Solid Analytical Geometry And Determinants

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SOLID ANALYTICAL GEOMETRY
AND DETERMINANTS

BY THE SAME AUTHOR

Plane Trigonometry

This book emphasizes the importance of the
function concept for elementary trigonometry.
Cloth; 6 by 9 inches; 110 pages; 58 figures.

PUBLISHED BY

JOHN WILEY & SONS, INC.
NEW YORK

SOLID 'ANALYTICAL GEOMETRY
AND DETERMINANTS

Professor of Mathematics , Swarthmore College

NEW YORK

JOHN WILEY & SONS, INC.

LONDON: CHAPMAN & HALL, LIMITED

1930

COPYRIGHT, 1930,
BY ARNOLD DRESDEN

Printed in U. S. A.

Printing Composition and Plates Binding

F. H. CELSON CO. TECHNICAL COMPOSITION CO. STANHOPE BINDEKY

BOSTON CAMBRIDGE BOSTON

PREFACE

A three-hour course in Solid Analytical Geometry is offered for
students in the junior year in many of the colleges and universities
in this country. Though books ,on Plane Analytical Geometry
frequently devote some chapters ^to the geometry of a space of
three dimensions, the material covered in these chapters is, with
few exceptions, not intended to do more than v prt)vide a general
introduction to the subject, so as to enable students tb understand
the references to it which have to be made in courses on the cll-
culus. But it rarely goes far enough to acquaint them with the,
more interesting and valuable methods of this field.

For many years, while teaching this subject at the University
of Wisconsin and at Swarthmore College, it has seemed to the
author that, in the study of Solid Analytical Geometry, the young
student of mathematics can find an excellent opportunity for an
introduction to methods and principles which have an important
part in various fields of advanced mathematics. Among these are
the methods based on the theory of determinants and on the con-
cept of the rank of a matrix. In more advanced mathematical
subjects these theories are developed and used with a great meas-
ure of generality; they find relatively simple application in the
subject to which this book is devoted. Unfortunately, there are
not readily accessible for use in undergraduate classes treatments
of these theories which are on the one hand adequate for the uses
to be made of them here and on the other hand not too advanced
to be available for an introductory purpose.

For these reasons, the first chapter of this book presents an
exposition of some of the properties of determinants and matrices,
followed in Chapter II by a treatment of systems of linear Aqua-
tions. The latter subject is not carried so far as to include the
most general case, but, it is hoped, far enough to serve in the later
chapters. The repeated applications of the results of the first
two chapters which are made in the subsequent work (as evidenced
by the numerous references to Chapters I and II) should indicate
their importance. With the basis thus provided it becomes pos-

iii

IV PREFACE

sible to deal with the geometrical questions of the later chapters
in a way which lends itself readily to extension to problems of a
more general character. Thus the discussion of the theory of
quadric surfaces in Chapters VII and VIII may be made to serve
as an introduction to the theory of quadratic forms in n variables.
Having studied these chapters, the reader should be able to proceed
to the well-known excellent books, by Bocher and by Dickson,
nientioned in the introductory paragraph of Chapter I. To these
books the author owes a large debt. The spirit which pervades
them has been a guide for him; and it would be a source of grati-
fic,tion if the present book were to lead its readers to more extended
sftidy of the subjects treated by these authors.

Chapters III to X deal with the loci of equations of the first
and second degree in three variables from the point of view of real,
metric geometry. Elements at infinity and complex elements
are considered as non-existent. This point of view has been
taken because, in the author's judgment, a satisfactory treatment
of the questions which arise through the inclusion of such elements
can only be made after the explicit adoption of adequate bases on
which projective geometry and complex geometry can be erected.
Since this would involve quite a different orientation than the
scope of the present book permits, it was deemed better to proceed
on the implicit assumptions of real metric geometry on which the
student's earlier work in geometry may be supposed to have been
founded.

It is only from this point of view that the detailed classification
of quadric surfaces made in Chapter VIII can be justified; and
only when it is kept in mind, can statements like the one in The-
orem 12 on page 175, to the effect that there are no lines on a
non-singular quadric with negative discriminant, be explained.
It must however be recognized that there occur instances in which,,
if only for the sake of emphasis, infinite elements and complex
elements must be mentioned.

In subject matter the last eight chapters follow largely the tra-
ditional content of introductory courses in Solid Analytical Geom-
etry. The treatment introduces modes of procedure and devices
which have been developed in the course of the many years during
which^the author has taught the subject and which have probably
also been used by other teachers.

PREFACE

It will usually be most satisfactory to work through the greater
part of Chapters I and II before Chapter III is started. But it
may be found desirable, as has frequently been the author's prac-
tice, to begin by spending one hour a week on the geometric part
of the book, beginning with Chapter III, while the remainder of
the time is given to the algebraic work of the first two chapters.

The exercises form an integral part of the course which this
book presents. The author has not hesitated therefore to refer,
in a number of cases (see e.g., pages 116, 222, and 228), to results
established in an exercise. A good many of the problems serve
no other purpose than that of illustrating the material in the text.
But there are other problems, and these are doubtless the nA>re
valuable ones, which require a certain amount of original thinking.

Thanks are due to other authors besides those which were men-
tioned in a preceding paragraph; but the uses which I have made
of their work are too indefinite in character to make explicit
references possible. It is a pleasure to acknowledge my indebt-
edness to the mathematical library of Brown University for allow-
ing photographs to be made of the models in its possession for
the illustrations of quadric surfaces which appear in this book.
Furthermore I wish to express my thanks to Mr. George B.
Hoaclley, a senior student in Swarthmore College, who has drawn
the figures, and to Miss Alice M. Rogers, research assistant at
the Sproul Observatory of Swarthmore College, who has given
valuable aid in the reading of proofs. This preface would not be
complete without a word of appreciation for the unfailing courtesy
and patience which the publishers and the printers have contrib-
uted to the production of this book.

ARNOLD DRESDEN.

SWAllTHMORE COLLEGE,

February, 1930.

CONTENTS

CHAPTER I

DETERMINANTS AND MATRICES
SECTION

1. Introduction 1

2. Definitions and notations 1

3 The value of a determinant 3

4. Exlrcises

5. Elementary theorems ' {

6. Exercises 10

T. Minors and cof actors 11

8. Exercises 15

0. Matrices. Rank of a matrix 16

10. Complementary minors. Elementary transformation of matrices . 17

11. Exercises 19

12. The Laplace development of a determinant 20

13. Exercises 24

14. The product of two determinants 25

15. Exercises 28

16. The adjoint of a determinant 28

17. The derivative of a determinant 31

18. Exercises 32

19. Miscellaneous exercises 33

CHAPTER II
LINEAR EQUATIONS

20. Definition and notation 35

21. The system of n linear non-homogeneous equations in n variables . . 36

22. The system of n linear homogeneous equations in n variables .... 38

23. The system of n + 1 linear non-homogeneous equations in n i

variables 38

24. Exercises 40

25. The system of n 1 linear homogeneous equations in n variables . . 41

26. The adjoint of a vanishing determinant. Symmetric determi-

nants 42

27. The system of n linear non-homogeneous equations in n variables,

continued 44

28. Exercises 47

vii

Vlll CONTENTS

CHAPTER III

POINTS AND LINES

SECTION PAGE

29. The cartesian coordinates of a point in three-space 49

30. The coordinate parallelepiped of a point 51

81. Exercises 52

32. Two points 53

33. Direction cosines of a line 55

J34. Three collinear points 57

35. Exercises 60

36. The angle between two lines. The projection method 61

37. Exercises 65

38. Miscellaneous exercises 65

CHAPTER IV
PLANES AND LINES

39. Surfaces and curves 67

40. Cylindrical surfaces. Systems of planes 68

41. The linear equation ax + by -\- cz -\- d = 71

42. Exercises 74

43. The distance from a plane to a point 75

44. The normal form of the equation of a plane 77

45. Exercises 80

46. Two planes 81

47. The line 83

48. Exercises 89

49. The pencil of planes. The bundle of planes 90

50. Exercises 94

51. Three planes. A plane and a line 95

52. The plane and the line, continued 97

53. Exercises 100

54. Four planes. Two lines 101

55. Exercises 105

56. Miscellaneous exercises 105

CHAPTER V

OTHER COORDINATE SYSTEMS

57. Spherical coordinates 108

58. Cylindrical coordinates Ill

59* Exercises 112

60. Oblique cartesian coordinates 113

61. Translation of axes 114

62. Transformation from oblique to rectangular axes 115

63. Rotation of axes 118

CONTENTS ix

SECTION PAGE

64. Exercises 121

65. Rotation of axes, continued 122

66. Linear transformation. Plane sections of a surface 125

67. Exercises 129

CHAPTER VI
GENERAL PROPERTIES OF SURFACES AND CURVES

68. Surfaces of revolution 131

69. Exercises 135

70. The shape of a surface determined from its equation. Contour

lines 1 35

71. Some facts from Plane Analytical Geometry 140

72. Some special surfaces 141

73. Exercises 148

74. The intersections of a surface and a line 149

75. Digression on Taylor's theorem 151

76. The intersections of a surface and a line, continued 152

77. Tangent lines and tangent planes. Normals 154

78. Exercises 156

79. The shape of a curve in space ^-r*, 157

CHAPTER VII
QUADRIC SURFACES, GENERAL PROPERTIES

80. The quadric surface and the line , 159

81. Tangent line; tangent plane; normal; polar plane. 161

82. Polar plane and pole. Tangent cone 165

83. Exercises 169

84. Ruled quadric surfaces ( 170

85. The centers and vertices of quadric surfaces 176

86. Exercises .-.:,;, ' 182

87. The asymptotic cone "T'^* wil ^r? [ 182

88. The diametral planes and the principal planes of a Quadric surface . 185

89. The discriminating equation . . 190

90. Principal planes and principal directions . . 194

91. Exercises 195

CHAPTER VIII
CLASSIFICATION OF QUADRIC SURFACES

92. Invariants 197

93. Invariants of a quadric surface with respect to rotation and trans-

lation of axes 199

94. Invariance of the discriminant of a quadric surface with resect to 9

rotation 202

X CONTENTS

SECTION PAGE

95* Exercises 205

96. Two planes 206

97. Translation of axes to the center of a quadric surface 210

98. Rotation of axes to the principal directions of a quadric surface.. . 211

99. Classification of quadric surfaces the non-singular cases 214

100. Classification of quadric surfaces the non-degenerate singular

cases 220

101. Classification of quadric surfaces the degenerate cases 227

102. The classification of quadric surfaces summary and geometric

characterization 229

103. Exercises 234

CHAPTER IX
QUADRIC SURFACES, SPECIAL PROPERTIES AND METHODS

104. The reguli on the hyperboloid of one sheet 235

105. Reguli on the hyperboloid of one sheet, continued 237

106. The reguli on the hyperbolic paraboloid 242

107. The straight lines on the singular, non-degenerate quadrics 243

108. Exercises 243

109. Circles on quadric surfaces, the general method 244

1 10. Circles on quadric surfaces, continued 251

111. Exercises 252

112. Tangent planes parallel to a given plane. The umbilics of a quad-

ric surface 253

113. The umbilics of a quadric surface, continued 263

114. Exercises 266

CHAPTER X
PROPERTIES OP CENTRAL QUADRIC SURFACES

115. Conjugate diameters and conjugate diametral planes of central

quadrics. Enveloping cylinder 268

116. Exercises 271

117. Conjugate diameters of the ellipsoid 271

118. Exercises 276

119. Linear families of quadrics 277

120. Focal curves and directrix cylinders of central quadrics 281

121. Focal conies and directrix cylinders, continued 285

122. Exercises 287

123. Confocal quadric surfaces. Elliptic coordinates 288

Appendix 296

Index 303

SOLID ANALYTICAL GEOMETEY
AND DETERMINANTS

CHAPTER I
DETERMINANTS AND MATRICES

1. Introduction. The Study of Solid Analytical Geometry, to
which this book is chiefly devoted, leads repeatedly to the problem
of solving systems of linear equations in several variables, in which
the number of variables may be less than, equal to, or greater than
the number of equations. The methods for dealing with this
problem which are found in books on elementary algebra and in
college algebra texts are not sufficiently general in character to
suit the needs of our subject. A more complete treatment of the
theory of determinants than is found in such books becomes neces-
sary. For this reason, and also on account of the manifold uses
of determinants in various fields of mathematics, finally because
of the great intrinsic interest of the subject, the first chapter of
this book will be devoted to an introduction to the theory of de-
terminants and to a few ideas concerning matrices. This will be
followed in Chapter II by a treatment of systems of linear equa-
tions. In this treatment the problem is not considered in its
complete generality, but in a form sufficiently inclusive to suit the
needs of the later chapters in this book. The reader who desires
to pursue this subject further can do so in two excellent books,
dealing with advanced topics in algebra, viz., B6cher, Introduction
to Higher Algebra, and Dickson, Modern Algebraic Theories.

2. Definitions and Notations.

DEFINITION I. A determinant is a square array of numbers to which
a single number, called the value of the determinant, is attached by
the method stated in Definition V.

Vertical bars are placed on either side of the array. The symbol
so obtained is used also to designate the number that is to be asso-

1

DETERMINANTS AND MATRICES

elated with the array. For example, the symbols

4-12

and

3-5
1 4

with each of which a single number is associated in accordance
with Definition V are determinants. The same symbols are used
to designate the values of these determinants.

DEFINITION II. The numbers In the square array which constitutes
the determinant are called its elements; the horizontal lines in the
array are called rows, the vertical lines columns; the diagonal which
runs from upper left to lower right is called the principal diagonal,
the other diagonal Is called the secondary diagonal.

DEFINITION III. The order of a determinant is the number of ele-
ments in any one row or column.

Remark. A determinant of order n is made up of n 2 elements.

Notations, In the general form of a determinant every ele-
ment has affixed to it two indices; the first of these designates
the row in which the element stands and is called the row index,
the second designates the column of the element and is called the
column index. The general forms of the determinants of the third
and fourth order will therefore be as follows :

#31 032

(2)

#11 #12 #13 #14

#21 #22 #23 #24

#31 #32 #33 #34

#41 #42 &43 #44

A determinant of the nth order, where n designates a positive
integer, in the most general form will appear as follows:

#11 #12
/Q\ #21 #22 #2 W

# n l

#

These are rather lengthy symbols; they make it desirable to
have more compact symbols which can be used when it is not
necessary to designate the elements of the determinant explicitly.
In such cases we frequently designate a determinant by merely
writing the elements of the principal diagonal. Thus the
symbols |#ii#22#33|, |#n#22#33#44| and \ana^ . . . a nn \ are used

THE VALUE OF A DETERMINANT 3

to designate the cleterminants (1), (2), and (3) respectively. A
still shorter way of representing the general determinant consists
in writing a single element with literal indices and indicating the
values which these indices a*e to take. In this notation the deter-
minants (1), (2), and (3) t&mld be represented by the notations
K>'l, i,i = 1, 2, 3; |oy|, i,j = 1, 2, 3, 4; and |a</|, i, j = 1, 2,
. . . , n respectively.
3. The Value of a Determinant.

DEFINITION IV. Whenever in a set of numbers, consisting: of Inte-
gers from 1 upward in arbitrary order, a larger integer precedes a
smaller one, we say that there is an inversion.

For example, in the row of indices

463251

there are 11 inversions: 3 inversions because the number 4 is
followed by the smaller numbers 3, 2 and 1 ; 4 inversions because
6 is followed by 3, 2, 5 and 1 ; 2 inversions because 3 precedes 2
and 1; 1 inversion because 2 precedes 1, and 1 inversion because
5 precedes 1.

DEFINITION V. The value of a determinant is the algebraic sum of
all possible products obtainable by taking one and only one factor
from each row and from each column, preceded by the plus or minus
signs, according as the number of inversions of the column indices
of the factors of a product are even or odd, when the row indices are
in the natural order 1, 2, 3, etc.

The indicated sum of these products is called the expansion
of the determinant.

Remark 1. We must remember that, although the same sym-
bol is used for a determinant and for the value of this determinant,
the concepts " determinant " and "value of a determinant" are
distinct concepts; the latter is a number; the former is a square
array of numbers with which a number is associated according to
Definition V.

Remark 2. The expansion of the general determinant of the
nth order (3) will therefore consist of terms of the form a^o^ . . .
anc n , in which Ci c 2 . . . c* is some permutation of the set of num-
bers 1,2,. . . n, this term will be preceded by the plus or minus
sign, according as the number of inversions of the set Ci C2 . . . c n
is even or odd. Since the number of permutations of the set of

DETERMINANTS AND MATRICES

integers 1,2,. . . n is nl, it follows that the expansion of the gen-
eral determinant (3) consists of n\ terms.
Examples.

1. The value of the second order determinant

011

021

012

is the algebraic

sum of two terms; each term must contain two factors, one and only one
from each row and from each column. If we take a\\ from the first row, we
must take 022 from the second; thus we get the product 011022. If we take 012
from the first row, we must take a>2i from the second, so that we obtain the
product 012021. In both these products the row indices are in the natural
order 1,2. In the first product the set of column indices is 1,2, which has no
inversions; the column indices in the second product form the set 2,1, which
has 1 inversion. Consequently the product n 022 is preceded by the plus
sign, and the product 012021 is preceded by the minus sign. Therefore

(in

a u

022

012021-

2. The value of the third order determinant (1) is obtained as the alge-
braic sum of 6 products; if we write the factors of each product in the order
of their row indices these products arc 011022033, 0,110290,32, 012021033, 012023031,
013021032, aisaajsasi. The numbers of inversions in the column indices of these
products are 0,1,1,2,2 and 3 respectively. We conclude that

011023032 ~ 012021033 + 0120*3031

011022033 I = 01102233

3. To determine the value of the determinant

013021^32 ~ 013022031.

1
3

G
10

1

4

10

20

we write

down every possible product of four factors, in each of which there is one and
only one element from each row and from each column. We order the factors
in each product according to the rows from which they are taken, and indicate
below them the columns to which they belong. The number of inversions
in the column indices then determines the sign to be prefixed to each product,
in accordance with the rule laid down in Definition V. Thus we obtain the
following expansion:

-4- 1

2-

6-20-1

2

10-

10

- 1

3-

3 20 + 1 3 10 4 + 1 -

4-

3- 10

i

2

3 4

1

2

4

3

1

3

24 1342 1

4

2 -3

1

. 4 .

6-4 -

1

1

6-

20

+ 1

1

10 10 + 1 3 1 20 - 1

10-1

i

4

3 2

2

1

3

4

2

1

43 2314 2

3

4 1

- 1

. 4 .

1 10 -

f

1 -

4 - 6

- 1

-f 1

1

- 3 20 - 1 I 10 - 4 - 1 -

2-

1-20

2

4

1 3

2

4 3

1

* 3

1

24 3142 3

2

1 4

-f 1

2

10-1

4-

1

4 J

t 1

4

3 1 - 1 1 3 - 10 + 1

1

6-4

3

2

4 1

3

4 ]

I 2 3

4

21 4123 4

1

3 2

-f 1

-2-

1 10 -

1

2

6- 1

1-3

1 -

4 + 1 3 3 1. It follows

that the

4

2

1 3

4

2

3 1

4 3

1

2 4321

* The symbol nl, called "n factorial " is an abbreviation for the continued
product 1 2 3 . . . (n 1) n.

ELEMENTARY THEOREMS

value of the given* determinant is equal to 240 - 200 - 180 + 120 + 120

- 96 - 120 -f 100 + 60 - 30 - 40 + 24 + 60 - 40 - 40 + 20 + 16 - 12

- 30 + 24 + 20 - 12 - 12 + 9 = 813 - 812 = + 1.

4. Exercises.

1. Determine the number of inversions in each of the following sequences of
numbers:

(a) 5 2 4 7 3 I G
(6) 3 6 1 5 4 7 2
(c) 7645321

2. How many terms are there in the expansion of a determinant of the 4th
order? Of the 5th order? Of the 6th order?

3. Prove that the number of inversions in a row of numbers is not changed
if all the numbers are increased or decreased by the same amount.

4. Show that if a row of integers is divided into two sections, such that
all the numbers in the left section are less than any number in the right section,
then the number of inversions in the original row is equal to the sum of the
number of inversions in the left part and that in the right part.

6. Generalize the theorem of the preceding exercise so as to cover the case
in which a row of integers is divided into more than two sections.

6. Determine the values of each of the following determinants by the method
explained and illustrated in Section 3:

(a)

4-1 2

3 -5

-2 1 4

(6)

-4
3

-3 2
1 -2
5 -2

4 -6 3

-6 1 2

3 2 5

7. Determine also the values of the following determinants:

(a)

1121

2-1 3-1

2 2 2 10

23-4
3210

; (6)

4 -2 -1 3
2-1-4 4

; (c)

0012
34-32

1-11 1

10 -5 -6 10

1-245

23 1-1

a 1 1 1

2003

1 a 1 1

(d)

4101

) \")

1 1 a 1

-12-2 1

Ilia

6. Elementary Theorems. The determination of the value of
a determinant by means of Definition V is quite laborious even
for a determinant of order 4, as will, have been discovered by the
reader who has done all the parts of Exercise 7. For determinants
whose order exceeds 4, this method becomes quite useless. Never-
theless it is important for the reader to do the exercises in the pre-
ceding set so that he may become thoroughly familiar with the
content of Definition V. In the next few sections we shall deriVe

6 DETERMINANTS AND MATRICES

from this definition a chain of theorems which will supply us with
the more useful methods for evaluating a determinant which will
be employed in our further work.

THEOREM 1. The Interchange of two adjacent numbers in a row of
integers which is an arrangement of the integers from 1 to n either
increases or decreases the number of inversions of the row by one.

Proof. Let p and q be adjacent. We have then to compare the
number of inversions of the sets

(1) ... pq . . . and (2) ... qp . . .

Let us suppose p < q. The inversions which arise in (1) from
any number preceding p or following q will also occur in (2), for
all such numbers are followed by the same numbers in (2) as in (1).
Furthermore p is followed in (2) by the same numbers which
follow it in (1), except q] but since q > p this change does not
affect the number of inversions, so that the number of inversions
due to p is the same in (2) as it was in (1). Finally q is followed
in (2) by the same numbers as in (1) and moreover by p, which is
less than g, so that the number of inversions due to q is one more in
(2) than it was in (1). We conclude that there is a gain of one
inversion in passing from (1) to (2), and consequently a loss of one
inversion in passing from (2) to (1). Since in (2) the larger one
of the two indices that are interchanged precedes the smaller one,
whereas in (1) the condition is the opposite one, the proof of the
theorem is complete.

THEOREM 2. The interchange of any two numbers in a row of in-
tegers which is an arrangement of the integers from 1 to n changes
the number of inversions in the row by an odd number.

Proof. Let there be k numbers between p and q and let us
compare the two arrangements

k k

(3) . . . p ... q . . . and (4) . . . q ... p . . . .

By interchanging p successively with each of the k numbers
which lie between p and q in (3), we obtain an arrangement which
may be represented by

k

ELEMENTARY THEOREMS 7

interchanging now q with p and with the k numbers preceding it,
we obtain the arrangement (4). Thus we obtain (4) from (3)
as the last of 2 k + 1 arrangements, each one of which is ob-
tained from the preceding one by the interchange of two adjacent
numbers. Hence the number of inversions in, (4) is obtained, in
virtue of Theorem 1, from that of (3) by 2 k + 1 changes of one,
some of which are losses and the others gains. Since 2 k + 1
is odd, no matter what integer k is, the net result will be the loss
or the gain of an odd number of inversions. For, if there are I
losses and g gains, and I > g, the net result will be I g = li losses;
but / + g = 2k + 1 and therefore 2Z = Zi + 2fc + l, from which
we conclude that Zi is odd. And if the number of gains exceeds
the number of losses it is shown in exactly similar fashion that
the net result consists of an odd number of gains.

Remark. By allowing the change in the number of inversions
to take negative as well as positive values we can say that every
interchange of two numbers in a row of integers which is an ar-
rangement of the integers from 1 to n changes the number of in-
versions by an odd number.

THEOREM 3. If an arrangement of the integers 1 . . . n can be
obtained from the natural order, or can be restored to the natural
order, by an even (odd) number of interchanges of a pah* of numbers,
it will have an even (odd) number of inversions.

Proof. The natural order presents no inversions and every inter-
change of a pair of numbers changes the number of inversions by
an odd number. The truth of the theorem follows therefore
from the fact that the sum of an even number of odd numbers is
even, while the sum of an odd number of odd numbers is odd.

THEOREM 4. The value of a determinant is not changed if the
columns are made into rows and the rows into columns.

Proof. Let the given determinant be |a#|, i, j = 1, 2, . . . , 77;
and let the determinant obtained by making the rows into col-
umns and the columns into rows be designated by |6#|, i,j = 1, 2,
. . . , n. Then &# = a#. An arbitrary term in the development
of |o#| has the form a^o^ . . . a^ n in which

(5) Ci C2 . . . Cn

is some arrangement of the integers 1, 2, . . . , n; and the si^n
of this term depends on the number of inversions in the row of

8 DETERMINANTS AND MATRICES

numbers (5) (see Remark 2 on page 3). Moreover, this term is

cqualto (6) Ms- &%

which is a term in the development of the determinant |&#|,
except possibly for sign. In order to determine the sign of (6)
we have to rearrange its factors so as to put the row indices in
natural order; in doing this we shall put the column indices in
irregular order and the sign of (6) will depend upon the number
of inversions in that order. Now this order is obtained from the
natural order by as many interchanges of pairs of numbers as it
takes to restore the arrangement (5) to the natural order; conse-
quently it will present an even or odd number of inversions ac-
cording as the number of inversions in (5) is even or odd. Con-
sequently the term (6) will appear in |6#| with the same sign that
the term a^c^ . . . a nCn had in |a#[. But this last term was an
arbitrary term in |o#|; therefore every term of the development
of \0ij\ occurs in the development of |6#| and with the same sign.
The same argument shows that the terms of |6#| are all reproduced,
in magnitude and in sign, in the development of |ay|. We have
therefore proved that |a,y| = |b#|.

COROLLARY. If a theorem has been proved concerning the rows of a
general determinant, we may conclude at once that a similar theorem
holds for the columns; and vice versa.

THEOREM 5. The interchange of two columns (rows) of a deter-
minant causes the value of the determinant to change sign.

Proof. Let the given determinant be |o#| and let the deter-
minant obtained from it by interchanging the columns whose
indices are c\ and C2 be designated by |6#|. Then from every term
of the former determinant, we can obtain one of the latter by
writing &'s in place of a's and interchanging the column indices
d and C2. It follows from Theorem 2 and Definition V that these
two terms will be opposite in sign while equal in numerical value.
Since moreover every term of |6#| can be obtained in this manner,
our theorem has been proved; the alternate form, indicated in the
parentheses, follows by application of the Corollary of Theorem 4.

THEOREM 6. If all the elements of a row (column) are multiplied
by the same number, the value of the determinant is multiplied by
that number.

This theorem is an immediate consequence of Definition V.

ELEMENTARY THEOREMS 9

THEOREM 7. If * two columns (rows) of a determinant are propor-
tional, the value of the determinant is zero.

Proof. Let us suppose first that the corresponding elements of
the columns whose indices are Ci and C2, are equal. Let the value
of the given determinant be A, and that of the determinant ob-
tained from it by interchanging the columns of indices Ci and c 2
be B. We conclude then from Theorem 5 that B = A; and
from the fact that the two columns which have been interchanged
are identical, that B = A. Therefore A = A and hence A = 0.
If now two columns of a determinant are proportional, its value
is equal, on account of Theorem 6, to a factor of proportionality
multiplied by the value of a determinant in which two columns
are identical; its value is therefore also equal to zero.

THEOREM 8. If the elements of a column (row) of a determinant are
binomials, its value is equal to the sum of the values of the two de-
terminants which agree with the given determinant in every element
except that the particular column (row) concerned consists in one of
them of the first terms of the binomials and in the other one of the
second terms.

This theorem is also an immediate consequence of Definition V;
the proof is left to the reader (see Section 6). For a determinant
of the third order the theorem asserts among other facts that

an i2 + ki 013
(hi 022 + & 2 023

31 032 + &3 033

011 012 013
021 022 023
031 032 033

0u ki ais

021 & 2 023
031 &3 033

THEOREM 9. To the elements of any row (column) of a determinant
may be added arbitrary multiples of the corresponding elements of any
other row (column) without affecting the value of the determinant.

Proof. It is a consequence of Theorem 8 that the value of the
new determinant is equal to that of the given determinant plus
the value of a determinant in which two rows (columns) are pro-
portional. From this remark the present theorem follows by use
of Theorem 7.

Remark 1. Theorems 7 and 9 are the first objectives of the
chain of theorems we are developing. The latter enables us to
derive from a given determinant another one which is equal to it
in value but in which all the elements but one of some one row

10

DETERMINANTS AND MATRICES

or column are equal to zero; such a change materially reduces
the labor involved in the evaluation of a determinant.

Remark 2. To abbreviate our terminology we shall speak of the
" addition of one row or column of a determinant to another row or
column " with the meaning " addition of the elements of one row
(column) to the corresponding elements of another row (column)/'

Examples.

4 -7
1. To calculate thf* . the determinant

1
-3

3

-2
2

we use

Theorem 9 as lc j o row 1 we add row 2 multiplied by 4; and to row

3 we add row 2 mui^plied by 3. Thus we find that A is equal to the value of
-27 11 ,

This determinant can readily be evalu-

the determinant 1 52
21 -4

ated by means of Definition V; in this way we find that A = (27
+ 11 - 1 21 = 123.

1--4)

2. To evaluate the determinant

5

-7
2

3

6

8

-1

2

-I
4
3

9

10

5

4

to column 1;
6

we, add column 3

obtain

4

-3
5

and we add column 2, multiplied by 1, to column 4; thus we
326
6-14

8
-1

4 -3
3 5

in which there are two equal columns. By

Theorem 7 the value of this determinant is zero; hence it follows from Theorem
9 that the value of the given determinant is also zero.

6. Exercises.

1. Show by an actual count of the inversions that the number of inversions
is changed by an odd number when the numbers 2 and 8 are interchanged in
the row 5 2 4 7 3 8 6 1.

2. Prove Theorem 1 for the case p > q without assuming it for the case

P < Q-

3. Write out a detailed proof of the Corollary to Theorem 4.

4. Write out a detailed proof of Theorem 6.
6. Also for Theorem 8.

6. Illustrate Theorems 4, 5, 7, and 9 by means of determinants of the 3rd
and 4th orders.

7. Evaluate each of the following determinants:

(a)

-523

264

-1 14 11

1-1 2-2
-32-23
-131-3

21-12

; (c)

2-231

-3 4 17 -2

5 -6 -9 3

7 8 25 -4

MINORS AND COFACTORS

11

8. Calculate the ^alue of each of the following determinants:

(a)

4323

5-210

-1431

214-3

(6)

3 -1 -2

3 -2 1

2 -2 1
-I 2

3 1

1
-4

7. Minors and Cofactors. The following theorems will enable
us to reduce still further the arithmetical work involved in the
evaluation of a determinant. They will moreover furnish a basis
for the application of determinants to the solution of systems of
linear equations.

THEOREM 10. The determinant obtained from a given determinant
by shifting rows and columns in such a way as to bring a certain ele-
ment in the upper left-hand corner, without changing the relative
position of the rows and columns which do not contain this element,
has a value equal to that of the given determinant or to its negative
according as the sum of the row and column indices of this element
is even or odd.

Proof. Let us consider first the determinant 1 011022033044 1
let us call its value A. The upper left-hand corner of the deter-
minant, now occupied by an will be called the " leading position. "
To form a new determinant in which 34 is in the leading position,
while the relative order of the 1st, 2nd, and 4th rows, and also of
the 1st, 2nd, and 3rd columns, remains unchanged, we interchange
the 3rd row successively with the 2nd and 1st rows; in virtue of
Theorem 5, this operation causes the value of the determinant to
change its sign twice, so that we can write

031 032 033 034

a\i 012 ai3 0i4

021 022 023 024
041 042 043 044

Now we interchange the 4th column successively with the 3rd,
2nd, and 1st columns; this leads to the desired result at the cost
of three changes of sign. Therefore, the new determinant

034 031 032 033

014 011 012 013

024 021 22 023

044 4 i 042 043

12 DETERMINANTS AND MATRICES

in which a^ occupies the leading position while the rows and col-
umns which do not contain a^ have the same relative order as in
the original determinant, has a value equal to A.

It should now be easy to understand the proof for the general
case. To bring the element a# of the nth order determinant |a#|
into the leading position without affecting the relative order of
the rows and columns which do not contain this element, we inter-
change the ith row successively with the (i l)th, (z 2)th, . . . ,
1st rows; then we interchange the jth column successively with
the (j l)th, (j 2)th, . . . , 1st columns. This is accomplished
by means of i 1 + j 1 = i + j 2 interchanges and there-
fore accompanied by i + j 2 changes of sign in the value
of the determinant. Consequently the determinant which we
obtain finally will have a value equal to that of \aij\ or to its nega-
tive according as i + j is even or odd ; this proves the theorem.

DEFINITION VI. The minor of an element of a determinant of order
n is the determinant of order n 1 obtained by deleting the row and
column in which this element stands.

DEFINITION VII. The cof actor of an element of a determinant is
equal to its minor or to the negative of its minor according as the sum
of the row and column indices of the element is even or odd.

Notation. The value of the cofactor of the element a% is desig-
nated by Atj.

THEOREM 11. All the terms in the expansion of the determinant
\aij\\ which contain a particular element as a factor are obtained, in
magnitude and in sign, by multiplying that element by its cofactor.

Proof. For the element an this theorem is an immediate con-
sequence of Definition V. Let us again denote the value of
| a^ | and also the determinant itself by A. To determine the sum
of all the terms in the development of A which contain the element
ay as a factor, we consider the determinant A' y obtained from A,
as in Theorem 10, by putting a# in the leading position without
affecting the relation of the rows and columns which do not con-
tain this element. Then A' = (- 1)'+M and hence A = (l)*+*A'.
Since a# occupies the leading position in A' the sum of the terms
in A' which contain a# as a factor is obtained by multiplying a,y
by its cofactor in A'. But the cofactor of a# in A f is equal to its
minor in A'; and the minor of ay in A f is the same as the minor of

MINORS AND COFACTORS 13

this element in A', because the relative order of the rows and col-
umns which do not contain the element a# has not been changed in
the transition from A to A f . Therefore the sum of the terms in
A f which contain the factor a# is obtained by multiplying this
element by its minor in A. Moreover the terms in A which con-
tain the factor a# are obtained by multiplying those in A f by
( I)*' 4 "'". Therefore the sum of the terms in A which contain the
factor ay is equal to ( 1 )*+*" X a# X the minor of a# in A = a,-.
X the cof actor of a# in A =

THEOREM 12. The value of a determinant is equal to the algebraic
sum of the products obtained by multiplying the elements of any
column (row) by then* cofactors.

Proof. In every term of the expansion of a determinant, there
is one and only one factor from each column (row). If therefore
we select a column (row) arbitrarily and take the sum of all the
terms which contain any one of its elements as a factor, we shall
obtain the value of the determinant. Hence the present theorem
is an immediate consequence of Theorem 11.

THEOREM 13. The algebraic sum of the products of the elements of
any column (row) by the cofactors of the corresponding elements of
another column (row) is equal to zero.

Proof. We observe that the cofactors of the elements of any
column (row) are not affected by changes made in that column
(row). The cofactors of the elements ai Cj , a^, . . . , a^ in |a#|
are therefore the same as those of the elements a\ c ^ o^, . . . , o^
in the first of the columns so designated in the determinant

021 ... 02* ... 02* ... 02

nl ^Wj ^fk; a n

This determinant is obtained from |a#| by replacing its column
Ci by its column c^ (we are taking ci < Cz) and leaving everything
else unchanged. On the one hand it follows from Theorem 7 that
the value of this determinant is zero; on the other hand we con-
clude from Theorem 12 and the remark made at the opening of
this proof, that its value is equal to

Qlc.Aic -f- QvcA%c. r i

14

DETERMINANTS AND MATRICES

We conclude therefore that

whenever c\ =|= c%. This proves our theorem.

Theorems 12 and 13 are the final objectives of the chain of
theorems which was started in Section 5. By means of Theorems
9 and 12, the value of any numerical determinant can be deter-
mined without an amount of arithmetical labor that is out of
proportion to the Border of the determinant; Theorem 7 can fre-
quently be used to reduce this labor still further. There are usu-
ally several effective ways in which Theorem 9 can be used; by
practice the reader will soon develop skill in applying it.

Examples.

2-13
1. To determine the value A of the determinant 4 3 2 we add

-3 2 1

row 1 multiplied by 3 to row 2; and row 1 multiplied by 2 to row 3. We
find then, by use of Theorem 9 that

2-1 3
-2 -7
1 7

It follows from Theorem 12 that A is equal to the sum of the elements of
the 2nd column, each multiplied by its cofactor; but this sum reduces to the
product of the element 1 by its cofactor. The row index of this element is
1, its column index is 2; therefore its cofactor is equal to the negative of its

-2 -7
1 7

minor. Therefore A = ( 1) X

-144-7 = -7.

Remark. The calculation of the value of a determinant by
means of Theorem 12, as illustrated in the above example, is
frequently called " developing the determinant according to a
column (row)." It reduces the evaluation of any determinant
to that of one of the next lower order.

2. To evaluate the determinant

1
2
3
4

1

3

6

10

1

4

10

20

(see Example 3, page 4),

we add the first column multiplied by - 1 in turn to the 2nd, 3rd, and 4th
columns; thus we find that the value A of the given determinant is equal

1000

1123

1259

1 3 9 19

to that of the determinant

and hence, by use of Theorem

MINORS AND COFACTORS

15

12, to that of the third order determinant

To evaluate this last

1 2 3

259

3 9 19

determinant, we add to the 2nd and 3rd columns respectively the 1st column
multiplied by 2 and by 3. It is then found that
1

A =

= 1

8. Exercises.

1. Illustrate Theorem 10 by means of determinants of the 3rd order and
of the 4th order.

5-70

2. Calculate the value of the determinant

by developing it,

23-1

642

without previous reduction, according to the 1st row; also according to the
3rd column. Verify that the two results are equal.

3. Verify, in the determinant of Exercise 2, that the sum of the products of
the elements of the 2nd column by the cofactors of the corresponding elements
of the 1st column is equal to zero.

4. Evaluate each of the

following

determinants by

the cofactor method:

(a)

5
4
-1

-2 3
1 -2
3 2

; (b

)

17
-5
6 -

8 3
7 2
11 -4

'>

(c)

2

(5
-4

3 4-1
-2 5 2
1-2 3 '
5 1 -3

6. Calculate the values of the

following determinants:

18

3 5

-3

-3 2 \

7

4-24

1

(a)

14

8

7 -4

-2 6

1
2

'

(b)

-1 J -4
-3 5

2
3

;

(c)

-2 1 -2 -2
4 -2 4 |

-10

11 1

5

2 4 -3

-10

1 -2 |

1

6. Also of the following:

2 8

6

14

12

2 -3

4-25

7

1

6 -4

3 -4

3 2 13

(a)

3 -6

3 -5

; (b)

3 -4

4 1 -2

.

1 4

3

7

7

4 -

-5

364

-1 6

11

10

23

5 -

-6

3 9 -2

7. Show that the cofactor of the element a rs in the determinant |o#|, i, j =
1, 2, . . . , w, is equal to the determinant obtained from \<HJ\ by replacing the
element OTS by 1 and all the other elements in the rth row (or in the sth
column, or in both) by zeros.

8. Prove that if the rows and columns of a determinant are shifted, as in
Theorem 10, so as to bring the element Of S into the leading position, then the
cofactor of any element in the new determinant is equal to the product of the
cofactor of this same element in the given determinant by ( 1 )*+*. (Hint:
Make use of the preceding exercise.)

16

DETERMINANTS AND MATRICES

9. Matrices. Rank of a Matrix. Before proceeding to the
application of the theory of determinants to the solution of sys-
tems of linear equations, we shall introduce some further concepts,
which, although perhaps not indispensable, will aid considerably
not only in the solution of such systems but in all our further work.

DEFINITION VIII. A matrix is a rectangular array of numbers.
The numbers composing the array are called the elements of the
matrix; the horizontal and vertical lines of the array are called re-
spectively rows and columns of the matrix.

Notation. In writing a matrix double vertical bars are placed
on either side of the array. Large parentheses are sometimes
used instead of the double vertical bars; we shall adhere to the
former notation. For example

5

3

2

3 -111

4

-1

2

-4

5 6||'

-3

2

-1

6

4

and

2

v/7
-4
-6

-3

6

2

-1
5

are matrices.

Abbreviated notations, similar to those used for determinants
(see Section 2) are also used for matrices; for example, ||o#||, i = 1,
. . . , 5; j = 1, . . . , 4 represents a matrix of 5 rows and 4 columns
and ||o$||, i, j = 1, 2, . . . , n represents a square matrix of n rows
and n columns. We shall also designate a matrix by the single
letters a or b.

Remark. We emphasize the fact that a matrix is merely an
array of numbers and that no numerical value is attached to it.
In particular it is important to notice the difference between a
square matrix and a determinant. Although both are square ar-
rays of numbers, the latter has a number associated with it,
namely, its "value," but the former has no number associated
with it. A determinant whose elements are identical with the
corresponding elements of a square matrix is called the "determi-
nant of the matrix." We also speak in such a case of the "matrix
of the determinant."

DEFINITION IX. The rank of a matrix is a positive integer or zero,
r, such that it is possible to form a determinant of order r whose value
is different from zero and whose rows and columns are obtained from
the rows and columns of the matrix, whereas it is not possible to form
a determinant of order r + 1 which satisfies the same conditions.

COMPLEMENTARY MINORS

17

Remark. It is an immediate consequence of this definition that
if the rank of a matrix is r, then the value of every determinant of
order r + 1, r + 2, etc., whose rows and columns are formed from
the rows and columns of the matrix, will be zero. And if the rank
of a matrix is zero, all its elements are zero.

10. Complementary Minors. Elementary Transformation of
Matrices.

DEFINITION X. Any determinant whose rows and columns are
formed from the rows and columns of a matrix (determinant) Is
called a minor of the matrix (determinant).

The terms " two-rowed minor," "three-rowed minor," etc.,
which we shall have frequent occasion to use, should be clear
without further explanation.

Remark. The definition of the rank of a matrix can now be put
in the following form: The rank of a matrix is an integer, r, posi-
tive or zero, such that the matrix has a non-vanishing r-rowed
minor but no non-vanishing minor of order higher than r.

DEFINITION XI. A principal minor of a square matrix (determinant)
Is a minor formed by using rows and columns of equal Indices only.

DEFINITION XII. If the rows and columns used In forming the minor
Af 2 of a square matrix (determinant) are those which were left unused
In the formation of the minor M 19 then Mi and M 2 are a pair of com-
pic mem ary minors of the matrix (determinant). Either is the com-
plement of the other.

DEFINITION XIII. The algebraic complement of a minor of a square
matrix (determinant) is equal to its complement multiplied by that
power of -1 whose exponent is equal to the sum of the indices of the
rows and columns used In the formation of the minor.

Remark. The one-rowed principal minors of a square matrix
(determinant) are the elements of its principal diagonal ; the comple-
ment of a single element is its minor; the algebraic complement of a
single element is its cofactor. (Compare Definitions VI and VII.)

Examples.

s OH a

1. The determinants

0*3
4 3

4 6

and

2 3
063

2 4

M

2 6
O&6

are two-rowed and

three-rowed minors respectively of the determinant \04j\ t i,j 1, . . . , 5.

2. The determinants

On

031

012
032

OH

034
064

and

2 3
043

2 6
4 6

are cornplementarv

18

DETERMINANTS AND MATRICES

minors of the determinant \aij\, i, j = 1, . . . , 5; and

principal minor of the same determinant.
3. The algebraic complement of the two-rowed minor

terminant |oj/ , i, j = 1, . . . , 5 is equal to ( 1)2+4+3+5

therefore equal to the complement of this two-rowed minor.

On QIS

Ois

31 33

035 is a

o a 63

O65

23 25 ^

the de-

4 3 46

On 012 o

14

0i 32

34 ; it is

4. The algebraic complement of the three-rowed minor

the determinant |a#|, i, j = 1, . . . , 6 is equal to (-1)
and therefore equal to the negative of its complement.

12 OH

a& 034

062

of

O21 O23 2
a 4 l #43 O 4

The operations upon the rows and columns of a determinant
which were discussed in Section 5 may also be performed upon the
rows and columns of a matrix. But since a matrix has no " value,"
the theorems on determinants which were there obtained will have
no exact analogues; we shall however be interested in the effect
of these operations upon the rank of the matrix. We introduce
first the following Definition.

DEFINITION XIV. An elementary transformation of a matrix Is one
which consists in performing upon it one of the following operations:
the interchange of two rows (columns); the multiplication of all the
elements of a row (column) by a non-vanishing multiplier; the addi-
tion to the elements of one row (column) of multiples of the corre-
sponding elements of another row (column).

Remark. It should be clear that if the matrix b is obtained from
the matrix a by an elementary transformation, then the matrix a
is obtainable from b by an elementary transformation.

THEOREM 14. The rank of a matrix is not affected by an elementary
transformation.

Proof. Let us consider the matrix a = ||a#||, i,j=l,.... 9 n
and let us suppose that its rank is r; it will then contain at least
one non-vanishing r-rowed minor, while every (r + l)-rowed mi-
nor vanishes. The interchange of two rows or columns of a and
Ihe multiplication of the elements of a row or column of a by a

COMPLEMENTARY MINORS 19

non-zero constant either have no effect whatever upon its minors,
or else they will multiply a minor by a non-zero constant; in
neither case will these operations kill off a non-vanishing minor of
a nor bring a vanishing minor back to life. These operations will
therefore leave the rank r of the matrix unchanged. If to the ith
row of a we add k times the jth row, an (r + l)-rowed minor of &
will not be changed in value if it does not contain the ith row, nor
if it contains both the ith and the jth rows. Let us suppose there-
fore that M is an (r + l)-rowed minor of a which contains the
ith row but not the jth row; and let us denote by M' the corre-
sponding minor of the matrix a' obtained from a by adding k times
the jth row to the ith row. Then it follows from Theorem 8 that
the value of M ' is equal to the value of M plus k times the value of
another (r + l)-rowed minor of a; but since every (r + l)-rowed
minor of a vanishes, it follows from this that the value of M ' is also
zero. Consequently every (r + l)-rowed minor of a' vanishes, so
that the rank of a' ^ r ; that is, the rank of a matrix is not increased
by any elementary transformation. But then it follows from the
remark preceding this theorem that the rank must remain un-
changed. For if it were decreased then the elementary transfor-
mation which carries the new matrix back to the original would
have to increase the rank; and we have just seen that this can not
happen. The theorem has therefore been proved.

COROLLARY. If a matrix a' is derived from another matrix a by a
succession of elementary transformations the ranks of the two ma-
trices are equal.

Remark. This theorem and its corollary can be used in the de-
termination of the rank of a matrix in the same way as Theorems
5, 6, and 9 are used in the evaluation of a determinant.

11. Exercises.

1. Write out the minors of \\atj\\, i, j = 1, . . . , 6 formed by using the
following sets of rows and columns: i = 1, 2, 5, j = 2, 3, 6; i = 2, 3, 4, 6,
j = 2,4,5,6; i = 3,6,,; =3,6.

2. Determine the algebraic complements of each of the minors of Exercise 1.

-1 23-5

3. Show that the rank of the matrix

3-45 2
5 -6 13 -1
2 14 -13

is 2.

20

DETERMINANTS AND MATRICES

4. Determine the rank of each of the following matrices:

(a)

(c)

-2 3 5
5 -1 -3
479

3 -4
6 -8
9 -12

(W

W)

2

3

-4

4

6

-5

6

9

-9

1

-3

5

-2

3

2

-4

7

-6

-13

25

1

-3

3

6. Prove that the algebraic complement of a principal minor of a square
matrix is equal to its complement.

6. Prove that one of two complementary minors of a square matrix is a
principal minor if and only if the other one is a principal minor.

7. Prove that if the minor MI of a square matrix is the algebraic complement
of the minor M 2} then M z is also the algebraic complement of MI.

12. The Laplace Development of a Determinant. In the re-
maining sections of this chapter we shall develop some further
interesting and important properties of determinants. These
properties will find application in the later chapters, but they are
not needed for the solution of systems of linear equations. The
reader can proceed therefore from this point immediately to
Chapters II, III, IV, and V, returning to the remainder of Chapter
I after he has completed these.

Our first objective is a generalization of the cofactor develop-
ment of a determinant, discussed in Section 7.

LEMMA 1. The determinant obtained from a given determinant by
shifting the rows and columns in such a way as to bring a specified
fc-rowed minor in the upper left-hand corner without changing the
relative position of the rows and columns not involved in this minor
has a value equal to that of the given determinant or of its negative
according as the sum of the indices of the rows and columns used in
this minor is even or odd.

Proof. Let the indices of the rows and columns used in the
fc-rowed minor under consideration be r i9 r 2 , . . . , r^ and ci, <% 9
. . . , cu respectively. To accomplish our purpose, we interchange
the nth row successively with each of the r\ 1 rows which lie
above it; next we interchange the r 2 th row successively with each
of the r 2 2 rows which lie above it but below the 1st row, the
rath row with each of the r 8 3 rows which lie above it but below
the 2nd row, etc., until we have interchanged the r*th row with

LAPLACE DEVELOPMENT OF A DETERMINANT 21

each of the rj& k tows which lie above it but below the (k l)th
row. Thus the rows whose indices are n, r 2 , . . . , n have been
placed in the positions of the first k rows, while the relative posi-
tion of the remaining rows has remained unchanged; and this
has been done by means of ri 1 + r 2 2 + . . + r* ft
interchanges of rows, so that the determinant we have obtained
has a value equal to that of the given determinant multiplied by
(_!)**.+ **-*<*>/. We p roc eed now to shift the columns
whose indices are fi, 02, . . . , Ck in such a way as to bring them in
the position of the first k columns without affecting the relative
order of the remaining columns; it should be easy to see that this
is accomplished by means of d 1 + <% "~ 2 + . . . + Ck k in-
terchanges of columns and therefore at the cost of c\ + c% + . . . +
c^ k(k + l)/2 changes of sign. The final result in which the
specified fc-rowed minor is in the upper left-hand corner and in
which the rows and columns not occurring in this minor have the
same relative order as in the given determinant has therefore a
value equal to that of the given determinant multiplied by a power
of 1 whose exponent is r\ + r 2 + . . . + r^ + Ci + 2 + - +
Ck k(k + 1). But, no matter what integer k may be, k(k + 1)
is always even. Consequently the value of the final determinant
is equal to that of the given determinant if r\ + r 2 + . . . + r* +
Ci + 02 + . . . + Ck is even, and equal to its negative if this sum
is odd.

Remark. Fork = 1, this lemma and its proof reduce to Theorem
10 and its proof.

LEMMA 2. All the terms In the development of a determinant which
contain as a factor any term in the development of a specified fc-rowed
minor are obtained in the product of this minor by its algebraic com-
plement; and this product contains nothing but such terms of the
development of the determinant.

Proof. We shall prove this proposition first for the principal
/c-rowed minor in the upper left-hand corner; and we shall denote
this minor temporarily by Ak- The algebraic complement of this
minor is equal to its complement (see Exercise 5, Section 11).
An arbitrary term in the development of this minor is ( l) c ai Cl O2<; 2
. . . a,kc k , where Ci, c&, . . . , Ck is a permutation of the numbers
1, 2, . . . , k and c is the number of inversions in this permutation;
an arbitrary term in the development of its complement is

22 DETERMINANTS AND MATRICES

(-1) a k+ i t y k+1 a k+2t y k+2 . . . a nyn , where y k +i t ik+z, , 7* rep-
resents an arbitrary permutation of the set of numbers
k + 1, k + 2, . . . , n and 7 is the number of inversions of this
permutation.* The product of these two terms is (-I)* 4 "* a^o^
. . . dkciflk+i, ^+i a *+2, y k+z a vn m Since the numbers of the set
Cij 02, . . . , Ck are all less than those of the set 7^+1, 7^+2, . . . , 7,
it follows from Exercise 4, Section 4 that the number of inver-
sions of the total set ci, 02, . . . , c*, 7^+1, 7^4-2, . . . , 7 W is equal to
c + 7; hence this product is a term in the development of
the original determinant. If, on the other hand, ( l^a^o^.
. . . a<nd n is a term in this development which contains as a factor
a term of A k) then its first k factors must be elements of A k and
therefore di, d 2 , . . . , d k and d^+i, d*+ 2 , . . . , d w must be permu-
tations of the sets 1, 2, . . . , k and k + 1, k + 2, . . . , n respec-
tively. Hence a^fr^ . . . &u k and a^+i. ^+^+2, d k + 2 . . . Und n will be
terms in the developments of Ak and of its complement re-
spectively, and the numerical factors, +1 or 1, will be such
that their product is equal to ( !)*. Our lemma has been
proved therefore for the principal minor A&.

To prove it for an arbitrary fc-rowed minor B k formed from the
rows and columns whose indices are n, r 2 , . . . , r* and d, (%,
. . . , Ck respectively, we form first, as in Lemma 1, the de-
terminant in which Bk occupies the upper left-hand corner. Let
us call this new determinant, and also its value, A'; then A' =
( 1)<*-KH- . . . +cj+n+n+ . . . r h A an( j ^ e m i n or Bk of A goes over
into the minor Ak of A'. Moreover the complement of Bk in A
is the same as the complement of Ak in A'. In virtue of these
facts and of the first part of this proof, we conclude that the sum of
the terms in A which contain a term of Bk as a factor is equal to
(- !)+*+ qk-Hi+rH- -'kXAk X the complement of A k ' in A'
= (-!)++ . . . -^H-n+n-f . . . +r k x ^ x t h e complement of B k in
A = BkX the algebraic complement of B*. This completes the
proof of the lemma.

* In the development of the algebraic complement the sign of this term is
determined by the number of inversions of the set of integers obtained from
^k r ^k v J 7 n ^ diminishing each of them by k; but it follows from
Exercise 3, Section 4 that this new set of integers has the same number of
inversions as the set y k+l , y k+z , . . . , y n .

LAPLACE DEVELOPMENT OF A DETERMINANT

23

Remark. The special case of this lemma which arises when k =
1 is identical with Theorem 11.

THEROEM 15. The value of a determinant is equal to the algebraic
sum of the products obtained by multiplying each of the /c-rowed
minors that can be formed from any k rows (columns) of the deter-
minant by their algebraic complements.

Proof. Let us consider the rows whose indices are r\ 9 r 2 , . . . ,r>.
Every term in the development of the determinant will contain as
a factor a product of k elements selected from these k rows, one
from each; and every such product will be a term in the develop-
ment of some one fc-rowed minor whose row indices are r\ y r 2 ,
. . . , Tk. Hence we shall obtain the value of the determinant if
we take the sum of all the terms which contain as a factor any
term in the development of any one of these fc-rowed minors.
But, since it was shown in Lemma 2 that for a given fc-rowed
minor all such terms are found, and without any additional terms,
in the product of this minor by its algebraic complement, we can
conclude that the sum of the products of all the &-rowed minors
formed from the fc rows, which were selected, by their algebraic
complements is equal to the value of the determinant.

Remark. The evaluation of a determinant by the method ex-
plained above is called the Laplace development of the deter-
minant. For the case k = 1, it reduces to the development accord-
ing to a row or column discussed in Theorem 12.

Example.

The determinant

-6,

a 3
63
3

can be evaluated in a very con-

venient way by means of the Laplace development,
columns, we find that its value is

If we use the first two

ai I
-ai I

I -ai
M -a 2 -6 2

c 3
-c 3

X

02

X

6 2 6 3
-f 3

-f

(a 2 6 3 - a 3 6 2 ) 2 -

a 3 6 3
bs - a 3 6 2 ) =

-a s -6 3

a 2 i
a 3 I

a 3 6 2 ) 2 .

X

6 2 6 3 1

a 3

24

DETERMINANTS AND MATRICES

13. Exercises.

1. Evaluate each of the following determinants by the Laplace develop-
ment, using the first two rows:

ai bi Ct

c\

-2
3

-5
1

53

41

2-4

-30

(6)

3

-6
4

-1
5
3
2

7

-2

-4
1

-5
3

(c)

/,

2. Prove that the sum of the products of the fc-rowed minors formed from
k columns of a determinant by the algebraic complements of the correspond-
ing fc-rowed minors of another set of k columns (that is, a set of k columns in
which there is at least one column that was not among the columns of the
first set) is equal to zero.

3. Evaluate each of the following determinants by Laplace's development:

(a)

3
-6
-3
4

1
-2

-5 7
4 -2
6 8
5 -3

''

(ft)

-2
3
5

-8
-4

5 4
1 -2
-7 10
4 -1
fi 3

6
8
3
12
-2

5

-9

1 3

-8 4

4 -5

2 7

-3

(c)

-5

1 2

5

6

4 -5

1

-2

-3 3

-4

4. Prove that the value of a determinant of order n, which contains for
any integral value of fc, a matrix of k rows (columns) and n k + 1 columns
(rows) of which every element is zero, is itself zero.

6. If a determinant of order n contains a matrix of k rows (columns) and
n k columns (rows) of which every element is zero, the value of the deter-
minant is equal to the product of the values of a single fc-rowed minor and its
algebraic complement.

6. Prove that the algebraic complement of a specified A>rowed minor of a
determinant is equal to the determinant obtained from the given one by re-
placing by 1 the elements in the principal diagonal of this fc-rowed minor and
by all the other elements of throws (columns) from which the minor is
formed.

7. Prove that if s is the sum of the row and column indices of a certain
fc-rowed minor of a determinant and if a new determinant is formed from the
given one in which this fc-rowed minor has been shifted to the upper left-
hand corner by the method of Lemma 1, Section 12, then the dofactors of any
element o# in the two determinants differ only by the factor ( I) 5 . (Hint:
Compare Section 8, Exercise 8.)

8. Prove that under the conditions of the preceding problem the algebraic
complements of two corresponding fc-rowed minors of the two determinants
differ only by the factor ( 1)*.

THE PRODUCT OF TWO DETERMINANTS 25

14. The Product of Two Determinants. The Laplace develop-
ment enables us to express the product of two determinants of
order n as a determinant of order 2 n. We denote by

p =

= 1,2,.

I' \bij\

the determinant of order 2 n whose first n rows consist of the rows
of |%-|, each extended by n zeros, while the last n rows consist of
the rows of |6#| preceded by an arbitrary square matrix of order
n; it should be easy to see, particularly in view of Exercise 5,
Section 13, that the value of P is equal to the product of the values
of \ciij\ and |6#|, that is, P = A B, where A and B designate
the values of the determinants |o#| and |6#| respectively.*

We choose now for the matrix /' the n-rowed square matrix in
which the elements in the principal diagonal are all equal to 1
and all the other elements are zero. We shall designate by
Ci, C 2 , . . . , C, C+i, . . . , Gin the matrices of one column and
2n rows each formed by the successive columns of P. Moreover
we shall introduce an abbreviation, current in all mathematical
writing and probably familiar to many readers, for a sum of terms
which differ in subscript only; namely, we shall write, in general,

n

V Uk for the sum u\ + w 2 + . . w-
*=i

We proceed now to apply Theorem 9 to change the form of the

determinant P as follows: we replace C n +i by C n+ i + bnCi + 621^2 +
. + b n iC nj that is, we add to the (n + l)th column of P each
of the first n columns after having multiplied them by &n, 6 2 i, . . . ,
b n i respectively. The (n + l)th column will then consist of the
following elements: an&n + 012621 + . . . + i&ni> &2i&n + 022621 +

, , . . . , 0; or,

using the abbreviation which has just been explained, of

n n

5) 2*&*i y > 2 nkbki, 0,0,. . . , 0. Next we replace the col-
*~i A=I

umn C n +2 by C w + 2 + fc^C^ + 622^2 + . . . + bniCn, the column
C w +, by C n + 3 + fciaC'i + & 2 3C 2 + + bnaCn, and so forth, until

* The reader is advised to write out in full the determinant P and to carry
out the operations described concisely and with abbreviated notation in the
following paragraphs.

26

DETERMINANTS AND MATRICES

we have replaced the last column of P, namely, C 2n , by C 2n +
binCi + & 2n <7 2 + . . . + b nn C n , Thus we obtain the following
result :

P = AB =

'21

22

tt<t n A

L 01*6*1 1

r . .- ^

*. #2*6*1

^ #1*6*2 . . . 2y
^ 02*6*2 2,

dlkbkn
(hkbkn

.i

fl2 .

. . a nn ^

L Q>nkbkl A

J a n ^* 2 ... 2

&nkbkn

-1

0.

. .

0-

...

-1 .

. .

...

0.

. .-1

0-

...

in which all the sums designated by the symbol 2) arc to be ex-
tended over the range k = 1, . . . , n. This determinant is de-
veloped by Laplace's development using the last n columns; in
accordance with Exercise 5, Section 13, the result will be equal to
the product of the n-rowed minor in the upper right-hand corner
by its algebraic complement. Thus we find that

P =

Thus we have shown that the product of the two nth order deter-
minants |a#| and |6#| is equal to an nth order determinant in which
the element in the ith row and jth column is equal to the sum

n

This result is stated in the following theorem.

THEOREM 16. The product of two determinants of order n is equal
to a determinant of order n in which the element of the ith row and
jth column is equal to the sum of the products of the elements in the
ith row of the first factor by the corresponding elements of the jth col-
umn of the second factor.

Remark. We know from Theorem 4 that the value of a determi-
nant is not changed if the columns are taken as rows and the rows
as columns. Hence if we rewrite A (or 5, or both) in such a way
as to interchange rows and columns and then determine their

THE PRODUCT OF TWO DETERMINANTS

27

product in accordance with Theorem 16, we obtain the following
extension of this theorem:

The product of two nth order determinants is an nth order de-
terminant whose element in the ith row and jih column is

(1) equal to the sum of the products of the elements in the

z'th row of the first by the corresponding elements of the
jth column of the second; or

(2) equal to the sum of the products of the elements of the

ith column of the first by the corresponding elements of
the jih column of the second; or

(3) equal to the sum of the products of the elements in the

z'th row of the first by the corresponding elements in the
jth row of the second; or

(4) equal to the sum of the products of the elements in the

ith column of the first by the corresponding elements in
the jih row of the second.

This statement is expressed symbolically by the following for-
mula:

AB =

A condensed form of the statement is that determinants may be
multiplied rows by columns, or columns by columns, or rows by
rows, or columns by rows. In most cases we shall use the first of
these methods for multiplying two determinants and, unless the con-
trary is explicitly stated, it is to be understood that this is the case.

Examples.

1. If |ay |, i,j 1, 2, 3 is used to denote the product of the two determi-

3-56 -17-3

nants 2 -1 and 6 2
-489 5-41

the products of the elements in the second row of the first of these by the
corresponding elements of the third column of the second; therefore a 2 s
2 . (-3) 4- o 2 + (-1) 1 = -7. Similarly we find a a2 = (-4) -7 + 8-
+ 9- (-4) = -16.
2. The product, rows by columns, of the two preceding determinants is

27 -33 -13

-7 18
49 -16

then a 2 3 is equal to the sum of

6

13

27 -33 -13

-1 39 -41

-7

=

-7 18 -7

-7 18 -7

37

110 -12

110 -12

-1 39 -41

-255 280

= 27,740.

110 -12

28

DETERMINANTS AND MATRICES

Their product, columns by columns, is

-23 49 -9
45 -67 23
39 -11

= 4X

1

-6
18 -55 -11

4 -9
48 23

Their product, rows by rows, is

-56 -18 41

1 -2 9

33 66 -43

-56 -130 545

100

33 132 -340

1

-6 72 -31
18 -127 151

-56 -130 545

100

-23 2 205

= 27,740.

= 27,740.

And their product, columns by rows, is equal to

23

-19
-40

4 3
16 -17
12 43

= 4 X

= 4X

1
-111 4
-109 3
2 63
109 -34

-29
34

= 27,740.

I 111 29
1109 -34

15. Exercises.

1. Multiply the determinants

(1) rows by columns;

(2) columns by rows.

2. Multiply the determinants

(1) rows by rows;

(2) columns by columns.

1
3
9

27

1

4

16

64

and

and

3. Determine the square and the cube of the determinant

1

2
2

-I

2

1

2
4

2
2

3

2
1

4. Form the products in each of the four possible ways of two general second-
order determinants and show that the results are equal in value.

5. Prove that a two-rowed minor of the product of two determinants is
equal to a sum of products of two-rowed minors of the two determinants.

6. Prove a theorem for three-rowed minors similar to the theorem of
Exercise 5.

7. Prove the general theorem of which the theorems of Exercises 5 and 6
are special cases.

8. Prove that the rank of the matrix of a product of two determinants can
not exceed the rank of the matrix of either factor."

16. The Adjoint of a Determinant.

DEFINITION XV. The determinant of which the element in the ith
row and j th column ( i, j = 1, . . . , n) is equal to the value of the co-

THE ADJOINT OF A DETERMINANT 29

factor Aij of the element o# of the determinant |o#|, i, j = 1, . . . , n
Is called the adjoint of that determinant.

We shall denote by A and by A' the values of the detenninant
\a,ij\ and its adjoint respectively. By means of Theorems 16, 12,
and 13 we should be able to see readily that the product rows by
rows of | a^ | and its adjoint forms a determinant in which the ele-
ments of the principal diagonal are all equal to A, and the remain-
ing elements are all zero; hence that A* A' = A n and therefore
that A' = A"" 1 , if A 4= 0. This result gives us the following
theorem.

THEOREM 17. If the value of the determinant \<HJ\, i 9 j = 1, . . . , n
is different from zero, then the value of the adjoint of this determinant
Is equal to the (n l)th power of the value of |a//|.

The theorem which we have just proved is a special case of the
following more general theorem.

THEOREM 18. The value of a fc-rowed minor of the adjoint of a de-
terminant Is equal to the product of the value of the algebraic com-
plement of the corresponding fc-rowed minor of the given determinant
by the (k - l)th power of the value of the given determinant, pro-
vided this latter value is different from zero.

Proof. We will prove this theorem first for the A-rowed prin-
cipal minor in the upper left-hand corner of the adjoint. We
begin by forming the product of |a#| by the determinant

An An

... A hl

...

AM A&

... A to

...

A\k Ay,

... A kk

...

Ai, k+i At, t+i

... At, k+\

1

...

A,, ^ A t , M

At, 4 +2

1 ...

A ltt A*,

... At.

...

1

The Laplace development of this determinant which uses the first
fc rows, together with Theorem 4, shows that its value is equal to
that of the fc-rowed principal minor which we wish to determine
(compare Exercise 5, Section 13) ; let us denote this value by V*.
The product of |a#| by this detenninant is formed in accordance

30

DETERMINANTS AND MATP Tri ES

With Theorem 16; if we make use again of Theorems 12 and 13,
we obtain the following result:

A

. . .

l, A+l

01, A+2 -

In

A

. . .

02, A+l

O2, A+2 0;

In

. . . A

0A, A+l

OA, A+2 . . . a

kn

. . .

OA+I, A+i

OA+1, A+2 ...

A+l> ft

...

0A+2, A+l

0A+2, A+2 ...

A+2,

. . .

On, A+l

Oft, A+2 .

nn

If the determinant on the right-hand side of this equation is de-
veloped by Laplace's development, using the first k columns, we
find, by thinking once more of Exercise 5, Section 13, that it re-

Ofc+lj A+l flA+li

duces to A*

#*, A+l a nn

pothesis, we conclude that

hence, since A =f= by hy-

V k

A+l

*+li

A+i

X

Now the determinant on the right of this equation is the algebraic
complement of the minor of |o#| which corresponds to the minor
of its adjoint which we are having under consideration; the theorem
has therefore been proved for this special case.

Let us now consider an arbitrary fc-rowed minor Mk f of the ad-
joint; let Mk be the corresponding minor of |a#| and let m^ be its
algebraic complement. Let us furthermore denote by s* the sum
of the row and column indices used in Mk (and therefore in M*')-
We know then from Lemma 1, Section 12 that the value of the
determinant obtained from \ay\ by a shifting of rows and columns
which brings the minor M k into the upper left-hand corner without
altering the relative order of the rows and columns not involved
in this minor is equal to (l)'kA ; and from Exercise 7, Section 13
we infer that the cofactors of the elements of this rearranged de-
terminant differ from the cof actors jrf the same elements in |a#|
by the factor ( l) f *. Therefore, if MI! denotes the fc-rowed prin-
cipal minor in the upper left-hand corner of the adjoint of the

THE DERIVATIVE OF A DETERMINANT 31

rearranged determinant, every element of M*' is equal to (jj
times the corresponding element of Mu f and hence MI! =
( 1)***M*'. To the minor M*' we can apply the conclusion
reached in the first part of the present proof; the algebraic com-
plement of the corresponding fc-rowed minor in the rearranged
determinant |a#| is identical with the complement of Af* and hence
equal to ( l)**w*. We have therefore the following result:
(-l^kMk* = (-l)*km k X [(-l) s kA] k ~ l , from which we conclude
that Mk = mkA k - 1 } this is the relation asserted in our theorem.
Remark. For k = n, Theorem 18 reduces to Theorem 17; for
fc = 1, it merely asserts that every element of the adjoint is equal
to the cofactor of the corresponding element of the given deter-
minant |%|. If we denote by a# the cofactor of the element Ay of
the adjoint, we obtain by putting k = n 1, the following
corollary.

COROLLARY. The cofactor aij of the element AIJ in the adjoint of the
determinant \aij\ is equal to aijA*~*.

17. The Derivative of a Determinant. The elements of the
determinants whose properties we have been discussing have been
constants. If these elements are functions of a single variable t }
let us call them tty(t), then the value of the determinant is also a
function of this variable. Denoting this function by the symbol
17(0 > we have

We inquire now for a convenient form in which to write the de-
rivative of U(t) with respect to t. To obtain such a form, we
recall two facts:

(1) That 7(0 is the sum of terms =bui^(0ti^(0 . . . w^CO*
in which ci, c&, . . . , c* are successively the different permutations
of the set of integers, 1,2,. . . , n and the sign of the term depends
on the character of the permutation;

(2) That the derivative of a product of two or more functions
is equal to the sum of all the products obtainable from the given
product by replacing one factor at the time by its derivative; for
example, if ' denotes differentiation with respect to t, (uiUju*)' =*

UiUjUs + UiUJUs + UiUjUs.

From these facts we conclude that U'() is equal to the sum
of the sums % Wi'^, . . . u^ 2) =fc u^^ . . . u*c n , . . ,

32

DETERMINANTS AND MATRICES

=fc ui c U2c t - u n c n , in each of which ci, 02, . . . , c are succes-
sively the different permutations of the set of integers 1,2,. . . , n
and the sign of the term is plus or minus according as the number
of inversions in the permutation is even or odd. But then it
follows from (1) that the first of these sums is the expansion of a
determinant obtainable from |w#(0| by replacing the elements in
its first row by their derivatives; also, that the second sum is the
expansion of the determinant obtainable from |w#(0| by replacing
the elements in the second row by their derivatives; and so forth.
We can therefore state the following answer to our inquiry.

THEOREM 19. The derivative with respect to t of the determinant
U(t] = \uij(t)\, i 9 j = 1, . . . , u is equal to the sum of the n deter-
minants obtained from U(t) by replacing the elements of one row
(column) at the time by their derivatives.

Examples.

1. The adjoint of the determinant

2 -1
1

is found to be

1 -2 -1
-7 1 5
-142

1 -3 2

The value, A, of the first determinant is 9; and the value, A', of the ad-
joint is 81 = ( 9) 2 . The cofactor of the element in the 3rd row and 2nd
_4

column of the adjoint is

-
/

; its value is 27, which is also the prod-

uct of the corresponding element in the original determinant, 3, by the 1st
power of the value of the determinant, 9.
2. The derivative of the value of the determinant

t 2 - 3 t 2t 2/ 2 - 7t + 4

3 t 2 + I 4 2 6 t - 3

2t + l 2 t* + 6 1 - 3 t 2 + 8 t - 7

is equal to the value of the sum

2t -3 2t 2 / 2 - 7 / + 4

6 / 4 / 2 t - 3

2

St-7

t 2 - 3 1 2 2

/ 2 -7/ +4

+ 3 / 2 + 1 8 /

6 / 3

21 + 1 4/ + 6

t*+8t- 7

2t 4J-7

4/2 6

t* -3t
3/ 2 + l
2t +1 2

18. Exercises.

1. Determine the jtdjoints of each of the following determinants:

1 A

1-4
-4 5

5 l

(6)

0-13

42-1

3-141

(c)

12-2 1
21 3-4
4 5-1-2
308-9

THE DERIVATIVE OF A DETERMINANT
2. Differentiate each of the following determinants:

33

(a)

| sin I cos t
I cos J sin t

-I* + 4
VC + 1)

ai cos 61 sin / C|

a-2 sin t b-i cos Z c 2

c t c 2

3. Determine the 1st, 2nd, and 3rd derivatives of the determinant

n - t
021

12

o 22 - t

032

033 -

4. A symmetric determinant being defined as one in which, for every pair
of indices i and j, y = cy,-, show that the adjoint of a symmetric determinant
is itself symmetric.

6. Show that the value of the adjoint of the determinant

tti2 Ol

Ol2 #2

13 2 3

is zero and the rank of its matrix is 1.

6. Prove that the value of the determinant

ait + pi ait + 7> 2 ait -f /> 3
ait -f- qi a-it -f o 2 a4 -f q*
a 3 t -f TI azt + ^2 o 3 + r 3

is a function of the first degree in f.

7. Verify, by direct computation, the Corollary to Theorem 18 for the case
of 3rd order determinants.

8. Work out a formula for the 2nd derivative of the determinant |w#(OI>
i, j = 1, . . . , n.

19. Miscellaneous Exercises.

1. Determine the value of each of the following determinants:

-4 2 3
5 -3 2

7 1 -6

; (6)

4-13 2

12 5 -7 4

-3 6 5 -9

14 15 9-10

3 2 -1 -2

2010

-11 3 2

-2020

2. Determine the rank of each of the following matrices:

(a)

2 -3 5 6
1 24-5
4 -13 7 28

; W

-321-2
2-1 3 4
1 11 8
, 1 70-5

; (c)

4-241
-2 1 -2 -2

4-24f
1 -2 * 1

3. Compute the adjoin ts of the determinants (a) and (c) in Exercise 1.

34

DETERMINANTS AND MATRICES

4. Determine, by inspection, the sum of the values of the determinants

-323
and 1-32
-6 1 -6

6. Show that

4

2

3

-1

-3

2

6

1

-6

1 1

1

1

a b

c

d

a' 2 b 2

c 2

d*

n* b*

c 3

d*

= (a-b) (b-c) (c-d) (a-c) (6-d) (a-d).

6. Determine the value of the determinant

1

On

7. Show that the square of the value of the determinant in Exercise 5

4 Si S 2 S 3
Si S 2 Sa 84

is equal to the value of the determinant

S4

, where sk = a k -\- b k

+ c* + d*, for fc = 1, . . . , 6.

8. Using the notation sk, introduced in the preceding exercise, set up an
wth order determinant which is equal in value to the square of the value of
the determinant in Exercise 6.

9. Show that if w 3 = 1,

1

1

0. What is the rank of the matrix

of this determinant?

10. Prove that if the elements a# of a determinant \aij\ are independent
variables then the partial derivative of its value with respect to a particular
element is equal to the cofactor of this element.

11. Prove that the second partial derivative of the value of a determinant
\aij\ with respect to the variables a t j and OTS is equal to the algebraic comple-

J "

ment of the two-rowed minor

dfj

12. Prove that the fcth partial derivative of the value of the determinant
\aij\ with respect to the elements a fiCv Or^t, , ^ r k c k ^ wmc h no two have
the same row-index nor the same column-index, is equal to the algebraic
complement of the fc-rowed minor whose rows and columns have the indices
r,, r 2| . . . , n and ci, C2, . . . , ck respectively.

* A determinant of this form is frequently referred to as a Vandermonde
determinant.

CHAPTER It
LINEAR EQUATIONS

20. Definition and Notation.

DEFINITION I. An equation In one or more variables Is called /iomo-
geneous If, after the right-hand side has been reduced to zero, the
terms on the left-hand side are all of the same degree in all the vari-
ables jointly.

Remark. It follows from this definition that a linear equation
(that is, an equation of the first degree in all the variables jointly)
is homogeneous if and only if it contains no term independent of
the variables.

Notation. We shall be dealing with systems of equations in n
variables; the variables will be designated by Xi, Xz, . . , x n . A
linear equation will therefore have the form :

+ 02X2 4- . . . + ax n = k.

It will be homogeneous if and only if k = 0.

A system of linear equations will be written in the form :

a p2 x 2 + . . . + a pn x n = k p .

The first subscript in each coefficient designates the equation in
which it occurs; the second subscript indicates the variable which
the coefficient multiplies. It will be a system of homogeneous
equations if and only if ki = k 2 = . . . = kp = 0.

It is convenient to designate the entire system of equations
briefly by writing

anxi + a i2 x 2 + . . . + a in x n = fc,-, i = 1, 2, . . . , p.

The coefficients o&, a,- w . . . , a,v, i = 1, 2, . . . , p form a ma-
trix of p rows and n columns; this matrix will be called the coeffi-
cient matrix (abbreviated c.m.) of the system of equations. If
p = n, this matrix will be a square matrix; the corresponding

35

36 LINEAR EQUATIONS

determinant is |o#|, i 9 j=l,. . . , /*>, which will be called the coeffi-
cient determinant of the system, and its value will be indicated,
as in Chapter I, by A. If we write not merely the coefficients of
the variables, but also the known terms which appear on the right-
hand sides of the equations, we obtain a matrix of p rows and n + I
columns; this matrix will be called the augmented matrix (abbrevi-
ated a.m.) of the system of equations. It will be a square matrix
if and only if p = n + 1, that is, if the number of equations in the
system is one greater than the number of variables.

21. The System of n Linear Non-homogeneous Equations in n
variables.

THEOREM 1. If the coefficient determinant of a system of n linear
equations in n variables has a value A, different from zero, then the
system has a single solution, consisting of one value for each of the
variables; these are equal to fractions whose denominators are all
equal to A, and whose numerators are the values of the determinants
obtained from the coefficient determinant by replacing the coefficient
of each variable in turn by the known terms as they appear on the
right-hand side of the equations.

Proof. Let the equations be written in the form
anXi + a 12 x 2 + . . . + a in x n = fci,

(1)

a n2 X 2 + . . .

If they are multiplied by the cofactors An, A 2i , . . . , A n i of the
elements an, 021, . . . , a n i in the first column of the coefficient
determinant and then added, it follows from Theorems 12 and 13
of Chapter I that the coefficient of x\ in the sum is equal to A,
while the coefficients of x 2 , x$, . . . , x n are all zero. Therefore we
find that

A xi = kiA n + kzA 2i + . . . + k n A ni .

If we use as multipliers the cofactors Aw, A 22 , . . . , A n2 of the
elements in the second column of the coefficient determinant and
add, we find that

A x 2 = kiAiz + ktAw + . . . + k n A m ;

and by the use of the cofactors of the elements in the third, fourth,
. , . , nth columns as multipliers, we obtain n 2 further equa-

LINEAR NON-HOMOGENEOUS EQUATIONS 37

tions of the same general form. They may be written simultane-
ously in the form :

(2) A Xi = kiA }i + k 2 A Z i + . . . + k n A ni , i = 1, 2, . . . , n.

Any set of values of the variables x\, x 2; . . . , x n which satisfy
equations (1) must satisfy these conditions. Since A 4= 0, there
is one and only one value for each x,-; and if we recall the observa-
tion made at the opening of the proof of Theorem 13 of Chapter
I (see page 13), we will recognize that the right-hand side of (2)
is the expansion according to its z'th column of the determinant
obtained from the coefficient determinant by replacing its Oh
column by the constants &i, 2, . . . , k n on the right-hand sides
of the given equations. Consequently, if A =J= 0, the system oi
equations (1) can not have more than one solution, namely, the
one given by the values

. . . + k n A ni

It remains to show that the values given by (3) actually do
satisfy the equations (1). Substitution of these values in the
left-hand side of the rth equation of this system gives, by repeated
use of the abbreviated notation for sums introduced in Section
14 (see page 25)

n n

V a H T kjAjt V kj V

A A

n

But it follows from Theorems 12 and 13 that V a^A^ is equal to

zero if j 3= T and equal to A if j = r. Hence the only term in the
sum which is different from zero is the one obtained for j = r, so
that it reduces to k,A and the values given by (3) do therefore
actually satisfy the given equations. This completes the proof of
the theorem.

Remark. The rule given by this theorem for writing down at
once the solution of a system of n linear equations in n variables
whose coefficient determinant has a value different from zero, is
known as Cramer's rule.

38 LINEAR EQUATIONS

22. The System of n Linear Homogeneous Equations in n
Variables. From the result obtained in the preceding section, we
derive some important consequences.

THEOREM 2. A system of n linear homogeneous equations In n
variables whose coefficient determinant has a value different from
zero, possesses the solution which consists of zero for each of the
variables, and no others.

Proof. It should be clear by inspection that, if ki = A* 2
= . . . = k n = 0, the equations of the system (1) are satisfied by
the values Xi = x% = =# = (). That the system has no
other solution if A =f= follows from the proof of Theorem 1.

Remark. It should be evident that every system of p linear
homogeneous equations in n variables Xi, x 2 , . . . , x n possesses
the solution x\ x 2 = = x n = 0; this solution of such a
system is called the trivial solution.

On account of the frequent use to be made of it in the later
parts of this book (see Sections 41 and 82) we state the following
corollary which is an immediate consequence of the preceding
theorem.

COROLLARY. In order that a system of n linear homogeneous equa-
tions in n variables shall have a non-trivial solution, it is necessary
that the value of its coefficient determinant shall be zero*

23. The System of n + 1 Linear Non-homogeneous Equations
in n Variables.

THEOREM 3. A system of n -f 1 linear equations in n variables,
whose coefficient matrix has rank n, possesses a solution if and only
if the determinant of its augmented matrix has the value zero; and
in this case the solution is unique.

Proof. Let the equations be written in the form:

= &;, Z = 1, 2, . . . , ft + 1.

Since the c.m. is of rank n, we can find in it at least one deter-
minant of order n, which has a non-zero value. And, because
the order in which the equations that constitute the system are
written is clearly a matter of indifference, we can suppose without
loss of generality that this determinant is the coefficient deter-
minant of the first n equations of the system. These equations
possess therefore a single solution, which can be written down

LINEAR NON-HOMOGENEOUS EQUATIONS 39

according to Cramer's rule; consequently the entire system of
n + 1 equations can not have more than one solution. It will
possess one solution if the values of the variables determined by
the first n equations also satisfy the (n + l)th equation. Now it
is possible to express these values in terms of the cofactors of the
elements in the last row of the a.m. If we use the familiar capital
letter notation to designate the cofactors of the elements of the
a.m., we find from the first n equations, since K n+i 4= 0:

n #12 . . . flj, i-i ki ai, 1+1 . . . ai, n -i CL\.

. . a ni n _i

l, w-l

tl fl n 2 . . . (l nj i-l MID i+1 Km n-1 &nn K>

The latter of these determinants is obtained from the former by
interchanging its ith column successively with each of the n i
columns which follow it. But the last written determinant is
clearly the minor of the element a w +i, t - in the a.m. and therefore
equal to ( l) n + l +* times the cofactor A n +i, , of this element. We
conclude therefore that

K X' = ( i)H t'+n+i+f A 4-\ ' A -LI i = 1 2 n.

Now we substitute the values of Xi obtained in this way in the last
equation of the system. We find then that the system possesses
a solution if and only if a n +\,iA n +iji + a n +i, 2 A n +i,2 + '
+ a rt+1 , n A n +i, n + k n +i K n +i = 0. But this is, in virtue of Theorem
12, Chapter I, the condition that the determinant of the augmented
matrix be equal to zero. We have therefore proved the theorem.

COROLLARY. A system of n -f 1 linear homogeneous equations In n
variables whose coefficient matrix has rank n possesses only the trivial
solution.

Examples.

1. The system of equations 3s-2?/ = 4, 2z + 32/ = 5, x - y = 2 ha,s
no solution. For, while the c.m. is clearly of rank 2 (the value of the determi-

Io o i

is different from zero), the determinant of the augmented
6 O I

40

LINEAR EQUATIONS

matrix has a value different from zero; this determinant is

3-24
2 35
1 -1 2

and

its value is 11.

2. Let us consider the system of equations 3 x 2y + z = 7, 2x -\- 3y
4 2 = g t x y + z*=4,x + 2y + 3z = 5. The rank of the c.m. is 3;
and the determinant formed of the coefficients of the first three equations,

has the value + 4. The determinant of the a.m. is

It should not be difficult to show that its value is zero.

This being done, it follows from Theorem 3 that the system has a single solu-
tion, which may be found by solving the first three equations of the system
by Cramer's rule. Thus we find, by use of Theorem 1, that

3 -2 1

namely,

23-4

1 -1 1

3-217

2 3-4-9

1-114

1235

7 -2 1
-9 3 -4
4 -1 1

4
4

3 -2 1
23-4
1 -1 1

3 7 1
2 -9 -4
1 4 1

3 -2 1
2 3 -4
I -1 1

-4
4

3-27
23-9
1 -1 4

3 -2 1
2 3 -4
1 -1 1

24. Exercises.

1. Give illustrations of systems of homogeneous and of non-homogeneous
linear equations in 2, 3, and 4 variables.

2. Determine the rank of each of the following matrices:

3 -2

(a)

3 -4

-1 2

5 -6

5 2

5 1

19 5

5 -2
-4 10
15 -6

(c)

1

-1 432
7-870
1 6 16 4

3. Solve each of the following systems of equations by Cramer's rule:

4. Proceed in the same way with the following systems of equations:
(a) 3x + y-4u=9, -5 y + 3z + 2u = 18, x - 6y + 7z =

(6)

= 20,
+ 5 v = 50.

6. Determine, for each of the following systems of equations, whether they
possess a solution; solve those systems for which a solution exists:
(a) 2x-y + l =0, s + 2y + 2 = 0, 15 x + 20 y + 24 = 0.
(6) x-2/ + 4=0, 3
(c) 5x-3y-7 =

LINEAR HOMOGENEOUS EQUATIONS

41

6. Also for each of the following systems:

(a) 6 + 5 ?/ -- 12 2= 11, 5s 2y

x + 18 y - Sz = -24.

(6) 2x + 3y + 3s = l, 33-y-4z =
-4z + ll2/ + 2l2 = -9.

(c) 3z-i/ + 2z= -3, -2z-f2?/-3

-3,
-4,

26. The System of FI 1 Linear Homogeneous Equations in
n Variables.

THEOREM 4. If the rank of the coefficient matrix of a system of
n 1 linear homogeneous equations in n variables is equal to n 1,
the system has a single infinitude of solutions; the ratios of the
variables are equal to the ratios of the cofactors of the elements in
the nth row of the determinant obtained by writing a row of arbitrary
elements pi, p 2 , . . . , p n under the coefficient matrix.

Proof. We write the equations in the form

aaxi + a i2 x<i + + ai n x n = 0, i = 1, 2, . . . , n 1.

Since the c.m. is of rank n 1, it contains a non-vanishing de-
terminant of order n 1, and there is no loss of generality if we
suppose that this is the determinant formed by the coefficients
of the variables xi, x 2 , . . . , x n -\. In virtue of Theorem 1, the
equations possess therefore a single solution for x\, x^ . . . , x n -\
for every definite value assigned to x n ] hence there is a single
infinitude of solutions.

We can apply Cramer's rule to solve the equations for
sci, #2, . . . , x n -i in terms of x n \ we find

an

021

022

t-l #2; +l

n-

, 1 n-l>2

= x n Pi y i = 1, 2, . . . , n, where PI, P 2 , . , P denote the co-
factors of the arbitrary elements pi, p 2 , . . . , p n in the determinant
obtained from the c.m. by writing a row of arbitrary elements

42 LINEAR EQUATIONS

under it. If we write this result in the form of a continued pro-
portion, we obtain the result stated in the theorem, namely:

xi : x 2 : . . . :x n = Pi :P 2 : . . . : P n .

COROLLARY. If the rank of the coefficient matrix of a system of p
linear homogeneous equations in n variables is n - 1, the system has
a single infinitude of solutions; the ratios of the variables are equal
to the ratios of the cof actors of the elements in some row of an n-rowed
minor of the coefficient matrix.

Proof. If the hypothesis is satisfied, it is clear that we must have
p ^ n 1 and that there must be at least one set of n 1 among
the p equations which satisfies the conditions of Theorem 4. The
order in which the equations of the system are written is imma-
terial and we may therefore suppose that the first n 1 equations
of the system furnish one such set. We will show now that the
solutions of this set also satisfy the remaining equations of the
original system of p equations. For, if we substitute kP iy i = 1 ,
2, . . . , n for Xi in the rth equation (r n, n + 1, . . . , p), the
left-hand side becomes k (a r fi + <7 f2 P 2 + + a rn P n } ; but the ex-
pression in parentheses is the expansion of the nth order determinant
obtained by writing the coefficients of the rth equation under the
c.m. of the first n 1 equations, so that its value is equal to zero.
Moreover the set of the first n 1 equations has, in virtue of
Theorem 4, no other solutions besides those of the single infinitude
indicated above; consequently solutions which are obtained by
using another set of n 1 equations selected from the given
system must be contained among this single infinitude. Hence
our corollary is proved.

26. The Adjoint of a Vanishing Determinant. Symmetric De-
terminants. By means of Theorem 4 we are able to obtain a
valuable extension of Theorems 17 and 18 of Chapter I. For in
these theorems we had to make the hypothesis that the value of
the determinant under consideration was different from zero;
it is this restriction which we are now able to remove. We begin
by proving the following theorem.

THEOREM 5. If the value of a determinant is zero, the rank of the
matrix of the adjoint of the determinant is equal to or 1.

Proof. If the cofactor of every element of the determinant
vanishes, the rank of the matrix of the adjoint is clearly equal to

ADJOINT OF A VANISHING DETERMINANT 43

zero; if this is not the case, let us suppose that Aj^ ^ 0. Then
for every i the equations

= o

+ 22^-i2 + ' ' ' + 2n^m

l*n =
+ CL n2 Ai2 + * + dnnAfn =

hold, in virtue of Theorems 12 and 13 of Chapter I. But these
equations may be looked upon as a system of n 1 linear homo-
geneous equations in the n variables AH, A&, . . . , A,- rt ; and since
the coefficient matrix surely contains the determinant Ak r , its rank
is n 1. We can therefore apply Theorem 4 and we find that

AH : A i2 : . . . : A in = A kl : A k z : . . . : A knj for i = 1, 2, . . . , n.

This means that the different rows of the adjoint are proportional,
so that every two-rowed minor vanishes; the rank of the adjoint
is therefore less than 2.

COROLLARY. The conclusions of Theorems 17 and 18 of Chapter I
still hold when the value of the determinant |o#| is zero.

We shall now prove a few important consequences of this theorem
which refer to symmetric square matrices (usually called symmet-
ric determinants) and which are needed in our later work (see
Chapter VIII).

DEFINITION II. A symmetric square matrix Is a matrix \\aij\\, i } = 1*
2, . . . , n, in which, for every i and every j, aij = a/,-.*

DEFINITION III. A singular square matrix is one whose determinant
vanishes.

THEOREM 6. If all the (n - l)-rowed principal minors of a singular
symmetric square matrix vanish, its rank is less than n - 1.

Proof. We have to show that under the hypothesis every ele-
ment of the adjoint vanishes. Now it should be easy to see that
the adjoint of the given symmetric square matrix is itself sym-
metric, so that Aij = Aji for every i and j.* Moreover it follows

* Compare Exercise 4, Section 18.

44 LINEAR EQUATIONS

from Theorem 5 that every two-rowed minor of the adjoint has

the value zero, that is,

= for every i and j. Hence,

= Aij 2 . But AH AJJ= 0, by hypothesis. Consequently
Aij = 0; and this is what our theorem asserts.

COROLLARY. If the rank of a singular symmetric square matrix of
order n is n 1, then it contains a least one no ti- vanishing (n 1)-
rowed principal minor.

THEOREM 7. All the (n - 1) -rowed principal minors of a singular
symmetric square matrix which do not vanish have the same sign.

Proof. If AH and A# are two non-vanishing principal minors,
then it follows from the hypothesis, as in the proof of Theorem 6,
that A#Ajj = Aij 2 > 0; hence AH and A$ have the same sign.

COROLLARY. If the sum of the (n - 1) -rowed principal minors of a
singular symmetric square matrix is equal to zero, the rank of the
matrix is less than n - 1.

This corollary follows immediately from Theorems 7 and 6.

27. The System of n Linear Non-homogeneous Equations in
n Variables, continued. In Section 21 we have seen that a system
of n linear non-homogeneous equations in n variables, whose c.m.
has rank n possesses a single solution; this solution may be de-
termined by means of Cramer's rule. We return now to such a
system of equations but under the hypothesis that the rank of the
c.m. is n 1, and we shall prove the following theorem.

THEOREM 8. If the coefficient matrix of a system of n linear non-
homogeneous equations in n variables is of rank n 1, the system will
have a single Infinitude of solutions or no solution, according as the
rank of the augmented matrix is equal to or greater than n - 1.

Proof. We write the equations in the form

+ aaxz + + a in x n = k if i = 1, 2, . . . , n.

Since the rank of the c.m. is n 1, there is at least one set of n 1
of the equations and at least one set of n 1 of the variables,
such that the c.m. of these variables in this particular set of equa-
tions is of rank n 1 ; these equations can therefore be solved by
Cramer's rule for n 1 of the variables in terms of the nth vari-
able, as soon as a value has been assigned to this variable. Con-

LINEAR NON-HOMOGENEOUS EQUATIONS

45

sequently this set of n 1 equations possesses a single infinitude
of solutions. It remains to determine whether these solutions will
also satisfy the single remaining equation. This question can be
answered by means of Theorem 3.

If the special set of n I variables consists of i, 2 , . . . , a^-i,
X/+1, . . . #, the determinant of the augmented matrix of the re-
lated system of n equations in n 1 variables is

Oil

021

022

0*1

On . .
021 .

fliw K,
. . Oi*

0,1 . .
On

. am

But since the rank of the c.m. of the given system is n 1, the
last term vanishes; therefore the determinant of the augmented
matrix of the related system reduces to the first term, which is an
n-rowed minor of the augmented matrix of the given system. If
the rank of this matrix is n 1, then every one of its n-rowed
minors vanishes; hence the augmented matrix of the related sys-
tem has rank n 1 and we conclude, by use of Theorem 3, that
all the solutions of the special set of n 1 equations also satisfy
the nth equation. But if the rank of this matrix is n, then, for
some one of the variables Xj, the determinant of the augmented
matrix of the related system will be different from zero, no matter
what value is assigned to xy 9 and in this case we conclude, again
by use of Theorem 3, that the given system of equations does not
possess a solution. This completes the proof of the theorem.

Remark. The theorems proved in Sections 21, 22, 23, 25, and
27 are special cases of a more general theorem which asserts that
a system of p linear equations in n variables possesses one or more
solutions if and only if the ranks of the c.m. and the a.m. are equal.
A proof of this theorem may be found in the books referred to in

46

LINEAR EQUATIONS

Section 1. The special cases dealt with here suffice for the appli-
cations to be made in the later chapters of this book; they furnish
moreover a suitable introduction to the study of the more general
cases. For this reason and also in order to avoid too elaborate
algebraic discussions, we have restricted ourselves to these special
cases. The reader is urged to make himself thoroughly familiar
with the content and the proofs of these theorems; they will
repeatedly be referred to in our further work.
Examples.

1. The system of equations

determines the ratios of the variables x, y, and z. For the value of the coeffi-
2-3 5

cient determinant

is readily found to be zero. Since the two-

2 1
4 7-7

rowed minor in the upper right-hand corner is different from zero, the rank
of the c.m. is 2. Consequently, we conclude from Theorem 4 and its corollary
that the ratios of the variables are equal to the ratios of the cof actors of the
elements in the last row. Thus we find that

x:y:z-

-3 5
2 -1

2 5

3 -1

2 -3

3 2

- -7 : 17 : 13.

It is easily verified that any three numbers which have these ratios satisfy the
given equations.

2. To determine the ratios of z, y, and z from the system of equations

x - 2 ;// + 3 z = 0, -3 z -h 6 ?/ 4- 2 =

we observe first of all that the rank of the c.m. is 2. Jn view of Theorem 4
we can conclude that the ratios of the variables arc equal to the ratios of the

1-23
cofactors of the elements in the third row of the determinant

hence we find that

-3

Pi

6 1

Pt Pa

x ' y . z =

-2 3
6 1

1
-3

1 -2
-3 6

= -20 : -10: = 2 : 1 : 0.

^ 3. In virtue of Theorem 8, the system of equations

2# 37/4-52 = 1, 3 # 4- 2 ?/ z = 4, 4x -\- 7 y 7 z 5

has no solution. For it should be easy to show that the rank of the c.m. is 2;

2 -3 1

and since the value of the determinant
-26, the rank of the a.m. is 3.

2 4

4 7

formed from the a.m. is

LINEAR NON-HOMOGENEOUS EQUATIONS 47

4. In the system of equations

the ranks of the c.m. and the a.m. are both equal to 2. It follows therefore
from Theorem 8 that the system possesses a single infinitude of solutions which
may be found by solving two of the equations for two of the variables in terms
of the third. If we solve the first two equations for x and y in terms of z,
we find that

14 - 7 z . 5 + 17 z

* = -T3- and = J5

It is readily verified that these expressions satisfy the three given equations
for all values of z.

28. Exercises.

1. Determine the ratios of the variables from each of the following systems
of homogeneous equations:

(a) x 4-3 y -z = 0, -2y +z = 0, 5x + y + 2z = 0.

GO 4x4- 67/4-82 -84^ =0, 2x + y+3z -48?; = 0, -2 x + y + z

- 12 v = 0, 4jc4-4y-2-24i;=0.
(c) 2 x - ?/ 4- 2 z = 0, -x 4- 2 ?/ 4- 2 z = 0, 2 x -f 2 y - 2 = 0.

2. Proceed similarly with each of the following systems:
(a) 2x- 7/4-22 =0, x + 2y + 2z =0.

(&) 5 2: - y + 3 2 = 0, 10 x - 2 y + 4 z = 0.

(c) 2aj-y-f2jg-t-5tt = l 3a:-f2/-^+2^^ = 0, 4^-2y-h3?/ = 0.

3. Show that none of the following systems of equations possess a solution:

(a) 3 x - 4 y = 5, 2 x + 5 y = 3, 6 x - 8 ?/ = 4.

(b) 2x + 3y - 4z = 3, s-2y + 3 = l, 4o: + 7/-2z=~2, 2o;

-4y + 5s = 3.

(c) 5 x - y 4- 2 z = 12, 2 x- + 3 y = 7, 3 x - 2 y - 4 z = -2, 4 s -f T/

-2 z = 5.

4. Determine which of the following systems possess a solution; solve
those for which a solution exists:

(a) z4-6
(6) 4x +
(c) 2x~

6. Determine the conditions under which the system of equations
aix -\- b\y -f CiZ -f ^ = 0, a 2 x + ^2?/ + C 2 2 + ^2 = 0,
a& + hy 4- c 8 2 4- ds = 0, o 4 x 4- &4?y 4- 42 4- rf 4 = 0,

possess one solution.

48

LINEAR EQUATIONS

6. Determine the ranks of the adjoints of the following square matrices:

-1

2
1

o

7. Prove:

2
3

-2
-3

1

-2
-1

2

-2
-3

2
4

(W

-1
2
1

-2

2
3

-2
-3

1

-2
-1

2

-2

-3

2

3

(a) If the rank of the matrix

p u
u q

is 2, then the values of no two

of the expressions pq u 2 , qr w 2 and rp v 2 are opposite in sign.

p u v

(b) If u q w
v w r
of the matrix in (a) is 1.

= and pq -f- qr -J- rp u 2 v 2 to 2 = 0, the rank

8. Prove that if the system of equations

(liiXi 4-

= fa, I = 1, 2, . . . , ?i

in which the value of the coefficient determinant |a#| is different from zero,
is solved for xi, x 2 , . . . , x in terms of k h k 2 , . . . , k n , then the value of the
determinant formed by the coefficients of ki, fc 2 , . . . , k n is equal to the recipro-
cal of the value of the determinant \a#\.

CHAPTER III
POINTS AND LINES

The primary object of Solid Analytical Geometry is the study of
the geometry of three-dimensional space by algebraic methods.
This end is accomplished by means of coordinate systems or frames
of reference. Such systems enable us, as in Plane Analytical
Geometry, to determine algebraic entities corresponding to various
geometric elements. We start this study with the simplest geo-
metric element, the point in three-space.

29. The Cartesian Coordinates of a Point in Three-space.
The simplest frame of reference is furnished by three mutually
perpendicular planes, called the coordinate planes. It is custom-

Fia. 1

ary to take one of these planes horizontal, the other two vertical.
The point common to the three planes is called the origin of co-
ordinates and is usually designated by the letter O. The lines of
intersection of the planes with each other are called the coordinate
axes, the X- and Y-axes being the intersections of the horizontal
plane with the two vertical planes, and the Z-axis the line in which
the two vertical planes meet (see Fig. 1).

It follows from elementary solid geometry that the three co-
ordinate axes are mutually perpendicular and that each of them
is perpendicular to the plane formed by the other two. In this
book we shall take the positive directions on the coordinate axes

49

4/0 POINTS AND LINES

as indicated in Fig. 1.* On d^PpPl^se axes a unit of measure-
ment is adopted; we shall use equal^||fc on the three axes.

If an arbitrary point || is now-t||||lin three-space, we drop
from it perpendiculars ^%e^8rdSfelanes. Let the feet of
these perpendiculars be designated by P^y, P yz , and P^, the sub-
scripts indicating the planes in which these points lie (see Fig. 1).
Now we lay down the following definition :

DEFINITION I. The x-coordinate of P is the measure of the line
PyzP, measured in accordance with the unit and the direction specified
for the X-axis; the ^coordinate is the measure of the line P ZX P 9
measured in accordance with the unit and direction specified for the
Y-axis; and the ^-coordinate is the measure of the line P xy P measured
in accordance with the unit and direction specified for the Z-axis.

Notation. It will frequently be found convenient to designate
the coordinates of the point P as XP, yp and ZP, particularly when
ready identification of the points is desired. When the reference
is to an arbitrary point of a specified group of points, we shall
usually omit the subscript; the coordinates of a point PI will be
denoted by Xi, y\ y z\\ those of a point P 2 by 2 , 2/2, 2 ; etc.

Remark 1. The coordinates of a point P are signed real numbers,
the signs depending upon the position of P with respect to the
coordinate planes. The ^-coordinate of P is positive or negative
according as P lies to the right or to the left of the 7Z-plane;
the ^/-coordinate of P is positive or negative according as P lies
in front of or behind the ZX-plane; the ^-coordinate is positive
or negative according as P lies above or below the XT-plane.

Remark 2. It should be clear from Definition I not only that
every point in three-space has a definite set of three real numbers
as coordinates, but also that an arbitrary set of real numbers, taken
in a definite order, determines one and only one point in three-

* The coordinate system which we have adopted is called a " left-handed
system " because the thumb, first and second fingers of the left hand can be
put in such a position as to suggest the positive directions along the X-,
Y-, and -axes, particularly by a person whose finger joints have not grown
stiff. If the positive direction along any one of the axes is reversed, we ob-
tain a right-handed system. It should be clear that any two left-handed
systems, and also any two right-handed systems, can be made to coincide;
but that a left-handed system and a right-handed system are symmetric with
respect to each other and can not be brought to coincidence if we are limited
to a three-dimensional space.

THE COORDINATE PARALLELOPIPED OF A POINT 51

space. The reader should^llfece himself that this point is
found as the point comnxj^^ three mutually perpendicular planes.
The point P whose ^-//Iffijbd ^-coordinates are a, 6, and c re-
spectively will be desig^g^Pby th$ g^tol P(a, 6, c).

30. The Coordinate Parallelepiped of ,a Point. The three per-
pendiculars dropped from P on the coordinate planes determine,
two by two, three mutually perpendicular planes. These three
planes together with the coordinate planes determine a rectangular
parallelepiped; we shall call this the coordinate parallelepiped
of P, a name which we shall frequently indicate by c.p.

The coordinate axes each meet the faces of the c.p. in and in
a second point; these points are designated by P x , P y , P z (see
Fig. 1). The twelve edges of the c.p. are equal, four by four, to
the lines whose measures are the coordinates of P. The four
pairs of opposite vertices of the c.p. are and P, P xy and P z , P yz
and P XJ P& and P y ; the body diagonals are the four lines which
join pairs of opposite vertices. The lines PP*, PP y , and PP 2 are
perpendicular to the X-, Y-, and Z-axes respectively; the points
P x , P y , and P s are therefore the projections of P on the coordinate
axes.

If we bear in mind the properties of the rectangular parallelepiped
which are proved in elementary solid geometry, we obtain at once
the following theorems.

THEOREM 1. The *-, y, and ^-coordinates of P are equal respec-
tively to the projections OP*, OP y , and OP Z of OP upon the X-, Y-,
and Z-axes.

THEOREM 2. The square of the distance OP Is equal to the sum of
the squares of the coordinates of Pi

OP 2 = * P 2 + y/ + z p \

THEOREM 3. The cosines of the angles which the line OP makes
with the positive directions of the X-, Y-, and Z-axes are equal re-
spectively to the quotients of xp, y p , and z p by OP.

If we designate these angles by a op , @ OP , and y op respectively,
and their cosines by A OP , ju op , and v op respectively, we have :

z p

52 POINTS AND LINES

Remark 1. The square roots in these formulas are to be taken
with the positive sign; if the sign of the square root is changed,
we obtain the cosines of the angles which the line OP makes with
the negative directions along the coordinate axes, or, what amounts
to the same, the cosines of the angles which the line PO makes with
the positive directions along the axes.

Remark 2. It is important that the reader should learn to draw
the coordinate parallelepipeds of points in various parts of space.
The following exercises are intended chiefly to develop skill in
doing this. *

31. Exercises.

1. Draw the coordinate parallelepiped for each of the following points;
determine their distances from the origin and the cosines of the angles which
the directed lines from the origin to them make with the positive directions
along the coordinate axes:

4(-2,5, -4); B(l, -2,3); C(-l, -2,3); D(0, 4, -2); #(4,6,7);
F(-4, -6, -7); 0(5, -2, -1); ff(-3, 4, 2); /(6, 0, -3);
K(, 3, -4); L(-5, -2, -3); M(-5, 3, 0).

2. Determine the loci of the points for which

(a) x = -2; (6) y = 4; (c) z = -5; (d) x = 4 and y = -3; (e)y = 2
and z = 6; (/) z = 3 and x = -3; (g) x 1 + y 2 -f z 2 = 9.

3. Determine the coordinates of the points which are symmetric with
P(a, b, c) with respect to

(a) the X-plane; (6) the F-axis; (c) the origin; (d) the Z-axis;
(e) the FZ-plane; (/) the X-axis; (g) the XF-plane.

4. Develop one or more algebraic conditions which are satisfied by the
coordinates of the points which lie

(a) on a sphere of radius 4 which has its center at the origin;

(b) on a plane which cuts the F-axis perpendicularly at a point 5 units

behind the origin;

(c) on a line parallel to the Z-axis and through the point A (3, 4, 1);

(d) on a plane parallel to the F-plane and passing through the point

4(-l, 2, 1);

(e) on a circle in the Z-X-plane whose center is at the point C(0, 0, 4)

and whose radius is 3;
(/) on a line perpendicular to the XZ-pl&ne and through the point

4(-2, -3, -1);
(g) on the line determined by the origin and the point A (2, 1, 1).

6. The point P lies on a line through the origin which makes with the posi-
tive directions on the coordinate axes angles whose cosines are equal to J,

i, and ^~, and the distance OP is equal to 3. Determine the coordinates of P.

TWO POINTS

53

6. Prqye that the line from to A(l, 1, 1) makes equal angles with the posi-
tive directions on the three coordinate axes.

7. Prove that the lines which join the origin to P(a, 6, c) and to Q(ka, kb, kc)
make equal or supplementary angles with the positive directions on the
coordinate axes.

8. Prove that the sum of the squares of the cosines of the angles which the
line OP makes with the positive directions on the coordinate axes is equal to 1.

32. Two Points. The c.p. of the point P is a rectangular
parallelepiped whose faces are parallel to, or lie in, the coordinate
planes and of which the origin and P are opposite vertices, while
the other vertices all lie in the coordinate planes. We construct
now a rectangular parallelepiped whose faces are parallel to, or
lie in, the coordinate planes, but of which two opposite vertices
are to be two arbitrarily assigned points P and Q. The constrvc-
tion, of which the result is shown in Fig. 2, is carried out most
readily as follows:

B

B

1

--

Q 1

A 1

Q

Fia. 2

Through the points P xy and Q xy we draw lines parallel to the
X- and F-axes, so as to form the rectangle P xy A xy Q xy B xy ] through
the four vertices of this rectangle we draw lines p z , a z , q t , and 6 a
parallel to the Z-axis which are the vertical boundaries of the side
faces of the parallelepiped. To complete the construction, we
draw through P a line parallel to the X-axis, meeting a s in A;
through A a line parallel to the 7-axis, meeting q z in Q'; through
Q' a line parallel to the X-axis, meeting 6, in 5; and through B

54 POINTS AND LINES

a line parallel to the F-axis, meeting p z in 1 J . Starting from Q,
we locate in a similar manner the vertices #', P' 9 and A' of the
parallelepiped.

Remark 1. The parallelepiped whose construction is described
above will be called the coordinate parallelepiped (c.p.) of P and
Q. It is important that the reader develop skill in carrying out
this construction; a number of valuable results can be obtained
readily by means of it.

Remark 2. The construction of the c.p. of two points P and Q
can be carried out equally well if we start with the points P yz
and Qysy or with the points P& and Q&.

Since the line P'A f is parallel to the X-axis, the segment P'A r
is equal to the segment P X Q X of the X-axis determined by the
projections of P and Q on the X-axis. Since 0, P x , and Q x are
points on the same directed line, namely, the X-axis, we know
moreover that

OP X + P X Q X + Q X =
that is,

XP + P X Q X -x Q = 0,

in virtue of Theorem 1, Section 30, page 51. Hence we conclude

that

Proj*PQ = P X Q X = x Q - x P .

Leaving the proof of similar formulas for the projections of the
segment PQ on the Y- and Z-axes to the reader, we state the fol-
lowing theorem.

THEOREM 4. The projections on the coordinate axes of a directed
segment of a straight line are equal to the differences between the
corresponding coordinates of the end point and those of the initial
point of the segment.

We observe now that the twelve edges of the c.p. of P and Q
have lengths equal, four by four, to the numerical value of the
differences between the coordinates of P and Q. If we make use
once more of the property of the body diagonal of a rectangular
parallelepiped which was brought forward in Section 30, we
obtain the following extension of Theorem 2.

THEOREM 5. The square of the distance PQ is equal to the sum of
the squares of the differences of the coordinates of P and of Q 9 that Is,

PQ 2 - (X Q - x p )* + (y Q - y p ) 2 + (M Q - * P )*.

DIRECTION COSINES OF A LINE 55

33. Direction Cosines of a Line. We recall that the angle
between two lines / and m which do not lie in the same plane is
defined as the angle between any two concurrent lines of which one
is parallel to / and the other to m. This extension of the concept
" angle between two lines" enables us to speak of the angles which
an arbitrary line makes with the coordinate axes and gives sig-
nificance therefore to the following definition.

DEFINITION II. The direction angles of a directed line are the angles
between -180 and +180 which the directed line makes with the
positive directions of the coordinate axes. The direction cosines of a
directed line are the cosines of its direction angles.

Notation. Whenever it is desirable to specify the directed
line to which reference is made, the direction angles of the line
PQ will be designated by the symbols a pQ , PQ , and y p ^ its
direction cosines by X M P and v pQ . Similar notations will be
used for the direction angles and the direction cosines of an un-
directed line /. When it is not essential to specify the line, the
subscripts will be omitted.

On the basis of this definition we obtain, from a consideration
of the c.p. of P and Q and by using some of the properties of the
rectangular parallelepiped mentioned in the second paragraph of
Section 30 (page 51), the following extension of Theorem 3.

THEOREM 6. The direction cosines of the directed line PQ are equal
to the quotients of the differences between the coordinates of Q and
those of P by the distance PQ ; that is, ! ^

= COS a pQ -

X Q ~ Xp

V(X Q - *

p) 2 + (VQ ~ Jp) 2 + (*Q - /) 2

PQ - COS 7pQ

Remark. The square root here, as in Theorem 3, is to be taken
with the positive sign; a change of sign in the square root leads
to the direction cosines of the line QP. In the case of an undirected
line I, the sign of the square root remains ambiguous, the two
signs corresponding to the two directions which may be specified

56 POINTS AND LINES

upon the line. The formulas of Theorem 6 lead to the following
very useful corollaries.

COROLLARY l. The direction cosines of an undirected line are pro-
portional to the differences between the coordinates of any two of its
points.

COROLLARY 2. The direction cosines of an undirected line through
the origin are proportional to the coordinates of any one of its points.

COROLLARY 3. The coordinates of any point on the line which Joins
the origin to a point P are proportional to the coordinates of P.

(Compare Exercise 7, Section 31, p. 53.)

Furthermore we obtain the following important result.

THEOREM 7. The sum of the squares of the direction cosines of a line
is equal to 1.

Proof. If we choose any two points P and Q on the line and
express the direction cosines of the line in terms of the coordinates
of P and of Q, as in Theorem 6, we find that

(XQ- XP)* + (y Q - VPY + (ZQ - ZP)* '

Remark. The formula proved in Theorem 7 will be used re-
peatedly in the sequel. We shall use the phrase "admissible values
of X, JUL, v" to indicate values of these variables for which X 2 + pt 2
+ i> 2 = 1. Its principal use will be to enable us to determine the
direction cosines of a line if we merely know numbers to which
they are proportional. This will in most cases relieve us of the
necessity of actually determining the direction cosines and make
it possible to operate with their ratios. A special case of Theorem
7 is contained in Exercise 8, Section 31, (page 53).

THEOREM 8. If the direction cosines of an undirected line are pro-
portional to three given numbers, then their actual values are equal
to these numbers, each divided by the square root of the sum of their
squares.

For, if X = ka, /* = kb and v = kc, then it follows from Theo-

rem 7 that fc 2 (a 2 + 6 2 + c 2 ) = 1, so that k = , 1

Vtt 2 + b 2 + c 2

The ambiguity of sign in the square root corresponds to the possi-
bility of two directions on the undirected line.

THREE COLLINEAR POINTS 57

34. Three Collinear Points. If A, B and P are points on a line,
the direction cosines of the segments AP and AB are either equal
or else equal numerically but opposite in sign, thus :

\4P = X AB> HAP = f *AB> V AP = V AB>

where the upper signs are to be used if the segments AP and AB
have the same direction and the lower signs if they have opposite
directions. And, if the points A, B and P are not collinear, not
all of these relations can hold. By means of Theorem 6 we derive
from these relations the following equations:

XP - x A = X B ~ x A VP ~ VA 'UB ~ VA

AB ' A~P AB

ZP - Z A = Z R - Z A m

AP AB '

or _

XP - X A = HP ~ HA = ZP ~ ZA = AP
XB ~ XA 1/B ~ VA Z B ~ ZA ~AB

Moreover, if the segments AP and A B have the same direction
we can take AP = +AP and AB = -\-AB; whereas if they have
opposite directions, we can take AP = +AP and AB AB.

AP AP AP

It follows therefore that we have-r-^ = + =or = according

At> AB AB

as the segments AP and AB have the same or opposite directions.
We obtain therefore the following important result.

THEOREM 9. The necessary and sufficient condition that the point
P(** y s) shall lie on the line through A and B is that the coordinates
x, y, s must satisfy the linear equations

*B-*A ys-yA S B-*A AB

Remark 1. It should be clear that by starting with the seg-
ments BP and BA we find the equations

x - X B = y - y B = z - Z B ^ BP m
XA - X B VA ~ VB Z A - Z B BA '

and if we use the segments AP and PB, the resulting equations are

x - XA = y ~ VA = * - ZA = AP ^
x B - x VB - V Z B - z PB'

58 POINTS AND LINES

Remark 2. The formulas established in Theorem 9, and also
those given in Remark 1, enable us to determine the coordinates
of the point P on the line AB as soon as the coordinates of the

AP
points A and B are known and also the ratio - of the segments

AP and PB in which P divides the segment AB. For, if AP : PB
= n : r 2 , then AP : AB = r\ : n + r 2 , and BP : BA = r 2 : r\ + r 2 .
Hence we have

x XA r\ x - X A n x - X B r 2

XB ~ x r* X B - X A n + r 2 ' X A - X B n + r 2
From either of these equations we find for the ^-coordinate of P:

_
Xp ~

and similar results are obtained for yp and zp. We have therefore
the following further result.

COROLLARY. If the points A, B, and P are in a straight line and if
AP : PB = ri : r 2 , then the coordinates x, y, s of P are given by the
formulas:

r& A -f r& B = r*y A + r,y g ^ r 2 s^ + r lgg

~" n + ^2 ' y n + r a ' ri -f r 2

Remark 3. The formulas of this corollary are very useful for
later developments. But they hide to some extent the simple
geometrical fact from which they have been derived and which
finds more direct expression in the formulas of Theorem 9 and
Remark 1. It is advisable therefore, particularly in the begin-
ning and in numerical problems, to go back to these earlier formulas
rather than merely to substitute in the formulas of the corollary.

Remark 4. The first three terms in the equations of Theorem
9 give two independent linear equations which the coordinates of
any point on AB must satisfy; the same statement may be made
for the equations in Remark 1. If we consider also the last term
in each of these cases and here the last set of equations in
Remark 1 is particularly useful they give us three equations
which express the coordinates of an arbitrary point P on the line
in terms of the coordinates of A and B, and of one parameter, or

auxiliary variable, namely, r = - , that is, in terms of the ratio r

7*2

THREE COLLINEAR POINTS 59

of the segments into which P divides the segment AB. This pa-
rameter r varies as the point P moves along the line AB. When
P coincides with A, r = 0; as P moves from A to B r increases.
When P lies outside the segment A B, either on the side of A or
on the side of B, r is negative; and as P moves off indefinitely along
the line in either direction, r tends towards 1. For

AP AB + BP

= _
PB ~~ PB '

since AB is fixed and PB increases numerically when P moves off
along the line, the first term on the right tends toward zero and
hence r tends toward 1. The reader will find it worth while to
make clear to himself in detail the manner in which the parameter
r varies as P occupies various positions on the line AB.

If we combine Theorems 6 and 9, we are led to the following
valuable theorem.

THEOREM 10. The necessary and sufficient condition that a point
P(x 9 y 9 z) shall lie on the directed line through A whose direction cosines
are A, /u, and v is that the coordinates x 9 y, and s must satisfy the
equations

For if P lies on the specified line and if B is another arbitrary,
but fixed, point on the line, then the equations of Theorem 9 hold.
But, in accordance with Theorem 6, we have

y B - y A = /z AB, Z B - Z A = v AB.

If these values are substituted in the equations of Theorem 9,
the desired result is obtained.

COROLLARY 1. The necessary and sufficient condition that a point
P(x 9 y, s) shall lie on the undirected line through the point A whose
direction cosines are proportional to 1 9 m and n is that the coordinates
x 9 y and s shall satisfy the equations

*-*A = y-y* = * ~*A = AP

I m n Vl* + m 2 + n 2

This corollary follows from Theorem 10 in combination with
Theorem 8. The ambiguity in the sign of the square root corre-
sponds to the possibility of two directions on the line of which only
the ratios of the direction cosines are given.

60 POINTS AND LINES

The observation made in Remark 4 (page 58) leads from Theo-
rem 10 and Corollary 1 to two further results of importance.

COROLLARY 2. The coordinates of a point P(x 9 y, a) on the line
through A whose direction cosines are equal to X, M v are given by the
equations

x = X A -f Xs, y = y A + us, z = Z A -f- vs,

where s designates the signed measure of the directed segment AP\
conversely, any point P whose coordinates are equal to these expres-
sions lies on the specified line at a directed distance from A equal to s.
COROLLARY 3. The coordinates of a point P(x 9 y, s) on the undirected
line through A whose direction cosines are proportional to /, m, n are
given by the equations

*=* A + lt 9 y=y A + mt, * - ^ + iu, where t = ^ + = ;

conversely, any point P whose coordinates are given by these ex-
pressions lies on the specified line at a distance from A equal to

t V> -f m 2 -f rt 2 .

Remark. The equations of Corollary 1 are frequently referred
to as the "symmetric equations of the line 7 '; those of Corollaries
2 and 3 as the " parametric equations of the line/' The variable
s, or t, which changes as P moves along the line, is the parameter.
The terminology "equations of a line" will be more fully justified
in Chapter IV (see Section 47, page 83).

The parametric equations of the line are used a great deal in the
sequel. The reader is urged to master thoroughly the methods
by which they have been obtained. It should be observed more-
over that the equations stated in the Corollary of Theorem 9 are
also parametric equations of the line, the parameter being the

ratio r (see Remark 4, page 58).
i

35. Exercises.

1. Construct the coordinate parallelepipeds of the following pairs of points,
and determine their distances and the direction cosines of the lines joining
them:

(a) 4(5, 2, -1) and B(-3, -4, 2); (6) 4(2, 4, 5) and 5(7, 1, 1);
(c) 4(2, 3, 4) and B(5, -2, 7); (d) 4(3, -2, -1) and B(-3, 4, 5);
(e)4(-4,3,5)and(-4, -2,0); (/)4(3,4,5)andB(-3, -4, -5);
(g) 4(0, 3, 6) and J5(4, -1, 6); (h) 4 (-2, 3, 5) and B(-2, 3, -1).

THE ANGLE BETWEEN TWO LINES 61

I. Determine which of the following sets of points are collinear:

(a) A(3, -1, 4), B(-2, 4, -1), (7(1, 1, 2); (6) A(Q, 0, 0), B(2, 5, -3),
C(4, 10, -6); (c) A(l, -2, 3), B(-l, 2, -3), C(-3, 5, 0);
(d) A(-2, 2, 3), B(l, -1, 0), C<7, -7, -3); () ^(0, -3, 1),
B(4, -2, -1), C(2, -4, 3); (/) 4(5, 2, 7), B(l, 5,5), (-3,8,3).

\Z. Determine the coordinates of the point at which the segment from
A(3, 2, 5) to B(5, 4, 2) is bisected; also the coordinates of the points at
which this segment is trisected.

& The line of the preceding problem is extended beyond B to a point C
such that (a) BC = AB; (6) BC = 3 AB; (c) BC = \ AB. Determine the
coordinates of C in each case.

*0. Find the coordinates of the center of mass of the homogeneous triangle
whose vertices are A(-2, 5, 4), (3, -1, -2) and C(8, -7, 4). (The center
of mass of a homogeneous triangle is the point of intersection of the medians.)

6. Determine the coordinates of the point in which the side AC of the tri-
angle of Exercise 5 is met by the bisectors of the interior and the exterior
angles at B. Find also the direction cosines of the bisectors.

7. Show that the points A(-3, 2, 5), B(l, 0, 1) and C(ll, -5, -9) are
collinear and determine the ratio of the segments AC : CB.

8. On a line through the point .4(5, 4, 2) whose direction cosines are pro-
portional to 2, 1 and 2, a point B is determined such that AB = 4.
Find the coordinates of B. How many positions arc possible for B?

9. Determine the center of mass of the homogeneous triangle whose vertices
are at Pi(xi, yi, z), * = 1, 2, 3.

10. Show that the three lines which join the midpoints of the three pairs of
opposite edges of the tetrahedron PiP 2 PsP4 have a common midpoint. This
common midpoint is called the center of mass of the homogeneous tetrahedron
PiP 2 PaP4. (Opposite edges of a tetrahedron are edges which have no point
in common.)

II. Prove that any vertex of a homogeneous tetrahedron, the center of
mass of the opposite face and the center of mass of the tetrahedron are col-
linear; determine the ratio of the segments in which the center of mass of the
tetrahedron divides the segment determined by the other two points.

12. Show that if and only if P(x, y, z) lies on the sphere of radius r, whose
center is at (7(a, 6, c), the coordinates x, y, z satisfy the equation

(x - a) 2 + (y - 6) 2 + (z - c)* = r 2 .

13. Determine the equation satisfied by the coordinates of all points which
are at the same distance from A ( 2, 1, 3) as from B(4, 2, 0).

14. Determine the equation satisfied by the coordinates of all points on the
surface of the sphere whose center is at C(3, 2, 3) and whose radius is 5.

15. Determine the equation satisfied by the coordinates of all points whose
distance from A(l, 3, 4) is twice as great as their distance from B( 2, 0, 2).

36. The Angle Between Two Lines. The Projection Method.

To calculate the angle between two lines whose direction cosines

62

POINTS AND LINES

are known, we shall use a method which will find frequent applica-
tion in the sequel and which is based on the following two theorems.

THEOREM 11. The projection of a segment AB of a directed line I
upon a directed line m is equal to the product of AB by the cosine of the
angle between the two directed lines.

Proof. We distinguish two cases, according as the lines I and
m do or do not lie in one plane.

(a) For the case when the lines I and m lie in one plane, the
proof can be found in most books on Plane Analytical Geometry
and in some books on Trigonometry.* For this reason we shall

not repeat the proof here.

(6) If the lines I and m are
not coplanar, let the angle be-
tween them be 6. We construct
a line m r through the point A on
Z, parallel to m (see Fig. 3); the
angle between m' and I will then
also be equal to 6. Through A
and B we construct planes per-
pendicular to m; these planes
will then also be perpendicular
to m'. If the points in which

these planes meet m and m' are C, D and A, D f respectively, then
CD = AD' (Why?). From these facts, in combination with part
(a) of this proof, we conclude that

Proj m AB = CD = AD' = AB cos 0.

This proves our theorem.

THEOREM 12. The sum of the projections upon a directed line m
of the segments of a closed broken line in space is equal to zero.

Proof. If the vertices of the broken line in space are A, B, C,
. . . , P and if their projections on the directed line m are .4', B', C',
. . . , P', we have to show that A'B' + B'C' + . . . + P'A'
= 0. That this is indeed the case follows from a fundamental
theorem, of which a proof is found in books on Plane Analytical
Geometry and on Trigonometry,! according to which the sum of

* See, for example, the author's Plane Trigonometry, p. 36.
t See, for example, the author's Plane Trigonometry, p. 4.

FIG. 3

THE ANGLE BETWEEN TWO LINES

63

the directed segments of a line, of which the end point of the last
segment coincides with the initial point of the first segment, is
equal to zero. We conclude therefore that

Pro} m AB + Pro] m BC + . . . + Pro] m PA = A'B' + B'C'
+ . . . + P'A' = 0.

We proceed now to the determination of the angle 6 between
two directed lines I and m. We construct first the c.p. of two
points A and B selected arbitrarily on one of the lines (see Fig. 4
in which A and B are taken
on Z); and we apply Theorem
12 to the closed broken line
ABDCA. Thus we obtain the
equation

Pro'] m AB + Proj w Z) +
Pro] m DC + Pro] m CA = 0.

The segment AB lies on I, the

segments BD y DC and CA on

lines which are parallel to the

Z-j Y-, and X-axes; the angles

which these lines make with the

line m are respectively equal to 0, y mj p m , and a m , the last three

angles being the direction angles of m. The last equation written

above leads therefore by means of Theorem 11 to the statement

that

AB cos 6 + BD v m + DC /% + CA X m = 0.

Moreover BD, DC and CA are equal to the projections of BA on
the Z-, F-, and X-axes respectively, so that it follows from Theorem
11 that

CA = BA

Fio. 4

BD = BA

DC = BA

and

If we substitute these values in the preceding equation and re-
member that BA = AB ^ 0, we obtain the following result:

THEOREM 13. The cosine of the angle between two directed lines is
equal to the sum of the products of their corresponding direction
cosines.

64

POINTS AND LINES

Remark. If we restrict angles between directed lines to lie be-
tween 180 and +180, the result contained in Theorem 13
determines the magnitude of angle 6 but leaves the sign of this
angle ambiguous. This ambiguity corresponds to the fact that
either of the two lines may be taken as the initial side of the angle.

COROLLARY 1. The necessary and sufficient condition that the lines
I and m arc perpendicular is that \i\ m + ^ m _j_ vp m = o.

COROLLARY 2. If the direction cosines of two undirected lines are
proportional to fi, mi, n t and 1 2 , m 2 , n 2 , the necessary and sufficient
condition for the perpendicularity of the lines is that lih +
= 0.

By means of Theorem 13 we can determine the numerical values
of all the trigonometric ratios of the angles between two directed
lines. On account of its special interest we develop a formula for
the sine of these angles. This is done most conveniently by
means of the following auxiliary theorem, which is of some interest
on its own account.

LEMMA. The following identity holds between any six numbers,
real or complex, a, 6, c and 01, 61, ci:

(a 2 +

+ c 2 )

b

+ *>i 2 + ci 2 ) - (aa,

c o I 2
* ml"

f-Wn + <
&
61

The proof of this formula requires merely that the indicated
operations on both sides of the equation shall be carried out; the
identity of the two sides will then at once become apparent.

If this lemma is applied to the direction cosines X /; jjL lf v l and
\ f Mm, VM of the lines I and m, we find that

(A/ 2 + M/ 2 + "/ 2 ) (X m 2 + ^ + O

Mm

v l

-h

X,
X*,

Mm

But in view of Theorems 7 and 13 this result leads to the equation

cos 2 B =

Mm v m

M/
Mm

THEOREM 14. The square of the sine of the angle between two di-
rected lines is equal to the sum of the squares of the two-rowed minors
which can be formed from the matrix constituted by the two sets of

THE ANGLE BETWEEN TWO LINES 65

direction cosines of the lines; that is,

2

sin 2 =

M/

M m "...

87. Exercises.

1. Determine the cosines of the angles between the lines whose direction
cosines Xi, MI, v\ and X 2 , M2, "2 are given by the following data:

(a) Xi : MI : "i = -2 : 1 : 2, X 2 : M2 : "2 = 2 : 6 : -3

(6) Xi : MI : "i = 2 : 3 : 6, X 2 : M2 : i* = 3 : 14 : 18

(c) Xi : MI : ?i = 3 : -2 : -6, X 2 : M2 : z = 1 : - 2 : -2

(d) X! : MI : "i = 4 : : 3, X 2 : M2 : "2 = 1 : 2 : 3

2. Determine the cosines of the angles formed by the sides of the triangle
whose vertices are the points A (3, 1, 4), B( 2, 4, 1) and (7(1, 1, 2).

3. Test for perpendicularity the pairs of lines whose direction cosines are
given by the following data:

(a) Xi : MI : *i = -2 : 1 : 2, X 2 : M2 : ^2 = 1 : -2 : 2
(6) Xi : MI : "i = 3: - 1 : 2, X 2 : M2 : *> 2 = 1 : 1 : -1
(c) Xi : MI : v\ = 2 : -3 : 6, X 2 : M2 : "2 = 1 : 2 :

4. Develop a formula for the tangent, the cotangent, the secant and the
cosecant of the angle between two lines.

5. Determine the direction cosines of a line which is perpendicular to the
two lines whose direction cosines are proportional to 3 : 2:4 and 1 : 3 : 2.

6. Determine the direction cosines of a line which makes equal angles with
the three lines whose direction cosines are proportional to 1, 4, 8; to 8, 1,
4; and to 1, 2, -2.

38. Miscellaneous Exercises.

1. Determine the conditions which the coordinates of a point must satisfy
in order to be equally distant from the points A (2, 1, 0), B( 3, 2, 1) and
C(l, 3, -2).

2. Solve the same problem for the points Pi(x{, yi, 2;), i = 1, 2, 3.

3. Determine a point which is equally distant from the four points A (2, 1, 3),
B(l, 1, 2), C(2, 0, 5), and D(2, 0, 3).

4. Establish the condition on the coordinates of the four points Pi(xi, yi, zi),
i = 1, 2, 3, 4, necessary and sufficient for the existence of a single point that
is equally distant from these four points.

5. Set up the equation which is satisfied by the coordinates of any point
on the sphere which passes through the four points A (3, -2, 4), B(2, -3, 2),
C(4, -2, 2) and D(5, -1,3).

6. Determine the direction cosines of a line perpendicular to the two lines
whose direction cosines are X,-, M, "*> i = 1, 2. Does this problem always have
a solution? Does it ever have more than one solution?

7. Remembering that a line is perpendicular to a plane if it is perpendicular
to two lines in the plane, determine the direction cosines of a line which is

66 POINTS AND LINES

perpendicular to the plane of the triangle whose vertices are the points
4(3, -2, 4), (4, 0, 2) and C(0, -4, -2).

8. Determine the condition which must be satisfied by the direction cosines
X,', /if, i/;, i 1, 2, 3 of three lines in order that there may exist one or more
lines perpendicular to these lines.

9. Show that if the coordinates of the three points Pi(xi, yi, zi),i = 1, 2, 3
satisfy an equation of the form ax + by + cz = 0, in which a, 6, c are not all
zero, then there exists at least one line which is perpendicular to the three
lines OP,, i = 1, 2, 3.

10. Determine the direction cosines of a line which is to make equal angles
with the three lines whose direction cosines are X,-, m, vi, i = 1, 2, 3. Does
this problem always have a solution? Can it have more than one solution?

11. Determine the direction cosines of a line which makes equal angles with
the three lines connecting the origin with the points 4 (2, 3, 6), 5(1, 8, 4),
and C( 1,2, -2).

12. The line which joins 4(1, 3, 5) to B(15, -15, 8) is produced beyond B
to a point P such that BP 5. Determine the coordinates of P.

13. Prove that every point whose coordinates satisfy the equation

z 2 + ?/ 2 + z 2 + 2 ax + 2 by + 2 cz + d =

lies on the surface of a sphere. Determine the center and the radius of this
sphere.

14. Determine the center and the radius of each of the spheres represented
by the following equations:

(6) x 2 -f t/ 2 + z 2 + 6 x - 2 y + 4 z - 2 =

(c) x 2 + y 2 -f z 2 -f 2 x - 6 y -f 8 z -f 26 =

(d) x 2 + ?/ 2 + z 2 - 8 x -f 4 y -\- 6 z - 33 =0

16. Two lines mi and m* meet at a point under an angle 6, = 180. A line
l f not necessarily in the same plane with mi and m 2 , makes angles cti and 2
with mi and m z respectively and an angle with a line in the plane of mi and
ma, perpendicular to m iy and on the same side of mi as w 2 . Prove that cos ft

C080

cos i cos a 2

of a frame of reference, the line m l as X-axis, and the pljfllHHand m 2 as
XF-plane. ^^^H

16. If, in the configuration of the preceding exercise, y I^Wmgle which
I makes with a line perpendicular to the plane of mi and m%, then cos y sin
= =fc [1 cos 2 i cos 2 a 2 cos 2 d + 2 cos i cos 2 cos 0]*

sin d = cos 2 ~ cos i cos 6 =

Hint: Take as the origin

= =fc

1 COS 6 COS ai
COS 1 COS 2
COS <Xi COS 2 1

the + or sign is to be used according as I and the perpendicular to the plane
of mi and w 2 point to the same side or to opposite sides of this plane.

CHAPTER IV

PLANES AND LINES

39. Surfaces and Curves. If a point P is to be chosen at
random in space, its determination will, in the coordinate system
which we have used thus far, depend upon three independent
choices of arbitrary real numbers, namely, of an x-, a ?/-, and a
^-coordinate, each of which can be selected without consideration
of the selections made for the other two. This fact is expressed
in the statement that a point in space has three degrees of freedom.

If a point is to be chosen at random on a surface such as we are
likely to meet in ordinary experience (we may think here of sur-
faces which limit the objects in our environment, such as tables,
bottles, lamp shades, trees, etc.)
the determination will depend
upon two independent choices
of arbitrary real numbers; for,
after two coordinates have been
chosen at random, the x- and y-
coordinates for example, the third
one, the 2-coordinate, must be so
chosen as to furnish a point on
the surface (see Fig. 5) ; and this
will leave, at least in the case of
surfaces of ordinary experience,
in general a choice among a finite
number ofvalues. For this rea-
son a poiiltt|ferface is said to have two degrees of freedom.

And if ^^Hlps to be chosen at random on a curve (again our
reference I51n the first place to curves of cornmOn experience)
only one coordinate can be selected arbitrarily; a point on a curve
is therefore said to have one degree of freedom.

It should be clear from this discussion that to restrict a point in
three-space to a surface, we have to impose one condition on its
coordinates; and to limit a point in three-space to a curve, we
shall have to subject its coordinates to two independent conditions.

67

KNJ.

68 PLANES AND LINES

These considerations, vague and inconclusive though they are,
suffice perhaps to indicate that there is some justification for the
following definitions.

DEFINITION I. A surface is the totality of all points in three-space
whose Cartesian coordinates satisfy one equation.

Remark. The equation which is thus laid down by definition
as the algebraic counterpart of the surface, is called the equation
of the surface ; and the surface is referred to as the locus of the
equation.

DEFINITION II. A curve is the totality of all those points in three-
space whose Cartesian coordinates satisfy two independent equations*

We shall speak of the "equations of a curve" and of the "locus
of a pair of equations/'

Remark. These definitions do not specify sharply the con-
cepts "surface" and "curve" because the word "equation" used
in them is left without specification. According as the class of
equations that is taken into consideration is widened or narrowed,
we shall presumably enlarge or restrict the concepts "surface"
and "curve." The remarks preceding the definitions are intended
to make clear that the surfaces and curves of ordinary experience
are included among the "surfaces" and "curves" introduced by
the definitions.

The surfaces and curves which are the loci of algebraic equa-
tions and pairs of algebraic equations, respectively, are called
"algebraic surfaces" and "algebraic curves." In this book we
shall be concerned exclusively with surfaces (and curves) which
are loci of the simplest types of algebraic equations in three vari-
ables (pairs of such equations), namely, of algebraic equations of
the first and second degrees. We shall, however, have to deal
occasionally with loci of equations of a more general character;
and we shall begin with some considerations of a general nature.

40. Cylindrical Surfaces. Systems of Planes. What can be
said about the space locus of an equation like x* + y z = 4, in
which only two of the variables are present? The plane locus of
such an equation consists of the points on the X F-plane, whose
coordinates satisfy the given equation; it is therefore a plane
curve. Since the equation does not restrict the ^-coordinate, a

CYLINDRICAL SURFACES 69

point P(a y 6, c) will belong to its locus if and only if the point
P xy (a, bj 0) does, that is, if and only if the projection of P on the
-XT-plane belongs to the plane locus of the equation. Conse-
quently the space locus of this equation will be generated by a
line which moves parallel to the Z-axis and which passes succes-
sively through the points of the plane locus of the equation.
We introduce now the following terminology:

DEFINITION III. A cylindrical surface is a surface generated by a
line which moves in such a way as to be always parallel to a fixed
line and in such a way as to pass through the points on a fixed plane
curve. Any position of the movfng line is called a generating line
(generatrix); the fixed plane curve is called the directrix. If the
generatrix is perpendicular to the plane of the directrix, we have a
right cylindrical surface; if not, an oblique cylindrical surface.

By the aid of this terminology we can express the result of the
foregoing discussion in the following form.

THEOREM 1. The locus of the equation /(*, y) = is a right cylin-
drical surface whose generating line is parallel to the Z-axis and whose
directrix is the plane locus of the equation.

Remark 1. Similar theorems hold of course concerning the loci
of equations from which the variable x or the variable y is absent.
It will be good practice for the reader to present in full the argu-
ment for these cases. The locus of the particular equation
#2 _|_ y2 _- 4 j s a right circular cylindrical surface, whose gener-
ating line is parallel to the Z-axis and whose directrix is a circle in
the XF-plane with center at the origin and radius equal to 2.

Remark 2. By means of this theorem the whole field of Plane
Analytical Geometry is seen to be a province in the domain of
Solid Analytical Geometry. For it shows that the determination
of the space loci of equations in two Cartesian variables depends
upon the determination of the plane loci of such equations.

In particular the space locus of a linear equation in two variables,
for example, of the equation ay + bz + c = 0, is a cylindrical
surface of which the directrix is a straight line and the generatrix
parallel to the Z-axis; the locus of this equation is therefore a
plane parallel to the X-axis.

If an equation contains only one Cartesian variable, its plane
locus is a set of lines, real or complex, parallel to one of the coordi-

70

PLANES AND LINES

nate axes. Its space locus is therefore a set u* planes, real or
complex, parallel to one of the coordinate planes. For instance,
the locus of the equation # 2 7# + 12 = consists of two planes
parallel to the FZ-plane, at distances of 3 and 4 units from the
FZ-plane.

We recall that the geometric statement "a point P belongs to a
certain locus" has as its algebraic equivalent "the coordinates of
P satisfy the equation of a locus/' With this fact in mind, we
turn to a particularly interesting case of an equation in two vari-
ables, namely, the equation obtained by eliminating one of the
variables from two equations in three variables. Suppose that
we eliminate z from the equations f(x, y, z) = and g(x, y, z) = 0,
and that the result of the elimination is the equation F(x, y) = 0.
What can be said about the loci of these three equations?

In the first place it follows from Theorem 1 that the locus of
the equation F(x, y) = is a cylindrical surface parallel to the
Z-axis. On the other hand, since it is satisfied (provided the
elimination has been carried out correctly) by all values of x, y,
and z which satisfy the equations /(x, ?/, z) = and g(x } y, z) = 0,

it must pass through all tho
points common to the two
surfaces which these two
equations represent, that is,
through their curve of in-
tersection (see Definitions
II and I). The locus of
the equation F(x, y) = is
therefore the cylindrical sur-
face which projects upon the
-XT-plane the curve repre-
sented by the equations

f(x, y, z) = and g(x, y, z) = 0, see Fig. 6. Moreover the plane
locus of the equation F(x, y) = 0, consisting of the points in the
-XT-plane whose coordinates satisfy the equation, is clearly the
projection of this curve on the .XT-plane. A similar statement
can be made concerning the equation obtained by eliminating x or
y from the two given equations. We have obtained therefore the
following theorem.

THE LINEAR EQUATION 71

THEOREM 2. The equation in two variables, obtained by eliminating
one variable from two equations in three variables, has as its space
locus the cylindrical surface which projects the curve represented by
the two equations upon the plane of the two remaining variables;
and as its plane locus the projection of the curve on the same coordi-
nate plane.

41. The Linear Equation ax + by + cz + d = 0.

DEFINITION IV. A plane is a set of points of such character that if
any two points A and B belong to the set, then every point on the line
Joining AB also belongs to it.

On the basis of this definition it is a simple matter to prove the
following theorem.

THEOREM 3. The locus of any equation of the first degree in *, y,
and a is a plane.

Proof. The most general equation of the first degree in x, y, and
z is ax + by + cz + d = 0. If an arbitrary point P is taken on
the line AB, its coordinates will be, according to the corollary to
Theorem 9, Chapter II (see Section 34, page 58)

_ _ r 2 y A

XP ~ ' yp ~ ' **

rt+r, ' ~ r, + r 2 ' " r, + r, '

when TI : r 2 = AP : PB. We have to show therefore that the
identities ax A + by A + CZ A + d = and ax B + by B + CZ B + d s=
have as a consequence the identity

, ^ A s , A i B , d s

"~

r 2 n r 2

If we write this equation in the equivalent form obtained by clear-
ing it of fractions (n + r 2 3p 0) and collecting the terms in r\ and
those in r 2 , namely,

by A + CZ A + d) + ri(ax B + by B + CZ B + d) ss 0,

it should be indeed evident that it results from the two given iden-
tities. The theorem has therefore been proved.
Next we shall establish its converse.

THEOREM 4. The equation of any plane is a linear equation in x, y,
and

72 PLANES AND LINES

Proof. We shall divide the proof into four parts.

(a) Suppose that the plane is parallel to two of the coordinate
axes, that is, to one of the coordinate planes; let us say for the sake
of definiteness that the plane is parallel to the ZF-plane at a dis-
tance k from it. Then for every point on the plane, and for no
other points, the x-coordinate is equal to fc. Therefore the equation
of this plane is x k = 0, which is obviously a linear equation.

(6) If the plane is parallel to one coordinate axis, for example,
the Z-axis, let the line in which the plane meets the Jf F-plane, re-
ferred to the X- and F-axes, have the equation ax + by + c = 0.
It follows then that the equation of the plane is also ax + by + c
= 0, see Theorem 1 and Remark 2 following it (Section 40, page
69) ; this is again a linear equation.

(c) Suppose that the plane cuts all three axes but does not pass
through the origin. Let it cut the X-axis in the point P(p, 0, 0),
the F-axis in the point Q(0, q, 0), and the Z-axis in the point
72(0, 0, r); then p y q, and r are all three different from zero. And

3* 77 Z

the given plane must be the locus of the equation (- ~ + - = 1.

For this equation is linear; its locus is therefore a plane, by virtue
of Theorem 3. Moreover, the points P, Q, and R clearly belong to
this locus since their coordinates manifestly satisfy the equation.
Hence the given plane and the plane which is the locus of this
equation have three points in common; therefore they coincide.

(d) Suppose finally that the plane passes through the origin.
We select in the plane two points PI(XI, y\, z\) and P 2 (#2, 2/2, 2) not
collinear with 0; and we seek to determine three numbers a, 6, and
c, not all zero, such that ax\ + byi + cz\ = and ax 2 + by 2 + cz z
= 0. Is this possible? Yes, for since the points 0, Pi, and P 2 are
not collinear, the coordinates x\ 9 y\, z\ are not proportional to the
coordinates x 2 , y^ 2 (see Corollary 3 to Theorem 6, Chapter III,
Section 33, page 56 and Exercise 7, Section 31, page 53) and

therefore not all the two-rowed minors of the matrix l ^ M

%2 2/2 3> II

can vanish, that is, the rank of this matrix is 2. But this fact
enables us to conclude, by means of Theorem 4, Chapter II
(Section 25, page 41) that we can indeed determine the numbers
a, 6, and c so as to satisfy the conditions mentioned above. The
locus of the equation ax + by + cz = has therefore the three

THE LINEAR EQUATION 73

points 0, PI, and P% in common with the given plane; since this
locus is moreover a plane (Theorem 3) it coincides with the given
plane.

This completes the proof of the theorem.

Remark. 1 The directed distances p, q, and r from the origin
to the points in which a plane cuts the coordinate axes are called

the intercepts of the plane. The equation - + - + - = 1 which

p q r

can be written down as soon as the intercepts of a plane are known,
is usually referred to as the intercept form of the equation of the
plane. It is a special case of the equation of the plane in terms of
the coordinates of three of its points.

COROLLARY. The intercepts of the plane which Is the locus of the
equation ax -f by ~h ex -f d = 0, In which neither a, nor 6, nor c are

equal to zero, are equal to , , , and .

a o c

Remark 2. It follows from this corollary that if d is a constant,
different from zero and a, 6, and c are variables which tend to
zero, then the intercepts of the plane represented by the equation
ax + by + cz + d = increase beyond all bounds and hence the
plane moves farther and farther from the origin the usual
phrase is " the plane moves to infinity." This is the sense in which
we are to understand the statement that the equation x + y
+ z + d = represents the " plane at infinity." A satisfac-
tory treatment of the question here hinted at belongs in the field
of Projective Geometry; we shall not undertake such treatment
in this book. Whenever it becomes desirable to recognize ex-
plicitly that the plane under discussion is not the "plane at
infinity" we shall speak of a plane at finite distance.

THEOREM 5. The plane which passes through the three non-colllnear

y
points Pi, P 29 and P 8 has the equation

y\
y*

Proof. The required equation is linear in x, y, and z, by virtue
Df Theorem 4. If therefore P(x t y, z) is any fourth point of the
plane determined by the non-collinear points PI, F 2 , and P 3 , there
tnust exist four numbers a, 6, c, and d which are not all zero, such

74 PLANES AND LINES

that

ax + by + cz + d = 0,

axi + byi + czi + d = 0,

ax 2 + by 2 + cz 2 + d = 0,
and axz + by* + C2 3 + d = 0.

This is a system of four linear homogeneous equations in the four
variables a, fr, c, and d. In order that this system of equations
may possess a non-trivial solution, it is necessary, in view of
Theorem 2, Chapter II and its corollary (Section 22, page 38), that
the value of the coefficient determinant must be zero. Hence the
coordinates of an arbitrary fourth point of the plane PiP 2 Ps must
x y z 1

2/1 2i 1

y-2, 2 1

?/3 23 1

satisfy the equation

- 0.

On the other hand this equation is linear in x, y, and z, in view of
Theorem 12, Chapter I (Section 7, page 13). It is moreover satis-
fied by the coordinates of the points PI, P 2 , and P 3 , as should be
evident by use of Theorem 7, Chapter I (Section 5, page 9) ; there-
fore its locus is the plane determined by PI, P 2 , and P 3 .

The equation established in this theorem is usually referred to
as the three-point form of the equation of the plane.

42. Exercises.

1. How many equations are required to specify a curve in a space of four
dimensions? To specify a surface in a four-space? To specify a three-space
in a four-space?

2. Formulate a general statement of which the answers to the preceding
exercise and the statements preceding the definitions in Section 39 are special
cases.

3. Determine the loci in three-space of the following equations:

(a) I _ |! = 1 (6) x * -f 2/2 + Z 2 16

( c ) 4 z = y i (d) x* - 5 x + 6 =

(e) 4 * 2 + 6 2/ 2 - 12 = (/) z 3 - 6 z 2 + 11 z - 6 -

4. Show that if PI, PI, and Pa are collinear points, the equation in Theorem
5 is satisfied identically, that is, for all values of x, y, and z.

6. Show that, if three points lie on a plane which passes through the origin,
the determinant whose rows are the coordinates of these points, all taken
in the same order, has the value zero.

THE DISTANCE FROM A PLANE TO A POINT 75

6. Determine the point on the line through the points A( 4, 2, 5) and
B(l, 3, 2) which also lies on the plane determined by the points Pi(0, 2, 1),
P 2 (-6, -2, 0) and P(-4, 0, 1).

7. Determine four points which lie on the plane 2x 4y + z + 7=Q.

8. Determine four points which lie on the locus of the equation 3 a: 2 4 y*
+ 5 * 2 = 22.

9. Determine three points which lie on the curve whose equations are

z + 2?/-32 = 5 and 2 x - 3 y + z = 3.

10. Determine three points on the curve which is the locus of the pair of

equations

2 x - y -h 2 z = 9 and x 1 -f y* + z 2 = 26.

11. Determine the equations of the projections on each of the coordinate
planes of the curve of the preceding exercise.

12. Determine the equations of the planes which pass through the following
sets of three points each; find the intercepts of each of these planes:

(a) P t (l, 2, 3), P 2 (2, 3, 4),

*M3, 5, 7)

(6) PiC-2,3,4), /M-1,2,5),

P 3 (7, 0, 2)

(c) P,(3, -2,5), P 2 (-2, 1, 3),

P 3 (8, -3, 7)

(d) P,(-4, 5, -2), P 2 (-4,3, 1),

P s (-4, -7,3)

(e) Pi (2, 4, -5), P 2 (-3, 1,2),

P a (-5, 11, -4)

(/) Pi(3, -4,2), P a (-2, -5, 1),

P 8 (-l, -2,4)

13. Set up the condition which the coordinates of three points must sa v
in order that the plane determined by them shall be parallel to (a) the F-pl
(b) the 2LY-plane; (c) the ^TF-plane; the Jf-axis; the F-axis; the Z-axihe

14. Determine the conditions which the coordinates of three points m^ r
satisfy in order that they lie on a line.

43. The Distance from a Plane to a Point. To determine the
distance from a plane to a point we make use of the projection
method explained in Section 36; we divide the discussion into
two parts.

(a) The plane does not pass through the origin.

Suppose that the direction cosines of the directed perpendicular
from the origin to the plane arc X, M, and v and that the positive
direction on this line is taken to be the direction from the origin to
the plane. * Let the unsigned length of the distance from the origin

* This agreement as to the positive direction on the perpendicular from
the origin to the plane is entirely arbitrary; if the opposite agreement were
made, the interpretation of some of the results obtained in the following
pages would be different, but equally useful. The convention adopted here
is in accord with general practice. It would be a good exercise for the reader,
after having thoroughly mastered the next few sections, to develop this part
of the work on the basis of the opposite convention.

76

PLANES AND LINES

to the plane be designated by p and the foot of the perpendicular
by H (see Fig. 7). Suppose furthermore that the given point is
P(a, ft, 7) and its projection on the given plane is Q.

FIG. 7

We consider now the projection on the directed line / of the closed
broken line which goes from to P along the edges of the c.p. of
and from P back to by way of Q and H. By virtue of Theorem
Chapter III (Section 36, page 62), we find that

Proj/OA + Proj/AJS + Proj/JSP + Proj/PQ + Proj,Q#
+ Proj/#0 = 0.

If we evaluate these projections by means of Theorem 11, Chap-
ter III (Section 36, page 62), noticing that QP \\ I and QH J_ I,
we conclude that

a\ + ftu + 7" + PQ + + HO = 0.

Here we have to bear in mind that the sign of PQ is to be taken
in accordance with the direction specified on Z, also that HO = p.
Accordingly we obtain the result given in the following theorem.

THEOREM 6. The distance from a plane which does not pass through
the origin and for which the perpendicular directed from the origin
to the plane has direction cosines X, /* and t> 9 and unsigned length />,
to the point P(, /?, 7) is equal to

<?P = a\ + fo + yv - p.

(b) In case the plane passes through the origin, the specifica-
tion of the positive direction on the line I becomes meaningless;
but, if we designate by X, /*> v the direction cosines of either direc-

THE NORMAL FORM OF THE EQUATION OF A PLANE 77

tion on a perpendicular to the plane, the proof goes through as in
part (a). We conclude therefore, since now p = 0, that the dis-
tance QP from a plane through the origin to the point P(a, 0, 7)
is equal to X + AU + yv, where X, /* and v are the direction cosines
of either direction on a perpendicular to the plane.

COROLLARY 1. The unsigned distance from a plane through the ori-
gin to the point P(, 0, 7) is equal to the numerical value of x -f
M _!_ 7l/> where x, M> and v are the direction cosines of either direction
on a perpendicular to the plane.

Remark. It follows from the above discussion that if the
distance QP from a plane not through the origin to P turns out to
be positive, then the direction of QP agrees with the positive di-
rection along /, that is, P lies on the side of the plane opposite to
that on which the origin lies; whereas, if the distance QP turns
out to be negative, P lies on the same side of the plane as the
origin. Furthermore, if for a plane through the origin, the ex-
pression a\ + #M + yv turns out to be positive (negative), the
point P lies on the side of the plane (on the side opposite to that)
indicated by the direction which the direction cosines X, ju, v
specify.

If, whether the plane passes through the origin or not, the
distance from the plane, calculated by means of Theorem 6 or
Corollary 1, turns out to be zero, the point P lies on the plane.
And conversely, it should be clear that if P lies on the plane, its
distance from the plane is zero. This simple fact enables us to
state an important further corollary of the theorem :

COROLLARY 2. The coordinates *, y, as of a point P on a plane satisfy

the equation

Xx + ny + vz - p = 0.

If the plane does not pass through the origin, then X, /i, and v are the
direction cosines of the perpendicular directed from the origin to
the plane and p is the unsigned distance from the origin to the
plane; if the plane passes through the origin, then p = and
X, ju, v are the direction cosines of either direction on a perpendicu-
lar to the plane. v

44. The Normal Form of the Equation of a Plane. A compari-
son of the equation established in Corollary 2 of the preceding
section with the general linear equation in x y y, and z yields a

78 PLANES AND LINES

valuable result. For we have seen in Theorem 4 (Section 41,
page 71) that every plane can be represented by an equation of
the form ax + by + cz + d = 0, in which a, 6, c, and d had no
particular significance; and in Corollary 2 of Section 43 we estab-
lished the fact that every plane can be represented by an equation
of the form \x + ny + vz p = 0, in which X, /x> v, and p have
the geometrical meanings stated in this corollary. But if these
two equations represent the same plane, they must be equivalent;
hence their coefficients must be proportional; thus there exists a
non-vanishing number k such that

a = k\j b = kfj,j c = kv and d = kp.*

From the first three of these equations we conclude (see Theorem
8, Chapter III, Section 33, page 56) that k = dbVa 2 + 6 2 + c 2 ;
from the last it follows, since p is an unsigned number, that the
sign of fc must be opposite to that of d. Thus k is completely de-
termined if the plane does not pass through the origin, whereas
its sign is left ambiguous if the plane passes through the origin.
We have therefore obtained the following geometrical interpreta-
tion of the coefficients in the general linear equation in x, y, and z.

THEOREM 7. The coefficients a, 6, and c of the variables x, y, s in
the equation of a plane, ax + by + cz -f r/ = 0, are proportional to
the direction cosines of a line perpendicular to the plane; if d 4= 0,
the quotients of a, 6, and c by that square root of the sum of their
gquares which is opposite in sign to c/, are equal to the direction co-
sines of the perpendicular directed from the origin to the plane, and
the quotient of d by the same square root gives the unsigned dis-
tance from the origin to the plane.

Remark 1. Corollary 2 of Section 43 gives us another form in
which the equation of a plane may be written. It is called the
normal form of the equation of a plane. This form of the equation
of the plane is characterized by the two facts that the sum of the
squares of the coefficients of the variables is equal to 1 and that
the constant term is negative or zero.

Remark 2. Division of the form ax + by + cz + d = of the
equation of a plane by +Va 2 + 6 2 + c 2 or by - Va 2 + 6 2 + c 2

* Two equations are equivalent if any values of the variables which occur
in it that satisfy either one of them also satisfy the other. It is a nice exer-
cise in algebra to show that, if two linear equations in x, y, and z are equiva-
lent, their coefficients arc proportional.

THE NORMAL FORM OF THE EQUATION OF A PLANE 79

according as d is negative or positive is called " reduction of the
equation of the plane to the normal form."

If we combine Theorems 7 and 6, we obtain the following corol-
laries.

COROLLARY 1. The distance from the plane ax + by + cs + d = 0,

d * 0, to the point P(, 0, 7) Is equal to the + or _

=b Va 2 -f 6 2 + c 2

sign being used according as d Is negative or positive; this distance
will be positive or negative according as P and the origin lie on oppo-
site sides or on the same side of the plane.

COROLLARY 2. The unsigned distance from a plane through the ori-
gin ax + by + cs = to the point P(, 0, 7) is given by the numerical

aa -f- bft -f- cy

value of

-f 6 2 -f c 2

If a definite choice of the sign of the square root in this last
formula is determined upon, then those points P for which the
distance turns out to be positive (negative) lie on the side of the
plane (on the side opposite to that) indicated by the direction
whose direction cosines are equal to the quotients of a, 6, and c
divided by that square root. Although in this case the parts of
space on opposite sides of the plane are not so readily character-
ized as when the plane does not pass through the origin, we still
have this essential fact that, once the sign of the square root has
been fixed in either way, two points P(a, , 7) and P'(a', /ft', 7')
will lie on the same or on opposite sides of the plane according as

aa + bp + cy , aa' + fe/3' + cy' . . . . .

, and ,, are equal or opposite in sign,

Va 2 + fr 2 + c 2 Va 2 + b' 2 + c 2

that is, according as aa + b0 + cy and aa' + 6/3' + cy' have the
same or opposite signs.

From Theorem 7, in combination with Theorem 13, Chapter III
(Section 36, page 63) we obtain moreover the following result.

COROLLARY 3. The angles between a line whose direction cosines
are x, /*, v and the plane ax + by -f cs + d = are determined by the

equation

\a-\-nb-i- vc

sin e =

6 2 + c 2

The angle between a line and a plane is the angle between that
line and its projection on the plane; the sine of this angle is there-
fore equal to the cosine of the angle made by the given line and a

80 PLANES AND LINES

line perpendicular to the plane. Corollary 3 follows from these
observations. The ambiguity of sign corresponds to the fact that
neither the direction on the given line nor that on the projection
have been specified.
Examples.

1. To find the distance from the plane II: 2 x + 3 y 4 z + 5 = to the
points A(-l, 2, 4), B(3, -2, 0), 0(0, 0, 0) and C (3, 3, 5), we determine the
direction cosines of the perpendicular directed from the origin to the plane;

2 3 _4

it is found that X = - 7=, M = ^7=, v = - 7=- The unsigned length
-\/29 -V29 -\/29

rj
of the perpendicular from the origin to the plane is * __ It follows that the

r* nA -2 + 6-16 + 5 7V29 .,. 6-6-0 + 5

distance II A = - 7 - = on , that II B

7 - on , - -= -

-V29 29 , -V29

-5 V29 ., . nn -5 V'29 . ., . 6+9-20 + 5 _ w

, that nO = and that 11(7 = - - = 0. We con-

clude that A and lie on opposite sides, B and O on the same side of the plane,
while C is on the plane. These are the geometrically essential facts concern-
ing the positions of these points and the plane; that the side of the plane
on which the origin lies happens to be the negative side is not of geometric
importance, but is a result of the convention made in Section 43 (see the foot-
note on page 75).

2. To find the distances from the plane II : 3 .c 12 y + 4 z = to the
points A(-3, 1, 4), 5(3, -12, 4) C(-3, 12, -4) and D(5, 1, 1), we determine
the direction cosines of the line perpendicular to the plane. We find that
X : M : ^ = 3 : -12 : 4, so that X = f 8 a , ju = =F}J, v = T \, in which either
all the upper signs or all the lower signs are to be used. If we choose the

upper signs, we find that UA = ~ 9 ~ 12 + 16 = - T 8 3 , HB = +13, IlC = -13

and nD = T \, from which we conclude that A and C lie on the side of the
plane opposite to that indicated by the direction whose direction cosines are
iV> it iV b ut B and D lie on the side indicated by that direction. If we
choose the lower signs, the signs of the four distances are reversed; this means
that A and C lie on the side of the plane indicated by the direction whose
direction cosines'are r \, {|, A, and B and D lie on the opposite side. These
conclusions are obviously identical in geometrical content with those stated
in the preceding sentence.

45. Exercises.

1. Determine the distances of the points A( 3, 2, 1), B(5 t 3, 1),
(7(2, 4, 2) and D(-l, 2, -4) from the plane 3z+2?/-6z-2=0, and
determine their positions relative to the plane.

2. Also the distances of the points A(l, 4, -3), B(3, -2, 2), C(-5, 1, 3)
and D(l, 0, 2) from the plane 2z-3t/ + z = 0.

TWO PLANES 81

3. Find the direction cosines of the lines which are perpendicular to the
following planes:

(a) 14 x - 3 y + 18 z + 1 = (c) 6 x - 2 y - 3 2 -f 2 =

(&)2a? + 3y-2+4=0 (rf) .c + 4 ?/ - 8 2 - 3 = 0.

4. Determine the distances:

(a) from the X-axis to the plane 3 <y 4 z + 7
(6) from the F-axis to the plane 5z 2z" 3=0
(c) from the Z-axis to the plane 5 x 12 ?/ 8 = 0.

6. Determine the coordinates of the point in which the plane 2 x y 2 z
-H 4 = is met by a line through A (3, 1, 2) perpendicular to the plane.
Find the distance from the plane to A in two ways.

6. Find the coordinates of the point in which the plane ax -\- by -\- cz ~\- d
= is met by the perpendicular from P(a, p, y) to the plane.

7. Through the point A (2, -2, 6) in the plane 3z-f2?/-z-f4=0,
a line is drawn whose direction cosines are proportional to 3, 6 and 2. Find
the angles which this line makes with the plane.

8. Set up the equation of a plane through the point A (2, 3, 1) and per-
pendicular to a line whose direction cosines are proportional to 3, 4, 2.

46. Two Planes. Two distinct planes either intersect in a line
or else they are parallel. Since two planes are perpendicular to
the same line if and only if they are parallel, it follows immedi-
ately from Theorem 7 that two planes

(1) aix + biy + CiZ + di = and (2) a 2 x + Iwj + o>z + r/ 2 =

are distinct and parallel if and only if ai : a 2 = &i : &2 = Ci : c* ^
d\ : dz y that is, if the rank of the coefficient matrix of the two
equations is 1, and the rank of the augmented matrix is 2 (see
Definition IX, Chapter I, Section 9, page 16 and Section 20, last
paragraph). If two equations represent the same plane, their co-
efficients are proportional (see footnote on page 78), the two-
rowed minors of the augmented matrix all vanish and the rank of

the a.m. is 1. We can therefore state the following conclusion.

. i
THEOREM 8. The planes represented by two linear equations are

(1) coincident If and only if the rank of the augmented matrix Is 1;

(2) parallel If and only If the rank of the augmented matrix is 2 and the
rank of the coefficient matrix Is 1; (3) Intersecting If and only if the
rank of the coefficient matrix is 2.

To determine the angles between two intersecting planes, we
make use once more of Theorem 7. These angles are the same
as the angles between two lines perpendicular to these planes

82

PLANES AND LINES

(see Fig. 8). Therefore the angles between the planes (1) and (2)
are equal to the angles between the lines whose direction cosines
are proportional to a it 61, c A and to a 2 , b 2 , c 2 . Consequently, if 6 is

Fia. 8

used to designate any one of these angles, we conclude, using also
Theorem 13, Chapter III (Section 36, page 03), that

bib 2 + CiC 2

(3) cos 6 =

c 2 2 )

THEOREM 9. The cosine of any of the angles between the planes
represented by two linear equations is equal to the sum of the prod-
ucts of the coefficients of the like variables in the two equations, di-
vided by the product of the square roots of the sums of the squares of
these coefficients.

Remark 1. The ambiguity of sign in the formula corresponds to
the fact that the different angles formed by two planes are related
in such a way that their cosines differ at most in sig^.

Remark 2. If the square roots in the denominator of formula
(3) are given the signs opposite to those of di and d 2 respectively,
we obtain the cosine of the angle between the perpendiculars to
the plane, directed in each case from the origin to the plane (see
Theorem 7, page 78), that is, the cosine of the supplement of that
angle between the planes in which the origin lies. We are sup-
posing in this statement that neither plane passes through the
origin.

COROLLARY. Two planes, a\x -f b\y + ci* -f rfi =0 and a*x -f 6 2 y +
c< 2 s -f- dz = 0, are perpendicular if and only if a { a> -f- b>b 2 + cic 2 = 0.

An equation of the form (a\x + b\y + c\z + di) (a 2 # + b^y
+ c%z + d 2 ) = is satisfied by values of the variables which cause

THE LINE 83

at least one of the factors of its left-hand side to vanish, and by
such values only. The locus of this equation consists therefore
of the two planes represented by the equations a\x + biy + c\z
+ di = and a^x + btfj + c 2 z + d 2 = 0. It should be clear that
this observation may be generalized as in the following theorem.

THEOREM 10. If F(x 9 y, z) =/i(, y, a) fi(x 9 y, )... fk(x 9 y, ),
the locus of the equation F(x 9 y, s) = consists of the loci of the
equations /i (*, y, s) = 0,/ 2 (*, y, s) = 0, . . . ,/*(*, y, s) = 0.

DEFINITION V. If the function F(*, y, s) is factorable into real
factors (that is factors which involve only real operations on the
variables), the locus of the equation F(x 9 y, s) = is called a degener-
ate locus.

47. The Line. The coordinates of every point on the line of
intersection of the two planes represented by equations (1) and
(2) of the preceding section satisfy, in case (3) of Theorem 8, these
two equations. Conversely, every point whose coordinates sat-
isfy these equations lies on the line of intersection of the planes.
We say therefore, in accordance with Definition II (Section 39,
page 68), that "two linear equations ciix + b^y + c\z + c?i =
and a 2 x + b' 2 y + c$z + cl 2 = 0, whose coefficient matrix has rank
2, are the equations of a line/'

Remark. A line, thus defined as the intersection of two
planes, has as its equations those of two planes passing through
it. But there is a single infinitude of planes which pass through a
given straight line (see Section 49) and the equations of any two
of these planes can be taken as the equations of the line. Thus it
is seen that one and the same straight line can be represented by
any one of an infinite number of pairs of linear equations. The
reader may at first be troubled by this lack of definiteness; he
will do well to think this question through until it has become
clear to him.

The results obtained in Chapter III, where the line was dis-
cussed as a locus of points, can now be interpreted in the light of
the point of view presented in the first paragraph of this section.
The equations found in Theorems 9 and 10, and in Corollary 1 of
Theorem 10, Chapter III (see Section 34, pages 57 and 59) are
all linear equations; and it is readily seen that in each case a pair
of equations can be selected whose coefficient matrix has rank 2.
For example, the equations of Theorem 9 may be written in the

84 PLANES AND LINES

following form:

(yB - VA)X - ( X B *A)y + yA*B - ysXA = 0,

(ZB - z A )y - tea - VA)* + z A ys - ZWA = 0, and

The second order determinants of the coefficient matrix of the first
two of these equations have the values

those formed from the coefficient matrix of the second and third
equations have the values

-(XB ~ X A ) (^B ~ Z A ), (y B - VA) (ZB - ZA), -(ZB - z A ) 2 ;

and those obtained from the third and first equations have the
values

-(XB - XA)*J (ys - VA) (XB - XA), -(XB ~ X A ) (Z B - Z A ) .

Now it should be clear that if A and B are distinct points at least
one of these second order determinants must have a value which
is different from zero; therefore a pair of equations can be selected
whose coefficient matrix has rank 2.

Similar arguments can be made for the equations of Theorem
10 and Corollary 1 of Theorem 10. A mere restatement of the
earlier results in terms of the terminology which was introduced
and justified at the beginning of the present section, leads to the
following theorems.

THEOREM 11. The equations of the straight line which passes
through the points A and B may be written in the form:

*-* y ~y * -

*B

also in the forms:

or

- x y-y * - *

THEOREM 12. The equations of the directed line which passes
through A and whose direction cosines are x, /* and v may be written
In the form:

* -* A _y ^y A ^*~*A

M

THE LINE

85

THEOREM 13. The equations of the undirected line which passes
through the point A and whose direction cosines are proportional to
f, m and n may be written in the form:

x - X A y -y A * - * A

I m n

Remark 1. It should be clear that in each of these theorems the
line is the intersection of three planes, any two of which suffice to
determine it. The three planes are, in each case, parallel to the
X-, F-, and Z-axes; they are indeed the planes which project
the line on the three coordinate planes (see Fig. 9), the diagonal
planes of the c,p. of any two points on the line.

Fio. 9

Remark 2. The equations of the line, established
11, 12, and 13, lose meaning whenever one of the denominators
vanishes. In spite of this disadvantage these forms for the equa-
tions of the line are, in general, more convenient than the extended
form obtained by equating two of the fractions at a time and then

86 PLANES AND LINES

clearing of fractions. This extended form becomes imperative if
one of the denominators vanishes. Frequently one finds the
condensed form used even in such cases; this is however not to be
recommended even though this apparently meaningless form is
intended to be symbolic for the extended equations. The diffi-
culty referred to here can be obviated by use of the parametric
equations, already obtained in Chapter III (see the Corollary of
Theorem 9, and Corollaries 2 and 3 of Theorem 10, Section 34,
pages 58, and 60), whose existence is formulated again as fol-
lows.

THEOREM 14. The equations of the line through the points A and B
may be written In the following form:

S

_

* ~

- 1+r ' ""14- r ' ~ 1 +r *

THEOREM 15. The equations of the directed line through the point
A whose direction cosines are equal to X, M and v may be written in the
form:

x = X A + Xs, y = y A + /us, * = S A -f vs.

THEOREM 16. The equations of the undirected line through the
point A whose direction cosines are proportional to 1 9 m and n may be
put in the form:

* = X A -f It, y y A + mt 9 * = S A + nt.

Remark. In each of the last three theorems the line is given by
means of three equations; but these equations involve four vari-
ables, namely the coordinates x y ?/, and z of the variable point
along the line, and the parameter r, s, or t. The locus of these sets
of equations has therefore one degree of freedom (see Section
39, page 67). The geometric significance of the parameters r, s,
and t was discussed in Corollaries 2 and 3 of Theorem 10, Chapter
III, and in Remark 4 following the Corollary of Theorem 9 (see
Section 34, page 58) ; it is desirable that the reader recall this in-
terpretation of the parameters at this point.

If an undirected line is given by means of two linear equations
in the general form, like equations (1) and (2) of Section 46, whose
coefficient matrix has rank 2, these equations can be reduced to
any one of the forms given in Theorems 11 to 16, as soon as the
coordinates of two points on the line and the ratios of its direction
cosines have been determined.

THE LINE 87

Since the coefficient matrix is of rank 2, the equations can be
solved for two of the variables in terms of the third. By assigning
values to this third variable arbitrarily, an infinite number of solu-
tions of the equations can be obtained; but each of these solutions
furnishes the coordinates of a point on the given line.

The direction cosines of the line are found by means of the fol-
lowing theorem:

THEOREM 17. The direction cosines of the line of intersection of
two intersecting planes are proportional to the two-rowed minors of
the coefficient matrix of their equations, taken alternately with the
plus and the minus signs.

Proof. The proof of this very useful theorem can be made in
various ways. We shall make use here of Corollary 1 of Theorem
6, Chapter III (Section 33, page 56). Suppose that P\ and P 2
are two arbitrary points on the line of intersection of the planes.
Then

+ biiji + CiZi + h = 0, a&i + 6 2 ?/i + c 2 2i + r/ 2 = 0,

Ci2 + r/i = and a 2 z 2 + 6 2 t/ 2 + c 2 2 2 + ^2 = 0.

If we subtract these equations in pairs, we find that the differences
of the coordinates of PI and P 2 satisfy the following two linear
homogeneous equations :

- 7/ 2 ) + ci(zi - z 2 ) = and 02(0?! - x 2 )
Zz) = 0.

Since the two given planes intersect, it follows from (3) in Theorem
8 (Section 46, page 81) that the rank of the coefficient matrix
of these equations is 2. Theorem 4, Chapter II (Section 25,
page 41) gives us the means therefore to determine from these
equations the ratios of the coordinate differences of PI and P 2 .
But we know from Corollary 1 of Theorem 6, Chapter III (Section
33, page 56) that these coordinate differences are proportional
to the direction cosines X, /i, v of the line. We find therefore that

bz (
This completes the proof of the theorem.

88 PLANES AND LINES

Examples.

1. The planes represented by the equations 2z-2/-f3z-4=0 and
2 x y + 5z + 3 = intersect in a line; for the rank of the matrix

2-13
2 -1 5

is clearly 2. To determine points on the line of intersection, we solve the
equations for x and z in terms of ?/ (we could equally well solve them for ?/ and z
in terms of x, but not for x and y in terms of z\ why not?) by Cramer's rule.
We find

By selecting values for y and calculating the corresponding values of x and
z from these equations, we can find. as many points on the line as we wish;
thus we locate the points A (8, , -}, #(*/, -2, -I) and C Y (% 0, - J) on the
line of intersection of the given planes. Having determined these points, we
can find the direction cosines of the line most simply by use of Corollary 1
of Theorem 6, Chapter III directly; it is found that

X : M : * = 8 - V : I + 2 : - J + J = I : I : = 1 : 2 : 0.

1 2

Consequently X = j-r, /* = ^, v - 0, so that the line makes an angle

=fc v 5 =fc v 5

of 90 with the Z-axis and is therefore parallel to the X F-plane; this could
have been foretold from the fact that all of its points have the same z-co-
ordinate, j.

The ratios of the direction cosines can also be found by applying the formula
proved in Theorem 17; this gives us \

\ '. fJL '. V =

-1 3
-1 5

12 3
2 5

2 -1
2 -1

= -2 : -4 : = 1 : 2 : 0.

In accordance with Remark 2, following Theorem 13 (page 85), the non-
parametric forms of the equations of the line as given in Theorems 11, 12, and
13 are not desirable in this case. The parametric forms of the equations are:

a: = 8 + f, y = 2 + 2 , 2 = - j, where t AP

-2+|r PB

---' *--*' where r = ;

29 , a 2s ,

x = -r TT , ?/ = ^^ , z = I, where s = 4P.
* V5 V5

We observe that the three sets of values of x, y, and z given by the different
parametric equations satisfy the equations of the two given planes identically
in t, r, or s respectively.

THE LINE 89

2. The planes represented by the equations 2x ?/ + 3 z 4 = U and
4 z 2 T/ + 6 z + 3 =()are parallel; for the rank of the c.m. is 1 and the
rank of the a.m. is 2. The parallelism or coincidence of two planes can readily
be recognized upon inspection of the equations.

3. The direction cosines of the line of intersection of the planes represented
by the equations 10z + 3?/-4z+8 = and 4u;+3?/-3z--4=0

are proportional to Ur ; <**Ct
7>-V. ***

3 -4
3 -3

" 4 -3

10 3
4 3

that is, X : /z : v = 3 : 14 : 18.

Since 3 2 + 14 2 + 18 2 = 529 = 23 2 , it follows that X = &, M = ii " = =*=it-

Solution df the equations for x and i/ in terms of z leads to x = -A ' 2,

y + 4 ; we arc now able to determine readily as many points on the line
J

as we wish.

The parametric equations of the line may be written in the following forms:

a: = -2 + 3 , ?/ = 4 -f 14 /, z = 18 <;
_ -2 -f 7r _ 4 + 46 r 54 r

* - 1 +r ' ^ ~ 1 -hr ' " ~ IT"r *

We can verify that these values of x, y, and z satisfy the given equations of
the planes identically.
The angles between the two planes are given by the equation:

10-4 -h 3-3 -f (-4) (-3) 61

COS0 =

V10 2 -f 3 2 + (-4) 2 X V4 2 -1- 3 2 -f (- 3) 2 5 Vl70

The direction cosines of the directed lines from the origin perpendicular to

.u T i* 2 3 4 ^ 4 3 3 ,

the plane are equal to T ., 7-, 7^ and p=, T=, 7=- There-

V5 5V5 5\/5 V34 V34 >/34

fore the cosine of that angle between the planes in which the origin lies is

i . 6l
equal to T=T

5V170

48. Exercises.

1. Determine by inspection which of the following pairs of equations repre-
sent intersecting, which parallel planes, and which coincident planes:

(a)3z-?/-h42 + l=0 and 2x-f?/-2z-f3=0

(b)2s + y-3;8 + 4 = and 2z + ?/-32-4 =

(c)z + 2y + 4z-3=0 and x-2y+4z + l=Q

(d) x -y + z = and 2ar-2?/ + 22 + 7 =

2. Determine the angles between the planes of the intersecting pairs of
planes in the preceding exercise.

90 PLANES AND LINES

3. Determine the distances between the following pairs of parallel planes:

(o)a:-8y + 4-3=0 and x - Sy + 4 z + 15 = 0;
(6) 2z-3?/-6z + 5=0 and 2 x - 3 y - 6 z -f 19 = 0;
(c) a; + y + z + 6 = and z + 2/-fz-8 = 0.

4. Write the equations of the lines which pass through the following pairs
of points:

(a) 4(-3, 5, 2) and 5(5, 4, -2), (c) A(5, 2, -3) and B(-l, -1, -1),
(6) A(4, -3, 1) and B(-8, 3, 5), (d) A(-2, 4, 1) and 5(3, -5, 2).

6. Write the equations of the line through A (3, 4, 1) and perpendicular
to the plane 2 x y -\- 2 z 5 =0. Determine the coordinates of the
point in which the plane is met by this line.

6. Set up the equation of the plane through the point A( 2, 3, 4) and

(a) parallel to the plane 3 x + y 5 z -f 7 =0;

(b) perpendicular to the line ^ = y = ^ .

7. Determine the parametric equations of

(a) the line of intersection of the planes 3 x y -J- 3 z 2 and

z+2?/-3z-f4=0;
(6) the line through the point .4(1, 3, 5) and parallel to the line of

intersection of the planes 3z + 2/ + 2z 3 = and 6 x -f- 3 y

+ 2 z + 5 = 0.

8. Determine a plane through the points A (2, 1, 4) and #( 1,3,2),
which intersects the plane 3 z ?/ 2 z = 4 in a line that makes equal angles
with the coordinate axes.

9. Find the equation of a plane through the points A(l, 3, 2) and
B(2, 4, 5) which is perpendicular to the plane 3 x + 6y 4 z 5 = 0.

10. Find the parametric equations of a line through the point A (2, 5, 3)
and parallel to the line of intersection of the two planes represented by the
equations 2 - 6 xy + 9 y* - 4 z* + 12 z - 9 = 0.

11. Set up the equation of a plane through the point A( 1, 4, 3) and
perpendicular to the line of intersection of the two planes represented by the
equation 9 x 2 4 y 2 -f z 2 6 xz 4 y - 1 =0.

12. Find the equation of a plane through the point A (3, 2, 1) and per-
pendicular to the two planes x y + z -f- 4 =0 and 2 x y 2 z -f 3 =().

13. Determine the coordinates of the point in which the plane 3 x - 4 y
-f z - 3 = is met by the line :c = 2-2J, ?/= - 1 3 J, z = 5 -H.

14. Determine the coordinates of the point in which the plane 4 x -f y 3 z
-|- 5 = is met by the line which joins the points A ( - 1 , 3, - 2) and B (4, - 3, 1 ) .

49. The Pencil of Planes. The Bundle of Planes. If the
left-hand sides of the equations

(1) aix + biy + ciz + di =

THE PENCIL OF PLANES 91

and

(2) Otx + b 2 y + c 2 z + d 2 =

arc denoted by EI and E 2 respectively* and if k t and k 2 are arbi-
trary constants, then the equation

(3) kiEi + k 2 E 2 =

is also a linear equation. Its locus is therefore a plane. Moreover
equation (3) will be satisfied by the coordinates of those points
which lie on both planes EI and E 2 , that is, by the points on the
line of intersection of these planes; and this last statement holds
true whether ki and k 2 are constants or not, because the coordi-
nates of the points on the line of intersection of the planes cause
both EI and 7 2 to vanish. Consequently, the equation represents
a plane through the line of intersection of the planes EI and E 2 for
any constant values assigned to &i and k 2 .

On the other hand the equation of every plane through this line
can, by suitable choice of the values to be given to the constants
ki and & 2 , be put in the form (3). We see, in particular, that for
fci = 1 and k 2 we obtain the plane E\\ and for k\ = and
k 2 = 1, we obtain the plane # 2 . And if any other plane through
the line of intersection, I, of the two planes is given, and if P(a, 0, 7)
is an arbitrary point in such a plane but not on J, then equation
(3) will represent the given plane, provided ki and k 2 are so chosen
that

kifaa + bi0 + Ciy + di) + k 2 (o^a + b 2 $ + C 2 y + d 2 ) = 0.

Since P does not lie on I and hence not on both planes, the two
expressions in the parentheses do not both vanish; consequently
the ratio ki : k 2 can always be determined in such a manner that
the last written equation is satisfied. If numbers which have
this ratio are substituted for ki and k 2 in equation (3), this equation
will indeed have the given plane as its locus. If we introduce now

* When this abbreviated notation is employed for the left-hand side of the
linear equation in x t y, and z, it is usually convenient to use the same letter to
designate the plane which is the locus of the equation. Thus we shall speak
of "the plane EI " instead of using the longer and more explicit phrase "the
plane whose equation is EI s a\x + b\y + c\z + d\ = 0"; this usage does
not frequently lead to confusion.

92 PLANES AND LINES

the expression " pencil of planes" to designate the set of all the
planes which pass through a line, we obtain the following theorem.

THEOREM 18. The pencil of planes through the line of Intersection
of the planes E { and E 2 Is represented by the equation kiEi -f k 2 E-> = 0,
In which k L and k> 2 are arbitrary constants not both zero.

Remark 1. Since the geometrical significance of equation (3)
is not altered when it is multiplied through by a non-zero constant,
the pencil of planes through the intersection of the planes E\ and
E% is also represented by the equation EI + kE 2 = 0, where

k

k = T- , except that in the latter form the plane E 2 which is ob-
KI

tained from equation (3) when ki = 0, is excluded. For this
reason the form (3) of the equation of the pencil deserves preference.

Remark 2. The lack of definiteness in the equations of a line
pointed out in the remark at the beginning of Section 47 (page 83)
can now, at least partially, be provided for, inasmuch as we can
say that the line which is given by the pair of linear equations
EI = and E z = is also determined by any two equations of the
form (3), that is, by any two planes of the pencil of planes through
this line.

Remark 3. The ratio ki : k% is a parameter in the equation (3)
of the pencil of planes. It is constant for any one plane of the
pencil, it varies as we pass from one plane in the pencil to another.
The pencil of planes is a "one parameter family of planes." It
will be instructive for the reader to compare the character of the
parameter ki : & 2 in equation (3) with that of the parameters r, s,
and t in the parametric equations of the line (see Section 47,
page 86).

Remark 4. The method used for determining the equation of
the pencil of planes finds frequent application throughout Ana-
lytical Geometry (see e. g. Exercise 3, Section 73, page 148 and
Section 82, page 168). In connection with one of the remarks
made in the opening paragraph of the present section, we observe
that equation (3) does not represent a plane, if ki and k z are not
both constants. The surface which it does represent will, how-
ever, still pass through the line of intersection of the planes EI
and Z? 2 . Furthermore if Si = and fi> 2 = are the equations of
two arbitrary surfaces, the equation fciSi + feS 2 = represents a

THE PENCIL OF PLANES 93

surface which passes through all the points common to the two
given surfaces, no matter what k\ and fc 2 may be.
Examples.

1. To determine the equation of a plane which passes through the line of
intersection of the planes 3 x 2y + z 4 =0 and x + 5y 2 z + 3 =0
and which is moreover perpendicular to the plane 2x + y 3z + l = 0,
we consider the pencil of planes through the given line. The equation of this
pencil is

fa (3 x - 2 y + z - 4) + kt(x + 5 y - 2 z + 3) = 0.

If a plane of this pencil is to be perpendicular to the plane 2x + y 3 z -f 1
= 0, fa and fa must satisfy the condition which follows from the Corollary
of Theorem 9 (Section 46, p. 82), namely, (3 fa + fa)2 + (-2 fa + 5 fa)
+ (fa - 2 fa) (-3) = 0. This leads to the condition fa + 13 fa = 0, that
is, fa : A~ 2 = 13 : 1. The equation of the required plane is therefore
13 (3 x - 2 // 4- z - 4) - (x + 5 y - 2 z + 3) = 0, or 38 x - 31 y + 15 z
- 55 = 0.

2. To determine a plane through the line whose parametric equations are

x = -4 + 3 , y = 5 - , z = 3 + 2 J

and through the point P( 4, 3, 3).

First solution. The parameter t may be eliminated between the first two of
the parametric equations of the line and also between the last two. This
furnishes the two linear equations x + 3 y 11 =0 and 2 y + z 13 = 0;
and the given line is the line of intersection of the planes which these equations
represent. The equation of the pencil of planes through the given line can
therefore be written in the form fa(x + 3 y 11) -f- fa (2 y + z 13) =0.
Since the point P( 4, 3, 3) must lie on the required plane, the constants fa and
fa must be so selected that fa(-4 + 9 - 11) + fe(6 + 3 - 13) = 0, or so
that 6 fa 4 fa = 0; hence fa : fa = 2 : 3. We conclude that the
equation of the required plane is 2(z-f3?/--ll) 3(2 y + z 13) =0
or 2 x - 3 z + 17 = 0.

Second solution. The required plane is determined by P and any two points
on the line. Such points can be found at once when the line is given by para-
metric equations, by assigning two values arbitrarily to the parameter.
The values t = and t = 1 yield the points A (-4, 5, 3) and B(-l, 4, 5).
The equation of the plane can now be written in the three-point form (see
Theorem 5, Section 41, page 73). Thus we find the equation

x y z 1
-4 3 3 1
-4531 u '
-1451

which, upon development, reduces to the form 2z 3z-}-17=0, found
by the first method.

94 PLANES AND LINES

If the plane ax + by + cz + d = is to pass through a fixed
point P(cx f j8, 7), its coefficients must satisfy the condition aa + bfi
+ cy + d = 0, so that d = aa b0 cy; and it should be
clear that if d has this value, the point P will lie on the plane.
Upon introduction of the term "bundle of planes" to designate
the set of planes which pass through a fixed point, we can state the
following theorem.

THEOREM 19. The bundle of planes through the point P(a, , 7) Is
represented by the equation a(* - ).+ b(y - 0) + c( - 7) = 0, in
which a, 6, and e are arbitrary constants, not all zero.

Remark. The ratios of the constants a, b, and c are the param-
eters in the equation of the bundle of planes; the bundle of planes
is a "two parameter family of planes.''

50. Exercises.

1. Determine the equation of a plane through the line of intersection of the
planes 3x-y + 2z-\-2=Qa.nd 2x + 4y-3s + l = 0, and

(a) through the point A ( 1,3, 2);

(6) perpendicular to the plane 4 x 5y + z 2 = 0;

(c) through the origin;

(d) parallel to the 7-axis;

(e) parallel to the Z-axis.

2. Determine the equation of a plane through the line x = 2 3 t, y I
+ 6 t, z = -3 - 2 t and through the line x = 2 + t, y = I - 2 t, z = -3
+ 2t.

3. Write the equation of a plane through the point A( 3, 4, 1) and per-

,. , . ,, ,. x -\- 2 z 4

pendicular to the line 5- - y 2 - =

j z

4. Prove analytically that every pencil of planes contains at least one
plane parallel to the X-axis, at least one parallel to the F-axis and at least one
parallel to the 2T-axis. Under what conditions will a pencil contain more
than one plane in such position?

6. Prove that every bundle of planes contains exactly one plane parallel
to the FZ-plane, one parallel to the ZJf -plane and one parallel to the XY-
plane.

6. Determine the equation of the plane through the line of intersection of the
planes 3x 6y 2z + 5 =0 and 2 x y 2z + 3 = which is per-
pendicular to the first of these planes.

7. Find the equations of the planes which bisect the angles between the
planes 2z-6?/-32-M=0 and 4*H-y-8z + 5=0. Hint: This
problem can be solved by observing that the bisecting planes belong to the
pencil of planes through the line of intersection of the given planes. Another

THREE PLANES 95

method of procedure is based on regarding the bisecting planes as the locus
of points whose distances from the two given planes are equal or equal nu-
merically but opposite in sign.

8. Find the equations of the planes which bisect the angles between the
planes atf + b\y -f- c\z + dt = and a 2 x + hy + c& + d 2 = 0.

51. Three Planes. A Plane and a Line. In this section we
shall be concerned with the question of determining from the
equations of three planes, E\, E%, and E 3 , how they are placed with
respect to each other, that is, with the problem of extending the
result stated for two planes in Theorem 8 (Section 46, page 81).
Let the equations of the planes be

#1 = aix + biy + ciz + di = 0, E 2 55 a 2 x + b^y + c 2 z + d 2 = 0,
#3 = azx + b*y + c 3 z + r/ 3 = 0.

It should be clear that if no two of the planes are parallel or
coincident, the ranks of the c.m. and of the a.m. of the system of
equations must be at least 2 (compare Theorem 8, Section 46,
page 81). We obtain further results by means of Theorems 1
and 8 of Chapter II (see Sections 21, page 36, and 27, page 44).
If the coefficient matrix is of rank 3, the system of equations has
a unique solution; in this case the three planes have a single point
in common. If the coefficient matrix has rank 2, the system of
equations possesses a single infinitude of solutions or no solution,
according as the rank of the a.m. is 2 or 3; in this case therefore the
planes will have a line in common or no point in common, according
as the rank of the augmented matrix is 2 or 3. If the rank of the
c.m. is 1, the rank of the a.m. can not exceed 2 (Why?). In case
it is 2, at least one pair of planes must be parallel; if it is 1, the
three planes must be coincident. In this case therefore the planes
are coincident or else they have no point in common, according as
the rank of the a.m. is 1 or 2. We have therefore obtained the
following conclusion.

THEOREM 20. Three planes will (1) have a single point In common
If and only if the rank of the coefficient matrix of its equations is 3;
(2) have a single line in common if and only if the ranks of the coeffi-
cient matrix and the augmented matrix are both 2; (3) be coincident
if and only if the ranks of the coefficient matrix and the augmented
matrix are both equal to 1; (4) have no point in common if and only
if the ranks of the coefficient matrix and the augmented matrix are
unequal.

96

PLANES AND LINES

Remark 1. If the planes have a single point in common, they
form a trihedral angle, see Fig. 10a; if they have a single line in
common, they are three planes of a pencil (see Fig. 106), unless
two of the planes coincide ; if they have no points in common, they
form a triangular prism (see Fig. lOc), unless there is a pair of
parallel or coincident planes among them. Since parallelism
and coincidence of planes are readily determined by inspection of
their equations (compare Exercise 1, Section 48, page 89), the
following corollary of Theorem 20 is of considerable use in nu-
merical cases.

FIG. 10a

FIG. 106

FIG. lOc

COROLLARY 1. Three planes of which no two are either parallel or
coincident will form (1) a trihedral angle if and only if the rank of the
coefficient matrix of their equations is 3; (2) a pencil of planes if and
only if the ranks of the coefficient matrix and the augmented matrix
are both equal to 2; (3) a triangular prism If and only if the rank of the
coefficient matrix is 2, while the rank of the augmented matrix is 3.

For future reference it is convenient to state separately the fol-
lowing immediate deduction from Theorem 20.

COROLLARY 2. Three planes have one or more points in common if
and only if the ranks of the augmented matrix and of the coefficient
matrix of their equations are equal.

If the rank of the c.m. is not less than 2, two of the three equa-
tions can be taken as the equations of a line. The corresponding
restatement of Theorem 20 leads to the following Corollary.

TIP* PLANE AND THE LINE 97

COROLLARY 3. A plane and a line will (1) meet in a point if and
only if the rank of the coefficient matrix of the three equations used
to represent them is 3; (2) be parallel if and only if the ranks of the
coefficient matrix and the augmented matrix of these equations are
2 and 3 respectively. The line will lie in the plane if and only if the
rank of each of these matrices is 2.

Remark. It is of interest to observe that these conditions must
continue to hold true when the equations of the line are replaced
by the equations of any two planes of the pencil of planes through
the line. Hence the rank of the matrices of the 3 linear func-
tions EI, EZ, E 3 is not changed if the functions EI and E 2 are
replaced by k\Ei + k z E 2 and l\Ei + l^E^ respectively. Thus we
obtain, for the special case n = 3, a geometrical interpretation of
a part at least of Theorem 14, Chapter I (Section 10, page 18).

62. The Plane and the Line, continued. The geometrical
content of Corollary 3 in the preceding section will become more
apparent if the conditions of that corollary are interpreted in
terms of the direction cosines of the line. Let the equations of
the line be given in the form stated in Theorem 12 (Section 47,
page 84) :

x - a. = y - 13 = z - y

X p. V

It will always be possible to select among the three equations here
represented two whose c.rn. has rank 2; these equations can then
be taken as the equations of the line. If we suppose that p = 0,
these equations may be taken to be the following two :

nx \y (XIJL + 0X = and vy M^ ~ &v + 7M = 0.

The condition that the given line shall meet the plane ax + by
+ cz + d = in a point, can therefore, in virtue of Corollary 3,
Section 51, (1), be written in the form:

0)

-X

v -

a b c

4=0.

If we develop this determinant and divide out the factor /u which
was supposed to be different from zero, we find the condition

(2) aX + bfji + CP 4= 0.

98 PLANES AND LINES

On the other hand the line will lie in the plane or be parallel to
it if and only if the determinant in (1) vanishes, that is, since
M 4 1 0, if and only if

(3) a\ + bn + cv =

while at least one two-rowed minor of the determinant in (1)
has a value different from zero.

To distinguish the case of parallelism from the case in which the
line lies in the plane, we have to consider the augmented matrix
of the system of equations

(4) nx \y oifjL + 0\ = 0, vy IJLZ $v + yn = 0,
ax + by + cz + d = 0.

There is no loss of generality, by virtue of the hypothesis that
the rank of the matrix of the determinant in (1) is 2, if we suppose
that not all the cofactors of the elements in the second column
of this determinant vanish; let us denote the values of these
cofactors by Ci, C 2 , and (7 3 and let us suppose that Ci 4= 0. We
know then from Theorem 14, Chapter I (Section 10, page 18),
that the rank of the augmented matrix of the system of equations
(4) is not changed if the first row is replaced by Ci times the first
row, plus C 2 times the second row plus Cs times the third row.
But if this operation is carried out, the first and third elements
of this row will reduce to zero by Theorem 13, Chapter I (Section
7, page 13), the second element vanishes on account of (3),
and the fourth element becomes

Consequently the augmented matrix of the system of equations
(4) will have rank 2 or 3 according as this last expression is or is
not equal to zero. Now, Ci = a/*, C 2 = c/z and C 3 = /x 2 - If
these values are substituted in the expression for the fourth
element, above, we find that the rank of the augmented matrix
is 2 or 3 according as the equation

- jSX) + cp(-fr + TM) + M 2 d =
is or is not satisfied. This condition reduces to

(act + cy + d)^ - (aX + ci/)ftu = 0.

THE PLANE AND THE LINE 99

But from (3) it follows that a\ + cv = fr/*; if this is used in the
preceding equation and if the non-vanishing factor p, 2 is divided
out, we are led to the condition

(5) aa + bp + cy + d = 0.

We have therefore obtained the following equivalent form of
Corollary 3 of Theorem 20.

THEOREM 21. The line I through the point P(, (3, ?) and with di-
rection cosines X, /*, v will (1) meet the plane ax -f by -f cs -f- d = in
a single point if and only if aX -f &/* + cv = 0; (2) be parallel to the
plane If and only if aX + b 4- cv = and aa + bft + cy + d 4= 0; or
lie in the plane if and only if a\ -f &M + cv = and aa -f bp + cy -f
d = 0.

Remark 1. The geometrical interpretation of these conditions
should be obvious. For it follows from Corollary 3 of Theorem
7 (Section 44, page 79) that the condition (3) requires that the
angle between the line and the plane shall be or 180, that is,
that the line shall lie in the plane or be parallel to it. And the
condition (5) clearly states that the point P(a, 0, 7) must lie in
the plane ax + by + cz + d = 0.

Remark 2. We have been interested in deriving Theorem 21
from Theorem 20 in order to illustrate the power of this general
theorem. It must be observed, however, that the conditions of
Theorem 21 are obtained more directly if the equations of the line
are taken in the parametric form of Theorem 15 (Section 47,
page 86), x = a + X,s-, y = + JJLS, z = y + vs. For if these
expressions are substituted for x, y, and z in the equation of the
plane, ax + by + cz + d = 0, we obtain the linear equation :

(a\ + &M + cv)s + (aa + h0 + cy + d) =

from which the value of s is to be ascertained, which determines
the point of intersection of the line with the plane. From this
equation it is evident that, if aX + 6/i + cv 4 1 0, the equation
has a single root and the line meets the plane in a single point;
if a\ + bfj, + cv and aa. + fc/3 + cy + d = 0, the equation has
no solution, and the line is parallel to the plane; if aX + fyu + cv
= and act + 6/3 + cy + d 0, the equation is satisfied by
every value of s and the line lies in the plane.

100

PLANES AND LINES

Examples.

1. Let it be required to determine the equation of a plane through the
point A( 4, 1, 2) and parallel to the lines

/ .^=J_?LJ~*_I_ 3 i i -*. + 3_y-_4_* + 2

-2

and

-1

-2

If P(;r, ?/, z) is an arbitrary point on the required plane, it must be possible
to determine four numbers a, b, r, and d, not all zero, such that

ax + by + cz -f- d = 0; such that

4 a + 6 -}- 2 c -|- rf =0, since -A is to lie in the plane; and such that
4 a - 2 b -f 3 c = 0, and

a + 3 6 2c =0, since h and J 2 are to be parallel to the plane

and their direction cosines are proportional to 4, 2, 3 and to 1, 3, 2
respectively. These four linear homogeneous equations in a, 6, c, and d possess
a non-trivial solution only if the value of the coefficient determinant is zero
(see Corollary of Theorem 2, Chapter IT, Section 22, page 38). Hence the

x y z 1

condition on the coordinates of P is that

- 0, or that

z

-4121
4-2 30
-1 3-20

x y 2 z + 9 = 0. The locus of this linear equation is a plane; it is easy to
verify that this plane meets the required conditions.

2. If it is required to show that the line of intersection of the planes x 2 y
-f z 4 = o and 3x + 5y 2 z + 4 = is parallel to the plane 7 x 3 y
-f 2 z 5 = 0, we can proceed in various ways. Using Corollary 3 of

Theorem 20 (Section 51, page 97), we can show that

and that the rank of the matrix

1 -2 1

that

3 5 -2

=

7 -3 2

is 3; thus the problem

1-2 1-4
35-24

7 o o r

i o <-/ j

is made to depend on the evaluation of determinants. We can also reduce the
equations of the line to the point-direction form, established in Theorem 12
(Section 47, page 84) and then apply Theorem 21.

53. Exercises.

1. Determine the relative positions of the planes in each of the following
sets:

(a) 3z-2i/-f42 = 0, 2s + 3y-z + 3=0, x~ 4y + 2z + 2 = 0;
(6) z-2^-42+3 =0, 3s + ?/-z + 2 = 0, 3x + 8y + Wz - 5 = 0;

(c) 2z + i/-3z + 4=0, 4z + 27/-6z + 5=0, 3z- Qy+z + 1 = 0;

(d) z-i/ + 2-3=0, 2z+?/ + 3z = 0, 3z+52/ + 3z-l=0;

(e) 4z + ?/-32:-f-2 = 0, 2z-3?y + z-4=0, 72/-5z + 4=0;
(/) * + 77 + 2 + 1=0, 2x + 2y+3z-4

(0) 5z-22/-7z + 3 =0, 10z-4i/- 14z-2 = 0,
-8 = 0.

FOUR PLANES 101

2. For those sets of planes in the preceding exercise which form a trihedral
angle, determine the point common to the three planes.

3. For those sets of planes in Exercise 1 which belong to a pencil of planes,
determine the direction cosines of the line common to the three planes.

4. Set up the equation of a plane through the point A (2, 1, 3) and paral-
lel to the lines x = 2 - 3t, y = l+2t,z = 3 t and z = 4 + 3f, y =
-3 + 5 t, z = 1 - 2 t.

5. Set up the equation of a plane through the point P(a, ft 7) and parallel
to the lines x = ai -f X/s, y = fo -f M*S, z = yi + i/,-s, i = 1, 2.

6. Prove that the ranks of the coefficient matrix and the augmented matrix
of the equations of three planes are 2, if these planes belong to the same pencil
of planes; in proving, use the results of Section 49 and Theorem 14, Chapter I.

64. Four Planes. Two Lines. The number of possible relative
positions of a set of planes increases quite rapidly when the set
contains more than three planes. The methods to be used in such
cases do not differ in any essential respect, however, from those
employed in the preceding sections. We shall not study this
general problem therefore; we shall restrict ourselves to the fol-
lowing special cases.

THEOREM 22. Four planes meet In a single point if and only if the
ranks of the coefficient matrix and the augmented matrix of their
equations are both 3.

Proof. If the planes meet in a single point there must be at
least one set of three among them which form a trihedral angle;
in that case at least one of the three-rowed minors of the c.m. has
a value different from zero and therefore the rank of the c.m. is
3. Moreover, it follows from Theorem 3, Chapter II (Section 23,
page 38) that the determinant of the augmented matrix vanishes;
hence the rank of that matrix is less than 4. But since the rank of
the a.m. can certainly not be less than that of the c.m., the rank
of the a.m. is 3.

Conversely, if the ranks of the c.m. and the a.m. are both 3, we
conclude by means of Theorem 3, Chapter II, that the equations
have a unique solution, that is, that the planes have a single point
in common.

COROLLARY. Two lines meet in a point if and only if the coefficient
matrix and the augmented matrix of the four linear equations used
to represent them, both have rank 3.

THEOREM 23. Four planes have a single line in common if and only
if the ranks of the coefficient matrix and the augmented matrix of
their equations are both equal to 2.

102 PLANES AND LINES

Proof. If the four planes have a single line in common then every
set of three of them have at least a line in common, and there must
be at least one set of three which have but a single line in common.
It follows therefore from Theorem 20 (Section 51, page 95) that
the ranks of the c.m. and the a.m. of any three of the four equations
is at most 2, and that there is one set of three equations among
them at least, whose c.m. and a.m. both have rank 2. We con-
clude from this that the ranks of the c.m. and the a.m. of the four
equations must both be 2.

Conversely, if the rank of both these matrices is 2, then for any
three of the four equations the ranks of both the c.m. and the a.m.
are at most 2, whereas there is at least one set of three equations
for which the ranks of the c.m. and the a.m. are exactly 2. It
follows therefore from Theorem 20 that there is at least one set
of three among the four planes which meet in a line, while the
fourth plane either passes through this same line, or else is parallel
to it. The latter alternative is ruled out by Corollary 3 of Theo-
rem 20, Section 51, page 97. Hence tho fourth plane passes
through the line common to the other three, as is required by the
theorem.

THEOREM 24. Four planes coincide if and only if the ranks of the
coefficient matrix and the augmented matrix of their equations are 1.

The proof is left to the reader.

From Theorems 22, 23, and 24 we obtain immediately the fol-
lowing corollaries:

COROLLARY 1. Four planes have one or more points in common if
and only if the ranks of the coefficient matrix and the augmented
matrix of their equations are equal.

COROLLARY 2. Four planes have no points in common if the aug-
mented matrix of its equations is non-singular. (Compare Definition
III, Chapter II, Section 26, p. 43.)

We return now to the Corollary of Theorem 22. The criterion
for deciding whether two lines meet in a point, which it supplies,
is not very convenient if the parametric forms of the equations of
the lines are used. We shall therefore develop this criterion in a
different form.

Let us consider the lines

h : x = ai + Xis, y = ft + ins, z = 71 + vis,

TWO LINES

103

and

I X =

= 72

is

Whether or not these lines have one or more points in common
depends upon whether or not it is possible to determine one or
more values of s and ,s' such that

that is, upon whether or not the system of equations

(1) XiS \2$' + Oil <*2 ~ 0, Ml* jJL'rt' + ft ~ ft = 0,
/ I f\

v\s v%8 ~r Ti T2 = "

possesses solutions.

The c.m. of this system of equations i

Xi >

Mi A

Vi V

its rank is 2, unless the lines are parallel or coincident. We con-
clude therefore, on the basis of Theorem 3, Chapter II (Section
23, page 38), that if the lines are neither parallel nor coincident
they will have one point in common or none according as the
rank of the augmented matrix of the system (1), that is, of the
matrix

Xi \2 Oil Oi2

(3) Ml M2 ft ft
v\ V<L 71 72

is 2 or 3. If two non-parallel lines have no point in common,
they are said to lie skew with respect to each other.

If the rank of the matrix (2) is 1, the lines are parallel or coin-
cident; and to distinguish between them we have to consider the
rank of the matrix (3). If this is 1, all its rows are proportional,
so that if any one of the equations is satisfied, the other two will
also be satisfied. This means that to any value of s there corre-
sponds a value of s' such that together they will satisfy the equa-
tions (1). Hence every point on l\ coincides with some point on
Z 2 ; in other words, the lines l\ and Z 2 coincide. If the rank of the
matrix (3) is 2, there is at least one pair among the equations
which do not possess a solution, in virtue of Theorem 3, Chapter

104

PLANES AND LINES

II; in this case therefore the lines can have no point in common.
Finally, it is clear that the rank of (3) can not be 3 if the rank of
(2) is 1 ; for in that case all the cofactors of the elements in the
last column of (3) vanish.

We summarize the conclusions in a theorem.

THEOREM 25. The two lines * = + \ts 9 y = fo -f #, * = yi -f vis
are skew if and only if the rank of the matrix (3) is 3; they meet in a
point if and only if the ranks of the matrices (2) and (3) are both 3;
they are parallel if and only if the rank of the matrix (2) is 1, while the
rank of the matrix (3) is 2; they are coincident if and only if the ranks
of the matrices (3) and (3) are both 1.

Remark. It should be clear that similar conclusions are ob-
tained if the equations of the lines are taken in the parametric form
of Theorem 16 (Section 47, page 86).

Examples.

1. To find the relative position of the lines l\, given by the equations
2 2 y -}- 3 z + 4 = 0, x -}- 2 y z 3 = 0, and / 2 , given by the equations
3x + 2?/ 2z-|-5=0, 2 x 3 y -}- z 4 = 0, we begin by finding the
direction cosines of each. We find, by use of Theorem 17 (Section 47, page 87)
that

-1 3

2 -1

2 -2

-3 1

13 -2

2 1

1:1, and

3 2
2 -3

4:7: 13.

It is evident from these results that the lines are neither parallel nor co-
incident. To decide whether or not they are skew, we determine a point on
each of the lines. The point PI( 2, 3, 1) lies on \i\ the point P 2 (l, 1, 5)
lies on Zg. And the determinant

-1 4 -3
1 7 2
1 13 -4

-1 4 -3
11 -1

17 -7

=4=0.

Hence the lines are skew.

2. The lines h : x = -4 -f t, y = 3 - 2 1, z = 2 + 3 t and 1 2 : x -2
4. 3 i' t y = 1 6 t', z = 8 + 9 /' are parallel or coincident, since their
direction cosines are proportional to each other. The augmented matrix
of the equations *-3*'-2=0, -2i-t-6i / -f4=Oand3-9< / -6 =

1 -3 -2
2 6 4

3 -9 -6
two lines coincide;
over into those of fe-

s

; and we see by inspection that its rank is 1. Hence the
the substitution I =* 3tf + 2 carries the equations of l\.

EXERCISES 105

55. Exercises.

1. Show that the four planes 3x -\- y z 5 = 0, a; 2y + 3z 2 = 0,
2 # + 4 ?/ 5 z 2 = 0, and 4 s + 3 y 7 2 + 3 =0 meet in a point.
Determine the coordinates of this point.

2. Show that the four planes 3z-?/ + 2z-3 = 0, 2x + 2y-3z
+ 4=0, 3 + 5 j/ -82 + 11=0 and Sy - 13 2 + 18 = meet in a line.
Determine the direction cosines of this line.

3. Show that the lines .c = 4 2 1, y = 3 + 2 , 2 = 5- 3 t and
x = t, y = I 41, z 1 + 3 meet in a point. Determine the co-
ordinates of this point.

4. Determine whether the lines # = 5 3, 2/ = 4-M, 2= 3 + 4 tf and
x = Q 6 t, y 2 + 2 <, 2 = 5 + 8 are parallel or coincident.

5. Prove that if four planes form a tetrahedron, the ranks of the coefficient
matrix and the augmented matrix of their equations are 3 and 4 respectively.

6. Prove that if four planes form a four-sided prism, the ranks of the
coefficient matrix and the augmented matrix of their equations are 2 and 3
respectively.

7. Prove analytically that if two lines are parallel there exists one and only
one plane in which they both lie.

8. Remembering that skew lines are non-parallel lines which have no point
in common, prove that if two lines are skew, there exists no plane in which they
both lie.

56. Miscellaneous Exercises.

1. Determine the distance of the point A (2, 3, 8) from the line I : x = 3
2 t, ?/ = 1 + 2 /, z = 6 t. Hint: The distance from the line to A
is equal to the product of the distance from A to the point #(3, 1, 6) on the
line by the sine of the angle between AB and /; use Theorem 14, Chapter III,
(Section 36, page 64).

2. Determine the distance of the point P(a\ t ft, 71) from the line x = a
+ As, y = + s, z = 7 -f vs.

3. Find the distances of the point A (4, 5, 3) from the planes 2 x 6 y
32 + 4 = and 3x -{- Qy 2z 5=0 and from their line of intersection.

4. Determine the relative positions of the planes in the following sets:

(a) 2z-?/4-3z-4=0, 3x + 2y - z + 2 = 0, s-42/ + 7z-10 = 0;
(6) 2z-2/ + 3z-4 = 0, 3x + 2y-z + 2=0, -42/4-72-6=0.

6. Find the points which

(a) the plane 3x 2y + z +2 = and the line x = l-\-t,y= 2

+ 2t, z = -3+4*;

(b) the plane 3 x - 2 y + z + 2 = and the line x = -1 + 2 *, y = 2

+ 2J, z = 3-2 t-,

(c) the plane 3 x - 2y + z + 2 = and the line x=-l+ t, y = 2

+ *, z = -3 - /
have in common.

106 PLANES AND LINES

6. The intercepts of a plane are a, 6, and c. On the axes of another rec-
tangular reference frame with the same origin as the original axes, the inter-

cepts of the same plane are a\, 61, and d. Show that -$ + r^ + -} = ~~2

CL 0" C Q i

7. Show that the locus of all points which are equally distant from two
given planes consists of two planes through the line of intersection of the
given planes. For the points on one of these planes the distances from the
two given planes are equal in sign as well as in magnitude; for those on the
other plane the distances from the two given planes are equal in magnitude,
but opposite in sign.

8. Prove that the two planes determined in Exercise 7 are perpendicular to
each other.

9. Prove that each of the planes determined in Exercise 7 makes equal
angles with the two given planes. The planes found in Exercise 7 are called
the bisecting planes of the dihedral angle formed by the two given planes.

10. Prove that if three planes meet in a point, the six bisecting planes of
the three dihedral angles formed by them meet three by three in a line. Hint:
Take the equations of the given planes in the normal form.

11. Write the equation of the plane through the origin determined by the
two lines through the origin whose direction cosines are Xi, MI, "i and X 2 , ^2, v*.

12. Show that three concurrent lines are coplanar (lie in one plane) if
and only if the determinant formed by their direction cosines vanishes. (Com-
pare Exercise 5, Section 42, page 74.)

Note. The determinant, mentioned in this exercise, whose rows consist
of the direction cosines of three concurrent lines, will be called the ori-
entation determinant of these lines.

13. A point moves in such a manner that its distances from two fixed lines
are always equal to each other. Determine the equation of the locus which
this point describes.

14. Determine the equations of the line which passes through the point
Pi(<*i, ft, 7i), is perpendicular to the line joining P 2 ( 2 , ft, 72) and I\(a z , ft, 73),
and lies in the plane determined by the points PI, P 2 , and P 3 .

15. Determine the distance of the point PI(I, ft, 71) from the line joining
the points P 2 (a 2 , ft, 72) and Ps(aa, ft, 73).

16. Show that four times the square of the area of the triangle whose ver-
tices are Pi(ai, ft-, 7;), i = 1, 2, 3 is equal to the sum of the squares of the

determinants

7i

1

ft 72 1
1

73

73 3

<*2

17. Find the distance of the point P(a, ft 7) from the plane determined
by the points Pi (a/, ft', 71), i = 1, 2, 3.

EXERCISES

107

18. Prove that the volume of the tetrahedron whose vertices are the points
Pi(<*i, fa, yi) i 1, 2, 3, 4 is equal to one sixth of the value of the determinant

1 /3i 71 1

2 & 72 1
/?3 73 1

* Pi 74 1

19. Determine the equations of the three planes, each of which passes
through one of three concurrent edges of a tetrahedron and is perpendicular
to the face opposite to the vertex in which these edges meet.

20. Show that the three planes determined in the preceding exercise meet
in a line which passes through one vertex and is perpendicular to the opposite
face.

21. Prove that the four perpendiculars from the vertices of a tetrahedron to
the opposite faces meet in a point.

22. Determine the equation of the plane through one edge of a tetrahedron
and through the midpoint of the opposite edge.

23. Prove that the six planes of the kind described in the preceding exercise
meet in a point.

24. Determine the equation of the plane through the midpoint of one edge
of a tetrahedron and perpendicular to the opposite edge.

26. Prove that the six planes of the kind described in the preceding exercise
have one point in common.

CHAPTER V
OTHER COORDINATE SYSTEMS

In the development of the subject up to this point we have used
a rectangular Cartesian coordinate system; and we have had no
occasion to change from one such system to another. Many of
the problems to be taken up in later chapters require such tran-
sitions; and for other purposes it is frequently desirable to use
reference frames different from that furnished by the rectangular
Cartesian coordinates. We shall therefore consider in the present
chapter some other reference frames in three-space, and also the
transition from one reference frame to another.

67. Spherical Coordinates. The reference frame consists of:
(1) a fixed plane II; (2) a fixed half -line, I, in this plane, called the

initial line ; (3) a fixed point
on the line /, called the origin;
and (4) a unit for linear meas-
urement and a unit for angular
measurement. To determine
the coordinates of a point P in
space with reference to this
FIG. 11 frame, we connect with P and

we drop a perpendicular from P

to the plane II; let P' be the foot of this perpendicular (see Fig. 11).
The spherical coordinates of P are then defined as follows.

DEFINITION I. The spherical coordinates of a point P in space are:
(1) the unsigned distance r from O to P, measured in terms of the unit
specified for linear measurement this is called the radius vector of
P; (2) the angle between -90 and 90 which the plane II makes with
OP, measured in terms of the unit specified for angular measurement
this is called the latitude; and (3) the angle e between and 360
which the line I makes with the projection of OP on the plane n,
measured in terms of the same unit this is called the longitude.

Remark 1. It follows from this definition that to every point
in space, except 0, there corresponds a definite set of three real
numbers, which are its spherical coordinates. But it is not true
in this case, as it was when Cartesian coordinates were used, that'

108

SPHERICAL COORDINATES

109

for every set of three real numbers there exists a point of which
these numbers are the spherical coordinates. The radius vector

is an unsigned real number, the latitude must lie between -

i

and x , and the longitude between and 2 ?r, if the radian is the
J

unit of angular measurement (between 90 and 90, between
and 360 if the degree is the unit).

Remark 2. There are various ways in which the definition of
spherical coordinates may be modified. The radius vector may
be defined as a signed number, with a possibility of its being either
positive or negative ; and the ranges of value for the latitude and
the longitude may be changed. Although there are some advan-
tages to be derived from such different agreements which the co-
ordinates, as defined above, do not possess, the present definition
has the desirable property, mentioned in Remark 1, of assigning to
every point in space, except 0, a single set of spherical coordinates.

FIG. 12

To establish a connection between spherical coordinates and
rectangular Cartesian coordinates, we make the point the origin
of a rectangular reference frame, the line I the positive X-axis
and the plane n the -XT-plane. Moreover we adopt the unit of
linear measurement as the unit on the three axes of this super-
imposed rectangular frame. It should now be easy to see (from
Fig. 12) that the Cartesian coordinates of a point P are connected
with its spherical coordinates by the following formulas:

x = OP X = OP' cos = r cos </> cos 0,
y = p x p f = OP' sin = r cos sin 0,
z = p'p = r sin <.

110 OTHER COORDINATE SYSTEMS

If these equations are squared and then added together, we find,
in accordance with the conventions laid down in Definition I, that

r = \Vx 2 + y' 2 + z 2 \ and hence that <t> = Arc sin. ,

' ' ^

+ I/ 2 + 2 2

Squaring and adding the first two equations leads to r cos < =

, . _ , y ,

|vV 2 + ?/ 2 |, and hence to the result that sin 6 = i / 2 . f[ alKl

#

cos = j . ; by means of these conditions the angle

|vV + j/2|

is completely determined between and 2 w. We state our results
as follows.

THEOREM 1. The transformation from spherical coordinates to rec-
tangular Cartesian coordinates, and vice versa, when in the two
reference frames the origins coincide, the initial line coincides with
the positive half of the X-axis, the initial plane with the XI -plane,
and the units of linear measurement are the same, is accomplished
by means of the equations:

x = r cos < cos 0, y r cos </> sin 0, z r sin </>;
and

4- Z 2 | f = Arc sin -.

sin e = ; -., cos e =

I ^x' 2 -f y 2 \

By means of the first set of formulas an equation in Cartesian
coordinates may be transformed into an equation in spherical
coordinates; and the second set of formulas enables us to trans-
form an equation in spherical coordinates into an equation in
Cartesian coordinates. In view of Definitions I and II of Chap-
ter IV (Section 39, page 68), we conclude from this that the
locus of a single equation in spherical coordinates is a surface, and
the locus of a pair of equations a curve. The geometrically

* We are using here the notation Arc sin u to indicate the " principal value "

of the angle whose sine is u, that is, the angle between and ~ which has

Z Z

its sine equal to u\ this function has a single real value for every real value

of u between 1 and 1. It should be clear that -. , is never

\Vx* + y* + z>\
more than 1 and the angle <t> as defined in the text always exists and lies between

7T j 7T

CYLINDRICAL COORDINATES 111

simplest surfaces, that is, the planes, are not represented by the
algebraically simplest equation when spherical coordinates are
used. By using Theorem 1, in connection with Theorem 4,
Chapter IV (Section 41, page 71), we find that the general
equation of a plane in spherical coordinates is

r(a cos </> cos + 6 cos <j> sin 6 + c sin 0) + d = 0.

On the other hand, the equation of a sphere whose center is at
the origin and whose radius is a has, in spherical coordinates, the
very simple equation r a.

58. Cylindrical Coordinates.
The reference frame now con-
sists of the initial plane II, the
initial line I, the origin 0, units
of linear and of angular meas-
urement, and besides of a di-
rected perpendicular to the
plane II at 0, called the z?-axis.
The cylindrical coordinates of

an arbitrary point P in space are then defined as follows (see
Fig. 13).

DEFINITION II. The cylindrical coordinates of a point P in space are:
(1) the perpendicular distance, f , from the initial plane to P, measured
in accordance with the unit of measurement and the direction speci-
fied for the -axis; (2) the undirected distance, p, from O to the pro-
jection P' of P on the initial plane, measured in terms of the specified
unit of linear measurement; and (3) the angle ^, between and 360,
which the initial line I makes with OP', measured in terms of the
specified unit of angular measurement.

Remark. The cylindrical coordinates of a point evidently com-
bine polar coordinates in the initial plane with a Cartesian f-
coordinate. For every point in space, except those which lie
on the 2-axis, there exists a unique set of 3 real numbers, which are
its cylindrical coordinates. But again it is not true that with
every set of three real numbers there is associated a point of which
these numbers are the cylindrical coordinates. The f -coordinate
is a signed real number, the coordinate p an unsigned real number,
and the coordinate ^ is restricted to the range 2 TT, if the radian
is the unit of angular measurement.

112 OTHER COORDINATE SYSTEMS

A rectangular Cartesian reference frame can be superimposed
on the reference frame used for cylindrical coordinates in the
manner used in the preceding section for spherical coordinates.
Thus we find the following relations between the rectangular
Cartesian coordinates z, y, z of a point and its cylindrical co-
ordinates p, \p, f (see Fig. 13) :

x = OP X = p cos ^, y = P X P' = p sin \f/, z = f ;
and

,-=*, . x- =-,, .

\Vx 2 + y 2 \ \Vx 2 + ?/|

Moreover, the reference frame for cylindrical coordinates con-
tains a reference frame for spherical coordinates. It is therefore
a simple matter to connect the spherical coordinates r, 0, of a
point with its cylindrical coordinates p, ^, f. We find:

p = OP 1 = r cos 0, ^ = 0, f = r sin <;
and

r = |Vp' 2 + H, * = Arc tan , = ^.

P

A repetition or the argument made in the last paragraph of
Section 57 should make it clear that the locus of a single equation
in cylindrical coordinates is a surface, and the locus of a pair of
equations a curve. The equation p = a represents a right cir-
cular cylindrical surface whose radius is a and whose axis is along
the *?-axis; the pair of equations p = a, f = b determines a circle
of radius a, in a plane parallel to the initial plane at a distance 6
from it and having its center on the 2-axis.

59. Exercises.

1. Determine the loci of each of the following equations:

(a)r = 2; (6) P - 3; (c)0 = ~; (</)f=-l; (e) 4 = -|; (/) * = ~

2. Write the equations in spherical coordinates of the surfaces whose equa-
tions in rectangular Cartesian coordinates are:

(a) x* -f 2/ 2 - 5; (6) y + * = 3; (c) 3 x - 2 y = 0;
(d) 3 z 2 4- 2 r/ 2 + 4 z 2 = 1; (e) 4 x 2 - y 2 = 1.

* The notation Arc tan ?z is used to designate the angle between and

2 2

whose tangent is u, that is, the " principal value " of the multiple-valued func-
tion Arc tan u; see the footnote on page 110.

OBLIQUE CARTESIAN COORDINATES 113

3. Determine the locus of each of the following pairs of equations:
(a)r = 3,0 = ~s (6)J"=4,* = *; (c) * = - *, * = ?^;

(d) P =5, r = -2; W r = 4, <#> = ~-

4. Transform the following equations into equations in rectangular Car-
tesian coordinates:

(a) r = tan d; (b) f = 2 <; (c) r(cos + sin tan 0) = 4 sec 0;
(df) P (3 cos ^ - sin t) + 2 r - 4 = 0; (e) P 2 + f 2 = 9;
(/) sin 2 + 2 sin 2 0-3 cos 2 (9=4.

60. Oblique Cartesian Coordinates. A reference frame for
a system of oblique Cartesian coordinates is furnished by any
three planes which meet in a
point, 0. These planes meet
two by two, in lines through 0;
we call these lines the X-, Y-,
and Z-axes and denote them
by OX, OF, and OZ respectively
(Fig. 14). On each axis we
specify a positive direction and
a unit of measurement. Through Fro. 14

an arbitrary point P in space, we

draw lines parallel to the coordinate axes, meeting the given
planes in the points P yz , P^, and P xy . We can now give the
following definition.

DEFINITION III. The (oblique) Cartesian coordinates of the point
P are the lengths of the lines PyzP, PzxP, and P xy P measured in accord-
ance with the units and directions specified for the X-, Y-, and Z-aies
respectively.

The coordinate planes, together with the planes determined by
the lines P yz P, P&P, and P xy P, taken two at the time, form an
oblique parallelepiped. This parallelepiped can be used con-
veniently to develop generalizations of some of the results obtained
in Chapter III, just as these results themselves were found by the
aid of the rectangular parallelepipeds, which we designated as the
c.p. of a point and the c.p. of a pair of points (see Sections 30 and
32).

Notation. The coordinate frame which we have just described
will be designated by the symbol 0-XYZ. If the angles between

114

OTHER COORDINATE SYSTEMS

the coordinate axes have to be specified, we shall use the symbol
O-XYZ-aQy, where a = / YOZ, ft = zZOX, and y = Z.XOY.

It will be supposed throughout that the units of measurement
on the three axes of any Cartesian reference frame, and also those
used on the axes of two such frames whose mutual relations are
under consideration, are equal to each other.

61. Translation of Axes. We consider now the relations exist-
ing between the two sets of coordinates of a point P with reference
to two Cartesian reference frames, whose axes are parallel; these
may be rectangular or oblique frames. Let the two reference

FIG. 15

frames be 0-XYZ and O'-X'Y'Z', and let the coordinates of 0'
with respect to 0-XYZ be a, b, and c. For an arbitrary point P(x,
y, z) we construct now the c.p. of P and 0' with respect to 0-XYZ.
Since the axes of the two frames are parallel, this parallelepiped
will also be the c.p. of P with respect to O'-X'Y'Z' and therefore
its edges will be equal in unsigned length to the numerical values
of the coordinates of P with respect to O'-X'Y'Z', that is, of
x', y 1 , and z'. The X-, F-, and Z-axes will meet the faces of this
parallelepiped in the points P x , <V; P y , O/; and P 2 , 0,' respec-

OBLIQUE TO RECTANGULAR AXES 115

lively; and the segments O X 'P X , O y 'P y , and 0/P 2 are equal to
x', y', and z r respectively (see Fig. 15). We have now, independ-
ently of the positions of P and of the reference frame O'-X'Y'Z',
the following relations:

0/0+ OP X + P X X ' = 0, 0/0 + OP y + PyO y ' = 0, and
0/0 + OP Z + P 3 2 ' =

and therefore

a + x x' = 0, b + y y' Q, and c + z z' = 0.

The result of the discussion can be summarized in the following
theorem :

THEOREM 2. The coordinates *, y, z of an arbitrary point P with
reference to a Cartesian frame of reference O-XFZ, and the coordi-
nates *', y' 9 z' of the same point with reference to a parallel Cartesian
frame of reference O'-X'Y'Z', whose origin has the coordinates a, 6, c
with respect to O-XYZ satisfy the relations

x' ~ x a, y' y 6, s' = z c.

Remark 1. The coordinates #', y' y and z f of the point 0' are all
0; hence we find from the theorem just stated, that the coordinates
of 0' with reference to O-XYZ are a, 6, c, as stated in the hypothe-
sis of the theorem; this simple fact serves as a check on the
formulas. Similarly, we find that the point whose coordinates
in the system O-XYZ are (0, 0, 0) has the coordinates ( a, 6,
-c) in the system O'-X'Y'Z'.

Remark 2. It should be noted that the formulas established in
Theorem 2 are the same, independently of whether the two refer-
ence frames are oblique or rectangular.

62. Transformation from Oblique to Rectangular Axes. Before
taking up the transformation of coordinates which results when we
pass from one arbitrary frame of reference to another, we shall
consider what happens when we change from an oblique frame of
reference to a special associated rectangular frame. A system of
rectangular axes can be superimposed upon a given oblique refer-
ence frame 0-XYZ-af$y by using in both systems the same XY-
plane, the same X-axis, the same origin, and the same units of
measurement. This is illustrated in Fig. 16, in which the axes of
the rectangular frame are designated by OX', OF', and OZ'. To
obtain the relations between the coordinates of an arbitrary point

116

OTHER COORDINATE SYSTEMS

P with respect to these two frames of reference, we make use of
the projection method (see Section 36). The coordinate parallele-
pipeds of P with respect to these two sets of axes furnish a closed
broken line, leading from to P along the edges OP*, P x P xy , P xy P,

X,X l

and back from P to along the edges PP xy ', P xy 'P x , P X 0. We
infer now from Theorem 12, Chapter III (see Section 36, page 62)
that

P,'0 = 0.

To evaluate these projections, we make use of Theorem 11, Chap-
ter III (see Section 36, page 62), remembering that the angles
YOZ, ZOXj and XOY formed by the original axes are equal to
a, 0, and 7 respectively, and that the X', Y', and Z' axes are mu-
tually orthogonal. In this way we find that

x + y cos 7 + z cos )8 x f 0.

By projecting the same path OP x P xy PP xy P x O upon the F'- and
Z'- axes, we find

y cos z YOY' + z cos z ZOY' - ?/ = and
2 cos z ZOZ' - z' = 0.

Clearly z F07' = ~ - Z-YOF, so that cos z F07 X = sin y.

t

To determine cos ^ZOY f and cos z^OZ', we make use* of the
result of Exercises 15 and 16, Section 38 (page 66), from which
we find that

cos a. cos /5 cos 7

sin 7

and

cos Z ZOZ' = -. Vl cos 2 a cos 2 ft cos 2 7 + 2 cos a cos cos 7,

sin 7

OBLIQUE TO RECTANGULAR AXES 117

in which the + or sign is to be used according as the Z-axis
and the Z'-axis point toward the same side or toward opposite
sides of the .XT-plane. Substitution of these values in the pre-
ceding equations leads to the following theorem.

THEOREM 3. If x, y, and z are the coordinates of an arbitrary point
P in the Cartesian frame O-XVZ-apy, and if *', y', and s f are the coo'r-
dinates of the same point with reference to the orthogonal Cartesian
frame O'-X'Y'Z' in which the units of measurement, the origin, the
A -axis, and the A Y-plane are the same as the corresponding elements
of the frame O-XYZ-apy, then

/ , / , , * (COS a COS COS 7)

* = x 4- y cos 7 + * cos 0, y' = y sin 7 H - - -fa - - -

f =fc*(l - COS 2 a - COS 2 - COS 2 7 + % COS a COS COS 7)*

* = - sin^

In which the plus or minus sign is to be used according as the two
reference frames are of the same or of opposite type (see footnote on
p. 50).

By means of the formulas of this theorem we can express the
distance of a point from the origin of a system of oblique axes in
terms of the oblique coordinates of this point. For

OP 2 = x'* + y'* + z' 2

= (x + y cos 7 + z cos /3) 2 +

S COS0COS7)! 2

- -

[
y si

.

sin 7 +

sin 7

2 2 (1 cos 2 a cos 2 ft cos 2 7 + 2 cos a cos cos 7)

sin 2 7
= x 2 + y* (cos 2 7 + sin 2 7)

2 2 [cos 2 /3 sin 2 7 + (cos a cos ft cos 7) 2 + 1 cos 2 a
__ cos 2 j8 cos 2 7 + 2 cos a cos ft cos 7] _

sin 2 7

+ 2 xy cos 7 + 2 xz cos
+ 2 2/2 [cos 7 cos j8 + cos a cos jS cos 7]
= rr 2 + i/ 2 + z 2 + 2 j/2 cos a + 2 zz cos jS + 2 XT/ cos 7.

COROLLARY 1. The square of the distance from the origin of a ref-
erence frame O-XYZ-<xpy to a point P(*, y, *) is equal to * 2 4- y 2 + * 2 +
2 y* cos a -f 2 ** cos -f 2 *y cos 7.

The expression for cos ZOZ' which was used in the proof of
Theorem 3 enables us moreover to obtain a convenient formula

118

OTHER COORDINATE SYSTEMS

for the volume of the c.p. of the point P(a, 6, c) in an oblique
reference frame. For the area of the base of this parallelopiped
is equal to a&sin 7 and its altitude is c cos ZOZ'. This leads
readily to the following result.

COROLLARY 2. The volume of the coordinate parallelopiped of the
point P(a 9 b, c) In the reference frame O-XYZ-afty Is equal to abc (1

1 cos 7 cos ft

COS 2 a COS 2 ft COS 2 7 -h 2 COS a COS ft COS 7]* = abc

COS 7

COS ft

1

COS a

COS a

1

63. Rotation of Axes. The projection method also enables us
to determine in a direct manner the relations which connect the
coordinates of one point in two Cartesian reference frames which
have the same origin. We shall first develop these equations for
the general case in which both systems are oblique and then ob-
tain as a special case the formulas for two rectangular systems.

Let the systems be 0-XYZ-a(5y and 0-X^YiZr-arfiy^ the
units being the same in the two. Let the cosines of the angles
formed by the axes of these systems be indicated in the following
table:

X

Y

Z

X,

h

nil

HI

Y l

k

W 2

n*

Zi

h

W.3

n.s

so that we have cos XOXi = h, cos XOYi = k, cos ZOYi = HZ,

etc.

We consider now the closed broken
line which leads from to P along
the edges OP*, P x P xy and P xy P of the
c.p. of P in the system 0-XYZ, and
which returns from P to along the
edges PP* iyi , PxwPxi, Pxfl of the c.p.
of P in the system 0-X \YiZi (see
Fig. 17). We project this closed
broken line in turn on the axes OX,
p IG 17 OYj and OZ, and then on the axes

OX ly OYi, OZi. If we make use in

each of these projections of Theorems 11 and 12 of Chapter III
(see Section 36, page 62) and if we employ also the notation in-

ROTATION OF AXES

119

troduced above for the angles between the two sets of axes, we
find:

Ix + y cos 7 + z cos ft - xili - 7/1/2 - Zi/ 3 = 0,
x cos 7 + y + z cos a x\m\ yim% z\m^ = 0,
x cos |3 + y cos a + 2 Zi/ii 7/1*12 Zin 3 = 0;
and

Iz/i + ymi + zni Xi 7/1 cos 71 Zi cos ft = 0,
xk + ym 2 + znz Xi cos y L yi z\ cos a t = 0,
xl s + 7/m 3 + zn 3 Xi cos ft ?/i cos i z\ = 0.

The system of equations (1) has a unique solution for x, y, z in
terms of Xi, yi, z\\ and the system (2) has a unique solution for
x\, 7/1, Zi in terms of #, T/, z. These solutions may be written down
by means of Cramer's rule (see Section 21, p. 37). For the coeffi-
cient determinants of these systems are oqual respectively to

1 cos 7 cos
cos 7 1 cos a
cos j3 cos a. 1

and

1

cos 71

COS ft

COS 71 COS ft
1 COS Oil

COS Oil 1

and, by virtue of Corollary 2 of Theorem 3 (Section 62, page 118),
the values of these determinants are equal to the squares of the
volumes of the c.p/s of the points (1, 1, 1) in the two systems.
If these volumes are denoted by v and vi respectively, we find from
the first system, that

Iixi+I 2 yi-+

Itfi

cos 7

cos/?

X

= X

niiXi+m 2 y

i+w 3 2i

1

cos a

=

V

*

niXi+n 2 yi

+ ^3^1

cos a

1

where

i

li COS 7

COS

1

^2

n

= X

?>l! 1

COS a ,

012 =

_ v/

m 2

V

z;

HI cos a

1

Ut

1

Zs COS 7

COS

#13

X

m 3 1

cos a

J

V

n 3 cos a

1

cos 7 cos
1 cos a
cos a 1

(see Theorem 8, Chapter I, Section 5, page 9). Similar results
are obtained for y and z, the trinomial elements now appearing in
the second and third columns. And from the system (2) we obtain

120

OTHER COORDINATE SYSTEMS

the solution

yi = - x

I

lix+miy+niz cos
l^x-^m^y-i-n^z cos

Vi

COS ft

hx+fthy+

tt;{2 1

where now

1

'''21 "~ /N

1

COS 7!

h eos ft

/2 COS i

]

1

COS ft

/3 1

V

fc, - 1 X

1 Hi

cos 7! ?1 2

cos/3i
cos ai

Vi

cos ft rig

1

1

cos 71

cos/3i

77? 1
?/?2

m 3

cos |9i

COS i

1

, with similar results for

x\ and 2i. The reader should have little difficulty in writing out
these further results. We state the following conclusion of our
discussion.

THEOREM 4. If *, y, z are the coordinates of an arbitrary point P
with respect to a reference frame O-XYZ-apy, and *i, y lf *i are the
coordinates of the same point in the reference frame O-XiYiZi-aifayi,
of which the A>, !>, and Ziaxes make with the axes of O-XYZ angles
whose cosines are /i, mi, /ii; / 2 , m 2 , n 2 and /a, m 3 , n 3 respectively, then

x a\\x\
y = 021*1
s = 031*1

> and

-f
4-

-f

Here aij(i,j = 1, 2, 3) is the value of the determinant obtained

1 cos 7 cos ft

from the determinant v =

cos 7

1

cos a

by replacing the

cos ft cos a 1
ith column by //, wj,-, n,-, divided by the value of v] and the co-
efficient by(i 9 j = 1, 2, 3) is the quotient by the value of the deter-
1 cos 71 cos ft

minant v\ =

cos 71

cos ot\

of the determinant obtained

cos ft cos i 1
from Vi by replacing the ith column by the jih column of the de-

terminant

Remark. The reader should write out in full the values of the
coefficients a# and &# in the form of the determinants, which in

ROTATION OF AXES 121

order to save space have merely been described in the statement
of the theorem.

The formulas established in this theorem take a particularly
simple form in case both reference frames are rectangular. For

in that case, a = |ft = 7 = ai = /3i = 7i=~, hence cos a = cos ft

2t

= cos 7 = cos ai = cos ft = cos 71 = 0; moreover the numbers
h> mi) ni] lz, nit, nz and Z 3 , 7/13, n^ become the direction cosines of
the axes OXi, OY lf and OZi with respect to 0-XYZ respectively.
The reader should have no difficulty in obtaining the formulas
for this special case which are stated in the following theorem; these
formulas, more than those of Theorem 4, arc the ones which we
shall have frequent occasion to use in our further work.

THEOREM 5. If x 9 y, z are the coordinates of a point P with respect
to a rectangular Cartesian frame of reference O-XYZ, and x l9 y\ 9 z\ the
coordinates of the same point with respect to another rectangular
frame O-XiYiZ i9 of which the Xi-, Yr, and Zi-axes have in O-XYZ
direction cosines \i, MU "i ^2, M2 ^ and X 3 , MS v* respectively, then

x = \ixi 4- X 2 yi 4- X 3 *i, and XL = Xi* + my -f ns 9
y = MI^I + M2ji + M3*i, y\ = X-2^ -f May H- 2S 9

x = VLXI + ^ 2 yi 4- ^si; -i = Xa^ 4- May 4- ^*

64. Exercises.

1. Set up the equations for the transformation of coordinates resulting from
translating the axes to a new origin whose coordinates in a system O-XYZ
are -3, 5, 2.

2. Determine the equation of the sphere x' 2 4" 2/ 2 + 2 2 = 9 with respect
to a new frame of reference obtained by translating the original axes to the new
origin O'(-2, -1,3).

3. Show that the planes determined by the equations 3 x 6?/4-2z=0,
2 x 4- y and 2 x 4 y 152 = are mutually perpendicular, and that
they pass through the origin. Establish the formulas for the transformation of
coordinates which results when these planes are taken respectively as the
FI#I-, the ZiXi-, and the XiFi-pIane of a new frame of reference.

4. Solve the same problem for the planes 2 x //4-2z = 0, x 2y
20 = 0, 2x + 2y -z = 0.

6. Apply the formulas obtained in Exercise 4 to determine the equations
in the new reference frame of the loci of the following equations:

(a) 2/ 2 4- z 2 = 3; (6) x 2 + y 2 + z 2 = 4; (c) 2 x 2 - 5 ?/ - 4 z 2 = 10;
(d) ax 4- 6?/ -f cz 4- d =*

6. Express the distance between two points in terms of their coordinates in
a system of oblique axes.

122 OTHER COORDINATE SYSTEMS

7. Determine the volume of the coordinate parallelepiped of two points in
an oblique frame of reference.

8. Prove that, if t , 2 , and 3 are the angles which a line I makes with the
axes of a reference frame 0-XYZ-apy, then

1 cos y cos cos O l
cos 7 1 cos a cos 2
cos /3 cos a 1 cos 3

COS 0i COS 2 COS 03 1

0.

Hint: If I coincides with one of the coordinate axes, the formula can
readily be verified. If I does not coincide with any of the axes, take a point P
on J, so that OP = 1 and project the closed broken line OP x P xy PO on the axes
and on /; from the resulting equations the desired formula should follow.

9. Show that, in case a. = 3 y = - , the formula of the preceding exercise

&

reduces to that given in Theorem 7, Chapter III (Section 33, page 56).

10. Show that if the formula of Exercise 8 reduces, for every line Z, to the
formula of Theorem 7, Chapter III, then a. = p = y = *

i

65. Rotation of Axes, continued. The formulas obtained in
Theorems 4 and 5 appear to contain a large number of parameters;
but these are not all independent parameters. For the a# and by
of Theorem 4, and the \, ^ and ^ of Theorem 5 can not be chosen
arbitrarily if the formulas are to represent a rotation of axes.
This can be seen most readily if we observe that the expressions
for the distance from to an arbitrary point P should be the same
in any two frames of reference which have the same origin. Hence
it follows that if the parameters in the formulas of Theorem 4 are
properly selected, then we must have, in view of Corollary 1 of
Theorem 3 (Section 62, page 117):

x 2 + y 2 + z 2 + 2 yz cos a + 2 zx cos ft + 2 xy cos 7

= Zi 2 + y\ 2 + Zi 2 + 2 y^i cos 71 + 2 z&i cos ft + 2 x^ cos 71

for all values of z, y, and z, if for xi, yi, and z\ we substitute the
expressions given in Theorem 4. If, in particular, both reference
frames are rectangular, we find by using Theorem 5 that

(XiX + Mi?/ + viz)* + (X 2 x + M 2 y + v&Y + (X 3 X + my + v&) 2
= x* + y* + z*

for all values of x, y, and z.

If we carry out the squaring of the trinomials on the left-hand
side, and equate the coefficients of like terms on the two sides, we

ROTATION OF AXES 123

are led to the conclusion that, if the formulas of Theorem 5 do
indeed represent a rotation of a rectangular reference frame, then
the parameters X,-, M, v^ i = 1, 2, 3, must satisfy the following
relations :

Xl 2 + X 2 * + X 3 2 = 1, Ml 2 + M2 2 + M3 2 = 1, *1 2 + "2 2 + "3 2 = 1

and MI^I + ^2^2 + M3^3 = 0, v\\i + ^2X2 + ^3X3 = 0,

+ X 2 M2 + X 3 /U :i = 0.

Conversely, in case these conditions hold, the equations of
Theorem 5 will carry a given rectangular frame over into another
rectangular frame with the same origin. For, by virtue of the first
three of the above relations, we can then take Xi, X 2 , Xs; jui> M2, Ms>
and *>i, j> 2 , J>3 as the direction cosines of three lines through the
origin; and it follows from the last three relations that these
lines are mutually perpendicular. The equations of Theorem 5
represent then the transformation to the new rectangular refer-
ence frame of which these three lines are the axes. Hence we have
established the following theorem.

THEOREM 6. If *i, y l9 i represent the coordinates of a point with
respect to a rectangular reference frame, the necessary and sufficient
conditions that the equations

xi = Xi* 4- my + viz, yi = \& -f M2y + v>s>> s\ = \& -f May + "a*

shall represent a transformation to another rectangular reference
frame, are that

Xi 2 + X 2 2 + X3 2 = 1, Mi 2 -f M2 2 -f Ma 2 = 1, vi* + *2 2 + "3 2 = 1

and that

Ml*'! H~ M2^2 ~f" M3**3 = 0, t>i\l ~f- J/2^2 ~f ^3X3 = 0. XiMl + ^2^2 4" XajUS = 0.

Remark 1. A transformation which satisfies the conditions of
Theorem 6 is called, with obvious justification, an orthogonal
transformation.

Remark 2. With the aid of Theorem 6, it becomes easy to verify
that the two sets of equations in Theorem 5 are equivalent. For
if we multiply those of the second set by Xi, X 2 , X 3 respectively and
then substitute the results in the first equation of the first set, we
find:

x = Xi(
=

= X]

124

OTHER COORDINATE SYSTEMS

and the other equations of the first set are verified in similar
manner.

Remark 3. It should be clear that the conditions of Theorem
6 can also be put in the equivalent form :

Xi 2 + Mi 2 + vi 2 = X 2 2 + M2 2 + *2 2 = A 3 2 + M3 2 + "3 2 = 1
and

+ iw = 0.

The equivalence of the two sets of equations in Theorem 5,
observed in Remark 2 above, leads to another interesting result.
Since neither set of axes consists of coplanar lines, their orientation
determinant (the determinant formed from their nine direction
cosines, see Exercise 12, Section 56, page 106) does not vanish.
Hence the equations of the first set can be solved for xi, T/I, and Zi
by Cramer's rule. If we denote the value of the orientation de-
terminant by D, we find, for example,

1 . . 2 " 1 1 1 M2 Ms

D

A,

But this value of x\ must be identical with the value furnished by
the first equation of the second set, for every value of x, y y and z.
Consequently, the coefficients of x, y, and z in the two expressions
for x\ must be equal, each to each. Hence we have

A 2 X 3

M2 M3

If these equations are squared and added, we find:

A2 AS

M2 Ms

But the right-hand side of this equation represents the square of
the sine of the angle between the lines whose direction cosines are
X2, M2, vt and X 3 , jt 3 , v 9f that is, between OFi and OZi] it is therefore
equal to 1 (see Theorem 14, Chapter III, Section 36, page 64).
We have therefore obtained the following result.

THEOREM 7. The value of the orientation determinant X 2 w ^

Xs M3 V*

of three mutually perpendicular directed lines is equal to -f 1 or to -1

LINEAR TRANSFORMATION

125

Remark. It follows from Theorem 7 that if two of the three
directed lines are interchanged, the value of the orientation de-
terminant changes sign; this will also happen if the direction on
one of the lines, or on all three, is changed. This suggests that
whether D is +1 or 1 depends upon whether or not the three
lines whose direction cosines are given by the elements in its rows,
taken in the order of these rows, form a reference frame of the
same type as the frame with respect to which their direction
cosines are taken (see footnote on page 50). This is indeed the
case, but a satisfactory proof of this fact can not be given without
a more extended discussion than can find a place in this book;
for it involves considerations of continuity. We shall therefore
not pursue this question.

We shall likewise omit a discussion of the conditions which the
parameters a# and % in the equations of Theorem 4 must satisfy
in order that these equations may represent a transformation from
one reference frame to another with the same origin.

66. Linear Transformation. Plane Sections of a Surface. If
we combine the results of Theorems 2 and 5 (see Sections 61,
page 115, and 63, page 121), we obtain formulas for the trans-
formation of coordinates which occurs when we pass from one
rectangular Cartesian refer-
ence frame to another, keep-
ing the units unchanged. For
such a change can always be
accomplished by a translation
and a rotation. Suppose that,
with reference to the frame
0-XYZ) the coordinates of
the new origin 0\ are a, ft, c;
and that the direction cosines
of the axes OiX ly OiY ly and
OiZi are Xi, /*i, *>i, \2, M2, ^ and

A 3 , /i3, v* respectively. Starting with 0-XYZ, we translate the axes
to the new origin 0\\ this leads to the reference frame Oi-XtYJli
(see Fig. 18). From this we make the transition to the frame
Oi-XiYiZi by a rotation of axes. It follows, from Theorem 2,
that x = x 2 + a, y = ?/ 2 + b, z = z 2 + c; and from Theorem 5,

FIG. 18

126 OTHER COORDINATE SYSTEMS

that X2 = Xi^i + X 2 ?/i + Xs2i, 2/2 = MI^I + M22/i + Ma^i, 22 =
+ v&i- We obtain therefore the following theorem.

THEOREM 8. If *, y, * and *i, yi, * are the coordinates of an arbitrary
point P with reference to the rectangular Cartesian frames O-XYZ
and Oi-XiYiZi respectively, and when, with respect to O-XVZ, the
coordinates of Oi are o, 6, c, and the direction cosines of OXi, O Yi,

and OZi are X if MI> "u X 2> /*2, "2, and \ 3 , jua, ^ respectively, then

Jt = Ai*i -h X2ji + Xa^i 4- >
y = MI*I + M2ji -h M3*i -h ^
* = ^1^1 -f ^yi 4- fa*i -f- c.

Remark 1. If we solve these equations for x\, yi, and zi, and
make use of the relations established in the proof of Theorem 6
(see Section 65, page 123), we find that

xi = \i(x - a) + Mi(y - 6) + vi(z - c),
2/i = X 2 (x - a) + /i2(j/ 6) + "2(2 - c),
zi = X 3 (x - a) + Ms(2/ - 6) + "sO - c).

It will be worth while for the reader to deduce this result by
direct application of Theorems 2 and 5.

Remark 2. A transformation of the frame of reference such as
we have discussed in the preceding paragraphs will be called a
rigid transformation. The algebraic transformation of coordi-
nates which corresponds to it is called a transformation of the
first degree, or a linear transformation.

COROLLARY 1. The degree of a polynomial in *, y, *, such as/to, y, s) 9
Is the same as that of the polynomial /i(#i, yi, s\) obtained from
/(*, y, s) by a linear transformation.

Proof. Since the expressions to be substituted for x, y, and z
are of the first degree in xi, y\, and zi, it should be clear that the
degree of fi can not exceed that of/. But since/ can be obtained
from /i by substituting for xi, T/I, and z l the linear functions of
x, //, and z stated in Remark 1, the degree of / can not exceed that
of /i. Therefore the degrees of the two polynomials arc equal.

Remark. The transformation of coordinates which corresponds
to a rotation of axes carries a homogeneous polynomial in x, y, z
over into a homogeneous polynomial of the same degree in x\, y\, z\.

We are now prepared to take up a question of interest and im-
portance, namely, to determine the character of the curve of

LINEAR TRANSFORMATION 127

intersection of a surface with an arbitrary plane. If one of the
variables, let us say by way of example y, is eliminated between the
equation of the surface f(x, y, z) = and that of the plane ax + by
+ cz + d = 0, we obtain an equation, say F(x, z) = 0, whose
space locus is the cylindrical surface parallel to the F-axis, which
projects the curve of intersection of surface and plane upon the
Z-ST-plane; and whose plane locus is the projection of this curve
upon the ZX-plane (compare Theorems 1 and 2, Chapter IV,
Section 40, pages 69, 71).

When the intersecting plane is parallel to one of the coordinate
planes the curve of intersection is congruent to its projection
upon that coordinate plane; in that case our question can be
answered immediately by the methods of Plane Analytical Geom-
etry. But when the intersecting plane is in a general position,
these two curves will not be congruent. The question can then
be answered, as is suggested clearly by the answer in the special
case, by first making a transformation of coordinates to a new
reference frame, of which one of the coordinate planes is parallel
to the given plane. How is such a transformation determined?

Let us propose so to transform a given frame of reference 0-XYZ
to a new frame Oi-XiYiZi that a plane whose equation in normal
form (see Section 44) is \x + ^y + vz p = shall be parallel
to the XiFi-plane. The necessary and sufficient condition for
this is that the direction cosines of the Zi-axis shall be X, /u, v (see
Theorem 7, Chapter IV, Section 44, page 78). It follows there-
fore from Theorem 8 that the desired transformation will be ac-
complished if we put

2 2/i + Xzi + a, y = MI^I + ^y\ + vz\ + 6,

+ Vtfji + VZ l + C,

where Xi, /i j; vi, X 2 , M2, v 2 , and a, 6, c are arbitrary, save for the re-
strictions imposed by Theorem 6. This arbitrariness in the choice
of some of the coefficients in the equations of transformation
corresponds to the fact that the position of the origin and that of
the axes OiJfi, OiFi have not yet been specified. When these
specifications have been made the equations of transformation can
be completely determined (see the Examples below).

In view of Corollary 1 (see page 126) the equation of the given
surface in the new reference frame will have the same degree as

128 OTHER COORDINATE SYSTEMS

the original equation of the surface. Since the equation of the
plane section of the surface is obtained from the new equation of
the surface by replacing one of the variables by a constant, the
degree of the equation of the curve of intersection will not exceed
the degree of the equation of the surface. For convenience of
reference, we record this fact as follows.

COROLLARY 2. The degree of the plane equation of the section of a
surface made by an arbitrary plane does not exceed the degree of the
equation of the surface.

Examples.

1. To determine the curve of intersection of the sphere x' 2 -f- y~ -f 2 2 = 9
with the plane 3 x 4 y + 12 z 2 = 0, we reduce the equation of the plane

, ,, , , 3 x 4 ?/ 12 z 2 .. IT . 3

to the normal form -7- ~ -f- - t - -- TTT = 0. Hence we have X = ,

lo 1J io lo lo

M = ~ TO ' v ~ 7v> From the conditions of Theorem 6 (see Section 65, page

lo lo

123), it follows that Xi, MI, "i and X 2 , /* 2 , y'2 must be so chosen that

(1) 3 Xi - 4 MI -f 12 vi = and Xi 2 -f- Mi 2 + "i 2 = 1
and that

(2) 3 X 2 - 4 M2 -h 12 v 2 = 0, X t X 2 -f MiM2 -f m^ = 0,

and X 2 2 -h M2 2 -f ^2 2 = 1.

If one of the coefficients Xi, /u t , or ^i is chosen arbitrarily and the others are
determined so as to satisfy the equations (1), then the remaining coefficients
X 2 , M2, ^2 are completely fixed by equations (2^ provided that Xi, MI, and ^
are so selected that the rank of the coefficient matrix of the first two of equa-
tions (2) is 2. If we take v\ = 0, we find Xi : /*i =4:3 and hence \i = ,
^ = jf ; Vl =o. For the determination of X 2 , ^2, ^2 we have then the conditions
3 X 2 4 /U2 + 12 >2 = and 4 X 2 -f 3 M2 = 0; we find therefore, by using
Theorem 4, Chapter II (see Section 25, page 41), that X 2 : ^ v 2 = -36 : 48 : 25
and hence that X 2 = -}j, ^ - ||, ^ 2 = jf. If, finally, we take a = 6 = c
= 0, we obtain the following equations of transformation:

_4xi 36 yi 3zi _3xj , 48 yi _4zi __ 5j/t , 12 2j
X " "F " "65" +13' IJ ~ 5 "*" 65 13' 13 "*~ 13 '

It follows now from the discussion at the beginning of Section 65 (see page
122) that these equations of transformation must carry the equation of the
sphere over into the equation rci 2 -f i/i 2 -f Zi 2 = 9. The equation of the
given plane 3x 42/ + 12z 2=0 becomes:

and this reduces to the simple equation 13 Zi = 2.

LINEAR TRANSFORMATION 129

Hence the curve of intersection of sphere and plane is congruent to the
curve in the A^Fi-plane whose equation is obtained by eliminating Zi from the
equations Xi~ -h y\ 2 -f- ^i 2 = 9 and 13 z\ = 2, that is, congruent to the plane
locus of the equation 169 xi 2 -f 169 yi 2 = 1517. The curve is therefore a
circle. That the result would be a circle was evident from the fact that every
plane cuts a sphere in a circle; the derivation which we have given serves to
illustrate the method which can be used also in cases in which the result
could not be so easily predicted.

2. The curve of intersection of the plane .3 x 4 y -f- 12 z 2 = 0, used
in Example 1, with the surface x 2 -h y 2 4- 2 z' 2 = is obtained by using the
same equations of transformation that were used above. The equation of the
surface now becomes

+ *)- 9.

If this equation is solved simultaneously with the equation of the plane,
13 z\ = 2, we find that the curve of intersection of the surface and the plane
is congruent with the plane locus of the equation

(re 7 , 24 \ 2
13 + I69

This equation reduces to 169 a?! 2 + 194 ?/i 2 -f ~JiL l _ 1513 100 = Q; the

240 yi
Lu;j xi- -p is*** /yr i

curve is therefore an ellipse.

67. Exercises.

1. Set up the equations for a transformation of coordinates which carries
the plane 2 x y -\- 2 z 5 = over into the ZjAVplane.

2. Determine the equations of transformation when the plane x + 2 y
- 2 z -f- 4 = is to become the Fi#i-plane; the line x + 2y 2z + 4 =0,
3 x ?/-}- z 3 = becomes the Zi-axis; and the point (?, \, 2) is the
new origin.

3. Set up the equations for a transformation of coordinates which will
carry the line 3 x + 4 y 2 z = 0, = into the AVaxis.

4. Determine the volume of the coordinate parallelepiped of the point P
whose coordinates are x 4, y 2, z 1 in the reference frame in
which the planes 2x 2y -f z = 0, y -\- z = and x z = are the YiZi-,
ZiXi-, and X^Yi -planes respectively.

6. Determine a plane equation of the curve of intersection of the plane
6 x 2y -{- 3 z 4 =0 with each of the following surfaces:

(a) 3 x 2 -f 2 y 2 - 3 z 2 + 18 = 0; (6) 4 x 2 - y 2 = 6 z;
(c) 3xy -f 4yz -2 zx 10.

6. Determine, for each of the axes of the new frame of reference introduced
in Exercise 3, Section 64 (page 121), such a direction that the new frame is of
the same type as the original frame; also such directions that the new frame
is of the opposite type.

130 OTHER COORDINATE SYSTEMS

7. Solve the corresponding problem for the reference frame introduced in
Exercise 4, Section 64.

8. Set up the equations of transformation for the transition from a refer-
ence frame O-XYZ to a new frame whose origin has in 0-XYZ the coordinates
5, 3, 2 and whose X-, F-, and Z-axes have in O-XYZ direction cosines
which are proportional to 4 ; 8, 1; 3, 2, 4, and to 34, 13 and 32 respectively.
Decide how to select the directions on the new axes to make the new frame
of the same type as O-XYZ.

9. Interpret geometrically the transformation of coordinates which is
determined by the equations 3 x = 2 oj ?/i 2 z\, 3 y = x\ 2 y\ -{- 2 z\
-6, 3 z = 2 xi + 2 ?/i 4- zi -f 3.

10. Prove that the curve in which an arbitrary plane meets the locus of an
equation of the second degree in x, y, and z is a conic section or a straight line.

CHAPTER VI

GENERAL PROPERTIES OF SURFACES AND CURVES

Before undertaking a somewhat detailed study of the loci of
equations of the second degree in x, y, and z, we shall consider in
the present chapter a few general properties of surfaces and curves.
Our attention will be restricted almost entirely to the loci of equa-
tions of the form /(z, y, z) = 0, whose left-hand side is a poly-
nomial in the three variables. The case in which not all three of
the variables are actually present in the equation has been con-
sidered in an earlier chapter (Section 40). We recall also Defi-
nitions I and II of Chapter IV (Section 39).

68. Surfaces of Revolution.

DEFINITION I. A surface of revolution is a surface that can be gen-
erated by revolving a plane curve about a line in its plane. The line a
around which the revolution takes place is called the axis of revolution
of the surface; the plane curve, in any
of Its positions, is called a meridian

curve.

It should be clear that all meridian
curves are congruent plane curves and
that the planes in which they lie con-
stitute a pencil of planes through the
axis of revolution.*

Every point P on the given curve,
see Fig. 19, describes a circle, whose
center is the projection P a of P on
the axis of revolution and whose
radius is P a P; these circles, which

lie in planes perpendicular to the axis of revolution, are called
parallel circles.

* The reader will notice that the simplest curved surfaces studied in ele-
mentary Solid Geometry, spheres, cones, cylinders, as well as the surfaces of
a large number of manufactured articles in daily use, such as teacups, lamp-
shades, hats, are, exactly or approximately, surfaces of revolution. Is there
a possible simple reason for this?

131

Axis

Meridian

FIG. 19

132 GENERAL PROPERTIES OF SURFACES AND CURVES

Our principal problem is to determine the equation of a sur-
face of revolution when we are given the equations of a meridian
curve and those of the axis of revolution. In case the axis of
revolution does not coincide with one of the coordinate axes, we
can always transform the frame of reference in such a manner as
to make one of the new coordinate axes coincide with the axis of
revolution; and there is evidently no loss in generality if we
assume that the meridian curve from which we start lies in one

of the coordinate planes through

i the axis of revolution. We shall

therefore suppose that the given
curve lies in the XZ-plane and
wo shall seek the equation of the
surface obtained by revolving
this curve about the X-axis (Fig.
20). Let the equations of the
meridian curve be f(x, z) 0,
y = 0; and let P(x, y, z) be an
F IO 20 arbitrary point on the surface.

The parallel circle through P
will cut the initial meridian in a point Q(x', 0, z'). It should now

be easy to see that x' = x, and z r = P X P =

Since

Q is a point on the meridian, its coordinates satisfy the equation
f(x, z) = 0. Consequently, the coordinates of P satisfy the
equation f(x, Vy 2 + z 2 ) = 0. Conversely, if the coordinates of
a point P f satisfy this equation, then the coordinates of the point
Q' in which the circle through P' with center at P x ' and radius
P x 'P f cuts the XZ-plane, will satisfy the equations f(x, z) =
and y = 0; the point Q' will therefore lie on the given curve and
the point P' on the surface of revolution which this curve generates
when revolving about the X-axis. We have therefore reached the
following conclusion:

THEOREM 1. The equation of the surface of revolution generated
when a plane curve in the ZX-plane revolves about the X-aiis, is ob-
tained by replacing z in the plane equation of this curve by Vy + * 2
and then rationalizing the equation.

It should be a simple matter to state similar conclusions for
the surface of revolution that is obtained when a curve in any co-

SURFACES OF REVOLUTION

133

ordinate plane revolves about either axis in that plane (see Section
69).
Examples.

1. The equation of the surface generated when a parabola in the X F-plane,
whose equations are y z = 4 ax and z = 0, is revolved about the X-axis, is
obtained by replacing y in the plane equation of the curve by V?/ 2 -f- z 2 and
then rationalizing. The surface is

called a paraboloid of revolution; its
equation is y 2 -f- z 2 = 4 ax (see Fig.
21).

2. Of especial interest to us are the
surfaces of revolution generated when
the conic sections are revolved about
one of the axes of the curve. The
shapes of these surfaces can easily be

pictured. By taking the equations of FIG. 21

the curves in the standard forms, fa-
miliar from Plane Analytical Geometry, we obtain readily the following
results:

Meridian curve
(1) x 2 -f ?/ 2 = a 2 , Circle

Axis of Equation of sur-
revolution fare of revolution

X-, or F-axis
X-nxis

a 2 ~*~ b* + 6 2

(5)

Name

Sphere.

Ellipsoid of revo-
lution ; prolate

spheroid.
Ellipsoid of revo-
lution; oblate

spheroid.

Hyperboloid of

revolution of two

sheets.

The surface that is obtained in this ease consists of two parts entirely
separate from each other; they are called the two sheets (nappes) of the sur-
face compare also the Remark on page 136.

_, ... -2 ..2 ,2 Hyperboloid of

revolution of one

sheet.

Paraboloid of
revolution.

a > b

(.3) ~2-f^ 2 = *. Ellipse
a > b

(4) ^ ~ iT2 = L Hyperbola

F-axis

JT-axis

~2 - ^2 = 1, Hyperbola

(6) |/ 2 = 4 oar, Parabola

(see Example 1)

(7) 2/ 2 ra 2 * 2 = 0, Pair of

intersecting lines.

(8) i/ 2 m 2 = 0, Pair of

parallel lines.

F-axis

-ST-axis

.XT-axis

2/ 2 -f- z
m 2 x 2 y 2

4 ax

Circular cone.

Circular cylinder.

3. Suppose that we wish to determine the equation of the surface obtained
when the line 2x + 6y 3^ + 1 =0, 3x y + 2z 3 =0 revolves about the

134 GENERAL PROPERTIES OF SURFACES AND CURVES

Iine2x + 6y -3*-f-l =0, x + y +z = 0. Evidently the two lines lie in the
plane 2x + 6?/ 32-f-l =0. Therefore we determine a transformation of co-
ordinates to a new frame of reference Oi-XiYiZi, in which this plane becomes the
XiFi-plane and the line in which it meets the plane x+y+ *> =0, that is, the axis
of revolution, becomes the Xi-axis. As origin Oi of the new frame, we shall take
the point in which the two given lines meet. In the notation of Section 66, we
have therefore a = 1, b - f , c = J. Since the XV-axis is the intersection of
the planes 2 z 4- 6 i/ 3 2 -f 1 =0 and x + y -f z - 0, we find, by use of
Theorem 17, Chapter IV (see Section 47, page 87), \i : MI : "i = 9 : -5 : -4;

954

and therefore \i = . _ , //i = -- ==, vi = -- ==. The Zi-axis is per-
V122 V122 V122

pendicular to the plane 2 x -f 6 y - 3 z -f 1 =0; its direction cosines are there-
fore proportional to 2, 6, -3 (see Theorem 7, Chapter IV, Section 44, page
78). Therefore X 3 = ?, MS = ?, "a = f. The direction cosines X 2 , M2, *2
of the Fi-axis must therefore satisfy the conditions 9 \2 5 M2 4 1/2 =
and 2 X 2 -h 6 M2 - 3 1/2 = (see Corollary 2 of Theorem 13, Chapter III,
Section 36, page 64). From this we find, by use of Theorem 4, Chapter

39
II (Section 25, page 41) that X 2 : M2 : "2 39 : 19 : 64, and that X 2 = ~~F= .

19 64

=- Hence the equations of transformation are, in

- - , 2 -

7\/122 7V122

view of Theorem 8, Chapter V (Section 66, page 126)

fl*i , 39 yi . 2 ^ nr 9(x - 1) - 5(y + }) - 4(g + t)

-- - -=- H- = - -

x -- 7= - -=- - , i

Vl22 7V122 7 Vl22

5si 19 yj 6zi

"r" l " 7V122

4 on , 64 y

The first of 'these two sets of formulas will carry the equations

2z+62/-3z + l' = over into 7zi = Q
x + y + z = Vl22 1/1 + 5 z, =

3z-2/-f 2z -3=0 84xi-f 113i/i -3VI222! =0.

Hence the equations of the given line may be written in the form zi = 0,
84 x\ -f- 113 2/1 = 0; and the equations of the axis of revolution can be put
in the form Zi - 0, 2/1 = (compare Remark 2, following Theorem 18, Chap-
ter IV, Section 49, page 92). The equation in Oi-XiYiZi of the required
surface of revolution is therefore obtained by rationalizing the equation
84 xi -f 113VV -j- Zl * = 0. This gives the equation -84 2 Zi 2 -f 113 2 (2/i 2 -h*i 2 )
= 0. To obtain the equation of the surface in 0-XYZ (from elementary
Solid Geometry we know that it is a circular cone), we now substitute for
xi t 2/1, and z\ the expressions in terms of x, y, z to which they are equal by the
second set of transformation equations. This gives us for the equation of
the required cone

-49-84 2 (9:c-52/-43- Y) 2 + H3 2 (39z + 190 + 64z - 5) 2
+ 122 113* (2 x + 6 y - 3 z + I) 2 0.

THE SHAPE OF A SURFACE 135

69. Exercises.

1. Prove that, if z = 0, g(x, y) = are the equations of a curve in the .XT-
plane, then: (a) g(x, V?/ 2 -f z 2 ) =0 is the equation of the surface of revolu-
tion obtained by revolving this curve about the A r -axis; (b) g( Vx 2 4- z 2 , y) =
is the equation of the surface of revolution obtained by revolving the curve
about the 7-axis.

2. Determine the equation of the surface which is generated when the circle
x = 0, t/ 2 + (z - 5) 2 = 9 in the FZ-plane, is revolved, (a) about the Z-axis;
(b) about the F-axis.

NOTE. The surface generated when a circle revolves about a line in its
plane not through its center, is called an anchor ring, or a torus. Which of
the surfaces described in this problem is a torus?

3. Determine the equation of the surface obtained by revolving the lem-
niscate (x 2 -f 2/ 2 ) 2 = 8 (x 2 ?/ 2 ), (a) about the X-axis; (b) about the K-axis.

4. Determine the equation of the circular cylinder which is formed when
the line z = 0, 3 x 4 ?/ -}- 2 = revolves about the line z 0, 3 x 4 y
-8 = 0.

6. Develop the general equation of a torus. (Take the axis of revolution
as one of the coordinate axes.)

6 Determine the equation of the surface generated when the equilateral
hyperbola xy = a 2 is revolved about one of its asymptotes.

7. Determine the equation of the surface of revolution obtained by re-
volving (a) the parabola z 2 4 ay about the Z-axis; (6) the semi-cubical
parabola x 2 = z 9 about the Z-axis; (c) the same curve about the X-axis;
(d) the curve y = sin x about the X-axis.

8. Determine the equation of the oblate spheroid obtained by revolving

the ellipse ( * ~ 3) * + (y ~ 2) " = 1, z = about the line y = 2, z = 0.
rr J

9. Determine the equation of the circular cone which is generated when the
line 8 z 4y -{- z 2 = 0, 2 x y 22 + 3=0 is revolved about the
line 8z-4 y + z -2 = 0, z + 2i/+22-4 = 0.

10. The hyperbola (x 4) 2 7 =1, x = is revolved about its

asymptote with positive slope. Determine the equation of the surface of rev-
olution which is generated.

70. The Shape of a Surface Determined from its Equation.
Contour Lines. One of the fundamental problems of Solid
Analytical Geometry is that of forming a clear picture of the shape
of the surface which is the locus of a given equation. We have
already obtained a number of partial solutions of this problem;
we will begin by summarizing these:

(a) the locus of an equation of the first degree is a plane.
(6) the locus of an equation from which one variable is absent
is a cylindrical surface.

136 GENERAL PROPERTIES OF SURFACES AND CURVES

(c) the locus of an equation which can be written in one of the
forms /(*, vV + z 2 ) = 0, f(y, Vz* + x 2 ) = or f(z, Vx* + y 2 )
= is a surface of revolution; the meridian curve of such a sur-
face can then be found by the methods of Plane Analytical Geom-
etry.

To these we now add the following further results.

THEOREM 2. The locus of an equation which can be reduced to the
form (x - a) 2 -f (y - 6) 2 4- (s - c) 2 = r 2 Is a sphere whose center Is at
(a, f>, c) and whose radius Is r.

The proof of this theorem is left to the reader.

DEFINITION II. A conical surface Is a surface generated by a line,
extending Indefinitely, which moves In such a way as to pass always
through a fixed point, called the vertex, and successively through the
points of a fixed curve, called the directrix.

Remark. It is a consequence of this definition that a conical
surface consists of two parts, one on each side of the vertex and
connected at the vertex; these parts are called the "sheets" or
"nappes" of the surface (compare Section 68, Example 2, part 4,
page 133). If the directrix is a plane curve and the vertex lies
in the plane of this curve, the conical surface is a sector of this
plane; if the directrix is a straight line, we obtain the entire plane.
In case the directrix consists of a pair of lines which do not both
lie in a plane with the vertex, the surface consists of a pair of
intersecting planes; if both lines lie in a plane with the vertex,
the surface reduces to a single plane counted twice. According
as the directrix is a curve of the first, second, third order, etc.,
the conical surface is said to be of the first, second, third order,
etc.; compare also Remark 1, Section 82, page 166.

THEOREM 3. The locus of an equation In x 9 y, and Is a conical sur-
face with vertex at the origin If and only If the equation Is homogeneous.

Proof. Recalling the definition of a homogeneous equation
(Section 20, page 35) we observe that if the equation f(x, y, z) =
is a homogeneous equation of degree n, then and only then will
all the terms in the polynomial f(x, y, z) be of the nth degree in the
three variables jointly, and hence we know that f(kx, ky, kz) =
k n f(x, y, z). Suppose now that P(a, , 7) is a point on the locus of
the equation /(x, y, z) = 0, supposed to be homogeneous of degree n.

THE SHAPE OF A SURFACE 137

The coordinates of an arbitrary point on the line OP are given by
the equations x = ta y y = tf$, z = ty, in which t is a parameter that
varies from point to point along the line (see Corollary 3 of Theo-
rem 6, Chapter III, Section 33, page 56). And it follows from the
homogeneity of the equation that f(ta, t$, ty) = t n f(a, 0, 7) = 0.
Consequently the entire line OP lies on the surface represented by
the equation f(x 9 y, z) = 0, if P does. The line OP will therefore
generate the surface if P passes successively through the points of
a curve in which the surface is cut by a plane not through the
origin. The proof of the converse of this theorem is left to the
reader.

Remark 1 . The locus of the linear homogeneous equation ax + by
+ cz = is a plane through the origin. In view of the remark
preceding Theorem 3 this is a special case of a conical surface

Remark 2. If an equation is homogeneous in x o, y 6,
z c, translation of the axes to the point A (a, b, c) as a new origin
will reduce it to an equation which is homogeneous in x' x a,
y' = y b, z' = z c (see Theorem 2, Chapter V, Section 61,
page 115). Its locus is therefore a conical surface whose vertex
is at A (a, b, c). Conversely, translation of axes to A as origin
puts the vertex of a conical surface with vertex at A at the origin
of the new reference frame. We can therefore state the following
corollary:

COROLLARY. The locus of an equation in x, y, a Is a conical surface
with vertex at the point (a, 6, c) if and only if it is homogeneous in
x a, y b, and z c.

THEOREM 4. The locus of the equation /(*, y, *) = is symmetric
with respect to the YZ-plane* if and only if the equations /(-*, y, *)
= and /(*, y, z) = are equivalent;! the locus of this equation is
symmetric with respect to the Z-axis* if and only if the equations

* Two points A and B are said to be symmetric with respect to a plane (or
line) if the segment AB is bisected perpendicularly by the plane (or line);
they are called symmetric with respect to a point if the segment AB is bi-
sected by that point. A surface is said to be symmetric with respect to plane
(line, point) if with every surface point A there is associated a surface point B
such that A and B are symmetric with respect to the plane (line, point).

t Two equations are called equivalent if any set of values of the variables
which satisfies either of the equations, also satisfies the other equation;
e.g., the equations x 2 - 2 y 3 + 4 z* + y = and (-x) 2 - 2 y 3 + 4 (-z) 2
4- y = are equivalent, also the equations x 3y + 2 z 3 = and 2 x -f 6 y
- 4 a 3 = 0.

138 GENERAL PROPERTIES OF SURFACES AND CURVES

/(-*, -y, *) = and/ r, y, z) = arc equivalent; this locus is sym-
metric with respect to the origin if and only if the equations f(x 9
-y> -*) = and/(*, y, s) = are equivalent.

This theorem and its obvious counterparts which assert sym-
metry with respect to the other coordinate planes and coordinate
axes are immediate extensions of well-known theorems of Plane
Analytical Geometry. The reader is entrusted with proving them.
We see, for example, that the locus of the equation z 2 2 y*
+ 4 z 2 + y = is symmetric with respect to the F-axis, and that
the locus of the equation x 3 ?/ + 2 z 3 = is symmetric with
respect to the origin. Compare Exercise 3, Section 31, page 52.

Of great value in determining the shape of a surface represented
by an equation are the contour lines of the surfaces; these are the
projections upon a fixed plane of the intersections of the surface
with a series of planes parallel to this fixed plane. We shall make
use only of sets of planes which are parallel to the coordinate
planes; accordingly we lay down the following definition:

DEFINITION 111. The X-contour lines of a surface are the projections
on the YZ-plane of the curves in which the surface is met by planes
parallel to the YZ-plane, that Is, by the planes whose equations are
x = a; the Y-contour lines (Z-contour lines) are the projections on
the ZX-plane (the A 1 -plane) of the curves in which the surface is met
by the planes parallel to the ZX-plane (the XI -plane), that is, by the
planes y = constant (the planes z = constant).

It follows immediately from this definition, in conjunction with
Theorem 2, Chapter IV (Section 40, page 71), that the .XT-con-
tour lines of the surface f(x, y, z) - are the curves in the FZ-
plane whose plane equations are /(fc, y, z) = 0, in which k is a
parameter taking all real values. Similarly the F-contour lines
and the Z-contour lines are given by the equations f(x, k, z) =
and f(x, y, k) = 0, respectively.

If we plot a set of contour lines, affixing to each of them the
value of the parameter k to which it corresponds, we obtain a
diagram which we shall call a contour map of the surface; such
contour maps should be of considerable aid in forming a mental
picture of the locus of an equation. The contour maps published
by the U. S. Geological Survey are approximately Z-contour
maps of portions of the earth's surface, if the level plane at some
point in the region is taken as the XT-plane.

THE SHAPE OF A SURFACE 139

Remark. In engineering practice various other methods are
used for representing space configurations in a plane drawing.
Of these we mention Descriptive Geometry and Perspective Draw-
ing. These subjects are treated in books especially devoted to
them. We shall not attempt to discuss them here; the interested
reader will have little difficulty in getting access to such books.

Example. To determine the J-contour lines of the locus of the equation

x 2 ?/ 2 2 2

-r - -f- ^ = 1, we consider the set of curves in the F-plane whose plane

4 y ID

2 2/2 ^2

equations are -r + TT; 1. There are three cases to be distinguished:
4 y 10

k 2
(a) \k\ < 2, that is, -2 < k < 2. In this case k 2 < 4, so that 1 - y > 0.

The equations of the X-contour lines for these values of the parameter k can
therefore be put in the form

= 1.

From this we conclude that these contour lines are hyperbolas whose center
is at the origin; that the transverse axis is along the Z-axis and the conjugate
axis along the F-axis. The semi-axes of the hyperbola are a = 2\/4 k 2

and b = ^ ~~ ; hence y- = 41 and all the hyperbolas of the set have the
2 06

4 V .5

same asymptotes, namely, the lines z = ^ , and the same eccentricity, '-.

(5\/4 k 2 \
0, 0, =b - ^ - y

(b) \k\ =2, that is, k = 2. The equation of the contour lines now

reduces to - -f ~ =0; the locus of this equation consists of a pair of inter-
y JLO

secting lines, namely the asymptotes common to the hyperbolas discussed
under (a).

(c) \k\ > 2, that is, k > 2, or k < -2. In this case k 2 > 4, so that

k 2 y 2 z 2

I - < and the equation takes the form / , - r -- -^- - r- = 1.

4 KT-O 16 (T-0

The contour lines are now hyperbolas with center at 0, the transverse axis
along the Y-axis and conjugate axis along the -axis. The semi-axes are

a = g ~ 4 and 6 =2 Vfc^=T; the foci are at the points (o, =fc 5 ^ " 4 ,
on the F-axis; the asymptotes are the same as those of the hyperbolas in (a),
namely, the lines z = -~ , and the eccentricity is again equal to -?

o 4

140 GENERAL PROPERTIES OF SURFACES AND CURVES

The X-contour map of this surface is sketched in Fig. 22, the plane of the
drawing being the ZF-plane. It follows from Theorem 4 that the surface is
symmetric with respect to the three coordinate planes, the three coordinate
axes and the origin. The part of the surface suggested by this contour map,

which lies to the right of the
ZF-plane is therefore duplicated
by a symmetric part to the left
of this plane.

It should be easy for the
reader to show that the Z-con-
tour lines of this surface are
also hyperbolas and that the
Z-contour map has the same
general character as in Fig. 22.
The F-contour lines are ellipses;
the reader should construct the
F-contour map. The surface
represented by this equation
is called a hyperboloid of one
Fi(i. 22 sheet. Of the axes of sym-

metry, the X- and Z-axes meet

the surface in real points; they arc called the transverse axes of the sur-
face. The F-axis does not meet the surface in real points; it is called the
conjugate axis of the surface.

71. Some Facts from Plane Analytical Geometry. In prepara-
tion for the construction of the contour maps of other surfaces,
we summarize in this section some facts from Plane Analytical
Geometry with which the reader should be familiar.

(1) the locus of the equation ~ + ~ = 1, a > fc, is an ellipse,

whose major and minor axes are along the X- and F-axis respec-
tively. The center is at the origin; the vertices at the points
(a, 0) and the foci at the points (dbc, 0), where c 2 = a 2 fc 2 .

The eccentricity e is equal to - , which is < 1 ; the directrices are

the lines x - . The ratio of the distances of any point on the

ellipse from a focus and from the corresponding directrix is equal
to e; the sum of the distances of any point on the ellipse from the
two foci equals 2 a.

The locus of the equation T^ + ~=l,a

6, is obtained from

SOME SPECIAL SURFACES 141

the ellipse just discussed by interchanging the roles of the X-
and F-axes.

x 2 y*

(2) The locus of the equation -^ ^ = 1 is an hyperbola with

center at the origin, transverse axis along the X-axis and conju-
gate axis along the F-axis. The vertices are the points (ita, 0),
the foci at (dbc, 0), where c 2 = a 2 + 6 2 . The eccentricity e is

(* CL

equal to - , which is > 1 ; the directrices are the lines x = -
H a' e

/j |y

The asymptotes are the lines - ~ = 0. The ratio of the dis-
tances of any point on the hyperbola from a focus and from the
corresponding directrix is equal to e\ the difference between the
distances of any point on the hyperbola from the two foci is equal
to db 2 a.

/Jr2 n 2

The properties of the locus of the equation -5 p = 1 are

analagous to those of the curve just discussed. The vertices and
foci lie on the F-axis, which is therefore the transverse axis;
the X-axis is the conjugate axis. The asymptotes, the number c
and the eccentricity are the same as the corresponding elements of
the first hyperbola. The two curves are called conjugate hyper-
bolas.

(3) The locus of the equation y 2 = 4 p(x a) is a parabola
whose axis is on the X-axis. The vertex is at the point (a, 0) ;
the focus at the point (a + p, 0). The curve will therefore open
in the direction of the positive or the negative X-axis, according
as p is positive or negative. The directrix is the line x = a p.
The ratio of the distances of any point on the parabola from the
focus and the directrix is equal to 1.

72. Some Special Surfaces. We should now be able to con-
struct readily the contour maps of the loci of a number of special
equations. The work will only be sketched; the reader should
work out carefully the details of the constructions.

X 2 IV 2 2

(a) the equation -5 + fj + -5 = 1 ; a > b > c.*

CL U C

* The case in which two or more of the numbers a, 6, c are equal has been
discussed in Example 2, Section 68 (page 133).

142 GENERAL PROPERTIES OF SURFACES AND CURVES

The Z-contour lines are given by the equation

6 2 ( 1 -

= 1. They are ellipses whose semi-axes are equal

to

aVc 2 - fc 2

and

aVc 2 -
The foci are at the points

The eccentricity is equal to
and therefore independent of fc.

Va 2 - 6 2 Vc 2 - fc 2

, 0, o).

For

fc = 0, the semi-axes have their largest values, a and 6; as |fc| in-
creases from to c, the semi-axes decrease; for fc = c, they are
and the ellipses shrink down to the points (0, 0, dbc) as fc tends
towards c; when |fc| > c, the ellipses are imaginary. We con-
clude that the contour map consists of a series of similar ellipses.
In Fig. 23 the contour lines for the part of the surface above

the .XT-plane are sketched.
The surface is symmetric
with respect to the XY-
plane; the contour map for
the part of the surface be-
low the -XT-plane is there-
fore identical with the one
here drawn. This surface
is called an ellipsoid; its
shape is suggested in per-
spective by Fig. 24. The segments a, 6, and c are called the semi-
axes of the surface; they are equal to one half of the segments
which the surface cuts off on the axes of symmetry.

FIG. 23

(6) the equations

Z 2 J/ 2

2 2

Z 2

2/ 2 z 2

2 + fe 2

C 2

' a 2

6 2+ c 2

2+^ =

1.

1 ; and

An equation of the second of these types has already been dis-
cussed in the Example of Section 70. We shall therefore leave the
discussion of these equations to the reader. Surfaces represented
by an equation of this form are called hyperboloids of one sheet;

SOME SPECIAL SURFACES

143

FIG. 24

FIG. 25

144 GENERAL PROPERTIES OF SURFACES AND CURVES

the segments a, 6, and c are its semi-axes. Figure 25 suggests the
appearance of an hyperboloid of one sheet.

(c) the equations^ - fL - 1 = i ; _-+L_!L = i

The loci of these three equations are of the same general charac-
ter. Interchanges among the coordinate axes will carry one of
them over into the others; we shall therefore make a discussion
of the first of these equations only, and we shall suppose that
6 > c.* The X-contour lines are determined by the equation

t/ 2 z 2
___ _j = i They are ellipses, whose semi-

the eccentricity is

, . bVk* - a 2 . cVk* - a 2
axes are equal to and ;

equal to

Vb* -

and is

therefore independent of k.
For \k\ < a, the semi-axes
are not real; therefore no
points of the locus are
found between the planes
x = a and x = a. For
k = =ta, the ellipses re-
duce to the points (a,
0, 0). For \k\ > a, we
obtain a series of similar
ellipses, which increase in-
definitely in size as \k\
increases. The surface
consists therefore of two
sheets, each of which
extends indefinitely, one
in the direction of the

positive X-axis, the other in the direction of the negative X-axis.

The surface is called an hyperboloid of two sheets (see Fig. 26).
* In case b c, the locus is a hyperboloid of revolution of two sheets, see

Example 2, Section 68, page 133.

FIG. 26

SOME SPECIAL SURFACES 145

Of the three axes of symmetry, the X-axis meets the surface in
real points; it is called the transverse axis. The other axes of
symmetry which do not meet the surface in real points are called
the conjugate axes.

.2 yZ L

(d) the equation ~2 + r^ + 1 = 1 .

(JL U C

It should be clear that there are no real points whose coordinates
satisfy this equation ; its locus is called an imaginary ellipsoid.

.2 y2 g2 .2 yZ g 2

(e) the equations-; + fr s == and -5 ffa = 0.

\ ' ^ a 2 o 2 c 2 a 2 o 2 c 2

It follows from Theorem 3 (Section 70, page 136) that the locus
of these equations consists of conical surfaces of the second order.
One contour map consists of similar ellipses, the other two of

FIG. 27

similar hyperbolas. In case the denominators in the two terms
with like signs are equal, the locus is a circular cone (see Example 2,
Section 68, page 133); otherwise it is an elliptic cone (see Fig. 27).

(/) the equation ^ + j + 5 = 0.

The only real point on the locus of this equation is the origin;
the surface is called an imaginary cone.

# 2 i/ 2
(0) the equation -^ + ~ = 2 pz 9 a > b.

146 GENERAL PROPERTIES OF SURFACES AND CURVES

The Z-contour lines are the similar ellipses

+

tr

1,

Jpfca 2 ' 2pfc6 2
whose semi-axes are equal to a V2 pk and 6 V2 pk, and whose

eccentricity is equal to . For fc = 0, the ellipse reduces

to the origin. If p > 0, the
ellipses are real f or fc > and
imaginary for fc < 0; if p < 0,
the situation is reversed. Con-
sequently the surface lies en-
tirely on one side of the -XT-
plane and extends indefinitely
on that side, on the side of
the positive Z-axis if p > 0,
on the side of the negative
Z-axis if p < 0. The Z-con-
tour lines and the F-contour
lines are parabolas. The sur-
face is called an elliptic para-
FIG. 28 boloid (see Fig. 28) . It should

be clear that the equations

v 2 z 2 z 2 x 2

~ + -5 = 2 px and -^ + -5 = 2 py also represent elliptic parabo-

c/ c c a

loids.

(h) the equation -g p = 2 pz.

The ^T-contour lines are the parabolas in the FZ-plane which are

y 2 fc 2
represented by the equation ~ = -5 2 pz, which can be written

<fc 2 \
z ^ 2 ) The axes of these parabolas
z pa /

(fc 2 \
0, 0, ^ r ).
^ pa /

Therefore, if p > 0, the parabolas will all extend in the direction of
the negative Z-axis, whereas their vertices will move upward along
the Z-axis as fc increases. But, if p < 0, the parabolas extend in
the direction of the positive Z-axis, and their vertices move down-

SOME SPECIAL SURFACES

147

ward along the Z-axis as k increases,
map has been drawn, for the case
p > 0, of the part of the surface
which lies on the right side of
the FZ-plane; since the surface is
obviously symmetric with respect
,to the FZ-plane, the X-contour
map of the other part of the sur-
face is identical with the one
drawn in this figure.

The F-contour map consists of
the parabolas

In Fig. 29, the Jf -contour

It should be clear that, if p > 0, the
contour lines are upward extending
parabolas which move downward
as k increases; and if p < 0, down-
ward extending parabolas which
move upward as k increases.

The Z-contour lines are hyperbolas.

Fio. 29
The surface is called an

FIG. 30

hyperbolic paraboloid (see Fig. 30). It should be clear that the
equations ~ = 2

D C

>lic naraholoids.

2 2 ~2

, <> a/

and -5 5 =

c 2 a 2

148 GENERAL PROPERTIES OF SURFACES AND CURVES

73. Exercises.

1. Show that the locus of each of the following equations is a sphere; de-
termine for each of them the radius and the coordinates of the center:

(a) z 2 + y 2 + * + 4 x - 6 y + 4 z - 8 - 0;

(6) z 2 + 2/ 2 + z 2 - 3 x -f 4 ?/ - 2 z + 3 - 0;

(c) * 2 -f 2/2 -f z 2 4. 6 s -f 4 T/ + 4 2 + 17 = 0;

(d) z 2 + y 2 + * 2 - 8 z - 6 ?/ - 4 z + 30 = 0;

(e) x 1 4- ?/ 2 + z 2 4- 2 az + 2 by -f 2 cz + d = 0.

2. Prove that the locus of the pair of equations 2 x 6?y + 3z 5 = o
and x 2 4- y 2 -f- z 2 4 x 6 ?/ 4- 8 z 7 = is a circle. Determine the
radius of this circle and the coordinates of its center.

Hint: The center of the circle is the foot of the perpendicular dropped on
the plane from the center of the sphere; the length of this perpendicular, the
radius of the circle and the radius of the sphere form a right triangle.

3. Show that the points common to the two spheres x 2 -f 2/ 2 4- z 2 6 x -f

6 y - 8 z 4- 3 = and z 2 4- 2/ 2 4- z 2 - 3 z 4- 2 ?/ - 5 z 4- 4 = lie in a
plane.

Hint: Write the equation of the pencil of spheres through the points com-
mon to the two given spheres and show that the pencil contains a plane.
(Compare Remark 4, after Theorem 18, Chapter IV, Section 49, page 92.)

4. Show that the pencil of spheres

ki(x* + ?/ 2 4- z 2 4- 2 aix 4- 2 b t y -f- 2 c& -f di) + k 2 (x* -f ?/ 2 -f z 2
- 2 b*y + 2 c 2 z + 4) =

contains a plane and prove that this plane is perpendicular to the line which
joins the centers of the two spheres, given by fci= 0, fc a = 1 and fci= 1, fc 2 = 0.
6. Prove that the curve determined by the equations x 7/+z = Oand
3 _|_ ^3 = 2 s j s symmetric with respect to the origin but not symmetric with
respect to any of the coordinate axes or coordinate planes.

6. Set up the conditions under which the plane ax + by + cz -f- d =
meets the sphere x 2 + y z + z 2 -f 2 aix -f 2 6iy -h 2 cis + di = in real
points.

7. Determine, supposing that the conditions of Exercise 6 are satisfied,
the radius and the coordinates of the center of the circle in which the plane and
the sphere meet.

8. Determine the equation of the conical surface of the second order whose
vertex is at the origin and whose directrix is the circle 2x ?/ 3z-f7 = 0,
* + y s + **-2-4y + 22-5 3S 0.

Hint: Two methods for solving this problem should be considered: First
method The required equation must be homogeneous of the second degree;
moreover it must be of the general form k\S\ -f &2$2 = 0, where $1 = and
$2=0 represent respectively the plane and the sphere, where k\ is a linear
function of x, y, and z, and where fe is a constant. Second method If

THE INTERSECTIONS OF A SURFACE AND A LINE 149

P(x, y, z) is an arbitrary point on the conical surface with vertex at the origin,
then there must exist a factor of proportionality r, such that rx, ry, rz satisfy
the equations of the plane and the sphere. Elimination of r leads to the re-
quired equation.

9. Find the equation of the circular conical surface whose vertex is at
F( 3, 2, 1) and whose directrix is the circle 3 x -\- y 2 z -f 4 =0,
x 2 + ?/ 2 4- z 2 + 4 x - 6 y - 4 z - 7 = 0.

10. Determine the equation of the conical surface with vertex at the origin
whose directrix is the curve of intersection of the plane x -}- 2 y 2 z 4=0

and the ellipsoid ^ + ~ + ^ = 1.
4 u

11. Write the equation of the circular conical surface whose vertex is at
V(a, /3, 7) and whose directrix is the curve of intersection of the plane ax -f by
+ cz + d = and the sphere x 2 + y z -f z 2 -f 2 a^x + 2 b } y + 2 r,z + 6?! = 0.

12. Construct

(a) the F-contour map of the locus of the equation xy = 2 z.

(b) the ^-contour map of the same surface.

(c) the ^-contour map of the surface xy + yz -f- zx 0.

74. The Intersections of a Surface and a Line. To find the
points in which a line meets a surface, we have to solve simul-
taneously three equations; two of these equations are linear,
namely, the equations of the line, the third may be of any degree.
The natural way to attack this algebraic problem would probably
seem to be to solve the linear equations for two of the variables in
terms of the third, to substitute these values in the third equation
and to solve the resulting equation for the third variable. For
example, if we wished to determine the points in which the line
2 x T/ + 2 4 = 0, + 2?/ 3z + 2 = meets the surface
x 2 2 xz + 7 y = 0, we would derive from the first two equations
that z = 5 x 6 and y ~ 7 x 10 ; substitution in the third
equation would then lead to the equation 9 x 2 + 61 x 70 = 0.
If this equation were solved for x, the coordinates of the desired
points could readily be found.

This method, although frequently useful in special cases, lacks
symmetry in the treatment of the variables, because it involves
the selection of one variable (in the example above this was x),
in terms of which the others are expressed; it casts the three vari-
ables in non-interchangeable r61es. It has been found that if
symmetry is maintained, greater elegance and clarity is intro-
duced in the treatment of algebraic problems. Without going
any further into a discussion of mathematical esthetics, to which

150 GENERAL PROPERTIES OF SURFACES AND CURVES

we are led quite naturally at this point, we shall proceed with the
problem of finding the intersections of a surface and a line in a
more symmetric form.

For this purpose, we take the equations of the line in the para-
metric form of Theorems 15 or 16, Chapter IV (Section 47, page
86). Let the equations of the line be

x = a + \s, y = + fjis t z = 7 + vs

and let the surface be represented by the equation f(x, y, z) = 0,
where /(#, y, z) represents a polynomial of degree n in x, y, and z.
To determine the points common to the line and the surface we
have now four equations in the four variables x, y, z and s. To
solve them, we substitute the expressions for x, y y and z given by
the equations of the line in the equation of the surface. This leads
to the equation

(1) f(a + Xs, + p.s, y + PS) = 0,

which has to be solved for s. When this solution has been ac-
complished, the coordinates of the required point can be deter-
mined at once. If we treat the example of the preceding para-
graph by this method, we derive first the parametric equations
of the line (see the Examples in Section 47, pages 88 and 89);
using the form of Theorem 16, Chapter IV, we find for them

x = 2 + t, y = 4 + 7t, ^==4 + 5 t*

Substitution of these values of x, y, and z in the equation of the sur-
face gives the quadratic equation 9 1 2 25 t 16 = 0. From this

.. . . 25 VI201 , , 61 + VI201

equation we obtain t = ^ and hence x\ =

lo

247 + 7A/1201 197 + sVlloT

247 - 7\/1201 197 - . . . . .

2/2 = 75 1 22 = TQ The points of mter-

lo lo

* If we put t -7= i we obtain the parametric equations in the s-form

of Theorem 15 (see Corollaries 2 and 3 of Theorem 10, Chapter III, Section
34, page 60). In numerical problems, the J-form of the parametric equations
is usually the more convenient; for theoretical discussions the s-form is
usually to be preferred.

DIGRESSION ON TAYLOR'S THEOREM 151

section of the line and the surface are approximately at PI (5.3, 27.2,
20.6) and P 2 (1.5, .2, 1.3).

75. Digression on Taylor's Theorem. The discussion of the
solution of equation (1) of the preceding section in the case of a
general polynomial, can be made in a very direct way by means
of Taylor's theorem,* which takes on a particularly simple form
for polynomials. This theorem tells us that if F(x, y, z) is a
polynomial, then

F(a + h, b + k, c + 1) = F(a, 6, c) + [hFi(a, 6, c) + kF 2 (a, 6, c)
+ lF*(a, fe, c)] + i (h*F u (a, 6, c) + fc 2 F 22 (a, 6, c) + / 2 F 33
(a, b, c) + 2 klFn(a, 6, c) + 2 lhF u (a, b, c) + 2 hkF 12
(a, 6, c)] + - + [AFi(a, 6, c)+kF 2 (a, 6, c)+ZF,(a,6, c)p.

In this formula Fi(a, &, c), F 2 (a, 6, c) and F 8 (a, &, c) are abbrevi-

dF I dF I

ations for the partial derivatives , ,

OX |x=a, y=b, zc vlj \ x = at ys -b t 2==c

dF I

respectively; F n (a, b, c), F&, F^, F&, F n , F 12 repre-

OZ \x=a,y=*b t z=c^

sent the second partial derivatives, for example, Fi 2 (a, 6, c) =

d 2 F I
--- ; similarly F 3 i2(a> &> c) will be used to designate

=a f y=6, z=*c

-a,y-6,i-c'

We observe that the expression in the second paif of brackets in
the formula closely resembles the square of that in the first pair
of brackets, which involves only the first derivatives of F. For
we see that [hFi(a, 6, c) + kF 2 (a, 6, c) + lF 3 (a, 6, c)] 2 = ft 2 Fi 2
(a, 6, c) + fcW(a, 6, c) + Z 2 ft 2 (a, 6, c) + 2 WF,(a, 6, c) ft(a, 6, c)
+ 2 toF 8 (a, 6, c)Fi(a, 6, c) + 2 WfcFi(o, 6, c)F 2 (a, 6, c). The second
order terms are obtained from this square if we replace F i 2 by FU,
FiF% by /^i 2 , etc., where these symbols designate second partial
derivatives in accordance with the notation explained above.

* For a fuller discussion of this important theorem the reader is referred to
hooks on the Calculus.

t Throughout our further work we shall use for partial derivatives the sub-
script notation which has here been introduced for the arbitrary function
F(*, V, *)

152 GENERAL PROPERTIES OF SURFACES AND CURVES

We shall therefore denote the set of second order terms by the
abbreviated notation [hFi(a, 6, c) + kF 2 (a, &, c) + lF 9 (a, 6, c)] (2) ;
and we shall call this a "symbolic square." Now the further terms
on the right side of the formula which states Taylor's theorem for
polynomials are the "symbolic cube" the " symbolic fourth power"
etc., to the "symbolic n-th power" of hFi(a, 6, c) + kF 2 (a, b, c) + IF 3
(a, b y c), divided by 3 !, 4 ! etc., n\ respectively. They are obtained
from the ordinary cube, fourth power, to nth power of this ex-
pression, if products of first partial derivatives are replaced by corre-
sponding higher partial derivatives; for example, F^FzF* should

be replaced by ^^ , etc.

Remark. We notice that each of these symbolic powers are
homogeneous functions of A, fc, and /.
Let us consider as an example the function

F(x, y, z) = ** - 3 x 2 y + 2 yz* + z 2 - 5 x + 3 y - 4.
Here

Fu = 6x-Qy, FM = 0, Fzz = 4 y + 2, F 23 = 4 z, F 31
= 0, F 12 = -6 x; Fm = 6, /^ 2 2 2 = F = 0, F m = ~6,

Fu3 = F 2n = F 223 = F 33 i = ^231 = 0, /^ 3 3 2 = 4. Hence
+ kF 2 + IFt = (3x 2 - bxy - 5)A + (-3o: 2 + 2z 2
+ (4^ + 2 z)Z,
+ Jfc^j + IF*) = (6 x - 6 ?y)/* 2 + (4 T/ + 2)Z 2 + 8

kF 2 + ZF 8 )^ = 6 /i 3 - 18 A 2 fc + 12 Z 2 fc.
We conclude therefore that

(a + fc) 3 - 3 (a + h)*(b + k) + 2 (6 + fc) (c + Z) 2 + (c + J) 2
-5(a + h) + 3 (b + fc) - 4 = a 3 - 3 a 2 6 + 2 6c 2 + c 2 +
3 6 - 5 a - 4 + [(3 a 2 - 6 a6 - 5)h + (-3 a 2 + 2 c 2 +3)
k + (4 be + 2 c)q + [(3 a - 3 fe)fe 2 + (26 + l)i! 2 + 4 cfcZ
-6a^] + ^~3/i 2 fc + 2Z 2 fc.

76. The Intersections of a Surface and a Line, continued.
We are now prepared for a further discussion of equation (1) of
Section 74 (see page 150). If we apply Taylor's theorem to the
left-hand member of this equation, it takes the following form:

THE INTERSECTIONS OF A SURFACE AND A LINE 153
/(, ft 7) + [Xtfi(a, ft 7) + fi(a, ft 7) + ?tfa(, ft 7)] + X

2 > =0.

But since each of the symbolic powers is a homogeneous function
(see Remark in Section 75), this equation reduces to an algebraic
equation in s of degree n :

/(a, j8, 7) + [Vi + M/2 + /], *, 7 + ^ [X/i + M/2 + "/ 3 P, 0, 7

+ + ^ ' [\/l + M/2 + /*] (w) ,0,7 = 0,

wliere the notation a, 0, 7 in the subscript position after a bracket
serves to indicate that a, ft and 7 are to be substituted for the
variables x, y, and z respectively within this bracket. We shall
state our result in the following form :

THEOREM 5. The parameter values of the points in which the locus
of the equation /(*, y, s) = 0, whose left-hand side is a polynomial, Is
met by the line x = a 4 Xs, y = ft 4 M, = T + , are the roots of the
algebraic equation

(1) a s n -f ais n ~ l 4- 4- aks n ~~ k 4 ... 4- a M _i* 4 a = 0,

where

(3) a n =/(a, 0, T)> n-i = [X/i 4 M/2 4 */j] af/8f7 ... a n -A = ^j
X [\/i + M/.+ ^C, 7 . , ao = ^ [X/i 4 Mfs + ^li%, 7 '

/19/29 and/3 are the partial derivatives of /(*, y, z) with respect to *, y, and
z respectively, and the symbolic powers of the trinomial are obtained
from the ordinary, non-symbolic powers by replacing products of the

dP+q+rf

form/i/ > / 2 9/3' by the partial derivatives d Pd qd , '

COROLLARY 1. A straight line has at most n points in common with
a surface which is the locus of an equation of degree n in *, y, and 2,
unless the entire line He in the surface.

For the equation (1), being of degree n in s, has at most n real
roots unless it be satisfied by every value of s; and to every real
root of (1) there corresponds one point that is common to the line
and the surface.

154 GENERAL PROPERTIES OF SURFACES AND CURVES

DEFINITION IV. A surface of order n Is the locus of an equation of
degree n in *, y, and *.

Remark. It follows from Corollary 1, in combination with this
definition, that if a surface is met by no line in more than n points,
its order is at most equal to n. The number of different points
which a line actually has in common with a surface depends on
the number of distinct, finite, real roots of equation (1); in every
numerical case this number can be determined by the methods
developed in the theory of algebraic equations. If two or more
roots of the equation are equal (let their common value be *)
we say that this number of points of intersection of the line and
the surface coincide at the point (a + Xs*, ft + jus*, y + vs*). Con-
versely, the algebraic meaning to be attached to the statement that
"a line x = a + Xs, y = ft + ps, z = y + vs meets a surface
f(x y y,z) = in two or more coincident points" is that the equation
(1) has a double, or a multiple root. To complex roots of the
equation (1) correspond complex points of intersection of the line
and the surface, that is, points whose coordinates are complex
numbers. To values of X, /*, and v for which a = 0, corresponds
an infinite root of the equation.

COROLLARY 2. The line whose equations are x = a + \s, y = ft + us,
s = y 4. vs lies entirely on the surface whose equation is/(*, y, s) =
if and only if

/(a, ft 7) = 0, X/i(a, ft y) + M/i(a, 0, y) + tf,(a, ft 7) = 0,

[X/l + M/2 + y f*\ahy = 0, . . . , [X/i -h M/2 + "/sll%, y = 0.

This corollary is an immediate consequence of Theorem 5.
77. Tangent Lines and Tangent Planes. Normals.

DEFINITION V. A line is tangent to a surface at a point A(a, ft 9 y) if
at least two of the points of intersection of the line with the surface
coincide at the point A.

This definition leads, by way of Theorem 5 (Section 76, page
153), to the following theorem:

THEOREM 6. If the point A(a, /s, 7) lies on the surface /(*, y, z) = 0,
the line * = a + Xs, y = + /*, s 7 4- vs will he tangent to this
surface at A if and only if the direction cosines x, M v satisfy the con-
dition:

(1) A/i(, A 7) + /i(, ft 7) + /a(a, ft 7) = 0.

TANGENT LINES AND TANGENT PLANES 155

Proof. Since the point A lies on the surface, the coefficient
a in equation (1) of Section 76 vanishes; hence s = is one root
of this equation and to this value of s corresponds the point A on
the line. The line is tangent to the surface at A if and only if at
least one other of its intersections with the surface falls at A,
that is, if the equation has as a root of multiplicity 2 at least
(see Remark in Section 76). But this is equivalent to the re-
quirement that a rt _i = [X/i + /i/2 + "/3] a ,&7 = 0; the theorem is
therefore proved.

We ask next for the locus of all points P such that the lines PA
which join P to a fixed point A on the surface shall be tangent to
the surface at A. If the coordinates of P are x, y, z, the direction
cosines of AP are proportional to x a, y /3, and z 7 (see
Corollary 1 of Theorem 6, Chapter III, Section 33, page 56). The
equation of the required locus is therefore obtained if x a,
y /3, z 7 are substituted for X, /*, v in condition (1) of Theorem
6. But the resulting equation is linear in x, y, and z; its locus is
therefore a plane. The discussion shows that the plane defined
in the next definition actually exists, if the function /(z, y t z)
possesses partial derivatives at the point A (a, 0, 7),* not all of
which are zero.

DEFINITION VI. A plane Is tangent to a surface at a point A If e?ery
line in the plane which passes through A is tangent to the surface at
A, and if every line which is tangent to the surface at A lies in this
plane* A line through a point A on a surface is normal to the surface
at A if It is perpendicular to a plane tangent to the surface at A.

We can now state the following results:

THEOREM 7. The plane tangent to the surface /(*, y 9 ) = at the
point A(a, 9 y) on the surface is the locus of the equation

(x - )/,(, ft r) 4- (y - //.(, ft 7) + (z - 7) Ma, ft 7 ) = 0.

* The possibility of non-existence of the partial derivatives of a function,
which is suggested here, need not disturb the reader at the present stage.
For every function considered in this book, we shall assume that partial deriva-
tives exist at every point. It must however be recognized that this involves
an assumption and that the reader is accepting its justifiability on faith.
For further treatment of this question the reader is referred to books on the
Theory of Functions of a Real Variable.

156 GENERAL PROPERTIES OF SURFACES AND CURVES

COROLLARY. The equations of the normal to the surface /(*, y, *)
= at the point on the surface A(a, ft, 7) are

x - a _ y ft = g 7
/i(i fa 7) /2(f ft T) /a(f ft 7)

and the parametric equations of the normal are

ft 7) ', ^ = 7 +/(, A 7) '

78. Exercises.

1. Determine the points in which the surface 2 a: 2 - y 2 -f 4 z 2 + 3 1/2 + 6 zz
-}- 4 x?/ 2x-\-y 4tZ-}-\ is met by the lines:

(a) x 9 - 4 , ?/ = 1 - , 2 - -7 + 3 J;

(6) x = -3 + 2 J, = -13 + 6 J, z = -3 + ;
(c) x = -3 - 2 , ^ = 9 -f- 5 , z = 7 + 4 J.

2. Determine the points which are common to the surface y' 2 10 xz 8 yz
- 12 x - 17 y + 16 z + 30 = and the lines:

(a) x = 3 - 2 J, y = -1+ 2, 2 = 2 - /;

(6) x = -1 -f 3 t, y = 3 - t, z = 2 ;

(c) x = 1 + 3 *, 2/ = -6 , 2 = -3 - 2 ;

(d) x = 1 - 2 f, ?/ = 1 -h 2 , 2=2-^.

3. Proceed similarly with the surface 3x 2 4 z* + 3 yz -\- 2 xy -}- 4 x 2y
+ 4 z + 2 = and the lines:

(a) a; = 1 + 3 , ?/ = 5 + 4/, 2=6 + 6 1;

(b) x = -2 + , y = 1 -*, 2 = 2;

(c) x = 10 *, ?/ = 2 t, z = I7t.

4. Expand by use of Taylor's theorem:

(a) 2 (x + h)* - 4 (x -f ft) (y + fc) + 6 (2 + Z) 2 - 3 (x + *) + 5;

(6) 3 (x - hY -h 5 (x - A) (2 - I) - 4 (y - fc) + 2 (x - A) + 3 (2 - I) - 7.

6. Determine the condition which the direction cosines of a line through
the point A(l, - 1, - 1) on the surface 3 x* - 4yz + 2 z 2 - 4x + 2 y + 5=0
must satisfy in order that the line may be tangent to the surface at A. Write
the equations for each of two mutually perpendicular tangent lines to the sur-
face at A.

6. Set up the conditions which the direction cosines of a line through the
point A(-l, 4, 3) on the surface 3 x 2 - 2y* -f 3 z 2 - 24 x - 4 y - 12 2
f 30 = must satisfy in order that the entire line may lie on the surface;
determine two lines through A which lie on the surface.

7. Determine the tangent plane and the normal to the surface 2 x 2 y 2
+ z* 3 zx -\-4xy-\-3x 2y 4 = 0at each of the following points on
the surface:

A(l, 1, 1), B(0, 0, -2), C(0, -2, 2).

THE SHAPE OF A CURVE IN SPACE 157

8. Write the equation of the tangent plane and the equations of the normal
to the surface:

x 3 + 7/ 3 4- z 3 = 1, at the point A(-l, 1, 1).

9. Determine the condition which the direction cosines of a line through
the origin must satisfy in order to lie entirely upon the conical surface z 2 4 i/ 2
-f 4 z 2 2 yz -f- 2 zx + 4 xy = (that is, to be a generator of this surface);
determine the generators of this surface which lie in the coordinate planes.

10. Prove that the origin lies on every plane that is tangent to a conical
surface whose vertex is at the origin. Extend this proposition so as to show
that the vertex of a conical surface lies on every plane that is tangent to the
surface.

79. The Shape of a Curve in Space. To determine the locus
of a plane curve in space, the methods developed in Chapter V
arc sufficient. For, by means of them we can transform the frame
of reference in such a manner as to make the plane in which the
curve lies one of the coordinate planes in the new frame. When
this has been accomplished the curve can be studied by the methods
of Plane Analytical Geometry.

For curves, whose points do not all lie in one plane (such curves
are usually called twisted curves, in French courbes gauches) a
representation by means of a plane drawing can be obtained by
the use of Descriptive Geometry or of Perspective Drawing. Some
idea of the shape of the curve can also be obtained from drawings
which show the projections of the curve on the three coordinate
planes.

Certain general properties of the curve can be detected by the
principles stated in Theorem 4 (Section 70, page 137). There
should be no difficulty in seeing, for example, that the curve de-
termined by the equations -j- + ~ + ~ = 1 and z 2 + y 2 = 1,

4 o y

which is the intersection of an ellipsoid and a circular cylinder, is
symmetric with respect to the three coordinate planes, the three
coordinate axes and the origin. The Z-projection of this curve is
given by the equations x 2 + y 2 = 1, z = 0; the F-projection by

x 2 2 z 2
the equations + = 1, y = 0; the -Sf-projection by the

1U 1O

equations-^- - ~ = 1.
For a more detailed study of the properties of curves and sur-

158 GENERAL PROPERTIES OF SURFACES AND CURVES

faces of general character, the reader is referred to treatises on
Differential Geometry. In the next Chapters we shall take up
the study of the loci of equations of the second degree in x, y,
and z.

CHAPTER VII
QUADRIC SURFACES, GENERAL PROPERTIES

Surfaces which are loci of equations of the second degree in
#, y> and z are surfaces of the second order, see Definition IV,
Chapter VI, Section 76, page 154; they are usually called quadric
surfaces or conicoids. A number of such surfaces have already
been discussed, as to their shape, in Chapter VI (see Sections 68
and 72). In proceeding to a more detailed study of these surfaces,
we shall apply to them the results obtained in Sections 76 and 77.

80. The Quadric Surface and the Line. We shall write the
general equation of the second degree in the form

Q(x, y, z) = a n x 2 + any* + a^z 2 + 2 a^yz + 2 a^zx +
+ 2 dux + 2 a^y + 2 a 34 z + a 44 = 0.

We shall find it convenient to use the symbol Q(x, y, z), or simply
Q, throughout as an abbreviation for this general form of the
function of the second degree in x, y, and z; we shall also use Q
to designate the quadric surface which is the locus of the equation
Q(x, y f z) 0. The notations a& and a 32 , si and a J3 , a u and 021,
an and 4i, 24 and a 42 , a 34 and a will be used interchangeably.
The partial derivatives of Q will be denoted by the subscript no-
tation, as already agreed upon in Section 75 (see footnote on page
151). Moreover we shall use q(x, y, z), or q, to designate the part
of Q which is homogeneous of the second degree; and q with sub-
scripts will be used for the partial derivatives of q. Thus we
have

q(x, y, z)

2(a n x + a&y + a 33 3 + a 34 );
2(a n x

Qn = 2 an, q& = Q 2 2 = 2 032, #33 = Qsa = 2 a 33 ,
= 623 2 025, &i = Qsi = 2 ou 9 0* = Qi2 = 2 a u .
159

160 QUADRIC SURFACES, GENERAL PROPERTIES

Furthermore we shall find it useful to use Q 4 as an abbreviation
for the linear expression 2(a^x + a^y + a 43 z + a 44 ), and corre-
spondingly # 4 as an abbreviation for the linear homogeneous part
of Q 4 , that is, for 2(a 41 x + a^y + a^z)\ and the derivatives of
these expressions will be denoted by the use of a double subscript
on Q or on q. We observe that the partial derivatives of Q and
of q of order higher than the second are identically zero.

From Theorem 5, Chapter VI (Section 76, page 153), we derive
therefore the following results.

THEOREM 1. The parameter values of the points In which the line

x = a + Xs, y = ft -f /us, z = y -f vs meets the quadrlc surface Q(x, y, )
= are the roots of the equation Los 2 + 2 LIS -f L 2 = 0, where

/ - o( H ^ r -

Lt = Q(a, (3, y), Li =

= X(a u a -f 0120 -f Oi

-f v(a z ict -f a 32 /3 -f 0337 4- 834),

and Lo = ^(X, /i, y) = flnX 2 + 22M 2 -f ass*' 2 -I- 2 a23Aw + 2 asii'X -f 2 a^X/x-

COROLLARY 1. The line * = a -f Xs, y = /3 -f /us, * = 7 -f ?s will (a)
meet the quadric surface Q in two distinct real points, if and only if
Li 2 - Lota > 0; (b) be tangent to Q if and only If Li 2 = L L 2 ; (c) not
have any real points In common with the surface if and only if
Li 2 - LoLs < 0.

COROLLARY 2. The line * = -fXs, y = ft + ius 9 z=y + vs will lie
entirely on the quadric surface Q if and only if L = L t = L 2 = 0.

Remark. Values of X, M, v for which L = q(\, n, v) = Q give
rise to at least one infinite root of the equation. Such values cor-
respond to lines which meet the surface in one or more infinitely
distant points. A line for which L\ = L = 0, but L 2 4 1 meets
the curve in two infinitely distant points. We lay down now the
following definition.

DEFINITION I. A direction determined by values of X, /* v which sat-
isfy the equation q(X, ^, v) =0 is called an asymptotic direction of the
quadric surface Q. A line which meets a quadric surface in two in-
finitely distant points is called an asymptote of the surface.

We can now state a further corollary.

COROLLARY 3. The necessary and sufficient conditions that the line

x = a + \s, y = ft + us, s = y + vs be an asymptote of the quadric
surface Q are that ?(x, ^, v) = and that \Qi(<x 9 , 7) +
7) = 0.

TANGENT LINE; TANGENT PLANE; NORMAL, ETC. 161

81. Tangent Line; Tangent Plane ; Normal; Polar Plane. If

the discussion of Section 77 be applied to the quadric surface Q,
the following results will appear without difficulty.

THEOREM 2. If the point A(a 9 (3, y) lies on the quadric Q, the line
x = 4- \s 9 y = -f M> * = T -f " will be tangent to the surface at ^
if and only if its direction cosines x, //, v satisfy the equation \Qi (, 0, 7)

+ /i<M 0> 7) -f *<?3(a, 0, 7) = 0.

Remark. This result can also be obtained from Corollary 1 of
Theorem 1.

THEOREM 3. The equation of the plane tangent to the quadric sur-
face Q at the point A(a 9 p, y) on the surface is (x - a) Q\(a 9 0, 7)
+ (y - 0) <M A 7) + (* - 7) <M 0, 7) = 0.

COROLLARY. The equations of the normal to the quadric surface Q
at the point A(<* 9 0, 7) are x = a -f (),( 0* 7) * y = + (M A 7) t,

* = 7 + <?i( A 7) ' t.

The equation of the tangent plane to the quadric surface, given
in Theorem 3, can be put in a form which is very convenient for
application in numerical cases. We observe first that Q(a, ($, 7)

= q(a, j3, 7) + 2 a^a + 2 034^ + 2 a^y + a 4 4 and that therefore
Qi = q l + 2 ai 4 , Qz = q* + 2 a 24 , Q 3 = g s + 2 a 34 .

And it follows from the notation introduced in the first paragraph
of Section 80 that Q 4 = q* + 2 a 44 . Furthermore, since q is a hom-
ogeneous function of the second degree, it follows* that aqi(a, 0, 7)
+ Pq*(ct f ft 7) + yq*(<* 9 ft 7) = 2 q(a, ft 7). If we add
2(2 a i4 a + 2 a 24 + 2 a 84 7 + 044) = 2 g 4 (a, ft 7) + 2 a 44 = 2 Q 4 (a, ft 7)
2 a 44 to both sides of the last equation, we find that

oQi(a, ft 7) + 0Q 2 (a, 0, 7) + 7<Me*, ft 7) + Q 4 (a f ft 7)
= 2<2(a,ft T ).

The left-hand side of the equation of the tangent plane, as given
in Theorem 3, may now be transformed as follows, remembering

* We are here making use of the following theorem, known as Euler's
theorem on homogeneous functions: If F(x, y, z) is a homogeneous function
of degree n in x, y, and z, then xFi -f- yF* + ^F 3 = nF. The proof of this theo-
rem may be made as follows: The homogeneity of F(x, y, z) tells us that
F(kx, ky, kz) = k n F(x, y, z) for every k. Differentiation with respect to A;
gives xFi(kx, ky, kz) + yF 2 (kx, ky, kz) + zF^kx, ky, kz) = nk~ l - F(x, y, z),
from which the formula of the theorem is obtained if we substitute 1 for k.

162 QUADRIC SURFACES, GENERAL PROPERTIES

that Q(, ft 7) = 0:

(x - )&(, ft 7 ) + (y - 0)Q 2 (a, ft 7) + (* - 7)Qs(, ft 7)

0220 + 0237 + ^24) + 2(a 3 ia + 320 + ^337

If we remember the convention concerning the coefficients a#,
which was made in the first paragraph of Section 80, we have the
following result :

THEOREM 4. The tangent plane to the quadrlc surface Q at the point
A (a, 0, 7) on the surface, has the equation

ana* 4- 0220y 4- 0337* 4- 023(0* + yy) 4- 031(7* 4- *) 4- oiz(ay -f fix)
+ ai 4 (* -f a) 4- a 2 4(y + /3) -h 34( -f 7) 4* 44 = 0.

Remark. The reader will observe that this form of the equation
of the tangent plane is obtainable from the equation of the quadric
surface by a process which consists in distributing the coefficients
of the equation among the variables x, y, z and the constants
a, ft 7 in equal shares; the technical name of this process is
" polarization. "

COROLLARY 1. If , 0, 7 are selected entirely arbitrarily, we have the
relation

<?i(f V* T) 4- <? 2 (, 0, 7) 4- 7<?( 0, 7) 4- <M> ft 7) = * (>(> ft 7).

The equation in Theorem 4 represents a plane whether A (a, 0, 7)
is on the quadric surface Q or not. For a general position of the
point A, this plane is called the polar plane of A with respect to
the surface.

DEFINITION II. The plane which Is represented by the equation

auax 4- <*vpy 4- a337* 4- 23(/3* 4- yy) 4- 031(7* 4- as) 4- ai 2 (ay 4- px) 4-
OH(* 4- a) 4- 24(y 4~ 0) 4~ 034 (* 4~ 7) 4- 044 = Is the polar plane of the
point A(a, 0, 7) with respect to the quadric surface Q; the point Is
called the pole of the plane with respect to the surface.

This definition enables us to state a further corollary of Theorem
4, namely,

COROLLARY 2. The tangent plane to the quadric surface Q at the
point A (, 0, 7) on the surface coincides with the polar plane of A
with respect to the quadric surface.

Thus we have obtained a geometrical interpretation of the polar
plane of a point on the surface Q with respect to this surface. We

TANGENT LINE; TANGENT PLANE; NORMAL, ETC. 163

proceed next to inquire as to the geometrical significance of the
polar plane with respect to the surface Q of a point which is not
on this surface.

We observe first that, in view of the proof of Theorem 4, of
Corollary 1 of this theorem, and of Definition II, the equation of
the polar plane can be written in the form

xQv(a, ft 7) + yOt(, 7) + *Qi(, ft 7) + <M, ft 7) =

or in the equivalent form

(1) (x - a)Qi(a, ft 7) + (if - &(, ft 7) + (* - 7)
Qi(a, ft 7) + 2 Q(a, ft 7) = 0.

Let us now consider an arbitrary line I through the point

A (a, ft 7):

x = a + \s, y = ft + ps, z = 7 + j/s;

let the points where the line i meets the quadric surface be A\ and

Az ; and let its point of intersection with the polar plane (1) of A

with respect to the surface be

A' (see Fig. 31). It follows

then from the geometrical

meaning of the parameter s in

the equations of the line I (see

Corollary 2 of Theorem 10,

Chapter III, Section 34, page

60) that the directed segments

AAi and AA 2 are equal to the

roots of the equation Lo 2 +

2LiS + L 2 = 0, established in FIG. 31

Theorem 1; and that the di-

rected segment A A' is equal to the root of the equation

XQi(, ft 7) + /*&(, ft 7) + *&(, ft 7) + 2 Q(a, 0, 7) = 0,

obtained by eliminating x, y, and z between the equations of the
line I and the equation of the polar plane (1). Now we have to
recall the meaning of the coefficients L , LI and L 2 stated in
Theorem 1, and also the relations between the coefficients and the

roots of an algebraic equation (the sum of the roots of the
equation ax 2 + bx + c = is equal to , their product is equal

164 QUADRIC SURFACES, GENERAL PROPERTIES

to - Y With the aid of these tools, we find the following result:

AA' = -

+ M& +

2L 2 L 2 /L

: = 2

+ AA 2

From this relation we derive two consequences:

(a) If we divide both sides by two and take the reciprocals of

211
both sides of the equation, we find that -r-j, = -r-r- + -r-r- , or that

1111

AA' AA 1 AA 2 '

This informs us that, independently of the direction of the line I,
the reciprocals of the segments AAi, A A', q,nd AA 2 form an arith-
metical progression, that is, the segments AAi, AA', and AA 2 form
a harmonic progression, so that A A f is the harmonic mean between
AAi and AA 2 .

(6) If we transform the relation by writing AA f = AAi + A\A' y
clearing of fractions and carrying out the indicated operations, we
find that

+ AA l AiA' + AA 2 A^' = AA l AA 2 ;

that is,

AAi(A 2 A + AA 1 + A,A') + AA 2 A,A' = 0,
A 2 A' + AA 2 - A,A' = 0,

fi i
or finally --.

This equation expresses the fact that the points A and A r divide
the segment AiA 2 in ratios which are equal numerically, but oppo-
site in sign. This is what is meant by the statement that A' and
A are harmonic conjugates with respect to the points AI and A 2)
according to the following definition:

DEFINITION III. If A, B, C, and D are collinear points and so situ-
ated that the ratio of the segments CA and AD, in which A divides the
segment CD, is equal numerically but opposite in sign to the ratio
of the segments CB and BD in which B divides the segment CD, then
A and B are called harmonic conjugates with respect to C and D.

POLAR PLANE AND POLE

165

It should be clear from the preceding discussion that if A and
B are harmonic conjugates with respect to C and D, then the
segment AB is a harmonic mean between the segments AC and
AD, and conversely. Furthermore, if B is the harmonic conjugate
of A with respect to the intersections of the line AB with the

2 Q 1

quadric surface Q(x, y, z) =0, then AB = ^ ; ^ ; 7r

in which X, /*, v are the direction cosines of the line AB.
from this that the coordinates of B are
2 XQ

It follows

+

+

ya = P -

2* = 7 ~
Consequently

+ M? 2 +

+

(, ft 7)

+

+ MQ 2 +

,ft7)+ (*B - 7)Q 3 (, ft 7)
= -2Q(a,0, 7),

from which we conclude that the coordinates of B satisfy the equa-
tion (1) of the polar plane. We can summarize our discussions
by the following theorem.

THEOREM 5. The polar plane of a point A with respect to a quadric
surface is the locus of the harmonic conjugates of A with respect to
the intersections of the surface with the lines through A.

82. Polar Plane and Pole. Tangent Cone. Preliminary to a
discussion of some further properties of the polar plane we raise
the question whether it is possible for a plane to have more than
one pole with respect to a quadric surface, that is, whether it is
possible for two different points A (a, 0, 7) and A* (a.', &', y') to
have the same polar plane with respect to the surface. In view of
Definition II this amounts to the question whether the equations
(ana + a^ft + a^y + a M )x + (a u a + a<np + 0237 + ^24) y +

044)

= 0,

(ana' + ai 2 /3' + 0137' +
y + (aisa' + 0280' +
+ 044) =

+ (a^af + 0%$' + a^y' + 024)
+ 034)2 + (ana' + <h&' +

166 QUADRIC SURFACES, GENERAL PROPERTIES

can represent the same plane. This will be the case if and only if
the coefficients of x, y, and z and the constant terms in the two
equations differ by a factor of proportionality, k (see the footnote
on page 78), that is, if and only if the numbers ka a', kft /3',
ky y' and k 1 satisfy the following system of linear homoge-
neous equations:

an(ka - a') + a 12 (/c/3 - 0') + a 13 (fc T - T') + u(fc - 1) = 0,
a 12 (ka - a') + a*(W - (?) + a*(ky - 7') + a,,(k - 1) = 0,

- cf) + a 23 (/c/3 - 0') + a,,(ky - 7') + <to(k - 1) = 0,

- a') + 024(fc0 - 0') + au(ky - 7') + a u (k - 1) = 0.

This system of equations will or will not possess a nontrivial
solution, according as its coefficient determinant has a value that
is equal to or different from zero (see Theorem 2, Chapter II, page
38). In the latter case, the only solution is the trivial one, so that
we find ka a' k/3 /3' = ky y' = k 1 = ; from this we
find k = 1, a = a', = 0', 7 = 7', so that no plane can have more
than one pole. The discussion leads to the following definitions
and conclusions:

DEFINITION IV. The determinant of the symmetric square matrix

1 1 ay 1 1, i, j = 1, 2, 3, 4, dij = aji, is called tht discriminant of the
quadrlc surface Q; we shall use A to designate the value of this
determinant.

DEFINITION V. A quadric surface whose discriminant vanishes is
called a singular quadric surface. (Compare Definitions II and III on
page 43.)

THEOREM 6. No plane has more than one pole with respect to a
non-singular quadric surface.

Remark 1. The homogeneous equation of the second degree in
x, y, and z represents a conical surface of the second order; it is
called a quadric cone. The quadric cone is a singular quadric
surface, for in its equation a J4 = 24 = 34 = #44 = 0; consequently
the value of the discriminant is zero.

Remark 2. While Theorem 6 asserts that no plane has more
than one pole with respect to a non-singular quadric surface, it
does not say that there exists a pole for every plane in space. The
question depends on whether the system of equations

n + #120 + ia7 + i4 = ka,

O>12<X + 220 "T" #237 + 24 = kb,

0240 + 0347 + 044 =

POLAR PLANE AND POLE 167

does or does not have a solution for a, /3, 7, and k for every set of
values of a, 6, c, and d. Since for a non-singular quadric the aug-
mented' matrix of this system is always of rank 4, it follows from
Theorems 1 and 8 of Chapter II (see Sections 21 and 27, pages 36
and 44) that a plane will have a single pole or none with respect to
a non-singular quadric surface, according as the value of the

determinant

0,12 #22 #23

is different from or equal to zero.

#13 #23 #33
#14 #23 #34

We derive now a number of further consequences from the defi-
nition of the polar plane.

THEOREM 7. If A(a 9 /3, 7) lies on the polar plane of A'(a' 9 ', 7') with
respect to a quadric surface, then A' lies on the polar plane of A with
respect to the same surface.

The proof of this theorem is left to the reader.
THEOREM 8. If A(a 9 0, 7) lies on its own polar plane with respect to
a quadric surface, then A lies on the surface; and conversely.

The first part of this statement becomes evident when we sub-
stitute the coordinat >l of A in the equation of the polar plane of A ;
the second part follows from Corollary 2 of Theorem 4 (Section
81, page 162).

Suppose now that from a point A not on the quadric Q tangents
be drawn to the surface ; let the points of contact of these tangents
be A', B', etc. Then, since A lies in the planes tangent to the
surface at A', B' y etc., A lies in the polar planes of A', B', etc., with
respect to the surface. It follows therefore from Theorem 7, that
the polar plane A passes through A 1 ', J5', etc. Conversely, if A'
is a point in which the polar plane of A meets the surface, then the
tangent plane to the surface at A' passes through A, that is, the
line A A' is tangent to the surface. We have therefore the follow-
ing theorem:

THEOREM 9. The points of contact of a quadric surface Q with the
tangents drawn from a point A not on the surface, are the points in
which the surface is met by the polar plane of A with respect to Q.

On the basis of this result we introduce the following definition:

DEFINITION VI. The tangent cone from a point A to a non-singular
quadric surface which does not contain A is the cone whose vertex is at
A and whose directrix is the curve common to the surface and to the
polar plane of A.

168 QUADRIC SURFACES, GENERAL PROPERTIES

Remark. This tangent cone is a quadric cone ; for the curve in
which a plane meets a quadric surface can not be of higher order
than the second; and since A does not lie on the surface, it 'can not
lie on its own polar plane.

To obtain the equation of the tangent cone, we shall use two
methods.

(a) In the first method, we translate the frame of reference to
A as a new origin, set up the equation of the cone in the new system
and then return to the original frame. Since the transformation of
coordinates has been discussed in Chapter V, we shall suppose that
the translation has already been carried out and we shall ask
therefore for the tangent cone from the origin to a quadric surface
Q which does not pass through the origin.

It follows from Theorem 9 that the cone passes through the points
common to the surface and the polar plane of the origin; conse-
quently its equation can be written in the form

(1) kiQ(x, y y z) + fc 2 (a 14 x + a^y + a 34 z + a 44 ) =

(see Remark 4, Section 49, page 92). The multipliers ki and k%
have not been restricted in any manner as yet; since we are look-
ing however for a quadric cone, they have to be selected in such a
way that equation (1) shall be a homogeneous equation of the sec-
ond degree in x, y, and z. Therefore we put k\ k and k%
= Ix + my + nz + p and we write down that the constant term
and the coefficients of the first degree terms in equation (1) must
vanish; this leads to the following conditions:

(2 k + p)a l4 + I(i 44 = 0, (2 k + p)a 24 + ma 44 = 0,
(2 k + p)a 34 + nau = 0, fca 44 + pa 44 = 0.

Since the origin does not lie on the surface Q, a 44 4= 0; hence the
last equation gives p = k. Substituting this value for p in the
other equations, we find Za 44 = fca i4 , ma 44 = fca^ and na 44
= fca 34 . We are therefore free to choose an arbitrary non-zero
value for fc, as could be expected from the fact that equation (1)
could have been divided through by k\ without essentially chang-
ing anything. Taking k = a^, we obtain I = a\\, m = 024,
n = 034, p = a 44 . Consequently the tangent cone from the
origin to the non-singular quadric Q is represented by the homo-
geneous equation a,uQ(Xj y, z) (dux + a^y + a^Z + a^) 2 = 0.

POLAR PLANE AND POLE 169

(6) In the second method w suppose that P(z, ?/, z) is an arbi-
trary point on the tangent cone. Then the line AP is tangent to
the surface and its direction cosines must therefore satisfy the
condition of Corollary 1 of Theorem 1 (Section 80, page 160),
namely:

1 [XQi(, 19, 7) + M<Ma, 18, 7) + KM, ft 7)] 2 = Q(, ft 7) X
(an\ 2 + a 2 2M 2 + 33^ 2 + 2 a 2 3M^ + 2 a 3 i^X + 2 Oi 2 X/i).

But for the line -AP, we have \:p:v x a :y ft :z y ; and
since the condition which we have just written down is homoge-
neous of the second degree in X, /z, and v, we may omit the factor of
proportionality. We obtain therefore for the required tangent
cone the following equation homogeneous of the second degree in
x dj y ft, and z 7 :

\ K* - )Qi(, ft 7) + (If - PXMa, ft 7) + (s - 7XM, ft 7)] 2
= Q(, ft 7) [au(x - a) 2 + a 22 (7/ - 0) 2 + 033(2 - 7> 2
+ 2 023(0 - 0) (2 - 7) + 2 031(3 - 7) (x - a) +
2 a w (x - a) (T/ - j8)].

Remark. For a: = = 7 = 0, this last equation should be
equivalent to the equation obtained by the first method.

83. Exercises.

1. Determine the equation of the tangent plane and the equations of the
normal for the surface 4 x- 6 xy -f 5 y 2 -f- 4 yz 3 z 2 -\- '2 zx 4 x -{- 3 y
4- 2 z -h 4 = at the point A(l, -1, 2).

2. Set up the condition which the direction cosines of a line through
P(2, 1,1) must satisfy in order to be tangent to the surface 3 x\ 2 t/ 2 +
5 zx -4 y 4-6^-3=0.

3. Set up the condition on X, ju, v under which the line x 1 + Xs,
y = 2 + jus, ^ = 2 4- J>s will lie entirely on the surface 4 z 2 6 y 2 -f- 8 z 2
= 12.

4. Determine the polar plane with respect to the ellipsoid 3 x 2 -f- 2 ?/ 2
-f 4 z 2 = 20 of the points A(-2, 2, 1), B(5, 5, 0), C(0, 4, -3), D(0, 0, 0).

5. Find the pole with respect to the surface 3 x 2 2 xy -\- y 2 -f 4 yz 6 x
-f 2 i/ + 7 = of the plane (a) z + y + - 3 = 0; (&)2s-t/-r-2z-f-
3=0.

6. Derive the equation of the tangent cone to the surface 4 x 1 -f 3 y 2 - 12 z
from the points A(0, 0, -6), (-4, 5, 3), C(0, 0, 4).

7. Determine the equation of the tangent cone from an arbitrary point

x 2 v 2 z 2
P(a, /3, 7) to (a) the ellipsoid i +#5 +-5 = 1; (b) the hyperboloid of one

x 2 v 2 z 2 a o c ^ 2 2

sheet -5 -h jr, i = lJ ( c ) the elliptic paraboloid + p = 2 pz.

170 QUADRIC SURFACES, GENERAL PROPERTIES

8. Show that the equation for the tangent cone obtained by the second
method of the last part of Section 82 reduces to the result found by the first
method if we put a = ft = 7 = 0.

9. Determine the asymptotic directions of the hyperboloid of one sheet

x 2 y 2 z*

7- TT + |T = 1, which lie in the plane 3 x 2 y = 0.

10. Show that the asymptotes of an hyperbola are also asymptotes of the
hyperboloid of revolution of one sheet which is obtained when the hyperbola
is revolved about the conjugate axis.

11. Show that the ellipsoid of Exercise 7 does not have any real asymptotic
directions.

~2 ? .2 2

12. Determine whether the hyperboloid of two sheets ~ ~ - 2 =1,

a o c

x 2 y 2
the elliptic paraboloid of Exercise 7 or the hyperbolic paraboloid ~ = 2 pz

have real asymptotic directions.

84. Ruled Quadric Surfaces. We have already met a few
examples of surfaces which contain every point of a line (see, for
example, Exercise 3 in Section 83). We proceed now to a sys-
tematic study of the question which quadric surfaces have straight
lines on them. We saw in Corollary 2 of Theorem 1 (Section 80,
page 160) that the line x=*a + \s 9 y = P + ns,z = v + vs will
lie entirely on the quadric surface Q if and only if L = q(\, M> v)
= flnX 2 + &22M 2 + 33^ 2 + 2 OKIJLV + 2 a 3 i*>X + 2 a^X/x = 0,

Ja = i [XQi(a, ft 7) + MQ 2 (, ft 7) + "Qs(, ft 7)] =
and L 2 = Q(a, ft 7) = 0.

Suppose now that we have a point A(a y ft 7) on the surface, so
that the condition L 2 = is satisfied. To determine lines through
A which lie entirely on the surface, X, /*, and v have to be so se-
lected that LI = Z/o = 0. These equations are homogeneous in
X, M, and v and of degree 1 and 2 respectively; if we solve them
for two of the variables, say X and M> in terms of the third variable
v, we shall in general be led to a quadratic equation, giving rise
to two values for the ratios X : n : v, that is, to two lines on the
surface through A. These two values may be real and distinct,
coincident or imaginary. We want to learn under which condi-
tions these different situations will arise.
We begin by proving the following auxiliary theorem.

THEOREM 10. A non-singular quadric Q contains no point at which
Qi = <?2 = Q* = 0.

RULED QUADRIC SURFACES 171

Proof. It follows from Corollary 1 of Theorem 4 (Section 81,
page 162) that at a point at which Q Qi = Q 2 = Q 3 = 0, we
must also have Q 4 = 0. Hence there would be at least one set of
three numbers a, 0, 7 which satisfy the four linear non-homogeneous
equations a^a + a^ + 8 7 + 0*4 = 0, i = 1, 2, 3, 4. If the
rank of the coefficient matrix of this system of equations is 3, a
solution exists if and only if the rank of the augmented matrix is
also 3 (see Theorem 8, Chapter II, Section 27, page 44). But the
determinant of this augmented matrix is the discriminant of the
quadric surface (see Definition IV, Section 82, page 166); hence
if the rank of the augmented matrix is 3, the surface is singular.
On the other hand, if the rank of the coefficient matrix of the
system of linear equations is less than three, then the cofactors of
the elements in the last column of the discriminant are all equal
to zero and the value of the discriminant is therefore zero. Thus
we have seen that if a quadric surface contains a point at which
Qb Qz an d Qz all vanish, then it is singular.

We shall have frequent occasion to refer to a point on a quadric at
which these conditions hold and we introduce therefore a name for it.

DEFINITION VII. A vertex of a quadric surface Q is a point at which
Q = <?i = <? 2 = <? 3 = 0.

In the terminology of this definition, we can then state the
following corollary:

COROLLARY. A non-singular quadric surface has no vertex; a singu-
lar quadric surface may have one or more vertices.

We proceed now with the problem of determining lines through a
point A (a, 0, 7) on a quadric which shall lie entirely on the surface;
and we shall divide our discussion of this question in two parts:
CASE I. The point A (a, 0, 7) is not a vertex.

In this case at least one of the partial derivatives of Q is different
from zero at A ; let us suppose that Qi(a, p, 7) =t= 0. We can then
solve the equation LI == for X in terms of M and v and substitute
the result in the equation L = 0. This will lead us to the follow-
ing quadratic equation in /* and v:*

(1)

dll 012 Ql
012 022 Ql

Qi Q 2

011 012 Ql
013 23 Q
Ql <?2

011 013 Ql
013 033 Q3

Qi Q 3

= 0,

* In order not to interrupt our main argument too much at this point, we
relegate the proof of this statement to the Appendix, I (see page 296).

172

QUADRIC SURFACES, GENERAL PROPERTIES

(2)

in which the partial derivatives Q\ 9 Q 2 , and Qz have the arguments
a, 0, and 7; it will be understood that this is the case throughout
our further argument, unless the contrary is definitely specified.

We observe now that the coefficients of v?, 2 pv, and j> 2 in this
equation are equal respectively to the minors of the elements
033, 23> and 022 in the determinant

011 01 2 013 *fcl

01 2 22 023 Qz

013 023 033 Qs

If the value of this determinant is denoted by A^(Q) and the co-
f actors of its elements a# by Aij(Q), it follows from Theorem 18,
Chapter I (see Section 16, page 29), that the discriminant of equa-
tion (1) is equal to ^4 23 2 (Q) A^(Q) X A^(Q)

\A^(Q) A&(Q) 3 Qi

Consequently the roots of the quadratic equation (1) will be real
and unequal, real and equal or complex according as the value of
the determinant (2) is positive, zero or negative. We will show
now that this determinant can be reduced to a simpler form. If
we add to the last column the products of the 1st, 2nd, and 3rd
columns by 2 a, 20, and 27 respectively, its elements will
become 2 an, 2 024, 2 as4 and 2 aQi 2 0Q 2 2 7^3. Next, we add
to the last row the products of the 1st, 2nd and 3rd rows by
2 a, 20, and 27 respectively; this transforms the elements
of the last row into 20 U , 2a 24 , 2034 and 2aQi 20Q 2 27Q 3
4 0i4 4 00554 4 7034. This last element, in the lower right-
hand corner, is equal to 2(aQi + 0Q 2 + 7^3 + Q\) + 4 a 44
= 4(044 Q), by use of Corollary 1 of Theorem 4 (Section 81,
page 162). Therefore the determinant (2) has been reduced to

011 012

013

2

014

0n

012

013

2014

01 2 022

023

2

024

012

022

023

20 24

013 023

033

2

034

013

023

033

2034

2014 20 24

2034

4(044

-Q)

2014

2024

2034

4044

011

012

013

01 2

022

023

013

023

033

2014

2024

2034

-4Q

RULED QUADRIC SURFACES 173

Therefore, if, in agreement with our general notation, we designate
by ^44 the cofactor of the element a 4 4 in the discriminant A of the
quadric surface, we obtain the interesting formula

(3) A 9 (Q) = 4A - 4^44 -Q(, 0,7).

In the particular case which we are having under consideration,
Q(ot y fi, 7) = 0, since the point A (a, 0, 7) lies on the surface and
therefore A 3 (Q) = 4 A. We state this preliminary result of our
discussion in the following theorem and corollary.

THEOREM 11. If the point A(a 9 , 7) lies on the quadric surface <?,
then the value of the determinant A*(Q) Is independent of the position
of A on the surface and equal to four times the value of the discrimi-
nant A of the surface.

COROLLARY. The matrices of the determinant A t (Q) and of the dis-
criminant A have equal rank, if the point A (a, 0, y) lies on the quadric
<?

Proof. This Corollary follows from Theorem 14, Chapter I, in
view of the fact that if Q(a, 0, 7) = 0, the matrix of the determi-
nant Az(Q) is transformed into that of the discriminant A by
means of elementary transformations (see Definition XIV, Chap-
ter I, Section 10, page 18).

The further discussion of our problem depends on the rank of
the matrix of the discriminant A ; henceforth we shall denote this
matrix by the symbol a 4 and its rank by r 4 . We consider now the
following possibilities:

(a) n = 4, that is, A 4= 0.

It follows from our discussion that in this case the quadratic
equation (1) will have two distinct roots, which are real if A >
and complex if A < 0; to each root of the equation (1) there
corresponds a set of direction cosines of a line through A which will
lie entirely on the surface. We can conclude therefore that
through every point on a quadric surface for which A > 0, there
pass two different real lines which lie entirely on the surface ; and
that through no point on a surface for which A < there are lines
which lie on the surface.

(b) n = 3.

Asa result of the Corollary of Theorem 11, the rank of the
matrix of the determinant (2) will also be equal to 3 in that case.

174 QUADRIC SURFACES, GENERAL PROPERTIES

We can show now that the coefficients of equation (1) can not all
vanish, by showing that if they did, then every three-rowed minor
of the determinant (2) would vanish.* Consequently in this case
the equation (1) has two coincident roots and through every point
of the surface, which is not a vertex, there will pass two coincident
lines which lie entirely on the surface.

(c) r 4 = 2 or 1.

It follows now from the Corollary of Theorem 11, that all the
coefficients of the equation (1) vanish. Consequently every line
through the point A whose direction cosines satisfy the condition
Li = lies entirely on the surface. But this carries with it that
every line through A which lies in the plane (x a)Qi(a, ft 7)
+ (y - 0)Q 2 (a, ft -V) + (* - 7)Q 3 O, ft 7) = must lie on the
surface. We conclude that the surface contains every point of this
plane; in virtue of Corollary 1 of Theorem 4 (Section 81, page
162) and because Q(a, ft 7) = 0, the equation of this plane may
also be written in the form xQi(a, ft 7) + yQ 2 + zQ s + $4 = 0.
CASE II. The point A (a, ft 7) is a vertex of the surface.

In this case, which can arise only on a singular quadric (see
Corollary of Theorem 10, page 171), the equation LI is satis-
fied by every set of direction cosines. And we shall show that the
condition L = q(\ M> v) = is satisfied by the direction cosines of
any line which joins A (a, ft 7) to another point A'(a f , /3', 7') on
the surface and by no others. For, in virtue of Taylor's theorem
(see Section 75, page 151) we have

Q(', 0', 7') = <?( + [' - ], + W - ft], y + [V ~ 7])
= Q(, ft 7) + [(' - )0i(a, ft T) + (0' - j8)Q a (a, ft 7)
-7XM, ft 7)] + (' - a, 0' - ft 7' - 7).

Therefore, if A (a, ft 7) is a vertex of the surface and if A'(a x , ($', 7')
is an arbitrary second point on the surface, then q(a f a, 0' ft
7' 7) = 0- But since the direction cosines of the line A A ' are
proportional to a' a, /3' ft and 7' 7, and since q is a homo-
geneous function, it follows that q(\, /z, v) 0. And it should be
an easy matter to show that this will not be the case for the di-
rection cosines of a line which connects the vertex A with any other
point in space.

* The proof is given in the Appendix, II (see page 296).

RULED QUADRIC SURFACES 175

The results of the discussion of our problem may now be sum-
marized as follows.

THEOREM 12. Through every real point A(a, 0, 7) on a non-singular
quadric surface with positive discriminant, there pass two and only
two lines which He entirely on the surface; through no real point on a
non-singular quadric surface with negative discriminant is there any
line which lies entirely on the surface. Through every point on a
singular quadric for which the rank of the matrix of the discriminant
Is 3, and which is not a vertex of this surface, there pass two coincident
lines which lie entirely on the surface, and no others. Through every
point on a singular quadric for which the rank of the discriminant
matrix Is less than 3, and which is not a vertex of this surface, there
passes a plane which belongs entirely to the surface. The lines joining
a vertex of a singular quadric to any other point on the surface lie en-
tirely on the surface.

COROLLARY 1. A singular quadric surface which possesses a vertex
is a conical surface; it is a quadric cone.

For, from the last part of Theorem 12, it follows that the surface
may be generated by a line through a vertex which moves so as to
pass through the points of the surface cut out by any plane which
does not pass through the vertex.

Remark. A vertex of a singular quadric is also a vertex of the
quadric cone which it represents.

COROLLARY 2. If the rank of the discriminant matrix of a quadric
surface is less than 3, the locus of the equation consists of two planes;
It is a degenerate quadric. (See Definition V, Chapter IV, Section 46,
page 83.)

Proof. For it follows from Theorem 12 that in this case there
is at least one plane all of whose points lie on the surface. Let
the left-hand side of the equation of this plane be E\ and let
Q = E Ei + R, where R is a function of y and z alone. Ob-
viously if R does not vanish identically we can determine particular
values of y and z for which R 4 1 0; and it will also be possible in
general to associate with these values of y and z a value of x such
that these values of x y y, and z cause E to vanish. But for these
values, we will have Q =f= 0; and therefore, we would have a point
on the plane E = which does not lie on the surface. This con-
tradicts our hypothesis. Consequently, R must vanish identically,
and Q = E EI. It is now easy to see that E\ is also a linear func-
tion and therefore we conclude by use of Theorem 10, Chapter IV

176 QUADRIC SURFACES, GENERAL PROPERTIES

(Section 46, page 83) that the locus of Q = consists of two
planes.

85. The Centers and Vertices of Quadric Surfaces. 'Among
the particular quadric surfaces with which we have already be-
come familiar are the sphere, the elliptic cylinder and the circular
cone. The center of a sphere is usually defined as the point from
which all the points on the sphere are equally distant; for an
elliptic cylinder, and even for a circular cylinder such a point does
not exist. If we take for the center of the sphere however the
property that it bisects every chord which passes through it, we
observe that every elliptic cylinder has points which possess the
same property, namely, the points on its axis. The axis of such a
cylinder could then be called a line of centers. But even on this
definition of a center, the cone, the elliptic paraboloid and
other quadrics do not possess any centers. We undertake there-
fore in the present section the inquiry as to the conditions under
which a quadric has a center; and we shall seek to develop con-
venient methods for the location of centers in the cases in which
they exist. Our work will be based on the following definition:

DEFINITION VIII. A center of a quadric surface Is a point which bi-
sects every chord drawn through It;* a proper center is a center which
does not lie on the surface, an improper center of a surface lies on the
surface.

It follows from this definition that, if A (a, 0, 7) is a center of the
quadric surface Q 9 then the two roots of the equation

L 2 =

which was established in Section 80, must be equal numerically
but opposite in sign for all admissible values of X, pt, and v (see
the Remark, following Theorem 7, Chapter III, Section 33, page
56) ; hence the sum of these roots must equal 0.

If Lo =1= 0, that is, if the line through A (a, 0, 7) does not have an
asymptotic direction (see Definition I, Section 80, page 160), the

O J

sum of the roots is equal to p- 1 ; it will be equal to zero therefore

JL/o

* A chord of a surface is a line which joins two of its points; it follows from
Corollary 1 of Theorem 1, Section 80, page 160, that a chord of a quadric surface
does not have any other points in common with the surface besides the two
points which it joins, unless it lie entirely on the surface.

THE CENTERS AND VERTICES OF QUADRIC SURFACES 177

if and only if LI = 0. And if L = 0, so that one of the roots is
infinite, the condition requires that the other root be also infinite,
which leads again to the condition LI = 0. Conversely, if LI = 0,
the two roots of equation (1) are equal numerically but opposite
in sign. Therefore the necessary and sufficient condition that
A(dj 0, 7) be a center is that L\ for all admissible values of
X, ju, v. In particular we must have LI s= XQi(a, , 7) +AtQ2(, , 7)
+ vQz(a, |8, 7) = for the sets of values 1,0, 0; 0, 1, and 0, 0, 1
of X, Hj v] these special sets lead to the conditions Qi(a, 0, 7) = 0,
Q 2 (a, 18, 7) = 0, Q 3 (a, )8, 7) = 0. Moreover it is easily seen that
if these conditions are fulfilled, then LI will vanish for every ad-
missible set of values of X, /*, v. We have therefore obtained the
following theorem :

THEOREM 13. The necessary and sufficient conditions that a point
shall be a center of the quadric surface Q Is that its coordinates satisfy
the three linear equations Qi(x 9 y 9 a) = 0, Qt(x 9 y 9 a) = 0, Qa(x 9 y 9 a) = 0.
If and only if the coordinates satisfy moreover the condition Q(x 9 j, 3}
=t= 0, the point is a proper center.

COROLLARY. An Improper center of a quadric surface Is a vertex of
the surface, and conversely.

For the further discussion of our problem we observe in the first
place that, in view of Corollary 1 of Theorem 4, the condition
Q = of Theorem 13 may be replaced by the condition Q 4 4= 0.
Consequently a proper center is a point common to the three
planes

~ = anx + a w y + aisz + M = 0, ~

+ 24 = 0, y = a 13 X + 0237/ + O& + 34 = 0,

but not on the plane

~ = a u x + any + 0342 + 044 = 0;

and a vertex is a point common to the four planes.

We shall denote the coefficient matrix of the first three equations
by as and its rank by r 3 ; the value of the determinant of a 3 has
already been designated by An (see equation (3), Section 84, page
173). The augmented matrix of the first three equations is

178 QUADRIC SURFACES, GENERAL PROPERTIES

lijll i = 1, 2, 3; j = 1, 2, 3, 4; we shall denote it by b. The
coefficient matrix of the equations of the set of four planes is
\\dij\\, i = 1, 2, 3, 4; j = 1, 2, 3; we shall denote it by b'; and the
augmented matrix of this set of four equations is the discriminant
matrix of the quadric, which we have already designated by a 4
(see page 173, proof of Corollary of Theorem 11) and whose de-
terminant is denoted by A.

We fall back now on Theorems 20, 22, 23, 24 of Chapter IV and
their Corollaries (Sections 51 and 54, pages 95, 101, and 102); ap-
plication of these theorems shows that if there is to be any center
the matrices a 3 and b must have the same "rank, and if there are
to be any vertices, the matrices b 7 and a 4 must have the same rank.

Now it should be clear: (1) that the ranks of the matrices b
and b' are equal, since either of these matrices is obtained from
the other if we write the columns as rows and vice versa; (2) that
the rank of a 3 can not exceed that of b, which in turn can not ex-
ceed the rank of a 4 ; (3) that the rank of a 4 can not exceed the
rank of b by more than 1 ; and (4) that the rank of b can not ex-
ceed the rank of a 3 by more than 1. Moreover, (5) if the ranks of
b and a 4 are equal, then the ranks of b and a 3 are equal.

An algebraic proof of this last statement may be somewhat
lengthy. It can be deduced very readily however from the theorems
of Chapter IV referred to above. For if the ranks of b and a 4
are equal, the four planes have at least one point in common;
consequently the first three planes have at least one point in
common and therefore the c.m. and the a.m. of the first three
equations have the same rank, i.e., the ranks of a 3 and b are equal.

We conclude from (3) and (4) that r 4 and r 3 can differ by 2 at
most. If r 4 r 3 = 2, then the rank of b is different from either.
If r 4 r 3 = 1, it follows from (5) that the rank of b is equal to
r 3 . And, if n = r s , the rank of b is of course equal to the same
number. It should be clear that the existence of proper centers
and vertices depends on the difference r 4 r 3 . If we draw on the
further content of the Theorems of Chapter IV, which were cited
above, we obtain the following result :

THEOREM 14. If the ranks of the matrices a 4 and as are equal the
quadric surface has a unique vertex, a line of vertices or a plane of
vertices, according as this common rank Is 3, 2 or 1. If the ranks of
these matrices differ by 1, the quadric surface will have a single

THE CENTERS AND VERTICES OF QUADRIC SURFACES 179

proper center, a line of proper centers or a plane of proper centers, ac-
cording as the lower of these ranks is 3, 2 or 1. If the ranks of these
matrices differ by 2, the quadric surface has no center at all.

Remark 1. The content of this theorem may conveniently be
put in the following tabular form :

7-3

r 4

The quadric surface has

3

4

a single proper center

3

3

a single vertex

2

3

a line of proper centers

2

2

a line of vertices

1

2

a plane of proper centers

1

1

a plane of vertices

n - r 3 > 1

no center

Remark 2. The reader should convince himself that the cases
indicated in this table include all possible cases for the ranks of
the matrices a 3 and a 4 and that therefore the conditions of Theorem
14 are sufficient as well as necessary.

Remark 3. A quadric surface with a single proper center is
called a central quadric. A quadric surface with a single vertex
is called a proper quadric cone.

Remark 4. A non-singular quadric surface is either a central
quadric or else a surface without any center.

We record moreover the following corollaries.

COROLLARY 1. The rank of the matrix a 4 can not exceed the rank of
the matrix a 3 by more than 2.

COROLLARY 2. The necessary and sufficient condition that a quadric
surface be a conical surface is that the ranks of the matrices a 4 and a 3
be equal.

We are able furthermore to complete in an essential way the
result contained in Corollary 2 of Theorem 12 (Section 84, page
175), as follows: If a quadric surface has a plane of vertices, it
consists of this plane, counted doubly. For, if it contained a point
A outside this plane it would have to contain every line which
connects a point of the plane with A (compare the last sentence
in Theorem 12, Section 84, page 175). This is obviously impos-
sible; therefore the surface can not contain any point outside the
plane of vertices. And it should be a simple matter to show that

180

QUADRIC SURFACES, GENERAL PROPERTIES

this plane muse be counted doubly. And if a quadric surface has
a line of vertices, it must consist of two planes through this line.
For, if A is any point of the surface outside the line I on which the
vertices lie, then the plane determined by I and A must be entirely
contained in the surface; and the argument used in the proof of
Corollary 2 of Theorem 12 (page 175) shows that then the surface
consists of two planes. If these two planes were coincident planes
the equation of the surface would be Q = (ax + 677 + cz + d) 2
= 0, from which we could conclude that r 3 = r 4 = 1, and there-
fore the surface would have a plane of vertices. We can therefore
state the following result :

COROLLARY 3. A singular quadric surface Is a proper quadric cone
if and only If r a = r 4 = 3, a pair of Intersecting planes If and only If
rj = r4 = 2, a pair of coincident planes If and only If n = r< = 1.

After the existence of centers or vertices has been established,
their position can be determined by solving the equations Qi = 0,
$2 = 0, Q 3 = 0. In the case of a central quadric, these equations
have an unique solution which is given by Cramer's rule (see
Theorem 1, Chapter II, Section 21, page 37). The solution may
be written in the following form:

012 013

x :y : z : 1 = -

034

011 012 014
12 022 024
013 023 034

011 014 013

012 024 023

013 034 033

011 012 013

012 022 023

013 023 033

It should be easy to see that the terms on the right are equal to
the cofactors of the elements in the last row of the discriminant A.
If these cofactors are designated in the usual manner, we have

x : y : z : 1 = An : A 2 * : AM : An.

COROLLARY 4. The coordinates of the center of a central quadric
surface are equal to ^ 4 , ^ 4 , ^-

A\\ Au A\i

Examples.

1. The coordinates of the possible centers of the surface 5 x- 4- 5 y 2 -f 8 z 2
Szx 2xy -{- 12x 12 y -f 6 =0 must satisfy the equations

THE CENTERS AND VERTICES OF QUADRIC SURFACES 181

|i = 5s-2/ + 4z+6=0,

6=0,

Q*

4
Moreover, ~ = 6 x <

The rank of the matrix a 3 =

5 -1 4
-1 54

4 48

3rd row is equal to the sum of the first two rows, whereas the two-rowed minor
in the upper left-hand corner does not vanish. And the rank of the matrix
5-14 6
4 -6
8
6

sum of the 1st and 2nd rows, whereas the matrix contains several non-vanishing
three-rowed minors. We conclude therefore that the surface has a line of

a 4 =

-1 5
4 4
6 -6

is readily found to be 2; for the

is found to be 3; for the 3rd row is equal to the

Q

centers in the line of intersection of the two planes ~ 5x y + 4z

and ~cT Z-J-5T/ + 42 6 = 0. The parametric equations of this line

are found to be x = 3 + tf, y 5 + t, z = 4 t.

2. To determine the possible centers of the surface 2 x 2 3 y 2 -f 4 ^z
5 zx + 4 x 3 ?/ + 5 = 0, we set up the equations Qi =4a; 52-f-4=0,
#2 = -6 y + 4 2 - 3 = 0, Q 3 = -5 x + 4 1/ = 0, and Q 4 = 4 x - 3 1/ -f- 10
= 0. The determinant of the matrix as has the value:

4 -5

-6 4

-5 4

43

and the discriminant

A== I6 X

40-54

0-64 -3

-5400

4 -3 10

861

Therefore r s = 3 and n = 4; consequently the surface has a single proper
center; its co6rdinates are A, *\, Jjj.

3. For the surface 2 x 2 + 20 1/ 2 4- 18 z 2 - 12 yz + 12 xy + 22 x + 6 y -
2 2 2 = 0, the matrices as and a 4 are

260
6 20 -6
-6 18

and

respectively. We find that r s = 2 and r 4
has neither a proper center nor a vertex.

2 6 11

6 20 -6 3

-6 18 -1

11 3 -1 -2

4; we conclude that the surface

182 QUADRIC SURFACES, GENERAL PROPERTIES

86. Exercises.

1. Show that through every point of the surface 4 x 2 6 y 2 12 z there
pass two real distinct lines which lie entirely on the surface.

x 2 y' 2 z 2

2. Prove that there are no real lines on the ellipsoid -- -f 75 + - = 1-

CL C

3.

Show that the ellipsoid ~ + ~ + ~ - 1, the hyperboloid of one sheet

x 2 y 2 z 2 x 2 y 2 z 2

I ~h r 2 2 = 1 an( l ^ c hyperboloid of two sheets ^ ^ 1 are

central quadric surfaces.

4. Show that through every point of the hyperboloid of one sheet of Exer-
cise 3 there pass two real lines which lie on the surface; and that no such lines
exist through any point of the hyperboloid of two sheets of Exercise 3.

6. Show that the locus of the equation : /- -f = is a proper quad-

a 2 b 2 c 2

ric cone; and prove that every tangent plane of this surface passes through
the vertex.

6. Determine the conditions which the direction cosines of a line through
the point A( 1, 1, 1) on the surface x 3 ?/ 3 -h 2 3 = 1 must satisfy in
order that the line may lie entirely on the surface.

7. Determine the centers, proper centers or vertices, of each of the follow-
ing surfaces:

(a) x 2 + 5 y 2 - 2 z 2 + 6 yz + 8 xy - 4 x + 6 y - 6 z -f 6 =
(6) 9 x 2 -f 49 y 2 -f 4 z 2 - 28 yz + 12 zx - 42 xy - 24 x -f 56 y - 16 z
+ 16-0

(c) 3 x 2 -f 5 y 2 -f- 9 z 2 + 2 yz -f 8 zx - 4 xy - 6 x + 4 y - 4 z + 3 =

(d) 5x 2 - y 2 - 16 z 2 - 20 yz + 4 ar - 8 xy - 6 * + 2 y - 8 z + 2 =
(c) 4 x 2 -f y 2 + 9 z 2 - 6 yz + 12 zz - 4 xy + 6 x - 3 y -f 9 2 - 4 =
(/) G x 2 - 2 y 2 - 2 z 2 + 5 ys - zz - 4 jy - 10 x - 6 y -f 9 z - 4 =
to) 3 :c 2 -f 3 y 2 + 3 z 2 - 2 yz - 2 zz - 2 x?y + 8 x - 4 2 + 6 =

(/O 2 x 2 -f- 5 ?/ + 2 z 2 - 6 2/2 + 4 ac - 6 xy -f 2 3 - 4 y + 2 z -f 2 = 0.

8. Show that the tangent lines from a point A (a, p, 7) to the elliptic cylinder

x 2 y 2

+ p = 1 lie on a pair of planes through a line parallel to the Z-axis.

9. Prove that if a quadric surface has a plane of centers the surface consists
of a pair of parallel planes.

10. Prove that the elliptic paraboloid ^-}-~^ = < 2pz and the hyperbolic

x z ? ,a a

paraboloid -5 ~-^ 2 pz do not possess a center.

87. The Asymptotic Cone. If C(a, 0, 7) is the center of the
central quadric Q, then Qi(a, ft 7) = 0, Q 2 (a, 0, 7) = 0, Q 3 (, ft 7)
= and Q(a, 0, 7) =t= 0. It follows that the equation of the

THE ASYMPTOTIC CONE 183

tangent cone from C to the surface reduces from the form given at
the end of Section 82 (page 169) to the simpler form:

(1) a n (x - aY + a*(y - ft 2 + (te(z - 7) 2 + 2 a n (y - ft)
(z - 7) + 2a 3 i(z -y)(x- a) + 2a 12 (rc - a) (y - 0) = 0.

Since the direction cosines of a generating line on this cone are
proportional to x a, y p, and z y, where x, y, z are the
coordinates of some point on the cone, it follows that the direction
cosines X, ju, v of such a line satisfy the equation anX 2 + a^v?
+ a^v 2 + 2 023M" + 2 anv\ + 2 a l2 \^ = 0, that is, the equation
q(\ fjLj v) = 0. Since moreover they evidently satisfy the equation
L! = \Qi(a, P, 7) + M& + J>#3 = 0, the generators of this cone are
asymptotes of the surface (compare Corollary 3 of Theorem 1,
Section 80, page 160).

DEFINITION IX. A cone of which every generator is an asymptote of
a surface is called an asymptotic cone of the surface.

We can therefore say that the center of a central quadric sur-
face is the vertex of an asymptotic cone of the surface. The same
argument shows that a proper center of any non-degenerate quad-
ric surface is the vertex of an asymptotic cone. And we raise the
question whether any other points, besides proper centers, can be
vertices of such cones of non-degenerate quadrics. If A (a, ft 7)
is such a point, we know from Corollary 3 of Theorem 1 (Section
80, page 160) that the equations q(\, /i, v) = and XQi(X, /i, v)
+ t*Qz + vQs = must have an infinite number of solutions which
are admissible values of X, /i, and p. Let us suppose now:

(a) that A (a, ft 7) is not a center of the surface. We can then
suppose that Qi(a, ft 7) 4= and proceed as in Section 84. The
quadratic equation (1) which was discussed in that section will
have more than two roots if and only if the rank r 4 of the dis-
criminant matrix is less than 3 (compare (b) on page 173), that is,
if the quadric surface is degenerate (see Corollary 2 of Theorem
12, Section 84, page 175). Hence for a non-degenerate non-
singular quadric a center is the only point which can be the vertex
of an asymptotic cone. And we suppose:

(b) that A (a, ft 7) is a vertex of the surface. In this case
Q, Qi, Qz, and Q 3 all vanish for x = a, y = ft z = y. If we make
use once more of Taylor's theorem as in Case II on page 174, we

184 QUADRIC SURFACES, GENERAL PROPERTIES
find that

Q(*> y, *) = Q(* + [*-], + [y - ffl, 7 + b - 7])

= Q(*, ft 7) + (x - )&(, ft 7) + fa - j8)Q 2 (a, ft 7)
+ (2 - 7)03(, ft 7) + q(x - , 2/ - ft ^ - 7),

so that the equation of the surface reduces to the equation q(x a,
y ft z 7) =0, which is the equation of the asymptotic cone.

If we observe furthermore that it follows from the Taylor's ex-
pansion written above that the equation of the asymptotic cone of
a central quadric can also be written in the form Q(x, y, z)
Q(ct, ft 7) = 0, we can put our results in the form of the follow-
ing theorem.

THEOREM 15. A non-degenerate quadric surface Q has an asymp-
totic cone if and only if It has a center. If It has a proper center at
A(a 9 9 y) the equation of the asymptotic cone is Q(x 9 y, z) Q(a 9 /3, 7)
= 0; if it has a vertex the asymptotic cone is identical with the surface
itself.

Obviously there is no further interest in considering the asymp-
totic cone of a surface which has vertices; therefore there remain
for consideration the non-degenerate quadrics which have proper
centers, that is, the cases in which r 4 and r 3 are 4 and 3, or 3 and 2
respectively.

CASE 1. 7*4 = 4, r 3 = 3. In this case there is a single center
and therefore a single asymptotic cone, which is a proper quadric
cone.

CASE 2. r 4 = 3, r 3 = 2. In this case there is a line of cen-
ters determined by the equations Qi(x, y, z) = 0, Q*(x 9 y, z) =
and Q 3 (#, y> z) = 0. The c.m. of these equations is as. We
shall henceforth denote the cof actors of the elements a#, i,j = 1, 2,
3 of this matrix by a#, i, j = 1, 2, 3. Since r 3 = 2, not all of these
cofactors vanish. Let us suppose a 33 4= 0; then the direction
cosines of the line of centers are proportional to ai 3 , 2 3, and a 3 s
(see Theorem 17, Chapter IV, Section 47, page 87). Therefore,
if a, ft 7 is an arbitrary point on the line of centers, the parametric
equations of this line may be put in the form x = a + ant,
y = ft + awl, 2 = 7 + asrf; and the equation of the asymptotic
cone which corresponds to an arbitrary center can be put in the
form:

Q(*> y, *) - Q(<* + irf, ft + "23*, 7 + aasO = 0.

matrix of this equation is

since in this case

DIAMETRAL PLANES OF A QUADRIC SURFACE 185

If the second term on the left-hand side is expanded by Taylor's
theorem, the equation reduces to

Q(x, V, *) ~ Q(, ft, T) ~ <[awQi(, ft 7) + 23 Q 2 (a, p, 7)
+ 0:3363(0;, ft 7)] - t 2 q(au, a 23 , ass) = 0.

The coefficient of t is obviously zero; and it is shown in the Ap-
pendix* that the coefficient of t 2 also vanishes. Consequently the
equation of the asymptotic cone, which corresponds to an arbi-
trary center, is independent of t] that is, there is only one asymp-
totic cone. If its equation is written in the form q (x a, y ft
* ~~ T) = 0, and we translate the axes to the point (a, ft 7) as
origin (see Theorem 2, Chapter V, Section 61, page 115), the
equation takes the form q(x', y', 2') = 0. The discriminant

i 2 ai3

022 #23

013 023 33

0000
r 3 = 2, the rank of this discriminant matrix is also 2, and there-
fore, the asymptotic cone consists of a pair of intersecting planes
(compare Corollary 3 of Theorem 14, Section 85, page 180) ; that
is, the asymptotic cone degenerates into a pair of asymptotic
planes. We have now obtained the following amplification of the
last theorem.

THEOREM 16. If the ranks r 4 and r 3 are equal to 4 and 3 respectively,
the quadrlc surface Q has a single proper quadrlc cone as asymptotic
cone; this cone may be real or Imaginary. If these ranks are equal to
3 and 2 respectively, the surface has a pair of asymptotic planes, which
may be real or imaginary. In either case the equation of the asymp-
totic cone may be written In the form Q(x 9 y, *) - <?(, /?, 7) = 0, where
, /3, 7 are the coordinates of a center of the surface.

88. The Diametral Planes and the Principal Planes of a Quadric
Surface. We return once more to the equation in Theorem 1
(Section 80, page 160) and inquire for the locus of points which
are midpoints of chords drawn through them in a fixed direction
given by the ratios X : n : v. The argument which led us to
Theorem 13 in Section 85 shows that if A (a, 0, 7) is a point of this
locus, then XQi(a, j8, 7) + vQz + vQ 3 must vanish for the specified
values of X, JJL, and v. Since d, Q, and Qs are linear functions of

* See III, page 297.

186 QUADHIC SURFACES, GENERAL PROPERTIES

a, P, y, we conclude that the locus is a plane. We obtain therefore
immediately the following theorem.

THEOREM 17. The locus of all points which bisect chords of the
quadric surface Q whose direction cosines are X, /*, v 9 is the plane
x(M* y ) + M<M* r> *) + "(M* j, *) = o.

DEFINITION X. The plane which is the locus of the midpoints of a
set of parallel chords is called the diametral plane of the direction of
these chords.

If we write out in full the expressions for Qi, Q 2 , and (? 3 in the
equation of the diametral plane and collect the terms in x, y y and
z, the equation of this plane takes the form :

= 0,

or, using the notation introduced in Section 80,

<7i(X, fjL 9 v)x + </ 2 (X, n, v)y + </ 3 (X, M, v)z + </ 4 (X, /x, ^) = 0.

For all values of X, ju, and *>, this equation represents a plane
(for those values for which #i(X, /*, i/) = g 2 (X, /i, v) = g 3 (X, /i, ^) = 0,
and g 4 (X, ju> v) 4= 0, this plane is the "plane at infinity/' see
Remark 2, Section 41, page 73), except for such values as cause
% #2, Qsj and #4 to vanish simultaneously; but this can not happen
for admissible values of X, ju> and v unless the rank of the matrix
b is less than 3 (see Theorem 2, Chapter II, Section 22, page 38;
compare also Section 85, page 178). We can therefore state the
following corollary.

COROLLARY. In a quadric surface for which the rank of the matrix
b is equal to 3, there exists a diametral plane corresponding to every
direction; In a quadric surface for which the rank of this matrix is
less than 3, this correspondence falls for those directions for which

|i(X f M> v) = q* = q 3 = q* = 0.

The correspondence between systems of parallel chords and di-
ametral planes which has been established for quadric surfaces, is
an extension to three-space of the correspondence between con-
jugate diameters in the theory of conic sections; for either of two
conjugate diameters is the locus of the midpoints of chords parallel
to the other. We recall that in the ellipse and the hyperbola
there is one pair of mutually perpendicular conjugate diameters,
namely, the axes of these curves. On account of the importance

PRINCIPAL PLANES OF A QUADRIC SURFACE 187

of these lines in the theory of these curves, we are led to inquire
whether there are directions in a quadric surface which are per-
pendicular to the corresponding diametral planes. To facilitate
the discussion, we introduce the following definition.

DEFINITION XL A principal direction of a quadric surface is a di-
rection which is perpendicular to the corresponding diametral plane;
a diametral plane which corresponds to a principal direction is called
a principal plane. *

According to Corollary 3 of Theorem 7, Chapter IV (Section
44, page 79), the angle between any direction X, /*, v and the cor-
responding diametral plane, when this is a "plane at finite dis-
tance," is given by the formula:

-

v qi* + g 2 2 + </s 2

Therefore the necessary and sufficient condition that the diametral
plane which corresponds to the direction X, M, v shall be a plane at

FIG. 32

finite distance and perpendicular to this direction, is that the
equation

Xffi(X, At, v) /igsCX, fJL, v) vq*(\ M, v) _ ,

2k "*" 2k ~*~ 2k

shall be satisfied by admissible values of X, M> and v, such that
fc = ^ 2 + g2 2 + ^ 2 ^ Q (gee Fig 32) ^ If wc multiply thia

* If fc = 0, we must have #1 = q* #3 = 0, so that wc would be dealing
with the plane at infinity if there were a plane at all; and if the diametral
plane were the plane at infinity, we would have q\ = 5-2 = q* and therefore
k = 0. Consequently the non-vanishing of fc is a necessary and sufficient
condition that the diametral plane shall be a plane at finite distance.

188

QUADR1C SURFACES, GENERAL PROPERTIES

equation by 2 and subtract the result from the sum of the equations

x 2 + ^ + ^ = i and rr 2 + rn + r& =1 ' we obtain the

4 fc 2 4 fc 2 4k 2
condition

which in turn is equivalent to the three equations

#i(X, /x, v) = 2 fcX, q 2 (\, M, ") = 2 fc/i, g 3 (X, M, v) = 2 fci/;
that is, to the equations

(1) (on ~ k)\ + a 12 M + auv = 0, Oi 2 X + (o 22 fc)/* +
= 0, Oi 3 X + o 23 /* + (a 33 fc> = 0*.

The condition for the existence of a principal plane, stated
above, is therefore equivalent with the condition that there exist
admissible values of X, /x, v which satisfy the equations (1) and for
which k 4= 0. But, since these equations may be looked upon as
linear homogeneous equations in X, /z, and v y it follows from
Theorem 2, Chapter II (Section 22, page 38) that their coefficient
determinant must vanish (since the trivial solution of these equa-
tions does not lead to admissible values of X, /x, and *>); that is,
we must have:

^12 O 22 fc O23

2l3 23 033

This equation is a cubic in fc; therefore it has 3 roots. To every
root fc*, for which the rank of the corresponding matrix

On fc* Oi 2

(2)

=0.

(3)

Ol 2

22 ~ fc*

3 3 fc

is 2, there corresponds (compare Corollary of Theorem 4, Chapter
II, Section 25, page 42) a single infinitude of values of X, /z, v deter-
mining uniquely the ratios X : n : v, and hence a single principal di-
rection. If the rank of the matrix (3) is 1, the three equations (1)

* It should be clear that these equations can be derived, independently of
the formula for sin 0, from the equation of the diametral plane by means of
Theorem 7, Chapter IV, Section 44, page 78. The derivation followed in the
text has the advantage of giving significance to the variable k.

PRINCIPAL PLANES OF A QUADKIC SURFACE 189

are equivalent; there is therefore only one linear condition on X, M,
and v t so that an arbitrary admissible value may be assigned to one
of these variables. Hence, to a value k* of k, for which the rank of
the matrix (3) is 1, there corresponds a single infinitude of principal
directions. If there is a k* which causes the rank of the matrix
(3) to become 0, then the direction cosines X, ju, v are entirely un-
restricted,t and hence every direction is a principal direction. In
order to facilitate the statement of our results we introduce the
following definition.

DEFINITION XII. The equation (2) is called the discriminating equa-
tion of the quadric surface Q, the discriminating numbers of the
surface Q are the roots of the discriminating equation.

THEOREM 18. Every quadric surface has three discriminating num-
bers; to each of these corresponds a single principal direction, a
single infinitude of principal directions, or all directions, according
as it gives the matrix (3) the rank 2, 1 or 0; with every discriminating
number which is different from zero, there Is associated a principal
plane at finite distance.

Remark. The direction cosines of a principal direction, associ-
ated with the discriminating number ki will be denoted by X,-, ju,-,
and vi, i = 1, 2, 3; they are found by solving any two of the
equations (1) for the ratios X,- : //, : v{.

Since q(x, y, z) is a homogeneous function of the second degree
in x, y, z, we find by application of Euler's theorem (see footnote
on page 161) that

2ff(Xi, MI, Vi) = A#i(X,-, /i,-, Vf) + Maftw &, vj) + ^(A;, ^, v$

= 2 fc,'(V + tf + V f) = 2 ki.

This leads us to the following useful corollary.

COROLLARY. If X;, & , \>i are the direction cosines of a principal direc-
tion t corresponding to the discriminating number ki of the quadric

f That such a situation may arise should be clear geometrically from the fact
that in a sphere every plane through the center bisects the chords which are
perpendicular to it, so that every direction is a principal direction. In an
ellipsoid of revolution, every plane through the axis of revolution bisects the
chords perpendicular to it, so that every direction perpendicular to this axis
is a principal direction; this furnishes an example of a surface in which the
direction cosines of a principal direction are subject to only one linear condi-
tion.

J It is to be understood here that more than one principal direction may bo
associated with a single discriminating number.

190

QUADRIC SURFACES, GENERAL PROPERTIES

surface Q 9 then

Mi> vi)
2, 3.

2 fa I'M

89. The Discriminating Equation. We proceed now to a fur-
ther discussion of the discriminating equation

11 k 012

012 022

023

013

023

#33 fc

= of a quadric surfacef

THEOREM 19. A root fc* of the discriminating equation is a single,
double, or triple root according as the rank of the matrix

fill ~ /C* Oi di2

(2)

is 2, 1 or 0; and conversely.

k*

k*

Proof. It follows from Theorem 19, Chapter I (Section 17,
page 32) that, if the left-hand side of equation (1) is designated by
A(k) and its derivatives with respect to k by means of accents,
then

-1

ft 11 k (l\2 $13

A' (k) = i2 022 k

023+0-1

013 023 033 ~~ k 013 023 033 ~~

- k

0n k (i 12

013

f 1 7

022 A,* 2 3

-)-

012 022 K>

023

= - n ]

-1

[1 023 033 k

0n k 013

a n ^ 012 |1

~*~

013 033 k

012 022 A' |p

A"(k) = 2[( n - k) + (, 2 - A-) + (033 - fr)], A'" (k) = -G.

f It should be clear how equations analagous to the one written above can
!>e formed for every symmetric square matrix ||i;||, i, j = 1, 2, . . . , n of any
order. Such an equation is usually called the characteristic equation of the
matrix. The equation treated in the text is therefore the characteristic
equation of the matrix a 3 . The characteristic equation of a matrix plays a
very important role in the theory of matrices. Many of the properties devel-
oped in the text for the characteristic equation of the matrix a 3 hold, with
appropriate changes, for the characteristic equation of the general symmetric
square matrix; and the methods of proof here used are readily adaptable to
the general case.

THE DISCRIMINATING EQUATION 191

Suppose now

(a) the rank of the matrix (2) is 0. Then clearly, A(k*) = 0,
A 9 (fc*)'= 0, and A 11 (fc*) = 0; therefore k* is a triple root of the
equation A (k) 0.

(b) the rank of the matrix (2) is 1. In this case A(k*) = 0,
A'(k*) = 0, but A"(k*) =f= 0. For, the rank of the matrix being
1, every two rowed-principal minor vanishes; that is, (a# k*)
(a,jj k*) = a# 2 ^ 0, i, j = 1, 2, 3; f 4 1 j. It follows from this
that if two of the elements in the principal diagonal of (2) vanish,
then all the elements outside the principal diagonal vanish also;
hence, the remaining element in the principal diagonal must be
different from zero and therefore A"(k*) 4 1 0. And it follows also
from this relation that any two principal diagonal elements of (2)
which do not vanish must be of the same sign, so that A"(k*)
can vanish only, if each of its terms vanishes, which has been shown
to contradict the hypothesis that the rank of the matrix (2) is 1.
Consequently fc* is a double root of the equation (1).

(c) the rank of the matrix (2) is 2. In this case we can apply
the Corollary of Theorem 7, Chapter II (Section 26, page 44);
and we conclude that A' (A;*) ^ 0. It follows therefore that k* is
a simple root of equation (1).

The converse follows from the fact that the three cases in the
hypothesis and in the conclusion both represent all possibilities.
For example, if k* is a double root of equation (1), the rank of the
matrix (2) is 1 ; for if the rank were not 1 , it would be 2 or 0, and
7c* would therefore be a simple root or else a triple root of the
equation.

If we expand the polynomial A (k) according to Taylor's theorem,

we find that

fr 2 fr 3

A(k) = A(0) + k A'(0) + ~ - A" (0) + ~ - A'" (0).

From the formulae developed in the proof of the theorem, we
find that

A (0) = |a y |, i, j = 1, 2, 3; that is, A (0) = A H (compare
page 173),

A f (o) = ~ n ^ a23 + an ai3 + an ai2 1 1

[J #23 #33 a !3 ^33 12 &22 | J '

= [n + <*22 + 0:33], (compare page 184)
A" (0) = 2(a n + a 22 + a 33 ), A"' (0) = -6.

192

QUADRIC SURFACES, GENERAL PROPERTIES

We shall now use the following abbreviations:

Ti = an + 022 + #33, T 2 = 11 + 22 + 33-

Our discussion yields then the following corollary.

COROLLARY 1. The expanded form of the discriminating equation
of the quadric surface Q is

(3) fc 3 - 7W 4- T*k - An = 0.

For convenience of future reference we record here also the
following results obtained by applying Theorem 18, Chapter I
(see Section 16, page 29; see also the Corollary of Theorem 5,
Chapter II, Section 26, page 43) and Theorem 7, Chapter II (see
Section 26, page 44) to the determinant A^.

COROLLARY 2. Between the value of the determinant An, the ele-
ments In its principal diagonal and the elements aij of its adjoint, the
following relations hold:

COROLLARY 3. If the determinant A vanishes, then those of its
principal two-rowed minors which do not vanish are of like signs.

THEOREM 20. The discriminating numbers of a real quadric surface
are real.

Proof. We will show first that, if the coefficients a# in the equa-
tion of the surface are real, then the discriminating equation can
not have a root which is a pure imaginary. Suppose that iq is a
root of equation (1). Then, according to a well-known theorem
of algebra, iq must also be a root of this equation; that is, we
will have

a u +iq
= 0, and

012

#23

012

012

022+ iq
023

023

= 0.

012 022 1

013 023 033 iq

But the product of the two determinants on the left-hand sides of
these equations must then also vanish; application of Theorem 16,
Chapter I (Section 14, page 26) gives us therefore the further
result

^01*03*

= 0.

THE DISCRIMINATING EQUATION 193

This last equation is of the same general form as equation (1) and

is obtained from it if we substitute q 2 for A; and

for ay.

Corollary 1 of Theorem 19, enables us to state therefore that the
expanded form of this equation is

where

and 183 =

It should be clear that S s = A^ y and therefore that S\ and S 3 are
both non-negative. To show that the same thing is true of S 2 , we
observe that to each determinant in S 2 we can apply the Lemma
preceding Theorem 14, Chapter III (Section 36, page 64) ; thus 2
is transformed in to the sum of the squares of 9 two-rowed de-
terminants. Consequently equation (4) has no negative coeffi-
cients and therefore, considered as a cubic in g 2 , no solutions for
which q 2 is positive. Therefore equation (2), the discriminating
equation of the quadric surface, can have no root of the form iq,
where q is real, unless q = 0.

Suppose next that equation (1) has a root of the form p + iq;
then the new equation obtained from (1) by replacing an, 022, 033

194 QUADRIC SURFACES, GENERAL PROPERTIES

by an p, a 2 2 P, 3'i P has the root iq. But the new equation
is of the same form as equation (1) and therefore it can not have a
root iq. Our theorem has therefore been proved.

THEOREM 21. Not all the discriminating numbers of a quadrlc sur-
face can lie equal to zero.

Proof. If all the roots of equation (3) were zero, then we would
have A 44 = 0, T<> = 0, and 7\ = 0. But from the last two of these
equations we could then conclude that

TV - 2 T 2 = (a u 2 + 22 2 + a 33 2 + 2 a 22 a 33 + 2 a 33 an + 2 a u a 22 )
2 (a 2 2fl33 23 2 ) 2(a 3 3n a J3 2 ) 2(a u a 2 2 i2 2 )
= an 2 + 22 2 + 33 2 + 2 aas 2 + 2 a 13 2 + 2 a 12 2 =

and from this it would follow that an = ^22 = ^33 = 23 = is = 012
= 0, which would mean that the equation of the surface contains
no terms of the second degree.

COROLLARY. The functions T and T 2 can not both vanish; If Ti = 0,
then T 2 < 0.

The first part of this corollary follows also from Theorem 20
(page 192), for, if TI = T 2 = 0, the discriminating equation re-
duces to fc 3 = A 44, which has only one real root.

90. Principal Planes and Principal Directions. The results of
the two preceding sections enable us to establish some further
properties of principal planes and principal directions.

THEOREM 22. Every quadrlc surface has at least one real principal
plane at finite distance.

This theorem is an immediate consequence of Theorems 18 (last
part) and 21.

THEOREM 23. The principal directions which correspond to two
distinct discriminating numbers are mutually perpendicular.

Proof. Let ki and k% be two discriminating numbers of the
quadric surface and let ki = Afe. Then, in the notation of the
Remark following Theorem 18 (see page 189) and in virtue of the
Corollary of this theorem, we have

Mi, Vi = iMi, (foi, Mi, vi =

M2, V 2 ) = fc 2 X 2 , ^2(X 2 , M2, 1*2) = ^2M2, ^(Xo, M2, ^2) =

PRINCIPAL PLANES AND PRINCIPAL DIRECTIONS 195

Now the reader should have no difficulty in showing, by writing
out the expressions in full, that

Ml, V\) ^2?si, Ml, ^l = l<?l'2, M2

Therefore, if we multiply the equations of the first set written
above by X 2 , M2, and v 2 respectively and those of the second set by
Xi, MI, vi respectively, we find that

(ki fe) (XiX 2 + MiM2 + VM) = 0.

Since we supposed that fci =(= fe, we conclude that XiX2 + MiM2
+ KM = 0, from which the theorem follows in virtue of Corol-
lary 1 of Theorem 13, Chapter III, Section 36, page 64.

THEOREM 24. If a quadric surface has three distinct discriminating
numbers, then there exist three mutually perpendicular principal
directions for the surface; if there are two distinct discriminating
numbers, one principal direction is defined, and the second and third
principal directions are any directions perpendicular to the first; if
there is only one discriminating number, the principal directions are
entirely arbitrary.

This theorem follows from Theorems 18, 19, and 23 and the ob-
vious facts that if the three discriminating numbers are distinct,
each of them is a simple root of the discriminating equation; if
there are two distinct discriminating numbers, one of them is a
simple root and the other a double root; and if there is only one
discriminating number it must be a triple root of the equation.

91. Exercises.

1. Determine for each of the following surfaces, wnether or not an asymp-
totic cone exists; set up the equation of this cone in the cases in which one is
present, and indicate whether the cone is proper or degenerate, real or imagi-
nary:

W) + | = 2z; (h) x*=pz.

2. Discuss the asymptotic cone for the surfaces in parts (a), (b), (c), and (d)
of Exercise 7, Section 86.

196 QUADRIC SURFACES, GENERAL PROPERTIES

3. Determine the diametral planes of the surface

which correspond to the following directions :

(a) X : M : v = 1 : 1 : 1, (6) X : M : v = 4 : -1 : 8, (c) X : M : v = 6 : -
2 : -3.

4. Are there any directions with which no diametral plane of the surface in
the preceding exercise is associated? Are there directions for which the asso-
ciated diametral plane is not at finite distance?

6. Determine for each of the following surfaces the directions for which
there is no diametral plane or no diametral plane at finite distance:

(a) 2z 2 -f 20?/ + 18z 2 - 12 yz + 12 xy + 22x + y - 2 z - 5 = 0;

(6) 2 x 2 + 20 1/ 2 + 18 z* - I2yz + I2xy + 6 x + 1677 + 62 - 3 = 0;

(c) 36 x* -f 4 2/ 2 -f z 2 - 4 yz - 12 zx -f 24 xy + 4 s -f 16 y 26 z -f- 1 = 0;

(d) 2 x 2 - 7 7/ 2 -f 2 z 2 - 10 yz - 8 zz - 10 xy + 6 x + 12 y - 6 z + 5 = 0.

6. Prove that a quadric surface which has a single center (proper center or
vertex) has a diametral plane at finite distance associated with every direction.

7. Prove that a quadric surface with a line of centers has a diametral plane
at finite distance for every direction except one.

8. Prove that for a quadric surface with a plane of centers there exists
an infinite number of directions with which no diametral plane at finite
distance is associated.

9. Set up the discriminating equation for each of the following surfaces:

(a) 2* 2 -72/ 2 -f 2z 2 - Wyz-8zx- Wxy + 4x - 2y + 3z - 7 = 0;

(6) 2z 2 +22/ 2 + 2* 2 +4a* + 42/-f-5 = 0;

(c) 2x 2 +22/ 2 +22 2 -h27/z + 22x-h2xi/ + 4x-}-42/-f 4z + 3 =0;

(d) x 2 -f 4 y 2 + z 2 - 4 xy - 12 yz -f 6 zx - x -f 2 y -f 5 z = 0.

10. Determine three principal directions for each of the following surfaces:

(a) 2* 2 -f 2?/ + 2z 2 + 2^ + 2*r + 2z2/ + 4o;--4y-h4e-f-3 = 0;
(6) x 2 + if + 2z* - 4xz + 2xy -f 1 = 0;

(c) x 2 + i/ 2 - 2 z 2 -f 4 7/2 + 4 zx + 8 ZT/ - G x -f 5 y - 4 z -f 6 = 0;

(d) x 2 -f z 2 -f 2 XT/ -f 2 zz - 2 t/z - 2 x + 4 y - 4 = 0;

(e) 13 x 2 -f 13 2/ 2 -f 10 z 2 + 4 yz + 4 zx -f 8 x?^ - 3 x - 4 1/ 4- 2 z - 6 0;

-5 = 0.

CHAPTER VIII
CLASSIFICATION OF QUADRIC SURFACES

92. Invariants. The properties of quadric surfaces which were
discussed in the preceding chapter, such as the existence of an
asymptotic cone, of straight lines on the surface, of centers and of
diametral planes, do not depend in any way on the frame of
reference which is used; they are intrinsic properties of the surface.
The algebraic magnitudes and relations involving the coefficients
of the equations of the surfaces, by means of which these proper-
ties were characterized, must therefore be preserved when a new
reference frame is introduced. This fact is expressed by the
statement that these expressions and relations are invariant with
respect to the transformation of coordinates which carry us over
from one reference frame to another. Conversely, it is to be ex-
pected that an expression or a relation involving the coefficients
of the equation of a surface which remains unchanged under such
a transformation of coordinates, has an important bearing on the
intrinsic geometrical properties of the surface. Indeed the search
for such expressions and relations furnishes a method for the
systematic study of these properties; it is therefore of fundamental
importance in the entire field of Analytical Geometry. We shall
undertake now to prove the existence of a number of such expres-
sions and relations. But before doing so, we shall give an exact
definition of the concept of invariance and we shall illustrate it by
a few familiar examples.

DEFINITION I. An expression Involving the coefficients In the equa-
tion of a surface in Cartesian coordinates, and numbers which depend
on these coefficients, are called invariants of the surface with respect
to a transformation of coordinates which leads to another Cartesian
reference frame, If they remain unchanged when the coefficients of
the equation are replaced by the corresponding coefficients of the
equation obtained from the given one by such a transformation of
coordinates; relations between the coefficients which are preserved
under such a transformation are called invariant relations with respect
to the transformation.

197

198

CLASSIFICATION OF QUADRIC SURFACES

Examples.

1. The unsigned distance from the origin to the plane ax + by + cz -f d =
d

is given by

Va 2

(compare Corollary 1 of Theorem 7, Chapter IV,

Section 44, page 79). This distance remains the same no matter what
Cartesian reference frame is used, so long as the origin is not changed. Con-
sequently, if we put

x = \iXi -h X 2 i/i 4- X 3 Zi, y = MiZi + M22A + MsZi, ^ = "1*1 + "22/1 -f "aZi

in the equation ax -f- by -f cz + d = and if we suppose that the equation of
the plane is thereby transformed into a'xi + b'yi -f c'zi -f d' = 0, then it must

be true that
I d

Va' 2 + 6' 2 -f

rf

6 2

If this is so, the expression

\Va 2 + 6 2 -f c 2

will be called an invariant of the plane with respect to the

linear homogeneous transformation indicated above. To verify this fact, we
observe that

a' - a\i -f km + cvi t b f = oX 2 + &M2 + ^2, c' = aX 3 + &M3 + cs, d f =d.
Therefore

-f- V* + c'* = (Xx 2
c 2 -f 2 (MH>I -f

X 3 2 )a 2

-f- M2

+ 2

"2X2 -f v 3 X 3 )ca -f 2 (Xi/*i

But it follows from Theorem 6, Chapter V (see Section 65, page 123) that the
coefficients of a 2 , 6 2 , and c 2 are each equal to unity and that the coefficients of

be, ca, and ab vanish; hence a' 2 -f- b' 2 -f- c' 2
d

-f- fc 2 +

Thus we have

proved that

is an invariant of the plane with respect to ro-

Va 2 + 6 2 -f c 2

tation of axes. Properly speaking this expression is an invariant with respect
to rotation of axes of the configuration consisting of the plane and the origin.
2. The angle between the planes a& -f biy + c\z + d\ =0 and a z x -f 6 2 ?/
-f c 2 z -h ^2 = is independent of the reference frame that has been used to
represent these planes. Therefore the expression for the cosine of this angle,
given in Theorem 9, Chapter IV (Section 46, page 82), should be an invariant
with respect to a transformation from one rectangular Cartesian system to
another. Such a transformation is given by the equations

(See Theorem 8, Chapter V, Section 66, page 126.)
If these expressions transform the equations of the given planes to
ai'si + 6/2/1 + ci'zi -f di' = and a*'xi + bfa + c^ + d/ =

we have

a\ = aiXi 4- bim -f- civi t bi f 01X2 + 61^2 -h CM, c\ = 01X3

efi' == aip -f 619 -f cir,
as' = a 2 Xi + bzfjii -h cji^i, 62' = ^2X2

INVARIANTS OF A QUADRIC SURFACE 199

It follows from example 1 therefore that

oi 2 + bi* + fi 2 = fli' 2 + 61" -f <V 2 and a 2 2 + b-. 2 + c-> 2 = a 2 ' 2 + V 2 + c^.
Moreover

iV -f &iV + d'c 2 ' = OiCuCV + X 2 2 + X 3 2 ) + M 2 <V + M2 2 + Ms 2 ) -f
eic 2 0i 2 -f- ^ 2 2 -f "3 2 ) + (aik -f 0261) (Xi/ii + X 2M2 + X 3/ u 3 ) + (6ic z -f

Consequently

Oia 2 -f bj) 2 -f

Vfl!* + V + c, 2 X Va 2 * -f 6 2 2 + r 2 *

is indeed an invariant with respect to the linear transformation of coordinates.
In particular we notice that the relation a\Qv -f- bj) 2 -f- r i^ 2 = is invariant;
the reader should see the geometric significance of this fact.

93. Invariants of a Quadric Surface with respect to Rotation
and Translation of Axes. We proceed now to the following im-
portant theorems.

THEOREM 1. The functions T t , T 2 , A and A of the coefficients of a
quadric surface are invariant under translation of axes.

Proof. Translation of axes is accomplished by means of the
transformation x = x' + 7?, y == y' + q, z = z' + r (see Theorem
2, Chapter V, Section 61, page 115). The equation of the quadric
surface Q(XJ y, z) = with respect to the new reference frame is
therefore Q(x f + p, y' + q, z' + r) = 0. But, in virtue of Section
75 and the notation introduced in Section 80, this equation may
be written in the form:

(1) Q(P, q, r) + x'Q^p, q, r) + y'Q z (p, q, r) + z'Q,(p, q, r)
+ q(x', y', z') = 0.

It follows from this that the coefficients of the second degree terms
in the new equation of the surface are the same as those of the
corresponding terms in the original equation; that is, if we differ-
entiate between the new and the original equation by the use of a ',

ll' = Oil, 012' = 012, Ol3 7 = 13, ^22' = 22, O^ = O 2 3,

033' = 033.
Moreover, we see at once that

OM' = I ' Qi(p, q, r), ^ = \ Q,(p, g, r) a 34 ' = \ Qs(p, g, r\
o 44 ' = Q(p, g, r).

200

CLASSIFICATION OF QUADRIC SURFACES

Therefore (see Section 89, page 192)

1 =

On + 2 2 + 3 3 = On + 022 + 33 = 1 i]

T /

2

022' 023'

^11 #13 . Ctn di2 O22 O 2 3 . On Oi3

023' 033'

Ois 033 Oi2 O22 O23 033 OM (133

+

On Oi2

3=5 ^;

Oi2 O22

Oi/ Oi 2 ' Oi 3 '

OH 012 Ois

144' =

n f n ' n '
Ol2 O 22 O23

=

(1 12 O-22 O23

= ^44-

/ t f

Oi3 O 2 3 033

Oi3 O 2 3 ^33

It remains to show that A' = A. From what we have already
proved, it follows that

Qi(p, q, r)

A' =

On

r)

2

o, 9, r)

ii **M l *'J.> Q

\ f f Q(P>V,r)
Since , L = a,-ip + a,- 2 (/ + a; 3 r + a,- 4 , i == 1, 2, 3, 4, and also

t

since 2 Q(p, 5, r) = pQi(p, q, r) + gQ 2 (p, 3, r) + rQ 3 (p, q, r)
+ Q4(P> 3> r ) ( see Corollary 1 of Theorem 4, Chapter VII, Section
81, page 162), it follows as in Section 84, page 172, that if to the
last row of this determinant are added the products of the first
three rows by p, 3, and r respectively, and then to the last
column, are added the products of the first three columns by
p, 3, r respectively, then this determinant reduces to the
discriminant A of the surface. This completes the proof of our
theorem.

COROLLARY 1. The discriminating numbers of a quadric surface are
Invariant under translation of axes.

This theorem follows from the fact that the coefficients of the
discriminating equation, namely, 1, 7\, T 2 , and A 44, are in-
variant under translation of axes.

COROLLARY 2. The rank of the matrix a 3 is invariant under transla-
tion of axes.

INVARIANTS OF A QUADRIC SURFACE 201

COROLLARY 3. The rank of the matrix a 4 is invariant under trans-
lation of axes.

Proof. The proof of the invariance of A shows that the matrix
a/ is obtained from the matrix a 4 by means of elementary trans-
formations (see Definition XIV, Chapter I, Section 10, page 18) ;
it follows therefore from Theorem 14, Chapter I, that these two
matrices have the same rank.

THEOREM 2. The functions T i9 T 2 , and A* are invariant under rota-
tion of axes.

Proof. The proof of this theorem and of the next could be
made by the direct method followed in the proof of Theorem 1,
which consists in first expressing the coefficients of the new equa-
tion in terms of those of the given equation and then substituting
these expressions in the function whose invariance we wish to
prove. But this method, besides being laborious, does not give
us any further insight into the geometric meaning of the theorem.
We shall therefore follow a method of proof which is apparently
less direct and which may impress the reader as being rather so-
phisticated, but which has the merit, apart from greater elegance
and brevity, of penetrating more deeply into the problem under
consideration.

We consider the function q(x, y, z, fc) defined as follows:
q(x, y, z, k) = q(x, y, z) - k(x 2 + y 2 + z*).

Let an arbitrary rotation of axes carry the function q(x, y, z) over
into the function q'(x', y', z'). Since the expression x 2 + y 2 + z 2
represents the square of the distance from the origin to the point
(x, ?/, 2), it is invariant under rotation of axes; that is, x 2 + y 2 + z 2
x' 2 + y' 2 + z' 2 . Hence, if the same rotation of axes changes
the function q(x, y y z, k) to q'(x' y y', z', fc), we have

q'(x', y', z', fc) = '(*', 2/', *') - k(x' 2 + y' 2 + z' 2 ).

The equation q(x, y, z, k) = 0, being homogeneous in x, y, and
z, for every value of fc, represents a quadric cone; this will be a
degenerate quadric cone (that is, a pair of planes) if and only if
k has such a value fc* that the determinant

(in fc* 012 013

012 022 ~~ fc* 023

013 023 033 fc*

the value of the determinant

202 CLASSIFICATION OF QUADRIC SURFACES

vanishes. If this determinant vanishes, the equation q(x, y, z,
fc*) = represents a pair of planes; therefore the equation

q'(x f , y', z', k*) = represents a pair of planes and consequently

/ ?/.* ^ t si '

11 K Cti2 #13

12 7 ' O22' k* 023'

/ / ~ t TLsfc

is also equal to zero. And it should be clear that the same argu-
ment holds in the opposite direction. From this we conclude, in
view of Corollary 1 of Theorem 19, Chapter VII (see Section 89,
page 192), that the two equations
fc 3 - Tik* + T 2 k - Au = and fc 3 - Ti'k* + T 2 'k - AJ =

have the same roots, that is, TI = TI, T 2 = TV, Au = A^', thus
our theorem is proved.

If r 3 , the rank of the matrix 83, is equal to 3, Au 4= 0; therefore
Au ^p and 7-3' = 3. Ifr 3 = 2, sothat-A 4 4 = 0, then^ 4 / = Oand
r 3 ' < 3. If r 3 = 1, all the two-rowed minors of a 3 vanish and
therefore 5T 2 = 0; hence, it follows from Theorem 2, that A^ ~
and TV = 0, and thence by use of the Corollary of Theorem 7,
Chapter II (Section 26, page 44) that r 8 ; < 2. If r 3 = 0, the
function Q(x, y, z) is of the first degree; therefore the function
Q*%&9 y r > 2') is also of the first degree (compare Corollary 1 of
Theorem 8, Chapter V, Section 66, page 126) and r/ = 0. But
this entire argument can be applied equally well to the transfor-
mation which carries Q' back into Q. It follows therefore that if
r 3 ' = 2, then r 3 < 3; if r 3 ' = 1, then r 3 < 2; and if r 3 ' - 0, then
r 3 < 1. We have therefore obtained the following important
corollary.

CQHOLLARY. The rank of the matrix a 3 is invariant with respect to
rotation of axes.

94. Invariance of the Discriminant of a Quadric Surface with
respect to Rotation. We shall begin by proving the following
theorem.

THEOREM 3. The singularity of a quadric surface is not affected by
rotation of axes.

Proof. In view of Definition V of Chapter VII (Section 82,
page 166) the statement of this theorem is equivalent to the
statement that if A = 0, then A ; = 0, where A' is formed from

INVARIANCE OF THE DISCRIMINANT

203

the equation obtained from Q(x, y, z) = by rotation of axes; or
again, to the statement that if r 4 < 4, then r 4 ' < 4, where r 4 '
designates the rank of the matrix a 4 ' formed from this same equa-
tion. All the cases in which r 4 < 4 have been specified geometri-
cally in Remark 1, following Theorem 14, Chapter VII (Section
85, page 179), excepting the case in which r 4 = 3 and r 3 = 1. In
this case, we know on the basis of the Corollary to Theorem 2
(Section 93) that r 3 ' = 1 and hence, in view of the discussion
preceding Theorem 14 of Chapter VII, that r/ can not exceed 3.
We conclude therefore that in every case in which r 4 < 4, we must
also have r/ < 4. Our theorem is therefore proved.

THEOREM 4. The discriminant of a quadric surface is invariant with
respect to rotation of axes.

Proof. The method of proof is similar to that used in the proof
of Theorem 2. We consider now the auxiliary function

Q(x, y, z, k) = Q(x, y, z) - k(x* + y* + z* + 1).
Rotation of axes will carry this function over into

QV, 7/, z', k) = QV, '/, ') - k(x'* + y'* + z'* + 1).

A value fc* of k for which the locus of the equation Q(x, y t z, k) =
is singular will, in virtue of_the preceding theorem, also be a vaidc
of k for which the surface Q'(x', y', z', k) = is singular. Hence
the roots of the equation

=

0ii - k

012

013

014

012

022 k

023

024

013

023

033 k

034

014

024

034

ot 44 k

will also be roots of the equation

11 - k

012'

013'

014'

f

012

013'

f
0J4

22 ' ~ k

023'

024'

023'

033' k

034'

024'

034 7

a 44 ' k

= 0,

and vice versa. These equations are therefore equivalent. Now
it should be obvious that they have the form

fc 4 + . . . + A = and k 4 + . . . + A' = 0; and therefore
that A = A'. This proves our theorem.

204

CLASSIFICATION OF QUADRIC SURFACES

It will be worth while to consider in further detail the equation
A (k) = 0, which is quite similar in form to the discriminating
equation considered in Section 89. If we use again Theorem 19
of Chapter I (see Section 17, page 32), we find

A'(fc) = -

#22 k #23 #24

# n -fc

#13 #14

#23 #33 k #34
#24 #34 #44 k

#13
#14

#33 ~~ k #34
#34 #44 k

#11 k #12 #14

#11 -fc

#12 #13

#12 #22 ~~ k #24

#12

#22 k #23 ,

#14 #24 #44 k

#13

#23 #33 k

| #M #44%

+

#22 ~~ k #24
#24 #44 '

c H

#11 k #14

#14 #44 ~~ k

#22 k #23
#23 #33 k

+

#11 ~fc #13
#13 #33 k

+

#11 fc #12 ll

#12 #22 fc 1 J

A" ' (k) = 6 [(#11 k) + (#22 ~ k) + (#33 A:) + (# 44 fc)],
A" " (fc) = 24.

Since, moreover, A(fc) = A(0) + A'(0) X fc + A" (0) X ~

+ A" ' (0) X || + A" " (0) X ^, the equation A (fc) =

can be written in the form

fc 4 - D x fc 3 + # 2 fc 2 ~ D 8 k + A = 0,

D 2 =

6

A" (0)

= #n + # 22 + #33 + #44,
4

and

- 2

It will be observed that D\, D 2 , and D 3 are respectively the sums
of the one-rowed, the two-rowed, and the three-rowed principal
minors of the discriminant A.
We have now the following Corollary of Theorem 4.

COROLLARY. The sums of the one-rowed, of the two-rowed, and of
the three-rowed principal minors of the discriminant of a quadric
surface are invariant with respect to rotation of axes.

INVARIANTS OF A QUADRIC SURFACE 205

It follows moreover from Theorem 4 that, if r 4 = 4, then r 4 ' = 4;
and that if r 4 = 3, then r 4 ' < 4. If r 4 = 2, all the three-rowed
minors -of a 4 vanish and therefore Z) 3 = 0; the corollary enables
us then to conclude that Z> 3 ' = and the Corollary of Theorem 7,
Chapter II (Section 26, page 44) establishes then the fact that
r\ < 3. Similarly it can be shown* that if r 4 = 1, then r/ < 2;
and it should be clear that if r 4 = 0, then r/ = 0. Moreover the
argument can be made equally well from the rank of A' to that of
A. We have therefore obtained the further result, stated in the
following theorem.

THEOREM 5. The rank of the matrix a 4 is invariant with respect to
rotation of axes.

The results obtained in Sections 93 and 94 may be summarized
in the following statement :

The values of the expressions A, /i< 4 , 7\, T 2 and the ranks of
the matrices a 3 and a 4 are invariant with respect to translation
and rotation of axes; the expressions A>,, />>, and D 3 are invari-
ant under rotation of axes.

95. Exercises.

1. Prove that the condition under which three planes have a single point in
common is invariant with respect to translation of axes, and also with respect
to rotation of axes.

2. Prove that the distance from the plane ax -f- by + cz -f d = to the
point P(XI, 2/1, 21) is invariant with respect to translation and rotation of axes.

x 2 ?/ 2 z 2

3. Show that for the surface -f ~ -f 1 =0, the sum A, of the

three-rowed principal minors of the discriminant is not invariant with respect
to translation of axes.

X 2 y 2

4. Show that for the surface 5 ^ 1 =0, the sum D 3 is invariant with

a 2 b 2

respect to translation of axes; also that the sum 7) 2 of the two-rowed prin-
cipal minors of the discriminant and the sum D\ of its one-rowed principal
minors are not invariant with respect to this transformation of coordinates.
6. Show that for the surface x 2 = a 2 , the sums D 3 and D> 2 are invariant with
respect to translation of axes; also that the sum A is not invariant.

6. Prove that if the axes are translated to the new origin P(a, 0, 7), the
sum ZY for the new equation is equal to Z> 3 plus multiples of three-rowed
minors of the matrix b (compare Section 85, page 178).

7. Prove that under the conditions of Exercise 6, the sum TV for the new
equation is equal to D 2 plus multiples of two-rowed minors of the matrix b.

* See Appendix, IV, p. 297.

206 CLASSIFICATION OF QUADRIC SURFACES

96. Two Planes, We have already met a number of instances
in which the equation Q(x, y, z) = represents two planes. In
the present section we undertake a more detailed study <tf these
cases; and we begin with the following theorem.

THEOREM 6. The necessary and sufficient condition that a quadrlc
surface consist of two planes is that the rank of r 4 of the discriminant
matrix a be less than 3.

Proof. The sufficiency of this condition has already been
proved in Corollary 2 of Theorem 12, Chapter VII (Section 84,
page 175). To prove the necessity of the condition, we observe
that if the locus of the equation Q(x, y, z) = consists of two planes
then, by the argument made in the proof of this corollary, the
function Q(x, ?/, z) must be factorable in two linear factors; that
is, there must exist numbers a, 6, c, d and ai, 61, ci, di such that

Q(x, y, z) = (ax + by + cz + d) (aix + biy + c\z + di).

If this is the case, the coefficients of the function Q can be ex-
pressed as follows:

an = aai, a 22 = 661, ass = cci, a 44 = ddi, 2 a i2 = a&i + a\b,
2 ais = aci + aic, 2 a H = adi + aid, 2 a 23 = bc\ + &ic,
2 a 24 = &di + &id, 2 a 34 = cdi + Cid.

Consequently the discriminant A is given by the equation

16 A =

2 aai ab\ + a\b ac\ + a\c ad\ + aid

ob\ + a\b 2 bhi bc\ + b\c bd + 6id

aci + aic 6ci + b\c 2 cci cdi + Cid

adi + aid &di + 6id cdi + Cid 2 ddi

It is not difficult to show that this determinant and its three-
rowed principal minors vanish (the details of this proof will be
found in Appendix, V, page 298). But since the matrix of this
determinant is symmetric, we can then conclude by use of Theorem
6, Chapter II (see Section 26, page 43) that the rank of the matrix
a 4 is less than 3.

It was proved in Corollary 3 of Theorem 14, Chapter VII (Sec-
tion 85, page 180), that if r 4 = r 3 = 2, the locus of the equation
Q(x y i/, z) = consists of two intersecting planes. We shall now
determine the equations of these planes.

TWO PLANES 207

Since ra = 2, it follows that at least one of the two-rowed prin-
cipal minors of the matrix a 3 is different from zero; let us sup-
pose that asa = flute 0i2 2 4= 0. From Theorems 12 and 13 of
Chapter I (see Section 7, page 13), we derive then the following
equalities :

^ <*33013_ = 0, 1 3 012 + 2 322 +..#33023 = ^j
#23023 + #33033 = 0, #1 3 014 + #23024 + #33034 = 0.

And if we denote by 0# the cofactors of the elements a# in the de-
an 012

terminant

012 022 023

, we have also (notice that 3 4 = # 33 4 1 0)

014 024 034

the equalities

013011 + 023012 + 034^14 = 0, 013012 + 023022 + 034024 = 0,
013013 + 023023 + 034034 = 0, 01 3 0H + 023024 + 034044 = 0.

If we multiply the equalities of the first of these sets by x, y, z, and
1 respectively and add, we obtain i 3 Qi + 0:23^2 + 33 Q 3 = 0; sim-
ilarly, we find from the second set the result 0i 3 Qi + 02sQ2 + 034$4
= 0. And from these equations we conclude (remembering that
3 4 = <* 33 4= 0), that

2

But, a 33 y 01232023 = (011022 012 2 )2/+ (011023 ~ 012013)^+ (011024

012014) = 011^2 012^1 ,'

and a33^-~Ctl3^ 013= (011022 012 2 )^+(022013~012023)2+(0 22 014

012024) = 022Ql 012^2-

Therefore, 2 a 33 Q(a;, y, z) = (anQ 2 ai2<2i)Q2+ (022^1 -

Since the discriminant of this quadratic function of Qi and Q 2 is
equal to 4 33 and is therefore different from zero, we conclude that
the equation Q(x, y,z) =0 is equivalent to the two linear equations
Qz XQi = and Q 2 /iQi = 0, where X and n are the distinct
roots of the quadratic equation a^t 2 2 12 ^ + 022 = 0. The
two planes represented by the equation Q = in this case are there-
fore distinct planes through the line of intersection of the planes
Qi = and Q 2 = (compare page 180). It follows moreover
that if 33 > 0, the two roots of the quadratic equation ant 2 2 a 12 t

208 CLASSIFICATION OF QUADRIC SURFACES

+ 0-22 = are complex, whereas they are real if #33 < 0. Finally,
we observe that the sign of 33 is the same as that of the invariant
7 7 2 , which is equal to the sum of the two-rowed principal minors of
the matrix a 3 , in virtue of Theorem 7 of Chapter II (see Section
26, page 44). We summarize the result in a theorem.

THEOREM 7. If the ranks of the matrices a 4 and a, are both 2, the
locus of the equation Q(x, y, s) = consists of two planes; these
planes are real if the invariant T 2 is negative, and imaginary if T 2 is
positive.

We consider next the case in which r 4 = 2 and r 3 = 1 ; it was
shown on page 178 that in this case the rank of the matrix b
is also equal to 1. Consequently the three rows of this matrix
are proportional; and if we suppose that a n ^ 0,* we can write

and Q 3 = !. Moreover Q 4 = q, + 2

#11 (hi #11

_i_ o a ' 4 v (n 9 \ _L o a nQi 2(a 14 2

+ 2 (7 4 4 = -- X (Ql 4 flu) + ^ #44 =

an #11 #11

Therefore

2(a H 2

#n

and the equation Q(x, y, z) = is equivalent to the equation
d 2 = 2(#i4 2 - 011044).

It is shown in Appendix, VI (page 299) that if r 4 = 2, the two-
rowed principal minors of the matrix a 4 can not all vanish and that
those which do not vanish all have the same sign and therefore the
sign of their sum D 2 . Since r 3 = 1, all two-rowed principal minors
of as vanish; the rank of b being 1, the other two-rowed principal
minors of a 4 differ by a factor and are therefore different from zero
and of the same sign of Z> 2 . We conclude that the locus of the
equation Q(x, y, z} = consists of two distinct parallel planes,
whose equations are Qi = dh V2(# u 2 #11044); these planes will be
real or imaginary according as Z) 2 is negative or positive.

* If all the elements in the principal diagonal of a 3 were zero, it would
follow since in this case an = a 22 = 3 3 = 0, that the elements au, a 2 a, and
an also vanish, so that the rank r 3 of a 3 would be zero; all the non-vanishing
elements of the principal diagonal have the sign of the invariant TV

TWO PLANES 209

Finally, the same discussion shows that if r 4 = r 3 = 1, then the
equation Q(x, y, z) = reduces to the form Qf = 0, so that its
locus consists of two coincident planes (compare Corollary 3 of
Theorem 14, Section 85, page 180).

THEOREM 8. If the rank of the matrix a 4 is and the rank of the ma-
trix a 3 is 1, the locus of the equation (X>, y, a) = consists of a pair of
parallel planes; these planes are real or imaginary according as the
sum D> of the two-rowed principal minors of the matrix a 4 is negative
or positive. If the ranks of the matrices a. and a. arf froth i f thr lonin
of the equation Q(x, y, z) = consists of the plane Q,(x f y r ) = 0,
counted doubly.

Remark. Since the hypotheses of Theorems 7 and 8 exhaust all
the possibilities as to the ranks of the matrices a 4 and a 3 , subject
to the condition of Theorem 6 that r 4 must be less than 3; and
since the conclusions of these two theorems include all the possible
relative positions of two planes, it follows that the converse of
each of these theorems also holds; that is, if a quadric surface con-
sists of two real intersecting planes, two imaginary intersecting
planes, two real parallel planes, two imaginary parallel planes, or
two coincident planes, the ranks of the matrices a 4 and a ;{ are 2
and 2(T 2 < 0), 2 and 2(T 2 > 0), 2 and 1(D 2 < 0), 2 and 1
(D 2 > 0), 1 and 1 respectively.

We state some further consequences of our discussion.

COROLLARY 1. If the rank of the matrix a 3 is 2, the locus of the equa-
tion q(x, y, z) = consists of a pair of intersecting planes, whose line
of intersection passes through the origin; if the rank of this matrix
is 1, the locus is a pair of coincident planes through the origin.

COROLLARY 2. A function Q(x , y, s) of the second degree is factorable
into two linear functions of*, y, and s with real or complex coefficients
if and only if the rank of its discriminant matrix is less than 3; it
is the square of a linear function of *, y, and z with real or complex
coefficients if and only if the rank of its discriminant matrix is 1.

COROLLARY 3. A homogeneous function q(x, y, z} of the second de-
gree in x 9 y, and z is factorable into two linear homogeneous functions
of x 9 y, z with real or complex coefficients if and only if the rank of the
matrix a 3 is less than 3; it is the square of a linear homogeneous
function of *, y, and z with real or complex coefficients if and only if
the rank of this matrix is 1.

Corollaries 2 and 3 are obviously restatements in algebraic form
of the results formulated in Theorems 6, 7, and 8.

210 CLASSIFICATION OF QUADRIC SURFACES

97. Translation of Axes to the Center of a Quadric Surface. If

a quadric surface has a center, its equation is materially simplified
when the axes are translated to the center as origin. For it should
be obvious from the definition of a center that the surface is sym-
metric with respect to such a point (compare Definition VIII of
Chapter VII, Section 85, page 176 and the first footnote on page
137) ; therefore, if a, fc, c are the coordinates of a point in the new
reference frame, then Q( a, 6, c) must vanish whenever
Q(a, 6, c) vanishes. Consequently Q(a, 6, c) Q( a, 6, c)
= for all sets of numbers a, fc, c for which Q(a, fe, c) = 0. But
Q(a, 6, c) Q( a, 6, c) = 2(a u a + a^b + a 34 c); if this linear
function is to vanish for all sets of values for which the quadratic
function Q(a, 6, c) vanishes, then a^ = #24 = #34 = 0. Conse-
quently the equation of a quadric surface referred to a reference
frame whose origin is a center of the surface does not have any first
degree terms.

We shall now reach this result in another way, which will disclose
some further properties. The translation of axes to the point
(a, b, c) as origin is accpmplished by means of the equations of
transformation

x = x' + , y = y f + b, z = z' + c.

The equation of the surface Q(x, y, z) = with reference to the
new system of coordinates is therefore

Q'(x', y', z')=Q(x'+a, y'+b, z'+c)=q(x', y', *')+*'<2i(, 6, <0
, b, c)+z'Q 3 (a, 6, c)+Q(a, 6, c) = 0,

(compare Section 93, formula (1), page 199).

But, if a, b, c are the coordinates of a center of the surface,
Qi(a, 6, c) = Q 2 (a, 6, c) = Q 3 (a, b, c) =0 (see Theorem 13, Chapter
VII, Section 85, page 177) ; in this case the equation of the surface
reduces therefore to the form

(*', 2/', *') + Qfo, ?>, c) =

and this equation is free from terms of the first degree in x' t y' f
and z'.

We observe, moreover, (1) that the second degree terms in the
new equation have the same coefficients as the corresponding
terms of the original equation; and (2) that the constant term
Q(a, 6, c) is equal to \[aQ\(a y 6, c) +bQz(a, 6, c)+cQ 3 (a, 6, c)

ROTATION OF AXES

211

b> c)] = ^Q^(d t by c) = a\\a + a^b + 0340 + 044. Further-
more the discriminant of the simplified equation is, in virtue of
Theorem 1 (Section 93, page 199) equal to the discriminant A of
the original equation; on the other hand it is equal to

Oil #12 #13 V
Q>12 #22 #23 vl
#13 #23 #33 **,, ( y

^

Therefore A = ^4 44 ~; if the surface has a unique center, and
&

in no other case, ^.44 4^ 0, so that the constant term in the reduced
equation can then also be put in the form -. . We summarize these

A 44

results as follows.

THEOREM 9. If the quadric surface Q has a center at the point
(a, &, c), its equation in a reference frame whose axes are parallel to
the original axes and whose origin is at the center, has the form

<?'(*' y', *')

', y', *') + a 44 ' = 0, where a 44 ' = Q(a, 6, c) =

if (a, 6, c) is the only center of the surface, we have, moreover, 044'

98. Rotation of Axes to the Principal Directions of a Quadric
Surface. It was proved in Theorem 24 of Chapter VII (Section
90, page 195) that for every quadric surface there exist three
mutually perpendicular principal directions; under some condi-
tions these directions can be determined in one and only one way;
under other conditions they can be determined in more than one
way. We will suppose now that for the quadric surface Q(x, y, z)
= three mutually perpendicular principal directions are given by
the three sets of direction cosines Xi, MI> v\\ \z, ^ 2 > v ^ anc ^ ^ 3 > ^ 3 > V3 >
and it is our purpose to determine the equation Q'(x', y', z 1 ) = of
the surface when it is referred to a reference frame whose origin
coincides with the origin of the original frame, but whose axes are
in these principal directions.

According to Theorem 5 of Chapter V (see Section 63, page 121),
the transformation is carried out by means of the substitution:

X =

y

212 CLASSIFICATION OF QUADRIC SURFACES

We have therefore
Q'(x f , y', z') =

' + X 2 7/' + X 3 z', fjnx' + My' + &&', v\x 9 + v 2 y r
X 2 7/ + X 3 z') + 2 a 24 ( M tz' + ny' + ^z')
v z z') + a 44 .

Since q(x, ?/, z) is a homogeneous function of the second degree
in x, y, and z, and since the expressions which have been substituted
for these variables are linear and homogeneous in #', y', and z 1 , it
should be clear that the function q(\ix' + , v\x' + ,
vix' + ) which constitutes the first term in the new equation
is homogeneous and of the second degree in x', y', and z'. The
terms of degree less than 2 in the new equation can be determined
readily; if we write that part of the new equation in the form
2 au'x' + 2 au'y' + 2 a 34 V + 44 ', we find

(134 = OwXs + 024M3 + 034*3 = ^ *' ^

It remains now to determine the coefficients of the second degree
terms in the new equation. For this purpose we expand
q(\\x' + , nix' + ji>ix' + ) by Taylor's theorem (com-
pare Sections 75 and 80). First we look upon Xi#' + X 2 i/', nix'+ &y'
and v\x 9 + v 2 y' as the (temporarily) fixed values of the variables in
the function q(x, y, z) and upon Xsz', vtft , and v$f as their incre-
ments. We find then

X 3 z', AUX' + wy f + &', v&' + v 2 y' + v&') =
q(\ix' + X 2 ?/, mx' + my', v\x' + v 2 y') + \ 3 z' qi(\ix' + \ 2 y',
ix' + My', v& r + v 2 y') + &' q 2 (\ix' + \ 2 y', mx f + toy',

\ 2 y' f . . . 9 . . . )

To the first four terms on the right we apply again Taylor's
theorem, remembering that % = 2 a# (see Section 80); we find

ROTATION OF AXES 213

then that

q'(x', y', z') = q(\ix', mx', vix') +

ix', vix f ) + 2 a n \ 2 y' + 2 a V2 ^y f + 2
' fe(Xio;', /Litre', vix 1 ) + 2 a 2i \2y' + 2 a^^y' + 2
ix', v&') + 2 a 3 iX 2 7/ / + 2 a^y' + 2

We recall once more that q is a homogeneous function of the
second degree and that q\, # 2 , and q$ are homogeneous functions of
the first degree; also the property of homogeneous functions of
which we spoke in the proof of Theorem 3, Chapter VI (see Section
70, page 136). If we make use of these facts, we should be able
to see that the second degree terms in Q'(x', y', z') reduce, under a
^i>c,rai rot^tv of axes, to

" ' / a) +x'y r [

y'z' [X 3 ^i

If in this expression we make use of the formulas established in
the Corollary of Theorem 18, Chapter VII (see Section 88, page
190) for the direction cosines of the principal directions, this ex-
pression reduces to

Finally we put into operation the hypothesis that the new coor-
dinate axes are mutually perpendicular and that therefore their
direction cosines satisfy, two by two, the condition of Corollary 1
of Theorem 13, Chapter III (see Section 36, page 64) ; our second
degree terms then become /bix' 2 + k z y' 2 + & 3 z' 2 . We have there-
fore obtained the result stated in the following theorem.

THEOREM 10. If the discriminating numbers of a quadric surface
Q are fci, & 2 , and fc 3 and if the frame of reference is rotated so that the
new X-, Y-, and Z-aies have the directions of the principal directions
determined by fci, k 29 and fc s respectively, then the equation of the
surface with respect to the new frame is

<?'(*', y', *') = fci* /2 4- fetf' 2 + k*z'* + q<(\ l9 Ml , *)
s> vz)s' + a 4 4 = 0.

214 CLASSIFICATION OF QUADRIC SURFACES

Remark. The phrase "the principal directions determined by
fci, fc 2 , and & 3 " used in the statement of this theorem is to be under-
stood in the same sense as in Theorem 24, Chapter VII (see
Section 90, page 195).

99. Classification of Quadric Surfaces the Non-singular
Cases. We are now in a position to analyze the general equation
of the second degree in x, y, and z, that is, to determine the types
of surfaces that can be represented by the equation Q(x, y y z) 0.
The problem of making this determination is usually referred to
as the " classification of quadric surfaces."*

The analysis of the equation Q(x, y, z) = will be based on the
ranks r\ and r 3 of the matrices a 4 and a 3 respectively; and we treat
first those surfaces for which r 4 = 4, that is, the non-singular
quadrics. In virtue of Corollary 1 of Theorem 14, Chapter VII
(see Section 85, page 179), this condition carries with it that
7*3 ^ 2; we have therefore to consider two cases, namely, n = 4,
7*3 = 3; and r 4 = 4, r 3 = 2.
CASE I. r 4 = 4, r 3 = 3.

We know from Theorem 14, Chapter VII (Section 85, page 178)
that the surface has a single proper center. If the axes are trans-
lated to this center as an origin, the equation becomes (see Theorem
9, Section 97, page 211)

(1) q& 9 y' 9 *)+-=0.

A 44

The three roots of the discriminating equation

(2) A: 3 - 7W + TJc -.444 =

are all real and different from zero. Since the first degree terms are
absent from equation (1), rotation of axes to principal directions
will carry the equation over into

fci*" 2 + W 2 + fcaz" 2 + -- = 0,

^44

* This problem concerns itself therefore primarily with the question of
determining what kind of surface is represented by given numerical equations
and not with that of locating the position of the surface with respect to a
frame of reference, nor with finding the particular numerical data which
serve to specify the surface as an individual of its type. The method of
treatment of OUT principal question is of such nature however as to develop
means for answering these further questions.

THE NON-SINGULAR CASES, 215

kiy kz, fc 3 being the roots of equation (2), (see Theorem 10, Section
98, page 213). Since A 4= 0, this equation may be written in the
form

r "2 ?y 2 "2

- + -=-T- = I.

This equation belongs to the types of equations whose loci were
discussed in Section 72. If we make use of the discussion of this
section, we reach the following conclusion :

(a) If ^ -r- , j -r~ , ; - A are all negative, the surface is an
/CiA 44 KzAu KsAu

ellipsoid.

* (6) If two of these numbers are negative, the surface is an
hyperboloid of one sheet.

(c) If one of these numbers is negative, the surface is an hyper-
boloid of two sheets.

(rf) If none of these numbers is negative, the surface is an
imaginary ellipsoid.

Remark. By reference to Example 2, Section 68, page 133, wo
see furthermore that, if the discriminating equation has a pair of
equal roots, the quadric surface will be a surface of revolution,
namely, an ellipsoid of revolution (real or imaginary), or a hyperbo-
loid of revolution (of one sheet or of two sheets), according as the
sign of the double root does or does not agree with that of the re-
maining root; if and only if the discriminating equation has a
triple root, the quadric will be a sphere (real or imaginary).

We observe that the complete determination of the character of
the surface depends in this case on the signs of A and ^4 44 , and on
the signs of the roots of the cubic equation (2), whose coefficients
are all invariant with respect to translation and rotation of axes.
Since the roots of this cubic are all real (compare Theorem 20,
Chapter VII, Section 89, page 192), Descartes' Rule of Signs
enables us to tell exactly how many positive and how many nega-
tive roots it has. If the signs of T\ and of ^4 44 are both changed,
all the roots of the cubic change sign, and therefore the numbers

? . preserve their signs for i = 1, 2, 3; hence we need consider

216 CLASSIFICATION OF QUADRIC SURFACES

only the sign of the product TiA^* We distinguish now the fol-
lowing cases:

(1) A > 0, T<t > 0. In accordance with the remark made
above, the sequences of sign in the cubic which have to be con-
sidered are the following:

When ^44 > 0, 7\ > 0, the signs are H 1 ;

and, when AU > 0, TI < 0, the signs are + + -\ .

If the first of these occurs, the equation has three positive roots
and the three " coefficients "T-T~ , * = 1, 2, 3 are positive; the sur-
face is therefore an imaginary ellipsoid; if the second sequence
occurs, there are two negative roots and one positive root and
hence two negative and one positive coefficients, so that the sur-
face is an hyperboloid of one sheet.

(2) A < 0, T 2 > 0. The sequences of sign are the same as
before, but since now A < 0, all the coefficients will have changed
their signs. Therefore we shall have an ellipsoid if A^Ti > 0,
and an hyperboloid of two sheets if A^Ti < 0.

(3) A > 0, T 2 < 0. If Au > 0, the sequence of signs will

be + H or H ,so that we have one positive and two

negative roots and also one positive and two negative coefficients.

If A 4 4 < 0, the sequences of signs are + H h or H h, so

that there are one negative and two positive roots but, since A 44
has changed sign, again one positive and two negative coefficients.
In this case therefore the surface is always an hyperboloid of one
sheet.

(4) A < 0, T 2 < 0. We have the same distribution of roots
as in (3), but, since A has the opposite sign, the coefficients will be
opposite in sign; the surface is therefore an hyperboloid of two
sheets.

It remains to consider the cases in which either T\ or T 2 vanishes;
that they can not vanish simultaneously was shown in the Corollary
of Theorem 21, Chapter VII (see Section 89, page 194). If either

* It should be clear that the signs of T\ and of Au can not be significant in
determining the character of the locus of the equation Q = 0. For, if this
equation is multiplied through by -1, TI and A 44 clearly change their signs,
but the locus of the equation is obviously not affected. This remark does not
apply to TI, A or 7\A 44 .

THE NON-SINGULAR CASES

217

Ti or 5T 2 vanishes, the roots can not all have the same sign; for
since TI = ki + k 2 + k 3 and T 2 = fcife + & 2 /c 3 + & 3 fci, the former of
these expressions would then have the sign common to the roots
and the latter would be positive. Moreover AU = fcifc 2 fc 3 ; hence, if
A> and A& > 0, there must be one positive and two negative roots
and also one positive and two negative coefficients, and if A >
and AM < 0, there are one negative and two positive roots, but
again one positive and two negative coefficients. In either case
the surface is an hyperboloid of one sheet.

But if A < 0, there will be one negative and two positive coeffi-
cients, so that the surface is an hyperboloid of two sheets.

We summarize the results in the following theorem.

THEOREM 11. If the ranks of the matrices a 4 and a 3 are 4 and 3 re-
spectively, the locus of the equation Q = will be determined by the
following table:

A>

A <0

T 2 > 0, ATi >

Imaginary ellipsoid

Ellipsoid

T 2 > o, AuTt ^ o

or
T 2 ^0

Hyperboloid of one
sheet

Hyperboloid of two
sheets

Remark. We observe that the character of the surface can be
completely determined in this case as soon as the signs of the in-
variants A, AM, T 2 and TI are known and that it is not necessary
for this purpose to solve the discriminating equation. Compare
also the Remark on page 215.
CASE II. r 4 = 4, r 3 = 2.

According to Theorem 14, Chapter VII, the surface does not have
a center in this case. Since AM is equal to zero, T 2 must be differ-
ent from zero, for otherwise we could conclude by means of the
Corollary of Theorem 7, Chapter II (see Section 26, page 44) that
r 3 < 2. Consequently, one and only one root of the equation (2)
vanishes; let it be k\. Rotation of axes to principal directions
will then reduce the equation Q = to the form

(3) fey' 2 + fc 3 2' 2 + #4(Xi, MI, vi}x' + g 4 (X 2 , M2, v^y' + q*(\3, MS, ?sX

+ 44 = 0.

218

CLASSIFICATION OF QUADRIC SURFACES

The discriminant of this equation is
000 a 14

a 24 '

a 24 '

where 2 ai/, 2 a 24 ' and 2
coefficients g 4 (Xi, MI, "0, ^

are used, as before, to designate the
2 , M2, ^2) and q(\ 3 , /i 3 , i> 3 ) of x', ?/', and 2'
respectively. But the discriminant of a quadric surface is in-
variant with respect to rotation of axes (see Theorem 4, Section
94, page 203) ; hence A = an /2 kjc^ and, since fe =j= and

& 3 ={= 0, an' = dby T-r ^p 0. The further reduction of the

equation is now made as follows; completing the square on the
terms in y' and z', it becomes
^2

/ fi '\ 2 / r, '\

, / . . a 24 y . 7 / , , 34 \

H y + ir) + H 2 + F)

We translate the axes now to the point

/ 044 , a 2 4 /2 , dU 2 __

V 2 ai/ "*" 2 a 14 '/c 2 "*" 2 ^4^3 '
as origin by putting

/ // ^44 , ^24 " , &.34 / // ^ 2 4

X' T* , I I /ij' I/

*^ o / i^ ?) TI~ i^ o // > / i/ *~7 >

^ (7] 4 u (Ij 4 A/ 2 M dj 4 n/3 rC 2

2 =2

This transformation carries the equation of the surface over into
the form

We reach therefore the conclusion, by means of the results of
Section 72, that in this case the locus of the equation Q = is an
elliptic paraboloid if fc 2 and fc 3 have the same sign, and an hyper-
bolic paraboloid if they are opposite in sign. But k\ and /c 2 are
the roots of the quadratic equation fc 2 Tik + T 2 = 0, and there-
fore the first or the second of these cases will arise according as

THE NON-SINGULAR CASES

219

T 2 is positive or negative. We shall replace this criterion by
another one; but before doing so we observe that the surface will
be a paraboloid of revolution if and only if fc 2 = & 3 .
We will prove now the following theorem.

THEOREM 12. If A = 0, then A is the square of a linear homoge-
neous function of a u , a 24 , and a 34 .

Proof. Since AM = 0, the development of A according to the
elements of its last column leads to the equation

#12 #22 #23
#13 #23 #33
#14 #24 #34

+ #24

#11 #12 #13
#13 #23 #33
#14 #24 #34

#34

#11 #12 #13
#12 #22 #23
tti4 O24 #34

If we develop each of these three-rowed determinants according to
the elements of their last row and use the notation ,y for the co-
factors of the elements a# of the matrix a 3 , as introduced on page
184 in the proof of Theorem 16, Chapter VII, we find that

A = ~#14(#1411 + #2412
+ #2423

2 ai 4 024ai2 + 2 a u a 34 ai 3 + a 24 2 a 22 + 2

#1412~#2422 ~

It follows from Corollary 2 of Theorem 19, Chapter VII (Section
89, page 192), since w~ are supposing that AM = 0, that a,-^
= a# 2 , for i, j = 1, 2, 3; therefore a// = Va^y, so that we
may write _ _ _

A = -(duVan a 24 V / o: 2 2 # 3 4V / a 33 ) 2

which proves our theorem, since the negative sign outside the
parentheses may be introduced under each of the radicals.

From this theorem we derive an important corollary. For the
discussion recalls, as might also be derived from Corollary 3 of
Theorem 19, Chapter VII (Section 89, page 192), that those of the
principal minors an, a 2 2, 0:33 which do not vanish have the same
sign as T z (and not all of them can vanish if r s = 2). Hence, if
Tz > 0, Van, V22, and Vo^ are real and A g 0; while if 7 7 2 < 0,
these square roots are pure imaginaries or zero (not all zero) and
A > 0.

COROLLARY 1.
site in sign.

If Au = 0, T 2 4= and A =|= 0, then T, and A are oppo-

220 CLASSIFICATION OF QUADRIC SURFACES

The method used to prove Theorem 12 enables us also to prove
the following important formula:

012

COROLLARY 2. The value of the determinant

Oi2 CI22 2
13 23 3

a 6 c

is

equal to -(ana 2 4- a 22 6 2 + assc 2 + ^ a 2 3&c 4- 2 a 13 co + 3 a^ab).

Returning now to the discussion which precedes Theorem 12, we
can state the following theorem.

THEOREM 13. If the rank of the matrices a 4 and a 3 are 4 and 2 re-
spectively, the locus of the equation Q = is an elliptic paraboloid if
A < 0, and an hyperbolic paraboloid if A > 0.

100. Classification of Quadric Surfaces the Non-degenerate
Singular Cases. If r 4 = 3, we can have r 3 = 3, 2 or 1.
CASE III. n = 3, r 3 = 3.

From Theorem 14, Chapter VII, we know that in this case the
surface has a single vertex and from Corollary 3 of this theorem we
know that it is a proper quadric cone. The reduction of the equa-
tion in this case is made in exactly the same way as in Case I,

except that we have now -r = 0, so that the final form of the

-A 44

equation is

/biz" 2 + hy" 2 + k 3 z" 2 = 0.

The cone is real if and only if the discriminating numbers do not
all have the same sign; this will always be the case unless the coeffi-
cients in the discriminating equation present either no variations
or three variations of sign, that is, unless T 2 > and A^Ti > 0.
In this case we have therefore the following result :

THEOREM 14. If the ranks of the matrices a 4 and a 3 are both equal
to 3, the locus of the equation Q = is an imaginary cone, if T 2 >
and AnTi > 0; in all other cases the locus will be a real quadric cone.

Remark. The surface will be a real circular cone if and only if
the discriminating equation has a simple root of one sign and a
double root of the opposite sign.

From Theorem 14 we shall derive an important algebraic
theorem. It is an immediate consequence of Theorem 14 that the
equation
q(x, y, z) = ana: 2 + 022^ + a&z 2 + 2 a^gz -r 2 a 3i zx + 2 a^xy =

THE NON-DEGENERATE SINGULAR CASES 221

represents a cone if the determinant A 44 = |ay-|, i, j == 1, 2, 3 is
different from zero; this cone will be imaginary if 7 7 2 = an + a<&
+ 0:33 > and A&TI = Au(an + 022 + #33) > 0, but in all other
cases it is real.

In the first case, the function q(x, y, z) is reducible to the form
kix 2 + &2?/ 2 + fc 3 z 2 , in which ki, & 2 , and & 3 are different from zero and
are of like sign; the function q(x y y, z) will be zero ifx = y z = Q
and it will be of one sign for all other sets of real values of the
variables, namely, of the sign of its coefficients, which will be the
sign of AM since AM is equal to k\k^. In the second case the func-
tion q(x, y t z) is also reducible to the form kix 2 + & 2 y 2 + & 3 2 2 , but
now the coefficients in this form are not all of the same sign, and
the function can therefore take negative, positive and zero values
for different sets of real values of the variables. We introduce now
the following definitions.

DEFINITION II. A homogeneous function of degree 2 in 3 variables
is called a quadratic ternary form.*

DEFINITION III. A positive (negative) definite form is one which
takes the value zero only when all the variables vanish and is positive
(negative) for all other sets of real values of the variables; an indefi-
nite form is one which can take positive, negative and zero values for
real values of the variables.

We can now state the following important algebraic theorem.

THEOREM 15. The quadratic ternary form q(x 9 y, *) for which the
determinant An does not vanish is definite if and only if an + "22
4- ass > and Au(au + 022 -f "33) > 0; it is positive or negative definite
according as A is positive or negative.

CASE IV. r 4 = 3, r 3 = 2.

It follows from Theorem 14, Chapter VII, that the surface has a
line of centers. We could therefore begin by translating axes to
one of the centers as origin; but the reduction of the equation is
accomplished more rapidly if we follow the method used in Case
II. Rotation of axes to principal directions leads again to equa-
tion (3) of Section 99 (see page 217); but since now A = and
since the discriminant is invariant under rotation, we conclude
from the discussion made in Case II (page 218) that a^ = 0.
The equation of the surface reduces therefore to the form
(1) fe/ 2 + fc 3 z' 2 + 2 a 24 y + 2 a 34 Y + a 44 = 0.

* A homogeneous polynomial of degree 3, 4, . . . , n is called a cubic, quartic,
. . . , n-ic form; a form in 2 variables is called a binary form, one in 4, 5,
. . . , n variables is called quaternary, quinary, . . . , w-ary.

222

CLASSIFICATION OF QUADRIC SURFACES

Completing the square on the terms in y f and on the terms in z'
and translating the origin to an arbitrary point on the line

y f = -1 y z' = ~- leads to the equation

where

y"

024,

fcT'

fez" 2 =

JL and

The discriminant of this last equation is
000
fc 2

-

It will clearly be of rank 2, unless a 44 " =t= 0. But, since n = 3 and
the rank of the discriminant matrix is invariant under rotation and
translation of axes (compare Corollary 3 of Theorem 1, Section 93,
page 201 and Theorem 5, Section 94, page 205), we conclude that
a 44 " rjz 0. The locus of the equation is therefore a cylindrical sur-
face ; it will be an hyperbolic cylinder if & 2 and fc 3 are opposite in
sign, a real elliptic cylinder if & 2 , fc 3 , and a 44 " are of like sign, an
imaginary cylinder if & 2 and fc 3 arc of like sign, opposite to that of
a 44 ".

As in Case II, we see that fc 2 & 3 = T 2 , so that fc 2 and & 3 will have
the same sign or opposite signs according as TI > or !T 2 < 0;
in the former case, they will have the sign of T\ & 2 + & 3 . To
determine whether or not, in case fc 2 and fe are of like sign, their
sign is the same as that of a 44 ", we consider the sum Z) 3 of the
three-rowed minors of the discriminant. Since equation (1) was
obtained from the original equation Q = by rotation of axes, we
know from the Corollary of Theorem 4 (see Section 94, page 204)
that Z> 3 ' = Z) 3 . The discriminant of equation (1) is

0000

fc 2 ttu/

h au'

Or /
#24 O 34 O 4 4

and we see that every three-rowed minor of the matrix b, associ-
ated with this discriminant, vanishes. We conclude therefore, by
making use of the theorem stated in Exercise 6, Section 95 (page

THE NON-DEGENERATE SINGULAR CASES 223

205), that JD 3 " = DJ = 3 . Now D 3 " = -a 44 "/c 2 /c 3 ; therefore
au"k 2 k 3 = Z) 3 . This relation enables us to say that if T 2
= k 2 k 3 > 0, 044" will be opposite in sign to Z) 3 . Since moreover
the signs of k 2 and k s are the same as that of Ti, we conclude that
fc 2 , k s , and 044" will have one sign if TiD 3 is negative, but the sign
of 044" will be opposite to that of k 2 and A; 3 if T\D^ is positive. We
have therefore reached the following conclusion.

THEOREM 16. If the ranks of the matrices a 4 and a 3 are equal to 3
and 2 respectively, the locus of the equation Q = will be a real elliptic
cylinder if and only if T 2 > and T { D 3 < 0, an imaginary cylinder if
and only if T 2 > and TiD 3 > 0, an hyperbolic cylinder if and only if
T 2 <0.

Remark 1. We observe that, as in Case II, T 2 must be different
from zero in this case.

Remark 2. The surface will be a circular cylinder if and only
if k 2 = k s .

CASE V. r 4 = 3, r 3 = 1.

The surface has no center in this case. Both T 2 and AM are
equal to zero, but TI is different from zero ; for otherwise it would
follow that r 3 = by means of an argument which is entirely sim-
ilar to the argument in earlier discussions and which is therefore
left to the reader. The discriminating equation is now

= 0.

Its roots are ki = k 2 = and fc 3 = TI 4= 0. In accordance with
Theorem 24, Chapter VII (Section 90, page 195), only one prin-
cipal direction is completely determined, namely, X 3 , /* 3 , v 3 ; the
other two principal directions are subject only to the condition of
perpendicularity to this first direction, and to mutual perpendicu-
larity. We are therefore free to impose one additional condition
on Xi, MI, PI or on X 2 , M2, v 2 .

It is easy to show that in this case X 3 : Ma : vz i : 2 : a#,
i = 1, 2, 3. For, since fc 3 = T\ = an + a 22 + a 33 , we can deter-
mine X 3 , ju 3 , and v s from any two of the three linear equations

~ (022 + 33)X 3 + a^Ma + i3^3 = 0, di 2 X 3 (an + a 33 )/z 3 +
= 0, ai 3 X 3 + 02 3 M3 ~ (an + 022)^3 = 0.

From the first two of these equations we find

X 3 : MS : ?3 = 012023 + 013(011 + 033) : 013012 + 023(022 +
: (0n + 033) (022 + 033) - 0i2 2 .

224 CLASSIFICATION OF QUADRIC SURFACES

But r 3 = 1 ; hence a i3 = #i20 23 #is#22 = 0, so that #12023 = #13022.
Also e*23 #is#i2 #n#23 = 0, so that #13012 ^ #11023.

And 0:33 = #n#22 #i2 2 = 0.

(Consequently we find that X 3 : MS ^ = Oia^i : #23? 7 i : #33^1. And
since TI ^ and r 3 = 1, so that the rows of the matrix 83 are pro-
portional, we reach the conclusion that X 3 : ^ : v$ = a,-i : #,2 : 0*3,
i = 1, 2, 3.

If we rotate axes to the principal directions determined in
accordance with these methods, the equation Q = will be carried
over to the form

/C 3 2' 2 + 2 # 14 V + 2 flju V + 2 34 Y + #44 = 0,

where, as before, 2 #,- 4 ' = # 4 (X,-, /x/, ?;), i = 1, 2, 3. The matrix
a 4 x of this reduced equation is

000 OH'
000 # 24 '
& 3 #34'

#14' #24' #34' #44

In virtue of the hypothesis r 4 = 3 and of Theorem 5 (Section 94,
page 205), the rank of this matrix must be 3; since the matrix is
obviously singular, it must contain, on the basis of the Corollary
of Theorem 6, Chapter II (Section 26, page 44), at least one non-
vanishing three-rowed principal minor. It should be easy to see
that the only three-rowed principal minors of this matrix which do
not vanish identically are those formed from 1st, 3rd, and 4th rows
and columns, and from the 2nd, 3rd, and 4th rows and columns;
also that the values of these are A; 3 #u' 2 and & 3 # 24 /2 . It follows
that at least one of the numbers #i/ and #2/ must be different from
zero. And now we make use of the freedom of choice left in the
determination of either Xi, jui, v\ or X 2 , M2, ^2 to effect a further simpli-
fication of the equation.

If we adjoin the condition # 4 (Xi, MI> ^i) = to the condition
XiX 3 + MiMs + ^1^3 = 0, which is imposed by the condition of per-
pendicularity of the principal directions, the direction Xi, juj, v\ is
determined; and then X 2 , /* 2 , vi will also be determined as the di-
rection perpendicular to the other two. In this manner we secure
the result that #14' = and therefore also the fact that # 24 ' 4= 0.
Since X 3 : MS : v* = #a : #12 : ##, i = 1, 2, 3, we can determine

THE NON-DEGENERATE SINGULAR CASES 225

Xi, MI, v\ from any one of the 3 systems of two equations eacli given
by #14X1.+ a 2 4jui + a 34 j>i = together with one of the equations
iXi + a^Mi + fl^i = 0. Hence Xi, /xi, and v\ are proportional to
the two-rowed determinants formed from one of the three matrices

II a * a '' 2 a& II, i = 1, 2, 3. This will always determine these di-
ll a M a 2 4 a 34 II

rection cosines, unless every two-rowed minor of the matrix b
vanished; but in this case the rank of the matrix b would be 1 and
this is incompatible with the condition r 4 = 3, in view of the ob-
servation (3) made in the proof of Theorem 14, Chapter VII (see
page 178). We conclude therefore that the principal directions
can in this case be so determined that a u ' = and 024' 4 1 0. The
equation of the surface thus takes the form

fc 3 z' 2 + 2 a* V + 2 au'z' + a 44 = 0.

If we complete the square on the terms in z, this equation finally
reduces to

/C 3 Z //2 = -2024V'

/*> /

i // in i i ^44 ^34 " // / i ^34

where x" = x', y" = *+^,- j^r , *' = *+-

The locus of this equation is a parabolic cylinder. We may there-
fore state the following conclusion.

THEOREM 17. If the ranks of the matrices a 4 and a 3 are 3 and 1 re-
spectively, the locus of the equation Q = is a parabolic cylinder.

Remark 1. It should be obvious that we might equally well
have determined the direction cosines \2, ^2, ^2 in such a way that
a 24 ' = and au 3r 0. In this case the final equation would be-
come fez" 2 = 2 ai 4 V, whose locus is also a parabolic cylinder.
Indeed this change merely amounts to an interchange of the X"-
and F"-axes.

Remark 2. It follows from Corollary 3 of Theorem 8 (Section
96, page 209) that in the case just treated the function q(x, y y z)
is the square of a linear homogeneous function of x, y, z with real
or complex coefficients. This observation is frequently useful for
recognizing whether or not the equation Q = represents a
parabolic cylinder.

Example.

To analyze the equation

4 x 2 + 7/ 2 + 4 z 2 - 4 xy - 4 yz + 8 zx + 2 x - 4 y + 3 z + 1 =0,

226

CLASSIFICATION OF QUADRIC SURFACES

we set up the matrices 84

4 -2

-2 1

4 -2

1 -2

4 1

-2 -2

4 I

1 1

and a 3 =

4-24
-2 1 -2

4-24

It is obvious that r$ 1; hence r 4 < 4; and since the three-rowed minor in

1 -2 -2

the lower right-hand corner of 4, namely, the determinant

-2

-2

has the value ^/, r 3 = 3.

In accordance with Theorem 17, we conclude therefore that the locus of the
equation is a parabolic cylinder. This settles the question as to the type of
surface represented by the equation. We proceed now to determine its po-
sition with reference to the given system of coordinates, partly in order to
exemplify and to verify the method used in the discussion of Case V, and
partly in illustration of the remark made in the footnote on page 214.

From the matrix a 3 we conclude furthermore that TI = 9 and we verify that
T 2 = 0. The discriminating equation is therefore fc 3 9 k 2 0, so that we
may take k\ k^ = and /c 3 = 9. To determine X 3 , pi 3 , and * 3 we have the
equations

5 Xs 2 us -f" 4 *s = 0, 2 Xs 8 jus 2 *3 = 0, 4 X 3 2 ^3 5 *3 = 0.

From any two of these three equations we obtain X 3 : Ai 3 : * 3 = 2 : 1 : 2, a
result which was predictable from the discussion in the first part of Case V.
Since ki = k 2 = 0, the conditions for X!, m, v\ (and also those for X 2 , ju 2 , z/ 2 )
reduce to the single equation 2 Xi /ui -f- 2 v\ = 0, which expresses the con-
dition of perpendicularity to the direction X 3 , /* 3 , *> 3 . To this condition we
adjoin the condition q*(\i, AH, *>i) = 2 X t 4 m 4- 3 *i = 0. From these two
conditions we find then that Xi : /*i : PI = 5 : 2 : 6. For X 2 , /u 2 , * 2 we have
now the conditions

2 X 2 - ^2 + 2 v-i = and 5 X 2 - 2 & - 6 *> 2 =

which express the condition of perpendicularity to the two directions already
determined; from them we find that X 2 : ^2 : v 2 = 10 : 22 : 1.

The rotation of axes to principal directions is therefore based on the fol-
lowing table (compare Section 63):

X

Y

Z

X'

5

V65

-2
V65

-G
V65

}n

10

22

1

3V65

3V65

3\/65

Z'

I

-t

2
3

THE DEGENERATE CASES 227

The equations of transformation are therefore

= 5x f 10 y' 2 z' __ 2^ 22 yV _ ^ 6s'

X V65 + 3V65 ~T' y ~ "" V65 3V65 3' * ~~" VS5

, 0' , 2 z'

3V65 T-

Hence*

The equation of the given surface with respect to the rotated axes may there-
fore be successively transformed as follows:

81V65/

27

This is the equation of the parabolic cylinder with respect to a frame of refer-
ence obtained from the original frame by first rotating the axes in accordance
with the table indicated above and then translating the rotated axes to a new
origin whose coordinates with respect to the rotated axes are x' 0, y' =

O f-

^ 81 VE^' z> ~ ~~ 27' ^ e P om * determined by these coordinates is the
vertex of a directrix parabola on the cylinder.

101. Classification of Quadric Surfaces the Degenerate
Cases. There remain to be considered the cases r 4 = 2, r 3 = 2;
r 4 = 2, r 3 = 1, and r 4 = r 3 = 1. These cases have already been
discussed in Section 96 (see Theorem 7, page 208 and Theorem 8,
page 209) and the results stated there completely settle the prob-
lem of classification for this case. It will however be instructive
to derive these results also by means of the methods of reduction
which were used in Sections 99 and 100.
CASE VI. r 4 = r 3 = 2.

The discriminating equation has the form fc 3 Tifc 2 + T z k = 0,
where T 2 4= 0. As in Case IV, rotation of axes to principal direc-
tions leads the equation Q = over into the equation

fey' 2 + fe' 2 + 2 otiV + 2 a 34 'z' + a* = 0.
* Compare Remark 2 following Theorem 17, page 225.

228 CLASSIFICATION OF QUADRIC SURFACES

Completing the square on the terms in y r and z f and translating
axes, we obtain the equation

W 2 + fez" 2 = 044",

whore a 44 " = -~ h ~r -- a ^> an( l the now or igi n ^ an Y point on

KZ A'a

,i i. / #24 / ^34

the line y ' = -- r- , z = -- 7 .
fc 2 fcs

It should now be easy to show that, since r 4 = 2 and since the
rank of the discriminant is invariant under rotation and transla-
tion of axes, a 44 " = 0. The final equation is therefore

W 2 + fez" 2 - 0;

and this equation represents a pair of intersecting planes, real if
T'2 = A; 2 A; 3 is negative, imaginary if T% is positive; this is the result
stated in Theorem 7, page 208.

CASE VII. 7-4 = 2, r 3 = 1.

The discriminating equation has the same form as in Case V
and rotation of axes to principal directions leads again to the
equation

/c 3 z' 2 + 2 a 14 V + 2 awV + 2 a 34 Y + a 44 = 0.

The argument used in the discussion of Case V (see page 224)
shows that, since now r 4 = 2, a^ = a 34 ' = 0. Completing the
square and translating the axes reduces this equation to the form

/c 3 z" 2 + 044" = 0,

where rj 44 " = r/ 44 -- ^- and where the new origin is any point on the

/fy

plane z f = --. The sum /V' of the two-rowed principal minors

A3

of the discriminant of this last equation is clearly equal to fcsa 44 "
and this is the only two-rowed minor of the discriminant which
does not vanish identically. Since r 4 = 2, this can not vanish and
therefore a 44 " 4= 0. Moreover, an argument similar to the one
used in the discussion of Case IV (see page 222) shows that the
sum Z) 2 " for the final reduced equation is the same as the sum D 2
for the original equation Q = 0. (The details of this argument
are left to the reader.) Therefore fc 3 a 44 " = D 2 , so that fc 3 and a 44 "
will be of like or of unlike signs according as D 2 is positive or nega-
tive. We conclude therefore that the locus of the equation Q =

SUMMARY AND GEOMETRIC CHARACTERIZATION 229

is, in this case, a pair of parallel planes which are real or imaginary,
according as D 2 is negative or positive; this result is stated in the
first part of Theorem 8 (see page 209).
CASE VIII. n = 1, r 3 = 1.

In this case rotation of axes to principal directions and transla-
tion of axes, as in Case VII, leads to the final equation

fc 3 z" 2 = 0,

which represents a pair of coincident planes, in accord with the
last part of Theorem 8.

From this discussion we derive some further algebraic theorems,
which complement Theorem 15 (see page 221).

If the rank of the matrix as is 2, the equation q(x, y, z) =
represents a pair of intersecting planes, which are real or imaginary
according as T% is negative or positive. In the latter case the
function q(x, y, z) is reducible to the form fcix 2 + & 2 i/ 2 , in which
fci and fc 2 have like sign, namely, the sign of TI, which is equal to
ki + & 2 ; the function is therefore a definite quadratic ternary
form, positive definite or negative definite according as T\ > or
< 0. If T 2 is negative, the function q(x, y, z) is reducible to
kix 2 + & 2 2/ 2 , and &i and & 2 will be opposite in sign; in this case the
function is an indefinite form.

If the rank of the matrix a 3 is 1, the equation q(x, y, z) =
represents a pair of coincident planes; the function q(x, y, z) is
therefore reducible to the form TVr 2 , which is a definite form, posi-
tive or negative, according as TI > or < 0.

We have therefore the following extension of Theorem 15.

THEOREM 18. The quadratic ternary form q(x, y, s) for which the
rank of the matrix a 3 is 2 is definite if and only if T 2 > 0, positive defi-
nite or negative definite, according as 7\ is positive or negative; if the
rank of the matrix a 3 is 1, q(x, y, s) is a definite form, positive definite
or negative definite according as 7\ is positive or negative.

102. The Classification of Quadric Surfaces Summary and
Geometric Characterization. The results which have been ob-
tained in Sections 99, 100, 101, in as far as they relate to the
classification of quadric surfaces, are summarized in the following
table, which specifies the type of surface represented by the equa-
tion Q(x f y,z) =0 in terms of the invariants of this equation. We
indicate once more the meaning of each of the symbols used in the
table.

230

CLASSIFICATION OF QUADRIC SURFACES

A = K'UW = 1,2,3,4;
^44 = \aij\,i,j = 1, 2, 3;

4 4

A =

u

= 2

r 4 = rank of matrix of A; r 3 = rank of matrix of AM.

, 3

\ r *

4

Singular

2 1

rsN,

Non-singular quadrics

non-degenerate

Degenerate quadrics

quadrics

A>0

A<0

Imagi-
T 2 >0; nary

Ellip,

Imaginary Cone

3

A 44 Ti>0 ellip-
soid

soid

Impossible

Impos-
sible

T 2 >0; Hyper-

Hyper-

^44^ ^0 boloid
or of one

boloid
of two

Real Cone

7^0 sheet

sheets

Imagi-

Hyperbolic

Ellip- -
tic

T*D\

nary
elliptic

Imagi-

cylin-

nary

der

T 2 >0

inter-

secting

2

paraboloid

parabo-
loid

f;&

Ellip-
tic cyl-
inder

planes

Impos-
sible

7 1 2 <0

Hyper-
bolic
cylin-
der

n<o

Inter-
secting
plane*

Imagi-

1

Impossible

Parabolic
cylinder

A>0

nary
parallel
planes

Coin-
cident
planes

Paral-

D 2 <0

lel

planes

SUMMARY AND GEOMETRIC CHARACTERIZATION 231

In Theorem 16, Chapter VII (see Section 87, page 185) we proved
that the quadric surfaces for which r 4 = 4 and r 3 = 3 have a single
proper asymptotic cone; and that the surfaces for which r 4 = 3
and 7*3 = 2 have a pair of asymptotic planes. In either case the
asymptotic quadric of the surface Q(x, y, z) = is given by the
equation Q(x, y t z) Q(a, 0, 7) = 0, where <*, 0, 7 are the coor-
dinates of a center of the surface. Since the equations of a quadric
and of its asymptotic cone differ therefore only in the constant term,
the invariants AM, jT 2 , and TI, which depend on the coefficients of
the second degree terms only, are the same for the two surfaces.
It is clear then from the above table that the asymptotic cone of
the ellipsoid is imaginary, whereas that of the hyperboloids is real ;
also that the asymptotic planes are real for the hyperbolic cylinder
and imaginary for the elliptic cylinder.

If these observations are combined with the results of Sections
84 and 85 we obtain the complete geometric characterization of
the real quadric surfaces indicated in the table on page 232.

Examples.

1. To determine the character of the surface represented by the equation

5 x 2 + 5 y 2 + 8 z 2 + 8 yz + 8 zx - 2 xy + 12 x - 12 y + 6 =
we set up the matrices 84 and as. We find

a 4 =

5 -1
-1 5

4 4

6 -6

4
4
8

G

-6

6

arid 3 =

-1
5
4

The determinants of these matrices are both found to vanish because in each
of them the third row is equal to the sum of the first and second rows. The
third order principal minor of a 4 which is formed from its last three rows and
columns, and the two-rowed principal minor of a s in its upper left-hand
corner are both found to be different from zero. We conclude therefore that
r 4 = 3 and r 8 = 2, and that the surface is a cylinder. Its axis, that is, its line
of centers, is determined by any two of the system of three linear equations
whose augmented matrix furnishes the first three rows of a 4 ; we can take for it
the equations 5 z y + 4 2 + 6 =0 and x + y +2 z = 0.
Moreover,

and

5-14
Z> 3 = -1 54

4 48
Hence, T 2 > and

lsr _ 5-11,154641
8-18, T, - _j s | + | 4 8 + 4 8 |

5-16

546

54-6

-(-

-1 5 -6

+

480

-j-

48

6-66

606

-60 6

= 72;

= -432;

< 0, so that the surface is an elliptic cylinder.

232

CLASSIFICATION OF QUADRIC SURFACES

Centers

Lines on
surface

Asymptotic
cone

Ellipsoid

Single proper
center

No lines

Imaginary,
proper

Hyperboloid of
one sheet

Single proper
center

Two lines through
every point

Real, proper

Hyperboloid of
two sheets

Single proper
center

No lines

Real, proper

Hyperbolic
paraboloid

No center

Two lines through
every point

Elliptic
paraboloid

No center

No lines

Cone

Single vertex

Elliptic cylinder

Line of proper
centers

Two coincident
lines through
every point

Imaginary,
degenerate

Hyperbolic
cylinder

Line of proper
centers

Two coincident
lines through
every point

Real, degenerate

Parabolic
cylinder

No center

Two coincident
lines through
every point

Intersecting
planes

Line of vertices

Parallel planes

Plane of proper
centers

Coincident
planes

Plane of vertices

The discriminating equation is k 3 18 k z + 72 k = 0; the discriminating
numbers are therefore 0, 6, 12. We put ki 0, k z = 6, k 3 = 12. We know
from the general discussion that rotation to principal directions will reduce the
equation to the form 6 y' 2 + 12 z' 2 + 2 a^'y' + 2 a 84 'z' +6=0. To verify
this fact, we proceed to determine the principal directions.

From ki = 0, we find 5 Xi m + 4 v\ = and Xi + 5 p\ + 4 ^ =0,
so that Xi : MI : f ! = 1 : 1 : 1 ; from fc 2 = 6, we find X 2 ^2 + 4 >> 2 =
and 2 X 2 + 2 /i2 + ^2 = 0, so that X 2 : /z 2 : ? 2 = 1 : 1 : 0; from k s = 12, we
find -7 X 3 - MS + 4 V9 = and ~X 3 - 7 ^3 + 4 v z = 0, so that X 3 : v* : t>s =
1:1:2. The equations for the rotation of axes to principal directions are

f' 11' y' T r ii f y f '*' 9 *'

thereforex=4-+4-+4=,2/ = 4--4- + -^, 2 = -4= + ^.

SUMMARY AND GEOMETRIC CHARACTERIZATION 233

If these expressions are substituted for x, y, and z in the original equation
of the surface, it becomes

6 2/' 2 + 12 z' 2 + 12 t/V2 +6=0;

completing the square on the terms in y', we obtain as the final form of the
equation ?/" 2 + 2 z" 2 = 1, where y" = y' + V2 and z" = z f . The locus of
this equation is indeed an elliptic cylinder; its axis is parallel to the X "-axis,
i.e., to the line y" = 0, z" = 0. But the equations of this line may also be
written in the form y' -f ^/2 = 0, z' = 0, or, with respect to the original frame
of reference, in the form x ?/ -f 2 = 0, x + y+2z=Q, which is equivalent
to the form of the equations of the axis of the cylinder given at the end of the
first paragraph. The directrix of the cylinder is the ellipse y" 2 -f 2 z" 2 = 1,
x" = 0; the equations of this ellipse with respect to the original axes are
5 z 2 + 5 y 2 + 8 z 2 + 8 yz + 8 zx - 2 xy + 12 x - 12 y + 6 = 0, x+y-z=Q.
2. We proceed in a similar manner with the equation

2 x 2 + 20 ?/ 2 + 18 2 2 - 12 T/Z + 12 xy + 22 x + 6 y - 2 z + 2 = 0.

We find a 4 =

2 6 11

6 20 -6 3

-6 18 -1

11 3-1 2

and 213 =

6

20 -6

-6 18

r 3 = 2, r 4 = 4; A = 33, 124 < 0. Therefore the surface represented by
the given equation is an elliptic paraboloid.

Furthermore, T\ 40 and T 2 364 and the discriminating equation is
/c 3 - 40 k 2 + 364 k = 0; its roots are 0, 14, and 26. From the discussion of

Case II (see Section 99, page 218), we know moreover that an' = 4 / ^

V * 2

= \/91. The equation of the surface is therefore reducible to the form 14 y" 2
+ 26 z" 2 = 2 V9I x".

This completes the determination of the type of surface represented by
the equation and also of the numerical data necessary to fix its individuality.
If we wish to determine its position with respect to the original frame of
reference, we have to find the principal directions and also the new origin to
which the axes have been translated.

From ki = 0, we find Xi : MI : "i = 9 : -3 : -1; from k 2 = 14, follows
\2 : M2 "2 = 1 2 : 3, and from 3 = 26, we derive X 3 : MS ^3 = 1 : 4 : 3.
If we base the rotation of axes to principal directions on the table

X

Y

Z

X'

9
V91

-3
V9l

-1
Vol

Y'

1

2

Vl4

3

vT3

Vl4

Z'

1
V26

4

V26

-3

V26

234 CLASSIFICATION OF QUADRIC SURFACES

the equation is transformed into 14 y'* + 26 2'* + 2 yUl x' + 2 VI? y' -f 2 V26 2'

+2 = 0: translation of axes to the point x' = 0, y' = 7^* P' = :=.

__ V14 V26

leads to the final equation 14 ?/" 2 + 26 2" 2 = -2V91 z". If the direction
cosines of the X'-axis are changed in sign, that is, if its direction is reversed,
the right-hand side of the final equation of the surface would be 2 V^l x".
The surface extends indefinitely on the negative side of the plane x" = 0,
that is, in terms of the original system of coordinates, on that side of the plane

Oaj S?/ 2 = which is determined by the direction cosines X = -7=,

VQl
g j j j

fji = T=i, v = -7=.; it has the point x' =0, ?/' = -7=., 2' = =, that
V91 VST Vl4 V26

10 27 9

is, the point x = ~qT>2/ =: ~~Q7 2:= ~~qT m common with this plane.

103. Exercises.

Determine the type of quadric surface which is represented by each of the
following equations and set up the reduced form of these equations; dis-
cuss their position in space in those cases in which the numerical work in-
volved does not become too laborious:

1. x 2 -f 4 y 2 + 9 2 2 + 4 xy -f 6 xz + 12 yz - x + 2 y + 5 z = 0.

2. x 2 -I- z 2 -f 2 xy -f 2 a* - 2 7/z - 2 x + 4 ?/ - 4 = 0.

3. x 2 4- 2/ 2 -I- 2 2 + 2/2 + zx + a^ -f 2 x + 2 T/ + 2 2 -f- 2 = 0.

4. 4 z 2 -f t/ 2 + 2 2 - 4 xy + 6 z + 8 = 0.

6. x* + y* + z* + yz -f zx + xy + x -f- y + z + 1 = 0.

6. z 2 4- 2/ 2 4- 2 z 2 - 4 X2 + 2 a;?/ + 1 = 0.

7. 2 2 - xy -h a; = 0.

8. 5z 2 -y 2 +z 2 + 6zz-Mo;?/ + 2z4-22/ + 23 = 0.

9. x 2 + y* + z* xy yz - y = 0.

10. 5x 2 + 13 2/ 2 -1-23 * 2 -f 36r/2 -f 22 a? -f 16^?/ + 10s + 167/4-222+5 = 0.

11. s 2 + 5 ?y 2 + 9 2 2 -f 4 x?/ + 6 yz - z - 3 x = 0.

12. 4 2/ 2 + 4 2 2 + 4 2/2 - 2 re - 14 ?/ - 22 z + 33 - 0.

13. 22/ 2 4-42z-f2 :r -42/ + 62-h5 = 0.

14. x 2 + 2/ 2 + 2^ - 6 xy -f 2 zx - 6 7/2 - 6 x - 2 1/ - 6 z + 1 = 0.
16. 3 x 2 -f 3 y 2 + 3 z 2 + 2 ?/2 + 2 2z + 2 x?/ -f 1 = 0.

16. x 2 + z?/ -f 2/2 + zx - 3 z - 2 ?y - 2 - 3 = 0.

17. x 2 + 3 2/2 - 2 2 -f 2 x = 0.

18. (x - 2 y + 2) 2 -f- 4 x - 8 y + 4 2 + 3 = 0.

19. 36z 2 H-47/ 2 + z 2 -4i/2- 12ac + 24xy + 4 + 16y -262 -3 = 0.

20. z 2 -f 4 ?/ 2 + 2 2 - 4 1/2 + 2 20; - 4 xy - 2 * + 4 y - 2 2 - 3 = 0.

21. 2x 2 - 22/2 -f 2zz-2;n/-z -2y + 82 - 2 0.

22. Determine the condition which must be satisfied by the discriminating
numbers of a quadric surface in order that it may be a surface of revolution;
also the conditions which will insure that the surface is a sphere.

23. Set up the conditions which the invariants of the surface Q(x t y, z) =0
must satisfy in order that the surface may be a sphere.

CHAPTER IX

QUADRIC SURFACES, SPECIAL PROPERTIES
AND METHODS

In this chapter we shall discuss some properties and methods
which are concerned with one or another of the classes of quadric
surfaces, with which we have become acquainted.

104. The Reguli on the Hyperboloid of One Sheet. We have
seen in Sections 84 and 102 that the hyperboloid of one sheet and
the hyperbolic paraboloid are the only real quadric surfaces
through every point of which there pass two real lines which lie
entirely on the surface. The general method developed in Section
84, by means of which the existence of these lines was demonstrated,
is not very convenient for actually determining the rulings through
a particular point on a given surface; we shall therefore in this
section take up a special method, for the hyperboloid of one sheet,
for the solution of this problem. And, having shown in the pre-
ceding chapter that every quadric surface can be represented,
with suitable choice of the frame of reference, by an equation char-
acteristic of the type to which the surface belongs, we shall hence-
forth make use of these standard forms of the equations of the
quadric surfaces.

The standard form of the equation of the hyperboloid of one
sheet is

It may be written in the form

It should be easy to see from this that the line which is determined
by the pair of equations

-O -

235

236 QUADRIC SURFACES

in which p\ and p 2 are arbitrary real numbers which do not both
vanish, lies entirely on the surface. Since the value of the ratio
PI : pi may therefore be chosen arbitrarily, we have here a single
infinitude of lines, or a one-parameter family of lines on the surface.
This family of lines is of the type known in Projective Geometry as
a regulus.* We shall use this term for convenience of reference
without entering further into its definition; and we shall refer to
the regulus whose lines are determined by equations (1) as the
p-regulus of the surface.

It is moreover clear that we can obtain a second regulus of the
surface in the form

we shall call this the q-regulus, it being again understood that q\
and #2 are arbitrary real numbers which do not both vanish.

A particular line I of the p-regulus is known as soon as the ratio
Pi : p 2 is given, and we shall designate this line by the symbol
l(pi> 2); similarly the symbol m(qi 9 g 2 ) will be used to designate
a line of the g-regulus. To determine the particular line of each
regulus which passes through a given point A (a, 0, 7) on the sur-
face, we substitute the coordinates of this point in one of the
equations (1) to determine pi : pz, and also in one of the equations
(2) to determine qi : q 2 .

Example,

To determine the lines through the point A (4, 2, 3) on the surface r-

-f X ~ <7 = *' we wr ^ e ^ ne equations of the reguli; for the particular sur-
face under consideration, these are:

and

Substituting the coordinates of A for ac, y, z in these equations gives p\ X
= p* X 2, p2 X 2 = pi X 0; qi X = q 2 X 0, ^ X 2 = qi X 2. Therefore
Pz and p\ is arbitrary and q\ = qi.

* For a definition and treatment of the regulus, see, for example, Veblen
and Young, Projective Geometry, Vol. 1, pages 298-304.

REGULI ON THE HYPERBOLOID OF ONE SHEET 237
Hence the required lines are given by the following pairs of equations:

105. Reguli on the Hyperboloid of One Sheet, continued. A
number of questions concerning the reguli on the hyperboloid of
one sheet, which suggest themselves naturally, will now be
considered:

(1) Will the determination of p\ : p 2 and of 51 : g 2 always be
possible in one and only one way?

(2) Do we get two distinct lines through every point on the
surface?

(3) Do two lines of one regulus ever lie in the same plane?

(4) Will every line of the p-regulus be coplanar with every
line of the g-regulus?

(5) What is the relative position of the plane determined by
the two rulings which pass through a point A on the
surface, and the tangent plane to the surface at PI

These questions, except the first, fall within the scope of Chapter
IV and can be answered by the methods developed there. We
shall take up these questions in some detail, because they furnish
an opportunity to illustrate the application of these methods.

(1) Will the determination of p\ : p% and of q\ : q 2 always be pos-
sible uniquely? From an equation of the form ap\ bp 2 the ratio
pi : pz is always uniquely determinable, unless a = 6 = 0. Hence
if the point A (a, /?, 7) lies on the hyperboloid of one sheet repre-
sented by the equation ^ ~ + ~i = 1 an d tf r > 1 >

a o c (to c

- + - , 1 + - are not all zero, then at least one and at most two de-
a b' c

terminations of each of the ratios pi : p% and qi : q% are possible.
But if these four expressions all vanished, it would follow that
a = & = 7 = 0, which is not a point on the surface. And if the
two equations (1) of Section 104 gave rise to two different values

of T>I : p 2 > it would follow that 1 - : r : fc- + ?:l+~>

^ ^ c a b a b c

cf B* *y 2

and hence that 2 ro^l -^so that the point A (a, 0, 7)

238

QUADRIC SURFACES

could not be on the surface. The same conclusion holds for the
equations (2). The question is therefore to be answered affirma-
tively; consequently the method of the preceding section is
always effective to determine the lines whose existence was
proved in Section 84.

(2) Will the two lines through A(a, )8, 7) determined by the method
of Section 104 olways be distinct?

If these lines are not distinct, then the four planes represented
by the equations (1) and (2) of Section 104 for the particular
values of pi : p 2 and qi : # 2 determined as in (1), must have a line
in common. According to Theorem 23, Chapter IV (Section 54,
page 101) this will happen if and only if the ranks of the coefficient
matrix and the augmented matrix of their equations are both
equal to 2.

Pi _pi P2

The coefficient matrix of these equations is

Pt
b

1 2? ^

\ b

its rank is not affected by elementary transformations (compare
Definition XIV and the Corollary of Theorem 14, Chapter I,
Section 10, pages 18, 19), so that we may multiply its 1st, 2nd,
and 3rd columns by a, fe, and c respectively, and add the 1st

Pi p 2

column to the 2nd. This leads to the matrix * ^ l

qi q 2

02 2 q 2 q\

It is found that the values of the third order determinants ob-
tained from this matrix by omitting the 4th, 3rd, 2nd, and 1st
rows respectively are equal to 2 p 2 (piq 2 + P*qi), 2 pi(p\q 2 + p^qi),
2q 2 (piq 2 + P2<?i) and 2qi(piq 2 + p 2 qi). Therefore, since, in
accordance with (1), pi, p 2j q\, and q 2 never vanish simultaneously,
the rank of the coefficient matrix can be 2 only if piq 2 + p 2 qi = 0.
Let us now consider the matrix formed from the 1st, 3rd, and
4th columns of the augmented matrix; after applying to it similar

REGULI ON THE HYPERBOLOID OF ONE SHEET 239

transformations as to the coefficient matrix we have to consider
Pi 2 p 2

the matrix

P2

pi

the values of its third order determi-

nants are found to be
(pzqz piQi) and 2#i(p 2 g 2 Piqi)- Hence if the rank of the
augmented matrix is also 2, we must have p 2 # 2 Piqi = as well
as PI& + P2<7i = 0. If we look upon these equations as linear
homogeneous equations in q z and q\, we find, by using Theorem 2,
Chapter II (Section 22, page 38), that either q\ = g 2 = or
Pi 2 + P2 2 = 0, that is pi = p 2 = 0. Neither of these cases can
arise, in virtue of the discussion in (1). The answer to our second
question is therefore also affirmative.

(3) Do two lines of one regulus ever lie in one plane?

Let us consider the lines I(p i9 p 2 ) and l'(pi, p 2 ') of the p-regulus.
They are given by the two pairs of equations

and

where pi : p 2 = Pi Pa'.

The value of the determinant of the augmented matrix of these
equations is

J)2

abc

Pi -Pi

P2 P2

ft' -ft'

ft' ft'

-Pi -Pi
ft' -ft'
-Pi' -Pi'

Pi

p 2 2 p 2
Pi'

P2

-Pi -5

The evaluation of this determinant is accomplished most con-
veniently by means of the Laplace development of the 1st and 3rd
columns (compare Theorem 15, Chapter I, Section 12, page 23);

Pi ;

we find then that its value is

Pi

X

2p 2 ' -2 Pi'

/ Pzpi) 2 t which is different from zero, in virtue of the
hypothesis p\ : p 2 4= Pi' p*- It follows from this, on account of
Theorem 22, Chapter IV, (Section 54, page 101), that the two lines

240 QUADRIC SURFACES

of the p-regulus can not have a point in common. Can they be
parallel?

There should be no difficulty in showing that the direction cosines
of the lines l(pi, p 2 ) and V(pi, 7)2') are given by the proportions

X : /i : v = a(pi 2 - p 2 2 ) : b(pi 2 + p 2 2 ) : 2 cpip 2 ,
and

X' : M ' : i/ = a(p x ' 2 - p 2 ' 2 ) : &(/>i' 2 + p 2 ' 2 ) : 2 cpi V-

Since pi, p 2 , p/ and p 2 ' are all real, both pi 2 + P2 2 and p/ 2 + p 2 ' 2
must be positive. Therefore, if the lines are to be parallel, there
must exist a positive factor of proportionality p 2 , such that

Pi 2 -p 2 2 = p 2 (pi' 2 -p 2 ' 2 ), Pi 2 +p 2 2 = p 2 (pi' 2 +p2 /2 ) and

From the first two of these equations we conclude, by addition
and subtraction, that pr = p 2 p/ 2 and p 2 2 = p 2 p 2 ' 2 , and hence
that pi = zb ppi and p 2 = dbpp/. If opposite signs were used in
these two equations, it would follow that pip 2 = P 2 p/p 2 '> which
is in conflict with the third of the above equations. Therefore
the lines can have the same direction cosines only if pi : p 2 =
Pi - pit fchat i s > ^ they coincide.

Our discussion has therefore brought us to the conclusion that
no two lines of the p-regulus can lie in the same plane.

The reader is urged to carry through the same argument for
the #-regulus. When he has done this, he will have completed
the proof that the answer to the third question is in the negative.

(4) Will every line of the p-regulus be coplanar with every line
of the q-regulus?

The discussion of this question will be based on Theorem 25,
Chapter IV (Section 54, page 104). If Z(pi, p 2 ) is an arbitrary
line of the p-regulus and m(gi, (? 2 ) an arbitrary line of the g-regulus,
these lines will be coplanar unless the determinant

X Xi a. i

M Ml 0-01
v 1/1 7 71

is different from zero; here X, /*, v and Xi, /n, vi are the direction
cosines of the lines I and m respectively, and (a, 0, 7) and (],
0i, 71) are arbitrary points on these lines. We have already
seen that X : /* : v = a(pi 2 p 2 2 ) : 6(pi 2 + p 2 2 ) : 2 cpip 2 (see (3)

REGULI ON THE HYPERBOLOID OF ONE SHEET 241

above) ; in similar manner we find that Xi : /*i : v\ = a(qi 2
<?2 2 ) : Ktfi 2 + <? 2 2 ) : 2 c^ 2 . It remains therefore to investigate
whether or not the determinant

(pi 2 ~ p 2 2 ) a(?i 2 - ? 2 2 ) a- o

has a value which is different from zero, when for a, ft 7 and i, ft, 71
are substituted sets of numbers which satisfy the pairs of equations

<Ml + )> ft/- + r ) = q\( 1 -- ) respectively. It turns out that

this determinant vanishes for every such choice of a, ft 7 and
i, ft, 7iJ the details of the proof of this statement are given in
Appendix, VII (page 299). Hence we conclude that every line of
the p-regulus is coplanar with every line of the g-regulus.

(5) What is the relative position of the plane determined by the two
rulings which pass through a point on the surface and the tangent
plane to the surface at this point?

The equation of the tangent plane to the surface at a point
A (a, j8, 7) on the surface is

OiX Vt 72

(compare Theorem 4, Chapter VII, Section 81, page 162). This
plane and the line of the p-regulus through A have at least the
point A in common; the line will therefore lie in the plane or meet
it in a single point, according as

- P2 2 ) X - b(pi 2 + ?2 2 ) X 2 + 2 cp lP t X

is equal to or different from zero (compare Theorem 21, Chapter
IV, Section 52, page 99). This expression is equal to pi 2 f - rj

7>2 2 ( ~ + v ) + 2 pip 2 - ; and this reduces, by virtue of the equa-
\ct o / c

tions of the lines of the p-regulus, to p\pji 1 - J p*pi( 1 + -J

242 QUADRIC SURFACES

+ 2pip 2 - = 0. Consequently, the line of the p-regulus lies in
c

the tangent plane; the reader should have no difficulty in proving
that the line of the g-regulus which passes through a given point
on the surface also lies in the tangent plane to the surface at that
point. These conclusions could also have been reached by a
geometric discussion, namely, by observing that any line which
connects A with a point on the plane determined by the lines
/(pi, pz) and m(qi, #2) meets the surface in two points which co-
incide at A, and is therefore tangent to the surface at A (compare
Definitions V and VI, Chapter VI, Section 77, pages 154, 155).
We summarize the results obtained in this section in a theorem.

THEOREM 1. A hyperboloid of one sheet contains two one-parameter
families of lines. Through every point of the surface passes one and
only one line of each family. No two lines of the same family are
coplanar; every line of one family is coplanar with every line of
the other family. The plane determined by the two lines which pass
through an arbitrary point on the surface is coincident with the
tangent plane to the surface at this point.

106. The Reguli on the Hyperbolic Paraboloid. The standard
form to which the equation of an hyperbolic paraboloid can be
reduced is

T 2 7/ 2

__ y _ 9 ~~

5 T-T & nz.
a 2 6 2

If we write this equation in the form

_AA + f) = 2n .s,

a b/ \a b/ '

it becomes clear that the surface contains the two one-parameter
families of lines represented by the following pairs of equations

Pl
bers, not both zero;

g \a ~~ b) = 2 nq *' ^ 2 (a + 1) = qiZf q
bers, not both zero.

With reference to these reguli on the hyperbolic paraboloid, the
same questions arise as were discussed for the hyperboloid of one

STRAIGHT LINES ON SINGULAR QUADRICS 243

sheet. The discussion of these questions is left to the reader
(see Section 108).

107. the Straight Lines on the Singular, Non-degenerate
Quadrics. It was proved in Theorem 12, Chapter VII (Section
84, page 175), that through every point on a non-degenerate singu-
lar quadric which is not a vertex of the surface, there pass two
coincident lines which lie entirely on the surface. This class of
surfaces includes the proper cone and the cylinders.

If the equation of the proper cone

is written in the form

a c a c

we recognize that the one-parameter families of lines, represented
by the pairs of linear equations

(x z\ y

---)= P2 -f,

j

and

z

in which pi, p^ qi, and q<* are real numbers and neither pi and p 2 ,
nor qi nor (j 2 vanishes simultaneously, lie entirely on the surface.
It should be clear however that the lines l(p\> p 2 ) and m(qi 9 # 2 )
are identical when qi = pi and g 2 = -p* Consequently, these
two families of lines are identical; the proper cone contains there-
fore two coincident reguli.

A similar argument shows that the elliptic cylinder, the hyper-
bolic cylinder, and the parabolic cylinder, also each contain two
coincident reguli.

108. Exercises.

1. Determine the equations of the straight lines on the surface -r ~

4 y

- 2z, which pass (a) through the point A(10, 9, 8); (b) through the point
B(-10, 9, 8); (c) through the point C(10, -9, 8); (rf) through the point
ZK-10, -9,8).

x 2 t/ 2

2. Determine the equations of the rulings of the surface -j- 4- fj- z 2 * 1

244 QUADRIC SURFACES

which pass (a) through the point A (4, 3, 2); (6) thro'.gh the point
(-4, 3, 2); (c) through the point C(4, -3, -2); (d) through the point
D(-4, -3, -2).

3. Determine the equations of the straight lines on the cone x 2 y 2 -f z a
= 0, which pass (a) through the point A (3, 5, 4); (6) through the point
B(-3, 5, 4); (c) through the point C(3, 5, 4); (d) through the point
/>(-3,5, -4).

4. Show that the elliptic cylinder, the hyperbolic cylinder and the parabolic
cylinder each contain two coincident one-parameter families of lines.

6. Prove that any two lines which belong to one regulus of the hyperbolic
paraboloid are skew.

6. Prove that every line of one regulus of the hyperbolic paraboloid is
coplanar with every line of the other regulus of that surface.

7. Prove that the plane tangent to the hyperbolic paraboloid at an arbitrary
point contains the two rulings of the surface which pass through that point.

8. Determine the cosine of the angles made by the two lines which pass
through an arbitrary point of the hyperboloid of one sheet and determine
the condition under which these two lines are perpendicular.

9. Discuss the corresponding question for the hyperbolic paraboloid.

10. Prove that to every point A (a, 0, 7) on the hyperboloid of one sheet
there corresponds a point A' on the surface such that the line of the p-regulus
through A is parallel to the line of the ^-regulus through A', and vice versa.

11. Determine the locus of all points on the hyperboloid of one sheet for
which the angle between the rulings of the surface which pass through them
is constant.

12. Show that it is possible to set up a correspondence between each of the
reguli of the hyperboloid of one sheet on the one hand, and the regulus of its
asymptotic cone on the other, such that corresponding lines are parallel.

109. Circles on Quadric Surfaces, the General Method. In

Section 66 we discussed the problem of determining the curve of
intersection of a plane and a surface. In accordance with Corol-
lary 2 of Theorem 8, Chapter V (Section 66, page 128), a plane
section of a quadric surface is a curve of degree not higher than the
second. It is of interest to inquire whether plane sections of
quadric surfaces can be circles. In answer to this question we
shall prove in the first place the following theorem.

THEOREM 2. The sections of a quadric surface made by two parallel
planes are either both circles or else neither is a circle.

Proof. The method developed in Section 66 for determining
the character of a plane section of a surface consisted in rotating
the axes in such a way as to make one of the new coordinate planes
parallel to the plane of the section. The rotation required by this

CIRCLES ON QUADRIC SURFACES 245

method is the same for the sections of a surface by each of two
parallel planes \x + py + vz p\ = and \x + ny + vz p 2 = 0.
Let us suppose that the rotation of axes is made in such manner
as to make X, /*, v the direction cosines of the new Z-axis; then
the equations of the two given planes will be, with reference to the
new axes, z' = p\ and z f = p 2 . Consequently the plane equations
of the two sections will be obtained when, in the new equation of
the quadric surface, z' is replaced by pi and p 2 . If the transformed
equation of the quadric surface is Q(x, y, z) = 0, then the plane
equations of the sections will be

2 a i2 xy + a 22 ?/ 2 + 2 (ai 3 pi + a^x + 2
33 pi 2 + 2 a 34 pi + 44 = 0,
2 a l2 xy + a 22 y 2 + 2 (a 13 p 2 + a u )z + 2
2 a 34 p 2 + a 44 = 0.

Since the condition that the plane locus of either of these equa-
tions shall represent a circle is that a u = #22 and a i2 = 0, the
theorem has been established.

Remark. To determine the circular sections of a quadric sur-
face, it suffices to consider the planes through some fixed point.

We could now proceed for the further discussion of our problem,
to determine the plane equation of the section of the quadric by
an arbitrary plane through the origin and then impose the con-
ditions which insure that the locus of this equation is a circle.
But we have had opportunity to observe before that the most direct
rhethod is not always the most convenient and that the end we are
seeking to accomplish is frequently reached in a more elegant and
more instructive way by a more sophisticated procedure. This
will be our program in the present case.

We begin by recalling from Elementary Plane Geometry that,
if lines Zi, Z 2 , . . . are drawn through a point P in the plane of a
circle, meeting the circle in pairs of points AI, BI, A 2j J9 2 , . . . ,
then the products PAi PBi, PA 2 PB 2 , . . . are all equal to each
other. It is true conversely, that if A\B\ 9 A 2 B 2 , ... are chords
of a conic section which pass through a fixed point P and the
products PAi PBi, PA 2 PB 2) ... are all equal for any fixed
position of P in the plane of the conic section, then this conic is
a circle.*

* A proof of this converse theorem will be found in Appendix, VIII, page 300.

246 QUADRIC SURFACES

Suppose now that the plane through the origin, whose equation is
(1) ax + by + cz =

cuts the quadric surface Q in a circle and that the lines Zi, Z 2 , h . . .
through the origin which lie in this plane cut the surface in the
pairs of points A\,B\] A^ 5 2 ; . . . . If the direction cosines of
an arbitrary one of these lines is designated by X, M, v, these di-
rection cosines satisfy the condition aX + fyu + cv = (compare
Theorem 21, Chapter IV, Section 52, page 99). And the dis-
tances from to the points in which the line meets the quadric
are given, in magnitude and direction, by the roots of the equation

q(\, n, v)s* + 2 (<z 14 X + a 24 ju + a z ^)s + a 44 = 0,

(compare Theorem 1, Chapter VII, Section 80, page 160 and re-
member that in this case a = /3 = 7 = 0); and the product of

these roots is equal to ^ 44 r It follows that if the section is a
ff(X, /*, ")

circle, then /x 44 r must have the same value for all those ad-

9(*, M, v)

missible values of X, /x, v for which aX + hp + cv = 0. This
means that there must exist a number k, which is independent of
X, M, and v, such that q(\ n, v) = fc, whenever aX + ?>M + cv = 0,
or again that the quadratic equation <z(X, /z, v) k == is satisfied
whenever the linear equation a\ + bfj, + cv = is satisfied.
From this we conclude, first that the quadratic function q(\, p. y v)k
must be factorable in two linear functions of X, /i, v with real or
complex coefficients (compare the argument made in the proof
of Corollary 2 of Theorem 12, Chapter VII, Section 84, page 175);
and secondly, that aX + bp + cv must be one of the factors.
Furthermore, the function <ft(X, ju, v) ss j(X, ju, v) k = g(X, /x, v)
fc(X 2 + M 2 + J' 2 ) is factorable, according to Corollary 3 of
Theorem 8, Chapter VIII (Section 96, page 209) if and only if
the rank of its discriminant matrix is less than 3, that is, if and
only if

CL\\ k dm #18

diz #22 "k flsa
#13 023 flaa k

0.

CIRCLES ON QUADRIC SURFACES 247

But this equation is the discriminating equation A(k) = of the
surface <3 (compare Sections 88 and 89). If, as before, we desig-
nate its roots by &i, & 2 , and & 3 , we conclude that if the plane
ax + by + cz = cuts the quadric Q in a circle, then ax + by + cz
must be a factor of one of the quadratic functions q(x, y, z)
MX* + 2/ 2 + z 2 ), i= 1,2 or 3.

Conversely, if ax + by + cz is a factor of g(z, y, z)
&*(# 2 + y 2 + z 2 )> for i = 1, 2, or 3, then aX + bp + cv is a factor
of q(\ /i, v) kt(\ 2 + M 2 + " 2 ), that is, of q(\ p, v) fc,-; conse-
quently g(X, /i, v) = & 4 * for all admissible values of X, M, v for which
aX + bfj, + cv = 0. If we take now an arbitrary point P(a, P, 7)
in the plane ax + by + cz = 0, the lines in the plane through P
will cut the quadric Q in pairs of points A, /? whose distances
from P are the roots of the equation Los 2 + 2 Li + L 2 = 0,
where L = q(\, M, *>) and L 2 = Q(a, ft 7) (see Theorem 1, Chap-
ter VII, Section 80, page 160); hence the product PA PB ==

Q i?' ^ y ) = Q(a> / >7) for all admissible values of X, M , v for
g(X, M, v) fc,-

which a\ + by -\- cv = 0. But this expresses the fact that the
product of these distances is constant for all lines through
P, no matter what point P is chosen, and therefore (see footnote
on page 245) that the plane cuts the quadric in a circle. We have
therefore established the following theorem.

THEOREM 3. The necessary and sufficient condition that the plane
through the origin represented by the equation ax 4- by -f cz =
shall be a plane of circular section of the quadric surface Q(x, y, s) =0
is that ax -f- by + cz must be a factor of the homogeneous quadratic
function q(x, y, *) - ki(x 2 -f y 2 + * 2 ) for i = 1, 2 or 3, where fei, k 29 and
fcs are the discriminating numbers of the surface.

The discriminating numbers are all real (see Theorem 20,
Chapter VII, Section 89, page 192), but they need not all be
distinct. Moreover, even though fci, fe, and fc 3 are real, it is not
certain whether the linear factors of q(x, y, z) k(x 2 + y* + z 2 )
are real for i = 1, 2, 3, that is, whether the planes of circular
section are real. In view of Theorems 2 and 3 we can state there-
fore that through every point in space there pass six planes which
cut an arbitrary quadric surface in circles; of these planes some
may be coincident and some may be imaginary. We proceed
now to a further study of the different possibilities.

248

QUADRIC SURFACES

We suppose first that the equation A(k) = has no multiple
roots; then the rank of the matrix

*(*,-) =

a 23

is equal to 2, for t = 1, 2, 3 (see Theorem 19, Chapter VII, Section
89, page 190) and hence the invariant T z (ki) of the quadric surface
represented by the equation q(x, y t z) ki(x 2 + y 2 + z 2 ) =0 is
different from zero. It follows therefore from the table in Section
102 (page 230) that this surface consists of two distinct real planes
or two imaginary planes according as T 2 (ki) < or T 2 (ki) > 0.
But

|

- AV) (ass
+ (an
2 T 7 ^ +

Cl\2 #22 "-" '

- #23 2 + (a n - ki) (a 33 ki) - #i 3 2

a i2 2

3 k? = -

K

Now A'(ki) represents the slope of the curve y = A(k) at the
points where it crosses the K-axis; since we are supposing that

the equation A(k) =0 has no
multiple roots, and since the co-
efficient of fc 3 in A(k) is 1, the
curve y = A(k) has the general
character indicated in Fig. 33.
From this it should be evident
that of the three numbers A'(ki),
i = 1, 2, 3, two are negative,
namely, those which correspond
to the least and to the greatest
of the numbers k\, k 2 , /c 3 , whereas
one is positive, namely, the one
which corresponds to the middle
discriminating number; conse-

* Notice, that in A(k) the coefficient of fc 3 is 1. This interesting formula
can be obtained directly from the proof of Theorem 19, Chapter VII (see
Section 89, page 190); the alternate proof given in the text does not make use
of the formula for the derivative of a determinant, but is not well suited for
extension to derivatives of higher order.

FIG. 33

CIRCLES ON QUADR1C SURFACES

249

qucntly, of the numbers T 2 (/Ci)> two are positive and one is
negative, and if the notation be so chosen that ki < k 2 < k 3 ,
T 2 (ki) > 0, T 2 (k 2 ) < and T 2 (fa) < 0. Therefore the equation
q(x, y, z) k 2 (x 2 + # 2 + 2 2 ) =0 represents a pair of real planes
of circular section through the origin, but the equations q(x, y, z)
- ki(x 2 + y 2 + z 2 ) = and q(x, y, z) - fa(x 2 + y 2 + z' 2 ) = rep-
resent pairs of imaginary planes.

If the equation A (k) = has a pair of double roots, let us say

fci = fc 2 , then the rank of the matrix a 3 (fci) is 1 (see Theorem 19,

Chapter VII) and therefore the equation q(x, y, z) ki(x 2 +y 2 +z 2 )

= represents a pair of coincident planes (see Corollary 3 of

Y

K

K

FIG. 34a

FIG. 346

Theorem 8, Section 96, page 209). From the discussion in the
preceding paragraph we conclude that in this case the graph of the
function A(k) has the character indicated in Figs. 34a and 34b,
from which it should be clear that A' (fa) < 0, therefore that
T 2 (fa) > and hence that the planes represented by the equation
q(x, y } z) fa(x 2 + y 2 + z 2 ) =0 are imaginary. In this case
there are therefore four coincident planes through the origin which
cut the surface in a circle; each of these planes is represented
by the equation [q(x, y, z) - ki(x* + y 2 + z 2 )]* = 0.

Finally, if fci is a triple root of the equation A(k) = 0, the rank
of the matrix a 8 (fci) is (see Theorem 19, Chapter VII) and there-
fore the function q(x, y, z) ki(x 2 + y 2 + z 2 ) vanishes identically,
so that every function of the form ax + by + cz is a factor of it.
In this case every plane through the origin is a plane of circular

250 QUADRIC SURFACES

section. In view of the remark on page 215 this constitutes a
proof of the well-known fact that the section of a sphere by an
arbitrary plane is a circle.

We should recall moreover that if the discriminating equation
has a pair of equal roots which are not zero, the quadric is a surface
of revolution, while if it has a pair of zero roots, the quadric is a
parabolic cylinder or a pair of parallel or coincident planes.
In view of these facts we can state the following conclusion :

THEOREM 4. The quadrics which are surfaces of revolution but not
spheres, the parabolic cylinder, and the pair of parallel or coincident
planes possess through every point of space A(<* 9 /?, 7) four coincident
planes of circular section; in these cases there exists a double root, fci, of
the discriminating equation, the function q(x 9 y, ) - k^x 2 -f y 2 -f * 2 )
is a perfect square and the planes of circular section through A are
given by the equation [g(x - , y - p, s ?)] 4 =0, where [g(x 9 y, *)] 2 =
q(x 9 y, z) - fci(* 2 -f-y 2 + 2 ). All other quadric surfaces possess
through every point of space A(a 9 (3, y) two distinct planes of circu-
lar section; no two of the discriminating numbers k l9 7c 2 , k 3 are
equal to each other and, if ki < k 2 < fc 3 , the function q(x 9 y, z)
kz(x* + y 2 + * 2 ) is factorable into two linear factors with real coeffi-
cients; if we write q(x 9 y, *) - fc 2 (* 2 + y 2 4- * 2 ) = gi(x 9 y, *) g 2 (x 9 y, *),
the planes of circular section through A are given by the equations

gi(* - , y , z - 7) =0 and g*(x - a, y - 0, s - 7) = 0.

Remark 1. The circles of these circular sections may be ordi-
nary circles with a finite center and finite radius, or they may
be "degenerate circles" (compare Appendix, VIII, page 301). It
should be clear that the circular sections of degenerate quadrics
are always degenerate circles. And it should be clear that this
will also be the case when the middle root or the double root of
the discriminating equation is equal to zero (compare also Sec-
tions 110 and 111).

Remark 2. If the notation for the discriminating numbers is
so chosen that in all cases ki g k 2 g & 3 , the planes of circular
section through the origin are always given by the equation
q(x, y, z) - fc 2 (x 2 + y 2 + z 2 ) = 0.

Example.

The locus of the equation

is an hyperboloid of one sheet; for A =

300-3

0-1 3 2

3-1-1

-3 2 -1 -2

= 99,

CIRCLES ON QUADRIC SURFACES 251

300

0-1 3 = -24, T 2 = -14, so that A > and T 2 < 0. More-

03-1

over we find that T\ = 1. The discriminating equation is k 3 k 2 14 k + 24
= 0; its roots are 2, 3, and 4. Therefore in the notation of Remark 2
above, /b 2 = 2, and the planes through the origin which cut the surface in
circles are given by the equation 3 x 2 y 2 z 2 -f 6 yz 2 (x 2 -{- y 2 + z 2 ) =0 f
that is, by x 2 - 3 (y - z) 2 = or by x - \/3 y -f V3 z = and x -f \/3 ?/ -
-s/3 2 = 0. The planes of circular section through an arbitrary point A (a, 0, 7)
are given by the equations (x a) V^(?/ 0) -h \/3(z 7) = and
(x a) -f- V3(y 0) V3(z 7) =0.

110. Circles on Quadric Surfaces, continued. To determine
the planes of circular section for a particular quadric surface, whose
equation is given in numerical form, we can proceed by the general
method developed in the preceding section. The work becomes
very simple if the equation of the surface has first been reduced
to the standard forms of Sections 100-102.

Examples.

X* V* Z 2

1. If in the equation of the ellipsoid i -h p + -5 = 1> a < b < c, the

middle discriminating number of the surface is r~ 2 . Therefore the planes
of circular section through the origin are given by the equation

that is, by

In virtue of the relative magnitude of a, b, and c, which has been presupposed,
the coefficients of z 2 and z 2 in this equation are both positive. Hence the
planes of circular section through the origin are represented by the equations

cxVb* - a 2 - azVc 2 - 6 2 = and czV& 2 a 2 + azVc 2 - 6 2 = 0;

and the planes of circular section through the arbitrary point A (a, 0, 7) are
given by the equations

cVb* - a?(x - a) = aVc 2 - b*(z - 7) and cVb 2 - a 2 (x - a) =
-aVc 2 - b*(z - 7).

2. For the parabolic cylinder y 2 - 4 pz, the discriminating numbers are
0, 0, 1; therefore fc 2 = and the planes of circular section are given by the
equation y* = 0. This equation represents the ZZ-plane counted fourfold.
The planes of circular section through the point A (a, 0, 7) are represented by
the equation (y 0) 4 = 0; therefore they are the planes y = ft counted
fourfold. The intersection of a plane of this family consists of the generating

252 QUADRIC SURFACES

s 2
line y = ft, z = ~- and of the infinitely distant line of the plane y = 0; the

circular section is therefore a degenerate circle (compare Appendix, page 301).
111. Exercises.

1. Determine the planes of circular section through the point A ( 2, 3, 1)
for each of the following surfaces:

2. Prove that the circular sections of a hyperbolic paraboloid are always
degenerate.

3. Prove that the two families of planes of circular section of a central
quadric are not affected when the surface is translated.

4. Determine the planes of circular section through the point A (3, 4, 1)
for each of the following surfaces:

(a) x 2 + 3 y 2 - z 2 = 0, (6) 4 x 2 + 9 if = 1, (c) 4 x 2 - 9 y* = 1.

6. Prove that the circular sections of the hyperbolic cylinder are always
degenerate.

6. Determine the angle between the two planes through the origin which

2 ? ,2 2 2

cut the ellipsoid -5 -f r? + = 1 in circles; and set up the condition under
a* o* c*

which these planes will be perpendicular.

7. Solve the similar problem for the hyperboloid of one sheet and also for
the hyperboloid of two sheets.

8. Determine the planes of circular section through the point A (2, 1, 1)
for each of the following surfaces:

(a) 4 x 2 + 6 ?/ 2 + 4 * 2 = 1, (6) x* - 2 y* + z 2 = 1,
(c) 2 z 2 - r/ 2 - z 2 = 1, (d) 4 x 2 + 4 ?/ 2 = 5 z,

(e) 6 x 2 - if 4- 6 z 2 = 0.

9. Prove that for an ellipsoid of revolution the planes of circular section
are perpendicular to the axis of revolution of the surface; prove the same prop-
erty for the hyperboloids of revolution of one and of two sheets.

10. Derive the equations of the planes of circular section through an arbi-
trary point for the hyperboloid of one sheet, and also for the hyperboloid of
two sheets with respect to a system of axes which are parallel to the principal
directions of the surfaces.

11. Solve the corresponding problem for the proper cone and for the elliptic
cylinder.

12. Determine the condition under which the two planes of circular section
of the elliptic cylinder which pass through a fixed point are perpendicular to
each other.

TANGENT PLANES PARALLEL TO A GIVEN PLANE 253

112. Tangent Planes Parallel to a Given Plane. The Umbilics
of a Quadric Surface. It may happen that, even though the
planes of circular section of a quadric are real, yet the sections
themselves fail to be real because the plane does not meet the
surface in real points; a limiting case arises when a plane of cir-
cular section is tangent to the surface. In that case, if we are
dealing with a family of real planes of circular sections, which are
non-degenerate, the circle of section reduces to a point; such a
point on a surface is called an umbilical point, or an umbilic.

DEFINITION I. An umbilic of a quadric surface is a point on the
surface at which the tangent plane is parallel to a plane through the
origin which cuts the surface in a non-degenerate circle.

Remark. It follows from this definition and from Sections 110
and 111 that umbilical points can exist at most on the central
quadrics, the cone, the elliptic paraboloid and the elliptic cylinder.

Since these quadrics have at most two sets of parallel planes of
circular section, the existence of umbilical points depends upon
the existence of points on the surface at which the tangent plane is
parallel to the planes of these sets. On account of the intrinsic
interest of the question we shall preface the further discussion of
umbilics by a treatment of the general question of determining
points on a quadric surface at which the tangent plane is parallel
to a given plane; and we shall discuss this problem for all real
non-degenerate quadric surfaces.

Let ax + by + cz = be an arbitrary plane through the origin
(a, 6, and c not all zero). Does there exist a point P(a, 0, 7) on the
surface Q such that the plane tangent to the surface at P is parallel
to the given plane? Since the tangent plane to the surface at P
is represented by the equation

(x - )Qi(, ft 7) + (V ~ PXMa, ft 7) + (z - 7)Q 3 (, ft 7) = 0,

the conditions of the problem require that there shall exist a non-
zero factor of proportionality 2 p, such that the coordinates of P
satisfy the equations

Qi(, ft 7) = 2 pa, Q 2 (a, ft 7) = 2 P 6, Q 3 (a, ft 7) = 2 pc;

and moreover these coordinates must satisfy the condition Q(a, ft 7)
= 0. The latter equation may be written in the form

oQi(a, ft 7) + j8Q 2 (a, ft 7) + 7Qs(, ft T) + Qi(<*> ft T) = 0,

254

QUADRIC SURFACES

which, by use of the first three equations, may be replaced by the
equation

Q 4 (, ft 7) + 2 p(aa + bft + 07) = 0;

and this equation has the advantage of being linear in a, ft and 7.
We find therefore that a, ft and 7 must satisfy the following four
linear equations:

ana + o 12 + ai 3 7 + OH - pa = 0,

Oi 2 a + 022/3 + a 2 37 + 024 P& = 0,
013CK + 023/3 + 0337 + 034 PC = 0,

(o 14 + po)a + (024 + pb)P + (o 34 + pc)7 + O 44 = 0.

(1)

If these equations are looked upon as forming a system of linear
equations in a, ft 7, it follows from Corollary 2 of Theorem 24,
Chapter IV (Section 54, page 102) that they possess no solution,
unless the rank of the augmented matrix is less than 4. We are
led therefore to the following equation for p:

R(p)

On

OH + pO

Oi 2

2 2

2 3

24 + pb

Oi 3 OH ~ PO

2 3 2 4 pb

33 3 4 PC

034 + PC 044

= 0.

We write this determinant as the sum of two determinants,
using OH, o 2 4, 034, o 44 as the elements of the 4th column in one and
pa, p&, pc, as the elements of the 4th column in the other;
each of these determinants is again written as the sum of two de-
terminants by making a similar distribution of the elements of the
4th row. The equation will then take the following form :

OH Oi 2

Oi3

OH

On

Oi2

Oi3

O

Oi2 O22

2 3

O24

Oi2

O22

23

&

P

Ois O 2 3

033

3 4

Oi3

23

033

C

b

C

OH

024

034

On

Oi2

Oi3 O

P 2

Ol2
Oi3

022
O23

023 b
033 C

= 0.

O

b

c

Now it should be an easy matter to see that the two middle terms
are equal numerically and opposite in sign; moreover the coeffi-

TANGENT PLANES PARALLEL TO A GIVEN PLANE 255

cient of p 2 is of the same form as the determinant which we denoted
by the symbol A*(Q) in Section 84 (see page 172), and will by
analogous notation be designated by Aa(a, b, c). The equation
for p can then be written in the simple form

(2) A = p 2 X A 3 (a, 6, c).

To each root of this equation there corresponds a single value for
each of the variables a, /?, y, to be determined from the equations
(1), provided the rank of the coefficient matrix of these equations
is 3. The discussion of the different possibilities, and also of the
cases in which the roots of the equation (2) are real and distinct,
real and equal, or complex is made most conveniently after the
equation of the quadric surface has been reduced to the standard
forms, discussed in Chapter VIII. The translation and rotation
of axes which are involved in this reduction will of course affect
the equation ax + by + cz = of the plane. It is therefore of
importance to establish first the following theorem.

THEOREM 5. The value of the determinant A 3 (a, 6, c) and the rank
of its matrix are invariants of the configuration consisting of the sur-
face Q and the plane ax + by +cs =0 with respect to translation and
rotation of axes.

Proof. The most general transformation of coordinates which
can be made by rotation of axes is carried out by means of the
equations

x = XiZi + \ 2 yi + XaZi, y =

+ vtfi

(compare Theorem 5, Chapter V, Section 63, page 121). If these
expressions are substituted for x, y, and z in the equation ax + by
+ cz = 0, it is carried over into the equation o!x\ + Vy\ + G'ZI = 0,
where

a' = a\i + bp,i + cvij V = aX 2 + 6ju 2 + cv^ c' = a\ 3 -f 6/x 3

We observe now that this transformation of the coefficients of
x, y, z in the equation of the plane is exactly the same as the trans-
formation of the coefficients a i4 , a 24 , a 34 of the linear terms in Q
under rotation of axes (compare page 212) ; consequently the de-
terminant -A 3 (a, b, c) is transformed by rotation of axes exactly

256

QUADRIC SURFACES

like the discriminant of the quadric surface q(x, y,z) + 2 ax + 2 by
+ 2 cz = 0. It follows therefore from Theorems 4 and 5, Chapter
VIII (Section 94, pages 203 and 205) that the value of the de-
terminant A 3 (a, 6, c) and the rank of its matrix are invariant with
respect to rotation of axes. That this invariance also holds with
respect to translation of axes becomes evident if we recall that the
coefficients of the second degree terms in Q are not changed by
translation of axes (compare proof of Theorem 1, Chapter VIII,
Section 93, page 199) and if we observe that the transformation

x = x' + p, y = y' + q, z = z' + r

carries the equation ax + by + cz = over into the function
ax f + by' + cz' + ap + bq + cr = 0, so that the coefficients a, 6,
and c are also invariant under translation of axes.

In the further discussion of our problem we shall now be able to
use the standard forms of the quadric surfaces.
CASE I. r 4 = 4, n = 3.

(a) Ellipsoid. The standard form of the equation is

We find that A = -

and A 3 (a, 6, c) =

-(33+33+^5)- The c q uation ( 2 )

1

a

p2

1

b

^

1

c

r 2

a

&

c

)ecomes therefore

P 2 (a 2 p 2 + & 2 # 2 + c 2 r 2 ) = 1. It has two real roots for every set'of
values of a, 6, and c and, since r 3 = 3, a single set of values of
a, /?, 7 is given by equations (1) for each root of (2). Therefore, to
every plane there correspond two points on the ellipsoid at which
the tangent planes are parallel to the given plane.

(6) Hyperboloid of One Sheet. From the standard form of the
equation, namely,

TANGENT PLANES PARALLEL TO A GIVEN PLANE 257

1 a 2 b 2 c 2

we find A = , and A 3 (a, 6, c) = -^ + -r-r -- =-= . In this
p2g2 r 2 ' \ j j / qt r z r 2 p 2 p 2 q 2

case A*> 0, although A 3 (a, 6, c) is positive, zero or negative ac-
cording as a 2 p 2 + b 2 q 2 c 2 r 2 is positive, zero or negative. Since
we have again r 3 = 3, we conclude that if a, 6, and c are so chosen
that a 2 p 2 + b 2 q 2 > cV 2 , there are two points on the surface at
which the tangent plane is parallel to the plane ax + by + cz 0;
if a 2 p 2 + b 2 q 2 < c 2 r 2 , there are no such points on the surface, and
if a 2 p 2 + b 2 q 2 c 2 r 2 , there is no finite point on the surface which
has this property.

/j2

~ 2

/ /j

(c) Hyperboloid of Two Sheets. Using the standard form -5 ~ 2

z* 1 a 2 b 2

2 = 1, we find A = -- r-y- and A 3 (a, 6, c) = -- =-3 + -7-5
r- 2 p 2 q 2 r 2 q 2 r 2 r 2 p 2

c 2
H 2"! 2* Now A < 0, so that equation (2) furnishes real values of

p only in case b 2 q 2 + c 2 r 2 < a 2 p 2 . In this case there will be two
real points on the surface of the desired kind; in no other case will
points of this kind exist at finite distance.
The conclusions for this case are therefore as follows:

THEOREM 6. On the ellipsoid there are, for every plane In space,
two finite points at which the tangent plane is parallel to the given
plane; on the hyperboloid of one or two sheets two such points exist
for certain planes but none for others.

CASE II. r 4 = 4, r 3 = 2.
(a) Elliptic Paraboloid. The standard form of the equation is

n 2 c 2

We find A = ^ and A^(a y 6, c) = ^ . Equation (2) be-
comes therefore n 2 = p 2 c 2 . If c = 0, there is no finite value of p
which satisfies this equation and hence no finite point on the sur-
face which satisfies the conditions of our problem. If c ^ 0, we

71. YL TL

have p = - ; let pi = - and p 2 = . If we substitute pi or p2

c c c

in the equations (1) for this case, we obtain a system of linear equa-
tions whose augmented matrix is

258

QUADRIC SURFACES

-, o o -<

3 - P1 6

-2n

or

1

-P2a

p 2

1

o

P2&

f

P2

P2&

2n

The rank of each of these matrices is clearly less than 4. The
three-rowed principal minors formed from the 1st, 2nd, and 4th

<a 2 6 2 \ /a 2 6 2 \

-r + -r ) and p2 2 ( -r + - I
g* p 2 / \q 2 p 2 /

respectively; and since pi and p2 are both different from zero, we
conclude that these two matrices are both of rank 3.

The rank of the coefficient matrix for the first of these systems
of equations is manifestly less than 3; for the second system the
coefficient matrix contains the non-vanishing three-rowed minor

o 4

whose value is

2n
Z>V

P2& -2 n

In accordance with Corollary 1 of Theorem

24, Chapter IV (Section 54, page 102) we conclude that for p = pi,
the system (1) has no solutions, while f or p = P2 it possesses one

a

solution. For p

y3
= 0,

P2, the equations (1) are
= 0, pz(aa + 6)3) 2 717 = 0. From these we obtain, since

P2 = -- , the following solution of our problem:
c

a =

anp*

y =

Thus, while the equation (2) has two real solutions in this case,
only one of them gives rise to a point (a, 0, 7) on the surface at
which the tangent plane is parallel to an arbitrarily given plane
through the origin, except when this plane has the equation
ax + by = 0.

TANGENT PLANES PARALLEL TO A GIVEN PLANE 259

(6) Hyperbolic Paraboloid. From the standard form of the
equation

T 2 7 ,2

x JL = 2 nz
p% q2

n 2 c 2

we obtain A = -y-^and At(a, 6, c) = 2 . The equation (2) re-
duces, as in (a) to the form n 2 = p 2 c 2 . For c = 0, there is no finite

solution of the problem; for c ={= 0, we find as before, pi = - and

c

P2 = . For these two values of p the augmented matrices of the
c

systems of equations (1) become

1

-Pia

1

-P20

1

1

p\b

and

pzb

9 2

<T 2

-2n

plC

i pib

P2<*

P2&

-2n

It is seen that, as in (a), both these matrices are of rank 3; also
that the rank of the coefficient matrix for the first system of equa-
tions is 2 and the rank of the coefficient matrix for the second sys-
tem of equations is 3. The conclusion is therefore the same as

for the elliptic paraboloid; we obtain the point a =

_ bnq* __ n(a 2 p 2
3 -~> y -

- 7>2,,2

THEOREM 7. On the elliptic paraboloid and on the hyperbolic parab-
oloid, there is for every plane in space, except for planes parallel to
the axis of the surface, one point at which the tangent plane is parallel
to the given plane.

CASE III. n r 3 = 3.

The on}y real quadric in this case is the proper cone, whose
standard equation may be put in the form

2CO

QUADR1C SURFACES

We find A = and

A 3 (a, b, c) =

1

a

_!

6

a

b

_!_

r 2
c

C

__ p 2 a 2 + & V - c 2 r 2

If -A 3 (a, b, c) = ^p 0, the only solution of equation (2) is p 0,
and equations (1) reduce to the equations for the vertex (compare
Theorem 13, Chapter VII, Section 85, page 177). Since at the
vertex Qi = $2 = Qa = 0, the tangent plane at this point does not
exist and our problem has therefore no solution in this case. On
the other hand, if A 3 (a, 6, c) = 0, i.e., if p 2 a 2 + 6 2 g 2 - c 2 r 2 = 0,

every value of p satisfies equation (2).

the system of equations (1) is

-4 o

The augmented matrix of

-pa

i -p6

pa p&

pc

-pc

; its deter-

minant is equal to p 2 As(a, 6, c) and vanishes. It should be clear
that the rank of this matrix is 3 and that the rank of the coefficient
matrix of the system of equations (1) is also 3. Hence, for every
value of p there is one point on the surface which satisfies the con-
ditions of our problem. By solving the system (1), we find

a

pap 2 ,

|3 = pbq 2 , 7 = per 2 .

As p varies, these equations are the parametric equations of a line
and it is readily seen that this line lies entirely on the cone. For,

since a 2 p 2 + b 2 # 2 - c 2 r 2 = 0, it follows that ^ + ^ - \ =

P 2 (a 2 p 2 + & 2 # 2 "~" c * r ^ ~ 0> independently of the value of p. And
the tangent plane to the cone at any point on this line is repre-

sented by the equation ^ H ^ -- ^ = 0, i.e. by the equation

TANGENT PLANES PARALLEL TO A GIVEN PLANE 261

ax + by + cz = 0. We have therefore reached the following
conclusion :

THEOREM 8. If and only if a, 6, and c are so chosen that a 2 p 2 + 6 V
c 2 r 2 = 0, the plane ax -f by + cs = will be tangent to the cone

*_ -|_2_ _ ?L = o; this plane touches the surface along the line on the

cone whose parametric equations are x = pap 2 , y = p6q 2 , z = per 2 ,
p being the parameter. For any other choice of a, b, and c there will
be no point on the cone at which the tangent plane is parallel to the
plane ax + by -f- cs = 0.

CASE IV. n = 3, r 3 = 2.

(a) Elliptic Cylinder. The standard equation is \ + ~ = 1 ;

P 5

c 2
A = and A 3 (a, 6, c) = ^-: 2 . If c = 0, the only solution of

equation (2) is p = 0; the system (1) is inconsistent for this value
of p and we have therefore no solution of the problem. If c 0,
the system of equations (1) reduces to

". = pa $- = pb p(aa + bft) = 1
p2 ^2 P y

Values of p can always be determined for which this system is
consistent; with these values of p, we find for a. and ft the follow-
ing results:

These equations determine a line parallel to the Z-axis, which lies
entirely on the cylinder; and the plane tangent to the cylinder at
any point on this line is represented by the equation ax + by
= Va*p* + 6y.

x 2 ?/ 2
(b) Hyperbolic Cylinder. From the standard equation ~

c 2
= 1, we find A = and Az(a, 6, c) = -j- 2 . Conditions are similar

to those in (a). If c ^ O y there is no tangent plane parallel to the
plane ax + by + cz = 0. If c = 0, the values of p and of a, /3, y
are to be determined from the equations (1) which reduce in this

case to

a = pap 2 , ft = -pbq\ p(aa + b/3) = 1.

262 QUADRIC SURFACES

Elimination of a and ft leads to the equation p 2 (a 2 p 2 6 2 # 2 ) = 1 ;
hence if and only if a 2 /? 2 ~ &V > 0, does there exist a real point
which satisfies the conditions of the problem. In this case, we find

ap 2 bq 2

a "~ > ~~

The line determined by these equations lies entirely on the surface >
and the equation of the plane tangent to the cylinder at any point
of this line is ax + by = db Va 2 p 2 6 2 j 2 .
CASE V. r 4 = 3, r 3 = 1.

In this case the locus of the equation Q - is a parabolic cylin-
der; the equation of this surface may be reduced to the standard
form y 2 4 px = 0. We have

a

A = and -As (a, b, c) =

0106
c
a b c

= 0.

Hence equation (2) imposes no restriction on the choice of p. The
system of equations (1) takes the form

-2 p - pa = 0, ft - pb = 0, -pc = 0, (-2 p + pa)a
+ pb/3 + PCT = 0.

If c 4= 0, the third equation requires that p == 0, which leads to a
contradiction with the first equation, since p ^ 0; in this case
there is therefore no solution. For a similar reason, we must have

2 v
a ^p ; and in this case we find p = , and hence ft = pb =

2 vb vb^

, a = ^~. These equations represent a line which lies en-
tirely on the cylinder; and the tangent plane to the surface at any
point on this line is given by the equation yft 2 p(x + a) = 0,

vb 2

that is, by the equation ax + by + = 0. There is no finite

a

point on the cylinder at which the tangent plane is parallel to the
plane ax -f by = 0, if a = 0; that is, there is no tangent plane
parallel to the plane y = 0.

We summarize now the conclusions reached in Cases IV and V
in the following theorem.

THE UMBILICS OF A QTTADRIC SURFACE 263

THEOREM 9. The tangent planes to the elliptic cylinder, the hyper-
bolic cylinder and the parabolic cylinder are all parallel to the gener-
ating line of the cylinder; they have contact with the surface along an

entire generator. The elliptic cylinder -- + ^ = 1 has two tangent

planes parallel to an arbitrary plane ax -f by = through its axis,
namely, the planes ax + by = V a 2 p 2 + 6 2 q 2 . The hyperbolic cylin-
der 2 - ^7, = 1 has two tangent planes parallel to an arbitrary plane

ax + by = through its axis, namely, the planes ax + by
= Va~p 2 - 6V, unless a 2 p 2 - &V :g 0, in which case no such plane
exists. The parabolic cylinder y 2 = 4px has one tangent plane parallel
to the arbitrary plane ax -j- by = through the Z-axis, namely, the

plane ax + by H- = 0, if a ^ 0; if a = no such plane exists.

113. The Umbilics of a Quadric Surface, continued. The
determination of the umbilics on the central quadrics, the cone,
the elliptic paraboloid and the elliptic cylinder can now be effected,
on the basis of Definition I (Section 112, page 253), by combining
the results of Theorems 6, 7, 8, and 9 with those of Theorem 4. We
shall take the equations of these surfaces in the standard forms
used in the preceding section.

(a) Ellipsoid.

If p < q < r, the equations of the planes of circular section,

through the origin, of the ellipsoid -2 + ^2 + -^! =0 are (see

Section 110):

rVq* - p 2 x pVf 2 - q 2 z = 0.

Hence, in the notation of Section 112, we have a = rVq*
b = 0, c = pVr 2 q 2 , and p = =b

g cr

, the double sign of p being independent of that of c.

prVr' 2 p 2

The first three equations of the system (1) of Section 112 are for

a 3 *v

this case z = pa, 2 = pb, -^ = pc. We find therefore

the double signs of a and 7 being independent of each other.
conclusion is given by the following theorem.

264 QUADRIC SURFACES

THEOREM 10. The ellipsoid ~+^ + ^j = I, in which p < q < r, has

four umbilics. Their coordinates are ( = __ , .

\ Vr 2 - p 2 Vr' 2 - p 2

they He on the ellipse in which the surface is cut by the XZ-plane.

(b) Hyperboloid of One Sheet. If the equation be taken in the

X 2 ?/ 2 2 2

standard form -^ + ^r -- -: 0. p < q. the central planes of cir-
p2 qt r 2 r i

eular section are (compare Exercise 10, Section 111, page 252):
rVq 2 - p' 2 x =FpVq' 2 + r 2 2 = 0.

Hence a = rVq 2 p 2 J b = 0, c = ^FpVq' 2 + r 2 ; and a 2 p 2

+ 6 2 g 2 - c 2 r 2 - p 2 r >2 (f - p 2 ) - r*-p*(q 2 + r 2 ) = - p 2 f 2 (p 2 + r 2 )

< 0. Therefore, the hyperboloid of one sheet has no umbilics.

(c) Hyperboloid of Two Sheets. We proceed as in the preceding

/v2 7/^ 2J

case, for the equation -7, ~ -- r > = l,q < r. The central planes

p2 g2 r 2 ^ * x

of circular section are represented by the equations

(j Vr' 2 + ;> 2 .r =h p\/r 2 - ^/ 2 ?y = 0.

We have a = ^V^ 2 + /> 2 , b = =t /;Vr 2 - <? 2 , c = 0; a 2 p 2 - 6 2 ^ 2
-c 2 r 2 = pV(r 2 + p 2 ) - 2 p 2 (r 2 - g 2 ) - p 2 g 2 (p 2 + g 2 )> 0. There-
fore there are umbilical points on this surface. Their actual de-
termination proceeds as in the case of the ellipsoid; the reader
should have no difficulty in completing the proof of the following
theorem :

THKORKM 11. On the hyperboloid of one sheet there are no umbilics.
On the hyperboloid of two sheets, which is not a surface of revolution,

X 2 V 2 -2

there are four umbilics; if the equation of the surface is -, - , --

pi q 2 r 2

( D \/r 2 4- n 2 n\/r' 2 n z \
p . ^" , L ~ "~ , o \ the
Vp 2 -f q 2 Vp 2 + q* )

two double signs being independent of each other.

/$ ^,2 ^2

(d) Cone. If the equation be written in the form -7, + ~ -- -

p z q z r~

= 0, p < q, the planes of circular section through the origin are:
r Vq* - p' 2 x pVg 2 + r 2 z = 0.

THE UMBILICS OF A QUADRIC SURFACE 265

Here a = rVq* - p*, b = 0, c = dbpVV + r 2 ; and a 2 p 2 + 6 2 g 2
- cV 2 *= pV 2 (g 2 - p 2 ) - p 2 rV + r 2 ) = - pV 2 (p 2 + r 2 ) 4= 0.
Therefore the cone has no umbilics.

x 2 ?y 2
(e) Elliptic Paraboloid. From the standard equation + -%

= 2nz,p < q, we obtain for the planes of circular section through
the origin, the equations Vq 2 p 2 x pz = 0. Hence

b = 0, c =

Tliere are therefore two umbilics on this surface; from the formulas,
given in Section 112, Case II, (a), page 258, we find that their co-
ordinates are

= 0, 7 =

(/) Elliptic Cylinder. We take the equation in the form H -

= 1, /; < g, and we find that the circular sections through the ori-
gin lie in the planes represented by the equations v q 2 p 2 x pz
= 0. Since these planes are not parallel to the generators of the
cylinder, it follows from Theorem 9 that there are no umbilics on
this surface.

The results found in cases (rf), (e), and (/) lead to the following
theorem.

THEOREM 12. There are no umbilics on the cone, nor on the elliptic
cylinder. There are two umbilics on the elliptic paraboloid which is
not a surface of revolution; if the equation of this surface be reduced

to the form ^ 2 -h ~ = 2 ns 9 p < q 9 the coordinates of the umbilics are

(g) Surfaces of Revolution. We have seen in Theorem 4 that for
a surface of revolution there is one plane of circular section through
every point of space; these surfaces can therefore have at most
two umbilics. The single central plane of circular section may in
these cases be obtained from the results already found, 'by setting
two of the discriminating numbers equal to each other. So, for
example, if in the ellipsoid, treated under (a), we put p = q y the
four umbilics reduce to two, namely to the points (0, 0, =hr), that

266 QUADRIC SURFACES

is, to the points in which the axis of revolution meets the surface.
The reader should have no difficulty in obtaining the corresponding
result for the hyperboloid of revolution of two sheets and for the
paraboloid of revolution. On a sphere every point is an umbilic.

THEOREM 13. On the ellipsoid of revolution, the hyperboloid of
revolution of two sheets and on the paraboloid of revolution, the urn-
bllics are the points in which the surface is met by the axis of revolu-
tion; on the other quadrics of revolution there are no umbilics.

We summarize the results of this section as follows:

Umbilics exist on the ellipsoid, the hyperboloid of two sheets and
the elliptic paraboloid. If these surfaces are not surfaces of rev-
olution, the number of umbilics on them are 4, 4, and 2 respectively;
if they are surfaces of revolution, the umbilics fall two by (wo into
the points where the surface is met by the axis of revolution.

114. Exercises.

1. Determine the planes parallel to the plane z 27/4-2=0 which are
tangent to the following surfaces:

(a) J 2 - 6 ^ - 3 z 2 = 1, (6) 2 x* + 4 7/ 2 + 5 2 2 = 1,

(c) x 2 4- 4 ?y 2 = 2 z, (d) 4 x* - ?/ 2 = 2 z.

2. Determine the umbilics on each of the following surfaces:

(c) 4- = 2*, (d) x* 4- y* + 4 z 2 = 12.

3. Prove that the planes through the point P(a, ft 7) which are parallel to

# 2 ?/ 2 2

the planes tangent to the cone ~ -\- ~ = are tangent to a cone whose

vertex is at P.

4. If u, v, w are called the " coordinates " of the plane ux -\-vy-\-wz + 1=0,
set up the equation which the coordinates of a plane must satisfy in order that

the plane be tangent to the cone ( ^~ + (V ""/^ - ( -^~ = 0.

X z 1/ z 2 2

6. Determine the umbilics on the surface -- 2^2 I = 1* P < r -

6. Prove that the planes of circular section of the quadric surface Q(x, y, z)
= are also planes of circular section of all the quadric surfaces whose equa-
tions have the form Q(x, y, z) + k(x* + y 2 + z*) = 0.

7. Prove that the four umbilics of an ellipsoid are the vertices of a rectangle;
and determine the condition under which they will be the vertices of a square.

8. Prove that the umbilics of the ellipsoids with the same center and the

EXERCISES 267

same principal directions, whose semi-axes are kp, kq, and kr, in which p, q, and
r are fixed while k is variable, lie on four lines through the common center;
and determine the direction cosines of these lines.

9. Determine the condition under which the tangent planes to an elliptic
paraboloid at the umbilical points are perpendicular to each other.

10. Prove that the planes of circular section of an hyperboloid of one sheet
are also planes of circular section of its asymptotic cone.

11. Prove that the tangent planes to an hyperboloid of two sheets at its
umbilical points are planes of circular section of its asymptotic cone.

12. Prove that the planes of circular section of a proper cone cut a tangent
plane of the cone in lines which make equal angles with the generator of the
cone along which the tangent plane touches it.

CHAPTER X
PROPERTIES OF CENTRAL QUADRIC SURFACES

115. Conjugate Diameters and Conjugate Diametral Planes
of Central Quadrics. Enveloping Cylinder.

DEFINITION I. A diameter of a central quadric surface is 21 chord
which passes through the center of the surface.

The common form of the standard equations of the central
quadrics is

(1) mix 2 + W2?/ 2 + m s z 2 = 1.

Here m\, 11^ and m^ are the quotients of the discriminating num-

bers of the surface by j ; the center of the surface is at the

^144

origin and the principal directions arc the directions of the co-

ordinate axes.

Let us now consider an arbitrary diameter d of the surface;

we shall designate its direction cosines by Xi, MI, v\- The diametral

plane corresponding to this di-
rection (see Definition X, Chap-
ter VII and Theorem 17, Section
88, page 186) in the surface (1)
is given by the equation

(2)

If a second diameter
rection cosines X 2 , M,
this plane,

(3)

= 0.

with di-
^ lies in

= 0,

(compare Theorem 21, Chapter
Fio. 35 IV, Section 52, page 99); and

the equation of the diametral
plane corresponding to the direction of c? 2 has the equation

(4) WiX 2 + miMJ + m$v& = 0.
268

CONJUGATE DIAMETERS OF CENTRAL QUADRICS 269

Equation (3) can now be interpreted as stating that the line
di lies in the plane (4). We have therefore proved the following
theorem (see Fig. 35).

THEOREM 1. If one diameter of a central quadric surface lies in the
diametral plane determined by the direction of another diameter,
then the second diameter lies in the diametral plane determined by
the direction of the first.

COROLLARY. The plane determined by two diameters is the diametral
plane which corresponds to the direction of the line of intersection of
the diametral planes of the first two diameters.

Proof. The diameter (/ 3 in which the planes (2) and (4) inter-
sect has direction cosines X 3 , MS, vs determined by the proportion

(5) X 3 :

Mi v\

M2 V'2.

MS MI

v\

,

A'2

MI MO

A! MI

X-2 M2

The diametral plane of this line* is represented by the equation

Xi ,

M'2

M2

= 0.

But this equation is equivalent to the equation

x

y z

XL Mi "i

X2 M2 ^2

which is indeed the equation of the plane determined by the lines
di and d%.

The corollary is also proved by the fact, that since rf 3 lies in the
diametral planes of d\ and d 2 , the diametral plane of rf 3 contains
the diameters d\ and d%.

Of the three diameters d\, d 2 , and e/ 3 any two determine the
diametral plane of the third; and of the three planes (d h d 2 ),
(di y d 3 ), and (d 3 , d\) any one is the diametral plane of the line of
intersection of the other two.

The correspondence between diameters and diametral planes is
a reciprocal one-to-one correspondence : not only is there one and
only one diametral plane of any line through the center, but there
is also one and only one diameter of which any given plane through

* Whenever it can be done without danger of confusion, we shall use the
phrase "diametral plane of a line V 1 in place of the more exact but less con-
venient expression "diametral plane corresponding to the direction of the
line L"

270 PROPERTIES OF CENTRAL QUADRIC SURFACES

the center is the diametral plane. For if ax + by + cz = is
an arbitrary plane through the origin, there is one and only one
set of values X, /x, v such that mi\ : w 2 M : ni$> = a :b : c, hence one
diameter d whose diametral plane mi\x + ^2M2/ + m & z co "
incides with the given plane.

We introduce now the following definition.

DEFINITION II. A set of three diameters of a central quadric, such
that the diametral plane of any one of them is the plane determined
by the other two, is called a set of conjugate diameters; and a set of
three diametral planes, such that any one of them is the diametral
plane of the line of intersection of the other two, is called a set of con-
jugate diametral planes.

We shall find it convenient to refer to such sets as a conjugate
set of diameters and a conjugate set of diametral planes.
In terms of this definition we have the following theorem.

THEOREM 2. For every diameter (diametral plane) of a central quad-
ric there exist an infinite number of conjugate sets; for every pair of
diameters (diametral planes), of which one lies in the diametral plane
of the other (passes through the diameter of the other) there exists one
conjugate set. For every diameter together with a diametral plane
passing through it, there exists one set of conjugate diameters and
one set of diametral planes, such that the diameters of the first set are
the lines of intersection of the planes of the second set.

Remark 1. The axes of symmetry of the ellipsoid furnish an
example of a conjugate set of diameters.

Remark 2. Any two of a conjugate set of diameters are " con-
jugate diameters" of the conic section in which their plane cuts
the quadric, where the words in quotation marks are to be under-
stood in the sense in which they are used in Plane Analytical
Geometry.

The coordinates (a, 0, 7) of the point P in which the diameter d
of direction cosines X, /i, v meets the surface, are proportional to
X, ^, v] that is, a = Xs, = ps, 7 = vs. The tangent plane to the
surface at this point may therefore be represented by the equation
wiXz + m 2 M2/ + m&z = 1. Comparison with equation (2) shows
that this plane is parallel to the diametral plane of d.

And if P(a, 0, 7) is a point of the conic in which the diametral
plane D of the diameter d cuts the surface, then m\a\ + w 2 ftu +
= 0; and the tangent plane to the surface at this point is

CONJUGATE DIAMETERS OF THE ELLIPSOID 271

represented by the equation m\ax + m$y + m^yz 1. Con-
sequently this tangent plane is parallel to the line d (compare
Theorem 21, Chapter IV, Section 52, page 99); hence there exists
also a tangent line through P parallel to d. If P moves along the
curve of intersection of the surface with the plane J5, these tangent
lines which are parallel to d generate a cylinder.

DEFINITION III. The enveloping cylinder of a quadrfc surface cor-
responding to a line d is the cylinder generated by the tangent lines to
the surface which are parallel to d.

We have therefore obtained moreover the following result.

THEOREM 3. The tangent planes to a quadric surface at the points
where it is met by a diameter d are parallel to the diametral plane of
d. The tangent planes at the points where it is met by the diametral
plane of d are parallel to d; the tangent lines drawn through these
points and parallel to d form an enveloping cylinder of the surface.

116. Exercises.

1. Show that the enveloping cylinder of an ellipsoid is an elliptic cylinder
for every direction of the generator. Obtain its equation.

2. Show that the enveloping cylinder of an hyperboloid of two sheets is an
hyperbolic cylinder for every direction of the generator; obtain its equation.

3. Show that the enveloping cylinder of an hyperboloid of one sheet is an
elliptic cylinder for some directions of the generator and an hyperbolic cylinder
for other directions of the generator; obtain the equation of the enveloping
cylinder.

4. Determine the directions of the generator for which the enveloping cylin-
der of an hyperboloid of one sheet will be an elliptic cylinder, and also the
directions for which it will be an hyperbolic cylinder.

X 2 ?/ 2 3 2

6. Determine the diameter of the surface T + 77 ~~ Tp ^ * which is con-
jugate to the diameters whose direction cosines are proportional to 2 : 1 : 2
and 1 : 9:4 respectively; determine also the diametral planes of these
diameters.

?/ 2 z 2

6. Solve the corresponding problem for the surface x 2 -f- 4- -h TT = 1 and

the directions 1:2: 3 and 2:3: 1.

7. Determine the enveloping cylinders of the ellipsoid in the preceding
problem for the directions of each of the three conjugate diameters.

8. Determine for the surface of Exercise 6 the diametral plane which is
conjugate to the planes x -f- 2 y -\- 2 z = and 4 x -{- y z = 0.

117. Conjugate Diameters of the Ellipsoid. We shall denote
the length of the chord determined by the diameter d of a quadric

272 PROPERTIES OF CENTRAL QUADRIC SURFACES

surface by 2 5, the points where d meets the surface by E(a, 0, y)
and #'(', jft', 7'); and the direction cosines of d by X, ju, v. If
several diameters are under consideration at the same time, we
shall distinguish between the numbers related to them by the use
of subscripts.
The parametric equations of d may be taken in the form

(1) x = Xs, y = us, z = vs.

The number 5 is the numerical value of the roots of the equation
(2) (mjX 2 + msM 2 + m& z )s* = 1,

provided these roots are real; and the coordinates of E and E'
are obtained by substituting these roots for s in equations (1).
It should be clear that the roots of equation (2) never coincide,
that for the ellipsoid they are always real and finite, while for
the hyperboloids of 1 or 2 sheets, they will be real and finite,
infinite, or imaginary according as raiX 2 + m^f + m^v 2 is positive,
zero, or negative.

We will show now that the ellipsoid is the only quadric for which
there exist sets of conjugate diameters, for each of which the roots
of the equation (2) are real and finite. That such sets do exist for
the ellipsoid follows from Theorem 2 (Section 115, page 270) in
conjunction with the observation in the preceding paragraph.
Suppose now that we have three diameters d\, c? 2 , and d 3 such that

(3) miXi 2 + w 2 Mi 2 + ^3*v > 0, ?ttiX 2 2 + m 2 ju 2 2 + m^ > 0,

WiX 3 2 + ?2M3 2 + 7ft 3 *> 3 2 > 0, and

(4) WiX 2 X3 + M 2 M2/*3 + m^va = 0, WiX 3 Xi + w 2 M3Mi + ^aWi = 0,

0.

From the first two of equations (4) we derive equations (5) of
Section 115; and from these we conclude that there exists a non-
zero constant fc, such that

Mi

MI

If we multiply the inequalities (3i) and (3 2 ) and subtract the
square of equation (4 3 ) from the result, we obtain, by an easy re-
arrangement of terms:

0.

CONJUGATE DIAMETERS OF THE ELLIPSOID 273
The equations just preceding enable us to replace this inequality by

and from this we conclude that

2 my 2

and finally on account of (3 3 ), that mim 2 m 3 > 0. Hence, either
nil > 0, M 2 > and ra 3 > 0, in which case our objective has been
reached, or else one of these numbers is positive, the other two
negative. Let us suppose m\ > and m 2 < 0, m^ < 0. We
derive then from (3i) and (3 2 ), the inequalities

miXi 2 > m 2 /zi 2 m 8 j>i 2 > and WiX 2 2 > m 2/ u 2 2 m-pf > 0;
multiplication of these inequalities leads to

mi 2 Xi 2 X 2 2 > m 2 2 MiW + rn^sCMi V + M2 V) + infrfyf.
If from the two sides of this inequality we subtract the equation
?Wi 2 Xi 2 X 2 2 = ?n 2 ViW + 2

obtained from (4 3 ) we reach the inequality
>

from which would follow that m 2 w 3 < and therefore that m 2 and
7M 3 are opposite in sign; but this contradicts the supposition which
we started from. We conclude therefore that mi > 0, w 2 >
and m 3 > 0; and we have the following theorem.

THEOREM 4. The ellipsoid is the only central quadric for which
there exist sets of conjugate diameters each of which meets the surface
in real Unite points.

In developing further properties of the conjugate sets of diam-
eters of the ellipsoid, we shall use equation (1), Section 115, with

the understanding that mi = ^ , m z = ^ and m 3 = -^

If di, d*, and d 3 are conjugate diameters of the ellipsoid, we derive
from the equations

+ waM,- 2 + m&i*)6t* = 1 and m^X/ + mww +
= 0, i,j = 1,2,3, ij,

274 PROPERTIES OF CENTRAL QUADRIC SURFACES

the interpretation that

Xi6i v Wi, jui5i v mo,

and

are the direction eosines of three mutually perpendicular lines.
If we apply to them the results obtained in Theorem 6, Chapter V,
the Remark 3 following this theorem, and Theorem 7, Chapter V
(see Section 65, pages 123 and 124), we find the following further
relations :

(5)
(6)

= 0,

= 0, <5rViXi + <5 2 2 j> 2 X2 + <5 3 2 *> 3 X 3 = 0;

3 2 ) = 1, 7/t 2 (iW+2W+3W)

3 3 V) = \ ;

(7)

Mi

From these formulas we derive the following interesting results.

THEOREM 5. The sum of the squares of the semi-diameters of a set
of conjugate diameters of an ellipsoid is the same for all conjugate
sets of diameters.

Proof. The semi-diameters of the conjugate set are 61, 5 2 , and
5 3 . If we divide the three equations in (6) by Wi, w 2 , and m 3
respectively and add the results, we find:

5i 2 (Xi 2 + Mi 2 + "i 2 ) + <5 2 2 (X 2 2 + M 2 2 + 2 2 ) + 5 3 2 (X 3 2 + ^ + ^ 3 2 )

that is

THEOREM 6. The volume of the parallelepiped of which the three
semi-diameters of a conjugate set are coterminous edges is the same
for all conjugate sets.

Proof. To determine the required volume we have at our dis-
posal the formula in Corollary 2 of Theorem 3, Chapter V (Section

CONJUGATE DIAMETERS OF THE ELLIPSOID

275

62, page 118). In the present case, the symbols used in this
formula have the following values:

a = Si, b = S 2 , c = 5 3 , cos 7 = XiX 2 + Miju2 +
XsXj. + MsMi + ^i; COS a = X 2 X 3 + M2Ms + ^2^3.

If we make use now of formula (7) above and observe that

cos ft =

Xi Mi '
X 2 M2 '
Xs Ms *

(compare part (3) of the Remark following Theorem 10, Chap-
ter I, Section 14, page 27), we obtain for the volume of the par-
allelopiped the expression

= pqr.

THEOREM 7. If the ellipsoid ~ + ~ -f ~ = 1 Is referred to a reference
frame O-X'Y'Z', whose axes are the lines of a conjugate set of diam-
eters, its equation is ~ -f ~ + f-r = 1, where 5i, 5 2 , and 5 3 are the semi-

Oi" O2 W3

diameters of the conjugate set.

Proof. Let P(a y (3, 7) be an arbitrary point on the ellipsoid,
as referred to the frame 0-XYZ, so that mia 2 + m 2 /3 2 + rn^y 2 = 1 ;
the coordinates a 7 , /3 X , y' of P with respect to O-X'Y'Z' are the
lengths of the segments P y > z 'Pj Pz'x'P, and P x yP of the lines
through P parallel to OX', OF', and OZ' respectively (compare
Fig. 14, page 113). The equations of the line through P parallel
to OX' are

The equation of the plane Y'OZ' is m{KiX + m^\y + m&iZ = 0;
and the distance PP y v is the value of s determined by the equation
which results when the expressions for x, y, and z just preceding
are substituted in this equation, that is, by the equation

s = 0.

276 PROPERTIES OF CENTRAL QUADRIC SURFACES
Therefore

In similar manner we find

Consequently we find

03

If the squares of the trinomials on the right side of this equation
are expanded, we obtain a homogeneous polynomial of the second
degree in <*, /3, and 7. The coefficient of a 2 is found to be
?Hi 2 (Xi 2 Si 2 + X 2 2 5 2 2 + X 3 2 <5 3 2 ) = mi, by virtue of formula (61); in
similar manner we find that the coefficients of /3 2 and 7 2 are w 2
and m s respectively. For the coefficient of /3y we find 2 morris
(Mivi5i 2 + jU2^ 2 5 2 2 + M3^5,3 2 ) = 0, on account of formula (5 2 ); and
it should be a simple matter to verify that the coefficients of ya
and aft are likewise zero. Hence we find

Since a', 0', and 7' are the coordinates with respect to 0-X'Y'Z f
of an arbitrary point on the ellipsoid, our theorem is proved.

118. Exercises.

1. Prove that a set of conjugate diameters of the hyperboloid of one sheet
mix 2 + m> 2 y 2 -f w 3 2 2 = 1 (mi > 0, w 2 > 0, m 3 < 0) is also a conjugate set
for the hyperboloid of two sheets mix 2 -\- m> 2 y 2 + msz 2 = 1.

2. Prove that of a set of conjugate diameters of the two hyperboloids of the
preceding exercise, two and only two meet one of these surfaces in real, finite
points, whereas the third diameter of the set meets the other surface in real,
finite points; and that this set of conjugate diameters for the two surfaces
determines three finite chords, two chords of one of the surfaces, and one of the
other.

3. Prove that if 2 81, 2 5 2 , and 2 5 3 denote the lengths of the diameters of a
conjugate set for the two hyperboloids of Exercise 1, then 5i 2 + 5 2 2 + 5s 2 =

-L + -L+-L.

nil wi* 7^3

LINEAR FAMILIES OF QUADRICS 277

v 2 z z

4. Determine the diameter of the surfaces z 2 ^- + = 1, which is

4 y

conjugate to the two diameters whose direction cosines are proportional to
1:6:9 and to 1 : 2 : 4 respectively.

6. Prove that three tangent planes of an ellipsoid which are parallel to
the three diametral planes of a conjugate set meet in a point. Determine the
locus generated by this point when all conjugate sets of diametral planes are
considered.

6. Determine the length of the diameter conjugate to two diameters which
lie in a plane of circular section through the center of an ellipsoid.

119. Linear Families of Quadrics. When two quadric surfaces
are given, as for instance by the equations Q(x, y y z) = and
Q'(x, y, z) = 0, the points common to these surfaces determine
a space locus with one degree of freedom, that is, a space curve.
The study of space curves constitutes an important part of the
field of Differential Geometry. Without knowing anything fur-
ther about the character of the curve of intersection of two quadrics,
we can affirm that, no matter what polynomials are represented
by the symbols A(x, y, z) and B(x, y, z), the locus of the equation
A(x, y, z) Q(x, y, z) + B(x, y, z) Q'(x, y, z) =

will be a surface which passes through this curve; for this equation
is surely satisfied by the coordinates of any point which belongs
to the locus of Q = and to that of Q' = 0. And if A (x, y, z)
and B(x, y, z) reduce to constants which are not both zero, the
equation represents a quadric through this curve.
Thus the equation

(1) k&(x, y, z) + k 2 Q'(x, y, z) =

represents, when the ratio ki : k< 2 varies, a family of quadric sur-
faces, all of which pass through the points common to the surfaces
Q = and Q f = 0. It is called a linear one-parameter family of
quadrics, also a pencil of quadrics (compare Section 49). For
fcj = o, fc 2 = 1, we obtain the surface Q'; for ki = 1, fc 2 = 0,

the surface Q. Upon division by ki and putting ~ = X, the

KI

equation (1) takes the form

(2) Q(x,y,z)-\Q'(x,y,z) = 0;

and this equation is equivalent to (1), except that it does not in-
clude the surface Q' for any finite value of X.

278 PROPERTIES OF CENTRAL QUADRIC SURFACES

The value of the parameter X may be so selected as to make the
surface represented by (2) satisfy a single condition, for example,
that of passing through one prescribed point, which does not
lie on the surface Q f . This particular problem has a unique solu-
tion. For if (a, ft 7) is the prescribed point, X must satisfy the
condition

Q(, ft 7) - AQ'(, ft 7) = 0,
so that

x = Q(a, ft 7) t

The condition that a surface of the pencil be a surface of revolu-
tion leads to an equation of higher degree in X; for it requires that
X be determined so that the equation

=

shall have a double root (compare the Remark on page 215).
We shall not pursue this problem further.

Of special interest is the question as to the singular quadrics
contained in the pencil of quadrics (2). A surface of this pencil
will be singular if and only if the rank of the matrix

Xon' 012 Xa^' 013 Xais' OH

2 ~ ^ a 12' 022 ~ Xa 22 ' 023 X023' O24

On Xon' k tti2 Xoi 2 ' a J3

012 Xai 2 ' 022 Xa 22 ' k a 23 Xa 23 '

a 2 s Xa 2 s' 033 \(IM k

/\ \ _

' 023 Xaas' o 33 Xo 33 ' a 34

O24 XO 2 4' O 3 4 XO34' 044 ~ XO4

is less than 4 (compare Definition V, Chapter VII, Section 82,
page 166); it will be a non-degenerate singular quadric, that is, a
proper cone or a cylinder, if and only if the rank of this matrix is 3.
In either case a necessary condition is that the determinant
A(X) of this matrix shall vanish. But the equation A(X) =
is of the fourth degree in X; there will therefore be at most four
singular quadrics in the pencil. A further discussion of the charac-
teristics of these surfaces, of the questions whether they are de-
generate surfaces, whether they are cylinders or cones leads into
a more extensive treatment than we can undertake here. The
interested reader is referred to Snyder and Sisam, Analytic Geom-

LINEAR FAMILIES OF QUADRICS 279

etry of Space, Chapter XI, or to Bocher, Introduction to Higher
Algebr^, Chapter XIII, for a consideration of these problems.

To determine the ratios of the ten coefficients which appear in
the general equation Q(x, y, z) = of a quadric surface, nine
points must be prescribed. Substitution of the coordinates of
these nine points in the general equation furnishes nine linear
homogeneous equations for an, a 22 , a., a 44 , a 12 , a 23 , a 3 4, i4, ai 3 , a 2 4
from which the ratios of these coefficients can in general be de-
termined. In order that this be possible, the 9 points (a,-, ft, 7,-),
i = 1, 2, . . . , 9 must have such coordinates that the rank of the
matrix, formed by the 9 rows

(3) a,- 2 , ft 2 , 7i 2 , 2ft T ,-, 2 7 W, 2a;ft, 2 a*, 2ft, 2 7,', 1

(i = 1, 2, . . . , 9) of 10 elements each, is 9. Whenever a set of
points (ai, ft, 7,-) is so constituted that the matrix, in which for
each of the points there is a row of 10 elements as indicated in (3),
has a rank equal to the number of points in the set, we shall say
that these points are in " general position." We can therefore
say that there passes a single quadric surface through a set of
nine points in " general position."

If only eight points are given, we have eight equations; if the
eight points are in general position, these equations will enable us
to determine 8 of the coefficients a y - as linear homogeneous func-
tions of the other two, by Cramer's rule; if these two be called k\
and fc 2 , the solution will be of the form Oy- = o/fci + a,/'/^, where
a,/ and a y -" are known. The equation of the quadric through the
given 8 points will then be Q(x t y, z) = 0, where Q(x, y, z) =
k\Q'(x, y y z) + fc 2 Q"(z, y, z). The surface Q = will therefore
belong to a pencil of quadrics determined by the quadrics Q' =
and Q" = 0, provided the latter two surfaces are distinct. It
will be worth our while to inquire further whether this will indeed
be the case. If we suppose that the value of the determinant of
8th order, which is formed by the first 8 elements in (3) for
i = 1, 2, . . . , 8, is different from zero (let us denote this deter-
minant by J5g), then we may take fci = a 3 4, fc 2 = a 44 ; and it follows
from Cramer's rule that an is equal to a fraction whose denominator
is equal to D 8 ; and the numerator is obtained from the denominator
by writing the column 2 a^y* + Q<u(i =1,2,. . . , 8) in place of
the column a,- 2 , i = 1, 2, . . . , 8. Hence if we denote by D 8 '

280 PROPERTIES OF CENTRAL QUADRIC SURFACES

and Z) 8 " the determinants obtained from D 8 by replacing its first
column by 2 7;, i = 1, 2, . . . , 8 and by a column of ones re-

Ds D 8 "

spectively, we find that an = -rr- X a 3 4 + -ry- X a 4 4 = kiQu + A; 2 an ,

D r D tf

where an' = ~- and an" = -~- . Similarly we find 22 = &ia 22 '

+ fc 2 a 22 / ', where a 22 ' and a 22 " are obtained from Z> 8 by replacing
the second column by the columns which were used in the first
column for the formation of TV and AJ" respectively, and dividing
by D 8 ; and so on for all the coefficients a# except a 34 and a 44 .
Finally, if we write a 34 ' = 1, a 34 " = 0, and a 44 ' = 0, a^" = 1, we
have also a 34 = k\a^ f + A; 2 a 34 ", and 44 = fcia 44 ' + & 2 a 44 ". Hence
we can write Q(x, y, z) = kiQ'(x, y, z) + k 2 Q"(x, y, z), where the
coefficients of Q 1 are not proportional to those of Q". It should
be clear that the same conclusion will be reached if we start from
the supposition that another 8th order determinant of the matrix
is different from zero. Therefore any quadric Q which passes
through the 8 given points belongs to the pencil of quadrics de-
termined by the quadrics Q' and Q" and passes through the curve
of intersection of these surfaces.

THEOREM 8. All quadrics which have In common eight points In
general position have a curve In common which passes through these
eight points.

A system of three quadrics Q(x, y, z) = 0, Q'(x, y, z) = and
Q"(x, y,z) = which do not belong to one pencil, that is, such
that the only values of fci, fe, and fc, for which the relation
kQ(x 9 y, z) = kiQ'(x, y, z) + k 2 Q"(x, ?/, z) holds identically are ki =
&2 = k 0, determines a linear two-parameter family of quadrics
kQ + k,Q f + k 2 Q" = 0, or Q + XQ' + M Q" = 0. Such a family
is called a bundle of quadrics. All surfaces of the bundle pass
through the points common to the three initial surfaces. These
equations will in general have 8 points in common, since their
equations are of the second degree in x, y, z. But these 8 points
are not in general position, since otherwise the three surfaces Q,
Q' and Q"' would, in accordance with Theorem 8, belong to one
pencil.

Suppose now that there be given 7 points in general position;
then seven of the coefficients a# can be determined in terms of the

FOCAL CURVES AND DIRECTRIX CYLINDERS 281

remaining three; if these be called fci, fc 2 , and fc 3 , we shall find
dij = k]fLi/ + k^dij" + kzdij" ', and therefore Q(x y y, z) = k\Q f
(x, y, z) + k<2,Q"(X) y, z) + kzQ'"(x, y, z). The determination of
the functions Q'> Q", and Q'" proceeds by a method analogous
to the one used in the proof of Theorem 8. On the supposition
that the determinant of 7th order whose rows are formed by the
first seven elements of (3) for i = 1, 2, . . . , 7 does not vanish,
we may take ki = a 2 4, fe = a 3 4, fc* = d^. Therefore if a# = k\(ii/ +
kidij" + k^dij" , we have d^ = 1, #24" = 0, #24"' = 0; d%t 0,
a 34 " = 1, a34 '" = 0; a 4 / = 0, a 44 " = 0, a 44 '" = 1. This shows
that the three surfaces Q' = 0, Q" = 0, and Q!" = do not
belong to the same pencil. Therefore any surface through the 7
given points belongs to the bundle of quadrics determined by these
three surfaces and will therefore pass through the eight points
common to them.

THEOREM 9. All quadrics which have In common 7 points In general
position also have an eighth point in common.

120. Focal Curves and Directrix Cylinders of Central Quadrics.

In Plane Analytical Geometry, the focus and directrix of a conic
section are defined as a point and a line such that the ratio of the
distances of any point on the curve from the focus and from the
directrix is the same as for any other point on the curve. A
corresponding definition of focus and directrix for the quadric
surfaces seems to have been given first by Chasles (in 1835) or
perhaps by Salmon (in 1862).

DEFINITION IV. A foeus and a directrix of a quadric surface are a
point and a line such that the ratio of the square of the distance of
any point P on the surface from the focus to the product of its dis-
tances from any two planes through the directrix is constant.*

Although the concepts defined in this manner are entirely defi-
nite, it is still an open question whether quadric surfaces possess
foci and directrices. We shall not deal with this question for the
general quadric but only for the central quadrics, which are not
surfaces of revolution.

* The planes through the directrix need not be real. The line of inter-
section of two planes may be real even though the planes themselves are not
real; for example, the planes x -f iy z = and x iy z = both contain
the line x = t, y = 0, z = t.

282 PROPERTIES OF CENTRAL QUADR1C SURFACES
If F(a, /3, 7) is a focus of the central quadric
(1) mix 2 + w 2 7/ 2 + nitf 2 = 1

(mi, w 2 , and m 3 being distinct and all different from zero), there
must exist two linear functions l(x, y, z) and /'(#, y> z), and a
constant c such that

(x - a) 2 + (y - 0)* + (z - 7) 2 = c X Z(a?, y, 0) X Z'fo

for every set of values of x, y, z which satisfy equation (1).

Consequently the functions mix' 2 + m 2 y 2 + m& 2 1 and (x a) 2
+ (y ~~ $Y + (^ T) 2 cZZ' must differ by a constant factor X;
that is, there must exist a factor X such that

niiX 2 + m 2 ?/ 2 + m&* - 1 - \[(x - a) 2 + (y - 0) 2 + (2 - 7)']
= \cl(x, y, z)l'(x, y, z).

Therefore the function on the left-hand side of this identity must
be equal to the product of two factors which are linear in x, y, and
z and the directrix which corresponds to the focus F(a, /3, 7) will
then be the line of intersection of the planes represented by the
equations which result when these linear factors are equated to
zero. According to Theorem 6, Chapter VIII (Section 96, page
206) the necessary and sufficient condition for the possibility of
thus factoring the function is that the rank of the discriminant
matrix of the quadric surface

mix* + 2 */ 2 + m*z* - 1 - \[(x - a) 2 + (y - 0) 2 + (z - 7) 2 ] =

is less than 3. The necessary and sufficient conditions which
will make certain that the point F(a, 0, 7) is a focus of the quadric
Q are therefore that the coordinates a, 0, 7 cause the determinant

i X 00 Xa

W2-X X0

w 3 -X X7

and all its three-rowed minors to vanish. But because the matrix
of this determinant is symmetric, the second condition may be
replaced, in view of Theorem 6, Chapter II (Section 26, page 43),
by the requirement that the principal three-rowed minors vanish.
If the principal three-rowed minor in the upper left-hand corner

FOCAL CURVES AND DIRECTRIX CYLINDERS 283

is to vanish, we must have (mi X) (m 2 X) (m 3 X) = 0,
that is, X = mi, or X = m 2 , or X = m 3 .

Let us consider the case X = mi. The principal minor formed
from the 1st, 2nd, and 4th columns and rows then becomes

m 2 mi m\& = mi 2 a' 2 (m 2 mi).

mifl -[l+m 1 (a 2 +^ 2 +7 2 )]

Since mi ^p and m 2 ^ m } , the vanishing of this determinant
requires that = 0; and when X = mi and a 0, the fourth-
order determinant and the three-rowed principal minor formed
from the 1st, 3rd, and 4th columns and rows obviously vanish
as well. There remains therefore the condition that the three-
rowed principal minor in the lower right-hand corner shall vanish,
that is, that

m 2 mi nii/3

m 3 mi

-[i-

0.

If we develop this determinant, we are led to the equation

(m 3 Wi)0 2 + WiW 3 (w 2 m\)y 2 + (ra 2 Wi) (m 3 Wi) = 0; we
write this equation in the form

(2)

J_

mi

+ 1 = 0.

This equation, in conjunction with the equation a = 0, determines
for varying /3 and 7 a real or imaginary conic section in the YZ-
plane; it is called a focal conic of the quadric surface.
If, conversely, (a, 0, 7) is a point on this conic section,

mix* + m 2 y* + m 3 z 2 - 1 - mjx 2 + (y - 0) 2 + (z - y) 2 ]

= (ra 2 mi)?/ 2 + (ra 3 mi)2J 2 + 2 mifty + 2 mi72 mi/3 2

[(m 2 - mi)y + mi/3] 2

1

m 3 mi

[(m 3

and this function is factorable into two factors linear in y and z
(the variable x is absent), with coefficients which are real or imagi-

284 PROPERTIES OF CENTRAL QUADRIC SURFACES

nary according as m 2 mi and m 3 mi are of opposite signs or
of like signs. In either case, the equations which are obtained by
equating these factors to zero represent planes which pass through
the line determined by the equations

(3) (m 2 m\)y + miff and (m 3 nii)z + m\y = 0.

This line and the point (a, 0, 7) on the conic section determined
above arc therefore a directrix and a focus of the quadric surface
(1). As the point (a, 0, 7) describes the conic (2), the line (3),
which is parallel to the X-axis describes a cylindrical surface,
whose equation is obtained when and 7 are eliminated from
equations (2) and (3). This cylindrical surface is called a direc-
trix cylinder of the quadric. We conclude therefore that from
the value X = mi, we obtain the focal conic

(4) - + -^-- + 1 = 0, x = 0,

and the directrix cylinder

77"

(5) m 2 (/ni w 2 )

i /LI ifi/i

The reader should have no difficulty in showing that from X = w 2
and X = m 3 , we obtain the focal conies represented respectively
by the pairs of equations

(6) -^- + -~- +1-0, y = 0,

and

x 2 ?/ 2

(7) j-+ i : j- + 1=0, z = 0;

ma mi m 3 m 2

and the corresponding directrix cylinders given respectively by
the equations

z 2 x 2

(8) m 3 (r?i 2 m 3 ) + mi(m 2 mi) = 1,

FOCAL CON1CS AND DIRECTRIX CYLINDERS 285

and

(9) Wi(w 3 wO h w 2 (w 3 w 2 ) = 1.

The results of this discussion are summarized in the theorem
which follows.

THEOREM 10. For every central quadric surface which is not a sur-
face of revolution there exist three real or imaginary focal conies and
three corresponding directrix cylinders; with every point on a focal
conic there is associated a generating line of a directrix cylinder, so
that point and line are focus and directrix of the quadric, as defined in
Definition IV.

121. Focal Conies and Directrix Cylinders, continued. We turn
now to a further consideration of the focal conies and directrix cyl-
inders for each type of central quadric, particularly with a view to
determining the conditions under which they are real.
CASE I. Ellipsoid.

If we take the equation in the standard form

71 + ^2 + ^2 = 1, P < q < r,

u 1 1 j 1 , 1 1 o 2

we have mi = , w 2 = -r, and w 3 = ; and = p- q 2

p 2 q 2 r 2 m\ w 2

< 0, = p 2 r 2 < 0. Hence equations (4) and (5) of

the preceding section give the real focal ellipse

V \Jy o i) O 9 -*->

q 2 p 2 r 2 p 2
and the corresponding real elliptic directrix cylinder

Equations (6) and (8) lead to the real focal hyperbola

z 2 x 2

and the associated real hyperbolic directrix cylinder

286 PROPERTIES OF CENTRAL QUADRIC SURFACES

Finally, equations (7) and (9) lead to an imaginary focal ellipse
and an imaginary elliptic directrix cylinder. We reach therefore
the following conclusion:

THEOREM 11. An ellipsoid possesses a real focal ellipse in the prin-
cipal plane determined by the two longer semi-axes, and a real focal
hyperbola in the principal plane determined by the two extreme semi-
axes. The associated directrix cylinders are real, elliptic and hyper-
bolic respectively, their generators being perpendicular to the planes
of the corresponding focal curves.

CASE II. Hyperboloid of One Sheet.
With the standard equation in the form

~2 9.2 ~2

-2 + ^ -~2 = 1 P<9>
p2 q2 r 2 r *7

we find

mi = - 2 , m 2 = -, m 3 = -- 2 -

From equations (4) and (5) of Section 120, we obtain the real
focal hyperbola

7/ 2 ? 2

r _ n y z = i

x " u ' q* - p 2 p 2 + r 2 '
and the real hyperbolic directrix cylinder

Equations (7) and (9) give the real focal ellipse

fy C\ _J J 1

& U, ; n ^T 01 O *

p 2 + r 2 q 2 + r 2
and the real elliptic directrix cylinder

The loci determined by equations (6) and (8) are imaginary in this
case. The following theorem states the results.

THEOREM 12. An hyperboloid of one sheet possesses a real focal
ellipse in the principal plane which cuts the surface in an ellipse, and
a real focal hyperbola in the principal plane determined by the con-
Jugate axis and the longer of the two transverse semi-axes. The as-
sociated directrix cylinders are real, elliptic and hyperbolic respectively ,
with generators perpendicular to the planes of the corresponding focal
curves.

FOCAL CONICS AND DIRECTRIX CYLINDERS 287

CASE III. Hyperboloid of Two Sheets.
The standard form of the equation

~2 7 /2 ~2

x v _ z = i, q<r

p2 q2 r 2

gives

1 1 1

mi = 5, m 2 - - -, w 3 = - ^-

Equations (6) and (8) yield the real focal hyperbola

__^ ___ z^___
2/ u > p* + q2 r 2 - q 2 '

and the real hyperbolic directrix cylinder

From equations (7) and (9) we obtain the real focal ellipse

/v2 /i2

^ ^J O i > I n i> J-

p 2 + T I r 2 q*
and the real elliptic directrix cylinder

The loci determined by equations (4) and (5) are imaginary in
this case. The conclusions are therefore as stated in the next
theorem.

THEOREM 13. An hyperboloid of two sheets possesses a real focal
ellipse In the principal plane determined by the transverse axis and
the shorter of the two conjugate semi-axes, and a real focal hyperbola
in the principal plane determined by the transverse axis and the
longer conjugate semi-axis; the associated directrix cylinders are real,
elliptic and hyperbolic respectively, with generators perpendicular
to the planes of the corresponding focal curves.

122. Exercises.

1. Determine the focal curves and the directrix cylinders of each of the
following surfaces:

, , x 2 2/ a z 2 ... x 2 ?y 2 z 2 '

(a) -9+7+6-1. (6) - + ---=1,

M ?! - nl _ l 2 - i

(c; 6 9 4

288 PROPERTIES OF CENTRAL QUADRIC SURFACES

2. Prove that an ellipsoid of revolution has a real focal circle in the prin-
cipal plane which cuts the surface in a circle and a real circular directrix
cylinder.

3. Prove that an hyperboloid of revolution of one sheet has a real focal
circle and a real circular directrix cylinder.

4. Prove that the focal ellipse of an ellipsoid is similar to the ellipse in which
the surface is cut by the plane of the focal ellipse if and only if the surface is
an ellipsoid of revolution.

6. Prove that the focal ellipse of an hyperboloid of one sheet is similar to
the ellipse in which the surface is cut by the plane of the focal ellipse if and
only if the hyperboloid is a surface of revolution.

6. Prove that the principal plane determined by the two shorter semi-axes
of an ellipsoid cuts the focal ellipse and the associated directrix cylinder in a
point and a line respectively which are the focus and the directrix of the ellipse
in which the surface is cut by this plane.

7. Prove that the semi-axes of the ellipse in which an ellipsoid is cut by the
plane of the focal ellipse are mean proportionals between the corresponding
semi-axes of the focal ellipse and of the directrix curve of the associated elliptic
directrix cylinder; also that the semi-axes of the ellipse in which an ellipsoid
is cut by the plane of the focal hyperbola are mean proportionals between the
corresponding semi-axes of the focal hyperbola and of the associated hyperbolic
directrix cylinder.

8. Prove theorems analogous to those of the preceding exercise for the
hyperboloid of one sheet and for the hyperboloid of two sheets.

9. Determine the distance from the origin of the points in which the focal
hyperbola of an ellipsoid is met by the planes of central circular section.

10. Determine the conditions under which the focal hyperbola of a central
quadric is a rectangular hyperbola.

11. Prove that the foci of the focal curves of an ellipsoid coincide with
the foci of the conic sections in which the surface is cut by the planes of these
focal curves.

12. Prove the corresponding theorem for the hyperboloids of one and two
sheets.

123. Confocal Quadric Surfaces. Elliptic Coordinates. It fol-
lows from formulas (4), (6), and (7) of Section 120 (see page 284)
that two central quadrics

= 1 and w/z 2 + m^y 2 + m 3 '2 2 = 1

will have their focal curves in common if and only if =

nii ntj

/ / > for i> 1 = 1> 2, 3.

m^ m/

This will certainly be the case therefore for all surfaces repre-

CONFOCAL QUADRIC SURFACES 289

sented by the equation

~2 ,,,2

in which X is a real parameter. For all these surfaces the focal
ellipse is given by the equations

+

;L ~~^ rt

p 2 r 2 p 2 '

and the focal hyperbola by the equations

z 2 x 2

y = 0, - - r -- - - = 1.
r 2 q 2 q 2 p 2

The family of surfaces represented by equation (1) is called a
confocal family of quadric surfaces.

If X < p 2 , all the denominators in equation (1) are positive;
the surface is therefore an ellipsoid. If p 2 < X < g 2 , the first
denominator is negative, the other two are positive, so that the
surface is an hyperboloid of one sheet, of which the X-axis is the
conjugate axis, If q 2 < X < r 2 , the first two denominators are
negative and the third one is positive; hence the surface is an
hyperboloid of two sheets, of which the Z-axis is the transverse
axis. Finally if X > r 2 , the surface is an imaginary ellipsoid.

For the critical values X = p 2 , X = g 2 , and X = r 2 , the equation
(1) has no meaning. If we multiply both sides of equation (1)
by p 2 X, we obtain the equation

which is equivalent to (1) except when X = p 2 . For this value of
X, equation (2) reduces to x 2 = 0, whose locus is the FZ-plane
counted doubly. If equation (1) is multiplied through by q 2 X
and by r 2 X, we obtain equations, which for X = q 2 and X = r 2
reduce respectively to the equations y 2 = and z 2 = 0. We
complete now the definition of the confocal family of quadrics
given by equation (1) by the statement that to the values X = p 2 ,
X = g 2 , and X = r 2 shall correspond the FZ-plane, thq ZX-plane,
and the .XT-plane respectively, each counted doubly. The
character of the surfaces in the confocal family is indicated dia-
grammatically in Figure 36.

290 PROPERTIES OF CENTRAL QUADRIC SURFACES

We shall now try to determine in what manner the surfaces of
the family change as X tends towards the critical values, passing
through values which remain steadily on one side of a critical
value. To indicate that X tends toward p 2 through values which
are greater than p 2 , we shall write X > p 2 + 0; to indicate that
X tends toward p 2 through values which are less than p 2 , we shall
write X p 2 0. Similar meanings are to be attributed to the
notations X - q 2 + 0, X -> q 2 - 0, X - r 2 + and X - r 2 - 0.

,Q p* 0* r^ a-axis

KlHpsoid Hyperboloid Hyperboloid Imaginary KUipsoid

I of 1 sheet I of 2 sheets I

Pu
N

FIG. 30

As X-p 2 0, the surface is steadily an ellipsoid; its semi-
axis along the Z-axis tends to zero. Since X < p 2 , the factor
p 2 X in the second term of equation (2) is positive; it should

y 2 z 2

therefore be clear from this equation that -~r + - - - 1< 0.

q 2 X r 2 X

But points in the FZ-plane for which this inequality holds lie

y 2 z 2

on the inside of the ellipse 9 x + -5 - - = 1.* Hence as

^ q 2 X r 2 X

X > p 2 0, the surface tends toward those points of the YZ-
plane which lie on the inside of the focal ellipse

i
T~

~ ,

q 2 p 2 r

* To be convinced of this fact, it is only necessary to observe that at the

v 2 z 2

origin, the function -~ r + -5 - - 1 reduces to 1 and that since the

C[ A T ~~~ A

function is a continuous function of X, y, and z for all values of X which differ
from q 2 and r 2 , it cannot change from negative values to positive values without
becoming zero. Since this can take place only on the ellipse, points for which
the function is negative lie on the same side of the curve as the origin, that is,
on the inside of the ellipse; and points for which it is positive lie on the out-
side of the ellipse.

CONFOCAL QUADRIC SURFACES 291

As X p 2 + 0, the surface is steadily an hyperboloid of one sheet,
whose semi-axis along the X-axis tends toward zero. But now

7/2 pj"

p 2 X is negative, and therefore -=-^ - + -5 - - 1 > 0.

q X T A

Therefore as X > p 2 + 0, the surface tends toward those points
of the FZ-plane which lie outside the focal ellipse.

As X > q 2 0, the surface is always an hyperboloid of one sheet,
whose semi-axis along the F-axis tends to zero. For those values
of X, q 2 X is positive; it follows then from an equation analogous

x 1 z 2

to (2) that 2 _ + 2 _ \ 1 is negative. An argument

similar to the one made in the footnote on the preceding page
shows that points for which this inequality holds lie on the same

x 2 z 2

side of the hyperbola -r - - + -= - - = 1 as the origin. If we
p 2 X r 2 X

call this the inside of the hyperbola, we conclude that as X g 2 0,
the surface tends toward the points of the ZJST-plane which lie
on the inside of the focal hyperbola

n * X * i

ffl II _ _ I

y v, t> n 4> i.
r 2 q 2 q 2 p 2

And the same reasoning shows that as X > q 2 + 0, the surface
tends toward the points of the ZX-plane which lie outside the
focal hyperbola.

Finally, as X r 2 0, the surface is an hyperboloid of two
sheets, whose semi-axis along the Z-axis (that is, the transverse axis)

x 2 y 2

tends to zero. Since r 2 X > 0, it follows that - - + -~ r

*p A q X

1 < 0; but now X > p 2 and X > q 2 and therefore this inequality
is satisfied by all points in the -XT-plane. Consequently as
X r 2 0, the surface tends toward the entire .XT-plane.
We summarize the discussion by a theorem.

THEOREM 14. The equation * , + ~~ - + ~ - = 1, p < q < r,

p* A q* A r* A

in which X Is a real parameter, represents a confocal family of quadrlc
surfaces. As X Increases from negative Infinity to p 2 , t.he locus of
the equation Is an ellipsoid which tends toward the inside of the
focal ellipse of the family; as X increases from p 2 to q\ the locus is
an hyperboloid of one sheet tending from the outside of the focal
ellipse to the inside of the focal hyperbola of the system; as X increases

292 PROPERTIES OF CENTRAL QUADRIC SURFACES

from q 2 to r 2 , the locus is an hyperboloid of two sheets, which tends
from the outside of the focal hyperbola to the entire XY-plane. For
x = p 2, x = q% an( | x = r % the locus is respectively the YZ-plane, the
ZX-plane, and the XY-plane, each counted doubly.

We shall now prove two properties of confocal families of
quadrics.

THEOREM 15. Through every point In space that does not lie on one
of the coordinate planes, there pass three surfaces of every confocal
family of quadrics, namely, an ellipsoid, an hyperboloid of one sheet
and an hyperboloid of two sheets.

Proof. Let P(a, 0, 7) be an arbitrary point of space that does
not lie on any coordinate plane; then a, 0, and 7 are all different
from zero. If P is to lie on a surface of the confocal family repre-
sented by equation (1), the parameter X must be so determined that

a 2 6 2 v 2

__rL_4_ p i T _ i = n-

p 2 - \ ^ q 2 - \ ^ r 2 - X
that is, X must be a root of the equation

F(\) = (X - p 2 )(X - ? 2 )(X - r 2 ) + (X - <? 2 )(X - r 2 ) a 2 + (X - r 2 )
(X - pW + (X - p 2 ) (X - <? 2 )7 2 = 0.

This is a cubic equation in which the coefficient of X 3 is +1;
consequently for large positive values of X, F(\) > and for large
negative values of X, F(\) < 0. Moreover

= (q 2 -r 2 ) (q 2 -p 2 )P 2 <0;

The graph of the function F(\) will therefore have the general
character indicated in Fig. 37.* And from it we conclude that
the equation F(\) = has three real roots, Xi, X 2 , and X 3 . Hence
there are three surfaces of the confocal family which pass through
the given point P(a, 0, 7). But we observe also from Fig. 37
that Xi < p 2 , p 2 < X 2 < <? 2 , and (f < X 3 < r 2 ; therefore, in virtue
of Theorem 14, one of these surfaces is an ellipsoid, one an hy-
perboloid of one sheet, and one an hyperboloid of two sheets.

* We are here assuming that the polynomial F(X) is a continuous function
of X, as in the argument in the footnote on page 290 we assumed that a rational
function is continuous except for a finite number of values of the independent
variable. A satisfactory proof of these facts is found in treatises on the
Theory of Functions of a Real Variable.

CONFOCAL QUADRIC SURFACES

293

If a = 0, the root Xi becomes equal to p 2 , so that in place of the
ellipsoid we have the FZ-plane counted doubly; similarly, if
ft = 0, the hyperboloid of one sheet is replaced by the double
ZX-plane, and if 7 = 0, the hyperboloid of two sheets is replaced
by the double XY-plane.

Fa)

FIG. 37

COROLLARY. If P(, p, 7) lies on one or more of the coordinate planes,
it is still true that three surfaces of every confocal family pass through
P; but one or more of the central quadrics of the family are then re-
placed by the coordinate planes in which P lies.

THEOREM 16. The three quadrics of a confocal family which pass
through an arbitrary point P(, p, 7) in space are mutually orthogonal
at P.

Proof. Suppose first that P(a, 0, 7) does not lie on any co-
ordinate plane. Then the three quadrics of the confocal family
(1) which pass through P have the equations

i = 1,2,3,

P' 2 ~ X,

where Xi, X 2 , and X 3 are the roots of the equation F(\) = 0, dis-
cussed above. The equations of the tangent planes to these
surfaces at the point P are

ax . fly

(3)

yz

= 1, i = 1,2,3.

- X f - ' q* - X,- ' r 2 - X,-
Since P lies on each of the three surfaces, we have moreover

X,- q~ X f -

r 2 _

= 1, i = 1, 2, 3.

294 PROPERTIES OF CENTRAL QUADRIC SURFACES

If we subtract any two of the last three equations from each other,
we find

___ 1

- \i g - Xj

- X,- p' - X,

' U- 1,2, 3; ,

A simple reduction transforms these three equations to the fol-
lowing form:

But since X,- 4= X/, we conclude from this last equation that

cP_ , ^ - T; __ n

(p 2 X,-) (p 2 X;) (q~ X,-) (g 2 X,-) (r 2 X,-) (r 2 X,-)

And this equation expresses the fact that any two of the tangent
planes represented by equations (3) are perpendicular to each
other (compare the Corollary of Theorem 9, Chapter IV, Section
46, page 82).

If P lies in a coordinate plane, one of the quadrics of the family
which pass through it is that plane itself. Let us suppose that
a = 0; then the surface in question is the FZ-plane counted
doubly. And let the equation

A. 2 72 2

I y I ^ ^

p~ \z q 2 \z T 2 ~~ Xa

be one of the non-degenerate quadrics passing through P; the
tangent plane to this surface at the point (0, 0, 7) is represented
by the equation

@y a. ** z i

This plane is parallel to the X-axis and therefore perpendicular to
the double FZ-plane. If P lies on a coordinate axis, two of the
quadrics degenerate into double coordinate planes; and these are
surely perpendicular. Our theorem has therefore been proved.

* * *

From Theorems 15 and 16, it follows that the quadric surfaces
of a confocal family cover the whole of space with a network of

ELLIPTIC SPACE COORDINATES 295

mutually perpendicular surfaces. To each of these surfaces a
number is attached, namely, the value of the parameter X to which
it corresponds; and for every point P in space there are three such
numbers. These numbers are called the elliptic space coordinates
of the point P. Our discussion has therefore shown that every

x 2 ?/ 2 2 2
ellipsoid -5 + H ^ ^ niay be made the basis of a system of

elliptic space coordinates. We have obtained a frame of reference
which generalizes in a remarkable way the Cartesian frames of
reference with which we began our study of Solid Analytical
Geometry in Chapter III.

And this return to our starting point provides a suitable stop-
ping point, ending in the key in which we began.

In our journey through this book we have examined a few ques-
tions in some detail and we have had a glimpse of many things
which lay outside our path. It is the author's hope that the reader
may have learned to appreciate the beauty and the power of the
theory of determinants and matrices, and that he may experience
the desire not only to continue the study of the subject to which this
book is primarily devoted, but also to enter some of the fields,
such as Projective Geometry and the Theory of Functions of a
Real Variable, to which we have had occasion to allude now and
then in the course of our work.

APPENDIX

(Compare Section 84, page 171)
To prove: If X Is eliminated from the equations

Li = 1 [\Qi(, ft 7) + M<M, ft 7) + *<?(i ft 7)] =
and Lo = ttnX 2 -f a 22 M 2 -f ass** 2 -H 2 azzjuip -f 2 a 3 n/X 4- 2 a^A/* = Of

the resulting quadratic equation In /* and v is

On 012

<?.
<?*

Oil

023

<?1

<?3

an

<?3

Proof. To simplify the writing we shall treat this problem in a slightly
modified form. The given equations are clearly equivalent to the non-homo-
geneous equations ax + by + c = and p\\x z + /?22?/ 2 + Pss H~ 2 p 2 s^ -f-

= obtained by writing x and y in place of - and - respectively,

and using general coefficients. We assume now that a =t= and solve the
linear equation for x in terms of y; substitution of the result in the second
degree equation leads to the following quadratic in y:

pii(fy/4-c) 2 -2 puay(by+c)+a 2 pMy 2 -2 p 13 a(by+c)+2 7> 23 a 2

upon reduction this equation becomes

(pub* 2 p u ab + z>2 2 a 2 )?/ 2 -f 2 (p n bc - p i2 ac - p n ab -f
2 p^ac -f Pasa 2 = 0.

Direct expansion of the third order determinants shows that the coefficients
differ in sign only from the respective determinants:

Pii

Pit

?>22

b

Pn
Pis
a

Pi2 a

P'2S C

b

and

Pn
Pi*
a

Pl3
P33
C

If we now return to the homogeneous form of the given equations and to the
coefficients as given, we have completed the proof.

II

(Compare Section 84, page 174)
To prove: If A*(Q) = and ^ 22 (^) = A 23 (Q) = A*(Q) = 0, then every

third order, minor of the determinant

a u

a 22 023
a 2 s ass

vanishes;

A,(Q) designates the value of this determinant and Aij(Q) the value
of the cofactor of the element aij.

296

APPENDIX 297

Proof. We shall use the notation Qi to designate the cofactors of the ele-
ments Qi in the determinant. Since A$(Q) = 0, it follows from the Corollary
of Theorem 5, Chapter II (Section 26, page 43) that A u (Q)An(Q) - A 12 Z (Q)
= and Aii(Q)Aw(Q) A\^(Q} = 0, so that the hypothesis leads at once to
the result that Aw(Q) = A\*(Q) 0. In the same way we find that ^4 22^44
Q 2 2 = 0_ and ^33^44 Q 3 2 = 0, so that also Q 2 -^ Q* = 0. Moreover
QiQi + QzQz -f QaQa = At(Q) 0; and therefore, since we are working on
the hypothesis that Qi =J= (compare page 171, opening paragraph of Case I),
it follows that Qi = 0. Finally we observe that, in virtue of ThftormrMS,
Chapter I_(Section 7, page 13), QiAn + QzAiz + QaAu - and auQ { -f-
aizQz + auQz 4- QiAu = 0; and from these equations we conclude that
An = Au = 0. This completes the proof of our statement.

Ill

(Compare Section 87, page 185)
To prove: q(a ls , 23, #33) = 0, If A u = 0.

Proof. Here aij are the cofactors of the elements 0$ in the third order
determinant ^.44 and q(x, y, z) is the homogeneous function of the second
degree introduced on page 159. By the use of Euler's theorem on homogeneous
functions (see footnote on page 161), we find

2 <K13, 23, 33) = Ofl3tfl(<*13i 23, Ot^) + a^falS, 23, 3s) + 33<?s(13, 23, 3s) ,*

and

, <x 2 3, ass) = 2 (anaw -f ai 2 a 23 + flisass) = 0,

, 23, ^33) = 2 (ai 2 13 + 22a 23 + 2333) = 0,
<7 3 (13, 23, ^33) = 2 (ai313 + 02323 + 3333) = ^44 = 0,

by Theorems 13 and 12, Chapter I (Section 7, page 13).

IV

(Compare Section 94, page 205)
To prove: If r 4 = 1, then r 4 ' <2.

Proof. Here r 4 is the rank of the discriminant matrix of the quadric surface
Q and r 4 ' is the rank of the discriminant matrix of the equation Q'(x', y', z')
obtained by rotation of axes from the equation Q(x, y, z) = 0.

If r 4 = 1, A, D 3 , and D 2 vanish and therefore, by Theorem 4, Chapter VIII and
its Corollary (Section 94, pages 203 and 204), A r = /V = /V = 0. It follows that
every three-rowed minor of A' vanishes and that the sum of the principal two-
rowed minors also vanishes. Since the three-rowed principal minors are them-
selves symmetric, we can apply to each of them Theorem 7, Chapter II (Section
26, page 44) ; hence the two-rowed principal minors of any one of the four three-
rowed principal minors are of like signs, and since any two of these three-rowed
principal minors have a two-rowed principal minor in common, all the prin-
cipal two-rowed minors have the same sign. It follows then from the fact
that ZV = that every two-rowed principal minor of A ; vanishes. Now we

298

APPENDIX

apply Theorem 6, Chapter II (Section 26, page 43) to each of the three-rowed
principal minors; and we conclude that every two-rowed minor of A' which is
also a minor of a three-rowed principal minor must vanish. It remains to
consider the two-rowed minors of A' which do not occur in any three-rowed
principal minor; the only ones of this kind are the minors

014

flu

34'

and

an'
043'

which do not have any element of the principal diagonal of A'. To show that
these minors vanish also, we consider the three-rowed minors:

a,/
024'
034'

and

a 23

a 44

These determinants vanish and all their two-rowed minors vanish, except
possibly the minors with which we are concerned; and of these, two occur as
minors in each of the two three-rowed minors. If we write down the de-
velopments of these determinants according to their last rows, we can conclude
that the first of these two-rowed minors also vanishes. In a similar way,
consideration of the pairs of three-rowed minors /I 2 i', A\i, and An', A 2 i'
shows that the remaining two-rowed minors vanish. This completes the
proof of the proposition.

(Compare Section 96, page 206)
To prove: The determinant

2a<u

abi -f- a\b
aci -f- a\c
CM /i + a ^/

abi + a^
2 661
bci + 6iC
6r/! -f- bid

aci + ic
6ci -f bic
2cc,

C/i -f Cid

atfi 4- aid
brfi 4- bid
cdi + <?irf

and Its three-rowed principal minors vanish.

Proof. Theorem 8, Chapter I (Section 5, page 9) enables us to write this
determinant as the sum of 2 4 fourth order determinants whose elements are
the product of one of the numbers a, 6, c, or d by one of the numbers a\ y 61, Ci,
or d\. A somewhat careful inspection shows that in every one of these 2 4
determinants at least two columns are proportional; for, after common factors
have been removed from the elements of the columns, these columns must
consist either of the numbers a, 6, c, d, or else of the numbers i, 61, Ci, d\.
Consequently, the value of the given determinant is zero. And every one of
the three-ro\yed principal minors can be written as the sum of 2 s three-rowed
determinants, in each of which there are at least two proportional columns.

The reader should have no difficulty in carrying out the details of this
proof; he is urged to write down explicitly a number of the simpler deter-
minants into which those under consideration are broken up.

APPENDIX 299

VI

(Compare Section 96, page 208)

To prove: If r 4 = 2, not all the principal two-rowed minors of the
matrix a 4 can vanish and those which do not vanish are of one sign.

Proof. The reader should have no difficulty in proving this statement on
the basis of Appendix IV.

VII

(Compare Section 105, page 241)

a(pi 2 p 2 2 ) a Ofi 2 rj 2 2 ) a

To prove: The determinant

4 p 2 2 ) fe(qi 2 4-

= 0,

if for a, 0, 7 there are substituted the coordinates of an arbitrary point
on the line

and for i, 0i, yi the coordinates of an arbitrary point on the line

Proof. If we substitute a, (3, y for x, ?/, z in the equations of the first line,
we obtain a pair of linear equations, which may be solved for and by
Cramer's rule; for the coefficient determinant of these equations with respect

to - and - is equal to p L 2 -f 7*2* =N 0. We find
a c

and = 2

= (Pi 2 - P2 2 ) + 2 p lpa 4- ( Pl -f

= [ 2
In a similar manner we obtain from the equations of the second line:

~ = [(<7i 2 - <?2 2 ) + 2 r M2 ] -s- fe' 4- 92 s ),
and -^ = I 27172-^ + q^ r/ 2 2 4- ((/r -f q.>~).

If we subtract the corresponding equations of those two sots, we can deter-
mine a ai and 7 71. We substitute those values of a a\ and 7 71
in the determinant D and make the obvious simplifications; thus we obtain
the following result:

D =

(Pi 2 4 P2 2 ) (?i 2 4- <72 2 )

Pi 2 - P2 2 ?1 2 - ?2 2 (qi 2 4- 72 2 ) (Pi 2 - 7>2 2 )0 - (Pi 2 4 P2 2 ) (7l

0i 4 2 6pip 2 (<7i 2 4 72 2 ) - 2 fytf 2 (pi 2 4
Pi 2 4- p 2 2 7i 2 4- ?2 2 (pi 2 + p 2 2 ) (7i 2 4 ?2 2 ) (0 - 00

P1P2 -qiq*

300 APPENDIX

To the third column of this determinant we add the product of the first
column by (qf + q< 2 *)0 and the product of the second column by (pi* -f-
then we add the second row to the first. Thus we find:

4abc

(Pi 2 4- 7>2 2 ) (<7l 2 4- ?2 2 )

Pi* </i 2 Pip2(gi 2 4- ?2 2 ) - <M2(/?i 2 4- P2 2 )

X

4- /> 2 2 ) (<7i 2 4- 7

- PiV) (/>i V - 7>*V) - /'iWfar 4- tt 2 ) 2 + <7i V(Pi 2 + P2 2 ) 2 ]
= 0.

VIII

(Compare Section, 109, page 245)

To prove: If li/?i, Ij/^ , . . . are chords of a conic section which pass
through a fixed point P and if the products PA l PD 19 PA 2 PB 2 , . . . are
all equal, no matter what point P is taken in the plane of the conic
section, then this conic section is a circle.

Proof. We take a plane Cartesian frame of reference in the plane of the
conic section. Let the equation of the conic with respect to this reference
frame be

C(x, y) = anx 2 + 2 a n xy + a^y 2 -\- 2 a^x + 2 a^y H- 33 = 0,

and let P(a, ff) be an arbitrary point of the plane. We write the equations
of an arbitrary line through P in the parametric form as follows:

x = a + Is, y = + ms;

here s is the parameter which designates the length of the segment of the line
from P to the variable point (x, /y); I = cos 0, m = sin 0, where is the in-
clination of the line. We find then that the distances from P to the points
A and B in which the line meets the conic arc the roots of the equation

s*c(l, m) + s[C,(, p)l + C 2 (, ft)m] 4- C(, 0) = 0,

where c(#, y) = a n x' 2 4- 2 a^?/ 4- a-wy 2 * and (?!, C 2 are the partial derivatives
of C(x, y) with respect to :c and y respectively (compare Sections 76 and 80).

Therefore PA PB = "' ! ; and we have to show that if c(Z, m) is inde-
c(L) ni)

pendent of I and m, then the locus of C(x, y) = must be a circle.
For = 0, we have I = 1, m = and c(Z, m) = an;
for = 90, we have I = 0, m = 1 and c(7, m) = a 2 2 ;

for $ = 45, we have I = ^?, m = ^ and c(/, m) = ^ + a 12 4- ~

&

Therefore, if c(l, m) is independent of the direction of the line, we must have
an = a 2 2, and a\ 2 = 0. The equation of the conic reduces then to the form

APPENDIX 301

Oii( 2 -h y z ) + 2 aisx H- 2 a^y + 44 = 0. And if an = 0, the locus of this
equation is indeed a circle. If an = 0, but a^ and a 2 a do not both vanish, the
locus is a straight line; and if an an a^ = 0, the equation has no finite
locus. From the point of view of Protective Geometry, the locus consists,
in these two cases of a finite line together with a line at infinite distance, and
of a line at infinite distance counted doubly. And these pairs are also
recognized as circles; we shall refer to them as degenerate circles.

INDEX

(The numbers refer to pages.)

Adjoint of a determinant, 28

, minor of the, 29
Adjoint of a vanishing determinant,

42

Admissible values, 56
Algebraic complement, 17
a.m., 36

Anchor ring, 135
Angle between line and plane, 79

between two lines, 63, 64

between two planes, 82, 198
Angles, direction, 55
Asymptote of a quadric surface, 160
Asymptotes of an hyperbola, 141
Asymptotic cone, 183, 185

, equation of the, 184

direction, 160
Augmented matrix, 36
Axes, coordinate, 49

of ellipse, 140

of hyperbola, 141

-, rotation of, 118, 211, 213
, translation of, 114, 210
Axis, conjugate, 140, 141, 145

of parabola, 141

of revolution, 131

, transverse, 140, 141, 145
, X-, 49
, Y-, 49
, Z-, 49

B

Bundle of planes, 94

Cartesian coordinates, 49

, oblique, 113
Center, improper, 176, 177

Center of quadric surface, 176, 177,

178, 210

Center, proper, 176, 179, 180
Central quadrics, 179, 180
Circles on ellipsoid, 251

on hyperboloid of one sheet, 252

on hyperboloid of two sheets,
252

on quadric surfaces, 244, 247,
250

Circles, parallel, 131
Circular cone, 133

cylinder, 69, 133

section, 244

Classification of quadric surfaces, 214,

220, 227, 229, 230
c.m., 35
Coefficient determinant, 36

matrix, 35

Cofactor of an element of a deter-
minant, 12

Coincident planes, 81, 102, 209
Collinear points, 57
Column index, 2

of a determinant, 2

of a matrix, 16
Complementary minor, 17
Cone, asymptotic, 183

, circular, 133

, elliptic, 145

, imaginary, 145, 185, 220

, quadric, 166, 175, 179, 180, 185,
220, 261, 265

, tangent, 167
Confocal family of quadrics, 289, 291,

292

Conic, focal, 283, 285, 286, 287
Conical surface, 136, 137, 175
Conicoid, 159
Conjugate axis, 140, 141, 145

303

304

INDEX

Conjugate diameters of ellipsoids, 271,

274

Conjugate, harmonic, 164, 165
Conjugate hyperbolas, 141

set of diameters, 270

set of diametral planes, 270
Contour lines, 138

Contour map, 138
Coordinate axes, 49

parallelepiped of a point, 51,
118

parallelepiped of two points,
54

planes, 49

systems, 49, 108

X-, 50
-, Y-, 50

, Z-, 50

Coordinates, cartesian, 49
Coordinates, cylindrical, 111

, elliptic space, 295

, oblique cartesian, 113

of a point, 50
, origin of, 49
, spherical, 108

, transformation of , 110, 112, 115,

120, 121, 123
Cosines, direction, 55
Courbes gauches, 157
c.p. of one point, 51
c.p. of two points, 54
Cramer's rule, 37
Curve, 68

, equation of, 68

, meridian, 131

, twisted, 157
Cylinder, elliptic, 223, 261, 263, 265

, enveloping, 271

, hyperbolic, 223, 261, 263

, imaginary, 223

, parabolic, 225, 262, 263
Cylindrical coordinates, 111
Cylindrical surface, equation of, 69

, oblique, 69

, right, 69
Cylindrical surfaces, 69

D

Degenerate locus, 83
Degree of a polynomial, 126
Degrees of freedom, 67
Derivative of a determinant, 31
Descartes' rule of signs, 215
Determinant, 1

, adjoint of a, 28

, adjoint of a vanishing, 42

, coefficient, 36

, cof actor of an element of a, 12

, columns of a, 2

, derivative of a, 31

, diagonals of a, 2

, expansion of a, 3

, Laplace development of a, 20

, minor of a, 17

, minor of an element of a, 12

, notations for a, 2

of a matrix, 16
, order of a, 2

, orientation, 106, 124

, rows of a, 2

, symmetric, 33, 42
Determinant, value of a, 3, 13
Determinants, product of two, 25

27

Development, Laplace, 20
Diagonals of a determinant, 2
Diameter of a quadric surface, 268
Diameters, conjugate, 270
Diametral plane, 186
Diametral planes, conjugate, 270
Direction angles of a line, 55

, asymptotic, 160
Direction cosines of a line, 55, 87
Directions, principal, 187, 189, 194,

211, 213
Directrix cylinder, 284, 285, 286, 287

of a conical surface, 136

of a cylindrical surface, 69

of an ellipse, 140

of an hyperbola, 141

of a parabola, 141

of a quadric surface, 281

INDEX

305

Discriminant matrix, 175

of a quadric surface, 166, 203, 204
Discriminating equation, 189, 190,

192
Discriminating numbers, 189, 192,

200, 213
Distance between two points, 51, 54,

117

from a plane to a point, 77, 198

Eccentricity, 140, 141
Element of a determinant, 2

, cof actor of an, 12

, minor of an, 12
Element of a matrix, 16
Elementary transformation of a ma-
trix, 18
Ellipse, 140

, axes of an, 140

, directrices of an, 140

, eccentricity of an, 140

, foci of an, 140

, vertices of an, 140
Ellipsoid, 142, 215, 217, 251, 256, 264,
272, 275, 286

, imaginary, 145, 215, 217

of revolution, 133, 215
, semi-axes of an, 142

Elliptic cone, 145

Elliptic coordinates, 295

Elliptic cylinder, 223, 261, 263, 265

Elliptic paraboloid, 146, 218, 220, 257,

265

Enveloping cylinder, 271
Equation, discriminating, 189, 190,
192

, linear, 71

, locus of an, 68

of an asymptotic cone, 184, 185

of a plane, 71

of a plane, intercept form of the,
73

of a plane, normal form of the,
78

Equation of a plane, three-point form

of the, 74

Equations, equivalent, 78, 137
, homogeneous, 35

of a curve, 68

of cylindrical surfaces, 69

of a line, 83, 84, 85, 86

of a line, symmetric, 60

of surfaces of revolution, 132
, parametric, 86

, systems of homogeneous, 38,

39, 41
, systems of non-homogeneous,

36, 38, 44

Equivalent equations, 78, 137
Euler's theorem on homogeneous

functions, 161
Expansion of a determinant, 3

F

Factorability, 209

Focal conic, 283, 285, 286, 287

Focal curves, 281

Focus of the ellipse, 140

of the hyperbola, 141

of the parabola, 141

of the quadric surface, 281
Form, binary, etc., 221

, cubic, etc., 221
, negative definite, 221, 229
, positive definite, 221, 229
, quadratic ternary, 221, 229

Four planes, 101

Frames of reference, 49, 108

Freedom, degrees of, 67

G

General position of a set of points,
279, 280, 281

Generating line of a cylindrical sur-
face, 69

Generatrix of a cylindrical surface, 69

Geometric characterization of quadric
surface, 232

306

INDEX

H

Harmonic conjugates, 104, 165
Homogeneous equation, 35, 136, 137
Homogeneous equations, system of,

38, 39, 41

Homogeneous function, 209
Homogeneous functions, Euler's the-
orem on, 161
Hyperbola, 141
, asymptotes of the, 141
, axes of the, 141
, directrices of the, 141
, eccentricity of the, 141
, foci of the, 141
, vertices of the, 141
Hyperbolas, conjugate, 141
Hyperbolic cylinder, 223, 261, 263
Hyperbolic paraboloid, 147, 218, 220,

242, 259

Hyperboloid of one sheet, 140, 142,
215, 217, 235, 242, 252, 256, 264,
286

of revolution, 133, 215

of two sheets, 144, 215, 217, 252,
257, 264, 287

I

Imaginary cone, 145, 220

cylinder, 223

ellipsoid, 145, 215
Index, column, 2

, row, 2

Infinity, plane at, 73
Intercept form of the equation of a

plane, 73

Intercepts of a plane, 73
Interchange of numbers in a row of

integers, 6

Interchange of two columns (rows), 8
Intersecting planes, 81, 208, 209
Intersection of 'two planes, 87

of a surface and a line, 150, 153
Invariant relations, 197
Invariants, 197

Invariants of a quadric surface, 199,

205, 255
Inversion, 3

Laplace development, 20
Latitude, 108
Left-handed system, 50
Line and plane, 97, 99

and quadric surface, 160

, angle between a plane and a, 79
, equations of a, 83, 84, 85, 86
, intersection of a surface and a,
150, 153

normal to a surface, 155, 156

of intersection of two planes, 87

of proper centers, 179

of vertices, 179

on a quadric surface, 160

, parametric equations of a, 86
, symmetric equations of a, 60

tangent to a surface, 154, 161
Linear equation, 71

families of quadrics, 277

transformation, 126
Lines, angle between two, 63, 64

, perpendicular, 64
, two, 104
Locus, degenerate, 83

of an equation, 68
, symmetric, 137

Longitude, 108

M

Matrix, 16

a 3 , 177, 178

a 3 , rank of the, 177, 178, 200,
208, 209, 217, 220, 223, 225

a 4 , 173, 178

a 4 , rank of the, 173, 178, 201,
208, 209, 217, 220, 223, 225

, augmented, 36

b, 178, 186, 225
, coefficient, 35
, column of a, 16

INDEX

307

Matrix, discriminant, 175
, elementary transformation of a,

18*

, elements of a, 16
, minor of a, 17
, notations for a, 16

of a determinant, 16
, rank of a, 16

, rows of a, 16

, singular square, 43

, square, 10, 43

, symmetric square, 43
Meridian curve, 131
Minor, algebraic complement of a,

17
Minor of a determinant, 17

of a matrix, 17

of an element of a determinant,
12

of the adjoint of a determinant,
29

, principal, 17
Minors, complementary, 17

N

Nappes of a surface, 133, 136
Non-homogeneous equations, system

of, 36, 38, 44
Normal form of the equation of a

plane, 78

Normal to a surface, 155, 156
Notations for determinants, 2

for matrices, 16
Numbers, discriminating, 189, 192

O

Oblate spheroid, 133

Oblique cartesian coordinates, 113

Order of a determinant, 2

of a surface, 154
Orientation determinant, 106, 124
Origin of coordinates, 49
Orthogonal transformation, 123

P

Parabola, 141
, directrix of the, 141
, focus of the, 141
, vertex of the, 141
Parabolic! cylinder, 225, 262, 263
Paraboloid, elliptic, 146, 218, 220,

257, 265

, hyperbolic, 147, 218, 220, 242,
259

of revolution, 133
Parallel circles, 131
Parallel planes, 81, 209
Parallelepiped, coordinate, 51, 54, 118

, volume of the, 118, 274
Parametric equations of the line, 86
Pencil of planes, 92, 96
Perpendicular lines, 64
Perpendicular planes, 82
Plane and line, 97, 99

, angle between a line arid a, 79

at finite distance, 73

at infinity, 73
, diametral, 186

, distance from a, 76, 77
, equation of a, 71
, intercepts of a, 73
, normal form of the equation of
a, 78

of proper centers, 179

of vertices, 179

, polar, 162, 165, 167
, principal, 187, 189, 194

section of a surface, 127, 128

tangent to a surface, 155, 161,
162

, three point form of the equation

of a, 74

Planes, angle between two, 82, 198
, bundle of, 94
, coincident, 81, 102, 209
, coordinate, 49 ,
, four, 101

, intersecting, 81, 208, 209
, line of intersection of two, 87

308

INDEX

Planes, parallel, 81, 209

, pencil of, 92, 96

, perpendicular, 82

, three, 90

, two, 82, 206
Point, distance from a plane to a,

77

Polar plane, 162, 165, 167
Pole, 162
p-regulus, 236

Principal diagonal of a determinant, 2
Principal directions, 187, 189, 194,

213

Principal minor, 17
Principal planes, 187, 189, 194, 211
Prism, triangular, 96
Product of two determinants, 25

, columns by columns, 27

, columns by rows, 27

, rows by columns, 27

, rows by rows, 27
Projection method, 61, 76, 116, 119
Prolate spheroid, 133
Proper quadric cone, 179

Q

q-regulus, 236

Quadratic form, ternary, 221, 229

Quadric, central, 179

Quadric cone, 166, 175, 179

Quadric surface, 159
, asymptote of a, 160
, asymptotic direction of a, 160
, center of a, 176, 177, 178
, circles on a, 244, 247, 250
, diameter of a, 268
, directrix cylinder of a, 284, 285,

286, 287

, directrix of a, 281
, discriminant of a, 166, 203, 204
, enveloping cylinder of a, 271
, focal conies of a, 283, 285, 286,

287

, focus of a, 281
, invariants of a, 199, 205, 255

Quadric surface, line tangent to a, 161

, normal to a, 161

, plane tangent to a, 161, 253

, polar plane of a point with re-
spect to a, 162

, pole of a plane with respect to
a, 162

, singular, 166, 175

, umbilics of a, 253

, vertex of a, 171, 177, 178

, a line and a, 160
Quadric surfaces, classification of, 214,
220, 227, 229, 230

, confocal family of, 289, 291,
292

, geometric characterization of,
232

, linear families of, 277

, ruled, 170

R

Radius vector, 108
Rank of a matrix, 16
Reference, frames of, 49, 108
Reguli on the hyperbolic paraboloid,
242

on the hyperboloid of one sheet,

235, 242 '
Regulus, 236
Relations, invariant, 197
Revolution, axis of, 131

, ellipsoid of, 133, 215

, hyperboloids of, 133, 215

, paraboloid of, 133

, surface of, 131, 266
Right-handed system, 50
Rigid transformation, 126
Rotation of axes, 118, 201, 202, 211,

213

Row index, 2
Row of a determinant, 2
Rows of a matrix, 16
Rule, Cramer's, 37
Rule of signs, Descartes', 215
Ruled quadric surfaces, 170, 235, 242

INDEX

309

s

Section, circular, 244

Section of a surface, plane, 127,

128

Semi-axes of the ellipsoid, 142
Sheets of a surface, 133, 136
Singular quadric surface, 166, 175,

^180, 202

Singular square matrix, 43
Solution, trivial, 38
Sphere, 133, 136
Spherical coordinates, 108
Spheroid, oblate, 133

, prolate, 133
Square matrix, 16, 43

, singular, 43

, symmetric, 43
Surface, 68

, conical, 136, 175

, cylindrical, 69

, invariants of a, 197

, line tangent to a, 154, 161

, nappes of a, 133, 136

, normal to a, 155, 156

of order n, 154

of revolution, 131, 266

of revolution, equation of, 132
, plane section of a, 127, 128
, plane tangent to a, 155, 161,

162

, quadric, 159

, shape of, 137, 138

, sheets of a, 133, 136
Surface and line, intersection of, 150,

153

Symmetric determinant, 33, 42
Symmetric equations of a line, 60
Symmetric locus, 137
Symmetric square matrix, 43
Symmetry, 137
System, left-handed, 50

, right-handed, 50
Systems of coordinates, 49, 108

of homogeneous equations, 38,
39,41

Systems of non-homogeneous equa-
tions, 36, 38, 44
of planes, 68, 70

Tangent cone, 167

Tangent line to a surface, 154, 161

Tangent plane to a surface, 155, 161,

162, 241, 253
Taylor's theorem, 151
Tetrahedron, 107
Theorem on homogeneous functions,

Euler's 161
, Taylor's, 151
Three-point form of the equation of a

plane, 74
Torus, 135
Transformation, linear, 126

of coordinates, 110, 112, 115,
120, 121, 123

of a matrix, elementary, 18
, orthogonal, 123

, rigid, 126

Translation of axes, 114, 200, 201, 210
Transverse axis, 140, 141, 145
Triangular prism, 96
Trihedral angle, 96
Trivial solution, 38
Twisted curves, 157
Two lines, 104

U

Umbilics, 253, 263, 266

on the cone, 265

on the ellipsoid, 264

on the elliptic cylinder, 265

on the elliptic paraboloid, 265

on the hyperboloid of one sheet,
264

on the hyperboloid of two sheets,
264

on the surfaces of revolution,
266

Units, 50, 108, 109, 110, 111, 113, 117,
118

310

INDEX

Value of a determinant, 3, 13
Vertex of a conical surface, 136, 137

of the ellipse, 140

of the hyperbola, 141

of the parabola, 141

of a quadric surface, 171, 175,
177, 178, 179

A^-axis, 49

A"-contour lines, 138
^-coordinate, 50

F-axis, 49

F-contour lines, 138
^-coordinate, 50

#-axis, 49

Z-eontour lines, 138
2-coordinate, 50