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ISH ADMT- ry View of the By R. Southey, nelsbyR.BeIl, . I7s. 6d. t BRITISH Esq. Barrister- . 38. 6d. lEN of the (LAND. With n the Popular By J. Forster, 1 of Pym, Eliot. 1 an Historical THiole. 6 vols. i:s. 6d. Mr. Forster 's STATESMEN. By Sir James Mackintosh, Rilfht Hon. T. P. Courtenay, and J. Forster, Esq. 7 vols 248. 6d. LIVES of the most Eminent ENGLISH POETS. By R. Hell, Esq. 3 vols. . 78. London : Longman, Brown, Green, and Longmans. r THEORY AND SOLUTION ALGEBRAICAL EQUATIONS THE HIGHER ORDERS. BY J. R. YOUNG, raoF^ssoR of mathematics in Belfast collegf. SECOND EDITION, ENLARGED. LONDON : SOUTER AND LAW, 131, FLEET STREET. 1843. Printed by C. Aolard, Bartholomew riose. PREFACE. The present work is an endeavour to exhibit a comprehensive view of the Theory and Solution of Algebraical Equations ; espe- cially in reference to that more important department of the subject, which relates to the practical analysis and computation of the roots when the coefficients of the equation are given numbers. I am not aware of the existence of any work in which so ample a discussion of the general problem of the solution of numerical equations is furnished as that which is attempted, however imperfectly, in the following pages. The main topics of this discussion are indeed but very recent contributions to science, made by different persons, at compa- ratively short intervals, within the last twenty or thirty years ; and are as yet too imperfectly known and estimated to have become generally incorporated into our systems of analysis. But the importance of these recent additions to our knowledge, in a branch of inquiry of so much practical consequence as the solution of numerical equations, in which all existing methods were felt to be so limited and imperfect, seemed sufficient to justify even reiterated endeavours to introduce them into general notice, and to urge their claims to general adoption. nTJf^n^rjOO. IV PREFACE. Under this conviction, the first edition of the present work, and the subsequent introductory volume on the Analysis and Solution of Cubic and Biquadratic Equations, were prepared : — I beHeve they have both been as instrumental as I had any right to expect, in accomplishing the objects of their publication. In the introductory work just alluded to, it was my more immediate aim to invite the attention of the young analyst to Sturm's infallible method of analysing numerical equations; and to show the efficiency and general practicability of that method, in conjunction with the subsequent development of Horner, to accomplish, with more ease and certainty than had hitherto been done, the solution of equations of the third and fourth degrees. It is a remarkable circumstance connected with the recent improvements adverted to above, that they have presented, in rapid succession, three independent methods for analysing a numerical equation ; thus affording a choice of means by which the most formidable obstacle that had hitherto stood in the way of a complete solution of the problem might be overcome. These methods are respectively due to Budan, Fourier, and Sturm : each is characterized by distinctive peculiarities, and each has its own advocates, who have perhaps sometimes allowed their partiahties, in favour of one method, to influence their judgment in estimating the merits of the others. What more especially distinguishes the method of Sturm from each of the other methods is its unfailing certainty, and its entire freedom from tentative operations : — two qualifications of the first importance in practical science. It was on these grounds chiefly that I had always regarded Sturm's method as the best ; and that I had ventured, more than once, to affirm its superiority even as regards equations of a higher degree than the fourth. Up to this point my conviction of the superior ehgibihty PREFACE. V of this method, when conducted conformably to the plan which I have recommended, remains unchanged. Nor should I now see any reason to modify my former views, as to the more ex- tended appHcation of Sturm's theorem, but for the im prQS[£ - ments which I think I have in this volume effected in other methods of analysis; more especially in that proposed by Fourier, which, however easy and concise in certain particular applications of it, had but very slender claims, in the state in which it was left by its distinguished author, to rank with the theorem of Sturm, whether we regard the certainty of its con- clusions, or the general practicability of reaching them. It is probable that the modifications here adverted to may contribute to bring the method of Fourier into more general use in the analysis of the higher equations, whenever, from the magnitude of the coefficients, Sturm's process might be ex- pected to involve numbers inconveniently large. It is not Hkely however that any future discovery will ever lead to the entire abandonment of Sturm's method. Within the limits above stated, it is still upon the whole the best that can be given. And the simplecharacter of its operations, and the undeviating certainty of its results, will always command for it a promi- nent position in every exposition of the doctrine of numerical equations. By far the greater part of the following work is devoted to the analysis and development of these theories. I have endeavoured to place each of the new methods before the reader in the best form I could ; and, by copiously illustrating them all, have afforded him the means of forming a correct estimate of their comparative merits, and of drawing his own conclusions, as to the preference to be given to any one in particular, in cases of more than ordinary difficulty. The critical examinations and discussions in which I have in- VI PREFACE. dulged will, I trust, be found to have been conducted with impartiality. In matters of science and demonstration it is ab- surd to entertain preferences and predilections. By unduly extolling a favorite author or a favorite method, a writer may, for a time, easily supplant the just claims of others ; and, as far as his influence extends, may actually impede the science he would wish to advance. In expounding the recent improvements in the doctrine of equations, I have attributed the first step in those improvements to BuDAN. This step consists in an extension of the theorem of Descartes; an extension which, although very obvious, and easily made, was nevertheless first publicly announced in the Nouvelle MHhode, &c. of Budan in 1807. The same theorem was afterwards given by Fourier, in his Analyse des Equations JDHerminees ; and pubHshed, after the death of the author, in 1 83 1 ; since which time it has been com- mon, with English writers, to call the extension in question the " Theorem of Fourier," without any regard to the prior claim of Budan. It would be difficult to discover upon what grounds this misappropriation is persisted in. It is certain that the leading men in France — Lagrange and Legendre, the colleagues and associates of Fourier — regarded the theorem as due to Budan ;* and there is no evidence to show that Fourier himself ever disputed the claim, although there are strong reasons for con- cluding that Fourier was led to the same thing by his own • The academicians appointed to examine the Memoire of Budan were Lagrange and Legendre, who make no mention in their report of any similar . theorem by Fourier. The report of the commissioners on Sudan's paper closes as follows : — " Nous croyons que le theoreme trouv^ par M. Budan m^rite I'attention de la classe, conime 6tant une extension de la regie de Descartes, et que son Memoire peut etre imprime dans le Recueil des Memoires presentes, accompagn6 du present rapport. Sign6 — Lagrange; i^ Legendre, rapporteur.'* See the Nouvelle M6thode of Budan, 2d edit. 1822. PREFACE. Vll independent investigations into the theory of equations. There is a passage in the Histoire de MatMmatiques of Montucla which justifies this conclusion. I have extracted it at page 151. It is very remarkable that Navier, the editor of Fourier's posthumous work, and all the other advocates of Fourier's priority, should have overlooked testimony so strongly confirma- tory of the position they have taken such pains to estabhsh. Perhaps the fairest way would be to consider the theorem in question as the common property of Fourier and Budan. But this theorem only partially accompHshes the object to which it is applied — the analysis of a numerical equation. Addi- tional principles were required to complete the decomposition thus partially effected. These were accordingly supplied both by Budan and Fourier : and at this point of the process the two methods become perfectly distinct and independent. This more advanced theorem of Fourier has, however, met with comparatively but little notice in this country. By the "Theorem of Fourier" is generally meant merely the pre- liminary theorem, noticed above, as common to Budan and/ Fourier; the additional principle, by which this theorem is perfected, and which in a peculiar manner displays the genius and resources of the author, being altogether overlooked. I have thought it necessary therefore to give a very full expo- sition of this second theorem of Fourier, which I have endea- voured to free from the principal imperfections which precluded its successful application beyond very narrow Hmits. These modifications are proposed and explained in the Ninth Chapter, and the importance of them practically illustrated in Chapters Eleven and Twelve. / I have also consigned to this twelfth chapter some new views and developments, which, as far as they extend, I beUeve to be Vlll PREFACE. accessions to our knowledge ; and to increase ^ur practical facili- ties in a difficult and delicate department of the subject. The methods proposed in this chapter for distinguishing real from imaginary roots, in cases of doubt, will be found to be simple in their theory and easy in their practical application. It is desirable that the doctrine of algebraical equations should be rendered independent of the more advanced principles of analysis. To establish Fourier's criterion for the testing of these doubtful cases, we must borrow assistance from the analytical theory of curves, or from the theorem of Lagrange on the limits of Taylor's series: both of which subjects involve the principles of the differential calculus. The methods here proposed effect the objects of Fourier's criterion by aid of only the common algebraic theories. Some apology may be necessary for a new term which I have ventured to introduce into these discussions : — the term im- perfect roots ; a name by which I have designated certain real values, by which a particular class of imaginary roots may be replaced, and which are shown to furnish real approximate solu- tions to the equation. Fourier had noticed the fact that imaginary roots divide themselves into two classes, distinguished by very marked peculiarities ; but I think he did not develop the theory of this interesting principle with sufficient fulness. I have considered it with more detail in the eighth chapter One of these two classes of imaginary roots suggests real values actually available in calculation, and having full claim to the character of approximate numerical solutions : these are the values that I have called imperfect roots, to mark their defect from that strict accuracy which belongs to the other real roots of the equation. The formulas given in this twelfth chapter for determining two roots of a numerical equation, after the others have been com- PRKFACE. IX puted, will be found useful on many occasions. One of these formulas has suggested a very simple investigation of a general expression for the solution of equations of the fourth degree : it will be found in the Eighteenth Chapter. This investigation might no doubt be derived from the general theory of symme- trical functions ; but the steps suppUed by that theory would be much more compUcated and difficult. In several other portions of the book I have also introduced particulars which appear to me to have claim to originahty. But I am anxious to express myself with all becoming caution and reserve in reference to these matters ; more especially as I have found, since the completion of the work, an anticipation of a principle which I had thought had hitherto escaped notice : I have made the proper acknowledgment in a note at the end. A cursory glance at the more advanced sheets of the present volume will show that the work abounds in calculations, several of which involve a good deal of numerical labour. It was neces- sary to enter upon these, in order to furnish a satisfactory view of the actual capabilities of the new methods. In these practical illustrations, as well as in the theoretical developments on which they are founded, I have tried to secure rigid accuracy : and although it is Ukely, that in such a large amount of calculation, one or two numerical errors may have escaped detection, I am persuaded that these, if any occur, are very few in number. It may be mentioned in conclusion, that in animadverting upon the methods proposed by Fourier and Budan, I have sought to establish the validity of my objections by practical illustrations, taken, for the most part, from the favourers of the methods themselves. These, however, I have considered it neces- sary carefully to recompute ; for I have found, from experience, that such calculations are not to be taken upon trust. Examples, incorrectly worked by the original proposers, or erroneously X PREFACE. copied by their immediate successors, are often transmitted from book to book, without any correction. The well known cubic equation, by which Newton illustrated his method of approxi- mation, is in this latter predicament : it is, in general, inaccu- rately calculated. I have given the correct figures at page 333. With this brief account of its general scope and intention, I submit the work to the impartial examination of those interested in the progress of the important department of analysis on which it treats ; I ought perhaps to say, to their indulgent examination ; since from the extent of ground gone over, much of which has been as yet but little trodden, and the innovations I have ven- tured to introduce into various parts of the subject, imperfections may have escaped me, which may render such indulgence neces- sary to secure to the book the favorable reception that was so largely given to the former edition. J. R. YOUNG. Belfast College; Oct. 1, 1842. CONTENTS. INTRODUCTION. ART. 1. Solution of equations the chief problem of algebra : difficulty of this solution by means of general formulas 2. General formulas investigated by the early Italian algebraists foi equations of the third and fourth degrees. These formulas not generally computable till developed in infinite series .3. Cardan's formula, when the values implied in it are all real, is in general only a compact symbolical expression for an infinite series ...... 4. Remarks on the inapplicability of Cardan's formula when the values of the unknown quantity are real 5. No advance made in the general solution of equations by algebraic formulas beyond the stages reached by Cardan and Descartes 6. Competency of arithmetical processes to effect the solution of nu merical equations of all degrees . 7. The solution of a quadratic reduced to the simpler case of the ex traction of the square root 8. Cardan's formula for the roots of a cubic analogous to Bernoulli' symbolical expression for a circular arc 9. Such forms useless in computation : but valuable for other purposes 1 0. The search after general formulas has perfected the theory of equa tions and prepared the way for present methods of numerical solution ..... 1 J. These methods purely arithmetical 12. Intention of the present work .... 13. Notation to be employed .... 14. Illustrations of the term /wwe^/ow 15. Derived functions .... 16. Algebraic functions divided into rational, irrational, integral, and fractional . . . . . . ib. ib. ib. ib, ib. 6 7 8 XU CONTENTS. ART. PAOE 17. These terms apply only to the quantities in reference to which the function is considered : the forms in which other quantities may combine with them not being regarded . . .9 18. Meaning of the term roo^ of an equation . . . . ib. 19. Meaning of the term solution of an equation :— meaning of the pro- blem of the general solution of algebraical equations . . 10 20. Meaning of the problem of the general solution of numerical equations . . . . . , ib. 21. Objects proposed to be accomplished in this treatise . . ib. CHAPTER I. Fundamental properties, preparatory to the general Theory of Equations. 22. Importance of Harriotts method of arranging the terms of an equa- tion : fundamental principle in the theory of equations . . 11 23. On rendering a polynomial numerically less than any assigned value 12 24. Any proposed term in a polynomial ascending by powers of x may be made numerically greater than the sum of all the terms that follow . . . . . . . 13 25. Any proposed term in a polynomial descending by powers of uc, may be made numerically greater than all the terms that follow . 14 26. Deductions from the preceding property useful in the doctrine of equations , . . . . .15 27. If in a polynomial /(j?), x vary continuously from x =^ a io x ■=. b, then /(a-) will vary continuously from /(a) to /(i) . . 1ft 28. One root at least always exists between two numbers which, when substituted for the unknown quantity in an arranged equation, furnish results with opposite signs . . .18 29. Every equation of an odd degree has at least one real root of sign contrary to that of the last term : and every equation of an even degree whose last term is negative has at least two real root«, one positive and the other negative . . . . ib. 30. Remarks on the impossibility of extending the theory beyond this point without the aid of additional principles . . . 19 CHAPTER II. On certain Imaginary Expressions. 31. Introductory observations . . . .21 32. Imaginary expressions spontaneously present themselves in analysis 22 CONTENTS. XUl ART. 33. Removable only by the operations indicated being neutralized by opposing operations . . . . . 34. Remarks on the office of signs of operation when connected with imaginary quantities ..... 35. Imaginary quantities connected with one another, and with real quantities by the law of continuity. Objections to the terms ima- ginary and impossible : instance of a really impossible algebraical condition in a problem . . . . . 36. Remark on the preceding digression 37. Imaginary quantities when combined together reproduce the same forms . . . . . . . 38. On the moduli of imaginary quantities 39. Cauchy's propositions respecting these moduli 40. Instance of the efficiency of imaginary expressions in the investiga- tion of the properties of real quantities 23 24 26 ib. 27 28 29 31 CHAPTER III. On the Property that every Equation has a Root. 41. Introductory lemma ..... 42. Investigation of the property that every equation has a root . 43. Proof that the real quantities which occur in the general expression for the root can never be infinite . . . . Remarks on an objection that might be made to the preceding reasoning" ...... Notice of the different versions that have been given of Cauchy's proof: — the preceding the most brief and simple 46. Certain irrational equations have no roots 44. 45. 33 34 37 38 39 40 CHAPTER IV. On the general Properties of Equations. 47. Introductory observations . . . .42 48. If a be a root of /(i) = 0, then/(a;; must be divisible by a; — a . ib. 49. On the importance of an expeditious method of executing this di- vision . . . . . .43 .50. Investigation of such a method . . . . 44 51. Examples of the operation . . . .46 52. Remarks upon the facilities of the preceding method . . 47 53. The remainder of the division expresses the value of the polynomial itself when a is put for x . . .48 L ^> XIV CONTENTS. ART. PAGE 54. An equation of the nth degree has n roots but not more . . 49 65. Deductions from this truth . . . .50 56. Every fractional power oi a-\-bs/—\ is of the same form . . 52 57. Imaginary roots enter equations in conjugate pairs . . 53 58 . On the decomposition of polynomials into real quadratic factors . 54 59. When the roots are all imaginary every substitution for a; must give a positive result . . . , , 55 60. Determination of the forms of the functions which the coefficients of an equation are of its roots . . . . ib. 61. The signs of all the roots are changed by changing the alternate signs of the equation . , . . . 58 62. Tf the leading coefficient be unity, and the others all integral, the equation cannot have a fractional root . . .59 63. On the importance of determining a priori the number of real roots in an equation . . . . . . 60 64. Demonstration of the rule of signs . . . .61 65. Use of the rule in incomplete equations . . . . 63 66. Certainty of the rule when all the roots are known to be real . 64 67. Generalization of the rule of signs so as to include incomplete equations . . . . . .65 68. On the indications of imaginary roots in incomplete equations . ib. 69. Completion of the rule of Descartes . . .67 CHAPTER V. On the Transformation of Equations. 70. The transformation of equations useful to facilitate their solution . 71 71. To change an equation into another whose roots shall be less or greater than those of the proposed equation by a given quantity . . . . . .72 72. Examples of the foregoing proposition . . . . 73 73. To change an equation into another whose roots shall be the reci- procals of those of the former . . . .76 74. On reciprocal equations . . . . . 78 75. To transform an equation into another whose roots shall be given multiples or submultiples of those of the original . . 79 76. Important connexion between fractional roots and the extreme coefficients of an equation . . . . . 80 77. To remove the leading coefficient without introducing fractions . 82 78. To effect the transformation when the roots are to be submultiples of the original roots . . , . .84 79. To remove the second term from an equation . . . 85 80. To remove any other term . . . .86 CONTENTS. XV ART. PAGE 81. Observations on the attempts to render equations solvible by means of removing intermediate terms . . . . 87 82. On the principal applications of the foregoing propositions . 88 CHAPTER VI. On determining Limits to the Real Roots of Equations. 83. Preliminary observations on limits . . . .91 84. Definitions of superior and inferior limits . . . 92 85. When the second term is negative, and all the other terms positive, the coefficient of the second term taken positively exceeds the greatest positive root of the equation . . .93 86-90. Limits of Maclaurin, Vene, Bret, &c. , . 94-100 91. If tvs'o numbers substituted for the unknown in an equation give re- sults with different signs, an odd number of roots lies between the numbers substituted : if they give results with like signs, an even number of roots lies between them . . .103 92. Method of detecting the places of the real roots by help of the pre- ceding proposition . . . . . 104 93. On limiting equations ..... 105 94. Method of determining the limiting equation . . . 106 95. Relation between the number of real roots in the limiting equation and the number in the proposed . . ' . .111 96. Consequences ;deduced by De Gua . . . . ib, 97. Theory of vanishing fractions .... 113 98. Theory of equal roots . . . . . 116 99. Method of detecting the existence and number of equal roots noticed byHudde . . . . . .121 100. Observations upon the practical difficulties of the common theory of equal roots . . . . . * ib. 101. Simpler methods proposed . . . .123 L CHAPTER VII. On the Separation of the Roots by the Method of Budan and Fourier, 102. Newton's method of finding a number greater than the greatest root of an equation . . . . . 127 103. Another mode of arriving at Newton's result . . 128 104. The same conclusions established otherwise . . . 131 132 133 134 135 136 138 139 141 142 147 148 XTl CONTENTS. ART. 105. By increasing the roots of un equation we may always obtain result presenting variations of sign only 106. Inferences from the foregoing principles 107. Theorem of Budan . . . • 108. Particulars involved in the theorem 109. On the indications of imaginary roots 110. Imaginary roots indicated by zero-coeflBcients in the transformed equations ..... 111. Fourier's rule of the double sign 112. Directions for ascertaining the nature and situations of the roots of an equation .... 113. Remarks on the doubtful intervals. Examples 114. Determination of the least number of imaginary roots in incom plete equations .... 1 ] 5. The exact number found in binomial equations . 116. The coefficients of the transformed equation arising from diminish ing the roots of the proposed by r are the several divided derived functions written in reverse order, and having r in place of x . ib. 117. Observations on the claims of Budan and Fourier to the theorem at (107) . . . . . .150 CHAPTER VIII. Analysis of Equations from Geometrical Considerations : Method of Fourier. 118. Connexion between algebraic polynomials and curves . .152 119. Mode of representing a polynomial geometrically . .164 120. Geometrical indications of the real roots of an equation . 156 121. Geometrical peculiarity of equal roots : of imaginary roots . 158 122. Connexion between real and imaginary roots. All the imaginary roots not indicated by the curve. Important deductions . . 160 123. On the analysis of doubtful intervals. Criterion of Fourier . 163 124. Consideration of the criterion of Fourier under certain re- strictions . . . . ... Itt5 125. Mode of applying it . . . . . 166 126. Practical examples . . . . .167 127. Consideration of the criterion when all restriction is removed . 169 128. Examination of the roots of the derived functions . .172 129. Examples of the general application of Fourier's method . 173 130. Summary of the necessary steps in the analysis of an equation . 177 CONTENTS. XVU CHAPTER IX. Remarks on the Method of Fourier ^ with Suggestions for its Improvement. ART. PAGE 131. On the general character, peculiarities, and defects of the method of Fourier . . . . . . 179 132. The principal imperfection in Fourier's method common to all preceding rules for analysing an equation • . .180 133. On the kind of equations practically unmanageable by Fourier's method . . . . . . 182 134. Features that might be expected to characterize any method that should in all cases involve the same amount of labour, whatever be the nature of the roots . . . . ib. 135. Extra assistance required in Fourier's method to render the sepa- ration of the roots alw^ays practicable : features common to the methods of Lagrange, Fourier, and Budan . . . 1 83 136. On a new method of ascertaining whether or not equal roots exist in an equation ..... 137. Application of this method to Fourier's examples . . .186 138. Instance of the great labour involved in the operation for the com- mon measure. The theorem of Sturm clearly discovers the im- perfection of Fourier's method . . . .187 139. Mode of applying the new precepts for equal roots in peculiar cir- cumstances ■■ , . . . . . 188 140. On the facilities thus introduced into the practice of Fourier's analysis . . . . . .189 141. Additional simplicity introduced . ... 190 142. Another obvious means of improvement overlooked by Fourier . ib. 143. Notice of a method for finding two roots of an equation when the others have been determined . . . .191 144. Recapitulation of the more important particulars connected with Fourier's process, improved as here suggested . .192 CHAPTER X. Method of Budan for Analysing Doubtful Intervals, 144. Statement and explanation of Budan 's method . .195 ib. Examples . . . . . . 198 145. Comment upon the preceding method : difficulties attendant upon it 202 146. Relative advantages of the methods of Fourier, and Budan in the actual solution of equations .... 205 XVm CONTENTS. CHAPTER XL Method of Sturm : comparison of it with the Methods of Budan and Fourier ART. PAOE 147. On the peculiarities of Sturm^s method . . . 207 ib. fnvestigation of the principles on which Sturm's theorem depends 208 148. Demonstration of the theorem . . . . 210 149. Mode of applying it : important deductions from it . . 213 150. Application of the theorem in the case of equal roots . , 220 151. Simplifications of the operation .... 221 152. Shortest and most convenient method of performing the numerical calculation . . . . . . 223 153. Example of Sturm's analysis .... 227 154. The same example analysed by the methods of Fourier and Budan ...... 228 ib. Second example analysed by Sturm's method . . . 229 155. Applications of the methods of Fourier and Budan to the same example ...... 231 156. Mode of reducing the extent of figures in Sturm's process: Example . . . , . . 232 157. Applications of the methods of Fourier and Budan to the preceding example . . . . . , 235 158. The three methods compared . . . .241 1 59. Practical illustration of the foregoing remarks : Example by Sturm's method . . . . . . 242 1 60. The same example by Fourier's method . . . 243 ib. The same by Budan's method very tedious . . . 245 161. Concluding remarks on the merits of the foregoing method . ib. CHAPTER XII. Solution of Equations of the Higher Orders. 162. Observations on Horner's general method . . . 248 163. Brief explanation of the operation . , . . 249 164. Useful remarks on the efficiency of the approximation : and on the best way of contracting the work . . . 2 30 165. Numerical examples . . . . . 252 166. On the mode of arrangement adopted by Mr. Horner himself . 257 ib. CONTENTS. XIX ART. I'AGE 167. Important particulars to be attended to when roots have leading figures in common . . . . • 259 168. The preceding case provided for (see Note at the end) . . 260 169. The foregoing inferences otherwise deduced . . .261 170. The tediousness of Fourier's method made apparent by these in- vestigations ....•• 263 171. Means of prosecuting the development when roots have leading figures in common . . . • . 172. New precepts for this purpose .... 265 173. On some peculiarities in the general theory of the trial-divisors . 266 174. Difficult example of the fifth degree . . . . 268 175. The same example differently solved . . • 272 176. Difficult example of the sixth degree, amply illustrating the efficacy of the precepts in delicate cases . . . • 277 177. Second solution to this example .... 288 179. These examples fully confirm what has been stated in reference to the efficacy of the methods of Sturm and Horner . . 296 180. Commentary upon the improvements here proposed . . ib. 181. Mode of proceeding in unusual cases . . . • 298 182. Method to be preferred in analysing a doubtful interval . ib. 183. The character of the interval may discover itself in various ways . 299 184. Efficacy of the new methods proposed in this chapter for distin- guishing imaginary roots from real . . . 300 185. Recapitulation of the preceding improvements in reference to Fourier's method . . . . . ib. 186. The combination of principles here expounded sufficient for every case . . . . . .302 187. No absolute necessity for Fourier's test . . . ib. 188. Examples analysed by aid of the preceding principles . . ib. 189. Second example . . . . . . 304 190. Further illustration of the theory . . . .305 ib. Remarks on the principle at p. 163 : imaginary roots replaced by real : apparent paradox . . . • . 307 191. Practical illustration . . . . . ib. 192. Exhibition of the real root which ought to replace the imaginary root . . . . . . . 309 193. Proposal to call this new class of roots imperfect roots . .311 194. Brief summary of the three efficient methods discussed in this chapter for determining the true character of doubtful roots . 312 195. The practical facility of these methods dependent mainly upon the ease with which a single isolated root may be developed . 313 196. Remarks on the abbreviations introduced into Sturm's analysis : imperfect roots determined by the abbreviated process . ib. 197. Proof of the adequacy of Sturm's method independently of every other theorem . . . . .317 .V -^ XX CONTENTS. ART. PAGE 198. Method of determining the remaining two roots of an equation after the others have been computed . . . . 320 199. Newton's method of divisors .... 324 200-201. Inferences and examples , . . . 326 202. Method of diminishing the number of trial-divisors . . 329 203. Example of this method . . . . . 330 204. Newton's method of approximating to incommensurable roots . 331 205. Process of Newton compared with that of Horner . . 335 CHAPTER XIII. Recurring and Binomial Equations. 206. Recurring equations always susceptible of reduction to inferior degrees . . . . . . . 339 207. Method of obtaining the reduced equations . . . 340 208. Examples . . .... 341 209. Equations of an even degree, whose equidistant terms are equal, but have unlike signs, may be converted into recurring equations 344 210. Binomial equations ..... 345 211. Some obvious properties of these equations . . . lb. 212. Every power of an imaginary root of the equation x'^ — 1 = is also a root ...... 347 213. Every odd power of an imaginary root of the equation x» + 1 =0 is also a root . . . . . • 348 214. Exhibition of the roots of x'* — 1 = when n is a prime number . 349 215. When p and q have no common measure, the equations ip — 1=0 and xi — 1=0 have no common root except unity . . 350 216-17. On the roots of a" — 1=:0 when n is a composite number . ib. 218. Roots of x" — 1=0 when n is the square of a prime number . 352 219. Determination of the roots of x** — 1 =0 by De Moivre's theorem 354 220. Determination of the roots of x" -j- 1 = by the same theorem . 358 CHAPTER XIV. On Continued Fractions. 221. Explanation of the term continued fraction . . . 360 222. Method of converting a rational into a continued fraction . 361 223. Application of the method . . . . . 363 224. On converging fractions . . . • . ib. CONTENTS. XXI ART. PAGE 225. Propriety of the term, converging fractions . . . 3tf6 226. Limit of error involved in these converging fractions . . 368 227. Another limit to the error . . . . . 371 228. Development of an irrational quantity in a continued fraction . 372 229. Application of continued fractions to the summation of infinite series : and to the solution of equations . . . 374 230. Remarks on the last application . ... 384 CHAPTER XV. Theory of Elimination. 231. Elimination between two equations containing two unknown quantities ...... 385 232. Discussion of the consequences arising from the suppression or introduction of factors . . . . . 388 233. Case in which a value oi y destroys a factor introduced . . ib, 234. Case in which a value oiy destroys a factor that has been suppressed 389 235. Conclusions from the preceding discussion . . . ib. 236. General examination of a pair of equations preliminary to actual solution . . . . . .390 237. Process of solution when no factor is necessary to render the divi- sion practicable . . . . . . 391 238. Process of solution when the introduction of a factor is necessary 392 239. Applications of the preceding principles . . . 393 240. On irrational equations . . . . . 399 241. Method of Tschirnhausen for solving equations . . 401 242. Application of this method to the general equation of the third degree . . . . . . . 402 243. Application of the method to the general equation of the fourth degree ...... 403 244. On the equation of the squares of the differences . . . 405 246. To find a limit less than the least of the differences . . ib. 246. Remarks on the proposed method of Lagrange . . . 406 247. Investigation of the equation of the squares of the differences . 407 248. Application of the method to the general equation of the third degree ...... 408 249. On the connexion between Sturm's final remainder and the num- ber of places in which nearly equal roots concur . .411 2.50. Connexion between the same remainder and the number of places in which roots of the derived functions concur . ,412 U 4 XXU CONTENTS. CHAPTER XVI. On the Symmetrical Functions of the Roots of an Equation. ART. PAGE 251. Symmetrical functions explained . . . .414 252. Determination of tlie sums of the powers of the roots of an equation 415 417 253. Extension of the preceding formulas 254. Expressions for the coeflBcients in terms of the sums of the powers of the roots of the equation . . . . 418 255. On double, triple, &c. functions of the roots . .419 256. To transform an equation into another whose roots shall be given functions of those of the original equation. Example . 421 25T. Example of other combinations . . , . 424 258. Determination of the equation of the squares of the diflFerences by this method . . . . . ib. 259. On the degree of the final equation resulting from the elimi- nation of one of the unknowns from two equations containing , two unknowns ..... 427 CHAPTER XVII. On the Determination of the Imaginary Roots of Equations. 260. Method of Lagrange by means of the equation of the squares of the differences . . . . .431 261. Mode of proceeding independently of the equation of the squares of the differences . . . . . . 437 CHAPTER XVIII. On the Solution of Cubic and Biquadratic Equations by General Formulte. A. 262. Introductory observations .... 440 263. Method of Cardan for cubic equations : remarks on the irreducible case . . . . . . . 441 CONTENTS. xxiii 264. Formula for the remaining two roots when one is determined : forms of the three roots in the irreducible case . . 44fi 265. Solution of the irreducible case by a table of cosines . . 44T 266. Euler's method of solving a biquadratic equation . . 449 267. Method of Louis Ferrari . . . . . 451 268. Method of Descartes . . . . .452 269. A new method . . . . . . 454 270. Euler's form deduced from the preceding . . . 456 CHAPTER XIX. Solution of Equations by Symmetrical Functions. 271. Equation of the third degree 272. Equation of the fourth degree 273. On the extension of the method to the higher equations 274. On an imperfection in Cardan's formula 275. A similar imperfection in Euler's formulas 276. A similar imperfection in the formulas of Ferrari 277. Remarks on a note of Poinsot 278. On De MoivTe's solvible form 458 464 468 470 471 472 ib. 47.3 \ Note on a principfe established in Chapter Xfll. . . .474 ERRATA. Page 31, line 19, for a and b read b and a. 60, to line 1 add " if the roots are all real." 168, lines 8 and 11, /or right read left, and /or left, right. 170, line 3, /or/, (J?) read /,(jr)=0. 177, head line, /or Sturm read Fourikr. 256, in second column dele 4 and /or 5 read 4. 266, line 15, after root-figure add " or that figure plus 1. 300, line 25 /or 300 read 298. 444, line 14, supply .... [5] THEORY AND SOLUTION ALGEBRAICAL EQUATIONS. INTRODUCTION. (Art. 1.) The great object of all computation is the determi- nation of numerical values for unknown quantities by help of the given relations which they bear to other quantities already known ; and it is the ojffice of algebra to express these relations in a sym- bolical form, by means of the marks and signs which constitute the notation of that science. Every such algebraical expression of the conditions which con- nect the known and unknown quantities together, in any mathe- matical enquiry, when reduced to its most convenient form, furnishes us with an equation; and it is thus that the solution of equations, as the evolution of the unknown quantities involved in them is called, becomes the chief business of algebra, and the grand problem to the fuU discussion of which all its rules and processes are merely subsidiary. But such is the difficulty of the subject that all the resources of algebra have been hitherto found inadequate to effect the solution of it in general and finite terms — that is, without the aid of infinite series — except in the simple cases where the unknown quantity rises to no higher a degree than the second. The rigorous solution of an equation of the second degree, by means 1 2 INTRODUCTION. of a general formula for the values of the unknown quantity, is readily accomplished by common algebra. But the discovery of a like formula for equations of the third degree has hitherto resisted every effort. The same may be said with respect to equations of the fourth degree, while for those of the fifth and more advanced orders it has been demonstrated that the existence of any such general formula is impossible. (2.) It is true that the early Italian algebraists, Ferrari, Tartaglia, and Cardan, investigated general methods for equations of the third and fourth degrees, and that these methods, expressed in the notation of modern algebra, furnish formulas which do really represent the sought values in terms of the coefficients of the equation and under a finite form. But, for certain relations among the coefficients, these formulas, and all others that have since been proposed for a like purpose, involve imaginary expressions, which, except in certain particular cases, render the actual computations impracticable till the formulas are developed in an infinite series ; and the imaginary terms, by mutually opposing one another, become exterminated. (3.) It is observable that the well-known formula for equa- tions of the third degree — usually called the formula of Cardan, and to which every other known method is reducible — is gene- rally in the predicament here described whenever the values of the unknown quantity are all real. The formula which represents these values, and from which they are to be evolved, turns out in nearly every such case to be merely a compact symbolical expression for an infinite series; and it is only from this series, and not immediately from the undeveloped formula itself, that the real values concealed under the imaginary expressions which enter the formula can be computed or approximated to. (4.) This failure of Cardan's formida to solve the cubic in a general form, when the values of the unknown quantity are all real, has always been regarded as a very remarkable and often indeed as an anomalous circumstance in the doctrine of equa- tions. But it will be shown hereafter, when we come to that INTRODUCTION. 3 department of our subject to which the discussion of the general formulas proposed for the solution of equations belongs, that this circumstance might have been anticipated, as a necessary conse- quence of the laws of algebra; and that the actual exhibition of the real values, in a finite computable form, by Cardan's method, would be a contravention of those laws, and therefore a really anomalous occurrence. Thus the method becomes essentially inapphcable as a general method whenever the values of the un- known quantity are all real. (5.) Every general form for equations of the fourth degree, involves in it the above-mentioned subsidiary expression for equa- tions of the third ; and thus, without extending our enquiries to equations of the higher orders, we are compelled to acknowledge that algebra has not as yet rewarded the labours of analysts with effective formulas for the solution of equations of the third and fourth degrees, by means of which the real values may always be computed without the aid of infinite series. As far as the dis- covery of such formulas is concerned, no advance of consequence has been made beyond the point reached by Cardan and Descartes ; and the search after general symbolical forms for the solution of the higher equations, as also the attempt to im- prove and rationaUze those already alluded to, are occupations which have always proved so barren of success that analysts have at length ceased to prosecute their enquiries in reference to these objects. (6.) But although the solution of equations above the second degree by general algebraical formulas competent to supply the unknown values under all relations of the coefficients has thus been found to be impracticable, yet the actual evolution of these values by common arithmetical processes may always be effected whenever such values really exist; so that, however the powers of algebra may in general transcend those of arithmetic, we have here, in the problem of the general solution of equations, a very remarkable example of the fact that we may arrive at numerical results by certain general and uniform arithmetical operations which, although very simple in their character, it is not within 4 INTRODUCTION. the power of algebra, with its present symbols of operation, to represent in any finite formula. In fact the operation by which the numerical solution of equations of all orders is now effected, is an inverse operation of so fundamental and purely elementary a character that it does not seem to admit of analysis into any simpler and subordinate processes, except in the single case of the quadratic; so that to represent the general operation symboH- cally it would seem that we must devise a new notation for that purpose, and not attempt the representation by combining to- gether other supposed component forms. (7.) The numerical evolution of the values of the unknown quantity in a quadratic equation is an operation of which the arithmetical extraction of the square root is but a particular case; so that this latter process may be regarded, theoretically, as one of a simpler and more elementary character; and to such a process we really can reduce the operation for solving a quadratic equation, as the common algebraical formula for quadratics shows. Our success in this particular class of equations is owing entirely to the power we possess of making the unknown member of every such equation a complete square, by the introduction of a certain known quantity. The case is very different with equations of the next higher and still more advanced degrees : Cardan's for- mula for cubics — admitting its general irreducibility when the values sought are real — so far from furnishing a finite algebraic expression for those values in terms of the coefficients, actually proclaims that the existence of such an expression is an impos- sibihty. (8.) Cardan's formula, if we except a few particular cases, effects no more for cubics whose roots are all real, than a cer- tain well-known imaginary expression for a circular arc effects for the rectification of the circle.* Each formula is but a compact symbolical expression for certain infinite series. • The expression here alluded to is the following, due to John Bernoulli, Yi5, ^ _ 2 log n/^ n/ITT INTRODUCTION. 5 (9.) Such forms, however, although utterly useless for the purpose of actual computation, are nevertheless valuable on other grounds; and although formulas, like the irreducible case of Cardan's, if they could be discovered for the higher equations, would be of no value in the numerical solution of such equations, yet they would be received as important acquisitions into other departments of analytical enquiry ; since they would enable us to exhibit, at least symbohcally, the elements of which every ra- tional and integral algebraic polynomial is composed, however difficult it might be to determine when these symbolical forms stand for real quantities, and when they are purely imaginary. (10.) The great labour which analysts have from time to time bestowed upon this research after general algebraic forms for the solution of the higher equations, although wholly unsuccessful, or very nearly so, as respects the ostensible object of enquiry, has been rewarded by the discovery of most of those interesting and important truths which constitute the general theory of equations, many of which have directly contributed to advance towards its present perfection the method of numerical solution^ which the recent efforts of Budan, Fourier, Horner, and Sturm, have rendered entirely effective and general. (11.) This method, as already noticed, is a purely arithmetical process, performed upon the numerical coefficients of the pro- posed equation, universally applicable without regard to the degree of the equation, and altogether independent of any such general algebraic model, or formula, as that which analysts have so long sought in vain to discover. (12.) It is our intention in the following pages to present a connected, and, as far as we are able, a perspicuous view of the researches just adverted to, with such modifications and additions as appear to us to be real improvements, calculated to increase our facilities in the analysis and solution of the higher equations. The more elementary details, as far as equations of the first four degrees are concerned, have already been discussed with sufficient 6 NTRODUCTION. copiousness in an introductory volume.* But in order to render the present work complete, as a comprehensive exposition of the modern theory and solution of algebraical equations, it will be ne- cessary to resume and generalize some of the theoretical investi- gations given in that Introduction ; the previous study of which, however, as an epitome of the leading topics in the present volume, is earnestly recommended to the student. f (13.) Before terminating these preliminary remarks, it will be necessary to say a word or two as to the notation to be generally employed in this treatise ; and to explain the sense in which cer- tain terms of frequent occurrence in the higher departments of analysis will be used in the following pages. The notation for the different classes of equations involving one unknown quantity will usually be as follows, although we shall occasionally depart from it in the advanced parts of the work. A simple equation will in general be expressed thus : Aa? + N = 0. A quadratic equation, A2 a?2 + Aa? -h N = 0. A cubic equation, Aga^^ Kx^ + Aa? -hN =0. A biquadratic equation, A4 a?4 + A3 a?3 + A2 a?^ + Aa; -f N = 0. And, in general, an equation of the wth degree will be written, A^ a?" . . . . + A3 ar^ -i- A2 a:'-^ H- Aa: -f N = 0; in which the absolute term N, and the coefficients A, Ag, A3, &c. • The Analysis and Solution of Cubic and Biquadratic Equations, 1842. t In tlie course of the following work we shall presume the reader to be in some degree acquainted with the introductory treatise alluded to, and shall occasionally refer to it for some practical details respecting equations of the third and fourth degrees, which need not be repeated here. INTRODUCTION. 7 usually represent real numbers, either positive or negative, in- tegral or fractional. The polynomial on the left of the sign of equality we shall frequently call the^r*^ side or the first member of the equation. (14.) It is common in algebraical enquiries, involving frequent reference to complicated expressions, to designate those ex- pressions by some more brief and commodious form ; and to faci- litate this abridgment, a new word, the word function, has been introduced into algebra, and represented symboHcally by the initial letter/, or F or ^, or/, ^^fv ^^• Thus any expression involving x, as, for instance, the left-hand member of either of the foregoing equations, is called, in brief, a function of x, and represented by one or other of the forms f{x), Y{x), {x), Hx), fix), r(x), fix),&c. when, however, one of these forms is fixed upon to represent any algebraical expression, it is plain that, in order to avoid con- fusion, we must adhere to that form of representation throughout the enquiry ; and must not employ the same form to characterize other expressions, or other functions. If, for example, we agree to represent the foregoing general equation of the nth degree by f(x) = 0, we are not afterwards at liberty to represent any other different function, occurring in the same enquiry, by the characteristic /, any more than we are at hberty to denote two different magnitudes by one and the same algebraical character. We see, therefore, that while the term function has the most extended signification, comprehending all algebraical combinations possible, yet, by varying the form of the initial letter, or characteristic, which stands for the word, the various forms of functions may all be represented in the proposed notation by distinctive symbols. (15.) The expression /(a?) =0, which we have just employed to denote, in short, the general equation of the nth degree, in- cludes in it, of course, all the particular equations written above, as n may be any positive and integral exponent whatever. The symbol f{^) or fi{x), &c. denotes, as already remarked, a func- 8 INTRODUCTION. tion of the same quantity, Xy although different from the function f{x) ; yet, as the preceding forms are derived from this last, by simply supplying an accent, or subscribed numeral, they are the forms usually employed to express functions derived from, or de- pendent on, a primitive function /(a?). For example, if the func- tion ax^ + bx^ be represented by /{x), and we have occasion to exhibit the successive quotients which arise from dividing this primitive function by x repeatedly, it would be convenient to use the following notation: f{x) =:ax^ -\- bxr> /j(a?) =aci^ -\-hx^ /2{x) = ax* + bx^ /^(x) = aa^ + bx^ &c. &c. where /(x) is the primitive, and the others the derived functions, each being derived from the preceding, by a repetition of a known process, viz. the process of division hy x. Again, suppose we had to deduce from the function, 3x* + 5a^ — 2x^ + 7x — 1 2, a series of others in succession, by the following uniform process, viz. each term in the derived function is to be deduced from the cor- responding term in the preceding function by multiplying that term by the exponent of x in it, and then diminishing the ex- ponent by unity ; the several functions would be as follows : primitive function, / (x) = 3x* + 5x^ — 2x^ + 7x — 12 1st derived function, /i(a?) = I2a^ + \5x^ — 4x + 7 2d derived function, f'X^) = S6x^ -f 30a? — 4 3d derived function, f^{x) = 72x + 30 4th derived function, /^(x) = 72 This last expression, 72, not containing x, cannot in strictness be regarded as a function of that quantity ; its symbolical repre- sentation, however, f^{x)y carrying the subscribed numeral 4, informs us that it has arisen from four repetitions of some uni- form process to a primitive function, f{x) . INTRODUCTION. 9 If in any function we change the quantity of which it is a function for any other, preserving however the form of the func- tion unaltered, then we must introduce a hke change in the abridged representation, merely altering the letter inclosed in the parenthesis, without changing the characteristic outside : thus, if /(a?) denote as in the last example, ihenf('i/),f{a)i &c. will be the respective representatives of 3y4 + 5y^ — 2f + 73/— 12, 3«4 + 5«3 — 2«2 + 7«— 12, &c. (16.) By a rational function of any quantity is to be under- stood an algebraical expression into which that quantity enters only in a rational form, that is, without the encumbrance of fractional exponents or radical signs. If the quantity enter the expression with any such appendage, that expression is an irrational function of the quantity. And by an integral function of a quantity, is meant an ex- pression into which the quantity enters only in an integral form ; that is, it never occurs in the denominator of a fraction: wherever it does so occur, the expression involving it is a, fractional function of that quantity. (17.) In thus classifying algebraical functions it is plain that we have regard only to the quantity, or quantities, in reference to which the function is considered; no attention being paid to the forms under which other quantities may enter the ex- pression. The left-hand member of the foregoing equation of the nth degree is a rational and integral function of x, whatever be the constitution of the coefficients with which x is connected. (18.) The expression root of an equation is apphed to every quantity, whether real or imaginary, which, when substituted for the unknown, actually reduces the first member of the equation to zero, thus satisfying the condition implied in the equation : so that if there exist p quantities, which when substituted for x in the polynomial f(x), reduce that polynomial to zero, then the equation /(a?) = has those |) quantities for its roots. 10 INTRODUCTION. (19.) The determination of all the roots is called the solution of the equation : and the problem which has for its object the determination of all the roots of an equation by a general for- mula, applicable to all particular values of the coefficients without restriction, aims at the discovery of such a function of those coefficients as will in itself embody all the values of x, both real and imaginary. This is called the problem of the general solution of algebraical equations. We have already observed that the search after such a function has been attended with complete success only as respects equa- tions of the first and second degrees. (20.) The problem which furnishes directions for evolving all the real roots, one after another, in numbers, by aid of the given numerical coefficients which any particular equation may ofier, is called the problem of the general solution of numerical equations. It is a problem which, like the former, has exercised the talents of the ablest analysts of all countries for the last two hundred years : but the satisfactory completion of it is an achievement of very recent date. (21.) In the present treatise we propose to ourselves the ac- compHshment, in moderate space, of the four following primary objects : We shall endeavour first to develop the theory upon which both the general problems just noticed equally depend : secondly, to explain the principles of the numerical solution, in connexion with the recent researches and improvements by which that solution has been perfected ; showing the practical efficiency of these principles by their successful appHcation to advanced equations of very considerable difficulty: thirdly, to discuss, with sufficient detail, the other and more general pro- blem, in so far at least as any real approach has been made towards a successful solution of it : and lastly, to blend with these leading and paramount topics, certain collateral and sub- sidiary enquiries usually expected to have a place in every treatise on the theory of equations. CHAPTER I. FUNDAMENTAL PROPERTIES, PREPARATORY TO THE GENERAL. THEORY OF EQUATIONS. (22.) The simple expedient, first adopted by Harriot, of arranging all the significant terms of an equation upon one side of the sign of equality, and leaving merely zero on the other side, has proved a preparatory step of considerable importance in the theory and analysis of equations. In the actual determination of the roots of an equation such a preliminary arrangement of its terms is not always necessary : but in the antecedent examination as to whether the things called roots necessarily exist for every equation; in the search after the number and nature of these roots; their connexion with the coefiicients; and, in short, in all en- quiries into the structure of equations, the preparation of Harriot must always form the initial step in the investigation. That every equation has a root, either real or imaginary, is a prin- ciple which Harriot and succeeding algebraists have, till lately, assumed. But, as this is the fundamental principle upon which nearly the whole theory of equations is based, it is of importance that it should be firmly established by a rigorous demonstration. Several attempts have accordingly been made, of late, to supply such a demonstration. Of these the most recent, probably the most satisfactory, and unquestionably the most simple and ele- mentary, is by Cauchy ; it is that which we shall adopt, in sub- stance, in the present exposition. We have observed that this principle is the foundation of nearly the whole of the present theory of equations. It is proper to make this slight quaUfication, because two or three interesting 12 GENERAL THEORY OF EQUATIONS: propositions belonging to this theory may be readily estabUshed without its aid. These it will be convenient to dispose of before entering upon a demonstration of the principle referred to : and it will be farther necessary to establish some prehminary theorems respecting polynomials in general, which theorems are in frequent request in analysis, and are indispensable here as lemmas to the principal proposition. They are as follow : PROPOSITION I. (23.) In any polynomial f{x) = k^x + k^'^ + k^oi^ + A^a?n in which all the coefficients are real and finite quantities, and which does not contain any term independent of x, it will always be possible to assign a value to a?, such as to render the entire expression less in value than any proposed quantity L. Let Ajj be the greatest coefficient without regard to sign ; then for every positive value of a?, the proposed polynomial will neces- sarily be less than Aj^ (a? + a?2 + a^ + . . . . a?") But if a; be less than unity, the series within the parenthesis will only be equal to , even should it go on to infinity (Alffebra, art. 76). Hence, when x is less than unity, we must have Consequently, the proposed condition f{x) < L will be fulfilled, provided either X X ^ 1 X 1 — X From the first of these conditions we have FUNDAMENTAL PROPERTIES. 13 from the second Hence the condition /(a?) < L may always be satisfied. The student must be careful not to infer from this proposition more than the reasoning warrants. The only object has been to show that a value of a? [1] exists, for which, and for every smaller value, [2], the condition /(a?) < L is necessarily satisfied. But whether or not other values exist, too large to come within the conditions [1], [2], which nevertheless satisfy the inequality f{x) < L, we are not authorized to say from anything that is proved above. Generally speaking such other values do exist ; but at present we are only interested in the fact that a value for X sufiiciently small may be assigned such that it and all values below it, down to zero itself, when severally substituted for a-, cause y(a?) to become smaller than any proposed quantity. (24.) It is obvious that the preceding demonstration applies to the case in which /(a?) is an infinite series of the proposed form, n being indefinitely great; provided, as above, that the coefficients are aU finite. Consequently, whether the series be finite or infinite, we may always give to a? a value sufiiciently small to render any proposed term in it numerically greater than the sum of all the terms which foUow : that is, Ap a?P being any proposed term, we may always satisfy the condition Ap^P > (Ap+i «^P-^^ + Ap^2^P+2 +....) For the series within the parenthesis, which may be written ^P(Ap-4-i^ + Ap+2^' + - • • •) may be rendered less than a?PL, L being any finite quantity. Let L be equal to Ap ; then we can fulfil the condition (Ap^,a.P+i+Ap^2^P+2 + ....) A^ (a?"-i . . . . a^ + x^ -\-x -{■ \) or, summing the geometric series, a?"— 1 A„a?" > A x-^ 1 or. A^ a?° — 1 A„ a^-1 And this is evidently satisfied provided that x be such as to render x—\ either equal to, or greater than, — ^- Hence to fulfil the proposed condition we have only to assume x so that a; = or > -^ + 1 K which we may of course always do. It thus appears that, the leading term of the polynomial /(a?) being positive, we can always give to a? a positive value a such that /(a) shall necessarily be positive, whatever be the values or signs of the subsequent coefficients ; or however we alter the signs in any proposed case. FUNDAMENTAL PROPERTIES. 15 Hence if the same value a be taken negatively instead of posi- tively, /( — a) will still be positive, provided n be even; because f{ — a) will differ from /(«) only as respects the signs of the terms after the first. But if n be odd, then when — « is put for X the leading term will be negative : and since, as just shown, this leading term will be numerically greater than the sum of all that follow, we infer that in this case/(— «) must be negative. (26.) If the polynomial considered in the present proposition were the first member of an equation, that is, if we had /(a?) = 0, we might remove the coefficient A^^ by division, without disturb- ing the condition implied in the equation : this is usually done in discussing the properties of equations, for the purpose of avoiding all unnecessary compUcation in expressing their general forms. Considering A^^ to be unity, conformably to this practice, A^ representing that coefficient which is numerically the greatest, as before, we may conclude, from what is shown above — 1. That the first member of the equation /(a?) = will always be positive if for x we put the positive quantity A^ + 1 or any greater value. 2. That the first member will in hke manner always be posi- tive if for X we put the negative value — (A^ +1), or any nega- tive value still greater, provided the equation be of an even degree. 3. And that the first member will be negative for the sub- stitutions in last case provided the equation be of an odd degree. These conclusions lead to important truths. They show that in every equation of an odd degree two values can always be found, which, when separately substituted for the unknown quantity, will furnish two results with opposite signs ; and that in every equation of an even degree, two such values can also be assigned whenever the final term, or absolute number, is negative. For in this case the substitution of zero for x will give a negative result, viz., the absolute number itself, and the substitution of either + (Aj^ -f- 1) or — (A^ -}- 1) will give a positive result. From these inferences it may be proved without difficulty that every equation of an odd degree without exception, and every equation of an even degree, provided its final term be negative. 16 GENERAL THEORY OF EQUATIONS must necessarily have a root. This conclusion we might indeed deduce immediately from what has just been established, provided it be conceded that every polynomial f{x)y which gives results of opposite signs when tvfo values a, 6, are successively given to x, passes from f{d) to f{b) continuously, through all intermediate values, as x passes continuously from a to b'. since, if this be admitted, f{x) cannot pass from plus to minus, or from minus to plus, without first becoming zero for one or more of the values of X intermediate between a and b. But this is a principle that requires demonstration. It is the object of the next proposition to estabhsh it with the necessary rigour. PROPOSITION III. (27.) If, in the polynomial JXx) =^" + Aj^_i x^-^ .... + A2 a?2 -f- Aa? + N X be supposed to vary continuously from a? = a, to a? = 6, then the function /(^) will vary continuously hova.f{a) to f{b). Let a' be any value intermediate between a and b. Substitute a' + A for a? in the polynomial, and it will become /(«' + A) = {a' + hy + A„_i {a' + A)n-i + A^ (a' + hf + A (a' + A) + N that is, actually developing by the binomial theorem, and arrang- ing the results according to the powers of h, A2 /{a' -^ h) = a'"" + wa'°-i + A„_ia'"-i+(/.-l)A„_,a /n-2 + A2 + N '2 +2A^a' + A H-(^,_l)(,^_2)A„_i«'n-3 + 2A, FUNDAMENTAL PROPERTIES, 17 Whence it appears that the development of f{a! + K) will be a series of the form where P, Q, R, &c. stand for the compound coefficients above. These coefficients, it will be observed, are all finite quantities, because the coefficients of the original polynomial, as well as the quantities a and w, are finite. Moreover, the first, P, is evidently no other than the function /(«'), the second, Q, is derived from P by multiplying each term of P by the exponent of d in that term, and then diminishing the exponent by unity; R is derived from Q, in a similar manner, and so on in succession, the law of derivation being that already adverted to and illustrated at article 15. Employing then the notation recommended in that article, replacing P by /(«')> ^ l>y /i(^0> ^ ^J /2(^')» ^c- we shall have A2 }fi h^ yi;«'+A)=A«')+/,(«')A+/2(«')Y+/,(«')273 /4(«')2:3:4 + - •*" the functions of a! which form the coefficients being all finite. Now by (23) a value so small may be given to h that the sum of the terms after /(a') shall be less than any assignable quantity, however small. Hence, whatever intermediate value d between a and h be fixed upon for x in f{x), in proceeding to a neigh- bouring value, by the addition to a! of a quantity h ever so minute, we obtain for f{a! + A) a like minute increase of the preceding value fici). In other words, in proceeding continuously from a to hy in our substitutions for a?, the results of those substitu- tions must be in like manner continuous, or all connected to- gether without any unoccupied interval ; for we have just seen that no such unoccupied interval adjacent to any result y(a') can possibly exist, however small the interval is supposed to be. 18 GENERAL THEORY OF EQUATIONS: PROPOSITION IV. (28.) If two real quantities be separately substituted for the unknown quantity in any arranged equation, and furnish results having different signs, that is, one plus and the other minus, then that equation must have at least one root of a value in- termediate between the values substituted. What this proposition affirms is this, viz., that if a quantity a be found which, when substituted for x in any rational and inte- gral polynomial f{x), gives a positive result; and another b be found which, when substituted, gives a negative result, then of necessity there exists some one value, at least, between the values a and b which, if substituted for ar, will render f{x) zero, and thus be a root of the equation/(a7)= 0. The truth of this immediately follows from the last proposition ; since it is there shown that in proceeding continuously from a to b in our substitutions for x, the results — ^which never become in- finite — ^proceed continuously from f{a) to f{b), leaving no un- occupied interval, but passing through every value between J\a) and/(6). But zero is one of these intermediate values, inasmuch as the results change sign somewhere in the interval, passing from positive to negative, or from negative to positive, which it is obvious a continuous series of finite quantities can never do without first becoming zero. Hence, there necessarily exists some value between a and b for which f{x) becomes zero ; that is, the equation /(a?) = has a root between a and b. PROPOSITION V. (29.) 1. Every equation of an odd degree has at least one real root of a contrary sign to that of its last term. 2. Every equation of an even degree, whose last term is nega- tive, has at least two real roots, one positive and the other ne- gative. -Let the equation be f{x) = a?" -f- .... k^x^ + A.^ + Aa- + N = ; FUNDAMENTAL PROPERTIES. 19 and let A^ be the coefficient which has the greatest numerical value, disregarding signs. Suppose first that n is odd, and that N is negative. Then for a? =: 0, f{x) is reduced to the last term, and is therefore nega- tive. But for a? = Ajj + 1, f{x) i» positive (26). Consequently (28) the equation f(x) = has at least one real positive root between and A^ + 1 . Suppose now, n being still odd, that 1^ is positive. Theny(0), being as before reduced to the last term, is positive. But (26) /{ — A^ — 1) is negative; hence, in this case also, the equation has a root (28) comprised between and — (A^ + 1), and there- fore negative. Again, let n be even, and N negative. Then, as in the first case,y(0) is negative, while both/(Aj^ + 1) andy( — Aj^ — 1) are positive (26). Consequently the equation f{x) = has at least two real roots (28): one a positive root between and Aj^ + 1, and the other a negative root between and — (Aj^ + 1). (30.) If this second part of the proposition could be readily generalized Hke the first part, that is, if we could now prove that an equation of an even degree must have a root, though the final term be positive, we might here complete the basis upon which the whole of the subsequent theory of equations is constructed. But to establish rigorously this particular case of the general proposition is by no means an easy task, although one which must necessarily be accomplished, unless we exclude from the general theory every equation of an even degree whose final term is positive. If by means of any algebraic transformation, or of any arith- metical operations performed upon the first member of an equa- tion, we could always convert it into another whose final term should be negative, the difficulty would be removed; but although a great variety of changes may in this way be effected upon an equation, yet no transformation can generally convert an equation of an even degree, whose last term is positive, into another whose final term shall have a contrary sign. It will be hereafter shown that such a change is impossible. And as the principles hitherto established are inadequate to meet the exigencies of this particular 20 GENERAL THEORY OF EQUATIONS. case of the problem, we must have recourse to other considera- tions and other arguments. These, as remarked at the outset (22), have been supplied by several modern analysts; and among others by Cauchy,* whose investigation, a little improved by STURM,f we propose to give in the third chapter. This investiga- tion accompHshes somewhat more than the case actually demands ; but it is well for the student to notice the exact amount of diffi- culty which stands in the way of a complete solution to the fun- damental problem by aid of only the ordinary elementary prin- ciples. But, before entering upon the proposed enquiry, we shall offer a few remarks upon the nature and signification of certain ima- ginary forms which the roots of equations sometimes assume. This seems to be the more necessary, since the answer usually made to the objections brought against expressions of this kind, viz., that the results reached through their aid have always proved valid when submitted to other tests, is far from satisfactory, as it can apply, at farthest, only within the limits of actual experience, and can afford no ground of confidence in any future extension of science which the employment of these expressions may effect. Among the earher algebraists it was common to reject all but the positive roots of an equation; those affected with the negative sign being csi^ed/alse roots, and those involving the symbol ^/ — 1, imaginary. The scruples about negative roots have long been removed ; J and the few observations which follow, on the other class of expressions, may tend to confirm their claim also to a place among the legitimate instruments of analysis. • Cours d'Analyse. f Traite d'Algebre par Mayer et Choquet. X Maseres and Frend were the last writers who stood out against the admission of negative roots. The objections of the former are of frequent recurrence throughout his voluminous productions : those of the latter will be found in his ably written work on The Principles of Algebra, 1796, and in his True Theory of Equations, 1799. CHAPTER II. ON CERTAIN IMAGINARY EXPRESSIONS. (31.) It has been proved in the preceding chapter, that every rational equation has at least one real root, provided the equation be not of an even degree with its last term positive. This is, in effect, the same as proving that the first member of every equa- tion, with the exception just mentioned, has at least one real binomial factor of the form a? — a. We have already adverted to the fruitlessness of every attempt that might be made to bring this case of exception under the same general conclusion by resorting to algebraical artifice for the purpose of changing the sign of the final term. No such artifice could succeed except in particular cases, since it is not generally true that an equation of an even degree with the final term positive has a real root, or is capable of division by a real binomial factor, without leaving a remainder. Our knowledge of the constitution of equations of the second degree — common quadratics — the theory of which is fully established by elementary algebra, is sufficient to authorize this assertion, since equations of this kind, when the final term is positive, often have only imaginary roots. Instead therefore of searching after real binomial factors in equations of an even de- gree with the last term positive, analysts have addressed them- selves to the enquiry whether or not every such equation admits of a real trinomial divisor of the form x^ -{■ px + q, and thence of at least two roots either real or imaginary. Demonstrations of the necessary existence of such a divisor 22 CERTAIN IMAGINARY EXPRESSIONS. under all circumstances, have been given by several algebraists. The most satisfactory and conclusive of these are by Gauss, Lagrange, Legendre, and Ivory. But none of them have a character sufficiently elementary to answer the purpose of in- struction, as they presuppose a command over the artifices and refinements of analysis but seldom acquired by the student till he has arrived at a stage of his progress far in advance of that at which a proof of the present proposition becomes necessary. The demonstration by Cauchy, that every equation must have a root of the form a -f 6 V — 1, where a and h are real, or one of them zero, and from which the necessary existence of a real trinomial factor in every equation belonging to the class referred to easily follows, is much more nearly commensurate with a stu- dent's capabiHties and previous attainments, and therefore better entitled to a place in any elementary exposition of the theory of which it forms so important a part. It is this which we shall introduce, with some modifications, in the next chapter. (32.) It wiU be observed, that the proposition which thus affirms that every equation without exception has a root, of which a -\- h \J — 1 is the general type, not only announces that every equation has a root either real or imaginary, but expressly de- clares the unvarying form of the latter. If h be zero, the symbol of impossibility will vanish, and the form will then express a real root, in all other cases the expression continues imaginary and unchanged in form. Such imaginary expressions, however, have often led to controversy, and have by some been altogether re- jected from the subject we are now discussing, as involving im- possibilities and contradictions irreconcilable with every rational system of algebra. But it should be remembered that these ex- pressions are not the invention of the analyst, arbitrarily and arti- ficially contrived to effect a purpose : he is involuntarily, and unavoidably led to them, by the recognized operations of the science, performed upon, or at least applied to, real quantities ; they naturally and necessarily arise out of these operations ; and therefore cannot be otherwise than consistent with them, however inexplicable they may seem to be. For example, the operations requisite for the solution of a quadratic equation ax^ — io? -f c = CERTAiN IMAGINARY EXPRESSIONS. 23 are all applied to the real coefficients «, b, c, and are all indi- cated symbolically by the general expression which spontaneously assumes the reprobated form whenever the relations among the real quantities entering it are such that 62 < 4ac, where a and c are both of the same sign. Under these circumstances the operation implied in the symbol \J cannot be performed, and to affirm the possibility of its performance would be to contradict established principles. But such an affirmation is never made. Whatever may have been the preparatory process whence the foregoing expression has issued, in proceeding with that expression through the reverse process, we shall undoubtedly arrive at the original equation; in other words, this expression, when substituted for Xy renders the first member zero, and is thus entitled to be called a root, or rather the roots, of that equation. The expression in question, under the circumstances supposed, is never regarded as referring to a final and complete result, in which all the operations implied in it terminate, but as necessarily coming short of such a result by the interposition of an insu- perable barrier — the demand of an impracticable process ; and this demand; be it remembered, must remain in full force, what- ever new operations the expression be submitted to, till it becomes counteracted and neutralized by another of directly op- posite import. And it is only thus that imaginary quantities can be rendered available as instruments of investigation in the doctrine of algebraic magnitude in general. Operations which it is not possible to execute are indicated: these must be ac- counted for ; and, whenever they disappear in the result of any investigation into which they have once entered, they do so solely by the similar entrance of counteracting operations directly the reverse of the former. (33.) It is by the repeated application of this same principle of reverse operations that we clear our way, step by step, to the solution of a simple equation, and to many other algebraic results. 24 CERTAIN IMAGINARY EXPRESSIONS. Transposition, removing fractions — radical signs — coefficient8,&c., are only so many reverse operations suggested at once by the step in which the direct operations occur. The operation of squaring necessarily removes /y/ from \Ja; or, without actually attempting to perform either process, we safely write down a, as the neces- sary result of hoth^ and that whether a be positive or negative. And it is in this way that imaginaries become converted into real effective quantities. (34.) In the higher analysis, imaginary quantities are fre- quently introduced into exponential, logarithmic, and other transcendental expressions, for the purpose of denoting, in a finite form, certain analytical developments. This employment of them is in some degree conventional, and requires our assent to certain extensions of notation naturally enough suggested by the individual cases before us. These extensions and general- izations of the ordinary notation do not come under consi- deration here ; our present remarks having reference only to the common imaginary form« -|- fiV — 1, and to its competency, when other operations are combined with the impracticable one impHed in it, to yield real results. There can be no more objection to the employment of the terms addition, subtraction, multiplication, &c., in reference to these expressions, than there is to the use of the same terms in reference to algebraical forms in general. In neither case is the actual execution of the operations necessarily implied, nor mere arithmetical results intended. Nothing more need be meant by these terms, than the simply linking together, by the signs + , — , X, &c., the expressions to be combined; remembering, however, the offices they are competent to perform when called into activity, but especially observing their neutrahzing in- fluences upon one another, with a view to the reduction of the combination to its utmost simplicity of form. The redundancies being thus removed, the simplified result may then be turned over to the processes of common arithmetic, as actually pointed out by the signs of operation. Should any of these imply an impossibihty, we then infer at once the impossibility of the arithmetical result sought ; a conclusion, however, which must CERTAIN IMAGINARY EXPRESSIONS. 25 not be come to, till we are sure that account has been taken of all the neutrahzing operations. This is, in fact, as observed above, all that is really done in the greater part of algebraical investigations. Signs of operation are abundantly employed throughout such investigations, which operations are thus implied, not performed. The final result embodies, and is accountable for, the aggregate of all these. Those that may have neutralized one another on the way leave no trace of their existence; and whether, if separately called into activity, they could have performed their offices or not, is matter of no moment ; we are not interested in the enquiry whether an obstacle could have been overcome, which extraneous assistance has removed out of the way. (35.) The vague denominations imaginary and impossible, as appHed to the pecuhar expressions here considered, and which convey no idea of their proper character, nor of their connexion with real quantity, have, no doubt, operated upon some minds in excluding them, as mere creatures of the imagination, from among the instruments of analysis. Yet these expressions are connected with each other, and with the real values furnished by any general result, by the same universally recognized prin- ciple that unites all the other particular cases of that result — the principle o? continuity. The continuous series of real values — values arising from the actual performance of all the operations — may terminate, and be succeeded by a continuous series of ex- pressions still involving an unsatisfied stipulation ; these again may arrive at an extreme limit, and there originate another continuous series of real values ; but throughout all these changes the law, impressed upon the general formula by the signs of operation which enter it, is uninterruptedly preserved ; and is impressed with equal distinctness upon each individual case. The law is continuous throughout, the effects of it are pre- sented to us in different continuous forms claiming a distinct classification, though having, in virtue of the common law, a common bond of connexion. These circumstances are well exemplified by the apphcation of 26 CERTAIN IMAGINARY EXPRESSIONS. analysis to curves: at present it will be sufficient to refer for such illustration to the hyperbola. The specific form which every imaginary result thus takes — every symbol in it, whether of operation or of quantity, being controlled by the same specific conditions as those that govern the real values, gives a character and definiteness to it, of which not the slightest notion is conveyed by the terms imaginary quantity and impossible quantity. We might in many ways alter the imaginary roots of a quadratic equation without making them in any respect more or less imaginary or impossible ; yet any such alteration, however minute, would violate specific con- ditions, and introduce error. The name imaginary or impossible is adequate to characterize fully only such objects of analytical research as cannot admit of expression, or algebraic representation, by any combination whatever of the symbols of algebra. For example, the problem which requires the determination of a value or algebraical expression for x that shall fulfil the condition (2x — 5) -f Va?2 — 7 = where the plus before the radical sign impHes the positive root of W oP — 7, is strictly impossible. No expression, either real or imaginary, can satisfy the condition, or represent a root of the proposed irrational equation. We shall have occasion to advert again to impossible relations of this kind in the next chapter; but for a full explanation of the circumstances to which such impossibihties are traceable, the student is referred to the treatise on Algebra, page 128. (36.) Our object in this digression has been to convey definite notions respecting a class of analytical expressions of frequent and unavoidable occurrence in the theory of equations. The preceding observations upon the meaning of the imaginary form — its claim to the character of a root, and the wide difference between every imaginary expression and such an algebraic im- possibility as that adduced above, may help to place these ex- pressions in a clearer hght before the mind of the student ; and CERTAIN IMAGINARY EXPRESSIONS. 27 to give him the same confidence in the real results derived from them that is so readily yielded to all the other deductions of analysis. (37.) It is a remarkable fact that, however we combine ex- pressions of this kind together, by the operations of addition, subtraction, multiphcation, and division, the results are always of the same form as the original expressions. 1 . Thus by addition and subtraction, {a±b V- 1) ± {a' ± b' V- 1) = (« ± a') ±{b + b') v^nr which is of the form A-}- bV^ like the original expressions : and the same form would of course be preserved if more terms were added or subtracted. 2. By multiplication, .^y {a±b\r^){a'±b'\/^) = {ad — bb') ± {a'b + ab') s/^ a result, as before, of the form A+B\/3T 3. By division, or rationalizing the denominator, a ±bs/^ _ {aa' + bb') ± {a'b — ab') V^ which is still of the same form. From the second of these conclusions we infer that any integral positive power of « + ^ V — 1 is of the same form ; so that, taking account of the first conclusion also, it follows that if 28 CERTAIN IMAGINARY EXPRESSIONS. a -\- b V— 1 be substituted for x in any rational and integral function of x, as for instance in the first member of an algebraic equation, the result will always be of the form P + Qs/— ^1, whether the coefiicients of x in the proposed polynomial be real, or imaginary quantities of the form A + B V— 1. This is a conclusion of importance. It might, if necessary, be rendered more comprehensive. In fact the third inference above justifies our affirming that the form would remain the same though negative integral powers entered the function; since these negative powers might each be replaced by unity divided by positive powers. And it will be an easy inference from the chief proposition in next chapter, that the same form is repro- duced when the power is fractional;* so that whatever ordinary algebraic operations be performed upon the quantity a -\- b V— 1 we are invariably conducted to the same form. The conclusion has indeed been extended even further than this, and operations not within the limits of ordinary algebra have been shown still to terminate in the same form. These general views were first propounded by D'Alembert : a brief account of them will be found in the Trait6 de la Resolution^ &c. of Lagrange, Note ix. (38.) We know from the theory of quadratic equations that if one root of such an equation be of the form a -\- b V— 1, there must necessarily be another, differing from it only in the sign which connects the imaginary part with the real, that is, there must be another root of the form a — b s/ — 1. We shall here- after find that this peculiarity has place in all equations. Such roots or expressions are called conjugate roots, or conjugate ex- pressions : and we thus say that the roots of a quadratic equation, when imaginary, are conjugate. Another term has been introduced by Cauchy into the arith- metic of imaginary quantities, the term modulus which it will be convenient to define here. * This inference also follows from the applicHtion of the bltiomial tlieorem to the proposed expn-ssioii ; but the form thence deduced is not finite. {Algcbrn, page 197.) CERTAIN IMAGINARY EXPRESSIONS, 29 The modulus of an imaginary quantity, a -\- b V^l, is the expression, wa^ + b^, formed by taking the square root of the sum of the squares of the real quantities which enter it. For example VQ + 16 or 5 is the modulus of 3 — - 4 V — 1 ; the same is also the modulus of 3 + 4 V — 1. Thus two conjugate expressions have the same modulus. (39.) The following properties respecting these moduli will be found useful in next chapter. 1. The sum of two quantities has a modulus comprised be- tween the sum and difference of the moduli of the quantities themselves. Let the two quantities be a + b V'—l, a' + b' V^ and let r, r' represent their moduli ; that is, let r2 = a2 4. 52^ /2 = ^/2 _^ 5/2 Let also R be the modulus of the sum of the proposed quan- tities ; then we shall evidently have R2 = (a + «')2 + (5 + b'y = ^2 _^ a'2 + 52 _^ 5/2 + 2 (««' 4- bb') = r2 + /2 + 2 (aa' + bb') Now multiplying 7*2, /2 together, we have ^2/2 = «2^/2 _,. 525/2 + ^25/2 _,_ ^/252 = (aa' + bb'y + (ab' - ba'y Hence the numerical value of aa' + bb' must be less than, or at most equal to rr' ; and consequently R2 must be comprised between y.2 ^ ^'2 ^ 2n'' and ^2 + r'2 -- 2rr' 30 CERTAIN IMAGINARY EXPRESSIONS. or, which is the same thing, between (r + /)2 and (r - r'y Therefore the modulus R is comprised between the sum and difference of the moduli r, r' ; and can never be less than the difference. 2. The product of two quantities has for modulus the product of their moduli. For by multipHcation {a-\-b V^) («' + b' V^) = aa' - hh' + {aV + ha!) sT^ and taking the modulus of this result, we have V { (««' - hh'Y + {aV + 5«')2 } = V { «V2 + h-^V^ + a%''^ + 62^'2 } = and the moduli of the original expressions are V a2 + 52 and V«'2 + i'2 Hence the product of any number of factors must have for modulus the product of aU the moduli of those factors ; so that when the factors are all equal, and in number w, we may express this conclusion by saying that the wth power of an imaginary quantity has for modulus the nth power of the modulus of that quantity itself. 3. The quotient of two quantities has for modulus the quotient of the modulus of the dividend by the modulus of the divisor. For by division, a j^hsT—i _ {ad -t- hh') + {a!h — ah') sT-J CERTAIN IMAGINARY EXPRESSIONS. 31 The square of the modulus of this last expression is («V2 ^ ^2^/2) ^ (^/252 ^ ^25/2) _ (a^2 _^ ^/2) (^2 ^ ^2) _ («/2 + J/2)2 - («'2^ 5/2)2 - «2 4-&2 «'2 + 5/2 which is the square of the modulus of the dividend divided by the square of the modulus of the divisor. It may be proper to add to what is here said respecting moduli, that the absolute values only of the expressions so called are recognized, signs being disregarded : that the modulus of a real quantity is the absolute value of that quantity itself; and that, in order for an imaginary expression to become zero, it is necessary and sufficient that its modulus be zero. For a + 6 V — 1 cannot be zero unless both « = and 6=0; and in these circumstances, and in these only, can Va2 -f 52 qj. ^2 _j_52 be zero. It is interesting further to notice, that the modulus of a + 6 V— 1 is no other than the expression for the radius of the circle, in reference to which a and b are the respective sine and cosine of the same arc 9. For putting « = R cos ^, and 6 = R sin 9y we have a J^b V^ 1 = R (cos + sin V ~ 1) and s/ a^ -^ h^ = R. (40.) We shall conclude these remarks with the following theorem, also from Cauchy, which shows, in a remarkable man- ner, the efficiency of imaginary expressions as instruments in the investigation of the properties of real quantities. If two numbers, of which each is the sum of two squares, be multiplied together, the product must be also the sum of two squares. Let the two numbers be «2 + 52 and a^ + V^. The first of these may be considered as the product of the factors a -^ b V — 1, and a — b v— 1, 32 CERTAIN IMAGINARY EXPRESSIONS. and the second as the product of the factors a' + b' \/^, and a' — b' V^, so that the product of the proposed numbers will be the pro- duct of the four factors a + 5\/iri, a-^bsT^l, a'^- b' sT^X, a! — V sT^l. Actually multiplying the first and third, and then the second and fourth, we have the following pair of conjugate expressions, viz. (W - bh') + {aV + ba!) sT^l, {ad - bb') - {aV ^-ba!) V^, of which the product is {aa! - bb'Y' + {ab< -f ba!)\ which is therefore equal to the product of the original numbers; and proves that that product must, like each of the proposed fac- tors, be the sum of two squares. If we interchange the numbers a and 6, or the numbers a', b'y the terms of the product just deduced will be different: thus putting d for 6', and b' for a', which produces no essential change in the proposed numbers, we have Consequently there are two ways of expressing, by the sum of two squares, the product of two numbers, each of which is itself the sum of two squares, thus : (52 -f 22) (32 -f 22) = 112 + 102 = 42 ^ 192 (22 -y 12) (32 + 22) = 42 -t- 72 = 12 -f 82 &c. &c. CHAPTER III. ON THE PROPERTY THAT EVERY EQUATION HAS A ROOT. (41.) In order to demonstrate the principal proposition of the present chapter in the most general manner, it will be convenient first to consider a particular and very simple case of it ; the case namely in which the equation is of the form ^™ ±; N = 0, where N may be limited to the values 1 and V — 1.* We shall give to this preparatory step the form of a lemma. PROPOSITION I. LEMMA. Each of the equations ^ a;"» = ± 1, a?"» = ± \/~—\. has a root comprehended in the general form a + 6 V— 1 . This is evidently the case with respect to the equation a?™ = + 1, whether the number m be even or odd ; since a? = 1 always satis- fies it. It is also as plainly true of the equation a?"™ = ~ 1 when m is odd, because then x ■=■ — 1 satisfies it. When m is even, it must either be some power of 2, or else some power of 2 multipHed by an odd number ; if it be a power of 2, then the value of x will be bbtained after the extraction of the square root repeated as many times in succession as there are units in the said power. Now the square root of the form * The previous consideration of this latter value is not absolutely necessary, but it may be included without adding much to the length or difficulty of the argument. 3 34 ON THE PROPERTY THAT a-\- bw — l is always of the same form.* Hence when w is a power of 2, each of the equations a?'"= — 1, a?™ = ±\/— 1 has a ' root of the form announced. When m is a power of 2 multipUed by an odd number, then, if we extract the root of this odd de- gree first, there will remain to be extracted only a succession of square roots. We have therefore merely to show, that when m is odd, a root of ±: V — 1 is of the predicted form. Now the odd powers, 1, 3, 5, &c. of + V— • 1, are + V^, - \/^, + n/ITT, .... and the same powers of — V— 1 are - V^ + >/=!, - \/^, .... Consequently, when m is odd, a root of ± V— 1 is either + V— 1 or — V— 1. Hence the predicted form occurs whether m be odd or even. It follows from this proposition that whatever positive whole 1 2, number m may be, (—1)"" and (V— l)™^will always be of the n n form a + b V— 1; or more generally (—1)'" and (V— 0" will always be of this form, n and m being any integers positive or negative (37). PROPOSITION II. (42.) Every algebraical equation, of whatever degree, has a root of the form a + b V — 1, whether the coefficients of the equation be all real, or any of them imaginary and of the same form. Let /(ar) = a;" + A„_ia:"-i-f . . , A^ + A^^^ -{-A^x + -^=0 ... [1] represent any equation the coefficients of which are either real or imaginary. • Algebra, p. 113. EVERY EQUATION HAS A ROOT. 35 If in this equation we substitute p -\- q \/ — 1 for x, p and q being real, the first member will furnish a result of the form P + Q \/^, P and Q being real (37). Should ^3 + qV^he a root of the equation, this result must be zero; or, which is the same thing, the modulus of P + Q V— 1, viz. VP^ + Q^, must be zero (39). And we have now to prove that values of ^ and q always exist that will fulfil this latter condition. In order to this it will be sufficient to show that whatever value of vP^ + QP, greater than zero, arises from any proposed values of p and q, other values of p and q necessarily exist, for which vP2 + Q2 becomes still smaller; so that the smallest value of which vP^ + Q? is capable must be zero ; and the particular ex- pression p + qsZ—l, whence this value has arisen, must be a root of the equation. For the purpose of examining the effect upon any function/(a?) of changes introduced into the value of a?, the development exhibited at (27) is very convenient. By changing x into x ■\- h the altered value of the function is thus expressed by where /(a?) is the original polynomial, SLXid.f^(x),f2(x) &c. contain none but integral and positive powers of a? (27). The first of these functions f(x) becomes P + Q, v— 1 when p -\- q V — 1 is substituted for x ; the other functions may some of them vanish for the same substitution, for aught we know to the contrary; but all the terms after y*(a?) cannot vanish; the last A", which does not contain x, must necessarily remain. Without assuming any hypothesis as to what terms of f(x-\-h) vanish for the value x=:p + ^v— 1 which causes the first of those terms, f{x), to become P + QV — 1, let us represent by h^ the least power of h for which the coefficient does not vanish when p + q V — 1 is put for x. This coefficient will be of the form R -H S V — 1, in which R and S cannot both be zero. When p -h q V — 1 is put for x, we have represented /(a;) by 36 ON THE PROPERTY THAT P H- Q V— 1. In like manner, when p -f qs/ —\-\-h'\s put for X we may represent the function by P' + Q'V— 1. The develop- ment [2] will then be F^- Q'>/I:T= (P + Q x/ITT) + (R + S \^^)A» + terms in h^^\ ^m+2^ .... A". Now h is quite arbitrary : — we may give to it any sign and any value we please, provided only it come under the general form a -^ b V— -1 . Leaving the absolute value still arbitrary, we may therefore replace it by either -\-koT — k,or-±.{—\)^k\ and thus render A™ either positive or negative, whichever we please, what- ever be the value of m; and we have seen that ( — 1)™ comes within the stipulated form (41). Hence we may write the fore- going development thus, the sign of k^ being under our own control : P'-h Q'\/^= (P + Q\/^) + (R + ^\r^)k^ + terms in /.m+l^ /.m+2^ _ _ ^n^ But in any equation of this kind the real terms in one member are together equal to those in the other; and the imaginary terms in one to the imaginary terms in the other. Consequently, F= P + RA;™ -F the real terms in k^^\ A;™+2^ . . . . k^ Q'= Q + SA;"™ + real terms involving powers above k^. Hence the square of the modulus of P' -f Q! V — 1 is P'2 + Q/2-p2+Q2 _,.2(PR+QS)^°» + real terms in ^™+i,/{;'n+2,,^ Now k may be taken so small that the sum of all the terms after P^ -|- Q? may take the same sign as 2(PR + QS)A:™ by (25), which sign we can always render negative whatever PR -|- QS may be, because, as observed above, k^ may be made either posi- tive or negative, as we please. EVERY EQUATION HAS A ROOT. 37 Hence we can always render P'2 -f Q'2 < P2 + Q2^ or VP'2 + a'2 < VP^ + QP. In other words, whatever values of p and q, in the expression p + q V-~r, cause the modulus VP^ + Q^ to exceed zero, other values exist for which the modulus will become smaller ; and consequently one case at least must exist, for which the modulus, and consequently the expression P + Q V — 1, must become zero. This conclusion presumes however that PR + QS is not zero. If such should be the case, then our having chosen the form of h as to secure a command over the sign of 2 (PR + QS) will have been unnecessary. The form must then be so chosen that a command may be secured over the sign of the first term after 2(PR + QS)^°», in the above series for P'2 + Q'2, which does not vanish ; when the preceding conclusion will follow. PROPOSITION III. (43.) The values of a and b in the expression a + 6 v— 1, which when put for x in /(x) cause that polynomial to vanish, can never be infinite. We may write /(a?) as follows, viz., /w=*"(i + ^ + V + — ^) •^ ^ ' a? x^ a?" or putting P + ClV— 1 for whaty((a?) becomes, when^ + q\/ — l is substituted for x, we have (^ + g\/^)"(l + N n — I I n—j I ^ \ Now the modulus of a quotient is the quotient of the modulus of the dividend by the modulus of the divisor (39). In each of the dividends A^-p Ajj_2, &c. above, the modulus is finite by hypothesis. Hence if either^ or q be infinite, and consequently 38 ON THE PROPERTY THAT the modulus of every denominator or divisor, also infinite, the modulus of each quotient must be zero. Hence in this case each of the above fractions must itself be zero (39), and therefore the modulus of the entire quantity within the parenthesis simply 1 ; and the modulus of a product is the product of the moduli of the factors, so that the modulus of the preceding product, viz., VP2 + OP, is the modulus o{ (p -\- q x/^)". But the wth power oip -\- q V — 1 has for modulus the wth power of the mo- dulus of ^ -f ^f V— 1, that is, the nth. power of \/jP -f q^ (39) which is infinite : consequently vPM-Q.^ must be infinite. But when p -\- q V— 1 is a root of the equation f{x) = 0, VP^ + Q? is zero. Hence in this case neither^ nor q can be infinite. (44.) An objection may be brought against the preceding reasoning that ought not to be concealed. It may be denied that the modulus of the product above referred to is simply the modulus of (^ -\- q V — 1)" in the case of ^ or g^ infinite; for it may be maintained, that although in this case aU the quantities within the parenthesis after the 1 become zero, yet the combina- tion of these with {p -H ^rV — l)", which involves infinite quantities, may produce quantities also infinite ; and thus the modulus of the product may differ from the modulus oi {p -^ q V— 1)° by a quantity infinitely great. It is not to be denied that there is weight in this objection. But it is not difficult to see that al- though the true modulus may thus differ from the modulus of {p -\- q V — 1)" by an infinite quantity, yet the modulus of {p -\- ^\/— 1)", involving higher powers than enter into the part neglected, is infinitely greater than that part. This part therefore is justly regarded as nothing in comparison to the part preserved, the former standing in relation to the latter as a finite quantity to infinity. But the proposition may be established somewhat diff*erently, as follows : Substituting {p -^ q V — l) for x in /(a?), we have (^ + r^s/ZT)"-i-A„_,(/>+?N/^)"-l-f ..Ai(y) + g\/^)-HN. EVERY. EQUATION HAS A ROOT. 39 Call the agregate of all these terms after the first, P' -f- Q!^^—\ ; then it is plain that the modulus of the first term, that is, (y/^^\-~^y\ must infinitely exceed the modulus VP'^ + Q!^, of the remaining terms whenever p or qis infinite ; because in this latter modulus so high a power of the infinite quantity p or q cannot enter as enters into the former. Now the modulus of the whole expression, that is, of the sum of (p + grV — 1)" and P' + Q'V— 1, is not less than the difference of the moduli of these quantities themselves (39), which difference is infinite. Hence, as before, Vp^ + Q^ must be infinite when p or q is infinite. (45.) Several different versions are given by the continental algebraists of Cauchy's argument to prove the existence of a root for every equation. Cauchy himself has given two distinct forms to his demonstration : one in the 18th number of the Journal de VEcole PolytechniquCy and the other in the lOtli chapter of his Cours d' Analyse. Of the other modifications that have been pro- posed, that contained in the Algebre of Lefebure de Fourcy, and the one in the treatise of Mayer et Choquet, for which the authors state themselves to be indebted to Sturm, are to be preferred on t^ie ground of involving fewer perplexities than the other demonstrations.* The method of investigation here employed has been mainly modelled upon this last form of proof ; but it differs from it in many respects; and from the departure thus made we think that the reasoning has not only been comprised in a smaller compass, but that it has also received additional simplicity. (46). It will have been noticed that this reasoning proceeds entirely upon the hypothesis that the equation is rational as respects the unknown quantity x-y the coefficients may be rational or not; the only condition with respect to them being that they • The student may also consult with advantage, Peacock's View of the Present State of Analysis, in the "Report of the Third Meeting of the British Association." pp. 297-305. 40 ON THE PROPERTY THAT must come under the form a -{■ b V— 1, where a and b are real and finite, but either rational or irrational. We have already adduced an example of an irrational equation which has no root (35) ; and it would be very easy to multiply such examples. Mr. Horner, in a paper in the Philosophical Magazine, *^ has commented at some length on equations of this kind. But the peculiarities there dwelt upon have been often remarked by preceding writers ; though very erroneous doctrines have sometimes been held respecting them. GARNiERf has in- stanced several examples of this kind, and has properly ac- counted for the foreign roots which are found to be involved in the equation when rendered rational. Thus the equation — \/x—\ = 1 — Va?— 4' when rendered rational, gives a? = 5. But this value, when put for Xy fails to satisfy the equation ; the equation really satisfied by it is 4- s/x—l =1 -f- Va?— 4 and no value or expression whatever can possibly satisfy the pro- posed equation : in other words it has no root. The mistake, however, of confounding the rational equation with the irrational from which it has been deduced, is often committed; and the roots of the former erroneously attributed to the latter equation. Rolle, in his attacks upon the Difierential Calculus, feU into this error more than once, from disregarding the restrictions which he himself had imposed upon the signs of his irrational terms. ^ When the signs thus prefixed to the radicals which enter an equation are regarded merely as links to connect the several terms together, and as exercising no control over the symbol of irrationahty, then it will involve no more restrictions than the ra- tional equation deduced from it ; and the roots of the one will be • Phil. Mag., 1836. t Analyse Alg<^brique, 181 J, p. 335. X See MoNTUcLA, Histoire des Mathematlques, torn, iii, p. 1 1 1. EVKRY EQUATION HAS A ROOT. 41 also the roots of the other. Some writers have contended for this view of the office of the signs in all such circumstances. But it is at variance with the general principle, conformably to which, analytical conditions are expressed; and when adopted, should be done so by special agreement in reference to the enquiry in hand. MoNTUCLA, in animadverting upon the errors of Rolle, fully recognizes the propriety of his original restrictions as to the signs of the irrational terms; and in accordance with the common practice of analysts always considers the symbol of irrationality, when forming part of the data of a problem, as controlled by the plus or minus sign prefixed to it. Yet when this symbol is in- troduced in the course of an algebraic investigation, and is thus unrestricted by the original conditions of the problem, the am- biguous sign is always to be regarded as involved in it. We must direct the student for a more complete discussion of the present topic to the source already pointed out.* It was proper thus briefly to invite his attention to it here, in order that he might clearly see the restrictions under which the general principle established in the present chapter is to be received; and be furnished with the reasons which render it necessary, in dis- cussing the theory of equations — of which theory the principle adverted to is the foundation — that irrational equations be ex- cluded, or at least that they be rendered rational by a preparatory process. • Algebra, Third Edition, p. 128. CHAPTER IV. ON THE GENERAL PROPERTIES OF EQUATIONS. (47.) The properties to be developed in the present chapter are those which belong to all rational equations containing but one unknown quantity, without regard to the degree of the equa- tion, and generally without any stipulation as to whether the coefficients be real or imaginary. Whenever the property an- nounced requires that the coefficients be real, we shall introduce the necessary restriction into the enunciation of it; otherwise we shall consider them to be indifferently either real or imaginary. In our occasional numerical illustration of particular truths we shall however always choose examples with real coefficients; be- cause in actual practice they generally present themselves in a real form. Moreover, we shall, for simphcity sake, usually sup- press the coefficient of the highest power of the unknown quan- tity, or rather shall regard this coefficient as unity, retaining it in a general form only when any practical operation to be deduced from the investigation would seem to be limited in generahty by suppressing it. It is plain that the coefficient of the highest power may in all cases be reduced to unity by division ; and that the new coefficients, thence resulting, will still preserve the pre- scribed form (39). PROPOSITION I. (48.) If « be a root of any equation f{x) = A,^a;" -h . . . . + Ayar^ ■\- K^ 4- Aa? -|- N = 0, then the first member of it, /(a?), will necessarily be divisible GENERAL PROPERTIES OF EQUATIONS. 43 by ^ — a. And conversely, if the polynomial f{x) be divisible \ by the binomial x — «, then a will be a root of the equation /W=o. 1 . Let the division of f{x) by a? — a be performed. Then, as the divisor contains only the first power of Xy the operation will evidently proceed till we arrive at a remainder independent Qi x\\ and the quotient, like the dividend, can contain only integral and positive powers of x. Call this quotient Q and the remainder R; then we have f{x) = (« — a) Q + R. Now if a be a root of the equation, the first member of this identity must become zero when a is put for x ; and consequently the second member also. But when a is put for a?, the second member is reduced to R; therefore R must be zero; that is, if « be a root, x — a will divide the polynomial without leaving a remainder. 2. Again, let a? — a divide f{x) without leaving a remainder; then will a be a root of the equation f{x) =0. For, caUing the quotient Q, which can contain none but posi- tive integral powers of x^ we have the identity /(«)=(x-«)Q. Put a for X in the second member, and it becomes reduced to zero; consequently the same substitution for x reduces the first member to zero ; that is, a is a root of the equation f{x) = 0. (49.) The actual operation of dividing the first member of an equation by a binomial of the form x — a, whether « be a root of that equation or not, is one of very considerable importance in the process of numerical solution. It is plainly an operation of the simplest character, belonging to the first rudiments of algebra, and requiring little or no ingenuity or address in the performance of it. Nevertheless, it is to an improved and more compact form of arranging the elements of this simple operation that the per- 44 GENERAL PROPERTIES OF EQUATIONS. fection at present attained in the solution of numerical equations of the higher degrees, is in a great measure attributable. Many of the accessions which, from time to time, abstract science has received, are clearly traceable to previous changes and improvements in the mere symbols of operation. A compact and significant notation has been a fertile source of modern ana- lytical discovery; and the recent advances in the solution of nu- merical equations show, in a striking manner, how much mere arithmetical arrangement may effect in facilitating the practical development of a comphcated theory. But for a few innovations upon the common methods of performing certain simple numeri- cal operations ; it is probable that the solution of equations — as far as practicability is concerned — ^would be now in the very state in which Newton and his immediate disciples left the problem. (50.) In the process for dividing f{x) by x — a, viewed in reference to its connexion with this important problem of the general solution of equations, it is the remainder that is more the object of search than the quotient; the determination of the quo- tient, however, always forms part of the operation for finding the remainder in the arrangement here to be recommended, although the remainder may be formed very readily without reference to this quotient. Thus caUing the quotient Q, and the remainder R, as before, we have A„a?" +.... + Aga?^ + k^'^ -j- Aa; + N = (^ — a) Q + R, and if in this identity we put a for x, we get the remainder at once, viz. A^a" -j- . . . . -j_ Agfl^ + Ago^ -t- A« -f N = R . . . . [1], showing, what is worthy of observation, that the remainder arising from dividing the polynomial hy x — a will always be exhibited by simply changing, in that polynomial, x ior a. As to the quotient Q, it will, as remarked above, contain only positive integral powers of x\ and it will moreover be in degree a unit lower than the proposed polynomial. In other words, if /(a?) be of the fifth degree, Q will be of the fourth; and ii f{x) be of the GENERAL PROPERTIES OF EQUATIONS. 45 fourth, Q will be of the third ; and so on. For simplicity we shall assume the polynomial to be of the fifth degree. It will be readily seen that the generality of the reasoning is not in the least impaired by this assumption. Putting then f{x) = Aga?^ + k^x^ + Aga^ + k^ + Aa? + N, the quotient Q will be a polynomial of the form Q = k!^x^ + k'^a^ + k!^ + k!x + N', from which we may return to the dividend, or original function /(a?), by multiplying it by a? — «, and then adding R to the pro- duct; that is to say, upon actually executing this multiplication, we shall have the identity k!^r> -+- {M^^ak!^x^ + {M^^ak!^^ + {M-ak!^x^ + (^'^aM)x-aW + R =k^x> + k^x^ + k^c^ + k^^ + Aa? + N so that by equating the coefficients of the like powers of x, we shall have the relations which connect together the coefficients of the dividend and those of the quotient; and from which the latter may all be determined from the former without going through the usual process of actual division. The relations are as follow : A 4 — A5 A'3 — aA'4 = A4 ; consequently, A'3 = «A'4 + A4 A'2 — «A'3 = A3 „ A'2 = ttA'g + A3 A' — ak'^ = A2 „ A' = aA'2 + A2 N' -ak' = A „ N' =ak' + A also R - aN' = N „ R = «N' + N. From the equations on the right, it appears that the first coefficient A'4, in Q, wiU be the same as the first iny(a?) ; that the second will be found by multiplying A'4 by a, and adding the second in /(a?) ; and that, generally, every coefficient in Q will be derived by the same uniform process of multiplying the pre- ceding coefficient of Q by a, and adding to the product the cor- 46 GENERAL PROPERTIES OF EQUATIONS. responding coefficient in f{x) ; which process, extended up to N', the coefficient of x^ in Q, furnishes also the remainder R. (51.) Thus is the operation of division, which sometimes by the common method extends itself to a tedious length, reduced to a form of remarkable compactness and simpHcity. We shall have merely to arrange the coefficients of the dividend in a hori- zontal row, to multiply the first by «, and add in the second ; then to multiply this result by a, adding in the third ; then again this new result by a, adding in the fourth ; and so on to the end. The last result wiU be the remainder ; the preceding results will be the several coefficients of the quotient in order, with their proper signs, commencing with the second coefficient ; the first being the same as the first number in the horizontal row with which we commence. We shall give an example or two for illustration ; premising that when terms are absent from the pro- posed polynomial, that is, when the expression is incomplete^ the absent terms must be supplied by zeros, and must occupy their proper places in the row of coefficients. 1 . Required the quotient and remainder arising from the divi- sion of the polynomial, a^ -i- 7a^ + 3a^ -M7ar2 -i- iq^ _ 14; by the binomial a: — 4. 1 + 7 +3 -f 17 + 10 -14 4 + 44-1- 188 +820 +3320 ^ 11 + 47 + 205 + 830 + 3306. Hence the quotient is ar4 + lla?3 _j_ 47a.2 ^ 205a? + 830, and the remainder + 3306. 2. Required the quotient and remainder arising from the divi- sion of 3a?6 — 6a^ + 4ar^ — ar — 45624, by the binomial a: + 5. GENERAL PROPERTIES OF EQUATIONS. 47 Supplying the vacant terms in this incomplete expression, we have 3+0—6 +4 +0 — 1 —45624 — 15 4- 75 — 345 + 1705 — 8525 + 42630 — 15 + 69 — 341 + 1705 — 8526 — 2994 Hence the quotient is 3^ — 15a?4 + 69a^ — 341a?2 + I705a? — 8526, and the remainder — 2994. (52.) It is this simple process for obtaining the coefficients of the quotient, and thence the remainder, due to the division of a polynomial f{x) by a binomial x — «, that gives to Horner's method of approximating to the roots of numerical equations much of that practical facility which distinguishes it from all other operations for the same purpose. The theoretical prin- ciple upon which it depends is too obvious not to have been long known before the discovery of the method referred to, and it is accordingly adverted to by several preceding writers. It is dis- tinctly enough stated by Garnier, in his Elemens ^ Algehre, 1811, p. 399, and still more so by Francosur, in his MathSmatiques^ 1819, tom. II. p. 37; but its practical bearing upon the solution of equations was not observed till the publication of Horner's researches.* Had Newton adopted, in his own method of approximation, the numerical arrangement suggested by the fore- going proposition, Horner's more perfect mode of proceeding would have been obviously so small an advance upon it, that it could scarcely have escaped being suggested to the minds of some * Mr. Horner's original investigation was published in the Philosophical Transactions for 1819. But it involved reasonings so intricate, and made so large a demand upon the more recondite departments of analysis, that, notwith- standing the subsequent modification of certain parts of it in Leyhourn's Repo- sitory, it attracted but little notice from mathematicians, and seemed likely to fall into unmerited neglect. We believe that the investigation owes the sim- . pie and elementary form in which it now appears, principally to the author of \\ the present Treatise. '^ 48 GENERAL PROPERTIES OF EQUATIONS. of the many persons who aimed at improving the Newtonian operation. This will be seen when we come to compare the two methods. (53.) The expression [1], deduced at (50) for the remainder arising from the division of a rational and integral polynomiaiy(ar) hj X — a, shows that the computation of this remainder is the same thing as the computation of the value of the polynomial itself for the particular value a of x : so that the operation de- scribed above, for finding the remainder, is that which furnishes the value of the polynomial for a given value of x. Thus, in the first of the preceding examples, 3306 is the numerical value of the polynomial when 4 is put for x. In the second example the value of the polynomial for a? = — 5 is— 2994. ^ It may be remarked, finally, that the remainder resulting from the division of any function of a? by a? — a, and the value of that function for x ■= a, will always be identical, provided only that the quotient, furnished by the division, do not become infinite for the proposed value of x. This is evidently the only condition necessary in order that the product of quotient and divisor may become zero when the divisor does, that is, when x ■=■ a ; the corresponding value of the dividend being then expressed by the remainder. PROPOSITION II. (54.) Every equation has exactly as many roots as there are units in the exponent of the highest power of the unknown quan- tity in it; that is, an equation of the wth degree, has n roots and no more. Let f{x) = a;" -1- Ago^ -i- k^ -f Aa; -f N = be any algebraical equation whatever. It necessarily admits of one root a,, either real or imaginary (42) ; and therefore (48), the first member, /(a?), is divisible by at least one binomial divisor GENERAL PROPERTIES OF EQUATIONS. 49 X — ay Hence, calling the quotient arising from this division, f{x)y we have y(x) = (^-a,)/'«, where f'{x) is a rational and integral polynomial of a degree in- ferior iof{x) by unity. If we equate this latter polynomial to zero, or assume the equation /'(^) = 0, a value for x will exist that will satisfy that equation (42) ; so that calling this value «2, it follows that there is at least one binomial divisor, x — «2, of f{x) ; that is, putting f'{x) for the quotient of the division, we shall have /(x) = (;.-«,)/'«, where f"{x) is a polynomial of a degree two units below the original one. In Hke manner the equation /''(a?) = has also a root a^ by the same general principle (42); so that/''(ir) has also a binomial divisor, x — a^ ; and the corresponding quotient will be in de- gree three units below the original polynomial. The continuation of this process would thus furnish a series of polynomials f'{x), f"{x)y &c., descending continually in degree by unity, and thus at length reaching a simple binomial x — a^^ there being of course as many such results as there have been divisions. Con- sequently f{a)^{x-a,)f'{x) = (ar — «i) {x — a,^f"{x) = (a? — «i) {x — a^ {x — a^f"'{x) = (^ — «i) (« — «2) (a? — %).•.. (^ — «n)- And thus the polynomial of the nth. degree /(a?) is decomposable into n binomial factors of the first degree. 4 50 GENERAL PROPERTIES OF EQUATIONS. In order that f{x) may become zero, it is sufficient therefore that any one of these simple factors become zero ; that is to say, it is sufficient that x take any one of the n values. fljj, fltgj % Consequently, every equation of the wth degree must have n roots. It is plain that a polynomial of the wth degree cannot have more than n factors of the form x — a; because if more than n factors be multiplied together, the product will be a polynomial of a higher degree than the nth.. It does not follow from the foregoing reasoning that the factors, and consequently the roots, are all necessarily unequal : any two or more may be equal to one another. All that the rea- soning shows is, that the number of simple factors in the first member of an equation in x, of the nth degree, is w, and that the number of roots is therefore n. There is no restriction as to the relative magnitudes either of the factors or of the roots. (55.) From this important proposition it follows, that we may always represent the first member of an equation of the wth de- gree under the form (x — ai) {x — flg) {x — a^)....{x- a„) or may consider such an equation as compounded (by multipli- cation) of as many simple equations, X — «j = 0, X — ^2 = 0» a? — ff^ = 0, &c. as there are units in the number denoting its degree, and of no more. We may infer also, when one of the roots («j) of an equation has been discovered, that we shall arrive at an equation containing all the remaining roots by dividing the first member of the pro- posed equation by a? — a^, and then equating the quotient to zero. The new equation, thus derived from the proposed, is called the GENERAL PROPERTIES OF EQUATIONS. 51 depressed equation. It is very easily obtained by employing the process at (51) instead of the common operation for division. In like manner if two roots, «j, «2> ^^ known, we may deduce the depressed equation containing the remaining roots by dividing the first member of it by the simple factors {x — «j), (x — ttg)* i^ succession, or by the quadratic factor (x — a^) x {x—a^ =zgi^ — («i +«2) a? + «i «2 ^* once, and then equating the quotient to zero. Hence, if we were in possession of a general method for determining a single root of every equation, we should have means sufficient for the discovery of all the roots of every equation ; and should thus be enabled to decompose every ra- tional and integral polynomial, with a single unknown quantity, into its constituent simple factors. Thus the component factors of the polynomial f{x) would be obtained by equating f{x) to zero ; then determining one after another, as above described, all the roots of f{x) = ; and finally writing down the binomials furnished by connecting each of these roots in succession, by a changed sign, with x. The actual determination of the roots of an equation constitutes the problem of the general solution of equations; and it is one, as already remarked in the Intro- duction, for which algebra has effected comparatively but Httle, except in the case of numerical equations; that is, equations whose coefficients are known numbers, and not general symboUcal ex- pressions or letters. In this latter case little or no advance has been made beyond equations of the fourth degree.* When the equation is numerical aU the real roots may be obtained by the combined methods of Sturm and Horner, as will be hereafter explained, so that the roots which these methods leave undeter- mined will all be imaginary. Lagrange has explained a method by help of which even these may be actually exhibited tf but the calculation of the real roots, and the discovery of the • And as observed in the Introduction, the symbolical expressions for the roots of equations of the third and fourth degrees are useless for the purposes of actual numerical computation except in particular cases. t Trait6 de la Resolution des Equations Num^riques, 1826, p. 19 and p. 177. The principles of this method will be explained hereafter. 52 GENERAL PROPERTIES OF EQUATIONS. number merely of those that are imaginary, comprise all that is usually sought to be accomplished in the numerical solution of equations.* But whatever difficulties may attend the decomposition of a given polynomial into its simple factors, the reverse operation, that of compounding the expression from given simple elements, or of constructing an equation that shall have given roots, is very easily accomplished. The roots being given, the simple factors, formed by connecting each, by a changed sign, with ar, are all given ; and the product of these is the first member of the re- quired equation, the other member being zero. (56.) It has already been proved (42) that every equation f{x) == has a root of the form a -\- h\/ — \\ therefore the de- pressed equation /(a:) = has also a root of the same form; in like manner the next depressed equation f'ipc) = has a root of this form, and so on, through the entire series. Hence all the n roots of an equation of the nth. degree are of the same form, « -I- J n/ _ 1 . It wiU be further shown in next proposition, that when the coefficients of the equation are aU real, and any of the roots are actually imaginary, b being different from zero, they must occur in conjugate pairs (38) ; that is, every ima- ginary root a-{-b V— 1, must be accompanied by another, a^b\/^-i. The conclusion just obtained, respecting the general form of all the roots of an equation, enables us to affirm with confidence that every fractional power of « ± 6 V — 1 must be of the same • Still the determination of the imaginary roots of numerical equations is not without its use in certain practical enquiries, more especially in the Integral Calculus. But it is a branch of the general doctrine of numerical equations which has not been cultivated with the success that has attended the researches of algebraists in reference to the real roots. The method of Lagrange, adverted to in the preceding note, is too laborious to be of much practical value. The same may be said of the method of Bernoulli, by recurring series ; so that a practicable process for calculating the imaginary roots is still a desideratum in the doctrine of numerical equations. GENERAL PROPERTIES OF EQUATIONS. form. For, representing this power by x, we have 53 The integral power m has already been shown to be of the form P + Q V— 1, whether m be positive or negative (37) ; so that the last equation may be written thus, a;° = P + QV^or^« — (P + QtsT-i) =0 And it has been shown above that the values of a?, in this equation, are all of the form a + ^ V— -1. Hence the principle announced is fully estabhshed. PROPOSITION III. (57.) Every equation whose coefficients are real, and which has one imaginary root, has necessarily another, conjugate to it : that is, imaginary roots enter into equations in conjugate pairs. Let the equation contain the imaginary root a + bw — 1 ; then if this root be substituted for x in /(a?), that is in N + Aa? + A2^2 + + a?° = we shall have N + A(« + i\/^) + A2(« + 6\/-T)2+----+(« + 5\/--T)"=0 Now it is obvious that if the several terms in the first member of this equation be developed by the binomial theorem, or by actual multiplication, we shall have a series of monomials, of which all those that involve odd powers of h v— 1 will be ima- ginary ; and all the others real : that is, calling the sum of the real terms P, and the sum of the imaginary terms Q V-~l , the equation may be written P + QV-1 =0 [1] 54 GENERAL PROPERTIES OF EQUATIONS. which equation can be satisfied only by the conditions P = 0, Q = ; for if these have not place an imaginary quantity would be equal to a real quantity. Now we know from the binomial theorem that the develop- ments of (p + q)^ and of (^ — gr)™ differ only in the signs of the terms involving the odd powers of q. Hence if a — b v — 1 had been substituted for x in /(a?), instead of a + 6 V — 1, we should have been led to a result P_QVzn" differing from the former [1] only in the sign of Q. And since P = and Q = 0, this result must hke the former be zero ; that is P — Q\/^ = .... [2] so that both a + 6 V — 1 and « — 6 v — 1 are equally roots of the proposed equation. The first member of that equation is therefore divisible by both of the simple divisors X — a — b V — 1 X — a -\- bw—l and consequently by their product, the real quadratic divisor ir2 - 2fl^ + «2 + 62^ or (a? - «)2 + 62 . . . . [3] If the depressed equation which results from this division have an imaginary root, then from what has just been shown, there must exist another real quadratic divisor, composed of another pair of conjugate roots. And in general there must exist as many real quadratic factors in /(a?) as there are pairs of imaginary roots in the equation /{x) = 0, besides those formed from pairs of real roots. (58.) It follows therefore that every equation of an even degree, 2m, with real coefficients, may be decomposed into n real quadratic factors; whatever be the character of the roots en- tering it. GENERAL PROPERTIES OF EQUATIONS. 55 And further, in conjunction with what is proved at (29), every equation of an odd degree, 2w + 1, with real coefficients, may be decomposed into w + 1 real factors ; n of these being of the second degree and one of the first. It is scarcely necessary to remark that, in both these cases, if more real roots than two enter the equation, they may be com- bined in pairs in different ways ; so that there may be different sets of n quadratic factors, all equally composing the original polynomial. And when we say that an equation may be decom- posed into its n quadratic factors, we merely imply the existence of these factors, in its first member. As remarked at {^5>) the actual determination of them requires the accomplishment of the general solution of equations. (59.) When the roots of an equation with real coefficients are all imaginary, then each of the real quadratic factors [3], into which it is decomposable being the sum of two squares, it follows that whatever number be substituted in the equation for Xy the result of that substitution must always ht positive; so that if a negative result arise from any substitution for x^ all the roots of the equation cannot be imaginary. That the final term of the equation, or that independent of x, must be positive is an inference froin (29) as well as from what is here estabhshed. By reasoning exactly similar to that employed above, may it be proved that if an equation with rational coefficients have a root of the form a-\- s/b, it must have another of the form a — \J h% so that quadratic surd roots, whether real or imaginary, always enter into equations in pairs. PROPOSITION IV. (60.) To determine the forms of the functions which the coeffi- cients in the general equation /(a?) = are of the roots. It has already been shown (55) that the first member of the general equation x^ + A„_i.t'»\+ A„_2^n-2 + A„_3a?"-3 -I- . . . -f- K.^ + K.^'^ + A^ + N = ' 56 GENERAL PROPERTIES OF EQUATIONS. is no other than the product of the n binomial factors X — Up X — a.2, X — ttg, X — flj, in which aj, «2» %> • • • • «n» *^^ *^® ^ roots of the equation. By the actual multiplication of these we shall arrive therefore at the polynomial f{x), and thus discover the manner in which the n roots enter into the formation of the coefficients. Two or three steps of this multiplication will be sufficient to make known the general law which connects the coefficients and roots together. X -a, X — «2 x^ — a^x -f- flj ^2 -«2 ^ — «3 a^-a, X^ -f «j ffg X - - «l«2«» — «2 + «i«3 -«3 :h«2«3 ^ -«4 x^ — a^\a^ -\- «j Og^ — ^1 ^^^2 ^3^^ + ''l ^2 ^3 ^4 — ttj' "H ^1 ^3 — ''^l ^2 ^4 -«8 + «2«'3 -«]«'3«4 ~«4 + «,«4 + «2«4 + «3«4 -«2«3«4 GENERAL PROPERTIES OF EQUATIONS. b7 Hence by continuing this process, we have, for the coefficients of the proposed equation, the values K-\ = — «i — «2 — «3 — • • • • — «n K-2 = «i «2 + «1 % + ^2 % + • • • • + <^n-l «n K-i = — «i «2 % — «1 «2 «4 — • • • • — «n-2 <^n-l «n N=ai«2«3«4 • • • • «n(— !)"• We infer, therefore, that in any equation in which the first term, or highest power of a?, has the coefficient unity, the coefficient of the second term is equal to the sum of the roots with their signs changed ; the coefficient of the third term is equal to the sum of the products of every two roots with their signs changed ; the coefficient of the fourth term is equal to the sum of the products of every three roots with their signs changed ; and so on : and the last term is equal to the product of all the roots with their signs changed. It is proper to observe however, that in the composition of the third, fifth, seventh, &c. coefficients, it is indifferent whether the signs of the roots be changed or not ; since the products that form these consist each of an even number of factors. It follows from this : 1 . That if the coefficient of the second term in any equation be 0, that is, if the term be absent, the sum of the positive roots must be equal to the sum of the negative roots. 2. When the coefficients are all whole numbers, and the first unity, every integral root of the equation will be found among the integral factors of the last term; for this last term, divided by a root, will express the product of the remaining roots. This product will therefore be the last term of the depressed equation involving those remaining roots ; and it is plain, from the process for deducing this equation, adverted to at p. 51, that all the resulting coefficients will be integral if the original coefficients, and the root by which they have been depressed, be all integral. This last consideration indeed justifies the more general inference that, whether the leading coefficient be unity or any other integer, still every integral root will accurately divide the last term. 58 GENERAL PROPERTIES OF EQUATIONS. 3. If the roots of an equation be all positive, the terms will be alternately positive and negative ; and if the roots be all negative, the terms will be all positive. It appears, moreover, that if one root only of an equation be changed, every one of the coefficients will be changed. For a different and very simple method of deriving the law of the coefficients, the student is referred to the Analysis and Solu- tion of Cubic and Biquadratic Equations, page 25. PROPOSITION V. (61.) If the signs of the alternate terms in an equation be changed, the signs of all the roots will be changed.* Let a?" + A„_ia?"-i + A„_2a?n-2 + A„_3a?"-3 + &C. =0 . . . . [1] be any equation, and a a root ; then, if a be substituted for x in the first member, the result will be zero ; and if we change the alternate signs, writing the equation thus, x^ — A„_i a?n-i + A„_2a?n-2 — k^_^x^-^ + &c. = . . . . [2], and instead of a substitute — a for x, the result, should n be even, will be the very same as before, and consequently zero ; but if n be odd, then the result will differ from what it was before only in this, viz. that all the signs merely of the polynomial wOl be changed, so that as it was zero before, it must be zero stiU. Hence, for every root a in [1], there is an equal root, with contrary sign, — «, in [2]. Thus the positive roots of any equation may be converted into negative, and the negative into positive, by simply changing the alternate signs of the equation, commencing at the second, and taking care to allow for the absent terms in incomplete equations. It is obvious that if the signs of all the terms are changed, the roots remain unaltered ; because whatever values of x cause the polynomial to become zero in one case, make it zero in the other also. • The equHtion is understood to be complete. If any term be absent it must be replaced by a cipher. GENERAL PROPERTIES OF EdUATIONS. 59 PROPOSITION VI. (62.) If all the coefficients of an equation be whole numbers, | and that of the leading term unity, the equation cannot have a '| fractional root. If possible, let the equation a?° + A^_ia?«-i + ....+ Ag^ + k^"^ + Aa; + N = 0, whose coefficients are all integral, have a fractional root ; and let the fraction in its lowest terms be -. Then, putting this for x^ we have 0" + A-O"-' + • • • + A3(i)' + A4)HA(|) +N =0. Or, multiplying by 6""^, b Now in this polynomial every term after the first is integral. Hence, transposing these to the other side, the first member, which will then be -— , must also be integral. But - being in its lowest terms, a and b have no factor in common ; and it is obvious that there can be no simple factors in a" that are not also in a itself : consequently a" and b have no factor in common ; that is the fraction — - is in its lowest terms : and yet it is equal to an o integer : which is absurd. Therefore the proposed equation can- not have a fractional root. Hence when the coefficients of an equation are whole numbers,H the first being unity, every real root of the equation must either be a whole number or an interminable decimal. The latter are called incommensurable roots. It follows from (60) that if there be/> 60 GENERAL PROPERTIES OF EQUATIONS. one of these in the equation, there must be one more at least ; whether the coefficients be integral or not, provided only that they be all rational; or that the second merely be rational. Moreover if, under the former hypothesis as to the coefficients, none of the divisors of the last term, whether taken positively or negatively, can satisfy the conditions of the equation, when they are severally substituted for x, we may conclude that the equation has neither an integral nor yet a fractional root : so that those roots which are not imaginary must be incommensurable. In next chapter this conclusion will be generalized ; for it will be proved that, whatever integer the leading coefficient A^ may be, every fractional root must have for its numerator a divisor of N, and for its denominator a divisor of A^. (63.) It is an important step towards the solution of a nume- rical equation to discover by a previous examination how many of its roots are real, and thence how many are imaginary. A more minute analysis indeed than this must be effected before the numerical value of the real roots can be actually developed : — we must know how many of these are positive, and how many are negative ; and lastly, if they be not whole numbers themselves, between what pair of whole numbers each is situated. Now if we happen to know that the roots of any proposed equation are all real, there is a rule — called the rule of Descartes — which will enable us to infer, from the mere inspection of the signs of the coefficients, how many of these roots are positive, and how many negative. The rule will do more ; it will show us, without any stipulation as to the character of the roots, the greatest number of each kind of real roots the equation can possibly have, consis- tently with the signs which connect its terms together ; and the knowledge of these limits, in certain cases to be hereafter noticed, will lead at once to the discovery of the number of imaginary roots entering the equation. The rule therefore is of considerable importance in the analysis of equations ; and that importance has been increased of late by the extension and greater efficiency which have been given to it by Budan and Fourier ; an account of whose researches will be found in a subsequent chapter. The rule is enunciated as follows : GENERAL PROPERTIES OF EQUATIONS. 61 PROPOSITION VII. (64.) A complete equation cannot have a greater number of positive roots than there are changes of sign from + to — , and from — to -f , in the series of terms forming its first member. Nor can it have a greater number of negative roots than there are 'permanencies^ or repetitions of the same sign in proceeding from term to term. To demonstrate this remarkable proposition, it will be necessary merely to show that, if any polynomial, whatever be the signs of its terms, be multiplied by a factor x — «, corresponding to a positive root, the resulting polynomial will present at least one more change of sign than the original ; and that if it be multipHed by a? + «, corresponding to a negative root, the result will exhibit at least one more permanence of sign than the original. This is the form to which the proposition was first reduced by Segner,* whose demonstration is distinguished from all others by its sim- plicity. It is in substance as follows : Suppose the signs of the proposed polynomial to succeed each other in any order, as + + __ + + + +, then the multiplication of the polynomial, by a? — a, will give rise to two rows of terms, which, added vertically, furnish the product. The first row will, obviously, present the very same series of signs as the original ; and the second, arising from the multiplication by the negative term — a, will present the same series of signs as we should get by changing every one of the signs of the first row. In fact, the two rows of signs would be ■h ^ + -+ + + + -++- + + + - and signs of prod. -\ ±H 1-±± — ±H • • Meraoires de Berlin, 17.')6. Lagrange, Traill, ^ (^7.) In the preceding investigation the equation is supposed to be rendered complete, when any terms are wanting, by the insertion of those terms with zero-coefficients. But, as these zeros / must be accompanied by the double sign i, their intervention merely adds to the number of ambiguities ; so that if a variation exist between two consecutive signs in the incomplete equation the ambiguities which intervene, however they be interpreted, cannot possibly destroy this variation, or convert it into a perma- nency. Hence no variations can be lost by these insertions; nor, if we give to all the intervening zeros the same sign, can any be gained. We may therefore suppress the condition ^as to the equation being complete, as far as the first part of the proposition is concerned, and conclude that, whether the equation be complete or not, the number of positive roots can never exceed the number of changes of sign. And as the negative roots of an equation are always convertible into positive by simply substituting — x for x, (61,) we may conclude farther, that the number of negative roots can never exceed the number of changes that would take place, if X were turned into — x. Thus the rule of signs may be freed from all restriction as to the complete form of the equation : and notice need be taken of the absent terms only with a view to detect the existence of imaginary roots. (68.) This method of searching after indications of imaginary roots is of considerable importance in the analysis of equations ; it is involved in the following general principles : 1 . The absence of an even number of consecutive terms from | an equation is an indication that there are at least that number of | imaginary roots in the equation. 5 66 GENERAL PROPERTIES OF EQUATIONS. Let the 2m absent terms in the equation of the nth. degree be replaced by 2m zeros ; and suppose, first, that the two terms between which they occur have like signs. Give to the 2m zeros these same signs. Then, considering only the two terms and the intervening zeros, we shall evidently have 2m + 1 permanencies ; so that if the terms omitted furnish k permanencies more, it wiU follow that the number of positive roots cannot exceed n — k — 2m— 1. Change now the sign of the first, and that of every alternate, zero ; then, their number being even, the first and last must evidently have unlike signs. Hence the last zero and the term following it have a permanence ; and this is the only permanence within the proposed limits : hence the number of negative roots cannot exceed ^ + 1 . Consequently the entire number of real roots cannot exceed {n — k — 2m — 1) + {k -\- \) =^n — 2m. But there are n roots altogether ; therefore there must be 2m imaginary roots at least. If the vacancies occur between unlike signs, then, when the zeros are aU plus, there will be only 2m permanencies ; and when they are alternately plus and minus, no permanencies, in the interval. Hence, supposing, as before, that the terms not taken into account furnish k permanencies, it will foUow from the first arrangement that the number of positive roots cannot exceed n — k — 2m, nor the number of negative roots, k ; so that the entire number cannot exceed n — k — 2m ■\- k-=n — 2m : therefore in this case also there must be 2m imaginary roots at least. 2. The absence of an odd number of consecutive terms from an equation is an indication that the equation has that number of imaginary roots, plus or minus one ; according as the vacancies occur between like or between unlike signs. When the extreme signs are Hke and the intervening 2m -f 1 zeros are written with the same signs, we shall have 2m + 2 per- manencies. And when the signs of the zeros are made alternately positive and negative we shall have no permanencies. From the first arrangement it follows, if k be the permanencies GENERAL PROPERTIES OF EQUATIONS. 67 furnished by the omitted terms, that the number of positive roots cannot exceed n — 2m — 2 — k; and from the second arrange- ment that the number of negative roots cannot exceed k ; so that the entire number of real roots cannot exceed n — 2m — 2. Hence 2m + 2 of the roots at least must be imaginary. When the extreme signs are unlike there will be but 2m + 1 permanencies under the first arrangement, and one under the second ; so that in this case the number of positive roots cannot exceed n — 2m — \ — k, nor the number of negative roots k -\- \. Hence, the number of real roots cannot exceed {n — 2m — 1 — k) -f (^ + 1) = w — 2m; so that 2m of the roots, at least, must be imaginary. (69.) By the aid of these principles we may examine the several intervals that occur in an incomplete equation, and thus determine a limit to the total number of its imaginary roots. And in order to this it will be unnecessary to keep in remem- brance the formal enunciations we have given to those principles above : for, from the character of the preceding investigation, and the conclusions deduced from it, it is plain that, when absent terms occur in an equation, we have nothing to do but to fill up each chasm by the requisite group of zeros, giving to the indivi- dual zeros of .each group the sign of the immediately preceding term, so tKat the greatest number of permanencies may be secured. We have then to write the several signs of the entire series of terms anew, giving however a changed sign to every alternate zero in each group, so that the least number of perma- nencies may be obtained. The number of permanencies thus lost in proceeding from the first row of signs to the second, or, which amounts to the same thing, the number of variations lost in proceeding from the second to the first, will be equal to the number of imaginary roots which the equation must have, at least. This gives the necessary completion to the rule of Descartes, and is included in the general theorem of Fourier and Budan, which will be discussed hereafter. It is usually ascribed to De Gua, who was the first to give a general demonstration of the 68 GENERAL PROPERTIES OF EaXJATIONS. rule of Descartes,* and who deduced several interesting conse- quences from it as respects the connexion between zero-coeffi- cients and imaginary roots. The leading principle, however, established by De Gua, in the paper referred to, is the important one that an equation can never have aU its roots real unless the equations of inferior degree derived from it, as at p. 17, have aU their roots real also ; from which he inferred that if for any value of cc one of these derived polynomials vanished, and at the same time caused the immediately preceding and succeeding polyno- mials to furnish results with like signs, imaginary roots neces- sarily existed in the proposed equation. f From the same gene- ral principle he further deduced an expression for the number of conditions necessary to be fulfilled in order that aU the roots of an equation may be real. Lagrange has given a condensed and able view of these researches of De Gua, in a note appended to his work on Equations, and has noticed the remarkable circum- stance, that the conditions of reality to which De Gua was led are the very same in number as those to which he himself was conducted by a very different route ; although these conditions were found to be unnecessarily numerous in equations of the third and fourth degree, and were suspected by Lagrange to be capa- ble of reduction in those of the higher orders, a suspicion which the discovery of Sturm has fully confirmed. To these observations we shall merely add, that the rule of Descartes was first pubhshed in his Geometry in 163/, but without demonstration. And as the work of Harriot, published in 1631 — ten years after the author's death, J had already exhi- bited the mode of effecting the composition of the first member of an equation from the simple factors involving the several roots — illustrating and confirming the method by numerical examples, it has been generally maintained by EngHsh algebraists that the rule • Mtmoires de VAcad^mie des Sciences, 1741. The general theorem of Fourier includes that of Descartes, as well as the deductions of De Gua noticed in the text. t These propositions will be proved in the chapter on the limits to the real roots of equations. % Artis Analyticx Praxis, GENERAL PROPERTIES OF EQUATIONS. 69 of Descartes was plainly set forth in Harriot's researches, and that it should bear Harriot's name. The learned Dr. Wallis strenuously supports Harriot's claim : but the precision with which the principle is announced by Descartes, although very probably but a mere conjectural inference from Harriot's composition of equations, will no doubt justify the common practice of giving his name to the rule.* • For further particulars respecting this subject reference may be made to Wa LLis's Algebra, to the before-mentioned paper of De Gua, to Hutton's History of Algebra, and to a note .at p. 220 of vol. i. of Hutton's Course, by Davies, 1841. It may be proper to add that De Gua, though maintaining the claims of Descartes to the discovery of the theorem, very naturally con- cludes, in the absence of all demonstration, that Descartes arrived at it only by induction : and we may here add, that for such an induction Harriot's examples were fully sufficient. Fourier, however, combats this opinion of De Gua, on the ground that a demonstration of the theorem was possible from the composition of equations, which composition however Harriot was the first to make known. See Fourier, Analyse ties Equations, p. v. CHAPTER V. ON THE TRANSFORMATION OF EQUATIONS. (70.) Algebraical EauATioNS do not always present them- selves in the most convenient forms for solution, and hence the expediency of being provided with the means of changing them from one form to another. Depriving the leading term of its coefficient, by division or otherwise, is the most simple change of this kind, and is a desirable preparative to the usual methods of solution, as it simplifies the form without affecting the roots of the equation. In most transformations, however, the roots them- selves become also changed, but still bear such known and simple relations to those of the original equation, as to render the deter- mination of these latter from them an easy operation. Generally indeed, to change the roots into others bearing given relations to them, is the direct object of the transformation; so that this part of the subject, in its full extent, involves the solution of the following comprehensive problem, viz. To transform an equation into another such that the roots of the latter shall be any given functions of those of the former. Under this form the subject will offer itself for discussion in a subsequent chapter; but we have no occasion to enter upon the investigation of so general a problem here, our attention at present being confined to those transformations which are useful or necessary in the actual so- lution of numerical equations, and which may be comprised in the four propositions following : PROPOSITION I. (71.) To transform an equation into another whose roots shaU differ, either in excess or defect, from the roots of the original by any given quantity. TRANSFORMATION OF EQUATIONS. 71 Let us suppose that the original equation is k^x^ + Ag^ + A2^2 -f A^ + N = . . . . [1], and that we wish to transform it into another whose roots shall be the same in number, but shall differ from them in magnitude each by the quantity r; then the relation between the x in the original equation and the x' in the transformed, will be a? = a?' + r, "or x' := X — r in which r will be plus or minus, according as the new roots, or values of x', are to differ from the original roots, or values of x, in defect or excess. If we actually substitute this value of x in the original, we shall obviously have the transformed equation, which will be of the form • k^x"^ -f A>'3 + A>'2 _^ AV + N' = . . . . [2]; and it is in this way that the result is generally obtained. But the method of actual substitution is unnecessarily laborious; and may be entirely superseded by a very simple process, which we shall now explain. Instead of xf put its value x — r m the equation [2]; we shall then have . ^ k^{x^rY-\rk'^{x^ry^-k'^{x-ry+k!{x-~r) + W=^Q . . [3]; an equation which, when reduced to a series of monomials by actually developing the terms, must be identical with the original; for, in fact, we have now returned from [2] to [1], by restoring to x' its value x — r. Hence we have the identity A^ {x - r)4 + A'g {x -ry + A'2 (^ - ^)^ + A' (^ - r) + N' = k^x^ + Aga^ + A2^2 + A^ + N. It is plain that if we divide the first member of this by a? — r, the remainder must be N'; but, the two members being identical, the division of either by a? — r must give the same remainder, and the same quotient. The division, therefore, of the second 72 TRANSFORMATION OF EQ,UATIONS. member, that is of the original polynomial, by a? — r, gives, for remainder N', and for quotient, A4 (x - rf + A'3 (x - r)2 + A^2 (^ - ^) + ^'' Also, dividing this by a? — r, we have for remainder A', and for quotient, A4 (x - ry + A'3 (a: - r) + A'2. Again, dividing this hy x — r, the remainder becomes A'^, and the quotient, A^{x — r) -\- A'3. And lastly, dividing this by a? — r, we have, for the final re- mainder, A'3; and, for the final quotient, A^; and in this manner may the coefficients in the transformed equation [2] be severally determined. Although, to avoid unnecessary complication, we have assumed the equation [1] to be of only the fourth degree, yet it is plain that the process by which the unknown coefficients in [2] have been derived one after another, in reverse order, is perfectly general. The division of the original polynomial by a? — r fur- nishes for remainder N' ; the division of the quotient by a? — r furnishes for remainder A'; the division of the second quotient gives for remainder Ag' ; and thus, by noting the remainders sup- plied by these successive divisions, we discover the several coeffi- cients of the transformed equation one after another, tiQ we finally arrive at the coefficient A'^^.^ ; the one preceding this, that is the leading coefficient, being the same as that in the original equation. Now we have exhibited at (51) a very easy way of performing the division of a polynomial, /(a?), by such a divisor as x — r; and, by employing that method in the present problem, the re- quired transformation may always be rapidly effected, as the fol- lowing examples will readily show. 1 . Transform the equation ^ + 5a^ + 4aJJ + 3a? _ 105 = 0, into another, whose roots shall be less than those of the proposed, TRANSFORMATION OF EQUATIONS. 73 by 2. Here the constant divisor is a? — 2, and the process directed by the above investigation, and conducted according to the plan at (51), will be as follows: A4 A., A2 A N 1 + 5 + 4 + 3 — 105 (2 =r 2+14+36+78 1 + 7 + 18 + 39 — 27 ,*. N' = — 27 2+18+72 1 + 9 -[-36 +111 .-. A'= 111 2 +22 1 +11 +58 .-. A'2 = 58 2 1 +13.-. A'3 = 13. Hence the transformed equation is a?'4 + I3a?'3 + 58^2 ^UW—27 = 0. j (72.) After what has been done in Proposition i. p. 42, it is presumed that the student will require no verbal explanation of the foregoing process. It will no doubt be sufficient to remark that, calling the numbers below the black lines results, each result is formed by adding r times the result immediately before it to the result immediately above it. We may observe, however, that the operation would be somewhat abbreviated by omitting the repe- tition of the first coefficient in the commencement of each row of results, by suppressing the plus signs, and by reserving the de- terminations of A'g, A'2, A', and N', till we come to the last result, thus : 74 TRANSFORMATION OF EQUATIONS. i 5 4 3 - - 105 (2 2 14 36 78 7 18 39 2 18 72 9 36 2 22 11 2 a/4 + 13a?'3 + 58a/2 -f- 1 1 1^ — 27 = 0. By suppressing also the several addends, and performing the addition operations mentally, we should, of course, abridge the space occupied by the process, very considerably. The whole would then, in fact, be reduced to this, viz. 1 5 4 3 — 105 (2 7 18 39 — 27 9 36 111 11 58 13 Other means might be easily contrived for shortening the apparent work; but we would recommend to the student the exhibition of the entire process rather than incur the risk of error by suppressing any of the steps. When r is 1, then indeed, as there is no effective multiplication, the process na- turally takes the form here given, as in the follovnng example. 2. It is required to transform the equation 2x^ — 13a?2 + 10a?— 19 = into another. whose roots shall be less than the roots of this equation by 1 . 2 0- 13 10—19(1 2 — 11— 1-20 .-. N' = — 20 4 — 7-8 /. A' = - 8 6- 1 .-. A'2= - 1 8 /.A', = 8. TRANSFORMATION OF EQUATIONS. 75 Hence, the transformed equation is 1x'^ + 8ar'3 — x'^ — Sx' — 20 = 0. 3. It is required to transform the preceding equation into another, whose roots are less by 3. 2 - - 13 10 - - 19 (3 6 18 15 75 6 5 25 6 36 123 12 41 6 54 18 6 trans, equa. 2a?'4 ^ 2Ax'^ + 95a?'2 -|- I48a?' + 56 = 0. 4. It is required to transform the equation \ 6a^ — 3a?2 + 4^' ~ 1 = 0, into another, whose roots shall exceed the roots of this by 3. Here the multipHer wiU be — 3. 6 - 3 4 - 1 ( - 3 — 18 ' 63 — 201 -21 ^1 — 18 117 -39 — 18 trans, equation, ^x'^ — hlx''^ + \^Ax' — 202 = 0. 7Q TRANSFORMATION OF EQUATIONS. The foregoing operation for diminishing or increasing the roots of an equation by any proposed number, deserves the student's special attention. The process is purely numerical; and arranges itself in a form that renders it of great value in the numerical solution of equations, which is conducted, step by step, by a series of operations of this kind. 5. Transform the equation a^ — 7x + 7 — Q into one whose roots shall be less than the roots of this by 2. The transformed equation is a^ -f 60?^ -j- 5a; + 1 = 0. 6. Transform the equation \9x^ — 220^ - 35a;2 — 16a? — 2 = into another, in which the roots shall be diminished by 3. The transformed equation is 1 9a^ + 206a;3 ^ 793^:2 _^ 1 232;^ + 580 = 0. 7. Transform the equation 3a?4 — 13a^ + 7a;2 — 8a? — 9 = into another, whose roots shall each be smaller than those of the proposed by ^. The transformed equation is 3ar*— 9a;^ — 4a;^— %^ a; — Y=t>. PROPOSITION II. (73.) To transform an equation into another whose roots shall be the reciprocals of those of the former. In the proposed equation N + Aa; 4- A^a-'-f A.^a>3 + .... A„a,'" = 0, TRANSFORMATION OF EQUATIONS. ^^ substitute - for a?, then the values of - will be the same as those y y . . of Xy and, consequently, the values of y wiU be the reciprocals of those of X ; that is, the roots of the equation y y^ y^ y° or, rather of %" + Ay"-i + k^y^-'^ + A33/"-3 + . . . . A„ = 0, will be the reciprocals of the roots of the proposed equation. Hence the transformed equation is deduced from the original, simply by reversing the order of the coefficients ; as many terms, therefore, as are absent in the original equation, so many and no more will be absent in the transformed. Hence we may transform an equation into another, whose roots shall be less or greater than the reciprocals of those of the proposed, by applying the process employed in last proposition to the coefficients of the given equation, written in reverse order. For example, let it be required to transform the equation x^— 12a?2 + I2a^— 3 = 0, into another, whose roots shall be equal to the reciprocals of those of the given equation, diminished by 1 . — 3 12—12 1 (1 9 _ 3 «.3 _ 2 6—30 3 6 Hence, the transformed equation is — 3/ ± 0y3 + 6y2 -h y — 2 = ; or rather 3y4 _ 6y2 4- 2 = 0. 78 TRANSFORMATION OF EQUATIONS. (74.) If the coefficients of the proposed equation be the same when taken in reverse order, as when taken in direct order, both in signs and numerical values, it is obvious that the reciprocals of the roots will furnish the same series of quantities as the roots themselves, seeing that the equation which involves the reciprocal roots will be the same as the original equation ; the roots of the original equation must, therefore, under such circumstances, be of the form 1 1 1 * a «2 % ~~^ y Ctny y ^Q) y OOCm of which the reciprocals produce the same series of quantities. The first row of roots, where each is accompanied by its reci- procal, will belong to equations of an even degree : the second to those of an odd degree, the signs of the coefficients observing the law just stated. It will appear presently that the single isolated root, in an equation of an odd degree, must be, as written above, — 1, the first term of the equation being, as usual, positive. If the equation be of an odd degree, and the coefficients, taken I in reverse order, be in magnitude the same as when taken in direct order, but in signs all diff'erent, then also will the roots of the transformed equation be identical with those of the original equation; for, by changing aU the signs of the transformed equation, which of course produces no change in the roots, the equation will become the same as the original, and must, there- fore, have the same roots. The same thing evidently has place in equations of an even degree, under like circumstances, pro- vided only the middle term be absent. If the middle term be present, then the signs taken in reverse order cannot all be contrary to those taken in direct order : the middle term will interfere with this arrangement of the signs. Equations whose coefficients exhibit either of these laws, and whose roots are, in consequence, of the above form, are called recurring equations, or reciprocal equations. They will be more fully treated of hereafter. TRANSFORMATION OF EClUATIONS. 79 In a recurring equation of an odd degree, one root will always be + 1 or — 1, according as the. sign of the last term is — or + ; for, as the roots of the transformed are always the same as those of the original in recurring equations, and yet at the same time the roots of the transformed are the reciprocals of those of the original, one of the odd number of roots must be + 1, or — 1 ; moreover, as the remaining roots consist of pairs, having the same sign, the last term of the equation, which is the product of all the roots with their signs changed, must take the opposite sign to the root unity, being — when that root is -f , and + when the root is — . PROPOSITION III. (75.) To transform an equation into another, whose roots shall be given multiples or submultiples of those of the proposed equation. Let the given equation be freed by division from the coefficient of the first term ;* then, in the resulting equation, the coefficient of the second term will be the sum of the roots with contrary signs; the next coefficient, the sum of the products, two and two ; the next, the sum of the products, three and three, signs being changed, and so on (prop. iv. p. 5.5) : hence, for the roots to be m times as great, we must multiply the second term by m, the third by m^, the fourth by m^, and so on. These multiplica- tions being effected we may introduce any additional factor into all the transformed coefficients without disturbing the roots : and thus the coefficient of the leading term of the equation, tempo- rarily removed by division, may now be restored by multiplication. In practice however we may evidently leave the original coeffi- cient undisturbed, and proceed at once with the multiplications here directed. * It is necessary to say freed by division, in order that the roots may be preserved unaltered. The present proposition furnishes other means of re- moving the first coefficient, but not without changing the roots. 80 TRANSFORMATION OF EQUATIONS. If, for example, it be required to transform the equation 2a;3_5^2+ 7a?- 12=0, into another, whose roots are three times as great, we shall merely ha\e to multiply the second term by 3, the third by 9, and the fourth by 27 ; the transformed will therefore be 2a^ — 15a^ + 63a? — 324 = 0. It is an obvious inference from the preceding rule, that if in an equation the coefficients of the second, third, fourth, &c. terms be divisible by m, m^, m^, &c., respectively, the roots will have the common measure m, ^^ ij^.) We may now easily prove the property mentioned at p. 60, viz., that when an equation, with integral coefficients, has a frac- tional root, the final coefficient, N, is divisible by the numerator of the fraction, and the leading coefficient, A^^, by the denomi- nator. For let - be the fractional root, a and h having no com- mon factor : then if the equation be transformed into another whose roots are h times as great, the final term of the transformed wiU be i"N. One root of this transformed equation will be the integer «, therefore (p. 57) « is a factor of &"N. But as a has no factor in common with 6, by hypothesis, a cannot be a factor of 5" : hence a must be a factor of N. Again, let the proposed equation be converted into another whose roots are the reciprocals of the original 'roots (73). The leading coefficient of the transformed equation will be N, and the absolute term A : also - will be a root of this equation. As a before, let the last equation be transformed into another whose roots are a times as great: the absolute term will then be «"A^, which (p. 57) is divisible by h. But, as before, h cannot be a factor of «" : hence it must be a factor of A . It thus appears that when the first member of an equation whose coefficients are all integral, is divided by the first member of the simple equation involving one of the roots, — numerical fac- tors common to both terms of the divisor being expunged, — the extreme terms of the quotient must be free from fractions. But TRANSFORMATION OF EaUATIONS. 81 this is only a particular case of a much more general property, viz., that if the first member of an equation be divided by the first member of any other equation whose roots all belong to the former — the coefficients of both dividend and divisor being free from fractions, and those of the latter free from common factors — then all the terms of the quotient will be free from fractions. This general proposition will be found of considerable use in the analysis of equations : it may be established as follows : Let the first member of the proposed equation, when the co- efficients are freed from common factors, be ax^ + hx^-"^ + c^°-2 -\- .... k ... . [1], and the first member of the equation involving m of its roots aJx"^ + b'x"'-'^ + .... ^ ... . [2], common factors being expunged. All the simple factors of [2] occur among those of [1], so that [1] is divisible by [2] without remainder. Suppose that, in order to preclude the entrance of fractions into the quotient arising from this division, it be necessary to multiply [1] by some factor P, and let the quotient whose coefficients are thus all ren- dered integral be '• a" xP -\- b" xi?-^ -h k" [3]. This quotient has no factor common to all its terms, otherwise P would involve this factor unnecessarily, and would therefore be needlessly large. Now since the operation terminates without a remainder, we must have [2] X [3] = P X [1]. The second member of this equation is divisible by the number P : hence the first member must be divisible by P. But neither [2] nor [3] is divisible by any number except unity. Conse- quently P = 1,* so that the division of [1] by [2] furnishes • We may prove that the first member of the above equation can have no numerical factor, as follows: — Suppose q to be any prime number, and let R represent the aggregate of those terms in [2], whose coefficients contain the 6 82 TRANSFORMATION OF EQUATIONS. a quotient free from fractions, without any previous pre- paration. It follows from this that, when the proposed equation is free from fractions, every depressed equation, arising from the eUmi- nation of any number of its roots — ^by dividing by the integral polynomial involving those roots— must also be free from fractions. Hence, when it is suspected that a polynomial, free from fractions and common factors, involves roots all of which belong to a pro- posed equation, we may proceed to divide the latter polynomial by the former ; and, if a fractional coefficient occur in the quo- tient, may discontinue the operation, and conclude that the sup- posed connexion between the polynomials has not place ; but if no fraction occur, and the operation terminate without remainder, then of course the suspicion is verified. The same conclusions follow although the suspected divisor be divided by any integer — as for instance by the leading coefficient in it — notwithstanding the fractional coefficients thence introduced; since the only effect upon the quotient will be to introduce this integer, as a factor, into all its terms. (77.) By help of the transformation in (75), the coefficient of the first term of an equation may be removed without intro- factor y, R' representing the remaining terms. In like manner let S be the sum of the terms in [3], whose coefficients all involve y, S' being the sum of the remaining terms ; then R4-R'=[2] S + S'=[3] Hence, multiplying these together, RS -f R'S -t- RS' -H R'S'= [2] x [3]. A nd since by hypothesis q divides R and S, it divides the first three terms of this equation : hence, that it may divide the product of [2] and [3], it must divide the fourth term R'S' : but this is impossible ; for let Ai', A V be the two terms in R' and S' respectively, affected with the highest power of x ; then since their product, hh' 1'^+ «', must form a distinct term of the product R'S', because all the other component parts of it involve inferior powers of r, it follows that hh' must be divisible by q ; which is absurd, because there is no factor either \x\h. or h' equal to q, seeing that all terms containing this factor have been excluded from R' and S'. TRANSFORMATION OF KQUATIONS. 83 ducing fractions ; for, if m be the coefficient of the first term, and we transform the equation into another, whose roots are m times those of the former, the factor m will then enter into all the other terms; dividing by it will therefore free the first term, and introduce no fractions. The transformed equation will therefore be obtained by expunging the coefficient of the first term, pre- serving the second term, multiplying the third by m, the fourth by w^2, &c. and the roots of the transformed will be m times those of the original. Thus, taking the equation which, completed, is 3a^-f0a?2 — 5^ + 2 = 0; we have for the transformed, whose roots are three times as great, the equation ^ + 0a?2 — 15a?+ 18 = 0, or, rather o^ - 15a? + 18 = 0. Fractions may be removed from an equation by transforming the equation into another, whose roots are those of the former, multiplied by the product of the denominators of the fractions, or by a common multiple of the denominators. For example, the equation a^ + ^x'^ — ^x-\- 2=0, wiU be transformed into a^ + 3a?2 — 12^7 + 432 = 0, by multiplying the terms, com- mencing at the second, by the successive powers of 6 ; and, if the roots of the former equation be «p ag* ^3> those of the latter will be 6«p 6a2J 6% This method will always effect the removal of the fractions, although not always without introducing a new inconvenience ; namely, very large coefficients. If the attainment of simplicity therefore be the ultimate object, we shall often find it better to clear the fractions in the ordinary way ; that is, by multiplying all the terms by the least common multiple of the coefficients. 84 TRANSFORMATION OF EQUATIONS. although we thus introduce a coefficient into the leading term. But which of these two methods is to be preferred in any parti- cular case, must he determined from the character of the deno- minators themselves, which may happen to be so related to one another that they may all be removed according to the first method by employing only a small submultiple of their common multiple. For instance, the equation a? x^ X 5 ^ ^2 3 36 648 will, agreeably to the first method, be converted into a?4 + 3a^ - 12072 - 6ar + 10 = 0, an equation whose roots are six times those of the former, by employing the multipliers 6, 6^, 6^, 6^ instead of the correspond- ing powers of the common multiple 648 ; 6 being a small sub- multiple of this. (78.) In what has preceded, the roots of the proposed equation are considered to be multiplied by m in the trans- formed : but if they be regarded as divided by m, then the terms of the equation, commencing with the second, will have to be divided by ?w, m^y m^, &c. respectively ; and as these divisors may all be removed by multiplying each of the terms by the highest power of m, it follows that the general equation of the wth degree Ana?" -h A„_i a?°-i + . . . k^a^ + k^oi^ -h Aa? -f N = will become Anw"a7"-f A„_imn-^ x^-^-\- . . . A^ni^a^+A^m^aP-\-Amx-{-'N=:0 when its roots are all divided by m. And it may be further remarked, that whatever powers of x be absent in the original equation, the same powers will be absent in the transformed, whether the roots be multiplied or divided. TR-ANSFORMATION OF EQUATIONS. 85 PROPOSITION IV. (79.) To transform an equation into another, in which any proposed term shall be absent. If the transformed equation is to be deprived of its second term, which is the term generally required to be removed, the transformation may be effected by the process in Problem I. p. 70, as it will be merely required to diminish the roots by such a quantity, r, as will cause the second coefficient in the resulting equation to vanish. Now, in the process of diminishing the roots, it is seen, that when the leading coefficient is unity, r is added to the second term n times ; so that for the result of these additions to be zero, r must be minus the nih. part of the second coefficient in the proposed equation. To illustrate this, let it be required to remove the second term from the equation a^ _ 12a?3 4- 17a^2 — 9;r + 7 = 0. Here r = ^2 = 3^ and the operation is as follows : ' 1 — 12 17—9 7 (3 3 —27 —30 —117 -9 — 10 -39 3 - 18 - 84 — 6 — 28 3 - 9 — 3 3 -f 0^ — 37*^ — 123^ - 110 =0. 86 TRANSFORMATION OF EQUATIONS. hence the transformed equation is aj4 — 37ar2 — I23a? — 1 10 = 0, the roots of which are those of the proposed diminished by 3. (80.) But in order to determine the value of r, necessary to cause any other coefficient to vanish, let us actually substitute a/ -H r for x, in the general equation a?° + K-i ^"~* + . . . . Aa? 4- N = 0, and then develop the several powers by the binomial theorem, arranging the result according to the decreasing powers of x' ; we shaQ thus have a/n + A n-l y^-J n(n— 1) + (w-l)A„_,r /"-2 4- + r» -HA + A n-l n-2 #n— 1 ,n-2 = -f Ar + N In order that the second term of this transformed equation may vanish, we must have the condition nr + A„_j = /. r = 2^, n as before determined. That the third term may vanish, we must have the condition n{n — 1) r2+(w-l)A„_,r + A„_2 = 0; TRANSFORMATION OF EQUATIONS. 87 which, being a quadratic equation, will furnish two values for r, each of which will cause the third term in the transformed equation to vanish. The determination of values for r, that will cause the fourth term to vanish, will require the solution of an equation of the third degree ; and, to remove the last term N, would require the solution of the following equation of the nth. degree in r, viz. the equation /•" + A„_i r"-i + . . . . At- + N = 0; which is no other than the proposed, x being replaced by r; so that the removal of the last term requires a preparatory process, equivalent to solving the original equation. (81.) It may be remarked here, that methods have been in- vestigated for removing from an equation as many intermediate terms as we please, with the view of reducing equations of the advanced degrees into soluble forms. This method was first pro- posed by TscHiRNHAusEN in 1683 ; and it has been recently re- vived and treated with much analytical skill and address by Mr. Jerrard, in his Mathematical Researches. But like all other attempts to extend the limits of the general problem of the solution of algebraical equations by finite formulae, beyond equations of the fourth degree, these methods have proved un- successful : the imperfection common to all of them being that, when applied beyond these bounds, their application requires the solution of equations higher in degree than that which is pro- posed to be solved. The only term which in the present state of algebra it is of any consequence to remove from an equation is the second; in most of the methods proposed for solving equations the absence of this term conduces to the simplicity of the operation, whether the solution be by an algebraical formula or by a process purely nu- merical. In the latter mode of treating the problem, however, the advantage of this absence is felt but in a trifling degree, and that chiefly in the preparatory analysis of the equation for the purpose of discovering the nature of the roots. We may further observe with respect to the transformed 88 TRANSFORMATION OF EQUATIONS. equation to which the removal of this term leads, that although the roots of this equation will diifer from those of the original, yet the differences of the roots of the original will be the same as the differences of the roots of the transformed ; so that when these differences are to be determined, with a view to the discovery of the roots themselves — as in the method of Lagrange, to be hereafter explained — we may substitute the transformed equation for the original, and this effects considerable saving in the labour attendant upon the method adverted to. By removing the second term from a quadratic equation, we shall be immediately conducted to the well-known formula for its solution. Thus, the equation being a;2 + Ai*? -f N = 0, the transformed in x' + r, will be ^2 + 2r 4-A x' -h 1 o~^ -f Arl = ; + nJ and, that its second term may vanish, we must have 2r + A = .-. r = — I A, which condition reduces the transformed to .-. a?' = ± V A A2 — N :,x = x' ^r = --^A± V|A2-N; which is the common formula for the solution of a quadratic equation. (82.) All the problems discussed in the present chapter, as well as that just commented upon, equally deserve notice, as furnishing the necessary operations preparatory to the actual TRANSFORMATION OF EQUATIONS. 89 solution of an equation. To facilitate the process of solution, or to convert the equation from an inconvenient form into another better adapted to certain rules and formulae for evolving the roots, is the only object proposed to be accomplished by the transformation of equations. The property inferred at (61), and which suggests the solution of the problem — ^to transform an equation into another whose roots shall be the same in value but opposite in sign — does in strictness belong to this department of our subject. It is indeed included in Proposition iii, as the roots of the transformed equation are no other than those of the original multiphed by — 1 . This transformation is useful in the analysis of equations ; as the examination need be applied only to the detection of positive roots, into which the negative are con- verted by simply changing the alternate signs. The more general transformation, of which this is a particular case, finds its appHcation as a preparatory step in the search after commensurable roots. We have already seen that if the leading coefficient be unity and the others integral, the commensurable roots must be integral also. But if the leading coefficient be other than unity, the commensurable roots may be fractional ; and cannot therefore, like the former, be found among the integral^ divisors of the absolute term. The transformation referred to, by changing these fractional roots into integral, will sometimes be found to facihtate the search after the com- mensurable roots. The same transformation may also be usefully apphed even in connexion with the very effective mode of solution that will be more especially dwelt upon in the practical part of this work — Horner's method : and which is independent of the preparatory operations that most other methods require. By means of this transformation however, each of the commensurable roots will always be presented by Horner's process in a finite form ; whether they be integral or fractional, instead of appearing, in the latter case, in the form of an incomplete or approximative decimal. An objection that has been hastily made against the method in question — viz. that approximate values only are fur- nished by it when the roots are fractional — is thus completely removed. 90 TRANSFORMATION OF EQUATIONS. The transformation at (73), by which roots are converted into their reciprocals, is another preparation essential to the success of certain methods of solution. More especially is this the case as respects the method of Budan, to be discussed hereafter ; and advantage may be occasionally derived from it even in the more perfect process of Horner, as wiU be sufficiently seen when we come to examine into the capabilities of that method. It is further worthy of notice that, by aid of this transfor- mation, the general proposition in (79) may be considerably simpHfied. In examining the different cases of that proposition, it was seen that the difficulty of applying it increased as the term to be removed approached nearer to the final term of the polyno- mial. But if instead of removing a term near the end, we were to remove the term as near to the beginning, and then transform the resulting equation into another whose roots are the reciprocals of it, an equation would be obtained in which the term at the proposed distance from the final term would be zero ; since the coefficients in the direct equation, reckoning from the first to the last, are the same as those in the reciprocal equation, reckoning from the last to the first. Hence, if any method of solution were proposed that would be facilitated by the removal of the last term but one, as certain existing methods are facilitated by the removal of the second term, the requisite preparation would be easily made by removing the second term, and then reversing the order of the coefiicients ; observing that the roots of the equation thus deduced are the reciprocals of those of the original equation, after the removal of its second term. The importance of the transformation proposed in Proposition i will be distinctly seen in connexion with Newton's rule for finding a superior limit to the roots of an equation ; and, as before observed, in connexion with the numerical operation for obtaining the roots themselves. CHAPTER VI. ON DETERMINING THE LIMITS TO THE REAL ROOTS OF EQUATIONS. (83.) In the concluding observations in last chapter we have briefly explained the objects which analysts have in view in attempting certain jchanges in the form merely of algebraical equations, preparatory to the apphcation of methods of solution. When these transformations are effected, wherever they are found necessary, the first step in the actual solution may be commenced; and this, as far at least as numerical equations are concerned, consists in the determination of the places in the numerical scale which the several real roots occupy. The satisfactory settlement of this important point constitutes what is called the analysis of the equation. And we are fortunately at length in possession of methods — the methods of Sturm and Fourier — by which this analysis may always be completed ; that is to say, when any two numbers are proposed, we can always discover whether or not roots lie between them, and how many roots are so situated ; and then, by narrowing the limits between which they are thus found to lie, we can find exactly between what two consecutive numbers each root, or each group of nearly equal roots, must be posited. In other words, we can completely analyse the equation. In commencing this analysis it would clearly be unwise to fix upon two numbers at random ; for these might be very remote 92 LIMITS TO REAL ROOTS. from the extreme limits of the entire series of roots ; and it is for the purpose of guiding us to a judicious selection of superior and inferior limits to the roots — thatjR, innmhftrH nhnvp thp grpAfpst root and helow J,he„Ji£!a&t — that the propositions in the present chapter are of any value. It is scarcely necessary to remark upon the advantage of such limits, in diminishing the range of divisors of the last term, in seeking for the commensurable roots. The remotest limits to every set of positive roots are obviously the extreme values and ^. The corresponding limits to the negative roots are and — a. . But the imaginary roots of an equation cannot be regarded with propriety as being comprehended within these ; because, in passing continuously from zero to infi- nity, that is from a? = to a? = ±: ^, we should never pass an imaginary value of a? ; so that imaginary roots have no r P (x""-^ + a?"-^-i -H . . . -\- x-\- \) or, which is the same thing, the condition a?«-»^+i — 1 x->V X — 1 • Maclaurin's Algebra, page 174. LIMITS TO REAL ROOTS. 95 Now, assuming x greater than unity, this latter condition is always satisfied when ^n-k+l X — 1 since the right-hand member of this inequality is greater than the right-hand member of the former. Multiplying each member by a? — 1, and then dividing each by ^n-k+i^ we have (a?— l)a:k-i>P which condition is satisfied whenever either of the following is, viz. {x — l)k = P, or {x — l)k > P, because the first member of either of these is less than the first member of the former. We have then to determine x so that 1 1 a? = Pk +1, ora?>P'»^-f. 1 1 Hence P^ + 1 exceeds the greatest root of the equation. The following examples will serve to show the application of this proposition. 1.'^ a^-_5^-|- 37a;2 — 3a?-4=0 1 /. P = 5, A: = 1 .*, P^ -|- 1=6 = superior limit. 2. r^ + 7^ _ i2ar^ - 49^2 + 52^ - 13 = .*. P = 49, ^ = 2 .-. P"k + 1 = 8 = sup, limit. 3. a?4 + lla?2_25^-67 = 1 /. P = 67, A; = 3 .*. P"*^ + 1 = 5 -t- . . . .*. 6 = sup. limit. 4. 3a^ - 2a?2- 11^ -H 4 =0 11 - 2 :.V = —,k—\ .-. Pi^ + 1 = 4 - .-. 5 = sup. limit. 90 LIMITS TO REAL ROOTS. PROPOSITION IV. (88.) If, in an equation, — P be the greatest negative coeffi- cient, and if, among those positive terms which precede the first ^ . P negative term, the greatest coefficient S be taken, then will -^ + 1 be a superior limit to the positive roots of the equation. The most unfavorable case will be that in which all the terms that follow the first negative term are also negative, and their coefficients equal to P. Under these circumstances, the equation may be written — P(l+a? + ar2+....^"') + Sa?'"+i-|-Tx"^+HUa?'"+3_j_,,,.=0....[l] Now the negative portion of this polynomial will be -P P which, by substituting — - + 1 for a?, becomes Also the positive portion, by a like substitution, becomes S{|-+l}-^' + T{|- + l}™+2^U{|-+l}-+3_^ . ... of which the first term alone exceeds the former portion ; there- fore the aggregate of both portions must be positive. If the coefficient S belonged to a term more advanced, it is obvious that the excess of the positive portion above the negative would be increased. It is easy to see that, when any value of x is found that will cause the positive part of [1] to exceed the negative, every higher value of x will have a similar effect ; for, if we divide both portions by .r"»'^^ the first will consist of a series of fractions LIMITS TO REAL ROOTS. 97 in X, and will consequently diminish as x increases ; while the P second part will continually increase with x. Hence -^ -f 1 is a superior limit to the positive roots of the equation. Applying this method of finding a limit to the examples in the preceding proposition, we have, for the limit in the first example, P 5 1 st, — + 1 = — — I- 1 .*. 6 = superior limit b 1 P 46 2d, -^+l=-^-4-l .*. 8 = superior limit P 0^7 3d, — -}-l=— -(- 1 .•.8 = superior limit o 11 p 11 4th, — --|-1=— -+ 1 .*. 5 = superior limit. IS 6 The limits given by this method are, in these examples, the same as those before determined, with the exception of that in the 3d example, to which the former method is applied with more success. In the following example, however, this latter method of finding a near superior limit has greatly the advantage : -. ^ a?4 + 16ar^ — 2^2 _ 12^ _ 6 = 0, P 12 ,*, — + 1 = — -f 1 .*. 2 = superior limit. b 16 By the former method the limit would be i I P*+ 1 = 122+1.*. 5= superior limit. The limit estabhshed in this proposition was first given by \ \\ M. ViNE in the M6moires de V Acadimie de Bruxelles, 1822. '^ PROPOSITION V. (89.) If each negative coefficient be taken positively, and divided, by the sum of all the positive coefficients which precede it, the 7 98 LIMITS TO REAL ROOTS. greatest quotient thus obtained, when increased by unity, will be a superior limit to the positive roots. The demonstration of this proposition depends upon the known expression for the sum of a geometrical series, from which we infer that x^^ ^— 1 X — \ and consequently that every power x^y of x, may itself be repre- sented by a polynomial of the form x'^ = {x — \){ x^-^ + a?"^-2 + x^-'^ +....+a? + l} + l where the series within the liraces involves all the powers of x inferior to the power represented. Let the positive terms of the equation A„a?"+ .... A2^2^A^ + N = .... [1] be each replaced by its equivalent polynomial : we shall have for the first of these A„a^"=A„(^-l)^"-i-f-A„(a?-l>"-2+A„(^~l)a^«-3 + ..A„(:r-l)+A„ and if the other polynomials be written under this, so that like powers of x may range vertically, we shall have for the sum of all these positive terms a polynomial of the form A„(a:-l>"-i+A'„_i(^-l>"-2^A'„_2(^_l)^n-3_|_..A'(a;_l)+F..[2] in which all the powers of x, inferior to the leading power a?", necessarily occur. Let the difFe?*ent jiegative terms of [1], that have been omitted, be now introduce^ in their proper places under the like terms of [2]. It is plain that more than one negative term cannot occur under any positive term of [2], since in different negative terms the powers of x are different. The negative terms of [1] being thus introduced into the sum of the positive terms of [1] as represented by [2], we shall of course obtain the original polynomial [1] under anew form; and it is from LIMITS TO REAL ROOTS. 99 contemplating it under this change of form that the proposition announced discovers itself. For it is plain that the polynomial in question will always be positive provided that x be greater than 1, and that moreover no negative term exceeds the positive term in [2] under which it is placed; since by the first condition all the terms of [2] are positive, and by the second, however many of these may be balanced by the negative terms written under them, none can be overbalanced; so that the aggregate must continue positive. These then are the only conditions we have to fulfil. Suppose the first negative term that occurs in [1] to be— A a?P; the proper place for this under [2] will be below the term A'p^j(a? — l)a?P; so that, with respect to these, the second condi- tion wiU be fulfilled, that is, the sum of these two terms will be positive, provided and the same sum will be zero if these two expressions be equal. In like manner, if the next negative term that occurs in [1] be — AqO?*!, the sum of it and the like term above it in [2], will be positive, provided A'q+i (^ - 1) > A^, or ^ > -^ + 1 and it will be zero if a? be equal to, instead of greater than, this expression. And by thus taking every negative term of [1], and comparing it with the Hke term of [2], we shall obtain the several partial conditions which, if simultaneously satisfied for any value of X, will render that value a superior limit to the roots of [1]: for the other condition, a? >• 1, is, we see, necessarily comprehended in each of the foregoing. Now it is obvious that each of the partial conditions referred to will be fulfilled if from among the fractions which enter them, viz.. A' A' P+l -^q+l we select the greatest^ and cause x to fulfil the condition involving 100 LIMITS TO REAL ROOTS. it alone. But the coefficient, A'^^,, of any term A'^^, (x-'\)x% in the development [2], is no other than the sum of all the posi- tive coefficients in [1] which precede the term A x^. Hence, if fractions be formed by taking each negative coefficient positively, and dividing it by the sum of all the positive coefficients that precede it, the greatest of these fractions, increased by unity, will be a superior limit to the positive roots of the equation. (90.) The preceding proposition was first given by M. Bret, in the sixth volume of the Annales des Mathematiques, and as a general principle, may be regarded as the most effective that has yet been proposed for finding a close superior limit to the positive roots of an equation. The limit of Maclaurin, given in Propo- sition II, and which, from the readiness of its application, is that which is most frequently employed, is evidently included in the limit of Bret as a particular case. The most unfavorable case for the application of the present proposition will be that in which the greatest negative coefficient is preceded by positive coefficients whose sum is comparatively small; when the method of V^ne may give a limit equally close, and that in Proposition ii, one stiU closer. It is plain that the limit determined by the former of these methods can never be closer than that found by the present proposition ; and hence we may infer that when our object is to obtain the closest limit in any case, we need apply only the two propositions (87) and (89). If we take the examples already given (87), the limits deter- mined by the present proposition are as follow : 5 49 1. — + 1 .-. limit = 6; 2. hi.*, limit = 8; 1 8 67 11 3. —r + 1 .-. Umit = 7; 4. — + 1 .*. limit = 5. 1 ^ o In the third of these examples the method of (87) gives a closer limit. Suppose the following examples were proposed, viz. 1. 4a;^ — 8a^ -h 2Zx^ + 105a^2 _ 80a: + 1 1 = 0, LIMITS TO REAL ROOTS. 101 8 then it is plain that the greatest fraction is — = 2 ; therefore 3 is a superior limit to the positive roots. 2. 2x^ ^-\\x^-' lOa?^ - 26a^4 + 31^ _|. 72a?^ — 230^ — 348 =0. The fractions are 10 26 230 348 13' 13' TTg' TTg' the last of which, equal to 3, is the greatest; therefore 4 is a superior limit. These propositions on limits refer exclusively to \h^^ 'positive roots of the equation ; the only class of roots which, as remarked at (82), need be attended to in the enquiry, on account of the facility with which the negative roots may be converted into positive roots. "When this conversion has been effected, and the superior limit determined by any of the preceding propositions, this limit taken negatively, will be numerically greater than the greatest negative root of the proposed equation. If the coefficients of the proposed be all positive, then it can have no real roots but nega- 1 tive roots; so that the foregoing propositions can apply only after/ the change adverted to, of the negative roots into positive, has! been made. With respect to the determination of inferior limits to the roots, it will be sufficient to remark that the propositions already established are applicable to the discovery of these after the equa- tion has, by (73), been converted into another whose roots are the reciprocals of those of the proposed. For the greatest root in the transformed equation will be the reciprocal of the least in the original equation, and vice versa; so that if a superior limit be found from the transformed equation, the reciprocal of this will be an inferior limit in reference to the original equation ; and this whether we regard the positive roots or the negative roots. It is sufficient therefore that the propositions for the determi- nation of limits to the roots of an equation, comprehend all that concerns the superior limits to the positive roots ; since everything else respecting the limits may be brought within the scope of these propositions by easy preparatory transformations. It 102 LIMITS TO REAL ROOTS. should be observed, however, that in many cases these propo- sitions may be altogether dispensed with, and a superior Hmit, closer even than they would give, inferred from roughly estimating the comparative importance of the positive and negative terms ; in which estimate we shall generally be assisted by bringing all the positive terms to the left of the sign of inequality >, and the negative terms to the right, and then seeking for the smallest value of a?, for which, and for all higher values, the assumed inequahty may be satisfied. The number sought will be most readily suggested if both members of the inequality be first divided by the highest power of x. Let example 4 at (87) be taken, viz. 3a?3 — 2a?2— lla?-|- 4=0; . then we have to satisfy the condition Zst^ ^A'>2x^ ^ Ux or 4 2 11 For a? = 2, the second member a httle exceeds the first; but for a? = 3, the inequahty is fulfilled, and obviously for every higher number, as the first member can never decrease below 3. Hence the Umit is 3. Again, let the equation be a^ + ^ _ i5a^ __ 19a. — 3=0. Then we must have af* -H a?3> 15a?2 + 19^? + 3 or 1 15 19 _3^ X x'^ X^ X!* which is readily seen to be satisfied for a? = 5, and for every higher number. Hence 5 is a superior limit. By (89) the hmit would be 1 1 ; and by (87) it would be 6. / LIMITS TO REAL ROOTS. 103 Besides the methods here explained there is another, that of Newton, by means of which a close limit to the positive roots may be obtained. It is however but seldom applied to this pur- pose, as it does not offer the facilities of the other methods. We shall postpone the consideration of it to next chapter; because it is intimately connected with the inquiries there to be discussed, and forms a suitable introduction to the methods of Budan and Fourier for separating the roots of equations. PROPOSITION VI. (91.) If the real roots of an equation, ranged in the order of their magnitudes, be «p a^i be substituted for x, the result will be negative ; if a number 63, between a^ and ^3, be substituted, the result wiU be positive, and so on. The first member of the proposed equation, after removing the leading coefficient, is the product of the simple factors {x — «i) {x — a^ (x — tt^) (x — a^ . . . . multiplied by the quadratic factors involving the imaginary roots. Omitting these latter for the present, let us examine the effect of our proposed substitutions upon the product of the real factors. Putting then 5j for a? in these factors, we have (&! — ttj) (6j — a^ {b^ — a.^ (6j — «^ = 9i positive number, because all the factors are positive. Putting ^2 for a?, we have (^2 ~ ^1) {h -~ ^2) {h -^ %) {h — «4) = a negative number, because the first factor is negative, and all the others positive. 104 LIMITS TO REAL ROOTS. Putting ^3 for x^ we have (^3 — «j) (^3 — a^ (^3 — a,^ (^3 — a J = a positive number, because the first two factors are positive, and the others negative. And by continuing these substitutions we shotdd obviously thus obtain as many changes of sign in the results as there are real and unequal roots, and no more. Now the quadratic factors which we have omitted always give a positive result for every real value of a? (59); consequently the introduction of these factors would cause no change in the foregoing results. Hence, we may deduce the following inferences, viz. 1 : — If two numbers be successively substituted for x in any^ equation, and give results affected with different signs, then there lie between those numbers, one, three, five, or some odd number of roots : And 2 : — If two numbers, when substituted successively for Xy give results afiected with the same sign, then there lie between those numbers, two, four, six, or some even number of roots, or else none at all. (92.) From the preceding proposition it is plain that the places which the real positive unequal roots of an equation occupy would all be detected provided we knew a number either equal to, or less than, the smallest of the differences of every pair of these roots. For if A be such a number, we could never pass over more than a single root at a time by employing for our successive substitutions the arithmetical progression 0, A, 2a, 3a, 4 a, &c. and thus the place for every positive and unequal root would be made known by a change of sign being furnished by the con- secutive substitutions between which it lies. It is obvious too that instead of commencing our series of substitutions with 0, we may begin with the inferior hmit to the positive roots ; never ex- tending them beyond the superior limit. These considerations led Waring, and afterwards Lagrange, to seek for general methods of determining in all cases a suitable value of A ; these were readily seen to be dependent upon a trans- formed equation of which the roots should be the differences of LIMITS TO REAL ROOTS. 105 the roots of the proposed equation : and hence the celebrity of the equation of the differences of the roots, upon which the method proposed by Lagrange for the analysis of numerical equations is founded; and of which an account will be given hereafter. It is plain that the actual solution of the equation of the differences is not necessary in order to furnish a proper value of A , since it will be sufficient that we know an inferior limit merely to the positive roots of the transformed equation, and that we put this limit for A. Theoretically, this method of separating the real unequal roots of an equation is perfect ; but the practical difficul- ties attendant upon the calculation of the coefficients of the trans- formed equation are insuperable in equations of a high degree. The method has however altogether yielded to those of Sturm, Fourier, and Budan, which when modified and improved, as hereafter proposed, will be found to leave but little to be desired in the practical solution of this important problem. (93.) The proposition next following is one of considerable im- portance in the analysis of equations. Its object is to discover an equation of a lower degree than the equation proposed, whose real roots shall have the remarkable property of lying either singly, or in groups, between every two real roots of the original equation, thus effectively separating all the latter by interposing themselves. Every equation whose roots thus separate those of another, by lying in the several intervals between them, is called a limiting equation to that other. If the roots of the original equation were known, limiting equations to it might be constructed in endless variety, for we should only have to assume values lying in the intervals which separate the given roots, and to construct an equation having these values for roots. But the roots of the original equation are supposed to be unknown; and our present object is to deduce at once a limiting equation whose roots, when determined, shall make known the situations of those of the pri- mitive equation. Such an equation may be derived with great ease from the coefficients merely of the equation proposed, by a uniform operation applicable to all cases. The derived equation being thus always connected with the primitive by a constant 106 LIMITS TO REAL ROOTS. law, is called the limiting equation, in order to distinguish it from all other equations whose roots may also separate those of the original. We shall now show how the limiting equation is to be deduced. PROPOSITION VII. (94.) An equation being given to determine another, a unit lower in degree, such that the real roots of the latter may sepa- rate all the real roots of the former. Let the proposed equation, deprived of its leading coeffi- cient, be y(a?)=;r"H-A„_ia?n-^-F -f-A3a?3-(-A2a?2-j-Aa?-i-N=0 [1], and let its real roots, taken in the order of their magnitude, com- mencing with the greatest and descending to the least, be fl!j, (l2i Ct^f Cl^t Let also the imaginary roots, if any, be represented by a: J, ATg, «^;^, A^, . . . then it is required to determine an equation of the n — \th de- gree, whose real roots shall arrange themselves in the several inter- vals between the real roots of the series above. It is evident that if the roots of the given equation be all dimi- nished by r ; that is, \S. x' -\- r be put for a?, the roots of the trans- formed will be _ ^ «i — r, a^ — r, a^ — r^ ; Atj — /•, Ajg — r, [2], the equation itself being of the same degree as that proposed ; that is, of the form /(^' + r)=:f'"-fA'n-ia/"-^ + A'3;lr'3^-A'2a/2^-AV-^-F=0. Now it is only the last term but one in this equation, as we shall presently see, that need occupy our attention ; and we know from (^), that the coefficient A', of this term, is found from the roots LIMITS TO REAL ROOTS. 107 exhibited in [2] by multiplying every n—l of them together with changed signs, and adding the several results ; that is, by taking the sum of all the partial products which the above factors, with changed signs, furnish, omitting each factor in succession. As only one factor is thus omitted in each partial product, and as every one is omitted in its turn, it follows that wherever a 5m- ffle imaginary factor enters without its conjugate, there must all the real factors occur ; for the omission of this conjugate necessi- tates the entrance into the product of all the other factors. It is of importance to observe this, because it authorizes us in con- cluding that, whenever we suppose any one of the real factors to become zero, thus causing all the partial products to vanish, ex- cept that from which this real factor has been omitted, the pro- duct preserved can contain only real simple factors, and the real quadratic factors formed from the different pairs of imaginary roots, the latter factors being always positive for every value ofr(59.) The last term but one in the above transformed equation being A! a/, the composition of A' will be as follows : A'= (/• - «i) {r - «2) (»• - %) (r- ^i) (»• - ^2) -i- (r - «j) (r - «2) (^ — «4) (^ - ^i) (^-^2) + {r- «i) (r — a.^ (r-a^) (r - k^) (r - k^) . . . . + (r- «2) (^ - %) (r -a^). . . . (r- k^) (r - k^) &c. &c. Now this same coefficient is easily obtained from the coefficients of the original equation by the process described at (50) ; and the result so obtained will be an arranged polynomial descending ac- cording to the powers of r.* The same polynomial may how- ever be obtained still more easily by a process which will be explained presently. But without seeking the arranged polyno- mial in r, to which the above expression for A' is equivalent, we may at once prove that the real roots of that polynomial equated * This method of obtaining A' is exhibited in full in the Analysis atid Solu- tion of Cubic and Biqitadratic EquationSy p. 63. 108 LIMITS TO REAL ROOTS. to zero, that is to say, the real values of r in the equation A' = 0, ■will separate all the real roots of the proposed equation /(^r) = 0; in other words, the equation A' = will be the limiting equation to the equation /(a?) = 0. For if in A' we put a-^ for r, each partial product must vanish, except one, since there is only one partial product from which {r — «j) is absent, and this will become («j — a,^ (ttj ~ ttg) («, — «4) . . . ., positive ; because, as ffj is the greatest root, all the real simple factors will be positive : and the real quadratic factors, as observed above, will be positive also. If we put «2 ^or ?•, all will vanish except the product from which {r — a.^ is absent, so that this will become («2 — «,) («2 — %) («2 — «4) . • . •, negative^ because a^ is the greatest root that enters, except «,, and the quadratic factors are positive. In like manner, putting a^ for r, we shall have («3 — «i) («3 — «2) (% — «4) . . . ., positive, and so on. But when a series of quantities «p «2» ^^ &c., sub- stituted for the unknown in any equation, give results alternately positive and negative, every pair of these quantities must com- prehend an odd number of the real roots of that equation. Con- sequently the real roots oif{x) = are necessarily separated, the interval between every pair being occupied by some odd number, at least one, of the real roots of A' = 0. Hence A' = is the limiting equation to /(a?) = 0. We have now to show how A' may be deduced from f{x) in an arranged form. In order to this we remark, that the trans- formed equation, whose roots are [2], will be obtained by substi- tuting x' -\- r ior X m \\\ and developing the several terms by the binomial theorem. The coefficient A' in this transformation will be that which multiplies the simple power a:'; and instead of being the last coefficient but one, it will become the second coeffi- cient, provided we develop /(r -\- x') instead oif{x' -\- r), because we shall thus reverse the order of the terms. LIMITS TO REAL ROOTS. 109 I i I + t ? ? u 1 1 1 1 1 1 1 I L so • 1 8 a < ^ < ^ y-^ ^N CO ^ «o i f 1 1 1 ^^ 1 1 1 s_i >>^ O C^ .^-«s ,^-v y^-^ •5 1 1 CO CO 1 CO 1 CO '< , 1 CO s C . '-' V f ^ 1 C < 1 c k ^ ^ T (M CO ■^ % ^1 ^ 1 1 1 ho C^ § '— ' ^-N '3» r,,, — ao when substituted for x in succession in the proposed equation, will furnish exactly as many changes of sign as there are real roots in the latter ; because never more than one of these roots can be passed over at a time, and all lie within the extreme limits + 00, -- 00. In those cases, therefore, where the real roots of the limiting equation can be found, the number and situations of the real roots of the primitive equation can always be determined. Thus a cubic equation can always be analysed by this method, since the derived quadratic can always be solved : but for farther details on this subject we must refer the student to the introduc- tory volume on the Analysis, ^c. of Cubic and Biquadratic Equations, pp. 67-70. Theory of Vanishing Fractions. \97.) From the principles established in the foregoing propo- sition, we readily derive the following consequences, viz. : Since f(x) = (a? - «i) (x - a^ {x - a^ {x-a^ and f{x) = {x—a^{x—a^{x—a.^....-{-{x—a^{x—a^{x—a^.,.-\-8ic. it follows that ^ = . . . . 4- + + .... [1]. j{x) oc—a^ x—a^ x—a^ x—a^ In like manner, for any other equation F(a?) = 0, we have F,(^) I 1 1 1 p^ = • • • • — r + — r + — T + — T • • • • [2] . r {x) X — 64 X — 63. 3C—o,^ X — 6j 8 114 VANISHING FRACTIONS. Suppose the two equations f(x)=0, F(^)=0, have a root in common, viz. a^ = b^ then, dividing [I] by [2], we have 1111 + + + /i{x) ¥(x) ' ' x—a^ x—a^ x—a^ x—a^ W' 7^) i i i r + a?— 64 x—o^ ^—62 x—b^ Hence, multiplying numerator and denominator of the second member by a; — «i, and then substituting for x, its value a? = Oj, we have . /iK) ^/K) from which we learn, that if any two equations have a common root a, and their hmiting equations be taken, the ratio of the original polynomials, when a is put for x, will be equal to the ratio of the limiting polynomials when a is put for x. This property furnishes us with a ready method of determining the value of a fraction, such as , when both numerator and )^{x) denominator vanish for a particular value of x, as, for instance, for X ■=■ a. For we shall merely have to replace the polynomials in numerator and denominator by their hmiting polynomials, and then make the substitution of a for x. If, however, the terms of the new fraction should also vanish for this value of a?, we must treat it as we did the original, and so on, till we arrive at a frac- tion of which the terms do not vanish for the proposed value of x. The following examples will sufficiently illustrate this method: 1 . Required the value of , when a; = «. VANISHING FRACTIONS. 115 Here ^—- = — = 2a, the required value. 2. Required the value of (l"=^^)^ ' when X = I. fi{x) n(n -\- l)a?" — n{7i + 1)^»- F,(^) -2(1-^) This still becomes - for a? = 1, . /2O) ^ K^ + 1 ) ••F^O) 2 ' the value sought. 3. Required the value of 1 — a?" when 07 = 1 . n. F,(l) _ 1 4. Required the value of h{a — wax) a — X for x-=- a, • This is the expression for the sum of n terms of the series 1 + 2a: -j- '^x^ + 4x3 + &c. IIG EQUAL ROOTS. We may here put yj x ■=]), and thus change the fraction into h{a — a^ y) a—y- fi(y) - ba^ , /i(«^) = - the vahic required. r,(y) - 2y ■ ■ r,(fl*) 2 5. Required the value of m m when X =-y. (See Algebra^ third edition, page 200.) Put a ■{- y =^ z^, then the fraction is changed into {a + xY — 2^™ X — z^ -\- a . /,w — mz"^~^ m 2-"* m m {a + yY •F,« — W2"-' n ^" n » + y and therefore the vahie, when x = y, is m m (« 4- a?)" n a + X Theory of Equal Roots. (98.) The foregoing proposition also readily leads to a method of freeing an equation from all repetitions of the same root, whenever such occur ; as also of ascertaining whether an equation has equal roots or not. For, as in the limiting equation /j (a^) =0, the polynomial /j (a') consists of the sum of the products arising from multiplying together every w — 1 of the factors of /(xjj each group of factors in /j (x) will differ from f(x) only by the EQUAL ROOTS. 117 absence of a single factor. Hence, if there be two equal factors in /(a?), that is, '\^ fix) = have two equal roots, one of these factors must occur in each of the groups which compose yj(x), so that/(a?) and/j ix) have this factor for a common measure. If there be three equal roots in/(^) =0, then will /(^) and/i(a?) have for a common measure the quadratic factor involving two of them, because more than one of the equal factors cannot be absent from any of the terms of f^(x). And generally if /(a?) = have p roots equal to a, then will (a? — a)P~^ be a common measure of f{x) and/i(cr) ; since in none of the component parts off^(x) can more than one of the p equal factors be absent. Again, if besides the p factors equal to (x — a), there also enter q factors equal to (x — b) in the composition of /(.r), then, besides the former common measure, the polynomials /(a?), /j(ir), will also have the common measure {x — b)^~^, for reasons similar to those which have already been assigned. And gene- rally, if the equation f(x) = have p roots equal to a, q roots equal to b, r roots equal to c, &c. then the greatest common measure of the polynomials /(a?), /^ (a?), wiU be {x - ay-^ (x — b)'i-'^ {x — cy-^ .... In order, therefore, to discover whether or not an equation f(^x) = has equal roots, we have only to ascertain whether or not f{x) and /j (x) have a common measure (a?) = cannot be all different ; hence ^ (a?) = will also contain equal roots. Let us therefore ascertain these. The limiting polynomial derived from ^ (a?) is 0,(a?) = 4a?3 + 9a?2 -f 2a? - 3, the common measure of

'(x), we have In like manner, calling the greatest common measure of \{x), o) . . . . + + «1) . . . . H- -H (>1) . . . . -f- + + . . . + + -H + + Hence, applying the rule of the double sign, we have (0){ (10) The first of these series gives four variations, and the second two, this loss of two variations indicates the existence of one pair of imaginary roots. The third and fourth series exhibit the same number of varia- tions ; hence the zero, produced by the transforming factor (1), does not arise from imaginary roots. Let us now examine the series (0) and (1), for which purpose we must compare the signs of (> 0) and (< 1), and we thus find that no root is comprised in the interval [0, 1], because there is no loss of variation. For the interval [1, 10], we must examine the series (> 1) and (10), which we find to indicate the existence of two roots, because two variations are lost, but whether they are real or not cannot as yet be ascertained ; this, however, is the only doubtful interval. 3. Let the proposed equation be x^ -^a!^-{- x^ — 25x — 36 = 0. The transforming factors — 10, —1, 0, 1, 10, give the following series of results : — iv; . . • • -| T — I (-1) . . . . + - + (0) . . . . + + + (1) . . . . + + + + — (10) . . . . + + + + + + 10 (0){ 146 BUDAN AND FOURIER ON Applying the rule of the double sign, we have (-1) — +- + « 0) . . . . + + - + -- (>0) ....+ + + + (1) — + + + + -- Comparing now these results, we find : — That all the real roots exist in the interval between — 10 and + 10. That two of these roots may lie between — 10, and — 1, because, in passing over this interval, two variations have dis- appeared ; the interval may, however, contain indications of two imaginary roots. That a pair of imaginary roots are indicated by (0), because the signs of « 0) and (> 0) differ by two variations. That no root exists between — 1 and 0, because the series ( — 1) and « 0) have the same number of variations. That no root exists between and 1, because the series (> 0) and (1) have the same number of variations. That one real root exists between 1 and 10, because one varia- tion has disappeared. The only doubtful interval here is that between — 10 and — 1 . "We shall give but one more example of the determination of the intervals of the roots. 4. Let the proposed equation be ^7_2a^_3a^ + 4^2_5^ + 6=0. The transforming factors — 10, — 1, 0, 1, 10, give the following results : (-10) (-1) (0) (1) (10) + - - + - + + + + + + — + ++++++++ THE SEPARATION OF ROOTS. 147 And, applying the rule of the double sign, we have (— 1) ....+- + - + + - + «0).... +-- + - + - + (>0)....+ + + - + (1) ....+ + + + + + (0){ We deduce, therefore, the following particulars : There is one root between the limits — 10 and — 1, and only one. The series (0) shows the existence of two imaginary roots in the equation, because the series « 0) and (> 0) differ by two variations. There is no real root between — 1 and 0. There may be two real roots between and 1, as two variations disappear between (> 0) and (1); but if there are not two real roots in this doubtful interval, there exists within it an indication of two imaginary roots. There may also be two more real roots between 1 and 10. The only intervals, therefore, in which we ought to seek for roots are those between — 10 and — 1, between and 1, and between 1 and 10 ; and we know also that the equation has two imaginary roots at least. (114.) It may not be improper to remark here, that when the equation proposed for examination has any of its terms wanting, as in the last three examples, we may always, by applying to it the rule of the double sign at once, determine the least number of imaginary roots that the equation can possibly have. Thus, in the last example, the signs of the proposed are +0-0- +- + instead of which, the rule of the double sign gives the two series, + + - + - + + + +~ + 143 SUDAN AND FOURIER ON which, because they differ by two variations, establish the ex- istence of at least two imaginary roots in the equation. (115.) In equations of the form this method makes known the exact number of imaginary roots. For example, suppose the equation is a;6 _ 1 = which gives the series + 00000- and, by the rule of the double sign, H- - +- + in which the upper series has five variations, and the lower but one. Hence, there are four imaginary roots in the equation, which is obviously the entire number ; the two real roots being + 1,-1. From a mere inspection of this upper and lower series, it is obvious that, in all cases, when m + 1 zeros intervene in an equation between unlike signs, there must exist at least tn ima- ginary roots ; and when m -\- I zeros intervene between like signs, there must exist at least m + 1 imaginary roots. These are the conclusions that have been otherwise deduced at (68). (116.) From what has already been shown at p. 110, it is evi- dent that the coefficients of the transformed equation /(a? + r) =0, to which we are conducted by diminishing the roots of a given equation /(a?) = 0, as in the preceding examples, by any number r, are no other than the successive functions when written in reverse order and r substituted for x. For the proposed equation being y(ar)=A„a?" + A„_i;r"-»+ . . . +A.^-{-A^-\-Ax-^'^=0.,. [2] THE SEPARATION OF ROOTS. 149 we have for the transformed function f{x + r), when the terms of the development are written in reverse order, the expression of which the coefficients are what those in [1] become when r, the factor of transformation, is substituted for x. For example, taking the first member of the equation proposed at page 142, and the several functions derived from it, we have f{x) = c^ — 3a?4— 24^+ 95a;2 — 46a? — 101 f^{x) = 5a^ - \2x^ — 72a?2 + 190^ — 46 ■-/^{x) = lOa^ — 18^2 _ >j2x + 95 ^/.W =10*2 -12* -24 2^/4W =5.-3 Putting now 1 for x and writing the results in reverse order, we have 1+2 — 26 + 15 + 65 — 78 and putting 10 for x we have all the vemHi^ positive. Also putting for x we have 1 _ 3 _ 24 + 95 — 46 — 101 the coefficients of the proposed polynomial. And thus, whatever general relations are shown to exist among the derived functions [1], a? being any value whatever, the same must exist among the coefficients of the original equation : for the functions [1] return 150 BUDAN AND FOURIER ON to these coefficients when a? = : in other words the original equation is •'^ ' 1-2-3 . . « ^ 1-2-3 . . . (»-l) ^ ^ 1-2-3 (117.) The examples by which we have illustrated the theorem at (107), or rather that at page 136, are from the Analyse des Equations of Fourier ; but the processes by which the several transformations involved in these examples are here effected are very different from those employed in that work. Fourier actually exhibits the several derived functions in every case, as above ; and as the signs of these only, for particular values of x, are required, he disregards the numerical divisors 2; 2-3; 2-3'4; &c. and thus encumbers the several expressions from which the series of signs are to be deduced, with coefficients unnecessarily large. The method of transformation uniformly employed in the present work is that exhibited at length in the analysis of the first example at page 143: it reduces the operation to the utmost simpHcity. In the preceding exposition of this method of partially ana- lysing a numerical equation we have united the names of Budan and Fourier, each of whom announced, independently it would seem of each other, the theorem at page 136, on which the method is founded. It is common with English writers to ascribe this theorem exclusively to Fourier — a singular pre- ference ; since the pubHcation of it by Budan preceded the work of Fourier by nearly a quarter of a century. In the advertise- ment prefixed to this work, the editor, Navier, adduces evidence in favour of Fourier's prior claim to the theorem. This evidence however consists of individual attestations to the fact that Fourier had developed his theory in manuscript so early as 1797 — ten years before the publication of Budan ; and that he had publicly expounded it in his lectures at the Polytechnic School in 1803. But testimony of this kind must always be deficient in that distinctness; as to the precise character and THE SEPARATION OF ROOTS. 151 extent of the communications made, which so eminently belongs to the printed publication of them. There is however no room to doubt that Fourier was reaUy engaged in researches upon numerical equations long before the appearance of Budan's work ; and that he had advanced in the enquiry beyond his pre- decessors. There is very conclusive evidence of this in a printed statement which seems to have escaped the notice of Navier, and the other advocates of Fourier's claims. We allude to a passage in Montucla's History of Mathematics^ which we quote below.* It is probable that Fourier was withheld so long from the publication of his researches — which after all were not printed till after his death — on account of the inefficiency of his theorem to make known the exact character of those doubtful intervals which, as we have already seen, frequently occur within the extreme limits of the real roots of an equation. Attempts were made both by Fourier and Budan to remove this defect, by help of certain supplementary operations apphed expressly to the intervals in question. In this further analysis of the equation the two methods are perfectly distinct. We shall discuss them separately in the next chapter. • In the passage referred to, Montucla, or rather Lalande, adverting to the previous inquiries of De Gua, and the general demonstration given by him of the rule of Descartes, proceeds as follows : *' Je ne puis passer sous silence un memoire sur la resolution des equations par le cit. Fourieb, ancien pro- fesseur de mathematiques au College de Tonnerre, qui s'est aussi specialement occupe de cette demonstration ; il en donne deux, I'une geometrique et fondee sur la consideration des courbes ci-dessus, I'autre purement analytique, et fondee sur des principes ditterens de ceux de 1' abbe De Gua. Ses recherches le conduisent a beaucoup d'autres verites utiles, qu'il est juste qu'il publie lui-merae le premier.'' — (Montucla: Hist, des Mathematiques, iom. iii. p. 39, 1802.) Fourier died just as his work on Equations was put to press : his mss. were consigned to the care of Navier, who published the first part in 1831. But the death of Navier himself, shortly afterwards, put a stop to the progress of the publication. "^-rri CHAPTER VIII. ON THE ANALYSIS OF EQUATIONS FROM GEOMETRICAL CONSIDERATIONS : METHOD OF FOURIER. (118.) From the investigations in the preceding chapter, it ap- pears that it is no difficult matter, when any numerical equation is proposed, to determine close inferior and superior limits, within the interval of which shall lie concealed not only all the real roots, but hkewise all the indications of imaginary roots. In fact, we can never be sure that all the real roots are actually comprehended within any proposed boundaries, till we have ascertained that the indications of the imaginary roots all lie be- tween the same limits ; so that when the extreme Hmits of the real roots are clearly determined, the complete analysis of the equation consists merely in a sufficiently minute subdivision of the interval between them. In the foregoing chapter such a searching scrutiny has not been attempted; and accordingly the character of some of the component intervals, which our partial analysis has furnished, often remained doubtful. In some cases it would be impossible to completely remove this doubt by simply narrowing the com- ponent intervals, or increasing the number of subdivisions ; that is, by making our factors of transformation less and less. Ex- traneous information would still be requisite before we could pronounce with confidence upon the character of an interval, however minute, in passing over which two changes of signs were lost : we should experience the same uncertainty as before, as to ANALYSIS OF EQUATIONS. 153 whether these two changes indicated a pair of imaginary roots, or a pair of real roots differing from each other so minutely as to he both within the small interval referred to. And we could resolve the doubt only by knowing, from some independent source, either the least of the differences furnished by every pair of real roots, or else a number less than the least difference, as already explained at (92). The determination of such a number, though theoretically possible, is an operation so laborious, in equations beyond the fourth degree, as to be practically useless in the analysis of equations. Indeed, roots sometimes differ by numbers so exceedingly small, that, even supposing a limit below this difference to be found, yet the labour of passing over an in- terval by such minute advances would be a very long and tedious process. It would be well if we could exhibit to the eye the continuous series of results which the first member of any equa- tion would furnish by substituting continuous values for x from the inferior up to the superior hmit of the roots : we should then perceive at a glance all the passages of the polynomial through zero, and thus become acquainted with the exact number of the real roots. Of course the practical difiiculties in the way of this we insuperable ; yet the idea obviously suggests the geometrical representation of an algebraic polynomial by means of a con- tinous curve hne, which shall unite all the isolated values of that polynomial resulting from individual substitutions. A contemplation of this curve would not only verify all the analytical results known to be imphed in the equation, but, from purely geometrical considerations, new truths might discover themselves which had escaped observation in the abstract alge- braical form. It was from examining in this manner the geome- trical representation of an algebraic polynomial, that Fourier was led to the method about to be explained for determining the character of the doubtful intervals occurring between the ex- treme Hmits of the roots of an equation, without having recourse to the problem for finding a number less than the least of the differences of the roots. The idea of converting an algebraic polynomial into a conti- nuous curve, embodying all the peculiarities of the symbolical ex- pression in a geometrical form, first suggested itself to Descartes, 154 ANALYSIS OF EQUATIONS and is a contrivance which has often been resorted to for clear illustration of certain particulars in the general theory of equa- tions, for which illustration we shall shortly see it to be well adapted : but the geometrical property noticed by Fourier enabled him to advance a step nearer than his predecessors towards the complete analysis of a numerical equation. (119.) If the student be already famihar with the principles of analytical geometry, he will readily perceive how this connexion between an algebraical equation and a geometrical curve subsists ; and how the properties of the one become convertible into those of the other : but for those who may be unacquainted with this important branch of analysis, it will be necessary to offer a few preliminary explanations. As usual let f{x) = represent any algebraical equation in x : this is a determinate equation^ because the unknown quantity x admits only of a determinate number of values to the exclusion of all other values. But if we remove the restriction which con- fines the first member to the single value zero, and imply that this value is arbitrary, by writing the equation in the more gene- ral form f{x) = y, we then render the equation indeterminate, since x admits of any value whatever ; to each of which, how- ever, there corresponds a certain determinate value of y, as im- plied in the sign of equality. We have then to exhibit the gene- ral law, which thus connects every value of x with the correspond- ing value of y, by means of a geometrical figure. From any assumed point A draw an indefinite straight line AX towards the right, and extend it indefinitely in the opposite direction AX'. In like manner draw from A a perpendicular to X'X of indefinite length, AY, and prolong it indefinitely in the opposite direction AY'. These two lines are called the axes, and the point A, where they intersect, is called the origin of the axes. Now if we assume any series of positive numerical values for x, and measure each of these values from A towards X, according to any unit of length chosen at pleasure, we must measure a like series of negative values of x in the opposite direction from A, that is from A towards X' ; for then the lengths, thus set off in each direction, will not only be correct representations of the ab- FROM GEOMETRICAL CONSIDERATIONS. 155 solute numerical values for which they stand, in reference to the Hnear unit previously agreed upon, but they will also imply, in the directions in which they are measured, the algebraic signs which those values take. In like manner, if positive numerical values of y are measured along the other axis, from A in the direction AY, then negative values must be measured in the oppo- site direction AY'. Y ^r\, i This being premised, let AB represent any value of x ; then the corresponding value of y will also have some Hnear repre- sentation, which may be set off upon the other axis, above X'X if y be positive, and below X'X if negative ; or, which is better, upon a parallel to this axis drawn through B, the termination of the linear value of x : let BP be this length. Of the point P, thus determined, AB is called the abscissa, and BP the ordinate: together they are called the coordinates of the point P. Another assumed value of x will furnish another abscissa AC, and for the corresponding y another ordinate CQ, represented in the diagram as negative, being drawn below the axis X^X. These new coor- dinates introduce a second point Q. And thus if it were possible to construct the continuous series of values for x and y, setting out with a? = 0, and proceeding towards a? = + oo on the right, and towards a? = — oo on the left, we should be furnished with a continuous series of points ; that is, with an uninterrupted curve line. We may therefore consider this curve as traced out by the extremity P, of an ordinate BP = y, moving parallel to itself, along the axis X'X, and varying in length as its distance x from A varies, the law of variation being expressed by the re- lation y ^=f(x). Conceiving the curve to be actually generated in this way, it is easy to see how, from any value of x being given, the corresponding value of y may be found, and vice versa. 156 ANALYSIS OF EQUATIONS Thus, if AC represent the given value of a?, then the perpen- dicular CQ, extended till it meets the curve, will represent the corresponding value of y, vrhich we shall know to be positive or negative according as Q is above or below XX'; — in the above diagram it is negative. In like manner, if the value of y be given, then setting off that value upon the axis YY', attending to the algebraic sign of it, in order to ascertain in which direction from A it is to be measured, and drawing from the extremity of this ordinate a parallel to X'X, the intercepted portions of this parallel, between YY' and the curve, will be so many abscissas, or values of x, corresponding to the single ordinate, or value of y, proposed. If, for instance, the given ordinate be AK, then TKW being parallel to X'X, the corresponding abscissas, or values of X, wiU be KL, KM, KR, KV, KW, positive, and KS, KT, negative: this is plain, because the points L, M, R, V, W, S, T, have these several abscissas, and one uniform length of ordinate, viz. the length AK. (120.) As observed above, it is not possible actually to con- struct this curve ; the utmost we could do would be to approxi- mate to its form by means of a series of isolated points determined from a series of successive values of x ; but, by making the inter- vals between these values very small, we could evidently form a tolerably accurate notion of the general character of the curve within any proposed limits taken for the values of x. Indeed, without any approximate construction at all, such a general no- tion may be formed from the nature of the polynomial whence it has been derived. Thus we may be quite sure that the undulat- ing curve above exhibits the general character of such a polyno- mial ; for the composition of the polynomial is such that to any value of X there corresponds but one value of y, as in the figure ; while for particular values of y there may exist several values of Xy as many indeed as there are units in the highest exponent of x. The curve, therefore, should be such that as many values as there are for x, corresponding to a given value AK for y, so many inter- sections must there be of the curve with the parallel to X'X through K. If y be zero, then the corresponding values of x must represent the real roots of the equation f{x) = : they will. FROM GEOMETRICAL CONSIDERATIONS. 157 therefore, be equal in number to the intersections of the undulat- ing Hne with the axis X'X : those intersections to the right of A will indicate the number of positive roots ; and those to the left the number of negative roots. In the diagram above three posi- tive roots are indicated, and two negative roots. As, in order to produce an intersection with the axis of abscissas, the curve must pass from one side of that axis to the other, it follows that the ordinates, immediately before and immediately after the intersec- tion, must have opposite signs ; that is, the polynomial f{x) changes sign while x passes through a single real root. Following the progress of the curve, after this intersection, it is plain that no second change can take place in the sign of the ordinate tiU the curve again crosses the axis ; that is, till another root is passed, when the ordinate emerges on the other side of the axis with a changed sign. Its length thence increases up to a certain limit, at which the curve again bends towards the axis, crosses it a third time, and gives rise to a new series of ordinates with signs opposite to those which vanished at the former point of intersection. Thus we see that two values of a?, which give for f{x) or y results with opposite signs, must intercept either 1, or 3, or 5, &c. real roots ; and two values which give results with like signs must intercept either 0, or 2, or 4, &c. real roots. When, in the polynomial f{x)y a number so great is substituted for x that the transformed equation, arising from diminishing the roots of the equation f{x) = by this number, has all its terms positive, then we know (102) that the number in question exceeds the greatest positive root of the equation ; and, moreover, that if a series of numbers, continually increasing beyond this, be succes- sively substituted, that the results f{x) or y, will also continually increase (103). In a similar manner will the results continually increase for substitutions for x continually tending towards — oo, after a certain limit is reached, viz., that which furnishes a trans- formed equation with its terms alternately positive and negative. Hence, the curve, after having furnished as many intersections with the axis of abscissas as there are real roots, continues its course interminably on each side of the axis of ordinates ; and. 158 ANALYSIS OF EdUATIONS after a certain limit, all undulations must cease, and the ordinates become continually longer and longer without termination. Whether the degree of the equation be odd or even, that part of the curve which is to the right of A will proceed on its un- limited course above the axis of abscissas ; since the ordinates, after the hmit referred to, must always be positive. But to the left of A the curve will extend above or below the axis of abscissas according as the degree of the equation is even or odd : this will appear from considering that when the terms are alternately positive and negative in an equation of an even degree, the final term, — which is that furnished by the polynomial /(a?) when the number by which the roots are diminished is substituted for x, — will be positive ; and in an equation of an odd degree the same term will be negative. This shows that for an equation of an odd degree there must be at least one intersection of the curve with the axis of abscissas : as the curve proceeds without limit on both sides of that axis, which it cannot do without crossing the axis, should it cross a second time, it must cross a third, otherwise it could not proceed on opposite sides of X'X : for a Uke reason if it cross a fourth time, it must also cross a fifth time, and so on : the number of intersections being necessarily odd ; that is, an equation of an odd degree must have an odd number of real roots. An equation of an even degree has not necessarily any real roots ; as the curve need not of necessity cross the axis, because it pro- ceeds without limit on one and the same side : but if there be one intersection, there must on this account necessarily be another ; and if a third, then a fourth, and so on ; so that when the degree is even the equation must have an even number of real roots, or else none at all. (121.) If in the equation /(a?) = y a succession of values be given to y, from 3/ = to y = «, we shall have a succession of equations from /(a?) = to/(a?) = «, or /(a?) — « = 0, difiering from one another only in the final or absolute term. If AK repre- sent one of these values of y, the intersections L, M, R, V, W, S, T, will show the number of real roots in the corresponding equation ; and by conceiving X'X to move parallel to itself, till it reach the FROM GEOMETRICAL CONSIDERATIONS. 159 distance a from its original position, the successive series of in- tersections will exhibit the number and character of the real roots of the several successive equations. It is easy to see that, by means of these changes, certain pairs of unequal roots will be separated more widely, while others will be brought nearer to equality : for instance, the two roots indicated originally by the intersections G, H, and which are separated by the interval GH, are changed into the roots indicated by the intersections S, L, which are wider apart, when the axis arrives at TW; whilst the two roots, differing originally by HI, now differ only by LM ; and these are actually brought together, and rendered equal, when the axis has advanced so far as to touch at V, rendering the separating interval LM zero. This then is the geometrical pecu- harity of a pair of equal roots : — instead of intersection, as in the case of a single root, there is contact with the axis. When the axis, by moving parallel to itself, advances still farther, the two roots that have been rendered equal no longer exist : as soon as the axis ceases to touch, a pair of intersections is lost ; and thus a pair of roots becomes unaccounted for by the intersections that remain. This, therefore, is the geometrical peculiarity of a pair of imaginary roots : — the curve approaches towards the axis, bends before arriving at it, and completes an undulation without meeting it. There is such a peculiarity at^; X'X being the position of the axis. It is worthy of notice that, by changing the value of the abso- lute term of an equation, without disturbing the other coefficients, we may always convert, as above, a pair of consecutive unequal roots into a pair of equal roots : but that that change will not generally suffice to render three unequal roots equal. In the curve above, no three intersections can be made to coalesce, and merge into a single point, by any change in the distance merely of the parallel TW from its original position. In order that three points of intersection may merge into one there must be a change in the coefficients of /(a?), such that the geometrical equivalent is not a mere traiisference of the axis parallel to itself, but a change of direction in that axis. Thus the three points P', P, P" merge into the single point P when X'X, by turning round P, takes the position of a tangent to 160 ANALYSIS OF EQUATIONS each of the portions ¥mP', FnP", of the curve, at the point of inflexion P. (122.) By thus giving a geometrical interpretation to the expressions of analysis, much clearness and distinctness may often be added to the ideas conveyed by our symbohcal forms, and many interesting analytical truths at the same time suggested. One great advantage of presenting algebraical expressions under this form is, that instead of our attention being confined to iso- lated individual values merely, we are enabled to contemplate the law of continuity that unites them all. It is, indeed, solely from this law being presented to our view, in the continuous curve which replaces the analytical formula, that the geometrical repre- sentation can supply anything in addition to our analytical de- ductions. Of course the algebraical form is competent to furnish all the inferences deduced from the curve which represents it ; but it often happens that what is so entirely concealed among the algebraic symbols, as to be evolved only by analytical artifice, may spontaneously offer itself to notice in the geometrical repre- sentation. From what is shown above, much light is thrown upon the connexion between real and imaginary roots ; and upon the fact of the necessary occurrence of the latter when the principle of continuity is carried fully out (35). The connexion here spoken of is not that between the roots of an individual equation ; but that between the successive series of roots of a continuous series of equations. Instead of considering an isolated equation, the geometrical form enables us to trace the connecting circum- stances of the entire series to which that one belongs ; and thus to ascertain how its imaginary roots arise, and what real values have given place to them, or have merged into them. Considering in this way any proposed equation as one of a series of others, in which the right-hand members pass continuously over a series FROM GEOMETRICAL CONSIDERATIONS. 161 of values from — a to + b, and, therefore, through zero, we clearly see how, by the operation of a uniform law, two unequal roots pass into equal ones, and thence into an imaginary pair ; and also how a minute change in the absolute term of an equation having a pair of equal roots, will convert those roots either into two unequal roots, lying very closely to one another, or into a pair of imaginary conjugates. For such a change corresponds to a slight movement of the axis parallel to itself. If contiguous ordinates, or values of f(x), one on each side of the ordinate corresponding to the equal roots, be found to have increased by this displacement of the axis, it will show that the axis must have receded from the curve : the two roots will then have become imaginary. But if the same ordinates have diminished, we may then infer that the roots have continued real, and have become unequal. We shall find these circumstances of consequence in the analysis of equations. It is of importance to observe, in reference to what has just been said respecting the geometrical indications of imaginary roots, that all the imaginary roots of any equation f(x) = are not necessarily thus indicated in the curve line which completely represents the general equation /(a?) = y. Only those are so dis- tinguished of which each pair unites continuously, as above de- scribed, with a pair of real values of the equation /(x) = a, which real values approach towards equality as a approaches towards zero ; or, referring to the geometrical representation, as TW ap- proaches towards X'X. After this equality is reached the values pass from a real into an imaginary form, which passage is indi- cated in the diagram by the undulation — which first gave a pair of intersections, and then by the union of these a point of contact p — -becoming altogether detached from the axis. It is essential therefore to the existence of this undulation, that the imaginary roots indicated by it be those into which two equal roots of f{x) — a = have merged by an alteration in the value of a. These equal roots are represented in the diagram by the line Aq, or rather by a line equal and parallel to Aq, touching at p. We know, from the theory of equal roots, that the re- peated root enters also once into the derived equation /j (x) = 0, which is the derived equation equally of /(x) — « = 0, and of 11 162 ANALYSIS OF EQUATIONS : f{x) = 0, since a change in the absolute number of the primitive, causes no change in the derived equation. Aq therefore repre- sents a root of /^(x) = 0. It thus appears that in passing over the interval nr, compre- hending the indication of a pair of imaginary roots, that is, in substituting continuous values for a?, from x = An, to a? := Ar, in f{x)y we necessarily pass over a root o{/^(x) = ; and the value of this (A^') is such, as to render /(a?), or pq, a minimum ; that is, less than the immediately preceding and succeeding values oif{x). The two changes of sign, lost in the interval nr, thus arise from the passage of f^{x) through zero. In the signs of the derived functions, from the last up to f^i^x) inclusive, only one change can be lost by this passage ; and as two are lost when the next following function fix) is included, it follows that the value which makes /j(^) zero, causes /(a?) and/gW to take like signs. Now not only has the equation f{x) = a pair of imaginary roots when these circumstances have place in the last three func- tions /(a?), /j (a?), f^i^^i hut also when similar circumstances have place in any three consecutive functions ; for wherever an inter- mediate function vanishes for a value which renders the signs of the functions contiguous to it, on each side, like, a pair of imagi- nary roots in the primitive equation will be implied (page 129). These latter imaginary roots are not indicated then in the curve referred to: they have their indications in other curves, those which arise from constructing every equation /^^ (a?) = y, of which the first member /^^ (a?) takes the same sign as/jj^^.2(^) for a value which causes the intermediate function /j^^j (a:-) to vanish.* This • These latter indications directly refer only to the imaginary roots of that derived function which immediately precedes the one that vanishes in the order of derivation, and the existence of imaginary roots also in the primitive equation is hut an inference from this. These latter roots are thus merely indicated, and nothing respecting them beyond the simple indication of their existence is furnished to us. It is not so with respect to the other class of imaginary roots, whose presence is immediately made known, as above explained, by the first and second derived functions. Each pair of such roots is not only indi- cated, but to a certain extent the real parts are actually represented or ex- pressed. In the diagram at page 155. Ay actually represents a portion of the root indicated, which portion becomes more and more important as pq dimi- nishes. The numerical value of hq is that which, when put for a in f{x), METHOD OF FOURIER. 163 distinction of the imaginary roots into classes, snggested by the different curves in which their indications occur, is of consi- derable importance in reference to the researches of Fourier; who, as we shall now see, is careful to observe the principles im~ plied in it in his analysis of those doubtful intervals which some- times occur within the limits of the real roots of an equation. (123.) An interval, anywhere within the extreme limits of the roots of an equation f{x) = 0, is doubtful, when the values of x which comprehend it produce no change of sign m.f{x), although an even number of changes are lost in the entire series of derived functions, in the passage of x from the smaller of those values to the greater. The indications lying in such an interval may imply real roots, either equal or unequal ; or they may belong only to imaginary pairs : our object at present is to discover criteria by which the true character of the interval may be ascertained. Suppose first that two roots only are indicated in the interval in question — the interval \_a, 6] ; and let the geometrical represen- tation of that interval be either that in figure 1 below, or that in more nearly satisfies the condition f{x) =0 than any neighbouring value : the defect of the result from zero is represented by pq : it can therefore be diminished, and finally annihilated, only by the p reced ing value taking an imaginary increment^ or one of the form a -j- /3 v — 1 , which, however, will become the more unimportant as the defect itself to be removed becomes smaller. Thus Aq, or the root of/,(i) = 0, will be an approximation to the real part of the imaginary pair. When actually put for x in /(x), the result will be nearer to zero than that given by any adjacent value : and if the defect from zero be so small as to warrant its disregard, in the inquiry in hand, the complementary imaginary part may unquestionably be rejected, and the real value taken for the root, or rather for one of two equal roots. It is easy to see how all trace of the existence of the other class of imagi- nary roots becomes lost in the curve. We have considered our proposed equation as one of a continuous series of equations differing from one another only in the final term ; and have taken note only of the intersections lost in passing over this series. But our equation may unite with an endless variety of varying equations, changing according to different laws. The curve at page 155 may have passed into that form through various preceding forms — forms which presented intersections that have gradually coalesced, and then disappeared. The manner of this disappearance, and of the passage from real values to imaginary, cannot of course be exhibited to the eye, although readily conceivable. 164 ANALYSIS OF EQUATIONS: figure 2. The former, from what is shown above, will indicate a pair of real roots, because the curve cuts the axis; the latter, where no intersection takes place, will correspond to a pair of imaginary roots. And, from knowing the interval ab, and the ordinates am, bn, at its extremities, we have to determine which of these two representations belongs to the case under examina- tion. „. ^ Fig. 1. P^S' 2. The following considerations suggested themselves to Fourier for this purpose : If the second figure represent the true construc- tion of the equation /(a?) = y in the doubtful interval, the follow- ing circumstances must have place, viz., if tangents be drawn from m and w, meeting the axis in a' and b', and then again tan- gents from m' and n', where the ordinates a'm! and b'n', meet the curve, or where ordinates still closer together would meet it ; and so on ; it is plain that we shall at length arrive at a pair of tan- gents that must cross one another before they reach the axis. Let these be the tangents from m', and w', which, after crossing, meet the axis m v, w; then the sum of the two portions a'v, b'w, must necessarily exceed the interval a'b'. Each of these portions is, in geometrical language, called a subtangent, and is defined as the part of the axis between the ordinate and tangent ; so that when the curve is that of figure 2, it is always possible, by bringing the ordinates am, bn, which bound the doubtful interval, closer to- gether, that is, by narrowing the interval [a, b'], to arrive at a pair of subtangents whose sum shall exceed the interval thus contracted. Now this can never be brought about in the first figure : nor can even the sum of the subtangents be rendered equal to the interval, as is evident; and in attempting to effect it by narrowing the interval [a, i], we should be led within the limits of the real roots, and thus to a value of y or f{x) opposite in sign to the values f{a) and f{b) ; so that the separation of the roots would METHOD OF FOURIER. 165 be accomplished.* We are thus informed of the desired criterion for testing the character of the proposed interval; and it only remains to convert the geometrical operations, involved in its application, into the processes of analysis. The well-known theory of curves at once suggests these. The subtangent corresponding to any abscissa a, is analytically expressed by ^ ;f so that, disregarding the algebraic signs, and actually adding the subtangents at the limits «, 5, the cri- terion of a pair of imaginary roots is And in seeking to fulfil this condition, by making the interval [«, 6] narrower and narrower, we shall either actually succeed in doing so, or be led to a value of y =/(«'), or y =/(&'), of opposite sign to that oif{a) and f{Tj) ; and thus to a separation of the roots. (124.) It must be observed, however, that it is all along pre- sumed in the foregoing reasoning, that the curve has no sinuo- sities or points of inflexion, as in the figure at page 155, through- out the interval between m and n : — the existence of such a point would be fatal to the preceding conclusions. This restriction requires that/gW preserves its sign unchanged throughout the interval [«, 5]; for the analytical indication of * It should be observed in this latter case, that if the new ardinates, from whose extremities the new pair of tangents is drawn, always spring from the extremities of the last pair of tangents, the limits will contract at a continually diminishing rate; and we shall never be able to bring them within the points of intersection, and thus separate the roots: every new ordinate, therefore, should be distinct from that last taken by an interval which exceeds the length of the last subtangent ; that is to say, in contracting the interval by assuming an intermediate value of r, this new value of oc should differ from each of the former values by a quantity greater than either of the subtangents which those former values furnish. It is of importance to remember this. t The proof of this, as well as of one or two other particulars in the next article, involves the elementary principles of the general theory of curve lines ; lor which the student may consult the second section of the author's Differen- tial Calcultis, chapters i and ii. 166 ANALYSIS OF EQUATIONS: a point of inflexion is f,J^x)^=Q. Moreover, the form assumed for the geometrical representation of the equation /(a?) =y, within the limits a? = «, a? = 5, is such as to imply the existence of a point P, at which the tangent is parallel to the axis : this implies the existence of a single value a' for a?, between a and 6, that will satisfy the analytical condition f^{a') = 0; so that while x passes over the interval [«, i>],f\{x) changes its sign. The conditions therefore implied in the preceding constructions, and in the analytical inferences drawn from them, are that the two changes of signs lost in passing over the interval [a, b\ are lost entirely in the passage of the last three of the series of func- tions /nW. /n-lW^ • • • • /3W> /2W. /lW» /W» that is, in the three functions f^{x), /i(a?), f{x), which give either the results x=a +— + — +— or, X-=:b + + 4. ___ the preceding terms of the series losing no changes within these limits. Hence in the application of the criterion [a] we must proceed as follows: (125.) Having substituted the two limits a and b in the series of functions above, and having compared the signs of the results, if we find that the second series of results has two changes of sign fewer than the first ; but that omitting the last two signs of each series the second has just as many changes of sign as the first ; then, in order to ascertain whether the two roots indicated are real or not, find the values of \,) . , and -rrJr'y and, disregarding their algebraic signs, see whether the sum of these fractions sur- passes, or is at least equal to h — a: if such be the case, we may be assured that the two roots indicated are imaginary. If the preceding condition have not place, the sum of the ft*ac- tions being less that b — a, we must narrow the interval [ff, 6] METHOD OF FOURIER. 167 by taking some intermediate number c; but, to avoid the endless subdivisions of the interval which would attend the attempt to separate in this way two roots that might eventually prove to be equal, we ought to examine whether /{x) and fi((v) have a com- mon measure ^(a?) ; and if so whether the equation (^7) = has a real root c comprised between a and b. If it have, the equation f(x) = has two real roots in the interval, each equal to c ; and thus the character of the interval becomes determined. But if the functions f(x), f\{x) have no common divisor ^ (x), or, having one, if the equation ^lust arise from the first and third of these taking the same sign for that value of x which renders the middle one zero ; and, therefore (109) this loss is permanent throughout the entire series ; that is, two roots of /(a?) = are wanting in the interval [a', b'] . Hence, in each of the indices cor- responding to the functions from /,n(a.') onwards to the last /(x) the 2, significant of the changes thus lost, necessarily enters. If this 2 be suppressed, account will then be taken, by the indices thus reduced, only of the roots the character of which still remains METHOD OF FOURIER. 173 to be examined; and the index corresponding to /^_j(a?) will become zero. Hence the index 1, which lies nearest to that fur- nished by /(a?), wiU occur in advance of f^_^(x)y or else zeros only will make up the remainder of the series. It follows, therefore, that in all cases, whether the two roots of /m-i W ^^ ^ ^^ ^^^^ ^^ imaginary in the interval \_a', 5'], the pre- ceding operations will give rise to new series of indices in which the index 1, nearest to the end, is still further advanced ; so that we must at length obtain series of which the last term in each is itself 1, or else zero. The following examples will suffice to illustrate the preceding precepts. (129.) The equation a?^— 3^4 _ 24a?3 + 95a?2 — 46^—101=0 partially analysed at (113) has indications between 1 and 10 of three roots, one of which is real and the other two doubtful. Let it be required to determine the true character of the latter. /5W /4W /3W /2W /iW /w (1) . . . + 120 + 48 — 156 +30 +65 - 78 0012 2 3 (10) ... + 120 + 1128 + 5136 + 15150 + 32654 + 54939 In proceeding along the series of indices from right to left w^e find the first 1 to correspond to the function /^(x): this index is followed by 2 and preceded by ; and, therefore, the nature of the two roots oif^(a;)= 0, thus indicated, may be ascertained by the rule at (123). Applying then the criterion [a] we find for the sum of the fractions -^^ + ^/x/e" ^ value less than the difference 9, between the limits ; so that the criterion is not satis- fied : narrower limits, however, must satisfy it, or else separate the roots, provided that is, that the roots are not equal, and there- fore inseparable. Agreeably to the rule we are to satisfy ourselves on this point first ; that is, we are to ascertain whether the func- tions /2(^) and/;j(a:), which are 20^ — 36^2 _ 144^ _|. 190, and 60^2 _ 72^ _ 144 have a common measure. Upon trial, we find that no common measure exists ; so that we may proceed to narrow the interval [1,10] with the certainty of eventually detecting the character of the roots. The intervals [1, 2], [2, 3] give the following results : 174 ANALYSIS OF EQUATIONS: (1) . . . . + 120 4- 48 ~ 150 + 30 + 05 - 78 O O O 1 o o (2) . . . . + 120 + 108 — 48—82 + 30 — 21 O 1 O 1 2 (3) . . . . + 120 + 288 + 180 — 20 — 43 — 32 As in the first of these intervals [1,2] the final index is 0, we infer that no real root exists in that interval. As in the second interval [2, 3] the index 1, nearest to the ter- mination of the series, is followed by 2, and preceded by 0, we have to apply to this interval the criterion [a], at page 105. The fractions f J, ff give a sum greater than the difference 1 between the limits ; so that the criterion is satisfied : and hence the roots indicated in the interval [2, 3] *re imaginary. The third root, which is of course real, must therefore lie between 3 and 10. 2. As a second example let us take the equation which has been partially analysed at page 144. The only doubtful interval here is the interval [1, 10], for which we have found the series (1) . . . . + 24IF0 - 12 — 11 +17 O O 1 1 2 (10) . . . . + 24 + 210 + 900 + 2797 + 5993 The index 1 nearest to the termination of the series of indices is followed by 2, but not preceded by 0. Hence the interval [1, 10] must be subdivided. Interposing the number 2 we have (2) . . . + 24 + 24 TO — 19 +1 O O O 1 2 (10) . . . + 24 + 210 + 900 + 2797 + 5993. In passing over the interval [1,2] no changes are lost, so that METHOD OF FOURIER. 175 no root exists in this interval, but two roots are indicated in the interval [2, 10]. And as the index 1, within a place of the end, is followed by 2 and preceded by 0, the character of these roots is to be tested by the criterion [a]. Writing the fractions -^, |-|^-|> ^^ s^^ t^^t ^^^i^ ^^^ is ^^s® ^^^^ ^> ^^® distance between the limits 2 and 10 ; hence a number must be employed interme- diate to these limits, unless the roots under examination prove to be equal; that is, unless the functions /(a?),/^ (a?) which are x'^—4a^ — 3x-\- 23, and 4^ — 12^:^ — 3 have a common measure. Upon trial we find that no common measure exists. Substituting then the intermediate number 3, we have the series of signs (3) . . . + + + - - and comparing this with the series (2) and (10) above, we find that the roots are real, one lying between 2 and 3, and the other between 3 and 10. We shall propose but one more example, of which we shall give the complete analysis; and, after the manner of Fourier, shall actually exhibit the several derived functions. 3. Let the equation be /(a?)==a^ +x^ + ar^ — 2^2 +2^ — 1 =0 .•./,(a;) =5a?4 +4a?3 +3^:2 — 4^4-2 /^{x) = 20a^ + 12a^2 ^_ 6^ _ 4 f^(x) = 60^2 ^_ 24a: +6 f^{a!) = l20x +24 (-.)...¥ 120. . (\< w ^-^ V ^^ -^ - 96 +42 — 18 + 10 — 6 o 1 2 2 2 2 (0) . . . + + 24+6 -4 +2 — 1 o O ©12 3 (1) . . . + + 144+90 + 34 + 10 + 2 176 ANALYSIS OF EQUATIONS : Comparing these results we find, 1 : That all the real roots are comprised within the limits — 1, -|- 1, as also the indications of imaginary roots: 2 : That indications of two roots occur between — 1 and 0, and of three between and 1 . In the interval [ — 1, 0] there is only one index equal to 1; this is followed by 2, and preceded by : applying therefore the criterion [a], we find the sum of the fractions ^|, ^\ to be less than 1, the difference between the limits; so that the criterion is not satisfied, the limits not being sufficiently close to enable us to determine, by a single operation, whether the roots are real or imaginary. Before narrowing the interval we ought to ascertain whether the roots indicated are equal; that is, whether /-j (a?) and f^{x)i which are 60a?2 + 24ar H- 6, and 120a? + 24 have a common measure. Upon trial we find that they have not. We must therefore contract the interval by interposing a number between — 1 and 0. If we employ — ^ we have the following results : <-i... . . + - 36 + 9 - ^ ^ 16 83 32 o 12 2 2 2 (0).. . + + 24 + 6-4 + 2 — 1. The first partial interval, that between — 1 and — i, cannot comprise any root, since no changes are lost in passing over it. Two roots are indicated in the second interval; and as the only index equal to 1 is followed by 2, and preceded by 0, we have to examine the fractions -^^ and -f-^^ in order to ascertain whether or not their sum surpasses, or is at least equal to |^, the difference between the limits. We at once see that the sum is equal to \y and consequently that the roots indicated are imaginary. The interval [0, 1] still remains to be examined. In this in- terval the series of indices are 000123: hence, applying the criterion [a] to the functions to which the indices 1, 2 belong, we find that f + ^^^ does not satisfy the condition [a] : the 4 + 2 — 1 2 2 9 2 + 25 T6 9 32 + + METHOD OF STURM. 1/7 interval must therefore be contracted, unless the functions /j (a?) and /aW, which are 5^ + 4^ + 3a?^ — 4a? + 2, and 20a?3 + 12a?2 + 6x —4 have a common measure. Upon trial we find that no common measure exists. Hence employing the intermediate number ^, and thus subdividing the interval [0, 1] into the two [0, ^] and [^, 1], we have the following results: (0) . . . + + 24 + 6 - O O O 1 (i) . . . + + 84 + 33 + (1) . . . + + + + The interval [^, 1] evidently comprehends a real root. The in- terval [0, J^] indicates two roots: the indices require the appli- cation of the criterion [a] to the fractions J and -|| -f- f , the sum of which surpasses ^ : therefore the criterion is satisfied, and the two roots of /i{aj) = are imaginary. Subtracting then the index 2, due to these imaginary roots, from the indices of /j(a?) and f{x)y the last index becomes 0, so that no real root exists in the proposed interval. Hence the proposed equation has but one real root: this lies between '5 and 1: the four remaining roots are imaginary. (130.) Such is the method which Fourier has proposed for discovering the character of the roots of which indications exist between given limits; and which, when taken in combination with the theorem at (107), completes the analysis of the equation. The theorem referred to shows that by substituting in/(a?), and in the several functions derived from it (1 12), a number increasing by insensible degrees from — ^ to -f ^, the series of signs fur- nished by these substitutions will continue to present variations only, till the number substituted arrives at a certain limit a: after this the variations will gradually disappear, till we reach a second limit b, when they will become replaced by permanencies. We are thus made acquainted with the limits a, h, between 12 178 ANALYSIS OF EQUATIONS. which all the real roots, and the indications of the imaginary roots, must be sought. It is evident that this loss of variations at any stage, can arise only from the number substituted causing one or more of the functions to become zero : and the evanescence of any one of the functions, not the last, /{x), must be attended with one or other 6f these circumstances, viz., the immediately preceding function will have either the same sign as that immediately succeeding, or a contrary sign. In the latter case it is plain that no variation can be lost in the passage of the intermediate function through zero. In the former case two variations must be lost; and as, by hypothesis, the number substituted does not render f{x) zero, this number is not a root of the primitive equation, but an indi- cator of two imaginary roots ; a fact which, as before remarked (69), had been previously taken notice of by De Gua. If a single real root exist in any of the partial intervals \a', b''], into which the whole interval [«, h] is divided, as in the exam- ple at p. 128, the theorem (107) will enable us to detect it. If more than one may exist, the same theorem will indicate the possibility. To remove the doubt, the second theorem at (125) is to be resorted to ; by aid of which, subdividing the interval as there directed, we shall arrive either at numbers which interpose themselves between the roots, and thus actually separate them, furnishing limits between which they severally lie, or else at numbers which are limits, not to real roots, but to indicators of conjugate imaginary roots.* And thus the character of any par- tial interval [«', 6'] may be discovered, and the analysis of the equation completed. We shall offer some remarks upon this method of completing the analysis of an equation in the next chapter, and shall suggest means for reducing much of the labour attending its application. • The numbers thus called indicators ure, as we have seen above, real roots of certain of the derived equations, each root being such that its substitution iti the function immediately preceding, and in that immediately succeeding the vanishing function, gives like signs. Hence in narrowing an interval in which only indications of imaginary roots exist, we approximate to a root, not of the primitive, but of one of the derived equations, the passage over which root is . attended with the loss of two variation.^. (See page Itf.'J.) CHAPTER IX. REMARKS ON THE PRECEDING METHOD^ WITH SUGGESTIONS FOR ITS IMPROVEMENT. (131.) The method which has just been explained is a very ingenious application of a simple property in the theory of curves to an important analytical purpose — the complete analysis of a numerical equation. The examples which have been given to illustrate it, and which are all taken from the work of Fourier, show that the method may frequently be employed with success, and with the expendi- ture of but Uttle trouble. But from these instances of its appli- cability it would be wrong to infer anything as to its merits as a general rule for the analysis of equations. The peculiar difficul- ties that have always opposed themselves to the success of every such general rule do not occur in the examples of Fourier, which have been so framed as not to involve the difficulties alluded to ; and by a similar adaptation of the end to the means, the most hmited and imperfect of methods might be exhibited to advantage. Theoretically, the process of Fourier is perfect ; and is characterized by great ingenuity and analytical address : but it is proper to speak undisguisedly of its jgrafitiral ijgfects, beyond certain limits, and a certain kind of equations; because^ some recent writers, in dwelling upon its merits, have altogether overlooked the imperfections which render it, in its present form, altogether impracticable even in equations of so low a degree as the fourth, except certain conditions, beyond our control, happen to have place. 180 IMPROVEMENTS IN THE METHOD OF FOURIER. If, by a preliminary examination, we could always ascertain whether an equation proposed for analysis involved these favor- able conditions or not, the value of Fourier's method would be gr&atly increased; for then we should never, as at present, be ex- posed to the risk of entering upon a series of calculations which terminate too remotely for ordinary patience and perseverance to carry us through them. The great blemish in the preceding method is, that it leaves us in doubt as to the character of contiguous roots, till by successive trials it either actually separates them, or else shows them to be imaginary. When they are real, and lie so closely together as to have four or five leading figures in common — a circumstance which at the outset we cannot foresee — it is plain, from the spe- cimens already given, that the indications which determine their character would be delayed to so remote a step, that our patience would become exhausted long before reaching it ; and the nature (of the roots would still be left in doubt. The example at p. 221 of the introductory volume on the Analysis and Solution of Cubic qnd Biquadratic Equations offers an instance of this kind. By the method proposed by Fourier, all the work of that example, as exhibited at pages 243, 244, 245, as well as the preliminary step at page 221, must be performed, in addition to the very tedious labour of narrowing the Hmits, by the successive steps above adverted to, till the roots, proceeding together to the extent of five places of decimals, are actually separated. It will be noticed that in pronouncing upon the impracticabiUty of Fourier's method, in cases such as this, we have altogether left the labour impUed in the process for the common measure out of consi- deration; because we regard the necessity of carrying on the i analysis by the tentative operations in last chapter, till the roots are actually separated, or till they are shown to be inseparable by being imaginary, as constituting the capital defect of Fourier's method. (132.) Every method which requires this separation to be accomplished is indeed essentially defective unless it also supply means by which the separation may be expeditiously effected. And it is solely because of this defect that all preceding rules IMPROVEMENTS IN THE METHOD OF FOURIER. 181 for analysing an equation have been abandoned one after another. Whatever other imperfections may attach to any proposed plan for detecting the character of the roots of an equation this, at all events, must be removed, before such plan can be generally apphcable. The presence of the defect here adverted to is the great drawback to the success of Lagrange's mode of analysis; for it will be found upon examination that the impracticability of this mode is traceable solely to the circumstance that the separa- tion of the real roots, or which is the same thing, the finding of a number less than the least of the differences of the roots, is indispensable to the discovery of the character of those roots by his method. The method of Lagrange, however, is far less practicable than that of Fourier; and it ought to be distinctly noticed, to the advantage of the latter method, that the labour of the separa- tion is only in proportion to the proximity of the roots, whether real or imaginary; whilst the principal step in the method of Lagrange involves pretty nearly the same amount of difficulty whether the roots are nearly equal or not. On account of this pecuharity in the method of Fourier, we should be encouraged to apply it, even in its present form, in cases where the method of Lagrange would appal us, from the known difficulties with which we should certainly have to contend. When the roots of the original equation, as well as those of the derived equations, within the limits under examination, do not approach nearer to equality than by a coincidence in the leading figure of each, or at most in the first two figures, the analysis of it by the preceding method may perhaps be accomplishable even in an equation of the sixth or seventh degree, whilst no one would think of applying to any such equation the process of Lagrange. If roots do not approximate so closely even as here supposed, then Fourier's method will be comparatively easy; but when the labour of the several operations for the common measure is viewed in connexion with that due to the successive subdivisions of the interval — even for the amount of proximity above sup- posed — ^we feel justified in speaking of the eligibility of the method with some hesitation. We shall shortly see, however, that this labour may be altogether dispensed with, or greatly reduced. 182 IMPROVEMENTS IN THE METHOD OF FOURIER. (133.) Besides the advantage of Fourier's method over others, when the real roots are moderately wide apart — that is, have no leading figures in common — there is another marked peculiarity in favour of it : it is, that it considerably reduces the embarrass- ment and uncertainty which the entrance of imaginary roots often occasion.* The considerations arising out of the ingenious con- trivance of the subtangents lead to a rapid detection of the existence of such roots whenever they are not in the pecuUar predicament of being rendered real by a very minute change in » - the absolute term of the equation involving them. In this latter case the imperfection before adverted to again discovers itself: for this case, and the case of nearly equal real roots, both involve the same peculiarities: — a minute change in the constitution of the equation being sufficient in either case to cause the doubtful roots to become equal. These then are the kinds of equations that are excluded from the rule of Fourier as it at present stands; namely, those whose roots are such that a very minute change in the final term, whether of the original or of any derived equation, suffices to ren- der a pair of imaginary roots real or a pair of unequal roots equal. This exclusion, however, does not arise from any theo- retical imperfection, but from the impracticability of performing the necessary calculations, without extraneous aid. (134.) Any method that should with equal facility apply to all cases, the amount of labour being independent of the peculiar * We regard this peculiarity as a most important feature of Fourier's method ; and we shall show hereafter how it may be rendered available, in combination with the methods of Sturm and Horner, in completing the analyiiis of equations of more than ordinary difficulty. The assistance which Cfhe method of Fourier is thus capable of affording to that of Sturm will be seen to effect a great saving of numerical labour. The two methods have hitherto been regarded as distinct and independent: — we shall endeavour to show that considerable advantage may result from uniting them together, after the defects, which have hitherto attached to Fourier's method — and which, as remarked in the text, preclude its application in extreme cases — are re- moved by means of the improvements suggested in the present chapter; these improvements will be brought into operation when we come to treat of the general solution of numerical equations. IMPROVEMENTS IN THE METHOD OF FOURIER. 183 relation of the roots just adverted to, must be expected, from this very comprehensiveness of character, to bring into operation, in the simpler examples to which more limited methods may be peculiarly adapted, a larger array of figures than those methods themselves ; for it is essential to the perfection of so general a pro- cess that, whatever be the example to which it is applied, provision be always made by it for the demands of such critical and extreme cases as those we have been considering, whether the example in question present one of these or not; seeing that it is this which it devolves upon the method itself to make known : for we cannot determine, from the mere aspect of an equation, whether peculiar relations exist among its roots or not ; nor, consequently, whether methods, which become inadequate under such relations, are apphcable or not. It would be very unfair therefore, for the purpose of instituting a comparison between a method but partially applicable and another of unfailing generality, to select an example within the powers of the former, place in juxta-position the processes of each, and thence draw our conclusions as to their relative merits : and the student requires to be cautioned against being misled, in this manner, when judging of the comparative claims of the theorems of Fourier and Sturm. (135.) We are of course, throughout these remarks, speaking of the method of Fourier as delivered by its author, and as ex- pounded in the preceding chapter; and are not here considering whether, by blending other principles with its own peculiar pro- cess, the aid adverted to above, for rendering the separation of the roots always practicable, can be readily commanded or not. In seeking to separate the roots occupying a doubtful interval by continually contracting that interval, according to the method of last chapter, we are left too much to mere conjecture as to the situation in this interval which the real roots, or the indicators of imaginary roots, occupy; and hence, in subdividing an interval, for the purpose of inclosing the roots within narrower limits, several trial operations, fruitless in their results, must always be calculated upon. It is to reduce this useless labour — a labour which needlessly multiplies the work several fold — that extra assistance is so much required. 184 IMPROVEMENTS IN THE METHOD OF FOURIER. / The method of Horner, modified by a certain improvement / to be hereafter explained, will be seen to furnish such assistance ; so that by introducing this method in combination with that of Fourier, in those cases where the character of the roots under examination unfolds itself only at a remote step of the analysis; the labour at present involved in Fourier's process may be very considerably diminished. The advantage of this combination will be seen when we come to treat of the calculation of the roots.* It may be observed, however, that the analysis of an /equation is thus made to depend upon its partial solution, instead ""•vpf being entirely preparatory to that solution. This remark equally applies to the methods of Lagrange, Fourier, and BuDAN; and must necessarily apply to every method that leaves the character of the roots undecided till those that are real are actually separated. The method of Sturm is the only one (which treats the analysis of an equation as a perfectly independent g^roblem; always making known the exact number of real roots in any proposed interval without trenching upon the subsequent problem of seeking the actual development of those roots. (136.) In the preceding observations upon the mode of ana- lysis expounded in last chapter, and upon the inadequacy of it in its present form as a general method, we have, as before remarked, disregarded that part of the operation which involves the finding -.^ of com mon measures. Fourier, and all who advocate this method, seem to entirely overlook the labour involved in these processes — two or three of which are sometimes required in the analysis of a single interval, (see example 3, page 1 75) . No attempt we believe has hitherto been made to remove this serious drawback to the application of the method, even in those cases where no remarkable proximity of roots occurs to preclude the hope of its success. We propose to show, however, that considerable improvement and reduction of labour may be intro- duced into this part of the operation, and that thus the apphca- tion of Fourier's method may in all cases of difficulty be greatly facilitated. The operations for the common measure so frequently occur- ring in Fourier's method of analysis, are always introduced for • See al*;o the foot-note at \ma;e 1H2. IMPROVEMENTS IN THE METHOD OF FOURIER. 185 the purpose of ascertaining whether or not a proposed equation has a single pair of equal roots in a given interval. It has been already proved that when an equation has a pair of equal roots, and only a pair, that these roots must be commensurable (101) ; and, moreover, that the repeated root must be such that the square of its numerator must be a divisor of the final term N, the square of its denominator a divisor of the leading coefiicient A^, the nume- rator itself a divisor of A, and the denominator a divisor of A^^j. (a.) Hence, in order that the suspected pair of equal roots may exist, A and N must have a common factor (the numerator of the repeated root) ; and N must be divisible by the square of this factor. Also A^ and Ajj_j must have a common factor (the de- nominator of the repeated root), and Aj^ must be divisible by the square of this factor. (b.) These tests will generally be found more than sufficient to determine the point in question. But if in any case they are all fulfilled by any fraction — , then we must proceed to divide the proposed polynomial by ir — -, according to the rapid process at (51). When b is unity the operation must be continued up to N, as at the article referred to ; and the number under trial will be a root, or not, according as the final result is zero or not.* • In the former case, a root - will have been determined, and the coeffi- o cients belonging to the depressed equation obtained ; if the operation, repeated a with these depressed coefficients, terminate in zero, the root - will enter the equation twice : if not, one root only will be - ; and thus the separation of the two will have been effected. It is worthy of observation that, in testing a suspected root in this manner, the doubt will often be more speedilj re- moved when the number under trial is a fraction than when it is an integer ; because in the latter case the operation must always be carried on up to the final result, or remainder ; whereas, in the former case, the occurrence of a fraction, which may take place anywhere between the first and last result, at once puts a stop to the work. However, in the case of a suspected integral root we may proceed somewhat differently : we may reverse the order of the operations by dividing the last coefficient by the supposed root, adding the quotient to the preceding coefficient ; dividing the result by the same number and adding the quotient to the next coefficient, and so on : a fractional quo- tient will, of course, stop the operation. 186 IMPROVEMENTS IN THE METHOD OF FOURIER. When b is not unity the same course must be adopted, and a corresponding conclusion drawn, unless we arrive at a fractional result ; in which case the process need not be continued ; for Under these circumstances the number under trial cannot be a root {7^). (137.) We shall now apply these precepts to Fourier's ex- amples, given in last chapter. In example 2 at page 168, it is required to determine whether the equation ar^ + iT^ + a?2 — 25a? — 36 = has a single pair of equal roots. Applying the foregoing tests, we have first to see whether any integral factor of 25, except 1, has its square for a factor of 36. A glance is sufficient to show that it has not : hence equal roots do not enter the equation. Again in example 1, page 173, we have to determine whether a pair of equal roots enters the equation 20a^—36a?2— 144^4-190=0, or rather 10^— ISa^- 72a; + 95=0 -Here neither of the numbers 95, 10 has a square factor except 1, so that equal roots cannot exist. In like manner the example at page 1 74 requires it to be ascer- tained whether the equation ^ _ 4a^ _ 3^ + 23 = has a pair of equal roots, which it evidently has not since 23 has no factor but unity ; which is not a root. Lastly, the example at page 1 7^, requires it to be determined whether either of the equations 10a?2 + 4a? + 1 = 0, 5a?^ + 4ar"> -j- 3a:2 — 4^? +2 = have a pair of equal roots ; and a simple inspection shows that they have not, since neither of the numbers 10, 1, 5, 2 has a square factor different from 1 ; which is not a root. In another of Fourier's examples, which we have not tran- IMPROVEMENTS IN THE METHOD OF FOURIER. 187 scribed, the analysis of a doubtful interval requires it to be ascer- tained whether the equation 360x^ - 1440a? + 1440 = or a?2 - 4^ + 4 = has equal roots ; which we at once see to be the case, each root being 2. (138.) The analysis of the equation a?4 + 312^3 -f 23337a?2 — 14874.^ + 2360 = would be interrupted by a doubtful interval between and 1 ; but we need not follow Fourier's directions, and seek the com- mon measure of the first member of this and the first derived function — ^which is a work of very considerable labour — in order to ascertain whether the roots in this interval are equal or not ; since we can confidently affirm at once that the roots cannot be equal ; because the equation, having its first coefficient unity, cannot have a pair of equal roots which are not integral. When, as in this last example, and in example 2, page 174, a common measure is to be sought, according to the directions of Fourier, between /(x) and fi{x), no other information is deduced from the process than is sufficient to remove all doubt as to the equality of the roots in the interval under examination. Fourier, indeed, was not aware that anything worth notice beyond this could be yielded by that process ; and, consequently, as soon as the non-existence of equal roots was ascertained, he applied himself to the discovery of the real character of the roots by the method of successive subdivisions already explained. But the more recent theorem of Sturm, to be given in Chap, xi, shows that all this additional labour is superfluous ; and that the process for the common measure does itself supply informa- tion, not only as to whether the roots in the proposed interval are equal or not, but also as to whether they are real or not. It is not remarkable that Fourier should not have recognized this, as the theorem of Sturm was a subsequent discovery : but it is singular that those who have since commented upon Fourier's method should not have adverted to the circumstance. It is cer- tainly worthy of being adverted to, since it not only shows the 188 IMPROVEMENTS IN THE METHOD OF FOURIER. superiority of Sturm's method over that proposed by Fourier, but discovers also the essential imperfection of the latter, in in- volving, as a mere auxiliary, an operation which is found to con- tain within itself the whole of the desired information. This operation, therefore, supphes much more than Fourier's method needs ; which is simply the resolution of the doubt as to whether or not equal roots exist in a given interval : and it would evi- dently be an improvement to exchange that operation for another of a less comprehensive, and therefore of a less laborious, cha- racter ; and adequate to meet no more than the demands of the case. (139.) The precepts (a) and (b), p. 185, by which this im- provement is to some extent effected, suppose that the doubtful interval to be examined is the only one, for the same function, between the extreme limits of the roots. If another doubtful interval occur for the same functions two pairs of equal roots may exist between the extreme limits ; and these need not necessarily be commensurable; because they may arise from the repetition of a quadratic factor which, when equated to zero, gives incom- mensurable roots. StiQ the conditions (a), omitting the paren- thesis, must be fulfilled, even in this case (101); so that in all the examples hitherto discussed it is matter of indifference whether the function furnishes other doubtful intervals or not. But when the conditions (a) are fulfilled, and yet from (b) we find that no equal commensurable roots exist, then if other doubt- ful intervals occur, for the same function, it is possible that there may be pairs of equal incommensurable roots entering as above stated. Generally speaking it will be best to put this to the actual test, by seeking the common measure between the function in question and the next derived function ; carrpng on the ope- ration till a quadratic remainder is obtained. If the supposed equal roots exist, this quadratic, factors common to all its coeffi- cients being suppressed, will accurately divide the original func- tion ; and fractions will be excluded from the quotient {j^^. We may either try if this be the case, or if the preceding cubic remainder be divisible by it without fractions : in either case the operation will be easy, because the squares of the extreme coeffi- IMPROVEMENTS IN THE METHOD OF FOURIER. 189 cients of the divisor must be smaller than, or at most equal to, the extreme coefficients of the original function, being factors of them (76). Thus the actual operation for the common measure need be entered upon only in those cases where the latter, and consequently the more laborious, steps of it become superfluous. It is scarcely necessary to remark that, when more than two equal quadratic factors enter into the composition of the proposed function, the operation for the common measure between it and its derived function will terminate at a still earlier stage. (140.) It thus appears that even in those rare instances, in which the process for the common measure must be actually entered upon, the work terminates before that part of the opera- tion is reached where by far the greater portion of the labour is accumulated.* But the calculation of even the earlier steps of the operation, by the common method of proceeding, is encum- bered with much unnecessary work ; and is apt to spread over an inconvenient extent of space. It would seem, therefore, in order to give to the method of Fourier all the practical facility which can render it available as a general rule of analysis, applicable to all cases, that, in addition to a more perfect method of narrowing the doubtful intervals, an easier and more compact form must be given to the numerical operation for the common measure. Such an improved form we have already proposed elsewhere,f and shall have occasion to recur to in the Chapter on Sturm's theorem. To these remarks we may add that if an equation of the fifth degree have two pairs of equal roots the remaining root must be commensurable ; and, generally, if an equation of the 2m -\- I degree have m pairs of equal roots, the remaining root must be commensurable. This may be useful in ascertaining whether so many pairs of equal roots are possible in any proposed case : when they exist, the depressed polynomial, resulting from elimi- nation of the aforesaid commensurable root, must be a complete square. * For an example of the great increase of labour involved in the computa- tion of the last two steps of the common measure, when large coefficients enter, see the Analysis of Cubic and Biquadratic Equations, page 2 13. t Mathematical Dissertations, pages 145 and 209. 190 IMPROVEMENTS IN THE METHOD OF FOURIER. (141.) It has been already obseired (117) that Fourier always employs the auxiliary derived functions, which enter into his pro- cess of analysis, encumbered with numerical factors, from which they maybe freed. (See page 175.) In the method of transfor- mation employed in the present work, these factors never make their appearance; and thus our several results involve smaller numbers than the corresponding ones deduced by Fourier. In Chapter vii. the signs only of these results were the objects of examination: but in the further analysis of the equation, by the method taught in last chapter, the actual results themselves are brought into requisition. The criterion [a] at page 165 involves numerical values only, signs being disregarded. By continuing, therefore, to employ the same mode of derivation as that adopted in the first stage of the analysis, additional simplicity vrill be in- troduced into Fourier's examination of the doubtful intervals. We shall only have to remember that in applying the formula [a] to two consecutive results /m-i(^)> fm('^)> ^^ must multiply the denominator /^^(r) by m, the number marking the place of the function in the series of derivations; as is plain from the ex- pressions at (1 1 6.) As multiplying the second member of the cri- terion [a] by m is equivalent to suppressing this factor in the denominator of each fraction in the first member, we may always adopt this latter course, which will in general be the easier, and employ the numerical results in the formation of the fractions, without any modification. Thus in the example at page 173 the series of results deduced as here proposed would have been as follow: (see page 143.) (1) . . . . +1+2 -26 +15 -{-65 -78 (10) . . . . +1+188 +856 +7575 +32654 +54939 The fractions submitted to the test would consequently have been ^^ and ^-^^5^ ; and the number to which their sum is to be compared, 27. (142.) The method of computing the derived functions ex- hibited at page 143 is always the most expeditious, even though these functions be written down before us, as at page 1 75. There IMPROVEMENTS IN THE METHOD OF FOURIER. 191 are cases, of frequent occurrence, in which we would recommend these functions, when freed from superfluous factors, to be thus written down ; not for the purpose of computing them, which is Fourier's object, but with a view to extract from them certain information which they readily offer, and which will sometimes enable us to dispense with a tedious analysis. Thus, whenever, as in example 3, at page 175, the index 1, which is followed by 2 and preceded by 0, falls under the function of the first degree, the 2 being under the quadratic function, we would always re- commend the derived functions to be actually written, before pro- ceeding to analyse the interval ; because the quadratic function would, in general, inform us, at a glance, whether the 2 under it referred to a pair of real roots, or to a pair of imaginary roots. In the example referred to, for instance, the quadratic function y^(a?) is seen at once to be such as to give imaginary roots when equated to zero ; since four times the product of the extreme co- efficients exceeds the square of the middle one. And thus all the trouble of examining the interval [0, — 1], at page 176, might have been spared. Had this relation among the coefficients of f^{oc) not subsisted, then we should have inferred that the index 2 indicated a pair of real roots ; and should have sought, in the usual way, to separate them, without taking at each step the un- necessary trouble of applying the criterion. Fourier evidently overlooked this means of simplifying his process.* (143.) In addition to these suggestions for simplifying the operations of the preceding chapter, we have only to remark, in conclusion, that when, as is usually the case, the analysis of an equation is merely preparative to the actual computation of its real roots, we may allow a single pair of roots to remain doubt- ful, provided all the others are real, and may proceed at once to the calculation of these latter; because, as will be shown hereafter, when all the roots but two are determined, these two, whether real or imaginary, may be derived from the former with compara- * There are two other examples in Fourier's work, which we have not transcribed, in which attention to this principle would have saved the trouble of analysing a doubtful interval. See pp. 148, 149, Jnalyse des Equatiom Dtterminte, 192 IMPROVEMENTS IN THE METHOD OF FOURIER. tive ease. This is a truth of considerable importance in connexion with Fourier's method of analysis; because it still further adds to the facility of its application to a class of equations that would otherwise be scarcely manageable without aid from Horner's process of approximation to the roots. The equation at page 187 is one of this class : the analysis of the doubtful interval, agreeably to the steps proposed by Fourier, is very tedious, on account of the great labour attendant upon the separation of the two roots in the interval [0, 1] by the slow and uncertain approach to them which those steps make. The remaining roots, however, are easily separated, and may therefore be expeditiously calculated; and thence, by the principle just stated, may the roots in the interval [0, 1] be readily and accurately determined without reference to any method for their separation. We shall exhibit the entire process in a future part of the work, when the value of the prin- ciple referred to will be more clearly seen.* (144.) It may not be amiss in conclusion to briefly recapitulate the more important particulars dwelt upon in the present and two preceding chapters. And first we may notice that the method of * In the treatise on the Analysis and Solution of Cubic and Biquadratic Equations the two roots of the equation adverted to, which lie in the interval [0, 1], are separated, by Horner's method of approximation, with comparative ease : the process supplying us with a series of transformed equations, such that if each, as it presents itself, be considered in conjunction with that which would arise from increasing the transforming factor (or root figure) by unity, we should have a pair of consecutive series of results at every step, to which Fourier's tests would be immediately applicable. Hence, every step of the work, in the volume referred to, is to be accompanied by another, considered as a bye- operation, arising from transforming by an additional unit. When we arrive at the sixth step the unit-transformation connected with it will not require completion : the change of sign in the final term will show that the roots separate at this step. In fact this unit-transformation may be dispensed with throughout, provided we test the roots, at each step, by the first fraction only of the criterion [A]. But a still better test will be given in Chapter xii. By thus blending Horner's method of narrowing the interval, with these contrivances for testing the character of it, and combining with this two-fold process the facilitating principles delivered in the present chapter, we shall always proceed with the perfect certainty that every step we take is a real ad- vance towards the removal of the doubt ; and that no part of our labour will have been needlessly expended. IMPROVEMENTS IN THE METHOD OF FOURIER. 193 Fourier divides itself into two principal parts — the theorem of BuDAN, and the subsequent process of Chapter viii. The first effects, in general, but a partial analysis of the equa- tion : it makes known the extreme limits within which all the the real roots as well as all the indications of imaginary roots must be sought; and it actually separates those of the former which are sufficiently wide apart to have no leading figures in common. But when either imaginary roots or nearly equal roots enter the equation, the indications of their entrance being the 1 same in both cases, we have no means of distinguishing the one | class of roots from the other in any given example; and thus I their character is left doubtful. Nevertheless, the theorem effectively prepares the way for the removal of this doubt by supplying the exact interval in which the necessary information must be sought. The second process, which is exclusively due to Fourier, addresses itself solely to this latter object; and aims at completing the analysis of the equation by extracting the desired information from the doubtful intervals. This involves a two-fold operation, or course of operations, viz. the continual narrowing of the in- terval, step by step, and the application of a certain test [a] at each contraction. If the concealed roots happen to be imaginary this test will sooner or later be satisfied; if they happen to be real and unequal, the continual diminution of the interval must at length separate them. The appHcation of the test is always easy: the separation of the roots often difficult and tedious. For as we are not always furnished with adequate means of knowing whereabouts in the doubtful interval the nearly equal roots may lie, supposing them to exist, or the indicators of the imaginary roots when real roots are wanting, we may become involved in tentative opera- tions of a very fatiguing extent and labour. In order to be certain that these will ever terminate we must be sure that the roots under examination are not equal. It behoves us therefore to put this matter beyond doubt by a preUminary investigation ; and thus has been combined with the trial-operations, adverted to above, the extra labour attendant upon finding common measures. The method of Sturm is wholly comprised in a single opera- 13 194 IMPROVEMENTS IN THE METHOD OF FOURIER. tion for the common measure; yet that method has been pro- nounced impracticable beyond very narrow limits. With much greater propriety might this be said of Fourier's process, in- volving sometimes two or three operations of this kind; in addition to the great labour of separating the roots by trial-trans- formations. To render then the method of Fourier generally practicable, the following improvements must be effected : 1. When roots are indicated in an interval, means must be furnished for guiding us to that particular subdivision of the in- terval which the indicated roots occupy; and we must not be left to find our way to this subdivision through trial-operations only. This improvement we propose to accomplish by extending Horner's method of approximating to the roots. 2. The necessity for the common measure must either be superseded, by some more readily applicable test for equal roots, or at least methods must be contrived for materially abridging the ordinary labour and extent of the operation. The former object is attained, in the generahty of cases, by the simple criteria marked (a) and (b); and the latter object — the curtailment of the process for the common measure, when instances of rare oc- currence render that process necessary — ^is effected by the consi- derations at page 186, which enable us to dispense with the more laborious steps of the work. 3. Lastly: when, as in cases like those just adverted to, the leading steps for the common measure must be performed, a compact method of working, purely numerical, like all the other parts of the process, and free from the needless encumbrances of the common method, is required to give additional facilities to the analysis in the more difficult cases. Such an improved mode of conducting the operation will be found in Chapter xi. Thus improved, the method of Fourier may with propriety be brought into advantageous comparison with that of Sturm ; and will probably henceforth take precedence of it in those cases where, from the magnitude of the coefficients, or of the leading exponent, very large numbers may be expected to enter the ope- ration for the common measure. CHAPTER X. METHOD OF SUDAN FOR DETERMINING THE CHARACTER OF ROOTS IN DOUBTFUL. INTERVALS. (144.) The method proposed by Budan for the analysis of a doubtful interval within the limits of the roots is theoretically much simpler than that of Fourier. It depends upon the obvious principle that, whatever be the number of roots of a pro- posed equation lying between the limits 0, r, the very same num- ber of roots of the equation whose roots are the reciprocals of the proposed, must lie between - and - ; for it is plain that to r whatever roots and r are the limits, in the proposed equation, to the reciprocals of those roots — and those only — ^will - and - r be Umits in the transformed equation. The inference to be drawn from this is, that as many real roots as there are in the proposed equation, lying between and r, so many changes of sign, at least, must there be in the reciprocal equation, after applying to it the transformation (-\ ; since be- tween this transformation and the final one by (-), this numbei' of changes, at least, must be lost, otherwise there could not be so many roots of the reciprocal equation, in the interval [-, -J, as in the proposed in the interval [0, r]. Should it happen therefore that, after applying the trans- 196 BUDAN ON THE CHARACTER OF formation - to the reciprocal equation, fewer changes appear r in the result than have disappeared by the tranformation (r) ap- plied to the direct equation, we may conclude that these losses cannot all indicate real roots: as many imaginary roots, at least, must be indicated in the interval [0, r] as the changes lost in the direct transformation exceed in number the changes left in the reciprocal transformation. But so long as the changes left in the latter transformation are the same in number as the changes lost in the former, or in fact so long as any changes at all are left, more than one, the interval [0, r] will remain doubtful. We may seek to remove this doubt by narrowing the interval [0, r\ in the direct equation; but the better plan will be to dimi- nish the unlimited interval [-, -lin the reciprocal equation, trans- forming successively by — h 1> - + 2, &c., till all the changes are lost. The single roots passed over in the course of these transformations will be indicated by so many changes of sign in the final term, and that number of real roots will thus be de- tected in the original interval [0, r]. If there are not as many single roots passed over as there are changes lost, then there will be doubtful intervals. To each of these the same treatment is to be applied as was applied to the interval [0, r] ; that is, each of the new doubtful intervals is to be considered as belonging to a new direct equation, and the interval, as before, exchanged for another indefinitely wide belonging to a new reciprocal equation, and so on. Whenever, in passing from a direct transformation to its collateral reciprocal transformation, more signs are lost by the former than are left by the latter, a number of imaginary roots equal to the excess will be accounted for ; and when, in narrowing the indefinite interval in a reciprocal equation, single roots are passed over, so many real roots will be accounted for. The ope- rations are to be continued till as many roots are thus detected as there are changes lost in the doubtful interval [0, r], when the analysis of that interval will be completed. The course of pro- ceeding may therefore be described by the following precepts : 1 . Let [«', h''\ be any doubtful interval occurring in the partial ROOTS IN DOUBTFUL INTERVALS. 197 analysis of an equation by the method of (108). The equation due to the transformation {a!) is the first direct equation; the in- terval between this and the transformation {b') — which is the doubtful interval — is b' — a' •=^r; that is, for the first direct equation, the interval to be examined is [0, r] . 2. Take the reciprocal of the equation {a!) ; that is, simply reverse its coefficients, and transform by (.-)• If no changes are left after this transformation, all the roots indicated in the doubt- ful interval are imaginary. 3. If as many changes are left as are lost in the direct trans- formation {b')y the original doubt remains undiminished. 4. If fewer changes are left than are lost in (&'), the difference will imply that number of imaginary roots, so that the doubt will be partially removed. 5. Whether as many changes are left or fewer, we must cause them all to disappear — or at least all but one — ^by continuing to transform onwards towards - : and every time a single sign dis- appears, or an odd number of signs, a single real root will be in- dicated, and thus the doubt will be still further reduced. 6. If the doubt be not entirely removed, on account of the signs not all vanishing singly in the preceding series of trans- formations, new intervals of doubt will occur in this series. Let [a", b"^ represent any such interval; and consider the transformation {a") as a new direct equation, and the passage from it to the transformation {b"), as the corresponding direct transformation, the doubtful interval for {a") being b" — a" -=7^ ; that is, [0, /], and proceed as with the former direct equation. And this process is to be continued till there are as many roots detected — real and imaginary — as there are changes of signs lost in the first direct transformation; that is, in the passage over the interval under examination. 198 BUDAN ON THE CHARACTER OF The following examples will sufficiently illustrate these precepts.* 1 . Let it be required to analyse the equation ^ - 7^ + 7 = 0, the ordinary superior limit to whose positive roots is 8. (0) 1+0-7 + 7 (1) 1+3-4+1 (2) . . . . 1 + 6 + 5 + 1 two variations lost. Here two variations are lost in the interval [1, 2]; or the trans- formation (1) loses two variations in the interval [0, 1]. This therefore is a doubtful interval. Hence taking the reciprocal of (1), and observing that 5 is a superior hmit to the positive roots, we proceed as follows: (0) 1-4 + 3+1 (1) . . . . 1 — 1—2 + 1 two variations left. (2) . . . . 1+2 — 1 — 1 (3) 1+5+6 + 1 The transformation (3) furnishes no variations, so that this step is terminated, and the doubtful roots are found to be real, both lying in the interval [1, 2]. The third root is found to lie in the interval [ — 3,-4]. 2. Let the proposed equation be a;6 _ i2a^ — 2ir3 _|- 37a;2 + lOa? -10=0. Here 13 is the ordinary superior limit to the positive roots: but if we diminish the roots by 10, all the resulting coefficients are • These examples are from a paper by Vincrnt in the Journal tie Math^ma- tiques for October, 1836. But the transformations have been differently, and somewhat more compactly arranged, and some errors corrected. The work of the fourth example, in the paper referred to, is altogether erroneous. ROOTS IN DOUBTFUL INTERVALS. 199 found to be positive: hence 10 is a superior limit. Therefore subdividing the interval [0, 1 0] we have the following results : (0) 1+ — 12— 2 + 37 + 10—10 (1) .... 1 + 6 + 3 — 30 — 26 + 36 + 24 owe var. lost. (2) 1 + 12 + 48 + 62-23-58+14 (3) .... 1 + + + + + + two var. lost. The interval [2, 3] is doubtful: hence taking the reciprocal of (2), and observing that 6 is a superior Hmit to the roots of the reci- procal equation, by (86), we proceed as foUows: (0)...14— 58- 23+ 62+ 48+ 12 + 1 (1)...14+ 26 — 103-330- 274- 4+56 twovar.left, (2) . . . 14+110 + 237—202 — 1412-1740—615 one var. left. This result is sufficient to show that the two roots in the pro- posed interval are both real ; since the variation still left must disappear between (2) and ( oo ) . To determine the places of the negative roots, let the alternate signs of the proposed equation be changed : then we have (0) 1+ — 12+ 2 + 37 — 10 — 10 (1) .... 1 + 6 + 3 — 26 — 14 + 28 + S one var, lost, (2) 1 + 12 + 48 + 66 + 1 - 30 + 6 (3) .... 1 + + + + + + I't'^o ^«^« ^0^1' We have, therefore, to discover the character of the interval [2, 3]. Taking the reciprocal of (2), and proceeding as before, we have (0) 6-30+ 1+ 66+ 48+ 12 +1 (1)....6+ 6— 59 — 110+ 42+196 +\()A two var. left. (2) . . . . 6 + 42+ 61 — 166-492—220 +185 (3) . . . . 6+78 + 361+618—114—1212—584 onevar. left. 200 BUDAN ON THE CHARACTER OF Hence, as there is only one more variation to lose, the two roots in the interval [2, 3] are real ; so that the situations of the real roots are as follows : one root in each of the intervals [0, 1], [0, — 1], and two roots in each of the intervals [2, 3], [-2,-3]. 3. Let the proposed equation be ajfi _ 6^ + 40a^ + 60x^ — a? -- 1 = of which aU the positive roots are included between and 7. (0) . . . 1-6 +0+40 + 60-1 — 1 (1) . . . 1+0 —15 + 0+135+215+93 onevar.lost (2) . . . 1+6 +0—40 + 60+431+429 (3) . . . 1 + + + +15+467 + 887 twovar. lost Recip. of (2) ... 429 + 431 + 60 — 40 + +6 +1 (1) ... 429 + + + + + + no var. left. Hence the two roots indicated in the interval [2, 3] are imaginary. Changing now the alternate signs of the equation and proceed- ing as before, we have (0) .... 1 + 6 +0 —40 + 60 + 1-1 (1) .... 1 + 12 + 45 + 40 + + + three var, lost Recip. of (0) .... 1 _ 1 —60+40-0 — 6 — 1 (l)....l+5 —50 — 90— — — one var. left. Hence, of the three roots indicated in the interval [0, — 1], two are imaginary, and one real. ROOTS IN DOUBTFUL INTERVALS. 201 4. Let the equation proposed for analysis be 4x7 — 6x^ — 7x^ + Sx^ -{-7x^ — 23^2 — 22a? — 5 = Here 3 is a superior limit to the positive roots (89) ; so that the highest transforming factor will, at most, be 3. (0) ... 4 — 6 -7 +8 +7 - 23 — 22 - 5 (1) ... 4 + 22 + 41 + 23 — 11 — 30 — 58 — 44 twovar. lost (2)...4H- + -f + + 4.~^ (3)* . . + one var. lost Recip. of (0)... 5 + 22 + 23 -7 —8 +7 + 6-4 (1) ... 5 + + + + + + + no var. left Hence, in the positive region there is but one real root : it Hes between 2 and 3. Changing the alternate signs of the proposed, we have (0) ..4 + 6 -7 —8 + 7 + 23 — 22 +5 (1)..4 + + + + + + +fourlost{A) Recip. of (0) . . 5-22 + 23 + 7 -8 -7 + 6 + 4 (1) ..5 + 13 —4 —33 — 15 + 16 + 14 -{■S two left (b) (2).. 5 + 48 +179 +317 +248 +33 —38 +4 (3).. 5+ + + + + + + two lost (a') * This transformation need not be computed, since we already know 3 to be tlie superior limit. 202 BUDAN ON THE CHARACTER OF Recip. of (2).. 4— 38+ 33+ 248+ 317+ 179+ 48 + 5 (ly.. 4- 10— 111- 17+ 1019+ 2462+ 2314 + 796 two left (b') (2)'.. 4+ 18- 87- 582- 219+ 4241+ 9640+6457 (3)'.. 4+ 46+ 105— 607—2917- 424+14838 + 19472 (4)'.. 4+ 74+ 465+ 748-3235-10993+3640 + 30517 (5)'.. 4+ 102+ 993+ 4323+ 6027-10366-22262+21220 {<^)'.. 4+ 130+ 1689 + 10958 + 35429+45197— 2016 + 41 (6-05y 4++ + + + + + two lost {a.") Taking now the reciprocal of (6)', and transforming by .^3^ = 20 (0) ..41—2016 + 45197 + 35429+ 10958 + 1689+ 130+4 (20).. 41+ + + + + + + none left {b'^ The results (a), (b) show that two of the four roots indicated by the interval [0, 1], are necessarily imaginary. The results (a'), (b') leave the remaining two roots still doubtful. The re- sults (a"), (b") show that these are also imaginary. Consequently the equation has no negative roots. (145.) These examples are amply sufficient to illustrate the method of Budan when exhibited in its most improved and con- venient form. The principle upon which it depends is very different from that upon which the operation of Fourier is founded ; its characteristic peculiarity being that, instead of pur- suing the doubtful interval through its successive contractions, till it can no longer conceal the character of the rooti»%ithin its narrowed limits, the process of narrowing the intervfd is stopped as soon as fractional numbers would become necessary for a fur- ther subdivision of it. The small interval is then exchanged for another indefinitely wide, but embracing indications, of precisely the same import in reference to the doubtful roots, as those in ROOTS IN DOUBTFUL INTERVALS. 203 the interval which it has replaced. This wide interval is in like manner gradually contracted, till the character of these indica- tions discovers itself ; or — ^if the information sought be delayed — till a further contraction would again introduce fractions, when the interval is as before replaced by another indefinitely wide ; and so on, tiU the indications, thus transmitted through these successive intervals, unfold their character, and the operation terminates. The proposition which justifies this transference of our opera- tions, from an interval inconveniently narrow, to another indefi- nitely wide, is simply this : viz., that as many real roots of any equation as lie in the interval [0, /•], so many, and no more, belonging to the reciprocal of that equation, must lie in the interval [-, oo] ; that is, if r be taken equal to unity, in the interval [1, oo]. By substituting, therefore, this latter interval for the former, fractional transformations are evaded ; and the proposed interval virtually narrowed without their aid. As in Fourier's method so here, the contraction of the original interval till the roots, if real, actually separate, must be accomplished before the character of those roots can be decided upon. Fourier effects this con- traction by the direct method of minutely subdividing the interval itself. BuDAN accomplishes the same thing by the indirect method above described, thus avoiding Fourier's minute sub- divisions : but the two methods proceed to their ultimate object pari passu ; and the same imperfection is common to both, each involving a vast amount of useless trial-operations in all cases where roots occur which a slight change in the coefficients of the equation would render equal. With the exception of such cases as these the method, like that of Fourier under the same restrictions, and independently of other aids, is readily practicable, as the foregoing examples suffi- ciently show ; since the superfluous trial transformations — un- avoidable where in our search after close limits so much is left to conjecture — are comparatively few ; because the roots entering those examples, having no remarkable proximity to one another, disclose their character after a few steps of the work. But if the 504 BUDAN ON THE CHARACTER OF method of Budan be applied to the example at page 187, we shall find, like as in the unmodified process of Fourier, that the number and extent of the operations will exceed all reason- able bounds. Another difficulty attendant upon the method, and the one which those who have cultivated it most have regarded as the most formidable, is that the case nf^g nal rnntia | s left unprovided for : and that, therefore, in order to ensure the success of the method it should previously be ascertained whether such roots enter the equation or not. This has been hitherto thought to involve the necessity of executing the operation for the common measure ; by which the numerical labour is often very seriously increased : as, for instance, in such an example as that just referred to ; and certain tests, all of which involve difficult and tedious operations, have been proposed, to supersede the necessity for the common measure.* But we have shown in the preceding • One of the most recent of these tests is given by Cauchy. It furnishes a limit to the number of steps towards the separation of the roots — whether those steps consist of successive transformations as above or not — within which, if the separation be not effected, we may infer the equality of the roots under trial. But this limit is in general so remote as to be practically useless. An- ^^pEBer limit for the same purpose has been since proposed by Vincent : but this is still more unsuited to any practical purpose. Both these limits may be seen in the Journal de Mathimatiques, (torn. ii. p. 235). Had the simple properties of equal roots, which we have established at (101), been known to these writers, the laborious investigations which conducted them to the useless forms adverted to would undoubtedly have been spared ; and a very for- midable obstacle to the success of the methods of Budan and Fourier would have been seen to yield to far more simple and practicable means. In reference to these means we may here observe that there is a peculiarity in the tests (a), (b) at page 185, which ought not to be left unnoticed, as it distinguishes them in a remarkable manner from all other tests for discriminating between equal and nearly equal roots. In these others the difficulty of applying the test, and the ambiguity attending it, if not pushed to its extreme limit, always increase with the proximity of the roots : thus the more nearly two roots ap- proach to equality the more nearly will the final remainder, in the process for the common measure, tend to zero ; and the greater will be the risk of con- founding it with zero — and thus inferring equality of roots — if any abbrevia- tions have been introduced into the work. And in like manner when other tests are applied any miiuite departure from strict accuracy in the numerical results would expose them to the same risk. But in applying the tests here ROOTS IN DOUBTFUL INTERVALS. 205 chapter that these laborious_^ontriiaiijce«-«ft€k^ j and that a few simple considerations, suggested by the general theorem which we have given at (76), and the inferences at (101) will either at once remove the doubt — which is their usual effect — or will so reduce the work for the common measure as to ren- der it comparatively trifling. An important objection to Budan's method is thus provided against. (146.) Considering each of the methods now discussed inde- pendently, and apart from the modifications which we have pro- posed to introduce into that of Fourier, we should, upon the whole, be inclined to give the preference to Budan's mode of examining intervals of doubt, on the score of superior convenience to that of Fourier ; as it is in general easier to employ integral than fractional Hmits. But advantage will often accrue from combining both methods together : or rather the process of Budan will often admit of simplification from the introduction of Fourier's criterion immediately before passing from the direct to its corresponding reciprocal equation. The necessity for this passage will thus sometimes be spared, from a very slight inspec- tion of the coefficients in the last pair of transformations. View- ing, however, the actual determination of the real roots of an equation as the ultimate object which the previous analysis of the equation is to subserve, we shall find the method of Fourier to unite, so much more readily than that of Budan, with Horner's process for developing the roots, as to claim, in general, a decided preference over the latter method, in connexion with the rapid mode of solution just mentioned. If Lagrange's method of approximating to the roots of equations, by means of continued referred to we are free from apprehension of this kind. However closely the roots may lie together, if they are not accurately equal, the tests of equality will in general be just as wide of fulfilment as if no such proximity existed. This is more especially seen to be the case when the leading coefficient of the equation is unity; for then the equal pair of roots, if such exist, must be in- tegral ; and it is plain that, however many decimals in a pair of nearly equal roots may coincide, each of these roots may comparatively be widely different from any whole number. The example at page 187, is an instance of this, the nearly equal roots being -316664 .... and '316665 .... 206 SUDAN ON ROOTS IN DOUBTFUL INTERVALS. fractions, and which Horner's method has superseded, were that which it were still proposed to employ in actual solution, then the analysis of Budan, combining more readily with that method, would have superior claims to adoption to that of Fourier, though, as remarked above, Fourier's test should be kept in view during the operation.* But the student will more fully understand the bearing of these remarks, the propriety of this distinction, and the grounds of our preference, when we come to examine the methods of solution as proposed by Lagrange and Horner. We shall only observe finally that when, as at page 202, it is readily seen that the last of a series of direct transformations will equally lose all its variations whether the transforming factor be unity, as usual, or some convenient decimal, as -5, '05, &c., it will often, as in the example adverted to, save much subsequent calculation to employ the decimal instead of the unit. It is plain, without actually effecting the transformation of (6)' by ('05)', that the results must be all plus : a glance at the last three terms of (6)' is sufficient to convince us of this ; so that all actual opera- tion by decimals is still avoided. For the analysis of the preceding equation by the method of Sturm reference may be made to the author's Mathematical Dissertations, page 195. * The remark, made at the close of last chapter, equally applies here, in reference to Budan's method of analysis : viz., that when the roots are to be computed, and a partial analysis of the equation has left but two roots in doubt —the others being real, the analysis of the doubtful interval need not be pur- sued : the values of the two roots indicated, whether they be real or imaginarj', may be determined from the values of the other roots previously found. The formulas for two roots of an equation, in terms of the remaining roots, will be given in a future chapter. CHAPTER XI. ON THE METHOD OP STURM! WITH A COMPARISON OF IT WITH THE METHODS OF BUDAN AND FOURIER. (147.) We now come to the theorem which Sturm has proposed for the analysis of an equation. It is distinguished from all other methods in three important particulars : — the simple character of its processes — the only operation involved in it being that for the common measure : the unfailing certainty of its results — it being entirely free from all tentative steps: and the basis fur- nished by it for the subsequent development of the roots — ^it being altogether independent of every method for their previous sepa- ration. This last is a striking peculiarity in the method of Sturm ; and gives to it a character to which no other method can lay the remotest claim, since whatever other mode of pro- ceeding we adopt we shall always be kept in suspense, as to the character of the roots indicated in any proposed interval, till those roots are actually separated, or till their separation is shown to be impossible. In the introductory volume on the Analysis of Cubic and Biquadratic Equations we have given a very simple and elemen- tary demonstration of this interesting theorem : the investigation which follows is more closely alUed to that furnished by Sturm himself in his original memoir.* * Memoires present^s par des Savans Etrangers, 942. Consequently the operation is terminated ; and the functions as far as neces- sary are X = a?'^ + Sx'^ +2a^ ^ 3x^ -2x-2 Xj = 5a?4 -f 12a^ + 6a?2 - 6a? - 2 X2 = \^o^ ^- 63a^2^ 22^^+44 X3 = - 301a?2-f- 94a? _ 196 These give, for a?= — 00...— + — — two variations. -|- 00 . . . + -f + — one variation. Hence there is only one real root, which must be positive, since the final sign of X is negative. A superior limit to this root (89) is 2 ; so that no number beyond 2 need be substituted in X in order to determine its situation. Putting for a: in X the result is minus : putting 1 for x the result is still minus : therefore the root must lie between 1 and 2. (154.) The analysis of the preceding equation is very easily accomplished either by the method of Fourier or by that of BuDAN : thus, transforming by (—1), (0), (1), (10), we have + ( — 1) . . . + — — -f — two imaginary roots. (0) ...+ + + -- - (1) . . . + + + + + -^ }on€ real root. (10) ...+ + + -f + +; From this partial analysis of the equation, we see that the WITH THAT OF FOURIER AND BUDAN. 229 only doubtful interval is [0,-1]. The transformations cor- responding to the limits of this interval are (_!)...+ _ _ _ + _ 1 1 O 1 2 (0)... 4-1+3 + 2-3 — 2 — 2 There is no necessity to contract the interval in order to deter- mine the character of the two roots indicated : for we may infer at once that they are imaginary, inasmuch as 7(0) =H= , /i(0) 2 which, alone, being equal to the entire interval [0,-1] we know from the criterion [a] at page 165, that the roots are ima- ginary. And as two other imaginary roots were indicated by the preliminary process above, we conclude that the equation has but the one real root lying, as determined by that process, between 1 and 10. By Bu dan's method we proceed as follows, for the same doubt- ful interval [0, — 1], or rather, changing alternate signs, for the interval [0, 1] : (0).... 1-3 + 2 + 3 — 2 + 2 (I)... .1+2 + + 1+3+3 four var. lost. Reciprocal (0)... .2 — 2 + 3+2 — 3 + 1 (1) .... 2 + 8 + + + + no var, left. Hence there are four imaginary roots. As a second example let us take the equation X = ^ - 10a?3 + 6a? + 1 = /. X, = 5a;4 - 30^2 + 6 and the remainder of Sturm's functions are found as follows : 230 THE METHOD OF STURM COMPARED 5 + — 30 + -f6 *0 + 20 — — 24 — 5 20—0 — 24 — 5 20 + +24+5 X 5 + — 30+0 + 6 24 + 5 — 120 + + 24 96 — 5 — 24 96 + 5 + 24 X 20 — 0-24 — 5 100 500 + 2400 920 46080 962 -221184-46080 174604 + 43680 or -;- 4 43651 + 10920 Hence the series of functions is X = a?s — 10a?3 + 6a? + 1 Xj = 5x* — 30a!2 + 6 X2 = 20a^ — 24^ — 5 X3 = 96a?2 — 5a? - 24 X4 = 43651a; + 10920 /. X5 = + * The advantage of having zero for the second coefficient is readily seen from the first two steps of this example. This advantage was alluded to at page 87. WITH THAT OF FOURIER AND BUDAN. 231 From the signs of the leading terms we infer that all the roots are real (149) : and from the signs of the final terms that two roots are positive, p. 216, and consequently that the remaining three are negative. From Maclaurin's limit we see that the numerical value of each of these must be below 11. To find their exact places we proceed as follows : X Xj Xg X3 X4 X5 x = 0,.. + + + + 1... -1- + + 2... + + + -f 3...- -f + -I- -H -h 4... -f + -f + H- -f X Xj X2 X3 X4 Xg a?=:0... + + + + - 1... + + - + -2... + + - -f -3... + + _ + - + -4...— + _ 4- _ + Whence the places of the five roots are as follow : [0, 1]; [3, 4]; [0, -1]; [0, -1]; [-3, -4]. (155.) The partial analysis of this equation by the theorem of BuDAN and Fourier (107) leaves the interval [0, — 1] doubt- fid, the corresponding transformations being (-1) + 1 -5 (0) . . . . +1 - 10 -1-20 —19 -f 4 012 + -f 6 -f I As -^ -\- -^ is less than the distance 1, between the limits we must contract the interval before we can determine the character of the roots. Previously to this there wiU be no occasion to seek the common measure of /(a:) and/j(a;), which would imply all the work in the foregoing process by the method of Sturm, for we know that the proposed equation cannot have equal fractional roots (101). Nor in narrowing the interval need we proceed en- tirely at random ; for we know, that if the roots in this interval 232 THE METHOD OF STURM COMPARED should happen to be real, they cannot both lie so near to the limit as — ^, nor so near to the other limit — 1, as — -^, else the criterion [a] would be fulfilled, and the roots would be ima- ginary. Consequently, — '2 is a suitable intermediate number, being somewhat greater than — ^. Transforming therefore by this number we have 1 -_o —10 -fO +6 +1 (—-2 — •2 -04 1-992 —-3984 — M2032 -•2 —9-96 1-992 5-6016 —-12032 We need not complete the transformation : the change of sign in the final term shows that —-2 separates the roots. Hence one root lies between —-2 and — 1, and the other between and — •2. Let us now apply the method of Budan to the same doubtful interval : (0) . . . . + 1 - - 10 + +6 +1 (-1) . . . . + 1 _ 5 - -H 20 - 19 + 4 Reciprocal (0) . . . . + 1 + 6 + — 10 — +1 (-1) -4 As a change of sign is produced in the final term, we need not complete this last transformation, but may conclude at once that the roots are real. (156.) We may remark here, that when Sturm's method is applied to an equation of a high degree, or when very large coeffi- cients enter an equation of even a low degree, the operation for the common measure will involve multiphcations by large num- bers, which numbers may be expected to increase very rapidly in the final steps of the work. But, as in order to discover the cha- racter of the roots, the accurate computation of a large array of figures in the several results is not necessary, it wiU in general be WITH THAT OF FOURIER AND SUDAN. 233 sufficient to secure the three or four leading figures merely of the final numerical result : those that follow may be rejected as re- dundant. In order to save the labour of computing these re- dundant figures, we recommend their progressive increase to be checked, as soon as they threaten to become uselessly large, by an easy reduction of the subsequent sets of multipliers. Thus, re- ferring to the general model at p. 225, if the numbers at the close of the second step, for example, are so large as to threaten unne- cessarily long multiplications in the computation of the third step : then in preparing for that step we should employ the mul- tiphers A^^, B^^, C^/, instead of A!', B'', Q>", these being a'" times the former. a"h"' M' =za"'b" ^a"b"' B'' = a" a'" a"b"' And in calculating — ~y only so many decimals are to be pre- served as will be sufficient to secure accuracy, upon the principle of contracted multiplication, in the proposed number of leading figures to be retained in the several products which enter the step. In the following example the last step of the work is cur- tailed in this way : Xj = 6a^ 4- 5a^ — 4a^ - 3x^ ■\-2x-\ K = b"- B„ = a" c„ = a'" 234 THE METHOD OF STURM COMPARED + 5 — 4 _3 +2 + 2 +3 -4 +5 — 6 5 -4 12+18- -3 +2 - 24 + 30 - — 1 -36 17 + 14- - 27 + 32 - -37 17_14 +27 —32+37 X 6+5 — 4 —3 + 2 1 102 172 — 14 +27 -32+37 2754 -3264 + 3774 -1156 -867 +578^289 — 1584 +4104 — 4320 + 252 or, -f-4 X 9 _44 +114 -120 +7 _44 -114 +120 -7 X 17 + 14 —27 +32-37 -1277 22 X 17 44 X 22 145578 - 153240 + 8939 -44880 + 2618 -26136 + 30976-35816 -74562 + 119646 + 26877 -74562-119646 —26877 X — 44+114 -120 + 7 - 21-69 2595 + 583 22 -591 37281 — 4474 + 261 2470 -844 or, -r-2 1235 -422 WITH THAT OF FOURIER AND BUDAN. 235 Consequently, the functions are as follow : X = x^ -\- x^ — x"^ — x^ -\- x^ — a? + l Xi= 6x^ -j- 5^4 _ 4^ _ 3a;2 + 2.r — 1 X2 = 17a?4 + [4x^ - 27a?2 + 32a; - 37 X3 = — 44^ + 114a?2 — 120a? + 7 X4= — 74562a?2 + 119646a? + 26877 X^= 1235a? — 422 .•.x,= - From the variations in the leading signs we infer that all the roots are imaginary (149). It may be proper to observe here, in reference to the foregoing method of abbreviating the work, that should roots approaching very nearly to equality enter the equation — thus causing a ten- dency in the final remainder towards zero — our curtailments, if too freely made, may deprive that remainder of all its significant figures ; and present us only with a row of zeros. The same would happen if the roots were imaginary, provided a minute change in the final term of the equation would render them real and equal. In such a case, the final sign would be doubtful; and could not be safely inferred from the preceding function of the first degree. We might, under such circumstances, disregard this function of the first degree, and ascend to that of the second; consider it the last of the series, and proceed with the analysis as explained in the introductory treatise, Chapter iv, and fully illustrated at pp. 214-226. But the peculiar case here noticed will come under special consideration in next chapter. (157.) Let us apply the method of Fourier to the pre- ceding example :— 1 + 1 - 1 — 1 + 1-14-1 (1 2 + 1 + + 1 + -f 1 3 + 4 + 4 + 5-1-5 4 4- 8 + 12 + 17 5 + 13 + 25 6 + 19 236 THE METHOD OF STURM COMPARED (0) . . . + 1 4- 1 — 1 — 1 -f 1 __ 1 4-1 four variations 112 3 4 (1) ... +1 4-7 4-19 4-25 4-17 +5 4-1 no variation. The index z corresponds to /gCa?) : we shall proceed to examine whether /gC^?) = can have a pair of equal roots in the interval [0, 1]. f{x)— a?6 4- otr"— x^—x^+oip— x-\-\ /i(x) = 6x^ 4- 5x^ -4:^ -3x^ -{-2x -1 -/2(^)=15a?4 4-10a^-6a?2— 3a? 4-1 ^ /3W = 20a;3 + 10^ _4^ -1 f^(x) = 15a;2 4- 5a? —1 2-3-4 1 2'3'4-5 -f,{x) = 6a? 4- 1 2.3.4.5.6/6W-I The equation /2(a?) = cannot have a pair of equal fractional roots, because its leading coefficient has no square factor (101). We may therefore proceed to subdivide the interval [0, 1]; and as our object is to interpose between the foregoing trans- formations, one that will change the first index 1 into o, we shall be guided to a suitable number by inspecting the fourth derived function above, viz., 15a?2 4- 5a? — 1, which we see becomes phis for a? = '3 : hence the transformation ('3) must produce the de- sired change in the leading index. WITH THAT OF FOURIER AND SUDAN. 237 Transforming therefore by (-3) we have I +1 —1 -1 +1 - 1 +1 (-3 •3 -39 —-183 —-3549 -19353 —-241941 -•80647 •758059 8355 r3 -•61 — M83 •6451 •3 1-6 •48 —•13 - 39 — r222 — '3666 •2785 •3 •57 •132 — •3270 1-9 •44 -r090 -•0485 •3 •66 •330 •72292 2^2 MO -^760 ^ • _75 2-5 1-85 2-8 (0)...+ l+l -1 -1 +1 -1 +1 O 1 2 2 (•3) . . . + 1 +2-8 +1-85 - ^760 - ^0485 -^72292 + '758059 1112 (1)... + 1 + 7 + 19 + 25 + 17 +5 + 1 Two imaginary roots are indicated in the interval [0, ^3], for !^ = - is greater than •3. To determine the character of the remaining two roots, indicated in the interval ['3, 1], we ob- serve, first, that f(x) = can have no equal roots in that inter- val (101) J we shall therefore seek to interpose a transformation that wiU change the second of the indices i, i, i, 2, into o. An inspection of the third derived function, viz., 20a^ + \0x^ — 4x — 1, suggests -5 as a suitable number; as the function changes from — to + in the interval ['3, -5]. Transforming therefore by ('o) we have 238 THE METHOD OF STURM COMPARED 1 + 1 — I — 1 +1 — 1 +1 (-5 •5 75 -125 —-5025 -21875 —-390625 1-5 -•25 -1-125 + ^4375 •5 1 •375 -•3750 2-0 •75 — •750 •0625 -5 1-25 2 1 •1250 2-5 •250 •1875 -5 1-5 1^75 3-0 3-5 2 -5 1-75 ' •78125 ^609375 3125 -•75 35 5^25 •5 4 (•5) • • . . + 1 + 0) . . . . + 1 -H Since •609375 + 1 + 4 + 5^25 + 2 + -1875 - -75 + -609375 O 12 + 1 + 7 + 19 -i- 25 + 17 + 5 + 1 - ... is greater than the distance 5 between the limits we infer that these two roots are also imaginary. It remains for us to examine the negative intervals; for this purpose we have, by changing alternate signs, and then changing those of the result, 1-1-1 + 1+1 + 1 + 1 (1 - - 1 1 2 1 1 3 2 3 2 5 2 7 3 4 9 5 WITH THAT OF FOURIER AND BUDAN. 239 (_1).... +i_ 5 4-9-7+3-3 + 3 112 2 2 2 (0) . . . . + 1 + 1 - 1 - 1 + 1 - 1 + 1 In order to reduce the first index to o the factor (— -2) will be suitable + 1 (-•2 246144 1 +1 ■ - 1 - ■ 1 + 1 - 1 + -•2 -•16 •232 •1536 -•23072 •8 -1-16 —•768 M536 -1-23072 — •2 -•12 + •256 •1024 —•25120 •6 -r28 -•512 1^2560 — r48192 — •2 -•08 + •272 480 •4 -1-36 — •240 1-3040 — •2 -•04 •280 •040 •2 -1^40 — •2 — — r40 — •2 ' —•2 (- 1) . . . . + - - + - - + - — + — + 1 2 2 2 + + — + (—2) + - - The character of the roots is still doubtful: the interval however is reduced to [ — •2, — 1]. Trying the intermediate number —G we have 240 THE METHOD OF STURM COMPARED 1 + 1 -•6 — 1 —•24 1 + 1 744 .1536 - 1 +1 (-•e — •69216 r015296 •4 -P24 — •256 1-1536 -•6 •12 •672 — •2496 —•2 — M2 •416 •9040 -•6 •48 •384 — •4800 -•8 -•64 •800 •4240 —•6 •84 -•120 -1^4 •20 •680 —•6 1-20 — 2^0 r40 -•6 -1-69216 2-015296 — •54240 -2^6 (-1).. + 1 -5 -3 2 + 3 2 (-6) + 9 ~ 7 + 3 O 1 2 + 1 -2-6 +1-4 +-68 +-424 -2-23156 +2^015296 424 /2(-i) . M-3. = A . /3(-l) /s(--6) 21 "^ 2^04 = -35 As this result is less than 'A, the distance between the limits, the interval [— ^6, —1] must be rendered still narrower. Trans- forming, therefore, by an additional unit, we have finally (— •7).. + l - 3-2 +2-85— -16 +-3315 —2-30592 +2-242479 O 1 2 2 2 (—•6).. +1 - 2-6 + 1-4 +-68 + -424 -2-23456 +2-015296 •3315 /2(-7) /8(-7) •48 = -69 . . . . This result being greater than -1, the distance between the limits, we conclude that the roots indicated are imaginary. WITH THAT OF FOURIER AND BUDAN. 241 Let us now apply the method of Budan to the intervals [0, 1], [0, -1]. (0) + + + - -h (1)....+ + + -f + + + four variations lost. l-l+l-l-l+l+l(] 1 0-1 1 12 2 11 '3 _f. A'2^/2 + AV + N' = 0, involving the remaining portion of the root; and, as this portion forms one of the entire roots of the transformed, if the first figure of it be found, we shall have the second figure of the original root, and, by a repetition of the process of transformation, we shall get a new equation, involving the following figures of the root. The evolution of any root would, therefore, be effected, by finding the first figure by trial, or by a previous analysis, and diminishing the roots by it ; then finding the first figure of this reduced root from the transformed equation, diminishing the roots by it, and so on till the proposed root be entirely evolved, or determined to any required number of decimals. It is evident that, after the determination of the first figure, and thence of the transformed equation, we shall not be left to con- jecture the value of the following figure ; for, as in the case of cubic and biquadratic equations, so fully developed in the intro- ductory volume, we may regard N', when transposed to the right, as a dividend; and, if the true first figure of the root x' be /, we shall have so to determine r' that, when the dividend is divided by A'^r'"-i -h . . . . AV-'2 + Ay + A', the quotient may be /; and we are evidently assisted in this de- termination of / by A', the known portion of the true divisor. The influence of this trial divisor will indeed be readily foreseen, after what has been done in the work referred to. When, by help of the trial-divisor, the new figure r' of the root is ascertained, and the divisor completed, we may proceed to the next transformation by diminishing the roots of the last transformed equation by /; we shall thus have an equation of the form k"y^ -H . . . . k\x"'^ 4- k".y = N"; 250 SOLUTION OF EQUATIONS the first figure r"^ in the root of which, must be such that, when N" is divided by A"^r""-i + . . . . AV' + ^'2^ ' the quotient must be r"\ and, for discovering r", we have the trial- divisor A"2, which is previously known. It is plain, therefore, that the determination of the several root figures, r, /, r" , &c. in succession, is effected by a continuous and uniform arithmetical process; the several trial-divisors A', k^\, &c., all presenting themselves as they are wanted, in passing from one transformation to another. (164.) These trial-divisors, it must be observed, although f always valuable aids towards suggesting the successive figures of / the root, must not be regarded as unerring guides in this re- spect : the influence of the preceding coefficients, in the trans- I formation which the anticipated figure is to complete, should i always be estimated and allowed for, just as in the common ope- I ration for the square root an estimate of the influence of the 1 suggested figure upon the divisor always operates in determining 1 that figure. { It is of importance to keep this in remembrance ; since the trial-divisor, if left unchecked by the consideration adverted to, may remain widely at fault through several leading steps of the development. It is further worthy of observation, that if two roots do not commence with the same figure, r, then, in approximating to the root whose first figure is i\ the first transformed equation cannot have more than one root, whose leading figure is below r in the numerical scale ; that one root being that to which our approxi- mation is at the outset directed. For if the transformed equation had two roots (A:, A/) below r in the numerical scale, then the ori- ginal equation woidd have had two roots a* = r -j- k^ and .2? = >• -j- A/, commencing with the same figure. Consequently, in passing through the successive transformations, in our pursuit of the single root fixed upon for development, we may be quite sure that no ambiguity can ever arise from two roots of any transformed existing in the numerical scale below that of the root-figure last OF THE HIGHER ORDERS. 251 reached. Similar reasoning shows that when two roots com- mence with the same figures, and not more than two, the trans- formed equations can have no other roots but these — dimi- nished by the common figures already determined — occupying places in the numerical scale below the last of these figures. And so on, whatever number of roots commence with the same figures. In approximating to these nearly equal roots, the theorem of BuDAN will always apprize us when any of them are inadver- tently overstepped; and when not only nearly equal, but also imaginary roots are indicated between the same limits, the crite- rion of Fourier, as also others, to be hereafter investigated, will always enable us to determine whether two real roots have been \ overstepped, or only an indicator of two imaginary roots. And 1 thus in all cases may we carry on the approximation without the I slightest ambiguity or embarrassment. We shall commence with an example or two of no more thaa the ordinary degree of difficulty, merely for the purpose of showing the general method of arranging the operation. This arrangement admits of some Httle variety: we shall give the two forms most to be recommended for compactness and ease of exe- cution. The plan, according to which the entrance of unneces- sary decimals into the several columns of the work is provided against, is that already so fully explained in the introductory trea- tise : the intention of it is to secure the greatest possible accuracy with the least possible expenditure of figures. A different mode of contraction was adopted by Mr. Horner ; in which we be- lieve he has been followed by every expositor of his method ; but we have always considered this mode of abridgment as involving the chief practical defect in the operation ; since in some parts of the process useless figures are retained, and in other parts really effective ones dismissed. The consequence is, that the final deci- mal of the root can never be depended upon, and not unfre- quently error sensibly affects both the last and the last but one. In the method of contraction here to be employed, accuracy is in general secured to the nearest unit in the last figure of the root ; and we therefore consider the change to be a real practical im- provement in the mode of working. As an illustration of the correctness with which all the roots of an equation may be com- 252 SOLUTION OF EQUATIONS puted under the guidance of this principle, reference may be made to the introductory volume, pp. 221, 231. (165.) 1. One root of the equation a^ ■\-Ax^ — 2^ + 10a?2 — 2a? — 962 = is found to lie between 3 and 4, required the development of it to seven or eight places of decimals. 4 -2 10 -2 962 (3-35484874 3 7 21 19 ^7 201 199 597 1 365 3 30 147 642 299-14833 10 49 214 1 841 2 3 39 88 264 1 478 156-1611 59-87805 3 5-97362 13 997-1611 3 48 1 136 42-537 169-4514 1166-6125 4-92583 , 4 16 520-537 1-04779 3 19-3 579 44-301 30-9484 -98760 s 14179 564-838 1197-56019 6019 3 5-88 46-092 z 610-930 31-353 3 1228-914 4940 19-6 147-67 1079 3 5-97 8-037 2-544 988 19'9 153-64 618-9617 1231-4518 91 3 6-06 8-09 2-54 86 20-2 159-70 627-016 1234-00 5 3 2 1-03 160-713 8-2 •50 5 2101-5 635-3 1234-510 1 -6 •5 ll213|5|-0 1611-7 1 613151-9 11613 OF THE HIGHER ORDERS. 253 2. One root of the equation a^ + 6x^— 10^ — 112^2 _ 207a? = 110 is found to lie between 4 and 5 : required the development of it. 1 [ 6 -10 -112 -207 110 (4-46410161 4 10 40 30 120 8 32 -175 -700 1 810 4 56 86 344 352 1408 1 1233 667-05984 14 142-94016 4 72 158 632 1 984 434-6496 1667-6496 133-46395 18 9-47621 4 88 246 102-624 477-4144 9-24089 22 1086-624 2145-0640 -23532 4 10-56 106-912 79-3352 •23158 26-4 256-56 1193-536 2224-39912 374 •4 26-8 10-72 267-28 111-264 2 1304-800 80-389 3 2304-788 232 142 •4 10-88 17-453 5-434 139 27-2 278-16 1322-2513 2310-2212 3 •4 27-6 •4 11-04 2 289-20 1-68 17-56 1339-811 17-6 _3 1357-4 5-44 2315-66 14 2315-8|0 2 1 2:8-10 290-818 1-7 1-2 1 292-16 1 1358-16 1 2|3|l|5-9 121914 1|316|0 In the two preceding examples the steps of the general investi- gation have been rigidly conformed to, and the arrangement which the operation thus takes is that which is in general to be preferred on the ground of simplicity ; it may however be varied 254 SOLUTION OF EQUATIONS. SO as to assume a form of greater compactness, and which may be the more convenient form to give to the work, when, from the magnitude of the coefficients, it would otherwise spread over too large a space. In the chapter on cubic equations in the intro- ductory volume (Chap, iii), we have shown how to reduce cer- tain portions of the work when the leading coefficient is unity : a similar reduction may be applied to the more advanced equations. This reduction chiefly affects the first column of figures. But Mr. Horner so arranged the process, that the subsequent co- lumns up to that which supplies the divisors inclusive, were dimi- nished in length: we shall exhibit the effect of both these princi- ples of abbreviation in the solution of the following equation of the fourth degree, of which the roots have been developed, as in the foregoing example, at page 211 of the introductory treatise, and with which the operation that follows may be compared : — 3. Let the equation be a:4 + 3ir* -h 2x^ -[- 6^ == 148*6, of which one root lies between 2 and 3. 117 131-8 3 2 6 148-6 (2-734400 10 24 60 — . ^ 1 12 30 88-6 14 28 82-9731 18 — 1 82 2 5-6269 44 8-19 36-533 4-8977 1 18-533 •7202 52-19 1, J. l_^ %J%JKJ 6-076 -6626 8-68 9-17 ^^ 161-142 2114 662 7004 •41 163-2516 1 4 70-415 -4 165-38 ^ 3 •28 70-9 -. 1 i^i •■ 1 /*\^ Il6l5|-6|6 OF THE HIGHER ORDERS. 255 In this process the first column on the left is formed, like that j in the operation for cubics ; the multiplier being the index of the | degree, in this case 4. In the second column each addend is I formed from the immediately preceding one, by adding to it the | square of the last root figure. The addends in the third column f are as usual formed by multiplying those of the preceding column I by the last root figure ; and every tiial divisor is the sum of the three numbers above it, as in cubic equations. Precisely in this way may the first and second columns always be formed, whatever be the degree of the equation ; but when there are intermediate columns of work, between the second and that which supplies the trial-divisors, as must always happen when the equation is above the fourth degree, the addends in these must each be formed, in the abridged method, by multiplying the corresponding addend in the preceding column by the last root figure, and, at the same time^ taking in the addend immediately above ; so that every addend in these intermediate columns helps to form the one immediately under it, by being incorporated with the product, which in the unabridged process, is carried from column to column. The following example, taken from Mr. Horner's paper in the Philosophical Transactions, with no further alteration than that which concerns the decimal contrac- tions, the abbreviation of the first column, and the preservation of the trial-divisors, will sufficiently illustrate our meaning : — 256 SOLUTION OF EQUATIONS 4. Required a root of the equation 12-6. 59 . 7-56 66-56 7-92 8-28 8-64 1 91-40 1151-013.. -4509 91-8509 •4518 •4527 -4536 2 93-2090 •1202 93-329|2 •120 •120 •120 150 39-936 189-936 44-688 49-656 1 284-280 2-755527 287-035527 2-769081 2-782662 1_2 292-587270 •746634 293-33390|4 •7475|9 •7485 15 3 294-830|0 56)4 9|4 3 294-886|4 5|6 5|6 2|9|5 201 113-9616 314-9616 26-8128 1 455-7360 8-61106581 464-34706581 8307243 2 473-04120405 2-34667123 475-3878752|8 -0059807 3 477-7405272 -1769318 477-917459|0 34 4 478-094425 1475 478-0959|00 s 478-0974 2 4|7|8-|0|9|7|6 207. 12 207 ( -638605803327 18-897696 18-02304 13-9304119743 4-0926280257 3-8031030022 •2895250235 •2867504754 27745481 23904795 3 3840686 3824781 15905 14343 1562 1434 128 96 32 33 Hence one root of the equation is -638605803327. The only objection that could be brought against this mode of arranging the numerical process, is, that in the third column of the work, such arrangement requires us to perform the operations j)f laiultiplication and addition simultaneously. But, in the case of a biquadratic equation, no such objection can apply, and, con- OF THE HIGHER ORDERS. 2^7 sequently, the foregoing arrangement of the work may perhaps be preferred on the score of practical facihty. (166.) We have observed above, that in the foregoing exam- ple, from Mr. Horner's paper, we have slightly modified the process, and it ought to be mentioned that, in so doing, we have, in fact, increased the length of the operation. This has arisen from our having actuall^ ^xhibited the trial-divisor derivable at everj^^teg from jthe last root-figure." ^'^^^^^^^^^^^'^^^^^^ ^'^ ^By dispensing, howeverpCTOhis, and merely writing under the true divisor the addend which is due to the formation of the next trial-divisor, we may, without actually performing the addi- tion, readily foresee what the leading figures in that divisor would be, and thence discover the new figure of the root. In the fore- going example, no inconvenience can arise from this suppression of the trial-divisors, even from the commencement of the ope- ration, on account of the smallness of the addends in the divisor column. But, where the addends are of considerable influence, we think it preferable always to exhibit the trial-divisors. The work of the last example stands in Mr. Horner's paper as be- low ; and it is easy to see, from the circumstances just adverted to, that after the first step, every true divisor may be safely taken as a trial-divisor for the next figure of the root, for the addends recede sufficiently far to the right to allow the leading figures of the divisors to continue constant. After a step or two, such will indeed usually be the case when but a single root lies in the interval under examination ; but not when two or more roots so lie, nor even when indicators of imaginary roots occupy the same interval, and occur in the divisor column, because this column will then tend to zero as well as the final column. In the case of nearly equal roots this is plain, since when roots accurately equal enter, diminishing by one of those roots reduces as many of the ad- vanced columns accurately to zero as there are repetitions of the root, page 140. And in the case of imaginary roots the state- ment is authorized by (122). Moreover, in aU cases of difficulty, where an appeal to the theorems of Budan and Fourier be- comes necessary, the entire row of transformed coefficients should always be exhibited. Nevertheless, the compact form given to the 17 258 SOLUTION OF EQUATIONS work in the following arrangement, may recommend it to adoption in cases where the length of the operation is the only difficulty. eo CO © 00 o 00 CO o <^ O C?i o o o t^ O Oi O 00 o eo 1>» r-l «0 Tf (N i« r^ ^ «0 CO O Tf iO (M CO »o 00 05 00 00 o ^ o t>» (M W Tf W ^o ^>. Oi CO o en ^5 ^5 o -^ O '^ O TT lO Tf o ^ 00 CO lO o tT O Tf d r—l f— 1 Tt r-i (M O C<1 «^ l><. Oi 00 00 O rf ^ ^ irj !>. !>. CO COC^ CO O (M CO Oi «o 0^ (N s X V CO Tf ^O CO I Oi «o 1 CO CO I CO CO \ ^ V ^- «o l-H CO Oi P-H CO o I-^ > § . 00 OD ' CO Tf CO O (M lO f* CO CO CO CO C-l CO 00 rr «o CO t^ »^ ' CO 05 00 *■>. CO O r-H CM CO O urs irj iO to tN 00 Tf Tt rf i^ 05 i^CO \ CO CO CO o> 05 OF THE HIGHER ORDERS. 259 (167.) In each of the preceding examples the root developed is the only one lying between the same two consecutive numbers in the arithmetical scale, that is, no second root exists having the same leading figures. When such is the case we shall usually find, as here, that the trial-divisors become efficient in suggesting the successive figures of the root from a very early stage of the approximation. But, as already observed, this will not happen under different circumstances. For let a be an approximate value of a root oif{x) = 0, that is, a number consisting of one or more of the leading figures of that root ; and let the remaining portion of the root be h. Then /(« + K) =f(a) +f,{a)h + ^ A2 + ^ ^ + &e'. =0....[l] • • Ma) - + 2fM + 2-3/,(«) + '"=-/,(«) Now as, by the increase of «, h diminishes and becomes less than unit, the terms involving h?, h^, &c. diminish also ; and if the coefficients of these also diminish with h, or even slightly increase, these terms must at length diminish more rapidly than h itself, and become with respect to it very small. Conse- N' quently the quotient will then give one or more of the /i(«) leading figures of h : that is, the trial-divisor fi{a) of N' will be fully efficient in determining the next figure of the root after a. But if/i(«) also regularly and rapidly diminish with A, whilst /2(a) does not, then it is plain that the second term ^^ k^, 2/j(«) of the preceding development, as well as the first, must form an N' important part of the whole ; so that ^ . , will no longer give /i(«) the leading figure of h but the leading figure of h -\- ^ - h^. What we have now described must evidently happen if the equa- tion /^ (a?) = has a root with several leading figures the same as those of the root a -\- h off{x) = ; or if this latter equation have two nearly equal roots in the interval under examination (98). 2G0 SOLUTION OF KCiUATIONS If there be three such roots the trial-divisor will be still more at fault : for then /i(«) will diminish as I?, and /^(a) will diminish as A : so that the third term as well as the second being thus of the same order of magnitude as the first, must be retained with it, as an approximation to the whole series. Thus, when nearly equal roots enter the equation, the difficulty of separating them will be very considerable if we depend entirely upon the trial-divisors for the discovery of the successive figures of the root. An example of the total inadequacy of these may be seen at page 227 of the introductory treatise. \1^8.) T[n order to remedy this defect let us take the limiting equation of [1] : then we shall have /,W +/2W^ +-^^2 + &c. = . . . . [2] This would be accurately true in conjunction with [1] provided two roots, A, of [1] were accurately equal (97) ; and it must approach nearer to the truth as the roots approach nearer to equality. The leading figure of the root h of this latter equation is given by — yV^* agreeably to the general principle. If three roots approach to equality, or continue the same for several decimals, then, continuing the derivation, we have for approximating to h the equation ./•2W +./;,(«)A + &c. = 0.. . .[3] and consequently for the leading figure of the repeated root ^^T^. And so on. The transformed coefficients - N', A', A'.,, A',, A'„ A'„. furnished by diminishing the roots by a, in Horner's process, are (116) no) f(.) m. f^ fM fnM J{^h fMh 2 ' 2.3' 2.3.4' • • • • 2.3.4... OF THE HIGHER ORDERS. 201 Hence, according as a single root, two roots, three roots, &c. lie in the same narrow interval should we employ, to obtain the leading figure of the root of the transformed equation, one or other of the expressions F A^ _ _A^ _ _A^ _ AT^ A" ~ ^A^' 3aV 4A'/ • ' * • nA'^' These conclusions supply a useful theorem for approximating to roots very nearly equal : cases of this kind are reduced by it to the same simplicity as the common case in which only a single root lies in the interval within which our search is directed. (169.) If we write the equations [1], [2], [3], &c, in the usual notation, the preceding conclusions may perhaps present them- selves with greater clearness : for we thus have /(x) = A>"+ A>4 + A>3 + A>2 + A'^-N'=0 ... [1] /i(^) = wA'„a?"-i+ 4A'4^ + 3A>2 + 2A>+A'=0 ... [2] /2(a;) = yi(?i-l)A>"-2 + ...3.4A>2 + 2.3A> + 2A'2=0 ... [3] f^(x) = w(w— l)(w— 2)A>«-3 + ...2.3.4A> + 2.3A'3=0 . . .[4] &c. &c. &c. where for two, three, four, &c. nearly equal roots the correspond- ing approximate equations are f,(x) = 0, /2W = 0, f,(x) = 0, &c. and the leading figure, common to these roots, is by the general principle, A' A' A' ^ _ _2_ 2A. &c 2A'2' 3AV 4A'/ It is plain that the process for computing the function /(a:) for any value of x supplies, in its progress, the computations of the subordinate functions fi(x), /2(a?), /^(x)^ &c. for the same value ; so that if there are m nearly equal roots, we approximate to the single root of the equation /^^^.j (a?) = having the same leading figures in common with them, in the act of approximating to the proposed root in/(^) = 0. We may consider this subject under a difierent aspect : — We know that when a single root only Hes in a given interval, we 262 SOLUTION OF EQUATIONS speedily reach a transformed equation, which, by neglecting all the terms except the last iwo,f^{a)x — N', furnishes a simple equa- tion /i{a)x — N' := 0, from which a correct figure or two of the sought root may be determined. This simple equation becomes more and more eflfective as the approximation advances ; till at length it, alone, is found sufficient to supply as many correct figures as are necessary to complete the stipulated amount of decimals in the root, and thus to close the operation. It is to this simple equation that the successive transformations tend in approximating to a single isolated root. But when instead of a distinct individual root we approach simultaneously to a pair of roots, then of course the process in like manner tends towards a quadratic equation, the leading co- efficient of which becomes less and less influenced by the addends to it accruing from the preceding columns of the work. Both roots of this quadratic, commencing, by hypothesis, with the same figure, this common figure will be found by dividing the second coefficient, with changed sign, by twice the first (see In- troductory Treatise, p. 143). When three roots are thus simul- taneously approximated to, the operation tends to merge into a cubic equation involving those roots diminished by the preceding root figures : and as by hypothesis these three roots commence with the same figure, it is plain that one third of their sum will commence with that figure. Hence, the common figure will be obtained by dividing the second coefficient of this approximate cubic equation, taken with changed sign, by three times the first, or simply minus the third by the second : and so on. And these are the conclusions established above. We thus see that the last two coefficients of the equation of the n — (m — l)th degree which furnishes an approximation to a sin- gle one of the contiguous roots, are the first two in the equation of the mth degree which is the approximate equation involving all these m roots. It is interesting to notice that in the case of the approximate quadratic the leading figure common to both roots will otherwise be obtained by dividing twice its final term (N') by the preceding coefficient A', as is obvious. In the case of the approximate cubic the leading figure common to the three roots will be found by dividing three times N' by the preceding coeffi- cient A' : and so on. OF THE HIGHER ORDERS. 263 (170.) We at once see, from these principles, how tedious it would be, when roots proceed together for several places of figures, to reach the place at which they separate by seeking each new figure through a set of trial-transformations com- mencing with — , or which is the same thing, with , as in the method of Fourier, and proceeding onwards from this, through all the intermediate transformations, till we arrive at twice, three times, four times, &c. this quantity, according to the number bf roots in the interval. By prehminary transformations Fourier finally renders the interval so narrow as to comprehend but two roots : if others are close to these their prehminary se- paration is very laborious : but even when the interval is made to comprehend only two roots proceeding together for five or six places of decimals, the separation of them, by commencing every new set of transformations with a trial-number only half the value it ought to be, thus rendering on the average three or four trial transformations necessary for every figure of the root, we say that even in this simplest case of contiguous roots the separation is so laborious as to justify the observations upon its impracticability which we ventured to make at (131). (171.) From what has now been shown, it appears that when roots have leading figures in common, the proper expressions for suggesting the true figures up to the place at which the roots se- parate are as foUow, the denominators of these expressions being the trial-divisors in reference to the numerators : TWO ROOTS THREE ROOTS FOUR ROOTS &C. A Af) A Ao„ .^, These, therefore, are to be employed in the cases under consi- deration, till a root becomes detached, when the ordinary trial- divisor may be brought into operation to carry forward the approximation of the single root thus separated. We shall generally be apprised of our arrival at this point by a discrepance between the figure given by the suitable expression above, and that given by the corresponding expression in the series 2N' 3N' 4N' „ 264 SOLUTION OF EQUATIONS But by continuing to pursue the root of /^,_j(a?) = 0, whose figures are given one after another by the proper expression in the former series, as long as these figures are the same as those of the roots to be separated, we shall of course always of neces- sity effect the separation desired, and be informed of it by the loss of variation in the series of transformed coefficients, agreeably to the theorem of Bud an. It is true that this theorem, as well as the discrepance adverted to above, may indicate the separation of a group of roots : but as we should then treat each group by itself, we may here suppose a single root to detach itself — the least of the group. Now when the separation is accomplished it will be advisable to pursue this smallest of the nearly equal roots, to at least one figure beyond that at which the separation takes place ; and to employ the resulting transformation for the nearly equal roots that still remain, remembering that these will present themselves each diminished by the figure by which the preceding approxi- mation was extended. For of the roots of this transformed equation, one will commence with zero, in consequence of the advanced step recommended, and the others will commence with a common figure, supposing the common figures of these to be not yet exhausted. Hence, as our approximate equation involving the roots in question, and of which the first co- efficient is Ajj^, and the second A^_j, has the sum of its m A roots expressed by — ~- ; and since the leading figure of one of m these roots is 0, and that of the others the same figure, it follows A' that '"~^^ ■ -, or this expression minus 1, will furnish that (m-l)A'^ figure. And thus the common leading figure of the m— 1 roots still remaining unseparated, may be easily discovered, and then the next root separated by the process above described, the trial-divisors, after the transformation by this leading figure, being advanced to the column next adjacent on the right, to that which supplied the divisors at first. But if after the separation of the first root, those that remain do not, as supposed, commence with the same figure, then the figure, determined conformably to that sup- position, will either be the first of the root next in order, or will effect a new separation : in the former case, the figure increased OF THE HIGHER ORDERS. 265 by unit will effect the separation. And in this uniform manner are we to proceed till all the roots are separated and severally developed as far as necessary. All that it is requisite to attend to in seeking the separation of roots one after another, which have several leading figures in com- mon, may be summarily stated in the following precepts: — (172.) 1. Find the leading figure common to all the nearly equal roots by a previous analysis of the equation, and effect with it the usual transformation. 2. Find the next figure common to the roots by the proper expression (c) ; the first being used if there be but two roots, the second if three, and so on. The figure thus determined will be i the same as that suggested by the corresponding expression | (d), as soon as (c) becomes effective for the root oif^_-^(x) = 0. 3. Continue to determine the successive common figures in this way, either till a change of sign in N' informs us that a root has separated — ^which will of course be the least root, supposing all to have been rendered positive, or till a discrepance between the figures determined by (c) and (d) announces a separation. If several roots separate at once, we must deal in this way with each distinct group, till the least in that group detaches itself. 4. By aid of the ordinary trial-divisor, extend the approxi- mation towards this least root one figure further, and employ the resulting transformed equation for the sepai*ation of another root. 5. To find the leading figure common to the remaining roots take the multiple of the divisor in (c), hitherto employed, a unit less ; and employ the expression, thus modified, to furnish the first transformation ; and then proceed anew as above, using, for future figures of the roots still unseparated, the expression iri (c) immediately preceding the one before employed, till an- other root separates : this root increased by the figures of the former root, will exhibit a second root of the proposed equation. The supposed common leading figure will, in all cases, either be the true figure of the next root — or it will effect a separation — or it will exceed the first figure of the remotest root by 1 . Or, instead of being guided by the last two precepts, we may proceed as follows. Having separated, as above directed, the 266 SOLUTION OF EQUATIONS least root of the group in the interval within which our approxi- mations are confined, we may take the transformed equation to which this separation is due ; and remembering that our ope- rations are now to be confined within the narrower interval com- prising only the remaining roots of the group, after these have been diminished by the figures resulting from the former set of transformations, we are to direct our approximation towards the root of /m-2W = ^» lyii^g ill tlie interval, just as at first it was directed towards the root of/j„_j(a?) = 0. The proper expression suggestive of the root-figures is that one of (c) which precedes that before employed. This expression will continue to suggest the correct figures of the root of /m-2(^) = 0, so long as these are the same as the figures in the roots still unseparated. A dis- crepance here will, as before, indicate that the figures have ceased to concur ; and the true root-figure will effect a second separation. And so on till the analysis is completed. (173.) We have only one more particular to notice in reference to the general theory of the trial-divisors : — It is, that in the earlier steps of our approximation to a root it will sometimes happen that the absolute number, which we seek to exhaust, will, instead of continually diminishing as true figures of the root become determined, increase and diminish alternately till a point is reached where all oscillations of this kind cease and the abso- lute number tends progressively to zero. A reference to the values assumed by N' in the leading steps of the approximation at pages 181, 218 of the treatise on Cubic and Biquadratic Equations will exemplify these circumstances. The cause of this pecuHarity may be easily discovered from an examination of the general expression for N' at page 259: for from that expression we have N' = /,(«){A + ^A^ + &c.} which will uniformly diminish with A provided neither f^{a) nor y!j(«) tends to zero — disregarding for the present the functions OF THE HIGHER ORDERS. 267 included under the &c. The consequences of /^(a) tending to zero with y(a) we have already adverted to: but if /^(a) alone tend to zero, then, as h diminishes, the coefficient of h^ also diminishes; and if this coefficient pass through zero, it must afterwards increase, with changed sign. It is obvious that, at an early stage of the approximation, be- fore h has become very small, this change in the term referred to, from additive to subtractive, or vice versa, may have sufficient influence upon N' to produce the effect mentioned. If several such changes occur during the process — that* is, if several roots of f2(a;) = are passed over in our approximation to the single root of/(a?)=0, then several of these oscillations of N' may take place. It is plain that similar reasoning will apply to the more advanced terms of the above general expression for N'; so that the pecu- liarities noticed, arise from derived functions vanishing, and reap- pearing with changed signs, during the development of a root of the primitive function. These fluctuations, however, become in- sensible when h has reached a certain small value. We shall now proceed to illustrate what has been established in these latter articles by two examples of peculiar difficulty. We shall prepare for the development of the roots by an actual ana- lysis of the equation by the method of Sturm, which we shall at first present without any abbreviation.* * These equations were publicly proposed for solution by Mr. Lock hart, a gentleman to whom we have been under obligations upon former occasions ; and who has furnished many other examples ^mirably adagted to Jj;^ the powers of Jhe modern methods of solution. The laborious computations in- volvedin the analysis and solution of the equations in the text were executed by two very promising young mathematicians— members of the senior mathe- matical class in Belfast College — Mr. William Finlay of Belfast, and Mr. Smylie Robson of Tobermore, in the county of Derry. The calcula- tions of Mr. Finlay are, for distinction, marked [F]; those of Mr. Robson with [R]. 268 SOLUTION OF EQUATIONS (174.) Required the analysis and solution of the equation y^ 4- 173^ _(_ 2356ar^ + 10468^^2 _ 14101a? + 4183 = 0. The function X, is 5a?4 + 692ar^ + 7068a?2 -|- 20936a; — 14101 I 5 173 + 692 + 7068 + 20936 — 14101 — 4712 — 31404 + 56404 — 20915 119716 + 1222764 + 3621928 — 2439473 -23560 — 157020 + 282020 - 104575 [F.] 96156 + 1065744 + 3903948 — 2544048 or, by dividing all the terms by the common factor 4x9, we have 2671 + 29604 + 108443 — 70668 2671 — 29604 — 108443 + 70668 X 5 + 692 + 7068 + 20936 - 14101 1700312 13355 7134241 1359477325 + 34080693500 - 19557716075 dividing all the terms by the common factor 5^, we have 54379093 + 1363227740 — 782308643 54379093 — 1363227740 + 782308643 X 2671 + 29604 + 108443 - 70668 — 50336036448 — 184386934216 + 120157648416 - 1448256265 + 943771140 50424815388 + 149362469576 — 100599932341 — 2031342624368 145246557403 2957085755502649 2769182614982857568320—1589136891937388812624 113627637222362534129 320675250583973765507— 208971336169861 199532 -3203485502789193867956+1798108228107250012156 or,- dividing by the common factor 1227089452, we have -2610637307303+1465344050653 - 26 1 0637307303 - 1 465344050653 X 54379093+1363227740-782308643 —3638577276801810382949 6815427150282258457133809 533 1 767565402726835634 1 207835 15697 — 533 1 7675654026706805756237882 11187 —561 550584969953045 1 OF THE HIGHER ORDERS. 2Gl) Hence, Sturm's functions are, X = a^ + 173^^4 + 2356^^3 _|_ i0468a?2 — 14101^ + 4183 Xi == 5x^ + 692^3 _^ 7068^2 _,_ 20936^^ — 14101 Xg = 2671^^ + 29604^2 _^ 1084430? — 70668 X.^ = 54379093^2 _^ 1363227740^ — 782308643 X^= — 2610637307303.T + 1465344050653 X. = - 56155058496995304510 As the signs of the leading terms of these functions present one variation, we infer that the equation has a pair of imaginary, a nd consequently three real, roots : and as the signs of the final terms present two variations more than those of the leading terms it follows, page 216, that two of these roots are positive, and consequently one negative. The situations of these are found as follows : X Xi X2 X3 X4 X5 Fora? = - 200 . . . — + — + + — /our vm'iations. - 100 . , . . 4- ■ — — + + — three „ . . . + — — — + — 1 . . . + + + + — — one ,, In developing the negative root we shall convert it into posi- tive by changing the alternate signs of the equation. 270 SOLUTION OF EQUATIONS Development of the root [150, 160.] 1 —173 100 -"73 100 "27 100 127 100 227 100 327 50 377 50 427 50 477 50 527 50 577 8 585 8 593 8 601 8 609 8 2356 -7300 -4944 2700 -2244 12700 10456 22700 33156 18850 52006 21350 73356 23850 97206 26350 123556 4680 128236 4744 132980 4808 137788 4872 142660 309 142969 31 143218 1 3 - 10468 —494400 —504868 —224400 — 729268 1045600 316332 2600300 2916632 3667800 6584432 4860300 11444732 1025888 12470620 1063840 13534460 1102304 14636764 71484 147082418 7164 14779819 718 -14101 4183 1 158-5612971 —50486800 -5050090100 —50500901 — 72926800 -123427701 145831600 22403899 329221600 351625499 99764960 451390459 108275680 559666139 7354124 56702026|3 738994 57441020 89162 5753018|2 8921 148517 86 5761939 149 576208|8 15 6 576224 3 14860|3 57|6|2|2|7 9 5050094283 1120194950 3929899333 3611123672 318775661 283510132 35265529 34518109 747420 576209 171211 115245 55966 51861 4105 4034 71 58 13 [F.] ^^- .W |4|8|6|9 .^ 6|1|7 ll4!3!6 OF THE HIGHJ:R ORDERS. 271 Development of the two roots [0,1]. 173 173-5 ^ 174-0 'b_ 174-5 ;5_ 175-0 _ii 175-5 •06 175-56 6^ 175-62 6^ 175-68 6_ 175-74 6 A3 2356 86-75 2442-75 87-00 2529-75 87-25 2617-00 87-50 ^ 2704-50 10-5336 2715-0336 10-5372 2725-5708 10-5408 2736-1116 10-5444 2746-6560 ^ •17580 A2 10468 1221-375 11689-375 1264-875 12954-250 - 1308-500 14262-750 ^ . 162-902016 14425-652016 163-534248 14589-186264 164-166696 14753-352960^ 2-7468318 14756-09979118 2-747008 14758-84680|0 2-74718 3 A 14101 5844-6875 -8256-3125 6477-1250 - N -4183 —4128' •5612971 15625 —54 — 54 1779-1875 865-53912096 -913-64837904 875-35117584 ^ -38-29720"320 14-756099792 — 23-541 10340|8 14-75884680 ^ —8-78225661 2-95242869 -5-8298279|2 2-9525386 —2-8772893 * 1-3287140 2746-8318|0 14761-59398 _i-548575|3 1758 54948 175-80 2747-007|6 -001 -176 1|7I5|-8|G|1 2747-18|4 •18 2747-36 ^ 4 2|7|4|7|-4|0 14762-1434|6 -5495 14762-693|0 -549 14763-242 -247 14763-48|9 •24 14763^7|3 •2 1-328736 -•219839 -103348 — -1164911 -10335 ^ 84375 8189027424 ^ -248472576 ' -2354110341 -130615419 — 116596558 -14018861 - 13937178 —81683 -81544 — 139 -117 —22' [F.] 147|6|4|-0 a •h ■J :tJ 7|6 -1314 1033 -28|1 103 -139 1-0000007 -197 58 'l-io^ 7^ 1/] \ - *"! %r k^iMr^^ V, 2/2 SOLUTION OF EaUATlONS The trial- divisors whence the several figures of the root in the preceding development are obtained, are all supplied by the column A2; the column A supplying the corresponding divi- .^ A' ^ a»^ dend. Thus : — The general form for the root figure is 2A'2' I agreeably to the precept at p. 263 ; so that when the transforma- 177 .. . ^*^ tion 1 is obtained the fraction supplies the cor- 2 X 14 . . . ^^ responding root-figure 6. In like manner, from the transforma- 000 tion 2 , we get for the next figure ' ' = 1 ; and so on, the trial-divisor being constantly 29. The place at which the roots separate is determined as soon as there is a discrepance A' 2N' between -r^ and — — - as stated at(171). 2A'2 A This discrepance in the above operation does not appear till the transformation 6 is reached ; when we have 1314 ^ ^2 X 139 = 4, and = 2. tw ^ 2 X 147... 131 Hence the roots separate after the 7. Carrying on the approxi- mation to the least root one figure beyond this, by aid of the ordinary trial-divisor — 1314, we find 1 for the seventh decimal. For 2 the absolute number in 6 becomes plus, and continues so for all numbers up to 7 inclusive ; but for 8 it is again minus. Hence 7 is the seventh decimal of the other root. But if, agree- ably to the precept 5 at p. 265, we had taken the transformation 7 and had divided — A' by A'g, that is, 102 by 14|7, we should have got 6 ; which increased by the advanced figure 1 of last root gives 7 for the corresponding figure of the second root. Thus the roots are •5612971 . . ., and -5612977 . . . (175.) In the following solution of the foregoing equation the analysis is conducted difierently in the final step, and the subse- quent developments are according to the more compact arrange- ment exhibited at (165). x^ -f 173.t4 4- 2356ar^ -|- 10468a'2 — 14l01.r -f 4183 = 5a?4 -f 692a^^ -\- 7068ar2 -|- 20936a? — 14101 OF THE HIGHER ORDERS. 273 P5 I I CO o o ^ + + 00 Tf to o O Tf t^ .—I +7 O? CM Oi —I +1 O CO CO lO CO o T ' 00 o Ci o r-H (M (N 00 + + + 1 to O .— I to !>. O Oi CO '^ I 00 o to I 00 Tf Oi CO o ^^ Oi CO CO + g o 00 X) to ^ to ^ CO 00 o I- + CO 2 + Tf CO Oi o + O Oi to o Oi I to ,— ( Tf S CO rf^ CO ?o Oi l^ Oi o Oi to o (M ^1 1—1 r— * + 1 O o ?o Tf t^ (M ^ to '^ l-H Oi CO l^ o Oi t^ Tf O CO (M 00 Tt ?o CO Oi CO s +g 1-H 1 + 00 to 00 ^ O 00 1 •Tf* (M CO ^ o to CO to '—i o (N 00 yD 00 Tf CO Tf (N CO Tf Tf o r-H O to ± •, to 2 I 00 UD o o 00 t^ ^ 1 . Oi (N ?M CO + ?o CO 1 1 CO 1—1 Oi l:^ o^l oi (mI w CO ^ to (N to ^ 1—1 to '^ CO CO C. Tf T ' + + O Oi l^ O Oi (M ilM O to 00 CO >— < lO Oi Oi 00 Tf lO t^ -« ^ to CO «o o -^ to to to !>. 00 .1. ^ W (M CO CO r 00 to o t^ Oi a ^ 00 CO Oi r-H O (M (M . ?o Tt* l^ O (M to r^ CO to o t^ O ?0 (M to CO (M !>. to to l-H 00 (M »^ 00 t^ ^ to 50 Tt< 00 Oi CO o CO <0 r-^ (N o ^5 l^ ^ CO (N 00 C^ CO rt O rf W C^ (M to O «o to to 1 1 lO O 3! ^ o o Tf CO t>. Oi OO lo CO -H CO 00 ^ CO 1 1 rr r-^ . !>. t>. CO 00 ^ p-^ r-H 276 SOLUTION OF EQUATIONS 377 585 If 1 be taken as the next figure of the root there will be no change of sign in the last column ; but if 2 be taken there will be a change of sign. If 7 be taken as the next figure, there will be a change of sign, but if 8 be taken, there will be no change of sign. We infer, therefore, that the first seven places of figures of the two positive roots are '5612971 and -5612977. It is plain that the results which complete each step in the third column when multiplied by 2 become the divisors, and the bracketed numbers in the next column the dividends for suggest- ing, as before, the successive figures of the root. In the operation below, the negative root is, for the sake of convenience, converted into a positive one, by changing the alter- nate signs of the equation. Development of the root in the interval [150, 160]. •173 2356 —10468 -73 —7300 —494400 —4944^, 2700 I , 12700 (* 22700J 18850 520061 21350 23850 26350 4680 128236^ 4744 I 4808 f* 4872j 617 309 1429691 309 I 309 f* 3091 37 1439331 37 37 r* 37j 144045 —5048681 —224400 ^^ 1045600-^ 2600300 29166321 3667800^2 4860300-' 1025888 124706201 1063840^3 1102304>' 71484 147082481 71639 >^ 71794-^ 8636 148603171 8638 > 5 8640^' 144 14877739>| 144 ;>6 144J 29 —14101 —50486800 — 5050090l\i — 72926800J 145831600 224038991. 329221600/^ 99764960 4513904591 3 108275680J 7354124 5670202631 7389944 J* 891619 5753018261 - 892137J 1487 8 5762088411 5 14878J 2976 576226695 4183(158-5612971 -5050090100 ^ 5050094283 1120194950 3929899333 3611123672 318775661 283510132 35265529 34518110 747419 576209 171210 115245 [R.] 55965 51860 4105 4034 71 14878056 OF THE HIGHER ORDERS. 277 rS I •§ -^ 3 ^ > ^ ^ ^ CO ^ ^ 1 Sg^l § o S + " oil K I .s ^ s " §> fe s " s J s .s ■- 11 lit. r.tiiiif •^ s s I '-3 S cff g f, ? I I ' ^ § S .s ^ ^COCC:§^c«SS>-S o^ CO CO OD .^cog O (N s ^ t ^W?^M^^ 1 + ^ ^% iO 3. CO + CO *?. 00 (N 00 l^ + 00 1. 10 CO (M 2 CO d"^oS^3 •^+co^cOiOOi^ S^co'S'pH^^^gio ^ >f X ^" ^^ ^-^ ><- -s 02 o . CO -^ --H (M Oi Oi W » O O rr CO cq P-^ CO C. (M Oi CO Tt 00 .-H lO to iO r-^ O 00 »o CI Tf o c<» Oi r-H ^ (N <=> O TP TT C<1 t^ Tf 'l ll (M LC C-< Tf r-( 50 CO r-< Ci CO t^ .—I TP t\ N TJH O Tt* O rP O O Tt o o o . CO -H (M O 00 r^ Oi o O -H (M 00 O 00 Oi o^ OS 00 00 (N (M lO o to »o Oi Tf CO Oi CO Oi (N O CO ^ rf (M W ^ C<> to 1^ t^ Tt (N (N O CO Tf 00 to TT O to (N tN. (N OS (M r^ ?0 00 o Tf <;o I- lO TP o o rr CO o:> 4^ «3 CO O Oi OS CO (M CO to CO Ol Ci O — (N r^ Tf CO Tj' 00 to Tj* to Oi O O o o '■ M 1 1— t r—i (N (M (M 04 (M C (M 00 o «o o^ CO o (N to CO T 00 o rp O ^ CO (M O ;o iNi (N O O 05 to ^ «3 O CO Tt* (^ O —H to l>. CO . + + ■> ^ 00 »n i^ TT CO Oi X X o (M «o F-H 00 1— 1 o 00 o iO r-H r-H + + Tf o lO o o ^o (N 00 t^ ?o lO o 1 7 (N 1 O l-H O . (M + C<1 00 ^ t^ o CO 1— t «o t^ l>» t^ + 1 to 00 Tf w Oi CO + 1 2 + - I 00 CO lO Oi + (M 00 + Oi to 1-H <0 ^ o o o ^ >-H 00 Oi Oi Tf » . O 00 ,_; r-H (N to (M o 1-H o o l>* I— 1 00 o 00 «o o lO •^ ■^ r>. . CO CO . (M (M t^ t>. 1 o 1 Oi + 4- 1 1 <£> «o 00 Tf Oi 00 W CO CO t^ o o CO 00 00 CO o Tt* 00 -T* CO CO o o 00 o (M CM t^ OJ 00 CO t^ + 1 + 9 OJ to 00 I— 1 i-H o w CO (M (M CO CO tT 00 (^ o t^ ?o o f-H «o «o + l>» 00 CO ^O CO CO tN» to ^5 o o -^ + Tf o (N » t^ to Oi O 00 «o <£> o 00 *^ o o «o to l-H o 1— « (M '^ Oi 00 . tN. 00 1 1 00 CM "oT Oi Oi (N o o Oi «o » CO Oi s CO CO .s OF THE HIGHER ORDERS. 289 Ol a> 00 1—4 CO QO CO 00 00 00 (M '^ o CO + o o CO 00 "^ as CO 00 CO )^ + I + 00 00 o 00 «o o to CO en (M ^ o Ci 00 00 23 ^ 00 O^ Tf iO gj ci 05 O 00 lO Tt* »0 Ci O ?o as 00 ^ (M l—H CO lO CO »o «o lO O (M IC trs (N cq CO 00 Tf CO 00 t^ "^f . CO t^ CO I. o CO ~6o~ »o t^ 00 CO o t^ lO o o t^ 00 (M o (M o o (N Tf ^ iO o ■^ «£5 o ^5 ^5 iO 00 Oi o 00 «o CO to 00 I— I o ^o as CO o 00 '^ o CO 00 CO CO 00 -^ «o to 00 OS CO t^ o (£> o^ C7S (N (M to t^ (N t^ to t^ CN (M as 00 T}^ CO 00 CO CO •-< 00 00 00 (M M* M* I CO -^ o 00 (N (M o o + 00 00 o 00 to to (M o •5f * * * * ■3f 00 t^ 00 00 o «o CO to to CO o ^ r— ( o 00 to c<» CTS o CO CO CO o o t^ w !>. t^ w t^ o (^ 00 00 Tf "^ CO CO 7 + ^ •?f ■X- * •x- * ■3f ^ ^ (£> CO ,—1 »>. < + + + + + + + -\- I I I 1 I I + ><^++ + + + -f + 4- >? \ \ \ M + + I + + -f + I + + + 4- + I + + + X! + I I + + I I + <» ^ '^ .S OOOOOOCOTf o <^ t^ '^ "^i —' * I I i I I X XI XI XI ><| X X OF THE HIGHER ORDERS. 291 In the following operations, the negative roots are, for the sake of convenience, converted into positive ones by changing the alternate signs of the function. Development of the root [190, 200.] to «o CO QO O CO '-*. (N O X o (M O C^ 1^ lO ^ Oi O CO to 00 t^ — • 00 iO o TT O lO (M CO Oi CO o (M O C-1 00 CO Tf OS -H CO O ^ C^ lO CO ^ 00 00 c^ to l^ 00 Oi 00 l->. . 00 o t^ «o o ?o TT Tt o O 40 Tf W O ^ O (M r^ to r-* to to ^ o «0 r^ ;o t^ 00 w r-H W CO 50 J:^ Tf C. Tf (M (M t^ t^ o (N Tf t^ o !>. -^ (M C^ C<1 f-H O to 00 tP Tf 00 to CO to r-t o o o 00 Tj< to r-H to 00 O to o to r-H CO CO o to r^ to r-H r-H r-1 |>» O 00 w 00 Tt* o 1 c^^ O O 00 (N to o C. (M to to Ci to t^ r-1 iM ^ 00 1^ C<1 00 Tf o Oi Tf O 00 CO to c» to 1 ^ 1 Tf O O Tt^ t^ 00 00 (N ^ to to t^ CO to ^ to l^ O r^ .. CO to CO 00 to CO 00 ^ ^ CO --H 1 O O to to 1 O to o^ 00 o t^ r^ 1 r-H CO O^ t^ O Tf to 1 . to CO Tt* I I 00 O O O . to Tf CO I to Oi Tt r-* Tf to to (M 00 to -rf (N (M to Tfi O >— I Tf (M Tt* Oi l-^ to CO O t^ O CO t^ to CO "^ rf O ^ r-H r-H CO to O C. t>. to Tf Tf Tj* ■^ l-H CO ^ -H r-H to CO Oi o 00 o r-H 00 00 t>. CO c^ 1 OiO o o o o 00 O O O O 00 CO 00 00 01 C. l>. (M t^ .-H to o Tf O Tf 00 ^ Oi W lO « Tt* C. 00 Oi »o CO C^ 1 1 t^ i^ «o -^ -^ ^ c<» CO (M 1 1 1 1 o O CO l^ CTi wo 1 1 1 1 -^ Tf O CO (M 1 1 1 1 1 1 40 TT 7"i 1 1 1 o o c^ o t^ 50 CO Tf .— * ;o (N O 1 00 1 o "^ o o o 1 O Tf 1 o o TT 00 1 00 Oi CO CO (M w o Oi rr «o CO ^ I 7 o o o> Tf lO 00 t^ o «o to O O CO 1 O CM 1 CO 'O (M Oi o Cl o CO to CTi o 1 to Ol to to ^ as Oi ^ ^ to TP 1 ^ crs 00 iO w O 1 O r-H 00 1 ^ r-H O Tf Tf Tf 1^ Oi to CO TT 00 «0 00 o to O r-H Oi i— ' to 00 o CM CO to 00 O 00 o wo ^ Tf 1— 1 00 to 00 o 1 rf O to o to (M » to o to 1 Tf 1 to to o o o Tf O O 00 to 00 00 to O to to CO 00 to to r-, O ^ to o TT Oi -H CO to r-H W C. Oi to — < o l^ — 4 r-H Tf to CO 00 CO OJ (M — H CO CO ^ Oi TP Tf i->. Tf l^ t>» o 00 00 o to o CO Oi (M 00 Oi CO TT O I -^ — I ^ 10 to r-H r-l rH to 'Tf Tf oq O to to l>. to CO Oi I I 00 O O O (M (N Tj^ Tf Tt O (M t^ Oi rl Oi O t^ '^ 00 Oi to Oi Tf O CO O ^ to CO l>. CM (N CO to Tf to to (M CO Oi 00 00 O Oi -rt* (M 00 l-^ CO O 1-H (M Tf r-H Tf to 00 O TT 00 1^ t^ 00 OS I— < Oi Tf O to to TT O to (M l^ Tt" to to CO O r-H ,-H ^ 00 CO '^ '^ '^ (N Oi Oi CO CO CO 00 to 00 00 00 CO to CO (M Oi 00 -H 00 00 w CO (M 1 Oi o o o o o 00 O O O O C. CO to to O -^ l^ rH CO ^ OI (N . CO OF THE HIGHER ORDERS. 293 Development of the root [10, 20]. —378 38189 —492368 —572554 —213720 26352(16-3666002 -368 —3680 345090 —1472780 —20453340 —206670600 34509 —147278 —3580 309290 —3480 274490 —3380 240690 —3280 113502 ■312 —1872 790694 18917 102486 —1836 91686 —1800 81102 —1764 3511 —1728 1069479 -2812 —85 3486 11704 3461 —85 3435 —84 681 —84 1080542 —84 680 -2180 —17 679 11350 678 —17 68 —17 1082647 —17 68 —16 m — 1 Q7 —2045334 1620120 4365020 4744164 8683970 5359080 5909196 320844 20273090 321890 322928 64832 20982740 64873 64914 6496 21119023 6496 6496 650 211132665 -20667060 —4252140 52103820 27184620 84258300 6081927 117524847 6178494 125896 4 124962305 1262857 12671 4 126351876 126753 1268 126491309 12680 4 1216503993 206696952 163107720 43589232 35257454 8331778 7497738 834040 758111 75929 75895 34 25 [R.] 111282 1081285 In explanation of the process on the next page it is necessary to remark that the three roots do not separate till after the fourth figure. The equation whose roots are (x — -3666) is that below the first series of double lines, from inspecting which it appears that any significant figure in the fifth or sixth place, or 3 in the seventh place would produce a change of sign in the last column showing that a root had been passed over. The least root is, therefore, -3666002. When 6 is put in the fifth place there is a change of sign in the last column, indicating the passage over the root already determined. Had 7 been put in the fifth place, three variations would have been lost, showing that the three roots had been passed over. Hence we infer that 6 is the fifth figure of both of the roots still sought, and the equation whose roots are (a?— -36666) is that below the second series of double lines. As any significant figure, either in- the sixth or seventh place, would cause the loss of more variations, it appears that the middle root is •3666600, and as 2 in the sixth place, or 1 in the sixth place and 1 in the seventh place, would make all the signs of the trans- formed positive, we have for the greatest root -3666610. 294 SOLUTION OF EQUATIONS 378-3 379-86 38189 113-49 38302-49 113-58 113-67 113-76 113-85 22-7916 38780-1416 22-7952 22-7988 22-8024 22-2806 380-166 2-280996 38873-624996 2-281032 2-281068 2-281104 2-28114 3801966 228118 38882-977458 228118 228119 228119 228119 38883-889933 22812 38883-912745 22812 22812 22812 22812 38884-003993 38 492368 11490-747 503858-747 11524-821 11558-922 1159305 2326-808496 540862-348496 2328-176208 2329-544136 2330-91228 233-241749976 548084-222869976 233-255436168 233-269122576 233-2828092 23-329786475 548807-360024395 23-329923346 23-330060217 23-330197088 548877-350205046 2-333034765 548879-683239811 2-333036134 2-333037502 2-333038870 548886-682352317 38884 548886-721236 Development of the three —572554 151157-6241 •421396-3759 154615-0704 158082-747 32451-74090976 -76246-81759024 32591-43148224 32731-2041304 3288-505337219856 — 7635-676640380144 3289-904869836864 3291-30448457232 329-284416014637 —725-182869956323 329-298413968644 329-312412004775 — 66-572043982904 32-932780994389 -33-639262988515 32-932920976557 32-933060958807 32-226718946848 5488867212 32-775605668 38884-00437 OF THE HIGHER ORDERS. 295 roots in the interval [0, 1]. 213720 -126418-91277 26352 26190-326169 -378 (0-3666 002 00 10 87301-08723 -80034-39165 —4574-8090554144 2691-8865245856 -2619-32316648 -45-814059842280864 26-749298263319136 —26-07463062325968 -435109721973794 239557918085662 —237530673592607 2027244493055 —2018355779311 8888713744 —42380520717 —33491806973 327756057 — 7162013 161-673831 161-513191475136 160639524864 160495789579914816 143735284085184 143734750851397 533233787 533322825 —89038 -716201 627163 [,R.] We need not subjoin to the preceding solution anything by way of comment or explanation ; the remarks which accompany the former solution at page 281, equally apply to this. The same precepts guide us to the successive common figures of the roots, and to the places at which they separate and become dis- tinct, whichever mode of arranging the work be adopted. In the method above, however, these precepts cannot be applied with the same facihty as in the former mode of arrangement ; but something, as regards facility of operation, must be expected to be sacrificed where brevity and compactness are the principal objects sought ; and these are very completely attained in the process just exhibited. 296 SOLUTION OF EQUATIONS (179.) The foregoing examples sufficiently confirm all that has been stated in the preceding parts of the present work as to the entire efficacy of the methods of Sturm and Horner in the analysis and solution of numerical equations, since they have been so framed as to put the competency of these methods to the severest test. The principal part of the labour involved in the solutions just given is that expended upon the preparatory ana- lysis : this, on account of the large coefficients which enter the equations, is very considerable, more especially in the second ex- ample, even when the entrance of redundant figures is provided against, as at page 289. It is easy to see that this labour is en- tirely attributable to the magnitude of the coefficients, conjointly with the number of steps in the process, and not at all to the proximity of the roots to each other : and we can thus at once infer what are the circumstances which in general set bounds to the practicability of Sturm's method. In the subsequent deve- lopment of the real roots, these circumstances affect the length and difficulty of the operation in but a comparatively small de- gree : the magnitude of the coefficients interfering but little with the progress of the work, after the first figure or two of the root has been correctly determined, and we are sure that our pursuit of the subsequent figures is in the proper direction. (180.) The difficulty that has hitherto retarded the process — affecting every step of it with uncertainty till verified by trial- operations — is that which has always been found attendant upon the separation of roots having several leading figures in common. In such cases the work of the true development, as exhibited to the eye, could convey but a very inadequate notion of the time, and cautious deliberation, expended upon each step of the work, up to the figure at which the nearly equal roots are found to se- parate. By means of the principles established at (1/1) this difficulty is now entirely removed, and roots having a great num- ber of leading figures in common may now be developed and separated with even more ease and expedition than so many roots already isolated. So long as the figures of the nearly equal roots actually keep together, the foregoing examples sufficiently illus- trate the ease and rapidity with which their development may be OF THE HIGHER ORDERS. 297 carried on : when their point of separation is reached we shall find indications of the circumstance spontaneously to present themselves, either at the step itself, or at that immediately be- yond : for if not at the place where the figures cease to be alike,' yet at the place immediately beyond it, we shall find a contra- diction between two expressions (c), (d), page 263, which on the principle of the non-separation of the roots must agree. The comparison of these tests, as we proceed from step to step, involves no trouble : it is made at a simple inspection. But we need not make the comparison at all, or may disregard it till it spontaneously offers itself to our notice. "We may continue to develope the root of the function /m-i(^) = 0, which lies be- tween the nearly equal roots (see page 261) till the changes of sign in the transformed coefiicients inform us that the separation is accomplished. It is true that the intervening root of/^_^(x) =0 spoken of, may itself concur with the figures of one of the sought roots, between which it lies, beyond the place at which these latter roots themselves separate ; but it must detach itself from both at last ; and thus make known, by the changes of sign re- ferred to, that the roots sought cannot concur beyond the place thus reached, although they may separate before reaching that place : yet actually up to it will the figures of one of the sought roots be correctly exhibited. To find the exact place at which they separate — ^whether we direct our attention to these latter indications of the separation, or to the former — we must start anew with the transformed coefficients with which we are fur- nished by the step immediately beyond that at which the se- paration was discovered, as explained at (171), and proceed to develop the remaining roots in the interval as there directed : the first significant figure, with which this new course of operations commences, points out distinctly the place where the preceding root has separated from the others. Generally speaking, the figure at which the root ofyjj^_j(a?)=0 separates the roots sought, will be that at which the roots them- selves separate, as in the preceding examples ; for it is improbable that a number, restricted only by the condition of lying between two others, should not become distinct from both at the point where they themselves separate from each other : but whether or 298 SOLUTION OF EQUATIONS not such be the case, the directions already given will infallibly make known whereabouts the separation must take place ; and it is impossible that we can be carried more than a figure or two beyond it without the discrepancy adverted to above forcing itself upon our attention. (181.) When, as in the improbable case just adverted to, the intervening root of the derived function continues to coincide with one of the two roots of the primitive between which it lies, after those roots themselves have separated, we should also become acquainted with the circumstance by increasing the last found figure of this intervening root by unity ; for we should thus obtain a new transformation, making known, by its loss of variations, that a root or roots had been passed over in the inter- val between it and the former transformation. We say roots, be- cause it is possible, where several lie close together, that the intervening root of which we speak may separate more than one on each side. Hence, if we continue our development till the discrepancy already mentioned presents itself, and then increase the last figure of the development by unity — should the roots remain unseparated — we shall assuredly separate them if real, or else discover that they are imaginary. (182.) The more nearly the doubtful roots in any interval approach to equality ; or to speak more accurately, the more . minute the change in the constitution of an equation which suffices to make these roots equal, the longer, of course, will the determination of their true character be delayed. Such must be expected to be the case, whatever tests be employed for distin- guishing imaginary roots from real. But, of all methods for effecting this object, that must be pronounced the best which, when the doubtful roots prove to be real, accomplishes the separation of them with most ease and expedition. For the tests in question become effective only when that point of the develop- ment is reached, the next step of which, to be correct, must involve an imaginary expression. Till this point is attained, the approximation is perfectly accurate, whether the roots ap- proached to, eventually prove to be real, or imaginary : since that OF THE HIGHER ORDERS. 299 must be regarded as a true approximation to a value of oc, in the equation f{x) = 0, which commences by furnishing for f{x) a value f{x) = a, and thence a continuous series of values, in which a, by losing its leading figures one after another, tends con- tinuously to zero. This is a sure criterion of a true approxima- tion ; and consequently, as far as it extends, we should seek in vain for any peculiarity among its steps that would enable us to predict whether this tendency of a towards zerocould continue indefinitely, by the continual accession of real increments to x ; or whether, beyond a certain point, the leading figures of a can be destroyed only by making the increment imaginary (p. 163.) It is at this point only that peculiarity can be first introduced. The operation by which this point is reached — supposing such to exist — is precisely that which separates the roots — supposing them to be real : so that, as stated above, the method which is competent to effect the separation of nearly equal roots with the greatest ease, is that which will also conduct us the most readily to the peculiarity alluded to when the roots prove to be imaginary. (183.) This peculiarity may manifest itself in various ways : — by satisfying the criterion of Fourier [a] at page 165 ; by ful- filling the conditions of Budan (144) ; by forbidding a tendency towards zero in the absolute number beyond a certain limit, as noticed above and explained at p. 163 ; or lastly, by causing the discrepancy, between the expressions (c), (d) at page 263, so much dwelt upon in the present chapter. All these may be regarded as concurrent — or, as very nearly concurrent — indica- tions of the same thing. Fourier's criterion keeps us in sus- pense during the steps of the approximation till a certain point is reached. This point is that at which the roots separate, if they are real : — it is that at which the approximation becomes defective, if they are imaginary. When they prove to be real, the operation, which has just discovered their character, has also supplied us with their approximate values ; when they prove to be imaginary, the same operation has furnished a real value which more nearly satisfies the equation than any other. The same observations apply to the transformations of Budan. These 300 SOLUTION OF EQUATIONS equally furnish the figures of the real roots, up to the point of separation, and those of the imaginary roots up to the point where the imaginary part becomes essential to the further con- tinuance of the approximation towards zero. (184.) It would seem then to be of but comparatively little moment what test we employ to distinguish nearly equal roots from imaginary ones. What is of real importance is that we take the safest and shortest path towards the point which must be reached before such test can be effective ; and that our arrival at this point be intimated to us by unequivocal marks or indications, so that we may be spared the labour of a continual appeal to the test at every step we make. These two important objects have, we conceive, been accom- plished in the present chapter : a rapid and certain means of separating the roots has been theoretically established, and prac- tically illustrated in examples well calculated to try its efficiency : and means, as ready and certain, for determining the point where I the indicator of imaginary roots Hes, have also been added. The I process of approximation is but an extension of Mr. Horner's Vown method to the case of nearly equal roots : — a case but in- adequately provided for by the general operation as applied to roots lying singly in distinct intervals. The additional precepts that have been furnished, for distinguishing between these nearly equal roots and those that are imaginary— as delivered at pages 299 and 300 , are, we believe, altogether new : and from the ease with which they may be applied," as the approximation proceeds, without requiring any actual calculation, they appear to be well adapted to general use, and to possess obvious advantages over the methods hitherto employed. (185.) In the researches of Fourier, so fully discussed in Chapters viii and ix, we have seen that the object first to be accomplished, in the analysis of a doubtful interval, is to reduce the last three of the indices to the values o, i, 2. Till we get three such consecutive indices, either for the three final functions, or for some preceding three, Fourier's peculiar method of pro- ceeding does not come into operation. In bringing about this OF THE HIGHER ORDERS. 301 arrangement of indices, the proposed interval must be narrowed, with a view to the detection of the situations of those roots of the derived equations which require to be passed over, or separated, before the desired arrangement can be presented. In the search after these extraneous roots, in the proposed interval, we are not suppHed by Fourier, with any guide ; but are left principally to mere random conjecture, as to the situations where they are most Hkely to be found ; so that this part of the analysis is purely tentative. When the desired arrangement of three consecutive indices is obtained, the trial-transformations, in the prosecution of the analysis, are limited within a certain definite range of values, fixed by the criterion [a] at page 165: and in our practical applications of Fourier's method, our substitu- tions have always been controlled by this formula; although Fourier himself never makes any distinct reference to it for this purpose. Now in carrying out Fourier's plan — in con- nexion with Horner's method of effecting the several transfor- mations, and of arranging the successive steps in one continuous process — we shall not be left without the necessary guidance to those roots of the derived equations, which occur in the same interval as those of the primitive, and which we wish to exclude from that interval. For the arrangement adverted to presents us with trial-divisors, suggestive of the figures which enter one after another into the development of the root to be excluded. These trial-divisors, it is true, may at first suggest figures wide of the truth : but the modifications which they undergo, in passing into the character of true divisors, are always to be more or less accu- rately anticipated from an inspection of the antecedent tributary columns of the work ; which inspection the arrangement pecuHar to the method greatly facilitates. The correct figures of the root, to which our approximation is more immediately directed, may thus be rapidly, and accurately, determined ; so that by increasing either of these figures by unity we can inclose the root spoken of within two consecutive numbers whenever we please, without expending any useless trial-transformations. And thus the desired arrangement in the advanced indices will be facilitated. When this arrangement is brought about, we have seen, by the foregoing precepts and 302 SOLUTION OF EaUATIONS examples, how we may pursue the analysis without any of that ambiguity attending our steps, which is unavoidable when those steps may range anywhere within the limits fixed by Fourier's criterion. The separation of the roots in the second of the pre- ceding examples, with nothing but this criterion to guide us, would be nearly impracticable. (186.) It is thus by combining the accurate and comprehen- sive theoretical views of Fourier with Horner's practical development, extended, as above, to the hitherto perplexing case of nearly equal roots, that the problem of the solution of nume- rical equations of the more advanced degrees may be reduced to a practicable form. The principles delivered in the present chapter, taken in con- junction with what is estabhshed at (101) respecting the criteria of equal roots, and combined with the theorem of Budan (107), contain all the essential elements of this solution. (187.) There is no absolute necessity for Fourier's formula [a], in order to complete the analysis which the theorem of Budan but partially accomplishes. The precepts (172) and the directions at pages 298-9 will, in general, effect the pbjects of that formula more readily, and with quite as much certainty. We should not in general, therefore, recommend any appeal to this test : unless indeed immediately before the initial step in the approximation for separating the roots suspected to be nearly equal — after the leading figure common to both has been found — or, which is the same thing, after the consecutive numbers between which they lie have been determined — by the theorem of Budan. But if in any case a further reference to Fourier's test be thought advisable, the foregoing directions will guide us to the exact spot where that reference should be made. (188.) We shall now show how the analyses of the foregoing examples are to be effected by aid only of the principles and pre- cepts here referred to. The first of those examples is a^ -f- 173a^ + 23r>6ar^ -|- 104683?'-^ — 14101a? -f 4183 = 0. OF THE HIGHER ORDERS. 303 And by (89) a superior limit to the positive roots is 3 : hence, in seeking the situations of these, we have merely to contract the interval [0, 3] : it is plain, from the variations in the signs of the terms, that there can be but two such roots. It is further ob- vious, without any calculation, that the transformation (1) will have all its terms positive. Hence, (0) . . +1 + 173 + 2356 + 10468—14101-1-4183 two variations. O 1 2 (1) . . + + + + + + no variation. Consequently two roots are indicated in the interval [0, 1]. In order to examine the negative intervals it will be convenient to change the alternate signs, or, which is the same thing, to substitute — x for x, and then change all the signs, writing the original polynomial thus —/(—a?) =^5— 173^4 ^ 2356^3 _ i0468a?2— 14101^ — 4183. As the superior Hmit here is so great as 1 74, it would be im- prudent to advance in our transformations merely by units. Advancing at first by 10, we see, without any calculation, that the transformation (10) must render the third, and all the fol- lowing coefficients, negative ; so that two roots will be indicated between and 10 ; or, in the proposed equation, between and — 10. Subdividing now this interval, in order to find between what consecutive numbers the roots are to be sought, we have the following results : (0) (1) (2) (3) (4) (5) (6) 1_ 173 + 2356- 10468- 14101 1_168+1674- 1_163+1012- 1_158+ 370 + 1_153_ 252 + 1 4428-28656- 404-33157- 1664-31576- 1836-27765- 4183 three var, 26568 58145 90856 120555 148- o 8.54 + 1 172-25456-146888 Z 2 1_143_1436_ 3268 — 28261 — 173173 one var. 304 SOLUTION OF EQUATIONS Hence, two roots are indicated between 5 and 6 ; or, in the original equation, between — 5 and — 6. As only one other root remains, the situation of it may be found from the proposed polynomial alone : we thus discover it to lie between — 100 and -200. Two doubtful intervals therefore remain to be examined ; viz., the intervals [0, 1] and [ — 5, —6]; which latter, in reference to the transformation (5), is also the interval [0, 1]. If the roots of the equation yj( — x) = ; in this last interval, are real, they will either commence with the same leading decimal, or will separate at this decimal. They cannot commence with the same 172 figure, because of the palpable discrepancy between - — — - and 25456 : and, by actually performing the first step of the approxi- mation to the root of /gC — ^) = ^i ^^ ^^^ ^^^^ the roots of /,(— a?) =0 do not separate. We conclude therefore that they are imaginary. The same conclusion would have immediately resulted from the criterion of Fourier ; which gives /i(5) _ 2546 ^ f,{5) 2x172 ^ Examining now the the interval [0, 1], in the original equation, . . , , . , . 14101 we find no inconsistency between the trial-expressions - — ■ 2 X 4183 and — . It is true that '6 is suggested by the former, and only *5 by the latter; but, upon trial, we find '6 to exceed the root of /, (x) = 0, to which our approximation is directed : there- fore '5 is the true figure. Setting out with this, we proceed, step by step, guided by the expressions (c), (d) at page 263, as at page 272 ; where we see that the roots separate at the seventh decimal. (189.) In the second example a?H378a^+38 1 89ar4+492368^''-5 72554a?2+2 1 3720a?- 26352 = 3 is a superior limit to the positive roots (89). But it is easily OF THE HIGHER ORDERS. 305 seen, from an inspection of the coefficients, that the transforma- tion (1) will render them all positive ; so that three variations will disappear in passing over the interval [0, 1]. We have, therefore, (0) . . . . + 1 +378 + 38189 + 492368-572554 +213720-26352 O 1 2 3 (1)....+ + + + + + + Now as there is a perfect consistency among the expressions 572554 213720 3x26352 3x492368* 572554' 213720 ' as far as the initial-figure in each quotient is concerned, we take this initial-figure, -3, for the first figure of the indicated roots ; and proceed, guided in like manner, to each successive figure, as at page 280 ; when we find the roots to separate as there shown. As the remaining roots are wide apart, their situations will be made known by Bu dan's transformations, without the aid of any additional principle. It may be proper to remark here, in reference to the doubtful interval [5, 6], in the former example, that if the roots of /j ( — ^) = had been real, we should have continued to develop the root of /gC"" ^) = ^> ^^ i^ ^^^ example, extending the work up to yi(— x), till the roots of /^( — x) = 0, or of /( — x) = 0, had separated. And thus should we proceed in all cases where several final indices are ahke, remembering to use for — N' the proper expression (169) due to the earhest of these indices. (190.) In order to illustrate still further the practical efficacy of the theory established in the present chapter we shall apply it to the equation I2a^ — 120ii?2 + 326^ — 127 = which has already been analysed, by help of the criterion of Fourier, at page 244. 20 306 SOLUTION OF EQUATIONS The doubtful interval is found, as at page 243, to be [4, 5] ; that is, the first figure of the root of /((a;) = 0, to which our ap- proximation is to be directed, is 4 ; or taking the equation already obtained from transforming by this 4 ; agreeably to the partial analysis at page 243, the first figure will lie between and 1 . The true first figure is to be found, as usual, by aid of the trial- divisor, in this case 2 x 24, conformably to the precepts at (172): the dividend — A' being 58. Regard being paid, however, to the influence of the preceding coefficient 1 2, it is easily seen that the figure sought is '7 12-1-24 — 58 — 25 (7 8-4 22-68 - 24-724 32-4 — 35-32 --276 8-4 28-56 40-8 -^6-76 8-4 49-2 It is obvious that from this step the trial-divisors become fully A' efiective j that is, •— — — - will henceforth always furnish the 2A^ correct root-figure. If the doubtful roots of f(x) = 0, in the interval under examination, are real, they have '7 for a common leading figure : they have also the number suggested by the second trial-divisor, that is, by 2 x 49-2, and which is the num- ber 6, for the common second figure, provided this same 6 is 2N' furnished by — y ; otherwise the roots, if real, must separate after A. the first figure (172). The latter alternative must have place, since the number given by the second expression is not 6 but 8. Transforming by -06, the roots do not separate : — no changes are lost. Hence, if they are real they must of necessity separate for the transformation -07 (181) : but this causes two variations to disappear. Consequently the roots are imaginai-y. OF THE HIGHER ORDERS. 307 Fourier's test applied to the transformation ('76) authorizes this conclusion immediately, as at page 244, and thus saves the additional transformation. The discrepancy, however, which it required this transformation to interpret, pointed out the exact place where Fourier's test ought to be introduced. But, in strictness, we should not regard the entire transformation, but only that portion of it which determines the absolute number, as expended in testing the character of the roots : we refer to the transformation ('07), which is deduced from the transformation ('06), by transforming by an additional unit. For where actual solution is the object, the process of develop- ment ought not to terminate as soon as the roots are ascertained to be imaginary. We ought, on the contrary, to continue this development till the absolute number becomes stationary in its leading figures, and converges towards zero only as respects its remote decimals : after which, it should be matter of delibera- tion whether we are to reject the root as imaginary, or to retain it as real, rejecting only the imaginary increment (p. 163). It is a very prevalent, but a very grave mistake, to admit only the ap- proximate values of the real roots of an equation, and to reject indiscriminately all those that are imaginary. It is an important, though we believe hitherto an unnoticed fact, that in the case of imaginary roots the real development, carried on up to the point adverted to above, is of far more practical consequence than would be the complete determination of the imaginary roots themselves ; since from these roots the real development could not be inferred. In other words, a more effective and available solution would be furnished to an equation by substituting the real development here mentioned for the imaginary roots, than by actually exhibiting those roots themselves in their complete form. This, doubtless, seems paradoxical ; but the truth of the statement will appear presently. (191.) We shall now return to our equation and pursue the development of the root of / (x) = till N' ceases to diminish in its leading decimals, and thus becomes convergent, not towards zero, but towards a finite constant. 308 SOLUTION OF EQUATIONS + 24 -58 -25 (-: 8-4 22-68 -24-724 i— M« 1 32-4 — 35^32 -•276 8-4 28-56 1 -•225888 2 40-8 —6-76 -•050112 8-4 . 1 2-9952 -•002564044 49-2 -3-7648 —•047547956 •72 3-0384 2 -•000000151 49-92 --7264 -•047547805 •72 •360108 -•366292 • 50-64 •72 2 •360696 9 51-36 -•005596 84 -002581 51-444 - -00301 15 84 26 4 51-528 --0004 84 3 (-76705 5|1-6|12 It is not necessary to extend the operation further : the abso- lute number, as far as the first four or five figures of it, will obviously remain constant, however far the root of /j (a?) = be carried. The necessity for an imaginary increment clearly dis- covers itself after the transformation 2. It may be remarked here, that after having obtained the trans- formation 2 above, wc might, agreeably to the directions before OF THE HIGHER ORDERS. 309 given, increase the root-figure '06, which led to that trans- formation, by unity, in order to decide whether the roots se- parate at the figure '7, or are imaginary, and might perform the following bye-operation : 51-36 —7264 - -050112 (-01 -5136 -002128 - -2128 — -047984 The final sign being minus shows that the roots do not separate, and that they are therefore imaginary. (192.) The development just exhibited when the 4, by which the roots were at first diminished, is introduced, is 4-76705. It would obviously be a real root of the proposed equation if the remainder — -047 . . . above were actually zero ; that is, if the original absolute number had been 127-047 . . . instead of 127. In other words the value 4" 76705 is accurately a root of the equation 12^:3 _ i20a?2 -f- 326^ — 127-047 . . . = . . . [1] as far as five places of decimals. And from the final result of the middle column of the work above, it is clearly also a root of the limiting equation 36x^ — 2A0x + 326 = ... [2] to the same extent of decimals. Thus the equation [1] has a pair of roots equal to 4-76705. And this same pair must be re- garded as approximate roots of the proposed equation 12a^— 120^:2 + 326^—127 =0, provided the left-hand member of [1] be regarded as an approxi- mation to the left-hand member of this last equation ; or pro- vided the nature of the inquiry be such that the 127 may be taken for 127*047 . • • without transgressing the limits of error which that inquiry may be permitted to involve, without appreciable in- jury to its conclusions. 310 SOLUTION OF EQUATIONS We know that in almost every physical inquiry the numhers furnished by observation, and of which the coefficients of the equations employed in that inquiry are functions, are more or less affected with error : and thus, on the principle of disregarding imaginary roots, certain real solutions of importance might be re- jected as imaginary, and on the other hand imaginary roots might be replaced by real. It would doubtless be the safer, as well as the more consistent plan to retain, as real approximate solutions, all those developments which, like that above, really indicate ima- ginary roots, but yet accurately solve equations which are close approximations to those under discussion. If we were actually to determine the complete imaginary roots of the equation here treated, we should find, as stated above, that they would afford us no clue to those approximate real values which, for a proposed extent of decimals, satisfy the equation more nearly than any other real values of like extent : for these real values are not, in general, the real parts of the imaginary roots. In the example before us, the real root of the equation, computed to seven places of decimals, is -4656774,* so that twice * The actual development of this root is as follows : 12 — 120 4-8 + 326 —46-08 279-92 —44-16 127 (•46567742 111-988 —115-2 4-8 15-032 13 768032 —110-4 4-8 235-76 -6-2928 1 J -263968 •113504 — 105-6 -72 229-4672 -6-2496 •150464 -133273 —104-88 •72 223-2176 — •5169 17191 15544 —104-16 •72 222-700|7 —•517 1647 1554 — 103-44 6 222^184 -62 03 89 — 103-3|H 6 222-12|2 6 4 .l|0|3|-3 2|2|2|-0|tt OF THE HIGHER ORDERS. 311 the real part of the imaginary pair is "4656774 : that is, the imaginary roots are 4-76716 . . . ± /3 V — 1 . And thus the imaginary increment a ± /3 \/ — 1 (foot-note, page 163), re- quired to complete the approximate root 4*76705, is -00011 . . . di ^ V — 1 ; but we have nothing to guide us to the real part of this imaginary increment, whether the imaginary part be determined or not : so that it is really true that the develop- ment 4-76705 furnishes a more accurate and effective real solution to the equation proposed than the complete imaginaiy roots them- selves could supply. (193.) It might not be amiss to call such developments imper- fect roots of the equation. As remarked at page 162 (foot-note), these developments are something more than mere indicators of imaginary roots : they would be perfect real roots of the equation were a shght correction introduced into the absolute term : and whether in reference to this correction, or in reference to the imaginary increment omitted, the designation of imperfect roots seems sufficiently expressive of the peculiarities by which they are distinguished : and thus, real numerical results, available in actual practice, become redeemed from the neglected mass of imaginary quantities. With respect to the second class of imaginary roots, com- mented upon at page 162, and which are merely indicated by the developments of roots of certain limiting equations inferior in degree to/j(a7) = 0, they are not to be replaced by real imperfect roots, like those of the first class : the imperfect roots which sup- ply these latter indications replace imaginary roots in the inferior limiting equations, and not those in the primitive equation. We shall merely remark further, that after it was ascertained, as above, that the roots under examination were imaginary, we might have developed the corresponding pair of imperfect roots to any extent of decimals without regarding the final column of the work, which is useless for that purpose. But, by completing this column, simultaneously with the others, we ascertain the amount of correction for N that would render the imperfect root perfect, I 312 SOLUTION OF EQUATIONS which correction it is necessary to know. We might thus express the general solution of the preceding equation as follows: One root, '4656774 . . . . : One pair of imperfect roots, 4*76705 : Correction of N = -047. (194.) From what has now been delivered it appears that we have at least three efficient and easily apphed methods for deter- mining the true character of doubtful roots. Each is indivi- dually fully competent to remove the doubt : but they are so related to one another that, as already observed at (183), each becomes conclusive at nearly the same stage of the analysis ; at which point either may be indifferently applied ; or, if need be, they may be concurrently employed as mutually confirmatory of one another. The chief matter of importance is not which criterion shall be used, but how we may approach, with most certainty and ease, the point where either becomes effective. And for this we have furnished ample directions in the present chapter. The first of the methods here adverted to is that of Fourier, taken in connexion with the improved method of subdividing the doubtful interval, explained and illustrated in the preceding rules and examples. The second method is that developed at (168-9), embodied in the precepts (172), and commented upon at pages 297-302 ; and which dispenses with the criterion of Fourier. The third method is founded upon the principle dehvered in the foot-note at page 163 and adverted to further, at (192-3). Agreeably to this principle we are to develop the intervening single root of the derived equation immediately inferior in degree to that in which the doubtful pair occurs in the same interval, just as in the other methods. This development we are to continue, as in the example above, being guided to the successive figures by the expressions [c] at page 263 either till it becomes obvious that the absolute number N' is converging, not to zero, but to a finite constant, and consequently can never change its sign however far that development be continued — a clear proof of imaginary roots (182), or till a change of sign actually presents itself in N', thus announcing that the roots have really separated. It is probable that this last method, being altogether inde- OF THE HIGHER ORDERS. 313 pendent of external tests or bye-operations, may be found, in practice, as useful and convenient as either of the others. For if the roots under examination turn out to be real, we shall thus have found the development of one of them at the same time as we have determined their character; and if they prove to be imaginary, we shall, in like manner, with the detection of their nature, have accomplished the development of the pair of im- perfect roots which supply their place. (195.) We are thus furnished with means, more than sufficient, for overcoming all the difficulties so long attendant upon the problem of the general solution of numerical equations : — there is no conceivable case of this problem which the methods now developed can be found inadequate to cope with. The practical facihty with which these methods may be brought into operation depends mainly, as we have abundantly shown, upon the ease and rapidity with which we can develop a single isolated root, situated in a known interval, or having a known leading figure. In further attempts to expedite the numerical process, attention should be chiefly directed therefore to this — the simplest of the various cases that can occur. The trial-divisors, which are to direct us to the successive figures of this isolated root, will become gradually more and more efficient as the preceding tributary coefficients in the successive trans- formations become more and more unimportant in numerical magnitude, in relation to those divisors. When therefore the proposed equation is such that the coefficients diminish consi- derably in magnitude from the first to the last, the case will, in general, be unfavorable to the early efficiency of the trial-divisors. But it may be converted into a favorable case by simply reversing the order of the coefficients ; thus changing the equation into another whose roots are the reciprocals of those in the original. This is the transformation which we adverted to at page 90, as occasionally useful in reference to Horner's method of developing the roots of equations. (196.) We might here terminate these researches: but the beautiful theorem of Sturm, which they in a great measure super- 314 SOLUTION OF EQUATIONS Xaede as respects equations beyond the fourth degree, demands a ^^ew additional observations, more especially in reference to the abbreviated form of computing the functions exhibited at pages 2/3 and 288-9. It is obvious, from inspecting these computations, that a very considerable saving, both of space and labour, is effected by means of the reductions there employed. The calculation of the com- plete coefficients that enter the advanced functions in Sturm's series is, as vre have sufficiently seen in the preceding examples, a work of great arithmetical labour in high equations vrith rather large coefficients : and the principal part of this labour is expended upon what invariably turns out to be mere useless redundancies. It is superfluous, in the computation of Sturm's functions, to aim at a greater degree of accuracy than is sufficient to secure correctness in the leading figure of the final constant, or in fact in the sign merely of this figure. We are not, however, in pos- session of any means by which all superfluous figures may be excluded ; all that we can do is to provide against the entrance, into the final result, of more than a predetermined extent of figures, by some such method of setting bounds to their increase, as that recommended at (156). We have acknowledged at p. 235 that this method is open to the objection of presenting the final result, in certain extreme cases, abridged of all its significant figures, leaving only a row of zeros, from which of course the character of one pair of roots would remain dubious. In the case of a pair of roots, accurately equal, the final result would be accurately a row of zeros ; and the repeated root would be accurately obtained by equating the preceding function, of the first degree, to zero : this function being the common measure of X and Xj, and hence the root of it a common root of X = and Xj = 0. But when significant figures occur in the final result, then there is only an approach to these circumstances ; which approach must evidently become more and more close, as the first significant figure recedes more and more to the right ; leaving a larger number of consecutive zeros, or unoccupied places, before it. It has been mentioned in the introductory treatise so often referred to, page 224, and will be proved hereafter, that OF THE HIGHER ORDERS. 315 the pair of nearly equal roots thus indicated must concur in their leading figures to about half the number which expresses these unoccupied places ; that is, supposing the roots indicated to be real and not imaginary. But in either case it is clear, that a slight variation in the final term of the original equation would convert the roots indicated into a real equal pair. Hence, although the character of the roots indicated by Sturm's final remainder, when preceded by several unoccupied places may be altogether unknown, from our curtailments having caused all but the zeros to disappear, yet we may confidently infer that an equa- tion, differing from that proposed only by a minute variation in its final term, will have two equal roots, each given by equating the function of the first degree to zero, its other roots being the same as those of the proposed. If six or seven zeros, or blank places, supply leading figures in the remainder, then, by the principle adverted to above, we may conclude that the two equal roots mentioned will agree with the two roots of the proposed equation to about three places of figures ; if eight or nine zeros occur, to about four places, and so on, when the latter roots are real. When they are imaginary, the same equal roots may never- theless be taken, to the same extent of places, as approximate roots of the proposed equation, or as what we have called real imperfect roots ; and which, as we have already seen, deserve to be taken into account in the practical solution of numerical equations. Suppose then that, in employing Sturm's method for the analysis of equations, we so regulate our abbreviations, where large numbers are involved, as to secure accuracy to the extent of ten or twelve places in the final remainder. Should these places all turn out to be blank, we may safely take the root of the simple equation, furnished by the preceding remainder, for one of a pair of equal roots of the proposed.* * It should be borne in mind that in all practical inquiries it is a waste of effort to attempt an accuracy in which even the data of our calculations is deficient : such an attempt is moreover as likely to lead us wrong as right. The following observations, by Professor Peacock, are so completely in harmony with some of the views set forth in the present chapter, that the author feels it obligatory upon him to quote them : *' If the root of an equation be determined approximately, the equation may 316 SOLUTION OF EQUATIONS If, however, for any other purpose, apart from practical utility, it be desired to determine the exact character of the doubtful roots then, without caring to secure half the above-mentioned number of true places in the final remainder, we may proceed to develop the root of Xj= 0, commencing our approximation with the leading figure furnished by the simple equation adverted to, and extending the process up to X, as at page 271, till, by aid of Fourier's criterion, or the other tests proposed in the present chapter, the nature of the roots is ascertained. If the function immediately preceding the last — the function of the first degree — have both its coefficients in Hke manner pre- ceded by blank places or zeros, we may infer the approach of three roots of the proposed towards equality, or else of two dis- tinct pairs of roots : for it is plain that the antecedent function of the second degree would, under these circumstances, require no change to be made in the leading figures of its coefficients to be depressed, and the general processes of solution, or of approximation, may be applied to find the roots of the quotient of the division. Thus in the equation a;3 - 3x + 2-0000001 = one of the roots is very nearly equal to 1 : if we divide the equation by x — 1, and neglect the small remainder which results from the division, we shall get the quotient x" — X — 2= (a: — l)(xH-2) = whose roots are 1 and — 2 ; or we may suppose one of the roots to be 1 '0001 , the second '9999, and the third — 2 ; or we may suppose two of the roots to be imaginary, namely, 1 +; '001 v — 1. All these roots are approximate values of the roots of the equation, which different processes, whether tenta- tive or direct, may determine : and it is obvious that when two roots are equal, or nearly so, an inaccuracy of the approximation to those roots which are employed in the depression of the primitive equation, may convert real roots into imaginary, or conversely. Such consequences will never follow when the limits and nature of the roots are previously ascertained, and every root is determined independently of the rest : but it is not very easy to prevent their occurrence when methods of approximation are applied without any pre- vious inquiries into the nature and limits of the roots, though the resulting conversion of imaginary roots into real, and of real roots into imaginary, may not deprive them of the character of true approximations to the values of the roots which are required to be determined." — Report of the Third Meeting of the British Association, p. 349. OP THE HIGHER ORDERS. 317 render it an exact divisor of the cubic that precedes it, or, which is the same thing, to render the coefficients of the functions of the first degree, referred to, accurately zero, and, consequently, the quadratic function accurately a common divisor of X, and Xj. Hence, if the roots of the quadratic are equal, three such roots must enter the equation X = ; if they are unequal, each must enter twice into X = 0. An examination of the leading figures of the roots of this quadratic must determine to which of these circumstances the case before us approximates. These leading figures will supply, as before, the first steps in the approxima- tions towards the doubtful roots. In this manner may all ambiguity that might otherwise attend the more advanced functions of Sturm, when extensively cur- tailed, be satisfactorily cleared up. And instead of thus comput- ing all the functions up to the last, we may, if we please, stop at the quadratic, as recommended in the treatise on Cubic and Biquadratic Equations ; and instead of examining the intervals thus left in doubt, by the method there taught, we may proceed to discuss them agreeably to the directions given in the present chapter. We thus see that by combining the methods of Fourier and Horner with that of Sturm, the calculations which would otherwise enter into this last method may be very considerably reduced. (197.) But before finally dismissing this method of Sturm it may be proper to show that it is in itself fully adequate, not only to determine the character of the roots of an equation under aU circumstances, but likewise to remove every ambiguity that might present itself in the course of their subsequent development, without the aid of any other theorem whatever — even the theorem of Descartes, called the rule of signs. The following example, taken from Sturm's MSmoire, is suffi- cient to show this ; and to point out the mode of proceeding whenever any such ambiguity occurs. Let the equation ^ + lla;2— 102a? + 181 = be proposed for analysis and solution. 318 SOLUTION OF EaUATIONS First, in order to ascertain the number and situation of the roots, we form the functions X = X, = a^ -f l\x^ - 102^ -f 181 3a^ + 22ar — 102 Xg = 854^ — 2751 from which, as all the leading signs are +, we infer that all the roots are real (page 214.) To determine the intervals of the positive roots, we make the substitutions a? = 0, which gives -f — X = I ... + — X = 2 ... + — X = 3 ... + — X =4 ... + + + + two variations + + + two variations 4- no variation. Hence the equation has two positive roots, both comprised between 3 and 4 ; so that the first figure, common to both, is 3. Therefore, by our method of approximation, the first step of the process will be as follows : 14 11 - 102 - 181 (3 42 - 180 - 60 - 1 9 20 . . OF THE HIGHER ORDERS. 319 and the resulting transformed equation, whose roots are those of the original, diminished by 3, is x''^ + 20^'^ - 9^' + 1 = . . . . [2]. The first figure of the root of this, or the second figure in the quotient above, appears to be '2, because, of all numbers occu- pying the place of the second figure, we find this to be the one which produces a result nearest to — 1 . Still we cannot always affirm, independently of all reference to any other principle, that the number which produces a result nearest to the absolute number, or which, when the terms are all arranged on one side, produces a result the nearest to zero, is necessarily the first figure of the root, unless the next figure in the scale produces a change of sign in the absolute number, which is not the case here. To test the figure "2, therefore, we transform all the other functions, as well as the first X, by diminishing the value of x in each, by 3, as above; and we find these results, viz. X' = x'''^ + 20^'2 _ 9^./ _|. 1 X\ = 3x''^ + 40^' — 9 X'^ = 854a?' — 189 X'. = + ; which, for x' = '2, gives the series + + two variations and for x' = S .... + + -|_ -f ^o variation ; so that two roots of [2] are comprised between -2 and -3, and thus -2 is the correct second figure of both roots of [1]. If the substitutions, -2, -3, had not given series of signs, differing by two variations, we should have concluded that the root figure, -2, was incorrect ; and should have continued to sub- stitute, in the transformed functions, the successive values, 0, -1, -2, -3, -4, . . . . 1, till such a difference of variations had been obtained, and should have taken the less of the two num- bers, to which the change was due, for the true second figure. 320 SOLUTION OF EQUATIONS We may now, therefore, complete the second step of our solution, and shall thus have 14 11 — 102 — 181 ( 3-2 42 — 180 20-2 — 60 9 — 9 . . 4-04 -4-96 4 — 1 — -992 — '008 - -88 20-6 and may thence proceed, without hesitation, to any extent, with our approximation; for the trial-divisor indicates -01 as the third figure of the root, and, trying '02, we find a change of sign in the absolute number, a sure indication that the two roots separate at the second figure, and become distinct. The roots, as far as three decimals, are 3-213 and 3-229.* To determine the remaining two roots of an equation after the others have been computed. (198.) Let the second term of the proposed equation be re- moved by (79) before any of the roots are developed; or, if this term be retained, let the developed roots be reduced to those belonging to the equation after the removal of the second term. We are to suppose that all the roots but two have thus been found, * For a further discussion of the theorem of Sturm, with examples of its application to equations of the seventh degree, the reader is referred to the Mathematical Dissertations, by the author of the present work. OF THE HIGHER ORDERS. 321 and our object is to obtain a formula for the determination of these two. It will be sufficient, to illustrate the method proposed, to confine our investigation to equations of the sixth and inferior degrees. Let then the equation of the sixth degree, deprived of its second term as proposed, be generally represented by x^ -\- px* + qx^ -\- . . . . = 0. and let its six roots be denoted by /y> /y> /y> ^ ^ *>» <*'p '*'2> '*'3» '*'4» '*'5' "'6' the sum of which must be zero on account of the absence of the second term of the equation; so that a?i + a?2 + ^3 + ^4 = - (x^ -h Xq) . . . . [1]. Now p being the sum of the products of aU the roots taken two and two, we have p = (X^ +a?2+^3+^4) (^5+^6) +^ia?2+^i^3+^1^4+^2^3+ ^4+^3^4+ Ve — '^~ v 1 "I "^2 "• S "■ 4/ "T" ^1 ^2 '" '^1 ^3 ~i "^\Xa "i XnXo "T" XqX^ ~\~ XnX» "j" XpJVq .•.p+(a?i + a?2 + ^3 + '^4)^— (^1^2 + ^1^3 • • • • ^3^4)=^5^6 P]- Equations [1] and [2] furnish expressions for the sum and product of the two roots x^, Xq ; and consequently for the coeffi- cients of the quadratic involving them. Hence, for the roots of this quadratic, that is, for the values of x^ and a?g, we have the formula which will give the remaining two roots of an equation of the sixth degree after four are determined. If the equation be only of the fifth degree then ^4=0; and the formula becomes X.-^Xo-\-X^ . , . „/^l4-^9 + ^Q\ , , ^ , 1 r.-i • ' 2 -^ "f -^( 2 ) -i'+^l(^2+^3)+^2^3 } ..[4]. 21 322 SOLUTION OF EQUATIONS If it be of the fourth degree, then x^=0, and for the two remain- ing roots, after two are determined, we have ^1 + ^9 . / f o /^] + ^' ±V{-3(^4^)-p+»,».}...[5] And finally when it is of the third degree, or a?2 = 0, we have for the required form the expression -^±V{-3(^)-^}...[6]. This last expression is, we find, also given by Garnier, in his Analyse Algebrique^ page 216. The others are, we beheve, new. It would be easy to vary their forms : indeed different forms for the third and fourth degrees have already been given in the introductory treatise, pp. 236-241. But the above are the most convenient on account of their simplicity, involving no division operations, nor any of the advanced coefficients of the original equation : so that if aU the roots but two are computed, without depriving the equation of its second term, then, in making the requisite change in it, for the application of the preceding for- mulas, we may stop the process of transformation when p is obtained. But the most convenient way of obtaining 'p is from the formula A^^ _ n{n-X) UA r which is readily deduced from the expressions at page 86. For introducing the coefficient Aj^ of the first term into those expressions, in order to give them the greater generaUty, we have for the third term of the transformed equation in which, that the second term of the transformed may vanish, we must have . _ An-, nA„ OF THE HIGHER ORDERS. 323 Putting therefore this in the third term above, that term becomes ^ A^2 _ n(n-l) fA„_iy A„ 2 XnAj As an example of the application of these formulas we may take the equation a;5 _ 41^ _ i2a?2 + 292^ - 240 = whose roots are 1, 2, 6, — 4, — 5; and any three of these sub- stituted in [4] wiU determine the remaining two. But the utility of these expressions is not confined to this ob- ject: a reference to them will often save many steps of calculation in the analysis of equations. Thus, in treating the equation of the fifth degree, at (188,) by the method of Fourier a good deal of calculation was found necessary in order to determine that two of the doubtful roots were imaginary : if these had been allowed to remain doubtful tiU the others had been developed, the character of the former might have been discovered, with but very little trouble, from the formula [4], after the determination of p, which is soon effected: thus, p = 2356 - 10(34-6)2 = 2356 — 11971-6 = — 9615-6 and it is obvious, from the formula, when the roots already deve- loped are each increased by 34*6, in order to reduce them to those belonging to the equation after the removal of the second term, that the quantity under the radical will be _3(££^) +9615-6- 123-96x70-326 + (35-16)2 = - 23. so that the two roots x^, x^ are imaginary. In equations of the fourth degree it will sometimes be found more convenient to substitute the following for the formula [5] above, viz. ^1 + ^'9 / f ^ /^i + ^' ±V{-2(^^0-(^-f-W 324 SOLUTION OF EQUATIONS As an example let the equation a?4 + 312ar^ + 23337^2 _ 1 4874a? + 2360 = be proposed. Two roots of this equation are found, in the introductory trea- tise, page 231, to be— 126-3166644731 and -186-3166651784. The determination of the remaining roots, in the treatise referred to, was attended with considerable trouble, on account of their character remaining long doubtful from their close proximity to each other. If we change the proposed equation into another of the form ^ + 0a?3 +^a?2-f- &c. = 312 by increasing the roots by -— - = 78, the two roots just exhibited will be changed into -48-3166644731 and -108-3166651784: and for p we shall have p = 23337 - 6(78)2 = - 13167 so that the preceding formula will become 78-3166648257±x/{ -2(78-3166648257)2-(30-0000003526)2+ 13167 } the two values given by which are 78-3166651 .... and 78-3166644 .... Consequently, subtracting the 78, by which the original roots were increased, we have for the two remaining roots of the proposed equation x^ — -3166651 . . . ., a?4 = -3166644 .... On the Determination of the Integral Roots by the Method of Divisors. (199.) It was demonstrated at (62) that no equation in which the coefficient of the first term is unity, and those of the other terms integers, can have a fractional root ; so that the roots of every such equation can comprise only whole numbers, and in- OF THE HIGHER ORDERS. 325 terminable decimals. These latter we have shown above how to approximate to as closely as we please ; and, although the same method will furnish us, figure by figure, with every integral root also, yet it is worth while to explain here a distinct process for the discovery and determination of every such root. The method we advert to was proposed by Newton, and is called the method of divisors. We may apply it to detect fractional roots by (82). Let N + Aa? + A2a?2 + k^s(^ + A4^ 4- .... a?" = be an equation of the wth degree, in which the coefficients are all whole numbers ; and let a be an integral root of it, then we must have N + A« + Agtt^ + k^a^ + k^a^ + .... a" = N .*. — =^— k^ k^a — k^a? — k^a^ — .... —a" ^ N from which we infer that — must be a whole number ; hence a every integral root must always be a divisor of the last term N. Call the quotient of this division Q, then, by transposing — k» and dividing by «, the last equation will become = — Ag — k^a — k^d^ —....— a"~2 ; consequently, is also a whole number, which, calling Qg, and transposing — Ag, we have, after division by «, ^±i2 = _A3-A,«-. . . . -e-3; Qo + A hence ^, or Qg, is also a whole number ; and, continuing this process, we shall obviously have the quotients Q, Q2, Q3, Q4, . . . . Q„ all whole numbers, and the last Q„, will be — 1 . 326 SOLUTION OF EQUATIONS (200.) We infer, therefore, that for a to he an integral root of an equation, the last term must be divisible by it, and so must the sum of the quotient and next coefficient; and throughout, the sum of each coefficient and preceding quotient must be divi- sible by a, the final quotient being always -- 1 ; which are con- clusions analogous to those at page 185. Hence, after having determined aU the divisors of the absolute term in an equation, we must submit all those of them which are between the hmits — L and + V of the roots, found by the rules in Chapter vi, to the foregoing tests, and retain only those divi- sors which satisfy them all. (201.) When, however, one divisor is found to succeed, we need not, in order to test the others, return to the original coeffi- cients, since, as it is easy to show, the quotients Q, Qg, Q^, &c., are no other than the coefficients of the depressed equation with their signs changed, or, which is the same thing, the coefficients in the quotient of N + Aa? + Ag^^ ^ ^ ^ ^ x^hy a — x; for, by actually performing the division, and recollecting that N = Q«, Q + A = Q2«, Qg -f Ag = (^a, &c [1], we have « _ ^) N -}- Aa? -H k^'^ -f k^ (Q ^. Oj^ 4- Qi^ Q^/ix -f- Aga?^ Q^ax — Q^ Q^ax"^ -f- AgO^ &c. OF THE HIGHER ORDERS. x 327 It follows, therefore, that a being a root of the proposed equa- tion, the equation Q + (^ + Q^^2 . . . . _ ^n-l = .... [2] will be the depressed equation involving the remaining roots, for changing the signs of all the terms does not change the roots. Hence the other integral roots of the original equation will also be roots of this ; so that, for the discovery of them, we may employ this depressed equation instead of the proposed. If we multiply every term of the depressed equation by «, keeping in mind the conditions [1] above, it will become N + (Q + A) a? + (Q2 H- Ag) «2 ^^n-i ^0 [3], the roots of which are, of course, the same as those of [2] ; so that, for the discovery of another integral root, we may, if we please, use the form [3] instead of [2], in which case the final quotient must be — «. As an example, let us take the equation x^ + bx^ +a^ — 16a?2 — 20a? — 16 = 0. The divisors of 16 are ±16, ±8, ±4, ±2, ±1. A superior limit to the positive roots is, by (87), 1 + VTe or 4; and, by substituting — a? for a? in the proposed, or, which is the same thing, by changing the signs of the alternate terms, the equation will be a^ -_ 5a;4 + ^ + 16^2 _ 20a? -f- 16 = 0, and a superior Hmit to its positive roots is, by (89), 6 ; but it is easy to see at a glance that 5 must also exceed the greatest 328 SOLUTION OF EaUATIONS positive root, therefore — 5 is a limit to the negative roots of the proposed. Hence the divisors not within the limits — 5, 4, that is, the divisors ±16, ±8, 4-4, must be rejected; we have, therefore, to try only the divisors ± 2 and — 4 : + 2)-16 -20 -16 + 1 + 5 +1 — 8 -14 -15 -7 -1(a) -2) -28 -30 -14 — 2 8 10 10 2{b) -4) -20 -20 — 4 4 4 4(c) — 16 —16 Hence +2, — 2, and — 4, are integral roots : the coefficients (a), (5), (c), are those of the successive depressed equations, re- versed : the final depressed equation is 4a?2 -f 4a? + 4 = 0, or rather a?2 4- a? + 1 = 0, the roots of which are imaginary. {Algebra, art. 107.) We have not appHed the method to the divisors + 1 and — 1, because it is easy to ascertain whether or not these are roots of the equation, and to depress the equation accordingly by (51). In fact the method of art. (51) will equally serve for the dis- covery of all the suitable divisors, and is perhaps on the whole but little inferior in facility to that above. We should indeed, by the method of (51), have in all cases to arrive at the final term of the transformation, before we could affirm that the num- ber under trial was a root or not ; whereas, in the method here explained, there is a chance of detecting the unsuitable divisors OF THE HIGHER ORDERS. 329 at every division, as the quotient may be fractional. It is scarcely necessary to observe that, when such quotients occur, the work is to be erased, and a new divisor tried; thus: suppose it were required to find whether the equation ^ _ 37a? + 72 = has any integral positive roots. "We readily see that 5 is a su- perior hmit to the positive roots ; so that the only divisors of 72 to be tried are 2, 3, and 4. Trying 2, we have 2) 72 —37 +1 36 — 1 the divisor 2 must be rejected, as the next quotient would be fractional. Trying 3, we have 3) 72 —37 +1 24 — 13 the divisor 3 is also unsuitable, as this gives Qg fractional. Lastly, trying 4, we have 4) 72 -37 +1 18 -19 which must be rejected for a like reason, so that there are no po- sitive integral roots. When the divisors of the last term between the limits — L and + L' are very numerous, the trials may become tiresome; but it is easy to devise a contrivance for diminishing the number of su- perfluous divisors thus: (202.) We have seen (201) that J-! = Q + Q.,^ -f Q.a?2 + . . . N -r Aa? + k^oc^ + k^oi^ + 330 SOLUTION OF EaUATIONS the second member being an integer for every integral value of x, because the coefficients are all integral; the simplest integral values of a? are + 1 and — 1 ; hence the first member shows that when + 1 is put for x, in the original polynomial /(a?), no divisor /(I) a can be admissible which does not render — ^^ an integer : a — 1 and, putting — 1 for x, we see that no divisor can be admissible /(— 1) which does not render — ^^ — an integer. The divisors between « + 1 the limits may, therefore, be advantageously submitted to these tests before those at (200) are apphed to them. We know from (53) that /(I) will be the last term of the transformed equation in (a?— 1), and /(—I) will be the last term of the transformed equation in (a? + 1) ; hence the best mode of proceeding will be, to effect one step of each transformation by (71), and to divide the final term in the first by each divisor minus 1, and the final term in the second by the same, plus 1 ; and then to employ only those divisors which furnish integral quotients. Should the final term in either transformation be 0, it will be a proof that the divisor unity is a root, and then we must employ the depressed equation for the other roots; the coefficients of this depressed equation will have been written down in proceeding to the final term, as at (51). (203.) Let the equation a^ __ 5a?2 _ 18a? + 72 = be proposed. The limit of the positive roots is 18 -f 1 = 19; and, changing the alternate signs, we have (88) 72 - (y + 1) = -- 15f for the limit of the negative roots; hence the only divisors of 72 which can comprise the integral roots, are 2, 3, 4, 6, 8, 9, 12, 18, -2, -3, -4, -6, -8, -9, -12. OF THE HIGHER ORDERS. 331 Let us, therefore, now proceed to determine /(I) and /(—I), 1 -5 -18 4-72(1,-1 —4 —22 50= /(I) -6 -12 84 = /(—I). Now those among the foregoing divisors, which, diminished by 1, divide 50, and which, increased by 1, divide 84, are 2, 3, -4, 6; and, by trying these in succession, we find 2 to fail ; but for 3, — 4, and 6, we have 3)72 -18 -5 + 1 24 2 — 1 -4) 6 -3 -18 3 6) - 12 12 hence the roots are all integral, and are 3, —4, and 6. Newton* s Method of approximating to the Incommensurable Roots of an Equation. (204.) The method proposed by Newton for approximating to the incommensurable roots which may still exist in an equa- tion, after the integral roots have been removed by the method of divisors, requires, like all other approximative methods, that we know the intervals in which the roots are situated. It requires, moreover, that before commencing the approximation to any 332 SOLUTION OF EQUATIONS root, we render the interval so narrow, that the extreme limits of it may not differ by more than -^^j ; in which case, either limit must he within the fraction -j\ of the value of the root. Call the initial value, thus obtained, a/, and its difference from the true root b : then, in the proposed equation /(a?) = 0, we have and, consequently, f(^)=/{^' + i) =f{?!) +/,(*0» +-^^*^ + &c. =0; and, since h is less than ^^q, S^ must be less than yj^, h^ less than t:oVoj ^^' '■> lience, rejecting the terms into which these dimi- nishing factors enter, we have, for a first approximation to the value of the correction 6, the expression ~ AW which will give the value true to two places of decimals : adding, therefore, this approximate correction to ^, we obtain a nearer value, x" i to the root, the error h' being below -j^^. For a second approximation, put ir = a/' + h\ then, proceeding as before, we have which will usually give the value of the correction, as far as four places of decimals, and this correction appUed to of' will give the more correct value x'" for x, being the true value, as far as about four decimals ; and, by repeating the operation, we shall get a new value, true to about eight decimals, and so on. OF THE HIGHER ORDERS. 333 The following is the example chosen by Newton to illustrate his method, viz. /(^) = ^ - 2^ - 5 = 0, /./iW = 3a;2 - 2. The root of this equation lies between 2 and 3 ; to narrow these limits, diminish the roots of the transformed in a? — 3, by '5, and we shall find no change of sign in the final term ; hence the root is between 2 and 2*5. Diminish the roots of this transformed by '4, and still the final sign is preserved; hence the root is between 2 and 2-1, so that the first two figures of it must be 2*0, that is. also. x=2 •0 + 5; i _ ^2-0) />(2-0) — 1 10 = -1 .*. X = 2-1 + b' yi2-i) /i(2-l) •061 11-23 ~ -•0054 .*. X = : 2-0946 /(2-0946) /,(2-0946) •0005416 11-16205 - -00004852. .-. X = 2-09455148. b" = - In this particular example the approximation is very rapid; this arises from the circumstance that, in the expressions for b', b", &c. the numerators are very small when compared with the denomi- nators ; such, however, will not be the case, when the root, to which we are approaching, diffiers but little from another root ; because, as the roots approach to equahty, the expression /^{x), when the value of one of these roots is put for x, approaches to 334 SOLUTION OF EQUATIONS zero (98) ; and hence the denominators of the foregoing frac- tions will be very small, as well as the numerators. In such a case, too, the terms rejected in the values of b', h", &c. might ex- ceed in magnitude those preserved, and thus no approximation to the true corrections would be obtained. These imperfections in Newton's process render its application unsafe, when the root sought differs by only a small decimal from any of the other real roots, unless, indeed, at each approximation, we test the value obtained, by actually substituting it in the proposed equation. As an illustration, let the equation, a^ -7x + 7 = 0, be proposed. After a few trials, a root is found to lie between 1*3 and 1*4, and to be nearer to 1*4 than to 1*3. Let us assume then a: = 1-4 + h, then we have f{\-i) _ -056 _ .-. a? = 1-35 + h'. To verify this approximation, let 1*35 and 1*36 be separately put for X in the proposed equation, the results are for X = 1-35, f{x) = + -010375 for X = 1-36, f(x) = — -004544 which, being of contrary signs, shows that our approximation is correct. For a secoofl approximation, we have i" = -£M = :^i^5ii = .oo68 /i(l-35) 1-5325 .*. x = 1-3568. ^ OF THE HIGHER ORDERS. 335 To verify this approximation, let 1-3568 and 1*3569 be substi- tuted for Xy in the proposed, the results will be for a? = 1 -3568, f{x) = + -000141586432 for X = 1-3569, f(x) = — -000006100991 ; which, being of contrary signs, proves the correctness of our ap- proximation : hence the root is between 1-3568 and 1-3569, the former number is, therefore, the true value, as far as four places of decimals. It will not escape the observation of the student, that the pro- cess for the determination of the successive values oif{x'), fix")^ f\{x'),f\{x")j &c. as also the operations for verifying the several approximations, may all be conducted with great advantage, agreeably to the method of arranging the transformations uni- formly employed throughout this volume. (205.) Had this been the arrangement adopted by the original cultivators of the Newtonian method, that method would, no doubt, soon have been perfected into the more general and com- pact process of Horner ; and Newton's divisor — ^which we see is nothing more than that which in the preceding pages has been employed as a trial-divisor — instead of being used to determine the figures of the root, would, in the early steps of the develop- ment, merely have been referred to to suggest those figures, and thence have been perfected into a true divisor before the opera- tion with it came to be actually performed. If we take Newton*s example at page 333, and arrange the steps of the work there given agreeably to the method of trans- formation referred to, the operation will stand thus : — 336 SOLUTION OF EQUATIONS 1 -2 5 (2-1,054485184577 2 4 4 / 2 2 — 1 1 2 4 8 _i 10 1-061 2 --061 2 — 1 6 •61 -55992625 3 10-61 -5007375 •1 61 -62 2 11-23 —4465824464 4 -541550536 •1 6-2 —31475 -446471845 11-198525 —95078691 •1 - 2 6-3 —31450 3 IM 67075 -89291957 6 -5786734 6-2846 -4 6-2842 —4 05 —251384 —5580721 6-295 11-16456116 —206013 —05 —251368 -111614 6-290 11-16204748 —94399 —05 -25135 -89291 3 6-285 11-1617961|3 -5108 —4 -2513 —4464 11-1615448 -644 -502 —558 11-161494|6 -86 —50 —78 |6|-2|8|3|8 11-161444 -8 -3 -8 1|1|-1|6|1|4|4|1 OF THE HIGHER ORDERS. 337 The root-figures after the comma are negative, since in the second transformation the root is overstepped ; and in this manner may we always proceed whenever, either by mistake or otherwise, too great a root-figure is employed in any step. By actually subtracting the negative portion of the root from the positive we find x — 2-0945514815423. On account of the smallness of the correction by which each trial-divisor is converted into a true divisor the preceding example is pecuUarly favorable to the method of Newton : yet the ope- ration conducted as above, and by which a single figure only is determined at each step, is even in this case far more brief and simple than the corresponding process of Newton, though several figures of the root are furnished at each step. This will be more clearly seen by comparing the foregoing work with the details of the calculation as given by Fourier, in pages 212-216 of the Analyse des Equations, according to what he considers an im- proved mode of conducting the Newtonian operation. Before closing this long chapter on the general solution of equations we have briefly to notice a new method of solving equations just published by Mr. Weddle of Newcastle-upon- Tyne. It is an ingenious and useful addition to the means we previously possessed for overcoming the practical difiiculties of this important problem ; more especially in reference to equations of advanced degrees in which several terms are absent. For it has the peculiarity of conducting its steps by aid of transforma- tions, through all of which the zero coefficients in the original equation are transmitted, and are never, as in other methods, replaced by significant numbers ; so that every zero coefficient in the proposed equation enables us to dispense with an entire column of the work, and thus to compress the solutions of cer- tain equations of a very high degree into a comparatively small space. The extent to which the researches in the present chapter 22 338 EQUATIONS OF THE HIGHER ORDERS. have been carried precludes our doing more than to make this brief mention of Mr. Weddle*s performance, and to recommend it to the perusal of those interested in the progress of improve- ment in the general solution of Numerical Equations.* * The work is published in a quarto tract under the title of " A new simple and general method of solving Numerical Equations of all orders. By Thomas Weddle;" and is sold in London by Hamilton, Adams, and Co. price 5s. CHAPTER XIIL SOLUTION OF RECURRING AND BINOMIAL EQUATIONS. Recurring Equations, (206.) It has been shown at (74) that every equation of an even degree, of the form i»2n + Aa;2n- 1 _j_ A2i»2n-2 _|_ A3a;2n-3 4. .... + A^^^^ + A2a;2 -I- Aa? + 1 = 0, in which the coefficients of any two terms, equally distant from the extremes, are alike both in magnitude and sign, has one half of the entire system of roots, the reciprocals of the other half; that is, if n of the roots be , . . . . ; «, «2 % «„ and, moreover, that even when the equidistant coefficients are like only in magnitude, and unlike in sign, the same relations will exist, provided only the middle term of the equation be absent. 340 SOLUTION OF It has also been shown, that if the equation is of an odd degree, then, whether the equal and equidistant coefficients have like signs or not, the same relations among the roots will have place, and that one root will always be + 1 or — 1, according as the sign of the last term is — or + ; so that a recurring equation of an odd degree may always be depressed to a recurring equation of a degree lower. On account of these pecuHar properties of recurring equations, they may always be reduced to others of inferior degrees ; in fact, every such equation of an odd degree may, as we have just re- marked, be at once reduced to the next inferior even degree ; and this, as we shall now prove, may be further reduced to an equa- tion of half the dimensions. (207.) Suppose the exponent 2w, in the general equation above, to be successively 2, 4, 6, &c. then dividing every term by a?", we shall have the several equations^ X -\ 4- A = 0, which may be written ^r -f- A = (^' + ^) + A{x+—)-^A,=0 .-. ^2__ 2 + A.- + A2 =0 X X (^ + ^) + A (x^ + 1) + A2 (^ + -i) + A3 = /. (^- 3^) 4- A (^2 _ 2) -f Ag^ H- A3 = &c. &c. These several equations in z are of a lower degree, by one half, than those from which they have been deduced ; and, if in either of these the value of 2 be found, x will be obtained by the solu- tion of a quadratic, from the condition 1 a? -I = z. X It is worthy of remark, that the depressed equations in z are RECURRING EQUATIONS. 341 formed according to a certain law, easily discovered from the general relation, which, by replacing a? + - by ^, gives X a formula from which the expression x^'^'^ + ^^^ is obtained in terms of the two preceding expressions ; hence we have X^ X x^ ^ + -T = (^' + -t) -^ -■ (^ + -1) = ^ _ 3^ ai^ x^ X x^ or x^ &c. &c. &c. the expressions in z, obviously forming a recurring series, of which the scale of relation is (— 1, z)y {Algebra, art. 172). (208.) Let now the recurring equation, Ax^ - 24a?5 + 57a?'* - 73^ + 57a?2 — 24a? + 4 = 0, be proposed for solution ; or, which is the same thing, the equa- tion 4 («^ + A) - 24 (0^2 + J_) H_ 57(a? + ■^) - 73 = 0, V ar'^ ^ x^ X 342 SOLUTION OF which, by putting X -\ — =c, or x^ — zx = — l, X and, taking account of the foregoing expressions, becomes 4^-24^2 + 45^-25 =0; an equation of a degree, lower by one half than the proposed. One root of this equation we find to be 1 ; thus 4-24 45 — 25 (1 4 -20 25 — 20 25 and, for the depressed equation, containing the other roots, we have 4^2 — 20^ + 25 = ; of which the first member is a perfect square, because the square of half the middle term is equal to the product of the extremes ; 5 its root is evidently 25- — 5 ; hence z has two values equal to — ; and, therefore, the six values of x are given by the three quad- ratic equations, a?2 — 07=— 1, a?2— -a; = — 1, a?2 a? = — 1; the roots of the proposed equation are, therefore. 1 ±\/-3 11 2 ' 2' ' 2' That the first pair of roots, viz. 1 -^ V^^ , 1 — \/~~3 , and RECURRING EQUATIONS. 343 are the reciprocals of each other, will be readily seen by multiply- ing the terms of the latter by 1 + V— 3. Again, let the equation a^ — lla^ + 17a?3 + \7x^ — l\x +1=0, be proposed for solution. Then, as this equation has necessarily the root a? = — 1, we immediately get the depressed biquadratic, 0^4 _ 12^"*^ + 29a?2 - 12^ + 1 = 0, or, dividing by cc^, and bringing the equidistant terms together, (^2 + _) - 12 (ar + i-) + 29 = 0, Ob X which, by means of the assumed relation, X -\ — =2", or a?2 — sx =■ — 1, X becomes .2_ x2z + 27 = 0. By solving this quadratic, we have, for z, the values 9 and 3 ; and consequently, the values of x in the preceding biquadratic equation are involved in the two quadratics following, viz. a?2 — 9a? = — 1, and a?^ — 3a? = — 1 ; these values are, consequently, hence the five roots of the proposed equation are _ 9 + VtZ 9-\/77 3 + V.5 3 - ^5 ' 2 ' 2 ' 2 * 2 * 344 SOLUTION OP or, if the terms of the second of these fractions be multiplied by 9 — sfrii and those of the last fraction by 3 -|- \/ 5, the four last roots will assume the following forms, viz. 9 + \fri 2 3 + V5 2 2 ' q + Vt;' 2 ' 3 + V5' each being accompanied by its reciprocal. (209.) It has been observed above that an equation of an even degree is recurring only when the equidistant coefficients are like in sign as well as magnitude ; if, however, the signs are unlike, the equation may be reduced to a recurring one, by dividing its first member by a? — 1 ; for it is plain that a root of the equation a?2n + A«2n-i ^ Ag a?2n-2 + _ Ag a?2 — Aa? — 1 = is 1, since the substitution of this for x renders the first member zero ; this first member is, therefore, divisible by a? — 1 ; and the resulting quotient must evidently be the same as that which we should get by dividing 1 + Aa? + k^x^ +...._ A2a?2n-2_ A3a?2n-i — a^n by 1 — X, because this dividend and divisor are no other than the former with changed signs ; the terms, however, of the latter quotient would be those of the former, reversed. The coefficients of the first quotient would, it is plain, be all obtained by dividing 1 -fA + A2+ — Ag — A — 1 by 1 — 1 ; and the coefficients of the second quotient would be obtained by dividing 1 -f A + A2 4- — A2 — A — 1 by 1 — 1 ; the same series of coefficients are, therefore, produced in both cases ; but this latter series is no other than the former. BINX)MIAL EaUATIONS. 345 taken in reverse order, therefore the coefficients in the quotient, arising from dividing the proposed polynomial by a? — 1, furnish the same series, whether taken in the direct or in the reverse order. The depressed equation, therefore, resulting from the elimination of the root 1, is a recurring equation of an odd degree, whose equidistant terms are equal in magnitude and sign. This depressed equation has, therefore, the root — 1, and, consequently, equations of the kind, here considered, have always two roots equal to + 1, and — 1, which may be eliminated, and the resulting equation lowered to one of half its degree. Binomial Equations. (210.) Binomial Equations are those which consist of merely two terms ; the one being some power of the unknown quantity, and the other the absolute number. The general form of such equations is, therefore, / ± «" = 0, in which a" represents a known quantity. By substituting ax for y, the form becomes a" a?" ± «" = ; or, more simply, a?^ ± 1 = 0, to which form we shall always suppose the equation to be re- duced. (211.) The following particulars respecting these equations, result from the simplest considerations. 1. If w be even, the equation a?^ — 1 = 0, or a?" = 1, has two real roots, viz. + 1 and — 1, and no more. That it has these two roots is plain, for an even power of it 1 is always + 1 ; and that it has no other real root is equally obvious, because no other number can, by its involution, produce 1 . Hence the binomial ;r" — 1 is divisible by (^ + 1) (a? — 1) = a?^ — 1. By actually 346 SOLUTION OF performing the division, we have the equation a?n-2 ^ a?"-4 + a?"-6 H- + ^ + a?2 -H 1 = 0, a recurring equation in which all the n — 2 roots must be ima- ginary. For example, the equation a?6 — 1 = 0, divided by ai?^ — i, gives a?4 + ;r2 + 1 = 0, whence ar = ± V — ; so that the six roots of the proposed equation are + 1, -1 ,-1+n/^ -3 2 ,-i~V^ -3 1 _ V - 3 + V n > - V- 2. If M be odd, the equation a?" — 1 =0 has only one real root, viz. + 1 ; for -h 1 is the only real number of which the odd powers are + 1 ; hence a? — 1 is the only real simple factor of a?" — 1 : dividing by this, we have the recurring equation ^n-l ^ ^n-2 ^ ^n-3 + 4. ^2 _|_ ^ ^ 1 _ Q^ of which all the n — 1 roots are imaginary. For example, the equation divided by a; — 1, gives whence — 1 ±\/ — 3 BINOMIAL EaUATIONS. 347 80 that the three roots of the proposed equation are i -1 -t-\/-3 - 1 --V -3 3. If « be even, the equation a?" + 1 = 0, or i»" = — 1, has no real root, for \/ — 1 is then impossible ; so that all the roots of this equation are imaginary. For example, the four roots of the equation a;4 -f. 1 = 0, as determined by the rules for recurring equations are ~ I +\/- 1 - 1 -V- I 1 + V- 1 i~\/~irT V2 ' V2 ' \/2 ' V2 4. If w be odd, the equation a?" + 1 = 0, or a?« = — 1, has one real root, viz. — 1, and no more, because this is the only real number of which an odd power is — 1 ; hence, if the equa- tion a?^ + 1 = be proposed, the first member being divisible by X -^ 1, we have the equation X^ — X -\-\=0; so that the three roots of the proposed equation are 1 -f- V--3 1 --\/-~3 2 ' r 5. The roots of the equation a?" ± 1 = are all unequal; for the limiting polynomial wa?"~' having no divisor in common with a?" di 1, the proposed cannot have equal roots (98). PROPOSITION I. (212.) If a be one of the imaginary roots of the equation a?" — 1 = 0, then any power of a will be also a root. 348 SOLUTION OF For, since a is one root of the equation a?" — 1 =0, therefore a" = 1, and consequently, a2n=l, a3n=l, a4n=l, &c., also a,-''=ly a-2n=l, a-3n = l, &c. the values a, a^, a^ • . . . , a~ ^, a,-^, a~^y . . . . , thus satisfying, the conditions of the equation are roots of it. Cor. 1. It hence appears that the roots of the equation a?" — 1 = may be represented under an infinite variety of forms, each term in the following series being a root, viz. ....a-3,a-2,a-i,l,a,a2,a3,....a"*i,a^a"+i,....a2",a2n+i, .... in which series, however, there cannot be more than n quantities essentially different, otherwise the equation would have more than n roots. PROPOSITION II. (213.) If a be one of the imaginary roots of the equation 0?" -f 1 = 0, then any odd power of a will be also a root. For, since a is one root of the equation a?" = — 1, therefore, ct^ = — 1 ; and, since every odd power, whether positive or negative, of — 1 is also — 1, therefore aSn = — 1, a^n _- __ i^ ^7n = — 1, &c., also a-3n = — 1, a-^" = — 1, a-7n = — 1, &c. ; so that the quantities a, a^, a^ • . • . , a"', a,-^, a~^, • • • •, are roots of the equation. These roots, therefore, assume an infinite variety of forms, although there cannot be more than ?i essentially different. BINOMIAL EQUATIONS. 349 PROPOSITION III. (214.) When w is a prime number, the roots of the equation a?" — 1 = are all exhibited in the series 1, a, a?y a^, .. .. a""^, or a^ a»+L an+2 fl,n+3 or.2n-l or a 2n cx?n+\ Qi2n+2 a^""''^ or 1, a-1, a-2, a-3, .... a-^«-i> or a-° a-°-^ a-n-2 a-^-^ a-°-^n-i> It has been shown in prop, i., that every one of the foregoing quantities is a root of the equation if a is a root ; if, therefore, no two of the n quantities in each series are the same under a different form, each series will exhibit all the n roots of the equation. Now, if we suppose any two of the roots in either series to be equal, as for instance aP and a*, in which ^ > ^, then by dividing the equation a^ = aP by aP, we have at~P = 1 ; and that this equation is impossible may be proved as follows : Because w is a prime number, and t — p necessarily less than n, therefore the numbers n and t —p are prime to each other ; and consequently two whole numbers, x' and y', may always be found such that (t^p) a/ =:ni/ +1'* and, as a'-P= 1, therefore a(t-P)^'= 1, and consequently a^y'^ ^ = 1 , that is, any' . a = 1 ; but ay' is a root (prop, i.) ; hence a"y' = 1, therefore, from the last equation, a = 1, which is impossible, because, by hypothesis a is imaginary. Hence, each of the series announced above, comprises the n roots of the equation under different forms. * See the Algebra, third edition, page 271. 350 SOLUTION OF PROPOSITION IV. (215.) When p and q have no common measure, then the equations x^ — I =0 and x^ — \ =0 have no common root except unity. If possible, let a be a root common to both equations, and different from unity, then we have aP = 1 and a p; » :^ Consequently, the six roots sought are the six products, arising from multiplying these three roots by 1, — 1, and are, therefore. PROPOSITION VII. (218.) To determine the roots of the equation a?" — 1 =0, when n is the square of a prime number^. Put a?P = z, then a?P — 5- = 0, and 2^p — 1 = 0, and let the roots of this last equation be 1, /3, /32^ /S^, . . . . /3p-^, then, by substitution, ra?p — 1 = 0, U;P-/32=0, Up _ /33= 0, &c. &c. hence the pp values of a?, in these p equations, will evidently be all different, and will be the roots of the equation a^PP — 1=0. To determine these roots, it will be sufficient to advert to art. (75)r which proves that the roots of a?P — /3 = are equal to the roots of a^P — 1 = multiplied by iy/3 ; and, in a similar manner, the roots of a?P — /S^ = are equal to the roots of BINOMIAL EQUATIONS. 353 a?P — 1 =0, multiplied by Vi^^, &c. ; therefore we immediately conclude that the roots of a.p-l=Oarel,/3,/32,/33 /S^-i \ ^ ^ ^„ ^ « , ., = the ?i roots of a;v-(3 =0 ... Vft/3pv/3,/32P//3 ^p" V/3 ;rP-/32=0 ... V/32, /3V/32,/32 V^2..^P-iP/^2J ^" 1 - 0, &c. &c. &c. For example, let it be required to find the roots o{ x^ ■— 1 =0. The roots of a:^ — 1 = are 2 hence the roots of a?^ — 1 =0 are - 1 + n/ - 3 ~ 1 - V-3 2 ' ~2 ' 3 . - 1 + s/~^3 ^ -i+x/"Z:3 3^^ -l + V^Ts From the preceding propositions we may infer, as at (42), that every root has as many values as there are units in its index; for, as there are n diiFerent quantities which satisfy the equation ^" = 1, it follows that ^ 1 has n different values ; and it is plain that if each of these values be multiplied by the common arithmetical value of V«> tli6 n products will all be different, and such that, if each be raised to the nth power, the result will always be a; hence the products of which we speak are so many different values of ^a. The determination, therefore, of the n roots of \/a requires that we are able to extract the wth arithmetical root of a, and to ex- hibit all the imaginary roots of V 1 . The foregoing propositions have been devoted chiefly to an examination of the properties and relations of these roots, and not to the actual exhibition of their 23 354 SOLUTION OF values, although, as in the proposition above, one or two exam- ples of the solution have been given to illustrate the reasoning. To obtain the imaginary roots, however, in their simplest form, that is, in the form a -\- b w — 1, and for all values of the ex- ponent, requires the aid of a theorem, borrowed from the science of Trigonometry. (219.) The theorem to which we refer, is the well-known formula of De Moivre given in most books on Analytical Trigo- nometry, viz. (see Trigonometry ^ second edition, page 59,) (cos a ±.Bma . V— 1)" = cos na ± sin na . V — 1 ; which, if the arc 2^7r, {k being a semicircumference, and k any integer,) be substituted for wa, becomes 2kjr . 2^ / ^^ ^, . . • , / — -. (cos ± sin , v — 1 )" = cos 2k7r ± sm 2hr . v — 1 ' n n that is, since cos 2hit = 1, and sin 2kTr = 0, 2kff . 2kTf , ^ (cos ± sm V — 1)"= 1; w n 80 that the expression 2kn' . 2hr , cos d: sm . v — 1, n n comprehends in it all the n roots of unity, or all the particular values of a?, which satisfy the equation ^" — 1 =0. Although, in this general expression, the value of k is quite arbitrary, yet, assume it what we will, the expression can never furnish more than n different values. These different values will arise from the several substitutions of 0, 1, 2, 3 . . . . up to the number , inclusively, if n is odd, and up to -, if n BINOMIAL EQUATIONS. 355 is even ; and for substitutions beyond these limits the preceding results wiU recur. To prove this, let us actually substitute as proposed: we shall thus have the following series of re- sults, viz. for A; = . . . . a? = cos ± sin . sT-i = 1 2Tf . 2Tt / a; = 1 . . . . X •=■ cos — lb sin — • v — 1 n ri Aft . Att / — " A: = 2 . . . . a? = cos — ± sm — • v — 1 n n 67r . 67r / ^ = 3 . . . . a? = cos — di sm — • v — 1 n n n—\ (n—l)^ . (n — l)7r / k = . . . ;r = cos ^^ — ± sm ^^ — • V — 1 Each of these expressions, except the first, involves two distinct values; so that, omitting the value given by ^ = 0, there are n — 1 values, and, consequently, altogether, there are revalues; and that they are all different, is plain, because the arcs 27r Aft 67r fn—l^it i — u, , , , . . . . , n n n n being all different, and less than tt, have all different cosines. The arcs which would arise from continuing the substitutions, are in + l^TT {n + 3)7r {n + 5)7r , , , «c. ; n n n or, which are the same, in — \)Ti ^ (n — 3)71' ^ (re — 5)7r ^ and the sines and cosines of these are respectively the same as the sines and cosines of the arcs 356 SOLUTION OF (w — l)7r (n — 3)'Tr (w — 5)7r. , , , &c., n n n which have already occurred.* n If n is an even number, the final substitution for k must be ~ fi 1 instead of , as above ; and therefore the final pair of con- jugate values for x will be X = cos tt ±: sin ti' . v — 1 = — 1, which values of x differ from all the other values, because in them no arc occurs so great as tt. The arcs which would arise from continuing the substitutions n -are {n + 2)7? (n + 4)7f {n + 6)7r beyond ^ = - are 2i n n or, which are the same. -, &c. ; {n - 2)ii ^ {n — 4)7r (w - 6)7r 27r _ -i <—, lit ^^ <—y 2'Tf — -^^ —, &c.. and the sines and cosines of these are respectively the same as the sines and cosines of the arcs {n — 2)7? {n — 4)7r {n — 6)7r n n ! which have already occurred.* It is easy to see that in every pair of conjugate roots, each is the reciprocal of the other. In fact whatever be ky 2k'n . 2hrt , ^ , 2Tat , 2kn' . ^ (cos f- sin V — 1) (cos sm . v — 1) = n n n n Jlht ^ . ^2hn cos^ 1- sm^ = 1, n n • The signs of the sines will, however, be different ; but the only effect of this difference is evidently to furnish each pair of conjugate roots in an inverse order. BINOMIAL EQUATIONS. 357 which shows that the two factors in the first member are of the form a, -. a We have proved (212) that every power of an imaginary root of the binomial equation is also a root ; but, unless n he sl prime number, we could not infer that all the roots would ever be pro- duced by involving any one of them. Such would not indeed be the case. There is always, however, one among the imaginary roots of which the involution will furnish all the others ; it is the first imaginary root, or that due to the substitution k = 1, in the foregoing series of values ; for, by De Moivre's formula, the powers of this produce all the others, thus : 2'7r . 27r / ^^ Am . Awhen n is odd. • • • i==J 1 1 1 ^\ a^ . a»-i ^ 1 1 J ) when n is even. and \_ a For further researches on the theory of binomial equations, the student may consult Lagrange's Traite de la Resolution des Equations Numeriques, Note 14; Legendre's Theorie des Nombres, Part v. ; the Disquisitiones Arithmeticae of Gauss ; Barlow's Theory of Numbers; and Ivory's article on Equations, in the Encyclopaedia Britannica. CHAPTER XIV. ON CONTINUED FRACTIONS. (221.) Let a represent either a fractional or an incommen- surable number ; and let a be the greatest integer below the value of a, and which, if a be less than 1, will of course be 0. Then, since a — a is less than 1, it follows that must be ereater a — a than 1. Put 1 o 1 a — a j3 and let b be the integer which in value is immediately below /3 ; then (3 — bis less than 1, and consequently -r — - must be greater p — o than 1. Put and let c be the greatest integer below the value of y ; then will y — c be less than 1, and therefore greater than 1. Put y - c° = .*. y = r + y — c b CONTINUED FRACTIONS. 361 Continuing this process, we obviously have, by substituting in the foregoing expression for a the values of /3, y, &c. in succes- sion, the following development of the value a, viz. = " + 1+1 7 1 — a -{- — 1 ^ b + — 1 ^ + -T &c. which development is called a continued fraction. If either of the quantities /3, y, b, &c. is a whole number, the development must of course terminate at that number; and this will necessarily be the case if a be rational, or a finite fraction ; but if a be irrational, then the fraction representing its develop- ment must be interminable. This is readily admissible ; it is, how- ever, an unavoidable conclusion from what follows. A (222.) If a be a rational fraction r^, we may very easily arrive at its equivalent continued fraction. For the first term a will be the quotient of A by B ; and, calling the remainder C, we shall have In like manner, the division of B by C gives b ; and putting D for the remainder, we have B ^ D C _ ^ 6 = - .-. y = -. Similarly the division of C by D gives c, and so on. Hence a, b, c, &c. are no other than the quotients which sue- 362 CONTINUED FRACTIONS. cessively arise in the process for finding the common measure of the terms of the proposed fraction — ; thus: B) A (a Ba Cb ^C (c Dc E &c. It is easy to see that when a is a rational fraction, the expres- sion deduced for it, in the preceding article, is readily derivable from this operation of the common measure; indeed the form of the continued fraction, as deduced from this process, will have greater generality than that given in last article. For without restricting the foregoing quotients to be integral and positive, we shall evidently have, in every case, A C 1 7 £ &c. &c. &c. so that A 1 —- = « + -- 1 B 6 + — J_ "" "^ ~d-\- &c. in which a, b, c, &c. are quotients, positive or negative, integral or fractional, derived by the foregoing operation. In most appli- cations of continued fractions, integral and positive quotients CONTINUED FRACTIONS. 363 only are employed; but it is useful to show that these restrictions are not essential to the form of the development ; which is pre- served, whatever be the character of the quotients. This is a truth that we shall have occasion to avail ourselves of at the close of the Chapter; at present, however, we shall require only posi- tive and integral quotients. (223.) As a particular apphcation, let the proposed fraction be 1103 , , , . , . , ; then, by applying the process lor the common measure, 887 the several quotients furnish the following development, viz. 1103 _ J_ 887- + 4+ _^ ^ + T+1 1 1171 and if the fraction be , we have the following equivalent de- velopment, viz. 1171 ^l_ 9743 8 + -^_^ 1 , "8"+ 4- 1 Since the process of seeking the greatest common divisor of two numbers always terminates, it follows that every rational fraction may be expressed in a finite continued fraction. (224.) We might obviously, by reduction, collect into one the successive portions 1 1 1 1 1 n a « H- -T- « + -T- . 1 o o -\- — c of a continued fraction, by putting for a, in the first, « + -T" » 364 CONTINUED FRACTIONS. then b -\ for b, and so on ; we should thus have the results c 1 b be + 1 — , , , &c. a ab + I a {ab + 1) + a'.-.a'R'>a-.-.^<^; hence the error committed by taking the converging fraction, -— , for the value of a?, must be less than -~- • From this expression for the limit of error, we can always de- termine a converging fraction, which shall approach as near to the true value as we please, or which shall differ from that value by less than any assigned quantity a ; for, in order that — may be the fraction, it will be sufficient that -— do not exceed A, or, that Ql be not less than sj ^ A The property estabhshed in (226), that FQ — Pa'= 1, wiU also furnish an expression for the inferior Hmit of the error, as well as for the superior limit ; for, in consequence of this pro- perty, we have (225) P F (QV + POF' 372 CONTINUED FRACTIONS. and, therefore, dividing the numerator and only part of the de- nominator by y, we have "" Y ^ {Q! + F) F P Hence, taking either of the converging fractions, — , for the true value of a?, we have the following limits to the error, viz. p/ ^ YQ! ^ F2 _Z. 1 F -^ (Q' + P') P' 2FQ' (228.) In the examples hitherto given of the development of a, in the form of a continued fraction, a has been considered to be a rational fraction, and the several quantities a, 6, c, &c. have been obtained by means of the operation to find the greatest common measure of the terms of the proposed' fraction. But, when a is an irrational quantity, it is obvious that we must de- termine tty b, c, &c. by some other means. Let us here recall the principles with which we set out at the commencement of the Chapter, and, from which, without any restriction as to the rationality of a, we arrived at the expressions y +-; and so on ; and let us follow the successive steps there pointed out, in order to effect the reduction of VTq into a continued fraction ; that is, let a ^ v 19. Now the greatest integer in Vl9 is 4, .*. a = 4, and conse- quently, = /3. In order to perceive more readily the >A9— 4 greatest integer in this, multiply both numerator and denominator CONTINUED FRACTIONS. 373 \/Tg -\- 4 by V19 + 4, then = (3, in which the greatest integer . , . , ^ - n/19 +4 ^ Vrp - 2 3 IS obviously 2 ; hence 2 = 3 3 \/T9 — 2 and, by proceeding in this way, we have a = \/l9 = 4 + , ""^ « = 4 ^^ 1 _ \/T9 + 4 \/l9 — 2 Vl9 — 4 3 3 " ~ 3 vTq + 2 \/l9 — 3 y = -p=z = r = 1 + ,\c = l VJ9 — 2 ^ 5 5 \/l9 + 3 „ \/l9--3 S=-7=. = ^ = 3+ .'.d = 3 V19 — 3 ^ 2 2 n/T9 + 3 ^ \/l9 — 2 = 1 H ;: .*. e = 1 \/l9 — 3 5 5 \/l9 + 2 VTg— 4 ^ = = = 2 H /. / = 2 \/T9 — 2 3 3 3 V19+4 /— V=-r=: = = 8 + Vl9-4/.^ = 8 VT9— 4 1 1 = \/T9 — 4 As we have now arrived at the same expression as that which we have already had for |S, it is plain that the series b, c, &c. 374 CONTINUED FRACTIONS. must recur; and that the continued fraction, as far as one period, will be VT9=4 +— 1 ' + T+-i 1 ^ 8 + &c. and the series of converging fractions, which may be carried to any extent, now that we have got «, 6, c, for a complete period, will be 2 13 12 8 2 &c. 1 1 4 5 14 117 248 &c. 2 3 11 14 39 326 691 /— 248 /. V19=4 4- —nearly; which does not differ from the truth by so much as (691)2 It is not only in the particular example which we have here chosen, that the continued fraction is periodical, for it is the property of all quadratic surds to give rise to these recurring fractions ; but, for the proof of it, we must refer the student to Barlow's Theory of Numbers, or to the Th^orie des Nombres of Legendre, page 43. Application of Continued Fractions to the Summation of Infinite Series. (229.) In our treatise on Algehra^ page 248, we promised to furnish, in the present volume, a direct and easy method of sum- ming every infinite series of which the generating function is rational. The method to which we alluded, is one of the many deductions from the doctrine of continued fractions, and may, therefore, without impropriety, be given in this place. CONTINUED FRACTIONS. 375 1 . Let the sum of the infinite series 1 - 3a? + 5a?2 __ 7^ _f. 9^4 _ 1 la^ ^ 13^6 _ &c. be required. Regarding this series as the numerator of a fraction, whose de- nominator is unity, and, dividing the denominator by the nume- rator, we obtain for quotient 1 and for remainder 3x — 5a?2 + 7a?3 — 9^ + 1 Ix^ — &c. dividing the former divisor, that is, the original series, by this re- mainder, we have, for quotient — , and for remainder, 3x 4 8 ^ 12 , 16 , 20 , „ dividing the last divisor by this, we obtain for quotient — -, and for remainder, a;2 _ 2^ -H 3a^ - 4^^ + 5^6 _ q^7 4. &c. and, lastly, dividing the preceding divisor by this, we get, for 4 quotient — — , and for remainder, zero. Hence the proposed series may be replaced by the continued fraction, 3x 9 1 ~4"^___J 3x We may get rid of the fractional denominators, one by one, in the usual way, thus : omitting the leading term, multiply nu- merator and denominator of the remaining fraction by 3x, then omitting the second term, multiply numerator and denominator 376 CONTINUED FRACTIONS. of the remaining fraction by 4, and, finally, omitting the pre- ceding terms, multiply numerator and denominator of the remain- ing fraction by 3x, and the continued fraction will then be or rather 1 + 3a.' 4x 1 — — X 3+-. This may be easily reduced to an ordinary fraction, by collecting the several terms, commencing at the last ; and we thus find, for the sum of the proposed series, the expression 1 —X 2. As a second example, let the series 1 - 2a? + 3a?2 — 4a?3 + 5a;4 _ 6^ ^ &e. be proposed. By proceeding as in the former example, we find the following series of quotients and remainders, viz. QUOTIENTS. REMAINDERS. 1 2x - 3x^ + 4a^ — 5a?4 + 6a?5 — &c. 1 2x X 2x^ 3x^ 4a^ 5a^ , ^2+2 2+2 2 + ^^ — 4 .r2 - 2ar^ + 3ari - 4a^ + &c 1 "" 2ar CONTINUED FRACTIONS. 377 Hence the equivalent continued fraction is 1 -f 2i + 4 -j- _L 2x1 or rather. 1 + 2x T — X 2 + 1* which, reduced to an ordinary fraction, is 1 of which the proposed series is the development. 3. Let the series be 1 + 5^ 4- 9ar2 + 13^ _|. 17-^4 + 21a?5 + &c. Then proceeding as in the last example, we have the following table of quotients and remainders, viz. QUOTIENTS. 1 1 25 "~ 16 I? 5a? -— 5x — 9x^ REMAINDERS. - 13^ — 17^— 21a?5 — &C. 16a? 32a;2 48a^ 64a?4 80a?5 . 5^5^555 ^2 + 23^3 4- 3a;4 4- 4x^ + &c. hence the equivalent continued fraction is 378 CONTINUED FRACTIONS. 1 T 5i + -25 I ~16+T6 ba:' or rather which, reduced to > a common fraction, is 1 -\-3x the development of which is the proposed series. By treating, in a similar way, the series 4a? + 15a?2 + 40^ + 85^-4 + 156a^ + &c. we find its generating rational fraction to be a?(l + a?2)(4 — a?) These examples are sufficient to show that the foregoing pro- cess, founded on the determination of the greatest common divisor, between unity and the proposed series, furnishes a direct and simple method of summing every infinite series of which the generating function is rational. We are indebted for it to a paper by M. Le Barbier, pubUshed in the Annales de MathSmatiques, for March, 1831.* Application of Continued Fractions to the Solution of Equations, The method of approximating to the incommensurable roots * It is proper to mention, that the preceding method is also embodied in a paper by Mr. Hobneii, '* On the use of continued Fractions in the Summa- tion of Series," published in the Annals of Philosophy for June, 1820, CONTINUED FRACTIONS. 379 of an equation, by continued fractions, is due to Lagrange. An example or two wiU suffice to illustrate it. 1 . Let the equation x^—2x — 5 =0, be proposed. It is soon seen that 2 is the first figure of the real root, the other two are imaginary, because 4( — 2)^ + 2/ x 5^ > (Introductory Treatise, page 106). Substitute, then, 2 + — j x^ for X, and we have the following transformed equation, in which the root a/ must necessarily exceed unity : x^—lOx'^ — ex'—l =0. Of course we effect this transformation, not by the actual sub- stitution of 2 -i — ;- for X, in the proposed equation, as Lagrange X did, but by operating as in (73), thus : 1 -2 -5(2 2 4 4 • 2 2 -1 2 8 4 10 2 6 and, consequently (73), the transformed equation is %'3- 10^'2 _ 6^ - ] I = 0. The first figure of the root of this equation, found by trial, is 10; putting, therefore. ^' = 10 + ^7' x' we have, for a new transformed, the equation 6la/'3 _ 94^"2 _ 20^" —1 = 0: 380 CONTINUED FRACTIONS. the first figure in the root of which is 1 . Put, therefore, ^'=' + ^" . and effect a third transformation, which will be 54a?''^ + 2hx"'^ - 89a/" — 61 =0; in which the first figure of the root is 1 . Continue this process, and we shall have, for the leading figures of the root of the original equation, the expression "^ 2 -h &c. which furnishes the converging fractions following, 112 13 1 1 12 21 23 44 HI j^ 576 731 1307 16415 ^' To' IT' 2l' 53 ' 74 * 275' 349' "62?' 7837"' and these are alternately below and above the true value of the root. The fraction is arreater than the true value : but 7837 the error being less than , by (227), that is, less than (7837) 16415 0000000163, it follows that the approximation, , will be 7837 true as far as the seventh decimal. The root is, therefore, 2*0945514, true to seven places. In each of the transformed equations, which occur in the fore- going operation, the root is necessarily greater than unity, and but one real root exists in each ; so that, in searching for the first figure, we are to limit our trials to the numbers 0, 1, 2, 3, ... . 10, 11, &c. When the equation has several real roots, they may all be separately evolved, as above, provided we know their number and CONTINUED FRACTIONS. 381 situation. This knowledge the apphcation of Sturm's theorem will always supply, and the method of Lagrange may thus be perfected. The same thing may also be effected by the transfor- mations of BuDAN, or by the methods expounded in Chap. xii. But, as remarked at (146), the method of Budan is that which best unites with Lagrange's process for developing the roots. We shall exemplify it in its application to the equation x^ - 7a? H- 7 = 0. By the first series of transformations exhibited at page 1 98, we find the interval [1, 2] to be doubtful: hence if the equation have any real positive roots, they must lie between 1 and 2, so that we must have ^ = 1 + ~, ar where a?' must be determined from the transformed equation ^'3 _ 4^/2 _|_ 3^/ _^ 1 3^ 0. Applying Budan's method to this, we are led to the second series of transformations at page 198, which resolves the doubt, and at the same time supplies the leading figure of each of the two positive values of x'\ that of one of these values being found to be 1, and that of the other 2. It is thus that Budan's trans- formations, for determining the character of a doubtful interval, facilitates the actual development of the roots under examination whenever they eventually prove to be real: for, as here, each series of transformations supplies an additional term in the con- tinued fraction by which the developments are expressed, till the roots separate and proceed singly, lying in distinct intervals. In order to approximate to the first of the above roots, which have now separated, put as before and we shall have the transformed equation ^"3 _ ^x""- - x" ^ 1=0; 382 CONTINUED FRACTIONS. to which there will be no necessity to apply the transformations of BuDAN, because we know that it has one root, and only one, greater than unity, so that two consecutive numbers in the series 0, 1, 2, .... , must, when substituted for a?, give results with contrary signs. These numbers are 2 and 3. To approidmate to the other positive root, we must put a/=2 + — , which will furnish the transformed equation ■ ^^f + ^,? - 2^/; -1 = 0; which has one, and only one, root greater than unity ; and, there- fore, its situation may be easily found by trial to be between 1 and 2. We have, therefore, now to make the substitutions -^'=2 + 4 x'" ^//= 1 + — ' and we thus have the new equations ^m __ 3aj///2 _ 4^/// _ 1 = ^/// ^^111 '^^iit 1=0, each of which has one, and only one, root greater than unity ; and the first figure of each is found by trial to be 4. The next transformation is consequently a/"'3 _ 20^'''2 __ 9^/// _ 1 = 0, the first figure of the root of which is 20 ; and therefore the next transformation is CONTINUED FRACTIONS. 383 the first figure of the root of which is 2 ; the next transformed equation is ig7x"^'"s _ 568x''""'^ - 695x'"'^' - 181 = 0, the first figure of the root of which is 3 ; and, by continuing these transformations, we have, for the values of a: sought, the following developments, viz. ^=^ + T+-L 1 ^ + T+1 1 ^ 20 + — 1 2 H ^3-1-, &c. 3 +,&c. The converging fractions deduced from these are , &c. —y , &C. Hence, for near values of a?, we have 3181 2745 2 4 20 2 3 1 T' 2 1 ' 5 22 13' 445 263' 912 539' 3181 1880 1 4 20 2 3 1 T 3 2' 4 19 14' 384 283' 787 580' 2745 2023 1880 2023' or, in decimals, X = 1-6920213, X = 1-3568957, which are true as far as six decimals, and are but a unit below the truth in the seventh place. 384 CONTINUED FRACTIONS. (230.) The two equations just solved are both given by Lagrange in illustration of the method, proposed by Waring and himself, for discovering the character of roots by means of an auxiliary equation, technically called the equation of the squares of the differences^ and also for the purpose of exhibiting, as above, the actual development of those that are thus ascertained to be real, by his peculiar process of continued fractions. The forma- tion of the auxiliary equation, which will be explained hereafter, is in general impracticable beyond very narrow Kmits, on account of the long computations required : and it is therefore advan- tageously replaced by the transformations of Budan, as in the example just considered. As before remarked, these transforma- tions actually supply the leading terms of Lagrange's develop- ment, up to the point where the roots are found to separate, when- ever those roots turn out to be real ; so that of all the methods of analysis that might be proposed, that of Budan would seem to claim a preference in connexion with Lagrange's process of de- velopment. But in those cases of difficulty, so fully discussed in the preceding chapters of the present work, where roots approach very near to equahty, Budan' s trial-transformations become in general too numerous to be available in actual practice. On the average four or five of these transformations must be calculated upon for each effective step in the analysis ; so that the separation of the two contiguous roots, in the equation of the fifth degree proposed at page 268, might be expected to involve about four times the labour which was expended upon the same object at page 271. (See articles 145, 146.) CHAPTER XV. THE SOLUTION OF TWO EQUATIONS, CONTAINING TWO UNKNOWN QUANTITIES I THEORY OF ELIMINATION. (23 1 .) Two equations, each containing two unknown quanti- ties, Xj y, together with known numbers, may be thus ex- pressed, viz. r(^,y)=0, /(^,y) = . . . . [1]; and their solution consists in determining the system of values for X and y, which simultaneously satisfy both equations. In order that y may have a value /3, which tv^ill equally belong to both equations, it is obviously necessary, and it is sufficient, that there exist a value for x, competent to satisfy the two equations F(^, /3)=0, yt^, /3)=0; that is, these two equations in x must have a common root, and therefore the polynomials Y{x, (3),/(x, (3), must have a common factor, or admit of a common measure in x. In order, therefore, to ascertain whether any proposed value, /3, for y is consistent with the conditions [1], we should have to perform the operation for the common measure upon the functions ¥(x, (5),f(x, /3). If . a common measure, which is a function of x, be found to exist, 25 386 THEORY OF ELIMINATION. the proposed value for y is admissible, and the common mea- sure, equated to zero, will be an equation whose roots will be the corresponding values of x ; but if no such common measure exist, we must then reject the assumed value of y, as being incom- patible with the conditions [1]. To assume different values for one of the unknowns, and, in this way, to try their eligibihty, would, in many cases, require an endless series of operations. The most direct and obvious mode of proceeding, in order to obtain values for y, which must necessarily cause the functions in x to have a common measure, would seem to be this, viz. to arrange the terms of each poly- nomial according to the powers of a?, and to operate upon them, for the common measure, till we arrive at a remainder inde- pendent of a?, and then to equate this remainder in y to zero. For the values of y, which satisfy this equation, are all such as to cause the remainder to vanish. It must be remembered, however, that, in the operation of find- ing the greatest common measure of two algebraical expressions, we have frequent occasion to suppress certain factors, and to intro- duce others, and, before we could affirm with confidence that the values of y, which cause the remainder to vanish, necessarily fulfil the proposed conditions, we must examine whether or not this remainder is affected by the factors, which may have been re- jected or introduced. If, however, the process for the common measure, in any particular case, requires neither the suppression nor the introduction of a factor, we may then safely infer that the final remainder, or that which is independent of x, will, when equated to zero, furnish all the values of y consistent with the proposed conditions ; because, if each value thus determined were to be put for y, in the original polynomials, and the common measure in each case found, we should, obviously, arrive at the very same series of collateral expressions for x. For example, suppose the equations -^ -f- 3ya;2 + (3y2 ^ y ^ \) cc ^ y'' - f + 2y = X^ -f 2ya? -\- y^ — y ■=.{) were proposed for solution. THEORY OF ELIMINATION. 387 As the polynomials are already arranged according to the de- creasing powers of a?, we may at once commence the operation for the common measure, which is as follows : o^-{-2y3iP+{ y'^—y)3c yx'^-\-{2y'^-\-\)x^-f—y^-\-2y yx^ + 2y 2^ 4- y3 _ y 2 x + 2y'\x^^-2yx-^f-^y\x x^-\-2yx Having now got a remainder, independent of Xy we have for the determination of all those values of y, which cause the pro- posed polynomials to have a common measure, the equation y2 _ 2/ = .*. y = 0, y = 1 ; and the values of a?, corresponding to these, are, of course, those furnished by equating the common measure to zero ; they are, therefore, a? = 0, X — — 2, It is plain that y = 0, and y = 1, are the only values which, when substituted in the proposed expression, will cause the pre- ceding operation to terminate ; if other values for y existed, the final remainder above would necessarily contain them. If a Hke process be performed with the two equations a^ + 2ya^ + (2y2 + 1)^2 ^ (^3 + 9y2 + ^ «. 81) a? + y2 = a?' + 2ya?2 ^ 2y2^ + 2/2 + Qy^ — 81=0, 388 THEORY OF ELIMINATION. we should find, without suppressing or introducing any factor, the expression 9y^ — 81, for the remainder in y, and the expres- sion a?2 -|- ya? + y^, for the corresponding divisor ; hence the final equations for determining x and y, are y2 _ 9 — 0, ar2 + ya? + y2 = 0, the solution of which will furnish the values which satisfy the proposed equations. But let us examine the consequences of introducing gr sup- pressing factors in the course of the process for finding the com- mon measure, or of arriving at a remainder Y independent of x. There are three distinct cases to consider, viz. 1 . The value attributed to y may reduce to zero neither of the factors which have been introduced or suppressed. 2. It may reduce to zero one of the factors which have been introduced. 3. The value may be such as to reduce to zero one of the factors which have been suppressed. (232.) 1. Suppose a value to be attributed to y that does not render any of the factors introduced or suppressed zero. If we substitute this value in the two polynomials, and perform the operation with the resulting functions of x, we shall obtain for remainder the same value that would be furnished by the sub- stitution of y in Y, or else a value equal to the result of this substitution multiplied or divided by a numerical factor. For every algebraic factor introduced or suppressed is, by the substi- tution of the proposed value for y, reduced to a number, because, by hypothesis, none of them are rendered zero ; these factors, therefore, affect only the numerical factors of the several remain- ders which arise in the course of the operation. Hence, in order that the value of y may satisfy the proposed equations, it is necessary and sufficient that it satisfies the equation Y = 0. (233.) 2. Let the value attributed to y destroy one of the factors introduced into a dividend to render the division possible ; THEORY OF ELIMINATION. 389 the dividend thus modified will, for that particular value of y, become zero ; so that, in order to carry on the division, we have introduced a factor that causes a dividend to vanish, which is of course not allowable ; for, with such a dividend, the process would always terminate, whether there was a common measure or not ; we cannot, therefore, affirm that the value of y, which causes one of the factors that have been introduced to vanish, satisfies the proposed equations, although it may fulfil the condition Y = 0. (234.) 3. Lastly, let the value attributed to y destroy one of the factors which have been suppressed, and yet not satisfy the condition Y = ; then, such a value of y causes the process to terminate at that remainder in which the factor has been sup- pressed, because, when the assumed value is put for y in the polynomials, this remainder becomes zero ; hence the preceding divisor is a common measure of those polynoniials, and thus a common measure may exist for values of y which do not satisfy the condition Y = 0. It must be remarked, however, that if in any part of the operation which precedes the suppression of the vanishing factor, a factor has been introduced which also vanishes for the same value of ?/, the above conclusion would not neces- sarily follow. (235.) From the foregoing considerations we see, that to obtain the values of y which belong to the proposed equations, we must equate to zero the remainder, which is independent of a?, as also each of the factors in y which have been suppressed in the course of the operation, and resolve each equation separately ; secondly, that among the values thus obtained, there may be found some which are extraneous, and which must therefore be rejected as not being consistent with the proposed conditions. If no factor has been suppressed in the course of the operation, the equation Y = alone will furnish all the suitable values of y, and may also contain values not admissible, provided factors have been introduced ; but when no factor has been either introduced or suppressed, then the values of y in the equation Y = all belong to the proposed equations, to the exclusion of all other values. Having thus examined the influence of the factors introduced 390 THEORY OF ELIMINATION. or suppressed in the course of the operation upon the final re- mainder in y, let us now return to the original polynomials F(a?, y), f{xy y), and analyse the process by which we must arrive at this remainder. (236.) The proposed functions being arranged according to the descending powers of a?, wiU each be of the form ax"^ + bx^-^ + ca?"»-2 . . . . = 0, where the coefficients a, hy c, &c. are all independent of x. As these coefficients may have a function of y for a common divisor, let us suppose that the greatest common divisor of the coefficients in F (a?, y) is ^,{y), and that the greatest common divisor of the coefficients in f{x, y) is ffy) ; also of these two divisors let the greatest common divisor be (y), which will therefore be the greatest divisor common to aU the coefficients of both equations. If now we represent by A the quotient of F (a?, y) by F/y) ; by B the quotient of /(a?, y) by /(y) ; by F'(y) the quotient of F/y) by (y) ; and by/' (y) the quotient ^^//(y) ^y (y)f ^^ ^^^ obviously have ¥(x,y) =0(y) X F(y) x A = /(^>y) =0(y) X f(y) X B =0, both of which equations will be satisfied by the condition (y) = 0, as also by either of the following pairs of conditions, viz. 1. F(y)=0, B = 0. 2. /(y)=0, A=0. 3. A = 0, B = 0. The conditions F'(y)=o, /'(y) = o. THEORY OF ELIMINATION. 391 it is evident, cannot exist, because tliey involve but one unknown quantity, and their first members have no common factor. As to the equation (3/) = 0, it furnishes certain values of y for which x is indeterminate ; for the proposed equations will evidently be satisfied for any value of x in conjunction with these values of y. To find the solutions of the system Y'{y) = 0, B = 0, we must resolve the first equation, which contains only y, and substitute the resulting values separately in B, and we shall thus have so many equations in x to determine the corresponding values. The system y^(y)'= 0, A = 0, requires similar treatment. This preliminary examination being disposed of, the equations will be thus reduced to the simplest form for the application of the general method, viz., to the system A = 0, B = 0, in which neither A nor B has any factor in y. To determine the solu- tions which satisfy this system, we must apply the process for finding the common measure. (237.) 1. Suppose that the first step of this process conducts to a remainder, R, of a lower degree in x than the divisor, with- out our being obliged to use any preparation to render the divi- sion possible, or to avoid the occurrence of y as a denominator in the quotient Q ; then, if A is the polynomial taken for the dividend, we shall have the identity A = BQ +- R, which shows that whatever values of x and y satisfy the equations A = 0, B = 0, the same must also satisfy the equation R = ; and that whatever values satisfy the equations B = 0, R = 0, satisfy also the equation A = ; so that the solutions of the proposed equations A = 0, B = 0, are exactly the same as those of the equations B = 0, R = 0, 392 THEORY OF ELIMINATION. which are more simple than the former system, inasmuch as one is of an inferior degree in x. The same conclusions evidently follow when the dividend A is multiplied at the outset by any numerical factor. It is easy to prove that the consequences just deduced could not have place if the quotient Q contained y in a denominator. TT For suppose the form of the quotient to be Q = -— , K being a K quantity containing y ; the identity above would then be A = — + R. If we gave to cc and y all the values which fulfil the conditions A = 0, B = 0, among these values there might be some for y which, for aught we know to the contrary, might render K zero, in which case BH -r^ would become §, which is not necessarily zero ; so that A = 0, B- = 0, would not necessarily imply R = ; and we could not therefore assert that all the solutions of the system A = 0, B = 0, were equally given by the system B = 0, R = 0. (238.) 2. Let us now suppose that, to avoid fractions in the quotient, it be necessary to introduce an algebraical factor into the dividend A : call it C, and let Q, R be the corresponding quotient and remainder, as before. We shall thus have «the identity CA = BQ + R. which shows that the solutions of the equations B = 0, R = 0, are the same as those of the equations CA = 0, B = 0. THEORY OF ELIMINATION. 393 Now this last system divides itself into two others, viz. A = 0, B = 0, and C = 0, B = 0. Consequently the equation B = 0, R = 0, will give all the solu- tions of the proposed system A = 0, B = 0, but they will give in addition, solutions to the system C = 0, B =0. These latter solutions we can separate from the others ; for C = 0, containing only y^ will furnish aU the values of y which are doubtful, and the values of a?, corresponding to these, are given by the solutions to B = 0, E = 0. Those pairs of these values which, substituted in the equation A = 0, satisfy its con- ditions are admissible, the others are to be rejected. (239.) From the preceding discussion it appears that the solution of the two equations proposed is reducible to the solu- tion of the two equations B = 0, R = 0. As the polynomial B contains no factors depending only on y, if R contain any such factors, we may of course suppress them ; but then we must take account of the solutions which reduce to zero these factors, connecting each value of y with that value of x which satisfies B = 0, when the said value of y is substituted in B. We shall now give an example or two of the application of this theory. EXAMPLES. 1 . Let the system of equations be a;2 _!_ (82^ _ 13) ^ _|- 2,2 _ 7^^ _l_ 12 = ^2 _ (4y _j_ 1) ^ _^ 2/2 + 5y = 0. Here the coefficients having no common measure, these equations may be regarded as the equations A = 0, B = 0, treated above; and from these we are to determine, agreeably to the general 394 THEORY OF ELIMINATION. theory, the system B = 0, R = 0, which will contain all the solu- tions required. Dividing A by B, we have a?^-(4y4-l)a?4-2/2^5y]a?2-|-(8y— 13>-fy2_7y^.i2 [l a;2— (4y+ l)a;-\-i/^-\-5i/ R=(12y— 12>-12y4-12=12(y-l)(^-~I), the equations which furnish the solutions are, therefore. .T-1=0 a?2 — (4y + 1 ) a? -f ?/2 4- 5y =0, and each of these systems may be solved without repeating the divisions; the solutions are y-1 3/ = l y = o y=-l x = 3 a? = 2 X=: 1 x= I. 2. Let the equations a?3 + 2ya?2 + 2y (y — 2) a? + 2/2 _ 4 = a?2 4- 2ya • +2f- 5y + 2 = be proposed. The coefficients having no common measure, we have, by dividing the first polynomial by the second, the following remainder, viz. R = (y - 2) a? + 2/2 _ 4 = (y - 2) (a^ 4- 2/ + 2); hence the solutions to the proposed equations are those of the systems y-2=o x2-\-2yx + 2f - 5y + 2 = a? + y + 2=0 a'2 + 2i/x -r 2f — 5y + 2 = 0. The first system furnishes the solutions y = 2 a? = y= 2 a; = — 4. THEORY OF ELIMINATION. 395 To solve the other system, proceed as at first ; that is, divide the second polynomial by the first, and there will result the remainder S^^ — 5y + 6 ; hence the system is replaced by the new system 2^2 _ 52/ + 6 = X + y + 2 = 0, which gives for solutions y = 2 y = 3 X = — 4 X = — 5; so that there are, in all, four solutions to the proposed equations. 3. Let the equations (y — I) x'^ + 2x — 5y + 3 = yx^ + 9c» — lOy = be proposed. Multiplying the first polynomial by 3/, to render it divisible by the second, and then performing the division, we have yx^ -\- 9x — lOy] (y — 1) t/x^ + 2i/x — Sy^ ^ Sy [y — 1 (y - 1) yx^ + (9y - 9) a? - 10y2 + ]0y (— 7y + 9) X + 5y2 _ 7y, As we have multiplied the dividend by the factor y, the equa- tion y = may be a solution to test. Substitute for y in the proposed equations; one, viz. the divisor, furnishes the value a? = 0, which value does not satisfy the other ; hence the factor introduced supplies no solution. We must now proceed with the polynomials B and R ; and, in order to this, must multiply 396 THEORY OF ELIMINATION. the dividend B by (— 7y + 9), and we shall have, for the re- mainder arising from the division, the polynomial 25y^ - 70/ — 126y3 + 414/ _ 243y. The final equations are, therefore, (- 7y + 9) ^ + 5y2 - 7y = 25/ - 70/ — 126/ 4- 414y2 _ 243y = 0; the roots of the second are 2/= 0, y — \y y = 3, y -3 ±3 VIO and to these correspond the following values of x, deduced from the first, viz. a? = 0, X — \y a? = 2, a; = — 5 q= VlO. No extraneous solution has been introduced by means of the factor — 7y+9, by which we have multiplied the second dividend, because none of the above values of y cause it to vanish ; but an inadmissible solution has been introduced by the factor y, which multiplies the first dividend, viz. the solution y ^ 0, a; = 0; re- jecting this, therefore, we have, for the entire number of true solutions, the four systems following, viz. y = i y = 3 — 3 -f3\/T0 — 3-3 \/T0 ^- 5 ^- 5 a? = 1 a? = 2 a?= — 5 — VTO a? = — 5 -f VTo 4. Let the equations proposed for solution be (/ - 1) a'2 + (2/ -2y)x A-y^- 2y^ -f 1 = (/ - 3?/ + 2) a;2 - / — 3/ -r 7/ + 15y - 18 = 0. The coefficients of the first polynomial admit of the common THEORY OF ELIMINATION. 397 divisor y^ — j . and those of the second admit of the common divisor y^ — 3y + 2; these two factors have themselves a common divisor, vrhich is y — 1 ; so that the proposed equations may be written thus : {y - 1) (y + 1) {^' 4- 22/^ + y2 -- 1) = (y - 1) (y - 2) (^2 _ y2 _ 6y- 9) = 0. These are satisfied by the values y = 1, combined with any value of a; whatever, as observed at page 391. They are also satisfied by the values which satisfy y + l=0, a;2-y2_6y_9=:0; which values are y =i — 1 y= — 1 x= 2 x=—'2 They are also satisfied by the values which satisfy y - 2 = 0, a?2 ^ 2ya? + 2/2 -. 1 = 0; which values are y = 2 x= - 1 y = 2 = — 2. The remaining solutions are involved in the equations A = 0, B = 0, (page 391), that is, in the equations a;2 ^ 2y^ + y2 _ 1 _ ^2 _ 2/2 _ 6y - 9 = ; to which we may apply the method of the common divisor ; but, as it is easy to see that the second equation gives a; = ± (y + 3), we may substitute these values in the first equation. The first value, y + 3, will reduce it to y2 + 3y + 2 = ; 398 THEORY OF ELIMINATION. which furnishes the values 2/ = — l,y = — 2; and, from the relation a? = 3/ + 3, we have, for the corresponding values of a?, a? = 2, a: = 1 . If we substitute, in the first equation, the other value, — (y 4- 3), for x, it will be reduced to 8 = ; this value, therefore, furnishes no solution. 5. As a last example, let the equations (y — 2) :r2 — 2a? + 5y — 2 = yx^ — bx -\- 4y •=■ Oy be taken. The coefficients having no common divisor, we at once com- mence the operation for finding R ; but, to avoid fractions in the quotient, we must prepare the dividend by multiplying it by y. ya?2 — 5a? + 4y] (y — 2) ya?2 — 2ya? + 5y2 — 2y [y — 2 (y-2)ya?2-(5y- 10)a? + 4y2-8y R = (3y-10)a? + y2+6y. It is necessary now to repeat the operation with B and R ; and, for- this purpose, we must multiply the dividend B by 3y — 10 ; the resulting remainder will be found to be y5 + 12y4 + 87y3 — 200y2 + lOOy ; so that the final equations are (3y — 10) a? H- y2 + 6y = y^ + 12y4 + 87y3 — 200y2 + lOOy = 0. The second of which is satisfied, for y = 0, to which cor- responds a? = 0, in the first ; but this is an inadmissible solution, as it does not satisfy the proposed equations. It is due to the factor y introduced in the first division. Suppressing this factor, the final equation in y becomes yA _,_ i2y3 _^ 87y2 _ 200y -f- 100 = ; IRRATIONAL EaUATIONS. 399 which cannot involve any inadmissible values of y^ because the only circumstance which could cause their introduction is the introduction of the factor, 3y — 10, in the second division, and this is reduced to zero, by the value y = — . But, as this value o is fractional, it cannot be a root of the equation above (62). We also see, from other causes, that the factor, 3y — 10, can intro- duce no solution ; the conditions 3y — 10 = 0, (3y — 10) a? + 2/2 + 6y = 0, are incompatible. The final equation in y has the root y = 1, to which cor- responds a? = 1, and the depressed equation in y is y3 + 13y2 + lOOy — 100 = ; the roots of which involve interminable decimals. Hence, the remaining solutions can be obtained only by approximation. In the Chapter next following, a method will be found of ob- taining the final equation in y, which shall comprise all the solutions to the proposed equations, and be unembarrassed with inadmissible values. We shall now proceed to one or two appHcations of the theory just delivered. On Irrational Equations. (240.) All the direct methods employed for the solution of equations suppose that the unknown quantities in them are not affected with any radical sign; when therefore, the unknown is found under a radical sign, it will be necessary, before applying the process of solution, to employ some preparatory method of rendering the equation rational. Such a method is at once suggested by the theory of elimination. For, if we equate each of the irrational terms with an unknown quantity, and remove the radical from each of these new equations by involution, we shall have a series of equations (including the original one, with its 400 THEORY OF ELIMINATION. irrational terms replaced by the new symbols,) without radicals, from which the quantities, temporarily introduced, may be ehminated, and thence a rational equation obtained, involving only the original unknown quantities. The following examples will fully illustrate the mode of pro- ceeding : 1 . Let the equation be X — sj x— 1 + ^x +1=0. Put ^ y ■=. sj X — 1, z ■=■ '\/ X -\- 1 ; and we then have the three following rational equations, from which we may eliminate y and z, viz. y'^ z=.x —\y ^■=- X -|- 1, a? — y + ^r = 0. From the last equation we get y •=.x •\- Zy and, by substituting this value in the first, y becomes eliminated, and we have these two equations in x and Zy viz. ^ — a? + 1 =0 z^ -V 2xz -^ x^ — X -\- \ = ; and, to eliminate z from these, we apply the process explained in the preceding articles, and thus get the final equation a^ — 3a;5 + 8a^ + ar* -h 7a?2 — 7a? + 2 = 0. 2. Let the equation be ^^4^? + 7 + 2 Va? — 4 = 1. Putting y=^4a? + 7, s'=\/a: — 4, we have the system of equations y3 = 4a? + 7, c2 = 0? - 4, y + 2.-= 1. METHOD OF TSCHIRNHAUSEN. 401 From the last we find y = \ — 2z, and this value of y, substi- tuted in the first, reduces the system to the two equations 8^3 _ 12^2 _,_ 25r + 6 = from which, by the process already explained, we obtain the final equation, 16x3 - 184x2 + 801a: - 1405 = 0. It will be remembered, in conformity with the remarks at (46) that the operations above, by means of which irrational equations are rendered rational, introduce foreign roots into the final result whenever the signs prefixed to the original irrational quantities are intended to indicate the character of the roots. The rational equation will always have a greater number of roots, or be satisfied for a greater number of values of a?, than the irrational equation which it is intended to replace, unless the signs of the irrational terms be perfectly unrestricted. Method of Tschirnhausen for Solving Equations. (241.) As another application of the theory of elimination we shall briefly illustrate the principle upon which Tschirnhausen proposed to accomplish the general solution of equations, but which, as observed at (81), was soon found to be of but very limited application, not extending beyond equations of the fourth degree; and even within this extent too laborious for general use. The principle consists in connecting with the proposed an aux- iliary equation of inferior degree with undetermined coefficients, and of as simple a form as possible consistently with the office it is to perform, but involving, besides the unknown quantity a?, a second unknown y. The unknown, common to both equations, is then eliminated according to the preceding theory, and a final equation in y thus obtained, of which the coefficients are functions of the undetermined coefficients in the auxihary equation. The arbitrary quantities, thus entering the coefficients of the final equation in y, are then determined so as to cause certain of these 26 402 THEORY OF ELIMINATION. coefficients to vanish ; by which means the equation is ultimately reduced to a prescribed form, supposed to be solvable by known methods. (242.) As an example, let it be required to reduce the cubic equation ix^ -\- ax^ -\- bx + c = . . . . [l} to the binomial form f -{- k = 0. Assume an auxiliary equation x^ + a'x + b' -\- y = . . . . [2] and eliminate x from [1] and [2] in the usual way. The re- mainder arising from dividing the first member of [1] by the first member of [2] is (^'2 _aa' -{-b- b'— y) x -f- {a' - a) (b' + y) + c which equated to zero gives. {a - a') Jb' + y) - c a'2 — aa' + b — b' --y and this value of Xy substituted in the proposed equation, trans- forms it, after reduction, into the form y^ + hy'' + iy\-k=0 . . . . \3-] where h = 3b' — aa' •\-a^-2b i = 36'2 _ 2b' {aa' - a^ + 2b) + a^b + (3c - ab) a' + ^2 _ 2ac k=b'^ — ab'-^a' + bb'a"^ ~ ca'^ + (a2 _ 2b) b'^ + (3e — ab) a'b' + aca''^ + (^2 — 2ac) b' - bca! -f . . . . L k k k k and to detect the negative roots to substitute in like manner the numbers _!,_?, _£,_!,.... _L' k k k k the former series will furnish as many changes of sign in the ab- solute number as there are positive roots, and the latter series as many as there are negative roots. But fractional substitutions may be altogether avoided by trans- forming the proposed equation into another whose roots are k y times as great (75) ; that is, by substituting - for x. The trans- formed equation in y will thus have the diiFerences of its roots k times the differences of the roots of the equation in x ; and as - is less than any of the latter differences, 1 must be less than any of the former differences, that is, the differences of the roots of the equation in y are all greater than unit; so that the roots will all be separated by means of the two series 1, 2, 3, 4, . . . . KL .-1, _2, —3, ^4, . . . . —kV As the squares of the differences of all the real roots are positive, it follows that if negative roots occur in the auxiliary equation they must arise from the imaginary roots in the proposed. And moreover, if any of the roots of the auxiliary equation are zero the proposed equation must have equal roots. (246.) It is easy to see that the foregoing method of separating the real roots, and, consequently, of discovering the number of imaginary roots in an equation is infallible; but, as we have before observed, the great length of the calculations which are necessary EaUATION OF SQUARES OF THE DIFFERENCES. 407 to the formation of the equation of the squares of the differences, when the proposed equation is above the third or fourth degree, renders the method nearly impracticable. This is now no longer a matter of regret, as the solution of the important problem of the separation of the roots has been rendered, by the researches to which the preceding chapters have been devoted, altogether in- dependent of the equation of the squares of the differences ; this latter problem, therefore, will henceforth be regarded with interest only on account of its connexion with the name of Lagrange, and with the history of algebraical research. We advert to the problem here merely to explain its meaning and object to the student, and to furnish an additional example in elimination. (247.) Let the proposed equation be /(*) = [1], and let a be any one indifferently of its n roots, aj, «2» % • • • ^n5 then, in order to obtain an equation whose roots may be the dif- ferences between those of the proposed, it will be sufficient to establish the relation y •=. X — «, or iJ? = « -|- y; which transforms the proposed into f{a -f y) = 0; of which the development is /(«) + /. («) y + /2 («) ^ +/3 w rfe + &«• = 0: but since, by hypothesis, a is a root of the proposed equation, /(«)=0; hence, suppressing this term, and dividing by y, we have /,(«) +/2W -[^2 +-^' W Y^3 + = «■•• [2] 408 THEORY OF ELIMINATION. the roots, or values of y, in this equation, are, by the condition above, the diiFerences between any assumed root, «, and the n — \ other roots of the proposed equation. By putting, in suc- cession, all the values for a — that is, in fact, all the values of x, deduced from [1] — in [2], the corresponding values of y, will, together, furnish all the possible differences between the roots of [1]. In other words, all the possible differences will be obtained by substituting the values of x, deduced from the equation /W = 0, in the equation /iW +/2W^ +/3 W iJs +••••■• = " ■' and this is tantamount to saying that the differences sought arise from the solution of this system of equations. It is easy to foresee the degree of the final equation in y arising from the elimination of x from these two equations ; for, as its roots are equal to the remainders obtained, by subtracting from each of the n roots of the proposed, all the other n — \ roots in succession, there are, obviously, in the whole, n{n — 1) roots or remainders, hence the final equation, furnishing these roots, is of the n(n — l)th degree. (248.) We shall apply this process to the following equation of the third degree xr^ -{- px -{• q =: 0, where /(a?) = x^ -\- px -\- q, f^{x) = 3x^ + p /2W = 6a?, f.^{x) = 6. Hence the two equations, from which x is to be ehminated, are a^ -^ px •\- q =^ 3^'-' + 3y^ 4-2/^+^=0, EQUATION OF SQUARES OF THE DIFFERENCES. 409 and the operation is as follows : 3x^ + 3y^ + y2 4.^] 3^ _^ ^px + 3g [a; — y 3^ + 3i/x^ + {f +p)x ■ — 3ya?2 — 3y2a? — ^3 _^2/ 2(2/2 + p)x -f- y3 _^ ^y _j. 3^ This remainder is now to be taken for a new divisor, and the former divisor for a dividend, and the operation continued ; but as the foregoing divisor, B, does not admit of division by the re- mainder, R, in its present form, we multiply it by the factor 2(2/2 4- ^), in order to render the division possible, and proceed with the operation as foUows : or, mult, by 2 (!/-'-\-p), 6{i/^-^p)(i/^^pi/—Sq)x-\-4{y^-{-py In the second division we have multiplied twice by ij^ + p, in order to render the division possible ; but this factor introduces no extraneous value of y, for the value which reduces it to zero, being given hjy^ -j- ^ =: 0, reduces the divisor, R, last employed, to qy and not to zero, as it ought, to be a solution. The final equation in y, involving the values sought, and those only, is, therefore. 410 THEORY OF ELIMINATION. which is the equation of the differences ; and, by putting z iory^, we have ^ + 6^^2 _|. 9p2^ + 4^)3 + 27^2 = for the equation of the squares of the differences. In the particular case of the equation ' a:3 — 7a: + 7 = . . . . [1] we have pz= ^7, q = 7, and therefore the equation in z is 2^ _ 42^2 + 441^ _ 49 = . . . . [2] In order to find an inferior limit to the positive roots of this equation, we change it into another, of which the roots are the reciprocals of those of [2], agreeably to the directions at p. 101. This reciprocal equation is 42 1 to the positive roots of which 10 is a superior limit (86) : we may ascertain whether this is the closest superior limit by trans- forming the equation successively by 9, 8, &c. We thus find that 9 is a limit still closer. Hence, in the present case X; = 3 ; and consequently a = ^. Transform now the proposed equa- tion into another whose roots are three times as great : that is, put - for x in [1], and we thus have tj 2^3 _ 632/ + 189 = ... . P] in which we have only to substitute in succession the numbers 1, 2, 3, &c. for y, and we thus find the only intervals within which the absolute number changes sign to be [4, 5] and [5, 6]. Hence the equation [3] has two positive roots, one in each of these intervals, and consequently the proposed equation has two positive roots, one between ^ and f and the other between f and ^. The situations of these roots have been otherwise determined at pp. 198,381. EQUATION OF SQUARES OF THE DIFFERENCES. 411 It is obvious that every equation of the differences, as well as that just deduced, will be of an even degree, and will contain only even powers of y ; because every root, as «j — «2> is accom- panied by another a^ — «p the roots being equal to the differences «j — «2' ^1 — «3, . . . «j — a^j «2 ~~ ^V ^2 — %» . . . «2 — ^n» ^^* of the roots of the proposed equation ; so that the polynomial in y is of the form (y - a) (y + a) (y - /3) (y + /3) , or of the form (y^-a2) (2,2^/32)...., and therefore involves only even powers of y. (249.) If in this expression we put y equal to zero, and change the sign of the result, we shall obtain an expression for the last of Sturm's functions, or for that function multiplied by a posi- tive numerical factor. This will appear evident from comparing the investigations from which the functions of Sturm, and the equation of the squares of the differences, respectively result. Hence, calling the numerical factor adverted to, P, the final func- tion of Sturm will be expressed by P(a/3y . . . .)2 where a, /3, y, &c. represent the differences of the roots of the proposed equation. If two of the roots «i, «2> <^8j &c. have leading figures in com- mon, then one of the preceding differences will have so many blank places, where otherwise there would have been significant leading figures ; and, consequently, the above-written square will have twice that number, or twice the number plus or minus 1, of leading places blank that would otherwise have been occupied by significant leading figures. We may infer therefore that when two roots have leading figures in common, Sturm's final remainder will, in general, be preceded by twice that number of blank places, or twice that number plus or minus 1 . s. 412 THEORY OF ELIMINATION. If three roots have any number of leading figures in common, then three of the dilierences, a, /3, y, will each have that number of places vacant, which would otherwise be occupied with leading significant figures : their product will therefore have in general about three times that number of unoccupied leading places, and the square of the product may, consequently, be expected to have six times that number — at least within about three or four places. Hence, in the case of two nearly equal roots, half the numbei* of blank places in Sturm's final remainder — or that half plus or minus 1 — will, in general, denote the number of places in which the two contiguous roots concur. In the case of three nearly equal roots one sixth the number of blank places — or that sixth plus or minus 1 — will, in general, denote the number of places in which the three contiguous roots concur. And, generally, in the case of m nearly equal roots, if the number of blank places, that would have been occupied by leading figures in Sturm's final remainder, had these roots difiered in their leading figures, be divided by m{m — 1), the quo- tient — or the quotient plus or minus 1 — will, in general, denote the number of places in which the m nearly equal roots concur. When m exceeds two, blank places will occur in one or more of the preceding remainders, according to the excess of m above 2 (p. 316) : these blanks are, of course, to be added to those addi- tional blanks by which the final remainder is preceded. (250.) In each of these cases it is plain that the function of the first degree, preceding the final remainder, will be an ap- proximate common measure of the original polynomial X and its derived function Xj ; and this approximate common measure must evidently have the character of a true common measure as far as the final blank places or zeros extend, since if a true common measure could exist, this approximate one would thus far coincide with it. In the case of two nearly equal roots, therefore, the function of the first degree, equated to zero, will give for x a value lying between the two roots of X = 0, and to as many places as there are blanks in the final remainder, actually coinciding with the EQUATION OF SQUARES OF THE DIFFERENCES. 413 root of Xj = lying in the same interval ; seeing that the simple equation actually divides this latter to the extent specified. As the function of the first degree is thus an approximate com- mon measure of X, Xj, it is of course equally so of the subse- quent functions of Sturm. This is the principle that we have applied to the analysis and solution of equations in the introductory treatise, page 224. In the case of three nearly equal roots, the root of the function of the first degree, being still the nearest to the common measure, within the extent mentioned, will not coincide throughout that extent with a root of either of the preceding functions, but must interpose itself between each of the two contiguous roots that enter into all these, up to Xj, inclusive ; just as in the case above, it interposed itself between the two contiguous roots of X =0. Similar remarks apply when the nearly equal roots exceed three in number.* • As observed in the introductorj' treatise the connexion between the blank places in Sturm's final remainder, and the concurring figures in two nearly equal roots, was first noticed by Mr. Ruthehford. We know not whether any proof of the principle in this particular case, or any generalization of it, has been given. CHAPTER XVI. ON TEIE SYMMETRICAL FUNCTIONS OF THE ROOTS OP AN EQUATION. (251.) A symmetrical function of the roots of an equation is any expression in which all the roots are similarly involved, so that any of them may he interchanged without affecting the form or composition of the function. The coefficients, for example, of every equation are each of them symmetrical functions of its roots ; for it has been shown (60) that if the roots of any equa- tion be ttp «2' «3 • • • • «n» *^® successive coefficients will be the following functions of them, viz. — («1 + «2 + «8 + • • • • «n)' ai«2 + ^1% + ^2% -h . . . . «„_! a„, — («1«2«3 + «1«'2«4 -+- • • • • «n-2«n-l«n)' and each of these is a symmetrical function, because, however we interchange the roots, the function itself will remain unchanged. SYMMETRICAL FUNCTIONS. 415 The preceding forms are, we see, immediately given by the co- efficients of the proposed equation ; and it is the object of the present Chapter to show that not only these, but every other rational and symmetrical function of the roots, may always be expressed in terms of the coefficients, without the aid of the roots themselves. Determination of the Sums of the Powers of the Roots of an Equation. (252.) As usual, let us represent the general equation of the nth. degree by f(x) =^" + A,^_, x^-^ + A„_2a?«-2 Aa? + N = [1], and its roots by fl!j, «2> ^3J ^41 .... a^. Then /j (a?), being the first derived function from the polyno- mial [1], we know (97) that f(x) 111 1 f (x) X — a, X — «2 ^ — % and, consequently, fix) fix) fix) f(x) '^ X — «, X — «o X — «o X — «.. Performing now the actual division for any one of these frac- f(x) tions, as, for instance, for the fraction , or, which is the X - «,„ same thing, depressing the original equation [1], by any one of its roots, a^y we shall get the polynomial which follows, the co- efficients being formed by the rule at (51), 416 SYMMETRICAL FUNCTIONS OF THE A n-l"^m x^-'^ + aJ ^n-l'*m -^11-2^111 A„-.s A n ^~^ -^11-2^111 N; and this will equally represent the development of either of the other fractions, by putting the corresponding value for m. Con- ceiving, therefore, m to be successively 1, 2, 3, &c. to w, and, putting for abridgment, ^m = «1™ + «2"' + %"' + ^n*"' we shall, obviously, have for the sum of all the developments, that is foryj(a?), the polynomial •n-l+X, i«!"-2+22 x"-»+Zj, x"-*.. ••^„-i «A„_, A„-,2i K-^\ K-^\-2 »A„_2 A„-2 2, ^lA„_, nN. . But the development of/j(a7) is also, p. 110, na?n-i + (w— 1) A„_ia?"-2 + (w— 2) A„_2;z*"-^ + . . . 2A2a: + A hence, by the method of indeterminate coefficients, 2, + ^A„_, = (w,~l)A„_, ^2 + A„_, Si -f ^A„_2 = (/I - 2) A„_2 I3 + A„_, I2 + A^.oS, + A„_3 = (w - 3) A„_3 2:„-, + A„_, I„_, + A.,_, I,_, + A,_,2.._, + . . . . .,N = A ; ROOTS OF AN EQUATION. 417 that is, ^1 + A„-i = ^3+An-i2:2tA„_,2j+3A„_3 = ^^n-l + A„_i 2„_2 + A^_2\_3 + .... (/I - 1) A = 0. By means of these equations the functions Ip Sg, I3, &c. may be easily calculated in succession up to the function 2„_j. (253.) The foregoing equations may be extended so as to in- clude the functions \, \^^, 2^^2» • • • • ^n+p; foi** from the original equation, we have «i" + K-i «i""' + A„_2 «i"-2 + A«, + N = «2" + A„_j a.,--^ + A„_2 «2"-2 + A«2 + N = «n" + A„_, «/-! + A„_2 «„"-2 + A«„ + N = ; and, by multiplying these equations respectively by AjP, a^ . . . , a„P, and adding them together, there results the equation ^^n+p + A„_i2„^p_, + A„_2 2:„^p_2 + . . . . A2p^, + N2p = 0, which, by putting 0, 1, 2, &c. for^, furnishes the following con- tinuation of the foregoing relations, viz. 2n + K-x \-i + A„-2 2„_2 + . . . . A2, + n^* = ^^n-.! +'A„_,2„ + A„_2 2„-, -H . . . . A22 + N2, = \^2+ K-i \-.i + A„_2l„ + . . . . A23 + N22 = ^^n+p + A„_i 2„^p_i -h A„_2 2„^p_2 -t- . . . . A2p^, -f N2p = 0. • It is plain that ^o = ^k + «2° + «3° + • • • • «n'' = ". 27 418 SYMMETRICAL FUNCTIONS OF THE Hence, by means of the coefficients merely, we may calculate the sums of the powers of the roots of an equation, in suc- cession, to any extent ; and it is plain, from the foregoing ex- pressions, that the several sums will all be integral functions of the coefficients. As a particular application of the preceding general formulas, let it be required to find the sum of the sixth powers of the roots of the equation a^ + a.3 — 7^2 _ ^ ^ 6 _- 0, 2, = -A„.,=-l 22=-A„_i Ij~2A„_2=l + 14 = 15 23=-A„_,22-A„_22,-3A„_3=-15-7 + 3 = -19 24=_A„_i23—A„_2 22—A„_32i—4A= 19 + 105-1 -24 = 99 25=-A„_i24-A„_2 23-A„_3 22-A2i = -99-133 + 15 + 6 = -211 26=-A„_i25-A„_2l4-A„_3 23-A22=211 +693-19-90=795. If the sums of the negative powers of the roots of an equation be required, we might derive suitable formulas from the general table above, by considering p to be negative; but it will be preferable in this case to transform the equation to another in - by (73), and then to employ the formulas in their present state.* (254.) By means of the general expressions in last article, we may find the values of the coefficients Aj^_i, A^^.g, &c. in terms of the sums of the powers of the roots; thus: * For another mode of investigating the expressions for the sunos of the powers of the roots of an equation, and for the use which Newton and Lagrange made of these sums for approximating to the greatest root, see the Author's Essay on the Computaiion of Logarithms^ page 97, second edition. ROOTS OF AN EQUATION. 419 A„-, = - 2:, ^2 + ^n-1 ^l -^11-2 2 K-3 - 3 » ^4 + ^n-i ^3 + An_2 Ilg + An-3 ^| An_4 — ^ &C. &C. Determination of any Combination of the Powers of the Roots of an Equation. (255.) By multiplying together the two expressions Ip = «iP + ^2^ +a.^v + «^P + , we have the two following series of partial products, viz. + ttl*" a^ + «i™ ^3? + «i'" ^4? + «2'" '^1^ + Each of these series is a symmetrical function of the roots ; the first being the sum of their m -\- p powers, and the second being the sum of the products of every two roots raised, the one to the power m, and the other to the power ^. This latter func- tion may be represented briefly by S {a{^ a^) ; so that we shall have Hence the function S (a™ a.^) is determinable in terms of the coefficients of the equation. 420 SYMMETRICAL FUNCTIONS OF THE Again, if we multiply together the expressions S(«i™ a./) = «!>" a^P + «j™ a./ + a,™ «/ + ^2*" Oj^ + .... and 2q = «,^ + aj"^ + ^s"^ + 04'* + .... we shall have a result consisting of three series of partial pro- ducts, the terms of each distinct series involving like combinations of the roots ; viz. The first series will consist of the products of every two roots raised, the one to the power m -j- q, and the other to the power ^, and which series may be denoted by S {a^^^^aj'). The second series will be formed of the products of every two roots raised, the one to the power p + q, and the other to the power m, which series may be expressed by S (a^^'^'^az^). The third series will be the products of every three roots raised, one to the power m, one to the power p, and one to the power q; and which will be represented by S {a^^ai a^). That is, we shall have S (« ™ a/) X 2y = S (ai"^+q «/) + S {a,^^^ a,^) + S {a,^ «/ a,^), and, therefore, by transposing and replacing the functions, by their values in last page, we have by which equation the triple function S {a^^ a^ a.^) may be obtained in terms of the coefficients. By continuing this process of deduction, we may obtain ex- pressions for the succeeding combinations. The functions thus determined are called the elementary symmetrical functions, and it is from the union of these that every complex, rational and in- tegral, symmetrical function is formed. We shall give a few examples of these combinations in the following article. Before proceeding to these, however, it may be proper to ROOTS OF AN EQUATION. 421 show how the above general functions become modified, when the exponents m,p, q, &c. are not unequal. The expression S («j™ a/) is truly the representation of «i"^«/ + ^i"" «3^ + ai™«/ 4- + a^'^a^^ + a^^a^^ + . . . . only when m and p are unequal ; for, when m = p^ this series consists of terms which are equal two and two ; so that, in that case, only half the entire sum will be expressed by S (aj^ag™). Hence sC^i'"^'") (SJ^-22. 2 For similar reasons, Lastly, when the exponents in this latter function are aU three equal, the terms represented will be equal six and six ; so that S («i™«2'»«3-) = (?HZ_:??2m?HL±^ Transformation of an Equation into another wJiose Roots shall be given Functions of those of the original Equation. (256.) Let it be required to form the equation whose roots are the sums of the roots of the equation f{x) = 0, taken two and two. If, as usual, we represent the roots of the proposed equa- tion by «j, «2> ^3» «4 . . . . any those of the transformed equation, F(y) = 0, will be «1 + «2» ^1 + '''3' «1 + «4» ^2 + ^3» ^^'i 422 SYMMETRICAL FUNCTIONS OF THE and will amount in number to the number of different com- binations which can be formed with the roots of the proposed, taken two and two. If each were to be combined with every one of the other roots, the whole number of combinations would obviously he n(n — 1) ; but it is plain that every combination would then occur twice; so that the correct number of com- binations must be — .* Hence the number 2 2 denotes the degree of the transformed equation. Let us proceed to the composition of its coefficients. The sums of the powers of the roots of the transformed equa- tion will be expressed by the formulas 2'i = («i + «2) + («i + %) + (^1 + «4) + («2 + %) + &c. = 2(«i-f«2+%+«4+^c.)=2S(«j) 2 («!* + «2'^ + ^'^ + «/ + &c.) -h 2(« jtts -f «ia3 + «i«4 -f- «203 -I- &c.) =2S(ai')-|-2S(«ia2) X'3= («i + (hf + («1 + «3)' + («1 + «4)'+ (%+ «3)3 +&C. = 2(«i^ + tti + ai + «/) + 3 {a^a^ -\- a^a^ + a^a^ + a^a^ + &c. =2S(«i')+3S(«>a) &c. &c. Hence the sums of the powers of the roots of the transformed equation may be obtained in terms of the sums of the powers of the original roots and their elementary combinations; the sums of the powers being thus known, for the transformed equations, the coefficients of this equation are found by the formulas, at (254), as in the following example : • The doctrine of combinations and permutations is given in almost every English treatise on Arithmetic. ROOTS OF AN ECIUATION. 423 Let it be required to transform the equation, x^ + Azx"" + A.r + N = 0, into another, f + Ay -f AV + N' = 0, whose roots shall be the sums of the roots of the former equation, taken two and two. Let us first calculate the values of 2i, Sg, I3, as at page 418; 2, = - A, ^2 — — ■'*-2 "1 ~~' •<"■'*- — Ag ~~" ^ix 23 = - As Sg - A2i - 3N = - A^^ + 3AA2 - 3N. The value of ^(a^a^ is, by p. 421, A; and for ^(ai'as), we have (255) S(«i''«2) = 2^ X 2i - 23 = - AA2 + 3N. Consequently, for the expressions on last page we have 2\ = 22i = - 2A2 2^2 = 2^2 + 2S (a,«2) = 2A22 - 2A 2^3 = - 223 + 3S («i2 ^2) = - 2A23 + 3AA2 + 3N. Finally, by the formulas (254), A'2=-2^=2A2 A' ^_^2 + A2^1^^2^^A y/ _L. A' T' -L \'Y/ N'= -r3±±2£2_+^Jli^A2A--N. 424 SYMMETRICAL FUNCTIONS OF THE Hence the transformed equation is 2/3 + 2A2y2 + (A22 4- A)y + A2A- N = 0. (257.) By proceeding in a similar manner, we may form an equation of which the roots are combinations of those of the original equation, of the form «j + Cg + ^«i«2' ^i + % + ^«i«»» &c. k being any given number. The degree of this equation will, of course, be the same as the degree of that formed from the sums of the roots ; it will, therefore, be denoted by the number Tiin — — 1^ — ^ . The expressions for the sums of the powers of the roots of the transformed equation are 2'j = («j + «2 4- ha^a^ -f («j + % + ka^a^ + &c. = (w — 1) S («i) + ArS («i«2) 2^2 = («i + «2 + ^«i«2)^ + (^1 + ^3 + ka^a^'^ + &c. = (w - 1) S («i2) + 2S («i«2) + 2AS {a^ai^ + A:2S (ajV) S'g = («j + «2 + ^«ia2)^ + (^1 + H + ^«i«3)^ + ^c. = (w - 1) S (ttj^) + 3S («j2«2) + 3^s (fli^g) + ^li^Wa^) + &c. &c. from which it is evident that the coefficients of the transformed equation may be expressed in functions of the coefficients of the given equation. (258.) As a second application, let it be required to form the equation of the squares of the differences of the roots of a given equation. The proposed equation being f{x) = 0, and its n roots as before, the roots of the transformed equation will be («1 — fl'a)^, (0, — flr.^)2, (ffj-fl4)2 _ _ («2--ff3)2, {(t^ — a^'^ .... ROOTS OF AN EQUATION. 425 the number of which, obviously, amounts to the number of com- binations, two and two, that can be formed with the n quantities, ttj, ffa, «3, &c. Hence the degree of the required equation is Ti(n — 1^ — ^^ —; and, to find its coefficients, we must, as before, first determine the values of S'j, 1/^, ^'3, &c. by the following for- mulas : i\ = («j — a^y + («i — a^y + («i — «4)2 -f &c. = (w - 1) S (ajS) _ 2S («i«8) = (/I - 1) S («i4) - 4S (a^^a^) + 6S (a^W) 1^3 = (a^ - a^y + (a, - a.^^ + (a^ - a^y + &c. = (w - 1) S («i6) - 6S («,^«2) + 15S (aiV) - 20s (a,3a,3) &c. &c. As a particular example, let the equation, already considered at page 410, viz. a^ - 7a? + 7 = 0, be proposed, in which A2 = 0, A=- 7, N=:7. By the formulas at p. 418 we have X, = 0, l,=z 14, 23 = ~ 21, I4 = 98, I5 = - 245, Xg = 833. Consequently, S («j2) = X, = 14 S («,4) = X4 = 98 S (ai6) = Xg = 833. 426 SYMMETRICAL FUNCTIONS OF THE Also, from (255), S («i«2) = A = — 7 S («i%2) = - 2:4 = — 98 SKS«2) =25^1-2:,=: -833 s«<) = (?^)!:^^ = l^^:^ = 49 S(aiV)= 245:2 -26 = 539 S («iV) = ^^'^'~^' = - 196. Therefore, making these substitutions in the formulas on last page, r, = 2S («i2) _ 2S {a^a^) = 42 X's = 2S {af) - 4S (a^^a^) + 6S («i'^«2^ = 882 ^3 = 2S («i6) « 6S («i5«2) + 15S (aiffg'') -20s (aiV) = 18669; and hence, finally. A'2=--ri=-42 A' = - ^'^^^^^'^ ^ 441 T' A- K' T 4- A'Y' N' = — 3 ^ ^2^1 ^ ^ ^1 _ _ 49 SO that the transformed equation is y3 __ 42y2 ^ 44iy _ 49 = 0. ROOTS OF AN EQUATION. 427 On the Degree of the Final Equation, resulting from the Elimina- tion of one of the Unknown Quantities from two Equations, containing two Unknowns. (259.) Let the two equations be F(a?)=a?°» + B„_ia;"^-i + B^_2a?™-2 + ....B3«2+Ba?+M=0....[2]; in which the coefficients, A, Ag, A^, . . . ; B, Bg, B3, . . . are functions of y. If we could resolve the first of these equations, we should obtain for a?, n values, «, b, c, &c. which would be functions of y, and which, when substituted in the second, would furnish the n equations, F(a)=«"» + B^_ia'»-i + B^-2«™"^+ • • • • B2a2 + Ba + M=0 F(5) = 6™ + B^_i6°'-i+B^_2 6'»-2+ .... B^b^+Bb + M^Q F(c) = c™+B„_iC"»-i+B^_2C™-2+ .... B2c2+Bc+M=0 and these equations being solved for y, would make known the corresponding values of this quantity. It is, however, in but few cases that we can actually solve the equation [1] for a? ; if we could, the determination of the corre- sponding values of x would not require the solution of the n sepa- rate equations, just obtained, because they may all be combined in a single equation, viz. the equation F(«) Y(b) F(c) . . . . = . . . . [3] ; and it is plain that the product, which forms its first member, undergoes no alteration, however we interchange a, b, c, &c. in 428 SYMMETRICAL FUNCTIONS OF THE the factors ; that is, this product will contain none but rational and symmetrical functions of the roots of the equation [1]. Hence, the first member of the equation [3] may be determined, by means of the coefficients of the equation [1], and the final equation in y, thus obtained. As an example, let us take the equations (y _ 2) a;2 — 2a; -t- 52/ — 2 = yx^ — 6x -f- 4y = 0. Let us represent the values of x in terms of y, which satisfy the second equation by a and b. These, substituted in the first, furnish the two equations, (3/ — 2) a2 — 2« + 5y — 2 = (y - 2) 52 __ 26 + 52/ - 2 = ; of which the product is (2/ - 2)2S(a252) - 2 (y - 2)S(«2^) + (2/ - 2)(52/ - 2)l2 - 2(52/ ~ ^i + 4S(«5) + (52/ - 2)2 = 0. The coefficients Ag, Aj, N of the terms in the second equation, are Ao = — — , A = 4, N = ; y consequently, we have (253, 255) 5 25 - 82/2 _ 125 - 6y2 . y y^ y^ ^{ah) = 4, S(«2i) = — , S(a2i2) ^ ig . y ROOTS OF AN EQUATION. 429 and, substituting these values in the preceding equation, there results yh 4. i2y3 _|_ 87^2 _ 200y + 100 = 0, which is the final equation sought. It must be confessed, however, that the foregoing method of deducing the final equation is usually very tedious ; yet it has the advantage of presenting that equation unencumbered with extra- neous roots. But the principal value of the foregoing investiga- tion consists in its readily leading to the establishment of this theorem, first demonstrated by Bezout, namely : The degree of the final equation, which results from the ehmi- nation of one of the unknowns, from two equations, of any degree whatever, involving two unknown quantities, can never surpass the product of the degrees of the two equations ; and it is exactly equal to that product when the proposed equations are in their most general form. In order to determine the degree in y, of the equation [3], we must consider that each term of the product [3], is formed by the multiplication of one term of the first factor, one of the second, one of the third, &c. Let, then, Ka^ YJh^', K"c^" . . . be terms, taken at random, in each of the n factors [3] ; the cor- responding term of the product, will be KK'K^' .... X 0)^1)^' c^" . . . .; moreover, the entire product is symmetrical in «, 6, c, &c. so that this term forms part of one of the symmetrical functions which enter into the composition of [3], which partial function may be represented by KK'K'' .... X S(«h6h'ch" ....)... [4]. It will, therefore, be sufficient to determine the highest degree in y of this function. Now, as by supposition, Ka^ is one of the terms in the poly- nomial r(«) of the wzth degree, it follows that the degree of y, in K, cannot exceed the m — h degree. In like manner, the degree 430 SYMMETRICAL FUNCTIONS. of y in K' cannot exceed m — h' \ the degree of y in K'' cannot exceed m — h!' ^ &c. Consequently, the product of the n polyno- mials KK'K" .... cannot exceed the degree mn—h^h'—h" .... Let us now ascertain the degree which the polynomial ^{a}^V^' c^" . . .) cannot surpass. Referring to the general expressions involving 1^, Sg* ^ ^c* at page 418, and recollecting that in our equation [1], page 427, the coefficient, A^_,, cannot exceed the first degree in y, the co- efficient, Aj^_2, cannot exceed the second degree, and so on, we shall immediately see that the expressions for 2p Sg, Sg, &c., deduced from our equation [1], cannot exceed the degree in y, denoted by the index suffixed to the symbol 2. Referring, in like manner, to the general expressions in (255), which exhibit the double, triple, &c. functions, we there also recognise that, in )^(a^b^'c^". . . . ), the degree in y cannot exceed h -\- K -{• h" . . . , Hence, in the expression [4], the degree in y cannot exceed mn. If the coefficients, A^_j, Aj^_2, A^.g, &c. in the equation [1], and those in the equation [2], are in their most general form, that is, if they exhibit a series of functions of y, regularly ascend- ing, in degree, the expressions 2)p Sg, Sg, &c. will have the degree denoted by their suffixed indices, and hence the degree of S(«J^ b^' c^" ), wiU be A + A' + F It is plain, too, that in this case, K, K', K", &c. being, in their most general form, their degrees in y will be exactly m — A, m — h', m — h", &c. Consequently, the degree of the final equation in y, will, under these circumstances, be exactly mn. CHAPTER XVII. THE DETERMINATION OF THE IMAGINARY ROOTS OP EQUATIONS. (260.) It has already been proved that imaginary roots always enter into equations in conjugate pairs of the form a di /3 V— 1. And this previous knowledge of the form which every imaginary root must take, suggests a method for the actual determination of the proper numerical values for a and /3 in any proposed case. The method is as follows ; Let a?" + A„_j a?"-i + . . . . Aa? + N = be an equation containing imaginary roots ; then, by substituting a + /3 \/ — 1 for a?, we have (a + /3\/^)" + A„_,(a+/3\/^)"-i + ..A(a + /3\/^) + N=0; or, by developing the terms by the binomial theorem, and col- lecting the real and imaginary quantities separately, we have the form M + N \/ - 1 = 0, an equation which cannot exist except under the conditions M = 0, N = . . . . [1]. 432 DETERMINATION OF THE From these two equations, therefore, in which M, N contain only the quantities a, /3, combined with the given coefficients, all the systems of values of a and /3 may be determined ; and these, substituted in the expression a -f /3 v — 1, will make known all the imaginary roots of the proposed equation ; those that are real corresponding to /3 = 0. It is obvious from the theory of elimination as developed in Chapter xv, and from the method of numerical solution explained in Chapter xii, that the labour of deducing from this pair of equations the final equation involving only one of the unknowns a, /3, and of afterwards solving the equation for that unknown, will in general be impracticable for equations above the third degree. Lagrange, by combining with the principle of this solution the method of the squares of the differences explained at (247), avoids both the ehmination and subsequent solution here spoken of. It is easy to see how this may be brought about if we have any independent means of determining one of the un- knowns /3 : for the adoption of these means would enable us to dispense with the elimination; and as the substitution of the value of /S in both of the equations [1] would convert those equations into two simultaneous equations involving but one un- known quantity, their first members would necessarily have a common factor of the first degree in a, which, equated to zero, would furnish for a the proper value to accompany /3 ; and thus, instead of solving the final equation referred to, we should only have to find the common measure between the two polynomials M, N containing the unknown quantity a. Now corresponding to every pair of imaginary roots a -f^v— T, a — /3 \/ — 1, there necessarily exists, in the equation of the squares of the differences, a real negative root, — 4 /B^ ; so that if all the negative roots of the latter equation be found, the quantity — 4 /32 must appear among them ; from which the value of (i would be immediately obtained, and thence, by aid of the com- mon measure as just explained, the corresponding value of a. But the equation of the squares of the differences may have a greater number of negative roots than there are pairs of imagi- nary roots in the proposed ; which however cannot happen except two non-conjugate imaginary roots have equal real parts. IMAGINARY ROOTS OF EQUATIONS. 433 or except a real root be equal to the real part of an^imaginary root. Lagrange discusses these peculiarities, and establishes the exactness and generality of the principle in question, as follows : When the real parts a, y, &c. of the imaginaries a + /3 V ■- 1, a - /3 V— 1 y + h V - 1, y — b V — 1 &c. &c. are unequal, as well when compared with one another as when compared with the real roots «, b, c, &c. it is evident that the equation of the squares of the diiferences cannot have any other negative roots than those furnished by the several pairs of conju- gate imaginary roots, and which are — 4/32, ^4b\ &c. All the other roots, not arising from the differences furnished by the real roots «, i, c, &c. will evidently be imaginary ; those between the real and imaginary roots supplying the forms (a — « + /3 V— 1)2, (a — a — /3 \/'^^y (a - 6 + /3 V ~ 1)2, (a — 6 - /3 V^^y &c. &c. and those between the non-conjugate roots the forms {(a-y) + (/3-6)V'=T}2, {(«-^)_(^_«)v/ZT}2 {(«-r) + (/5+*)V^}2, {(a_y)_(/3+8)V3r}2 SO that in this case every negative root in the auxiliary equation will indicate a pair of imaginary roots in the proposed, and will moreover supply the value of the imaginary part. But if it happen that among the quantities a, y, &c. there be found any equal among themselves or equal to any of the quantities a, b, c, &c. then the auxiliary equation will necessarily have negative 28 434 DETERMINATION OF THE roots, corresponding to which there can be no imaginary pair in the proposed equation. For let «=a ; then the two imaginary roots (a— « -f ^ V — 1)^, (oc — a — (is/ — 1)2 will become — /S^ and — /32, and conse- quently real and negative ; so that if the proposed equation con- tain only two imaginary roots, a + /3 v — 1 and a — /3 v — 1, then, in the case of « = a, the equation of the squares of the differences wiU contain, besides the real negative root — 4/3^, the two — /32, — /32, both negative and equal. We thus see that when the equation of the squares of the di£ferences has three negative roots, of which two are equal to one another, the proposed may have either three pairs of ima- ginary roots, or but a single pair. If the proposed contains four imaginary roots, a -{■ /3 v — 1, a — - /3 V — 1, y-f-SV— 1, y — SV— 1, then the equation of the squares of the differences must contain the two negative roots — 4p^y and — 4b^ ; if a = «, it must also contain the two equal negative roots — /S^, — fi^ ; and if moreover y = ^ it must contain, in addition to these, the negative pair — b'^, —b'^ : and lastly, if a = y the four imaginary roots {(a~y) + (/3-6)>/i:T}2, {(a-y)-(/3-S)V'^}2 { (a - y) H- (/3 + S) V^ } 2, { (a - y) - (/3 + ^) V^ } 2 will be converted into the two negative pairs -(3 -by, -(/3-6)2; -(^ + 5)2, ■^((i-{-by Hence we may deduce the following conclusions, viz. 1. When aU the real negative roots of the equation of the squares of the differences are unequal, then the proposed will necessarily have so many pairs of imaginary roots. If in this case we call any one of these negative roots — w, we shall have /3 = ; and if this value be substituted for /3 in the two equations [1], and the operation for the common measure of their first members be carried on till we arrive at a remainder IMAGINARY ROOTS OF EaUATIONS. 435 of the first degree in a, the proper value of a will be obtained by equating this remainder to zero. Thus each negative root — w will furnish two conjugate imaginary roots a + /3 V— 1, and a - /5 V — 1. 2. If among the negative roots of the equation of the squares of the difierences equal roots are found, then each unequal root, if any such occur, will, as in the preceding case, always furnish a pair of imaginary roots. Each pair of equal roots may, how- ever, give either two pair of imaginary roots or no imaginary roots, so that two equal roots will give either four imaginary roots or none ; three equal roots will give either six imaginary roots or two ; four equal roots will give either eight imaginary roots, or four, or none : and so on. Suppose two of the negative roots — w, — w, are equal : then putting, as above, ^ = ^^, we shall substitute this value of /3 in the two polynomials [1], and shall carry on the process for the common measure between these polynomials till we arrive at a remainder of the second degree in a ; since the polynomials must have a common divisor of the second degree in a, seeing that the equations [1] must have two roots in common, on account of the double value of /3. Equating then this quadratic remainder to zero, we shall be furnished with two values for a : these may be either both real or both imaginary. In the former case call the two values a! and a!' ; we shall then have the four imaginary roots a! + /3 V^', a! - flV^^, a!' + /3 V"^, a!' - fW - 1 In the second case, the values of a being imaginary — contrary to the conditions by which the fundamental equations [1] are governed — we infer, that to the equal negative roots — w, — w, there cannot correspond any imaginary roots in the proposed equation. If the equation of the squares of the differences have three equal negative roots, — w, — w, — - w\ then, putting as before (S= , we should operate on the polynomials [1], for the 436 DETERMINATION OF THE common measure, till we reach a remainder of the third degree in a : this remainder equated to zero will furnish three values of a, which will either be all real, or one real and two imaginary. In the first case six imaginary roots will be impHed : in the second only two ; the imaginary values of a being always re- jected, as not coming within the conditions imphed in [1], The foregoing principles are theoretically correct : but the practical application of them, beyond equations of the third and fourth degrees, is too laborious for them to become available in actual computation. We give the following illustration of them from Lagrange. To determine the imaginary roots of the equation ^ __ 2a; — 5 = 0. Computing the equation of the squares of the differences from the general formula for the third degree at (248), viz. s^ + 6/?^2 _|_ 9^2^ _^ 4^3 + 27^2 = in which ^ = •— 2 and ^r = — 5, we have ^ _ 12^2 _|_ 36^ + 643 = In order to determine the negative roots of this equation, change the alternate signs, or put z ■=■ — w^ and then change all the signs, converting the equation into ^«;3 4- 12^2 + 36^ — 643 = and seek the positive root, which is found by trial to lie between 5 and 6. Adopting Lagrange's development, p. 379, this root proves to be M) = 5 + — - 1 ^ + 6 + &c. from which we get the converging fractions 31 160 991 ^' ^' ~Z\' 192' Knowing thus an approximate value -of w, we know /3 = IMAGINARY ROOTS OF EQUATIONS. 437 In order now to get the equations [1], page 231, substitute a + /3\/ — 1 for X in the proposed equation ; and form two equations, one with the real terms of the result, the other with the imaginary terms : we shall thus have the equations [1] re- ferred to, viz. a? _ (3/32 + 2) a - 5 = 3a2 _ /32 - 2 = in which /3 is known. Seeking now the greatest common measure of the first mem- bers of these equations, stopping the operation at the remainder of the first degree in a, and equating that remainder to zero, we have — 15 "" 8/J2 + 4 and thus both a and /3 are determined in approximate numbers. (261.) There is another method of proceeding for the determi- nation of imaginary roots, somewhat different from the preceding, being independent of the equation of the squares of the diffe- rences. It is suggested from the following considerations : Since the quadratic involving a pair of imaginary conjugate roots is always of the form sc'i — 2a,x + a2 + ^52 = 0, every equation into which such roots enter must always be accu- rately divisible by a quadratic divisor of this form : that is, the proper values of a and /3 are such that the remainder of the first degree in a?, resulting from the division must be zero. This fur- nishes a condition from which those proper values of a and /3 may be determined ; the condition, namely, that the remainder spoken of, kx — B, must be equal to zero, independent of parti- cular values of x ; and this implies the twofold condition A = 0, B = 0, from which a and /3, of which A and B are functions, may be determined. 438 DETERMINATION OF THE As an example let the equation proposed be a?4 ^ 4a^ _j- 6a:2 + 4a? + 5 = 0. Dividing the first member by a;2 _ 2oLX + a2 -I- /32 we have for quotient a,2 ^ (4 ^. 2a) a? + 6 + 8a + 3a2 — fP- and for the remainder of the first degree in x (4 + 12a + 12a2 + 4a3 — 4a/32 — 4/32)a? — (a2 + /32) (6 + 8a + 3a2 - /32) + 5 which, being equal to zero whatever be the value of x^ furnishes the two equations 4 + 12a + 12a2 + \a? — 4aj32 — 4/32 = (a2 + /32) (6 + 8a + 3a2 - /32) -f 5 = From the first of these we get /32 = (1 + a)2 and this, substituted in the second, gives 4a4 + 16a3 + 24a2 + 16a = two roots of which are and — 2 : the other two are imaginary, and must consequently be rejected as contrary to the hypothesis as to the form of the indeterminate quadratic divisor. The two real values of a, substituted in the expression above for /32, give for a = 0, /32 = 12 /. /3= + 1 a=-2, /32 = (_i)2... ^ = _i IMAGINARY ROOTS OF EQUATIONS. 439 and consequently the component factors of the original quadratic divisor, viz. the factors ^ _ a — /3 V— 1, a? — a + /3 V — 1, furnish these two pairs of imaginary roots, viz. V — 1, X — — V — 1 anda? = — 2 — V— 1, a? = — 2 + \/ — 1 Tliis method, like that before given, is impracticable beyond very narrow limits, because of the high degree to which the final equation in a, usually rises. And it is further to be observed of both, and indeed of all methods for determining imaginary roots by aid of the real roots of certain numerical equations, that when- ever, as is usual, these real roots are obtained only approximately, our results may, under peculiar circumstances, be erroneous. For instance, in the two methods just explained we have two equa- tions /(a) = 0, F(/3) = 0, where the coefficients of a in the first are functions of /3, and the coefficients of (i in the second func- tions of a ; hence, whichever of these symbols be computed ap- proximately, in order to furnish determinate values for the coeffi- cients of the other, these coefficients must vary sHghtly from the true coefficients ; and^ consequently, under this slight variation of the coefficients, real roots may become converted into imagi- nary and imaginary into real, — see page 315. CHAPTER XVIII. ON THE SOLUTION OF CUBIC AND BIQUADRATIC EQUATIONS BY GENERAL FORMULA. (262.) In the former chapters of the present treatise ample instructions have been given for the complete solution of every algebraical equation whose coefficients are expressed in known numbers. It still remains for us to give a concise account of the labours of mathematicians, as far as they have been successful, in the solution of equations with literal coefficients. The problem we now pro- pose to consider is therefore this, viz. to determine finite expres- sions for the roots of an equation in functions of the coefficients; a problem long regarded as the most important in Algebra, because of its involving the complete solution of numerical equa- tions. But the recent discoveries, unfolded in the former part of the present work, have reduced this celebrated problem to one of comparative insignificance ; and have removed that regret, which was so long and so universally felt, on account of the failure of every attempt to extend the solution of literal equations beyond the first four degrees. We shall, therefore, content ourselves with briefly explaining the principal formulas which have been proposed for the solution of cubic and biquadratic equations. CUBIC EaUATIONS : METHOD OF CARDAN. 441 Solution of a Cubic Equation by the Method q/* Cardan. (263.) Let the proposed equation be first deprived of its second term by the rule at (79), it will then have the form a^ ^-px -\- q=zO . . . . [1] Assume x equal to the sum of two other unknown quantities ; that is, put X — y -\- z, we shaU then have oc^ =1 y^ -\- s^ -\r ^yz {y -\r z) ; that is, replacing y -\- zhy x, and transposing, a^ — 3yzx — y^ — c^ = 0, and, in order that this may be identical with the proposed equa- tion, we must determine y and z so as to satisfy these conditions, viz. P yz= ——y y'^ -^ ^= —q. The problem is therefore reduced to the determination of y and z from these two equations. From the first we have hence, combining this with the second, we have the sum of two quantities, y^ + s^, and their product, y^ s^, given to determine the quantities : a problem which we know may be solved by help of a quadratic equation {Algebra^ p. 156), viz. the equation "' + '/"- ^ = ... . [2] 442 CUBIC EQUATIONS : of which the two roots, or values of v, will be the expressions for y^ and s^ sought. Hence, solving the equation, and separating the two roots, we have ^ = -| + v{| + |}....r.3] and consequently, since x =:y + z, there results the following general expression for the roots of the proposed equation, viz. which is the formula of Cardan. Since the cube root of y^ may be represented indifferently by either of the three expressions, page 352, 1 H-V — 3 — 1 —V -3 -y* "' 2 "^ 2 and the cube root of s^ by either of the expressions - 1 +V-3^ - 1 -V — 3 ""' 2 2 ^' it would seem that x = y + z admits of nine values, or that the proposed equation has nine roots. It must be remembered, how- ever, that in all cases when we assign the root of any expression, a tacit reference is made to the generation of the proposed power, the root being in fact assumed to be the expression from which the given power has been actually produced. When we speak of any proposed power having a multiplicity of roots, we merely refer to the various expressions from either of which that power mic/ht be generated ; and as many of these as prove inconsistent with the conditions involved in the production of the power, are METHOD OF CARDAN. 443 of course to be rejected. Now one of the conditions in virtue of which y^ and ^ have been produced, is 'P that is to say, the products of the roots y, z^ must be possible ; but of the nine products which the preceding expressions for y and z are competent to furnish, six will be found to be imaginary: such a combination of values must therefore be rejected, as incon- sistent with the conditions to be fulfilled ; the other three pro- ducts are possible. Hence the only admissible solutions are the three following, where the product of y and z fulfils the condition above y -V Zy -1 + v^^ 7. y + a ^- -1-V^ -3 2 -x^sT- -3 _ 1 -V — 3 — 2 — y^ But a form will be given to Cardan's formula hereafter, that shall be unencumbered with superfluous values. The equation [2], upon the solution of which that of the pro- posed cubic is made to depend, being of the next inferior degree, is called the reducing equation. The roots of this reducing equa- tion are exhibited in [3] and [4] . If these are real, the formula for X will consist of the cube roots of two real quantities : but if the roots of the reducing equation are imaginary, then the expres- sion for X will be the cube roots of two imaginary quantities ; and, consequently, each root must itself be imaginary : that is to say, if the relation between p and q^ be such that the expression for x will consist of two parts, each of which is imaginary. But the sum of these parts, that is, the complete expression 444 CUBIC EQUATIONS : for Xy must of necessity be real ; because the preceding relation between p and q is that which necessitates the reality of all the roots of the equation. (See Introductory TreatisCy page 106.) Still, with the exception of a few particular cases, it is impossible to deliver the compound expression for the roots from its imagi- nary symbols, without developing each part separately, and then by incorporating the two, representing the aggregate by an infinite series, from which the imaginary symbol shall have dis- appeared. The cases of exception referred to are those in which [3] and [4] are each complete cubes. For then the roots involved in the expression for x will be expressible in finite terms : and as they must be of the forms, page 54, P 4- Q V"^ and P - Q V"^ it follows that x will be expressed, without series, by the finite quantity 2P.* Moreover, in what may be regarded as the ex- treme case of the above condition respecting p and q^ viz. the case in which • 4 ^ 27 which implies two equal roots in the reducing equation, and also two equal roots in the cubic, the formula for x becomes real and finite, because the quantity under the quadratic radical vanishes. With these exceptions we may lay it down as a general rule, that Cardan's formula is incompetent to furnish either of the roots of a cubic equation, in a finite form, when all three are real. The formula can present a real root in a finite form only when the remaining two roots are imaginary, or the expression under the quadratic radical positive. The case in which the three roots are real has hence been • In the above observations we refer merely to the fact that, when the ex- pressions in question are complete cubes, x must admit of an expression in finite terms : whether the roots of these complete cubes can be actually deter- mined in all cases, and thence the finite value of x deduced, is a distinct inquiry. It is considered in the foot-note at the bottom of next page. METHOD OF CARDAN. 445 called the irreducible case of cubic equations, and many attempts have been made to convert the irreducible expressions in this case to a finite real form. It is obvious that these attempts can succeed only to the ex- tent mentioned in the instances of exception adverted to above. The two parts of the formula for x can never be replaced by finite expressions such as [5], unless [3] and [4] be perfect cubes. It has been regarded as anomalous and paradoxical that the value of oc should not appear under a real finite form whenever the roots represented by x are known to be real ; and yet to expect such to be the case is to expect that every expression of the form a ± v — h must be a perfect cube.* * In order to deduce a general expression for the cube root of a -|- *y b, whether h be positive or negative, put V(a + x/*) = A + ^/B....[aJ then, cubing each member, we have a + ^ & = A3 + 3A'V B + 3 AB + B V B ; and equating the rational terms with the rational, and the irrational with the irrational, we have the following equations of condition for determining A and B, viz. a=A3-|-3AB Vi=(3A2+B) v'B, ori = (3A2+B)2B. If from the square of the first of these we subtract the second, we shall have a=— i=(A2 — B)3- It appears therefore that, in order that in any case the proposed extraction may be possible, in finite terms, a^ — b must be a perfect cube. This condition, it will be observed, is fulfilled by each of the expressions [3], [4], in the text ; otherwise the cube roots of these could never be obtained in finite terms. Putting for brevity a^ — i = c3 , we have A2 — B=e.-. B = A2 — pi — 4^ Q = ^2^72 _j_ ^2^2 ^ ^22^2 _. _- ^ 222 ^ = __ orz.2^,2^^2=__ 16 2 29 450 BIQUADRATIC EQUATION : The conditions show that the quantities w^, v^^ w^y must be the roots of the cubic equation or, putting z 2^=4' the roots of the equation ^ + 2^^2 ^ (^2 _ 4^) ^ _ ^2 = . . . . [3]. Call these roots 2^, z" ^ z'", then the roots of [2] will be ^ ^ z^ 4' 4' 4' and hence the expression [1] for a? takes the following forms, viz. 22"^2' 2 2 2' 2'*'2 2' 2 2"^2' V^ , ^^' , v^''' V^' V^" V^' 2 2 2 2 2 2 yy yy^ ^z'" v^_v^_N/f^ 22 2 * 2 2 2 • But some of these values are inadmissible, since a necessary condition is, that wm = — ■- ; hence, we must preserve only those of the foregoing trinomial expressions of which the product of the radicals gives always a sign contrary to that of gr, and these are, when q is positive, X = METHOD OF FERRARI. 451 X = 2 2 2 2 and when g' is negative X = 2 ^= 2 «l? — —————— And these formulas exhibit the four roots of the proposed equation : they will be given in a better form hereafter. Solution of an Equation of the Fourth Degree, hy the Method of Louis Ferrari. (267.) Taking the same general form as before, viz. x^ + px"^ + 5'^ -f r = ; we have, by transposition, ocf^ i=. — px^ — qx — r. Add the quantity 2kx^ + J^, 452 BiaUADRATIC EQUATION : to both sides, and we shall then have (x^ + kf = (2k — ^) x^ — qx + (k' — r) ; and it remains to determine k, so that the second member of this equation may be a complete square. In order to this, k must fulfil the condition (2A__p)(F-r)=-J, since, in every perfect square, four times the product of the ex- treme terms is equal to the square of the middle one. Actually multiplying the two factors, and dividing by the co- efficient, 2, of ^, we have, finally, the cubic equation, k^-l-k'-^rk^^ - -^; k^^-k -^k+- - -. and a root of this being determined by the rule of Cardan, or otherwise, the solution of the proposed biquadratic is reduced to that of the two quadratics following, viz. v^ + k=\/2k^p.x—\/~^ x^ + k= — s/2k—p.x-\- \J ]^ — r For a better form, see the Scholium at the end. Solution of the Biquadratic by the Method of Descartes. (268.) Taking the same general form as before, viz. a?4 + ^a?2 -f gra? + /• = . . . . [1] let the first member be decomposed into two quadratic factors, by assuming it equal to the product (x^ + kx+f) (x^-kx-\-ff) and then determining the unknown coefficients k, /, ff, so as to bring about the desired identity between the two expressions. For this purpose we have by actually executing the multiphcation ^ + if + ff - ^') ^"^ + iff^ -M^ +fff ... .[2] METHOD OF DESCARTES. 453 and equating the coefficients of the Hke terms in [1] and [2] f+g-k^=.p, gk -fk ^ci,fg-=r . . . .\Z-\ Multiplying the first of these by k, and then adding and subtract, ing the second, we have 2gk = k^ -\- pk ^ q 2fk = k^ +pk^q Lastly, multiplying these together and introducing the third con- dition, viz. 4/g =4r, we get finally k^ + 2pk^ + (^2 _ 4^) ^2 _ ^2 = or, putting s for A;^, the reducing cubic ^ + 2p^2 ^ (^2 -.4r)s — q^z=z0 which is the same as that in Euler's solution. When z is determined from this equation, k becomes known, being the square root of s : and from the first two of the equa- tions of condition [3] we have f^g=.k^+p^ g-f^l. and, consequently, the two quadratics for determining the four values of x are a?2 + A;^ + ^ (A;2 + jp _ _!)= ^2 --. kx ^ \{k^ -^ p Jr ■T-) = k All the preceding methods of investigation involve particular analytical artifices. That which follows, and which we beheve to be new, has the peculiarity of being altogether independent of these artifices. It furnishes at the same time the results both of 454 BIQUADRATIC EQUATION : Descartes and Euler, which are shown to be immediate in- ferences from the known composition of the coefficients of the biquadratic, without the aid of any additional principle. New Method for Equations of the Fourth Degree. (269.) As in the preceding cases let the proposed equation be oc^ + px^ -\- qx -{- r ^=i and let its four roots be represented by a?j, x^i x^^ x^ : then, since the sum of any two of these is equal to the sum of the repaaining two, the quadratic equation involving the roots x^y x^, wiU be x^ + (^1 + ^2) ^ "^ V4 ^ ^ and, consequently, the roots themselves will be expressed by This expression, as remarked at (198) may be converted into a variety of different forms from attending to the known composi- tion of the coefficients p and q. Two of these forms are exhibited at (198) ; a third is obtained as follows : Since J) = (a?i + a?2)(^3 + ^4) + ^1^2 + V4 ^ — yX'^ + Xsj^ -r x-^x^ 4- x^x^ it follows that the expression under the radical is ^— 2~"; V4~-2"" V"~2~^ ^ 2 ^^ But g = — (.«?i H- X^ a?s^4 ~ (^8 + ^4) ^1^2 = {x^ 4- x^ (a?ia?2 — x.^x^ q __ x^x^ — x^x^ x^x^ 4- x^^ •*• 2 {x, + ^2) " 2~^ - 2 ""^4 NEW METHOD. 455 SO that, substituting this in the expression referred to, we have a?2 + X.2 ^/f-l- {^L±f2\' _JL_\ If the other two roots, x>^, x^ be substituted for these, then since x.^ -\- x^=- — {x^ + x.^ we shall have for all four of the roots the formula _ "^-i 30-1 "Y" Oun n 2 v-T-)-2-(irT^i 2 or putting m for x^ -\- x^', 2 2 1 m) where, it is to be observed, that the double sign of the radical ap- plies whether the upper or the lower sign of m be taken. Now if we multiply all these four values together the product must evidently be r. This multiplication is very easily performed in consequence of the double sign adverted to furnishing for each sign of m a pair of expressions, forming the one the sum and the other the difference of two quantities, so that, taking the upper sign of m, we have for the product of the corresponding pair of values 4 4\^ mi 2 2 2m In like manner, for the product of the other pair of values, we have 441^ mj 22 JL 2m These two products form also the sum and difference of two quantities : hence the final product is V 2 2 ^ V 2m ^ that is m^ pw? p^ cp 456 BIQUADRATIC EQUATION ; or finally m^ -\- 2pm'^ + (i?^ — 4r) m^ -^ (f =. And this, when z is put for m^ becomes ^ -\- 2pi^ ■]- {p" — Ar) z — q^ = . . . . [2] the reducing cubic of Euler and Descartes. If a root of this cubic be found, and the square root of it be put for m in the general formula [1], the four roots of the biquadratic will be exhibited. (270.) These four roots are evidently those given by the pair of quadratic equations in the solution of Descartes. If all three of the roots of the reducing cubic be found, then representing them by z'^ z", z'" ; and observing, from the coeffi- cient of the second term of that cubic, that — 2p =^' ^- s" + jn and from the final term that f = ^z"z"\ the general expression [1] for the four roots of the biquadratic may be written thus : q: -^ ± I V { ^' + ^'' + ^"' - -^ ± 2\/?V^ } = + -^ ± IV { «" ± 2 VTv" + y } which expression resolves itself into the four expressions of Euler for the roots of the biquadratic. It should be remarked that, as in [1], the second of the am- biguous signs in this result always preserves its double significa- tion, giving a pair of values corresponding to each single value of the preceding term. But the third ambiguous sign, that of >/y, is not equally arbitrary. It must always be fixed by the condi- tion that the product of the three radicals in each of the four results may furnish a sign opposite to that of ^?,i the values of these quantities might be easily deter- mined by elimination. EQUATION OF THE THIRD DEGREE. 459 Let US assume the relation la -f m«2 + wflg = z ; then, as there is nothing to distinguish one root from either of the others, the relation which we have just assumed may be in- differently one or the other of the six following, viz. la + m«2 + w«.^ = z la H- ma^ + w^g = z la^ -^ ma -\- na^ = z /«2 + ^^<^3 -{• na = z la.^ -\- ma -\- na^ = z la^ + ?w«2 + ^^ = ^ ; and these could all be given by the solution of an equation of the sixth degree, in z. But, in order that such an equation might be solved as a quadratic, it must be of the form ^6 + A^ + B = . . . . [1] ; which, if we put u for 2^, becomes w'' -f- Am + B = hence, putting „ = -- ^'3 = _4-- V4-'-B; and recollecting that the three cube roots of unity are 1, a, a?, we have, for the six values of z, the following expressions, viz. z\ a/-, aV, z\ OLz", a V ; 460 EQUATION OF THE taking, then, any two of the six expressions above, for / and z"y as, for instance, la + ;w«2 + ^^3 = ^'> ^« + ^«3 + ^^2 = ^' y the four others must fulfil the following conditions, viz. Z«2 "T '^via + na.^ = a (/« + ^w^g + na^ la^ + //2«3 + w« = o?{la -\- ma^ + w^) ^^3 + ?w« + w«2 = a (^« H- mag + na,^ la^ + w^«2 -\- na =. a?{la + ma^ + na^ which must be formed so that the coefficients of «, a^, a.^y in one member of each, shall be different from those in other members, in order to avoid contradictory conditions. The four equations, just deduced, are transformable into the following, viz. (/ — an)a2 + (m — ccl)a -\- (n — am)a^ = {I — a%)«2 + (m — a?n)a^ -f- {n — o?l)a = (/ — aw)«3 + {in — oLX)a + {n — am)a2 = (/ — a?m)a^ -{■ {m — a?n)a2 + (w — o?P^a = ; which will evidently be satisfied if we can fulfil the conditions , I ■=. OLTiy m •= cdy n ■=. am I = a^Tw, m = a^w, n = a^/, which are reducible to the two following, viz. m = a/, and n = a?l\ for, from a^ = 1, we have a = — -, and a- = — ; so that m = a/ is the same as m — —r, /, whence / = oChn, In like manner, THIRD DEGREE. 4G1 n = ocH is the same as w = — I; whence / = an. Lastly, the a, relations, m = al, n = olH, divided the one by the other, give — = — ,'.m =■ — n ■=: oihi, and n = ccm ; hence it will be n cc a, sufficient to consider the two relations, m ■=olI, n ■=■ olH ; from which, as we have just seen, all the others are deducible. We thus have m. and n expressed in terms of /, which, being arbitrary, put it for simplicity equal to unity ; then we shall have w^ = a, n ■=■ ofi, and thus the three values, /, m, n, are no other than the three cube roots of unity. Substituting these values in the expressions la + m«2 + w«3 = s-', la + ma^ + na^ = z" ; they become a + aag + a^ a^ = y, a + a^.^ + c^a^ = z" . We may, in like manner, substitute the same values in the four remaining equations, and afterwards form, by multiplication, the equation in z ; as, however, we know that this equation is to be of the form [1], its six roots must be comprised in the two equations, r^ = z'^ = (« + a«2 + <^^^3)^ ^ = z"^ = (« -j- a«3 + ci?a^^ or, which is the same thing, in the single equation, { ^ — (« + a«2 + a2«3)3 } { 2^ _ (a + a«3 + a^a^^ } = 0. By actually performing the multipUcation here indicated, and comparing the coefficients of the resulting terms with those of the corresponding terms in [1], we have these conditions, viz. A = — {(«+ a«2 + a2«3)3 + (« + a«3 + a?a^^ } B = {a ■\- 0La^-\- ci?a^^ . {a + a^ + ci?a^^. 462 EaUATION OF THK Hence the coefficients of the equation, ^« -f A^ + B = 0, are symmetrical functions of the roots proposed. If we develop these values of A and B, and keep in mind that a? =1, a4 = a, a^ = a?, a^ = 1, &c. and, because the coefficient of the second term in ^ -r- 1 = is zero, that 1 + a + 0.2 = 0, and, therefore, a -{- a^ = — 1, we shall have these values, viz. (see page 422), A = - { 2S(«3) + 3(a + a2) ^{a^a^ + Xlaa^fl^ = — { 2S(a3) - Z%{aH^ + \2aa^(H B = { S(a2) + (a + a?) ^{aa^ } 3 = {S(«2)-S(«a,)}3. Now, from the original equation, oj^ + i>i*^ + 5' = 0, we obtain, in terms of the coefficients, ^, 5^, the following values for these symmetrical functions (255), viz. S(fl2) = 2, = - 2i), S(«3) = 23 = - 3fy, S(aa,) = ^, S(A,) := s^Xj __ 23 = -. is = 3gr. Moreover, ««2®3 ^^ — ? 5 hence, substituting these values in the preceding expressions for A and B, we have A = - { — 6^ — 9g — 12^ } = 27g B= { -2j9~i)}3=:(-3p)3= -27i>8; consequently, the equation in z is now fully determined ; it is s^ + 27 ; therefore, — (« 4- «2 — % ~ «4)^ = 4^ — 4 {aac^ + a-^ a^). In like manner, ^ (« + % — «2 — ^4)'^ = 4^ — 4 (««3 -f- «2 ^^4) — (a + «4 — ffg — f^zY = 4^ — 4 (aq^ -{- a^a^); putting, therefore, for abridgment, z^ -\- 4p =z 4u, the equation in z will be transformed into the following, viz. { w— (a<3f8+«3a4) } { u — {aai+aiai) } { U'-{aat-{-a3as) } =0: which is of the form, * u^ + A'ti^ + B'w + C = ; FOURTH DEGREE. 46/ its coefficients being A' = — (««2 + ««3 + ««4 + flfg «3 + «2 «4 + «3 ^4) = — i? B' = (a«2 + «3 «4) (««3 + «2 ^4) + (a«2 + «3 ^4) (««4 + «2 %) + (««3 .+ «2 «54) (««4 + «2 ^3) = S (a' «2 ^s) C = — (««2 + %«4) («f«3 + «2«4) (««*4 + «2«3)- Now, from the formula at page 421, we have, in which 11 = 0, l2 = -2i?, l3 = -3s' 24 = —pl^ — 4r = 2/ — 4r, Is = — 2/ 4- 4^;- + 3q^ + 2^r = — 2/ + 6^r + 3^. Hence, substituting these values in the expressions for B' and C, we have C = — ^2 _j_ 2^/' + 2jor = 4^r — g^ . and thus the equation in u is v?> — pv? — 4ru + 4pr — q^ = ; z -f- 4i9 z^ and replacing w by its value = —- + ^, or, for simpli- city, by Zj + p, we have, for the final cubic, the equation z^^ + 2jp^j2 + (p2 — 4r) ^j - ^2 := 0. Calling the roots of this equation /, 2-", 5^'', we shall have 4^, 4^', 4z'" for the squares of the expressions a -\- a^ — a^t — a^, 468 EQUATION OF THE ^J! + % — ^2 — '^4> a -\- a■^ — «2 — <353 ; that is, these ex- pressions are « + «2 — «3 0^4 = ih 2 \/2' a + a-i — a^ — ^4 = d: 2 s/z" a -\- a^i — ttz — tti = ±L 2 sjs^" also, « + «2 + «3 + «4 = 0. By adding these, we find 4« = ± 2 v^' ± 2 v^' ± 2 v^"; :,a= ±\sj z' ±\^ z" ±\s/ ^"' Again, adding the first and fourth, and subtracting the sum of the other two from the result, we have 4«2 = ± 2 V ^ H= 2 V ^" =P 2 V ^" Similarly, «3 = q: 1^^' ± I yjz" q: | s/z"' «4= +i\/^'±i\/^" + i\/^''- As to the proper signs of the radicals \/ ^, \/ z", \Jz'"i we must observe that since (« + ttg — «3 — «*) (a + «3 — «2 — «4) (a + 04 — «2— «3) = — Sg', these signs ought to be such as to render their product positive if 2 is negative, and negative if j is positive. The values thus deduced are the same as those otherwise determined in the pre- ceding chapter, and commented upon in the Scholium following. (273.) The above method of investigating analytical expressions for the roots of equations by means of symmetrical functions, may be extended to equations of higher degrees than the fourth ; but the auxiliary equation in z^ to which the investigation leads, is, after the fourth degree, of a higher order than the proposed. In equations of the fifth degree the auxiliary one rises to the 1 20th FOURTH DEGREE. 469 degree, which, by means of certain artifices, is, however, capable of depression. But no method has yet been devised, whereby an equation of the fifth degree can be solved by help of an auxiliary equation below the sixth. It has indeed been the object of ana- lysts of late to demonstrate the utter impossibility of such a solution by any combination of algebraic symbols ; on which sub- ject the student is referred to the profound paper of Sir William Hamilton in the Transactions of the Royal Irish Academy, vol. xvi. It is of course to be understood that the impossibility referred to does not exclude particular classes of equations from becoming solvible in general symbols. It would, indeed, be easy to assume particular irrational forms for the roots, and, by eliminating the radicals, thence to deduce an equation of any degree we please, the coefficients of which should be literal quantities, ready to receive any numerical values we may choose to give them. But these would only be equations of a particular class ; the coeffi- cients of which would be related to one another by a fixed law, virtually imphed in the form assumed for the roots ; and could, therefore, be only of very rare apphcation in actual practice. Such particular forms have, however, been investigated by De Moivre, Waring, and Euler : an account of the mode of deducing them may be seen in the second volume of the Algebra of Meyer Hirsch.* The formula of De Moivre, for a certain class of equations of the nth. degree, we shall give in the scholium below ; because of the direct and simple manner in which it may be investigated without assuming the irrational form of the roots, and also because of the celebrity which it has acquired. • This work of Hirsch contains a very comprehensive discussion of the theorj- of the symmetrical functions of the roots of equations : a subject upon which we have been precluded from dwelling at any great length in the present volume on account of the space occupied by more useful inquiries. In addi- tion to this work of Hirsch, we would also direct the attention of the student to the researches of Mr. Murphy, as contained in his Theory of Equations, and in his papers in the Cambridsa Transactions for ^ ^;?1 » «"'^ '" thn.w nf th^ Royal Society joT 1837^ A view of the method of approximating to the roots of equations by the method of Rgaurriag SerJeaJMay. he^een in Eu^titi'j. . Algebra ; and in a more extended form in Murphy's Equations adverted to above. 470 EaUATlON OF THE Scholium. (274.) The formula of Cardan for the roots of a cubic equa- tion, and the expressions of Euler for the roots of a biquadratic, furnish exemplifications of the doctrine at pp. 26, 40 respecting the Hmited sense in which irrational results must be taken when the signs of the radicals are brought under any control by stipulated conditions. One of the irrational terms in Cardan's formula takes a fixed and determinate sign as soon as the sign of the other irrational term is determined upon, on account of the controlling condition at page 443. And in like manner any one of the three irrational terms in Euler's forms becomes fixed in sign as soon as the other two radicals, which are entirely independent and uncontrolled as to sign, become fixed, because of the overruHng condition Suvw = — §», page 449. In the expressions for the four roots of the biquadratic at p. 45 1 the signs prefixed to the radicals thus point out the character of the roots, and limit the generality that would otherwise belong to them. But it is easy to avoid all inadmissible generahty in the formulas referred to, and to deduce them at once free from extraneous values : and it is undoubtedly better to do this than to encumber our results with what must afterwards be rejected upon adjusting the formula to the case in hand. The necessity for any such supplementary adjustment does in fact imply some imperfection in our method, or some mistake in our reasoning : unless indeed the original restrictions of the problem are such as unavoidably to Hmit the generality of the ordinary symbols of operation from the outset. This is not the case in either of the problems referred to : the limitation to which the result ought to conform has been introduced in the course of the investigation, and has then been improperly disregarded, leaving the final result uncontrolled by it. The condition z •=■ , introduced into the investigation of Cardan's formula, ought to have had its influence upon every J9 \ 3 subsequent step : s^ should have been replaced by ( — x") and if » the final result should accordingly have been written thus : FOURTH DEGREE. 471 from which all superfluous values are excluded. (275.) The formulas of Euler are open to the same objection. The condition V^ V^' sjz'" = -q,ov v.- = -^ introduced into the investigation, has been disregarded ; and thus the result is uncontrolled by its restrictions, and involves a su- perfluous radical, like the formula of Cardan. When freed from this imperfection, the result of Euler's investigation will take the form ^v^-'^Vf:^ -q 2 2' 2^z'^z"' which comprehends all that is impUed in the four distinct ex- pressions at page 451, and involves no superfluous values, the radicals being unrestricted in generality. It will have been observed that in the formulas of Euler at page 451 the signs prefixed to the radicals to designate the cha- racter of the roots are sufficiently explicit for that purpose : but in the formula of Cardan, at pages 442 and 463, similar preci- sion cannot be impressed upon the expressions by means of the prefixed signs, so that in applying the formula it is essential that the overruhng condition Zyz = — ^ be always borne in remem- brance. (276.) Like observations apply to the formulas of Ferrari at page 452 : they are not only faulty inasmuch as they preserve no trace of the restrictions that have been introduced into the in- vestigation, but, as in the case of Cardan's expressions, they are also incapable of the necessary adjustment by putting any restric- tions upon the signs of the radicals ; so that as these signs are entirely inoperative, it is matter of indifference whether they be plus or minus. The requisite precision is only to be attained by conforming the values to the condition 472 EaUATION OF THE But if, as in the cases discussed above, we actually introduce this condition into the results themselves, the expressions become freed from all ambiguity, and take the explicit form s/{2k-.p) where the sign of the radical is entirely unrestricted. And it is in this form that the expression should always be written. (277.) It may be remarked here that Lagrange, at the end of the last edition of his work on Equations, incidentally notices the above mode of writing Euler's form as that in which it mai/ be expressed. But he does not insist upon this as the form in which it always ought to be written, and which is the only form strictly in accordance with the conditions impressed upon the symbols in the course of the investigation. Poinsot, in a note appended to the above-mentioned work, animadverts somewhat upon Lagrange's remark, and proposes what he regards as a preferable mode of interpretation as respects Euler's ambi- guous forms. We have not space to transcribe this note of Poinsot, but must express our entire dissent from the doctrine there propounded. (278.) It may be observed, in conclusion, that the formula of Cardan is only a particular case of a much more general ex- pression, an expression which in fact exhibits under a similar form the roots of an equation of the T^th degree. But the practical utility of this general form ceases with the equation of the third degree, because for the higher degrees it is in general impossible to transform the proposed equation into another belonging to the particular class of equations comprehended under the general form. This class of equations was first solved by De Moivre, and every equation belonging that class is said to come under De Moivre's solvible form. The general investigation may be conducted m a manner ana- logous to that which has been employed in the particular case ; FOURTH DEGREE. 4/3 the problem to be solved being, indeed, simply this, viz.. To find two quantities, y and z, such that the sum of their ?^th powers may be a and their product h. The value y determined by these conditions {Algebra^ third ed. p. 156), is 2/ = { i « + V («' - 46-) } -^ and consequently, since s- = - we must have for s the expression h { i « + V («^ - 4^") } "^ And {Algebrtty p. 157), 3/ + ^ is the value of x in the equation ^n_„J,„-2 + »i^Zi) i2^-4_»("-3)(''-4)^,„-6+. . . . = „. When w = 3 the equation is a^ — 3bx — « = 0, a form to which every cubic may be reduced, as a and b may be any values whatever. Making « = — q', and 36 = — ^, we have the general form at (263), and for a? = y -\ — the expres- sion already deduced. When w = 5 the general form becomes x^ — 5bx^ + 5b^x — a = 0, of which the roots are represented by the formula a'= {^a-\-^(a^-4b')}i b + {i«+ V(«'-46^)}^ Many attempts have been made to reduce the general equation of the fifth degree to the above solvible form, but they have all been fruitless. We shall merely add, that by substituting in the preceding equation c for 5b, we may write the form thus, x^ — cx^ -{- ^ c'^x — a ^ 0, where a and c are unrestricted. NOTE. The principle established at pp. 260-3 and there affirmed to remove an imperfection in Horner's method of appoximation was, we find, known to Mr. Horner himself. The author of the present wcrk did jiot discover this till long after the sheet referred to was printed. Mr. Horner announces the principle as follows : " When an equation has a extensive or incommensurable equal roots, the following indications will be observed as the work pro- ceeds : simultaneous and regular diminution in the last a terms ; the preceding terms constant in regard of signs, and, to an in- creasing extent, in their first digits also ; the last a + 1 signs alternately + and — , or — and + ; that mutual relation among the last a + 1 terms which the expansion of h{x — r)" indicates. " From that law of expansion, also, the value of the approxima- tion r may be readily found, and in various ways. E. g. (a) If the last term but one be made the search divisor, the dividend must be a times the last term. (b) The ath term, reckoning backward, may be the dividend, and the search divisor be a times the preceding, (c) The last term being the dividend, and the a -f- 1th term, reckoning backward, the search divisor, the ath root of the quotient may be tried for r. " The ordinary rules, which are only adapted to the case a = l, of course fail when appUed to other cases ; but by attending to the indications here made out, the new method is divested of perplexity, and the transformations may proceed under every circumstance without interruption.? {LeyhourvLS Repository, No. 19, page 63.) The preceding observations apply to the case of equal roots : when the roots are nearly equal Mr. Horner proceeds as follows : NOTE. 47;) " When two or more roots of an equation agree in their first digit or digits, and are at the same time smaller or greater than any other root, abstracting the signs, the evolution of the portion in which they agree will necessarily be attended with indications very similar to those which we have traced to the influence of equal roots. Consequently they present the same resistance to the Newtonian and similar methods ; a fact which Lagrange has not failed to dilate upon. But they also share in the facili- ties pointed out in our remarks above ; to such a degree at least, that an approximation may thence be generally obtained which shall at once place the remaining solution within the reach of the ordinary methods." {Ibid, page Q7 .) Mr. Horner prosecutes the subject no further than this : he illustrates the preceding observation by means of the equation of Lagrange, viz. a^ — 7oc -\- 7 =^0 ; from which it would appear that his view of the principle is, after all, very different from ours ; as he employs it to suggest the second figure, 6, of one of the roots, which figure is widely diiFerent from the second figure of the other root. We employ the principle, only so long as the figures in the contiguous roots concur, to suggest those figures : and we have shown its competency to detect the place at which this concurrence ceases. The suggestion of the true root-figure, in Lagrange's example, is mere matter of chance; and proves nothing as to the value of the principle in question. But had such examples as those developed, by aid of this principle, in the text, occurred to Mr. Horner, there is no doubt that he would have prosecuted the developments in the same manner. The difficulty, however, of ascertaining the exact place at which a root separates, and the method of determining the leading figure of those that remain — ^particulars of much importance, would still have remained unprovided for.* • It may be proper to add to these remarks, that Mr. Lockhart, by exa- mining a great variety of equations of the fifth degree, each having a pair of roots very nearly equal — or concurring in several leading figures — found from 2N' experience, that -— - always separated those roots ; so that the successive A values of this expression, as the approximation advanced, always furnished the concurring figures. See Resolution of Equations by means of Inferior and Superior Limits, 1842. 476 NOTE. The precept 5 at page 265 accomplishes the latter of these particulars in a very direct and easy manner : the other mode of proceeding, explained in the directions which follow this precept, is purely tentative ; and of course far less eligible. We may here remark that in comparing Fourier's method of analysis with that exemplified at p. 306, it is implied at p. 307, that the former method dispenses with a transformation neces- sarily involved in the latter : it must not be overlooked, however, that the transformation (4.8), employed in Fourier's analysis at p. 245, is equivalent to the additional transformation adverted to ; and, therefore, that the desired conclusion is reached in both methods by the same number of steps. THE END, PIUNTRD BV C. ADLARD, BARTHOI4IMRW Cl.nsK ANALYTICAL COURSE OF MATHEMATICS, BY J. R. YOUNG, PROFESSOR OF MATHEMATICS, ROYAL COLLEGR, BELFAST. 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