xt7ghx15n28p https://exploreuk.uky.edu/dips/xt7ghx15n28p/data/mets.xml Smith, William Benjamin, 1850-1934. 1886  books b92-276-32008276 English Ginn, : Boston : Contact the Special Collections Research Center for information regarding rights and use of this collection. Geometry, Analytic. Elementary co-ordinate geometry for collegiate use and private study  / By William Benjamin Smith. text Elementary co-ordinate geometry for collegiate use and private study  / By William Benjamin Smith. 1886 2002 true xt7ghx15n28p section xt7ghx15n28p 



ELEMENTARY



CO-ORDINATE



GEOMETRY,



FOR



    COLLEGIATE USE AND PRIVATE STUDY.



                   BY

WILLIAM     BENJAMIN SM1IITH, PAi.D. (GOrrINGEN),
     PROFESSOR OF PHYSICS, MISSOURI STATE UNIVERSITY.
                 MISSOURI.



MAXIMUM REASONING, MINIMUM RECKONING.







       BOSTON:
  GINN & COMPANY.
         1886.

 





























Entered according to Act of Congress, in the year 1885, by
         WILLIAM BENJAMIN SMITH,
in the Office of the Librarian of Congress, at Washington



J. S. CUSHING & CO., PAtTwmts, BoSTON.

 







PREFACE.



IN the study of Analytic Geometry, as of almost anything
     else, either or both of two ends may be had in view:
gain of knowledge, culture of mind. While the first is in
itself worthy enough, and for mathematical devotees all suffi-
cient, it is certainly of only secondary importance to the
mass of college students. For these the subject can be
wisely prescribed in a curriculum only in case the mental
drill it affords be very high in order of excellence.
  The worth of mere calculation as an exercise of reason
can hardly be considerable, for reason is exercised only in
a tread-mill fashion. Even the solution of problems by al-
gebraic processes is a very inferior discipline of reason, for
onlv in forming the analytic statement does the reasoning
rise clearly into consciousness; the operations that follow
conduct one to the conclusion, but -with his eyes shut. In
this respect Geometry is certainly a better discipline than
Algebra, and the Euclidean than the Cartesian Geometry.
But not in any kind of reasoning is the very best discipline
found. No argument presents difficulty or calls for much
mental effort to follow it, when once its terms are clearly
understood; for no such argument can be harder to under-
stand than the general syllogism of whice it is a special
case, and that is of well-known simplicity. The real diffi-
cultv lies in forming clear notions of things; in doing this
all the higher faculties are brought into play.  It is this
formation of concepts, too, that is the really important part
of mental training. He who forms them clearly and accu-
rately may be safely trusted to put them together correctly.

 

PREFACE.



Logical blunders are comparatively. rare.   Nearly every
seeming mistake in reasoning is really a mistake in concep-
tion. If this be false, that will be invalid.
  It is considerations like the above that have guided the
composition of this book. Concepts have been introduced
in abundance, and the proofs made to hinge directly upon
them. Treated in this way, the subject seems adapted as
hardly any other to develop the power of thought.
  The correlation of algebraic and geometric facts has been
kept clearly and steadily in view. While each may be taken
as pictures of the other, the former have generally been
treated as originals, lending themselves much more readily
to classification.
  Only natural logical order has been aimed at in the devel-
opment of the subject; no attempt has been made to keep
up the distinctions of ancient and modern, analytic and
synthetic.
  With every step forward in Geometry the difficulty and
tedium of graphical representation increases, while more and
more the reasoning turns upon the form of the algebraic
expressions. Accordingly, pains have been taken to make
the notation throughout consistent and suggestive, and Deter-
minants have been used freely.
  By all this effort to make the book an instrument of
culture, its worth as a repertory of mathematical facts has
scarcely suffered; in this regard, as in others, comparison
with other texts is invited.
                                                 AUTHOR.
CENTRAL COLLEGE, MO.
   March 25, 1885.



1Y

 









                TABLE OF CONTENTS.




                      INTRODUCTION.
                      DETERMII4ANTS.
Article.                                                     Page.
  1. Permutations, Straight and Circular, of n Things ............  xiii
  2. Inversions in Permutations .........  ......................  xiv
  3. Exchange of Elements in a Permutation ......  .............  xiv
  4. Definition of Determinant .........  .......................  xv
  5. Ways of writing Determinants .......   .....................  xvi
  6. Simplest Properties of Determinants ......  ................  xvii
  7. Co-factor of an Element of a Determinant .................. xviii
  8. Products of Elements by their Co-factors ................... x ix
  9. Decomposition of a Determinant .......    ...................  xx
10. Evaluation of a Determinant. Examples ......  ............  xxi
11. Multiplication of Determinants ............................ xxiii
12. Determinant of the Co-factors ............................. xxiv
13. Solution of a System of Linear Equations ...... ............  xxv
14. Conditions of Consistence of Equations ...... .............. xxvi
15. Elimination between Equations of Higher Degree ..... ...... xxvii



                  PART I. THE PLANE.


                         CHAPTER I.

                       FIRST NOTIONS.
 1. Definition of Functions ...................................... 1
 2. Algebraic Expression of a Functional Relation. Argument ..... 2
 3. Classification of Functions .............   ......................  3
 4. Definition of Continuity. Illustrations ..... .................. 4
 5. Numbers pictured by Tracts; Addition and Subtraction ........ 6
 6. Pairs of Numbers pictured by Points .......  .................. 7
 7. Equations pictured by Curves ..........................:.. 8

 

Vi                  TABLE OF CONTENTS.
Article.                                                       Page.
8. Determination of Position on a Surface ......    ................  9
  9. Points pictured by Pairs of Numbers. Co-ordinates ........... 10
  10. Polar Co-ordinates ................... .................    11
  11. Polar Equations pictured by Curves .......     ..................  12
  12. Degrees of Freedom. The Picture changes with the System .. . 12
  13. Summary................................................. 14
  14. Definition of Co-ordinate Geometry .......    ...................  14
    N.B. As to Real and Imaginary Numbers .................... 15



                          CHAPTER Il.

                       THE RIGHT LINE.
 15. Distance between Two Points ........    ................. .....  16
 16. Intersection of Curves ..........   ...........................  17
 17. Division of a Tract in any Ratio ............................. 18
 18. Parallel Projection of a Tract ........   .......................  20
 19. Projection of a Polygon ...................    ................  21
 20. Co-ordinates as Projections ........    .........................  22
 21. Formulke of Transformation ................................. 22
 22. Note on the Formula .......... .............................  25
 23. Linear Substitution..................                       25
 24. Equation of 1st Degree pictures a Right Line ............. ... 28
 25. Special Forms of the Equation .......     .......................  29
 26. Angle between Two Right Lines ............................. 32
 27. Distance from a Point to a Right Line ......    ................. 34
 28. Families of Right Lines .........     ..........................  35
 29. Pencils of Right Lines ............................ ......... 38
 30. The Equation M1L1+tL2+:,3L3=0. TriangularCds .......... 39
 31. Constant Relation between Triangular Co-ordinates ............ 40
 32. Geometric CMeaning of the Parameter A ......  ....... ........  42
 33. Halvers of the Angles between Two Right Lines .....   .........  43
 34. Illustrations of Abridged Notation..................... ..... 43
 35. Polar Equation of the Right Line .......     ....................  45
 36. Area of a Triangle given by its Vertices ................   ...  46
 37. Area of a Polygon given by its Vertices ...................... 47
 38. Area of a Triangle given by its Sides ......   ..................  48
 39. Ratio in which a Tract is cut by a Right Line .....  ........... 48
 40. Theorems on Transversals ..................................   49
 41. Cross Ratios of Ranges and Pencils ................... ...... 50
 42. Cross Ratios of Rays given by their Equations ........ ........  54
 43. Equation of Condition that Four Rays be Harmonic .....  ...... 55

 

                     TABLE   OF CONTENTS.                       Vii
Article.                                                       Page.
44. Relative Position of Four Harmonics .......      ................. 55
45. Homographic Pencils ....................................... 56
46. Common Harmonics ...........      ............................  57
47. Involution of Rays (Points) ........................... ..... 57
48. Centre and Foci of Involution ............................... 58
49. Cross Ratio in Involutions ............s.....................    59
50. Homogeneous Equation of Nth Degree ......      ................  60
51. Resolution of Equations of Higher Degree ......      ............. 60
52. Angles between the Pair kx2 + 2hxy +jy2 = 0 .................. 62
53. Halvers of these Angles ..........      .......................... 63
54. Intersection of Right Line and Curve of 2d Degree .....   ...... 63
55. The Right Line as a Locus. Examples ......        ................ 64
66. Families of Right Lines through a Point ......     ...............  69



                          CHAPTER III.
                          THE CIRCLE.
 57. Equation of the Circle ...................................... 71
 58. Determination of a Circle ................................... 71
 59. Normal Form of the Equation of a Circle ..................... 72
 60. The Circle conditioned ...................     .................  73
    N.B. On the Quadratic Equation .......         .................... 74
61. Axial Intercepts of the Circle .      .............................. 75
62. Polar Equation of the Circle .. ............................. 76
63. Intersection of Circle and Right Line .......    .................  76
64. Coincident Points.    .  .   ............                   77
65. Tangent to the Circle ...........     ...........................  79
66. Tangent to Curve of 2d Degree ........       .....................  80
67. Tangents from a Point to Curve of 2d Degree .....      ...........  81
68. Outside and Inside of a Curve ............................... 82
69. Polar and Pole. Conjugates ................................ 84
70. Construction of Polar ...........     ...........................  85
71. Polar as Locus of a 4th Harmonic .......       .................... 85
72. Equation of a Pair of Tangents ........      .....................  87
73. Diameter and Normal ...........       ...........................  88
74. Power of a Point as to a Circle .............................. 89
.5. Relative Position of Pole and Polar .......................... 89
76. Power-Line...............................                   92
77. Power-Centre ........         .......................       93
78. Power-Line of a System ................................         94
79. Equation of a System .    .....................................  94

 


Viii                TABLE    OF CONTENTS.
Article.                                                      Page.
80. Limiting Points ...........................................       94
81. Orthogonal Circles ............     ............................  95
82. Similar Figures .................. 99
83. Circles are Similar ...........     .............................  100
84. Chords of Contact ...........        ............................  101
85. Axes of Similitude ...........      .............................  101
86. Centrode of a Circle cutting 3 Circles under W a .............   102
87. The Taction-Problem proposed .............................       103
88. The Taction-Problem solved .......        ........................  104
89. The Circle as Locus. Examples ..........          ................  106



                          CHAPTER IV.

              GENERAL, PROPERTIES OF CONICS.
 90. Results recalled ...........................................     113
 91. Distance from a Point to a Conic .....        .....................  114
 92. Diameter and Centre ...........       ...........................  114
 93. Centre and Diameter..................                 ..  115
 94. Direction of Diameters .......       ............................  116
 95. Central Equation of the Conic .......       ......................  117
 96. Axes of the Conic ............      ............................  117
 97. Criterion of the Conic ..........     ...........................  118
 98. Asymptotes of the Conic .........       ..........................  119
 99. Central Distances of Pole and Polar ......       ..................  119
 100. Equation of P reduced.............    ........ ........   120
 101. Ratio of Distance Products ........       .......................  120
 102  Constant Functions of k, hj,          .............................  123


                          CHAPTER V.

              SPECIA.L PROPERTIES OF CONICS.
103. Equations of Centrics in Intercept Form  .....      ...............  125
104. Relation of a Centric to its Axes ......      .....................  126
105. Central Polar Equations of Centrics   ......      .................  127
106. The Centrics traced ...........      ............................  127
107. The Equations solved as to y .......        .......................  129
108. Direction of Conjugate Diameters ......        ....................  1,0
109. Co-ordinates of Ends of Conjugate Diameters ........ .......      131
110. Squared Half-Diameters .      ..................................  132
111. Central Distance of Tangent .......        ........................  132

 

                     TABLE OF CONTENTS.                        ix
Article.                                                     Page.
112. Tangents, Subtangents, Normals, Subnormals .134
113. Perpoles and Perpolars .136
114. Foci and Directrices .137
115. Polar and Perpolar as Bisectors .138
116. Focal Radii .138
117. Asymptotic Properties .141
118. Asymptotic Equation of H .142
119. Polar Equations of E and H .143
120. P as the Limit of E and H .145
121. Properties of P as such Limit .146
122. Tangent and Normal to P .148
123. Interpretation of 4q' ..................................... 149



                         CHAPTER VI.

             SPECIAL METHODS AND PROBLEMS.
124. Magic Equation of Tangent .151
125. Magic Equation of Normal .152
126. Illustration of Use of Magic Equations .153
127. Eccentric Equation of E .153
128. Eccentric Equation of Chord, Tangent, Normal .154
129. Quasi-Eccentric Equation of H .155
130. Hyperbolic Functions .157
131. Supplemental Chords .158
132. Auxiliary Circles of E and H.                            158
133. Their Correspondents in P .159
134. Vertical Equation of the Conic .160
135. Tangent Lengths from a Point to a P .161
136. Areas in the E .162
137. Areas in the P ....................................... .... 162
138. Areas in the H .163
139. Varieties of Conics .166



                         CHAPTER VII.

      SPECIAL METHODS AND PROBLEMS.-(Continued.)
140. Conditions fixing a Conic .168
141. Conic through 5 Points .169
142. Conics through 4 Points.                                 170
143. Pascal's Theorem .171

 

X                    TABLE OF CONTENTS.
Article.                                                       Page.
144. Construction of the Conic by Pascal's Theorem .....     .........  173
145. Elements of the Centric ........       ...........................  173
146. Elements of the Non-Centric .......       ........................  176
147. Construction of Conics ...............       ....................  177
148. Confocal Conics ...........      ...............................  179
149. Confocals as Co-ordinate Lines .......      ......................  180
1.50. Similar Conics ............................................    181
151. Central Projection ..........      ..............................  183
152. Projection of 4 s into 4 s given in Size .....     ...............  184
153. The Conic a Central Projection of a Circle .....     .............  185


                         CHAPTER VIII.

                  THE CONIC AS ENVELOPE.

154. Homogeneous Co-ordinates .................................      187
155. Line-Co-ordinates .........................................     188
156. Equations between Line-Co-ordinates ......       .................  189
157. Interchange of Equations ......  ...........................     190
158. Tangential Equation of 2d Degree ..........................     191
159. Double Interpretation .........      ............................  193
160. Brianchon's Theorem ......................................      193
161. Loci of Poles and Envelopes of Polars .....       .................  195
    Note on Points and Lines at X ......       .......................  196
    Examples ................................................       197




                   PART II. OF SPACE.



                          CHAPTER I.

 1. Triplanar Co-ordinates .........      ...........................  223
 2. Cylindric Co-ordinates .........       ...........................  224
 3. Spheric Co-ordinates ...............      ......................  225
 4. Direction-Cosines .........................................      226
 4 Projections with Oblique Axes ......      .......................  228
 5. Division of a Tract ........................................     231
 6. Transformation of Co-ordinates ......       ......................  231
 7. General Theorems ..........        .............................  233
 8. Equation of the Right Line.      ................................  234

 

                     TABLE OF CONTENTS.                         xi
Article.                                                       Page.
9. Intersecting Right Lines .................................... 235
10. Common Perpendicular to Two Right Lines ................... 237
11. Co-ordinates of a Right Line ............... ................ 238
12. Equation of the Plane ......................................  2 38
13. Normal Form of the Equation ............................... 239
13 The Normal Form with Oblique Co-ordinates .... r   ............ 239
14. The Triangle in Space ...................................... 240
15. Position-Cosines ..   ......................................  241
15 Position-Cosines with Oblique Co-ordinates ....... ............ 241
16. Intersecting Planes .................      ............... 241
17. Lines on a Surface.................                         243
18. Three Planes ..................   ............................  243
19. Four Planes .  ...............................................  243
20. Meaning of X......................2 ....................... 244
21. Right Lines halving 4 between Two Right Lines .............. 244
22. Pencils and Clusters of Planes .......     ......................  245
23. Distance between Two Right Lines ........................... 246
24. Tetraeder fixed by Planes .............. .................... 247
25. Meaning of Co-ordinates of a Right Line .247


                           CHAPTER II.

26. Generation of Surfaces .249
27. Cylindric Surfaces .250
28. Conic Surfaces ........................................     . 250
29. Surfaces of Revolution.                                     251
30. Discriminant of the Quadric.                                251
31. Centre of the Quadric.                                      252
32. Tangents to the Quadric.                                    253
33. Tangent Cones .                                             254
34. Poles and Polars.                                           254
35. Diameters ............................................      . 255
36. The Right Line and the Quadric.                             255
37. Perpendicular Conjugates.                                   256
38. Rectangular Chief Planes.                                   258
39. Special Case.                                               259
40. Classification of Quadris    ..........  ............. -260
41. Ratios of Distances to the Quadric .......    ...................  261
42. Invariants of the Quadric .........     ........................  261
43. Geometric Interpretation .........     ..........................  262
44. The Ellipsoid .......................---.-..... 263
45. Cyclic Planes ..............    ....-.-.-.. 264

 

Xii                  TABLE OF CONTENTS.
Article.                                                        Page.
46. The Ellipsoid a Strained Sphere .......    .....................  265
47. Eccentric Equation of the Tangent Plane .....   ...............  266
48. Normal Equation of the Tangent Plane ......................  266
49. The Simple Hyperboloid ................... ................ 267
50. The Double Hyperboloid ........     ...........................  267
51. The Hyperboloids as Strained Equiaxials ..................... 268
52. The Elliptic Paraboloid ..................................... 268
53. The Hyperbolic Paraboloid ........................ ........ 269
54. Imaginary Cyclic Planes .................................... 269
55. The Quadric as ruled ...........  ............................  270
56. Right Lines on the Simple Hyperboloid ......   ................ 271
57. Imaginary Right Lines on the Ellipsoid ......  ................ 272
58. Right Lines on the Hyperbolic Paraboloid .....  ...............  272
59. Foci and Confocals ........ ...................--. 273
60. Confocal Co-ordinates .................................... 274
61. Cubature of the Hyperboloids ............................... 274
62. Cubature of the Ellipsoid .................................. 276
63. Cubature of the Elliptic Paraboloid ..-.-..-.-.-..          276
64. Cubature of the Hyperbolic Paraboloid ..........   .....    277
65. Determination of the Quadric ..........................     278

 






                    INTRODUCTION.

                  DETERMINANTS.

                      Permutations.

  1. Two things, as a and b, may be arranged straight, in the
order of before and after, in but twco ways: a b, b a. A third
thing, as c, may be introduced into each of these arrangements
in three ways: just before each or after all. Like may be said
of any arrangement of n things: an (71 + 1)th thing may be
introduced in n + 1 ways, namely, before each or after all.
Hence the number of arrangements of n + 1 things is n + 1 times
the number of arrangements of it things. Or

               Pn+l = P.- it + 1.
  Writing n ! for the product of the natural numbers up to n,
we have
               n + 1 ! = i n + 1
  Hence, if   P,,, = n!,  P.-,= it + 1!,  and so on.
  Now         P2=2=1-2=2!;
hence,        P3 = 1 .2 - 3 = 3 !,and  P, = it!

  The various arrangements of things in the order of before
and after are called straight Permutations or simply Permuta-
tions of the things. The number of permutations of n things is
n ! (read factorial it or n factorial).
  If the things be arranged not straight but around, in a ring,
we may suppose them strung on a string; if there are n of
them, there are also ut spaces between them. We may suppose
the string cut at any one of the 7i spaces and then stretched
straight; this will turn the circular permutation into a strigltt
one; and since we may make n different cuts, each yielding a

 
CO-ORDINATE GEOMETRY.



distinct straight permutation, the number of straight permuta-
tions of it things is n times the number of circular ones. Or

                  P.,- C",  n ; .-. C", =n n-1 !

   2. The things, whatever they be, are most conveniently
 marked or named by letters or numbers. Of letters the alpha-
 betic order is the natural order; of numbers the order of size is
 the natural order; as: a, b, c, .. z; 1, 2, 3, 4, ... n.
   If any change be made in either of these orders, say in the
last, then some less number must appear after some greater,
some greater before some less. Every such change from the
natural order is called an Inversion. The number of inversions
in any permutation is found by counting the number of numbers
less than a number and placed after it, and taking the sum of
the numbers so counted.
   A permutation is named even or odd, according as the number
of inversions in it is even or odd. Thus 2 5 3 1 6 4 is an even
permutation containing 6 inversions; 3 1 2 .5 4 6 is an odd
permutation containing 3 inversions.  The natural order,
1, 2, 3, -- n, contains 0 inversions and is even; the counter

order, n ... 3, 2. lcontains 1 + 2 + 3 + -- + n-  or " 1.2  
inversions and is even when the remainder on division of n by 4
is 0 or 1, odd when the remainder is 2 or 3.
  It is plain that in any permutation any thing, symbol, or ele-
ment may be brought to any place or next to any other one by
exchanging it in turn with each of the ones between it and that
other one. Thus, in 3 7 4 5 1 6 2, 7 may be brought next to 6
by exchanging it in turn with 4, 5, 1. Hence any permutation
may be produced from any other by exchanges of odJacents.

  3. By an exchange of any two adjacents, as p), q, the rela-
tions of each to all the others, and the relations of all the others
among themselves, are not changed; only the relation of those
two is changed. Now if pq be an inversion, qIP is not; and if
pq be nlot an inversion, qp is one; hence in either case, by this



Xiv

 

INTRODUCTION.



exchange of two adjacents, the number of inversions is changed
by 1; hence the permutation chcnges from even to odd or from
odd to even.
  If p and q be non-adjacent, and there be k elements between
them, then p is brought next to q by k exchanges in turn with
adjacents, and then q is brought to p's former place by c + 1
exchanges with adjacents: p is carried over k elements and q
over k + 1; thus p and q are made to exchange places by 2 k + 1
exchanges of adjacents. The permutation meanwhile changes,
from even to odd or from odd to even, 2 k+ 1 times; and an
odd number of changes back and forth leaves it changed. Hence,
an exchange of any tzco elements in a permutation changes the
permutation from even to odd or from odd to even.
  Plainly all the permutations may be parted into pairs, the mem-
bers of each pair being alike except as to p and q, which are
exchanged in each pair; hence. one permutation of each pair
will be even, and one odd; hence, of all the permutations, half
are even, half odd.


                      Determinants.

  4. It is plain that n2 things may be parted into n classes of n
each. We may mark these classes by letters: a, b, c, ... n, where
it is understood that n is the nth letter; the rank of each in its
class may be denoted by a subscript; thus p, will be the kth
member of class p. Clearly all members of rank k will also
form a class of n members. The whole number may be thought
arranged in a square of n rows and n columns, as in the special
case n = 5, thus:
                        a, b, Cl dl el
                        a2 b. C9 c12 e2
                        a. b3 C3 d., e3
                        a4 b4 (4 d4 e4
                        a5 b5 c. d5 e5

  This arrangement is not at all necessary to our reasoning, but
is quite convenient.



TV

 
Xvi              CO-ORDINATE GEOMETRY.

  Suppose we pick out of these n- things n of them, taking one
of each class and one of each -anlk (clearly, then, we take only
one of each). This we may do in ni! ways; for we mav write
off the n letters in natural order, a, b, c, .. n, and then suffix
the subscripts in as many ways as we can permute them, i.e.,
in n! ways.
  Now suppose these n2 things symbolized by letters to be mag-
nitudes or numbers, and form the continued product of each set
of n picked out as above: write off the sum of these products,
giving each the sign + or - according as the permutation of
the subscripts be even or odd: the result is called the Deter-
mint of the magnitudes so classified. Accordingly a Deter-
minant may be defined as
  A sum of products of n2 symbols assorted into n classes of
n ranks each, formed of factors taken one from each class and
each rankt each product marked + or - according as the order of
ranks (or classes) is an even or an odd permutation, the order
of classes (or ranks) being natural.

  The symbols are called elements of the Determinant; each
product, a term; the number of the degree of the Determinant is
the number of factors in each product. The classes may be
denoted bv letters anld the ranks by subscripts, or vice rersa.
The definition shows that classes and ranks stand on exactly
like footing; in any reasoning they may be exchanged.

  5. There are several ways of writing Determinants. In the
square way, exemplified in Art. 4, the classes are written in
columns and the ranks in rows, or vice versa. Hence rows and
columns are always interchangeable.  This is a very vivid way
of writing them, but is tedious. It is shorter to write simply
the diagonal term, thus:

                       Y. : alb2c,  n..

  The sign of summation E refers to the different terms got by
permuting the subscripts, -there are a! of them; the double

 
INTRODUCTION.



sign  means that each term is to be taken + or - according
as the permutation of the subscripts is even or odd.
  Still another way is to write the diagonal between bars:
                        a al2 c3 ... n,

  This is very convenient when there can be no doubt as to
what are the elements not written: otherwise, the square form
is best.

  6. To exchange two rows in the square form would clearly
be the same as to exchance in every term the indices or sub-
scripts that mark those rows; but by Art. 3 this would change
each permutation of the subscripts from even to odd or from
odd to even; and this, by the definition, would change the sign
of each term, and hence of the whole Determinant. Moreover,
since rows and columns stand on like footing, the same holds
of exchanging two columns: hence,
  To exchange two columns (or rows) changes the sign of the
Determinant.

  If the two rows (or columns) exchanged be identical or con-
gruent, i.e., if the elements corresponding in position in the
two be equal each to each, clearly exchanging them can have
no effect on the value of the Determinant, although it changes
the sign; now the only number whose value is not changed by
changing its sign is 0: hence,
  T7he value of a Determinant with two congruent rows (or col-
umns) is 0.

  Every term of a Determinant contains one and only one
element out of each row and column; hence a common factor
in every element of a row (or column) must appear as a factor
of every term of the Determinant and hence of the Determinant
itself; hence, we may divide each element of the row (or col-
umn) by it, if at the same time we multiply the whole Determi-
nant by it; i.e., any factor of every element of a row (or col-



Xvii

 

CO-ORDINATE GEOTMETRY.



umn) of a Determinant may be set out aside as a factor of the
whole Determinant.
  It is equally plain that any factor may be introduced into each
element of a row (or column), if at the same time the whole
Determinant be clivided by that factor.

  7. If we will find all the terms that contain any one element of
the Determinant, say a,, we may suppose all the other elements
in its column and row to be 0; this will make vanish no term
containing a, and all terms 7tot containing a,. The Determi-
nant, say of 5th degree, will then be
                          a10 0 0 0
                          o b2 C2 (11 e2
                          o b3 c3 d3 e3
                          o b4 c4 (14 e4
                          o b; c, c. e3

  Setting aside ac as the first factor in each product, we find all
the part-products by holding the order bede fast and permuting
the subscripts 2 3 4 ; but this is the way we form the Deter-
minant j