An elementary treatise on solid geometry

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GIFT

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SOLID G-EOMETBY.


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WORKS BY CHAKLES

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BY CHARLES SMITH, M.A. AND SOPHIE BRYANT,

D.Sc.

Euclid s Elements of Geometry. Second Edition.
Books I IV, VI and XI. Globe 8vo. 4s. Gd.

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ELEMENTAEY TREATISE
ON

SOLID GEOMETRY
DY

CHARLES SMITH,

M.A.

MASTEB OF SIDNEY SUSSEX COLLEGE, OAilBBIDQK

ELEVENTH

EDITION.

Hontion

MACMILLAN AND

CO.,

LIMITED

NEW YORK: THE MACMILLAN COMPANY
1907
All rights reserved.


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Agtron.

First Edition, 1884.
Edition, 1893.
i/w/i,

1

899.

Second Edition, 1886.

/7/M

^//M Edition,

Tenth Edition, 1905.

1

90 1

6Y.rM Edition, 1897.
.

M/A ,ff^Vw,

Eleventh Edition, 1907.

A

^y,

Third Edition, 1891.

Edition, 1895.

:

1

903.


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PREFACE.
THE

following

work

is

intended as an introductory text

book on Solid Geometry, and I have endeavoured

to present

the elementary parts of the subject in as simple a manner as
possible.

Those who desire

fuller information are referred to

the more complete treatises of Dr Salmon and Dr Frost, to
both of which I am largely indebted.
I have discussed the different surfaces which can be
represented by the general equation of the second degree at

an

earlier stage

this

than
is

is

for

sometimes adopted. I think that
many reasons the most satisfactory,

arrangement

and I do not believe that beginners will find it difficult.
The examples have been principally taken from recent

Examination papers; I have
included many interesting theorems of M. Chasles.
University and College

I

Mr

am

S. L.

indebted to several of

Loney, B.A., and

to

Mr

my

also

friends, particularly to

R H. Piggott, B.A., Scholars

of Sidney Sussex College, for their kindness in looking over

the proof sheets, and for valuable suggestions.

CHARLES SMITH.
SIDNEY SUSSEX COLLEGE,
April, 1884.


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CONTENTS.
CHAPTER

I.

CO-ORDINATES.

PAGE
1

Co-ordinates
Co-ordinates of a point which divides in a given ratio the line joining

3

two given points
Distance between two points

.

Direction- cosines

*

Relation between direction-cosines .

.

.

.

....

.

.

.

,

4
5

.

.

5

Locus of an equation

.

6

Projection on a straight line

7

8

Polar co-ordinates

CHAPTER

II.

THE PLANE.
An

9

equation of the first degree represents a plane

of a plane in the form Ix + my + nz =p

Equation
Equation
Equation
Equation

9

of a plane in terms of the intercepts made on the axes
of the plane through three given points

.

.

10
11

of a plane through the line of intersection of two given

....

.
planes
Conditions that three planes

may have

11

a

common

line of intersection

Length of perpendicular from a given point on a given plane
Equations of a straight line
Equations of a straight line contain four independent constants
Symmetrical equations of a straight line

.

.

11

.12
14

.

.

14
15


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CONTENTS.

Vlll

...

PAOK

....
....

1C

line

19

Equations of the straight line through two given points
Angle between two straight lines whose direction-cosines are given
Condition of perpendicularity of two straight lines

Angle between two planes whose equations are given
Perpendicular distance of a given point from a given straight
Condition that two straight lines may intersect
Shortest distance between two straight lines

16

.

17

18

.

19

20

.22

Projection on a plane
Projection of a plane area on a plane
Volume of a tetrahedron

23

....

Equations of two straight lines in their simplest forms
Four planes with a common line of intersection cut any straight
in a range of constant cross ratio

24
25

line

26

Oblique axes

26

Direction-ratios

26

Relation between direction-ratios

27

Distance between two points in terms of their oblique co-ordinates
Angle between two lines whose direction-ratios are given
Volume of a tetrahedron in terms of three edges which meet in a

28

point, and of the angles they make with one another
Transformation of co-ordinates
Examples on Chapter II

28

...
...
.

CHAPTER

28

29
34

III.

SURFACES OF THE SECOND DEGREE.

Number

of constants in the general equation of the second degree

All plane sections of a surface of the second degree are conies
.
.
.
Tangent plane at any point of a conicoid
.

Polar plane of any point with respect to a conicoid
Polar lines with respect to a conicoid

A

chord of a conicoid

is

.

.

37

.

38

.38

....
.

40

cut harmonically by a point and

its

polar

plane
Condition that a given plane may touch a conicoid
.
.
Equation of a plane which cuts a conicoid in a conic whose centre
.

40

.41
is

43

given

Locus of middle points
Principal planes

39

of a

system of parallel chords of a conicoid
,

.

44
44


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CONTENTS.

IX

PAGE
Parallel plane sections of a conicoid are similar

and similarly situated
45

conies
Classification of conicoids

The
The
The
The
The
The

A

.

.

.

..-.-.

ellipsoid

,

:

.

.

.

.

hyperboloid of one sheet

.

hyperboloid of two sheets
cone

.

.

.

.

.

.

.

.

.

-

.

.

.49
.

.
.
.
paraboloids
paraboloid a limiting form of an ellipsoid or of an hyperboloid

.

.52
52

.

.

.

54
54

Cylinders
The centre of a conicoid

56

.......
.

The discriminating cubic

50

.51
51

...

asymptotic cone of a conicoid

Invariants

46

Conicoids with given equations
Condition for a cone

.

.

.

.

.

,

.

.

.

.

58
59

.

.

60
66

Conditions for a surface of revolution

66

Examples on Chapter

67

III

CHAPTER

IV.

CONIOOIDS REFERRED TO THEIR AXES.

The sphere
The ellipsoid
Director-sphere of a central conicoid
Normals to a central conicoid

Diametral planes
Conjugate diameters

69
".71
.

.

.

.

.

.

.

72
73

.

.

.

.74
75

Relations between the co-ordinates of the extremities of three conjugate
diameters

75

Sum

76

The

of squares of three conjugate diameters is constant
.
.
i
parallelepiped three of whose conterminous edges are conjugate
semi-diameters is of constant volume

Equation

of conicoid referred to conjugate diameters as axes

.

.

76
78

The paraboloids

80

Locus of intersection of three tangent planes which are at right angles
Normals to a paraboloid

81

Diametral planes of a paraboloid

81

80


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X

CONTENTS.
PAGE

Cones

.

,

83

Tangent plane at any point of a cone

83

.

Reciprocal cones
Eeciprocal cones are co-axial

Condition that a cone

84
85

may have
may have

three perpendicular generators
Condition that a cone
three perpendicular tangent planes
Equation of tangent cone from any point to a conicoid
Equation of enveloping cylinder

.

.

....
.

85
86
86

88

Examples on Chapter IV.

90

CHAPTER

V.

PLANE SECTIONS OF CONICOIDS.
Nature of a plane section found by projection
Axes and area of any central plane section of an ellipsoid or of an
hyperboloid

.

98

....

99

.

Condition that a plane section may be a rectangular hyperbola
Condition that two straight lines given by two equations may be at
.

Two

.

.

CHAPTER

99

101
101

.

102

.

103

102

common have

also another

103

circular sections of opposite systems are

Circular sections of a paraboloid
Examples on Chapter V.

97

......

Area of any plane section of a central conicoid
Area of any plane section of a paraboloid
Area of any plane section of a cone
Directions of axes of any central section of a conicoid
Angle between the asymptotes of a plane section of a central conicoid

right angles
Conicoids which have one plane section in
Circular sections

96

on a sphere

.

.

-.

"..

.

.

.

105

.

.

.

105

108

VI.

GENERATING LINES OF CONICOIDS.
Ruled surfaces defined
Distinction between developable and skew surfaces
Conditions that

all

....

points of a given straight line may be on a surface .
to a conicoid at any point on a generating line

113

113
113

The tangent plane

contains the generating line
a generating line of a conicoid touches the surface

Any plane through

115
.

115


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XI

CONTENTS.

PAGE

Two generating lines pass through every point of an hyperboloid of
.
r
.
.
.
sheet, or of an hyperbolic paraboloid

one

116

.

lines
systems of generating
lines
All straight lines which meet three fixed non-intersecting straight
of the same system of a conicoid, and the three fixed
are

11G

system of the same conicoid .
of
Condition that four non-intersecting straight lines may be generators

117

Two

generators

lines are generators of the opposite

H?

the same system of a conicoid
to
lines through the angular points of a tetrahedron perpendicular

the opposite faces are generators of the same system of a conicoid
three lines joining
If a rectilineal hexagon be traced on a conicoid, the
its opposite vertices meet in a point
Four fixed generators of a conicoid of the same system cut all generators

The

of the opposite system in ranges of equal cross-ratio

118

.118

.

.

118

Angle between generators
of an hyperboloid of
Equations of generating lines through any point
one sheet
/
of an hyperbolic
Equations of the generating lines through any point

***
12

l"-1

paraboloid

Locus of the point of intersection of perpendicular generators
.
.
Examples on Chapter VI.

CHAPTER
SYSTEMS OP CONICOIDS.
All

12-4

VII.
RECIPROCATION.

TANGENTIAL EQUATIONS.

conicoids through eight given points have a
intersection

124

.

.

common

curve of
I 28

.

Four cones pass through the intersections of two conicoids

.

.

.

129

Self-polar tetrahedron

129

Conicoids which touch at two points
All conicoids through seven fixed points pass through another fixed

130

.

point

.

130

.131

Rectangular hyperboloids

Locus of centres of conicoids through seven given points

.

Tangential equations
Centre of conicoid whose tangential equation is given
Director-sphere of a conicoid
Locus of centres of conicoids which touch eight given planes
Locus of centres of conicoids which touch seven given planes
.

.

132

.133
.

.

134

.135
.

.

.

.

136
137


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CONTENTS.

Xll

PAGE
Director- spheres of conicoids which touch eight given
planes, have a
common radical plane

The

......

director-spheres of all conicoids which touch six given planes are
cut orthogonally by the same sphere

Reciprocation

.

.

137

137

.

The

degree of a surface is the same as the class of its reciprocal
.
.
Eeciprocal of a curve is a developable surface
Examples of reciprocation

...
.

.

138

138
140

.

Examples on Chapter VII

141

CHAPTER
CONFOCAL CONICOIDS.

137

VIII.

CONCYCLIC CONICOIDS.

Foci OP CONICOIDS.

Confocal conicoids defined

144

Focal conies. [See also 158]
Three conicoids of a confocal system pass through a point
One conicoid of a confocal system touches a plane

145

....
.

.

.

Two

conicoids of a confocal system touch a line
Confocals cut at right angles

The tangent planes through any

line to the

145
146

146
147

two confocals which

it

touches are at right angles

148

Axes of central section of a conicoid in terms of axes of two confocals
Corresponding points on conicoids
Locus of pole of a given plane with respect to a system of confocals

Axes of enveloping cone of a conicoid
Equation of enveloping cone in its simplest form
Locus of vertices of right circular enveloping cones

.

149
151

.

152
153

....

Coney clic conicoids

153

155
155

Reciprocal properties of confocal and concyclic conicoids
Foci of conicoids

Focal conies

.

.

.

156

156
.

.

.

158

.

160

Focal lines of cone

159

Examples on Chapter VIII

CHAPTER

IX.

QUADRIPLANAR AND TflTRAHEDRAL CO-ORDINATES.

Definitions of Quadriplanar

and of Tetrahedral Co-ordinates
Equation of plane
Length of perpendicular from a point on a plane

.

.

164

165
167


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CONTENTS.

xiii

PAGE

Plane at infinity

167

Symmetrical equations of a straight line
General equation of the second degree in tetraliedral co-ordinates

168
.

169

Equation of tangent plane and of polar plane

170

Co-ordinates of the centre

170

Diametral planes
Condition for a cone
Any two conicoids have a

171
171

.

.

common

self-polar tetrahedron

.

.

The circumscribing conicoid
The inscribed conicoid
The circumscribing sphere

.

172

172

.172

.

173

Conditions for a sphere

173

Examples on Chapter IX

175

CHAPTER

X.

SURFACES IN GENERAL.

The tangent plane

at

any point

of a surface

178

Inflexional tangents
The Indicatrix

179

Singular points of a surface
.
.
.
Envelope of a system of surfaces whose equations involve one arbitrary

180

180

parameter

.

.

of regression of envelope
Envelope of a system of surfaces

.

181

.182

Edge

whose equations involve two arbitrary
parameters
Functional and differential equations of conical surfaces
.
.
Functional and differential equations of cylindrical surfaces .
.
.
Conoidal surfaces
.

Differential equation of developable surfaces
Equation of developable surface which passes through two given curves
conicoid will touch any skew surface at all points of a

A

183
184
185
186

188
190

generating

line

.

Lines of striction

.

.

.

.

.

.

8

,

Functional and differential equations of surfaces of revolution
.
.
.
.
,
Examples on Chapter X.
,
,

191

,.

191
.

.

192

.

.

194

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CONTENTS.

XIV

CHAPTER

XI.

CURVES.
PAGE

Equations of tangent at any point of a curve
Lines of greatest slope

197

....

Equation of osculating plane at any point of a curve
Equations of the principal normal
.
Radius of curvature at any point of a curve
Direction-cosines of the binormal
.

.

.

.

......

Measure of torsion at any point of a curve

,

.

.

.

may

202
203
203

206

Kadius of curvature of the edge of regression
Curvature and torsion of a helix

Examples on Chapter

201
202

204

be plane
Centre and radius of spherical curvature

Condition that a curve

198

of the polar developable

.

207
208

XL

210

CHAPTER

XII.

CURVATUBB OF SuKFACES.
Curvatures of normal sections of a surface

.

.

...",".

.

.

Euler

s

213

214

Principal radii of curvature

Theorem
s Theorem

.214

.

Meunier

215

Definition of lines of curvature

217

The normals

to

any surface

at consecutive points of a line of curvature

217

intersect

217

Differential equations of lines of curvature
Lines of curvature on a surface of revolution

218

.
.218
*
.
.
Lines of curvature on a developable surface .
219
Lines of curvature on a cone

on both
If the curve of intersection of two surfaces is a line of curvature
220
the surfaces cut at a constant angle
221
s Theorem

Dupin

To

.

.

...

find the principal radii of curvature at

any point

Principal radii of curvature of the surface
Gauss measure of curvature

Geodesic lines

of a surface

.

.

222
223

Umbilics

z=f(x,

y)

224

225
22 6


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CONTENTS.

XV
PAGE

Lines of curvature of a conicoid are

its

curves of intersection with con-

focal conicoids

Curvature of any normal section of an ellipsoid
.
.
.
.
The rectangle contained by the diameter parallel to the tangent at any
point of a line of curvature of a conicoid, and the perpendicular

The

from the centre on the tangent plane at the point is constant
.
rectangle contained by the diameter parallel to the tangent at any
point of a geodesic on a conicoid, and the perpendicular from the
centre on the tangent plane, is constant

227
228

228

228

Properties of lines of curvature of conicoids analogous to properties of
confocal conies

229

Examples on Chapter XII.

230

Miscellaneous Examples

.

.

.

.

.

.

.

.

.

.

.

237

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SOLID GEOMETRY.
CHAPTER

I.

CO-ORDINATES.

THE

position of a point in space is usually determined
it to three fixed planes.
The point of inter
section of the planes is called the origin, the fixed planes are
called the co-ordinate planes, and their lines of intersection
1.

by referring

The three co-ordinates of a point are
the co-ordinate axes.
its distances from each of the three co-ordinate planes,
measured parallel to the lines of intersection of the other
When the three co-ordinate planes, and therefore the
two.
three co-ordinate axes, are at right angles to each other, the

axes are said to be rectangular.
The position of a point is completely determined when
2.
be
co-ordinates are known.
For, let YOZ, ZOX,
the co-ordinate planes, and
OX, Y OY, OZ be the axes,
and let LP, MP, NP, be the co-ordinates of P. The planes
are parallel respectively to YOZ, ZOX,
MPN, NPL,
if therefore they meet the axes in Q, R, S, as in the
is a
figure, we have a parallelepiped of which
diagonal;
and, since parallel edges of a parallelepiped are equal,

XOY

its

X

Z

LPM

XOY\

OP

LP = OQ,MP =

OR, and

NP = 08.

Hence, to find a point whose co-ordinates are given, we have
only to take OQ, OR,
equal to the given co-ordinates,

OS

S. S. G.

1


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CO-ORDINATES.
Q, R, S parallel respectively
then the point of intersection of

and draw three planes through
to the co-ordinate planes

;

these planes will be the point required.

Z

If the co-ordinates of

tively be a,

3

b, c,

then

P

is

P

to OX, OY,
parallel
said to be the point (a,

OZ respec
6, c).

To determine the position of any point P it is not
absolute lengths of the lines
merely to know the
the directions in which
know

MP, NP, we must also

sufficient

LP

drawn
those drawn

If lines

they are drawn.

sidered as positive,

as negative.
shall consider that the directions

must be considered

We

in one direction be con
in the opposite direction

OX, 0\

,

OZ

are

,

positive.

The whole

of space

is

divided by the co-ordinate planes

UAY /,,
namely OXYZ, OX Y /,,
YZ
OX
and
YZ
OXYZ,
OX
2
OXYZ, OXY
there is a

into eight compartments,
,

P

,

be any point in the first compartment,
other compartments whose absolute
point in each of the
to those of P\
distances from the co-ordinate planes are equal
are
a,
(a, -b, c)
be),
(b, c) the other points
and, if P be
- 6, c) and (- a,-b-c)
a,
6,
c),
(a,
c), (b,
c), (a,
(a, b,
If

(a>

respectively,

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00-OKDINATES.
4.
To find the co-ordinates of the point which divides the
straight line joining two given points in a given ratio.

Let P,
divides

Let

N.

Q

be the given points, and R the point which
the given ratio m l ra 2
be (a?, y, *).
fo,
^), $ be (a?2 ya *a ), and

PQ in
Pbe

:

^

.

,

^

,

Draw PL, QM, EN parallel to OZ meeting .XO F in L, M,
Then the points P, ft ^, Z, M,
are clearly all in one

N

LM will

and a line through P parallel to
plane, and will therefore meet QM, EN,

be in that
in the points K,

plane,

suppose.

Then

PR

HU

m
m
+
PQ

-

77^=

l

But Z/P = z

1

/

H

!
.

9?^

2

~&

*

-*,

=
tt^

2-=

+ m,

*
?;

oc

Similarly

and y

When Py

is

=

^

=

~

divided externally,

m

z

is

negative.

ll


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CO-ORDINATES.

4

The most useful case is where the line
co-ordinates of the point of bisection are
The above

We

may

is

bisected

:

the

whatever the angles between

results are true

the co-ordinate axes

PQ

be.

shall in future consider the axes to be rectangular in
when the contrary is expressly stated.

all cases except

To express

5.

the distance between two points in terms

of

their co-ordinates.

Let

Pbe

the point (xiy y lt zj and Q the point (a?a y2 s ).
2
and Q planes parallel to the co-ordinate
planes, forming a parallelepiped whose diagonal is PQ.

Draw through

,

P

,

z

Let the edges PL, LK, KQ be parallel respectively to
OY OZ+ Then since PL is perpendicular to the plane

OX,

QKL,

}

the angle

Now PL

is

from the plane

PLQ is

a right angle,

the difference of the distances of
= x^
YOZ, so that we have

PL

P

and Q
x^ and

LK and KQ.
Hence PQ 1 = (, - *,) + (y, + (,, - ztf
(i).
The distance of P from the origin can be obtained from
the above by putting #2 = 0, y^ = 0, z^ = 0. The result is
similarly for

y,)"

<

+

2
i/ 1

+*

2
1

(ii).


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CO-ORDINATES.
Ex.

1

.

The co-ordinates of the

points are (x lf y v zj,
and i(*i + * 2 + 2a).

(a?2 , t/ 2 ,

z,,),

5

centre of gravity of the triangle whose angular
y s 2 8 ) are $ (x l + x^ + x^, i (yi + y* + y 8 ),

(x9 ,

,

Ex. 2. Shew that the three lines joining the middle points of opposite
Shew also that this point is on the
edges of a tetrahedron meet in a point.
line joining any angular point to the centre of gravity of the opposite face,
and divides that line in the ratio of 3 1.
:

Ex.

Find the locus
and (3, 2, - 1).

of points

3.

(1, 2, 3)

Ex. 4. Shew that the point (,
passes through the four points (1, 2,

which are equidistant from the points
Ans. x -22=0.

0, |) is

the centre of the sphere which

3), (3, 2,

-!),(-

1, 1, 2)

and

(1,

-

1,

-

2).

Let a, 0, 7 be the angles which the line PQ makes
6.
with lines through
parallel to the axes of co-ordinates.

Then, since in the figure to Art. 5 the angles PLQ, PMQ,
are right angles, we have
vv

P

PNQ

/

PQ cos a = PL,

PQcos/3 = PM,
and

PQcosy=PN.

Square and add, then

PQ

2

2

{cos a

+

+ cos 7 = PU + PM* + PN = PQ

cos a + cos /? 4- cosfy = 1.

cos

2

2

2

/8

2

Hence

2

}

.

2

The cosines of the angles which a straight line makes
with the positive directions of the co-ordinate axes are called
its direction-cosines, and we shall in future denote these
cosines

by the

letters I, m, n.
the above we see that any three direction-cosines
2
are connected by the relation f +
+ n2 = 1. If the
direction-cosines of
be I, m, n, it is easily seen that those
of QP will be
n and it is immaterial whether we
I,
m,
consider I, m, n, or the same quantities with all the signs

From

m

PQ

;

changed, as direction-cosines.
If

we know

cosines of
cosines.

that a,

some

For we have
.

6,

c are proportional to the direction-

we can

line,

-

a

at once find those direction-

=T=
b

c

hence each

;

1

V(^~+

6*

+

2

c )

is

equal to

a

..,_
*

V(a*

+ &* + (?)