a treatise on the analytic geometry of three dimensions vol i 1914

7/24/2019 A Treatise on the Analytic Geometry of Three Dimensions Vol I 1914

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A

TREATISE

ON

THE ANALYTIC

GEOMETRY

OF

THREE

DIMENSIONS


6/507

BY

THE

SAME

AUTHOR,

TREATISE

ON CONIC

SECTIONS.

CONTAINING

AN

ACCOUNT OF

SOME

OF

THE

MOST

IMPORTANT

MODERN

ALGEBRAIC

AND

GEOMETRIC

METHODS,

8vo,

125.

LONGMANS,

GREEN

AND

CO.,

LONDON,

NEW

YORK, BOMBAY, CALCUTTA,

AND

MADRAS,


7/507

A

TREATISE

ANALYTIC

GEOMETRY

OF

THREE

DIMENSIONS

BY

GEORGE

SALMON,

D.D.,

D.C.L.,

LL.D.,

F.R.S.

LATE

PROVOST

OF TRINITY

COLLEGE,

DUBLIN

REVISED

BY

REGINALD

A.

P. ROGERS

FELLOW

OF TRINITY

COLLIiGE,

DUBLIN

SIXTH

EDITION

VOL.

I

LONGMANS,

GREEN

AND

C0.

39

PATERNOSTER

ROW,

LONDON

3

FOURTH

AVENUE

&

SOrn

STREET,

NEW

YORK

7:

BOMBAY, CALCUTTA,

AND

MADRAS

HODGES,

FIGGIS

AND

CO.,

LTD.

104

GRAFTON

STREET,

DUBLIN

1914


8/507


9/507

EDITOR'S

PREFACE

TO FIFTH

EDITION.

VOLUME

I.

IN

order to

avoid

delay

it

has

been

thought

advisable

to

publish

this

edition

in two volumes.

While

pre-

serving

the substance

of

the fourth

edition,

I

have

added some

new matter

(generally

enclosed in

square

brackets

and

in

different

type) giving

brief

accounts

of methods

or

points

of

view

which

appear

to

me

to

be of

interest

and

to

fit

in with

the

rest

of

the

work.

The additions

include

illustrations

of

models

of

most

of the

different

species

of

quadrics,

with

gener-

ators

or lines of

curvature

(Chapter

V.)

;

articles

or

paragraphs

on

the

analytical

classification

of

real

quadrics

(880),

on

projection

and

Fiedler's

projective

coordinates

(144e),

on

the non-Eucliclean

theory

of

distance

and

angle

(144^),

and

on

the

expression

of

twisted

cubics

and

quartics

by

rational

or

eUigtic

parameters

(3330,

3470,

348,

349).-

In

differential

geometry

my

aim

has

been

to

form

a

closer

connecting-link

between

Salmon's

book

and

the more

extensive and

more

purely

analytical

methods

used

by

Bianchi,

Darboux

and others.

I

have

there-

fore

added

articles

on the

now

well-known Frenet-

Serret

formulae,

with

some

applications

(3680),

on


10/507

vi

EDITOE'S

PEEPACE

TO

FIFTH

EDITION.

the

intrinsic

equations

of

a twisted

curve

(368J),

on

Bertrand

curves

(368c),

and

on

the

application

of

Gauss's

parametric

method

to

conformal

represent-

ation,

geodesic

curvature

and

geodesic

torsion

(3960,

3965).

To

the

portion

dealing

with

the

differential

geometry

of

curves

on

quadrics,

I have

added

Staude's

thread-construction

for

ellipsoids

(4210),

which

is

the

three-dimensional

analogue

of

Graves's

theorem

;

and

his

definitions

of confocal

quadrics

(421ft)

by

means

of

broken

distances

these

are

the

analogues

of

the

ordinary

definitions

of conies

by

means

of

focal

radii.

In

the

Golden

Age

of

Euclidean

geometry,

analogues

of

these

types

were

of

great

interest

to

men

like

Jacobi,

MacCullagh,

Chasles

and

M.

Roberts,

but

Staude's

constructions

have

virtually

brought

the

subject

to

a

conclusion.

Staude's

treatment

is

also

an excellent

illustration

of

the

elementary

and

visible

meaning

of

elliptic

and

hyper-elliptic

in-

tegrals.

New

matter

is

also

contained

in

Arts.

8Qa,

80/;,

880,

1590,

172,

173, 1760,

261,

304,

384,

and

in

various

paragraphs

throughout

the

book.

Most

of

these

additions

are

of

the

nature

of

commentaries.

About

100

examples

are

added

many

of

them

being

solved

in

order

to illustrate

the

principles

of

the

articles

to

which

they

are

appended.

The

numbering

of

the

chapters

and

of

the articles

is

the same

as in

the

fourth

edition,

except

in

Chapter

III.,

where

the

order

has been somewhat

changed,

and

in

Arts.

172,

173.

The

present

edition

has

been

published

by

the


11/507

EDITOB'S

PBEFACE

TO

FIFTH

EDITION.

vii

direction of

the

Board

of

Trinity

College,

who

appointed

me as

Editor

in

November,

1910.

REGINALD

A.

P.

ROGERS.

TRINITY

COLLEGE,

DUBLIN,

November,

1911.

The Sixth

Edition

(1914)

of

Vol.

I.

is

reprinted

from

the

Fifth,

with

a

few

corrections,

of

which

the

most

important

are

in

Arts

88$,

338

(Ex.),

344,

357*


12/507


13/507

PREFACE

TO

THE

THIRD

EDITION.

IN

the

preface

to

the

second

edition

of

my

Higher

Plane

Cnrres,

I

have

explained

the

circumstances

under

which

I

obtained

Professor

Cayley's

valuable

help

in

the

preparation

of

that

volume.

I

have

now

very

gratefully

to

acknowledge

that

the

same

assistance

has

been

continued

to

me in

the

re-editing

of

the

present

work.

The

changes

from

the

preceding

edition

are not

so

numerous here

as

in

the

case

of

the

Higher

Plane

Curves,

partly

because

the

book

not

having

been

so

long

out of

print required

less

alteration,

partly

because

the

size

to

which

the

volume

had

already

swelled

made

it

necessary

to

be

sparing

in

the

addition

of

new

matter.

Prof.

Cayley

having

read

all

the

proof

sheets,

the

changes

made

at

his

suggestion

are

too

numerous

to

be

particularized

;

but

the

following

are

the

parts

which,

on

now

looking

through

the

pages,

strike

me

as

calling

for

special

acknowledgment,

as

being

entirely

or

in

great

measure

derived

from

him

;

Arts.*

51-53 on

the

six

coordinates

of

a

line,

the account

of focal lines Art.

146,

Arts.f

314-322

on

Grauss's

method

of

representing

the

coordinates

of

a

point

on

a

surface

by

two

parameters.

The

discussion

of

Orthogonal

Surfaces

is

taken from

a

manuscript

memoir of

Prof.

Cayley's

Thesejarticles^have

been

altered

in

the

present

edition^

^

Arts.

377-884.

_

'

*

'


14/507

X

PEEFACE

TO

THE

THIRD EDITION.

Arts.*

332-337

nearly

without

alteration,

and

the

following

articles

with

some modifications

of

my

own.

Prof,

Cayley

has

also

contributed

Arts, t

347

and

359

on

Curves,

Art.$

468

on

Complexes,

Arts.

567

to

the

end

of

the

chapter

on

Quartics,

and

Arts.

600

to

the

end.

Prof.

Casey

and

Prof.

Cayley

had each

supplied

me with

a

short

note

on

Cyclides,

but

I

found

the

subject

so

interesting

that

I

wished

to

give

it

fuller

treatment,

and

had

recourse

to

the

original

memoirs.

I

have

omitted

the

appendix

on

Quaternions

which

was

given

in the

former

editions,

the work of

Professors

Kelland

and

Tait

having

now

made

information

on

this

subject

very easy

to

be

obtained.

I

have

also

omitted

the

appendix

on

the order

of

Systems

of

Equations,

which has

been transferred

to

the

Treatise

on

Higher

Algebra.

I

have,

as

on

several

former

occasions,

to

acknow-

ledge help

given

me,

in

reading

the

proof

sheets,

by

my

friends

Dr.

Hart,

Mr.

Cathcart

and

Dr. Fiedler.

Owing

to

the

continued

pressure

of other

en-

gagements

I

have

been

able

to

take

scarcely

any

part

in

the

revision

of

this

fourth edition.

My

friend,

Mr.

Cathcart,

has

laid

me

under

the

great obligation

of

taking

the work

almost

entirely

oft*

my

hands,

and

it

is

at

his

suggestion

that

some

few

changes

have

been

made

from

the

last

edition.

TBINITY

COLLEGE,

DUBLIN,

Sept.,

1882.

*

Arts.

476-479

in

fourth edition.

Art.

453

in

fourth

edition,

flirts,

316

$md

328

ip

fourth

edition,

Art.

020

ip

fourth


15/507

CONTENTS OF

VOLUME

I.

The

following

selected

course is

recommended

to

Junior

Readers

:

The

Theory

of

Surfaces

of

the

Second

Order,

Arts.

1

1446,

omitting

articles

specially

indicated in

footnotes.

Confocal

Surfaces,

Arts.

157

170.

The

Curva-

ture

of

Quadrics,

Arts.

194199.

The

General

Thewy

of

Surfaces,

Chap,

xi.

The

Theoj

y

of

Curves,

Arts.

314

323,

358376.

CHAPTER

I.

COORDINATES.

PAGE

Method of

coordinates

1

Properties

of

projections

3

Coordinates

of

point

cutting

in

a

given

ratio the

distance

between

two

points

5

Coordinates

of

centre of

a

tetrahedron

6

Distance

between

two

points

(rectangular

coordinates)

....

6

Direction-cosines,

polar

coordinates,

cylindrical

coordinates ...

7

Area

of

a

figure

in

terms

of

areas

of

its

projections

8

Angle

between

two lines in

terms of

their

direction-cosines ...

8

Perpendicular

distance

of

a

point

from

a line

......

9

Direction-cosines of

the

perpendicular

to

the

plane

of two lines

.

.

9

TRANSFORMATION

OP

COORDINATES

10

Distance between two

points

(oblique

coordinates)

..... 11

Degree

of

an

equation

unaltered

by

transformation

.....

12

CHAPTER

II.

IOTERPRETATION OP

EQUATIONS.

Meaning

of

a

single

equation

j

of

a

system

of

two

or three

equations

. 13

Every

plane

section

of

a

surface of the

n

tl1

degree

is

a

curve of the

n

th

degree

15

Every

right

line meets

a

surface of

the

h

ih

degree

in

n

points

.

. 15

Order

of

a

curve

in

space

defined

16

Three

surfaces

of

degrees

m,

n,

p

intersect

in

mnp

points

...

16

Cylindrical

surfaces

Defined

.

IT


16/507

xii

CONTENTS.

CHAPTEB

III,

THE

PLANE

AND EIGHT

LINE.

PAGE

THE

PLANE

18

Every

equation

of the

first

degree

represents

a

plane

....

18

Equation

of

a

plane

in terms

of

its

direction-cosines and

perpendicular

from

origin

18

Angle

between

two

planes

19

Condition

that

two

planes

may

be

mutually

perpendicular

... 19

Equation

of

plane

in terms

of

intercepts

made

on

axes ....

20

Equation

of

plane

through

three points

20

Interpretation

of

terms

in

this

equation

21

Value

of

determinant

of

the

direction-cosines

of

three

right

lines

.

.

22

Length

of

perpendicular

from

a

given

point

on

a

given

plane

.

.

23

Coordinates

of

intersection

of three

planes

23

Condition

that four

planes

may

meet

in

a

point

24

Volume

of tetrahedron

in terms

of

coordinates

of

its

vertices

...

24

Volume

of

tetrahedron,

the

equations

of

whose

faces

are

given

.

.

24

Equations

of surfaces

passing

through

intersection

of

given

surfaces

,

25

The

equation

of

any

plane

can be

expressed

in

terms

of

those

of

four

given

planes

26

QtJADRIPLANAR COORDINATES,

AND

THEIR DUAL

INTERPRETATION

.

.

26

Anharmonic ratio

of

four

planes

27

General formulae for

linear

transformation of

quadriplanar

coordinates

,

28

Volume

of

tetrahedron

in

terms

of

homogeneous

coordinates

of

vertices .

29

THE

BIGHT

LINE

SO

Equations

of

a

right

line

include

four

constants

80

Condition

that

two lines

may

intersect

31

Direction-cosines

of

a

line whose

equations

are

given

.... 32

Equations

of

perpendicular

from a

given

point

on

a

given

plane

.

.

38

Direction-cosines

of

the

bisectors

of

the

angle

between

two

lines

.

.

S3

Equation

of

plane

bisecting

the

angle

between

two

given

planes

.

.

34

Angle

between

two

lines

34

Conditions

that a

line

may

lie in

a

given

plane

.

...

85

Number

of

conditions

that

a

line

may

lie

in

a

given

surface

.

.

.35

Existence

of

right

lines

on

surfaces

of

the

second

and

third

degrees

.

85

Equation

of

plane

drawn

through

a

given

line

perpendicular

to

a

given

plane

g$

Equation

of

plane

parallel

to

two

given

lines

86

Equations

and

length

of

shortest

distance

between

two

given

lines

.

38

THE Six

COORDINATES OF A

LINE

39

Identical

relation

connecting

them

40

Moment

of

two

right

lines

.

.

.

.

...

.

.

.41


17/507

CONTENTS.

Xlll

PAGE

Homogeneous

coordinates

in

terms

of

right

line

as

ray

and

as

axis

. 41

Passage

from one

system

to the other

43

Condition

of

intersection of

right

lines

.......

4:5

Lines

meeting

four

right

lines

45

Transformation

of

homogeneous

coordinates

of line

46

SOME

PROPERTIES

OF

ETRAHEDRA

46

Relation

connecting

the

mutual distances of four

points

in

a

plane

.

46

Volume

of

a

tetrahedron

in

terms of its

edges

47

Relation

connecting

mutual distances

of

four

points

on a

sphere

.

.

48

Radius of

sphere

circumscribing

a

tetrahedron

49

Shortest

distance

between two

opposite

sides

50

Angle

of

inclination

of

two

opposite

sides

50

Examples

.

.

50

CHAPTER

IV.

PROPERTIES

OF

QUADRICS

IN GENERAL.

Number

of

conditions

necessary

to

determine a

quadric

....

51

Result

of

transformation

to

parallel

axes

52

Equation

of

tangent

plane

at

any

point

53

Equation

of

polar plane 54

Cones

defined

tangent

cone

.55

Cylinder

the

limiting

case

of

a cone

55

Locus

of

extremities

of harmonic

means

of

radii

through

a

point

.

. 55

Every

homogeneous

equation

in

03,

y,

z

represents

a

cone

whose vertex

is

the

origin

56

Polar

plane

of

a

point

with

regard

to

a

cone

57

Discriminant

of

a

quadric

;

condition

for

a cone

57

Coordinates

of

centre

59

Conditions

that

a

quadric

may

have

an

infinity

of

centres

...

59

Equation

of

diametral

plane

60

Conjugate

diameters

61

A

quadric

has

three

principal

diametral

planes

62

Formation

of

equation

representing

the

three

principal

planes

.

. 62

Similarity

of

parallel

sections

62

Rectangles

under

segments

of

intersecting

chords

proportional

to

the

rectangles

under

the

segments

of

a

pair

of

parallel

chords

.

63

Equations

of

tangent plane

and

cone,

&c.,

derived

by

JoachimsthaFs

method

^

Condition

that

a

plane

may

touch the

surface

65

Coordinates

of the

pole

of

a

given

plane

66

Condition

that

a line

may

touch

the

surface

66

This

condition

a

quadratic

function

of

the

six

coordinates

of

a line

.

67

Condition

that

point

of

intersection

of

three

planes

may

lie

on

quadric

.

67

Condition

that

line

joining

two

given points

may

lie

wholly

on

surface

.

67

Condition,

that

line

of intersection

of

two

given

planes

may

lie

wholly

on

surface

^


18/507

XIV

CONTENTS.

PAGE

LINE

COORDINATES

AND

QUADRICS

68

Coordinates

of

polar

line

of

a

given

line 68

Condition

that a

line

may

touch

quadric

70

Determination

of

points

of

contact

of

tangent

planes

through

a

given

right

line

70

Complex

of

lines

72

Conditions

that

a

right

line

lie in

a

quadric

73

CHAPTER

V.

CLASSIFICATION

OF

QUADRICS.

Functions

of

coefficients

which

are

unaltered

by

rectangular

transfor-

mation

............

75

Discriminating

cubic

76

Cauchy's

proof

that

its

roots

are

real

77

CENTRAL

SURFACES

77

Ellipsoids

78

Hyperboloids

of one

sheet

79

Asymptotic

cones

79,

80

Hyperboloids

of

two

sheets

80

NON-CENTRAL

SURFACES

82

Paraboloids,

cylinders,

pairs

of

planes

84=

Actual

reduction

of

equation

of a

paraboloid

84

ANALYTICAL

CLASSIFICATION

OF

QUADRICS

86

CHAPTER

VI.

PROPERTIES OF

QUADRICS

DEDUCED FROM SPECIAL

FORMS

OF

THEIR

EQUATIONS.

CENTRAL SURFACES

88

Equation

referred

to

axes

88

Length

of normal

89

Sum

of

squares

of

reciprocals

of

three

rectangular

diameters

is

constant

89

Locus

of

intersection

of

three

tangent

planes

which

cut

at

right

angles

.

90

CONJUGATE

DIAMETERS

90

Sum of

squares

of

three

conjugate

diameters

is

constant

....

91

Parallelepiped

constant

whose

edges

are

conjugate

diameters

91

Sum

of

squares

of

projections

of

conjugate

diameters

constant

on

any

line

or

plane 93

Locus

of

intersection

of

tangent

planes

at

extremities

of

conjugate

diameters

93


19/507

CONTENTS.

xv

PAGE

Quadratic

which

determines

lengths

of

axes

of

a

central

section

.

.

93

One section

through

any

semi-diameter

has

it

as

an

axis

...

94

Axes found

when

the

quadric

is

given

by

the

general

equation

. .

95

CIRCULAR

SECTIONS

96

Form

of

equations

of con

cyclic

surfaces

97

Two

circular

sections

of

opposite

systems

He

on

the

same

sphere

.

.

98

Umbilics defined

98

RECTILINEAR

GENERATORS

.

99

Two

lines

of

opposite

systems

must

intersect

101

No two

lines of the

same

system

intersect

101

Buled

surfaces

defined

103

Distinction between

developable

and

skew

surfaces

104:

A

right

line whose

motion is

regulated

by

three conditions

generates

a

surface

104

Surface

generated

by

a

line

meeting

three director

lines

....

105

Ditto,

when

lines

are

given by

their six

coordinates ....

106

Four

generators

of

one

system

cut

any

generator

of

the

other

in a

con-

stant

anharmonic

ratio

107

Surface

generated

by

lines

joining

corresponding

points

on two

homo-

graphically

divided

lines

107

NON-CENTRAL SURFACES

108

Circular

sections

of

paraboloids

108

Bight

lines on

hyperbolic paraboloid

109

Method

of

constructing

hyperbolic

paraboloid

110

Conditions

for

SURFACES

OF

BEVOLUTION

112

EXAMPLES OF

Loci

114

Orthogonal

hyperboloids,

equilateral

hyperboloids

116

Locus

of

intersection

of

three

rectangular tangent

lines

to a

quadric

.

116

Method

of

finding

equation

of

cone,

given

its

vertex,

and

a

curve

through

which

it

passes

117

Beciprocal

cones

117

CHAPTEB

VII.

RECIPROCATION,

DUALITY,

ABRIDGED

NOTATION

AND

PROJECTION.

BECIPROCATION

AND

THE

PRINCIPLE

OF DUALITY

119

Degree

of

the

reciprocal

of

a

surface,

how

measured

....

119

Beciprocal

of

a

curve

in

space

120

Osculating

plane

of

a

curve defined

120

Plane section

of

one

surface answers

to

tangent

cone

of

reciprocal

.

.

121

BECIPROOATION

WITH

REGARD

TO

SPHERE

121

Beciprocal

of

quadric,

when

ellipsoid,

hyperboloid,

or

paraboloid

. .

121


20/507

XVI

CONTENTS.

PAGli

Reciprocal

of ruled

surface

is

ruled

surface

of

same

degree

.

. .

122

Reciprocal

cones defined

122

Sections

by

any

plane

of

reciprocal cones

are

reciprocal

....

122

Focal

lines

of

a

cone defined

122

Reciprocal

of

sphere

123

Properties

of surfaces

of

revolution

obtained

by

reciprocation

.

.

.

124

Equation

of

reciprocal

of

a

quadric

125

TANGENTIAL

EQUATIONS

126

Tangential

equation

of

a

quadric

127

ABRIDGED NOTATION

.128

Equation

of

system

of

quadrics

having

common

curve

....

129

All

quadrics

through

seven

points pass

through

an

eighth

.

.

.

130

Locus of

centres

of

quadrics

touching eight

planes

130

5J

passing

through

eight

points

.

.

.

131

Four

cones

pass

through

the

intersection

of

two

quadrics

.

. .

132

The

self

-conjugate

tetrahedron

of

two

quadrics

132

Four

conies

are

touched

by

tangent

planes

common

to two

quadrics

.

133

Properties

of

systems

of

quadrics

having

double

contact

....

133

Properties

of three

quadrics

having

one

plane

curve

common

. .

. 134

Similar

quadrics

135

Geometrical

solution

of

problem

of

circular

sections

....

135

Twelve

umbilics

lie

three

by

three

on

eight

lines

135

Definition

of

'

axes

'

and

'

perpendicular

'

by

imaginary

circle

at

infinity

136

Quadrics

touching

along

a

plane

curve

136

Properties

of

their

sections

137

Two

quadrics

having

plane

contact with

the same

third

intersect

in

plane

curves

137

Form

of

equation

referred

to

self-con

jugate

tetrahedron

....

137

Lines

joining

vertices of

tetrahedron

to

corresponding

vertices

of

polar

tetrahedron

all

generators

of

same

hyperboloid

.

.

.

.188

Analogues

of

Pascal's

theorem

142

Hexagon

of

generators

of

a

hyperboloid

142

Examples

145

PEOJEOTIVE

COOEDINATBS,

GOLLINEATION,

RBCIPKOCATION

.

.

.

145

Linear

projection

defined

146

Projection

determined

by

five

elements

and

their

correspondents

.

.

146

Euclidean

congruence

147-148

The

Euclidean

absolute

143

Generalised

congruence

and

the

absolute

148

Generalised

definition

of

distance

148

Elliptic

and

hyperbolic

spaces

149

Parabolic

space,

Euclidean

distance

150

Generalised

definition

of

angle

150


21/507

CONTENTS.

CHAPTER

VIII.

FOCI

AND

CONFOCAL

SURFACES.

PAGE

Foci

AND

FOCAL

CONICS

(first

method)

151

Focus

and

directrix

of

a

quadric

defined

,

151

Focus

defined

as

intersection

of

two

generators

of

circumscribed

develop-

able

through

circle

at

infinity

152

Foci

in

general

lie on

a

curve

locus

153

Surfaces

when confocal

153

Tangential

equations

of

quadrics

of

confocal

system

153,

Two

kinds

of

foci

153

Focal

conies

154

Analysis

of

species

of focal

conies for

each

kind

of

quadric

. .

,

155

Foci of

sections normal

to a

focal

conic

157

Focal

lines

of a cone

157

Focal

conies

of

paraboloids

159

Focus

and

directrix

property

of

quadrics

160

Tangent

cone,

whose

vertex

is

a

focus,

is

a

right

cone

....

162

Beciprocal

of

a

quadric

with

regard

to

a

focus

is

a

surface

of

revolution

.

162

Property

of

umbilicar

foci

obtained

by

reciprocation

with

regard

to

sphere

163

Focal

properties

of

quadrics

obtained

by

reciprocation

with

regard

to

sphere

168

Spheres

of

given

radius

having

double

contact

with

quadric

. .

. 164

FOCAL CONICS AND

CONFOCAL

SURFACES

(second

method)

.

.

. 165

Focal

conies

as

limits of

quadrics

167,

Three

confocals

through

a

point

all

real

and

of different

species

.

.

167

Reciprocal

of

confocal

system

168

Coordinate's

of

intersection

of

three confocals

168

Coordinates

of

umbilics

169

Two

confocals

cut at

right

angles

169

Axes

of

central

section

in terms of

axes

of confocals

through

extremity

of

conjugate

diameter

.

170

Expression

for

length

of

perpendicular

on

tangent

plane

.

. .

171

pD

constant

along

the intersection

of

fcwo

confocals

....

172

Locus

of

pole

of fixed

plane

with

regard

to

a

system

of

confocals

.

.

173

Axes of

tangent

cone are

the

three

normals

through

its

vertex

. .

173

Transformation

of

equation

of

tangent

cone

to

the

three

normals

as

axes

of

coordinates

(three

methods)

175-177

Cones

circumscribing

confocal

surfaces

are

confocal

....

179

The

focal

lines

of

these

cones are

the

generators

of

the

hyperboloid

through

the

vertex

179

Tangent

planes through

any

line

to the

two

confocals

which

it

touches

are

mutually

perpendicular

180

Two confocals

seen from

any

point appear

to

cut

at

right

angles

.

.

180

b


22/507

XV111

CONTENTS.

Eeal

and

apparent

intersections

of

confocals

180

Normals

to

tangent

planes

through

a

given

line

generate

a

hyperbolic

paraboloid

.

181

Chasles's

method

of

obtaining

equation

of

tangent

cone

....

183

Locus of

intersection

of

three

rectangular

planes

each

touching

a

confocal

183

Intercept

on bifocal chord

between

tangent plane

and

parallel

central

plane

184=

Given three

conjugate

diameters

of

a

quadric,

to

find

the

axes

.

.

185

Locus of

vertices

of

right

cones

enveloping

a

quadric

:

which

stand

on

given

conic

...........

186

Either of

two

focal

conies

may

be

considered

as

a

locus

of

foci

of

the

other

187

Locus

of

intersection

of

three

mutually

perpendicular

tangent

lines

,

189

Corresponding

points

on

confocals

189

Elliptic

coordinates

191

Projections

of curves

on

ellipsoid

on

planes

of

circular

section

. .

191

Ivory's

theorem

as

to

the

distance

of two

corresponding

points

.

.

192

A

mechanical

property

of

confocal

hyperboloids

of

one sheet

.

.

.

193

Jacobi's

analogue

to

the

plane

theorem

that

the

sum

of

focal

distances

is

constant

193

JbachimsthaPs

analogue

for construction

of

normals

....

194

Locus

of

points

of contact

of

parallel

planes

touching

a series

of

confocals

.

195

Purser's theorem

on

intersecting

normals

196

CURVATUEE

OF

QUADBICS

196

Eadii of

curvature

of

a

normal

and

of

an

oblique

section

....

197

Line

of

curvature defined

199

Construction

of

principal

centres

of

curvature

199

Surface

of

centres

;

its sections

by principal

planes

200

Equation

of

its

reciprocal

201

CHAPTEE

IX.

INVARIANTS

AND

COVARIANTS

OF

SYSTEMS

OF

QUADRIC8.

SYSTEM

OF

TWO

QUADEICS

202

Fundamental

invariants

of

a

system

of

two

guadrics

....

202

Geometrical

meaning

of

vanishing

of

invariants

204

Hesse's

theorem

as

to

vertices

of

two

self-conjugate

tetrahedra

.

.

204

Property

of

spheres

circumscribing

self-conjugate

tetrahedra

.

. .

205

Condition

that

two

quadrics

should

touch

,

205

Equation

of

surface

parallel

to

a

quadric

;

expressed

in

parameters

. 206

Point

of

contact of

two

surfaces,

a

double

point

on their curve

of

inter-

section

207

Stationary

contact

defined

208

Equation

of

surface

of

centres

formed

209

Surface

of

centres defined

by

two

parameters

211


23/507

CONTENTS.

'

XIX

PAGE

Condition

that

a

tetrahedron

can be

inscribed

in

one

quadric

having

two

pairs

of

opposite

edges

on

another

211

Equation

of

quadric

touching

four

planes

212

Invariants of

a

cone

and

a

quadric

213

Two

kinds

of

equilateral hyperboloids

214

Orthogonal

hyperboloids

214

Tangential

equation

of

imaginary

circle at

infinity

215

Two

planes

at

right

angles

conjugate

with

regard

to

imaginary

circle

.

215

Tangential

equation

of

curve

in

space

.......

215

How

to

form

reciprocal

of

tangential

equation

of

a conic

. .

.

216

Equation

of

cone

touching

a

quadric

along

a

given

plane

section

.

. 216

Contravariants

of

a

system

of

two

guadrics

216

Two

principal

covariant

guadrics

of

a

system of

two

quadrics

.

. .

218

Equation

of

developable

circumscribing

two

quadrics

....

220

Its

sections

by

the

principal

planes

.

220

The

developable,

its

cuspidal

edge

and

its

tangent planes

defined

by

parameters

222

Complex

of

lines

meeting

curve

of

intersection

of two

quadrics

.

. 222

Condition

that

a

line should

be

cut

harmonically

by

two

quadrics

.

.

223

Equation

of

developable

generated

by

intersection

of

two

guadrics

.

.

223,

Intersection

of

this

developable

with

either

quadric

.....

224

Coordinates

of

a

tangent

to

the

common

curve

expressed

by

a

parameter

226

Section

of

developable

by

principal

planes

227

Coordinates

of

point

on

curve and of

osculating

plane,

how

expressed

.

228

A

system

of confocals

is

touched

by

a

common

developable

.

.

.

228

Locus

of

points

whence three

rectangular

lines

or

planes

can be

drawn

to

a

quadric

229

Two

quadrics

having

a

common

curve

can

be

described

to

touch a

line,

and

three

to

touch

a

plane 230

Properties

of confocals

deduced from

those of

a

system

inscribed

in

a

common

developable

230

Method

of

finding

equations

of

focal

conies

of

quadric

given

by

general

equation

232

Quadriplanar equation

of

system

of confocals

232

Theorems

respecting

perpendiculars,

how

generalised

....

233

Tangential equation

of

sphere

inscribed

in

a

tetrahedron

.

.

.

234

Equation

of

circumscribing

sphere

234

Conditions

that

quadriplanar

equation should

represent

a

sphere

. .

235

Condition

that

a

plane

section

should be

a

parabola

or

equilateral

hyperbola

236

Coordinates

of foci

of

a

plane

section

236

Locus

of foci of

central

section

of

a

given

quadric

237

Jacobian

of

a

system

of four

quadrics

238

Locus

of

vertices

of

cones which

pass

through

six

given

points

.

.

239

Jacobian

of

four

spheres

240

Reduction

of

two

quadrics

to

their

canonical

form

.

240


24/507

XX

CONTENTS.

PAGE

SYSTEM

OF THBEE

QUADBICS

242

Invariants

of

a

system

of three

quadrics

242

Condition

for

three

quadrics

to

be

each

the

sum

of

five

squares

same

as

for

each to

be

polar

quadrio

of a

point

with

regard

to

a

cubic

.

243

Tact-invariants

245

Discriminants:

of

discriminants

........

246

Jacobian

curve

of

three

quadrics

246

Complex

of

lines cut in

involution

by

three

quadrics

....

247

CHAPTER

X.

CONES

AND

SPHEBO-CONICS.

A

cone

may

have sheets

of two kinds 249

Spherical

coordinates

..........

250

Protective

generalisation

of

sphero-conics

254

Cyclic

arcs

of

sphero-conics

analogous

to

asymptotes;

analogous

theorems 255

Sum

of

focal

distances

constant

........

256

Focus

and

directrix

property

of

sphero-conics

......

257

Confocal

sphero-conics

257

Any

generator

of

hyperboloid

makes

with

planes

of

circular section

angles

the

product

of

whose

sines

is

constant

258

Difference

of

squares

of

reciprocal

of axes

of

central

section

of

a

quadric,

proportional

to

product

of

sines

of

angles

it

makes

with

cyclic

planes

259

Graves

's

theorem

260

Locus of

intersection

of

rectangular

tangents

to

a

sphero-conic,

&c.

.

260

Equation

in

spherical

coordinates

of

imaginary

circle

at

infinity

.

.

261

Equation

of

sphere

inscribed

in

a

tetrahedron

262

Equation

of

a

right

cone

.........

262-263

Investigation

of

Hart's extension of

Feuerbach's

theorem

. .

. 264

Extension of

Hart's

extension

267

CHAPTER

XI.

GENERAL

THEOBY

OF

SUBFACES

(iNTBODUCTOBY

CHAPTEB),

CURVATUBE,

CHABACTEBISTICS

AND

LOCAL FOBM

OF

SUBFACES

Number of

terms in

general

equation

208

Surfaces

depending

on

one or

two

undetermined

constants

.

.

.

269

Section

of

surface

by

tangent

plane

has

point

of

contact for

a

double

point

269

A

surface

in

general

has

triple

tangent

planes,

and

an

infinity

of

double

270

Inflexional

tangents

defined

271

The

indicate

;

elliptic,

hyperbolic,

and

parabolic

points

.

.

.271

Synclastic

and

anticlastic

regions

273

Asymptotic

curves

on

a

surface

374

The

three

kinds

of

point

illustrated

by

y

i

*

?

-L

* ,

y' ,

* ',

1

Expanding

this

by

the

common

rule,

the

equation

is

+

y

{/

(a?

-

a;' )

+

*

(a;'

-

z')

+

/

(x'

-

a; )}

+

*

{x'

(if

-

y' )

+

x

(y

fff

-

?/)

+

x'

(y

f

-

y )

}

=

x'

(y z

tft

-

if'z )

+

(^'V

-

y

y )

+

a-

w

^'

-

?J

V)

.

If

we

consider

x,

y,

z

as the

coordinates

of

any

fourth

point,

we

have

the

condition

that

four points

should

lie

in

one

plane.

32.

The

coefficients

of

a?,

y,

z

in

the

preceding

equation

are

evidently

double

the

areas of the

projections

on

the

co-

ordinate

planes

of

the

triangle

formed

by

the

three

points*

If

now

we

take

the equation

(Art.

27)

x

cos

a+y

cos

/3-f

#

cos

y=p,

and

multiply

it

by

twice

A

(A

being

the

area

of

the

triangle

formed

by

the

three

points),

the

equation

will

become identical

with that

of

the last

article,

since

A

cos

a,

A

cos

/3,

A

cos

7,

are

the

projections

of

the

triangle

on

the

coordinate

plane's

(Art.

4).

The

absolute

term

then

must

be

the

same

in

both

cases.

Hence

the

quantity

%

(y * '

-

0 '* )

+

of'

(y

Y

-

Y )

+

x'

(y'z

-

#V)

represents

double the

area

of

the

triangle

formed

by

the

three

points

multiplied by

the

perpendicular

on

its

plane

from the

origin

;

or,

in

other

words,

six

times the

volume

of

the tri-

angular

pyramid^

whose base is that

triangle,

and

whose

vertex

is

the

origin.

We

can

at

once

express

A

itself

in

terms

of

the

coordi-

nates

of

the

three

points

by

Art.

12,

and

must

have

44

2

equal

to

the

sum

of

the

squares

of

the coefficients

of

x,

y,

and

#,

in

the

equation

of the

1

last

article.

Ex.

1.

To

express

the

volume

of

a

tetrahedron

in terms

of

three concur-

rent

edges

and

the

angle

between

them.

If

in

the

preceding

values we

substitute

for

x

f

,

y',

z

f

the

values

p'

cos

a',


50/507

ANALYTIC

GEOMETBY

OF

THBEE

DIMENSIONS.

p'

cos

',

p'

cos

7',

&c.,

we find

that

six

times

the

volume

of

this

pyramid

=

p'p p (

multiplied

by

the

determinant

cos

a',

cos

',

cos

y'

cos

a ,

cos

j8 ,

cos

y

cos

a' ,

cos

|8' ,

cos

7'

Now

let

us

suppose

the

three

radii

vectores

cut

by

a

sphere

whose

radius is

unity,

having

the

origin

for its

centre,

and

meeting

it

in

a

spherical triangle

R'R'

f

R

f

.

Then

if a

denote the side

E'E ,

and

p

the

perpendicular

on

it

from

E

f

,

six

times

the

volume

of

the

pyramid

will

be

p'py

sin

a

sin

p

;

for

p'p

sin

a

is

double

the area

of

one face

of

the

pyramid,

and

p'

sin

p

is

the

perpendicular

on

it

from

the

opposite

vertex.

It

follows

then

that the

determinant

above

written

is

equal

to double

the

function

*/(sin

s sin

(5

-

a)

sin

(s

-

b)

sin

(s

-

c)J

of

the

sides

of

the

above-mentioned

spherical

triangle.

The

same

thing

may

be

proved

by forming

the

square

of

the

same

determinant

according

to

the

ordinary

rule

;

when

if

we

write

cos

a

cos

a'

+

cos

&

cos

j8'

4-

cos

7

cos

7'

=

cos

a,

&c.,

we

get

1,

cos

c,

cos

b

cos

c,

3

,

cos

a

cos

Z>,

cos

a,

1

,

which

expanded

is

1

+

2

cos

a

cos

b cos

c

-

cos

e

a

-

cos-6

-

cos-c,

which is

known

to have the value

in

question.

It is useful to

remark

that

when

the

three

lines

are at

right angles

to

each

other

the

determinant

cos

a',

cos

j3',

cos

7'

cos

a'',

cos

# ,

cos

7

cos

a' ,

cos

8' ,

cos

7'

has

unity

for

its

value.

In

fact we

see,

as

above,

that

its

square

is

1,

0,

0,

1,

0, 0,

1 .

[Ex.

2.

If the

coordinates

are

oblique

six times

the

volume

of the

pyra-

mid

is

equal

to

x',

y',

z'

x ,

?/',

*

x

f

,

y ',

z'

multiplied by

the

sine

of

the

angle

between

the

axes of

y

and

z,

multiplied

by

the sine

of

the

angle

between

the

axis

of

x

and

the

plane

of

y

s.

Ex.

3.

If the

direction

cosines

of

the

fixed axes

(oblique

or

rectangular)

re-

ferred

to

three

rectangular

axes

are

A

15

p

lt

Vl

\

A

21

/^,

v%\

A

3

,

/*.,,

y

st

prove

that

the determinant

has

the

same

value for

all

such

rectangular

axes,

and

find

its

geometrical

meaning.

(It

is the

coefficient of

the

determinant

in

Ex.

2).]


51/507

THE PLANE

AND

THE

BIGHT

LINE.

23

33.

To

find

the

length

of

the

perpendicular

from

a

given

point x'y'z

on

a

given

plane,

x

cos

a

+

y

cos

(3+z

cos

y=p.

If

we

draw

through

x'y'z'

a

plane

parallel to

the given

plane,

and let

fall on the

two

planes

a

common

perpendicular

from

the

origin,

then

the

intercept

on

this

line

will be

equal

to

the

length

of the

perpendicular

required,

since

parallel

planes

make

equal

intercepts

on

parallel

lines.

But

the

length

of

the

perpendicular

on the

plane

through

x'y'z'

is,

by

defini-

tion

(Art.

5),

the

projection

on that

perpendicular

of the

radius

vector

to

x'y'z'

>

and

therefore

(Art.

27)

is

equal

to

x'

cos

a

+

y'

cos

ft

+

z'

cos

7.

The

length

required

is

therefore

x'

cos

a+y'

cos

/?

+

#'

cos

y

-p.

N.B.

This

supposes

the

perpendicular

on

the

plane

through

x'y'z'

to be

greater

than

p

; or,

in

other

words,

that

x'y'z'

and

the

origin

are

on

opposite

sides

of

the

plane.

If

they

were

on

the

same

side,

the

length

of

the

perpendicular

would

be

p

-

(x

f

cos

a

+

y'

cos

/5+

/

cos

7).

If the

equation

of

the

plane

had

been

given

in

the form

Ax+

By

+

Cz+

D

=

0,

it

is

reduced,

as

in Art.

28,

to

the

form

here

considered,

and

the

length

of

the

perpendicular

is

found

to

be

Ax'+By'+Cz'+D

It

is

plain

that all

points

for

which

Ax'+By'+Cz'+D

has

the same

sign

as

D,

will

be on

the same

side

of

the

plane

as

the

origin

;

and

vice

versa when the

sign

is

different.

34. To

find

the

coordinates

of

the

intersection

of

three

planes.

This

is

only

to

solve

three

equations

of

the

first

degree

for

three

unknown

quantities

(Higher

Algebra,

Art.

29,

or

Burnside

and

Panton,

Art.

144)

.

The

values

of

the

coordinates

will

become

infinite

if

the determinant

(AB'O )

vanishes,

or

A

(B'C

-

B C')

+

A'

(B C

-

BC }

+

A

(BC

r

-

B'C)

-

0.

This

then

is the

condition

that

the

three

planes

should

\>e

parallel

to

the

same

line.

For

in

such

a

case

tt^e

lii^e

of


52/507

ANALYTIC

GEOMETRY

OF

THREE

DIMENSIONS.

intersection

of

any

two would

be also

parallel

to this

line,

and

could

not

meet the

third

plane

at

any

finite

distance.

35.

To

find

the condition

that

four

planes

should

meet

in

a

point.

This

is

evidently

obtained

by

eliminating

x,

y,

z

between

the

equations

of

the

four

planes,

and

is

therefore

the

de-

terminant

(A&CrD' ),

or

A,

B,

C,

D

A',

JB',

C

f

,

D'

\ j3 ,

C ,

D

=

0.

36.

To

find

the

volume

of

the

tetrahedron

whose

vertices

are

any

four

given

points.

If we

multiply

the area

of

the

triangle

formed

by

three

points,

by

the

perpendicular

on

their

plane

from

the

fourth,

we

obtain

three

times

the

volume.

The length

of

the

per-

pendicular

on

the

plane,

whose

equation

is

given

(Art.

31),

is

found

by

substituting

in

that

equation

the

coordinates

of the

fourth

point,

and

dividing

by

the

square

root of

the

sum

of

the

squares

of

the

coefficients

of

x,

y,

z. But

(Art.

32)

that

square

root is

double

the

area

of

the

triangle

formed

by

the

three

points.

Hence

six

times

the

volume

of

the tetrahedron

in

question

is

equal

to

the

determinant

*',

y',

*,

1

a ,

y ,

* ,

I

x'

ff

,

y' ,

*' .

1

*/' ,

y ,

* ,

1

Ex.

The

volume

of the tetrahedron

formed

by

four

planes,

whose

equa-

tions

are

given,

can be

found

by

forming

the

coordinates

of

its

angular

points,

and

then

substituting

in

the

formula

given

above. The

result

is

(HigJier

Algebra,

Art.

30,

or

Burnside or

Panton,

Art.

146),

that

six

times

the

volume

is

equal

to

_

AB'C )

(A

f

B G

fl1

)

(A B

f

C)

(

where

R

is

the

determinant

(A&C & )

t

Art.

35,

and

the

factors

in

the

de-

nominator

express

the

conditions

(Art.

34)

that

any

three

of

the

planes

should

be

parallel

to

the

same

line

f


53/507

THE PLANE AND THE

BIGHT

LINE.

25

37.

It

is

evident,

as

In

Plane

Geometry

(see

Conies,

Art.

40),

that

if

8,

S',

8

represent

any

three

surfaces,

then

aS+

IS',

where

a

and

b

are

any

constants,

represents a

surface

passing

through

the

line

of

intersection

of

S

and 8'

;

and

that

aS

bS'+

cS

represents

a

surface

passing through

the

points

of intersection

of

S, S',

and

S .

Thus

then,

if

L, M,

N

de-

note

any

three

planes,

aL

+

bM

denotes

a

plane

passing

through

the

line of

intersection

of

the

first

two,

and

aL

+

bM

+

cN

denotes

a

plane

passing

through

the

point

common

to

all

three.*

As

a

particular

case

of

the

preceding

aL

+

b

denotes

a

plane parallel

to

Z/,

and

aL

+

bM+

c denotes

a

plane parallel

to

the

intersection

of

L

and

M

(see

Art,

30).

So

again,

four

planes

L,

M,

N,

P

will

pass

through

the

same

point

if

their

equations

are

connected

by

an identical

relation.

aL

+

bM+

cN+ dP

=

0,

for

then

any

coordinates

which satisfy

the

first

three

must

satisfy

the

fourth.

Conversely,

given

any

four

planes

inter-

secting

in a common

point,

it

is

easy

to

obtain

such

an

identical

relation.

For

multiply

the

first

equation

by

the

determinant

(A'E G

f

},

the

second

by

-

(A B' C),

the

third

by

(A^BC ),

and

the fourth

by

-

(AB'C ),

and

add,

then

(Higher Algebra,

Art.

7,

or

Burnside and

Panton,

Art.

145)

the

coefficients of

x,

y,

z

vanish

identically

;

and

the

remain-

ing

term

is the

determinant

which

vanishes

(Art.

35),

because

the

planes

meet

in

a

point.

Their

equations

are

therefore

connected

by

the

identical relation

L

(A'B C' )-~M (A B' 0)

+

N

(A^'BC')

-P

(AB'C )

=

Q.

[This

equation

may

be obtained

by

eliminating

1,

x,

y,

z,

from

the four

identical

equations

of

the

form L-D^Ax

+

By

+

Cm,

and

equating

to zero

the determinant

of

Art.

35.

By

the

same

method

it

follows

that

the

equations

of

any

*

German

writers

distinguish

the

-system

of

planes

having

a line

common

by

the name Biischel from

the

system

having

only

one

point

common,

which

they

call

BiindeL

[The

corresponding

English

terms

are

generally

pencil

and

sheaf.']


54/507

26

ANALYTIC

GEOMETBY

OF

THBEE

DIMENSIONS.

four

planes

are

connected

by

an

identical

relation

38.

Given

any

four

planes

L,

M,

N,

P

not

meeting

in

a

point,

it

is

easy

to see

(as

at

Conies,

Art.

60)

that

the

equa-

tion

of

any

other

plane

can

be

thrown

into

the form

[This

relation

may

be

found

by

eliminating

x

9

y,

2, 1,

A:

from

the

five

equations

kL,

&c.,

where

A

=

\x

+

py

+

vz

+

p

=

0,

is

the

plane.

Thus

the

equations

of

any

five

planes

are

connected

by

a

linear

homogeneous

relation.]

In

general

the

equation

of

any

surface

of

the

degree

n

can be

expressed

by

a

homogeneous

equation

of

the

degree

n

between

L,

M,

N,

P

(see

Conies,

Art.

289),

For

the

number

of

terms

in

the

complete equation

of

the

degree

n between

three

variables

is the same

as

the

number

of terms

in

the

homogeneous

equation

of

the

degree

n between

four

variables.

Accordingly,

in

what

follows,

we shall

use

these

quadri-

planar

coordinates,

whenever

by

so

doing

our

equations

can

be

materially simplified

;

that

is,

we

shall

represent

the

equa-

tion

of

a

surface

by

a

homogeneous

equation

between

four

coordinates

x,

y,

$,

w

;

where

these

may

be

considered

as

denoting

the

perpendicular

distances,

or

quantities

propor-

tional to

the

perpendicular

distances

(or

to

given

multiples

of

the

perpendicular

distances)

of

the

point

from

four

given

planes

#

=

0,

?/

=

0,

#

=

0,

w

=

0.

It

is

at

once

apparent

that,

as

in

Conies,

Art.

70,

there is

also

a

second

system

of

interpretation

of

our

equations,

in

which

an

equation

of

the first

degree

represents

a

point,

and

the variables are the coordinates of

a

plane.

In

fact,

if

L'M'N'f denote

the

coordinates

of

a

fixed

point,

the

above

plane

passes

through

it

if

aL'+bM'+cN'+dP'=(),

and

the

coordinates of

any

plane

through

this

point

are

subject only

to

this relation.

The

quantities,

a, b,

c,

d

may

be

considere4


55/507

THE

PLANE

AND

THE

BIGHT

LINE.

27

as

denoting

the

perpendicular

distances,

or

quantities

pro-

portional

to

the

perpendicular

distances

(or

to

given

multiples

of

the

perpendicular

distances)

of

the

plane

from

four

given

points

a

=

0,

b

=

0,

c

=

0,

d

0.

[For

the

point

whose

equation

in

plane

coordinates

is

a

=

is

the

intersection

of

the

three

planes

whose

equations

in

point

coordinates are

lf

=

0,

JV =0,

P=0;

if

we

substitute

the

coordinates

(x

l9

y

lt

^)

of

this

point

in

aL

+

bM+cN+dP

we

get

aL

lt

which

must be

proportional

to

the

perpendicular

from

x

lt

y

l9

^

on

the

plane

aL +

bM+ cN+dP

=

Q

;

hence

a

is

proportional

to

this

per-

pendicular

since L

1

is

constant.]

Ex.

1.

To

find

the

equation

of

the

plane

passing

through

x'y'z',

and

through

the

intersection

of

the

planes.

Ax

+

By

+

Cz

+

D,

A'x

+

B'y

+

C'z

+

D

f

(see

Conies,

Art.

40,

Ex?.

3).

Ans.

(A'%

1

+

B'y*

+

C'z'

+

D')(Ax

+

By

+

Ce

+

D)

=

(Ax

1

+

By'

+

C*'

4-

D)(A'x

+

B'y

4-

C's

+

')

Ex.

2.

Find

the

equation

of

the

plane

passing

through

the

points

ABC,

figure, p.

2.

The

equations

of

the

line

BC

are

evidently

~

=

1, f

+

-

=

1.

Hence

ob-

ct>

be

viously

the

equation

of

the

required

plane

is

-

+

~

+

~

=

2,

since

this

passes

CL

C

through

each

of

the

three

lines

joining

the

three

given point*.

Ex.

3. Find the

equation

of

the

plane

PEP

in

the

same

figure.

The

equations

of

the line

JEF

are x

=

0,

^

+

--

=1

;

and

forming

as

above

u C

the

equation

of

the

plane

joining

this line to

the

point abc

t

we

get

39.

If

four

planes

which

intersect in a

right

line be

met

by

any

plane,

the

anharwionic

ratio

of

the

pencil

so

formed

will

be

constant. For we

could

by

transformation of

coordinates

make the

transverse

plane

the

plane

of

xy,

and

we

should

then

obtain

the equations

of

the

intersections

of

the

four

planes

with

this

plane

by

making

#=

in the

equations.

The

resulting equations

will

be

of

the form

aL

+

M,bL

+

M,

cL

+

M,

dL

+

M

(each

equated

to

zero)

whose anharinonic

ratio

(see

Conies,

Art.

59)

depends

solely

on

the

constants

a,

6, c,

d

;

and

does

not

alter

when

by

transformation of

coordinates

L

and

M

come

to

represent

different

lines.

[This

anharmoiUQ


56/507

28

ANALYTIC

GEOMETRY

OF THEEE DIMENSIONS.

ratio

is

evidently

the

same

as

the

anharmonic

ratio

of

the

four

points

in which

any

line

meets

the

planes.]

[Ex.

The

preceding

is

equivalent

to

the following

theorem

in

spherical

trigonometry

:

If

P,

Q,

R,

S

are

four

points

on

a

great

circle

and

any

other

point

on

the

sphere,

then

sin

PQ

sin RS

_

sin

POQ

sin

EOS

sin

PR sm

QS

~

s n~POE

sin

~QOS

where

POQ,

&c.,

represent

the

angles

between

the

great

circles

PO, OQ,

&c.]

40.

[The

homogeneous

coordinates

(x,

y,

z,

w)

of

a

point

are

linear

functions

of

its

Cartesian

coordinates

and

are

either

proportional

to

the

perpendiculars

from

the

point

on

the

faces

of

the tetrahedron

or are

proportional

to

given

multiples

of

these

perpendiculars.

If

we transform

from

the

homogeneous

coordinates

(#,

y

t

z,

w]

to another

set

(f

,


57/507

THE

PLANE

AND

THE

EIGHT LINE. 29

It

is

evident

that

the

transformation of

coordinates is

quite

similar

for

the

coordinates

of

planes.

41.

If

we

denote

by

ax

Q9 %

,

7#

,

Sw the

perpendiculars

from

the

vertices

on the

opposite

sides of

the

original

tetra-

hedron, a,

fi,

7,

8

being

constants,

we

have

obviously,

if

Ay

By C,

D be

the

areas

of those

faces,

Aax

Q

-

Bfa

(]

=

C


58/507

30

ANALYTIC

GEOMETRY

OF

THREE

DIMENSIONS.

to

perpendiculars

on the coordinate

planes.

Hence we

may

write

the new

coordinates

=

X

sin

p,

y

Y sin

#,

z

=

Z sin

r,

where

p,

#,

r

are

the

angles

the

axes

of

X,

1 ,

Z

make

with

the

planes

YZ,

&c.

;

therefore

x

r

,

y',

0',

1

^n

,.n

y

tr

i

y

s

}

i

ft

,*

iy

'

,,,'

^

5

if

*

j

or

by

the

relations

=

6TF

sin^

sin

q

sin

r sin

(XYZ),

y',

z',

w'

y ,

z ,

w

,

y

m

,

z

m

,

w'

sin

#

sin

r sin

(XYZ).

We

may

give

this

another

form

by

remarking

that

the

determinant

re-

duces

for

the

tetrahedron

of

reference

to

the

continued

product,

which

is

its

leading

term,

hence

^oI/oVo

=

6F?

sin^;

sin

g

sin

r

sin

(XYZ),

whence,

dividing

the

former

equation by

this,

(ayW

K

)

W

[If

a?,

y,

^,

?^

are

aw-^ homogeneous

coordinates

that

is,

any

linear

func-

tions

of

Cartesian coordinates

or

of

another

set

of

homogeneous

coordinates

-

then it

is

still

true

that

the

volume

of

the

tetrahedron

is

proportional

to

the

determinant

(X

>

IJ Z

>

IQ'

V

).

Hence

the

above

result

still

holds

good.]

The

Right

Line.

42.

The

equations

of

any

two

planes

taken

together

will

represent

their

line

of

intersection,

which

will

include

all

the

points

whose

coordinates

satisfy

both

equations. By

eliminat-

ing

x and

y

alternately

between

the

equations

we

reduce them

to

a

form

commonly

used,

viz.

The first

represents

the

projection

of

the

line

on the

plane

of

xz

and

the

second that

on

the

plane

of

yz.

The

reader will

observe

that

the

equations

of

a

right

line

imlude

four

inde-

pendent

constants.

We

might

form

independently

the

equations

of

the line

joining

two

points;

for

taking

the

values

given

(Art.

8)

of

the

coordinates of

any

point

on

that

line,

solving

for

the

ratio

m

: I

from

each

of

the

three

equations

there

given,

and

equa-

ting

results,

we

get


59/507

THE

PLANE

AND

THE

BIGHT

LINE.

31

x -x'

__

y

j^ y^ _z

-

z

x

f

-

x

~

y

f

-

y

/f

~~

d~s{

for

the required

equations

of

the

line.

It

thus

appears

that

the

equations

of

the

projections

of

the

line

are the same as

the

equations

of

the

lines

joining

the

projections

of

two

points

on

the

line,

as

is

otherwise

evident.

43.

Two

right

lines

in

a treatise on the analytic geometry of three dimensions vol i 1914

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