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