THE BOOK WAS DRENCHED a: DO DQ THE THEORY OF EQUATIONS. VOLUME I. DUBLIN UNIVERSITY PRESS SERIES TIIK THEORY 01- EQUATIONS: INTRODUCTION TO THH THHORY OF BINARY ALGKBRAIC FORMS. BY WILLIAM SNOW BURNSIDK, M.A., D.Sc., LAIE SENIOR I-KLIOW OF TRINI1Y COII.EGK, DUBLIN; AND ERASMUS SMUH's PROFESSOR OP MATHBMAIICS IN IHK UNIVBRSI'IY OF DUBLIN; AND ARTHUR WILLIAM PANTON, M.A., D.Sc., I'KIIOW AND 1U1 OR, IRINI'lY COLLEGE, DUBLIN; DONhOAl I.KCirRI<K IN MA1HEMA1ICS. VOL. I. EIGHTH EDITION (Second hsue). DUBLIN: HODGES, FIGGIS, & CO., NASSAU STREET. LONDON: LONGMANS, GREEN Sc CO., PATERNOSTER ROW. 1924. Printed by PONSOVBV & GIBBS, University Press, Dublin. PREFACE TO SEVENTH EDITION OF VOLUME I. HAVING been deprived of Dr. Pan ton's co-operation, I have thought it inadvisable to make any change in this volume by the introduction of new matter, as I wished as far as possible to preserve the original design, and also the limits fixed for this volume as indicated in the Preface to previous editions. In coining to this decision I have been influenced to a great degree by the favourable reception already given to this work. W. S. F>. June !//, 1912. Dr. linrnside died llth March, 1920. Dr. Panton died 18th December, 1906, vi Preface. under the title of Modern Higher Algebra, a Chapter on Determinants. It has been our aim to make this Chapter as simple and intelligible 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 kindness of Mr. Cathcart, Fellow of Trinity College. We have approached the consideration of Covariants and Invariants through the medium of the functions of the differences of the roots of equations. This appears to be the simplest and most attractive mode of presenting the subject to beginners, and has the advantage, as will be seen, of enabling us to express irrational oovariants rationally in terms of the roots. 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 trans- formation of algebraic forms. Of the works which have afforded us assistance in the more elementary part of the subject, we wish to mention particularly the Traite d'Alyebre of M. Bertrand, and the writings of the late Professor Young 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 indebted mainly, among published works, to the Lessons Introductory to the Modern Higher Algebra of Dr. Salmon, and the Theorie der bindren of algebraischen Formen Clebsch ; and in some degree to the Theorie des Formes binaires of the Preface. yii Ohev. F. Fad 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 University Examinations. 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, Cayley, Hermite, and Sylvester. We have added also in this and the preceding edition, to what was contained in the earlier editions of this work> a new Chapter on the Theory of Substitutions and Groups. Our aim has been to give here, within as narrow limits as possible, an account of the subject which may be found useful by students as an introduction to those fuller and more systematic works which are specially devoted to this depart- ment of Algebra. The works which have afforded us most assistance in the preparation of this Chapter are Serret's Traitt des Cours (PAlgebre supMeure ; Substitutions et des Equations algebriques by M. Camille Jordan (Paris, 1870); Netto's Substitutionentheorie und ihre Anwendung auf die Algebra (Leipzig, 1882), of which there is an English translation F. N. Cole by (Ann. Arbor, Mich., 1892) ; and Legons sur la Resolution algtbrique des Equations, by M. H. Vogt (Paris, 1895). COLLEGE, DUBLIN, May, 1904. TABLE OF CONTENTS OF VOL I. INTRODUCTION. A rt Page 1. Definitions, 1 2. Numerical and algebraical equations, . * ... 2 3. Polynomials, ........... 4 CHAPTER I. GENEKAL PROPERTIES OF POLYNOMIALS. 4. Theorem relating to polynomials when the variable receives large values, 5 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 8. 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. Maximum and minimum values of polynomials, * . 17 CHAPTER II. GENERAL PROPERTIES OF EQUATIONS. 12. 13, 14. Theorems relating to the real roots of equations, . .19 15. Existtjnce of a root in the general equation. Imaginary roots, .21 16. Theorem determining the number of roots of an equation, . .22 17. Equal roots, 25 18. Imaginary roots enter equations in pairs, . ..... 26 . 28 19. DescRrtes' rule of signs for positive roots, ... x Table of Contents. Art. 30 20. Descartes' rule of signs for negative roots, . 30 21. Use of Descartes' rule in proving the existence of imaginary roots, for the 22. Theorem relating to the substitution of two given numbers 31 variable, 32 Examples, 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 36 24. Applications of the theorem, 42 26. Depression of an equation when a relation exists between two of its roots, " 43 26. The cube roots of unity, 46 27. Symmetric functions of the roots, 48 Examples, 53 28. Theorems relating to symmetric functions, Examples, . ... 54 CHAPTER 1Y. TRANSFORMATION OF EQUATIONS. 29. Transformation of equations 60 30. Roots with signs changed, 60 31. Roots 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 84. Removal of terms, 67 35. Binomial coefficients, , . 68 36. The cubic, 71 37. The biquadratic, . 73 38. transformation 76 Homographic , 89. Transformation by symmetric functions, 76 40. Formation of the equation whose roots are any powers of the roots of the proposed equation, 78 41. Trans foimation in general, ......... 80 42. Equation of squared differences of a cubic . .81 43. Criterion of the nature of the roots of a cubic, j . 84 44. Equation of differences in general, ......*. 84 Examples, 86 Table of Contents. xi CHAPTER V. SOLUTION OF RECIPROCAL AND BINOMIAL EQUATIONS. Art. Page 90 45. Reciprocal equations, 46-52. Binomial equations. Propositions embracing their leading general properties, 92 - 53. The special roots of the equation #> 1 = 0, 96 54. Solution of binomial equations by circular functions, .... 98 Examples, 100 CHAPTER VI. ALGEBRAIC SOLUTION OP THE CUBIC AND BIQUADRATIC. 105 55. On the algebraic solution of equations, 66. The algebraic solution of the cubic equation, 108 109 67. Application to numerical equations, 68. Expression of the cubic as the difference of two cubes, . .111 59. Solution of the cubic by symmetric functions of the roots, . .113 Examples, 114 60. Homographic relation between two roots of a cubic, .... 120 61. First solution by radicals of the biquadratic. Euler's assumption, . 121 Examples, 126 62. Second solution by radicals of the biquadratic, 127 63. Resolution of the quartic into its quadratic factors. Ferrari's solution, . 129 7 64. Resolution of the quartic into its quadratic factors. Descartes solution, 133 65. Transformation of the biquadratic into the reciprocal form, . 135 66. Solution of the biquadratic by symmetric functions of the roots, . 139 67. Equation of squared differences of a biquadratic, 142 68. Criterion of the nature of the roots of a biquadratic, .... 144 Examples, ............ 146 CHAPTER YIL PROPERTIES OF THE DERIVED FUNCTIONS. 69. Graphic representation of the derived function, 154 70. Theorem relating to the maxima and minima of a polynomial, . .155 71. Rolle's Theorem. Corollary, 167 72. Constitution of the derived functions, 167 73. Theorem relating to multiple roots, . .168 74. Determination of multiple roots, 159 75. 76. Theorems relating to the passage of the variable through a root of the equation, 161, 162 Examples, 163 xii Table of Contents. CHAPTER VIII. SYMMETRIC FUNCTIONS OF THE ROOTS. Art. Page 77. Newton's theorem on the sums of powers of roots. Prop. I., . 166 78. Expression of a rational symmetric function of the roots in terms of the coefficients. Prop. II., 167 79. Further proposition relating to the expression of sums of powers of roots in terms of the coefficients. Prop. III., 169 80. Expression of the coefficients in terms of sums of powers of roots, . 170 81. Definitions of order and weight of symmetric functions, and theorem relating to the former, 173 82. Calculation of symmetric functions of the roots, . * .174 83. Homogeneous products, * . 178 CHAPTER IX. LIMITS OF THE HOOTS OF EQUATIONS. 84. Definition of limits, 180 85. Limits of roots. Prop. I.. 180 86. Limits of roots. Prop. II., 181 87. Practical applications, 183 88. Newton's method of finding limits. Prop. III., .... 185 89. Inferior limits, and limits of the negative roots, 186 90.