UC-NRLF B M 2SD M3b AN ELEMENTARY TREATISE THEORY OF EQUATIONS SAMUEL MARX BARTON, Ph.D. PUOFESSOR OF MATHEMATICS. UNIVF.KSITY OF THE SOUTH BOSTON, U.S.A. D. C. HEATH & CO., PUBLISHERS 1899 A "B Copyright, 1899, By D. C. heath & CO. CAJORl. .T. S. Gushing & Co. - Berwick & Smith Norwood Mass. U.S.A. PREFACE. Ix this treatise it is my aim to give the elements of Deter- minants and the Theory of Equations in a form suitable, botli in amount and quality of matter, for use in the undergraduate courses in our colleges and universities. To this end I have endeavored to make the work in every part readily intelligible to the average student who has become proficient in algebra and the elements of trigonometry. All use of the calculus has purposely been avoided. While the presentation of the sub- ject has necessarily been condensed to suit the requirements of college courses, great pains has been taken not to sacrifice clearness to brevity. It is a short treatise, but not a syllabus. Part I treats of Determinants. The first two chapters give the fundamental theorems, with examples for illustration. The third chapter consists* of applications and special forms of determinants, followed by a collection of carefully selected examples. These three chapters on determinants should serve as a helpful introduction to the study of this interesting class of functions. Part II treats of the Theory of Equations proper. The principal elementary theorems concerning algebraic and nu- merical equations are deduced. After a brief introduction, giving definitions, etc., there follows a chapter on Complex Quantities, a subject which seems worthy of more space than is usually allotted to it in so elementary a treatise. This chapter, however, is given not so much for use in the chai)ters that follow, as with the hope tliat it may prove useful to the iii 911343 iv PREFACE. student who pursues later in liis course the study of the Theory of Functions. As all the theorems considered have become classic, no special references to authors consulted seem neces- sary in the body of the book. After Chapter IV I have fol- lowed quite closely Burnside and Panton, though in some places the general arrangement has been altered to make the necessary abridgments Avhile securing clearness, and, wherever it seemed desirable, the method of proof has been changed. Almost every theorem is elucidated by the complete solution of one or more representative examples. I desire to call spe- cial attention to this feature of the book, which will surely commend itself alike to teacher and pupil. In Chapter XI I have striven to make the rather complicated process of the solution of numerical equations as simple as possible. It would defeat the object of this treatise were much space devoted to these methods, which are laborious and of no great practical value, but what is given is complete in itself. Hor- ner's method is explained in detail. The following works have been most helpful in the prepa- ration of the treatise, Muir and Burnside and Panton in particular furnishing many examples: Baltzer, Theorie mid Aiiivendung der Deterniiuantem, 1881; Burnside and Panton, Theory of Equations, 1892 ; Carnoy, Cours d'Alg^bre Supe- rieure, 1892; Hoiiel, Conrs de Calcul Infinitesimal, 1878; Klempt, Lehrbuch zur Einfalirunq in die Moderne Algebra, 1880; Muir, A Treatise on Determinants, 1882. Todhunter's Theory of Equations, Chrystal's Algebra, \o\. 1, Scott's Theonj of Determinants, and that excellent little American work by Professor L. G. Weld {A Short Course in the Theory of De- terminants) should also be mentioned; and the author has consulted with profit the well-known works of Serret, Peter- sen, Biermann, Matthiessen, and others. PREFACE. V The author gratefully acknowledges his indebtedness to Dr. 1). E. Smith, of the State Normal School, at Brockport, N.Y., to Professor William H. Echols, of the University of Virginia, who have read the manuscript and made suggestive criticisms, and to Professor R. D. Bohannan, of the Ohio State Universit}^, and Dr. J. H. Gore, of the Columbian University, Washington, who have kindly read the revised proof sheets, and given many valuable suggestions, though he does not wish to hold them in the least responsible for the method followed in the treatment of the subject, nor for any errors that may have crept into the work. SAMUEL M. BARTON. Sewanek, Tenn., 1899. TABLE OF CONTENTS. PART I. DETERMINANTS. CHAPTER I. Origin, Notation, and General Definition of Determinants. ARTK LE l-AOE Historical note 1 1. Permutations 2 2. Permanences and inversions 2 3. Cliange of class of permutations 3 4. Number of even and odd permutations in a group ... 4 5. First definition of a determinant 4 6. Second definition of a determinant 5 7. General rule for the expansion of a determinant ... 6 8. Rule of signs 9. Determinants as the result of elimination .... 7 10. Interpretations of a determinant array 8 11. Values of unknowns of two simultaneous equations ... 8 Examples 12. Three simultaneous equations 10 13. Diagram for expanding determinants 11 Examples 12 14. Values of x, y, and z from three simultaneous equations . . 12 15. Four simultaneous equations !•"> 16. Definitions concerning elements and rows . . . .14 17. Other notations 14 Examples 16 Vlll CONTENTS. CHAPTER II. Properties of Determinants. AKT. PAGE 18-21. Elementary theorems relating to determinants ... 18 Examples 20 22. Determinant minors 2.3 23. Development of a determinant 25 24. Application of these principles 27 25-28. Special theorems concerning determinants .... 29 29. Development of a determinant 30 30. Change of order of a determinant 31 31'. Evaluation of determinants 32 Examples 32 32. Laplace's development 34 33-35. Theorems concerning the addition of determinants . . 38 Examples 40 36. The product of two determinants 43 37. Euler's theorem 45 38. Rectangular arrays 46 39. Reciprocal determinants 48 Examples 49 CHAPTER III. Applications and Special Forms of Determinants.- AppUcatioiis of Determinants. Solution of simultaneous linear equations, — number of unknowns same as number of equations .... 50 Examples 53 42-43. Number of equations greater than the number of imknowns 55 44-46. Homogeneous linear equations 58 Determinants of Special Forms. 47. Symmetrical determinants 63 48. Skew symmetric and skew determinants 65 Miscellaneous examples 65 CONTENTS. PART II. THEORY OF EQUATIONS. INTRODUCTION. ART. PAGE Historical note 70 49. Elementary principles 77 50. Functions defined 7H 51. Equations defined 79 52. Classification of equations 80 CHAPTER IV. Complex Numbers. 54. Definitions of an imaginary number 82 55. The complex number 82 56. Successive powers of i 82 57-61. Elementary theorems concerning the complex number . 83 62. Conjugate imaginaries 85 -i-SS. Theorem concerning conjugate roots 85 64. Definition of norm and modulus 85 65-66. Theorems concerning moduli 86 67. Graphic representation — Argand's diagi-am .... 87 68-69. Exponential form of x + iy 88 70. DeMoivre's theorem 90 1 71. Values of (e'S)'' 92 72. Solution of the equation x" - 1 = 92 73. Solution of the equation a;" + 1 = 93 74. Addition of complex numbers 94 75. Subtraction 95 76. Multiplication and divi.siou '•'0 CONTENTS. CHAPTER V. Properties of Polynomials. ART. 77. Reduction to the form /(x) = 78. Theorem relating to polynomials when the variable receives large values 79. Similar theorem when the variable receives small values 80. Derived functions. Change of form of a polynomial spending to an increase or decrease of the variable 81. Continuity of a rational integral function 82. The remainder theorem 83. Tabulation of functions 84. Graphic representation of a polynomial Examples PAGE 97 98 100 101 102 103 106 107 109 89. CHAPTER VI. General Properties of Equations. Theorems relating to the existence of a root in special cases 112 . 114 . 115 . 116 . 117 . 118 . 119 Existence of a root. Imaginary roots Theorem concerning the number of roots of an equation Examples Equal roots Imaginary roots occur in pairs Descartes' Rule of signs Examples 122 CHAPTER VII. Relations between Eoots and Coefficients. — Symmetric Functions. 94. Eelations between the roots and coefficients of an equation . 124 95. Applications of the preceding theorem 125 Examples 120 90. Derived functions 128 97. Multiple routs, — theorem ....... 129 CONTENTS. Xi ART. PAOP, 98. Determination of multiple roots ]2".t Examples l;50 99. Theorem relating to the passage of the variable through a root of the equation l;Jl 100. The cube roots of unity 132 101. Symmetric functions of the roots i;33 Examples 134 102. 103. 104. 105. 100. 107. 108. 109. 110. 111. 112. 113. 114. 115. IIG. CHAPTER VIII. Traksformation of Equations. Roots with signs changed .... Roots multiplied by a given number Examples . 13G . 137 . 138 Reciprocal roots 139 . 139 . 140 . 141 . 143 . 144 Conditions for infinite roots Reciprocal, or recun-ing, equations .... Examples Roots diminished or increased by a constant difference Examples Removal of terms 140 Algebraic solution of the cubic equation 147 Application to numerical equations 149 Algebraic solution of the biquadratic equation . . .150 Examples 152 CHAPTER IX. Limits of the Roots of an Equation. Definition of limits .... Limits of roots, Prop. I. . Limits of roots. Prop. II. A third method of getting the limits Examples Inferior limits and limits of the negative roots Examples 153 153 154 150 150 158 159 Xll CONTENTS. Separation of the Roots of Equations. ART. PAGE 117. Separation of the roots 159 118. Sturm's theorem 160 119. Application of Sturm's theorem 163 Examples 164 CHAPTER X. Elimination, 120. Review of methods of Chapter III. 121. Resultant by simple elimination 122. Euler's method of elimination , 128. Sylvester's dialytic method 124. Other methods of elimination . • Examples 166 1C6 1G7 169 171 172 CHAPTER XI. Solution of Numerical Equations. 125. Difference between algebraic and numerical equations . .175 126. Theorem concerning commensurable real roots oif(x)—0 . 175 127. Integral roots determined by trial 176 128. Newton's method of divisors 177 129. Application of the method of divisors . . ... .178 . 178 . 181 . 183 . 184 . 187 Examples 131. Newton's method of approximation . - . 132. Horner's method of solving numerical equations Examples 133. Principle of tlie trial divisor Examples 188 134. Negative roots 191 Examples 191 Miscellaneous examples 191 Appendix 197 THEORY OF EQUATIONS. PART I. — DETERMINANTS. CHAPTER I. THE ORIGIN, NOTATION, AND GENERAL DEFINITION OF DETERMINANTS. As an introduction to tlie Theory of Equations, it seems proper that Ave should devote a few chapters to the discussion of the important class of functions known as determinants. Historical Note. The first notion of Determinants we owe to Lt'il)nitz, who iu l(i93 had observed the peculiarity of the expressions which arise from tlie solution of linear equations. These functions were first called "determinants" by Cauchy, this name being adopted by him from the writings of Gauss, who had applied it to certain special classes of the.se functions ; namely, the discriminants of binary and ternary quadratic forms. After Leibnitz no furtlier advance in, the sub- ject was made until Cramer, iu 1750, was led to the study of such functions in connecti(m with the analysis of curves. During the latter part of tlie eighteenth century, tlie subject was further enlarged by the labors of Bezout, Laplace, Vandermonde, and Lagrange. In the present century the first mathe- maticians who were prominent in developing this branch of mathematics were Gauss and Cauchy, and the subject was also studied by Binet in France and Wronski in Italy. We are indebted to Cauchy for the first formal treatise on the subject. A great impetus was given to tlie study of these functions by the writings of Jacobi in Vrellc's Jouriial in 1841. Among more recent writers who liave advanced the subject may be mentioned Hermite, Hesse, Joachims- tlial, Cayley, Sylvester, and Salmon. Text-books on Determinants were written by Spottiswoode (1851), Brioschi (1854), Baltzer (18,-)7), Giinter (1875), Do.stor (1877), Baraniechi (1871»), Scott (1880), Muir (1882), Weld (lS«t:3), and others.* * The general text-books on Higher Algebra that devote n chapter or two to Determi- nants are numerous. Some of these are referred to in the author's rreface. Gordon's Vorlt- 1 ! *'2' ::>.:/•*•. •.•' theory of EqUATIONS. Art. l 1. Permutations. Let there be a group of elements a, b, c, d, e, •••, represented by different letters or the same letter affected with indices arranged in order of increasing magnitude. If we assemble these elements by placing them in any order, the group thus obtained is called a permHtcUion. It is proved in algebra that the number of permutations of the members of a group of n things is 1 • 2 • 3 • • • >i, or M ! * If the members of a group are arranged alphabetically, or, when represented by a single letter, if the indices of that letter occur in order of increasing magnitude, tliey are said to be written in the natural order. In one, and in only one, of the permutations of the members of a group, the members are arranged in their natural order. In every other permutation the natural order is more or less deranged. 2. Any two members of a group arranged in their natural order constitute a permanence. Thus, the pairs ab, ac, be, bd, 12, 13, 23, are permanences. Any two members of a group arranged in an order which is the reverse of the natural order constitute an inversion. Thus the permutation eadcfb presents eight inversions, ea, ed, ec, eh, dc, db, cb, fb ; sungen vher Immrianfentheorie I. Bd. mipht be mentioned in tliis connection. For extended bibliographical notice, see Gunther's Lehrhtich der />eterm>ii(infe>i-T/ieoiie, pp. 208 and 200, Muir's Theorij of DeierminatifK, and Scott's Theory of Detetmi.nants. * The symbol \v_, read "factorial «," is also used to denote the product of the first n whole numbers, but in printed work n I is the most convenient symb<»l. Art. 3 GENERAL DEFIXITION OF DKTEliMIXAyrs. 3 the permutation a-,aia^a^a2 presents seven inversions of siibscri})ts, 51, 54, 53, 52, 43, 42, 32. The permutations of the members of a group are diviikMl into two classes, the even or positive permutations, and tlie odd or negative permutations. Even permutations are those which contain an even number of inversions. Odd permutations are those which contain an odd number of inversions. The permutations deabc, 32541 are even (positive), because each contains an even number (six) of inversions ; while the permutations daehc, 32451, are odd (negative), because each contains an odd number (five) of inversions. 3. Theorem. A permutation changes its class, from even to odd or from odd to even, token an// tiro of its members are infi'i- changed. /^q - Let a, b, c, ■■■,q be the indices of tlie elements of a certain permutation. Now form the product F^(b-a)(c-a)(d-a) . . . . ('/-«)■ (c - b){d -b) . . . . ('I - b) (d- c) . . . . (7- '•) ■Uj-P) of the differences two and two of these indices, taken by sul> tracting each of them from all those which come after it in this permutation. To each inversion there will correspond a nega- tive difference ; therefore, P will be positive or negative accord- ing as the permutation belongs to the even or to the odd class. This assumed, let g, k be the indices of two rb-mcnts whitdi 4 THEORY OF EQUATIONS. Art. 3 are to be interchanged ; the product P, relative to the original permutation, can be put under the form 1 2 3 4 P= ± l(6-a)(c-a)-.. X (g-a)(g-b)-'- x {k-a){k-b)--- x (k-g), the group (1) embracing all the factors not contained in the groups (2), (3), (4). If we interchange g and k, the group of factors (1) undergoes no change ; the groups (2) and (3) will only be interchanged, the one for the other ; the factor (4) alone will change its sign, and, therefore, the permutation will change its class, which was to be proved. The sign of the product P, which determines whether the permutation is even or odd, is called, for brevity, the sign of the permutation, and hence the name positive and negative is given to the even class and odd class respectively. 4. TiiKOKEM. Of all possible penmitations of the members of a group, one-half are even and one-half are odd. Supjiose all the possible permutations written down. Now, let a new set of permutations be formed by fixing upon any two of the members and interchanging them in each permu- tation. The even permutations will thus be clianged to odd, and the odd to even. That is, for every even permutation in the old set there is an odd one in tlie new, and vice versa. But, as is evident, the new set of permutations is the same as the old, only differently arranged. Hence in either set there are as many even as there are odd permutations, or one-half the permutations are even and oncvhalf are odd. 5. Definition. ^Ve can now give our lirst definition of a dotcn-minant. A determinant of order n is the algebraic sum of the permutations of a product of >i elements aib.jc^ ••• k„_il„ obtained either by the interchange o^ letters, or by the inter- change of subscripts^. Z^''"' ' ^' ' If Art. G GENERAL DEFIXITIOy OF UETEUMIXAyrs. ,j [We must of course remember that permutations of the even class have the + sign ; those of the odd class, the — sign.] The function aib., — a.,bi of the four quantities a„ bi, «2, b.,, is obtained by assigning to a and b written in alpliabetical order, the suffixes 1, 2, and 2, 1, corresponding to the two per- mutations of the numbers 1, 2 (th.e second term being minus, because 2, 1 is odd) ; and adding the two products so formed. Similarly the function chh<^3 — "1^2 + (iA<-\ — ^QhCi 4- a-Ac, — aJj.^Ci (1) of the nine quantities cii, bi, Cj, €12, bo, Cg, C'S) ^3) ^3; is obtained by adding algebraically all the products abc which can be formed by assigning to the letters (retained in their alphabetical order) suffixes corresponding to all the permutur tions of the numbers 1, 2, 3. In like manner, we could form a similar function of the 4th order, of the sixteen quantities «1, K Cl, du a,, h, Co, d,, «3, K c„ (h, a^, K Ci, ch- These functions are Determinants according to our first defini- tion. In these functions, the quantities a^ bi, c,, dy, a^ etc., are called elements, or constituents. 6. Second Definition. We see from the foregoing that a determinant embraces a square number of elements, and this leads us to a notation — a square array of the elements between two vertical lines, thus : THEORY OF EQUATIONS. Art. 6 A = ctj b., C2 ttg 63 C3 (1) and we give as our second definition, embodying this notation, — fa" determinant of a system of n^ elements, which are arranged / in n rows of n elements each, or n columns of n elements each, is the algebraic sum of alL_possible products of n of these elements, no two of which belong to one row or to one column, the sign of any product being +, if the term is an even per- mutation ; — , if the term is an odd permutation. 7. It follows at once from this definition that a general rule for the expansion of a determinant array is : Write down all the products which can be formed by taking as factors one, and only one, element from each column and each row of the array. Of these products, the number of which is n ! , one half involve the even permutations, and the other half involve the odd permutations of the subscripts 1,2,3, ...n. Now give to those products the 2^osHive sign, if the permuta- tions of the subscripts are even; the negative sign, if the per- mutations are odd, and take their algebraic sum. The result is the expanded form of the determinant array. This method of expansion is, however, of little practical value. 8. Rule of Signs. The diagonal aib-yC^ ••• l„ is called the principal diagonal of the determinant. The product afi^c^ ••• /„ of the letters in the principal diagonal always has the sign +, because in this permutation the letters as well as the suffixes occur in the natural order. [ To determine the sign of any other product : first, put its I letters in alphabetical order ; then count the interchanges necessary to bring the subscripts into the order, 1, 2, 3, 4, -••, ' of the svibscripts in the principal diagonal. If they make an Art. 9 GENERAL DEFINITION OF DETEUMl S AS I. >. 7 even number, the term is affected with + ; if they nuike an odd number, the term is affected with — . A better rule is : To determine the sign of any term, count it.f number of inversions, making the sign j)lus or minus according a^ that number is even or odd. EXAMPLES. 1. What sign is to be attached to the term aj)-cjli\f^f., in the determinant of the seventh order ? Here four interchanges put the subscript 1 first, tlien five interchanges put 2 in second i)lace, then three put 4 in fourth pLace, then two put 5 in fifth place, and finally one inter- change puts 6 between 5 and 7 ; hence in all there are fifteen interchanges, consequently the sign of the term is — . Or, more simply, the number of inversions is 15, an odd number, therefore the sign of the term is minus. 2. In a determinant of the fourth order, find the signs of the iQVTCi^-.^a^iCid^; aib^Csd.,; ctjbf'id^; a^)jp-/li. 3. In a determinant of the fifth order, find the signs of the terms : a-jb^Cod^ei ; aJyoC^die-j ; biaod^c^e^ ; eia-^dib^Ci- 9. Determinants most frequently occur as the result of elimination from linear equations. For example, solving the two simultaneous linear equations, ciiX + &i.y = miy ttoX -\-b-^ = m2, we readily get * x = —^;= j-^ (1) O162 — t'/'i 1 a,m., — a..mi ,,,-. and y = -^ '-r (-) ttiWo — api 8 rriEORY OF EQUATIONS. Art. 9 Tlie common denominator of these two values of x and y is a function of the coefficients of x and y in the given simul- taneous equations. This function a,b., - a A ■ ■ (3) is the determinant of the coefficients ctj, bi, cu, h.^, and is com- monly expressed by the symbol Cly hi iU bo (^) This symbol is called a determinant array, and the quantities a I, hi, a.2, b-i are called elements. (Art. 5.) The polynomial (o) is called the e.vpansion of the determi- nant. Since each term of this expansion is the product of tico elements, the deternnnant is said to be of the second order. 10. From our general theory, as given in Arts. G, 7, and 8, we recognize the equation Oo bo ai&2 — ^2^1 as an identity ; but, in practice, we find more suitable rules for expanding determinants, and these we shall examine later. The first member of the above identity is thus interpreted : The determinant array of the second order must be under- stood to mean that the product of the elements on the diagonal ]iassing from the lower left-hand corner to the upper right- hand corner of the array is to be subtracted from the product of the elements of the other diagonal. Here and elsewhere we use the sign = to denote identity. 11. The numerators of the fractions in equations (1) and (2) of Art. 9 may also be written in the form of determinant arrays. Thus the values of x and y in the given simultaneous equations are Art. 11 GENERAL DEFINITION OF DETKIiMINANTS. m^ bi tiu b. Cly bi a. b.. 1. Expand I b a 1 -2 2. Evaluate 4 3 3. Evaluate 4. Evaluate 5. Evaluate 6. Expand and reduce 7. Expand and reduce 8. Expand and reduce 9. Expand and reduce and y = EXAMPLES. a. 7H, Oi Vli a I b. «2 b. Ans. a- — h-. Ans. 11. - 11 9 1' 1 4 100 50 50 25 cos X sni X sin?/ Qosy — 1 sin a I sin « — 1 I 1 — tan X tan X 1 \ x + y x-y Ans. cos {x 4- y). X + y X — y I Solve, by Arts. 9 and 11, the following simultaneous ciC3 + «3^i<'"2 — '^^s'-'i^l The common denominator of these three fractions, which express the values of x, y, and z, is the determinant «! ^1 Ci ct2 b, C2 (4) «:, b., c. The function ciibXs — aiZ>oC2 -|- 02^3^1 — ciobiCs + tts&iCo — O3&2C1 • 0"5) is called the expansion of the determinant (4), and since each term of this expansion is the product of three elements, the determinant is said to^be of the third order. Note. Such examples as the one given here are simply to sliow that flo- terminants often occur as the result of elimination. The reader will learn in Ciiapter III that the process of elimiuatiou is much simplified by the use of determinants. 13. Since the determinant (4) is identically equal to the function (5), we have, arranging the terms of (5) in a' epn- venient order. Art. 13 GENERAL DEFINITION OF UKTEUMIXANTS. H «1 ^'1 Ci «2 i>2 c., «3 h ^'3 If a line be drawn tlirough each triad of letters forinin 1 Srd term. \''V '^ ^% ^X and for the negative terms 6lh term'' 5th term/ Ulh term/ X. / In making practical use of these diagrams, it is customary to carry out the multiplication as each stroke is made. No similar diagram exists for a determinant of higher order than the third. 12 THEORY OF EQUATIONS. Art. 13 EXAMPLES. Expand by tliis method the following determinants ; X y z V to u t r s {. X -2z -2/2 — y —2x z"^ -z 2y —x^ Evalnate the determinants, . 4 5 2 -1 2 -3 6-4 5 f_ 4 -1 -2 3 3 -7 4 9 8 7 6 5 4 3 2 1 10. a -2a 6 36 -c 4d 2c 3 J -4 6 a a a 6 a 6 c c a 2 3 1 3 5 2 6 10 15 4 -3 1 2 10 5 3 7 -1 4 2 3 5 4 6 14. The numerators of the fractions in Art. 12 may also be written in the form of determinant arrays, and thus we have for the values of x, y, and z in the given simnltaneous equa- tions : 7Jll hi c, «! Wl Ci Wi2 bo c. tto V12 c, m^ h. C3 y = (h m^ C-l a, h, C] Cli hi Ci a,. K Cl' 02 b. c. «;j h. c-i ^3 h. C3 Art. 15 GENERAL DEFINITION OF DETFUMINANTS. Vi and «1 fh Ml a.. h vu Ch h tih Cll 61 Cy «2 h c. «3 b. Cs Note. It may be observed that the numerator of the fraction ex- pressing the value of x may be furmeil from tlie denominator of the same fraction by replacing rti, n.y, 03, the coefficients of x, by the absolute terms mi, m-2, J/13, respectively. Similarly for y and z. 15. The solution of the four simultaneous equations UiX + bill + CyZ + cliiv = ?Ui a^ + b.,)/ + c.jZ + (Iw — m2 ci^x + b.^i + c^z + f's?^ = ^'^3 UiX + Z>4.?/ + C42; + d^^o = vii (1) would shoAv that the values of x, y, z, and w are expressed by fractions having a common denominator which is a function of the sixteen coefficients Oi, &i, Cj, d„ a^ is a determinant of the fourth order, identical equation : • ••, etc. This function We have tlie following ttl &i Ci f?, tto h C2 ^2 «3 h C3 C?3 a* 64 C4 ch = (iibx./li — a^h^^d^ — cifb^c^i + afiiC/l^ + aihfiiil2 — afiiC-fh — aJ)iC./h + fhh''A'h + a^JiC^di — UibiC/ls — (t^b^c/h + ciibic/h + cQ)^Cidi — aJjiCtd.^ — o.jMidi -f a^ijCifi's + ajbiCyd^ — ctibsCido — ap-jp^y -]- a.j643''/^ (2) The solution of five simultaneous linear equations involving five unknoAvn quantities would give rise to a determinant of the fifth order, the expanded form of which contains 120 (=5!) terms. 14 THE our OF EQUATIONS. Alt. 15 tto ^2 f"2 ^^2 ^2 <-h h ("3 C?3 ^3 (f4 64 C4 CZ^ 64 f?-. Oi^2<^3f?4^5 ± etc. (3) Similarly, the solution of n sinmltaueous linear equations involving n unknown quantities would give rise to a determi- nant of the ?jth order, the expanded form of which contains n ! terms. The determinant array may be written thus : (4) ttl h Cl • • h 02 b. C2 • • h Kg h Ci • ' k 16. It is evident from the foregoing that the determinant of the nth order involves nr elements, which agrees with Art. 6. The horizontal ranks of elements are called rows of the determinant, and the vertical ranks are called columns. The rows are numbered from the top row downward, and the col- umns from the left-hand column to the right. A line is either a row or a column. In any determinant, the diagonal from the upper left-hand corner to the lower right-hand corner is called the priitcipcd diagonal, as we have had occasion to remark, and the other is called the secondary diagonal The terms of the expansion, which are the products of the elements on these diagonals, are called respectively the principal term ^.nd the secondary term. Thus in the determinant (4), of the preceding article, aih-f.^ ••• ?„ is the principal term, o„ft„_iC„_2 ••• Zj is the secondary term. 17. Another notation for the determinant of the Jitli order is the following : Art. 17 GENERAL DEFINITION OF DETEUMIN ANTS. 15 «! «1 a/" ••• a/"> Oo' a.r a. J" ••• a,,<"> O3' Ch" (I3" ... a3<'" «,.' «„ (1) iu which the number of the nm is indicated by the subscript, and the number of the column by the sujyerscriiJt. Another notation, and one that is very much used, is the following : (2) a„ "12 «13 • • «ln «21 a.r.2 «03 • • as,. «31 Ch2 «33 • • «3,. Here the number of the roiv is indicated by the Jirst of the two subscripts, and the number of the column by the second. Thus, the element a^ of the above array is in the third row, and the fifth column. There are several simpler methods for writing determinants, when it is perfectly well understood what the elements of the determinants are. Thus, if A denotes the determinant (4) of Art. 15, it may, for brevity, be represented in the following ways : A = (a,V3---Q (3) A = \aAcs-h\ (-4) that is, simply by placing the principal term within brackets. The notation 2 ± a,V3 ••• ^.. is also used to represent A; this expressing its constitution as consisting of the sum of a num- ber of terms (with their proper signs) formed by taking all possible permutations of the n suffixes. With this notation determinant (2) of this article would be expressed by 2 ± «n«2-/'33---« (^) 16 TUEOllY OF EQUATIONS. Art. 17 EXAMPLES. 1. Expand o-j ?/i 1 ^2 1/2 1 2. Evaluate 3. Evaluate 4. Evaluate 5. Expand ^ns. — 15. 3 2 4 7 G 1 5 3 8 1 4 6 -3 ^ns. 20 + 8 + 96 -(-60) 5 4 2 0-(-12) lOtr- 1 a a 1 6 c In the following examples express tlie values of x, y, and z in the notation of deterniinauts, as in Art. 14, and then evaluate these determinants by the method of Art. 13. 6. Solve the simultaneous equations, X + y —z = l, ?>x-\-oy — Qz = l, — Ax-y-\-'Sz = \. Art. 17 GENERAL DEFINITION OF LETEHMlNANrs. 17 Here, by Art. 14, 1 1-1 1 3-6 1 -1 3 1 1 -1 8 1 -6 -4 1 3 1 1 -1 8 3 -G -4 -1 3 y = ■^, !/ = 1 1 8 1 -4 _1 1 1 -1 8 -6 -4 -1 3 3, z. -1 • -1 7. Solve tlie simultaneous equations, 3x + 2y — 4:Z = 15, 5a; -3// + 22 = 28, - ic + 3 y + 4 2 = 24. Ans. X = 8. Solve the simultaneous equations, 4x-'Sy + 2z= 9, ■^1 2x + 5y — 8z 6y-2z + 5x 2 = 4. = 4, = 18. Alts 2,y 3, 2 = 5. 9. Solve the simultaneous equations, 3a; + 2?/+ z = 23, 5x + 2y + iz = 4C), 10 .K -H 5 ?/ -f- 4 2 = 75. 10. Solve the simultaneous e(iuations, 2 a; -7// + 42= 0, 3a; -3//+ 2= 0, 9x + oy + 3z = 2S. CHAPTER II. PROPERTIES OF DETERMINANTS. From our definitions of a determinant, as given in Articles 5 and 6, we readily deduce the folloAving important theorems : 18. Theokem. The value of a determinant is not changed by substituting the columns for corresponding rows and the roivs for corresponding columns; that is, «1 &i Ci ' • h «i «2 «3 • • a,. 02 b. C2 ' • h ^1 b. bs • • K ttg h C3 • ■ h ~ q Co Cs • • c„ a„ K c„ • •• L /i h Is ■ • l„ YoY, the two determinants having the same principal term, they will be identical on account of the way in which all the other terms are deduced. (Art. 8.) It follows that any theorem true in regard to the rows of a determinant is also true in regard to the columns, and vice versa. 19. Theorem. Interchanging any tico roios (or columns) of a determinant, simply changes the sign of the determinant; that is, a, &i Ci . •• ^1 a-> b2 C.2 • • h h Cd Ci . • h ch bi Co • • ^2 cii bi c'l • • ^x &2 ftj ('2 • ■ h «3 ^3 Cg • • k = — «3 ^3 C3 • • /3 = - l>3 «3 Cg • • h a„ 6„ c„ • • In «„ b„ c„ • 18 • In ^„ a„ o„ • • L Art. 21 PROPERTIES OF DETERMINAXTS. i:» For this modification amounts to changing tlie index 1 with the index 2, or the letter a with the letter b in the different terms of the determinant, and we know that in this case the corresponding permutation changes its sign, and hence all the terms of the last two determinants will have the same absolute value as that of the first, but their signs will be different. 20. Theorem. If two rows (or cohimns) of a determinant are identical, the determinatd is equal to zero; thus A = «! ^1 Ci • • h «r K ^r • . /, a^ K c,. • /, On h„ c,. • lu For, by interchanging the two identical rows, we obtain Whence A = -A, 2A = 0. A = 0. 21. Theorem. If each element in any line be mxltiplied by the name factor, the determinant is multiplied by that factor ; thus : mcii b, <-'i ■ • ^1 ma2 b. C2 • • h ma^i bs C3 • • h a, ^ c, • /. a.. b. Co • • /. CJg bs C3 • • k ia„ b„ c„ ••• In For every term of the determinant must contain one, and only one, element from any row or any <-()lumii. 20 TIlEOIiV OF EQUATIONS Art. 21 Cor. I. If the elements in any line are the same multiple of the corresponding elements of any other parallel line, the determinant vanishes. bi mbi 1)2 a I «i a. «3 b. ^1 h. bs t'l t'l C'2 ("3 rfi dy (h , it is sufficient to multiply the columns in order by 10, 5, 2, 4 ; we thus obtain 20 20 20 20 o 8 12 70 10 10 20 30 2 IC 2 • 4 . 10 Taking out the multiplier 20 from the first row, 10 from the third row, and 2 from the fourth row, we get finally A = 1111 5 8 12 7 11 2 15 1 8 10. Reduce the following determinant to one in which the firsl column shall consist of units : A = 2 2 1 3'\ 4 6 ^^6 7- 6 8 4 4 5 22. Determinant Minors. It is evident from the notation, in a s(puire array, of a deteiminant of the Hth order, that the suppression of }> rows and of p columns leaves a square con- 24 THEORY OF EQUATIONS. taining no more than /t —j) rows and n —p columns. We thus obtain a series of determinants of lower order, which we call minors of the primitive determinant. For example, in «1 hi Ci • • ^-1 h (U b. Co ' • h k Ch Ih C3 • • A-3 k let us suppress the row and the column which contain the element a^ ; it will become the determinant of the {n — l)th order, Ay. h c-i • • A-, h h. C-i ' • A3 h K f« • • A-,. In ^±boCs-'-l„ which is called the first muior of A with respect to the ele- ment Cli- As we can repeat this operation on each element, a deter- minant of the 7->^ 0.^^ the rows and tlie columns suppressed have in common the elements of the determinant «i and the determinant of the (/i — 2)th order which precedes is the second minor of A with respect to a de terminant of t\^ second. pr.d.er. ^ In general, by the suppression of j) rows and of j) columns, we get a determinant of the {n — 7))th order which we call the l^th minor of A corresponding to a determinant of the ^)th "ordeF formed by the elements common to the rows and col- umns suppressed. The minor thus formed is said to bex(>^H- ^>/'eH^e»^a/•^/ to the determinant formed by elements common to the suppressed rows and columns. .# v'23. Development of a determinant according to the elements of a row and of a column. Take the determinant of )(th order. A = «1 ^ 'h • • h /, «2 h. C, • h ^.' «3 h ('3 • ■ h hi A-.. /.. 2±o,''/'3---A-„_i^« The different terms of A which contain the element ((, are obtained by forming all the possible permutations of the other .-J^p ■J 26 THEORY OF EQUATIONS Art. 23 elements, which gives the deteiiiiinant of {n — l)th order, 2 ± 62C3 • • • ^',,-1^,., resu'king/froiu the suppression of the first row and the first columiy. The determinant A will contain then, first, a series of ternfs having ctj as a factor represented by ai2 ± h^'^U -In- Interchanging a and h, this becomes for all the terms containing by) the sign — is caused b}'^ the permutation of a and h. The sum 2 designates the determi- nant obtained by omitting the first row and the second column. By changing b into c, we get, similarly, the expression which woidd represent the series of terras having Cj as a fac- tor; the sum 2 designating the determinant arising from the suppression of the first row and of the third column, and so on. All the terms containing /j would be represented by (-l)"/,2±«A'"4--^„ ^^ Now, by definition, the sums which accompany the elements «„ — 6|, -f Cj, ••• (— 1)"/, are the fird^minors with respect to these elements. Therefore, we have the following formula: A = a,A, + b,B, -K r.C, -f - + 7.A, .^. (1) with the conditioA that, according to the cojnposition of the determinant, we mu^t atb-ibute rs the minors Signs alternately positive and negative •^"t^^^^^)^ -vv^'^m o-vaa, J^- "K v^^^^' The number of terms in the second member of this equality is evidently n(l •2.3...y_^J)= v\, the same as the number of terms in the determinant. Art. 24 PROPERTIES OF DETERMINANTS 27 Following the same reasoiiiiig, and interchanging successively the indices two and two, we arrive at the similar relation ^ = aiA,^a,A, + a,A,'T '■■ -h^A,,, . . . (2) ■where we must give to the minors the signs alternately + and — . This formula gives the development of A according to the elements of the first column. It is evident that there exists a similar development for each row and for each column. Finally, to fix the sign of the minors in each formula, we move the row, or the column that we are considering, to the first place by the interchanges of the rows or columns, in observing that the determinant changes only its sign for an odd number of interchanges, while it pre- serves the same sign for an even number. Thus, A =7i2A., + b.B. 4- c.a + ••• + hLo ... (3) A = a,A, + ^^3^3 + CsC, + • • • + I,Ls ; . . . (4) in formula (3) we would alternate the signs commencing with the sign — for A.2; for, to lead the second row to tlie first place, one interchange of two rows suffices ; in formula (4) it is necessary to commence with the sign + for A^, since two interchanges of rows are required to lead the third row to the first place ; and so on. 24. Let us apply these principles to a determinant of the third order : A = ttj ^1 c'l a., bo Co 0,5 63 c^ Expressing the minors with their proper signs, we have A = a^A, - &1B1 -\-c,Ci=- cioAo -f ho B, - c.,a = ff,. I3 - k' Ih + '-.Oi, A = a,^li -(^12-1-% .13= - h, B, + b,B,-b,B,=Ci(-\-i\C,+c,C,, 28 THEORY OF EQUATIONS. or, replacing the minors by their vakies, A = Art. 24 a, b, C2 ^3 -^ a, «3 C2 C3 + c, tto «3 &2 &3 a.. + h C3 — Co h h «3 bi C2 -h «2 Cj Co + C3 «1 bi b2 «i &2 &3 C3 -a. &3 C3 + a3 ^2 Cj ^\ O2 Co + Z>2 Cti Ci -&3 tti ^1 Ch C3 a,, (•3 02 C2 Cj as «3 — Co «i «3 &1 h + C3 62 By virtue of Avhat precedes, we can operate as follows to ascertain the sign of a minor with respect to any element. For example, let it be proposed to find the sign of the minor of d^ in a determinant of the «th order. Proceeding on the first row from cii, alternating the signs until we get to the column of the cl elements, we reach c?i with the sign — ; we descend tlien the column of the d elements, changing the sign each time that we cross a row until Ave arrive, in this manner, at d^ with the sign + ; therefore the minor of d^ ought to be affected with the positive sign. Again, let the determinant be represented by «21 O12 • a22 . . a.2, «H a*2 • ■ ■ (iu ■ ■ ■ «*, a , a..., . . a Art. 26 FliOPERTIES OF DETEliMINANTS. 29 and let us seek the sign of the minor relative to the element Ua- To this end, we must by the interchange of rows and columns lead this element to the first place. By ^ — 1 inter- changes of two consecutive columns, the element a„ will occupy the first position in the ^th horizontal line; then by k — 1 interchanges of rows, this element will take the first place in the first row. All these operations amount to multi- plying the determinant by (— l)*+'-2 or (— 1)*+'. The sign of the minor A^i will therefore be positive if the sum of the indices of the element a« is even, and negative in the con- trary case. It is useful to observe that the first minors of the elements of the diagonal are all positive. The preceding developments lead to iinportant consequences which we shall now give, for brevity making use of simple determinants in illustration. 25. Theorem. When all the elements of a row or of a col- umn are zero, the determinant is equal to zero. Thus = 0, «1 b, Ci = 0, &, c «2 1)2 C.2 b, c. 63 c, for all the terms of the development according to the elements of these lines become zero by the presence of the factor zero. 26. Theorem. When each of the elements of a row (or column) is zero except one of them, the order of the determinant is lowered by unity. We have, for example, Also «, = 0,^1, + .jU + O.A,= «i bt c. a.i 62 C2 i, C3 «s 63 Cs- hi = -b, a, c. a.2 62 (',, "a fs «3 ^3 C3 30 THEORY OF EQUATIONS. Art. 27 27. Theorem. A determinant is reduced to its princijjal term ivheu each of the eJements on one side of the diagonal is zero. For, taking a determinant of the fourth order, we have successively aib^Csd^. a, = ttj &, = ajb2 cs a^ b, h C3 c, d. Cfg h cs b, Ci di "4 h '-i , it suffices to 'mnltiplij the elemevds of a row or of a column by this factor. We have P«I P^l l^Cl p- A= pa^Ai + 2AB1 + iX'iC 29. If, in one of the developments «,.4, + b,Bi + CiC\ + ••• + hLi, we replace the elements, Oj, b^, Cj ••• l^, which appear here by those of any other row, the result is zero; the same is the case, if in one of the expressions rt,yli + 0*2^42 + «3^3 + ••• + «„-l„, we replace the elements by those of another column. For, in substituting, for example, in the place of the elements «i, &j, c'l, ••• /i, those of the second row a.2, b.,, c.^, ••• l.j, the expression a,A, + b,B, + c,C,+ --+hLy, represents the determinant obtained by this substitution, the coefficients A^, Bi ••• />, being always the minors of I he first Alt. 30 PliOPEHriES OF DETEliMl.\AyTS. r,\ row; and this determiuaut is zero, since it contains two identical rows. That is, we have Oj.li + bjii + CiC'i + ... + /,/., = A, a^A, + a.. A, + a.^A^ + ••• + ('„/-/,. = A, but ttoAi + b.Jii + c..Ci H + l.,Li = 0, lind other similar relations. In general, the expression represents the determinant A, if J = i; and is zero, if j is different from /. Cor. With the notation witli two subscripts, for a determi- nant of the //til ortler, this property- is expressed thus: tiie developments ciijAu + o,jA,, + ••• + a„jA„,, cij^Aii + aj.xl,2 + -■■ + cij^A,,,, represent the determinant A, when _/ is a number of the series 1, 2, 3, ••• n, and equal to /; while, if^ is different from /, they equal zero. 30. We can always raise the order of a determinant without changing its value. Thus, after the preceding properties, we have the equalities «! ^ = 1 = 1 X y = 10 a^ b. X <'i h a, h^ x 1 y th h a, b. .'/ t ih f>i z II ((., b., and so on. The elements .r, ?/, z, t, ii being any quantities whatever. 32 THEORY OF EQUATIONS. Art. 31 31. Development and Evaluation of Determinants. The fuucla- meutal formula A = rt,.4i + h,B, + Cid + ••• + hL, enables us to replace a determinant of the ni\\ order by an expression containing only determinants of the {n — l)th order; in this last we can substitute for y1„ Bi, Cj, ••• expressions con- taining only determinants of the (n — 2)th order ; in continuing in this way we finally arrive at the value of the determinant A. It is necessary to give some applications to indicate the steps in the different cases that may present themselves. As we have seen, determinants of the second order are calcu- lated directly. We have «! «2 ^2 3.-4-1.2 = a-^h^ — (uhy. 14, For a determinant of the third order, of which all the ele- ments are different from zero, we would take, for example, the formula Uo bo «3 h \ = «1 b. Co -a. b, Oi + % &i Cl *3 bs Cs bs Cs b. C2 EXAMPLES. = 1 2 3 + 3 -l-2(-2) + 3(-l) = 0. •'^•l .'/l 1 = X, ?h 1 -X, ?/l 1 + a-3 ^1 1 Xi y-2 1 ys 1 y?. 1 y2 1 a^s Vz 1 ^•i (y- - 2/3) + x,(ys - ?/,) + X., (?/i - ?/o). Art. 31 PROPERTIES OF DETERMIXASTS. ■.V6 Note. In examples where certain elements are zero, we oufjlit to employ the development aceordinjj to the line which contains the greatest number of zero elements. Thus : = 3. 10 3 = 1 1 1 + 3 2 1 2 11 1 4 2 1 2 14 12 4 10 13 2 = 2 I 1 2 I = • 4 1 14. 5. Develop 6. Develop 2 3 1 3 a b e b c f e f 9 12 Develop b -c i- -1 8. Develop the determinants : 2 1 3 5 () 7 2 14 3 2 1 4 3 1 -1 Ans. 1 -\- a- + b- + r. 2 5 10 4 , 3 6 4 2 6 8 1 3 5 9. Develop 1 4 3 2 7 8 5 6 2 34 THEOBY OF EqUATlONS. 10. Develop c d Art. 31 1 sin a sin « 1 32. Laplace's Development, — Development of a Determinant according to the Elements of Two Rows or of Two Columns. Take the determinant of the ?;th order: a I ^1 fi • • h Ch h (-2 • • ^2 as h f-s • • /3 f'4 h Ci • • ^4 n &„ e,. ... /,, Let us consider the principal term of A, aib.>c./Ii •••l„; to this term there corresponds another, —ciobiCsdi '•'!„, arising from the interchange of the letters a and b. Uniting these two terms so as to put their common factor in evidence, we have ¥h-l.. Let a and b be fixed, and form all the possible permutations of the other letters, c, d, ••• I, then the terms of A, which have as a factor the determinant (ai/>2), will be represented by The coefficient of (ai^j) is therefore the second minor of A obtained in suppressing the rows and columns which contain these elements. We reach an analogous conclusion for the coefficients of the determinants of the second order («A), (<'A)--("J^„), («A) •••("„- A.), ^^hi(•h result from the combinations two and two of the ele- ments of the first two columns of A. In calling the second minors Bf,, Z?,,;, TJ^, etc., we find the following development : Art ^2 PROPERTIES OF DETEinfiyANrS. 8.' A = (afi2)B,, + {aJj,)B,, + - + (<^,6,.)/i„. + {aJ>,)R,., + ... + (i + I3i Ci a.2 + «2 ^2 + ^2 f 2 «3 + «3 ^3 + A C3 is equal to the sum of the four determinants {afi^C^ + («l&2f3) + («1^2C3) + («]M)- In like manner it follows that if each of the elements of one column consists of the algebraical sum of any number of terms, the determinant can be resolved into the sum of a correspond- ing number of determinants. Tor example : ai-«i + «i' b, C, tti &i Cx «1 ^'l C\ «/ ^1 Ci O2 — «2+«2' b.2 C.2 = a.2 ^2 c.2 - «2 f>2 <^2 + «2' b.2 c. «3-«3+«3' h C3 Ch ^3 C3 «3 h C3 "3' h C3 ^^.r^ Art. 35 PROPERTIES OF DKrERMINANTS. 30 And, in general, if^one column (or row) consists of the alge- braic sum of m otheili, a second column (or row) of the sum of 71 others, a third of the sum of p others, etc., the determinant can be resolved into the sum of vinp •••, etc., others. 34. Theobem. If the elements of one row (or cohnnn) are equal to the sums of the corresponding elements of the other rows (or columns) multiplied by constant factors, the determinant van- ishes. For it can then be resolved into the sum of a number oi determinants which separately vanish. For example, mai + nbi Oj bi rtj tti bi bi Oi bi m«2 + ^^^2 0^2 b, = m a.. a, b. + n &2 «■.' ^2 ma^ + ?t&3 «3 h. 19 — 7 12 -2 5 3 5 2 3 4 12 3 9 ■972. Here the first transformation is obtained by adding to the second row three times the first, subtracting the first from the third row, and adding the first to the fourth row. Calculate the determinant 1 15 14 4 A = 12 6 7 8 10 11 9 5 13 3 2 16 Art. 35 PliOPEIiTIES OF DETEIiMINANTS. 41 Tlie first sixteen natural numbers are arranged here in wluit is called a " magic square," i.e. the sum of all the figures in any row or any column is constant. In general, for a square of the first n^ numbers, this sum is \n(n^ + 1). Determinants of this kind can be at once reduced one degree. Here adding the last three columns to the first, and sub- tracting the last row from each of the others, we have A = 34: 1 15 14 4 16 7 9 1 10 11 5 = 34 1 3 2 16 12 12 -12 3 5 rr 7 9 -11 13 2 16 = -34x12 11-1 3 5-7 7 9 -11 and subtracting the second row from the last row, it is evident that the reduced determinant vanishes : hence A = 0. 5. Calculate the determinant formed by the first nine natural numbers arranged in a magic square: 4 9 2 3 5 7 8 16 Ans. 360. 6. Calculate 2 3 8 4 6 4 6 12 4 Ans. 72. 111 10 1 z^ y^ 1 2^ y2 1 z^ .^•2 1 z- -z^ x'-z' 1 y- XT 1 f o?-f -f 1 z- f 1 -z^ x»-z* 1 x^-f -.V' Here, to obtain the second determinant, we subtract the second column from each of the following ones. In the re- 42 THEORY OF EQUATIONS. Art. 35 ducecl determinant, subtracting the first row from each of the foHov/ins, ^ve find A = 1 z- f -2z- o(r — z- — y- = - x'-.f-z'' -2f 1 z- if+z--x- y' + z^-x" 2/ = {y"" + Z-- x-f - 4 y-z- = {f + 2^ _ .^.2 _j. 2 2/^) (^2 + 2-2 _ ^.2 _ 2 y^;) = - (a- + ?/ + 2) (?/ + 2 - a;) (2 + x - y) {x + y-z). 8. Evaluate the determinant 1 2 L 1 4 4 1 8 4 12 13 2 4 2 11 9. Evaluate 2 2 2 10 1-1-1 5 3 _3 3 _15 1 1-1-5 10. Evaluate a b c d a & c d -a b a /5 26 c+a d-\-(3 -a -b c 7 2c d + y -a -b — c fZ ■2d 11. Evaluate A = 111 1 a ?> 1 a r 1 ?> c 4» s. A = a- + 6" + c^- -2 he- A71S. —15. 2hibcd. Alt. 36 PROPERTIES OF DETEUM LXANTS. 4;{ MULTIPLICATION OF DETKU.MIXAXTS. 36. TiiEOKEM. The jrrodnct oftico defermiiuutts of any order is itself a determinant of the same order. AVe shall prove this for two determincants of the third order, and, from the nature of the proof, it will be evident that it is equally applicable in general. "We propose to show that the product of the two deteruiinants A = {ciih.f.-^ and B = (ui/Soy-i) is P = «!«! + ^1^1 4- Ciyi «i«2 + ^lA + ("iTi «i«3 + ^A, + Ciys (f2«i + ^2/8i + C'2y, (t2«2 + ^I'iSo + Cryo a.M3 + b.J3i + C'sya «3«1 + ^3^1 + ^'syi «3«2 + ^3i82 + ('3y2 <'3«3 + ^Si + ^^m 0) whose elements are the sums of the products of the elements in any row of (aiboCs) by the corresponding elements in any row of (fiiftoy-i)- The determinant P can evidently (Art. 33) be expanded into the sum of twenty-seven others. The following proof of this tlieorem is derived from Laplace's method of development already explained (Art. 32). The product of the two determinants. A, B, is (see Ex. 5, Art. 32) plainly equal to the determinant «1 ^1 Ci (k b. ^"2 Ch h ^-3 X «1 (i.j "3 -1 /?. /5-. /?3 -1 yi 72 73 (2) In this determinant add to the fourth column the sum of first multiplied by «„ the second by fi^, and the third by add to the fifth column the sum of the first multiplied by the second by /S., and the third by y.; and add to the si the 71; xth 44 THEORY OF EQUATIONS. Art. 36 column the sum of the first multiplied by «;j, the second by ^3 and the third by yg. The determinant (2) becomes then (12 62 ^"2 02«1 + &A + C2yi a2«2 + &2/324-C2y2 a2«3 + &2/33 + C273 tts &3 C3 a3rti + &3/?i + C3yi «3«2+&3/?2 + C3y2 «3«3 + ^-^3 A + ^373 -10 0-100 0-10 And this is, by Ex. 5, Art. 32, equal to the product, with the proper sign (which in this case is evidently — ), of the determinant (which is equal to — 1) 1 -1 -1 by the complementary minor, which is the P of this article. Hence, the theorem AxB=P. Cor. Two determinants of different orders may be multi- plied together by raising the lower determinant to the order of the higher (Art. 30), and then applying the above rule. Thus : 6i c, xi Vi bo C2 X = bs Cg a^2 2/2 «i &i Cl ^2 &2 C2 X tta b. Ca 10 .Ti y, x., ?/2 ai b^x^ + Ci?yi b^x^ + c{ijo a^ &2^i + C22/1 b.^2 + C2?/2 as &3.r, + C3?/i &3.r., + G32/2 Art. 37 PROPERTIES OF DETERMINAyi'S. 45 37. Eulek's Theorem. TJie jyrodud of two numbers, each the sum of four squaj-es, is itself the sum of four squares. By Laplace's method of development, we readily prove the following identity : = (a' + b' + c- + d-)- (1) Similarly, (a' + l3' + r + 8y . (2) a b c d —b a —d c — c d a —b —d — c b a a 13 y S -(3 a -8 y -y B a -/3 -8 -y (3 a Now multiply equations (1) and (2) together, member for member. Letting «« + b/3 + cy-\- d8 = A, — a/? + ba - c8 + dy = B, — ay + b8 + ca — f?/? = C, — a8-by + cft + da = D, the product of the left-hand members may be written : A B C D = {A' + B'+C' + D'f . (3 -B A -D C -C D A -B -D -C B A Therefore (a2 + 62 + c^ + d') (a' + (3' + y'-\- 8") = (.P + B' + C + D^, which is the theorem.* * This theorem is due to, and natncd after, the Swiss mathematician I.eonhanl Kul» (n07-lTS3). 46 TIIEOliY OF EQUATIONS. Art. 37 EXAMPLES. 1. Find the product of the two determinants 1 1 1 Xi X2 a?3 Vi y-2 2/3 2. Find the value of «ii ai2 o, a,, «« a.. 3. Find the product of the two determinants 1 3 10 3 2 3 1 2 5 1 ' 2 2 2 3 1 2 2 1 1 2 9 1 38. Rectangular Arrays. Arrays in which the number of rows is not equal to the number of columns are called reo- taiif/nJar. The common notation for rectangular arrays, or matrices, as they are called, is : ttl &1 tto b. } (f.J h b. Co Eectangular arrays do not themselves represent any defi- nite function; but if two such arrays of the same dimensions are given, we can derive from them by the multiplication theorem of Art. 36 a determinant whose value we proceed to investigate. (1) When the number of columns exceeds the number of roivs. Theorem.* TJie "2^^'oduct " of two rectangular arrays of the same dimensions is equal to the sum of the jiroducts of cdl possi- * By the so-cullod "jirodtR't" hc-ro and the multiplication of two rectangular arrays in the following tlioorcm, we simply mean that the procfxn of Art. 36 is employed ; of course, as matrices are not functions, they cannot really be multiplied together. Art. 38 PBOPERTIES OF DETKIiMlNAyTS. 47 ble deter)ni)iaHts ichich can be formed fruni one army [In/ tnkimf a number of columns equal to the number of j-ows) multijilied by the correspondimj determinants formed from the other array. To prove this, take any two rectangular arrays, «! &i t'l (?i ... «i A yi ^1 ttj 62 *^-2 (h ' «■■• ^2 72 82 (2) and perform on these a process simihir to that employed in multiplying two determinants. We thus obtain the deter- minant o.«i + 60^1 + c.yi + fUi UM.. 4- ^2^3. + f.y^ + dA, The vahie of this is easily found to be («.^2) («iA) + («iC2) («iy2) + ((hd-d («iS2) + (^c•2) OS.y,,) Hence the theorem. This proof can be easily generalized. (2) When the number of roivs exceeds the number of columns. Theorem. In this case, the determinant resultinrf from the multiplication (so called) of the two arrays vanishes. Take, for example, the two arrays, «1 hi a^ b. as h (1), «1 /?. «2 ^2 (^ ^S (2). Performing the process of multipli(!ation, we have the determinant «,«1 + b^/Si «i«2 + ^1)^2 «1«3 + Wf^s a^fCi + b.,(3i a.2(i-2 + ^Si cue, + b.fi^ a3«l + h(^\ «.i«2 + ^3/^2 ".!'<3 + Ms 48 THEORY OF EQUATIONS. Art. 38 This determinant is obviously the same as would arise if a column of ciphers were added to each of the given arrays, and the determinants so formed then multiplied. It follows that the determinant vanishes. In an exactly similar way, we can prove the general theorem. I ^ EXAMPLES. 1. From the two arrays 111' a (3 y (1), 111 « /3 7 (2), prove 3 a-\-(3 + y a + (3-hy a' + l3- + y- 2. By squaring the array ^(a-(3y- + ia-yy + (f3-yy a b c d e f prove {a- + b--Y-c') (d-+e^+f-) = (ad + &e + c/)-+(ae - bdy-\-(cd - a/y + (bf-cey. 39. Reciprocal Determinants. The first minors (with their proper signs) xl„ Bj, Ci, ••• A2, B.^, etc. (Art. 22), which occur in the ex})ansion of a determinant are called inverse elements; and the determinant formed with them as elements is called the inverse or reciproccd of the original determinant. The following theorem gives a useful relation connecting the two determinants : Theorem. The reciprocal of any determinant of the ni\\ order is equal to the {n — l)th power of the given determinant. Let the reciprocal of A be denoted by A', and multiply the two determinants Art. PROPERTIES OF DETERMIXAXTS. 4!» A = «1 h C-i a. b. Co a. h Cs A': A ^1 c, -I2 ^2 C'i A, B, 6-3 All the elements of the resulting determinant except those in the diagonal vanish (Art. 29) ; and the result is AA' AGO A () A whence From the nature of the above proof, it is evident that the process here employed in a particular case is equally appli- cable in general ; giving for a determinant of the nt\i order AA' = A'', or A' = A"-^ EXAMPLES. 1. If A' = the reciprocal of the determinant 12 3 -11 9 6 3 14 , show that A' = 2 -13 8 6 4 5 5 5 -5 and, hence, verify the fornmla A' = A-. 2. Form the reciprocal of the determinant A = a h 9 h h f 9 f c CHAPTER III. APPLICATIONS AND SPECIAL FORMS OF DETER- MINANTS. APPLICATIONS OF DETERMINANTS. In Arts. 9, 11, 12, and 14, we have seen how the work of solving simple linear equations of two or three variables may be abbreviated by the use of the determinant notation. We shall now extend these principles, and proceed to investigate some of the fundamental properties of systems of equations. 40. First, taking a special case, let it be required to solve the simultaneous linear equations, cii'x' + ai"x" + ai"'x"' = u^ a.,'x' + 02";c" + a.,"'x"' — n.^ tts'x' + 0,3 "ar" + a3"'x"' = u^ Cli' Cli" a, a.J a.," «2 a' a," a. (1) (2) is called the determinant of this system of equations. By Art. 29, we have : A.'ni' + ^aV + .43'a.3' = A, A.'cii" + AJaJ' + AJcis" = 0, (3) A,'a,"' + AJaJ" + A,'a,"' = 0, If now we add the equations (1) after having multiplied them respectively by J/, A^, A^', the coefficient of x' would 60 Alt. 41 become A, and those of x" and have iPPLICATIOyS OF DKTEUMIXAXTS. 51 become zero. Hence we Ax' = Ai'ui + A,'u., + A^'u^ «1 0," 0/" »2 «.," a,'" "3 %" a-J" »1 «l" or »2 a.," a,'" «3 «3" a,'" a/ 0/' ar 0,' CI2" a.,'" «3' «3" a.J" The values of x" and x'" may be found in the same manner. We proceed in exactly the same way to solve the general case, as follows. 41. Let the given system of simultaneous linear equations be a^x' + 0/ V -\ \- o/'V" H f- ai'">a;("> = «, a.^x' + a^'x" + ••• + «2<''*.f"' H h o,.*"'.r<"' (1) rt„'.x-' + a„"a;" -\ 1- a,/".i-<" H \- a„<"'.i-<"' = »„ where the number of unknown quantities is the same as the number of equations. Let us form the determinant of this system of equations A = ai a, a J a., (-') «,. '^H •" ''.. ••• "n and let A,^'^ be the coetficicnt of a^'" in this doterminant. 52 The sum THEORY OF EQUATIONS. + A,^'W^> + - + A/W Art. 41 (3) is equal to A for j = i, and is zero for all values of j different from i. (Compare Art. 29, Cor.) If now we add the equations (1), after having multiplied them respectively by A,^'\AJ'\--AJ'\ the coefficient of a;'"' is equal to A, and those of all the other unknown quantities vanish. We have therefore • Ao;"' = ^i"'«i + A^^'hio -\ 1- A"*"n Ul «! «/ (i) the second member being what A becomes, when we replace the coefficients of x''* by the second members of the corresj)ondiug given eqiiations As long as A is different from zero, this formula gives for the n unknown quantities finite and determinate values. Cor. If, for brevity, we denote the numerator of the frac- tion giving the value of x''^ (Equation 4) by 8*", then, with this notation, we would have : 8' „ 8" — , x" = — a' a 8(n) Now, if A = 0, and 8', 8", ••• 8'"' are not zero, then the values of the unknowns are infinite. Art. 41 APPLICATIONS OF DETEIlMiyANTS. 53 If A = 0, and at the same time S' = 0, 8" = 0, etc., tlicn tlie values of the unknowns are indeterminate. This would be the case if Ui = 0, ?<2 = 0, ••• u„ — 0. EXAM PLES. 1. Solve the equations x + rj + z + t+tc= 5 x+y+z+t+v= 3 x+y+z+u+v= 1 x + y + t-}-u-i-v= 7 x + z + t + ic + v= 9 y-{-z + t+u + v = 11 Here there are six equations and six unknowns, and as A is not zero, as we find by calculation, there is a solution. We first calculate A, and then the determinants which we may call 8., 8„ 8,, 8,. S,„ 8, ^ ^ \AK,S^ y ' - r: 111110 ^ 11110 1 1 1 1 1 1 110 111 10 1111 1 111^ ^N 1 -1 1 -1 1 -1 1 -1 ' 1 -1 1 1 1 1 1 -1 -1 -1 -1 -1 5 5 4 3 o 1 + 54 THEORY OF EQUATIONS. Art. 41 In reducing this determinant, we have employed the princi- ple of Art. 35. As a still further illustration of the ready- application of this principle, we give the steps in the calcu- lation of 8,. K = 1 1 1 1 1 1 = 1 1 1 1 5 1111 2 1-1 6 0-11 8-1 1 10 1 13 1 8 -1 10 5 11110 2 1-11 1 6 0-11 1 8-1 1 1 10 1 1 5 1111 7 112 6 0-110 8-1 10 10 10 7 1 13 1 8 -1 10 13 1 3 21 4 10 1 = -19. 15 1 13 1 111 17 19 1 11 1 1 1 1 = -9 ■^ ^.1 o Art. 42 APPLICATION.'S OF UETEIiMiyANTS. 55 Similai'ly 8, = + l, 8, = + 31, 8„= + 21, 8„ = + ll Therefore we have 2. Solve the system of equatiuus, — Xi + a;,, + Xs + x-4 = 8, a'l — .To + .Tg + x^ = 6, cc, + X, — a-g + Xi = 4, Xi + x.^ + x^ - a:^ = 2. 3. Solve the simultaneous equations x-2y-{-Zz= 6, 2x + 3y-4z = 20, 3x-2y + 5z = 26. A)is. x = 8, ,v = 4, 2 = 2. 42. Number of Equations Greater than the Number of Un- knowns. In this case where the number of equations in a given system is greater than the number of unknowns, it will not, in general, be possible to solve the system. Whenever values may be assigned to the unknowns which \vill simul- taneously satisfy all the equations, the system is said to be consistent. The consistency of any such system must obviously depend upon some relation among the coefficients. We shall first find what this relation is for the simple case where we have three simultaneous equations involving only two unknowns. Let the given equations be aix' + bix" = k^ (1) a^' + b^x" = k. (2) a3x' + b,x" = k, (3) 56 THEORY OF EQUATIONS. Art. 42 Since the above system is to be consistent, the values of the unknowns obtained by solving any two of the equations must satisfy the third equation. Solving equations (^2) and (3), we get h. b. b. k. 1 _ h h = - A^ h a.2 b. Ch ~h «3 hs «3 h a.2 k2 «3 h a.2 h «3 h S'ubstituting these values of x' and x" in equation (1), and reducing, we get «i b.2 h -b 1 1 ^2 ^'2 + k. ao b. h ks 1 , 1 «3 ks as 6 «i bi ki «2 &2 ^'2 = 0, «3 ^3 ^ 3 0, which is the condition of consistency of the three given equa- tions. For example, the system of equations 6a;'+ x" = — l, 5.x' -10 a;" = 5, 4.r'+ 3a;" =-7 is consistent, because we have = 0. 43. We shall now take up the general case, and investigate this relation in the case of {n + 1) linear equations involving n unknowns. Consider the following system : Art. 43 APPLICATION.'^ OF DETERMINANTS. 57 a^x' + a, .r a.,'x' + a.,"x' + ••• + a,"'>x<"> =«, a„'.f' + ((J'x" + ••• + a,/"'.f*"' = u„ a) Since the above system is to be regarded as consistent, the values of the iinknowns obtained by solving any ?t of the equations must satisfy the remaining equation. Solving the last n equations by the method of Art. 41, we obtain, after permuting the column, ?<,, »3 •••«„+!, till it occu- pies the last position, and having regard to the proper signs : x'=i-ir-^ x"=(-iy- .(n) a. a 1 1+1 «2 ' «„+! a.i' a," -a,*'" n) ««+ ,' «,.+i" •••««+■ 58 THEORY OF EQUATIONS. Art. 4.3 Substituting these values in the first equation of system (1), clearing of fractions, and reducing, we obtain a/ ••• a/"' «i tta' ... «,'»> ''2 aj ... «„'"> u„ a„+i' "• «„+/" w„+l = (2) which is the condition of consistency, giving the required rela- tion among the coefficients. AVhen the equations are consistent, this determinant is called the eliminont or resnltant of the system, because it is the result obtained by eliminating the unknowns from the given equations. We should observe that the resultant, in this case, is the determinant of the coefficients and absolute terms. Example. Test the consistency of the system a; + 15?/ + 14 2= 4, x+ 6t/+ lz= 9, Here X + 102/ + 11 2! = 5, x-\- 3y+ 22 = 16. 1 15 14 4 16 7 9 " 1 10 11 5 1 3 2 16 and the system is consistent. HOMOGENEOUS LINEAR EQUATIONS. 44. If in equations (1) of the preceding article, the absolute terms (it's) become zeros, we have a system of homogeneous linear equations, and, in this case, the numerators of the frac- Art. 44 HOMOGENEOUS LINEAR EQUATIONS. r)9 tions giving the values of llie unknown quantities vanisli. This shows, as we know from other considerations, that such a homogeneous system can always be satisfied by giving to each unknown the value zero. It often happens, however, that such equations may be simultaneously satisfied by assigning to the unknowns values other than zero. We shall now consider the case of a system of n homo- geneous linear equations involving n unknowns. Let ai'x' + cii'x" + ••• -f o/ = a^'x' + a.J'x" + ••• + o,,("'.^<"' = a„'x' -f a„"x" H h a,/"'.r'"' = • • (1) be any system of n homogeneous linear equations involving n unknowns x', x", -■' x^"\ in which the coefficients are so related that a I cii" ■" cf/ aJ a.>" ••• a.7* a,' a„ a,. (2) Applying the method of Art. 41 to the system (1), we can obtain the values of the unknowns only in the indeterminate form -. (Compare Art. 41, Cor.) Though it is thus impossible to determine the absolute values of the unknowns in such a system as (1), it is possible to find the ratios of any (n - 1) of the unknowns to the remaining one. For, dividing each of the equations (1) by a;<", and repre- senting the ratios x^*^ x"* ^, ...^_ by v',v", 60 THEORY OF EQUATIONS. Art. 44 respectively, rememberiug that i'^'' = 1, we obtain the system (3) 'y" + •••+«„ '+a„ ' + ••• + «/ '=-a('> This is a system of n non-homogeneous linear equations involving (?i — 1) unknowns, v', v", ••• v'-'~'^^, v^''^^'', ••• t'^"^, and, so long as the condition of consistency (2) holds good, as in ■1), v^'^ may Art. 43, the values of the ratios v', v", generally be obtained by solving any n — 1 of equations (3). Hence the system (1) will be satisfied by any values of x', x", • •• x^"^ among which we have the ratios v', v", ••• v''"\ as deter- mined by any n — 1 of equations (3) ; that is, if equati8fei-(l) are satisfied by the values Xq', Xq", ••• a;o^"', they will be equally satisfied by Xxq, Xxq", ••• AaV"', A being any factor. The relation (2) gives the condition of consistency of the system (1), and A, tJie determinant of the coefficients, is the elimi- minant or resultant of the system. To illustrate, let us solve the homogeneous system 2 a; + 4^ + 52 = 01 3x-]-5y + 6z = \ 4,x -{- 6y + 7 z = ) Here the determinant of the coefficients 2 4 5 A= 3 5 6 =0. 4 6 7 An attempt to solve the system (1) by Art. 41 gives 0' which are indeterminate. But, since A = 0, the system is con- sistent, and we can obtain definite values for the ratios ?, ^- (1) 0' z = -, I Art. 45 IIOMOGEXEOUS LINEAR EQUATIoys. 01 Dividing the equations by z, we have 2- + 4'l = -5 z z 3^ + 5-^ = -6 z z 4^ + 6-^: z z (2) Solving any two of these, we get m z~ 2' 'z~ 2' "^ ' ll x:}j:z::l:-3:2, tities having these ratios will satisfy the given ecjiiations. 45. By the last article the system of homogeneous linear equations a^'x' + ai"x" + ••• + a/"\t;("' = a.,'x' + aj'x" + ••• + 02<"'.«<"' = . . (1) ajx' + a„"x" + ••• + a,.<"\i;'"> = is consistent if A = 0, and in this case we can determine the values of the (n — 1) ratios x^ ' x^ ' x^ ' Now, since A = 0, we have (Art. 29) the n equations a/A' + (h"A" + "• + ",'"'^1*"" = 0, a; A' + a," A," + - + a*'"'.-l*<"' = A = 0, a,: Ax' + a„"J^" + ••• + a,;"».l<^"" = 0, 62 TEEOUY OF EQUATIONS. . Art. 45 which give for the ratios values identical Avith those which the proposed equations (1) give for the ratios x[_ x" -r'" ''. therefore A^', A^" ••• ^4'"' are proportional to x', x" •■• a;*"' what- ever may be the index Jc, so that we have the proportions x':x": ...:a;("> = A':^": " = Ao':A'':- = A,::AJ': - Hence, in any determinant lohich equals zero, the mino^of the elements in any row (or column) are jwoportional to the^kivors of the corresponding elements in any other row (or column). 46. Among the proportions of Art. 45, let us consider those of the last line, for example x':x":-: .7j(») = A„' : A," : - : AJ'^K ... (1) The coefficients of the last of the proposed equations ((1) of Art. 45) a„'x' 4- a„"x" + ••• + a,/"'.c("' = 0, not appearing in the expressions for A J, AJ', •••,AJ"\ there results that the proportions (1) determine the ratios -of the, unkiiowns a;', x", •••, x'"\ which will satisfy the n — 1 equations «i'.i;' + ai"x" -\ h a/"'a;(") = Alt. n DKTEIiMINANTS OF SPECIAL FOI{^fS. 63 expressed by means of the niiiiovs wliich can be formed with the »(h — 1) coefficients of these equations, in suppressing in turn each of the vertical lines. Therefore having given n homogeneous equations between u + 1 unknowns ajx' + aj'x" H 1- a„<"+"x"'+" = ,' •••o/'-^' a/'^" ••.o,<" if we put a '■" «« the solution of the proposed equations would be given by the proportions x' :x":--: .t'"+" = B' : R" : ■■• : /2<"+»>. For example, the two equations -4:x + y + z = x-2y+z=0 give x:y:z = 1 1 2 1 = 3:5:7. -4 1 1 1 4 1 1 -2 y DETEinilXAXTS OF SPECIAL FORMS. 47. Symmetrical Determinants. Two elements of a deter- minant so situated, that one occupies with reference to the leading element the same position in the rows as the other does in the columns, are called ro>>jin/((te elements. For ex- ample, in the common form of determinant, d^ and /v^ are con- jugates, one occui)ying the fourth i)lace in the second row, and the other the fourth place in the second column. 64 THEORY OF EQUATIONS. Art. 47 Each of the leading elements (that is, the elements of the X)rincipal diagonal) is its own conjugate. Any two conjugate elements 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 each element has itself for a conjugate element. Examples of symmetrical determinants are the following : a h g h b f g f c (1). y X (2) In a symmetrical determinant the first minors complementary to any two conjugate elements are equal, since they differ only by an interchange of rows and columns. The corresponding inverse elements are also equal, the signs to be attached to the minors being the same in both cases. It follows that the recip- rocal of a symmetrical determinant is itself synimetrical. The leading minors are all symmetrical determinants. The principal diagonal is called the axis of symmetry. EXAMPLES. 1. Form the reciprocal of the symmetrical determinant a h g h b f 9 f c Using the capital letters to denote the reciprocal elements (Art. 39), the reciprocal determinant may be written thus : A H G hc-f- fg-ch hf~bg A' = H B F = fg - rh ca - f/ {/'' - «/ G F C hf-bg gh-af ab-h? 2. Prove by means of the proposition of Art. 36, that the square of any determinant is a symmetrical determinant. Art. 18 DETERMINANTS OF SPECIAL Foil MS. r,5 48. Skew-Symmetric and Skew Determinants. A skeir-st/m- metric determinant is one in which each element is its conju- gate with sign changed. Since each leading element is its own conjugate, it follows that in such a determinant all the elements of the principal diagonal are zero. For example, the determinant A = a h a d b -d -c -e -f is skew-symmetric. A slcew determinant is one in which each element, except the leading elements, is its conjugate with sign changed. Thus, while a skew-symmetric determinant is zero-axial, a skew determinant is not. Thus b c I 'nl z n n 10 is a skew determinant. MISCELLANEOUS EXAMPLES. Evaluate the following determinants : 2 1 10 2. 13 3. 25 5 10 3 6 2 5 15 3 9 4 5 7 4 6 7 12 3 10 4 5 5. 10 5 6. 1 3 4 15 5 6 15 3 6 2 4 5 20 6 8 20 4 7 3 5 6 66 THEORY OF EQUATIONS. Art. 48 10. 13. 3 4 5 I 4 -1 -2 3 3-7 4 15 13 10 12 17 10 16 11 19 11. 14. 4 5 2 -1 2 -3 6 -4 5 -1 -1 1 -3 1 -4 2 -3 -5 20 15 25 17 12 22 19 20 16 12. 15. 1 - -1 1 4 - -3 3 9 _ 5 15 17 16 12 18 14 19 17 13 30 36 35 33 31 37 38 34 32 16. Expand and simplify the determinant a a + 3 a + 6 a + 1 « + 4 a + 7 a + 2 a + 5 a + 8 Evalnate the following determinants : 17. Ans. 0. 19. 21. Ill: L 18. 2 3 -1 . 5 1-1-1 1 6 -5 -3 1-1 1-1 1 1 1 1 1 1-1-1 Ans + 16. 1 -1 1 -1 An& . -74. 2 2 2 10 20. 6 3 2 1 12 5 8 7 2 3 4-32 4 o 8 4 1-14 5 Ans . +16. 3 6 3 3 A7 IS. +660 1 3 5 2 22. 4 12 5 1 1 5 2 4 4 3 3 2 4 5 2 2 1 6 2 3 Art. 48 3/ 23. 3 1 4 2 2 8 1 6 4 3 9 5 MISCELLANEO US EX A MPL ES. 25. 27. 29. 31. 3 4 ■3 1 6 -2 5 9 1 -3 24. 4hs. -101. 2 1 1 2 3 4 5 2 2 5 7 5 2 3 4 1 4 Ans. 172. 1 1 1 5 3 1 7 9 1 11 ^)iS. ^ 26. 30. 32. 3 7 4 3 7 4 3 5 2 19 4 8 G 4 7 vl>/.s. -33(5. 10 8 9 14 1 17 15 18 11 15 19 10 13 16 17 18 10 Ans. -2660. 5 -1 4 6-2 -1 4 6 -2 5 4 6 -2 5 -1 6 -2 5-14 -2 5 -1 4 6 Ans. +22,692. 3 7 9 5 -3 4 2 1 -7 -4 11 101 -9-2 -11 2 -5 -1 - -101 -2 2 4 3 14 3 -4 2 - -3 2-1 2 5-1 6 2-1 5 1 1 1 —2 —2 —2 7 -3 - -5 1 4 2 3 1 2 _ 1 2 3 ^ns. +14,940. 68 THEORY OF EQUATIONS. Ai-t. 48 12 22 14 17 20 10 16 -4 7 1 —2 15 10 -3 o 3 —2 8 7 12 8 9 11 C 11 2 4 -8 1 9 24 6 6 3 4 22 33. .^jis. 12,228. 34. Find the number of inversions in the series h a c f i g d h e. 35. Find the number of inversions in the foHowing permu- tations : 3, 6, 4, 1, 5, 2 ; 7, 1, 6, 5, 3, 4, 2 ; 2, 4, 1, 3, 6, 7, 5 ; 4, 8, 6, 7, 2, 5, 3 ; 3, 1, 8, 9, 2, 5, 6, 7, 4. Develop the following determinants : 36. 38. X y X y - X - -y d d d a a a b h b c c c 37. 39. a -1 1 c 1 1 d Ans. abed -{- ab + orf + cd + 1. 40. 1+a 111 1 1+6 1 1 1 1 1+c 1 111 1+d Ans. abcd(l -\- a~^ + b-' + C-' + d-'). '^'' Art. 48 MI SC ELLA NEO US EXAMl'LES. 69 41. k I X h X .T X e f g 42. tti a., a^ a^ 6, c, C-, c, f/o iU X a b c cl Ans. x' Alts. a^b■Jr./l^. 43. Write the expauded form of the determinants : 1 a-^aj'tt^' 1; 1 an022«K« vif^oo 1 ; 2 ± ciyb.:^ Yh^s- Find the values of x in the following equations : 44. X -i 1 45. 1 1 1 -6 3 -2 = 0. a X c = 0. a; 2 1 b b X 46. a + bx c d 47. X 3 e +fx g h = 0. 1 -X 4=0. i + kx I in 2 5-6 Ans. x = 1 f<^»*' 1 1 bgm 1 ^ L>(S. x = -l 48. "What effect is produced on a determinant of the jjth degree by multiplying all its elements by — 1 ? 49. 50. Show that 1 1 1 1 1 l+o; 1 1 1 1 1+y 1 1 1 1 1+2 Show that a.^ «3 oil bi &2 b, = 1 f/;,///-, aM,c, Cl C2 Cs 1 tta^^iCj a. 'V3 70 TUEORY OF EQUATIONS. Art. 48 51. Prove that . a + c b + d a + c b + d b+d a + c b+d a+ c a + b b + c c +d d+a c -\-d d + a a + b b + c 52. Showth It a b c a c b b c a " c b a 1 1 1 1 c' 6^ 1 c' a^ 1 b' a' = 0. Resolve into simple factors the two determinants ; 63. a; 1 1 1 1 CB 1 1 11x1 1 1 1 x Ans. {x+3){x-iy 54. a a a a a b b b a b c c a b c d ins. —a{a—b)(b—c){c—d). a b c d b c d a c d a b d a b c 55. Transform so as to have the principal diagonal composed (1) of the four a's, (2) of the four b's, (3) of the four c's, (4) of the four d's. Prove the following identities : 56. a + b a b c c b + c a b c + a Aabc. Art. 48 MISCELLA NEO US EXAMPLES. 71 57. 58. c a c c b' + '•- a b b a -^b — c c a b + c b b a C- + a- 59. Find the value of 12 11 3 14 2 1-1 2 3 0-4 4 abc. c a b c a a = b c a c^ a — b c a b -1 9 4 2-1 _ 1 3 -2 2-1 1 3 4-1 Perform the following multiplicatious, giving the results as determinants : 60. 61. b d e f Ans. a b d / ab ac ae + bf Ans. bd c' + d' ce et< ae + bf df ef a' -a 1 b' -b 1 c'-c 1 a' (r - ab + W a^ -((C + (^ ^2 _ ab + b-' b- U^ _ 6c 4- c» a- — ac -\ - c ' 6-' - - 6c + c* . c^ 72 THEORY OF EQUATIONS. Art. 48 a a a a -1 1 a h b b -1 1 a h c c -1 1 a h c d 1 1 1 -1 Solve, by means of determinants, the following equations : 63. 64. 65. 67. 68. 3 a; + 5 3/ = 17 2 a; + 3?/ = 11 4:X + 7y-10= 7x — 4:y-{-l= 3x-4:y + 2z= 11 2x + 3y^3z = -l 1 Ans. 1, 2, 3. 5x — 5y + iz= 7J 4:X-7y+ 2 = 161 3.^+ y-2z = 10\ Ahs. 5, 1, 3. 5x-6y-3z = 10) 5x-4.z= 42 3z + 5y= 1 A ns. 6, 2, -3. Ay-3x = -10 4x + 7y -\-3z — 2iv= 9 2x— y-4:Z + 3 IV = 13 3x + 2y — 7z — 4:iv= 2 Ans 1,3, -1,3. 5x — 3y+ 2 +5 10 = 13 69. 3 .T + 2 ?/ + 4 z — ?c' = 13 5x+ y — z + 2w= 9 2 a; + 3 // - 7 2 + 3 ?o = 14 4:X — Ay-{-3z — 5io= 4 ^ns. 2, 4, -1,-3. Art. MTSCEL L A NEO US EX A MPL ES. 70. What relation must exist between a, h, c, d if the equa- tions ax + by-{- cz + d= 0, bx + ay + dz + c = 0, ax+ cy+ bz + d= 0, cx + ay + dz+b= 0, be simultaneously true ? 71. Test the consistency of the system x + 10y + Uz= 3 2x- 6y+ 7z= S x + 12y + nz= 4 X- 3y+ 2z = 12) 72. Test the consistency of the system 2x-3y + 10z= 4 x + 4:y- 82= 2 3x+ y+ 2z= 6 4.'i; + 5?/+ z= 8 73. Solve the homogeneous equations x + 2y + 3z= 2x + 3y-\-4.z= 3a; + 4y + 52;= 74. A skew-symmetric determinant of odd order vanishes. For any skew-symmetric determinant, A (see Art. 48) is unaltered by changing the columns into rows, and then diang- ing 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. Fur example, a 6 -a c =0 -b -c 74 THEORY OF EQUATIONS. Art. 48 We give as our last example a special determinant as the product of differences. Exercises 7 and 8, after Art. 21, have afforded examples of the resolution of this particular form of a determinant, of which we now consider the general case. 75. Take any n quantities, a, b, c, •••, I', I, and form a de- terminant containing as rows (or columns) the powers of these quantities from to n — 1 ; thus : 1 1 1 ... 1 1 a b c • • A; I cC- b' c' ... k- I- a' W c« ... A;^ P ^•"-^ I" This determinant possesses the property of vanishing when any two of the n numbers are equal, for example, if we put : a = b, a = c, ..., a = I, b = c, b = d, etc., since then two columns become identical. It results that A ought to contain as factors all the differences which can be formed with the series a, b, c, ... k, I, ill subtracting from each letter all the letters that follow it. The product P of these differences would be P = (a - b) (a - c) (a - r?) • • • (a - k) (a - I) (b-c)(b-d)-'(b-k)(b-l) (c-d)--(c-k)(c-l) (h-k)(h-I) (k-T). Art. 48 MISCELLANEOUS EXAMPLES. 75 The detenninaut A is equal to P iu absohite value. For the degree of A with respect to a, b, c, •••, I, is equal to l + 2+3 + ... + (n-l) = ?^^5^ as we see from its principal term ; this is also the degree of P, which embraces " !^ ~, ^ diiferences ; therefore P and A can differ from each other only by a numerical factor. Finally, to determine this factor, we remark that the principal term of A is the expression 1 ■b-c''Cp---k"---l"-\ The corresponding term of the product P, obtained in con- sidering the columns, we find to be (- 1)6 . (- i)V . (- i)\p ... (- i)''--r-2 . (- iy-H"-\ This has for coefficient "C-D (_ l)'+2+3+-+('»-l) _ / 1) 2 Therefore, we have A = ±P, according as Ill^Lp — I is even or odd. PART IL — THEORY OF EQUATIONS. INTRODUCTION. Historical Note. While we cannot, in this brief notice, go back to the beginnings of algebra, a few historical notes may prove of interest to the reader.* The first comprehensive algebra was published in 1494 by Lucas Pacioli, an Italian mathematician. Scipio Ferro (Professor of Mathematics at Bologna from 149() to 1525) first solved a cubic equation of the form a:^ + mx = n. His method is not known. A second solution of cubics was given by Nicolo, called Tartaglia (150(5- 1557). This solution, known as Cardan's Solution, was stolen by Hieronimo Cardano (1501-1570) and published in 1545 in Cardan's Ars Magna. Ferrari (a pupil of Cardan's) discovered a general solution of bi-quadratic equations, which was also published in the Ars Ma(/)ia, a work far in advance of any algebra previously printed. About the middle of the sixteenth century nega- tive roots were receiving considerable attention, but it seems impossible to say who first fully comprehended them. Bombelli, in his algebra published in 1572, opened the way to the recognition of imaginary I'oots. Here, too, prog- ress was slow. Michael Stifel was the greatest German algebraist of the six- teenth century. Vieta (1540-1(;03), the most eminent French mathematician of the sixteenth century, enriched algebra by innovations in notation, and by numerous discoveries in the Theory of Equations. Thomas Harriot (1500-1621), of England, made further improvement in notation, and did much to establish the Theory of Equations on a scientific basis. After this it was enriched by the fruitful discoveries of Descartes, Newton, Lagrange, Argand, Gauss, Abel, Hermite, Kronecker, Cayley, Sylvester, and others. The solution of numeri- cal equations was particularly advanced by Fourier, Budau, Horuer, and Sturm. There ai'e many text-books in which the subject is discussed, among them we may mention : Burnside and Panton's Theory of Equations ; Todhunter's * An excellent history of mathematics, and perhaps the one most easily accessible to the reader, is Gajori's A IlUtorij of MalhemuUca, Macmillan & Co. Interesting historical notes may be found in Fine's The yumher-Sijsteni of Algebra. 76 Art. 49 PA R T II. — IN TR 01) UCTION. ( i An Elementary Treatise on the Tliconj of Equatiunx; Serret's Coitrs d' Al- gebra Sap^rieure; Carnoy's fours d'Algebre Superieure ; Bieriuaiurs Kle- mente der Iloheren Matheniatik; Matthiessen's Grundzdije der Antiken nnd Modernen Algebra der Litteralen Gleichungen;* Petersen's Algebraisehe Gleichungen. 49. In elementary algebra the student has solved equations of the iirst and second degrees, and has become somewhat familiar with the meaning of the word root as applied to an equation ; and some of the definitions given in these pages, as well as some of the processes described and employed, will not be entirely new to him. Let us consider the theorem : An integral equation of the first degree in one unknon-n has one and only one solution. For example, take the equation a.v + h = (1) One solution of this is .r = — -. To prove that this is the a only root, let us suppose that there are two distinct solutions, X — u, and x = /3, of (1). Then we must have aa 4-6 = 0, ap + h = 0. From these, by subtraction, we derive a(« -fi) = 0. Now, by hypothesis, a is not = 0, therefore we must have a — (3 = 0, that is, « = )8; in other words, the two solutions are not distinct. Hence there is only one root, and it is a function of the coefficients.t y * Matthiessen develops the subject historipally, and on padres 0(V4-l(ioi may be r.miiil a very extended bibliograidiical list. t .\s is well known, the constant term b is called a coefficient, and it is the cotllblent otafi. 78 THEORY OF EQUATIONS. ■ Art. 49 The quadratic equation ax- + bx + c = (2) has two roots, namely, h +Vb- — 'iac -, — b — V6- — 4ac , 2 a and with respect to these roots, we know that their sum is b c , and their product is -; that is, their sum is equal to the u' ^ a coefficient of the second term of the equation a a Avith its sign changed, and their product is equal to the last term of this equation. Thus the student has seen that the root of an equation of the first or second degree may be ex- pressed in terms of its coefficients. The general object of this treatise is to establish results with respect to equations of a higher degree than the second, similar to those that have been established in elementary algebra respecting equations of the second degree. In fact, the science of the Theory of Equations seeks to discover gen- eral methods for the solution of equations of any degree. The limitations to this search will appear later (see Art. 53). 50. Definitions. Any algebraic expression that depends upon any quantity as x for its value is said to be a function of x. Thus 3 a;- — 4 a: + 16 is a function of x, so also is V«" — x'. An algebraic function involves the operations of addition, subtraction, multiplication, and division applied only a finite number of times.* All other functions are called transcendental functions, such as logarithmic, exjjonential, trigonometric, and in- * Thi.s of course includes involution .and evolution with constant exponents. See Appendix A. Art. 51 PART II. — IM'liODL (Jloy. T«J verse trigonometric. In this work, when we use the word fuuc- tion, we mean an algebraic function, unless it is expressly stated or shown by the form that the function is transcendental. A function of x is, for brevity, represented by F{x), f(x), (x), or some such symbol. Thus, for example, F(x) = 3 .ir - 4 .i- + 16, f(x) = a log .v, 4> (x) = sin 3 x. A rational fund ion of a quantity is one that contains the quantity in a rational form only ; that is, a form free from fractional indices or radical signs. An integral function of a quantity is a rational function in which the quantity enters in an integral form only ; that is, never in the denominator of a fraction. A rational integral function of x, as discussed here, is one that can be put in the form ax'^ + 6.«"~^ + c.i-"~- + ••• + 'kx + /, in which n is a positive whole number, and a,h, c •••I denote any real expressions not containing x. It will be observed that the coefficients may be irrational or fractional. Algebraic symbols are numerals, letters of the alphabet, or conventional signs to denote certain operations or relations, such as — , +, X, H-, =, >, or <, etc. An algebraic expression is any combination of algebraic sym- bols which represents a quantity. A term is an expression whose parts are not separated by the signs + or — , as 4 a^, 3 ahc, or — A monomial is an algebraic expression of one term ; a itobj- nomial is one of two or more terms. 51. An identical equation is the statement of equality be- tween mathematical expressions which are either the same, initially, or become the same by the apidication to one or Ixith of the allowable mathematical operations; for example, 80 THEORY OF I^QUATIONS. Art. 51 a^ _ ^2 _ (^_^ _ y-^ (^j. ^ yj^ sij;i 2 yl = 2 sin ^4 cos A, are identical equations. If one algebraic expression containing x is equal, for certain values of a;, to another differently constituted, the equality thus formed is called an equation of condition. Whenever an equa- tion of condition is meant, we shall use the single word equation. An equation, then, is the statement of an equality, which is true only for certain values of the unknown quantity. Any value of x which satisfies this equation is called a root of the equation. The determination of all possible roots con- stitutes the, complete solution of the equation. By bringing all the terms to one side, we may obviously arrange any equation according to descending powers of x in the following way : ao^;" + ttiX"'^ + aa"-"^ + a.x"-" + \- a„_iX -f- a„ = . (1) An equation is not altered if all of its terms be divided by any quantity. Dividing (1) by a^, and thus making the co- efficient of x" equal to unity, it may be written in the form : a;" + p^uf-' + pox^'-- H h p„_iX + p„ = . . (2) The highest power of x in this equation being n, it is said to be an equation of the nth degree in x. An equation is complete when it contains terms involving x in all its powers from n to 0, and incomplete when some of the terms are absent ; that is, when some of the coefficients, aj, a,, tts, etc., are equal to zero. The term a,„ which does not contain x, is called the ahmlnte term. 52. A numerical equation is an equation in which the co- efficients are represented by figures only ; a literal equation is one in which the coefficients are represented wholly or in part by letters. ^t. 53 PART II. — lyrnoDUCTION. 81 A linear equation is one of tlie lirst degree. A quadratic equation is one of the second degree. A cubic equation is one of the third degree. A biquadratic, or quartic equation is one of the fourth degree. A quintic equation is one of the fifth degree. A sextic equation is one of the sixth degree. Equations above the second degree are called higher equiUions. 53. In both mathematical and ph3'sical researches, we frequently meet with problems that involve the solution of equations. As the equations thus met with are often liigher than the second degree, it becomes a matter of importance to find, if possible, some general method for the solution of higher e'■•'■■■• '' ! I CHAPTER IV COMPLEX NUMBERS. 54. In the solution of quadratic equations, the student has frequently met with the square root of a negative quantity. Such a number is said to be imaginary or unreal, for the square of no real quantity is negative. The imaginary unit V— i is denoted for brevity by i, and integral powers of i beyond the first can always be reduced by the relation /- = — 1. All the operations that we perform on the unit i must, then, be subject to this definition, i^ = — 1, and to the general laws of algebra. For example, yi^iy, yi+y'i=(y-\-y')i=i{y + y'), etc., exactly as if i were a real quantity. 55. If we combine, by addition, any real quantity a with a purely imaginary quantity bi, there arises a mixed qiumtity a -f bi, a form frequently met with. Such an expression, consisting of a positive or negative real units and b positive or negative imaginary units, is called a corapiex number, or quantity. (Throughout this book we make no distinction between the words "number" and '* quantity.") lieal and purely imaginary numbers are both included in the expression a + ib, the former being obtained when 6 = 0, and the latter when a = 0. Of course, in such expressions, a and b are considered real. 56. The successive powers of i are periodic. We have : i^ = /, i^ = — Ij i^ = i^ . i = — i^ i* = i- . r = -f 1, i^ = i^ • i — -\- i, etc. 82 Art. 60 COMPLEX yUMIlEUS. 83 Beginning with the fifth power, nil the results repeat them- selves in the same order. There are only four different values, namely : + /, — 1, — i, + 1- 57. If X + ill — 0, then must x=0, y = 0. Otherwise we should have x= - i>/; but x is real by hyputhesis, and hence X cannot equal — iij, which is imaginary. 58. If X + ii/ = a + ib, then x = a, y = h. Otherwise we should have x — a= i(b — y), which cannot be, since x — a is real. 59. The alr/ebmic sum of any number of com2)lex q nan titles is a complex quantity. Suppose we have, say, three complex luimbers, .r, -f y,/, X2-\-yJ, Xs+y^i, then (xi+yii) + (xo-tyJ)-(x3+y/) = (Xi + .r.,-.rs) + {yi + y2— yi)h by the laws of algebra already established. But a?! + x'2 — .1-3 and ?/i + ^^2 — Vi are real, since x^, x.>, x-j, y,, y.^ ys are real. Hence (Xi + X2 — x^) -f (?/i 4- ?/, — .Vs)*" is a complex number. The conclusion obviously holds, however many terms there may be in the algebraic sum. For special case where the sum is real see Art. 04. 60. The 2-)roduct of any number of complex numbers is a complex number. Consider the product of two complex numbers, .r, + yii and X2 + yoi- We have (^1 + z/iO (-^'2 + yJ) = -^'la.'!. + yiy/- + ^\yJ + a-.^/J- Hence, bearing in mind the definition of /, we have (^i + .'/lO (-'^2 + y-/) = (^i->'*2 - yi.V2) + (-''LVi + av/,)', Avhich proves that the product of two comi)lex numbers is a complex number. The proposition is easily extended to a product of three or more complex numbers. For special ease where the product is real see Art. 04. L )C L :^ Q 84 THEORY OF EQUATIONS. Art. 61 61. TJie quotient oftico complex mmibers is a complex number. We have X2 + yd xi - (jj4f {x^x., + y^y.^ -(^i2 h. - x&iY xi + 2/2^ XxX2 + y,yo xi + yi + yir which proves the proposition. Cor. I. Since every rational function involves only the operations of addition, subtraction, multiplication, and divi- sion,- it follows from the above theorems that every rational function of two or more complex numbers can be reduced to a complex number. Cor. II. If f{x -f- yi) be any integrcd function of x-\- yi, having all its coefficients real, and if f{x + yi) = P + Qi, then f{x - yi) = P- Qi, ivhere P and Q are real. For it is obvious that P can contain only even powers of y, and Q only odd powers of y. If, therefore, we change the sign of y, P will remain unaltered, and Q will simply change its sign. Hence the theorem. Cor. III. If (x -{- yi) be any rational function of x + yi, having all its coefficients real, and if <}>(^x + yi)^X+Yi, then {x — yi) = X — Yi. Art. 64 COMPLEX ^'rMnEll.'i. 85 EXAMPLES. 1. 3(3 + 2 - 2(2 - 3 + (<•» + 8 /) = 11+20 i. 2. (2 + 3 0(2 -30(3 -5/) = (4 + 1)^(3-50 = 30 -05/. 3 3 + 5/ ^ (3 + 5 0(2 + 30 ^ 10. • 2-3i 4 + 9 13 13'" 4. {x + yiy = (.«•• - 6 ayy' + ^^) + (4 x'y - 4 .r/) /. 62. Two complex numbers which differ only in tlie si«,Mi of their imaginary part ai-e said to be co»J"gote. Thus - 3 - 2 ?: and - 3 + 2 « ; - 4 1 and + 4 t ; x -\- >/i and X — yi, are C(jnjugate. The student has met with conjugate imaginaries in tlie solution of quadratic equations, where if one root is imaginary, the other is also imaginary, and is conjugate to the first. 63. If a + ib is a root of an cdgebraic equation, then also is a — ib a root of the same equation. For, let f(x) = be the equation. If a + ib is a root, we must have /(a + ib) = 0. This may be written f(a + ib) = 0= P+iQ = 0; and this requires P= 0, Q = (Art. 57). Hence P-iQ = 0, and f(a — ib) = P — iQ=z{)-^ hence a — ib is a root of f(x) = 0. 64. The sum of the conjugate imaginaries, x + ///, x — iy, is the real quantity 2 x ; their difference is the pure imaginary 2 iy. Their product x- + y' is called the norm of either of them. noi-ni (x + iy) = norm (x — iy) = ar + y-. 86 THEORY OF EQUATIONS. Art. 64 The modulus of a complex quantity is the positive square root of the norm. Thus, employing the usual symbol, mod {x + iy) = Va;'^ + y'\ mod (x — iy) = VX' + y-. Rein. When y = 0, that is, if the complex number be wholly real, then the modulus reduces to +Va;-, or x, that is simply the numerical value of x. For example, mod (- 3) = + V(- 3f = + 3, mod (+ 5) = + 5. EXAMPLES. norm (- 3 + 4 = (- 3)^ + (4)' = 25. norm (4 — 5 i) = 41. mod(-3 + 4i) = 5. mod (2-5 = V29. mod (1 + i) = V2. mod (6 + 8 = 10. 65. If a complex number vanish, its modulus vanishes; and conversely, if the modxdus vanish, the complex number vanishes. For, if X + yi = 0, then a; = 0, and y = 0. Hence Vx- + if = 0. Again, if Vx^ -{-y- = 0^ then x"^ -\-y" = 0, hence, since x and y are real, a; = and y — 0. 66. If tu:o complex numbers are equal, their modidi are eqmd. For, if X + yi = x' + y'i, then x = x', y = y' ; hence V.r^ + y- = Va;'- + y''\ The converse is obviously not true. Art. 6/ COMPLEX NUMBERS. 67. Graphic Representation. — Argand's Diagram.* "We shall consider now tlie giai)liu' method of reprcsLMiting complex numbers originally suggested by Argand. We have seen that the usual representation of positive or negative quantities is by means of distances measured along a straight line, positive quantities being represented by distances measured to the right, negative quantities by distances to the left. For some reasons it is best to say that positive quanti- ties are represented by distances measured to the right, and that the effect of multiplying any quantity by — 1 is to reverse the direction; that is, if the quantity is multii)lied twice by i, the direction is reversed. If now the factor /-, or i • i, changes the direction by 180°, then it seems natural to consider * a factor that changes the direction by 90°. It is customary, to say that the effect of multiplying by i is to tuni the line through an angle of 90° in the positive direction (counter-clockwise). It is evident that the repetition of the operation of using i once as a factor, reverses the direction. jS^ow, let XOX', YO Y' be two rectangular axes. We shall Fig.l X'- * So called because to Arpand is due the credit of first gMng this peometriral construc- tion in his KMmi mir une mtniieri' de reprenenier lex quauliliH iinngintiirm tlnnn let coustruetions gevmHriqueH (l^)ii). SfC ChrysUl's Al(/ebrti, Vol. I., p -•»" < "> Appendix B. 88 THEORY OF EQUATIONS. Art. 67 call XOX' the axis of real quantity, YO Y' the axis of purely imaginary quantity. To represent the complex number x + iy, we lay off on the a>axis tlie distance OM — x, and on MP, perpendicular to the X-axis, the distance 3IP = y. Thus the poiut P is definitely located by the quantity x -{- iy. The distance OP =r = Vx- + y- = mod (x -j- iy), and we have cos MOP = cos ^ = -, sin = t r r Hence the expression x + iy may be written in the form ?*(cos + i sin 6). The quantity r is called the modulus* and the angle the argument of the complex nnmber x + iy. • The modulus and argument of a; + iy are for brevity repre- sented by the notation mod (x + iy), arg {x + iy). Example : To write 3 + 4 i in the trigonometric form r(cos 6 + i sin 6), we have r = V32 + 4- = 5, cos ^ = f , sin 6 = |, and .-. 3-f 4i = 5(f + i4). Cor. Of course x—iy, or r(cos 6 — i sin 0), represents the point P', the y in this case being measured downward because it is negative. If the argnment of x + iy is $, the argument ^i x — iy is 2 7r — ^, or we may say that two conjugate numbers have the same projection on the a;-axis. 68. The Exponential Form of jr + //. The following develop- ments for cos $, sin 0, and e'', which are deduced in works on trigonometry and elementary calculus, are supposed to be known : * fiennan writers uso " ab solute value" instead of "modulus," and denote it by tlie symbol | x + it/ 1. Thus '•^j-^ + //" = | ^r + /(/ | = absolute value of the complex number ir + iy. So also 6 is often called the "amplitude" instead of the "argument." Alt. 08 COMPLEX ^UMliERS. 89 e' = l+a; + — + — + — +•-, 2!3!4! 008^ = 1--^ + -^--^+..., o . o . < ! From the last two we have cos e + i sill ^ = 1 + ^^ _ -^ _ /-^ + -^ + /il . 2! 3! 4! 5! If we define a function e'^ by the series 2\ J I 4 ! 5 ! 2! 3! 4! 5!^ which is entirely analogous to the form for e', where x is real, then we have e''« = cos ^ + i bin 0, and, consequently, x+ iy = r(cos ^ + ?' sin 9) = re'*. Similarly, x* — iy = 9-(cos ^ — 1 sin ^) = re~*^. Cor. I. The following fornmlce are sometimes useful : e'" = cos IT + i sin tt = — 1, e-'> = cos TT — ?■ sin tt = — 1, .-rr TT , . • TT e'2 = cos - + I Sin - = i, e-'f = cos J - i sin J = - «• 90 THEORY OF EQUATIONS. Art. 68 Cor. II. \i 6 = ^, and r = 1, then x + iy becomes ' ' 1 ■ (cos^ + i sm~\ = l ' i = e+'l. Hence e"*"''? = i, is the operator which turns the direction through 90°. 69. Expressing e'^, e'"* in their respective trigonometric forms, and performing the operations of multiplication and division, we can readily prove the relations : Hence, the function e'^, defined in the last paragraph, obeys the same laws of multiplication and division as the function e'', where a; is real. 70. De Moivre's Theorem. First, for n a positive whole number. If in the equation (x + iy) (a + ib) = ?jtre'^^+"^,* we let a -\- ib = X+ iy, it becomes (x + iyf = r'e- • 2« = ,^ (cos 2 ^ + ? sin 2 ^) ; similarly, (x + iy)" = r" • e' • "* = r" (cos nO + i sin nO). Hence, for n a positive whole number, (cos 6 -j-i sin 6)" — cos n9 + i sin ?i^. Second, for ?i a negative whole number. We know that * Here a + ib = we*". . d Art. 70 COMPLEX NUMIiEIiS. 91 ■ e" - a + to m If in this we make (a + lb) = (;« + ii/)"+^ = ,-n+ie'(n+i)« we shall have Hence [r(cos + i sin ^)]-" = r-''[cos (- nO) + i sin (- «^)]. .-. (cos ^ + i sin ^)-" = cos n6 — i sin h$. Hence (e'^)" = e'"^, where u is any positive or negative wliole number. Third, n any number. Suppose that $ = ^, then e'^ — e^j, and (e**)' = (e'7)'=>»=g'*. That is, the tth power of e'r is e''i> ; conversely, one of the ^th roots of e'* must be e'Y; hence (cos ^ + i' sin 6f = cos - + i sin — ^ t t Finally, if s and t are any wliolauumbers^we have (e'"). = e*^ 9 = cos - ^ + i sin ^ ^ ; b ut as s and t are any j mmbers, wJiatfivei-, - may repr esent anx . rational or irrfttiinnil 'Tiunrberj hence, when n is any number whatever, integer, fractional, or liiaLiuim^ we have (e'9)" = e"'", or (cos 6 + i sin 6y = cos nO + t sin ud, which is De Moivre's Theorem* * Abraham de Moivre (ir)6T-n54). The discovery of this theorem by De Muivre revohi- tionized analytical trigonometry. 92 THEORY OF EQUATIONS. ■ Art. 71 71. The Values of (e'^)", for integer value of n. By definition, we have e"'" = cos 2 TT + t sin 2 TT = 1 ; hence e'^ • e-''" = e'^ = e'(9+2,r)^ or, more generally, gte _ (_j(e+-2kn)^ where k is any whole number whatever. 1 1 , e+gA- r Hence (e'«)" = (e-(e+2*>r))n ^ e' " ', whence (e'^Y = cos 1+1^ + ,• gin 1±1^ ?i n where k may be any whole number. While from this equation the number of values of (e*^)" is apparently infinite, there are really only n different values, for Avhen k has run through the 1 numbers 0, 1, 2, 3, •••, n — 1, the values of (e'^)" begin to repeat themselves, as may be readily shown. 72. Solution of the Equation a-" — 1 = 0. This is a special form of the binomial equation, the general form of such equations being a;" = a + 6v — 1, where a and b are real quantities. To find the roots of (1) we have X" — e-"^ = e" Hence ic = e^^^ = cos 5-^^+1^ + i sin ?iil±l^ . (2) n n Yov k = 71 — 1, we have 2mr , . . 2mr ^ . ■ ^ X = cos h i sm = cos 2 TT + ; sm 2 tt = 1. n n Tlierefore + 1 is a root of the ecpiation, .r" = 1. Alt. 73 COMPLEX SUM HERS. y^ If 11 is even, we may make A- = ^ — 1, then we have 7i7r , . . Uir . . » = COS h I sm - - = cos TT + I sin tt = — 1, Hence, if n is even, both + 1 and —1 are roots of x" = 1. But if /i is odd, + 1 is the only real root. This is evident from the fact that for all values of k, other than " — 1 for n even, o _i_ *? z- "^ and n — 1 for n even or odd, sin "^ ^ "*" *" is not zero, and II, therefore the root is imaginary. 73. Solution of the Equation jr" + l = 0. To find the roots of X" = -!..., Ave have a;"= e''<"+-*'^', since e'"' = — 1. Hence a; = e' » = cos — f- 1 sin ^!^ n n If n is even, the roots are all imaginary, since no even power of a real quantity can be negative ; but if n is odd, we may make A: = ~ ; then we find x = cos tt -\- i sin tt = — 1. "We conclude that when n is odd, there is one and only one real root, — 1. EXAMPLES. 1. rind the cube roots of +,1. Here ar^=l, and in equation (2), Art. 72, A" may be made equal successively to 0, 1, 2, while n = 3. We thus get for the roots X = cos f TT + ' sin I TT = — J- + ^ V3 i, X = cos ^TT + i sin | tt = — | — ^ V3 i, X — cos 2 TT + / sin 2 TT = 4- 1- 2. Solve the equations x* = 1, and x* = — 1. ^ 94 THEORY OF EQUATIONS. 3. Solve the equation x^ = 1. The roots are : -i(V5+i)+iva<>-2V5)i, -i(V5 + l)-iV(10-2V5)/, .^J KV5-i)-iva^> + 2V5)i. 4. Solve the equation x^" ~ 1. 74. Complex Numbers. — Addition. Let rectangular axes be taken and a point P representing a-\-ih', that is, Art. 07, OM^cu PM=b, and 4^ c^j X' - ■ Art. 73 (At- OP = Va- + &- = ^ = mod (a + ih), and MOP^ a = arg (a + /6). F rig B- Let a second complex number a' + ib' be represented by the poinr A, so that 0.4 = mod (a' + (6'), XOA = arg (a' + ib'). Now the sum of these two complex numbers is a + ib -f- a' + ib', Art. 75 COMPLEX Xi'MBEIiS. 95 which may be written in the form a + a' + i(b + b'), and we observe that this sum is represented by the point whose coordinates are a + a', b + b'. To find this point draw PP' parallel and equal to 0A\ since PC, PC are equal to a', b', P' is the required point, and we have «^ 0P'= mod \ a + a' + / {b + b')\, XOP = arg * a + a'+i {b + b')l. Therefore, to add two complex numbers, represented by the points A and P, Ave draw PP* equal and parallel to OA ; then P' represents the sum of the two complex numbers. Since OP' is not greater than OP + PP', it follows that the modulus of the sum of two complex numbers is less than {or at most equal to) the suui of their moduli. To add a third complex number a" + ib", represented by B, we draw P'P" parallel and equal to OB. Then P" repre- sents a + a' + a" + i{b + b' + b"), which is the sum of the three given complex numbers. As this mode of representation may be extended to the addition of any number of such quantities, it is evident that, in general, the modulus of the sum of aivj number ofcouij>le.c quan- tities is less than (or at most equal to) the sum of their moduli. 75. Subtraction. Subtraction can be represented in a similar way. Since /-*' represents the sum of /-* and .1, 1* will repre- sent the difference of P' and A. To subtract two complex numbers, therefore, we draw from the point repr(>senting the minuend a line parallel and equal to the line from the origin to the point representing the subtrahend, but in the opposite direction. We join O to the extremity of this line to find the 96 THEORY OF EQUATIONS. Alt. 76 modulus of the point which represents the difference of the two given complex numbers. 76. Multiplication and Division. The theorems of Arts. 60 and 01 may readily be proved by De Moivre's Theorem, as follows : To multiply the two complex numbers a + ib, a' + ib', we write them in the form (a + ib) = /A (cos « + i sin a), a' + ib' = fjt.' (cos o&' + i sin a'). Then (a + ib) (a' + ib ') = ixfx' \ cos (« + «') + i sin (a + «') I , which proves that the product of tico comjilex nvmbers is a com- plex number, whose modulus is the product ofthetico moduli, and ichose argument is the sum. of the two arguments. Similarly, we may prove that the product of any number of complex quantities is a complex quantity whose modulus is the product of all the moduli, and whose argument is the sum of all the arguments. To divide a + ib by a' + ib', we have similarly fi±JJL = H I cos (a - a') + / sin (a - a') | , a' + ib' fji' which proves that the quotient of two complex nmnbe7-s is a coni- plex number whose modulus is the quotient of the two moduli, and ■whose argument is the difference of the two arguments. Cor. Similar theorems for involution and evolution are derived at once from De Moivre's Theorem.* * From the formula It is evident that, in invohition, 8 increases by arithmetical projrression, while r increases by jreometrical progression. CHAPTER v.* PROPERTIES OF POLYNOMIALS. 77. Reduction to the Form f{^x)=0. Any rational integral function of x, J\x), uuiy, as we have seen, be put in the form f(x) = o,,i-" + ciix"-^ + (uv"-'- + OyV" ■■' H h ((„_,.»• + f(„. Any equation in x having rational coefficients can be trans- formed into an equation of the form f(x) — 0, as the following example will show. Example. Eeduce '''^ ~ ^^ = ^-^ - to the form Fix) = 0. 1+x ^.i_^2 Clearing the given equation of fractions, we obtain x^-x + 2 xi - 2 x' = .r-i + 1 - 3 - 3 .r, or, multiplying by x to free of negative exponents, x"^ -x''-\-2x^ -2x^ = 1 -2x-3x' . . . (1) To transform (1) into another equation with integral ex- ponents, put x = ]f, 6 being the least common multiple of the denominators of the fractional exponents of .i-. Thus we get 2y2 + 7/"-2y« + 2y'-f 2/-1 =0 . . . (2) which is the required form, the roots of (1) and (2) holding the relation x = ?/'. * In this and subsequent chapters, we shall consider mainly the r. al vrtluos i.f x. and shall not enter upon the general discussion of the theory of the complex variable. 97 98 THEORY OF EQUATIONS. Art. 77 EXAMPLES. Keduce the following expressions to the form f{x) = : 1. -^+2x-^x^-x'=l. 2. "^1^ = 2 + x-\ 1 + x^ 3. ^4:-5x = l-3x^. . 4. ViK^^-5a? = Vl — 2x — X. 5. (x^ - 3 x^) (1 - x) = (.'^-2 + 1) (x-'^ - 2). 78. We shall now give two theorems concerning the relative importance of the terms of a polynomial when values very- great or very small are assigned to x. AVriting the polynomial in the form , ( rfi 1 a. 1 a„_i 1 a„ 1 ) ( Cfo X «o ^ f'o ^ «0 •*' ) it is plain that its value tends to become equal to aox" as x tends toward co. 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 affic" ex- ceed the sum of all the others. In Avhat follows we suppose ttu to be positive; and, in general, in the treatment of poly- nomials and equations the highest term is supposed to be written with the positive sign. Theorem. If rn the polynomial «o-«" + «i-^-""^ + aiicc"-^ + ••• + a„_ia? -f- a„ - d the value -*-+ 1, or any greater value, be substitutecl for x, ivhere a,, is that one of the coefficients a^, a.,, Og, ••• a„ whose numerical value is greatest irrespeciive of sign, the term containing the high- est p)0wer of X will exceed the sum of all the terms tvhich follow. Art. 78 PEOPERTIKS OF POLYNOMIALS. 99 The inequality ttox" > a^x"-^ + (M'"-- + ••• + o„_i.r + a„ is satisfied by any value of x whii-h makes cioX" > Ot(a;"-' + x" - + ... + .r + 1), Avhere a^ is the greatest among the coeftieients cr,, a.^, ••• «„_|. a„, without regard to sign. Summing the geometric series -within the brackets, we have ao.^•" > «*^^^-=^ or X" > /'* ^ (x'' - 1), x—1 a^,{x — l) which is satisfied if aa(x — 1) be > or = o^ ; that is, a; > or = -* + !. This theorem 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, 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 negative value of x, such that for any value nearer to — oc, the polynomial will retain con- stantly a positive sign, if n be even, and a negative sign, if n be odd. As illustrative of the use of this theorem, consider the polynomial 10 .v^ — 17 x- -f- •'" + 6. Here, substituting 10 for (/„ and 17 for a^, the test formula becomes a; > or = J-l + 1, or a; > or = 2.7, which shows us that the function 10 a^ - 17 x-^ -f a; +-6 retains positive values for all positive values of x greater than L'.7, and negative values for all values of x nearer to — x than 2.7. 100 THEORY OF EqUATlOyiS. Art. 79 79. We next consider a theorem which shall enable ns to determine what term controls the sign of a polynomial when the value of x is indefinitely diminished. Theorem. If in the polynomial C(,fX" + ClyX"-'^ + • • • + ttn-l^ + a« the value — ^-^ — , or any smaller value, he substituted for x, where (f,, + «A a„ is the greatest coefficient exclusive of a^, the term a„ ivill be numerically greater than the sum of all the others. To prove this, let x = -; then by the theorem of Art. 78, a^ y being now the greatest among the coefficients cIq, ctj, ••• a„ i, without regard to sign, the value ^ + 1, or any greater value of y, will make «„7/" > «»-i2/"~ '+«, -22/"-'+ •• . + a^y + tto, that is a„ > a J + y r ••+«.A; yU hence the value - a„ or an y less value of x will «n + «* ff„ > a„_iX + a„_.2X- + ••• -{-aQX\ Cor. I. This proposition may be stated as follows : Values so small may he assigned to x as to make the polynomial a„_iX + a„ oX- -\- ••• + Oyii;" less than any assigned quantity. This statement of the theorem follows at once from the above proof, since a„ may be taken to be the assigned quantity. Cor. II. Another useful statement of the theorem is as ollows : Art. 80 PliOPEHTIES OF POLYNOMIALS. IQl'.':' Wheii the variable x receives a very small value, the sign of the poUjnomial a„_ix + «„_2a^ H h ooo;" is the same as the sign of its first term a„_^^x. This is evident, if we write the expression in the form ^ 1 ^',.-1 + (',1-2-''' + • • • + <'o-'-"~' \ ■ 80. Derived Functions. Change of form of a pol tinomial cor- respoiidiitg to an increase or dimi)iutio)i of the variable. We shall now examine the form assumed by the polynomial when X + h is substituted for x. Here the resulting form will correspond to an increase or diminution of the variable x, according as h is positive or negative. The polynon)ial /(.r) = UoX" + Oi.^"-' + a.jX"-- + o..^"-'' -\ h o„_,.r -|- a„ . (A) becomes, when x is changed to x + h, f{x + h), or ao{x + hy + a,(x + hy-' + 0^(0) + hy-^ + ••• + «„_,(.r + h) -f a„. Expanding each term of this expression by the binomial theorem, and arranging the result according to ascending powers of h, we have a^x" + tti-x'""^ + a-yX"'^ + ••• + a„_2^^ + a„_ia; + a„ +/i|«ao.'i^"~^+(n — l)o,a;"--+(?i— 2)a2.i;"-'H h -",.-?»; +«,.-i I + JLln(n - l)ao.r»-2+ (n - l)(n - 2)a,r"-' + - + 2a„_2| + We observe that the part of this expression in(lo])endent of h is /(.-<;), and that the successive coefficients of the different powers of h are functions of x of degrees dimiiiisliing by unity. * . i bS* "••**'•'• * THEOR Y OF EQ UA TIONS. Art . 80 We also see that the coefficient of h may be obtained from f(x) by multiplying each term of f(x) by the exponent of x in that term and, diminishing the exponent of x by unity, the sign being retained. .. The sum of all the terms of f{x) treated in this way will constitute a polynomial, one degree lower than .m- This polynomial is called the first derived function of f{x), and is usually represented by the notation f'{x). The co- efficient of -^ is gotten from /'(a-) in exactly the same man- ner as f{x) is derived from f{x), or by the operation twice performed ou f{x). This coefficient, denoted by fix), is called the second derived function. In a similar way the third derived function, f"'(x), is obtained from f"(x), and so on; so that the expression, B, may be written as follows : fix+h)=f{x)+f<{x)h+'fp^h'+f^li'+--' + a,h'^ . (C) EXAMPLE. Find the result of substituting x + h for x in the polynomial 5 x" - G .x-2 -f 8 .T + 4. Here f(x) = 5x'-6x'' + Sx + A, f'(x) = 15 x" - 12 .r 4- 8, /"(a;) = 30.^-12, /"'(x) = 30, and the result is 5.x.3_Ga;2+8a- + 4 + (15a;2_i2a;+8)/i + (30a;-12)^^-f30.p^. 81. Continuity of a Rational Integral Function. Theorem. If in a rationed and integnd function f{x) the value ofx be made to vary, by indefinitely small increments, from any quantity a to a greater quantity b, then loill f{x) at the same time vary also by indefinitely small increments; that is, f(x) varies continuously ivith X. . Art. 82 PROPERTIES OF POLYNO^fIALS. 103 Suppose X to increase from a to a + h. The correspouding increment of f{x) is /(« + /') -/(«), and, by Art. 80, this is equal to /'(a)/i+/"(a)^+...+ca", in which expression all the coefficients /'(a), /"(a), etc., are finite quantities. Now, by Art. 79, Cor. I, this latter expression may, by tak- ing h small enough, be made to assume a value less than any assigned quantity ; so that the difference between /(a + h) and /(a) may be made as small as we please, and will ultimately vanish with h. The same is evidently true during all stages of the variation of x from a to 6; thus the theorem is proved. We should, observe that it is not here proved that f(x) in- creases continuously from f(a) to f(b), but simply that it varies continuously, for it may sometimes increase and at other times decrease. 82. Form of the quotient and remainder tchen a polynomicd is dicided by a binomial. Divide c(qX" + a^x"'^ + a-pf'^ + ••• 4- f'n-i-i^ 4- «„ by ic — h, and let the quotient be 6o.x"-' + b^x"-^ + M""'' + ••• + b„_.p: + 6„_i. This we shall represent by Q, and the remainder by R. We have then f(x) = {x-h)Q-\-R. The meaning of this equation is, that when Q is multiplied hj X — h, and. R added, the result must be identiad, term for term, with./'(;c)- The rigM hand side of the identity is 104 THEORY OF EQUATIONS. Art. 8-2 + h. + R Equating the coefficients of x on both sides, we get the fol- lowing series of equations to determine h^, bi, b.,, ••• &„_i, B : h = Clo, hi = boh + ttj, 62 =bih-j- Clo, bs = bih + ttg, bn-l = bn_2ll -\- Cln_i, R = h„_-Ji + a„. These equations supply a ready method of calculating in succession the coefficients h^, b^, bo, etc., of the quotient, and the remainder R. For this purpose we write the series of opera- tions in the following manner : tt'jj Cfgj ^25 •*' ^'7i,_i) ^^'n? boh, bih, b.Ji, ••• hn_Ji b„_ih, bi &2 R In the first line are written down the successive coefficients of f(x). The first term in the second line is obtained by mul- tiplying tto (or 605 which is equal to it) by h. The product bji is placed under a^, and then added to it in order to obtain the term 6, in the third line. This term, thus obtained, is multi- plied in its turn by h, and placed under a,. The product is added to a., to obtain the second figure &2 in the third line. Tlie repetition of this process furnishes in succession all the coefficients of the quotient, the last figure thus obtained being the remainder. Tliis process, called Horner's Method of Syn- thetic Division, will be made plain by a few examples. Art. 82 PROPERTIES OF POLYNOMIALS. 10; The theorem of this article is known as the " Remaimler Theorem." EXAMPLES. 1. Find the quotient and remainder wlien 2x' + 4 x" - X- - 16 X - 12 is divided by a; + 4. Write the coefficients with — 4 at their right and proceed as below 2 4 -1-16 -12 [ - 8 16-60 304 -4 2 - 4 + 15 - 76 + 292 Thus the quotient is 2 or' — 4 x-^ + 15 x — 76, and the remainder is 292. 2. Find Q and E when 3 a;* - 27 x^ + 14 x + 120 is divided hyx~G. 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 calcula- tion is as follows : 3 0-27 14 120 [6 18 108 486 3000 3 + 18 + 81 + 500 + 3120 Hence Q = 3x^ + lSx^ + 81 x + 500, and E = 3120. 3. Divide x* - 4 x" - 8 .i- + 32 by x- 4. 1_4 0-8 32 [4 4 - 32 i 0-8 In this case, therefore, Q = x^ — 8 and 7? = 0, or the division is exact, and 4 is a root of the equation f{x) = 0. 4. Find Q and E, when ar' - 4 a;* + 7aT^- 11 a; - 13 is divided by x — 5. Aas. Q = x' + .r' + 12 .i-^ + 60 .c + 289 ; E = 14:52. 106 THEORY OF EQUATIoy^S. Art. 82 5. Find Q and E Avlieu x^ + 3 a;' — 15 x^ + 2 is divided by x-2. 6. Find Q and li wlien .x-' + o.-^ — 10.« + 113 is divided by a; + 4. 83. Tabulation of Functions. Horner's synthetic nietliod of division aifords a convenient practical method of calculating the numerical value of a polynomial, with numerical coefficients, when any number is substituted for x. For, since f{x) = {x-h)Q + R is an identical equation, it is satisfied by any value whatever of X. Let X = h, then f(Ji) = R, x — h being equal to zero, and Q remaining finite. Hence the result of substituting h for x in fix) is the remainder when f(x) is divided by x — h, and can be calculated rapidly by the method of the preceding article. For example, the result of substituting — 4 for x in the polynomial of Ex. 1, Art. 82, viz., 2x^ + 4: x" - ar - 16 x - 12, is 292, this being the remainder after division by x + 4. This can be verified by actual substitution. Again, the result of substituting 5 for x in x^-4x* + 7 a;3 - 11 x - 13 is 1432, as appears from Ex. 4, Art. 82. We saw in Art. 81 that as x receives a continuous series of values increasing from — co to + oc, f(x) Avill pass through a corresponding continuous series. If we substitute in succession for X, in a polynomial whose coefficients are given numbers, a series of numbers such as 5, - 4, - 3, -2,-1, 0, 1, 2, 3, 4, 5, •••, Art. 81 PROPERTIES OF POLYNOMIALS. 101 and calculate the corresponding values of f{x), the process may be called the tabulation of the function. EXAMPLES. 1. Tabulate the trinomial 2.r + values of x: _ 4, _ 3, _ 2, - 1, 0, 1, 2, 3, 4 Values of ic . . Values of f{x) . G for the following -4 -3 _ 2 -1 1 2 3 4 oo 9 -5 -6 -3 4 15 30 2. Tabulate the polynomial a;* — 4 .if' — 8 a; + 32 for the same values of x. 3. Tabulate af' - G or' + 11 x - 6. Values of x . Values of f{x) 84. Graphic Representation of a Polynomial. The values of f(x) corresponding to the different real values of x may be con- veniently exhibited to the eye by a graphic representation which we shall now explain. t -4 -3 — 2 -1 1 2 3 4 -210 -120 -GO -24 -G +6 Let two straight lines OX, OY (T\g. 3) cut one another at right angles, and be produced indehnitely in both directions. 108 THEORY OF EQUATIONS. Art. 81 These lines are called tlie x-axis and y-axis respectively. Lines, such as Oxi, measured on the a>axis, to the right of the y-a,xis, are regarded as positive; and those, such as OA', measured to the left, as negative. Lines parallel to YY', and above the ic-axis, such as ^P or B'Q', are positive; and those below XX', such as AS or A'P', are negative. The student of Trigonometry or Analytic Geometry is already acquainted Avith these conventions. 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'. Inf(x), give to x the value a and let OA = a ; calculate /(a) ; from A draw AP parallel to Y to represent /(a) in magnitude on the same scale as that on which OA represents a, and to represent by its position above or below the line XX' the sign of /(a). OB = b, and BQ =f{b), would determine another point Q. Thus, corresponding to the different values of x represented by OA, OB, OC, etc., we shall have a series of points P, Q, R, etc., which, when we suppose the series of values of x indefinitely increased so as to include all numbers between — co and + oo, 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 /(aj). The process here explained is also called tracing the function fix), and the curve itself is often called the cjraph of the function. In the practical application of this method it is well to begin by laying down the points on the curve corresponding to cer- tain small integral values of x, positive and negative. A curve drawn through these points will give at least a general idea of the character of the function. If we wish, at any particular locality, to examine the curve more minutely, we must take several intermediate fractional values of x, and, of course, the closer together such points are taken, the more accurately will the function be delineated. Art. 84 PROPERTIES OF POLiWOMIALS. 109 EXAMPLES. 1. Trace the trinomial — or — 2 j; + 4 ; that is, find its ffmjJi. The unit of length taken is one-fourth of the line OE in Fig. 4. The values of f(x) corresponding to integral values of x, within the limits of the figure, are as follows : Values of x, Values of /(a-). 4 I -3 4 +1 + 4 -1 + 5 + 4 + 1 + 1 + 2 -4 By means of these values we obtain the positions of seven points on the curve, A, B, C, D, E, F, G. This done, we draw as smooth a curve as we can througli these points, whicli curve is the required yraph. 110 THEORY OF EQUATIONS. Art. 84 2. Trace the polynomial 10x^ — nx- + x + 6. Tabulating the pol^'nomial, we have Values of x, —3 _2 -1 + 1 + 2 + 3 Values of /(.i-), -420 -144 — 22 6 20 12G We have found, Art. 78, that this function retains positive values for all positive values of x greater than 2.7, and nega- tive values for all values of x nearer to — co than — 2.7. The graph 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 between — 2.7 and + 2.7 ; so, if we wish simply to examine the position of the roots of the equation f(x) = 0, the tabula- tion may be confined to the interval between — 2.7 and + 2.7. This is a case in which the substitution of integral values only of X gives little help toward the tracing of the curve, and where, consequently, smaller intervals have to be examined. It would be well to tabulate the function for intervals of one-tenth between the integers — 1, ; 0, 1 ; 1, 2, This tabulation and the tracing of the curve is left as an exercise for the student. 3. Trace the trinomial 2 x^ + x — 6. 4. Trace the polynomial x* — 15 a;^ + 10 x + 24. The graph in Ex. 1 cuts the axis of x in two points (a num- ber 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 — a^ — 2a;-l-4 = 0. It will be found that the graph of the polynomial in Ex. 4 cuts the axis of x in four points, corre- sponding to the roots of the equation x' - 15 .^•2 + 10 .V + 24 = 0, viz. - 4, - 1, 2, 3. The graph of a given polynomial may not cut the axis of x at all, or may cut it in a number of points less than the degree Art. 84 PROPERTIES OF POLYNOMIALS. Ill of the polynomial. Such cases corresi)uiul to the imaginary roots of equations, as will appear more fully in a subsequent chapter. For example, the graph of the polynomial 2x'+x-\-2 will be found to lie entirely above the axis of x. It is evident, by the solution of the equation 2 jt + cc + 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 x falls short of the degree of the poly- nomial, it is customary to speak of the curve as cuttinrj the line in imaginary points. ^^0.^^ CHAPTER VI. GENERAL PROPERTIES OF EQUATIONS. 85. We shall first prove some theorems which establish the existence of a real root iu an equation in certain cases. TiiKOKEM. If two real numbers substituted for x in a rational integral expression f(x) give results ivith contrary signs, one root at least of the equation f(x) = lies between those values of x. Let a and b denote the two numbers ; then /(«) and /(&) have contrary signs. By Art. 81, as x changes gradually from a to b, the expression f(x) passes without any interruption of value from /(a) to f(J)) ; but since /(a) and/(&) are of contrary signs, the value zero lies between them, so that f(x) must be equal to zero for some value of x between a and b ; that is, there is a root of the equation f(x) = between a and b. We do not say that there is only one root; and w^e do not say that if f(a) and f(b) are of the same sign there will be no root of the equation f(x') = betAveen a and b. Reference to the graphic method of representation will assist our conception of this theorem, and will enable us to make it more general. It is evident that if there exist two points of the graph of f(x) on opposite sides of the axis XX\ then the curve between these points must cut that axis an odd number of times, and if the two points are on the same side of the axis, the curve must cut that axis either not at all or an even number of times ; thus several values may exist between a and b for which f{x) = 0, that is, for which the graph cuts the axis. For example, in Ex. 2, Art. 84, x = — l gives a negative value (— 22), and x = + 2 gives a positive value (20), and 112 Art. S« GEN Eli AL rRoPEHTIES OF EQUATlnSS. 113 between these points of the curve there exist t/irei' points of section with the .r-axis, as can be easily shown. 86. Theorem. Every equation of an odd degree //a.s at le<(nt one real root of a sign opposite to that of its last term. This is evident at once from the theorem of the last article. Substitiite in succession — oc, 0, oo for x in the polynomial f{x). The results are, )i being odd (see Art. 78), for X = — cc, f(x) is negative ; for X = 0, sign of /(.r) is the same as that of «„; for X = + oc, f{x) is positive. If a„ is positive, the equation must have a real root between — 00 and 0, i.e. a real negative root ; and if o„ is negative, the equation must have a real root between and cc, i.e. a real positive root. The theorem is therefore proved. 87. Theorem. Every equation of an even degree, trhose last term is negative, has at least two real roots, one jjositive and the other negative. The results of substituting — oo, 0, oo are in this case -cc, +, 0, -, +^, +; hence there is a real root between — oo and 0, and another between and + <» ; i.e. there exist at least one real negative and one real positive root. 88. To prevent mistakes, it is well to call attention to exactly what has been proved in the last two articles. In Art. 86 it is proved that the equation considered has at least one real root: it is not i)roved that it h:is only one. In Art. 87 it is proved that the etpiation considered lias at li-a.^t two real roots ; it is not proved that it has only two. 114 THEORY OF EQUATIONS. Art. 89. Existence of a Root. Imaginary Roots. We have now proved the existence of a real root in the case of every equa- tion, except one of an even degree whose last term is positive. Such an equation may have no real root at all. We must then examine whether there may not be cases where the equa- tion has imaginary roots, or whether there may not be in cer- tain cases both real and imaginary values of the variable which satisfy the equation. In Chapter IV we have assumed that such is the case. Let us take a simple example by way of illustration. In Art. 84 we have seen that the graph of the polynomial f(x) ~2x'-\-x + 2 lies entirely above the axis of x, as in Fig. 5. The equation f(^x) = has no real roots ; but it has the two imaginary roots as is evident by the solution of the quadratic We observe, therefore, though there are no real roots, there are in this case two imaginary expres- sions which reduce the polynomial to zero. The corresponding general propo- sition is that every rational integral equation has a root, real or iniagi- iiary. Such a root has the general form a and /3 being real finite quantities. This form includes both real and imaginary roots, the former corre- sponding to the value /3 = 0. Figr. 5. Art 90 GENERAL PROPEllTIES OF EqUATIoSS. Ho The proof of this fundamental theorem, involving principles too intricate to be introduced in an elementary treatise, will not be given, and we shall simply assume the i)ropo.siti()ti as true, referring the student for the proof to Hurnside and Pan ton's Theory of Equations, or Serret's Cours il'AJtjNne Siiperieure, or any advanced work on the subject.* 90. Every Equation of the nth Degree has n Roots and No More. It is evident from Art. ^o that if any nunibor h is a root of the equation f{x) = 0, then f{x) is divisible by x — h without a remainder ; for if /(/t) = 0, i.e. if h is a root of /(x) = 0, K must = 0. Let the given equation be /(.r) = .x>" + /JiX"-' +;ur"-- + ••• +p„.iX+p„ = 0. This equation must have a root, real or imaginary (Art. 80), which we shall denote by «i. Let the quotient, when f(x) is divided by x—a^, be <^i(.i'); we have then tlie identical ecpiation /(.r) = (.c-«,)<^,(.r). Again, the equation <^,(.i') = 0, which is of the (?i — l)th degree, must have a root, which we represent by rc^. Let the quotient obtained by dividing (f)i(x) hy x— a^ be <^2(-''*)- Hence <^,(.r) = (x - «,,) <^.(.r), and .-. fix) = (x - «,) (x - «.,) .^v), where 2{x) is of the (u — 2) th degree. Proceeding in this way, we prove that /(.r) consists of the product of n factors, each containing x in the tirst degree, and a numerical factor <^„(a*)- If, in the identical equation f(x) = (:x- «,) (x - a,) ••• (x - «„) „i.v), • See also Fine's XnmberSijHtfm <>/ Aijehni, Artt. .V.>-,'4. 116 THEORY OF EQUATIONS. Art. 90 we compare the coefficients of a*", it is plain that <^„(.t) = 1. Thus we prove the identical equation fix) = (x - «,) (x — «2) (x — «3) ••• (x - «„_i) (x - a„). It is evident that the substitution of any one of the numbers «!, «2 ••• «„ for X in the right-hand member of this equation will reduce that member to zero, and will, consequently, reduce f(x) to zero ; that is, the equation f(x) — has for roots the n quantities «j, «2, «3 ••• a„_i, «„. And it can have no other roots ; for if any number other than one of the numbers «i, «2, «3 ••• «„ be substituted in the right-hand member of the above equation, the factors will all be different from zero, and, therefore, the product cannot vanish. This theorem, while of no assistance in the solution of the equation f(x) = 0, enables us to solve the converse problem ; that is, 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 subtracting from x each of the given roots. It follows also from the present theorem that, when any (one or more) of the roots of a given equation are known, we can obtain the equation containing the remaining roots by dividing the given equation by the given binomial factor or factors. The quotient will be the required polynomial com- posed of the remaining factors. EXAMPLES. 1. Find the equation whose roots are 2, -1, -4, -f 3. Ans. x* -15x' + 10x + 24: = 0. 2. Two of the roots of the equation X* - 5 x^ -ISx"" + 53x + m = are —3, +4; find the other roots. Use the method of division of Art. 82. Art. 91 GENERAL PROPERTIES OF EQirATlONS. 117 3. Find the eciuatioii whose roots are -2, 0, +1, +5. 4. In the equation x'-Sa^-lGx + iS, one root is — 4 ; find the other roots. 5. Solve the equation X* - IG x^ + 86 .r' - 176 x + 105 = 0, two roots being 1 and 7. Ans. Other roots 8, o. 6. Form the equation whose roots are -|, 2, +4. Ans. 16x^ + 37x^ + 12x--i = 0. 7. Solve the equation ^.4_4a-3_8.f + 32 = 0, two roots being — 1 + V— 3, — 1 — V— 3. 8. Solve the cubic equation x' -1 = 0. Here it is evident that x = 1 satisfies the equation. Divide by X — 1, and solve the resulting quadratic to get the other two roots. 9. Solve the cubic equation ar* + 1 = 0. 91. Equal Roots. It is evident that the n factors of which a })()lyiioniiaI f(x) consists need not be all different from one another. The factor x — u, for example, may occur in the second or any higher powder not superior to n. In this ca.so tw^o or more of the n roots of f(x) are equal to one another, and the root a is called a multiple root of the equation, — double, triple, etc., according to the number of times the factor is repeated. 118 THEORY OF EQUATIONS. Art. 91 Equal roots form the connecting link between real and imaginary roots. A reference to the graphic construction (Art. 84) will make this plain. Or, returning to the equation given in Art. 49, we know that the two roots of the equation ax^ + bx + c = 0, are real, if ¥ > 4 ac, equal, if 6^ = 4 ac, and imaginary, if Z>- < 4 ac. 92. Theorem. Li an equation ivith real coefficients, complex roots occur in pairs. Let f(x) be a rational, integral functi on of x in which the coefficients are all real; then if a + ^V— 1 is a root of the equation f(x) = 0, a — ftV^^ will also be a root. For when a + ^V— 1 is put for x, the function f{x) takes the form P + Q/3^/—l, where Pand Q involve even powe rs of ^. Now as the coefficients in f{x) are supposed real, V— 1 can- not occur except with some odd power of /3. If then a— ySV — 1 be substituted for x in f(x), the result will be obtained by changing the sign of ^ in the result obtained by substituting « _(_ /3V^^ for x; the result is therefore P — Q^V^^. (Art. 61, Cor. 2.) Now if a + /3V— 1 is a root of f{x) = 0, then and, therefore, Art. 57, since (3 is not zero, P=0, and Q = 0. Hence P-QJ^ V^l = 0, and a — I3\ — _ is also a root of f(x) = 0. Thus the total number of imaginary roots in an equation with real coefficients is always even. Note. A proof exactly similar to that above given shows that surd roots, of the form a ± Vy, enter in pairs equations ivhose coefficients are rational. Art. 93 GENERAL PROPERTIES OF EQUATIOXS. 110 EXAM PLES, 1. Form a rational cubic equation which shall liave for two of its roots 1, 3-2V^ri. 2. Form a rational ecjuation which shall have for two of its roots Ans. X* - 12 x" + 72 x' - 312 x + G7G = 0. 3. Solve the equation X* - x^ - 8 .r^ + 8 = 0, which has a root 1 + VS. 4. Solve the equation 2^? — o(? — Q>x-{- 77, one root being 2 + V— 7. Ans. 2 ± V— 7, — |. 93. Descartes' Rule of Signs. This celebrated theorem of Descartes* establishes an interesting and useful relation between the number of changes of sign of the first member of an equation, f(x) = 0, and the number of real roots, and, thereby, enables us to find a superior limit to the number of positive and negative real roots of an equation. Definition. "When each term of a set of terms has one of the signs + or — before it, then in considering the terms in order, a contimmtion is said to occur when a sign is the same as the immediately preceding sign, and a change^ is said to occur when a sign is contrary to tlie immediately preceding sign. Thus in the expression a.-8 _ 2 .1^ - 3 .f« + 4 or' + x^ + 2 .1-^ - 3 jr^ - x + 1 * Rene Descartes (l.'>!)f»-ir..50). + Instead of "continuation" and "change" the terms permanenct and rariation aro often used. 120 THEORY OF EQUATIONS. Art. 93 there are four continuations and four changes. It is obvious that in any comi)lete equation the number of continuations to- gether with the number of changes is equal to the number which expresses the degree of the equation. If in any com- plete equation we put - x for x, the continuations and changes in the original equation become respectively changes and con- tinuations in the new equation. (a) Positive Roots. Theorem. No equation can have more positive real roots than it has changes of sign from + to —, and from — to +, in the terms of its first member. Let the signs of a polynomial taken at random succeed each other in the following order : -f + - + -- + + + - + - + In this there are in all eight changes of sign. It is proposed to show that if this polynomial be multiplied by a binomial whose signs, corresponding to a positive root, are -\ — , the resulting polynomial will have at least one more change of sign than the original. Writing down only the signs that occur in the operation, we have + - -- + - + + + - + +±-+-T+±±-+-+- Here, in the result, the ambiguous sign ± is placed wher- ever there are two terms with different signs to be added. We readily see that in this case, and in any other arrangement, the effect of the process is to introduce the ambiguous^ sign wherever the sign + follows -f , or — follow^s — , in tlie orig- inal polynomial. The number of variations of sign is never diminished, and there is always one variation added at the Art. 93 GEXERAL PliOPKliTIES OF EQUATIONS. llil end. By trying different arrangements of signs, it is easy to convince ourselves that, in even the most unfavorable case — that, namely, in "svhich the continuations of sign in the original remain continuatious in the resulting polynomial, — there is one variation added. We may conclude in general that the effect of the multiplication of a polynomial by a binomial x — a is to introduce at least one change of sign. Now suppose we have 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 — «, x — (3, x — y, etc., corresponding to the positive roots «, /3, y, etc., is to introduce at least one change of sign for each ; so that when the complete product is formed containing all the roots, we conclude that the resulting poly- nomial has not more positive roots than there are changes of sign. (6) Negative Roots. Theorkm. Xo equation can have a greater ninnber of necja- tive roots than there are changes of sign in the terms of the yolij- nomial f(—x). Now, if — X be substituted for a; in the equation f(x) = 0, the resiilting equation will have the same roots as the original, except that their signs will be changed; for, from the identical equation /(.r) = (.f - «i) (:c - «.,) (x - «,) ••• (.r - «„), we derive /(- x) = {- 1)" (x + «,) (x + «-.) (x + Us) ■■'(x + «„). From this it is evident that the roots of /(— x) = are — «i, — a-,, — «3, ••• — u„. Hence the negative roots oif(x) are positive roots of /(— x), and our theorem for negative roots is true. 122 THEORY OF EqUATIONS. Art. 98 EXAMPLES. 1. If the coefficients in /(.«) are all positive, the equation f{x) = has no positive root. 2. If the coefficients in any complete equation be alter- nately positive and. negative, the equation cannot have a nega- tive root. 3. If an equation consist of a number of terms, whose coefficients are positive followed by a number of terms whose coefficients are negative, it has one positive root and no more. Apply Art. 85 and Art. 93. 4. If an equation contain only even powers of x, and if all the coefficients have positive signs, it cannot have a real root. 5. If an equation contain only odd powers of x, and if all the coefficients have positive signs, it has the root zero and no other real root. 6. Find an inferior limit to the number of imaginary roots of the equation x" - 3 .^•2 - a; + 1 = 0. Here, Art. 93, the arrangement of signs for f(x) = + + exhibits two changes of signs, hence there cannot be more than two positive roots ; and, examining the arrangement for /(— a;) = 0, -f- — + -f, we find again two changes of sign, so there cannot be more than two negative roots. As there are six roots in all, it follows that there must be at least two imaginary roots. 7. Find an inferior limit to the n\imber of imaginary roots of the equation x^ + 3x'-{-4x'+2x-6 = 0. At least four imaginary roots. Art. 98 GENERAL PROPEIiTIES OF EQUATIONS. l'2:\ 8. Find the nature of the roots of the equation x' + 15 x" + 7 a; - 11 = 0. Ans. One positive, 1 negative, 2 imaginary. 9. Show that the equation x^ + qx 4- ?• = 0, where q and r are essentially positive, has one negative and two imaginary roots. 10. Find the nature of the roots of the equation x^ — qx + ?• = 0. 11. Show tliat the equation X" -1 = has, when n is even, two real roots, — 1 and + 1, and no otlier real root ; and, when n is odd, the real root 1, and no other real root. 12. Show that the equation X" + 1 = has, when n is even, no real root ; and, when n is odd, the real root — 1, and no other real root. CHAPTER VII. RELATIONS BETWEEN ROOTS AND COEFFICIENTS.— SYMMETRIC FUNCTIONS. 94. Relations between the roots and coefficients of an equation. Kepresenting the n roots of the equation X" + Pi.T"-! + 2hx"-^ -\ 1- Pn-iX +Pn ' ' ' (1) by «i, «2, «3, ••• «„, Ave have the identity Ct-" + piX"-^ + j9oX"-- H h 2)n-lX + Pn = (x-a{)(x — a.,)(^x-a.;)--'(x — a„) ... (2) When the factors of the second member of this identity are multiplied together, the highest power of x in the product is X", and the coefficient of this term is unity. The coefficient of the second term, a;""^, is — a^ — a.^— a^— a^--- — a„; that is, the sum of the roots with their signs changed ; the coefficient of x""'^ is the sum of the products of the roots taken two and two ; the coefficient of x"'^ is the sum of the products of the roots taken three at a time, with their signs changed ; and so on, the last term being the product of all the roots Avith their signs changed. Therefore, equating coefficients of like powers of X on each side of the identity (2), we have 2h = - («i + «, + «3 + ••• + a,,) ] ^2 = («i«2+ «i«.i+ •••) I 2h = — (filfij(^ + «l'<.;«4 + •••) J 24 (3) Art. 95 ROOTS AND COEFFICIENTS. 125 These results give us the following relations between the roots and coefficients : In every ahjebndc equation, the coefficient ofichose highest term is unity, the coefficient jJi of the second term, icith its sign changed, is equal to the sum of the roots. The coefficient jh of the third term is equal to the sum of the products of the roots taken two by two. The coefficient 2h of the fourth term, loith its sign changed, is equal to the sum of the jyroducts of the roots taken three by three, and so on, the signs of the coefficients being alternately negative and positive, till finally that function is reached which consists of the product of the n roots. When the coefficient Uq of x" is not unity (Art. 51), we must divide each term of the equation by it. Cor. I. Every root of an equation is a divisor, whole or fractional, of the absolute term of the equation. Cor. II. If the roots of an equation be all positive, the coefficients (including that of the highest power of x) will be alternately positive and negative ; and if the roots be all negative, the coefficients will be all positive. 95. It might perhaps be supposed that the relations given in the preceding article Avould enable us to find by elimination the roots of any proposed equation; for they furnish equa- tions involving the roots, and the number of these equations is the same as the number of the roots. But this is not the case, for, on attempting this elimination, we merely reproduce the j)roposed equation itself, as the following example will show : Let «, p, y be the roots of the cubic equation ar^+iV^'' + i>L'^- + ^'3 = 0) We have, by Art. 94, p,^-(a + f3 + y), p, = (c(3 + ay + (3y. Pi = - «^y- 126 THEORY OF EQUATIONS. Art. 95 Multiplying the ftrst of these equations by a^, the second by a, and adding the three, we find or rr + jh"-' + i^2« + Ih = 0> which is the given cubic with a in the place of x, and, there- fore, we are no nearer the solution of (1) than v/e Avere at first. Thus, although the equations (3) afford no aid in the general solution of the equation, they are often useful in facilitating the solution of nun^erical equations when any particular rela- tions among the roots are known to exist, as will be made apparent by the following examples. EXAMPLES, 1. Solve the equation two of its roots being equal. Let a, a, /3 be the three roots. We have 2a + (3 = 3, «2 + 2 «/3 = 0, from which we find a = 2, |8 = — 1. The roots are 2, 2, — 1. 2. Solve the equation a,-3_ 5.^2 _ 16a; + 80 = 0, the sum of two of its roots being zero. Let the roots be «, jS, y. We have then « + /8 + 7= 5, a{3 + ay + (3y = - 16, (i/3y = - 80. Taking ^ + y = 0, we get « = 5, /S = 4, y = - 4. Thus the three roots are 5, 4, — 4. Art. 95 ROOTS AND COEFFICIENTS. V21 3. The equation x' _ 4 x" - 12 .r-' + 32 a; + 64 = has two pairs of equal roots ; find them. 4. Solve the equation x'-9x' + Ux + 2-1 = 0, two of whose roots are in the ratio of 3 to 2. Let the roots be a, /?, y, with the relation 2a = 2 ^. Ans. The roots are 6, 4, — 1. 5. Solve the equation X* -f 2 ar' - 21 .x-2 - 22 a; 4- 40 = 0, whose roots are in arithmetical progression. Assume for the roots a — 3 8, a — 8, a + 8, a + 3 8. 6. Solve the equation 8 .K-* - 30 •r' + 35a^ - 15 x + 2 = 0, whose roots are in geometrical progression. Assume for the roots -, -, up, ap\ Ans. \, \, 1, 2. P P 7. Solve the equation x^-3x^-x + 3 = 0, whose roots are in arithmetical progression. 8. Solve the equation 24.r^-2Grr + 9.r-l = 0, whose roots are in harmonic progression. Ans. ^, J, }. 9. Solve the equation X* + 15 x" + 70 ar + 120 x + CA = 0, whose roots are in geometric progression. 10. The equation 3 X* - 25 ar' + 50 x^ _ r,o x + 12 = has two roots whose product is 2 ; find all the roots. 128 THEORY OF EQUATIONS. Art. 9G 96. Derived Functions. In order to examine an equation for equal roots, it will be found convenient to express the derived functions (Art. 80) in another form. Let the roots of the equation f(x) = be «i, «2) «35 • • • ««• We have /(:^•) = (.« - «i) (x - «2) (x - «3) ■•■(x- «„). In this identical equation substitute h + x for x: ■X (h + x) = (h -\- X — «i) (h + x — a^ • • • (h -\- x — «„) ~" = A" + gi/i"-i + g2^t''-2 -I \- q„_ih + g,„ where gi = ;^- — «i + a; — «2 + -t" — «3 + h S'' — «„, g, = (x - «i) (.^• - «2) + (-c - «i) (x- - «3) H \-(x — «„_,) (.r - «„), g„_i = (.^• — «,) (x — fta) •••(.« — «„) + (ic — «i) (a; — «3) • • • (.r — «„) +" • . • + (a; - tti) (.T - «2) • • • (it- - «„_i), g„ = (x — ai)(x — a^) (x — Wg) ••• (x — «„). Also we have, by Art. 80, f(h + a-) =/(.r) +./"(.i-)7i +-y~^/r + - + /i". Equating the two expressions for f(h + a:), we obtain f(x) = (w - «i) (x — «2) • • • (a; — «„), f'(x) = (x — a.,) (x — (is) ••• (x — «„) + •••, as above written, f"(x) ^ — rr — tlie similar value of g„_2 in terms of a; and the roots. The value of/'(.c) may be conveniently written as follows: n.)=i<±+i(-'L+...+i(^.. X — Ui X — a.2 x — a„ 0^ Alt. 98 ROOTS AND CO EFFICIENTS. \-l\) 97. Multiple Roots. TiiEouKM. .1 multiple root of the order p of the etjiiation f{x) = is a multiple root of the order p — 1 of the first dirired equation f (x) = 0. This follows at once from the expression given for f (x) in the i)receding article; for, if the factor {x — u^Y occnrs in fix), that is, if «i = Uo = • • • = «^,, we have Each term of this will still have {x — «, p as a factor, ex- cept the first, Avhich will have (.r — «,)''-' as a factor ; lience {x — «i)^~^ is a factor in/'(a'). Cor. I. Any root which occnrs p times in the eqnation f(x) = occurs in degrees of multiplicity diminishing by unity in the first p — 1 derived equations. Since /"(.t;) is derived from f'(x) in the same manner as f'{x) is from f(x), it is evident by the above theorem that f"{x) Avill contain {x — a^Y'- as a factor. Tlie next derived function, /'"(ic), will contain (x — UiY'^; and so on. 98. Determination of Multiple Roots. From the preceding article it is obvious that if f(x) antl f'(x) have a common factor (x — «/"', (x — «V will be a factor inf{x); hence a is a root of f(x) of multiplicity p. In the same way, it appears that if f(x) and/'(j:) liave other com- mon factors (x — /3)'~^ (x — y)*""', {x — 8)' ', etc., the efpiation /(.c) =. will have q roots equal to /?, r roots equal to y, .s i-oots equal to 8, etc. Hence, in order to examine an equation /"(.r) for e(|ual roots and to determine these roots, if such exist, we must find the highest common factor of /(.r) and./*'(.r). Let this ho F,(.r)=0. The solution of Fi{x) — will give the equal roots. 130 THEOBY OF EQUATIONS. Art. 98 EXAMPLES. 1. Find the multiple roots of tlie equation x" - 7 .r + 16 X - 12 = 0. Here the H. C. Y. of f(.v) = or' - 7 x- + 16x- 12, and /' (.r) = 3 X- - 14 X + 16 is .i; - 2 ; hence {x - 2)^ is a factor in /(.«). The other factor is x — 3, hence the roots of the equation are 2, 2, 3. Whenever, after determining the multiple factors of f(x), we wish to get the remaining factors, it will be convenient to apply Horner's method of division (Art. 82). In this example we would divide twice by ^ - 2, the calculation being repre- sented as follows 1 -7 +16 -12 2 - 10 +12 1-5 6 9 _ A 1-3 Thus 1 and — 3 being the two coefficients left, the third fac- tor is X — 3. This operation verifies the previous result, the remainders after each division vanishing as they ought. 2. Find the multiple roots, and the remaining factor of the equation a,-5 _ 10 :v- + 15 a; - 6 = 0. The H. C. F. of f(x) and /' (x) is x''-2 x + 1. Hence (;x-lf is a factor in f(x). Dividing three times in succession by x — 1, we obtain f(x) = (x-iy(x' + 3x-\-6). Find the multiple roots of the following equations : 3. o!^ + x'-16x + 20 = 0. 4. x*-2 x" - 11 a;^ + 12 x + 36 = 0. Art. 99 ROOTS Ay D VOEFFICIENTS. 1:}1 5. x^-llar + 18;«-8 = 0. 6. X* - 11 af* + 44 X- - 76 x + 48 = 0. A„s. f(x) = (x - 2y(x -3){x- 4). 7. 2x*-12x' + ldx--ex + 9 = 0. Ans. The roots are 3, 3, + jV— -, — .VV— 2. 8. Show that the binomial equation x" —p" — cannot have equal roots. 9. Apply the method of Art. 08 to determine the condition that the cubic .r' -f 3 ZTx- + (^ = should have a pair of equal roots. Ans. C^ + 4 H^ = 0. The ordinary process of finding the H. C. F. of /(a*) and/'(x') may often become very laborious. It is chiefly in connection with Sturm's theorem (Art. 118) that the operation is of any practical value. Multiple roots of equations of degrees inferior to the sixth can be determined more readily by trial. 99. Theorem. — In jyassing continnonsJ}/ from a value a — // of X a little less than a real root a of the erpiation f(x) :=0 to a valne a + h a little greater, the pob/iwmials f(x) and f'(x) hare unlilce sights immediately before the jmssage through the roof, and like signs immediateli/ after. Substituting a — h in /(.r) and ,/''(•'•)> ^"<^ expanding, we have /(« - h) =/(«) -f'(u)h +-^~'4fr - ..., f'(a-h)= f\u) -f"(u)h +-. Now, since ./(«) = ^i the signs of these expressions, dejiend ing on those of their first terms, are unlike. Wliou the .sign of h is changed, tlie signs of /(«< + h) and /'(« + h) are like. Hence the theorem. 132 THEORY UF EQUATIONS. Art. 100 100. The Cube Roots of Unity. Equations of the forms x" — p = 0, .t" -|- j; = 0, are called hinomial equations. We shall see later that such equations are intimately connected with the more special forms a;" - 1 = 0, X" + 1 = 0, the roots of the first of which are called the n nth roots of unity. We shall here consider the simple case of the binomial cubic. We have seen (Ex. 1, Art. 73) that the roots of the cubic ar' -1 = are 1, -i + |V^^, - i - i V^=^. (See also Ex. 8, Art. 90.) If either of the imaginary roots be represented by w, the other is easily seen to be 7) (« + W-/3 + wy) is rational. 3. Form the equation whose roots are m + ?j, mm 4- -h, (ji'ni -\- 0)71. 101. Symmetric Functions of the Roots. Symmetric func- tions of the roots of an equation are those which are not altered if any two of the roots be interchanged. For example, if «, /8, y are the roots of a cubic equation, « + ^ + y, «)8 + «y + j8y, afiy are symmetric functions, for all the roots are involved alike. The functions p„ p^, 2^3, etc., of Equation 3, 134 THEOBY. OF EQUATIONS. Art. 101 Art. 94, are the simplest symmetric functions of the roots, each root entering in the first degree only in any one of them. We can often, as shown by some examples appended to this article, obtain the values of a great variety of symmetric functions in terms of the coefficients of the equation whose roots we are considering. A symmetric function is usually represented by the Greek letter 2 attached to one term of it, from which, by analogy, the entire expression may be written down. Thus, in the case of a cubic, whose roots are «, (3, y, where all possible products in pairs are taken, and all the terms added after each is separately squared. Again, 2«-/3 = a-/3 + ay + (By + /?-« + y-« + y'(3, where all possible permutations of the roots, two by two, are taken, and the first root in each term then squared. In the case of a biquadratic, we have 2«^/3- = «'/32 + ay + a^W + fiY + [B-h- + y-8l We give a few examples, which may serve to give the stu- dent some insight into the formation of this class of functions. EXAMPLES. 1. Find the value of 2«"/3 of the roots of the cubic equation or' + px^ -\- qx -\- r = 0. Multiplying together the equations ySy 4- y« + a(i = q, we obtain ^a-(3 + 3 «/3y = - 2)q ; hence 2«-/3 = 3 r - ^)g. Alt. 101 BOOTS AND COEFFICIENTS. 135 2. Find for the same cubic the value of a^ + fi^ + y-. Ans. 2«- = p^ — 2 q. 3. Find for the same cubic the value of Multiplying the values of 2« and 2«-, we obtain «3 _^ ^3 + ^3 _^ 2(r/8 = -i/ + 2iJ5; hence, by Ex. 1, Sk"* = — j>^ + 3 jxj — 3 >•. 4. Find for the same cubic the value of 2«-)8-. 5. If a, (3, y, 8 are the roots of the biquadratic equation X* +ixr'' + (J.,^ + r.v + s = 0, find the value of the symmetric; function 2«2y8y = «-/8y + (ciS^ + «'yS + l^'uy + /3'-'«8 + /3-yS + y-u(3 + y-(<8 4- y'/5S -f 8'«/3 + 8-«y + S-^y. Multiplying together a + (3 + y + 8 = -p, apy + a(3B + «y8 + /8y8 = - r, we obtain 2«-^y + -1 «/3y8 = i)r ; hence 2a^)8y = i^r - 4 .„ = f>- Ans. la-" = ]>r - 2 p.. CHAPTER VIII. TRANSFORMATION OF EQUATIONS. In many cases the discussion and solution of an equation is facilitated by some algebraic transformation that will change it into a form more convenient for investigation. We shall now consider some simple and useful cases of transformation. 102. To transform an Equation into Another, the Roots of which are those of the Proposed Equation with Contrary Sign. Let «i, u.,, Us •■• "« l^Je the roots of the equation .r" + i>i;«'*-i + i^-C'-^H +})» = 0. We have then the identity X" + pi.r"-^ + 2)-2X''-- + . • • + p„ = (.^ - «i) (X - Uo) (x -Us)--- (x - «„). Changing x into — y, we have, whether n be even or odd, 2/" - 2hU"~' + i\'2/""' T Pn-iy ± P^ = (y + «i) (y + «2) (y + «3) •••(?/ + «„) = o. The roots of the last equation are — «i, — Uo, — a^--- — «„, and thus the transformed equation may be obtained from the given equation by changing the sign of the coefficient of every other term beginning icith the second. In applying this rule to an equation that is not complete, we must first supply the missing terms by writing them down, each in its proper place with zero for a coefficient. 136 Art. 103 TRANSFORMATIOX OF EQUATIONS. 137 EXAMPLES. 1. Find the equation whose roots are the roots of a;*' — 4 X' + 3x^ + x^ + 7 jr + 2x-i-5 = witli their signs changed. 2. Change the signs of the roots of the equation cc» + 2 a;« + 4 x* + x' + o x- + G = 0. Ans. x^ + 2x^ + 4:X* — x^ + ox^ + (j = 0. 103. To transform an equation into another, the roots of ichich are equal to those of the proposed equation multiplied by a r/icen quantity. Let «i, a.2, «3, ••• «„ be the roots of an equation f(x) = 0, and let it be required to transform the proposed equation into another, the roots of which shall be ka^, ka.^, ku^, ••• ka„. Assume x = ■-, and substitute in the identity of the preced- Ic ing article. After multiplying by k'\ we have ?/" + A-/)!^"-! + k^p--y"~- + ••• + l^"~hh,-^I/ + ^•"i>n = (M-ku,)(y-ka.;)-'(>/-ku„). Hence, to multiply the roots of an equation by a given quantity k, we have only to multiply the successive coefficients, beginning ivith the second, by k, k^, k"^, ••• k". Any missing power of x must be written with zero as its coefficient before the rule is applied. This transformation is very useful for removing the coeffi- cient of the first term when it is not unity, and, in general, for removing any fractional coefficients. When thi-re are frac- tional coefficients, we get rid of them by using a multiplier k which may be determined by inspection. 138 THEORY OF EQUATIONS. Art. 103 EXAMPLES. 1. Change the equation 2 x^ - 3 .v" + 5 or^ - 4 X + 6 = into another the coefficient of whose highest term will be unity. We jnultiply the roots by 2. Ans. x'-3 x" + 3 .ar - 16 x + 48 = 0. 2. IMake a similar transformation for the equation 3 x^ + a;^ - 5 ic^ + 2 ar - 7 a; + 5 = 0. 3. Kemove the fractional coefficients from the equation x'-\x'' + ^x-l = Q. Here we multiply the roots by 6, thus a.-3-i(6)x^ + |(r,)-\'«-(6>^=:0. Am. x^ - 3 ;«- + 24 a; - 21 6 = 0. 4. Remove the fractional coefficients from the equation supply missing term, and use 10 as a multiplier. Ans. X* + 30 a;2 + 520 a; + 770 = 0. Eemove the fractional coefficients from the following equations : 5. a;* - y2_ x- + ^x + l = 0. 6. cc3-|a;2 + |a;-| = 0. 7. a-3-ix^-J^.T + ^i3=0. - 8. x'-\x^-^lx'+\x--,l,^^0. Art. 105 TRANSFOliMAriOX OF EQUATIOyS. 131> 104. To Transform an Equation into Another the Roots of which are the Reciprocals of the Roots of the Proposed Equation. Here we substitute - fur x in the identity of Article 1(12. !f Making this substitution and reducing, we have i Pi P> Pn 1 yn yn-l yU . ^ ^ " Hence, if, in the given equation, we replace x by - and naul- y tiply by y", the resulting equation will have for roots the reciprocals of «i, Uo, ••• «„. EXAMPLES. Find the equations Avhose roots are the reciprocals of the roots of Alls. 2y*-5f-7y^ + 3y-l = 0. 2. x'-T x" + 4 .r- - 7 a; + 2 = 0. 3. x« - 5 x' - or' + 5 x2 + 7 X + 10 = 0. 4. a^-'Sar-6 = 0. 105. Infinite Roots. If j\ = 0, one root of /(x) = is zero, and, therefore, by Art. 104, the corresponding root of ^Q-'^J 140 THEORY OF EQUATIONS. Art. 105 That is, if in an equation the coefficient of a;" (the highest poiver of x) is 0, one root is infinity. Thus, one root of the equation {m - n) X'- 3 mx- + 2 a; - 10 = is infinite, if m = n. In like manner, if the coefficients ofx" and o;""^ are both 0, two roots are infinity, and so on. 106. Reciprocal Equations. Reciproccd or recurring equations are tliose which remain unaltered, when x is changed into its reciprocal. The conditions that must hold among the coefficients of an equation in order that it should, belong to this class are, by Art. 104, as follows : l)n-l Pn-2 Pi 1 — =p„—=p, ...-=„„_„ --=j,.. The last of these conditions gives p^ = 1, or p,^ = ± 1. Eeciprocal equations are divided into two classes, according as Pn is equal to + 1, or to — 1. (a) In the first case, we have Pn-l = Ih, Pn-2 =P2, " " Pi = Pn-l ', and these relations determine the first class of reciprocal equa- tions, in which the coefficients of the corresponding terms taken from the beginning and end are equal in magnitude, and have the same sign. (h) In the second case, when p„ = — 1, we have Pn-l = ~ Pi. Pn-2 = -P2, "• Pi = - Pn-l', which relations give the second class of reciprocal equations, in which corresponding terms taken from the beginning and end Art. lOG TRANSFOTiMATIOX OF EQUATIOXS. Ml are equal in magnitude, but different in sign. In thi.s case, when the degree of the equation is even, say n = 2 m, one of the conditions becomes j)„ = — p„., or p,„ = 0, so that in recipro- cal equations of the second class, whose degree is even, the middle term is absent. It is evident that tlie roots of reciprocal etiuations occur in pairs, «, - ; I3,~; etc. Wlien the degree is odd, there must be a (3 a root which is its own reciprocal, and it is obvious that in this case — 1 or + 1 is a root according as the equation is of the first or second class. In either case we can divide by the known factor (x + 1 or x — 1), and what is left is a reciprocal equation of even degree and of the first class. In equations of the second class of even degree .r^ — 1 is a factor, and, by dividing by .i-^ — 1, this is reducible to a recijv rocal equation of the first class of even degree. Hence all reciprocal equations may be reduced to those of the first class of even degree, Avhich, therefore, may be regardeil as the stand- ard form of reciprocal equation.^. We append a few examples, with some hints as to the method of solving such equations. EXAMPLES. 1. Solve the reciprocal equation X* + x^-i X- + x-\-l = 0. Dividing through by x-, this becomes x- + x-4: + - + \ = 0. X XT Adding and subtracting 2, this may be put in the form and (. + iJ+(.+ l) + Uf: 142 THEORY OF EQUATIONS. Art. 106 therefore x-\ 1-- = ±-. X 2 2 ... ;,; + ?:^2or -3. a; Solving this quadratic, the first value gives x = l, and the 1 • — 3 ± VS second gives x — 2. Solve the equation x^-l = 0. This is a reciprocal equation of the second class. Dividing by X — 1 (since x = 1 is evidently a root), we reduce it to the reciprocal equation of the first class of the fourth degree. x'^ + x^ + x' + if + 1 = 0, or, dividing by x^ and arranging terms, Therefore (a^' + H +(x + -]=l. Solving this as in the preceding example, we get finally a- = ill T V5 ± V^=T(10 ± 2 V5)^|, which expression gives the four values of x. 3. Keduce to a reciprocal equation of even degree and of first class a;« + 1^5 _ 2_2 ^4 _,_ 2^2 a,2 _ 5 3. _ 1 = 0. 4. Solve the reciprocal equation 2a;6 + a:^ _ i3_^4 ^_ 13^,2 _ 3. _ 2 = 0. Divide the left-hand member by x"^ — 1. Alt. 107 TliANSFOUMATlOX OF EQUATIOXS. 113 107. To transform an Equation into Another, the Roots of which shall be Less ( or Greater; than those of the Proposed Equa- tion by a Constant Difference. Let /(.1-) = be the proposed equation. In this equation we change x into y + k. The resulting equation in y will have roots each less or greater by k than the given equation in x, according as k is positive or negative. The resulting equation is (Art. SO) /W +/(A-)y + 4^^ / + - + 3/" = 0. The following mode of formation of this equation is, for practical purposes, much more convenient than the direct cal- culation of the derived functions and the substitution in them of A-. Let the proposed equation be f(x) = .-«" + 2)iX"-^ + 2)-2X"-^ + • • • -f Pn-ix -f 2\ = 0, and suppose the transformed polynomial in y to be PoU" + P^f-' + P-2U"-' + - + Pn-lV + Pni since y =:x — k, this is equivalent to p,(x - ky + p,(x - ky-' -f ... -h P„_,(.r - A-) + P„ which must be identical with the given polynomial. AVe con- clude that if the given polynomial be divided by x — k, the remainder is P,„ and the quotient p,{x - ky-' + p,(x - ky-' +...-}- P,._o(x - k) + p,._, ; if this again be divided by a; — A", the remainder is P„_i, and the quotient p,(x - ky-' + p,(x - ky-'' + "' + /'„_2. Proceeding in this way we can, by a repetition of the opera- tions explained in Art. 82, calculate in succession the several 144 THEORY OF EQUATIONS. Art. 107 coefficients P,„ P„_i, etc., of the transformed equation ; the last, Po, being equal to unity, as we know from other con- siderations. We shall find, when we give in Chapter IX. an explanation of Horner's Method, that the best practical method of solving numerical equations is only an extension of the process here indicated. A few examples will make the process plain. EXAMPLES. 1. Find the equation whose roots are the roots of a;'' + x3-29ic2-9ic + 180, each diminished by 6. The calculation is best exhibited as follows : 1 1 -29 -9 180 6 42 13 78 91 114 78 69 546 414 7 6 594 13 6 615 19 205 6 25 Here the first division of the given polynomial by cc — 6 gives the remainder 594 (P4), and the quotient a;^ + 7 X- + 13 .T + 69 (compare Art. 80). Dividing this again by x — 6, we get the remainder 615 (P.^) and the quotient x^+13x + 91. Dividing again, we get the remainder 205 (P3) and the quotient x + 19, and dividing this Art. 107 TRANSFOUMATWX OF EQUATIOys. 145 we get Pi = 25, and 1^ = 1; hence the recpiii-ed transformed equation is i/ + 2oif + 205 y- + G15 f/ + 594 = 0. 2. Find the equation whose roots are the roots of af + 4ar'-x2 + ll = 0, each diminished l>y 3. Ans. f + loy' + di7f + 305 y- + 507 y + 353 = 0. 3. Find the equation whose roots are the roots of 4 x^ - 2 ar^ + 7 a; - 3 = 0, each increased by 2. Here we divide by x + 2, as follows : 0-2 8 16 7 56 - 3 -126 - 8 14 - 28 63 -129 - 8 32 - 92 240 -16 46 -120 303 - 8 48 -188 -24 94 -308 - 8 64 -32 158 - 8 -40 The transformed equation is therefore 4 ^ _ 40 7/^ + 158 / - 308 / + 303 y - 129 = 0. 4. Increase by 5 the roots of the equation 3x* + 7ar''-15.r + J^-2 = 0. 5. Diminish by 20 the roots of tlie equation r,y._ i:5.,^_ 12x4- 7=0. 146 THEORY OF EQUATIONS. Art. 108 108. Removal of Terms. The solution of an equation is often facilitated by the removal of a certain specified term, which can be done by the transformation of Art. 107, as we shall now show. If f{x) = be expressed in the form ci()X"- + ciiX"'^ + a^x"^^ -}-... + a„ = 0, and the transformed equation be written in descending powers of y, we have chVn + {nuok + a,) y""^ + j ^^^."9 "'^'^ + ^'^ ~ -^) "1^" + ^*2 j- 2/""^ + ... = 0. If we give Jc such a value that naok + aj = 0, the transformed equation will be wanting in the second term. If k be either of the values which satisfy the equation n(7i — 1) .„ , ^ . , r. — ij — -^y^ aji- + (w — 1) a-ji + cu = 0, the transformed equation will Avant the third term. To remove the fourth term, a cubic equation will have to be solved; and so on. The following examples will illustrate the method : EXAMPLES. 1. Transform the equation x^-6 X- + 12 a: + 19 = into one wanting the second term. ?m,/c + ai = gives k — 2; therefore we must diminish the roots by 2. A7}S. f + 21^ 0. 2. Transform the equation a;^_4.r"^-18a;2-3a; + 2 = into one wanting the third term. Art. 109 TRAXSFOIiMATION OF EQUATIOys. 147 The quadratic for k is 6 k- -12k- 18 = 0, giving ^- = 3, ^ = - 1. Thus there are two ways of effecting the transformation. Diminishing the roots by 3, we get y' + Sf-iny-19(J = 0. Increasing the roots by 1, we get y^-Sf + 17y-S = 0. 3. Transform the equation x'-\-Sx^ + x-5=0 into one wanting the second term. 4. Transform the equation x^-6x' + 9x-10 = into one wanting the third term. 109. The Algebraic Solution of the Cubic Equation. Let the general cubic equation he written in the form x' + Sp,x- + Si^^x + p^ = (1) We first simplify this by transforming it into an equation lacking the second term. To do this, we replace x by y + k (Art. 107), where k is determined by the equation (Art. 108) 3^• + 37), = 0, which gives k = — j). Then (1) becomes iy-ihf + ^ih{y-ihf-\-^ih(.y-ih)+ih = ^ • (2) which reduces to the form if + 3IIy + G = (3) where H = ih— Pi and G = 2 p^ - 3 ;>, p., + ;>,. 148 THEORY OF EQUATIOXS. Ai't. 109 To solve (3), assume y ^j.i + si; ... t/3-3?-y?/-(r + .s) = (4) Comparing coefficients in (3) and (4), we have o^h^ = -H, r+s = -G; from which equations we obtain r = |(-G=+VG^TT^) (5) s = ^(-G-VW+Tii^) (6) J TT and, substituting for s^ its value — —, we have y = r^ + ^ (7) 9-3 the value of r being given in (5). AVe observe that if r be replaced by s, this value of y is unchanged, as the terms are then simply interchanged; also, since r^ has the three values -Vr, m-Vr, w^vV, obtained by mul- tiplying any one of its values by the three cube roots of unity (Art. 100), we obtain three, and only three, values for ?/; namely, We have then x+2h = '>'^ + ^^ (8) as the comi^lete algebraic solution of the cubic equation ar' + S2hx- + 3 j72X + 2h = 0, the square root and cube root involved being taken in their entire generality.* * This solution is known as Cnrdan\ Solution, because it was first published by him in IW!). See Uistorical Note, page 76. Art. no TRANSFORMATION OF EQUATrOXS. 149 110. Application to Numerical Equations. The solutiuu of the cubic- ubtaiiird in the last artirk- is of little practical value, wlien the equation has numerical coefticieut.s. For, when the roots are all real and unequal, G' -\- A IP <_ (this may i>e shown by Sturm's Theorem, see Chapter IX), wlience }• is imaginary, and the roots involve the square root of an imagi- nary number, Avhich in general we cannot solve. If the equa- tion has equal roots, it can be solved; and if it has a pair of imaginary roots, it likewise can be solved, for in this case G'- + 4 IP is positive. In the first case, namely, when the roots are all real, the roots may be computed by the use of Trigonometry.* To illustrate this method by an example, let \is solve the equation a^-lSx-3o = (1) Put x=r^ + s^', ... ar - 3 rM-K - (r -f .s) = (2) .-. 7-y = 6, r + s = 3o, r^ = 3, s^ = 2 ; * Throuphoiit a trontiso „{ the >.Tak = r + ab, k- = s + b-. . . (4) *The solution here given is due to Ferrari. (See Ilistoriciil Note, pajre Tfi.) This and the solutions of Descartes, Euler, Laplace, Lajrraiitjc, and others, all involve the solution of a cubic by Cardan's method, and will of course fail when that fails. We would then employ a trigonometrical solution. Alt. Ill TliANSFOIiMATION OF EQUATIONS. lol Eliminating a and b from (4), we have or 2A-•'-7^•^'4-20>r-s)^•^;)2^•f 7.s-/-2 = 0. . . (5) From this cubic we find, if possible, a real value of /,• b}' the method of Art. 109. The values of a and b are then known from (4). Subtracting (2) from (3), we have (a^'+^,.v + 7,)2_ (ax + 6)2=0, which is equivalent to the two quadratic equations x" + (i) - a) X + (k - 6) = 0, x'+(2^ + a)x + (k + b) = 0, the roots of which are readily obtained. As an example of this method, let us solve the equation ;^.4 + 2.1'-' -7 a- -8.^ + 12 = (1) Adding (ax + b'f to both members, Ave obtain x*+2 x'+(a'-7)x''+2(ab-4:)x+b' + 12 = (cuv+b)-. . (2) Since p = -\-l, assume a'+2a^+(a'-7)x'+2(ab-4)x+b'+l2 = (x'+x+ky . (3) Equating coefficients, Ave have a'-T = 2k + l, ab - 4 = A-, b' + 12 = A.- . . (4) .-. (2k + S)ik'-12) = (k + 4y, ... 2 Jc' + 7k-- :V2 A- - 112 = 0. Whence k = 4; hence a- = IH, ab = 8, b'- = 4, and .-. a = 4, 6 = 2. 152 THEORY OF EQUATIONS. Art. Ill Therefore, from (2), (3), and (4), we olDtain (or' + .T + 4)2 -(4a; + 2)2 = 0, which is equivalent to the two equations ar - 3 X + 2 = 0, a;^ + 5 .r + 6 = ; and, therefore, the four roots are 1, 2, — 2, — 3. EXAMPLES. 1. Solve X* - 6 x" -f 12 x^ - 14 x- + 3 = 0. 2. Solve X* + 4 af + 3 x' - 44 .r - 84 = 0. 3. Solve a;*-6x-2-8x-3 = 0. Ans. -1, -1, -1, 3. 4. Solve X* - 3 x2 - 42 a; - 40 = 0. r CHAPTER IX. LIMITS OF THE ROOTS OF AN EQUATION. 112. Definition of Limits. In attempting to find the real roots of numerical ecpiations, it is very advantageous to nar- row the limits within which such roots must be sought. iJes- cartes' Kule of Signs gives us the limit of the number of real roots, but tells us nothing as to the limit of the value of such roots. The closing remarks of Art. 78 suggest that there are means of getting the limits between which the roots of a given equation must lie, and we shall now proceed to give some of the methods for doing this. A superior limit of the j^ositice roots is any positive number greater than the greatest of the roots, that is,- nearer + x ; an inferior limit of the positive roots is any positive number smaller than the smallest of them. A superior limit of the negative roots is any negative number greater in absolute value than the greatest of them, that is, nearer to — co than the greatest ; an inferior limit of the nega- tive roots is any negative number smaller in absolute value than the smallest of them. In the next three articles we liave three rules for the determination of the superior limits of the positive roots. 113. PiioposiTiox I. /// an equation f(x) = .V" -f />,.»•"-' + y^a-"-- + ••• + 7',,-i-r + Pn = 0, if the first negative term he — p^c"'', and if lite greatest negative coefficient be — Pt, then ■\/pi,-\-\ is a superior liuiit of the pnsilive roots. 163 154 THEORY OF EQUATIONS. Ait. 11 -J Now fix) is certainly positive for any value of x, which makes a^ ^.n-r+l _ J .', f y' ^ X'" > Ihi^'^"' + iv""""^' H \-x-\-l)>2h ' z — X — 1 But this inequality is true, taking a; > 1, if X — 1 or a;"+^ — cc" > Pi^""''"'"^ or a; — 1 > jJ^.*"''"^^, or aj' — a?*""^ > jikj that is a;'-'(a; — 1) > p^. Bat, since a;''"^ > (x — 1)''~\ a;'-'(a; — 1) is > j)^, if (x — iy-'^{x — 1) >p^, or (a;-l)'->j>,. Hence /(a;) will always be positive, if a; = or > 1 + Vp.t. Hence A/j^t + 1 is a superior limit of the positive roots. 114. Proposition II. If in any equation each negative coefficient he taken positively, and divided by the sum of all the positive coefficients ivhich precede it, the greatest quotient thus formed increased by unity is a superior limit of the positive roots. Let the equation be OoX"+aia;''-^+ a^a;"-'-^- a.^x"'^-\ a,a;"-'■^ [- «„= (1) in which we regard the fourth coefficient as negative, and we consider also a general negative coefficient ; namely, — «,.. Now, since a;™— 1 = x'"-' + a;"'-^ -^ \-x-\-l, X — 1 we have a;"" = (x — 1) (x"-' + x""-^ + 1- .x + 1) -f 1. Art. 114 ROOTS OF AN EQUATIoy. l.");> Let us now develop each positive term of equation (1) by the formula o„.x'" = «„.(-^' - 1) (.^•'""^ + x""'- H \-x + l)+ a„, the negative term remaining uut-hanged. The polynomial /(j;) becomes then: Oq (^ — 1 ) .r""* + ao(x - l)a;"-2+ a^ix - l)x^-'^-\ h ao{x - l)x'-'+ •■•+a„, + Oi(.c - l)a;"-2+ ai(x — \)x"-^-\ f- a^{x — l)x"-^-\ h a„ + a.2(x —l)x" ^+ ••• + a./x — 1) j;"~''4- ••• + a,, a^x , + In the new polynomial thus formed, representing the left- hand member of the transformed equation, the successive co- efficients of x"~\ a;" ^, etc., are ao(x — 1), («o + tti) (x — i), («o + «i + Ol>) (^* — 1) — «3' etc. Any value of x greater than unity is sufficient to make posi- tive every term in which no negative coefficient O;,, o^ etc., occurs. To make the latter terms positive, we must have (ao + ai + «,) (x - 1) > (h, (tto -f Cfi + «2 + ••• «r-l) (•» — 1) > «r> etc. Hence a; > — ; — ^ h 1 tto + «! 4- "2 a;> 1-1, etc. a„ + rti + «.+ ••• 4- a,_i 156 THEORY OF EQUATIONS. Art. 114 If now we take for x the greatest of all these quantities, the first member will be positive (for this value and for all greater values of x) ; and this will be a superior limit of the roots. 115. Limit obtained by grouping Certain Terms. It is usually possible to determine, by inspection, a limit closer than that given by either of the preceding propositions. In this method we arrange the terms of an equation in grcmps having a posi- tive term first, and then observe what is the lowest integral value of X, which will have the effect of rendering each group positive. Such a value of x will be a superior limit of the roots. The form of the equation will suggest the arrangement into groups in each case. Of the propositions in the two preceding articles, sometimes one will give the closer limit, sometimes the other. In most cases Prop. II will give the closer limit. Of course the smaller the number found, the better. We consider the inte- ger next above the numerical value found by either rule as the limit. EXAMPLES. 1. Find a superior limit of the positive roots of the equation x*-5a^ + A0x^-Sx + 23 = 0. Art. 113 gives 8 + 1, or 9, as a limit, Art. 114 gives |^ + 1, or 6, as a limit. Hence 6 is a superior limit. 2. Find a superior limit of the positive roots of x^ + 4 x'' - 3 x' + 5 .«" - 9 x-^ -llx' + 6x-8 = 0. Art. 113 gives 5 as limit. Of the fractions 3 9 11 8 1+4' 1+4 + 5' 1+4 + A' 1+4 + 5 + 6' tlie third is the greatest, and Art. 114 gives the limit 3. In this case Art. 114 gives the closer limit. Art. 115 nOOTS OF AX FAjU AVION. 157 3. Find the superior limit of the positive roots of x-' + 8 x* - 14 .it' - 53 .ir + 56 x -18 = 0. Here, Art. 113 gives 9 as a limit, ami Art. 114 gives 7 as a limit. 4. Find the superior limit of the positive roots of x^ + 20 X' + 4 .f« - 11 x-^ - 120 X* + 13 X - 25 = 0. The methods of Arts. 113, 114 both give the limit G. In this case we can find a much closer limit by ajtplying the method of Art. 115. The equation may be arranged as follows : X' (ar' - 11) + 20 a-* {t? - 6) + 4 .i-« + 13 .r - 25 = 0. Here a; = 3, or any greater number, renders each group posi- tive ; hence 3 is a limit. 5. Find a superior limit of the roots of the equation a;* -x^-2 y? - 4 .K - 24 = 0. "When there are several negative terms, and the coefficient of the highest term is unity, it is convenient to multiply the whole equation by such a number as will enable us to distribute the highest term among the negative terms. Here, multiply- ing by 4, we can write the equation as follows : r^{x - 4) + .^-' i?r - 8) 4- .^• ix" - 10) + x' - 90 = 0, and 4 is a sujjerior limit. Find a superior limit of the positive roots of the following equations : 6. 4 ar* - 8 .t:^ + 22 x^ + 98 .» ' - 73 .r -f 5 = 0. 7. 5 or^ - 7 x' - 10 x^ - 23 x^ - 90 x -317= 0. 8. ar>-a-*-2,,'''-|-2.r-f-a'-l=0. 9. x'*-8jr' + 12.t;- + lGx-39 = 0. I 158 THEORY OF EQUATIONS. Art. 116 116. Inferior Limits, and Limits of the Negative Roots. To find an inferior limit of the positive roots, we must transform tlie equation into another whose roots are the recip- rocals of those of the first by the substitution x = - (Art. 104). Find then the superior limit I of the positive roots of the equation in y. The reciprocal of this, -, will be the required inferior limit ; for since y -, i.e., a; > -. y I I Yov example, take the equation of example (3) under the last article a^ + Sx*-Ux^-53x- + 56x-lS = 0. . . (1) Putting x= -, (1) becomes y f-Hy' + nf + \V/--j%y-T\--(^, • . (2) and a superior limit of (2), by Art. 114, is ■&! + 1 =||, and, therefore, if is an inferior limit of the positive roots. To find limits of the negative roots, we have only to trans- form the equation by the substitution x = — y. This transformation (Art. 102) changes the negative into positive I'oots. If I and V be the superior and inferior limits of the positive roots of the equation in y, then — I and — /' are the limits of the negative roots of the proposed equation. For example, take the equation x*-2x'-13x''-Ux-\-24: = 0. . . . (1) Putting X = — y, this becomes y'-\-2f-13y'- + Uy + 24. = (i. ... (2) By the method of Art. 115, we readily find a superior limit of the positive roots of (2) to be 5 ; therefore — 5 is a superior limit of the negative roots of equation (1). Alt. 117 ROOTS OF AX EQUATION. loO EXAMPLES. >^ 1. Find limits to the positive and negative roots of a;« _ 5 x^ + x' + 12 .»;■' - 12 x- + 1 = 0. Show that the real roots of the following eqnatious lie be- tween the limits respectively given : V 2. x^-a-3_(_4x-'-3.f + l = 0; ^ andl. 3. X* + o:'^ - 10 .IT - a- + 15 = ; - 4 and 3. 4. • ar> + 5 x* + .^•' - 10 ar _ 20 .r - IG = ; - 5 and 3. 5. (x- --ix- 2)- - 43 = ; - 2 and 6. 6. x^ + 2 x^ + 3 .1-3 + 4 .r + 5 x- - 54321 ; - 3, 9. Separation of the Eoots of Equations. 117. Having found the limits within which the real roots of an equation lie, the next step in the solution of an equation is to discover the intervals in' which the separate roots lie. The two most useful theorems for determining the number of real roots between any two arbitrarily assumed values of the variable are the Theorem of Fourier and Buchiu, and tlie Theorem of Sturm. For a proof of the first, we refer the reader to Buru.side and Paiito)i's Tlieonj of Equations. The theorem of Sturm,* wliich Ave shall consider in the next article, has the advantage of being unfailing in its application, giving always tlie exact number of real roots between any two proposed quantities; * .1. C. F. Sturm (lS<«-IsV.). 160 THEORY OF EQCfATIONS. Art. 117 whereas the theorem of Fourier and Ijudan gives only a cer- tain limit which the number of real roots in the proposed equation cannot exceed. 118. Sturm's Theorem. Let f(x) = a-" + 2hx''-' + • • • + 2\-iX +Pn = . . . (1) be an equation from which the multiple roots have been re- moved (Art. 98).* To find the equal roots we have employed the common operation of finding the H. 0. F. of a polynomial f(x), and its first derived function, f'(x). Sturm has employed the same operation for forming the auxiliary functions which are iised in this method for separating the root's of an ecpiation. Let the process of finding the H. C. F. of f{x) and f'{x) be performed. The successive remainders will go on diminishing in degree, and, as f(x) has, by hypothesis, no multiple roots, f(x) and f'(x) have no common divisor except unity, and we finally ()l)tain a remainder, /„(a;), independent of x; that is, which is numerical. Dividing f{x) by f'(x), we shall obtain a quotient f/j, Avith a remainder of a degree lower than that of f(x). Denote this remainder, with its sign changed, by /aO^), and divide /'(j-) by f-Ax), and so on ; the operation being precisely the same as lluit of finding the H. C. F. of f{x) and f{x), except that the signs of each remainder must be changed, Avhile no other changes of sign are permissible. In the process of finding ./i(-^)j fi{^), etc., any j^ositive numerical factor may be omitted or introduced, in order to avoid fractions, for the sign of the result is not affected thereby. * Tliis limitation is not neepssnry. but for simplicity we consider the equation cleared of equal roots, as this can always be done by the method of Art. DS. Art. 118 ROOTS OF AX EQUATIOX. IT.l The expressions f(x), /'(x), /^(x), /,(x; ■- f„(x) are called Sturm's Finictious. Keeping in mind the above explanations and definitions, we may now state Stunn's Theorem : TiiKoKK.M. //■ (Uiy tico real numbers a mid h he substituted for X ill IStunns Functions A^), f{^), M^) -/.-.(^-j. /nW, and the signs noted, the difference heticeen the number of chmirfes of sign in the series tvhen a is substituted for x, and the vuniber when b is substituted for x, expresses exactly the number of real roots of the equation f(x) = between a a)id b. From the •way in which Sturm's Functions are formed, we derive the following series of equations, in which represent the successive quotients in the operation : f{x)^qj\x)-f{x) fiix) = q,f{x)-f(x) fr-i(^) = (JrfX-r)-fU^) /„-2(^-) = 9,.-../;-i(-^)-./:.WJ Having regard to these relations, we observe : (1) The last of the functions f„(x) is not zero; for by sup- position it is independent of x, and if it were zero, the equa- tion f(x) = would have equal roots by Art. 1>S, whii-h is contrary to the hypothesis. (2) No two consecutive functions in the series can have a connnon factor; for, if they could, all the succeeding func- tions would vfttrrsh, i H u luding f„{^), and riiis is impossible 162 THEORY OF EQUATIONS. Art. 118 (3) When any auxiliary function vanishes, the two adjacent functions have contrary signs. Suppose, for example, that /^(■x) = 0, then from the second of the above system of rela- tions we have /i(.t) = —fz{x). In examining, therefore, what changes of sign can take place in the series during the passage of x from a to h, we may exclude the case i^f two consecutive functions vanishing for the same value (fK the variable ; therefore the different cases in which any change of sign can take place are the following : (a) When x passes through a root of the equation f{x) = 0. (6) When x passes through a value which causes one of the functions /', f^, /g •••/„_! to vanish. (c) When x passes through a value which causes two or more of the functions /', /a, fs---fn-i to vanish together; no two of the vanishing functions, however, being consecutive. (a) When x passes through a root of f(x) = 0, it follows from Art. 99 that one change of sign is lost, since immediately before the passage f(x) and f'(x) have unlike signs, and imme- diately after the passage they have like signs. (6) Suppose X to take a value a which is a root of the equa- tion /r(.i-) = 0. From the equation we have /r-i(«) = -/r+i(«)' which proves, as we have seen, that this value of x gives to /. ,(.i') and fr+i(x) 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 sm^ll that it con- tains no root of X_i(a;) or fr+i{x) ; hence, throughout the inter- val under consideration, these two functions retain their signs. ^Ve conclude that just before x, varying continuously, reaches tlie value «, the signs of fr^i(x), f^x), fr+\{^) must be -f ± — or — ± +, and just afterwards they must be + T — or — T + ; that is, f^{x) changes sign as x passes through the Art. 119 ROOTS OF AN EQrATloy. KIS value u, and the other two do not. Biit^iough the sign of X{-?0^ changeSj_no Yarijitipn_jjf^^gn_^ is eithei;jrqst "or gai ne'd^ thereby in the group of three; because, on ac-count of the difference of signs of the two extremes ./j-il'*") ^^^^^ /mi-'')' there will exist both before and after the passage one variii- tion and one permanency of sign, whatever be the sign of the middle function. For in the change from + ± — to -f- T — , or from — ± + to — T +, a permanency and a variation are changed into a variation and a permanency, or a variation and a permanency into a permanency and a variation ; but no variation of sign is lost or gained on the whole. (c) It follows at once that if two or more of the auxiliary functions vanish for the same value of x, since no two adjacent ones can vanish, the same reasoning that was employed in {b) holds good here, and, therefore, if f(x) 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 l>asses through a root of f{x) = 0, one change of sign is lost, and under no other circumstances is a change either lost or gained. Hence the theorem : the number of changes of sign lost while X varies horn a to i is equal to the number of real roots of the equation between a and b. 119. Separation of the Real Roots. The substitution of + oo and — v: for .'; in Stururs Fuurtiuns determines the number of real roots of f(x) = 0. The number of imaginary roots would, of course, be the difference between the degree of the equation and the number of real roots thus determined. The substitution of -f- x and for X determines the number of positive real roots, and the substitution of — oo and determines the numl)er of negative real roots. In applying Sturm's theorem, it is convenient in practici' tt> substitute first — cc, 0, +00 in Sturm's Functions, so as to obtain the whole number of negative and of positive roots. 164 THEORY OF EQUATIONS. Art. 119 To separate the negative roots, the integers —1, —2, —3, etc., are to be substituted in succession till we reach the same series of signs as results from the substitution of — co ; and to sep- arate the positive roots we svibstitute 1, 2, 3, etc., till the signs furnished by + co are reached. A few examples will illustrate the application of the theorem. EXAMPLES. 1. Find the number and situation of the real roots of the equation /(a-) = x'-2x-5 = 0. We find /' (x) = 3 x' - 2, /. (x) = 4 .^ + 15, f^ (x) = - 643. Corresponding to the values — co, + co of x, we have (-^) - + - - (0) - - + - (+^) + + + - Hence there is only one real root, and it is positive. Again, corresponding to values, 1, 2, 3 of x, 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 /(.r) = x' - 6 x" + 5 x' + 14 X- 4: = 0. Here /' (x) = 2 or^ - 9 .-c- + 5 x + 7, omitting a factor 2. f2{x)=:17x'-57x-5, f^{x) = 152 X- 4:57, Mx) = +. In this exami)le it will be found that the calculation of fi{x) is somewhat complicated; it is sufficient for our purpose^ Alt. 119 ROOTS OF AX EQUATION. 165 however, to know the aign, and thus when we ascertain that it is positive we need not calculate it exactly, but merely put down J\(x) = +. Here we have the following series of signs: (-co) + - + - + (0) - + - - + (+x) + + + + + Hence all the roots are real : one negative and three positive. We have further the series of signs : (—2) + — + — +, 4 variations. (— 1)— — + — +, 3 variations. (0) — + — — +,3 variations. (1) + + — — +,2 variations. (2) + — — — +, 2 variations. (3) + — — — +,2 variations. (4) + + + + +,0 variations. There is one change of sign lost between — 2 and — 1, one between and 1, and two between 3 and 4. If we put 3|- for x, the succession of signs is — + + +, and thus there is only one change of sign, so that one root of the equation lies between 3 and 3^; therefore another root lies between 3.V and 4. Find the number and situation of the real roots of the equations : 3. a;^-3.r--4.T-f 13 = 0. 4. x-''- 7x + 7 = 0. 5. a;*-4ar'-3a; + 23 = 0. Ans. Two real positive roots, between 2 and 3, and 3 and 4, respectively. 6. a;*-4ar'' + ic2 + 6a; + 2 = 0. 7. a-* -|- .r-^ + .r - 1 = 0. 8. ar'- 6x2 + 8 a; + 40 = 0. CHAPTER X. ELIMINATION. 120. Under the head of Applications of Determinants, in Chapter III, we have considered, as the student will recall, several cases of elimination whereby a system of equations may be solved. In Art. 41 there was given the method of solving a system of simultaneous equations where the number of unknown quantities is the same as the number of equations. In Arts. 42 and 43, the case where the number of equations is greater than the number of unknowns was considered, and the condition of consistency of such a system was obtained. In such a case the eliminant, or resultant, which is the determinant obtained by eliminating the unknowns from the given equation, is the determinant of the coefficients and absolute terms. We next considered homogeneous linear equations (Art. 44), and found that for a system of w homogeneous linear equations involving n unknowns the eliminant is the determinant of the coefficients, and that if this determinant vanishes, the ratios of the unknowns may be determined, but not their absolute values. There are various ways of determining the resultant, R, of a system of equations. We shall give some of the best methods of eliminating a single unknown from two consistent equations of any degree. 121. The method that naturally presents itself is as follows : The resultant of two linear equations ax + h = 0, a'a; + 6' = is evidently ab' — ba' = 0. 166 Art. 122 ELIMINATION. 107 If now we have two quadratic equations ax^ + bx-\-c = Q . . (1) a'x^ + h'x-^c' = . . (2) multiplying the first by a', the second by a, and subtracting, Ave get (o?y')x + («c') = (3) where (ah') = a b and (etc') = ", ^, [See Art. 17, (3)], and, again, multiplying the first by c', the second by c, sub- tracting, and dividing by x, we get (ac> + (6c') = (4) The problem is now reduced to elimination between two linear equations, and the result is {ac')-+{ba'){bc') = (5) This method of forming the resultant is practically very limited in application, as it becomes very tedious for equations higher than the fourth degree. 122. Euler's Method of Elimination. Having given two equations of the Hith and ?ttli degrees respectively, F{x) = b^'^ + b,x"-' + ... + Z;,. = j * * ' ^' we propose to eliminate .r, or to find their resultant. If these equations admit a common root r, we may assume f{x) = {x-r)f,{x), F{x) = {x-r)F,{x\ /,(.f) = «,.c"'-' + «2.r"-- + . • • + «,„i I ,^. ^^''' i^.(.r)^;8,.- + ^^-+- + ^J • • • ^-^ the coefficients being undetermined quantities depending on r. 168 THEORY OF EQUATIONS. Art. 122 "Whence we have an identical equation of the (m -\-n — l)th degree. Now, equating the coefficients of like powers of x on both sides of the equation, we have m + n homogeneous equations of the first degree in the m + n quantities Jh, Ih, '•• Pm, Qi, q->, ••• Qn', and eliminating these quantities by the method of Art. 44, we obtain the resultant of the two given equations in the form of a determinant. The method will be made clear by a few ex- amples. EXAMPLES. 1. Find the resultant of the two equations ax- + bx -j-c — 0, GiX^ + bjX -f Cj = 0, supposing thein to have a common root. We have identically (q^x + (h) (ax- + bx + c) = (pix +2h) (ckx^ + biX + Cy), or (i«i = 0, gi^ + Q'2<* — Pi^i — P2«i = 0) g^c + g^b — PiCi —2hh = 0, gaC — P2C1 = 0, and, eliminating pi, jh^ g^ % "^^e obtain the resultant in the form a «i b a &i a, c b Ci bi c Cl Art. 1-23 ELIMIXA TIOX. Kj'J The student can easily verify that this result is the same as that of Art. 121. 2. Find the resultant of the equations a^ + «i-x + «3 = 0, 6o.r + biX + 62 = 0. Euler's identity («oX^ + ctyV- -f a.^ + cis) (ftox + /?i) — (b(ix^ + b^x + b.;) («oX-^ + UiX + ag) = 0, gives the following five equations : «o,5o — ^i«0 = 0, Oi^O — Oo/?1 — ^i«o — ^u«i = 0, «3i3o + Oji^i — b.Ui — Z*i«.., = 0, a-ifSi — b-^Ui = ; whence 22 = 123. Sylvester's Dialytic Method of Elimination. This method leads to the same determinants for resultants as Euler's method ; but it is simpler in its application and has an advantage over Euler's method in point of generality, sinc^e it can often be applied to form the resultant of equations involving several variables. To find the resultant of the two equations f(x) = ciox"" + a^x^-' + a.^^-'- + ••• + a« = 0, F{x) = ftoOf + &,x"-^ + 6.-C"-' + •.. + i„ = 0, «o -bo tti-ttu -bi -K a-i -Oi -h -b, -bo a^ cu -b. -61 a eta -b. 170 THEORY OF EQUATIONS. Art. 123 of degrees m and n, with one nnknown, we multiply the first successively by U/, *^.l '^J "^J ***? 3 and the second by aP, «', .t^, or', •••, a;"'"^ We obtain thus the system of equations fix) = 0, xf(x) = 0, :v^(x) = 0, . . . .x"-y (a-) = 0, J'(a-) = 0, xF(x) = 0, x-F{x) = 0, .•• x^^-^F{x) = 0. There are m + « equations, and the highest power of x is m + ?i — 1. If there is a common root, it will satisfy all the equations of this system. And, in taking for unknowns, the different powers of x, the preceding equations form a system of m -f- n linear equa- tions with m -\-n — \ unknowns. Hence, by Art. 43, we can eliminate these unknowns and get a resultant, i?, which is equal to zero, if the equations are consistent. EXAMPLES. 1. Find the resultant U of two quadratic equations ax- -f &.» -f- c = 0, Oio;- + \x -f Cj = 0. We have xfix) = ax^ -\- hx? + ex — 0, /(.!•)= a:^\hx^c =0, ■ xF{x) = ciiX^ + b^x- + c^x = 0, F(x) = a,x^ + b^x + Ci = ; from which, eliminating .r"*, ai', x, we get the same determinant as in the preceding article, columns now replacing rows : Art. 124 ELIMlXAriON. 171 a b c a b c «i ^ t'l "i ^ Ci R 2. Find the resultant of the two equations /(.r) = a^^x* + Oior' + "l-*-'' + "y^ + 04 = 0, F(a-) = 60^-' + 6iX- + b, = 0. We have the following system : f{x) = •x^ + ttox* + GiX^ + o,,.r + a^x + a* = 0, SK/(a;) = (tox^ + ciix* + cuv" + a^x^ + «4^ + = 0, F(x) = 0-x^ + 0'x'4-0'X^+boX^+ bix + b., = 0, xF(x) = • af + . x' + ^o-^'' + ?>!•'>-' + ^2^^' + = 0, x^F(x) = . of + M' + ^i-^'' + b-^x^ + . a- + = 0, x^F(^x) = b^py' + Z^i-x-* + ft.!'"' + • .x-2 + • a; + = 0. Therefore, we have for the resultant, ffo «! Oo ttg a^ f'o f^i f'2 f's «4 &o ^1 ^2 ■^^0 60 ^ ^2 bo b, 60 bo b, b, 124. There are other methods of elimination, notably the nu'lhud by Symmetric Functions and Bezout's Method, for an explanation of which we refer the student to a higher work on the subject, such as Burnside and Pantou's Theory of E'juatious. 172 THEORY OF EQUATIONS. Art. 124 We shall close this chapter by giving some examples illustra- tive of the methods that we have considered in the foregoing articles. EXAMPLES. Ic Eliminate, by the method of Art. 122, x from the two qiiadratic equations 0? -I- 4 ;f - 21 = 0, ar - 13 x' + 30 = 0, and show that R = 0, and thus prove that the equations have a common factor. 2. Apply the same method to find the resultant of the two cubic equations ax^ + bx^ + ex 4- d = 0, a'x^ + b'x- + c'x + cV = 0. 3. To solve, making use of Euler's method, the equations : 3 2/2 + 4 xy ^3x^ — 9y-15 x = f'-2xy + x^-\-2y-10x==0.} • ' ' ' (^) Rearranging the terms according to descending powers of x, e have 3 x^ + (4.y - 15)x + 3 y- - 9 y = 0, x'-(10 + 2y)x + f- + 2y = 0. } .... (2) These are equations of the second degree with respect to x, of which the coefficients a, b, c, Ui, b^, Cj (see Art. 122, Ex. (1)) are respectively 3, 4?/ -15, 3f--9y; 1, -(10 + 2./), f + 'iy. Therefore, by substitution in the value of R of Ex. (1), Art, 123, we have Vit. 124 ELIMIXATION. 3 10 4?/ -15 3 _(10 + 2v/) 1 oy--9y 4I/-15 r + 2^ -(10 + 27/) 3r-9y ^f + 2y 173 = . (3) or, ill developing, //?^y(y' + 2y^-9y-18) = .... (4) The solution of this equation gives for the roots y = 0, y=?,, y = -3, y = -2. Then, to calculate the corresponding values of x, in this example, we simply eliminate ar between the proposed e0 (1) and the last two will be imaginary when r<0 >-; 174 THEORY OF EQUATIONS. Art. 124 Now we have, Art. 94, (3), 2af3 + (3'-y' = 3H, (3) «^' — ay- = — G. To eliminate a and (3 from (3), we substitute the value of a from the iirst in the second and third, and then multiply the second by (3 twice, and the third by /3 once, thus forming the hve equations: 3^- + (r + Si7)=0, 3fi' + (f+'SH)(3 = 0, 3{S* + iy' + 3H)(3' = 0, 2(3'-2y'l3-G = 0, 2 ;8^ - 2 y'/B' -G(3^0, whence, the determinant 3 3 (y' + 3H) -2y2 This reduces to 72 2 (y' + SII) (y' + 3II) 27 G- + 4H^ G 0. 4 (iy' + dliy which, compared with (1), shows that the roots are all real when G' + 4H'<0, the required condition. When G' + 4 W > 0, the two conjugate roots are imaginary. The function Gr^+4iZ'^ is called the dhcriminant of the cubic — ' x^ + 3 IIx +(7 = 0. CHAPTER XI. SOLUTION OF NUMERICAL EQUATIONS. 125. Tliere is an essential difference between the solutions of algebraic and numerical equations. In the former we have a general result expressed in symbolic characters, and it has been proved to be impossible to carry this solution beyond equations of the fourth degree (Art. 53). But it is possible to solve numerical equations of a much higher degree, and to obtain at least appi'oximate values of the roots accurate enough for all practical purposes. To this end, we determine the roots separately, and Ave must first separate the roots ; for, before attempting the approxima- tion to any individual root, it is generally necessary that it should be situated in a known interval which contains no other real root. In Chapter IX. certain methods of separating the roots of an equation have been explained. Real roots of numerical equations are either commensurable or incommensurable. Commensurable roots include integers, fractions, and repeating decimals which can be reduced to fractions; incommensurable roots consist of interminable deci- mals. The roots of the former class can be found exactly, and those of the latter, as we have just intimated, approximated to with any degree of accuracy. In this chapter we shall con- sider the solution of numerical equations. 126. Theorem. If the coefficient of the first term of f(x) is unitii and all the other coefficients are irhole numbers, an;/ com- mensurable real roof of f(x) = is a whole number anil an exact divisor of p„. 176 176 THEORY OF EQUATIONS. Art. 120 For, if possible, let ~, a fraction in its lowest terms, be a root of the equation f{x) = X'' +2hx''-'^ -\-2hx'"'^ + ••• +2^n-ix +Pn = 0; we have then from which, multiplying by 6""\ we obtain h Now, since — is a fraction in its lowest terms, this equation h is impossible, for an integer cannot be equal to a fraction. Hence - cannot be a root of the equation. The real roots of b the equation, therefore, are either integers or incommensurable quantities. It is evident, by Art. 94, that 'any commensurable root is an exact divisor of 2^,i- Every equation with finite coefficients can be reduced to the form in which the coefficient of the first term is unity, and those of the other terms whole numbers by the method of Art. 103. 127. Knowing that the integral roots of f{x) are factors of p,„ we can often determine them by trial. To do this, we must first find the limits within which the roots lie (Chap. IX). For example, take the equation af»_4.T2 + a; + 6 = 0. Here the real roots lie between + 4 and — 2. The possible commensural)le roots, being integral factors of 6, are ±1, + ?. + 3, and we easily find that the roots are — 1, +2, -|- 3. Art. 128 NUMERICAL EQUATIONS. 177 We shall in the next article explain a general method of obtaining the integral roots of an equation whose coellicieuts are all integers. 128. Newton's Method of Divisors. Suppose h to be an integral root of the equation ((o-r" -^ ciiX" '+..•+ a„_i.T + o„ = 0. . . . (1) Let the quotient, when the polynomial is divided by x — h, be b^i"'-^ + ^i.f"-'' + ••• + 6,._2.^• + 6„_i, in which h^, hi, etc., are all integers. Proceeding as in Art. 82, we obtain Oo = b^„ rtj = &j — /<&„, a.^ — ho — hbi ••• a„_2 = 6„_2 — '*^n-3> C*,.-! = ^n-l — ^*^-.-2> «,. = — '*^h-1' The last of these equations proves that «„ is divisible by h, the quotient being — b„_i. The second last, which is the same as a,._i + ^ = -//6,._,, h proves that the sum of the quotient thus obtained and the second . last coefficient is again divisible by /*, the quotient being — b„_2 ; and so on. Continuing the process, the last quotient obtained in this way will be — b^, which is equal to — tto- In this way we can test all the divisors of a„ and see Avhether they are roots of the equation. They must, at each step of the above process, give integral quotients and a final quotient equal to — a^. As soon as a fractional quotient is met with, the number that we are trying must be rejected, for it cannot be an integral root. This is called Newton's* Method of Divisors. 178 THEORY OF EQUATIONS. Art. 129 129. Application of the Method of Divisors. In applying this method it is convenient, after a manner analogous to Art. 82, to write the series of operations as follows : a„ a„_i a„_2 ••• ag «! a,, — b„_i — 6„_2 — ^2 — ^1 — *o — hi>„_2 — hb„_3 — hbi — hbo The first figure in the second line (— 6„_i) is obtained by- dividing a„ by h. This is to be added to a„_i to obtain the first figure in the third line (— hb„_2). This is to be divided by h to obtain the second figure in the second line (— 6„_2); 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 — cIq. AVhen we have proved in this manner that h is a root, the next operation with any divisor may be performed, not on the original coefficients a„, a„_i, •••, but on those of the second line Avith their signs changed, for these are the coefficients of the quotient when the original polynomial is divided by x — h. We need not include the numbers 1 and — 1 in the number of trial divisors. It is more convenient to determine before- hand by trial whether either of these numbers is a root. EXAMPLES, 1. Find the integral roots of the equation x^ + 6 r^ + X- - 24 X - 20 = 0. We observe that all the roots lie between + 3 and — 6. Hence, the following divisors of 20 are possible roots: -5, -4, -2, -1, +1, +2. By trial we find that — 1 is a root, and + 1 is not. We commence with + 2. Alt. 129 NUMERICAL KQUATIONS. ITD _ 20 -24 +1 +6+1 _ 10 -17 -8 -1 Hence 2 is a root. We next try — 2, making use of the coefficients of the second line with the sign changed. 10 17 S 1 - 5 -() -1 + 12 +^ ~0 Hence — 2 is a root. We proceed next with — 4. As this does not divide 5, it is not a root, so we try — 5. 5 6 1 -1 -1 ""5 ~~0 and — 5 is a root. One step more in the process would show ns, as we already know, that — 1 is also a root. Hence the roots of the equa- tion are — 1, — 2, — 5, 2. 2. Find the integral roots of the equation x' + 11 .r + 41 x"" + ()1 X + 30 = 0. It is evident that there is no positive root. By trial we find that the limit of the negative roots is — 6. Hence the possible integral roots are -1, -2, -3, -4, -5. We commence with — 5. 30 61 41 11 1 - 6 -11 - 6 -1 55 30 5 is a root. 180 THEORY OF EQUATIONS. Art. 129 As 4 will not divide 6, — 4 is not a root (as we knew in the beginning, for it does not divide oO), so we try — 3, and then — 2, and lastly — 1, as follows : 6 11 6 1 -2-3-1 9 ~3 ~0 Hence, — 3 is a root. 2 3 1 -1 -1 2 Hence, — 2 is a root. 1 1 -1 Hence, — 1 is a root, and the roots are all integral. 3. Find the integral roots of af -4x*- 16 x-s + 46 x' + 63 x - 90 = By trial we find that + 1 is a root ; we therefore depress the equation by dividing through by a; — 1, which gives x' - 3 x" - 19 .T- + 27 a; + 90 = 0. Ans. 1, 3, 5, - 2, - 3. 4. Find all the roots of X* -Sx"- 11 x"" + 19 a; + 42 = 0. Here limits of the roots are + 4 and — 3 ; and the possible integral roots are +3, +2, +1, — 1, — 2. Ans. + 3, - 2, 1 + 2 V2, 1 - 2 V2. 5. Find all the roots of the equation x' + ar^ - 2 a;2 ^ 4 a; - 24 = 0. 6. Find the integral roots of the equation 15 .^•^ - 19 x' + 6 x^ + 15 x- - 19 .r + 6 = 0. Art. 131 NUMERICAL EQUATIONS. 181 7. Fiud all the roots of the equation x*-2 x' - 19 X" + 68 a; - 60 = 0. The roots lie between —6 and 6. We find that 2, 3, —5 ai-e roots, and that the factor left after the final division is a- — 2 ; hence 2 is a double root, and the polynomial is there- fore equivalent to (.^•-2j^•-3)(.^• + 5). 130. Determination of Multiple Roots. The j\retliod of Divisors, as shown by Ex. 7 of the last article, determines multiple >oots when they are commensurable. In applying the method, Avdien any divisor of a,„ which is found to be a root, is a divisor of the absolute term of the reduced poly- nomial, it may also be a root of the latter. If it is, it will be a double root of the proposed equation. If it is found to be a root of the next reduced polynomial, it will be a triple root of the proposed equation, and so on. It is often a saving of labor to seek for multiple roots in this way, rather than by the laborious method of the H. C. F. (Art. 98). EXAM PLES. Find the commensurable and multiple roots of 1 . 2 x" - 31 .t-- + 1 12 .i- -h 04 = 0. 2. x' - x" - 30 .«■- - 76 X - 50 = 0. 3. .c' _ 8 x' + 22 .*;■' - 20 x- + 21 .t- - 18 = 0. 131. Newton's Method of Approximation. AVe shall now proceed to the determination of incommensurable roots, giv- ing iirst Newton's Method. In any method of approximation, the root that we are seek- ing is supposed to be separated from all other roots and to be contained within close limits. Let f(x) = be the given 182 THEORY OF EQUATIONS. Art. 131 equation, and let a be a known number differing by a small quantity (a decimal fraction), h, say, from the root a -f h. We have then f(a + h)=f(a)+f'{a)h+'^^h'+-=0.. . (1) In the first approximation, since h is small, we neglect the terms which contain /r and higher powers. Hence (1) becomes /(a) +f(a)h = 0, which gives, as a first approximation to the root, the value Kepresenting this root by b, and applying the same process a second time, we have for a second approximation to the root b — ^Wj and so on. The oftener this process is repeated, the more accurate is the approximation. In general the approximation is rapid, but this method has been entirely superseded by Horner's IMethod, which we take up in the next article. To illustrate this method, consider the equation /(.T) = .r^-4.^'2-2.^• + 4 = (1) We find that the three roots are comprised respectively in the intervals (- 1, - 2), (0, 1), (4, 5). Let us first calculate the last one. Narrowing the limits, we find that the root is comprised between 4.2 and 4.3. We find, then, /(g) _/(4.2)_- 0.872 _ /'(a) "/'('i.2) -+17.32- ^•^''^^- Art. 132 NUMERICAL EQUATIONS. 183 A first approximation is, therefore, 4.2504. Calling this b, we have f(b) _ ./•(4-2504) _ 0.014825 fW) -/'(4.2504) - 18.1840 "O-OOOSl^- A second approximation is 4.2504 - 0.000815 = 4.249585, which will be found to be correct to the third decimal place. In like manner the root between and 1 is found to be 0.853G3, and that between —1 and —2, to be —1.102775. Here, as the example is given simply to illustrate the method, no pains has been taken to carry the approximation beyond the third decimal place. As the coefficient of x- in equation (1) is 4, the 3 roots added together should give 4. 132. Homer's Method of Solving Numerical Equations. By Horner's method both the commensurable and the incom- mensurable roots can be obtained. The root is evolved tigure by figure ; first the integral part (if any), then the decimal part till the root terminates if commensurable, or to any number of places if incommensurable. This inethod is really an extension of the principles of the method of Art. 107, which involves the diminishing of the roots by known quantities. A X'oot Avliich has several figures is obtained by continued applications of that method, the successive transformations being exliibitod in a compact form, as will be made apparent by the examples given below. The first step in the solution of a numerical equation is to find the Jirst figure of the root. This can usually be done In- trial, though sometimes it may be necessary to resort to one of the methods of Chapter IX to separate the roots. 184 THEORY OF EQUATIONS. Art. 132 EXAMPLES. 1. Find the positive roots of tlie equation 8 .r* - 260 .^■2 - 546 x - 207 = 0. There can be only one positive root ; and it is found by trial to lie between 30 and 40. Thus the first figure of the root is 3. We now diminish the roots by 30. The transformed equar tion will have one root between and 10. It is found to lie between 4 and 5. We next diminish the roots of the trans- formed equation by 4, so that the roots of the proposed equation will be diminished by 34. 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 ; that is, the diminution of the roots of the proposed equation by 34.5 reduces its absolute term to zero. Therefore, 34.5 is a root of the given equation. The method of calculation is exhibited as follows : 8 -260 - 546 - 207 1 34.5 240 - 600 - 34380 - 20 -1146 - 34587 240 6600 29688 220 5454 -4899 240 1968 4899 460 7422 32 2096 492 9518 32 280 524 9798 32 556 4 660 Art. 132 NUMERICAL EQUATIONS. 185 The broken lines mark the conclusion of each transforma- tion, and the ligures in dark type are the coefficients of the successive transformed equations. (See Art. 107.) Thus 8 x" + 4G0 x" + 5454 x - 34587 = is the first transformed ec^uation, whose roots are less by 30 than the roots of the proposed equation, and are found to lie between 4 and 5. And 8 r* + 556 x^ + 9518 x - 4899 = is the second transformed equation. If this second transformed equation had not an exact root .5, we should find the limits between which the root lies, and then proceed as before, and so on. 2. Find the positive root of the equation 4ar'-13a;2-31cc-275 = 0. . Here the arithmetical calculation is as follows ; (1) -13 24 ~11 24 -31 66 ~35 210 6.25 210 65 51.392 35 245 -13.608 24 11.96 13.608 59 256.96 .8 12.12 69.8 .8 269.08 3.08 60.6 272.16 .8 61.4 .2 186 THEORY OF EQUATIONS. Art. 132 We find by trial that the proposed equation has its positive root between 6 and 7. The first figure of the root is, there- fore, 6. Diminish the roots by 6. The transformed equation 4 a^ + 59 ar + 245 a; - 65 = has a root between and 1 . It is found by trial to lie between .2 and .3. Diminish the roots again by .2. The transformed equa- tion 4 x3 4_ 61.4 r" + 269.08 x - 13.608 = 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 .abc •••) is about to appear, multiply the roots of the corresponding transformed equation by 10; that is, 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 (Art. 103). 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 b.c •••, and the process is continued as before. To illustrate this we repeat the above operation, omitting the decimal points. In subsequent examples in this book this simplification will be adopted, and the student is advised to make use of this principle in the solution of all such examples. Art. 133 4 ■13 24 11 24 35 24 NUMERICAL EQUATIONS. -31 6G 187 590 8 598 8 606 35 210 24500 111)6 25696 1212 2690800 30800 2721600 65000 5l;;i)2 -13608000 136(18000 6140 20 6160 In the examples here considered the root terminates at an early stage. When there are many more figures in the root, the process would become very laborious, if it were not for a simplification which we shall explain in the next article. This introduces to us one of the most valuable practical advantages of Horner's Method, which is, that after the second or third (sometimes even after the first) figure of the root is found, the transformed equation itself sucjrjests, by mere inspection, the next figure of the root. 133. Principle of the Trial-divisor. We have seen in Art. 131 that when an e(|uation is transformed by the substitution of a + /< for a, a being a nundjer dilforing from the true root by a quantity h, small in proportion to a, an approximate numerical value of /i is /(«_) /'(«)' 188 THEORY OF EQUATIONS. Art. 133 Now, as in the successive transformed equations of Horner's method, the last coefficient is /(a) and tlie next to the last is f'(a), we would evidently get the next figure of the root by- dividing /(o) by/'(o); that is, by dividing the last coefficient by the coefficient next to the last. This will, in general, give the correct figure only after two or three steps in the process have been completed, and the part of the root to be found bears a small ratio to the part already evolved. We might, therefore, if we pleased, at any stage of Horner's operations, apply Newton's method to get a further approximation to the root. The second last coefficient of each transformed equation is called the trial-divisor. It is evident that the application of this principle will greatly facilitate the work, but we must use due care not to apply Newton's method too soon. Thus, in the second example of the last article, the number 5 is correctly suggested by the trial-divisor 2690800, for 2090800 into 13608000 goes 5 times (and something over, of course). In this example, indeed, the second figure of the root is correctly suggested by the trial-divisor of the first transformed equation ; although, in general, such is not the case. In practice the student must estimate the probable effect of the leading coefficients of the transformed equation. To illustrate, consider the following examples : EXAMPLES. 1. Find the roots of the equation a.*^ — 7 cc + 7 = 0. We first separate the roots by Sturm's Theorem (Ex. 4, Art. 119). We find that there are two positive roots between 1 and 2, and a negative root between — 3 and — 4. Trans- forming the equation by diminishing the roots by 1, we find that of the two positive roots, one lies between 1.3 and 1.4, and the other between 1.6 and 1.7. We shall find the first root to five deeimal places, and leave it as an exercise for the student to find the root between 1.6 and 1.7 and the negative root. I It. l:};3 NU MEltIC AL EqrATIONH. IHU The calculation is written as follows : -7 +7 1 1.35680 1 1 -6 1 -6 1000 1 -^ J -90;; 2 -400 97000 1 1)9 -301 - 8(;(;i'5 30 10375000 3 108 — 90 IS! IS 1 33 -19300 1326016000 3 1975 -173L>5 -llSllL".).-,(iS 3G 141586432000 3 2000 390 -1532500 o 395 24;5;!6 -15(»S1(;4 5 2i;;72 400 -148379200 r> 3L>55()4 -MS(r)360(; 4050 6 ',V2~^7A\)> 4056 -14772812800 6 4062 6 40680 8 40688 8 )6 4061 8 407040 Here we first diminish tlie roots by 1. As the decimal i»art is about to appear, attach ciphers to the coetKcieuts of the transformed equation, which thus becomes 190 THEORY OF EQUATIONS. Art. 133 ips + 30 X- - 400 X + 1000 = 0. We next diminish the roots by 3, as we have already found that 3 is the next figure of the root souglit. After multiplying the roots by 10, the second transformed equation is x" + 390 X- - 19300 X- + 97000 = 0. The trial divisor now becomes effective; 19300 into 97000 goes 5 times, and 5 is found to be the next figure of the root. If we had adopted the figure 6, the absolute term would have become negative, the change of sign showing that we had gone beyond the root. V\'e must take care that, at least after the first transformation, the absolute term preserves its sign throughout the operation. The figure to be adopted in every case as part of the root is that highest number which in the process of transformation tcill not change the sign of the absolute term. If Ave were to take by mistake a number too small, the error would show itself, just as in ordinary division or evolu- tion, by the next suggested number being greater than 9. After diminishing by 5 the roots of the second transformed equation (and multiplying the roots of the resulting equation by 10), the next figure of the root is 6, for 1532500 goes into 10375000 6 times. And so we proceed, as indicated in tlie above operation. Of course the process can be continued indefinitely, and the root obtained correct to any number of decimal places. 2. I^'ind, to 5 decimal places, the positive root of the equation which lies between 2 and 3. 3. Find the two positive roots of the equation x' + 4 ar'' - 4 a;2 - 11 X + 4 = 0. Art. l;5t NUMERICAL EQUATIONS. 191 There are several abbreviatiuns of Horner's process, by ■which, after three or four phices of decimals have been calcu- lated as above, several more may be correctly obtained by a contracted process, for an explanation of which we refer the reader to Buruside and Panton's 77 = 0. A».'i. (x-2y(x-S)\x+lf. 25. Find the equation wdiose roots are the roots of a-' - 6a,-« - X' + 2 ar' - X + 7 = with their signs changed. 26. Change the equation 2 .r* _ 4 x" + 5 x"^ - 7 a- + 3 = into another, the coefficient of wliose liighest term will l>e unity, and the coefficients of the other terms integers. 194 THEORY OF EQUATIONS. Art. 134 Remove the fractional coefficients from the equations : 27. x*-:^x' + ^a^-5x + 2 = 0. 28. x^ + ^jx'--j\^0. 29. x' + \x'-^x + 3 = 0. 30. Find the equation whose roots are the reciprocals of the roots of x' - 91 x~ - 910 x + 1000 = 0. 31. Give condition that the following equation should have, (1) one inhnite root, (2) two infinite roots. (a^ - 4) x^ + (c - 7) x^ + ax- - ex + 20 = 0. 32. Increase by 5 the roots of the equation 2 x' - r' + 6 .T- + 3 .T - 10 = 0. 33. Increase by 3 the roots of the equation a^-3x^ + x-7 = 0. 34. Diminish by 2 the roots of the equation x^-x'^ + x'^ — x-{-5 = 0. 35. Diminish by 6 the roots of the equation x'-3x^-2x + 16 = 0. 36. Diminish by 1 the roots of the equation x^-2x^ + x^ + 4 ar' - a:* + 7 .r- - 16 = 0. 37. Increase by 10 the roots of the equation 3 x' -Qx^-4x^ + 2x-5 = 0. Transform each of the following three equations into an- other wanting the second term : 38. x" - 3 x" + 4 .^• - 4 = 0. 39. x^-8.r^ + 5 = 0. Art. i;M MISCELLA XEO Us EX A Ml'L KS. 1 : t.j 40. 2 x" + 12 x' - 3 .r + 5 = 0. 41. Remove the tliird term in the equation x*-8 .r' + 18 X- - 15 X + 14 = U. Remove the secoml term and solve the two euhie equations (Art. 110) : 42. x^ - 18 X- + 157 X - 510 = 0. Aits. 6, 6 ± T^AH". 43. x" - 7 .^-' + 11 -v = 20. .1,;.^. 5, 1 + V^3, 1 - V^^. Find a superior limit to the positive and negative roots of the equations : 44. x' - 5 x" + 37 a.-2 - 3 .r + 39 = 0. 45. CL^ + 7 a;* - 12 x" - 49 x- + 52 x - 13 = 0. Apply Sturm's theorem to determine the number and situa- tion of the real roots of the following five equations : 46. x*-4:X^ + 7 ar - 6 .« - 4 = 0. 47. X* - 5 ar' + 10 .«2 - G x -21=0. 48. a.-* - 10 x' + 6x + l= 0. ^l//.s'. Roots all real; one in the interval J — 4, — 3|; two in the interval | — 1, 0|; and i)()sitive roots in tlie intervals |0, IJ, 13, 4(. 49. x' - 2 .r' _ 4 .i; + 10 = 0. 50. x" _ 4 .^•2 - 4 a; + 20 = 0. 61. Show, by Sturm's tlieorem, that all the roots of the equation a^ -h 3 x^-jr- 3 x + H = <> are imaginary. 196 THEORY OF EQUATIONS. Art. 134 Find the integral roots of the following equations : 52. X* - 5 X''' + 25 .7; - 21 = 0. 53. 9 x' 4- 30 x' + 22 x' + 10 0;=^ + 17 a;^ - 20 a; + 4 = 0. 54. x' + 6 r' - 22 af - 33 x- + 54 = 0. 55. Find the commensurable and multiple roots of X* + 12 a;3 + 32 a.-2 - 24 a- + 4 = 0. A)is. The equation has two pairs of equal roots, both incommensurable. 56. Find the commensurable and multiple roots of a;« - 8 af + 20 x' - 32 x' + 68 x- - 32 a; + 64 = 0. Ans. (x - 4)- (if2 + 2)- = 0. 57. Find, by Horner's method, to six decimal places, the root between 2 and 3 of the equation ar^ _ 49 X' + 658 x - 1379 = 0. 58. Find the two real roots of the equation a;* - 11727 » + 40385 = 0. Ans. 3.45592, 21.43067. Find all the roots of the three equations : 59. x^+x--2x-l = {). Am. -1.80194,-0.44504,1.24698. 60. x^ - 315 x'' - 19684 x + 2977260 = 0. 61. x''-10x'-\-C^x + l = 0. - 3.065315791, - 0.691576280, Ans. \ -0.175674799, + 0.879508708, + 3.053058162. APPENDIX A. The definitions of algebraic and transcendental functions given in Art. 50, page 78, are somewhat broader than those found in our elementary text-books on Algebra. That these definitions are exact and cover the entire ground, is evident from the following considerations : In mathematics there are only four fundamental operations, namely : addition, subtraction, multiplication, and division. If two quantities, x and y, are so related that when one of them is given the other can be calculated, the one is said to be a mathematical /»«c