CORNELL UNIVERSITY LIBRARY MATHEMATICS Cornell University Library QA 155.C55 1904 V.I Algebra; an elementary text book for the 3 1924 001 533 102 The original of tiiis book is in tine Cornell University Library. There are no known copyright restrictions in the United States on the use of the text. http://www.archive.org/details/cu31924001533102 ALGEBRA ELEMENTAEY TEXT-BOOK Uniform with Part I PAKT n Completing Ihe ^Vork and containing an Index to both Parts. 640 pp., post 8to, 'prici' 1 2s. dd. ALGEBRA AN ELEMENTAEY TEXT-BOOK FOR THE HIGHER CLASSES OF SECOI^DAEY SCHOOLS AND FOE COLLEGES G. CHRYSTAL, M.A, LL.D. HONORARY FELLOW OF CORPUS CHRISTI COLLEGE, CAMBRIDGE ; PROFESSOR OP MATHEMATICS IN THE UNIVERSITY OF EDINBURGH PAET L FIFTH EDITION LONDON ADAM AND CHARLES BLACK MCMIV '• I should rejoice to see , . , morphology introduced into tho elements of Algebra. '— SyLVESTER FublisJied July 1886 Reprinted, with corrections ami additions, 1889 ,Veuj impressions 1893 and 1898 Reprinted with corrections and additions 1904 PEEFACE TO THE FIFTH EDITION. In this Edition considerable alterations have been made in chapter xii. In particular, the proof of the theorem that every integral equation has a root has been amplified, and also illustrated by graphical considerations. An Appendix has been added dealing with the general algebraic solution of Cubic and Biquadratic Equations ; with the reducibility of equations generally ; and with the possibility of solution by means of square roots. As the theorems established have interesting applications in Ele- mentary Geometry, it is believed that they may find an appropriate place in an Elementary work on Algebra. G. CHKYSTAL. 29th June 1904. PEEFACE TO THE SECOND EDITION. The comparatively rapid sale of an edition of over two thousand copies of this volume has shown that it has, to some extent at least, filled a vacant place in our educational system. The letters which I have received from many parts of the United Kingdom, and from America, containing words of encouragement and of useful criticism, have also strengthened me in the hope that my labour has not been VI PREFACE iu vain. It would be impossible to name here all the friends who have thus favoured me ; and I take this oppor- tunity of offering them collectively my warmest thanks. The present edition has been thoroughly revised and corrected. The first chapter has been somewhat simplified ; and, partly owing to experience with my own pupils, partly in consequence of some acute criticism sent to me by Mr. Levett of Manchester, the chapters on Indices have been recast, and, I think, greatly improved. In the verification and correction of the results of the exercises I have been indebted in a special degree to the Eev. John Wilson, Slathematical Tutor in Edinburgh. The only addition of any consequence is a sketch of Horner's Method, inserted in chapter xv. I had originally intended to place this in Part II. ; but, acting on a sugges- tion of Mr. Hayward's, I have now added it to Part I. To help beginners, I have given, after the table of contents, an index of the principal technical terms used in the volume. This index will enable the student to turn up a passage where the " hard word " is either defined or other- wise made plain. G. CHEYSTAL. Edinburgh, lU/i October 188.9. PKEFACE TO THE FIRST EDITION. The work on Algebra of wliich this volume forms the first part, is so far elementary that it begins at the beginning of the subject. It is not, however, intended for the use of absolute beginners. The teaching of Algebra in , the earlier stages ought to consist in a gradual generalisation of Arithmetic ; in other words. Algebra ought, in the first instance, to be taught as Arithmetica Universalis in the strictest sense. I suppose that the student has gone in this way the length of, say, the solution of problems by means of simple or perhaps even quadratic equations, and that he is more or less familiar with the construction of literal formula, such, for example, as that for the amount of a sum of money during a given term at simple interest. Then it becomes necessary, if Algebra is to be any- thing more than a mere biindle of unconnected rules, to lay down generally the three fundamental laws of the subject, and to proceed deductively — in short, to introduce the idea of Algebraic Form, which is the foundation of all the modern developments of Algebra and the secret of analy- tical geometry, the most beautiful of all its applications. Such is the course followed from the beginning in this work. Vlll PREFACE As mathematical education stands at present in this country, the first part might be used in the higher classes of our secondary schools and in the lower courses of our colleges and nniversities. It will he seen on looking through the pages that the only knowledge re([uired outside of Algebra proper is familiarity with the definition of the trigonometrical functions and a knowledge of their funda- mental addition-theorem. The first object I have set before me is to develop Algebra as a science, and thereby to increase its usefulness as an educational discipline. I have also endeavoured so to lay the foundations that nothing shall have to be im- learned and as little as possible added when the student comes to the higher parts of the subject. The neglect of this consideration I have found to be one of the most important of the many defects of the English text -books hitherto in vogue. Where immediate practical application comes in question, I have striven to adapt the matter to that end as far as the main general educational purpose would allow. I have also endeavoured, so far as possible, to give complete information on every subject taken up, or, in default of that, to indicate the proper sources ; so that the book should serve the student both as a manual and as a book of reference. The introduction here and there of historical notes is intended partly to serve the purpose just mentioned, and partly to familiarise the student with the great names of the science, and to open for him a vista beyond the boards of an elementary text-book. As examples of the special features of this book, I may ask the attention of teachers to chapters iv. and v. With respect to the opening chapter, which the beginner will PREFACE ix doubtless find the hardest in the book, I should mention that it was written as a suggestion to the teacher how to connect the general laws of Algebra with the former experience of the pupil. In writing this chapter I had to remember that I was engaged in writing, not a book on the philosophical nature of the first principles of Algebra, but the first chapter of a book on their consequences. Another peculiarity of the work is the large amount of illustrative matter, which I thought necessary to prevent the vagueness which dims the learner's vision of pure theory; this has swollen the book to dimensions and corresponding price that require some apology. The chapters on the theory of the complex variable and on the equivalence of systems of equations, the free use of graphical illustrations, and the elementary discussion of problems on maxima and minima, although new features in an English text -book, stand so little in need of apology with the scientific public that I offer none. The order of the matter, the character of the illustra- tions, and the method of exposition generally, are the result of some ten years' experience as a university teacher. I have adopted now this, now that deviation from accepted English usages solely at the dictation of experience. It was only after my own ideas had been to a considerable extent thus fixed that I did what possibly I ought to have done sooner, viz., consulted foreign elementary treatises. I then found that wherever there had been free considera- tion of the subject the results had been much the same. I thus derived moral support, and obtained numberless hints on matters of detail, the exact sources of which it would be difficult to indicate. I may mention, however, as specimens X PKEFACE ut' the class of treatises referred to, the elementary text- books of Baltzer in German and Collin in French. Among the treatises to which I am indebted in the matter of theory and logic, I- should mention the works of De Morgan, Pea- cock, Lipschitz, and St-rret. Many of the exercises have been either taken from my own class examination papers or constructed expressly to illustrate some theoretical point discussed in the text. For the rest I am heavily indebted to the examination papers of the various colleges in Cam- bridge. I had originally intended to indicate in all cases the sources, but soon I found recurrences which rendered this difficult, if not impossible. The order in which the matter is arranged will doubt- less seem strange to many teachers, but a little reflection will, I think, convince them that it could easily be justified. Tbere is, however, no necessity that, at a first reading, the order of the chapters should be exactly adhered to. I think that, in a final reading, the order I have given should be followed, as it seems to nie to be the natural order into which the subjects fall after they have been fully com- prehended in their relation to the fundamental laws of Algebra. With respect to the very large number of Exercises, I should mention that they have been given for the con- venience of the teacher, in order that he might have, year by year, in using the book, a sufficient variety to prevent mere rote-work on the part of his pupils. I should much deprecate the idea that any one pupil is to work all the exercises at the first or at any reading. We do too much of that kind of work in this country. I have to acknowledge personal obligations to Professor PREFACE XI Tait, to Dr. Thomas Muik, and to my assistant, Mr. E. E. Allaedice, for criticism and suggestions regarding the theoretical part of the work ; to these gentlemen and to Messrs. Magkay and A. Y. Fkaser for proof reading, and for much assistance in the tedious work of verifying the answers to exercises. In this latter part of the work I am also indebted to my pupil, Mr. J. Mackenzie, and to my old friend and former tutor. Dr. David Eennbt of Aberdeen. Notwithstanding the kind assistance of my friends and the care I have taken myself, there must remain many errors both in the text and in the answers to the exercises, notification of which either to my publishers or to myself will be gratefully received. G. CHRYSTAL. EDiNBaRGH, 2Gi/i June 1886. CONTENTS. CHAPTEE I. FUNDAMENTAL LAWS AND PROCESSES OF ALGEBRA. PAGE Laws of Association and Commutation for Addition and Subtraction 2-7 Essentially Negative Quantity in formal Algebra . 8 Properties of . . . 11 Laws of Commutation and Association for Multiplication 12 Law of Distribution ...... 13 Laws of Association, Commutation, and Distribution for Division 14-19 Properties of 1 . . 17 Synoptic Table of the Laws of Algebra 20 Exercises I. 22 Historical Note . 24 CHAPTER II. MONOMIALS LAWS OF INDICES DEGREE. Laws of Indices ... . 25-29 Theory of Degree, Constants and Variables 30 Exercises II. ... . , 31 CHAPTEE III. THEORY OF QUOTIENTS FIRST PRINCIPLES OF THEORY OF NUMBERS. Fundamental Properties of Fractions and Fundamental Operations therewith . . 33-36 Exercises III. . 36 Prime and Composite Integers . 38 Arithmetical G.C.JI. . . 39 Theorems on the Divisibility of Integers . 41 Kemainder and Residue, Periodicity for Given Modulus , 42 Arithmetical Fractionality .... 43 The Resolution of a Composite Number into Prime Factors is unique 44 General Theorem regarding G.G.M., and CoroUarie.'; 44 The Number of Primes is infinite 47 Exercises IV. , 48 CONTENTS CHAPTER IV. DISTRIBUTION OP PRODUCTS ELEMENTS OF THE THEORY OP RATIONAL INTEGRAL FUNCTIONS. PAOE Generalised Law of Distribution .... 49 Expansion by enumeration of Products ; Classification of the Products of a given set of letters into Types ; S and 11 Notations . , 51-54 Principle of Substitution . . 51 Tlieorcni regarding Sum of Coefficients 55 Exercises V. ...... 56 General Tlieorems regarding the Multiplication of Integral Functions 57 Integi'al Functions of One Variable . 59-69 Product of Binomials . 60 Binomial Theorem . . 61 Detached Coefficients ... 64 Addition nile for calculating Binomial Coeflicients, with a General- isation of the same . 66 j-n±yn as a Product . . 68 Exercises VI. . 69 Exercises VII. . . 70 Homogeneity . . 71-75 General forms of Homogeneous Integral Functions . 72 Fundamental Property of a Plomogeneous Function 73 Law of Homogeneity . . . .74 Most general form for an Integral Function . 75 Symmetry ... . . 75-79 Properties of Symmetric and Asymmetric Functions 76 Rule of Symmetry . ... 77 Most general forms of Symmetric Functions 78 Principle of Indeterminate Cui'lficients 79 Table of Identities 81 Exercises VIII. , 83 CHAPTER V. TRANSFORMATION OP THE QUOTIENT OF TWO INTEGRAL FUNCTIONS. Algebraic Integrity and Fractionality 85 Fundamental Theorem regarding Divisibility . . 86 Ordinary Division-Transformation, Integral Quotient, Remainder . 86-93 Binomial Divisor, Quotient, and Remainder ... 93 Remainder Theorem ... 96 Factorisation by means of Remainder Theorem ... 97 Maximum number of Linear Factors of an Integral Function of a; . 98 New basis for the Principle of Indeterminate Coefficients . 99 Continued Division . 102 CONTENTS XV PAGE One Integral Function expressed in powers of another . . . 105 Expansion in the form Ao + Ai{x-ai) + A2{x-ai){x-a2) + As{x-a-i) {x~a2){x-a3)+ . . . . . . . ' . 107 Exercises IX. .... 108 CHAPTER VI. GREATEST COMMON MEASUEB AND LEAST COMMON MULTIPLE. G. CM. by Inspection .... . 112 Ordinary process for Two Functions . . .113 Alternate destruction of highest and lowest terms . . . 117 G. CM. of any number of Integral Functions . . . . 119 General Proposition regarding the Algebraical G.C.M., with Corollaries regarding Algebraic Primeness . , , . 119 L.C.M. . . 1,22 Exercises X. . . . . . . . , 124 CHAPTEE VII. FACTORISATION OF INTEGRAL FUNCTIONS. Tentative Methods ...... General Solution for a Quadratic Function of a; Introduction of Surd and Imaginary Quantity Progression of Keal Algebraic Quantity Square Root, Rational and Irrational Quantity Imaginary On it ... Progression of Purely Imaginary Quantity . Complex Quantity ...... Discrimination of the different cases in the Factorisation of ax' + ix + c . .... Homogenepus Functions of Two Variables Use of the Principle of Substitution .... Use of Remainder Theorem ...... Factorisation in general impossible . . . . Exceptional case of ax^ + 2hxy + hy' + 2gx + 2fy + c Exercises XI. ... CHAPTER VIII. RATIONAL FRACTIONS. General Propositions regarding Proper and Improper Fractions Examples of Direct Operations with Rational Fractions Inverse Method of Partial Fractions . General Theorem regarding decomposition into Partial Fractions . Classification of the various species of Partial Fractions, with Methods for determining Coefficients . Integral Function expansible in the form 'S{aij + ai{x-a)+ + ar-i{x-a)'-^}{x-l3)'(x-yy .... Exercises XII. ..... 126 128-137 130-133 130 132 132 133 133 134 136 136 138 139 140 142 144-147 147-150 151-159 151 153 155 159 XVI CONTENTS CHAPTER IX. FURTHER APPLICATIONS TO THE THEORY OP NUMBERS. PAGE Expression of an Integer as an Integral Factorial Series 163 Expression of a Fraction as a Fractional Factorial Series 165 Scales of Arithmetical Notation . lti7-\7'> Expression of an Integer in a Scale of given Radix 167 Avithmetical Calculation in various Scales 169 Expression of any Fraction as a Radix Fraction . . .170 Divisibility of a Number and of the Sum of its Digits by r - 1 ; the " Nine Test " ' 174 Lambert's Theorem ] 76 Exercises XIII. 177 CHAPTER X. IRRATIONAL FUNCTIONS. Interpretation of aWs ..... .180 Consistency of the Interpretation with the Laws of Indices Examined 182 Interpretation of a;" 185 Interpretation of x"" . . . 186 Examples of Operation with Irrational Forms 187-189 Rationalising Factors ..... 189-198 Every Integral Function of \/p, \/q, \/r, &c., can be expressed in the linear form &. + V>\/p + C\/q + 'D\/r+ . +'E.\/pq+. . + F\/jJ3r+ . . . . , ■ 193 Rationalisation of any Integral Function of \/p, \Jq, \/r, &c. 195 Every Rational Function of \/p, \/q, «y;-, &c., can be expressed in linear form . . 196 General Theory of Rationalisation 197-198 Exercises XIV. . 199 Historical Note 201 CHAPTER XI. ARITHMETICAL THEORY OF SURDS. Algebraical and Arithmetical Irrationality 203 Classification of Surds . . . 204 Independence of Surd Numbers 205-207 Expression of \/{a + \/i) in linear form 207 Rational Approximations to the Value of a Surd Number 210-215 Extraction of the Square Roi. I . 211 Square Root of an Integral Function of./- 215 Extraction of Roots by means of Indeterminate Coefficients 217 Exercises XV. . 218 CONTENTS XVll CHAPTEE XII. COMPLEX NUMBERS. PAGE Independence of Real and Imaginary Quantity . . . 221 Two-foldness of a Comples Number, Argand's Diagram . . 222 , lix + yi = x'+'y'i, t'heiix=x', y=y' ..... 224 Every Rational Function of Complex Numbers is a Complex Number 224-227 If 0(a!+2/i) = X + Yi, th.enip{x-yi) = X-Yi; ii (p{x + yi)=0, then (p{a;-yi) = ....... 226 Conjugate Complex Numbers ...... 228 Moduli. .... ... 229 If x + yi = 0, then \x + yi\ = ; and conversely . . .229 1 0(a! + j/i)|= (y{0{a; + 2/i)i/>(a;-2/t)}; Particular Cases . . . 230 The Product of Two Integers each the Sum of Two Squares is the Sum of Two Squares ...... 230 Discussion by means of Argand's Diagram . . . 232-236 Every Complex Number expressible in the form r(cos 6+i sin 6) ; Definition of Amplitude ...... 232 Addition of Complex Numbers, Addition of Vectors . . 233 \z^ + Z2+ . . . +s!„|«K| + |a2|+ . . . +|z»l . . .234 The Amplitude of a Product is the ^um of the Amplitudes of the Factors ; Demoivre's Theorem ..... 235 Root Extraction leads to nothing more general than Complex Quantity ....... 236-244 Expression of ij{x + yi) as a Complex Number . . .237 Expression of ^(aj + 2/i) as a Complex Number . . . 238 Every Complex Number has n rath roots and no more . . 240 Properties of the mth roots of ± 1 . . . . . 240 Resolution of a:"± A into Factors ..... 243 Every Integral Equation has at least- one root ; Every Integral Equation of the »th degree has n roots and no more ; Every Integral Function of the mth degree can be uniquely resolved into n Linear Factors ..... 244-250 Upper and Lower Limits for the Roots of an Equation . . 247 Continuity of an Integral Function of s . . . 248 Equimodular and Gradient Curves of/(«) .... 248 Argand's Progression towards a Root ... . . 249 Exercises XVL . . ' . • • • .251 Historical Note ...... 253 CHAPTEE XIII. RATIO AND PROPORTION. Definition of Ratio and Proportion in the abstract Propositions regarding Proportion Examples ..... Exercises XVII. .... VOL. I 255 257-264 264-266 267 h xvm CONTENTS Ratio and Proportion of Concrete Quantities , Definition of Concrete Ratio Difficulty in the case of Incommensurables Euclidian Theory of Proportion Variation . . . . . Indefiendent and Dependent Variables Simplest Cases of Functional Dependence Other Simple Cases ... Propositions regarding "Variation'' Exercises XVIII. . . . , 268-273 269 270 272 273-279 273 274 275 276 279 CHAPTEE XIV. ON CONDITIONAL EQUATIONS IN GENERAL. General Notion of an Analytic Function . . 281 Conditional Equation contrasted with an Identical Equation . 282 Kno^\^l and Unknown, Constant and Variable Quantities . . 283 Algebraical and Transcendental Equations ; Classification of Integral Equations ........ 283 Meaning of a Solution of a System of Equations . . 284 Propositions regarding Determinateness of Solution . 286-288 Multiplicity of Determinate Solutions .... 289 Definition of Equivalent Systems ; Reversible and Irreversible Derivations ....... 289 Transformation by Addition and Transposition of Terms . . 291 Multiplication by a Factor ...... 292 Division by a Factor not a Legitimate Derivation . . . 293 Every Rational Equation can be Integralised .... 294 Derivation by raising both sides to the same Power . . . 295 Every Algebraical Equation can be Integralised ; Equivalence of the Systems, Pi = 0, Po = 0, . . . P„ = 0, and LjPj + L2P0 + . . . + L„P„ = 0, P2 = 0, . . . P„ = .... .296 Examination of the Systems P = Q, E = S; PR = QS, R = S . . 297 On Elimination . ..... 298 Examples of Integralisation and Rationalisation . . 299-301 Examples of Transformation ...... 302 Examples of Elimination . . . . . . 304 Exercises XIX. ....... 305 Exercises XX. ........ 306 Exercises XXI. ....... 308 CHAPTER XV. VARIATION OP A FUNCTION. Graph of a Function of one Variable ..... 310 Solution of an Equation by means of a Graph , . , 313 Discontinuity in a Function and in its Graph . . . 315 CONTENTS XIX PAGE Limiting Cases of Algebraic Operation . 318-322 Definition of tlie Increment of a Function .... 322 Continuity of the Sum and of the Product of Continuous Functions . 323 Continuity of any Integral Function ..... 324 Continuity of the Quotient of Two Continuous Functions ; Ezceptiou 324 General Proposition regarding Continuous Functions . . 325 Number of Roots of an Equation between given limits . . 326 An Integral Function can change sign only by passing through the value ; Corresponding Theorem for any Eational Function . 326 Sign of the Value of an Integral Function for very small and for very large values of its Variable ; Conclusions regarding the Number of Roots ........ 328 Propositions regarding Maxima and Minima .... 330 Continuity and Graphical Representation of f{x, y) ; Graphic Surface ; Contour Lines ....... 331 f{x, 2/) = represents a Plane Curve . . . . .334 Graphical Representation of a Function of a Single Complex Variable 335 Horner's Method for approximating to the Real Roots of an Equation ....... 338-346 Multiplication of Roots by a Constant . . 338 Increase of Roots by a Constant . . . . .339 Approximate Value of Small Root ..... 340 Horner's Process ... ... 341 Example ....... 842-346 Extraction of square, cube, fourth, . . . roots by Horner's Method . 346 Exercises XXIL .... .347 CHAPTER XVI. EQUATIONS AND FUNCTIONS OF FIRST DEGREE. Linear Equations in One Variable . . . 349, 350 Exercises XXIIL ....... 351 Linear Equations in Two Variables — Single Equation, One-fold Infin- ity of Solutions ; System of Two, Various Methods of Solution ; , System of Three, Condition of Consistency . . 352-364 Exercises XXIV. ....... 364 Linear Equations in Three Variables — Single Equation, Two-fold In- finity of Solutions ; System of Two, One-fold Infinity of Solu- tions, Homogeneous System ; System of Three, in general Deter- minate, Homogeneous System, Various Methods of Solution ; Systems of more than Three .... 365-372 General Theory of a Linear System .... 373 General Solution by means of Determinants . . 374-376 Exercises XXV. ....... 376 Examples of Equations solved by means of Linear Equations . 379-383 Exercises XXVI. . . .... 383 XX CONTENTS PAGE Graph of OKC + 5 . . ... 385 Graphical Discussion of the cases 6 = 0; «=0 ; a=0, b = 388 Contour Lines oi ax + by + c . . . . . 389 Illustration of the Solution of a System of Two Linear Equations . 390 Cases where the Solution is Infinite or Indeterminate discussed graphically ....... 391 Exceptional Systems of Three Equations in Two Variables . . 393 Exercises XXVII. . . . . 394 CHAPTER XVII. EQUATIONS OF THE SECOND DEGREE. aar' + Ja; + c=0 has in general just two roots . . . .396 Particular Cases .... . . 397 General Case, various Methods of Solution . . . 398 Discrimination of the Roots . . . 400 Exercises XXVIII. . ..... 401 Equations reducible to Quadratics, by Factorisation, by Integralisa- tion, by Rationalisation ..... 402-406 Exercises XXIX., XXX. ...... 406 Exercises XXXI. ...... 407 Reduction by change of Variable ; Reciprocal Equations . 408-413 Rationalisation by introducing Auxiliary Variables . . . 413 Exercises XXXII. ....... 413 Systems with more than One Variable which can be solved by means of Quadratics ... . . 414-427 General System of Order 1x2 . . . . . 415 General System of Order 2x2; Exceptional Cases . . 416 Homogeneous Systems ... . 418 Symmetrical Systems ...... 420 Miscellaneous Examples ...... 425 E.xercises XXXIII. . . . . 427 Exercises XXXIV. . .... 429 Exercises XXXV. . ..... 430 CHAPTER XVIII. GENERAL THEORY OP INTEGRAL FUNCTIONS. Relations between Coefficients and Roots .... 431 Symmetric Functions of the Roots of a Quadratic . . . 432 Newton's Theorem regarding Sums of Powers of the Roots of any Equation ..... . . 433 Symmetric Fmictions of the Roots of any Equation . . 438 Any Symmetric Function expressible in terms of certain elementary Symmetric Functions ...... 440 COKTENTS Xxi PAGE Exercises XXXVI. .... .445 Special Properties of Quadratic Functions . . 447-453 Discrimination of Roots, Table of Results .... 447 Generalisation of some of the Results . . . . 449 Condition that Two Quadratics have Two Roots in common . 450 Lagrange's Interpolation Formula . . . . . 451 Condition that Two Quadratics have One Root in common . 452 Exercises XXXVII. .... . 453 Variation of a Quadratic Function for real values of its Variable ; Analytical and Graphical Discussion of Three Fundamental Oases, Maxima and Minima ... . 454-458 Examples of Maxima and Minima Problems .... 458 General Method of finding Turning Values by means of Equal Roots . 461 Example, 2/=a;3- 9x2 + 24a; + 3 . ..... 462 Example, j/=(a;2-8a; + 15)/a; ...... 463 General Discussion of y={ax' + bx + c)l{a'x^ + i'x + c'), with Graphs of certain Particular Cases ..... 464-467 Finding-of Turning Values by E.tamination of the Increment . 468 Exercises XXXVIII. ...... 469 CHAPTEE XIX, SOLUTION OP PROBLEMS BY MEANS OF EQUATIONS. ChoioB of Variables ; Interpretation of the Solution . . . 471 Examples ....... 472-476 Exercises XXXIX . . 476 CHAPTER XX. ARITHMETIC, GEOMETRIC, AND ALLIED SERIES. Definition of a Series ; Meaning of Summation ; General Term . 480 Integral Series ...... 482-488 Arithmetic Progression ... . 482 Sums of the Powers of the Natural Numbers . 484-487 Sura of any Integral Series ... . 487 Arithmetic-Geometric Series, including the Simple Geometric Series I as a Particular Case ..... 489-494 495 496-502 496 497 497 499 499 Convergency and Divergency of Geometric Series Properties of Quantities in A. P., in G.P., or in H.P. Expression of Arithmetic Series by Two Variables Insertion of Arithmetic Means Arithmetic Mean of n given quantities Expression of Geometric Series by Two Variables Insertion of Geometric Means Geometric Mean of n given quantities . . 500 XXll CONTENTS PAGE Definition of Harmonic Series ...... 500 Expression in terms of Two Variables .... 501 Insertion of Harmonic Means ..... 501 Harmonic mean of )i given quantities .... 601 Propositions regarding A.M., G.M., and H.M. . . . 501 Exercises XL 502 Exercises XLI. ........ 505 Exercises XLII. . . ... 507 CHAPTEE XXI. LOGARITHMS. Discussion of a' as a Continuous Function of a; . . . 509 Definition of Logarithmic Function .... 511 Fundamental Properties of Logarithms . . . 512 Computation and Tabulation of Logarithms . . . 513-519 Mantissa and Characteristic ...... 514 Advantages of Base 10 ..... . 515 Direct Solution of an Exponential Equation , . . 516 Calculation of Logarithms by inserting Geometric Means . . 517 Alteration of Base ....... 519 Use of Logarithms in Arithmetical Calculation . 519-523 Interpolation by First Differences ..... 524 Exercises XLIII. . . .... 527 Historical Xote ...... 529 CHAPTER XXII. THEORY OF INTEREST AND ANNUITIES. Simple Interest, Amount, Present Value, Discount . . 531-532 Compound Interest, Conversion - Period, Amount, Present Value, Discount, Nominal and Effective Eato . . . 533-535 Annuities Certain, Accumulation of Forborne Annuity, Purchase Price of Annuity, Terminable or Perpetual, Deferred or Undeferred, Xumber of Years' Purchase .... 536-540 Exercises XLIV. ... ... 540 APPENDIX. Commensurable Roots, Eeducibility of Equations Equations Soluble by Square Roots Cubic . .... Biquadratic, Resolvents of Lagrange and Descartes Possibility of Elementary Geometric Construction Exercises XLV. .... RESULTS OF EXERCISES. 543-546 546-548 649 550 561 553 556 INDEX OF PEINCIPAL TECHNICAL TEEMS USED IN PAET I. Addition Rule for binomial coefficients, 66 Affine of a complex number, 223 Algebraic sum, 10 Algebraical function, ordinary, 281 Alternating function, 77 Amount, 532 Amplitude of a complex number, 236 Annuity, certain, contingent, termin- able, perpetual, immediate, deferred, forborne, number of years' purchase, 5S6 et seq. Antecedent of a ratio, 255 Antilogarithm, 518 Argand-diagram, 222 Argand's progression, 249 Argument, 524 Arithmetic means and arithmetic mean, 497 Arithmetic progression, 482 Arithmetico-geometric series, 491 Association, 3, 12 Auxiliary variables, 380 Bask of an exponential or logarithm, 511 Binomial theorem, 62 Chahacteeistio, 514 GoefBcient, 30 Commensurable, 203 Common Measure and Greatest Common Measure (arithmetical sense), 38, 39 ; algebraical sense, 111 Commutation, 4, 12 Complex number or quantity, 133, 221 Conjugate complex numbers, 228 Consequent of a ratio, 255 Consistent system of equations, 288 Constant, 30 Continued division, 102 Continued proportion, 256 Continuity of a function, 317, 323, 324, 336 Contour lines of a function, 333 Convergency of a series, 493 Conversion-period (for interest), 533 Degeee, 30, 58 Degree of an equation, 284 Derivation of equations, 290 Detached coefficients, 63, 91 Determinateness of a system of equa- tions, 286 Difl'erences (first), 521 Discontinuity of a function, 317 Discount, 632 Discriminant, 134, 141 Distribution, 13, 49 Divergency of a series, 493 Divisibility (algebraical sense), 85 Divisibility (arithmetical sense), 38 Elimikaht (or resultant), 415, 430 Elimination, 298 Equation, conditional, 282 Equation and equality (identical), 22 Equimodular curves, 248 Equiradical surds, 204 Equivalence of systems of equations, 289 Exponent, 25 Exponential function, 609 Exponential notation (exp „), 511 Extraneous solutions, 294 Extremes and means of a proportion, 256 Factok (arithmetical sense), 38 Fractional (algebraical sense), 30, 85 Fractional (arithmetical sense), 43 Freehold, value of, 539 Function, analytical, 281 Function, rational, integral, algebraical, 68 Geometeio means and geometric mean, 499 Geometric series, 489 Gradient curves, 248 Graph of a function, 312, 333 Graphical solution of equations, 313 Greatest Common Measure (algebraical sense). 111 Harmonic means and harmonic mean, 501 Harmonic series, 500 Homogeneity, 71 Homogeneous system of equations, 418 Identity, identical, 22 Imaginary unit and imaginary quantity, 132 XXIV INDEX Increment of a function, 322 Indeterminate coefficients, 79, 100 Indeterminate forms, 318, 319, 320 Indeterminateness of a system of equa- tions, 286 Index, 25 Infinite value of a function, 315 Infinitely great, 318 Infinitely small, 318 Integral (algebraical sense), 25, 85 Integral (arithmetical sense), 37 Integral function, 58 Integral quotient (algebraical sense), 86 Integral series, 484 Integralisation of equations, 296 Integro-geometrio series, 492 Interest, simple and compound, 531, 533 Interpolation, 524 Inverse, 5, 14 Irrationality (algebraic), 203, 240 Irrationality (aritbmetical), 203 Irreducible case of cubic, 549 Irreducible equation, 545 Irreversible derivation, 290 Laws of Algebra, fundamental, 20 Least Common Multiple (algebraical sense), 122 Limiting cases, 318 Linear (algebraic sense), 138 Linear irrational form, 193 Logaritbmic function, 511 Manifoldness, 496 Mantissa, 514 Maxima values of a function, 330 Mean proportionals, 256 Minima values of a function, 330 Modulus (arithmetical sense), 43 Modulus of a complex number, 229 Modulus of system of logarithms, 519 Monomial function, 30 Negative quantity, 9 Nine-test, 175 Operand and operator, 4 Order of a symmetric function, 439 Partial fractions, 151 Pascal's triangle, 67 Periodicity of integers, 43 Il-notation, 53 Prime (arithmetical sense), 38 Primeness (algebraical), 120 Principal, 532 Principal value of a root, 182 Proper fraction (algebraical sense), 86, 144 Proportion, 266, 269 Proportional parts, 526 Quantity, ordinary algebraic, 130 Rate op interest, nominal and eflFective, 535 Ratio, 255, 268 Rational (algebraic sense), 144 Rational fraction, 144 Rationalisation of equations, 296 Rationalising factor, 190, 197 Reciprocal equations, 410 Reducibility of an equation, 545 Remainder (algebraical sense), 86 Remainder (arithmetical sense), 42 Remainder theorem, 93 Residue (arithmetical sense), 42 Resolvent of a biquadratic, 550 Resultant equation, 415 Reversible derivation, 290 Root of an equation, 284 Roots of a function, 313 Scales of notation, 168 Series, 480 Similar surds, 204 S-notation, 53 Solution of an equation, formal and approximate numerical, 284 Substitution, principle of, 18 Sum (finite) of a series, 481 Sum (to infinity) of a series, 493 Surd number, 132 Surd number, monomial, binomial, &c., 203, 204 Symmetrical system of equations, 420 Symmetry, 75 Term, 30, 58 Transcendental function, 282 Turning values of a function, 330 Type (of a product), 52 Unity (algebraical sense of), 17 Variable, 30 Variable, independent and dependent, 273 Variation of a function, 311 "Variation" (old sense of), 273, 275 Weight of a symmetric function, 434 Zero (algebraical sense of), 11, 14 CHAPTEE I. The Fundamental Laws and Processes of Algebra as exhibited in ordinary Arithmetic. § l.J The student is already familiar with the distinction between abstract and concrete arithmetic. The former is con- cerned with those laws of, and operations with, numbers that are independent of the things numbered; the latter is taken up with applications of the former to the numeration of various classes of things. Confining ourselves for the present to abstract arithmetic, let us consider the following series of equalities : — 2623 1023_2623v 3 + 1023 X 61 61 "*" 3 ~ 61x3 70272 „„, = -183- = ^«^- The first step is merely the assertion of the equivalence of two different sets of operations with the same numbers. The second and third steps, though doubtless based on certain simple laws from which also the first is a consequence, nevertheless require for their direct execution the application of certain rules, of a kind to which the name arithmetical is appropriated. We have thus shadowed forth two great branches of the higher mathematics: — one, algebra, strictly so called, that is, the theory of operation with numbers, or, more generally speaking, with quanti- ties ; the other, the higher arithmetic, or theory of numbers. These two sciences are identical as to their fundamental laws, but differ widely in their derived processes. As is usual in elementary text-books, the elements of both will be treated in this work. VOL. I B 2 REPRESENTATIVE GROUPS ohap. § 2.] Ordinary algebra is simply the general theory of those operations with quantity of which the operations of ordinary abstract arithmetic are a particular case. The fundamental laws of this algebra are therefore to be sought for in ordinary arithmetic. However various and complex the operations of arithmetic may seem, it appears on consideration that they are merely the result of the application of a very small number of fundamental principles. To make this plain we return for a little to the very elements of arithmetic. ADDITION, AND THE GENERAL LAWS CONNECTED THEREWITH, § 3.] When a group of things, no matter how unlike, is con- sidered merely with reference to the number of individuals it contains, it may be represented by another group, the individuals of which are all alike, provided only there be as many individuals in the representative as in the original group. The members of our representative group may be merely marks (I's say) on a piece of paper. The process of counting a group may therefore be conceived as the successive placing of I's in our representa- tive group, until we have as many I's as there are individuals in the group to be numbered. This process of adding a 1 is represented by writing + 1. We may thus have + 1, +1 + 1, +1 + 1 + 1, +1 + 1 + 1 + 1, &c., as representative groups or " numbers." As the student is of course aware, these symbols in ordinary arithmetic are abbreviated "^^° 1, 2, 3, 4,&c. Hence using the symbol " = " to stand for " the same as," or "replaceable by," or "equal to,'' we have, as definitions of 1, 2, 3, 4, &c., 1= +1, 2= +1 + 1, 3= +1+1 + 1, -4= + 1 + 1 + 1 + 1, (fee. I ASSOCIATION IK ADDITION 3 And there is a further arrangement for abridging the repre- sentation of large numbers, which the student is familiar with as the decimal notation. With numerical notation we are not further concerned at present, but there is a view of the above equalities which is important. After the group +1 + 1 + 1 has been finished it may be viewed as representing a single idea to the mind, viz. the number " three." In other words, we may look at +1 + 1 + 1 as a series of successive additions, or we may think of it as a whole. When it is necessary for any purpose to emphasise the latter view, we enclose +1 + 1 + 1 in a bracket, thus ( + 1 + 1 + 1) ; and it will be observed that pre- cisely the same result is attained by virriting the symbol 3 in place of + 1 + 1 + 1, for in the symbol 3 all trace of the for-, mation of the number by successive addition is lost. We might therefore understand the equality or equation 3= +1 + 1 + 1 to mean ( + 1 + 1 + 1)= +1 + 1 + 1, and then the equation is a case of the algebraical Law of Association. The full meaning of this law will be best understood by con- sidering the case of two groups of individuals, say one of three and another of four. If we wish to find the number of a group made up by combining the two, we may adopt the child's process of counting through them in succession, thus, + 1 + 1 + 1 I +1 + 1 + 1 + 1 = 7. But by the law of association we may write for +1 + 1 + 1 (+1 + 1 + 1), and for +1 + 1 + 1 + 1 (+1 + 1 + 1 + 1), and we have +(+1+1 + 1) + ( + 1+1 + 1 + 1) = 7, or +3 + 4 = 7. It will be observed that we have added a + in each case be- fore the bracket, and it may be asked how this is justified. The answer is simply that setting down a representative group of three individuals is an operation of exactly the same nature as i COMMUTATION IN ADDITION chap. setting down a group of one. The law of association for addition worded in this way for the simple case before us would be this : To set down a representative group of three individuals is the same as to set down in succession three representative individuals. The principle of association may be carried further. The representative group +3 + 4 may itself enter either as a whole or by its parts into some further enumeration : thus, + 6 + ( + 3 + 4)= +6 + 3 + 4 ' is an example of the la* of association which the student will have no difficulty in interpreting in the manner already indi- cated. The ultimate proof of the equality may be regarded as resting on a decomposition of all the symbols into a succession of units. There is, of course, no limit to the complication of associations. Thus we have + [( + 9 + 8) + { + 6 + ( + 5 + 3)}] + { + 6 + ( + 3 + 5)} = +( + 9 + 8) + { + 6+( + 5 + 3)} + 6 + ( + 3 + 5), = +9 + 8 + 6 + ( + 5 + 3) + 6 + 3 + 5, = +9 + 8 + 6 + 5 + 3 + 6 + 3 + 5, each single removal of a bracket being an assertion of the law of association. The student will remark the use of brackets of different forms to indicate clearly the different associations. § 4.] It follows from the definitions 3=+l + l + l, 2=+l + l, that +3 + 2= +2 + 3; and by a similar proof we might show that + 3 + 4 + 6= +3 + 6 + 4= +4 + 3 + 6, &c.; in other words, the order in which a series of additions is arranged is indifferent. This is the algebraical Law of Commutation, and it will be observed that its application is unrestricted in arithmetical operations where additions alone are concerned. The statement of this law at once suggests a principle of great importance in algebra, namely, the attachment of the " symbol of operation " or " operator " to the number, or, more generally speaking, " subject " or "operand," on which' it acts. Thus in the above equations I SUBTEACTION DEFINED 5 the + before the 3 is supposed to accompany the 3 when it is transferred from one part of the chain of additions to another. The operands in + 3, + 4, and + 6 are already complex ; and it may be shown by a further application of the reasoning used in the beginning of this article that the operand may be complex to any degree without interfering with the validity of the com- mutative law ; for example, + {+3 + ( + 2 + 3)} + (+6 + 8) = +(+6 + 8) + { + 3 + ( + 2 + 3)}, of which a proof might also be given by first dissociating, then commutating the individual terms +6, +8, +3, &c., and then reassociating. The Law of Commutation, thus suggested hy arithmetical considera- tions, is now laid down as a general law of algebra ; and forms a part of the definition of the algebraic symbol + .* SUBTEACTION. § 5.] For algebraical purposes the most convenient course is to define subtraction as the inverse of addition; or, as is more convenient for elementary exposition, we lay down that addition and subtraction are inverse to each other, t By this we mean that, whatever the interpretation of the operation + b may be, the operation - 6 annuls the effect of + 5 ; and vice versa. Thus, - is defined relatively to + by the equation + a-b + b= +a (1), or +a + b-b= +a (2). These might also be written ft * See the general remarks in § 27. t Here we virtually assume that if x + a=y + a, then x=y. See Hankel, VorUsungen ii. d. Oomplexen ZahUn (Leipzig, 1867), p. 19. tt It may conduce to clearness in following some of the ahove discussions to remember th?.t the primary view of a chain of operations written in any order is that the operations are to be carried out successively from left to right ; for example, if Ve think merely of the last addition, + 2 + 3 + 5 + 6 in more fully expressive symbo"!^ means + ( + 2 + 3 + 5) + 6, that is, +10 + 6; +a + 6 + c means + ( + a + S) + c; + d— b + c means + ( + a - 5) + c ; and so on. "We may here re- mind the reader that, In ordinary practice, when + occurs before the first member of a chain of additions and subtractions, it is usually omitted for brevity. 6 LAWS OF COMMUTATION AND ASSOCIATION ohAP. + { + a-h) + b= +a (1'), + ( + a + li)-b= +a (2'). From a quantitative point of view we might put the matter thus : the question, What is the result of subtracting b from a 1 is regarded as the same as the question, What must be added to + b to produce + a t and the quantity which is the answer to this question is s)'mbolised by + a- b. Starting with the defini- tion involved in (1) and (2), and putting no restriction upon the operands a and b, or, what is the same thing from a quantitative point of view, assuming that the quantify + a-b always exists, we may show that the laws of commutation and association hold for chains of operations whose successive linlis are additions and sub- tractions. We, of course, assume the commutative law for addi- tion, having already laid it down as one of our fundamental laws. § 6.] Since + a - c + c= + a \)j the definition of the mutual relation between addition and subtraction, we have a+b-c=a-c+c+b-c; = a~c + b + c-c, by law of commutation for addition ; = a-c + b (1), by definition of subtraction. Also a-b-c = a-c + c-b-c, by definition ; = a-c-b + c -c, by case (1) ; = a-c-b (2), Ijy definition Equations (1) and (2) may be regarded as extending the law of commutation to the sign - .* We can now state this law fully as follows : — ± a±b = ±b ± a; * It might be objected here that it has not been shown that - c may come into the iirst place in the chain of operations. The answer to tliis would be that +a-c-b may either be a complete chain in itself or merely the latter part of a longer chain, say p + a~c-b. In the second case our proof would show that p + a-c-b =p -c + a-b ; and the nature of algebraic generality I FOE ADDITION AND SUBTRACTION 7 or, in words, In any chain of additions and subtractions the different members may be loritten in any mder, each with its proper sign attached. Here the full significance of the attachment of the operator to the operand appears. Thus in the following instance the quantities change places, carrying their signs of operation with them in accordance with the commutative law : — + 3-2 + 1-1= +3 + 1-1-2, = +3-1 + 1-2, = - 2 - 1 + 1 + 3. § 7.] By the definition of the mutual relation between addi- tion and subtraction, we have a + ( + b-c)= +a + ( + b-c) + c-c, = a + b-'c (1). Again, by the definition, p + b- c + c-b =p + b-b, =p. Hence a-( + b-e) = a-( + b-c) + b-c + c-b; = a-{ + b-c) + ( + b-c) + c-b, by case (1) ; = a + c-h, by the definition ; = a-b +c (2), by the law of commutation already established. § 8.] The results in last paragraph, taken along with those of § 3 above, may be looked upon as establishing the law of associa- tion for addition and subtraction. This law may be symbolised as follows : — ± ( ± a ± J ± c ± &c.) = ± ( ± a) ± ( ± J) ± ( ± c) ± &c., with the following law of signs, + { + a)= +a, -{ + a)= -a, + (-«)=-«, - ( - a) = + a. requires that +a-c-b should not have any property in composition which it has not per se. As to all questions of this kind see § 27. 8 NON-AEITHMETICAL CASES CHAP. The same may be stated in words as follows : — If any number of quantities affected with the signs + or - occur in a bracket, the bracket may be removed, all the signs remaining the same if + precede the h-acket, each + being clianged into - and each - into + if -precede the bracket. In the above symbolical statement double signs ( ± ) have been used for compactness. The student will observe that with three letters 2x2x2x2, that is, 16, cases are included. Thus the law gives + ( + a + b + c)= + a + b + c, + (-a + b + c)= - a + b + c, -(-a + i + c)= + a-b - c, &c. § 9.] It will not have escaped the student that, in the as- sumption that + a -b is a quantity that always exists, we have already transcended the limits of ordinary arithmetic. He will therefore be the less surprised to find that many of the cases included under the laws of commutation and association exhibit operations that are not intelligible in the ordinary arithmetical sense. If a = 3 and b = 2, then by the law of association and by the definition of sub- traction + 3-2= +1 + 2-2, = +1, in accordance with ordinary arithmetical notions. On the other hand, if a = 2 and i = 3, then by the laws of commutation and association and by the definition of subtraction + 2-3= +2-( + 2 + l), = +2-2-1, = -1 + 2-2, = - 1. Here we have a question asked to which there is no ordinary arithmetical answer, and an answer arrived at which has ho meaning in ordinary arithmetic. I ESSENTIALLY NEGATIVE QUANTITIES 9 Such an operation as + 2 - 3, or its algebraical equivalent, - 1, is to be expected as soon as we begin to reason about operations according to general laws without regard to the appli- cation or interpretation of the results to be arrived at. It must be remembered that the result of a series of operations may be looked on either as an end in itself, say the number of in- dividuals in a group, or it may be looked upon merely as an operand destined to take place in further operations. In the latter case, if additions and subtractions be in question, it must have either the + or the - sign, and either is as likely to occur and is as reasonably to be expected as the other. Thus, as the results of any partial operation, + 1 and - 1 mean respectively 1 to be added and 1 to be subtracted. The fact that the operations may end in results that have no direct interpretation as ordinary arithmetical quantities need not disturb the student. He must remember that algebra is the general theory of those operations with quantity of which ordinary arithmetical operations are particular cases. He may be assured from the way in which the general laws of algebra are established that, when algebraical results admit of arithmetical meanings, these results will be arithmetically right, even when some of the steps by which they have been arrived at maj'' not be arithmetic- ally interpretable. On the other hand, when the end results are not arithmetically intelligible, it is merely in the first instance a question of the consistency of algebra with itself. As to what the application of such purely algebraical results may be, that is simply a question of the various uses of algebra ; some of these will be indicated in the course of this treatise, and others will be met with in abundance by the student in the course of his mathematical studies. It will be sufScient at this stage to give one example of the advantage that the introduction of algebraic generality gives in arithmetical operations. +a-h asks the ques- tion what must be added to -i- 6 to give + a. If a = 3 and J = 2, the answer is 1 ; if a = 2 and 6 = 3, then, arithmetically speaking, there is no answer, because 3 is already greater than 2. But if we regard + a-h &s, asking what must be added to or subtracted 10 REDUCTION OF AN ALGEBRAICAL SUM chap. from +b to get + a, then the evaluation of + a-b in any ease by the laws of algebra will give a result whose sign will indicate whether addition or subtraction must be resorted to, and to what extent; for example, if a = 3 and b = 2, the result is + 1, which means that 1 must be added j if a = 2 and i = 3, the result is - 1, which means that 1 must be subtracted. § 10.] The application of the commutative and associative laws for addition and subtraction leads us to a useful practical rule for reducing to its simplest value an expression consisting of a chain of additions and subtractions. We have, for example, + a-b + c + d-e-f+r/ = +a + c + d + g-b-e~f, = + (a + c + d + g) - (b + e + f), = + { + {a + c + d + g)-{b + e+f)} (1), = - { + (b + e+f)-{a + c + d + g)} (2). If a + c + d + g be numerically greater than b + e+f, (1) is the most convenient form ; if a + c + d + g be numerically less than b + e + f, (2) is the most convenient. The two taken to- gether lead to the following rule for evaluating a chain of additions and subtractions : — t* Add all the quantities affected with the sign + , also all those affected with the sign - / take the difference of the two sums and affix the sign of the greater. Numerical example : — +3-5+6+8-9-10+2 = +(3 + 6 + 8 + 2) -(5 + 9 + 10), = +19-24, = -(24-19)= -5. § 11.] The special case +a-a deserves close attention. A special symbol, namely 0, is used to denote it. The operational definition of is therefore given by the equations + a ~ a= - a + a = Q. In accordance with this we have, of course, the results, * Such a chain is usually spoken of as an " algebraical sum." I PKOPEETIES OF 11 & + = J = 6-0, and + = - 0, as the student may prove by applying the laws of commutation and association along with the definition of 0. § 12.] It will be observed that 0, as operationally defined, is to this extent indefinite that the a used in the above definition may have any value whatever. It remains to justify the use of the of the ordinary numerical notation in the new meaning. This is at once done when we notice that in a purely quantitative sense stands for the limit of the difference of two quantities that have been made to differ by as little as we please. Thus, if we consider a + x and a, + (a + x) -a= + a - a + X = i: If we now cause the x to become smaller than any assignable quantity, the above equation becomes an assertion of the identity of the two meanings of 0. MULTIPLICATION. § 13.] The primary definition of multiplication is as an ab- breviation of addition. Thus + a + a, + a + a +.a, +a + a + a + a, &c., are abbreviated into + a y. 2, -i-ax3, + a x i, &c. ; and, in accordance with this notation, + a is also represented by + a x 1. ffi X 2 is called the product of a by 2, or of a into 2 ; a is also called the multiplicand and 2 the multiplier. Instead of the sign X , a dot, or mere apposition, is often used where no am- biguity can arise. Thus a x 2, a. 2, and a2 all denote the same thing. § 14.] So long as a and b represent integral numbers, as is supposed in the primary definition of multiplication, it is easy to prove that a y. b = b X a ; or, adopting the principle of attachment of operator and operand, with full symbolism (see above, § 4), X. a X b = X b X a. 12 COMMUTATION AND ASSOCIATION chap. The same may be established for any number of integers, for example, xax6xc= X a X c X b = i^ b x c x a, &c. In other words, TJie order of operations in a chain of multiplication is indifferent. This is the Commutative Law for multiplication. § 15.] We may introduce the use of brackets and the idea of association in exactly the same way as we followed in the case of addition. Thus in x a x ( x & x c) we are directed to multiply a by the product of b by c. The Law of Association asserts that this is the same as multiplying a by b, and then multiplying this product by c. Thus xax(x&xc)= xaxjxc. The like holds for a bracket containing any number of factors. In the case where a, b, c, &c., are integers, a proof of the truth of this law might be given resting on the definition of multi- plication and on the laws of commutation and association for addition. § 1 6.] Even in arithmetic the operation of multiplication is extended to cases which cannot by any stretch of language be brought under the original definition, and it becomes important to inquire what is common to the different operations thus com- prehended under one symbol. The answer to this question, which has at different times greatly perplexed inquirers into the first principles of algebra, is simply tliat what is common is the formal laws of operation which we are now establishing, namely, the commutative and associative laws, and another presently to be mentioned. These alone define the fundamental operations of addition, multiplication, and division, and anything further that appears in any particular case (for example, the statement that f X I is I of §) is merely a matter of some interpretation, arithmetical or other, that is given to a symbolical result demon- strably in accordance with the laws of symbolical operation. Acting on this principle we now lay down the laws of com- mutation and association as holding for the operation of multi- plication, and, indeed, as in part defining it. I FOE MULTIPLICATION LAW OF DISTEIBUTION 13 § 17.] The consideration of composite multipliers or com- posite multiplicands introduces the last of the three great laws of algebra. It is easy enough, if we confine ourselves to the primary definition of multiplication, to prove that + ax( + b + c)=+axb + axc, + ax( + b-c)= + a x b - a X c, { + a-b)x( + c-d)= +axc-axd-bxc + bxd. These suggest the following, which is called the Distributive Law : — The product of two expressions, each of which consists of a chain of additions and subtractions, is equal to the chain of additions and subtractions obtained by multiplying each constituent of the first expres- sion by each constituent of the second, setting down all the partial products thus obtained, and prefixing the + sign if the two constituents previously had like signs, the - sign if the constituents previously had unlike signs. Symbolically, thus : — (±a±b)x(±c±d) = {±a)x{±c) + {±a)x(±d) + (±b) X (±c) + {±b)x(±d), with the following law of signs : — ( + a) X ( + c)= + ac, { + a) X (-c)= - ac, {- a) X ( + c)= - ac, {- a) X (- c)= + ac. There are sixteen different cases included in the above equation, as will be seen by taking every combination of one or other of the double signs before each letter. Thus ( + a-b){ + c + d) = + ac + ad-bc-bd ; {-a-b){-c + d) = +ac-ad + bc-bd; and so on. There may, of course, be as many constituents in each bracket as we please. If, for example, there be m in one 14 PKOPEETY OF chap. bracket anil n in. the other, there will be mn partial products and 2'"+" different arrangements of the signs. Thus { + a-h + c){-d + c) = - ad + hd - cd + ae-be + ce ; and so on. The distributive law, suggested, as we have seen, by the primary definition of multiplication, is now laid down as a law of algebra. It forms the connecting link between addition and multiplication, and, along with the commutative and associative laws, completes the definition of both these operations. § 1 8.] By means of the distributive law we can prove another property of 0. For, if b be any definite quantity, subject without restriction to the laws of algebra, we have + la-ha- +bx( + a-a) = { + a-a)x( + b), = -bx( + a-a) = ( + a-a)x{-b), whence = { + b) x = x ( + b) = {- b) x = x (- b) ; or briefly bxO = Oxb = 0. DIVISION. § 19.] Division for the purposes of algebra is best defined as the inverse operation to multiplication : that is to say, the mutual relation of the symbols x and -=- is defined by X a-h-b xb= X a (1), or* X a xb^h = x a (2). From a quantitative point of view, this amounts to defining the quotient of a by b, that is, a-^b, as that quantity which, when multiplied by b, gives a. In a-i-b, a is called the dividend and b the divisor. Some- times a is called the antecedent and b the consequent of the quotient. Another notation for a quotient is very often used, namely, j or a/b. As this is the notation of fractions, and therefore has a meaning already attached to it in the case where a and b are integers, it is incumbent upon us to justify its use in another * See second footnote, p. 5. I QUOTIENT AND FRACTION 15 meaning. To do this we have simply to remark that h times j, that is, h times a of the Jth parts of unity, is evidently a times unity, that is, a ; also, by the definition ot a~b, b times an-i is a. HeJice we conclude that j is operationally equivalent to a-^b in the case where a and b are integers. No further justification is necessary, for when either a or b, or both, are not integers, j loses its meaning as primarily defined, and there is no obstacle to regarding it as an alternative notation for a~b. In the above definition we have not written the signs + or - before a and b, but they were omitted simply for brevity, and one or other must be understood before each letter. We shall continue to omit them until the question as to their manipulation arises. § 20.] Since division is fully defined as the inverse of multi- plication, we ought to be able to deduce all its laws from the definition and the laws of multiplication. We have * X a X b-i-c = xa-i-cx.cxb-7-c, by definition ; = X a-^c X b X c-^c, by law of commutation for multiplication ; - X a-^c X b (1), by definition. Again, x a-^b-.-c= x a-^c x c^b-i-c, by definition ; = xa-T-c-^bxc-^c, by case (1) ; = X a^c-^b (2), by definition. In this way we establish the law of commutation for division. * Here again tlie remark made in the tHrd note at tlie foot of p. 5 applies, namely, axb-tc primarily means, if we think only of the last operation, the same as (a x S) -^ c ; a-i-ixc the same as (a -h 6) x c ; and so on. As in the case of +a, when xa comes first in a chain of operations, x is in practice usually omitted for brevity. 16 COMMUTATION AND ASSOCIATION IN MULTIPLICATION chav. Taking multiplication and division together and attaching the symbol of operation to the operand, we may now give the full statement of this law as follows : — • In any chain of multiplications and divisions the order of Hie constituents is indifferent, provided the proper sign be attached to each constituent and move with it. Or, in symbols, for two constituents, ^a^b = ^b^a, there being 4 cases here included, for example, -i-a X b — X b-~a, -i-a-^b = -i-b-i-a, and so on. § 21.] By the definition of the mutual relation between multiplication and division, we have xax{xb-7-c)= xa>^(^b-i-c)xc-7-c, - X axb-h-c (1). Again, since x'pxb-^exc-h-b= X p xb-^h, = xp, therefore xa-f-(x?)-He)= x a-^(^xb-~c) xb-i-c x c-^b ; = xa^{xb-^c)x(^xb~c)xc-i-b, by case (1) ; = X a X c-i-b, by definition ; = X a~b X c (2), by the law of commutation already established. These are instances of the law of association for division and multiplication combined, which we may now state as follows : — When a bracket contains a chain of multiplications and divisions, the bracket may be removed, every sign being unchanged if x precede the bracket, and every sign being reversed if -H precede the bracket. Or, in symbols, for two constituents, l(lalb) = l{la)l{lh), 1 AND DIVISION PROPERTIES OF 1 17 with, the following law of signs : — X ( X a) = X fl, X ( -^ a) = -f- a, -^ ( X a) = -^ a, ~-(^a) = x a. In the above equation eight cases are included, for example, X ( -f- (i X J) = -i-a X b, -r-(-=-axi)= y. a-=rh, -T-(-^a■^i)= x a x b, and so on. § 22.] Just as in subtraction we denote the special case + a -ft by a separate symbol 0, so in division we denote x a-i-a by a separate symbol 1 . From this point of view, 1 has a purely operational meaning, and we can prove for it the following laws analogous to those established for in § 11 : — x«-Ha= -;-ax« = l, bx 1 =b = b-hl, X 1= -^1. Like 0, 1 has both a quantitative and a purely operational meaning. Quantitatively we may look on it as the limit of the quotient of two quantities that differ from each other by a quantity which is as small a fraction as we please of either. For example, consider a + x and a, then the equation (a + x) ~- a = a -^ a + X ^ a ■ = 1 + x-^a becomes, when x is made as small a fraction of a as we please, an assertion of the compatibility of the two meanings of 1. It should be noted that, owing to the one-sidedness of the law of distribution (that is, owing to the fact that in ordinary algebra 6 + ( x a -^ c) = x (J + a) -h (J + c) is not a legitimate trans- formation), there is no analogue for 1 to the equation 6x0 = 0, which is triie in the case of 0. § 23.] If the student will now compare the laws of commuta- tion and association for addition and subtraction on the one hand and for multiplication and division on the other, he will find them to bo formally identical. It follows, therefore, that so far as these VOL. I C 18 PRINCIPLE OF SUBSTITUTION CHAP. laws are concerned there is virtually no distinction between addi- tion and subtraction on the one hand and multiplication and division on the other, except the accident that we use the signs + and — in the one case and x and -^ in the other, — a conclusion at first sight a little startling. This duality ceases wherever the law of distribution is concerned. § 24.] We have already been led to consider such expressions as + ( + 2) and + ( - 2), and to see that + a may, according to the value given to a, be made to stand for + ( + 2), that is, + 2, or + ( - 2), that is, - 2. The mere fact that a particular sign, say + , stands before a certain letter, indicates nothing as to its reduced or ultimate value ; the sign + merely indicates what has to be done with the letter when it enters into operation. In ^^'hat precedes as to division, and in fact in all our general formulae, we may therefore suppose the letters involved to stand for positive or negative quantities at pleasure, without affecting the truth of our statement in the least. For example, by the law of distribution, (a -V) (c + d) = ac + ad -lie -hd ; here we may, if we like, suppose d to stand for — d'. We thus have {a -V){c + {- d')} = ac + a( - d') -^hc-b{- d'), which gives, when we reduce by means of the law of signs proper to the case, (a -h){c- d') = ac - ad' -bc + hd', which is true, being in fact merely another case of the law of distribution, which we ha^e reproduced by a substitution from the former case. This principle of substitution is one of the most important elements in the science ; it is this that gives to algebraic calculation its immense power and almost endless capability of development. § 25.] We have now to consider the effect of explicit signs attached to the constituents of a quotient. As this is closely bound up with the operation of the distributive law for division, it will be best to take the two together. I DISTRIBUTION OF A QUOTIENT 19 The full symbolical statement of this law for a dividend having two constituents is as follows : — ( ± a ± J) -^ ( ± c) = ( ± a) ^ ( ± c) + ( ± i) -^ ( ± c), with the following law of signs, ( + a)-^( + c)= +a-Hc, ( + a)^(-c)= -a-Hc, (-a)-h( + c)=-a4-c, {- a)-^{- c)= + a^c. Or briefly in words — In division the dividend may he distributed, the signs of the partial quotients foUomng the same law as in multiplication. The above equation includes of course eight cases. It will be sufficient to give the formal proof of the correctness of the law for one of them, say { + a-h)^{ — c)=-a^c + h-\rc. By the law of distribution for multiplication, we have ( — ftH-(- + iH-c)x( — c)=+(a-Hc)xc-(&-;-c)xc; = + a-h, by the definition. Hence ( + a— J) — ( — c) = ( — a-^c + J-^c)x(-c)-=-( — c); = — «-^c + i-^c, by the definition. § 26.] The law of distribution has only a limited application to division, for although, as just proved, the dividend may be distributed, the same is not true of the divisor. Thus it is not true in general that a -^ {h + c) = a -^h + a -^ c, or that a ^ (h — c) = a -^l — a ~ c, as the student may readily satisfy himself in a variety of ways. § 27.] As we have now completed our discussion of the fundamental laws of ordinary algebra, it may be well to insist once more upon the exact position which they hold in the science. To speak, as is sometimes done, of the proof of these laws in all their generality is an abuse of terms. They are simply laid down as the canons of the science. The best evi- dence that this is their real position is the fact that algebras are 20 POSITION OF THE FUNDAMENTAL LAAVS CHAP. in use whose fundamental laws diifer from those of ordinary- algebra. In the algebra of quaternions, for example, the law of commutation for multiplication and division does not hold generally. What we have been mainly concerned with in the present chapter is, 1st, to see that the laws of ordinary algebra shall be self-consistent, and, 2nd, to take care that the operations they lead to shall contain those of ordinary arithmetic as particular cases. In so far as the abstract science of ordinary algebra is con- cerned, the definitions of the letters and symbols used are simply the general laws laid, down for their use. When we come to the application of the formulae of ordinary algebra to any particular purpose, such as the calculation of areas, for example, we have in the first instance to see that the meanings we attach to the symbols are in accordance with the fundamental laws above stated. AVhen this is established, the formulae of algebra become mere machines for the saving of mental labour. § 28.] We now collect, for the reader's convenience, the general laws of ordinary algebra. Definitions connecting the Direct and Inverse Operations. Addition and subtraction — + a — !i + h = + a, + a + h — h = + a. Multiplication and division — X a-^h X b = X. a, X a X h-^h = X a. Law of Association. For addition and subtrac- For multiplication and divi- tion- sion- \{lalb) = l{la)l{lh), ±{±a±h)= ± ( ± a) ± ( ± J), with the following law of signs : — The concurrence of like signs gives the direct sign ; The concurrence of unlike signs the inverse sign. SYNOPTIC TA.BLE OF LAWS 21 Thus— + { + a)= + a, + ( - a) = - ft, - ( - a) = + a, - ( + a) = - a. ( X a) = X a, X ( -f- a) = -^ as, - ( -=- a) - X a, -i- ( -x a) = -i- a. Law of Commutation. For multiplication and divi- For addition and subtrac- tion — ±a±b= ±h±a, the operand always carrying its own sign of operation with it. ^a^h = '^h'^a. Properties of and 1. = + a- a, ±J + 0= ±6-0= ±h, + = - 0. 1 = X ffi -^ a, JJx 1 = JJ-1 = J6, X 1= --1. Law 01' Distribution, For multiplication — {±a±h)x{±c±d)= +{±a)x{±c) + {±a)x{±d) + ( ± 6) X ( ± c) + ( ± /;) X ( ± fZ), with the following law of signs : — If a partial product has constituents with like signs, it must have the sign + ; If the constituents have unlike signs, it must have the sign - . Thus— + ( + a)x( + c)= +axc, +( + a)x(-c)= -axe, + (-a)x(-c)= +a X c, +(-a)x( + (;)= -axe. Property of 0. 0x5 = 5x0 = 0. For division — ( ± a ± 5) -^ ( ± c) = + ( ± a) -r ( ± c) + ( ± J) 4- ( ± c), with the following law of signs : — 22 EXERCISES I chap. If the dividend and divisor of a partial quotient have like signs, the partial quotient must have the sign + ; If they have unlike signs, the partial quotient must have the sign- Thus— + { + a)~( + c)= +a^c, + ( + a) -^ ( - c) = - a-rC, + ( - rt) -=- ( - c) = + « -^ c, + ( - a) -^ ( + c) = - a -f- c. N.B. — The divisor cannot be distributed. Propertij of 0. -H 6 = 0. N.B. — Nothing is said regarding 6 4-0. This case will be discussed later on. The reader should here mark the exact signification of the sign = as hitherto used. It means "is transformable into by applying the laves of algebra, vt'ithout any assumption regarding the operands involved." Any "equation" which is true in this sense is called an " Identical Equation," or an "Identity"; and must, in the first instance at least, be carefully distinguished from an equation the one side of which can be transformed into the other by means of the laws of algebra only when the operands involved have particular values or satisfy some particular condition. Some writers constantly use the sign = for the former kind of equation, and the sign = for the latter. There is much to be said for this practice, and teachers will find it useful with beginners. We have, however, for a variety of reasons, adhered, in general, to the old usage ; and have only introduced the sign = occasionally in order to emphasise the distinction in cases where confusion might be feared. Exercises I. [Ill workiug this set of examples the student is expected to avoid quoting derived formula that he may happen to recollect, and to refer every step to the fundamental principles discussed in the above chapter.] I EXEECISES I . 23 (1.) Point out in what sense the usual arrangement of the multiplication of 365 by 492 is an instance of the law of distribution. (2.) I have a multiplying machine, but the most it can do at one time is to multiply a number of 10 digits by another number of 10 digits. Explain how I can use my machine to multiply 13693456783231 by 46381239245932. ' (3.) To divide 5004 by 12 is the same as to divide 5004 by 3, and then divide the quotient thus obtained by 4. Of what law of algebra is this an instance ? (4.) If the remainder on dividing N by a be R, and the quotient P, and if we divide P by 6 and find a remainder S, show that the remainder on dividing N by ah will be aS + E. Illustrate with 5015-^12. _ (5.) Show how to multiply two numbers of 10 digits each so as to obtain merely the number of digits in the product, and the iirst three digits on the left of the product. Illustrate by finding the number of digits, and the first three left-hand digits in the following : — 1st. 3659893456789325678 x 342973489379265 ; 2nd. 2«^. (6. ) Express in the simplest form — -(.-(-(-{ ■ • • (-1) • )))), 1st. "Where there are 2w brackets ; 2nd. Where there are 2b + 1 brackets ; » being any whole number whatever. (7.) Simplify and condense as much as possible — 2a- {3a-[a-{&-a)]}. (8.) Simplify— 1st. 3{4-5[6-7{8-9.10-ll)]j, 2iid. Mi-i[i-|C^-^A^)]}- (9.) Simplify— l-(2-(3-(4-. . . (9-(10-ll)) . . . ))). (10.) Distribute the following products: — 1st. (a + b)x(a + b) ; 2nd (ffi-6)x{a + 6); 3rd. (3(i-66) x {3a + 66) ; 4th. (Ja- J6) x (!« + J6). (11.) Simplify, by exjianding and condensing as much as possible^ \(m + l)a+(n+\)b]{(m-l)a + {n-\)b] + {(m + l)a-(n + l)b}{{m-l)a-{n-l)b]. (12.) Simplify- (13.) Simplify— (14.) Expand and condense as much as possible — 24 HISTORICAL NOTE chap, i Historical JVnfc. — Tlie separation and classification of tlie fundamental laws of algebra has been, a slow process, extending over more than 2000 years. It is most likely that the first ideas of algebraic identity were of geometrical origin. In the second book of Euclid's Elements (about 300 B.C.), for example, we have a series of propositions which may be read as algebraical identities, the operands being lines and rectangles. In the extant works of the great Greek algebraist Diophantos (350 ?) we find what has been called a syncopated algebra. He uses contractions for the names of the powers of the variables ; has a symbol ^ to denote subtraction ; and even enunciates the abstract law for the multiplication of positive and negative numbers ; but has no idea of independent negative quan- tity. The Arabian mathematicians, as regards symbolism, stand on much the same platform ; and the same is true of the great Italian mathematicians Ferro, Tartaglia, Cardano, Ferrari, whose time falls in the first half of the sixteenth century. In point of method the Indian mathematicians Aryabhatta (476), Brahmagupta (598), Bhaskara (1114), stand somewhat higher, but their works had no direct influence on Western science. Algebra in the modern sense begins to take shape in the works of Regiomon- tanus (1436-1476), EudolfT (about 1520), Stifel (14S7-1567), and more particularly Viete (1540-1603) and Harriot (1560-1621). The introduction of the various signs of operation now in use may be dateil, with more or less certainty, as follows : j and apposition to indicate multiplication, as old as the use of the Arabic numerals in Europe; -I- and -, Rudolff 1525, and Stifel 1544; =, Eecorde 1557 ; vinculum, Vlite 1591 ; brackets, first by Girard 1629, but not in familiar use till the eighteentli century ; <; >■ , Harriot's Praxis, published 1631 ; X , Oughtred, and -=-, Pell, about 1631. It was not until the Geoinetry of Descartes appeared (iu 1637) tliat the im- portant idea of using a single letter to denote a quantity which might be either positive or negative became familiar to mathematicians. The establisliment of the three great laws of operation was left for the present century. The chief contributors thereto were Peacock, De Morgan, D. F. Gregory, Hankel, and others, working professedly at the philosophy of the first principles ; and Hamilton, Grassmann, Peirce, and their followers, who threw a flood of light on the subject by conceiving algebras whose laws ditt"er from those of ordinary algebra. To these should be added Argand, Caucliy, Gauss, and others, who developed the theory of imaginaries in ordinary algebra. CHAPTEE 11. Monomials — Laws of Indices — Degree. THEORY OF INDICES. § l.J The product of a number of letters, or it may be num- bers, each being supposed simple, so that multiplication merely and neither addition nor subtraction nor division occurs, is called an integral term, or more fully a rational integral monomial (that is, one-termed) algebraical function, for example, ftx3x6xa;xa xxxxxyxbxb. 'By the law of commutation we may arrange the constituents of this monomial in any order we please. It is usual and con- venient to arrange and associate together all the factors that are mere numbers and all the factors that consist of the same letter ; thus the above monomial would be written (3 X 6) X (a X a) X (J X &) X (x X a; X a;) X y. 3x6 can of course be replaced by 1 8, and a further contrac- tion is rendered possible by the introduction of indices or ex- ponents. Thus a X a is written a^, and is read " a square," or " a to the second power." Similarly J x & is replaced by b^, and X X. X X xhy of, which is read " x cube," or " x to the third power." We are thus led to introduce the abbreviation »'* for a; x a; x a; x . . . where there are n factors, n being called the index or exponent,* while a;" is called the mth power of x, or x to the nth power. § 2. J It will be observed that, in order that the above defini- tion may have any meaning, the exponent n must be a positive * In accordance with this definition x^ of course means simply x, and is usually so written. 26 RATIONAL INTEGRAL MONOMIAL I'HAP. integral number. Confining ourselves for the present to this case, we can deduce the following "laws of indices." I. (a) a™ X a" = «'"+", and generally a'" x a" x a^" x . . . = «»+''+?+ ■ ■ • a™ (/3) — = ffl"'-™ if TO>n, a" 1 r,n~iii if m < n. a" II. (a'")" = a™" = (a")" III. (a) (aJ)™ = a'«i'", and generally {abc . .)'"■ = a"'/y"c"' . . iH) (j) =r-- To prove I. (a), we have, by the definition of an index, a^ y. ci'^ = {ct x. a X a . . . m factors) x (a x a ^ ii ■ n factors), = a X a X a . . . m + n factors, by the law of association, = fl™+", by the definition of an index. Having proved the law for two factors, we can easily extend it to the case of three or more, for (('" X a" X aP = {a™- x «"■) x a^, by law of association, = ffi™+" x a^, by case already proved, = a^m+n)+p^ ^jy g^gg already proved, and so on for any number of factors. In words this law runs thus : The product of any number of powers of one and the same letter is equal to a power of that letter whose exponent is the sum of the exponents of these powers. To prove I. (Ji), w - = (a X a X . . . m factors) -i-(a x ax . . n factors), by definition of an indeix, II LAWS OF INDICES 27 = a X a X a . . . m factors -H a H- a -^ . n divisions, by law of association. Now if m > n we may arrange these as follows ; — gm — = (a X ffi X . . . ni — n factors) x (n -7- a) x. {a -¥ a) . . n factors, by laws of commutation and association, = a X a x . . .111-11 factors, by the properties of division. If m n, so that m - n is positive, gm-n X a«. = a'™-"'+" by I. (a), Hence a"*-" X ft" ^ a™ = a™ -h a". Therefore, by the definition of x and -f- , a™-™ = a"» -f- a". Again, if m < n, so that n - m is positive, a™ X a"-™ = ffl'i+i" -">•), by I. (a), = a", by the laws of + and - Hence a™ X ««-™ -^ a"-™ = a" ^ a"-™. Therefore, by the definition of x and ~ , a"» = a"H-a"-»». Hence, by the laws of x and -h , am ^ a» = a" -4- a""™ ~ a", = (a"^o")-^a"-™, = l-^a"-'". 28 LAWS OF INDICES OHA.P. To prove II., (a'")" = «'■' X ft"'' X . . . n factors, by definition, = (a X a X . . . m factors) x. (a x a x . . m factors) X . . ., n sets, by definition, = a X (I X . . . mn factors, by law of association, = a"™, by definition. To prove III. (a), (ab)™' = {ah) x (ah) x . . . m factors, by definition, = (ffl X ffl X . . . m factors) x (b x b x . . m factors), by laws of commutation and association, = ffi^J™, by definition. Again, (aJc)™= {(a&)c}'", = {ab)™-c'^, by last case, = (a™'J'»)c™, by last case, = a^b^c™, and so on. Hence the mih. power of the product of any number of letters is equal to the product of the mth powers of these letters. To prove III. {(i), i-\ = (a-^b) X {a-i-b) X . . m factors, by definition, = (a X a X . . m factors) ^ (b x b x . . . m factors), by commutation and association, = ((.■'"■ -^ b'"-, _ «™ ~h"'' In words : The mth power of the quotient of two letters is the quotient of the mth powers of these letters. The second branch of III. may be derived from the first without further use of the definition of an index. Thus l") xb^=Qxb^ , by III. (a), = a™, by definition of x and -^ . Hence Q'\ j™^ j^ = ^m^j™ that is, /a\^ _a^ \b) ~ i™" II LAWS OF INDICES 29 § 3.] In so far as positive integral indices are concerned, the above laws are a deduction from the definition and from the laws of algebra. The use of indices is not confined to this case, however, and the above are laid down as the laws of indices generally. The laws of indices regarded in this way become in reality part of the general laws of algebra, and might have been enumerated in the Synoptic Table already given. In this respect, they are subject to the remarks in chap, i., § 27. The question of the meaning of fractional and negative indices is deferred till a later chapter, but the student will have no diffi- culty in working the exercises given below. All he has to do is to use the above laws whenever it is necessary, without regard to any restriction on the value of the indices. § 4.] The following examples are worked to familiarise the student with the meaning and use of the laws of indices. At first he should be careful to refer each step to the proper law, and to see that he takes no step which is not sanctioned by some one of the laws of indices, or by one of the fundamental laws of algebra. Example 1. = aVbVc^c^^ -^a* -^ b^ -^ c^^, by commutation and association, = a3+5J2+6(;5+ii -:- a* -^ J3 _^ ^15^ i,y j^^,, ^f jndices, I. (a), = (a'+5 -=-«") X {6^+6 -^ 53) X {c=+" -h c'5), by commutation and association. ^„3+5-4 X 6^6-3 X cHii-is, by law of indices, I. (/3), = a^bh. Example 2. = 152(a;2)2(2/S)2(25)2 x j^jL'' ^y ^^'^^'■'^ of indices, III. (a) and III. (/3), = 32x4yai0 ' °y I- (") '^^°- II- > = 32 H- 3^ X 52 -^ 42 X a;8 X i/i' -^ ?/ X z" -=- z^", = 5- — i" xafi -^y-, ^5X2,,8 30 ALGEBRAICAL INTEGRALITY AND FRACTIONALITY OHAP. THEORY OF DEGREE. § 5.] The result of multiplying or dividing any number of letters or numbers one by another, addition and subtraction being excluded, for example, 3 •/. a -^ x x b-^c-i-y x d, is called a (rational) monomial algebraical function of the numbers and letters involved, or simply a term. If the monomial either does not contain or can be so reduced as not to contain the operation of division, it is said to be integral ; if it cannot be reduced so as to become entirely free of division, it is said to he. fractional. In drawing this distinction, division by mere numbers is usually disregarded, and even division by certain specified letters may be disregarded, as will be explained presently. § 6.] The number of times that any particular letter occurs by way of multiplication in an integral monomial is called the degree (or dimension) of the monomial in that particular letter ; and the degree of the monomial in any specified letters is the sum of its degrees in each of these letters. For example, the degree oi Q x a x a ■< x x x y. x x y x y, that is, of Qic/if, in a is 2, in X 3, in y 2, and the degree in x and y is 5, and in a, x, and y 7. In other wm-ds, the degree is the sum of the indices of the named letters. The choice of the letters which are to be taken into account in reckoning the degree is quite arbitrary ; one choice being made for one purpose, another for another. When certain letters have been selected, however, for this purpose, it is usual to call them the variables, and to call the other letters, including mere numbers, constants. The monomial is usually arranged so that all the constants come first and the variables last; thus, X and y being the variables, we write 32a%cx'y' ; and the part 32a^bc is called the coefficient. In considering whether a monomial is integral or not, division by constants is not taken into account. § 7.] The notion of degree is an exceedingly important one, and the student must at once make himself perfectly familiar with it. He will find as he goes on that it takes to a large extent in algebra the same place as numerical magnitude in arithmetic. II NOTIONS AND LAWS OF DEGEEE 31 The following theorems are particular eases of more general ones to be proved by and by. The degree of the product of two or more monomials is the sum of their respective degrees. If the quotient of two monomials be integral, its degree is the excess of the degree of the dividend over that of the divisor. For let A = ci'ij^^u'^ . A' = c'xV /m^' . . . where c and d are the coefficients, x, y, z, u . . . the variables, and I, m, n,p. . ., r, m', n',p' . . . are of course positive integral numbers. Then the degree, d, of A is given hjd = l + m + n+p + . . ., and the degree, d', of A' by d' = 1' + m' + n' + p' + . . . But A X A' = {cx^y'^z^uP . . .) x (c'/y'/w^' . . .) = (c X c')x'+'V™+'" «"+"'«*'+*'' . . . the degree of which is (I + V) + (m + m!) + (n + n') + {p + p') . . ., that is, {l + m + n+p . . .) + (l' + m' + n' +p' + . . . ), that is, d + d', which proves the first proposition for two factors. The law of association enables us at once to extend it to any number of factors. Again, let Q = A H- A', and let Q be integral and its degree S. Now we have, by the definition of division, Q x A' = A. Hence, by last proposition, the degrees of A and A' being d and d', as before, we have d= S + d', and thence S = d- d'. As an example, let A = 6x'y\ A' = 7x'y^, then A x A' = 42a;'y, and A -r A' = fxY. The degree of A x A' is 24, that is, 14 + 10 ; that of A-^A' is 4, that is, 14 - 10. The student will probably convince himself most easily of the truth of the two propositions by considering particular cases such as these ; but he should study the general proof as an exercise in abstract reasoning for on such reasoning he will have to rely more and more as he goes on. Exercises II. [Wherever it is possible in working the following examples, the student should verify the laws of degree, §§ 5-7.] (1.) Siraplify- 5^ x 12^x 32^ x (3^x 4^x 5)' (3xl5x23)i» 33 EXERCISES IT OHAP n. (2. ) Which is greater, (2-)'- or 2- ? Find the diflference between them. (3.) Simplify— 2^ (4.) Simplify— SeVb'(^(i- (5.) Express in its simplest form — (6.) Simplify— fi5aVc^y f2i2aWxy- V 27a?bh J ^[ ISOa'bc J ' (8.) Simplify— { xY^z^V x ('/"■-'.-■-■-)^ x {-■b"-,ry (9.) Simplify— /■^\m-h> ^ /^ny+i ^ /■ af^y+m (10. ) Simplify— / {a''-')' x (a'-')'-)' l ifr I (oi'+'O'-'' J (11.) Simplify- ^^-,^^-c)a^ (a?" X:!/)"-^ (,./'+'•)<' (12.) Simplify— / j^y+'' ^ / xPy^yVi (V.;.) Simijlify— { (^j "" (:?; |--{(a^')'x(x»)»l X {(rr»)'x(a;')'"}. (14.) Prove that- {^Jz)'>^(zxy'ixyy'^ _ [xxjz)'>+''+^ (15.) Distribute the product — {j:''-' - of-" + a^-») ( -L + J_ + J_ y (16.) Distribute— /„ 1\/ 1 \2 (17.) Ifm = a'^, n = c(J', a' = {ni"n'Y ; show tliat a;2/« = l. CHAPTEE III. Fundamental Formulae relating to Quotients or Fractions, with Applications to Arithmetical Fractions and to the Theory of Numbers. OPERATIONS WITH FRACTIONS. § 1.] Before proceeding to cases where the fundamental laws are masked by the complexity of the operations involved, we shall consider in the light of our newly-acquired principles a few cases with most of which the student is already partly familiar. He is not in this chapter to look so much for new results as to exercise his reasoning faculty in tracing the opera- tion of the fundamental laws of algebra. It will be well, how- ever, that he should bear in mind that the letters used in the following formulae may denote any operands subject to the laws of algebra; for example, mere numbers integral or fractional, single letters, or any functions of such, however complex. § 2.] Bearing in mind the equivalence of the notations -, a/5, and a-^h, the laws of association and commutation for multiplication and division, and finally the definition of a quotient, we have that is, VOL. I pa pb --{jpay. Hpb)- --p^ < a-. -p- ^i, : = a- = a- ~b- ^b; '■^P xp, pa pb a 34 ALGEBRAICAL ADDITION OF FRACTIONS chap. Read forwards and backwards this equation gives us the important proposition that we may divide or multiply ilie numerator and denominator of a fraction Jjy the same quantity withoid altering its value. § 3.] Using the principle just established, and the law of distribution for quotients, we have b q qh qb' _ ±qa±pb that is. To add or subtract two fractions, transform each by multiplying numerator and denominator so that both shall have the same denomi- nator, add or subtract the numerators, and write underneath the common denominator. The rule obviously admits of extension to the addition in the algebraic sense (that is, either addition or subtraction) of any number of fractions whatever. Take, for example, the case of three : — (( c e adf cbf cbd -. „ _ ±-±-±-=±— ^±^i-± — bv^2, b d f bdf bdf bdf ^ ^ ' ± adf ± cbf ± ebd , , r t , •, , . = —TTF > oy '^'^ °^ distribution. > hdf The following case shows a modification of the process, which often leads to a simpler final result. Suppose b = lc, q = ir; then, taking a particular case out of the four possible arrangements of sign, a p a p b q Ic Ir' _ ar pc Icr Ire _ ar - pc Icr Here the common denominator Icr is simpler than bq, which is Par. The same result would of course be arrived at by following Ill MULTIPLICATION AND DIVISION OF FRACTIONS 35 the process given above, and simplifying the resulting fraction at the end of the operation, thus : — a p air - pic , (ar -pc)l by using the law of distribution in the numerator, and the laws of association and commutation in the denominator ; ar - cp ^ „ „ § 4.] The following are merely particular cases of the laws of association and commutation for multiplication and division : — ©x©=(''H-S)x(.^./), = a-^h -x. c-^d, = axc-irb-i-d, = (ac) -^ (M), ac or, in words, To multiply two fractions, multiply their' numerators together for the numerator, and the denominators together for the denominator of the product. Again, = a-^l-^cx d, = axd-^b-i-c, = (ad) -H (be), ad ^Fc' also =G)'^t)> by last case. In words : To divide one fraction by another, invert the latter and then multiply. § 5.] In last paragraph, and in § 2 above, we have for 36 EXERCISES III CHAP. simplicity omitted all explicit reference to sign. In reality wc have not thereby restricted the generality of our conclusions, for by the principle of substitution (which is merely another name for the generality of algebraic formulae) we may suppose the p, for example, of § 2 to stand for - lo, say, and we then have ( - (i>)a a that is, taking account of the law of signs, — a a and so on. Exercises III (1.) Express in its simplest form — y"~ X - 1/ y - X (2.) Express in its simplest form — a h f + £ — a-0 D-a (3.) Simplify— P+Q P-Q P-Q P + Q' where 'P=x + y, Q=x-v. (4.) Simplify— 1 ^a-y) x+y , 1-'/ 1 + — - x + y (5.) Simplify— 1 1 1 ab ac be a'-ib- cf a (6.) Simplify— ("-«+&) ''(''+«-0- (7.) Simplify— 1 1 2x ^ _| iB + 2/ x-y x^-\-y- (8.) Simplify— (i-i)g-9-e-i)(f-9- (9.) Simplify— \b a) ■ \b' a?)- ARITHMETICAL INTEGRALITY AND DIVISIBILITY 37 (10.) Simplify— a{a-b)-b{a + b) a b a+b a-b (11.) Simplify— 1 -x 1+x (12.) Simplify— _x (13.) SimpUfy- l+x + x' 1-x+x"' X T+x . {x + lf-a? 1 ■ x^ + x + 1 ■ »; + =—— 1+x a^ + i^ ab ib\a bj (14.) Show that x* [x'^-a^f (x^-Vf is independent of x. (15.) Simplify— aV^a\o?-b^) 6V-6') b ■'— , e f (16.) Simplify— 1 a-2b-- a-2b- a-lb (17.) Simplify— a + b 1 a + b+— 1 a + 6 APPLICATIONS TO THE THEORY OF NUMBERS. § 6.] In the applications that follow, the student should look somewhat closely at the meanings of some of the terms employed. This is necessary because, unfortunately, some of these terms, such as integral, factor, divisible, &c., are used in algehra generally in a sense very different from that which they bear in ordinary arithmetic and in the theory of numbers. An integer, unless otherwise stated, means for the present a positive (or negative) integral number. The oi-dinary notion of greater and less in connection with such numbers, irrespective of their sign, is assumed as too simple to need definition.* When * This is a very different thing from the algebraical notion of greater and less. See chap. xiii. , § 1. It may not be superfluous to explain 38 PRIME AND COMPOSITE INTEGERS CHAP. an integer a can be produced by multiplying together two others, h and c, b and c are called fadms of a, and a is said to be exactly (lirisible by b and by c, and to be a multiple of b or of c. Since the product of two integers, neither of which is unity, is an integer greater than either of the two, it is clear that no integer is exactly divisible by anotlw greater than itself. It is also obvious that every integer (other than unity) has at least two divisors, namely, unity and itself; if it has more, it is called a composite integer, if it has no more, a prime integer. For example, 1, 2, 3, 5, 7, 11, 13, . . are all prime integers, whereas 4, 6, 8, 9, 10, 12, 14 are composite. If an integer divide each of two others it is said to be a common factor or common measure of the two. If two integers ha\'e no common measure except unity they are said to be piime to each other. It is of course obvious that two integers, such as 6 and 35, which are prime to each other need not be themselves pime integers. We may also speak of a common measure of more than two integers, and of a group of more than two integers that are prime to each other, meaning, in the latter case, a set of integers no two of which have any common measure. § 7.] If we consider any composite integer N, and take in order all the primes that are less than it, any one of these either will or will not divide N. Let the first that divides N be a, then N = uN,, where N, is an integer ; if N, be also divisible by a we bavc 'S^ = al^^, and 'S = a{a^,^ = a^^^; and clearly, finally, say X = a^N^, where X„ is either 1 or no longer divisible by a. N„ (if not = 1) is now either prime or is divisible by some pi-ime ->a and , <, 1>, < ; they mean res]iectively "is not equal to," "is greater than," "is less tlian," "is not greater than," "is not less than." Instead of J> , «t wo may use <, 5* which may b3 read "is equal to or less than," "is equal to or greater than." Ill ARITHMETICAL G.C.M. 39 is to be observed that a" if, . . are powers of primes, and therefore, as we shall prove presently, prime to each other. It is therefore always possible to resolve every cornposite integer into factors that are powers of primes ; and we shall presently show that this resolution can be effected in one way only. § 8.] If a he divisible by c, then any integral multiple of a, say ma, is divisible by c; and, if a and b be each divisible by c, then the algebraic sum of any integral multiples of a and b, say ma + nb, is divisible by c. For by hypothesis a = ac and b = fie, where u. and f3 are in- tegers, hence ma = mac = {ma)c, where ma is an integer, that is, ma is divisible by c. And ma + nb — mac + m/3c = {ma + nP)c, where ma + Ji/S is an integer, that is, ma + nb is divisible by c. The student should observe that, by virtue of the extension of the notion of divisibility by the introduction of negative integers, any of the numbers in the above proposition may be negative. § 9.] From the last article we can deduce a proposition which at once gives us the means of finding the greatest common measure of two integers, or of proving that they are prime to each other. If a=pb + c, where a, b, c, p are all integers, then the G.C.M. of a and b is the G.C.M. of b and c. To prove this it is necessary and it is sufficient to show — 1st, that every divisor of h and c divides a and b, and, 2nd, that every divisor of a and b divides 6 and c. Since a = pb + c, it follows from § 8 that every divisor of b and c divides a, that is, every divisor of b and c divides a and b. Again, since a = ph + c,it follows that c = a —pb ; hence, again by § 8, every divisor of a and b divides c, that is, every divisor of a and b divides b and c. Thus the two parts of the proof are furnished. Let now a and b be two numbers whose G-.C.M. is required ; they will not be equal, for then the G-.C.M. would be either of them. Let b denote the less, and divide a by 6, the quotient being j3 and the remainder c, where of course cb, b>c, od, d>c, &c., it is clear that the re- mainders must diminish down to zero. We thus have the following series of equations : — a=pb + h = qc + d c = rd + e ■III = till. Hence the G.C.M. of a and h is the same as that of h and c, which is the same as that of c and d, that is, the same as that of d and e, and finally the same as that of m and n. But, since m = wn, the G.C.M. of //( and n is n, for n is the greatest divisor of n itself. Hence the G.C.M. of a and b is the divisor corresponding to the remainder in the chain of divisions above indicated. If n be different from unity, then a and b have a G.C.M. in the ordinary sense. If n be equal to unity, then they have no common divisor except unity, that is, they are prime to each other. § 10.] It should be noticed that the essence of the foregoing process for finding the G.C.M. of two integers is the substitution for the original pair, of successive pairs of continually decreasing integers, each pair having the same G.C.M. All that is necessary is that J), q, r, . . be integers, and that a, b, c, d, e, . . . be in decreasing order of magnitude. The process might therefore be varied in several ways. Taking advantage of the use of negative integers, we may some- times aljbreviate it by taking a negative instead of a positive remainder, when the former happens to be numerically less than the latter. For example, take a=4323, & = 1595, we might take 4323 = 2x1595 + 1133 (.1- 4323 = 3x1595-462; the latter is to be prefen'ed, because 462 is less than 1133. In practice the negative sign of 462 may be neglected in the rest of the ojieration, wliich may be arranged as follows, for the sake of comparison \\ith the ordinary process already familiar to the student : — in PEIME DIVISOES 41 1595)4323(3 4785 462)1595(3 1386 209)462(2 418 44)209(5 220 11)44(4 44 G.C.M. = 11. By means of the process for finding the G.C.M. we may prove the following proposition, of whose truth the student is in all probability already convinced by experience : — If a and h he prime to each other, and h any integer, then any common factor of ah and h must divide h exactly. For, since a and h are prime, we have by § 9, a=pb + c b = qc + d c = rd + e l = vm+ 1 (1). Hence < ah=pbh + ch bh = qch + dh ch = rdh + eh Ih = vmh + h J r-(-)- Now, since any common factor of ah and 5 is a common factor of ah and bh, it follows from the first of equations (2) that such a common factor divides ch exactly, and by the second that it also divides dh exactly, and so on ; and, finally, by the last of equations (2), that any common factor of a/i and b divides h exactly. In particular, since i is a factor of itself, we have Cor \. If b divide ah exactly and be prime to a, it must divide h exactly. Cor. 2. If a' be prime to a and to b and to c, <&c., then it is prime to their product abc . . . For, if a' had any factor in common with abc . . ., that is, with a{bc . . .), then, since a' is prime to a, that factor, by the proposition above, must divide be . . . exactly ; hence, since a 42 REMAINDER AND RESIDUE OHAP. is prime to b, the supposed factor must divide c . . exactl}', and so on. But in this way we exhaust all the factors of the pro- duct, since all are prime to a'. Hence no such factor can exist, that is, a' is prime to ahc . . . An easy extension of this is the following : — Cor. 3. If all the integers a', b', c', . . he piime to all the integers a,b,e,. . ., then the product a'Vc' . . is prime to the product ahc . . A particular case of which is Cor. 4. If a' he prinie to a {and in particular if both he primes), then any integral power of a' is prime to anij integral power of a. § 11.] It is obvious that, if a and h be two integers, we can in an infinite number of ways put a into the form of qh + r, where q and r are integers, for, if we take q any integer whatever, and find r so that a - qh = r, then a = qh + r. There are two important special cases, those, namely, where we restrict r to be numerically less than b, and either (1) positive or (2) negative. In each of these cases the resolution of a is always possible in one way only. For, in case 1, if qb be the greatest multiple of h which does not exceed a, then a - qh = r, where rb, ajh is called in this case an improper fraction ; if a < i, a proper Hence ru> true fraction, proper w improper, can be equal to an integer. Every improper fraction ajh can be expressed in the form q + rjb, where q is an integer and rjb a proper fraction. For, if r be the least positive remainder when a is divided by i, a = qb + r, and a/b = (qb + r)/b = q + rjb, where q and r are integers and r b), ivhere A and V, are jjositive integral numbers. The vpper sign being used fm- the \d, 3rd, 5th, t&c, and the lower for the 2nd, ith, <&c., remaimhrs. For, by the equations in § 9, we have successively — c= +{a-pb] (1); cl = b - ip; = b - q(a - pb), = -V-(l+^#} (2); e = c - rd, = {a - pb] + r{qa - (1 + pq)b], = + {(l + qr)a - (p + r + p.ir)l,'f (3); in THEOREMS REGARDING G.C.M. 45 and so on. It is evident in fact that, if the theorem holds for any two successive remainders, it must hold for the next. Now equations (1), (2), and (3) prove it for the first three remainders; hence it holds for the fourth ; hence for the fifth ; and so on. In the chapter on Continued Fractions, a convenient process will be given for calculating the successive values of A and B for each remainder. In the meantime it is sufficient to have established the existence of these numbers, and to have seen a straightforward way of finding them. Cor. 1. Since g, the G.C.M. of a and b, is the last remainder, we can always express g in the form — g=±{Aa--Bb) (4), where A and B are positive integers. Cor. 2. If a be prime to b, g = 1 ; hence, If a and b be two integers prime to each other, two positive integers, A and B, can always be found such that — Aa - B6 = ± 1 (5). N.B. — It is clear that A must be prime to B. For, since afg and big are integers, I and m say, we have, from (4), 1 = ± (A? - Bm) ; hence, if A and B had any common factor it would divide 1 (by § 8 above). Cor. 3. From Cor. 1 and § 8 we see that every common factor of a and 6 must be a factor in their G.C.M. A result which may be proved otherwise, and will probably be considered obvious. Cor. 4. Hefnce, To find the G.C.M. of more than two integers a, b, c, d, . . ., we must first find g the G.C.M. of a and b, then g' the G.C.M. of g and c, then g" the G.C.M. of g' and d, and so on, the last G.C.M. found being the G.C.M. of all the given integers. For every common factor of a, b, c must be a factor in a and b, that is, must be a factor in g ; hence, to find the greatest com- mon factor in a, b, c, we must find the greatest common factor in g and c ; and so on. From Cor. 2 we can also obtain an elegant proof of the conclusions in the latter part of § 10. 46 EXAMPLES CHAP. Example 1. To exprpss the G.C.M. of 565 and 60 in the form A565 - B60. AVe have 565 = 9x60 + 25, 60 = 2x26 + 10, 25 = 2x10 + 5, 10 = 2x6. Hence the G.C.M. is 5, and we have successively 25 = 565-9x60; 10 = 60-2{665-9x60} = - {2x565-19x601 ; 5 = 25-2x10 = 565 -9 x60 + 2{2x 665-19x60} = 5x665-47x60. Example 2. Show that two integers A and B can be found so that 5A-7B = 1. We have 7 = 1x5 + 2, 5 = 2x2 + 1; whence 2 = 7-5, 1 = 5-2(7-5) = 3x5-2x7. Hence A = 3, B = 2 are integers satisfying the requirements of the question. Example 3. If a, h, c, d, . . he a series of integers whose G.C.M. is g, show that integers (positive or negative) A, B, C, D, can be found such that g=Aa + 'Bb + Cc + 'Dd+ . . . (Gauss's Disquisitioncs Arithmeticse., Th. 40). Find A, B, C, D, when a=36, J = 24, c = lS, d=SO. This result may he easily arrived at by repeated application of corollaries 1 and 4 of this article. Example 4. The proper fraction p/ab, where a is jjrime to b, can be de- composed, and that in one way only, into the form a' V , a b where a' and b' are both positive, a' Q = ma-a' (a' positive <:a). Then (1) becomes p ■, a' b' ab a b ^ ' Now, since p/ai is a proper fraction, the integral part on the right-hand side of (2) must vanish ; hence, since the integral part of a'/a + b'/b cannot exceed 1, we must have l-m=0, or l-m= - 1. ni NUMBER OF PRIMES INFINITE 47 If the lower sign has to be taken in (1), we have merely to take the resohitions pA = lb-b' (V positive ^'^~'^ + ■ ■ ■ +PiX+p„, where Po, Pi, ■ • • , Pn are the various coefficients and n is a posi- tive integral number, which, being the index of the highest term, is the degree of the function. The function has in general n+ \ terms, but of course some of these may be wanting, or, which amounts to the same thing, one or more of the letters Po,Pi,- -tPn may have zero value. § 9.] When products of integral functions of one variable have to be distributed, it is usually required at the same time to arrange the result according to powers of x, as in the tjrpical form above indicated. We proceed to give various instances of this process, using in the first place the method described in the earlier part of this chapter. The student should exercise himself by obtaining the same results by successive distribution or otherwise. In the case of two factors {x + a) (x + b), we see at once that the highest term is x', and the lowest ab. A term in x will be obtained in two ways, namely, ax and bx ; hence (x + a){x + b) = af + (a + b)x + ab (1). This virtually includes all possible cases ; for example, putting - re for a we get (x + (- a)) {x + li)=x^ + {{ - a) + b)x+{ - a)b, = x^+{-a + b)x-ab. 60 DISTRIBUTION OF (.V-«i) (.t:-(^,) . .(,t-((„) CHAP. Similarly [x-a){j:-b) = .r- + [-a-i)x + ab, = x" ~ {a + b),e + ab. {x-a)(x-a)=x- + { -a-a)x + a-, =x--2ax + a-, &o. Cases in wliich numbers occur in place of a and b, or in which x is affected with coefficients in the two factors, may he deduced by specialisation or other modification of formula (1), for example, (i--2)(a; + 3) = j;- + (-2 + 3)a; + (-2)( + 3), = x^-i-x-6. {lKC + q)(rx + s)=24-':+j\rU + -^ I \2> r) p r I ( q s\ IK r| i + - ]x+pr-!-, \2> r) pr =pTx--\-pr\ =prx- + (rq +ps)x + qs, which might of course be obtained more quickly by directly distributing the product and collecting the powers of x. Ill the case of three factors of the first degree, say (x + a,) {.r. + a^ (x + a.,), tlie higliest term is x^ ; terms in x' are obtained by taking for the partial products x from two of the three brackets only, then an a must be taken from the remaining bracket ; we thus get ttyi:', aX, a.j)? ; that is, (ffii + «„ + a.^jT is the term in of. To get the term in x, x must be taken from one bracket, and a's from the two remaining in every possible way ; this gives ((iitta + flifta + ciM-^x. The last or absolute term is of course a^a^i,^. Thus (.c + a,) {x + ((^ (x + a^) = x^ + (ffli + a. + «j)j° + (((,((;, + a,a3 + a.j(j)x + a^a^a^ (2). By substitution all other cases may be derived from (2), for example, {x-ii,){x-a,){x-a,) = »,■■' - (a, + ttj + ((3),';' + {ij^n.. + a^a^ + a2a3)x - a^cui^ (3); (,« + 1) {x + 2) (a; - 3) = x' - Ix - 6, and so on. After what has been said it is easy to find the form of the dis- tribution of a product of n factors of the first degree. The result is {x + II,) (x + it„) . . . (x + a,) = ./;" + P,:«"-i + P,.v;"-2+ . . +P„_ja; + P„ (4), IV BINOMIAL THEOREM 61 where P, signifies the algebraic sum of all the a's, P^ the alge- braic sum of all the products that can be formed by taking two of them at a time, P3 the sum of all the products three at a time, and so on, P„ being the product of them all. § 10.] The formula (4) of § 9 of course includes (1) and (2) already given, and there is no difficulty in adapting it to special cases where negative signs, &c., occur. The following is par- ticularly important : — (x - «i)(a; - flSj) . . . (a; - «„) = a" - P,x''-i + P^'^-2 _... + (_ l)"-ip„_,a; + ( - 1)'^P„ (1). Here Pi Pg, &c., have a slightly different meaning from that attached to them in § 9 (4) : Pg, for example, is not the sum of all the products of -a^, - a^, . . ., - a„, taken three at a time, but the sum of the products of + a,, + a^, . . ., + a^, taken three at a time; and the coefficient of a;""^ is therefore -P3, since the concurrence of three negative signs gives a negative sign. As a special case of (1) let us take (x -«)(«- 2a) (x - 3a) {x - 4a) = x' - P,x' + P^' - F^x + P,. Here P, = a -1- 2a -1- 3a -1- 4a = 10a, P, = 1 X 2a' H- 1 X 3a' -I- 1 X 4a' + 2 X 3a' -I- 2 X 4a' -t- 3 x 4a' = 35a', P3 = 2 x 3 X 4a' -H 1 X 3 X 4a' + 1 X 2 X 4a' -f 1 X 2 x 3a' = 50a', P, = 1 X 2 X 3 X 4a' = 24a'. So that (x - a) (x - 2a) (x - 3a) (x - 4a) ^ =a;'-10ax'-H35aV-50a'a;-H24a'. § 11.] Another important case of § 9 (4) is obtained by making «; = aa = aj = . . . = a,j, each = a say. The left-hand side then becomes {x + a)™. Let us see what the values of P,, P^, . . ., P„ become. Pj obviously becomes na, and P„ becomes a". Con- sider any other, say P^; the number of terms in it is the number of dififerent sets of r things that we can choose out of 11 things. This number is, of course, independent of the nature of the things chosen ; and, although we have no means as yet of calcu- lating it, we may give it a name. The symbol generally in use 62 BINOMIAL COEFFICIENTS CHAP. for it is „C,., the first suffix denoting the number of things chosen from, the second the number of things to be chosen. Again, each term of P,. consists of the product of r letters, and, since in the present case each of these is a, each term will be a^. All. the terms being equal, and there being „C,. of them, we have in the present case P, = ,Sj./f. Hence (,x + a)" = x" + maa;"-i + ,A»^a;''-2 + ^C3aV-» + . . . + «"; or, if we choose, since ,jCi = n, „C„ = 1, we may write (« + aY = v? + nC,aa;"-i + .S^A''-'^ + . .. + „C„_,a™-ia; + „C,ia" (1). Tlih is the "binomial theorem" for positive integral expoiients, and the numbers ,iCi, „C„, nCj, . . . are called the binomial coefficients of the nth order. They play an important part in algebra ; in fact, the student has already seen that, besides their function in the binomial expansion, they answer a series of questions in the theory of combinations. When we come to treat that subject more particularly we shall investigate a direct expression for ,,0^ in terms of n and r. Later in this chapter we- shall give a pro- cess for calculating the coefiicients of the different orders by successive additions. By substituting successively - a, +1, and - 1 for a in (1) we get {x - ay = a;« - ^C.fflx" " i + „C/t=^(;" " ^ - ^G.cfix'' " ^ + . . . + ( - 1)\C„«- (2) : (a;+l)'' = a;» + „C,a;»-i + „ax"-2 + . . + ,fi„ (3): (a;-l)» = a;"-„C,a"-i + „C^"-2-. . + ( _ l)'»^C„ (4); and an infinity of other results can of course be obtained by substituting various values for x and a. § 12.] In expanding and arranging products of two integral functions of one variable, the process which is sometimes called the long rule for multiplication is often convenient. It consists simply in taking one of the functions arranged according to descending powers of the variable and multiplying it successively by eich of the terms of the other, beginning with the highest and proceeding to the lowest, arranging the like terms under one another. Thus we arrange the distribution of I,ONG MULTIPLICATION 63 (x' + 2a;' + 2a;'+ l){x'-x+l) as follows : — x" + 2a;' + 2a; + 1 a;^- X +1 x' + Sx' + 2x' + x' - a;" - 2a;' - 2a;' - x + a;' + 2s' + 2x + 1 x' + xU ■ x' + x' + x + 1.. or again (^x' + qx + r){rx'' + qx +p) px^ + qx + r rx' + qx +p pr/ + qrx^ + pqx'' + ? V + q%' +pV + qrx +pqx +pr pi'x' + (pq + qr)x^ + {p^ + ; + 3){a;+l) + (a; + l)(a; + 2)}. (5.) {x+a}{a? + (b + c)x + bc}{x'-{a + b + c)a!? + {be + ca+ai)x-aic}. (6.) {{x+p){x-g){x + l)}i{x-p){x + g){x-l)}. (7.) {0!?- y^) (3? - 2y"-) {7? - 3/) {x^ - ii/) (a? - 67/). (8.) {ax + {b~c)y} {bx + {c-a)y] {cx+[a-b)y} ;* and show that the sum of the coefficients of x'^y and y^ is zero. (9.) Show that (k + |a)* - 10a[x + 4(8)3 + s^a^^. + ^f _ 50a3(a; + |a) + 24a* = (x=-ia2)(x^-|a=). (10.) Show that (^ + f2/)(-+j2/)(-+f 3/) - (- + Jy)(»=+f2/)(-+f2/) _ tei/(a -y){q- r) (r-p) (p-q ) pqr Distribute and arrange according to powers of x, the following : — (11.) {(b + e)x' + {c + a)x + {a + b)} {(b-c)3? + (c-a)x + {a-h)}. (12.) (3?-x + l){ii? + x + l){a?-2x + l){s? + 2x+l). * In working some of these exercises the student will find it convenient to refer to the tabl^ of identities given at the end of this- chapter. 70 EXERCISES VI, VII OHAP. (13. ) { 5a^ - ix{x -y) + {x-yf}{2x + Zy). (14.) (2a?-Zxy + 1y-)('ix'^ + Zxy + 2y-). (15.) {(3? + x+l)(x' -x + \){x' -1))\ (16.) (D?-3? + x-lf(x^ + a? + x + lf. (17.) (i:>?-\7? + \x + l)(lx> + ^x'- Ix + i). (18.) (.t-* - as?y + abxhj- + hxy^ + j/*) (aa;- - abxy + by"). (19. ) (x' + ax + b-f + (x^ + ax- b^ + (x' -ax + by + {x'^ -ax- Vf. (20.) (x*-2aV + a<)«. (21.) {.r'-a^f. (22.) (Zx+iy. (23.) (a + 6x2)8. (2i.) {(x^ + f)(afi-f)y. (25.) (l + a; + a;2 + a;3 + 3vi)3_ (26.) Calculate the coefficient of .c^ in the expansion of (l + x+x^)'. (27.) Calculate the coeflicient of jj" in {l-Zx + Zx^ + ix'-Ti^f. (28.) Show that [a + J)3(ffi= + JS) + 5ab(a + b^a* + b*) + 15a-b^[a + b){a^ + ¥) + Z5a%\a? + b-) + 1(ia%^ = (a + hf. (29.) Show that «Ci + „C= + „C3+ • +„C„ = 2«-1 ; 1 + nC2 + mC4 + . . = i.Ci + nCs + ^Cs + . . . ; nOr = n-aCr + 2„_2C,._i + n-^Or-^. (30. ) There are five boxes each containing five counters marked with the numbers 0, 1, 2, 3, 4 ; a counter is drawn from each of the boxes and the numbers drawn are added together. In how many difTcrent ways can the drawing be made so that the sum of the numbers shall be 8 ? (31.) Show that ''y+ . . +xy''~^ + y"-')=x''-x"-^y-xy"~'^ + y". Exercises VII. Distribute the following, and arrange according to descending powers of x: — (1.) (3x + 4)(4x + 5)(5x + 6)(6x + 7). (2.) {px + q-r){qx + r-2}){rx+p- rj). (3. ) {x-a)[x- 2a) {x - 3a) {x - ia) {x + a){x + 2a) {x + Za) [x + 4a). (4.) (x^ + Zx'' + 3x + l){3^-Zx' + Zx-l). (5.) iW + i^' + ix + Di^a^ + ix'^ + hx+l). (6.) {X - i){x^-ix + i){x + i){x'' + ix + i). ,. , ( I „ in »\ /'» , « T'\ (ii- t T- in\ /. -x= + -x+y -r? + ~^x^-- -^x?-^-x + -\. yni n IJ \'it I in J \l m n/ (8.) (2x-3)ii. (9.) {(x + y)(x^-xy + y'^)]>. (10.) (x--lY[x + \y''. IV HOMOGENEOUS FUNCTIONS 71 (11.) In the product {x + a){x + b){x + c), sfi disappears, and in the product {x-a)[x + b){x + c), X disappears; also the coefficient of x in the former is equal to the coefficient of x" in the latter. Show that a is either or 1. Prove the following identities : — (12.) {b-c){x-a)'' + {c-a)(x-b)^ + {a-b){x-c)^ + l,b-c](c-a){a-b) = 0. (13. ) S(2ffi - S - c) [h -b)(h-c) = 2(6 - cf[h - a). (14. ) (.9 - af + (s - bf + (s - cf + Zabc = ^, where 2s=a + b + c. (15.) {s-a)* + {s-b)^ + {s-c)* = 2(s - bf{s - c)2 + 2(s - cf{s - cs)2 + 2(s - af{s - bf, where 3s=a + b + c. (16.) {as + bc)(bs + ca){cs + ab) = {b + cf{c + af{a + bf, where s=a + 6 + c. {17.) s{s- a- d){s-d-b){s- c-d) = {s-a){s-b){s-c){s-d)- abed, where 2s = a + b + c + d. (18. ) 16(s - a) (s - 6) (s - c) (« - cZ) = i(bc + adf - {i' + c^ - a^ - d'f, where 2s=a + b + c + d. (19.) S(6-c)«=3n(J-c)2 + 2(Sa2_2ic)5. (20.) If U„=(6-c)'' + (c-a)« + (a-J)'', then U„+3 - (as2 + 62 + c2 - 6c - ca - a6)U„+i - (6 - c) (c - a) (a - 6)U„ = 0. (21.) I{pi = a + b + c,pi=bc + ca + ai,p3 = aic, 6-„ = a" + 6'> + <;'», show that Sl=^l, S2=piSi-2p2, S3=PiS2-p2Si + 3p3, Sn =PlSn-l -P2Sn-2 +PsS„-s ■ (22.) If^2=(6-c)((;-a) + (c-a)(a-6) + (a-6)(6-c), ps={b-c){c-a)(a-b), s„ = (6 - c)» + (c - a)" + (a - 6)", show that S2 = - 2^2 , S3 = Sps , Si = 2p2% S5= - bpips , Se= -2p/+Sps^, Si=7p2^Pst 25s7S3=21s>i\ Homogeneity. § 17.] An integral function of any number of variables is said to be " Homogeneous " when the degree of every term in it is the same. In such a function the degree of the function (§ 6) is of course the same as the degree of every term, and the number of terms which (in the most general case) it can have is the number of different products of the given degree that can be formed with the given number of variables. If there be only two variables, and the degree be n, we have seen that the number of possible terms is n+ 1. 72 HOMOGENEOUS FUNCTIONS CHAP. For example, the most general homogeneous integral functions of x and y of the 1st, 2nd, and 3rd degrees are * Ax + By (1), Ax- + 'Bxy + Cy'' (2), Aa? + Bxhj + Cxy^ + 'Dy'> (3), &c., A, B, C, &o. , representing the coefficients as usual. For three variables the corresponding functions are Ax + By + Cz (i), Ax' + By^ + Cz- + Dyz + 'Ezx + ¥xy (5), Asfl + Bf + Cs3 + Pj,z2 + F'y-z + Qzx^ + Q'z^x + lixr/ + RVj/ + Sxyz (6), &o., As the case of three variables is of considerable importance, we shall in- vestigate an expression for the number of terms when the degree is n. We may classify them into — 1st, those that do not contain x ; 2nd, those that contain x ; 3rd, those that contain x'; . . .; m + 1th, those that contain x". The first set will simply be the terms of the nth degree made up with y and 2, ra + l in number ; the second set will be the terms of the (»-l)th degree made up with y and z, n in number, each with x thrown in ; the third set the terms in y and z of {n - 2)th degree, m - 1 in number, each with x^ thrown in ; and so on. Hence, if N denote the whole number of terms, N = (» + l) + « + («.-l)+ . . .+2 +1. Reversing the right-hand side, we may write N= 1 4-2-1- 3 -1- . . . +n + {n + l). Now, adding the two left-hand and the two right-hand sides of these equali- ties, we get 2T^ = {n + 2) + (n + 2) + (7i + 2)+ . . . +{n + 2) + {n + 2) ; = (n + l){n + 2), since there are «-t-l terms each =n + 2. "Whence N = J(9H-l)(?i-l-2). For example, letii=3; ]Sr = i(3-Hl)(3-l-2) = 10, which is in fact the number of terms in (6), above. In the above investigation we have been led incidentally to sum an arithmetical series (see chap, xx.) ; if we attempted the same problem for 4, 5, . . ., m variables, we should have to deal with more and more complicated series. A complete solution for a function of the ?ith degree in m variables will be given in the second part of this work. * Homogeneous integral functions are called binary, ternary, &c., accord- ing as the number of variables is 2, 3, &c. ; and quadric, cubic, &c., accordinf as the degree is 2, 3, &c. Thus (3) would be called a binary cubic ; (5) a ternary quadric ; and so on. IV HOMOGENEOUS FUNCTIONS 73 The following is a fundamental property of homogeneous functions : — If each of the variables in a homogeneous function of the nth degree be multiplied by the same quantity p, the result is the same as if the function itself were multiplied by p". Let us consider, for simplicity, the case of three variables ; and let F = AxPyiz'^ + A'xP'yi'z'" + . . ., where p + q + r =p' + q' + r' = &c., each = n. If we multiply x, y, z each by p, we have Y = K{pxy{py)i{pzy + A;{pxY{pyY{pzY+ . . .■ = ApP+i+^'xPy^z'' + A'pf'+i'+'-'xP'yrz'" + . ., by the laws of indices. Hence, since p + q + r =p' + q' + r' = &c. = n, we have F' = p" { AxPy^z" + A'xP'y^Y' + . . .}, = p"F, which establishes the proposition in the present case. The reasoning is clearly general. "* * * This property might be made the definition of a homogeneous function. Thus we might define a homogeneous function to be such that, when each of its variables is multiplied by p, its value is multiplied by p" ; and define n to be its degree. If we proceed thus, we naturally arrive at the idea of homo- geneous functions which are not integral or even rational ; and we extend the notion of degree in a corresponding way. For example, {^-y^)l{x + y) is a homogeneous function of the 2nd degree, for ( {px)^ - (pyf)l( (px) + [py) ) =f?(^ - y^)l(x + y). Similarly \/{3? + j/'), l/{a^ + y'^) are homogeneous functions, whose degrees are f and - 2 respectively (see chap. x. ) Although these ex- tensions of the notions of homogeneity and degree have not the importauce of the simpler cases discussed in the text, they are occasionally useful. The distinction of homogeneous functions as a separate class is made by Euler in his InirodMctio in Analysin Infinitorwm (1748), (t. i. chap, v.), in the course of an elementary classification of the various kinds of analytical functions. He there speaks, not only of homogeneous integral functions, but also of homogeneous fractional functions, and of homogeneous functions of fractional or negative degrees. 74 LAW OF HOMOGENEITY CHAP. Example. Consider the homogeneous integral function Zn^ - 2zy + y", of the 2n(i degree. We have 3[px)- - 2{px) ipy) + (pyf = 3pV - 2p-xy + p-y^, =p^(3x'-2xy + y\ in accordance with the theorem above stated. The following property is characteristic of homogenems integral functions of the first degree. If for the variables x, y, z, . . . we substitute Aaii + fux^, Xy^ + fxy^, A^i + /i«2, . . . respectively, the result is the same as that obtained by adding the results of substituting Xj, «/,, «i, . . and :i\, ^2,^2, . . respectively for X, y, z, . . . in the function, after multi^ plying these results by A. and jj. respectively. Example. Consider the function Ax + B)/ + Cz. We have A(Xa!i + pxi) + B(X3/i + jxy^) + C(\zi + p-z^) = AXzi + BX?/i + CXjj + A/xx-2 + B/ij/2 + 0,022 = X(Aa:i + Bj/i + C^i) + ,u( Axa + Bi/a + Cz-i). This property is of great importance in Analytical Geometry. § 1 8.] Law of Homogeneity. — Since every term in the product of two homogeneous functions of the mth and «th degrees re- spectively is the product of a term (of the mth degree) taken from one function and a term (of the mth degree) taken from the other, we have the following important law : — Tlie product of two homogeneous integral functions, of the mth and nth degrees respectively, is a homogeneous integral function of the (to + n)th degree. The student should never fail to use this rule to test the distribution of a product of homogeneous functions. If he finds any term in his result of a higher or lower degree than that indicated by the rule, he has certainly made some mistake. He should also see whether all possible terms of the right degree are present, and satisfy himself that, if any are wanting, it is owing to some peculiarity in the particular case in hand that this is so, and not to an accidental omission. The rule has many other uses, some of which will be illus- trated immediately. IV SYMMETRICAL FUNCTIONS 75 § 19.] If the student has fully grasped the idea of a homo- geneous integral function, the most general of its kind, he will have no difficulty in rising to a somewhat wider generality, namely, the most general integral function of the nth degree in m variables, unrestricted by the condition of homogeneity or otherwise. Since any integral term whose degree does not exceed the mth may occur in such a function, if we group the terms into such as are of the 0th, 1st, 2nd, 3rd, . . . , wth degrees respectively, we see at once that we obtain the most general type of such a function by simply writing down the sum of all the homogeneous integral functions of the m variables of the 0th, 1st, 2nd, 3rd, . . ., mth degrees, each the most general of its kind. For example, the most general integral function of x and y of the third A + Bx + Cy + 'Dx'' + 'Exy + Fy^ + Grs? + 'Hx'y + Ixy'' + Jy^. The student will have no difficulty, after what has been done in § 17 above, in seeing that the number of terms in the general integral function of the mth degree in two variables is i{n+l){n+2). Symmetry. § 20.] There is a peculiarity in certain of the functions we have been dealing with in this chapter that calls for special notice here. This peculiarity is denoted by the word " Symmetry "; and doubtless it has already caught the student's eye. What we have to do here is to show how a mathematically accurate definition of symmetry may be given, and how it may be used in algebraical investigations. 1st Definition. — An integral function* is said to be symmetrical with respect to any two of its variables when the interchange of these two throughout the function leaves its value unaltered. * As a matter of fact these definitions and much of what follows are applicable to functions of any kind, as the student will afterwards learn. According to Baltzer, Lacroix (1797) was the first to use the term Symmetric Function, the older name having been Invariable Function. 76 Various kinds of symmetry ohap. For example, 2a + 3b + 3c becomes, by the interchange of b and c, 2a + 3c + 3b, which is equal to 2a + 3b + 3c by the commutatiTe law. Hence 2a + 36 + 3c ia symmetrical with resjicct to b and c. The same is not true with respect to a and 6, or a and c ; for the interchange of a and b, for example, would produce 2b + 3a + 3c, that is, 3a + 26 + 3c, which is not in general equal to* 2a + 3i + 3c. 2nd Definition. — An integral function is said to he symmetrical {tluit is, symmetrical with respect to all its variables) when the interchange of anij pair whatever of its variables would leave its value unaltered. For example, 3x + 3y + 3z is a symmetrical function of x, y, z. So are yz + zx + :nj and 2{x- + y- + z-) + 3i:yz. Taking the last, for instance, if we interchange y and z, we get 2{x'' + z' + y^) + 3xzy, that is, 2(:r" + y'' + z-) + 3xyz, and so for any other of the three possible interchanges. On the other hand, x"y + y"z + z"x is not a symmetrical function of x, y, z, for tlio three interchanges x with y, x with z, y with z give respectively yhi+a?z + z^y, z-y + y'^x + x"z, a?z + z^y + y^x, and, although these are all equal to each other, no one of them is equal to the original function. It will be observed from this instance that asymmetrical functions have a property — which symmetrical functions have not — of assuming different values when the variables are interchanged : thus x^y + yh + z'x is susceptible of two different values under this treatment, and is therefore a two-valued function. The study of functions from this point of view has developed into a great branch of modern algebra, called the theory of substitu- tions, which is intimately related with many other branches of mathematics, and, in particular, forms the basis of the theory of the algebraical solution of equations. (See Jordan, Traite des Substitutioiis, and Savret, Cours cl'Algibre fSiqjerieure. ) All that we require here is the definition and its most elementary con- sequences. 3rd Definition. — A function is said to be collaterally symmetrical in tn-o sets of variables] " " ' '' " r, each of the same number. {a„a,,. ..,«„) • It may not be amiss to remind the student that for the present " equal to " means " transformable by the fundamental laws of algebra into." IV RULE OF SYMMETRY 77 when the simultaneous interchanges of two of the first set and of the corresponding two of the second set leave its value unaltered. For example, a'x + Vy + A and (6 + c)x + (c + a)i/ + (a + J)s are evidently symmetrical in this sense. Other varieties of symmetry might be defined, but it is needless to perplex the student with further definitions. If lie fully master the 1st and 2nd, he will have no difficulty with the 3rd or any other case. At first he should adhere somewhat strictly to the formal use of, say, the 2nd definition ; but, after a very little practice, he will find that in most cases his eye will enable him to judge without conscious effort as to the symmetry or asymmetry of any function.* § 21. J From the above definitions, and from the meaning of the word " equal " in the calculation of algebraical identities, we have at once the following Eule of Symmetry. — The algebraic sum, product, or quotient of two symmetrical functions is a symmetrical function. Observe, however, that the product, for example, of two asymmetrical functions is not necessarily asymmetrical. Tlius, a + i + c and ic + ca + ai being both symmetrical, their product, (a + h + c)[ho + ca + ab) = Vc + Jc^ + i?a + ca' + a?h + aV^ + Zahc, is symmetrical. Again, a^Jo and a5V are both asymmetrical functions of a, 6, c, yet their product, (a26c)x(a6V) = a'JV, is a symmetrical function. § 22.] It will be interesting to see what alterations the resti'iction of symmetry will make on some of the general forms of integral functions written above. Since the question of symmetry has nothing to do with degree, it can only affect the coeflScients. Looking then at the * There is a class of functions of great importance closely allied to sym- metrical fuuctions, which the student should note at this stage, namely, those that change their sign merely when any pair of the variables are interchanged. Such functions are called " alternating. " An example is {y - s) (2 - a;) {x-y). Obviously the product or quotient of two alternating functions of the same set of variables is a symmetric function. The term Alternating Function is due to Cauchy (1812). 78 APPLICATION OF THE RULE CHAP. homogeneous integral functions of two variables on page 72, we see that, in order that the interchange of x and y may produce no change of value, we must have A = B in § 17 (1) ; A = C in (2) ; A = D and B = C in (3). Hence the symmetrical homogeneous integral functions of a; and y of 1st, 2nd, 3rd, &c., degrees are Ax + Ay (1), Ax- + yj.qi + Ay^ (2), Ax' + Ex'^y + Bxy'^ + Af (3), &c. The corresponding functions of x, y, s are Ax + Ay + Az (4), Ax°- + Ay'' + Az- + 'Byz + Bzx + 'Bxy (5), Ax^ + A?/' + A== + Pj/z^ + Fy''z + Pza;^ + Ps=a: + Voq/^ + Vx°y + Sxyz (6), &c. Tlie most general symmetrical integral function of x, y of the 3rd degree will be the algebraic sum of three functions, such as (1), (2), and (3), together with a constant term, namely, F + Ax + Ay + Bx' + Cxy + 'By'' + 'Dsf + 'Ex'y + 'Em/ + 'Dy^. And so on. If the student find any difficulty in detecting what terms ought to have the same coefficient, let him remark that they are all derivable from each other by interchanges of the variables. Thus, to get all the terms that have the same coefficient as x^ in (6), putting y for x, we get g' ; putting z for x, we get / ; and we cannot by operating in the same way upon any of these produce any more terms of the same type. Hence x^, y, / form one group, having the same coefficient. Next take y^ ■ the inter- changes X and y, x and g, y and z produce x/, yaf, yi ; applying these interchanges to the new terms, we get only two more new terms — z'^, xif ; hence the six terms y/, y^z, zx', /.c, zy', ^y form another group j xyz is evidently unique, being itself symmetrical. § 23.] The rule of symmetry is exceedingly useful in abbre- viating algebraical work. Let it he required, for example, to distribute the product {a + h + c) {a^ + b- + 1? - ic - ca - ab), each of whose factors is symmetrical in a, b, v. The distributed product will be symmetrical in a, b, c. Now we see at once that the term a^ occurs with the coefficient unity, hence the same must be true of 6^ and (fl. Again the term b'c has the coefficient 0, so also by the principles of symmetry must each of the five other terms, ic^, c'a, ea?, ab', a'b, belonging to the same type. Lastly, the term -abc is obtained by taking a from the first bracket, hence it must occur by taking b, and by taking c, that is, the IV INDETERMINATE COEFFICIENTS 79 oftc-term must have the coefficient - 3. We have therefore shown that {a + b + c){a^ + b^ + c^-bc-ca-ab) = a^ + b^ + c^-Sabc; and the principles of symmetry have enabled us to abbreviate the work by about two-thirds. PRINCIPLE OF INDETERMINATE COEFFICIENTS. § 24.J A still more striking use of the general principles of homogeneity and symmetry can be best illustrated in conjunction with the application of another principle, which is an immediate consequence of the theory of integral functions. We have laid down that the coefficients of an integral function are independent of the variables, and therefore are not altered by giving any special values to the variables. If, therefore, on either side of any algebraic identity involving integral functions we determine the coefficients, either hy general considerations regarding the forms of the functions involved, or by considering particular cases of the identity, then these coefficients are determined once for all. This has (not very happily, it must be confessed) been called the principle of inde- terminate coefficients. As applied to integral functions it results from the most elementary principles, as we have seen ; when infinite series are concerned, its use requires further examination (see the chapter on Series in the second part of this work). The following are examples : — {x + yY = {x + y)(x + y), being the product of two homogeneous symmetrical functions of x and y of the 1st degree, will be a homogeneous symmetrical integral function of the 2nd degree ; therefore {x + yf = Az' + Bxy + A.y' (1). We have to determine the coefficients A and B. Since the identity holds for all values of x and y, it must hold when x=\ and y = 0, therefore (l+0)' = Ar-i-Bl xO + AO^ 1=A. We now have {x + yf = x^ + Bxy + y" ; this must hold when x=l and y = - I, therefore (1 - 1)' = 1 + B.l.(- 1) + 1, that is, = 2 - B, whence B = 2. Thus finally (x + yf = x^ + Ixy + /. 80 INDETERMINATE COEFFICIENTS CHAP. This method of working may seem at first sight somewhat startling, but a little reflection will convince the learner of its soundness. "We know, by the principles of homogeneity and symmetry, that a general identity of the form (1) exists, and we determine the coefficients by the consideration that the identity must hold in any particular case. The student will naturally ask how he is to be guided in selecting the particular cases in question, and whether it is material what cases he selects. The answer to the latter part of this question is that, except as to the labour involved in the calculation, the choice of cases is immaterial, provided enough are taken to determine all the coefficients. This determination will in general depend upon the solution of a system of simultaneous equations of the 1st degree, whose number is the number of the coefiicients to be determined. (See below, chap, xvi.) So far as possible, the particular cases should be chosen so as to give equations each of which contains only one of the coefficients, so that we can determine them one at a time as was done above. The student who is already familiar with the solution of simultaneous equations of the 1st degree may work out the values of the coefficients by means of particular cases taken at random. Thus, for example, putting x=2, y = 3, and x=l, y=i successively in (1) above, we get the equations 25 = 13A + 6B, 25 = 17A + 4B, which, when solved in the usual way, give A = 1 and B = 2, as before. We give one more example of this important process : — By the principles of homogeneity and symmetry we must have {x + y + z)(x' + y^ + z'^-ysi-2x-xy) = A{3? + y^ + z^) + B(yz^ + y\ + za? + ^x + xy"^ + cc-j/) + Gyxz. Putting x=\, y — 0, z=0, we get 1 =A. Using this value of A, and putting x = l, y = l, 2=0, we get 2xl = 2 + Bx2, that is, 2 = 2 + Bx2, therefore 2B = 0, and therefore B = 0. Using these values of A and B, and putting x=1, y=l, z=l, we get 3xO = 3 + C, that is, = 3 + C, therefore C = - 3 ; and we get finally {x + y + z) {x' + y^ + z^-yz-zx-xy)=a? + y^ + z'-Sxyz (2), as in § 23. § 25.] Reference Table of Identities. — Most of the results given below vv'ill be found useful by the student in his occasional calcu- lations of algebraical identities. Some examples of their use TABLj; OF IDENTITIES 81 }iave already been given, and others will be found among the Exercises in this chapter. Such of the results as have not already been demonstrated above may be established by the student himself as an exercise. (x + a){x + h) = y? + {a + h)x + ah ; (x + a)(z + h) {x + c) = £' + {a + h + c).if + (be + ca + ab)x + abc ; and generally (x + aj(x + a^ . ..{x + ttn) = X" + Pia;"-i + P^k^-^ + . . . + P„_iX + P„(see §9). {x±yy = x'±2xtj + f; (x ± yf = x'± Saftj + Zxy" ± f ; &c.; the numerical coefficients being taken from the following table of binomial coefficients : — HI) Power. Coefficients. 1 1 2 2 1 3 3 3 1 4 4 6 4 1 5 5 10 10 5 1 6 6 15 20 15 6 1 7 7 21 35 35 21 7 1 8 8 28 56 70 56 28 8 1 9 9 36 84 126 1 26 84 36 9 1 10 1 10 45 120 210 252 210 120 45 10 1 11 11155 165 330 462 402 330 165 55 11 1 12 1 12 66 220 495 792 924 792 495 220 66 12 1 &c. Wnr * This table first occurs in the Arithmetiaa Integra of Stifel (1544), in connection with the extraction of roots. It does not appear that he was aware of the application to the expansion of a binomial. The table was dis- cussed and much used by Pascal, and now goes by the name of Pascal's Arithmetical Triangle. The factorial formulae for the binomial coefficients (see the second part of this work) were discovered by Newton. VOL. I G 82 TABLE OF IDENTITIES CHAP. (III.) HIV.) Wv.)* (,c ± yf T 4.ry = (x T yf- (x + y){x-y) = x^ -f ; (a ± 2/) (x' T «2/ + 2/') = k' ± 2/' ; and generally (.,;_y) (,,■"-! +a;«-2y + . , . + X?/" " ^ + y'' " 1) = x" - ?/" ; (x + 2/)(x"-i-x"-'-;/ + . . . TJi/""-± «/""■') = 3'"*J'"' upper or lower sign according as n is odd or even. {i + y^) {!•:' + y") = (xx' t !/«/')' + (a^y' ± F')"; ^ (■'" - f) {^'\ - y") = (■«' ± 2/2/T - (^y ± y^T; (x' + 2/° + -')(*" + y" + 2;'') = (xx' + yy' + «s')^ + {ys' - y'zf + (»x' - «'x)° + (xy' - x'y)^ ; (x" + 2/^ + ^'^ + M^)(x"' + y"* + «'° + w'") = (xx' + yy' + «»' + mm')^ + (^y' ~ y^' + ^"' ~ "^T + (x2' - yv! - zx' + uy'y + (xm' + y3' - zy' - Mx')^ (x° + x?/ + 2/^) (x^ - xy + /) = .«' + xy + 2/^ (VI-) (a + J + c + rf)' = a" + i" + c" + (/' + 2ai + 2ac + 2acZ + 2Jc + 2M + 2crf ; and generally (a, + a„ + . . + ff„)' = sum of squares of a„ fto, ...,«; + twice sum of all partial products two and two. (a + J + c)" = •«' + J' + c' + 3&'c + 36c' + 3cVt + 3cft= + Za% + 3tti^ + 6«Jc = a^ + J'' + c" + 3ic(i + c) + 3ca(c + a) + 3ffiJ(a + i) + 6«ic. (aA-h + c) (a' + V + c^ -hc-ca- ah) = a" + b" + c" - 3ahc. (IX.) {h - c) {c-a)(a-b)= - a''(b - c) - b^{c -a)- c'(a - b), = a{b' - c) + b{c' - a') + c(a' - b% = - bc(b - c) - ca(c -a)- ab{a - b). = +bc' - Vc + a^ - c'a + aV - a%. * These identities furnish, inter alia, proofs of a series of propositions in tlie theory of numbers, of which the following is typical : — If each of two integers be the sum of two squares, their product can be exhibited in two ways as the sum of two integral squares. y(vii.) (VIII.) >(X.) IV EXERCISES VIII 83 {b + c){c + a){a + b) = a\b + c) + b\c + a) + c\a + b) + 2abc, \ = bc(b + c) + ca{c + a) + ab(a + b) + 2abc, l(XI.) = bc^ + Vc + ca' + c% + ab' + a'b + 2abc. I {a + b + c) (a' + b' + c') = bcQ> + c) + ca{c + a) + ab{a + b) \ + a' + b' + c\ I (XII) }(XIII.) }(XIV.) (XV.)* {a + b + c) {be + ca + ab) = a^(b + c) + b\c + a) + c-(a + b) + 3abc. {b + c-a)(c + a-b)(a + b-c) = a\b + c) + b\c + a) + cXa + b)-a? -h^ -c^ - 2abc. (a + b + c){- a + b + c){a - b + c)(a + i - c) = 2JV + 2cV + 2a'b' -a'-b'- c\ (b-c) + {c-a) + (a-b) = ; \ a(b -c) + b(c - a) + c{a - S) = ; i (XVI.) (b + c){b -c) + {c + a){c - a) + (a + b)(a - b) = 0. ) Exercises VIII. (1.) Write down the most general rational integral symmetrical function of X, y, z, u of the 3rd degree. (2.) Distribute the product {x''y + '!fz + z'^x){xif + yz'' + zx'). Show that it is symmetrical ; count the number of types into which its terms fall ; and state how many of the types corresponding to its degree are missing. (3.) Construct a homogeneous integral function of x and y of the 1st degree which shall vanish when x=y, and become 1 when x=t and y=2. (4. ) Construct an integral function of x and y of the 1st degree which shall vanish when x=x!, y=y', and also when x=x", y=y". (5.) Construct a homogeneous integral function of x and y of the 2nd degree which shall vanish when x=x', y=y', and also when x=a^', y=y", and shall become 1 when x=l, y=l. (6.) If A(a;-3)(a;-5) + B(a!-5)(a;-7) + C(a;-7)(aj-3) = 8a;-120 for aU values of x, determine the coefficients A, B, C. (7.) Show that 5a;^ + 19a; + 18 can be put into the form l(x-i){x-3) + m.{x-S){x-l) + n{x-l)(x-2); ani find I, m, n. (8.) Assuming that (x-l)(x-i)(x-Z) can be put into the form l(x-l)(x + 'i){x + Z) + 7n{x-2){x + Z)(x + l) + n(x~2,)(x + l){x + i), determine the numbers I, m, n. * Important in connection with Hero's formula for the area of a plauel triangle. 84 EXERCISES VIII CHAP. IV (9.) Find a rational integral function of x of the 3rd degree wliieh shall have the values P, Q, B,, S when x = a, x=6, x = c, x=d respectively. (10.) Find the coefficients of y'h and yz" in the expansion of (ax + hy + cz) {a'x + Vy + c-z) {a?x + h^y + (?z) . (11.) Expand and simplify 'Z{y'^ + z' - x') {y + z - x). Prove the following identities : — (12.) (ad+bcf + (a + h + c-d)(a + h-c + d)(l + d)(b-d) = (i--d- + ah + cdf. *(13.) J,(V + c^-a- + bc + ca + ab)-(c"-h-) = i{V-(?)(c'^-a'^){.d--h-). (14.) S(OT - 62) (ab - c-) = {Zbc) (SSc - Sa-). (15.) 2(ftc' - h'c) (6c" - 6"c) = Xa^'Sn'a" - ^aa"Zaa". (16. ) Sn{y+z) - e-Zyz=Xz{Xx - 1) (S..- - 2) - ri{a: - 1) {x-2). (17.) ^{b^ + c''-a'')l2bc = {ipip2-Pi'-6p3)l22>3, where ^i= - 2a, i)2 = 26c, P3= -abc. (18. ) n{y + z)"- + 2x"-yh"- - ^-Av + -)== 2(2t/2;)'. (19.) ^{,- + y-z){{y-z)^-{z-9f){x-y)}=X.,''-Zxyz. (20. ) n{a±b±c±d) = 2a' - i'Za^b^ + 62a*6^ + i'SaVc"- - iOaV-c''dK (21.) Show that (a;5 + j/3 + t'' - 3s!/s)= = X" + Y^ + Z» - 3XYZ, where X = x'^ + 2)jz, &c. ; also that (2a? - 3xj/z) CSx'' - Sx'y'z') = 2{xx' + yz' + y'zf - 3n(a:cc' + yz' + j/'s). (These identities have an important meaning in the theory of numbers. ) (22. ) Show that, if m he a positive integer, then 1-i + i-. . .-l(,i,even) = 2('^ + ^+. . .+i\; n \» + 2 m + 4 2m/ 1-HJ-- ■ . + i(modd) = 2C^+_l- + . . .+1 (i??tssarfZ). * In this example, and in others of a similar kind, 2 is not used in its strict sense, hut refers only to cyclical interchanges of a, 6, c ; that is, to interchanges in which a, 6, c pass into 5, c, a respectively, or into c, a, 6 respectively. Thus, 2a2(6-c) is, strictly speaking, =0; but, if S be used in the present sense, it is a^(6 - c) + 6^(0 - a) + c2((i - 6). CHAPTER V. Division of Integral Functions— Transformation of Quotients. , § l.J The operations of this chapter are for the most part inverse to those of last. Thus, A and D being any integral functions of one variable x* and Q a function such that D X Q = A, th«n Q is called the quotient of A by D ; A is called the dividend and D the divisor. We symbolise Q by the nota- A tion A -;- D, A/D, or =-, as explained in chap. i. The operation of finding Q is called division, but we prefer that the student should class the operations of this chapter under the title of transformation of quotients. A and D being both integral functions, Q will be a rational function of x, but will not necessarily be an integral function. When the quotient can he transformed so as to become integral, A is said to he exactly divisible hy D. WTien the quotient cannot he so transformed, the quotient is said to he fractional or essentially fractional. It is of course obvious that an essentially integral function cannot he equal, in the identical sense, to an essentially fractional function. § 2.] When the quotient is integral, its degree is the excess of the degree of the dividend over the degree of the divisor. For, denoting * For reasons partly explained below, the student must be cautious in applying many of the propositions of this chapter to functions of more vari- ables than one ; or at least in such cases he must select one of the variables at a time, and think of it as the variable for the purposes of this chapter. 86 THEOEEM EEGAEDING DIVISIBILITY CHAP. the degrees of the functions rejaresented by the various letters by suffixes, we have therefore, by chap, iv., § 7, m =p + n, that is, ^ = m - m. § 3.] If the degree of the dividend be less titan that of the divisor, thr quotient is essentially fnictwnnl. For, m being + q) {a+p) + q (a+p)-{a''+ap) {a'' + ap + q)+r (^a' + ap + q)- (a' + a'p + aq) {a^ + a'p + aq+r) Hence the integral quotient is x'' + {a+p)x + (a'' + ap + q) ; and the remainder is a' + a?p + aq + r. The student should ohserve the use of brackets throughout to preserve the identity of the coefficients. Example 3. 1st. Let us consider a as the variable. Since the expressions are homo- geneous, we may omit the powers of b in the coefficients, and use the numbers merely. 1-3+6-3+1 j 1-1+1 1-1 + 1 11- •2 + 3 -2 + 5- -2 + 2- -3 a^ -2 -2a5 + 362 3- 3- -1 + 1 -3 + 3 2-2 2a63-26* BINOMIAL DIVISOR 93 whence a* - 3a^ + Ga^^ - 3ab^ + b* -2ab + 3l^- 2rtJ3 _ 264 a' - ab + b" We must then arrange according 2nd. Let us consider b as the variable, to descending powers of b, thus — {b* - Sab^ + 6aV - Za% + a*) -=- (6^ -ab + a'-). Detach the ooeificients, and proceed as before. It happens in this particular case that the mere iramerical part of the work is exactly the same as before ; the only difference is in the insertion of the powers of a and b at the end. Thus the integral quotient is J^-26a + 3a^, and the remainder is 2ia'-2a^, whence a* - Za% + 6a2p _ Sajs + ^4 = 3a^-2ab + b''- + 2a^b-2a'' -ab + b'^ "" ' " ' a'-ab + b''' § 12.] The process of long division may be still further abbreviated (after expertness and accuracy have been acquired) by combining the operations of multiplying the divisor and sub- tracting. Then only the successive residues need be written. Thus contracted, the numerical part of the operations of Example 3 in last paragraph vi'ould run thus : — 1-3 + 6-3 + 1 I 1-1 + 1 -2+5-3+1 1 1-2+3 3-1 + 1 2-2 BINOMIAL DIVISOR — REMAINDER THEOREM. § 13.] The case of a binomial divisor of the 1st degree is of special importance. Let the divisor he x — a, and the dividend p^'^+p,x'^-^+p^'^-^ + . . .+pn-,x+p„. Then, if we employ the method of detached coefficients, the calculation runs as follows : — Po+Pi Po -Po^ + P2 + - + Pn-,+i {PoO-+Pl}+P2 (Poa + Pi) - (Poo. + p,y 1-, Po + {p,a+p) + {p„a +p,a +p.^ (p^a + p^a + p^) + p^ (poa +p,a +p^) - {p^a + p,a + p^)a (Poa + p,a + p,a + p^) 9-4 RULES FOR COEFFICIENTS OF chap. The integral quotient is therefore The law of formation of the coefficients is evidently asfulJnica. — Thefird m the first coefficient of the dicidciul ; The second is oUained by multiplying its predecessor by u, and adding the second coefficient of the dividend ; The third by muUiphjing the second just obtained by a and adding the third coefficient of the diridend ; and so on. It is also obvious that the remainder, which in the present case is of zero degree in x (that is, does not contain j:), is obtained from the last coefficient of the integral quotient by multiplying that coefficient by a and adding the last coefficient of the diridend. The operations in any numerical instance may be con- veniently arranged as follows : — * Example 1. (2.(r' - 2,3- + 6^ - 4) -=- [x - 2). 2+0-3+ 6- 4 + 4 + 8 + 10 + 32 2 + 4 + 5 + 16 + 28 Integral quotient = 2a,-'' + 4 j- + 5a; + 1 6 ; Remainder =28. The figures in the first line are the coefficients of the dividend. The first eoellicient in the .second line is 0. The first coefficient in the third line results from the addition of the two above it. The second figure in the second line is obtained by multiplying the first coefficient in the third line by 2. The second figure in the third line by adding the two over it. And so on. • Example 2. If the divisor be x+2, we have only to observe that this is the same as * The student should observe that this arrangement of the calculation of the remainder is virtually a handy method for calculating the value of an integi-al function of x for any particular value of x, for 28 is 2 x 2* - 3 x 2" + 6x2-4, that is to say, the value of 2.',''- dx'' + 6x-i when x = 2 (see § 14). This method is often used, and always saves arithmetic when some of the coefficients are negative and others positive. It was employed by Newton ; see Horsley's edition, vol. i. p. 270. V DIVISOR, AND FOE REMAINDER 95 ^~(~2); and we see that tlie proper result will be obtained by operating throughout as before, using - 2 for our multiplier instead of +2. (2a:!* - 3a;2 + 6a; - 4) -=- (a; + 2) = (2x^ - 3a? + ex- i) ^ {x- (-2)). 2+0-3+ 6-4 0-4+8-10+8 2-4 + 6- 4 + 4. Integral quotient = 2x^ - ix'^ + 6x - i ; llemainder =4. Example 3. The following example will show the student how to bring the case of any binomial divisor of the 1st degree under the case of x- a. 3a*-2a?' + 3!g^-2a; + 3 _ 3a*-2af' + 3a:^-2a; + 3 3a! + 2 ~ S(a; + |) _^ f 3x'-2Q? + Sx^-2x+S -\ Transforming now the quotient inside the bracket { } , we have 3-2+ 3-2+3 0-2+ f -¥ + W- 3-4H -5T-- Integral quotient = 3k?- ix^ + ^-x - ^^. Remainder = '//-. Whence 3a;'-2x->+3K2-2a; + 3 3a; + 2 ' 3»s - ix> + V-K - V- + ^-^2 ) } 3a; + 2' Hence, for the division originally proposed, we have — Integral quotient = x* - ^x^ + ^^x - 4f ; * Kemainder = V"- The process employed in Examples 2 and 3 above is clearly applicable in general, and the student should study it attentively as an instance of the use of a little transformation in bringing cases apparently distinct under a common treatment. § 14.] Eeverting to the general result of last section, we see that the remainder, when written out in full, is 96 EEMAINDEE THEOREM CHAP. Comparing this with the dividend p„a;"+^,a;"-i + . . .+Pr,-iX+Pn, we have the following " remainder themem " : — JFhen an integral function of x is dioided hy x- a, the remainder is obtained hy substituting a for x in the function in question. In other words, the remainder is the same function of a as the dividend is of x. Partly on account of the great importance of this theorem, partly as an exercise in general algebraical reasoning, we give another proof of it. Let us, for shortness, denote ^oa;''+p,a;"-i + . . . + Pn-^x + p^hj f{.c), f{a) will then, naturally, denote the result of substituting a for X in /(.'•), that is, ^oa"+_Pia''-l + . . .+Pn-,o.+Pn. Let ^((x) denote the integral quotient, and R the remainder, when /(.(■) is divided by a; - a. Then xC') is an integral function of X of degree n-1, and R is a constant (that is, is independent of a), and we have X- a ^ X- a whence, multiplying by x - a, we get the identity /(a;) = (a;-a)x(«) + R. Since this holds for all values of x, we get, putting x = a throughout, /(a) = (a - a)x(a) + R, where R remains the same as before, since it does not depend on X, and therefore is not altered by giving any particular value to X. Since x(a) is finite if a be finite, (a - a)x(a) = x yfa) = ; and we get finally /(a) = K> which, if we remember the meaning of /(a), proves the "re- mainder theorem." V FACTORISATION BY KEMAINDER THEOEEM 97 Cor. 1. Since x + a = x-{-a), it follows that The remainder, when mi integral function f(x) is divided by x + a, is f{- a). For example, the remainder, when a^-Z3? + '23?-5x + Q is divided hy K + IO, is (-10)*-3(-10)3 + 2(-10)3-5(-10) + 6 = 13266. Cor. 2. The remainder, when an integral function of x, fix), is divided by ax + b, is /( - bja). This is simply the generalisation of Example 3, § 13, above. By substitution we may considerably extend the application of the remainder theorem, as the following example will show : — Consider pmit'^)'" +Pm-i{<'i^)'^~^ + . . . +Pi{x'') +^o and af - a". Writing for a moment J in place of x", and u. in place of a", we have to deal with Pm^+Pin-i^~^+ ■ . • +Pil+i'o and i-a. Now the remainder, when the former of these is divided by the latter, is jJina™ H-^m-ia™"^ + . . . +pia+po. Hence the remainder, when pn,(ai^)'" +^,„_i(a;")"'-' + . . . +pi!>i" +po is divided by a;" - a", is ^„(a'')"' +^„_i(a")"-i + . . . +pia"+p(i. APPLICATION OP REMAINDER THEOREM TO THE DECOMPOSITION OF AN INTEGRAL FUNCTION INTO LINEAR FACTORS. § 15.] i/ a„ U2, . . . , Or be r different values of x, fm which the integral function of the nth degree f(x) vanishes, where r < n, then f(x) = (x- aj (k - ttj) . . . (a; - n-ri^) being an integral function of x of the (n - r)th degree. For, since the remainder, /(a,), when f{x) is divided by x - a„ vanishes, therefore f{x) is exactly divisible by a; - a^, and we have f{x) = {x-a.,)<^n.lx), where 4>n-i{^) is an integral function of x of the (n - l)th degree. Since this equation subsists for all values of x, we have that is, by hypothesis, = (as - a,)(^„_i(a2). Now, since aj and a^ are different by hypothesis, a^ - a; =t= ; therefore 4>n-i{°-2l = 0- Hence, 4>n-i{^) is divisible by {x~ a^), that is, <^n-i(a:) = (a;-a2)<^n-2(a;); whence f{x) = (x- a^) (x - a^)n-2{x). VOL. I H 98 FACTORISATION BY MEANS OF CHAP. From this again, which gives, since a,, a^, a^ are all unequal, <^„-2(a3) = ; whence n-s{^) = (« - a3)^„-3(a;) ; so that f{x) = {X- a,) {x - a,) {x - a^)(j>n-s{^)- Proceeding in this way step by step, we finally establish the theorem for any number of factors not exceeding n. Cor. 1 . If an integral function he divisible by the factors x - a^, X - a„ . . ., x-Oy, all of the \st degree, and all different, it is divisible hy their product ; and, conversely, if it is divisible by the product of any number of such factors, all of the \st degree and all different, it is divisible by each of them separately. The proof of this will form a good exercise in algebraical logic. Cor. 2. The particular case of the above theorem where the number of factors is equal to the degree of the function is of special interest. We have then f{x) = {x- a,) (x-a^) .. .{x- a„)P. Here P is of zero degree, that is, is a constant. To determine it we have only to compare the coefficients of x" on the left and right hand sides, which must be equal by chap, iv., § 24. Now /(a;) stands for _poa;" +pia''"i + . . . +^„_,a;+^„. Hence P=^o) and we have f{x) =p,{x - a,) (x-a,). ..{x- an). In other words — If n different values of x can be found for which the integral function f{x) vanishes, then f(x) can be resolved into n factors of the 1st degree, all different. The student must observe the "if" here. We have not shown that n such particular values of x can always be found, or how they can be found, but only that if they can be found the factorisation may be effected. The question as to the finding of ii„ a^, . . . , &c., belongs to the Theory of Equations, into which we are not yet prepared to enter. § 16.] The student who has followed the above theory will naturally put to himself the question, " Can more than n values V THE REMAINDER THEOREM 99 of X be found for which an integral function of x of the mth degree vanishes, and, if so, what then ? " The following theorem will answer this question, and complete the general theory of factorisa- tion so far as we can now follow it. If an integral function of x of the nth degree vanish for nwre than . n different values of x, it must vanish identically, that is, each of its coefficients must vanish. Let a,, a,, . . . , a,i be w of the values for which f{x) vanishes, then by § 15 above, if Po be the coefficient of the highest power of X in /(«), we have /(«) =^^0(3= - ai) {x-a,).. .{x- a„) (1). Now let /? be another, value (since there are more than n) for which /(a) vanishes, then, since (1) is true for all values of x, we have =/(/?) =MI3 - «.) (/? - a,) . . . (^ - a„) (2). Since, by hypothesis, a„ uj, . . . , a„ and /3 are all diiferent, none of the differences y8 - «!, P - o.2> ■ ■ ■ , P - n, be equal in value for more than in different values of x, a fortiori, if they he equal for all values of x, that is to say, identically equal, then the coefficients of like powers of x in the two must be equal. We may, without loss of generality, suppose the two functions to be each of degree m, for, if they be not equal in degree, this simply means that the coefficients of a^+i, a;"+^, . . . , x™ in one of them are zero. We have therefore, by hypothesis, j?„a;™+j7,a;'"-i+ . . + p,n = q {)i + 2)a^+i-(»H-l)3;"+2 TT -, = l + 2a; + 3a:2 + . . .+(n + l)x'^ + - '—- \n (1-a;)-' [l-xY (14). EXPRESSION OF ONE INTEGRAL FUNCTION IN POWERS OF ANOTHER. § 21. J We shall have occasion in a later chapter to use two particular cases of the following theorem. If P and Q be integral functions of the ruth and nth degrees respectively (ni > n), then P may always he put into the form P = E„ + E,Q + E,Q^ + . . . + E^QP (1), where E„, E, , . . ., Ej, are integral functions, the degree of each of which is n - I at most, and p is a positive integer, which cannot exceed mjn. Proof. — Divide P by Q, and let the quotient he Qj and the remainder E„. If the degree of (^o be greater than that of Q, divide Q„ by Q, and let the quotient be Q, and the remainder Ej. Next divide Qi by Q, and let the quotient be Q^ and the remainder Eg, and so on, until a quotient Q,j,_i is reached whose degree is less than the degree of Q. Qj,-i, for con- venience, we call also Ep. We thus have P = Q„Q + E„ Qo = Q,Q + E, 106 EXPANSION THEOREMS chap. Xow, using in the first of these the value of Q„ given by the second, we obtain P=(Q.Q + E.)Q + R„, = E„ + E,Q + Q,Q^ Using the value of Q, given by the third, we obtain P = R„ + E,Q + E,Q^ + Q,Q^ (3). And so on. "We thus obtain finally the required result; for, E„, E„ . . ., Ej, being remainders after divisions by Q (whose degree is n), none of these can be of higher degree than n - 1 ; moreover, since the degrees of Q„, Q,, Qo, . . ., Qj,_i are m-n, m-2n, m - 3«, . . ., m- np, p cannot exceed to/?i. The two most important particular cases are those in which Q = a - a and (l=3? + ^x + y. We then have P = ao + ai(3;-a) + . . .+a„(a;-a)", where ao» ^u • • • > ^« ^^® constants ; T = {ao + box) + {ai + iix){x- + ^x + y) + . . +{aj, + bj^){x^ + ^x + y)P, where ao, ai, . ., ttj,, ho, 5i, . . , bp are constants, and p )> in/2. Example 1. Let P = 5r'-lU~ + 10«-2, The calculation of the successive remainders proceeds as follows (see § 13) :- 5 -11 +10 -2 0+5-6+4 5 - 6 + i\ + 2 + S - 1 5 - 1| + 3 + 5 5|+ i ITS; and we find 52?-llir=+10a;-2 = 2 + 3(£c-l) + 4(K-lf + 5(a;- If. Example 2. F=afi + 3x' + ix'' + ix^ + 3x + l, Q=x'-x + l. The student will find E<,=llic, Ri=-22:(; + 7, E2 = 19a;-22, K3 = 7x + 15, R4=l ; V EXPANSION THEOREMS 107 so that F = nx + (-22x + '!){x^-x + l) + (lSx-22){x'-x + l)'' + I7x+15)(x'-x+lf + {x^-x + l)\ § 22.] If a^, a„, . . ., a^ he n constants, any two or more of which may he equal, then any integral function of x of the nth degree may he put into the form A„ + A,(a; - a,) + KJ^x - a^ (x — a^ + KJ^x - a^ (x - a^) (x- a^) + . . . + An(x - a,) (x-a^). . .{x- a„) (1 ), where A,,, Ai, A^, ■ . ., A^ are constants, any one of which except' A^ may be zero. Let Pa be the given integral function, then, dividing P^ by a; - a, , we have P« = P«-.(«-ai) + A„ (2), where A„ is the constant remainder, which may of course in any particular case be zero. Next, dividing P„_i by (x - a^), we have Pm- 1 = P)i-2(3i ~ a^) + A, (o) ; and so on. Finally, P, = AJx - «„) + A,^ _ , (» + 1 ) . Using these equations, we get successively P„ = A„ + AfyX - a,) + (a; - a^{x - a^Y^-^, = A„ + A,(a; - a,) + AJ^x - a,) (x - a^ + (a; - a,) (a; - a^ (x - a^V^-^, P„ = Ao + Ai(a; - a^ + A„(a - a^ (x - a^) + A^(x - a,) (a; - a^ (x - a,) + . . . + A,i(a; - a,) (x-a^) . . . (x- a^). This kind of expression for an integral function is often useful in practice. Knowing a priori that the expansion is possible, we can, if we choose, determine the coefEcients by giving particular values to x. But the most rapid process in general is simply to carry out the divisions indicated in the proof, exactly as in Example 1 of last paragraph. Thus, to express a^-1 in the form Ao + Ai(a;-l) + A2(a;- 1) (ii:-2) + Asix - 1) (x - 2) (a; - 3), we calculate as follows : — 108 EXERCISES IX '1+0+0 -1 1 0+1+1 +1 1 +1 +i|o +2 +6 3 1 +3| + 7 +3 l|+6 Hencea^-l=0 + 7(^-l) + 6(a;-l)(a:-2) + (a;-l)(a;-2)(x-3). Exercises IX. Transform the follomng quotients, findin, remainder where the quotient is fractionah («= - 5^ + 5.r= - l)l{x'' + 3x + l). (6a;« + 2r- - 19^* + Slar' - 37»= + 27a; - 7)/(2d' ( 4.>;'' - 2x' + 3a?-x+1)l{x^-2x + 1). {.c--Sx + \5)(rc- + 8x + 15)l{x^-25). {[x-l){x-2]{x-3)(x-i)(x-b)-760{x- both integral quotient and (1- (2. (3. (4. (5. U-7). (6. (7. (8. (9-: (10. (11. (12. (13. (14. (15. (16. (17. (18. (19. (20. (21. (22. (23. (24. (25. (26. (27. (28. (29. (30. -3x + l). 6) + 120(a;-7)}H-(a!-6) {x' + ia~'-3x*-16T^ + 2x'' + x + S)l[a^ + 4x- + 2x + l). {27x^ + 10x^ + l)l(3x^ -2x + l). {.Tp - 9x- + 23a; - 15) (a; - 7)/(a;2 - 8a; + 7). (x' + lix> + ^^x^ + i\x + i)/(a;2 - ^a; + 1). (a;* + \x^ + ia;= + ^a; + i)/(a;2 + 2a; + 1 ). (a;' + Ti^)/(2a; + l). (a;2 - a; + 1) (a^ - l)/(,'>;* + x2 + 1). [xy'-if - afii/'')l(x - y). (9aJ + 2crU'- + b^)l(3a? + 2ah + J-). (a7 + b')l(a + h). (.«* + ?/ - 1xhf)l{x'' + 3xy + y-). [x' - 2x*y + 4a;y - 8x-y' + 16xy* - 32t/)l{a? - 8j/'). (a:-* + .^.r^// + 7x-y- + ISa-j/" + 12y*)l(x + iy). (l+a; + ai^ + a;^ + a:* + a;'' + a:'' + a;' + a;' + a;i'')/(l -a;^ + a;''). (x'^-Zj-? + ?,)I(x''- + x + 2). {aba? + {ac - bd)x? - {af+ cd)x + df] /(ax - d). { a-b^' + aV + bV + 2jbc - 2ab\ - 2abc^} -^ {a= - (a - 6) (a - c) } . {1 + b + c-bc-b-c- bc-)/{l - be). {{ax + bijf + {ax-by)^-(mj-bx)^ + [ay + bxf} / {{a + bfx^-3ab{a?-y^)} {{a? + iy + bV}l{{a + bf-ba}. { [a? + xy + y-Y - [j" -xy + y"^)^} / { a;* + 3.ry + y^}. {(x + yy -x^ -y''}l(x? + xy + yY- {(x + \f-x^-\]l{3? + x + l}. {ab(x? + y"-)+xy(a^ + b'^)]j{ab(x^-y^)-xy(a'-b'')}. (a^ + 2aV + 2aV - 3¥)l{a' - 2ab + b\ V EXERCISES IX 109 (31.) {x*-Sx''~2x + i]l{x + 2). (32.) (a:*-4aJ'-34a^ + 76a! + 105)/(a!-7). (33.) Find the remainder when k' - 62:^ + 8a; - 9 is divided by 2x + 3. (34.) Find tlie remainder when ^k^^- 30:2 + ja:+^ is divided by aj- 1 ; and find the condition that the function in question be exactly divisible by a;^ - 1. (35.) Find the condition that Ax^'" + Ba""!/" + Ci/^" be exactly divisible by Pa;™ + Q3/". (36.) Find the conditions that a!^ + na:^ + 6a: + c be exactly divisible by a? +px + q. (37. ) If a; - a be a factor of x^ + 2ax - 36^, then a= ±6. (38.) Determine X, /j,, j, in order that 3^+?>^-V\x- + ia + v be exactly divisible by (a;'^-l)(a: + 2). (39.) If sf^ + i3^ + 6px'^ + iqx + r be exactly divisible by x' + 3a;2 + 9cc + 3, find^, q, r. (40.) Show that pa? + {p'^ + q)x^ + {2pq + r)x + q^ + s and pa? + {p'-q)x- + rx-q^ + s either both are, or both are not, exactly divisible by x" +px + q. (41.) Find the condition that (a;™ + a:™-^ + . . . + 1 )/(;£" + a;"-! + . . . +1) be integral. (42.) Expand l/(3a; + l) in a series of ascending, and also in a. series of descending, powers of x ; and find in each case the residue after n + 1 terms. (43.) Express 1/(0^ - aa; + a:^) ;„ ^-j^g fop^ A + 'Bx+0a? + 'D3? + 'R, where A, B, C, D are constants and R a certain rational function of x. (44.) Divide l+a; + ^ + ^--2-3 + . . .byl-a:. (45.) Show that, if t/-=l, then approximately 1/(1 + ;/) = 1 - y, 1/(1-2/) = l + j/, the eiTor in each case being lOOj/^ per cent. Find similar approximations for 1/(1+2/)" ^"'1 1/(1 -Z/)"; where » is a positive integer. (46.) If a>l, show that «">l + m(a-l), n being a positive integer. Hence, show that when n is increased without limit a" beconies infinitely great or infinitely small according as a> or <1. (47. ) Show that when an integral function f(x) is divided by (x - ai) (a; -02) the remainder is {/(a2)(a;-ai) -/(ai)(a;-a2)}/(o2-ai). Generalise this theorem. (48.) Show that f(x)-f{a) is exactly divisible by x-a; and that, if /(a:)=^o«"+i'ia?'~'+P2a:""^+. . . +Pn, then the quotient is x(») =i'oa?'~' + (poa+pi)3^-'^ + (paa?+Pi.a.+p.i)ai^-^+ . . . + (^Joa""^ +i'ici"-^ + . . +p„-i). Hence show that when f(x) is divided by (x - of the remainder is X(a)(a;-a)+/(a), where /(a) =^oa" +ft"''"^ + • • ■ +P-n< X(o) = wpoa"-^ + (»-l)i'ia"~^+- ■ ■ +Pn-i- (49.) If K"+i)ia?'-i+. . . +pn and x"-^ + qi3i^-' + . . . +qn-i have the same linear factors with the exception of a: - a, which is a factor in the first only, find the relations connecting the coefiicients of the two functions. (50.) If, when y + c is substituted for x in x" + OiXf^'^ + . . .+a„, the 110 EXERCISES IX OiiAP. V result is i/" + &ii/"-' + . . . +J„, show that 6„, 6„_i, . ., Jx are the remainders when the original function is divided byx-c, and the successive quotients hy x-e. Hence obtain the result of substituting j/ + 3 for x in x^-lSar* + 20^3 -17a- -a; + 3. (51.) Express {x^ + 3a; + l)'' in the form A + B(a; + 2) + C(a; + 2)H&c., and also in the form Ax + 'B + {Cx + 'D){3? + x + l) + {'Ex + 'F){x^ + x+l)- + kc. (52.) Express a^ + a::* + .i" + a; + l in theform Ao + Ai(a; + 1) + A2(x + l)(x + 3) + A3{x + l){x + 3){x + 5) + Ai{x + l){x + S){x + 5){x+7). (53.) If, when P and P' are divided by D, the remainders are R and R', show that, when PP' and RR' are divided by D, the remainders are identical. (54.) When P is divided by D the remainder is R ; and- when the integral quotient obtained in this division is divided by D' the remainder is S and the integral quotient Q. R', S', Q' are the corresponding functions obtained by first dividing by T>' and then by D. Show that Q = Q', and that each is the integral quotient when P is divided by DD'; also that SD + R = S'D' + R', and that each of these is the remainder when P is divided by DD'. CHAPTEK VI. Greatest Common Measure and Least Common Multiple. § l.J Having seen how to test whether one given integral function is exactly divisible by another, and seen how in certain cases to find the divisors of a given integral function, we are naturally led to consider the problem — Given two integral functions, to find whether they have any common divisor or not. We are thus led to lay down the following definitions : — ■ Any integral function of x which divides exactly two or more given integral functions of x is called a common measure of these functions. The integral function of highest degree in x which divides exactly each of two or more given integral functions of x is called the greatest common measure (G.C.M.) of these functions. § 2.] A more general definition might be given by suppos- ing that there are any number of variables, x, y, z, u, &c. ; in that case the functions must all be integral in x, y, z, u, &c., and the degree must be reckoned by taking all these variables into account. This definition is, however, of comparatively little importance, as it has been applied in practice only to the case of monomial functions, and even there it is not indispensable. As it has been mentioned, however, we may as well exemplify its use before dismissing it altogether. Let the monomials be i32a%Vy*z, 270a'iVyV, 90a'bx'yV, the variables being x, y, z, then the G.C.M. is x'y\ or Gsfy^ where C is a constant coefficient (that is, does not depend on the variables x, y, z). The general rule, of which the above is a particular case, is as follows : — 112 G.C.M. BY INSPECTION CHAP. The G.C.M. of any number of monomials is the product of the variables, each raised to the lowest power * in tvhich it occurs in any one of the given functions. This product may of course be multiplied by any constant coefficient. G.C.M. OBTAINED BY INSPECTION. § 3.] Eeturning to the practically important case of integral functions of one variable .r, let us consider the case of a number of integral functions, P, P', P", &c., each of which has been re- solved into a product of positive integral powers of certain factors of the 1 st degree, say x - a, x - /3, x - y, &c. ; so that P = p{x - aY(x - /3f(x -y)'..., ¥'=p-(,:-ar{x-/3)'"ix-yr..., -p"^p"{x^ar"{x-fir(x-yr..., By § 15 of chap, v., we know that every measure of P can contain only powers of those factors of the 1st degree that occur in P, and can contain none of those factors in a higher power than that in which it occurs in P, and the same is true for P', P", &c. Hence every common measure of P, P', P", &c., can contain only such factors as are common to P, P', P", &c. Hence the greatest common measure of P, P', P", cfec, contains simply all the factoi's that are common to P, P', P", (&c., each raised to the lowest jyower iit which it occurs in any one of these functions. Since mere numbers or constant letters have nothing to do with questions relating to the integrality or degree of algebraical functions, the G.C.jVI. given by the above rule may of course be multiplied by any numerical or constant coefficient. Example 1. F=2x^-ej:r+ 4 = 2(a:-l)(a;-2), P' = 6»2-6.e-12 = 6(a; + l)(a;-2). Hence the G.C.M. of P and P' is k- 2. Example 2. P =x^-53i^+7x' + x'-8x + i = {x-l]''{x + l)(x-2Y', P' =a«-7a^ + I7x*-I3ai3-l0a;2 + 20a;-8 = (a;-l)=(a; + l)(a;-2)=, V" = a?-Sx^-s? + 7x--i = {x-l)(x+l)-{x-2)\ The G.C.M. is (ai-l) (a;+l)(a;-2)'-, that is, ar* - 42^ + 3a;- + 4x - 4. * If any variable does not occur at all in one or more of the given func- tions, it must of course be omitted altogether in the G.C.M, VI CONTKAST BETWEEN ALGEBRAICAL & ARITHMETICAL G.C.M. 113 § 4.] It will be well at this stage to caution the student against being misled by the analogy between the algebraical and the arithmetical G.C.M. He should notice that no mention is made of arithmetical magnitude in the definition of the algebraical G-.C.M. The word "greatest" used in that definition refers merely to degree. It is not even true that the arithmetical G.C.M. of the two numerical values of two given functions of x, obtained by giving x any particular .value, is the arithmetical value of the algebraical G.C.M. when that particular value of x is substituted therein ; nor is it possible to frame any definition of the algebraical G.C.M. so that this shall be true.* The, student will best satisfy himself of the truth of this remark by study- ing the following example : — The algebraical G.C.M. of a!^-3a!+2 and o?-x-1 is x-l. . Now put a;=31. The numerical values ofthe two functions are 870 and 928 respectively ; the numerical value of a; - 2 is 29 ; but the arithmetical G. C. M. of 870 and 928 is not 29 but 58. LONG RULE FOR G.C.M. § 5.] In chap. v. we have seen that in certain cases in- tegral functions can be resolved into factors ; but no general method for accomplishing this resolution exists apart from the theory of equations. Accordingly the method given in § 3 for finding' the G.C.M. of two integral functions is not one of perfectly general application. The problem admits, however, of an elementary solution by a method which is fundamental in many branches of algebra. This solution rests on the following proposition : — If K- BQ + E, A, B, Q, R heing all integral functions of x, then the G.C.M. of A and B is the same as the G.C.M. of B and E. To prove this we have to show — 1st, that every common * To avoid this confusion some writers on algebra have used instead of the words "greatest common measure" the term "highest common factor." "We have adhered to the time-honoured nomenclature because the innovation in this case would only be a partial reform. The very y/OTi factor itself is used in totally different senses in algebra and in arithmetic ; and the same is true ofthe yiot:As fractional and integral, with regard to which confusion is no less common. As no one seriously proposes to alter the whole of the terminology of the four species in algebra, it seems scarcely worth the while to disturb an old friend like the G.C.M. VOL. I ■ I 114 LONG RULE FOK G.C.M. chap. divisor of B and E divides A and B, and, 2nd, that every common divisor of A and B divides B and E. 'Now, since A = BQ + E, it follows, by § 4 of chap, v., that every common divisor of B and E divides A, hence every common divisor of B and E divides A and B. Again, E = A - BQ, hence every common divisor of A and B divides E, hence every common divisor of A and B divides B and E. Let noio A aTid B he two integral functions whose G.C.M. is required; and let B he the one whose degree is not greater than that of the other. Divide A hy B, and let the quotient he Qi, and the remainder Ei. Divide B iy E,, and let the quotient he Qj, and the remainder E^. Divide El hy B..,, and let the quotient he Q3, and the remainder E3, and so on. Since the degree of each remainder is less hy unity at least than the degree of the corresponding divisor, Ej, E2, E3, <&c., go on diminishing in degree, and the process must come to an end in one or other of two ways. I. Either the division at a certain stage hecomes exact, and the remainder vanishes ; II. Or a stage is reached at which the remainder is reduced to a constant. Now we have, by the process of derivation above described, A =BQ, +E, B = E,Q, + E, E, = E,Q3 + E3 Ul). Hence by the fundamental proposition the pairs of functions ^ 12 15' ] S'l ■ • • 5"'' 15"" lall have the same G.C.M. B j Ej J Ej J E3 J E„_i J K„ J In Case I. E„ = and E,i_2 = QnE„_,. Hence the G.C.M. of E,j_2 and E„_„ that is, of Q„E„_, and E^-,, is E„_,, for this divides both, and no function of higher degree than itself can divide E„_i. Hence E„_, is the G.C.M. of A and B. In Case II. E,t = constant. In this case A and B have no G.C.M., for their G.C.M. is the G.C.M. of E„-, and E„, that is, their G.C.M. divides the constant E„. But no integral function VI MODIFICATIONS OF LONG RULE 115 (other than a constant) can divide a constant exactly. Hence A and B have no G.C.M. (other than a constant). If, therefore, the process ends with a zero remainder, the last divism- is the G.C.M.; if it ends with a constant, there is no G.C.M. § 6.] It is important to remark that it follows from the nature of the above process for finding the G.C.M., which con- sists essentially in substituting for the original pair of functions pair after pair of others which have the same G.C.M., that we may, at any stage of the process, multiply either the divism' or the remainder by an integral function, p-ovided loe are sure that this function and the remainder or divisor, as the case may be, have no common factor. We may similarly remove from either the divisor or the remainder a factor which is not common to both. We may remove a factor which is common to both, provided we introduce it into the G. C. M. as ultimately found. It follows of course, a fortiori, tluit a numerical factor may be introduced into or removed from divisor or remainder at any stage of the process. This last remark is of great use in enabling us to avoid fractions and otherwise simplify the arithmetic of the pro- cess. In order to obtain the full advantage of it, the student should notice that, in what has been said, "remainder"' may mean, not only the remainder properly so called at the end of each sepa- rate division, but also, if we please, the " remainder in the middle of any such division,'' or "residue," as we called it in § 18, chap. v. Some of these remarks are illustrated in the following examples : — Example 1. To find the G.C.M. ois?-'i3^-2y? + ix^-1x + 2 and rr^ - 4a; + 3. af-23d^--2K?+ &x^- Ix + i I x*~ix + Z X +\ X +1 3? - ix^ + 3a! 1 !) -2rt''- -2a? + 12x2- lOx + 2 f s?- &x^ + 52:-! ix + ^ x^ 7?- 6a;2 + te-4 4a; + 3 4a! 3) Qa?- gx^i -1-3 3a:2 -Hi lix' + Wx-i 9) 9x2- 18a! -1-9 231+1 116 EXAMPLES CHAP. e,-iP + 9,1- - 2x''+ X - 4) -4a^ + 8:e - 4 X +1 Hence the G. C. M. is ci~ - It must be observed that what we have written in the place of quotients are not really quotients in the ordinary sense, owing to the rejection of the numerical factors here and there. In point of fact the quotients are of no importance in the process, and need not be written down ; neglecting them, carrying out the subtractions mentally, and using detached coefficients, we may write the whole calculation in the following compact form : — -e-3 ^9 1 2 -2-2+ 8- 7+2 -2-2 + 12-10 + 2 1 + 1- 6+ 5-1 1- 6+ 9-4 1+0+0- 4+3 6-9+ 0+3 2-3+ 0+1 9-18 + 9 4 - 4+ 8-4 1- 2 + 1 1- 2 + 1 ! G.C.M., *2_ 2a; +1. Example 2. Eequired the G.C.M. -8a; -32. of ix^ + 26x> + ili---2x-2i and S.t" + 20ar'' + 32x2 Bearing in mind the general principle on which the rule for finding the G.C.M. is founded, we may proceed as follows, in order to avoid large num- hers as much as possible : — -53 The G.C.M. isa^ + eai + S. Here the second line on the left is obtained from the first by subtracting the first on the right. By the general principle referred to, the function a;* + 6ar' + 9a;2 + 6ir + 8 thus obtained and 3a;* + 20^?* + 32a;2 - 8a; - 32 have the same G.C.M. as the original pair. Similarly the fifth line on the left is the result of subtracting from the line above three times the second line on the right. 4 + 26 + 41- 2- 24 x2 1+ 6+ 9+ 6+, 8 3 + 20 + 32- 8-32 2+ 5- 26- 56 2 + 12 + 18+ 12+ 16 7 + 44+ 68+ 16 1 + 29 + 146 + 184 -53-318-424 1+ 6+ 8 -h23 23 + 138 + 184 1+ 6+ 8 VI SECOND RULE FOE G.C.M. 117 If the student be careful to pay more attention to the prin- ciple underlying the rule than to the mere mechanical application of it, he will have little difficulty in devising other modifications of it to suit particular cases. METHOD OF ALTERNATE DESTRUCTION OF HIGHEST AND LOWEST TEEMS. § 7.] If I, m, p, q he constant quantities {such that Iq - mp is iwt zero), and if P = ZA + mB (1), Q, = pA + qB (2), where A and B, and therefore P and Q, are integral functions, then the G.C.M. of P and Q is tlie O.G.M. of A and B. For it is clear from the equations as they stand that every divisor of A and B divides both P and Q. Again, we have gP - mQ = q(JA. + toB) - m {pA. + gB) = (Iq - mp)A. (3), -pP + iq= -p{lA + mB) + l{pA + qB) = (Iq - mp)B (4) ; hence (provided Iq - mp does not vanish), since I, p, m, q, and therefore Iq - mp, are all constant, it follows that every divisor of P and Q divides A and B. Thus the proposition is proved. In practice I and m and p and q are so chosen that the highest term shall disappear in ZA + mB, and the lowest in pA + qB. The process will be easily understood from the follow- ing example : — Example 1. Let A=4x* + 26«= + 41a;2-2rr-24, ' B = Sx* + 20z;3 + 32x2 - Sk - 32 ; then -3A + 4B = 2r' + 5a;2-26x-56, 4X-3B = 7x^ + ii3? + e8ix? + 16x. Rejecting now the factor x, which clearly forms no part of the G.C.M., we have to find the G.C.M. of A' = 7!B' + 44a2 + 68K + 16, B' = 2a? + 5z^-'26x-56. Repeating the above process — 2A'-7B' = 53K= + 318a;+424, 7A' + 2B' = 5Sa? + 31 8a;2 + 424a;, 118 TENTATIVE PROCESSES ohap. the G.C.M. of which is 53a^ + 318a; + 424. Hence this, or, what is equivalent so far as the present quest is concerned, afi + 6x + 8, is the G.C.M. of the two given functions. When the functions differ in degree, we may multiply the one of lower degree by such a power of x as shall make its degree equal to that of the other, and use it thus modified in order to destroy the two highest terms, using it in its original form when the two lower terms have to be destroyed. When detached coefficients are employed, this merely amounts to shifting the coefficients one or more places to the right or left. For example : Example 2. To find the G. C. .M . of K^ - Sa?* + 2.>- + a; - 1 and k' - x? - 2.(- + 2, we have the following calculation : — 1-3+2+ 1-1 l-lj-2 4^2_ 2-4+ 1+1 2-5+ 3+0 A B A' = - A + Ba; B' =_ 2A +B A" A' -B' B" = 3A' X A"'= 6A"-?," B'" = 8A" - B" The G.C.M. is 2,.' -2, ora;-l. 1- 2 + 1 6-14 + 8 2-2 2-2 There is a certain restriction to be attended to here, which the student will readily discover by going over the theory again, with the necessary modi- fications introduced. The failing case of the original process, where lq-inp = 0, may be treated in a similar manner, the exact details of which we leave to be worked out as an exercise by the learner. § 8.] The following example shows how, by a semi-tentative process, the desired result may often be obtained very quickly : — Example. B = 2a;' - jf' - ^3? + 4a: + 4. Every common divisor of A and B divides A - B, that is, - 2x^ + G.'T - 4.f, that is, rejecting the numerical factor - 2, x{x' - 3x + 2), that is, x{x- 1) (a; - 2). We have therefore merely to select those factors of x(x - l){x~ 2) which divide both A VI PROPOSITIONS REGARDING G.C.M. 119 and B. x clearly is not a common divisor, but we see at once, by the remainder theorem (§13, chap, v.), that both x-l and X- 2 are common divisors. Hence the G.C.M. is (x - l)(x- 2), or x' -dx+ 2. § 9. J The student should observe that the process for finding the G.C.M. has the valuable peculiarity not only of furnishing the G.C.M., but also of indicating when there is none. Example. B=sP-ix + 6. Arranging the calculation in the abridged form, we have 1-3 + 1 2 + 1 1-4 + 6 -1 + 6 11 The last remainder being 11, it follows that there is no G.C.M. 6.C.M. OF ANY NUMBER OF INTEGRAL FUNCTIONS. § 10.] It follows at once, by the method of proof given in § 5, that every common divisor of two integral functions A and B is a divisor of their G. CM. This principle enables us at once to find the G.C.M. of any number of integral functions by successive application of the process for two. Consider, for example, four functions. A, B, C, D. Let Gi be the G.C.M. of A and B, then Gi is divisible by every common divisor of A and B. Find now the G.C.M. of G, and C, Gj say. Then G„ is the divisor of highest degree that will divide A, B, and C. Finally, find the G.C.M. of G^ and D, G3 say. Then G3 is the G.C.M. of A, B, C, and D. GENERAL PROPOSITIONS REGARDING ALGEBRAICAL PRIMENESS. § 11. J We now proceed to establish a number of propositions for integral functions analogous to those given for integral numbers in chap, iii., again warning the student that he must not confound the algebraical with the arithmetical results ; 120 PROPOSITIONS REGARDING G.C.M. OHAV. although he should allow the analogy to lead him in seeking for the analogous propositions, and in devising methods for proving them. Definition. — Two integral functions are said to he prime to each other when the// have no common divism: Proposition. — A and B being any two integral functions, there exist always two integral functions, L andM., prime to each other, such that, if A andB have a G.C.M. , G, then LA + MB = G ; and. If A and B he prime to each other, LA + MB = 1. To prove this, we show that any one of the remainders in the process for finding the G.C.M. of A and B may be put into the form PA + QB, where P and Q are integral functions of x. We have, from the equalities of § 5, R, = A-Q,B (1), R, = B-Q,R, (2), E3 = Pw ~ Q;,R, (3), R„ = R„_,-Q„K„_, (4). Equation (1) at once establishes the result for R, (only here P = 1, Q = - Q,). From (2), using the value of Ri given by (1), K = B - Q,(A - Q,B) = ( - Q,)A + ( + 1 + Q,Q,)B, which establishes the result for R2. From (3), using the results already obtained, we get R3 = A - Q,B - Q3{( - Q,)A + ( + 1 + Q,Q,)B | = (1 + y,Q,)A + ( - Q, - Q,3 - Q,QA)B, which establishes the result for R3, since Qi, Q^, Q,, are all in- tegral functions. Similarly we establish the result for R^, R^, &c. Now, if A and B have a G.C.M., this is the last remainder which does not vanish, and therefore we must have G = LA + MB (I.), VI ALGEBRAIC PEIMENESS 121 where L and M are integral functions ; and these must be prime to each other, for, since G divides both A and B, A/G ( = a say) and B/G ( = J say) are integral functions ; we have therefore, dividing both sides of (I.) by G, 1 = La + Mi ; so that any common divisor of L and M would divide unity. If A and B have no G.C.M., the last remainder, E„, is a constant ; and we have, say, E„ = L'A + M'B, where L' and M' are integral functions. Dividing both sides by the constant E,,,, and putting L = L'/E„, M = M'/E„, so -that L and M are still integral functions, we have 1 = LA + MB (II.). Here again it is obvious that L and M have no common divisor, for such divisor, if it existed, would divide unity. The proposition just proved is of considerable importance in algebraical analysis. We proceed to deduce from it several con- clusions, the independent proof of which, by methods more analogous to those of chap, iii., § 10, we leave as an exercise to the learner. Unless the contrary is stated, all the letters used denote integral functions of x. § 12.] If Khe prime to B, then any common divisor of AH and B must divide H. For, since A is prime to B, we have LA + MB = 1, whence LAH + MBH = H, which shows that any common divisor of AH and B divides H. If A and B have a G.C.M. a somewhat different proposition may be established by the help of equation (I.) of § 11. The discovery and proof of this may be left to the reader. Cor. 1. IfR divide AH and be prime to A, it must divide H. Cor. 2. If A' be prime to each of the functions A, B, C, tfec, it is prime to their product ABC . . . 122 ALGEBRAIC PEIMENESS Cor. 3. If each of the functions A, B, C, . . . he prime to each of the functions A', B', C, . . . , then the p-oduct ABC ... is prime to the product A'B'C. . . . Cor. 4. If A he prim.e to A', then A" is prime to A'"', a and a heing any positive integers. Cor. 5. If a given set of integral functions he each resolved into a product of powers of tlie integral factors A, B, C, . . . , which are prime to each other, then the G.G.M. of the set is found by witing doiun the product of all the factors that are common to all the given functions, each raised to the lowest power in which it occurs in any of these functions. This is a generalisation of § 3 above. After what has been done it seems unnecessary to add de- tailed proofs of these corollaries. LEAST, COMMON MULTIPLE. § 13.] Closely allied to the problem of finding the G.C.M. of a set of integral functions is the problem of finding the integral function of least degree which is divisihle hy each of them. This function is called their least common multiple (L.G.M.). § 14.] As in the case of the G.C.M., the degree may, if we please, be reckoned in terms of more variables than one ; thus the L.C.M. of the monomials Zx'ys?, QxS/z', Sxyzu, the variables being x, y, z, v., is x^y^z'u, or any constant multiple thereof. The general rule clearly is to write down all the variables, each raised to the highest power in which it occurs in any of the mono- mials. § 15.] Confining ourselves to the case of integral functions of a single variable x, let us investigate what are the essential factors of every common multiple of two given integral functions A and B. Let G be the G.C.M. of A and B (if they be prime to each other we may put G = 1) ; then A = aG, B = JG, where a and b are two integral functions which are prime to each VI LEAST COMMON MULTIPLE 123 Other. Let M be any common multiple of A and B. Since M is divisible by A, we must have M = PA, where P is an integral function of x. Therefore M = PaG. Again, since M is divisible by B, that is, by JG, therefore M/6G, that is, PaG/JG, that is, Vajh must be an integral function. Now h is prime to a; hence, by § 12, 5 must divide P, that is, P = Q6, where Q is integral. Hence finally M = QaJG. This is the general form of all common multiples of A and B. Now a, b, G are given, and the part which is arbitrary is the integral function Q. Hence we get the least common multiple by making the degree of Q as small as possible, that is, by making Q any constant, unity say. The L.C.M. of A and B is therefore abG, or (iiG) (6G)/G, that is, AB/G. In other words, the L.C.M. of two integral functions is their product divided by their 6.C.M. § 16.] The above reasoning also shows that every common multiple of two integral functions is a multiple of their least common The converse proposition, that every multiple of the L.C.M. is a common multiple of the two functions, is of course obvious. These principles enable us to find the L.C.M. of a set of any number of integral functions A, B, C, D, &c. For, if we find the L.C.M., Li say, of A and B; then the L.C.M., L^ say, of L^ and C; then the L.C.M., L3 say, of L^ and D, and so on, until all the functions are exhausted, it follows that the last L.C.M. thus obtained is the L.C.M. of the set. § 17.] The process of finding the L.C.M. has neither the theoretical nor the practical importance of that for finding the G.C.M. In the few cases where the student has to solve the problem he will probably be able to use the following more direct process, the foundation of which will be obvious after what has been already said. If a set of integral functions can all be exhibited as powers of a 124 EXERCISES X CHAP. set of integral factors A, B, C, d:c., which arc either all of the 1st degree and all different, or else are all pime to each other, then the L.C.M. of the set is the product of all these factors, each being raised to tJie highest power in which it occurs in any of the given functions. For example, let the functions be (ir-l)V + 2)V + a: + l), {x-lf{x-2f{x-3y{x" + x + lf, then, by the above rule, the L.C.M. is {x-l)Hx-2)%c-3y{x- + 2)'{x'' + x + lf{x'-x + l)". Exercises X. Find the G.C.M. of the following, or else show that they have no CM. (1.) (x"-lf, x^-1. (2.) a«-l, ai^-2a^ + Sr-2x + l. (3.) a^-a^ + 1, x' + x' + l. (4.) a? + l, x^ + Tl. {5.) sfi-x'^-8x + 12, x' + ix^-Sx-lS. (6.) a''-7.i:'-22.i° + 139x + 105, a*- Sa;' - llK^ + ngK + yO. (7.) a;*- 286.^2 + 225, a^ + 14a?'-480a:2- 690a;-225. (8.) x^-jr*-8.r- + l2, a!« + 4aT*-3..~-18. (9. ) ar' - 2^-" - 2a^ + 4a;= + a; - 2, x' + 2.x-' - 2a;'' - Sx- - 7,.j - 2. (10.) afi + 63fi-83i*+l, x^'' + 7x^<'-S}?-3x--2. (11.) 120)3 + 1.3.iy- + 6.r + l, J6x> + iex^ + 7.,; + l. (12.) 5arH38f'-- 195a; -600, ix>-15x^~SSx + 6r,. (13.) 16a;* -56a;'-88.-<,-2 + 278a; + 105, 16a;*- 64x^ - 44a:= + 232a; + 70. (14. ) 7*" + 6:c' - 8a;- - 6a; + 1, 11a:* + ISa,-^ - 2x' - 5x + 1. (15.) .i'' + 64a*, (a; + 2ai)''-16a''. (16.) 9ar' + 4a;2 + l, 3sy2ar' + .i;- + 1. (17.) x'' + 3px--{l + 3p), ^r'-3(l+3?7)a; + (3 + 8i)). (18. ) x^-3{a- b]x^ + (4a2 - 3aJ>)a: - 2a-(2a - 36), ai^-{ia + byjy' + {5a^ + 2ab)x^-a^5a + 3b)x + 2a\a + b). (19.) ma;"+i- (?i + l)a;»+l, a;" - ?ia; + (»i - 1 ). (20.) Show that a;^+^ar + ja; + l, a? + qx-+px + l cannot have a common measure, unless cither ;j = g' or ^^ + g + 2 = 0. (21.) Show that, if aa;' + 6a; + c, ca;^ + 6a; + a have a common measure of the 1st degree, then «±J + c = 0. (22.) Find the value of a for which {a;'-aa;^ + 19a;-ft - 4}/{a;'- (a + l)a;^ + 23x-a-7\ admits of being expressed as the q^uotient of two integral functions of lower degree. (23.) If asfi + Shz' + d, tx' + 3dx + e have a common measure, then {ae-ibdf = 27{a(P + bhy. VI EXERCISES X 125 (24. ) Aa;^ + Hxy + Cy^, Ba;^ - 2( A - C)xy - Bj/^ cannot have a common measure unless the first be a square. (25.) aa? + la? + cx + d, da? + ex^ + hx + a\v\\\ have a common measure of the 2nd degree if nhn- n?h - Vd + acd _ ru" - bed -a' + ad- _ d(ni: -hd) ac - bd ab - cd a- - d' show that these conditions are equivalent to only one, namely, ac-bd = d--d^. (26.) Find two'integral functions P and Q, such that P(a2-3a; + 2) + Q(x2 + a; + l) = l. (27.) Find two integral functions P and Q, such that ,P(2a;"-7a:2 + 7a;-2) + Q(2a^ + a52 + a;_i) = 2x-l. Find the L.C.M. of the following : — (28.) aP-aV, a?\a%, a^ + h^ + a?b\a? J^-b^ (29.) a?- K^- 14a; + 24, s^ - 2a;" - 5a: + 6, a;=-4a; + 3. (30.) 3ar' + a;2-8a; + 4, 3ar' + 7a;--4, ar^ + 2a;" - .« - 2, Sa?' + 2a;2 - 3a; - 2. (31.) a!=-12a! + 16, ar»-4ar'-a;2 + 20a;-20, a;* + 3a;^- llai^- 3a; + 10. (32. ) 7?\ 'lax' + aW + 5a^a; + a*, a? + a'x - ax- - a^. (33.) If x^-hax + b, x- + a'x + b' have a common measure of the 1st degree, then theii' L.C.M. is .. nh — a'V {»''':C-^'/}^ b-b' \. , \a-a'J ) h-V (34.) Show that the L.C.M. of two integral functions A and B can always be expressed in the form PA + QB, where P and Cj are integral functions. CHAPTEE VII. On the Resolution of Integral Functions into Factors. § 1.] Having seen how to determine whether any given integral function is a factor in another or not, and how to deter- mine the factor of highest degree which is common to two in- tegral functions, it is natural that we should put to ourselves the question, How can any given integral function be resolved into integral factors ? TENTATIVE METHODS. § 2.] Confining ourselves at present to the case where factors of the 1st degree, whose coefficients are rational integral functions of the coefficients of the given function, are suspected or known to exist, we may arrive at these factors in various ways. For example, every known identity resulting from the distri- bution of a product of such factors, when read backwards, gives a factorisation. Thus {x + y){x- y)=x' - y' tells us that x' - y' may be re- solved into the product of two factors, x + y and x- y. In a similar way we learn that x + y + z\s,& factor in x^ + y^ + z^ — Zxyz. The student should again refer to the tables of identities given on pp. 81-83, and study it from this point of view. When factors of the 1st degree with rational integral coefficients are known to exist, it is usually not difficult to find them by a tentative process, because the number of possible factors is limited by the nature of the case. CHAP. VII TENTATIVE FACTOEISATION EXAMPLES 127 Example 1. Consider x^- 12a; + 32, and let ns assume that it is resolvable into (x-a) (x-h). Then we have x^ --12x + Z2 = a? - (a + h)x + ah, and vpe have to find a and i so that db=+Z2, a + h=+12. We remark, first, that a and h must have the same sign, since their pro- duct is positive ; and that that sign must be + , since their sum is positive. Further, the different ways of resolving 32 into a product of integers are 1 X 32, 2x16, 4x8; and of these we must choose the one which gives a + 5= +12, namely, the last, that is, a = i, i = 8. So that a;5 - 12a; + 32 = (cc - 4) (a; - 8). Example 2. af-ix^- 23a; + 60 = (a; - a) (x - i) (a; - c) say. Here -a6c=+60. Now the divisors of 60 are 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60 ; and we have therefore to try a;±l, a; ±2, a: ±3, &c. The theorem of remainders (chap, v., § 14) at once shows that a; + 1, a;- 1, x + 2, x- 2, are all inadmissible. On the other hand, for a; -3 we have (see chap, v., § 13) 1-2-23 + 60 + 3+ 3-60 1 + 1-20+ that is, X- 3 IS a factor ; and the other factor is x^ + a;- 20, which we resolve by inspection, or as in Example 1, into (x- 4) (x + 5). Hence ar*- 2s=-23x + 60 = (x-3)(x-4) (x + 5). \ Example 3. 6a;2-19x + 15 = (ax + 6)(cx + rf). Here ac= +6, M= +15 ; and we liave more cases to consider. "We might have anyone of the 32 factors, x±l, x±3, x±5, x±15, 2a; ±1, 2x±3, 2x± 5, 2x± 15, &c. A glance at the middle coefficient, - 19, at once excludes a large number of these, and we find, after a few trials, 6x2 - 19a;+ 15 = (o^, _ 3) (3^, _ 5)_ § 3.] In cases like those of last section, we can often detect a factor by suitably grouping the terms of the given function. For it follows from the general theory of integral functions already established that, if P can be arranged as the sum of a series of groups in each of which Q is a factor, then Q is a factor in P ; and, if P can be arranged as the sum of a series of groups in each of which Q is a factor, plus a group in which Q is not a factor, then Q is not a factor in P. 128 FACTOKISATION OF OIIAP. Example 1. a^-2.r--23a: + 60 =x'-{x-2)-2Z{x-2) + li, that is, a; - 2 is not a factor. x^-2a?-23x + 60 = x-i.r - 3) +0^ - 23.r + 60 = a^{x-3) + x{x-3)-20.r. + 60 = x^{x-3) + x(x-3)-20(x-3), that is, K - 3 is a factor. Example 2. px' + (1 +2>(l)xy + qy^ =})x- + xy +pqxy + qxj'' = x{px + y) + qy{px + y), that is, 2'x + y is a factor, the other being x + qy. Example 3. a^ + (m + » + 1 )x-a + {m + n + m7i)xa^ + mna? = .I-' + x^a + (m + Ji.) (x"a + xa-) + nin [xa? + a') =x?(x + a) + {'m, + n)xa{x + a) + ■mna?(x + a) = {a? + (m + n)xa + m nd^} (x + a) = {x(x + ma)+'na(x + ma)}(x + a) = (x + ma) (x + iia) {x + a). GENERAL SOLUTION FOR A QUADRATIC FUNCTION. § 4.] For tentative processes, such as we have been illustrat- ing, no general rule can be given ; and skill in this matter is one of those algebraical accomplishments which the student must cultivate by practice. There is, however, one case of great im- portance, namely, that of the integral function of the 2nd degree in one variable, for which a systematic solution can be given. We remark, first of all, that every function of the form .r'+px + q can be made a complete square, so far as x is con- cerned, by the addition of a constant. Let the constant in question be a, so that we have x^ + ju + q + a = {x + fSf = x" + 2fix + ji", j8 being by hypothesis another constant. Then we must have ^ = 2/3, q + a = ji\ The first of these equations gives /3=|j/2, the second a = /3' -q = (jy/^y - q- Thus our problem is solved by adding too? +px + q the constant (p/2)' - q. A QUADRATIC FUNCTION 129 .It is c aa;° + bx + c, as follows : — - The same result is obtained for the more general form, 2 7 f 2 ^ c ax + bx + c = a\x +-« + - \ a a. Now, from the case just treated, we see that x^ + (hja)x + cja is made a complete square in x by the addition of {bj'laf - cja, that is, (Jf - 4:ac)/4:a'. Hence ax^ + bx + c will be made a com- plete square in x by the addition of a(b'' - iac)jia^, that is, (6^ - iacj/ia. We have, in fact, „ , b^ - iac / b ax +bx + c-\ : -a\x + — ia \ 2a. § 5.] The process of last article at once suggests that ax^ + bx + c can always be put into the form a{(x + Ff - m'}, where I and m are constant. In point of fact we have ax^ + bx + c = a\ x^ + -x+ - f I a a ) V ^L'^ii) -(i) ^1} In other words, our problem is solved if we make / = bj2a and find m, so that m' = (b" - iac)jia\ This being done, the identity X' - A' = (X - A) (X + A) at once gives us the factorisation of ax^ + bx+ c; for we have ax^ + bx + c = a {{x + If— ni } = a {{x + l) +m]{{x + l)-m}. Example 1. Consider Bx''- 19a; + 15 ; ive have , and «i2 = I — ) ; so that our problem is solved if wo take m = — • HeroZ= -J|, ....... - ^^y , --- - j^ VOL. I 130 REAL ALGEBRAICAL QUANTITY CHAP. We get, therefore, ..=-*«.=,{(.-!|)4}((,-a)-i) = (2x-3){Sx-5); tlie same result as we obtained above (in § 2, Example 3), by a tentative process. Example 2. Consider .u^-Bx' + e. "We may regard this as {x>)^ - 5{a?) + 6, that is to say, as an integral function of x? of the 2n(l degree. We thus see that xf^-5x? + 6 = {.(?f-5{3?) + 6, = (x^ - 3) (ar> - 2). INTRODUCTION INTO ALGEBRA OF SURD AND IMAGINARY NUMBERS. § 6.] The necessities of algebraic generality have already led us to introduce essentially negative quantity. So far, algebraic quantity consists of all conceivable multiples positive or negative of 1. To give this scale of quantity order and coherence, we introduce an extended definition of the words greater than and less than, as follows : — a is said to be greater or less than h, according as a - b is positive or negative. Example. ( + 3)-( + 2)=+l therefore +3>+2; (-3) -(-5)= +2 therefore -3> -5 ; ( + 3)-(- 5)= +8 therefore +3>-5; (-7) -(-3)= - 4 therefore -7< -3. Hence it appears that, according to the above definition, any negative quantity, however great numerically, is less than any positive quantity, however small numerically ; and that, in the case of negative quantities, descending order of numerical magni- tude is ascending order of algebraical magnitude. We may therefore represent the whole ascending series of algebraical quantity, so far as we have yet had occasion to con- sider it, as follows : — -OO...-l...-|...0...+|. .+1...-I-O0.* * The symbol m is here used as an abbreviation for a real quantity as great as we please. VII RATIONAL AND SURD QUANTITY 131 The most important part of the operations in the last para- graph is the finding of the quantity m, whose square shall be equal to a given algebraical quantity. We say algebraical, for we must contemplate the possibility of (If - 4oc)/4a', say Ic for shortness, assuming any value between - oo and + oo . WTien m is such that m^ = Jc, then in is called the square root of h, and we write m = -Jk. We are thus brought face to face with the problem of finding the square root of any algebraical quantity ; and it behoves us to look at this question somewhat closely, as it leads us to a new extension of the field of algebraical operations, similar to that which took place when we generalised addition and subtraction and thus introduced negative quantity. 1st. Let us suppose that A is a positive number, and either a square integer = + k, say, or the square of a rational number = + (kIXY, say, where k and A are both integers, or, which is the same thing [smce (k/XY = k'/X^], the quotient of two square integers. Then our problem is solved if we take m= + K, OT m= - K in the one case, or TO = + k/X, or m = - k/X in the other ; for m' = {± k)' = K = k, -x)-[x)-'' which is the sole condition required. It is interesting to notice that we thus obtain two solutions of our problem ; and it will be afterwards shown that there are no more. Either of these will do, so far as the problem of factorisation in § 5 is concerned, for all that is there required is any one value of the square root. More to the present purpose is it to remark that this is the only case in which m can be rational ; for if m be rational, that is, = ± kJX where k and A are integers, then m^ = (k/XY, that is, k = {k/XY, that is, k must be the square of a rational number. 2nd. Let k be positive, but not the square of a rational 132 DEFINITION OF THE IMAGINARY UNIT i CHAV. number ; then everything is as before, except that no exact arithmetical expression can be found for m. We can, by the arithmetical process for finding the square root, find a rational value of m, say v, such that m^ = {± t)^ shall differ from k by less than any assigned quantity, however small ; but no such rational expression can be absolutely exact. In this case m is called a surd number. When h is positive, and not a square number, as in the present case, it is usual to use \/h to denote the mere (signless) arithmetical value of the square root, which has an actual existence, although it is not capable of exact arith- metical expression; and to denote the two algebraical values of m by ± v//v'. Thus, if /c = + 2, we write m = ± ^/2. In any practical application we use some rational approximation of sufficient accuracy ; for example, if /; = + 2, and it is necessary to be exact to the 1/10, 000th, we may use m- ± 1-4142. A special chapter will be devoted to the discussion of surd numbers ; all that it is necessary in the meantime to say further concerning them is, that they, or the symbols representing them, are of course to be subject to all the laws of ordinary algebra, f 3rd. Let k be negative = — k', say, where /j' is a mere arith- metical number. A new difficulty here arises ; for, since the square of every algebraical quantity between — co and + co (ex- cept 0, which, of course, is not in question unless h' = 0) is positive, there exists no quantity m in the range of algebraical quantity, as at present constituted, which is such that m^ = -H. If we are as hitherto to maintain the generality of all algebraical operations, the only resource is to widen the field of algebraical quantity still farther. This is done by introducing an ideal, so-called imaginary, unit commonly denoted by the letter i* whose definition is, that it is such that r= - 1. It is, of course, at once obvious that i has no arithmetical existence whatsoever, and does not admit of any arithmetical expression, approximate or other. We form multiples and sub- multiples of this unit, positive or negative, by combining it with * Occasionally also by i. t See vol. ii. chap. xxv. § 28-41. vir COMPLEX NUMBERS 133 quantities of the ordinary algebraical, now for distinction called real, series, namely, -00... -1.. -J ..0 ..+J.. +1...+00. We thus obtain a new series of purely imaginary quantity : — -ooi. . - i . . . - ^i. . . Oi. . . + ^i . . . + i . . . +ooi. These new imaginary quantities must of course, like every other quantity in the science, be subject to all the ordinary laws of algebra when combined either with real quantities or with one another. All that the student requires to know, so far at least as operations with them are concerned, beyond the laws already laid down, is the defining property of the new unit i, namely, i^ = - 1. When purely real and purely imaginary numbers are com- bined by way of algebraical addition, forms arise like p + qi, where. p and q are real numbers positive or negative. Such forms are called complex numbers ; and it will appear later that every alge- braical function of a complex number can itself be reduced to a complex number. In other words, it comes out in the end that the field of ordinary algebraical quantity is rendered com- plete by this last extension. The further consequences of the introduction of complex numbers will be developed in a subsequent chapter. In the meantime we have to show that these ideal numbers sufiice for our present purpose. That this is so is at once evident ; for, if we denote by >Jk' the square root of the arithmetical number k', so that Jh' may be either rational or surd as heretofore, but certainly real, then m= ±i Jh' gives two solutions of the problem in hand, since we have m^ — {± i rJk'Y = (±i JM) X ( ± i sih'), upper signs going together or lower together, = (iO X ( x/^T = (-l)x(A') = -h'. § 7.] We have now to examine the bearing of the discus- sions of last paragraph on the problem of the factorisation of an? + bx + c. 134 FACTORISATION OF QUADRATIC chap. It will prevent some confusion in the mind of the student if we confine ourselves in the first place to the supposition that a, b, c denote positive or negative rational numbers. Then I = bj2a is in all cases a real rational number, and we have the following cases : — 1st. If b^ - iae is the positive square of a rational number, then TO has a real rational value, and a/ + bx + c = a{z + I + m){x+ I - m) istheproductof two linearfactorswhose coefficients are real rational numbers. Example 1, § 5, will serve as an illustration of this case. 2nd. If W - 4ac is positive, but not the square of a rational number, then m is real, but not rational ; and the coefficients in the factors are irrational. Example 1. ..~ + 2.---]=a;2 + 2.); + l-2, = {x + l + s/2){x + l-'^2). 3rd. If b^ — iac is negative, then in is imaginary, and the coefficients iu the factors are complex numbers. Example 2. x^+2x + 5=x^ + 2x + l + 4, = {x + lf-{2ir, = {x + l + 2i){x + '[-2i). Example 3. ;z~ + 2a; + 3 = a:2 + 2.->! + 1 + 2, = (a: + l)2-(iV2)2, = {x + l+i's/2){x + l-i'^2). 4th. There is another case, which forms the transition between the cases where the coefficients in the factors are real and the case where they are imaginary. If b' - iac = 0, then m = 0, and we have a.r° + bx + c = a[.r. + Tf ; in other words, ax^ + bx + c is a complete square, so far as x is concerned. The two factors are now x + I and x + I, that is, both real, but identical. We have, therefore, incidentally the important result that ax' + bx + c is a complete square in x if b^ - iac = 0. Example 4. 3a;=-3x + | = 3(a;2-2.irc + J), = 3(a;-.i)2. * b'^-iac is called the Disorirainaut of the quadratic function ax' + bx + c. VII FUNCTIONS RESUMED 135 § 8.] There is another point of view which, although usuallj' of less importance than that of last section, is sometimes taken. Paying no attention to the values of a, b, c, but regarding them merely as functions of certain other letters which they may happen to contain, we may inquire under what circumstances the coefficients of the factors will be algebraically rational functions of those letters. In order that this may be the case it is cleaily necessary and sufficient that b^ - iac be a complete square in the letters in question, = P° say. Then aa^ + bx + c = a\ ( a; + -- | - (-t-) \, \ \ 2a J \2aJ j / b P\/ b = a(x + -— + — ) [X + -— \ 2a 2aJ \ 2a which is lutional, since P is so. If b^ — iac = - P^, where P is rational iii the present sense, then az^ + bx + c = a^(x + ^^ - gi) }, / b P .X / b Pa V 2a 2a J \ 2a 2a ) where the coefficients are rational, but not real. Example 5. px^ + (p + g)x + q =ph x + l )(a; + 2 P. = {x + l){px + q) ; a result which would, of course, be more easily obtained by the tentative processes of §§ 2, 3. 136 HOMOGENEOUS QUADRATIC FUNCTIONS chap. § 9.] It should be observed that the factorisation for ax^ + hx + c leads at once to the factorisation of the homogeneous function as^ + hxy + ctf of the 2nd degree in two variables ; for ao? + hxy + cif { \yJ u\yJ a J , r .r b //? ^4«- ) f a; b /b' - iuc ) . ( fb /i' - 4ac\ If /b lb 4(r By operating in a similar \\k\ any homogeneous function of two variables may be factorised, provided a certain non-homo- geneous function of one variable, having the same coefficients, can be factorised. Example 1. From tc- + 2a; -h 3 = (a: 4-1 -I- i V2) (» -H - i\/2), we deduce K^ + 2x1/ + 3j/-= {a; + (1 -I- i V2)i/} {rr -h (1 - iV2)2/} . Example 2. From .t' - 2..J- - 23ic -I- 60 = (a; - 3) (a - 4) {a;-|- 5), Te deduce ,1-^ - 2,//-;/ - lixf + 601/3 = (a: - 3^) (^, _ 4^^) (3, _,_ 5^)^ § 10.] By using the principle of substitution a great many apparently complicated cases may be brought under the case of the quadratic function, or under other equally simple forms. The following are some examples : — Example 1. a;' -(- xhf -h 1/' = (j;2 + iff - {xyf, = (a:- -I- )/« -h xy) (./- ^xf- xy), ( / 1 \^ 3 ,1 r / 1 \2 3 , I = {(-+/) +ri\{"-^)+i'')' VII EXAMPLES 137 Here the student should observe that, if resolutiou into quadratic factors only is required, it can be effected with real coefficients ; but, if the resolutiou be carried to linear factors, complex coefficients have to be introduced. Example 2. !i!? + y^ = {x + y) [x^-xy + y^) Example 3. = (x' + y^f-(-^2xyf = (k^ + \l^y + 2/2) (a;2 - ^-Jxy + y\ Again x'- + \J1xy + y'^=lx+'^y\ +--f 2 ^+^y ] -\^^y) :{a; + ^^l + %}{x+^(l-i)j/}. The similar resolution for a?— f^jlxy + y^ will be obtained by changing the sign of \/2. Hence, finally, = [x+^{l+i)y]{x + ^(l-i)y){x-^{l + i)v){x-^Ci-i)y]. Example 4. a:12- 2/12= (a;6)2_ (2,6)2 = {(x>f-{y''f} {(cc2)3 + (2/2)3} = (a;2 - 3/2) (a^ + a;22/2 + y*) (x' + 1/) (»« - xh/ + y^) = {x + y){x~ y) {x + iy) {x - iy) (ic' + 3;2j/2 + j/*) {x'^ - x^y^ + J/"*), where the last two factors may be treated as in Example 1. Example 5. 262c2 + 2c2a2 + 2a2j2 _ ^ii _ JJ - c^ = 4J2c2 - {a;2 - J2 - (?f = (26c + «« - i2 _ c2) (26c - a2 + J2 + (.2) = {a^-(b-cf} {(6 + c)2-a2} = (a + b-c){a-h + c)('b + c + a)(b + c-a). * The student should observe that the decomposition x'^ + y^ + xy = (x + y+\/xy) ix+y- \/xy), which is often given by beginners when they are asked to factorise x' + y^ + xy, although it is a true algebraical identity, is no solution of the problem of factorisation in the ordinary sense, inasmuch as the two factors contain \/xy, and are therefore not rational integral functions of X and y. 138 USE OF REMAINDER THEOREM RESULTS OF THE APPLICATION OF THE REMAINDER THEOREM. § 11.] It may be well to call the student's attention once more to the use of the theorem of remainders in factorisation. For every value a of x that we can find which causes the integral function f(x) to vanish we have a factor x- a off(x). It is needless, after what has been shown in chap, v., §§ 13-16, to illustrate this point further. It may, however, be useful, although at this stage we cannot prove all that we are to assert, to state what the ultimate result of the rule just given is as regards the factorisation of integral functions of one variable. If f{x) be of the wth degree, its coeffi- cients being any given numbei-s, real or imaginary, rational or irrational, it is shown in the chapter on Complex Numbers that there exist n values of x (called the roots of the equation /(.);) = 0) for which f{x) vanishes. These values will in general be all different, but two or more of them may be equal, and one or all of them may be complex numbers. If, however, the coefficients of /(a) be all real, then there will be an even number of complex roots, and it will be possible to arrange them in pairs of the form A. ± fii. It is not said that algebraical expressions for these roots in terms of the coefficients of /(a:) can always be found ; but, if these coefficients be numerically given, the values of the roots can always be approximately calculated. From this it follows that f{x) can in all cases he resolved into n linear * factors, the coefficients of which may or may not he all real. If the coefficients of f{x) he all real, then it can he resolved into a product of p linear and q quadratic factors, the coefficients in all of which are real numbers ivhich may in, all cases he calculated approxi- mately. JVe have, of course, p + 2q = n, and either p or q may he zero. The student will find, in §§ 1-10 above, illustrations of these statements in particular cases ; but he must observe that the • "Linear" is used here, as it often is, to mean "of the 1st degree." VII QUADRATIC FUNCTION WITH TWO VAKIABLES 139 general problem of factorising an integral function of the nth degree is coextensive with that of completely solving an equation of the same degree. When either problem is solved the solution of the .otlier follows. FACTORISATION OF FUNCTIONS OF MORE THAN ONE VARIABLE. § 12.] When the number of variables exceeds unity, the problem of factorisation of an integral function {excepting special cases, such as homogeneous functions of two variables) is not in general soluble, at least in ordinary algebra. To establish this it if sufficient to show the insolubility of the problem in a particular case. Let us suppose that x^ + y^ + 1 is resolvable into a product of factors wliicli are integral in x and y, that is, that x^ + y'^ + l= (px + qy + r) [p'x + q'y + r'), then .x" + y'^ + l=pp'x^ + C[i^y^ + rr' + [ijq' +p'q)xy + (pr' +p'r)x + {qr' + q'r)y. Since this is, by hypothesis, an identity, we have pp'=-i (1) pq' + p'q = W ??' = ! (2) pr' + p'r = (5) rr' = l (3) 1 qi' + q'r = Q (6). First, we observe that, on account of the equations (1) (2) (3), none of the six quantities^ j rp' q' r' can be zero ; and further, p' = -, (/ = -, r'=-. Hence, as logical consequences of our hypothesis, we have from (4) (5) and (65 — £+2=0 (7) 5+r=o ' (8) r p 2 + r = (9); r q and, from these again, if we multiply by pq, rp, and qr respectively, we get i)H2= = (10) / + r2=0 (11) j2 + J-2=0 (12). Now from (11) and (12) by subtraction we derive i)2-22=0 (13); 140 QUADRATIC FUNCTION CHAP, and from (10) and (13) by addition from this it follows tliat^ = 0, which is in contradiction with the equation (1). Hence the resolution in this case is impossible. § 13.] ISTevertheless, it may happen in particular cases that the resolution spoken of in last article is possible, even when the function is not homogeneous. This is obvious from the truth of the inverse statement that, if we multiply together two integral functions, no matter of how many variables, the result is integral. One case is so important in the applications of algebra to geometry, that we give an investigation of the necessary and sufficient condition for the resolvability. Consider the general function of x and y of the 2nd degree, and write it "^ = ax^ + 2hxy + 'by'' + 2gx + yy + c. AVe observe, in the first place, that, if it be possible to resolve F into two linear factors, then we must have '? = {\/ax + ly + 'm) {\/ax + l'y + m'), = [\/ax+{i{l + l') + i{l-l')}y + l(m + in') + i{m-m')'] X l-^ax + {i{l + i') - ie - Z')} 2/ + 4(™ + ™') - 4(™ - ™')], = {^ax + i{l + l')y + i{m + m')}^- (i(Z -r)2/ + i(m-m')}2. Hence, when F is resolvable into two linear factors, it must be expressible in the form L^ - M^, where L is a linear function of x and y, and M a linear function of y alone ; and, conversely, when F is expressible in this form, it is resolvable, namely, into (L + M) (L-M). Let us now seek for the relation among the coefficients of F which is necessary and sufficient to secure that F be expressible in the form L^ - M^. 1st. Let « + 0, then F =a[a;2 + 2{hy + g)xla + (lif + Ify + c)la\ = fi[ {a + {liy + g)la) ^ - {{hy + gf -a(hy'' + Ify + c)} /a=], = a[ {x + [hy + g)la] " - {(h^ - ab)y"' + 'i{gh~ af)y + (g"- - ac)} ja?]. Hence the necessary and sufficient condition that F be expressible in the form L^ - JI- is that (h? - ah)y'^ + 2(gh - af)y + {g''- - ac) be a complete square as regards y. For this, by § 7, it is necessary and sufficient that i (.9'' - <■'/)" - 4( A= - al,) {g^ -ac) = 0; that is, - a {abe + '2fgh - ap - hg' - ch?\ = 0. Now, since a =1=0, this condition reduces to abc + 2fgh-ap-bg^-ch^=0 (1). 2nd. If = 0, but 6=t=0, we may arrive at the same result hy first arranging F according to powers of y, and proceeding as before. VI r OF TWO VARIABLES 141 3rd. If (1 = 0, i = 0, and /t + O, the present method fails altogether, but F now reduces to F = 21ixy + 2gx + Ifxj + c, and it is evident, since o? and j/^ do not occur, that if this be resolvable into linear factors the result must be of the form 'ni{x+p){y + q). "We must therefore have Ig = 2hq, 2f^2hp, c = 2Jtpq. Now the first two of these give fg = fi?j^q, that is, 2hpq='~- ; whence using the third, ch=2fg, or, since A 4= 0, 2fgh -ch^ = (2) ; but this is precisely what (1) reduces to when a = 0, i = 0, so that in this third case the condition is still the same. Moreover, it is easy to see that when (2) is satisfied the resolution is possible, being in fact 2hxy + 2gx + 2fy + c=^2hfx + Qfy + S\ (3), which is obviously an identity if c = 2fg/h. 4th. If a = 0, t = 0, h = 0, F reduces to 2gx+2fy + c. In this case we may hold that F is resolvable, it being now in fact itself a, linear factor. It is interesting to observe that in this case also the condition (1) is satisfied. Returning to the most general case, where a does not vanish, we observe that, when the condition (1) is satisfied, we have, provided h^-ab + 0, ^{{h?-ab)^f + 2{gh-af)y+{g''-ac)} = \/f?^^h{y + ^^;^^y so th.^t the required resolution is V=a\x+ y + -+ 7n, — T-,VA2-a6 \ I a a a{k- - ab) J To the coeflicients in the factors various forms may be given by using the relation (1) ; but they will not be rational functions unless K' - a6 be a com- plete square, and they will be imaginary unless h^-ab is positive. If A2_a6 = 0, then (1) gives {gh-aff = Q, that is, gh-a/=0; and the required resolution is F=a\x + -y '9'-ao\\^n.g \Jg^ . . „- _ . (5)- The distinction bet^veen these cases is of fundamental importance in the analytical theory of curves of the 2nd degree. The function abc + 2fgh -af'-bg'^-ch?, whose vanishing is the condition for the resolvability of the function of the 2nd degree, is called the Diserimi- nant of that function. 142 EXERCISES XI It should be noticed that, if 'E=ax'+2hxy + b%f + '2gx + 'ify + c =(lx + my + n)(l'x + m'y + n') (6), then ca? + 2hxy + by'^'^{lx + my) {I'x + m'y), so that the terms of the 1st degree in the factors of F are simply the factors of CO? + 21\xy + by'^. "We have therefore merely to find, if possible, values for » and n' which will make the identity (6) complete. Example. To factorise 3a:' + 1xy - y^ + ix - 2y - 1. "We have 3a;' + 2xy - y- = {3x-y) (x + y). Hence, if the factorisation be possible, we must have 3x^ + 2xy-y^ + 2x-2y-l={Sx-y + n){x + y + n') (7). Therefore, we must have n + 3n' = 2 (8), n-n'=-2 (9), nn'=-l (10). Now, from (8) and (9), we get n= -1, and «.'= + 1. Since these values also satisfy (10), the factorisation is possible, and we have 3a:' + 2a:!/ - J/- + 2a: - 22/ - 1 = (3a: - 3/ - 1 ) (k + 2/ + 1). It should be noticed that the resolvability of F = ax^ + 2hxi/ + hif + Igx +1fy + c carries with it the resolvability of the homogeneous function of three variables having the same coefficients, namely, Y = ax^ + hif + cz' + 2fyz + 2gzx + Ihxij, as is at once seen by writing xjz, yjz, in place of x and y. EXEKCISES XL Factorise the following functions : — (1- (3. (5. (7. (10 (13. (16. (18. (20. (22. (24. (21 (27. (28. (29. (30. (a + bf + (a + cf - (c + df - (J + df. (2. ) 4a'6' - (a' + b" - e'f. (a'-2b'-c'f-i{l!^-c'f. (4.) (Sx^- lla: + 12)' -(4x'- 15a: + 6)'. {x^-(P + y)x + ^yY--(,x-ynx-a.)\ (6.) x'>-y\ 7?-y^. (8.) a:' + 6a:i/ + 9!/2-4. (9.) 2a:' + 3a:-2. a;' + 6x-16. (11.) a;' - lOx + lS. (12.) x'+ic-30. ic' + 14x + 56. (14.) a::' + 4a:+7. (15.) 2a;' + 5a:-12. .j:' + 2a;V(/J + ?) + 2?. (17.) a;'- 2fa/(i> + c) + (6 -c)/(6 + c). (x'+n?-{P + i)-^?- (19-) ab{x^-y''-) + xy[a?-V). pq{x + yf -(p + q)(ii? -y'^) + (x-yf. (21.) ar' -15a:' + 71a:- 105. x''- 141'+ 148a:. (23.) i? -lix^ + Zix-I'l. ar* - 8,?;' + a; - 8. (25. ) ar^ + 3j3a;' + (3;)' - q'')x +p{p^ - q% {p + qW + {p-q)x''-(p + q)x-(p-q). x^-{'i.+p+p^)x^ + {p+p^+p^)x-p^. ar" - (a + 6)ar> + (a'6 + ab^)x - a?V^. ^+x'a-\- a:^a' - a:'(i'' - a:a' - a". (l + x)'(l+ii/')-(l+i/)'(l + a:'). (31.) a,-' + a,y + 2/4. VII EXERCISES XI 143 (32.) Asswming X* + y* = {x' + pxy + y'') (x^ + qxy + y^), determine ^7 and q. (33.) Factorisea^+2/*-2(K2 + 2/2) + l. (34. ) Determine r and « in terras ol a, p, and q in order that sfi -a? may be a factor in x* + px' + qx^ + rx + s. Factorise (35. ) (a;™+")2 - (x'^a'^f - (x''a'"f + {a™+")2. (36. ) (x^ + a?f{X^ + a?!? + a-") - (a? + ai^a"' + a^). (37.) xy'^-2xy-%/ + x + 2y-l. (38.) ix^ + xy + lx + Zy + Z. (39.) ix^- + xy -Zy"^ -x-iy -\. (40.) xy + lx + Zy + il. (41.) a;2-2(/2-322+72/s + 2sa + a;2/. (42.) Determine Xso that (3: + 6!/ -l)(6a! + j/-l) + X(3a; + 2j/ + l) (20; + 3j/ + 1) may be resolvable into two linear factors. (43.) Find an equation to determine X so that a.r- + Sj/^ + 2to3/ + 2g'a; + 2/1/ + c + 'Kxy may be resolvable into two linear factors ; and iind the value of X when c = 0. (44.) Find the condition that (ax + §y + ys){a.'x + §'y + y'z)-{a"x + §"y + y"zf break up into two linear factors. (45.) If (x+p) (x+2q) + (x + 'ip)(x + q) be a complete square in a;, then (46. ) li(x + V)(x + c) + (x + c)(x + a) + (x + a){x + h) be a complete square in X, show that a=b = c. Factorise (47.) a^ + b^ + e>-Sabc. (48.) s? + Saxy + y^-A (49.) {x-3?)'< + {x^-lf + {l-xf. Factorise the following functions of x, y, z : — * (50.) X(y'^ + x^) (z^ + x^)(y-z). (hi.) J.(a? + y^){x-y). [5%) Jafi(y"--z\ (53.) (Zs)^- Sari. (54.) Simplify {■2,{x'' + y'^-z^){x^ + z^-y'')}l-n.(x±y±z). (55.') Show that S(2/'"2?" - !/"s") and Sa!"(j/"*»^ - jz-^z™) are each exactly divisible by (y -z){z- x) (x-y). (56.) Show that «a;"+i - (» + 1 )»" + 1 is exactly divisible by (x-Vf. (57.) Show that 'Zx'iy + z-xf is exactly divisible by "Lx^-^Xyz. (58.) Showthat (a; + 2/ + 2)^''+i-:c2»+i-2/*'+^-K^+^ is exactly divisible by {y + z)(z + x)(x + y). (59. ) (y- zf"+'^ + (z- a:)^+^ + {x- j/)^+^ Is exactly divisible by {y - 2) (s - x) (a; - ?/)• (60.) If»be of the form &m-\, then (y - 2)" + (s - a)" + (aJ - !/)" is exactly divisible by Ja^-ljxy ; and, if » be of the form 6m + 1, the same function is exactly divisible by (Za^ - '^xy)^. (61.) Prove directly that xy-\ cannot be resolved into a product of two linear factors. (62.) If a and 6 be not zero, it is impossible so to determine p and q that x-\-py + qz shall be a factor of x' -^ at/ -{-}>!? . * Regarding the meaning of S in (50), (51), &c., see the footnote on p. 84- CHAPTER VIII. Rational Fractions. § l.J By a rational algebraical fraction is meant sim{ply the quotient of any integral function by any other integral function . Unless it is otherwise stated it is to be understood that we are dealing with functions of a single variable x. If in the rational fraction A/B the degree of the numerator is greater than or equal to the degree of the denominator, the fraction is called an improper fraction, if less, a proper fraction. GENERAL PROPOSITIONS REGARDING PROPER AND IMPROPER FRACTIONS. § 2.] Every improper fraction can be expressed as the sum of an integral function and a proper fraction '; and, conversely, the sum of an integral function and a proper fraction may be exhibited as an improper fraction. A, For if in ^ the degree m of A,„ be greater than the degree n of B„, then, by the division-transformation (chap, v.), we obtain ^-O + — B~ ^m^-n ^ -r> ' n ^n which proves the first part of our statement, since Qm-n is integral, and the degree of E is w. Examples of these transformations have already been given under division. It is important to remarlc that, if two improper fractions he equal, then the integral parts and the properly fractional parts must he equal separately. Am _ ^ jj^ j3 -SJm-„ + 3, For let ^" = Q,^.,^ + , A' ' E' and r,r=Q'm'-n' + w- by the above transformation. A A' , B„ B'„' ' Then, if we have Qm-n + =5- = Q'm' - ji' + =5^- -D,i -D ri TT n O' R'B« - EB'„.' xience \^ -n~ ^ m' - n' B„, B'„,' Now, since the degrees of E' and E are less than n' and n respectively, the degree of the numerator on the right-hand side of this last equation is less than n + n! . Hence, unless Qm-n ~ Q'm'-n' = 0) ^6 have an integral function equal to a proper fraction, which is impossible (see chap, v., § 1). We must therefore have Qm-n = Q'm'-n'. and Consequently ^ = ^-. N.B. — From this of course it follows that m-n = m! - n'. As an example, consider the improper fraction (s? + 9.x^ + Zx + i)j(x''- + x + l), and let us multiply, both numerator and denominator by a^ + 2* + 1 ; we thus obtain the fraction (x'' + 4a5 ' + 8 A'H 1 23;2 + 1 la; + 4 )/ (x* + 3a;'' + to^ + 3k + 1 ) , which, by chap, iii., § 2, must be equal to the former fraction. Kow transform each of these by the long-division transformation, and we obtain respectively x + Z x + l + -„ , , ,, x^ + x + 1 vor.. I 146 DIRECT OPERATIONS The integi-al parts of these are equal ; and the fractional parts are also equal (see next section). The sum of two proper ahjehmkal fractions is a proper algebraical fractiuii. After what has been given above, the proof of this proposition will present no difficulty. The proposition is interesting as an instance, if any weie needed, that fraction in the algebraical sense is a totally different conception from fraction in the arithmetical sense ; for it is not true in arithmetic tl}at the sum of two proper fractions is always a proper fraction ; for example, J + 1 = 4, which is an improper fraction. § 3.] Since by chap, iii., g 2, we may divide both numerator and denominator of a fraction by the same divisor, if the nu- merator and denominator of a rational fraction have any common factors, we can remove tliem. Hence every rational fraction can be so simplified that its numerator and denominator are algebraically prime to cadi other ; when thus simplified the fraction is said to be at "its lowest terms." The common factors, when they exist, may be determined by inspection (for example, by completely factorising both numerator and denominator by any of the processes described in chap, vii.) ; or, in the last resort, by the process for finding the G.C.M., which will either give us the common factor required, or prove that there is none. E.\ample 1. x'^ + Zx^ + ix^ + Sx + l' By either of the processes of chap. vi. the G. C. M. will be found to be a;^ + 2a; + 1. Dividing both numerator ami denominator by this factor, we get, for the lowest terms of the given fraction, a; + .3 a;2 + a;+i' The simplification might have lieen effected thus. Observing that both numerator and denominator vanish when x= - 1, we see that x+ 1 is a com- mon factor. Kemoving this factor we get x^ + ix + S .ii? + 2x'' + 2x + l' Here numerator and denominator both vanish when a:= - 1 , hence there is the comnmn factor x+1. Removing this we get VIII WITH EA.TIONAL FRACTIONS 147 x + 3 x^ + x+1' It is now obvious that numerator and denominator are prime to each other ; for the only possible common factor is x + 3, and this does not divide the denominator, which does not vanish when x= -S. § 4.J The student should note the following conclusion from the above theory, partly on account of its practical usefulness, partly on account of its analogy with a similar proposition in arithmetic. If two rational fractions, P/Q, P'/Q' ^^ eqital, and P/Q be at its lowest terms, then P' = XP, Q' = AQ, where \ is an integral function of X, which will reduce to a constant if V/Q,' be also at its lowest terms. To prove this, we observe that p p Q'"Q' whence P' = —^y , that is, Q'P/Q must be integral, that is, Q'P must be divisible by Q ; but P is prime to Q, therefore by chap, vi., § 12, Q' = AQ, where A is an integral function of x. We now have P' = '^ = AP; so that P' = AP, Q' = AQ. If P'/Q' be at its lowest terms, P' and Q' can have no com- mon factor ; so that in this case A must be a constant, which may of course happen to be unity. DIRECT OPERATIONS WITH RATIONAL FRACTIONS. § 5.] The general principles of operation with fractions have already been laid down ; all that the student has now to learn is the application of his knowledge of the properties of integral functions to facilitate such operation in the case of rational fractions. The most important of these applications is the use of the G.C.M. and the L.C.M., and of the dissection of functions by factorisation. 148 EXAMPLES OF DIRECT OPEEATIONS CHAP. No general rules can be laid down for sucli transformations as we proceed to exemplify in this paragraph. But the following pieces of general advice will be found useful. Never make a step that you cannot justify by reference to the fundamental laws of algebra. Subject to this restriction, make the freest use of your judg- ment as to tlie order and arrangement of steps. Take the earliest opportunity of getting rid of redundant members of a function, unless you see some direct reason to the contrary. Cultivate the use of brackets as a means of keeping composite parts of a function together, and do not expand such brackets until you see that some- thing is likely to be gained thereby, inasmuch as it may turn out that the whole bracket is a redundant member, in which case the labour of expanding is thrown away, and merely increases the risk of error. Take a good look at each part of a composite expression, and be guided in your treatment by its construction, for example, by the factors you can per- ceive it to contain, by its degree, and so on. Avoid the unthinking use of mere rules, such as that for long division, that for finding the G.C. M. , &c. , as much as possible ; and use instead pro- cesses of inspection, such as dissection into factors ; and general principles, such as the theorem of remainders. In other words, use the head rather than the fingers. But, if you do use a rule involving mechanical, calculation, be patient, accurate, and systematically neat in the working. It is well known to mathematical teachers that quite half the failures in algebraical exercises arise from arithmetical inaccuracy and slovenly arrangement. Make every use yon can of general ideas, such as homogeneity and sym- metry, to shorten work, to foretell i-esults without labour, and to control results and avoid errors of the grosser kind. Example 1. Express as a single fraction in its simplest form — T, — z r, — T = F say. Transform each fraction by division, then F = (2.0+4) +-^-(2.+ J)- 2.'' ■ ~ 2.'" — 2 ^ 2(..-'-x°- -l) x'-l ■ Example 2. Express as a single fraction P^ 1 1 1 _1 X" - Sx- + ;3,/j - 1 x^- x- - J- h 1 K* - 2;/-' + 2.r - 1 a^ - 2x3 + 23'^'- ix + V -a- + 2 VIII WITH EATIONAL FRACTIONS 149 We have x^-x'^-x+l =x' + l-x{x + l) = (x + l) {x^-x + l-x), = {x + l){x-lf; x/^-23i? + 2x-l=x^-l-2x{jr-l), = {x^-l){x-lf, = {x-lf{x + l) ; K* - 23?* + 2a;2 - 2a; + 1 = (a;= + 1 )= - 2a;(a;= + 1 ), = {x^ + l){x-l)''. Whence 111 1 F = = 2 {x-if {x + l){x-lf (x-lf{x + i) (x' + lHx-lf (a;+l)-(a;-l) _ (a^ + l) + (a;- 1) (a!+l) ■ (rc + lXaj-l)" (i<;-l)^(a; + l)(a;2 + l)' 2 2,1" ' (x + lT{x^Vf ~ (a;-l)^(a; + l)(a!-^ + l)' x^ + l-x^ {x + l)(x' + l){x-lf 2 '{x^-l){x-lf 2 x^-2x^ + x*-x^ + 2x-l Example 3. ^i^'}/3\ fx + y ir^+if^' -y x' + y'/ \x-tj .i^-y'i /x-y x<-y3 \ ^ /x + y ^ .,■' \.r + y x^ + y"/ \x-y ,'•' _( ^-y \ (-1 x^ + sai + y^ \ (x+jj\ f -, x ^-x;i + y ^\ "Kx + y/y x^-xy + y'^J \.v-y) \ X' + xy + y^)' \x'-xy + y-J \x' + xy + y-J _ 4.c)/(.v- + 7/°) .ij-'+.c-y- +?/-'■ Example 4. -,2 b-c 2 c-a 2 a-b ^=T — + ; w ^^ + + ; rm — ; + — i + b-c (c-a){a-b) c-a (a-b) {b-c) a-b (b-c)(c-a) _ 2{c-a) {a-b) + {b-cf + 2(a - b )(b-c) + (c-a)'^ + 2(b-c){e-a) + {a- b) ^ ~ {b-c) (c-a) (a-b) _{{ b-c) + (c-a) + {a-b)y - k^. ' = ii= 2 =0 &o. (b-c){c-a)(a-b) ' it being of course supposed that the denominator does not vanish. 150 DIRECT OPERATIONS WITH K.ATIONAL FRACTIONS CHAP. Example 5. a^ f (^ ^(a-b)(a-c)^ (b^c) (b -n)'^{c- a) {a-b)' _ -a!'{b - c)-b'{o-a)~r^{a-b) (b -c){c- a) (a - b) Now we observe that when b = c the numerator of F becomes 0, hence h-c is a factor; by symmetry c-a and a-b must also be factors. Hence the numerator is divisible by {b -c){c- a) {a - h). Since the degree of the numer- ator is the 4th, the remaining factor, owing to the symmetry of the expression, must be Pa + P6 + Pf. Comparing the coefficients of a?b in - n'(6 - c) - b^c -a)- c^(a - b) and V(a + b + c){b-c){c-a){a-b), we see that P= +1. Hence, finally, V=a + b + c. Example 6. „_ a^+pa + q b^+pb + q c^ + pc + q ~(a-b){a-c)(x-a) (b-a)(b'-c)(x-b) {c-a)(c-b)(x-c) „ _ {b -c){a''+xia + q){x-b)(x-c) + kc.+kc. (b -c)(c-a)(a- b) {x - a) {x - b] (x - c) _{i-c) (a? + pa + q ) {■>■- -{h + c) x + be) + he. + &c. &c. Now, collect the coefBcients of x", x, and the absolute term in the numerator, observing that the two &c. 's stand for the result of exchanging a and b and a and c respectively in the first term. We have in the coefficient of x" a part independent of ^ and q, namely, a?ifi - c) + b-{c ~a) + C'{a -b)= -{b-c){e- a) {a -b) (1). The parts containing p and q respectively are {a{b -c) + b[c-a) + c{a-b)]p = and {{b-c) + {c-a) + {a-b)}q-0. The coefficient of .c^ therefore reduces to (1). Next, in the coefficient of x we have the three parts, - (a2(62 - c2) + b^e^ - a^) + (fi(a^ - 6=)} =0, - {a{b^ -(-) + i{c2 - ffi2) + c{a^ _ ly^p = -{b-c){e-a){a-b)p (2), and -{(S2_c2) + (c=-a2) + (a2_j2)jj = 0. Finally, in the absolute term, abc{a{b-c) + b{c-a)+c(a-b)} =0, abc {'b -c) + {c-a) + {a- 6)}p— 0, {bc{b-c) + ca{c - a) + ab{a - b)} q = -{b-c){c-a){a-b)q (3). VIII INVERSE METHODS 151 I Hence, removing the common factor (6 - c) {c - a) (a - b), which now appears both in numerator and denominator, and changing the sign on bolli sides, we have X'+px + q F = (x - a) (x - b) (x - c) The student should observe here the constant use of the identities on pp. 81-83, and the abbreviation of the worlc by two-thirds, effected by taking advantage of the principle of symmetry. In actual practice the greater part of the reasoning above written would of course be conducted mentally. INVERSE METHOD OF PARTIAL FRACTIONS. § 6.] Since we have seen that a sum of rational fractions can always be exhibited as a single rational fraction, it is naturally suggested to inquire how far we can decompose a given rational fraction into others (usually called "partial fractions") having denominators of lower degrees. Since we can always, by ordinary division, represent (and that in one way only) an improper fraction as the sum of an integral function and a proper fraction, we need only consider the latter kind of fraction. The fundamental theorem on which the operation of dissec- tion into " partial fractions " depends is the following : — If A/PQ be a rational pivper fraction whose denominator contains two integral factors, P and Q, which are algebraically prime to each other, then we can always decompose A/PQ into the sum of two proper fractions, P'/P -I- Q'/Q- Proof. — Since P and Q are prime to each other, we can (see chap, vi., § 11) always find two integral functions, L and M, such that LP 4- MQ = 1 (1). Multiply this identity by A/PQ, and we obtain _A^_AL AM ,^. PQ" Q * P ^^'' In general, of course, the degrees of AL and AM will be higher than those of Q and P respectively. If this be so, transform AL/Q and AM/P by division into S + Q'/Q and T 4- P'/P, so that 152 PARTIAL FRACTIONS CHAP. S, T, Q', and P' are integral, and the degrees of P' and Q' less than those of P and Q respectively. We now have where S + T is integral, and P'/P + Q'/Q a proper fraction. But the left-hand side of (3) is a proper fraction. Hence S + T must vanish identically, and the result of our operations will be simply PQ P Q ^ ' which is tlie transformation required. To give the student a better hold of the above reasoning, we work out a particular case. Consider the fraction F=7 '{o.^ + Sx' + ix + l) (a;^ + x+l)' Here A = .i'' + 1, V = x' + 3x^ + 2x + l, Q=x'' + x + l. Carrying out the process for finding the G.C.M. of P and Q, we have l-rl + l)l + 3 + 2 + l(l + 2 2 + 1 + 1 -l-l)l + l + l(-l + + 1 + 1 whence, denoting the remainders by Ri and Rj, P = (a;+2)Q + Ri, Q= -xRi + Ra. From these successively we get R, = P-(j' + 2)Q, l=Rj = Q + ,.Ri, = Q + .fP-,--j(x + 2)Q, 1 =(-a;2-2x+l)Q + aP (1). In this case, therefore, U=-x^-'2x + l, L = x. Multiplying now by A/PQ on both sides of (1), Ave obtain (putting in the actual values of P and Q in the present case) {x' + l){-x'--2z + l) {x' + n,: . x^ + Sx' + ix + l ic^ + x + l' -j'''-2^'^ + sc*-x--2x + l x'> + x X ' + 'ix' + 2k + 1 x' + x + l' or, carrying out the two divisions. vm SPECIAL CASES 153 --^ + ---1+ ^ + 3^^1 + 1 +-^ -^-^^-^^^Vl' or, seeing that the integral part vanishes, as it ought to do, F= ^'+.'^ I -1 a? + Sx^ + 2x + l x'^ + x + l which is the required decomposition of F into partial fractions. Cor. i/' P, Q, E, S, . . . he integral functions of x which are prime to each other, then any proper rational fraction A/PQES . . . can he decomposed into a sum of proper fractions, P'/P + Q'/Q + E7E + S7S + . . . This can be proved by repeated applications of the main theorem. § 7.] Having shovirn a priori the possibility of decomposition into partial fractions, we have nov7 to examine the special oases that occur, and to indicate briefer methods of obtaining results which we know must exist. We have already stated that it may be shown that every integral function B may be resolved into prime factors with real coefficients, which belong to one or other of the types (x - a)'', (/ + /3x + yY. 1st. Take the case where there is a single, not repeated, factor, X- a. Then the fraction F = A/B may be written (a; - a)Q say, where x - a and Q are prime to each other. Hence, by our general theorem, we may write X — a ^5 each member being a proper fraction. In this case the degree of P' must be zero, that is, P' is a constant. It may be determined by methods similar to those used in chap, v., § 21. See below, Example 1. P' determined, we go on to decompose the proper fraction Q'/Q, by considering the other factors in its denominator. 154 SPECIAL CASES chap, 2nd. Suppose there is a repeated factor {.r - a)*" ; say B = {x - a)''Q, where Q does not contain the factor x - a. We may, by the general principle, write {X - ay Q P' is now an integral function, whose degree is less than r: hence, by chap, v., § 21, we may put it into the form P' = «„ + ai(a; - a) + . . . + a,._j(a; - a)''~\ and therefore write F = ^_ + ":^ + + -'*-^' + 9l. (2) {x-aY {x-af-^ a;-a Q ^'' where a„ a„ ., ffi,._i are constants to be determined. See below. Example 2. 3rd. Let there be a factor {o? + /3x + y)", so that 'B = {x' + fSx + yYQ, Q being prime to of + fix + y. Now, we have r- ^- +q: {x' + jix + yy Q" P' is in this case an integral function of degree 2s - 1 at most. We may therefore write, see chap, v., § 21, P' = (a„ + 5„a;) + {a^ + h,.r) {x' + /3x + y) i- (rt^ + h^x) {a? + /3x + yY + ((h., + h,_,,:) (x? + fir. + yY-\ We thus have ^ «o + h„r. «. + h,x a,., + b„_,x Q' {x' + fSx + yy (/ + ^a; + y)«-i ■ • x' + px + y^ q ^•^>' where the 2s constants a„, b„, &c., have to be determined by any appropriate methods. See Examples 3 and 4. In the particular case where s= 1, we have, of course, merely x' + /3x + y q (*)■ VIII EXPANSION THEOREM 155 By operating successively in the way indicated we can decompose every rational fraction into a sum of partial fractions, each of which belongs to one or other of the two types Pr/(x - a)'', (ftg + bsx)/{x^ + j3x + yY, where a, /3, y, p„ ag, bs are all real constants, and r and s positive integers. It is important to remark that each such partial fraction has a separate and independent existence, and that if necessary or convenient the constant or constants belonging to it can be determined quite independently of the others. Cor. If F be an integral function of x of the nth degree, and a, a, . . . , a; fi, jB, . . . , j8 ; y, y, . . . , y, . . . constants not less than n+ I in number, r of which are equal to a, s equal to f3, t equal toy,..., then we can always express P in the form P = 2{(i„ + «i(a; - a) + . . . + ar_i(x - a)'"-i}(x - /3)*(a; - y)* • • , where «„, «,,..., a^_j, ... are constants. In particular, if r = 1, s = I, t= 1, . . . , ive have P = ^a„{x -j3){x-y)... These theorems follow at once, if we consider the fraction Yjix - ay{x - py(x - y)* . . . There is obviously a corresponding theorem where x - a, X- /3, X - y are replaced by any integral functions which are prime to each other, and the sum of whose degrees is not less than n+ I. § 8.] We now proceed to exemplify the practical carrying out of the above theoretical process ; and we recommend the student to study carefully the examples given, as they afford a capital illustration of the superior power of general principles as contrasted with " rule of thumb " in Algebra. Example 1. It is required to determine the partial fraction, corresponding to a; - 1, in the decomposition of {ix^~iex^ + 17x^-Sx + 7)/{x-l){x-2]''{x'+l). We have ix*-16x^ + '\.'!x'^-8x + 7 _ p Q' ,j, {x-l)(x-2)^{x^ + l) ~x-l {x-2f{x^+i) ^ '' and we have to find the constant^. 156 EXAMPLES CHAP. From the identity (1), multiplying both sides by [x-l) {x-2)-{x'' + l), we deduce the identity ix-* - 16x^ + 17.1" -8x+7 =p(x - 2)2(a;2 + 1) + q'{x - 1) (2). Now (2) being true for fil values of .>', must hold when x=X ; in this case it becomes 4 = 2p, that is, ^ = 2. Hence the required partial fraction is 2/(a; - 1). If it be required to determine also the integral function Q', this can be done at once by putting _p = 2 in (2), and subtracting 2(«-2)^(,7;^ + l) from both sides. We thus obtain 2.--'-S.r' + 7.'^-l = Q'(a:-l) (3). This being an identity, the left-hand side musi he divisible by x-\.* It is so in point of fact ; and, after carrying out the division, we get 2ar''-G.r- + a- + l = Q' (4), which determines Q'. The student may verify for practice that we do actually have ^,,■^ _ 16,>-= + 17a:=- 8r + 7_ _2_ 23?- 6.'-- + .e+1 "(.•■-\)(x-2f(x^ + \) x-l'^ \x - 2 )- (:>/+ 1 ) ■ Example 2. Taking the same fraction as in Example 1, to determine the group of partial fractions corresponding to (x - 2)-. 1°. We have now 4x '-16.>-' + 17.)"-8j; + 7 _ «o ai (}' ... ~(x-l)(x-2f{,x' + \) ~{x~2.)-' (x-2y (x- V,%f+i) ^ '' whence ix* - 16a:3 ^_ i7-,.-.- _ g_-,. ^_ 7 = „^(,. _ i) ^^.s + j) _j. ^^j,^ -2)[x- 1) (.y- + 1) + Q'(K-2)= (2). In the identity (2) put x=2, and we get — 5 = 5aoi that is, ao~ ~\, Putting now ci„=-l in (2), subtracting (- 1) (.c-1) (a;^ + l) from both sides and dividing both sides by x - 2, we have ix' - Ix- + 2x-i = ai(.c - 1 ) (a!^ + 1 ) + Q'(i- - 2) (3). Put x=2 in this last identity, and there results + 5 = 5ai, that is, «i= +1. The group of partial fractions i'C'(|uired is therefore -ll{x-2f + ll{x-2). If required, Q' may be determined as in Example 1 by means of (3). 2°. Another good method for determining ojo and cii depends on the use of "continued division." If we put is = 2/ + 2 on both sides of (1), we have the identity 4(i/ + 2)^-16(y + 2)^ + 17(?/ + 2)^-8(y + 2)+7 _ao ffii _ Q" (2/ + i)2/M(2/ + 2p+i} y" y (y+i){{.y + 2f+iY If it is not, then there has been a mistake in the working. yni EXAMPLES 157 tliat is, - 5-4y + &c. _ao ai Q" ,.. 5y'' + 9y' + kc. 7/ y (l + y){S + iy + if) ^ '' Now, by chap, v., § 20, the expansion of a rational fraction in descending powers of 1/y and ascending powers of y is unique. Hence, if we perform the operation of ascending continued division on the left, the first two terms must be identical with ao/z/^ + ai/2/ ; for Q"/(l + 2/) (3 + 41/ + !/'') will obviously furnish powers of y merely. "We have -5-4+ . . . I 5 + 9+ . . + 5+ ... 1 -1 + 1+ .^"" therefore ao= - 1) «i= +1- The number of coefficients which we must calculate in the numerator and denominator on the left depends of com-se on the number of coefficients to be determined on the right. Example 3. Lastly, let us determine the partial fraction corresponding to x^ + l in the above fraction. "We must now write ix' - 16s? + 17x' - Sx + 7 _ ax + b Q' . . (a;-l)(a!-2)2(a;2 + l) " a? + l^ {:x-l){x-2)^ ^ '' V. "ftHience, multiplying by {x-\)(x-'2lf, ix^^l&x' + l'x'-ix + l _ (ax + b) [x-l)(x-2) a-'+l ~ a^+'l whence + Q' (2); ix'-\%x+l^ + ^^-^ = {ax + b)(x-t> + '^.^-]\ + Q', X^-tI \ X^ + IJ ,,, ,, 1ax'^ + {1b + a)x + b _, = (ax + b){x- 6) ^ ^--iTl ^ ' = (ax + b)(x-i) + 1ri + ' i^-p^ ^' + Q (3). Now the proper fractions on the two sides of (3) must be equal — that U, we must have the identity (lb + a)x + {b-7a) = ?,x- 6, therefore Tb + a = &, b-7a=-6. Multiplying these two equations by 7 and by 1 and adding, we get 60& = 60, that is, J = l. Either of them then gives o = l, hence the required partial fraction is ix + -i)/{x^ + l). 2°, Another method for obtaining this result is as follows. Remembering tha,tx'' + '\. = {x + i]{x-i) (see chap, vii.), we see thatx= + l ' vanishes when x = i. Now we have ix^-16x? + lTx''-8x + 7 = (ax + b) (cc- 1) (a:- 2)2-j-Q'(k^ + 1) = {ax + b){x'-l,x"- + 8x-i) + q'{x^- + l) (4). 158 EXAMPLES CHAP. Put ill this identity x=i, and observe tliat jJ = t2xi? = (-l)x(-l)=+l, 'p='p>t.i = (-l)'x.i— - i ; and we have 8i-6 = (csi + 6) (7i + l), = (lh + a)i + {b~1a) ; whence (76 + a-8)i= -6 + 7a-6, an equality which is impossible" unless both sides are zero, hence 7i + a-S = 0, -J- + 7a-6 = 0, from which a and b may be determined as before. 3°. Another method of finding a and b might be used in the present case. We suppose that the partial fractions corresponding to all the factors except x^ + 1 have already been determined. We can then write „ 2 1 1 cix + h .... F = 1 \-—. — (•')■ x-l (x-if x-l X-+1 From this we obtain the identity 4,.''-lG..-' + 17a"-S,f' + 7 = 2(a;-2)-(,<° + l)-(.c-l)(K= + l) + (a:-l)(x-2)(a2 + l) + ((Kc + S) (.« - l)(x-if; whence 3-* - 4.r'' + ix'^+ix-i = (ax + b)(x-l)(x-2)-; and, dividing by (x - 1) (x - 2)^, x + \ = ax + b. This being of course an identity, we must have ,(, = 1, i!, = l. Another process for finding the constants in all the partial fractions depends on the method of equating coefHcients (see chap, v., § 16), and leads to their determination by the solution of an equal number of simultaneous equations of the 1st degree. The following simple case will sufficiently illustrate this method. Example 4. To decomiiose (3:f - i)l(x -l){x-2) into partial fractions. We have 3jj-4 _ a b (aT- 1 Ki^ 2 ) " zT^ "*" K^ ' therefore ?iX-i = a(x-2) + b(x-\), = (a + b)x-(1a + b). Hence, since this last equation is an identity, we have a + b = Z, 'i.a + b-i. Hence, solving these equations for u, and b (see chap. xvi. ), we find a = l, b = 2. * For no real multiple (ditfering from zero) of the imaginary unit can be a real quantity. See above, chap, vii., § 6, The student should recur to this case again after reading the chapter on Complex Numbers. vm EXERCISES XII 159 Example 5. We give another instruotive example. To decompose y_ x^+px + q _ {x -a){x- b) {x-c)' we may write x^+px + q _ A B C {x-a){x-b){x-c)~x-a x-b x-c ^ '' where A, B, C are constants. Now a;2 +px + q = ^{x -b){x-c] + B(a; - c) (x -a) + C{x-a){x-h) (2). Herein pnt x = a, and there results a^+pa + q^ .\{a-b){a-c) ; , , a'+pa + q whence A = ; ~ -, . (a - b) [a - c] By symmetry We have therefore x^+px + q b'^+pb + q ~~{b-a){b-c)' c'+pc + q {c-a){c- b)' {x -a)(x- b) {x - c) _ a^+pa + q h'^+pb + q r?+pc + q ~{a-b){a-c){x-ay (b-c)(b-a)(x-b)'^ (c~a)(c~b)(x-c) ^'' an identity already established above, § 5, Example 6. It may strike the student as noteworthy that it is more easily established by the inverse than by the direct process. The method of partial fractions is in point of fact a fruitful source of complicated algebraical identities. Exercises XII. Express the following as rational fractions at their lowest terms. (1.) (ii? + 'ix'^-x + G)l{ai^-x' + ix-i). (2.) (9K3 + 53K2-9a;-18)/(4a:2 + 44a3 + 120). ai^ + 2x?-1x-\ ai> + x^-3x''-5x~2 ^^■' 5I+5*~'3a;2-5K-2 "^ + 2a? - 2ir - 1 " (4.) (3s?-x^-x-l)l{3a? + 5x^ + Sx + l) + (a? + Sx^ + 5x + 3)/{af + x^ + x-Z). (5.) (a:5-2a? + l)/(a;2-2a; + l) + (a;H2a7* + l)/(a;H2a; + l). (6. ) {6a? + 13rf - Sa'^x - Wa^)l{9T> + Vlax^ - \\a?x - lOffi^). (7.) {\-a']l{{\ + axf-{a + xf}. (8.) {(w + a; + 3)(w + a;)-2/(2/ + «)}/{(M' + a: + s)(M' + 2)-2/(a! + 2/)}. '^•^ (l-K)(l-a;2)V 1(1^?" (1-*^)(1 -»')"*" (T^^'/' (:0.) {{fll + bmf + {a'm,-blf)l{[ap + bqf'\[aq-bpf]. 160 EXERCISES XII (11. (12. (14. (15. (16. (ir. (18. (19. (20. (21. (22. (23. (24. (25. (26. (27. (28. (29. (30. (31. (32. (33. (34. (35. {px- + {kj-s)j^ r }°- { ]a ~ + {k + s) x + r } ^ {px^ + {k + t}x + r}^ - {pi^ + {k-i)x + r\-' x-y (13.) 1 2.S -iy 2y-2x \l(a-2h- Ilia - 26 - 1/(« - 2b) ) ). l/(6i; + 6) - 1/(2a' - 2) + 4/(3 - 3a-). xr'-y' o:-y ( x + y I "1 k'-i/' x--y- -\x- + y^ -e + y]' I X 1 - .r\ / / X 1-x \ \\ + x X ) I \1 +x x I' -b-(a,-b)x a + b + (a + b)x (,::■ 30a;- + 4.r 3.f - 2 ~ 9a- + 4 " 2 1 2 4^-_ "3j;'T2' 2 2x + l x (x + lf (x + lf (x+iy 1 5 1 1 24(..'-l)"'"8(x + l)"'"4(j- + l)- 2(:^' + l)'' 3(y- + .<- + lV 1 12 2 (x + l)-[x + 2f (x + 2f x + 1 x + 2' (a + b)j{x + a) + {a- b)l{x -a)- 2a{x + b)/(x^ + a-}. {{x-y)l{x + y]} + {{x-y)l{x + i/)y-+ {[x-i/yix + y]]". / .■-■- 3,7' + 2 \ ^ /,ij ^ + 2..+ l\ \.r' + -2y- + 2x+l/ \X^-5x+iJ' /(i- + .y- , \ ax- ia(a + x, -y- + x-y' + y^ 1 \("^s;)(r5-)^ {ii,-c)(x-a) 2a[a + c](x + a) (c' - a^) [x + c)' f 20 180 420_ 280^ / 1 _ _?" i^" 1?2_ 2^" \ \ +.-F+T 1^ + 2 + ^ + 3 x+iJ^'X »,'-l+..-2"..--3+x-4j' [(xy-T.)'' + {x + y-2){x + y-2xy))/{{.ry + l)"-{x + yr-}. {l + y^ + ^-Syz)l{l + y + z]. {u{,i + 2b)+b{h + 2c) + c{c + 2a)}/{a'~h'~c''-2hc}. (a + lf + {b + cY-{a + 2b + cf {a + b)(b + c)(a + 2b + c) :/' + r(" a-x- {i-'' + a''){x^-a-) + aV^i _ „*) "*" .7» - «» - aV-"(«- wt-J ft = + ( 2ac - &- l.f ^ + f- J '' a^ + ( ac - J^ ).i'^ ^Sca;' «- + 2ubj- + {2ac + b'')x^ + 26^ + c'-.t"' a' + (ac - b-)x-' + bcx'' EXERCISES XII 161 ' x^ + y'^+x + y-xy + \ x^- + y^ + x-y + xy + l x-y-l x+y-1 ,„„ , ( a° - 1 Ox?y^ + 5xy ^f + { 5x*y -Wx"y' + f^f (S + c)H2(52-c5) + ( .^2 (*^-2.-+^^){,i!,).+i.!,.+(,-:hp}- (39.) 2{b'^ + c'-~a^)/{a-b){a-c). (40.) (i:x){Sx?)/xyz-X{y + zyx. (41.) S(6 + c)/(c-a)(«-6). (42.) S6c(a + A)/(a - ft) (a - c). (43.) 2(&2 + Jc + c2)/(a-J)(a-c). (44.) {Tia-x^) + Il{x-yz)}/{l-xyz). (45.) {S(J + c)3-3n(& + c)}/{2a3-3a5c}. (46 ) !-=« I '"-y I y-1 I (l-a^)(«^-2/)(2/-l) ■^ 1+K x + y y + 1 {l+x){x + y)(y + iy (2/ - 3) (s - x) (03 - 2/) \y-s z-x x-y) iAa\ ft - c , c - tt , a - S , (6 - e) (c - a) (a - ft) x-a x-b x — c {x — a){x — bj{x — c) (49.) 2(a+^)(a + g')/(a-&)(a-c)(a + 7i). (50.) 2a2/(a-6)(a-c)(A-a). (51.) ■Sa''j{a?-b^){a'-c^){h^ + a^). (52. ) 2(2/2 + z2 _ x^)lyz{x - y) [x - z). . . a{b-cf + iic-af + c{a- bf + {b^-c^){b-c) + {c-'~a^)(c-a) + (a'-b^)ia- b) ^ '' " a\b-c) + b\c-a) + c\a-b) {(x + yf + (y + zy }{{z + xf + {x + wf} (54.) {{x + tj){s+x) + {y+z){x + w)}^+{{x+y){x + w)-{y + z){z + x)]^ Prove the following identities : — (55.),2a3/(a-J)(a-c) = Sa. (56.) c{u^-v)=au(l-uv), c(v'-u) = bv{l-uv), where u = (ab - c^)/{bc -a^), v={ab-c^)/{ca- V). (57. ) 2 (a + o) (a + ^) (a + 7)/a(a - 6) (a - c) (a -d)=- a^yjabcd. abed {b-cf + {c-af + {a-bY {ab - cd) {a? -b^ + c'-(P) + {ac - bd) (a" + b^-c-- cP) ^ ' (a^ - J2 + c2 - d') (a^ + 62 - t;2 _ d2) ^. 4(„J _ ^,^) ((,(. _ Jrf) (i!i + c)(a + t;) ~(J + c)^ + (a + rf)2" (60.) !e^^MfLz£}+£!M=_Za^-26c=-a6c. ^ ' (c-6)(a-c)(o-a) (61.) {2(2/-2)8}/{2(2/-«)2}-4n(i/-2)=={2^2-22/=}3. Decompose the following into sums of partial fractions : — (62.) (a?'-l)/(a;-2)(a;-3). (63.) x''-l{x~l){x-2){x-Z). (64.) 30a?/(K2_i)(a,2_4)_ (g5_) (a;2 + 4)/((Z! + l)2(a:-2)(a^ + 3). VOL. I M 162 EXEKCISES XII CHAP. VIII (G6.) {x'-2)l(siP~l). (67.) {x' + x + l)l{x + l){x" + l). (08.) (2x-S)l{z-l){x^ + l)-. (69.) lj{x-a){x-b){x"-2px + q), p^? + x7-x> -!>?). (74.) E.xpress (3a^ + a;+l)/(a::*-l) as the sum of two rational fractions whose denominators are ar"- 1 and X^ + 1. (7.5.) Expand 1/(3 -a;) (2 + a;) in a series of ascending powers of x, using partial fractions and continued division. (76.) Expand in like manner lj{l-xy{l+x^). (77.) Show that 2 {h + c + d)j{b-a){c - a)(d-a){x -a) = {x-a-b-c-d)j{x - a){x - h){x - c)(x - d). . CHAPTEE IX. Further Application to the Theory of Numbers. ON THE VARIOUS WAYS OF REPRESENTING INTEGRAL AND FRACTIONAL NUMBERS. § 1.] The following general theorem lies at the root of the theory of the representation of numbers by means of a systematic scale of notation : — Let i\, r^, 1\, . . ., r„, r^+i, . . . denote an infinite series of integers* restricted in no way except that each is to he greater than 1, then any integer N may he expressed in the finite form — where po'-i +P2'r,r^ + . . . +Pn'rj-^ ■ ■ ■ ''„) (C). But the two brackets on the right and left of (0) contain integers, and po/r, and Po/'>\ are, by hypothesis, each a proper fraction. Hence we must have p„/?-, -pali\ ; that is. Pi +P2'r2 +P2r„i\ + . . . +PnV3 ••■»•» = p,' +p,'r, + 2h'i:^'3 + - ■ ■+P,ir.n---rn (D). Proceeding now with (D) as we did before with (C), we shall ])rove Pi=2-h''j and so on. In other words, the two expressions (A) and (B) are identical. Example. Let 1^ = 719, and let the numbers ri, r.., r^, . . . be the natural series 2, 3, 4, 5, . Carrying out the divisions indicated above, we have 2 )719 3 ;j.:9 . . 1 4 )119 . . 2 6)^. . 3 5 . 4. IX FACTORIAL SERIES FOR A FRACTION 165 Hence pi, = l, pi—2, P2 = S, P3=i, Pi=5 ; and we have 719 = 1 + 2x2 + 3x2.3 + 4x2.3.4 + 5x2.3.4.5, § 2.] There is a corresponding proposition for resolving a fraction, namely, i\, r^, . . ., r„, &c., being as before. Any proper fraction A/B can be expressed in the form B r, r.r, r,r,r^ r,r, ...?•„ where pi+7 2 2.4 2. 4^ 2.4" 2. 4". 3' Example 5. 7 1,1 + i 29 5 5.5 5.5.29' 1111 1 1 1 1 . ^^'° =5 + 6:6 + 5^6 + 5rP + 55:65 + 5r65 + 5T:6»+i;r6i + *'=- _1 J_ 1 1 ~6"'"6.3"''6.3.3"'"6.3.3.29' an di § 3.] The most important practical case of tlie proposition in § 1 is that where r,, r^, . . . are all equal, say each =?'. Then we have this result — Every integer N can be expressed, and that in one way only, in the fm'm whereto, j7,, . . ., ^„ are each < r. In other words, detaching the coefficients, and agreeing that their position shall indicate the power of r which they multiply, and that apposition shall indicate addition (and not multiplica- tion as usual), wo see that, r being any integer whatever chosen 168 SCALES OF DOTATION, INTKGERS ciiaP. as the radix of a scale of notation, any integer whatever may be represented in the form PuPn - 1 • • • PiPo \ where each of the letters or digits f^-, P\^ • • ■> Vn must have some one of the integral vakies 0, 1, 3, 3, . . .,r -\. For example, if ?•= 10, any integer may be represented by PnPn-\ • ■ • PiPo where Po,Pi, ■ ■ -jPn have each some one of the values 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. The process of § 1 at once furnishes us with a rule for finding successively the digits j>„,j?,,j)2, . . ., naxaely, Divide the gircn integer N hj tlie cJiosen radix r, the remainder will he p„ , divide the integral quotient of last division hy r, the remainder uill be p^ , a7id so on. Usuallj-, of course, the integer N will be given expressed in some particular scale, say the ordinary one whose radix is 1 ; and it will be required to express it in some other scale whose radix is given. In that case the operations will be carried on in the given scale. The student will of course perceive that all the rules of ordi- nary decimal arithmetic are applicable to arithmetic in any scale, the only difference being that, in the scale of 7 say, there are only 7 digits, 0, 1, 2, 3, 4, 5, 6, and that the "carriages" go by 7's and not by lO's. If the radix of the scale exceeds 10, new symbols must of course be invented to represent the digits. In the scale of 1 2, for example, digits must be used for 10 and 11, say t for 10 and e for 1 1 . Examiilc 1. To convert 136991 (radix 10) into the scale of 12. 12 )136991 1 2)11415 . . e 1 2)951 ... 3 12]79 . . 3 6 ... 7 The result is 6733f. Example 2. To convert 6733e (radix 12) into the scale of t. T)6733f T)7fl7 ... 1 r)961 ... 9 rjrf . . 9 t]!! . 6 1 ... 3 The result is 136991. IX EXAMl'LES OF ARITHMETICAL OPERATION 169 Although this method is good practice, the student may very probably prefer the following ; — 6733e (radix 12) means 6x12' + 7x123 + 3x122 + 3x12 + 11. Using the process_.of chap, v., § 13, Example 1, we have 6+ 7+ 3+ 3+ 11 + 72 + 948 + 11412 + 136980 6 + 79 + 951 + 11415 + 136991. § 4.] From one point of view the simplest scale of notation would be that which involves the fewest digits. In this respect the binary scale possesses great advantages, for in it every digit is either or 1. For example, 365 expressed in this scale is 101101101. All arithmetical operations then reduce to the addition of units. The counterbalancing disadvantage is the enormous length of the notation when the numbers are at all large. With any radix whatever we can dispense with the latter part of the digits allowable in that scale provided we allow the use of negative digits. For let the radix be r, then whenever, on dividing by r, the positive remainder p is greater than r/2, we can add unity to the quotient and take - (r - p) for a negative remainder, where of course r-p = «, +^-+-^ + . . . r r =p, + (/-i say. Now (/i, is a proper fraction, hence 'p^ is the integral part of r. Again rcl>,=j),+^- + . . . =p^ + cl>^ say. So that j?2 is the integral part of r^„ and so on. It is obvious that a vulgar fraction in any scale of notation must transform into a vulgar fraction in any other ; and we shall operational use of the decimal point ; and in his Oonstructio (written long before his death, although not published till 1619) it is frequently used. See Glaisher, Art. "'Sa.^i^T," EncydnjisalUi Briiannica, 9th ed. ; also Rae's recent translation of the Conslrudia, p. 89. EXAMPLES 173 show in a later chapter (see Geometrical Progression) that every repeating radix fraction can be represented by a vulgar fraction. Hence it is clear that every fraction which is a terminating or a repeating radix fraction in any scale can be represented in any other scale by a radix fraction which either terminates or else repeats. It is not, however, true that a terminating I'adix fraction always transforms into a terminating radix fraction or a repeater into a repeater. Non-terminating non-repeating radix fractions transform, of course, into non-terminating non-repeating radix fractions, otherwise we should have the absurdity that a vulgar fraction can be transformed into a non-terminating non-repeating radix fraction. It is obvious that all the rules for operating with decimal fractions apply to radix fractions generally. Example 1. Reduce 3'168 and 11'346 to the scale of 7, and multiply the latter by the former in that scale ; the work to be accurate to 1/lOOOth throughout. The required degree of accuracy involves the 5th place after the radical point in the scale of 7. •168 •346 7 7 1)-176 2)-422 7 7 l)-232 2)^954 7 7 l)-624 6)^678 7 7 4)-368 4)^746 7 7 2)-576 5)^222 3 = 3-11142 (radix 7). 11^346 = 14^22645. 14-22645 3-11142 46-01601 1-42265 14227 1423 632 32 60-64146 174 REMAINDER ON DIVIDING BV r~ 1 CHAl'. On account of the duodecimal division of the English foot into 12 inches, the duodecimal scale is sometimes convenient in mensuration. ICxaniple 2. Find the number of square feet and inches in a rectangular carpet, whose dimensions are 21' Zy by 13' llf". Expressing these lengths in feet and duodecimals of a foot, we have 21' 34"= 19-36. 13'llJ"=ll-e9. If, following Oughtred's arrangement, we reverse the multiplier, and put the iinit figure under the last decimal place which is to be regarded, the calculation runs thus — 19-36 9611 19360 1936 1763 13e 209-78 209 (radix 12) = 288 + 9 = 297 (radix 10) feet. -78 (radix 12) = 7 x 12 + 8 = 92 square inches. Hence the area is 297 feet 92 inches. § 7.] If a number N he expressed in the scale of r, and if we diriile N and the sum of its digits by r - 1, or by any factor of r - 1, the rrmaimler is the same in both cases. Let N =^o+i^i'" +i'2'"^ + ■ • ■+i')i'''"'- Hence H -{2}o+p^ + . . .+/<„) =|Ji(r - 1) +^2(/ - 1) + . . . + i',.(r"-l) (1). Now, m being an integer, r" - 1 is divisible by r - 1 (see chap, v., § 17). Hence every term on the right is divisible by r - 1, and therefore by any factor of r - 1. Hence, p being r - 1, or any factor of it, and fi some integer, we have N-0'„ +!>, + • • .+Pn) = H'P (2). Suppose now that the remainder, when N is divided by p, is cr, so that N = i/p + tr. Then (2) gives Po+lJ> + ■ ■ ■ +Pn = {v-p)p + st — 2722nCG■2-^568 = 47923 + 398^-568, 01- 27220662 = 47923x568 + 398. Here L'- P' = 0-2= -2, Jl'. N' = 7xl = 9-2. The test is therefore satisficiL § 9.]* The following is another interesting method for ex- panding any proper fraction A/B in a series of fractions with unit numerators : — Let q^, q^, q^, . . ., qn, «'"'' '"n r.,, r.„ . . ., r„, be the quotients and remainders respectively ivhen B is divided by A, r,, r.,, . . ., 7-„_i re- specti.rely, then _ = + ..+^ ^ +F (1), vJtere F = (- l)"i'n/l\1i- ■ ■ 'ii/,13) "'"' *s, F is numerically less than i/q^q. ■ ■ ■ q,i- For we have by hypothesis B = Aq, + r,, therefore A/B = 1/q, - r,/g,B (2), B = r/i^ + r„, therefore rjl) = l/qs- r^/qji (3), B = r.,q.^ + r.„ therefore r^/B = Ijq^ - rJq^B (4), and so on. From (2), (.3), (4), we have successively A_1_J_ _!__/»■ ~1i~q, q,q., J.^Ab - i _ J_ ^ 1 _ J^ h\ and so on. Since r,, r,, . ., f,„ go on diminishing, it is obvious that, if A and 13 be integers as above supposed, the process of successive division must come to a stop, the last remainder being 0. Hence * In liis Essai d' Analyse Xumirique sur la Transformation des Fractions {aSurres, t. vii. p. 313), on wliich the present chapter is founded, Lagrange attributes the theorem of § 9 to Lambert (1728-1777). Heis, Sainmlung von Beispielen und Aufgaten aus der allgemeinen Arithmetik mid A Igebra ( 1 882), p. 322, has applied series of this character to express incommensurable numbers such as logarithms, square roots, &c. In the same connection see also Syl- vester, American Jour. Math., 1880. Sec also Cyp. St(5phanos, Bull. Sac. Math. Fr. 7 (1879), p. 81 ; G. Cantor, Zeilseh. f. Math. 14 (1869), jj. 124 ; J. Liiroth, Math. Ann. 21 (1883), p. 411. IX EXERCISES XIII 177 every vulgar fraction can be converted into a terminating series of the form 1 1 1 Example. 113 1 1 7 + 244 2 2.13 2.13.24 2.13.24.61 From tliis resolution we conclude that 1/2-1/2.13 represents 113/244 within l/26th, and that 1/2-1/2.13 + 1/2.13.24 represents 113/244 within l/624th. ■ EXEECISES XIII. (1.) Express 16935 (scale of 10) in the scale of 7. (2.) Express 16 '935 (scale of 10) in the scale of 7. (3.) Express 315-34 (scale of 10) in the scale of 11. (4.) Express T7e9ee (scale of 12) in the scale of 10. (5.) Express lT8e54 (scale of 12) in the scale of 9. (6.) Express 345 '361 (scale of 7) in the scale of 3. (7.) Express 112/315 (scale of 10) as a radix fraction in the scale of 6. (8.) Express 3169 in the form ^ + 2'3 + ?'3.5 + s3.5.7 + &c., where p{{x''''y]i, X VERIFICATION OF THE LAWS FOR X^'^'^ 183 all by the laws for positive integral indices, regarding which there is no question. Now, by the meanings assigned to a;^'* and a;'''', we have (xPlt)i = xP and {w""'")" = x''. Hence g?s = (xpy{xy, = xJ?'+i'; by the laws for positive integral indices. It now follows that it is the qsih. root of xP^+'i^' ; and, since z is real and positive, it must be that jsth root which we denote by 3^ps+, q, r, s are posi- tive integers, and let z = {xPliy'". Then, since, by hypothesis, xpI^ is real and i^ositive, therefore (xPlqyis^ that is z, is real and positive. Also = [{{xpiiy^yy, by laws for positive integral indices ; = [{xP'iyy, by definition of a fractional index ; = {xi'l''}i^, = [{xP'iy]''; by laws of positive integral indices ; by definition of a fractional index ; = xP'; by laws of positive integral indices. Hence .? is a g'sth root of xP'', and, since z is real and positive, we must have Z = XP'^l1'', that is, z = x(Pl'i>'-''l'\ 184 PARADOXES ruAP. Lastly, to prove Law III. (a), let Then, since, by hypothesis, .rP^'J and y**'' are each real and positive, z is real and positive. Also z^ = (xpIi yP'!')5', = {xPl^)i(ijPli)i, by laws for positive integral indices ; = xPif, by definition for a fractional index ; = (,,7/)P by laws for positive integral indices. Hence a is a qth root of {xi/)p ; and, since z is real and positive, we must have z = {xyyii. The proof is obviously applicable where there is any number of factors, X, y, . . . § 4.] Although it is not logically necessary to give separate proofs of Laws L (yS) and III. (/?), the reader should as an exercise construct independent proofs of these laws for himself. It should be noticed that in last paragraph we have supposed both the indices pjq and rjs to be fractions. The case where either is an integer is mot by supposing either g = 1 or s = 1 ; the only eflfcct on the above demonstrations is to simplify some of the steps. § 5.] Before passing on to another case it may be well to call attention to jsaradoxes that arise if the strict limitation as to sign of xvli be departed from. By the interpretation of a fractional index 4/2 2/4 1 2 x' = ijx = ±x. But x'>- = x\ which is right if we take a;'*'' to stand for the positive value of -c/;/:'' ; but leads to the paradox x^ = - xJ if we admit the negative value. A similar diflBculty would arise in the application of the law, (,(■'")" = a;""' = (.1;")'" ; 185 for example, (4^)' = (4')^ would lead to ( ± 2)^ = ± 4, that is, 4 = ± 4, if both values were admitted. Such difBculties are always apt to arise with a-^'^' where the fraction _p/2 is not at its lowest terms. The true way out of all such difficulties is to define and discuss a" as a continuously varying function of n, which is called the exponential function. In the meantime fractional indices are introduced merely as a convenient notation in dealing with quantities which are (either in form or in essence) irrational ; and for such purposes the limited view we have given will be sufficient. § 6.] Case of a". This case arises naturally as the extreme case of Law I. (/3), when n = m; for, if we are to maintain that law intact, we must have, provided a; 4= 0,* that is, x^ =\. This interpretation is clearly consistent with Law I. (a), for simply means a;™ X 1 = cc™, which is true, whatever the interjjretation of x™ may be. Again, a;™" = (a™)", that is rsO = (a;™)", simply means 1 = 1 by our interpretation ; and a;™^ = (a;")'"', or x" = (x")'", gives 1 = 1™, which is right, even if m be a positive fraction, provided we adopt the properly restricted interpretation of a fractional index given above. The interpretation is therefore consistent with II. The interpretation x" = 1 is also consistent with III. (a), for xy = {xyf simply means 1x1 = 1. * This provision is important since the form 0" is indeterminate (see chap. XXV.) 186 NEGATIVE INDICES rii.vi'. § 7.] Case of .r-'", where m is any real positive (or signless) number, and .r. 4= 0. Let z = x''"^, then, since ,)•■" =t= 0, we have if Law L (a) is to hold for negative indices. Whence z = a;7a;™ = 1/.,;'" by last paragraph. In other words, .c"'" is the reciprocal of »;'"'. As an example of the reconciliation of this with the other laws, let us prove I. (a), say that ^j. - JH J. - It _ y- 111- It. By our definition, we have x-'^x-"- = {\j',i-™){llx"), = 1 /,»:'" j:", = l/x'"+", the last step by the laws already established for all positive indices ; „ -(m+ii) , by definition of a negative index. Hence k -"» a; ""■ = ,«-'"-". In like manner we could show that The verification of the other laws ma}' be left as an exercise. § 8.] The student should render himself familiar with the expression of the results of the laws of indices in the equivalent forms with radicals ; and should also, as an exercise, work out demonstrations of these results without using fractional indices at all. For example, he should prove directly that K/^r^x^'l/y-^i (1); 4/1 Vr^^'V = V'"' = V{ i/A^' (2) ; yx-^//^,:'= V('"/-) (3); ;/a-/q) + n{ \/qf, are integral functions of sjp, Jq, tjr, of the 1st and 2nd degrees respectively, provided I, m, n do not contain Jp, Jq, sir. Again, (Z \/p + m \/q)/{l \/q + m \/r) is a rational, but not integral, function of these irrationals. \/(/ \lp + m \/q), on the other hand, is an irrational function of \/p and \/q. The same ideas may also be applied to higher irrationals, such as J)!/™, q^''\ &c. § 11.] Confining ourselves for the present to quadratic irrationals, we shall show that every rational function of a given set of quadratic irrationals, v^p, \/q, \/r, &c., can be 190 RATIONALISING FACTORS CHAP. reduced to a linear integral function of the square roots of p, q, r, and of their products, pq, pr, qr, pqr, &c. This reduction is effected mainly by means of rationalising factors, whose nature and use we proceed to explain. If T he any integral function of certain given irrationals, and Q another integral function of the same, such that the product QP is rational so far as tlie gken irrationals are concerned, tlien Q is called a rationalising factor of P with respect to the given irrationals. It is, of course, obvious that, if one rationalising factor, Q, has been obtained, we may obtain as many others as we please by multiplying Q by any rational factor. § 12.] Case of Monomials. 1°. Suppose we have only quadratic irrational forms to deal with, say two such, namely, p^ and q^. Then the most general monomial integral function of these is I = AOi)2"»+i(2i)2«+i, where A is rational. There is no need to consider even indices, since ^i)2m_^m jg rational. Now I reduces to I = (Ap™g«)j9''2i, where the part within brackets is rational. Hence a rationalising factor is jo%-, for we have which is rational. Example. A rationalising factor of 16 . 2" . 3* . 5" is 2-3"0", tliat is, (30) . 2°. .Suppose we have the irrationals j>^'*, g^'*, ?•■'/", say, and con.sider I = Aio"' o'"'* r"'" * which is the most general monomial integral function of these. A rationalising factor clearly is gnl - lis pi - in It n.1 - nfu or «(s - 0/s o(* - m)lt ,.(« - n)ln * ^YIlere of conr.se lJq. For, if I = a v/p + b ijq, = c^p - b'q, which is rational The two fwms a \/p + I \/q and a ^/p - b \/q are said to be con- jugate to each other with reference to \/q, and we see that any binomial integral function of quadratic irrationals is rationalised hy multiplying it by its conjugate. 2°. Let us consider the forms op"/'' ± bq^l^, to which binomial integral functions of given irrationals can always be reduced. ■' Let X = ap"l^, y = I = op^/y - = x-y. Let m be the L.C.M. of the two integers, y, &. Now, using the formula established in chap, iv., § 16, we have Here x™ - y™' = {iC^ p'^'^ly - b™ g™*^'*), where majy and OTy8/8 are integers, since m is divisible by both 7 and 8, that is, a™ - y"^ is rational. A rationalising factor is therefore a'^'^ + x™--'^y + . . + xy^''^ + y"^'^, in which x is to be replaced by ap"'^, and y by hq^'^ Tartaglia's problem. See Cosmli Storia delV Algebra (1797), vol. ii. p. 192 RATIONALISING FACTORS CHAP. The form af'''^ + 6/'^ may be treated in like manner by means of formulse (4) or (5) of chap, iv., § 16. E.xample. I = 3.2*-4.3*. Here m=6, re = 3. 2*, 2/=4.3*; and a rationalising factor is = 3^2^ + 3-'.4.2*.3* + 33.4=.2.3* + 3".4^2^.3^ + 3.4^.2\3^ + 4''.3'', = 3'.2.2' + 3-'.8.2*.3* + 3".32.3* + 32.43.2l3* + 3.4-'.2*.3' + 4'>.3'. § 14. J Trinomials with Quadratic Irrationals. This case is somewhat more compHcated. Let I = Vp + \/q+ V'' ; * and let us first attempt to get rid of the irrational >/?•. This we can do by multiplying by the conjugate of \/ji + s/q+ \/r i\'ith respect to V'', namely, \/p + \/2 - \/''- We then have (Jp+ ^/q- ^r)l = {Jp+ ^/qY - ( ^.'rf, =p + q-r+2^/{pq) (1). To get rid of ij{pq) we must multiply by the conjugate of p + q-r + 2 \/{pq) with respect to \/{pq)- Thus finally {2} + q-r-2 s/{pq)}{ ^/p + Jq- v/r)! = (p + q- rf - {2 s/{pq)Y, = 'f + ^''!/)■ First of all, proceeding as before, , and attending to Jp alone, we get <^( n'j", Vg) = F + (iJp, wliere P and Q are rational so far as p is concerned, but are irrational as regards q, being each rational integral functions of \/q. Eeducing now each of tliese with reference to Jq we shall obtain, as in (1), P = P' + Q\/fi, Q = P" + Q"v/2, and, finally, A( sV: -^q) = P' + Q' v/'i + (P" + Q" Jq) -./p, = P' + P" ^/p + g; Jq + Q" ^{pq) (2), which proves the proposition for two irrationals. If there be three, we have now to treat P', P", Q', Q" by means of (1), and we shall evidently thereby arrive at the form A + B ^/p + G ^/q + B ^/r + Y. J(qr) +F J{rp) + G '^{pq) + Ti ^'{pqr), and so on. Cor. It follows at once from the process by which we arrived at (1) that <^( - -Jp) = P - Q -Jp- Hence if ( \lp)) he any integral fvudioii. of Jp, { \^p) , and, more gencndlij, if c/)( ^/p, Jq, ^/r, . .) he an integral functiini of \'p,\/q, Jr, . , then, if we take any nai- of than, say Jq, awl rliavge its sign, the product 4>{\''p, Jq, \^'i', ■) ^ */>( '-^P' ~ ^^'/• ^''''> • ■) '*' ndiunal, so far as ijq is concerned. Example 1. If ip(j)-,i? + x" + x + \, find tlie values of cp{\ + ^2,) and 0(1 - V3) and 0(1 + V-) x 0(1 - \/3)- 0(l + V'3) = (l + V3r + (l + \/3)= + (l + \/3) + l, = l + 3V3 + 3. 3 + 3x73 +1+2V3+3 + 1 + V3 + 1, = 1C + '.I\'3 (f(l - \/3) is deduced by writing - \/3 i" place of + \JZ everywhere in the aliove calculation. Hence 0(1-V3) = 16-9V3; 0(1 + V3) X 0(1 - ^i) = {\ef-(9^/i)-, = 2.56-24.3, = ia. X LINEAR FUNCTION OF SQUARE ROOTS RATIONALISED 195 Example 2. Find the value of a^ + 1/^ + z» - .r;/;, when x=\/q-\//; l/ = \/r~\/p, z=^p~^q. Since «: + y + z=\/q- \/r + \/r- s/p + ^Jp~ ^q = 0, we have (chap, iv., § 25, IX.) Xa? - Zxijz = ZxCZx' - Z.rj/), = 0. Therefore "Z.i? - xyz = iryz, = 2( V? - V*-) Wr - ^p) (^p - V?), = 2(!? - i-)^Jp + 2(r -i)) V? + '^{P - ?)\/'-- Examples. Evaluate (l+y + z)(l+z + x)(l+x + y) when x=\/1, y = \/S, {l+y + z){l+z + x){l + x + y) = l + 2{x + y + z) + x^ + {y + s)x + yz + kc.+kc. + x{ff + z") + &c. + &c. + Ixyz, = l+a:2 + 2/2 + := + C2 + j/2 + :^)a: + (2 + s= + a:"))/ + (2 + a!^ + y")z + Zyz + 1,zx + Sas/ + 2:)-7/;, = 11 + 10V2 + 9V3 + 7V5 + 3\/15 + 3Vl0 + 3^/6 + 2^30. § 16.] We can now prove very easily the general proposition indicated above in § 14. If V he the sum of any number of square roots, say Jp, y/q, Jr, . . ., a rationalising factor Q is obtained for P by multiplying together all the different factors that can be obtained from P as follows : — Keep the sign of the first term unchanged, and take every possible arrangement of sign for the following terms, except that tvhich occurs in P itself. For the factors in the product Q x P contain every possible arrangement of the signs of all but the first term. Hence along with the + sign before any term, say that containing \/q, there will occur every possible variety of arrangement of all the other variable signs ; and the same is true for the - sign before ,^^q. Hence, if we denote the product of all the factors containing + sjq hy ( - \/q). Hence we may write Q X P = ^( Jq) x <^( - ,^q), which, by § 15, Cor. 1, is rational so far as \/q is concerned. The like may of course be proved for every one of the irrationals \/q, Jr, . . . Also, for every factor in Q X P of the form -Jp + Je there is evidently another of the form 196 EATIONAL FUNCTION REDUCED TO LINEAlt FORM chap. Jp -k; SO that Q, x P is rational as regards Jp. Hence Q x P is entirely rational, as was to be shown. § 17.] Bveri/ rational function, whether integral or not, of any mimhcr of square roots, \/p, \/q, ^//■, . . ., can he expressed as the sum of a rational part and rational rimUiplcs of Jp, .^Iq, Jr, A-c, and of their products ^/(2xj), \/{pr), ^'{qr), J{pqr), dr.* For every rational function is the quotient of two rational integral functions, say li/P. L('t Q be a rationalising factor of P (which we ha\'e seen how to find), then E EQ, P"PQ- But PQ is now rational, and RQ is a rational integral function of Vy, ^q, \'i', . ., and can therefore be expressed in the required form. Hence the pi'oposition is established. Example 1. To express 1/(1 + V2 + \/3) as a sum of rational multiples of square roots. Rationalising the denominator we obtain by successive steps, 1 ^ 1 +y2 - V3 i + V^ + Vs {i + V2)'-(\/y)'' _l + \/2- V3 2V2 ^ V2(l + V2-V 3) 2x2 = i(V2 + 2-V6), = 4 + iV2-iV6- Example 2. Evaluate {x''-x + l)l{x- + x + l), where .i = ^3 + \/5. x -~x + l ^ 9 + 2V 1. -, ~V3- V5 .;- + x + l~9 + 2V1.5 + V3 + V5' _ (9 + 2V15)- - 2(9 + 2 V15)( V3 + ys ) + W3+^Af (g + avisy-^-TVS + Vs)' 149 - 38 V3 - soys + 38 V15 133 + 34V15 _ ( 14 9 - 38V3 - soys + 38 yiS) ( 1 33- 34 Vl5 ) 13.32-34=^x15 " ' _ +437 + 46V3- 1 14v'5-12V1 5 3l9 * Besides its theoretical interest, the process of reducing a rational func- tion of quadratic irrationals to a linear function of such irrationals is important from an arithmetical point of view ; inasmuch as the linear function is in general the most convenient form for calculation. Thus, if it be required to calculate the value of l/(l + y2 + \/3) to six places of decimals, it will be foimJ more convenient tu deal with the equivalent form i + i\/'2 - ,l\/6. THEORY FOE lEEATIONALS OF ANY OEDEE 197 GENERALISATION OF THE FOEEGOING THEOEY. § 18.] It ijiay be of use to the student who has already made some progress in algebra to sketch here a generalisation of the theory of §§ 13-17. It is contained in the following jDro- positions : — I. Every integral function of ^■'■'"' can he reduced to the form A„ + A,//" + A2/'"+ . . . +A„_y'-i5'", where A„, A,, . . ., A,j_, are rational, so far as p^l^'- is concerned. After what has been done this is obvious. II. Every integral function of p^l^, 2^'% r^'", &c., can he ex- pressed as a linear function of p'-^\ ^^", . . ., _p('-i)/^y gi/"», j^/in-^ _ _ _^ gOii-i)/m^. ^i/ra ^2/K _ _ _^ ,,Oi-i)/»^. ^f.^^ find of the products of these quantities, two, three, dtc, at a time, riamely, ^^" g^'™, p^l^ qV'^, &c., the coefficients of the tinear function being rational, so far as p^l\ g^''", r^l"; &c., are concerned. Proved (as in § 15 above) by successive applications of I. III. A rationalising factor of A,, + A,jbV" + A^j^^'" + . . . + A,i- ,^(»-i)'» can always he found. We shall prove this for the case w = 3, but it will be seen that the process is general. Let P = A„ + K,p^ + A^p* (1), then p^V =pk._ + Kp^ + K^pi (2), and piY =pK + MsP* + -'^oP* (3). Let us now put x for jjJ, and y ior p'\ on the right-hand sides of (]), (2), and (3); we may then write them (A„ - P) -H A,* -f A,2/ = (1'), {pK-piV) + K^ + K,y = (2'), {;pk,-piV)+pK^ + Ky=Q (3'), whence, eliminating x and y, we must have (see chap, xvi., § 8) (A„ - P) (A„= -pA.A.) + {pk, -p'-'V) (M/ - A„A,) + (^.A,-i^iP)(A;-A„A,) = (4). 198 THEORY FOR IRRATIONALS OF ANY ORDER Whence ;(A/ -7)A,A,) + 0 (7'), and will have no difficulty in writing down the rationalising factor and the result of rationalisation in the general case. 1\'. A raliunalisiii(j fdctiir cim be found for any rational integral fiindum of p^^^, (/i'""', r^'", . . . , &c., hj first rationalising with respect to p^^'; then rationalising the result with respect to q^l™, and so on. V. Enrii rational function of ji^l'-, q^l™, r'^l'"', whether integral or 'iKil, oni be expressed as a linear fuiuilon of p^l^, p?^^, . . , p<.i-mi ^ (^i/m /» "^ " \/a:° x " ^ + y)={(x)4,{y)} ; ¥{x + y) = I'{xmy)l{l + {xMy) } . (18.) Uxi'lyi = l, thenx''-'"ly^-" = x"''li-^=y"-'"ilp. (19.) If ?»=«■', n=i-(H, in«n'=d-'', then a;2/;=l. Transform the following into sums of simple irrational terms : — (20.) sJal{-s/a + \/h) + 's/bl{\Ja- \Jh). (21.) (2V5-3v'a + 4\/6)'- (22. ) (a; + 1 - V2 + VS) (k + 1 + V2 - V3) (a; + 1 - \/2 - V3)> arranging ac- cording to powers of x. 200 EXKECISES XIV chap. (23.) (l/V^' + l/V") (.'■- -a*)/l(\/«+ V'')''- ( v'" - \/xf]- (24 \ \/(''''+ {a(l -m)/3Vro}-) + a(m - l)/2V»i. ^ ■■^ V(aH{aCl-m)/3Vm}^)-a(m-l)/2\/"'-' (25.) V(^/«±M+y(W^), ,vhere«..= ,-M,. (26.) h^^ — j~- — r, where a;=\/(«°)- (27.) i/{v'(?'-?) + Vi'+V?}+V{\/(i'-!?)-Vi'-v'?} + i/(vi'-V?)- , , /J' l-^-V(2.-' + .'-'-) t , /f l-ai + V(2x + ^^) l. ^ ■■' V ll-.r+v'(2.'-+.f-)i V ll-a;- V(2-b + k")/ (29.) {^(a + h + c) + ^{a-b + c)fl(~J{a + l + c)-'s/(a-Tj + c)f. (30.) {(V(;'-?)+Vi^-V?)/(VO^-?)-Vp-V?)}- (31.) (2..-'^6.T + 5)/(s/2.-«+ i.U + 1). (32. ) (ar'= + 3 x/2,.- + 1 )l(x + ^2 - 1 ). (33.) {l/{a + l)- l/{a-h)}{[^{a + h)f + il/(a-b)f- 1/ (ct- - b'')] . Show that (^*' ' li^vcr- .'-■)'' + vi+v(i-')7 - "^^ ^^'' ~ ^^- " ^ '' • (35.) (v'(r+i) + V(/'--i))-'+(V(?''+i)-\/(^'-i))-'=(^'-iM?''+i)- (36.) V[v'{«'+ v/(«*')} +\/{i'+ '^{a?b')}'\ = {a^ + b^f- Express in linear form — (37.) {^x-t/y)l(l/x+tjy). (38.) (H-V3 + \/5 + \/7)/(l- V3-\/5 + \/7)■ (39. ) i;( v + \/<:)l(\/b + Vc - V«) - 4Sa( V&+ \/';)/n( V^ + Vc - V«)- Rationalise the following : — (40.) 3.5^-4*. (41.) SV(;' + c-«). (42.) VS + VS + V-l-Ve- (i3.) 3.2' + 4.2-'-l. (44.) «-^ + 6* + c^. (45.) 2^ + 2- + l. (46.) l(u=x~^{l+y"-) + ysJ{l + x%then^/{l + u-)=xy + 's/{{l+x'){l + y'')). 2 (47.) Show that —r, ; r, ^, — 77 ^, ^ ( y - c)^ + (;. - x)' + {x - ^)' + (y - t O^(« - aO^(a: - y )^ x^ + y^ + z^ — yz-zx — xy (48.) If x = ll{'s/b + ^Jc-^a), 2/ = l/(Vc + \/«-\/*). = = l/(v'«+\/* - y'c), •!6=1/(V«+ \/&+ \/«); then n(-a; + !/ + ~ + ?i)/(2'''-*')^=n(6 + c-a)/8aic. Jlis/on':;il Xi'lr. — Tlie use of exponents liL'gan in the works of the (leniian 'Cssists," Undoltf (1525) and Stifel (1544), who wrote over the contractions X HISTORICAL NOTE 201 for the names of the 1st, 2nd, 3rd, . . powers of the variable, which had been used in the syncopated algebra, the numbers 1, 2, 3, . . . Stifel even states expressly the laws for multiplying and dividing powers by adding and subtracting the exponents, and indicates the use of negative exponents for the reciprocals of positive integral powers. Bombelli (1579) writes ^, i, ^, Jt where we should write a", K, a:^ a^, . . . Steviu (1585) uses in a similar way ©, ©, ® J Cil) and suggests, although he does not practically use, fractional powers such as (J) , (1) , which are equivalent to the a; , a; , of the present day. ViJte (1591) and Oughtred (1631), who were in full possession of a literal calculus, used contractions for the names of the powers, thus, Ag, Ac, Kqq, to signify A^, A^, A*. Harriot (1631) simply repeated the letter, thus, aa, aaa, aaa(t, for ct^, o?^ a^. Herigone (1634) used numbers written after the letter, thus. A, A2, A3, . . . Descartes introduced the modern forms A, A^, A^, . . . The final development of the general idea of an index unrestricted in magnitude, that is to say, of an exponential function a^, is due to Newton, and came in company with his discovery of the general form of the binomial coefficients as functions of the index. He says, in the letter to Oldenburg of 13th June 1676, "Since algebraists write a', a^, a*, &c., for aa, aaa, aaaa, &c., so 1 write ".", «-, a', for \/a, \/a^, \/c.d'; and I write a~^, a--, a-', &c., for -> — , 1 a aa —, &c." aaa The sign \J was first used by Rudollf; both he and Scheubel (1551) used yw/ to denote 4th root, and -yyy/ to denote cube root. Stifel xised both ^y§; and \/ to denote square root, sf^^. to denote 4th root, and so on. Girard (1633) uses the notation of the present day, \/, v , &c. Other authors of the 17th century wrote ^/i :, ^3 :, &c. So late as 1722, in the second edition of Newton's Arithvictica Universalis, the usage fluctuates, the three forms \JZ : /= 3/ „ ■ V : , \/ all occurrmg. In an incomplete mathematical treatise, entitled De Arte Lo'jistica, &c., which was found among the papers of Napier of Merchiston (1550-1617 ; pub- lished by Mark Napier, Edinburgh, 1839), and shows in every line the firm grasp of the great inventor of logarithms, a remarkable system of notation for irrationals I I illllll is described. Napier takes the figure _J , and divides it thus 4] ["5l [6 . He then uses _|, |_l, I , &c. , which are in effect a new set of symbols for the nine digits 1, 2, 3, &c., as radical signs. Thus UlO stands for \JW, l_10 for \/lO, JIO for \/lO, J] 10 for '*i/lO, _J Zl or 3 for \/lO ; and so on. Many of the rules for operating with irrationals at present in use have come, in form at least, from the German mathematicians of the 16th century, more particularly from Scheubel, in whose Algebne Covipendiosa Fadlisque Descriptio (1551) is given the rule of chap. xi. , § 9, for extracting the square root of a binomial surd. In substance many of these rules are doubtless much older (as old as Book X. of Euclid's Elements, at least) ; they were at all events more or less familiar to the contemporary mathematicians of the Italian school, who did so much for the solution of equations by means of radicals, although in symbol- ism they were far behind their transalpine rivals. See Hutton's Mathematical Dictionary, Art. "Algebra." The process explained at the end of next chapter for extracting the square or 202 HISTORICAL NOTE ouap. x cube root by successive steps is fouml in. the works of the earliest European writers on algebra, for example, Leon.irdo Fibonacci (t.-. 1200) and Luca Pacioli (r. 1500). The first indication of a general method appears in Stifd's Ar/'fhmc/ica Integra, where the necessary table of binomial coefficients (see p. 81) is given. It is not quite clear from Stifel's work that he fully understood the nature of the process and clearly saw its connection with the binomial theorem. The general method of root extraction, tojijether with the triangle of binomial coefficients, is given in Napier's I>c Arte Lniil.Uica. He indicates along the two sides of his triangle the powers of the two variables (prrecedens and succedens) with which each coefficient is associated, and thus gives the binomial theorem in diagi'am- matic form. His statement for the cube is — " Supplementnm triplicationis tribus constat numeris : prinnis est, duplicati prfficedentis tri]ilum multiplicatum per succedens ; secundus est, praecedentis triplnm multiplicatum per duj^licatum suc- cedentis ; tertius est, ijjsum triplicatum succedentis." In modern notation, CHAPTEE XL Arithmetical Theory of Surds. ALGEBRAICAL AND ARITHMETICAL IRRATIONALITY. § l.J In last chapter we discussed the properties of irra- tional functions in so far as they depend merely on outward form ; in other words, we considered them merely from the algebraical point of view. We have now to consider certain peculiarities of a purely arithmetical nature. Let p denote any commensurable number ; that is, either an integer, or a proper or improper vulgar fraction with a finite number of digits in its numerator and denominator ; or, what comes to the same thing, let p denote a number which is either a terminating or repeating decimal. Then, if n be any positive integer, li^p will not be commensurable unless p be the «th power of a commensurable number y' for if li/p = k, where h is commensurable, then, by the definition of ^p, p = /j", that is, p is the mtli power of a commen- surable number. If therefore j? be not a perfect mth power, ^i/p is incommensur- able. For distinction's sake ^p is then called a surd number. In other words, we define a surd number as the incommensurable root of a commensurable number. Surds are classified according to the index, n, of the root to be extracted, as quadratic, cubic, biquadratic or quartic, quintic, . , . m-tic surds. The student should attend to the last phrase of the definition of a surrl ; because incommensurable roots might be conceived which do not come under * This is briefly put by saying that 2> is a perfect «th power. 204 CLASSIFICATION OF SURDS chap. the above definition ; and to tliem the demonstrations of at least some of tlie propositions in this chapter would not apply. For example, the number c (see the chnpter on the Exponential Theoi'em in Part 11. of this work) is incom- mensurable, and \/c is incommensurable ; hence \/e is not a surd in the exact sense of the definition. Neither is VCV- + l)i foi' \/2 + 1 is incommensurable. On the other hand, \/(\/2), which can be expressed in the form ^2, does come under that definition, although not as a quadratic but as a biquadratic surd. He should also observe that an algebraically irrational function, say \/x, may or may not be arithmetically irrational, that is, surd, strictly so called, according to the value of the variable x. Thus sji is not a surd, but \J2 is. cljVSSification of surds. § 2.] jV single surd number, or, what comes to the same, a rational multiple of a single surd, is spoken of as a sinrplc mono- wial surd umnhcr ; the sum of two such surds, or of a rational numljer and a simple monomial surd number, as a simple binomial surd number, and so on. The propositions stated in last chapter amount to a proof of the statement tliat every rational function of surd numbers can be expressed as a simp)le surd number, monomial, binomial, trinomial, &c., as the case may be. § 3 ] Two surds are said to lie similar when they can be expressed as rational multiples of one and the same surd ; dissiudlar when this is not the case. For example, \/l8 and v^S can be expressed respectively in the forms 3 J2 and 2 J2, and are therefore similar ; but Jd and \/2 are dissimilar. Again, ^/'>i and ^/16, being expressible in the forms 3^2 and 2 ^/2, are each similar to ;|/i]. yill the sttrds that arise from the extraction of the same nth root (ire said to be cquiradiral. Thus p", j;5, j)», jj^ are all cqiiiiadical with 7'". There are n—\ distinct surds eqiiiradical with p^l'\ namely, ]i^l"', p~^", . . ., j/""-'-)'", ami no more. For, if we con.sider j/"/" where m>n, then we have p™!'"- = ■p"-+'l>'-, where /..- anil v are integers, and v = {A + BsJq + C Jr + D J{qr)Y, which would either be a rational equation connecting two dis- similar quadratic surds, which is impossible, as we have just seen ; or else an equation asserting the rationality of one of the surds, which is equally impossible. An important particular case of the above is the following : — A quadratic surd cannot be the sum of two dissimilar quadratic surds. It will be a good exercise for the student to prove this directly. § 7.] The theory which we have established so far for quadratic surds may be generalised, and also extended to surds whose index exceeds 2. This is not the place to pursue the matter farther, but the reader who has followed so far will find the ideas gained useful in paving the way to an understanding of the delicate researches of Lagrange, Abel, and Galois regarding XI ■ INDEPENDENCE OF SUED NUMBERS 207 the algebraical solution of equations whose degree exceeds the 4th. § 8.] It follows as a necessary consequence of §§ 5 and 6 that, if we are led to any equation such as A + B ^/p + C ^/g + D ^J{pq) = 0, where Jp and Jc[ are dissimilar surds, then we must have A=0, B = 0, C = 0, D = 0. One case of this is so important that we enunciate and prove it separately. If X, y, z, u he all commensurable, and Jy and Ju incommen- surable, and if X + Jy = z + >Ju, then must 'x = z and y = u. For if X =t= s, but =a + z say, where a 4= 0, then by hypothesis a + z + ^/y = z + ^,hi, whence a + ^/y = Jii, a^ + y + 2a \,/y = u, ^y = (u -a'- y)/2a, which asserts that ^Jy is commensurable. But this is not so. Hence we must have x = z ; and, that being so, we must also have Jy = Ju, that is, y = u. SQUARE ROOTS OF SIMPLE SURD NUMBERS. § 9. J Since the square of every simple binomial surd number takes the form p + Jq, it is natural to inquire whether Jij) + ijq) can always be expressed as a simple binomial surd number, that is, in the form \lx + \/y, where x and y are rational numbers. Let us suppose that such an expression exists ; then sj{p + s/g) = x/a; + ^/y, whence p + \/q = x + y +2 \/{xy). If this equation be true, we must have, by § 8, x + y=p (1), 2 ^(xy) = V2 (2) ; and, from (1) and (2), squaring and subtracting, we get {x + yy-ixy=p'-q, that is, (x-yT=p'-q (3). 208 LINEAR EXPRESSION FOR \'{a + \fh) chap. Now (3) gives either x-ij= + ^(ir-q) (4), or a; -2/= - s'\f - i) (4*). Taking, meantime, (4) and combining it with (1), we have {x + y) + {.r-y)=2}+ x^ip'-q) (5), (•'■ + 2/) - (■'■ -y)=P' \ '{/ - 2) (C) ; whence ■2.r=p+\/(p'-q), 2i/=p- x/{f-q); that is, x=l\p+ J{p'-q)\ (7), If we take (4*) instead of (4), we simply interchange the vahies of X and y, which leads to nothing new in the end. Using the values of (7) and (8) we obtain the following result : — Sinri', by (2), 2 \''x x \/y = + \/g, we must take either the two uppei' signs together or the two lower. If we had started with \/(p - sjq), it would have been necessary to choose \/:r and \^y with opposite signs. Finally therefore we have s'{p + \/'/) - ■ /it .yi?'!- •> -M . vr-- 2 (9), {'-* 2 '-"]- vr --^]) J{p- Jq)= ± ^ (9*). The identities (9) and (9*) are certainly true ; we have in fact already verified one of them (see chap, x., § 9, Example 14). They will not, howex'er, fiu'nish a solution of our problem, unless the values of x and y are rational. For this it is necessary and sufficient that p^ - q he a, positive perfect square, and that^:i be XI EXAMPLES 209 poaith'e. Hence the square root of p + ijq can he expressed as a simple binomial surd number, provided p be positive and p" — qbe a positive perfect square. Example 1. Simplify V(19 - W-'^)- Let V(19-iV^l)=\/»+V2/- Then a; + »/ = 19, (a;-j/)-=361-336 = 25, x — y= +5 say, a; + 2/=19 ; whence a; = 12, y = 7, V'.r=±\/12, V2/=tV7. so that V(19-*V21) = ±(V12-v'7). Example 2. To find the condition that VlVP+Vs) ™^y l^e expressilile in the form {\/x+ \/y) ij'p we have V( Vp + V?) = \/^ X V {1 + V(?/P)} • Now v'{l + V( N. Now (chap, iv., § 1 1) (1 + af = 1 + 2"a + a series of terms, which are all positive. Hence it will be sufficient if we make 1 + 2"a > N, that is, if we make 2"a > N - 1 , that is. 21-1: EXAMPLES CHAP. if we make 2">(N - l)/a, which can always ho done, since by making n sufficiently great 2" may be made to exceed any quantity, however great. Example. How many times must wo extract the square root beginning with 51 in order tliat the final result may tlilfer from 1 by less than -001 ? Wo must have 2" > (51-1)/-001, 2" - 50000. Xow 2" = ."276S, 218 = 65536, hence we must make ?i = 16. In other words, if we extract the square root sixteen times, beginning with 61, the result will be less than 1 '001. § 15.] It follows from S 14 that we can approximate to any surd whatever, say j»"", by the process of extracting the square root. For (see chap, ix., § 2) lot l/« be expressed in the binary scale, then Ave shall have l/7. = „/2+^/2= + y/2' + . . .+/., where each of the numerators a, /?, y, is either or 1, and /x is either absolutely or ,„, + 2px^ + {p^ + 2q)x^ + {2pq + 2r)x' + {2pr + q^ + 2s)x^ + . . . (3). Now this must be identical with (1) ; hence we must have 2^ = 6, ;j= + 2g'=13, 2^2 + 2»-=4, 2pr + q^ + 2s= -18. 218 EXERCISES XV chap. The first of these eqiiations gives ju = 3 ; ^i being thus known, the second gives ([=2 ; p and q being known, the third gives r= -4 ; imdj), y, 7- being known, tlie last gives s = l. We couhl now find t in like manner ; but it is obvious from the coeflieient of x that t= - 1. Hence one value of the square root is .y.B. — The equating of the coeflScients of the remaining terras of .(1) and (3) will simply give equations that are satisfied by the values of p, q, r, s already found, always supposing that the given radicand is an exact square. A process exactly similar to the above will furnish the root of an exact cube, an exact 4th power, and so on. EXEECISES XV. Express the following as linear functions of the irrationals involved. (1.) l/isyn + s/S + sJU). (2.) V12/(l + \/2)(\/i;-V3). (3.) (1 - V2 + \/3)/(l + \/2 + VS) + (1 -- V2 - V3)/(l + \/2 - \/3)- (4.) (3 - V5)/(\/3 + V5y' + (3 + n/SWVS ~ VS)'"'- (5.) V5/( V3 + VS - 2x72) - \/2/(\/3 + \/2- V5). (6.) (7 - 2 v'fi)(5+ v''')(31 + 13v'5)/(6 - 2V7)(3 + V5)(ll + i\/7). (7.) V(25 + 10V6). (8.) V(3/2 + v'2). (9.) V(123-22V2). (10.) V(44V2 + 12V26). (11.) V{(8 + i\/10)/(8-4v'10)}. (12.) V(7 + 4v'3) + \/(5-2v'6)- (13.) v'(15-4V14) + l/\/(15+4Vl'')- (14.) lMl6 + 2v'63) + l/V(16-2V63). (15.) l/V(16v'3 + 6v'21) + V(16V3-6V21)- (16.) Calculate to five places of decimals tlio value of f\/(5 + 2\/6) - V(5 - 2x76)}/ !\/(5 + 2V6) + V(5 - i\'(>)\ ■ (17.) Calculate to seven places of decimals the value of \/(\/\f) + \Jli) + V'(\/16-V13)- Simplify — (18.) V{3 + V(9-.?'-)} + \/{3-\/(9-iJ=)}. (19.) ^\a + b~c + 1^{t{a-c))\. (20.) V{o-^2 + «v'(«'-4)}- (21 >yf('-fJ^)(r.U-) (22.) Showthat V[2 + V(2-V2)}= y{2 + xA|±V2)J (23. ) Express in a linear form \/{f> + \J& + sJW + \JIZ). (24.) „ ,, V'-'.0-4v'3-12V2 + 6V6). ^y|-W(2W2)|. XI EXERCISES XV 219 (25.) If ffi^d = 5c, then \/{a + sjb + \/c + \/d) can always be expressed in the form {\/x+ \/y){\/X+ \/Y). Show that this will be advantageous if w'-b and a'- c are perfect squares. (26.) If ^/{a + \/b)=x + \/y, where a, b, x, y are rational, and \/b and V?/ irrational, then ^(a- \Jb) = x- \Jy. Hence show that, if a?-b=z^, where a is rational, and if x be such that ia? - 3.i'; = a, then ^{a + \Jb) = X+\/{3?-~). (27.) Express in linear form 4^(99 -35V8)- (2S.) ,, ,, 4^(395 + 93 V18). (29.) „ „ ^(n7V2 + 74V5). (30.) Show that v'(90 + 34V7)- ^(90 -34x77) = 2s, 7. (31.) If a:= \/(p + q)+ \/{p-ci), and 2^°-i' = i^, show that x?-Zrx-%p = 0. (32.) lipy +qy' + r = <), where ^, q, r, y are all rational, and y irrational, then p = 0, g'=0, ?-=0. Hence show that, if x, y, z be all rational, and X , y , s all irrational, then neither of the equations x +y =z, x' + y =z is possible. (33.) Find, by the full use of the ordinary rule, the value of s/l" to 5 places of decimals ; and find as many more figures as you can by division alone. Use the value of \/10 thus found to obtain Ay'004. Extract the square root of the following : — (34.) {yz + zx + ssy^-ixijz^z + x). (35.) 25x^+9y'' + z^ + 6yz-10zx-30xy. (36.) 9a:* + 24a?' + 10a;2-8a; + l. (37.) x^-'i}^ + 2x'^ + ix + l. (38. ) 4ar^ - 12x'y + 26x"y° - 24a;?/ + IB?/*. (39.) x'^-6x^ + ix> + 9x''-12x + 4. (40.) ix<^ - 12s^ + 5a;^ + 22a^ - 2Sx- - 8a: + 16. (41.) 27{p + q)\p^- + q^)--2{p"-+ipq + q^f. (42.) x^-2x's/x + 3x-2\/x + l. (43.) Extract the cube root of 8x^ -123? + 6a;' - 37a;6 + 36ar* - 9a;i + 54ar' - 27a;'- - 27. (44.) Extract the cube root of l?,{p<+p\+pq^ + (x,, + y^i, x., + yd, . . , a;„ + y„i) be any rational function of n complex numbers, liaving all its coefficients real, anil if {^i + yii, •'■.. + yi, • • , ^n + yJ) = x + Yi, then 'K^>-y(z) be real, and if {x + yi) = 'K + Yi where X and Y are real. Hence, if <^(a; + yi) = 0, we have X + Yi = 0. Hence, by § 3, X = and Y = 0. Therefore <^(x - yi) = X - Yj = - Oi = 0. Cor. 5. If all the coefficients of the integral function <^{z„ «2, . . . , a„) he real, and if the function vanish when «„ z.., . . , »„ are equal to x^ + y-j,,- x^ + yJ., ■ ■ • , Xn + yj, respectively, then the function will also vanish when «„ g,^, . . . , «„ are equal to x^ - y^i, x^ - y.,i, . . ■ , Xn- yj, respectively. Example 1. 3(3 + 2i)-2(2-3i) + (6 + 8!;) = 9 + 6i-4 + 6i + 6 + 8i, = ll + 20i. Example 2. (2 + Zi) (2 - 5i) {3 + 2j') = (2 - 5i) (6 - 6 + 9i + ii), = (2-5i)13i:, = 26i; + 65, = 65 + 26i. Example 3. (h + c-ai)(o + a-hi)(a + 'b- ci) = {n{6 + c)-26c(i + c)} + {dbc-''Za(a + h){a + c)]i, = 2ahc + {abc - Za^ - 'S.a?{h + c)- 3aSc} i, = iabc- {a!' + b^ + (.■?= -i, i^=i^y.i'^= +1 ; and, in general, j-4n+l _ j_ i4ii+2 = _ 1, i4n+3 = - i, ■i''(«+l) = + 1 . Example 5. Z + U (3 + 5t)(2 + 3t) _6-15 + 19i_ 9 19. 2^r3i- i^ - 13 - 13"'"13*' Example 6. = {si"-nG2X''-h/ + „CiX''-Y- ■ •) + („Ci x»-iy - „Cs xo-hf + „C6 a;"- V - ■ )»• In particular (x + yiY = (3i^- &xhf + y*) + (is?y - ixy^)i. Example 7. If 2= T": — i -,, 3^ + 2+1 then 0(2 + 3z)=^2 + 3i)^ + (2 + 3i) + l' -5 + 12t-2-3i; + l ■~ -5 + 12i + 2 + 3i!+l' 228 CONJUGATE COMPLEX NUMBERS - -^ + S' _ 3i-2 + Si) { -2-15i) ~ -2 + 15i 229 ' ~ 229 "''229*' From this we infer that ^^ ' 229 229 a conclusion which the student should verify by direct calculation. CONJUGATE COMPLEX NUMBERS, NORMS, AND MODULI. § 6.] Two complex numbers which differ only in the sign of their imaginary part are said to be (onjugate. Thus - 3 - 2i and -3 + 2/ are conjugate, so are - 4i and + ii ; and, generally, X + yi and x - iji. Using this nomenclature we may enunciate Cor. 3 of § 5 as follows : — If the coefficients of the rational function {x + yi) be real, we have seen (§ 5, Cor. 2) that if ^{x + yj) = X + Yi, where X and Y are real, then 4>{x - yi) = X - Yi. Now \(x + yi)\= J{r+T)= J{{X + Yi){X-Yi)}, = ^/{{x + yi) {x-yi)\ (1). 230 MODULI OHAP. In like manner it follows from § 5, Cor. 3, that = + v/[<^(a;i + 2/A x^ + yj., . . ., Xn + yJ) X cf>{x, - y,i, X, - yd, . , x^- y„i)] (2). The theorems expressed by (1) and (2) are very useful in practice, as will be seen in the examples worked below. It should be observed that (1) contains certain remarkable particular cases. For example, I (a^i + 2/i») {x, + y^i) ■ ■ ■ {xn + yJ) I = + \/[{Xi + y,i) {x, + y^i) . . . {Xn + yj) ^ («i - yi*) i^2 - yJ) . . . {x„- yni)], = + \/(a;/ + «/,") (x/ + y„!) . . . (a:,/ + y^'), = I (a;i + Vii) I X I (*2 + y^i) I >^ • • ■ ''\{^n + yJ) I (3). In other words, the modulus of the product of n complex numbers is equal to the p-odiict of their moduli. Also K, + y,i v'O^i' + yi') ^ I «! + y,i I x„ + y^i s/(a;/ + «//) | x^ + y^i | (4). In other words, the modulus of the quotient of two complex numbers is the quotient of their moduli. § 10.] The reader should establish the results (3) and (4) of last paragraph directly. It may be noted that we are led to the following identities : — (a;,' + y^) (a;/ + y/) = {xfl^ - y.y^f + {x,y, + x,y,f. If we give to a;,, y,, x^ y^ positive integral values, this gives us the proposition that the product of two integers, each of which is the sum of two square integers, is itself the sum of two square integers ; and the formula indicates how one pair of values of the two integers last mentioned can be found. Also (x," + y,') (x/ + y') {xi + y-f) = x,x^^ - x,y,y^ - x,y,y, - x^y.y^y + (jji^^jjc^ + y^'z^x + ^aJi*!. - 2/12/2^3)'- XII MODULI 231 This shows that tlie product of three sums of two integral s/pmres is the sum of two integral squares, and shows one way at least of finding the two last-mentioned integers. Similar results may of course be obtained for a product of any number of factors. Example 1. Find the modulus of (2 + 3i) (3 - 2i) (6 - ii). |(2 + 3i)(3-2i)(6-4i)| = |(2-|-3i)|xl(3-2i)jx|(6-4i)|, = j{13)x V(13)x ^(52), = 26^(13). Example 2. Find the modnXus o! {sj2 + iJS) {^S + ij5)lis/2 + ij5). I { j2+isJZ){ JS + isJ5) I I J2 + ij6 I /r/ (V2 + tV3)(V3 + W5) \,,/ (x/2-W3)(N/3-'\/5) \-1 - V L 1 j2 + i^5 I \ ^/2-'i^5 /J' Example 3. Find the modulus of {((3 + 7) + ((3 - 7)1} {(7 + a) + (7 - a)i} {(o + /3) + (a - (3)i} . The modulus is ^'{{{l3 + 7T + {P-yf}{{y + a)^ + {y-a.f}{{a + p)'' + {a-p)''}) = ^/{8(/32 + 7-^)(7' + «^)(a2 + ^-')}. Example 4. To represent 26 x 20 x 34 as the sum of two integral squares. Using the formula of § 10 we have 26 X 20 X 34 = (12 + .52) (2^ + 4^) (S^ + 5^), = (1.2.3 -1.4.5 -2.5.5 -3.5.4)2 + (5.2.3 + 4.3.1 + 5.1. 2 -6.4.5)2, = 1242 + 432. § 11.] The modulus of the sum of n complex numbers is never greater than the sum of their moduli, and is in general less. This may be established directly ; but an intuitive proof will be obtained immediately from Argand's diagram. § 1 2.] We have seen already that, when PQ = 0, then either P = or Q = 0, provided P and Q be real quantities. It is natural now to inquire whether the same will hold if P and Q be complex numbers. If P and Q be complex numbers then PQ is a complex number. Also, since PQ = 0, by § 8, | PQ | = 0. But | PQ | = |Pix|Q|, by § 10. Hence |P|x|Q| = 0. Now |P| and 232 argand's diageam €hap. I Q I are both real, hence either | P | = or 1 Q | = 0. Hence, by § 8, either P = or Q = 0. We conclude, therefore, that if PQ = 0, then either P = or Q= 0, whether P and Q be real qvanlifies or complex numbers. discussion of complex nl'mbeus by means of argand's diagram. § 13.] Returning now to Argand's diagram, let us consider the complex number x + yi, which is represented by the radius vector OP (Fig. 1). Let OP, which is regarded as a signless magnitude, or, what comes to the same thing, as always having the positive sign, be denoted by r, and let the angle XOP, measured counter-clock-wise, be denoted by 6. We have seen that if OP represent x + yi, then x and y are the projections of r on X'OX and Y'OY respectively. Hence we have, by the geometrical definitions of cos d and sin 9, r= + sj{x' + f) (1), xjr = cos 6, yjr = sin 6, (2). From (1) it appears that r, that is OP, is the modulus of the complex number. The equations (2) uniquely determine the angle 6, provided we restrict it to be less than Stt, and agree that it is always to be measured counter-clock-wise from OX.* We call 6 the amplitude of the complex number. It follows from (2) that every comple.x; number can be expressed in terms of its modulus and amplitude ; for we have x + yi = r(cos 9 + i sin 9) (3). This new form, which we may call the iv.rmal forrti, jjossesses many important advantages. * Sometimes it is more convenient to allow 6 to inerease from - x to -f-ir ; that is, to suppose the radius OP to revolve counter-clock-wise from OX' to OX' again. In either way, the amplitude is uniquely determined when the coefficients x and y of the complex number are given, except in the case of a real negative number, where the amplitude apart from external considera- tions is obviously ambiguous. COMPOSITION OF VECTORS 233 Since two conjugate complex numbers ditl'cr only in the sign of the coefficient of i, it follows that the radii vectores which re- present them are the images of each other in the axis of x (Fig. 3). Hence two such have . the same modulus, as we have already shown ^ analytically ; and, if the amplitude of the one be 0, the amplitude of the other will be Stt - 6. In other words, the amplitudes of two con- jugate complex numbers are con- jugate angles. Example. § 14.] If OP, OQ' (Fig. 4) represent the complex numbers Y X -{■ yi and x -v y'i, and if PQ be drawn parallel and equal to OQ', then OQ will represent the sum of x + yi and x' + y'i. For the projection of OQ on the K-axis is the algebraic sum of the Y Fig. 4. projections of OP and PQ on the same axis, that is to say, the projection of OQ on the x-axis is X + X. Also the projection of OQ on the y-axis is, by the same reasoning, y -t- «/'. Hence OQ represents the complex number (a; + «') + («/ + y')% which is equal to {x -h //«) -t- («' -H y'i). By similar reasoning we may show that if OPj, OPj, OP3, OP4, OP5, say, represent five complex numbers, and if PjQ^ be parallel 234 \z, + ::,+ . . . +«J = cos (^1 + 6^) and sin (j> = sin (0, + 6^), and we then have ^, + 6^ = 2mr + 4>- This last result is clearly general ; for, if we multiply both sides of (1) by an additional factor, r3(cos O^ + i sin ^3), we have ri(cos 6^ + i sin ^i)r2(cos d^ + i sin ^2)?'3(cos d^ + i sin ^3) = r,r2{cos (0, + ds) + i sin (^1 + O^)} rJ,cos 63 + 1 sin ^3), = Tjr^r^lcos (6, + 6lg + ^3) + i sin {6, + 6^+ 6^)}, by the case already proved, = 7-,V3{cos {e, + e^ + d;,) + i sin (d, + 6^ + O^)}. Proceeding in this way we ultimately prove that j-,(cos di + i sin ^,)r2(cos 6^ + ^ sin 6^ . . . r„(cos 9n + i sin 0„) = ri?-2 . . . r„{cos(0i + 6, + . . . + 6n) + i sva.{di + $,_ + ... + 6IJ} (2). This result may be expressed in words thus — The product of n complex numbers is a complex number whose modulus is the product of the moduli, and whose amplitude is, to a multiple of 2ir, the sum of the amplitudes of the n complex numbers. If we put ri = r^ = . = r,„ each = 1 say, we have (cos 61 + i sin ^i) (cos O^ + i sin 0^ . . . (cos 6^ + i sin 0^ = cos{d, + 0^ + . . . + dn) + i sin (61 + 6^+ . + O^) (3). This is the most general form of what is known as Demoivre's Theorem. If we put 9i = 6^ = . . . = On, each = 6, then (3) becomes (cos 6 + ism 5)" = cos nd + i sin n6 (4), which is the usual form of Demoivre's Theorem.* It is an analy- tical result of the highest importance, as we shall see presently. * This theorem was first given in Demoivre's Miscellanea Analytica (Lend. 1730), p. 1, in the form a; = iv'{Z+V(^^-l)}+i/\/{Z + \/(^^-l)/. wherea=cos^, l = cosne. 236 QUOTIENT OF COMPLEX NU.11BERS CHAP. Since COS ^ - i sin 6 - cos (fiir - 0) + i sin (27r - 0), we have, by (3) and (4), n(cos e, - i sin 6,) = cos (2^,) - i sin (26',) (3') ; and (cos 6 -is,m 0)^ = cos nO - i sin nO (4'). The theorem for a quotient corresponding to (1) may be obtained thus — r(cos 61 + 4 sin 6) ?'(cos 6 + i s in 6) (cos 6' -i sin 6') r'(cos 0' + i sin &) ~ r'{cos '6' + sin ^6') T = - {(cos 6^ cos 0' + sin 9 sin 6') r ^ + (sin cos 0' - cos 6 sin 5')i}, = ^{cos(^-^) + isin(6>-6>')} (5). Hence the quotient of tioo complex numbers is a complex number whose modulus is the quotient of the moduli, and whose amplitude is to a multiple of 2?r the difference of the amplitude of the two complex numbers. IRRATIONAL OPERATIONS WITH COilPLEX NUMBERS. § 16.] Since every irrational algebraical function involves only root extraction in addition to the four rational operations, and since we have shown that rational operations with complex numbers reproduce complex numbers and such only, if we could prove that the wth root of a complex number has for its value, or values, a complex number, or complex numbers and such only, then we should have established that all algebraical operations with complex numbers reproduce complex numbers and such only. The chief means of arriving at this result is Demoivre's Theorem ; but, before resorting to this powerful analytical engine, we shall show how to treat the particular case of the square root without its aid. Let us suppose that ^{x + yi) = X + Yi (1). XII SQUARE EOOTS OF A COMPLEX NUMBER 237 Then z + «/i = X'-Y'+2XYJ. Hence, since X and Y are real, we must have, by § 3, X'-Y' = a; (2), 2XY = 2/ (3). Squaring both sides of (2) and (3), and adding, we deduce (r + Yy=.x= + /; whence, since X'' + Y^ is necessarily positive, we deduce X' + T= + ^(x' + tf) (4). From (2) and (4), by addition, we derive 2X^ = + ^/{a? + if) + X, , + ^/{af + f) + x that IS, X = c, We therefore have X= ± ^7 | ±^(?^±ll±-*| (5). In like manner we derive from (2) and (4), by subtraction, &c., Y=W|^^^^^^f^^'} (6). Since x^ + 'if is numerically greater than x^, + ^(af + f) is numerically greater than x. Hence the quantities under the sign of the square root in (5) and (6) are both real and positive. The values of X and Y assigned by these equations are therefore real. Since 2XY = y, like signs must be taken in (5) and (6), or unlike signs, according as y is positive or negative. We thus have finally m. if y be positive ; if y be negative. Example 1. Express \/{8 + 6i) as a complex number. Let \J{fi + &i)=x + yi. Then x--y'^=i,, 2xy=e. Hence {3? + y'-f=6i + 36 = 100 Hence x'' + y^ = lO; and x^-y^=8; (8), 238 EXAMPLES CHAP. therefore 2.'c- = 18, 2ij- = 2. Hence x=±S, j/=±l. Since 2.)'7/ = G, we must liave either l);=+3 and y=+'i, or 3 = -3 and y=-l. Finally, therefore, we have V(8 + 6i)=±(3 + i); the correctness of which can be immediately verified by squaring. Example 2. v<.-.>=.{,y(*r=)-^/(-«-')}- Example 3. Express ^y( + s) and \/( - i) as complex numbers. Lot \/{ + i) = x + yi; then t = ,r--2/2 + 2a;i/i. Hence .r--)/- = (a), 2.n/ = l (;8). From (a) we have (k + !/)(.)'- 1/) = ; that is, either y= -x or 2/ = ^'- The former alternative is inconsistent with (/3) ; hence the latter must be accepted. \Vc then have, from (^), 2.i,- = l, whence .e'- = 1/2 and x=±\j\/2. Since 7/ = .'-, we liave, finally, v+'=±:^ (7). Similarly we show that V--=tl^- (5). Example 4. To express the 4th roots of + 1 and - 1 as complex numbers. ^ + l = V(\/+l) = \/±l = V + lor v'-l = ±lor ±»- Hence we obtain four 4th roots of + 1, namely, +1, -1, +i, -i. 4/ Again ■\' -1 = \J{\/ -\) = \J±i. Hence, by Example 3, '/-!-+ „, § 17.] We now proceed to the general case of the wth root of any complex number, r(cos ^ + i sin 9). Since r is a positive number, \/r has (see chap, x., § 2) one real positive value, which we inay denote by r^/'l Consider the n complex numbers — ■2TV + e . . 27r + V - 1 - ±' , or ± - cos - + i sin - (l\ u 111 ^ " T^ln COS + isin -— ) (2), n n WTH ROOTS OF ANY COMPLEX NUMBER 239 }■!/» I cos + t sin j (3), r^'"i cos 2sw + . . 2s7r + ( + "'"--^7^J (« + l). rVn / COS ^ ^ + I sm -!: ^ (?i). No two of these are equal, since the amplitudes of any two differ by less than 2ir. The wth power of any one of them is j-(cos 6 + isind); for take the {s + l)th, for example, and we have "v r^'" I cos + I sm ■ 3s7r + 6- n r AIn 2s7r + e . . 2s7r + e' cos + I sm 'ft TO 2s7r + 6 . . 2s7r + : r I COS 71 + I sm //. \ n n by Demoivre's Theorem, = r(cos (2sir + 0) + i sin (Sstt + 0) ), = r(cos ^ + i sin 6). Hence the complex numbers (1), (2), . . ., (n) are n different mth roots of r(cos ^ + i sin 9). We cannot, by giving values to s exceeding w - 1, obtain any new values of the nth root, for the values of the series (1), (2), . . ., (n) repeat, owing to the periodicity of the trigonometrical functions involved. We have, for example, ?-'/"(cos.(2M?r + 0)ln + i sin.(2m7r + d)jn) = rl'^{cos.9/n + i sin. Ojn) ; and so on. We can, in fact, prove that there cannot be more than n values of the Mth root. Let us denote the complex number r(cos 6 + i sill 0) by a, for shortness ; and let 2 stand for any mtli root of a. Then must z" u., and therefore ,:;" - a 0. Hence every mth root of u., when substituted for s in a" - a, causes this integral function of n to vanish. Hence, if z,, z„, . . ., 240 JITH ROOTS OF ± 1 CHAP. Zg be s nth roots of a, s- z„ z - :„, . . ., s - Sg ^vill all be factors of «" - a. Now «" - a is of the nth degree in z, and cannot have more than n factors (see chap, v., § IG). Hence s cannot exceed n ; that is to sa}', there cannot be more than n nth roots of u. "We conclude therefore that ever;/ complex number has n nth roots and no mwe ; and each of these nth roots can he cqin'ssed as a complex number. Cor. 1. Since every real number is merely a complex num- ber whose imaginary part vanishes, it follows that ercry real number, whether posiiire or negatiic, has n nth routs and no mm~e, earh of whirh is e.rpressihlc as a complex number. Cor. 2. The imaginarij nth roots of any real number can be arranged in conjiignic pairs. For wo have seen that, if x + t/i be any «th root of a, then (,« + iji)'"- - a = 0. Hence, if a be real (but not otherwise), it follows, by § 5, Cor. 4, that (,r - yi)" - a = ; that is, X - j/i is also an nth root of i*. N.B. — This does not hold for the roots of a complex number generally. § 18.] Every real positive quantity can be written in the form r(cos + * sin 0) (A) ; and every real negative quantity in the form r(cos TT + « sin ir) (B) ; where r is a real positive quantity. Hence, if we know the n nth roots of cos + isin 0, that is, of +1, and the n nth roots of cos TT + i sin TT, that is, of - 1, the problem of finding the n nth roots of any real quantity, whether positive or negative, is reduced to finding the real positive value of the wth root of a real positive quantity r (see chap, xi., § 15). By means of the nth roots of ± 1 we can, therefore, com- pletely fill the lacuna left in chap, x., § 2. In addition to their use in this respect, the mth roots of ± 1 play an exceedingly im- XII nTH ROOTS OF ±1 241 portant part in the theory of equations, and in higher algebra generally. We therefore give their fundamental properties here, leaving the student to extend his knowledge of this part of algebra as he finds need for it. Putting r = 1 and = in 1, . . ., m of § 17, and remem- bering that 1'/"= 1, we obtain for the n nth. roots of + 1, cos + ? sm 0, cos — + i sm — , , n n 2(TO-l)7r . . 2(jl-l)7r cos + 4 sm n n Putting r= 1, = ^, we obtain for the ftth roots of - 1, COS - + 4 sm -, cos — + » sm — , , n n n n (271 -IV . . (2w-l)7, cos ^^ — + I sm ' Cor. 1. Since cos . 2(ra - \)irjn = cos . iirjn, sin . 2(?i. - \)Trjn = -sin . 2ir/«; cos . 2(n- 2)?r/7i = cos . 4ir/«, sin . 2(w- 2)ir/m = - sin . 47r/m, and so on, we can arrange the roots of +1 as follows : — Tjth roots of + 1, n even, =2m say, 27r . . StT 477 ,. . 47r + 1, cos — ±t sm — , cos — ±4 sm — , . . ., n n n n cos ^ — ± 4 sin —, - 1 (C) ; nth. roots of + 1, to odd, = 2m + 1 say, 27r . . 27r 4;r , . . 47r + 1, cos — ±4sm — , cos — ±4sm — , . . ., n n n n 2rmr , . . 2nnr ,.^. cos ± 4 Sin (D). n n ^ ' VOL. I R 242 «TH ROOTS 0F± 1 CHAP. Similarly we can arrange the roots of - \ as fnllowg : — Mth roots of - 1, « even, = '2iii sa.y, IT . . TT 3— . . Stt cos - ± I sm -, cos — ± ; sin — , . , n 7/ 11 n (2m- l)7r^ . . (2m- l)7r ,„ cos --- -^-±«sin^ -- (E); n n ^ nth roots of - 1, w odd, = 2w + 1 say, 7r . . 7r Btt . . Stt COS - ± 1 sm -, cos — ± ( sm — . . , // // H II (2m- l)7r . . (2iii - ly cos> ' ±tsm^-- -' , -1 (F). 11 11 ^ ' From (C), (D), (E), (F) we see, in accordance with chap, x., § 2, that the nth root of + 1 has one real value if n be odd, and two real values if n be even ; and that the mth root of - 1 has one real value if n be odd, and no real value if n be even. We have also a verification of the theorem of § 17, Cor. 2, that the imaginary roots of a real quantity consist of a set of pairs of conjugate complex numbers. Cor. 2. The first of the imaginary roots of + 1 in the series (1), . . ., (m), namely, cos. 27r/« + 'i sin. 2x/?i, is called a jjrimiWw* mth root of + 1. Let us denote this root by w. Then since, by Demoivre's Theorem, 27r . . -JwY 2.577 . . 2,s-7r COS h I sui — I = cos h 1, sm , « n / 11 11 and, in particular. 2,r . . 2,r\" COS — + t sm — = COS 271 n n J = 1, * By a primitive imaginary ;itli root of + 1 in general is meant an iith root which is not also a root of lower order. For example, cos.27r/3 + isiu.27r/3 is a 6th root of +1, but it is also a cube root of +1, therefore cos.27r/3 + ?'sin.27r/3 is not a primitive 6th root of +1. It is obvious that cos.27r/;i + j sin . 27r/jj is a primitive ';ith root ; but there are in general other's, and it may be shown that any one of these has the property of Cor. 2. xii FACTORISATION OF a" ± A 243 we see that, if (a he a primitive imagiuary nth root of + \, then the n nth roots of +\ are CO , co', Oj' , . . , 0)" (G). Similarly, if w = cos. it jn + i sin.ir/w, which we may call a primitive imaginary nth root of - I, then the n nth roots of - 1 are V 'V2 -^v'-i )' .... (\/3 + l) + (\/3-l)* -1+i ( V3-l) + (V3 + l)i Llln.t' IS, ;, J •' j^ J . . 2* 2* 2^ Here it will be observed that the roots are not arran<,'('d in conjugate pairs, as they would necessarily have been had the radicand been real. Example 3. To find approximately one of the imaginary 7th roots of +1. One of the imaginary roots is cos 51°25'43" + i sin 51°25'43" By the table of natural sines and cosines, this gives •6234893+ -78183181 as one approximate value for the 7th root of + 1. Example 4. If w be one of the imaginary cube roots of +1, to show that 1 + u + m^ = 0, and that (ax + Lo-y) {u-.c+unj) is real. We have 1 + co + ^^ = (1 - m'*)/(1 - w) = 0, since w'=l and 1-w + O. Again, {ux + oi-y) {uflx + u>y) = w-',r- + (w* + ur)j:y + w-y . Now tu^ = l ; and w'* + 01^ = 01^0) + a)- = 07 + 01^= —1, since l + w + w° = 0. Hence (ux + (J^y) (di^x + wy) =x^-xy + y^. FUNDAMENTAL PROPOSITION IN THE THEORY OF EQUATIONS. § 20.] If f{z)=K + K,-: + KJ + . . +A^» he an in function of z of the, nth degree, whose coej/irienls A„, A,, . ., A,j are (j'lrcu complex, numbers, or, in particular, real numhers, where, of course. A;, =t=0, then f{z) can ahvui/s be c.qjressnl as the product of 71 factors, each of the l.st dcr/r/r in ::, say z - z^, ::-::.,, z - ::,,, . ., z - Zn, «i, «2, ., '-n, being in general complex numbers. XII FACT0K1SA.TI0N OF ANY INTEGRAL FUNCTION 245 It is obvious that this proposition can be deduced from the following subsidiary theorem : — One value of z, in general a complex number, can always be found ■which causes f{z) to vanish. For, let us suppose that /(»,) = 0, then, by the remainder theorem, f(z) =fi{z) (s - z^, where f^{z) is an integral function of z of the {n - l)th degree. Now, by our theorem, one value of z at least, say z^, can be found for which /i(») vanishes. We have, therefore, /[(^z) = ; and therefore fj{z) ^fsiz) (z - z^), where /.(a) is now of the {n - 2)th degree ; and so on. Hence we prove finally that f{z) = A(Z - «,) {Z-Z^)...{Z- Zn), where A is a constant. § 2 1 .] We shall now prove that there is always at least one finite value of z, say z = a, such that by taking z sufficiently near to a, that is by making | « - ft | small enough, we can make \f{z) I as small as we please. So that in this sense every integral equation /(«) = has at least one finite root. Let I « I = E. Then, since \f{z)\ = \A^\W^\l+A^.,IA^z+ ... +A„/A,-|, we have, by § 14, \f{z) I > I A„ I E"{1 - I A„.,/A„« + . . . + A„/A„;" I }, provided | z |, {i.e. E), be large enough ; therefore |/(«)|>|A„|Eni-C(l/E+ . . . +1/E-)}, where C is the greatest of [A„.,/A,i|, . . ., |Ao/AJ. There- fore, taking provisionally E>1, we have \f{z) I > I A„ I E"{1 - C(l - l/E-)/E(l - 1/E)}, >|A^|Eni-C/(E-l)} (1), provided E > C + 1 . Hence, hy taking \ z \ sufficiently large, we can make | f(z) \ as large as we please ; and we also see that there can be no root of f(z) = whose modulus exceeds C + 1. Let now w be the value of z at any finite point in the Argand Plane, so that \f{w) \ is finite. It follows from what has 246 EXISTENCE OF A ROOT CHAP. just been proved that wc can describe about the origin a circle S of finite radius, such that, at all points on and outside S, |/(~) I > l/C'") I • Then, since \f{w) \ is real and positive, if we consider all points within S, we see that there must be a finite lower limit L to the value of |/(«') | ; that is to say, a quantity L which is not greater than any of the values of \f{w) \ within S, and such that by properly choosing u we can make |/(w) | = L + £, where e is a real positive quantity as small as we please. We shall show that L must be zero. For, suppose L > 0, and choose w so that \f{w) | = L + e. Let h be a complex number, say r(cos 6 + i sin ff). Then f{w + h) = Ao + A{w + h)+ . . . + Kn{iv + A)", =f{ii)) + B,A + BJi" + . + A„/i'* (2), where A„ is independent both of w and h, and by hypothesis cannot vanish, but B,, . ., B,j_i are functions of w, one or more of which may A-anish. Suppose that B,„ is the first of the B's that does not vanish, and let i,„(cos a,„ + i sin a,,,,), etc., be the normal forms of the complex numbers B„j//(to), etc. Then, since \f{w) \ is not zero, 5,„, etc., are all finite. Also we have, by Demoivre's Theorem, f{w + h)lf{w) = 1 + b,„r^e^ + &^+,r™+ie,„+, + . . . + J„r"e„ (3), where 9^ = cos (mO + a,„) + i sin (m9 + a^), etc. We have h, and therefore both r and 6 at our disposal. Let us first determine so that cos (in9 + a,„,) = - 1, sin (mO + a^) = 0; that is, give 6 any one of the m values (tt - a,„,)/m, (Stt - am)lm, . . ., {2rii - 1 . 77 - a„^!ni, say the first. Then we have 0,„ = - 1 ; and 0,„,-)-„ etc., assume definite values, say, Q'm+u etc. We now have f{w + h)lf(w) = I - h^r^^ + Z/,„+,r"'+ie'™+, + . . . + J„r"e'„ (4). Considering the right hand side of (4) as the sum of 1 - 5,„7-™ and &,„+,r"'+'e'm+, + . . + //„r"e'„, we see, by § 14, that the modulus of f{iv + h)lf{iv) lies between the difference and the sum of the moduli of these two. Also XII EXISTENCE OF A ROOT 247 I K+.r^+^e'm+, + . . . + h^Q',f' I < J,„+,r'"+i + . . + J„r", 7-"i+2> . >,•«_ Therefore we have 1 - Kr^ -{n- m)br'»+^ < |/(w + li)IJ{w) \ < 1 - h^r'^ + (ra - m)hf''+^ (5) ; provided r be so chosen that 1 - Jmr™ and 1 - &,„r™ - (m - ni)br™+'^ are both positive. Let us further choose r so that (m - m)br'^+''^ h^j{n-m)h. (6). When r is thus chosen, \f{w + h)jf{w) \ will lie between two positive proper fractions, so that |/(w + h)jf{w) \=\ - jx, where jx is a positive proper fraction ; and we have |/(«; + /0| = (l-/.)|/HI = (l-/z)(L + .) = L + e-;.(L + .)(7). which, since e may be as small as we please, is less than L by a finite amount. L is therefore not a finite lower limit as sup- posed ; in other words, L must be zero. Our fundamental theorem is thus established. By reasoning as above we can easily show that, if f(z) = J„ + A^+ . . . +A,,z'\ then \K\{l-{n-s+\)d\z\^]<\f{z)\ <\K\{\+{n-s+l)d\zy] (8), where d is the greatest of | Aj/Ao | , . . ., | A„/Aj, | , provided | z \ is less than the lesser of the two quantities 1, l/{(m - .s + l)rf}i'^ Combining this result with one obtained incidentally above, we have the following useful theorem on the delimitation of the roots (real or imaginary) of an equation. Cor. 1. The equation A,, + A^s* + . . . + A„«" = can have no root whose modulus exceeds the greatest of the quantities 1 + 1 AJAn | , 248 GRADIEXT AXD EQUIMODULAR CURVES CHAP. 1 + I As/A„ I , . . ., 1 + j A,j_i/A„ ! , or whose modulus is less than the hast of 1, !/{(« -s+l)\ A,/A„ | }V», . . ., ■!/{(« -s+\)\ A,/A„ | fK Cor. 2. JVa can aliuays assign a positive quantity -q, such that, if |/i|<7;, 1/(5 ■(- A) -f{z) |/(m;i)> f{w^ > . . , This process we may call Argand's Progression towards the root of an equation. Since 6„<6, 6m''™ + (t-OT)6r"'+iJSl2), which are the afiixes of the two imaginary roots. The diagram of chap. xv. § 19 will also furnish a curious illustration by taking initial points on one or other of the two dotted lines. It should be noted that the question as to whether |/(s) | actually reaches its lower limit is not essential in Argand's proof, if we merely propose to show that a value of z can be found such that \f{z) | is less than any assigned positive quantity, however small. Nor do we raise the question whether the root is rational or irrational, which would involve the subtle question of the ultimate logical definition of an irrational number (see vol. ii. (ed. 1900) chap. XXV. §§ 28-41). § 22.] We have now shown that in all eases f{z) = A{z -z){z-z,) . . .{z- «„), where A is a constant. ' «„ 22, . . ., 2,1 may be real, or they may be complex numbers of the general form x + yi. They may be all different, or two S-'iO EQUATION OF VTA DEGREE HAS n ROOTS CHAP. or more of them may be identical, as may be easily seen by considering the above demonstration. The general proposition thus established is equivalent to the following : — If f{z) be an integral fnmiion of rs of the nth dei/rre, there are n values of z for which f{z) vanishes. These values may he real or complex numbers, and may or may not he all unequal. AVe have already seen in chap, v., § 16, that there cannot be more than n values of z for which f{z) vanishes, otherwise all its coefficients would vanish, that is, the function would vanish for all values of z. "We have also seen that the constant A is equal to the coefficient A„. "We have therefore the unique resolution /(s) = A„(.-.,)(.-g. ,(.-r„). § 23.] If the coefficients of f{z) be all real, then we have seen that if f{x + yi) vanish f{x - yi) will also vanish. In this case the imaginary values among s„ ~j, . ., z,i will occur in conjugate pairs. If a + fSi, a - j3i be such a conjugate pair, then, correspond- ing to them, wo ha\'e the factor {z-a- pi) (z-a + j3i) = (Z- af + ji\ that is to say, a real factor of the 2nd degree. It may of course happen that the conjugate pair a ± /3i is repeated, say s times, among the values z„ ,"_,, . ., g^. In that case we should have the factor (2 - a)^ + fi^ repeated s times ; so that there would be a factor {(? - a)^ + fS^Y iti the function f(z). Hence, erery integral function of v, whose coefficients are all real, can he resolved into a product of real factors, each of which is either a positire integral povcr of a real integral function of the \st degree, or a 'positive integral power of a reed integral function of the 2nd degree. This is the general proposition of which the theorem of § 19 is a particidar case. EXERCISES XVI 251 Exercises XVI. Express as complex numbers — (1.) {a + Uf + {a-Uf. (2.)-l±i+l::i. ^ ' l + 2i l-2r , 2 + 36i; 7-26i \p-qij \p + qij ' ,5 > 69-7V(15) + (V3-6V5) t (6.) Show that 2 j +V 2 j = -1, if ?t be any integer which is not a multiple of 3. (7. ) Expand and arrange according to the powers of x {x-l-i^2){x-l + i~^2){x-2 + i^J3){x-2-i^JB). (8. ) Show that {(2a-6-c) + i(J-c)V3}^={(2i-c-a)-l-i(c-e)V3}». (9. ) Show that {(v'3 + l) + (\/3-l)»P = 16(l + »)- (10.) If f + 7;i be a value of x for which ax^ + tx + c — 0, a, b, c being all real, then 2o(|ij + Jt; = 0, ai)^=aJ^ + 6J + c. (11.) If i/{x + yi) = X + Yi, show that 4(X2-Y2)=rr/X + 2//Y. (12.) If » be a multiple of 4, show that l + 2i + 3i^+. . .+(«. + l)i"=J(?i + 2 -wi). (13.) Show that |a|) + ffiiS + . . .+a„z"|*|an| |z|"(l - mc/|3|), provided |s| exceeds the greater of 1 and nc, and c is the greatest of | oso/o^m l> • • • > I a-n-'ilan I- (14.) Find the modulus of (2-3t)(3 + 4i) (6 + 4«)(15-8i)' (15.) Find the modulus of {x+sJix' + yH)]^ (16.) Find the modulus of hc(fi - ei) + ca(c - ai) + ab{a - hi). (17.) Show that \\ + ix + i'^x'^ + 'Pa? + . . . ad co| = l/^(l+a;2)^ -vyliere a;<:l. (18.) Find the moduli oi(x + yiY and {x + yi)"j{x-yi)". Express the following as complex numbers : — (19.) \/-7 + 244. (20.) J e + iJlS. (21.) v/ - 7/36 + 2i/3. (22.) Jiab + 2(o? - l}^)i. (23.) 'Jl+2xsKx'^\)i. (24.) 'J\+iJ{x'^~i). (25.) Find the 4th roots of - 119 + 120J. 252 EXERCISES XVI I'HAP. (26. ) Resolve 3?-a^ into factors of the 1st degree. (27. ) Resolve a^ + 1 into real factors of the 1st or of the 2nd degree. (28.) Resolve a^ + afi+x^ + sfi+x^+x + l into real factors of the 2nd degree. (29.) Resolve x^-2 cos to^K^ + a^'" into real factors of the 1st or of the 2nd degree. (30.) If u be an imaginary mth root of +1, show that l + w + u" + . . . + u»-^ = 0. (31.) Show that, if a be an imaginary cube root of +1, then a? + if + z^- 'i.rir- = {x + y + z){x + oiy + w"i) (x + (iihj + uz), and {x + oiy + u'zf + (x + f (cos 29 + « sin 29) (cos 0-ts i n 4>f cos~(9 + .^) + isin(9 + 0) ^ cos (9 + 0)- i sin (9 + 0) (35.) If ,y(a + i«)+ ;^/(c + rfi)= sl(x + yi), show that (x-a-cf + (y-^b-df = i^J{[a- + V')(c- + d'^)}. (36.) Prove that one of the valuus of ^{a 4- hi) + ^(a - hi) is J[ V{2a + 2v'(a= + i')} +2 ^{a.- + h''-)l (37.) If ■«i = cos 7r/7 + i sinir/7, prove that [x-to) (x + w^) {x-v^) {x + w'') {x-icfi) {x + w^) = x^-x^ + x^-afl + x--x + l. (38.) Find the value of w''i + w''2+ . . . +w''„ ; Wj, w„ . . . w„ being the »ith roots of 1, and r a positive integer. What modification of the result is necessary if r is a negative integer ? Prove that 1/(1+ Wire) + 1/(1 + ■!02a;) + . . . +1/(1 +tt>,„cc) = m/(l -a;"). (39.) Decompose l/{l+x + x^) into partial fractions of the form al{bx + c). Hence show that xl(l + x + x^)=x-x'' + x'-afi + x''-:ifi + . . . + x^+^ - x^+^ + II, where a^, a;", etc. , are wanting ; and find R. (40.) Find the equation of least degree, having real rational coefBeients, one of whose roots is i^2 + i. One root of a^ + 3ar'-30a:2 + 366a; -340 = is 3 + 5i, find the other three roots. (41.) If a be a given complex number, and z a complex number whose afl5xe lies on a given straight line, find the locus of the affixe of a + z. (42.) Show that the area of the triangle whose vertices are the affixes of 2l, S2, =3 is S{(32-%) I % IV-i^-i}- (43.) If » = (a + 70os 9) + i(/3 + 7 sin 9), where a, /3, y are constant and 6 variable, find the maximum and minimum values of \z\; and of amp z when such values exist. xn HISTORICAL NOTE 253 (44.) If the affixe o{x + yi move on the line Sx+iy + 5=0, prove that the irtininram value of \x + yi\ is 1. (45.) If u and v are two complex numbers such that u=v + llv, show that, if the affixe of v describes a circle about the origin in Argaud's diagram, then the affixe of m describes an ellipse (a;'^/a^ + 2/^/6^=1) ; and, if the affixe of ■M describes a circle about the origin, then the affixe of v describes a quartic curve, which, in the particular case where the radius of the circle described by the affixe of u is 2, breaks up into two circles whose centres are on the i-axis. (46.) If K and y be real, and x + y = l, show that the affixe of xz^ + yz^ lies on the line joining the affixes of Sj and Cj. Hence show that the affixe of' xZi + yz2 lies on a fixed straight line provided lx + my=l, I and m being constants. (47.) If J + '/i be an imaginary root of a:^ + 2a! + l = 0, prove that (?, ri) is one of the intersections of the graphs of 17^=3^^ + 2 and i)^=l/2 + 3/8f. Draw the graphs : and mark the intersections which correspond to the roots of the equation. If a be the real root of this cubic, show that the imaginary roots are i{-a + iJ{2-Sla)}. (48.) If f+i)i be a pair of imaginary roots of 3fi-px + q = 0, show that (f, 17) are co-ordinates of the real intersections oti^''-Ti'-p = 0, S^-rf + ip^- 32=0. Hence prove that the roots of the cubic are all real, or one real and two imaginary, according as 4jp'< >27j^. What happens if 4;5'=27}^ ? (49.) If a^ + qx + r=0 has imaginary roots, the real part of each is posi- tive or negative according as r is positive or negative. (50.) The cubic a^ - 9a;^ 4- 33a; - 65 = has an imaginary root whose modulus is ;y/13 ; find all its roots. (51.) Find the real quadratic factors of x-" + x-"-''^ + ... -1-1 ; and hence prove that 2" sm 3— --t: sm ;r — — - . . . sm -^ r = V(2« H- 1). 2ji -f 1 2» -I- 1 271 -I- 1 ^ (52.) Find in rational integral form the equation which results by eliminating 6 from the equations a: = a cos ^-f 6 cos 3^, y=asin0 + bsmS6. (Use Demoivre's Theorem.) Give a geometrical interpretation of your analysis. SistoHcal Note. — Imaginary qu.intities appear for the first time in the works of the Italian mathematicians of the 16th century. Cardano, in his Artis Magnsa sive de Regulis Algebraicis lAier Unus (1545), points out (cap. xxxvii., p. 66) that, if we solve in the usual way the problem to divide 10 into two parts whose product shall be 40, we arrive at two formulae which, in modern notation, may be written 5 + ij - Ih, 5 - J - IZ. He leaves his reader to imagine the meaning of these "sophistic" numbers, but shows that, if we add and multiply them in formal accordance with the ordinary algebraic rules, their sum and product do come out as required in the evidently impossible problem ; and he adds " hucusque progreditur Arithmetica subtilitas, cujus hoc extremum ut dixi adeo est subtile, ut sit inutile." Bombelli in his Algebra (1522), following Cardano, devoted considerable attention to the theory of complex numbers, more especially in connection with the solution of cubic equations. 254 HISTORICAL NOTE CHAP xri. There is clear indication in the fragment De Arte Logislka (see above, p. 201) that Napier was in possession to some extent at least of the theory. He was fully cognisant of the independent existence of negative quantity ( " quantitates delectiva^ minores nihilo"), and draws a. clear distinction between the roots of positive and of negative numbers. He points out (Napier's Ed., p. 85) that roots of even order have no real value, either positive or negative, when the radicand is negative. 8uch roots he calls " nugacia " ; and expressly warns against the error of supposing that l_l - 9 — - j_l 9. In this passage there occurs the curious sentence, ''Hujus arcaiii magni algebraici fundamentum superius, Lib. i. cap. 6, jecimus : quod (quamvis a nemine quod sciam revelatum sit) quan- tum tamen emolumenti adferat huic arti, et cEeteris mathematicis postea patebit." There is nothing farther in the fiaguient De A rte Logistica to show how deeply he had penetrated the secret which was to be hidden Jrom mathematicians for 200 years. The theory of imaginaries received little notice until attention was drawn to it by the brilliant results to which the use of them led Euler (1707-1783) and his contemporaries and followers. Notwithstanding the use made by Euler and others of complex numbers in many important investigations, the fundamental principles of their logic were little attended to, if not entirely misunderstood. To Argand belongs the honour of first clearing up the matter in his Essai sur une maniere de reprisenter les quantitis imaginaires (hms les constructions giometrii/ues (1806). He there givi^s geometrical constructions for the sum and product of two complex numbers, aud deduces a variety of conclusions therefrom. He also was one of the fir^t to thoroughly understand and answer the question of § 21 regarding the existence of a root of every integral function. Argand was an- ticipated to a considerable extent by a Danish mathematician, Caspar Wessel, who in 1797 presented to the Koyal Acadenjy of Denmark a remarkable memoir Om iJirekiionens analytiske Jjiirf/ning, et Forsog, anvendt foniemmelig til plane og sphaeriske Polygoners Oplijsning, which was published by the Academy in 1799, but lay absolutely unknown to mathematicians, till it was republished by the same body in 1897. See an interesting address by Beniau to Section A of the American Association for the Promotion of Science (1897). Even Argand's results appear to have been at first little noticed ; and, as a matter of history, it was Gauss who first initiated mathematicians into the true theory of the imaginaries of ordinary algebra. He first used the phrase coviplex mimhrr, and introduced the use of the symbol i for the imaginary unit. He illustrated the twofold nature of a complex number by means of a diagram, as Argand had done ; gave a rfiasterly discussion of the fundamental principles of the subject in his memoir on Bi- quadratic Residues (1831) (see his Works, vol. ii., pp. 101 and 171) ; and furnished three distinct proofs (the first published in 1799) of the proposition that every equation has a root. From the researches cf Cauchy (1789-1857) and Riemann (1826-1866) on complex numbers has sprung a great branch of modern pure mathematics, called on the Continent function - theory. The student who wishes to attain a full comprehension of the generality of even the more elementary theorems of algebraic analysis will find a, knowledge of the theory of complex quantity indispensable ; and without it he will find entrance into many parts of the higher mathematics impossible. For further information we may refer the reader to Peacock's Algebra, vol. ii. (1845) ; to De Morgan's Trigonometry and Double Algebra (1849), where a list of most of the English writings on the subject is given ; and to Hankel's Vorlesungen iiAer die complexen Zahlen (1867), where a full historical account of Continental researches vrill be found. It may not be amiss to add that the theory of complex number'; is closely allied to Hamilton's theory of Quaternions, Grassmann's Ausdehnungslehre, and their moilern developments. CHAPTEE XIII. Ratio and Proportion. EATIO AND PROPORTION OF ABSTRACT QUANTITIES. § 1.] The ratio of the abstract quantity a to the abstract quantity b is simply the quotient of a by b. When the quotient a~ b, or ajb, or y is spoken of as a ratio, it is often written a:b; a is called the antecedent and b the con- sequent of the ratio. There is a certain convenience in introducing this rmw name, and even the new fourth notation, for a quotient. So far, how- ever, as mere abstract quantity is concerned, the propositions which we proceed to develop are simply results in the theory of algebraical quotients, arising from certain conditions to which we subject the quantities considered. If a > ft, that is, if a - J be positive, a : i is said to be a ratio of greater inequality. li ab, the ratio a:b is diminished by adding the same positive quantity to both antecedent and consequcni ; and increased by subtracting the same positive quantity (< b) from both antecedent and consequent. If a i, b - a is negative ; and x, b, b + x are all positive by the conditions imposed ; hence x(b - a)/b(b + x) is negative. .-r a + X a . Hence t is negative, b + x b =" a + x a that IS, :, < T. b + x b a-x a x(a - b) ° ' Ij - X b l>(b - :r,) But, since a>b, a-b is positive, and x and b are positive, and, since x -. h -X h' The rest of the proposition may be established in like manner. The reader will obtain an instructive view of this proposition by comparing it with Exercise 7, p. 2G7. § 4.] Permidatinns of a Proportion. If a:h = c:d (1), then b:a = d:c (2), a:c = b:d (3), and c:a = d:h (4). For, from (1), we have Hence l/^^l^ that IS, - = - ; a c that is, h : a = d:c, which establishes (2). Again, from (1), - = -, multiplying both sides by -, we have a h c h c d c 4.1, i ■ * ^ that is, - = -, : c d that is, a:c = h:d, which proves (3). (4) follows from (3) in the same way as (2) from (1). § 5.] The product of the extremes of a proportion is equal to the product of the means , and, conversely, if the product of two quantities he equal to the product of two others, the four form a proportion, the extremes leing the constituents of one of the products, the means the constituents of the other. VOL. I g 258 RULE OF TI-IKEE For, if a:l) = c:d. that is, a c h ~ d' then - X M = -, X hd, h d whence ad = he. Again, if ad = he, then adjhd = hejbd, whence a c b~d' Cor. If three of the terms of a proportion be given, the remaining one is uniquely determined. For, when three of thequantities a, b, c, d are given, the equation ad = be, which results by the above from their being in jjroportion, be- comes an equation of the 1st degree (see chap, xvi.) to deter- mine the remaining one. Suppose, for example, that the 1st, 3rd, and 4th terms of the proportion ^re \, f, and f ; and let x denote the unknown 2nd term. Then *:^=|:t; whence ^ X .T — 5 X f. Multiplying hy •£, we have a;=Jxf X = A- § 6.] Relations connecting quantities in continued proportion. If three quantities, a, b, c, be in continued proportion, then a:c = €^:'\f = lf:i; and b = \''(«c). If four quantities, a, h, c, d, be in continued proportion, then a:d = a^:h^ = V:c^ = c^: cl\ and b = l/{a\l), c = l/{ad?). For the general proposition, see Exercise 12, p. 267. DETERMINATION OF MEAN PKOrORTIONALS 259 For, if a:b = b:c, then a b b~'c Therefore a b b b T X - = -x -, c c c whence a V a' (!)• Also ac = b\ whence b = ^/(ac) (2). Equations (1) and (2) establish the first of the two proposi- tions above stated. Again, if a:b = b:e = c:d. then abac b c b d' Also a a b^b' hence a a a a b c b b b c d that is, a" a F'd' therefore a ft" b" c" d~b'~?''d' (3). Further, since a a" ~d~f' b' = a'd; whence b=l/{a'd). (4). Also, since a i d~d" i = a^ ■ whence c = lj{ad,') (5). It should be noticed that the result (2) shows that the finding of a mean pi'oportional between two given quantities a and c depends on the extraction of a square root. For example, the mean proportional between 1 and 2 is V{lx2) = V2 = l-4142 . . . 260 DELIAN PROBLEM CHAP. Again, (i) and (5) enable us to insert two mean proportionals between two given quantities by extracting certain cube roots. For example, the two mean proportionals between 1 and 2 are 4/(1x2)= ^2 = 1-2599 . . . and v'(lx 2=) = ^ =1-5874 . . V2 Converselj', of course, the finding of the cube root of 2, which again corre- sponds to the famous Delian problem of antiquity, the duplication of the cube, could be made to depend on the finding of two mean proportionals, a result well known to the Greek geometers of Plato's time. § 7.] After what has been done, the student will have no difficulty in showing that (1)- (2). if a:'b = c:d, then ma : mb = nc:nd and ma : nh = mc : nd §8.] if Also that a,:\ = c^: d^, then afl.2 . . . a„ : bfi^ - . • i„ = cfi^ . . .Cn -.d^d.^ Cor. If a:h = c:d, then a" : J" = c" : d". (!)• (Here n, see chap, x., may be positive or negative, integral or fractional, provided a"^, &c., be real, and of the same sign as a, &c.) § 9.] 7/" a:b = c:d, then a±b:b = c±d:d (1), a + b:a-b = c + d:c-d (2), la + mh: pa + qb = Ic + md '.pc + qd (3), to'" + mb'' : pa'' + qb'"' = k'' + md" : ptf + qd'' (4), where I, m, pi, q, r are any quantities, positive or negative. ?III CONSEQUENCES OF PEOPOETION 261 Also, if a,:i, = a^:h = a^:bs= . . . = a„ : 5„, then each of these ratios is equal to a, + cig + . . . + a„ : J, + 5, + . . . + S„ (5) ; ttTid also to :y{ka^ + Ifii + . . .+ l^a^) : l/q,},,^ + li)i + . . + Z„V) (6). Thougti outwardly somewhat different in appearance, these six results are in reality very much allied. Two different methods of proof are usually given. FIRST METHOD. Let us take, for example, (1) and (2). Since t = -,, a therefore 7 ± 1 = -= ± 1 : d a±b c±d whence —. — = — r- : a this establishes the two results in (1). Writing these separately we have a+l _c+d a-h c- d (a + b) I (a-b) _{c + d) I (c - d) whence b d , . a+b c+d that is, i = ;, a-b c-d which establishes (2). Similar treatment may be applied to the rest of the six results. 262 EXPRESSION IN TERMS OF FEWEST VARIABLES SECOND METHOD. Let US take, for example, (2). Since a/b = c/d, we may denote each of these ratios by the same symbol, p, say. We then have a c rp' rp' whence a = pb, c = pd (a). Now, using (a), we have a + b pb + b a- ■b 'ph-h' b{p+\) b{p-iy p+l -p-r exactly the same way, we have c + ■d _ pd + d c-d pd - d' P-V a+b p+l c+d Hence , = - - = ,. a- b p- I c-d Again, let us take (5). A\ e have t = t = ir= ■ ■ ■ =i-i each = p, say, Oi O2 O3 o» hence ai = ph^, a^ = ph^, . . ., a„ = pb^; «! + a^ + . . . + an pbi + pb^ + . . . + pbn therefore hence 61 + J2 + . . +bn b^ + h^+ . . . +bn ^ P{h + h + . . . + hn) a, a^ . ffli + a, + . . . 4- a„ : &c. = p = 6, 6, ■ b, + h+ . . . +b„ GENERAL THEOREM 263 Finally, let us take (6). Since a,'" = (pb.Y = /d'-J,'", we have a/ = (pb^y = p^b/, &o. (see chap, x., § 4). It follows that 1/{IA' + W+ ■ ■ . +lnbn') ~''~ b,' b~ Of the two methods there can be no doubt that the second is the clearer and more effective. The secret of its power lies in the following principle : — - In establishing an equation between conditioned quantities, if we first exp-ess all the quantities involved in the equa.tion in terms of the fewest quantities possible under the conditions, then tlie verification of the equation involves merely the establishment of an algebraical identity. In establishing (2), for instance, we expressed all the quantities involved in terms of the three b, d, p, so many being necessary, by § 5, to determine a proportion. A good deal of the art of algebraical manipulation consists in adroitly taking advantage of this principle, without at the same time destroying the symmetry of the functions involved. § 10.] The following general theorem contains, directly or indirectly, all the results of last article as particular cases ; and will be found to be a compendium of a very large class of favourite exercises on the present subject, some of which will be found at the end of the present paragraph. If cj>{xj, x^, . . ., x,^ be any homogeneous integral function of the variables x^, x.^, . . ■, x^ of the rth degree, or a homogeneous function of degree r, according to the extended notion of homogeneity and degree given at the foot of p. 73, and if at:bi = a^:bs= . . . =an:bn, then each of these ratios is equal to !:/4>{a„ a,, . . ., an):^4>{b„ b„ . . ., J„). 264 EXAMPLES ciiAl'. This theorem is an immediate consequence of the property of homogeneous functions given in chap, iv., p. 73. Example 1. Which is the greater ratio, x" + y'' : x + y, or x^-y'^-.x-y, x and y being each positive ? rr- + 1/- x'-y-_ {x- + y^){x-y)-{x^-y-){x + y) x + y x-y~ {x + y)ix-y) _ Ixy'^ - t^y ~{x + y){x-y)' _ 2xy(x - y) {x + y){x-y) x + y Now, if X and y be eacli positive, -ixyjix + y) is essentially negative. Hence x^ + y'-.x + y^x^-y-'.x-y. Example 2. If a:h=c:d, and A:B = C:D, then a^J^.-h\/'R■.c^JG-d\JTl = a^JK + l'^B:cs/G + d\/D. Let each of the ratios a : h and c : d=p, and each of the two A : B and C : D = ff, then a = pb, c=pd ; A = crB, C = crD. Wo then have a\/A - 6v 'E_ p6\/((rB) - &VB cVC-fi\/D~P''V( b-c ~ c-a a-b then {a + b + c){x + y + z) = ax + by + cz (2). Let each of the ratios in (1) be equal to p, then bz + cy = p{b-c) (3), cx + az=p{c-a) (4), ay + bx=p(a-b) (5). From (3), (4), (5), by addition, {b + c)x+(c + a)y + {a+b)z=p{{b-c) + {c-a) + {a-b)], = pO, = (6). If now we add ax + by + cs to both sides of (6) we obtain equation (2). * Examples 4, 5, and 6 illustrate a species of algebraical transformation which is very common in geometrical applications. In reality they ai-e ex- amples of a process which is considered more fully in chap. xiv. 266 EXAMPLES Example 6. If cy + bz _ az + c.r _ h.,- + ay__ qb + rc- pa re +pa - qb 2)a + i/li~ re show that X a{pa{a + b + c)-qb{a + b-c)-rc{a-b + c)} y (1), 'b[qb(a-\-b + c)-pa(a + b-c)-rc( - a + b + (,)]' (2). c{rc(a + b + c)-qb{-a + b + c) -pa(a -b + c)} Let each of the fractions of (1) be =p; and observe that the three equations, cy + bz = {qb +rc -pia)p (a) ~1 az + ex = (re + 2ia - qb)p (/3) 1- (3), bx + ay = {2>a + qb - rc)p (y) ) which thus arise are symmetrical in the triple sct-^ abc K so that the simul- [ijqrj taneous interchange of the letters in two of the vertical columns simply changes each of the equations (3) into another of the same set. It follows, then, that a similar interchange made in any equation derived from (3) will derive therefrom another equation also derivable from (3). Now, if we multiply both sides of (/3) by b, and both sides of (7) by c, we obtain, by addition from the two equations thus derived, 2bcx + a [cy + bz} =p [b(rc+2}ci.- qi) + c{2Ja + qb -re)] (4). Now, using the value of cy + bz given by (a), we have 2bcx + pa{qb + re -p>ci) = p {pa{b + e)-qb(b-e) -rci-b + e)] (5). Subtracting pa{qb + re - 2j(') from both sides of (5), wc have 2bex=p{pa{a + b + c) - qb{a + b - c) - re{a -b + c)} (6). From (6), we have X _ p a {2)a{a + b + c)- qb{a + b-c) - rc(a - 6 + c) [ 2abe (7). /xuji\ /xap\ We may in (7) make the interchange (into), or (into), and we shall \ ybq / \ zcr / obtain two other equations derivable from (3) by a process like that used to derive (7) itself. These interchanges leave the right-hand side of (7) un- altered, but change the left-hand side into the second and third members of (2) respectively. Hence the three members of (2) are all equal, each being in fact equal to pj2abe. > This is a good example of the use of the principle of symmetry in compli- cated algebraical calculations. EXERCISES XVII 267 Exercises XVII. (1.) "Which is the greater ratio, 5 : 7 or 151 : 208 ? (2. ) If the ratio 3 : 4 be duplicated by subtracting x from both antecedent and consequent, show that x=lf. (3.) What quantity x added to the antecedent and to the consequent of a : b will convert this ratio into c-.dl (4.) Find the fourth proportional to 3^, 5-|, 6f ; also the third proportional to l + \/2and 3 + 2V2. (5.) Insert a mean proportional between 11 and 19 ; and also two mean proportionals between the same two numbers. (6.) Find a simple surd number which shall be a mean proportional be- tween ^y7 - \/5 and IIV^ + ISVS. (7.) If X and y be such that when they are added to the antecedent and consequent respectively of the ratio a:b its value is unaltered, show that X :y=a -.b. (8. ) If X and y be such that when they are added respectively to the ante- cedent and consequent and to the consequent and antecedent of « : 6 the two resulting ratios are equal, show that either x=y or x + y= -a- b. (9. ) Find a quantity x such that when it is added to the four given quan- tities a, b, c, d the result is four quantities in proportion. Exemplify with 3, i, 9, 13 ; and with 3, 4, IJ, 2. (10.) If four quantities be proportional, the sum of the greatest and least is always greater than the sum of the other two. (11.) If the ratio of the difference of the antecedents of two ratios to the sum of their consequents is equal to the difference of the two ratios, then the antecedents are in the duplicate ratio of the consequents. (12.) If the n quantities ai, a-i, . . ., a„ be in continued proportion, then «! : a„=ai"-^ : a2"~''=a2"~-' : 03""^ = &c. ; and 02= "A/(ai'>-%„), a3="v^(ai"-%2„)_ . . ., a,= "y(ffli"-'-a„'-i). (13.) If {pa + qb + rc + sd){pa-qb-rc + sd) = {pa-qb + rc-sd)[pa + qi-rc- sd), then ba:ad=ps:qr ; and, if either of the two sets u, b, c, d or p, q, r, s form a proportion, the other will also. (14.) lia:h=c:d=c:f, then a^ + Sa% + y^:J ace : ^Jbdf = V(a^ -c^ + c" + 2ac) ■.\/{b''- d? +f + 2bd) (7). 268 EXERCISES xvir chap. (15.) Ua:a' = b:b', then a"'+" + (i'»J" + 6"+" : «'""+" + a'"'b'" + 6'"+" = (a + 6)™+" : {a' + 6')™+". (16.) Ua:b = c:d, anda:/3 = Y:5, then a,V + {a^b + ab^)ap + b^^ : {a? + b^) (a^ + /S^) = c'y- + {cH + al-]yS + cPd^ : [c^ + d^) {y^ + S'). (17.) 11 a: b = b ■.c = c:d, then (a:- + b'- + c-){b'' + c' + d-) = {ab + bc + cdf (a); (i-c)= + (<;-o)2 + (rf-6)- = (a-rf)"- (;S) ; ab + cd + ad = {a + b + c){b-c + d) (7); a + 6-c-(« = (a + J)(i-rf)/6 (5); (ct + 6 + c + rf) [a-b-c + d) = 2{ab-cd]{ac-bd)l{ad + bc) (e). (18.) If a, J, c he in continued proportion, then a' + ab + b^:b'' + bc + c-=a:c (a); »-(a-6 + (;)(a + 6 + c) = a* + a=62 + 6*' (/3) ; (6 + cflib -c) + {c + afl(c -a) + (a + bfl(a -b) = ib{a + b + c)l(a - c) (y). (19.) If a, b, c, d he in continued proportion, then (a-c){b-d)-(a-d)(b-c) = {b-c)- (a); ^{ab) + sj{bc) + ^[ed) = '^{{a + b + c)(b + c + d)\ (/3). (20.) Uab = cd = ef, tlien {ac + ca + ea)ldbf(d + 6 +/) = (a" + c^ + e-)j[b"d- + cC-f +fl>^). (21.) K (a-b)j(d~e) = (b-c)l(e-f), then each of them = {b{f-d) + (cd-af)\le(f-d). (22.) miyx = vlx'' = ^lyz, thena!/|,, = j,/f =s/',f. (23.) If 2x + 3ij:3ij + iz:is + 5x=:ia-5b:Sb-a:2b-3a, then 7.(- + 62/ + 8« = 0. (24.) If ax + cy.by + dz = aij + cz -.bz + dx = az + ex -.bx + dy, and if x + y + z^O, ab-cd + 0, ad-bc + 0, then each of these ratios =» + <; :6 + d ; and x^ + y^ + z- = yz + zx + xy. (25.) li {a-ny + viz)ll' = (b-lz + nx)lin'=(c-'mx + ly)ln', then / m'c - n'b \ ,, / n'a -I'c \ , ( Vb - m'a IV + mm' + nn! ,, ( na — Ic \ I f lb -ma \ , , 1= v-rr, -, ; )/™= I ~- iT, i ; M \^^ U +7nm +nn /' \ U +mni +nn J' RATIO AND PROPORTION OF CONCRETE QUANTITIES. § 11. J "We have now to consider how the theorems we have (-■stablished regarding the ratio and proportion of abstract num- bers are to be applied to concrete quantities. We shall base * Important in the theory of the central axis of a system of forces, &c. XIII CONCRETE RATIO AND PROPORTION 269 this application on the theory of units. This, for practical pur- poses, is the most convenient course, but the student is not to suppose that it is the only one open to us.- It may be well to recall once more that any theory may be expressed in algebraical symbols, provided the fundamental principles of its logic are in agreement with the fundamental laws of algebraical operation. § 12.] If A. and B he two concrete quantities of the sane hind, which are expressible in terms of one and the same unit by the com- mensurable numbers a cmd h respectively, then the ratio of A. to 'B is defined to be the ratio or quotient of these abstract numbers, namely, a : h, or a/b. It should be observed that, by properly clioosing the unit, the ratio of two concrete quantities which are each commensurable with any iinite unit at all can always be expressed as the ratio of two integral numbers. For ex- ample, if the quantities be lengths of SJ feet and 4| feet respectively, then, by taking for unit gth of a foot, the quantities are expressible by 26 and 35 respectively ; and the ratio is 26 : 35. This follows also from the algebraical theorem that (3 + i)/(4 + f ) = 26/35. If A, B he two concrete quantities of the same hind, whose ratio is a : b, and C, D two other concrete quantities of the same hind (hut not necessarily of the same hind as A and B), whose ratio is c:d, then A, B, 0, D are said to he proportional when the ratio of A to B is equal to the ratio of C to D, that is, when a:b = c:d. We may speak of the ratio A : B, of the concrete magnitudes themselves, and of the proportion A : B = : D, without alluding explicitly to the abstract numbers which measure the ratios ; but all conclusions regarding these ratios will, in our present manner of treating them, be interpretations of algebraical results such as we have been developing in the earlier part of this chapter, obtained by operating with a, h, c, d. The theory of the ratio and proportion of concrete quantity is thus brought under the theory of the ratio and proportion of abstract quantities. There are, however, several points which require a nearer examination. § 13.] In the first place, it must be noticed that in a concrete 270 SPECIAL POINTS IN CONCKETE PROPORTION CHAP. ratio the antecedent and the consequent must be quantities of the same kind ; and in a concrete proportion the two first terms must be alike in kind, and the two last alike in kind. Thus, from the present point of view at least, there is no sense in speaking of the ratio of an area to a line, or of a ton of coals to a sum of money. Accordingly, some of the propositions proved above — those regarding the permutations of a proportion, for instance — could not be immediately cited as true regarding a proportion among four concrete magnitudes, unless all the four were of the same kind. This, however, is a mere matter of the interpretation of algebraical formulae — a matter, in short, regarding the putting of a problem into, and the removing of it from, -the algebraical machine. § 14.] A more important question arises from the considera- tion that, if we take two concrete magnitudes of the same kind at random, there is no reason to expect that there exists any unit in terms of which each is exactly expressible by means of com- mensurable numbers. Let us consider, for example, the historically famous case of the side AB and diagonal AC of a square ABCD. On the diagonal AC lay off AF = AB, and draw FE perpendicular to AC. It may be readily shown that BE = EF = FC. Hence CF = AC - AB (I), CE = CB - CF (2). Now, if AB and AC were each commensurably expressible in terms of any finite unit, each would, by the remark in § 12, be an integral multiple of a certain finite unit. But from (1) it follows that if this were so, CF would be an integral multiple of the same unit ; and, again, from (2), that CE would be an integral multiple of the same unit. Now CF and CE are the side and diagonal of a square, CFEG, whose side is less than half the side XIII COMMENSURABF.ES AND INCOMMENSUKABLES 271 of ABOD; and from CFEG could in turn be derived a still smaller square whose side and diagonal would be integral mul- tiples of our supposed unit ; and so on, until we had a square as small as we please, whose side and diagonal are integral multiples of a finite unit; which is absurd. Hence the side and diagonal of a square are not magnitudes such as A and B are supposed to be in our definition of concrete ratio. § 15.] The difiiculty which thus arises in the theory of con- crete ratio is surmounted as follows : — We assume, as axiomatic regarding concrete ratio, that if A' and A" be two quantities respectively less and greater than A, then the ratio A : B is greater than A' : B and less than A" : B ; and we show that A' and A" can be found such that, while each is commensurable with B, they differ from each other, and therefore each differs from A by as little as we please. Suppose, in fact, that we take for our unit the nth part of B, then there will be two consecutive integral multiples of B/m, say mB/m and (m + l)B/m, between which A will lie. Take these for our values of A' and A" ; then A" - A' = (m + l)B/m - mB/?!, = B/n. Hence A" - A' can, by sufficiently increasing n, be made as small as we please. We thus obtain, in accordance with the definition of § 12, two ratios, m/n and (to + l)/n, between which the ratio A : B lies, each of which may be made to differ from' A : B by as little as we please. Practically speaking, then, we can find for the ratio of two incommensurables an expression which shall be as accurate as we please. Regarding this matter, see vol. ii., chap, xxv., §§ 26-41. Example. If B be the side and A the diagonal of a square, to find a rational value of A : B which shall be eorreot to 1/lOOOth. If we take for unit the 1/lOOOth part of B, then B = 1000, and A''= 2,000,000. Now 1414-2=1999396, and 1415^=2002225. Hence 1414/1000 Hence we have A/B = l-414, the error being <1/1000. ■ili EUCLIDIAN THEORY OF PROPORTION niAP. § 16.] The theory of proportion given in Euclid's Elements gets over the difficulty of inconiniensurables in a very ingenious although indirect manner. No working definition of a ratio is attempted, but the proportionality of four magnitudes is defined substantially as follows : — If there be four magnitudes A, B, C, D, such that, always, mA>, =, or oiB, according as mC > , = , or < mD, m and n being any integral numbers whatsoever, then A, B, C, D are said to be proportional. Here no use is made of the notion of a unit, so that the difficulty of in- commensurability is not raised. On the other hand, there is substituted a somewhat indirect and complicated method for testing the subsistence or non- subsistence of proportionality. It is easy to see that, if A, B, C, D be proportional according to the algebraical definition, they have the property of Euclid's definition. For, if a : b and c ; d be the numerical measures of the ratios A : B and C : D, we have a _c i~d' , ma mc lience — r=~ji no iia from which it follows that ma^-, =, or , =, or oii, according as mo, =, or ^nd, m and 71 being any integers whatever, then we must have a_c b~cf For, if these fractions (which we may suppose to be commensurable by virtue of § 15) differ by ever so little, it will be possible to find another fraction, n/m say, where ?!. and m are integers, which lies between them. Hence, if ajb be the less of the two, we must have -r<— , that is, mcKiib ; b m -;> — , that is, niond. a m In other words we have found two integers, m and ii, such that we have at once ma<7ib and •moiid. But, by hypothesis, when ina y = a{x + h) (6). * The use of the word " Variation " in the present connection is unfortunate, tecause the qualifying particle "as" is all that indicates that we are here concerned not with variation in general, as explained in § 17, but merely with the simplest of all the possible kinds of it. There is a tendency in uneducated minds to suppose that this simplest of all kinds of functionality is the only one ; and this tendency is encouraged by the retention of the above piece of antiquated nomenclature. 276 OTHEE SIMPLE CASES CHAP. The corresponding forms of equation (1) would then be y:y' = ^: x'" (a), y:y' = llx:llx' (A y:y'=\l.y':\l.rr- {y'), y:y' = z + b:x' + b (S'). y is then said to vary as, or be proportioned to, x\ l/x, l/.r', x + b. In cases (/S) and (y) y is sometimes said to vary inversely as X, and inversely as the square of x respectivelj^. Still more generally, instead of supposing the dependent variable to depend on one independent variable, we may suppose the dependent variable u to depend on two or more independent variables, -x, y, z, &c. For example, we may have, corresponding to (2), and, corresponding to (1), M = axy (^). u = a:ry~ (0, u = a(x + y) iv), u = axjy (0); u:u' = xy: x'y' (^'), u:u' = xyz : x'y'z' (a u:u' = x + y:x' + y' (VX u:u' = xjy : x' jy' (n In case (e) u is sometimes said to vary as x and y jointly ; in case (6) directly as x and inversely as y. § 20.] The whole matter we are now discussing is to a large extent an affair of nomenclature and notation, and a little attention to these points is all that the student will require to prove the following propositions. We give the demonstrations in one or two speciinen cases. (1 .) If zo^y and y ^x, then Z'xx. Proof. — By data z = ay, y = bx, where a and b are constants ; therefore z = abx. Hence Z'^x, since ab is constant. (2.) If y^'^x^ and y^^x^, then y^y^.'^x^x.,. Proof. — By data y^ = a^^, y^^a^^, where a, and a^ are con- PEOPOSITIONS EEGAEDING VAEIATION 277 stants. Hence y.;y^ = a-fl^^x^, which proves the proposition, since ftiffia is constant. In general if y, = Xi, y^^Xi, . . ., «/„»=«:„, then y^y^-'-Vn °= XA . . . a„. And, in particular, if y^^ a-, then «/" = «". (3.) If y=z/x. (5.) If z depend on x and y, and on these alone, and if za=x when y is constant, and z<^y when x is constant, then z°^xy when both X and y vary. Proof. — Consider the following system of corresponding values of the variables involved. Depeiideiit Variable. Independent Variables, X, y. x',y. x;y'. Then, since y has the same value for both z and z^, we have, by data, e X Again, since x' is the same for both z^ and z', we have, by data, «' y'' From these two equations we have z z^ X y z^ z X y z xy z'~ x'y" that is, which proves that z <= xy. A good example of this case is the dependence of the area of a triangle upon its base and altitude. 278 EXAMPLES CHAP. We have Area, oc base (altitude constant) ; Area oc altitude (base coustant). Hence area oc base x altitude, when both vary. (6.) In a similar manner we may prove that if z depend on Xy , x,, . , ;>•„, and on these alone, and vary as any one of these when the rest remain constant, then z = x^x^ ■ ■ -x^ when all vary. (7.) If z°=x (y constant) and 2=1/?/ (x constant), then z^^x/y lohen both vary. For example, if V, P, T denote the volume, pressure, and absolute tem- perature of a given mass of a perfect gas, then V K 1/P (T constant), V oc T (P constant). Hence in general V oc T/P. Example 1. If .s oc f when/ is constant, and s oc/when t is constant, and 2s=/when t = 1, find the relation connecting s, /, t. It follows by a slight extension of § 20 (5) that, when/ and t both vary, s xfp. Hence s=affi, where a is a constant, which we have to determine. Now, when t = \, s = iif, hence 4/=qA^ that is, ^f=af ; in other words, we must have a= ^. The relation required is, therefore, s = iifP. Example 2. The thickness of a grindstone is unaltered in the using, but its radius gradually diminishes. By how much must its radius diminish before the half of its mass is worn away ? Given that the mass varies directly as the square of the radius when the thickness remains imaltered. Let m denote the mass, r the radius, then by data, m=ar'', where a is constant. Let now r become ?'', and, in consequence, m become ^m, then iin = ar'-, hence «r'- _ ^m ar- m ' that is, ~s = i ; r- '■' 1 whence —=-7-. /■ V- It follows, therefore, that the radius of the stono must be diminished in the ratio 1 : yji. Example 3. A and B are partners in a business in which their interests are in the ratio a : b. They admit G to the partnership, without altering the whole amount of capital, in such a way that the interests of the three partners in the business are then equal. C contributes £c to the capital of the firm. XIII ^ EXERCISES XVIII 279 How is the sum £c which is withdrawn from the capital to he divided between A and B ? and what capital had each in the business originally ? Solution. — Since what C pays in is his share of the capital, they each have iinally £c in the business ; let now £x be A's share of C's payment, so that £(c - x) is B's share of the same. In efiect, A takes £x and B £(c - x) out of the business. Hence they had originally £{c + x) and £(c + c - k) in the business. By data, then, we must have c + x a Ic-x V hence h{c + x) = a{2c-x) ; we have, therefore, ic + l)x = 2f(c - ax. From this last equation we derive, by adding ax - be to both sides, {a + h)x={2a-i)c. Hence, dividing by a + 6, we have _{2a-b)c a + b {2a,~b)c (1). Hence c-x=c-- a + o ' (2). a + b _{'ib-a)c a + b It appears, then, that A and B take £(2a - 6)c/(a + b) and £(26 - ffi)c/(a + b) respectively out of the business. C's payment must be divided between them in the ratio of these sums, that is, in the ratio 2a-b:2b-a. They had in the business originally £3ac/{a + b) and £Zbcj(a + b) respectively. Exercises XVIIl. (1.) \iyxx, and if 2/=3| when x=6\, find the value of j/ when a;=|. (2.) 2/ varies inversely as cc- ; and s varies directly as a;-. 'Whenc(;=2, 2/ + z = 340 ; when x=l, y~z = 1275. For what value of a; is 2/ = «? (3.) zxu — v; uxx; vxx^. When x=2, ^=48; when x=5, s=30. For what values ofa;is^ = 0? (4.) If xyoix^ + y'^, and x=3 when y = i, find the equation connecting y and x. (5.) It x + y xse-y, then x^ + y^ xxy a.nd afl + y^{x, y, z, . . .). § 2.] Consider any two functions whatever, say c/)(;», y, z, a, b, c, . . .), and ip{x, y, z, . . . a, b, c, . . .), of the variables X, y, z, ., involving the constants a, b, c, . . . If the equation {x, y,z,. . .) = 0, 284 WEANING OF SOLUTION chap. ■where <^ is a rational integral function. Such equations are therefore of great analytical importance ; and it is to them that the " Theory of Equations," as ordinarily developed, mainly applies. An integral equation of this kind is described by assigning its degree and the number of its variables. The degree of the equation is simply the degree of the function <^. Thus, :)'.' + ^xij + y^ - 2 = is said to be an equation of the 2nd degree in two variables. § 4.] Equations of condition may occur in sets of one or of more than one. In the latter case we speak of the set as a set or system of simultaneous equations. The main problem ivhich arises in connection with every system of equations of condition is to find a set or sets of values of the variables which shall render every equation of the system an identity literal or numerical. Such a set of values of the variables is said to satisfy the system, and is called a solution of the system of equations. If there be only one equation, and only one variable, a value of that variable which satisfies the equation is called a root. We also say that a solution of a system of equations satisfies the system, meaning that it renders each equation of the system an identity. It is important to distinguish between two very different kinds of solution. When the values of the variables which con- stitute the solution are closed expressions, that is, functions of known form of the constants in the given equations, we have what may be called a formal solution of the system of equations. In particular, if these values be ordinary algebraical functions of the constants, we have an algehmcal solution. Such solutions cannot in general be found. In the case of integral algebraical equations of one variable, for example, if the degree exceed the 4th, it has been shown by Abel and others that algebraical solutions do not exist except in special cases, so that the formal solution, if it could be found, would involve transcendental functions. "\A^ien the values of the variables which constitute the solution are given approximately as numbers, real or complex, the solution is said to be an approximate numerical solution. In this case the XIV EXAMPLES OF SOLUTION 285 words " render the equation a numerical identity " are understood to mean "reduce the two sides of the equation to values which shall differ by less than some quantity which is assigned." For example, if real values of the two sides, say P and P', are in question, then these must be made to differ by less than some given small quantity, say 1/100,000; if complex values are in question, say P + Qi and P' + Q'«, then these must be so reduced that the modulus of their difference, namely, \/{(P - P')^ + (Q - QT}> shall be less than some given small quantity, say 1/100,000. (Cf. chap, xii., § 21.) As a matter of fact, numerical solutions can often be obtained where formal solutions are out of the question. Integral alge- braical equations, for example, can always be solved numerically to any desired approximation, no matter what their degree. Example 1. 2a; + 2 = 2. a:=0 is a solution, for this value of x reduces the equation to 2x0 + 2=2, which is a numerical identity. Strictly speaking, this is a case of algebraical solution. Example 2. x=b''ja reduces the equation to a~-b^=0, a which is a literal identity ; hence x=i^/a is an algebraical solution. Example 3. k2-2 = 0. Here x= + \/2 and x= - \/2 each reduce the equation to the identity 2-2=0; these therefore are two algebraical solutions. On the other hand, a;=+l'4142 and x= - 1'4142 are approximate numerical solutions, for each of them reduces a;^ - 2 to - ■00003836; which differs from by less than -00004. Example 4. {x-lf + 2 = 0. x=l + i\J2i and x=\-\J1% are algebraical solutions, as the student will easily verify. a;=l '0001 + 1 '41421 and a; = l'0001 -1 '41421 are approximate numerical solutions, for they reduce (a; - 1 )= + 2 to '00003837 + '00028284^ and '00003837 - -000282841 respectively, complex numbers whose moduli are each less than '0003. 286 CONDITIONAL EQUATION A HYPOTHETICAL IDENTITY CIIAP. Example 5. x-y=l. Here x=\, y = 0, is a, solution ; so is a; = l'5, y= '5 ; so is x=2, y = l ; and, in fact, so is x=a + l, y=a, where a is any quantity whatsoever. Here, then, there are an infinite number of solutions. Example 6. Consider the following system of two equations : — x-y = l, 2x + y=5. Here x=2, y=l is a solution ; and, as we shall show in chap, xvi., there is no other. The definition of the solution of a conditional equation suggests two remarks of some importance. 1st. Eoery conditional equation is a hypothetical identity. In all operations with the equation we suppose the variables to have such values as will render it an identity. 2nd. The ultimate test of every solution is that the values which it assigns to the variables shall satisfy the equations when substituted therein. No matter how elaborate or ingenious the process by which the solution has been obtained, if it do not stand this test, it is no solution ; and, on the other hand, no matter how simply obtained, provided it do stand this test, it is a solution.* In fact, as good a way of solving equations as any other is to guess a solution and test its accuracy by substitution.! § 5.] The consideration of particular cases, such as Examples 1-6 of § 4, teaches us that the number of solutions of a system of one or more equations may be finite or infinite. If the number be finite, we say that the solution is determinate (singly determin- ate, or multiply determinate according as there are one or more solutions) ; if there be a continuous infinity of solutions, we say that the solution is indeterminate. The question thus arises. Under what circumstances is the solution of a system of equations determinate ■? Part at least of the answer is given by the following fundamental propositions. Proposition I. The solution of a system of equations is in general determinate (singly, or midtiply according to circumstances) when the number of the equations is equal to the number of the variables. * A little attention to these self-evident truths would save the beginner from many a needless blunder. + This is called solving by "inspection." xiy PROPOSITIONS AS TO DETERMINATENESS OF SOLUTION 287 Eightly considered, this is an ultimate logical principle which may be discussed, but not in any strictly general sense proved. Let us illustrate by a concrete example. The reader is aware that a rectilinear triangle is determinable in a variety of ways by means of three elements, and that consequently three condi- tions will in general determine the figure. To translate this into analytical language, let us take for the three determining elements the three sides, whose lengths, at present unknown, ve denote by X, y, z respectively. Any three conditions upon the triangle may be translated into three equations connecting x, y, z with certain given or constant quantities ; and these three equations will iu general be sufficient to determine the three variables, X, y, z. The general principle " common to this and like cases is simply Proposition I. The truth is that this proposition stands less in need of proof than of limitation. What is wanted is an indication of the circumstances under which it is liable to excep- tion. To return to our particular case : What would happen, for example, if one of the conditions imposed upon our triangle were that the sum of two of the sides should fall short of the third by a given positive quautit}' ? This condition could be expressed quite well by an equation (namely, x + y = z - q, say), but it is fulfilled by no real triangle, f Again, it might chance that the last of the three given conditions was merely a con- sequence of the two first. We should then have in reality only two conditions — that is to say, analytically speaking, it might chance that the last of the three equations was merely one derivable from the two first, and then there would be an infinite number of solutions of the system of three variables. Such a system is x + y + z= 6, 3x+2y + z=l0, 2x + y = 4, for example, for, as the reader may easily verify, it is satisfied hy X = a - 2, y = S - 2a, z = a, where a is any quantity whatsoever. * A name seems to be rec[iiired for tliis all-pervading logical principle : the Law of Determinate Manifoldness might be suggested, t See below, chap. xix. 288 PROPOSITIONS AS TO DETERMINATENESS OF SOLUTION chap. It will be seen in following chapters how these difficulties are met in particular cases. Meantime, let us observe that, if we admit Proposition I., two others follow very readily. Proposition II. If the number of equations be less than the number of variables, tlie solution is in general indeterminate. Proposition III. If the number of independent equations be grea/i'r than the number of variables, there is in general no solution, and the system of equations is said to be inconsistent. For, let the number of variables be n, and the number of equations m, say, where m n. If we take the first n equations, these will in general, by Proposition I., give a determinate set, or a finite number of determinate sets of values for all the n vaiiables. If we now take one of these sets of values, and substitute it in one of the remaining m — n equations, that equation will not in general be satisfied ; for, if we take an equation at random, and a solution at random, the latter will not in general fit the former. The system of m equations will therefore in general be inconsistent. It may, of course, happen, in exceptional cases, that this proposition does not hold ; witness the following system of three equations in two variables : — x-y=l, 2x + y = 5, 3x+2y=8, which has the common solution x = 2, y = I. § 6.] We have also the further question, When the system is determinate, how many solutions are there ? The answer to this, in the case of integral equations, is furnished by the two following propositions : — XIV MULTIPLICITY OF DETEEMINATE SOLUTIONS 289 Proposition I. An integral equation of the nth degree in one variable has n roots and no more, ^ohich may be real or complex, and all unequal or not all unequal, according to circumstances. Proposition II. A determinate system of m integral equations with m ■ variables, whose degrees in these variables are p, q, r, . . . respect- ively, has, at most, pqr . . . solutions, and has, in general, just that number. Proposition I. was proved in the chapter on complex num- bers, where it was shown that for any given integral function of X of the nth degree there are just n values of x and no more that reduce that function to zero, these values being real or complex, and all unequal or not as the case may be. Proposition II. will not be proved in this work, except in particular cases which occur in chapters to follow. General proofs will be found in special treatises on the theory of equa- tions. We set it down here because it is a useful guide to the learner in teaching him how many solutions he is to expect. It will also enable him, occasionally, to detect when a system is indeterminate, for, if a number of solutions be found exceed- ing that indicated by Proposition II., then the system is certainly indeterminate, that is to say, has an infinite number of solu- tions. Example. The system x^ + ip=l, x-y=l has, by Proposition II., 2x 1 = 2 solutions. As a matter of fact, these solutions area; = 0, j/=,-l, anda;=l, 2/=0. EQUIVALENCE OF SYSTEMS OF EQUATIONS. § 7.] Two systems of equations, A and B {each of which may con- sist of one or more equations), are said to be equivalent when every solution of A is a solution of B, and every solution of H a solution of A. From any given system. A, of equations, we may in an in- finity of ways deduce another system, B ; but it will not necessarily be the case that the two systems are equivalent. In other words, we may find in an infinity of ways a system, B, of equations which will be satisfied by all values of the VOL. I U 290 DEFINITION OF EQUIVALENCE CHAP. variables for whicli A is satisfied ; but it will not follow con- versely that A will be satisfied for all values for which B is satisfied. To take a very simple example, a; - 1 = is satisfied by the value x=l, and by no other; a;(a;-l) = is satisfied by x=l, that is to say, x(x - 1) = is satisfied when .i; - 1 = is satisfied. On the other hand, a-{x - 1) = is satisfied either by a; = or by .'T = 1, therefore a; - 1 = is not always satisfied when x{x - 1) = is so ; for a; = reduces a; - 1 to - 1, and not to 0. Briefly, x(x - 1) = may be derived from a; - 1 = 0, but is not equivalent to a; - 1 = 0. a:(.(;-l) = is, in fact, more than equivalent to a; - 1 = 0, for it involves a; - 1 = and a; = as alternatives. It will be convenient in such cases to say that x{x - 1) = is equivalent to a; = 0] -1 = 0J When by any step we derive from, one system another which is exactly equivalent, we may call that step a reversible deriva- tion, because we can make it backwards without fallacy. If the derived system is not equivalent, we may call the step irreversible, meaning thereby that the backward step requires examination. There are few parts of algebra more important than the logic of the derivation of equations, and few, unhapisily, that are treated in mure slovenly fashion in elementary teaching. No mere blind adherence to set rules will avail in this matter ; while a little attention to a few simple principles will readily remove all difficulty. It must be borne in mind that in operating with conditional equations we always suppose the variables to have such values as will render the equations identities, although we may not at the moment actually substitute such values, or even know them. We are therefore at every step, hypothetically at least, applying the fundamental laws of algebraical transformation just as in chap. i. The following general principle, already laid down for real quantities, and carefully discussed in chap, xii., § 12, for com- plex quantities, may be taken as the root of the whole matter. XIV ADDITION AND TRANSPOSITION OF TEEMS 291 Let P and Q he two functions of the variables x, y, z, . . ., vMch do not become infinite * for any values of those variables that we have to consider. IfPx Q = and Q 4= 0, then will P = 0, and i/P x Q = and P + 0, then will Q = 0. Otherwise, the only values of the variables which make P x Q = are such as make either P = 0, w Q = 0, or both P = and Q = 0. § 8.] It follows by the fundamental laws of algebra that if P = Q (1), then P ± E = Q ± E (2), where E is either constant or any function of the variables. "We shall show that this derivation is reversible. For, if P ± E = Q ± E, then P±E=fE = Q±EtE, that is, P = Q ; in other words, if (2) holds so does (1). Cor. 1. If we transfer any term in an equation from the one side to the other, at the same time reversing its sign of addition or subtrac- tion, or if we reverse all the signs on both sides of an equation, we deduce in each case an equivalent equation. For, if P + Q = E + S, say, then P + Q-S = E + S-S, that is, P + Q - S = E. Again, if P + Q = E + S, then P + Q-P-Q-E-S = E + S-P-Q-R-S, that is, - E - S = - P - Q, or -P-Q=-E-S. Cor. 2. Every equation can be reduced to an equivalent equation of the form — E = 0. For, if the equation be . P = Q, * 111 all that follows all functions of the variables that appear are supposed not to become infinite for any values of the variables contemplated. Cases where this understanding is violated must be considered separately. 292 MULTIPLICATION BY A FACTOR chap. we have P - Q = Q - Q, that is P - Q = 0, which is of the form R = 0. Example. Subtracting x~ -x-S from both sides, we have the equivalent equation -3x' + 3? + ix + 3 = Q. Changing all the signs, we have 3x3-0:^-40:- 3 = 0. In this way an integral equation can always be arranged with all its terms on one side, so that the coeliicient of the highest term is positive. § 9.] It follows from the fundamental laws of algebra that 'f P = Q (1), tJien PE = QE (2), the step being reversible if E is a constant differing from 0, bid not if E be a function of the variables.* For, if PE = QE, an equivalent equation is, by § 8, PE - QE = (3), that is, (P - Q)E = (4). Now, if E be a constant 4= 0, it will follow from (4), by the general principle of § 7, that P - Q = 0, which is equivalent to P = Q. But, if E be a function of the variables, (4) may also be satisfied by values of the variables that satisfy E = (5); and such values will not in general satisfy (1). In fact, (2) is equivalent, not to (1), but to (1) and (5) as alternatives. * This is spoken of as " multiplying the equation by R. " Similarly the process of § 8 is spoken of as " adding or subtracting E to or from the equa- tion. " This language is not strictly correct, but is so convenient tli-at we shall use it where no confusion is to be feared. XIV DEDUCTION OF INTEGKAL FEOM RATIONAL EQUATION 293 Cor. 1. From the above it follows that dividing loth sides of an equation ly any function other than a constant not equal to zero is not a legitimate process of derivation, since we may thereby lose solutions. fP-Q = 0\. Thus PE = QR is equivalent to whereas PE/E = QE/E* gives P = Q, which is equivalent merely to P-Q = 0. Example. If we divide both sides of the equation (a;-l)a;2 = 4(x-l) (a) by x— 1, we reduce it to x'^=i {P), which is equivalent to {x-2)[x + 2) = 0. (a), on the other hand, is equivalent to {x-l){x-2){x + 2) = 0. Hence (a) has the three solutions x=l, x=2, x= - 2 ; while ((3) has only the two x=2, x= -2. Cor. 2. To multiply or divide both sides of an equation by any constant quantity differing from zero is a reversible process of derivation. Hence, if the coefficients of an integral equation be fractional either in the algebraical or in the arithmetical sense, we can always find an equivalent equation in which the coefficients are all integral, and have no common measure. Also, we can always so arrange an integral equation that the co- efficient of any term we please, say the highest, shall be + 1. Example 1. Sx + 2 ex + 3_2x + i 4 5 ~ 8 gives, on multiplying both sides by 40, 10{3x + 2) + 8(6a; + 3) = 5(2x + 4), that is, 30x + 20 + 48a; + 24 = 10a; + 20, whence, after subtracting 10x4-20 from both sides, 68x + 24 = ; * As we are here merely establishing a negative proposition, the reader may, to fix his ideas, assume that all the letters stand for integral functions of a single variable. 294 DEDUCTION OF INTEGRAL FROM RATIONAL EQUATION CHAl-. whence again, after division of both sides by 68, Example 2. If we multiply both sides by pq{p -q){p + j), that is, by pq{p' - j"), we derive the equivalent equation {{p'-S-)x+Pqy} {{p''-q^)x+pqy]=2pq{p"-q'')xy, that is, {p^ - q')V + 22Jq{p' - q')xy +p^q''y' = 2pq{p'' - q'')xj/, which is equivalent to {p--q-)-x''+pf'q''y"=0. Cor. 3. F7-om every rational algebraical equation an integral equa- tion can he deduced ; hut it is possible that extraneous solutions may be introduced in the process. Suppose we have P = Q (a), where P and Q are rational, but not integral. Let L be the L.C.M. of the denominators of all the fractions that occur either in P or in Q, then LP and LQ are both integral. Hence, if we multiply both sides of (a) by L, we deduce the integral equation LP = LQ (;8). Since, however, the multiplier L contains the variables, it is possible that some of the solutions of L = may satisfy (/?), and such solutions would in general be extraneous to (a). We say possible ; in general, however, this will not happen, because P and Q contain fractions whose denominators are factors in L. Hence the solutions of L = will in general make either P or Q infinite, and therefore (P - Q)L not necessarily zero. The point at issue will be best understood by studying the two following examples : — Example 1. If we multiply both sides by (x - 2) (a; - 3), we deduce the equation (2a;-3)(a:-2)(a;-3) + (3r-6a: + 8)(a;-3) = (a:-2)2 (;3), which is integral, and is satisfied by any solution of (a). We must, however, examine whether any of the solutions of (a5-2)(a;-3) = satisfy (/3). These solutions are ic=2 and x=:3. The second of these obviously does not satisfy (/3), and need not be considered; but x=2 does .satisfy (/3), and we must examine (a) to see whether it satisiies that equation also. XIV EAISING BOTH SIDES TO SAME POWER 295 Now, since x' -6x + 8 = {x-2){x- i), (a) may be written in tlie equivalent form 23;-3 + a;-4=^, x-o which is obviously not satisfied by a; = 2. It appears, therefore, that in the process of integralisation we have intro- duced the extraneous solution x=2. Example 2. 2a; -3 + — = r (a'. a;-2 a;-3 ' ' Proceeding as before, we deduce (2a;-3)(x-2)(!s-3) + (2a;--6a3 + 8)(a;-3) = (a!-2)= (^')- It will be found that neither of the values a;=2, a;=3 satisfies (^'). Hence no extraneous solutions have been introduced in this case. N'.B. — The reason why x=2 satisfies (p) in Example 1 is that the numer- ator ar - 6a; + 8 of the fraction on the left contains the factor a; - 2 which occurs in the denominator. Cor. 4. Baising both sides of an equation to the same integral power is a legitimate, but not a reversible, process of derivation. The equation P = Q (1) is equivalent to P - Q = (2). If we multiply by P^-i + P^-^Q + P"-3Q2 + . , . . + Q"-\ ^ deduce from (2) P» - Q» = (3), which is satisfied by any solution of (1); (3), however, is not equivalent to (1), but to r p=Qi It will be observed that, if we start with an equation in the standard form P - Q = 0, transfer the part Q to the right-hand side, and then raise both sides to the wth power, the result is the same as if we had multiplied both sides of the equation in its original form by a certain factor. To make the introduction of extraneous factors more evident we chose the latter process ; but in practice the former may happen to be the more convenient.* If the reader will reflect on the nature of the process described in chap. x. for rationalising an algebraical function by means of a rationalising factor, he will see that by repeated operations of this kind every algebraical equation can be reduced to a rational * See below, § 12, Example 3. 296 EVERY ALGEBKAICAL EQUATION CAN BE INTEGKALISED UHAi'. form ; but at each step extraneous solutions may be introduced. Hence Cor. 5. From every algebraical equation we can derive a rational integral equation, which will he satisfied ly any solution of the given equation , but it does not follow tliat every solution^ w even that any solution, of the derived equation will satisfy the original one. Example 1. Consider the equation '.J{x + \} + >^{x-l) = l (a), where the radicands are supposed to be real and the square root to have the positive sign. * (a) is equivalent to \/{x + l) = l - \J{x- 1), whence we derive, by squaring, x + l = l+x-l-2\/{x~l), which is equivalent to 1 = - 1\/(x - 1 ). From this last again, by squaring, we derive l = i{x-\), which is equivalent to the integral equation 4:r-5 = (;8), the only solution of which, as we shall see Jicreaftcr, is x=\. It happens here that a;=:f is not a solution of (a), for ^y(| + 1) + ^y(J - ] ) = 5 + 4=2. E.xamplc 2. V(*- + l)-\/(a:-l) = l (a). Proceeding exactly as Ijcfore wo have x + l = l+x-l + 2's/{x-l), \=+2^J{x-\), l = 4(a;-l), 4:e-5 = (/3'), Here (/3') gi\'cs x = l, which happens this time to be a solution of the original equation. We conclude this discussion by giving two propositions applicable to systems of equations containing more than one equation. These by no means exhaust the subject ; but, as our object here is merely to awaken the intelligence of the student, the rest may be left to himself in the meantime. § 10.] From the system P, = 0, P, = 0, . . ., P„ = (A) we derive L,P,+LP, + . . +L„P„ = 0, P, = 0, . . ., P„ = (B), and the tim will he equivalent if Lj he a constant differing from 0. * Wheu \/.c is imaginary, its "principal value" (see chap, xxix.) ought to be taken, unless it is otherwise indicated. XIV EXAMPLES OF DEKIVATION 297 Any solution of the system (A) reduces P,, Pg, . . ., P„ all to 0, and therefore reduces LjPi + L^Ps + . . . + L,iP,i to 0, and hence satisfies (B). Again, any solution of (B) reduces P2, P3, . . ., P,i all to 0, and therefore reduces LjPi + L^Pg + . . . + L,jP„ = to LjPj = 0, that is to say, if Lj be a constant =)= 0, to Pi = 0. Hence, in this case, any solution of (B) satisfies (A). If Li contain the variables, then (B) is equivalent, not to (A) simply, but to j'Pi = 0, P, = 0, . . ., P„=o-j I.L, = 0, P, = 0, . . , P„=0) As a particular case of the above, we have that the two systems P = Q, E = S; and P + E = Q + S, E = S are equivalent. For these may be written P-Q = 0, E-S = 0; P-Q + E^S = 0, E-S = 0. If I, V, m, m' be constants, any one of which may le zero, hut which are such that Im' - I'm 4= 0, tlien the two systems U = 0, U' = 0, and W + VW = 0, mU + wi'U' = are equivalent. The proof is left to the reader. A special case is used and demonstrated in chap, xvi., § 4. § 11.] Any solution of the system P = Q, R = S (A) is a solution of the system PE = QS, E = S (B); but the two systems are not equivalent. From P = Q, we derive PE = QE, which, since R = S, is equivalent to PE = QS. It follows therefore that any solution of (A) satisfies (B). 298 EXAMPLES OF DEEIVATION CHAP. Starting now with (B), we have PR = QS (1), E = S (2). Since R = S, (1) becomes PR = QR, which is equivalent to (P-Q)R=0, that is, equivalent to fP-Q = o\ R = 0J" Hence the system (B) is equivalent to 'P = Q, R = S\ R = 0, R = SJ' that is to say, to fP = Q, R = S-| ^E=0, S = 0J' In other words, (B) involves, besides (A), the alternative system, R = 0, S = 0. Example. From o:~2 = l — y, x=l+y, a system which has the single solution x=1, j/ = l, we derive the system x{x - 2) = 1 - J/-, x = l + y, which, in addition to the solution x=2, j/ = 1, has also the solution a: = 0,y= -1 belonging to the system a;=0, l + 2/ = 0. § 12.] In the process of solving systems of equations, one of the most commonly-occurring requirements is to deduce from two or more of the equations another that shall not contain certain assigned variables. This is called " eliminating the variables in question between the equations used for the purpose." In per- forming the elimination we may, of course, use any legitimate process of derivation, but strict attention must always be paid to the question of equivalence. Example. Given the system ».' + '/=l (1), x + y=l (2), it is required to eliminate y, that is, to deduce from (1) and (2) an equation involving x alone. XIV EXAMPLE OF ELIMINATION 299 (2) is equivalent to y=l-x. Hence (1) is equivalent to a;2 + (l-a;)2=l, that is to say, to 2a!2-2a;=0, or, if we please, to x''-x=0 ; and thus we have eliminated y, and obtained an equation in x alone. The method we have employed (simply substitution) is, of course, only one among many that might have been selected. Observe that, as a result of our reasoning, we have that the system (1) and (2) is equivalent to the system x'-x=0 (3), x + y=l (4), from which the reader will have no difficulty in deducing the solution of the given system. § 13.] Although, as we have said, the solution of a system of equations is the main problem, yet the reader will learn, especially when he comes to apply algebra to geometry, that much information — very often indeed all the information that is required — may be derived from a system without solving it, but merely by throwing it into various equivalent forms. The derivation of equivalent systems, elimination, and other general operations with equations of condition have therefore an im- portance quite apart from their bearing on ultimate solution. We have appended to this chapter a large number of exercises in this branch of algebra, keeping exercises on actual solution for later chapters, which deal more particularly with that part of the subject. The student should work a sufficient number of the following sets to impress upon his memory the general principles of the foregoing chapter, and reserve such as he finds difficult for occasional future practice. The following are worked out as specimens of various artifices for saving labour in calculations of the present kind : — Example 1. Reduce the following equation to an integral form : — ax' + bx + c ax + b , . — 5 = (a-)- puf + C[X + r px + q We may write (a) in the form x{ax + i) + c ^ax + i . , x(px+'q) + r~px + q 300 EXAMPLES OF INTEGEALISATION CHA) Multiplying (;8) by {jyx + q) { x(px + q) + r}, we obtain x[ax + b)(px + q) + c{px + q)=x{ax + b) [px + q) + r(ax+h) (7). Now, (y) is equivalent to c{px + q)=:r{ax + b) (5), which again is equivalent to [cp-ra)x + {cq-rh) = f) (e). The only possibly irreversible step here is that from [§) to (7). Observe the use of the brackets in (/3) and (7) to save useless detail. Example 2.. Integralise (a -x){x+ m) _{a + x){x- m) (a). x+n x-n Since x + m={x + n) + {m-n), x-m={x-ii)-{m-n), (a) may be written in the equivalent form, ia-x)(l+'-^)={a + x)(l-'^ m, ^ '\ x+nj \ x-n/ whence the equivalent form / s , ^ / ^/ci-x a + x\ ^ {a-x)-{a + x) + {m-n){ ——+ =0, \x+n x-n/ that is, _2^+?(!iz^M!L±f^='=o (7). x'-n^ Multiplying by - J(a;^ - n-), we deduce from (7) the integral equation x{x- - n- - {in - n) {n + a)) =0 (5). In this case the only extraneous solutions that could be introduced are those of x^ -n- = 0. Note the preliminary transformation in (/3) ; and observe that the order in which the operations of collecting and distributing and of using any legitimate processes of derivation that may be necessary is quite unrestricted, and should be determined by considerations of analytical simplicity. Note also that, although we can remove the numerical factor 2 in (7), it is not legitimate to remove the factor x ; a:=0 is, in fact, as the student will see by inspection, one of the solutions of (a). Example 3. X, y, Z, U denoting rational functions, it is required to rationalise the equation v'X±VY±v'Z±VU = (a). "We shall take + signs throughout ; but the reader will see, on looking through the work, that the final result would bo the same whatever arrange- ment of signs be taken. From (o), ^/X+^/Y=-^/Z-^/U, whence, by squaring, X + Y + 2V(XY) = Z + U + 2V(ZU) (^). XIT EXAMPLES OF RATIONALISATION ' 301 From (P), X + Y-Z-U=-2V(XY) + 2V(ZU), whence, by squaring, (X + Y-Z-U)2=4XY + 4ZU-8V(XYZU) (7). We get from (7), X2 + Y2 + Z2 + U=-2XY-2XZ-2XU-2YZ-2YU-2ZU= -8V{XYZU), whence, by squaring, { XH Y2 + Z= + U^ - 2XY - 2XZ - 2XU - 2YZ - 2yXJ - 2ZU } = = 64XYZU (S). Since X, Y, Z, U are, by hypothesis, all rational, (5) is the required result. As a particular instance, consider the equation V(2a; + 3) + V(3« + 2) - VlSa; + 5) - V(3«) = (a'). HereX = 2a; + 3, Y=3a; + 2, Z = 2a; + 5, l! = Sx; and the student will find, from (S) above, as the rationalised equation, (48a!2 + 112a; + 24)2=64(2K + 3)(3a; + 2)(2a: + 5)3j; (S'). After some reduction (5') will be found to be equivalent to (K- 3)2 = (6-). It may be verified that x=S is a common solution of (a') and (e'). Although, for the sake of the theoretical insight it gives, we have worked out the general formula (5), and although, as a matter of fact, it contains as particular cases very many of the elementary examples usually given, yet it is by no means advisable that the student should work particular cases by merely substituting in (3) ; for, apart from the disciplinary advantage, it often happens that direct treatment is less laborious, owing to intervening simplifications. Witness the following treatment of the particular case (a') above given. From (a'), by transposition, V(2a; + 3) + -^{Sx + 2) = ^/(2x + 5) + ^J{Sx), whence, by squaring, 5x + 5 + 2sJ{63? + lZx + 6) = 5x + 5 + 2^(63? + 15x), which reduces to the equivalent equation ^J{6a? + 13x+6)=^/{Sx' + 15x] (^')- From (B'), by squaring, 6a^ + 13a; + 6 = 6a2 + 15a!, which is equivalent to a;-3 = (6")- Thus, not only is the labour less than that involved in reducing (5), but (5") is itself somewhat simpler than (5'). Example 4. If a; + i/ + 2=0 (a), show that 2(j/2 + yz + z'f = 3n(j/2 + yz + z^) (§). 302 EXAHrPLES OF TRANSFORMATION CHAi We have y- + yz + z- = y^ + z(y + z), = {-z-xf + z{-x),hy{a), = z- + zx + x^, = a?+xy + rj'^, by symmetry. It follows then that i:(f- + yz + z^f = ?.(y^ + yz + z^f (7), and Za(f- + yz + z') = i(f- + yz + z^f (S). From (7) and (5), (fi) follows at once. E-xample 5. If x + y + z = (a), show that From (a), 2/ + ~= -» (7), whence, squaring and then transposing, we have J/' + 2' = x^ - 2yx (5). Similarly z + x= -y (7'), z" + X- = y-- Izx (5'). From the last four equations we have 2 (?/^ + y)(2- + a;'- ) ^ ^ {'3? - lyz) ()/ - 2:3;) {»/ + =) (s + a-) "^ a^!/ ' _ ^ a^2/^ - 2iC^z - ly^z + 4,»;?/t^ "'^ 5^ ' -( Now, from (a), by squaring and transposing, 2a;-= -22.JJ!/ (f). Also 2a:=(j/Hi^) = S3;y(a; + j/), = - 2a;Y2, by (a), = - xyz2.xy (ij). If we use (f) and (ij), (e) reduces to (i/ + s)(z + x) which is equivalent to {/3). The use of the principles of symmetry in conjunction with the S notation in shortening the calculations in this example cannot fail to strike the reader. Example C. If yz-T?_z:,--y^ y + z- z + x' <"'• XIV EXAMPLES OF TRANSFORMATION 303 and if X, y, z be all unequal, show that each of these expressions is equal to {xy - 2^)/(x + y), and also to x + y + z. Denote each of the sides of (a) by U. Then we have yz- y + z zx-y- = U (7). z + x Since j/ + : = and z + x=0 would render the two sides of (a) infinite, we may assume that values of x, y, z fulfilling these conditions are not in ques- tion, and multiply (/3) and (7) hy y + z and z + x respectively. AVe then deduce yz-x'-iy + zW^O (S), zx-tf-{z + x)l! = (e). From (S) and (e), by subtraction, we have z{x-y) + (x^ - y^ -(x- 2/)U = 0, that is, {x+y + z-\5){x-y) = (f). Now a;-2/ = is excluded by our data ; hence, by (f), we must have x + y + z-XJ = (i, (yi), that is, 'n = x + y+z {B). We have thus established one of the desired conclusions. To obtain the other it is sufficient to observe that (17) is symmetrical in x, y, a. For, if we start with (17) and multiply hy x-z (which, by hypothesis, + 0), we obtain y{x-z) + {x^-z^)-{x-z)l!=0 ; and, coihbining this by addition with (5), xy-z"-{x + y)l!=0; which gives (since a; + 2/ + 0) U = xy- x+y The reader should notice here the convenient artifice of introducing an auxiliary variable U. He should also study closely the logic of the process, and be sure that he sees clearly the necessity for the restrictions x-y=^=0, x + jz + O. Example 7. To eliminate x, y, z between the equations y^ + z^ = aijz {a), z^ + x^=hzx (^), a!^ + 2/^=ca!J/ (7), where a;=f=0, 2/=f=0, s + 0. In the first place, we observe that, although there are three variables, yet, since the equations are homogeneous, we are only concerned with the ratios of the three. We might, for example, divide each of the equations by a;^ ; we should then have to do merely with yjx and z\x, each of which might be regarded as a single variable. There are therefore enough equations for the purpose of the elimination. 304 EXAMPLES OF ELIMINATION CIIAP. From (a) and (^) wo deduce, by subtraction, x--y^={bx-ay)z (S). We remark that it follows from this equation that h.v - ai/ #= ; for te - a?/ = would give x'=y', and hence, by (7), a' = (at least if we suppose c=f=±2). This being so, we may multiply (/3) by (bx-ay)". We thus obtain 2-(ia; - oj/)- + x-(fa - ftJ/)' = ia;s(6a; - ay)-, whence, using (5), we have (a;^ - y-)^ + X'{bx- ay)'' = bx{ix - ay) {.r- - y-), which reduces, after transposition, to {:rr-y')^=xy{ax-by) (bx-ay), that is to say, (x'^ + y'^)-- i.v-i/- = xii{a.v-by)(b,c~a7j) (e). Using (7), we deduce from (e) (c- - i)x-y- = ,ry(ax - by) (bx - ay), whence, bearing in mind that 3'7/=|=0, we get (c- - i)xy = ab(x- + y") - [a" + b-):i'y, which is equivalent to (a'' + b- + c''-i)xy = ab(x- + y-) (f). Using (7) once more, and transposing, we reach finally (a^ + i- + c- - 4 - abc)xy = 0, whence, since xy + G, we conclude that d- + b- + c- -i- abc = (S (1;), so that (-q) is the required result of eliminating x, y, z between the equations (a), (/3), {7). Such au equation as (tj) is often called the eliminarit (or re- sultant) of the given system of equations. Example 8. Show that, if the two first of the following three equations be given, the third can be deduced, it being supposed that a; =1= 2/ =1= = + 0. o-(i/2 + 2/3 + ,-=) - ayz(y + z) + y-z- = (a), a'^(z" + ZX + X-)- azx(z + x) + zhr = {/3), «-(.«" + xy + y-)- axy(x + y) + xhf- =0 (7). This is equivalent to showing that, if we eliminate z between (a) and (j3), the result is (7). Arranging (a) and ((3) according to powers of z, we have ahf -a{- ay + y"-)z + [a" - ay + 1/^)2^ = (5), a-x- -a(-ax + x-)z + (n^ - ax + a;")s^ = (e). Multiplying (S) and (e) by k^ and y" respectively, and subtracting, wc get a^xy(x - y)z + {a^ix + y)- axy] {x - y)z''=Q, whence, rejecting the factor a(x-y)z, which is permissible since x^y, z + 0, axy+{a(x + y)-xy)z = (f). Again, multiplying (5) and (e) by a'-ax + x'^ and a--ay + %f respectively, and subtracting, we get, after rejecting the factor «-, u(x^y)-xyJ!-\a-{x-Vy)\z = !i {■,,). XIV EXERCISES XIX 305 Finally, multiplying (f) and (ij) by a{x + y)-xij and axy respectively, and subtracting, we get, since s =)= 0, {a(x + y)- xy} ^ - aocy {a-{x + y)} = 0, which gives a^{x^+xy + y'')- axy{x + y) + 3?y^ = 0, the req^uired result. EXEKOISES XIX. (Ore (he Reduction of Equations to an Integral Form.) Solve by inspection the following systems of equations : — (1.) a;2_4_3(^ + 2) = 0. (2.) a + h_ 25 x-b x-a (3.) {a-b)x-a^ + b-=0. (4.) x{b-c) + y{c-a) + {a-b) = 0, ax{b-c) + by{c-a) + c{a-b) = 0. (5.) _ x + y + z=a + b + c, ax + by + cz=a^ + V^ + c^, bx + cy + az = bc + ca + ab. (6. ) For what values of a and b does the equation {x-a){Zx-1) = Zx'' + bx + W become an identity ? lutegralise the following equations ; and discuss in each case the equiva- lence of the final equation to the given one. (7.) a+2^ 2a; + ll^^^ x-i x~2 (8.) _l-_3/(a; + l) ^ l + Zj{x,-\) a! + l + l/(a; + l) a;- 1 + 1/(2:- 1)' (9.) ^+J_ = J_ + _^. x+a x-c x-a x+c (10.) ^+JL=_^+^'. x-a x-b x-a x-b (3- a:) (a; + 10) _ (3 + a;)(g-10 ) K+ll ~ X-ll x^+2JX + q _ x^+px + t X- + rx + 2q~ x- + rx + 2t (11.) (12.) nai (x-af (x-b)^ jx-cf _ ^ '' {c-a)(a-b) (a-b)(b-c) {b-c){e-a) (14) a + T + U ^ a: + T-U a;= + (2-i!)a; + s(2-s-<) x^ + {i-s)x + t[2-s-t)~ when 21=s + t-s^-st-t-. VOL. I (22, 306 EXERCISES XIX, XX chap. ,, p. , x- + ax + b x- + cx + d_x- + a.r + 1)' X' + cx + d' x + a x + c " x + a x + c Rationalise the following equations and reduce the resulting ec[uation to as simple a fomi as possible : — (16.) ^yX + \/Y + ^yZ = 0, where X, Y, Z are rational functions of the variables. (17.) \/{x + a) + ^(x + b) + '^{x + c) = 0. (IS.) V(l+a;) + V(* + »)-V(9 + a;) = 0. (19.) [x-c+{{x- c)- + 1/! -]/[a.- + C+ {{x- cf + if] -] = m. (20.) x-a=^J{a--'s/{,a?x^-a^)}. (21.) V.'-+V(.-'--7)-21M^-7). (23 N V(2+a;) _ V(2-a) ^ ■'^ V'2 + V(2 + a^) V2-V(2-a:)' (24.) v'(» + a) + V(»-») + V'(* + »=) + \/(*-a:) = 0. (25 1 \JO.+x + x'^) + sJ(l-x + x-} ^ '' V(l+a:) + V(l-*0 (26.) {y-z){ax + bf + {z-x){ay + lf+{x-tj)(az + b)^=0. (27.) S-y(i/-s) = 0; and show that I,x= \/{S'Syz) (throe variables x, y, z). (^«-) V{.^.=3i)y + v{^,3^(5^=Vl2(^.i)}. (29.) a* + 5.1;^ -22 = 0. (30. ) \V +^) ^ \^ {a + x) ^ \/;i' ,t a c (31 . ) \/(a + V-f) + \/(a - V*:) = ^^''■ (32.) xi + yi + zi = 0, where x + y + z=0. Exercises XX. {On the Transformation of Systems of Equations.) [In working this set the student should examine carefully the logic of every step he takes, and satisfy himself that it is consistent with his data. He .should also make clear to himself whether each step is or is not reversible. ] (!■) If ' l+? + '-i±^ + '' + y + 2 = 0, x + y + z=t=0, X y z then 1 + 1+1 = „JL_. y z x+y+z XIV EXERCISES XX 307 (2.) If a^ + 2/S=2?, (3.) Ux, y, z be real, and it ai^(y-z) + i/'{s-x)+s*{x~ij) = 0, then two at least of the three must be equal. (i-) If ix + y + z)^=x^ + y^ + ^, then {x + y + z)^+'^ = x^''+^ + j/^^+i + s2«+i. (5.) If {ph: + 2pry + A) {q^x + 2qsy + s%) = {pqx + {ps + qrjxj + rsz] \ then either y'^ -zx=Q or ps-q7-=Q. (6.) If ^ + ^„+J!?L = o, a^_,.i J^-z-^i c'-r' where r'-'=a? + 2/2 + »^ Vc'-p' e'a'-p' a-'b'-p' where p^ = aV + jy + ch^. (Important in the theory of the wave surface. — Tait. iH > IS V + z z + x x + y J , , „ (7.) If ^-^ = _! — =— ^, and a + 2/ + a=0, b—c c-a a-o show that each of them is equal to ^J \^x'^j2['S,a'^ - S&c)}. (8. ) If a(hy + cz- ax) = h{cz + ax- by) = c{ax + by- cz), a + b + c = 0, x + y + z=0. x + 2y _y + 2z_z + 2x 2a + b~2b + c'~2c + a' \XaJ ~'Zab~^'' 2a& + ft^ a^ - &^ , , , (a-J)^ a + b ab x=a + b + ] ' y = — — + -—-, i(a + b) '^ i a + b (x-af-(y-hf=b\ a = ax + by + cz + dw, P—bx + ay + dz + cw, y=cx + dy + az + bii>, S = dx + cy + bz + aw, and if yi", ft 7, 5) = (a + ^ + 7 + S)(«-|3 + 7-S)(a-/3-7 + «)(a + /3-7-5), then /(a, ft 7> «)=/(«, *> c, (^)/(a;, y, z, w). (13.) If a; + 2/ + »=0, then Sl/a;2=(Sl/a;)2. and if then (9.) If then (10.) If then (11.) If then (12.) If 308 EXERCISES XX, XXI (14.) If (rKr^fi)'"-=(r-c Cf"''^" h^tl-\^n n77t-|-ft* show that (Sa;"°''l'»+»)) (Sx^^^'"-**)) = d™. (15.) If a-''^=i-- = c-^, x + 0, J/ + 0, r + 0, then a + ^S^ = 6 + =c + f- o-c c-a a—b (16.) If a; + (j/2 - a;^)/(a;^ + 5/" + ~") be unaltered by interchanging x and y, it ■\\'ill be unaltered by interchanging x and c, provided x, y, z be all unequal ; and it will vanish ii x + y + z = l. (17.) If zxyl(y + z)-x''=xyzl{s + x)-y'', and x^y, then each of these is equal to xyzj{x + y)-z^ and also to yz + zx + xy. (18.) Of the three equations __5 y+z x'-io' {m + l)w'-(n + 1)yz' y _ z + x y'^ - w^~ (in+l)w' - (n+ l]zx' z x + y where x^y^z, any two imply the third (Cayley). (19.) Given ^— -^^ + ^J^^ = 1, 1+x + xz 1+y + xy 1+z + yz a ^ M/ ^ 1 ^j 1 + x + xz 1 + y+xy 1+z + yz ' none of the denominators being zero, then x—y=z. (20.) Given 2(?/ + k)7x = 32x, Sx + d, prove S(2/ + s-a;)3 + n(j/ + ;:-a;) = 0. (21.) Given Sa; = 0, prove S(aJ' + 2/»)/(ic + j/) + 5a:?p2(l/a;) = 0. (22.) Given 22;= 0, prove that 'Zsf'Lx'^l'Zs? is independent oix, y, z. (23.) If Sx»= -5xyzXxy, then Sa;=0, or 'Zxf^ - :Sx'y + Xxhf + 2'Sx'^yz=0. (24.) If n(K2+l) = a2 + l, n(a52-l) = a2-l, and 2as/=0, then a; + j/ + 3 = or = ±n. (25.) If x + 2/ + « + M=0, then 42ar'' + 32(2/ + z)(m + 2/)(m + =) = 0, where the S refers to the four variables x, y, z, u. EXEKOISES XXI. (On Elimination.) (1. ) Eliminate x between tlie equations x + llx=y, a? + l/x'=z. (2.) \i z=\J(mf-a'jy), y=\]{ax?-a?lx>), express \J(az^-a?lz) in terms oix. XIV BXEKCISES XXI (3.) If is (J=c= + a=rf2)y + 2abcW{V-c^ + a'cP - 2a?b''-)p' + a-b-cH^a? - c=) (6^ - d-) = 0. (7.) Eliminate x, y, x', y' from ax + by = c^, x^ + y'=c'', , , _ a'x' + b'y'= c'^ x'^ + 2/'= = c'=, ^2/ + »= 2/ - "• (8.) If ll[x + a) + ll(y + a) + ll(z + a)=:\ja, with two similar equations in which b and c take the place of as, show that S(l/«) = 0, provided a, b, c he all different. (9. ) Show that any two of the following equations can he deduced from the other three : — ax + ie = zu, by + ca = uv, cz + db='vx, du + ec=xy, ev+ad=yz. (10.) Eliminate x, y, z from the three equations {z+x-y)(x + y-z)=ayz, (x + y-z)(y + z-x) = bzx, {y + z-x)(z + x-y) = cxy; and show that the result is abc=(a + b + c-i.)-. CHAPTEr. XV. Variation of Functions. § l.J The view which we took of the theory of conditional equations in last chapter led us to the problem of finding a set of values of the variables which should render a given conditional equation an identity. There is another order of ideas of at least equal analytical importance, and of wider practical utility, which we now proceed to explain. Instead of looking merely at the values of the variables x, y, z, . . which satisfy the equation /(x, y,z,. . .) = 0, that is, which render the function /(x, y, z, . . .) zero, we consider all possible values of the variables, and all possible corresponding values of the function ; or, at least, we consider a number of such values sufficient to give us a clear idea of the whole ; then, among the rest, we discover those values of the variables which render the function zero. The two methods might be illustrated by the two possible ways of finding a particular man in a line of soldiers. "We might either go straight to some part of the ranks where a preconceived theory would indicate his presence ; or we might walk along from one end of the line to the other looking till we found him. In this new way of looking at analytical functions, the graphical method, as it is called, is of great importance. This consists in representing the properties of the function in some way by means of a geometrical figure, so that we can with the bodily eye take a comprehensive view of the peculiarities of any individual case. THE GRAPHICAL METHOD 311 GENERAL PROPOSITIONS REGARDING FUNCTIONS OF ONE REAL VARIABLE. § 2.] For the present we confine ourselves to the case of a function of a single variable, /(.r) ; and we suppose that all the constants in the function are real numbers, and that only real values are given to the variable x. We denote, as in chap. xiii., § 17, f(x) by. y, so that 2/ =/(■'■) (1), and we shall, as in the place alluded to, speak of x and ij as the independent and dependent variables ; we are now, in fact, merely following out more generally the ideas broached there. Y To obtain a graphical representation of the variation of the function /(.>:) we take two lines X'OX, Y'OY, at right angles to each other (co-ordinate axes). To represent the values of x wc measure x units of length, according to any convenient scale, from the intersection along X'OX to the right if x have a positive value, to the left if a negative value. To represent the values of y we measure lengths of as many units, according to the same or, it may be, some otlier fixed scale, from X'OX parallel to Y'OY, upwards or downwards according as these values are positive or negative. For example, suppose that, when we put x= - 20, x= - 7, x= +18, x= +37, the corresponding values of 312 are CONTINUITY OF FUNCTION AND OF GRAPH /(-20), /(-7), /(+18), /(+37) + 4, -10, +7, -6 respectively ; so that we have the following scheme of corre- sponding values : — X y -20 - 7 + 18 + 37 + 4 -10 + 7 - 6 Then we measure off OM, (left) = 20, OM, (left) = 7, OM, (right) = 18, OM, (right) = 37 ; and M,P, (up) = 4, M,P, (down) = 10, M;P, (up) = 7, M,P, (down) = 6. To every value of the function, therefore, corresponds a re- presentative point, P, whose abscissa (OM) and ordinate (MP) represent the values of the independent and dependent variables; that is to say, the value of x and the corresponding value of f{x). Now, when we give x in succession all real values from - oo to + ao , y will in general * pass through a succession of real values withont at any stage making a sudden jump, or, as it is put, tviihout becoming discontinuous. The representative point will therefore trace out a continuous curve, such as we have drawn in Fig. 1. This curve we may call the graph of the function. § 3.] It is ob\ious that when we know the graph of a function we may find the value of the function cwresponding to any riiluc of the iiidcpcndcnt variable x with an accuracy that depends merely on the scale of our diagram and on the precision of our drawing instruments. All we have to do is to measure off the ^'alue of x in the proper direction, OM, say ; then draw a parallel through Mj to the axis of y, and find the point 1', where this parallel meets the graph ; then apply the compasses to MjP„ and read off the number of units in MjP, by means of the scale of ordinfites. This number, taken positive if P, ha above the ^\^J shall return to the exceptional cases immediately. X7 GRAPHICAL SOLUTION OF AN EQUATION 313 axis of X, negative if below, will be the required value of the function. The graph also enables us to the same extent to solve the converse problem, Given the value of the function, to find the corre- sponding value or values of the independent variable. Suppose, for example, that Fig. 1 gives the graph of f{x), and we wish to find the values of x for which f{x) = +7. All we have to do is to measure ON, = 7 upwards from on the axis of y ; then draw a line (dotted in the figure) through N, parallel to the axis of x, and mark the points where this line meets the graph. If P, be one of them, we measure N^P, (obviously = OMj) by means of the scale of abscissae, and the number thus read off' is one of the values of x for which f{x) = + 7 ; the others are found by taking the other points of intersection, if such there be. Observe that the process we have just described is equivalent to solving the equation /(^)=+7. In particular we might look for the values of x for which f(x) reduces to zero. When f(x) becomes zero, that is, when the ordin- ate of the graphic point is zero, the graph meets the axis of x. The axis of x, then, in this case acts the part formerly played by the dotted parallel, and the values of x required are - OMj, - OM3, + OM, + OMs, + 0M,„, where 0M„ OM3, &c., stand merely for the respective numbers of units in these lengths when read off upon the scale of abscissse. Hence By means of the graph of the function f{x) we can solve the equation f{x)^0 (2). The roots of this equation are, in point of fact, simply the values of X which render the function f(x) zero ; we may therefore, when it is convenient to do so, speak of them as the roots of the function itself § 4.] The connection between the general discussion of a function by means of the graphical or any other method and the problem of solving a conditional equation will now be apparent to the reader, and he will naturally ask himself how the graph 3U EXAMPLE is to be obtained. "We cannot, of course, lay down all the infinity of points on the graph, but we can in various ways infer its form. In particular, we can assume as many values of the independent variable as we please, and, from the known form of the function /(a-), calculate the corresponding values of y. We can thus lay down as many graphic points as we please. If care be taken to get these points close enough where the form of the curve appears to be changing rapidly, we can draw with a free hand a curve through the isolated points which will approach the actual graph sufficiently closely for most practical purposes. When the form of the function is unknown, and has to be determined by observation — as, for example, in the case of the curve which represents the height of the barometer at different times during the day — the course we have described is the one actually followed, only that the value of y is observed and not calculated. Before going further into details it will be well to illustrate by a simple example the above process, which may be unfamiliar to many readers. Example. Let the function to be discussed be l-'c", then the equation (1) which determines the graph is 2/= 1 - x". AVe shall assume, for the present without proof, what will probably be at once admitted by the reader, that, as x increases without break from up to + 00 , a:" increases without break from up to + oo ; and that x-> = <\, according as a;> = <1. Consider, in the first place, merely positive values of x. When x—d, ij — \ ; and, so long as x\, then a;^>l and 1 -x^ is negative. Hence from x = Q until a- = l, l-x' continually decreases numerically, but remains always positive. When x=\, 1"X- becomes zero, and when x is further increased 1-a;^ becomes negative, and remains so, but continually increases in numerical value. We may represent these results by the following scheme of corresponding values : — ■ Ji y 1 -c+l + +1 -->+l - + CO - CO EXAMPLE 315 The general form of the graph, so far as the right-hand side of the axis of y is concerned, will be as in Fig. 2. As regards negative values of X and the left-hand side of the axis of J/, in the present case, it is merely necessary to notice that, if we put x = -a, the re- sult, so far as 1 - £c^ is concerned, is the same as if we put x= +a\ for l~{-af = l-(+ of. Hence for every point P on the curve, whose abscissa and ordinate are -fOM and -I- MP, there will be a point P', whose abscissa and ordinate are - OM and + MP. P and P' are the images of each other with respect to Y'Y ; and the part AP'B' of the graph is merely an image of the part APB with respect to the line Y'Y. Let US see what the graph tells us regarding the function 1 -x^. First we see that the graph crosses the a;-axis at two points and no more, those, namely, for which x= + 1 and a;= - 1. Hence the function X-x^ has only two roots, + 1 and - 1 ; in other words, the equation has two real roots, x= +l,x= -\^ and no more. Secondly. Since the part BAB' of the graph lies wholly above, and the parts C'B', CB wholly below the a;-axis, we see that, for all real values of x lying between -1 and +1, the function l-x^ is positive, and for all other real values of x negative. Thirdly. We see that the greatest positive value of 1 -a;^ is 1, correspond- ing to a;=0 ; and that, by making x sufficiently great (numerically), we can give \ — 3? a, negative value as large, numerically, as we please. All these results could be obtained by direct discussion of the function, but the graph indicates them all to the eye at a glance. § 5.] Hitherto we have assumed that there are no breaks or discontinuities in the graph of the function. Such may, how- ever, occur ; and, as it is necessary, when we set to work to discuss by considering all possible cases, above all to be sure that no possible case has escaped our notice, we proceed now to consider the exceptions to the statement that the graph is in general a continuous curve. I. The function f(x) may become infinite for a finite value of x. 316 DIFFERENT KINDS OF DISCONTINUITY Example 1. Consider the function 1/(1 - x). When a; is u. very little less than + 1, say a;= -99999, then 2/ = l/(l -a) gives ?/= +1/-00001 = +100000 ; that is to say, y is positive and very large ; and it is obvious that, by bringing x suffi- ciently nearly up to + 1, we can give ij as large a positive value as we please. On the other hand, if x be a very little greater than + 1, say x— +1 '00001, then y = \j(- -00001)= -100000 ; and it is obvious that, by making x ex- ceed 1 by a sufficiently small quantity, y can be made as large a negative quantity as we please. The graph of the function 1/(1 - x) for values of x near -(- 1 is therefore as follows : — The branch BC ascends to an infinite distance along KAK' (a line iiarallel to the j/-axis at u, distance from it = +1), continually coming nearer to KAK', but never reaching it at any fiinite dis- tance from the K-axis. The branch DE comes up from an infinite distance along the other side of KAK' in a similar manner. Here, if we cause cc to in- crease from a, value OL very little less than -I- 1 to a value Fig- 3- OM very little greater, the value of y will jump from a very large positive value + LC to a very large negative value - MD ; and, in fact, the smaller we make the increase of x, provided always we pass from the one side of + 1 to the other, the larger will be the jump in the value of y. It appears then that, for x= +\, 1/(1 -k) is both infinite and discon- tinuous. Example 2. 2/=l/(l-a;)=. ^\'e leave the discussion to the reader. The gi'aph is as in Fig. 4. The function becomes infinite when x= + 1 ; and, for a very small increment of x near this value, the increment of y is very large. In fact, if we increase or diminish x from the value + 1 by an infinitely small amount, y will diminish by an in- finitely great amount. Here again we have infinite value of the function, and accompanying discontinuity. Y C B K M O L <' ^ X r Fio. 4. DIFFEEENT KINDS OF DISCONTINUITY 3r, M X Fio. 5. II. The value of the function may make a jump mthmit becoming infinite. The graph for the neighbourhood of such a value would be of the nature indicated in Fig. 5, where, while x passes through the value OM, y jumps from MP to MQ. Such a case cannot, as we shall immediately prove, occur with in- tegral functions of x. In fact it cannot occur with any algebraical function, so that we need not further consider it here. The cases we have just considered lead us to give the follow- ing formal definition. A function is said to he continuous when for an infinitely small change in the value of the independent variable the change in the value of the function is also infinitely small ; and to be discontinuous when for an infinitely small change of the independent variable the change in the value of the function is either finite or infinitely great. III. It may happen that the value of a function, all of whose con- stants are real, becomes imaginary for a real value of its variable. Example. This happens with the function -I- ^(1 - x^). If we confine ourselves to the positive value of the square root, so that we have a single-valued function to deal with, the graph is as in Fig. 6 : — Y a semicircle, in fact, whose centre is at the origin. For all values of a; > -I- 1, or < - 1, the value oiy=+ \J(\ - x^) is imaginary ; and the graphic points for them cannot be constructed r^ in the kind of diagram we are now using. The continuity of the function at A can- not, strictly speaking, be tested ; since, if we attempt to increase x beyond +1, y becomes imaginary, and there can be no question of the magnitude of the increment, from our present point of view at least.* No such case as this can arise so long as f(x) is a rational algebraical function. Fia. See below, § 18. 318 LIMITING CASES chap. We have now enumerated the exceptional cases of functional variation, so far at least as is necessary for present purposes. Graphic points, at which any of the peculiarities just discussed occur, may be generally referred to as critical points. ON CERTAIN LIMITING CASES OF ALGEBRAICAL OPERATION. § 6.] We next lay down systematically the following propo- sitions, some of which we have incidentally used already. The reader may, if he choose, take tliem as axiomatic, although, as we shall see, they are not all independent. The important matter is that they be thoroughly understood. To secure that they be so we shall illustrate some of them by examples. In the meantime we caution the reader that by " infinitely small " or " infinitely great " we mean, in mathematics, " smaller than any assignable fraction of unity," or "as small as we please," and "greater than any assignable multiple of unity," or "as great as we please." He must be specially on his guard against treating the symbol oo , which is simply an abbreviation for "greater than any assignable magnitude," as a definite quantity. There is no justification for applying to it any of the laws of algebra, or for operating with it as we do with an ordinary symbol of quantity. 1. If F he constant or variable, provided it does not become infinitely great when Q becomes infinitely small, then when Q becomes infinitely small PQ becomes infinitely small. Observe that nothing can be inferred without further examin- ation in the case where P becomes infinitely great when Q becomes infinitely small. This case leads to the so-called inde- terminate form 00 X 0.* E.xample 1. Let us suppose, for example, that P is constant, =100000, say. Then, if we make Q = I/IOOOOO, we reduce PQ to 1 ; if we make Q = 1/100000000000, we reduce PQ to 1/1000000 ; and so on. It is abundantly evident, therefore, that by making Q sufficiently small PQ can be made as small as we please. * Indeterminate forms are discussed in chap. xxv. XV LIMITING CASES 319 Example 2. LetP=a; + l, Q=a:-1. Here, when x is made to approach the value +1, P approaches the finite value + 2, while Q approaches the value 0. Suppose, for example, we put x=l + 1/100000, then PQ = (2 + 1/100000) X 1/100000, = 2/100000 + l/10i'>, and so on. Obviously, therefore, by sufficiently diminishing Q, we can make PQ as small as we please. Example 3. P = l/(a;2-l), Q=a;-1. Here we have the peculiarity that, when Q is made infinitely small, P (see below, Proposition III. ) becomes infinitely great. We can therefore no longer infer that PQ becomes infinitely small because Q does so. In point of fact, PQ=(!K- l)/(a!'^-l)=l/(a; + l), which becomes 1/2 when x=l. II. If P be either constant or variable, provided it do not become infinitely small when Q becomes infinitely great, then when Q becomes infinitely great PQ becomes infinitely great. The case where P becomes infinitely small when Q becomes infinitely great must be further examined ; it is usually referred to as the indeterminate form x co . Example 1. Suppose P=l/100000. Then, by making Q = 100000, we reduce PQ to 1 ; by making Q = 100000000000 we reduce PQ to 1000000 ; and so on. It is clear, therefore, that by sufficiently increasing Q we could make PQ exceed any number, however great. The student should discuss the following for himself :— Example 2. F=x + 1, Q=l/(a;-l). PQ = c>o whena;=l. Example 3. F={x-lf, Q = l/(a-l). PQ = 0whena;=l. III. IfPbe either constant m- variable, p-ovided it do not become infinitely small when Q becomes infinitely small, then when Q becomes infinitely small P/Q becomes infinitely great. The case where P and Q become infinitely small for the same value of the variable requires further examination. This gives the so-called indeterminate form q- 320 LIMITING CASES chap. Example 1. Suppose P coiisUnt=l/100000. If we make Q = l/100000, P/Q becomes 1 ; if we make Q = 1/100000000000, P/Q becomes 1000000 ; and so on. Hence we see that, if only we make Q small enough, we can make P/Q as large as we please. The student should examine arithmetically the two following cases : — Example 2. P=!B + 1, Q=a;-1. P/Q = 00 whena:=l. Example 3. P = a;-1, Q=a:-1. P/Q = l when x=l. TV. If T be either constant or variable, provided it do not become infinitely great when Q becomes infinitely great, then when Q becomes infinitely great P/Q becomes infinitely small. The case where P and Q become infinitely great together re- quires furtlier examination. This gives the indeterminate form CO Example 1. Suppose P constant = 100000. If we make Q = 100000, P/Q becomes 1; if we make Q = 100000000000, P/Q becomes 1/1000000 ; and so on. Hence by sufficic-'ntly increasing Q we can malce P/Q less than any assignable quantity. Example 2. P = ,c + 1, Q = 1/{k-1). P/Q = when x=\. Example 3. P = ]/(a;-l)^ Q = l/(a;-l). P/Q = 00 when x = l. V. -// P and Q each become infinitely small, then P + Q becomes infinitely small. For, let P be the numerically greater of the two for any value of the variable. Then, if the two have the same sign, and, a fortiori, if they have opposite signs, numerically P + Q < 2P. Now 2 is finite, and, by hypothesis, P can be made as small as we please. Hence, by I. above, 2P can be made as small as we please. Hence P + Q can be made as small as we please. VI. If either P or Q become riifi,nitcly great, or if P and Q each XT LIMITING CASES 321 become infinitely great and both have finally the same sign, then P + Q becomes infinitely great. Proof similar to last. The inference is not certain if the two have not ultimately the same sign. In this case there arises the indeterminate form 00 - 00 . Example 1. P=a^/(a;-l)=, Q = (2a;-l)/(a:-l)". Whena3=l, wc have P = 1/0=+ oo, Q = l/0= + <». Also 2a;-l a^ + 2a;-l P+Q = 7 '(x-lf (x-lf (x-lf ' Example 2. = ^=oo, wliena;=l. ¥=x'l{x-lf, Ci=-{2x-l)l{x-i)-. Here x=l makes P=+oo, Q=— oo, so that we cannot infer P + Q = oo. In fact, in this case, p,n__^! 2x-i_{x-il_ '^^~(x-lf {x-lf~[x-lf for all values of x, or, say, for any value of x as nearly = + 1 as we please. In this case, therefore, by bringing x as near to + 1 as we please, we cause the value of P + Q to approach as near to + 1 as we please. § 7.] The propositions stated in last paragraph are the funda- mental principles of the theory of the limiting cases of algebraical operation. This subject will be further developed in the chapter on Limits in the second part of this work. In the meantime we draw the following conclusions, which will be found useful in what follows : — I. ij^ P = PiPj . . . Pm, then P ivill remain finite «/ Pj, Pa, . . ., P„ all remain finite. P ivill become infinitely small if one or more of the functions P, , Ps, . . . , P„ become infinitely small, provided none of the remain- ing ones become infinitely great. P will become infinitely great if one or more of the functions Pj, P2, . . . , P„ become infinitely great, provided none of the remaining ones become infinitely small. II. 7/ S = Pi + P2 + . . . + P,i, tlien S will remain finite if P,, Pg, . . ., 7 n each remain finite. VOL. I Y 32-3 LIMITING CASES OHAl'. ^ will hecome infinitely small if T^ I, V.,, . . ., P^ each become in- finitely small. S will become infinitely great if one or more of the functions P,, P„, . . ., P„ become infinitely great, provided all those that become infinitely great have the same sign. III. Consider the quotient P/Q. p J- 'icill certainly be finite if both P and Q be finite, may be finite «/ P = 0, or ?/ P = CO , Q = 0, Q=oo. p — will certainly =0 if F = 0, or if P+ CO, Q. + o, Q=co; may =0 if V = 0, 0?- i/ P = CO , P . . p- will certainly = co if T? = co , ^ m- if -p + 0, may = co «/ P = 0, or «/ P = CO , Q = 0, ON THE CONTINUITY OF FUNCTIONS, MORE ESPECIALLY OF liATIONAL FUNCTIONS. § 8.] We return now to the question of the continuity of functions. By the increment of a function f(x) corresponding to an increment h of the independent variable x we mean f{x + h) -/(:«). For example, iif(x)—x'^, the increment is [x + h)- - x-^ixh + li?. lff(x) = l/x, the increment is ll{x + h)-llx= -hjx(x + h). The increments may be either positive or negative, according partly to choice and jjartly to circumstance. The increment of the independent variable x is of course entirely at our disposal ; but when any value is given to it, and when x itself is also assigned, the increment of the function or dependent variable is determined. XV CONTINUITY OF A SUM OR PEODUOT 323 Example. Let the function be 1/a;, then if a;=l, h=S, the corresponding increment otljxis -3/1(1 + 3)= -3/4. Ifa;=2, A=3, theincrementofl/Kis -3/2(2 + 3) = - 3/10, and so on. If P be a function of x, and p denote its increment when x is increased from x to x + h, then, by the definition of ^, P +j? is the value of P when x is altered from x to x + h. We can now prove the following propositions ; — I. The algebraic sum of any finite number of continuous functions is a continuous function. Let us consider S = P - Q + E, say. If the increments of P, Q„ R, when x is increased by h, be p, q, r, then the value of S, when X is changed to a; + A, is (P +p) - (Q + j) + (R + »■) ; and the increment of S corresponding to h i% p - c[ + r. Now, since P, Q, R are continuous functions, each of the increments, p, q, r, becomes infinitely small when A becomes infinitely small. Hence, by § 7, I., p — q + r becomes infinitely small when h becomes infinitely smaU. Hence S is a continuous function. The argument evidently holds for a sum of any number of terms, provided there be not an infinite number of terms. II. The product of a finite number of continuous functions is a continuous function so long as all factors remain finite. . Consider, in the first place, PQ. Let the increments of P and Q, corresponding to the increment A of the independent vari- able X, be p and q respectively. Then when x is changed to x + h PQ is changed to (P +p)(Q, + q), that is, to PQ +pQ,. + qP +pq. Hence the increment of PQ corresponding to h is j?Q + gP +pq. Now, since P and Q are continuous, p and q each become in- finitely small when h becomes infinitely small Hence by § 7, I. and II., it follows that p(^ + qP+pq becomes infinitely small when h is made infinitely small ; at least this will certainly be so, provided P and Q remain finite for the value of x in question, which we assume to be the case. It follows then that PQ is a continuous function. Consider now a product of three continuous functions, say PQR. By what has just been established, PQ is a continuous 324 ANY INTEGRAL FUNCTION CONTINUOUS chap. function, which' wc may denote by the single letter S ; then PQR = SR where S and li are continuous. But, by last case, SR is a continuous function. Hence PQR is a continuous function. Proceeding in this way, we establish the proposition for any finite number of factors. Cor. 1. If A be constant, and P a continuous function, then AP is a continuous function. This can either be established independently, or considered as a particular case of the main proposition, it being remembered that the increment of a constant is zero under all circumstances. Cor 2. A.i:"', where A is constant, and m a positive integer, is a continuous function. For X™ = x y. z y. . . . x x (m factors), and x is continuous, being the independent variable itself. Hence, by the main proposi- tion, a;™ is continuous. Hence, by Cor. 1, Ax'" is a continuous function. Cor. 3. Evmj integral function of x is continuous ; and cannot become infinite for a finite value of x. For every integral function of a; is a sum of a finite number of terms such as Ax'". Now each of these terms is a continuous function by Cor. 2. Hence, by Proposition I., the integral func- tion is continuous. That an integral function is always finite for a finite value of its variable follows at once from § 7, 1. III. If P and Q be integral functions of x, then P/Q is finite and continuous for cdl finite values of x, except such as render Q = 0. In the first place, if Q 4= 0, then (see § 7, HI.) P/Q can only become infinite if either P, or both P and Q, become infinite ; but neither P nor Q can become infinite for a finite value of x, because both are integral functions of x. Hence P/Q can only become infinite, if at all, for values of x which make Q = 0. If a value which makes Q = makes P 4= 0, then P/Q certainly becomes infinite for that value. But, if such a value makes both Q = and also P = 0, then the matter requires further investi- gation. Next, as to continuity, let the increments of P and Q corro- XV CONTINUITY OF A EATIONAL FUNCTION 325 spending to A, the increment of x, be p and q as heretofore. Then the increment of P/Q is T+p V pQ-qP Q + q Q~Q(Q + 2)' Now, by hypothesis, p and q each become infinitely small when h does so. Also P and Q remain finite. Hence pQ - qP becomes infinitely small. It follows then that (^Q - gP)/Q(Q + q) also becomes infinitely small when h does so, provided always •(see § 6) that Q does not vanish for the value of x in question. Example. The increment of l/(a;-l) corresponding to the increment, A, of x is lj{x + h-l)-ll{x- 1)= -h/{x-l){x + h- 1). Now, if a;=2, say, this becomes - A/(l + A), whicli clearly becomes infinitely small when h is made infinitely small. On the other hand, if x=l, the increment is -hjOh, which is infinitely great so long as h has any value ditfering from by ever so little. § 9.] When a function is finite and continuous between two values of its independent variable x = a and x = b, its graph forms a continuous curve between the two graphic points whose abscissae are a and b ; that is to say, the graph passes from the one point to the other without break, and without passing any- where to an infinite distance. From this we can deduce the following important pro- position : — If fix) he continuous from x = a to x = h, and if f{a) =p, f{h) = q, then, as x passes through every algebraical value betv)een a and b, fix) passes at least once, and, if more than once, an odd number of times through every algebraical value between p and q. Let P and Q be the graphic points corresponding to a; = a and x = b, AP and BQ their ordinates ; then A'P=p, BQ = g. We have supposed p and q both positive ; but, if either were negative, we should simply have the graphic point below the a;-axis, and the student will easily see by drawing the corresponding figure that this would alter nothing in the following reasoning. Suppose now r to be any number between p and q, and draw a parallel UV to the iK-axis at a distance from it equal to r units of the scale of ordinates, above the axis if r be positive, below if r be negative. The analytical fact that r is 326 LIMITS FOK THE ROOTS OF AN EQUATION intermediate to f and q is represented by the geometrical fact that the points P and Q lie on opposite sides of UV. Y U p I i....^ ...-.Q - L O A M, M2 Ma 3 X Fig. 7. Now, since the graph passes continuously from P to Q, it must cross the intermediate line UV ; and, since it begins on one side and ends on the other, it must do so either once, or thrice, or five times, or some odd number of times. Every time the graph crosses UV the ordinate becomes equal to r ; hence the proposition is proved. Cor. 1. If f{a) ie negative andfih) he positive, or vice versa, then fix) has at least one root, and, if more than one, an odd number of roots, between x = a and x = h, provided f(x) be continuous from x = a to x = b. This is merely a particular case of the main proposition, for is intermediate to any two values, one of which is positive and the other negative. Hence as x passes from a to bf(x) must pass at least once, and, if more than once, an odd number of times through the value 0. In fact, in this case, the axis of x plays the part of the parallel UV. Observe, however, in regard to the converse of this proposition, that a function may pass through the value ivithout changing its sign. For the graph may just graze the a^axis as in Figs. 8 and 9. 12 7T Fig. 8. Fro. 9. XV VALUES FOE WHICH A FUNCTION CHANGES SIGN 327 Cor. 2. If f{a) and f(h) have like signs, then, if there he any real roots of f{x) between x = a and x — h, there must he an even numher, provided f(x) he continuous between x = a and x = i. Since an integral function is always finite and continuous for a finite value of its variable, the restriction in Cor. 1 is always satisfied, and we see that Cor. 3. An integral function can change sign only hy passing through the value 0. Cor. i.IfV and Q he integral functions of x algebraically prime to each other, P/Q can only change sign hy passing through the values or CO . With the hint that the theorem of remainders will enable him to exclude the ambiguous case 0/0, we leave the reader to deduce Cor. 4 from Cor. 3. Example 1. When x=0, l-a;^=+l; and wlien x=+2, l-a;'=-3. Hence, since 1 - K^ is continuous, for some value of x lying between and + 2 1 - a;^ must become ; for is between + 1 and - 3. In point of fact, it becomes once between the limits in question. E,\ample 2. y=x^-6x'' + Ux-6. When x=0, y= -6; and when x=+i, y=+6. Hence, between x=0 and x= +i there must lie an odd number of roots of the equation a?-6a;2 + llu:-6 = 0. It is easy to verify in the present case that this is really so ; for x^ - 6x^ + ll!K-6=(x-l)(a3-2)(a;-3); so that the roots in question are a;=l, £c= 2, x = B. The general form of the graph in the present case is as follows : — Fig. 10. Example 3. When x=0, 1/(1 -a!)= +1 ;. and when x=+2, 1/(1 -a;)= - 1 ; but since 1/(1 -x) becomes infinite and discontinuous between a; = and x= +2, namely, when x=l, we cannot infer that, for some value of x between and +2, 328 SIGN OF /(O) AND /( ± oo ) chap. 1/(1 -a) will become 0, although is intermediate to +1 and -1. la fact, 1/(1 -x) does not pass through the value between x=0 and x= +2. § 10.] It will be convenient to give here the following pro- position, which is often useful in connection with the methods we are now explaining. If f(x) be an integral function of x, then hy iiinling x small enough we can always cause f{x) to have the same sign as its loioest term, and hy making x large enough we can always cause f(x) to have the same sign as its highest term. Let us take, for simplicity, a function of the 3rd degree, say y =p? + qx' + rz + s. If we suppose s =t= 0, then it is clear, since by making x small enough we can (see § 7, II.) make %n'^ + ^.i;" + rx as small as we please, that Ave can, by making x small enough, cause y to have the same sign as s. If s = 0, then we have y =i}x^ + qx' + rx, = (px^ + q.r, + r)x. Here by making x small enough we can cxas&pv? + qx + r io have the same sign as r, and hence y to have the same sign as rx, which is the lowest existing term in y. Again, we may write 3 f q r s 2' = -"|^ + x-^? + ? Here by making x large enough wc may make qjx + rja? + sj'j? as small as we please (see § 6, IV., and § 7, II.), that is to say, cause p + qjx + r/af + s/a? to have the same sign as ]/. Hence by making x large enough we can cause y to have the same sign as px. If we observe that, by chap, xiv., § 9, we can reduce every integral equation to the equivalent form /(a;)=a:»+p„_,a;"-i + . . .+po = 0, and further notice that, in this case, if n be odd, /( + CO ) = + CO , /( - CO ) = - CO , and, if n be even, /( + -r. ) = + CO , /( - CO ) = + CO , we have the following important conclusions. MINIMUM NUMBER OF SEAL BOOTS 329 Cor. 1. Every integral equation of odd degree with real co- efficients has at least one real root, and if it has mwe than one it has an odd number. Cor. 2. If an integral equation of even degree with real coefficients has any real roots at all, it has an even number of such. Cor. 3. Every integral equation with real coefficients, if it has any complex roots, has an even number of such. The student should see that he recognises what are the cor- responding peculiarities in the graphs of integral functions of odd or of even degree. Example. Show that the equation has at least two real roots. Let y=x*-Ga? + llx^-x-i. We have the following scheme of corresponding values : — X V — CO + CO + 00 1 -4 + CO Hence one root at least lies between - oo and 0, and one at least between and + 00 . In other words, there are at least two real roots, one negative the other positive. We can also infer that, if the remaining two of the possible four be also real, then they must be either both positive or both negative. When the real roots of an integral equation are not very close together the propositions we have just established enable us very readily to assign upper and lower limits for each of them ; and in fact to calculate them by successive approxima- tion. The reader will thus see that the numerical solution of integral equations rests merely on considerations regarding con- tinuity, and may be considered quite apart from the question of their formal solution by means of algebraical functions or otherwise. The application of this idea to the approximate determination of the real roots of an integral equation will be found at the end of the present chapter. 330 MAXllIA AND MINIMA ALTERNATE GENERAL PROPOSITIONS REGARDING MAXIMA AND MINIMA VALUES OF FUNCTIONS OF ONE VARIABLE. § 11.] IFhenf(x) ?';(, ^;«ss/h|7 through any value, f (a) say, ceases to increase and her/ins to decrease, f{a) is called a maximum value of f(x). JFJien f{x) in passing through the value f(a) ceases to decrease and begins to increase, f (a) is called a minimum value of f{x). The points corresponding to maxima and minima values of the function are obviously superior and inferior culminating points on its graph, such as Pa and Pg in Fig. 1. They are also points where, in general, the tangent to the graph is parallel to the axis of x. It should be noticed, however, that points such as P and Q in Fig. 11 are maxima and minima points, according to our present definition, although it is not true in any proper sense that at them the tangent is parallel to OX. It Fig. 11. Fio. 12. should also be observed that the tangent may be parallel to OX and yet the point may not be a true maximum or minimum point. "Witness Fig. 12. We shall include both maximum and minimum values as at present defined under the obviously appropriate name of turning values. § 12.] By considering an unbroken curve having maxima and minima points (.see Fig. 1) the reader will convince himself graphically of the truth of the following propositions : — I. So long as f{x) remains continuous its maxima and minima values succeed each other cdternately. II. If x = a, x = b be two roots of f{i) {a alg.■, y) has always either the sign + or the sign - , and S always furiiis the boundary between two regions in which f(x, y) has opposite signs. If we draw a continuous curve from a point in a + region to a point in a - region, it must cross the boundary S an odd number of times. This corresponds to the analytical statement that iff{a, h) be positive and f{a', V) be negative, then, if (x, y) vary continuously from (a, b) to {a', V), f{x, y) will pass through the value an odd number of times. The fact just established, that all the " variable points '' for which f(x, y) = lie on a continuous curve, gives us a beautiful geometrical illustration of the fact established in last chapter, that the equation f(x, y) = has an infinite number of solutions, and gives us the fundamental idea of co-ordinate geometry, namely, that EXAMPLE 335 a 'plane, curve can he analytically represented by means of a eguation connecting two variables. Example. Consider the function z^x' + ip-l. If we describe, with as centre, a circle whose radius is unity, it will be seen that for all points inside this circle z is negative, and for all points outside z is positive. Hence this circle is the zero contour line, and for all points on it we have 3:2 + 7/2-1 = 0. Y Pig. 16. INTEGRAL FUNCTIONS OF A SINGLE COMPLEX VARIABLE. § 17.] Here we confine ourselves to integral functions, but no i restrict either the constants of the function or its independent variable X to he real. Let us suppose that x = ^ + iji, and let us adopt Argand's method of representing ^ + -rji gtaphically, so that, if AM = f , ^p MP = rj, in the diagram of Fig. 1 6, then P represents ^ + lyi. If P jnove continuously from any position P to another P', the . complex variable is said to vary continuously. If the values of (^, ij) at P and P' be (a, /3) and (a', /?') respectively, this is the same as saying that ^ + -rji is said to vary continuously from the value a + /3i to the value a + (S'i, when ^ varies continuously from a to a, and Tj varies continuously/ from jS to j3'. There are of course Fig. 16. 3.36 CONTINUITY OF COMPLEX FUNCTION an infinite number of ways in which this variation may be accomplished. § 18.] Suppose now we have any integral function of x whose constants may or may not be real. Then we have f{x) =/(f + rfi); but this last can, by the rules of chap, xii., always be reduced to the form ^' + rj'i, where ^ and r{ are integral functions of ^ and -q whose constants are real (say real integral functions of (£, Tj) ). Now, by § 14, ^ and t) are finite and continuous so long as (^, -q) are finite. Hence f{^ + rji) varies continuously when ^ + 1)1 varies continuously. A graphic representation of the function /(^ + -qi) can be obtained by constructing another diagram for the complex number ^' + -qi. Then the continuity of /(^ + ?;/) is ex- l)ressed by saying that, when the graph of the independent vari- able is a continuous curre S, the graph of the dependent variable is another continuous curve S'. Example. Let 2/= + V(l-a~'). H 1 J B' C, ii A c Fig. 17. Fig. 18. Foi- simplicity, we shall coiiliue ourselves to a variation of x wliicli admits only real values ; in other words, we supjiose t; always =0. The path of the independent variable is then lACBJ, the whole extent of the f-axis. In the diagram we have taken CA = CB = 1 ; so that A and B mark the points in the path for which the function begins to have, and ceases to have, a real value. Let Fig. 18 be the diagram of the dependent variable, y = ^' + ■>]'{. If A'C' = 1 (A', B', and il' are all coincident), then the pnth of the dependent COMPLEX ROOTS DETERMINED GEAPIIICALLY 337 variable is the whole of the i/'-axis ahove fi, together with A'C, each reckoned twice over. The pieces of the two paths correspond as follows : — Independent Variable. Dependent Variable. lA AC CB BJ I'A' A'C CB' B'J' § 19.] ^' and r{ being functions of ^ and -q, vi^e may represent this fact to the eye by writing If vs^e seek for values of (£, rj) that make ^ = 0, that is the same as seeking for values of (^, r/) that make (£, t]) = 0. All the points in the diagram of the independent variable corresponding to these vi^ill lie (by § 1 6) on a curve S. Similarly all the points that correspond to rj = 0, that is, to i'iii l) = 0, lie on another curve T. The points for which both ^' = and i/ = 0, — in other words, the points corresponding to roots of f($ + rji), — must therefore be the intersections of the two curves S and T. Example. If we put x=i + vi, and y=i'+n'i, we have Hence ' |' = 2(4-|7;), V = l'-T- Hence the S and T curves, above spoken of, are given by the equations 2(4-f7,) = (S), f2-^==0 (T). 4 These are equivalent to 57=7 (S), (T). VOL. I 338 HORNER S METHOD The studeut should have no difficulty in constructing these. The diagram that results is H .J 'a M The S curve (a rectangular hyperbola as it happens) is drawn thick. The T curve (two straight lines bisecting the angles between the axes) is dotted. The intersections are P and Q. Corresponding to P we have | = 2, ?7 = 2 ; corresponding to Q, f = - 2, ^=-2. It appears therefore that the roots of the function are + 2 + 2i and - 2 - 2fi. The student may verify that these values do in fact satisfy the equation 1x2 + 8 = 0. HORNER'S METHOD FOR APPROXIMATING TO THE VALUES OF THE REAL ROOTS OF AN INTEGRAL EQUATION. § 20.] In the following paragraphs we shall sho%v how the ideas of §§ 8-10 lead to a method for calculating digit by digit the numerical value of any real root of an integral equation. It will be convenient in the first place to clear the way by estab- lishing a few preliminary results upon which the method more immediately depends. § 21. J To deduce from the equaiion pX'+Pv'''-~^+- ■ ■+Pn-v'' + Pn=0 (1) iinother equation each of whose roots is m times a corresponding root XV DIMINUTION OF ROOTS 339 of (1). Let X be any root of (1) ; and let ^ = mx. Then x = ^jm. Hence, from (1), we have pl^jmY +i),(^/™)""^ + ■ • • +Pn-ii$H +Pn = 0. If we multiply by the constant m", we deduce the equivalent equation i'oP+PiWp"^ + - ■ . +Pm-im""i^+^„m" = (2), which is the equation required. Cor. The eguation whose roots are those of (1) with the signs changed is i'oP -^ip-^ + ...+(- )^-^Pn-.^ + (-)>» = (3). This follows at once by putting m = - 1 in (2). We thus see that the calculation of a negative real root of any equation can always be reduced to the calculation of a positive real root of a slightly different equation. Example. The equation whose roots are 10 times the respective roots of Sari- 15a;2+5a; + 6 = is 3k«- 150^2 + 500a; +6000 = 0. § 22.] To deduce from the equation (!) of § 21 another, each of whose roots is less by a than a corresponding root of (I). Let X denote any root of (1); ^ the corresponding root of the required equation ; so that ^ = x- a, and x = ^ + a. Then we deduce at once from (1) Po(^ + af +p,{i + ffl)"-i + . . . +Pn-.{^ + a)+p^ = (4). If we arrange (4) according to powers of f, we get ^0^ + ?.?""' + ■ ■ • + ?»-i^ + 2« = (5), which is the equation required. It is important to have a simple systematic process for calculating the coefficients of (5). This may be obtained as follows. Since ^ = x- a, we have, by comparing the left-hand sides of (1) and (5), _P„X"+^,X*'-1 + . . .+Pn-iX+p„ =p^{x-aY + qy{x-aY-'^ + . . . + qn_i(x- a) + q„. 340 APPEOXIMATION TO ROOT CHAr. The problem before us is, therefore, simply to expand the function/(a;) =j?„a;" +2h^^^'^ + ■ • . + Pn-i^+Pn'^^ powers of {x - a). Hence, as we have already seen in chap, v., § 21, q^ is the remainder when fix) is divided by x-a; q^-x the remainder when the integral quotient of the last division is divided by x-a\ and so on. The calculation of the remainder in any particular case is always carried out by means 'of the synthetic process of chap, v., § 13. It should be observed that the last coeflBcient, (j^, is the value of/(«). Example. To diminisli the roots of 5a?-lla!2 + 10a:-2 = by 1. We simply reproduce the calculation of § 21, Example 1, in » slightly modified form ; thus — 5 -11 +10 -2(1 5 -6 -6 4 4 1 2 5 -1 -1 1 3 5 1 i Hence the required equation is 5« + 4J2 + 3J + 2 = 0. § 23.] If one. of the roots of the equation (1) of % 2\ he small, My between and + 1, then an approximate value of that root is -PnjPn-v For, if X denote the root in question, we have. by(i), ^ i^n - 1 trn - 1 .+p^^^-^) (6). Hence, if x be small, we have approximately a- = - p„jpn-i- It is easy to assign an upper limit to the error. We have, in fact, *= -Pn/Pn-i-^, where mode l, we have mode<(ft- l)PrlPn-i- XV HORNER'S PROCESS 341 It would be easy to assign a closer limit for the error ; but in the applications which we shall make of the theorem we have indirect means of estimating the sufficiency of the approxima- tion ; all that is really wanted for our purpose is a suggestion of the approximation. Example. The equation a;3 + 8192^2 + 16036288a;- 5969856 = has a root between and 1, find a first approximation to that root. By the above rule, we have for the root in question a;=5969856/16036288-e = -37227-6 where e<2 x 8192/16036288<: -00103. Hence «= -372, with an error of not more than 1 in the last digit. In point of fact, since a; < -4, we have e< {(-4)3 + 8192 X (-4)=}/16036288, < 1311/16036288 < 1600/16000000, <-0001 ; so that the approximation is really correct to the 4th place of decimals. § 2 4. J Horner's Method. Suppose that we have an equation f{x) = 0, having a positive root 235-67 . . . This root would be calculated, according to Horner's method, as follows : — First we determine, by examining the sign of /(cc), that f{x) = has one root, and only one,* lying between 200 and 300 : the first digit is therefore 2. Then we diminish the roots of /(a;) = by 200, and thus obtain the first subsidiary equation, sa,j /^(x) = 0. Then /i(a;) = has a root lying between and 100. Also since the absolute term of /,(a;) is /(200), and no root of f{x) = lies between and 200, the absolute term of f(x) (that is, /(O)) and the absolute term of /,(a;) must have the same sign. By examin- ing the sign of /,(«) for x = 0, 10, 20, . . ., 90, we determine that this root lies between 30 and 40 : the next digit of the root of the original equation is therefore 3. The labour of this last process is, in practice, shortened by using the rule of § 23. Let us suppose that 30 is thus suggested ; to test whether this * For a discussion of the precautions necessary when an equation has two roots which commence with one or more like digits, .see Burnside and Panton's Theory of Equations, § 104. 342 HORNER'S PROCESS ohap. is correct we proceed to diminish the roots of /i(a:) = by 30,' — to deduce, in fact, the second subsidiary equation /„(i) = 0. Since the roots oifix) have now been diminished by 230, the absolute term oi fJix) is/(230). Hence the absohite term of /^{x) must have the same sign as the absolute term of /,(a'), unless the digit 3 is too large. In other words, if the digit 3 is too large, we shall be made aware of the fact by a change of sign in the absolute term. In practice, it does not usually occur (at all events in the later stages of the calculation) that the digit suggested by the rule of § 23 is too small ; but, if that were so, we should become aware of the error on jiroceeding to calculate the next digit which would exceed 9. The second subsidiary equation is now used as before to find the third digit 5. The third subsidiary equation would give '6. To avoid the trouble and possible confusion arising from decimal points, we multiply the roots of the third (and of every following) subsidiary equation by 10; or, what is equivalent, we multiply the second coefficient of the equation in question by 10, the third coefficient by 100, and so on; and then proceed as before, observing, how- ever, that, if the trial division for the next digit be made after this modification of the subsidiary equation, that digit will appear as 6, and not as '6, because the last coefficient has been multiplied by 10"' and the second last by lO'''^. The fundamental idea of Horner's method is therefore simply to deduce a series of subsidiary equations, each of which is used to determine one digit of the root. The calculation of the coefficients of each of these subsidiary equations is accomplished by the method of § 22. After a certain number of the digits of the root have been found, a number more may be obtained by a contraction of the process above described, the nature of which will be easily understood from the following particular case. Example. Find an approximation to the least positive root of f(x)='j? + 2.c"-Zx-1 = (1). Since /(0)=-7, /(l)=-9, /(2)=-l, /(3)=+23, the root in question lies between 2 and 3. The first digit is therefore 2. XV EXAMPLE 343 We now diminish tlie roots of (1 ) by 2. The calculation of the coefficients runs thus : — 1 +2 -5 -7 (2 2 8 6 4 3 '1-1 2 12 6 ^1 15 where the prefix ^| is used to mark coefficients of the first subsidiary equation. The first subsidiary equation is therefore Since the next digit follows the decimal point, we multiply the roots of this equation by 10. The resulting equation is then ar! + 80a:^ + 1500a; -1000 = (2). Since 1000/1500 <1, it is suggested that the next digit is 0. We there- fore multiply the roots of (2) by 10, and deduce ar* + 800a;2 + 150000a -1000000 = (3). Since 1000000/150000 = 6- . . ., the next digit suggested is C. We now diminish the roots of (3) by 6. 1 +800 +150000 -1000000 (06 6 4836 929016 806 154836 "I -70984000 6 4872 812 1 15970800 __ 6 1 8180 The resulting subsidiary equation, after the multiplication of its roots by 10, is ijr' + 8180a:- + 15970800aj - 70984000 (4). Since 70984000/15970800 = 4- . . ., the next digit suggested is 4. The reader should notice that, owing to the continual multiplication of the roots by 10, the coefficients towards the right increase in magnitude much more rapidly than those towards the left : it is for this reason that the rule of § 23 becomes more and more accurate as the operation goes on. Thus, even at the present stage, the quotient 70984000/15970800 would give correctly more than one of the following digits, as may be readily verified. We now diminish the roots of (4) by 4 ; and add the zeros to the coefficients as before. 344 + 8180 4 8184 4 EXAMPLE + 15970800 32736 16003536 32752 ''I 1603628800 -70984000 (4 64014144 '1-6969856000 8188 4 1 81920 Then we have the subsidiary equation 3^ + 81920x2 + 1603628800a; -6969856000 (5). It will be observed that throughout the operation, so far as it has gone, tlie two essential conditions for its accuracy have been fulfilled, namely, that the last coefficient shall retain the same sign, and that each digit shall come out not greater than 9. It will also be observed that the number of the figures in the working columns increases much more rapidly than their utility in determining the digits of the root. All that is actually necessary for the suggestion of the next digit at any step, and to make sure of the accuracy of the suggestion, is to know the first two or three figures of the last two coefiicients. Unless, therefore, a very large number of additional digits of the root is required, we may shorten the operation by neglecting some of the figures in (5). If, for example, we divide all the coefficients of (5) by 1000, we get the equivalent equation * •OOli-s + 81 -920;= + 1603628 Sx - 6969856 = (5'). Hence, retaining only the integral parts of the coefiicients, we have Os!? + 81a;2 + 1603628a; -6969856 = (5"). It will be noticed that the result is the same as if, instead of adding zeros, as heretofore, we liad cut off one figure from the second last coefficient, two from the third last, and so on.t Since 6969856/1603628 = 4- . . ., wo have for the next digit 4. "We then diminish the roots of (5") by 4. In the necessary calculation the first working column now disappears owing to the disappearance of the coefficient of a;' ; we have in its place simply 81 standing unaltered. It is advisable, however, in multiplying the contracted coefficients by 4 to carry the nearest number of tens from the last figure cut off (just as in ordinary contracted multiplication and division and for the same reason). * If the reader find any difficulty in following the above explanation of the contracted process, he can satisfy himself of its validity by working out the above calculation to the end in full and then running his pen through the unnecessary figures. + In many cases it may not bo advisable to carry the contraction so far at each step as is here done. We might, for instance, divide the coefficients of 5 by 100 only. The resulting subsidiary equation would then be Oar* + 819..r + 16036288a; - 09698560, with which we should jiroceed as before. EXAMPLE 345 The next step, therefore, runs thus :- 81 +1603628 328 1603957 328 -6969856 {i 6416828 -55i028 (6); (6'). ^1 1604285 The corresponding subsidiary equation is 81a;' + 16042850a; - 55402800 = or, contracted, 0a;2 + 160428a;-554028 = The next digit is 3 ; and, as the coefficient of x', namely 0'81, still has a slight effect on the second working column, the calculation runs thus : — +160428 -554028 (3 2 481293 7|- 160431 2 - 72735 T 160433 The resulting subsidiary equation after contraction is 16043a;- 72735 = (7). The rest of the operation now coincides with the ordinary process of contracted division ; it represents, in fact, the solution of the linear equation (7), that is (see chap, xvi., § 1), the division of 72735 by 16043. The whole calculation may be arranged in practice as below. But the prefixes ^|, ^|, &c., which indicate the coefficients of the various equations, may be omitted. Also the record may be still farther shortened by performing the multiplications and additions or subtractions mentally, and only recording the figures immediately below the horizontal lines in the following scheme. The advisability of this last contraction depends of course on the arithmetical power of the calculator. + 2, -5 -7 (2-064434533 2 8 3 6 4 ■'1 - 1000000 2 12 929016 6 '■'■ 1 150000 ■'1 -70984000 2 4836 64014144 ""1 800 154836 "1 - 6969856 6 4872 6415828 806 ■*l 15970800 "1-554028 6 32736 481293 812 16003536 1-72735 6 32752 64173 1 8180 i 1 1603628^ 328 1 - 8562 8022 8184 1603957 1-540 4 328 481 8188 4 1 160428^ 2 ^1-59 48 'IJ^W^ 160431 2 -11 346 EXERCISES XXII chap. The number of additional digits obtained by the contracted proeess is less by two than the nnmber of digits in the second last coefficient at the beginning of the contraction. Owing to the uncertainty of the carriages the last digit is uncertain, but the next last will in such a case as the present be abso- lutely correct. In fact, by substituting in the original equation, it is easily verified that the root lies between 2-064434534 and 2-064434535 ; so that the last digit given above errs in defect by 1 only. The number of accurate figures obtained by the contracted process will occasionally be considerably less than in this example ; and the calculator must be on liis guard against error in this respect (see Horner's Memoir, cited below). § 2,5.] Since the extraction of the square, cube, fourth, . . . roots of any number, say 7, is equivalent to finding the positive real root of the equations, x' + O,*; -7 = 0, % + 0.«° + Oa: - 7 = 0, .r.' + 0,i;' 4 O.i;" + 0.« - 7 = 0, . . respectively, it is obvioiis that by Horner's method -we can find to any desired degree of approximation the root of any order of any given number whatsoever. In fact, the process, given in chap, xi., § 13, for extracting the square root, and the process, very commonly given in arithmetical text-books, for extracting the cube root will be found to be contained in the scheme of calculation described in S 24.* * Horner's method was first published in the Transactions of the Philoso- 2>hical Society of London for 1819. Considering the remarkable elegance, generality, and simplicity of the method, it is not a little surprising that it has not taken a more prominent place in cvirrent mathematical text-books. Although it has been well expounded by several English writers (for example, De iMorgan, Todhunter, Burnside and Panton), it has scarcely as yet found a recognised place in English curricula. Out of five standard Continental text- books where one would have expected to find it we found it mentioned in only one, and there it was expounded in a way that showed little insight into its true character. This probably arises from the mistaken notion that there is in the method some algebraic profundity. As a matter of fact, its spirit is purely arithmetical ; and its beauty, which can only be appreciated after one has used it in particular cases, is of that indescribably simple kind which distinguishes the use of position in the decimal notation and the arrangement of the simple rules of arithmetic. It is, in short, one of those things whose invention was the creation of a commonplace. For interesting historical details on this subject, see De Morgan — Companion to British Almanaclc, for 1839; Article "Involution and Evolution," Pcnmi/ Cyclopmdia; imd Budget of Paradoxes, pp. 292, 374. EXERCISES XXII 347 Exercises XXII. [The student should trace some at least of the curves requh-ed in the following graphic exercises by laying them down correctly to some convenient scale. He will find this process much facilitated by using paper ruled into small squares, which is sold under the name of Plotting Paper.] Discuss gi-aphically the following functions : — (l.)2/=i. (2.) 2/=^. (3.) 2, = . ^ 'x' + r ' ' " {x-lf (^■) ^=(^- (^-^ y="^2- («•) y=^r (7.) Construct to scale the graph of 1/= -a;^ + 8x- 9 ; and obtain graphic- ally the roots of the equation x--Sx + S = to at least three places of decimals. (8.) Solve graphically the equation a?-16a;2 + 71a;-129 = 0. (9. ) Discuss graphically the following question. Given that y is a. con- tinuous function of x, does it follow that a; is a continuous function of i/ ? (10.) Show that when h, the ihcrement of x, is very small, the increment of p„X''+Pn-lX"-^ + . ■ .+P1X+P0 is {np„x''--^ + {n-T.)p„,-iX'^~^ + . . .+l.pi)h. (11. ) If A be very small, and a; = 1, find the increment ol 2o[?-9x' + 12x + 5. (12. ) If an equation of even degree have its last term negative, it has at least two real roots which are of opposite signs. (13.) Indicate roughly the values of the real roots of 10a!3-17a!2-l-i2;H-3 = 0. (14.) What can you infer regarding the roots of r'-5a:-l-8 = 0? (15.) Show by considerations of continuity alone that af-l = cannot have more than one real root, if n be odd. (16.) If/(a;)bc an integral function of cc, and if/(a)= -p:f(l>)= +?, where p and q are both small, show that x= {qa +2Jb)l{p + q) is an approximation to a root of the equation_/(a;) = 0. Draw a series of contour lines for the following functions, including in each case the zero contour line : — (17.) z=xy. (18.) z=~. (19.) ~^=x'>-y^ (20.) z=^±^\ Is the proposition of § 16 true for the last of these ? Draw the Argand diagram of the dependent variable in the following oases, the path of the independent variable being in each case a circle of radius unity whose centre is Q : — (21.) 2/=^. {22.)y= + ^/x. {2Z.)y="^x (2i.) y=l-xK 348 EXKRCISES XXII chap, xv Find by Homer's method the positive real roots of the following equations in each case to at least seven places of decimals : — * (25.) a:3_2 = o. (26.) x"-2,i'-5 = 0. (27.) 3fi + x-lOOO = 0. (23.) ai3-46.j;2- 36a; + 18 = 0. (29.) .r^ + x' + X' + x - I2769i = 0. (30.) .j^-80r'+24.i--6a;-80379639 = 0. (31.) afi- i3^ + 7a?-86S = 0. (32.) .r'-7 = 0. (33.) ir^ - 4a; - 2000 = 0. (34.) ix'^ + 7r' + Sx-^ + ex' + 5x^ + Sx-792 = 0. (35.) Find to twenty decimal places the negative root of 2a/'* + Sx' - 6a; - 8 = 0. (36.) Continue the calculation on p. 344 two stages farther in its unoon- tracted form ; and then estimate how many more digits of the root could be obtained by means of the trial division alone. * Most of these exercises are taken from a large selection given in De Morgan's Elmnenis of Arilhmetic (1854). CHAPTER XVI. Equations and Functions of the First Degree. EQUATIONS WITH ONE VARIABLE. § 1.] It follows by the principles of chap. xiv. that every integral equation of the 1st degree can be reduced to an equiva- lent equation of the form ax + 6 = (1); this may therefore be regarded as a general form, including all such equations. As a particular case h may be zero ; but we suppose, for the present at least, that a is neither infinitely great nor infinitely small. Since a=l= 0, we may write (1) in the form «{- (-1)1=0 (2); whence we see that one solution is a; = - hja. We know already, by the principles of chap, xiv., § 6, that an integral equation of the \st degree in one variable has one and only one solution. Hence we have completely solved the given equation (1). It may be well to add another proof that the solution is unique. Let us suppose that there are two distinct solutions, x=a. and a;=/3, of (1). Then we must have ffla + 6=0, a/3 + J = 0. From these, by subtraction, we derive a(a-;8) = 0. Now, by hypothesis, a 4= 0, therefore we must have a - ;8 = 0, that is, a = /3 ; iu other words, the two solutions are not distinct. 350 TWO LINEAR EQUATIONS IN ONE VARIABLE chap. § 2.] Tivo equafions of the \st degree in one variahle will in general be inconsistent. If the equations he ax + h = (1), a'x + h' = Q (2), the necessary and sufficient condition fm- consistency is ab' -a'b = (3). The solution of (1) is x = - bja, and the solution of (2) is x= - b'ja'. These will not in general be the same ; hence the equations (1) and (2) will in general be inconsistent. The necessary and sufficient condition that (1) and (2) be consistent is -^^-? (4). a a Since a =(= 0, a' 4= 0, (4) is equivalent to a'b = ab', or ab' - a'b = 0. Obs. 1. If J = and b' = 0, then the condition of consistency is satisfied. In this case the equations become ax = 0, a'x = ; and these have in fact the common solution x = 0. Obs. 2. When two equations of the 1st degree in one vari- able are consistent, the one is derivable by multiplying the other by a constant. In fact, since a + 0, if we also suppose 5 =t= 0, we derive from (3), by dividing by ab and then transposing, ^-: = ?:, each = May; a b ' J ' hence a' = ha, b' = Icb, so that a'x + b' = kax + Icb, = ]c(ax + b). If, then, (3) be satisfied, (2) is nothing more or less than k{ax + 6) = where k is a constant. This might have been expected, for, transpositions apart, the only way of deriving from a single equation another perfectly equivalent is to multiply the given equation by a constant. EXERCISES XXIII 351 Exercises XXIII. Solve the following equations : — ,, , 18X + 7 /„ 2k-1 (1.) -2 (2^--7- (2.) (3.) l + (l-a;)/2 — A-=l. 51 62 3a;- 1 12a!+5' 2 29 (4.) '\-r^~^' i — x (5.) •68(-32a:--5) + ~g = 3-694a:, find X to three places of decimals. (6.) al{l-bx) = b/{l-ax). (7. ) (a+x){i + x)-a{b + c) = {ca^ + fa^)/6. x-a x-c „ ^ ' ffi-a a:-o x-c (10.) (a3 + 63)!K + a3_J3^„4_j4 + a6(a2 + j a? -a? ! ^-(? (12.) (ij;-l)(a: + 2)(23;-2) = (2a;-l)(2a; + l)(a;/2 + l). _1 ?___^ i^ a; + l x + 2~a; + 3 x + \' (11-) TT^--6T-=''-'^- (13.) (14.) x-\ x-3_x-2 x-i x-2 x-i x-S x-5' 11 5 7 (^^■) 12a; + ll'^6a; + 5~4x + 7' 3-aj 5-a;_ a!^-2 2 14 _ 10 6 (^^■) a; + 2"''^+T0~a; + 6"''a; + 14" (^^•' a;- + 6a + 10 Va; + 3y ~ 352 ax + llj + c=0 HAS co^ SOLUTIONS (19.) J~ + -^ ^('' + ^' _«_6r + 17 K + l i; + 2 (x + l)(a; + 2) a: + 2 a + 2« a - 2g 4a6 (20.) (a + 6)a; + e (a-&)a; + e _ 4a6 ^ •' (,«-J)x + d (a + 6)a+/~(a + i)(«-6) a+6 a- 6 a h (22.) + -,= ,. ^ ' x-a x-,0 x-a x-b (23.) -^ + a b a- + b' x-a x-b x{x-a-b) + ab' (24.) —^+-7^ = 2. ^ ' x+a-b x+b—c (25.) 2 , 1 {x-a) (x-b) {x-a){x — c) {x — b){x — c) 1 '{x + a){x + b) {x + a){x + c.) {x + b){x + c)' EQUATIONS WITH TWO VARIABLES. § 3.] A single equation of the Isi degree in two variaUes has a one-fold infinity of solutions. Consider the equation ax + hy + e = (1). Assign to y any constant value we please, say fi, then (1) becomes M + i/? + c = (2). We have now an equation of the 1st degree in one variable, which, as we have seen, has one and only one solution, namely, X = - {hji + c)ja. "We have thus obtained for (1) the solution x= - (hj3 + c)/a, y = P, where ^ may have any value we please. In other words, we have found an infinite number of solutions of (1). Since the solution involves the one arbitrary constant /3, we say that the equation (1) has a one-fold infinity (sometimes symbolised by oo ') of solutions. Example. 3x-2y + l = 0, the solutions are given by 2^-1 TWO LINEAR EQUATIONS IN TWO VARIABLES 353 we have, for example, for /3= - 2, /3= - 1, ;3=0, /3= +-, /3= +1, j3= +2, the following solutions : — ,3 -2 -1 ^\ +1 + 2 5 ^ -3 -1 1 3 *\ + 1 y -2 -1 -\ +1 + 2 And so on. § 4.] We should expect, in accordance with the principles of chap, xiv., § 5, that a system of two equations each of the 1st degree in two variables admits of definite solution. The process of solution consists in deducing from the given system an equivalent system of two equations in which the variables are separated ; that is to say, a system such that x alone appears in one of the equations and y alone in the other. "We may arrive at this result by any method logically con- sistent with the general principles we have laid down in chap, xiv., for the derivation of equations. The following proposition aifords one such method : — If I, r, m, m' he constants, any one of which may be zero, but which are such tlmt Im! - I'm 4= 0, tJien the two systems ax +by + c =0 (1), a'x + b'y + c' = (2), and l{ax + by + c)+ l'{a'x + b'y + c') = (3), m{ax + by + c) + m'ia'x + b'y + c') = (4), are equivalent. It is obvious that any solution of (1) and (2) will satisfy (3) and (4) ; for any such solution reduces both ax + by + c and a'x + b'y + c' to zero, and therefore also reduces the left-hand sides of both (3) and (4) to zero. Again, any solution of (3) and (4) is obviously a solution of VOL. I 2 A 354 SOLUTION BY CEOSS MULTIPLICATION CHAP. m'\ l{a:i- + hi/ + c) + l'{a'x + h'y + c')} - r {in{ax + by + c) + in'{a'x + h'y + c')} = (5), - m { l{ax + by + c)+ l'{a'x + b'y + c')} + 1 {m{ax + by + c) + 7n'{a'x + b'y + c')} = (6). Now (6) and (6) reduce to {Im' - I'm) (ax +by +c) = (7), (Imf - I'm) (a'x + b'y + c') = (8), and, provided Im' - I'm 4= 0, (7) and (8) are equivalent to ax + by + c =0, a'x + b'y + c' = 0. We have therefore shown that every solution of (1) and (2) is a solution of (3) and (4) ; and that every solution of (3) and (4) is a solution of (1) and (2). All we have now to do is to give such values to I, V, m., m' as shall cause y to disappear from (3), and x to disappear from (4). This will be accomplished if we make 1= +b', r = -b, m= - a', m' = + a; so that Im' - I'm = ab' - a'b. The system (3) and (4) then reduces to {ab' - a'b)x + cV - c'b =0 (3'), (ab' - a'b)y + c'a - ca' = (4') ; and this new system (3'), (4') will be equivalent to (1), (2), provided ab' - ffl'J 4= (9). But (3') and (4') are each equations of the 1st degree in one variable, and, since ah' - a'b =t= 0, they each have one and only one solution, namely — ch' - c'b y (10). ab' - a'b ac' - a'c ab' - a'b XVI MEMORIA TECHNICA 355 It tlierefore follows that the system ax +ly +c =0 (1), a'x + h'y + c' = (i (2) has one and only one definite solution, namely, (10), provided aV - a'6 4= (9). The method of solution just discussed goes by the name of cross multiplication, because it consists in taking the coefficient of y from the second equation, multiplying the first equation therewith ; then taking the coefficient of y from the first equation, multiplying the second therewith ; and finally subtracting the two equations, with the result that a new equation appears not containing y. The following memoria technica for the values of x and y will enable the student to recollect the values in (10). The denominators are the same, namely, ab' - a'b, formed from the co- efficients of X and y thus ■-b the line sloping down from left to right indicating a positive product, that from right to left a negative product. The numerator of x is formed from its denominator by putting c and c' in place of a and a! respectively. The numerator of y by putting c and c! in place of h and V. Finally, negative signs must be affixed to the two fractions. Another way which the reader may prefer is as follows : — Observe that we may write (10) thus, 6c' - Vc „„._ 2/ = „-77rv7, (11). imon denominator and t to the scheme aH - a'b' ab' - a'b where the common denominator and the two numerators are formed according a \ / 6 \ /c \ ya It is very important to remark that (1) and (2) are col- laterally symmetrical with respect to I a,h ], see chap, iv., § 20. \a', b'J Hence, if we know the value of x, we can derive the value of y by putting everywhere b for a, a for b, V for a', and a' for U. In 356 EXAMPLES OHAP. fact the value of y thus derived from the value of x in (10) is - {ca! - c'a)j(ha' - b'a) ; and this is equal to - {ad - a'c)/{ab' - a'b), which is the value of y given in (10). Example 1. 3a; + 2t/-3 = (a), Proceeding by direct application of (11), we have -9X\ + 4 -'^^-^ + 5--'^^ -9 _ 10 + 12 _11 ^ 27-15 2 ^~12 + 18"'15' ^■"12 + 18~5' Or tlius : multiply (a) by 2, and we have the equivalent system 6a; + 4i/-6 = 0, - 9a: + 4i/ + 5 = ; whence, by subtraction, 15k -11 = 0, which gives ^~T^' Again multiplying (a) by 3, and then adding (j3), we have 102/-4 = 0, which gives 4 2 Example 2. y=v>=i- -+l=i. a 2/_l Multiplying the first of these equations by -, and subtracting the second, we obtain /I j-V -I I 7 that is, — ,,„ x= -5 — , whence x= — 7(a-/3)- (X y \ ' I we get the value of y by interchanging a and /3, namely, ■^ 7(/3-«)' XTi SPECIAL CASES 357 Sometimes, before proceeding to apply the above method, it is convenient to replace the given system by another vrhich is equivalent to it but simpler. Example 3. a-x + b^y=2ab{a + b) (a), b{2a + b)x + a{a + 2b)y=a'''+ a?b + aft^ + ft-' {j3). By adding, we deduce from (a) and (/3) {a + bfx + [a + bfy=(a + bf, whicli is equivalent to x + y—a + b (7). It is obvious that (a) and (7) are equivalent to (a) and (/3). Multiplying (7) by 6^ and subtracting, we have (a2-ft2)a = 2a2ft + ffift2_j3_ = ft(2a-6)(a + 6). TT 6(2(1-6) Hence x=-- r-^. a-b Since the original system is symmetrical in ( ' ^ j, we have a(26 - a) " b-a § 5.] Under the theory of last paragraph a variety of par- ticular cases in which one or more of the constants a, b, c, a', V, c' involved in the two equations aa; + Sy + c = 0, a'x + b'l/ + c' = become zero are admissible ; all cases, in short, which do not violate the condition ab' - a'b 4= 0. Thus we have the following admissible cases : — a=0 (1), 6' = (4), J = (2), a = and 5' = (5), a' = (3), a' = and J = (6). The following are exceptional cases, because they involve ab' - a'b a = and «' = (I.), a, b, a', V all different a = and J = (H.), from 0, but such that J' = and a' = (III), ab' -a'b = Q (V.). J' = and 6 = (IV.), 358 HOMOGENEOUS SYSTEM CHAP. AVe shall return again to the consideration of the exceptional cases. Ill the meantime the reader should verify that the formulae (10) do really give the correct solution in cases (1) to (6), as by theory they ought to do. Take case (1), for example. The equations in this case reduce to Jy + c=0, a'x + b'y + c' = 0. The first gives y= - cjh, and this value of y reduces the second to ax- -J- + C = 0, , . , . Vc-bc' which gives x= r, — ■ ab It will be found that (10) gives the same result, if we put a=0. There is one special case that deserves particular notice, that, namely, where c = and c' = ; so that the two equations are homogeneous, namely, ax +hj =0 (a), a'x + h'ij=Q {(3). If ah' - a'b =1= 0, these formulae (10) give x=Q, y -Q as the only possible solution. If ab' - ah = 0, these formulas are no longer applicable ; what then happens will be understood if we reflect that, provided y 4= 0, (a) and (/3) may be written az +b =0 (a'), a's + b' = (/S'), where z = x/y. We now have two equations of the 1st degree in z, which are consistent (see § 2), since ah' - a'b = 0. Each of them gives the same value of z, namely, s = - h/a, or z = - b'ja' (these two being equal by the condition ah' - a'b = 0). If then ab' - a'b =(= 0, the only solution of (a) and (/3) is x = 0, y - ; if ab' - a'b = 0, x and y may have any values such that the ratio x/y = -b/a = - b'/a'. § 6.] There is another way of arranging the process of solu- tion, commonly called Besout's method* which is in reality merely a variety of the method of § 4. * For an account of Bezout's methods, properly so called, see Muir's papers on the " History of Determinants ;" Froc. R.S.E., 1886. XVI USE OF UNDETERMINED MULTIPLIER 359 If \ he any finite constani quantity w/iatever,* then any solution of the system ax + iy + c=0, a'x + b'y + c' = (1) is a solution of the equation (ax + liy + e) + \{a'x + Vy + c') = (2), that is to say, of {a + \a')x + (li + \b')y + [e + 'Kc') = fi (3). Now, since X is at our disposal, we may so choose it that y shall disappear from (3) ; then must X5' + 6 = (4), and (3) will reduce to (a + Xa')a; + (c + Xc') = (5). From (4) we have X= - hjb', and, using this value of X, we deduce from (5) _ c 4- Xo' _ i'c - 6c' a + Xa' ah' - a'h' which agrees with (10). The value of y may next he obtained hy so determining X that x shall disappear from (3). We thus get Xa' + a=0 (6), (h + \b')y + (c + \e') = (S (7), and so on. To make this method independent and complete, theoretically, it would of course be necessary to add a proof that the values of x and y obtained do in general actuallj' satisfy (1) and (2); and to point out the exceptional case. § 7.] There is another way of proceeding, which is inter- esting and sometimes practically useful. The systems ax +hy + c =Q \ ,^ , a'x + h'y + c' = Q] ^ ^ and y=—ir\ (2) a'x + h'y + c' = ) are equivalent, provided 6 =1= 0, for the first equation of (2) is derived from the first of (1) by the reversible processes of trans- position and multiplication by a constant factor. Also, since any solution of (2) makes y identically equal to - {ax + c)/b, we may replace y by this value in the second equation of (2). We thus deduce the equivalent system, * So far as logic is concerned X might be a function of the variables, but for present purposes it is taken to be constant. A letter introduced in this way is usually called an " indeterminate multiplier " ; more properly it should be called an ' ' undetermined multiplier. " 360 SOLUTION BY SUBSTITUTION (l:r + c 1 h'(nx + c) , ax ^-; + c =0 (3). b Now, since 6=1= 0, the second of the equations (3) gives (a'b - ah')x + Q>c' - b'c) = (4). If a'b - aV =t= 0, (4) has one and only one solution, namely, be' - b'c ,^s X = -r, 7 (5), ab - ab this value of x reduces the first of the equations (3) to \^ a{bo'-b'c) \ y- ~b\ ab'-a'b +'r abc' - a'bc b{ab' - a'b)' that IS, to 1/ = —r, ri {">■ ■' ab - ab The equations (5) and (6) are equivalent to the system (3), and therefore to the original system (1). Hence we have proved that, if ab' - a'6 =t= and J 4= 0, the system (1) has one and only one solution. We can remove the restriction J =t= ; for if J = the first of the equations (1) reduces to ax + c = 0. Hence (if a 4= 0, which must be, since, if both a = and i = 0, then ab' - a'b = 0) we have a; = - cja, and this value of x reduces the second of equations (l)to + by + c =0, a ^ which gives (since b' cannot in the present case be without making ab' - a'b = 0) y = (ca' - c'a)jab'. Now these values of x and y are precisely those given by (5) and (6) when 6 = 0. The excepted case 6 = is therefore included ; and the only exceptional cases excluded are those that come under the condi- tion aV - a'b = 0. XVI SYSTEM OF THREE EQUATIONS IN X AND y 361 The method of this paragra^jh may be called solution by substitution. The above discussion forms a complete and independent logical treatment of the problem in hand. The student may, on account of its apparent straightforwardness and theoretical simplicity, prefer it to the method of § 4. The defect of the method lies in its want of symmetry ; the practical result of which is that it often introduces needless detail into the calculations. Example. Sx + 2y-3 = (a), -9x + iy + 5 = O). From (a) we have y = — (7). Using (7), we reduce (^) to -9a; + 2(-3a3 + 3) + 5 = 0, that is, -15a; + ll = 0; whence k=tt- 15 This value of x reduces (7) to -3x11 + 3 ^2 The solution of the system (o) and (p) is therefore 11 2 =^=15' y=6 § 8.] Three equations of the 1st degree in two variables, say ax + hy + c = 0, a'x + b'y + c' = 0, a"x + b"-y + c" = (1), mil not be consistent unless a" {be'-- b'c) + b"(ca' - c'a) + c"{ab' - a'b) = (2) ; and they will in general be consistent if this condition be satisfied. We suppose, for the present, that none of the three functions ab' - a'b, a"b - ab", alb" - a"b' vanishes. * This is equivalent to supposing that every pair of the three equations has a deter- minate finite solution. If we take the first two equations as a system, they have the definite solution * See below, § 25. 363 CONDITION OF CONSISTENCY he - h'c cii — c'a ab' - a'b' ah'-a'b' The necessary and sufficient condition for the consistency of the three equations is that this solution should satisfy the third equation ; in other words, that „hc' -Vc .„ca'-c'a a -J-, + V^, 77 + c = 0. ab - ab ab - ab Since ab' - a'b 4= 0, this is equivalent to a"{hc' - h'c) + h"(m! - c'a) + c"{ab' - a'b) = (3). The reader should notice that this condition may be written in any one of the following forms by merely rearranging the terms :— a{b'c'- - b"c') + b{c'a" - c"a') + c{a'b" - a"b') = (4), a'(bc" - b"c) + b'{ca" - c"a) + c'(ab" - a"b) = (5), a{b'c" - h"c') + a'{h"c - he") + a"(hc' - h'c) = (6), b{c'a'' - c"a') + h'{c"a - ca") + h"{ca' - c'a) = (7), c{a'b" - a"V) + c'{a"b - ab") + c"{ab' - a'b) = (8), ab'c" - ah"c' + he' a" - he a' + ca'h" - ca"h' = (9). The forms (4) and (5) could have been obtained directly by taking the solution of the two last equations and substituting in the first, and by solving the first and last and substituting in the second, respectively. Each of these processes is obviously logic- ally equivalent to the one actually adopted above. The forms (6), (7), (8) would result as the condition of the consistency of the three equations iix + a'y + a" = Q, bx + b'y +b" = Q, ex + e'y + c" = (10). We have therefore the following interesting side result : — Cor. If the three equations {I) he consistent, then the three equa- tions (10) are consistent. If the reader will now compare the present paragraph with § 2, he will see that the function ah' - a'V plays the same part for the system ax + h=0, a'x + lj' = XVI DETERMINANT OF THE SYSTEM 363 as does the function a(6'c" - b"o') + b{c'a" - c"a') + c{a'b" - a"b') for the system ax + by + c=0, a'x + b'y + c' = 0, a"x + b"y + c"=0. These functions are called the determinants of the respective systems of equa- tions. They are often denoted by the notations \a h I , for aV -a'b (11) ; a b c a' b' c' a" b" c" V for ab'd' - aV'c' + bc'a" - bd'a' + m'b" - ca"b' (12). The reader should notice — 1st. That the determinant is of the 1st degree in the constituents of any one row or of any one colnmn of the square symbol above introduced. 2nd. That, if all the constituents be considered, its degree is equal to the number of equations in the system. A special branch of algebra is nowadays devoted to the theory of deter- minants, so that it is unnecessary to pursue the matter in this treatise. For the sake of more advanced students we have here and there introduced results of this theory, but always in such a way as not to interfere with the progress of such as may be unacquainted with them. The reader may find the following memoriiR technicae useful in enabling him to remember the determinant of a system of three equations : — For the form (4), a b c J' \ / c' \ / o' \ / 6' to be interpreted like the similar scheme in § i. For the form (9), b 6"/ -Xc"/ \«" \i", where the letters in the diagonal lines are to form products with the signs + or - , according as the diagonals slope downwards from left to right or from right to left. Example. To show that the equations 3cc-l-52/-2 = 0, ix + 6y-l = Q, 2x + iy-S = are consistent. Solving the first two equations, we have x= - 7/2, y = 5/2. These values 364 EXERCISES XXIV reduce 2a; + 4)/ -3 to -7 + 10-3, which is zero. Hence the solution of the first two equations satisfies the thii'd ; that is, the three are consistent. "We might also use the general results of the above paragraph. Since 3x6-5x4= -2, 5x2-3x4= -2, 4 x 4-2 x 6= + 4, each pair of equations has by itself a definite solution. Again, calculating the deter- minant of the system by the rule given above, we have, for the value of the determinant, -64- 10 -32 + 24 + 12 + 60 = 0. Hence the system is consistent. + 3+5 2+3+5 Exercises XXIV. Solve the following : — (1.) u+ki=^, J»+42/=e (2.) 2x + iij=n, 3x-2y = (3.) ■123.'c+-6852/=3-34, -8933;- find X and y to five places of decimals. ■5932/ = 3-71, (4.) (5.) (6.) (7.) (8.) (9.) (10.) (11.) (12) (13.) (X x + y:x-y = 5 :3, x + 5y=3S. Sx + l=:2y + l = Zy + 2x. 3){y + 5) = {x~l){y + 2), 8x + 5 = x + y = a + b, {x + a)l{y + h) = bja. X y — (--^ = 1, la nib 2ma y = 1. 3lb ax + by=0, {a-b)x + {a + b]y=2c {a + i)x- {a-b)y = c, {a-b]x + {a + h)y=c. {a, + b)x + {a-b)y = a'' + 2ab-b^, {a-b)x+{a-b)y = w' + b\ y 7;,2 = «*. , + - y 2-2)2 a^ + ab + b'^ ' ci' + lf a^-ab + b- iiiji"' + brf)x + (op^+i + bcr^^)y = ap'"+^ + bq"^+'', (a2}" + bq'^)x + {ap"+'^ + b, be consistent, t\\ea a?-\-b^ + i?-3abc = 0. (17.) Find the condition that ax + by = c, a?x + ll'y = c'^, d^x + b^y=ifi be consistent. XVI SYSTEMS OF ONE AND OF TWO EQUATIONS IN X, y, Z 365 (18.) Find an integral function of x of the 1st degree whose values shall be + 9 and + 10 when x has the values - 3 and + 2 respectively. (19.) Find an integral function of x of the 2nd degree, such that the coefficient of its highest term is 1, and that it vanishes when a; =2 and when x= -3. (20.) Find an integral function of x of the 2nd degree wliich vanishes when x=Q, and has the values -1 and + 2 when x= -\ and a; =+3 respectively. EQUATIONS WITH THREE OR MORE VARIABLES. § 9.] A single equation of the \st degree in three variables admits of a two-fold infinity of solutions. For in any such equation, say ax + hy + CZ + d=Q, we may assign to two of the variables any constant values we please, say y = P, » = y, than the equation becomes an equation of the 1st degree in one variable, which has one and only one solution, namely, b/3 + cy + d a We thus have the solution hB + cy + d x= -^ , y = p, z = y. Since there are here two arbitrary constants, to each of which an infinity of values may be given, we say that there is a two-fold infinity (oo ") of solutions. A symmetric form is given for this doubly indeterminate solution in Exercises xxv., 27. § 10.] A system of two equations of the 1st degree in three vari- ables admits in general of a one-fold infinity of solutions. Consider the equations aa; + Jy + C2 + fZ = 0, a'x -^ b'y + c'z -{- d' = (1). We suppose that the functions be' - b'c, ca' - da, ab' - alb do not all vanish, say ab' - a'b =1= 0. If we give to z any arbitrary constant value whatever, say z = y, then the two given equations give definite values for x and y. We thus obtain the solution 366 HOMOGENEOUS SYSTEM OF TWO EQUATIONS chap. _ {he' - h'c)y + {hd' - h'd) _ (ca - c'a) y + (da' - d'li) _ ,g. Since we have here one arbitrary constant, there is a one-fold infinity of solutions. Cor. There is an important particular case of the above that often occurs in practice, that, namely, where c^ = and d' = 0. We then have, from (2), ca' - c'a ■■7- be'- '' ah' -b'c -'a'b'>' , y-- ca' - c'a ah' -a'h'^' ' This result can be written as follows : — X 7 be' -h'c »// - a'V y y ca' -c'a ah' -a'V z 7 aV - a'b aV - a'h Now, y being entirely at our disposal, we can so determine it that yj{ah' - a'b) shall have any value we please, say p. Hence, p being entirely arbitrary, we have, as the solution of the system, ax +by +cz =0) a'x + b'y + c'z = ) ^ '' X = pihc' - h'c), y = pica' - c'a), e = p(ab' - a'b) (4). It will be observed that, although the individual values of X, y, z depend on the arbitrary constant p, the ratios of x, y, z are perfectly determined, namely, we have from (4) X ■.y:z = (be' - b'c) : {ca' - c'a) : (ah' - a'b). Example 1. 2x + 3y + iz=0, Sx-2y-6z=0, -2 -6 3 -2 give x:y:z= -10:2i: -13; XVI SYSTEM OF THREE EQUATIONS IS X, 1J, Z 367 or, which is the same thing, a;=-10p, y=2ip, s=-13p, p being any quantity whatsoever. Exampla 2. ax+by + cz = 0, give x = {bc^- b\)p = - bcp{b - c), y = (ca? - c'a)p — - cap{c - a) , z=(ab'^ - a?b)p= -abp{a-h). If we choose, we may replace -abcp by -l _ hx + iy + iz + 'i.ii + v + 2 ~ 4 ~ 9 (22.) ax + by = l, cx + dz=\, cz-Vfu=\, gu + hv=\, x + y + z + u + v=0. (23.) Prove that, with a certain exception, the system U = 0, V = 0, W = 0, and \U + /iV + vW = 0, \'U + /i'V + ^'W = 0, X"U + /i"V + ;/" W = are equivalent. (21.) If x=hy + cz + du, y = ax + cz + du, z=ax + by + du, u = a.r + by + cz, ,, a b c d ^ then - + , — -H -+-; — -=1. os + l 6 + 1 c + 1 d + 1 (25.) Show that the system ax + by + cz + d=0, a'x + b'y + c'z + d'=0, a"x + b"y + c"z + d"=0 will be equivalent to only two equations if the system ax + a'y + a"=0, bx + b'y + b"=0, cx + c'y + c" = 0, dx + d'y + d"=0 be con- sistent, that is, if bV^'Y _ V'c-bc" _ be' - b'c a'd" - a"d' a"d - ad" ad' - a'd' Show that in the case of the system x + y + z = a + b + c, - + y + - = l, ^ + |^, + 4, = 0, the above two conditions reduce to one only, namely, bc + ca + ab = 0. (26.) Show that the three equations x=A + A'u + A"v, 2/ = B + B'tt + B"'V, ; = C + C'i6 + C"i;, where it and v are variable, are equivalent to a single linear equation con- necting X, y, z; and find that equation. (27.) I{ax + by + cz + d=0, show that ^-"V.-c)-|, 2/=(J + 2)(c-«)-i <^') where 2> and q are arbitrary constants. XVI EQUATIONS REDUCIBLE TO LINEAR SYSTEMS 379 (28.) I{ax + ly + cs + d=0, a'x + i'y + c'z + d'=0, show t'ha.t x=p{bc' -b'c) + {{b' - c')d - (b ~ c)d'} / {a{b' - c') + b{c' ^ a') + c{a' -b')} , y=p(oa' -c'a) + {(d - a')d - (c - a)d']l{a(b' -c') + b(c' -a') + c(a' -b')\, z=p(ab' -a'b)+ {{a' -b')d-(oi,-b)d'}l{a(b' -c') + b(a' -a') + c{a' -b')}, where p is au arbitrary constant. EXAMPLES OF EQUATIONS WHOSE SOLUTION IS EFFECTED BY MEANS OF LINEAR EQUATIONS. § 17.] We have seen in chap. xiv. that every system of algebraical equations can be reduced to a system of rational integral equations such that every solution of the given system will be a solution of the derived system, although the derived system may admit of solutions, called " extraneous," which do not satisfy the original system. It may happen that the derived system is linear, or that it can, by the process of factorisation, be replaced by equivalent alternative linear systems. In such cases all we have to do is to solve these linear systems, and then satisfy ourselves, either by substitution or by examining the reversibility of the steps of the process, which, if any, of the solutions obtained are extraneous. The student should now re-examine the examples worked out in chap, xiv., find, wher- ever he can, all the solutions of the derived equations, and examine their admissibility as solutions of the original system. We give two more instances here. Example 1. (Positive values to be taken for all tlie square roots.) If we rationalise the two denominators on the left, we deduce from (a) the equivalent equation, From ((3) we derive, by squaring both sides, 2a; + 2V{K^-(a:^-l)} =2(2^+1), that is, 2a: + 2 = 2x3 + 2 (7). Now (7) is equivalent to a?-x=Q, thatis, to ii;(x-l)(a; + l) = (S). / a; = 0" Again (S) is equivalent to the alternatives f a; - 1 = that is to say, its solutions are 3^=0, a;=l, x= - 1. 380 EXAMPLES F = V2, Since, however, the step from (fi) to (7) is irreversible, it is necessary to examine which of these solutions actually satisfy (a). Now a;= Ogives \/ -i+\/ + i=\/2, that is (see chap, xii., § 17, Example 3), l-i \+i V2 V2 ■ which is correct. Also, x=l obviously satisiies (a). But x= - 1 gives 2i = 0, which is not true, hence x= - 1 is not a solution of (a). Remark that x= -1 is a solution of the slightly different equation, Example 2. x'-y^=x-y, 2a; + 3j/-l = (a). Since the first of these equations is equivalent to {x-y){x+y-Vj = .)-^J{x + n)=\. (4.) V«+V(« + 3) = 12/\/('« + 3)- 384 EXEECISES XXVI (5.) V(* + 2) + \/{^'-2) = 5/VGb + 2). (6.) V-' + \/('' + /3)-V(c'-ffl = \/C»= + 2/3). (7.) \'{j+q-r) + \/{x + r-p) + '^{x+p-q) = 0. (8-) V(.'-i-Vi'"*"V(^)TVi'=^'"-^'- (9. ) x='^{a'-X'^(b- + x--a,'')}+a. (10.) VCV^ + V'^) + %/(%/»! -\/«) = V'C2\/« + 2 V*)- (11.) '^x + ■^{3-^^{2x + x"-)]=^/3. (12.) v(^-v(j^)4. V(.-2)=^V(-3). (13.) \J',a-x)-sJ(y-x) = sJy, \/{b-x) + ^{i/-x)=-.^i/. 1 17 (14.) \/x-\/y=^, ^-2/=^- (15.) (a;-a)2-(i/-i)2 = 0, {x-h){i/-a)=a{2b-a). (16.) a;-2/=3, .r--7/^=45. (17.) xly = alb, x'-y^=d. (18.) a; + ay + a°; + «%t + a*=0, K + 6)/ + i^:; + 6^M + 6'' = 0, x + cy + A + (Pu + c'=0, x + dy + (F:: + axis. Let OM represent any positive value of x, and MP the cor- ^responding value of y. VOL. I 2 Fib. 2. Through B draw a line parallel to the 386 THE GRAPH 0¥ OX + 'b By the equation to the graph, we have (y - b)/x = a. Now, since h= + OB = + MN in Fig. 1, = - OB = - MN in Fig. 2, we have 2/ - i = PM - MN = PN in Fig. 1, J/ - 6 = PM + MN = PN in Fig. 2. Hence we have in both cases PN _ PN _ y - & _ BN~OM"~ar "''■ In other words, the ratio of PN to BN is constant ; hence, by elementary geometry, the locus of P is a straight line. If a be positive, then PN and BN must have the same sign, and the line will slope upwards, from left to right, as in Figs. 1 and 2 ; if a be negative, the line will slope downwards, from left to right, as in Figs. 3 and 4. The student will easily complete the discussion by considering negative values of x. y Y O MX B F N Fio. 3. Fio. 4. § 20.] So long as the graphic line is not parallel to the axis of X, that is, so long as a =t= 0, it will meet the axis in one point, A, and in one only. In analytical language, the equation ax + b = has one root, and one only. Also, since a straight line has no turning points, a linear function can have no turning values. In other words, if we increase x continuously from -co to + oo , aa; + 6 either increases continuously from - qo to + oo , or decreases continuously from + 00 to - 00 ; the former happens when a is positive, the latter when a is negative. Since ax + b passes only once through every value between THE GEAPH OY OX + b 387 + 00 and - 00 , it can pass only once through the value 0. We have thus another proof that the equation ax + b = has only one root. A purely analytical proof that ax + b has no turning values may be given as follows : — Let the increment of x be h, then the increment oi ax + b is {a{x + h) + b}- {ax + b} = ah. Now ah is independent of x, and, if h be positive, is always posi- tive or always negative, according as a is positive or negative. Hence, if a be positive, ax+b always increases as z is increased ; and if a be negative, ax + b always decreases as x is increased. § 21. J We may investigate graphically the condition that the two functions ax + b, a'x + V shall have the same root ; in other words, that the equations, ax + b=Q, a'x + 6' = 0, shall be con- sistent. Denote ax + b and a'x + b' hy y and y' respectively, so that the equations of the two graphs are y = ax + b, y' = a'x + V. If both functions have the same root, the graphs must meet OX in the same point A. Now, if P'M PM be ordi- nates of the two graphs corresponding to the same abscissa OM, and if the graphs meet OX in the same point A, it is obvious that the ratio P'M/PM is constant. Conversely, if P'M/PM is constant, then P'M must vanish when PM vanishes ; that is, the graphs must meet OX in the same point. Hence the neces- sary and sufficient analytical condition is that {a'x + b')/{ax + b) shall be constant, = k say. In other words, we must have a'x + b' = Jc(ax + b). From this it follows that a' = Tea, b' = hb, and aV - alb = 0. These agree with the results obtained above in § 2. § 22.] By means of the graph we can illustrate various limiting cases, some of which have hitherto been excluded from con- sideration. 388 GRAPHICAL DISCUSSION OF LIMITING CASES I. Let b = 0, adrO. In this case OB = 0, and B coincides with ; that is to say, the graph passes through (see Figs. Y Y Fio. 6. Fio. 7. 6 and 7). Here the graph meets OX at 0, and the root of aa; = is a; = 0, as it should be. 11. Let & =t= and ft = 0. In this case the equation to the graph is y = Z), which represents a line parallel to the a;-axis (see Figs. 8 and 9). In this case the point of intersection of the Y B O X Pio. 8. Fio. 9. graph with OX is at an infinite distance, and OA = co . If we agree that the solution of the equation ax + b = shall in all cases be a; = - h/a, then, when 6 =1= 0, « = 0, this will give a; = oo , in agreement with the conclusion just derived by considering the graph. This case will be best understood w by approaching it, both geometric- r ally and analytically, as a limit. Let us suppose that J = - 1, and that a is very small, = 1/100000, say. Then the graph correspond- 1 is something like Fig. 10, where Fig. 10. ing to y = a;/l 00000- XVI INFINITE EOOT 389 the intersection of BL with the axis of x is very far to the right of ; that is to say, BL is nearly parallel to OX. On the other hand, the equation -1 = 100000 gives a;=100000, a very large value of x. The smaller we make a the more nearly will BL become parallel to OX, and the greater will be the root of the equation ax + & = 0. If, therefore, in any case where an equation of the 1st degree in z was to be expected, we obtain the paradoxical equation 5 = 0, where b is a constant, this indicates that the root of the equation has become infinite. III. If a = 0, J = 0, the equation to the graph becomes y=Q, which represents the axis of x itself. The graph in this case coincides with OX, and its point of intersection with OX becomes indeterminate. If we take the analytical solution of ax + b = to be a; = - bja in all cases, it gives us, in the present instance, X = 0/0, an indeterminate form, as it ought to do, in accordance with the graphical result. § 23.] The graphic surface of a linear function of two inde- pendent variables x and y, say z = ax+by + c, is a plane. It would not be difficult to prove this, but, for our present pur- poses, it is unnecessary to do so. We shall confine ourselves to a discussion of the contour lines of the function. The contour lines of the function g = ax + by + c are a series of parallel straight lines. For, if k be any constant value of z, the corresponding con- tour line has for its equation (see chap, xv., § 1 6) ax + by + c = k (1). Now (1) is equivalent to f a\ h-c ,„. But (2), as we' have seen in § 19 above, represents a straight line, which meets the axes of x and y in A and B, so that 390 CONTOUR LINES OF OX + by + C 0B = 0A = k- c la h ■ ta K - I Fio. 11. Let k' be any other value of z, then the equation to the corre- sponding contour line is ax + ly + c = k' (3), 2,= (-|)x + ^' (4). Hence, if this second contour line meet the axes in A' and B' respectively, we have OB' = '"-^, Ok! ■■ Hence h' - c OA ob' k' - c is given by the equa- (5). 6_0A' a "OB" which proves that AB is parallel to A'B'. The zero contour line of z = ax + by ■ tion ax+hy + c = This straight line divides the plane XOY into two regions, such that the values of x and y corresponding to any point in one of them render ax + hy + c positive, and the values of x and y cor- responding to any point in the other render ax + by + c negative. § 24.J Let us consider the zero contour lines, L and L', of two linear functions, z = ax + hy + c and z' = a'x + b'y + c'. Since the co-ordinates of every point on L satisfy the equation ax + by + c = (1), XVI GRAPHIC ILLUSTRATION OF INFINITE SOLUTION 391 and the co-ordinates of every point on L' satisfy the equation a!x + h'y + c' = Q (2), it follows that the co-ordinates of the point of intersection of L and L' will satisfy both (1) and (2); in other words, the co- ordinates of the intersection will be a solution of the system (1), (2). Now, any two straight lines L and L' in the same plane have one and only one finite point of intersection, provided L and L' be neither parallel nor coincident. Hence we infer that the linear system (1), (2) has in general one and only one solution. It remains to examine the two exceptional cases. I. Let L and L' (Fig. 11) be parallel, and let them meet the axes of X and Y in A, B and in A', B' respectively. In this case the point of intersection passes to an infinite distance, and both its co-ordinates become infinite. The necessary and sufficient condition that L and L' be parallel is OA/OB = OA'/OB'. Now, OA = - cja, OB = - cjh ; and OA' = - c'ja', OB' = - c'/6'. Hence the necessary and suffi- cient condition for parallelism is Ija - b'/a', that is, ab' - a'b = 0. We have thus fallen upon the excepted case of §§ 4 and 5. If we assume that the results of the general formulae obtained for the case ab' - a'b 4= 0, namely, be' - b'c _ ca' - da, " ab' - a'b' ^ ~ ab' - a'V hold also when ab' - a'b = 0, we see that in the present case neither of the numerators be' - b'c, ca' - c'a, can vanish. For if, say, be' - b'c = 0, then -c/b= - e'/b', that is, OB = OB' ; and the two lines AB, A'B', already parallel, would coincide, which is not supposed. It follows, then, that be' - b'c ca' - c'a and the analytical result agrees with the graphical. II. Let L and L' be coincident, then the intersection becomes indeterminate. The conditions for coincidence are 392 INDETERMINATE SOLUTION CHAP. OA = OA', OB = OB', whence -■c/a= - c'ja', - c/b = - c'jV. Ihese sive — , = f7 = -> which again give Ic' - b'c = 0, ca' - da = 0, ah' - aH = 0. "We thus have once more the excepted case of §§ 4 and 5, but this time with the additional peculiarity that Id -h'c = Q and ca' - da — 0. If we assert the truth of the general analytical solution in this case also, we have that is, the values of x and y are indeterminate, as they ought to be, in accordance with the graphical result. § 25.] Since three straight lines taken at random in a plane have not in general a common point of intersection, it follows that the three equations, ax + by + c = 0, a'x + Vy + d = 0, a'x + h"y + d' =^Q (1), have not in general a common solution. When these have a common solution their three graphic lines, L, L', L", will have a common intersection. We found the analytical condition for this to be aVd - aV'd + Ida" - hd'a' + ca'h" - ca"h' = (2). In our investigation of this condition we left out of account the cases where any one of the three functions, ab' - a'b, a"b - ab", a'h" - a"b', vanishes. We propose now to examine graphically the excepted cases. First, we remark that if two of the functions vanish, the third will also vanish ; so that we need only consider (I.) the case where two vanish, (II.) the case where only one vanishes. I. ab' - a'b = 0, a"b - ah" = 0. This involves that L and L' are parallel, and that L and L" are parallel ; so that all three, L, L', L", are parallel ; and we have, in addition to the two given conditions, also a'b" - a"b' = 0. XVI EXCEPTIONAL SYSTEMS OF THREE EQUATIONS 393 Hence, since the condition (2) may be written c{a'h" - a"h') + c'{a"b - ab") + c"{aV - a'b) = 0, it appears that the general analytical condition for a common solution is satisfied. This agrees with the graphical result, for three parallel straight lines may be regarded as having a common intersection at infinity. In the present case is of course included the two cases where two of the lines coincide, or all three coincide. The corre- sponding analytical peculiarities in the equations will be obvious to the reader. II. ab' - a'b = 0. Here two of the graphic lines, L and L', are parallel, and the third, L", is supposed to be neither coincident with nor parallel to either. Looking at the matter graphically, we see that in this case the three lines cannot have a common intersection unless L and L' coincide, that is, unless a' = ha, V = hb, d = he, where k is some constant. Let us see whether the condition (2) also brings out this result, as it ought to do. Since ah' — a'b = 0, 1, a' V , we have - =-- = h say. a b ' ^ Hence a' = ka, V = hb. Now, by virtue of these results, (2) reduces to a"{bc' - b'c) + b"{ca' - c'a) = 0, that is, to a"(bc' - hbc) + b"{cha - c'a) = 0, that is, to {a"b - ab") (c' - he) = 0, which gives, since a"b - ab" 4= 0, c' -kc = 0, that is, c' = he. 394 FXERCISES XXVII CHAP. Hence the agreement between the analysis and the geometry is complete.* § 26.] It would lead us too far if we were to attempt here to take up the graphical discussion of linear functions of three variables. We should have, in fact, to go into a discussion of the disposition of planes and lines in space of three dimensions. We consider the subject, so far as we have pursued it, an essential part of the algebraic training of the student. It will help to give him clear ideas regarding the generality and coherency of analytical expression, and will enable him at the same time to grasp the fundamental principles of the application of algebra to geometry. The two sciences mutually illuminate each other, just as two men each with a lantern have more light when they walk together than when each goes a separate way. Exercises XXVII. Draw to scale the graphs of the following linear functions of x : — (1.) y=x + l. (4.) y = 2x + S. (2.)y=-x+l. (5.)y=-ix-l {3.)y=-x-l. (6.)y=-3{x-l). (7.) Draw the graphs of the two functions, 3x-5 and 5a: + 7; and by means of them solve the equation 3a! - 5 = 5j; + 7. (8.) Draw to scale the contour lines ol z=2x-3y + l, corresponding to s=-2, 2=-l, 2=0, 2=+l, s=+2. (9. ) Draw the zero contour lines oi z = 5x + Gy-3 and ^ = 8x-Sy + l; and by means of them solve the system 5k + 6j/-3 = 0, 8x-9y + l = 0. * It may be well to warn the reader explicitly that he must be careful to use the limiting cases which we have now introduced into the theory of equations with a proper regard to accompanying circumstances. Take, for instance, the case of the paradoxical equation 6=0, out of which we manu- factured a linear equation by writing it in the form Ox + b = 0; and to which, accordingly, we assigned one infinite root. Nothing in the equation itself prevents us from converting it in the same way into a quadratic equation, for we might write it Ox^ + Oa; + J=0, and say (see chap, xviii., § 5) that it has two infinite roots. Before we make any such assertion we must be sure beforehand whether a linear, or a quadratic or other equation was, generally speaking, to be expected. This must, of course, be decided by the circum- stances of each particular case. XVI EXERCISES XXVII 395 Also show that the two contour lines divide the plane into four regions, such that in two of them (5a; + 61/ - 3) (8a: - 9i/ + 1) is always positive, and in the other two the same function is always negative. (10.) Is the system 3a;-4y+2=0, 6x-8y + Z = 0, x-iy+l = consistent or inconsistent ? (11.) Determine the value of c in order that the system 2x+y-l = 0, ix+2y + 3 = 0, {c + l)x + {c + 2)y + 5 = may he consistent. (12.) Prove graphically that, if ab' -a'b = 0, then the infinite values of x and y, which constitute the solution of ax + by + c=0, a'x+i'y + c' = 0, have a finite ratio, namely, x/y= {be' - b'c)l{ca' - c'a). (13.) If {ax + by + c)l{a'x + b'y + c') be independent of x and y, show that ai' -a'b = 0, ca' -c'a=0, to' -b'c=0; and that two of these conditions are sufficient. (14. ) Illustrate graphically the reasoning in the latter part of § 5 of the preceding chapter. (15. ) Explain graphically the leading proposition in § 6. CHAPTER XVII. Equations of the Second Degree. EQUATIONS OF THE SECOND DEGREE IN ONE VARIABLE. § 1.] Every equation of the 2n(i degree (Quadratic Eqiiation) in one variable, can be reduced to an equivalent equation of the form ax^ + bx + c = (1). Either or both of the coefficients 6 and c may vanish ; but we cannot (except as a limiting case, which we shall consider presently) suppose « = without reducing the degree of the equation. By the general proposition of chap, xii., § 23, when a, b, c are given, two values of x and no more can be found which shall make the function a^ + hx + c vanish ; that is, the equation (1) lias always two roots and no more. The roots may be equal or unequal, real or imaginary, according to circumstances. The general theory of the solution of quadratic equations is thus to a large extent already in our hands. It happens, however, that the formal solution of a quadratic equation is always obtainable ; so that we can verify the general proposition by actually finding the roots as closed functions of the coefficients a, b, c. § 2.] We consider first the following particular cases : — I. c = 0. The equation (1) reduces to aaf + bx = 0, CHAP, xvn EOOTS EQUAL AND OF OPPOSITE SIGN 397 that is, since a + 0, axl X + -\ = 0, which is equivalent to x = 0~ a Hence the roots are x = 0, x= - b/a. II. 6 = 0, c = 0. The equation (1) now reduces to ax X x=0, which, since a =1= 0, is equivalent to {:::}■ Hence the roots are a; = 0, a; = 0. This might also be deduced from I. Here the roots are equal. We might of course say that there is only one root, but it is more convenient, in order to maintain the generality of the proposition regarding the number of the roots of an integral equation of the mth degree in one variable, to say that there are two equal roots. III. 6=0. The equation (1) reduces to ax^ + c = 0, that is, since a =# 0, to "(-y-i)(^v-3="- which is equivalent to x + ■-s/--a = ' Hence the roots are a; = - v/( - cja), x= + sl{- c/a) ; that is, the roots are equal, but of opposite sign. If c/a be negative. 398 GENERAL CASE CHAP. both roots will be real ; if cja be positive, both roots will be imaginary, and we may write them in the more appropriate form x= -i \/{c/a), z= +i \/{c/a). § 3.] The general case, where all the three coefficients are different from zero, may be treated in various ways ; but a little examination will show the student that all the methods amount to reducing the equation ax' + bx + c = (1) to an equivalent form, a(x + A)" + /t = 0, which is treated like the particular case III. of last paragraph. 1st Meflwd. — The most direct method is to take advantage of the identity of chap, vii., § 5. We have «af + 5a; + c E 2a j hence the equation (1) is equivalent to b- Jjh'-iac)'] 2a -b+ x/(y-4ac) 1 f -b- s/{V- 4ae) 2fli that is, to 7 ■b+ v/(5 {'- 2a ■ 4ac) 2a ■b- ^l{V - 4ac) y. 2a = and The roots of (1) are therefore {-b+ \/(i^ - 4ac)}/2a, {-&- s/{b'-4:ac)}/2a. 2nd Method. — We may also adopt the ordinary process of " completing the square.'' We may write (1) in the equivalent form X +-x = a (2), and render the lef<>hand side of (2) a complete square by adding (b/2ay to both sides. We thus deduce the equivalent equation / b\' b' c 4«^ a' 4»' (3). xvn VARIOUS METHODS OF SOLUTION 399 The equation (3) is obviously equivalent to 6 //V-iac\ 'W - iac x+ • ■ 2a V V y, id' from which we deduce a; = { - 6 + s/{b' - 4ac) }/2a, x={-b- ^/(5= - 4ac) } /2a, as before. 3rd Method. — By changing the variable, we can always make (1) depend on an equation of the form a/ + a! = 0. Let us assume that x = z + h, where h is entirely at our disposal, and z is to be determined by means of the derived equation. Then, by (1), we have a{z + hy + b{g + h) + c = (4). It is obvious that this equation is equivalent to (1), provided x be determined in terms of z by the equation x = z + h. Now (4) may be written «/ + {2ah + l)z + (aA' + JA + c) = (5). Since h is at our disposal, we may so determine it that 2ah + & = ; that is, we may put h= - b/2a. The equation (5) then becomes as' + «( - ^ ) +b{ - 7^ ) + c = 0, i-Y^'-^-i) „ V - Aac that is, ffl/ j^ = (6). From (6) we deduce z= + n/(5^ - 4fflc)/2a, z= - \/(&' - 4ac)/2a. Hence, since x = z + h= - b/2a + z, we have x = {-b+ ^/{b' - iac)}l2a, x = {-b- V(J' - 4ac)}/2fli, as before. In solving any particular equation the student may either quote the forms {-b± s/{b^ - 4«c)}/2a, which give the roots in all cases, and substitute the values which a, b, c happen to have in the particular case, or he may work through the process of 400 DISCEIMINATION OF THE ROOTS chap. the 2nd method in the particular case. The latter alternative will often be found the more conducive to accuracy. § 4.] In distinguishing the various cases that may arise when the coefficients a, h, c are real rational numbers, we have merely to repeat the discussion of chap, vii., § 7, on the nature of the factors of an integral quadratic function. "We thus see that the roots of ax' + bx + c= 0, (1) Will be real and unequal if b^ - iac be positive. (2) Will be real and equal if b' - iac = 0. (3) Will be two conjugate complex numbers if b^ - iac be negative. The appropriate expressions in this case are {-b + i ^{iac - V)}IU, {-b-i x/(4ac - b'}}/2a. (4) The roots will be rational if 5" - 4f(C be positive and the square of a rational number. (5) The roots will be conjugate surds of the form A ± \/B in the case where &" - 4ac is positive, but not the square of a rational number. (6) If the coefficients a, b, c be rational functions of any given quantities j), q, r, s, . . . then the roots will or will not be rational functions of p, q, r, s, . . . according as b^ - iac is or is not the square of a rational function of p, q, r, s, . . . It should be noticed that the conditions given as characterising the above cases are not only sufficient but also necessary. The cases where a, b, c are either irrational real numbers, or complex numbers of the general form a + a'i, are not of sufficient importance to require discussion here. Example 1. 2x=-3x=0. By inspection we see that tlie roots are x=f), x=3/2. Example 2. 2a:2 + 8 = 0. This equation is equivalent to x^ + 4 = 0, whose roots are a!=2i, x= -2i. Example 3. 35a;2-2a:-l = 0. The equation is equivalent to „ 2 _ 1 XVII EXAMPLES, EXERCISES XXVIII 401 that is, to L_Ly=± + l-^. \ Z5) 352*35" 352 Hence 1 ^^-35^ -4, Hence X 1±6 35 ■ The roots are, therefore, + ■1/5 and - -1/7. Example 4. x"- 2jc-2 = 0. The roots are l + sJZ and l-VS- Example 5. 3a;= + 24a; + 48 = 0. The given equation is equivalent to 352 + 8a; + 16 = 0, that is, to '(a + 4)2=0. Hence a;= — 4db0; that is to say, the two roots are each equal to —4, Example 6. iK2-4a;+7 = 0. This is equivalent to x^-ix + i=-Z, that is, to (x-2f=Zi\ Hence the roots are 2 + \/Si, 2 - \/Zi. Example 7. x'~2[p + qfx + '2p^ + \2fq^ + 2 + ab + b']x + ab{2a + b) = 0. (30.) {c + a-2b)x^ + {a + b-2c)x + (b + c-2a) = 0. (31.) {a'-ax + c^){a^ + ax + c'')==a* + w'c' + e'. (32.) a:2-2(a2 + J2 + c2)a; + aHJ' + c* + J2c2 + cV + a262=2aic(a + 6 + c). (33.) (b-c){x-aY + (c~a){x-b)^ + {a-b){x-cf=0. (34. ) Evaluate V(7 + V{7 + \/(7 + \/(7 ■ • ■ ad « . . . )))). EQUATIONS WHOSE SOLUTION CAN BE EFFECTED BY MEANS OF QUADRATIC EQUATIONS. § 5.] Reduction hy Factorisation. — If we know one root of an integral equation /(«=) = (1), say a; = a, then, by the remainder theorem, we know that f(x) = {x - a)4>{x), where <^(x) is lower in degree by one than f(x). Hence (1) is equivalent to (a;-a = 0) I (x) = Oj (2). The solution of (1) now depends on the solution of <^{x) = 0. It may happen that <^{x) = is a quadratic equation, in which case it may be solved as usual ; or, if not, we may be able to reduce the equation <^{x) = by guessing another root ; and so on. Example 1. To fiud tlie cube roots of - 1. Let X be any cube root of - 1, then, by the definition of a cube root, we must have o?= - 1. We have therefore to solve the equation We know one root of this equation, namely, a;= - 1 ; the equation, in fact, is equivalent to {x + l)(x'-x + V) = 0, that is, to ( „ '^ + 1 = I The quadratic a;--3; + l = 0, solved as usual, gives a; = (l±t"y3)/2. XVII INTEGRALISATION AND RATIONALISATION 403 Hence the three cube roots of -1 are -1, {l + i\/i)j2, (\-i\/Z)l2, which agrees with the result already obtained in chap. xii. by means of Demoivre's Theorem. Example 2. 7a!3-13a;2+3!K + 3 = 0. This equation is obviously satisfied by x=l. Hence it is equivalent to (7a;2-6a!-3)(a;-l) = 0. The roots of the quadratic 7a;^ - 6a: - 3 = are (3 ± VSO)/?. Hence the three roots of the original cubic are 1, (3 + VSO)/?, (3 - V30)/7. It may happen that we are able by some artifice to throw an integral equation into the form PQE . . . = 0, where P, Q, R, . . . are all integral functions of x of the 2nd degree. The roots of the equation in question are then found by solving the quadratics P = 0, Q=0, E = 0, ... Example 3. p(aa2 + l)x + cf - q{dx' + ex +f Y = 0. This equation is obviously equivalent to {■s/p(ax'' + l3X + e) + s/q(dx^ + ex+f)} {\Jp(a3? + hx + c) - sjqidx^ + ex+f)} =0. Hence its roots are the four roots of the two quadratics {a\/p + d\Jq)x'^ + (h\/p + es/q) x + {fi-sjp +f\/q) = 0, {a\/p - d's/q)x^ + (i\/p - e\/q)x + (c\/p -f\/q) = 0, which can be solved in the usual way. § 6.] Integralisation and Rationalisation. — We have seen in chap. xiv. that every algebraical equation can be reduced to an integral equation, which will be satisfied by all the finite roots of the given equation, but some of whose roots may happen to be extraneous to the given equation. The student should recur to the principles of chap, xiv., and work out the full solutions of as many of the exercises of that chapter as he can. In the exer- cises that follow in the present chapter particular attention should be paid to the distinction between solutions which are and solutions which are not extraneous to the given equation. The following additional examples will serve to illustrate the point just alluded to, and to exemplify some of the artifices that are used in the reduction of equations having special peculiarities. 404 EXAJIPLES CHAP. Example 1. 1 1 1 1 x+a+b x-a+b x+a-b x-a-b~ If we combine the first and last terms, and also the two middle terms, we derive the equivalent equation 2.r 2x . x'-[a + bf x:-'-(a-bf If we now multiply by {x^ - (a + 6)^} {x^ -(a- 6)^) we deduce the equation 2r<2,t°-2(a= + &2)}=0; and it may be that we introduce extraneous solutions, since the multiplier used is a function of x: The equation last derived is equivalent to f x=0\ Hence the roots of the last derived equation are 0, + \/{a" + 6^), - \/{(i? + 6^). Now, the roots, if any, introduced by the factor {x- -{a + bf] {x'^ -{a- b)'} must be ± (a + 6) or ± (a - 6). Hence none of the three roots obtained from the last derived equation are, in the present case, extraneous. Example 2. a-x a+x _ . , > \/a+~^{a~x) V« + \/(« + »-')~ If we rationalise the denominators on the left, we have («-a;){\/a-V(ffl-a:)} ^ (a + x){'^a- \/{a + x)} ^ , .V -X ^ ^"'' From (/3), after multiplying both sides by x, and transposing all the terms that are rational in x, we obtain {a + x)^-{a-x)^ = 3x\/a (7). From (7), by squaring and transposing, we deduce 2a'-3aa;2=2(ffi2_a;2)3 (5). From (5), by squaring and transposing, we have finally the integral equation {ix^-Sa')x'* = (e). The roots of (e) are {repeated four times, but that does not concern us so far as the original irrational equation * (a) is concerned) and ±a\/3/2. It is at once obvious that a;= is a root of (a). If we observe that ^/{l±\j3/2) = {\/5±l)/2, we see that ±a\/3/2 are roots of (a), provided that is, provided 2h=\/3 2it:V3 2:i:l + V3 2±1 + V3 ' 2- V3 , 2 + V3 _ 1 + V3"^3 + V3~ ' which is not true. Hence the only root of (a) is a; = 0. * For we have established no theory regarding the number of the roots of an irrational equation as such. XVII EXAMPLES 405 Example 3. \/{a + x) _ V(g-a;) 's/a + \/{a + x) \Ja- \/{a-x) ^"•'' By a process almost identical with that followed in last example, we deduce from (a) the equation 4a;*-3aV=0 (;8). The roots of (/3) are 0, and ±a\/3/2 ; but it will be found that none of these satisfy the original equation (a). Example 4. s/(2x^ - 4a; + 1) + \/(x^ - 5a; + 2) = V(2k^ - 2ii; + 3) + \/{x^ - 3a; + 4) (a). The given equation is equivalent to \/{2x^-ix + l)-s/{x'-3x + i) = ^{2x''-2x + S)-'s/{x^-5x + 2). From this last, by squaring, we deduce 3x3 - 7x + 5 - 2V(2x2 - 4x + 1) (x^ - 3a; + 4) = 3x2 - 7x + 5 - 2\/(2x2 - 2x + 3) (x2 - 5x + 2), which is equivalent to v'(2x*-10x3 + 21x2-19x + 4) = v'(2a;*-12x3 + 17x2-19x + 6) C;8). From (;8), by squaring and transposing and rejecting the factor 2, we deduce x' + 2a;2-l = (y). One root of (7) Is x= - 1, and (7) is equivalent to (x + l)(x2+x-l) = 0. Hence the roots of (7) are - 1 and ( - 1±\/5)/2. Now x= - 1 obviously satisfies (a). We can show that the other two roots of (7) are extraneous to (a) ; for, if x have either of the values {-l±\/5)j2, then x' + x-l = 0, therefore x^=-x + l. Using this value of x\ we reduce (a) to '\/{-Sx + 3)=\/{- ix + 5). This last equation involves the truth of the equation - 6x + 3 = - 4x + 5, which is satisfied by x = - 1, and not by either of the values x={- l±^5)/2. N.B. — An interesting point in this example is the way the terms of (a) are disposed before we square for the first time. Example 5. l-V(l-x°) ^ V(l + x) + V(l-x) l + V(l-a') W{l+x)-'s/(l-x) ^"■>- Multiply the numerator and denominator on the left by 1 - ^y(^ - x^), and the numerator and denominator on the right by \/{l+x) - \/(l -x), and we ob- tain the equivalent equation (l-\/r^)° _ X x2 ~ 'l-V(l-!c')' Multiply both sides of the last equation by x^(l - V 1 - x'), and we deduce {l-V(l-^^)}^=27x3 (p). 406 EXERCISES XXIX, XXX CHAP. If 1, w, w^ (see chap, xii., § 20) be the three cube roots of +1, then (|3) is equivalent to |l-V(l-a;=) = 3u.a y (7). {l-\/{l-x^) = 3ui^xj By rationalisation we deduce from (7) the three integral equations (l + 9uV-6ra;=0 J- (5). The roots of these equations (5) are 0, 3/5 ; 0, 6u/(l + 9w=) ; 0, 6w7(l + 9w)- The student will have no difiBculty in settling which of them satisfy the original equation (a). Exercises XXIX. (I.) a;6-l = (a;= + f)2{x2_i). (2.) x<-{a + b + c)x^-(a^ + b'' + c'-bc-ca~ab)x + a^ + P+c'-3abc=0. (3.) K^-40a; + 39 = 0. (4.) x!*+2{a-2)x' + {a-2fx' + 2a''{a-2)x + a*=0. (5.) 2a^-a;--2a;-8 = 0. (6.) aa? + x + a + l = 0. (7.) x^-Sx' + ix--3x + l = 0. (8.) ?.%-+^=l?+^+y. p p'' p' X'' x (9.) a;i-6a? + 10ri_8^ + lg^0. (10.) sd^-6 = 5x{x'-x-l). (11.) {3fi + 6x + 9){x'' + 8x + 16) = {x'^ + ix + i){x''-12x + 36). Exercises XXX. (2.) " +-i.i, 0+x a+x 2x^ - a; - 1 2.r--3a;-8 _ 8a:--8 * ' " a:-2 "^ a^^^S ~ 2a; - 3 ' 9.>; + 5 4n--2_ 12a: + 3 4a; + 3 11 ^ ' "T2~^7.!;-l~~l6 73^+9"*" 48' (5.).-3 = 5^^ c CB + S 2c c ^ ax + b ix + a_{a + b){x + 2) cx + b cx + a cx + a + b „ .x + a ..x + b x + c ax bx 2cx (»■) 4r— :+4r-i:-;;r-=;s-:r2+; ■6 x-c x?-a? ^-V 3?- a x—a x-b EXERCISES XXX, XXXI 407 „., a-c b-c _a + b-2c ^ '' W+x 2a + x~ a + b + x' x + a x-a x^ + a? (11.) ^^^ + — — =V^2 + (12.) ,.,„ , /«M-_(ra+a^Y_a \a^-ctx + x^J ~a (14.) (15.) x-a x + a x^-a^ x' + a^' {x - a){x - b) _{x - c){x - d) x-a-b x-c—d + 2x 2x' 4a^+4a;^+8a! + l_2a:° + 2a; + l 2x2 + 2a: + 3 ^ x + 1 X 7i 2a;-15 x' + ix-^h a;=+10a; + 21 . , 2x + Za 2x-Sa_a+b a-b ^ '' 2x-Sa'^2x + Sa~a^'^a + b' ^ (a;-6)(a;-c) {x-c){x-a) {x-a){x-b) ExEEcrsBS XXXI. (!•) x+\/x_x{x- -^/x (3.) X+7—7-, — ; — ^ 7-T2=2«- (4.) (a^ + 5a!)v'(«= + c^) = (a^ + 6c) V(«^ + x'). (5.) 6a;(K-l)-3v'{3(x-2)(a; + l)-2(K-5)}=4(a:+3). (6. ) \/{x + \Jx) + \J{x - 's/x) = asjxl-^{x + sjx). (7.) {\+x)^/(l-x') + (x-l) = 0. (8.) {x-■i)|^J{x^-ex + ^&) = {x-i)j^J{x^'-?,x + U). (9. ) (2a! - a)l's/(x'^ -asc + a^) = (2x - b)l\/{x^ -hx + V). <«yG4,)-{v/(i^)-x/c;;)}- (11.) V(»^+6x' + l)-V'(^^ + 6» + 4) + V{^ + 6a!-3) = 0. (12.) 's/{a''' + bx) + sJ{b'^ + ax) = Z{a + b). (13.) V{a(6a;-a2)/6} + \/{*(«^-**)/«} =<*-*• (14.) V(a+a!) + v'(J + a;) = 2V(a + 6 + a=)- Consider more especially the case where a=b. (15.) 's/[x + i:)-'s/(x-\) = \/{x-l). (16.) 2xsJ{x' + a^) + 2x^J{s? + V')=a^ - bK (17.) V(^' + ^'»+3)-V(»^ + 3a:+2) = 2(a;+l). Two solutions, x= -X and another. (18.) x' + a^ + \f{s^ + a'^) = 2x\/{x^+\J{s^ + a.yi. 408 CHANGE OF VARIABLE (19.) x=\J {ax + x^ -a\/{ax+x^)}. 7 12 (20.) ' ■ -^^ 7+- T + - ; + - r = 0. \/{x-Q) + i ' ^(a;-6) + 9 '\/{x~Q)-i'\/(x-%)- (21.) V(a^ + ^) + \/(:iax) = v'(a^ + Sos) + \/{*' + 3a«). (22.) ^ \ ^ = ''Hl \J(a + x)- \Ja ^Ja + sJ(a + x) \J{a + x)- \/{a-x)' (23.) ^a + '^{a + x)-sj{a-x)= ^{a^-x'). (24. ) msj{a+x) + nsj{a -x) = V(m^ + »') .^(a^ - x')- (25. ) Rationalise and solve 2\/{x -b-c) = \/x. (26.) V{(«' + a')(«^ + 2'')}+a;{\/(»' + «^)-\/(^^ + ''')}=«^^ + a''- (27.) a + {x + b)s/{{x'' + a')l{x' + ?f')}=b + ix + aW{{x' + b^)l{x^ + a^)}. § 7.] Reduction of Equations hy cliange of Variable. If we have an equation which, is reducible to the form {/(«:)r +!'{/(*)} + ?=0 (a), then, if we jDut ^ =f{x), we have the quadratic equation to determine ^. Solving {(S), we obtain for ^ the values { -p ± s/(/ - 4j)}/2. Hence (a) is equivalent to (r)- tf the function /(a) be of the 1st or 2nd degree in x, the equations (y) can be solved at once ; and all the roots obtained will be roots of (a). Even when the equations (y) are not, as they stand, linear or quadratic equations, it may happen that they are reducible to such, or that solutions can in some way be obtained, and thus one or more solutions will be found for the original equation (a). In practice it is unnecessary to actually introduce the auxiliary variable ^. We should simply speak of (a) as a quadratic in f(x), and proceed to solve for f(x) accordingly. Example 1. K2^'« + 4a;J''?-12 = 0. We may write this equation in the form (xi'^if + iixKl") -12 = Q. -p + V(/ - iq) 2 -p - ^J{p' - iq) 2 XVII EXAMPLES 409 It may therefore be regarded as a ctnadratio equation in a;*'*. Solving, we find xPl). The original equation is therefore equivalent to the two quadratics x'^ + {2a + Zb)x + a? + 3ab + b''=±-J{¥ + c''). § 8.] Beciprocal Equations. — A very imporbant class of equa- tions of the 4th degree (biquadratics) can be reduced to quadratics by the method we are now illustrating. Consider the equations ax* + b3? + cx' + bz + a=0 (1), ax* + bx^ + cx^ -bx + a=0 (I.), where the coeflBcients equidistant from the ends are either equal, or, in the case of the second and fourth coefficients, equal or numerically equal with opposite signs. Such equations are called reciprocal* If we divide by of, we reduce (1) and (I.) to the forms a[x + + c = a\x + (2), (II.) These are equivalent to ■I) 1 a\x + - + c-2a = + b{x 2a = + b{x + - XJ XJ \ XJ 3 and III. are quadratics in x+l/x and x- l/x respectively. If their roots be a, fi, and y, S respectively, then (3) is equivalent to (3), (III.) X + - = a X X + -=B X Y; * If in equation (1) we write 1/^ for x, we get an equation which is equiva- lent to a^ + b^' + cp + b^ + a = 0. Hence, if J be any root of (1), 1/f is also a root. In other words, two of the four roots of (1) are the reciprocals of the remaining two. In like manner it may be shown that two of the roots of (I. ) are the reciprocals of the remaining two with the sign changed. XVII GENERALISED EECIPEOCAL BIQUADRATIC 411 that is, to ("af- CUE + 1 = 0) ,, [x'-l3x+l=o\ ^ ^" Similarly, III. is equivalent to The four roots of the two quadratics (4) or (IV.) are the roots of the biquadratic (1) or (I.) Generalisation of the Beciprocal Equation. — If we treat the general biquadratic ax* + hof + CO? + dx + e = in the same way as we treated equations (1) and (I.), we reduce it to the form Now, if eja = cf /J^, this last equation may be written which is a quadratic in a; + djhx. Cor. It should he noticed that the following reciprocal equations of the 5 th degree can be reduced to reciprocal biquadratics, and can therefore be solved by means of quadratics, namely, ax^ + bx* + CO? ± caf ± bx ± a = 0, where, in the ambiguities, the upper signs go together and the lower signs together. For the above may be written a(x' ± 1) + bxix" ± 1) + cx\x ± 1) =0, from which it appears that either x+ I orcc-lis a factor on the left-hand side. After this factor is removed, the equation becomes a reciprocal biquadratic, which may be solved in the manner already explained. The roots of the quintic are either + 1 or - 1, and the four roots of this biquadratic. 412 EXAMPLES CHAP. In an appendix to this volume is given a discussion of the general solution of the cubic and biquadratic, and of the cases where they can be solved by means of quadratics. Example 1. To find the fifth roots of +1. Let a; be any fifth root of +1 ; then x' = l. Hence we have to solve the equation a^-l = 0. This is equivalent to / a;-l = 0\ \x!' + s^ + x^ + x + l = OJ' The latter equation is a reciprocal biquadratic, and may be written After solving this equation for x + l/x, we find 1 1 + V5 1 1-V5 X 2 X 2 These give the two quadratics These again give the following four values for x : — - (1 + V5)/4±i-V(10 -2V5)M, - (1 - ^/5)|i±^^/(10 + 2^/5)|i, these, together with 1, are the five fifth roots of + 1. This will be found to agree with the result obtained by using chap, xii., § 19. Example 2. {x + aY + {x + hY=17{a-by. This equation may be written {x + a)* + ix + bf = T.7{ix + a)-{x + b)}\ from which, by dividing by {x + by, we deduce \x+bj Ix+b J or ?Hl = 17(f- •1)^ where ^={x + a )l{x + b). This equation in ^ is reciprocal, and may be written thus — H D'-?(<^l yi'- Hence <^H or »|4 From this last pair we deduce i=% or I ; and f= k^—^- XVII EATIONALISATION BY MEANS OF AUXILIARY VAKIABLES 413 Hence we have the four equations a; + a _o x + a_. x + a _7 ±i\/{15) x+b~ ' x+h~^' x+i 8 ■ From these, four values of x can at once be deduced. The real values are x=a-2b and x=b- 2a. § 9.] By introducing auxiliary variables, we can always make any irrational equation in one variable depend on a system of RATIONAL equations in one or more variables. For example, if we have \/(a; + a) + sj{x + J) + sl(x + c) = d, and we put u = + 29ai2 + 42a; + 8 = 0. (12.) aar* + Ja:2 + 6K + a=0. (13.) ax' + bx^-bx-a=0. (14.) aa^ + 6a^ + c=0. (15.) aK* + 6a^-te-a=0. (16.) rt=.-c* + 2a6x5 + 6W-c2 = 0. (17.) a;5 + l = 0. (18.) a^ + 7!K* + 9r'-9a^-7a;-l = 0. (19.) 12a^ + x^ + 13a:S-13a:=-a;-12 = 0. (20.) Show that the biquadratic ax* + b3? + cx'' + dx + e=0 can be solved by means of quadratics, prorided 'bj'2a=iadl(iac-'b'^). (21.) a;» + 10a^' + 22ari-15a; + 2 = 0. (22. ) ai^ + 1(p- qW + {f + sV + ^P^iP - i)^ +PQ{P^ +PQ + , x' + y^=i21i, is equivalent to y = l?,-x, ^3 + (18 -a)' = 491 4. The second of these two last equations reduces, as it happens, to x^- 18a; + 17 = 0. Hence the finite solutions of the given system are x=n, 1 ; y=l, 17. § 12.] A very important class of equations are the so-called Ilomogenems Systems. The kind that most commonly occurs is that in which each equation consists of a homogeneous function of the variables equated to a constant. The artifice usually em- ployed for solving such equations is to introduce as auxiliary variables the ratios of all but one of the variables to that one. Thus, for example, if the variables were x and y, we should put y = vx, and then treat v and x as the new variables. XVH HOMOGENEOUS SYSTEMS EXAMPLES 419 Example 1. x^+xy=12, xy-iif = l. Put y='vx, and the two equations become x\l + v) = l-i, a:>-2i;2) = l. From these two we derive x\l + v)-\1x^[v-1i?) = (), that is, a;= {241)2 -Ilt> + 1}=0. Since a;=0 evidently affords no solution of the given system, we see that the original system is equivalent to a;2(l + ii) = 12, 24i)2-lli; + l = 0. Solving the quadratic for v, we find 'i)=l/3 or 1/8. Corresponding to i)=l/3, the first of the last pair of equations gives a:^ = 9, that is, x= ±3. Corresponding to 'y=l/8, we find in like manner x= ±4V(2/3). Hence, bearing in mind that y is derived from the corresponding value of x by using the corresponding value of v in the equation y=vx, we have, for the complete set of solutions, x=+2,, -3, +4V(2/3), -iVCS/S); 2/=+l, -1, H-l/VS, -1/V6. Example 2. x^ + 'iyz = l, y'^ + izx = m, z- + 2xy=n. Let x=uz, y=vz, then the equations become (v? + 2v)z^ = l, (v^ + iu)z^='m,, (l + iuv)z^=n. Eliminating z, we have, since s=0 forms in general no part of any solution, We have already seen how to treat this pair of equations (see § 11, Example 2). The system has in general four different solutions, which can be obtained by solving a biquadratic equation (reducible to quadratics when 71 = 1). If we take any one of these solutions, the equation {l + 1uv)z^=n gives two values of z. The relations x=uz, y=i)z, then give one value of x and one value of y corresponding to each of the two values of z. We thus obtain all the eight solutions of the given system. There is another class of equations in the solution of which the artifice just exemplified is sometimes successful, namely, that in which each equation consists of a homogeneous function of the variables equated to another homogeneous function of the variables of the same or of different degree. Example 3. The system cKc'^ + bxy + cy''=dx + ey, a'x^ + b'xy + c'y^ = d'x + e'y (1) is equivalent to {a + bv + cv^)x'=:{d + ev)x, (a' + li'v + cV)x'' = {d' + e'v)x (2) where y=vx. 420 EXAMPLES CHAP. From this last system we derive the system ar {[a + 6t) + CT-) (d' + c'v) - {a' + b'v + c'v'){d + ev)\ =0\ ,„, x{{a + bv + cv-)x-{d + ev)} =0 f ^ '' which is equivalent (see chap, xiv., § 11) to (2), along with (a + bv + cv')a? = (4), {d + cv)x = (5). If we observe that x=0, y = is a solution of the system (1), and keep account of it separately, and observe further that values of v which satisfy both (4) and (5) do not in general exist, we see that the system (1) is equivalent to (a + bv + cv') (d' + e'v) - (a! + b'v + c'v^) (d + cv) = (6) along with (a + bv + cv^)x-(d + ev) = (l (7) and s = 0, y=0. The solution of the given system now depends on the cubic (6). The three roots of this cubic substituted in (7) give us three values of x, and y=vx gives three corresponding solutions of (1). Thus, counting x = 0, y = 0, we have obtained all the four solutions of (1). The cubic (6) will not be reducible to quadratics except in particular cases, as, for example, when ad' - a'd = or ce' - c'c = 0. For example, the system 3x'^ -2xy + 3y^=x + 12y, ej~ + 3xy-2y-=2x + 2Sy, is equivalent to x=0, y = 0, together with ?)(111d2-86i) + 8) = 0, {S - 2v + Sv-)x=1 + Uv. The values of v are 2/3, 4/37, and 0. Hence the solutions of the system are x = Q, 3, 185/227, 1/3; 2/ = 0, 2, 20/227, 0. § 13.] Symmetrical Systems. — A system of equations is said to be symmetrical when the interchange of any pair of the variables derives from the given system an identical system. For example, x + y = a, x^ + if = h ; x^ + y = a, y^ ■¥ x = a; x + y + z = a, x' + y^ + z^ = 1), yz + zx + xy = c, are all symmetrical systems. There is a peculiarity in the solutions of such systems, which can be foreseen from their nature. Let us suppose in the first place that the system is such that it would in general have an even number of solutions, four say. If we take half the solutions, say 2/ = /3i, ft, then, since the equations are still satisfied when the values of x and y are interchanged, the remaining half of the solutions are x = ft, /?„ y = a,, a^. XVII SYMMETRICAL SYSTEMS 421 If the whole number of solutions were odd, five say, then four of the solutions would be arranged as above, and the fifth (if finite, which in many cases it would not be) must be such that the values of x and y are equal ; otherwise the interchanges of the two would produce a sixth solution, which is inadmissible, if the system have only five solutions.* These considerations suggest two methods of solving such equations. 1st Method. — Eeplace the variables by a new system of vari- ables, consisting of one, say x, of the former, and the ratios to it of the others, u, v, . . . say. Eliminate x, v, . . . and obtain an equation in u alone ; then this equation will be a reciprocal equation ; for the values of u are <^l Pl f^2 H2 / ^ • 1 T \ M= „-, — , -jj, —, &c. (and, it may be, m= 1), Pi "i Pz "-2 that is to say, along with each root there is another, which is its reciprocal. The degree of this resultant equation can therefore in all cases be reduced by adjoining a certain quadratic, just as in the case of a reciprocal biquadratic. ind Method. — Eeplace the variables x, y, z, . . . by an equal number of symmetric functions of x, y, g, . . ., say by 2a;, Xxy, "Szyn, . . . , &c., and solve for these. The nature of the method, its details, and the reason of its success, will be best understood by taking the case of two variables, x and y. Let us put u = x + y, v = xy. After separating the solutions, if any, for which x = y, we may replace the given system by a system each equation of which is symmetrical. "We know, by the general theory of symmetric functions (see chap, xviii., § 4), that every integral symmetric function can be expressed as an * We have supposed that for all the solutions (except one in the case of an odd system) »#=!/. It may, however, happen that x=y for one or more solutions. Such solutions cannot be paired with others, since an interchange of values does not produce a new solution. This peculiarity must always arise in systems which are symmetrical as a whole, hut not symmetrical in the individual equations. As an example, we may take the symmetrical system a? + y=a, ■tf + x=a, three of whose solutions are such that x = y. 422 SYMMETRICAL SYSTEMS CHAP. integral function of u and v. Hence it will always be possible to transform the given system into an equivalent system in u and v. "We observe further that, in general, u and v will each have as many values as there are solutions of the given system, and no more ; but that the values of u and v corresponding to two solutions, such as x = a,, y = P„ and a; = /3n y = <^\, are equal. Hence in the case of symmetrical equations the number of solu- tions of the system in u and v must in general be less than usual. Corresponding to any particular values of u and v, say u = a, V- fS, we have the quadratic system x + y = a, xy = fi, which gives the two solutions a; = {a ± v/(a= - 4/3)}/2, y = {a t ^{a - 4/3)}/2. If we had a system in three variables, x, y, z, then we should assume u = x + y + z, v = yz + zx + xy, w = xyz, and attempt to solve the system in u, v, w. Let u = a, v = (3, w = y, he any solution of this system ; then, since we see that the three roots of constitute a solution of the original system, and, since the equations are symmetrical, any one of the six permutations of these roots is also a solution. In this case, therefore, the number of solutions of the system in u, v, w would, in general, be less than the corresponding number for the system in o; y, z. The student should study the following examples in the light of these general remarks : — Example 1. A (a;^ + 2/^) + Ba:j/ + C (a; + t/) + D = \ . If we put y='m, and then eliminate x by the method employed in § 11, the resultant equation in v is {(D'A) + (D'B> + (D'A)^)=j==(D'C)(l-l-y)={(C'A) + (C'B)i. + (C'AK} (2), where (D'A) stands for D'A - DA', (D'B) for D'B - DB', and so on. The biquadratic (2) is obviously reciprocal, and can therefore be solved by means of quadratics. The solution can then be completed by means of the equation { (D'A) + (D'B)i; + (D'Ay }a; + (D'C) (1 + v) = (3). xvii EXAMPLES 423 As an instance of this metliod the student should worlc out in full the solution of the system 2{x^+jf) -3xy + 2{x + y) -S9 = 0, 3(x^ + y^)-ixy+ {x + y)-50 = 0. Wo may treat the above example by the second method of the present paragraph as follows. The system (1) may be written A{x + yf + {B -2A)xy + C{x + y) + 'D =0, A'{x + yf + (B' - 1k:)xy + Q,'{x + «/) + D' = ; that is A»=^+(B-2A)t, + C»,+D=01 ^ '^'^' AV + (B'-2A')?J + C'it + D' = o/ ^*'' Eliminating first \C^ and then i>, we deduce the equivalent system (A'B)1J + (A'C)M + (A'D) = 01 (A'BK+{(C'B)-2(G'A)}m+{(D'B)-2(D'A)}=0/ * '■ where (A'B), &c., have the same meaning as above. The system (5) has two solutions, say, corresponding to which we find for the original system a;={a±V(a^-i(3)}/2, {a'± V(a'=- 4^')}/2, ,y={a=FV'C«'-4ffl}/2, {a'T\/(«"- */3')}/2, in all four solutions. This method should be tested on the numerical example given above. Example 2. x* + 2/*=82, x + y = i. We have sii^ + y''={x+y)*- ixy{x^ + y^) - 6a;y , = {x+y)'^-4xy{{x + y)^-2xy} -6x-y-, = 14*- iu^v + 2i^. Hence the given system is equivalent to li* - iuH + 21)2 = 82, „ _ 4, Using the value of u given by the second equation, we reduce the first to d2-32« + 87 = 0. The roots of this quadratic are 3 and 29. Hence the solution of the u, v system is v,=i, 4, v=3, 29. From x + y=i, xy=29, we derive (x-yf= -100, that is, x~y=±10i; combining this with x + y=i, we have x=2±5i, y=2^5i. From x + y=i, xy=S, we find x=S, y = l ; x=J, y=3. All the four solutions have thus been found. Examples. x:^ = mx + ny, y*=nx + my (!)• Let us put y=vx; then, removing the factor x in both equations, and noting the corresponding solution, x = 0, y=0, we have These are equivalent to a?=m + nv, «*(m + rei))=mu+« (2). 424 EXAMPLES chap. The second of these may be ^^Titten »(j;5-l) + mr(^-s-l) = (3), and is therefore equivalent to 11-1=0 s The second of these is a reciprocal biquadratic. Hence all the five roots of (3) can be found without solving any equation of higher degree than the 2nd. To the root v = l correspond the three solutions, of the original system, where (ro + ?i)"^ is the real value of the cube root, and w, u" are the imaginary cube roots of unity. In like manner three solutions of (1) are obtained for each of the remain- ing four roots of (3). Hence, counting x = 0, y=0, we obtain all the sixteen ■solutions of (1). The reader should work out the details of the numerical case !i^ = 2x + 3y, y'^=3x + 2y, and calculate all the real roots, and all the coefficients in the complex roots, to one or two places of decimals. Example 4. yz + zx + xy = 26, y:(y + :) + zx{z + x) + xy{x + y) = 162, 2/:(7/2 + .2) + 2a;(22 + x^) + xy{x- + j/^) = 538. H we put u = x + y + z, v = ys + zx + xy, w = xyz, the above system reduces to i) = 26, ■ uv-iw = \62, {w'-2v)v-uw = 53S. Hence 26M-3it' = 162, 26u^-tiw = 1890. Hence 26^2 + 81m- 2835 = 0. The roots of this quadratic are u=9 and zt= - 315/26. We thus obtain for the values of u, v, w, 9, 26, 24, and - 315/26, 26, - 159. Hence we have tlie two cubics |3_9|2 + 26f-24 = 0, f' + ^?H26| + 159 = 0. Twelve of the roots of the original system consi.st of the six permutations of the three roots of the first cubic, together with the six permutations of the roots of the second cubic. The first cubic evidently has the root f =2 ; and the other two are easily fouud to be 3 and 4. Hence we have the following six solutions : — x = 2, 2, 3, 3, 4, 4; 2/=3, 4, 4, 2, 2, 3; z = i, 3, 2, 4, S, 2. Other six are to be found by solving the second cubic. § 14.] We conclude this chapter with a few miscellaneous e.xamples of artifices that are suggested merely by the peculi- XVII MISCELLANEOUS EXAMPLES 425 arities of the particular case. Some of them have a somewhat more general character, as the student will find in working the exercises in set xxxiv. A moderate amount of practice in solv- ing puzzles of this description is useful as a means of cultivating manipulative skill ; but he should beware of wasting his time over what is after all merely a chapter of accidents. Example 1. ax hy (a + h)o a+x o+y a+b+o Let a + a;=(ffl + 5 + c)|, i + y = (a + l + c)7) ; the system then reduces to a'/^ + b^lr, = {a + bf, 1 + 77 = 1. This again is equivalent to {(a + 6)|-a}==0, J + i)=l. Hence we have the solution ^=aj(a + b), ■q = bl(a + b) twice over. The solutions of the original system are therefore x=acl(a + b), y= bc/ia + b) twice over. Example 2. ax^ + bxy + cy'' = bx^+cxy + ay''=d (1). This system is equivalent to (a-b]x^ + {b-c)xy + {c-a)y'' = (2), ax'' + bxy + cy^=d (3). The equation (2) (see chap, xvi., § 9) is equivalent to x^ = (ca- + l)p, xy=(a(r + l]p, y^ = {b(T + l)p (4), where p and cr are undetermined. Since x^y- = (xyY, we must have (c(T+\)(ba + l) = (.a - j/^) = I, x'lp!' - y^/q^ = 0. a + 3 y x-S y+3 3_, x-3 y-3 ^ 2)/ + 3 6xy=16. 2,r + 3 2(x -y) + xy = 3.ry -{x-y) = 7. {x + y)l7 = 8/[x + y + l), xy = 12 x+lly = '\Ojx, y + l/x=10x. 3{3? + y'^)-2xy = 27, i{x' + y'-)- a^-2/5=208, x-y = L x'y + xy''=162, 9:? + y^ = 2i3. x"y + xi/ = 30, K^i/2 + a^2/^ = 468. x' + y' = {a + b){x-y), x- + xy + y-=a x^ + x-y'' + y* = 7il, x^-xy + y- = 19. xy{x + y) = i8, a^ + y^=72. x* + y* = 97, x + y = 5. x> + y*={p^ + 2)x^y'', -y'^ = 2xy + x + y, x + i (31.) a^ + y^=a*, x + y = b. (34.) x^ + y^=33, x + y=3. i=a. y^ = Sxy{x + y). ix + y){x'^-y^) = 819, (k-j/) (a?' + i/) = 399. x-/y + y^lx=2, x + y=5. x"y-{3^ - y^) = «■*, xy{x' + y^){x'-y-) = a- x*-3? + y^-y'' = 8i, x'^ + x''y'^ + y'^ = i^i. x'^ly = a^ - xy, y^/x =b--xy. x + y + \/{xy) = li, x'' + y^ + xy=8i. ^/(l- a/a;) + \/(l-«/3/) = V(l + «/*)> x + y = b. x + y+\/{x--y') = a, 2y^{x?-y^) = b\ V(a;- + 12j/) + v'(2/' + 12;!;) = 33, x + y = 23. ^{x/y) + sjivlx) = 5/2, ^{x^ly) + ^{y^x) = 9 V2/2. V(a; + a)+\/{y-a) = i^Ja, V(^ -a) + \J{y-a)= f V"- x* + / = a*, {x' + y'')^ + (2xyf = b. o" + n"- + y'^ + b- = ^2{x{a + ij)-b(a-y)}, x^-a-- 5/ + b'^=sJ2\ x{a -y) + b{a + y)]. [x- + a-) (t/2 + 6^) = m{xy + abf, (x' - a?) {y" - i^) =n{bx- ay)'', a a a h b b y + ::rr r.T zrr ■ ■ ■■ y=^+: y+ y+ y + .r'» I/" = (!)'""'', x"y'^- X'+n = yia^ yX+li—^_ x+ x+ x + =(*)" EXERCISES XXXIV 429 Exercises XXXIV. *(1.) Sa!=0, 2ax = 0, Za'3i^=SIl{b-c). (2.) {y-a){z-a) = bc, (z-h){x-l) = ca, {x-c){y-c)=ab. (3.) 2/2 + 2(2/ + 2) = 11, 2x + 2(2 + a;) = 8, an/ + 2(K + 2/) = 16. (*•) Z + - + - = ^' yz + zii: + xy=^xyz, 2zx + 3yz=2xu. j^ y z (5.) a;(2/ + 2) = 24, y{z + x) = li, z[x + y) = W. (6.) x[y + z)=y{z+x)=z{x+y) = l. (7.) (3+a;)(a!+2/)=(i2, {x + y)[y + z) = li^, {y + z){z + x)=c'^. (8.) »» + 2/2 = 32/ + 23: = £82 + 0:^=^2. (9.) a;2 + 22/2=128, 2/^ + 22a:=153, 22^.2an/=128. (10.) a2(2, + 2)8=aV + l, l\z + xf=lY- + l, c\x + yf=cH^ + l. (11.) a(2/ + 2-K) = (a! + 2/ + 2)2-262/, J(» + ^ - 2/) = (« + 2/ + 2)'" - 2c2, c{x+y - z) = (x + y + z)^ - lax. (12.) 2(a:?-2/s)-2(x2-2/2)=a^ 2(a;2 - 2/2) - 2(2/^ - 2x) = i^, S(a:^ - 2/2) - 2(s^ - a;2/) = (?. (13.) SJot=0, Sa2/2=0, 2x^=1. (14.) a(a; + 2/2) = 6(i/ + 2a;)=c(2+a;2/), 0^ + 2/^ + 2^ + 2x2/2=1. (15.) x(a + 2/ + s)=2/(a + 2 + a;)=2(a + x + 2/) = 3a(x + 2/ + 2). (16.) x2 + 2/^ + 2-=a2 + 2x(2/ + 2)-x^, and the two equations derived from this one by interchanging -j ■, \- (17.) ax2=- + i, V = ---, C22=- + -- (18.) 2/^22 + 2V + x22/2=49, x2 + j,2^_j,2^i4_ x(2/ + 2)=9. (19.) (yz-o?)la'h:={zx-y'^)lhHi={xy-z^)l(^z=llxyz. (20.) a^^(2/+2)^=(a^ + x^)2/V, and the two derived therefrom by inter- changing {^^^}. (21.) Sx3=ffl(Sx-2x)=S(Sx-22/) = c(Sx-22). (22.) (x-l)(2/ + 2-5) = 77, (2/-2)(s + x-4) = 72, (2- 3)(3; + 2/- 3) = 65. (23.) u{y-x)j{z-u) = a, z(y-x)/{z-u) = b, y{u-z)l{x-y)=c, x{u - z)l{x -y) = d. (24.) If !t? + 2/3 + j3 + 6a;i/2 = a, ^y'^z + z'x + xhj) = b, Z{yz' + z3? + xy^)=c, show that x+y + z=(a + l + c) , x + ojy + ofiz = {a + ab + u^c) , x + 0)^2/ + u2 = (a + w^6 + wc) , where u^ + c , x + y + z^O, - + T + - = 0, Ax-p) = -{y-q) = -(z-r). (6. ) x,y, z from SAjr=0, SA'2;'- = 0, Zax = 0, and show that the result Is 21/ {b-(Gk') + c=(AB') - «-{BC') } = 0, where (CA') = CA' - C'A, ha. (7.) Show that the following system of equations in x, y, z are inconsistent unless r^ - p^ = 3r j^, and that they have an infinite number of solutions if this condition be fulfilled. 2a^ - 3xyz =jfi, ^yz= q^, 2a; = r. Eliminate (8. ) X and y from {a-x)(a-y)=p, (b-x)(h-y) = q, (a-x)(b-y)l(b-x){a-y) = c. (9.) X, y, s from x + y-z=a, x^ + j/^ - s^ = 6', si? + y^-!? = c', xyz=a?. (10.) X, y, z from ax + ijz = bc, by + zx = ca, cz + xy = ab, xyz=abc. (11.) X, y, z from Xx^=p^, 2.1^ = y'', 2x' = 7^, xyz=s^. (12.) a', y, z from {x + a)(y + b)(z + c) = abc, (y-c)(z-b)=o?, (z-a)(x-c) = P, (x-b)[y-a)=cK (13.) The system XxX2 + yiyi=ki^, Xnsuz-Vyiyi^'ki, . . ., x„Xi + y„yi = kr?, xi' + y-? =x?->ryi=. .= x^ + y^ = a-, either has no solution, or it has an infinite number of solutions. * The eliminant is in all cases to be a rational integral equation. CHAPTEE XVIII. General Theory of Integral Functions, more particularly of Quadratic Functions. RELATIONS BETWEI;N THE COEFFICIENTS OE A FUNCTION AND ITS ROOTS SYMMETRICAL FUNCTIONS OF THE ROOTS. § 1.] By the remainder theorem (chap, v., § 15), it follows that if a,, Oj,, . . ., Oa be the n roots of the integral function ^oX™+^,a;"-i+p2x"-2 + . . .+Pn-,x+pn (1), that is to say, the m values of x for which its value becomes 0, then we have the identity ^jX^+j^ja;""^ +^2X""^ + . . ■+Pn =p„{x - a,){x - a,) ... (a; - a„) (2). Now we have (see chap, iv., § 10) (x - a,){x -a,)., .{x- an) = x" - P^a;"-! + P^a;'*-^ -... + (- 1)™P„, where Pi, Pj, . . ., P^ denote the sums of the products of the n quantities a,, a^, . . ., u„, taken 1, 2, . . ., » at a time re- spectively. Hence, if we divide both sides of (2) byj?,,, we have the identity Po I>o Po = a;''-P,a;"-i + P2a;»-2-. . . + ( - 1)"P„ (3). Since (3) is an identity, we must have pJPo = - Pu P2/P. = p., • • , Pn/Po = ( - 1)''P» (4). In particular, ii Po= 1, so that we have the function a:"+^ia;"-^+^2a!"-^+ . .+Pn (5), then i',= -P„ P2 = 'P,, ■ ■ ■, Pn = {-irYn (6). 432 SYMMETRIC FUNCTIONS OF TWO VARIABLES CH.vr. Hence, if tve consider the roots of the function X^ + p^X"'-^ + p^'^-^ + . . . +Pn-i^+Pn, or, what comes to the same thimj, the roots of the equation then -pi is the sum of the n roots ; p^ the sum of all the products of the roots, taken two at a time; -p<, the sum, of all the products, taken three at a time, and so on. Thus, if u, and j3 be the roots of the quadratic function ax^ + bx + c, that is, the values of x which satisfy the quadratic equation cui' + hx + c = 0, then a + /S = - hja, a/3 = cja (7). Again, if a, /3, y be the roots of the cubic function ax^ + Ix" + ex + d, then a + yS + y = - hja, /3y + ya + afS = c/a, a/3y = - d/a (8). § 2.] If s,, Sj, S3, . . ., s^ stand fm- the sums of the 1st, 2nd, 3rd, . . . , rth powers of the roots a and (3 of the quadratic equation a?+p^x+p„ = (1), we can express s^, «2, . . ., s^ as integral functions of p^ and p^. In the first place, we have, by § 1 (6), s, = a + l3= -p, (2). Again S, = a' + 13' = (a + I3y - 2a/3, =p--2p, (3). To find S3 we may proceed as follows. Since a and f3 are roots of (1), we have a +p,a+ps = 0, /T +p,/3 +p2=0 (4). Multiplying these equations by u, and /? respectively, and adding, we obtain Ss+PA+pA = (5). Since s^ and s^ are integral functions of Pi and p^, (5) determines S3 as an integral function oi p, and ^2. We have, in fact, «3= -pXp'-'2l'2)+P2Pi, = -P'+ Sp,p, (6). xviri SYMMETEIC FUNCTIONS OF TWO VARIABLES 433 Similarly, multiplying the equations (4) by a^ and f^ respectively, and adding, we deduce S4+i?,S3+^A = (7). Hence S4 may be expressed as an integral function of p^ and p^, and so on. IVe can now express any symmetric integral function whatever of the roots of the quadratic (1) as an integral function of p^ and p^. Since any symmetric integral function is a sum of sym- metrical integral homogeneous functions, it is sufficient to prove this proposition for a homogeneous symmetric integral function of the roots a and jB. The most general such function of the 7'th degree may be written A(a'- + P^) + Ba/3(a'-2 + /?^-2) + Ca'/^^a'-* + /S*-*) + . . . , that is to say, ASr - Bpg S^_2 + Cp/ Sr-i + . . . (8), where A, B, C are coeflS.cients independent of «, and ji. Hence the proposition follows at once, for we have already shown that s^, s^-s, s^^^, . . . can all be expressed as integral functions of ^1 andjSg. It is important to notice that, since a and /3 may be any two quantities whatsoever, the result just arrived at is really a general proposition regarding any integral symmetric function of two variables, namely, that any symmetric integral function of two variables a, /3 can he expressed as a rational integral function of the two elementary symmetrical functions p^= - (a + /3) and p^ = afi. There are two important remarks to be made regarding this expression. 1st. If all the coefficients of the given integral symmetric function he integers, then all the coefficients in the expression for it in terms of j?i and p, will also he integers. This is at once obvious if we remark that at every step in the successive calculation of s^, s^, S3, . . ., &c., we substitute directly integral values previously obtained, so that the only possibility of introducing fractions would be through the co- efficients A, B, C, ... in (8). VOL. I 2 F 431 EXPRESSION IN TERMS OF 2^ AND p., chap. 2nd. Since all the equations above written become identities, homogeneous throughout, when for j3, and j)o we substitute their values - (a + ^) and a/3 respectively; and since 2h is of the 1st and ^2 of the 2nd degree in a and p, it follows that in every term of any function of p^ and p^ which represents the value of a homogeneous symmetric function, the sum of the suffixes * of the p's must be equal to the degree of the symmetric function in a and /?. Thus, for example, in the expression (6) for S3 the sum of the suffixes in the term -^^i^ that is, -p^p^Pi, is 3 ; and in the term ?>Pip.^ also 3. This last remark is important, because it enables us to write down at once all the terms that can possibly occur in the ex- pression for any given homogeneous symmetric function. All we have to do is to write down every product of j», and p.,, or of powers of these, in which the sum of the suffixes is equal to the degree of the given function. Example 1. To calculate a'' + ^ in terms of^i and jl)2. This is a homogeneous symmetric function of the 4th degree. Hence, by the rule just stated, we must have a-* + /3' = A^/ + 'Spi-p-2 + C'p-?, where A, B, C are coefficients to be determined. In the first place, let j3=0, so that pi= -a, p2 = 0- We must then have the identity a''=Aa^ Hence A = l. We now have ai + /3* = (a + /3)4 + B(a + j3)=a/3 + Ca2/3=. Observing that the term a^jS does not occur on the left, we see that 1) must have the value - 4. Lastly, putting a= -(3 = 1, so that^i = 0, ^2= - 1, we see that C = 2. Hence a'+|S'=;V-4i'l'?'2 + 2p2^. The same result might also be obtained as follows. We have Si+PlS3+2}2S2 = 0- Hence, using the values of S2 and S3 already calculated, we have Si = -Pii -Pi^ + Spi}).) -ihil'i- - 2iJ2), =Pi^ - ipi'l>-2 + ipi'- Example 2. Calculate a' + /3' + o'/3- + a^/3"' in ti'rnis of pi and^)o. * This is called the weight of the symmetric function. See Salmon's Higher Algebra, § 56. XVIII NEWTON S THEOREM 435 We have a? + j35 + a?f- + a?^^ = Api" + B^Ji^jjo + Opipi. Putting ^ = 0, we find A= - 1 ; considermg the term a% we see that B = 5; and, putting a=/3 = l, we find C= - 6. Hence Since any alternating integral function * of a, /?, say /(a, /3), merely changes its sign when a and /S are interchanged, it follows that we have /(a, j3) = -f{(i, a). Hence, if we put p = a, we have /(a, a) = -/(a, a); that is, 2/(a, a) = 0. Therefore /(a, a) = 0. It follows from the remainder theorem that /(a, ;S) is exactly divisible by a - /3. Let the quotient be g(a, p). Then g{a, /?) is a symmetric function of a, jB. For g{a, jS) =f(a, /S)/'(a - /?), and ^(A a) =/(A «)/(^ - a) = -/(a, /S)/(;8 - a) =/(a,^)/(a - fi); that is, g'(a, /?) = g(fi, a). Hence any alternating integral function of a and (3 can be expressed as the product of a - /? and some symmetric function of a and yS. Hence any alternating function of a and /3 can be expressed without difficulty as the product of ± -JiPi^ - ipa), omd an integral function of p^ and p^. Example 3. To express a'''^ - a^^ in terms of ^i and ^2. We have = (a-/3)a,3(a + ^)(a' + n = ±\/(j!'i'^ - ^P-i){PiVi{p{'- - '^Pifi- Every symmetric rational function of a and /? can be ex- pressed as the quotient of two integral symmetric functions of a and /?, and can therefore be expressed as a rational function of ^1 and p.^. Example 4. ~1>jPi P-i § 3.] The general proposition established for symmetric functions of two variables can be extended without difficulty to symmetric functions of any number of variables. * See p. 77, footnote. 436 NEWTON S THEOEEM We shall first prove, in its most general form, Newton's Theorem that the sums of the integral poweis of the roots of any integral equation, X^+p^X^-'^ +2).X'^-^+ . . .+J7„=0 (1), can be expi-essed as integral functions of Pi, Ps, . ■ ., Pni whose co- efficients are all integral numbers. Let the n roots of (1) be a,, a„, . . ., a^, and let the equation whose roots are the same as those of (1), with the exception of u.,, be a;"-i + ,p,x"-2 + ,;?„a;"-3 + . . . + ,^„_i = (2); also let the equation whose roots are the same as those of (1), with the exception of Oj, be a:"-i + 2^7,a;™-2 + ,p^x''-^ + . . . + ^jJ^-, = (3), and so on. Then a;""^ + ,^,a;"~^ + ,;?2a;"~' + . . . + ,^„_, = (a;" + p^x"' ~ 1 + p^a;" ~ - + . . + p,^l{x - a,), = x'^''^ + {a^+p^)x^~'^ + {ai + p^a^+p^X^'^ . . . H-(a/+^,ai'-i + . . .+^r)a;"-'-i + . . . by chap, v., § 13. Hence, equating coefficients, we have ,p, = a, +p„ \ iP2 = a.i +P,«-i +lh. ,Pr = o.! +p,ai' .+Pr, ,Pn-, = <-^+P.<-^- + Pn- (3'). sPn-i Similar values can be obtained for „pi, ,p^, ^p^, . in terms of a^ and^,, jjg, . . ., Pn', a-nd so on. Taking the (r - l)th equation in the system (3'), and multi- plying by tti, we have Similarly and so on. XVIII Newton's theoeem 437 Adding the n equations thus obtained, we have ,Pr-iai +2i'r-i<»2 + - • ■+nPr-i when expressed in terms of a,, a,, . ., u.„, will introduce the power ai^i+^=+' ' ■ + ^" with the coeflScient {- iy\Pi\p2^^ ■ ■ ■ ,Pn-i^- Now, since there are no terms of higher degiee than p^^^p^^ . . . pj^", if the power ai'^i + ^2+- ■ •+^" occur again, it must occur as the highest power, resulting from a different term of the same degree ; that is to say, it will occur with a different coefficient and cannot destroy the former term. Hence the index of the highest power of any letter in the symmetric function must be equal to the degree of the highest term in its expression in terms of PoP^, ■ ■ ■, PnA Although, in establishing the leading theorem of this para- graph, we have used the language of the theory of equations, the result is really a fundamental principle in the calculus of algebraical identities ; and it is for this reason that we have introduced it here. "We may state the result as follows : — Let us call 2»,, 2x,:?'2, '^-XiX^s, . . ., XiytvXj . . . a:„ the n elementary symmetric functions of the system of n variables Xi,:?;,, . . ., a;„. Then we can express any symmetric integral function of X,, x^, . . ., Xn as an integral function of the n elementary symmetric functions ; and therefore any rational symmetric function of these variables as a rational function of the n elementary symmetric functions. On account of its great importance we give a proof of this * They are, in fact, the functions of oo , 03 , . . . , u„ defined in § 3. See Exercises xxxvi., 51. + Salmon, Higher Algcbni, § 58. XVlil PROOF OF GENERAL THEOREM 441 proposition not depending on Newton's Theorem (which is itself merely a particular case).* Let „ji, „(/2, . . ., nin denote the n elementary symmetric functions of the n variables x^, x.^, . . ., .t„, that is to say, 2/^iCij *^.,j^ii^3, . . ., X^Xq . . . X^f and let n-iflu n-i^'sj • ■ •? n-i^n-i denote the m — 1 elementary symmetric functions of k,, x^, . . ., a;^i, that is, 2^,, 2 a;,a!„, . . ., XiX^. . . x^i^^. It is obvious that, n-l n-1 when Xn = 0, „y,, ,,5-3, ., ^q^-i become „_,2,, n-i n1n-i) is a symmetric integral function of a;,, .Tg, . . ., «„, ^{x^, x^_, . . ., a;,; _,,«„) is obviously a symmetric integral function of these variables. If we put «„ = on both sides of the identity (7), then /(•"^ij ^2) • • -J *«-ij 0) = („_,q„ „_,g„ . . ., „_,2„_,)- Hence, by (8), ^(zy, x^, . . ., Xn-i, 0) = 0. Therefore the integral function \p{Xi, x^, . . ., Xn-i, x^) is exactly divisible by some * This proof is taken from a paper by Mr. R. E. AUardice, Proc. Edinb. Math. Soc. for 1889. 442 PKOOF OF GENERAL THEOREM iHAr, power of .)■„, say .c," ; hence, on account of its symmetry, also by ^i" ^^■, ■! ■''m-i"- ^^ ni^y therefore put where /, is a symmetric integral function of .t,, ;i\,, . ., x,^ of lower degree than /. We can now deal with /^ in the same way as wo dealt with /; and so on. We shall thus resolve /(''i, :'%, ■ • ■, ■'-'n-i, !>',i) into a closed expression of the form where i, ■nSii ■ ■ ■> nin wUl cilso be integral numbers. We now give a few examples of the calculation of symmetric functions in terms of the elementary functions, and of the use of this transformation in establishing identities and in elimination. Example 1. If a, (3, 7 be the roots of the equation express /3'7 + /S-y' + yV + 7a' + a^/3 + aj3^ in terms oi pi, j)-^, p^. Here we have^i = Sa, ^j2=Sa/3, ^3=a/37. Remembering that no term of higher degree than the 3rd can occur in the value of 2a'/3, we see that xviu EXAMPLES 443 Sa3^ = Ai7i2p2 + BpiiJ3 + Crf (1), where A, B, C are numbers which we have to determine. Suppose 7 = ; thenpi = a + (3, ^2 = a/3, ^3=0 ; and (1) becomes a3;8 + a/33=A(a + /3)V + C!a''^2 5 that is to say, a^+'^^=A(a + ^f + Ca^. Hence A = l, C=-2. We now have 2a^j3=^i^^2 + Bpi^s - 2^2^- Let a = ;8=7=l, so that^i = 3, P2=S, ps = l. We then have 6 = 27 + 3B-18. Hence B=-l. Therefore, finally, Sa5^=^i2^2 - PiPs - 2^3". In other words, we have the identity 2a3j3=(Sa)2Sa|8 - a^yXa - 2(Sa(3)2. Example 2. To show that {yz-xu){zx- yu) {xy - zu) = {yzu + zux + vscy + xyzf - xyzu(x + y + z + uf (2). The left-hand side of (2) is a symmetric function of x, y, z, u. Let us calcu- late its value in terms of pi=2a:, pi = '2iXy, p3='Sxyz, pi=xyzu. Since the degree of Xi(yz - xu) in x, y, z, u is 6, and the degree in x alone is 3, we have Ti(yz -xu)= Api'pi + Bpipips + Cp^^ + Dp^Vi + Eps^ (3). If we put «s=0, then ^1 = 233;, p2=23CCT/, ps=xyz, Pi=0, where the suiEx 3 under the S means that only three variables, x, y, z, are to be considered. If Pit Pit Pa have for the moment these meanings, then (3) becomes the identity p^ ='BpiPiPi + Cpi + 'E.p^. Hence B = 0, C = 0, E=l. Hence THiyz - xu)= A-p-^pi + 'DpiPi+ pi' (4). Now let a;=2/=l, and s=m= - 1, so that ^i=0, ^2= -2, ^3=0> i'4=l- Then (4) becomes 0=-2D. Hence D = 0. We now have 'n.{yz-xu)=Ap^pi+p^. In this put x=y=z = u=l, and we have = 16A + 16. Hence A= - 1. Hence, finally, 'n.(yz-xu)=p^-p-?pi, which establishes the identity (2). Example 3. lix + y + z=(i, show that k" + 2/" 4- s^ ' _ 3? + y^ + >? 7? + 'f + ^ _ (a?+y^+^f x- + y''+z ^ 11 ~ 3 2 9 2 (^WolstenJiolme.) (5). 444 EXAMPLES chap. Ify)i = -,'■, Pa = 2.17/, ps — yijz, So = 2.1", S3=2.r'', &c., then we are required to prove that .'11 S3S8 S^Si ,,,. We know that Su is a rational function of ^1, p^, 2h- In the present case ^1 = 0, and we need only write down those terms which do not contain pi. We thus have Sn=A.P'iY + ^Pm^ (6), provided x + y + z — 0. A may be most simply determined by putting z= -(x + ij), writing out both sides of (6) as functions of x and y, dividing by xy, and comparing the coefBcients of x^. We thus find A = ll. We have therefore Si_i=np2^P3 + '&p«pi. In this last equation we may give ;>■, y, z any values consistent with .i' + 2/ + c = 0, say a:=2, )/= - 1, ;= - 1. We thus get B= - 11. Hence Sn=np.2%-llp2pi^ (7). In like manner we have Ss = Ap2' + 'Bpnp3\ Putting in this equation first x=l, y= -1, z=0, and then x = 2, y= -1, z= -1, we find A = 2, B= -8. Hence Se = 2p2* - Sp^ps" (8). We also find 83 = 8^33 (9), S2=-2p2 (10). From (8), (9), and (10) we deduce _Sn ~ii' which is the required equation. Since we have four equations, (7), (8), (9), (10), and only two quantities, P2, Ps, to eliminate, we can of course obtain an infinity of different relations, such as (5) ; all these will, however, be equivalent to two independent equa- tions, say to (5), and 72S8=9S2''+4S2S3= (11). Example 4. Eliminate x, y, z from the equations x + y + z=(i, 3? + 'i^ + z^=a, x' + y^ + ^ = b, x' + y'' + z' = c. Using the same notation as in last example, we can show that S3 = Zp3, Si=~5p2P3, ST=7pi'p3- Our elimination problem is therefore reduced to the following : — To eliminate p^ and ps from the equations 3/'3 = a, - 5p2P3 = i, Iv^Vi = C" This can be done at once. The result is 21&2-25ac=0. XVIII EXERCISES XXXVI 445 Exercises XXXVI. o and /3 being the roots of the equation x^+px + q=0, express the follow- ing in terms of ^ and q : — (1.) a= + /35. (2.)(a'' + ^6)/(a-/3)2. (3.) a-= + ;8-=. (4.)a-=-^-=. (5. ) (a^ + /33)-i + {a?- ^3)-i. (6. ) (1 - af^- + (1 - /3)'a=. (7. ) If the sum of the roots of a quadratic be A, and the sum of their cubes B', find the equation. (8.) If s„ denote the sum of the »th powers of the roots of a quadratic, then the equation is (S„S„_2 - S„-i^)x^ - (S„+1 S„_2 - S„S„-i)x + {Sn+lS„-i - sj) = 0. (9.) If a and /3 be the roots of x^+px + q=0, find the equation whose roots are (a - Kf, ((3 - A)'. (10.) Prove that the roots of !(y'-{2p- q)x +p'-pq + q'^=0 are p + (iiq, p + 01% lo and w^ being the imaginary cube roots of 1. (11.) If a, /3 be the roots of x'^ + x + l, prove that a» + )3"=2, or = -1, according as n is or is not a multiple of 3. (12.) Find the condition that the roots of ax^ + tx + c=0 may be deduc- ible from those of aV + h'x + c' = by adding the same quantity to each root. (13.) If the difi'erences between the roots of x''+px + q = and x^ + qx+p = be the same, show that either p=q ov p + q + i = 0. What peculiarity is there when p=q% Calculate the following functions of a, ^, y in terms of ^i=Sa, ^2=Sa;8, ps=apy:— (14.) a^l^y + ^/ya + y^lap. (15.) a-« + ;8-5 + 7-=. (16.) (/32 + 7^)(7^ + a=)(a2 + /32). (17.) X{a^ + I3y)l{a^ - ^). (18.) Z((3-7)2. (19.) X{a-mP-y?- (20.) W + jf- Calculate the following functions of a, ^, 7, 5 in terms of the elementary symmetric functions : — (21.) Sa*. (22.) Xa-'. (23.) 2^'^'^. (24.) 2o=/37. (25.) S(a + ;8)^ (26. ) If a, /3, 7, S be the roots of the biquadratic a^ +piii? -Vpi£' +p^ +p^ = 0, find the equation whose roots are |37 + a5, 7a + (35, 0^ + 78. (27.) If the roots of y?-pxx-vpi='^, a^ - ^jK + g'3 = 0, x^- na: + r2=0, be /3, 7 ; 7, a ; a, (3 respectively, then a, /3, 7 are the roots of ^'-\{'Pi + 1\ + '■1)^^ + (^2 + ?2 + ri)X - \{p\q\rx -P1P2 - giffa - nr^) = 0. (28.) If a, ft 7 be therootsof a^+pa; + j = 0, show that the equation whose roots are a + (37, § + ya, 7 + aft \s3i?-px^+{p + Zq)x + q-(p + qf=(). (29.) If a, /3, 7 be the roots of pl{a + x)+ql{h+x) + rl{c + x) = l, show that p = [a + a){a + §)(a + y)l{a -b){a-c). If a, p be the roots of a?+pix+p2=0, and a', ^' the roots of x^+piX+p^ -0, express the following in terms of^j, ^2, Pi, Pi •— ' (30.) (a'-a)(/3'-/3) + (a'-/3)(0'-a). 446 EXERCISES XXXVI CHAP. (31.) {a'-ar-+{p'-p)^ + {a'-pr-+{p'-ar. (32.) (ci + a')(^ + (3'){a + ^')(;8 + a'). (33.) 4(a-a')(a-/3')(^-a')(/3-/3'). [The result in this case is 4(^2-/2)' + i{Pi-Pi') {piP2 -P1P2) = (2^2 + 2p2' -pipif- (V- iPi) (K^- iPi)- ] (34.) A, A' and B, B' are four points on a straight line whose distances, from a fixed point on that line (right or left according as the algehraio values are positive or negative), are the roots of the equations ax^ + hx + c = 0, a'x"- + Vx + c'=(i. If AA'.BB' + AB'.BA' = 0, show that 2ca' + Ida -bb' = ; and if AA'.BA' + AB'.BB'=0, that 2ca'--2c'aa' + ab'--a'W = 0. (35.) a, /3 are the roots of x^-2ax + i''=0, and a', j3' the roots of X- -2cx + d'-=0. If aa,' + ^^' = 4?i', show that (a2-J2)((;2-(?-!) = (ac-2?l2)2. (36.) a, p, a', /3', being as in last exercise, form the equation whose roots are aa'+^/S', a^' + a'^. (37.) If the roots of ax- + bx + c=0 be the square roots of the roots of aV + b'x + c' = 0, show that a'b^ + a-b' = 2aa'c. (38.) Show that when two roots of a cubic arc equal, its roots can always be obtained by means of a quadratic equation. Exemplify by solving the equation 12a;'' - 56a;2 + 87a; - 45 = 0. (39.) If one of tlie roots of the cubic a?+pjx'+P'^+p3=0 be equal to the sum of the other two, solve the cubic. Show that in this case the coefificients must satisfy the relation p^-ipiPi, + ?>P3 = Q. (40.) If the square of one of the roots of the cubic x^+piU?+p2X+Pi=Q be equal to the product of the other two, show that one of the roots is -p^jpi ; and that the other two are given by the quadratic PiP-iX^ +Pi( Pi- -P2)x +pi-p3 = 0. As an example of this case, solve the cubic a^-9x-~63x + Zi3 = 0. (41.) If two roots of the biquadratic x!^+pix'+P'^''+2>3X+Pi=0 be equal, show that the repeated root is a common root of the two equations iaP + Spix'^ + 2p.2X +ps=0, ar* +pi3? +pi^^ +P3X +pi = 0. (42.) If the three variables x, y, z be connected by the relation 'Zx=xy«, show that S23;/(l - x^) = 112x1(1 - x-). (43.) If 2j; = 0, show that 2Xx' = 7 X7/-::L:cK (44.) If 2x = 0, show that Xafi =2{Xyz)* - SxhjVSyz. (45.) If 2(j = (three variables), then CZ(b'' -e'')/a^)C2a^j{b^-c')) = il6-i(Sa-'){:L,ir->). (46.) If S.r' = 0, ^x* — (three variables), show that Zx^ + xyz{{y-z){s-x) {x-y)}^ = 0. XVIII DISCRIMINATION OF THE ROOTS OF A QUADRATIC 447 (47.) lix + ij + z + u=0, show that (2aJ')2=-.9(2a;!/s)2 = 9n(2/s-ra). (48.) Under the hypothesis of last exercise, show that ux(u + xf + ys{u-xf + uy{u + yf + zx{u -yf + uz{u + zf + xy(u - zf + ixyzu = 0. Eliminate x, y, z between the equations <„,zg)=,, .g)-.„, 0. Both roots negative cla + , bla + . Roots imaginary . 62-4a(;<0. Roots of opposite Roots equal . b"-iac=0. signs . cla - . Roots equal with One root =0 c=0. opposite signs b = 0. Two roots = 6 = 0, c=0. Both roots positive c/a + , h/a - . One root =oo a = 0. Two roots = oo J = 0, a=0. § 6.] The reader should notice that some of the results em- bodied in the table of last paragraph can be easily generalised. Thus, for example, it can be readily shown that if in the equation PoX"' +PiX'^ +p„ = (1) the last r coefficients all vanish, then the equation mil have r zero roots ; and if the first r coefficients all vanish it will have r infinite roots. Again, if p,, = 0, the algebraic sum of the roots will be zero ; and so on. It is not difficult to find the condition that two roots of any equation be equal. We have only to express, by the methods already explained, the symmetric function n(a, - 02)^ of the roots in terms oi Po, Pi, ■ ■ -, Pn, and equate this to zero. For it is obvious that if the product of the squares of all the differences of the roots vanish, two roots at least must be equal, and con- versely. For example, in the efce of the cubic a?+Pix''+p2X+ps=0 (2), whose roots are a, /3, 7, we find ((3 - 7)2(7 - a)\a. - §f = - W^s +PiV + ISpifti's - ipi - 'I'lpi. The condition for equal roots is therefore - Wft + Pi W + 1 8piP2ft - W - SVpa^ = 0. Further, if all the roots of the cubic be real, ((3 - 7)^(7 - of [a. - (3)" will be positive, and if two of them be imaginary, say p = \ + )ii, y — X-fd, then (/3-7)'(7-«)'(a-(3)'=-V{(X-a)' + Ai'}'. that is, 03-7)2(7-0)^-/3)2 is negative. Hence the roots of (2) are real and unequal, such that two at least are equal, or such that two are imaginary, according as VOL. 1 2 G 450 TWO QUADRATICS, CONDITION OF EQUIVALENCE chap. - IpiVs +P-?P2^ + ISyiPaPs - ipi - 27p3^ is positive, zero, or negative. The further pursuit of this matter belongs to the higher theory of equa- tions. § 7.] ijf the two quadratic equations ax" + hx + c = 0, a!^ + 6'a; + c' = he equivalent, then hja = b'/a' and c/a = c'/a'. For, if the roots of each be a and /?, then b/a = - (rt + yS) = b'/a', c/a = a/3 = c'/a' ; and this condition is obviously sufficient. The above proposition leads to the following : A quadratic function of x is completely determined when its roots are given, and also the value of the function corresponding to any value of % which is not a root. This we may prove independently as follows. Let the roots of the function y be a and /8 ; then y = A{x - a) (x- /?). Now, if V be the value of y when x = X., say, then we must have V = A(A - a) (X - /3). This equation determines the value of A, and we have, finally, ^ {X-a){X-/3) ^'' The result thus arrived at is only a particular case of the following : An integral function of the nth degree is uniquely deter- mirwd when its n+\ values, V,, V,, ., Vn+u corresponding re- spectively to the n + I values A,, A^, ., A^+i of its variable x, are given. To prove this we may consider the case of a quadratic function. Let the required function be an? + bx + c ;- then, by the con- ditions of the problem, we have «A,^ + JA, + c = V,, aX.2 +bX^ + c = Y„, aAj'' + iAj + c = V,,. These constitute a linear system to determine the unknown coefficients a, b, c. This system cannot have more than one definite solution. Moreover, there is in general one definite solution, for we can construct synthetically a function to satisfy the required conditions, namely, xvni Lagrange's interpolation formula 451 Y {x-X,)(x-X,) (a; -A,) (a -A3) ( x-\,)(x~X,) ' (A. - A,) (A. - A3) ' (A, - A.) (A, - A3) ^ ' " (A3 - A,)(A3 - A,) (2). The reasoning and the synthesis are obviously general. "We obtain, as the solution of the corresponding problem for an in- tegral function of x of the nth. degree, (Ai - A2)(A, - A3) . . . (Ai - A„+,) This result is called Lagrange's Interpolation Formula. Example 1. Find the quadratic equation with real coefficients one of whose roots is 5 + 6i. Since the coefficients are real, the other root must be 5 - 6i. Hence the required equation is {x - 5 + 6?) {x - 5-6i) = 0, that is, {x-6f + e^ = 0, that is, a;2-10x + 61 = 0. Example 2. Find the quadratic equation with rational coefficients one of whose roots is 3 + V^. Since the coefficients are rational, * it follows that the other root must be 3 - \/7. Hence the equation is {x-3 + '^7){x-S-sJ7) = 0, that is, a''-6a' + 2 = 0. Example 3. Find the equation of lowest degree with rational coefficients one of whose roots is \/2 + \JS. By the principles of chap, x.* it follows that each of the quantities \/2-\/3, -\/2 + \/3,-\/2- \/3 must be a root of the required equation. Hence the equation is (b- V"2-\/3)(»-V2 + V3)(k + \/2- V3)(» + \/2 + \/3) = 0> that is, ar»-10a;2 + l = o. Example 4. Construct a. quadratic function of x whose values shall be i, 4, 6, when the values of x are 1, 2, 3 respectively. * This we have not explicitly proved ; but we can establish, by reasoning similar to that employed in chap, xii., § 5, Cor. 4, that, ita + b\Jp be a root of /(a) = 0, and if a and I and also all the coefficients of /(x) be rational so far as \Jp is concerned, then a - l\/p is also a root of/(a;) = 0. 152 TWO QUADKATICS, CONDITION FOR COMMON EOOT chap. The required function is { x-2)(x-Z) (...■-l)(.T-3) (.<;-l)(:r-2) (l-2)(l-3) (2-l)(2-3) (3-lj(3-2)' that is, x^ - 3,c + 6. § 8.] The condition that the two equations axf + Ja; + c = 0, aV + 5'a; + c' = have one root in common is the same as the condition that the two integral functions y = la? + hx + c, y' = aV + h'x + d shall have a linear factor in common. Now any common factor of y and y' is a common factor of dy - cy', and ay' - a'y ; that is, if we denote ac' - ale by (ac'), &c., a common factor of {ac')x' + (bc')x, and (a,b')x + (ac') ; that is, since x is not a common factor of y and y' unless c = and c' = 0, any common factor of y and y' is a common factor of iac')x + (be'), and (ab')x + (ac'). Now, if these two linear functions have a common factor of the 1st degree in x, the one must be the other multiplied by a constant factor. Hence tJie required condition is (ac') (he) (aV) {ac')' or {lie' - a'ef = {be - b'c) {aV - a'b). The common I'oot of the two equations is, of course, be' - b'c ac' - a'c ad-a'c aV - a'b' By the process here employed we could find the r conditions that two integral equations should have r roots in common. It is important to notice that the process used in the demon- stration is simply that for finding the G.C.M. of two integral functions — a process in which no irrational operations occur. Hence XVIII EXERCISES XXXVII 453 Cor. 1. If two integral equations have r roots in common, these roots are the roots of an integral equation of the rth degree, whose co- efficients are rational functions of the coefficients of the given equations. ' In particular, if the coefficients of the two equations be real rational numbers, the r common roots must be the roots of an equation of the j'th degree with rational coefficients. For example, two quadratics whose coefficients are all rational cannot have a single root in common unless it be a rational root. Cor 2. We may also infer that if two integral equations whose coefficients are rational have an odd number of roots in common, then one at least of these must be real. EXEKOISBS XXXVII. Discriminate the roots of the following quadratic equations without solving them : — (1.) ix^-8x + S = 0. (2.) 9a;2-12a;-l = 0. (3.) ix^-ix + 6 = Q. (4.) 9x2 -36k + 36 = 0. (5.) ix''-ix-3 = 0. (6.) ix^ + 8x + 3 = 0. (7.) (x-3){x + i) + {x-2){x+S) = 0. (8.) Show that the roots of {b''-iac)x'' + i{a + c)x- 4 = are always real ; and find the conditions — 1° that both be positive, 2° that they have opposite signs, 3° that they be both negative, 4° that they be equal, 5° that they be equal but of opposite sign. (9.) Show that the roots oix^ + 2{p + q)x + 2{p^ + q^) = are imagjnarj-. (10.) Show that the roots of {(?-2hc + i^}x''-2{l, and has an algebraically least value when x = l ; for (« - tf, the only variable part, being essentially positive, cannot be less than zero. When x is diminished beyond the value x = l,{x- Tf continually increases in numerical value. We conclude, therefore, that in all three cases the quadratic function y has an algebraical minimum or maximum value when x = l, according as a is positive or negative; and that the function has no other turning value. In Case L, where the roots are real and unequal, y will have the same sign as a or not, according as the value of % does not or does lie between the roots. For y = a{x- a){x - /S) ; and (x -a){x- /?) will be positive if X be algebraically greater than both a and /3, for then x- a and X- (3 are both positive ; and the same will be true if x be alge- braically less than both a and yS, for then x- a and x- (3 are both negative. If x lie between a and /?, then one of the two, x-a, X- /3, is positive and the other negative. In Cases II. and III, where the roots are either equal or imaginary, the function y mil have the same sign as a for all values of x. For in these cases the function vdthin the crooked brackets has clearly a positive value for all real values of x. § 10.] The above conclusions may be reached by a different but equally instructive method as follows : — ■ Let us trace the graph of the function y = ax' + bx + c (1); and, for the present, suppose a to be positive. 456 GRAPH OF A QUADRATIC FUNCTION To find the general character of the graph, let us inquire where it cuts a parallel to the axis of z, drawn at any given distance y from that axis. In other words, let us seek for the abscissae of all points on the graph whose ordinates are equal to y. We have y - Wi? -^ hx A- c, that is, ax" + bx + (c- y) = (2). We have, therefore, a quadratic equation to determine the abscissEe of points on the parallel. Hence the parallel cuts the graph in two real distinct points, in two coincident real points, or in no real point, according as the roots of (2) are real and unequal, real and equal, or imaginary. Since a is positive, it follows that when x= - co , y= +oo; and when x- + ao , y = +co. Moreover, the quadratic function y is continuous, and can only become infinite when x becomes infinite. Hence there must be one minimum turning point on the graph. There cannot be more than one, for, if there were, it would be possible to draw a parallel to the a;-axis to meet the graph in more than two points. The graph therefore consists of a single festoon, beginning and ending at an infinite distance above the axis of x. The main characteristic point to be determined is the minimum point. To obtain this we have only to diminish y until the parallel UV (Fig. 1) just ceases to meet the graph. At this stage it is obvious that the two points U and V run together ; that is to say, the two abscissa3 corresponding to y become equal. Hence, to find y, we have simply to express the condition that the roots of (2) be equal. This condition is 4:a(c -y) = 0. *' - 4cc y= — ^- 3)- 4a V Hence THKEE FUNDAMENTAL CASES 457 The corresponding value of x is easily obtained, if we notice that the sum of the roots of (2) is in all cases - bja, and that when the two are equal each must be equal to - b/2a. Hence the abscissa of the minimum point is given by 2a ^ ' There are obviously three possible cases — I. The value of y given by (3) may be negative. Since a is supposed positive, this will happen when b' - iac is positive. In this case the minimum point A will lie below the axis of T, and the graph will be like the fully drawn curve in Fig. 1. Here the graph must cut the x-axis, hence the function y must have two real and unequal roots, namely, x = OL, x = OM ; and it is obvious that y is positive or negative, that is, has the same sign as a or the opposite, according as x does not or does lie between OL and OM. II. The value of y given by (3) will be zero, provided 6' - 4«c = 0. will be In Fig. 3. In this case the minimum point A falls on the axis of x, and the ^ graph will be like the fully drawn curve in Fig. 2. Here the two roots of the function are equal, namely, each is equal to OA. It is obvious that here y is always positive, that is, has the same sign as a. III. The value of y given by (3) positive, provided b^ - 4:ac be negative. this case the graph will be like the fully drawn curve in Fig. 3. 458 EXAMPLES CHAP. Here the graph does not cut the axis of k, so that the function has no real roots. Also y is always positive, that is, has the same sign as a. If we suppose a to be negative, the discussion proceeds exactly as before, except that for positive we must say negative, and for minimum maximum. The typical graphs in the three cases will be obtained by taking the mirror-images in the axis of X of those already given. These graphs are indicated by dotted lines in Figs. 1, 2, 3. For simplicity we have supposed the abscissae of the points L, M, N, A to be positive in all cases. It will of course happen in certain cases that one or more of these are negative. The cor- responding figures are obtained in all cases simply by shifting the axis of y through a proper distance to the right. Example 1. To find for what values of a; the function y^ix?- 12a; -f 13 is negative, and to find its turning value. We have y = 2{x'-&x + 9)-5, = 2{{x-?,f-i], = 2{x-(^-^Ji)}{x-{S + ^Ji)]. Hence y will be negative if x lie between 3 - \/{5j2) and 3 + \/(5/2), and will be positive for all other values of x. Again, it is obvious, from the second form of the function, that y is algebraically least when (a- 3)^=0. Hence t/= - 5 is a minimum value of y corresponding to a: =3. Example 2. To find the turning values of (x" - 8a; 4- 15)/a;. 15 We have ii=x-\ 8. X First, suppose x to be positive, then we may write from which it appears that j/ has a minimum value, - 8 + 2^/l 6, when \Jx- \/(15/i») = 0, that is, when x=\Jl:j. Next, let X be negative, = - J say, then we may write 2/=-f-y-8, = -8-2V15-(vi- y^V U' xviii MAXIMA AND MINIMA 459 from which we see that - 8 - 2\/15 is a maximum value of y corresponding to {=\/16, that is, to x= - \/15. Example 3. If a; and y he hoth positive, then — li x + y be given, the greatest and least values of xy correspond to the least and greatest values of (x - y)' ; so that the maximum value of xy is obtained by putting x=y, if that be possible under the circumstances of the problem. If xy be given, the greatest and least values o{ x + y correspond to the greatest and least values of [x-yf ; so that the minimum of a; + ?/ is obtained by putting x=y, if that be possible under the circumstances of the problem. These statements follow at once from the identity {x + y)^ - (x-yf=4xy. ixy = a'-(x- y)^. For, ifx + y=c, then And, ifxy=d^, then (x-^y)- = id- + (x-yf. Hence the conclusions follow immediately, provided x and y, and therefore xy and x+y, be both positive. These results might also be arrived at by eliminating the value of y by means of the given relation. Thus, if x+y=c, then xy=x(c-x) = cx-x''' = (?li - (c/2 - x)^. Hence xy is made as large as possible by making x as nearly = c/2 as possible, and so on. Many important problems in geometry regarding maxima and minima may be treated by the simple method illustrated in Example 3. Example 4. To draw through a point A within a, circle a chord such that the sum of the squares of its segments shall be a maximum or a minimum. Let r be the radius of the circle, d the distance of A from the centre, x and y the lengths of the segments of the chords. Then, by a well-known geometrical proposition, xy=r^-d^ (1). Under this condition we have to make %i,=x'^ + y'' (2) a maximum or minimum. Now, if we denote x^ and y'^ by J and % then f and 17 are two positive quantities ; and, by (1), we have ^■n=[r'-d"-f (3). Hence, by Example 3, f + 17 is a minimum when =17, and is a maximum when (f - 1;)^ is made as great as possible. If we diminish 1;, it follows, by (3), that f increases. Hence (^--nf will be made as great as possible by making f as great as possible. 160 GEOMETRICAL MAXIMA AND MINIMA CHAP. Hence the sum of the squares on the segments of the chord is ia minimum when it is bisected, and a maximum wlien it passes through the centre of the circle. Example 5. A and B are two points on the diameter of a circle, PQ a chord through B. To find the positions of PQ for which the area APQ is a maximum or a minimum. Let bo the centre of the circle. The area OPQ bears to the area APQ the constant ratio OB : AB. Hence we haye merely to find the turning values of the area OPQ. Let OB=a, and let x denote the perpendicular from on PQ. Then, if M denote the area OPr>, 'U,=x\l{i^ -x-). We have therefore to find the turning values of u. Since M is positive, this is the same thing as finding the turning values of ir. Now V? = x'W'^ -x') = — - 4 There are two cases to consider. First, suppose a>rj\/2. Then, since the least and greatest values of x allowable under the circumstances are and a, we have, confining ourselves to half a revolution of the chord about A, three turning values. If we put a;=0 we give to [x^-r^jl)'' the greatest value which we can give it by diminishing x below r\\l2. Hence a; = gives a minimum value of OPQ. If we put x=rl\J2, we give (afi-r^ji)- its least possible numerical value. Hence, for x=rl\J1, OPQ is a maximum. If we put x=(i, we give (x^-r-jl)" the greatest value which we can give it by increasing x beyond rjsji. Hence to x=a corresponds a minimum value of OPQ. Next, suppose arl\J2. Hence, corresponding to x=0, we have, as before, OPQ a minimum. But now (a^ - r-/s/2)- diminishes continually as x increases up to a. Hence, for x=a, OPQ is a maximum. Remark. — This example has been chosen to illustrate a peculiarity that very often arises in practical questions regarding maxima and minima, namely, that all the theoretically possible values of the variable may not be admissible under the circumstances of the problem. E.'^ample 6. riiven the perimeter of a right-angled triangle, to show that the sum of the sides containing the right angle is greatest when the triangle is isosceles. Let X and y denote the two sides, p the given perimeter. Then the hypotenuse is p~x-y ; and we have, by the condition of the problem. (1). 0^- {x + y)}--- =x' + y Hence xy -p{x + y)z 2' This again may be written {p~ 2){p-y]-- P' 2 MAXIMA AND MINIMA, GENERAL METHOD 461 Under the condition (1) we have to make ii^x + y a maximum. If we put i=p-x, ■n=p-y, we have (2) (3); and we have to make u = 2p-(^ + -q) a maximum ; this is, to make i + ti a. minimum. Now, under the condition (3)i 1+'? is a minimum when i=-q. Hence x + y is a maximum when x=y. § 11.] The method employed in § 10 for finding the turning points of a quadratic function is merely an example of the Fig. 4. general method indicated in chap, xv., § 13. Consider any function whatever, say y=f(f) _ _ (1). Let A be a maximum turning point on its graph, whose abscissa and ordinate are x and y. If we draw a parallel to OX a little below A, it will intersect the graph in a certain number of points, TUVW say. Two of these will be in the neighbour- hood of A, left and right of AL. If we move the parallel up- wards until it pass through A, the two points U and V will run together at A, and their two abscissEe will become equal. If we move the parallel a little farther upwards, we lose two of the real intersections altogether. HeTice ■ to find y we Jiave simply to express the condition that the roots of the equation f(x)-y=0 (2) 462 EXAMPLES be equal, and then examine whether, if we increase y by a small amount, we lose two real roots or not. If we do, then y is a maximum value. If it appears that two real roots are lost, not by increasing but by diminishing y, then y is a minimum value. Example 1.. To find the turning values of The values of x corresponding to a given ordinate y are given by !c3-9a;H24a; + (3-2/) = 0. If D denote the product of the squares of the differences of the roots of this cubic, then all its roots will be real, two roots will be equal or two imaginary, according as D is positive, zero, or negative. Using the value of D calculated in § 6, and putting ^i= -9, ^2 = 24, P3 = ^-y, we find D=-27(y-19)(i/-23). Hence 3/= 19, 2/ = 23 are turning values of y. If we make y a little less than 19, D is negative, that is, two real roots of the cubic are lost. Hence 19 is a minimum value of y. If we make y a little greater than 23, D is again negative ; hence 23 is a maximum value of y. It is easy to obtain the corresponding val- ues of X, if we remember that two of the roots of the cubic become equal when there is a turning X value. In fact, if the two equal roots be a, a, Pio. 5. and the third root 7, we have, by § 1, 2a + 7 = 9, a2 + 2aY = 24. Hence a^- 6a + 8 = 0, which gives a = 2, oro = 4. ¥ia.i. It will be found that x=i corresponds to the minimum value !/ = 19 ; and that x=2 corresponds to the maximum i/=23. EXAMPLES 463 Remark. — The above method is obviously applicable to any cubic integral function whatsoever, and we see that such a function has in general two turning values, which are the roots of a certain quadratic equation easily ob- tainable by means of the function D. If the roots of this quadratic be real and unequal, there are two distinct turning points, one a maximum, the other a minimum. If the roots be equal, we have a point which may be regarded as an amalgamation of a maximum point with a minimum, which is sometimes called a maximum-minimum point. If the roots be imaginary, the function has no real turning point. If the coefficient of a? be positive, the graphs in the first two cases have the general characters shown in Figs. 5 and 6 respectively. Example 2. To discuss the turning values of y= — - — (1). The equation for the values of x corresponding to any given value of y is a;2-(2/ + 8)a;-l-15 = 0. Let D be the function V - Aac of § 5, whose sign discriminates the roots of a quadratic. In the present instance we have D = (;/-f8)2-60={i/-(-8-V60)}{j/-(-8-)-V60)} (2). Hence the turning values of y are y= _8- v'(60), and 2/= -84-V(60)- If y has any value between these, D is negative, and the roots of (1) are imaginary. Hence the algebraically less of the two, namely, - 8 - V(60), is a maximum ; and the algebraically greater, namely, - 8 -I- V(60), a minimum. The values of a; correspond- ingto these are atonce obtained from the equation a; = (s/ -I- 8)/2. They are a;= - \l^^) and x= + \l{yS) respectively. The reader should examine carefully the graph of this function, which has a discon- tinuity when a:=0 (see chap, xv., § 5). "We have the following series of corresponding values : — ^■=-00, -1, -0, -1-0, -1-3, ^g, 5, >5, +CO, y—-co, -24, -to, -1-00, 0, - 0,-1- +x. Hence the graph is represented by Fig. 1. Fig. r. 464 GENERAL DISCUSSION OF chap. Example 3. To discuss generally the turning values of the function _ aie' + h.r + c The equation which gives the values of x corresponding to any given value of 2/ is {a - a'y)oi:' + {b-b'y)x + {c- c'y) = 0. Let 'D = [b~b'yy- i{a-a'y){c-c'y), = [b'" - ia'c')y- + 2(2a'c + 2ac' - bb')y + (i^ - iac), = Ay2 + Bj/ + C, say. Then we have ^^ -(&-yy)±VD (2). 2(a-a'y) The turning values of y are therefore given by the equation Ar + Bi/ + C = (3). I. If B--4AC>0, this equation will have real unequal roots, and there will be two real turning values of y. If A be positive, then, for real values of .t, y cannot lie between the roots of the equation (3). Hence the less root will be a maximum and the greater a minimum value of y. If A be negative, then, for real values of a-, y must lie between the roots of (3). Hence the less root will be a minimum and the greater a maximum value of y. II. If B^- 4AC<0, the equation has no real root, and D has always the same sign as A. In this case the sign of A must of necessity be positive ; for, if it were not, there would be no real value of x corresponding to any value of y whatever. Hence there is a real value of x corresponding to any given value of y whatever ; and y has no turning values. III. If B" - 4AC = 0, we may apply the same general reasoning as in Case II. The present case has, however, a special peculiarity, as we shall see im- mediately. The criteria for distinguishing these three cases may be expressed in terms of the roots a, (3 and a', ^' of the two funotions ax^ + bx + c and a'x' + b'x+c', and in this form they are very useful. We have B2 - 4AC = 4(2ac' + 2a'c - bb'f - 4(6^ - iac) {b'^ - ia'c'), = 4aV{f2£: + 2i-^^)=-f?,-4^)f*;:-4i:)}, I \ a a a a) \a' a) \a ^ a! J } = 4aV2 { [2a'/3' + 2a/3 - (a + /3) (a' + ^') ? - (a - ffl' K - /3?} , = 4aV2{2a'/3' + 2a/3-(a+/3)(a' + |8')-(a-^)(a'-|3')} X )2a'^' + 2a^-(a + (3)(a' + /3') + (a-/3)(a'-^')}, = \%ahi'\a. - a') (a - ;8') (/3 - a') (^ - /3'). Hence it appears that the sign of B^ - 4AC depends merely on the sign of E = (a-a')(a-/3')(P-a')(/3-^') (4) XVI" {ax^ + hx + c)/(a'x' + b'x + c') 465 Since a, h, c, a', V, c' are all real, the roots of cu? + bx + c and of a'x^ + Vx + c', if imaginary, must be conjugate imaginaries. Hence, by reasoning as in § 6, we see that, if the roots of ax" + hx + e, or of a'a? + Vx + c', or of both, be imaginary, E is positive. The same is true if the roots of either or of both of these functions be equal. Consider, next, the case where a, /3, a', jS' are all real and all unequal. Since the sign of E is not altered if we interchange both a with a' and /3 with /3', or both a with j3 and a' with /3', we may, without losing generality, suppose that a is the algebraically least of the four, u, ft a', (3', and that a' is algebraically less than /3'. If we now arrange the four roots in ascending order of magnitude, there are just three possible cases, namely, a, ft a', ft ; a, a', ft, /3 ; a, a,', ft ft. In the iirst case, a -a', a -ft, /3-a', j3- ft have all negative signs ; in the second, two have negative signs, and two positive ; in the third, three have negative signs, and one the positive sign. It is, therefore, in the third case alone that E has the negative sign. The peculi- arity of this case is that each pair of roots is separated as to magnitude by one of the other pair. We shall describe this by saying that the roots inter- lace. Lastly, suppose E = 0. In this case one at least of the four factors, a - a', iS - /3', /3-a', (3 - (3', must vanish ; that is to say, the two functions ax^ + I>x + e and a'x^+b'x + c' must have at least one root, and therefore at least one linear factor in common. * Hence, in this case, (1) reduces to a(£-a) ^ a'{x-a') ^^'' say. Hence we have x — a' + a' — a a aia! — a) y=a. — j^ =- + ^1 1 6 . a{x-a) a a\x-a.) ^ ' From (6) it appears that y has a discontinuity when x=a', passing from the value + oo to - oo , or the reverse, as x passes through that value ; but that, for all other values of x, y either increases or decreases continuously as X increases. Hence y has no real turning values in this case, unless we choose to consider the value y=aja', which corresponds to »= ±oo , as a maximum- ' minimum value. The graph in this case, supposing aja', a, and a' - a to be both po.sitive, is like Fig. 8, where OA = a, OA' = a', OB = aja'. To sum up — Case I. occurs when the roots of either or of both of the functions ax^ + bx + c, a'x'^ + Vx + c' are imaginary or equal, and when all the roots are real but not interlaced. Case II. occurs when the roots of both quadratic functions are real and interlaced. * In the case where they have two linear factors in common, y reduces to a constant, a case too simple to require any discussion. VOL. I. 2 H 46G («.(■" + hx + c)/{a'.v' + h'x + c') Case III. occurs when the two quadratic functions have one or both roots in common. In this case y reduces to tlie quotient of two linear functions, or to a constant, and has no ma.ximum or minimum value properly so called. In the above discussion wo have assumed that neither a nor a' vanish ; in other words, that neither of the two quadratic functions has an infinite root. The cases where infinite roots occur are, however, really covered by the above Y B V ^ "\ X •\ A' statements, as may be seen either by considering them as limits, or by work- ing out the expression for B^ - 4AC in terms of the finite roots in the particular instances in question. In stating the above conclusions so generally as this, the student must remember that one of the turning values may either become infinite or corre- spond to an infinite value of x ; otherwise he may find himself at a loss in certain cases to account for the apparent disappearance of a turning value. A great variety of particular cases are included under the general case of this example. If we put «' = 0, c' = 0, for instance, we have the special case of Examjile 2. As our object here is merely to illustrate methods, it will be sufficient to give the results in two more particular cases. Example 4. To trace the variation of the function _ s?~7x + e The quadratic for x in terms of y is (1 - 2/>2 - (7 - Sy)x + {e - 15j/) = 0. IiGllCG D = (7-82/)=-4(l-z/)(6-152/) = 4{j/-a-V6)}{j/-(J-fV6)}. Hence 7/2 - \/6 and 7/2 -1- \/6 are ma.ximum and minimum values of y re- spectively. The corresponding values of x are given by SVIII Graphs foe particular cases 467 x=l 1-2/ and are 9 + 2\/S and 9 - 2\/6 respectively. "We observe farther that y is dis- continuous when x=S and when x=5 ; that when k=+oo or = -oo,2/=l; and that the other value of x for which y=l is K=9. We have thus the following table of corresponding values : — x=-oo, 0, +1, +3-0, +3 + 0, +4-1, jy=+l, +-4, 0, - 00, + », +5-9, min. a!=+5-0, +5 + 0, + 6, + 9, + 13-9, +00 y= +°°> - °°, 0, + 1, + 1-05, +1. The graph has the general form indicated in Fig. 9, which is not drawn to scale, but dis- torted in order to bring out more clearly the maximum point B. Example 5. To trace the variation of the function Y ! max. B /- i X ^ M ( " y a^-5x + i Fig. 9. a;^-8a; + 15' {5-8y)x + {i-l5y) = 0. The quadratic for x is {l-y)x^ Here we iind r>=4{(2/-4)'+2} Hence there are no real turning values. The gi-aph will be found to be as in Fig. 10. \ "O^ \ X Fio. 10. 468 MAXIMA AND MINIMA, METHOD OF INCREMENTS CHAP. Example 6. To find the turning values o{x=x' + y', given that ax- + ix!/ + ct/=l. We have, since cafi + hxij + cy- = 1, x' + y^ ^ 1^ + 1 ~ax- + bm/ + cy-~a^'^ + b^ + c' where ^=xjy. We have now to find the turning values of z considered as a function of f. The quadratic for t is (a2-l)|2 + fc| + (cs-l) = 0. Hence the turning values of z are given by &V=4(as-l)(c2-l), that is, by [h- - iac)z- + 4((» + c)3 - 4 = 0. The result thus arrived at constitutes an analytical solution of the well- known problem to find the greatest and least central radii (that is, the semi- axes) of the ellipse whose equation is aii? + lxy + cy^=\. Semai h — The artifice used in this example will obviously enable us to find the turning values of u=f{x, y), when 0. (22.) Show that {ax- + bx + c)/{cap+bx+a} is capable of all values if 470 EXERCISES XXXVm CHAP. XVIII 6->((i + c)^; that there are two vahies between which it cannot lie if (a + c)^ > b^ >■ iac ; and that there are two values between which it must lie if J' < 4(T« ("Wolstenholme). (23. ) If ra >pb, then the turning value o{{ax + b)l(px + rf is a"lip(ra -pb). Find the turning values of the following ; and discriminate maxima and minima : — (24.) (a;-l)(iK-3)/a:2. (25.) (x-2,)Hfl? + X'?,). ( 26. ) l(wx, + bf + V{a'x + b'f + l"(a"x + b")". (27.) aa; + %, given a;° + 2/^=A (28.) a^x- + jy, given a; + i/=a. (29.) xy, given a'/x- + b^/y^=l. (30.) a^y + xri/- + xy^, given xy=a-. (31.) ax^ + 2hxy + by", given Acc° + 2Ha!2/ + B2/^=l. (32.) xyj's/ix' + y"-). (33.) {2x - 1) (3x - i) (x - Z). (34.) l/\/x + lj'^y, given x + y = c. (35. ) To inscribe in a given square the square of minimum area. (36.) To circumscribe about a given square the square of maximum area. (37.) To inscribe in a triangle the rectangle of maximum area. (38.) P and Q are two points on two given parallel straight lines. PQ subtends a right angle at a fixed point 0. To find P and Q so that the area POQ may be a minimum. (39.) ABC is a right-angled triangle, P a movable point on its hypotenuse. To find P so that the sum of the squares of the perpendiculars from P on the two sides of the triangle may be a minimum. (40.) To circumscribe about a circle the isosceles trapezium of minimum area. (41.) Two particles start from given points on two intersecting straight lines, and move with uniform velocities u and v along the two straight lines. Show how to find the instant at which the distance between the particles is least. (42.) OX, OY are two given straight lines ; A, B fixed points on OX ; P a movable point on OY. To find P so that AP' + BP- shall be a minimum. (43. ) To find the rectangle of greatest area inscribed in a given circle. (44. ) To draw a tangent to a given circle which shall form with two given perpendicular tangents the triangle of minimum area. (45.) Given the aperture and thickness of a biconvex lens, to find the radii of its two surfaces when its volume is a maximum or a minimum. (46. ) A box is made out of a square sheet of cardboard by cutting four equal squares out of the corners of the sheet, and then turning up the flaps. Show how to construct in this way the box of maximum capacity. (47.) Find the cylinder of greatest volume inscribed in a given sphere. (48.) Find the cylinders of greatest surface and of greatest volume in- scribed in a given right circular cone. (49.) Find the cylinder of minimum surface, the volume being given. (50. ) Find the cylinder of maximum volume, the surface being given. CHAPTEE XIX. Solution of Arithmetical and Geometrical Problems by means of Equations. § 1.] The solution of isolated arithmetical and geometrical problems by means of conditional equations is one of the most important parts of a mathematical training. This species of exercise can be taken, and ought to be taken, before the student . commences the study of algebra in the most general sense. It is chiefly in the applications of algebra to the systematic investi- gation of the properties of space that the full power of formal algebra is seen. All that we need do here is to illustrate one or two points which the reader will readily understand after what has been explained in the foregoing chapters. § 2.] The two special points that require consideration in solving problems by means of conditional equations are the choice of variables, and the discussion or interp-etation of the solution. With regard to the choice of variables it should be remarked that, while the selection of one set of variables in preference to another will never alter the order of the system of equations on whose solution any given problem depends, yet, as we have already had occasion to see in foregoing chapters, a judicious selection may very greatly diminish the complexity of the system, and thus materially aid in suggesting special artifices for its solution. With regard to the interpretation of the solution, it is im- portant to notice that it is by no means necessarily true that all the solutions, or even that any of the solutions, of the system of equations to which any problem leads are solutions of the 4(2 INTERPRETATION OF THE SOLUTION chak problem. Every algebraical solution furnishes numbers which satisfy certain abstract requirements ; but these numbers may in themselves be such that they do not constitute a solution of the concrete problem. They may, for example, be imaginary, whereas real numbers are required by the conditions of the con- crete case ; they may be negative, whereas positive numbers are demanded ; or (as constantly happens in arithmetical problems involving discrete quantity) they may be fractional, whereas integral solutions alone are admissible. In every concrete case an examination is necessary to settle the admissibility or inadmissibility of the algebraical solutions. All that we can be sure of, a priori, is that, if the concrete problem have any solution, it will be found among the algebraical solutions ; and that, if none of these are admissible, there is no solution of the concrete problem at all. These points wll be illustrated by the following examples. For the sake of such as may not already have had a suflSciency of this kind of mental gymnastic, we append to the present chapter a collection of exercises for the most part of no great difficulty. Example 1. There are three bottles, A, B, C, containing mixtures of three substances, P, Q, li, in the following proportions : — A, aP + a'Q + a"R; B, 6P+i'(J + i"K; C, cP + c'Q + c"R. It is required to find what proportions of a mixture must be taken from A, B, C, in order that its constitution may be dP + d'Q, + d"R (Newton, Arithinelica Universalis), Let 'j\ y, z be the proportions in question ; then the constitution of the mixture is (ax + 6j/ + c:)P + (a'x + h'y + c'«)Q + {d'x + V'y -)- c"2)R. Hence we must have cvX'\-'by-\-cz=d, a'x + Vy + c'z=d', a"x + b"y + c"z=d". The system of equations to which we are thus led is that discussed in chap, xvi., § 11, with the sole difference that the signs of d, d', d" are re- versed. If, therefore, ab'c" -ab"c' + bc'a" -lc"a' + ca'b" -ca'V^O, we shall obtain a unique finite solution. Unless, however, the values of x, y, s all come out positive, there will be no proper solution of the concrete problem. It is in XIX EXAMPLES 473 ' fact obvious, a priori, that there are restrictions ; for it is clearly impossible, for instance, to obtain, by mixing from A, B, C, any mixture which shall contain one of the substances in a proportion greater than the greatest in which it occurs in A, B, or C. Example 2. A farmer bought a certain number of oxen (of equal value) for £350. He lost 5, and then sold the remainder at an advance of £6 a head on the original price. He gained £365 by the transaction ; how many oxen did he buy ? Let X be the number bought ; then the original price in pounds is 350/x. The selling price is therefore 350/a; + 6. Since the number sold was k- 5, we must therefore have (a; - 5)^?^ + 6^ -350 = 365. This equation is equivalent to 6x2- 395x- 1750 = 0, which has the two roots x=lQ and m= -25/6. The latter number is in- admissible, both because it is negative and because it is fractional ; hence the only solution is a; =70. Example 3. A'OA is a limited straight line such that OA=OA' = a. P is a point in OA, or in OA produced, such that 0P=^. To find a point Q in A'A such that PQ^ = AQ . QA'. Discuss the different positions of Q as ^ varies from to its greatest admissible value. ' Let OQ = a;, x denoting a positive or negative quantity, according as Q is right or left of 0. Then PQ=±(a!-^), A'Q = o-l-a;, AQ=a-a;; and we have in all cases {x-pf={a + x){a-x) = a^-x^ (1). Hence x^-px + l(p'^-w') = Q (2). The roots of (2) are 4i)± V(K - J?^)- These roots will be real if p^<'ia? ; that is to say, confining ourselves to positive values of^, if^< \/2a. From (1) we see that in all cases where x is real it must be numerically less than a. Hence Q always lies between A' and A. When^=0, the roots of (2) are ±a\/2 ; that is to say, the two positions of Q are equidistant from 0. So long as^ is a, then both roots are positive, and the points will be as in Fig. 2, where 0Q3=QiP. 4"i EXAMPLES I I I M I II A' Q2CQ1A PB Fio. 2. If 0'B = \/2a, then B is the limiting position of P for which a solution of the problem is possible. When P moves up to B, Qi and Q2 coincide at G (OC being JOB). Example 4. To find four real positive numbers in continued proportion such that their sum is a and the sum of their squares b'-. Let us take for variables the first of the four numbers, say .)', and the common value, say y, of the ratio of each number to the preceding. Then the four numbers are x, xij, xy", x)f. Hence, by our data, X + ./■;/ + xy'^ + ;ry^ = a, .r- + xrij- + x'lf + x-y^ = h" ; that is to say, o-{\+y)(l+y'^) = a (1), a;2(l +,/)(! +,/) = J2 (2). From (1) we derive x\\ + yf{\ + y^'^=a? (3), and from (2) and (3), rejecting the factor j/^ + l, which is clearly irrelevant, we derive a=(l + //-') = J=(l + 2/=)(l+2/f (4). The equation (4) is a reciprocal biquadratic in y, which can be solved by the methods of chap, xvii., § 8. For every value of 2/ (1) gives a corresponding value of x. The student will have no difficulty in showing that there will be two proper solutions of the problem, provided a be >h. Since, however, the two values of y are reciprocals, and since ^(l+j/) (l+j/^)=a;j/'(l + l/2/)(l + l/)/^), these two solutions consist merely of the same set of four numbers read for- wards and backwards. There is, therefore, never more than one distinct solution. Newton, in his Arithmelica Universalis, solves this problem by taking as variables the sum of the two mean numbers, and the common value of the product of the two means and of the two extremes. He expresses the four numbers in terms of these and of a and 6, then equates the product of the second and fourth to the square of the third, and the product of the first and third to the square of the second. It will be a good exercise to work out the problem in this way. Example 5. In a circle of given radius a to inscribe an isosceles triangle the sum of the squares of whose sides is 26^ Let X be the length of one of the two equal sides of the triangle, 2y the length of the base. If ABC be the triangle, and if AD, the diameter through A, meet BC in E, then, since ABD is a right angle, we have AB^ = AD . AE. Hence a;2 = 2aV(«'-2/') (!)• XIX EXAMPLES 475 Again, by the conditions of the problem, we have that is, a? + 2t/=h^ (2). From (1) and (2) we derive a;'-6aV + 2a262=0 (3). The roots of (3) are and tlie corresponding values of y are given by (2). The necessary and sufficient condition that the values of x and of y be real is that b -c 3a/\j2. When this condition is satisfied, there are two real posi- tive values of x, and if 6 > 2a there are two corresponding real positive values oiy. It follows from the above that, for the inscribed isosceles triangle the sum of the squares of whose sides is a maximum, b = 3a/\/2. Corresponding to this we have x=\JZa, 2y = \/Za ; that is to say, the inscribed triangle, the sum of the squares of whose sides is a maximum, is equilateral, as is well known. Example 6. rind the isosceles triangle of given perimeter 2p inscribed in a circle of radius a ; show that, if 1p be less than 3^3, and greater than 2a, there are two solutions of the problem ; and that the inscribed triangle of maximum perimeter is equilateral. Taking the variables as in last example, we find x^ = 2a\/[x^-y'') (1), x+y=p (2). Hence s^-%a?px + ia?p'^=0 (3). "We cannot reduce the biquadratic (3) to quadratics, as in last example ; but we can easily show that, provided p be less than a certain value, it has two real positive roots. Let us consider the function y=:c^ - 8a^px + ia'p" (4) ; and let I be the increment of y corresponding to a very small positive incre- ment (A) of X. Then we find, as in chap, xviii., § 12, that l = i{a?-2a^p)h (5). Hence, so long as a:^<:2ffi^^, I is negative ; and when si?>2w'p, I is positive. Hence, observing that y= +oo when a;= ±oo , we see that the minimum value of y corresponds to x= i^{2a^p), and that the graph of (4) consists of a single festoon. Hence (3) will have two real roots, provided the minimum point be below the K-axis ; that is, provided y be negative when x= ^lla^p) ; that is, provided / 3 - -N ia-pi I — r,rt5 +p^ j 476 EXEECISES XXXIX chap. be negative ; that is, provided 2p-^Z\/Za. It is obvious that both the roots are positive ; for when 3'=0 we have y=ia?p-, which is positive ; lience the graph does not descend below tlie axis of x until it reaches the right-hand side of the axis of y. From the above reasoning it follows that the greatest admissible perimeter is 3\/3a. When ip has this value, the minimum point of the graph lies on the axis oix, and x= ,^{ia?p)= ^ {Q\/Za}') = \JZa corresponds to two equal roots of (3). The corresponding value of iy is giv.en by 2y='2,p - 2x=Z\JZa - 2\/3a = \JZa; in other words, the inscribed isosceles triangle of maximum peri- meter is equilateral. Another interesting way of showing that (3) has two equal roots is to dis- cuss the graphs (referred to one and the same pair of axes) of the functions y=x^, &u+l)^ Hence .s.-{'^y (2). Cor. 1. „S3 «s (ZK integral function of n of the ith degree. Cor. 2. The sum of tlie cubes of the first n integers is the square of the sum of their first powers. § 8.] Exactly as in § 7 we can show that (« + 1)" -(«+!) = 5„.s, + 10,A + 10„S2 + 5„s, (1); and from this equation, knowing „s, , ^Sj, ^s^, we can calculate nSi. The result is n{n + \) {%n^ + ^n' + n - I) »'^ 30 ^'^>- § 9.] This process may be continued indefinitely, and the functions „s,, ^s^, . . ., n^r-i ■ ■ ■ calculated one after the other. Suppo.?e, in fact, that ^Si, ^s^, . . ., nSr-i had all been cal- culated. Then, just as in §§ 5-8, we deduce the equation (n+lY+'^-(n+\) = ^+,C,,,Sr + r+iG2nSr-i+- ■ ■ + r+fir nSi (1), where r+fii, r-i-iCs, &c., are the binomial coefficients of the r + 1th order. The equation (1) enables us to calculate „s,. Cor. 1. n^r w an integral function of n of the r + Ith degree, so that we may write inyr ■ qX+^ + q^n' + q,n'-^ + . . .+ qr+, and it is obvious from (1) that XX SUM OF ANY INTEGRAL SEEIES 487 _ 1 _ 1 Cor. 2. „s,. is divisible by n{n + 1), so that we may write ,^Sr = n{n+l)\- — +pfl''-'^+p^'nT'^ + . . .+Pr for this is true when r = 1, r = 2, r = 3, r = 4 ; hence it must be true for r = 5, for we have (w + 1)° - (71 + 1) = eCinSj + fj^nSi + S^2,nh + S^2nS2 + S^mh, and (w + 1)° - (» + 1) is divisible by n{n + 1) ; and so on. § lO.J "We can now sum any series whose mth term is re- ducible to an integral function of n. By § 4 and § 9, Cor. 1, we see that the sum of n terms of any series whose nth term is an integral function of n of the rth degree is an integral function of n of the r + 1th degree. We may, therefore, if we choose, in summing any such series, assume the sum to be Am''+i + Bm'' + . . . + K ; and determine the coefficients A, B, . . ., K by giving particular values; to TO. If S,, Sj, . . ., S^+a be the sums of 1, 2, . . ,r + 2 terms of the series, then it is obvious, by Lagrange's Theorem, chap, xviii., § 7, that the sum is '■+2 {n-l)(n-2)...{n-s+\)(n-s-l)...{n-r-2) -'Xs-l){s-2) 1 (-1) ...(s-r-2)" The following are a few examples : — Example 1. To sum the series S = «. + (« + J) + (« + 26) + . . . + {« + (»- 1)5}. The 7ith term is {a-b) + nb. The « - 1th term is (a - 6) + ()i - 1 ) b. The 2nd term is {a -b) + 2b. The 1st term is {a-b) + lb. Hence S=(a-5)ji + 5„si, -=(a-b)n + b — !^— , ^~{2a,+ {n-l)b}. as was found in § 3. 488 EXAMPLES CHAP. Example 2. 2 = 1 - + 3^ + 5- + to ?!. terms. The nth term is {In -l)- = iii- - in + l. Hence 2= 4k- -in +1 + 4(7i-iy--4(ji-l) + l + 4.22 _4_2 +1 + 4.12 _4.i +1, Hence, adding in vertical columns, we have 'Z = i„Si-inS\ + n, = 4 '^ ^-4-5 -^ — ' + n, _( 2«-l)?t(2» + l) 3 Example 3. 2 = 2.3.4 + 3.4.5 + 4.5.6+. . to ?i terms. The mth term \s{n + l){n + 2){n + S) = n^ + 6'n?+lln + Q. Hence 2: = „.S3 + 6„S2 + ll„Si + 6?i., = \ {n* + 10«3 + 35»- + 50?i). Example 4. A wedge-shaped pile of shot stands on a rectangular base. There are m and n shot respectively in the two sides of the lowest rectangular layer, m- 1 and ra- 1 in the two sides of the next rectangular layer, and so on, the upper- most layer being a single line of shot. Find the whole number of shot in the pile, m being greater than n. The number in the lowest layer is inn ; in the next (m - 1) (« - 1) ; in the next (m- 2) (a -2), and so on; the number in the last layer is (m-?i-l) (»-«-! ), that is, {m-n + l). Hence we have to sum the series 2 = »i7i+()ra-l)(/!-l) + (OT-2)(m-2) + . . .+{in-n-l){n-n-l), in which there are n terms. The rth term of the series is (m- »■- 1) (m- )•- 1), that is, (m + l-r) (n + l-r), that is, (m + \)(n + \)- (m + n + 2)r + r-. Hence we may write the series as follows : — 2= (m + l)()i + l) -(m + n + 2)n +n^ + {m + l){ii + l) -(m + « + 2)(7i-l) +{n~lf + (m + l)(;f + l) -(m + m + 2)2 +2^ + (m + l)(-rt + l) ~{in + n + 'i)\ +1-, = n{m + l){n + l)-{m + n + 2)„si +„«2, = {m + \)n{n+l)-i{m+n+2)n{n+\) + ^n{n + l){in + \), = J»(» + l)(3m-';i+l). Raimrh. — In working examples by this method the student must be care- ful to see that the series is complete ; in other words, that there are exactly n terms, all formed according to the same law. If any terms are wanting, or if there are redundant terms, allowance must be made by adding or subtract- ing terms, as the case may be. XX GEOMETRIC SERIES 489 SERIES WHOSE MTH TERM IS THE PRODUCT OF AN INTEGRAL FUNCTION OF n AND A SIMPLE EXPONENTIAL FUNCTION OF W. § 11. J The typical form of the »th term in the class of series now to be considered is {Poii^ + jj,w*~i + . . . + Ps)''"> where Po, Pi, ■ • •, Ps, i" are all independent of n, and s is any- positive integer. The simplest case is that in which the integral function re- duces to a constant. The mth term is then of the form p^r'^, or say^j?' . »'"~\ that is, ar^'^, where a=pgr is a constant. The ratio of the 7ith to the (n - l)th term in this special case is ar'^jar'^'^ = r, that is to say, is constant. A series in which the ratio of each term to the preceding is con- stant is called a GEOMETRIC series or geometric PROGRESSION ; and the constant ratio in question is called the COMMON ratio. If the first term be a and the common ratio r, the second term is ar ; the third {ar)r, that is, ar' ; the fourth {ai^r, that is, ar^ ; and so on. The mth term is ar'^~'^. A geometric series is therefore neither more nor less general than that particular case of the general class of series now under discussion which introduced it to our notice. § 12.] To sum a geometrical series. Let 1 = a + ar + ar'' + . . .+ar'"--^ (1). Multiply both sides of (1) by 1 - r and we have (1 - r)S = a + ar + ai^+. . . + ar^~'^ - ar - ar' - . . . - ar^'^ - ar"; = a- af^ (2). Hence 2 = «^-^ (3). 1 -r ^ ' Since the number of operations on the right-hand side of (3) is independent of n* we have thus obtained the sum of the series (1). Cor. If I be the last term of the series, then Z = ffir""i and (ir™ = rl. Hence (3) may be written ■^ Here we regard the raising of r to the nth power as a single operation. <^>- 490 EXAMPLES CHAP. Example 1. :i: = | + J + | + . .tolOterms. In this case a=f, ?' = i. Hence v_,i-(4)^°_ 1- = 3 1 --)■ Example '2. 3 = 1-2 + 4-8 + 10. . to » terms. Here a=l, r= - 2. Hence ^^ l-{-2)"_l^{-ry^ ■^ '■ l-(-2) 3 = 1(1-2"), if m be even, = i(l + 2"), i(nhe odd. Example 3. ^ = {x + y) + {x^ + xy+y-) + {a? + x-y + j:y- + y^) + . . . to m terms, .r- - 11- x?-ifi x^-y* x"+^ - 1/"+' = ' + — + +. . . + — , x—y x-y x-y x-y = — (,K- + s? + . . .+a;«+i)--,^(y' + 2/3 + . . .+^f^^), X >j -'^ y = ''-' (l+a; + . . .+K"-i)--^(l + )/ + . . .+»/"-'). x-if ' x-y^ ■' Now l + a; + . . +x"-i = (l -a;")/(l -^), and 1+2/ + . . ■ +^"' = (1 -r)/(l "J/). hence -- ^^d"^) 2/^(1 -r) — (,«-;/) (1- a-) (a;-2/)(l-y)- § 13.] "W^e next proceed to consider the, case where the integral function which multiplies ?'" is of the 1st degree. The general term in this case is {>i + bn)r"' (IX where a and b are constants. It will be observed that a term of this form ■would result if we multiplied together the mth term of any arithmetic series by the mth term of any geometric series. For this reason a series whose wth term has the form (1) is often called an arithmetico- geometric series. The series may be summed by an extension of the artifice employed to sum a G.P. Let i; = (a + i. l)r' + (tf + b.2y + (a + &. 3)r' + . . . + (a + b. «>". XX ARITHMETICO-GEOMETKIC SERIES 491 Multiply by 1 - r, and we have (l-r)S = {a + h. \y + (a + 5. 2)r + {a + h.zy + . . . + (a + 6«)r" -{a + l}.iy-{a + h.2y- . .-{a + bir^iy" - (a + Jra)r»+\ = (ffi + 6.1)r + 1 Jr' + ir' + . . . + Jr" | - (a + 6m)?'™+i (1). Looking merely at the terms within the two vertical lines, we see that these constitute a geometric series. Hence, if we multiply by 1 - r a second time, there will be no series left on the right-hand side ; and we shall in effect have found the required expression for 2. We have, in fact, (l-'f2 = (l -r){a + h)r + br^ + br" + . . + br'" -br^- . . .-br'^- Jr^+i - (1 - r) (a + hny^+\ = (1 - r) (a + b)r + br' - &r"+i - (1 - r) (a + J«)r"+i, = (ffi + b)r -(a + by + br' -{a + {n+ l)i}r'*+i + {a + bny'+^ (2). Hence „ (a + b)r - (a + by + br' -{a + {n+ l)6}r"+i + (a + bny^+^ ,,, 2 ^^^^^ (3). § 14.] If the reader has not already perceived that the artifice of multiplying repeatedly by 1 - r will sum any series of the general form indicated in § 11, probably the following argu- ment will convince him that such is the case. Let fs(n) denote an integral function of n of the sth degree ; then the degree of fs{n) -fs(n - 1) is the {s - l)th, since the two terms in to* destroy each other. Hence we may denote /s(m) -fs{n - 1) by /s-i(m). Similarly, /«_,(«) -/,_,(» - 1) will be an integral function of n of the {s - 2)th degree, and may be denoted Consider now the series 2 =Uiy +fs{^y + ■ . . +fln)r- (1). Multiply by 1 - ?•, and we have 492 INTEGEO-GEOMETEIC SERIES CHAP. (l-r)2 =u\y+ my+ fpy. . . .+/»■" =/.(iy + l/.-.(2K+/.-,(3K + . . . +/,-,(«>■"! -/«(«K+M2). The series between the vertical Hnes in (2) is now simpler than that in (1) ; since the integral function which multiplies »'" is now of the (s - l)th degree only. If we multiply once more by 1 - r we shall find on the right certain terms at the beginning and end, together with a series whose nth term is now /;_„(«)?■". Each time we multiply by 1 - r we reduce the degree of the multiplier of r" by unity. Hence by multiplying by (1 - r)''+i we shall extirpate the series on the right-hand side altogether, and there will remain only a fixed number of terms. It follows that any series wJiose nth term consists of an integral function of n of the sth degree multiplied hy f" can be summed hy simply multiplying hy (1 - r)*+i. This simple proposition contains the whole theory of the sum- mation of the class of series now under discussion. Example 1. 2 = IV H- 2 V -f- 3 V 4- . . .-HiV". Here the degi-ee of the multiplier of r» is 2. Hence, in order to effect the summation, we must multiply by (1 - rf. We thus find (l-r)'S = l-/--t- 2-r'+ 3V+ 4V-I-. . .-f »V -3.lV-3.2V-3.3-r'-. . .-3{n-lfr"- 3«.V+i -I- 3. IV -I- 3.2V- .-|-3(m-2)V-l-3(m-l)V'+'-l- 3»iV+2 - IV -. .- (w-3)V»- (?i-2)V"+i-(?i-l)V"+* -™v+', ~r + r^- (re-f- l)V"+i -f {2n'' + 2ii - l)r^+'^ - 9i,V+^. Hence ^^r + r"' -{n + 1 )V»+i + (2n'' + 2n-l )r"+2 - «,V'+' '^~ {1-rf Example 2. 2=1 -2r-f3r=-4r'-l-. . . - 2jir-«-'. Multiply by {l + rY, and we have (l-l-rfS = l-2r+ 3r-2- ij^ + . . .- 2nr^"-' + 2r-2.2r''- + 2.Zr^-. . .+2{2n-l)r-''-'-- 2.2nr^ + ^=_ 2rH. . .- (2re-2)r2"-'-l-(2»-l)r=^-2^^r'»+^ = 1 - (271 -I- 1 )r-2" - 2nr''''+\ Hgiicg ^ 1 -(2»,-t-l)?-^"-2)tr^+' XX CONVERGENCY OF GEOMETRIC SERIES 493 If we put r=l, we deduce 1-2 + 3-4 . . . -2n=-ii, which agrees with § 3, Example 3, above. CONVERGENCY AND DIVERGENCY OF THE ABOVE SERIES. § 15.] We have seen that the sum of n terms of a series whose mth term is an integral function of n is an integral function of n ; and we have seen that every integral function becomes infinite for an infinite value of its variable. Hence the sum of n terms of any series whose nth term is an integral function of n may be made to exceed (numerically) any quantity, however great, by sufBciently increasing n. This is expressed by saying that every sucli series is divergent. § 16.] Consider the geometric series '2 = a + ar + ar^+ . . . + ar^'\ If r = 1, the series becomes 'Z = a + a + a + . . . + a = na. Hence, by sufficiently increasing n, we may cause 2 to surpass any value, however great. If r be numerically greater than 1, the same is true, for we have a(7-"-l) 2 = r-1 r-1 r-1 Now, since r > 1, we can, by sufficiently increasing n, make r", and therefore ar'^jir - 1), as great as we please. Hence, by suffi- ciently increasing ii, we can cause 2 to surpass any value, how- ever great (see Ex. ix. 46). In these two cases the geometric series is said to be divergent. If r be numerically less than 1, we can, by sufficiently increas- ing n, make r" as small as we please, and therefore ar"/(l - r) as small as we please. Hence, by sufficiently increasing n, we can cause 2 to differ from a/ (I - r) as little as we please. This is often expressed by saying that when r is numerically less than 1, the mm to infinity of the series a + ar + ar' + . . . is a/(l - r). 494 EXAMPLE OF INFINITE GEOMETRIC SERIES CHAl'. In this case the series is said to be convergent, and to converge to tlie value a/{l -r). There is yet another case worthy of notice. If r = - 1, the series becomes Hence the sum of an odd number of its terms is always a, and the sum of an even number of them always 0. The sum, there- fore, does not become infinite when an infinite number of terms are taken ; but neither does it converge to one definite value. A series having this property is sometimes said to oscillate. Example 1. Find the limit of the siim of an infinite number of terms of the series ^111 S = --f 1-- -f- " 2 2^ 2^ For n terms we have ,._, !- 1/2" 1 ^ ^ 1-4 2"- Hence, when n is made infinitely great, 2 = 1. This case may be illustrated geometrically as follows : — Let AB be a line of unit length. I i j j [— I Bisect AB in Pi ; bisect PiB in P2, A Pi Po I's P4K PsBinPg; and so on indefinitely. It is obvious that by a sufficient number of these operations we can come nearer to B than any assigned dis- tance, however small. In other words, if we take u sufficient number of terms of the series APl-^PlP2-^P2P3-^-P3P4-^ . ., we shall have a result differing from AB, that is, from unity, as little as we please. This is simply a geometrical way of saying that 111 2 + 22 + ^3+ • • ««<» =1. Example 2. To evaluate the recurring decimal •34. Let v_ h:_ 34 34 34 "- ^*-ioo + ioo5+ioP+ • • • «<^"- Then 2 is obviously a geometric series, whose common ratio, 1/100, is less than 1. Hence v^ 34 1 _34 ~"1001-TfT,~99' PROBLEMS ON ARITHMETIC PROGRESSION 495 PROPERTIES OP QUANTITIES WHICH ARE IN ARITHMETIC, GEOMETRIC, OR HARMONIC PROGRESSION. § 17.] If a be the first term, h the common difference, n the number of terms, and 2 the sum of an arithmetic progression, we have 2 = |{2a + (»-l)J) (1). This equation enables us to determine any one of the four quan- tities, 2, a, b, n, when the other three are given. The equation is an integral equation of the 1st degree in all cases, except when n is the unknown quantity, in which case the equation is a quadratic. This last case presents some points of interest, which we may illustrate by a couple of examples. Example 1. Given 2 = 36, a = 15, 6= -3, to find to. We have by the formula (I) above 36 = I {30 -(71-1)3}. Hence n^-lln + 2i = 0. The roots of this equation are n = S and to = 8. It may seem strange that there should be two different numbers of terms for which the sum is the same. The mystery is explained by the fact that the common difference is negative. The series is, in fact, lo + 12 + 9| +6 + 3 + 0-3-61 -9-. . .; and, inasmuch as the sum of the part between the vertical lines is zero, the sum of 8 terms of the series is the same as the sum of 3 terms. Example 2. 2 = 14, « = 3, 6 = 2. The equation for to in this case is n^ + 2n = U. Hence »= - 1± V(15)= +2'87 .,or-4'87... The second of the roots, being negative, has no immediate reference to our problem. The first root is admissible so far as its sign is concerned, but it is open to objection because it is fractional, for, from the nature of the case, to must be integral. It may be conjectured, therefore, that we have set our- selves an impossible problem. Analytically considered, the function to* + 2to varies continuously, and there is in the abstract no difficulty in giving to it any value whatsoever. The sum of an arithmetic series, on the other hand, varies per saltiim ; and it so happens that 14 is not one of the values that S can assume when a = 3 and b = 2. There are, however, two values which 7i 496 DETERMINATION OF ARITHMETIC SEEIES BY TWO DATA chap can assume between which 1 4 lies ; and we should expect that the integers next lower and next higher than 2 '87 would correspond to these values of S. So, in fact, it is ; for, when m = 2 ^ = S, and when n=3 S = 15. § 18.] An arithmetic progression is determined wlien its first term and common difference are given ; that is to say, when these are given we can write down as many terms of the pro- gression as we please. An arithmetic pi-ogression is therefore what mathematicians call a twofold manifoldness ; that is, it is determined by any two independent data. Bearing this in mind, we can write the most general arith- metic progressions of 3, 4, 5, &c. terms as follows : — a — /3, u, a + /3, a — 3/3, a — P, o, + p, a+ 3^, a - 2/3, a- 13, a, a + /3, a + 2/3, &C., where a and /3 are any quantities whatsoever. It will be observed that in the cases where we have an odd number of terms the common difference is /3, in the cases where we have an even number 2/3. These formulse are sometimes useful in establishing equations of condition between quantities in A. P. Example 1. Given that the ^th term of an A. P. is P, and that the qth term is Q, to find the A.P. Let a be the first term and i the common difference ; then the^th and qfh terms are a + {p-l)b and a + {q-l)b respectivelj'. Hence a + {p-l)b = V, a + {q-l)b = Q. These are two equations of the 1st degree to determine a and b. We find b = {P-Q)l{p-q), a={{p-l)Q-{q-l)?}l{p-q). Example 2. If a, b, c be in A.P. , show that 2 a-(b + c) + 6-(c + a) + c\a + b)=-{a + b+cf. We may jjut a = a-/3, 6 = o, c = a + p. The equation to be established is now Since a and j3 are independent of one another, this equation must be an identity. The left-hand side reduces to XX ARITHMETIC MEANS 497 2a {{a -pf + {a + ^f\+p{{a~ ,3)2 - (a + /3)=} + 2a^ = 2a |2a- + 2(32} + ;8 { - 4a/3} + gaS^ = 6a3. Hence the required result is established. § 19.] If three quantities, a, h, c, be in A. P., we have b - a = c-b by definition. Hence b = (c + a)/2. In this case b is spoken of as the arithmetic mean between a and c. The arithmetic mean between two quantities is therefore merely what is popularly called th§ir average. If a and c be any two quantities whatsoever, and Aj, A^, . . ., A,j n others, such that a, Aj, A^, . . ., A,i, c form an A. P., then Ai, A^, . . ., A,i are said to be n arithmetic means inserted between a and c. There is no difficulty in finding Aj, A^, . . ., A„ when a and c are given. For, if b be the common difference of the A.P., a, Ai, Aa, . . ., A,i, c, then Ai = a + i, As = a + 25, . . ., Kn = a + nb, and c = a + {n + \)b. From the last of these we deduce b = (c - a)/{n +1). Hence we have A, = a + , A, = ffi + 2 -, &c. n+ 1 « + 1 N.B. — By the arithmetic mean or average of n quantities a,, a,, . . ., a„ is meant (a, + ffj + . . . + an)/n. In the particular case where two quantities only are in question, the arithmetic mean in this sense agrees with the definitions given above ; but in other cases the meanings of the phrases have nothing in common. Example 1. Insert 30 arithmetic means between 5 and 90 ; and find the arithmetic mean of these means. Let h be the common difference of the A.P. 5, Aj, Aa, . . ., A-u, 90. Then J = (90-5)/(30 + l) = 85/31. Hence the means are &c. : -!■ --S' '*-'i that is, 240 325 31 ' 31 ' 410 , 31 TOL. I 498 EXAMPLES ,,, , A1 + A0+. . . +A„ If .\i + A„"l \\ e have =-i w — ;, — N _ Ai + A,. = (5 + 90)/2 = 95/2. Ilnnarl-. — It is true generally tliat the arithmetic mean of the 71. arith- metic means between a and c is the arithmetic mean between a and c. Examjjle 1. The arithmetic mean of the squares of n quantities in A. P. exceeds the square of tljeir arithmetic mean by a quantity which depends only upon ?i and upon tlieir common dill'erence. Let the n quantities be a + h, a + lb, ., a + nb. Then, by §§ 5 and 6, (a + hf + i. a + 'a))- +. . . +{a + nbf n = \a-n + ain(n + l) + b^ ~ r f > = a- + ab{n + 'l.) + -{2n- + Sn+l). ({a + b) + {a + 2b)+ . . . +{a + nb)\^ ( 7i + l,\„ Again, 1^ -_^A___ ^_ ^ | =(^„ + ._^^j. = a^ + rt,i(7t + 1 ) + -An^ + In + 1 ). Hence A. 11. of sr[Uares - square of A.M. = -y^— ft", which proves the proposition. § 20.] If S be the sum of n terms of a geometric progression whose first term and common ratio are a and r respectively, we have ,■■" - 1 1 (!)• "When any three of the four, S, a, r, n, are given, this equation determines the fourth. "When either 2 or a is the unknown quantity, we have to solve an equation of the 1st degree. "When r is tlie unknown quantity, we have to solve an integral equation of the 7ith degree, which, if n exceeds 2, will in general be effected by graphical or other approximative methods. If n be the unknown quantity, we have to solve an exponential equation of the form r" = .5, where r and a are known. This may be XX DETEEMINATION OF GEOMETKIC SERIES BY TWO DATA 499 accomplislied at once by means of a table of logarithms, as we shall see in the next chapter. § 21.] Like an A.P., a G.P. is a twofold manifoldness, and may be determined by means of its first term and common ratio, or by any other two independent data. In establishing any equation between quantities in G.P., it is usual to express all the quantities involved in terms of the first term and common ratio. Since these two are independent, the equation in question must then become an identity. Example 1. The ^th term of a G.P. is P, and the qth term is Q ; find the first term and common ratio. Let a be the first term, r the common ratio. Then we have, by our data, From these, by division, we deduce rJ'-9 = P/Q, whence r=(P/Q)i'(''-9t. Using this value of r in the first equation, we find „_py(pyQ)(2)-l)/(p-S) = P(l-?)/(j)-!)Q(l-ii)/(3-P). Hence we have a = pii-alflp-slQIi-rf/fe-J'), r = pV(3'-!)Qi'(«-i'). Example 2. If a, b, c, d be four quantities in G.P., prove that 4:{a^ + b'' + ifi + d^)-{a + b+-c + df={a~bf + {c-df + 2{a,-df. If the common ratio be denoted by r, we may put b=ra, c^r'a, d=7^a. The equation to be established then becomes ia'{l + r^ + r*+r'^)-a\l+r + r^ + r^f=a?{l-r)' + aV{l-rf + 2a\l-r^f, that is, = l-2r + r'- + r»-2r5 + r<' + 2-4r5 + 2r^ which is obviously true. § 22.] When three quantities, a, b, c, are in G.P., b is called the geometric mean between a and c. We have, by definition, c/b = b/a. Hence b" = ac. Hence, if we suppose a, b, c to be all positive real quantities, 6 = + \/{ac).. That is to say, the geometric mean between two real positive quantities is the positive value of the square root of their product. If a and c be two given positive quantities, and G,, Gj, . . ., G„ n quantities, such that a, G,, G„, . ., G„, c form a G.P., then G„ Gj, . . ., G„ ar« said to he n geometric means inserted between a arid c. 500 GEOMETRIC MEANS CHAP. Let r be the common ratio of the supposed progression. Then we have G, = ar, G^ = ar', . . ., G„ = ar", c = ar"+'^ From the hist of these equations we deduce r = (c/a)i'("+i), the real positive value of the root being, of course, taken. Since r is thus determined, we can find the value of all the geometric means. The geometric mean of n positive real quantities is the positive value of the nth root of their product. This definition agrees witli the former definition when there are two quantities only. Example. The geometric mean of the n geometric means between u and c is the geometric mean between a and c. Let the n geometric means in question be ar, ar-, . ,, ar", so that c = rtr"+^ Then {ar. ar' . .. rtr»)i'" = (a")-'+2+ • • • +'')i'«, = («V"+1)1'2, which proves the proposition. § 23.] A series of quantities which are such that their reciprocals firm an arithmetic progression are said to he in harmonic progression. From this definition we can deduce the following, which is sometimes given as the defining property : — If «., h, c he three consecutive terms of a harmonic progression, then alc = (a-h)/{b-c) (1). For, by definition, 1/a, 1/b, 1/c are iu A.P., therefore 1 1 _1_1 bach TT a~b h — c Hence = . ab he jT a -h ah a Hence = — = - , h - e be c which proves the property in question. § 24.] A harmonic progression, like the arithmetic progression, from which it may be derived, is a twofold manifoldness. The following is therefore a perfectly general form for a harmonic XX HARMONIC PROGRESSION, HARMONIC MEANS 501 series, 1 /(a + i), l/{a+^h),. l/{a+3h), . . ., l/{a + nb), . . ., for it contains two independent constants a and b ; and the reciprocals of the terms are in A.P. The following forms (see § 18) are perfectly general for harmonic progressions consisting of 3, 4, 5, . . . terms respect- ively : — Ilia- 13), 1/a, l/{a + /3); l/(a-3/3), l/(a-^), l/(a + /3), l/(a + 3/3) ; I /{a -2/3), l/{a-/3), 1/a, l/(a + ^), ll{a+2/3), i\;c. The above formulse may be used like those in § 18. % 25.] If a, b, c be in II.P., b is called the harmonic mean between a and c. We have, by definition, 1/c- 1/b = 1 /b - l/a- Hence 2/6 = 1/a + l/c, and b = 2ac/{a + c). If a, H,, H„, . . ., H,j, c form a harmonic pivgression, H, , H„, . . ., H„ are said to be n harmonic means inserted between a ami c. Since 1/a, 1/H,, 1/Hj, . . ., 1/H^, l/c in this case form an A.P., whose common difference is d, say, we have d = (l/c - l/a)/(w + 1) = (a - c)/(w + l)ac. Hence 1/H, = 1/a + (a - c)/(ra + l)ac, l/H^ = 1/a + 2 (a - c)/(ft + l)ac, &c.; and Hi = (n + l)ac/{a + nc), H^ = (w + l)ac/(2a + (w - l)c), &o. If a quantity H be such that its reciprocal is the arithmetic mean of the reciprocals of n given quantities, H is said to be the harmonic mean of the n quantities. It is easy to see, from the corresponding i)roposition regard- ing arithmetic means, that the harmonic mean of the n harmmiic means between a and c is the harmonic mean of a and c. § 26.] The geometric mean between two real positive quantities a and c is the geometric mean between the arithmetic and the harmonic means between a and c ; and the arithmetic, geometric, and harmonic means are in descending order of magnitude. Let A, G, H be the arithmetic, geometric, and harmonic means between a and c, then A = {a + c)/2, G= + J(ac), H = 2ac/(a + c). 502 ARITHMETIC, GEOMETTJC, AND HARMONIC MEANS chap. i TT « + C lilC ^n Hence AH = --- x =ac = G , 2 a + c which proves the first part of the proposition. Again, A - G = -^ J(ac) = -^( v/« - \/c)'. 9 'Jac _ ij(ac). G - H = V(ac) - ^ - ' = ^^^^-^( x/« - Vc)^ ^ ' « + c a + c Now, since a and c are both positive, \'r6 and Jc are both real, therefore ( \/a - Jcf is an essentially positive quantity ; also J{ac) and a + c are both positive. Hence both A - G and G - H are positive. Therefore A >G>H. The proposition of this paragraph (which was known to the Greek geometers) is merely a particular case of a more general proposition, which will be proved in chap. xxiv. § 27.] Notwithstanding the comparative simplicity of the law of its formation, the harmonic series does not belong to the cate- gory of series that can ha summed. Various expressions can be found to represent the sum to n terms, but all of them partake of the nature of a series in this respect, that the number of steps in their synthesis is a function of n. It will be a good exercise in algebraic logic to prove that the sum of a harmonic series to n terms cannot be represented by any rational algebraical function of n. The demonstration will be found to require nothing beyond the elementary principles of algebraic form laid down in the earlier chapters of this work. EXEKOISES XL. Sum the following arithmetical progressions : — (1.) 5 + 9 + 13+ to :,5 terms. (2.) 3 + 34 + ^+- ■ • to 30 terras. (3.) 13 + 12 + 11+ . . to 24 terms. (i.) 1 + 1+ . . . to 16 terms. I 91-1 (5.) - + 1- . . . to 51 terms. II 11 (6. ) (as - iiy + {a' + n-) + {a + n)- + . . . to n terms. (''•) i^7 + i~7"2+ • ■ • to Z terms. (8.) The 20tli ternn of an A. P. is 100, and the sum of 30 terms is 500 ; find the sum of 1000 terms of the progression. XX EXERCISES XL 503 (9.) The first term of an A. P. is 5, the number of its terms is 15, and the sum is 390 ; find the common difference. (10.) How many of the natural numbers, beginning with unity, amount to 500500 ? (11.) Show that an infinite number of A.P.'s can be found which have tile property that the sum of the first 2m terms is equal to the sum of the next m terms, m being a, given integer. Find that particular A. P. having the above property whose first term is unity. (12.) An author wished to buy up the whole 1000 copies of a work which he had published. For the first copy he paid Is. But the demand raised the price, and for each successive copy he had to pay Id. more, until the whole had been bought up. What did it cost him ? (13. ) 100 stones are placed on the ground at intervals of 5 yards apart. A runner has to start from a basket 5 yards from the first stone, pick up the stones, and bring them back to the basket one by one. How many yards has he to run altogether ? (14.) AB is a straight line 100 yards long. At A and B are erected per- pendiculars, AL, BJI, whose lengths are 4 yards and 46 yards respectively. At intervals of a yard along AB perpendiculars are erected to meet the line LM. Find the sum of the lengths of all these perpendiculars, including AL and BM. (15.) Two travellers start together on the same road. One of tliem travels uniformly 10 miles a day. The other travels 8 miles the fir.st day, and increase.s his pace by half a mile a day each succeeding day. After how many days mil the latter overtake the former ? (16.) Two men set out from the two ends of a road which is I miles long. The first travels a miles the first day, a + b the next, a + 26 the next, and so on. The second travels at such a rate that the sum of the number of miles travelled by him and the number travelled by the first is always the same for any one day, namely c. After how many days will they meet ? (17.) Insert 15 arithmetic means between 3 and 30. (18.) Insert 10 arithmetic means between —3 and +3. (19.) A certain even number of arithmetic means are inserted between 30 and 40, and it is found that the ratio of the sum of the first half of these means to the second half is 137 : 157. Find the number of means inserted. (20.) Find the number of terms of the A.P. 1 + 8 + 15+ . . . the sum of which approaches most closely to 1356. (21.) If the common difference of an A.P. be double the first term, the sum of TO terms : the sum of n terms =m" : iv'. (22.) Find four numbers in A.P. such that the sum of the squares of the means shall be 106, and the sum of the squares of the extremes 170. (23.) If four quantities be in A. P., show that the sum of the squares of the extremes is greater than the sum of the squares of the means, and that the product of the extremes is less than the product of the means. (24.) Find the sum of » terms of the series whose rth term is J(3r + 1). (25. ) Find the sum of n terras of the series obtained by taking the 1st, rth, 2rth, 3rth, &c. terms of the A.P. whose first term and qommon difference are a and h respectively. 504 EXERCISES XL chap. (26.) If the sum of n terms of a series be always n{n + 'i,), show that the series is an A.P. ; and find its first term and common diflference. (27.) Show by general reasoning regarding the form of the sum of an A.P. that if the sum of p terms be P, and the sum of j terms Q, then the sum of n terms is 7n(n - q)/p{p -q) + Qji(?i -p)li(q ~P)- (28.) Any even square, (in)-, is the sum of n terms of one arithmetic series of integers ; and any odd square, (2re + l)^, is the sum of ii terms of another arithmetic series increased by 1. (29.) Find n consecutive odd numbers whose sum shall be Jiv. Show that any integral cube is the difference of two integral squares. (30.) Find the rath term and the sum of the series 1-3 + 6-10 + 15-21+ . . (31.) Sum the .series 3 + 6+ ... +3™. (32.) If Si, So, . . ., s^ be the sums of ^ arithmetical progressions, each having n terms, the first terms of which are 1, 2, . ., p, and the common differences 1, 3, . . ., 2p-l respectively, show that S1 + S2+ • . . +Sj, is equal to the sum of the first ii^j integral numbers. (33.) Tlie series of integral numbers is divided into groups as follows : — 1, I 2, 3, I 4, 5, 6, I 7, 8, 9, 10, ] . . , show that the sum of the ?ith group is ^-(?i' + n). If the series of odd integers be divided in the same way, find the sum of the nth group. Sum the following series : — (34.) 42 + 7=+ . . . +(3?i + l)=. (35.) S„(»»-l)(?i-l). (36.) 2„{p + q(^n~l)}{p + q{n-2)}. (37.) P- 22 + 32- . . . +(2«-l)2-(2»)2. (38.) a? + {a + bf+. . . +(a + «nj)^. (39.) (l3-]) + (23-2) + (3-'-3)+. . . tomterms. (40.) 1.22 + 2.32 + 3.4-+. . . tomterms. (41.) 1 + 2. 32 + 3. 5"- + 4. 72 + 5. 92+ to Ji terms. (42.) 1.3.7 + 3.5.9 + 5.7.11+ . . . to w terms. (43.) l2 + (P + 22) + (12 -;- 22 + 32)+ . . tomtei-ms. (44, ) A pyramid of shot stands on an equilateral triangular base having 30 shot in each side. How many shot are there in the pyramid ? (45.) A pyramid of shot stands on a square base having m shot in each side. How many shot in the pyramid ? (46.) A symmetrical wedge-shaped pile of shot ends in a line of m shot and consists of I layers. How many shot in the jiile ? (47.) If„S,-=l'- + 2'-+ . . . +n',t'beu„S,.=l)on'-+^+pin'-+ . . . +^;^i, where f 0, Pi, ■ ■ • can be calculated by means of the equations r+lOipo = 1, ,.+lC2?Jo + ,-Ci^i=rCi, r+lCs po + r^-2Pl + r-lC'i p2 = rCo , „C, denoting, as usual, the j'th binomial coeUicient of tlie nth. rank. XX EXERCISES XLI 505 (48.), Show that A=«(»+i)(«-i)(»-2). . . o^-o{ ,(,^i)(;!;)(,_i), - (r-l)7-(™-r+l)l!(r-2)! (r- 2) ()•-!)(?!,- r + 2)2!(r- 3)! 1.2(»-l)(r-l)!j ' ■where r ! stands for 1.2.3 . . . r. (49.) If P,. denote the sum of the products r at a time of 1, 2, 3,- . . ., n, and Sr denote !'■ + 2' + . . .+?i'', showthatrP^sSiPr-i-SaP.-a + SsPr-s-. . . Hence calculate Pj and P3. (50.) If/(a;) be an integral function of x of the (r- l)th degree, show that f{x) - rCif{x-l)+,.C2f{x-2) ...{- Yf{n -r) = 0, rCi, ^Co . . . being binomial coefficients. EXEEOISBS XLI. Sum the following geometric progressions : — (1.) 6 + 18 + 54+ . . . to 12 terras. (2.) 6-18 + 54- . . . to 12 terms. (3.) -3333 ... to n terms. (4.) l-j + p-- ■ to ra terms. (5.) 6-4+. . . to 10 terms. (''•'v'3TiVlT2+---^°2°*^™^- (7.) l+g + g2+. . to?iterms. (8.) l-^ + ga" • • ■ to «= • (9.) l-a; + K^-a^+. . . tooo, a;,a;0(r + !r')• (21.) 2"(3''-i + 3«-H. . .+1). (22.) (-l)"a^''. (23.) Sum ton terms the series (?'" + l/r")- + (r«+i + l/r«+i)2+. . . . (24. ) Sum to m terms (1 + 1/rf + (1 + l/r^f + . . . (25.) Show that (a+6)»-J" = a6'-^ + aft"-V + &) + - • ■ +a{a + b)"-'^. (26.) If St denote the sum of n terms of a G.P. beginning with the ith term, sum the series S1 + S2+ . . . +S(. (27.) Show that ^(-037)= -3. 506 EXERCISES XLI chap. Sum the following series to n terms, and, where admissible, to infinity: — (28.) l--J>- + 3.t--4At3+. . . (29.) 1 --;;-+ «-!+.. . (30.)!-^+;-^:+ . (3i.)n-^.?^+i^+. . (32.) l'< + 2^x + Z^x'' + 12 3 12 3 (33.) ;; + ^, + „, + _-; + .. + .,.+ . . to oo , where the numerators recur ( I- r I* r' i" with the peiiod 1, 2, 3. (34. ) - + - + J + -j + -3 + -g + to 3)1 terms, where the numerators recur with the period a, b, c. (35.) A servant agrees to serve his master for twelve months, his wages to he one farthing for the first month, a penny for the second, fourpence for the third, and so on. "What did he receive for the year's service ? (36.) A precipitate at the bottom of a beaker of volume Y always retains about it a volume v of licjuid. It was originally precipitated in an alkaline solution ; find what percentage of this solution remains about it after it has been washeil 11 times by filling tlie beaker with distilled water and emptying it. Xeglect the volume of the precipitate itself. (37.) The middle points of the sides of a triangle of area A] form the vertices of a second triangle of area Ao ; from A.> a tliird triangle of area A3 is derived in the same way ; and so on, ad infinituin. Find the sum of the areas of all the triangles thus formed. (38.) OX, OY are two given straight lines. From a point in OX a perpend- icular is drawn to OY ; from the foot of that perpendicular a perpendicular on OX ; and so on, ad iiifiiiilum. If the lengths of the first and second per- pendicular be a and i respectively, find the sum of the lengths of all the per- pendiculars ; and also the sum of the areas of all the right-angled triangles in the figure whose vertices lie on OY and whose bases lie on OX. (39. ) The population of a certain town is P at a certain epoch. Annually it loses d per cent by deaths, and gains h per cent by births, and annually a fixed number E emigrate. Find the population after the lapse of m years. 2, a, y. It,, Z, having the meanings assigned to them in § 12, solve the following problems : — (40.) Express S in terms of a, n, I ; and also in terms of r, n, I. (41.) 3=440(1, rt = ll, ?i = 4, find ;■. (42.) 2 = 180, r=3, m = 5, findu. (43.) 2 = 95, «=20, « = 3, findr. (14.) 2 = 155, ft = 5, '/!, = 3, findr. (45.) 2 = 605, « = 5, '/• = 3, flndm. (46. ) If the second term of a G.P. be 40, and the fourth term 1000, find the sum of 10 terms. XX EXERCISES XLI, XLII 507 (47.) Insert one geometric mean between \/3/\/2 and 3\/3/2V2- (48. ) Insert three geometric means between 27/8 and 2/3. (49. ) Insert four geometric means between 2 and 64. (50.) Find tlie geometric mean of 4, 48, and 405. (51.) The geometric mean between two numbers is 12, and the arithmetic mean is 25J : find the numbers. (52.) Four numbers are in G.P., the sum of the first two is 44, and of the last two 396 : find them. (53. ) Find what common quantity must be added to a, t, c to bring them into G.P. (54.) To each of the first two of the four numbers 3, 35, 190, 990 is added X, and to each of the last two y. Tlie numbers then form a G.P. ; find x and y. (55.) Given the sum to infinity of a convergent G.P., and also the sum to infinity of the squares of its terms, find the first term and the common ratio. (56.) IfS = ai + a2+. . . +»„ be a G.P., then S' = l/ai + l/ffi2+ . . . +l/an is a G.P., and S/S'=aia„. (57.) If four quantities be in G.P., the sum of the squares of the extremes is greater than the sum of the squares of the means. (58.) Sura 2™ terms of a series in which every even term is a times the term before it, and every odd term c times the term before it, the first term being 1. (59.) If x=a + a/r + alr^+ . . . adm, y = b—h/r+ hjr- - ad oo , z = c + e/r^ + cjr* + ad oo , then xij/z=ai/c. (60.) Find the sum of all the products three and three of the teims of an infinite G.P., and if this be one- third the sum of the cubes of the terms, show that r=i. Exercises XLII. (1.) Insert two harmonic means between 1 and 3, and five between 6 and 8. (2.) Find the harmonic mean of 1 and 10, and also the harmonic mean of 1, 2, 3, 4, 5. (3.) Show that 4, 6, 12 are in H.P., and continue the progression both ways. (4. ) Find the H. P. whose 3rd term is 5 and whose 5th term is 9. (5.) Find the H.P. whose ^th term is P and whose g'th term is Q. (6.) Show that the harmonic mean between the arithmetic and geometric means of a and b is 2(a + &)/{((i/J)* + (i/a)^}=. (7.) Four numbers are proportionals ; show that, if the first three are in G.P., the last three are in G.P. (8.) Three numbers are in G.P. ; if each be increased by 15, they are in H.P. : find them. (9. ) Between two quantities a harmonic mean is inserted ; and between each adjacent pair of the three thus obtained is in.serted a geometric mean. 508 EXERCISES XLII CHAP, xx It is now found that the three inserted means are in A. P. : show that the ratio of the two quantities is 7 - 4^3 : 1. (10.) The sides of a right-angled triangle are in A. P. ; show that they are proportional to 3, 4, 5. (11.) a, b, c are in A.P., and u, b, d in H.P. : show that c/tZ = 1 - 2(a - b)-/ab. (12.) If X be any term in an A. P. whose two first terms are a, b, y, the term of the same order in a H. P. commencing with the same two terms, then {x-a)l{y~x) = bl{y-b). (13.) If a^, b"', c2 be in A.P., then ll{b + c), \l{c + a), ll{a + b) are in A.P. (14.) If P be the product of n quantities in G.P. , S their sum, and S' the sum of their reciprocals, then P^ = (S/S')". (15.) If a, b, c be the ^th, jth, and rth terras both of an A.P. and of a G.P., then a»-''J'-«c^-»=l. (16.) If P, Q, R be the^jth, ^th, rth terms of a H.P., then I,{PQ{p-q)} = 0, and |Z(,/- - r-yV'-] = = 4 [I.{q - r)/P=} {Xqr{q - r)/F'} . fl7. ) If the sum of m terms of an A.P. be equal to the sura of the next n, and also to the sum of the next^, then (m + ») (l/u- 'i.ll') = {n+p) {Ijm-l/n). (IS.) If the squared differences of p, q, r bo in A.P., then the differences taken in cyclical order are in H.P. (19.) If f6 + 6 + c, a- + i- + <;'^ a^ + 6^ + c' be in G. P. , prove that the common ratio is ^Sa - 3abc/2I,bc. (20.) If :r, y, z be in A. P., ax, by, c; in G.P., and a, b, c in H.P., then H.M. ofa, c:G.M. of a, c=H.M. of.r, =:G.M. of a;, z. (21.) lia? + b'^-c^, b- + c--a-, e' + a'-b'' be in G.P., then d^ji^ + c'lV^, i'/c- + c"/b'', a?jb''- + b'^lc' are in A.P. (22.) If a, b, c, d be in G.P., then abcdfx ^y = {^a)", and (23.) The sura of the n geometric means between it and k is (a"("+'»i- - aii''(''+i))/(F/("+i) - a;i'("+i)). (24.) If Ai, Aa, . . ., A„ be the « arithmetic means, and Hi, Ha, . . ., H„ the n harmonic means, between a and c, sum to n terras the series whose rth term is (A, - a) (Hr - «)/Hr. (25.) Ifai, (12, . . ., "„ be in G.P., then ("i + a2 + «3)" + (a2 + fe3+«4)^+ • • ■ +{an-i + ''i„-i + a-nf = («r + aido + a2-i-(i({-""' - a^'"-~*)l(a'i' - a^)ai^-\ (26.) \iar, br be the arithmetic and geometric means respectively between [ir-i and i,— 1 , show that a„-2={«J±(a„^-6„')*}^ b„-2=\a}zf:(a,?-b^-y'}-. (27.) If (ii, 02, ■ • ., On be real, and if {ai^^-ar+. . . +an-r){ch' + a3-+ . . . +an^) = {aiCh + a-za3+ . . . +a„_ia„),^ then Oi, «2, ...,«» are in G.P. CHAPTEE XXI. Logarithms. § 1.] It is necessary for the purposes of this chapter to define and discuss more closely than we have yet done the properties of the exponential function a^. For the present we shall sup- pose that a is a positive real quantity greater than 1. What- ever positive value, commensurable or incommensurable, we give to X, we can always find two commensurable values, m/n and {m+ l)/n (where m and n are positive integers), between which X lies, and which differ from one another as little as we please, see chap, xiii., § 15. In defining «* for positive values of x, we suppose X replaced by one (say m/n) of these two values, which we may suppose chosen so close together that, for the purpose in hand, it is indifferent which we use. We thus have merely to consider a™'"; and the understanding is that, as in the chapter on fractional indices, we regard only the real positive value of the mth root ; so that a'"/" may be read indifferently as ( ^a)™, or as ;/a™. For negative values of x we define a* by the equation a* = Ija'", in accordance with the laws of negative indices. § 2.] We shall now show that a", defined as above, is a continu- ous fimction of x susceptible of all positive values between and + oo . \st. Let y be any positive quantity greater than 1, and let n be any positive integer. Since «> 1, «!'"> 1 ; but, by suffi- ciently increasing n, we may make a^'" exceed 1 by as little as we please. Also, when n is given, we can, by sufficiently in- creasing m, make a™'" as great as we please.* Hence, whatever * See chap, xi., § 14. 510 (^ CONTINUOUS ITS GRAPH may be the value of y, we can so choose n that a}-'^ 1, we can find a value of x such that a* shall be as nearly equal to y as we please, Ind. Let y be positive and < 1 ; then l/y is positive and greater than 1. Hence we can find a value of x, say x', such that a^' = 1 Jy as nearly as we please. Hence a'''' = y. "We may make y pass continuously through all possible values from to + 00 . Hence a^ is susceptible of all positive values from to + CO . It is obviously a continuous function, since the difference of two finite values corresponding to a: = m/« and x = {m+l)ln is a"''"(ai/'' - 1), which can be made as small as we please by sufficiently increasing n. Cor. We have the following set of corresponding values : — a;=-co, -, -1, 0, +, 1, +<»; jf = a^ = 0, < 1, 1/a, 1, > 1, a, + co . Y 1 1 /.y / B /^ ^^ —-rrrT:^^^^''"'^ i\ i\ X In Fig. 1 the full-drawn curve is the graph of the function y = 10^ ; the dotted curve is the graph of y = 100*. XXI DEFINITION OF LOGARITHMIC FUNCTION 511 It will be observed that the two curves cross the axis of y at the same j)oint B, whose ordinate is + 1 ; and that for one and the same value of y the abscissa of the one curve is double that of the other. § 3.] The reasoning by which we showed that the equation y = ft% certain restrictions being understood, determines y as a continuous function of x, shows that the same equation, under the same restrictions, determines a as a continuous function of y. This point will perhaps be made clearer by grapiiical considera- tions. If we obtain the graph of ?/ as a function of x from the equation y = «*, the curve so obtained enables us to calculate x when y is given ; that is to say, is the graph of x regarded as a function of y. Thus, if we look at the matter from a graphical point of view, we see that the continuity of the graph means the continuity of y as a function of x, and also the continuity o.f x as a function of y. When we determine x as a function of y by means of the equation y = »=", we obviously introduce a new kind of transcend- ental function into algebra, and some additional nomenclature becomes necessary to enable us to speak of it with brevity and clearness. , The constant quantity a is called the base. y is called the exponential of x to base a (and is sometimes written expa^).* X is called the logarithm of y to base a, and is usually written ^og„2/. The two equations y = a^ (1), « = log«y (2), are thus merely different ways of writing the same functional relation. It follows, therefore, that all the properties of our new. logarithmic function must be derivable from the properties of our old experiential function, that is to say, from the laws of indices. * This notation is little used in elementary text-books, , but it is con- venient when in place of x we have some complicated function of x. Thus expa(l -t-K-f ic^) is easier to print than 0^+'^+''^. 51 3 FUNDAMENTAL PROPERTIES OF LOGARITHMS CHAP. The student should also notice that it follows from (1) and (2) that the equation y = d°Say (3) is an identity. § 4. J If the same base a be understood throughout, we have the following leading properties of the logarithmic function : — I. The logarithm of a product of positive numbers is the sum of the logarithms of the separate factors. II. The logarithm of the quotient of two positive numbers is the excess of tlie logarithm of the dividend over the logarithm of the divism: III. The logarithm of any power {positive or negative, integral or fractional) of a positive number is equcd to the logarithm of the number multiplied by the power. Let yi,y.2, . . ., y-n be n positive numbers, a;,, :r:^, . ., r.^ their respective logarithms to base a, so that «i=log«2/i, «2 = log«?/2, . ., a-„ = log„y„. By the definition of a logarithm wo have ?/, = «="!, «/2 = «^-, • • •, 2/« = «*'- ' Hence ?/,//, . . . y„ = a^ia""^ ...«'*' = a^i+^J+ ■ • ■ +^, by the laws of indices. Hence, by the definition of a logarithm, X,+X,_+. . . +Xn = logaiy^y^ ■ ■ ■ yn), that is, log„yi + loga.% + . + log ^4/„ = log ^(y, «/„... 7/„). We have thus established I. Again, y^jy^ = «^i/a^2 = a'^i -==2^ by the laws of indices. Hence, by the definition of a logarithm, x,-x^ = \oga(y,ly.^, that is, log^y, - log^?/, = logail/Jy^), Avhich is the analytical expression of II. Again, y^'' = (a^i)'' = a"\ hy the laws of indices. XXI EXAMPLES 513 Hence, by the definition of a logarithm, ra, = log a «/,'-, that is, r \ogay, = log^y/, which is the analytical expression of lit. Example 1. log 21 = log (7 X 3) = log 7 + log 3. As the equation is true for any base, provided all the logarithms have the same base, it is needless to indicate the base by writing the suffix. Example 2. log (113/29) = log 113 -log 29. Example 3. log (540/539) = log (2=33. 5/72.II), = 2 log 2 + 3 log3 + log5-2 1og7-logll. Example i. log ^(49)/ ;/(21)=log(72/V3W7W), = f log7-f log3-f log7, =Hlog7-flog3. COMPUTATION AND TABULATION OF LOGARITHMS. § 5.] If the base of a system of logarithms be greater than unity, we have seen that the logarithm of any positive number greater ■ than unity is positive , and the logarithm of any positive number less than unity is negative. The logarithm of unity itself is always zero, whatever the base may be. The logarithm of the base itself is of course unity, since a = a\ The logarithm of any power of the base, say (f, is r ; and, in particular, the logarithm of the reciprocal of the base is - \. The logarithm of + 00 is + , and the logarithm of is - . It is further obvious that the logarithm of a negative number could not (with our present understanding) be any real quantity. With such, however, we are not at present concerned. The logarithm of any number which is not an integral power of the base will be some fractional number, positive or negative, as the case may be. For reasons that will appear presently, it is usual so to arrange a logarithm that it consists of a positive VOL. I 2 L 514 DETERMINATION OF CHAEACTEEISTIC fractional part less than unitj-, and an integral part, positive or negative, as the case may be. The positive fractional part is called the mantissa. The integral part is called the characteristic. For example, the logarithm of '0451 to base 10 is the negative number - 1-3458235. In accordance with the above understand- ing, we should write logi„-0451= -1-3458235= - 2 + (1 --3458235), = -2 + -6541765. For the sake of compactness, and at the same time to prevent confusion, this is usually written log ,„-0451 = 2-6541765. In this case then the characteristic is 2 (that is, - 2), and the mantissa is -65417G5. § 6.] To find the logarithm of a given number y to a given base (( is the same problem as to solve the equation "■'■ = 2/> where a and y are given and x is the unknown quantity. There are various ways in which this may be done ; and it will be instructive to describe here some of the more elementary, although at the same time more laborious, approximative methods that might be used. In the first jslace, it is always easy to find the characteristic or integral part of the logarithm of any given number y. We have simply to find by trial two consecutive integral powers of the base between which the given number y lies. The algebraically less of these two is the characteristic. Example 1. To find the characteristic of logs 451. We liave the following values for consecutive integral powers of 3 : — Power 1 2 3 4 5 6 "\'alue ' 9 27 81 243 729 XXI LOGAEITHMS TO BASE 10 Hence 3° < 451 < 3". Hence log3451 lies between 5 and 6. Therefore logs 451 = 5 + a proper fraction. Hence char, logs 451 = 5. Example 2. Find the characteristic of log3'0451. We have 515 Powers of Base -1 -2 -3 Values 1 ■333 . . . ■111. . . •037 . . . Hence 3-'< '0451 < 3"^ ; that is to say, log 3'0451 = - 3,+ a proper fraction. Hence char. Iog3'0451=3. When the base of the system of logarithms is the radix of the scale of numerical notation, the characteristic can always he obtained by merely counting the digits. For example, if the radix and base be each 10, then If the number have an integral part, the characteristic of Us logarithm is + {one less than the number of digits in the integral part). If the number have no integral part, the characteristic is - (one more than the number of zeros that follow the decimal point). The proof of these rules consists simply in the fact that, if a number lie between 10" and 10"+^, the number of digits by which it is expressed is n + \ ; and, if a number lie between lO'C^+i) and 10"", the number of zeros after the decimal point is n. For example, 351 lies between lO^and 10'. Hence char. Iogio351 =2 = 3-1, according to the rule. Again, "0351 lies between ■Ol and 1, that is, between 10"== and 10"'. Hence char, logic "0361= -2= -(1 + 1), which agrees with the rule. The rule suggests at once that, if Id be adopted as the base of our system of logarithms, then the characteristic of a logarithm depends merely on the position of the decimal point , and the ma/ntissa is in- dependent of the -position of the decimal point, but depends merely on the succession of digits. 516 COMPUTATION OF LOGARITHMS BY SOLUTION OF a^ = y chap. "We may formally prove this important proposition as follows : — Let N be any number formed by a given succession of digits, c the characteristic, and in the mantissa of its logarithm. Then any other number which has the same succession of digits as N, but has the decimal jjoint placed differently, will have the form 10*N", where i is an integer, positive or negative, as the case may be. But log.JO'N = logiolO^ + log,„N, by § 4, = j + log,„N = (i + c) + m. Now, since by hypothesis m is a positive proper fraction, and c and i are integers, the mantissa of log,„10'N is m, and the characteristic is i + c. In other words, the characteristic alone is altered by shifting the decimal point. § 7.] The process used in § 6 for finding the characteristic of a logarithm can be extended into a method for finding the mantissa digit by digit. Example. To calculate logic 4 "21 7 to tliree places of decimals. The characteristic is obviously 0. Let the three first digits of the mantissa be xyz. Then we have 4•217 = 10»•''^'^ hence (4-217)1''= 10»^•»^ We must now calculate the 10th power of 4 '217. In so doing, however, there is no need to find all the significant figures — a few of the highest will sufl^ce. We thus find 1778400 = 10^-;». We now see that x is the characteristic of logiol778400. Hence »= 6. Dividing by 10*, and raising both sides of the resulting equation to the 10th power, we find (l-778)i« = 10i'-' ; hence 315-7 = 10»-=. Hence y = 1. Dividing by 10", and raising to the 10th power, we now find (3-16)"=10'; hence 99280 = 10'. Hence ~ = 5 very nearl}'. We conclude, therefore, that log 10 4 -21 7= '625 nearly. This method of computing logarithms is far too laborious to be of any practical use, even if it were made complete by the addition of a test to ascertain what effect the figures neglected in the calculation of the 10th powers produce on a given decimal place of the logarithm ; it has, however, a certain theoretical XXI COMPUTATION BY INSERTING GEOMETEIC MEANS 517 interest on account of its direct connection with the definition of a logarithm. By a somewhat similar process a logarithm can be expressed as a continued fraction. § 8.] If a series of numbers he in geometric progression, their logarithms are in arithmetic pvgression. Let the numbers in question be y^, y„, y^, . . ., «/„. Let the logarithm of the first to base a be a, and the logarithm of the common ratio of the G.P. yi, ya, yi, ■ • ■, yn to the same base be (3. Then we have the following series of corresponding values : — Vl, 2/2, Vs, ■ ■ ; tin, II II II II from which the truth of the proposition is manifest. As a matter of history, it was this idea of comparing two series of numbers, one in geometric, the other in arithmetic pro- gression, that led to the invention of logarithms ; and it was on this comparison that most of the early methods of computing them were founded. The following may be taken as an example. Let us suppose that we know the logarithms x,^ and a^ of two given numbers, y^ and 2/9 ; then we can find the logarithms of as many numbers lying between y^ and y^ as we please. We have 2/1 = a"^', 2/9 = a"^- Let us insert a geometric mean, y^, between y^ and y^, then 2/5=(Mo)^ = «^^^+'^^'' = *^say, where q\ is the arithmetic mean between x^ and x^. We have now the following system : — Logarithm x^ z^ x^, Number y, y, y^. Next insert geometric means between y,, y^ and y^,, y^. The logarithms of the corresponding numbers will be the arithmetic means between x^, x^ and x^, a-g. We thus have the system — Logarithms x^, x^, x^, Xj, a-g ; Numbers y„ y^, y„ y,, y,. 518 COMPUTATION BY INSERTING GEOMETRIC MEANS CHAP. Proceeding in like manner, we derive the system — Logarithms x^, x^, x^, a\, x^, x^, x^, Kg, »„; Numbers y„ y„ y.„ y„ y„ y„ y,, y^, y^; and so on. Each step in this calculation requires merely a multi- plication, the extraction of a square root, an addition accompanied by division by 2, and each step furnishes us with a new number and the corresponding logarithm. Since 9',, rcj, . . ., Xn form an A.P., the logarithms are spaced out equally, but the same is not true of the corresponding num- bers which are in G.P. It is therefore a table of antilogarithms * that we should calculate most readily by this method. It will be observed, however, that by inserting a sufficient number of means we can piake the successive numbers differ from each other as little as we please ; and by means of the method of interpola- tion by first differences, explained in the last section of this chapter, we could space out the numbers equally, and thus con- vert our table of antilogarithms into a table of logarithms of the ordinary kind. As a numerical example we may puta = 10, j/i = l,j/9 = 10; thenai = 0, X9 = l. Proceeding as above indicated, we should arrive at the following table : — Numtier. Logarithm. Nuinber. Logarithm. 1-0000 0-0000 4-2170 0-6250 1-3336 ■ 0-1250 5-6235 0-7500 1-7783 0-2500 7-4990 0-8750 2-37] 4 0-3750 10-0000 1-0000 3-1622 0-5000 § 9.] In computing logarithms, by whatever method, it is obvious that it is not necessary to calculate independently the logarithms of composite integers after we have found to a suffi- cient degree of accuracy the logarithms of all primes up to a certain magnitiide. Thus, for example, log 4851 = log 3^.7^11 = 2 log 3-1-2 log 7 + log 11. Hence log 4851 can be found when the logarithms of 3, 7, and 11 are known. * By the antilogarithm of any number N is meant the number of which N is the logarithm. XXI CHANGE OF BASE 519 Again, having computed a system of logarithms to any one base a, we can without difficulty deduce therefrom a system to any other base b. All we have to do is to multiply all the logarithms of the former system by the number /t = 1/loga J. For, if x = \ogi,y, then y = i^. Hence \ogay = logab^, = X log J, by § 4. Hence logj?/ = a: = loga2//log^6 (1). The number /x is often called the modulus of the system whose base is b with respect to the system whose base is a. Cor. 1. If in the equation (1) we put y = a, we get the following equation, which could easily be deduced more directly from the definition of a logarithm : — logi,a = l/log„&, or logab log^a = 1 (2). Cor. 2. The equation y = i^ may be written Hence the graph of the exponential If can be deduced from the graph of the exponential a" by shmiening or lengthening all the abscisses of the latter in the same ratio 1 : log^b. This is the general theorem corresponding to a remark in § 2. We may also express this result as follows : — Give7i any two exponential graphs A and B, then either A is the orthogonal p-qjection of B, or B is the orthogonal projection of A, on a plane passing through the axis of y. USE OF LOGARITHMS IN ARITHMETICAL CALCULATIONS. § 10.] AVe have seen that, if we use the ordinary decimal notation, the system of logarithms to base 10 possesses the im- portant advantages that the characteristic can be determined by inspection, and that the mantissa is independent of the position of the decimal point. On account of these advantages this system is used in practical calculations to the exclusion of all others. 520 SPECIMEN OF LOGARITHMIC TABLE. Ko. 1 2 3 4 5 6 7 8 9 DifF. 3050 484 2998 3141 3283 3426 3568 3710 3853 3995 4137 4280 51 4422 4564 4707 4849 4991 5134 6276 5418 5561 5703 52 5845 5988 6130 6272 6414 6557 6699 6841 6984 7126 53 7268 7410 7553 7695 7837 7979 8121 8264 8406 8548 54 8690 8833 8975 9117 9259 9401 9543 9686 9828 9970 55 485 0112 0254 0396 0539 0681 0823 0965 1107 1249 1391 56 1533 1676 1818 1960 2102 2244 2386 2528 2670 2812 57 2954 3096 3239 3381 3523 3665 3807 3949 4091 4233 142 58 4375 4517 4659 4801 4943 5086 5227 5369 5611 6653 1 14 2 28 3 43 4 57 59 5795 5937 6079 6221 6363 6505 6647 6788 6930 7072 60 7214 7356 7498 7640 7782 7924 8066 8208 8350 8491 3061 8633 8775 8917 9069 9201 9343 9484 9626 9768 9910 5 71 62 486 0052 0194 0336 0477 0619 0761 0903 1045 1186 1328 6 85 63 1470 1612 1764 1895 2037 2179 2321 2462 2604 2746 7 99 3 114 9 128 64 2888 3029 3171 3313 3455 3596 3738 3880 4021 4163 65 4305 4446 4588 4730 4872 5013 5155 5297 5438 5680 66 5722 6863 6003 6146 6288 6430 6571 6713 6855 6996 67 7138 7279 7421 7563 7704 7846 7987 8129 8270 8412 68 8554 8695 8837 8978 9120 9261 9403 9544 9686 9827 69 9969 0110 0252 0393 0535 0676 0818 0959 1101 1242 70 4871384 1525 1667 1808 1950 2091 2232 2374 2515 2657 3071 2798 2940 3081 3222 3364 3505 3647 3788 3929 4071 72 4212 4353 4495 4636 4778 4919 5060 5202 6343 6484 73 5626 5767 5908 6050 6191 6332 6473 6616 6766 6897 74 7039 7180 7321 7462 7604 7745 7886 8027 8169 8310 75 8451 8592 8734 8875 9016 9167 9299 9440 9681 9722 76 9863 0004 0146 0287 0428 0569 0710 0852 0993 ll34 77 4881275 1416 1557 1698 1839 1981 2122 2263 2404 2545 78 2086 2827 2968 3109 3261 3392 3533 3674 3815 3956 79 4097 4238 4379 4520 4661 4802 4943 5084 5226 5366 80 5507 5648 5789 5930 6071 6212 6353 6494 6635 6776 141 1 14 2 28 3081 6917 7058 7199 7340 7481 7622 7763 7904 8045 8186 82 8326 8467 8608 8749 8890 9031 9172 9313 9454 9594 3 42 83 9735 9876 0017 0158 0299 0440 0580 0721 0862 1003 4 66 84 4891144 1285 1425 1566 1707 1848 1989 2129 2270 2411 5 71 85 2552 2692 2833 2974 3115 3256 3396 3537 3678 3818 6 85 7 99 8 113 86 3959 4100 4241 4381 4522 4663 4804 4944 5085 5226 87 5366 5507 5648 5788 5929 6070 6210 6351 6492 6632 9 127 88 6773 6914 7064 7195 7335 7476 7617 7757 7898 8038 89 8179 8320 8460 8601 8741 8882 9023 9163 9304 9444 90 9585 9725 9866 0006 0147 0287 0428 0569 0709 0850 3091 490 0990 1131 1271 1412 1552 1693 1833 1973 2114 2254 92 2395 2535 2676 2816 2957 3097 3238 3378 3518 3669 93 3799 3940 4080 4220 4361 4501 4642 4782 4922 5063 94 5203 6343 5484 5624 5765 5905 6046 6186 6326 6466 95 6C07 6747 6887 7027 7168 7308 7448 7589 7729 7869 96 8010 8150 8290 8430 8571 8711 8851 8991 9132 9272 97 9412 9552 9693 9833 9973 0113 0253 0394 0534 0674 98 4910814 0954 1094 1235 1375 1515 1655 1795 1935 2076 99 2216 2356 2496 2636 2776 2916 3057 3197 3337 3477 3100 3617 3757 3897 4037 4177 4317 4457 4597 4738 4878 XXI USE OF LOGARITHMIC TABLE 521 In printing a table of logarithms to base 10 it is quite un- necessary, even if it were practicable, to print characteristics. The mantissse alone are given, corresponding to a succession of five digits, ranging usually from 10000 to 99999.* A glance at p. 520, which is a facsimile of a page of the logarithmic table in Chambers's Mathematical Tables, will show the arrangement of such a table. To take out the logarithm of 30715 from the table, we run down the column headed "No." until we come to 3071 ; the first three figures of the mantissa are 487 (standing over the blank in the first half column) ; the last four are found by running along the line till we reach the column headed 5, they are 3505. The characteristic is seen by inspection to be 4. Hence log 30715 = 4-4873505. To find the number corresponding to any given logarithm we have of course simply to reverse the process. To find the logarithm of '030715 we have to proceed exactly as before, only a different characteristic, namely 2, must be pre- fixed to the mantissa. We thus find log -030715 = 2-4873505. If we wish to find the logarithm of a number, say 3-083279, where we have more digits than are given in the table, then we must take the nearest number whose logarithm can be found by means of the table, that is to say, 3-0833. We thus find log 3-0833 = 0-4890158f nearly. Greater accuracy can be at- tained by using the column headed "Difi".," as will be explained presently. Conversely, if a logarithm be given which is not exactly coincident with one given in the table, we take the one in the table that is nearest to it, and take the corresponding number as an approximation to the number required. Greater accuracy can ■ be obtained by using the difference column. Thus the number whose logarithm is 1-4872191 has for its first five * For some purposes an extension of the table is required, and such ex- tensions are supplied in various ways, which need not be described here. For rapidity of reference in calculations that require no great exactness a short table for a succession of 3 digits, ranging from 100 to 999, is also usually given. t The bar over 0158 indicates that these digits follow 489, and not 488. 522 NUMBER OF FIGURES REQUIRED chap. significant digits 30705 ; but, if we wish the best approximation with five digits, we ought to take 30706. Since the character- istic is 1, the actual number in question has two integral digits. Hence the required number is 30'706, the error being certainly less than -0005. § 11.] The principle which underlies the application of logarithms to arithmetical calculation is the very simple one that, since to any number there corresponds one and only one logarithm, a number am he identified by means of its logarithm. It is this principle which settles how many digits of the mantissa of a logarithm it is necessary to use in calculations which require a given degree of accuracy. Suppose, for example, that it is necessary to be accurate down to the fifth significant figure ; and let us inquire whether a table of logarithms in which the mantissse are given to four places would be sufficient. In such a table we should find log 3-0701 = 0-4871, log 3-0702 = 0-4871 ; the table is therefore not sufficiently extended to distinguish numbers to the degree of accuracy required. Five places in the mantissa would, in the present' instance, be sufficient for the purpose; for log3'0701 = 0-48715, log 3-0702 = 0-48716. Towards the end of the table, however, five places would scarcely be sufficient ; for log 9-4910 = 0-97731 and log 9-4911 = 0-97731. § 12.] The great advantage of using in any calculation logarithms instead of the actual numbers is that we can, in accordance with the rules of § 4, replace every multiplication by an addition, every division by a subtraction, and every operation of raising to a power or extracting a root by a multiplication or division. The following examples will illustrate some of the leading cases. We suppose that the student has a table of the loga- rithms of all numbers from 10000 to 100000, giving mantissse to seven places. Example 1. Calculate the value of 1-6843 x -00132-=- -3692. If A = 1-6843 X -00132^- -3692, log A = log 1 -684.3 + log -00132 - log -3692, EXAMPLES 523 log 1-6843 = -2264194 log -00132 = 3 3 loff-3692 =1 1205739 3469933 5672617 logA = 3-7797316. Hence A= -0060219. Observe that the negative characteristics must be dealt with according to algebraic rules. Example 2. To extract the cube root of -016843. Let A = (-016843)W then log A = J log -016843, = 1(2-2264194), = K3 + 1-2264194), = f -4088065. A= -25633. Example 3. Calculate the value of A = (368)™/(439)5'9. LogA={log368-flog439. J log 368=-J(2-5658478) = 5-9869782 ^ log 439 = 1(2-6424645) = 1 -4680358 logA = 4-5~189424 A= 33033, Example 4. Find how many digits there are in A = (1-01)'™°". logA = 100001ogl-0], = 10000 X -0043214, = 43-214. Hence the number of digits in A is 44. Example 5. To solve the exponential equation 1-2»^=1-1 by means of logarithms. We have logl-2» = logl-l. Therefore a!logl-2 = log 1-1. _ log 1-1 _ ;041 3927 ^™''® ^~logl-2~.-0791812' Hence log k= log -0413927 - log -0791812, = 1-7183059. Therefore x= -52276. iJcmarA:.— It is obvious that we can solve any such equation as a^''-'"'-^ = b, where ^, q, a, b are all given. For, taking logarithms of both sides, we have {x^ -px + q)loga=logb. We can now obtain the value of x by solving a quadratic equation. 524 INTERPOLATION BY FIRST DIFFERENCES INTERPOLATION BY FIRST DIFFERENCES. § 13.] The method by which it is usual to find (or "interpo- late ") the value of the logarithm of a number which does not happen to occur in the table is one which is applicable to any function whose values have been tabulated for a series of equi- different values of its independent variable (or " argument "). The general subject of interpolation belongs to the calculus of finite differences, but the special case where first diflferences alone are used can be explained in an elementary way by means of graphical considerations. We have already seen that the increment of an integral function of x of the 1st degree, y = Ax + 'B say, is proportional to the increment of its argument ; or, what comes to the same thing, if we give to the argument x a series of equidifferent values, a, a + h, a + 2h, a + 3/t, &c., the function y will assume a series of equidifferent values Aa + B, Aa + B + Ah, Aa + B + 2 Ah, A,' + B + 3Ah, S:c If, therefore, we were to tabulate the values of Ax + B for a series of equidiff'erent values of x, the differences between suc- cessive values of y (" tulmlar differences ") would be constant, no matter to how many places we calculated y. Conversely, a function of x which has this property, that the differences between the successive values of y corresponding to equidifferent values of x are absolutely constant, must be an in- tegral function of x of tlie 1st degree. If, however, we take the difference, h, of the argument small enough, and do not insist on accuracy in the value of y beyond a certain significant figure, then, for a limited extent of the table of any function, it will be found that the tabular differences are constant. Eef erring, for example, to p. .520, it will be seen that the difference of two consecutive logarithms is constant, and equal to '0000141, from log 30660 up to log 30S99, or that there is merely an accidental difference of a unit in the last place ; that is to say, the difference remains constant for about 240 entries. LIMITS OF THE METHOD 525 A similar phenomenon will be seen in the following extract from Barlow's Tables, provided we do not go bej^ond the 7th significant figure : — Number. Cube Root. Diff. 2301 13-2019740 19122 2302 13-2038862 19117 2303 13-2057979 19111 2304 13-2077090 19105 2305 13-2096195 Let us now look at the matter graphically. Let ACSDQB be a portion of the graph of a function y = f{x) ; and let us suppose that up to the mth significant figure the diff'erences of y are constant for equidifferent values of x, lying between OE and OH. This means that in calculating (up to the nth significant figure) values of y corresponding to values of X between OE and OH we may replace the graph by the straight line AB. Thus, for example, if a; = OM, then /(OM) = MQ ; and PM is the value calculated by means of the straight line AB. Our state- ment then is that PM - QM, that is PQ, is less than a unit in the n significant place. If this be so, then, a fortiori, it will be so if we replace a portion of the graph, say CD, lying between A and B by a straight line joining C and D. In other words, if up to the nth place the increment of the func- tion for equidifferent values of x he constant, between certain limits, then, to that degree of accuracy at least, the increment of the function leill he proportional to the increment of the argument far all values those limits. § 14.] Let us now state the conclusion of last article under fl-6 RULE OF PROPORTIONAL PARTS chap. an analytical form, all the limitations before' mentioned as to constancy of tabular (or first) difference being supposed fulfilled. Let h be the difference of the arguments as they are entered in the table, D the tabular difference f{a + h) -f(a), a + h' a, value of the argument, which does not occur in the table, but which lies between the values a and a + h, which do occur, so that h';„_ia^, then Xix^. . . x„ = l. Historical Note. — The honour of devising the use of logarithms as a means of abbreviating arithmetical calculations, and of publishing the first logarithmic table, belongs to John Napier (1550-1617) of Merchiston (in Napier's day near, in our day in, Edinburgh). This invention was not the result of a casual inspira- tion, for we learn from Napier's Habdologia (1617), in which he describes three other methods for facilitating arithmetical calculations, among them his calculat- ing rods, which, under the name of " Napier's Bones," were for long nearly as famous as his logarithms, that he had devoted a great part of his life to the con- sideration of methods for increasing the power and diminishing the labour of arithmetical calculation. Napier published his invention in a treatise entitled "Miriflci Logarithmorum Canonis Descriptio, ejusque usus, in utraque Trigo- nometria ut etiam in omni Logistica Mathematica, Amplissimi, faoillimi, et expeditissimi explicatio. Authore ac Inventore loanne Nepero, Barone Merchis- tonii, &c., Scoto, Edinburgi (1614)." In this work he explains the use of logarithms ; and gives a table of logarithmic sines to 7 figures for every minute of the quadrant. In the Canon Mii-ificus the base used was neither 10 nor what is now called Napier's base (see the chapter on logarithmic series in the second part of this work), Napier himself appears to have been aware of the advantages of 10 as a base, and to have projected the calculation of tables on the improved plan ; but his infirm health prevented him from carrying out this idea ; and his death three years after the publication of the Canon Mirificns prevented him from even publishing a description of his methods for calculatiag logarithms. This work, entitled Mirifici Logarithmorum Canonis Constructio, &c., was edited by one of Napier's sons, assisted by Henry Briggs. To Henry Briggs (1556-1630), Professor of Geometry at Gresham College, and afterwards Savilian Professor at Oxford, belongs the place of honour next to Napier in the development of logarithms. He recognised at once the merit and seized the spirit of Napier's invention. The idea of the superior advantages of a decimal base occurred to him independently ; and he visited Napier in Scotland in order to consult with him regarding a scheme for the calculation of a logarithmic table of ordinary numbers on the improved plan. Finding Napier in possession of the same idea in a slightly better form, he adopted his suggestions, and the result of the visit was that Briggs undertook the work which Napier's declining health had interrupted. Briggs published the first thousand of his logarithms in 1617 ; and, in his Arithmetica Zogarithmica, gave to 14 places of decimals the logarithms of all integers from 1 to 20,000, and from 90,000 to 100,000. In the preface to the last-mentioned work he explains the methods used for calculating the logarithms themselves, and the rules for using them in arithmetical calculation. VOL. I 2 M 530 HISTORICAL NOTE chap, xki While Briggs was engaged in filling up the gap left in his table, the work of calculating logarithms was taken up in Holland by Adrian Macq, a bookseller of Gouda. He calculated the 70,000 logarithms which were wanting in Briggs' Table ; and published, in 1628, a table containing the logarithms to 10 places of decimals of all numbers from 1 to 100,000. The work of Briggs and Vlacq has been the basis of all the tables published since their day {with the exception of the tables of Sang, 1871) ; so that it forms for its authors a monument both lasting and great. In order fully to appreciate the brilliancy of Napier's invention and the merit of the work of Briggs and Vlacq, the reader must bear in mind that even the exponential notation and the idea of an exponential function, not to speak of the inverse exponential function, did not form a part of the stock-in-trade of mathe- maticians till long afterwards. The fundamental idea of the correspondence of two series of numbers, one in arithmetic, the other in geometric progression, which is so easily represented by means of indices, was explained by Napier through the conception of two points moving on separate straight lines, the one with uniform, the other with accelerated Aelocity. If the reader, with all his acquired modern knowledge of the results to be arrived at, will attempt to obtain for him- self in this way a demonstration of the fundamental rules of logarithmic calcula- tion, he will rise from the exercise with an adequate conception of the penetrating genius of the inventor of logarithms. For the full details of this interesting part of mathematical history, ami in particular for a statement of the claims of Jost Biirgi, a Swiss contemporary of Napier's, to credit as an independent inventor of logarithms, we refer the student to the admirable articles "Logarithms" and "Napier," by J. W. L. Glaisher, in the Encyclopwdia Brilannica (9th ed.). An English translation of the Canstructio, with valuable bibliographical notes, has been iiublished by Mr. W, R. Maodonald, F.F.A. (Edinb. 1889). CHAPTEE XXII. Theory of Interest and Annuities Certain. § 1.] Since the mathematical theory of interest and annuities affords the best illustration of the principles we have been dis- cussing in the last two chapters, we devote the present chapter to a few of the more elementary propositions of this important practical subject. What we shall give will be sufficient to enable the reader to form a general idea of the principles involved. Those whose business requires a detailed knowledge of the matter must consult special text-books, such as the Text-Booh of the Institute of Actuaries, Part I., by Sutton.* SIMPLE AND COMPOUND INTEREST. § 2.] When a sum of money is lent for a time, the borrower pays to the lender a certain sum for the use of it. The sum lent is spoken of as the capital or principal ; the payment for the privilege of using it as interest. There are various ways of arranging such a transaction ; one of the commonest is that the borrower repays alter a certain time the capital lent, and pays also at regular intervals during the time a stated sum by way of interest. This is called paying simple interest on the borrowed capital. The amount to be paid by way of interest is usually stated as so much per cent per annum. Thus 5 per cent (5 %) per annum means £5 to be paid on every £100 of capital,- for * Full references to the various sources of information will be found in the article " Annuities " (by Sprague), Emyclopxdia Britannica, 9th edition, vol. ii. 532 AMOUNT PRESENT VALUE DISCOUNT chap. eveiy year that the capital is lent. In the case of simple interest, the interest payable is sometimes reckoned strictly in proportion to the time ; that is to say, allowance is made not only for whole years or other periods, but also for fractions of a period. Sometimes interest is allowed only for integral multiples of a period mutually agreed on. "We shall suppose that the former is the understanding. If then r denote the interest on £1 for one year, that is to say, one-hundredth of the named rate per cent, n the time reckoned in 3ears and fractions of a year, P the prindpal, I the whole interest paid, A the amount, that is, the sum of the principal and interest, both reci<:oned in pounds, we have I = ?M-P (1); A = I + P = P(1 +nr) (2). These formulae enable us to solve all the ordinary problems of simple interest. If any three of the foiu' I, n, r, P, or of the four A, n, r, P, be given, (1) or (2) enaljlcs us to find the fourth. Of the various problems that thus arise, that of finding P when A, 11, r are given is the most interesting. We suppose that a sum of money A is due n years hence, and it is required to find what sum paid down at once would be an equitable equivalent for this debt. If simple interest is allowed, the answer is, such a sum P as would at simple interest amount in n years to A. In this case P = A/(l + nr) is called the present value of A, and the difference A - P = A{1 - 1/(1 + mr)} = Anr/(1 + nr) is called the discount. Discount is therefore the deduction allowed for immediate payment of a sum due at some future time. The discount is less than the simple interest (namely knr) on the sum for the period in question. When n is not large, this difference is slight. Example. Find tlie difference between the interest and the discount on £1525 due nine months hence, reckoning simple interest at 3J "/o- The difference in question is given by Anr - Anr/d + nr) = AnV/{\ + nr). xxn COMPOUND INTEREST 533 In the present case A = 1525, ■ m = 9/12 = 3/4, r=3-5/100=-OS5. Hence Difference = 1525 x {•02625)2^(1-02625), = £1 :0:5i. § 3.] In last paragraph we supposed that the borrower paid up the interest at the end of each period as it became due. In many cases that occur in practice this is not done ; but, instead, the borrower pays at the end of the whole time for which the money was lent a single sum to cover both principal and interest. In this case, since the lender loses for a time the use of the sums accruing as interest, it is clearly equitable that the borrower should pay interest on the interest ; in other words, that the interest should be added to the principal as it becomes due. In this case the principal or interest-bearing capital periodically increases, and the borrower is said to pay compound interest. It is important to attend carefully here to the under- standing as to the period at which the interest is supposed to become due, or, as it is put technically, to be convertible (into principal) ; for it is clear that £100 will mount up more rapidly at 5 % compound interest convertible half-yearly than it will at 5 % compound interest payable annually. In one year, for in- stance, the amount on the latter hypothesis will be £105, on the former £105 plus the interest on £2 : 10s. for a half-year, that is, £105 : 1 :3. In what follows we shall suppose that no interest is allowed for fractions of the interval (conversion-period) between the terms at which the interest is convertible, and we shall take the conversion-period as unit of time. Let P denote the principal, A the accumulated value of P, that is, the principal together with the compound interest, in n periods ; r the interest on £1 for one period ; 1 + r = R the amount of £1 at the end of one period. At the end of the first period P will have accumulated to P + Pr, that is, to PE. The interest-bearing capital or principal during the second period is PE ; and this at the end of the second period will have accumulated to PR + PRr, that is, to PR^. The principal during the third period is PE°, and the amount at the end of that period PR^, and so on. In short, the 534 EXAMPLES CHAP. rimmint increases in a geometrical progression whose common ratio is R ; and at the end of n pcriuds we sliall luive A = PE" (1). By means of this equation we can solve all the ordinary problems of compound interest ; for it enables us, when any three of the four quantities A, P, R, n are given, to determine the fourth. In most cases the calculation is greatly facilitated by the use of logarithms. See the examples worked below. Cor. 1.7/1 denote the whole compound interest on P during the n piei'iods, we have I=A-P = P(R"-1) (2). Cor. '2. If V denote, the present valim of a sum A dae n periods hence, compound interest being cdlowed, then, since F must in n periods amount to A, we have A = PR", so that P = A/R» (3). The discount on the present understanding is therefore A(l - 1/R") (4). Example 1. Find the amount in two years of £2.350 : 5 : 9 at 3i "/o compound interest, convertible quarterly. Here P=2350-2875, a = 8, r = 3-.5/400= -0087.5, R = l-00875. log A = log P + ?i log R, log P = 3-3ni210 »logE,*= -0302684 3-4013894 A = £2519-936 = £2519 : 18 : 8. Example 2. How long will it take £186 : 11 : 9 to amount to £216 : 9 : 7 at 6 % com- pound interest, convertible half-yearly. "When n is very large, the seven figures given in ordinary tables hardly afford the necessary accuracy in the product n log R. To remedy this defect, supplementary tables are usually given, which enable the computer to find very readily to 9 or 10 places the logarithms of numbers (such as R) which differ little from unity. XXir NOMINAL AND EFFECTIVE KATE 535 Here P= 186 7375, A = 216-4792, R = l-03. 71= (log A - Tog P)/log E ■0128372 Hence the required time is five half-years, that is, 2A years. Example 3. To find the present value of £1000 due 50 years hence, allowing compound interest at i 7„, convertible half-yearly. HereA = 1000, «.=100, E = l-02. "We have P = A/E". log P = log 1000 - 100 logl-02, = 3-100x -0086002, = 2-1399800. P = £138-032 = £138 : : 8. § 4.] In reckoning compound interest it is very usual in practice to reckon by the year instead of by the conversion- period, as we have done above, the reason being that different rates of interest are thus more readily comparable. It must be noticed, however, that when this is done the rate of interest to be used must not be the nominal rate at which the interest due at each period is reckoned, but such a rate (commonly called the effective rate) as would, if convertible annually, be equivalent to the nominal rate convertible as given. Let r„ denote the effective rate of interest per pound which is equivalent to the nominal rate r convertible every 1/nth part of a year ; then, since the amount of £1 in one year at the two rates must be the same, we have (l+ry'=l+r,„ that is, r„ = (l -t-?-)"- 1 (1), and r = (1 + r^Y'^ - 1 (2). The equations (!) and (2) enable us to deduce the effective rate from the nominal rate, and vice versa. Example. The nominal rate of interest is 5 "/oi convertible monthly, to find the effective rate. Here r= -05/12 = -004166. Hence n2=(l-004166)i2- 1, = 1-05114-1. ?-i2= -05114. Plence the effective rate is 5-114 7„- 536 ANNUITIES TERMINABLE OR PERPETUAL ANNUITIES CERTAIN. § 5.] When a person has the right to receive every year a certain sum of money, say £A, he is said to possess an annuity of £A. This right may continue for a fixed number of years and then lapse, or it may be vested in the individual and his heirs for ever ; in the former case the annuity is said to be terminable, in the latter perpetual. A good example of a terminable annuity is the not uncommon arrangement in lending money where B lends C a certain sum, and C repays by a certain number of equal annual instalments, which are so adjusted as to cover both principal and interest. The simplest example of a perpetual annuity is the case of a freehold estate which brings its owner a fixed income of £A per annum. Although in valuing annuities it is usual to speak of the whole sum which is paid yearly, yet, as a matter of practice, the payment may be by half-yearly, quarterly, &c. instalments ; and this must be attended to in annuity calculations. Just as in compound interest, the simplest plan is to take the interval between two consecutive payments, or the conversion-period, as the unit of time, and adjust the rate of interest accordingly. In many cases an annuity lasts only during the life of a cer- tain named individual, called the nominee, who may or may not be the annuitant. In this and in similar cases an estimate of the probable duration of human life enters into the, calculations, and the annuity is said to be contingent. In the second part of this work we shall discuss this kind of annuities. For the present we confine ourselves to cases where the annual payment is certainly due either for a definite succession of years or in perpetuity. § 6.] One very commonly occurring annuity problem is to find the accumulated vcdue of a forborne annuity. An annuitant B, who had the right to receive n successive payments at n suc- cessive equidistant terms, has for some reason or other not received these payments. The question is, What sum should he receive in compensation 1 XXII ACCUMULATIOK OF FORBORNE ANNUITY 537 To make the question general, let us suppose that the last of the n instalments was due m periods ago. It is clear that the whole accumulated value of the annuity is the sum of the accumulated values of the n instalments, and that compound interest must in equity be allowed on each instalment. Now the mth instalment, due for m periods, amounts to AR™, the n - 1th to AE^+i, the n - 2th to AE™+2, and so on. Hence, if V denote the whole accumulated value, we have V = AR™ + AE™+i+ . . +AR"'+"-i (1). Summing the geometric series, we have V = AR'"(R"-1)/(R-1) (2). Cor. If the last instalment be only just due, in = 0, and the accumulated value of the forbwne annuity is given hy V = A(R»-1)/(R-1) (3). Example. A farmer's rent is £166 per anmim, payable half-yearly. He was unable to pay for five successive years, the last half-year's rent having been due three years ago. Find how much he owes his landlord, allowing compound interest at 3 %. HereA = 78, R = l-015, m=6, «=10. V = 78 X l-OlS^Cl-OlSi"- 1)/-015. 10 logl-015= -0646600, l-015i» = l-16054. V=78xl-015''x-16054/-015. log 78 =1-8920946 6 log 1-015 =_ -0387960 log -16054 = 1-2055833 1-1364739 log -015 =2-1760913 logV =2-9603826 V=£912-814=£912:16:3. § 7.] Another fundamental problem is to calculate the purchase price of a given annuity. Let us suppose that B wishes, by paying down at once a sum £P, to acquire for himself and his heirs the right of receiving n periodic payments of £A each, the first pay- ment to be made m periods hence. We have to find P. P is obviously the sum of the present values of the n pay- 538 PURCHASE PRICE OF AX ANNUITY ohap. ments. Now the first of these is due m years hence ; its present value is therefore A/B,™. The second is due m + 1 years hence ; A's present value is therefore A/R'"+-', and so on. Hence Hence p = A A A -1 p= ^I^(' '(■ ■■s). A Vm+n n(E"- 1)/(R-1) (!)• (2). Cor. 1, The ratio of the purchase price of an annuity to the annual payment is often spoken of as the number of years' purchase which the annuity is worth. If the "period " understood in the above investigation be a year, and p be the number of years' purchase, then we have from (2) p = (R" - l)/R™+"-i(R - 1) (3). If the pier iod be l/qth of a year, since the annual payment is then ijA, ive have p = (R'^ - l)/qW+"'-\R - 1 ) (4). Cor. 2. If the annuity be not "deferred," as it is called, but begin to run at once, that is to say, if the first payment be due one period hence,* then m=l, and tve have P = A(R''-1)/R"(R-1), = A(l-R-«)/(R-l) (5). Also j5 = (R«- l)/rv"(R- 1), = (l-R-»)/(R-l) (6); or p = (l-R-")MR-l) (7), accoj-ding as the period of conversion is a year or the qth part of a year. Cor. 3. To obtain the present value of a deferred perpetual annuity, or, as it is often put, the present value of the reversion of a perpetual annuity, we have merely to mccJce n infinitely great in the equation (2). We thus obtain * This is the usual meaning of "beginning to run at once." In sotne cases the first payment is made at once. In that case, of course, m = 0. NUMBER OF YEARS PURCHASE OF A FREEHOLD 539 = A/R™-i(E-l) ' (8). Hence, for the number of years' purchase, we have ^=l/R™-i(R-l) (9), or p=l/qR"<-\'R-l) (10), according as the period of conversion is a year or 1 jqth of a rjear. gins to run at once the Putting ni '= 1 we have (11). (12); (13), Cor. 4. When the perpetual annuity formulcB (8), (9), (10) become very simple. P = A/(R-1), = A/(l+r-l) = A/r For the number of years' purchase p = llr or p = 1/qr according as the period of conversion is a year or 1/qth of a year. If the period be a year, remembering that, if s be the rate per cent of interest allowed, then r = s/100, we see that p=100/s (U). Hence the following very simple rule for the value of a perpetual annuity. To find the number of years' purchase, divide 100 by the rate per cent of interest corresponding to the hind of investment in question. This rule is much used by practical men. The following table will illustrate its application : — Rate % 3 3i 4 4i 5 54 6 No. of years' purchase . 33 28 25 22 20 18 17 Example. A sum of £30,000 is borrowed, to be repaid in 30 equal yearly instalments which are to cover both principal and interest. To find the yearly payment, allowing compound interest at 4J %. Let A be the annual payment, then £30,000 is the present value of an 540 INTEREST AND ANNUITY TABLES CHAP. annuity of £A payable yearly, the annuity to begin at once and run for 30 j'ears. Hence, by (5) abovp, 30, 000 = A(l - 1 ■045-™)/-045, A = 1350/(1-1 ■045-3"), -30 log 1-045 = 1-4265110, l-045-*»= -267000. A = 1350/-733, = £1841 : 14 :11. § 8.] It would be easy, by assuming the periodic instalments or the periods of an annuity to vary according to given laws, to complicate the details of annuity calculations vei-y seriously ; but, as we should in this way illustrate no general principle of any importance, it will be sufficient to refer the student to one or two instances of this kind given among the examples at the end of this chapter. It only remains to mention that in practice the calculation of interest and annuities is much facilitated by the use of tables (such as those of Turnbull, for example), in which the values of the functions (1 + r)", (1 + r)"", {(1 + r)" - 1 }/r, {1 - (1 + r)"''}/?-, 'r/{l -(1 +r)'^}, &c., are tabulated for various values of lOOr and n. For further information on this subject see the Text-Book of the Institute of Actuaries, Part I., p. 151. Exercises XLIV. (1.) The difference between the true discount and the interest on £40,400 for a period x is £4, simple interest being allowed at 4 % ; find ^• (2.) Find the present value of £15,000 due 50 years hence, allowing 4J 7o compound interest, convertible yearly. (3.) rind the amount of £150 at the end of 14 years, allowing 3 % com- pound interest, convertible half-yearly, and deducting 6d. per £ for income- tax. (4.) How long will it take for a sum to double itself at 6 7<, compound interest, convertible annually ? (5. ) How long will it take for one penny to amount to £1000 at 5 "/„ com- pound interest, convertible annually ? (0.) On a salary of £100, what difference does it make whether it is paid quarterly or monthly ? Work out the result both for simple and for compound interest at the rate of 4-2 7o- (7.) A sum £A is laid out at 10 "/o compound interest, convertible annually, and a sum £2A at 5 7o compound interest, convertible half-yearly. After how many years will the amounts be equal ? XXII EXERCISES XLIV 541 (8.) Show that the difference between bankers' discount and true discount, simple interest being supposed, is A.nV {1 - «?• + »V - nV +. . . ad <» } . (9.) If r> 5/100, MJ>10, find an upper limit for the error in taking 100(1 + „Cir+„C2r2'+„C3r') as the amount of £100 in n years at lOOr % com- pound interest, convertible annually. (10.) If £lc and £1, denote the whole compound and the whole simple interest on £P for n years at lOOr %) convertible annually, show that Ic-I. = P(nC2r2 + „C37^+. . . + r»). (11.) A man owes fP, on which he pays lOOr "/„ annually, the principal to be paid up after n years. What sum must he invest, at lOOr' 7„i so as to be just able to pay the interest annually, and the principal £P when it falls due ? (12.) B has a debenture bond of £500 on a railway. When the bond has still five years to run, the company lower the interest from 5 7„i which was the rate agreed upon, to 4 7o> and, in compensation, increase the amount of B's bond by x °/^. Find x, supposing that B can always invest his interest at 5 7„. (13.) A person owes £20,122 payable 12 years hence, and offers £10,000 down to liquidate the debt. What rate of compound interest, convertible annually, does he demand ? (14. ) A testator directed that his trustees, in arranging his affairs, should set apart such sums for each of his three sons that each might receive the same amount when he came of age. When he died his estate was worth £150,000, and the ages of his sons were 8, 12, and 17 respectively. Find what sum was set apart for each, reckoning 4 7o compound interest for accumulations. (15.) B owes to C the sums Ai, As, . . ., A,, at dates ni, n^, . . ., n^ years hence. Find at what date B may equitably discharge his debt to C by paying all the sums together, supposing that they all bear the same rate of interest ; and 1st. Allowing interest and interest in lieu of discount where discountis due. 2nd. Allowing compound interest, and true discount at compound interest. (16.) Required the accumulated value at the end of 15 years of an annuity of £50, payable in quarterly instalments. Allow compound interest at 5 7o- (17.) A loan of £100 is to be paid off in 10 equal monthly instalments. Find the monthly payment, reckoning compound interest at 6 7o- (18.) I borrow £1000, and repay £10 at the end of every month for 10 years. Find an equation for the rate of interest I pay. What kind of interest table would help you in practically solving such a question as this ? (19.) The reversion after 2 years of a freehold worth £168 : 2s. a year is to be sold : find its present value, allowing interest at 2 °/^, convertible annually. (20.) Find the present value of a freehold of £365 a year, reckoning com- pound interest at 3^ °/^, convertible half-yearly, and deducting 6d. per £ of income-tax. (21.) If a perpetual annuity be worth 25 years' purchase, what annuity to S42 EXERCISES XLIV CHAP. XXII Continue for 3 years can be Bought for £5000 so as to bring the same rate of interest ? (22. ) If 20 years' purchase be paid for an annuity to continue for a certain number of years, and 24 years' purchase for one to continue twice as long, find the rate of interest (convertible annually). (23.) Two proprietors have equal shares in an estate of £500 a year. One buys the other out by assigning him a terminable annuity to last for 20 years. Find the annuity, reckoning 3J % compound interest, convertible annually. (24.) The reversion of an estate of £150 a year is sold for £2000. How long ought the entry to be deferred if the rate of interest oh the investment is to be 4f 7o> convertible annually ? (25.) If a lease of 19 years at a nominal rent be purchased for £1000, what ought the real rent to be in order that the purchaser may get 4 % on Ms investment (interest convertible half-yearly) ? (26.) B and C have equal interests in an annuity of £A for 2); years (pay- able annually). They agi'ce to take the payments alternately, B taking the first. What ought B to pay to C for the privilege he thus receives ? (27. ) A farmer bought a lease for 20 years of his farm at a rent of £50, payable half-yearly. After 10 years had run he determined to buy the free- hold of the farm. AVhat ought he to pay the landlord if the full rent of the farm be £100 jjayable half-yearly, and 3 "/o be the rate of interest on invest- ments in land ? (28.) What annuity beginning ■» years hence and lasting for n years is equivalent to an annuity of £A, beginning now and lasting for 7i years ? (29.) A testator left £100,000 to be shared equally between two institu- tions B and C ; B to enjoy the interest for a certain number of years, C to have the reversion. How many years ought B to receive the interest if the rate be 34 7o> convertible annually ? (30.) If a man live in years, for how many years must he pay an annuity of £A in order that he may receive an annuity of the same amount for the rest of his life ? Show that, if the annuity to be acquired is to continue for ever, then the number of years is that in which a sum of money would double itself at the supposed rate of interest. (31.) A gentleman's estate was subject to an annual burden of £100. His expenses in any year varied as the number of years he had lived, and his income as the square of that number. In his 21st year he spent £10,458, and his income, before deducting the annual burden, was £4410. Show that he ran in debt every year till he was 50. (32.) A feu is sold for £1500, with a feu-duty of £18 payable annually, and a casualty of £100 payable every 50 years. AVhat would have been the price of the feu if it had been bought outright ? Reckon interest at 4J %■ (33.) Find the accumulation and also the present value of an annuity when the annual payments increase in A.P. (34.) Solve the same problem when the increase is in G.P. (35.) The rental of an estate is £inX to begin with ; but at the end of every q years the rental is diminished by £A, owing to the incidence of fresh taxation. Find the present value of the estate. APPENDIX ON THE GENERAL SOLUTION OF CUBIC AND BIQUAD- RATIC EQUATIONS; AND ON THE CASES WHERE SUCH EQUATIONS CAN BE SOLVED BY MEANS OF QUADRATIC EQUATIONS. § 1.] Since cubic and biquadratic equations are of frequent occurrence in elementary mathematics, and many interesting geometric problems can be made to depend on their solution, a brief account of their leading properties may be useful to readers of this boot. Incidentally, we shall meet with some principles of importance in the General Theory of Equations. COMMENSURABLE ROOTS AND REDUCIBILITY. § 2.] We shall suppose in all that follows that the coefficients Po • • ■, Pn oi any equation, Po«^'^+I>,x'"'^ + ■ ■ .+Pn = (1), are all real commensurable numbers. If, as in chap, xv., § 21, we put X = ^/m, we derive from (1) the equivalent equation ^„£"+j3,mp-i + . . . +^;„_,m"-i^+^„m" = (2), each of whose roots is m times a corresponding root of ( 1 ). If we then choose m so that mp^jp^, . , 'ni^~^p„-t/fim in^Pn/Po are all integral — for example, by taking for m the L.C.M. of the de- nominators of the fractions p,//i„, . . ., Pn-t/Po, Pn/Po — we shall reduce (2) to tJieform f' + 2,^-1+ . .+g„ = (3), in which all the coefficients are positive or negative integers, and the 544 APPENDIX coefficients of tlie highest term unity. \Ve may call this the Special Integral Form. § 3.] If, as in chap, xv., § 22, we put x = ^ + a, we transform the equation (1) into i'o?' + !?ip-^ + - • . + 2,1 = 0, where (ii = np„a+p^. Hence, if we take a= -Pi/np,,, the trans- formed equation becomes Pci" + q\i^-' + ■ . . + qn=0 (4), wherein g'j, . , q'„ have now determinate values. It follows that By a proper linear transformation, we can always deprive an equation of the nth degree of its highest term hut one. We can, of course, combine the transformations of §§ 2, 3, and reduce an equation to a special integral form wanting the highest term but one. § 4.] If an equation of the specicd integral form Ms commensurable roots, these roots must he integral, and can only he exact divisors of its absolute term. For, suppose that the equation (3) has the fractional commensurable root ajh, where a may be supposed to be prime to h. Then we have the identity {ajhY + q.iajhy^-^ + . . .+J,= 0, whence a"/i = - j,a" - 1 - q./i'" -%-. . - 2„i™ - 1, which is impossible, since the left-hand side is a fraction and the right-hand side an integer. Also, if z = a be any integral root, we must have g„/a= - ffl"-i - g,ffi"-2 - ... - q^_^. Hence, since the right-hand side is obviously an integral number, a must be an exact divisor of j„. T]>.e a/nimensurahle roots of an, equation, if any exist, can therefore always be found hy a limited number of arithmetical operations. We have merely to reduce the equation to an equivalent special integral form, and substitute the divisors of its absolute term one after the other in the characteristic. The number of trials may in most cases be reduced by obtaining upper and APPENDIX 545 lower limits for the roots by means of the theorem of chap, xii., § 21, by graphical methods, or otherwise. Example. x^ - Wx^ + 31x-30 = 0. This equation is already in the special integral form. Hence the only possible commensurable roots are ±1, ±2, +3, +5, +6, ±10, +15, ±30. It is obvious that the equation has no negative roots ; and, since a;(a;^ + 31) = 10(a;^ + 3), it is useless to try a; = 10 or any larger nuuiber. Of the remaining possible numbers, +1, +2, +3, +5, it is immediately found that +2, +3, and +5 are roots. § 5.] An equation is said to be reducible when it has a root or roots in common with an equation of lower degree (having of course commensurable coefficients) ; irreducible if it has no root in common with any equation of lower degree. If an equation f{x) = have roots in common with an equation of lower degree <^{x) = 0, then the product of the linear factors corresponding to all such common roots, say g{x), is the G.C.M. of the characteristics /(«) and (^(a;), which can be deduced from these fimctions by purely rational operations. It follows tha,t f{x) = g(x)h(x), where g{x) and h{x) are integral functions of X having commensurable coefficients. The roots of J{x) = are therefore the aggregate of the roots of the two equations g{x) = 0, h(x) = 0, each of lower degree than the original equation. Each of these new equations may be reducible or irreducible ; but it is obvious that at last we must arrive at a series of irreducible equations the aggregate of whose roots are the roots of f(x) = ; and the characteristics of these equations are the irreducible factors of f(x), irreducible in the sense that they cannot be decomposed into integral factors of lower degree having com- mensurable coefficients. § 6.] The folloviring theorem, often appealed to in the theory of equations, is an immediate consequence of tho notion of irre- ducibility explained in the last paragraph. If an irreducible equation A have a root in common with an equation B (reducible m- irreducible), then every root of K is a root o/B. For, if only some of the roots of A were roots of B, then a common commensurable factor of the characteristics of A and B could be found of less degree than A itself ; and A would not be irreducible. § 7.] PFhen an equation is reducible it can be reduced by a finite number of arithmetical operations. VOL. I - ^ 546 APPENDIX Consider the equation /(«)=*" +^,a;"-i + . . .+pn=0 (5), whose roots are a„ a^, . . . a„. If (5) be reducible, then f(x) must have a commensurable factor of the 1st, or 2nd, . . ., or/th degree, where (/) is the greatest integer in ujl. If /(x) has a commensurable factor of the 1st degree, (5) has a commensurable root, which can be found as explained in § 4. If /(.)•) has a commensurable factor of the sth degree, let a„ Oj, . ., z). It follows that the roots of the equation x''-{Zyz)x + {f + ^) = (2) are -y - z, - wy - az, - toy - wz. Now, if Hence, if .= -g/3, / + / = r (3), J2) are identical. lat y" and / are the roots of the quadra' e-ri-qy27 = (4). the equations (1) and (2) are identical. From (3) we see that y" and / are the roots of the quadratic ^ = 2- Al-ITJ- ^ = 2-^4-17; (^)- then the roots of (1) are -4/L-4/M, -(o-5/L-/B + \/C + D a/B \/C ; or it may happen that two or four of the roots may be commensurable. § 14.] If a biquadratic be reducible, it may reduce (i.) to a linear equation and a cubic ; (ii.) to two quadratics ; (iii.) to two linear equations and a quadratic ; (iv.) to four linear equations. It should be noticed as regards a biquadratic that reducibility and solubility by means of square roots {i.e. by means of quadratic equations) alone are not the same thing. For example, a biquadratic may have one commensurable root, and its other roots may be the roots of an irreducible cubic. Again, (,r + 3a; + 1)^ - 2(2/ - 2a; + 1)' = is evidently soluble by means of square roots ; but it is not reducible, for we cannot factorise the characteristic without introducing the surd v/2. § 15.] There is another way of solving a biquadratic which is often convenient in practice. Suppose the biquadratic reduced to the form x^ + qx^ + rx + s=Q (1). Then j\ + X2 + Xs + Xi = Q ; and we can reduce the characteristic to the form {x^ - (s, + Xj)x + Xjxjla" + (a-, + x.^x + x^x^] (2). Hence, if we put ..-, + ,)■. = ^Jp, x,x, = /3- Jyp, x.^^ = j3 + s/yp (3), we have (^r + l3y-p{x + yy = x' + qX' + rx + s (4).* Therefore 2ft-p = ,j,-2py = r,/3'-py' = S (5), which are equivalent to ■':;,,: P = iq + p)l-2, y=~rl2p (6); * The whole of the present process is a natural application of the last remark in § 9. APPENDIX 553 together with p' + 2qp' + {q^-is)p-r'=0 (7), which is Descartes' Cubic Sesolvent. When any root of (7) is known, the values of /3 and y are given by (6) and the roots of the biquadratic are given by X^-s/pX + /3- Jyp = 0, X'+'JpX + P+ Jyp = (8). As before, the necessary and sufficient condition for solubility by means of square roots is that the resolvent shall have at least one commensurable root. Cor. 1. The roots of Descartes' resolvent are three of the six quantities {x^ + x^y, {x^ + x^f, {x, + x^\ {x^ + x,f, {x^ + x,)\ {x^ + x^\ which are equal in pairs. Cor. 2. If the biquadratic be reducible to two quadratics, one of the roots of Descartes' resolvent must be a perfect square ; and this condition is sufficient. § 16.] If the solution of a geometric problem be expressed by a series of equations, the necessary and sufficient condition for solubility by means of the ruler and compass alone is that these equations either are, or are replaceable by a series of linear and quadratic equations (see Introduction, § 240). The foregoing considerations often enable us to settle the possibility or impossi- bility of such a solution. For example, the abscissa, or ordinate, of the intersection of two conies is in general the root of a bi- quadratic equation : hence the intersections of two conies cannot be constructed by the ruler and compass alone, unless the cubic resolvent of this biquadratic have a commensurable root. EXEKCISBS XLV. (1.) Prove that the biquadratic xi'+pa^ + qx'' + rx + s=0 is soluble by- square roots ii^-ipq + 8r—-0. (2.) Discuss the Lagrangian resolvent of a;*+^!i;^ + }a!'+^a; + l = 0. Solve the following biquadratics : — (3.) ai^+10a? + 22x'-15x + 2=0. (4.) x* + 10oi? + 35a!' + 50x+i = 0. (5. ) x*+2{p- qjoi? -(- (p" + c[V + 2i'2'(p - q)'^ +pq{v'^ +P1+ f) = 0. (6.) 2a!*-K5-9a!'2-H4a! + 3 = 0. 554 APPENDIX (7.) 2a^ + ar'-3*--8a;-12 = 0. (8.) 2j^ + 5x^ + 6x=-a;-6 = 0. (9.) 2x'' + 3a? + 16x + 6 = 0. (10.) a;^-4ar>-4a? + 16a;-8 = 0. (11.) ar'-6a:2+8a:-3 = 0. (12.) If an equation of the special integral form of § 2 reduce to equations of lower degree, prove that each of these equations is also of the special in- tegral form (see Weber's Algebra, § 2). (13.) Show that, without solving equations of higher degree than the second, we can determine u, ft y so that the substitution y = a + px + yx' shall transform any cubic equation into the form j/' + A = 0. (14.) Show that, without solving equations of higher degree than the third, we can determine a, /3, 7, S so that the substitution y=a + px + yx' + Sa;* shall transform any biquadratic into the form y* + Aj/^ + B = 0. (15.) Show that i/{rj2+ ^'{r^/i + g^/27)] is expressible in the form ^+^y where x and y are rational, when, and only when, the cubic u? + qx+T=0 has a commensurable root. "What bearing has this on the solution of a cubic equation ? (16.) Show that one root of Descartes' resolvent of the biquadratic 3^ + qx^ + rx + s = is always real and positive; and that the roots of the biquadratic are 1° all real, if the other two roots (pj, p^ of the resolvent are both real and positive ; 2° all imaginary, if p^ and p^ are both real, negative, and unequal ; 3° two real and two imaginary, if /)2 and pz are both real, negative, and equal, or if p2 and p^ are both imaginary. (17.) If the roots of Lagrange's resolvent be all real and unequal, show that the roots of the biquadratic are either all real, or else all imaginary ; and that, if only one root of the resolvent is real, then two roots of the biquadratic are real and two imaginary. (18.) Show that a regular heptagon cannot be inscribed in a circle by means of the ruler and compasses only (see Ex. xxxii., 33). (19.) Show that the inscription in a circle of a regular polygon of 11 sides depends on an irreducible quintic equation. (20.) Show that, if A, B be two given points in a straight line, we can by the ruler and compass alone find a point on the line such that AP + KP:AP-BP = AP3:BP3. (21.) Starting from the remark at the end of § 8, prove that, if a biquadratic equation be soluble by square roots alone, then its Lagrangian resolvent must have at least one commensurable root. (22.) Show that, if the roots of the biquadratic sc^ + qx^ + rx + s = be rational functions of two quadratic surds, then the cubic y'''-iqy^ + 4(g'--4s)j/ + 8r^=0 must have a commensurable root, say y = m; and (5'/2-m/8)^ + ?'72m must be the square of a rational number. Are these conditions sufficient ? RESULTS OF EXEECISES. I. (5.) 1st. The number of digits is 34 ; for the best approximation the first three digits are 126. 2nd. The number of digits is 20 ; the first three 184. (6.)-l, +1. (7.) a -6. (8.) 1707 ; 30521/415800. (9.) 6. (10.) aa + 2ab + bb;aa-bb;9aa-3ebb;iaa--s\bb. (11.) 2{mm-l)aa + 2{nn-l)bb. (12.) 2xy + 2/x2/. (13. ) 2yz/x + 2zxly + 2xy/s + 2/xyz. (14. ) ixx + Jot - -^y + ^\zz. II. (1.) 1/2^.312.5 = 1/42515280. (2.) The second is greater by 65280. (3.) 1/2. (4.) Wa^^cP. (5.) c^la>V3?y'^\ (6.) (81/16)a«6V2a;2. (7.) yH'^jx'. (8.) (xyz)^. (9.) 1. (10.) af^-^'-. (11.) k(«-<^) (i-=). (12.) IfxP^^K (13.) 1. (15.) l+a;2«-»-«-x-2»+'+° + a;-''+2'-° + a;''-2''+°-K-«-»+2o + a;a+i-2i:. (16.) aP+^ + 2ai>+^lbP + aPJb^ + a^jbi + 2a^/bP+!> + l/i^jH^. III. (1.) x + y. (2.) 1. {3.) {x^-y^)lxy. (i.) y. (5.) l/6c( -a + ft-c). (6.) (a*-a--'i2 + 2aJ3-J-')/(a2-i2). (7.) ix/l{x^-y*). (8.) 2(b^ii^ + aY)labxy. {9.) abKa' + V). {10.) a^-i^. (11.) -{ix + 2x^)/{l+x^ + x*). (12.) 1. (13.) 1. (14.) The function is =1. (15.) (adf-ae)/(bdf -be- cf). {16,)(a^-iai + ib--l)l{a^-6a^b + 12ab^-8i'-2a + ib). (17.) (a?-b^ + l)l(a^-b^+2). IV. (7.) 2.33.7.11^; 35.55.72. (8.) 53. V. (1.) 120. (2.) a;'-2a;V + 2/'- (3-) '^-'f- (4.) x'^-Za^k/^ + Zx^y^-jf. (5.) afi-Uafiy^ + 96!)A/-256xhf + 256f. (6.) bV - ¥c^ + c'a* - c^a' + a^b* -a%^. (7.) a;6 + 3ar' + 6a:'' + 7a^ + 6a;2 + 3a; + l. (8.) 27a' + 863-l + 54a26 + 36«&2_l262 + 66 + 9a-27a^-36a5. (9.) a;^ + 2ar' + Sa;^ + 4a! + 5 + 4/a; + S/r" + 2/a;3+l/a!*. (10. ) + a* + 6* H- r-" + ia?b + iaV^ + 46% + ihi? + ica? + i(?a + Qa%^ + 66V +'6cW + 12a26c + 1 2as6% + 12a6c2 ; + + + - - + + -- + + + + (11.) Four types; 3 like ar* ; 6 like 3?y ; 3 like x^yz; 3 like xV- (12.) (3. 556 RESULTS OF EXERCISES Three types; 4 like a?; 12 like 0,% ; i like abc. (13.) (2 + 3 + 4) '=729. ( 14. ) aPjib^ - C-) + &c. + &c. + 2(ca - ab) yz/{a^ - b-){c- - a=) + &c. + &c. (15. ) 2(r'- + J/- + C-). (16.) 0. (17.) ax + bij + c:. (18.) - Za'^ + 2Sa'^b + ^a^b' - WSa*bc -iXa'>V> + 8-2aVc-18a?lf'e^. (22.) aic. (23.) Sabc. (24.) 2(6%-ic= + c'a -ra' + a'J-rei^). (25.) (a=- J2),i'' + 2«J-x-'V + (2a2 + 2i=- a-h-)rr-,f--2ab-xi/ + ia'-b')'>/. VI. (l,)(a)2nd. (/3) Fractional. (7) 4th. (5) 2nd. (2.) 2j;- + 10a; + 14. (: ix^+p;. (4.) 9ar»-7S.r + 121. (5.) .c<''-x^:2a-+x'-Za-b^-a-tf^c'. (6.) ..« (i)- + 3Hl).)-* + (i)^j'=+^^ + j^)a:^-^^?^. (7.) x^'>-15a?y^ + 85ofiy*-225x*y^ + 2nx-f - 120?/i». (8. ) aba? -{b-c){c- a) (a - b)x\j + CZa'b - Sa' - 3abc)xif + {b-c){c-a){a-b)if. {11.) (V-<-")x^ + 2c{b-a)x^ + {c-a){c + a-2b)y- + 2a(c-b)x + {a?-b-). (12.) ^-a^-a:- + l. {IZ.) i.i? + lQx-y + 8xy'' + Z%/. (14.) \3:^-:.',p + ii/. (15.) a:ii=-2,." + l. (16.) x^^ + 2x^'' -a? - io^-x'^ + 2x^ + l. (17.) Ai"-i.i'* + TrV'-'-iiV-' + Tr- (18.) ft,''«-o(o + i)a;«!/ + i(2a2 + l)3^2/2-o=62K3y3 + (u;y-i(a-i)a:t/= + 6y'. (19.) 4a*+ 12(i-.f* + 126%~'. (20.) a;2*-12aV2 + eea^ar^" - 220««xis + 495a8a;" - 792a" a:^^ + 924a'2a;i2 - &c. (21. ) x^"' - Sa^a^ + Sar'ft" - a»5. (22.) 2187x' + 1701^ + 567x:''' + 1053i* + -%^ - {q - r) {r - j}) {p-q)x^ + (-'Zf + :2p^q~Spqr)x + {q-r){r-p){p-q). (3.) afi - SOa'afi + 273a-'a^-820aHK= + 576a8. (4.) x^-Sa^ + Sx^-l. (5.) iofi + W'-^ + iP-^ + m^ + iix' + ia: + -l (6.)af>--^\. {!.) x'> + {XPImn)x' + 2S[mnlF')xf^+{i + Pj-m? + ■rn^l'n? + ■n?IP)x? + 2CZPImn) x^ + (XmnlP)x + 1. (8.) 2048xii - 33792k" + 2534403^ - 1140480a^ + 3421440a;' - 7185024a;8 + 10777536«5 - 11547360*^ + 8660520ar' - 4330260^2 + 1299078a; - 177147. (9.) x^ + Sx'^f + 28x^^y'> + 56a;'V + '!Ox^Y"- + &<=• (10-) a.-" + 10x" + 41a;" + 80a;i= + 36,r" + 168a;" - 364a;i2 - 208a;ii + 286a:i'' + i:;72.v» + &e. VIII. (1.) A + B2a;+CSa;2 + DS.r!/ + E2ar' + F2a=!/ + G2a;j/;. (2.) Zx?if + I.x>!j:. + 3x-y-:P. Three types present, four missing, viz. , 3;^, x!'y, af'j/^, afiy^z. (3. ) ^x-iy. (4.) F{{y"-y')(a:-x')-{sii'-x')(y-y')}, wheve P is any constant. This may also be written V {{y" - y')x - (a^' - x')y + x'y' - x'y"] . (5.) (y'x-x'y) {y"x-x\j'jl{x'-y'){a^'-y"). (6.) A= -8, B= -12, C = 20. (7.)Z=21, ?ft=-76,n = 60. (8.) ?=6, m=-15, ?i = 10. (9.) SP(x-6) (a;-c) (a;-d)/ (a - 5) (a - c) {a - d). (10. ) frV + bV + b% bV + h-c* + b(fi. (11. ) SZ,?-' - -^.v°-y. IX. [1.) Q=o;'-33? + Zx-l,R=0. (2.) Q = 3a;*+J5>-»^-V«^ + -»/x--JL R = »3a._4|_ (3.) Q = 4ar'' + 6a;= + lla; + 16, R = 20a;-15. (4.)a;2-9. (5.) The function=a;»-2a;2 + 17a; + 80-40/(a;-7). (6.) Q=a;3-6a; + 3, R = 0. (7.) Q RESULTS OF EXERCISES 557 = 9a^ + 6a;5 + a^ + 2a;+l, R=0. (8.) Q=a:--8.i+15, R=0. (9.) Q=ar' + 4!C^ + Ja5 + i, R=0. (10.) Q=a?-^+i^, R= _J^ia;_s3_ (^j Q=4a:6-Ja^ + ia!*-Aa^ + -^a^-sija; + xfj, R = 0. (12.)a!-l. (13.) a^2/V + a!°2/+a^y'' + . . .+j/«). (14.) 3a2-2a6 + 62. (15.) a« - a^j + a4j2 _ _ _ _.,_j6. (le.) a;^ -3aM/ + 2/2. (17.) x^-2xy + iy\ (18.) a::^ + aj^j/ + 3ot/ + 3i/. (19.) l+s+a^ + . . .+x^. (20.) a*-ar!-x=--2A' + 4. (21.) bx^ + ex-f. (22.) a6 + ac-6c. (23.) 1 + h + c. (24.) 2(a + 6)x. (25.) a^ - a^J + SaSja - ajs + 64_ (je.) 8Ky(a;= + 2/-). (27.) 7a;)/(a; + 2/). (28.) Q = 6a33 + 9a;2 + 5a; + l, R= -1. [ZS.) [bx + ay)l (bx -ay) = l + 2ayl{bx -ay)=-l + 2bx/{bx - ay). (30. ) If a be variable, the transformed result is a" + 2a?b + 5ab^ + lOftS + (15a6* - lSl^)/{a^ - 2ab + 5-). (31.) Q=ii^-2x'+x-i,'R=12. (32.) a^ + 3a;5i-13x-15. (33.) -303/8. (34.) 2(^^ + 2); p + q=0. (35.) AQ2-BPQ + CP2=0. (36.) p"'~ap-q+b=0, pq~aq + c = 0. (38.) X = l, /t= - 3, v= - 2. (39.) ^ = 2, g' = 3, r = 3. (40.) The remainder in each case is rx + s. (41.) m + 1 must be amultiple of « + l. (42.) l-Zx + Sx'-27a? + . . . +(-3)"a:''-( -3)»a;»+V(3a; + l) ; ^"9^ + 27^ +x^). (44.) l + 2K + 5a;V1.2 + 16ar*/1.2.3 + 65K*/1.2.3.4 + . . . (45.)l-«i/; l+ny. (50.) a.'S- 70^3 -377a?' -778k -585. (51.) P8-4P'' + 2P6+8P5-5P' -8P3 + 2P3 + 4P + 1, where P=a! + 2; Q* + (8x + 24)Q3 + (8a:-40)Q2 + (-32a: + 16)Q + 16x, where Q=x^ + x + l. X, (1.) x^-l. (2.) a^-K + l. (3.) No CM. (4.) a + l. (5.) aj^+a-e. (6.) a^- 12a! + 35. (7.) a;^- 16a:- 15, use § 7. (8.) a;* + a;2- 6 ; compare with Examples. (9.) a;3_3a,_2. (io.)a;2_i_ (11.) 4a!2 + 3a;+l. (12.)a;-5. (13.) 4a,-^-24x + 35. (14.) a;^ + 2a; + l. (15.) 3? + iaa + 8a?. (16.) 33^- VSai + l. (17.)a'-l. (18.) ar-aa; + 2a=. (19.) (a;-l)=. (20.) The G.C.M. would be a measure of {j)-q)x{x-l), neither of the factors of which is in general a measure of either of the given functions. If, however, p + q + 2 = 0, then a;- 1 is a measure of both. (22.) a =8, there is then a factor a,'^-4a: + 3 common to numerator and denominator. (23.) Use § 7. (25.) Use §§ 6 and 7 ; the first gives the conditions in the first form, the second gives the single condition. (26.) P=-^a! + JV. Q=-!rT3' + ¥T- (27.) P=Aa;+A! Q=--^\x+ii. {28.) a'{a-b){a + b)ia^ + b^'){a*+b*). (29.) (a:- 1) (a;- 2) (k- 3) (a: + 2) (a; + 4). (30.) (a:-l)(a; + l) (a; + 2)(3x-2) (3.1 + 2). (31.)(a;=-l) (a;- 2)^(a; + 4) {x+ 5) {a? - 5). (32.) The product of the given functions. XI. (1.) 2{a-d)ia + b + c + d). (2.) {a + b + c){-a + b + c){a~b + c){a + b-c). (3. ) (a' - Sc") (a? - W + c^). (4. ) i(a! + 2 + V2s) (a; + 2 - V2i) (9a: - 1 3 + V7) (Ba^-lS-V?). (5.) (a-^)(2a!-a-/3)(x-7)2. (6. ) (.a; + 1/) (a; - 2/) {x - (i+i'f)y}{^-{i-i^)y}{-(-i+if)y}{^-(-i-i^)y}-W 558 RESULTS OF EXERCISES' (x + y)(x-y){x + iy){x-iy){x + y{l + i)l^i\{x + y{\-i)j^2}{x-y[\ + i)l's/'i} {x-y(\-i)l^i\. {i.) {x + Zy + 'i){x + Zy-1). (9.) 2(x + 2)(a=-i). (10.) {x + %)(x-2). (ll.)Ca---5 + V7)('--5-V7). (12.)(a: + 6)(a;-5). (1.3.){x + 1 + is/7)(x + 7-i\/7). J14.) {x + 2 + i-s/S){x + 2-i^3). (15.) 2{x + i){x-i). (16.) {x + \/p + q + '^p-g){x + \/p + q-\/p-q). (17.) {x-l){x-{b-c)/ {b + c)}. (18.) (.j;+^)(a; + g')(a;-^)(:c-(?). (19.) {ax-by){bx + ay). (20.) {(l-ij)a:-(l+p)j/};(l-g')»:-(l + ?)2/}. (21.) (a;-3)(a;-5)(K-7). (22.) x{x-7 + Si\/U){x-7 -3W11). [23.){x-3){x-i){x-e). (24.) (a- 8)(a; + i) (s-i). {25.)(x+p)(x + p + q){x+p-q). (26.) {x + l)(x -l){{p + q)x+ [p-q)]. [27.) {x-l){.);'P){x-p''). {2S.) (x- a) {x-b){x + \/ah){x-\/ab]. (29.) (.K^ - «■•) (x^ + xa + (»-) = &c. (30. ) 2{x-y){l- :ni). (31. ) (.i- + xy + y'^) (x- - xy + y-) = &c. (32. ) _;< = + \/2, ? = - V2- (33. ) (.V- + ^J2xy + y--\){x'- sjixy + y--l). (3i.) r=-pii-, s=-qa"- (I*. (35.) (.u™ + a™) (x™ - a™) (x" + a") (»"-«"). (36.) 3«">~(.J'- + acc + a2)(x'--o,i' + «2) = &c. (37.) (x-l)(y~lf. (38.) (.« + 3)(2..+2/ + l). (39.) (2x + Zy + \)(x-y-l). (40.) (a; + 3) (2/ + 7). (41. ) (x + 2y~ .;) (x - ?/ + 3e). (42. ) Equate the disoi-imiuaiit of tlie function to zero and thus obtain a cubic equation for X. (43. ) When c = 0, X = [a/- + by" -2fgh)lfg. (44.) a((3V-/3"7') + /3(ya"-7"a') + 7(a'j3»- ay) = 0. (47.) (a + b-^-c)(a- + b'' + c--bc-ca-ab). (48.) (x + y-a){x- + y^ + a?-xy + ax + ay). (49.) ?,x(..+\)(x-\f. (50.) -(■Z.,:- + -S.yz)(y-z){z-x)(x-y). (51.) (x + y + z){y^ z) (z - x) (X - y). (52.) - (y + z){z + ,>■) (x + y) (y -z){z- x) U - y). (53.) •i{,y + z)(z + x)(,- + y). (54.) -1. (60.) If p,= -^(y~ z) (z - ,:) = 2(2,,''-' - ^xy), and ps = (y-z){z-x){x-y), s„=S(//-;)", then it may be shown (see chap, xviii., § 4) tliat S6„._i = Apsp-J''"-" + B;j3 W'""''"" + • ■ + Lys-^'V:, where A, B, . ., L are numerical coefficients. Hence tlie theorem follows. XII. (1.) (.c + 3)/(ar + ,c-2). (2.) (to=-.ij- 3)/4(a; + 5), (3.) {2x-3)l{x"- -3x + 2). (4.)2(;»- + l)/(a,-2-l). (5.) 2ar» + 6.ij= + 2. (6.) (2a; + 5ffi)/(3* + 5a). (7.) 1/(1-.?-). (8.){w + x-y)Hw + z-y). (9.) 1/(1 -."). [10.) {P + m"-)l{p^ + q^). {11.) -s/t. {12.) yl2(x-y). {13.) 2{a + bx)l{a^ -b-){l-x% (11.) (a^ - iab + ib--l)l{a?-6a^ + 12aV^-Sb^-2a + ib). (15.) (6 + a!)/3(l- a~). (16.) 0. (17.) 1/(2,/;-- 1). (18.) (240a;2 + 32a;)/(81x*-16). (19.) {3x + 2)/x{x + l)^. (20.) x~/{x-l){x + l)^{x- + x+l). (21.) l/(x + l)2. (22.) iax{-bx + a,^)/ ic'-a*). (23.) {3x? + y^){x-y)j{x + y)'K (24.) (a; + 2) («=- l)/(a;2+a; + l) (a;^ + X-4). (25.) x/Sa. (26.) 4ar?//(a:« - 2/«). (27.) -1. (28.) (2ra + c--l-a2)/ (a5-c2)(,.--«-Kx + c). (29.) 1. (30.) (.>-l)(2/-l)/(,x:+l)(2/ + l). (31.) l+y^ + z'-y-z^yz. {32.) {a + b + c)/{a-b-c). (33.) -3. {3i.) lj{x^ - a"). {35.) {a-bx)l{a + bx). {36.) 2{x? -1)/ {{x-1)"- - y"'}. {31.) {x-' + y^f. (38.) 1. (39.)- 2. {iO.)Z:c'l:njz. (41.) 0. (42.) A. (43.) 0. (44. ) 1 - Sar^ + 2,t»/j. (45.) 2. (46.) 0. (47.) 0. (48.) 0. {i9.) {h-p){h-q)jn{h + a). {50.) h''in{h-a). (51.^ ~A='n'/--H «■=). ^52.) -224vi/:, (53.) 1 - a- 6- c. (54.) L (62-1 'EBSULTS OF EXERCISES 559 1 - S/{x - 2) + 8/(a; - 3). (63.) l/2(a; - 1) - i/{x - 2) + S/2{x - 3). (64. ) 30x - 5/ {x + l)~5/{x-l) + S0/(x + 2) + S0/{x-'i). (65.)17/36(a! + l)-5/6(!)! + l)2 + 8/45 ■ (a;-2)-13/20(a! + 3). (66.) {ix + 5)la{x'' + x + l)-ys{x-l). (67.) l/2(a;+l) + {x + l)/2{aP + l). (68.)-l/4(»-l) + (a + l)/4(a;2 + l) + (x + 5)/2(a2 + l)2. (69.) V(a-b) {a^-2pa + q) {x-a) ~J/{a-h){b^~2pb + q) (x-i) + {{a + b ~ 2p)x + {2p-a){2p-b)-q}l{a^-2pa + q) {b''-2pb + q) {x'-2px + q). (70.) 3/4(:c-l)= -3/8(a;-l) + l/8(ffi + l) + (a;-l)/4(ir2 + i). (71.) 2/(a;+l) + 3/(a; + l)2- (2a:-3)/ (a;=-2x+3). (72.) l/(a;-l)-l/2(a;+l)-(xH-3)/2(a;2 + l). (73.) -l/x + l/x^ ~V''^ + yS(x-l) + 9/S(x + -l) + l/i{x + lf-{x+l)mx^ + l). (74.)(3a!= + !K+l)/ 2(a^-l)-(3a;2 + ..; + l)/2(a*+l). (75.) l/6 + S{l/15.3'' + (-l)»/10.2''}a!". (76.) 1 + 11 {{n + 2)k" + ( - 1 )«s=«+i). . XIII. (1.) 100242. (2.) 22-g354. (3.) 267'3861249. (4.) 2653919. (5 ) 788001. (6. ) 20200-1122212. . . (7.) -204. (8.) 1 + 1 x 3 + 1 x3. 5 + 3 x 3.5.7 + 3x3.5.7.9. (9.) l/2! + 0/3! + 3/4! + l/5! + 2/6! + l/7! + 2/8! + l/9i + 6/10i + 4/11!.* (10.) l/3 + 0/3.5 + l/32.5 + 3/3''.52 + l/33.52 + 4/33.53 + l/34.53+. (11.) 2466411243. (12.) 100-1431. (13.) 12-74i;50. (14.) 18-355. (15.)l-i + J 1—+—^ 1 1 5.47 5.47.60 5.47.50.367 5.47.50.367.56l"^5. 47.50. 367.55lTl0"3' (16.) 1 + 2 + 25. (17.) 4.1 + 1.5 + 1.5^ + 1.53 + 1.5^-1 + 2.5 + 1.52 + 1.58 + 1.5^. (18.) 831'80|''. (19.) 300'64". (20.) 53-617 cubic ft. (21.) 11'9H". (22.) r=7. (23.)»-=4,&e. (24.)503. (25.) x{x + l)(x + 2)(x + 3) + l={x' + 3x+lf. Since a;=10m+^ where p = 0, 1, 2, . . ., 9, we have only 10 different cases to consider. It -will be found that the last digit is 5 when j) = l or =6 ; in all other cases the last digit is 1. (28.) Since Sp(10'--10»)=SplO»(10'--»-l) is always divisible by 10-1 = 9. (31.) Since 10" and all higher powers of 10 are divisible by 2", it follows that^o+i5ilO + . . . +p„_jlO"-^ must be divisible by 2". (32.) po+Pi'i.O+p2lO^=Po + 2pi + ip.2 + Sipi + 12p,) must be divisible by 8, therefore, &c. (34.) If the digits in the period otp/n be qi, q-i, . . ., q„ those in the period of 1 -^;/« are 9 - ji, S-q., . . ., 9-q,. (36.) Since any number ]nay be written llm+^ where i) = 0, ],..., 10, we have merely to show that 0^ 1^, 2^ . . ., t^ all end in one or other of the digits 0, 1, t. We proceed to test thus : 9" ends in 4 ; therefore 9* ends in 5 ; therefore 9^ ends in 1; and so on. (38.) pqr2)qr={W + l)2)qr=7.n.l3.pqr. (39.) 0' ends in 0, 1^ in 1, 2^ in 8, 3= in 7, 4? in 4, 5^ in 5. Hence the theorem ; for every number can be written 10m±^, ^J> 5. (40.) I" = (12to+^)-, hence j9=0 or 6. The latter only is admissible. Hence P=(12m + 6)2=0 + 3.12 + /i.l22; I-"=(12m + 6)3=0 + 6.12 + .'.122. (41.) N-^o-i'if»- . . . -Pn-^a"-^ -(p,^a''> +p„^ia'^^+ . . .)=pi{r-a)+p.2{r'-a^)+ . . .=/i(r-a)=/ta'» ; therefore, &c. (42.) (pia) is simply the result of casting out the nines in the sum of the digits of a, that is, 0(a) is the remainder when a is divided by 9. Hence, &c. (43. ) We must have 2xr^ + 2yr + 2; = cr^ + i/r + a; ; whence {2x - z)r + y + (2a - x)j * 3! stands for 1.2.3, 4! for 1.2.3.4, &c. 560 RESULTS OF EXERCISES r=0; therefore (2s - a;)/?- must be an integer. Now 2;-.)<2j-, hence cither 2;-.r = 0or=r. The former can be shown to be inadmissible. The latter leads finally to z=2t + l, y=3t + l, x=t, r=St + 2, where i is any positive integer. Hence the theorem. XIV. (1.) 18^3. (2.) 1. (3.) 7"™. (4.) y (5''. 331). (5) („4-3n(„-l)Ji+3«(„-l)) i/te(,.-i)_ (6.) 1. (7.) Each =a;2/«(«+i)("+2). (8.) »-='' + a;-i'3)/-i'3 + j,-2/3. (9.) j;= + 2a; + 3 + 2x-^ + x-^. (10. ) x"^ - Sx'^l- + 5x"" - 7x^i' + 7.'-i - Sx-^l" + Zx-'^i'--x~-''-. (11.) x' + x + l. (12.) a;W»-2a:>V'- + 3.^'"iV'^ + K-'"i»2/ -Sx^yi". (13.) x^^ - 2x^lh/l^ + 2j/. (14.) 2(2i'-'' + 2= •'). (15.) a-*" + X^P«yV"-n - 2,,-= "J/1 " - 3k5'="1/3/2» + 3K3/2nj^5/2n ^ ^X^InySIn _ r^ll2„y7r2„ -yiln_ (JO. ) {a + b)l{a-b). (21.) 39* - 6^3 - 12V10 + 2V30. (22.) x^' + (3 - V2 - V3)a:=+(-2-2v'2-2V3 + 2V6)!K + (-4-2V2 + 2v'6). (23.) {x-ayiax. (24.) (13m2 + 10m + 13 + 12(m - 1) V«r + 7m + !)/( - 5in- + i6m - 5). (25.)l/j. (26.) 0. (27.) {q\/p + q\/q + q'^P^q-^pq(p-q))k( p-q). (28. ) 2[(1-.t)/ (l-4a;)}v'(l-4«)- (29.) \2(a + cf-b'' + 2{a + c)\/(a + c)"-h^\lh\ (30.) { - ^{P -q) + {p- q)\/pq+p\/q{p -q)- q'^pip - q)}lq{q -p)- (31.) 2-%- -(2W + 2).b + (2-'3-2-''3 + 1). (32.) a:- - (21''' - l)a; + (2" = + 2W + 1). (33.) 2i - 2 ^ {(a + h)-{a - i)| + 2 ^ {(« - h)\a + b)]. (37. ) (x - 2a2'Y'-' + 2a:i'-)/i'2 - 2..i'V^ + )/)/(a;-2/). (38.) - | ;il + 6V3 + 5V5 + 4V7 + 4V15 + 3V21 + 2V35 + V105}. (39.) 2Srt(Sa + 2sVic)/(Sa=-22fe). (40.) The rational- ised product is S^^S'^ - 4'. (41. ) A rationalising factor is 2(3a - 6 - c)\/(& + c - a) -2\/{n(6 + c-a)} ; theresultis -52a- + 6Sa6. (42.) A rationalising factor is (2V5-3v'2)(v'5 + 2-V3 + V6); tlie result 4. (43.) A rationalising factor is 19. 22'3 + 22 . 21 3 - 23 ; the result 307. (44. ) The result Za? + Zl,a?b - 21abc. 45.) A rationalising factor is 2''-' + 2.2-^- 3.2W-t-l ; the result 7. XV. (1.) ^(llV3 + 3Vll-\/462). (2.) 2. (3.)-2V2. (4.)12 + 5V3. (5.) JV6- (6.) 14J. (7.) ±(V10 + V15). (8.) ±(1 + 1/V2). (9.) ±(11 -V2). (10.) 23'*(3 + 13'H. (11.) ±i{sje + s/15)/3. (12.) 2-V2 + 2V3. (13.) 4V2-2V7. (14.) 3. (15.) (3 + 6V3- v'7-2V21)/2y3. (16.) iLv'6= '81649. (17.) 3-2518293. {IS.) '^(6 + 2p). (19.) V(a-c) + V*- (20.) {a + V(«=-4)}/V2- (21.) a;/(l-a;=). (23.) 1 + v'(3/2) + v'(5/2). (24.) 2-^3-3^/2. (27.) 3-2V2. (28.) 5 + V18- (29.) 2i'«(3 + 10i'-). (33.) 3-162277660; -0632455532. (34.) ±(yz-zx + xy). (35.) ±{5x-3y-s). (36.) ±{?jx- + ix - 1). (37.) ±(a;=-2x-l). (38.) ±(2x--3xy + ii/). (39.) ±{t'-3x + 2). (40.) ±(2x^-3x--x + i). (41.) ±{5p^ + 3p^q-3pq^ - 5rf). (42.) ±(x-^Jx + l). {i3.) {23i>-x^-3). (44.) {(3± V3);' + (3t\/3M/ ^3. (45.)X = 1; the square root is ±(ar-l-3a;-l). (46.) -6, 92,105; or 38, -92,137. ^47.) 3, 4, 12; or 27, 108, 108. (49.) 7.c2-2x + l. (50.) The cube root is x^ + dx+k; e = 3k + 3dr, f=6dk + d\ g=3iP + 3k(P, n=3kH. KESULTS OF EXEECISES 561 (51.) It is {52-J(c + a)}3. (52.) l+^x + ix^ + ^^afi + ^-^ai^ (53.) 1- lx--f^^-^ir>-Tili^x^- . . . (54.) \Jx{l + \/2x - l/8a:= + l/Uofi - 5/128a!^ + 7/256ci^ . . }. XVI. (1.) 2(a8-28aW + 70ai'M-28ffl26« + 68). (2.) 6/5. (3.) 8. (i.) {Spqip'^ - q^)l (y2 + j2)2}i. (5.) 2 + (V3-4\/5)i- {^.) x*-6a? + lSx^-26x+21.. (14.)/,:. (15.) a?+^2x■^{^/{3i' + ^/)+x^}+s/{ai' + y''). (16.) V(2a^*^ + 2a6cSa='6). (18.) {x^ + y"-)"l'- ; 1. (19.) ±(3+4i). (20.) ±(v'13 + i)/V2. (21.) ±K3 + 4i). (22.) ±{{a + b) + {a-b)i}. (23.) ±{cc + V(a;'- !)»}• (24.) ±[V{(»^ + l)/2}+i\/{(a:'-l)/21]. (25.) ±(3 + 2i), ±(2-3i). (26.) (x + a) {x-a) (a-wa) (x-a'a) (cc-w-a) (a;-u'%), where w = ( - l + \/3i)/2, a,' = (l + V3i)/2. (27.) (a; + l) (a;=-a;(V5 + l)/2 + l) (a;2 + a(v'5-l)/2 + l). (28.) {a;2 - 2x cos. 2t/7 + 1} {a;=- 2a; cos. 47r/7 + 1} {a;^ - 2a; cos. 6x/7 + 1} . (29.) "u' rx'-2axcostt^ + a^'\. Tc=o L m J XVII. (1.) 151/208 > 5/7. (3.) {ad-bc)l[c-d). (4.) 10|| ; 7 + 5v'2. (5.) 14-456 . . ., 13-198 . . ., 15-835 . . (6.) \/1 + \J5. (9.) {ad-bc)/ (b + c-a-d); - 1 ; 0. XVIII. (1.) 145/416. (2.) The real values of a; are ± 4. (3.) 0, 6. (4.) 25xy = 12{x^' + y% (10.) 19:16. (11.) 29i5 mm. past 10. (13.) ?->2/t. (14.) -01875 111. (15.) 6-373 ft. XIX. (1.) -2,5. (2.) a + 26. (3.) a + ft. (4.) 1, 1. (5.) a, b, c. (6.) a = 5, 6=_17. (7.) lla;2-87a; + 160 = 0. (8.) a;'^ + l = 0. (9.) re= + ac=0. (10.) (x-a){x'' + {a-b)x-ab + a^}=0. (11.) a;(a;--107) = 0. (12.) a;(a; + 2p -r) = 0. (13.)3a;-(a + 6 + c) = 0. (li.) a?~Us + t){s + t-s'- + st-t^} +2st + Vs-Vt = 0. (15.) '(b + d-b' + d')x + c{b-b') + a{d-d') = 0. (16.) X2 + Y5 + Z=-2YZ-2ZX -2XY = 0. (17.) 3x^ + 2{a + b + c)x-{a- + b^ + c^-2ab-2be-2ca) = 0. (18.) 3a;2 + 28a;=0. (19.)4cm(l-m)a; + (l -TO)V=4mc-.' (2O.)a;'^(2a;"-4aa; + 3a2) = 0. (21.) X- 16 = 0. (22.) 49a;- 1936 = 0. (23.) a;*- 3a;2=0. (Zi.) 5ai^ - ibx> -2a:'x'-ia''bx + a^ + ia'r- = 0. (25.) a!'' = 0. (26.) ^ {x\y - z)* - 2yz{z - xf {x-yf}=0. (27.) x^ + y^ + z'^-xy-yz-zxr^O. (28.) a;(a;^- 1) = 0. (29.) 625a:^- 24641a; + 234256 = 0. (30.) c»(a; + «)»+' = aVH (31.) 27S(a'-a-) = (J - 2af. (32. ) xy{x + y) (a;^ + xy + 1/) = 0. XXI. (1.) z=y^-5if + 5y. (7.) {aV + a'bf={ac'^a'cf + (bc'±l'ef. VOL. I 2 562 RESULTS OF EXERCISES XXII. (7.) 1-3543, 6-6457. (8.) 2-0508. (11.) -67(1 (13.) +-55826 . . ., --35826 . . ., and 1-5 exactly. (14.) One between -2 and -3, namely, -2-8025 ; the rest imaginary. (25.) 1-259921049894873. (26.) 2-094551481542326. (27.) 9-96666679. (28.) 46-7616301847, -3471623192. (29.) 18-64482373095. (30.) 123. (31.) 4-5195507. (32.) 1-475773161. (33.) 4-581400362. (34.) 2-0520421768796. (35.) -1-4142135623730950488. XXIII. (1.) m- (2.) 21. (3.) -m- (4.) 2. (5.) -1-455. (6.) l/(a + J). {T.) acfb. (8.) J. (9.) 0. (10.){{a? + V)(a?-V- + ab)-a? + l^]j{a?+'l)^). (11.) ±b. (12.) -2, |. (13.)0,-|. (14.)}. (15.) -i. (16.) -I. (17.) 4. (18.) -i. (19.) J. (20.) at/{a + b). I„,^ iabfd-{a^-b''){cf-de) ^ ' ' (a2 - b-) {(c +f) {a + b)-{c + d){a-b)\ - iab {dia + b) +/{a -b)}' (22.) {n? + b^)l[a + b). (23.) a + b. (24.) 2{a-b){b - c)l{c-a). (25.) 0, ± \/{ - ab - be - ca). XXIV. (1.) 6, 9. (2.) H, ^. (3.) 6-60485, 3-68993. (4.) 16, 4. (5.) f, f 1 6) -Vif, -%'-• (7.) 26-a, 2ffi-6. (8.) 2lm{m-3l)a/{2m^-3l-), 3lm{2m -Z)6/(2m2-3Z2). {9.)-2bcl{a- + !^), 2ac/{a^ + b^). {10.) ac/ {a? + b-), bcl{a? + V). {11.) a-b, 2abl{a-b). {12.) a'^-b*, a'> + a'b^ + b\ {13.) -2>q, p + q- (14.) \ = a-b-c, /j.= -ab + bc-ca. {n.) abc{b~c){c-a){a-b) = 0. (18.) ia; + ^''-. (19.) x' + x-S. (20.) -^x- + iix. XXV. (1.) 60, 40, 36. (2.) 2, 3, 4. (3.) -7293, -8039, - -0269. (4.) 12, 8, 6. 5.)A=i, B = i C=J. (6.) {(i/-y')x-{:^'-o^)y-xy + x!'y'}l{{,f~j/')x"' -{3;'-x')y"'-xY + .-r"tj",. (7.) -,V- + ix-TV {S.)x=i{b + c-a),kc. (9.) x=2bc{(P-a-){a + b)/{bc- + ca'' + ab'^ + abc), &c. (10.) XA{x-b){x-c)/{a-b) {a-c). {11.) a, b, c. {12.) .r = {a + a){a + §){a + y)l{a-b){a- c), kc. (13.) ■'■ = a{a + b + c)l{a-b){a-c), kc. (14.) x=ij = z='Sa'-Zbc. (15.) x=(m2 + 7i^ - 2lmn -li-n? + Pm - ?w= + Pn)l2{'ZP - Xmn) = (m + ?i)/2, &e. {16.)x = {p'' + mn)/ {l-m){l-n), kc. {n.) x:y ■.z=b + c-a:c + a-b:a + h-c. (18.) Put x+j/ + z = p, then a;=pa/(a + & + c)-(5f-/i,)/(a + 6 + c). (19.) - W, " fr. Jf > sf • (20.) E.xpi-ess p, q, r in terms of j:, y, -., s, then eliminate p, q, r, and there results a system of three equations in x, y, x : z=g{ad + be)/ {bce + bde-bcf - bdf- bef+ cde + cef- cdf- abc - aid - abe - acd - ade - adf- acf- aef} . (21. ) 1, 3, '), 9, 11. (22.) By means of the first four equations express all the variables in terms of z ; the last equation then gives c = {nfh + beg -bef- cfh - bfh - beh)-=r{adfh + beeg + befh - bfdh - bceh). EESULTS OF EXERCISES 563 {H.) x:y:z:u=ll{l + a):1l{l + i):ll{l + c):ll{l + d). (25.) The required equation is (B"C')a: + (A'0")2/ + (A"B')s=A(B"C) + A'(BC") + A"(B'C). where (B"C') = B"C'-B'C", &c. XXVI. (1.) b{a + ib + i\/ab)/a^ (2.) TOV/(m-re)2. (3.) 14. (4.) V"- (B.) ¥■ (6.) a-/3. (7.) x=±V{|(2p2_2py)}, This solution is extraneous if all the radicals be taken positively. (8.) ^J,' 2^5 + 2. (9.) (5a^-&-)/4ffl. (10.) a + 5. (11.) 0, a. (12.) W-> 11. (13.) ablja + i), ia + i) /i. (14.) t|, ^i. , , fa:=a-5, a + 6, a±Va(a-2i) ; 1 2/= 0, 26, 5=pVffi(a-2S). (16)9 6 (17) |^=<^4/W(»'-^')}' "^^^ "'^-' (18.) a;=a6c£?, 3/= -SfflSo, s=SaS, ii= -2a. (19.) a= -(6^-c=)(a + 5)(a + c), &c. (20,) K=a/m% &c. (21.) a:=i2a(2a + 6), &c. (22.) -a, -5, -c. (23.) !K= (6 - c) (6^ - c^)/(22a3 - 2a-i), &c. (24.) In order that the system be consistent we must have l/Jc = l/{k-a) + lj(h-b) + l/{k-c) + 'i./{k-d) ; then x : y: z : u = l/{k-a): 1/(A - b) : l/( J - c) : l/(^ - («). (25. ) x = i{a + c + d- 2b), &o. (26. ) 6, 8. (27.) 0, 0; and fj, -|^. (28.) ^V, 18- (29.) a;=±V{4(X'' + /^*) {ll\^ + l/lJ?)}y=±^{i{\*+/i*)il/\''-l//i^)}. (30.) a, S; and a-a{a-b)/ c{e-b), b-b{b-a}/c{c-a). (31.) -b-c, -a-c. (32.) a;=±V2«> &c. (33.) x=±\/ibcla, &o. (34.) -ffl, -b, -c. XXVIII. (1.) 0, -1. (2.) 1, |. (3.j_0, 0. (4.) 1, -2. (5.) i(3±i;). (6.) -1±V6. (7.) {-p^ + ?/3±VM(/3-a)}/(p-2). (8.) j -2i,g±(|,2-22)i}/ (y^ + ^S). (9.) ^(_i±Vir). (10.) -4, -7. (11.) a, If. (12.) 4(10±VW)- (13.) ll±7i. (14.) 200, 1. (15.) -53, -49. (16.) 53, -49. (17.) -3V7±2V2. (18.) l + ^/2±^/3. (19.) - 11 - 5i, -12-7*. (20.) 7 + 4i, l-6i. (21.) 2±i^J3. (22.) (l±8i)/13. (23.) f, |. (24.) 0, a + b. (25.) -2a±{b + c). (26.) a + c, -a-b. (27.) \/(W"), \/{nlm). (28.) {a + b)/ab, -2/{a + b). , (29.) a, J(2a + J)/(ct- J). (30.) 1, (b + c~2a)/ {c + a-2b). (31.) c, -c. (32.) Sa^iSffiS. (33.) (a + J + e)/3. (34.) i(l + \/29). XXIX. (1.) ±1, ±\/%i. (2.) ±V(2ffi^-Sa5), 2a. (3.) -2±3i, 1, 3. (4.) i[-(a-2)±aV2 + V{4-4a-a2=p2V2a(«-2)}], J[ - (a - 2) ±aV2 - Vji - 4a - a2T2V2a(a - 2)}]. (5.) 2, i(-3±iV23). (6.) -1, (Va±iV(3a + 4))/2Va. (7.) -w, -u\ 1,1. (S.) wlp,u'lp,p\-f. (9.) J{3±V(\/« + 4)}+4{-l±\/(V'tl-4)}«. H3± V(V41 + i)} - 4{ - 1± V(Vil - 4}}*. (10.) 2,3, 1, -1. (11.) -n, 0, -|. VOL. I 2 2 564 EESULTS OF EXERCISES XXX. (1.) l/a + yi, -2/a. (2.) ±^J{a^-ab + b"-). (3.) (l±V19)/2- (*•) 3. (5.) 3, -J. (6.) {bd-2bc)l{2ca-cd). (7.) 0, {c{a- + b'-)-{a' + b«)}l{c[a'' + b-) -^c-(n+b)}. (8.) ±^{C2a-b-'-c-a'-b-'c')l{a^ + b"--2c^)}, 0. (9.) (a' + b^)/ {a + b), a + b, 0. (10.) -■2(a + b) + 2c. (11.) 0, 0. (12.) {ab(c + d)-cd (a + b)}l{ab-cd). (13.) ±\/2«', 0. (14.) 2. (15.) 5,-34. (16.) ±V/*- (17.) i^a. XXXI.* {1.) i,H-3±^7i). (2.) 4(-6 + 4V3). (3.) Ml-\/5)«- (4.) c, (2a=6 + b"-c-w'c)J(a' + 2bc-b"-). (5.)[M5±\/52), -|],3. (6.) (a2-2a + 2)74(a- 1)^. (7.)0, 1, [K- 3±V7»)]. (8.)[V^], 0. (9.)[ab/{a + b)l0. (10.) (3±V22)/2. (11.) -.3±\/¥+*\/37- (12.) 4(« + J), [8(a + i)(2a + J)(a + 26)/(a-6)2]. (13.) [(a3 + J3)/aj], (a + j) {a? + Sab + b-)/ab. (14.) [-(9a2 + 14a6 + 962)/8(a + 6)]; if i = a, x=-2a, which d oes not satisfy the equation. (15.) 5, [--V-]- (16.) +{a''--b-)l2\/2{a- + b-). (17.) -fi, -1- (18.) +2a/V3. (19.) 0, co . (20.) 42, 15. (21.) Reduces to x = 0, along with a reciprocal biquadratic whose roots a, a, {9±i\/o)a are all extraneous. (22. ) ia{m - 2)/(m^ - 4m + 8). (23.) 0, [±4V5a/?]. (24.) a[-(m + )i)/2(m-5i)±W{l-4W(™^ + «°)}]- (25.) CL'-ibT-jiabc. (26.) ±(ffl2-)i26-)/2V W»- 1) (a^- «2>-)}- (27.) - \J{ab), [ + V(«2'), 0]. XXXII. (1.) b, c, &e. (2.) |logei{^±V(?^'-42)}. (3.) {a + 5)/(a- J)}«^-«l, {(a-S)/(a + 6)}=J'«"^-«', &c. (4.) 2, -1. (5.) 2 + llogV-/log 3. (6.) -\ + h{l^±'^(lJ.--i)]. (7.) ±«, ±V(«'' + 2)J. (8.) 2, 1/2, 3, 1/3. (9.) |, |, i(-3±V5)- (10-) 4(-3±V5), i(-l±\/15i:). (11-) 4, -1/4, 2, -1/2. ('l2.) {a-J±V(*'-2ffl6-3a-)}/2-i, -1. (13.) {-(a + b)±'s/(b'' + 2ab-Za?)]l 2a, 1. (14.) ±V{(-J±Vi'--4ac)/2a}. (IB.) {- J± V(6^- 4a2)}/2a, ±1. (16.) {-6±V('''-±4f«;)}/2a (4 solutions). (17.) i{l±V5 + V(10T2V5)i}, J!l±V5-\/(10T2V5)i}, -1. (18.) 4(-3±V5), 4(-5±V21), 1. (19.) 3:Vi-13±\/73 + v'(-2062T26V73)}, jV{&c- -&o.}. (21.) i(-5±v'33), i(-5±v'29). (22.) The equation is equivalent to 'j:- + (2^-q)x+pq=:iii\J ]n(f + '!")]■ (23.) ±V{tV(221±V48241)}. (24.) - Y, 3, K - l±\/251i). (25. ) Reduces to a reciprocal biquadratic, the roots of which are extraneous. (26.) 0, [±\/24]- (27.) -4. (28.) Put {= {(k- a)/(a; + a)}* ; the equation then becomes a reciprocal cubic. (29.) -2±J{V(v'*5 + 4)±\/(\/45-4)i}. (30.) 4(-7±V77), [4(-7±v'53)]. (31.) 16, [Wi*-]- (32.) l{-p±'s/[v'' + 4j)[, where g' = ^{-Sa±2\/(2a--SKj)}. (34.) Reduces to a reciprocal biquadratic, all the roots of which are extraneous except 4\/(2 + 2\/2)- (35.) ±1, 2±V3, 1(\±\/1U). (36.) 4{a + J±V3(«-«')i|, o, S, °°. (37.) ±2/ V(20V6-4o), ±2i/V(20\/6 + 4o). (38.) -^±i\/229, -V±iv'21- "* 'When extraneous solutions are given at all, they are in most cases dis- tinguished by enclosing them in square brackets, thus [ - V']- RESULTS OF EXERCISES 565 XXXIII. (1.) 12,18; (2.) 7, -4; (3.) 7,3,-7,-3; (4.) §,5; (5.) 0,Z, i{-l±^7i) ; 18,12. 4,-7. 3,7,-3,-7. -V, 3. 0, 3, i-( - 1 =F V7i). (6.) 1, 2, J{-ll±\/209); (7.) 4 16±a*(26-a)'*} ; 2, 1, H-11TV209). i{b^J{2b-a)-}. (8. ) • 0, {bq - ap) (q> -p^)l{{a^ + b^) {p^ + q^) - iab23q] ; 0,(bp- a q){q^-p'-)l{{a^ + b'-) (f + q'-) - iabpq} . (9.) {l+ab±\^{a^-l){b^-l]}/{a + b) ; {1 - ab±\/(a^ - 1) (b^ - !)}/(« - 6) ; two solutions. (10.) {c±V(;(crf-4a6)/d!}/2a; {cT &c. }/26. (12.) a:=±V{(325±3Vll721)/68}, 2//a;=(-107±\/11721)/2. (13.) 7,-7; (14.) 5,-3; (15.) ±a-^(a.^ + b-)/b ; 5, -5. -3, 5. ±6\/(a^ + 6-)/a; two solutions. (16.) 4(V5±l)y; (17.) ±4V3(2J-a), ±rti; 4(V'5T1)?. ±|V3(2a-6). =F&i- (18.) 0, 4a6(Ju + au2); 0, ^ab{ba — aoi^) ; where w^ = l. (19.) ±{2^(aV + 6'?')/(y-2*)}-; (20.) f, -f. (21.) 3, -1; ±>{2q'-{a^p^ + b^q^)/{p<'-q*}}^; 4 solutions. 1, -3. (22.) 4, 3, -6, -2; (23.) ±{^/S±-^2), ±(9V3±ll\/2) ; 4 solutions. 3, 4, -2, -6. (24.) ±3, ±2; (25.) 6,-2; (26.) 3, 6, 3u, 6m, Sm^, 6u= ; ±2, ±3; 4 solutions. 2, -6. 6, 3, 6w, 3u, 6u-, 3w=. (27.) 2, 3, 2w, 3u, 2u^, Sui^ ; (28.) ±a*(a*-"5*)^ ±u6*(a*- u6*)^. 3, 2, 3w, 2u, Su^, 2w2. (29.) 5, 2, -5, -2; (30.) 2, 4, 2w, 4u, 2w=, 401^; 2, 5, - 2, - 5. 4, 2, 4w, 2u, 4w2^ 2m2. (31.) 4[J±v'{-3'''±2V(2ci^ + 2J^)i]; 4[6=fV{-3*'±2V(2«-» + 2J^)}]. (32.) 3, 2, 4(5±v'151i:) ; ,„- , 2a 2, 3, 4(5=FV15U-). '^^■' ''-2±y±V(p^ + 4)' *'■ (34.) 1, 2, 4(3±V19«) ; (35-) (2* + 2^u + u2)/2* ; 2, 1, 4(3TVl9i). w-M (36.) 5v, 2v' ; where u*= +1, (37.) -j\(17± V51i) ; 2u, 5u'; !;'■'= -1. A(17=F VSl*')- (38.) If v=y/x, then u(l + u^) = a'(l + v*), a reciprocal biquadratic. (39.)±3, ±2;„ , ,. ,„ . , a; + 2/=±v'(-li±6V7); ±2; ±31^-1*1°"= ; andSmore given by ^_ j,^±,^(_ i4^6V7). (40.) ±a\'abl{a' + b-), ±ia'Vabl{a^ + b-); (41.) 2, 8 ; ±b\/ab/{a- + b-), :^ib\/abi{¥+¥). 8, 2. (42.) Eationalise the first equation, using the second in the process, and thus find a quadratic for xy. (43. ) {a? - b^-)l2a ; {a- + r-± V(a^ - ea%^ + fi*) } /4a. 566 RESULTS OF EXERCISES (44.) 10, 13; (45.) 2, 8; (46.) |a ; 13, 10. 8, 2. la. (47. ) We can derive (a; - yf - 2a{x + y) +a- = ; {x-yf~\/2b{x + y) + sjiab - J^ = 0. (48.) ±6s, \/2a + l[his); ^ai, \J2h + a (his). (49.) I{u=xyjab, v=x'la,--¥y^/b^, we can derive (m-2)u^ + 2{m-n)u + {m-2) + nv=0 ; mu^ + 2{m + n)u + m,- {n + 2)v = 0. (50.) ±als/{a + b); (51.) B; (52.) One real solution is i{ia + l±^{8a + l)} ; 4{ -1=fV(8« + 1)}- Another is given by y' + 2ay'' + l = 0, x = y-'^. XXXIV. {\.)x=+{b-c)l{abcf, y=+(c-a)l(abcf, z = &c.; x=u{b-c)l{abc) , y = a{c-a)l{abcf, : = &c. ; x = i^-{b^c)/{abc), i/ = &c. :=&o. ; where w^= +1. (2.) Eliminate z between the iirst two equations, and piit | = K- c, ri=y-c. The following are solutions : — x = i + c, y = c + a, z=a + i, x={b'' + c'^-a{b + c)\lib + o-a), y = ka., c=&c. (3.) a;=2, t/ = 3, z = l ; or x=-6, 2/=-7, s=-5. (4.) a: =3, i/ = 2, ; = 1. (5.) x=±}\J{WOl), t/=±fijV(1001). z=±TT\/(1001). two solutions. (6.)a:=2/=z=±V2/2. (7.) ij:=±(a2S2 + aV-6V)/2a6c, 2/=±&c., c=±&c., two solutions. (8.) We derive by subtraction from the first two equations (i- - 2/) (« - z) = 0, and from the first and third (x-2){a-y) = Q, Combining these two with one of the original equations, we obtain the following five solutions (the last three twice over) : — x=a, ^, (i^^-a^)/«, «, «; y = a, /3, a, {p^-a-)ja, «; z = a, )3, a, a, {p'-a^)la ; where a and (3 are the roots of x'' + ax-p' = 0. (9.) Eight solutions, as follows : — x = z={±5 + \/m)IS, 1/ = (::]= 10 + \/409)/3; x = z = {±:i-\/i09)/S, 2/ = (±10-V409)/3; a: = (±Vi635 + i3V33)/6, z=(±\/l635-i3\/33)/6, 2/=±V(1635)/6; _ ,r = (±V'l635-i3v'33)/6, s = (±\/l635 + s3v'33)/6, 2/=±v'(1635)/6; upper signs together and lower together throughout. (10.) '• = ±(p + 35)/2(^2 + p + ^2)*. 2/=&c., - = &c, (11.) If we add the EESULTS OF EXERCISES 567 ■three equations, we obtain the equation {a + b + c){x + y + z) = S{x + y + z)'. Hence x + y + z=0, or =(a + J + c)/3. The three equations can therefore be replaced by three linear equations : x = 0, y=0, 2=0, and x={Sbe + 2ca + ai + 6^-c^)/2(5c+ca + aJ), &c., are solutions. (12.) The equations are linear in x'-yz, y^-zx, z^-xy. Solving, we obtain, x^-yz=p, y^-zx=q, z^-xy=r, say. If we now put x=uz, y=vz, we obtain the following biquadratics for u and V : — (r^ -pq)u* -{p'- qr)v? - (r^ -pq)ii,+{p'^ - qr) = 0, (r^-pq)ii^-(q^-rp)ifi -{r^-pq)v + {q' -rp) = 0. "We thus find the following values for u and v : — M=l, w, w2, ip^-qr)/{r^-pq), v=l, w^, a, {q^-i'p)l{r^-pq)- The first three pairs give !c=oo, y = oo, s=(». The last pair gives x=±{p^-qT)l\/{p^ + ^ + r^-3pqr),y=±kc.,z=±kc. (13.) From the first equation we see that x=pa{b-c){a- + a), y=pb{c-a)((T + h), s=&c., where p and (T are arbitrary. Tlie second equation gives the following quadratic for a, {2a2 - 26c} 0-2 + {Za^{b + c) - &abc] o- + {26 V _ abcla] = 0. When u is known, the third equation gives /) = ± 1/ VSa^(5 - cf{a- + of. Hence we obtain four sets of values for x, y, z. (14.) From the first three equations we have x + yz =pla, &c. From these, squaring and using the last equation, we deduce (l-y^){l-z^) = p-/a^, &c. From these last we deduce x=±sj{l±pa/bc), ±&c. ±&c. Substituting these values in the last equation we find p=0 and p=±(Sa^-2S6V)/4a6c. Hence x=y=z=-l ; and x= - {a^ - b^ ~ c^)/ 26c, y= -&c.. 2= -&c. (15.) x=0, ia, fa, -a, -a; y=0, ia, -a, fa, -a; 2=0, ia, —a, -a, fa. (16.) The given equations may be written {x - yY + {y — z)^=a?, &c. Hence wehave(2/-2)= = (62 + c2-a2)/2, &c. Hence y-z=±^(b' + (^- a'-')/2, &c. The system is therefore insufficient to determine the three variables ; in fact it will not be a consistent system unless Sa* - 26%^ = 0. (17. ) If p = xyz, x=uz,y = vz, we may write the equations ap = {v + l)lu, bp = {u-l)lv, cp=u + v. Elim- inating u and V we find p^ = {b + c-a)/ahc ; and so on. (18.) If x be eliminated, the resulting equations may be written 2^77 + 178 - 147; - 28f - 81 = 0, where ^=yz, ■q=y^ + z^; one set of solutions is x=Z, y=l, 2=2; another x=Z, y=2, 2=1; &c. (19.) From the given equations we can deduce (J}hj-ch)l{y-z) = kc.=ka.=(T, say. Whence (a*-o-)a;=(6''-i7)2/ = (c''-(r)2 =T, say. We can then determine tr and r by means of the given equations. Result, ij; = /){n(a8-6*c'')}*/(a8-64c*)(2ffii2-3a^6*c'»)*, where p is any one of the 4th roots of +1. (20.) The equations can be written xy + xz — yz =p/a^, &c., where p=xYz^/{yz + zx + xy). Result, a;=±26c(26V)*/a(62 + c2), 2/=±&o.,z=±&c., two solutions. (21.) x=±\/2Jbh\b + c)/{Sa\b + c)''}\ 568 EESULTS OF EXERCISES 2/=±&c., ;=±&c. [22.) x=y = z = 8; &ndx= -6, y=-i, ::= -2. (23.) x=d(b-a)/{i:-d), y=e(h-a)j(c-d), z=b{c-d)j{b-a), u=a{c-d)j{b-a). (24.) The real solutions are .v=3, 1, 2 ; 2/ = 2, 3, 1; c = l, 2, 3. (25. ) We have (a,-^ — yz)- - {y' -zx){z-~xy) = a*- IrcK Hence x—p(a*-b'^c'), y = p{b^-c-a-), kc. x=±{a*-bh-)l\JCZa^-^a-b\'^), &c. XXXV. (1.) [(ac'){{ad") + {bc")]-{ac"){{ad') + (bc')]} x [{bd") {{ctd') + {be')} -(bd') {{ad") + (bd')}'\ = [{ac')[bd") - (ac")(bd')f. (2.) {P + vi'-lfia + bf-iP + vi'-l) {{P-m'f-P-m-}{a"-by- + p7)v'{a-i)- = 0. (3.) a^ + b''=c''{a^ + b^). (4.) 8d'^ = 1I{b- + c''-a-). {5.)p + q + r = Q. {a.) c{a~by-2c{p + q){a-b)- + {cp-q) (2)-cq) = 0. (9.) Eliminate z, and put ^ = x + y, ■q=xy, in the three resulting equations, then eliminate {, and there results two quadratics in ri, &c. (10.) 2(t'P = 5a-6^A (11.) Put « = 2.>.', ■!; = 2.)'i/, w=xyz, eliminate v and w, and reduce the resulting equations to two quadratics in «. (12.) Let ^ = x + a, Tj=y + b, ^=z + c, then ^T]i=dbc. "We have Ti^-(b + c)(Ti + i) = a''-(b + c)-, &c. These give J, t;, f in terms of ijf, ^|, |-);. Again multiplying the last three equations by |, ij, f, we have abc-(b + c)(^i) + ^i). These give |, ?;, f again in terms of ijf, f|, I?;. We thus get three linear equations for f, -q, f, &c. (13.) We can deduce a:^K„ + j/2j/„ = const. = oi'„" say; av-'n + 2/3j/>i = s^n^ . . .; and finally a;„_ia!„ + j/„_ii/„ = „_i7^„^. Now either „_i^„2=A„^, in which case the system is indeterminate, or „_ii„^=t=fc„^, in which case the system is inconsistent. XXXYI. (1.) -f + 5p^q-hpq'^. (2.) (i)"- 6;^^ + V?'- W(^^- 4?). (3.) ( -^' + 5p'q-5pq^)lq^ (4.) ±{p'-3fq + q^W{p^-4q)/q\ (5.) {-^,(^,2-39) Mp--q)s/{p''~iq)}lzfPif-(l){p"--^W{p^-i. (20.) 2pi^-Zpip.^-Zp3. {21.) p^^ - ip-?p^ + 2p.J^ + ipiP3-ipi. (22.) {SpiPr-Sp2P32)i+23s^)/pi^. {23.) pr - 2piP3 + 2pi. {2i.) pips-ipi. {25.) S2)i*-8pi^p2 + W-iPiP3 + '^6pi. (26.) x' -p^^ + {PiP3- ipi)x-pi^Pi+ ip«Pi-p.i' = 0. (30.) 22}2+2p2-piPi. (31.) = 2( ;;r + V') - 4( ;)2 +i^3') - 2/)iPi'. (32. ) ( JJs -p4f + PiPi{p2 +P-2) + p-^pi +I>i-'2h. (36.) 'j:--iacx + id-

%a+ \/{2{b'-W)}- {i5.)r + h[l±sj{2-{l + 2rlh)'}y2,r={^2 -l)A/2. XL. (1.) 495. (2.) 307i. (3.) 36. (4.) -\\ (5.) l{n'-Sn + i). (6.) n{w> + n{n-S)a+n^}. (7.) {Sl + P + 5P-l*)l2{l-P). (8.) 8998148/^. (9.) 3. (10.) 1000. (11.) Any A. P. whose first term is a and whose CD. is 2a/(m-l- 1) has the required property. (12.) £2131 : 5s. (13.) 50,600 yards. (14.) 2525 yards. (15.) 9. (16.) Z/c. (17.) f|, Jj!^, &c. {18.) - ^, - H, ka. (19.) 20. (20.) 20. (22.) 1, 5, 9, 13. (24.) i?i(3)i + 5). (25.) b + {a-b -lrb)n + lrbn\ (26.) 3, 2. (28.) (2)1)^=4 + 12 + . .. + (8ji-4), (2)1 + 1)2-1 570 EESULTS OF EXERCISES = 8 + 16 + . . .+8?i. (29.) The first odd number is m''-^- 51 + 1. The second part follows from § 7. (30.) T„=4(~l)"-i-rt(™ + l); 2„= -i?i(»i + 2)ori(w + l)2, according as n is even or odd. (31.) |)i(?i + l). (33.) »^ (34. ) J)i(6re2 + 15» + 11). (35.) „S4-„S3-„Si + m. (36.) {p-q){2}~2q)n + g{2p-Sq)„Si + q\s2. (37.) -2)i2_M. (38.) {a-tfn + 3b{a-by-„Si + Zb-{a-b)nSa + b\ss. (39.) i{n-l)n{n + l){n + 2). (40.) ^ji{n + l){n + 2){37i + 5). (41.) in{n + l)(6n^ -271-1). (42.) M6™' + 32?i2 + 33re-8). (43.) xV«(» + l)-(™ + 2). (44.) 4960. (45. ) im(m + 1) (2m + 1). (46. ) il{l + 1 ) (3m + 21-2). XLI. (1.) 313-3. (2.) -i(3"-3). (3.) J(l-1/10"). (4.) |(l-(-l)"/4"). (5.)a^{l-(2/3)i»}. (6.) i(3 + v'3){l-(V3-in- (7.) |(1 -1/3"). (8.) |. (9.) l/(l+a:). (10.) 2^/2. (11.) h (12.) (a + xf/2{a-x){aHx'). (13.) {l + x)/{l + x-). (14.) (a;="-t/2")/(a; + 2/)a.-2"-2. (15.) a;(l -K")/(a;-i/) (1 -a) -j/(l-r)/(a:-2/)(l-2/). (16.) i!V{9»-V10 + lA0"+H. (17.) V(6"-l). (18.) a;V"-l)/(a^-l)-W™ + l)(2'H-l)- (19.) {(a:i/)"+'- (a;i/)-»}/(a:j/- 1) - {x(xlyY-y{,ylxY}l{x-y). {20.)p'ip'>-l)/{23--l)-q\q-''-l)l{q^ -1). (21.) »6"- 2" + l. (22.) -a''(l - ( - 1 )"«'")/( 1 +a5). (23.) 2n + {r-" + l/r^-^){r^"-l)/ (r^-l). (24.)?i + 2(l-?-")/(r-l) + (l-r-2")/(r^-l). (26.) a(r"-l) (r'- 1)/ (r-l)2. (28.) :„={!-(- )"(re + l)x»- (- )'ta"+i}/(l + a:)2, S„ =l(l + a:)2. (29.) 2;„ = f + (-)»-i(67i + l)/9.2"-i, 2,^=f (30.) 2;„ = /^ + (-)'-i(97i2 + 12« + 2)/ 27.2"-i, S,^=A- (31.) 2„ = 1 + 2x7(0: -1)3- {(7i= + m)a;=- (2h=- 2)a: + (»2-?!.)}/ a--'(''-l)^S„=l + 2ai'7(a:-l)'. (32.) 2„={l + ix + x')/{l- x)*. (33.) S„ = 11/57. (34.) (a?-2 + 5r + c)(l-?-3'')/(j-3-l). (35.) £5825 :8 : 5j. (36.) 100(7)/Y)». (37.) JAi. {38.)a^l{a-b). (39.) a"?- (1 -a")E/(l - a), wherea = l -{d-b)llOO. (41.) 7. (42.) 180/121. (43.)-for|. (44.)5,or-6. (45.) 5. (46.) 2(5"-l), or 1(510-1). (47.)|. (48.) 3, f, 1. (49.) 4, 8, 16, 32. (50.) 12^45. (51.) 3, 48. (52.) 11, 33, 99, 297; or -22, 66, -198, 594. (53.) {ac-b^)l(2b-a-c). (54.) 5, 10; or -25/3, -970/3. (55.) If ^ and 2 be tlie given sums, the results are 2jiqj{j}- + q), (p--q)l(p- + q)- (58.) (1 + a) {].-(acT}l(\-ac). (60.) a;V/(l + r)(l-r)=(l-r'). XLII. (1- ) ^, f ; ¥A Y/, &o. (2. ) -11, lif . (3. ) w, j^s a/, -V-, V-, V-. - ¥, &o. (4.) For the corresponding A. P. a = \^, b= -jV (5.) For the correspond- ing A.P. a={{p-l)V-{q-l]Q}l{p-q)7Q, 6 = (Q-P)/(jB-g')PQ. (8.) i{y- V(i''-900)},- 15, i{p + \/{p^-900)}, where^ is arbitrary. XLIII. (l.)4, 5, 3, 4. (2.) 4, 3, 3, 1 (3.) 3J. (4.) 5-1871 .. . (5.) 32617-105. (6.) -00794818. (7.) 3144-973. (8.) 1-3800812. (9.) -0 . . . (22 cyphers) 433352. (10.) -2674734. (11.) -979467. (12.) 7-7794. (13.) 38-37. (14.) 48-3. (15.) 4-81213. (16.) 282351500. (17.) 3-560625. (18.) The common ratio is 1-079188. (19.) 1-2921592. (20.) 36-833432. (21.) 20. EESULTS OF EXERCISES 571 (22.) 35. (23.) 1-041393. (24.) 2432, number of digits 19. (25.) 5. (26.) 1373454. (27.) f. (28.) '98397. (29.) "47320. (30.) 2-10372, -1-10372. (31.) a;= -3-313811, y = -000527696. (32.) 1-49947. (33.) a'=-76028, 2/=-02060. (34.) 1-24207. {35.)\og{a^-b^)l2\og(a + b). (36.) » + j/=±2a, x=yl (37.) 2-793925. XLIV. (1.) 3 montlis. (2.) £1660 : 12 : 10. (3.) £225 : 4 : 10. (4.) 12 years. (5.) 254 years. (6.) 7s., 7s. 6d. (7.) 15 years. (11.) P(l+m?-)/(l+7i?-') if the surplus interest be not reinvested ; otherwise P{(l-7-/r')/(l+ /)" + ?-//}. tl2.) 4-526%. (13.) 6%. (14.) £41,746, £48,837, £59,417. (15. ) SA,.7v/2A„ (logSA,-logSA,E>»,.)/logR. (16.) £1107:3:7. (17.) £10 : 5 : 6. (19.) £8078. (20.) £11,231. (21.) £1801 : 14 : 10. (22.) 4%. (23.) £479 : 14 : 11. (24.) 10 years. (25.) £75 : 12 : 10. (26.) A(l-R-2")/2(R + l). (27.) £2904:2:5. (28.) AR". (29.) 20 years. (30.) {log2-log(l + R-"')}/ logRyears. (32.)£1912:8: 11. (33.) Present value=a/(R-l) + J(l -K-»+^)/ {B.-1)-- {a + {n-l)b]'R-»l{R-l). (34.) Present value=a{l -(6/R)"}/(R- J). (35.) A{mR('»+')«-(m+l)R'»' + l}/R'»«(R«-l) (R-1). XLV. (X and p denote roots of the resolvents of Lagrange and Descartes.) {l.}X = 2rlp. (2.)X=±2. (3.)\=-3. (4.)X = 10. {B.) \=pq. (6.) 3/2 is a, root. (7.) 2 and -3/2 are roots. (8.) The equation reduces to (,x^ + 2x + 3){2x' + x-2) = 0. (9.) \ = 4't'566. (10.) a; = l + v'2 + V3, etc. (11.) p=4. END OF PART I. Printedby R. & R. Clakk, Limited, Edinburgh.