugate Triangle of equilateral Hyperbola
341
lies on Curve Locus of intersection of Perpendiculars of Triangle inscribed in one Conic and circumscribed about another Condition that such a Triangle should be possible
341
'
.
.
.
.
Tangential equation of four Points
common
.
.
two Conies
to
.
.
Equation of four common Tangents Their eight Points of Contact lie on a Conic Covariants and Contravariants defined
Discriminant of Covariant F, when vanishes How to find equations of Sides of self -conjugate Triangle
common
to
(see also p. 347)
Vertex of a Polygon all whose sides touch one Conic, and all whose Vertices but one move on another Condition that Lines joining to opposite vertices, Points where Conic meets Triangle of reference should form two sets of three meeting in a Point . free
through an imaginary Circular Point, perpendicular to itself Condition for Equilateral Hyperbola and for Parabola in Trilinear Coordinates
Every
Coordinates of Foci of Curve given by general Equation Extension of relation between perpendicular Lines
.
.
.
.
.
352
852
.
.
.
.
.
.
. four Points to touch a given Conic Jacobian of three Conies having two Points common, or one of which reduces
two coincident Lines
.
.
.
common
360 3fiO
to
361
361 361
.
.
359 359
.861
.
To draw a Conic through
Equation of Circle cutting three Circles orthogonally To form the equation of the sides of self -con jugate Triangle Conies
356
360
Lines joining corresponding Points cut in involution by the Conies General equation of Jacobian . . .
to
366 366
and touching three
.
.
852 853
364
Four Conies having double contact with 8, and passing through three Points, or . touching three Lines, are touched by the same Conies Condition that three Conies should have double contact with the same conic Jacobian of a system of three Conies Corresponding points on Jacobian .
361
362
...... one,
850
.
.
.
other such Conies
349
.
.
Equation of reciprocal of two Conies having double contact Condition that they should touch each other
To draw a Conic having double contact with a given
849
351
....
Equation of Directrix of Parabola given by Trilinear Equation
348
.
line
General Tangential Equation of two Circular Points at infinity General Equation of Director Circle
844 845 346
two Conies
Envelope of Base of Triangle inscribed in one Conic, two of whose sides touch another
Locus of
342
842 343
two .
362
CONTENTS.
XV
Area of common conjugate Triangle of two Conies Mixed Concomitants
.
MOD .362
.
.....
.
.
.
368
.
Condition that they should have a common Point . Condition that X + p. V 4- v can in any case be a perfect Square Three Conies derived from a single Cubic, method of forming its Equation .
W
U
.362
.
.
Condition that a line should be cut in involution by three Coriics Invariants of a system of three Conies
365 365
.
.366
.
368
.
CHAPTER XIX. THE METHOD OP INFINITESIMALS.
..... ..... ..... ..... ......
Direction of Tangents of Conies Determination of Areas of Conies
.
.
.371
.
.
372
Tangent to any Conic cuts off constant Area from similar and concentric Conic Line which cuts off from a Curve constant Arc, or which is of a constant length where met by its Envelope
373
Determination of Radii of Curvature
374
.
Excess of
sum
of
374
two Tangents over included Arc, constant when Vertex moves 377
on Confocal Ellipse Arc and Tangent, constant from any Point on Confocal Hyperbola
Difference of
Fagnani's Theorem
377 378
Locus of Vertex of Polygon circumscribing a Conic, when other Vertices move
on Confocal Conies
.
.
.
.
.378
,
NOTES.
..... .
379
systems of Tangential Coordinates Expression of the Coordinates of a Point on a Conic by a single Parameter . On the Problem to describe a Conic under five conditions
.
386
On
.
Theorems on complete Figure formed by
On
six Points
on a Conic
.
,
.
,
,
387
.
.
systems of Conies satisfying four Condition Miscellaneous Notes
383
.
389 391
ANALYTIC GEOMETRY, CHAPTER
I.
THE POINT.
THE following method of determining the position of any a plane was introduced by Des Cartes in his Geometric, on point 1637, and has been generally used by succeeding geometers. We are supposed to be given the position of two fixed 1.
right lines
through
draw
XX',
any
PM,
YY' and
YY
point
PN XX,
1
intersecting in the
P
parallel it
is
Now,
if
N
plain
we knew
PM, PN', or, vice versa. we knew the lengths PM, PN, we should know
rallels
^
that if
of
0.
to
the position of the point P, we should know the lengths of the pathat, if
point
we
M
/
the position of the point P. Suppose, for example, that
we are given PN= a, PM = b, we need only measure OM = a and parallels PM, PN, which will intersect It is usual to
and to
PM
2.
The
P
a,y = b.
PN
are called the coordinates of the parallels PM, is often called the ordinate of the while
PM
PN, which
and draw the
OY
by be determined by the two equations x
point P. is
denote
PN parallel to OX
ON=b,
in the point required. parallel to by the letter y, the letter x, and the point is said
is
equal to
OM the
point
intercept cut off
P;
by the ordinate,
called the abscissa.
B
THE
2
The
XX'
fixed lines
POINT.
and YY' are termed the axes of coin which they intersect, is called the
and the point 0,
ordinates,
The axes are said to be rectangular or oblique, according as the angle at which they intersect is a right angle or oblique.
origin.
M
readily be seen that the coordinates of the point that those of the point ; itself are x 0. 0, y
It will
on the preceding figure are x = a, y = are x = 0, y = b ; and of the origin
N
In order that the
3.
be satisfied by one point, only
the
to
magnitudes,
x
equations is
it
but
=
y = b should only
a,
necessary to also
to
the
pay
attention, not
signs
of the
co-
ordinates.
If we paid no attention a and might measure and any of the four points
OM=
P, P,,
P P
8
a,
the equations
would
distinguish
satisfy
x = a, y
b.
however, to
It is possible,
between
to the signs of the coordinates, we on either side of the origin,
ON= b,
algebraically
the
lines
OM,
OM'
(which are equal in magnitude, but opposite in
by giving them
direction) different
down a
rule
measured be
site
in
considered
lines
measured direction
sidered direction
consider
We
signs.
as
that,
if
one direction as
positive,
in the
oppomust be con-
negative.
we measure
OM
lay lines
It
is,
positive
of
course,
lines,
but
arbitrary it
is
(measured to the right hand) and upwards) as positive, and OM', ON' (measured directions) as negative lines.
in
which
customary to (measured
ON
in the opposite
P P
Introducing these conventions, the four points P, P,, 2 , are easily distinguished. Their co-ordinates are, respectively,
8
THE These
distinctions of sign
who
learner,
trigonometry.
supposed
can present no difficulty to the be already acquainted with
to
=
= 6, or oj a, y points whose coordinates are are generally brieflj designated as the point (a, 6),
The
N.B.
x
is
POINT.
= x, y = y,
or the point x'y. It appears from what has been said, that the points (-f a, -f &), _ He on a right line passing through the origin that (_ a? they are equidistant from the origin, and on opposite sides of it. ;
)
To express
4.
two points x'y\ x'y")
the distance between
the
axes of coordinates being supposed rectangular.
Euclid
By
I.
47,
PQ = P8* + SQ*, l
but
PS= PM- QM' = y- y",
QS=OM-
and
OM' = x'- x"
hence
;
p
Q To
express the distance of any point from the origin, we in must make x" 0, y" = the above, and
5.
we
we
find
In the following pages seldom have occa-
shall but
make use of oblique general, much simplified by
coordinates, since formulae are, in the use of rectangular axes; as
Similarly, the square of the distance of a point, n the origin = x'* + y + 2x'y' cos o>.
to.
xy\ from
THE
4
POINT.
In applying these formulae, attention must be paid to the If the point for example, were , signs of the coordinates. in the angle XOY', the sign of y" would be changed, and the line
PS
The
would be the sum and not the
learner
no
find
will
difficulty,
difference of
y and
y".
written
having
if,
the
PS
coordinates with their proper signs, he is careful to take for and QS the algebraic difference of the corresponding pair of coordinates. Ex.
1.
Find the lengths of the sides of a triangle, the coordinates of whose x'" = - 3, y'" = -G, the ares being x' = 2, y' = 3 ; x" = 4, y" = - 5
vertices are
;
Ans. J68, J50, J106.
rectangular.
Ex.
2.
Ex.
3.
Express that the distance of the point xy from the point (2, 3) is equal Ans. (x - 2) 2 + (y - 3) 2 = 16
4.
Express that the point xy Ans. (x -
Find the lengths of the sides of a triangle, the coordinates of whose vertices are the same as in the last example, the axes being inclined at an angle of 60. Ans. J52, J57, J161.
to 4.
Ex.
equidistant from the points
is
2)
2
+
(y
- 3) =
(*
- 4) 2 +
(y
(2, 3), (4, 5). 2
-
5)
Ex. 5. Find the point equidistant from the points (2, 3), (4, 5), have two equations to determine the two unknown quantities x, y.
x
Ans.
6.
The
distance
then
the
ib
or
x+y=
(6, 1).
and the common distance
Ilere
7.
we
|/CA\ v
,
--
is
P
between two points, being expressed
the form of a square root, If the distance PQ, sign. positive,
VS y
;
in
necessarily susceptible of a double measured from to be considered ,
is
P
distance
QP,
measured from
Q
to
P,
If indeed we are only concerned negative. with the single distance between two points, it would be unmeaning to affix any sign to it, since by prefixing a sign we in fact direct that this distance shall be added to, or subtracted
is
considered
But suppose we are given three from, some other distance. in a right line, and know the distances PQ, points P, Q,
R
QR, we may now given,
R
infer
PR = PQ +
this
equation
P
QR.
remains
And
with tho explanation even though the
true,
PQ
and Q. For, in that case, lie between and point are measured in opposite directions, and P/?, which is their arithmetical difference, is still their algebraical sum. Except in the case of lines parallel to one of the axes, no convention
QR
has been established as to which shall be considered the positive direction.
THE POINT. To find
7.
m
ratio
:
coordinates of the point cutting in a given
the
two given points
the line joining
ra,
xy,
Let o?, y be the coordinates of the point to determine, then
m:n::PR:RQ m
in
::
x"y".
R
which we seek
MS SN :
9
x x-x"> mx"=nx
x
::
or rax
:
hence
x = W^JL^
m+n
/VB
.
~~M
In like manner
my"+ny' If the line were to be cut
should have
m r
,
,
x=
and therefore
externally
n
:
: :
mx"
x
x
x
y=
my" y
x
nx
m-n
,
in
:
we
ny
mn y
.
be observed that the formulas for external section
It will
are obtained from those for internal section
sign of the
ratio
In
in
the
in
the
fact,
given ratio
the
measured
;
that
case
same
:
:
is,
of
by changing the
- n. by changing m + n into m internal section, PR and RQ are
and their ratio (Art. 6) is to But in the case of external section are measured in opposite directions, and their direction,
be counted as positive.
PR
and
ratio
is
RQ
negative.
Ex.
1.
To
find the coordinates of the middle point of the line joining the points
Ex.
2.
To
find the coordinates of the middle points of the sides of the triangle,
the coordinates of whose vertices are
(2, 3), (4,
5),
(-
3,
6).
Atu.'(l,- V), (-*,-*). (3,~1). Ex. 3. The line joining the points (2, 3), (4, - 5) is trisected ; to find the coAns. x = ordinates of the point of trisection nearest the former point. y = $. ,
The
coordinates of the vertices of a triangle being x'y', x"y", x'"y'", to find the coordinates of the point of trisection (remote from the vertex) of the line
Ex.
joining
4.
any vertex
to the
middle point of the opposite Ans. x =
side.
TRANSFORMATION OF COORDINATES.
6 Ex.
5.
To
find the coordinates of the intersection of the bisectors of sides of the
whose
triangle, the coordinates of
vertices are given in
Ex.
Ans.
2.
x=\, y = -
.
m
: Ex. 6. Any side of a triangle is cut in the ratio n, and the line joining this to + n : I; to find the coordinates of the point the opposite vertex is cut in the ratio
m
_ kf "~+ mx" +
ofBection.
naf"
~ ~' y _
ty
+ my" + ny'" l+m + n -'
TRANSFORMATION OF COORDINATES.*
When we know
8.
one pair of axes,
the coordinates of a point referred to
frequently necessary to find its coThis operation is ordinates referred to another pair of axes. called the transformation of coordinates. it
is
We
shall consider three cases separately; first, we shall the origin changed, but the new axes parallel to the suppose old; secondly, we shall suppose the directions of the axes
changed, but the origin to remain unaltered ; and thirdly, we shall suppose both origin and directions of axes to be altered.
Let the new axes be parallel to the
First.
old.
Let Ox, Oy be the old
axes,
O'X^
Y
new
axes.
0'
the
'
i
/
/
/
p
Let the coordinates of the new origin referred to the old be
*',>, or 0'S=x', 0'R = y'. Let the old cc,
be
coordinates
y, the
new X, Y,
we have
then
OM=OR + that
x = x' 4- X, and y -y' -f
is
These formulae
Y.
are, evidently, equally true,
whether the axes
be oblique or rectangular. 9.
Secondly,
the origin *
is
let
the directions of the axes be changed, while
unaltered.
The beginner may postpone
of Art. 41.
the rest of this chapter
till
he has read to the end
TRANSFORMATION OP COORDINATES. Let the original axes be O.r, Oy, PQ = y. Let the new axes be OX, OY, so that we have
ON=X, PN=Y.
OY a,
make
/3,
we have
OQ =
OX,
angles respectively the old axis of an
with
and angles
Let
so that
a',
with the old
/3'
and
if the angle of y; old axes be the between xOy = a>, have we obviously a + a' &), since JfOa? + -3T% = xOy; and in
axis
n.
a
manner
like
The
formulae of transformation are most easily obtained by on the original axes, in expressing the perpendiculars from
P
terms of the
new
coordinates and the old.
Since
PM=PQ PM=y sino>. But also PM=NE + PS = ON smNOB + PN sin PN8. Hence y sin = X sin a + Y sin smPQM, we have
&>
/3.
In like manner
= X sina' + x smta = X sin (a>
x or
sin
In the figure the angles
y sin/3' a)
-H
;
1^ sin
(ft)
$).
measured on the same side of Ox\ and a', /3', &> all on the same side of Oy. If any of these angles lie on the opposite side it must be given a, /8,
&>
are
all
OY
a negative sign. Thus, if lie to the left of Oy, the angle yS) is negative, and therefore greater than &>, and ft' (= u> in the expression for x sin to is the coefficient of negative. /3 is
Y
This occurs
in
the
following
special case, to which, as the in practice, we a
one which most frequently occurs
give
separate
figure.
To transform from a system of rectangular axes to a with the old. rectangular system making an angle
Here we have
and the general formulae become
y
X sin
+ Y cos 0,
x*= XcosO- Fsin0;
MR
new
TRANSFORMATION OF COORDINATES.
8
the truth of which
may
also be seen directly, since
y
x=OR-SN,vf\u\e There is only one other case of transformation which often occurs in practice. To transform from oblique coordinates to rectangular, retaining the old axis
We
of x.
may
use the general for-
mulae making
But have y,
it is more simple to investhe formulae directly.
We
tigate
OQ
and
PQ
x and
for the old
OM and PM for the new Y=y
sin
;
<>
and, since
o>,
PQM=
X. = x + y
o>,
we have
coseoi
while from these equations we get the expressions for the old coordinates in terms of the new
ysinw=F, x
sinco
= X sin&>
Y cosa>.
Thirdly, by combining the transformations of the two preceding articles, we can find the coordinates of a point referred to two new axes in any position whatever. first find 10.
We
the coordinates (by Art. 8) referred to a pair of axes through the new origin parallel to the old axes, and then (by Art. 9)
we can The and
find the coordinates referred to the required axes.
general expressions are obviously obtained by adding the values for x and y given in the last article.
Ex.
1.
The
coordinates of a point satisfy the relation a;
what
will this
become
if
2
+ y2 - 4x -
Gy
The coordinates of a point to a 2 x2 = 6 ; what will this become y angles between the given axes ? Ex.
2.
Transform the equation 2or2 other at an angle of 60 to the right Ex.
3.
-
18
;
the origin be transformed to the point
relation
Ex
4.
(2,
3) ?
set of rectangular if
axes satisfy the
transformed to axes bisecting the Arts.
XY = 8.
2 bxy + 2y = 4
lines
from axes inclined to each which bisect the angles between the Ans.
given axes.
o* x.
x
?/ to
A - 27T 2 + 2
12
=
0.
Transform the same equation to rectangular axes, retaining the old axis Ana.
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Pluto. 57.9. 108.2. 149.6. 227.9. 778.3. 1427. 2869. 4497. 5900. 10. Designing a Satellite Disli The reflector of a televi- sion satellite dish is a paraboloid of revolution with diameter. 5 ft and a depth of 2 ft. How far from the vertex should the.