OHM! ITY OF CALIFORNIA LIBRARY OF THE UNIVERSITY OF CALIFORNIA LIBRARY ITY OF CALIFORNIA LIBRARY OF THE UNIVERSITY OF CALIFORNIA LIBRi m m m ITY OF CALIFORNIA LIBRARY BF THE UNIVERSITY OF CALIFORNIA Mm LIBRARY W I UNIVERSITY OF CALIFORNIA ^ ™4 \ 1 1 LIBRARY OF THE UNIVERSITY OF CALIFORNIA VERSITY OF CALIFORNIA LIBRARY OF THE UNIVERSITY OF CALIFORNIA UNIVERSITY OF CALIFORNIA LIBRARY OF THE UNIVERSITY OF CALIFORNIA : 3 \ l ° i — I 1 j )4mUu^ l/sruv&xu&f THEORY OF EQUATIONS. / DUBLIN UNIVERSITY PRESS SERIES. THE THEORY OF EQUATIONS WITH AN INTRODUCTION TO THE THEORY OF BINARY ALGEBRAIC FORMS. BY WILLIAM SNOW BURNSIDE, M.A., FELLOW OF TRINITY COLLEGE, DUBLIN ; ERASMUS SMITH'S PROFESSOR OF MATHEMATICS IN THE UNIVERSITY OF DUBLIN: AND ARTHUR WILLIAM PANTON, M. A., FELLOW AND TUTOR, TRINITY COLLEGE, DUBLIN; DONEGAL LECTURER IN MATHEMATICS. SECOND EDITION. DUBLIN: HODGES, FIGGIS, & CO., GRAFTON-STREET. LONDON : LONGMANS, GREEN, & CO., PATERNOSTER-ROW 1886. 3? ) f PHYSICS DEPT. ' M DUHLIN : PRINTED AT THE UNIVERSITY PRESS BY PONSONISY AND WELDRICK. PEEFACE TO THE FIRST EDITION. We have endeavoured in the present work to combine some of the modern developments of Higher Algebra with the subjects usually included in works on the Theory of Equations. The first ten Chapters contain all the propositions ordinarily found in elementary treatises on the subject. In these Chapters we have not hesitated to employ the more modern notation wher- ever it appeared that greater simplicity or comprehensiveness could be thereby obtained. Eegarding the algebraical and the numerical solution of equations as essentially distinct problems, we have purposely omitted in Chap. VI. numerical examples in illustration of the modes of solution there given of the cubic and biquadratic equations. Such examples do not render clearer the conception of an algebraical solution ; and, for practical purposes, the algebraical formula may be regarded as almost useless in the case of equations of a degree higher than the second. In the treatment of Elimination and Linear Transformation, as well as in the more advanced treatment of Symmetric Func- tions, a knowledge of Determinants is indispensable. We have 66G356 vi Preface to the First Edition. found it necessary, therefore, to give a Chapter on this subject. It has been our aim to make this Chapter as simple and intelli- gible as possible to the beginner ; and at the same time to omit no proposition which might be found useful in the application of this calculus. For many of the examples in this Chapter, as well as in other parts of the work, we are indebted to the kind- ness of Mr. Cathcart, Fellow of Trinity College. We have approached the consideration of Covariants and Invariants through the medium of the functions of the diffe- rences of the roots of equations — this appearing to us the sim- plest mode of presenting the subject to beginners. We have attempted at the same time to show how this mode of treatment may be brought into harmony with the more general problem of the linear transformation of algebraic forms. In the Chapters on this subject we have confined our attention to the quadratic, cubic, and quartic ; regarding any complete discussion of the covariants and invariants of higher binary forms as too diffi- cult for a work like the present. Of the works which have afforded us assistance in the more elementary part of the subject, we wish to mention particularly the Traite d' Aigebre of M. Bertrand, and the writings of the late Professor Toung* of Belfast, which have contributed so much to extend and simplify the analysis and solution of numerical equations. In the more advanced portions of the subject we are in- debted mainly, among published works, to the Lessons Intro- ductory to the Modern Higher Algebra of Dr. Salmon, and the * Theory and Solution of Algebraical Equations, London, 1835; Analysis and Solution of Cubic and Biquadratic Equations, London, 1842 ; and Theory and Solution of Algebraical Equations of the Higher Orders, London, 1843. Preface to the First Edition. vii Theorie der Unarm algebraischen Formen of Clebsch ; and in some degree to the Theorie dcs Formes binaires of the Chev. F. Faa De Bruno. We must record also our obligations in this department of the subject to Mr. Michael Roberts, from whose Papers in the Quarterly Journal and other periodicals, and from whose professorial lectures in the University of Dublin, very great assistance has been derived. Many of the examples also are taken from Papers set by him at the Uni- versity Examinations. In the Chapter on the Complex Variable we have followed closely the treatment of imaginary quantities given by M. Briot in his Legons d'Algebre. In connexion with various parts of the subject several other works have been consulted, among which may be mentioned the treatises on Algebra by Serret, Meyer Hirsch, and Rubini, and papers in the mathematical journals by Boole, Cay ley, Sylvester, Hermite, and others. We have, in the last place, to express our thanks to Mr. Robert Graham, of Trinity College, Dublin, who has read the proof sheets, and verified most of the examples. His thorough acquaintance with the subject has been invaluable to us, and many improvements throughout the work are owing to sug- gestions made by him. Teinity College, September, 1881. PREFACE TO THE SECOND EDITION. The chief alterations in the present edition will he found in the Chapter on Determinants, which has been considerably en- larged ; and in Chap. XVI., on Transformations, to which two new sections have been added. The former of these contains a discussion of Hermite's theorem relating to the limits of the roots of an equation ; and in the latter we have given an account of the method of transformation from a system of two to a system of three variables, by means of which the Theory of Covariants and Invariants of Binary Forms receives a simple geometrical illustration. Trinity College, December, 1885. Note. — The first ten Chapters of this work may be regarded as forming an elementary course. In reading these Chapters for the first time, Students are recommended to omit Art. 53 of Chap. V., and to confine their attention in Chap. VI. to Arts. 55, 5G, 57, 61, 62, and 63. TABLE OF CONTENTS. INTRODUCTION. Art. Pa g e 1. Definitions, 1 2. Numerical and algebraical equations, 2 3. Polynomials, 3 CHAPTER I. GENEBAL PBOPEETLES OF POLYNOMIALS. ■1. Theorem relating to polynomials when the variable receives large values, 5 ■") . Similar theorem when the variable receives small values, ... 6 6. Change of form of a polynomial corresponding to an increase or diminu- tion of the variable. Derived functions, ...... 8 7 . Continuity of a rational integral function, 9 S. Form of the quotient and remainder when a polynomial is divided by a binomial, ............ 10 9. Tabulation of functions, 12 10. Graphic representation of a polynomial, .13 11. Maxima and minima values of polynomials, 17 CHAPTER II. GENERAL PBOPEETLES OF EQUATIONS. 12, 13, 14. Theorems relating to the real roots of equations, 15. Existence of a root in the general equation. Imaginary roots, 16. Theorem determining the number of roots of an equation, 17. Equal roots, ......... 18. Imaginary roots enter equations in pairs, . 19. Descartes' rule of signs for positive roots, . . . . 19 21 22 25 26 23 x Table of Contents. Art. Page 20. Descartes' rule of signs for negative roots, 30 21. Use of Descartes' rule in proving the existence of imaginary roots, . . 30 22. Theorem relating to the substitution of two given numbers for the variable, 31 Examples, 32 CHAPTER III. RELATIONS BETWEEN THE ROOTS AND COEFFICIENTS OF EQUATIONS, WITH APPLICATIONS TO SYMMETRIC FUNCTIONS OF THE ROOTS. 23. Relations between the roots and coefficients. Theorem, . . .35 24. Applications of the theorem, ........ 36 25. Depression of an equation when a relation exists between two of its roots, 42 26. The cube roots of unity, . 43 27. Symmetric functions of the roots, 46 Examples, 48 28. Theorems relating to symmetric functions, 53 Examples, 54 CHAPTER IV. TRANSFORMATION OF EQUATIONS. 29. Transformation of equations, .60 30. Eoots with signs changed, ......... 60 31. Eoots multiplied by a given quantity, . . . . . . .61 32. Reciprocal roots and reciprocal equations, ...... 62 33. To increase or diminish the roots by a given quantity, . . . .64 34. Removal of terms, 67 35. Binomial coefficients, 68 36. The cubic, 71 37. The biquadratic, 73 38. nomographic transformation, 75 39. Transformation by symmetric functions, ...... 76 40. Formation of the equation whose roots are any powers of the roots of the proposed equation, .......... 78 41. Transformation in general, 80 42. Equation of squared differences of a cubic, . . . . . .81 43. Criterion of the nature of the roots of a cubic, 84 44. Equation of differences in general, 84 Examples, 86 Table of Contents. XI CHAPTER V. SOLUTION OF RECIPROCAL AND BINOMIAL EQUATIONS. Art. Pa f e 45. Reciprocal equations, 90 46-52. Binomial equations. Propositions embracing their leading general 09 properties, ....•••-••• 53. The special roots of the equation x" - 1 = 0, 95 54. Solution of binomial equations by circular functions, . . ■ .98 Examples, 10 ° CHAPTER VI. ALGEBRAIC SOLUTION OF THE CUBIC AND BIQUADRATIC. 55. On the algebraic solution of equations, .... 56. The algebraic solution of the cubic equation, 57. Application to numerical equations, . 58. Expression of the cubic as the difference of two cubes, . 59. Solution of the cubic by symmetric functions of the roots, Examples, nomographic relation between two roots of a cubic, First solution by radicals of the biquadratic. Euler's assumption, Examples, .....•••• Second solution by radicals of the biquadratic, Resolution of the quartic into its quadratic factors. Ferrari's solution, Resolution of the quartic into its quadratic factors. Descartes' solution, 65. Transformation of the biquadratic into the reciprocal form, . 66. Solution of the biquadratic by symmetric functions of the roots, 67. Equation of squared differences of a biquadratic, 68. Criterion of the nature of the roots of a biquadratic. Examples, .....•■••• 60. 61. 62. 63. 64. 103 106 107 109 111 112 118 119 123 125 127 131 133 137 140 142 144 CHAPTER VII. PROPERTIES OF THE DERIVED FUNCTIONS. €9. Graphic representation of the derived function, lo2 70. Theorem relating to the maxima and minima values of a polynomial, . 153 71. Rolle's theorem. Corollary, 155 72. Constitution of the derived functions, 155 73. Theorem relating to multiple roots, 156 74. Determination of multiple roots, 157 75. 76. Theorems relating to the passage of the variable through a root of the equation, l59 > 160 Examples, Xll Table of Contents. CHAPTER VIII. Art. 77. 79. 80. 81. 82. 83. LIMITS OF THE BOOTS OF EQUATIONS. Page Definition of limits, 163 Limits of roots. Prop. I., ........ 1G3 Limits of roots. Prop. II., ........ 164 Practical applications, . . . . . . . . .166 Newton's method of finding limits. Prop. III., .... 168 Inferior limits, and limits of the negative roots, . . . . .169 Limiting equations, .......... 170 Examples, 171 CHAPTER IX. SEPARATION OF THE ROOTS OF EQUATIONS. 84. General explanation, ..... 85. Theorem of Fourier and Budan, .... 86. Application of the theorem, .... 87. Application of the theorem to imaginary roots, 88. Corollaries from the theorem of Fourier and Budan, 89. Sturm's theorem, ...... 90. Sturm's theorem. Equal roots, 91. Application of Sturm's theorem, 92. Conditions for the reality of the roots of an equation, 93. Conditions for the reality of the roots of the hiquadrath Examples, ........ 172 172 175 177 180 181 186 189 193 194 195 CHAPTER X. SOLUTION OF NUMERICAL EQUATIONS 94. Algehraical and numerical equations, 95. Theorem relating to commensurable roots, . 96. Newton's method of divisors, 97. Application of the method of divisors, 98. Method of limiting the number of trial-divisors, 99. Determination of multiple roots, 100. Newton's method of approximation, 101. Horner's method of solving numerical equations, 102. Principle of the trial-divisor in Horner's method, 103. Contraction of Horner's process, 104. Application of Horner's method to cases where roots are nearly equal, 105. Lagrange's method of approximation, 106. Numerical solution of the biquadratic by Descartes' method. Examples, 197 198 199 200 203 204 207 209 213 217 220 223 224 227 . 232 -IV., 234 . 239 . 239 . 244 Table of Contents. xiii CHAPTER XL DETERMINANTS. Art. Pa S e 107. Elementary notions and definitions, 108. Rule with regard to signs, 109-112. Elementary propositions relating to determinants. Props. I. 113. Minor determinants. Definitions, ..... 114. Development of determinants, I 115. Laplace's development of determinants, .... 116. Development of a determinant in products of leading constituents, . 245 / 117. Expansion of a determinant by products in pairs of the constituents of a row and column, 247 US. Addition of determinants. Prop. V., . ... 248 V 119-120. Further propositions. Prop. VI. and Prop. VII., . . . 250 121. Multiplication of determinants. Prop. VIII., 255 122. Second proof of Prop. VIIL, 256 123. Rectangular arrays, 259 124. Solution of a system of linear equations, 262 125. Linear homogeneous equations, ........ 263 126. Reciprocal determinants, 264 127. Symmetrical determinants, 267 . 128. Skew-symmetric and skew determinants, 269 V 129. Theorem relating to a determinant whose leading first minor vanishes, . 273 Miscellaneous Examples, . ' • • • • • • • 276 CHAPTER XII. SYMMETRIC FUNCTIONS OF THE ROOTS. 130. Newton's theorem on the sums of powers of roots. Prop. I., . • 289 13^, Expression of a rational symmetric function of the roots in terms of the coefficients. Prop. II., 291 132. Further proposition relating to the expression of sums of powers of roots in terms of the coefficients. Prop. III., 293 133. Expression of the coefficients in terms of sums of powers of roots, . 294 134. Definitions of order and weight of symmetric functions, and theorem relating to the former, ......... 299 135. Calculation of symmetric functions of the roots, 300 136. Brioschi's differential equation connecting the sums of the powers of the roots and the coefficients, 304 137. Derivation of new symmetric functions from a given one, . . . 306 138. Equation of operation, 307 139. Operation involving the sums of the powers of the roots. Theorem, . 309 Miscellaneous Exainiiles, ......... 310 XIV Table of Contents. CHAPTER XIII. ELIMINATION. Art. Page 140. Definitions, 318 141. Elimination by symmetric functions, . . . . . . .319 142. Properties of the resultant, 320 143. Euler's method of elimination, ........ 322 144. Sylvester's dialytic method of elimination, 323 145. Bezout's method of elimination, 324 146. The common method of elimination, ....... 329 147. Discriminants, ........... 331 148. Determination of a root common to two equations, .... 333 149. Symmetric functions of the roots of two equations, .... 334 Miscellaneous Examples, 335 CHAPTER XIV. COTAEIANTS AND INVARIANTS. 150. Definitions, ...... 151. Formation of covariants and invariants, 152. Properties of covariants and invariants, Examples, ....... 153. Formation of covariants by the operator D., 154. Roberts' theorem relating to covariants, 155. Homographic transformation applied to the theory of covariants, . 156. Reduction of homographic transformation to a double linear transforma- tion, ............ 157. Properties of covariants derived from linear transformation, . 158-161. Propositions relating to the formation of invariants and covariants of quantics transformed by a linear transformation, .... Miscellaneous Examples, ......... 338 339 341 343 344 346 347 349 351 354 360 CHAPTER XV. COVAEIANTS AND INVAEIANTS OF THE QUADRATIC, CUBIC, AND QUAETIC. 162. The quadratic 163. The cubic and its covariants, . . . . 164. Number of covariants and invariants of the cubic, 165. The quartic ; its covariants and invariants, . 166. Quadratic factors of the sextic covariant, 366 366 369 371 372 Table of Contents. xv Art. p age 167. Expression of the Hessian by the quadratic factors of the sextic co- variant, 374 168. Expression of the quartic itself by the quadratic factors of the sextic covariant, 374 169. Resolution of the quartic, ......... 376 170. The invariants and covariants of kU — \H f , ..... 377 171. Number of covariants and invariants of the quartic, .... 379 Miscellaneous Examples, 382 CHAPTER XVI. TRANSFORMATIONS. Section I. — Tschirnhausen's Transformation. 172. Theorem, 385 173. Formation of the transformed equation, 387 174. Second method of forming the transformed equation, .... 388 175. Tschirnhausen's transformation applied to the cubic, .... 389 176. Tschirnhausen's transformation applied to the quartic. Theorem, . 390 177. Reduction of the cubic to a binomial form by Tschirnhausen's transfor- mation, 391 178. Reduction of the quartic to a trinomial form by Tschirnhausen's trans- formation, 392 179. Removal of the second, third, and fourth terms from an equation of the n th degree, 392 180. Reduction of the quintic to the sum of three fifth powers. Sylvester's theorem, 395 181. Quartics transformable into each other, ...... 396 Miscellaneous Examples, 398 Section II. — Hermite's Theorem. 182. Homogeneous function of second degree expressed as sum of squares, . 401 183. Hermite's theorem, 403 184. Sylvester's forms of Sturm's functions, 406 Examples, 409 Section III. — Geometrical Transformations. 185. Transformation of binary to ternary forms, 411 186. The quartic and its covariants treated geometrically, .... 414 187. Determination of the proper ternary form, 416 Example?, 418 xvi Table of Contents. CHAPTER XVTI. THE COMPLEX VARIABLE. Art. Page 188. Graphic representation of imaginary quantities, 422 189. Addition and subtraction of imaginary quantities, .... 423 190. Multiplication and division of imaginary quantities, .... 424 191. The complex variable, 425 192. Continuity of a function of the complex variable, .... 427 193. Variation of the argument of the function corresponding to the descrip- tion of a small closed curve by the complex variable, .... 428 194. Cauchy's theorem relating to the number of roots comprised within a plane area, 430 195. Proof of the fundamental theorem that every equation of the n th degree has n roots real or imaginary, 431 NOTES. A. Algebraic solution of equations, 433 B. Solution of numerical equations, 437 C. Determinants, 441 D. The proposition that every equation has a root, 442 THEOKY OF EQUATIONS. INTRODUCTION. 1. Definitions. — Any mathematical expression involving a quantity is called a function of that quantity. We shall be employed mainly with such algebraical func- tions as are rational and integral. By a rational function of a quantity is meant one which contains that quantity in a rational form only ; that is, a form free from fractional indices or radical signs. By an integral function of a quantity is meant one in which the quantity enters in an integral form only ; that is, never in the denominator of a fraction. The following expres- sion, for example, in which n is a positive integer, is a rational and integral algebraical function of x : — ax n + bx n ^ -v cx ll ~ 2 + + he + I. It is to be observed that this definition has reference to the variable quantity x only, of which the expression is a function. The several coefiicients a, b, c, &c, may be irrational or fractional, and the function still remain rational and integral in x. A function of * is represented for brevity by F(x),f(x) , $ (x), or some such symbol. The name polynomial is given to the algebraical function to express the fact that it is constituted of a number of terms containing different powers of x connected by the signs plus or 2 Introduction. minus. For certain values of the variable quantity x one poly- nomial may become equal to another differently constituted. The algebraical expression of such a relation is called an equa- tion ; and any value of x which satisfies this equation is called a root of the equation. The determination of all possible roots constitutes the complete solution of the equation. It is obvious that, by bringing all the terms to one side, we may arrange any equation according to descending powers of x in the following manner : — a x n + a l x n ~ l + a 2 x n ~ 2 +....+ cin^x + a n = 0. The highest power of x in this equation being n, it is said to be an equation of the n th degree in x. For such an equation we shall, in general, employ the form here written. The suffix attached to the letter a indicates the power of x which each coef- ficient accompanies, the sum of the exponent of x and the suffix of a being equal to n for each term. An equation is not altered if all its terms be divided by any quantity. "We may thus, if we please, dividing by a , make the coefficient of x n in the above equation equal to unity. It will often be found convenient to make this supposition ; and in such cases the equation will be written in the form ■ i" + 2hX n ~ l +paX 7 '- 2 + . . . . +p n -\X + p» = 0. An equation is said to be complete when it contains terms involving x in all its powers from n to 0, and incomplete when some of the terms are absent ; or, in other words, when some of the coefficients p y , p 2 , &c, are equal to zero. The term p n , which does not contain x, is called the absolute term. An equa- tion is numerical, or algebraical, according as its coefficients are numbers, or algebraical symbols. 2. Ulunierical and Algebraical Equations. — Iu many researches in both mathematical and physical science the final mathematical problem presents itself in the form of an equation on whose solution that of the problem depends. It is natural, therefore, that the attention of mathematicians should have been Numerical and Algebraical Equations. 3 at an early stage in the history of the science directed towards inquiries of this nature. The science of the Theory of Equa- tions, as it now stands, has grown out of the successive attempts of mathematicians to discover general methods for the solution of equations of any degree. When the coefficients of an equation are given numbers, the problem is to determine a numerical value, or perhaps several different numerical values, which will satisfy the equation. In this branch of the science very great progress has been made ; and the best methods hitherto advanced for the discovery, either exactly or approximately, of the nume- rical values of the roots will be explained in their proper places in this work. Equal progress has not been made in the general solution of equations whose coefficients are algebraical symbols. The stu- dent is aware that the root of an equation of the second degree, whose coefficients are such symbols, may be expressed in terms of these coefficients in a general formula ; and that the nume- rical roots of any particular numerical equation may be obtained by substituting in this formula the particular numbers for the symbols. It was natural to inquire whether it was possible to discover any such formula for the solution of equations of higher degrees. Such results have been attained in the case of equa- tions of the third and fourth degrees. It will be shown that in certain cases these formulas fail to supply the solution of a numerical equation by substitution of the numerical coef- ficients for the general symbols, and are, therefore, in this respect inferior to the corresponding algebraical solution of the quadratic. Many attempts have been made to arrive at similar general formulas for equations of the fifth and higher degrees; but it may now be regarded as established by the researches of modern analysts that it is not possible by means of radical signs, and other signs of operation employed in common algebra, to ex- press the root of an equation of the fifth or any higher degree in terms of the coefficients. 3. Polynomials. — From the preceding observations it is b2 4 Introduction. plain that one important object of the science of the Theory of Equations is the discovery of those values of the variable quan- tity x which give to the polynomial f(x) the particular value zero. In attempting to discover such values of x we shall be led into many inquiries concerning the values assumed by the poly- nomial for other values of the variable. We shall, in fact, see in the next Chapter that, corresponding to a continuous series of values of x varying from an infinitely great negative quan- tity (-00 ) to an infinitely great positive quantity (+00), f(x) will assume also values continuously varying. The study of such variations is a very important part of the theory of poly- nomials. The general solution of numerical equations is, in fact, a tentative process ; and by examining the values assumed by the polynomial for certain arbitrarily assumed values of the variable, we shall be led, if not to the root itself, at least to an indication of the neighbourhood in which it exists, and within which our further approximation must be carried on. A polynomial is sometimes called a quant ic. It is convenient to have distinct names for the quantics of various successive degrees. The terms quadratic (or quadric), cubic, biquadratic (or quartic), quintic, sextic, &c, are used to represent quantics of the 2nd, 3rd, 4th, 5th, 6th, &c, degrees ; and the equations obtained by equating these quantics to zero are called quadratic, cubic, biquadratic, &c, equations, respectively. CHAPTER I. GENERAL PROPERTIES OF POLYNOMIALS. 4. In tracing the changes of value of a polynomial correspond- ing to changes in the variable, we shall first inquire what terms in the polynomial are most important when values very great or very small are assigned to x. This inquiry will form the subject of the present and succeeding Articles. Writing the polynomial in the form a x n 11 + — + — i+ . • . + =-:+ — - , ( a x a %- a x n ~ x a x n ) it is plain that its value tends to become equal to a x n as x tends towards go . The following theorem will determine a quantity such that the substitution of this, or of any greater quantity, for x will have the effect of making the term a x n exceed the sum of all the others. In what follows we suppose a to be positive; and in general in the treatment of polynomials and equations the highest term is supposed to be written with the positive sign. Theorem. — If in the polynomial a x n + a x x n ~ l + a>x n - 2 + . . . + a n ^x + a n the value — + 1, or any greater value, be substituted for x, where au is that one of the coefficients a i3 a 2 , . . . a n whose numerical value is greatest, irrespective of sign, the term containing the highest power of x will exceed the sum of all the terms which follow. The inequality a x 1i > a A x n ~ l + flotf" -2 + . . . + a n -\X + a n (! General Properties of Polynomials. is satisfied by any value of x which makes > a k (#" _1 + x"-- + . . . + x + 1), where a k is the greatest among the coefficients a lt a k — , or x" > — — h" - 1), x-1 ajx-l) which is satisfied if a (x - 1) be > or = a k> that is x>ot = —'+1. The theorem here proved is useful in supplying, when the coefficients of the polynomial are given numbers, a number such that when x receives values nearer to + oo the polynomial will preserve constantly a positive sign. If we change the sign of x y the first term will retain its sign if n be even, and will become negative if n be odd ; so that the theorem also supplies a nega- tive value of x, such that for any value nearer to - oo the polynomial will retain constantly a positive sign if n be even, and a negative sign if n be odd. The constitution of the poly- nomial is, in general, such that limits much nearer to zero than those here arrived at can be found beyond which the function preserves the same sign ; for in the abovej proof we have taken the most unfavourable case, viz. that in which all the coefficients except the first are negative, and each equal to a k ; whereas in general the coefficients may be positive, negative, or zero. Several theorems, having for their [object the discovery of such closer limits, will be given in a subsequent Chapter. 5. "We now proceed to inquire what is the most important term in a polynomial when the value of x is indefinitely dimi- nished ; and to determine a quantity such that the substitution of this, or of any smaller quantity, for x will have the effect of giving such term the preponderance. Theorem. — If in the poll nomial a x n + a v x"~ x + . . . + a„^x + a„ Theorem. 7 the value — , or any smaller value, be substituted for x, where a* is the greatest coefficient exclusive of a„, the term a„ will be nume- rically greater than the sum of all the others. To prove this, let x = - ; then by the theorem of Art. 4, dk being now the greatest among the coefficients a , a u . . . a,,.,, without regard to sign, the value — + 1, or any greater value of y, will make ('nf 1 > «n-iy n ~ l + «n-2y"-- + . . . + «i tj + «o, that is, a„ > r/„_! - + o»_a -r + . . . a Q — j y y~ >/ hence the value — - — , or any less value of x, will make a„ + a k ' J (t,i > a n _ y x + a n _ 2 x 2 + . . . + a x n . This proposition is often stated in a different manner, as follows : — Values so small may be assigned to x as to make the polynomial cin^x + a„_ z x 2 + . . . + a „.> ■ ' less than any assigned Quantity. This statement of the theorem follows at once from the above proof, since a n may be taken to be the assigned quantity. There is also another useful statement of the theorem, as follows: — When the variable x receives a very small value, the sigi of the polynomial a n _ { x - a n _ 2 x- + . . . + ((qX 71 is the same as the sign of its first term a^iX. This appears by writing the expression in the form x[a„„i + a H _ 2 x + . . . + a x n ~ 1 } ; for when a value sufficiently small is given to x, the numerical value of the term «,»_i exceeds the sum of the other terms of the expression within the brackets, and the sign of that expression will consequently depend on the sign of a n _ x . 8 General Properties of Polynomials. 6. Change of Form of a Polynomial corresponding to an increase or diminution of the Variable. Derived Functions. — We shall now examine the form assumed by the polynomial when x + h is substituted for x. If, in what follows, h be supposed essentially positive, the resulting form will corre- spond to an increase of the variable ; and the form corresponding to a diminution of x will be obtained from this by changing the sign of h in the result. When x is changed to x + h,f{x) becomes/^ + h), or a u + h) n + a, (:r li)"-' + a, (x + //)"-- + . . . + a n _ y (x + h) + a n . Let each term of this expression be expanded by the binomial theorem, and the result arranged according to ascending powers of h. We then have a x n + a x x n ~ l + a 2 x n ' 2 + . . . + cin-2% 2 + c? n -\£ + ). The sign of f'{a) will determine whether, /'(.r) is increasing or diminishing; for it appears by Art. 5 that when h is small enough the sign of the total increment will depend on that of f'(a)h. We thus observe that vchenf'ip) is positive f(x) is increasing vnth x ; and when f '(a) is negative f(x) is diminishing as x increases. 8. Form of the Quotient and Remainder when a Polynomial is divided by a Binomial. — Let the quotient, when a x n + ciiX"' 1 + a 2 x n ~ 2 . . . + a n _ v c + a n Form of Quotient and Remainder. 1 1 is divided by x - h, be b x n ~ l + biX n ~' 1 + . . . + b n _ 2 x + b n _i. This we shall represent by Q, and the remainder by R. AVe have then the following equation : — f(x)--{x-h) Q + R. The meaning of this equation is, that when Q is multiplied by x - h, and_R added, the result must be identical, term for term, with/(#). In order to distinguish equations of the kind here explained from equations which are not identities, it will often be found convenient to use the symbol here employed in place of the usual symbol of equality. The right-hand side of the identity is b x n + h \x n ~ l + b 2 )x n - 2 + ... + b n _, )x + R -hb ) -hb v ) -hbnJi - hb n .i. Equating the coefficients of x on both sides, we get the fol- lowing series of equations to determine b , 6,, J 2 , . . . b n _ 1} R: — b = «<>, 61 = bji + a 1, b 2 = b x h + a 2 , bi = boji + a>„ 6„_i= b n _o.h + a,^, R = b n _Ji + a n . These equations supply a ready method of calculating in succession the coefficients b , b ly &c. of the quotient, and the remainder R. For this purpose we write the series of operations in the following manner : — a , (hi (hi (hi • • • • On-u o n , bj), bji, b z /i, .... b n _oJi, b n -ih, hi bo, hi, .... &»_!, R. In the first line are written down the successive coefficients 12 General Properties of Polynomials. of f{x). The first term in the second line is obtained by multi- plying Oq (or b , which is equal to it) by h. The product b h is placed under <7,, and then added to it in order to obtain the term b x in the third line. This term, when obtained, is multi- plied in its turn by h, and placed under a 2 . The product is added to a 2 to obtain the second figure b 2 in the third line. The repetition of this process furnishes in succession all the coef- ficients of the quotient, the last figure thus obtained being the remainder. A few examples will make this plain. Examples. 1. Find the quotient and remainder when 3a; 4 - 5x 3 + 10a; 2 + lis - 61 is divided by a; -3. The calculation is arranged as follows : — 3-5 10 11 - 61. 9 12 66 231. 4 22 77 170. Thus the quotient is 3a; 3 + 4a; 2 + 22a; + 77, and the remainder 170. 2. Find the quotient and remainder when x 3 + 5a; 2 + 3x + 2 is divided by a-- 1. Am. Q = a; 2 +6a; + 9, J2=ll. 3. Find Q and B when x 5 - ix* + 7a; 3 - 1 la; - 1 3 is divided by x - 5. N.B. — When any term in a polynomial is absent, care must be taken to supply the place of its coefficient by zero in writing down the coefficients oif(x). In this example, therefore, the series in the first line will be [I -4 7 -11 -13. Am. Q = a; 4 4- a; 3 + 12a; 2 + 60a; + 289; iJ=1432. 4. Find Q and R when x 9 + 3a; 7 - 15a; 2 + 2 is divided by a;- 2. Am. (2 = a;8 + 2a;7 + 7a; 6 +14a; 5 + 28a; 4 + 56a; 3 +112a; 2 + 209a; + 418; iJ = 83S. 5. Find Q and R when x 5 + x- - 10a;+ 113 is divided by x + 4. Am. Q = a; 4 - 4a; 3 + 16a; 2 -63a; + 242 ; i? = -855. 9. Tabulation of Functions. — The operation explained in the preceding Article affords a convenient practical method of calculating the numerical value of a polynomial whose coef- ficients are given numbers when any number is substituted for x. For, the equation /(*)-(*- h) Q + M, since its two members are identically equal, must be -satisfied when any quantity whatever is substituted for x. Let x = h, Tabulation of Functions. 13 then/(^) = R,x- h being = 0, and Q remaining finite. Hence the result of substituting k for x in f{x) is the remainder when f{x) is divided by x - /i, and can be calculated rapidly by the process of the last Article. For example, the result of substituting 3 for x in the poly- nomial of Ex. 1, Art. 8, viz., 3# 4 - 5a; 3 + 10^ + 11^-61, is 170, this being the remainder after division by x - 3. The student can verify this by actual substitution. Again, the result of substituting - 4 for x in a* + g» _ 10s + 113 is - 855, as appears from Ex. 5, Art. 8. We saw in Art. 7 that as x receives a continuous series of values increasing from - oo to + co , f{x) will pass through a corresponding continuous series. If we substitute in succession for x, in a polynomial whose coef- ficients are given numbers, a series of numbers such as ...-5,-4,-3,-2,-1, 0, 1, 2, 3, 4, 5,..., and calculate the corresponding values of fix), the process may be called the tabulation of the function. Examples. 1. Tabulate the trinomial 2x 2 + x - 6, for the following values of x :- -4,-3,-2,-1, 0, 1, " 2, 3, 4. Values of x, -4 — 3 -2 -1 1 2 3 4 „ „ /(*), 22 9 -5 -6 -3 4 15 30 2. Tabulate the polynomial l(te: 3 — 17a; 2 + x + 6 for the same values of x. Values of x, -4 1 -3 _2 - 1 1 2 3 4 » >, /(*), -910 -420 - 144 — 22 6 20 12G 378 10. Graphic Representation of a Polynomial. — In investigating the changes of a function fix) consequent on any 14 General Properties of Polynomials. "i ip O A series of changes in the variable which it contains, it is plain that great advantage will bo derived from any mode of repre- sentation which renders possible a rapid comparison with one another of the different values which the function may assume. In the case where the function in question is a polynomial with numerical coefficients, to any assumed value of x will correspond one definite value of /(#). We proceed to explain a mode of graphic representation by which it is possible to exhibit to the eye the several values of f(x) corresponding to the different values of x. Let two right lines OX, OY (fig. 1) cut one another at right angles, and be produced indefinitely in both directions. These lines are called the axis of x and axis of y, respectively. Lines, such as OA, x measured on the axis of * at the right-hand side of 0, are regarded as positive ; and those, such as OA', measured at the left-hand side, as negative. Lines parallel to OY which are above XX', such as AP or B'Q', are positive ; and those below it, such as AT or AP', are negative. These conventions are already familiar to the student acquainted with Trigonometry. Any arbitrary length may now be taken on OX as unity, and any number positive or negative will be represented by a line measured on XX' : the series of numbers increasing from to + oo in the direction OX, and diminishing from to - oo in the direction OX'. Let any number m be represented by OA ; cal- culate f(m) ; from A draw AP parallel to OY to represent /(w) in magnitude on the same scale as that on which OA represents ui, and to represent by its position above or below the line OX the sign of f(ni). Corresponding to the different values of m represented by OA, OB, OC, &c, we shall have a series of points P, Q, R, &c, which, when we suppose the series of values of Fisr. 1. Gi 'aph /■-• Represen ia Hon . 15 m indefinitely increased so as to include all numbers between - oo and + ce , will trace out a continuous curved line. This curve will, by the distances of its several points from the line OX, exhibit to the eye the several values of the function /(ar). The process here explained is also called tracing tin function f(x). The student acquainted with analytic geometry will observe that it is equivalent to tracing the plane curve whose equation is y=f(x). In the practical application of this method it is well to begin by laying down the points on the curve corresponding to certain small integral values of oc, positive and negative. It will then in general be possible to draw through these points a curve which will exhibit the progress of the function, and give a general idea of its character. The accuracy of the representation will of course increase with the number of points determined between any two given values of the variable. When any portion of the curve between two proposed limits has to be examined with care, it will often be necessary to substitute values of the variable separated by smaller intervals than unity. The following ex- amples will illustrate these principles. ExAJHTLES. 1. Trace the trinomial 2x- + x — 6. The unit of length taken is one -sixth of the line OD in fig. 2. In Ex. 1, Art. 9, the values off{x) are given corresponding to the integral values of x from — 4 to + 4, inclusive. By means of these values we ohtain the positions of nine points on the curve ; seven of which, A, B, C, D, E, F, G, are here represented, the other two correspond- ing to values of f(x) which lie out of the limits of the figure. The student will find it a useful exercise to trace the curve more minutely between the points C and E in the figure, viz. by calculating the values of f(x) corresponding to all values of x between - 1 and 1 separated by small intervals, say of one-tenth, as is done in the following example. 16 General Properties of Polynomials. 2. Trace the polynomial 10* 3 -l7* 2 +a:+6. This is alnady tabulated in Art. 9 for values of X between —4 and 1. It ma)- he observed, as an exercise on Art. 4, that this function retains positive values for all positive values of x greater than 2-7, and negative values for all values of x nearer to — co than — 27. The curve will, then, if it cuts the axis of x at all, cut it at a point (or points) corresponding to some value (or values) of x hetween — 2 - 7 and + 2 - 7 ; so that if our object is to determine, or approximate to, the positions of the roots of the equation f{x) = 0, the tabulation may be con- fined to the interval between - 2*7 and 2 - 7. This is a case in which the substitution of integral values only of x gives very little help towards the tracing of the curve, and where, consequently, smaller intervals have to be ex- amined. We give the tabulation of the func- tion for intervals of one-tenth between the integers - 1 , ; 0, 1 ; 1, 2. From these values the positions of the corresponding points on the curve may be approximately ascertained, and the curve traced as in fig. 3. Values of x -1 -•9 -•8 -•7 I--6 1- •5 -•4 -•3 — - 2 -■1 ,, „ f ' -22 -15-96 -10-8 -6-46|-2-88| 2-21 3-9 5-04 5-72 Values of x , -1 G 5-94 •2 5-6 •3 5-04 •4 4-32 •5 3-5 •G 2-64 •7 1-8 •8 1 -9 1-04 j -42 Values of a; 1 1-1 1-2 1-3 1-4 1-5 1-6 1-7 1-8 j 1-9 -•16 •54 1-52 3 5-04 7-7 11 -04 J 15-12 2 '20 The curve traced in Ex. 1 cuts the axis of x in two points (a number equal to the degree of the polynomial) : in other words, there are two values of x for which the value of the given polynomial is zero ; these are the roots of the equation 2x 2 + x - 6 = 0. viz., - 2, and 1'5. Similarly, the curve traced in Ex. 2 cuts the axis in three points, viz., the points corresponding to the roots of the cubic equation lOr 5 - Ytx z + z + 6 = 0. The curve Maxima and Minima Values. 17 representing a given polynomial may not cut the axis of a; tit all, or may cut it in a number of points less than the degree of the polynomial. Such cases correspond to the imaginary roots of equations, as will appear more fully in the next Chapter. For example, the curve which represents the polynomial 2x- + x + 2 will, when traced, lie entirely above the axis of x ; in fact, since this function differs from the function of Ex. 1 only by the ad- dition of the constant quantity 8, each value of /(.<•) is obtained by adding 8 to the previously calculated value, and the entire curve can be obtained by simply supposing the previously traced curve to be moved up parallel to the axis of y through a distance equal to 8 of the units. It is evident, by the solution of the equation 2x 2 + x + 2 = 0, that the two values of x which render the polynomial zero are in this case imaginary. Whenever the number of points in which the curve cuts the axis of * falls short of the degree of the polynomial, it is customary to speak of the curve as cutting the line in imaginary points. 11. Maxima and Minima Values of Polynomials. — It is apparent from the considerations established in the pre- ceding Articles, that as the variable x changes from - go to + co , the function f(x) may undergo many variations. It may go on for a certain period increasing, and then, ceasing to increase, may commence to diminish ; it may then cease to diminish and commence again to increase ; after- which another period of diminution may arrive, or the function may (as in the last example of the preceding Art.) go on then continually in- creasing. At a stage where the function ceases to increase and commences to diminish, it is said to have attained a maximum value ; and when it ceases to diminish and com- mences to increase, it is said to have attained a minimum value. A polynomial may have several maxima, or several minima values, or both : the number depending in general on the degree of the function. Nothing exhibits so well as a graphic representation the occurrence of such a maximum or minimum value ; as well as the various fluctuations of which the values of a polynomial are susceptible. c 18 General Properties of Polynomials. A knowledge of the maxima and minima values of a func- tion, giving t he position of the points where the curve bends -with reference to the axis, is often of great assistance in tracing the curve corresponding to a given polynomial. It will be shown in a subsequent chapter that the determination of these points depends on the solution of an equation one degree lower than that of the given function. CHAPTER II. GENERAL PROPERTIES OF EQUATIONS. 12. The process of tracing the function /(.r) explained in Art. 10 may be employed for the purpose of ascertaining approximately the real roots of a given numerical equation ; for when the cor- responding curve is accurately traced, the real roots of the equa- tion f(x) = can be obtained approximately by measuring the distances from the origin of its points of intersection with the axis. With a view to the more accurate numerical solution of this problem, as well as the general discussion of equations both numerical and algebraical, we proceed to establish in the present Chapter the most important general properties of equa- tions having reference to the existence, and number of the roots, and the distinction between real and imaginary roots. By the aid of the following theorem the existence of a real root in an equation may often be established : — Theorem. — If two real quantities a and b be substituted for the unknown quantity x in any polynomial fix), and if they furnish results having different signs, one plus and the other minus ; then the equation f(se) = must have at least one real root intermediate in value between a and b. This theorem is an immediate consequence of the property of the continuity of the function f(x) established in Art. 7 ; for since. f{x) changes continuously from/(«) to f{b), and therefore passes through all the intermediate values, while x changes from a to b ; and since one of these quantities, f(a) orf(b), is positive, and the other negative, it follows that for some value of x inter- mediate between a and b, fix) must attain the value zero which is intermediate between/(«) and/(6). c2 20 General Proper/if* of Equations. The student will assist his conception of this theorem by reference to the graphic method of representation. What is here proved, and what will appear ohvious from the figure, is, that if there exist two points of the curved line representing the polynomial on opposite sides of the axis OX, then the curve joining these points must cut that axis at least once. It will also be evident from the figure that several values may exist between a and b for which f{x) = 0, i. e. for which the curve cuts the axis. For example, in fig. 3, Art. 10, x = - 2 gives a nega- tive value (- 144), and x = 2 gives a positive value (20), and be- tween these points of the curve there exist three points of section of the axis of x. Corollary. — If there exist no real quantity which, substituted for x, makes f(x) =0, thcnj\x) must be positive for every real value of x. For it is evident (Art. 4) that x= cc makes f(x) positive ; and no value of x, therefore, can make it negative ; for if there were any such value, the equation would by the theorem of this Article have a real root, which is contrary to our present hypo- thesis. With reference to the graphic mode of representation this theorem may be expressed by saying that when the equa- tion f(x) = has no real root, the curve representing the poly- nomial f(x) must lie entirely above the axis of x. 13. Theorem. — Every equation of an odd degree has at least one real root of a sign opposite to that of its last term. This is an immediate consequence of the theorem in the last Article. Substitute in 'succession - oo , 0, go for # in the poly- nomial/^). The results are, n being odd (see Art. 4), for x = - oo , f(x) is negative ; „ x = 0, sign of f(x) is the same as that of a n ; ,, x = + go ,f(x) is positive. If a n is positive, the equation must have a real root between - go and 0, i. e. a real negative root ; and if a n is negative, the equa- Imaginary Roots. 21 tion must have a real root between and co , i. e. a real positive root. The theorem is thus proved. 14. Theorem. — Every equation of an even degree, whose last term is negative, has at least two real roots, one positive and the other negative. The results of substituting - go , 0, oo are in this case CO 0, + 00 , +, + ; hence there is a real root between - oo and 0, and another be- tween and + co ; i. e. there exist at least one real negative, and one real positive root. We have contented ourselves in both this and the preceding Articles with proving the existence of roots, and for this purpose it is sufficient to substitute very large positive or negative values, as we have done, for x. It is of course possible to narrow the limits within which the roots lie by the aid of the theorem of Art. (4), and still more by the aid of the theorems respecting the limits of the roots to be given in a subsequent Chapter. 15. Existence of a Root in the General Equation. Imaginary Roots. — We have now proved the existence of a real root in the case of every equation except one of an even degree whose last term is positive. Such an equation may have no real root at all. It is necessary then to examine whether, in the absence of real values, there may not be values involving the ima- ginary expression v - 1, which, when substituted for x, reduce the polynomial to zero ; or whether there may not be in certain cases both real and imaginary values of the variable which satisfy the equation. We take a simple 32 General Properties of Equations. example <<> illustrate the occurrence of such imaginary roots. As already remarked (Art. 10), the curve corresponding to the polynomial / [.<■) .'.' - . x + 2 lies entirely above the axis of :c, as in fig. 4. The equation f(jr) = has no real roots ; but it has the two imaginary roots 4 + 4 v i? 4" r^ -1 ' as is evident by the solution of the quadratic. We observe, therefore, that in the absence of any real values there are in this case two imaginary expressions which reduce the polynomial to zero. The general proposition of which this furnishes an illustra- tion is, that Every rational integral equation a x n + a L x n ~ i + a,.v"' 1 + . . . + a„_i% + a„ = must have a root of the form « and /3 being real finite quantities. This statement includes both real and imaginary roots, the former corresponding to the value j3 = 0. As the proof of this proposition involves principles which could not conveniently have been introduced hitherto, and which will present themselves more naturally for discussion in subsequent parts of the work, we defer the demonstration until these principles have been established. For the present, therefore, we assume the proposition, and proceed to derive certain consequences from it. 16. Theorem. — Every equation of n dimensions has if roots, and no more. We first observe that if any quantity // is a root of the equa- tion/^) = 0, then /(a*) is divisible by x-h without a remainder. This is evident from Art. !) ; for if f(h) = 0, i. e. if h is a root of f(x) = 0, E must be = 0. Imaginary Roots. • ) -'> Let, now, the given equation be f x .'■" +p l x n ~* +p 2 x"-- + . . . +pn„iX +p n = 0. This equation must have a root, real or imaginary (Art. 15), which we shall denote by the symbol o t . Let the quotient, when f(x) is divided by x - a u be (pi (x) ; we have then the identical equation /(*)■(*- ox) 01 («). Again, the equation i (x) = 0, which is of n - 1 dimensions, must have a root, which we represent by a 2 . Let the quotient ob- tained by dividing 0i (x) by x - a, be (f> 2 {x). Hence 0i(^)-('-'-a 2 )^(.r), and .*./(») = (x - «i) (/• - a 2 )

, . . . a n be substituted in the right-hand member of the above equation, the factors will be all different from zero, and therefore the pro- duct cannot vanish. Corollary. — Tico jiolynomiak of the n ih degree cannot be equal to one another for more than n values of the variable without being completely identical. For if their difference be equated to zero, we obtain an equ i - tion of the n th degree, which can be satisfied Vy n values only of the variable, unless each coefficient be separately equal to zero. 2-i General Properties of Equations. The theorem of this Article, although of no assistance in the solution of the equation /(•*") = 0, enables us to solve completely the converse problem, i. e. to find the equation whose roots are any n given quantities. The required equation is obtained by multiplying together the n simple factors formed by subtract- ing from a- each of the given roots. By the aid of the present theorem also, when any (one or more) of the roots of a given equation are known, the equation containing the remaining roots may be obtained. For this purpose it is only necessary to divide the given equation by the product of the given bino- mial factors. The quotient will be the required polynomial composed of the remaining factors. Examples. 1. Find the equation whose roots are -3, -1, 4, 5. Am. a-'-OA- 3 -13r- + 53.r + 60 = 0. 2. The equation ,1- Q X 3 . S ,: 17.,; +10 = has a root 5 ; find the equation containing the remaining roots. Use the method of division of Art. 8. Am. X* -./•-' 3a;- 2 =0. 3. Solve the equation .<•'-! 6a; 3 -l- 86a? - 176a; + 105 = 0, two roots being 1 and 7. Ans. The other two roots are 3, 5." 4. Form the equation whose roots are 7 Ans. 14a?-23a; 2 -60a;4-9 = 0. 5. Solve the cubic equation a 3 -l =0. Here it is evident that x= 1 satisfies the equation. Divide by x— 1, and solve the resulting quadratic. The two roots are found to be 1 1 / — 1 1 ,— - C. Form an equation with rational coefficients which shall have for a root the in a ional expression \/p + \/g. Equal Roots. 25 This expression has four different values according to the different combinations of the radical sign*, viz. \Zp+\/g, -Vp-Vi^ \/p-y/q, -^/p+Vq- The required equation is, therefore, (x - •/p-'/q) (x + \/p + y/q) {.v - «/p + ' | ■2z i -5.c 2 + 6x+2 = 0, -2 + ^/3. Ans. The roots are — 2 + \/ 3, 1+v^-l- 3.f 3 -4* 2 + *+88 = 0, 2 1 x/-"f. Ans. The roots are 2±^f — ~, — -. 19. Descartes' Itule of Signs — Positive Roots. — This rale, which enables us, by the mere inspection of a given equa- tion, to assign a superior limit to the number of its positive roots, may be enunciated as follows : — No equation can have more positive roofs than it has; changes of sign from + to -, and from- to +, in the terms of its first member. AVe shall content ourselves for the present with the proof which is usually given, and which is rather a verification than a general demonstration of this celebrated theorem of Descartes. It will be subsequently shown that the rule just enunciated, and other similar rules which were discovered by early investigators relative to the number of the positive, negative, and imaginary roots of equations, are immediate deductions from the more general theorems of Budan and Fourier. Let the signs of a polynomial taken at random succeed each other in the following order : — + + - + + + - + -. In this there are in all seven changes of sign, includiug changes from + to -, and from - to +. It is proposed to show Descartes' Rule of Signs. 29 that if this polynomial be multiplied by a binomial whose signs, corresponding to a positive root, are + -, the resulting poly- nomial will have at least one more change of sigh than the original. We write down only the signs which occur in the operation as follows : — + + - + + + - + - H + + + H h + ± !-- + + + + - + h Here in the third line the ambiguous sign ± is placed wher- ever there are two terms with different signs to be added. We observe in this case, and it will readily appear also for every other arrangement, that the effect of the process is to introduce the ambiguous sign wherever the sign + follows +, or - follows -, in the original polynomial. The number of variations of sign is never diminished. There is, moreover, always one variation added at the end. This is obvious in the above instance, where the original polynomial terminates with a variation ; if it terminate with a continuation of sign, it will equally appear that the cor- responding ambiguity in the resulting polynomial must furnish one additional variation either with the preceding or with the superadded sign. Thus, in even the most unfavourable case : that, namely, in which the continuations of sign in the original remain continuations in the resulting polynomial, there is one variation added ; and we may conclude in general that the effect of the multiplication of a polynomial by a binomial factor x - a is to introduce at least one additional change of sign. Suppose now a polynomial formed of the product of the factors corresponding to the negative and imaginary roots of an equation ; the effect of multiplying this by each of the factors x - a, x - j3, x - y, &c, corresponding to the positive roots a, j3, 7, &c, is to introduce at least one change of sign for each ; so that when the complete product is formed containing 30 General Properties of Equations. all the roots, we conolude that the resulting p Lalhasat Least as many changes of sign as it has positive roots. This is Descartes' proposition. 20. Descartes' Rule of Signs — \egative Hoot*. — In order to give the mosi advantageous statement to Descartes' rule in the case of negative roots, we first prove that if - x be substi- tuted for x in the equation f(x) = 0, the resulting equation will have the same roots as the original except that their signs will be changed. This follows from the identical equation of Art. 16 f(x) = (x- ai ) (x - a 2 ) (x - a.) .... (0 - a„), from which we derive /(-^)-(-l)" (./■ + «,) (x + a 2 ) (z+a,) {x + a n ). . From this it is evident that the roots of/(-.r) = are Hence the negative roots of f(x) are positive roots of/(-:r), and we may enunciate Descartes' rule for negative roots as fol- lows : — No equation can have a greater number of negative roots than there are changes of sign in the terms of 'the polynomial /(- x). 21. Use of Descartes' Rule in proving the existence of Imaginary Roots. — It is often possible to detect the existence of imaginary roots in equations by the application of Descartes' rule ; for if it should happen that the sum of the greatest possible number of positive roots, added to the greatest possible number of negative roots, is less than the degree of the equation, we are sure of the existence of imaginary roots. Take, for example, the equation x s + 10x 3 + x-4: = 0. This equation, having only one variation, cannot have more than one positive root. Now, changing x into - x, we get a»-10a*-x-4 = 0; and since this has only one variation, the original equation can- not have more than one negative root. Hence, in the proposed Theorem. :;l equation there cannot exist more than two real roots. It has, therefore, at least six imaginary roots. This application of Descartes' rule is available only in the case of incomplete equations ; for it is easily seen that the sum of the number of variations in /(./■) and /(-a?) is exactly equal to the degree of the equation when it is complete. 22. Theorem. — If two numbers a and b, substituted for x in the polynomial fix), give results with contrary signs, an odd number of real roots of the equation fix) = lies between them ; and if they give results with the same sign, either no real root or an even num- ber of real roots lies between them. This proposition, of which the theorem in Art. 12 is a par- ticular case, contains in the most general form the conclusions which can be drawn as to the roots of an equation from the signs furnished by its first member when two given numbers are substituted for x. "We proceed to prove the first part of the proposition : the second part is proved in a precisely similar manner. Let the following m roots oi, a 2 , .... a m , and no others, of the equation f(x) = lie between the quantities a and b, of which, as usual, we take a to be the lesser. Let

. Form the equation which has for roots the different values of the expression whore 0- = 1. If no restriction had been made by the introduction of 0, this expression would have 8 values. The *f 7 must now be taken with the same sign where it occurs under the second radical and free from it. There are, therefore, only four values in all. Ana. x*- 8x 3 -12x 2 + 84z-63 = 0. 1 9. Form the equation which has for roots the four values of - 9 + vA37 + 3 \/u - 20 Vl37, where 8- = \. Am. z 4 + 36s 3 -400.r 2 - 3168*+ 7744 = 0. 20. Form an equation with rational coefficients which shall have for roots all the values of the expression 0iV / P + 02V / !? + 03\A> where 0i 2 = l, 2 2 = 1, 03 2 =1. There are eight different values of this expression, viz., V / J P + V / ? + \/>', -V i>-V q-V r, -v / p + v / 'j-V / >; V 1>-V i + V r, -VP-^/q + i/r, VP + s/ 9.- *y r. Assume x = 0i *S p + 02 \/q + 03 \/r. Squaring, we have x* =p + q + r + 2 (0o 3 */qr + 3 0i \/ rp + 0i 2 «/pq). Transposing, and squaring again, (x- -p-q-r)^ = 4(qr + rp+pq) + 8ei 0203 '/pqridi V'p + Ozx/g + ezv'r). Transposing, substituting x for 0! */ p + 2 \/q + 3 \/ r, and squaring, we obtain the final equation free from radicals { x i -2x 2 (p + q + r) + p- + q* + r* - 2qr - 2rp -2pq}- = 6ipqrx 2 . This is an equation of the eighth degree, whose roots are the values above writ- ten. Since 0i, 2 , 0s have disappeared, it is indifferent which of the eight roots ± V^iv?iV /, ■ is assumed equal to a; in the first instance. The final equation is that which would have been obtained if each of the 8 roots had been subtracted from x, and the continued product formed, as in Ex. 6, Art. 16. CHAPTER III. RELATIONS BETWEEN THE ROOTS AND COEFFICIENTS OF EQUA- TIONS, WITH APPLICATIONS TO SYMMETRIC FUNCTIONS OF THE ROOTS. 23. Relations between the Roots and Coefficients. — Taking for simplicity the coefficient of the highest power of * as unity, and representing, as in Art. 16, the n roots of an equa- tion by oi, a 2 , a 3 , . . . . an, we have the following identity : — x n +piX n ~^ + p*x n ~°~ + . . . . -vp^xX +p n = (x- ai) {x - a 2 ) (x-a 3 ) (x- a«). (1) When the factors of the second member of this identity are multiplied together, the highest power of x in the product is x 11 ; the coefficient of x n ~ l is the sum of the n quantities -a Xi -a 3 , &c, viz., the roots with their signs changed ; the coefficient of x n ~ 2 is the sum of the products of these quantities taken two by two; the coefficient of x n ~ 3 is the sum of their products taken three by three ; and so on, the last term being the product of all the roots with their signs changed. Equating, therefore, the coefficients of x on each side of the identity (1), we have the following series of equations : — Pi = - (ai + a a + a 3 + . . . . + a„), "^ p 2 = (ai a 2 + ai a 3 + a 2 a 3 + . . . . + a n -\ a„), Pi = - (ai a 2 a 3 + a x a 3 a t + . . . . + a„_ 2 a„_i a») , )> (2) p n = (- 1)" ai a 2 a 3 . . . . a»_i a n , which enable us to state the relations between the roots and coefficients as follows : — d2 36 Roots and Coefficients of Equations. Theorem. — /// every algebraic equation, the coefficient of whose highest term is unity, the coefficient pi of the second term with its sign changed is equal to the sum of the roots. The coefficient p 2 of the third term is equal to the sum of the products of the roots taken two by tteo. The coefficient p 3 of the fourth term with its sign changed is equal to the sum of the products of the roots take)) three by three; and so on, the signs of the coefficients being taken alternately negative and positive, and the number of roots multiplied together in each term of the corresponding function of the roots increasing by unify, till finally that function is reached which consists of the product of the n roots. When the coefficient a of a? 1 is not unity (see Art. 1), we must divide each term of the equation by it. The sum of the roots is then equal to — ; the sum of their products in pairs is equal to — ; and so on. Cor. 1. — Every root of an equation is a divisor of the abso- lute term of the equation. Cor. 2. — If the roots of an equation be all positive, the coef- ficients (including that of the highest power of x) will be alter- nately positive and negative ; and if the roots be all negative, the coefficients will be all positive. This is obvious from the equations (2) [cf. Arts. 19 and 20]. 24. Applications of the Theorem. — Since the equations (2) of the preceding Article supply n distinct relations between the n roots and the coefficients, it might perhaps be supposed that some advantage is thereby gained in the general solution of the equation. Such, however, is not the case ; for suppose it were attempted to determine by means of these equations a root, tii, of the original equation, this could be effected only by the elimination of the other roots by means of the given equations, and the consequent determination of a final equation of which a! is one of the roots. Now, in whatever way this final equa- tion is obtained, it must have for solution not only a,, but each Theorem. 37 of the other roots a 2 , a 3 , . . . a n ; for, since all the roots enter in the same manner in the equations (2), if it had been proposed to determine a z (or any other root) by the elimination of the rest, our final equation could differ from that obtained for a x only by the substitution of a 2 (or that other root) for ai. The final equa- tion arrived at, therefore, by the process of elimination must have the n quantities ai, a 2 , a n for roots ; and cannot, consequently, be easier of solution than the given equation. This final equation is, in fact, the original equation itself, with the root we are seeking substituted for x. This we shall show for the particular case of a cubic. The process here employed is general, and may be applied to an equation of any degree. Let a, /3, 7 be the roots of the equation x 3 +piX 2 +p 2 x+p 3 = 0. We have, by Art. 23, i?i = -(a + /3 + y), p z = a/3 + ay + j3y, p 3 = -a(3y. Multiplying the first of these equations by [ar, the second by a, and adding the three, we find p x a~ +Pza +2h = - a 3 , or a 3 +p x a~ +p 2 a+p 3 = 0, which is the given cubic with a in the place of x. The student can take as an exercise to prove the same result in the case of an equation of the fourth degree. In the corre- sponding treatment of the general case the successive equations of Art. 23 are to be multiplied by a n ~\ a' 1 ' 2 , a n ~\ &c, and added. Although the equations (2) afford, as we have just seen, no assistance in the general solution of the equation, they are often of use in facilitating the solution of numerical equations when any particular relations among the roots are known to exist. They may also be employed to establish the relations which must obtain among the coefficients of algebraical equations cor- responding to known relations among the roots. 38 Roots and Coefficients of Equations. Examples. 1 . Solve the equation a?-6afi-16z + 80=0, the sum of two of its roots being equal to nothing. Let the roots be a, B, y. We have then o+/3 + 7= 5, aj8 + a7 + £7 = -16, a/8-y = -80. Taking $ + y = 0, we have, from the hist of these, o = 5 ; and from either the second or third we obtain $y = - 16. We find for £ and y the values 4 and - 4. Thus the three roots are 5, 4, — 4. 2. Solve the equation x z - 3s 2 + 4 = 0, two of its roots being equal. Let the three roots be a, a, &. We have |2a + = 3, a 2 + 2aj8 = 0, from which we find o = 2, and /3 = - 1. The roots are 2, 2,-1. 3. The equation x i + 4z 3 - 2s 2 - 12z + 9 = has two pairs of equal roots ; find them. Let the roots be o, a, £, £ ; we have, therefore, 2a + 2£=-4, o 2 + 2 + 4a0 = - 2, from which we obtain for a and # the values 1 and - 3. 4. Solve the equation a^- 9a* +14* + 24=0, two of whose roots are in the ratio of 3 to 2. Lit the roots be o, j8, 7, with the relation 2a = 3£. By elimination of a we easily obtain 5/3 + 27 = 18, 30 2 + 507 = 28, from which we have the following quadratic for j3 : — 19/3 2 — 90>3 + 56 = 0. Examples. 39 14 The roots of this are 4, and — ; the former gives for a and y the values 6 and — 1. The three roots are 6, 4, — 1. The student ■will here ask what is the signi- 14 ficance of the value — of /3 ; and the same difficulty may have presented itself in the previous examples. It will he observed that in examples of this nature we never require all the relations between the roots and coefficients in order to deter- mine the required unknown quantities. The reason of this is, that the given con- dition establishes one or more relations amongst the roots. "Whenever the equations employed appear to furnish more than one system of values for the roots, the actual roots are easily determined by the condition that they must satisfy the equation (or equations) between the roots and coefficients which we have not made use of in determining them. Thus, in the present example, the value /3 = 4 gives a system satisfying the omitted equation a#7 = -24; 14 while the value P = t~. gives a system uot satisfying this equation, and is therefore to be rejected. 5. Solve the equation a; 3 -9x 2 + 23x-15 = 0, whose roots are in arithmetical progression. Let the roots be a - 5, o, a + 8 ; we have at once 3a = 9, 3a 2 -8 2 = 23, from which we obtain the three roots 1, 3, 5. 6. Solve the equation x i + 2x i - llz 2 - 22x + 40 = 0, whose roots are in arithmetical progression. Assume for the roots a — 38, a— 5, a+ 8, a + 38. Ans. -5,-2, 1, 4. 7. Solve the equation 27z 3 + 4:2z 2 -28x-8 = 0, whose roots are in geometric progression. Assume for the roots ap, a, -. From the third of the equations (2), Art. 23, we P 8 2 have « 3 = — , or a = -. Either of the remaining two equations gives a quadratic for p. £ I o 2 -2 Ans. -2, -, _. 8. Solve the equation 3^ 4 - 40x 3 + 130* 2 - 120a; + 27 = 0, 40 Roots and Coefficients of Equations. whose roots arc in geometric progression. a a .mi for the roots — . -, ap, ap 3 . Employ tho second and fourth of the equa- P 3 P tions (2), Art. 23. Ans. -, 1, 3, 9. o 9. Solve the equation x i + lbx 3 + 70x 3 + 120* + 64 = 0, whose roots are in geometric progression. Ans. — 1, —2, - 4, - 8. 10. Solve the equation 6a.- 3 - 11a; 2 + 62-1 = 0, whose roots are in harmonic progression. Take the roots to he a, (3, y. We have here the relation 1 1 2 a y p hence @y + 7a + afi = 3ya ; &c. 1 1 Ans. 1, -, -. ' 2' 3 1 1 . Solve the equation 81z 3 -18.i; 2 -36s + 8 = 0, whose roots are in harmonic progression. 12. If the roots of the equation x 3 — px z + qx — r = he in harmonic progression, show that the mean root is — . 1 13. The equation x* - 2x? + 4a: 2 + 6* - 2 1 = has two roots equal in magnitude and opposite in sign ; determine all the roots. Take a + £ = 0, and employ the first and third of equations (2), Art. 23. Ans. -v/3, -\/3, l±\/~6. 14. The equation 3** - 2ox 3 + 50* 2 - 50z + 12 = 2 2 _2 9' 3' ~3' has two roots whose product is 2 ; find all the roots. Ans. 6, -, 1+v/- 1 - 15. One of the roots of the cuhic x 3 - px- + qx — r = is double another ; show that it may he found from a quadratic equation. Examples. 41 16. Show that all the roots of the equation Z»+p 1 Xn-l+p 2 X n - Z + .... +p n -l.V+p n = -can be obtained when they are in arithmetical progression. Let the roots be a, a + 8, a + 25, . . . . a + (» - 1) 8. The first of equations (2) gives -Pi=na+{1 +2 + 3+ . . . . +(»-l)}8 n(n — 1) = „«+_}__' 5 . (1) Again, since the sum of the squares of any number of quantities is equal to the square of their sum, minus twice the sum of their products in pairs, we have the equation P\- - 2/; 2 = or + (a + S) 3 + (a + 25) 3 + . . . ,, « »(»-l)(2n-l) = /tu- + n(n- l)a8 + — — -S 2 . (2) 6 ' Subtracting the square of (1) from n times the equation (2), we find 8 2 in terms of pi and p 2 . "We can then find a from equation (1). Thus all the roots can be expressed in terms of the coefficients p\ and jy 2 . 17. Find the condition which must be satisfied by the coefficients of the equa- tion x 3 -px 2 + qz-r=0, when two of its roots a, £ are connected by a relation a + 13 = 0. Ans. pq — r = 0. 18. Find the condition that the cubic x 3 -px~+ qx — r=0 should have its roots in geometric progression. Ans. p 3 r — q 3 = 0. 19. Find the condition that the same cubic should have its roots in harmonic progression (see Ex. 12). Ans. 27r 2 -9pqr + 2q 3 = 0. 20. Find the condition that the equation x i +px 3 + qx 2 + rx + * = should have two roots connected by the relation a + £ = ; and determine in that case two quadratic equations which shall have for roots (1) a, fi ; and (2) y, 5. Ans. pqr-p 2 s-r 2 = 0, (1) px 2 + r = 0, (2) x-+px+— = 0. T 21. Find the condition that the biquadratic of Ex. 20 should have its roots con- nected by the relation + y = a + 8. Am. p 3 - ipq + 8r = 0. 22. Find the condition that the roots a, 0, y, 8 of x i +px 3 + qx 2 + rx + s = should be connected by the relation aft = 78. Ans. p 2 s-r 2 = 0. 23. Show that the condition obtained in Ex. 22 is satisfied when the roots of the biquadratic are in geometric progression. 4 2 Roots and Coefficients of Equations. 25. Depression of an Equation when a relation exists between two of its Hoots. — The examples given in the preceding Article illustrate the use of the equations con- necting the roots and coefficients in determining the roots in particular cases when known relations exist among them. We shall now show in general, that if a relation of the form (3= (p(a) exist between tico of the roots of an equation f[x) = 0, the equation may be depressed tiro dimensions. Let (p [z) be substituted for x in the identity f(x) = a x n + a x x n ~ x + . . . + a n , then f{ ix) ) m a ({*)) n + «i (tf>0*) )' !_1 + + ««-i0(«O + ««• "We represent, for convenience, the second member of this identity by F{x). Substituting a for x, we have *W-/(*(«))-/0)-O; hence a satisfies the equation F{x) = 0, and it also satisfies the equation /(a?) = ; hence the polynomials f(x) and F(x) have a common measure x - a ; thus a can be determined, and from it 0(a) or /3, and the given equation can be depressed two dimen- sions. Examples. 1 . The equation x z - bx 2 - 2 ). Changing x into - x, we get the following identity also : — x* + 1 = (x + l)(x + U))(X + w 2 ), which furnishes the roots of a? + 1 = 0. Whenever in any product of quantities involving the imagi- nary cube roots of unity any power higher than the second presents itself, it can be replaced by w, or w 2 , or by unity ; for example, to 4 = to 5 , to = to, w 5 = to 3 , to 2 = to 2 , to G = to 3 . to 3 = 1, &C. The first or second of equations (2), Art. 23, gives the fol- lowing property of the imaginary cube roots : — 1 + to + to'- = 0. By the aid of this equation any expression involving real quantities and the imaginary cube roots can be written in either of the forms P+ a>Q, P+ w 2 Q. Examples. 1. Show that the product (tarn + urn) (u 2 m f un) is rational. Ans. m 2 — mn + » 2 . 2. Prove the following identities : — m 3 + n 3 = (m + ») (aim + o>-«) (a>-m + un), ni 3 — n 3 = (m — n) {aim — u'-n) (arm — wn). 3. Show that the product (a + ai/3 + ary) (a + w 2 + 017) is rational. Ans. a~ + j8 2 + 7 2 -/3-y — 70— afi. 4. Prove the identity (a + £ + 7) (o + u& + a> ? 7) (a -f ur$ + wy) = a 3 + /8 3 + 7 3 - 3a£7. 5. Prove the identity (a+ a>P + wry) 3 + (a + w 2 )8 + a>7) 3 = (2a - £ - 7) (20 - 7 - a) (27 - a - 0). Apply Ex. 2. The Cube Roofs of Unity. 45 6. Prove the identity (a + «j3 + oryf - (a + ur& + ay) 3 = -B */~Z ()3 - 7) (7 - a) (a - j8) . Apply Ex. 2, and substitute for w - ur its value v - 3. 7. Prove the identity a' 3 + 3' 3 + 7' 3 - 3o'^'7' = (« 3 + /3 3 + 7 3 - 3 a/37) 2 . where a' = a 2 + 2fiy, &=&~ + 2ya, y' = y 2 + 2a&. 8. Form the equation whose roots are m + n, u))i+w 2 n, arm + wn. Ans. x 3 - 3>nnx - (m 3 + n 3 ) = 0. 9. Form the equation whose roots are l + m+n, l + um i-u 2 )i, l + a» 2 m + un. Ans. x 3 - Six 2 + 3(P- mn) x-{l 3 + m 3 + n 3 - Zlmn) = 0. It is important to observe that corresponding to the n n th roots of unity there are n n th roots of any quantity. The roots of the equation x n -a = Q are the n n th roots of a. The three cube roots, for example, of a are v a, wva, uyvci, where 1/ a represents the real cube root according to the ordinary arithmetical interpretation. Each of these values satisfies the cubic equation x z - a = 0. It is to be observed that the three cube roots may be obtained by multiplying any one of the three above written by 1, o>, or. In addition, therefore, to the real cube root there are two imaginary cube roots obtained by multiplying the real cube root by the imaginary cube roots of unity. Thus, besides the ordinary cube root 3, the number 27 has the two imaginary cube roots --+-a/— 3, ~o~oV -3> 2 2 V ' 2 2 as the student can easily verify by actual cubing. 10. Form a rational equation which shall have wVq+. VCP + P 3 + w 2 V Q- -JQ 2 + F i for a root ; where co 3 = 1 . Compare Ex. 8. Ans. x 3 + 3Px-2Q = 0. 46 Roots and Coefficients of Equations. 1 1. Form an equation with rational coefficients which shall have 0^1/1+ e 2 yQ for a root, where 0i 3 = 1, and 0o 3 = 1. Cuhing hotb sides of the equation and substituting x for its value on the right-hand side, we get x 3 -P-Q=3e l 6zl/FQ.x. Cubing again, we have (z 3 -P-Q)3=27 PQx 3 . Since 0\ and 02 may each have any one of the values 1, w, or, the nine roots of this equation are l/f+ t/Q, a>Z/P+a>yQ, orZ/P+orZ/Q, o>l/P+or\/~Q, orl/~P + Z/Q, wl/P+ Z/Q, o,-Z/P+ o>l/~Q, yP+orl/Q, l/P\wt/Q. We see also that, since 0\ and 0* have disappeared from the final equation, it is indifferent which of these nine roots is assumed equal to a; in the first instance. The resulting equation is that which would have been obtained by multiplying together the nine factors of the form x -1/ P-%/ Q obtained from the nine roots above written. 12. Form separately the three cubic equations whose roots are the groups in three (written in vertical columns in Ex. 11) of the roots of the equation of the pre- ceding example. We can write these down from Ex. 8, taking first m and n equal to ^/P, \/ Q ; then equal to o>%/P, oil/ Q ; and finally equal to oryP, or\/Q. Am. a*- 3 yp~Q x-P-Q = 0, x 3 -3a>-Z/PQx-P-Q=0, x*-3w yp~Qx-P-Q = Q. 27. Symmetric Functions of the Roots. — Symmetric functions of the roots of an equation are those functions in which all the roots are alike involved, so that the expression is unaltered in value when any two of the roots are interchanged. For example, the functions of the roots (the sum, the sum of the products in pairs, &c.) with which we were concerned in Art. 23 are of this nature ; for, as the student will readily perceive, if in any of these expressions the root oi, let us say, be written in Symmetric Functions of the Roots. 47 every place where er 2 occurs, and a 2 in every place where a, occurs, the value of the expression will be unchanged. The functions discussed in Art. 23 are the simplest sym- metric functions of the roots, each root entering in the first degree only in any term of any one of them. We can, without knowing the values of the roots separately in terms of the coefficients, obtain by means of the equations (2) of Art. 23 the values in terms of the coefficients of an infinite variety of symmetric functions of the roots. It will be shown in a subsequent Chapter, when the discussion of this subject is resumed, that any rational symmetric function whatever of the roots can be so expressed. The examples appended to this Article, most of which have reference to the simple cases of the cubic and biquadratic, are sufficient for the present to illus- trate the usual elementary methods of obtaining such expres- sions in terms of the coefficients. It is usual to represent a symmetric function by the Greek letter S attached to one term of it, from which the entire ex- pression may be written down. Thus, if a, /3, y be the roots of a cubic, 2a 2 /3 2 represents the symmetric function a 2 /3 2 + a 2 7 2 + /3 2 y 2 , where all possible products in pairs are taken, and each term separately squared. Again, in the same case, 2a 2 (5 represents a 2 (3 + a 2 y + j3 2 y + fi 2 a + y'a + y 2 /3, where all possible permutations of the roots two by two are taken, and the first root in each term then squared. As an illustration in the case of a biquadratic we take "2a 2 ft-, whose expanded form is as follows : — o 2 j3 2 + a 2 7 2 + a 2 S 2 + 0V + /3 2 S S + 7 2 S 2 . By the aid of the various symmetric functions which occur among the following examples the student will acquire a facility in writing out in all similar cases the entire expression when the typical term is given. 48 Boots and Coefficients of Equations. Examples. 1. Find the value of 2a' of the roots of the cubic equation a -3 + p.r- + qx + r = 0. Multiplying together the equations a + & + y = -p, 07 + ya + a/3 = q, we obtain 2a 2 + 3a0y = -pq ; hence 2a 2 = 3r — pq. 2. Find for the same cubic tbe value of a 2 + 2 + y 2 . . fws. 2a 2 = ;; 2 - 2 3 + 2/>? ; hence, by Ex. 1, 2a 3 = - p 3 + Zpq - Zr. 4. Find for tbe same cubic the value of 2 7 2 + 7 2 a 2 + a 2 2 . "We easily obtain 2 7 2 + 7 2 a 2 + a 2 2 + 2a0y(a + + 7) = ? 2 , from which 2a 2 2 = 0 Roots and Coefficients of Equations. From the second last, and last of the equations of Art. 23, we have B8O3 . . . . an+aias . . . . On+ • ■ • • + 0102 . ■ • . a„.\ = (- l)"' [ p„.\, ai 02 03 • • • • a„ = (— 1)"7 ; " : ilividing the former by the latter, we have 111 1 -pn-i - + - + -+ +- = -*—, a\ 03 as On p„ or 1 -jp»_i a\ p n In a similar manner the sum of the products in pairs, in threes, &C. of the reciprocals of the roots can be found by dividing the 3rd last, or 4th last, &c. coef- ficient by the last. 13. Find for the cubic equation a x? + 3«i z z + 32 — (>i 2 )x 2 .+ {ao an( i subtracting 5 from each root in turn, we have the three other differences o-8, /3 - 8, 7 — 5. We combine these in pairs as follows: — O8-7) (a -8), ( 7 -a)(j8-8), (a-flft-S). The symmetric function in question is the product of the differences of these three taken as iisual in circular order. Employing the values of \, ft, in the preceding example, we have -H+v = (P-y)(a-d), -p + \ = (y-a)(0-B), - \ + /x= (a- 0) (7- 5). We have, therefore, to find the value of (2\-fi-r)(2n-i>-\)(2v-\-n), (3A. - 2aj8)(3ji- 2a£);(3j/ - 2aj3), in terms of the coefficients of the biquadratic. Multiplying this out, substituting the value of 2a0, and attending to the results of Ex. 17, we obtain the required expression as follows : — a 3 (2\-/ji.-v) (2/x-v-A)(2»'-A.-^) = -432{ao« , 2«i+2ffi«2fl'3-« , off3 2 -«'i 2 «'4-« r 2 :i }. The function of the coefficients here arrived at, as well as those before obtained in Examples 13, 15, and 16, will be found to be of great importance in the theory of the cubic and biquadratic equations. 19. Find, in terms of tho coefficients of the biquadratic of Ex. 16, the value of the symmetric function (a - j3)2 + (o - 7) 2 + (a - S) 2 + (j8 - 7) 2 + (j3 - S) 2 + (7 - 8) 2 . This may be represented briefly by 2(a-j8) 2 . Ans. ff 2 S(o - fi)~ = 48 {ct\- - a «2). 20. Prove the following relation between the roots and coefficients of the biqua- dratic of Ex. 16 : — «o 3 (0 + 7 - a - 8) (7 + a - £ - 5) (a + £ - 7 - 5) = 32 (a 2 a 3 - 3ff «i «2 + 2«i 3 ). Theorems relating to Symmetric Functions. 53 28. Theorems relating to Symmetric Functions. — The following two theorems, with which we close for the present the discussion of this subject, will be found useful in many in- stances in verifying the results of the calculation of symmetric functions. (1). The sum of the exponents of all the roots in any term of any symmetric function of the roots is equal to the sum of the suffixes in each term of the corresponding value in terms of the coefficients. The sum here spoken of, which is of course the same for every term of the symmetric function, and which may be called the degree in all the roots of that function, will be subsequently defined (see Ch. XII.) as the weight of the symmetric function. The truth of the theorem will be observed in the particular cases of the examples 13, 15, 16, 17, &c. of the last Article ; and that it must be true in general appears from the equations (2) of Art. 23, for the suffix of each coefficient in those equations is equal to the degree in the roots of the corresponding function of the roots ; hence in any product of any powers of the coefficients the sum of the suffixes must be equal to the degree in all the roots of the corresponding function of the roots. (2). When an equation is written with binomial coefficients, the expression in terms of the coefficients for any symmetric function of the roots, which is a function of their differences only, is such that the algebraic sum of the numerical factors of all the terms in it is equal to zero. The truth of this proposition appears by suppos- ing all the coefficients a , a x , a-,, &c. to become equal to unity in the general equation written with binomial coefficients, viz., a x n + na 1 x n - x + n ^ ~ ' a z x n - 2 + + a n = 0. JL . /O The equation then becomes (x + l) n = 0, i. c. all the roots be- come equal ; hence any function of the differences of the roots must in that case vanish, and therefore also the function of the coefficients which is equal to it ; but this consists of the alge- braic sum of the numerical factors when in it all the coefficients #o, «i, a*, &g. are made equal to unity. In-Exs. 13, 15, 16, 18, 20 of Art. 27 we have instances of this theorem. Ans. ^-3. ■ ) i Roots and Coefficients of Equations. Examples. 1 . Find in terms of />, q t r the value of the symmetric function 02 + yi 7 2 + a 2 a > + yg3 JT" ~ + ~ + 7T- » 07 70 a0 where o, 0, 7 are the roots of the cubic equation x 3 +px 2 + qx + r -0. 2. Find for the same equation the value of (£ + 7 - o) 3 + (7 + o - 0) :i + (a + - 7) H . Ans. 2ir-p :i . 3. < ulculate the value of 2a 3 3 of the roots of the same equation. Here 2a02cr0 2 = 2a 3 3 + a0y2a 2 ; hence &c. -4«s. (jr 3 — 3pqr + or-. 4. Find for the same equation the value of the symmetric function (0 3 -7 3 )'- + (7 3 -a 3 )'- + (a 3 -0 3 )-. 2o fi is easily obtained by squaring 2o 3 (see Ex. 3, Art 27). Ans. 2p* - 12 p*q + 12 p 3 r + 18 p-q 2 - 18 pqr - 6 q z . 5. Find for the same equation the value of 02 + yi 7 2 _|_ g 2 g 2 +j9 2 0+7 7 + o a + 2p 2 q-4pr-2q 2 Ans. . r-pq 6. Find for the same equation the value of a 2 + 07 2 + 7a 7 2 + a + 7 7 + a a + " p i - Sp^q + opr + q* Ans. . r - pq 7. Find for the same equation the value of 207 -q- 27a - 0- 2 a0 - 7- + 7-a 7 + a-0 a + 0-7' p i -2p i q + 14pr-8q z Ans. . 4pq —p J — or Examples. 55 ( _ Q\ 2 I for the same cubic equation. 3p 2 q 2 — 4p 3 r — 4g 3 — Ipqr — 9r 2 Ans. (r -i?j) 2 oj3 9. Calculate in terms of p, a, r, s the value of 2 — - for the equation T x* +px 3 + qx 2 + rx + s = 0. 1 a a$ 1 a. Here 2a£2- = 2- + 2 — ; and 2a2 - = 4 + 2-. a- £ 7^ a /3 or 2 — 2tf 2 s - »>'« + 4** Ans. 10. Find the value of 2 — of the roots of the equation X n +P1X"' 1 + poX n '~ + . . . . +p n -\X+p n = 0. . Pn-lPn-piP>i 2 -l + 2PlPn-2Pn Ans. ; . 11. Find for the biquadratic of Ex. 9 the value of (j8y-aB) (7a- J88) (0)8-78). Compare Ex. 22, Art. 24. Ans. r 2 - p 2 s. 12. Find the value of 2(«oct + «i) 2 {&-y)- in terms of the coefficients of the cubic equation aoX 3 + 3aix 2 + 3«2£ + «3 = 0- Ans. — - (ao«2 — «i 2 ) 2 . «o" 13. Find the value r of the symmetric function 2 — of the roots of the oi az equation X n +PIX"' 1 +p 2 X n - 7 + +Pn-lZ+Pn = 0- The given function may be written in the form ai ( 1 1 -+-+. • (Ol 02 a n ) + az - + -+• • 1, Ol 02 ..♦ifl a n ) + . . + a n ( l l -+-+• • [Ol 02 o„) 1 or 2cti 2 — ■ Ol -«.; hence &c. , P\Pn-\ Ans. Pn 14. Clear of radicals the equation «/t - a 2 + y/t-p + 's/t-y 2 = ; 56 llnoh awl I'm jjic irn Is <>f Equations. and express the i oefficients of the recruiting equation in t in terras of the coefficient* of the cubic of Ex. 1. Ans. 3t--2(p 2 -2q)l-p i + Ap 2 q - 8 jor= 0. 15. If a, fi, y, 5 be the roots of the biquadratic of Ex. 9, prove (a 2 + 1) (i8* + 1) (y~ + 1) (5* + 1) = (1 - q + «)» + (p - r) 2 . Substitute in turn each of the roots of the equation x 2 + 1 = in the identity of Art. 16, and multiply. 16. Trove tho following relation between the roots and coefficients of the general equation of the n' h degree : — (ai- + 1) (a; 2 + 1) . . . . (a„ 2 + 1) = (1 -fr+Pi - . . -) 2 + (/>i -J* + • • -) 2 - 17- Find the numerical value of (a 2 +2)03 2 + 2)( 7 - + 2)(5- + 2), where a, 0, y, 5 are the roots of the equation **- 7* 3 + %x~-bx + 10=0. Substitute in turn for x each root of the equation x 2 + 2 = 0, and multiply. Ans. 166. 18. If a, fi, y, 5 be the roots of the equation a ti x i + iaix 3 + 6 a 2 x 2 + ia-iX + , where \, p., v have the values of Ex. 17, Art. 27. 19. Calculate the value of the symmetric function 2 (o - £) 4 of the roots of the biquadratic equation of Ex. 9. Am. Zp*--\§p 2 q + 2§q 2 +±pr-\§s. 20. Show that when the biquadratic is written with binomial coefficients, as in Ex. 18, the value of the symmetric function of the preceding example may be ex- pressed in the following form : — «o 4 2 (a - j8) 4 = 16 [48 («o«2 - «i 2 ) 2 - c - («off4 - 4 aia 3 + 3<7 2 2 )} . 21. The distances on a right line of two pairs of points from a fixed origin are the roots (a, £) and (a, &') of the two quadratic equations ax 2 + 2bx + c = 0, a'x 2 + 2 b' x + c' = ; prove that when one pair of the points are the harmonic conjugates of the other pair, the following relation exists : — ar' + a'c- 2bb' = 0. 22. The distances of three p i its A, B, Con a right line from a fixed origin O on the line are the roots of the equation ax 3 + 3!>.c 2 +3cx+d = ; Examples. 57 find the condition that one of the points^, B, C should bisect the distance between the other two. Compare Ex. 15, Art. 27. Ans. a 2 d-3abc+2b 3 = 0. 23. Retaining the notation of the preceding question, find the condition that the four points 0, A, B, C should form a harmonic division. Ans. ad 2 -3bcd+2c 3 = Q. This can be derived from the result of Ex. 22 by changing the roots into their reciprocals, or it can be easily calculated independently. 24. If the roots (a, fi, y, 5) of the equation ax 1 + ibz 3 + 6cz 2 +idz + e = are so related that o— 5, £-5, 7-5 are in harmonic progression, prove the relation among the coefficients aee+ 2bcd — ad- - J 2 e-c 3 = 0. Compare Ex. 18, Art. 27. 25. Form the equation whose roots are fiy + wya + ur afi fiy -r (o 2 ya + wafi a + w/3 — u-y a + w 2 fi — oiy •where &> 3 = 1, and a, fi, 7 are the roots of the cubic az 3 +3bz 2 + 3cz + d=0. Ans. (ac - b 2 ) x 2 + (ad— be) x + (bd- c 2 ) = 0. Compare Exs. 13 and 14, Art. 27. 26. Express (2 fiy - 70 - afi) (2 7a - afi - £7) (2 afi - fiy - ya) as the sum of two cubes. Ans. (fiy + wya + w 2 afi) 3 + (fiy + w 2 7a + inafi) 3 . Compare Ex. 5, Art 26. 27. Express (z + y + z) 3 +(z + wy + w 2 :) 3 +(z + u> 2 y + u>z)i in terms of z 3 + y 3 + z 3 and zyz, where w 3 — 1. Ans. 3 (ar 5 + y 3 + z 3 ) + 18 zyz. 28. If (x i + y 3 + z 3 -3zyz)(z' 3 + y' 3 + z' 3 -3z'y'z')=X 3 + Y 3 +Z 3 -3XYZ, find X, Y, Z in terms of z, y, a ; z', y, z'. Apply Example 4, Art. 26. Ans. X=zz' + yy' + zz', Y=zy' + yz' + zz', Z-zz' + yz' + zy'. 29. Eesolve (a + fi + y) 3 afiy - (fiy + ya + afi) 3 into three factors, each of the second degree in a, fi, 7. Ans. (a 2 - fiy) (fi 2 - ya) (y 2 - afi) . Compare Ex. 18, Art. 24. 58 Roots and Coefficients of Equation*. 30. Resolve into simple factors each of the following expression.? : — (1). (0 - y) 2 ($ +y-2a) + (y - a) 2 (y + a-20) + (a- |8) s (o + j3-2 7 ). (2). (0-y)($ + y-2a) 2 +(y-a)(y + a-20) 2 +{ a -0)(a + $-2y) 2 . Ant. (1). (2a-j3-7)(2j3-7-o)(27-a-j8). (2). -9(/8- 7 )(7-«) (a-*). 31. Find the condition that the cuhic equation x* —px 2 + qx — r = should have a pair of roots of the form a + a^/— 1 ; and show how to determine the roots in that case. If the real root is b, we easily find, by forming the sum of the squares of the roots, p 2 — 2 q — b 2 . The required condition is {p 2 -2q)(q 2 -2pr)-r 2 = 0. 32. Solve the equation a; 3 -7x 2 + 20z-24 = 0, whose roots are of the form indicated in Ex. 31. Am. Eoots 3, and 2 ± 2 »/-l. 33. Find the conditions that the biquadratic equation x* -px z + qx 2 — rx + s = should have roots of the form a + a \/ - 1 , b ±b y/ — 1 . Here there must be two conditions among the coefficients, as there are only two independent quantities involved in the roots. Am. p'--2q = 0; r 2 -2qs = 0. 34. Solve the biquadratic x i + ix* +8x 2 -120a; + 900 = 0, whose roots are of the form in Ex. 33. Ans. 3 + 3 /-l, -5 + 5v/^T. 35. If o + \/~ 1 be a root of the equation x z + qx + r = 0, prove that 2a will be a root of the equation £ 3 + qx — r = 0. 36. Find the condition that the cubic equation x 3 +j)x 2 + q.v + r = should have two roots a, $ connected by the relation a@ + 1 = 0. Ans. 1 +q +pr + r 2 = 0. Examples. 59 37. Find the condition that the biquadratic x* +px 3 + qx- + rx + s — should have two roots connected by the relation a& + 1 = 0. The condition arranged according to powers of s is 1 + q +pr + r~ + (p- + pr - 2 q - 1 ) s + (q - 1) s- + s 3 = 0. 38. Find the value of 2 (ai — 02)'- 0304 ... . a» of the roots of the equation x"+pix»- 1 +p2X n -~+ . . . . +p n =0. This is readily reducible to Ex. 13. Ans. (-1)" {jnPn-i-n-pn) . 39. If the roots of the equation , m(m-1) a^x n + na\x n - 1 + -± — ~ azx"-- + ....+«*»= be in arithmetical progression, show that they can be obtained from the expression _«i ± r | 3( gl 3_ fl «o «o \j n + 1 ('0 ai) by giving to r all the values 1, 3, 5, ....«- 1, when n is even ; and all the values 0, 2, 4, 6 .... n — 1, when n is odd. 40. Representing the differences of three quantities o, /8, 7 by ai, j8i, 71, as follows : — ai=)3-7, fii = y-a, 71 = a - j8 ; prove the relations ai 3 + £i 3 + 7i 3 = 3ai/3i7i, ai 4 + 0i 4 + 71 4 = \ {ar + /81 2 + 71 2 } 2 , ai 5 + 0i 5 + 71 5 = I { af + £i 2 + 71 2 } ai £1 71 . These results can be derived by taking a\, &\, 71 to be roots of the equation x i + qx — r = (where the second term is absent since the sum of the roots = 0), and calculating the symmetric functions 2ai 3 , 2ai 4 , 2ai 5 in terms of q and r. The process can be ex- tended to form 2ai 6 , 2ai~, kc. The sums of the successive powers are, therefore, all capable of being expressed in terms of the product aiRiyi and the sum of squares ai 2 + £i 2 + 71 2 ; the former being equal to r, and the latter to — 2 (£1 71 + 71 ai + ai/3i), or — 2q. These sums can be calculated readily as follows : — By means of x 3 = r — qx, and the equations derived from this by squaring, cubing, &c, and multiplying by x or x-, any power of x, say xp, can be brought by successive reductions to the form A + Bx + Cx-, where A, B, C are functions of q and r. Substituting a\, /81, 71, and adding, we find 2ai? = 3 A — 2qC. The student can take as an exercise to prove in this way 2ai 7 = 7y n - 2 + 0. If h be such as to satisfy the equation na Q h + a^ = 0, the trans- formed equation will want the second term. If h be either of the values which satisfy the equation J. . £ the transformed equation will want the third term ; the removal of the fourth term will require the solution of a cubic for h ; and so on. To remove the last term we must solve the equation f(h) = 0, which is the original equation itself. f2 G8 Transformation of Equations. Examples. 1. Transform the equation *S-6a: 2 + 43:-7 = into one which shall want the second term. na h + «i = gives /; = 2. Diminish the roots hy 2. Am. y 3 - 8y - 15 = 0. 2. Transform the equation z 4 + 8.r 3 + z-5 = into one which shall want the second term. Increase the roots hy 2. Am. y i - 24 y- + 66 y - 55 = 0. 3. Transform the equation x i -4x 3 -\8.v--Zx + 2 = into one which shall want the third term. The quadratic for h is 6k--Uh-l8 = 0, giving A=3, A=-l. Thus there are two ways of effecting the transformation. Diminishing the roots by 3, we obtain (1) y«+8y 3 -llly-196 = 0. Increasing the roots by 1 , we obtain (2) y 4 -8y 3 +17y-S=0. 35. Binomial Coefficients. — In many algebraical pro- cesses it is found convenient to write the polynomial/^) in the following form : — , n(n-l) , n(n-l) a x n + naxx n - x + \ a z x n ~ 2 + . . . + v a n . 2 x' + na^x + a n , in which each term is affected, in addition to the literal coef- ficient, with the numerical coefficient of the corresponding term in the expansion of (x + l) n by the binomial theorem. The student will find examples of equations written in this way on referring to Article 27, Examples 13 and 16. The form is one to which any given polynomial can be at once reduced. We now adopt the following notation : — }i (ji 2 ) JJ n a a x n + na^x"' 1 + a 2 x n ~' i + . . . + na„. l x + a„, JL • /O thus using JJ with the suffix n to represent the polynomial of the n th degree written with binomial coefficients. Binomial Coefficients. 69 "We have, therefore, changing n into n-1, &c, C^n-i s aox"- 1 + (n - 1) Oi^ n " 2 + ... + («- 1) an_oZ + o,^, Ui = a cc 3 + SciiX 2 + ScizX + a 3 , U 2 = a Q ar + 2a l x - r/ : , Ui = a x + #i, U = a . One advantage of the binomial form is, that the derived functions can be immediately written down. The first derived function of U n is, plainly, n L^" 1 + (n - 1) a.x 11 - 2 + in" "-^ + ' * " + a *-\ ; or n U n .i ; so that the first derived function of a polynomial re- presented in this way can be formed by applying to the suffix of £7 the rule given in Art. 6 with respect to the exponent of the variable. Thus, for example, the first derived of C 4 is formed by multiplying the function by 4, and diminishing the suffix by unity ; it is, therefore, 4 Z7 3 , as the student can easily verify. We proceed now to prove that the substitution of y + h for x transforms the polynomial U n , or a x n + na^x' 1 ' 1 + — ± — ~- a i ar~ 1 + . . . + na n ^x 4 a n , 1.2 into A y n + nA,y n - 1 + " ^ ~ 2 A 2 y n ~ 2 + . . . + »A-iy + 4«, where A U} A\, A?, . . . -4;i_l, -4« are the functions which result by substituting h for x in Z7o, 271, V* • • • ^«-i, tf, ; i.e. ^ = «o, -4i = fl , o/i + «i, A 2 = a h' t + 2aih + a 2 , &c. Representing the derived functions oi /(h) by suffixes, as 70 Transformed ion of Equations. explained in Art. G, we may write the result of the transfor- mation, viz. f[y + It) , in the following form: — f{h) is the result of substituting h for x in Z7„; it is, therefore, A n ; its first derived f\ (Ii) is, by the above rule, nA„^ ; the first derived of this again is n(n~ 1) A n _ 2 ; and so on. Making these substitutions, we have the result above stated, which enables us to write down without any calculation the transformed equation. Examples. 1. nd the result of substituting y + h for x in the polynomial Ant. a y 3 + 3 (a h + «i) y 1 + 3 {a n h? + 2ai h + «,) y + a h 3 + Za\ K 1 + 3« 2 /< + «3- The student will find it a useful exercise to verify this result by the method of calculation explained in Art. 33, which may often be employed with advantage in the case of algebraical as well as numerical examples. 2. Remove the second term from the equation a x z + 3a\x 2 + 3a 2 x + 0^ = 0. We must diminish the roots by a quantity h obtained from the equation «oA + (?i = 0, i. e.,h = . «o Substituting this value of h in A-z, and A3, the resulting equation in y is y 3 + - 2 y + ; = 0. 3. Find the condition that the second and third terms of the equation U n = should be capable of being removed by the same substitution. Here A\ and A* must vanish for the same value of h ; and eliminating h be- tween thrm we find the required condition. Ans. «o«i-«i 2 = 0. 4. Solve the equation z 3 + 6z 2 +12*-19 = by removing its second term. The third term is removed by the same substitution, which give* y 3 -27 = 0. The Cubic. 71 The required roots are obtained by subtracting 2 from each root of the latter equation. 5. Find the condition that the second and fourth terms of the equation U n = should be capable of being removed by the same transformation. Here the coefficients A\ and A 3 must vanish for the same value of h ; eliminat- ing h between the equations a h + ai = 0, a h 3 + 3aih 2 + 3azh + (i 3 = 0, we obtain the required condition a 2 a 3 — 3a O 0i a 2 + 2ai 3 = 0. N.B. — "When this condition holds among the coefficients of a biquadratic equa- tion its solution is reducible to that of a quadratic ; for when the second term is removed the resulting equation is a quadratic for y 1 ; and from the values of y those of x can be obtained. 6. Solve the equation x i + 16 x* + 11x 2 + 64* - 129 = by removing its second term. The equation in y is y 4_ 24^-1=0. 7 Solve in the same manner the equation x i + 20* 3 + UZz* + 430.r + 462 = 0. Am. The roots are - 7, -3, -5±\/3. 8. Find the condition that the same transformation should remove the seoond and fifth terms of the equation U n = 0. Am. ffo 3 «4 — 4tfo 2 «i#3+ 6«o«i :, «2 — 3«i 4 = 0. 36. The Cubic. — On account of their peculiar interest, we shall consider in this and the next following Articles the equa- tions of the third and fourth degrees, in connexion with the transformation of the preceding Article. When y + h is sub- substituted for x in the equation (toz* + 3diX 2 + 3a 2 % + a 3 = 0, (1) we obtain a y 3 + 3Aiy* + 3A 2 y + A 3 = 0, where A t , A 2 , A 3 have the values of Art. 35. If in the transformed equation the second term is absent, Ai = 0, or h = — • . 72 Transformation of Equations. Substituting this value for h in A 2 and A i} we find, as in Ex. 2, Art. 35, a A 2 = a tf2 - "i 2 , fl\M s = «7 V/ S - 3fl^flifl2 + 2fl! 3 ; hence the transformed cubic, wanting the second term, is 3 1 y* + —{a»a 2 - a?)y + — {a 2 a 3 - 3tf tfi«2 + %a x *) = 0. The functions of the coefficients here involved are of such importance in the theory of algebraic equations, that it is custo- mary to represent them by single letters. "We accordingly adopt the notation tf «2 - d\ - S, a 2 a 3 - 3ff «i j3, 7, S, &c., which is a function of their differences only, can be expressed by the functions of the coefficients which occur in the transformed equation wanting the second term. This is obvious, since the difference of any two roots a, /3' of the transformed equation is equal to the difference of the two corre- sponding roots a, /3 of the original equation ; and any symmetric function of the roots a, /3', 7', &', &c, can be expressed in terms of the coefficients of the transformed equation. For example, in the case of the cubic, all symmetric functions of the roots which contain the differences only can be expressed as functions of a , H, and G. Illustrations of this principle will be found among the examples of Art. 27. 37. The Biquadratic, — The transformed equation, want- ing the second term, is in this case a i/ i + QA.jf + 4:A 3 i/ +Ai=0, where A 2 and A 3 have the same values as in the preceding Article ; and where Ai is given by the equation a v 3 Ai = «o 3 «4 - 4tf 2 #i#3 + Qa Q a^a 2 - Sa^. The transformed equation is, therefore, 6 4 1 !/'+ — i Hy i + — Gy+ — (a 3 a i -4:a 2 a 1 a 3 + 6a ai i a 2 -3a 1 i ) =0. &o CIq O/q We might if we pleased represent the absolute term of this equation by a symbol like H and G, and have thus three func- tions of the coefficients, in terms of which all symmetric func- tions of the differences of the roots of the biquadratic could be ex- pressed. It is more convenient, however, to regard this term as 74 Transformation of Equations. composed of IT and another function of the coefficients deter- mined in the following manner : — We have plainly the identity a 3 rr l - 4a 2 a x Y- This involves «„, 7Z", and another function of the coefficients, viz., ff a i -4a l a x + 3fl 2 2 , which is of great importance in the theory of the biquadratic. This function is represented by the letter I, giving a 3 « 4 - 4ff 2 tf,ff 3 + Qa a^a., - Saf s a. "When this product is expanded, the successive coefficients'of y will be symmetric functions of the roots a, (5, y, &c, of the given equation ; and may therefore be expressed in terms of the coefficients of that equation. Examples. 1. The roots of x 3 +px 2 + qx + r = are a, /3, 7 ; find the equation whose roots are a 2 , /3 2 , y 2 . Suppose the transformed equation to he y* + Py 2 +Qy + R = Q; then -P=a 2 +/3 2 + 7 2 , Q = 2a 2 ( 3 2 , - R = a /3 2 y 2 ; and we have to form the symmetric functions 2a 2 , 2a 2 /3 2 , a 2 /3 2 7 2 , of the"given equa- tion. We easily ohtain 2a*=jr-2«?, '2a 2 (S i = q 2 -2pr, a 2 fi i y 2 =r 2 ; the transformed equation is, therefore, V 3 ~ (P 1 - 2?) y~ + (? 2 - 2pr) y-r 2 = 0. 2. Find in the same case the equation whose roots are a 3 , /3 3 , y 3 . Am. y 3 + (p 3 - Zpq + 3r) y 2 + (q 3 - 3pqr + 3r 2 ) y+r 3 = 0. 3. If a, /3, 7, 8 he the roots of x i +PX 3 + qx 2 + rx + s = ; find the equation whose roots are a 2 , £ 2 , 7 2 , 8 2 . Let the transformed equation he y i +Py 3 +Qy 2 + Ry + S=0, then -P=2a 2 , Q = 2a 2 /3 2 , -i? = 2a 2 /3 2 7 2 , S=x - az) . . . (mx-an), (2) F+ Equation of Squared Differences of a Cubic. 83 in the above. The result is _ 18R a SIH 2 27 ... a 2 a 4 « 6 ^ ; v ; which has for roots 03 -y) 8 , (y-a) 2 , (a-/3) 2 . The equation (4) can be written in a form free from fractions by multiplying the roots by a 2 . It becomes then x* + l&Hx 2 + 8Lff*a + 27 {G 2 + 4# 3 ) = 0, (5) whose roots are ffo 2 (/3-7) 2 , «o 2 (r-o) 2 ,- ff 8 («-/3) 2 - We can write down from this an important function of the roots of the cubic (3), viz. the product of the squares of the diffe- rences, in terms of the coefficients : — ^(/3- 7 ) 2 (7-«) 2 («-i3) 2 = -27(^ 2 + 4S- 3 ). (6) It is evident from the identity of Art. 37 that G 2 + 4iP contains a 2 as a factor. We have in fact GP+4JP = a 2 [a 2 a 3 2 - 6« «i«2^3 + 4tf «2 3 + 4a 1 3 tf 3 - 3^*0/) . The expression in brackets is called the discriminant of the cubic, and is represented by A ; giving the identities G 2 + ±H Z = a 2 A, HI - a J = A. Examples. 1. Form the equation of squared differences of the cubic z 3 -7x + 6 = 0. Ans. z 3 -42x z + Ulz- 400 = 0. 2. Form the equation of squared differences of x 3 + 6x* + 7x + 2 = 0. First remove the second term. Ans. z 3 -30z 2 + 225.<;-68 = 0. 3. Form the equation of squared differences of x* + 6z 2 + 9z + 4 = 0. Ans. z 3 -l8z 2 + 81z = 0. 4. What conclusion with respect to the roots of the given cubic can be drawn from the form of the resulting equation in the last Example ? g2 84 Transformation of Equations. £ 43. Criterion of the Nature of thejRoots of a Cubic. — "We can from the form of the equation' of differences obtained in Art. 42 derive criteria, in terms of the coefficients, of the na- ture of the roots of the algebraical cubic. For,, if the equation (5) of Art. 42 has a negative root, the cubio ((3) Art. 42) must have a pair of imaginary roots, in order that the square of their difference should be negative ; and if (5) has no negative root, the cubic (3) has all its roots real, since a pair of imaginary roots of (3) would give rise to a negative root of (5) . In what follows it is assumed that the coefficients of the equation are real quantities. Four cases may be distinguished : — (1). Wlien G 2 + 4J2" 3 is negative, the roots of the cubic are alt rea l — For, to make this negative H must be negative (and 4H° > G") ; the signs of the equation (5) are^then alternately positive and negative, and, therefore (Art. 20), (5) has no negative root ; and consequently the given cubic has all its roots real. (2). When G 2 + 4iP is positive, the' t \cubic has tico imaginary roots. — For the equation (5) must then have a negative root. (3). When G 2 + 4H 3 = 0, the cubic has tu-o equal roots. — For the equation (5) has then one root equal to zero. In this case A = 0, it being assumed that a does^not vanish. We may say, therefore, that the vanishing of the discriminant (see Art. 42) ex- presses the condition for equal roots. (4). When 6r = 0, and H = 0, the cubic has its three roots equal. — For the roots of (5) are then all equal to' zero. These equa- tions may also be expressed, as can be ^ easily seen, in the form Oo _ tfi _ (h Oi a, cts which relations among the coefficients are therefore the conditions that the cubic should be a perfect cube. 44. Equation of DifTerenees in'CSeiieral. — The general problem of the formation, by the aid of symmetric functions, of the equation whose roots are the differences, or the squares of the differences, of the roots of a given equation, may be treated as follows : — Let the proposed equation be f{x) m (ar- a x ) [x - o») [x-a 3 ) (x -a H ) = 0. Equation of Differences in General. 85 Substituting x + a/ for x, and giving r the values 1, 2, 3, . . . n, in succession, we have the equations f[x + ai)=x{z + ai-az){x + ai-a 3 ) • • • • [z + fn-a n ), ~) f{x + a 2 )=x{x + a 2 -ai){x+a2- a 3 ) . . . . [x + a 2 -a n ), ! ,,* f [z + a»)^z[z+ a n - ai){z + an- a 3 ).. . . > + a n -« n _,;. J Also, employing the expansion of Art. 6, and observing that J\ar) = 0, we hud the equation \/(z + « r ) =/(«,) + j^rw + n7T3 /,,(ar) + ■ • • • + * n ~ 1 ' Denoting the second side of this equation by (f> (x, a r ), and multiplying both sides of the identities (1), we obtain y + a 3 £3> ^ + jp ~ 7 3- -4ws. y 3 + 12y 2 - 172y - 2072 = 0. 3. The roots of the cubic x 3 + qx + r = are a, )8, 7 ; form the equation whose roots are )8 2 + £7 + 72, 7- + 7a + a 2 , a 2 + a£ + 2 . ^«*. (y + j) s = 0. 4. The roots of the cubic x* + pz 2 + qz + r = being a, j8, 7; form the equation whose roots are J8 2 + 7 2 - a 2 , 7 2 + a 2 - 2 , a 2 + 2 - 7 2 . Ans. y 3 - (p 2 -2q)y-- (p* -ip z q + 8pr) y + p 6 -Gp i q + 8p 3 r + 8p 2 q 2 - 16pqr + Sr" = 0. 5. If a, )8, 7 be the roots of the cubic z 3 -3(l + a + a-)x + 1 + 3a + 3a 2 + 2a 3 = ; prove that (0 - 7) (7 - o) (o - fi) is a rational function of a. Ans. ±9(1 + a + a 2 ). 6. Find the relation between G and H of the cubic ac^ 3 + Za\x z + 3a» 2 qr - m 3 r 2 = 0. 19. If a, (3, 7 be the roots of the cubic a z 3 + 3«i£ 2 + Za-ix + a 3 = 0, find the equation whose roots are (a-j8)(a-7), (i3 - -y) (j3 - a), {y-a)(y-0). 20. Form, for the cubic of Ex. 19, the equation whose roots are 03- 7 ) 2 (2a-£-7) 2 , (7- a) 2 (20 -7 -a) 2 , (a - P)' ( 2 7 - «" 0) 2 ' The required equation can be obtained by forming the equation of squared diffe- rences of the cubic (4) of Art. 42, since (7 - a)-- (a - /B)»=0 -7)(2a-0- 7)- 21. Form, for the cubic of Ex. 16, the equation whose roots are o(/3-7) 2 , 0(y-«) 2 , 7(«-^) 2 - Let the transformed equation be a; 3 + Px 2 + Qj; + i2 = 0. .4ms. F = pq-9r, Q = q 3 -9pqr + 27r*+p 3 >; Jl = -r (iq 3 + 27 r- + 4/> 3 r -p 2 q 2 - iSpqr). 22. Form, for the same cubic, the equation whose roots are a 2 + 2/37, £°" + 2 7 a » 7 2 + 2 «0- Am. P=-J> 2 , Q = q(2p 2 -3q), -P =4p 3 r - ISpqr + 2q 3 + 21r\ CHAPTER V. SOLUTION OF RECIPROCAL AND BINOMIAL EQUATIONS. 45. Reciprocal Equations. — It has been shown in Art. 32 that all reciprocal equations can be reduced to a standard form, in which the degree is even, and the coefficients counting from the beginning and end equal with the same sign. We now proceed to prove that a reciprocal equation of the standard form can always be depressed to another of half the dimensions. Consider the equation a x 2m + a x x 2m ~ l + . . . + a m x m + . . . + a x x + a - 0. Dividing by x m , and uniting terms equally distant from the extremes, we have a ° if + x^) + ai v + x^) + -'- + Um - x (* + ;) + *■ = °- Assume x + - = z, and let x v + — be denoted for brevity by V p . "We have plainly the relation V p + \ — v pZ — Vp-\. Giving p in succession the values 1, 2, 3, &c, we have F 3 = F 2 s-F 1 -s 3 -33, V^V.z-V, = z*-4z 2 + 2, V, = V, z - V z = z 5 - 5z> + 5z ; and so on. Substituting these values in the above equation, we get an equation of the m th degree in z ; and from the values of 2 those of x can be obtained by solving a quadratic. Examples. 91 Examples. 1 . Find the roots of the equation x s + x l + x z + x 2 + x + 1 = 0. Dividing hy x + 1 (see Art. 32), we have z 4 + z- + 1 = 0. This equation may he depressed to the form z- - 1 = 0, giving z = ±l; whence x + - = l, x + - = - 1, x x and the roots of these equations are 1 ±jv/£3 -1 fyZ-3 2 2 2. Find the roots of the equation z 10 - 3.r 8 + 5a; 5 - 5z* + Zx 2 - 1 = 0. Dividing hy x 2 — 1, which may be done briefly as follows (see Art. 8), 1 _3 5-5 3-1 1-2 3-2 1 -2 3-2 1 0, we have the reciprocal equation X B _ 2*6 + 3a 4 _ 2x 2 + 1=0, (1) (^)- 2 (** + L) + 3 = 0. Substituting for V\, s 4 — 4s 2 + 2 ; and for F2, s 2 - 2, we have the equation z 4 - 6z 2 + 9 = 0, or (a 2 - 3) 3 = 0, whence s 2 = 3, and z - ± */%, giving x + - = -v/3, a; + - = - */z ; x x and the roots of these equations are */z ± */^\ - \/3 + »/- 1 2 ' 2 These roots are double roots of the equation (1). 3. Solve the equation x 5 - 1 = 0. Dividing by x — 1 we have x K + x 3 + x 2 + x + 1 = ; from which we obtain t 2 + * - 1 = 0. 92 Solution of Reciprocal and Binomial Equations. Solving this equation, wo have the quadratics * 2 + 2U +\/»)x+ 1=0, ** + $(! -\/b)z + l = 0, from which we obtain X = \{-1 +0^5 ±(10 +20 -v/5) 1 /^l}, where 2 = 1 . This expression gives the four values of x. 4. Find the quadratic factors of z G + 1 = Transforming this, we have z 3 - 32 = 0, whence 2 = 0, and z = ±\/3. The quadratic factors of the given equation are, therefore, x- + 1 = 0, z 3 + v^S * + 1 = 0. 5. Solve the equations (1). (1 + xf m fl (l + *«), (2). (1 + x) 5 = a (1 + x 5 ). 6. Reduce to an equation of the fourth degree in z (1 + *) 5 A (l-*) 5 „ ^*w. (1 - a)s* + (7 + 3a)z 2 - (4 + a) = 0. 46. Binomial Equations. General Properties. — In this and the following Articles will be proved the leading general properties of Binomial Equations. Prop. I. — If a be an imaginary root of x n - 1 = 0, then a m also mil be a root, m being any integer. Since a is a root, a n = 1, and therefore (o n ) m = 1, or (a m ) n = 1 ; that is, a m is a root of x n - 1 = 0. The same is true of the equation x n + 1 = 0, except that in this case m must be an odd integer. 47. Prop. II. — If m and n be prime to each other, the equations x m - 1 = 0, x n - 1 = have no common root except unity. Binomial Equations. 93 To prove this we make use of the following property of numbers : — If m and n be integers prime to each other, integers a and b can be found such that mb - na = ±l. For, in fact, when — is turned into a continued fraction, — is the approximation pre- n o ceding the final restoration of — . ° n Now, if possible, let a be any common root of the given equations ; then a m = 1, and a n = 1 ; also a mb = 1, and a na = 1 ; whence a ('" 6 -" a ) = 1, or a ±l = 1, or a = 1 ; that is, 1 is the only root common to the given equations. 48. Prop. III. — I/Jcbe the greatest common measure of two integers m and n, the roots common to the equations x m - 1 = 0, and x n - 1 = 0, are roots of the equation x k - 1 = 0. To prove this, let m = km', n = Jen. Now, since m' and n are prime to each other, integers b and a may be found such that mb - na = ± 1 ; hence mb - na= + k. If, therefore, a be a common root of x m - 1 = 0, and.e n -1 = 0, O (m6-no) = 1, ora*= 1; which proves that a is a root of the equation #* - 1 = 0. 49. Prop. IV. — When n is a prime number, and a any imaginary root ofx n -l = 0, all the roots are included in the series 1, a, a 3 , . . . a n '\ For, by Prop. I., these quantities are all roots of the equa- tion. And they are all different ; for, if possible, let any two of them be equal, a? = a q , whence a ^'^ = 1 ; but, by Prop. II., this equation is impossible, since n is neces- sarily prime to (p - q), which is a number less than n. 94 Solution of Reciprocal and Binomial Equations. 50. Trior. V. — When n is a composite number formed of the factors p, q, r, &o., the roots of the equations x p - 1 = 0, x q - 1 = 0, x r -1 = 0, &c, all satisfy the equation x n - 1 = 0. For, consider a root a of the equation x p - 1 = ; then a p = 1 ; from winch we derive ( Q p)qr • • = 1 ; or «" -1 = 0; which proves the proposition. 51. Prop. YI. — When n is a composite number formed of the prime factors p, q, r, &c, the roots of the equation of 1 — 1 = are the n terms of the product (1 + a + a 2 + . . . + a?' 1 ) (1 + j3 + . . . +/3*" 1 ) (1 + y + . . . + j*' 1 ) where a is a root ofx p - 1 = 0, /3 ofx q -1 = 0, y of x r - 1 =0, &c. We prove this for the case of three factors^?, q, r. A similar proof applies in general. Any term, e. g. a" (5 b y c , of the product is evidently a root of the equation x n -1 = 0, since a an = 1, fi bn = 1, y cn = 1, and, therefore, (a a /3 6 y c ) n = 1. And no two terms of the product can be equal ; for, if possible let a"(3 h y c be equal to another term a a ' /3 6 ' y c ' ; then a a '~ a = /3 6 " 6 ' y™'. The first mem- ber of this equation is a root of x p - 1 = 0, and the second member is a root oix qr -1 = 0. Now these two equations cannot have a common root since p and qr are prime to each other (Prop. II.) ; hence a a j3V cannot be equal to a a 'P h 'y c '. 52. Prop. YII. — The roots of the equation x n - 1 = 0, where n = p a q b r c , and p, q, r are the prime factors ofn, are the n products of the form a/3y, where a is a root of xP a = 1, j3 a root of x q = 1, and y of x r ° = 1. This is an extension of Prop. YI. to the case where the prime factors occur more than once in n. The proof is exactly similar. Any such product a)3y must be a root, since a" = 1, (5" = 1, y n = 1, n being a multiple of p a , q h , r c ; and a proof similar to that of Art. 51 shows that no two such products can be equal, since p a , q h , r c are prime to one another. We have, for convenience, stated this proposition for three factors only of n. A similar proof can be applied to the general case. From this and the preceding propositions we are now able to derive the following general conclusion : — The Special n th Roots of Unity. 95 The determination of the n th roots of unity is reduced to the case where n is a prime number, or a poiver of a prime number. 53. The Special Roots of the Equation x n - 1 = 0. — Every equation x n - 1 = has certain roots which do not belong to any equation of similar form and lower degree. Such roots we call special roots* of that equation, or special n th roots of unity. If n be a prime number, all the imaginary roots are roots of this kind. If n = p a , where p is a prime number, any n th root of a lower degree than n must belong to the equation x p -1 = 0, since every divisor of p a is a divisor of p a ' x (except n itself) ; hence there are p a ( 1 — ) roots which belong to no lower degree. If, again, n = p a q h , where p and a are prime to each other, there are/ 7 (l- -), andWl ) special roots of xP a - 1 = 0, and x q h _ 1 = 0, respectively. Now, if a and (5 be any two special roots of these equations, a/3 is a special root of x n - 1 = ; for if not, suppose (a/3) m = 1, where m is less than n ; we have then a m _ fi->n . i^t a m j g a r00 t f x p a - \ = 0, and (S~ m is a root of x qh -1 = 0, and these equations cannot have a common root other than 1, as their degrees are prime to each other ; conse- quently m cannot be less than n, and a[3 is a special root of x" -1 = 0. Also, as there are 'KM'-;>-('-i)(-i> such products, there are the same number of special n th roots. This proof may be extended without difficulty to any form of n. All the roots of x" -1 = are given by the series 1, a, a 3 , . . a"~ l ; where a is any special n th root. For it is plain that a, a 2 , &c, are all roots. And no two are equal ; for, if a? = a q , a^ = 1 ; and therefore a is not a special n th root, since p - q is less than n. When one special n th root a is given, we may obtain all the other special n' h roots of unity. * The term "special root " is here used in preference to the usual term "pri- mitive root," since the latter has a different signification in the theory of numhers. 96 Solution of Reciprocal and Binomial Equations. Since a is a special root, all the roots 1, a, a 2 , . . . a n_1 are different n th roots, as we have just proved ; and if we select a root a? of this series, where p is prime to n, the roots cP, u-p, . . . aC"^, a"P(= 1) are all different, since the exponents of a when divided by n give different remainders in every case ; that is, the series of numbers 0, 1, 2, 3, ...» - 1 in some order ; whence this series of roots is the same as the former, except that the terms occur in a different order. To each number p, prime to n and less than it (1 in- cluded), corresponds a special n th root of unity ; for a mp cannot be equal to 1 when m is less than n, for if it were we should have two roots in the series equal to 1, and the series could not give all the roots in that case ; therefore a p is not a root of any binomial equation of a degree inferior to n : that is, a p is a special )i' h root of unity. What is here proved agrees with the result above established, since the number of integers less than n and prime to it is, by a known property of numbers, nil — Hi — J when n = p a q h , which is also, as above proved, the number of special roots of x n - 1 = 0. Examples. 1. To determine the special roots of x 6 — 1 = 0. Here, 6 = 2x3. Consequently the roots of the equations x 2 — 1 = 0, and a? — l = are roots of z 5 — 1 =0. Now, dividing x 5 — 1 by x 3 — 1 we have x z + 1 ; x"-- 1 and dividing x 3 + 1 by -, or x + 1, we have z 2 — x + 1 = 0, which determines the special roots of x 6 — 1 =0. Solving this quadratic, the roots are 1 +y~s o = , oi also since aai = 1 = a' a\ = a 5 , which may be easily verified. The special roots are, therefore, o, o 5 ; or oi s , oi ; or o, 1 Examples. 97 2. To discuss the special roots of x 12 — 1 = 0. 12 12 Since 2 and 3 are the prime factors of 12, and — = 6, — = 4, the roots of x e - 1 = 0, and .r 4 -1=0, are roots of x lz — 1 = ; now, dividing x 12 — 1 by x i - 1, and x 6 - 1, and equating the quotients to zero, we have the two equations x B + x i + 1 = 0, and a; 6 + 1=0, both of which must be satisfied by the special roots of x 12 - 1 = ; therefore, taking the greatest common measure of x* + x i + 1, and x 6 + 1, and equating it to zero, the special roots are the roots of the equation x i - x 2 + 1 = 0. The same result would plainly have been arrived at by dividing .r 12 - 1 by the least common multiple of x* — 1 and x 6 — 1. Now, solving the reciprocal equation x* - x 2 + 1 = 0, we have x + - = + v 3 ; whence, if a and oi be two special roots, x («l)-^¥^- {*•$--- are the four special roots of x n — 1 = 0. "We proceed now to express the four special roots in terms of any one of them a. Since a + - + a\ + — = 0, or (a + m) ( 1 + — ] = 0, a a\ \ aaij we take aa\ = - 1 (as consistent with the values we have assigned to a and a\) ; and since a and a\ are roots of x 6 + 1 = 0, o 6 = - 1, and a 5 = = ai. The roots a a> ai; _ } - may therefore be expressed by the series a, a 5 , a", a 11 , since a 12 = 1. a\ a Further, replacing a by a 5 , a 7 , a 11 , we have, including the series just determined, the four following series, by omitting multiples of 12 in the exponents oi a: — a, a 5 , a 7 , a 11 , a 5 , a, a 11 , a", where the same roots are reproduced in every row and column, their order only being changed. We have therefore proved that this property is not peculiar to any one root of the four special roots ; and it will be noticed, in accordance with what is above proved in general, that 1, 5, 7, and 11 are all the numbers prime to 12, and less than it. We may obtain all the roots of x 12 — 1 = by the powers of any one of the four special roots o, a 5 , a 7 , a 11 , as follows : — a, a 2 , a 3 , a 4 , a 5 , a s , a 7 , a», a 9 , a ln , a 1 ', 1, a 5 , a 10 , a 3 , a*, a, a c \ a 11 , a\ a*, a 2 , a 7 , 1, a 7 , a 2 a 9 , a 4 , a 11 , a 6 , a, a s , a 3 , a 10 , a 5 , 1, 1. II 98 Solid in, i of Reciprocal and Binomial Equations. 3. Prove thai I mots of x lb - 1 = are roots of the equation a; 8 - x~ + x 5 - x* + x 3 - x + 1 = 0. 4. Show thai the eight roots of the equation in the preceding example may be obtained by multiplying the two roots of x 2 + x + 1 = by the four roots of x 4 + x 3 + x 1 + x + 1 = 0. 54. Solution of Binomial Equations by Circular Functions. — We take the most general binomial equation x 11 = a + by/- 1, where a and b are constants. Let a = R cos a, b = R sin a ; then x n = R (cos a + y/- 1 sin a) ; now, if r (cos + «/- 1 sin 0) he a root of this equation, we have, by De Moivre's Theorem, r" (cos nd + -v/- 1 sin nd) = R (cos a + i0 = cos a, sin nO = sin a ; and, consequently, nO = a + 2A-7T, k being any integer ; whence the assumed n th root is of the general type „ / ,-J a + 2k it / — ; . a + 2/. - 7r\ V R cos + */ - 1 sin I. V n n J Giving to k in this expression any n consecutive values in the Solution by Circular Functions. 99 series of numbers between - oo and + go , we get all the n th roots ; and no more than n, since the n values recur in periods. We may write the expression for the n th root under the form if/ R\ cos- + >/- 1 sin - j ( cos + y~ 1 sin — - If we now suppose R = 1, and a = 0, the equation x" = a + b v /- 1 becomes x" = 1 + \/- 1 ; the general type, therefore, of an n th root of 1 + v/- 1, or unity, is 2A-7T /— . 2h /— . ZKTT cos — -f v/ - 1 sm . n n If we give k any definite value, for instance zero, v" R ( cos - + *Y- 1 sin - \ n n is one n 1h root of a + b ^/- 1. The preceding formula shows, therefore, that all the n th root* of any imaginary quantity may be obtained by multiplying any one of them by the n th roots of unity. Taking in conjunction the binomial equations x n = a + b ^/- 1, and x n = a - b */- 1, we see that the factors of the trinomial x 2n - 2R cos a.x n + R 2 are " /T. ( ° + 2&7T / r . a + 2klT ) V R cos ± a/ - 1 sm } , ( n n ) where k has the values 0, 1, 2, 3 . . . n - 1. h2 100 Solution of Reciprocal and Binomial Equations. Examples. 1. Solve the equation x 1 - 1 = 0. Dividing by x — 1, this is reduced to the standard form of reciprocal equation. Assuming z = x + -, we obtain the cubic x z 3 + z- - 1z - 1 = 0, from whose solution that of the required equation is obtained. 2. Eesolve (x + I) 7 - x 1 — 1 into factors. Ans. 7x{x + l)(x 2 + x + l) 2 . 3. Find the quintic on whose solution that of the binomial equation x n — 1 = depends. Ans. z 5 + z*- 4z 3 - 3z 2 + 3z + 1 = 0. 4. "When a binomial equation is reduced to the standard form of reciprocal equation (by division by x - I; x + 1, or x- — 1), show that the reduced equation has all its roots imaginary. (Cf. Examples 15, 16, p. 33.) 5. "When this reduced reciprocal equation is transformed by the substitution z = x + - ; show that the equation in z has all its roots real, and situated between x - 2 and 2. For the roots of the equation in x are of the form cos a + v — 1 sin a (see Art. 54) ; hence x + - is of the form 2 cos a, and the value of this is real and be- x tween - 2 and 2. 6. Show that the following equation is reciprocal, and solve it : — 4(x* -x + l) s -27x-(x- 1) 2 =0. -4ms. Eoots: 2, 2, -£-, £, - 1, - 1. 7. Exhibit all the roots of the equation a; 9 - 1 = 0. The solution of this is reduced to the solution of the three cubics x 3 - 1 = 0, x* - u = 0, x 3 - or = ; where w, or are the imaginary cube roots of unity. The nine roots may be repre- sented as follows : — 1, co. 1 ;, (ti'i, w, wl, oij, co' J , co', co.^. Excluding 1, to, w- ; the other six roots are special roots of the given equation ; and are the roots of the sextic ..'■ i .i :i +1=0. 8. Eeducing the equation of the S ih degree in Ex. 3, Art. 53, by the substitution z *-z3_4s 2 + 42+l = 0; x + -, we obtain Examples. 101 prove that the roots of this equation are 2w 4tt 8tt 14ir 2 cos — , 2 cos — , 2 cos — , 2 cos —3-. 15 lo lo 15 9. Reduce the equation 4Z 4 - 85a; 3 + 357* 2 - 340a; + 64 = to a reciprocal equation, and solve it. x 2 Assume z — --\ — Ans. Roots: 4-, 1, 4, 16. 2 x 10. Solve the equation x i + mpx 3 + m-qx 2 + m 3 px + m* = 0. Dividing the roots by m, this reduces to a reciprocal equation. 11. If a he an imaginary root of the equation x n - 1 = 0, where n is a prime number ; prove the relation (1 - a) (1 - a«) (1 - a 3 ) ... (1 - a"" 1 ) = ». 12. Show that a cubic equation can be reduced immediately to the reciprocal form when the relation of Ex. 18, Art. 24, exists amongst its coefficients. 13. Show that a biquadratic can be reduced immediately to the reciprocal form when the relation of Ex. 22, Art. 24, exists amongst its coefficients. 14. Form the cubic whose roots are a + o fi , a 3 + a 4 , a 2 + a , where a is an imaginary root of x" — 1 = 0. Ans. x 3 + x 2 - 2x — 1 = 0. 15. Form the cubic whose roots are o + o 8 + a 12 + o 5 , 1 Algebraic Solution of the Cubic and Biquadratic. Subsequently we shall reduce the cubic to the form (Ix + m)' - (I'x + m'Y, or u 3 - v 3 , and obtain its solution from the simple equations u - r = 0, u - wv = 0, u - w 2 r = 0. It will be shown also that the biquadratic may be reduced to either of the forms (/>- + mx + n) 2 - (I'x 2 + mx + n'f, (x 2 + px + q) (x 2 +p'x + q'), by solving a cubic equation ; and, consequently, the solution of the biquadratic completed by solving two quadratics, viz., in the first case, /x 2 + mx + n = ± {l'x 2 + mx + ri) ; and in the second case, x 2 +px + <7 = 0, and x 2 + p'x + q' = 0. (2). Second method of solution : by assuming for a root a general form invoicing radicals. Since the expression p + and rationalizing, we have x 2 - 2px + p 1 - q = 0. Now, if this equation be identical with x 2 + Px + Q = 0, we have 2p = -P, p 2 -q=Q, ,- -p±yp 2 -4Q giving x = p + y q = _ t which is the solution of the quadratic equation. In the case of the cubic equation we shall find that Vp + 37=, and Vp Vq (Vp + V q) vp are both proper forms to represent a root ; these formulas having each three, and only three, values when the cube roots involved are taken in all generality. In the case of the biquadratic equation we shall find that Vl } - V'l + /-- / ' Vq y>- + y/r Vp + y/p Vq V r vi Algebraic Solution of Equations. 105 are forms which represent a root ; these formulas each giving four, and only four, values of x when the square roots receive their double signs. (3). Third method of solution : by symmetric functions of the roots. Consider the quadratic equation x 2 +Px+Q=0, of which the roots are a, (5. We have the relations a + j3 = - P, aj3 = Q. If we attempt to determine a and /3 by these equations, we fall back on the original equation (see Art. 24) ; but if we could obtain a second equation between the roots and coefficients, of the form la + mfi =/(P, Q), we could easily find a and /3 by means of this equation and the equation a + (5 = - P. Now in the case of the quadratic there is no difficulty in finding the required equation; for, obviously, (a - j3) 2 = P - 4Q ; and, therefore, a - j3 =

V P, <*> 2 VP, obtained by multiplying any one of its values by the three cube roots of unity, we obtain three, and only three, values for z, namely, V- ~E ,/- ,-E %%/ - -E Vp + ~r^i wVp + io- — , to'Vp + u> -77= , Vp VP v p the order of these values only changing according to the cube root of p selected. Now, if s be replaced by its value ax + b we have, finally, ax + b = vp + -jy= vp (where p has the value previously determined in terms of the coefficients) as the complete algebraic solution of the cubic equation ax z + 3bx* + 3cx + d= 0, the square root and cube root involved being taken in their entire generality. 57. Application to Numerical Equations. — The solu- tion of the cubic which has been obtained, unlike the solution of the quadratic, is of little practical value when the coefficients of the equation are given numbers ; although as an algebraic solu- tion it is complete. For, when the roots of the cubic are all real, Gr + 4ZP = - K\ an essentially negative number (see Art. 43) ; and, substituting for p and q their values *(-'0±jV=I) 108 Algebraic Solution of Ike Cubic and Biquadratic. in the formula Vp + \/q i we have the following expression for a root of the cubic : — -g+jzy^i y f-G-Ky~\ Now there is no general arithmetical process for extracting the cube root of such complex numbers, and consequently this formula is useless for purposes of arithmetical calculation. But when the cubic has a pair of imaginary roots, an ap- proximate numerical value may be obtained from the formula ) + \ 5 > since G 2 + 4iP is positive in this case. As a practical method, however, of obtaining the real root of a numerical cubic, this process is of little value. In the first case ; namely, where the roots are all real, we can make use of Trigonometry to obtain the numerical values of the roots in the following manner : — Assuming 2E cos

oosf, -2(-J2>oos^; o 3 from which formulas we obtain the numerical values of the roots Expression of the Cubic as Difference of two Cubes. 109 of the cubic by aid of a table of sines and cosines. This process is not convenient in practice ; and in general, for purposes of arithmetical calculation of real roots, the methods of solution of numerical equations to be hereafter explained (Chap. X.) should be employed. 58. Expression of the Cubic as the Difference of two Cubes. — Let the given cubic ax 3 + 3 bx 2 + 3 ex + d = (x) be put under the form z 3 + ZHz+G, where 2 s ax + b. Now assume Z 3 + dHz + G = — — [fX (s + v)' - v (z + fx) 3 }, (1) where ju and v are quantities to be determined : the second side of this identity becomes, when reduced, Z 3 — 3/XV Z — fXV (fX + v). Comparing coefficients, fxv = - H, fxv {jx + v) = - G ; therefore G ay/A f* + v = g, fi-v= —j£-\ where a 2 A = G z + 4iP, as in Art 42 ; also (z + n)(z + v) = 2 2 + — z-R. (2) Whence, putting for z its value, ax + b, we have from (1) ... (G + a&\( G-aMV fG-aAh\f . G + atf\* which is the required expression for Algebraic Solution of the Cubic and Biquadratic. solved into its simple factors, and the solution of the equation completed. We proceed to obtain expressions for the roots of the equation /3 + w~y, M = a + or/3 + toy. "We have then (OLf = A + Bto+ Cw\ (0W) 3 = A + Bu 2 + Cw, where ^ = a 3 + /3 3 + 7 3 +6a/3 7 , £=3(a 2 /3 + /3 2 7 + 7 2 a), C= 3(aj3 2 + j3 7 2 + ya 2 ) ; from which we obtain D + M> = 2 Sa 3 - 3 2a 2 |3 + 12 a/3 7 = - 27 -. (Cf. Ex. 5, p. U ; Ex. 15, p. 50.) Again, (0L) (0W) = LM= a 2 + /3 2 + r - 07 -7a-a|3 = - 9 — ; whence (a + w/3 + u> 2 yY, (a + 2 W) ( U+ a, 2 V+ w W). 2. Prove that the several equations of the system (j8 - y)« (s - a)^ = (7 - a) 3 (s - 0) 3 = (a - 0) 3 (s - yf have two factors common. Examples. 113 Making use of the notation in the last Example, we have U 3 =V*=JF Z ; whence j73_ v* = (u-v)(u* + uv+ r 2 )=i-(u-v){u 2 + v 2 + w 2 ), since U+V+W=0; therefore {$ - y) 2 (x - a) 2 + (y - a) 2 (* - 0) 2 + (a - 0) 2 (x - y)- is the common quadratic factor required. 3. Resolve into factors the expressions (1). 08 - yf (x - «)» + (7 - o)3 (x - $)* + (a - j8)» (a; - y)», (2) . (3 - y)P {x-aY+(y-aY(x-&y+{a-W{x-yY, (3). ()3 - y)' (X - a)- +(y- a) 1 (X - f3)' + (a - $? (x - yf . These factors can he -written down at once from the results established in Ex. 40, p. 59. Using the notation of Ex. 1, and replacing a\, Pi, 71, in the example referred to, by U, V, W, we obtain the following : — Am. (1) ZUVW; (2) ${TP+ V 2 + W 2 )UVW; (3) i(U 2 + V 2 + W 2 ) 2 UVW. 4. Express (* -a)(x- 0) (x - y) as the difference of two cubes. Assume (x-a)(x-H)(x-y)= Ui*-V?; whence Vi - Vi = \(x-a), wT7i-aPVi = p(z-f3), » 2 Z7"i — w Vi = v(x — y). Adding, we have A. + ju + v = 0, \a + fj.fi + v y = ; and, therefore, \ = p(t3-y), fi=p(y-a), v=p{a-&); but \fx.v = 1 ; whence - = 03 -7X7 -«)(«-£). p Substituting these values of \, ix, v, and using the notation of Ex. 1, Ui-Vi = pU, u)Ui-orVi = pV, w 2 Ui-uVi = plF; whence 2Ui = p{U+w 2 V+w W), -3Fi = P (U+w Y+w-TF); and XJ\ and V\ are completely determined. 5. Prove that L and M are functions of the differences of the roots. We have L = a + u>& + w 2 7 = a - h + « (£ - h) + a 2 (7 - h) for all values of A, since 1 + w + w 2 = ; and giving to h the values o, )8, 7, in suc- cession, we obtain three forms for L in terms of the differences f3 -y, y— a, a — (3. Similarly for M. I 114 Algebraic Solution of the Cubic and Biquadratic. 6. To express the product of the squares of the differences of the roots in terms of the coefficient-. We have L + 2T=2a-0-y, Z + w i M={2$-y-a)a>, L + aM=(2y-a- /3)o» 2 j and, again. X- M= (0-y)(o>- co-) ( co 2 i,-«jJ/=(7-o)(ctf- w 2 ), ul- co-J/= (a-/3)(co- w 2 ), from which we ohtain, as in Art. 26, £3 + JJ/ 3 = ('la - 0-y) (2j3 - y - a){2y - a - 0), £3 _ jp = _ 3 yTjj (# - 7) (7 - a) (a - £) ; and since (Z 3 - MY = (Z 3 + if 3 ) 2 - 4Z 3 JZ 3 , we have, substituting the values of X 3 + M 3 and ZJZ obtained in Art. 59, ««(6 - 7) 2 (7 " «) a (» " 0) 2 = - 27(6° + 4# 3 )- (Cf. Art. 42.) 7. Prove the following identities : — [? + M 3 = i{(2« - jS - 7) 3 + ('^ - 7 - «) 3 + (27 - « - &?} , Z 3 - J/ 3 = (y - a) 2 + a. 2 (a - £)'-. In a similar manner, we find from these expressions - Z* = (p-y)*{2a-0-y)* + u (y - a'r(20-y- a)- + a-(a- 0)- (2y- a- 0) 2 , -M* =(j3-7) 2 {2a-)8-7) > 4 w 2 ( 7 - a) 2 (20 - 7 -o) 2 + a. (a-0)- (2 7 - o - j8) 2 . Also, without difficulty, we have the following forms for LM, and Z 2 JZ- : — ■1 IM = - y)~ +(y- a) 2 -f (a - 0f, I. -M- = (a - /8)- (a - 7 ) 2 + (J3 - 7)- (0 - a) 2 + (7 - a, - (7 - j8) 2 . 'J. There are six functions of the type of Z or M, viz., a + u)0 + or 7, o>o + ary8 + 7, (u'*a + + wy, a + w 2 + 107, wa + + w'7, to a + &>;8 + 7, to foim the equation whose roots are these pix quantities. Examples. 115 These functions may be expressed as follows : — L, (aL, ui-Z, 3f, wM, w-M; hence they are the roots of the equation {

L) ( - oo-Z) (<(> - M)(f- wM) {

— Zbb' . Substituting for the roots of the transformed equations their values expressed by radicals, we have + « 2 \/ '/ ) ( w \/p' + <° 2 V '/) 4 (or '/p 4 hi 1/ 0 = 3 {\Yp~q' 4 \Vp'q) • Cubing this, we find 4> 3 - Vt\/pqp'q' ' and by previous results \p = 3(l + p + p*), A 3 + M 3 = -27p(l + p); substituting these values, we have the required equation a- A (l + p + p-) 3 - 21H 3 (p + r) 2 = 0. 13. Find the relation between the coefficients of the cubics ax 3 + 3bx- + Zcx + d = 0, ax' 3 + 3b'x' z + 3c x + d' = 0, when the roots are connected by the equation «(0' -y') + e (/ - «') + y (<*■' - ft = °- Multiplying by co - « 2 , this equation becomes iJf ' = L'M. Cubing, and introducing the coefficients, we find G 2 S' 3 = G' 2 H 3 , the required relation. 14. Determine the condition in terms of the roots and coefficients thai the cubics of Ex. 13 should become identical by the linear transformation x' = px + q. In this case a'=pa + q, 0' = p$ + q, y' = Py + 'j- Eliminating p and q, we have fiy' - $'y + ya - y'a + afl' - a'fi = 0, which is the function of the roots considered in the last example. This relation, moreover, is unchanged if for o, /3, y ; o', /3', y', we substitute la + m, IP + m, ly + m, Va + m\ V& \ m', l'y' + mf; 118 Algebraic Solution of the Cubic and Biquadratic. e we may consider the cubits in the last example under the simple forms 2 s + 3fls + (7 = 0, s ' 3 + 3.HV + 0' = 0, obtained by the linear transformations z = ax + b, z' = a'x' + V ; for if the condition holds for the roots of the former equations, it must hold for the roots of the latter. Now putting ;' = /■:, these equations become identical if //' /-//, Q' = &G; whence, eliminating k, <;- //■ - (/- U is the required condition, the same as that obtained in Ex. 13. It may be observed that the reducing quadratics of the cubics necessarily become identical by the same transformation, viz.. Q , {a'x + b') = — (ax + b). 60. nomographic Relation between two Roots of a Cubic. — Before proceeding to the discussion of the biquadratic we prove the following important proposition relative to the cubic : — The roots of the cubic are connected in pairs by a homographu relation in terms of the coefficients. Referring to Exs. 13, 14, Art. 27, we have the relations a* ( (0 _ y f + ( y _ a y + (a - i3) 2 } = 18K- - *«), flo 8 {a (/3 - yf + (5 (y - a) 2 + y (a - j3) 2 } = 9(a,a, - / + \/b 2 - ac+ a 2 3 . If this formula be taken to represent a root of the biquadra- tic in z, it must be observed that the radicals involved have not complete generality; for if they had, eight values of z in place of four would be given by the formula. The proper limitation is imposed by the relation -/i V? y/q> v^i that their product may maintain the sign deter- mined by the above equation ; thus, Vv y~-, a$ + b = -

-, ay + b = - y/-p - y<[ + y/r, a$ + b = yj) + y/q + y/r ; from which may be immediately derived the following expres- sions for p, q, r the roots of Euler's cubic : — # = jg(/3 + Y-«- o)% q = ^ (7 + a - - S) 2 , (4) Subtracting in pairs the equations (3), and making use of the relations above written between p, q, r and 0„ 2 , 3 , we easily establish the following useful relations connecting the differences of the roots of the cubics (1) and (2) with the diffe- rences of the roots of the biquadratic :- 4{q- r) = 4 + - \/p + \/p V'l) = (\/p + V<1 + '/rf -p-q- r, which shows that the number of distinct values of the radical expression of the present Article is the same as the number of values of (\/p + \/ q + y/r) 2 , namely four. In order to express p, q, r in terms of the roots a, j3, 7, 8 of the biquadratic, we have, giving to x the four values a, j3, 7, S, Zi = (la + b = V ' q v/V - y/r

/p + »/p */q, Zi = a$ + b = y/q */r + y/r >/p + y/p y/q. The student may easily satisfy himself that no combination of the signs of the radicals can lead to any value different from these four. From the values of z 2 + z 3 - Zi - s 4 , and S2S3 - 8i9 4 , we obtain , 3 , we obtain the three pairs of quadratic factors of the original quartic, and the problem is completely solved. In order to make clear the connexion between the present solution and the solution by radicals, let us suppose that the roots of the quadratic factors in the order above written are /3, 7 and a, $ ; and that the roots of the remaining pairs of qua- dratic factors are similarly y, a and |3, S ; a, /3 and y, S. We have, therefore, CI CI CI a + S = --(b + M 1 ) y B + B = --(b+' , M i ), y + $ = --{b + M>), a a a where M x = y/b 2 -ac + a?0i, M 2 = y/b 2 -ac + a 2 d 2 , M 3 = y/b 2 -ac + a*0 3 > Subtracting the last equations in pairs, we find /3 + y-a-d = 4 — , y + a-Q- 6 = 4 — , a + 6 - y - 6 = 4 — ;. a ' ' a a and since + 6 + 7 + 8 = -4-, a we obtain aa + b = - Mi + M 2 + J/ s , aj3 + b = M 1 -M« + M 3 , ay + b = M x + M«- M i} ad+b = -M 1 -M 2 -M 3 . Resolution of the Quartic into its Quadratic Factors. 129 It appears, therefore, that the roots of the biquadratic are here expressed separately by formulas analogous to those of Art. 61. The values of M 2 , viz. M 2 , M 2 , M 3 2 , are in fact identical with the roots of Euler's cuibc in the preceding Article. There exists also with regard to the signs of the radicals involved in M u M 2 , M 3 a restriction similar to that of Art. 61 ; since, in virtue of the assumptions above made with respect to the roots of the quadratic factors, we have the equation fl 3 (/3 + 7-a-S)( T + a-i3-g)(a + /3-7-8)=64Jf 1 Jf 2 Jf 3 , which implies the following relation (see Ex. 20, p. 52) : — M x MiMi=\Q\ and by means of this relation the signs of Mi, M 2 , M 3 are re- stricted in the manner explained in the previous Article. By aid of the equation last written we can eliminate M :i from the expressions for the roots, and thus obtain, as in Art. 61, all the roots of the biquadratic in a single formula, viz., G ax + b = jlf i + M 2 2MM in which the radicals Mi = */b 2 - ac + a 2 1} and M 2 = ^b 2 - ac + cPB-, are taken in complete generality. Examples. 1 . Form the equation whose roots are \, n, v, viz., (3y + aS, ya + £5, a£ + ?5. Adding the last coefficients of the quadratic factors of the quartic, we have )8y + a5 = 40i +2-, a , c ya + 08= 402 + 2-, a aj8 + >5 = 40 3 + 2 -, a where 0i, 0>, 03 are the roots of the reducing cubic ; hence the required equation. Am. {ax - 2c) 3 - il(ax - 2c) + 16 /= 0. (Compare Exs. 4, 5, Art. 39.) K loO Algebraic Solution of the Cubic and Biquadratic. 2. Express, by means of the equations of the preceding example, the roots of the reducing cubic in terms of the roots of the biquadratic. 1c . Substituting for — its value in terms of o, £, y, 8, we find immediately 12 0i = 2 A. - yu - v = (7 - a) (0 - 5) - (a - (3) (y - 5), 120> = 2 M - v - A = (a- (3)(y- 8) - (0 - 7) (a - 5), 120 3 = 2v _\_ M =(/3_ 7 )( a -5)- (7-o)(j3 - S). (Cf. (6), Art. 61.) 3. Verify, by means of the expressions for 0\, 62, 63 in Ex. 1, the conclusions of Ex. 10, Art. 61, with respect to the manner in which the roots of the biquadratic and reducing cubic are related. 4. Form the equation whose roots are the functions £(j87-a8)G3 + 7-a-8), i(7a-/35){7+a-/S-8), J-(«j8 - 78) (0 + £ -7- 5). From the quadratic factors of the quartic we find 4M 1 2Ni =/3+7 — o— 8, — ■ = £7 - o5 ; a a also M1N1 = be - ad + 2abS\ = - a 2 (pi, the roots of the required cubic being represented by ], 02, 03- We obtain, therefore, the required equation by a linear transformation of the reducing cubic. Ans. (a 2 cp +bc - ad) 3 - b 2 I(a 2

3 + (a 2 e + 6b 2 c - 9ac 2 + 2abd)

, we find that the required equation may be obtained from the reducing cubic by means of the homographic transformation 2 bed- ad*-eb* + 4abd0 = — . d- — ce + aed This result may be derived from Ex. 5 by changing the roots into their recipro- cals, and making the corresponding changes in the coefficients. 64. The Resolution of the Huartic into Quadratic Factors. Second Method. — Let the quartic ax i + 4bx 3 + 6cx 2 + 4dx + e be supposed to be resolved into the quadratic factors a (or + 2px + q) (x 2 + 2p'x + q'). We have, by comparing these two forms, the equations P+p'=2-, q + q + ±pp=§^, pq' + pq = 2-, qq=-. (1) a ci a a If now we had any fifth equation of the form F(p, q, p, q') = , we could eliminate p, p', q, q; and thus find an equation giving the several values of (p. The fifth equation might be assumed to be pp = ty, or q + q' =

pq+pq + a° a and eliminating p, p\ q, q, by means of the identical relation (f +p*W + f) - (pq -p'vY + iP'i + v''' - Ictcp + J = 0, which is the reducing cubic obtained by the previous methods of solution. Having thus found pp\ or q + q', we may complete the resolution of the quartic by means of the equations (1). The reason for the assumption above made with regard to the form of the fifth equation is obvious. From a comparison of the assumed values of

3 - Ia

is determined by a cubic equation when it has all possible values corre- sponding to each of the following types : — q-q pq'-p'q pq'-p'q Q + 1 > „ — > r> p-p p-p q-q (p-p) 2 , (p -p')(q-q')> (q-q') 2 , (pq'-p'q) 2 ; and by an equation of the sixth degree when it has all values corresponding to p, q, p - p', q - q', pq' -p'q, or p 2 - 4?. Expressing these functions in terms of the roots, the number of possible values of each function becomes apparent. 65. Transformation of the Biquadratic into the Reciprocal Form. — To effect this transformation we make the linear substitution x = ky + p in the equation ax* + 4bx z + 6cx* + 4dx + e = 0, which then assumes the form afc'f + 4^/,-y + 6UJ: 2 y 2 + 4U,ky + U, = 0, where Ui = ap + b, U 2 = ap 2 + 2bp + c, U z = ap 3 + 3bp 2 + 3cp + d, &c. (Cf. Art. 35.) If this equation be reciprocal, we have two equa- tions to determine k and p, viz., ak*= U it k*U 1 = klT 3 ; 134 Algebraic Solution of the Cubic and Biquadratic. eliminating /.-, we have the following equation for p : — all,; - U;U x = 0; and since „ _ U» _ ap 3 + Sbp 2 + 3cp + d ~£7"i~ ap + b ' there are two values of k, equal with opposite signs, correspond- ing to each value of p. The equation aUi- UcU, = Q, when reduced by the substitutions (Arts. 36, 37) a 2 U^ Ui+ZHU,+ G, tfUi m Uf + SHU? + 4:GU 1 + a t I- 3H\ becomes 2GUS + {crl- 12.fi 5 ) U? - 6GEU, - G> = 0, (1) which is a cubic equation determining Ui = ap + b; and if we put ±G 9 is determined by the standard reducing cubic 4r/ :! ;! -IaB + J=0. This transformation* may be employed to solve the biqua- dratic ; and it is important to observe that the cubic (1) which here presents itself differs from the cubic of Art. 62 only in having roots with contrary signs. We proceed now to express k and p in terms of a, /3, 7, S, the roots of the biquadratic equation. Since the equation in y, obtained by putting x = ky + p, is reciprocal, its roots are of the form y x , i/ 2} — , — ; hence we may write « = *yj + p, /3 = tyi + p, y = /.•-+ p, S = k—+p; * This method of solving the biquadratic by transforming it to the reciprocal form was given by Mr. S. S. Greatheed in the Camb. Math. Journ., vol. i. Transformation of Biquadratic into Reciprocal Form. 135 and, therefore, (a -p)(S -p) = ((5 -p)(y-p) = k 2 , from which we find /3y - aS P = + 7 - a -li' imd _ y .(7-^-»)(-ffl(r-»), (j3 + 7 - a - 8y An important geometrical interpretation may be given to the quantities k and p which enter into this transformation. Let the distances OA, OB, OC, OB, of four points A, B, C, B, on a right line from a fixed origin on the line be determined by the roots a, (5, y, $, of the equation ax i + 4 bx s + 6 ex 2 + 4 dx + e = ; also let Oi, 2 , 3 be the centres ; and F lt F{\ F 2 , F: ; F„ F/, the foci of the three systems of involution determined by the three following pairs of quadratics : — (•-0)(«-7)-O, («-a)(*-8)-0; (z-y)(z-a) = 0, («-0)(*-8)-O; (a-a)(#-/3) = 0, (a>-y)(a>-S) = 0. We have then the equations o^ . ac = X A . 0,B = 0,F X 2 , &c, which, transformed and compared with the equations ((5- P )(y-p) = (a-p)(S-p) = Jr, &c, prove that the three values of p are OOi, 00., 00 o , the distances of the three centres of involution from the fixed origin 0. Also since OiF 2 = k 2 , k has six values represented geometrically by the distances b 1 F li o 1 F 1 , ; o 2 f_,o 2 f; ; 0^,0^;, where OiF x + X F{ = 0, &c, as the distances are measured in opposite directions. 136 Algebraic Solution of the Cubic and Biquadratic. We can from geometrical considerations alone find the posi- tions of the centres and foci of involution in terms of a, (5, y, S, and thus confirm the results just established, as follows : — Since the systems [F^BFiC] and [F^AF^'D] are harmonic, 2 1111 F,F X ' F V B F^C F,A F,D y and if .r represent the distance of jP, or F v ' from the fixed origin 0, we have 1111 ^ + = + x- (5 X - y x - a x - $' Solving this equation, we find = __gy--ag_ + S-( 7 - a) (|3 - 8) (a - fl) (y - 8) + y-a-S - |3 + y _ a _g or x =-- p ± /»•, OF, + OF: , + OF, - Off' whence /r> = „ , k = ± = ± (A-P,. Example. Transform the cubic ax 3 + Zbx 2 + Zex + d to the reciprocal form. The assumption x — ky + p leads to the equation - G TJ X 3 + 3U 2 Ui 2 + H 3 = 0, where Ui = ap + b. The values of p are easily seen to he fiy — a 2 ya — )3 2 a/3 — y 2 £ + y - 2a' y + a - 2/it' a + £ - 2-/ The geometrical interpretation in this case is, that if three points A', Ji', C be taken on the axis such that A' is the harmonic conjugate of A with respect to B and C, B' of B with respect to C and A, and C" of C with respect to A and 5 ; then we have the following values of p and k : — OA + OA' , 0^ - CM' v 2 ' 2 For the values of 0^', 0.B', OC", in terms of a, 0, y, see Ex. 13, p. 88. Solution of the Biquadratic by Symmetric Functions. 137 66. Solution of the Biquadratic by Symmetric Func- tions of the Roots. — The possibility of reducing the solution of the biquadratic to that of a cubic by the present method de- pends on the possibility of forming functions of the four roots a, /3, 7, S, which admit of only three values when these roots are interchanged in every way. It will be seen on referring to Ex. 2, Art. 6-4, that several functions of this nature exist. These, like the analogous functions of Art. 59, possess an important pro- perty to be proved hereafter, viz., any two such sets of three are so related that any one function of either set is connected with some one function of the other set by a rational homographic relation in terms of the coefficients. For the purposes of the present solution we employ the functions already referred to in Art. 55, since they lead in the most direct manner to the expressions for the roots of the bi- quadratic in terms of the coefficients. We proceed accordingly to form the equation whose roots are the three values of when the roots are interchanged in every way, and 9= -1. These values are t .p±iz£z»y, ..( ^-/-s j, fc .(«-±fciz« and since ((5 + y - a - Sj 2 - 2a 2 + 2X - 2fx - 2v, 2 (a - j3) 2 = 3 Sa 2 - 2A - 2fi -2 V — 48 — , we find the following values of h, t 2 , t z : — 2\-n-v _ H 2fi - v - X _ H 2v-\- u _ H 12 a 3 ' 12 a- ' ~1^~ a 2 ; whence ti + t% + t 3 = - 3 — . a- 138 Algebraic Solution of the Cubic and Biquadratic. Again, since Q S(2/i-i;-A)(2i;-X-/i)=-3(A 3 + /i'-+i; a -/Liv-i'X-A)ii) = -^S(jU- v)\ and SU-v) 2 =24-, a we have £ 2 also kkU = -^- Hence the equation whose roots are t lt (■>, / 3 becomes (any + 3fi-(«^) 2 + (W - ^ («-0 - f - ; or, substituting for G~ its value from Art. 37, 4 (a 2 * + ITf - a*I(aH +R) + « 3 J"= 0, which is transformed into the standard reducing cubic by the substitution a~t + H = a 2 9. To determine a, /3, y, B we have the following equations : — -a + /3 + 7-b = 4//,, a-j3 + y-S = 4v/T, a + /3 - y - S = 4 -//~ along with ci + |3 + 'y + S = -4-; from which we find a = y/t t + \/S + >A, 6 /3 = - - + -vA - vA - -vA, a y = - - + y7, + yA - v/tfj, s = - - - y/h - yj, - yr 3 . Solution of Biquadratic by Symmetric Functions. 139 We have also from the above values of «/t x , ^/ 1>, \Zts the equation Vh vh vt* = ^, by means of which one radical can be expressed in terms of the other two, and the general formula for a root shown to be the same as those previously given. It is convenient, in connexion with the subject of this Article, to give some account of two functions of the roots of the biqua- dratic which possess properties analogous to those established in Art. 59 for corresponding functions of the roots of a cubic. Adopting a notation similar to that of the Article referred to, we may write these functions in terms of X, n, v in the follow- ing form : — L = (0y + ad) + u) (ya + 08) + uT (a0 + yS), M - (07 + aS) + u>* {ya + 08) + w (a/3 + y 8) . By means of the equations of Ex. 1, Art. 63, these functions can be expressed in terms of the roots of the reducing cubic in the form They may also be expressed, by aid of the equation of the pre- sent Article connecting t and 9, in terms of the values of t u t 2 , t 3l as follows : — lL = t l + U)t i + U) 2 t i , \M = ti+ (D"t 2 + wty The functions L and M are as important in the theory of the biquadratic as the functions of Art. 59 in the theory of the cubic. The cubes of these expressions are the simplest functions of four variables which have but two values when the variables are interchanged in every way ; they are the roots of the re- ducing quadratic of the reducing cubic above written, and underlie every solution of the biquadratic which has been given. 1 [() Algebraic Solution of the Cubic and Biquadratic. Examines. 1 . Show that L and M are functions of the differences of a, P, y, 5. Increasing a, P, y, 5 by li, L and M remain unaltered, since 1 + o> + u>- = 0. 2. To find in terms of the coefficients the product of the squares of the diller- ences of the roots a, P, 7, 5. From the values of L and M in terms of 0i, 2 , 3 , we find easily 125, = i+ M, L- M = (p - y) ( a - 8) (a, 2 - «), 1202 = w 2 Z + w.V, «-Z - w,l/ = (7- a) (^ - 5) (w 2 - w), 12 03 = aiZ + ull/, wi - co'-\Tf = (a - j3) (7 - 5) (or - »). Again, from these equations, multiplying the terms on both sides together, and remembering that 6\, 6>, 63 are the roots of 4a 3 3 - lad + /= 0, we find Z 3 + jtf 3 =-432-, ^i3_ j J/3 = 3v /r^ ()8 _ 7 )( 7 _ a ) (a _ J g )(a _ 5)(; g_5 )(7 _ 5); also, adding the squares of the same terms, we have 2LM = 24 - = (p - y)- (a - 5) 2 + (y - a) 2 (P - 5) 2 + (a - 0) 2 (y - 5) 2 ; and, since (X 3 - Jtf 3 ) 2 ■ (Z 3 + Jf 3 ) 2 - 4Z : UP, substituting for these quantities their values derived from former equations, we have finally « fi ()3 - 7 ) 2 (y - a) 2 (a - 0) 2 (a - 5) 2 (j8 - 8) 2 (7 - S) 2 = 256 (7 3 - 27 Z 2 ). 3. Show by a comparison of the equations of the present Article and Art. 59 that the results of the previous Article may be extended to the biquadratic by changing P-y, y-a, a-P into - (P~ y){a-5), _( 7 _ a )(j8-8), -(a-P)(y-5), 4 respectively ; and, consequently, H into I, and G into 16/. 67. Equation of Squared Differences of a Biqua- dratic. — In a previous chapter (Art. 44) an account was given of the general problem of the formation of the equation of dif- ferences. It was proposed by Lagrange to employ this equa- tion in practice for the purpose of separating the roots of a given numerical equation ; and with a view to such application Equation of Squared Differences of a Biquadratic. 141 he calculated the general forms of the equation of squared dif- ferences in the cases of equations of the fourth and fifth degrees wanting the second term (see Traite de la Resolution des Equa- tions Nitmeriques, 3rd ed., Ch. v., and Note ill.). Although for practical purposes the methods of separation of the roots to be hereafter explained are to be preferred ; yet, in connexion with the subjects of the present Chapter, the equation of squared differences of the biquadratic is of sufficient interest to be given here. We proceed accordingly to calculate this equation for a biquadratic written in the most general form. It will appear, in accordance with what was proved in Ex. 17, Art. 61, that the coefficients of the resulting equation can all be expressed in terms of a, H, I, and J. The problem is equivalent to expressing the]; following product in terms of the coefficients of the biquadratic {^-( J 8- 7 )2}{^-( 7 -«)2}{4,-( a - i 3) 2 }{^-(«-S) 2 }{*-(/3-8) 2 }{^-(')'-8) 2 }- The most convenient mode of procedure is to group these six factors in pairs, and to express the three products (which we denote by rii, II2, n3) separately in terms of the roots of the reducing cubic, and finally to express the product rii n 2 Ui in terms of a, H, I, J. rii = 2 -{ (3 -T) 2 + (a- 8) 2 } + 4 --480203- \ a' J a- Introducing now for brevity the notation l6S=a 2 P, U=cfiQ, 16J=a*fi,

- 480 2 03- Reducing the product ni n> n3 by the result of Example IS, page 89, we obtain ¥ 5 + 3QV 2 - {±Qcp' + l81i 3 + 12Q 2 tf> 2 + 36QE

6 + ZP

< + (P 3 + 8PQ - 26R) f + (6P 2 Q- 7Q 2 - 18PP) f + 9Q {PQ - 6P)i

i<- and Biquadratic. We give for convenience of reference the result also in terms of a, II, I, J* : — a 6 e + 48«*J5ty 5 + 8a 2 (9G7P + a-I)

3 + 1 1 52 [2 // / 3 «J) fy+256(.P - 27/-) = 0. It should be observed that the value above obtained for IIi can be expressed as a quadratic function of Q\ by aid of the equation 02 :i = Or — — , and the subsequent ■id" calculation might have been conducted by eliminating 9\ between this quadratic and the reducing cubic. 68. Criterion of the IVature of the Roots of the Biquadratie. — Before proceeding with this investigation it is necessary to repeat what was before stated (Art. 43), that when any condition with respect to the nature of the roots of an algebraic equation is expressed by the sign of a function of the coefficients, these coefficients are supposed to represent real numerical quantities. It is assumed also, as in the Article re- ferred to, that the leading coefficient does not vanish. Using as before A to represent that function of the coef- ficients (called the discriminant) which, when multiplied by a positive numerical factor, is equal to the product of the squares of the differences of the roots, we have, from the results estab- lished in preceding Articles, the equation « s 03 - t) 2 (7 - «) 2 (« - ma - TO - S) 2 ( 7 - S) 2 = 256A, where A = P - 27 J\ It will be found convenient in what follows to arrange the discussion of the nature of the roots under three heads, according as — (1) A vanishes, or (2) is negative, or (3) is positive. (1) Wlien A vanishes, the equation has equal roots. This is evident from the value of A above written. Four distinct cases may be noticed — (a) when two roots only are equal, in which case Zand J do not vanish separately; (/3) when three roots are equal, in which case 1=0, and J = 0, separately (see Ex. 7, Art. 61) ; [y) when * The equation of squared differences was first given in this form by Mr. M. Roberts in the Noutelles Annates de Mathematiques, vol. xvi. Criterion of Nature of Roots of Biquadratic. 143 two distinct pairs of roots are equal, in which case we have the conditions G = 0, a 2 I-12H Z = (Ex. 8, Art. 61). It can be readily proved by means of the identity of Art. 37 that these conditions imply the equation A = ; hence these two equations, along with the equation A = 0, are equivalent to two indepen- dent conditions only. Finally, we may have — (§) all the roots equal; in which case may be derived from Art. 61 the three independent conditions M = 0, 1=0, and J = 0. These may be written in a form analogous to the corresponding conditions in case (4) of Art. 43. (2) When A is negative, the equation has two real and two ima- ginary roots. — This follows from the value of A in terms of the roots ; for when all the roots are real A is plainly positive ; and when the proper imaginary forms, viz. h±k ^ -\,h' ± k' \/-l, are substituted for a, (3, y, $, it readily appears that A is positive also when all the roots are imaginary. (3) When A is positive, the roots of the equation are either all real or all imaginary. — This follows also from the value of A, for we can show by substituting for a, )3 the forms h ± k */ - 1 that A is negative when two roots are real and two imaginary. In the case, therefore, when A is positive, this function of the coefficients is not by itself sufficient to determine completely the nature of the roots, for it remains still doubtful whether the roots are all real or all imaginary. The further conditions necessary to discriminate between these two cases may, however, be obtained from Euler's cubic (Art. 61) as follows : — In order that the roots of this cubic should be all real and positive, it is necessary that the signs should be alternately positive and negative ; and when the signs are of this nature the cubic can- not have a real negative root. We can, therefore, derive, by the aid of Ex. 9, Art. 61, the following general conclusion appli- cable to this case : — When A is positive the roots of the biquadratic are all imaginary in every case except when the following conditions are fulfilled, viz. H negative, and a % I - 12IP negative ; in which case the roots are all real. 144 Ah/chraic Solution of the Cubic and Biquadratic. Examples. 1 . Show that if II be positive, or if II = (and G not = 0), the cubic will have a pair of imaginary roots. 2. Show that if IT be negative, the cubic will have its roots — (l)all real and unequal, (2) two equal, or (3) two imaginary, according as G- is — (1) less than„ (2) equal to, or (3) greater than - 4ZP. 3. If the cubic equation aoX 3 + %a\x z + 3a->x + aj = have two roots equal to a ; prove where ao«2 - «i 2 = H, 2 - 2M 2 tf> + c\c-if- - 4HiH 2

= J (ai - + -BTis) 2 - 2i/" 2 ^ + HiH>} 2 = 4H X H 2 (a-ia 2

S, — , 91 92 93 where 91 — — Examples. 147 (a - P) (y - 8) = \-n _ 0i - 02 (7 - o) {$ - 8) ~ A. - v ~ 0i - 03 _ Q3 - 7) (g - 8) = /t-y _ 02-03 f* ~ ~ (a - J3) (7 - 8) ~ m - \ ~ 02 - 0i' _ (7 - a) (|3 - 8) = v- \ _ 03 -01 . ** ~ "(£ - 7) (« ~ 5 ) ~ " " M ~ 3 - 02 " also the equation whose roots are (/3-7)(«-8), (y-a)(fi-8), (a - j3) (7 - 8) is one of the cubics «o 3 * 3 - I2oolt ± 16 v// 3 - 27/- = 0. The equation whose roots are the ratios, with sign changed, of the roots of either of these cubics is 4a(9 2 -9 + 1) 3 -27IV(- 1) 2 = (see Ex. 15, p. 88), where A = J 3 - 27/ 2 . The roots of the equation in 9 are the six anharmonic ratios. This equation can be written in a more expressive form, as will appear from the following propo- sitions : — (a). The six anharmonic ratios may be expressed in terms of any one of them, as follows : — From the identical equation (0 - 7) (a - 8) + (7 - a) (0 - 5) + (a - 0) (y - 8) s we have the relations 01 + — =1> 4>2+- = l, 03 + - =1, 93 91 92 which determine all the anharmonic ratios in terms of any one of them. (b). If two of the anharmonic ratios become equal, the six values of 9 are — o> and — or, each occurring three times ; and in this case 7=0. For suppose 91 = 92 ; we have then from the second of the above relations 91 2 ~ 91 + 1 = 0, whence 91 = — »> or — w 2 ; and substituting either of these values for 9 in (a), we find all the anharmonic ratios. Also, since A, — u a — v „ . ._ - — - + = = 0, or 20* -yf = 0, A — v \ — /j. we have I = rto«4 — 4(?i «3 + 3«2 2 = 0. l2 148 Algebraic Solution of the Cubic and Biquadratic. (r). If one of the ratios is harmonic, the six values of

y writing the sextic in (p under the following form : — 4/ 3 {( + 1)(* " 2 )(4> -£)} 2 = 27.T-{( + w)(

, 03 ; and ultimately in terms of the coefficients of the quartic. / 2H\ 96 Am. - 128 2(0 2 - e-i) 2 f 0i + —J = - ^(4-ff/ + 3aJ). 19. Express (j3 2 _ y Y-( a - - 5 2 ) 2 + ( 7 2 - aW - 5 2 ) 2 + (a 2 - /3 2 ) 2 (r - 5 2 ) 2 as a rational function of 0i, 02, 03- This symmetric function is equivalent to (H- - v-y- + {iT- - \-)~ + (A- 2 - M 2 ) 2 = 256 5 (0 2 - 3 ) 2 (01 - *-) . 20. Form the equation whose roots are the several products in pairs of the roots of a hiquadratic. The required equation is the product of three factors of the type (

2 - \

2 (tup 2 - 2c

+--0i. Ans. 4 (a

- 0i) 2 + (0 3 -0i) 2 + (01-02) 2 )' which may be expressed in terms of a, H, I, J, as above. 23 Prove 2 (^r ' if / = 0, and m of the form Zp or Zj> + \,p being a positive integer. 24. Prove that U=ax 2 + ey 2 + ez 2 + 2dyz + 2czx + 2bxy can be resolved into the sum or difference of two squares if J ' = ace + 2Sc<2 — «<2 2 - eb 2 — c 3 = 0. Here ac7= (oa;+5y+<») 2 + («c- b 2 )y 2 + 2(ad-bc)yz + (ae-c 2 )z 2 , and (ac - b 2 ) y 2 +2 (ad - be) yz + (ae - c 2 ) z 2 is a perfect square if (ae — b 2 ) (ae — c 2 ) = (ad — be) 2 , or J=0. 25. If a, /8, y, 8 be the roots of the equation «o«* + iaix 3 + 6«2^ 2 + 4a3# + «4 = 0, solve, in terms of the coefficients a , a\, &c, the equation V .>• - a + v x - P + v x — y + y/x —8 = 0. When \A + \/)3~ + %/? + a/s"= ° is rationalized, and the coefficients substituted for o, /8, y, 8, \re have (3«0«2 — 2«i 2 ) 2 = 2 C-i — #0 «3 2 — 2<*2 3 )- Now, replacing « 2 , «3, «4 by A 2 , A 3 , Ai, and substituting for the latter quanti- ties the values of Art. 37, we obtain the result. — Mr. M. Eoberts. 31. "When a biquadratic has two equal roots, prove that Euler's cubic has two equal roots whose common value is ZoJ- 2EI 21 and hence show that the remaining two roots of the biquadratic in this case are real, equal, or imaginary, according as 2EI- 3«/ is negative, zero, or positive. 32. Prove that when a biquadratic has — (1) two distinct pairs of equal roots the last two terms of the equation of squared differences (Art. 67) vanish, giving the conditions A = 0, 2-57 - 3a J = ; and when it has — (2) three roots equal, the last three terms of this equation vanish, giving the conditions I = 0, J = ; and show the equivalence of the conditions in the former case with those already obtained in Ex. S, Art. 61, and Ex. 12, p. 146. Prove also that the equation of squared dif- ferences reduces in the former case to + . . . . or But w. /w+ m 4+ h f(x + h) -f(x) 1.2 (1) 9^ = 9^ = tan QPS = tan PRJY. MN PS Now, when h is indefinitely diminished, the point Q approaches, and ultimately coincides with, P; the chord PQ becomes the Maxima and Minima Values. 153 tangent PT to the curve at P ; the angle PEN becomes PTM. Also all terms of the right-hand member of equation (1) except the first diminish indefinitely, and ultimately vanish when h = 0. The equation (1) becomes therefore tan PTM =/(«); from which we conclude that the value assumed by the derived function f'(x) on the substitution of any value of x is represented by the tangent of the angle made with the axis OX by the tangent at the corresponding point to the curve representing the function fix) . 70. Maxima and Minima Values of a Polynomial. Theorem. — Any value of x which renders fix) a maximum or minimum is a root of the derived equation fix) = 0. Let a be a value of x which renders f{x) a minimum. We proceed to prove that/'(.r) = 0. Let h represent a small incre- ment or decrement of x. We have, since /(a) is a minimum, f(a), p, the tan- gent to the curve will be parallel to the axis OX, and, consequently, tan PTM = f(x) = 0. Fig. 6 represents a polynomial of the 5th degree. Correspond- ing to the four roots of /'(#) = (supposed all real in this case), viz. OM, Om, OM', Om', there are two maxima values, MP, M'P, and two minima values, mp, m'p', of the function. Fig. 6. Examples. 1. Find the max. or min. value of f(x) = 2x 2 + x - 6. /»= 4* +1, /» = 4. 1 — 49 x = — - makes f'(x) = -, a minimum. (See fig. 2, p. 15.) 2. Find the max. and min. values of f{x) = 2x 3 - Zx* - 36* + 14. /'(*) = 6 (a* - x - 6), f"(x) = 6 (2x - 1) . x = — 2 makes f(x) = 68, a maximum. x = 3 makes f(x) — — 67, a minimum. 3. Find the max. and min. values of f(x) = 3s 4 - 16.r 3 + 6x 2 - 48a; + 7. Here f'(x) = has only one real root, x = 4 ; and it gives a minimum value, /(*) = - 345. 4. Find the max. and min. values of f{x) = lO.r 3 - nx°- + x+ 6. The roots of f'(x) are, approximately, -0302, 1-1031. The former gives a maximum value, the latter a minimum. (See fig. 3, p. 16.) Constitution of the Derived Functions. 155 71. Rolle's Theorem. — Between tiro consecutive real roots a and b of the equation f[f) = there lies at least one real root of the equation fix) = 0. For as x increases from a to b, f(x), varying continuously from/(rt) tof(b), must begin by increasing and then diminish, or must begin by diminishing and then increase. It must, therefore, pass through at least one maximum or minimum value during the passage from f{a) to f{b). This value (/(a), suppose) corresponds to some value a of x between a and b, which by the Theorem of Art. 70 is a root of the equation /(•) = 0. The figure in the preceding Article illustrates this theorem. We observe that between the two points of section A and B there are three maximum or minimum values, and between the two points B and C there is one such value. It appears also from the figure that the number of such values between two consecutive points of section of the axis is always odd. Corollary. — Two consecutive roots of the derived equation may not comprise between them any root of the original equation, and never can comprise more than one. The first part of this proposition merely asserts that between two adjacent zero values of a polynomial there may be several maxima and minima values ; and the second part follows at once from the above theorem ; for if two consecutive roots of f(x) = comprised between them more than one root of f{x) = 0, we should then have two consecutive roots of this latter equation comprising between them no root of f(x) = 0, which is contra- dictory to the theorem. 72. Constitution of the Derived Functions. — Let the roots of the equation f(x) = be cti, a 2 , a 3 , . . . a». We have f(x) = (x - di) - a 2 )(x - a 3 ) . . . (x - u n ). In this identical equation substitute y + x for x ; f(y + x) = {y + x- ai ){y + x-a 2 ) . .. (y + x-a n ) = y n + q x y 11 - 1 + q 2 y n ~ 2 + . . . + q n _ x y + q n , 156 Properties of the Derived Functions. where <7i = x - m + x - a 2 + x - a 3 + • • . + X - a n , q* = (x - ai) (x - a 8 ) + {x- a,) (a? - o 8 ) + . . . + [x - a„_,)(# - a„), ?«-i ■ (» - a,) (a? - a s ) ... (a? - a„) + (a? - a,) (a? - a 8 ) . . . [x - a») + . . + (a; - oi) (aj- o») . . . (x - a 7l _,), ft = (a> - ai ) (a; - « 2 ) (a; - a 3 ) . . . (x - a n ). "We have, again, f{y + *) - /(•) +/W * + y^r ^ + • • • + f. Equating the two expressions for/(y + a?), we obtain /(*) = («-oi)(af-o,) . . . (aj-o*), /"(a?) = (a; - a 2 ) (a? - « 3 ) ... {x- a n ) + ...., as above written, f"{x) " --> — g = the similar value of q n _ 2 in terms of a; and the roots, The value of f(x) may be conveniently written as follows : — a?— ai a?-a 2 X-a n 73. Multiple Roots. Theorem.— ^1 multiple root of the order m of the equation f(x) = is a multiple root of the order m-1 of the first derived equation f'[x) = 0. This follows immediately from the expression given for/' (x) in the preceding Article ; for if the factor {x-a,)" 1 occurs in /(a:), i. e. if ai = a 2 = . . . = a m ; we have f( x ) = m ' f ^ + - f ^ + . + /(;r) X - a x X- er,,, +1 X - cu, Each term in this will still have (x - ai)' n as a factor, except the first, which will have (x - ai)'" -1 as a factor ; hence (x - ai) m_1 is a factor in f'{x). Determination of Multiple Roots. 157 Cor. 1. — Any root which occurs m times in the equationf{x) = occurs in degrees of multiplicity diminishing by unity in the first m-1 derived equations. Since f"{x) is derived from /'(if) in the same manner as/'(#) is from/(#), it is evident by the theorem just proved that,/" (x) will contain [x - ai) m ~ 2 as a factor. The next derived function, /'"(#), will contain (x - ai) m ~ 3 ; and so on. Cor. 2. — Iffif) audits first m-1 derived functions all vanish for a value a of x, then (x - a) m is a factor inf{x). This, which is the converse of the preceding corollary, is most readily established directly as follows : — Representing the derived functions by f(x),f (x), . . . .fm-i(x) (see Art. 6), and substituting a + x - a for x, we find that/(#) may be expanded in the form + ,-4^- C* - «)■ + ■ • • + r^ (• - «)"- 1 .2 . . . m 1 . 2 . . . n ' from which the proposition is manifest. 74. Determination of Multiple Roots. — It is easily inferred from the preceding Article that if f(x) and/'(#) have a common factor (x - a)'" -1 , (x - a) m will be a factor in /(a?) ; for, by Cor. 1, the m - 2 next succeeding derived functions vanish as well &sf(x) and/'(,r) when x = a; hence, by Cor. 2, a is a root oif(x) of multiplicity m. In the same way it appears that if f{x) and /'(#) have other common factors (x - py-\ (x - y )**, (x - dy-\ &c, the equation f{x) = will have p roots equal to /3, q roots equal to 7, r roots equal to $, &c. In order, therefore, to find whether any proposed equation has equal roots, and to determine such roots when they exist, we must find the greatest common measure off(x) and f[x). Let this be , AAA are and for a value a + h of x they are for before the passage through the root the sign of f must be different from that of / 5 ; the sign of f 3 must be different from that of j\, and so on ; and after the passage the signs must be all the same. It is of course assumed here that h is so small that no root of f 5 {x) = is included within the interval through which x travels. Examples. 161 Examples. 1 . Find the multiple roots of the equation f(.r) = x i + 123? + 32z 3 - 24.r + 4 = 0. Ans. f(x) = (x* + 6s - 2) 3 . 2. Show that the binomial equation x n - a n = cannot have equal roots. 3. Show that the equation x™ — nqx + (n — 1 ) r = will have a pair of equal roots if q n = »•""' . 4. Prove that the equation z 5 + 5jot 3 + 5p 2 x + q = has a pair of equal roots when q- + ip 5 = ; and that if it have one pair of equal roots it must have a second pair. 5. Apply the method of Art. 74, to determine the condition that the cubic s 3 + 3fl"s + G = should have a pair of equal roots. The last remainder in the process of finding the greatest common measure must vanish. Ans. G 2 + 42T 3 = 0. 6. Apply the same method to show that both G and H vanish when the cubic has three equal roots. 7. If a, P, y, 8 be the roots of the biquadratic f(x) = 0, prove that /»+/'(# +/'(?)+/' (8) can be expressed as a product of three factors. Ans. (a + 0-7- 8) (a + 7-0- 5) (a+8-j8--y). 8. If a, P, 7, 8, &c, be the roots of f{x) = 0, and a', ff, y , &c, of f'(x) = ; prove /'(«) /'(*)/' (7) /(») • • • •=«"/(«') /(£')/ (7) , and that each is equal to the absolute term in the equation whose roots are the squares of the differences. 9. If the equation x n + p\ x»- 1 + pi x n - n - +....+ p H -i % + p n = have a double root a ; prove that o is a root of the equation pi x n ~ l + 2p 2 x n - 2 + 3/>.i3"- 3 + .... + np n = 0. M 162 Properties of the Derived Functions. 10. Show that the max. and min. values of the cubic ax 3 + Zbx- + 3cx + d arc the roots of the equation '■ p 2 - 2Gp + A = 0, where A is the discriminant, If the curve representing the polynomial / (a;) be moved parallel to the axis of y (see Art. 10) through a distance equal to a max. or min. value p, the axis of x will become a tangent to it, i. c. the equation /(.r) — p = will have equal roots. Hence the max. and min. values are obtained by forming the discriminant of f(x) — p, or by putting d-p for p in £« + 4E 3 = 0. 11. Prove similarly that the max. and min. values of ax 1 + 45a; 3 + 6cx z + idx + e arc the roots of the equation a 3 p 3 - 3 {a-I - 9H 2 ) p 2 + 3 {al 2 - 18JS7) p - A = 0, where A is the discriminant of the quartic. 12. Apply the theorem of Art. 76 to the function f(x) = «* - 7x* + 15a; 2 - \3x + 4. "We have /i (x) = ix 3 - 21a; 2 + 30a; - 13, f 2 (x) = 2{6x--2lx+ 15), / 3 <*) = 2(12z-21), /*(*) = 24. Here/3 (x) is the first function which does not vanish when x = 1 ; and/3 (1) is negative. "What the theorem proves is, that for a value a little less than 1 the signs of /, fu ft, /; are H — + - , and for a value a little greater than 1 they are all negative. We are able from this series of signs to trace the functions/, /, &c, in the neighbourhood of the point x = 1. Thus the curve representing f{x) is above the axis before reaching the multiple point a; = 1, and is below the axis immediately after reaching the point, and the axis must be regarded as cutting the curve in three coincident points, since (x- l) 3 is a factor in/(a;). Again, the curve corresponding to f (x) is below the axis both before and after the passage through the point x = 1 . It touches the axis at that point. The curve representing f> (x) is above the axis before, and below the axis after the passage, and cuts the axis at the point. CHAPTER VIII. LIMITS OF THE ROOTS OF EQUATIONS. 77. Definition of Limits. — In attempting to discover the real roots of numerical equations, it is in the first place advan- tageous to narrow the region within which they must be sought. We here take up the inquiry referred to in the observation at the end of Art. 4, and proceed to prove certain propositions relative to the limits of the real roots of equations. A superior limit of the positive roots is any greater positive number than the greatest of them ; an inferior limit of the posi- tive roots is any smaller positive number than the smallest of them. A superior limit of the negative roots is any greater ne- gative number than the greatest of them ; an inferior limit of the negative roots is any smaller negative number than the smallest of them : the greatest negative number meaning here that nearest to - oo . When we have found limits within which all the real roots of an equation lie, the next step towards the solution of the equation is to discover the intervals in which the separate roots are situated. The principal methods in use for this latter pur- pose will form the subject of the next Chapter. The following Propositions all relate to the superior limits of the positive roots ; to which, as will be subsequently proved, the determination of inferior limits and limits of the negative roots can be immediately reduced. 78. Proposition I. — In any equation x n + p^"- 1 + p 2 x n ~ z +....+ p n . x x+p n = 0, if the first negative term be - p r #""'', and if the greatest negative m2 1G4 Limits of the Roots of Equations. coefficient be -p*, thenl/p% + 1 is a superior limit of the posit in roots. Any value of x which makes x n >p k (x n - r + .r"-'" 1 + ... + »+ 1) >p k — will, a fortiori, niake/(.r) positive. Now, taking x greater than unity, this inequality is satisfied by the following : — x n ~' +1 or tf w+1 -.r n >2uz"- r *\) or of 1 [x- 1) >^a, which inequality again is satisfied by the following : — (x - l) r_1 (x - 1) = or > p ki since plainly ^" _1 > (x - l) r_1 . "We have, therefore, finally (x - l) r = or > p A , or .r = or > 1 + \/p*- 79. Proposition II. — 7/' in any equation each negative coef- ficient be taken positively, and divided by the sum of all the positive coefficients which precede it, the greatest quotient thus formed in- creased by unity is a superior limit of the positive roots. Let the equation be aoX n + a x x n - x + tf 2 #"- 2 - a 3 x n - z + ....- a r x n -'' + + a* = 0, in which, in order to fix our ideas, we regard the fourth coef- ficient as negative, and we consider also a negative coefficient in general, viz. - a r . Let each positive term in this equation be transformed by means of the formula a m x'" = a m [x - 1) {x m ~ x +x m -* + ... + » + !) + a m Propositions. 165 which is derived at once from x m - 1 — = x m ~ x + x m ~ 2 + . . . + x + 1 : x-1 the negative terms remaining unchanged. The polynomial /(.r) becomes then, the horizontal lines of the following corresponding to the successive terms of f{x) : — ■i,\.c-l)x n - l + a (x-l)x n - 2 + a (x - l)x n ~ 3 + . . . +a {x-l)x n - r + ... + a 0y + a l (x-l)x n ~ 2 + a l (x-l)x n ~ 3 + . . . +a 1 [x-l)x n ~ r + . . .+a ly + « 3 , (a + cii + a 2 + . . . + a r _i) (x - 1 j > a n &c. Hence + 1 x> — ~+l, + 4.r s - 3.r 5 + 5a; 4 - 9a; 3 - 1 la; 3 + 6a; - 8 = 0. Of the fractions 3 9 11 8 1 + -i' 1+4 + 5' 1 + 4+5' 1+4 + 5 + 6' the third is the greatest, and Prop. II. gives the limit 3. Prop. I. gives 5. 4. Find a superior limit of the positive roots of z s + 20a; 7 + ix* - 11a; 5 - 120a; 4 + 13a; - 25 = 0. Ans. Both methods give the limit 6. 5. Find a superior limit of the positive roots of 4* 5 - 8* 4 + 22a; 3 + 98a 2 - 73a; + 5 = 0. Ans. Prop. I. gives 20. Prop. II. gives 3. Examples, 167 It is usually possible to determine by inspection a limit closer than that given by either of the preceding propositions. This method consists in arranging the terms of an equation in groups having a positive term first, and then observing what is the lowest integral value of x which will have the effect of render- ing each group positive. The form of the equation will suggest the arrangement in any particular case. 6. The equation of Ex. 2 can be arranged as follows : — x- (x 3 - 8) + x (3.r 3 - 51) + x 3 + 18 = 0. x — 3, or any greater number, renders each group positive ; bence 3 is a superior limit. 7. Tbe equation of Ex. 4 may be arranged tbus : — x 5 (x 3 - 11) + 20a 4 (x 3 - 6) + ijfi + 13* - 25 = 0. x = 3, or any greater number, renders eacb group positive ; bence 3 is a limit. S. Find a superior limit of tbe roots of tbe equation xt - ix 3 + 33.r 2 - 2x + 18 = 0. This can be arranged in tbe form x 2 {x- - 4x + 5) + 28a; (x - &) + 18 = 0. Now the trinomial x- - ix + 5, having imaginary roots, is positive for all values of x (Art. 12). Hence x = 1 is a superior limit. The introduction in this way of a quadratic whose roots are imaginary, or of one with equal roots, will often be found useful. 9. Eind a superior limit of the roots of the equation ox 5 - 7x i - 10a 3 - 23x 2 - 90a- - 317 = 0. In examples of this kind it is convenient to distribute tbe highest power of x among the negative terms. Here the equation may be written xS (x - 7) + x 3 {x- - 10) + x- {x 3 -23) + x{x i - 90) + x 5 -317 = 0, so that 7 is evidently a superior limit of the roots. In this case the general methods give a very bigh limit. 10. Find a superior limit of tbe roots of tbe equation & - x 3 - 2x- - ix - 24 = 0. When there are several negative terms, and the coefficient of the highest term unity, it is convenient to multiply the whole equation by such a number as will enable us to distribute tbe highest term among the negative terms. Here, multiply- ing by 4, we can write the equation as follows : — x 3 (x - 4) + x 2 (x- - 8) + x (x 3 - 16) + x i - 96 = 0, and 4 is a superior limit. The general methods give 25. 168 Limits of the Roots of Equations. 81. Proposition III. — Any number which renders positive the polynomial /(./■) and all its derived 1 functions fi{x),f_{x), .. -f n (f) is a superior limit of the positive roots of the equation f{x) = 0. This method of finding limits is due to Newton. It is much more laborious in its application than either of the preceding methods ; but it has the advantage of giving always very close limits ; and in the case of an equation all whose roots are real the limit found in this way is, as will be subsequently proved, the next integer above the greatest positive root. To prove the proposition, let the roots of the equation f{x) = be diminished by h ; then x- h = y, and f{y + h) =f(h) + f [h) y -f^| f + . . . + l f ^ n |T. If now h be such as to make all the coefficients f(h),Mh),Mh),...f n (h) positive, the equation in y cannot have a positive root ; that is to say, the equation in x has no root greater than h ; hence h is a superior limit of the positive roots. Example. f(x) = x*- 2x 3 - 3x- - lo* - 3. In applying Newton's method of finding limits to any example the general mode of procedure is as follows : — Take the smallest integral numher which renders fn.i(x) positive ; and proceeding upwards in order to f\ (x), try the effect of substi- tuting this numher for x in the other functions of the series. When any function is reached which becomes negative for the integer in question, increase the integer successively by units, till it makes that function positive ; and then proceed with the new integer as before, increasing it again if another function in the series should become negative ; and so on, till an integer is reached which renders all the functions in the series positive. In the present example the series of functions is / {£) = x* - 2x s - 3.r°- - 15* - 3, fi(.r) = 4z z - 6x- - 6x - 15, hfi(x) = 6.K- - 6x - 3, $f s [x) = 4,/ - 2, Inferior Limits, and Limits of the Negative Roots. 169 Here x = 1 makes f$ (x) positive. "We try then the effect of the substitution x = 1 mfi(x). It makes /a (x) negative. Increase hy 1 ; and x = 2 makes / 3 (x) positive. Try the effect of x = 2 in /i (#) ; it gives a negative result. Increase by 1 ; and x = 3 makes f\ (x) positive. Proceeding upwards, the substitution x = 3 makes f(x) negative ; and increasing again by unity, we find that x = 4 makes f(x) posi- tive. Hence 4 is the superior limit required. It is assumed in this mode of applying Newton's rule, that when any number makes all the derived functions up to a certain stage positive, any higher number will also make them positive ; so that there is no occasion to try the effect of the higher number on the functions in the series below that one where our upward progress is arrested. This is evident from the equation A 2 (a + h) = {a) — - + . . . (taking "(a), . . . are all positive, and h also positive, -, i.e. x > -. To find limits of the negative roots, we have only to trans- form the equation by the substitution x = —y. This transfor- mation changes the negative into positive roots. Let the su- perior and inferior limits of the positive roots of the equation in y be h and //. Then - h and - ti are the limits of the negative roots of the proposed equation. 170 Limits of the Hoots of Equations. 83. Limiting Equations. — If all the real roots of the equation f'(z) = could be found, if would be possible to determine the number of real roots of the equation f(x) = 0. To prove this, let the real roots oif'{x) = be, in ascending order of magnitude, a, (3', y', . . . A' ; and let the following series of values be substituted for x'va.f{x): — - go , a, /3', y\ ... A', + go . When any successive two of these quantities give results with different signs there is a root of/ (x) = between them ; and by the Cor., Art. 71, there is only one ; and when they give results with the same sign there is, by the same Cor., no root between them. Thus each change of sign in the results of the successive substitutions proves the existence of one real root of the proposed equation. If all the roots of f(x) = are real, it is evident, by the theorem of Art. 71, that all the roots of f(x) = are also real, and that they lie one by one between each adjacent pair of the roots of f{x) = 0. In the same case, and by the same theorem, it follows that the roots of /"(.?) = 0, and of all the successive derived functions, are real also ; and 1 the roots of any function lie severally between each adjacent pair of the roots of the function from which it is immediately derived. Equations of this kind, which are one degree below the degree of any proposed equation, and whose roots lie severally between each adjacent pair of the roots of the proposed, are called limiting equations. It is evident that in the application of Newton's method of finding limits of the roots, when the roots oif{oc) = are all real, in proceeding according to the method explained in Art. 81, the function /(#) i s itself the last which will be rendered positive, and therefore the superior limit arrived at is the integer next above the greatest root. Examples. 171 Examples. 1. Prove that any derived equation/,,, (x) = cannot have more imaginary roots, but may have more real roots, than the equation f(x) = from which it is derived. From this it follows that if any of the derived functions be found to have imaginary roots, the same number at least of imaginary roots must enter the primi- tive equation. 2. Apply the method of Art. S3 to determine the conditions that the equation z 3 — qx + r = should have all its roots real. 3. Determine by the same method the nature of the roots of the equation x n — nqx + (n - 1) r = 0. Ans. When v is even, the equation has two real roots or none, according as q» > or < f\ "When n is odd, the equation has three real roots or one, according as q" > or < r"" 1 . 4. The equation x n (x - 1)" = has all its roots real ; hence show, by forming the n th derived function, that the following equation has all its roots real and un- equal, and situated between and 1 : — n , n In - 1) n («— 1) „ „ x n - n — xfi-l + _J 1 5 L. x n-1 _ Sec, = o. In 1.2 In (2m - 1) 5. Show similarly by forming the n th derived of (x 2 - 1)" that the following equation has all its roots real and unequal, and situated between - 1 and 1 : — n(n-l) , »(»-l) »(»-l)(»-2)(»-3) , . n aP-n — — a^'" 2 + — ■ — — — a:"- 4 - &c. = 0. 2»(2»-l) 1.2 2«(2«-l)(2»-2)(2«-3) 6. If any two of the quantities I, m, n in the following equation be put equal to zero, show that the quadratic to which the equation then reduces is a limiting equa- tion ; and hence prove that the roots of the proposed are all real : — (s _ a ) (x -b){x-c)-r-{x-a)- m? (z-b)- « 2 (* - e) - 2lmn = 0. CHAPTER IX. SEPARATION OF THE ROOTS OF EQUATIONS. 84. By the methods of the preceding Chapter we are enabled to find limits between which all the real roots of any numerical equation lie. Before proceeding to the actual approximation to any particular root, it is necessary to separate the interval in which it is situated from the intervals which contain the remain- ing roots. The present Chapter will be occupied with certain theorems whose object is to determine the number of real roots between any two arbitrarily assumed values of the variable. It is plain that if this object can be effected, it will then be possible to tell not only the total number of real roots, but also the limits within which the roots separately lie. The theorems given for this purpose by Fourier and Budan, nlthough different in statement, are identical in principle. For purposes of exposition Fourier's statement is the more con- venient, while with a view to practical application the statement of Budan will be found superior. The theorem of Sturm, although more laborious in practice, has the advantage over the preceding that it is unfailing in its application, giving always the exact number of real roots situated between any two proposed quan- tities ; whereas the theorem of Fourier and Budan gives only a certain limit which the number of real roots in the proposed interval cannot exceed. 85. Tneoreni of Fourier and Budan. — Let tivo numbers a and h, of which a is the less, be substituted in the series formed by f(x) and its successive derived functions, viz., Theorem of Fourier and Budan. 173 the number of real roots which lie between a and b cannot be greater than the excess of the number of changes of sign in the series when a is substituted for x, over the number of changes when b is sub- stituted for x ; and when the number of real roots in the interval falls short of that difference, it mil be by an even number. This is the form in which Fourier states the theorem. It is to be understood here, as elsewhere, that, when we speak of two numbers a and b, of which a is the less, one or both of them may be negative, and what is meant is that a is nearer than b to - qo . We proceed to examine the changes which may occur among the signs of the functions in the above series, the value of x being supposed to increase continuously from a to b. The fol- lowing different cases can arise : — (1). The value of x may pass through a single root of the equation f{x) = 0. (2). It may pass through a root occurring r times inf(x) = 0. (3). It may pass through a root of one of the auxiliary functions f m (x) = 0, this root not occurring in either f m ^{x) = or f m+l (x) = 0. (4). It may pass through a root occurring r times in/„,(^) = 0, and not occurring inf m _i(x) = 0. In what follows the symbol x is omitted after / for con- venience. (1). In the first case it is evident, from Art. 75, that in passing through a root of the equation f(x) = one change of sign is lost ; for / and j L have unlike signs immediately before, and like signs immediately after, the passage through the root. (2). In the second case, in passing through an ^-multiple root of f(x) = 0, it is evident that r changes of sign are lost ; for, by Art. 76, immediately before the passage the series of func- tions St fi J-1 ' • •/''-!) fr have signs alternately + and -, or - and +, and immediately after the passage have all the same sign as f r . 174 Separation of the Roots of Equations. (3). In the third case, the root olf m (x) = must give to/,„_i and/„ 1+ i either like signs or unlike signs. Suppose it to give like signs ; theu in passing through the root two changes of sign are lost, for Lefore the passage the sign of /,„ is different from these like signs, and after the passage it is the same (Art. 76). Sup- pose it to give unlike signs ; then no change of sign is lost, for before the passage the signs of / m _i, ./,„, f m+l must be either + + - , or - - + , and after the passage these become + - -, and - -i- +. On the whole, therefore, we con- clude that no variation of sign can be gained, but two variations may be lost, on the passage through a root of f m {x) = 0. (4). In the fourth case x passes through a value (let us say a) which causes not only/,,, but also/„ i+1 ,/,„ +2 , . . . ,/»i + »-i to vanish. It is evident from the theorem of Art. 76 that during the passage a number of changes of sign will always be lost. The definite number may be collected by considering the series of functions ,/m-l) /)«) ./m+l> . . . . , t /„ (+ ,-_i, Jni+r' "We easily obtain the following results : — (a). When/,„_i(a) and/„, +r (a) have like signs: If r be even, r changes are lost. If r be odd, r + 1 changes are lost. (b). "When fm-i (a) and/„, + ,.(a) have unlike signs: If r be even, r changes are lost. If r be odd, r - 1 changes are lost. We conclude, therefore, on the whole, that an even number of changes is lost during the passage through an r-multiple root Of f m (x). It will be observed that (1) is a particular case of (2), and (3) of (4), ?'. e. when r = 1. Since, however, the cases (1) and (3) are those of ordinary occurrence, it is well to give them a sepa- rate classification. Reviewing the above proof, we conclude that as x increases from a to b no change of sign can be gained ; that for each Application of the Theorem. 175 passage through a single root of /(») = one change is lost ; and that under no circumstances except a passage through a root of J' (%) = can an odd number of changes be lost. Hence the number of changes lost during the whole variation of x from a to b must be either equal to the number of real roots of / (x) = in the interval, or must exceed it by an even number. The theorem is therefore proved. 86. Application of the Theorem. — The form in which the theorem has been stated by Budan is, as has been already observed, more convenient for practical purposes than that j ust given. It is as follows: — Let the roots of an equation f(x) = be diminished, first by a and then by b, where a and b are any two numbers of which a is the less ; then the number of real roots be- tween a and b cannot be greater than the excess of the number of changes of sign in the first transformed equation over the number in the second. This is evidently included in Fourier's statement, for the two transformed equations are (see Art. 33) — /(«) +/, (a) y + f£y> + ... + j^fL y » = , /(*) +/i (b) V +-f§ *» + ... + j^^ V n - ; from which, assuming the results of the last Article, the above proposition is manifest. The reason why the theorem in this form is convenient in practice is, that we can apply the expeditious method of dimi- nishing the roots given in Art. 33. Examples. 1 . Find the situations of the roots of the equation x 5 - 3z 4 - 24z 3 + 95a: 2 - 46* - 101 = 0. We shall examine this function for values of x hetween the intervals -10, - 1, 0, 1 10; these numbers being assumed on account of the facility of calculation. Diminution 170 Squmttiou <>/ the Hoots of Equations. of the roots by 1 gives the following series of coefficients of the transformed niiiatioii : — 1, 2, - 2G, 15, 66, -78. In diminishing the roots by 10, it is apparent at the very outset of the calculation that the sitrns of the coefficients of the transformed equation will he all positive; so that then i m ex ion to complete the calculation in this case. In diminishing the roots hy - 10 and - 1, it is convenient to change the alter- nate signs of the equation, and diminish the roots by + 10 and + 1 ; and then in tlic result change the alternate signs again. The coefficients of the transformed equation when the roots are diminished by — 1 are 1, -8, -2, 139, -291, GO. In diminishing by - 10 we observe in the course of the operation, as before, that the signs will lie all positive in the result, i.e. when the alternate signs are changed they will be alternately positive and negative. Hence we have the following scheme : — (-10) +_ + _ + - (-1) +-- + - + (0) + — - + --, the equation itself. (1) + + - + + - (10) + + + + + + These signs are the signs taken by/(.r) and the several derived functions f\, f>, f'h /■*> fs on the substitution of the proposed numbers ; but it is to be observed that they are here written, not in the order of Art. 85, but in the reverse order, viz., /s, fi, fz, fi, fh /• From these we draw the following conclusions : — All the real roots must lie between - 10 and + 10 ; one real root lies between — 10 and - 1, since one change of sign is lost ; one real root lies between — 1 and 0, since one change of sign is lost ; no real root lies between and 1 ; and between 1 and 10, since three changes of sign arc lost, there is at least one real root ; but we are left in doubt as to the nature of the other two roots : whether they are imaginary, or whether there are three real roots between 1 and 10. We might proceed to examine, by further transformations, the interval between 1 and 10 more closely, in order to determine the nature of the two doubtful roots ; but it is evident that the calculations for this purpose might, if the roots were nearly equal, become very laborious. This is the weak side of the theorem of Fourier and Budan. Both writers have attempted to supply this defect, and have given methods of determining the nature of the roots in doubtful intervals ; but as these methods are complicated, we do not stop to explain them ; the more especially as the theorem of Sturm effects fully the purposes for which the supplementary methods of Fourier and Budan were invented. Application of the Theorem to Imaginary Roots. 177 2. Analyse the equation of Ex. 1, p. 100, viz., x 5 + x- -2z-l=0. The roots of this are all real, and lie between — 2 and 2 (see Ex. 5, p. 100). "When- ever the roots of an equation are all real, the signs of Fourier's functions determine the exact number of real roots between any two proposed integers. We obtain the following result : — The roots lie in the intervals (-2, -1); (-1,0); (1,2). 3. Analyse the equation of Ex. 3, p. 100, viz., x 5 + x* - 4Z 3 - 3z~ + 3x + 1 = 0. Ans. Two roots in the interval (—2, — 1), and one root in each of the intervals (- 1, 0) ; (0, 1) ; (1, 2). 4. Analyse the equation x* - SO* 3 + 1998* 2 - 14937a: + 5000 = 0. The equation can have no negative roots. Diminish the roots by 10 several times in succession till the signs of the coefficients become all positive. We obtain the following result : — (0) + - + - + (10) + - + + - (20) + - + + (30) + + + - + (40) + + + + + Thus, there is one root between and 10, and one between 10 and 20 ; no root between 20 and 30. Between 30 and 40 either there are two real roots, or there is an indication of a pair of imaginary roots. That the former is the case will appear by diminishing the roots of the third transformed equation by units. This process will separate the roots, which will be found to lie between (2, 3) and (4, 5) ; so that the proposed equation has a third real root in the interval (32, 33), and a fourth in the interval (34, 35). 87. Application of the Theorem to Imaginary Roots. — Since there exist only n changes of sign to he lost in the passage of x from - co to + oo , if we have any reason for knowing that a pair of changes is lost during the passage of x through an interval which includes no real root of the equation, we may be assured of the existence of a pair of imaginary roots. Circumstances of this nature will arise in the application of Fourier's theorem when any of the transformed equations con- tain vanishing coefficients. For we can assign by the principle of Art. 7G the proper sign to this coefficient, corresponding to N 178 Separation of the Hoofs of Equations. values of x immediately before and immediately after that value ■which causes the coefficient to vanish ; the whole interval being so small that it may be supposed not to include any root of the equation f(.r) = 0. Examples. 1. Analyse the equation f(x) = x* - 4s 3 - Sx + 23 = 0. We shall examine this function between the intervals 0, 1 , 10. The transformed equations are M (0).-' 4 t \ft (0) x* + ifz (0)z~ +/! (0) x +f (0) = 0, &f i (l)Z l + $f 3 (l)x* + hMl)3?+fl(l)x+f(l) = 0, hfi{10)x* + $f 3 (10)a?+$f 2 (10)z*+M10)x+f(lO) = 0, the first of these being the proposed equation itself. Making the calculations by the method of the preceding Article, we find that the coefficient /3(1) = 0, and we have the following scheme : — (0) + - - + (1) + - - + (10) + + + + + We may now replace each of the rows containing a zero coefficient by two, the first corresponding to a value a little less, and the second to a value a little greater, than that which gives the zero coefficients ; the signs being determined by the principle established in Art. 76. It must be remembered that in the above scheme the signs representing the derived functions are written in the reverse order to that of the Article referred to. The scheme will then stand as follows, using h to repre- sent a very small positive quantity : — 1 [- h + _ + - + (0). I 1 [ + h + — — — + 1 [i -h • + _ _ + (i); i i li + h + + - - + (10) + + + + + In this scheme the signs corresponding to - h and + h are deteiinined by the condition that the sign of the coefficient which is zero when x = must, when x = — h, be different from that next to it on the left-hand side ; and when x = + h it must be the same. The signs corresponding to 1 — h and 1 + h are determined in a similar manner. Rule of the Double Sign. 179 Now since a pair of changes is lost in the interval (— h, + h), and since the equation has no real root between — h and + ft, we have proved the existence of a pair of imaginary roots. Two changes of sign are lost between 1 + ft and 10, so that this interval either includes a pair of real roots, or presents an indication of a pair of imaginary roots. "Which of these is the case remains still doubtful. 2. If several coefficients vanish, we may be able to establish the existence of several pairs of imaginary roots. This will appear from the following example : — x 6 - 1 = 0. The signs corresponding to — A and + h are, by the theorem of Art. 76, (-A) + - + - + - - (+ h) + + + + + + - Hence, since no root exists between - h and + ft, and since 4 changes of sign are lost in passing from a value very little less than to one very little greater, we are assured of the existence of two pairs of imaginary roots. The other two roots are in this case plainly real (see Art. 14). The number of imaginary roots in any binomial equation can be determined in this way. 3. Find the character of the roots of the equation x* + Kb: 3 + x - 4 = 0. In passing from a small negative to a small positive value of x we obtain the following series of signs : — (-ft) + - + - + + - + (0) + 000 0+0 + (+ h) + + ++ + + + + Since six changes of sign are here lost, there are six imaginary roots. The remaining two roots are, by Art. 14, real : one positive, and the other negative. The negative root lies betwen - 2 and — 1, and the positive between and 1. 4. Analyse completely the equation x 6 - Zx- - x + 1 = 0. There are two imaginary roots. "Whenever, as in the pi - esent instance, the roots are comprised within small limits, it is convenient to diminish by successive units. In this way we find here a root between and 1, and another between 1 and 2. Proceeding to negative roots, we find on diminishing by — 1 that — 1 is itself a root, and writing down the signs corresponding to a value a little greater than — 1, we observe an indication of a second negative root between — 1 and 0. 5. Analyse the equation x 5 + z 4 + x 2 - 2bx - 36 = 0. There are two imaginary roots ; one real positive root between 2 and 3 ; and two real negative roots in the intervals (- 3, - 2), (- 2, - 1). n2 180 Separation of the Roots of Equations. 88. Corollaries from the Theorem of Fourier ami lludan. — The method of detecting the existence of imaginary roots explained in the preceding Article is called The Rule of the Double Sign. A similar rule, due to De Gua, was in use before the discovery of Fourier's theorem. This rule and Descartes' Rule of Signs are immediate corollaries from the theorem, as we proceed to show. Cor. 1. — De Gua's Rule for finding Imaginary Roots. The rule maybe stated generally as follows : — When 2m suc- cessive terms of an equation are absent, the equation has 2m imaginary roots; and when 2m + 1 successive terms are absent, the equation has 2m + 2, or 2m imaginary roots, according as the two terms be- tween which the deficiency occurs have like or unlike signs. This follows, as in case (4), Art. 85, by examining the number of changes of sign lost during the passage of x from a small nega- tive value - h to a small positive value h. Cor. 2. — Descartes' 1 Rule of Signs. When is substituted for x in the series of functions fnif),f n -x{x), . . .f(x),Ji(x),f(x), the signs are the same as the signs of the coefficients a , a u a 2 , . . . a n _ x , a,„ of the proposed equation ; and when + co is substituted the signs are all positive. Fourier's theorem asserts that the number of roots between these limits, viz., the number of positive roots, cannot exceed the number of variations lost during the passage from to + co , that is the number of changes of sign in the series a , a x , a 2 . . . a„. This is Descartes' rule for positive roots ; and the similar rule for negative roots follows in the usual way by changing the negative into positive roots. Cor. 3. — Newton's Method of finding Limits. When a number h has been found which renders positive each of the functions /n(*),/«-i(«), • • • A(^),f{^), /(#) 5 since + cc also renders each of them positive, it follows from Fourier's theorem that there can be no root between h and + co , that is to say, h is a superior limit of the positive roots ; and this is Newton's proposition (Art. 81). Sturm'' s Theorem. 181 89. Sturm*;* Theorem. — We have already shown (Art. 74) that it is possible by performing the common algebraical operation of finding the greatest common measure of a polynomial f (x) and its first derived polynomial to find the equal roots of the equation f{x) = 0. Sturm has employed the same operation for the formation of the auxiliary functions which enter into his method of separating the roots of an equation. Let the process of finding the greatest common measure of f(x) and its first derived be performed. The successive re- mainders will go on diminishing in degree till we reach^finally either one which divides that immediately preceding without remainder, or one which does not contain the variable at all, i. e. which is numerical. The former is, as we have already seen, the case of equal roots. The latter is the case where no equal roots exist. It is convenient to divide the discussion of Sturm's theorem into these two cases. We shall in the present Article consider the case where no equal roots exist ; and pro- ceed in the next Article to the case of equal roots. The per- formance of the operation itself will of course disclose the class to which any particular example is to be referred. The auxiliary functions employed by Sturm are not the remainders as they present themselves in the calculation, but the remainders with their signs changed. In finding the greatest common measure of two expressions it is indifferent whether the signs of the remainders are changed or not : in the formation of Sturm's auxiliary functions the change is essential. It is convenient in practice to change the sign of each remainder before making it the next divisor. Confining our attention for the present, therefore, to the case where no equal roots exist, Sturm's theorem may be stated as follows : — Theorem. — Let any two real quantities a and b be substituted for x in the series of n + 1 functions consisting of the gken polynomial f (x) y its first derived f^x), and 182 Separation of the Roots of Equations. the successive remainders {with their signs changed) in the process of finding the greatest common measure off[x) andf(x) ; then tht difference between the number of changes of sign in the series when a is substituted for x, and the number when b is substituted for x expresses exactly the number of real roots of the equation f '(x) = between a and b. The mode of formation of Sturm's functions supplies the following scries of equations, in which q l3 q 2 , . . . q n -\ represent the successive quotients in the operation : — /(*) - ft /i(«) -/.(*), 1 fr-i (■<■) = 'lrfrix) -,/' > (1) fn-i {X) = (?„_i/ M _i(.r) -f n {x). These equations involve the theory of the method of finding the greatest common measure ; for it follows from the first equa- tion that if f{x) and/^a?) have a common factor, this must be a factor in/ 2 (^) ; and from the second equation it follows, by like reasoning, that the same factor must occur in/ 3 (^) ; and so on, till we come finally to the last remainder, which, when/(.r) and/j(.r) have common factors, will be a polynomial consisting of these factors. In the present Article, where we suppose the given polynomial and its first derived to have no common factor, the last remainder f n (x) is numerical. It is essential for the proof of the theorem to observe also, that in the case now under consideration no two consecutive functions in the series can have a common factor ; for if they had we could, by reason- ing similar to the above, show from the equations that this fac- tor must exist also inf(x) and/i(#) ; and such, according to our hypothesis, is not here the case. In examining, therefore, what changes of sign can take place in the series during the passage of x from a to b, we may exclude the case of two consecutive functions vanishing for the same value of the variable ; and the Sturm'' s Theorem. 183 different cases in which any change of sign can take place are the following : — (1). When x passes through a root of the proposed equation /(*)= 0: (2). When x passes through a value which causes one of the auxiliary functions /^/o, . . ./«_i to vanish: (3). When x passes through a value which causes two or more of the series/,/'., . . .f n _ x to vanish together; no two of the vanishing functions, however, being consecutive. (1). When x passes through a root of f(x) = 0, it follows from Art. 75 that one change of sign is lost, since immediately before the passage fix) andyi(^) have unlike signs, and immediately after the passage they have like signs. (2). Suppose x to take a value a which satisfies the equation f,-{x) = 0. From the equation fr-i(x)=qrfr{x)-f r +i{x) we have /r-i(a) = -/r+i(a), which proves that this value of x gives to/,-i(#) and/ r+1 (.r) the same numerical value with different signs. In passing from a value a little less than a to one a little greater, we can suppose the interval so small that it contains no root of/,-i(.#) OTf rn (x) ; hence, throughout the interval under consideration, these two functions retain their signs. If the sign of /,-(.?) does not change (as will happen in the exceptional case when the root a is re- peated an even number of times) there is no alteration in the series of signs. In general the sign of f r (x) changes, but no variation of sign is either lost or gained thereby in the group of three ; because, on account of the difference of signs of the two extremes f r -i(x) and/,. +1 (,*■), there will exist both before and after the passage one variation and one permanency of sign, whatever be the sign of the middle function. If, for example, before the passage the signs were + — ; after the passage they are + + -, i.e. a variation and a permanency are changed into a perma- nency and a variation ; but no variation of sign is lost or gained on the whole. 184 Separation of the Roots of Equations. (3). Since the recasoning in the previous cases is founded on the relations of the function to those adjacent to it only ; and since those relations remain unaltered in the present case, be- cause no two adjacent functions vanish together, we conclude that if/(^) is one of the vanishing functions, one change of sign is lost, and if not, no change is either lost or gained. We have proved, therefore, that when x passes through a root of f(x) = one change of sign is lost, and under no other circumstances is a change of sign either lost or gained. Hence the number of changes of sign lost during the variation of x from a to b is equal to the number of roots of the equation be- tween a and b* Before proceeding to the case of equal roots, we add a few simple examples to illustrate the application of Sturm's theo- rem. It is convenient in practice to substitute first - oo , 0, + co in Sturm's functions, so as to obtain the whole number of negative and of positive roots. To separate the negative roots, the integers -1,-2, - 3, &c, are to be substituted in succession till we reach the same series of signs as results from the substitu- tion of - oo ; and to separate the positive roots we substitute 1, 2, 3, &c., till the signs furnished by + oo are reached. Examples. 1. Find the number and situation of the real roots of the equation f(x) ~ a; 3 - 2x - 5 = 0. We find /, (.r) = Zz- - 2, f 2 {x) = 4.x + 15, f 3 {x) = - G43. Corresponding to the values - oo , 0, + co of x, we have (-oo) - + , (0) - - + -, (+ » ) + + + -. Hence there is only one real root, and it is positive. * The student often finds a difficulty in perceiving in what way a number of changes of sign can be lost in Sturm's series, since the only loss of sign takes place between the first two functions, /(.*;) and f\ (x). It may tend to remove this diffi- culty to observe, that as x increases from one root a of f(x) — to a second £, although no alteration takes place in the number of changes of sign, the distribution of the signs among f\ (x) and the following functions alters in such a way that the signs of f(x) and/i(.c), which were the same immediately after the passage of x through o, become again different immediately before the passage through £. Examples. 185 Again, corresponding to values 1, 2, 3 of #, we have (1) - + + - (2) - + + -, (3) + + + -. The real root, therefore, lies between 2 and 3. 2. Find the number and situation of the real roots of the equation x 3 - 7x + 7 = 0. "We easily obtain /,(*) = 3*2 -7, Mx) = 2* - 3, /a(*) = i; whence (-«) - + - +, (0) + +, (+'oo) + + + +. Hence all the roots are real : one negative, and two positive. We have, further, the following results : — (-4) - + - +, (-3) + + - +, (-2) + + - +, (-1) + - - +, (1) + - - +, (2) + + + +. Here — 4 and + 2 give the same series of signs as — oo and + ■*> ; hence we stop at these. The negative root lies between — 4 and - 3 ; and the two positive roots between 1 and 2. This example illustrates the superiority of Sturm's method over that of Fourier. The substitution of .1 and 2 in Fourier's functions gives, as can be immediately verified, the following series of signs : — (1) + - + +, (2) + + + +. From Fourier's theorem we are authorised to conclude only that there cannot he more than two roots between 1 and 2. From Sturm's we conclude that there are two roots between 1 and 2. If we have occasion to separate these two roots, we must, of course, make further substitutions in f(x). 3. Find the number and situation of the real roots of the equation x* - 2x 3 - 3x 2 + 10.r - 4 = 0. 18G Separation of the Roots of Equations. We obtain, removing the factor •_' from the derived, /i (.<•) = 2j- 3 - 3^ 2 - 8* + 6, /:{■<■) = 9x l - 27*+ 11, /:,(*) = -8* -3, A{x) = -1433. [N.B. — In forming Sturm's functions it is allowable, as is evident from the equations (1), Art. 89, to introduce or suppress numerical factors just as in the proi ess of finding the G. c. M. ; taking can ' rer, that these are positive, so that the signs of the remainders are not thereby altered.] We have the following series of signs : — (-00) + - + + (0) - + + - (+00) + + + - Hence there are two real roots, one positive, and one negative and two imaginary roots. To find the position of the real roots, it is sufficient to substitute positive and negative integers successively inf(x) alone, since there is only one positive and one negative root : we easily find in this way that the negative root lies between — 2 and — 3, and the positive root between and 1. 90. Sturm's Theorem. Equal Roots. Let the opera- tion for finding the greatest common measure off(x) and/'(.r) be performed, the signs of the successive remainders being changed as before. The last of Sturm's functions will not now be numerical, for since fix) and/('#) are here supposed to con- tain a common measure involving a?, this will now be the last function arrived at by the process. Let the series of functions be:— /(*),/i(«)./.M, ,/■(•). During the passage of x through any value except a multiple root of / (x) = 0, the conclusions of the last Article are still true with respect to the present series, since no value except such a root can cause any consecutive pair of the series to vanish. When x passes through a multiple root olf(x) = 0, there is, by the Cor., Art. 75, one change of sign lost between/ and f ; and we pro- ceed to prove that no change of sign is lost or gained in the rest of the series, viz./u/o, . . . . /,.. Suppose there exists an wi-mul- tiple root a oif(x). It is evident from the equations (1) of Art. 89, Sturm'' s Theorem. 187 that (x - a)" 1 ' 1 is a factor in each of the functions/!, f. . . . ./,-. Let the remaining factors in these functions be, respectively, 0i, 02, ... .

- By dividing each of the equations (1) by [x - a)'" -1 , we get a series of equations which establish by the reasoning of the last Article that, owing to a passage through a, no change of signs is lost or gained in the series 1} 2 , . . . . 0,-. Neither, therefore, is any change lost or gained in the series /i*/ft • • •/• 5 f° r the effect of the factor {as- a) m_1 in the passage of x from a value a - h to a value a + h is either to change the signs of all (when m - 1 is odd) or of none (when m - 1 is even) of the functions 1} 2 , . . . . 0,- ; and changing the signs of all these functions cannot increase or diminish the number of variations. We have therefore proved that when x passes through a multiple root of f{x) = one change of sign is lost between / and /, and none either lost or gained in any other part of the series. It remains true, of course, that when x passes through a single root of/ (x) = a change of sign is lost as before. We may thus state the theorem as follows for the case of equal roots : — The difference between the number of changes of sign when a and b are substituted in the series ff\-,f-i • • • •/>•> the last of these being the greatest common measure off and f u is equal to the number of real roots between a and b, each multiple root counting only once. Examples. 1 . Find the nature of the roots of the equation x i - 5a^ + 9a; 2 - lx + 2 = 0. We easily obtain fi(z) = ix 3 - \bx- + 18* - 7, f 2 {z) = z* - 2x + I ; fi(x) divides f\ (x) without remainder; hence in this case Sturm's series stops at fi{x), thus establishing the existence of equal roots. 188 Separation of the Roots of Equations. To find the number of real roots of the equation, we substitute - oo and + oo fora in lb' i tea of functions/, f\, f>. The result is (-=0) + - +, (+ cc ) + + + . Ilence the equation has only two real distinct roots ; but one of these is a triple root, iileut from the form of/>(s), which is equal to (s - 1)'-. 2 . Find the nature of the roots of the equation .>;* - 6s 3 + 13s 2 - 12s + 4 = 0. II. IV /i (s) = 4s 3 - 18s 2 + 26s - 12, / 2 (s) = s 2 - 3s + 2; fz(x) is the last Sturmian function ; so the equation has equal roots. (-co) + - +, (+ oo ) + + + • There are only two real distinct roots. In fact, since f% (x) = (x - l)(x - 2), each of the roots 1, 2 is a double root. 3. Find the nature of the roots of the equation x 5 + 2s 4 + x 3 - s 2 - 2x - 1 = 0. Here /i = 5x i + 8s s 4- 3s 2 - 2x - 2, f% = 2s 5 + 7s 2 + 12a; + 7, f 3 = — x 2 — 6s — 5, fi = -x-l, /s = 0. Since /5= 0, x+ 1 is a common measure of/and /], and/(s) has a double root — 1. We have also (-cc) - + _-+, (+oo) + + + __. Hence there are two real distinct roots. The equation has, therefore, beside the double root, one other real root, and two imaginary roots. 4. Find the nature of the roots of the equation s 6 - 7s 5 + 15s 4 - 40s 2 + 48s -16 = 0. Here /i (s) = 6s 5 - 35s* + 60s 3 - 80s + 48, / 2 (s) = 13s 4 - 84z 3 + 192s 2 - 176s + 48, / 3 (s) = s 3 - 6s 2 + 12s - 8 = (s - 2) 3 . Ans. There are three real distinct roots, one of them being quadruple. Application of Sturm's Theorem. 189 91. Application of Sturm's Theorem. — In the case of equations of high degrees the calculation of Sturm's auxiliary functions becomes often very laborious. It is important, there- fore, to pay attention to certain observations which tend some- what to diminish this labour. (1). In calculating the final remainder when it is numerical, since its sign is all we are concerned with, the labour of the last operation of division can be avoided by the consideration that the value of x which causes /„_! to vanish must give opposite signs tofn-2 and/,,. It is in general possible to tell without any calculation what would be the sign of the result if the root of /«-*(#) = were substituted inf n . 2 (x). Thus in Ex. 3, Art. 89, if 3 the value - ^, which is the root of fjx) = 0, be substituted o for x in 9x 2 - 27# + 11, the result is evidently positive; hence the sign of/, (x) is -, and there is no occasion to calculate the value - 1433 given for/, (a?) in the example in question. (2). When it is possible in any way to recognize that all the roots of any one of Sturm's functions are imaginary, we need not proceed to the calculation of any function beyond that one ; for since such a function retains constantly the same sign for all values of the variable (Cor. Art. 12), no alteration in the number of changes of sign presented by it and the following functions can ever take place, so that the diiference in the number of changes when two quantities a and b are substituted is indepen- dent of whatever variations of sign may exist in that part of the series which consists of the function in question and those following it. With a view to the application of this observation it is always well, when we arrive at the quadratic function (ax 2 + bx + c, suppose), to examine, in case the term containing x 2 and the absolute term have the same sign (otherwise the roots could not be imaginary) , whether the condition 4a c > b 2 is ful- filled ; if so, we know that the roots are imaginary, and the cal- culation need not proceed farther. Similar observations apply to the case where one of the functions is a perfect square, since such a function cannot chaDge its sign for real values of x. 190 Separation of the Hoots of Equations. EXAMrLKS. 1. Analyse the equation y 3a; 3 + 7a; 2 + 10* + 1 = 0. We find /:(.<) =- 29*8 -78*+ 14, f 3 (x) = - 1086a; -481, Here we see immediately that the value of x given by the equation / 3 (x) = 0, which differs little from - !, makes fn(x) positive; hence f t (x) is negative. Tliere are two real, and two imaginary roots. The real roots lie in the intervals {-2,-1}, {-1,0}. 2 . Analyse the equation x* - 4a; 3 - Zx + 23 = 0. "We find /_>(.r) = 12a; 2 + 9x - 89, / 3 (.r) = -491.r+ 1371, it ., v A ■ 1371 1371 „ mA 5 5 Here fj(x) = gives x = -j^r-> -rr-- > 2-74 > -, and x = - makes ■±1/1 OUU w ^ /2(a;) positive ; hence the root of/3(a;) makes it positive also. There are two real and two imaginary roots. The real roots lie in the intervals {2, 3}, {3, 4}. 3. Analyse the equation 2x i - 13s 2 + 10a; -19 = 0. Here; /i (x) = 4a; 3 - 13a; + 5, f 2 (x) = 13a; 2 - 15a; + 38. Since 4 x 13 x 38 > 15 2 , the roots of /j (x) are imaginary, and we proceed no farther with the calculation of Sturm's remainders. Substituting — co , 0, + oo , we obtain (-») + - +, (0) - + +, (+ oc ) + + +. There are two real roots, one positive, the other negative. 4. Analyse the equation f(x) = x 5 + 2x i + a; 3 - 4a; 2 - 3a; - 5 = 0. Here f\ (x) = 5a; 4 + 8a? + 3a; 2 - 8x - 3, f 2 ix) = 6a; 3 + 66a;2 + 44a; + 119, f 3 (x) = - 116a; 2 -57a;- 223. Examples. 191 Since 4 x 116 x 223 > 57-, we may stop the calculation here. "We find, on substituting — go , 0, + oo , (-00) - + - -, (0) - - + -, (+00) + + + -. There are four imaginary roots, and one real positive root. 5. Find the number and situation of the real roots of the equation x* - 2x 3 - 7.r 2 + 10* +10 = 0. Ans. The roots are all real, and are situated in the intervals {-3,-2}, {-1,0}, and two between {2, 3}. 6. Analyse the equation x 5 + 3^ + 2x 3 - 3x 2 - 2x - 2 = 0. It will be found that the calculation may cease 'with the quadratic remainder. Ans. There is only one real root : in the interval (1, 2}. 7. Analyse the equation x 3 + ll* 2 - 102* + 181 = 0. We find /> (x) = 854* - 2751, / 3 (s) = 441. In some examples, of which the present is an instance, it is not easy to tell immediately what sign the root of the penultimate function gives to the preceding function. "We have here calculated fz{x), and it turns out to be a much smaller number than might have been expected from the magnitude of the coefficients inf%{x) . In fact when the root of f-i(x) is substituted in/i (x) the positive part is nearly equal to the negative part. This is always an indication that two roots of the proposed equation are nearly equal. There are in the present instance two positive roots be- tween 3 and 4. Subdividing the intervals, we find the two roots still to lie between 3*2 and 3 - 3 ; so that they are very close together. "We see here another illustra- tion of the continuity which exists between real and imaginary roots. If fz{x) turned out to be zero, the roots would be actually equal. If it turned oit to be a small negative number, the two nearly equal roots would be imaginary. 8. Analyse the equation x 5 + x* + x 3 - 2x 2 + 2x - 1 = 0. The quadratic function is found to have imaginary roots. Ans. One real root between {0, 1}; four imaginary roots. 192 Separation of the Roots of Equal ions. 9. Analyse the equation x r> _ c* 5 - 30s 2 + 12s- 9 = 0. We find f, (x) = ox* + 20s 2 + 7 ; and as this has plainly all imaginary roots, the calculation may stop here. Am. Two real roots ; in the intervals {-2, - 1 }, {6, 7}. 10. Analyse the equation 2x* - 18s 5 + 60s 4 - 120s 3 - 30s 2 + 18s - 5 = 0. We find ft (x) = 5s 1 + 220s 2 + 1 ; and the calculation may stop. Ans. Two real roots ; in the intervals {—1,0}, {5,6}. 1 1 . Examine how the roots of the equation 2s 3 + 15s 2 - 84s - 190 = are situated in the several intervals between the numbers — oo , - 7, 6, + oo . Here /i (s) = s 2 + 5s - 14, /.(s) = 27s + 40, /s (*) = +• The substitution of the above quantities gives (-«,) - + - + , (-7) + - + , (6) + + + + , (+•) + + + + • Whenever, as in this example, any quantity makes one of the auxiliary functions vanish (here — 7 satisfies /i(s) = 0), the zero may be disregarded in counting the number of changes of sign in the corresponding row ; for, since the signs on each side of it are different, no alteration in the number of changes of sign in the row could take place, whatever sign be supposed attached to the vanishing quantity. The roots are all real. There is one root between — oo and - 7 ; and two be- tween - 7 and 6. 12. Analyse the equation 3s 4 - 6s 2 - 8s - 3 = 0. We find /i (x) = 3s 3 - 3s - 2, / 2 (s) = (s+l) 2 . As/2(s) is a perfect square the calculation may cease. Ans. Two real roots ; in the intervals {— 1, 0}, { 1, 2}. Conditions for Reality of Roots of an Equation. 193 92. Conditions for the Reality of the Roots of an Equation. — The number of Sturm's functions, including f{x),f'{x) and the n - 1 remainders, will in general be n + 1, In certain cases, owing to the absence of terms in the proposed function, some of the remainders will be wanting. This can occur only when the proposed equation has imaginary roots ; for it is plain that, in order to insure a loss of n changes of sign in the series of functions during the passage of x from - go to + go (namely, in order that the equation should have all its roots reaV, all the functions must be present. And, moreover, they must all take the same sign when x = + go ; and alternating signs when x = - go . Since the leading term of an equation is always taken with a positive sign, we may state the condition for the reality of all the roots of any equation (supposed not to have equal roots) as follows : — In order that all the roots of an equation of the n th degree should be real, the leading coefficients of all Sturm's remainders, in number n - 1, must be jjositive. Examples. 1. Find the condition that the roots of the equation ax- + 2bx + c = Ans. b 2 — ac > 0. should he real and unequal. 2. Find the conditions that the roots of the cubic s 3 + 3Hz + G=0 should he all real and unequal. . When this cubic has its roots all real, it is evident that the general cubic from which it is derived (Art. 36) has also its roots all real; so that, in investigating the conditions for the reality of the roots of a cubic in general, it is sufficient to discuss the form here written. We find f 1 (z)=z 2 + H, Mz)=-2Kz-G, M=) =-(e?2 + 4iP). [In calculating these, before dividing f\ (z) by/ 2 (c), multiply the former by the positive factor 2.ET 2 .] Hence the required conditions are, iTnegative and G 2 + 4IZ" 3 negative. These can be expressed as one condition, viz., G 2 + iH 3 negative, since this implies the former (cf. Art. 43). O 194 Separation of the Roots of Equations. 3. Calculate Sturm's remainders for the biquadratic ; 4 + 6-He 3 + iOz + //'-; and when the re- mainder is found, remove the positive factor a-. Before dividing/2 by /a, multiply by the positive fai b >t (1111 — 3«/) 2 ; and when the remainder is found, remove the positive factor a' 1 Jl "'-'. 93. Conditions for the Reality of the Root's of a BSi quadra tie. — In order to arrive at criteria of the nature of the roots of the general algebraic equation of the fourth degree by Sturm's method, it is sufficient to consider the equation of Ex. 3 of the preceding Article. By aid of the forms of the leading coefficients of Sturm's remainders there calculated, we can write down the conditions that all the roots of a biquadratic should be real and unequal in the form H negative, 2HI- 3a J negative, P - 27 J 2 positive. It will be observed that the second of these conditions is different in form from the corresponding condition of Art. 68. To show the equivalence of the two forms it is necessary to prove that when H is negative and A positive, the further con- dition 2HI - 3a J negative implies the condition a 2 1 - 12 B? negative, and conversely. From the identity of Art. 37, written in the form - H(a 2 I - 12H 2 ) - a 2 {2EI-3aJ) - 3G\ it readily appears that when H and 2HI - 3a J are negative a 2 I - 12H 2 is necessarily negative. And to prove the converse we observe that when a J is positive 211 I - 3a J is negative, since / is positive on account of the condition A positive ; and when aJ is negative 2HI - 3a J is still negative, since the negative part 2HI exceeds the positive part - 3a J, as may be readily shown by the aid of the inequalities 12 R~ > a 2 1 and P > 27 J 2 . The student will have no difficulty in verifying, by means of Sturm's functions, the remaining conclusions arrived at in the different cases of Art. 68. Examples. 195 Examples. 1. Apply Sudan's method to separate the roots of the equation x* - 16.c 3 + 69a- 3 - 70L>; -42 = 0. Ans. Roots in intervals {- 1, 0}, {2, 3}, {4, -5}, {9, 10}. 2. Apply Sturm's theorem to the analysis of the equation x* - ix 3 + Ix 1 — Qx — 4 = 0. In analysing a biquadratic of this nature which has plainly two real roots, when a Sturmian remainder is reached whose leading coefficient is negative, the calculation may cease, since the other pair of roots must then be imaginary, and the positions of the real roots can be readily found by substitution in the given equation. Ans. Two roots imaginary ; real roots in intervals {- 1, 0}, {2, 3}. 3. Analyse in a similar manner the equation x* - ox 3 + 10.e- - 6x - 21 = 0. Ans. Two roots imaginary; real roots in intervals {- 1, 0}, {3, 4}. 4. Apply Sturm's theorem to the analysis of the equation x i + 3x 3 - X- - 3x + 11 = 0. Ant. Roots all imaginary. 5. Find by, Sturm's method the number and position of the real roots of the equation x 5 - 10x 3 + 6x + 1 = 0. Ans. Roots all real; one in the interval {—4, — 3}; two in the interval {-1, 0}; and positive roots in the intervals {0, 1}, {3, 4}. 6. If, in the following, the sequences of .'signs are those of the leading coef- ficients of Sturm's remainders for a biquadratic, prove + + - ) + + + four real roots : + - - > two real roots ; - + + no real root ; / + - + - + - cannot occur. 7. If the signs of the leading coefficients of the first two of Sturm's remainders for a quintic be — + , prove that the number of real roots is determined. Ans. One real root only. 8. If R and /are both positive, prove that all the roots of the biquadratic are imaginary ; and that under the same conditions the quintic written with binomial coefficients has only one real root. Mr. M. Roberts, Lublin Exam. Papers, 1S62. o2 196 Separation of the Roots of Equations. 9. Prove that, if < has any value except unity, the equation ( .1 X \ _ 202^3 A 2x -1=0 has a pau of imaginar] roots. 10. Prove thai the roots of the equation a* _ ( tt 2 + }2 + C 2) a; - 2ah are all real, and solve it when two of the quantities a, b, c become equal. 11. Prove that when the biquadratic /(.r) - ax* + 4 bx 3 + 6 ex 2 + 4dx + e has a triple factor, it may be expressed in the form a i ■■.,■) ^ {«z + l> + -J^Hy^ax+b-Z^^rfl}. VI. Verify by means of Sturm's remainders the conditions which must be ful- filled when the biquadratic of the previous example is a perfect square, and prove in that case e^f(x) = {(ax+by~+3H}°~. 13. If an equation of any degree, arranged according to powers of x, have three consecutive terms in geometric progression, prove that its roots cannot be all real. These three terms must be of the form hx r + kax"' 1 + ko?x r - 2 . Let the equation be multiplied by x - a. The resulting equation will have two consecutive terms absent, and must therefore have at least two imaginary roots ; but all the roots of this equation except o are roots of the given equation. 14. If an equation have four consecutive coefficients in arithmetic progression, prove that its roots cannot be all real. This can be reduced to the preceding example. "Writing down four terms of the proper form, and multiplying by x - 1, it readily appears that the resulting equation has three consecutive terms in geometric progression. CHAPTER X. SOLUTION OF NUMERICAL EQUATIONS. 94. Algebraical and \ nine tic a 1 Equations. — There is an essential distinction between the solutions of algebraical and numerical equations. In the former the result is a general for- mula of a purely symbolical character, which, being the general expression for a root, must represent all the roots indifferently. It must be such that, when for the functions of the coefficients involved in it the corresponding symmetric functions of the roots are substituted, the operations represented by the radical signs *y ^/ become practicable ; and when the square and cube roots of these symmetric functions are extracted, the whole expression in terms of the roots will reduce down to one root : the different roots resulting from the different combinations ± y/ of square roots, and %/, w £/, ur %/ of cube roots. For a simple illustration of what is here stated we refer to the case of the quadratic in Art. 55. In Articles 59 and 66 we have similar illustrations for the cubic and biquadratic. It is to be observed, also, that the formula which represents the root of an algebraic equation holds good even when the coefficients are imaginary quantities. In the case of numerical equations the roots are determined separately by the methods we are about to explain ; and, before attempting the approximation to any individual root, it is in general necessary that it should be situated in a known interval which contains no other real root. The real roots of numerical equations may be either com- mensurable or incommensurable ; the former class including integers, fractions, and terminating or repeating decimals which L98 Solution of Numerical Equations. are reducible to fractions; (he latter consisting of interminable decimals. The roots <>fthe former class can be found exactly, and these of the latter approximated to with any degree of accuracy, by the methods we are about to explain. We shall commence by establishing a theorem which reduces the determination of the former class of roots to that of integral roots alone. 95. Theorem. — An equation in which the coefficient of thi Jirst term is in/if//, and the coefficients of the other terms whole numbers, cannot hare a commensurable root which is not a whole number. For, if possible, let -, a fraction in its lowest terms, be a root of the equation x 11 + 2) l x n ~ l + p, z ll ~- + . . . . + Pn-i) ; this to be added to a„_ 2 ; and so on. If h be a root, the last figure in the second line thus obtained will be- We commence with 4 : - 24 38 -13 -6 8 Examples. 1 . Find the integral roots of the equation & - 1x* - 13x- 2 + 38* - 24 = 0. By grouping the terms (see Art. 79) we observe without difficulty that all the roots lie between — 5 and + 5. The following divisors are possible roots : — -2 32 - 5 The operation stops here, for since — 5 is not divisible by 4, 4 cannot be a root. "We proceed then with the number 3 : - 24 38 - 13 - 2 1 - 8 10 - 1 - 1 30 - 3 - 3 ; hence 3 is a root ; and in proceeding with the next integer, 2, we make use, as above explained, of the coefficients of the second line with signs changed : 8-10 1 1 4 -3 - 1 - 6 -2 0; hence 2 also is a root ; and we proceed with — 2 : -4 3 1 •1 hence - 2 is not a root, for it does not divide 5. — 3 is plainly not a root, for it does not divide - 4. [We might at once have struck out - 3 as not being a divisor of the absolute term 8 of the reduced polynomial. This remark will often be of use in diminishing the number of divis >rs* 202 Solution of Numerical Equations. We proceed now to the lasl divisor, - 1 : - 4 3 1 1 - 1 4 Thus - 1 i> a root. The equation has. therefore, the integral roots 3, 2,-4; and the lasl stage of the operation shows that when the original polynomial is divided by the binomials x — 3, x — 2, x-\ I, the result is x— 1 ; so that 1 is also a root. Hence the original polynomial is equivalent to (x- l)(s-2) (x -3) (x + 4). 2. Find the integral roots of 3x* - 23z 3 + 35a; 2 + 31.r - 30 = 0. The roots lie between — 2 and S ; hence we have only to test the divisors 2, 3, 5, 6. We find immediately that 6 is not a root. For 5 we have - 30 31 35 - 23 3 - 6 5 S - 3 2.") 40 - 15 ; henee 5 is a root. For 3 we have 6-5-8 3 2 - 1 -3 -3 -9 0; hence 3 is a root ; and we easily find that 2 is not a root. The quotient, when the original polynomial is divided by (x - 5) (.r — 3), is, from the last operation, 3-r- + x - 2 : of this 1 is not a root, and - 1 is a root. Hence all the integral roots of the pro- posed equation are — 1, 3, 5. 2 The other root of the equation is-. It is a commensurable root ; but, not being integral, is not given in the above operation. 3. Find all the roots of the equation x* + x 3 - 2a? -f 4.c - 24 = 0. Limits of the roots are — 4, 3. Am. Koots - 3, 2, + -l*/~. Method of Limiting the Number of Divisors. 203 4 . Find all the roots of the equation x i - o x s _ 19^2 + 6Sx _ 60 = 0. The roots lie between — 6 and 6. "We find that 2, 3,-5 are roots, and that the factor left after the final division is x — 2 ; hence 2 is a double root. The polynomial is therefore equivalent to (*- 2) 2 (3-3) (a: + 5). In Art. 99 the case of multiple roots -will be further considered. 98. Method of Limiting the \umber of Divisors. — It is possible of course to determine by direct substitution whether any of the divisors of a„ are roots of the proposed equa- tion ; but Newton's method has the advantage, as the above examples show, that some of the divisors are rejected after very little labour. It has a further advantage which will now be explained. When the number of divisors of a,, within the limits of the roots is large, it is important to be able, before proceeding with the application of the method in detail, to diminish the number of these divisors which need be tested. This can be done as follows : — If h is an integral root of f(x) = 0,f(x) is divisible by x - //, and the coefficients of the quotient are integers, as was above explained. If therefore we assign to x any integral value, the quotient of the corresponding value of /(•'') by the correspond- ing value of x - h must be an integer. We take, for convenience,, the simplest integers 1 and - 1 ; and, before testing any divisor //, we subject it to the condition that ,/'(l) must be divisible by I - // (or, changing the sign, by // - 1) ; and that/(- 1) must be divisible by - 1 - h (or, changing the sign, by 1 + h). In applying this observation it will be found convenient to calculate/ (1) and/(- 1) in the first instance: if either of these vanishes, the corresponding integer is a root, and we proceed with the operation on the reduced polynomial whose coefficients have been ascertained in the process of finding the result of substituting the integer in question. 204 Solution of Numerical Equations. Y.\ \\W\A 8. 1 . x b - 23.r> + 1 6< i . ■ 281a: 2 - 257* - 440 = 0. The roots lie betw een - 1 and 2 1. We have the following divisors : — 2, 1. •">, 8, 10, 11, 20, 22. We easily find /(l) = -840, and /(- 1) = - 648. We therefore exclude all the ahove divisors, which, when diminished by 1, do on I divide 840; and which, when increased by 1, do not divide 648. The first condition excludes 10 and 20, and the second 4 and 22. Applying the Method of Divisors to the remaining integers 2, 5, 8, 11, we find that 5, 8, and 11 are roots, ami that the resulting quotient is x~ + x + 1. Hence the given polynomial is equi- valent to (x - 5) (a; - 8) (a: - 11) (x n - + x + 1). 2. x 5 - 29.c 4 - 31./; 3 + 31a? - 32a; + 60 = 0. The roots lie between - 3 and 32. Divisors: - 2, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30. /(I) = : so 1 is a root. /(— 1) = 124 ; and the ahove condition excludes all the divisors except — 2, 3, 30. We easily find that - 2 and 30 are roots, and that the final quotient is x- + 1. The given polynomial is equivalent to (x — 1) (x - 30) (x + 2) (x 2 + 1). 99. Determination of Multiple Roots.— The Method of Divisors determines multiple roots when they are commen- surable. In applying the method, when any divisor of a n which is found to be a root is a divisor of the absolute term of the re- duced polynomial, we must proceed to try whether it is also a root of the latter, in which case it will be a double root of the proposed equation. If it be found to be a root of the next reduced polynomial, it will be a triple root of the proposed ; and so on. Whenever in an equation of any degree there exists only one multiple root, r times repeated, it can be found in this way ; for the common measure of f{x) and ,/"(•'') will then be of the form (.'■ - o) r_1 , and the coefficients of this could not be com- mensurable if « were incommensurable. Determination of Multiple Roots. 205 Multiple roots of equations of the third, fourth, and fifth degrees can be completely determined without the use of the process of finding the greatest common measure, as will appear from the following observations : — (1). The Cubic. — In this case multiple roots must be com- mensurable, since the degree is not high enough to allow of two distinct roots being repeated. (2). The Biquadratic. — In this case either the multiple roots are commensurable or the function is a perfect square. For'the only form of biquadratic which admits of two distinct roots being repeated is (x - a y(x - /3) 2 , i. e. the square of a quadratic. The roots of the quadratic may be incommensurablo. If we find, therefore, that a biquadratic has no commensurable roots, we must try whether it is a per- fect square in order to determine further whether it has equal incommensurable roots. (3). The Quintic. — In this case, either the multiple roots are commensurable, or the function consists of a linear commensurable factor multiplied by the square of a quadratic factor. For, in order that two distinct roots may be repeated, the function must take one or other of the forms (0 - af{x - (5Y(x - y), (» - a)\x - /3) 3 . In the latter case the roots cannot be incommensurable ; but the former may correspond to the case of a commensurable factor multiplied by the square of a quadratic whose roots are incom- mensurable. If then a quintic be found to have no commen- surable roots it can have no multiple roots. If it be found to have one commensurable root only, we must examine whether the remaining factor is a perfect square. If it have more than one commensurable root, the multiple roots will be found among the commensurable roots. 206 Solution of Numerical Equations. Examples. 1. Find ;til the commensurable roots of ■■'Ax- + 112.r + 64 = 0. Tin' roots lie between the limits - 1, 1G. The divisors are 2, 4, 8. 64 112 -31 2 8 15 -2 120 -16 0; S is therefore a root. Proceed now with the reduced equation : -8 -15 2 - 1 - 2 - 16 0; . 8 is a root again, and the remaining factor is 2x + 1 . Am, f(x) = (2x + 1) (x - S) 2 . 2. Find the commensurable and multiple roots of x* - x 3 - 30z 2 - 76* - 56 = 0. The roots lie between the limits - 6, 12. (Apply method of Ex. 10, Art. 80). An*. f(x) = (x + 2)*{x-7). 3. Find the commensurable and multiple roots of 9x i - V2x 3 - 71s 2 - iOx + 16 = 0. The roots lie between the limits - 2, 5. The equation as it stands is found to have no integral root ; but it may still have a commensurable root. To test this we multiply the roots by 3 in order to get rid of the coefficient of x*. We find then jr 4 - 4./- 3 - 7U- 2 - 120.C + 144 = 0. Limits : — 6, 15. We iind - 4 to be a double root of this, and the function to be equivalent to {x- - \2x + 9) (x + 4) 2 . The original equation is therefore identical with the fol- lowing : — (z 2 -4* + 1)(3* + 4)'- = 0. 4. Find the commensurable and multiple roots of x i + 12* 3 + 32a; 2 - 24a; + 4 = 0. Neivtorfs Method of Approximation. 207 The roots lie between — 12 and 1. The only divisors to he tested are, therefore, — 4, — 2, — 1. We find that the equation has no commensurable root. We pro- ceed to try whether the given function is a perfect square. This can be done by extracting the square root, or by applying the conditions of Ex. 8, p. 123. We find that it is the square of s 2 + 6x — 2(cf. Ex. 1, p. 161). Hence the given equation has two pairs of equal roots, both incommensurable. •5. Find the coniniensurable and multiple roots of f(x) ^x 5 -^- 12s 3 + 8s 2 + 28s + 12=0. The limits of the roots are — 4, 4. We find that — 3 is a root, and that the reduced equation is s 4 - 4s 3 + 8x + 4 = 0, and that there is no other commensurable root. The only case of possible occurrence of multiple roots is, therefore, when this latter function is a perfect square. It is found to be a perfect square, and we have f(x) B (s 2 - 2x - 2)2 (* + 3). 6. Find the commensurable and multiple roots of /(s) = x 5 - 8s 4 + 22s 3 - 26s 2 + 21s - 18 = 0. Ans. f(x) = (s 2 + 1) (s - 2) (s - 3) 2 . 7. The following equation has only two different roots : find them : — s 5 - 13s 4 + 67s 3 - 171s 2 + 216s - 108 = 0. In general it is obvious that if an integral root h occurs twice, the last coefficient must contain h % as a factor, and the second last h ; if the root occurs three times, h z must be a factor of the last, h 2 of the second last, and h of the third last coef- ficient. The last coefficient here = 2 2 . 3 3 . Hence, if neither — 1 nor 1 is a root, the required roots must be 2 and 3. That these are the roots is easily verified. S. The equation 800s 4 - 102s 2 - s + 3 = has equal roots : find all the roots. In this example it is convenient to change the roots into their reciprocals before applying the Method of Divisors. Ans. f(x) = (10s - 3) (5s - l)(4s + I) 2 . 100. Xewton's Method of Approximation. — Having shown how the commensurable roots of equations may be ob- tained, we proceed to give an account of certain methods of obtaining approximate values of the incommensurable roots. The method of approximation, commonly ascribed to Newton,* which forms the subject of the present Article, is valuable as * See Xote B at the end of the volume. 208 Solution of Numerical Equations. being applicable to numerical equations involving transcendental as well as those involving algebraical Functions only. Although when applied to the latter olassof functions Newton's method is, for practical purposes, inferior in form to Horner's, which will be explained in the following Articles, yet in principle both methods are to a great extent identical. In all methods of approximation the root we are seeking is supposed to be separated from the other roots, and to be situated in a known interval between close limits. Let/(.r) = be a given equation, and suppose a value a to be known, differing by a small quantity h from a root of the equation. We have then, since a + h is a root of the equation, /'(a + /?) = ; or /{a)+f\a)h+- f -^/r+ .... = 0. Neglecting now, since h is small, all powers of h higher than the first, we have f(a)+f(a)h = 0, giving, as a first approximation to the root, the value * A") Representing this value by b, and applying the same process a second time, we find as a closer approximation By repeating this process the approximation can be carried to any degree of accuracy required. Example. Find an approximate value of the positive root of the equation x 3 - 2x - 5 = 0, The root lies between 2 and 3 (Ex. 1, Art. 89). Narrowing the limits, the root is found to lie between 2 and 2-2. We take 2-1 as the quantity represented by a. It cannot differ from the true value a + h of the root by more than 0-1. "WVfind easily /(*) /(2-1) -061 />) = /(2a) = lT23 = 000543 - Horner's Solution of Numerical Equations. 209 A first approximation is, therefore, 2-1 - 0-00543= 2-0946. Taking this as *, and calculating the fraction -zr~, vre obtain 5-^ = 2-09455148 for a second approximation ; and so on. The approximation in Newton's method is, in general, rapid. When, however, the root we are seeking is accompanied by an- f(a) other nearly equal to it, the fraction'-^- is not necessarily small, since the value of either of the nearly equal roots reduces/' (a-) to a small quantity. A case of this kind requires special precau- tions. We do not enter into any further discussion of the method, since for practical purposes it may be regarded as entirely superseded by Horner's method, which will now be explained. 101. Horner's Method of Solving Numerical Equa- tions. — By this method both the commensurable and incom- mensurable roots can be obtained. The root is evolved figure by figure : first the integral part (if any), and then the decimal part, till the root terminates if it be commensurable, or to any number of places required if it be incommensurable. The pro- cess is similar to the known processes of extraction of the square and cube root, which are, indeed, only particular cases of the general solution by the present method of quadratic and cubic equations. The main principle involved in Horner's method is the suc- cessive diminution of the roots of the given equation by known quantities, in the manner explained in Art. 33. The great ad- vantage of the method is, that the successive transformations are exhibited in a compact arithmetical form, and the root obtained by one continuous process correct to any number of places of decimals required. Tbis principle of the diminution of the roots will be illus- trated in the present Article by some simple examples. In the p 210 Solution <>f Numerical Equations. following Articles we shall proceed to certain considerations which tend to facilitate the practical application of the method. Examples. 1. Find the positive root of the equation 2a: 3 - 85s 2 - 85* - 87 = 0. The first step, when any numerical equation is proposed for solution, is to find the first figure of the root. This can usually be done by a few trials ; although in cer- tain cases the methods of separation of the roots explained in Chap. IX. may have to be employed. In the present example there can be only one positive root ; and it is found by trial to lie between 40 and 50. Thus the first figure of the root is l. We now diminish the roots by 40. The transformed equation will have one root between and 10. It is found by trial to lie between 3 and 4. We now diminish the roots of the transformed equation by 3 ; so that the roots of the proposed equa- tion will be diminished by 43. The second transformed equation will have one root between and 1. On diminishing the roots of this latter equation by - 5, we find that its absolute term is reduced to zero, i. e. the diminution of the roots of the pro- posed equation by 43 - 5 reduces its absolute term to zero. We conclude that 43 - 5 is a root of the given equation. The series of arithmetical operations is represented as follows : — (43-5 -85 -85 -87 80 -200 -11400 - 5 - 285 -11487 80 3000 9594 75 2715 -1893 80 483 1893 155 3198 6 501 161 3699 6 87 167 3786 6 173 1 174 The broken lines mark the conclusion of each transformation, and the figures in dark type are the coefficients of the successive transformed equations (see Art. 33). Thus 2s 3 + 155a; 2 + 2715s - 11487 = Examples. 211 is the equation whose roots are each less by 40 than the roots of the given equation, and whose positive root is found to lie between 3 and 4. If the second transformed equation had not an exact root -5 ; but one, we shall suppose, between -5 and -6, the first three figures of the root of the proposed equation would be 43-5 ; and to find the next figure we should proceed to a further transformation, diminishing the roots by -5 ; and so on. 2. Find the positive root of the equation 4a; 3 - 13a; 2 - 31a; - 275 = 0. "We first write down the arithmetical work, and proceed to make certain observations (6-25 it:— 4 - 13 24 - 31 66 -275 210 11 24 35 210 -65 51-392 35 24 245 11-96 -13 608 13-608 59 256-96 •8 12-12 59 -S •8 26908 3-08 60-6 •8 272-16 61-4 •2 61-6 We find by trial that the proposed equation has its positive root between 6 and 7. The first figure of the root is, therefore, 6. Diminish the roots by 6. The equation x 3 + 59a; 2 + 245a; - 65 = has, therefore, a root between and 1. It is found by trial to lie between -2 and *3. The first two figures of the root of the proposed are therefore 6-2. Diminish the roots again by -2. The transformed equation is found to have the root -05. Hence 6-25 is a root of the proposed equation. It is convenient in practice to avoid the use of the decimal points. This can easily be effected as follows :— When the decimal part of the root (suppose -ale . . .) is about to appear, multiply the roots of the corresponding transformed equation by 10, i.e. annex one zero to the right of the figure in the first column, two to the right of the figure in the second column, three to the right of that in the third ; and so on, if there be more columns (as there will of course be in equations of a degree higher than the third). The root of the transformed equation is then, not -abc . . . , but a-bc . . . Diminish the roots by a. The transformed equation has a root -be . . . Multiply the roots of this equation again by 10. The root becomes he..., and p2 212 Solution of Numerical Equations. the process is continued as before. To illustrate this we repeat the above operation, Omitting the decimal points. In all subsequent examples this simplification will be adopted : — 13 24 11 24 31 66 -275 210 (6-25 35 210 -65000 51392 25 24 24500 1196 - 13608000 13608000 590 8 25696 1212 598 8 2690800 30800 606 8 2721600 6140 20 6160 3. Find the positive root of the equation 20z 3 - 121a; 2 - 121z - 141 = 0. The root is easily found to lie between 7 and 8. It is, therefore, of the form ab . . . When the roots are diminished by 7, and multiplied by 10, the resulting equation is 20Z 3 + 2990a; 2 + 112500a; - 57000 = 0. The positive root of this is a . b . . .; and as the root plainly lies between and 1, we have a = 0. "We therefore place zero as the first figure in the decimal part of the root, and multiply the roots again by 10, before proceeding to the second trans- formation. 5 is easily seen to be a root of the equation thus transformed. Ans. 7*05. In the examples here considered the root terminates at an early stage. When the calculation is of greater length, if it were necessary to find the successive figures by substitution, the labour of the process would be very great. This, however, is not necessary, as will appear in the next Article ; and one of the most valuable practical advantages of Horner's method is, that after the second, or third (sometimes even after the first) figure of the root is found, the transformed equation itself suggests by mere inspection the next figure of the root. The principle of this simplification will now be explained. Principle of the Trial-divisor. 213 102. Principle of the Trial-divisor. — We have seen in Art. 100 that when an equation is transformed by the substitution of a + h for x, a being a number differing from the true root by a quantity h small in proportion to a, an approximate numeri- cal value of A is obtained by dividing /(a) hy f'(a). Now the successive transformed equations in Horner's process are the results of transformations of this kind, the last coefficient being /((/), and the second last /"(«) (see Art. 33). Hence, after two or three steps have been completed, so that the part of the root remaining bears a small ratio to the part already evolved, we may expect to be furnished with two or three more figures of the root correctly by mere division of the last by the second last coefficient of the final transformed equation. We might there- fore, if we pleased, at any stage of Horner's operations, apply Newton's method to get a further approximation to the root. In Horner's method this principle is employed to suggest the next following figure of the root after the figures already obtained. The second last coefficient of each transformed equation is called the trial-divisor. Thus, in the second example of the last Ar- ticle, the number 5 is correctly suggested by the trial-divisor 2690800 In this example, indeed, the second figure of the root is correctly suggested by the trial-divisor of the first trans- formed equation, although, in general, this is not the case. In practice the student will have to estimate the probable effect of the leading coefficients of the transformed equation ; he will find, however, that the influence of these terms becomes less and less as the evolution of the root proceeds. Examples. 1 . Find the positive root of the equation j? + x- + x - 100 = correct to four decimal places. It is easily seen that the root lies between 4 and 5. "We write down the work, and proceed to make observations on it : — 2H Solution of Numerical Equations. 1 4 1 20 - 100 84 5 4 21 36 - 16000 11928 9 4 5700 264 -4072000 3788.37G 130 2 5964 268 - 283624000 256071744 132 2 623200 8196 - 27552256 134 2 631396 8232 1360 6 63962800 55136 1366 6 64017936 55152 1372 6 64073088 13780 4 13784 4 13788 4 (4-2644 13792 First diminish the roots hy 4. As the decimal part is now ahout to appear, at- tach ciphers to the coefficients of the transformed equation as explained in Ex. 2, Art. 101. Since the coefficient 130 is small in proportion to 5700, we may expect that the trial divisor will give a good indication of the next figure. The figure to he adopted in every case as part of the root is that highest number which in the pro- cess of transformation will not change the sign of the absolute term. Here 2 is the proper figure. In diminishing hy 2 the roots of the transformed equation a; 3 + 130a; 2 + 5700a: - 16000 = 0, the absolute term retains its sign (- 4072). If we had adopted the figure 3, the absolute term would have become positive, the change of sign showing that we had gone beyond the root. We must take care that, after the first transformation (the reason of this restriction will appear in the next example), the absolute term pre- serves its sign throughout the operation. If we Mere to take by mistake a number too small, the error would show itself, just as in ordinary division or evolution by the next suggested number being greater than 9. Such a mistake, however, will rarely be made. The error which is most common is to take the number too large, Principle of the Trial-Divisor. 215 and this will show itself in the work hy the change of sign in the absolute term. In the above work it is evident, without performing the fifth transformation, that the corresponding figure of the root is 4, so that the correct root to four decimal places is 4-2644. 2. The equation x* + 4^ - 4z 2 - 1U- + 4 = has one root between 1 and 2 ; find its value correct to four decimal places. 4 -4 - 11 4 (1-6369 1 5 1 -10 5 1 -10 -60000 1 6 7 50976 6 7 -3000 - 90240000 1 7 11496 72690561 7 1400 8496 - 175494390000 1 516 14808 152131052016 80 1916 23304000 -23363337984 6 552 926187 86 2468 24230187 6 588 935601 92 305600 25165788000 6 3129 189387336 98 308729 25355175336 6 3138 189766488 1040 311867 25544941824 3 3147 1043 31501400 3 63156 1046 31564556 3 63192 1049 31627748 3 63228 10520 31690976 6 10526 6 10532 6 10538 6 10544 216 Sulnllmi of Nurru Heal Equations. \W Bee without completing the fifth transformation thai 9 la the next figure of ot. The root is, therefore, 1-6369 corn decimal places. The trial-divisor becomes effective after the second transformation, suggesting ■ tly the number 3, and all subsequent numbers. Ti. Formed equation its last two terms negative. We may expect, therefore, that the influence of tlic preceding coefficients is greater than that of the trial-divisor, as in fact is here the ci ! C), the second figure of the root, must be found by substitu- tion. We have to determine what is the situation between and 10 of the root of the equation .-<•' + 80.T 5 + 1400.C 2 - 3000.T - 60000 = 0. A few trials show that 6 gives a md 7 a positive result. Hen<-i- the root lies between 6 and 7 ; and 6 is the number of which we are in search. In the quent trials we take those greatest numbers 3, 6, 9, in succession, which allow the absolute term to retain its negative sign. In the first transformation, diminishing the roots by 1, there is a change of sign in the absolute term. The meaning of this is, that we have passed over a root lying between and 1, for gives a positive result, 4 ; and 1 gives a negative result, — 6. In all subsequent transformations, so long as we keep below the root, the sign of the absolute term must be the same as the sign resulting from the substitution of 1 . This supposes of course that no root lies between 1 and that of which we are in search. This supposition we have already made in the statement of the question. In fact the proposed equation can have only two positive roots ; one of them lies between and 1, and therefore only one between 1 and 2. When two roots exist between the limits employed in Homer's method, i.e. when the equation has a pair of roots nearly equal, certain precautions must be observed which will form the subject of a subsequent Article. 3. Find the root of the preceding equation between and 1 to four decimal places. Commence by multiplying by 10. The coefficients are then 1, 40, -400, - 11000, 40000; the trial-divisor becomes effective at once in consequence of the comparative small - ness of the leading coefficients. The positive sign of the absolute term must be pre- served throughout. Am. -3373 4. Find to three places of decimals the root situated between 9 and 10 of the equation z 4 - 3.r 2 + 75.* - 10000=0. [Supply the zero coefficient of x 3 .] Am. 9-886. In the examples hitherto considered the root has been found to a few decimal places only. We proceed now to explain a method by which, after three or four places of decimals have been evolved as above, several more may be correctly obtained with great facility by a contracted process. Contraction of Horner's Process. 217 103. Contraction of Homer's Process. — In the ordi- nary process of contracted Division, when the given figures are exhausted, in place of appending ciphers to the successive divi- dends, we cut off figures successively from the right of the divisor, so that the divisor itself becomes exhausted after a num- ber of steps depending on the number of figures it contains. The resulting quotient will differ from the true quotient in the last figure only, or at most in the last two figures. In Horner's contracted. method the principle is the same. We retain those figures only which are effective in contributing to the result to the degree of approximation desired. When the contracted process commences, in place of appending ciphers to the succes- sive coefficients of the transformed equation in the way before explained, we cut off one figure from the right of the last coef- ficient but one, two from the right of the last coefficient but two, three from the right of the last coefficient but three ; and so on. The effect of this is to retain in their proper places the im- portant figures in the work, and to banish altogether those which are of little importance. The student will do well to compare the first transformation by the contracted process in the first of the following examples with the corresponding step in the second example of the last Article, where the transformation is exhibited in full. He will then observe how the leading figures (those which are most important in contributing to the result) coincide in both cases, and retain their relative places ; while the figures of little im- portance are entirely dispensed with. In addition to the contraction now explained, other abbrevia- tions of Horner's process are sometimes recommended ; but as the advantage to be derived from them is small, and as they increase the chances of error, we do not think it necessary to give any account of them. The contraction here explained is of so much importance in the practical application of Horner's method of approximation that no account of this method is complete without it. 218 Solution of Numerical Equations. Ex.mi'i.i s. 1. Find tin- root between 1 and '_> of the equation in Ex. 2 of the last Article cmiri i tn seven or eight decimal p] Assuming the result of the Example referred t", we Bhall commence the con- tracted pro< i 3fl after tin third transformation has been completed. The subsequent work stands as follows: — 3150\*. 6 25165788 18936 - 17549439 15213090 (1-636913576 3156 6 2535515 18972 -2336349 2301597 3162 6 255448^ 285 - 34752 25601 - 9151 7680 :;n;.s 255733 285 $X 256018 - 1171 1280 - 191 179 12 Here the effect of the first cutting off of figures, namely, 8 from the second last coefficient, 14 from the third last, and 052 from the fourth last, is to banish alto- gether the first coefficient of the biquadratic. "We proceed to diminish the roots by 6 as if the coefficients 1, 3150, 2516578, - 17549439 which are left were those of a cubic equation. In multiplying by the corresponding figure of the root the figures cut off should be multiplied mentally, and account taken of the number to be carried, just as in contracted division. After the diminution by 6 has been completed, we cut off again in the transformed cubic 7 from the last coefficient but one, 68 from the last but two, and the first coefficient disappears altogether. The work then proceeds as if we were dealing with the coefficients 31, 255448, -2336349 of a quadratic. The effect of the next process of cutting-off is to banish altogether the leading coefficient 31. The sub- sequent work coincides with that of contracted division. "When the operation ter- minates, the number of decimals in the quotient may be depended on up to the last two or three figures. The extent to which the evolution of the root must be carried before the contracted process is commenced depends on the number of decimal places required ; for after the contraction commences we shall be furnished, in addi- tion to the figures already evolved, with as many more as there are figures in the trial-divisor, less one. 2. Find to seven or eight decimal places the root of the equation ar> - 12* +7 = which lies between 2 and ','>. Examples. 219 This equation can have only two positive roots : one lies between and 1, and the other hetween 2 and 3. For the evolution of the latter we have the following : — 2 4 4 8 - 12 8 7 (2-047275671 -8 2 2 -4 24 - 100000000 83891456 4 2 12 12 20000000 972864 - 16108544 15493401 6 2 240000 3216 20972864 985792 -615143 446262 800 4 243216 3232 21968656 17478 - 168881 156226 804 4 246448 3248 2213343 17478 - 12655 11159 808 4 249696 2496 223082\ 49 - 1496 1338 812 4 223131 49 - 158 156 ^ 22318Q On this we remark, that after diminishing the roots by 2, and multiplying the roots of the transformed equation by 10, we find that the trial-divisor 20000 will not "go into" the absolute term 10000 ; we put, therefore, zero in the quotient, and mul- tiply again by 10, and then proceed as before. 3. Find the root of the same equation which lies between and 1. Am. -593685829. 4. Find the positive root of the equation x 3 + 24-84a: 2 - 67-613:*; - 3761-2758 = 0. [When the coefficients of the proposed equation contain decimal points, it will he found that they soon disappear in the work in consequence of the multiplications by 10 after the decimal part of the root begins to appear.] Am. 11-1973222. 5. Find the negative root of the equation z* - 12a- 2 + \2x - 3 = to seven places of decimals. When a negative root has to be found, it is convenient to change the sign of x and find the corresponding positive root of the transformed equation. Am. - 3-9073785. 220 Solution of Numerical Equations. li'l. Application of Horner** Method to Case*, where ■tool* are nearly Kqual. — Wo have seen in Art. 100 that the method of approximation there explained fails when the pro- posed equation has two roots nearly equal. Examples of this nature are those which present most difficulties, both in their analysis (see Ex. 7, Art. 91) and in their solution. By Horner's method it is possible, with very little more labour than is neces- sary in other cases, to effect the solution of such equations. So long as the leading figures of the two roots are the same certain precautions must be observed, which will be illustrated by the following examples. After the two roots have been separated, the subsequent calculation proceeds for each root separately, just as in the examples of the previous Articles. It is evident, from the explanation of the trial-divisor given in Art. 102, that for the same reason as that which explains the failure of Newton's method in the case under consideration (see Art. 100), it will not become effective till the first or second stage after the roots have been separated. Examples. 1. The equation a? - 1x + 7 = has two roots between 1 and 2 (see Ex. 2, Art. S9) ; find each of them to eight de- cimal places. Diminishing the roots by 1, we find that the transformed equation (after its roots are multiplied by 10), viz. x 3 + 30x 2 - 400x- + 1000 = 0, must have two roots between and 10. We find that these roots lie, one between 3 and 4, and the other between 6 and 7. The roots are now separated, and we pro- teed with each separately in the manner already explained. If the roots were not rated at this stage, we should find the leading figure common to the two, and, having diminished the roots by it, find in what intervals the roots of the resulting equation were situated ; and so on. Ans. 1-35689584, 1-69202147. 2. Find the two roots of the equation x 3 - 49.s 2 + 658* - 1379 = which lie between 20 and 30. "We shall exhibit the complete work of approximation to the smaller of the two roots to seven places ; and then make certain observations which will be a guide to the student in of the kind. Examples. 221 -49 20 658 -580 - 1379 (23- 1560 -29 20 78 - 180 181 -180 -9 20 -102 42 1000 - 992 11 3 -60 51 8000 -6739 14 3 -900 404 1261000 - 1217403 17 3 -496 408 43597 -34183 200 2 -8800 2061 9414 -6786 202 2 -6739 2062 2628 -2372 204 2 - 467700 61899 256 -236 2060 1 -405801 61908 20 2061 1 -34389$ 206 2062 1 -34183 206 20630 3 20.6, -33977 4 20633 3 -3393 4 20636 3 2, -338§ 206^9 The diminution of is an indication that a sent concerned. The the roots hy 20 changes the sign of the ahsolute term. This root exists hetween and 20, with which we are not at pre- roots of the first transformed equation z 3 + llz 2 - 102z + 181 = are not yet separated, lying both hetween 3 and 4. The substitution of each of these numbers gives a positive result, so that we have not here the same criterion to guide us in our search for the proper figure as in former cases, viz., a change of sign in the absolute term. We have, however, a different criterion which enables J ; J*J Solution of Numerical Equations. w< to find by more substitution the interval within which the two roots lie. If we diminish the mots of < ! I U/ : - 102x4- 181 = by 4, the resulting equation is x 3 + 2 • -f 13 = 0, which has no change of sign. Hence the two roots must lie between and 4. It we diminish its roots by .'!, the resulting equation (as in the above work) has the same number of changes of .sign as the equation itself. Hence the two roots lie between 3 and 1. They are, therefore, not yet separated ; and we proceed to diminish by 3. The next transformed equation ar 3 + 200a; 2 - 900a; + 1000 = is found in the same way to have both its roots between 2 and 3 : the diminution by 2 leaving two changes of sign in the coefficients of the transformed equation (as in the above work) , and the diminution by 3 giving all positive signs. So far, then, the two roots agree in their first three figures, viz. 23*2. We diminish again by 2. The resulting equation x 3 + 2060a; 2 - 8800a; + 1201000= has one root only between 1 and 2 ; 1 giving a positive, and 2 a negative result : its other root lies between 2 and 3 ; 3 giving a positive result. The roots are now separated. We proceed, as in the above work, to approximate to the lesser root, by diminishing the roots of this equation by 1 ; the trial divisor becoming effective at the next step. To approxi- mate to the greater root, we must diminish by 2 the roots of the same equation, taking care that in the subsequent operations the negative sign, to which the pre- viously positive sign of the absolute term now changes, is preserved. The second root will be found to be 23-2295212. So long as the two roots remain together, a guide to the proper figure of the root may be obtained by dividing twice the last coefficient by the second last, or the se- cond last by twice the third last. The reason of this is, that the proposed equation approximates now to the quadratic formed by the last three terms in each transformed equation, just as in previous cases, and in Newton's method it approximated to the simple equation formed by the last two terms, this quadratic having the two nearly equal roots for its roots ; and when the two roots of the equation ax- + bx + e = -2c -b are nearly equal, either of them is given approximately by — — - or — . Thus, in the above example, the number 3 is suggested by , and the number 2 bv . In this way we can generally, at the first attempt, find the two integers between which the pair of roots lies. We shall have, also, an indication of the separation of the roots by observing when the numbers suggested in this way by the last three 2e b coefficients become different, i.e. when — suggests a different number from — . b °° 2a 3. Calculate to three decimal places each of the roots lying between 4 and 5 of the equation a; 4 + 8a; 3 - 70a 2 - 144a; + 936 = 0. Am. 4-242; 4-246. 4. Find the two roots between 2 and 3 of the equation 64a; 3 - 592a; 2 + 1649a; - 1445 = 0. Ana. The roots are both = 2-125. Lagrange's Method of Approximation. 223 Here we find that the two roots are not separated at the third decimal place. When we diminish by 5 the absolute term vanishes, showing that 2*125 is a root ; and proceeding with this diminution the second last coefficient also vanishes. Hence 2-125 is a double root. When an equation contains more than two nearly equal roots, they can all be found by Horner's process in a manner similar to that now explained. Such cases are, however, of rare occurrence in practice. The principles already laid down will be a sufficient guide to the student in all cases of the kind. 105. I^agrange's Method of Approximation. — Lagrange has given a method of expressing the root of a numerical equa- tion in the form of a continued fraction. As this method is, for practical purposes, much inferior to that of Horner, we shall content ourselves with a brief account of it. Let the equation f{x) = have one root, and only one root, between the two consecutive integers a and a + 1. Substitute a + - for x in the proposed equation. The transformed equation y in y has one positive root. Let this be determined by trial to lie between the integers b and b + 1. Transform the equation in y by the substitution y = b -. The positive root of the equation in s is found by trial to lie between c and c + 1. Con- tinuing this process, an approximation to the root is obtained in the form of a continued fraction, as follows : — 1 a + 6 + 1 c + 1 Examples. 1 . Find in the form of a continued fraction the positive root of the equation x z - 2x - 5 = 0. The root lies between 2 and 3. To make the transformation x = 2 + -, we first employ the process of Art. 33, y diminishing the roots by 2. We then find the equation whose roots are the reci- procals of the roots of the transformed. 224 Solution of Numerical Equations. The equation in y i< in this way found to he y 3 - 10y 2 -6y - 1 = 0. This has a root between 10 and 11. Make now the substitution y = 10 + -. The equation in e is 61z 3 - 94c 2 - 20z - 1 = 0. This has a root between 1 and 2. Take 2 = 1 + -. u The equation in u is 5iu 3 + 25m 2 - 89m -61=0, which has a root between 1 and 2 ; and so on. We have, therefore, the following expression for the root x = 2 + 2. Find in the form of a continued fraction the positive root of x z - 6x - 13 = 0. Ans. 3 + i + iT.. 106. Numerical Solution of the Biquadratic. — It is proper, before closing the subject of the solution of numerical equations, to illustrate the practical uses which may be made of the methods of solution of Chap. VI. Although, as before observed, the numerical solution of equations is in general best effected by the methods of the present Chapter, there are certain cases in which it is convenient to employ the methods of Chap. VI. for the resolution of the biquadratic. When a bi- quadratic equation leads to a reducing cubic which has a com- mensurable root, this root can be readily found, and the solution of the biquadratic completed. We proceed to solve a few examples of this kind, using Descartes' method (Art. 64), which will usually be found the most convenient in practice. Examples. 225 Examples. 1 . Eesolve the quartic x i - 6x 3 + 3x 2 + 22a; - 6 into quadratic factors. Making the assumption of Art. 64, we easily obtain p+p'=-3, q + q' + ±pp'= 3, pq' + p'q = 11, qq'=-6. 1.1,. Also

is A <, 1H 225 « Multiplying the roots by 4, we have, if 4<^> = ^, * 3 - UK -450 = 0. By the Method of Divisors this is easily found to have a root — 6 ; hence 3 . . = --, giving PP=2, q + (J =-5. From these, combined with the preceding equations, we get p = -2, p'=-l, q=l, q'=-6. "When the values of q and q are found, the equation giving the value of pq' +p'q determines which value of q goes with p, and which with p', in the quadratic factors. The quartic is resolved, therefore, into the factors {x-- ix + 1) (z 2 -2x -6). By means of the other two values of

3 - 195 - 475 = 0, which is found to have a root = — 5. Am. f{x) = (x 2 - 2x - 3) (z 2 - 6x - 21). Q 226 Solution of Numerical Equations. 3. Resolve into factors f(x) = x* - 17* 2 - 20* - 6. The reducing cubic is found to be . 217 3185 „ 4d> 3 + = ; 9 12 v 21G or, multiplying the roots by 6, 4i 3 - 651j5 + 3185 = 0. 7 This has a root = 7 ; hence = -. 6 Am. f(x) s (* 2 + 4* + 2) (* 2 - ix - 3). 4. Resolve into factors /(*) = x i - 6* s - 9* 2 + 66* - 22. The reducing cubic is „ 335 897 „ v 4 V 8 3 hence

x = 0. Again, the result of eliminating x, y, z from the equations a^ + b x y + CiZ = 0, a 2 x + b 2 y + c 2 z = 0, a z x + b z y + c 3 z = 0, is, as the student will readily perceive by solving from two of the equations and substituting in the third, ctib 2 c 3 - (iihcn + Ozb z Ci - OzbiCs + a 3 biC z - a 3 b 2 Ci = ; (2) and this function differs from (1) above written only in having three of its terms negative, instead of having the six terms posi- tive. Elementary Notions and Definitions. 231 Similarly the elimination of four variables from four linear equations gives rise to a function of the sixteen quantities Oi, 61, dt c h, «2s h, &C-, which differs from the function above represented by (abed) only in having twelve of its terms negative. Expressions of the kind here described are called Determi- nants* The notation by which they are usually represented was first employed by Cauchy, and possesses many advantages in the treatment of these expressions. The quantities of which the function consists are arranged in a square between two vertical lines. For example, the notation #2 #2 represents the determinant aib 2 - a 2 b u Similarly, the expression on the left-hand side of equation (2) is represented by the notation 61 Ci h Ci b 3 c 3 I And, in general, the determinant of the n 3 quantities «i, b h d . . . h> 02, b 2 , &c, is represented by 0>n On C/i 'n (3) By taking the n letters in alphabetical order, and assigning to them suffixes corresponding to the n(n - l)(n - 2) . . . 3. 2. 1 permutations of the numbers 1, 2, 3, . . . n, all the terms of the * See Note C at the end of the volume. 232 Determinants. determinant can be written down. Half of the terms must receive positive and half negative signs. In the next Article the rule will be explained by which the positive and negative terms are distinguished. The individual letters a h b lf c h . . . a,, . . . &c, of which a determinant is composed, are called constituents, and by some writers elements. Any series of constituents such as a 1} b u c u . . . /,, arranged horizontally, form a rote of the determinant ; and any series such as o&i, a 2 , a 3 , . . . a n , arranged vertically, form a column. The term line will sometimes be used to express a row or column indifferently. 108. Rule with regard to Signs. — It is evident from the preceding Article that each term of the determinant will, since it contains all the letters, contain one constituent (and only one) from every column ; and will also, since the suffixes in each term comprise all the numbers, contain one constituent (and only one) from every row. We may thus regard the square array (3) of Art. 107 as the symbolical representation of a function con- sisting in general of n(n-l)(n - 2) ... 3. 2. 1 terms, comprising all possible products which can be formed by taking one con- stituent, and one only, from each row ; and one constituent, and one only, from each column, All that is required to give perfect definiteness to the function is to fix the sign to be attached to any particular term. For this purpose the following two rules are to be observed : — (1). The term aib 2 c 3 . . . Informed by the constituent* situated m the diagonal drawn from the left-hand top corner to the right- hand bottom corner, is posit ire. This is called the leading or principal term. In it the suffixes and letters both occur in their natural order ; and from it the sign of any other term is obtained by means of the following rule. (2) Any interchange of two sujfi.res, the letters retaining their order, alters the sign of the term. Rule ivith regard to Signs. 233 This rule may be otherwise expressed thus : — Any interchange of txco letters, the suffixes retaining their order, alters the sign of a term. For if two letters be interchanged, and the two corre- sponding constituents interchanged, the process is equivalent to an interchange of suffixes. If, for example, in rti6 2 c 3 G^5 the letters b and e be interchanged, we get ch. e-nCzdJ)^ which is equal to (iibsCidiCi, and this is derived from the given term by an in- terchange of the suffixes 2 and 5. In applying this rule it is evident that an even number of interchanges will not alter the sign of a term, and that an odd number will. Examples. 1. What sign is to be attached to the term azb±cidf,e\ in the determinant of the 5th order ? The question is, How many interchanges will change the order 123-15 into 34251 ? Here, when 3 is interchanged with 2, and afterwards with 1, it comes into the lead- ing place, the order becoming 31245. Again, the interchange in 31245 of 4 with 2, and afterwards with 1, presents the order 34125. The interchange of 2 with 1 gives the order 34215 ; and finally the interchange of 5 with 1 gives the required order 34251. Thus there are in all six interchanges ; and therefore the required sign is positive. The general mode of proceeding may plainly be stated as follows : — Take the figure which stands first in the required order, and move it from its place in the natural order 1234 . . . into the leading place, counting one displacement for each figure passed over. Take then the figure which stands second in the required order, and move it from its place in the natural order into the second place ; and so on. If the number of displacements in this process be even, the sign is positive ; if it be odd, the sign is negative. 2. "What sign is to be attached to the term aihc&d- 3 eifig^ in the determinant of the 7th order ? Here two displacements bring 3 to the leading place ; five displacements then bring 7 to the second place ; four then bring 6 to the third place ; three then bring 5 to the fourth place ; the figure 1 is in its place ; and finally, one displacement brings 4 into the sixth place. Thus there are in all fifteen displacements ; and the required sign is therefore negative. 3. Write down all the terms of the determinant «2 fa Ci do «3 #3 c 3 d-A <*i hi rj di 234 Determinants. The six permutation in which tlio figure 1 occurs first are 1234, 1243, 1321, 1342, 1423, 1432. The six corresponding terms are, as the student will easily see hy applying the Rule (2), as in the previous examples, 2<'z:;<; Determinants. number of displacements of rows necessary to bring it into the leading diagonal in the first case is the same as the number of displacements of columns necessary in the second case. ExAJlH.i:. rti h x C^_ d l rt"2 bi h cz d 3 oil b± Ci di «l «2 «3 ai h h h h c\ c-i cz a d\ d% dz di Here the sign of any term, e.g. a 2 hci d 3 , is the same in both determinants. For three displacements of rows are required to bring this term into the leading position in the first determinant ; and the same number of displacements of columns is re- quired to bring the same constituents into the leading position in the second deter- minant. 112. Prop. IV. — If every constituent in any line he multiplied by the same factor, the determinant is multiplied by that factor. For every term of the determinant must contain one, and only one, constituent from any row or any column. Cor. 1. If the constituents in any line differ only by the same factor from the constituents in any parallel line, the de- terminant vanishes. Cor. 2. If the signs of all the constituents in any line be changed, the sign of the determinant is changed. For this is equivalent to multiplying by the factor - 1. Examples. ka\ bi Cl «i h ci 1ca% b% c% mh 0,1 b% c% ka 3 b 3 Cz «z bz cz ai ma\ 02 a\ ai a> /3i mj8i 02 = m J3i 0, & 71 myx 72 71 7i 72 Examples. 237 3. Show that the following determinant vanishes : — 3 15 2 6 5 7 3 9 1 4 15 21 9 When the constituents of the last row are divided hy 3, they become identical with those of the second row. 4. Prove the identity be a a 2 1 a 2 « 3 ca b b°~ = 1 b 2 P ab e c 2 1 c 2 i — 1) differences which can be formed with these n quantities. 113. Minor Determinant*. Definitions. — When in a determinant any number of rows, and the same number of columns, are suppressed, the determinant formed by the remain- ing constituents (maintaining their relative positions) is called a minor determinant. If one row and one column only be suppressed, the corre- sponding minor is called a first minor. If two rows and two columns be suppressed, the minor is called a second minor ; and so on. The suppressed rows and columns have common con- stituents forming a determinant ; and the minor which remains is said to be complementary to this determinant. The minor complementary to the leading constituent a* is called the leading first minor, and its leading first minor again is the leading second minor of the original determinant. It is usual to denote a determinant in general by A. We shall denote by A a the first minor obtained by suppressing in A the row and column which contain any constituent a ; by A 0; p the second minor obtained by suppressing the two rows and two columns which contain a and j3 ; and so on. Thus A ai repre- sents the leading first minor, and A ai ,j 2 or ^ a „,b 1 the leading second minor. The determinant A, formed by the constituents a u b l} c h &c, is often denoted for brevity by placing the leading term within brackets, as follows: A = («i bzfy . . . . /„). The notation 2 ± a x b 2 c 3 . . . /„ is also used to represent A ; this expressing its constitution as consisting of the sum of a number of terms (with their proper signs attached) formed by taking all possible per- mutations of the n suffixes. 114. Development of Determinants. — Since every term of any determinant contains one, and only one, constituent from each row and from each column, it follows that A is a linear ami homogeneous function of the constituents of any one rote or any one 240 Determinant*. column. Thus we may write A = ctiA\ + a 2 A, + a 3 A 6 + etc. A - b,B { + b 2 B 2 + b 3 B> + &c. ; or, again, A = a x A x + b x B x + ^Ci + &c. A = a 2 A 2 + b 2 B : + CaC 2 + &C. The student, on referring to Ex. 3, Art. 108, will observe that the determinant of the fourth order there written at length is constituted in the way here described, namely A =a. 0\ C\ (h b 3 c 3 d 3 + a. bi r, d 4 tu c t d t b 3 q , where each term is the product of two complementary determinants (see Art. 113). To prove this, we observe that every term of the determinant must contain one constituent from the column a and one from the column b. Suppose a term to contain the factor a p b q , there must then (interchanging p and q) be another term containing the factor - a q b p ; hence, the determinant can be expanded in the form 2 (a p b q ) A p , q ; and A p , q is plainly the sum of all the terms which can be obtained by permuting in every possible way the n - 2 suffixes of the letters c, d, e, &c., viz., ± A p , q , the sign being determined in any particular instance by the rule of Art. 108. This reasoning can easily be extended to the case where any number p of columns are taken, and all possible minors formed by taking p rows of these columns. Each minor is then multiplied by the complementary minor, and the determinant expressed as the sum of all such products with their proper signs. Examples. 1. Expand the determinant {a\bic 3 d^ in terms of the minors of the second order formed from the first two columns. Employing the bracket notation, we can write down the result as follows : — (aih)(c3fk) - (aibz)(c2di)+ {aih){c 2 d 3 ) + {a 2 b 3 ){cidi) - fabi) (ntf,) + (a 3 bi)(cidn) ; where the sign to be attached to any product is determined by moving the two rows involved in the first factor into the positions of first and second row. Thus, for example, since three displacements are required to move the second and fourth rows into these positions, the sign of the product {a 2 bi){c\d 3 ) is negative. Development of Determinants. 245 2. Expand similarly the determinant {aib-ic^dies). Am. (rtii 2 ) fadies) - («ih) (^i^s) + {(iifa) (c2d 3 e 5 ) - ( 72 «S h c 3 t»3 03 73 03 03 73 This appears by expanding the determinant in terms of the minors formed from the first three columns, for it is evident that all these minors vanish (having one row at least of ciphers) except one, viz. (aifo's). In general it appears in the same way that if a determinant of the 1ni th order contains in any position a square of m 2 ciphers, it can he expressed as the product of tWO dete rmin ants of the m ih Order. 4. Expand the determinant a h 9 \ A.' h b f M / 9 f c V V X M- V \' m' V in powers of a, (3, 7, where a = fxv — fi'v, f$ = v\' — v'\, 7 = Ayu.' — \'jj.. Arts, (la 2 + bf5 l + cy 2 + 2/0-y + 2ffya + 2ha$. 5. Verify the development of the present Article by showing that it gives in the general case the proper number of terms. Consider the first r columns of a determinant of the n th order. The number of minors formed from these is equal to the number of combinations of n things taken r together. This number multiplied by 1 . 2 . 3 . . . r (the number of terms in each minor), and 1.2.3 . . . n — r (the number of terms in each complementary minor), will be found to give 1.2.3. . . n, viz. the number of terms in the determinant. 116. Development of a Determinant in Products of the leading Constituents. — In the present and next fol- 240 DeteriiiiiKoils. lowiug Articles will bo explained two additional modes of deve- lopment which will be found useful in the expansion of certain determinants of special form. The application which follows to the determinant of the fourth order will be sufficient to explain how any determinant may be expanded in products of the lead- ing constituents — It is required to expand the determinant A h As c\ (72 B t'2 d> a 3 b 3 C di «4 bi Ci D according to the products of A, B, C, B. In order to give prominence to the lead- ing constituents we have here replaced «i, b 2 , c 3 , di by A, B, C, B. When the ex- pansion is effected it is plain that the result must be of the form A = Ao + 2\A + IK'AB + ABCB, where Ao consists of all the terms in which no leading constituent occurs ; ~S.\A is the sum of all the terms in which one only of these constituents occurs ; ~S,K'AB is the sum of all in which the product of a pair of the leading constituents is found ; and ABCB, the leading term, is the product of all these constituents. It will be observed that the expansion here written contains no terms of the form \"ABL\ and it is evident in general that the expanded determinant can contain no terms in which products of all the leading constituents but one occur, since the coefficient of any such product is the remaining diagonal constituent. It only remains to see what is the form of Ao, and of the undetermined coefficients A, /x, . . . A.', fi,... &c. Putting A, B, C, B all equal to zero in the identity above written, we have A„ = Cl ii cj (7 2 CjJ «3 h 10 rf 3 | tf 2 ez rfi ex 1 I fli +^c +^u + BC +£D + CD \ Ci \ J 4 bz rt 4 «3 1 «2 +^ + ^^ A determinant whose leading constituents all vanish has been called zero-axial. The result just obtained may be stated as follows :—Any determinant may be ex- panded in products of the leading constituents, the coefficient of every product in the result being a zero-axial determinant. 117. Expansion of a Determinant by Products in Pairs of the Constituents of a Row and Column. — In what follows we take the first row and first column as those in terms of which the expansion is required. This is plainly suf- ficient, since any other row and column may be brought by displacements into these positions. It will be found convenient to write the determinant under consideration in the form «o o j8 y . a a x 6i C\ |3' a 2 b 2 c 2 . y (iz b z c 3 ■ . I 3 Determinants. Let this be denoted by A', and its leading first minor { + a 3 ) An + (a 3 + a 3 ) A 3 + &c. = (ij.Ai + «oAo + a 3 A 3 + . . &c. + a x Ai + a 2 An + a 3 A 3 + &g. ; or, h Cn + a 3 h = m ma 3 + nb 3 a 3 b 3 h + n f Here, to obtain the second determinant, we subtract the second column from each of the following ones. In the reduced determinant, subtracting the first row from each of the following, we find A = - !'■ ■2:- •r 2 S 2 yt + z 2- X 2 y 2 + z 2 - x 2 2y 2 x 2 -y 2 -z- ~2f- = (y 2 + s 2 - * 2 ) 2 - 4/ z- = (y 2 + s 2 - x 2 + 2yz) {y* + z 2 - a 2 - 2yz) = {(y + z) 2 -x 2 }{(y-z) 2 -x 2 } = - (x + y + z)(y + z - x) (z + x - y) {x + y - z). 10. Prove the identity (b + c) 2 a 2 a 2 I As b 2 (c + a) 2 b 2 = 2abc{a + b + cf. c 2 c 2 (a + b) 2 j Subtracting the last column from each of the others, (a + b + c) 2 may be taken out as a factor. Calling the remaining determinant A', and subtracting in it the suni of the first two rows from the last, we have \b + c — a a 2 A'=| c+a-b b 2 e — a-h c — a-b (a + b) 2 b+ c — a a 2 c + a-b b 2 -2b -2a 2ab a(b + c-a) « 2 b (c + a-b) b 2 -2( - 2ab 2ab 254 Determinants. A' Adding the Inst column to each of the others, we obtain a (b + c) a 2 a 2 b 2 b[e + a) b 2 2ab a[b + c) a 2 b 2 b(c + a) = 2ab b + c a b c + a 1abc(a + b + c). Eence, A = A> + b + c)-= 2abc{a + b + c) 3 . 1 1 . Prove the identity 1 1 1 a fi 7 =(iB-7)(y-a)(a-j8)(a + /3 + 7). a 3 j8 s 7 3 Subtracting the first column from each of the others, - a and y - a become factors. In the reduced determinant, subtract the first row multiplied by a- from the second row. 12. Resolve into simple factors the determinant 1111 A = Proceeding as in Ex. 1 1, we easily find that (/8 - a) (y - a) (5 - o) is a factor, and that the reduced determinant is 1 1 1 7 5 JB= T 8 2 P 7 4 S* 5+a 8 3 + 5 2 a+5o 2 + o 3 7 +a 7 3 + 7 ! a -f- 7a 2 + a 3 Subtracting the first column from each of the others, (7 — j3) (8 — £) comes out as a factor, and the remaining factor is easily found to be (8 - 7) (a + $ + 7 + 8). Hence, finally, A = -(jB-7)(a-8)(7-o)(/8-8)(a-j3)(y-8)(a+j8 + 7+8). 13. Resolve into linear factors the determinant a b Multiply the second column by u, and the third by or ; and add to the first. The factor a + 2 b + ojc). Multiplication of Determinants. 255 14. Kesolve into linear factors the deteri ninant a b c d b a d c A = c d a b d c b a The result is as follows : — A = - (a+b + c + d)(b + c-a-d)(e + a-b-d)(a + b-c-d), since each of the factors here written is a factor of the determinant ; for example, a + b — e — d is shown to be a factor by adding the second column to the first, and subtracting the third and fourth. By comparing the sign of a 4 it appears that the negative sign must be attached to the product. It may be observed that the determinant of Ex. 9 is a particular case of the de- terminant here considered, viz., that obtained by putting a = 0, as will appear by comparing the equivalent forms of Ex. 9, Art. 114. 121. Multiplication of Determinants. — Prop. VIII. — The product of two determinants of any order is itself a determinant of the same order. We shall prove this for two determinants of the third order. The student will observe, from the nature of the proof, that it is equally applicable in general. "We propose to show that the product of the two determinants (tfi& 2 e 3 ), (ctij3 2 y 3 ) is «iai + &i/3i + Ciyi «ia 2 + Jij3 2 + C172 «ia 3 + 5i/3 3 + Ciy 3 # 2 ai + & 2 j3i + C 2 ji a 2 a 2 + 6 2 /3 2 + C 2 7 2 a 2 a 3 + 6 2 /3 3 + C 2 y 3 «3<*i + 6 3 j3i + C 3 y l a 3 a 2 + & s /3 3 + c 3 y, a 3 a 3 + b 3 (3 3 + c 3 y 3 whose constituents are the sums of the products of the con- stituents in any row of (a x b 2 c 3 ) by the corresponding constituents in any row of (ai/3 2 y 3 ). Since each column consists of the sum of three terms, this determinant can be expanded into the sum of 27 others (Art. 118). Now it will be observed that when any one of these is written down, a common factor can be taken off each column ; and that several of the partial determinants will, when these factors are removed, have two (or more) columns identical. The determinants which do not vanish in this way can be easily selected. Taking, for example, the first vertical line of the first 25G Determinants. 01 h Ci a 2 h C2 + aij 2 (5i (h b 3 c 3 j as /3 3 72 S 2 j s2 260 Determinants. and, performing on these a process similar to that employed in multiplying two determinants, we obtain the determinant tfia! + bi^i + Ciyi+ diSi rt,a 2 + £>i/3 2 + Ciy 2 + d x d 2 a 2 a x + £> 2 /3i + c.yi + f/ 2 Si a 2 a 2 + bjfi* + e 2 y 2 + d 2 $ 2 The value of this is easily found to be (a l b z )(a l (5 2 ) + («iC 2 )(ai7 2 ) + (M 2 )(aiS 2 ) + (61 ossible determinants which can be formed from one array {by taking a number of columns equal to the number of rows) multiplied by the corresponding determinants formed from the other array. The student will have no difficulty in extending this proof to any two arrays of the kind here treated. (2). When the number of rotes exceeds the number of columns the resulting determinant vanishes. Take, for example, the two arrays, a 2, ((z, &c., and must therefore vanish. By the aid of these relations we can write down the solution of a system of linear equations. The following application to the case of three variables is sufficient to explain the general process. Let the equations be a,\X + b\y + d% = m u a 2 x + b 2 y + c-,z = m., a 3 x + b 3 y + c 3 z = m s . Multiply the first equation by A ly the second by A 2 , and the third by A 3 ; and add. The coefficients of y and z vanish, in virtue of the relations above proved ; and we obtain or (aiAi + a 2 A 2 + a 3 A 3 ) x = m t | Wi b,. A Y + Ci n % A%+ m 3 A 3 , Ax = m 2 b> r = ' cf 2 x + b 2 y + c 2 z + d 2 ic = a x - b y - '• s + 4 , or, A, B t C 4 D t A* [ ' The first three of these equations express the ratios of the four variables in terms of the coefficients in the three given equations. And, in general, the variables are proportional to the coefficients in the expansion of A of the constituents of the n th row supposed added to the n — 1 rows resulting from the given equations. We can now express the condition that n linear homogeueous equations should be consistent with one another ; for example, that the equation (2) should, when A = 0, be consistent with the equations (1). We have only to substitute in (2) the ratios de- rived from (1), when we obtain aiAi + biBi + c 4 d + diDi = 0, or A = 0. The same thing appears from the equations (3) , for if A = 0, and if the variables do not all vanish, A must vanish. What has been proved may be expressed as follows: — The result of eliminating the variables between n linear homogeneous equations in n variables is the vanishing of the determinant formed by the coefficients of the given equations. 126. Reciprocal Determinants. — The quantities Ai, B u d . . . Ao, B,, &c. (Art. 114), which occur in the ex- pansion of a determinant (/. e. the first minors with their proper signs), may be called inverse constituents; and the determinant formed with them the inverse or reciprocal determinant. We Reciprocal Determinants. 265 proceed to prove certain useful relations connecting the two de- terminants. (1). To express the reciprocal in terms of the given determinant. Let the reciprocal of A be denoted by A', and multiply the two determinants A = All the constituents of the resulting determinant except those in the diagonal vanish (Art. 124) ; and the result is tfl 6, Ci #2 h C-2 , A'- «3 h c 3 Ar B> e, A, B z C 2 A, B 3 c 3 AA' = whence A A A A' = A 2 . = A 3 The process here employed in the particular case of two de- terminants of the third order is equally applicable in general ; giving A A' = A n , or A' = A" -1 . Hence the reciprocal determinant is equal to the (n - l) th power of the given determinant. (2). To express any minor of the reciprocal determinant in terms of the original constituents. "We take, for example, the determinant of the fourth order, and proceed to express the first minors of its reciprocal. Multi- plying the two determinants on the left-hand side of the follow- ing equation, and employing the identical equations of Art. 124, we obtain ai by Ci d t a-t b-i c. d z a s h c 3 d z ai b t c^ di 10 A 2 B 2 C, A .0.3 X>3 G3 -I/3 A, B, C, A ffi (h A a 3 A a, A 266 Determinants. whence B 2 B 3 C, B t C 3 B 3 = tf,A 3 or (^C 3 A)=«iA 2 , thus expressing the first minor of A' complementary to A v . Again, to express the second minors of a', we have, by an exactly similar process, th bi Ci d, 10 a, ^ (h h Co d 2 10 on cling minor of the original determinant A multiplied by the (m - l) th fioicer of A. For example, in the case of a determinant A of the fifth order a minor of the third order is expressed in the following manner : — as the student can easily verify by a method exactly similar to the proof above given. If the original determinant A vanishes, it is plain that not only the reciprocal determinant itself, but also all its minors of any order vanish. The vanishing of the minors of the second order may be expressed in the following form : — When a deter- minant vanishes, the constituents of any row of its reciprocal are Symmetrical Determinants. 267 proportional to those of any other row, and the constituents of any column proportional to those of any other column. 127. Symmetrical Determinants. — Two constituents of a determinant are said to be conjugate when one occupies with reference to the leading constituent the same position in the rows as the other does in the columns. For example, d 2 and hi are conjugates, one occupying the fourth place in the second row, and the other the fourth place in the second column. Each of the leading constituents is its own conjugate. Any two con- jugate constituents are situated in a line perpendicular to the principal diagonal, and at equal distances from it on opposite sides. A symmetrical determinant is one in which every two con- jugate constituents are equal to each other. For examples of such determinants the student may refer to Art. 114, Exs. 2, 9, 10, and Art. 115, Ex. 4. In a symmetrical determinant the first minors complementary to any two conjugate constituents are equal, since they differ only by an interchange of rows and columns. The correspond- ing inverse constituents are also equal, the signs to be attached to the minors being the same in both cases. It follows that the reciprocal of a symmetrical determinant is itself symmetrical. The leading minors are plainly all symmetrical determi- nants. The mode of expansion of Art. 117 is especially useful in the case of symmetrical determinants, as will appear from the examples which follow. Examples. 1. Form the reciprocal of the symmetrical determinant a h g A= h b f it f o Using the capital letters to denote the reciprocal constituent?, as explained in Art. 114, so that A may he expanded in any one of the forms a A + hH + gG, :V,x DflcfniiiKiiiis. fill + hB + JF, gG + /F + cC, we may write the reciprocal determinant A' as follows : — A' .; n g II B F G F C 2. Form similarly the reciprocal of a h g h b f bc-f- fg-ck hf-bg fg - ch ca - g- gh - af hf— bg gh — af ab — h- A = f c m n Using a notation similar to that of the preceding example, so that A may be ex- panded indifferently in any of the forms aA+ hH + gG + IL, hH + bB +/F + mM, &c, the reciprocal determinant A' is obtained by replacing in A the constituents by the corresponding capital letters. The student will find no difficulty in writing out, if necessary, the expanded form of any of the reciprocal constituents ; for example, F is the minor complementary to/ with its proper sign (the negative sign in this case), and F is therefore obtained from the expansion of I f 3. Expand the determinant A of Ex. 10, Art. 114, by the method of Art. 117. Bringing the last row and last column into the position of first row and first column, and using the notation of Ex. 1 for the inverse constituents of the leading minor, the result can be written down at once in the form - A = A\* + B/j? + 6V + 2Ffi» + 2Gv\ + 2E\fi. Since a determinant is unaltered when both rows and columns are written in re- verse order, if the expansion of a determinant be required in terms of the last row and last column (as in the present example), it is not necessary to move them in the first instance into the position of first row and first column. The expansion can be written down from the determinant as it stands, replacing in the rule of Art. 117 the leading constituent and its minor by the last diagonal constituent and its com- plementary minor. 4. Expand the determinant A of the above Ex. 2, in terms of the last row and column, by the method of Art. 117. Attending to the remark at the end of the preceding example, and using Skeiv- Symmetric and Skezv Determinants. 269 A, B, C, F, G, H, to represent the same quantities as in Exs. 1 and 3, the result may be written down as follows : — A = d 9 h 9 b f f c AP - B»i- - Cn- - 2Fmn - 2Gnl - 2m»i. "When a symmetrical determinant of any order is bordered symmetrically (i. e. by the same constituents horizontally and vertically) the result is plainly a symmetri- cal determinant of the next higher order. The result of Art. 117 shows in general that the expansion of the bordered determinant consists of the original determinant multiplied by the constituent common to the added row and column, together with a homogeneous function of the second degree of the remaining added constituents. 5. Expand the determinant h 9 I a b f m £ f c n 7 m n d S This is the determinant of Ex. 2 bordered symmetrically, the common consti- tuent of the added lines being zero. The result is plainly a homogeneous function of the second degree of a, ft, y, 8 ; and, by aid of the notation of Ex. 2, may be written down at once in the form - A = Aa? + Bft- + Cy- + D5- + 2Ffty + 2Gya + 2Haft + 2LaS + 2M(38 + 2NyS. 6. Prove by means of the Proposition of Art. 121, that the square of any deter- minant is a symmetrical determinant. 128. Skew-Symmetric and Skew Determinants. - A skew-symmetric determinant is one in which every constituent is equal to its conjugate with sign changed. Since each leading constituent is its own conjugate, it follows that in a skew-sym- metric determinant all the leading diagonal constituents are zero. A determinant in which all except the leading constituents are equal to their conjugates with sign changed is called a skeic determinant. Thus, while a skew-symmetric determinant is 270 Determinants. zero-axial, a skew determinant is not. The calculation of a skew determinant may plainly be reduced to that of skew- symmetric determinants by the method of Art. 116. The remainder of this Article will be ocoupied with the proof of oertain useful properties of skew- symmetric determinants. (1). A skew-symmetric determinant of odd order vanishes. For any skew-symmetric determinant A is unaltered by changing the columns into rows, and then changing the signs of all the rows. But when the order of the determinant is odd, this process ought to change the sign of A ; hence A must in this case vanish. For example, a b A = - a -b -c -0. (2). The reciprocal I of a skeio-symmetric determinant of the n th order is a symmetric determinant when n is odd, and a skew-symmetric determinant when n is even. In any skew-symmetric determinant the minors correspond- ing to a pair of conjugate constituents differ by an interchange of rows and columns, and by the signs of all the constituents. Hence the two minors are equal when their order is even, namely when n is odd ; and equal with opposite signs when n is even. In the former case, therefore, the reciprocal determinant is sjrmmetric ; and in the latter case it is skew-symmetric, its leading diagonal constituents all being skew-symmetric deter- minants of odd order. (3). A skew-symmetric determinant of even order is a perfect square. This follows from the principles established in Art. 126. Take, for example, the determinant of the fourth order, a b c -a d e - b -d / -c -e -f Examples. 271 and let the inverse constituents forming its reciprocal be de- noted by A 1} £i, . . . A 2 , &c. "We have then, by (2), Art. 126, / A X B, - A,By = A /* / 2 A. Now Ai, and B 2 , being skew-symmetric determinants of odd order, vanish ; and A 2 = - B x , since these are conjugate minors ; hence /*£ = A-r, which proves that A is a perfect square. Similarly, for the determinant of the sixth order, it is proved that the product of A by a skew -symmetric determinant of the fourth order is a perfect square ; and since the latter determi- nant has been just proved to be a perfect square, it follows that A is also. By an exactly similar process, assuming the truth of the Proposition for the determinant of the sixth order, it follows for one of the eight ; and so on. Examples. 1. Verify the following expression for the skew -symmetric determinant of the fourth order : — a b c — a Ode -b -d / -c -e -f 2. Expand in powers of x the skew determinant b (af— be + cd)-. x — a -b — c -f WTien the expansion of Art 116 is employed to calculate a skew determinant, it is to be observed that the determinants of odd order in the expansion all vanish, and those of even order may be expressed as squares. Here the coefficients of the odd powers of x plainly vanish, and the result takes the form A = x* + {a- + b- + c 2 + d z + e 2 +p) x- + (af- be + cd) 2 . 272 Di'lcrmiiKiiits. 3. Expand the skew determinant A a b c d -a B e f 9 -b -e C h i • -c -f - h B J -d -g -i - -J E The result may be written in the form ABODE + 2r ABC + 2( fj- fi + ghfA, where the first 2 includes ten terms similar to the one here written, and the second 2 includes five terms. The terms involving the products in pairs of the leading con- stituents vanish, as also the term not involving these quantities. 4. The square of any determinant of even order can he expressed as a skew- symmetric determinant. The following method of proof is applicable in general. The square of («i b 2 czdi) is obtained by multiplying the two following determi- nants : — 01 bx C\ di -h «i -0*1 Cl 02 h C2 d 2 -b 2 02 -d 2 c 2 a-i bz c% d 3 -b 3 03 -d 3 C 3 di bi a di -bi di -di a and the product of these is 0, - (aib 2 ) - (cid 2 ), - (01*3) - (cidz), - (0-1*4) - (cidi), (01*2) + (cid 2 ), 0, - (a 2 b 3 ) - (e 2 d 3 ), - (a 2 h) - (c 2 di), (aib 3 ) + (cidz), {a 2 b 3 ) + (c 2 d 3 ), 0, - (a 3 £ 4 ) - (c 3 di), (aih) + (cidi), (a 2 bi) + (c 2 di), (azh) + (c 3 di), 0, which is a skew-symmetric determinant. 5. Form the reciprocal of a skew-symmetric determinant of the third order. Using for A the form in (1) of the present Article, the result is easily found to be the symmetric determinant - be -be b 2 -ab ■ ah Theorem. 273 6. Form the reciprocal of the skew-symmetric determinant A of the fourth order in Ex. 1. Kepresenting hy (p the function af—be + ed whose square is equal to A, and by A' the required reciprocal, we easily find A' The value of this skew- symmetric determinant may be written down by aid of the result of Ex. 1. It is thus immediately verified that A' = (af— be + cd) % (£* = A 3 . 7. Form the reciprocal of the skew- symmetric determinant A of the fifth order, obtained by making the leading coefficients all vanish in the determinant of Ex. 3. Since the reciprocal is a symmetric determinant (see (2), Art. 128), and since also it must be such that the constituents of any line are proportional to those of any parallel line (Art. 126), it appears that the required determinant must be of the form Z b

i 5 3 % 53, i, li2/3 + 2i& + —, To, the determinant reduces to the form a- b- c- d- = (&- y)(a- S)(y- a)(p-8)(a- 0)(y-8). 2 7 2 +a 2 8 2 07+08 1 7 2 a 2 + 2 8 2 ya + 05 1 o-0 2 + 7 2 8 2 a0 + 78 1 Add the last column multiplied by 2a078 to the first. The determinant be- comes then of the form of Ex. 9, Art. 112. Miscellaneous Examples. 277 6. Prove (0 + 7 - a - 8) 4 (£ + y - a - 8) 2 1 (7 + a - - 5) 4 (7 + a - £ - 8) 2 1 (a + 0-7-8) 1 (a + j3-7-5) 3 1 7. Prove a Z> a.r + £ b c bx + c «£ + 6 fo + e s64(j3-7)(a-8)(7-a)(j8-8) (a-/8)( 7 -S). = - (ae - b 2 )(ax 2 + 2ta + c) . Subtract from the third row the second row plus the first multiplied by x. 8. Prove similarly a b c ax 1 + 2bx + c b c d bx 2 + lex + d c d e ex 2 + 2dx + e ax 2 + 2bz + c bx 2 + 2cx + d ex 2 + Idx + e a b c I J c tf ! («z 4 + ibx 3 + 6cx 2 + idx + e). c d e 9. Given prove the identity fi (x) = aix 3 + Zb x x 2 + Zmx + d h fz(x) = a%x z + Zfax 2 + Zc-ix + d 1 -2/3' P 2 y- y 1 1 -27 7"- Here and these two determinants may be resolved as in Ex. 9, Art. 112. 16. Eesolve into factors the determinant (a - a') 3 (0 - ay (7-«', :i (a - 0V (a - 7 T (jB - 0') ;i (0 - 7 '} 3 (7-0? (7-7') 3 •J,si» Determinants. Multiplying the two rectangular arrays a s a- a 1 \ 1 — 3a' 3a' 2 — a fr fr- /3 1 [ (1), 1 -3/3' 3/3' 2 -£'* y ! -y- -y 1 / 1 — 3-y' 3-y' 2 — y r a- 1 2 , 1 7 2 1 A becomes equal to the sum of four terms, from each of which we can take out as a fa. tor the product of the two d< terminants 1 a 1 J8 1 y The remaining factor is 3 { 3 a@y - 2)37 2a' + 2 fry 2a - 3 a' fr y' J , which can be written also in the foim Z{(a-a)(0-fr)(y-y') + ( a -fr)(0-y')(y-a) + (a-y')(fS-a)(y-fr)}. 1 7. Prove the expansion 1+fli 1 1 1 + «2 1 1 1 1 1 1 1 1 1 + as 1 1 1 + (ii I 1111] |l + - + - + - + - f «l «2 «3 «4 This is easily proved by subtracting the first column from each of the others, and then expanding the determinant as a linear function of the constituents of the first column. It will be apparent from the nature of the proof that the value of the similar determinant of the n th order is a\a^az . . . a„\ 1 + 2— j. 18. Prove the relation =/(*)-*/'(*), where /{,-) = (x - a) (x-fr(x-y)(x-S). This can he derived from the pi-evious example, or proved independently in a similar way. As in the last example, the determinant of this form of the n th degree can b ■ similarly expressed. Miscellaneous Examples. 281 19. Each of the coefficients of any equation can be expressed in terms of the roots as the quotient of two determinants. The student can easily extend to any degree the following application to the equation of the third degree. From Ex. 10, Art. 112, we have X 3 X 2 x 1 a 3 a" 0« a 1 J3 1 > - - y) {y - « )(a-/3)(. 7 3 7' 7 1 Expanding the determinant, this identity can be written a 2 a 1 a 5 a 1 o 3 a 2 1 F /3 1 x 3 - 3 )3 1 x- + £ 3 £ 2 1 £ — y- y i 7 3 7 1 7 3 7 2 1 a 2 a 1 = /3 2 1 7 2 7 1 7" 7 {s 3 -^ 2 -!-.^*-/*}, from which the above proposition follows ; pi, pz, pi being the coefficients of the equation whose roots are a, /3, y. 20. To express as a determinant the reducing cubic in the solution of a biqua- dratic. Employing Descartes' method, and substituting from equations (1), Art. 64, in the identity 1 1 p p' , g. 4 o we find the equation 1 1 p' p q' q 12 i? + p q + q / = \ p +p' 2pp' pq'+p'q 1 q + q' pq' + P'q Iqq a b c + 2a<(> b c — a

, which is easily seen to be identical with the cubic 4a 3 o? + eza+f*, f 3 (a) = «3a 5 + i 3 a 4 + c 3 o 3 + (ha? + e^a +fs. Now taking a, P, 7 to be the roots of the equation x 3 + px 2 + qx + r = 0, and forming the product of the following determinants : — a 3 a 4 a 3 a? a 1 ffl h Cl dx e\ /1 P & 3 & (3 1 (72 h c-i ck £2 fi y 5 7 4 7 3 7 2 7 1 «3 h e 3 dz ez A 1 1 P Q r 1 1 P «\ «2 «z «i at "., Cl 02 «:i «3 ff 4 Ob «1 «2 Here in all the rows tlie constituents are the same five quantities taken in cir- cular order, a different one standing first in each row. A determinant of this kind is called a circulant. It is convenient to write a circulant in the form here given, viz., such that the same element occupies the diagonal place throughout. Taking 6 to he any root of the equation x 5 - 1 = 0, and adding to the first column the sum of the constituents of the remaining columns multiplied by 6, d 2 , 3 , 4 , re- spectively, M'e observe that the following are factors of the determinant : — ai + a 2 +a 3 +ai +a 5 , «l + Q«2 + 2 «;s + s «4 + 1 « 5 , ,+ $a i + 3 n-l -1 (ln-2 bn-2 -1 On.3 4»-3 • in which all the constituents are zero except those which lie in the diagonal and in lines adjacent to it on either side and parallel to it, one of these latter sets consisting of constituents each equal to — 1. Expanding in terms of the first column, we have the following relation connect- ing three determinants of the kind here considered whose orders are n, n — 1, n - 2 :— A„ = a H A„-i + b H A„_2. Miscellaneous Examples. 285 By aid of this equation the calculation of any determinant is reduced to that of the two next inferior to it in the series A,„ A,,-i, A H -2, . . . A2, Ai ; and the values of Ai and A2 are plainly ct\ and ctia\ + bo, respectively. Dividing the equation just given by A„-i we have A„ _ b„ An-l ~ " A„-l ' A,,-2 replacing by a similar value the quotient of A n ~i by A ; ,_2, and continuing the pro- cess, it appears that the quotient of any determinant by the one next below it in the series can be expressed as a continued fraction in terms of the given consti- tuents. On account of this property determinants of the form here treated are called continuants. When each of the constituents b„, b»_i, . . . b 3 , J 2 (in the line above the diagonal) is equal to+ 1 the resulting determinant is a simple continuant. 29. Calculate the determinant of the n th order 1 £ a 1 a 1 fl a 1 A„ whose only constituents which do not vanish are a, $, 1, lying in the diagonal and the lines adjacent and parallel to it as here represented. The calculation is readily effected for any particular value of n, in a manner similar to that of the last example, by aid of the equation A,j = aA,i-i — j8A„_2, the values of Ai and A2 being a and o 2 - 0, respectively. By examining the formation of the successive values of A, the student will readily observe that the terms contained in the result are a->; a 2r -=£, a 2 '" 4 /8 2 , . . . a 2 fS r -\ P r , when n is even and of the form 2r ; and a 2 " 1 , a~ r - l 0, a 2 - 3 ^ 2 , . . . a 3 ^- 1 , afJ', when n is odd and of the form 2r + 1 . For the purposes of a subsequent investigation, in which the results just stated will be made use of, it is not necessary to know the general forms of the numerical coefficients which enter into these expressions ; but such forms can be arrived at without difficulty, and the following general expression obtained for A,» : — A„= a»- {>!- I)a»" 2 j8 + {n - 3) (n - 2) 1T2 (n - 5) (n - 4) (n - 3) 1.2.3 a''- G 3 +&c. •js<; Determinant*. 30. When a polynomial U is divided by another V of lower dimensions, the coefficients of the quotient, and of the remainder, can be expressed as determinants in terms of the coefficients of £7 and U'. The method employed in the following particular case is equally applicable in general. Let U be of the 5th, and I" gf the 3rd degree; the quotient and re- mainder can then b follows : — qox 2 + q\x + qz, R = r x 2 + r\X + r 2 . Also, let U= a x 5 + aix i + a 2 x 3 + dsx 2 + a t x + « 5 , V = «o'^ 3 + «i'x 2 + d 2 'x + aj. From the identity U = QU' + Jl we have the following equations : — tfo = qoCo', #i = qo&i + qiao', d 2 = qod 2 + q\ «i' + ?2«o', «3 = ?o«3'+ q\az + ? 2 «i'+ r 0) - o we have from the first four equations = 0, or a ' 3 r = 31. Find the general forms of the coefficients of the quotient, and of the re- mainder, when a polynomial of even degree 2m is divided by a quadratic. Taking x 2 + ax + fi as the given quadratic function, we have the identity «o' -co «l' oo -a\ '■'/ a\ «o' — d 2 C>3 az a\ - a 3 + r Co CO a\ do a\ Oz «i' «o' a 2 <';' «2' a\ 03 «o^ + a x x 2m -' 1 + d 2 x 2m ' 2 + . . . + a-2,n_ 2 x 2 + a> m .ix + a 2 - (qoX 2m - 2 +qiX 2m - 3 + . . . + qom-sx + qo nu . 2 ) (x 2 + ax + 0) + r x + n. Writing down the first r + 1 equations, formed as in the preceding example, to solve for qo, qu qz, • • • Qr, it is easily seen that the value of q r thence derived is a determinant of the r th order of the form treated in Ex. 29, bordered at the top with the constituents 1, 0, ... 0, a,,, and at the right-hand side with do, a\, ... d r . Expanding this determinant in terms of the last column, it is immediately seen that any quotient is expressed by means of a series of the determinants of Ex. 29 in the form q r - d r — d r .\ Al + rtr-2 A2 — &C. . . . + d\ A r _l + A,-, Miscellaneous Examples. 287 a + x h 9 h b + x f 9 f c + X the upper or lower sign to be used according as r is even or odd. To obtain the coefficients of the remainder, we have the equations qzm-3 + aq-im-z + r = <72>«-l, j3?2m-2 + n = (l-im. Expressing the values of qz, n -3, ?2».-2 by the formula just proved, and attending to the results of Ex. 29, we derive the following general forms for > - o and n : — r = Azm.1 + A 2m _ 3 & + A 2m -o j8 2 + . . . + A 3 p-»-- + ^ijS'"- 1 , J-l = a 2m + Bo m .n j3 + B 2m .i & + . . . + Bifrn- 1 + B & m , in which the coefficients A, B are all functions of a, the highest power of a in any- coefficient A or B being represented by the suffix attached to the coefficient. 32. If the leading constituents of a symmetric determinant be all increased by the same quantity x, the equation in x, obtained by equating to zero the determinant so formed, has all its roots real. Let the determinant of the n th order under consideration be denoted by An and written in the form A„ Let the detenninant obtained from this by erasing the first row and first column, i.e. the leading first minor of A„, be denoted by A„_i ; again, the leading first minor of A„-i by A n -2 ; and so on, the last function Ai obtained in this way being of the form 1 + x. To these we add the positive constant Ao = 1, which may be re- garded as completing the series of minors and obtained by the same process, since A„ is not altered by affixing a last row and a last column consisting entirely of zero- elements, with the exception of the constituent + 1 in the leading diagonal. We have now a series of n + 1 functions — An, An-l, A„-2, . . . As, Ai, Ao, whose degrees in x are represented by the suffixes. When + oo is substituted for x the signs are all positive ; and when — oo is substituted, the signs (beginning with Ao) are alternately positive and negative. Hence, if x be regarded as increasing con- tinuously, n changes of sign must be lost in this series during the passage from — oo to + oo . Now it appears by the theorem of Art. 129, that a value of x which causes any function (excluding A„, Ao) in this series to vanish gives opposite signs to the functions adjacent to it on either side. Ao retains its sign throughout. It follows, exactly as in (2), Art. 89, that a change of sign can never be lost except when x passes through a real root of A„ = 0. There must, therefore, exist n real roots of this equation in order that n changes may be lost during the passage of x from — oo to + oo . 288 Determinants* Any equation in the series, being of the same form as A„ = 0, has all its roots real. It is plain also that each of these equations is a limiting equation (see Art. 83) with reference to the equation next above it in the series ; since, in order that a change of sign may be lost between A„ and A„.i at the passage through each of two consecutive roots of the former, the value of A H -i must change sign between these two values of x. The equation A„ = may have equal roots, and by what has been just proved it appears that when this equation has r roots equal to a, the equation A„-i = has r — 1 roots equal to a, the equation A»-2 = has r — 2 roots equal to a, and so on. The determinant here discussed occurs in several investigations in pure mathe- matics and physics. The proof here given of the property above stated is taken from Salmon's Higher Algebra (Art. 46), to which work the student is referred for other proofs of the same theorem. 33. If the determinant of tbe preceding example have r roots equal to a ; prove that every first minor has r — 1 roots equal to a ; every second minor r — 2 roots equal to a, and so on. Employing the notation A, H, G, . . . for the elements of the reciprocal deter- minant, we have the equation AB- H 2 = A„. 2 A„. Now it is easily seen by proper transpositions of rows and columns that every leading first minor contains the multiple root r — 1 times. It follows from the equation just written that the minor H must contain this root r — \ times ; and H may be taken to represent any first minor. 34. Find the conditions that the equation a + x h (I h b + x f 9 f e + x should have equal roots. Since each first minor must contain the double root, we readily derive the required conditions in the following form : — „ gk , - kf fg a - —=b -c- — . 1 9 h This and the preceding example are taken from Kouth's Dynamics of a System of Big id Bodies, Part n., Art. 61. CHAPTER XII. SYMMETRIC FUNCTIONS OF THE ROOTS. 130. Newton's Theorem on the Sums of the Powers of the Roots. — We now resume the discussion of symmetric functions of the roots of an equation, of which a short account has been previously given (see Art. 271 ; and proceed to prove certain general propositions relating to these functions : — Prop. I. — The sums of the similar powers of the roots of cm equation can be expressed rationally in terms of the coefficients. Let the equation be fix) = x n + p x x n ~ x + p 2 x n ~ 2 + . . . + p n = {x-a 1 ){x-a 2 )(x-a 3 ) . . . (x - a n ) = 0. "We proceed to calculate 2a 2 , 2a 3 , . . . 2a m ; or, adopting the usual notation, s 2 , s 3 , . . . s m , in terms of the coefficients Pi, Jh, ■ • • P»> We have, by Art. 72, /(*) , /(•) , A*) /(< x — a\ x — a 2 % — a.n - nx"-* + (n - l)p x x^ + (n - 2)p 2 x- 3 +... + 2p, l _ 2 x +p n _ l ; and we find, dividing by the method of Art. 8, x-a x n ~ l + a + Pi + a~ + Pia +p 2 X n ~ 3 + a 3 + Pia 2 + Pia U . + a"" 1 +p l a 1l ~ 2 +lha n ~ 3 + . . . + Pn-aa + P»-i. 290 Symmetric Functions of the Roots. If in this equation we replace a by each of the quantities «„ a 2 , . . . a„ in succession, and put s p = 2a p = af + a/ + . . . + a/, we have, by adding all these results, the following value for f{x) = nx n ~ l + s, iC" -2 + s 2 x 11 - " 3 + s 3 x"- -' + . • • + 8n-i + np x + Pi Si + np 2 + j?l*2 + /),«! + np 3 + PlSn-2 +i>2«*-3 + p ft -zS 1 + np^ ; whence, comparing this value of f\x) with the former, we obtain the following relations: — Si +pi= 0, s 2 + P1S1 + 2p 2 = 0, s 3 + i>iS 2 + P2S1 + 3/; 3 = 0, St +pis 3 + ]hs 2 + PzSx + 4pi = 0, S«-l + PiS„--> + PzSn-s + ... +2hi-2Si + (n - l)p n -i = 0. The first equation determines s y in terms of p u 2h, . . . p n ; the second s 2 ; the third s 3 ; and so on, until s„_i is determined. We find in this way Si = - Pn s 2 = p? ~ 2pz, s 3 = - 2h z + 3pip 9 - Bp 3i Si = pf - 4p?pz + 4^i p% + 2p* - 4p i} s s = - Pi 5 + ^2h 3 2h - 5p*P3 - 5 (jtv - Pa) Pi + 5 (p 3 p s - p s ) ; &c. Having shown how s lt s 2 , s 3 , . . . s n _i can be calculated in terms of the coefficients, we proceed now to extend our results New tori s Theorem on Sums of Poivers of Roots. 291 to the sums of all positive powers of the roots, viz., s n , s u+1) . . . s m . For this purpose we have a?™f(x) = x" 1 + 2h% m ~ x + P2X m ~' + . . . p n x"™. Replacing, in this identity, x by the roots cti, a 2 , . . . a„, in succes- sion, and adding, we have S m + piSm-i +P%S m ^ i + . . . +p n Sm-n = 0. Now, giving m the values n, n + 1, » + 2, &c, successively, and observing that s = n, we obtain from the last equation Sn + PiS,i-i + P-.S,^ + . . • + np n = 0, Sn+i+PiS„ +2hS,i-i + . . . + J) n S l = 0, Sn^ + PlS n+1 +2hS n +...+ p n S 2 = 0, &C. Hence the sums of all positive powers of the roots may be expressed by integral functions of the coefficients. And by transforming the equation into one whose roots are the reci- procals of oi, a 3 , a 3 , . • . a„, and applying the above formulas, we may express similarly all negative powers of the roots. 131. Prop. II. — Every rational symmetric function of the roots of an algebraic equation can be expressed rationally in terms of the coefficients. It is sufficient to prove this theorem for integral symmetric functions, since fractional symmetric functions can be reduced to a single fraction whose numerator and denominator are integral symmetric functions. Every integral function of <*i, a 2 , . . . a w is the sum of a number of terms of the form Na^ a 2 q a 3 r . . . , where JSf is a numerical constant ; and if this function is symmetrical we can write it under the form iVSa^ a 2 q a/ . • . , all the terms being of the same type. Therefore, if we prove that this quan- tity can be expressed rationally in terms of the coefficients, the theorem will be demonstrated. We shall first establish the following value of the symmetric function Sa/W : — Sa, p a/' = S p 8 q - 8pm (1) u2 292 Symmetric Functions of the Roots. To prove this, we multiply together s p and 8 q , where S p - Oi" + a? + a/' + . . . a„", S 9 = af + a>* + as 7 + . . . a,,'; whence 8 p 8 q = a/** + ar q + . . . + a/ +? + O^O g « + a x '> a-f + &0., or s^s ? = V(? + Sa^cta 4 , which expresses the double function "2a? a 2 q in terms of the single functions s p , s q , s p+q in the form above written. We proceed now to prove a similar expression for the triple function, i.e., Set)' do Oi = SpSqS? — Sq+rSp — S r+p Sq — S p +qS r + /ZS p +q +r . \ZJ Multiplying together 'Zafa-f and s r , where Zafaf = a? a?. + a J of + afa-f + . . . S r = af + af + af + . . . + a,, r , we obtain an expression consisting of three different parts, viz., terms of the form 2af +r af, Sa^a/, and So^a 2 «a 3 r . Hence SvWa.9 = W + ''a,« + Sa^a/ + W af af , a formula connecting double and triple symmetric functions. But, by (1), **d\ af = $ p+ r*, s 3 , &g. In the same manner the quadruple function 2af af af a* Neivtorfs Theorem on Sums of Powers of Roots. 293 can be made to depend on the triple function Sor" a{> a/, and ultimately on s u s 2f s 3 , &c. ; and so on. Whence, finally, every rational symmetric function of the roots may be expressed in terms of the coefficients, since, by Prop. I., s x , s 2 , s 3 , &c, can be so expressed. The formulas (1) and (2) require to be modified when any of the exponents become equal. Thus, if p = q, afa-fl = a/a^, and the terms in (1) become equal two and two ; therefore 2ai p a 2 q = 2Sai /, a/ ; whence Wo/ = hiSp^-thp). Similarly, if p = q = r in ^a^'a^a/, the six terms obtained by interchanging the roots in ai p a 2 q a/ become all equal ; hence 2ai p asW = 73-^ (V - 38 p 8y, + 2s 3p ). And, in general, if t exponents become equal, each term is repeated 1 . 2 . 3 ... £ times. Examples. 1. Prove 2ai^a2«a3 r 04 s = S p S q S r S s — '2s p S q S ri s + 2~Zs p S q+r +s + 2s P 4. q S rts — 6Sp+q+ r +c 2. Prove 24 2ai'"03" , a3 , "ai'" = «m 4 — 6*n» 2 *2»i + Ss,„S3 m + 3«2m 2 — 6*4m- 132. Prop. III. — The value of s r , expressed in terms of Pi,p 2} . . . p n , is the coefficient off in the expansion by ascending powers of y of- r log #"/(-)• Since x n + p x x n ~ l + p 2 x n ~ 2 + . . . + p n = (x - 01) [x-a 2 ) . .. (x- a„), putting - for x in this identical equation, we find 1 +lhy +ihf +jhy 3 + . • • + pny" - (1 - a,y) (1 - a 2 y) ... (1 - a n y). 294 Symmetric Functions of the Roots. Now, taking the Napierian logarithms of both sides, Piy + P* 2 Pl if - Jh f + Pi y i +Ps -Pipi -PiPz -2hP* 1 , 2 ?h + P1 2 P2 +PiPi +pah - PiPi - P*Pz 1 5 + 5 Ih y 5 +&c +P r tJ r +&C. ^ o r Therefore, equating coefficients of if in both expansions, s r = - rP n where P r is the coefficient of y r in log y n f[-\ From the above identical equation it may be seen that s r (;• less than n) involves the coefficients p u p z , ]h, • • ■ Pr only ; and, therefore, p r+l , p r+i , . . . p n may be made to vanish without affecting the form of the expression of s r in terms of the coef- ficients. 133. To express the coefficients in terms of the sums of the powers of the roots. Since 1 +Pi-!/ + P*P*+ ...+p n y n =(l-a 1 y)(l-a i y) ... (1 -a n y), we have log (1 +p iy + ... +p n y n ) ^-ys,-- y 2 s 2 -...-- fs r -...; (1) Z r and, therefore, 1 + p,y + p 2 if + . . .+p n y n ^ery^-^^-^ s - , Coefficients in Terms of Sums of Powers of Boots. 295 which becomes by expansion 1-Siy-gSa + H2 Sl , 1 + T7T Sl * 1 .2.3 ^-4 S * + C «1 «3 -i* J * + 2T4 ,Si " /- 2.3.4 l Now, comparing the coefficients of the different powers of y, we obtain values for^i, p 2 , ih, . . . p,i, in terms of s x , s 2 , . . . s„ ; and we see that p r involves no sum of powers beyond s r . If the identity (1) be differentiated with regard to y, the equations of Art. 130 connecting the coefficients and sums of powers may be derived immediately from the resulting identity. It is important to observe that the problem to express any symmetric function of the roots in terms of the coefficients or any coefficient in terms of the sums of the powers of the roots is perfectly definite, there being only one solution in each case. We add some examples depending on the principles estab- lished in the preceding propositions. Examples. 1. Determine the value of (a„), where 01, as, 03, . . . a„ are the roots oij'(x) = 0, and <£>(.»•) is any rational and integral function of x. We have f(x) X — 01 ' X — 02 and + ... + /'(*)*{*) = *(*) + 4>(*) + > + »(*) m f(x) X — 01 x — 02 X — a„ 290 Symmetric Functions of the Roots. Performing the division, and retaining only the remainders on both sides of this equation, we have i?,,r"-' + Riz»-* + . . . + i?„-i _ (a„) , fix) x — 01 a - — a > a; — a )t whence RoX"- 1 + Biz"-- + . . . + 7J„-i = 2^(ai)(j--aj)(.).--o:i)...(^-a, 1 ) ; and, comparing the coefficients of .r»- 1 on both sides of this equation, Eo= 2(ai). 2. Prove that s p is the coefficient of — -z in the quotient of the division oif'(x) hyf(x) arranged according to negative powers of x. 3. Prove that s. p is the coefficient (with sign changed) of xp- 1 in the same quo- tient arranged according to positive powers of x. 4. If the degree of (p(x) does not exceed )i — 2, prove i-=n where 2 denotes the sum obtained by giving r all values from 1 to n inclusive. r=l "We have, by partial fractions, (p(x) _ A\ A 2 A n + — 1- • ■ • + f(x) X — a\ x — as X — a„ ' and, multiplying across by f(x), and putting x equal to oi, 02, • • . in succession, + TT, — \ — r • ■ • + -j, f{x) /'(ai) x - a\ /'(a 2 ) x - a 2 /' (o„) x - a„ ' whence ^ (- r ) 7c When

l-I I rt'l+r-l /(«) /'(«)i-«y 2 /» yr; "whence, comparing coefficients of y'" in these two expansions, 6. To express the coefficients of an equation in terms of the homogeneous pro- ducts of the roots, and vice versu . From the equation of the preceding example 1 (l-a v y){l-aiy) . . • (1 - a»y) we have {l+Piy+poy 2 +. . .+Pny")(l + Tliy + Tl2if+.. . ) = 1, which gives the following relations : — pi + IIi = 0, i?2 + n 2 +i?in 1 = 0, P3 + n 3 +P1II2 + P2TI1 = 0, &c. These equations (in -which, for values of r not greater than n, p\, p%, . . . p„ and IIi, II2, . . . n r are interchangeable) determine p\, p%, . . . p n in terms of IIi, IT2, . . . n,j, and vice versa. By means of this and the preceding example the values of the following symme- tric functions may be found in terms of the coefficients : — a"' 1 a» a'"i 2 7w' 2 />)' 2 /W &c - 7. To express n r by the sums of the powers of the roots. Eepresenting by - the product (1 - aiy) (1 - a 2 y) ... (1 - a n y), and differentiat- ing, we find 1 dn a - -j- = 2 = Si + s 2 y + s 3 y- + ...; u ay 1 — ay also u = 1 + U\y + Uoy 2 + . . . 298 Symmetric Functions of the Boots. We b ivc. therefore, (l + niy + u 2 y- + . . .)(«i + *2y+«sy*+.. .) b iii + 2n 2 y + 3n 3 y 2 + . .. X(iw comparing the several coefficients of the different powers of y, we have a number of equations by means of which the sums of the homogeneous products 111, II2, II3, . . . may be expressed in terms of s\, s 2) «3, &c. 8. To find a general expression for s„, in terms of the coefficients p\, p 2 , . . . p,„ of an equation of the 11 th degree. We have r= °°(- 1)'' - \og e (l + pi y+p 2 if +. . .+p, t y n )= 2 — — (piy+Pzf- + ---+Pny n ) r = *i y + n s *y 2 + o s^y 3 + ••• + — *». y™ + .. . 2 3 m Now, making use of the known form of the coefficient of y m in the expansion of (Piy + Pzy" + ■ • ■ +Pn!/") r by the multinomial theorem, and comparing coefficients of y m in the above equation, we find v (-l)«-«?r(ri + r 2 +... +r„) S "' = ■ E r(r i+ l ) T(r 2+ l)...T { r n + l) ^^ " ' ' ** in which r\ + r 2 + r 3 + . . . + r„ = r, n + 2r 2 + 3r 3 + . . . + nr n = m ; and ri, r 2 , T3, . . . r„ are to be given all positive integer values, zero included, which satisfy the last of these two equations. Also, representing by n any of these in- tegers, r (>-,- + 1) = 1 . 2.3 ... n, with the assumption that r(l) = 1 when n = 0. 9. To find a general expression for any coefficient p m in terms of the sums of the powers of the roots «i, s 2 , • • • s,„. We have 1 +pyy +p 2 y~+ . . . +p,„y m +. . . + p n y» = e' VSl . e' 1 *'* 2 . e' iy3ss . . . When the factors on the right-hand side of this equation are developed, and the coefficients of y m on both sides compared, we find, employing the notation of the last example, -^ (- 1)>-i + >-2+- ■ •♦>•„, 8\ rl S 2 r * . . . S„/m Pm= 2, r(>-i + l) r(?- 2 + 1) . . . r(>„, + 1) 2 r * fr* . . »%' in which n, r 2 , . ■ . r„, are to be given all positive values, zero included, which .satisf y the equation >"i + 2r 2 + 3/'3 + . . . + mr m = m. Theorem. 299 134. Definitions. Theorem. — The weight of any sym- metric function of the roots is the degree in all the roots of any term in the function. For example, the weight of SajS 2 ? 3 is six. The order of any symmetric function of the roots is the highest degree in which each root enters the function. For example, the order of 2a/3 3 7 3 is three. It has been proved (see Art. 28), that the weight of any symmetric function of the roots, when expressed by the co- efficients flo, #i, f'2, • • • (in, is the same as the sum of the suffixes of each term in the expression. We now prove another im- portant theorem, viz. : If any symmetric function be expressed in terms of the coefficients Pn Pa • • • Pn, the degree in the coefficients is the same as the order of the symmetric function. For example, Sa z j3 2 =p 2 2 -2pi£>3 + %4, no term being of higher degree than the second in the coefficients, and the order of the symmetric function being two. The student may easily satisfy himself of the truth of this theorem by observing that in the equations (2) of Art. 23, the value of each coefficient in terms of the roots contains each root in the first power only ; hence the highest degree in the co- efficients will be the same as the highest degree of the cor- responding symmetric function in any individual root. We add the following formal proof, as it is in accordance with the proofs of certain general propositions to be given subsequently. Replace the coefficients p u p«, . . . p n by — , --,... — . Now, if

, and >// an integral function not divi- sible by the product of all the roots ; (aia>a 3 . . . a„) p being the lowest common denominator of all the terms. Substituting in (1), we have aj>^ (a u a : , . . . a n ) = ± a,P-^F(a >l , «„_„ . . . a ). From this equation it follows that p is equal to w ; for if p were greater than w, xp(a u a 2 , . . . a„) would be divisible by the product a^tii . . . a n , and if it were less, the function of the coef- ficients F(a„, a„_i, . . . «o) would be divisible by a n , both of which suppositions are contrary to hypothesis. 135. Calculation of Symmetric Functions of the Roots. — Any rational symmetric function can be calculated by the method of Art. 131. In practice, however, other methods are usually more convenient, as will appear from the examples given at the end of the present Article, and from the following Articles, in which we shall give certain general propositions which in many cases facilitate the calculation of symmetric functions. The number of terms in any symmetric function of the roots is easily determined. For example, the number of terms in 2ai 3 a 2 2 a 3 of the equation of the n th degree is n(n-l) (n-2), this being the number of permutations of n things taken three together. If the exponents of the roots in any term be not all different, the number of terms will be reduced. Thus, 2a-f3y for a biquadratic consists of twelve terms only (see Ex. 6, p. 48), and not of twenty-four, since the two permutations a/3y, ay (5 give only one distinct term, viz., a"/3y, in 2a 2 j3y. The student Calculation of Symmetric Functions. 301 acquainted with the theory of permutations will have no diffi- culty in effecting these reductions in any particular case. When two exponents of roots are equal, the number obtained on the supposition that they are all unequal is to be divided by 1.2; when three become equal this number is to be divided by 1 . 2 . 3 ; and so on. In general, the number of terms in Sai^a 2 ? a 3 r ... of the equation of the n th degree, each term con- taining m roots, and v of the indices being equal, is n (n - 1) (n - 2) . . . (n - m + 1) 1.2.3. . .v When the highest power in which any one root enters into the symmetric function is small, i. e., when the order of the function (see Art. 134) is low, the methods already illustrated in Art. 27 may be employed with advantage for the calculation of the symmetric function of the roots in terms of the co- efficients. It is important to observe that when any symmetric function whose degree in all the roots (i. e., its weight) is n, is calculated in terms of the coefficients p u 2h • • • Pn for the equation of the n th degree, its value for an equation of any higher degree (the numerical coefficients being all equal to unity) is precisely the same ; for it is plain that no coefficient beyond p n can enter into this value, and the equations of Art. 130, by means of which the calculation can be supposed to be made, have precisely the same form for an equation of the n th degree as for equations of all higher degrees. It is also evident that the value of the same symmetric function for an equation of a degree m (lower than n) is obtained by putting p m+l , p M+z , . . . p n all equal to zero in the calculated value for an equation of the n th degree, since the equation of lower degree can be derived from that of the n th by putting the coefficients beyond p m equal to zero ; and the corre- sponding symmetric function reduces similarly by putting the roots o„ i+ i, am+z. • • • a» each equal to zero. •'i'l.' Symmetric Functions of the Roots. Kx V M 1 I.I 8. 1. Calculate 2ara.-a.i of the roots of the equation x» + pi x"- 1 + p 2 x"-* + ....+ p„-\x + p„ = 0. .Multiply together the equations 2oi =~Pi- 2aiaja3 = — J?s- In the product the term or 02 03 occurs only once; the term 01020304 occurs four times, arising from the product of ai hy 020301, of 02 by 010304, of 03 by 010204, and of 04 hy 010203. Hence 2ara2 03 + 4 2aia2 03a4 = P\P3 ', therefore 2ara2a;; = pipz - 4pi. (Compare Ex. 6, Art. 27.) If the calculation were conducted by the method of Art. 131, we should have 2oi 2 0203 = |*2*1 2 — Si*3 — |s 2 2 + *4, •vrhich leads, on substituting the values of Art. 1 30, to the same result ; but it is evident that in this case the former process is much more simple, since the values of vi, «2, &c, introduce a number of terms which destroy one another. 2. Calculate 2oi 2 a2 2 for the general equation. Squaring 2oi 02 = P2, we have 20l 2 O2 2 + 22ai 2 02 03 + 02aiO2O3O4 =P2 2 - In squaring it is evident that the term 01020304 will arise from the product of 0102 by 03 04, or of 01 03 by 02 ai, or of ai 04 by a> 03 ; hence the coefficient of 01 02 03 04 in the result is 6, since each of these occurs twice in the square. The result differs from the similar equation of Ex. 8, Art. 27, only in having 2 before the term 01020304. Hence, finally, 2ai 2 a2 2 =p 2 2 - 2p\pz + 2pi_ 3. Calculate 2oi 3 02 for the general equation. "We have, as in Ex. 9, Art. 27, 2oi 2 2aia2 = 2ai 3 aj + 2ai 2 02 03. Hence, employing previous results, 2ai 3 o 2 =pvpz - 2pr-pip 3 + 4pi. 4. Calculate 2ai 2 02 2 03 for the general equation. The result will be the same as if the calculation were made for the equation of the fifth degree. Examples. 303 To obtain the symmetric function we multiply together 2aia2 and 2010203 ; and consider what types of terms, involving the five roots 01, 02, 03, 04, as, can result. The term ai 2 a2 2 03 will occur only once in the product, since it can only arise by multiplying 0102 by 010(203. Terms of the type ai 2 02 03 04 will occur, each three times ; since or 02 03 04 will arise from the product of ai 02 by a\ 03 04, of 01 03 by 010304, or of 0104 by 010203 ; and it cannot arise in any other way. The term 01 o2 03 04 05 will occur ten times in the product, since it will arise from the product of any pair by the other three roots, and there are ten combinations in pairs of the five roots. We have then, for the general equation, 2ai02 2ai02 03 = 2ai 2 02~03 + 3 2ai 2 02 03 04 + 1020102030405. [We can verify this equation when n = 5, just as in Ex. 9, Art. 27 ; for the product of two factors, each consisting of 10 terms, will contain 100 terms. These are made up of the 30 terms contained in 2ai 2 02 2 03, along with the 20 terms con- tained in 2ai 2 020304, each taken three times, and the term 0102030405 taken 10 times.] Thus the calculation of the required symmetric function involves that of 2ar 02 03 04 ; for which we easily find 2ai 201020304 = 2ai 2 020304 + 520102030105. Hence, finally, we obtain 2ai 2 a 2 2 03 = -P2P3 + Spipi - bp b . The process of Art. 131 would involve the calculation of «s; and many terms would be introduced through the values of «i, so, &c, which disappear in the result. 0. Find the value of 2ar aj 2 03 04 for the general equation. We multiply together 2aia2 and 201020304, and consider what types of terms can arise involving the six roots ai, o 2 , 03, 04, 05, a6. The term ar a2 2 ay 04 can occur only once. Terms of the type ai 2 02 03 04 05 will each occur four times, this term arising from the product of 0102 by 01030405, or of 0103 by 01020405, or of 0104 by 01030305. or of 0105 by 01030304. The term 010303040501 will occur 15 times, this being the number of combinations in pairs of the six roots. Hence 2ai 02 2ai as 03 04 = 2ar 03 s 03 04 + 4 2ai 2 02 03^04 as + 1 •"> 2oi 02 03 04 as as- We have again, for the calculation of 2ai 2 ja2 03 04 05, 2ai 20102030405 = 2ai 2 020304 as + 62010203040,05. Hence, finally, 2ai 2 a2 2 a 3 04 =p?Pi- 4piPo + 9p a . 6. Find the value of 2ai 2 02 2 as 2 in terms of the coefficients of the general equa- tion. Here, squaring 2010203, we have 2ai 02 03 2ai as 03 = 2ai 2 a2 2 03 2 + 2 2ai 2 02 2 aj 04 + 6 2ar 02 03 ai as + 20 2ai 02 03 04 05 a6, from which we obtain 2aro2-a3 2 -pr - 2/Jipi + 2pij) b - 2p c . 30-4 Symmetric Functions of the Hoots. 13G. Brloschl's Differential liquation. — M. Brioschi has given the following differential equation connecting the coefficients and sums of powers of the roots of an equation: — dpr+k 1 ds r r To prove this we have, as in Art. 132, and differentiating, d n whence, comparing the coefficients of the different powers of //, --y *' d — (1 +2h1J +P2lf+ • • • +PnV n ) - - (1 +PlV +P:f + . • • +Pn?/ n ) ~ dpq = ds r 0, when q < r } dp r _ ds r 1 dp r ^ k r ' ds r i -p- We can now express the result of differentiating with respect to s r any function of the coefficients Since F(Pl,P2,Pi, • • • Pn). dp i dp 2 dp r _i ds^ ds r ' ds r all vanish, — Fl «, v* r> v)- — d ^+ dF *** + + dF *- ds r dp r ds r dpr+i ds r dp n ds r and, applying the formula given above, this reduces to 1 (dF dF dF dF\ By means of this result symmetric functions can often be calculated with great facility, as will appear from the following examples : — Examples. 305 Examples. 1. Calculate the value of the symmetric function 2ar a 2 2 ay ai 2 of the roots of the equation x" + pi z"- 1 + pzx"- 2 + . . . + p n = 0. Knowing the order and 'weight of any symmetric function, we can write down the literal part of its value in terms of the coefficients. Here 2 is of the second order, and its weight is eight ; hence 2 = t ps + tip-;pi + t 2 p 6 p 2 + t 3 p 5 p 3 + hpi 2 , where to, t[, h, &c, are numerical coefficients to he determined. Terms such as ptP\ 2 , PipiPi, PoPi 3 , &c, although of the right weight, are of too high an order, and therefore cannot enter into the expression for 2. Again, 2 expressed in terms of the sums of the powers of the roots is of the form F(s2, si, s 6 , S&) ; for, in general, 2ai? a-2 q 03 r . . . , expressed in terms of the sums of the powers of the roots, is made up of terms such as s p , s p+9 , s p , q + r , . . . sk p , . . . all of which are sums of even powers when p, q,r, . . . are even ; therefore in this case none hut even sums of powers enter into the expression for 2. Also, since — =0, and — =0, we have, using the formula above given for — , ds 3 ds^ ds r tops + tipiPi + t 2 p 3 p 2 + t 3 (p 2 p 3 +Po) + 2hPiP± =I°> toPi + hpi = 0. From these equations we infer t + h = 0, h + h = 0, h + t = 0, h + 2ti = ; but U=l, since for a quartic 2 = pi 2 ; therefore ti = -2, t = 2, t 3 = -2, t 2 = 2; and, substituting these values of to, h, t 2 , t 3 , ti, 2ai 3 a2 2 03 2 ai 2 = 2p s - 2p-,p\ + 2p^p 2 - 2p$p 3 + Pi 2 - 2. Calculate 2oi 2 o-ra3 2 for the same equation. Ans. - 2p% + 2p\po - 2p 2 pi + p 3 2 . (Compare Ex. 6, Art. 135.) 3. Calculate for the same equation the symmetric function 2ai 3 a; 2 o 3 . Here the weight is six, and the order three ; hence 2ai 3 a 2 2 a 3 = t pe + t\p 5 pi + tiPiPz + hpipv + hp 3 2 + t b p x p 2 p 3 +[t 6 p 2 3 . Also 2, expressed in terms of *i, s 2 , s 3 , &c, is (see Art. 131), SiS 2 S 3 — SiSo — S 3 2 — «2*1 + 2*6- Now, differentiating by means of Brioschi's equation these two values of 2 with regard to *6> and comparing differential coefficients, we have *£ — £-«, or fc=-12. (isk 6 30G Symmetric Functions of the Roots. Differentiating with regard to .'5, we have toPi + t\p\ = 5*i = -'>/>] ; .-. ti-7. Differentiating with regard to 3 if top 2 + h pr + hp 2 + dpr = 4.s- 2 = 4 (pr - 2p 2 ) ; ■whence to + C 2 = - 8, h + to = 4 ; and to = _ 3, to = 4. Again, to = ; for 2 vanishes if n — 2 roots vanish. And we find to and to hy taking the particular case when n — 3 roots vanish ; for in this case 2ai 3 ot2-a3 = ai02a3 2ai 5 a2 = - Pz{- pipz + 3j33) =pip2Ps — Sps 2 , and tlierefore to = - 3, to = l ; whence, finally, 2ai 3 a 2 2 a 3 = - 12j» 6 + 7i?iJ»5 + 4piP2 - Spipi 2 - 3pr + pipzp 3 . 137. Derivation of new Symmetric Functions from a given one. — From any relation such as a^ 20 (ai, a 2 , . . . a„) = F(a , a l} a 2 , . . . a„), where is an integral function, of the order vs i of some or all of the roots of the equation a x n + na x x n ~ l + — - — 1 - 1 a % x n ~~ + . . . + a n - 0, X . £ we may derive a number of other symmetric functions and their equivalents in terms of the coefficients. For this purpose diminish each of the roots by any quantity .r, and consequently change any coefficient a,, into U r (see Art. 35). When this is done the original relation becomes rto^Stf) (a, -X, a 2 -X, . . . On-X) = F{U , U h U 2 , . . . U n ) ; and comparing the coefficients of the different powers of x on both sides of this equation, we have a number of symmetric functions of the roots expressed in terms of the coefficients as required. It should be observed, however, that this method leads to no new symmetric functions when the given function

(ai, a 2 , . . . a„) = F(a 0) a h a 2 , . . . a„), as in the last Article. Adopting the notation . d d\ d ddi da 2 aa n d O d Q '' (I Z) = a — + 2a, — + 3ar 2 — + . . . + na n -i -j-, da x aOi da z aa„ we have the following equation of operation : — da-o"^ (ai, a t , . . . a„) = DF[a , «i, . . . a n ). To prove this, we have, as in Art. 137, [ac> + - — = 8 2 02, • • • a») . Again, omitting all powers of x higher than the first, F{ U , U, ... Un) becomes F[a , and DF may take the place of and F in this equation, ff,rS 2 becomes D 2 F, &e. It may be noticed, moreover, that if Scp vanishes, S 2 0», <$ 3 o , &c, all vanish ; and thus that x disappears in the expansion of cp(ai - X, a 2 - X, . . . a n - x). Now this can happen only when $ is a function of the differences of a„ a 2 , .... a !t ; whence we conclude that if a^Fya,,, a x , a 2 , . . . a,) is the value in terms of the coefficients of a function of the differences of the roots, then DF(a n , %, 02, • • • B ) vanishes identically. This identical relation is often sufficient to determine the numerical coefficients in a function of the differences expressed by the coefficients, when the order and weight are known. It is not sufficient for this purpose when there exist more than one function of the differences of the required order and weight. "We add examples of functions of the differences determined in this way. Examples. 1. Determine a function of the differences -whose order and weight are hoth three. Assume

= {1A + B) rtcr«2 + ('2B + 3C) a{-a s 0. Hence 3A + B=0, and 2# + 3C=0; and putting A = l, we have B = -3, and 0=2; whence, finallv, ._ , nB . = dd> deb dd> dF _ dF n dF dF -j- + 7- + j-+...+~ = 5 or +2s 1 -+3« i — + . . . + rs r ^-~. clai aa 2 da 3 da n asi ds 2 ds 3 ds r For, let the roots be increased by h ; and comparing the coefficients of h on both sides of the equation (1), when Si + h s , s 2 + 2/iSi, . . . s r + rhsr.it are substituted for Si, s 2 , . . . s r , we have the required relation. Employing the results of the last Article, we have, therefore, the following equation of operation connecting the coefficients and the sums of the powers of the roots : — _. / d ft d n d d\ _ -D = af\8»— + 2sy — + 3s 2 — + ... + rsr-i 3- = a a A, \ asi ds 2 ds 3 ds r J where D s represents the result of substituting s for a in the operator D. :$10 Symmetric Functions of the Roots. From this it follows that if f(a , o t , a., . . . a„) is a function of the differences, /(s , «i, 82, • • . »») is a function of the diffe- rences also; for it is plain that when JD/(a , a ly a>, . . . a n ) = 0, A/(« , 81, s 2 , . . . s n ) = 0, and therefore Df(s , s { , « 2 , . . . s n ) = 0, since D. s = - ai^D. Ex. 1. a '2 •^ *'4 which is therefore a function of the differences. Miscellaneous Examples. 1. Prove, hy squaring the determinant of Example 10, Art. 112, the following relation between the roots a, j8, 7, 8, of the biquadratic : — *0 *1 S2 V, A'l «2 *'3 Si »2 *3 «4 »S S3 «4 *5 «G (/3- 7 r-(a-S)2( 7 -a) 2 (/3-S)2(«-/3r-( 7 -8) 2 . The student will have no difficulty in writing down for an equation of any de- gree the corresponding determinant in terms of the sums of the powers of the roots which is equal to the product of the squares of the differences. 2. Prove, for the general equation, So Si = 2(o ' ft 2 - *1 *2 This appears by squaring the array 1 1 1 a & 7 1 1 . 5 € . • 1 1 . ) (See Miscellaneous Examples. 311 3. Prove similarly, for the general equation, so Si tz 1 H s z =-2(&-yy-(y- a y-(a-0)-. So S 3 Si I By the process of Art. 123, a series of relations of this kind can be established ; and when the number of rows in the array becomes equal to the degree of the equation, the value of the determinant is the product of the squares of the differences, as in Ex. 1. When the number of rows exceeds the degree of the equation the value of the corresponding determinant vanishes. For example, the value of the determinant of Ex. 1 is zero for equations of the second and third degrees. 4. Prove, by means of the equations of Art. 130, that the sums of the powers can be expressed in terms of the coefficients, or vice versa, in the form of determi- nants, as follows : — &e. 2p 2 = Pi 1 Pi 1 , *3=~ 2p 2 Pi 1 2 P 2 Pi Zpz Pi Pi «l 1 *1 1 , 6pz=- *2 Si 2 «2 Si S3 *2 «i lipl Pi 1 2p2 Pi 1 3^3 Pz Pi 1 4pi PS Pz P\ Sl 1 *2 Sl 2 *3 «2 ■n 3 Si Si *2 Sl , &L-. ). Resolve into factors the determinant «3 X *2 X si X so 1 1 where s , sj, * 2 , &c, are the sums of the powers of three quantities, o, /3, 7. This determinant is the product of the two determinants a 3 P ? 3 X* a 2 P y- x 2 a 7 ■ 1 1 1 1 1 7 3 y : y 1 'J y y 1 1 1 and each of the latter can be resolved into simple factors. 312 Symmetric Functions of the Roots. 6. Prove, for the general equation, *o *i »a 13 *1 «2 S3 Si *2 *3 *1 S 5 1 * X 2 X Z Multiplying the two arrays 111.. a /3 7 a- 0- 7 2 . we show that 2 is equal to S X — Sl = 3(li-yy*(y-a)Ha-fSy-(x-a)(x-fi)(z-y). x - o o (.<- - o) a- [x — a) S\X — s% six — s 3 Si x — s 2 «2^' - S3 S3X - 54 z-0 (* - J8) 2 (x-/3) *2^ - S3 S3 X — «4 S42; - s 6 *-7 7 (* - y) y*{x-y) which is easily transformed into the proposed determinant. It appears in like manner in general that the determinant of similar form of order p -f 1 is equal to the corresponding symmetric function, each of whose terms contains p factors of the original equation multiplied by the product of the squared differences of the p roots involved therein. 7. Prove that'the leading coefficients of Sturm's functions (i.e.f(x),f'(x), and the n - 1 remainders) differ by positive factors only from the following series of determinants : — So Sl S2 S3 so Sl *2 Sl S2 S3 Si «0 Sl I Sl So S3 s> S3 S4 S5 Sl *2 1, *2 «3 54 > S3 Si S5 S6 1, So, Sl S 2 I, S2 S3 Si , S3 Si S 5 S C , . . . . (S S 2 S4 . . . *2n-2) Representing Sturm's remainders by i? 2 , B 3 , . . . Bj, . . . B n , and the successive quotients by Q h Q 2 , Q 3 , &c, we have (see Art. 89) H2=Q l /'(x)-f(x), B 3 = Qo.Z* -f{x) = (Q, Q z - l)f(x) - Q 2 f(x), Hi = Q3B3 - i?2 = (Qi Q2Q3-Q1- Qz)f'{x) - (QiQ 3 - l)f{x), &c. Proceeding in this manner, we observe that any remainder Bj can be expressed in the form Bj = Ajf(x)-B s f(z). (1) The degree of Bj is n -j ; and since Q h Q 2 , &c, are all of the first degree in#, it appears that the degrees of Aj and Bj are./ - 1 and> - 2, respectively. Miscellaneous Examples. 313 Assuming, therefore, for Rj and Aj the forms Rj = r +r\X + r 2 x 2 ■+.'..+ r„.jX n -J, Aj = \ + M x + A.2.C 3 + . . . . + a>.i xJ- 1 ; and substituting in (1) any root a of the equation f(x) = 0, we have Ao + Aia -f \za- + + A.-io-'- 1 = /'(«) Multiplying by o, a 3 , . . . a-*'- 2 , aJ-\ in succession ; making similar substitutions of the other roots ; and adding the equations thus derived, we obtain by aid of the relations of Ex. 4, p. 296, the following system of equations : — Ao«o + Kisi + . . . + A/-2 sj-2 + \j.\ Sj-i = 0, Ao«i + A1S2 + . . . + Aj-2«,--i + Am Sj - 0, Ao«;-2 + \\Sj-l + . . . + Aj-2 Sy-4 + \j-l $2j-3 A0*>-1 + \\Sj -f . . . + \j-2Soj. 3 + \j-\S2j.2 ■■ From these equations we have, without difficulty, So «1 . r »-j = yj so *1 *1 *2 Sj-i »'l Sj.2 Sj-l Sj.1 Sj Aj = Jj Sj-l Sj S2J-2 Sj-Z Sj-i . . . S 2 jA S 2 j-3 1 x . . . xi- % xJ- 1 the value of yj being so far arbitrary. It appears therefore that the coefficient of the highest power of x in Rj differs by this multiplier only from the determinant (50*2*4 • • • • *2/-2). We proceed to show that the sign of yj is positive. For this purpose we make use of the following relation connecting the successive values of the functions R and A : — A M R k - & ktl A k =f(x). (2) To prove this ; substituting for R k¥ \ Rk, Rk-i their values in terms of A and B in the relation R k +\ = Q k R k - Rk-i, we derive A k+ i = QkA k - Ah-i, £ k+ i = Qk B k - Bk-i ; by aid of which we readily obtain the following relations connecting the successive functions : — Ak+iBk-Ak£kn = A k Bk-i-A k - l B k = . . . =A 1 B -A B 1 = - 1, A k ^R k - A k R k +i = A k R k -i - A k .\R k = ... = A X R - A Ri =/(#), in which -Si =f'(x), R =f(x). Now, comparing the coefficients of the highest powers of x in (2) ; observing that *" occurs only in A k+ i R k , and making use of the determinant forms above obtained, we have 7*+I (*o s 2«4 • • • S2J.-2) 74 (*o*2*4 •••»»-»)= 1, y k y k *\ = («oS2*4 • • • *2*-s)" 2 :;i I Symmetric Functions of the Roots. Also, calculating the value of if: in the ordinary manner, we easily find A« = — s - whence it is seen that the value of 72 is — . •Si- lt follows, from the relation just established between any two successive values of 7, that 73, 74, ... 7;, &c, are all positive squares ; and therefore, finally, that >•„-,, the coefficient of the highest power of x in fij, has the same sign as the deter- minant (SoSzSi . . . ftty-2). It may be observed that by aid of the preceding example the value of the quotient of Aj by 7, may be written as a symmetric function involving the roots and the variable. For example, when,/ = 4, we have =± = Mft-7) 2 (7-a)Ha-0) 2 (*-a)(z-fi){x-y). 7« 8. Determine p = T . 4>i0i + p e p >>- 1 = T p .i. Atts. , H, I, J the value of the symmetric function a J 2 (3a - £ - 7 - 8)- (3/3 - 7 - 8 - a)- (3y - 5 - a - j8) 2 . Here ,/ 6 2 = 4 6 2:r; 2 2 --3 2 , where z\, z 2 , z 3 , z± are the roots of the equation e* + 6Hz- + Wz + « -J - 3^=0. (See Art. 37.) Hence, by Ex. 2, Art. 136, Ann. 4" { - 7J2" 3 + a - HI - 4ff„V} . Miscellaneous Examples. 315 y-(y-sy-=ip+mr-, 10. Prove that n s= a 6 (j8 - y) - (7 - a) 2 (a - £) 2 (a - 8)'- (j8 where m = - 27/. The weight of this function of the roots is 12, and the order 6. We now make use of a proposition which will be proved subsequently, namely, that any even, rational, and integral symmetric function of the roots, of the order w, and involving the differences only of the roots, is, when multiplied by a^, a rational and integral function of «u, H, I, J. (Compare Ex. 17, p. 124.) Hence, expressing the function whose order is 6, and weight 12, in terms of nio, -H, /, /, it is easy to see from the table Order. Weight. J 3 6 I 2 4 E 2 2 that H cannot enter, for the terms of the sixth order containing H, viz., H 3 , IP I, MP, have not the proper weight. Therefore n must be of the form IP + mJ-, where I and m are numerical coefficients. Now put «3 and a^ equal to zero, and n will vanish, since in that case the quartic will have equal roots ; hence, employing the reduced values of / and /, = /(3a 2 2 ) 3 + m{- a 2 3 ) 2 , and m = — 111. 11. Calculate the symmetric function of the roots of a biquadratic 203- 7 ) 2 ( 7 -a) 2 (a-j8) 2 . Since the order of this symmetric function is four, and its weight six, we may assume a 4 2(/8 - if (7 - a) 2 (a - 0) 2 = ISI + ma a J. (1) The values of I and m may be found by putting a 3 = 0, a± = 0, as in the pre- ceding example, and calculating the value of the reduced symmetric function (when 7 = 0, 5 = 0) in terms of the coefficients of the quadratic equation aoX" + iaix + 6(72 = 0. Identifying then this value with the reduced value of IHI + maoJ, we obtain two simple equations to determine I and m. Or we may proceed as follows by taking two biquadratics whose roots are known, and calculating in each case the syni- :316 Symmetric Functions of the Roots. metrio function by actually substituting the roots, and then comparing both sides of the equation when II, I, J are replaced by their values calculated from the numerical coefficients. First we take the biquadratic equation 6x 4 - 6x- = 0, whose roots are 0, 0, 1, - 1 ; whence 2 = 8, 2f=-6, 1=3, J=l. Substituting in equation (1), we have 1728 = -3; + m. Proceeding in the same way with the biquadratic equation x i — 6x 2 + 5 = 0, whose roots are + \/ b, ± 1, 2 = 768, S = -l, 1=8, J=-i; - 192 = 21 + m, I = - 2 x 192, m = 3 x 192 ; «o* 2 = 192 (- 2HI+3a J). 12. Calculate the determinant we find whence and and, finally, *0 -1 S 2 Si .v; »3 «2 «3 «4 4{8P 2 Pi-2P 2 3 -9P 3 2 }; in terms of the coefficients of a quartic. This determinant is a function of the differences of the roots (see Ex. 2, Art. 139); we may therefore remove the second term of the quartic before calculating it ; and if the equation so transformed be .V 4 + P 2 y- + Pzy + Pi = 0, 4 - 2P 2 - 2P 2 - 3P 3 -2P 2 -3P 3 2P 2 2 -4P 4 but « 2 P 2 = 6JET, aJP 3 = 4G, a *P i = a 2 I - 3S 2 . Substituting for P 2 , P3, Pi these values, we have a 4 A= 192 {- 2HI + 3a J) : the same result as in the preceding example. (Compare Ex. 3, p. 311.) 13. If o, 0, 7, 8 be the roots of the equation «o** + ictix 3 + 6a 2 x 2 + 4a 3 x + en = 0, Miscellaneous Examples. 317 express H„ I s , J t , G, of the equation Sox* + 4si# 3 + 6s 2 z 2 + is 3 x + S4 = 2 (* + a) 4 = ia terms of H, I, J, G. .4m. -—=-3— r, — = j , -5 = -3— ; *o- <*o 2 s * «o 4 s s <*o and by the aid of the relations G- + 4tf 3 = a'(a), $'(0), '(5), where o, j8, 7, 8 are the roots of the equation 4> (x) = aoX i + 4ai x 3 + 6« 2 x 2 + 4«3 x + ' 4 + — ' 3 + — 1 -. — - d>' 2 + — ~ R = 0. «o 3 flo 4 «o 6 16. If 2 (a - /8) 2 (0 - 7 ) 2 (7 - a) 2 (* - 8) 4 , when expanded, becomes Kox* + 4:Kix 3 +6Kzx 2 +4K 3 x + Ki ; prove that KpaPy + .ffi ($y + ya + afi) + JT 2 (q + j3 + 7) + £3 _ + 16 ^/a ()3-7)(7-«)(«-/3) = «o 3 ' where A = J 3 - 27/ 2 . 17. Prove that a 4 2 (0 + y - a - 5) 2 (0 - 7 ) 2 (a - S) 2 = 192 (3« /- 2.H7). 18. Prove that «o 8 2 (j8 + 7 - a - S) 4 (jS - 7) 2 (a - 5) 2 =512 (« 2 i 2 - 36 a HJ + 122Z" 2 /). CHAPTER XIII. ELI M IN ATION. 140. Definitions. — Being given a system of n equations, homogeneous between » variables, or non-homogeneous between ;/ - 1 variables, if we combine these equations in such a manner as to eliminate the variables, and obtain an equation R = 0, containing only the coefficients of the equations ; the quantity R is, when expressed in a rational and integral form, called their Resultant or Eliminant. In what follows we shall be chiefly concerned with the dis- cussion of two equations involving one unknown quantity only. In this case the equation R = asserts that the two equations are consistent ; that is, they are both satisfied by a common value of the variable. We now proceed to show how the elimination may be performed so as to obtain the quantity R, illustrating the different methods by simple examples. Let it be required to eliminate x between the equations ax 2 + 2bx + c = 0, ax 2 + 2b'x + c = 0. Solving these equations, and equating the values of x so obtained, the result of elimination appears in the irrational form b y&-ac V yb' 2 -a'c — + = - - + -, « a a a a Multiplying by ad we obtain ab' - ab = a t/V* - ac - a'^b 2 - ac. Squaring both sides, and dividing by a a' (for R does not vanish when ad vanishes), and then squaring again, we find R = 4(ffe - b 2 ) [aV- b' 2 ) - (ac + a'c - 2bb') 2 . Elimination by Symmetric Functions. 319 This method of forming the resultant is very limited in ap- plication, as it is not, in general, possible to express by an algebraic formula a root of an equation higher than the fourth degree. Other methods have consequently been devised for determining the resultant without first solving the equations. We now proceed to explain the method of elimination by sym- metric functions of the roots of the equations. 141. Elimination by Symmetric Functions. — Let two algebraic equations of the m th and n th degrees be

(x) = be en, a 2 , . . . a m . If the given equations have a common root it is necessary and sufficient that one of the quantities should be zero, or, in other words, that the product i>[ai) ^(a 2 ) i//(a 3 ) . . . i/»(a m ) should vanish. If, now, we transform this product into a rational and integral function of the coefficients, which is always possible as it is a symmetric function of the roots of the equation [x) = 0, we shall have the resultant required. Further, if /3i, /3 2 , . . . j3„ be the roots of the equation \p (x) = 0, we have xf,{a 1 )=b {a 1 -^ l ){a l -^)... ( ai -/3„), *(«*) = 6o(«,-0i)(a,- 0.) ...(«.- j3„), ^W = b (a m - /3i) (a m - /3 2 ) ... (a„, - /3„). If we change the signs of the m n factors, and multiply these equations, taking together the factors which are situated in the same column, we find a " + ( Ql ) i| (« 2 ) . . . 4 («.) = (- 1)'"" C ^ (3i) * (/3 8 ) ... (0«). 320 Elimination. We may therefore take 12= (-l) BM »*o m ^O3i)0(]3 2 )...«(l3„)=«o ,, ^(a 1 )^(a 8 )...^ (a*), (1) for both these values of R are integral functions of the coef- ficients of cp(x) and $(*), which vanish only when (.r) and ?// (a?) have a common factor, and which become identical when they are expressed in terms of the coefficients. 142. Properties of the Resultant. — (1). The order of the resultant of two equations in the coefficients is equal to the sum of the degrees of the equations, the coefficients of the first equation entering R in the degree of the second, and the coefficients of the second entering in the degree of the first. This appears by reviewing the two forms of R in (1), Art. 141 ; for in the first form a , a u . . . a m enter in the n a degree, and in the second form b , b x , . . . b„ enter in the m th degree. Also it may be seen that two terms, one selected from each form, are (- 1)'"" b m a m n and a n b n m . (2). If the roots of both equations be multiplied by the same quantity p, the resultant is multiplied by p mn . This is evident, since any one of the mn factors of the form a p - fi q becomes p (a p - (5 g ), and therefore p" m divides the result- ant. From this we may conclude that the weight of the resultant is mn, in which form this proposition is often stated. (3). If the roots of both equations be increased by the same quantity, the resultant of the equations so transformed is equal to the resultant of the original equations. For we have ±R = a H m IL(a p -P s ), where n signifies the continued product of the mn terms of the form a p - fi q ; and this is unaltered when a p and (3 q receive the same increment. (4). If the roots be changed into their reciprocals, the value of R obtained from the transformed equations remains unaltered, except in -sign when mn is an odd number. Making this transformation in R = a n bo m n(ap-(3 l] ), Properties of the Resultant. 321 we have *-*»* < l > («!«,...«,,)« (ft/3,... ft)-' but **..'. «.-(-l)-£, ftft...0.-(-l)*jh "o Oo substituting, we obtain R' = a J' b m (- 1)"'" n (op - 13,) = (- l) ffln iJ. From this it follows that in the resultant of two equations the coefficients with complementary suffixes of both equations, e.g. a , a m ; «u «m-ij &c, may be all interchanged without alter- ing the value of the resultant. (5). If both equations be transformed by /tomographic transfor- mation ; that is, by substituting for x Xx + fi XwTT" and each simple factor multiplied by \'x + //, to render the new equations integral ; then the new resultant R' = (Xf/ - X'fx) mn R. To prove this, we have (x) = a (x - ai) (x-ao) ... (x - a m ), *(*) = *,(• -ft) (*-0O ... (»-/3,); also .c - a r becomes (A - A'a,) ( # - ^ — tt — )> \ A - A a r J Multiplying together all the factors of each equation, tf becomes a u (X - A'ai)(A - A'a 2 ) ... (A - A'o TO ), 6 „ ft, (A - X'/3i) (A - A'p\) ... (A - A'j&O . Also, since a,, p\- are transformed into ;~ r , , (* ' , -, A - A a, A - A p, * - 0' ^omes (A . V „ r)(A _ yp r ) ' 322 Elimination. whence a J' b,r n (or - fir) becomes a n b '" (A/u' - XV)'"" n (a r - /3 r ), that is, the resultant calculated from the new forms of $ (x) and xP (.r) is (V- *V)""-B- This proposition includes the three foregoiug ; and they are collectively equivalent to the present proposition. 143. Euler's method of Elimination. — When two equations „„ ^(flj) a gi**" 1 + q t af* + ... + q n , the coefficients being undetermined constants depending on 8. Whence we have (j>(x) i/,,(a?) =4>{x) 0i(ar), an identical equation of the (w + n- l) th degree. Now, equating the coefficients of the different powers of x on both sides of the equation, we have m + n homogeneous equations of the first degree in the m + n constants p l9 p %9 . . . p m , q i9 q 2 , . . . q n ; and eliminating these constants by the method of Art. 125, we obtain the resultant of the two given equations in the form of a determinant. Example. Suppose the two equations ax 2 + bx + c = 0, ct\ a? + b\ x + c 1 = to have a common root. We have identically (qix + qi)(az 2 +bz + c) = (pix + p2)(aix' i + hx + Ci), or (qia-piai)x i + (qib + q2Ct-pih-p2ai)x- + (qic + q2b-piCi-p 2 bi)x + q 2 c-p2Ci = 0. Sylvester's Dialytic Method of Elimination. 323 = o. Equating to zero all the coefficients of this equation, we have the four homo- geneous equations q x a -P\(ti =0, q\b + q 2 a - pibi - p 2 ai = 0, qic+ q-J) -pici - pzbi = 0, qic — pzc\ = ; and eliminating the constants pi, p-z, qi, q%, we obtain the resultant as follows : a ai b a b\ o,\ c b c\ Si c ci The student can easily verify that this result is the same as that of Art. 140. 144. Sylvester's Dialytic Method of Elimination. — This method leads to the same determinants for resultants as the method of Euler just explained ; but it has an advantage over Euler's method in point of generality, since it can often be applied to form the resultant of equations involving several variables. Suppose we require the resultant of the two equations (x) = a x m + a x x m ~ l + a z x" 1 '- + . . . + a m = 0, \P(x) m b x n + & 1 ar M + b 2 x n ~ 2 + ... + &« = 0, we multiply the first by the successive powers of x, and the second by g.n-1 gJl-Z /gtn-i x m ~ 2 thus obtaining m + n equations, the highest power of x being m + n - 1. TVe have, consequently, equations enough from which to eliminate ,,m+n-l ~,m+n-2 x-, X, considered as distinct variables. y2 324 Elimination. we have Example. In the case of two quadratic equations ax 1 + bx + e = 0, a\X- + b\x + C\ = 0, ■ + bx- + ex = 0, ax 2 + bx + c = 0, aix 3 + b\x- + c\x = 0, a\x- + b\.x +c\ = ; from which, eliminating a; 3 , x 2 , x, we get the same determinant as before, columns now replacing rows : I a b c a b c bi d <7l h Cl 145. Bezout's Method of Elimination. — The general method will be most easily comprehended by applying it in the first instance to particular cases. We proceed to this applica- tion — (1) when the equations are of the same degree, and (2) when they are of different degrees. (1). Let us take the two cubic equations ax 3 + bx 2 + ex + d = 0, a x x 3 + b x x 2 + c v x + d y = 0. Multiplying these two equations successively by #1 and a, a x x + bi „ ax + b, a x x 2 + b Y x + c x o.r + bx + c, and subtracting each time the products so formed, we find the three following equations : — [ab x )x 2 + (aci) x + (adi) = 0, [ac x )x 2 + {{ad x ) + [bc x )}x + (bd x ) = 0, (ad x )x 2 + (bd x )x + (cd x ) = 0. Bezoufs Method of Elimination. 325 By eliminating from these equations x 2 , x, as distinct variables, the resultant is obtained in the form of a symmetrical determinant as follows : — (aii) (aci) (ad,) (ac,) (ad,) + (fci) (Wi) (adi) {bdi) (cd,) To render the law of formation of the resultant more ap- parent, the following mode of procedure is given : — Let the two equations be ax A + bx 3 + ex 2 + dx + e =0, «i#* + b,x 3 + c,x 2 + d,x + e, = ; whence, following Cauchy's mode of presenting Bezout's method, we have the system of equations a bx 3 + ex 2 + dx + e a, b,x 3 + C,X 2 + d,x + e, ax + b ex 2 + dx + e a,x + b, C,X 2 + d,x + e x ax 2 + bx + c dx + e a,x 2 + biX + c x d,x + e? ax 3 + bx 2 + ex + d e a,x 3 + b,x 2 + c,x + d x 6i which, when rendered integral, lead, on the elimination of x 3 , x 2 , x, to the following form for the resultant : — (ab,) (flc,) (ad^ (««0 (ad) (adi) + (be,) (a*) + (M,) (be,) (ad,) (aex) + (bd,) (be,) + (cd,) (ce,) (ae x ) (be,) (ce,) (de,) 326 Elimination. If, now, we consider the two symmetrical determinants, (a&i) [aci) (adi) (ae,) {ac,) {ad,) {ae,) {be,) {b Cl ) (bd,) (ad,) (ae,) {be,) (ce,) ' {bd,) {ca\) (ae,) (be,) {ce,) (dc,) \ the formation of which is at once apparent, we observe that R is obtained by adding the constituents of the second to the four central constituents of the first. Similarly in the case of the two equations of the fifth degree ax 5 + bx % + cx z + dx 2 + ex + f = 0, tfia? 5 + b,x* + c,x z + d,x 1 + e,x + f = 0, the resultant is obtained from the three following determi- nants : — (b Cl ) (bd,) (be,) (bd,) (be,) (ce,) (be,) (ce,) (de,) , (Cdy), (ab,) (ac,) (ad,) (ac,) (a/,) (ac,) (ad,) (ae,) (a/,) (bf,) (ad,) (ae,) (a/,) (bf,) (cf,) (ae,) (a/,) (bf,) (cf) (df) (*/i) (bf) (cf,) (df) (ef) by adding the constituents of the second to the nine central constituents of the first, and then adding the third to the central constituent of the determinant so formed. The student will have no difficulty in applying a similar process of superposition to the formation of the determinant in general. (2.) We take now the case of two equations of different dimensions, for example, ax* + bx z + cx~ + dx + e = 0, a,x~ + b,x + c, = 0. Multiplying these equations successively by a, and ax 1 , a,x + b, ,, (ax + fyx 1 , Bezoufs Method of Elimination. 327 and subtracting each time the products so formed, we find the two following equations : — (ab^x* + {acijx* - da x x - ea x = 0, (aCi)x 3 + {(Jci) - da x }x 2 - {db x + ea^x - cbi = 0. If, now, we join to these the two equations ciiX 3 + bxX* + c x x = 0, dix 2 + biX + Ci = 0, we shall have four equations by means of which x 3 , x 2 , x can be eliminated; whence we obtain the resultant in the form of a determinant as follows : — (abi) (aci) da i ea L (aci) (bCi) - da x dbi + ea x eb x a x bi - Ci «! -b x - Ci This determinant involves the coefficients of the first equa- tion in the second degree, and the coefficients of the second equation in the fourth degree, as it should do ; whence no extraneous factor enters this form of the resultant. "We now proceed to the general case of two equations of the m th and n th degrees. Let the equations be n ; and let the second equation be multiplied by x m ~". We have then b x m + bi x'"- 1 + boX m - 2 + ... + b n x m - n = 0, an equation of the same degree as the first. This equation has, however, in addition to the n roots of \p (x) = 0, m - n zero roots ; 328 Elimination. so that we must be on our guard lest the factor a m m ~" (i.e. the result of substituting these roots in +... + b n x m -> 1 ' a x n - 1 + tf,.r"- 2 + . . . + a t! _ x a n x m ~ n + a n ^x m - n ~ l + . . . + a K b x n ~ l + b^ 2 + . . . + b n _y b n x m ~ n which, when rendered integral, are all of the (m - l) th degree ; whence, eliminating x m ~\ x m ~ 3 , . . . x as independent quantities between these n and the m - n equations, b x m - 1 + b x x m - z + box" 1 - 3 + . . . =0, b x m - z + b,x m ^ + . . . = 0, b x n + b l x n ~ 1 + ... + b n =0, we obtain the resultant in the form of a determinant of the m th order, the coefficients of the first equation entering in the degree ji, and the coefficients of the second equation entering in the degree m ; whence it appears that no extraneous factor can enter ; and that the resultant as obtained by this method has not been affected by the introduction of the zero roots. If It be the resultant of two equations, $ (x) = 0, xfj (x) = 0, whose degrees are both equal to m, the resultant R' of the system A (.r) + n\P (x) = 0, A> (*) + fx\P (x) = is {\fi'-\'fi) m Il; for each of the minors (a r b s ), which in Bezout's method con- Besoufs Method of Elimination. 329 stitute the determinant form of R, becomes in this case \a r + fib n \'a r + f/b r -(V-*70 (**■); \a s + fxb s , \'a a + jxb s whence R' = (X/x - X'/j.)' n R, since R is a determinant of order m. 146. "We conclude the subject of Elimination with an ac- count of a method which is often employed, but which has the disadvantage, when applied to equations of higher degree than the second, of giving the resultant multiplied by extraneous factors. The process about to be explained is virtually equiva- lent to that usually described as the method of the greatest common measure. In forming by this method the resultant of the two quadratic equations ax 2 + bx + c = 0, aiX~ + b x x + Ci = 0, we multiply these equations successively by «i and a, Ci and c, and subtract the products so formed. We thus find the two linear equations (abi) x + (aci) = 0, (aci) x + (bcx) = ; from which, eliminating x, we have {actf- (a6x)(5 Cl ) = 0. As the degree of this expression is four, and its weight four, it can contain no extraneous factor, and is a correct form for the resultant. To form by the same process the resultant of the cubic equations ax z + bx 2 + ex + d = 0, a^x z + biX 2 + c x x + d x = 0, we multiply these equations successively by a t and a, r/, and d, and subtract each time the products so formed. We have then (ab^x 2 + {aci)x + (adi) = 0, [ad^x 1 + (bdi)x+ [cd x ) = 0. 330 Elimination. Now, eliminating z between these two quadratics by means of the formula above obtained, we find for their resultant (ah) (ad,) 2 j (a&i) (aci) (ac,) (ad,) — X (ad,) (ed\) | {ad,) (bd,) (bd,) (ed\) an expression whose degree is 8 and weight 12, in place of de- gree 6 and weight 9 ; whence it appears that it ought to be di- visible by a factor whose degree is 2 and weight 3. This factor must therefore be of the form l(bci) +m(adi). We proceed now to show that it is (adi) ; and to find the quotient when this factor is removed. For this purpose, retaining only the terms which do not directly involve (ad,), we have (ab 1 )(cd l ){(ab l )(ed 1 ) + (ea 1 )(bd 1 )}, which is divisible by (adi), since (be,) (ad,) + (ca*) (ba\) + (ah) (ca\) = 0. Expanding the determinants, and dividing off by (ad,), we find ultimately the quotient (aa\y - 2 (ah) (ed\) (ad,) + (bd,) (ca,) (ad,) + (ca,y(cd<) + (ah) (bdtf - (ah) (bd) (ed 1 ), which, being of the proper degree and weight, is the resultant. If we proceed in a similar manner to form the resultant of two biquadratic equations, by reducing the process to an elimi- nation between two cubic equations, we shall have to remove an extraneous factor of the fourth degree, which is the condition that these cubics should have a common factor when the biquad- ratics from which they are derived have not necessarily a com- mon factor ; and in general, if we seek by this method the resultant of two equations of the n th degree, eliminating between two equations of the (n - l) th degree, we shall have to remove an extraneous factor of the- order 2n - 4. This method there- fore is inferior to all the preceding methods ; and it cannot be Discriminants. 331 conveniently used except when, from the nature of the investi- gation, extraneous factors can be easily removed. 147. Discriminants. — The discriminant of an equation in- volving a single variable is the simplest function of the coeffi- cients, in a rational and integral form, whose vanishing expresses the condition for equal roots. We have had examples of such functions in Arts. 43 and 68. "We proceed to show that they come under eliminants as particular cases. If an equation t\.r) = has a double root, this root must occur once in the equation f(x) = ; and subtracting xf(x) from nf(x), the same root must occur in the equation nf(x) — xf'[x) = 0. This is an equation of the (n - l) th degree in x ; and by eli- minating x between it and the equation f (x) = 0, which is also of the (n- l) th degree, we obtain a function of the coefficients whose vanishing expresses the condition for equal roots. The degree of this eliminant in the coefficients of f{x) is 2 (n - 1) ; and its weight is n (n - 1), as may be seen by examining the specimen terms given in section (1), Art. 142. Expressed as a symmetric function of the roots of the given equation, the discriminant will be the product of all the differences in the lowest power which can be expressed in a rational form in terms of the coefficients. Now the product of the squares of the differences n (cti - a 2 ) 2 can be so expressed ; and since it is of the 2(w - l) th - degree in any one root, and of the n(n - l) th degree in all the roots, it follows that the discriminant multiplied by a numerical factor is equal to a 2(n_I) n (t^ - a 2 )~ ; and is, moreover, identical with the eliminant just obtained. If the function f(x) be made homogeneous by the introduc- tion of a second variable y, the two functions whose resultant is the discriminant of f{x) are the differential coefficients of f(x) with regard to x and //, respectively. In the same way, in ge- neral, the discriminant of a function homogeneous in any num- ber n of variables is the result of eliminating the variables from the n equations obtained by differentiating with regard to each variable in turn. 332 Elimination. = o. Examines. 1 . Finx + 03 = 0, a^x z + 2a%x + ai = 0. Hence the condition = J=0. That the other condition for a triple root is / = may be inferred from Ex. 10, p. 315 ; for when three roots are equal the discriminant must vanish, and it is of the form ll z + mJ*. 5. Prove that the discriminant of the product of two functions is the product of their disc rimin ants multiplied by the square of their eLLminant. This appears by applying the results of Art. 141 and the present Article ; for the product of the squares 01 tie differences of all the roots is made up of the product of the squares of the differences of the roots of each equation separately, and the square of the product of the differences formed by taking each root of one equation with all the roots of the other. 148. Determination of a Root common to two Equations. — If R be the resultant of two equations U '= a m x m + dm^x" 1 ' 1 + ... + Oo = 0, V^b n x n + b,^x n -* +... + 6 = 0, and a any common root, then dR dR dR _ da x _ da 2. _ da^ _ „ a ~dR~dR = 7R = '°' da da x da z 334 Elimination. To prove this we substitute in R, for a and b , rr lt - 77 and b - r, and obtain an identical equation connecting 77, V which is satisfied by every value of x, and which is of the form R = 77

T _ dxL fltop+i da p+1 da p+1 and when o is a common root of the equations V = 0, and V= 0, we have, substituting this value for x in the preceding equations, dR_ dR da p da 1H x which proves the proposition. A double root of an equation can be determined in a similar manner by differentiating the discriminant A. 149. Symmetric Functions of the Roots of two Equations. — If it be required to calculate a symmetric func- tion involving the roots a t , a 2 , a 3 , . . . a,„, of the equation (x) = a x m + a x x m - x + a«x m -- + ... + a m = 0, (1) along with the roots j3 b /3s, j3 3 , • . • /3„, of the equation ^ (//) - \t + i>i v n ~ x + hy n ~ 2 + ... + &„ = 0, (2) we proceed as follows : — Assume a new variable t connected with x and y by the equation t = \x + (iy ; and let //be eliminated by means of this equation from (2). The result is an equation of the n th degree in x whose coefficients in- volve A, n, and t in the n th power. Now let x be eliminated by any of the preceding methods from this equation and (1). We obtain an equation of the mn th degree in t, whose roots are the mn values of the expression Xa + fi(5. Miscellaneous Examples. 335 If, now, it be required to calculate in terms of the coefficients of cj) (x) and \p (y) any symmetric function such as "2a p (3 q , we form the sum of the (p + q) ih powers of the roots of the equation in t. We thus find the value of S (\a + /J.fi) p+q expressed in terms of the original coefficients and the several powers of A and fx. The coefficient of X p n q in this expression will furnish the required value of '2a p /3 ? in terms of the coefficients of '/3) 3 . 2a'j8' /3' 3 This appears by eliminating x and y from the equations ax + fry = 0, a'x + fry = ; for from these equations we derive (a* + $yf = 0, («* + frj) (<*'* + 0'y) = 0. («'* + 0» 2 = °- The determinant above written is the result of eliminating x 2 , xy, and y 2 from the latter equations ; and this result must be a power of the determinant derived by eliminating x, y from the linear equations. 5. Prove similarly 3cr/3 3a/3 2 /3 3 a 2 /3'+2aa'/3 2a/30' + a'jS- /3 2 /8' (a/3' - a'/3)<\ a' 2 /3 + 2oa'/3' 2a'/3j8'+ a/3' 2 0/3' 2 3o' 2 j8 3a'/3' 2 )8' 3 Miscellaneous Examples. 337 This appears by deriving from the linear equations the following equations of the third degree : — {cue + Py) 3 = 0, {ax + fit/) 2 {a'x + &i/) = 0, &c, and eliminating x z , x-y, xy-, y 3 . 6. Prove the result of Ex. 12, p. 278, by eliminating the constants \, fx., A.', (*.', from four equations , \a + /u. A.0 + n „ a = —7~ ,1 P ~ ~T7, ,> &c -> A0+/1 \ p + fl connecting the variables in homographic transformation. 7. Given V = -4m 2 + 2Buv + Cv-, V = A'u 2 + 2B'uv+ CV, u s ax- + 2bxy + cy 2 , v = a'x 2 + 2b' xy + c'y 1 ; determine the resultant of U and V considered as functions of x, y. Since U=A{u- av){u- p~v), V ' = A'(u-a'v){u- ffv), if U and V vanish for common values of x, y, some paii - of factors, as u - av and u - a'v, must vanish ; whence forming the resultant of u - av and u - a'v, and re- presenting the resultant of u and v by R{u, v), we have R{u — av, u - a'v) = (o — o') 2 R{u, v) ; and multiplying all these resultants together, we find R{U X , Vm) = A^A'^a - a') 2 (j3 - /3') 2 {a - 0') 2 {& - a') 2 {R{l(, v) }*, or R(U„ V X ) = {R{U, F)} 2 {R(u, v)}\ 8. Prove that the equation whose roots are the differences of the roots of a given equation /(.r) = may be obtained by eliminating x from the equations /(*) = 0, />) +/"(*) j^ + ^'"^ nfrs + &c - = ° j and determine the degree of the equation in y(cf. Art. 44). Z CHAPTER XIV. ("OVA HI ANTS AND INVARIANTS. 100. Definitions. — In this and the following Chapters the notation (oo, «i, (h, ■■ ■ a n )(x, y) n will be employed to represent the quantic a u x n + na x x n ~'y + "^ a 2 x n ~ 2 y 2 + . . . + na^xif 1 + a n y n , which is a homogeneous function of x and y, written with bino- mial coefficients. If we put y = 1, this quantic becomes U n of Art. 35. Let be a rational, integral, and homogeneous symmetric function, of the order zj, of the roots a„ a 3 , a 3 , . . . a„ of the equation TJ n = (a , (h, a 2 . . . a„)(x, l) n = 0, this function involv- ing only the differences of the roots ; then if di - x a 2 - x a n — x be substituted for a { , a., . . . a n , respectively, the result multi- plied by TJ,? (to remove fractions) is a covariant of TJ n if it in- volves the variable x, and an invariant if it does not involve x. From this definition of an invariant we may infer at once that a„'

in the orders ■&, zs\ zs", &c. . . , respectively. We may substitute for each root a, as before, and remove a - x fractions by the multiplier TJfTJf'TJ,?" ' . . . . &c. If the result involves the variable x, we obtain a covariant of the system of quantics U p , U q , U n &c. ; and if it does not, (j> is an invariant of the system. 151. Formation of Covariants and Invariants. — We proceed now to show how the foregoing transformations may be conveniently effected, and covariants and invariants cal- culated in terms of the coefficients. With this object, let the symmetric function of the differences of the roots be expressed in terms of the coefficients as follows : — a ~, a x , # , we have r/ 2 S( 7 - j3) 2 (8 - a) 2 = 24 (a^- 4^ + 3ff 2 2 ). These transformations, therefore, do not alter equation (1) : again, since in this case \p (a, |3, 7, S) is a function of the diffe- rences of the roots, \p is unchanged when a - r, j3 - x, &c. . . . , are substituted for a, /3, 7, §. We infer that a^cti - 4^3 + 3^. 2 is an invariant of the quartic 27" 4 . We observe also, in accordance with, what was stated in Art. 150, since c/>-(j3-7) 2 (a-S) 2 + (7-«) 2 (i3-S) 2 + (a-/3) 2 (7-S) 2 , that each of the three terms of which is made up involves all the roots in the degree zr, which is here equal to 2. In a similar manner it may be shown that a | y-«)(j3-S)-(«-j3)(y-S)}{(a-/3)(y-S) - leads to an invariant, = ±\p, that is is unchanged (except in sign, when its type term is the product of an odd number of differences of the roots, i. e. when its weight is odd), when for the roots their reciprocals are substituted, and fractions removed by the simplest multiplier (aia 2 a 3 ... a„)~. From another point of view an invariant may be regarded as a covariant reduced to a single term. 152. Properties of Covariants and Invariants. — Since is a homogeneous function of the roots, the covariant derived from it may be written under the form a.i — x a 2 - x a„ - x where ts is the order, and k the weight of (p. Also, as

is an invariant, rm = 2k. For, in this case $ and \p are the same function, and conse- quently their weights k and rm - k also the same. (2). Ail the. invariants of qualities of odd degrees are of even order. For if n be odd, it is plain from the equation ms = 2k that zs must be even, and k a multiple of n. (3). All covariants of qualities of even degrees are of even degrees. For in this case nzs - 2/c is even. (4). The resultant of two covariants is always of an even degree in the coefficients of the original quantic. Examples. 343 For, the degree of the resultant expressed in terms of the orders and weights of the covariants is zs(im'- 2k) + -zs'{nzs - 2k) = 2{nzszs r - tssk'- ts'k). "We add some examples in illustration of the principles ex- plained in the preceding Articles. Examples. 1. Show that the resultant of two equations is an invariant of the system. 2. Show that the discriminant of any quantic is an invariant. 3. Prove directly that any function of the differences of the roots of the cova- riant \ai — x az — x 03 — x a n — x/ equated to zero is a function of the differences of aj, a-z, 03, . . . a«. 4. If a, #, 7 ; and a, £' be the roots of the equations U = ax 3 + 3bx 2 + 3cx + d = 0, V" = a'x 2 + 2b'x + c = ; express in terms of the coefficients the function (13 - 7 y- (a- a ') («- j3') + (7 - a)- {$ - a') (0-0) + (a- 0) 3 (7 - «') (y-/3')- Denoting this function by = 9 { a' (bd — c 2 ) — b' (ad — be) + c (ac — b 2 ) } . Attending to the definition at the close of Art. 150 we observe that this function is an invariant of the two equations ; for it involves all the roots of the cubic in the second degree, and all the roots of the quadratic in the first degree. If, in fact, we make the substitutions of Art. 150, and render the function integral by multiply- ing by IT 2 V, the result will not contain x, and is therefore an invariant of the system. The geometrical interpretation of the equation = is that the quadratic V should form with the Hessian of the cubic V a harmonic system. 5. If a, P, 7; a, |3', 7' be the roots of the equations ax 3 + Zbx 2 + ocx + d = 0, a'x 3 + Zb'x 2 + Zc'x + F n + . . . + — ^ D>F» + . . . , 1 . I 1 . Z .6 . . . r Formation of Covariants hj the Operator D. 345 where F is the result of making x = in F(U„, U n . u . . U ), viz., F = F[a ni a n _„ . . . a ), and IJ = a —- + 2a 1 -— +3ao -— + ... + na,^ —— . ««! «rf : rfa 3 oa re In forming a covariant by this process, the source jF with which we set out is altered by the successive operations D till we arrive at the original function F(a , a h . . . a n ), from which the source was formed. Since this is a function of the differences, the coefficient derived by the next operation D vanishes, and the covariant is completely formed. The corresponding operations § on the symmetric function i// have the effect of reducing the degree in the roots by one each step, the final symmetric func- tion containing the differences only. Thus the successive operations supply between the roots and coefficients a number of relations equal to the number of coefficients in the covariant. The degree m of the covariant is plainly equal to the number of times 8 operates in reducing [-, -, ...— j \cti a> a n J the weight of \f> is rns - k, where k is the weight of <£ (a x , a 2 , . . . a„) ; hence the degree of the covariant whose leading coefficient is a ~

m - 2k, the same value as before obtained. We add two simple examples in illustration of this method. Examples. 1 . Form the Hessian of the cubic OqX 3 + 3(71 X 2 + 3(1-2 x + (lz = 0. Taking the function H ' s= a a 2 - «r, we find, as in Art. 151, « o 2 2a 3 03 - 7 ) 2 = 18(« 2 2 _ ax a z ). Operating on the left-hand side by 5, and on the right-hand side by D, we obtain -< ?0 2 22af)3-7) 2 =18(«,ff2-fl'o^) \ 346 Covariants cytd Invariants. ■\" rating in the Bame way again, S2(jB-7)» = 36(«i 8 -aoaa). The next operation causes both sides of the equation to vanish. Hence the re- quired oovarianl is, as in Art. 151, (n\(ii - ay) + (0003 - aiaz)x + (a a.: - (ir)x-. We tind at the same time the corresponding expression in terms of x and the roots. 2. Form the Ilessian of the biquadratic a,,* 4 + 4aix 3 + 6anx 2 + iazx + a± = 0. The covariant whose leading coefficient is JI = a a-z - «i 2 is called the Ilessian of the biquadratic. Its degree is 4, since rs = 2, and k - 2 ; and .-. n~ZJ - 2k = 4. Changing the coefficients into their complementaries, the source of the covariant is <74<7-2 — <73 2 , and wo easily find -H!r= (a a2 — ai i )x i + 2(a a 3 -aiO2)x 3 + (a n± + 2aia i - 3a-r)x- + 2((7i« 4 -(7 2 «3)a;+ (a2«4-«3 2 ). 154. Theorem.* — In the discussion of covariants through the medium of the roots, as in the previous Articles, the following proposition, due to Mr. Michael Roberts, is of importance : — Any function of the differences of the roots of two covariants is a function of the differences of the roots of the original quantic. Let (#„ B lf B,, . . . B p ){.r, i/Y = B [x - fry) [x - fry) ...(•- fry), (Co, C h C* ... 0,)(*,y)*- C {x- 7 iy)(*- 73 y) . . . (*-y ff y) be two covariants of the quantic («r„, «i, ff 2 , • • . a„)(.r, y) n . Operating with D or S on the identical equation Bo(3r P + PB^^ + P \ P " l B,(3 r p - 2 + . . . + B, = 0, Quarterly Journal of Mathematics, vol. v., p. 4S. Homographic Transformation applied to Covariants. 347 and remembering that, in general, Df = a * S<£, where f( a w ( '\, Os, • • • a n ) = Oo'Qiai, a;, • • ■ cr»), we have p {B u p r i>- 1 + (p - 1) B^l*-* +...+ B p .i) (1 + Sfr) = ; and, therefore, S/3, = - 1 ; similarly 87* = - lj whence Sf/3, - Ts ) = 0, proving that /3,- - y s is a function of the differences of the roots ai, a 2 , a 2, . . . a n . 155. nomographic Transform a tion applied to the Theory of Covariants. — Hitherto we have discussed the theory of covariants and invariants through the medium of the roots of equations. We proceed now to give some account of a different and more general mode of treatment, by means of which this theory may be extended to quantics homogeneous in more than two variables, such as present themselves in the numerous important geometrical applications of the theory. Although this enlarged view of the subject does not come within the scope of the present work, we think it desirable to show the connection between the method of treatment we have adopted and the more general method referred to. With this object we give in the present Article two important propositions. Prop. I. — Let any quantic TJ n be transformed by the tomo- graphic transformation _ \x + ft X = A' ' , ' > AX + fl if I and I' be corresponding invariants of the two forms, we have 1'= Qtf-XuYI. To prove this, let I = flo" 2 (a, - a,)" (a, - a a ) 6 ... (a, - a„)', each root entering in the degree ■&. 3 I s CovarianU and Invariants. Now, transforming the similar value of I', since x = - — ^-r, A — A x we have /_ , _ (V - \'fx) (a P - a g ) "'' ° 9 "(A-AVXA-A'a,)* Again, trrmsforniing U ni and rendering the result integral, V I akes the form '/,,'(.'•'- «i'j(.r'- a,') . . . (#'- a,/), where r/ ' = a (A - A'(«i) (A - A'a-) ... (A - A'a„) ; making these substitutions for all the differences, and for aj, the denominators of the fractions which enter by the transfor- mation disappear ; and we have, finally, I' = (A|t' - \'n) K I. Prop. II. — If(.<) thus derived; since the factors A - Xa u A - A'a 2 , . . . all enter iu the same degree zs in the denominator (for each root enters the source in the degree w), they will all be removed by the multiplier a { '~, and the transformed value of $ (x) is [\fi-XmY^(x). Reduction of Homograplric Transformation. 349 156. Reduction of nomographic Transformation to a Double Linear Transformation. — With a view to this reduction let the quantic be written under the homogeneous form TI n = a x n + na v x n ~ l y + a 2 x n ~~ if + . . . + a n y n ; J. . £ and, in place of putting as before x = ^j— f „ and removing A x + ju *y* \x I tin x fractions to make TJ n integral, let now - = w , , „ where - y Xx+fxy y x' and — are the variables in the ordinary sense. The transfor- mation may therefore be reduced to a linear transformation of both the variables x and y, and can be effected by putting in the original quantic x = \x f + fxif, y = XV + fx y, the introduction of fractions being in this way avoided. Thus we pass from a homographic transformation of functions of a single variable to the linear transformation of homogeneous functions of two variables. The determinant A// - X/x, whose constituents are the coef- ficients which enter into the transformation, is called the modulus of transformation. We are now enabled to restate Propositions I. and II. of Art. 155, in the following way : — Prop. I. — An invariant is a function of the coefficients of a quantic, such that when the quantic is transformed by linear trans- formation of the variables, the same function of the neio coefficients is equal to the original function multiplied by a power of the modulus of transformation. Prop. II. — A covariant is a function of the coefficients of a quantic, and also of the variables, such that when the quantic is transformed by linear transformation, the same function of the neio variables and coefficients is equal to the original function multiplied by a power of the modulus of transformation. :!.■)() Covariants and Invariants. The definitions contained in the preceding propositions are plainly applicable to quantics homogeneous in any number of variables, and form the basis of the more extended theory of covariants and invariants referred to in the preceding Article. "We give among the following examples an application in the case of a quantic involving three variables. Examples. 1 . Performing the linear transformation x = \X+(iY, y = \iX+ ml', if ax- + lb xy + cy 2 = AX 2 + 2BXY + CY 2 , prove that AC-B 2 = (A M i - Mufiac - b 2 ). 2. Performing the same transformation, if (a, b, c, d, e){x, y)* = [A, B, C, D, £)(X, T)*, prove that AE-4BB + 3C 2 = (\m - Aim) 4 (** - ^d + 3c 3 ). 3. Performing the same transformation, if ax* + 2bxy + cy 2 = AX 2 + 2BXY + CY 2 , and B] x 2 + 2b x xy + ny 2 = Ai X 2 + 2B 1 XY + C, Y\ prove that ACi + AiC- 2BB X = {\fjn - Aiju) 2 («o + aic- 2bh). This follows from Ex. 1, applied to the quadratic forms (a+Kai)x 2 + 2{b + Kb l )xy+(c + KC 1 )y 2 = {A+KA 1 )X 2 +2(B + K Bi)XY+[C + KC i )Y 2 , hy comparing the coefficients of k on hoth sides. Whence we may infer that, if two quadratics determine a harmonic system, the new quadratics obtained hy linear transformation also form an harmonic system. For their roots being a, & and en, £i, we have «*l { (a - ai)(j8 - 0i) + (a - >3i) (j8- oi)} = 2 (w, + a y c-2bb{). 4. If the homogeneous quadratic function of three variables ax 2 + by 2 + cz 2 + 2fyz + 2gzx + 2h xy be transformed into AX 2 + BY 2 + CZ 2 + 2FYZ + 2GZX + 2HXY by the linear substitution * = \iX + I l 1 Y+viZ, y = \2X + p 2 Y+v2Z, z = \ 3 X + w r+ V3 Z; Linear Transformation. 351 prove the relatior L A H G a h 9 R B F = (A1/U2V3) 2 h b f G F C (1 f c where the determinant (A-i/^s) is the modulus of transformation. This is easily verified by multiplying the proposed determinant of the original coefficients twice in succession by the modulus of transformation written in the form Ai M M Mi M2 Ms v\ Vl Vi and comparing the constituents of the resulting determinant with the expanded values of the coefficients of X 2 , Y 2 , &c, in the new form. It appears therefore that the determinant here treated is an invariant of the given function of three variables. 157. Properties of Covariants derived from Linear Transformation. — We proceed now to show, taking the second proposition of Art. 156 as the definition of a covariant, that the law of derivation of the coefficients given in Art. 153 imme- diately follows ; that is, given any one coefficient, all the rest may be determined. For this purpose, performing the linear transformation x = X + hY, y = 0X+Y, whose modulus is unity, the quantic (ff , «i, (a , a t , K fli, 02} • • • «n, *, y) = (A A,, A 2 , . . . A, t ,x-hy, y). Expanding the second member of this equation, and con- h 352 Covariants and Invariants. fining our attention to the terms which multiply A; observing also that —77^ = ra, 1 when terms are omitted which would be ah multiplied in the result b} r lr, P, &o., we have t + h(-y d £ + l)+yh*( ) + &o.. which must hold whatever value h may have ; hence dd> d(j) d<\> d(j) li ,=a -j L + 2a 1 -j- + 3a, -/-+...+ na^ -~- , (1) (/.c da x (in. da. aa» and, substituting for the value {B ,B u B 2 ,...B m ){x,y) m , we have mB x m - x y + m (m - 1) BiX m ~ 2 y 2 + . . . + mB„,^y m = BB x m +mDB 1 x m ~ l y + . . . + BB m y m ; whence, comparing coefficients, we have the following equations : BB„ = 0, BB, = B , BB 2 = 2B U . . . BB m = mB m _ lf which determine the law of derivation of the coefficients from the source B m ; the leading coefficient B being a function of the differences, since BB^ = 0. The calculation of the coefficients is facilitated by the follow- ing theorem which has been proved already on different prin- ciples : — Tiro coefficients of a covariant equally removed from the extremes become equal {plus or minus) when in either of them a , a h . . . a n are replaced by a m a„^, . . . a n , respectively . To prove this, let the quantic be transformed by the linear substitution x = OX + Y, y =X + 0Y, whose modulus = - 1. Thus (flo, «i, o 2 , . . . a n )(x, y) n = (a n , «„_„ a n _ 2 , . . . a )(X, Y) n , and, by definition, any covariant {«n, "/«-!, «n-2, • • • do, X, Y) = (- 1) K (fl , «i, ff s , . . . Cl n , X, 1j) = (- 1)" (j>(a u , ft,, a 2 , ... a n , Y, X) ; Linear Transformation. 353 whence it follows that the coefficients of the covariant equally removed from the extremes are similar in form, and become identical (except in sign when k is odd) when for the suffixes their complementary values are substituted. We may infer similarly that a covariant satisfies the diffe- rential equation d(t> d _ d d(b dy da n . x da n _ 2 da„^ da as well as the equation (1) already given. Again, if a n - 7 J -- + 2a n .i -~- + 3a n - 2 - — + . . . + na x ~ = 0, da n _i dcin-a aa»i-3 da n either of which may be regarded as contained in the other, since if we make the linear transformation x = T, y = X (whose modulus = - 1), we have from the definition of an invariant (a„, cin-i, a n _ 2 , . . . «o) = (- 1)

~ M\" dX dYf du_dUdX dUdY dy dX dy dY dy l_/_ dU ,dU\ + dY dy~ M\ ^dX* dt} which equations may be put under the form dy~ A \MdYJ +,x \ MdXf dx A \MdYj +fX \ MdXj' and since /(AX + fiY, XX + fx'Y) = F(X, Y), Linear Transformation. 355 changing X and Y into jf^y, an respectively, the proposition is proved. In an exactly similar manner, changing X and Y into it may be proved that The results (1) and (2) may he applied to generate cova- riants and invariants as we proceed to show. Suppose /(a?, y) and u to be co variants of any third quantic v, where v may become identical with either as a particular case ; also, denoting by F C (X, Y) and U c the same 'covariants ex- pressed in terms of the X, Y variables and the new coefficients of .v after linear transformation, we have, by Prop. II., Art. 156, the identical equations MpF{X, Y) = F e (X, Y), and M*U = U c ; whence, substituting from these equations in (1), proving that /( — , - -7-) is a co variant of v. And in a similar manner it is proved from (2) that leads to an invariant or covariant of v, according as u is of the n ih or any higher order. We add some applications of this method of forming inva- riants and covariants. 2a2 356 Oovariants and Invariants. EXAMPLES. 1. If — , be substituted for z and y in the quartic [a, b, e, d, e) (z, y) x m V, dy ax :tnd tin resulting operation performed on the quartic itself, show that the invariant /is obtained. We find (a, />, r, d, e) (j, - ~\ U= 48 (ac - ibd + 3c 2 ). 2. Trove, by performing the same operation on H x , the Hessian of the quartic (see Ex. 2, Art. 153), that the invariant /is obtained. Here we find (a, b, c, d, e) (—, -~) H x =12 (ace + Ibcd - ad* - eb* - c 3 ). \dy dzj 3. Trove that (a, b, c, d) (—, -— Yg x =: - I2(a 2 d n - ~ 6abcd+ iac 3 + iPd-3b 2 c 2 ), \dy dzj where O z is the cubic covariant of the cubic (a, b, c, d) (z, y) 3 . 4. Find the value of ./ ,„ /du\ 2 , du du ... *./du\- where u = (a, b, c, d) (z, y) 3 . Am. - 9H t 2 . 159. Prop. II. — If tj>{a , tfi, a 2 , . . . a n ) be an invariant of the form [a , a x , a 2) . . . a n ) (x, y) n , and u any quantic of the n th or any higher degree, (d»u d n u d"u d n u\ Vte"' dx> l - l uy dx n - 2 d>f ' " ' dy") is an invariant or covariant of u. To prove this, let x^XX+fiY, x'=\X'+tiY', y = XX+^Y, y'=XX' + f/Y'; and, transforming as in the last Proposition, r d , d d , d dx J dy dX dY Linear Transformation. 357 also, transforming u, we have whence X dX +J dt) u '{** + *lj)*t and writing this equation when expanded under the form (Do, A, A, • • • D»){X' t Y') n = ,,/,„ d u f 4, • • • d n ){x\ y')\ we have, from the definition of an invariant, 0(2)„, A, A, • • • Da) =M*(do, d u d i} ... d n ), showing that (d n , d lf d 2 , . . . d n ) is an invariant or co variant. When x, y and x', y are transformed similarly, as in the present Proposition, they are said to be cogredient variables. Examples. 1 . Let the quadratic aox- + 2a\xy + a^y 1 become A X- + 2A\XT + A^T 2 . "We have then, as in Ex. 1, Art. 156, A Az — A£- = M 2 (a Q ai — a x 2 ). Now since r, 2 d-U t ov>Tr , d 2 U v ,d 2 U n&u^c. . ,&u ,,d 2 u X *JX* +2XY dXlfr+ Yi dYi = ^d*+ Uy d^ + i ''w> it follows from the last result, considering X', T' and x', y as variables, that d-Ud^U I d 2 U y_ {d-udhi I d-u \ 2 ) dX 2 dY 2 ~ \dXdY) ~~ J [dx* df ~ \dHdy) )' This covariant is called the Hessian of U. 2. When u has the values (a, b, c, d)(r, y) 3 , and (a, b, e, d, e)(x, y) A , what covariants are derived by the process of the last example ? Ans. (1). (ac - V) x' + (ad - bc)xy + (bd - c 2 )y 2 . (2). {ae - 0-) x* + 2(ad- bc)&y + (ae + 2bd - 3c 2 ) x 2 ,f- + 2 (be - cd)xy 3 + (ce - d 2 )y*. 358 Covariants and Invariants. 160. Prop. III. — If any invariant of the quantic in x, y, U+k{xy'-x'y) n be fanned, the coefficients of the different poivers of k, regarded as homogeneous functions of the variables x', y', are covariants of U. For, transforming U by linear transformation, let (a , a„ a-., . . . a n ) [x,y) n = (A , A x , A 2 , . . . A n ) {X, Y)"; also if x, y and x', y be cogredient variables, xy' - x'y = M{XY' - X'Y). Whence (a , a 1} a 2 , . . . a n ) (x, y) n + k{xy - xyf becomes when transformed [A ,A l3 A Si . . . A n ){X, Y) n + kM n (XY'-X'Y) n ; and forming any invariant

r is a covariant. When [xy - x'y) n is replaced by (b e , b lf b 2 , . . . b n )(x, y) n , we have the following Proposition which is established in a similar manner : — If <{> { a o> flu a 2 , . . . a n ) be an invariant of(a , «,, a 2 , . . . a n ) (x, y) n > all the coefficients of k in [Oo + kb , ai + kb ly . . . a n + kb n ) are invariants of the system of qnantics (a , «i, flfe, . . . a n ) [x, y) n , (b , b l} b it ... b n ) (x, y) n ; or, which is the same thing, ( b — + bi — + . . . b n — ) , &c, &c, \ da da x da n ) are invariants of the system. If, further,

d(j> dx dy dp dp dx dy is a covariant of these quanttcs. For, transforming ^ and p by the linear substitution x = \X + iuY, y^XX + fxY, we have giving *(X,F) = *(*,?), ¥(X, F)-*(*,y), dX dx dy ' d^¥ _ dp dp dX dx dy' d$ dY d(f> ,d d^r dip ,dp ^Tx^ Whence dif dY = /* d

, d dx dy dip ,dip dx dy i^y «y ^ »y »y i, 2, ■ ■ ■ 3 y" 1 -xy X 2 X 2 2xy y % Miscellaneous Examples. 361 we have 111 Il2 il3 u 4 hi In III IiZ hi I33 V w U V w 3 0, where 2I pq = a p o q + a q c p - 2b p b q Expanding this determinant we have (/22/3S-/23 2 ) tP+ (Tszlll-hl 2 ) r 2 + (/l, J 33 -/i2 2 ) W* + 2 (Jsi Ix% - Iu In) VW + 2(J23li 2 - Jsa/si) WV+2{h 3 I 31 - I 3 3il2) 0T ■ 0. (1) There are two particular cases worth noticing : — (1). When the three quadratics are mutually harmonic. — In this case ^23 = 0, /31 = 0, J12 = ; and the identical equation assumes the following simple form: — y/iu) VW VW (2). When one of the quadratics W=0 determines the foci of the involution of the points given by the other two, JJ = 0, and V — 0. — In this case /13 = 0, and J33 = ; and making this reduction in the general equation (1), we have (ii 2 2 - J11 Inn) W- = J33 (Jza U- - 2 J r . UV+ In F 2 ) ; but from the equations /13 = 0, and J23 = 0, we find «3 = k(«i£ 2 ), - 2*3 = *c(cia 2 ), c 3 = K(biC2.) ; 4 (« 3 c A - bi-) = k 1 { 4 («i J 2 ) (Ji c 2 ) - (ci a-i) • } , /33 = « 2 {/ll/22-/l2 2 }, and reducing, when k = 1, or W=J(U, V), -{J{V, V)Y = I ii lP-2I 1 oUV-t- InV*. 6. Determine the invariants of the quartic M(x- oi) 4 + A 2 (^-a2) 4 + ... +\ n {x- a n y. Ans. 1= 2A.iA>(ai - a 3 ) 4 , /= 2A 1 A 3 A.3 V (ai , a 3 , 03), where v(oi, a 3 , ... a,) represents the product of the squared differences of «i> «2, • • • a, . 7. Prove that the condition that four roots of an equation of the n th degree should determine on a right line a harmonic system of points may be expressed by whence or equating to zero an invariant of the degree (m-1)(m-2)(«.-3) 8. If y) = Similarly, where J is the ternary invariant of F. Again, since Dx' D X t E>r* b c 1 A B C c d B C D Dth fo •ms of V, we have a' b' c b' c' d' a b c * + a b c y = b c d b c d a b c b c d a' b' c' X + a b' c' y = b' c i' b' e d' JY if*' JX and _*(/)„ -D,)=— Dx, performing the operation Q(D y , -D x )${x,y), or +{D y , - D x ) -

, ))\ Q 3 V=(B , Co, Vo,Eu)()\ and, therefore, ldF Q 3 U = also 4 d

. A tf 2 . Number of Covariants and Invariants of the Cubic. 369 For, from Ex. 6, p. 114, «o 6 (0-y) 2 (y-a) 2 (a-/3) 2 = -27(£ 2 + 4JT 3 ) = - 27« 'A, and transforming this equation as before, «'o 6 (/3-7) 2 (7-«) 2 (a-/3) 2 (^-a) 2 ^-/3) 3 (^- 7 r=-27(^ 2 + 42; 3 ); whence A U~ = G 2 + ±H X \ (5). Solution of the Cubic. The expression {tt is a linear factor of U. For, from the relations in (2) and (3), we have 2« S {Lx + X,) 3 = 27 ( U*/Z - G x ), - 2ao 3 {Mx + M 1 y = 27 ( U yi + G x ) ; and since {Lx + L,) - {Mx + M x ) is a factor of U, the proposition follows. This form of solution of the cubic is due to Prof. Cayley. 164. \ ii in bo i- of Covariants and Invariants of the C'tibio. — The following method of determining the number of covariants and invariants of the cubic is similar to that employed by Professor Cayley for the same purpose : — The cubic has only two covariants, their leading terms being H and G ; and only one invariant, viz., the discriminant A, where a 2 A = G 2 + 4:E 3 , or A = a 2 d 2 + 4«c 3 - 6abcd+ 4db* - 36V. To prove this, let {a, (5, y) be any integral symmetric function of the differences of the roots (of order ar), expressible by the coefficients in a rational form. We have then (Art. 36), ar(a,P,y)=F(a,II,G) (1) (where r remains to be determined) ; and, in the first place, if be an even function of the roots, G can enter this equation in even powers only, since JET is an even function of the roots. 2b 370 Covariants and Invariants of Quadratic, Sfc. Eliminating the even powers of G by means of the relation G 2 + 4ff s = a 2 A, we show therefore that in the case of an even function of the roots equation (1) takes the form fl-*(a,ft y)=F(a,H, A), which may be written <*•*(«, ft y)-F.{a, H, A) ^ ^ ,A) , (2) where st is the order of (a, j3, 7), and F an integral funotion. It is now necessary to prove the following Lemma : — No function of H and A exists which is divisible by a. For, suppose F P (IT, A) to be divisible by a; then making a vanish, we have F P {H\ A') s 0, where H' = - b 2 , A' = 4:db 3 - 3b 2 c 2 , the values of H and A when a vanishes. This equation is plainly impossible ; for, eliminat- ing b by means of the equation H' = - b 2 ,c and d remain in the equation connecting H' and A'. Wherefore equation (2) must assume the form a°{a,(5,y) = F {a, E,A); for the first side of the equation is expressible as an integral function of the coefficients ; therefore so must the second side also, and consequently the fractional part disappears. Now, to extend this result to odd functions of the roots, we have only to multiply the first side^of the equation by fl 3 (2a - |3 - 7) (2/3 - 7 - a) (2y - a - /3), and the second side by 27 G, for]Cr must be a factor of every odd function, since H is even. We are now in apposition to prove the original proposition as to the number of invariants and covariants. For since «" $ is of the form GF(a,H,A), or E (a, H, A), according as ^> is an odd or even function of the roots, it follows Covariants of the Quartie. 371 in the first place that there cannot be an invariant of an odd degree in the roots, since GF (a, H, A) does not remain the same function when a, b, c, d are changed into d, c, b, a, respectively ; and the only invariant of an even degree must be a power of A, since if F(a, H, A) contained a or H besides A, it could not remain the same function when the coefficients are similarly interchanged. Again, the cubic has only two distinct covariants ; for it has been proved that every function of the differences a^cp is of one of the forms F(a, R, A), or GF{a, H, a). Now, considering these forms as the leading terms of cova- riants, every covariant must be expressible as F{U,H X ,±), or G X F(U,H X ,A); that is, every covariant is expressible in a rational and integral form in terms of H x and G x , along with U and A ; or in other words, there are only two distinct covariants. 165. The duartie. Its Covariants and Invariants. — We have shown already that the quartie has two invariants, 7" and J (see Art. 151). From the functions H and G of the diffe- rences of the roots we can derive two covariants H x and G x , whose leading coefficients are H and G ; for from the relation 0o 2 2(a-|3) 2 =48(tfo0 2 -O we derive, by the process of Art 151, a;- S (a - (3y (x - y y {x - S) 2 = 48 ( U U z - UJ) ; and, expanding UU»- U-i, we have M x ■ (a a 2 - a?) x* + 2 (a a 3 - a x a 2 ) a? + (a a A + 2a x a 3 - 3a 2 2 V + 2(«i« 4 - aza 3 ) x + (a 2 a t - a 3 ). In a similar manner, since G = a-«)»; combining the terms in pairs,*and noticing that S(/3- 7 )(a-8)Z7 s 0, 2 (a -j3) 2 (*- 7 ) 2 (z-g) 2 -S{0- 7 )(*-«)f*-J)+ (a-S)(*-/3)(*- 7 )}», the quantities between brackets being w, 0, w, we have . Q Hz a- which is the required expression for H x . 168. Expression of the Quartio itself by the Qua- dratic Faetors of G x . — From equations (3) a symmetrical value may be obtained for U; for, substituting in those equa- tions in place of A, p, v their values in terms of the roots P\, /o 2 , ps of the equation 4p 3 - Ip + J= 0, we find a~ {v* - w*) = 16 [p 2 - Pi ) U, « 2 ( w 2 - M ») = 16 (p, - Pl ) U, a*(u* -v~)= 16 ( Pl -p 2 )Zr, from which equations, by means of the value of H x in the pre- ceding Article, we obtain (««)*= 16 {ptU-H*), (atf= 16 { P ,U-H X ), (4) {awf = 16 (p z U-H x ). Expression of the Quartic, 6fc. 375 We now [make the substitutions u 2 s ^X 2 , v 2 = A 2 F 2 , w 2 m A 3 Z 2 , where A l} A 2 , a 3 are the discriminants of u, v, w ; thus replacing u, v, 10 by three quadratics X, F, Z whose discriminants are each equal to unity. By means of this transformation the for ms of the quadratics are further fixed, and the identical re- lation connecting their squares (see (1), Ex. 5, p. 361) is ex- pressed in its simplest form. Calculating the discriminants, we find A 1 =(j3 + 7-a-8){/3y(a + S)- 7 o(/3 + 8)}-G3y-a8)«, with similar values of A 2 and A 3 ; whence we have A 1 = -(X- j u)(X-v), A 2 = -0u-v)0u-A), A 3 =-(v-A)(v- / u). Making these substitutions, the preceding equations become (pi - p^ (pi - p 3 ) X 2 = H x - px U, (p 2 -p 3 )(p*-pi)Y 2 =H x -p 2 U, (5) G°3 - pi) (/»3 - pi) Z 2 = H x - p 3 U; from which are easily deduced the following values of U and H x , and the identical equation connecting X, F, Z: — H x = p l 2 X 2 + pSY 2 + p z 2 Z 2 , -U = Pl X 2 + P iY 2 +p 3 Z 2 , (6) = X 2 + Y 2 + Z 2 ; where, as has been proved, X, Y, Z are three mutually harmonic quadratics whose discriminants are reduced to unity in each case. The value of G x may be expressed in terms of X, F, Z as fol- lows. Since u~v 2 w 2 = (fi- v) 2 (v -A) 2 (X- p.) 2 X 2 Y 2 Z 2 - — (P -27 J 2 ) X 2 Y 2 Z\ it we easily find G x =\ 2 - pi) \/k -pi\ */H x - pi U+ (p 3 - pi) 2/o 3 - P3/01). 378 Covariants and Invariants of Quadratic, 6fc. On account of the similarity of the forms Pl X 2 + fnY* + p 3 Z 2 and B, X* + R 2 Y 2 + R 3 Z 2 , which are of a fixed type, we calculate the invariants and cova- riants of kU- \II x by simply changing Pl , p 2 , p 3 into J2„ R 2 , R 3 in the expressions for the invariants and covariants of U. Therefore, since I= iiip2-p3) 2 +{pz-plf+(pi-p2f}, J=-4p 1 p 2 p 3 , and -Bs - & = {p% - p-i) {k - \p x ), R 3 - R r = [p» - pi) (k - \pt), Ri-R 2 = ( pi - p 2 ) ( K - \p 3 ), we find the following values for the invariants of nU-\H x : — p I(k,X) = I*' ~ 3t/)cA + ZT-) ^ 2 > r „. u ., 54j 2 -p v(k,K) = e/K - -5- K~A + — kA" 7^-7; A 3 . If we form the covariants H (K> A) , and G r ( (Cj K) , of 4Q - 4 K 3 - IkX 2 + JX 3 (the reducing cubic rendered homogeneous in k, A), we find, as M. Hermite has remarked, I(k,\) = - 12H( KtK ), J{k,\) = 4 G( Kt a) Again, to calculate the Hessian of kU - \H X , we reduce R X 2 X 2 +R 2 2 Y 2 + R 3 2 Z 2 by the substitutions P SX 2 + p.?Y 2 + p 3 3 Z 2 - - J-(i77+ JV) - - i/^7, ^x 2 + P s r 2 + p ,«z* - j- (75, + jct), the first of which follows from the equations pi = pi pi + \I, pi 1 = p 3 p , + \I, p 3 2 = Pl p 2 + \I, multiplying by Pl X\ p 2 Y\ p 3 Z 2 , respectively; and the second from the first by changing X\ Y 2 , Z 2 into Pl X 2 , p 2 Y 2 , p 3 Z\ Number of Covariants and Invariants of Quartic. 379 In this way we find the following form for the Hessian of kU-\H x :— which may be expressed in the form 1/ dQ TT da\ 3\ dn dkj Again, since P - 27 J* = 16 (p 3 - pa) 2 (/o 3 - pi) 2 {pi - /o 2 ) 2 , and G X = ^I 3 -27J 2 .XYZ; transforming p u p 2 , p 3 into Ri, JR 2 , R.\, we find ij, A) - 27^, A) = G 2 (P - 27 J 2 ), G {K> K)x = QG X . We have therefore expressed the invariants and covariants of kU- \H x in terms of the invariants and covariants of TJ. 171. Xunifoer of Covariants and Invariants of the Quartic. — We proceed to prove the following proposition, which determines the number of these functions : — The quartic has only the two distinct invariants I and J, and tico distinct covariants tvhose leading coefficients are H and G. This proposition asserts that every invariant is a rational and integral function of I and J, and every covariant a rational and integral function of U, H x , G„ I, J. The following discussion is founded on principles similar to those already employed in the case of the cubic. Attending to the observations in Arts. 36, 37, it is plain that if (a, /3, y, S) be any integral function of the differences of the roots expressible by the coefficients in a rational form, we have, in general, considering the equation with the second term removed, a>-

-(a, j3, 7, B) can be expressed as a rational and integral function of a, H, I, J. To prove this, the following lemma is necessary : — There exists no function ofH, I, J which is divisible by a. For, suppose F(H, I, J) to be divisible by a. Making a vanish, we have F[H', /', J') - 0, where H' = - b\ I'=- 4bd + 3r, J' = 2bcd - eb 2 - c 3 (the values of H, I, J, when a = 0) ; and as it is impossible to eliminate b, c, d, e, so as to obtain a relation between H', I' y J', we conclude that no relation such as F(II', I', J') = exists ; and therefore there is no function of the form F(H, I, J) which is divisible by a. We now proceed with the proof of the proposition ; and since, as has been already proved in the case of an even function of the roots, a'^{a, /3, 7, 8) = F(a, H, /, J), Number of Covariants and Invariants of Quartic. 381 ■we have, dividing by a r_5r , *•*(«, (3, y, 8) = JP.fo #, I, J) + 2 FAR >J> J l Again, since the first side of this equation is expressible as a rational and integral function of the coefficients not divisible by a, the second side must be a similar function of the coefficients ; and this, by the lemma just established, is impossible unless such ^FJE, /, J) ,. terms as 2 — ■ disappear. Wherefore a*cp(a,(3,y,E) = F («,H,I,J); and, finally, we have proved that a"$(a, j3, 7, B) may be ex- pressed by the forms GF[a, E, I, J), or F(a, E, /, J), according as

comparing the coefficients of the different powers of A. we can render the last equa- tion an integral function of the roots, which again, expressed in terms of the coef- ficients, takes the form {3P} 3 -27(P 3 -27Q) =0, or Q = 0. 7. If "a quantic have a square factor, prove that the same square factor enters its Hessian. 8. If a quartic have a square factor, the covariant G x has that factor as a quin- tuple factor. 9. If f{x) and (#) be two quartics with unequal roots, the roots of f{x) being a, j8, 7, 5 ; prove that the condition that a quartic of the system Xf{x) + /j. (x) can have two square factors may be expressed as follows : — 1 a or y/ (p (a) 1 £ j3* \/*(JBJ 1 7 7 2 / **•» K M ; tnus obtaining an invariant In of the fourth degree in the coefficients of both equa- tions. = 0. 384 Covarianfs and Invariants of Quadratic, 6fc. 11. Prove that the resultant of two quartics becomes a perfect square when the invariant Ju vanishes. Rendering rational the determinant in Ex. 9, and dividing by the product of the squares of the differences of the roots, we find, introducing the coefficients, Iu = h"? - 64i?; whence, &c, &c. 12. Prove that the sextic covariant G x of the quantic 2, ■ • ■ an) [x, 1)" = arranged in powers of a„ heing « . p — 1 . tp = A p + pA p -\a„ H Ap.ia n l + . . . + A a n P ; prove that DAj = - na n .\jAj.\, and hence show that if \p[a , »i, «2, • • • a r) is a function of the differences so also is ty[Ao, A\, A2, . . . A r ). 16. If the discriminant of a biquadratic be written under the form [Ao, Ai, A 2 , A 3 )[a i} l) 3 , prove that the discriminant of this cubic is 27 2 e7 2 A 3 3 , where A3 is the discriminant of [ao, «i, #2, «3) [x, l) 3 . 17. Form the equation whose roots are [ai), (t>[a 2 ), [a„), where 01, 02, 03, . . . a n are the roots of f[x) = 0, the resultant R oif[x) and -i ; and therefore by the present proposition reducible to a rational and integral function of «//! of the degree p -1, since i// has only ^ values considered as a function of ai, a 2 , . . . a„. Now considering the special cases referred to — (1), when p = 2, and n = 3, it is proved that a linear relation connects

[a 2 ), 0(a 3 ), .... 0(a»), where [x) is a rational and integral function of z of the degree n - 1. Let

~ 1 . Eaising [x] to the different powers 2, 3, ...» in succession, and reducing the exponents of x in each case below n (by dividing by f(x) and retaining the remainder), we have 3 = c + c x x+ c,ar +.... + c n _ l x n ' 1 , (j> n = l + kx + l 2 x x + . . . . + l„-iX n -\ 2c2 388 Transformations. Substituting for .v in these equations each of the roots of the equation /(a;) = 0, and adding, we find, if Si, S>, 8 Z , &c, denote tin sums of the powers of the roots of the required equation, Si - na + fli«i + o 2 s 2 +.... + ff,,_is„-i, 8% = nb + Mi + b 2 s 2 + .... 4 i„_i6„_,, S n = nl Q + liSi + %8 t + . . • . + /,;-!«„_,. Now, expressing 8 i9 s 2 , . . . s„. x in terms of the coefficients of f[x), we have Si, 8 2j . . . S n determined in terms of the coeffi- cients of

{ai), by elimination, which we now give. Since a - + OiX + a 2 x 2 + . . . + a n ^x n ~ x = 0, if this equation be multiplied by x, x z , . . . x"~ l , and the expo- nents of x reduced below n by means of the equation f(x) = 0, we have in all n equations to eliminate dialytically the n - 1 quan- tities x, x 2 , . . . x n ~ l . We thus obtain the transformed equation in the form of a determinant of the n th order,

Although these methods of performing Tschirnhausen's transformation appear simple, yet if they be applied to j ar- Tschirnhausetfs Transformation applied to Cubic. 389 ticular cases the result usually appears in a complicated form. Professor Cajley, hy choosing a form of the transformation suggested by M. Hermite, was enabled to take advantage of the theory of covariants, and thus to complete the transforma- tion for the cubic, quartic, and quintic. We shall content our- selves with showing in an elementary way how Professor Cayley's results for the cubic and quartic may be obtained. 175. Tschirnhausen's Transformation applied to tbe Cubic. — Let the cubic equation ax 3 + 3bx 2 + 3cx + d = be written under the form z 3 + 37/3+ G=0; and let it be transformed by the substitution y = A + ks + z 2 . If Si, Sg, z 3 be the roots of the cubic, and y u y 2 , y* the correspond- ing values of y, we have y 2 - y z = (z 2 - 2 s )(k - 8i), y3-yx = («3-8i)(K-s»), (1) yi-Vi= (Sl-S 2 )( K -2g), and consequently, 2^i - y»- Vs = (2«i - z 2 - s 3 ) k + (2z 2 z 3 - z 3 Zi - Si z 2 ), 2y 2 -y*-yi = (2s 2 -3 3 -z 1 )fc+(2z 3 Zi -ZiZ2-Si«a)t (2) 2y z - yi — y%= (2z 3 -Zi-z 2 )k + (2z x z 2 -z 2 z 3 -z 3 Zi). Wherefore, if the equation in y with the second term removed be r 3 + 37J'r+£'=o, we have from equations (1) and (2) H.' = 27*, G = G K , .'590 Transformations. where H K and G K are the Hessian and cubic covariant of k 3 + ZH K + G ; and the transformation is therefore completed, since y x + y 2 + y 3 can he easily determined. 17G. Tsehirnhausen's Transformation applied to the Unartie. — In this case we do not attempt to form directly the transformed quartic, hut prove the following theorem, which shows how this transformation may be resolved into two others. Theorem. — Tschirnhausen's transformation changes a quartic U into one having the same invariants as ITT + mH x , and therefore in general reducible to the latter form by linear transformation. To prove this, let the quartic x i + p x x z + p 2 x 2 + p 3 x + p t = be transformed by the substitution y = a + a x x + a 2 x 2 + a 3 x z . If Xi, x 2 , x 3 , Xi be the roots of the quartic, and y Xi y 2 , y*, y* the corresponding values of y, we have &2 ~ 2-3 = a { + a 2 (x 2 + x 3 ) + a 3 (ay + x 2 x 3 + x 3 2 ) , = , /1 / , A 7 T7 : = P + Qo(x 2 x 3 + x^) ; (x 2 - X 3 ) (A - Xi) whence (y* - ih) (yi - yO = (v - fi) (P + QoX) . Now, introducing p l} p- 2 , p 3 , in place of X, p, v, this and the similar equations preserve their forms ; whence, altering P and Q into similar quantities, we obtain the equations it/* - V*) (yi - y 4 ) = 4 (p 3 - p 2 ) (P - Q pi ), (ys - 2/1) (y* - y*) = 4 ( Pl - p 3 ) (P - Q^), (yi - y») (ys - yO = 4 ( P , - Pl ) (P - q P3 ), which lead at once to the invariants of the transformed quart ic ; and comparing their values with the invariants of kU-\H x given in Art. 170, the theorem follows at once. 177. Reduction of the Cubic to a Binomial form by Tschirnbausen's Transformation. — Let the cubic ax 3 + 3bx 2 + 3cx + d be reduced to the form y 3 - V by the transformation y = q + px + x 1 . If x lf x 2 , x 3 be the roots of the given cubic, and y x a root of the transformed cubic, we have the following equations to deter- mine p and q : — x 2 + px x + q = //„ x 2 + px 2 + q = toy i f x 2 + px 3 + q = u> 2 y x ; from which we find x* + loX 2 + io 2 .c 2 . p = — , q = - i (*2 + psi). Xi + wX 2 + to Xi ;]92 Transformations. Adding x { + x 2 + x 3 to this value of p, we have a*.,r, + (,>.r v /\ + u> 2 x x x 2 p + x x + x 2 + x 3 = ; ; X\ -t- w.c, + w"X 3 it follows (see Ex. 25, p. 57) that there are only two ways of completing this transformation, as the values of p, q ultimately depend on the solution of the Hessian of the cubic. 178. Reduction of the (fcuartic to a Trinomial Form by Tsehinihausen's Transformation. — Let the quartic ax 1 + 4bx z + 6cx 2 + 4dx + e be reduced to the form y* + Py 2 + Q, in which the second and fourth terms are absent, by the transformation y = q + px + x 2 . If x lt x 2f x 3 , Xi be the roots of the quartic ; also y x , y 2 two distinct roots of the transformed quartic, we have the follow- ing equations to determine p and q : — *i +pxx + q = y h x 3 2 +px 3 + q = y 2 , x 2 2 +px 2 + q = - y x , Xi 2 + pxi + q = - y 2 ; from which we find Xi + X 2 — X 3 — Xi . . . p = i q = - I (** + psi). Xi + x 2 - x 3 - #4 And, adding x x + x 2 + x 3 + x t to this value of p, we have 2(x l x 2 -x 3 x i ) p + Xi + X 2 + X 3 + Xi = OCi'Y {€% X§ CC^ hence, by Ex 5, p. 130, it follows that there are three ways of reducing the quartic to the proposed form, the determination of which ultimately depends on the solution of the reducing cubic of the quartic. 179. Removal of the Second, Third, and Fourth Terms from an Equation of the n tk Degree. — We begin Removal of 2 nd , 3 rd , and ± th Terms of an Equation. 393 by proving the following proposition, which we shall subse- quently apply : — A homogeneous function V of the second degree in n quantities %\, x-z, #3 5 ... %n can be expressed in general as the sum of n squares. To prove this, let V, arranged in powers of x u take the fol- lowing form : — V = P 1 x l * + 2Q l x 1 + R 1 , where P t does not contain x u x 2 , . . . . x n ; also Qi and R v are linear and quadratic functions, respectively, of x 2 , x 3} . . . x„. We have then also, assuming V^R l -~ = P 2 x* + 2Q 2 x 2 + R 2 , where P 2 is a constant, and Q 2 and R 2 do not contain x y and x s , we have, similarly, so that Proceeding in this way, we arrive ultimately at R,^i - p— > ■l n-i which is equal to P n ^n ; and the proposition is proved. Now, returning to the original problem, let the equation be x" + p x x n ~ l + p 2 x n ~ z + . . . + p n = ; and, putting y = ax* + fix 3 + yx 2 + $x + e, let the transformed equation be y n +Q>g n - } + Q*sT s + ...+ Q n = 0, '*94 Transformations. where, by Art. 173, Q h Q,, ... Q r , ... are homogeneous functions of the first, second, . . . r a degrees in a, /3, 7, S, £. Now, if o, /3, 7, S, s can be determined so that Q. = 0, Q 2 = 0, Q 3 = 0, the problem will be solved. For this purpose, eliminating £ from Q. and Q iy by substituting its value derived from Qi = 0, we obtain two homogeneous equations, of the second and third degrees in a, /3, 7, S ; and by the pro- position proved above we may write M z under the form U' - V 2 + IV 2 - t~, which is satisfied by putting u = v and w = t. From these simple equations we find 7 = la + t»fi, and S = ha + w?,/3 ; and substituting these values in Q 3 = 0, we have a cubic equation to determine the ratio j3 : a. "Whence, giving any one of the quantities a, /3, 7, $, a a definite value, the rest are determined, and the equation is reduced to the form y n + &2T* + Q,y^ + . . . + Q n = 0. In a similar way we may remove the coefficients Q u Q 2 , Qi, by solving an equation of the fourth degree. Applying this method to the quintic, we may reduce it to either of the trinomial forms* x 5 + Px + Q, x 5 + Px- + Q ; or again, changing x into -, to either of the forms x 5 + Px' + Q, x 5 + Px* + Q. In this investigation we have followed M. Serret (see his Cours d'Algebre Suptrieure, Yol. I., Art. 192). * See Note A. Reduction of Qnintic to three Fifth Pozuers. 395 180. Reduction of the Quintic to the Sam of Three Fifth Powers. — This reduction can be effected by the solu- tion of an equation of the third degree, as we proceed to show. Let (rt , «i, « 2 , fls, a i} a 5 ) (x, ij) 5 = b x {x + j3i?/) 5 + b 2 {x + fayY + b 3 (x + /3j*/) \ where j3i, j3 2 , /3 3 are the roots of the equation p 3 x z + p 2 x 2 + p x x + p = 0. Now, comparing coefficients in the two forms of the quintic, a = bx + b 2 + b ?J , ch. = &i/3i + b 2 fi 2 + b 3 fi 3 , a 2 = bfi? + bffi + 6 3 j3 3 2 , fla = J^i 3 + & 2 /3 2 8 + & 3 j3 3 3 , a, = bfif + & 2 j3 2 4 + 6 3 |3 3 4 , « 5 = W + 6 2 j3 2 5 + W ; whence p a +piOi +p 2 a 2 + p 3 a 3 = 0, Podi + PiCi 2 + p 2 a 3 + pidi = 0, p a 2 +pi(h + p 2 (ii +p 3 a 5 = 0. When these equations are taken in conjunction with the equation p + p x x + p 2 x 2 + p 3 x 3 = 0, we have the following equation to determine /3i, /3 2 , |3 3 : — 1 On = 0. X X~ X" ffi a 2 a 3 a x a z a 3 o 4 a 2 a 3 o 4 a 5 Also, bi, b 2 , b 3 are determined by the equations b x + b 2 + b 3 = a , &i/3i + 6 2 j3» + 6 3 /3 3 = «i, Ji/3 1 2 +* 2 j3 2 2 + b 3 (3 3 * = a 2 ; :>9G Transformations. whence the question is completely solved when /3i, (Si, /3 3 are known. This important transformation of the quintic is a particular case of the following general theorem due to Dr. Sylvester : — Any homogeneous function of .r, y, of the degree 2n - 1, can be ml ileal to the form bi(z + fry) 2 "- + h(x + /3,//) 2 "" 1 + ... + b n (x+ (5 n y) 2n - 1 by the solution of an equation of the n th degree. The proof of the general theorem is exactly similar to that above given for the case of the quintic. 181. Huartics Transformable into each other. — We proceed to determine under what conditions two quartics can be transformed, the one into the other, by linear transformation. Let the quartics be U={a, b, c, d, e)(x,yy = a{x-ay){z-py)(x- y y)(x-$y), V = (a, b\ c, d', e) (x, y'Y = d{x- ay) (x- fi'y) (x-y'y) {x- S'y') 5 and if they become identical by the transformation x = Xx + ny, y = Xx + fxy, we have, by Art. 38, (flW)(q-r) = (y -*')&'-*) _ {a'-P){y'-V) (0- 7 )(a-«) ' ( 7 -a)U3-S) " (a-t3)(y-8)' showing that the six anharmonic ratios determined by the roots must be the same for both equations. From these equations we have also the following relations between the invariants of the two forms : — /' = r% J' = r«J; (1) whence r 3 r Quartics Transformable into each other. 397 The quantity — , being absolutely unaltered by transforma- tion when the quartic is linearly transformed, is called the absolute invariant of the quartic. The condition expressed by equation (2) is, therefore, that the absolute invariant should be the same for both quartics. The condition here arrived at agrees with the result of Ex. 6, p. 146, where it is proved that the sextic which determines the anharmonic ratios of the roots involves the absolute invariant, and no other function of the coefficients, of the quartic. The conditions expressed by the equations (1), (2), are always necessary; but not always sufficient, as we proceed to illustrate by two exceptional cases. Suppose, in the first place, U^n-vic, V=u' 2 r' : , where u, r, u; u, r, are of the linear form Ix + my. I 3 I' s . Although the condition -=j = — 7t is satisfied in this case, the common value of these fractions being 27, it is impossible to transform U into V, since it is impossible to make vw a perfect square by linear transformation. Secondly, if TJ = u?v, V = u'* ; although the equations /' = ; ,4 J, J' = > S J are satisfied, since I' = 0, 1=0, J' = 0, J = 0, it is, nevertheless, impossible to transform £7 into V. In both these cases it would be impossible to identify the six anharmonic ratios depending on the roots of the quartics. In general, it may be stated that it is impossible to transform one quantic into another by linear transformation when any relation exists between the invariants of one of them which does not exist between the invariants of the other (see Clebsch's Theorie der Binaren Algebraischen Formen, Art. 92). 398 Transformations. Miscellaneous Examples. 1 . Transform two given quadratics in x, y to the forms au 2 + bv 2 , a'u 2 + b'v 2 , where m and v are linear functions of x and y. 2. If the coefficients of three quadratics a\x 2 +2bixy + ciy 2 , a 2 x 2 + 2b 2 xy + c 2 y 2 , a 3 x 2 + 2faxy + c 3 y 2 he connected by the relation a\ bi ci az b 2 c% =0; «3 fa e 3 prove that they may be reduced by linear transformation to the forms A x X 2 +C l Y 2 , A 2 X 2 +C 2 Y 2 , A 3 X 2 +C 3 Y 2 . The determinant here written is the condition that the three quadratics should determine a system of points or lines in involution. 3. Reduce (a, b, c, d)(x, y) 3 to the sum of two cubes by the method of Art. 180. 4. Prove that two cubics can in general be transformed one into the other by linear transformation. 5. Express three cubics, U, V, W, by means of three cubes. Assuming \U+fiV+ vW={x-py) 3 , (1) and comparing coefficients, we have A«i 4- ixa-i + va 3 = 1, \bi + ufa + vb 3 = — p } ACi + fJ.C 2 4- vcz = p 2 , \di -f fidi -f vd 3 = — p 3 . These equations, by eliminating A, /u, v, give three values of p, and corresponding values of A, n, v : in this way we obtain three equations of the form (1) to deter- mine U, V, W in terms of (s-piy) 3 , {x-p 2 y) 3 , (x-p 3 y) 3 . Miscellaneous Examples. 399 It is easy to see that p is given by the equation aip + b\ a-ip + bi a 3 p + i 3 bip+ci hp + d b 3 p + cz =0. eip + di czp + di czp + - - 3J

* (cos a - / sin a). Denoting for shortness x x + a r x z + a r ~x s + . . . + a r n ' l x n by Y n and substituting these values in F, and Y 2 , we find Y^U+iV, Y 2 =U-iV, where U andj V are real ; also putting = r(cos (p + i sin 0), = r (cos*# - i sin 0), a x - p a t -p the part of the function F depending on cti and a 2 , viz., Fx 2 r 2 2 + — -, a x - p a>-p 2d2 40 \ Transformations. becomes r jf cos | + i sin | j ( U + i Vf + ( cos | - i sin | Y ( £7 - i Vf which may be also written as the difference of the squares 2,-(c7cos^-rsin|J-2r(^ S m| + rcos| N proving that two imaginary conjugate roots introduce into F t w( i r< >al squares, one of which has a positive and the other a negative coefficient. We now state Hermite's theorem as follows : — Let the equa- tion f{x) = (#- ai)(x - a 2 ) . . . {x- a„) = have real coefficients and unequal roots : if then by a real substitution ice reduce rr- y:- yz yj + — + — + ...+ — , (i) a\- p 02-/0 a 3 -p a where Y r = x Y + a r cr 2 + a,?x z + . . . + a r n ~ l x n , to a sum of squares, the number of squares having positive coefficients mil be equal to the number of pairs of imaginary roots of the equa- tion f{x) = 0, augmented by the number of real roots greater than p. This theorem follows at once from what has preceded if we consider separately the parts of the function (1) which refer to real roots and to imaginary roots, for obviously there is a posi- tive square for every root greater than p, and we have proved that every pair of conjugate imaginary roots leads to a positive and negative real square, without affecting the other squares independent of these roots. The number of real roots between any two numbers pi and (>, may be readily estimated. For, denoting in general by P, the number of positive squares in F when p = pj, by Nj the number of roots of the equation f(x) = greater than pj, and by 21 the number of imaginary roots, we have P, = N, + I, P, = N, + I; whence N, - iV* = P x - P 2 , Hermitc's Theorem. 405 proving that the number of real roots between pi and p 2 is equal to the difference between the number of positive squares when p has the values pi and p 2 , respectively. The number here determined may be shown to depend on a very important series of functions conuected with the given equation. In order to derive these functions we consider F under the form (Art. 182) AiXr-+ ^xj+^xj +....+ ~*x a \ A, A o A„_i J The number P expresses the number of coefficients in this form which are positive, or, which is the same thing, the number of the following quantities which are negative : — -^1 -^1 _^ 3 __f^L o\ V Ai' A.;"-' A„_i' w We proceed now to calculate Ai, A 2 , . . . Ay, . . . A rt in terms of p and the roots of the equation f{x) = ; and as the method is the same in every case it will be sufficient to calculate A 3 , i. e. the discriminant of the original form of F when all the variables except x lt x 2 , x 3 vanish. Writing for shortness v,- = , we have in this case a, - p F 3 = 2v r (#i + a,x 2 + a r 2 x 3 y. The discriminant of this form is 2v 2ov 2a 2 y A 3 = 2ai> 2a"v 2a 3 v 2«"y 2a 3 v 2a 4 v which may be written as the product of the two arrays 1 1 . . . 1 \ I'l v 2 • v„ \ Oi a 2 . . a,, ) > aiVi a 2 i'j Oh >'>i \ 5 a: 2 er 2 2 . . . a n z ) CIi'Vi a 2 2 v 2 . . • OnV m J 4 G Tran sforma lions . and, consequently, 1 1 1 A J = 2 l'l Vi V;\ a 3 a? (a 2 - a 3 ) 3 (a 3 - aQ 2 (ai - a 2 ) 2 (ai-p)(a 3 -p)(o 3 -/>) In an exactly similar manner we find Ay-S V (qd Q2, «3> • » . ay) >i-p)(a 2 -p) . . . (aj-p)' where the notation y (m, a 2 , a 3 , . . . ay) is employed to represent the product of the squares of the differences of o l5 a>, a 3 , . . . ay. Hence the quantities Ai, A 2 , Ay . . . A„ are all determined. Now, multiplying the numerator and denominator of each of the fractions in the series (2) by / (p), each value of A is rendered integral, and the series becomes where P, v, V, V n V FV V,, ' ' ' ' v n .: V = (p - oi) (p - a 2 ) ... • (p - a«)» Vi = % (p - a,) (p - a s ) . . . . (/> - a„), V% - 2V (ai, a 2 ) (/o - a 8 ) . . . . (p - a„), (3) F 3 = 2V (a„ a 2 , a 3 ) (p - a 4 ) (p - a„), F« = v(«i> «2, a 3 , .... a„). Since negative terms in the series (3) correspond to varia- tions of sign in the series V, Vi, V 2 , V 3 , . . . . V,„ it is proved that the number of variations lost in the series last written, when p passes from the value p x to the value p 2 , is exactly equal to the number of real roots of the equation / (p) = comprised between pi and p.. 184. Sylvester's Form* of Sturm's Functions. — It will be observed that the functions V, Vi, V 2 , &c, arrived at in the preceding Article have the same property as Sturm's Sylvester's Forms of Sturm's Functions. 407 functions ; from which in fact they differ by positive multipliers only, as was observed by Sylvester, who first published these forms in the Philosophical Magazine, December, 1839. The identity of the two series of functions may be established as follows : — We make use of the notation already employed in Ex. 7, p. 312, and we propose to show that the Sturmian remainder Rj differs only by the positive factor jj from the function Vj. From the example referred to, we have Rj - Ajf\x) - Bjf{x), (1) where Rj = >' + )\x + r 2 x 2 + . . . + r„_/ .?""•>, Aj = Ac, + \\X + A 2 ;r + . . . + XjLia^" 1 , Rj = no+ fj.\x+ fx.x- + . . . + fjj-2^' 2 ; and from the value of >•„_/ there given we have immediately r n-j = fj 2V (a„ a 2 , a 3 , aj), showing that the leading coefficients in Rj and Vj differ only by the factor yy. We now proceed to prove that the last co- efficients in these functions differ only by the same factor. For this purpose, dividing the identity (1) hjf(x), substituting in it from the equation f(x) S Si s 2 f(x) x x' ar and comparing coefficients, we find Ho = Xi«o + A 2 S! + A 3 s> + .... + A,_i Sj. 2 , jui = A 2 s + A 3 «i + — + Af-i*/-3, fij-2 = \j-\ So. Also, putting x = in (1), we have and, substituting for /m,, in terms of Ai, A 2 , A 3 , &c, = Ao s_i + Aj •<-',! + A 2 Si + + A,_i-sv_2 ; Pn 408 Transformations. whence, giving to A,„ X^ .... A/_, tlie same values as in the calculation of r«_,-, we find ro=(-l) j Pnjj S-l -So So So . . . S;_ ;-i •V-2 *H Si Now, referring to the calculation of A,- in Art. 183, and put- ting p = 0, or v r = — , in the value of A,- there found, we find a r for the determinant just written the value ^a V (ai, a 2 , a 3 , . . . aj) ^ ^ aiajan . . . aj hence, giving p n its value in terms of the roots, we have r = (- l) n ~*yj 2v(ai, a 2 , a 3 , . . . aj) ay+ia/^ . . . a n , whioh was required to be proved. The first and last coefficients of Rj, when divided by yy, having been thus shown to be the same as in the form V}, it follows that all the intermediate terms must be similarly related ; for, in the first place, Rj is a function of the diffe- rences of the quantities x, a u a 2 . . . a n , as may be seen by transforming f(x) before calculating Rj by the substitution z = a x + a u as in Ex. 3, Art. 92. When this transformation is completed, every coefficient in Rj, as well as 2, is a function of the differences ; consequently, Rj satisfies the differential equation d d d d \ dRj + — + ... + — )Rj = 0, or - - DR, = 0, dx dx da.i da 2 da n j showing, as in Articles 138 and 157, that all the coefficients may be obtained from the last by a definite law. The same conclusions plainly holding also for the function Vj, it is there- fore proved, finally, that Rj^jjVj. Sylvester's Forms of Sturm's Functions. 409 Examples. 1. To reduce two quadrics in three variables to the sums of the same three squares with proper coefficients. Let U=ax i + by- + cz- + 2fyz + 2yzx + 2hxy, V=aix n -+ hy-+ Ciz 2 + 2/iya+ 2yizx + 2/nxy, #h*0-HT + r, X=if, T-& *.j" ax ay dz We have then identically F=- A(a) A« + a\ \h+ hi \y + yi X AA+Ai \b + h Kf+fi Y ty + ffi V + /i \c + ci Z X Y Z *'A) where A (A) is the discriminant of \U + V; and *(a) is a function of the 2nd degree in A, the symbols X, Y, Z being retained in it for the present, and not replaced by the values involving A. Kesolving into partial fractions, we have F _ *(\i) 1 (A 2 ) 1 »(A 3 ) 1 ~A'(Ai) A-Ai A'(A 3 ) A-A2 A' (A3) A - A3' (1) in which *(\i), * (A2), * (A3) are all perfect squares, since they are obtained by bordering the vanishing determinants A (Ai), A (Aj), A (A3). (See Art. 129.) Now, replacing X, Y, Z by their values, \Vi + Vi, &c, * (\j) is easily re- ducible to the form -(A-A,f A>« + ai \jh + hi \jy + yi JJi \jh+hi \jb + bi \jf+fi TJz N9 + 9\ Mf+fi V + c i #3 Ui u 2 u 3 {\-\j)W, where j = 1, 2, or 3, and tij is independent of A. Substituting these values in (1), we find * v + T *-*iitt**-*-Bk3 + *-**m3. 410 Transformations. Equating the coefficients of A, we have U = >'\- + ":;- A'(Ai) T A' fa) A' (A3)' Ml 2 M2 2 W3 2 — r = Ai — 77 — r + A2 —7-7 — r + A3 — r; — r , A'(Ai) A'(A 2 ) A' (As)' which was required to he done. It is to he ohserved that this problem has only one solution. The mode of reduction here given is due to Darhoux ; and is plainly applicable whatever he the number of variables. 2. Prove that a quadrie in n variables may be reduced by a real orthogonal transformation to a sum of n squares. An orthogonal transformation is a linear transformation such that, when the modulus written as a determinant is squared the terms in the principal diagonal are each equal to 1, and all the other terms vanish. In a transformation of this kind it follows that the sum of the squares of the new variables is equal to the sum of the squares of the old. 3. Writing as before one of Sturm's remainders in the form prove that where Bj = 7, Hj = Aj{x), */-2 Sj-1 T x *M Tj-x Tj = s xr i + sixj- z + sixi-* + . . . + sj.i. 4. Denoting by TJ n S {X - Or) (*1 + OX2 + a?x% + . . . + a n ' l x n )-, prove that the discriminant of Uj may be determined by the equation Aj = — , n where Aj and jj have the same signification as before ; and show directly that if Aj = for a certain value of x, Aj.\ and Aj k \ have opposite signs for the same value of x. Note. — Hermite's theorem holds where a r — p is changed into (a r — p) m in the enunciation on p. 404, m being any odd integer, positive or negative. Transformation of Binary to Ternary Forms. 411 Section III. — Geometrical Transformations.* 185. Transformation of Binary to Ternary Forms. — We think it desirable, before closing the present Chapter, to give a brief account of a simple transformation from a binary to a ternary system of variables, whereby a geometrical interpre- tation may be given to several of the results contained in the preceding Chapters. The applications which follow in connexion with the quadratic and quartic will be sufficient to explain this mode of transformation ; and will enable the student acquainted with the principles of analytic geometry to trace further the analogy which exists between the two systems. Denoting the original variables, i.e. the variables of the bi- nary system, by x , y m we propose to transform to a ternary system by the substitutions x = x , y = ^x y , z = y ~. For example, taking the simple case of a quadratic whose roots are a, /3, viz., •V - (a + /3) x y + a(3y 2 = 0, and transforming, we obtain ■-*(« + P)y + a{5z = 0. (1) "We have also the identical equation y 2 - 4zx = 0. This is the equation of a conic, which we call V , and (1) is plainly the equation of a chord of this conic joining the points a and )3, the point determined by the equations x y i -*"u — = -— = z, where

x y (t + c 3 t/ 2 , it is seen that the determinant («i& 2 r a ) is an invariant in both systems, its vanishing being the condition in the binary system that the quadratics should form an involution (Ex. 2, p. 398), and in the ternary system that the three corresponding lines should meet in a point. As a final illustration, we consider a system of three qua- dratics connected in pairs by the harmonic relations aiCo + a 2 c l - 2b x b 2 = 0, &c. Transforming the quadratics, we obtain three lines X, Y, Z, which form a self-conjugate triangle with regard to the conic V. The theorem relating to three mutually harmonic quadratics, viz., that their squares are connected by an identical linear relation (see Ex. 5, p. 360, and Art. 166), is suggested by a well-known property of conies ; for V expressed in terms of X, Y, Z is of the form - V^X* + Y*+ Z 2 ; whence, restoring the original variables x , y a , V vanishes iden- tically, and X, Y, Z become the original quadratics, each divided by a factor which may be seen to be the square root of its dis- criminant (see (1), Ex. 5, p. 360). 414 Transformations. 186. The Quartic and its Covariants treated geo- metrically. — It will appear from the remarks to be made in the next Article that in applying the transformation now under consideration to the quartic U = (a, b, c, d, e)(x 0) y a y, the term 6cx " y 2 will be replaced by 2cxz + cy 2 , so that the quartic will be replaced by the two following conies : — U = nx 2 + cy 2 + ez 2 + 2dyz + 2czx + 2bxy = 0, V ^ f - 4zx = ; the form of U here selected being connected with V by an invariant relation. The invariants of U and V are invariants of the original binary form, for the discriminant of U - p V is 4p 3 -Ip + J, and the invariants of the ternary system are A' = -4, 9' = 0, 6 = 7, A = <7; where i" and J are the invariants of the quartic, and the dis- criminant of U - p V is written as usual under the form a - pe + ^e' - (o'a'. Let the conies U and V intersect in the points A, B, C, D ; these points being determined by the equations x y tf = 2$ = S ' when has the four values a, /3, y, 8, the roots of the binary quartic ; and let the points of intersection of the common chords BC, AD; CA, BD; AB, CD be E, F, G, respectively, the triangle EFG being self-conjugate with regard to both conies. Now, denoting by (a/3) = the equation of the line AB, and using a similar notation for the remaining chords, we have by the theory of conies U- Pi r=(Py)(a$), U-p 2 V=(ya)(pS), U- p 3 V= (a/3)( 7 S), where pi, p t , p ;i are the roots of the equation 4p 3 - Ip + J= 0. The Quartic audits Covariants treated geometrically. 415 On restoring the original variables x , y in these equations, Vo vanishes identically, and we have U resolved into a pair of quadratic factors in three different ways, depending on the solution of the reducing cubic of the quartic. "Whence it appears that the resolution of a quartic into its pairs of quad- ratic factors, and the determination of the pairs of lines which pass through the intersections of two conies, are identical prob- lems, each depending on the solution of the same cubic equa- tion. "We now proceed to show that the sides of the common self- conjugate triangle of U, V correspond to the quadratic factors of the sextic covariant in the binary system. Since the side FG is the polar of E, the co-ordinates a?', y of E are found by solving the equations (j3y) = 0, (aS) = ; we have, therefore, j3y(a + S)-aS(/3 + y) 2(j3y-a8) /3 + y - a - S' and, substituting for x\ y\ z the values thus determined in the polar of E, viz., xz - ^- + x'% = 0, we express this equation in the form (j3 + 7-o-S)*-2(/37-a8)y+(/3y(a + S)-a8(/3 + 7 ))3=0. On restoring the original variables x , y , this is seen to be one of the quadratic factors of the sextic covariant (see Art. 166). It is therefore proved that the points where FG meets V are determined by the quadratic equation (/3 + 7-a-8)0 3 -2( i 3y-ag)0 + )3y(a+g)-a3(/3 + y)-O; and consequently the six points on V which correspond to the roots of the sextic covariant are the points where this conic meets the sides of the common self- conjugate triangle of U and V. To determine the points on V which correspond to the roots of the Hessian, we calculate for the conies £7" and F"the co- 416 Transformations. variant conic JF (Salmon's Conic Section*, Art. 378) ; thus finding - j JP = (ac - b 2 ) or + (bd - (Salmon's Conies, Art. 377) which is the envelope of a line cut harmonically by the conies TJ and V. 187. When the transformation of Art. 185, viz., w — %o ■> y = £%$y§, s = 7/u", is applied to a quantic /{x n , y ) of even degree 2m, it is plain that the roots of this quantic will be represented geometrically by the points of intersection of a curve of the m th degree with the conic section V. If the degree of the quantic is odd, it must be squared before the transformation is effected ; and the roots will then be represented geometrically by the points of contact of the corresponding curve with the conic. In transforming the quantic f(x , yo), we may obtain an indefinite number of ternary forms by varying the mode of transformation ; for if TJ denote any one of these forms, U + tf> m _ 2 V, in which the coefficients of 0„,_ 2 are arbitrary, would equally well be a transformation of /(.r , y ), since this form would on restoring the original variables return to the quantic / (*o, yo) • Moreover, every possible transformation is included in the foregoing. Among these innumerable ternary forms there is one, and only one, such that its invariants and co- variants are invariants and covariants of the binary quantic The Qaartic and its Covariants treated geometrically. 417 also. To determine this form, take the tangential form of V, and let n be the operator obtained by substituting D x , D y , D. for the variables therein ; operating then with n on JJ + 3-5 in the manner ClXo uXq Cll/o (it/ just explained, and multiplying by x, y, z, respectively, we obtain a ternary cubic U of the proper form. In a similar manner the transformation of the octavic is made to depend on that of the sextic ; and proceeding in this way step by step we may transform any binary quantic u of even degree to a ternary quantic JJ of half the degree, such that n ( JJ) ■ 0. The following examples are given to illustrate the trans- formation explained in the present section. 2e 418 Transfer ma tions . Examples. 1. If f(xo, y ) becomes U {x, y, z) by the transformation x = x z , y = 2x yo, z = yo 2 ; prove in general that 3 -C-D §-«-(»). ,Jlf,2(n-l) i? + 2j,n(I7), ax o a i/o "'J d H =2(»-l) ^-4*n(ir), rfy - " z ,, ,_ rf 2 cT rf«r/ where n is the degree of f{x, y), and n( U) = — - - -^. "Whence, in particular, if 11(17") = 0, prove that Id d \ 2 I , dU , dU dU\ where zo, yo> and *o', y<>' are cogredient variables. 2. If 2 2 ;!£-*<**')■ ^ = * a(a: ' y ' 2) ' &-*«**<>. prove that n(xi) = 0, n (<£ 2 ) = 0, n (<£ 3 ) = 0. Since d 2 z dz dy dx ' but dd>i d(t>z _ rf2 and therefore vanishes by what precedes. We have thus a formal proof of the .statement at the end of Art. 187. "When n(i), U{

i + y<^2 + z3) = (»-l){*n(^i) + yn( 2 ) + =n(4> 3 )}. 3. If two quantics u and w be transformed ; prove that the Jacobian of u, w in the binary system becomes the Jacobian of U, V, W in the ternary system. Express J(u, w) in terms of xo*, Xoyo, yo 2 and the second differentials of u and w, and then transform by Ex. 1. 4. Prove that the quartics (aiz'- + 2j8izy + 7iy 2 )(a3r : + 2foxy + y 3 y*) - (o 2 x 2 + 2faxy + yzy 2 ) 2 , (1) (ai x* + 2a 2 xy + 03 y 2 ) (71a; 2 + 2y 2 xy + 73/) - (j8i x 2 + 2$ 2 xy + 3 y 2 ) 2 (2) have the same invariants. Examples. 419 Transforming (2) to the ternary system, we have the conic (aix + a 2 y + o 3 z) (71* + 72*/ + 73-) - (0iZ + 2 y + /3 3 z) 2 , which for shortness we write as LN — M~, where L = aix + a 2 y + a 3 z, M= fax + fay + faz, X = y\x + y 2 y + 73:. (3) Now, when the discriminant of LN- jr- + M 02 72 2A 03 03 73 -4A 1 ai 02 03 _1 0i 02 03 1 71 72 73 Il2 1 21 + A ^23 7l3 - 2A J23 ^33 where 21pq = a P y q + a q y P - 2 fa fa. This determinant becomes when expanded 4A 3 + 4(/ 2 2-/l3)A 2 -{7ll/33-/l3 2 +4(/l3/22-/l2/23)}A- every coefficient of which is the same for both quartics, as may be verified directly. 2e2 In r« 1X3 In J22 J23 /13 J* IzZ 420 Transformations. 5. Determine the condition that three quadratics should hy linear transfor- mation be reducible to the forms (I- t!-

+ ci, py = « 2 * + h

+ i sin ) ; we have then, if f(z) be the given function, /(«)-/(* + h) -/(*) +/ ( S9 )h+ f ^h*+ &c. 428 The Complex Variable. and the increment olf{z), being equal to/(s + h) -f(z ), is / w /, +/^ # + Z54 a 3 + &o. 1.2 1.2.3 In this expression the coefficients of the powers of h are all imaginary expressions of the usual form ; and if their moduli be a, b, c, &c, the moduli of the successive terms are ap, bp 2 , cp*, &c. ; and since, by Art. 189, the modulus of a sum is less than the sum of the moduli, it follows that the modulus of the increment of f(z) is less than ap + bp" 1 + cp 3 + &e. Now a value may be assigned to p (A.rt. 4), such that for it, or any less value of p, the value of this expression will be less than any assigned quantity. It follows that to an infinitely small variation of the complex variable corresponds an infinitely small variation of the function ; in other words, the function varies continuously at the same time as the complex variable itself. 193. Variation of the Argument of f (z) corresponding to the Description of a small Closed Curve by the Com- plex. Variable. — Corresponding to a continuous series of values of z we have a continuous series of values oif[z), which can be represented, like the values of z itself, by points in a plane. We represent these series of points by two figures (fig. 10) side o' x" 10. by side, which, to avoid confusion, may be supposed to be drawn on different planes. To each point P, representing x + iy, cor- Variation of the Argument off(z), Sfc. 429 responds one determinate point P' representing /(s). When P describes a continuous curve, P / describes also a continuous curve; and when P returns to its original position after describ- ing a closed curve, P' returns also to its original position. Our present object is to discuss the variation of the argument of/(z) corresponding to the description of a small closed curve by P. Let A be any determinate point whose co-ordinates are *Ko, y , i. e. 2 = «b + iy a . "We divide the discussion into two cases : — (1). When To + iy is not a root of f(z) = 0, i.e. when /(s„) is different from zero. (2). When ,r + iy is a root of/(z) = 0, or/(«„) = 0. (1). In the first case, to the point A corresponds a point A' representing the value of /(s ), and O'A' is different from zero. Let z = z + h, where h - p (cos + i sin ) ; and suppose P, which represents z, to describe a small closed curve round A. Let P r represent /(s) ; then A'P* represents the increment of f(z) cor- responding to the increment AP of z. By the previous Article it appears that values so small may be assigned to p, that the modulus of the increment of f{z), namely A'P', may be always less than the assigned quantity O'A'; hence P may be supposed to describe round A a closed curve so small that the correspond- ing closed curve described by P' will be exterior to 0'. It fol- lows, by Art. 191, that corresponding to the description by P of a small closed curve, which does not contain a point satisfying the equation f(z) = 0, the total variation of the argument of f(z) is nothing. (2). In the second case, suppose x + iy is a root of the equa- tion /(s) = repeated m times, and let then f(z) = h m \f,(z) = p m (cos ?n + i sin miO), 9 is increased by 2tt, and, therefore, org. z n is increased by 2mr. It follows from Cauchy's theorem, Art. 194, that the muni nr of roots comprised within the circle described by z, i.e. the total number of roots of the equation f(z) = 0, is n ; and the theorem is proved. The proposition whose proof was deferred in Art. 15 is thus shown to be an immediate consequence of Cauchy's theorem, which may therefore be regarded as the fundamental preposi- tion of the Theory of Equations. It is proper to observe, how- ever, that the theorem of Art. 15, viz., that every equation has a root, can be proved directly, and independently of Cauchy's theorem, by aid of the principles contained in Art. 193 and the preceding Articles, as we proceed now to show. If possible, let there be no value of z which makes f(z) vanish; and let the value s„, represented by A, fig. 10, corre- spond to the nearest possible position, A', of P' to the origin (/. Now, giving s a small increment h, and considering the first term /'(~ ) h of the corresponding increment of/(s ), it is seen that the directions in which these two small increments take place are inclined at a constant angle. It is possible therefore, by properly selecting the direction of the increment h, to cause the increment of f(z ) to take place in the direction A'O', and thus to make A' approach nearer to the origin, which is con- trary to hypothesis. It follows that the minimum value of the modulus of/(s) cannot be different from zero, and therefore that some value of z exists which makes /(s) vanish. In note D will be found some further observations on the subject of this Article. NOTE S. NOTE A. ALGEBRAIC SOLUTION OF EQUATIONS. The solution of the quadratic equation was known to the Arabians, and is found in the works of Mohammed Ben 3Iusa and other writers published in the ninth century. In a treatise on Algebra by Omar Alkhayyami, which belongs probably to the middle of the eleventh century, is found a classification of cubic equations, with methods of geometrical construction ; but no attempt at a general solution. The study of Algebra was introduced into Italy from the Arabian writers by Leonardo of Pisa early in the thirteenth century ; and for a long period the Italians were the chief cultivators of the science. A work, styled I? Arte Maggiore, by Lucas Paciolus i^known as Lucas de Burgo) was published in 1494. This writer adopts the Arabic classification of cubic equations, and pronounces their solution to be as impossible in the existing state of the science as the quadrature of the circle. At the same time he signalizes this solution as the problem to which the attention of mathematicians should be next directed in the develop- ment of the science. The solution of the equation x 3 + mx = n was effected by Scipio Perreo ; but nothing more is known of his discovery than that he imparted it to his pupil Plorido in the year 1505. The attention of Tartaglia was directed to the problem in the year 1530, in consequence of a question proposed to him by Colla, whose solution depended on that of a cubic of the form x 3 + pxr = q. Florido, learning that Tartaglia had obtained a solution of this equation, proclaimed his own knowledge of the solution of the form x 3 + mx = n. Tartaglia, doubting the truth of his statement, challenged him to a disputation 2f 43 -A Notes. in the year L535 : and in the mean time himself discovered the solu- tion of Ferreo's form .r 3 + mx = n. This solution depends on assuming for x an expression l/l - ^/u consisting of the difference of two radi- cals ; and, in fact, constitutes the solution usually known as Cardan's. Tartaglia continued his labours, and discovered rules for the solution of the various forms of cubics included under the classification of the Arabic writers. Cardan, anxious to obtain a knowledge of these rules, applied to Tartaglia in the year 1539; but without success. After many solicitations Tartaglia imparted to him a knowledge of these rules ; receiving from him, however, the most solemn and sacred pro- mises of secrecy. Regardless of his promises, Cardan published in 1545 Tartaglia' s rules in his great work styled Ars Magna. It had been the intention of Tartaglia to publish his rules in a work of his own. He commenced the publication of this work in 1556 ; but died in 1559, before he had reached the consideration of cubic equations. As his work, therefore, contained no mention of his own rules, these rules came in process of time to be regarded as the discovery of Cardan, and to be called by his name. The solution of equations of the fourth degree was the next problem to engage the attention of algebraists ; and here, as well as in the case of the cubic, the impulse was given by Col la, who proposed to the learned the solution of the equation x^ + 6x 2 + 36 = 60a;. Cardan appears to have made attempts to obtain a formula for equations of this kind ; but the discovery was reserved for his pupil Ferrari. The method employed by Ferrari was the introduction of a new variable, in such a way as to make both sides of the equation perfect squares ; this variable itself being determined by an equation of the third de- cree. It is, in fact, virtually the method of Art. 63. This solution is sometimes ascribed to Bombelli, who published it in his treatise on Algebra, in 1579. The solution known as Simpson's, which was pub- lished much later (about 1740), is in no respect essentially different from that of Ferrari. In the year 1 637 appeared Descartes' treatise, in which are found many improvements in algebraical science, the chief of which are his recognition of the negative and imaginary roots of equations, and his " llule of Signs." His expression of the biqua- dratic as the product of two quadratic factors, although deducible immediately from Ferrari's form, was an important contribution to the study of this quantic. Euler's algebra was published in 1770. His solution of the biquadratic (see Art. 61) is important, inasmuch Notes. 435 as it brings the treatment of this form into harmony with that of the cubic by means of the assumed irrational form of the root. The methods of Descartes and Euler were the result of attempts made to obtain a general algebraic solution of equations. Throughout the eighteenth century many mathematicians occupied themselves with this problem ; but their labours were unsuccessful in the case of equations of a degree higher than the fourth. In the solutions of the cubic and biquadratic obtained by the older analysts we observe two distinct methods in operation : the first, illus- trated by the assumptions of Tartaglia and Euler, proceeding from an assumed explicit irrational form of the root ; the other, seeking by the aid of a transformation of the given function, to change its factorial character, so as to reduce it to a form readily resolvable. In Art. 55 these two methods are illustrated ; together with a third, the concep- tion of which is to be traced to Yandermonde and Lagrange, who pub- lished their researches about the same time, in the years 1770 and 1771. The former of these writers was the first to indicate clearly the necessary character of an algebraical solution of any equation, viz., that it must, by the combination of radical signs involved in it, represent any root indifferently when the symmetric functions of the roots are substituted for the functions of the coefficients involved in the formula (see Art. 94). His attempts to construct formulas of this character were successful in the cases of the cubic and biquadratic ; but failed in the case of the quintic. Lagrange undertook a review of the labours of his predecessors in the direction of the general solution of equations, and traced all their results to one uniform principle. This principle consists in reducing the solution of the given equation to that of an equation of lower degree, whose roots are linear functions of the roots of the given equation and of the roots of unity. He shows also that the reduction of a quintic cannot be effected in this way, the equation on which its solution depends being of the sixth degree. All attempts at the solution of equations of the fifth degree having failed, it was natural that mathematicians should inquire whether any such solution was possible at all. Demonstrations have been given by Abel and Wantzel (see Serret's Cours d'Algebre Supe- rieure, Art. 516) of the impossibility of resolving algebraically equa- tions unrestricted in form, of a degree higher than the fourth. A transcendental solution, however, of the quintic has been given by M. Hermite, in a form involving elliptic integrals. Among other 2 f2 436 Notes. contributions to the discussion of the quintic since the researches of Lagrange, one of leading importance is its expression in a trinomial form by means of the Tschirnhausen transformation (see Art. 179). Tschirnhausen himself had succeeded in the year 1683, by means of the assumption y = P + Qx + x z , in the reduction of the cubic and quartic, and had imagined that a similar process might be applied to the general equation. The reduction of the quintic to the trinomial form was published by Mr. Jerrard in his Mathematical Researches, 1832-1835 ; and has been pronounced by M. Ilermite to be the most important advance in the discussion of this quantic since Abel's demonstration of the impossibility of its solution by radicals. In a Paper published by the Rev. Bobert Harley in the Quarterly Journal of Mathematics, vol. vi. p. 38, it is shown that this reduction had been previously effected, in 1786, by a Swedish mathematician named Bring. Of equal importance with Bring's reduction is Dr. Sylvester's transformation (Art. 180), by means of which the quintic is expressed as the sum of three fifth powers, a form which gives great facility to the treatment of this quantic. Other contributions which have been made in recent years towards the discussion of quantics of the fifth and higher degrees have reference chiefly to the invariants and cova- riants of these forms. For an account of these researches the student is referred to Clebsch's Theorie der bindren algebraischen Formen, and to Salmon's Lessons Introductory to the Modem Higher Algebra. There has also grown up in recent years a very wide field of in- vestigation relative to the algebraic solution of equations, known as the; " Theory of Substitutions." This theory arose out of the researches of Lagrange before referred to, and has received large additions from the labours of Cauchy, Abel, Galois, and other writers. Many important results have been arrived at by these investigators ; but the subject is of too great extent and difficulty to find any place in the present work. The reader desirous of information on this subject is referred to Serret's Cours d' Algcbre Superieure, and to the Traite des Substitu- tions et des Equations Algcbriques, by M. Camille Jordan. Notes. -137 NOTE B. SOLUTION OF NUMERICAL EQUATIONS. The first attempt at a general solution by approximation of nume- rical equations -was published in the year 1600, by Vieta. Cardan had previously applied the rule of "false position" (called by him •'regula aurea") to the cubic; but the results obtained by this method were of little value. It occurred to Yieta that a particular numerical root of a given equation might be obtained by a process analogous to the ordinary processes of extraction of square and cube roots ; and he inquired in "what way these known processes should be modified in order to afford a root of an equation whose coefficients are given numbers. Taking the equation / (x) = Q, whore Q is a given number, and / (x) a polynomial containing different powers of x, with numerical coefficients, Yieta showed that, by substituting m. f (x) a known approximate value of the root, another figure of the root expressed as a decimal) might be obtained by division. When this value was obtained, a repetition of the process furnished the next figure of the root ; and so on. It will be observed that the principle <»f this method is identical with the main principle involved in the methods of approximation of Newton and Horner (Arts. 100, 101). All that has been added since Vieta' s time to this mode of solution of numerical equations is the arrangement of the calculation so as to afford facility and security in the process of evolution of the root. How great has been the improvement in this respect may be judged of by an observation in Hontucla's Histoire des Mathematiques, vol. i. p. 603, where, speaking of Vieta's mode of approximation, the author regards the calculation (performed by Wallis) of the root of a biquadratic to eleven decimal places as a work of the most extra- vagant labour. The same calculation can now be conducted with great ease by anyone who has mastered Horner's process explained in the text. Newton's method of approximation was published in 1669 ; but before this period the method of Vieta had been employed and sim- plified by Harriot, Oughtred, Pell, and others. After the period of Newton, Simpson and the Pcrnoullis occupied themselves with the 4:}$ Notes. same problem. Daniel Bernoulli expressed a root of an equation in the form of a recurring series, and a similar expression was given by Euler ; but both these methods of solution bave been shown by Lagrange to be in no respect essentially different from Newton's solution (Traite de la Resolution des Equations Numeriques). Up to the period of Lagrange, therefore, there was in existence only one distinct method of approximation to the root of a numerical equation ; and this method, as finally perfected by Homer, in 1819, remains at the present time the best practical method yet discovered for this pur- pose. Lagrange, in the work above referred to, pointed out the defects in the methods of Vieta and Newton. "With reference to the former he observed that it required too many trials ; and that it could not be depended on, except when all the terms on the left-hand side of the equation f{x) = Q were positive. As defects in Newton's, method he signalized — first, its failure to give a commensurable root in finite terms ; secondly, the insecurity of the process which leaves doubtful the exactness of each fresh correction ; and lastly, the failure of the method in the case of an equation with roots nearly equal. The problem Lagrange proposed to himself was the following: — " Etant donnee une equation numerique sans aucune notion prealablc de la grandeur ni de l'espece de ses racines, trouver la valeur numerique exacte, s'il est possible, ou aussi approchee qu'on vouclra de chacune de ses racines." Before giving an account of his attempted solution of this problem, it is necessary to review what had been already done in this direction, in addition to the methods of approximation above described. Harriol discovered in 1631 the composition of an equation as a product of factors, and the relations between the roots and coefficients. Vieta had already observed this relation in the case of a cubic; but he failed to draw the conclusion in its generality, as Harriot did. This discovery was important, for it led to the observation that any integral root must be a factor of the absolute term of an equation, and New- ton's Method of Divisors for the determination of such roots was a natural result. Attention was next directed towards finding limits of the roots, in order to diminish the labour necessary in applying the method of divisors as well as the methods of approximation previously in existence. Descartes, as already remarked, was the first to recog- nise the negative and imaginary roots of equations; and the inquiry Notes. 439 commenced by him as to the determination of the number of real and of imaginary roots of any given equation was continued by Newton, Stirling, De Gua, and others. Lagrange observed that, in order to arrive at a solution of the problem above stated, it was first necessary to determine the number of the real roots of the given equation, and to separate them one from another. For this purpose he proposed to employ the equation whose roots are the squares of the differences of the roots of the given equa- tion. "Waring had previously, in 1762, indicated this method of separating the roots ; but Lagrange observes {Equations Num&riques, Note in.), that he was not aware of Waring's researches when he composed his own memoir on this subject. It is evident that when the equation of differences is formed, it is possible, by finding an inferior limit to its positive roots, to obtain a number less than the least difference of the real roots of the given equation. By substi- tuting in succession numbers differing by this quantity, the real roots of the given equation will be separated. When the roots are sepa- rated in this way Lagrange proposed to determine each of them by the method of continued fractions, explained in the text (Art. 105). This mode of obtaining the roots escapes the objections above stated to Xewton's method, inasmuch as the amount of error in each suc- cessive approximation is known ; and when the root is commensurable the process ceases of itself, and the root is given in a finite form. Lagrange gave methods also of obtaining the imaginary roots of equations, and observed that if the equation had equal roots they could be obtained in the first instance by methods already in existence (see Art. 74). Theoretically, therefore, Lagrange's solution of the problem which he proposed to himself is perfect. As a practical method, however, it is almost useless. The formation of the equation of differences for equations of even the fourth degree is very laborious, and for equa- tions of higher degrees becomes well nigh impracticable. Even if the more convenient modes of separating the roots discovered since Lagrange's time be taken in conjunction with the rest of his process, still this process is open to the objection that it gives the root in the form of a continued fraction, and that the labour of obtaining it in this form is greater than the corresponding labour of obtaining it by Homer's process in the form of a decimal. It will be observed also that the latter process, in the perfected form to which Horner 440 Notes. has brought it, is free from all the objections to Newton's method above stated. Since the period of Lagrange, the most important contributions to. the analysis of numerical equations, in addition to Horner's improve- ment of the method of approximation of Vieta and Newton, arc those of Fourier, Budan, and Sturm. The researches of Budan were pub- lished in 1807; and those of Fourier in 1831, after his death. There is no doubt, however, that Fourier had discovered before the publica- tion of Budan' s work the theorem which is ascribed^to them conjointly in the text. The researches of Sturm were published in 1835. The methods of separation of the roots proposed by these writers are fully explained in Chapter IX. By a combination of these methods with that of Homer, we have now a solution of Lagrange's problem far simpler than that proposed by Lagrange himself. And it appears impossible to reach much greater simplicity in this direction. In" extracting a root of an equation, just as in extracting an ordinary square or cube root, labour cannot be avoided ; and Horner's process appears to reduce this labour to a minimum. The separation of the roots also, especially when two or more are nearly equal, must remain a work of more or less labour. This labour may admit of some reduc- tion by the consideration of the functions of the coefficients which play so important a part in the theory of the different quantics. If, for example, the functions H, I, and J, are calculated for a given quartic, it will be possible at once to tell the character of the roots see Art. 68). Mathematicians may also invent in process of time some mode of calculation applicable to^numerical equations analogous to the logarithmic calculation of simple roots. But at the present time the most perfect solution of Lagrange's problem is to be sought in a combination of the methods of Sturm and Horner. Notes. 441 NOTE C. DETERMINANTS. The expressions which form the subject-matter of Chapter XI. were first called "determinants" by Cauchy, this name being adopted by him from the writings of Ganss, who had applied it to certain special classes of these functions, viz. the discriminants of binaiy and ternary quadratic forms. Although Leibnitz had observed in 1693 the peculiarity of the expressions which arise from the solution of linear equations, no further advance in the subject took place until Cramer, in 1750, was led to the study of such functions in connexion with the analysis of curves. To Cramer is due the rule of signs of Art. 108. During the latter part of the eighteenth century the subject was further enlarged by the labours of Bezout, Laplace, Vandermonde, and Lagrange. In the present century the earliest cultivators of this branch of mathematics were Gauss and Cauchy; the former of whom, in addition to his investigations relative to the discriminants of quadratic forms, proved, for tbe particular cases of the second and third order, that the product of two determinants is itself a determinant. To Cauchy we are indebted for the first formal treatise on the subject. In his memoir on Alternate Functions, published in the Journal de VKcole Polytechnique, vol. x., he dis- cusses determinants as a particular class of such functions, and proves several important general theorems relating to them. A great impulse was given to the study of these expressions by the writings of Jacobi in Crelle's Journal, and by his memoirs published in 1841. Among more recent mathematicians who have advanced this subject may be mentioned Hermite, Hesse, Joachimsthal, Cayley, Sylvester, and Salmon. There is now no department of mathematics, pure or applied, in which the employment of this calculus is not of great assistance, not only furnishing brevity and elegance in the demonstration of known properties, but even leading to new discoveries in mathematical science. Among recent works which have rendered this subject accessible to students may be men- tioned Spottiswoode's Elementary Theorems relating to Determinants, London, 1851 ; Brioschi's La teorica dei Determinant!, Pavia, 1851 Baltzer's Theorie unci Anwendung der Determinanten, Leipzig, 1864 Doctor's Elements de-la theorie des Determinants, Paris, 1877; Scott's 442 Xotes. Theory of Determinants, Cambridge, 1880; and the chapters in Salmon's Lessons introductory to the Modern Higher Algebra, Dublin, 1876. For further information on the history of this subject, as well as on that of Eliminants, Invariants, Covariants, and Linear Transforma- tions, the reader is referred to the notes at the end of the work last mentioned. NOTE D. THE PROPOSITION THAT EVERY EQUATION HAS A ROOT. It is important to have a clear conception of what is established, and what it is possible to establish, in connexion with the proposition discussed in Art. 195. If in the equation a x n + a^" -1 + . . . a n = the coefficients a , a u . . . a n are used as mere algebraical symbols without any restriction ; that is to say, if they are not restricted to denote numbers, either real, or complex numbers of the form treated in Chapter XVII., then, with reference to such an equation it is not proved, and there exists no proof, that every equation has a root. The proposition which is capable of proof is that, in the case of any rational integral equation of the n th degree, whose coefficients are all complex (including real) numbers, there exist n complex numbers which satisfy this equation; so that, using the terms number and numerical in the wide sense of Chapter XVII., the proposition under consideration might be more accurately stated in the form — Every numerical equation of the n th degree has n numerical roots. With reference to this proposition, there appears little doubt that the most direct and scientific proof is one founded on the treatment of imaginary expressions or complex numbers of the kind considered in Chapter XVII. The first idea of the representation of complex num- bers by points in a plane is due to Argand, who in 1806 published anonymously in Paris a work entitled Essai sur une maniere de repri- senter les quantites imaginaires dans les constructions geometriqucs. This writer some years later gave an account of his researches in Gergonne's Annates. Notwithstanding the publicity thus given by Argand to his new methods, they attracted but little notice, and appear to have been discovered independently several years later by Warren in England and Mourey in France . These ideas were developed by Gauss in his Notes. 443 works published in 1831 ; and by Cauchy, who applied them to the proof of the important theorem of Art. 194. "With reference to the proposition now under discussion, the proof which we have given at the close of Art. 195 is to be found in Argand's original memoir, and is reproduced by Cauchy with some modifications in his Exercices d 'Analyse. A proof in many respects similar was given by Mourey. Before the discovery of the geometrical treatment of complex numbers several mathematicians occupied themselves witli the pro- blem of the nature of the roots of equations. An account of their researches is given by Lagrange in Note IX. of his Equations Numi- riques. The inquiries of these investigators, among whom we may mention D'Alembert, Descartes, Euler, Foncenex, and Laplace, re- ferred only to equations with rational coefficients ; and the object in view was, assuming the existence of factors of the form x - a, x - {3, &c, to show that the roots a, /?, &c, were all either real or imagi- nary quantities of the type a+ by/ - 1 ; in other words, that the solution of an equation with real numerical coefficients cannot give rise to an imaginary root of any form except the known form a + b */— 1, in which a and b are real quantities. For the proof of this proposition the method employed in general was to show that, in case of an equation whose degree contained 2 in any power 1c, the possibility of its having a real quadratic factor might be made to depend on the solution of an equation whose degree contained 2 in the power 1c - 1 only ; and by this process to reduce the problem finally to depend on the known principle that every equation of odd degree with real coef- ficients has a real root. Lagrange's own investigations on this sub- ject, given in Note X. of the work above referred to, related, like those of his predecessors, to equations with rational coefficients, and are founded ultimately on the same principle of the existence of a real root in an equation of odd degree with real coefficients. As resting on the same basis, viz., the existence of a real root in an equation of odd degree, may be noticed two recently published methods of considering this problem — one by the late Professor Clif- ford (see his Mathematical Papers, p. 20, and Cambridge Philosophical Society's Proceedings, II., 1876), and the other by Professor !Malet (Transactions of the Royal Irish Academy, vol. xxvi., p. 453, 1878). Starting with an equation of the 2m th degree, both writers employ Sylvester's dialytic method of elimination to obtain an equation of the degree m(2m - 1) on whose solution the existence of a root of the I I I Notes. proposed equation is shown to depend ; and since the number m [2m - 1) contains the factor 2 once less often than the number 2rn, the problem is reduced ultimately to depend, as in the methods above mentioned, on the existence of a root in an equation of odd degree. The two equations between which the elimination is supposed to be effected are of the degrees m and m - 1 ; and the only difference between the two modes of proof consists in the manner of arriving at these equations. In Professor Malet's method they are found by means of a simple transformation of the proposed equation, while Professor Clifford ob- tains them by equating to zero the coefficients of the remainder when the given polynomial is divided by a real quadratic factor. The forms of these coefficients are given in Ex. 31, p. 286 ; and it will be readily observed that the elimination of (3 from the equations obtained by making r and >\ vanish will furnish an equation in a of the degree m(2m - 1). INDEX Abel, 435. Addition of determinants, 248. Algebraical equations, 2, 197. their solution, 103. solution of cubic, 106. of biquadratic, 110. Alkbayyanii, 433. Alternants, 283. Approximation to numerical roots : Newton's, 207. Horner's, 209. Lagrange's, 223. Arabians, 433. Argand, 442. Argument of complex variable, 422. variation of, 428. Arrays, square, 232. rectangular, 259. Ball, on quadratic factors of sextic co- variant of quartic, 372. Baltzer, 441. Ben Musa, 433. Bezout's method of elimination, 324. Binary forms transformed to ternary, 411. Binomial coefficients, 68. Binomial equations : solution of, 90. leading general properties of, 92. solution by circular functions, 98. Biquadratic, 73. Euler's solution of, 119. Ferrari's, 127. Descartes', 131. transformed to reciprocal form, 133. solved by symmetric functions, 137. equation of differences, 140. nature of its roots, 112, 194. Bombelli, 434. Bring, 436. Brioschi's differential equation, 304. Budan : theorem of Fourier and, 172. Cardan: solution of cubic, 106. his relations with Tartaglia, 434. Cauchy : his theorem, 431. on determinants, 441. Cayley : solution of cubic, 369. number of covariants and inva- riants of cubic, 369. solution of quartic, 376. results of Tschirnhausen's transfor- mation, 389. Circulants, 284. Clebsch, referred to, 359, 397, 436. Cogredient defined, 357. Colla, 433. Commensurable roots : theorem, 198. Complex variable, 425. Continuants, 285. Continuity, rational integral function, 9. function of complex variable, 427. Covariants : definitions, 338. formation of, 339. properties of, 341. formation by operator D, 344. Roberts' theorem relating to, 346. homographic transformation ap- plied to, 347. properties derived by linear trans- formation, 351. propositions relating to their for- mation, 354, 356, 358. of cubic, 366. their number, 369. of quartic, 371. their number, 379. mi; Index. Cube routs of unit} . 43. Cubic, 71. equation of differences, 81. criterion of nature of its roots, 84. Cardan's solution of, 106. as difference of two cubes, 109. solved by symmetric functions, 111. nomographic relation between two roots, 118. covariants and invariants, 366, 369. transformed by Tschimhausen's method, 389. De Gua : rule for finding imaginary roots, 180. Derived functions, 8. graphic representation of, 152. in terms of the roots, 155. Descartes: rule of signs, 28, 30, 180. solution of the biquadratic, 131. his improvements in Algebra, 431. Determinants : definitions, 229. propositions relating to, 231-255. minor determinants, 239. development of, 239, 244, 245, 247. addition of, 248. multiplication of, 255. reciprocal determinants, 264. symmetric determinants, 267. skew and skew-symmetric, 269. miscellaneous examples in, 276. note on their history, 441. Dialytic method of elimination, 323. Discriminants, 331. Divisors, Newton's method of, 199. Elimination, 318. by symmetric functions, 319. Euler's method, 322. Sylvester's method, 323. Bezout's method, 321. the common method, 329. Equal roots, 25. condition for in cubic, 84. in biquadratic, 142. determination of, 157. Equation of squared differences : of cubic, 81. of the general equation, 84. of biquadratic, 140. Equation whose roots are any powers of roots of given equation, 78. Equations linear, solution of, 202. linear homogeneous, 263. Kiilor: solution of biquadratic, 119. method of elimination, 322. publication of his AUjcbra, 434. Ferrari : solution of biquadratic, 127. Florido, 433. Fourier: his theorem, 172,440. the theorem applied to imaginary roots, 177. corollaries from the theorem, 180. Galois, 436. [13. Graphic representation : of polynomial, of derived functions, 152. of imaginary quantities, 422. < rreatheed : solution of biquadratic, 134. Harley, 436. Ilermite : his theorem relating to limits of roots, 403. on the reduction of the quiutic, 436. Hessian : of cubic, 340, 367. of quartic, 346, 371. its form in general, 357. of quartic expressed by factors of sextic covariant, 374. Homogeneous : linear equations, 263. quadratic expressed as sum of squares, 401. Homographic transformation, 75. reduced to double linear transfor- mation, 349. Homographic relation between roots of a cubic, 118, 399. Horner : bis method of solving nume- rical equations, 209. contraction of the process, 217. process applied to cases where roots are nearly equal, 220. his improvements in solution of numerical equations, 439. Imaginary roots, 2 1 . enter equations in pairs, 26. Imaginary quantities : graphic representation, 422. addition and subtraction of, 423. multiplication and division of, 424. Index. 447 Invariants : definitions, 33S, 349. formation of, 339. properties of, 341. modes of generation, 355, 356. of cubic, 369. of quartic, 371, 379. of the form k U — \H Z , 377. Jacobian, defined, 359. Jerrard, 436. Jordan, 436. Lagrange : method of approximation, 223. on equation of differences, 140. on solution of equations, 435. his treatise on Numerical Equatiom referred to, 141, 438, 439, 443. Laplace : development of determinant, 244. proof that equation has a root, 443. Leonardo, 443. Limits of roots : definitions, 163. propositions relating to superior limits, 163, 164, 168. Newton's method, 168. inferior limits, and limits of nega- tive roots, 169. Limiting equations, 170. Linear equations : solution of, 262. homogeneous, 263. Lucas de Burgo, 433. Maxima and minima, 17, 153. Minor determinants, 239. Modulus : linear transformation, 349. of complex numbers, 422. Multiple roots, 156, 157. determination of, 157. Multiplication of determinants, 255. of complex numbers, 424. Newton's method of finding limits, 168, 180. method of divisors, 199. of approximation to numerical roots, 207. theorem on sums of powers of roots, 2S9. Numbers, complex, 108, 42-5. Numerical equations, 2, 197. commensurable roots of, 198. multiple roots of, 157, 204. methods of approximation, 207, 209, 223. note on the solution of, 437. note on the proposition that every equation has a root, 442. Order of Symmetric Functions, 299. Polynomials: general properties, 5, 6. change of form of, 8. continuity of, 9. graphic representation of, 13. maxima and minima, 17. Quartic: covariants and invariants, 371. expressed by quadratic factors of scxtic covariant, 374. resolution of, 376. number of its covariants and inva- riants, 379. transformed by Tschirnhausen, 390. two transformable into each other, 396. Quintic : solution of special form, 102. reduction to trinomial form, 394. to sum of three fifth powers, 395. impossibility of its solution by ra- dicals, 435. Quotient and remainder: when poly- nomial is divided by binomial, 10. when one polynomial is divided by another, 286. general forms of the coefficient* when polynomial of even degree is divided by a quadratic, 286. Reality of roots : of cubic, 84. of biquadratic, 142. in general, 193. Reciprocal determinants, 264. Reciprocal roots and reciprocal equa- tions, 62. solution of reciprocal equations, 90. transformation of biquadratic to reciprocal form, 133. Rectangular arrays, 259. Removal of terms, 67. of three terms by Tschirnhausen's transformation, 393. I is Index. Resultant of two equations, 318. properties of, 320. B " rts : on an equation derived from two culiics, 116. on equation of squared differences of biquadratic, 142. identical relation, 161. example on quart ic and quintic, 195. on source of covariant, 339. theorem on covariants, 346. on product of covariants, 362. multinomial theorem, 362. Rolle's theorem, 155. Roots: theorems relating to, 19. imaginary, 21. number of, 22. equal, 25. Descartes' rule for positive, 28. for negative and imaginary, 30. how related to coefficients, 35. rube roots of unity, 43. symmetric functions of, 46, 2S9. multiple, 156, 204. limits of, 163. separation of, 172. commensurable, 198. common to two equations, 333. Hermite's theorem on limits of, 403. Cauchy's theorem concerning, 431. Routh : examples in determinants, 28S. Rule : Descartes', 28. of signs for determinant, 232. DeGua's, ISO. Salmon's Modern Higher Algebra re- ferred to, 288, 359, 436, 442. ' Scipio Ferreo, 433. Reparation of roots, 172. Serret's Algebra referred to, 394, 399. Sextic covariant of quartic, 372. Simpson, 434. Skew and skew-symmetric determi- nants, 269. Special roots of binomial equations, 95. Sturm: his theorem, 181. for equal roots, 186. application of theorem, 189. leading coefficients of functions ex- pressed as determinants, 312. Sylvester's forms of his remain- ders, 406. Sums of powers of roots : New ton's the irem on, 289. in terms of coefficients, 293. coefficient 1 by, 294. operation involving, 309. Sylvester: meth id of elimination, 323. reduction of quintic to sum of three fifth powers, 396. theorem relating to quadric ex- pressed as sum of squares. 402. forms of Sturm's remainders, 406. Symmetric functions: definitions, 40. theorems relating to, 53. transformation by means of, 70. expressed rationally in terms of coefficients, 291. order and weight of, 53, 299. calculation of, 300. by Brioschi's equation, 305. derived from given one, 306. applied to elimination, 319. of the roots of two equations, 334. Tabulation of functions, 12. Tartaglia, 433. Transformation : of equations, 60. of cubic, 71. of biquadratic, 73. homographic, 75. by symmetric functions, 76. in general, 80. reduction of homographic to double linear, 349. linear, applied to covariants, 351. theorem relating to, 385. Tschirnhausen's, 387. geometrical, 411. of binary to ternary forms, 411. Tschirnhausen's transformation, 387. applied to cubic, 389. to quartic, 390. reduction of cubic to binomial form, 391. of quartic to trinomial form, 392. of quintic to trinomial form, 394. Vandermonde, 435, 441. Variable, complex, 425. Vieta, 437. Wantzel, 435. Weight of symmetric functions, 53, 2 A LI ■it- )) = — tzl As y RETURN PHYSICS LIBRARY TO— #► 351 LeConte Hall LOAN PERIOD 1 -OUARWR- 642-3122 2 / - Mo\/ T ^ ALL BOOKS MAY BE RECALLED AFTER 7 DAYS Overdue books are subject to replacement bills DUE AS STAMPED BELOW JftfP3-J383 HA848J983 D EC>fr-^ FORM NO. DD 25 5m, UNIVERSITY OF CALIFORNIA, BERKELEY 78 BERKELEY, CA 94720 M •s^. s . ^w%gg 2 /.- JN I VERS ITT M^dmKlT X: LIBRARY OF THE UNIVERSITY OF CALIFORNIA — IP — P NIVERSITY OF CALIFORNIA LIBRARY OF THE UNIVERSITY OF CALIFC", NIVERSITY OF CALIFORNIA LIBRARY OF THE UNIVERSITY OF CALIFORNIA mw, ::'"'■■ mm