iv?) ^v^i,'^ ^fW^^u-; Mx.*'ri'*V( (cp) X LP ^%^ y SonHon: C. J. CLAY and SONS, CAMBRIDGE UNIVERSITY PRESS WAREHOUSE, AVE MARIA LANE. ©lasgofa): 263, AEGYLE STREET. Eeipjtg; F. A. BEOCKHAUS. i^eh) govk: THE MACMILLAN COMPANY. iSomfiag: E. SEYMOUR HALE. Pitt prfs^e? iMatftmatiral ^nit^ ELEMENTARY ALGEBRA EDITED FOR THE SYNDICS OF THE UNIVERSITY PRESS BY W. W. ROUSE BALL, FELLOW AND TUTOR OF TRINITY COLLEGE, CAMBRIDGE; AUTHOR OF A HISTORY OF MATHEMATICS, A HISTORY OF THE STUDY OF MATHEMATICS AT CAMBRIDGE, MATHEMATICAL RECREATIONS, ETC. SECOND EDITION. CAMBRIDGE AT THE UNIVERSITY PRESS 1897 [All Rights resell) ed] PKINTED BY J. AND C. F. CLAY, AT THE UNIVEKSITY PKESS, Clf\lsz p^ PREFACE. THIS work on Elementary Algebra has been written at the request of the Syndics of the Cambridge University Press, and is intended to. include those parts of the subject which most Schools and Examination Boards consider as covered by the adjective elementary. The discussion, herein contained, of Permutations and Combinations, the Binomial Theorem, and the Expo- nential Theorem — subjects which are sometimes in- cluded in Elementary Algebra, and sometimes excluded from it — should be regarded as introductory to their treatment in larger text-books. I have in general followed the order of arrangement and method of presenting the subject which are tra- ditional in England. I hope that the Table of Contents wdll enable the reader to find with ease the articles in which any particular part of the subject is discussed. It may assist a student who is reading the subject for the first time, without the aid of any one to explain his difficulties, if I add that the propositions here given fall naturally into five groups, and that in the text these groups are divided one from the other by col- lections of miscellaneous questions or examination papers which have been set recently by various repre- sentative Examining Bodies. All articles and examples which are marked with an asterisk (*) may be omitted by any one who is reading the subject for the first time. vi PREFACE. I am indebted to the kindness of the Secretaries of the Cambridge Local Examinations Syndicate and of the Oxford and Cambridge Schools Examination Board for permission to use the papers and questions which have been set in the examinations held under their authority. A large number of the examples inserted at the end of each chapter are, except for a few verbal alterations, derived from one or other of these sources, and indicate the tests of a knowledge of the subject which are usually applied : those questions which are marked with an asterisk are intended for the more advanced students only. The numerous examples in- terspersed in the text of each chapter are in most cases easier than those placed at the end of the chapters, and can be solved by a direct application of the rules given in the text. I shall be grateful for notices of misprints, cor- rections, or criticisms on the work which may occur to any of my readers. W. W. ROUSE BALL. Teinity College, Cambridge. TABLE OF CONTENTS. 10]. Chapter I. Definitions and Notation. Subject-matter of Algebra [Art. 1] Symbols [Art. 2] . Symbols of Operation [Art. 3] Expressions [Art. 4] . Addition [Art. 7] . . Subtraction [Art. 8] . Positive and Negative Quantities [Art Multiplication [Art. 12] Factors [Art. 13] Coefficient [Art. 15] . Division [Art. 16] Brackets [Art. 17] Common Symbols of Operation. List of [Arts. Powers, Indices, Exponents [Arts. 21, 22] . Koots, Surds [Art. 23] Terms [Art. 24] Simple and Compound Expressions [Art. 25] Degree or Dimensions of a quantity [Art. 26] Degree or Dimensions of an expression [Art. 27] Homogeneous Expressions [Art. 28] . Formula. Identity. Definitions of [Art. 29] 18, 19] PAGE 1 1 2 2 3 3 4 5 5 6 6 7 7 10 12 14 15 15 16 16 16 VIU CONTENTS. Chapter II. Addition and Subtraction Addition and Subtraction of Simple Expressions [Art Combination of like terms [Art. 34] Simplification by collection of like tern: Addition of Multinomials [Art. 36] Process of addition [Arts. 37, 38] . Subtraction of Multinomials [Art. 39] Extension of results [Arts. 40, 41] Process of subtraction [Art. 42] Removal and Introduction of Brackets [Arts. 43 Algebraical Sum [Art. 46] . Inequalities. Definitions [Ai'ts. 47, 48] s [Ar -45] 32] ..35] PAGE 18 19 20 21 21 22 23 24 25 26 26 Chapter III. Multiplication. Order of Multiplication immaterial [Art. 50] . . . 29 Product of Simple Expressions [Arts.. 51, 52] ... 29 Product of a Simple Expression and a Multinomial [Art. 53] 31 Product of two Binomials [Art. 54] 33 Extension of results [Art. 55] . . . .34 Rule of Signs [Art. 56] 35 Process of multiplication [Arts. 60 — 62] . . 36 Product of two Multinomials [Arts. 63—67] ... 39 Product of a number of expressions [Arts. 68 — 70] . . 42 Index Law I. To shew that a-™ x .'r»=A^'» + » [Arts. 72—74] 43 Theorem. To shew that (a6)"=a'^6" [Art. 75] ... 44 Index Law II. To shew that (.i-'")'i=.v»"* [Art. 76] . . 45 Chapter IV. Division. Division by Simple Expressions [Art. 78] . . . .48 Rule of Signs [Art. 79] 49 Quotient of a Multinomial by a Simple Expression [Art. 80] 50 Quotient of a Multinomial by a Multinomial [Arts. 81 — 85] 52 Index Law. To find the value of .r'"-r.^^" [Art. 86] . . 57 CONTENTS. ix Chapter V. Simple Equations involving one unknown quantity. PAGE Identity. Definition of [Arts. 87—88] .... 60 Equation. Root. Unknowns. Definitions of [Arts. 89 — 92] 60 Classification of Equations [Art. 93] 61 Axioms used in the theory of Equations [Art. 94] . . 62 Transposition of Terms [Art. 95] 62 Method of Solving Simple Equations [Art. 97] . . . 63 Illustrative Examples [pp. 64—66] 64 Chapter VI. Problems leading to Simple Equations. Definition of a Problem [Art. 98] 69 Illustrative Examples [Arts. 99—100, pp. 69—76] . . 69 Formulae used in certain problems [Art. 101] . . . 76 Examination Questions. Pages 81—84. Chapter VII. Factors. Exx3ressions having a factor common to every term [Art. 103] 85 Known Forms [Arts. 104—111] (i) Difierence of two squares [Arts. 105—106] . 87 (ii) Perfect squares [Art. 107] .... 88 (iii) Expressions of the third degree [Art. 108] . 89 (iv) Expressions of the fourth degree [Art. 109] . 90 Table of known forms [Art. 110] . . . .91 Division by inspection [Art. Ill] . . . . 91 Factors of Quadratic Expressions [Arts. 112 — 118] . . 92 Illustrative Examples [Arts. 113—114] . . 92 Any Quadratic Expression [Arts. 115— 118]. . 95 Imaginary factors [Art. 119] 97 Method of determining by inspection whether an expression is divisible by a linear factor [Arts. 120—122] . . 98 Integral values of n for which a^" + «" is divisible by A' +« or .r- a [Art. 122] .... 99 X CONTENTS, Chapter VIII. Highest Common Factors. Highest Common Factor. Definition of [Art. 123] . H. C. F. of Simple Expressions [Art. 125] . H. C. F. of Expressions resoluble into linear factors [Arts 126—127] H. C. F. of Compound Expressions [Arts. 128—132] . PAGE 101 101 102 104 Chapter IX. Lowest Common Multiples. Lowest Common Multiple. Definition of [Art. 133] . .110 L. C. M. of Expressions resoluble into linear factors [Art. 135] 110 >/ L. C. M. of Compound Expressions [Arts. 136 — 137] . .111 Chapter X. Fractions. Fractions. Reciprocals. Definitions of [Arts. 1 38 — 140] . 114 Properties of Fractions [Arts. 141 — 147] . . . .114 Simplification of Fractions [Art. 148] 117 Addition and Subtraction of Fractions [Arts. 149 — 150] . 118 Multiplication by Fractions [Arts. 151—152] . . .123 Division by Fractions [Art. 153] 124 Simplification of Expressions involving Fractions [Art. 154] 126 Chapter XI. Simple Equations Continued. Illustrative Examples 132 Chapter XII. Simultaneous Simple Equations. Simultaneous Equations. Definition of [Ai-t. 157] . . 139 Simultaneous Simple Equations [Arts. 158—159] . .139 Principle of method of Solution of two equations [Art. 160] 139 First method of solution [Arts. 162—163] , . 141 Second method of solution [Art. 164] . . .142 Illustrative Examples [Art. 165] . . . . 142 Three Simultaneous Simple Equations [Art. 166] . . 146 Indeterminate Equations [Arts. 167— 168]. . . .147 CONTENTS. xi Chapter XIII. Problems leading to Simultaneous Simple Equations. PAGE Illustrative Examples 153 Chapter XIV. Miscellaneous Propositions. Expression of theorems in algebraic notation [Art. 1 72] Substitutions [Art. 173] . Elimination [Ai-t. 174] Symmetrical Expressions [Arts. 175 — 179] . Cyclical Interchanges [Art. 176] . Comparison of unequal quantities [Ai'ts. 180 — 181] Examination Papers and Questions. Pages 178—183. 165 167 169 170 170 173 Chapter XV. Evolution (Square Roots and Cube Roots). Rational and Irrational Quantities. Definitions of [Art. 182] 184 Roots of Simple Quantities [Art. 185] . . . .185 Roots of Products [Art. 186] 185 Square Roots of Compound Expressions [Arts. 187 — 195] . 186 Trinomials [Art. 188] 187 Quadratic Expressions [Art. 189] . . . .188 Multinomials [Arts. 190—196] . . . .189 Process of extracting a square root [Arts. 191 — 192] 191 Square Roots by inspection [Arts. 193 — 195] . 192 Cube Roots of Multinomials [Arts. 197—200] . . .195 Expressions of foiu* terms [Art. 198] . . . 195 Expressions of degree not higher than nine [Art. 199] 196 Any multinomial [Art. 200] 197 Chapter XVI. Quadratic Equations. Definitions. Nomenclature [Arts. 201—202] . . .198 Methods of Solution [Arts. 203—211] Resolution into Factors [Arts. 203- Completing the Square [Arts. 208- Substitution [Art. 211] . Equations involving Fractions [Art. 212] . -207] . . 199 210] . . 201 . . 204 . . 206 xii CONTENTS. Chapter XVII. Simultaneous Equations, of which at least one is of a degree higher than the first. PAGE A Simple and a Quadratic Equation [Arts. 215 — 216] . 212 Two Simultaneous Quadratic Equations [Arts. 217—219] . 216 Chapter XVIII. Problems leading to Quadratic EcLuations. Illustrative Examples 224 Chapter XIX. Equations reducible to a Quadratic Form. Quadratics in a power of x [Arts. 224 — 225] . . . 238 Cubic equations, of which one root is known [Art. 227] . 241 Keciprocal Equations [Arts. 228—229] . . . .242 Chapter XX. The Theory of Quadratic Equations. Conditions that the roots may be real, equal, imaginary, rational [Art. 231] 245 Sum and Product of the roots [Art. 232] . . . .246 Factors of quadratic expressions [Arts. 233 — 235] . . 248 There cannot be more than two unequal roots [Art. 236] . 250 Formation of equations whose roots are connected with those of a given equation [Art. 237] . . . .251 Application to Maxima and Minima [Art. 238] . . . 254 ^Chapter XXI. Indeterminate Equations. Indeterminate Equations. Definition of [Art. 239] . . 257 Illustrative Examples [Art. 240] 258 Examination Questions. Pages 261—264. CONTEXTS. xiii Chapter XXII. Fractional and Negative Indices. PAGE Meanings assigned to a". The Index Laws [Arts. 241 — 247] 265 Applications of the index laws [Ai*ts. 248 — 249] . . . 269 Fourth and sixth roots of expressions [Art. 250] . 272 Chapter XXIII. Logarithms. Logarithm. Base. Definitions of [Art. 252] . . . 275 Determination of certain logarithms [Art. 253] . . . 275 Properties of logarithms [Arts. 254—259] . . . .276 Theorem. To shew that log«m=log;,mxIogf(6 [Art. 260] . 281 Common Logarithms. Their advantages [Arts. 261 — 262] 282 Mantissa and Characteristic. Definitions of [Art. 263] . 283 Characteristics can be determined by inspection [Art. 264] 284 Logarithmic Tables [Art. 267] 285 Chapter XXIV. Surds. Definitions [Arts. 269—270] 289 Simple Expressions involving Surds [Arts. 272 — 279] . . 290 Compound Expressions involving Surds [Ai'ts. 280 — 284] . 293 Equations involving Surds [Arts. 285—286] . . .298 Chapter XXV. Ratio and Proportion. Ratio. Definitions and Notation [Arts. 287—290] . . 305 Ratios are measured by Fractions [Arts. 291 — 294] . . 306 Incommensurable Quantities [Arts. 292 — 293] . . . 306 Comparison of Ratios [Arts. 295—296] . . . .307 Proportion. Definitions and Notation [Arts. 297 — 298] . 311 Properties of a Proportion, and Examples [Arts. 299 — 303] 312 Continued Proportion [Arts. 304—306] . . . .316 xiv CONTENTS. Chapter XXVI. Variation. PAGE Direct Variation. Definition and Notation [Arts. 307 — 308] 323 Properties and Examples [Arts. 309—310] .... 323 Inverse Variation [Arts. 311—312] 325 Joint Variation [Arts. 313—317] 327 Illustrative Examples [Art. 318] 329 Chapter XXVII. Arithmetical Progressions. Definition of. Condition for an A. P. [Arts. 319—321] . 335 Expression for the n\h term of an A. P. [Art. 322] . . 336 Sum of n terms of an A. P. [Arts. 323—324] . . .338 Arithmetic Means [Arts. 325—328] 341 Chapter XXVIII. Geometrical Progressions. Definition of. Condition for a G. P. [Arts. 330—332] . 348 Expression for the nth. term of a G. P. [Art. 333] . . 349 Sum of n terms of a G. P. [Art. 334] 350 Sum of an infinite number of terms in G. P. [Art. 336] . 353 Infinite Series [Art. 337] 354 Convergency. Divergency [Arts. 338 — 339] . 354 Geometric Means [Arts. 340—344] 358 Chapter XXIX. Harmonic and other Series. Harmonical Progressions [Arts. 347 — 356] .... 365 Sum of l2 + 22+...4-?i2 = i'/K?i + l)(2?i + l) [Arts. 358—359] . 371 Sum of series \^ + ">?-{■... -\-n^=\n^{n+ If [Art. 360] . .374 Sum of series 1 4- 2.^' + 3.^2 + 4^-3 + . . . [Art. 363] . . .376 Sum of series a + (« + 6) r + (« + 26) r2 + . . . [Ai-t. 364] . .376 Examination Papers and Questions. Pages 380—385. CONTEXTS. XV Chapter XXX. Permutations and Combinations. PAGE Definitions and Notation [Arts. 366—367] . . . .386 Permutations [Arts. 369—375] 387 Combinations [Arts. 376—381] 392 Chapter XXXI. The Binomial Theorem. Proof for Positive Integral Exponents [Arts. 383—386] . 401 Propositions about [Arts. 387—395] 405 Multinomial Theorem [Art. 396] 408 Proof for Fractional and Negative Exponents [Arts. 398 —403] 410 Propositions about [Arts. 404—406] 417 Application to Approximations [Art. 405] . . . .418 Homogeneous Products [Art. 407] 421 Chapter XXXII. The Exponential Theorem. Definition and Value of e [Arts. 408— 409] . . . .426 The Exponential Theorem [Arts. 410— 411] . . .427 The Logarithmic Series [Art. 412] 429 Natural Logarithms [Arts. 413—414] 431 Calculation of [Art. 414] 431 Common Logarithms [Arts. 415—417] .... 432 Calculation of [Art. 416] 432 Table of [Art. 417] 433 Rule of Proportional Parts [Arts. 418—423] . . .434 Examination Papers. Pages 439—443. Chapter XXXIII. Properties of Numbers. Scales of Notation [Arts. 425 — 434] 444 Properties of Numbers [Arts. 435—437] . . . .453 CHAPTER I. DEFINITIONS AND NOTATION. 1. Algebra is a science which treats mainly of numbers. The distinction between arithmetic (which is also concerned with numbers) and algebra may be stated as follows. In arithmetic, every number is distinguished from every other number by the use of a certain figure or figures ; but in algebra, we use symbols (such as the letters of the alphabet a, b, x, y, &c.) to represent any number whatever. We may, in a special problem, give a particular numerical value to one or more of the letters introduced, but our processes are general, and the letters may usually stand for any numbers whatever. Numbers represented by letters are often called quantities. It is customary to employ not only the letters of the English alphabet, such as a, h, c,..., or A^ B, C,..., and those of the Greek alphabet, such as a, /5, 7,..., but also letters with accents, like a\ a!\ . . . , or with suffixes, like »!, ao,..., each of which may represent any number. Thus the numbers represented by letters, like a, will generally be different, and will have no connection one with the other. 2. Symbols. Each of the quantities like a, 6, Li , b\ tti, 61, is termed a symbol. B. A. 1 2 DEFINITIONS AND NOTATION. The numbers denoted by algebraical symbols are abstract numbers. Every concrete quantity, such as a length, an area, a time, a weight, &c,, is measured by the number of units of its own kind \vhich it contains. Thus a length may be expressed as \ foot, 6 inches, &c., according as a foot or an inch is the unit of length. The numerical measure of the quantity, that is, the numljer of times the unit is contained in it, is called an abstract number ; such as | or 6 in the case above given. 3. Symbols of Operation. The operations or processes of algebra are denoted by certain signs which are known as symbols of operation. With many of these symbols, such as +, — , x, -^, &c. the student has already become acquainted in arithmetic. The words "of operation " are often omitted, and, for brevity, these symbols of operation are called symbols. They cannot be well confused with the symbols defined in Art. 2, and no difficulty is found to arise from this double use of the word. 4. Expressions. Any combination of symbols by algebraical processes is called an algebraical expression. In other words, any combination of letters (which denote numbers) by means of symbols of operation is called an algebraical expression. 5. It follows from the definition given in Art. 1, that algebra may from one point of view — and this is the best way of present- ing it to one who is reading it for the first time — be regarded as a generalization of arithmetic. This is historically the origin of the science, which indeed was once known as universal arithmetic. The description of algebra as a universal arithmetic may be illustrated by shewing how its notation enables us to express various arithmetical relations in a concise and general manner. For example, if a man walk for 4 hours at the rate of 3 miles an hour, he will walk 3x4 miles ; if he walk for 2 hours at the rate of 4 miles an hour, he will walk 4x2 miles. Now these and all similar conclusions may be included in a single state- ment or formula. Let us say that a man walks at the rate of v miles an hour (where the letter v stands for 1, 2, 3, 3^, or any DEFINITIONS AND NOTATION. 3 number, whether fractional or not), and let us suppose that he walks for t hours (where the letter t also stands for any number whatever), then the number of miles he will walk will be the product of V and t. If we represent this number of miles by the letter s, then s = vxt. This algebraical relation includes in a single statement every particular numerical example of the kind. 6. We shall begin by describing some of the pro- cesses, and defining some of the terms, used in algebra. It will be noticed that in many cases these definitions are the same as those with which the student is familiar in arithmetic. 7. Addition. The result of adding two or more numbers together is called their sum. The operation of addition is denoted by the word plus, which is represented by the symbol +. This symbol, when ^Titten between two numbers, signifies the operation of adding the number placed after the symbol to the one placed before it. Thus 2 + 3 (read as two plus three) indicates that we are to add 3 to 2. So a + 6 (read as a plus b) indicates that we are to add the number denoted by the letter b to that denoted by a. Similarly x-\-7/-\-z (read as a; plus y plus z) indicates that we are first to add y to x\ and then to add z to that sum. 8. Subtraction. The result of subtracting a smaller number from a greater number is called the difference of the two numbers. The operation of subtraction is denoted by the word minus, which is represented by the symbol — . This symbol, when written between two numbers, signifies the operation of subtracting the number placed after the symbol from that placed before it. Thus 7-3 (read as seven minus three) indicates that we are to subtract 3 from 7. So a — b (read as a mimis b) indicates that we are to subtract the number denoted by the letter b from that denoted by the letter a. Similarly a-\-b — c indicates that we are first to add b to «, and then from the sum to subtract c. 1—2 4 DEFINITIONS AND NOTATION. 9. Signs. Many symbols of operation are em- ployed in algebra, but the word sign, when used alone, is generally taken to refer only to the symbols + and — . 10. Positive and Negative Quantities. It is evident that addition and subtraction are processes opposed to one another. If + a means increasing a quantity by a, then — a must mean decreasing it by a ; and if these operations be performed in succession, no effect will be produced. If, for instance, a certain length measured along a line from a fixed point (estimated say in feet) be given, then + a will repre- sent a feet in that direction. Hence — a must represent a feet in the opposite direction, since the effect of the two taken in succession is to be nothing. So, if we are considering a man's income (reckoned say in pounds sterling), then +a will signify an addition of £a, and —a will signify a decrease of £«. If however we are considering his expenditure, then +a will signify an increase of £a, while —a will refer to what decreases his expenditure by £«. Thus, if I earn 20 shillings and then lose 5 shillings, the result may be stated either by saying that I have gained 15 shillings, or that I have lost — 15 shillings. Similarly, if 4- a represent a distance of a miles to the north, then - a will represent a miles to the south ; and vice versa. Thus, if I walk 10 miles to the north, I may be said to have walked - 10 miles to the south. So again, if a man be x years older than a boy, the fact may also be expressed by saying that the boy is — x years older than the man. 11. The quantities -fa and —a are in fact always equal in magnitude but opposite in character, and we use the signs -f and — to signify this difference in their nature or quality, without any regard to whether the quantities to which they are prefixed are actually added to or subtracted from any other quantity. DEFINITIONS AND NOTATION. 5 12. Multiplication. The number obtained by multiplying two or more numbers together is called their product. Where more than two numbers are multiplied together their product is sometimes called their continued product. The operation of multiplication is denoted either by a dot {.) ov by the symbol x . Either of these symbols, when placed between two numbers, signifies the opera- tion of multiplying the number placed before the symbol by the number placed after it. Thus 7 X 5 or 7 . 5 (either of which is read as five times seven or seven midtiplied by five or seven into five) indicates that we are to multiply 7 by 5. The latter form,' namely 7 . 5, might be mistaken for the decimal fraction 7*5 ; and it is therefore better to avoid using it, if there be any chance of confusion. So the product of 2 and the number denoted by a can be represented either by 2 x a, or by 2 . « ; it is also often written as 2a, the dot between the 2 and the a being left out. Similarly the result of multiplying the number denoted by a by the number denoted by h is represented either by a x 6, or by a . 6, or by placing the symbols side by side ; thus, ah. It is evident that this latter form of re- presenting the result cannot be used where both the quantities are arithmetical numbers : thus the symbol 75 is used to denote seventy-five and therefore cannot be also used to denote five times seven. Of the methods of denoting multiplication of alge- braical quantities which are above described, that of placing the symbols (which represent the quantities) side by side is the most common. 13. Factors. Each number in the product of several numbers is called d^ factor of the product. If the factor be a number expressed in figures, it is called a numerical factor : if it be a number denoted by a letter or letters, it is called a literal factor. 14. Where several literal factors occur in the same product, it is usual to write them in their alphabetical order and to place the numerical factors first. Thus we generally write the product of 5, a, h, and c as bahc^ and not a x 5 x c x 6 (or any similar 6 DEFINITIONS AND NOTATION. order). Similarly it is usual to write the product of a and x as ax, and not as xa, though it is always permissible to use the latter form if it be more convenient. We shall see later that we infer from arithmetic that it is immaterial in what order the numbers which form a product are multiplied together. 15. Coefficient. Each expressed factor in a pro- duct (or the product of some of the factors) is called the coefficient of the product of the remaining factors. A coefficient may be a product of an arithmetical number and a number denoted by a letter or letters. If the coefficient be a number expressed in figures, it is called a numerical coefficient: if it be a number denoted by a letter or letters, it is called a literal co- efficient Where the coefficient is unity it is usually omitted. For example, we write x and not 1 x x. Thus in the product Qabx, the coefficient of ahx is 6, which is a numerical coefficient ; the coefficient of ax is 66 ; the coefficient of X is Qah ; and so on. Similarly in the quantity y, the coefficient of y is miity. 16. Division. The result of dividing one number by another number is called the quotient of the first by the second. The number divided is known as the dividend, and the number by which it is divided is called the divisor. If there be no remainder, then the dividend is said to be exactly divisible by the divisor. The operation of division is denoted either by the symbol -^, or by the symbol /. Either of these symbols, when placed between two numbers, signifies the opera- tion of dividing the number placed first (the dividend) by the number placed after it (the divisor). The symbol / is called a solidus: it is desirable that the student should know its signification, but except in this chapter we shall use it but rarely. DEFINITIONS AND NOTATION. 7 The operation of division may also be represented by a fraction having the dividend for numerator and the divisor for denominator. Thus either 35 -^ 7 or 35/7 (read as iJiirty-five divided hy seven or thirty -live hy seven) indicates that we are to divide thirty-live (the dividend) by seven (the divisor). The operation can also be indicated by the use of a fractional form, as '^-. So any of the forms a-^h, a/b, or j indicates that we are to divide the number denoted by a by the number denoted by b. 17. Brackets. We sometimes want to isolate a particular set of quantities, and treat them for the moment by themselves as if they were a single quan- tity. This is effected by placing them within a pair of brackets. The same result may be otherwise denoted by drawing a line, called a vi7icidum, over the quantitj^ it is desired to isolate. Brackets of various shapes are used, such as ( ), {},[]. The methods of treating brackets, and of removing or insert- ing them, will be fully explained in chapters ii and in ; but their use and meaning may be here illustrated by considering an expression such as a—{b + c). Here the part 6 -i- c is enclosed in a bracket, it is therefore to be treated as a single quantity : thus the sum of b and c is to be subtracted from a. The same result might also be denoted by drawing a vinculum over the 6-f-c: thus, a-b-\-c. Similarly {a-^h){c-d) signifies the product of the sum of a and h and the quantity obtained by subtracting d from c. So again [a — (6 + c)] \b — c\ indicates the product of the expres- sion in the brackets { } by the expression in the brackets [ ] : the quantity in the brackets { } is found by first adding c to 6 and then subtracting their sum from «, the quantity in the brackets [ ] is found by subtracting c from b. 18. Equality. The symbol = represents equality, and stands for the words is equal to. Thus a = b (read as a is equal to b or a equals b) indicates that the number denoted by a is equal to the number denoted hyb. ■ ^ 8 DEFINITIONS AND NOTATION. 19. Other Symbols of Operation. The follow- ing symbols of operation are used as abbreviations for the words against which they are placed. The symbol > stands for is greater than. < is less than. 4= '^^ ^^^^ equal to. :[> is not greater than. <(: is not less than. ± plus or minus. • therefore. •.• because. - when placed between two numbers, stands for the difference hetiueen them. Thus 7 X 5 = 35 signifies that five times seven is equal to thirty-five. Again {a-h)>c signifies that the result of subtracting the number denoted by h from the number denoted by a is greater than the number denoted by c. So a — & indicates the difference between the numbers de- noted by a and h. 20. We shall now give a few examples to illustrate the above notation. The beginner will find it desirable (i) to write every step in a line by itself, (ii) to place each fresh line below the one last written. (iii) to keep the symbols for equality in a vertical line. Should it be necessary to explain how one step is derived from the one immediately preceding it, the explanation should be written between the two lines. To save space, explanatory statements and successive steps are often printed in the same line, but in luriting his work the student is recommended to follow the above rules. DEFINITIONS AND NOTATION. 9 Ex. 1. What is the numerical value of x-^a. u-hen .r = 6, a = 2 In this case, .^• -^ « = 6 -r 2 = 3. Ex. 2. If m = 2, n=l^ ^=3, y = \^find the numencal values of (i) 2m + n {x - 2y) ; (ii) (2m -f n) {x -2y); (iii) {m - n) {x - 3). These are three separate examples. We shall take them in their order. (i) Here 2w = 2x2 = 4. Also, A'-2y = 3-(2xl) = 3-2 = l, and .'. n{x- 2y) = 1x1 = 1. .'. 2wi + ?i(.^-2y) = 4+l = 5. (ii) Here 2m + ?i = 4 + l = 5, and .r-23/ = 3-2 = l. . •. (m + 2n) {x - 2y) = 5 X 1 = 5. (iii) Here m-n = 2-l = l, and .'t;- 3 = 3 -3 = 0. But the product of two numbers, one of which is zero, is itself zero. Hence {m — n) {x - 3) = 1 x = 0. Note. The student should remember that the sum of a number of quantities, each of which is zero, is necessarily equal to zero ; and therefore the product of a number of quantities, of which one is zero, must be equal to zero. EXAMPLES ON THE ELEMENTARY PROCESSES. I. A. 1. Write down the continued product of «, .r, 3, and h. 2. ^^Hiat is the numerical value of lax when a = ^V ^i^^id x = 3 ? And what is the numerical value when a = 2 and x=^^ 1 Write dow^n the coefficient of x in the following quantities, numbered 3 to 6 ; and state whether the coefficient is literal or numerical. 3. 7.V. 4. 23a.r. 5. xy. 6. x. 10 DEFINITIONS AND NOTATION. What are the numerical vakies of the following quantities, numbered 7 to 10, when a = 1, .i' = 2, y = 3 ? 7. 'f. 8. y-^.^•. 9. g. 10. .ly. What are the numerical values of the following quantities, numbered 11 to 14, when « = 2, 6 = ^, c = \, o?=l? 11. ahcd. 12. a^h. 13. h^a. 14. ^. cd 15. AVhat is the numerical value of the quotient of (« - h) by c, when a = ll, 6 = 2, and c=3? 16. State in words the meaning of the expression {{a-h)-{c-d))^{a-c). 17. Find the numerical value of 5a -I- 3& - {(c + cT) ^ (c - c^)} + 2e, when « = 1, 6 = 5-, c = |, c^=^, e = \. 18. If J. gained £7 and lost 125. how much did he lose as the result of the whole transaction ] 19. If A walked 7 miles in a S.W. direction, how far did he walk in a N.E. direction ? 21. Powers. When a quantity is multiplied by itself a number of times the resulting product is called a poiuer of the quantity. Thus ocoo is called the second power of x, or the square of x, or x squared ; xxx is called the third power of X, or the cube of x, or x cubed ; xxxx is called the fourth poiver of x ; and so on. 22. Indices. Exponents. The square of x is usually denoted by x^ instead of by xx, the small number placed above and to the right of x shewing the number of times the factor x has been repeated to form the pro- duct. The cube of ^ is similarly denoted by ^ instead of xxx. And generally, if the factor x be repeated n times, the result is written as .z'". The small number or letter placed above and to the right of the symbol, and which DEFINITIONS AND NOTATION. 11 denotes the number of times the symbol is repeated in the product, is called the index, or exponent. When the number x is taken by itself, it might be called the first power of x and denoted by x^, but if the index be unity it is usual to omit it. Thus, the fifth power of .i-, or xxxxx, is denoted by x' ; and in this case, 5 is the index, exponent, or power. So x''^ denotes the n^'^ power of x, and is usually read as x to the power n. A similar rule applies to more complicated products. Thus a?-})^ is written instead of aahhb. So lao^y^ is written instead of laxxxyyyy. Note. Beginners are sometimes apt to confuse the index (which denotes the power to which a quantity represented by a letter is raised) with a suffix (which is merely used to dis- tinguish the quantity from other quantities as explained in Art. 1). They have no connection. The student wall also notice that the number which represents an index is written above and to the right of the symbol to which it refers, while the number which represents a suffix is usually written below the symbol to which it refers. Example. If a = 2, 6 = 3, x = 4, y = l, find the nu7nerical values of (i) 36/; (ii) .r2_^^; (iii) .r« -6??; (\v) av-^y. (i) 36y2 = Sxbxyxy =3x3x1x1 = 9. (ii) and Here .vi/z ; but the terms 2aj;'^ and 7a'^x (though they contain the same letters) are unlike, since the powers to which a and x are respectively raised are dififerent. The terms 2clv^, la\v, 46^, and xyz are positive terms : the terms - 36^ and — bxyz are negative terms. 25. Simple and Compound Expressions. A simple expression consists of one term only. A com- pound expression contains more than one term. A simple expression is sometimes called a monomial. If a compound expression consist of two terms, it is called a binomial', if of three terms, a trinomial; and if of more than three terms, a multinomial or polynomial. Thus oab'^c^ is a simple expression, a + b is a binomial, x^+y^-\-z^ is a trinomial, and so on. Again 2x^ — Sax'^ — bx -\- 4: + a^ /b is a compound algebraical expression, made up of five terms. The first term is 2.^^, the second is 3ax'^, the third is bx, the fourth is 4, and the fifth is the quotient of a^ by b. The first, fourth, and fifth terms are positive terms, the second and third terms are negative terms. No two of the terms are like. As one more illustration, consider the expression [^f{jj'+^W+x']-a]{x + b). Although this looks complicated, it is a simple expression ; it consists of the product of two quantities, namely that enclosed in the brackets ( ) and that enclosed in the brackets [ ]. The quantity enclosed in the brackets ( ) is a binomial expression, consisting of the sum of two terms. The quantity enclosed in the brackets [ ] is also a binomial expression, consisting of the result of subtracting a from the square root of the expression enclosed in the brackets { } . This last expression, namely that enclosed in the brackets { }, consists of the sum of x'^ and of the square root of the sum of b'^ and x'^. 26. Degree or Dimensions of a quantity. A quantity which is the product of n letters is said to be of the /I*'' degree or of n dimensions. 16 DEFINITIONS AND NOTATION In reckoning dimensions, numerical factors are not counted. Thus each of the quantities abc, 2a-b, and - 3.v^ is of the third degree, or of three dimensions. Sometimes we speak of the dimensions in a parti- cular letter, and we then confine ourselves to that letter. Thus 2a^b is of two dimensions in a and one dimension in b. 27. Degree or Dimensions of an expression. The degree of an expression is the degree of the term of highest dimensions in it. In estimating it, it is usual to confine ourselves to only one letter. An expression which is of the ?i*'^ degree is said to be of n dimensions. An expression which is of the first degree in a letter is said to be linear in that letter. Thus x^-y -\- xy- -\-y'^ is of the second degree in x^ but it may be said to be of the third degree if all the letters are taken into account. 28. Homogeneous Expressions. A compound expression is said to be homogeneous when every term in it is of the same dimensions. Thus the expression x-y + xy^ + j/^ is homogeneous in x and y, and is of the third degree. 29. Formula. Identity. When an algebraical expression can be written in two ways the result ob- tained by equating one to the other is said to be an algebraical yb7*??ii<^a or identity. Examples will be given later [see ex. gr. Art. 87]. 30. It is desirable to warn the beginner to be careful to observe both the order and the extent of the operations indicated in an algebraical expression. Thus a + b'Xc ov a-\-bc signifies that the product of b and c is to be found and then added to a, while either {a + b)xc or {a + b)c or a + b , c signifies that the number which is formed by adding b to a is to be multiplied by c. DEFINTTIONS AND NOTATIOX. 17 So again Ja + b signifies that b is to be added to the square root of a, but J{a + b) or V^ + ^ signifies the square root of the sum of a and b. It is obviously undesirable to use forms of notation which re- semble one another so closely as those above written, and which are liable to be mistaken the one for the other ; and in practice we shall avoid the use of forms which a careless reader might regard as ambiguous. EXAMPLES ON ALGEBRAIC NOTATION. L D. 1. What are the numbers seven and three called respectively in the expression Ta^ ? 2. Define what is meant (i) by ^, and (ii) by 3.r. Which is the gi-eater when ^ = 1, and which is the greater when :r=2 ? 3. Write down the continued product of 3, b, b, c, and y. 4. Write down the sum of the five quantities given in Ex. 3. What are the dimensions of each of the following quantities ? 5. X1/Z. 6. Sax\ 7. 2aKv. 8. bx^. 9. .v. 10. .r". 11. What are the dimensions in x of each of the quantities given in Exs. 5, 6, 7, 8, 9, 10 ? State the dimensions in x of the expressions numbered 12 to 14. 12. 2-3.t\ 13. 3«.t'2-7^ + 2a%. 14. A-»-a". 15. Are any of the terms in the following expression like terms ? 2a^3 _ ^,2 + 2aKv^ + 363 _ ^abx. 16. If a = 8, 6 = 5, c = l, find the numerical value of c V 1 Oab + 6 V 8ac + a \/456c. 17. If x=2, find the numerical values of the expressions ... 2^-1 , 3^+4 2U'-6 ,.., , ,—, — — 18. If a = l, 6 = 3, c=4, c?=0, find the numerical values of (i) 3a62 - d [6c2 + 2 (c - 6)] + ac (63 - cd) + 16 ; ... \/« + 2c 3(a + 26) ^^^^ a + ^b^'d 2c -a 19. If a = 3, 6 = 4, c=5, c?=6, find the numerical values of 2\/^2 + 62-f-c/- c (6- fc)(cg+a)-(a + 6)(c+^) ^^^ r/-r+b-a ' *^"^ ab + rd-br-drt B. A. 2 CHAPTER II. ADDITION AND SUBTRACTION. 31. The first thing that we have now to do is to learn how to add, subtract, multiply, and divide alge- braical expressions. These operations will constantly occur in all our subsequent investigations, and it is necessary to know how to effect them with accuracy and facility before we proceed any farther. We shall deal in this chapter with the rules for adding and subtracting algebraical expressions, and shall consider first the case of simple expressions and next the case of compound expressions. 82. Addition and Subtraction of Simple Ex- pressions. The addition of a number of simple quan- tities is indicated by writing them down in succession, each preceded by the sign of addition, namely -f [see Art. 7]. If any one of them has to be subtracted, it must be preceded [Art. 8] by the sign of subtraction, namely — . Thus to indicate the addition of 6 to a and then the sub- traction of c from the result we write a-^-h — c. We cannot simplify this until we know what numbers or expressions are represented by a, 6, c, 38. Order of addition is immaterial. In arithmetic it is shewn that the sum or the difference of several numbers is the same in whatever order the additions or subtractions are made. We assume that ADDITION AND SUBTRACTION. 19 the same will therefore be true of numbers when they are represented by algebraical symbols. Thus the sum of a and h may be written indifferently as a-\-b or b-\-a. It is easy to verify this statement by giving a and h numerical values, or considering some particular case (such as one in which a and h stand for a number of shillings, or for lengths measured in some given direction) ; and all similar refer- ences to arithmetic can be tested in like manner. By similar reasoning the expression a-h+c-d may also be written (among other ways) as a + c-h — d, or a — d-h-\-c, or c-\-a-d-h. Any difficulty that may arise from negative quantities will be explained later. 34. Combination of like terms. Where like terms occur, they can he combined into a single term. (i) Where the like terms are of the same sign they can be replaced by a single term of the same sign, like either of them, and having a numerical coefficient equal to the sum of the numerical coefficients of the separate terms. For example, to add 2a to any quantity and then to add 3« to it is equivalent to adding 5a to it. That is, 2a + 3a = 5a. Similarly, to subtract 2a from any quantity and then to subtract 3a from it is equivalent to subtracting 5a from it. That is, - 2a - 3a = - 5a. (ii) Where two like terms are of opposite signs we take the difference of the numerical coefficients and affix the sign of the gi'eater. For example, 3a-2a = 2a+ a-2a=-fa, 2a-3a = 2a-2a- a=—a. The terms may cancel one another. For example, 2—2 20 ADDITION AND SUBTRACTION. (iii) Where there are several like terms, some positive and some negative, first, as in case (i), we combine all the positive terms into one term, and all the negative terms into another term ; and then, as in case (ii), we combine these two terms into one term. For example, 4r( + 2(7 - 3« + « - 5« = 4a + 2a + a - 3« - 5a = la - 8a 35. Simplification of Expressions by Col- lection of like terms. There may be different sets of like terms in the quantities to be added. In such a case all the like terms of each kind can be collected together, and then combined into one term. The final result will be the sum or difference of the terms so formed. For example, the expression 3a + 6 + 2a-3Z>-4a + 6 = 3a + 2a-4a + 6 + 6-36 = 5a-4a + 26-36 = a — b. EXAMPLES. II. A. Find the values of the following expressions by combining like terms. 1. -a + 2b + 'dc-2a-2c + 2b + Sa. 2. — la + c- 26+71 — |a + 4?i-^c. 3. .r2 + 2f -z^- 3y2 + 2^2 _ 7^2 ^^2_ 4. la-b + lb-la-\-lh-\a. 5. 2^2 + 3^2_i^.+ i^2_|^^2 + 2r2-^r. 7. s^a + 2^b-^a-Jb. 8. 2V«+c-3^/6 + Va + 2^/6-3c. ADDITION AND' SUBTRACTION. 21 36. Addition of Multinomials. To add to any quantity a multinomial expression write down every term of the multinomial with its own sign prefixed. To prove this rule, let us consider the addition to any quantity of a binomial expression like {a-\-h) or (a -6). In the case of a + 6 both the terms are positive. Now we know from arithmetic that to add a to any quantity and then to add h to it is equivalent to adding their sum (a + h) to it. That is, + (« + Z>) = + « + 6 (A). In the case of a — h the terms are of opposite signs, and we will suppose for the present that a>h. Then, if we add a to the given quantity we shall have added h too much, and therefore must subtract h from the result. That is, + (a-6) = + a-6 (B). A similar proof evidently applies to the case of any multinomial, and hence the rule given at the head of this article follows. Thus, for example, + (a + 6 + c)=+a + 6+6', and -\-{a — h-c)— -\-a-h-c. Ex. Fhid the sum of {a + b - c), (2a +4c), and (3a — 26 — 3c). ^ The sum =(a-f 6-c) + (2a + 4c) + (3a-26-3c) = a + h - c + 2a + Ac + Sa- ^b - 'Sc. Collect like terms, .-. the sum =a + 2a + 3a-\-b-2b-c-{-4:C-3c = 6a-b. 37. Process of addition. It is often convenient to write the expressions so that the like terms come in vertical columns, and then add them as in arithmetic. ADDITION AND SUBTRACTION. Thus the above example would be written a+ 6- c 2a -f4c 3a - 26 - 3c Add, 6a- 6 38. The numerical coefficients may be fractional as in the following example. Ex. Add together \x -Vy — Iz, ^x - ^y, and \y + z. Here we have \x-\- y — \^ 39. Subtraction of Multinomials. To subtract from any quantity a multinomial expression turite doiun every term of the multinomial with its sign changed; that is, change every plus into a minus, and change every minus into a plus. To prove this rule, let us consider the subtraction from any quantity of a binomial expression like (a + h) or {a — h). In the case of a + 6 both the terms are positive. Now we know from arithmetic that to subtract a from any quantity and then to subtract h from it is equiva- lent to subtracting their sum (a + h). That is, -{a + h) = -a-h (C). In the case of a — 6 the terms are of opposite signs, and we will suppose for the present that a>h. Then, if we subtract a from the given quantity we shall have subtracted 6 too much, and we must therefore add h to the result. That is, -(ct-6) = -a + 6 (D). A similar proof applies to the case of a multi- nomial, and hence the rule given at the head of this article follows. ADDITION AND SUBTRACTION. 23 Thus -{a + b + c)= -a-h-c, — {a-b-c)= -a + b + Cf and -{a + b-c)= -a-b + c. 40. Extension of results. The results (B) and (D) of Arts. 3G and 39 were proved true on the hypo- thesis that a was greater than b. We shall now see under what conditions we may regard them as true for all values of a and b. If they be true for all values of a and b, we may put a = in them. We shall then get + (-6) = -6, -{-b) = + b. The first of these results shews that the addition of a negative quantity must be taken as being equivalent to the subtraction of a positive quantity of the same magnitude. The second shews that the subtraction of a negative quantity must be taken as being equivalent to the addition of a positive quantity of the same magnitude. Neither of these operations is discussed in elemen- tary arithmetic, but if we take them to have the meanings above given, then it will be found on trial that the equations (B) and (D) are true whatever be the numerical values that we give to the symbols. Therefore, on this h}^othesis, we may consider those equations to be true for all values of the quantities involved. For example, to subtract x-y-{-z from 2.v + 3y, we have {2x^-Zij)-{x-y-\-z) = 2x+Zy-:c-{-y-z = x-\-^y-z. Hitherto we have supposed that our algebraical symbols denoted positive numbers only, for if a symbol had denoted a negative number we could not have subtracted it from any other number. We now know the meaning to be assigned to the addition or sub- traction of a negative number, and henceforth we shall consider that our symbols may stand for negative as well as positive numbers. 24 ADDITION AND SUBTRACTION. *41. The meanings obtained, in the List article, for the addition and the subtraction of a negative quantity will be found on consideration to be a natural extension of the results of arithmetic, and to be consistent with the description given in Arts. 10, 11 of negative quantities. But the method by which we have found a meaning for these operations is worthy of close attention, since it is one of which we shall make frequent use, and on it large parts of algebra are founded. The following account of the method may help the student to understand it better. We have two relations, (B) and (D), which we have proved in Arts. 36, 39 to be true in every case in which they are arith- metically intelligible, namely, in every case in which the number denoted by a is greater than the number denoted by 6. But if certain numbers {ex. gr. whenever a is less than h) are sub- stituted in these relations they involve operations (such as the subtraction of a negative number) which are arithmetically un- intelligible. Such operations have" no meaning, and we can make them stand for any thing we please, or define them in any manner we like, provided that our subsequent use of them is always consistent with the meaning so selected, and leads to results which are not inconsistent with the meaning of operations already employed. Now the use of algebra depends largely on employing relations which are true whatever be the numbers for which the symbols stand. We therefore try to find a meaning for these operations, not in an arbitrary way, but by extending the arithmetical results so that our new meanings shall be con- sistent with all the results already obtained, and shall thus allow of our symbols having any numerical values. 42. Process of subtraction. When one ex- pression is to be subtracted from another, it is often convenient to write the expressions so that the like terms come in vertical columns, and then to subtract them as in arithmetic — changing mentally the sign of every term in the quantity subtracted, but not altering the signs on the paper. Thus the example given at the end of Art. 40, where x-y-\-z is subtracted from 2.r -}- 3y, would be written thus : 2^-f3y Subtract, .r -f 4?/ - 2 ADDITION AND SUBTRACTION. 25 43. Removal of Brackets. A plus sign before a bracket enclosing a quantity merely signifies that the quantity is to be added ; while a minus sign before a bracket enclosing a quantity signifies that the quantity is to be subtracted. Hence the results of Arts. 36, 39, 40 may be stated thus. If, in a given expression, lue have a quantity in a bracket preceded by the plus sign, we can remove the bracket provided we write doiun the terms inside it each with its own sign py-efixed : if the bracket be preceded by the minus sign, tue can remove the bracket jyrovided we lurite down the terms inside it each with its sign changed. For example, a + {b-c) + {c-d) = a + b-c + G-d = a-\-h — d. a + {b-c-d)-{-a-b-^d) = a + b-c-d+a-\-b-d = 2a-h2b-c-2d. 44. Sometimes the quantities contained within brackets are themselves compound expressions involv- ing other brackets. In this case the brackets are made of different shapes so as to enable us to pick out each pair. Thus a-{b — [c-{d- e)]} means that we are to subtract from a the quantity enclosed within the brackets { } . This quantity is itself formed by subtracting from b the quantity within the brackets [ ] ; and so on. To find the value of such an expression, the beginner will find it best to remove only one pair of brackets at a time, and to begin with the innermost brackets. Thus a-{b-[c-{d- e)]}^a - {b -[c-d+e]} = a— {b — c + d-e} — a — b-\-c—d+e. 45. Introduction of Brackets. Conversely, we can introduce brackets. Any of the terms of an expression can be placed within a pair of brackets. 26 ADDITION AND SUBTRACTION. preceded by the plus sign, provided no alteration is made in the signs of the terms inside the brackets. Similarly, any of the terms of an expression can be placed within a pair of brackets, preceded by the minus sign, provided the sign of every term within the brackets is changed. For example, we may write a-\-h-c\\\ the form a-\-{h-c) or a-{-h-\-c). Other ways of writing it are the following : h + {a-c), b-{c-a), {a + b)-c, -{-a-h)-c. Each of these forms reduces to a + h-c when the brackets are removed. 46. Algebraical Sum. We have shewn [Art. 40] that the subtraction of a positive quantity is equivalent to the addition of a negative one of the same numerical magnitude ; and also that the subtraction of a negative quantity is equivalent to the addition of a positive one of the same numerical magnitude. Hence, subtraction may be regarded as equivalent to an algebraical sum- mation; and the result of subtracting one quantity from another is often described as their algebraical sum. Thus the algebraical sum of 5a, - 2a, and — 4a is equal to 5a — 2a - 4a = 5a - 6a = — a. This extension of the word sum to cover subtraction as well as addition is convenient and saves much cir- cumlocution. NOTE. When hereafter we speak of the sum of two or more quantities, or of adding certain quantities together, we shall in all cases mean the algebraical sum. 47. Inequalities. A similar extension of arith- metical language has been given to the phrases greater than and less than. ADDITION AND SUBTRACTION. 27 In arithmetic, we say that the excess of a number a over a number 6 is a — 6, or that a number a is greater than a number h by a — h, where, in both cases, a is supposed to be greater than h. So long as a is greater than h this is intelligible, but if a be less than h it is arithmetically unintelligible. We therefore extend the meaning of the phrase "greater than," and say that (whatever numbers a and h may denote) a is greater than h hy a — b. Thiis 5 is greater than 3 by 5-3, that is, by 2; similarly 3 is gi'eater than 4 by 3 — 4, that is, by -1. So -2 is greater than 7 by - 2 - 7, that is, by - 9, Again ~ 3 is greater than -Iby -3-(-l), that is, by — 2 ; and so on. 48. We say also that a is greater than b, if a — b be positive ; and a is less than b,ii a—b be negative. This enables us to compare the magnitudes of any two numbers, whether they be positive or negative. For example, - 2 is greater than - 3, because - 2 - ( - 3) is equal to + 1 , which is positive. MISCELLANEOUS EXAMPLES. II. B. 1. Add 3 to -2. 2. Add - 3 to 2. 3. Add - .V to - 3/. 4. Find the sum of .v^, i/\ - 3x^, 7^^, z% and - Sy^. 5. Find the sum of a\ 3a,-^, - 2a%, - 2a% — djb^, and 2aiv. 6. Subtract 7 from 2. 7. Subtract 7 from - 2. 8. Subtract b from a. 9. Subtract — x from — ?/. 10. If A gained 10s. and then lost £2, how much did he lose ? How much did he gain ? 11. To a + h- X add - 2a - h +y. 12. Add together .1+^ and .(.—y. 28 ADDITION AND SUBTRACTION. 13. Find the sum of \a-\-x, -\ci- 2y, and .^■^-3j/. 14. From x-\-y subtract x-y. 15. From the sum of 2a 4- 35 and 26 - 3a subtract x + a. 16. What is the difference between a?- h^ and a'^+.v^ ? 17. From 6c + 2a6 - 3ca subtract 26c -f a6 — 3ca. 18. Add together 3a - (6 - c), 36 - (c - a), 3c — (a — 6) ; and find the nimierical vakie of the result when a = Q, 6 = 3, c=2. 19. Add together 463 + (3a2c- 662c), 562c - (a^ + ,3a2c), a^ -(263 + 262c). 20. Subtract (^ + 2y + z) from {x -'2.y + z)-\-{2y-z). 21. Simplify [a + 6-{a4-6+c-(a + 6 + c + c^)}]-a. 22. Simplify (a - 6) - {3a - (a + 6)} + ((a - 26) - (5a - 26)}. 23. Add together ^a -|- 16 + ic, ^a + 16 - -^-c, - ^a - ^6 + Jc. 24. From f ^'2 _j_ 1.^2 g^ btract ^2 _ ^2 _j. 2^. 25. From the sum of ^l + ^m and — -^m + ^^i subtract the sum of 1^ — ^n and ^m — %n. 26. A man is now a years old, how long will it be before he is 6 years old ? What would be the meaning of the result if 6 were less than a 1 27. A walks 10 miles due north, then 15 miles due south, and then 7 miles due north. What is the total distance towards the south that he has gone ? 28. What quantity must be added to a in order that the sura may be 6 ] 29. What quantity must be subtracted from a in order that the result may be a + 6 ? 30. What quantity must be added to a3 + 63 in order that the sum may be x^ ? 31. What quantity must be subtracted from x'^-y'^ in order that the result may be 2xy ? 32. By how much is x greater than y ? 33. By how much is - 6 greater than - a ? 34. By how much is 11 greater than — 11 ? 35. Shew that, if .^'=2a + 36, y = 3a — 26, 5; = 6 •-4a, then x-\-y-\-z = a + 2h. CHAPTER III. MULTIPLICATION. 49. We proceed now to the consideration of the multiplication of algebraical quantities, and shall dis- cuss successively (i) the product of two or more simple expressions, (ii) the product of a compound expression and a simple expression, and (iii) the product of two or more compound expressions. 50. Order of multiplication is immaterial. In arithmetic it is shewn that the product of one number a by another h is the same as the product of b by a, that is, ah = ha. Similarly, ahc = ach = hca = hac = cah = cha. Thus it is immaterial in what order we multiply the different numbers together. We assume that the same is true of all algebraical expressions (whether they represent positive or negative quantities). 51. Product of Simple Expressions. The pro- duct of two or more simple quantities is indicated by writing them down in a line separated, each from the one next to it, by the symbol x, or by a dot (.), or, in the case of literal factors, side by side without any intermediate symbol [see Art. 12]. Where the result contains the same quantity re- peated more than once, the result can be somewhat simplified by using the index notation [Art. 22]. 30 MULTIPLICA TION. Ex. 1. Find the product of 2abx and Zaxy. {2abx) {Zax2/) = 2xaxbxxx3xax.vxi/ = 2 X 3 X a X a X b X X X X X 1/ = ^aVix^y. With a little practice, the result can be written down by inspec- tion : and as soon as possible the student should solve such questions by inspection only. Ex. 2. Find the prodvxt of Za^xy"^ and ^ax'^. (da^xy^) (^a.^) = 3 x ^ x a^a x xx^ x y^ = a\v^y^. 52. Product of a positive and a negative quantity. The explanation of the multiplication of two numbers which is given in arithmetic is only in- telligible if both the numbers be positive. If one of them be negative, we have to find the meaning of an expression like (— a) b or like a (— 6). The expression (— a) b signifies that — a has to be taken b times. .' . (- a) 6 = (- a) -f (- a) + (- a) +. . . . ..[b terms] = — a — a — a — [6 terms] = — (a + a-fa+ ) [6 terms] = -(ab) = — ab. The above proof applies only to cases where 6 is a whole number, but, by the principle explained in Art. 41, the rule can be extended to cases where 6 is a positive fraction. The expression a (— b) may be written as (— b) a, since the order in which the multiplication is made is immaterial. But, by the above reasoning, (— 6) a = — bctj which is the same as — ab. Therefore a{—b) = — ab. We shall defer for the present the consideration of the meaning to be attached to the product of two nega- tive quantities [see Art. 55]. MUL TIP Lie A TION. 31 For example, the product of Zah cand - 4co? is - l^ahcd. Here «ill the letters are different, and no further simplification is possible. Similarly, the product of - 2xi/ and 3./; is - 6.i'-^. Similarly, the continued product of a6, 6c, and - ac is {ah) (be) {-ac) = {alf-c) ( - ac) = - a^b'^cl EXAMPLES. III. A. Write down the product of the following expressions. 1. 2a.v and 36y. 2. Sab and cd. 3. 7al^ and 7b\v\ 4. habc and 2a6V. 5. 4a6, 76c, 11 CO?, and 8da. 6. a.T, b.v, ex, dx\ ex^, and /^. 7. ai/z, bzXy and c^y. 8. a^i/z, abzx, and 62^y. 9. ^a2^, ^6%, and 63a6.r3/2. 10. ^l^m, |m%, and i?7i2, 11. 7ax and — 6.^. 12. — Ta^^? and 6x 13. abx, box, and - acx. 14. ^^^j,^^^^ _ 9?i2^j,y^ ^nd Sx'^i/'^. 15. - Sa^aa^j ^^2^^^^^^ 2a^a^a^. 16. - 1-^^^^, i^^z, and 6.^^^. 53. Product of a Simple Expression and a Multinomial. The product of a multinomial and a number denoted by a letter is found by multiplying every term of the multinomial by the number, and taking the algebraic sum of these products. Consider first the case of the product of a number and a binomial. We know from arithmetic that to add to any quantity n times the sum (that is, the algebraic sum) of two numbers, a and 6, is equivalent to first adding na and then adding nb. That is, n {a + b) = na + nb, and n(a — b)= na, — 7ib. 32 MULTIPLICATION. These results may also be proved directly by a method similar to that given in Art. 52. For n{a+h)= (a + 6) + (a + 6) + [n terms] = a-{- a -[- [?i terms]) + h+ h -\- [71 terms]) = na + nh. Similarly, n{a — h)= (a - 6) + (a - 6) + {n terms] = a + rt + \_n terms]] — h — h — \ii terms]) = na — nh. Moreover, since the order of multiplication is im- material, we shall also have {a -^h)n = an + hn, and {a — h) n — an — hn. The same method is applicable to multinomials, and hence the rule given at the head of this article follows. E.TC. 1. Multiply la~2h + 3c hy x. x{ja-'^h + ^c) = x . la-x . 2b + x . 3c = 7 ax — 2b.v + dcx. Ex. 2. Multiply 7x+3y + llzby - 3x. (7x + 3y + Uz){- 3x) = (7.r) ( - 3x) + {3y) ( - 3x) + (l 1^) ( _ 3^-) = -2lx^--9xy-33xz. Ex. 3. Multiply ax-hy — cz hy xyz. {ax -hy- cz) {ooyz) = {ax) {xyz) — hy {xyz) - cz {xyz) = ax^yz - hxyh — cxyz^. EXAMPLES. III. B. Find the product of the following expressions. 1. 2.r4-33/+2 and .r. 2. al + bm + cii and - 7a7i. 3. lyz - mxz — nxy and Iny. 4. Find the coefficient of x in ^ {x + a)- {2a -h{c- x)}. MUL TIPL ICA TIOK. 33 Eeduce the following expressions (numbered 5 to 10) to their simplest forms ; and state in each case what is the co- efficient of x^ and whether the form is a binomial, trinomial or multinomial. 5. 3a-2(6-^-)-{2(a-6)-3Cr + a)}-{9^-4(.r-a)}. 7. '^\x-a-Z\cc — ^{x—hx-a—a)-a\\ 8. 6a - 2 [5- 4 {Zc-^x)^Z{a - (4c + ^)}]. 9. 2(3^-3/)-4{2:r-(.r-3/)}-3{(^-4y) + 2(3.r-y)}. 10. ,(.-|)-e[(..-|)-i.(|-?^)]. 11. Express a?-{x''-'-\-x-\)-oh{\-\-x''--x)^-}p-i:\^ x- x'^) with numerical coefficients, when a = 4 and 6 = 8; collect like terms, and arrange the result according to powers of x. 12. Simplify 62 + [«(«- 6) - {ah - 6 (a + 6)}]. 13. Simplify xy — {y3a -'ix — a (4y — x) — x{a — 2y)}. What is the coefficient of y in the result ? 54. Product of Two Binomials. The product of two hinomials is found by multiplying evei^y term in the one by every term in the other, prefixing the proper sign according to the rule of signs enunciated in Article 56, and taking the algebraic sum of these products. To prove this rule, we have to find the product of two binomial expressions, one like a + 6 or a — b, and the other like x + y or x — y. (i) To find the value of (a -}- 6) (a? -f- y). Denote a 4- 6 by n, .-. (a 4- 6) (^ 4- y) = n{x + y) = nx -\- ny [Art. 53 = (a + b)x + {a + b)y = ax H- bx -\- ay -{- by. [Art. 58 B. A. 3 34 MULTIPLICATION. (ii) To find the value of {a ■\-h){x — y). Denote a + hhy n, .'. {a-]rh){x-y) = n{x-y) — nx — ny [Art. 53 = {a + h) X — {a -{-h) y = {ax + hx) — {ay + hy) [Art. 53 = ax + hx — ay — hy. [Art. 39 (iii) To find the value oi{a — h){x — y). Denote a — hhy n, and let us assume that a is greater than b, .-. {a-b){x-y) = n{x~y) = 7ix — ny [Art. 53 = {a — b)x—{a—b)y = {ax — bx) — {ay — by) [Art. 53 = ax — bx-ay + by. [Art. 39 All these results are included in the rule enunciated at the head of this article. 55. Extension of results. The results of the last article have been proved on the assumption that we are finding the product either of two positive quan- tities, or of a positive and a negative quantity. We now proceed to find the meaning which must be given to the product of two negative quantities so that those results may be true for all values of the symbols a, b, X, y. To do this, we use a method analogous to that employed in Arts. 40, 41. If, in the result of (iii), we put b=0 and y = 0, we obtain a x x = ax, which is of course true. If, in the same result, we put a = and y = 0, we obtain {^b)x = — bx. MULTIPLICATION. 35 Similarly, if we put ^ = and ^ = 0, we obtain These give us the rule for forming, the product of a positive and a negative quantity, which we had previously obtained in Art. 52. If, in the same result, we put a = and a; = 0, we get (- ^) (- 2/) = + H which gives us the meaning to be attached to the product of two negative quantities. If we agree that the product of two negative numbers shall be taken as a positive number equal to the product of the numbers, then it will be found on trial that the results (i) (ii), (iii) of Art. 54 are true whatever be the numerical values (positive or negative) that we give to the symbols, and any one of these results may be deduced from either of the other two of them. For example, if we take the first result, namely, {a + b) {x+y) =ax-^ hx -{•ay-\- hy, and put - c for 6, and - z for y, we obtain {a-c){x-z)==ax-{-{-c)x-\-a{-z) + {-c){-z) =ax — ex — az + czj which is equivalent to the result (iii). Similarly, from any one of the three results of Art. 54 the other two can be deduced. 56. Rule of Signs. The results of the last article are known as the rule of signs^ which may be stated in the following form. The product of two quantities of the same sign {either both positive or both negative) is a positive quantity ; while the product of two quantities of opposite signs (one positive and the other negative) is a negative quantity. The rule is sometimes enunciated in the following form. Like signs produce plus, and unlike signs pro- duce minus. 3—2 36 MULTIPLICATION. 57. The application of the rule gives the results (-l)(6) = -6, and (- 1)(- 6) = + ^j. Also, by Art. 40> we have -(b) =-b, and -(-b)=-\-b. Hence to multiply a quantity by — 1 is equivalent to subtracting it. This result is sometimes assumed ; but it is desirable that the student should notice the different meanings which are attached to expressions like (-1)(6) and —(b), though it happens that the results are equal to one another. 58. As particular instances of the rule of signs, we have (— a) (— a) = + a^, and also (+ a) (+ a) = + ct\ Thus the square of any quantity, whether positive or negative, is itself positive. The special case that (-1)(-1)=+1 is worthy of notice. 59. The continued application of the rule of signs enables us to write down the product of several factors of different signs. Thus {-a)(-b){-c) = {-a)( + bc)= -abc, ^_ar = (-a){-a){-a)=-a^ {-ay={-a){-aY = {-a){-a^) = aS {-ay = {-a){-ay={-a){a^)=-a^ The method is general, and the product of a number of factors will be positive or negative according as the number of factors which are negative is even or odd. 60. Process of multiplication. It is usual to perform the process of multiplication in the manner shewn in the following example, where the product of a? + 8 and ^ — 2 is determined. a; +3 X -2 A'^ + 3^ (i), -2x-e (ii), MULTIPLICATION. 37 The multiplier, x - 2, is written in a line under the multipli- cand, .^' + 3, and a horizontal line is then drawn. Each term of the multiplicand is then multiplied by the first term (in this case, x) of the multiplier, and the results with their proper signs are written down in a line, which is denoted above by (i). Next, each term of the multiplicand is multiplied by the next term in the multiplier (in this case, ~2), and the results with their proper signs are written in a second line, which is denoted above by (ii) ; any term in this product which is " like " one of the terms in row (i) is put immediately under it. The final result is obtained by adding these rows together. The fact that the like terms are in vertical columns facilitates the addition. This method of multiplying is sometimes called "cross multiplication." Note. The lines of dots and the numbers (i) and (ii), in the work given above, are inserted only to facilitate the explanation, and they form no part of the process itself. 61. The numerical coefficients of the terms in the expressions which are multiplied together may be fractions. In such a case, we may either multiply the expressions together in a manner similar to that given above, or we may reduce each expression to a fraction ^vith a numerical denominator (which is the arithmetical L.c.M. of the denominators of the fractional coefficients) and in which all the coefficients in the numerator are integers. The beginner will find the latter course the easier. For example, to find the product of \x-^y and ^A' + jy we may proceed in a manner similar to that given in Art. 60 ; thus t^ + h 1^2 _ 1^ + l^y- -ht \x^^r-h^y- 'jht Or we may put i-^-iy = H3^-2y), and ^x + ^ = -^^{4.x-\-\ 38 MULTIPLICATION. Hence i^x-ly) (b+|y) = ^ (3.r- 2y) J^ (4.^ + 3y) the product of 3^-2y and A:OG-\-'^y being found in the manner explained in Art. 60. 62. The following examples are important, and will serve to illustrate the above remarks. Ex. 1. Find the square of the binomial x+a. X + a X + a x^+ ax ax+a^ x'^ + 2a^^ + a^ Ex. 2. Find the square of x - a. X — a X — a x^— ax — ax+a^ a^-2ax+cfi Ex. 3. Find the product of x-\-a and x ^2 We see, from this example, that the product of the sum and the difference of two quantities is equal to the difference of their squares. Thus, 98 X 102 = (100 - 2) (100 + 2) = 10000 - 4 = 9996. Ex. 4. Find the product of x-\-a and x + h. X +a X +b x^ + ax + bx-\-ab x^-{-{a + b)x+ab MUL TIP Lie A TION. 39 EXAMPLES. III. C. 1. Write down the product of (i) — ax and - hy ; (ii) ax and - by ; (iii) - x and - x. 2. Multiply a; + 2 by y - 3. 3. Multiply ^' - 1 by x - 2. 4. Find the product of x - a and x + 6. 5. Multiply Ix - 1 by a; - 7. 6. Multiply 7a - m by m + 7a. 7. Multiply 3.r - ^a by ^^ - 3a. 8. Multiply ax — h^ by hx — a?: 9. Find the product of x^ - a^ and x^ + a^. 10. Multiply a.r + hy by .i-a - 23/6. 11. Multiply 2a^2 _ 3^„^2 ^y 36? - 2am. 12. Find the product of 40.^^ - Zhyz and 36y2 - 2a^^:. 63. Product of Two Multinomials. The pro- duct of any two expressions is found hy midtiplymg every term of the one hy every term of the other, prefix- ing the proper sign according to the rule of signs, and taking the algebraic sum of these products. To prove this rule, we have only to apply the same method as that given in Art. 54. Consider the product of two multinomials, such as a + 6 + c + . . . and x-^y -\-z-{- ,.., where the +... in these expressions is intended to shew that there may be a number of other terms similar to these written down. Denote (a + 6 + c + . . . ) by 7?, .•. the product (a + 6 + c+ ...) {x -j- y -\- z ■{- ...) = n(x-{-y + z+ ...) = nx + ny + nz-\- ... =(a+6 + c+...)^+(a+6+c+...)3/+(a+6+c+...)-^+... =ax-hhx-\-cx+...-i-ay+hy-\-cy -{-... +az-\-hz-\-cz+... If some of the terms be negative, we can still write each expression as an algebraical sum, and then, by the rule of signs, find the corresponding terms in the product. 40 MULTIPLICATION. Thus the product {a — h){x — y — z) = (a + (-6)}{a; + (-2/) + (-^)} = cix-^a{-y)^a{-z)^{-h)x-^{-h){-y)\{-}}){-z) = ax— ay — az — bx-\-by+ hz. 64. It is usual to perform the process of multipli- cation in a manner similar to that given in Art. 60. Thus to multiply x^-\-x+\ hj x^-x-\-\yfQ, proceed as follows : x^+x +1 x^ — x +1 oc^ + a^+x'^ -ofi — x'^-x ■\-x'^ + x + l X* +x^ +1 As another example, let us multiply 2a;2-3a;+l by 3x-4. 2^2 _ 2x + 1 3x - 4 6^- 9x^+ 3x - 8^2 + 1207-4 6x^-l7x^ + 16x-4: 65. If an expression consist of several terms, and these contain different powers of the same letter, it is usual and convenient to arrange the terms so that the term containing the highest power of the letter shall be the one to the extreme left, the term containing the next highest power of the letter shall be next to it, and so on. When so written the expression is said to be arranged in descending powers of that letter. Thus each of the expressions x^-\-{a + h)x + c% x"^ - 4:a^x^ - 'Sa\v + a^ is arranged in descending powers of x. The latter of them is also arranged in ascending powers of a. The process of multiplication is facilitated if the expressions are arranged in descending powers of the same letter. MULTIPLICA TION. 41 EXAMPLES. III. D. 1. Find the product of 3^-2 + 7^ + 2 and %x^ - 4:X + 3. 2. Multiply 3^2 _ 5^ _ 7 by 7^^^ -5^-3. 3. Multiply ofi - ax + a^ hy X -\- a. 4. Multiply x'^ + ax + a^ by x — a. 5. Find the product of x"^ + ax + a^ and x^ — ax + a?. 6. Multiply a^-\-W--\-c^-hc- ca -ah hj a->th + c. 66. A judicious use of brackets will often enable us to write down the product of two expressions by inspection. Thus the three following examples are included in Art. 62, Ex, 3, where it is proved that {x-a) {x + a) = x'^ — a'^. Ex. 1. (a2 + 62) {a^-h'^) = {a^f- (b^f = a^-b\ Ex. 2. (a-6 + c) {a-\-h-c) = {a-{h-c)]{a + {h-c)} = a^-{h-cf = a^-{b'^-2bc + c) = a^-b^ + 2bc-c^ Ex. 3. {x^ + ^^ + 1} {^2 _ ^ + 1| = {(^2 4- 1) -I- ^'} {(^^2 + 1 ) _ jtj = (^2+l)2_^2 = (^'4+2^2+l)_^2 = ^4 + ^-2+1. 67. Again, if we want to find the square of any multi- nomial, for example of a + b + c + d, we may multiply a-\-b + c + d by a + b + c + d in the manner indicated in Art. 64, or we may find the product by repeated applications of Art. 62, Ex. 1. Using the latter method, we have {a + b+c + d]^ = {a + {b + c + d)]^ = a'^ + 2a{b + c + d) + {b + c + df = a^ + 2ab + 2ac + 2ad+{b + [c + d]f = a^ + 2ab-{-2ac + 2ad+b'^ + 2b[c + d] + [c + df = a^ + 2ab + 2ac-{-2ad+b^ + 2bc-\-2bd+c^ + 2cd+d^. The required square consists therefore of the following groups of terms, (i) the square of the first term, namely a^ ; (ii) twice the product of the first term and the second term, twice the product of the first term and the third term, &c. — every term that folloivs a being multiplied by 2a to give one in this group of 42 MULTIPLICATION. terms ; (iii) the square of the second term, namely h^ ; (iv) a group of terms, every term that folloivs h in the original ex- pression being multipHed by 26 to give one in this group of terms ; and so on. The method is general, and enables us to write down the square of any multinomial by inspection. For example, the square of {a^ + h'^-c^-(P+ e^) = (a2)2 + 2^2 (62 _ c2 - cZ2 + e2) + (62)2 + 2^2 ( - c2 - cf=i + e2) + ( _ c2)2 + 2(-c2)(-£^2 + g2)^_(_^2)2 + 2(-C^2)(e2) + (e2)2 = a^ + 2a262 _ 2a2c2 - 2a?(P' + 2a2e2 + 64- 262c2 - ^bH^ + 262^2 + (A + 2c2t^2 _ 2c2e2 + # - 2c^2e2 ^_ q\ EXAMPLES. III. E. \^The results of the following examples can he reduced to one of the three following forms : {x-\-a){cG-a) —x^-a^^ {x-\-d){x^ — ax-\-a^)—3(^-\-a^^ {x - a) (^2 + ax+a^)=x^ - a^ ; and the student ought to be able to write doion the answers by in- spection^ in a manner analogous to the examples given in Art. ^^^ Write down the product of the following expressions : 1. .^'2 4.7and^2_7^ 2. 77 and 83. 3. ^ + 3 and ^^2- 3^ + 9. 4. 2^ + 11 and 2:^7- 11. 5. a-6 + c and -a-6-fc. 6. 3/2_62 and ?/ + 62j/2 + 6*. 7. y-(a + l) andy2 + (a + l)3/ + a2^2a + l. 68. Product of a number of expressions. The product of more than two expressions can be deter- mined by first finding the product of two of them, then multiplying that by another of the factors, and so on in succession. 69. For example, the cube oi x-\-a is denoted by {x + a)^, and its value is obtained by multiplying (^'+a)2 by x-^a. The value of the former of these quantities, namely of (.r + a)2, has been determined in Art. 62, Ex. 1, and is x'^-^'iax-^-a^. Hence to find the value of {x-\-af we have the following work. .'^2 + 2a^' + a? X + a x^ + 2ax'^+ a^x + ax'^ + 2a^x+a^ x^ + 3ax^ + da^x+a^ MULTIPLICATION. 43 Similarly, the cube of x-a will be found by multiplying {x - ay by {x - a). The value of (x - cif is determined in Art. 62, Ex. 2 ; and the product of this by {x - a) will be found to be a^ — dax^ + 3a^x - a\ This result can however be found at once from the result of the last example. For, if in it we put ( - a) for +a we obtain {x+{-a)f=x^ + 3{-a)x'^ + 3{-afx + {-af, that is, {x-aY=x^- Zax'^ + ^a^x - a^. These results can be combined in a single formula, thus : {x±df-= x^ ± ZaaP- + %a^x ± a^, where either the upper sign is taken throughout, or the lower sign is taken throughout. 70. Similarly, it can be shewn that {x^aY= x"^ + 4a^^ + 6a%2 + ^a?x + a\ and {x -af^3^- A:ao(? + ^aW - 4a% + aK Both results are contained in the formula {x±af^x/^± ^aofi + 6a2^2 + 4,3^3^ + ^K *71. We have hitherto confined ourselves to cases where the indices were definite numerical numbers ; and we have assumed that results, such as x y. x^ — a? and {aP-Y —cuf, were obvious from the definition of indices given in Art. 22. The indices may however be them- selves denoted by algebraical symbols, which may stand for any integral number: thus x'"^ will mean the product of m factors, each equal to x. We now proceed to prove two rules of which the above instances are special cases. We shall for the present assume that the indices are positive integers. *72. Index Law I. To shew that x'"" xx'' = x"^-^'', where x is any quantity , and m and n are positive integers. We have, by the definition of indices [Art. 22], x'^ xx'*^ = {xxx ... m factors) (xxx ... n factors) = xxx ... (m + n) factors = x ,m+n 44 MULTIPLICATION. Thus the index of the product of two factors is the sum of their indices, provided the factors are powers of the same quantity. For example, x^'^ x ^'25 = ^8^ *73. A similar rule holds if we have the product of several factors, provided each factor is some power of the same quantity. Thus *74. It follows that the product of two homogeneous expressions is itself homogeneous. For, if the sum of the indices of every term in the first expression be m, and the sum of the indices of every term in the second expression be n, then the product of any term of the one expression and any term of the other will be of m-\- 71 dimensions. Should the student make a mistake in one term of such a product, this test may enable him to detect his blunder. For example, if it were asserted that the result of multiplying aj^-\-2a.v^ + 3a\v — a^ by x^-2aj; + 5a'^ contained a term Sa^v, we know there must be a mistake, since every term in the product must be of (3 + 2) dimensions, and 3^3.^ is of only 4 dimensions [Art. 26]. As a matter of fact 3a^x^ is one term in the product. It follows similarly that the product of a number of homogeneous expressions is itself homogeneous, and therefore that any power of a homogeneous expression is itself homogeneous. *75. Theorem. To shew that (aby' = a"b''\ It is obvious from the definition that {ahY = aW. This is a particular instance of the general theorem that (a6)'*=a"6". MULTIPLICATION. 45 We have, by the definition of indices [Art. 22], (a6)" = ah xab xab X ...n factors = (aaa ...n factors) x {hhh . . . ?? factors) = a" X 6^ = a"6«. Similarly, [ahcY = [a {hc)Y = «« (&c)« = a*^6^c". And similarly, the n^^ power of a product is the product of the 7i^^ powers of its factors. *76. Index Law II. To shew that {x"')''' = x>'^''\ where x is any quantity, and m and n are positive integers. We have, by the definition of indices, {x^Y = ^'" X ^"' X *•''' X . . . n factors __ /y,m+7n+m+...n terms fArt 72 __ rjQtnn For example, {oi^Y=x^\ ♦EXAMPLES ON THE INDEX LAWS. III. F. Write down the product of the following expressions num- bered 1 to 9. *1. x^^ and x^\ *2. la^\v'>^ and - Sa^s^^. *3. - 4a5y9 and - .3a^oi5206y43, *4^ ^n_^ 1 and .^^" - 1. *5. ^" + a and A'" + 6. *6. x'^-\-x-\ a.n& x'^-x-l. *7. 2.z^ + 3.r« + 1 and x'^ - 2.r" + 3. *8. a" - 6", 5" - c", and a" - c". *9. {ahy - (^2)n^ ^n _ ^n and 6" - ^". *10. Write down the continued product of x to the power of a million, of x to the power of a thousand, and of x. 46 MULTIPLICATION. MISCELLANEOUS EXAMPLES. III. G. [The beginner should take care that all the expressions in each example are arranged in descending powers of some letter before he begins to multiply them together.'] 1. If ^=0, and 3/= - 5, find the value of 7 {x-7j) {2x-y)-S{2x-y) {x+y) + 5 {x+yf. 2. Find the value of ^ {a-bf + {b-cf+(a-b){b-c) + 5c% when a = l, 6= -2, c=|. ^^ 3. Subtract (2a - 36) (c - d) from (2a + 36) (c + d). 4. Simplify {x - of + (^ - 6)^ + 2 {{x -a)(b-x)^ ab]. 5. Prove that (a-6)(a + 6-c) + (6-c)(6 + c-a) = (a-c)(a+c-6). 6. Simplify (i) (a-36)2 + 6(a-6)(6-c) + (36-c)2 + 6(ac-262); (ii) (a + 6 + c)2-(a-6+c)2 + (a + 6-c)2-(-a + 6 + c)2. 7. Reduce to its simplest form (a + 6 + c-cZ)(a + 6-c + o?) + (a-6 + c + ) _ 2 (^2 _ 2^ _ 6)}. , 9. Multiply ^2 _ 5^y + ^yi by ^ - 4y. 10. Multiply a5 + 3a%+9a3^2 + 27a2^ + 81a:t^ + 243^ by a -3.^. 11. Find the product of 1 - 2^^ + 4:^2 and 1 - 1 + ^ - ^ . 12. Multiply ^ — a^2 _ 2^2^ 4- ofi by :^?2 + ^x:?? - a2^ 13. Multiply ^ - 2a^2 ^. 2^2^ _ 3^3 by ^2 _ 3^^ 4- 2a2. 14. Multiply 0(? + 6^2y _|. 1 2^3/2 + g^^ by ^r^ - 3^2^ + Zxy'^ - y\ 15. From (2a - 6)2 + (a - 26)2 subtract the square of 2 (a - 6). 16. Multiply x'^+y'^+z^ — xy-\-xz+ yz by x +y — z. 17. Multiply together 3a + 6 + 2c, 2a + 26 + c, and a - 26 - 3c. 18. Multiply together 3a - 6 + 2c, 2a - 36 + c, and a + 2b- 3c. 19. Find the continued product of x+y + z, x+y-z, x-y + z, and —x+y-\-z. MULTIPLICATION. 47 20. Multiply Ja2 ^ :^ah - ^h^ by ^a^ -\ah^ §62. 21. Find the continued product of 2^- 1, 2.^2+^, and \x-\-\. 22. Find the continued product of \ (.r2 + 307 + 2), ^ (^2 _ 5^ + 6)^ and J (^2 + g^ - 3). 23. Prove that {x + 3)3 - (^ + 2)3 = 3 (^ + 2) (^ + 3) + 1. 24. Simplify (a-6)(^+a)(^+5)+(6-c)(j^+6)(^ + c)4-(<3-a)(^+c)(^4-a). 25. Prove that {a-hf-\-{c-a){G-h) = {h- cf + {a-b){a- c). 26. Shewthaty(y-l)(3/-2)(3/-3) + l = (3/2_33/ + l)2. 27. In the expression ^ — 2^2 + 3^ — 4^ substitute a — 2 for ^, and arrange the result according to descending powers of a. 28. Simplify (i) (a + 6-fc)2-(6 + c)2-(c4-a)2-(a+6)2+a2 + 62 + c2; (ii) la + b + cf-{b + cf-lc + af-{a-hbf + a^-\-b^ + c\ 29. If ^ = a2 - 6c, 3/ = 62 _ ca, and = c2 - a6 ; prove that (i) ax+b7/ + cz = {a;+^ + z){a + b+c); (ii) be (^2 _ y^;) = ca (^2 _ ^^^ = ab {f — xy). *30. Simplify the expression ax'^-\-by^-\-cz^-\-xyz^ where x—b->rc-a^ y = G + a-b^ and z = a + b-c. *31. Prove the following identities, by multiplying out each side of the equality. (i) (l+a)2(l + 62)_(i4.a2)(i + 6)2=2(«-6)(l-«6); (ii) (a + 6 + c)3 = a3 + 53 + c3+3 (5 + ^) (c + a) (a-\-b) ; (iii) (2/ + Z- 2xY + {z + x- 2y)3 + (-c+y- 2z)^ = Z{y+z-2x){z + x-2y){x+y-2z)', (iv) (^ -y){x- 2y) {x - Zy) + ^y{x -y){x- 2y) + l^y\x-y) + 6^/3 = .'?;(^4-:?/)(^ + 2y); (v) {x-{-y + zf = {x-\-y-zf-\-{x-y-{-zY^{-x+y-\-zy-\-24:Xyz; (vi) (y - 2) (y + 2)3 + (2 - .r) (2 + ^')3 + (^ - y) {x+yf = 2^2 (3/2 - 02) + 22^ (^2 - X^) + 2.^7/ (^'2 _ y2)_ *32. Find the coefficient of x^ in the product of (1-^^)2 and 1 + 2^+30724-4^ + 5^. CHAPTER IV. DIVISION. 77. We proceed next to the consideration of the division of one algebraical expression by another al- gebraical expression; and shall discuss successively division by simple expressions and division by multi- nomials. We may add that we can always test the correctness of the result of a division by seeing whether the product of the quotient and the divisor when added to the remainder is equal to the dividend. For this reason, division is sometimes said to be the inverse of multi- plication. 78. Division by Simple Expressions. The quotient of an expression (the dividend) when divided by a simple quantity (or by a product of simple quan- tities) is indicated by writing the divisor after the dividend and preceded either by the symbol -^ or by a solidus / [see Art. 16]. The operation may also be represented by a fraction having the dividend for numerator and the divisor for denominator. For example, we may represent the operation of dividing ahc by X either by the form ahc-^x, or by abc/x, or by — . In using either of the first two of these forms it must be noticed that all the quantities that come immediately after the symbol -^ or / are to be regarded as a divisor. Thus either abjxc or ab-^xc would mean that ab was to be divided by xc, and not that ab was to be divided by x and then the result DIVISION. 49 multiplied by c : the latter could be represented cither by {ah-=rx)c, or by {ab/x)r^, or by — xc. Similarly we can express the operation of dividing .v^-\-^^ by a by any of the following forms, {x^ + i/^)-^a, or {x^-\-y'^)la^ or —. But an expression like x'-+y'^^a would signify that y^ only was to be divided by a, and the quotient added to x^ ; while x'^-^a+y^ would signify that x'^ was to be divided by «, and that y^ was to be added to the quotient : we shall however avoid the use of forms such as these which a careless reader might regard as ambiguous. Where the same quantity appears as a factor of both divisor and dividend it may be cancelled, since to multiply any expression by a certain quantity and then to divide it by the same quantity cannot alter it. Ex. 1. Divide 2lax^ by Zx. Here -— — = ^ ; Zx 3/ = lax. Ex. 2. Divide 6a*Pc^- by bah^c\ „ QaW-_6M^ ^ea^b " 5c ' Ex. 3. Divide 2a% (c^+c^^) by Zax\ jj^^^ d^bil+d^^^oMl+dF)^ dax"^ x^ Note. Examples such as those given above can generally be solved by inspection, and it is only to explain the reason of the method used that any steps are inserted in the work here printed. 79. Rule of Signs. One or more of the factors may be negative. We may, by Arts. 55, o6, regard a negative factor as the product of — 1 and an equal positive factor. The following results, where A and B B. A. 4: 50 DIVISION. stand for any algebraical expressions, follow at once : -A_ (-1)A A_ A W _^ {-l)B ^ ^B B' -A_ {-1)A _^A ^'''^ -B~{-l)B B' Thus the rule of signs in division is similar to that in multiplication, as given in Art. 56, and may be stated in the following form. The quotient of any quantity by another quantity of the same sign (that is either when both are positive or both are negative) is positive, but its quotient by a quantity of the opposite sign is negative. This is sometimes enunciated in the form, Like signs produce plus, and unlike signs produce minus. Ex. 1. Divide lla^bc by TT lla^c 11 ab Here — - — 5= -— . Ux. 2. Divide — Zxy by — 2a6. Here =^^ = 1% - 2ab 2 ab EXAMPLES. IV. A. 1. Divide x^ by x^. 2. Divide 18a^ by Zx. 3. Divide ^^ah\^ by 46c2. 4. Divide h\lmh by Vllz. 5. Divide QSa/agW by la^a^a^. 6. Divide 56^2.^3 by - lax'^. 7. Divide - 52^3^ by 13^2^. 8. Divide - 81^7 by - 9^. 9. Divide hXaW'cx''-' by Zaby. 10. Divide — 4p%'3 by Zpqx'^. 80. Quotient of a Multinomial by a Simple Expression. The quotient of a midtinomial quantity by a simple quantity is the algebraic sum of the quotients obtained by dividing each term of the multinomial by the simple quantity. DIVISION. 51 We know from arithmetic that the quotient ob- tained by dividing the sum of two or more numbers by a divisor is the sum of the quotients obtained by dividing the separate numbers by that divisor. The same proposition will therefore be true of numbers represented by algebraical symbols : the principle ex- plained in Art. 41 allowing us to extend its application to negative quantities. This is equivalent to the rule printed at the head of this article. Thus, if a -I- 6 be divided by x, the result is the sum of ajx and hjx^ which we may express in any of the forms {a-\-h)-^x = {a-^x)-^{h-^x) ; a + h a . h or = - + - ; X XX or (a + b)lx=-a!x + b/x. We have a similar result if there are more than two terms. Thus a+b-c _a ^ __o W OC %h 0(j which may also be written (a + 6 — c)/^ = a\x -f b\x — c/x. Ex. 1. Divide 7ab^ + 3a?b by ab. ^ 7a62-|-3a26 ^ 7a62 3a26 ab ab ab = 1b+3a. We might also have proceeded thus : 7ab-^ + 3a'-b ab{7b + 3a) ^, , _ ab ab Ex. 2. Divide Umhi^ - Zlhn by - 2lm^. ilnv^n^ - Slhn _ _ ( Umhi^ _ Uhn \ 2 m 4—2 52 DIVISION. EXAMPLES. IV. B. 1. Divide Ia%^-I4.ah'' by lah. 2. Divide '2.ah:—2ah^ by 4a. 3. Divide UPt/ - Iblmy^ by -3li/, 4. Divide ^a^'^-\-^ba^-lcx^ + 19x by -^x 5. Divide - ^x^ - ^x^ - '^x by — ^f^. 81. Quotient of a Multinomial by a Multino- mial. We proceed now to consider the division of one compound expression by another compound expression. We want to obtain an expression such that the product of it and the divisor shall be equal to the dividend, and our method of finding the expression is equivalent to gradually building up an expression which when multiplied by the divisor will give the dividend. To do this, we arrange both dividend and divisor in descending powers of some common letter, which let us suppose is x. Then, if we divide the first term in the dividend by the first term in the divisor we shall get one term of the quotient. Multiply the whole of the divisor by this, and subtract the result from the quotient ; we shall then get a remainder which is at least one degree lower in x than the original dividend. This remainder must then be treated as the dividend, and we shall get another term in the required quotient and a new expression to divide which is of still lower dimensions in x. Finally, we shall get a number of terms in the quotient, and either no re- mainder, or a remainder which is of lower dimensions than the original divisor. 82. Thus to divide x^ + x'^ + 1 by 1 + ^ + ^^2 ^e write the expressions in descending powers of x as shewn herewith. Divisor. Dividend. Quotient, x'^ + x+l )^+^Hl ( The first term of the dividend, viz. x^, when divided by the first term of the divisor, viz. x^, gives x^ ; hence the fii'st term in the quotient is x'^. Write i>f- in the space reserved for the DIVISION. 63 quotient; then multiply the divisor by it, writing the result below the dividend, and subtract. It is often convenient to place all like terms (if any) in the same vertical column ; thus -x^ +1 Next, we have to divide this remainder by x^+x + l. The highest powers of x in the remainder and the divisor are respec- tively — ^ and .r^, and the result of dividing — x^ by x^ is — x. This therefore is the next term of the quotient. We write this as another term of the quotient, and then multiply the divisor by it, writing the result below the quantity from which we have now to subtract it. The operation so far will therefore be as follows. ^2 + ^+1)^ +x^ +\{x'^-x x/^+ofi + x"^ -^ +1 -x^-x^-x x^-\-x-\-l Next, we have to divide this remainder by x^-{-x-\-\. The first term of this new remainder, ^r^, when divided by the first term of the divisor, viz. x"^, gives 1 ; hence the next term in the quotient is 1. Proceeding as above, we get ^2 + ^+1)^4 _|_^2 _|_l(^2_^,+ l x^-^-x^+x"- -cfi +1 — x^ — x'^ — x ^2 + ^+1 x'^-\-x + l Therefore the quotient is .^2 - .r + 1, and there is no remainder. Hence (x^ + x^ + l) is exactly divisible by (x^+x + l) [Art. 16]. The student should verify this result by forming the product of :j?2^^ + 1 and x'^-x + l; and he will find [see Art. 66, Ex. 3] that {x'^+x + l){x'^-x-{-l)=x^+x'^ + l. 83. The above process is really equivalent to writing the dividend in the successive forms x^+x^-}-!, x'^{x'^+x+l)-x^ + l, x'^{x^+x+l)-x{x^-{-x+l) + {x^+x+l), which are only different ways of writing the same expression. The last form shews that the dividend is exactly divisible by ^^2 + ^+1, and that x^-x+1 is the quotient of the division. 54 DIVISION. 84. Note. The successive terms of the quotient are sometimes written by the side of the steps from which they arise, thus : ^2 + ^+1)^ +^2 4-1(^2 a^+a^ + x^ -x^ +1 ( —X — a^ — x^ — x ^2+^+1 ( +1 ^24^+1 85. The following are additional examples. Ex. 1. Divide x^ — a^hyx-a. We have x-a)x'^ -a^{x + a x^-ax ax - a^ ax — oi? Hence the quotient is x-\-a^ and there is no remainder. The result of this division ought to have been obvious, since we have already shewn that {x-\-a){x — a)=-x'^ — o?', and hence the quotient of .r2-a2 by ;r — a must be x-\-a. It should be noticed that, if we can resolve the dividend into factors, we can write down by inspection the result of dividing it by any one of those factors. Ex. 2. Divide 2.^2 - Zax +a^ by x + 2a. We have x + 2a ) 2.^2- 3a^+a2 ( 2x- la 2x^+4ax — 7ax+ a2 -lax-Ua"^ 15a2 The quotient therefore is 2x-7a, and there is a remainder I6a^. This result is equivalent to the formula 2^2 _ Sax + a2 = (2x - la) {x + 2a) + 15a2, which the student can verify by multiplication. Note. Here we wanted the remainder to be independent of X. If however we had arranged the expressions in descending powers of a, we should have had a quotient \a — ^x, and a remainder ^-x'^ which is independent of a. This is shewn by the accompanying work. 2a+x)a^-Sax+2x'^{^a-lx a^+^ax -^ax+ 2x^ -^ax— 1^2 DIVISION. 55 This result is equivalent to the formula a2 - Zax + 2.^2 = (la - fr) (2a + x) + ^x\ Ex. 3. Divide ^x"^ + f i^ - ^y Ix - J. We shall treat this by a process analogous to the second method given at the end of Art. 61. We have |.t;2 + 1^^-^ = ^(3^/724-11^ -4), and |^-^=i(3.r- 1). XT ix'^ + ^x-l _ ^-^{^x'^-\-l\x-A) _^ Zx'^ + l\x-4 . ^^^^^ |.r-i - i(3^-l) "^ 3a;- 1 ' Now dividing 3:i;2 4_ii-^_4 by 3^- 1 we have a quotient ^ + 4, as shewn herewith. 2x-\ ) 3.r2 + ll.r-4(^^ + 4 3.^2 _ X 12^-4 12:^-4 Therefore the required quotient is ^ (^+4). Ex. 4. Divide a^ + h^-\-c^- Zabc by a + h + c. Here it will be convenient not only to arrange each expression in descending powers of a, but also to arrange the coefficients of every power of a in powers of 6 : in short, to arrange primarily in powers of «, next, as far as is consistent with this, in powers of 6, and so on. To save space, we shall not concern ourselves with arranging the expressions so that only like terms are in the same vertical column. a-{-h + c ) a?-Zahc + ¥-\-c^ ( a'^-ah-ac + h'^-hc + c'^ — a^h — a^c — Sahc + b^ + c^ — a% — aW— ahc -a^c + ab^-2abc+ b^ + c^ — a^c — abc — ac^ ab^— abc + ac^ + b^ + c^ a62 +b^ +62c — abc + ac^-b'^c+ c^ — abc - b^c — bc^ ac^ + bc^+ c^ Hence the quotient is a^-ab-ac + b'^ — bc+c^, and there is no remainder. 56 DIVISION. We might also have collected the coefi&cient of each power of a in a bracket, thus : -a2(5-i-c)-3a6c + 63 + c3 -a^{h-\-c)-a{h-\-cf a(62-6c+c2) + &3+c3 Here we write down the difference of a (6 + cf and Zahc in its simplest form, either performing the w'ork mentally, or doing it as a subsidiary example. Similarly we write down the product of (6 + c) and {b^—hc-\-c^) as h^-^-c?, without inserting the details of the multiplication. ^Ex. 5. By tuhat expression must x^ + x^ — '^ he divided to give ;r*+2;r2 + 2 as quotient, with no remainder'^ We always have the relation dividend = (divisor) . (quotient) + remainder. Here there is no remainder. . •. .r<5 + .^4 _ 2 =: (divisor) . {pi^ + 2.^2 + 2). If we divide both sides of this equality by x'^ + 2x^ + 2, we see that the required divisor is equal to the quotient of x^-^x^-2 by x^-\-2x^ + 2. This will be found to be x^-l, which is therefore the required expression. EXAMPLES. IV. 0. Divide .r2-l7.:^+16 by x-l. Divide x^ + 24:X^-l1x-8 hj x-l. Divide 4:X^-5x^ + 6x-^ by 2x-3. Divide x^ + a^ by x+a. Divide x^—a^ by x—a. Divide x^-{-y^-{-^-2yh^-2z^x^-2x^f by x+y+z. Divide x'^-lZx-SO by x'^-2x+3. Divide 2x^ - ^x'^y - lOi/^ by ^x-y. Divide 2^-3^-4 + 1 by x^-{-2x-\-l. Divide x^ + 3ax^ + a^x+3a^ by x^+4ax + 3a\ l^ 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. DIVISION. 57 11. Divide a^x {x+y) + Za {x~ - 2xy - y"^) + 9y (y - x) by ax - 3y. 12. Divide by {a — c) the sum of the four quantities 3a3 _ a2 (7^, _|. gc) + Zah\ b^ + ab{4a- 56) - 7bc% 2a^c-b^ + a^Zb-2a), and 6a26 + 6c2+2aZ>2_a3. 13. Multiply ^x^-^.^ + ^x^-2x + l by i.'ir2-^+l, and divide the product by ^x^-^x^ + ^x-l. 14. Divide abx^ + (ac - bd) x"^ - {af+ cd)x + df by ax — d. *86. Index Law 1. We proceed now to consider the case of the quotient of a quantity raised to a literal power divided by the same quantity raised to some other literal power. The method of treating this ques- tion is analogous to that given in Art. 72, and we shall for the present assume that the indices are positive integers. We want to find the value of x'^^ -^ x*^ when m and n are positive integers, and we shall consider (i) the case when m > n, and (ii) the case when tn < n. First. If m > n, then —- = x^-'K x^ _x X X {m factors) x''^ X X X (?^ factors)* Now, if m > n, then each factor in the denominator^vill cancel with a factor in the numerator, and therefore all the n factors in the denominator will cancel with n of the factors in the numerator. This will leave (m — n) factors in the numerator, each equal to x, and the product of these is denoted by x'^~'^. Thus we have Second. If n> m, then — - = For x'^ X X X (m factors) x^ X X X (?i factors) * Now, if ?i > m, then each factor in the numerator will cancel with a factor in the denominator, and therefore 58 DIVISION. all the m factors in the numerator will cancel with m of the factors in the denominator. This will leave {n — m) factors in the denominator, each equal to x, and the product of these is denoted by x^-"^. Thus we have x^ _ 1 ^EXAMPLES ON THE INDEX LAWS. IV. D. •*!. Write down the quotient of l2la^W by lla"^'*. *2. Write down the quotient of lla^^x^-^ hy -^a^\v'^. •^3. Write down the quotient of 27a2" by 3a". ■^4. Divide .r^'^-l by ^" + 1. •^5. Divide .r2" + ^'^ + l^-.^^-l by .r" + l. MISCELLANEOUS EXAMPLES ON DIVISION. IV. E. 1. Divide l-5x*-\-4x^ by l-2x-\-x\ 2. Divide 3(^-\-'^xy-\-y^-\ by x-\-y-\. 3. Divide a^ + '^a% + ^ah'^-\-P-\-c^ by a + 6 + c. 4. Divide 3.^^ + 2^4 + 13^-10 by ^ + 2^H^-2. 5. What is the remainder when x^-2x^ + 2x^-x-l is divided by ^2 + 3^.1? 6. Divide a'^+4:a^b + 6a%'^ + 4ab^ + ¥-c^ by a + b + c. 7. Divide x^-^27f by ^2 + 3^^^ 4. 3^2^ 8. Divide 2^ + 5x^y - 1 e^Y + 35.^^/3 -12y^ by x^ - 2xy + Zif. 9. Divide 2^* - lO.^y + 25^%2 _ 3i^^^3 + ^Qy\ by .-^2 _ -^^.y + 4^2^ 10. Divide 126* - Zd" - ^aW - 8ab {a? + W) by 3a2 + 662 + 2ab. 11. Divide x'^ -2y^'^-1x^y'^-lxy^'^ + \4:X^y^ by x-2y\ 12. Divide a6(.^2 4.^2) + (^2 4.53),^^4.(Qj_5)(.^_^)_l by ax + by- 1. 13. Divide Sx^ - lOx^y + 1 Bx^y^ - 1 2.r2y3 -i- _^yi _|. 2^/5 by (.^^ - y)\ DIVISION. 59 14. Divide .r4-(6-2).^^-(26- l).r2-(62 + 26-8) .r + 36+ 3 by ^2_j.3a; + 6 + l. 15. Divide 1 + 3^7 by 1 - 2cc to 5 terms in the quotient. 16. Divide 14a'* + 15a36 + 33^262 + 36a6H28M by 7a2-3a6 + 1462. 17. Divide ^X^-l^T^y + lZx^f-Axy^ + y^ by ^x^-Zxy+y"^. 18. What is the remainder when a'^ — 2a?h-\-2aP'lf--¥ is divided by a^-ah^2b'^1 19. Divide y^+y^-hy'^-'^y hj y'^ + Zy-\-\. 20. Divide / - 26y + h^ by y^ -f by"- + Wy + h^. 21. Divide xf^-^{a-\) o? -{^a^r\) x^-\-{a^-^^a-^) x^Za^-% by ^'2_3.^.^(t^ + 2). 22. By what expression must x + 3 be multiphed to give ^7 + 2187? 23. By what expression must Zd^ - 8a^b + 4a262 _ g^js _ 1 254 be di\'ided to give the quotient 3a2 — 2ab + 662 ? *24. By what expression must ^3 + 6^2 _ 4^_ 1 be divided to give ^2 + 5^^-9 as quotient, with 8 as remainder ? 25. The quotient and divisor in a certain example are a-b—c and a^ + b^ + c^ + ab-{-ac— be. What was the dividend ? 26. The product of two factors is (x + 2yy + {3x+zy, and one of them is 4^ + 2y + : find the other. 27. The quantity ^+^4^2^ + 3^2_^.^4 jg ^^le product of two algebraical expressions. One of them is x^-x+l. Find the other. 28. If the product of two expressions be x^ + x^y^+y^^ and one of them be x^ - xy +y^ ; find the other. *29. Prove that if n'^ = 7i + l, then x^+ax^-\- a'^x^ + a^x + a^ is exactly divisible by x^ + 7iax + a^. *30. Find the quotient by a + 6 + c + 0? of {a-\-b){b + c){c-\-d){d-\- a) {a -{-c){b + d) + {abc + bed + cda + dabf. *31. If x^-\-lx-\-e be exactly divisible by .i' + 4, what is the value of c ? CHAPTER V. SIMPLE EQUATIONS INVOLVING ONE UNKNOWN QUANTITY. 87. Identity. When two algebraical expressions are equal for all values of the letters involved, the state- ment of their equality is called an identity or di, formula or an identical equation [see Art. 29]. In an identity, the expressions which are separated by the symbol of equality are merely different ways of expressing the same quantity. Thus (a+6)2 = a24-2a6 + 62 is an identity, for it is true for all values of a and h. 88. Some writers use the symbol = to represent the words is identically equal to^ and would write the above identity as (« + 6)2 =a^+ 2ab + b\ The symbol = has however another and different meaning, which is in common use in the higher parts of mathematics ; and as it is important that every mathematical symbol should be unambiguous and definite it is undesirable to use = as a symbol of identity. 89. Equation. When two algebraical expressions are equal to one another only for certain values of some of the letters involved, the statement of the equality is called an algebraical equation. The quantities separated by the sign of equality are called the sides of the equation. Thus ^+7 = 11 is an algebraical equation, for it is true only when X is put equal to 4. The expression a; + 7 is the left-hand side of the equation; the number 11 is the right-hand side of the equation. SIMPLE EQUATIONS ETC. 61 90. Root. The values of the letters which make the two sides of an equation equal to one another are called the roots of the equation. The roots of an equation are said to satisfy it. To solve an equation is the process of finding the roots. The word solution is often used to denote the roots, which are the results of this process. 91. Unknowns. The letters which represent the unknown quantities are often called the imhiowns, — an abbreviation for unknown quantities ; and the equa- tions are said to be equations in {i.e. involving) the un- known quantities. 92. Notation. It is usual to represent the value of known quantities (other than mere numbers) by the letters near the beginning of the alphabet, such as a, b, c, &c.; and to represent the unknown quantities, whose values we want to determine, by the letters near the end of the alphabet, such as x, y, z, &c. 93. Classification of Equations. Equations which involve (beside the unknown quantities) only numbers are called numerical equatio7is. Equations in which some or all of the known quantities are re- presented by letters are called algeh^aical equations. Equations are further classified according to their dimensions in the symbols which represent the un- known quantities. Simple Equations. An equation which contains only one unknown quantity, say x, is said to be a simple equation, or an equation of the first degree, or a linear equation, when only first powers of x occur in it. Quadratic Equations. An equation which con- tains only one unknown quantity, say x, is said to be a quadratic equation, or an equation of the second degree, when the only powers of x which occur in it are the first and second, namely, x and x\ 62 SIMPLE EQUATIONS INVOLVING ■^Similarly, an equation in x is called a cubic equation, or an equation of the third degree, when the only powers of x involved are x, x'^ and x^. It is called a biquadratic equation, or an equation of the fourth degree, when the only powers of x involved are x, x-, o^, and x^. 94. Axioms. We assume the following axioms, (i) If equal quantities be added to equal quan- tities the sums will be equal. (ii) If equal quantities be taken from equal quan- tities the differences will be equal. (iii) If equal quantities be multiplied by the same number (or by equal numbers) the products will be equal. (iv) If equal quantities be divided by the same number (or by equal numbers) the quotients will be equal. In other words, we assume that we may multiply or divide both sides of an equation by the same quantity ; and also that we may add to or subtract from each side of an equation the same quantity. 95. Transposition of terms. Any term may he moved from one side of an equality to the other pro- vided its sign is changed. Suppose that we have an equality such as a + h — c=^d. Add c to each side. This, by axiom (i), is permissible. .*. a + h — c-\- c = d + c, that is, a-{-h = d-\-c. Thus c has been moved from the left-hand side of the equality to the right-hand side, but it appears on the right-hand side with the opposite sign to that which it had when on the left-hand side. Next, returning to an equality like a + h — c=^d, let us subtract h from each side, or (in other words) add — h. This, by axiom (ii), is permissible. .*. a-\-h — c — h^d — h, that is, a — c^d — h. ONE UNKNOWN QUANTITY. 63 Thus h has been moved from the left-hand side of the equality to the right-hand side, but it appears on the right-hand side with the opposite sign to that which it had when on the left-hand side. Terms which are thus removed from one side of an equality to the other with a change of sign are said to be transposed. If we transpose all the terms from each side of an equation to the other side, the sign of every term will be changed. This is equivalent to multiplying both sides by — 1, which we already know, by axiom (iii), is permissible. We can thus change the sign of every term of an equation. 96. We shall confine ourselves in the rest of this chapter to some of the easier examples of simple equa- tions which involve only one unknown quantity. 97. Method of Solving Simple Equations. The method of solving a simple equation will be readily understood from the following examples. The usual process is as follows. (i) First, we clear the equation of fractions, remove brackets, and perform any other algebraical operations which are indicated. (ii) Next, we transpose all the terms involving the un- known quantity to one side of the equation (usually to the left-hand side), and all the other terms to the other side. (iii) Then, we simplify each side of the equation as far as is possible, combine like terms, and in par- ticular collect all the terms which are multiplied by the unknown quantity into one term with it as a factor. (iv) Lastly, we divide both sides of the equation by this coefficient of the unknown quantity : and we then have the required root, which must involve nothing but numbers and known quantities. It is desirable to write each successive step in a separate line, and also to indicate how it is deduced 64 SIMPLE EQUATIONS INVOLVING from the preceding step. The beginner should also state his answer at the end of the work, so as to make it quite clear what is the result arrived at. Ex. 1. Solve the equation Zx-^==x — % (i) Transpose the term on the right-hand side which involves X to the left-hand side ; and transpose the term on the left-hand side which does not involve x to the right-hand side. .'. 3^-^=8-2. (ii) Collect like terms, - .*. 2x = 6. (iii) Divide by 2, .-. ^^=3. Thus the required root is 3. This value of x will therefore "satisfy" the given equation, that is, make it identically true. We can verify this result by seeing whether this root does so satisfy the given equation. If we put a" = 3 in the given equation it becomes 3x3-8 = 3-2, that is, 9-8 = 3-2, which is true. In the following examples we shaU not in general verify our results, but a student who is doubtful as to the correctness of his work can always test the final result in the manner above described. Ex.2. Solve the equation iy — 4=3/ + 2• Clear of fractions. The l.c.m. of the denominators is 12; multiplying both sides of the equation by 12, we have 4y-3 = 12?/-i-6. Transpose terms, . • . 4^/ - 1 2y = 6 -f- 3. Collect like terms, . • . - 8y = 9. Change the sign of every term, .*. 8j/= - 9. Divide by 8, .*. y= -f. Therefore the root of the given equation is — f . Ex. 3. Solve the equation lx + 4:{2x-l)-2{x-\-Z)^-b = b{x-\). Remove the brackets, . • . 7^ -f 8^ - 4 - 2^ - 6 + 5 = 5.r - 5. Transposing, .-. lx-\-^x-2x-bx= -b-\-'i + Q-b. Collect Hke terms, .*. 8.^=0. .-. ^^=0. Therefore the root of the given equation is 0. ONE UNKNOWN QUANTITY. 65 E.v. 4. Solve the equation •2.i' + 3 = l - Zx. v, Transposing, . ' . •2.t' + •3.^• =1-3. Collect like terms, .*. -5.1'= -2. 2 2 Divide by -5, . .-. .r= - — =--= -4. Therefore the root of the given equation is - 4. [We might also have at once expressed the decimals as vulgar fractions, or have multiplied both sides of the equation by 10, and so got rid of the decimal fractions ; and then solved.] Ex. 5. Solve the equation ax + h = ex + d. Transposing, .•. ax — cx = d—h. Collect like terms, . *. {a — c)x = d — h. Divide by a- c, .'. x = . -^ ' a-c Therefore the root of the given equation is (d?- h)l{a — c). The following examples are rather harder. We shall follow the above procedure, but only indicate the more important steps. Ex. 6. Solve the equation 7 {x-l) (.r + 2) = 11^2_4 (.^,_,_ 1) (.r- 2). Multiplying out the products, we have l{x'^ + x-'2.) = l\x'^-A{x'^-x-% that is, T^^HV^'- 14 = 11.^2-4^2 + 4^ + 8. Transposing, . '. 7.^2 -11^-2 + ^^2. + 7^ _ 4^^ = g + 1 4. Collect like terms, .-. (7 + 4-ll)A'2+(7-4).r=:22. .-. 3.^' = 22. .-. ^ = ^ = 7^. Ex. 7. Solve the equation -6.27 + ^ = • 1 6.^ + 2 • 1 , Express the decimals as vulgar fractions, • • To •^"'3— 9 ^^ \{)' . . 5 •t' -r 3 — (}'<' T^ 10* B. A. 5 66 SIMPLE EQUATIONS INVOLVING The L. c. M. of the denominators is 30. Multiplying by 30, we have 18^+ 10 = 5^*; + 63. .-. iar-5^ = 63-10. .'. 13^ = 53. . ^_ 53 — 4 11 • • •*^ — 13 — ^13- EXAMPLES ON SIMPLE EQUATIONS. V. 1. Is 2 a root of any of the following equations ; and if so, of which is it a root ? (i) .r-3 = 2.r-5; (ii) 2.y-3 + .r = 7^-4; (iii) ^2_2 = 2. 2. Shew that 2, ^, and — § are roots of x^ {Qx — 11 ) = 4 (.^^ - 1). Solve the following equations, numbered 3 to 45. 3. l3b, then x is a positive number, and there is no difficulty in the solution. The event takes place x years after the present time. For example, a = 36, 6 = 10. (ii) If a = 36, .'. x = 0; that is, the father is at the present time three times as old as his son. For example, a = 30, 6 = 10. SIMPLE EQUATIONS. 75 (iii) But if cKZhy the answer is a negative number of years. This will mean that the event took place before the present time [see Art. 10]. For example, a = 28, 6=10. Thus our result sums up all the cases in a single statement, which we must interpret according to the usual algebraic rules. Ex. 12. Tioo couriers, A and B, travelling in the same direction along a certain road, start at the same time from two stations, H and A", whose distance apart is n miles. A travels at the uniform rate of a miles an hour, B travels at the uniform rate of b miles an hour. A starts behind B, lohen will he overtake B ? A starts from R, B from K. Let A overtake B at the end of t hours after they start, and suppose that A overtakes B at P. Then the distance travelled by A is equal to HP=at, and „ „ „ „ B „ KP=bt; .'. at-bt = HK=n. a — b (i) If a>b, then t is a positive number, and there is no difficulty in the solution. (ii) If a<6, then Ms a negative number. In fact, as A travels more slowly than B, he can never overtake B. But, if the couriers be assumed to have been travelling in the same direction and at the same rates before they reached H and A', then at some time before that at which A is at H and B at A", they were together, and it is this time that is given by the answer. Moreover, since HP = at and KP^bt and t is a negative number, it follows that HP and KP are both negative, and therefore are to be taken as measured in the opposite direction to that in which motion is taking place. (iii) If a = 6, then 1 = -^^. Now we know from arithmetic that the smaller the denominator of a fraction the bigger does the value of the fraction become (if the numerator remain the same), and when the denominator vanishes the value of the fraction becomes indefinitely large. In this case it will take an infinite time before A overtakes B, which is the same as saying that A never overtakes B. In fact, as they travel at equal rates, they will always be a distance n miles apart. 76 PROBLEMS LEADING TO Ex. 13. As one more example, consider again the problem of filling a cistern by means of four pipes which is given on p. 72, Ex. 7. If we had supposed that the second, third, and fourth pipes had let the water escape, and the first had supplied water we should have had the equation V _V V V V from which we obtain x= — 12. To determine what meaning can be attached to a negative answer, we notice that if we had tried to find the number of days in which the cistern, supposed full, would have been emptied when the last three pipes were emptying it and the first pipe filling it, we should have obtained the equation V V V v V and therefore ^'=12. Hence the negative answer to the problem as stated in the first part of this Example shews us that the problem as it stands is impossible, but that the analogous problem of emptying a cistern which is already full is capable of solution. 101. The following rules may help the beginner to form correctly the equations of the problems hereafter given. First. In problems concerning distance, time, and (uniform) velocities, we have [see Art. 5] s = vt, where s is the distance traversed (i.e. the number of units of length in it), t is the time occupied in traversing it (i.e. the 7iumber of the units of time occupied), and V is the velocity with which it is traversed. But in all cases the same unit of length should be used throughout the same question ; that is, all distances should be expressed either in feet, or in yards, or in miles, and so on — whichever unit is most convenient being chosen. Similarly all durations should be expressed as multiples of the same unit of time ; and so for other quantities. SIMPLE EQUATIONS. 77 Second. In problems concerning Avork done by men, taps filling cisterns, &c., we notice, (i) that if a man do a piece of work in a days, he does - of it in each day; if a tap fill a cistern in a hours, it must fill - of it in each hour ; a (ii) that if cc men complete a piece of work in one day, then each man must do - of it in 1 day ; (iii) that if oo men complete a piece of work in a d'ciys, then xa men would do it in 1 day, and therefore 1 man would do — of it in each day. xa -^ Should the student feel any difficulty in writing down the foregoing results, a numerical illustration (such as putting « = 2, A' = 3) will probably guide him aright. EXAMPLES. VI. [Some additional examples on problems leading to simple equations will be found in Chapter XI.] 1. Divide the number 46 into two parts, such that when the one is divided by 7 and the other by 3, the quotients to- gether may amount to 10. 2. Find a number such that the sum of it and of another number m times as great may be a. 3. Divide the number 237 into two parts, such that one may be contained in the other 1^ times. 4. A person has 264 coins of two kinds, and 4^ times as many of one sort as the other. How many has he of each sort ? 5. The difference between the third and eighth parts of a certain number exceeds five times the difference between the eighth and ninth parts by 10 ; find the number. 6. The difference between the fourth and ninth parts of a certain number exceeds four times the difference between the ninth and tenth parts by 34 ; find the number. 78 PROBLEMS LEADING TO 7. Find a number which when multiplied by 6 exceeds 35 as much as 35 exceeds the original number. 8. Find a number such that if it be subtracted from the sum of its half, third, and fourth parts, the remainder may be 1. 9. An army is defeated, losing ^gth of its numbers in killed, and 4,000 prisoners. It is then reinforced by reserves amounting to 5,000 troops ; but, in retreating, its rear-guard, which consists of jth of the numbers now with the colours, is completely cut off. There remain 21,150 men. What was the original force 1 10. A man sold a horse for £35, and half as much as he gave for it. He gained £10. 10s. ; what did he pay for the horse? 11. A fortress has a garrison of 2,600 men ; among whom are nine times as many foot soldiers, and three times as many artillery soldiers as cavalry. How many of each corps are there ? 12. I think of a certain number. I multiply it by 7, add 3 to the product, divide this by 2, subtract 4 from the quotient, and obtain 15. What number did I think of? 13. Gun metal is composed of 90 per cent, of copper and 10 per cent, of tin. Speculum metal contains 67 per cent, of copper and 33 per cent, of tin. How many cwt. of speculum metal should be melted with 4 cwt. of gun metal in order to make an alloy in which there is foiu* times as much copper as tin ? 14. An express train runs 7 miles an hour faster than an ordinary train. The two trains run a certain distance in 4 h. 12m. and 5h. 15m. respectively. What is the distance? 15. A man drives to a certain place at the rate of 8 miles an hour. Returning by a road 3 miles longer at the rate of 9 miles an hour he takes 7^ minutes longer than in going. How long is each road ? 16. A tom^ist finds that, if he spends sixteen shillings a day, the money at his disposal will enable him to go on for two days longer than if he spent eighteen shillings a day. How much money has he ? 17. Find two consecutive numbers such that the sum of the fifth and eleventh parts of the greater may exceed by 1 the sum of the sixth and ninth parts of the less. 18. The difterence of the squares of two consecutive numbers is 17. Find the numbers. SIMPLE EQUATIONS. 79 19. In an examination paper one boy A got three marks more than half of the full marks, and another boy B got six marks less than one- third of the full marks. The marks ob- tained by ^1 were twice as many as those obtained by B. What were the marks that each obtained ? 20. I rode one-third of a journey at the rate of 10 miles an hour, one-third more at the rate of 9 miles an hour, and the rest at the rate of 8 miles an hour. If I had ridden half the journey at the rate of 10 miles an hour, and the other half at the rate of 8 miles an hour, I should have been half a minute longer on the way. What distance did I ride ? 21. A train travelling at the rate of 37| miles an hoiu* passes a person walking on a road parallel to the railway in 6 seconds ; it also meets another person walking at the same rate as the other, but in the opposite direction, and passes him in 4 seconds. Find the length of the train. 22. A cash box contains three equal sums of money, one in sovereigns, one in shillings, and one in pennies. If the total number of coins is 1305, find how much money is in the box.<^r-^ 23. A person wishes to raffle a gold watch, and for that purpose sells a certain number of tickets. If he sell each ticket for 5s. he would lose £5, because the watch cost him more than he would in this case get ; but if he sell each ticket for 65., then he gains £4. How many-tickets did he sell ? 24. A father's age is equal to the united ages of his five children. In 15 years his age will be only one-half their united ages. How old is the father 1 25. A father's age is three times that of his younger son. In seven years' time he will be twice as old as his elder son, who is 5 years older than the younger son. What are their present ages? 26. The prices of the stalls, pit and gallery of a theatre are respectively ten shillings, half-a-crown and one shilling. The pit holds twice as many and the gallery three times as many people as the stalls. If the receipts are £90 when all the seats are full, find the number of people present. 27. A train travels for 35 miles, completing the whole dis- tance in 46| min. The first 11 miles are accomplished at a uniform rate ; the speed is then increased 25 per cent, and remains constant for the next 10 miles, after which it is diminished in the ratio of 14 to 15, at which rate the rest of the journey is performed. Find the original speed of the train. 80 PROBLEMS. 28. What number is that to which if 13 be added then one- third of the sum will be equal to 13 ? 29. A man leaves home with a certain sum of money in his pocket; he spends one-eighth of it in travelling expenses, one- half of the remainder in purchases, and the rest, amounting to 21s., he loses. How much did he start with? 30. A bankrupt pays a dividend of 7s. Qd. in the pound. If his debts had been £1200 more he would have paid only Q>s. 8d in the pound : what were his debts ? 31. In the same time A can do twice as much work as C, B one and a half times as much work as C. The three work together for two days, and then A works on alone for half a day. In what time could A and C together do as much as the three will have thus performed 1 32. A purse contains 28 coins which together amount to £7. A certain number of the coins are shillings, one-fifth of that number half-sovereigns, and the rest are sovereigns. Find the number of each. 33. At a cricket match the contractor provided dinner for 24 persons, and fixed the price so as to gain 12|^ per cent, upon his outlay. Three of the cricketers were absent." The remaining 21 paid the fixed price for their dinner, and the contractor lost Is. What was the charge for dinner ? 34. A cattle-dealer spends all his money in buying sheep, and sells them at a profit of 20 per cent. He spends the money which he now has in the same way, and gains 25 per cent. He again spends all his money, and makes a profit of 16 per cent. If the profit which he made upon the last transaction be £300, find how much money he had originally. 35. A has 6 more shillings than J?, but if A gives B half his money, and then B gives back to A one quarter of his increased sum, they will each have the same sum : find what each had at first. 36. At what time between 4 and 5 o'clock are the hands of a watch together ? 37. At what time between the hours of 10 and 11, will the hands of a clock be equally inclined to the vertical ? EXAMINATION QUESTIONS. [7%e following questions are taken from papers on Elem&atary Algebra set in recent years to the lower forms of various Public Schools under the authority of the Oxford and Cambridge Schools Examination Board.'] 1. Find the numericcal values, when a = 3, 6 = 2, c = l, of (i) a'^-\-bc-a; (ii) {a^b){a-b)-ac-\-b(c-a). 2. Add together 2a + 3b- 4c, 3a-2b + 3c, a-b, and 4a - c. 3. Subtract x-y'^+zx from xy-{-3x-\-y^. What is the co- efl&cient of x in the result ? 4. Solve the equation 2{x-l)-3{x-2) + 'i{x-3) = 4. 5. A father makes the following agreement with his son. Por every day on which the boy is among the first ten boys in his class he is to get three pence ; but if he is lower than tenth he is to give his father a penny. At the end of 12 days the boy has Is. 4o?. During how many days was he among the first ten ? 6. Define factor, term, coefficient, power, index. Given a = 3, 6 = 2, c= - 7, find the value of (i) 3a^b - 5ab^c - b^c^ ; (n) t-ttt • ^ ^ ^ ^ a a-c 6 + 2c 7. Remove the brackets from the expressions (i) 4a + {a-(3«-26) + 26}: (ii) 7 [3a -4 {a -6 + 3 (a 4- 6)}]. 8. Multiply ^y^+^xy-x"^ by 3x^{x-y)-2xy(x+y)+y\ 9. Divide x^-{a-b)x — ab by x-a. 10. When A and B sit down to play, B has two-thirds as much money as A. After a while A wins 15.?., and then he has twice as much money as B. How much had each at first ? B. A. 6 82 EXAMINATION QUESTIONS. 1 1 . Subtract - ±%^ + ^x^y + bf - ^xjf- from 4.ry2 _ ^ _ ^3 ^ \x^y. 12. Simplify the following expressions : (i) 2a- (6 -c) -(3a- 26- c) -(a- 36 -4c); (ii) - 2 [^- 3 {^- 4 (^^- 5.r- 1)}] ; ^-2^4-3_2.v-3y-f4 2^-y + 12 (111) g —^ + ^ . 13. Enunciate the rule of signs in multiplication. Multiply 3a2-a6 + 262 by Zd^^-cib-W--, and verify your result when a = l, 6 = 2. 14. Solve the equation 2^-+l 4^- -3 5a---H 3 ~~b~ 6 = -6. 15. A coach travels between two places in 5 hours ; if its speed were increased by 3 miles an hour, it would take Z\ hours for the journey: what is the distance between the places? - 16. If ^ = 0, y= - 1, 2; = 2, find the values of ^.3 +y3 + .3 _ -^xyz • (^^2 _ ^2) ^ (^2 _ ^;^) . sl%if-Zyz^\\x^y-\-^z\ 1 7. Simplify (i) 7a. - [2a -f 3 (a + 1 )] ; (ii) 3.r - [2^ - 3 {x - {x + 1 )}] ; (iii) (a-6)[(a + c) + {6-c}]-a2 + 6(6 + c)-cx6. 18. Form the continued product of a + 6, a + 26, 2a + 6, and a — 6. What are the dimensions of the product ? 19. Distinguish between an equation and an identity. Solve the equations : .. x+Qa 2x-3a_5x 7x-\-6a ^^^ "~3 f~ ■" "6 14~ ' (ii) 5^(^^ + 3)-7(^^ + 2)(a^-11) + 2(^2_3^, + 7)^0. 20. In an examination paper one boy. A, gets 8 marks more than the third part of full marks, while another, B, gets 11 marks less than the half of full marks. The marks obtained by A are eight-elevenths of those obtained by B. Find the full marks for the paper. EXAMINATION QUESTION'S. 83 2 1 . Multiply a + 6 by 2a + 36 ; Zd^ + 2(< 6 + 6'^ by a^ -2ab-b'^; cind a + b-chj a—b + c. 22. Divide 6cd)c by 2a ; a^ - x^ by a — x^; and 3.i;'-^-4/-3s2_4^.^^3^.~^8y2 by ^•-2?/ + 35. 23. Prove thcat 24. Solve the equation 3.1--8 _ x-l 1-x _ 8-2x _ 8-5^- 5 4 ■*" 3 ~ 6 ~T0~' 25. A piece of work could be done by 240 men in 20 days, but when it is half completed 144 workmen strike, and the work has to be finished by the remainder. How many days' delay are thus caused ? 26. Multiply (i) x"^ + 2.v - 1 by x + 2; and (ii) multiply together a + b — Cj b + c — a, and c+a-b. 27. Divide the sum of ^+17, 14.^'^-18, and .^•2 + 15 by 10 + ^-; and test your result by putting x=2. 28. Divide 4:X^-8x^ + x^+x^ + 4: by 2x^-x-2; and (a2 + 62)3-(2a6)3 by {a-b)\ 29. Shew that a term may be moved from one side of an equality to the other side, provided its sign is changed. Solve the following equations : (i) {.V - 3) {2x + 5) - (.r + 4) {7x + 1 ) = (.r + 2) (3 - 5^^) - 48 ; (ii) (5^ + 3)(7.^-4)-(3.r + 5)(8.v-ll) = (ll^-2)(.*;+l). 30. A man walks up a mountain at the rate of two miles an hour and down again by a way six miles longer at the rate of 3-| miles an hour. He is out eight hours altogether. How far has he walked ? 6—2 84 EXAMINATION QUESTIONS. 31. Simplify (i) a(26 + 3c)-[c(2a+6)-6(c-2a)]; (ii) 2\x-a-^ {x -^{x-bx — a)- a}]- Explain what is meant by a coefficient. Write down the coefficient of x in the second of the examples in this question. 32. Multiply ^2-862 by ^2+862; subtract 176* from the result, and then divide by a - 36. 33. Divide Sa^ + 7^3^ _ 42^^252 + 47^53 _ 1454 by a? + iSab - W. 34. li x—y = 2a, shew that x^ — 6ax + dw^ = (y - «)^- 35. A man starts to walk from A to B at the rate of 4 miles per hour, and on the way is overtaken by a dog-cart ; if the cart had started an hour later than it did, he could have walked 8 miles further before he was overtaken : find the rate at which the dog-cart travels. 36. If a = 0, 6 = 1, c = 2, d=3, find the numerical value of {2abc-{-2bcd)^a^bc+c^bd+3. 37. Multiply ^2 + (^2- 1)^'-1-1 by x^-{^2 + l)x + l. 38. If a = ^'2 -^2, 6 = ?/2 _ 2^^ c = z^-xy\ prove that ax ■\-by-\-cz={cfi — bc)\x. 39. If a-i-6 + c + cZ=0, prove that a^ + 63-f- c3 -I- c^^ = 3 Q)cd -f cda -f- dab -\- ahc). 40. The gross income of a certain man was £40 more in the second of two jmrticular years than in the first, but in con- sequence of the income-tax rising from 4o?. in the pound in the first year to Qd. in the pound in the second year, his net income (after paying income-tax) was unaltered. Find his income in each year. CHAPTER VII. FACTORS. 102. We proceed now to the consideration of some of the simpler algebraical processes. This chapter is devoted to the discussion of factors. Our investigations will often be greatly facilitated if we can resolve a given expression into factors. This cannot always be effected, and we shall here confine ourselves to a few expressions where it is possible to obtain the factors by inspection. We shall consider in succession (i) expressions which have a factor common to every term, (ii) certain known forms, (iii) quadratic expressions, and (iv) lastly, a general theorem which enables us to tell by inspection whether any given expression of the first degree is a factor of another expression. 103. Factor common to every term. If a certain letter or quantity divide every term in a given expression, it will divide the whole expression ; and it will therefore be a factor of the expression. In general we confine our attention to literal factors. All the numerical factors are usually combined into a single numerical coefficient, which is prefixed to the rest of the expression, but is not reckoned as a factor of it. 86 FACTORS. Ex. 1. Find the factors of la^-^ax. Here a is a factor of each term, .-. la'^-Zax = a{1a-Zx). Therefore a and la - 3x are the required factors. Fx. 2. Find the factors of 1 5abh^ - 1 2a^bc^ - 2 1 ac\ 15a6V - Ua^bc^ - 21ac* = Sac^ (5b'^c - 4:a^b - Tc^). . Fx. 3. Find the factors of a {2x - 3) + 6 (2x - 3). a(2^-3) + 6(2^^-3) = (a + 6)(2^-3). IFx. 4/ Resolve Aax -I2bx- Qay + 1 86y i?i^o factors. "3a^ - 126.r - 6ay + 186y = 2 {2,ax - Qbx - 3a?/ + 96y} = 2 {2^ (a - 36) - 3y {a - 36)} = 2(2.r-3y)(a-36). This last example is less obvious than the others, but we shall see later how we can obtain the required factors in another way. EXAMPLES. VII. A. Resolve the following expressions into factors. 1. ax+bx. 2. 5.^-20^2^ 3. a^-Mb. 4. Zlm^-^lhn. 5. x'^y + xy^ + xy. 6. 1 \p^ — '^p^q - ^pq^. 7. 8xyh'^+Ux^yz'^-16x^z^. 8. a{x-y)-b{x-y). 9. ac^-ad+bc-k-bd. 10. x'^ + lx-Zx-U. 11. ax->ray + az+px-\-py+pz. 12. ay2^(^_ i)y_2^ 104. Known Forms. We proceed next to the consideration of certain known forms, and under this head shall discuss in succession (i) expressions which can be written as the difference of two squares; (ii) trinomial expressions, which are perfect squares; (iii) certain expressions of the third and fourth degree. FACTORS. 87 105. DIFFERENCE OF TWO SQUARES. We have already proved [Art. 61, Ex. 3] the formula a-— b^=z(a — h) {a + b). Hence the difference of the squares of two expressions can always be resolved into factors. Ex. 1. Resolve .r^ — 64 into factors. .r2-64=.r2-82 =:Gv-8)(.r + 8). Thus .r — 8 and x + 8 are the required factors. Ex. 2. Resolve 49.t'2- 64?/^ into factors. 4a^•2-64y2 = (7.^^)2_(8y)2 = {1x-^y){1x^^y). Thus lx—d>y and lx-\-Sy are the required factors. Ex. 3. Resolve 100 - 9 zw^o factors. 100-9 = 102-32 = (10-3)(10-f-3) = 7xl3. Thus 7 and 13 are the factors of 91. Ex. 4. Find the factors of x'^-Z. ^^2 _ 3 =.^2 _ (^3)2 = {x-JZ){x-\-^Z). That is, x-jZ and .r + v'3 are the required factors. A factor involving the square root of x is not a factor such as we want, but ^3 is a number and may properly form a term in a factor. Similarly, if a be a given quantity or a number, then the factors of x^ — a are x - sja and x + s]a. 106. The squares of compound expressions are subject to the same rule, for a and 6 in Article 105 may stand for any expressions. Ex. 1 . Find the factors of {%v -\-yf- {2y - xf. {2x +y)2 - (2y - xf = {{2x+y) - {2y - x)} {{2x ^y) + {2y - x)] = (3.r-y)(.^ + 3y). Ex. 2. Find the factors of {x^ +1)2- x^ (x'^ + l)^-.v'^ = {(x^-^l)-x}{{x^ + l)+.'^, that is, x*-\-x'^ + l = {x^ - x+l) (x^+x+l). 88 FACTORS. EXAMPLES. VII. B. Write down by inspection the factors of the following expressions, each of which is the difference of two squares. 1. a2-l21. 2. 81- ■144. 3. xY-l2l. 4. 1-6462. 5. ^2- 1. 6. ^2_5, 7. 4 -^^2, 8. X^- 4. 9. 9-4^2. 10. 121^2 -25m2. 11. Sljo 2-a2?,2, 12. a2J2_^2^2. 13. aW-xhf. 14. ^•2- 144^2^12. 15. .^-144m2^^2^ 16. a^-h\ 17. 16.t^' i-81a4. 18. ^a^^c^-dxA 19. {x+yf-l^l. 20. (a-6)2-9c2. 21. ^cc+yf-{x-y)' 2^ 22. (a2 + 62)2-a2^ 23. (a2_62)2_52. 24. («4-6 + c)2-(a + &-c)2. 25. (3.r-4y)2-(4^- -^yf 26. a2-4(a-6)2. 27. (2a- 1)2- (a -f 1)2. 28. a^-{x^ + 2xy+f). 29. {l + my-{m-\-n] )K 30. (2.^' + l)4-(^-2)4. 107. PERFECT SQUARES. We know [Art. 61, Ews. 1, 2] that a2 4- 2ab -^¥ = (a + by, and a- — 2a6 -\-¥ = (a — by. Thus any expression which can be put in either of these forms (where a and b may stand for any compound ex- pressions) can be resolved into factors. These examples can however generally be solved more simply by the methods given later. Arts. 118,114, and Chap. xv. Ex. Fi7id the factors of 4x^ - 4^ + 1 . The first term is the square of 2x, and the last term is the square of 1. Hence, if the given expression be of one of the above forms, it can be written {2xf-'2{2x) + l. This is the case, .*. the given expression = (2.1- — 1)2. FACTORS. 89 EXAMPLES. VII. 0. Find the factors of the following expressions, each expression being a perfect square. 1. .r2 + 2a; + l. 2. x^-Qx^-^. 3. 4'6'2-12.a'?/ + %2. 4. x^-Qx'+lQ. 5. a^-Mb + lVK iS) (a + &)2-4(a + 6)(a-6)+4(rt-6)2. 108. EXPRESSIONS OF THE THIRD DE- GREE. The factors of the following expressions of the third degree are known. a=^ - 6^ = {a - h) (rt^ -f a6 + 6') a=^ + Sa^i + 3a6- + 6^ = (a + 6)=^ a^ - Zci?h + 3a62 - 6^ = (^ _ 5)3 a^ -{- }f + c^- Sahc = (a+b -\-c)(cv'+b'^+c^-ah-ac-hc). Hence an expression which can be written in any one of these forms can be resolved into factors. Ex. 1. Fiiul the factors of ^-27. ;z;3_27 = :r'5-33 = (.i'-3)(^2 + 3^^ + 9), Ex. 2. Find the factors of {2x - yf -f {x - 2yf. This expression is of the form a^ + 6^, hence, it = {(2^; -y) + (x - 2y)} {(2^ - yf - {2x - y) {x - 2y) + {x- 2yf}. Simplifying, = {Zx - 3y} {3.2;2 _ 3^^ 4. 3^2} = ^{x-y){x'^-xy-^y'^). Ex. 3. Find the factors of ^a^ -\- 1 2^2/, + 6«7>2 4. 53_ %a^-\-\2a%^-Qa¥ + h-^ = {2af + ^{2afh + Z{2a)h'^ + h^ = {2a + hf. 90 FACTORS. EXAMPLES. VII. D. Find the factors of the following expressions. 1. .%^-l. 2. a^ + \. 3. d,aH^-\. 4. {^x-yf-{x-2yf. 5. a^ + ^x^ + Zx-\-l. 6. l-33/ + 3?/2_y3, rj x^ + a^+l-^ax. 109. EXPRESSIONS OF THE FO URTH DE- GREE. The factors of the following expressions of the fourth degree are also known. a'-h' = {a" - h') (a' + h') = {a - h)(a + 6)(a2 + 6^) a' + 4>a% + Ga^' + 4fa¥ + 6^ = (a + by a' - 4a=^6 + Qa%'' - 4a63 + h' = (a- by a' + a^b^ + b' = (ct^ - ah + b%a' + ab + ¥). Hence an expression which can be written in any one of the above forms can be resolved into factors. Kv. 1. Find the factors of x^-{^yy. .r* - (2y)4 = {^2 _ (2y)2} {.^.2 + (2j/)2} = Gr-2y) (^' + 2y).(.^•H4y2). Ex. 2. ^i?ic^ i;Ae /acifors o/ 81 a^ + 1 44a262 + £566*. 8 1 a-t + 1 44^262 + 2566* - {Zaf + (3a)2 (46)2 + (46)^ = {(3a)2 - (3a) (46) + (46)2} ^(3^)2 + (3^) (4^,) ^ (4^,)2| = {9a2 = I2a6 + 1662} {9^2 + 12^6 + 1662}. EXAMPLES. VII. E. Find the factors of the following expressions. 1. l-a^. 2. .1^-16. 3. (3.r + 4?/)4-(A' + 2y)4. 4. l+4a + 6a2 + 4a3 + a4. 5. y^- 4?/3 + 6?/2 - 4j/ + 1 . 6. (a + l)H(« + l)2(6-l)2 + (6-l)4. FACTORS. 91 110. The beginner may tind the following table, containing the foctors of some of the forms above considered, useful for reference. A and B may stand for any expressions. A'^-B'^ = {A-B){A+B). -42 + i?2 j^ag 1^0 real factors, A^±B'^ = {A±B){A'^^AB + B''^). A^-B^ = (A-B){A+B){A^+B^). A'^±2AB + B'- = {A±Bf. A^±3AW + 3AB^±B^ = {A±By\ 111. It may be noticed that, if we can resolve an expression into factors, the result of dividing it by one of those factors is obvious. This is sometimes useful if the expression be of one of the forms given in the last article. Ei\ Divide {.v - y )^ + (.r - zY hy 2,v —y — z. We know that {x-y)^^{x-zf = {{x-y)^{x-z)]\{^x-yf-{x-y\x-z)^{x-zf\ = {2x-y-z]{{x-yf-{x-y){x-z)^-{x-z)\ Hence the result of dividing the left-hand side by 2.^-3/ -^ is {x - yf - {x - y) {x -z) + {x - zf, which reduces to x'^ — xy — xz+y'^—yz-\-z^. EXAMPLES. VII. F. The results of the following examples can be written down by inspection. 1. Divide {%vf + {^f by 2x + Zy. 2. Divide (3a -6)4 -(2a -6)4 by 5a -26. 3. Divide (4r + 3y - 2^)2 - (3^ - 2y -}- 82)2 by x + by-bz. 4. Divide {Zx-ir2y+zf^-{x-\-2y-\-Zzf by x+y-{-z. 5. Divide (2« + 36 + 4c)3-(a + 6+c)3 by a +26 + 3c. 6. Divide {ac-hdf-{hc-adf by (a-6)(c + o?). 92 FACTORS. 112. Factors of Quadratic Expressions. We proceed next to the consideration of the factors of quadratic expressions. A quadratic expression in x is one which is of the second degree in x, and therefore is of the form px"" -\-qx ■\-r, where p, q, r are any numbers, positive or negative. The factors of such an expression (if they exist) can always be found. Before proceeding to the general rule, we shall consider a few cases where they are obvious by inspection. 113. We know, by multiplication, that x"" -\-{a + h) X + ah =^ (x -\- a)(x + h). Hence, if we have to factorize an expression like x^+ qx-\- r, and we can guess or find two numbers, a and h, such that their sum is q and their product is r, then the required factors will be « + a and x-\-h. For, \i a-\-h = q and ah = r, then X- j^ qr -\- r = X" -\- {a -\-h) X + ah = {x -\- a) {x -\r h). Under the same conditions, x^ — qx + r — x^— {a + h) X -\- ah = (x — a) (x — h). These results can be expressed in one formula, thus : x^ ±qx + r = x~ ±(a-[-h)x + ah =(x ±a)(x ± h), where, in the ambiguity +, either the + is to be taken throughout, or the — is to be taken throughout. Ex. 1 . Find the factors of x'^ + ^x+12. We want to find two numbers whose product is 12 and whose sum is 8. Now the only pairs of positive integral factors of 12 are 12 and 1, G and 2, 4 and 3; and the only one of these pairs whose sum is 8 is 6 and 2. Hence .r2 + 8^ + 1 2 = (.t? + 6) (.^^ + 2). Similarly, .r^ - 8.j«7-|- 12 = {x- 6) {x - 2). FACTORS. 93 Note. An exactly similar rule enables us to determine the factors of x^ + Sxy + 12?/'-^ and x^ - 8x1/ + 12^'^, namely, x^ + 8xi/ + l2f={x + Qy){x + 2//\ and x^ - 8x7/ + 1 2f = {x - 6^) (x - 2}j). Ex. 2. Find the factors of a--lZa + \2. We want two numbers whose product is +12 and whose sum is - 13. These are clearly - 1 and - 12. Hence we have a2- 13a + 12 = (a- 1) {a- 12). Ex. 3. Fiiid the factors of x'^ 4- 1 Ixy + QOif. We want two numbers whose product is +60 and whose sum is +17. These are clearly 5 and 12. . • . x'^ + nxy + 60/ = [x + 5?/) {x + 1 2y). Ex. 4. Find the factors of x'^ - Qx + 9. We want two numbers whose product is + 9 and whose sum is - 6. These are - 3 and — 3. ... x'^ _ 6.C + 9 = (^ - 3) {x - 3) = {x - 3)2. This example might have been treated as a known form [Art. 107]. 114. In the cases considered in the last article, the term independent of w, which is often called the absolute term, (i.e. r), was taken as positive ; and thus the numbers a and b were of the same sign. But had 7' been negative, they would necessarily have been of opposite signs : in such a case, we want to find two numbers, a and 6, so that the given expression may be the same as x- -\-(a — b)x — ab, that is, as (x + a)(x — b). Thus the coefficient of x must be equal to the difference of the two numbers selected. For example, to find the factors of x^+x-l2, we want to find two numbers whose product is 12 and whose difierence is 1. The only pairs of integral factors of 12 are 12 and 1, 6 and 2, 4 and 3, and the only one of these pairs whose difference is 1 is 4 and 3. Hence x^ + x -I2 = {x + i) {x - S). Similarly, .r^ - ^ - 1 2 = (^ - 4) (^ + 3). If however we regard the rule given in Art. 113 as referring to an algebraical sum and an algebraical 94 FACTORS. product, it will cover all the cases. The only practical points to be observed being that, if the term inde- pendent of X be positive, the numbers found must be of the same sign, and the sign of each of them must be the same as the sign of the coefficient of x ; but if the term independent of x be negative, the numbers found must be of opposite signs, and the sign of the greater number must be the same as the sign of the coefficient oi X. Ex. 1. Find the factors of x^ — x — %). Here we want to find two numbers whose product is 20 and whose difference is 1 ; that is, using algebraical notation, whose product is - 20 and whose algebraical sum is - 1. The fact that the product is negative shews that the numbers must be of opposite signs, and since the sum is negative the one which is arithmetically the greater must be negative. The required numbers are -5 and +4. .-. x^-x-20 = {x-b){x-\-4.). Ex. 2. Resolve 1 + 6a — 7a^ into factors. Here we want two numbers whose product is - 7 and whose sum is + 6. These are — 1 and + 7. .-. l + 6a-7a2 = (l-a)(l + 7a). EXAMPLES. VII. G. Write down the factors of the following expressions. T. .t;2 + 9^ + 18. 2. a2_7^ + i2. 3. l2 + ly^-y\ ^'~^ ^2_i9^. + 88. 5. n'^-Un + 4Q. Q. y'^-l^y + M. Tf) a^-bah-\-Qh\ 8. ^y-29A'y + 54. 9. l-lZl + 22l\ 10. 22^-4 + 1^^2 + 1^ ^ a? - 19«6 + 8862. 12. a'2 - lla'h' + 666'2. 13. p^-Up + l^. 14. 1-8^-84^2. -^ .^'2 + ^_2. i^ a2-a-2. Cth ^2_ar-20. ^^) y2_^_42. 't^i a2 ^ 3^5 _ 2862. 20. \-bm-Mm\ ^^ x'^-Qx-ZQ. (^ 62^5 _ 42. 23. ^-9^'2_9o. ^[4; a2 + ^_110. f5. :r2-.r-42. ^1 x'^ + Qxy-lQ7f.C^. y^-\\y-lO± '28: .^'2-12J^;-85. 29. a^^-lQa^-bl. 30. a2^2 + ii^^_242. FACTORS. 95 115. Method of luriting a quad7^atic expression as tlie diffh^ence of two squares. We know that the factors of a- — ¥ are (a — 6) and (a + b). If therefore we can write a quadratic expression as the difference of the squares of two compound expressions we can resolve it into factors. To write a quadratic expression as the difference of two squares, we try to find some quantity which when added to the terms involving x^ and x will make a perfect square. Adding that number to the terms involving x^ and x, and at the same time subtracting it from the rest of the expression, will make no difference to the value of the expression ; and if the result of this latter subtraction be a negative quantity, we can apply the rule. In the following examples the coefficient of .r^ is unity, and in such cases the required number (which we add to the terms involving x^ and a;, and subtract from the other terms) is the square of half the coefficient of .v. Kv. 1. ^Har + 12 = (.^2 + ai' + 16) + 12-16 = (^• + 4)2-4 = {(^' + 4)-2}{(^^ + 4) + 2} = G^ + 2)(.^•^-6). E.v. 2. x-H ^^■ - 20 = .1-2 + x' + (1)2 - 20 - 1 ={(^+i)-f}{(^+i)+t} = (^-4) (^ + 5). *Fx\ 3. Similarly ^•2 + 2a- - 1 = (.r2 + 2.V + 1) - 1 - 1 = (^•+1)2-2 = (A^+l-V2)(^+H-x/2). These factors are rational so far as x is concerned. *116. If we want the factors of j^^'^ + qx + r, we can write the expression in the form p Ix"^ -{•- x -{- ~] . The 96 FACTORS. factors of the terms in the bracket can then be obtained as described above, and thus the factors of the given expression are known. For example, to find the factors of 2^;^ — 5a' + 2. 2jt;2 _ 5^^ + 2 = 2 {^-2 _ |.^, ^ i| = 2{^2_|^+(5)2+l_|5} =-2{(^^-f)-|}{(^-|) + |} = 2{^-2}{^-i} = (^-2) (2^-1). *117. Lastly, consider the case of any quadratic expression, such as x^ -\- qx + r, where q and r stand for any numbers whatever. The form ax^-\-bx+c is reducible to this form by the method given in the last article. The result of adding (^g)- to x- + qx is x'^ -{-qx + Jg-, which is the square of (x — ^q)'\ Hence x"- + qx -\- r = [x^ + qx + {^qy} - iq'' + r = (x + iqf - (jq' - r) = {x + lq- Jlf -r\{x + ^q + Jlf-r\. This is a general formula which includes all the cases given above ; and if we write for q and r the values which they have in any particular expression, we obtain its factors. For example, consider the expression x^ + 2x-l given above ; here q=2, and r=l, and we at once obtain (^^ + 1 - ^2) and {.v+l+/^2) as the factors required. Similarly, to find the factors of x^ + ^-x+l, put q = ^^- and r = l, and we find that the factors are / ,. /81 X 49\ , / ,. /81 X 49\ that is, are (.t' + |^) and (^ + 8). *118. In the analysis given in Art. 117, we took the square root of iq"^ — r. This is always possible (though the result may be a surd whose value cannot be exactly FACTORS. 97 found) provided \q^ - ?• is a positive quantity. If how- ever \q- — r be negative, we are landed in a difficulty, since it is impossible, by any rules given in arithmetic, to take the square root of a negative quantity. In fact, a negative quantity, no less than a positive one, when mul- tiplied by itself gives a positive quantity; and hence no real quantity (whether positive or negative) when squared can be a negative quantity. Thus it is only possible to find real factors of x^ + qx + r when iq^> r, that is, when q- > 4r. *119. For example, if we want to find the factors of x'^ + 2x + 2, we put it in the form (A''' + 2.r + l) + 2- 1, that is, (^' + 1)^ + 1 ; and we can proceed no further. If we choose to write this result as (a'+ 1)" — «, where a stands for —1, we can express this as (^- -f- 1 — V^) (^ -f 1 + \/a)- The product of these factors is {x +l)^ — a, that is, (jc + l)'^ + 1, provided a is supposed to stand for — 1. A quantity like \/( - 1) is unknown in arithmetic ; but if we define it to be a new kind of quantity, which is such that its square is - 1, then we can find the factors of a:^-\-2a^ + 2. A quantity, such as ^/(-l), is said to be imagi7iary, and expressions involving such quantities are called imaginary. We shall not concern ourselves in this book with the meaning or the properties of imaginary quantities ; but as the student will find them occur constantly in the more advanced parts of mathe- matics, it is desirable to call his attention to their existence. It may be added that they are not the merely artificial quantities which would be suggested by the arithmetical interpretation of them given above. There are other branches of mathematics, besides the arithmetic with which alone the student is here supposed to be famihar, and a study of those Ijranches enables us to assign a definite, intelligible, and useful meaning to a quantity like sj{ - 1 ). EXAMPLES. VII. H. Write down the following expressions as the difi'erence of two squares, and thence resolve them into factors. 1. ^2_^_2. 2. }/-1y + l2. 3. ?i2-24w + 95. 4. a2 + «_42. 5. ^2_2Lr-|-104. 6.1-126-8562. 7. a^-\\ah-2W-. 8. ^.^ -f- 8.^2 4- 7. 9. 98a2_7a6-62. 10. 10.^2 + 79^-8. 11. Ila2-|-75a6-1462. 12. Uy^'^-^byz+QzK *13. ^'2 + 1. *14. x'^-'^x + b. n5. .i'2 + 3.i--f3. B. A. 7 98 FACTORS. 120. It is always possible to tell whether a quan- tity, such as ^ - a, is a factor of a given expression by dividing the expression by it ; because, if there be no remainder, then x — a is a factor of the expression. The same conclusion can however be obtained at once by the aid of the following theorem. Theorem. If an algehraical expression involving X vanish ivhen a is put for x wherever x occurs, then the expression is exactly divisible by x — a. Let us denote the given expression by X. If we divide it by x — a, we shall get a certain quotient, which we will denote by Q, and a remainder, which we will denote by R. Hence, X = Q(x-a)i-R. Now this equation is true for all values of x, and therefore will be true if we put a for x wherever x occurs in it. This will leave R unaltered ; for R does not contain x, since if it did we could continue the division. Also, by hypothesis, the effect of putting a for ^ in X is to make X vanish. Hence, putting a for X, we obtain = -hR. Therefore R = 0, that is, there is no remainder, and .•. X — a h Si factor of X. Note. The reader will remember that we are here confining ourselves to algebraical expressions in which all the numbers denoting powers of x are positive integers, and in which no roots of quantities involving x are involved. Such expressions are said to be rational and integral. It is only of rational integral algebraical expressions that the above theorem is necessarily true. 121. For examjDle, x^-1x-\-^ vanishes if x be put equal to 1. Therefore x—\ is a factor of it. Dividing by .t'— 1, we find that the other factor is .2; - 6. Again, .^'* - bx^ — lofl + 5.>; + 6 vanishes if x be put equal to -1, since it then becomes 1 + 5-7-5 + 6, .*. it is divisible FACTORS. 09 by A'+l. Dividing by .r + 1, we find that the quotient is x^-Qx^-x-\-Q. Again, this latter expression vanishes when x is put equal to -1, .-. .I'+l is a factor of it. Dividing by x+\, we find that the quotient is x'^-lx-\-Q. The factors of this last expression are x - 1 and x - 6. Hence the given expression = {x + \){x + \){x-\){x-Q) = {x^-lf{x-\){x-Q). 122. As another example of this theorem, we will take the following important application to determine w^hether .^" ± a" is divisible by x-\-a or hj x-a^ where n is any positive integer. (i) To see if .r"-a" be divisible by x-a we put x=a in it. It then becomes a^-a"- which obviously is zero. Therefore ,^»i_ Qj»i {g always divisible hy x-a^ whatever integer n may be. (ii) Again, to see if .t-"-(X'* be divisible by x-\-a^ we put ^=- a in it. It then becomes (-a)'^-a^ If n be even, this is equal to «"■ - «"■ [Art. 59], which is zero. If n be odd, it is equal to -a" -a'* [Art. 59], which is not zero. Therefore x^ — aJ^ is or is not divisible by x + a, according as n is even or odd. (iii) Next, let us see whether .r'^ + a" is divisible by x-a. If it be divisible by ^ - a, it must become equal to zero when X is put equal to a ; but if x = a it becomes a'^ + a", which is not zero. Therefore ^'"■ + «"■ is never divisible by x-a. (iv) Lastly, to see if ^^» + a" be divisible by x + a, we put X equal to -a. Now if x=-a, A-'^ + a'^ becomes (-«)" + «"• If n be odd, this is equal to -«'' + «'* [Art. 59], which is zero; but if n be even, it is equal to a'^ + «"-, which is not zero. There- fore ^'* + a" is or is not divisible by x + a, according as n is odd or even. EXAMPLES. • VII. I. 1. Determine by inspection whether x—\ is a factor of each or of any of the following expressions ; and if so, find the other factors. (i) a^ + ^x^-x-2', (ii) x'^-Qx + b; (iii) x^-4:X+2. 2. Determine by inspection whether x + 1 is a factor of each or of any of the following expressions ; and if so, find the other factors. {[) x^ + ox-^ + lx + S', (ii) A-^-3^'-' + l ; (in) x-^-x^ + U- + 6. 7—2 100 FACTORS. 3. Each of the following expressions has either ^ - 1 , .r + 1, or x-2 as a factor. Determine all the factors of each expression, (i) x^-2x'^-x + 2\ (ii) x^-lx^ + Qx) (iii) ofi + {a-Z)x'^ + {^-'^a)x-\-2a', (iv) x^-\-bx'^ - ^x-^0 ; *( v) x^ + 3.^'3 - 3,^2 - 1 1^ - 6. MISCELLANEOUS EXAMPLES ON FACTORS. VII. J. Resolve into the simplest possible factors the following exj)ressions, numbered 1 to 18. 1. .2^3-36^. 10. a^ — ya? + z^a — z^y. 2. (2.r + 3)2-(^--3)2. 11. (^2 + 4a) -(62 -46). 3. {x-2yf+y\ 12. 6^2 + 7or^ _ 2^^ - 3a2 _ 3a. 4. 72a^«-/. 13. a2-62 + 86c-16c2. 5. (3^2 _ 6-2)2 _ (^2 _ 3^2)2. 14. a2 + 62 + 2(a6 + ac + 6c). 6. aW-a?-h^-\-l. 15. 9a2 + 6a6-16c2-86c. 7. y3+^2_^_l_ 16. 72(^2_i)_i7^,_ 8. x\x+yf-{x^^+yy^. 17. (a + 26 + 3c)2-4(a + 6-c)2 9. 9a2 + 6a6-4c2 + 4c6. 18. (.^2^4^,)2_2(^2 + 4^,)_15_ 19. Resolve into four factor s x^- -6.^2+1^ 20. Resolve into factors of the first degree 21. cfi (6 + c) + 62 (c + a) + c2 (a + 6) + 2a6c. Prove that a* + wia262 + 6^ is divisible by o?' + ah sj^m + 62, and find the other factor. Resolve a^ + 6^ into two factors. 22. Find the continued product of x'^-\-ax-^a^^ x^ — ax + a^, x"^ — a~x^ + a* ; and deduce (without division) the quotient of x^ + l6a'^x^ + 266a^ when divided by A'2 4-2a.r + 4a2. 23. Find the factors of {x^ -{y- zf) {y2 - (^ - ^•)2} {^2 _ (^ _ y^^ *24. Prove the following identity a3(c-6) + 63(a-c)+c3(6-a) = (a-6)(6-c)(c-a)(a + 6 + c). Find the factors of the expressions, numbered 25 to 27. *25. (6c + ca + a6)2 - (62c2 + 02^2 + ^262), *26. (a - d) (62 - c2) + (6 - d) (c2 - ciF) + {c-d) {a' - 62). *27. (a-f6)(a + 26)(a-|-36)-96(«-f-6)(a + 26) + 1862(a + 6)-663. CHAPTER VIII. HIGHEST COMMON FACTORS. 123. The Highest Common Factor of two or more algebraical expressions is the expression of the highest dimension which will exactly divide each of them. The letters H. C. P. are often used as an abbreviation for highest common factor. Some writers call this factor the highest common divisor, and denote it by h. c. d. ; others call it the greatest common measure, and denote it by g. c. m. The latter name is properly used in arithmetic, but in algebra our symbols stand for any numbers, and we cannot with correctness speak of one symbol being greater or less than another. 124. We shall first consider the rule for finding the H. c. F. of simple expressions and of expressions which can be resolved into factors on inspection, and then the rule for finding the H. c. F. of compound expressions of which the factors are not obvious. 125. Rule for finding the H. C. F. of Simple Expressions. The h. c. f. of two or more simple expressions is the product of every factor common to them all, each such factor being raised to the lowest power which it has in any of them. This is obvious from the definition, and is illus- trated in the accompanying examples. If the expressions have numerical coefficients, it is usual to find the G. c. M. of these coefficients, and prefix it to the H. c. F. as a numerical coefficient. 102 HIGHEST COMMON FACTORS. ^ Be:. ,1. Fi7zd the h.c.f. o/a*, a^, and a^. The H. c.F. required must be a power of «, and obviously is a^. Ex. 2. Find the h. c. f. of 4x7/^ and 2x^^. Here ^' will divide both quantities, but no power of x higher than the first will divide both. Similarly, ^ will divide both quantities, but no power of ?/ higher than the first will divide them both. Lastly, the g. c. m. of 4 and 2 is 2, Therefore the H. c. F. of the given expressions is 2xy. Ex. 3. Find the h.c.f. of 6ab^chl\ da^^cd^ and -Za%^c\ The only letters common to all these expressions are «, 6, and c. The first expression involves a to the first power only, and therefore is not divisible by any higher power of a. Similarly, 6^ will divide them all, but any higher power of h will not divide the first or second of them. Similarly, no power of e higher than the first will divide them all. The g.c.m. of 6, 9, and 3 is 3. Therefore the h. c. f. of the given expressions is Zah'^c. EXAMPLES. VIII. A. Find the h. c. f. of the following quantities. 1. 3a62c3, 4a56c3, Qa¥c\ and ba'^hh'^. 2. Ibx^yh'^, -20x^yH\ haP-yh^, and - 25^V^2. 3. 16jo"g8r9, 12j9Yrs6, and -22p^qh^t^. 126. Rule for finding the H. C. F. of Com- pound Expressions which can be resolved into factors of the first degree. This case is covered by the rule just given; since, if we place a factor of the first degree in brackets, we may treat it as if it were a simple quantity. Hence the required H. C. F. is the product of every factor common to the expressions, each such factor being raised to the lowest power it has in any of them. Numerical coefficients are treated in the manner ex- plained in Art. 125. Ex. 1 . Find the h. c. f. of 6 («2 - ^2) and 9 (a - h)\ The given expressions must be resolved into factors of the first degree. They then become respectively 6 (a -6) (a +6), and Q{a-hf. The H. c. F. required is 3 (a - 6), since a — 6 is the only factor common to the two expressions, and the g.c.m. of 6 and 9 is 3. HIGHEST COMMOX FACTORS. 103 Ex. 2. Find the H. c.F. of (^2_a2)2^ (.r-a)3, and {x-aY{x-h)\ The first of these expressions can be resolved into factors, {x— a)- (.r+a)-. All the quantities are now expressed as products of ftictors of the first degree. Their h.c. f. is {x-aY^ since it is clear that the only factor common to them all is a power of {x - a), and no power of it higher than the second will divide them all. Ex. 3. Find the h. c. f. of 6a46 + 6a362-36a263 and 9 {aW--^4.a'^h^-\-^a¥). Here Qa'^h + 6a^b'^ - 36a^b^ = 6^26 {a - 2h) (a + Sh), and ^{a^b^ + 4a'-b^ + 3ab^) = 9ab^{a + b){a + 3b). Hence their h. c. f. is Sab {a + 3b). EXAMPLES. VIII. B. Find the h. c. f. of the following quantities. 1. x^ - xy and x'*- — y'^. 2. a^ —2ab and a^ _|- 2ab. 3. X* - «4 and {x + a)^. 4. x'^ - 4 and x^'+ x - 6. 5. x'^-Sx + 2,x'^-l,a,ndx^- + 6x-7. 6. 2^-2 _ 3,^ _ 2, and 4:X^ + 8x + 3. 7. 3a262 (a3 ^ b^\ Ba'^b {a^ - b^), and 9ab^ (a - bf. 127. It is worth noting that if one of the ex- pressions can be resolved into factors of the first degree, we can tell by trial whether each of these factors is also a factor of the other expressions; and we can thus obtain the H. c. F. of the given expressions. Example. Find the h. c. f. of 2 (a -6) (11a -216) and 209a3-399a2& + 407a62-7776^ Here we can try whether the factor a — b divides the second expression, and on trial we find that it does not do so ; hence it cannot form part of the h.c. F. Next, in the same way trying the factor 11a -216, we find that it divides exactly into the second expression. Therefore 11a - 216 is the required h. c. f. EXAMPLES. VIIL C. Find the h. c. f. of the following expressions. L 3 (a + 6)2 and 17a''+19a26-5a62-763. 2. 7a (a2 - 62) and 29a4 _ 60a36 + 1 2a262 + 1 9a63. 3. 3.^2 - 3x - 6 and 3.r^' - 6.r^ + 3.i?^ - 3.v^ -Ux + 4. 104 HIGHEST COMMON FACTORS. 128. Rule for finding the H. C. F. of two Compound Expressions. We confine ourselves, in the first instance, to the case where there are only two compound expressions. To determine their H. c. F. we begin by removing all the simple factors involved. The H. c. F. of the factors so removed must be found, and will be a factor of the final H. c. F. required. The remaining part of each expression must then be arranged in descending powers of some letter, say x. Next, the expression of higher dimensions (or if both be of the same dimensions, then either of them) must be divided by the other. The remainder, if any, will be of lower dimensions than the expression used as a divisor. This remainder is now used in turn as a divisor and the former divisor as a dividend. Con- tinuing this process until there is no remainder, the last divisor will be the H. c. F. of the two expressions. If there be a final remainder which does not involve x^ then there is no factor involving x common to the expressions. The H. c. F. of these expressions (if any), when multiplied by the H. c. F. of the factors removed, will form the H. C. F. required. 129. Before proving the rule, we will illustrate the method by finding the h. c. f. of cfi - 'ix^ + 1 and 'ix^ ■\-x^-\- 4x - 7. The first step is to divide one expression by the other, ^3 _ 2.^^2+1 ) 2x^+ x^ + Ax-1 ( 2 2;^-4.a?2 +2 5.^2 + 4^-9 We have now to divide x^ - 2x" + 1 by 5.^•2 + 4^v - 9. To avoid the introduction of fractional coefficients, we will multiply the first of these expressions by 5. This cannot affect the factor we are seeking, since this factor is to be a compound expression common to the two given expressions. We may therefore at any stage of the process multiply or divide either the divisor or the dividend by any number or simple factor without altering our result. This will often enable us to simplify the working of an example. HIGHEST COMMON FACTORS. 105 The next step therefore will be as follows, 5.^2 + 4.r - 9 ) 5^ - 10a'2 + 5 ( .i? 5^+ 4r2_9^. -14^2 + 9^+5 Multiply the dividend again by 5 so as to avoid fractions, .*. we have 5.r2 + 4.i--9 ) -V0.r2+ 45.r+ 25 ( -14 -70.^-2- 56a; +126 101 A^- 101 We have now to divide 5.1-2 + 4.y- 9 by lOl.r- 101, that is, by 101 (.r-1). In the same way as before, we may reject the numerical factor 101, since it can form no part of the required H. c. F. The next step will then be, x-l ) 5^2 + 4^-9 ( 5.^ + 9 bx^ — bx 9^-9 Qx-Q There is no remainder, and there is no numerical factor common to the two expressions, therefore x-\ (which is the last divisor used) is the h. c. f. required. 130. Proof of the rule. The rule enunciated in Art. 128 depends on the principle that any quantity which is a factor of A and B will also be a factor of iiiA + nB, where A and B are two algebraical expres- sions, and m and n stand for any quantities (except fractions involving the factor in the denominator). If we suppose the operations described in Art. 128 to terminate in (say) three steps, the process will be represented as follows : A) B{Q, AQ, G)A{ Q, CQ, DQz That is, Q, is the quotient, and C the remainder when B is divided by A. Next, A is to be divided by 106 HIGHEST COMMON FACTORS. C\ here, ft represents the quotient, and B the remainder. Next, G is to be divided by D; here, ft represents the quotient, and if the process terminate in three steps, there will be no remainder. Thus we have 5 = ^ft+(7, that is, B-AQ^=C (i); A = CQ,+D, that is, A-GQ, = D (ii); C = DQ,, (iii). From (i), we see that any quantity, which is a factor both of B and of j4, is also a factor of G. The required H. c. F, is a factor both of B and of A ; there- fore it is also a factor of G. Similarly from (ii), we see that any quantity, which is a factor both of A and of (7, is also a factor of D. The required H. c. F. is a factor both of A and of G ; therefore it is also a factor of D. Thus the required H. c. F. is a factor both of G and of Z), that is, by (iii), is a factor both of DQ, and of B. It is clear that the highest factor common to these two quantities is B, Hence the required H. c. F. is B. 131. A direct application of the principle will sometimes enable us to write down at once the H. c. F. of two expressions. Ex. 1. We might thus have obtained the h.c.f. of the two expressions given in Art. 129 directly by the use of this principle. The argument in that article was that any quantity which is a factor of {x^ — 'ix-\-l) and (2x^+x^-\-4:X — l) is a factor of (2a^3 + ^2 + 4^ _ 7) _ 2 (^^3 _ ^^^ + 1)^ that is, of 5.372 + 4^-9. This was the first step in Art. 129. Now the factors of 5^^ _|_ 4^ _ 9 ^j-e readily seen to be x-1 and bx + 9. The latter factor does not divide the given expressions, the former factor does ; and since there are no other factors of bx^ + 4^ - 9, it follows that x - 1 is the required H. c. F. ^Ex. 2. Find the h. c. f. of ax'' + 1 - (a + 1 ) .1^'^ + 1 and x'^ -ax + a-l, n being any positive integer. Any factor of ax''^ ■*" ^ - (a + 1) x''^ + 1 and x'^ - ax-{- a— I is a factor of {«.v" + 1 - (a + 1 ) .r« + 1 } - {ax -{a + l)] [x^'-ax+a-l ] . HIGHEST COMMON FACTORS. 107 The last expression reduces to ct^^x-iy. The coefficient a"^ is rejected in the same way as a numerical coefficient, not common to the two expressions, would be rejected. Hence the required H.c.F. is (^-1)2. 132. H. C. F. of several Compound Expres- sions. The rule given in Art. 128 for finding the H. c. F. of two expressions can be readily extended to include the case of the H. c. F. of several expressions. Let A, B, C\ ... stand for the expressions. Find the H. c. F. of two of them, say of A and B : this will include every factor common to A and B, The H. c. F. of this quantity and of C will therefore include every factor common to ^, -B, and C: and so on. Example. Find the H.c.F. of x^-lx'^-x + 1, afi^-Zx'^-x-Z, and^3_^^2_5_^.^.5. The H. c. F. of the first two expressions can be found either by the method of Art. 126 or by that of Art. 128. The student should perform the analysis, and he will find that x'^ — 1 is the H.c.F. of these two expressions. He must then determine the H. c. F. of ^2 — 1 and x^ — x'^ — bx + 5. This h. c. f. will be found to be ^—1. Hence ^—1 is the required H.c.F. of the three given expressions. Note. The student ought to be able to resolve the above expressions into factors, in which case the answer is at once obvious. MISCELLANEOUS EXAMPLES ON H.C.F. VIII. D. [/n working the following examples, the student is recommended to resolve every expj'ession into factors — lohen these are obvious — in which case he will generally he able to write down the answer hy inspection.'] Find the Highest Common Factor of the following expressions numbered 1 to 36. 1. 4r2 + 3.r - 1 and 4^3 + 7.^2 _ 3.^; - 1 5. 2. a34.^_2 anda3_3«^_2. 3. a3_2a + 4 and a3 + «2^4 4. x^ - 6^' + 9 and x^ + 4a-2 - 9. 108 HIGHEST COMMON FACTORS. 5. 8^-10.^2 + 7^-2 and Qa^-\\x'^-\-^x-'2.. 6. x^ - la^'2 + 101.^' - 99 and x^-l Qx'^ + 72.r - 81. 7. 8.^3 + 18.^2 - 1 l.r - 30 and Qa^ - 1 1.^2 - lAx + 24. 8. 9.r3 + 2 Lr2 - 1 Ix + 3 and 3.^^ + 1 7.^2 + 2 l.i- - 9. 9. 7.^3 - 10.^2 _ 7^^. ^ 10 and 2^3 _ ^2 -2x+l. 10. 7.2;3 - 3^^2 _ 7^^ + 3 and 2^3 _ 5.^2 -2:^7 + 5. 11. 4^1-9 and 2x^ - 2.^3 + .1^2 _ 3^, _ 3^ 12. x^ + 3.^3 + 12.1^ - 16 and x^ - IZx + 12. 13. ^3 _ 4^2 + 2^^ + 3 and ^x^ - .^2 - bx - 3. 14. 2^-4 _ _^3 _ 10.^2 _ 1 u^ + 8 and 2.^3 - 3.^2 _ 9^, _^ ^ 15. 3^'3_^.2_g^_|.2 and ^ + .^-2a-'-4. 16. .r* - 15a^2 _,_ 28^^ - 12 and 2^ - 15.i^ + 14. 17. x^-Zx + 20 and bx^ -^x^-^t 64. 18. x^-bafi + bx^^-x-l^ and a-* - 2.^3 - 12.^^2 + 11^ _,.20. 19. 3.^ + 7.^3-2^^2 + ^4.1 and .^4 + . ^3 _ 2^2^ 7^, + 3_ 20. 2.^^4 4. 2.^3 - 1 1^2 + 1 3^ _ 3 and 2.^* - 2.^^ - 5a;2 + 1 1.7; - 6. 21. 2.2;4 + .273 - 4.t^2 + 1 i.y _ 4 and 2x^ - Ix"^ + 14.^-2 _ 1 4.^ + 8. 22. 2.2;*-3^3_2^2 + 6.r + 3 and 2^*- 7^3 _ 10^2+,^ 4 2. 23. 2.r4 - 1x^ + 12^2 _ 1 1^. + 4 and ^a^ - 8.^^ + 5^2 + 2^-2. 24. 3a;4 + 5.^3 - 7.:«;2 + 2.^+2 and 2.^^4 ^_ 3^3 _ 2^2 + 1 2.^- + 5. 25. x^ + 4^2 4 1 6 and 2.^* - .r^ + 1 6^' - 8. 26. .r^ - 8ji^3y2 _,_ 1 7^2^3 _ I g ^y 4 9^5 and .2^ - 9.r4?/ + 26.2;3^2 _ ^Q^^-f + 27.ry . '27. x^y - xY - l^xhf + Zd>xy^- 14 f and x^ - 7;r4y + 2 1.1-3^2 _ 34^2^3 4. 28.1^. 28. 6.^3 _ 5^y2 + lo.r - 3 and 6.^t^3 + ^^2 _ iq,^. + 3^ 29. x^ - 2a.i-2 - a\v + 2a3 and x^ + 2a:z;2 _ ^2 ^ _ 2a\ 30. 8^3 + 1 6a2.^ - 40aj*;2 + 1 ^^^,3 and 8a'^ - 1 2«3^ _ 8a.r3 + 1 2x^. 31 . 1 2a3 _ 24a^x - 60ax^ - 24.^3 and 1 2^4 4 1 8a3^ + 1 2a.^ + 1 8x^. HIGHEST COMMON FACTORS. 109 32. x^ + 2.i'2y + 4.x f + 3^/3 and .v^ + x^y + 4t%2 + .^.^3 + 3^4, 33. .r" + l aiid^-i2 + ^^.-}^ 34. 1+X + x^ - x^ and 1 - ^-^ - x^ + ^7. 35. 2^-3 + 3^2 _ 1 u^ _ 6, 4^3 _ 1 6^-2 + 1 L^■ + 10, and 4r3+4jt'2_29^'-15. 36. a6 (.r2 +y2) _f_^^^ («2 _,_ ^,2) and ah {x"- +f) + xy {a?x + 62y). 37. What value of x will make both the quantities in ex- ample 4 above vanish ? *38. What value (other than zero) must be given to a in order that x^ — x — a and x'^ + x — a may have a common factor ; and what is their highest common factor when a has this value ? 39. Find the h. c. f. of the expressions 4a3 _ 47«62 ^_ 753 and 6a3 + lla2^-31a62 + 1453. and the g.c.m. of their numerical values when a = 4, b = l. 40. Find the h. c . f. of 2x^ + x'^-x-2 and x^ - x^ - 2^-2 + 2x ; and shew that its square is a factor of the latter expression. *41. Find a value of x (other than zero) which will make a^ + lx^- 49^2 + 8^ + 2585 exactly divisible by x'^- - 7^'+ 1. CHAPTER IX. LOWEST COMMON MULTIPLES. 133. The Lowest Common Multiple of two or more algebraical expressions is the expression of the lowest dimensions which is exactly divisible by each of them. The letters L. C. M. are often used as an abbreviation for lowest common multiple. Some writers call this expression the least common multiple. The name is properly used in arithmetic, but in algebra our symbols stand for any numbers, and usually we cannot with correctness speak of one expression being less than another, 134. We shall first consider the rule for finding the L. C. M. of simple expressions, and of expressions which can be resolved into factors of the first degree on inspection, and then the rule for finding the L.c.M. of compound expressions of which the factors are not obvious. 135. Rule for finding the L. C. M. of Simple Expressions or of Compound Expressions which can be resolved into factors of the first degree. The L. c. M. of two or more simple expressions (or of expressions which can be resolved into factors of the first degree) is the product of every factor which occurs in them, each such factor being raised to the highest power it has in any of them. If the expressions have numerical coefficients, it is usual to find the numerical L.C.M. of these coefficients, and prefix it to the required L.c.M. as a numerical coefficient. LOWEST COMMON MULTIPLES. Ill Ex. 1. Find the l.c.m. of x^^ ^-2, and x^. The required l. c. m. must be some power of x ; and, since it is to be exactly divisible by each of the given quantities, it must be X^. Ex. 2. Fuid the l.c.m. ofSab'^c^, Sa^iV, and Ga^b'^c. The L.C.M. required must contain a^ as a factor, otherwise a^b-c will not divide exactly into it. Similarly, it must contain b- and c^ as factors. The L. c. M. of 3, 2, and 6 is 6. Hence the required l.c.m. is 6a^6V. Ex. 3. Find the l. c. m. of {x - a){7/ - bf{z - c)\ {x - a)%7/ - b)\z - c)2,and {x - af{y - bf{z - c). This, by the same method, will be {x — of (y — b)^ {z — c)^. Note. The l.c.m. of expressions like the above can be ob- tained in a manner analogous to that used in arithmetic; but the method described above is so easy of application that it is unnecessary to use a more elaborate process. EXAMPLES. IX. A. Find the l. c. m. of the followmg expressions. 1. Zx"^, 2x^, and x-\ 2. 27abc^x, 24arbc7/, 60«c%, and Ibahcxy. 3. ha^a^a^x'^., la'^a^a^xy, and Za^^a^^a^y-. 4. ^-2 - y\ {x +yf, and {x - yf. 5. x''- - 4/, {x + 2y)2, and {x - 2yf. 6. 9 (a^2 _ y2)^ 8 {x - yf, and 12 {x^ +y^). 186. Rule for finding the L. C. M. of two Compound Expressions. The L. c. M. of two com- pound expressions, whose factors are not obtainable by inspection, is found by dividing one of them by their H. c. F. and multiplying the quotient so obtained by the other of them. For let A and B stand for the expressions, and let H be their H. c. F, Then A and B are each exactly divisible by H. Suppose the quotients to be respectively a and h. Therefore A = Ha, and B = Hh. Now a and h have no common factor, because otherwise we 112 LOWEST COMMON MULTIPLES. could obtain an expression of dimensions higher than H which would divide A and B. Therefore the L. c. M. of Ha and Hh is Hah. But Hah = {Ha){b) = Ab, also Hab = (Hb)(a) = Ba; and these results are a statement (in algebraical nota- tion) of the rule given at the head of this article. JVote. We can also obtain the L. c. m. of A and B by dividing their product by their H. c. F. For we have the L. c. M. of A and B = Uab = ^ -^ — - = -^ . 137. Rule for finding the L. C. M. of several Compound Expressions. The l.c.m. of three or more algebraical expressions can be obtained by first finding the L.C.M. of two of them; next, finding the l.c.m. of that result and of the third of the given expressions ; and so on. The final result is clearly the L.C.M. re- quired. MISCELLANEOUS EXAMPLES ON L.C.M. IX. B. [Li working the following examples^ the studeyit is recommended to resolve every expression into factors^ when possible^ and to make use of the method given in Art. 136 only when the factors of the expressions cannot he otherwise obtai7ied.] Find the l. c. m. of the expressions given in examples 1 to 18. 1. .^'2 - 1, x^ + 1, x^ - 1, x^ + l, and x^ + x^- + l. 2. {x^ +y^)y (3^^ + ^xy - y2) and {x^ - x^y+xy^). 3. 2^2 _ 5^ + 3^ 2x^- X - 2, and 6x^ -5x-6. 4. x^^ + x^ and x^'^+x'^. 5. x^-3x'^ + 3x-l and x^-x^-x + l. 6. x^-lbx^^ + Gbx-nsindx^-lSx^^ + Qlx-SS. LOWEST COMMON MULTIPLES. 113 7. 9a;3_^,_2 and Zc(^-\Ox^-lx-^. 8. ^-2.1-2 + 2^+5 and ^+5.r3 + 12.i-2 + 13.^- + 5. 9. ^-.r3 + ar-8 and oi? -V \x^ - %x ^-1^. 10. :r4 - .t-3 + 2.r2 + .r + 3 and %v^ - 1 4.1-2 + 26.r - 30. 11. .t-3-8.r+3 and .^ + 3.^5 + ^ + 3. 12. x'^-\-'^x'^-y-\-^xy'^-\-y^ and .27* -oT^y- ^3/^+3/'*. 13. 3.1-4 + 7.373 + 13.1-2+7^+6 and 6.r4 + 11.273 + 10.r2 + 7.i- + 2. 14. .r3-.i-2-4.^ + 4, .?73_2.^2__3._|_2^ and x'^^^x'^-x-% 15. 12.?73 + 8.r2-27.r-18, 12^-3 _ 8.^2-27.^+18, and 18x"5 + 27.r2-8.2;-12. 16. x{x-y)\ x'^y{x+y)\ xy{x'^-2/\ x^-x'^y+xy\ x'^y + xy'^-\-y^^ and x^ — y^. 17. x^{x-yf, y'^ix+yf, x^-xy\ x^y+y\ x'^-xy+y\ and x'^ + xy+y'^. 18. .2733/- 3.272^2 ^5_j,^3_6y and x^-Zx'^y + 2xy''-. 19. The L.c.M. of two quantities is x^ - ba^x'^ + ^a\ and their h.c.f. is x'^- - a^. One of the quantities is x^ - ^ax"- - a?x + 2a3 Find the other quantity. 20. The H. c. F. of two expressions is .^ — 7, and their l, c. m. is .^-3- 10^.-2 + 11.^ + 70. One of the expressions is x'^-hx- 14. Find the other. B. A. CHAPTER X. FRACTIONS. 138. Fraction. The fraction, denoted either by Y or by ajh, is defined as such a quantity that, if it be multiplied by h, then the resulting product is a. 139. In arithmetic, a fraction is sometimes defined as above. But it is also sometimes otherwise defined by saying that if unity be divided into h equal parts and a of them be taken, the result will be the fraction -^ ; and it is thence shewn that its value is the quotient of a by h. This latter definition requires that a and h shall be positive integers, but the one given in the last article w^ill allow us to assign any values whatever (simple or compound expressions, positive or negative numbers, integral or fractional numbers, &c.) to our symbols a and 6, and is therefore better suited for the purposes of Algebra, since we there suppose the symbols to be unrestricted in value. 140. Reciprocal Fractions. Two fractions are said to be reciprocal when the numerator of the one is the denominator of the other, and vice versa. Thus, -7 and - are reciprocal fractions, and each is said to ha be the reciprocal of the other. Similarly, - and - are reciprocal; thus - is the reciprocal of a, and a is the reciprocal of - . 141. Properties of Fractions. We now proceed to consider some of the properties of fractions. FRACTIONS. 115 142. The fraction r is the quotient of a luhen J divided by b. The student may perhaps regard this proposition as obvious ; but we must deduce it as a consequence of the definition which we have selected. By definition, - x b= a. Divide each side by 6, .'. , = a h- 6. Thus :^ = 4^ ; for if 4x-3 be divided by x-l, the quotient is 4 and the remainder is 1. Similarly, — = ^ + a-c^ ; . X -p C X -\- c 143. The value of a fraction is unaltered if both its numerator and its denominator be multiplied by the same quantity. Let us denote the fraction -r by x, and let m be the multiplier. We want to prove that a _ma b mb ' By definition, [Art. 138], ocxb = Txb = a. Multiply both sides by 7n, ,'. xmb = ma. Divide both sides by mb, .'. x = —^ • 144. Since we may multiply the numerator and denominator of a fraction by the same quantity without altering the value of the fraction, we can (by taking — 1 as the multiplier) change the sign of every term in both numerator and denominator. ^ . -2a2 + 3a?)-462 2a2-3a6 + 462 For example, 33^2:1^6 4- 2^ = 3^^+^6^2p ' 8—2 116 FRACTIONS. 145. Multiplying up. Again, if we have a relation like A_G_ where A, B, C, D stand for any expressions, we may multiply each side of the equality by BD. We thus obtain AD = GB. This process is known as multiplying up. For example, if '^^i = ^^, then (.^; + l)(^'+4) = (^' - 3)(^- 2). JU Zi Ob *|~ 4r 146. The value of a fraction is unaltered if both its numerator and its denominator he divided by the same quantity. Let us denote the fraction j- by x, and let m be any quantity. We want to prove that _a _ ajm b bjm * By definition [Art. 138], Divide both sides by m, Divide both sides by — , xb = - a. xb a — — — — m m a m . X ■=■ — ^ h m 147. Removal of factors common to the numerator and the denominator of a fraction. It follows from Art. 146 that if the numerator and the denominator of a fraction have any common factor, and both of them be divided by it, the value of the fraction will be unaltered. Ft 1 a^h _{ah)a_a ab^~{ab)b b' (x -\- a) {a; — a) .v-\-a Ex. 2. {x -of X- FRACTIONS. 117 148. Simplification of Fractions. The last article enables us to reduce a fraction to its lowest terms, that is, to express it in such a form that the numerator and the denominator have no common factor. To do this, we have only to find the h.c.f. of the numerator and the denominator, and divide each of them by it. Wherever any factors common to each are obvious, it will generally be better to divide by them first ; but the beginner must remember that a factor cannot be thus cancelled unless it is a factor of every term in both numerator and denominator. By thus successively removing factors we can often reduce the fraction to its lowest terms without having to find the H.C.F. of the numerator and the denomina- tor ; and even when the complete simplification cannot be thus effected, we can so simplify the fraction that it is comparatively easy to find the h.c.f. of the numera- tor and the denominator of the fraction thus simplified. Ex. I. Simplify the fraction -^ ,. 2_h2\ ' Eesolving the numerator and denominator into factors [Art. 110], we have a^bta'^-b'^)~ a%{a-h){a-\-h) ' Cancelling the factors common to both numerator and denomi- nator, the latter form becomes h i^d' + ah + h-^) a(a + b) Since a + b and a^ + ab-\- b'^ have no factor common to them, the fraction is incapable of further reduction. Ex. 2. Simplify the fraction Zx^-^x Eesolving the numerator and denominator into factors, we have {x+lf-{,c-lf ^ 2 (3.y2 + l) ^ 2 3a^ + x ~ X (3.f- -\-l)~ x' 118 FRACTIONS. EXAMPLES. X. A. Reduce the following fractions to their lowest terms. ^_7^2y_i8y4 (a + l)^-(Q^-l)^ ^-8l3/4 • *• (a + l)3-(a-l)3- lOjt^-7^-3+^'2 ^4-15^2 + 28^-12 4^-2^-2:^ + 1* • 2^-15^+14 2^^-9^2^ + 10^3/2-33/3' • ab + a + b + l 9. 10. 12a^ - 4^36 - 23a262 + 9ab^ - 96^ 8a4-14a262-964 ((^2 + 52_c2_^2 4_2a6 + 2cc^(a2 + 62-c2-c^2-2a6-2cc^) ((^2 ^ c2 - 62 _ ^2 + 2ac + 26c^) (a^ + c2 - 62"^^^ac - 26^ 149. Addition or Subtraction of Fractions. If two or more fractions have the same denominator, their sum (or difference) will be equivalent to a fraction having the same denominator, and having for numera- tor the sum (or difference) of the numerators of the separate fractions. The method of proof will be sufficiently illustrated by finding the value of -^ ± -^ . Multiply by c^, .-. rf g + ^) = cZ^ + cZ^ = a±h. Divide each side by c?, -'.-^ ±-1= 1 • a2 52 c2 a2_52 + c2 For example, --- + _ = — ^^—. FRACTIONS. 119 150. If the given fractions have not the same de- nominator, we must begin by expressing them as equi- valent fractions having a common denominator. This is always possible. For the value of a fraction is un- altered if its numerator and its denominator be each multiplied by the same quantity [Art. 143]. If then we find the l.c.m. of the denominators of all the fractions, the denominator of any one of the given fractions will divide exactly into it; and if both the numerator and the denominator of this fraction be multiplied by this quotient, the fraction will be changed to an equivalent fraction having this l.c.m. for its denominator. Thus all the fractions can be reduced to equivalent fractions having a common denominator; and these equivalent fractions can be added by the rule already given in Art. 149. Note. The converse problem of resolving a given fraction into the sum of a number of simpler fractions (called partial fractions) will not be considered in this book. Ex. 1. Find the value of --\-\ + - . -' a h c Here the l. c. m. of the denominators is ahc. X y z xbc yea zah ' ' a b c ahc ahc ahc _xhc-\-yca-\-zah ahc Ex. 2. Find the value of a+- — I — ax Here the l. c. m. of the denominators is ax. x^ a^ a'^x x^ a^ a^x + x^ -{■ a^ ,-. a+- +- = — + — + — = 2^~- a X ax ax ax ax 120 FRACTIONS. Ex. 3. Fmd the value of . '' x-a x + a Here the l.c.m. of the denommators is (x-\-a) (x-a). 1 1 _ x + a x—a ''x-a x + a (x — a){x+a) {x + a){x-a) _(x+a)-{x-a) x^—a^ 2a (a) X^ Ex. 4. Simplify ' + ' ' {a -b){b- c) {b -c){c- a) (c -a){a — b)' First, arrange each factor in every term in descending order of a, 6, c. The expression then takes the form 1 1 1 {a — b){b — c) (6 — c){a — c) {a — b){a-c)' The L. c. M. of the denominators is [a -b){b- c) {a - c). ,, . {a-c)-{a-b)-{b-c) . . .'. the expression = ^-^ — -ixtl fr-^ — ^ (a) ^ {a - b) {b - c) {a- c) ^ ' _ a—c-a+b-b+G (a — b){b~c)(a — c) {a-b){b-c){a — c) = 0. Ex.5. Si,rplify^^^ + ^^ + ^^-^^. The denominators of the component fractions are (l-^)^, (l-x) (1 +x), and (1 +xf. Their l. cm. is (1-^)2 (1+^)2, which is equal to (1 —x^)"^. Hence the given expression = ^ ~ -^ — -~ — ^ (a) (1 — ^ )" ^{l+2x+x^) + {2-2 x'-) + {l-2x+x^) (1^)2 _ 4 ~{l-x^y' FBACTIONS. 121 EXAMPLES. X. B. \^In solving the following examples, the beginner will find it convenient, when he is adding together the numerators of the com- ponent fractions, to insert a step in which the separate numerators are enclosed in brackets and added together, before he commences to simplify their sum. This step is shewn in the lines marked (a) in the Examples 3, 4, and 5 above. The lines of dots and the letter (a) are only inserted to facilitate this explanation, and form no part of the process of simplification.'] Simplify the following expressions by reducing them to single fractions in their lowest terms. „2_^ ^5,^.52 ^ «2_^5 + 52 a-\-b a — b a + b b-\-c c-\-a _{a-{-b)(b-\-c){c+a) cab abc x-a a^ + 3ax x + a x+a a^-x^ x-a 11. 12. 2a , 2(a2_4a^) Za x + a a^ — x^ X- 1 2 8. (^-1)(^+1) (x+l)(^ + 3) (^ + 3)(^'-l) _3 2 x-6 x^~ x-4:~ {x-2){x-5)' ^+.y ^ 4by ^ b-y h-y b--y^ b+y' a-3b a-\-3b bab a + 2b a-'2b a^ + ^b'-' ^' that is, X = Q-. a Thus X is equal to the product of Q and the re- ciprocal of Y . E^.l. Divide ^-^^hy ^^. The quotient = ^ ;,., x — ^ . Cancelling the factors common to both numerator and denomi- nator, this ^aY aa^ — a* , x^ -\- ax -^ a^ x^-ax-\-a^ "^ x^ + a^ ax^ — a* x^ + a^ Ex. 2. Divide -^ — ^^j^^.t ^V The quotient Kesolve into factors .^2 — ax -\- a?- x^ 4- ax + a^ a{x — a) {x^ -\- ax ■\- a^) (^x + a) {x^ — ax + a^) x^ — ax -^-a^ x^-\-ax + a^ Cancelling the factors common to both numerator and denomi- nator, = a{x—a){x-\-a) = a{xP' — a^). 126 FRACTIONS. EXAMPLES. X. D. Reduce to sim]5le fractions in their lowest terms the following expressions which are numbered 1 to 6. x'^ + bxy + Qy^ ' x^ + xy-2y'^ ' 1. ■ 272 + 2^-15 . x'^ + ^x+^0 /_x g \ x'^ + a'^ .172+ 8a; -33 • ^^.2 + 7^ _ 44' • \a;-a x+a) ' x^-ax' [y^-c2 d^-a^ a^-m (b-c c-a a-b] 6^y2 ^ r s{m-n)x ^ ( 4:{r-s) ^ r^-s^ ] "1 ^' Tn^+n' [_ 7(r+s) ' [ 2lxy^ ' 4:(m^-n^)fj' -7. Divide 49p--7^ + T--^ + l42by^-5 + J. ■^8. Shew that the first four terms in the quotient of a+dc divided by a + 6c are l-ih-d)l + ib-d)^^-{b-d)^^. 154. Simplification of Expressions involving Fractions. The rules given above frequently enable us to reduce expressions of considerable complexity to a simpler form. The following are a few examples. 9. 1/ X Ex. 1. Simplify 7— r; +— «^- ^— . i_l i-l i_^ X y X y This expression = — — + ■■ — ' ^ y--^ ''*^ — 3/ 3/ — ''^ = 2^^+v-^ ^'^^ [Art. 153. ?/ — .r ^ x—y y-x ■■ 2.27^ ^ xy ^ xy x—y x—y x — y _ —'2xy-Yxy + xy _ ~ x-y = 0. FRACTIONS. 127 E.. 2. Simplify [^-^ - ^^ x [^^ " ^^l] ' _ r 5(.^+l)-3(^ + 3) -| r 5(^-l)-4(^-2) -] L 2(^' + l)(.r + 3) JL (.^-2)(.r-l) J 2.r-4 .r + 3 2 (^-4-1) (^- + 3) (^-2)(a;-l) 2 (^--2) (^ + 3) 1 1 ^:r. 3. Simplify a" x-\- X X A fraction of this form is known as a Continued Fraction. The best way of simplifying a fraction of this form is to begin with the last denominator, and work up, as illustrated by the following analysis. The fraction a' x + a^ x'^-a? X a2 00 + dKT^ x^-a^ a2 " -a^x+d^x x^ — a? 155. Additional examples on fractions, especially on the comparison of unequal fractions, will be found in Arts. 181, 296. 128 FRACTIONS, MISCELLANEOUS EXAMPLES ON FRACTIONS. X. E. 1. Find the value of - lac - {2c {a - 36) - 3a (5c - 26)}, when c= -^ . n -rr, b-c c-a a-b ,, ^ 2. ii c^ = , ^= — r- , z = , prove that 3. Prove that the sum of two quantities divided by the sum of their reciprocals is equal to the product of the quantities, 4. Shew that the expression \- - — , has the same x-a oc~b value when x = a-\-h as it has when .r=2«6/(a + 6). 5. Resolve into their simplest possible factors each of the expressions ^^ — (t — \ x-\; x'^-ia + ~\ x + \. Reduce to their simplest forms the expressions given in Examples 6 to 48. ^' [^{x-'^) 2(^-l)J^L^ + 2 x + \\' \ \+x \-x^ ) V4^-iy' ^x'^-\-xy+y^ x^-X7/ + 7/^\ x^ — y^ \x^ — xy-{-y'^ x^ + xy-\-y^J x^-y^' to + l a~]~ jo + l ~a~j ' V y-^J\ y-x ) \y-^J 10 ' \a 6 cj abc 12. _^ + J.^_ 2 a , 1 1 1+6 l+a a+6 / y 3^ \ 3^-.y , / 1 1 \ * \Zx-y Zx+y) ^ 9.r2+3/2 • \3.r-?/ 3.r+?// ' FRACTIONS. 129 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. -xy 0,2 ~o,"^^ •^""'"17 3/^ y JL 2 2^ 2^2 2^2 8^ 1 +.t^2 + (1 -^)(1 +^2) (1 +.^;)(1 +^^2) 1 _^ /.r2+y2 ^ ^2_y\ ^ /^+y ^ .r-y X 1 1 {x-\-yY — x^ — y^ x'^ y y' / 3a 2b \ 3a + 26 . / 1 1 VBa - 26 3a + 267 ^ 9^'^ + ^^^ * V3a - 26 "^ 3a + 26 (6-c)2 (c-a)2 (a-6)2 (a-6)(a-c) (6-c)(6-a)"^(c-a)(c-6)' a2-62 + 26c-c2 _ c2 + 2ca + a2_52 62-c2 + 2ca-a2 62 + 26c + c2-a2- {a-hf-{h-cf {h-cY-{c-af {c-af-{a-hf a^j^ah-hc-c^ ^ b'^ + bc-ca-a^ c^+ca-ab-b'^ ' (6 + cKfHa5 (c + a) ( x'^ + b^ ) (a + b) (x^ + c'-) (c-a){a-b) "•" (a-6)(6-c) "^ (6-c)(c-a) ' a - 6 a62 a'' V aJV « + , „ -yo-^) 111 (a-6)24-(6-c)2 + (c-a)2 a-b'^b-c'^c-a'^ 2{a-b){b-c){c-a) ' a + 2 _ 4a + 5 l+x _ 1+^2 2a + 3 5a + 6 g^ 2a + 3 3a + 4* 3a + 4~4a + 5 B. A. 14. .^2 1^^ l+X^ l+P' l+x-^ l+X^ 130 FRACTIONS. ah \ \ f o?- }fi a — h a^ 62 ^ a^ + b^ a b a + b'^a-b 2 \ /->? 01 \ /I 1\2 3/2 ^2 \^ ^j 30. ^^^^ , -^^ . 31. 6 c b-c h+c a—b b—c l + ah l + bc 33. 62 (a - 6j ( 6-c) ^ (l + a6)(l + 6c) (a-b)(b-c) ' — rx-^ — TT -«• 2/2 f l+x^ _l+x^ ■^"'■^ ^^2 1+^2 1+^.3 «6 _1_1 -'^+#L'--r "' 1+-+^ -3-1 a?-¥ a b y X y 36. il^x:^-^. 37. ^ •^^; /^ + 38. r-r- 39 a^ — - 1 40. 1 , 1 a+1 a-\ + 1) ^-1 ^+1 2' ^ f q q p 41. ^ ^x^^,x<+^, 3/ ^ v-^y a;2-(23/-3g)2 43/2 -(3^ (^ + 23/)2- 9^2 "^(23/4-32; 43 i_^ ^ I ^3^' Ix^^ ;r2-(23/-30)2 43/2-(3^-:f)2 9 ^2_(^_2y)S *^' (^ + 23/)2 - 9^2 "^ (23/ + 3^)2 - ^2 ^- (32 + ^)2 _ 4^2 44, FRACTIONS. 131 2y ^ 2 {x^-y) (^ + 2y) ^ (^+y) (^+3y) ^ (;r + 2y) (^+%) ^+33/' .r f, 4 12 1 r 4 12 ] ^5- {1-.T31 + .— alr+^i-.Hral- 1-^+S -1-1 — 47. ^JL^JLxt- : ^^ y V if" x^ , ' /I 1\^ X x^ y^ \x y 48. ix-y i \x---- *49. If ^ = ^ and y = / , , , // , 1 ,\ j V^^yq that 26c '^ (a + 6 + c) (-« + 6 + c)' ^ (.r+l)(y+l) = 2. *50. If^^^ = a, ^— = 6, ?^^=c, shew that (1 -a) (1 - 6) (1 - c) = (l +a) (1+6) (1 +c). 9—2 CHAPTER XI. SIMPLE EQUATIONS CONTINUED. 156. We return now to the further discussion of simple equations. We shall give in this chapter some more examples of simple equations — particularly of those involving fractions — and of problems leading to simple equations. We shall follow the procedure ex- plained in Chapter V without specifically indicating every step. 12 3 Ex. 1. Solve the equation H = . ^ x-l x+l x + 2 The L.c.M. of the denommators is (x-l) {x+l){x + 2). Multiplying each side of the equation by this l. c. m., we have {x+l){x + 2) + 2{x-l){x + 2) = 3{x-l){x+l). ... a;'^ + 3x-\-2 + 2x'^ + 2x- 4 = 2x^-3. .'. x^ + 2x^-3x'^ + 3x+2x= -3-2 + 4. .-. 5x=-l. Multiply- .'.x=-^. Ex. 2. ^oCve the equation = -; . The L. c. M. of the denominators is (x-l) {Sx ing each side of the equation by this l.c.m,, we 1 -2). lave {x+2){3x-2) = {3x+l){x- ... 3.r2 + 4.r-4 = 3.r2_2.r-l. ... 3x^-3x''- + 4.x + 2x = 4-\. :. 6a; =3. ,.x=l. 1). SIMPLE EQUATIONS CONTINUED. 133 r, ^ Ci 1 1 .' -^'-2 X-\ ^'-10 ^-11 Ex. 3. Solve the equation , - — -= _ ^ . ^ x-\ .r - 3 ^ - 8 ^ - 9 We will begin by dividing the numerator of each fraction by the denominator [Art. 142]. The equation may therefore be written (-A)-(-..:!-3) = (-,.^s)-(-.!o)- 2 2 _ 2 2 '*^-4 x-Z~ ^-8 ^-9' Divide each side by 2 ; and then, instead of at once bringing all the fractions to a common denominator, simjilify each side of the equation by itself, (■2;-3)-(.y-4) ^ -(.a;-9) + (.3;-8) •*• (^-4)(^-3) ~ (.^•-8)(.r-9) ' 1 1 •'• (^-4)(^-3)~(^-8)(^-9)* Multiply up [Art. 145], .-. (.r-8)(^-9) = (^-4)(.x'-3). .-. ^•2-17.r + 72 = jt;2_7^+12. .-. -10.r=-60. Ex. 4. Solve the equation x^ + 2x^ + x^--1x-3 _ x* + 6x^ + 2.v^-l6x-4 . x^ + 3x + 5 ~ x'^ + lx+10 Here the numerators are of higher dimensions than the denominators, and it will be convenient to begin by simplifying each fraction by dividing the numerator by the denominator [Art. 142]. The equation then becomes x+2 x+6 x^-x-l+ o- "^ — --Z = x^-x-l + x^ + 3x + 5 ^2 + 7^ + 10 Q. ,.» . ^ + 2 x+6 Multiply up, .-. {x + 2) {x^ + 7x+lO) = (x + 6) (.^'•^ + 3^- + 5). Simplify, . ' . x^ + 9x^ + 24.r + 20 = jp^ + 9^2 + 23x + 30. .•..^^=10. 134 SIMPLE EQUATIONS CONTINUED. „ ^ CY 7 .7 .' a-h . ax ^ 4a^(b-ax) Ex. 5. /Soa-e the equation x -, = 4 -, + j- — , ; . ., — ^ . ^ a-{-h a + b {a + b){a^ + b^) The L. c. M. of the denominators is (a + b) {a- + b^). Multiply- ing every term by this l.c. M., we have x{a + b) (a2 + 62) - (a _ 6) {a^ + 62) = 4.ax {a^ + 62) + 4^2 (6 - ax). /.x{a + b) (a2 + 62) - 4ax {p? + 62) + ^a^x = 4^26 + (a - 6) (^2 + 62). .'.x{a?-^ a% - Zab^ + 6^) = a3 + 3^25 4. ^52 _ 53, .-. .^(a-6)(a2 + 2a6-62) = (a + 6)(«2 + 2a6-62). .•. x{a-b) = a^-b. _a + 6 a — b' Ex. 6. -4 certain number is added both to the numerator and to the denominator of the fraction § . The value of the resulting fraction is^^. What was the number added? Let the required number be x^ 2 + x 8 ' ' Z+x~ l3" Multiply up, .-. 13(2+jp) = :8(3 + ^). .-. 5x= = -2. .'. x = ■■-i- This is negative. Hence the answer is that | must be sub- tracted from both the numerator and the denominator of § to make it equal to ■^^. It is easy tp verify that this answer satisfies the conditions of the question. Ex. 7. Of the candidates in a certain examinatio7i 36 per cent, failed. If there had been 11 more candidates, and if of these 11 candidates 2 had passed, the total number of failures would have been 37 "5 per cent. How many candidates were there ? Let X be the number of candidates. The number of those who failed was 36 per cent, of x, that is, was -{^^x. If there had been (^ -|- 11 ) candidates, there would have been 9 more failures, that is, the number of failures would have been This latter number would, by hypothesis, have been 37"5 per cent, of a total of {x+l\) candidates. Hence, by the question, SIMPLE EQUATIONS CONTINUED. 135 36 ^ 37-5, .'. 36.i- + 900 = 37-5:1; + 412-5. .-. 1-5^^ = 487-5. Therefore the number of candidates was 325. Ex. 8. A can row a mile in ^ of a mimite less time than B. In a mile race, B gets 250 yards start, and loses hy \A yards. Find the time A and B take to row a mile, on the assumption that they row at the same pace throughout. Suppose that A can row a mile in x minutes. .. B can row a mile in {x-\-^) minutes. Now B rows (1760-250-14) yards, that is, 1496 yards, in the same time that A rows 1760 yards. But B rows 1760 yards in (^+|) minutes. .'. . 1496 „ if|g(.*- + |) „ Hence, by the question, ytI o (-^ + 1) = ■^'• .'• U{JO+l)=x. .-. 68^ + 51 =80.t\ .-.^=.11 = 41. Hence A can row a mile in 4|^ minutes, that is, in 4 min. 15 sees. EXAMPLES. XI. Solve the following equations numbered 1 to 22. frjj-+^=-^ .-'2-^ _L . _^=_A_ ^^ x-\ X x + V 'w- x-\-2 x-'2 x-^' ^, 1 ^^2 ^ ^ ^^1 ^ x-b x + 5 X \y 2x-5 3a'-7 x > s x + 3 3.r-3 ^ - X Sx ^ _ ^'^ x-2^ x-h2 ' -• ^4-1 x + 2 s x+Z x + l ^-^ 2x + 7 3^--5 5.r + 9 (!) x-l x-3 ^' x+l x+2 x+S x'^ + lx-e _ x+l -Q ^ ^2-6.r+1 _ { x-3Y _ 072 + 5^- 10 ~A--1' ^y x^ + 8x-\-n {x + 4f~ 136 aiMPLE EQUATIONS CONTINUED. K^_ x-h x-a ^-^ a b b ^ J^=3+__:^. (^ -1 ^=.^. _y a — 26 2a — 6' ^^ x — d x + c x j^x+a ^^^2. 16. f+^ + p:^=2{^^-cd). \ J X + C x-a c+x d + x 111 , 4J)x 4:a , X X-T X 1-1 7 hi 17. ^+_S + -r^=0. 18. -J- = ^J^. cab bx a 19. ^+^^-_-; = 2 2 ?_ be ac ab a b c 20. 21. *22. X^-2x^ + x'^-1x + 4: _x/^-\-^x^- 10^2 _ 6^^ + 9 ^2 + ^ + 3 ~ ~ ^2_,_5^,_,_4 • ^3 4.e^2^.i3^^.10 ^^2 + 3^_t_3 5^^2_i.26^+38 " "5^+11 • h + c — a c + a—b a + b — x^ — {b + c)x + bc x^-{c + a)x + ac x^ — (a + b)x + ab 23. A person sells 100 acres more than the third part of his estate, and there remain 2 acres less than the half. What was the extent of his estate ? 24. Find a number such that if we divide it by 12 and then divide 12 by the number and add the quotients, we obtain a result which is equal to the quotient of the number increased by 9 when divided by 12. 25. A man's income rises £10 a year. But, owing to a change in the income-tax from 4d. to 8d. in the pound, he finds his net income the same in two successive years. What was it ? 26. A man can walk from A to B and back in a certain time at the rate of 4 miles an hour. If he walk at the rate of 3| miles an hour from ^-1 to B, and at the rate of 4^ miles an hour from ^ to ^, he requires 3 J minutes longer for the double journey. What is the distance AB1 27. The time required to walk from A to B, at a uniform rate of 3^ miles an hour, is 5 minutes less than that required to walk half the distance at 3 miles an hour and the other half at 4 miles an hour : what is the distance ] SIMPLE EQUATIONS CONTINUED. 137 28. If you heave can hour and ,y = ^. The corresponding value of x can then be obtained. Ex. 6. Solve tJie eqiiations x—l y+l If we treat (.^-1) and (y + l) as the unknown quantities, we have, by Ex. 5, x-l=^, and y + l=i. .'. x = ^, and y=—^. B. A. 10 146 SIMULTANEOUS EQUATIONS Ex. 7. Solve the eqiiations {x-a){y + h)-{x + a)(^-h)^2a\\ {jj-h){y^a)-{x + b){y-a)=^h\] Perform the multiplications indicated, and simplify. Hence we have .*. hx- ay = a?' (i), and ax-hij = h'^ (ii). Multiply (i) by a, and (ii) by 6, .•. ahx — a^y = ahx — IP-y = h'^\ Subtract, .'. -{a?--h'^)y = a?-h\ a2 + a6 + 62 a + h Similarly, x= t • a-\-o 166. The method for solving three simultaneous equations of the first degree involving three unknown quantities is as follows. Suppose X, y, z to be the unknown quantities. Take one pair of the equations (say, for example, the first and second), and eliminate z between them ; this can be effected in either of the ways described in Arts. 162, 164. The result will be an equation involving x and y only. Next, take another pair of the equations (say for example the first and third), and eliminate z between them also. The result will be an equation involving X and y only. We now have two simultaneous equations involving only X and y ; and these can be solved in the manner already described. OF THE FIRST DEGREE. 147 For example, to solve X- y-2z= (i), -x + 2y- z=-\ (ii), 2^- y- z= 5 (iii). First, eliminate z between (i) and (ii). [To effect this, multiply (ii) by 2, and subtract from (i) ; or find z from (ii), and substitute the value in (i).] The result is 3^-5y = 2 (iv). Next, eliminate z between (ii) and (iii). The result is .-. -3.^ + 3y=-6 (v). AYe have now to solve (iv) and (v). Add, .-. -2j/=-4. .-. y = 2. Substituting this value of y in (iv), we obtain 3x'-10 = 2. .*. ^ = 4. Lastly, substituting ^^=4 and y = 2 in (i), we obtain 4-2-2^ = 0. .-. -22=:-2. .-. 2=1. Hence the required solution is .2; = 4, 3/ = 2, z = \. 167. It is however clear that, if the elimination of z between the second and third of the given equations had led to the same equation as the elimination of z between the first and second equations, we should have only had one equation between x and y, and this could have been satisfied by an infinite number of pairs of roots. In such a case, the equations are said to be indeterminate or not independent. A system of three equations is indeterminate when- ever from two of the equations the third can he deduced. Thus the equations 3^-23/ + 22=-l 2x- y-\- z= 0\ X- y-\- z= -1 10—2 148 SIMULTANEOUS EQUATIONS are not independent. In fact, if the second be subtracted from the first, we obtain the third. The given equations are thus equivalent to only two independent equations. *168. The above remarks will enable us to complete the discussion of the solution of the equations ax + hy = c, a'x+h'y = c' w^hich is given above in Ex. 4 on p. 144, and which includes all possible cases. If —, = - = -, then one of the equations is a multiple of the a be other, and is therefore deducible from it ; that is, the equations are indeterminate because they are not independent. In this case, the solution found in Ex. 4 on p. 144 becomes •^ = qj V^q- Again, if -7 = 77 , and if the value of each of them be not equal ^ ' a b to - , the equations cannot be solved, because they are inconsistent: in fact, the second equation can ( since — = — j be written in the form ax-\-by = c' — , and since c' — is (by hypothesis) not equal to c, this is inconsistent with the first equation. In this case, the solution found on -p. 144, Ex. 4, becomes x = — - — , EXAMPLES. XII. [In solving the folloiving questions^ the beginner should remember that he must find the values of each of the unhiovm quantities^ and also that each root must contain nothing but numbers and known constants. In using the method of solution given in Art. 162, it is generally better to eliminate that unhioion which has the simpler coefiicients. When the value of one of the unknowns is determined, it is better to select the simpler of the original equations as the one in which it is to be substituted."] Solve the following systems of simultaneous equations, numbered 1 to 74. 1. Il3/-.r=10, 1 2. 3^^ + 2y = 43,| ll^-101y=110.J A- + 5y = 49.J OF THE FIMST DEGREE. U9 3. 9.r-41y = 40,l 4. 1x- y= 3,| 4r-19j/= 3. J 5^-f43/=10.J 5. 7.r-2y= 1,\ 6. 7.^-3y= 3,1 3a- + 5y = 59.J 5^' + 73/ = 25.1 7. :r + 17y = 53,] 8. ^+ 5y = 49,| 8.r+ ?/ = 19.J 3.t7- 11^ = 95./ 9. 33^- + 35y= 4,| 10. 7.^^^-lly = 68,■l 55.r - 55y = - 1 6. J 9^' - ^y = 33. J 11. 7.r-9y = l7,] 12. 14.^-9^ = 5,1 ay-7y = 71.J 35.r + 6y = 3.J 13. ^hx- ay= a6,l 14. ^ahx+ y= 86,1 bx + 2ai/ = 3ab.} 4a6^-3y = 156.j 15. «.r +63/ =2, 1 16. ax -{-by = 2, 1 17. 5x-4y = 3x + 2y = l. 18. .r + 19^ = 79, 9.^^+y = 31. 19. 4:X-6i/-3 = 7x + 2y-4:= -2.v + 3?/ + 24. 20. 5.^ + 2y-l = 3.i;-y + 14 = .a? + 19?/ + 6. 21. 12.^ + 13^ = 19, 13.^ + 123/ = 31. 22. 77^--153/ = 8, 14r + 21y = 6-2. 23. 91.27 + 8?/=15, 21.2;-35y=-5-75. 24. |-% = 1, •23.2; + -26^ = 2. ^^- ^"~ 6 ~ ' 6 3~^- 26. .^ + ^^^=15, ^ + 2y=.36. ^„ ^ 5y + 9.r ,„ ^ ll?/ + 9.r ^, 27. 7.r + -^ -17, 9,r + — ^^^y— = 21. 28. -^ + 37/ = 13, 4.r+-^'^ = 1.3. 29. ^^ + 4v/ = 13, 3.r+^ = 13. 30. i{^-y) + ^{^+y)=^, ^+2/-^=n(^-^)- 150 SIMULTANEOUS EQUATIONS 61. 7 + 8-^7, 8 + 7-'^*'- ^^- 6 + 5-"' 9 + 2-^*- 35. -+f = |V| = l. 36. |.|=13, 1-1=3. 3^_ |_|/=3. 1-1=1. 38. f-|=-l,|H-|.U. "^^•4 7~ ^' 3 2-^- ^"- 3 + 2-^' 5 + 3"^- 41 ^ + ^-43 ^4-^-42 42 -4-^-15 ^ ^-4 43. ^^ + 1=7,2^-1 = 4. 44. f^-i3/ = 3, §^-|3/=2. 45. f^-iy = l, |^ + 33/ = 26. 46. f.r + f3/ = 22, i^-iy=-7, 4^' by ^^ X y , ^„ 2.t; + l 3y + 2 ^ 49. 3*+^ = ^ + | = -. 50. 3+1=6, y-^=7. 51. !-? = 2, ?-i + U. 52. ?-i=8, llVI=101. X y y X Q x y x y 53. l-? = 3, ? + ? = 32. 54. 2-5 = 1, ? + ? = 26. X y X y X y X y 55. ? + 5 = 19, ^-? = 3. 56. ? + ^ = 29, 5_5 = 2. X y X y X y x y 57. ax~hy = 0, c{x-y) = a-h. 58. hx-ay = ax-hy = ah. 59. a.r + 63/=-.c2, a(a+^)=6 (6 + ?/). 60. .'??+3/ = a + 6, a(^ + a) = 6(y + 6). 61. y=^\{x + a) + lh, x=\{2, + h)+la. 62. {a—b)x-{b-c)y = c-a, {c — a)x—{a — h)y — h-c. OF TUE FIRST DEGREE. 151 63. {a-\-c)x-{h-\-c)7j = {a-h){a-\-h + ^c\\ {a-c)x-{h-c}y = w^-h'^. J 64. -4-1- + ^-T = 2, ax-hy=a'^^V\ a + b a-b ' ^ 65. i(.^+y) = H^-y), 3.i' + 17y = 2. 66. ^-^+1 + 2^ = 6, ^^ + -^^ = 2. o «3 7 5 67. i{^v + 7j) = l{x-7/), '3x + Uy = 4:. 68. i(.^'-ll) + ^/=18, 2.r + Hy-13) = 29. 69. 2y + ?-4 = 5y + L%2=y-? + 4. 70. 9(i-4-i-) = 8a+l) = 2. \3.r by J \5x 3y/ 71. + 1=0, -+ — =«- 6. 72. .'r + 2;/ + 32=32, 4^- 5?/ + 6^=27, 7a' + 8?/-92 = 14. 73. x = S(i/ -z), ^ = 4(y — ^), :^^ + 2 = 2y — 5. *74. a(y + 2) = 6(2: + .t;) = c(.r+y) = o?2^ *75. Shew that the following system of equations is indeter- minate. 2.i^ + ?/=10, x+z-3y + 4^0, 2y + 6^3x + z. CHAPTER XIII. PROBLEMS LEADING TO SIMULTANEOUS SIMPLE EQUATIONS. 169. We discussed in Chapters VI. and XI. the solution of problems which could be expressed alge- braically by simple equations. We shall now treat of problems involving more than one unknown quantity, and such that the given relations between the unknown quantities can be expressed by two or more algebraical equations of the first degree. The chief difficulty is the translation of the ex- pression of certain relations from ordinary language into algebraical language. As soon as the given re- lations are expressed by algebraical equations, the methods given in the last chapter will enable us to solve the equations. It is however worth remembering that we can only solve a system of equations when we have as many equations as there are unknown quantities. To ensure this, the beginner will generally find it advisable to begin by writing down all the equations without attempting to simplify them, and not to commence the actual solution of the equations until he has seen that he has as many independent equations as un- known quantities. To facilitate this, it is convenient to number the equations as they are written down. SIMULTANEOUS SIMPLE EQUATIONS 153 170. The following examples are typical of some of the more common problems. Ex. 1. A number of tico digits is equal to four times the sum of the digits, and it exceeds the sum of the digits by twenty -seven. What is the number? Let X be the digit in the tens' place and y the digit in the units' place. Then the number is \Ox-\-y. Then, by the question, {\Ox+y) is equal to 4 times {x+y) ; .-. 10^+y=4(.r + ?/) (i). Again, by the question, {lOx+y) is greater than {x-\-y) by 27 ; .-. 10^+^=^+3/4-27 (ii). We have now two equations involving two unknown quantities. Simplifying these equations, and collecting like terms, (i) becomes Qx = Zy, and (ii) becomes 9x = 27. Hence x = 3, and y = 2x = 6. .-. the required number is (3 x 10) + 6 = 36. Ex. 2. When unity is added to the numerator and denomina- tor of a certain fraction, the result is f , and when unity is sub- tracted from its numerator and de7iominator, the result is 2. Eind the fraction. Let X be the numerator of the fraction and y be its deno- minator. Then we have, by the question, x + l 3 ... yTT = 2 W, and '^---^ = 2 (ii). y-l We have therefore two equations involving two unknown quantities. Multiplying up, they become respectively 2x-3y = l (iii), and x-^y= -I (iv). The equations (iii) and (iv) can be solved by any of the methods given in the last chapter. For example, multiply (iv) by 2, and subtract from (iii), ••• y=3. Substituting y = 3 in (iv), we obtain x^^b. .-. the required fraction is §. 154 PROBLEMS LEADING TO Ex. 3. A and B together can do a 'piece of work in fifteen days. After working together for six days., A went away, and B finished it hy himself ticenty-four days after. In what time would A alone do the whole. ^ Suppose that A working alone would take x days to do the whole : and that B working alone would take y days. Then in one day A does - of the whole, .'. in 15 days ^ ... — 15 Similarly, in 15 days ^ ... — But in 15 days A and B would finish it; .■.i^+i5=i (i). X y Again in 6 days they had done - + - of the whole. X y 24 In the next 24 days B did — of the whole. 1/ This served to finish the whole, 6 6 24 , .-. - + - + — = 1 n. X y y We have thus two equations involving two unknown quantities. Simplifying, they become respectively X y 1 A 6 30 f • and --\ = 1 X y f Multiply the first of these by 2, and subtract the second, X X X .-. .^ = 24. Hence A working alone would take 24 days to do the piece of work. We are not asked to find y ; but if we substitute this value of X in either of the equations, we shall find y = 40. SIMULTANEOUS SIMPLE EQUATIONS 155 Ex. 4. A nude and a donkey were going to market laden loith wheat. . The mule said ''Hf you gave me one measure, I should carry twice as much as you ; but if I gave you one, we should hear equal bu?'de7is." Tell me what were their burdens. [This problem is said, by tradition, to have been given by Euchd in his lectures at Alexandria, circ. 280 B.C., and is perhaps one of the earliest problems of this kind ever asked.] Let X be the number of measures carried by the mule ; and let y be the number of measm-es carried by the donkey. Then if the mule had received one measure from the donkey, the mule would have carried .r + 1 measures and the donkey would have carried y—\ measures ; hence, by the question, ^ + l=2(y-l) (i). But if the mule had given one measure to the donkey, the mule would have carried x—\ measures and the donkey would have carried y + 1 measiu*es ; hence, by the question, A'-1=V + 1 (ii). Thus we have two equations involving two unknown quanti- ties. From (ii), ^=2/ + 2 (iii). Substitute this value of x in (i), .-. (y + 2) + l=2(y-l). ••• y=5. Therefore from (iii), x = l. Therefore the mule carried 7 measures, and the donkey carried 5 measures. Ex. 5. A man starts to walk a certain distance in a certain time, but after a time being obliged to diminish his pace by one- fifth, he is four minutes late in reaching his destination. If he had walked another mile before diminishing his pace, he would have been only one minute late. What was his original pace, and how far from the end of his journey did he slacken his speed? Suppose that he starts at the rate of v miles an hour ; and let the distance from the end of his journey at which he begins to slacken speed be x miles. Now, if he had not slackened his pace, he would have walked X miles in - hours [Art. 101]. But when he diminished his pace by one-fifth, he walked at the rate oi \v miles an hour, and he 156 PROBLEMS LEADING TO therefore took t- hours to cover x miles. The difference between these times is stated in the question to be 4 minutes, thcat is, y\ hour ; 00 X _l " ry~^~i5 ^^^' Again, if he had walked a distance of (^-1) miles at these rates, the difference of times would by the question have been 1 minute, or Jq of an hour. Hence x — 1 X- 1 _ 1 ~Y^ V ~m ^"^" We have therefore two equations involving two unknown quantities. Simplifying them, they become respectively and 15(^-1)= vj' Subtracting, we obtain 15 = 3y, .*. v = 5. Hence ' 15.r = 20, .*. x = %=^\\. Therefore the man started at the rate of 5 miles an hour, and slackened pace at a distance of a mile and a third from the end of his journey. Ex. 6. The value of 112 coins^ con&istmg of half-crowns, shillings, and sixpences, amounts to £5. 16s. Qd. If there were twice as many sixpences, half as many half-crowns, and three times as many shillings, the total value of the coins would he £16. 3s. How many coins are there of each kind'? Let X be the number of half-crowns, y the number of shillings, and z the number of sixpences. The number of coins was 112 ; .-. x+y + z=l\<2. (i). The value of the x half-crowns was ^x shillings, since 2s. 6(/.= I shillings. The value of the y shillings was y shillings. The value of the z sixpences was \z shillings. The value of the whole was £5. 16s. Qd., which is equal to 11 6^^ shillings; .-. ^x+y + \z = \\^ (ii). If there had been ^x half-crowns, Zy shillings, and 'Hz six- SIMULTANEOUS SIMPLE EQUATIONS 157 pences, the value of the whole would have been £16. 36\, which is equal to 323 shillings ; ••• f(^^) + 33/ + ^(2s) = 323 (iii). We have therefore three equations involving three unknown quantities. Getting rid of fractions, these reduce to bx+ 23/+ 2 = 233 [ . 5.r + 12y + 4s=1292J We shall proceed to eliminate z. Subtracting the first of these equations from the second, we get 4jp + ?/=121 {a). Multiplying the first by 4, and subtracting from the last, we get ^ + 8y = 844 (6). We have now two equations involving two unknown quantities. To solve these, multiply (6) by 4, and subtract (a), .-. 31?/ = 3255. .-. y=105. Substitute this value of y in (6), .-. ^ + 840 = 844. .-. ^' = 4. Substitute these values of x and y in (i), .-. 4 + 105 + 2=112. .-. 2 = 3. Therefore there were 4 half-crowns, 105 shillings, and 3 sixpences. Note. Several of the examples given in Chapters VI. and XL might have been treated as simultaneous equations. For example, in Art. 99, Ex. 4 (p. 71), we might have supposed that B was x years old, and that A was y years old. We should then have had the simultaneous equations y = ^x, and ?/ + 20 = 2 (« + 20). But in this instance, and in all other similar cases in those chapters, it was not necessary to introduce more than one unknown quantity. 158 PROBLEMS LEADING TO Similarly, many of the problems given in this chapter can be solved by the introduction of only one unknown quantity. EXAMPLES. XIII. 1. Find two numbers, whose difference is 1, such that the sum of the fifth and the seventh parts of the less is less by 1 than the sum of the fourth and the ninth parts of the greater. 2. A number of two digits is equal to four times the sum of its digits : shew that one digit is double the other. 3. A number of two digits is added to another consisting of the same digits reversed, and the sum is 55. The difference of the numbers is 27. Find the numbers. 4. A number, consisting of two digits, is such that -when divided by the sum of its digits, the quotient is 7 and the remainder is 3. The number formed from the given number by reversing the digits is less than the given number by 36. Find the number. 5. A number consists of two digits, one of which is treble the other. Another number is formed from the first by revers- ing the digits. The difference between the numbers is equal to 18. Find the numbers. 6. In a certain proper fraction, the difference between the numerator and the denominator is 12, and if each be increased by 5 the fraction becomes equal to f . Find the fraction. 7. What is that fraction which becomes § when its numerator is doubled and its denominator is increased by 1, and becomes | when its denominator is doubled and its numerator increased by 4? 8. If 1 be added to the numerator of a fraction it becomes equal to |, if 1 be added to the denominator it becomes equal to J : find the fraction. 9. The sum of three numbers is 21. The greatest exceeds the least by 4, and the other number is half the sum of the greatest and least : find the numbers. 10. Two workmen save one-third and one-fourth of their daily earnings respectively. At the end of a year their united savings amount to £20, and the total amount of their earnings was £67. 10s.? What did each earn during the year? 11. A and B have £70 between them; but if A were to lose half his money, and B were to lose one-quarter of his, they would then have only £43. How much has each ? SIMULTANEOUS SIMPLE EQUATIONS. 159 12. A has 6 more shillings than B, but if A give B half his money, and then B give back to A one quarter of his in- creased sum, they will each have the same sum; find what each had at first. 13. Three rabbits cost Qs. 4d. The second was worth lOd. more than the first, and the third M. less than the second. Required the cost of each. 14. When A and B sit down to play, B has two-thirds as much money as A. After a time A has won 155., and then he has twice as much money as B. How much had each at first? 15. A person bought 40 yards of cloth for £18, some at 10*. a yard, and the rest at 7s. 6d. a yard ; how many yards of each kind did he buy ? 16. A grocer buys a quantity of tea at 35. a lb. and also an equal weight of tea at 25. 6d. a lb. If he had divided his money equally between the two kinds he would altogether have bought one lb. more of tea. What amount did he buy, and how much did he spend? 17. A father's age is four times that of his elder son, and five times that of his younger son : when the elder son has lived to three times his present age, the father's age will exceed twice that of his younger son by three years. Find their present ages. 18. A boy is one- third the age of his father, and has a brother one-sixth his own age ; the ages of all three amount to 50 years. AVhat is the boy's age ? 19. Seven years ago, the eldest of three sisters, who is three years older than the next, was twice as old as the youngest, and their united ages were 22. What are their present ages, and how long is it since the age of the eldest sister was equal to the sum of the ages of the two younger ones ? 20. A boy is a years old ; two years after his birth, his mother was 25 years old ; his father is now half as old again as his mother was when the boy was b years old. Find the present ages of his father and mother. 21. In a certain community, consisting of p persons, a per cent, can read and write. Of the males alone, b per cent, can read and write; and of the females alone, c per cent, can read and write. Find the number of males and females in the com- munity. 22. An income of £120 a year is derived from a sum of money invested, partly in a 3^ per cent, stock, and partly in a 4 per cent, stock. If the stock be sold out when the 3| per cents. are at 108 and the 4 per cents, at 120, the cai)ital realised is £3672. How much stock of each kind was there? 160 PROBLEMS LEADING TO 23. £1000 is divided between A^ B, C, and D : B receives half as much as A ; the excess of C's share over i)'s share is equal to one- third of A's share ; if ^'s share were increased by £100, B would have as much as C and I) have between them ; find how much each receives. 24. Two trains, 92 feet and 84 feet long respectively, moving with uniform velocities, on parallel rails, in opposite directions, pass each other in 1^ seconds. If they move in the same direction, their respective velocities being the same as before, the faster train passes the other in 6 seconds. Find the rate at which each train moves. *25. In a contested election 728 votes were polled ; the first of three candidates obtained only 10 less than the second and third obtained together, and he could have given enough votes to the third candidate to have brought him in above the second. How many votes were recorded for each candidate ? 26. A person buys 9 oxen and 20 sheep for £230 ; he sells the oxen at a gain of 25 per cent., and the sheep at a loss of 20 jyer cent., gaining by the transaction £35. Find the price he gave for each. 27. A man spent £200 in buying heifers and lambs, pur- chasing in all 20 animals. If the animals that he bought had all been heifers, he would have paid £160 more than he did ; if they had all been lambs, he would have paid £160 less than he did. How many were there of each kind ? 28. The rent of a farm consists of a fixed money payment, together with the value of 325 quarters of corn, partly wheat and partly barley. When wheat is at 56s. per quarter and barley at 40^., the rent is £900 ; but when wheat is at 485. and barley at 36s., the rent falls to £810. Find the amounts of money, wheat, and barley payable as rent. 29. By investing a certain sum in railway shares paying 3 per cent, per annum, at a certain discount per cent., an income of £315 is obtained. If the same sum be invested in the shares of another railway, paying 4 per cent, per annum at a premium equal to the former discount, the income is increased by £65. Find the amount invested, and the prices of the shares. 30. Divide the number 28 into 4 parts such that, if the first part be increased by 2, the second diminished by 4, the third multiplied by 3, and the fourth divided by 2, the results shall all be equal. ^IMULTAXEOUS SIMPLE EQUATIONS. IGl 31. Find the number of three digits which is equal to i\ times the number formed by its first and third digits, and also is equal to 12 times the sum of its digits, and also is equal to half the number formed by adding 1 to each of its first and second digits and subtracting 2 from its third digit. 32. Divide 48 into three parts such that the first part shall exceed the second part by 2, and be less than the third part by 3. 33. A certain number of 4 digits is unchanged when the digits are reversed : the sum of the digits is equal to the number formed by the first two digits. What is the number i 34. The number of candidates who entered for a certain examination was a number of four digits. The sum of the digits was 20, the two middle digits were ahke, and when the whole number was divided by 100 the remainder was 70. Find the number of candidates. 35. A train travelling from A to C, at a uniform rate of 54 miles an hour, accomplishes the distance in the same time as a train which travels from ^ to a station B (between ^1 and C) at the uniform rate of 60 miles an hour, and without stopping at By proceeds from B to C at the uniform rate of 50 miles an hour. If the distance between B and C be 3 miles greater than that between A and i?, find the number of miles between each pair of stations. 36. A and B run a long-distance race round a J-mile course ; they start together, but when A has completed his first lap, B is 40 yards behind; in 15 minutes after starting .1 overtakes B by overlapping him. How long does B take to run a mile ? 37. A and B run a mile race. If B receive 12 seconds' start, he is beaten by 44 yards. If B receive 165 yards' start, he arrives at the winning post 10 seconds before A. Find the time in which each can run a mile. 38. A and B work together upon a piece of work for six days, A then leaves ofi:' work, and after B has worked alone for two days more, it is found that the work is half done. B then leaves ofi" work, A resumes work, and is joined by a third workman who can do in one day twice the excess of the work done by A in one day over that done by B in one day; and the work is completed in eight days. Find the time in which each workman could do the whole work, and the proportions in which they should be paid. B. A. 11 162 PROBLEMS LEADING TO 39. ^1 and B .start simultaneously from two towns to meet one another : A travels 2 miles per hour faster than B, and they meet in 3 hours : if B had travelled one mile per hour slower, and A had travelled at two-thirds his previous })ace, they would have met in 4 hours. Find the distance between the towns. 40. A man has £100 in sovereigns, half-crowns, and shillings ; the weight of the coins is 235 ounces, and the number of coins is 852. How many coins of each kind has he ? [Assume, for the purpose of this question, that a sovereign weighs j ounce, a half-crown ^ ounce, and a shilling 4 ounce.] 41. A has a certain number of coins, all being sixpences ; B has eight coins less, all being half-crowns ; C has the same number of coins as A and B together have, all of them l)eing shillings. The value of C's coins is the same as the sum of the values of ^'s coins and ^'s coins. What sums have they respectively ? 42. A sum of £2000 is divided into two unequal portions, and these are lent out at rates of interest which differ by one per cent, per annum. It is observed that the income arising from the portion lent at the higher rate of interest, is twice that arising from the other portion ; also that the whole income arising from the £2000 is twice that which would be obtained by lending out at the lower rate the portion which is lent at the higher rate. Find the rates of interest, and how much is lent at each rate. 43. The change for a shilling consisted of a certain number of pence together with twice as many half-pence and some three- penny pieces, making 11 coins in all ; how many coins were there of each sort ? 44. Nineteen shillings' worth of silver consisted of a certain niunber of florins, twice as many sixpences, and the rest half- crowns, making 13 coins in all ; how many of each sort were there? 45. A boy spent his week's pocket money in oranges : if he had got 5 more for his money, each orange would have cost a halfpenny less ; if 3 fewer, a halfpenny more. How much was his week's pocket money \ *46. A gardener took to market two baskets of the same size, one filled with currants and the other with raspberries. He sold all the fruit, except 6 quarts, for 8 shillings, obtaining 4c?. per quart for the currants and Id. per quart for the raspberries ; and then found that the value of the raspberries sold was seven times that of the currants unsold. Find how many quarts each basket contained. SIMULTANEOUS SIMPLE H^UATIONS. 163 47. A qucantity of land, partly amble and partly pasture, is sold at the rate of £60 an acre for the })astiu-e, and £40 an acre for the arable; and the whole sum obtained is £10,000. If the average price per acre had been £50, the sum obtained would have been 10 percent, greater. How much of the land is arable, and how much is pasture? 48. .1 and B enter into partnership with unequal sums of money. They agree that each shall receive interest on his capital at the rate of 4 per cent, per annum, and that all re- maining profits shall be equally divided. At the end of the first year they receive £428 and £508 resi)ectively; and it is found that A thus receives £77 more than he would have gcjt if they had shared in proportion to the capital invested by each. Find each man's capital. 49. An income of £196 is derived from two sums invested, one at 4 per cent., the other at 7 i^er cent. : if the interest on the former had been 5 per cent., and on the latter 6 per cent., the income derived would have been £212. Find the sums invested. 50. The gross income of a certain man was £30 more in the second of two particular years than in the first, but in consequence of the income-tax rising from bd. in the pound in the first year to 8o?. in the pound in the second year, his net income after paying income-tax was unaltered. Find his income in each year. 51. A ship, provisioned for a certain voyage, encounters a storm 16 days after starting, which it is calculated will delay it for 8 days: the daily rations are therefore reduced to ; of the original quantity: a boat is subsequently picked up containing 9 men withcjut provisions, in consequence of which the daily rations are reduced to 5 of the original quantity. What was the number of men at starting, and how long was the voyage expected to last? *52. A newspaper proprietor finds that his receipts are reduced by a shilling in the pound through his town customers paying for their penny papers in foreign bronze and his country subscribers in postage stamps. The bronze pence are pur- chased from him by the Post Office at the rate of 13 for a shilling, and the stamps at a charge of 2^ per cent. The number of his country subscribers exceeds the nimiber of his town customers by a thousand. Find the number -of each. 11—2 164 SIMULTANEOUS SIMPLE EQUATIONS. 53. A cask contains a certain number of gallons of water, and another contains twice as many gallons of wine : six gallons are drawn from each, and what is drawn from the one cask is then put into the other. If the mixture in each cask be now of the same strength, find the amounts of water and wine which they originally contained. 54. A, B, and C walk from P to Q, a, distance of 30 miles; A starts 2^ hours before B, and B 1^ hours before C, and they arrive at Q together. If B had started ^ an hour earlier, he would have passed A 2 hours before A reached Q. Find the rates at which A, B, C walk. 55. A man takes five times as long to run a quarter-mile as he does to run a hundred yards ; but if he could run the quarter- mile at the same pace as the hundred yards he would do it in 6f seconds less time than he does. How long does he take to run each? 56. A journey is performed in a certain time. By travelling 2 miles an hour faster, it would be performed in half-an-hour less time; by travelling 2 miles an hour slower, it would take one hour longer. Find the length of the journey, 57. A person with a sum of £2,596 to invest finds that he can obtain £750 more nominal stock in the 3 per cents, than in the 4 per cents., and lO^j. greater income : what is the price of each stock? 58. If each of the two greater sides of a rectangle be increased by 3 yards, and each of the two smaller sides be increased by 2 yards, its area is doubled : if each of the greater sides be diminished by 3 yards and each of the smaller sides be increased by 2 yards the area is unaltered. Find the sides. 59. A passenger is anxious to reach his destination 21 miles distant at the earliest possible time. Two steamers go there, a slow one which starts at 6 a.m. and a more rapid one which starts at 8 a.m. Sixteen hoiu-s after the latter passes the former they are 80 miles apart ; but if the slow steamer after being passed had increased its pace by one-fourth, and at the same time the quick one its pace by one-fifth, they would have then been 92 miles apart. Which is the better steamer for him to choose ? CHAPTER XIV. MISCELLANEOUS PROPOSITIONS AND EXAMPLES. 171. We here add a few miscellaneous examples which require the application of algebraic processes, but do not come exactly within the range of the preceding chapters. EXPRESSION OF THEOREMS IN ALGEBRAIC NOTATION. 172. We can often render theorems involving pro- perties of numbers immediately obvious by the use of the notation of algebra. The following are a few simple examples. Ex. 1. Sheiu that if the number 4 he divided into any two "parts., their product is less than 4 hi/ the square of half their difference. Let X be one part, .'.4 — ^18 the other part. "VVe want to shew that the product x (4 - x) is less than 4 by the square of half the clifFerence between x and 4 — x We want therefore to shew that ..(4-.) = 4-{^^4^)}\ that is, 4.r - ^2 _ 4 - {.r - 2f = 4-(^2_4^+4) = —x^ + Ax, a result which obviously is true. 166 MISCELLANEOUS PROPOSITIONS. Ex. 2. Shew that the difference between the squares of two consecutive integers is equal to the sum of the Let .V be any integer, .*. the next higher integer is .r + 1. We want to shew that {x-^\f-x' = {x + l)-\-.v, that is, (.r2 + %v + 1 ) - .r2 = 2^- + 1 , a result which obviously is true. EXAMPLES. XIV. A. [/?i this set of examples, the word number refers only to integers.] 1. Prove that, if the sum of two numbers be equal to 3, then their difference is equal to one-third of the difference of their squares. 2. A boy is told to think of a number ; to subtract 1 from It ; to multiply the result by any number, n ; then to subtract 1 ; and finally to add the original number. Shew that the number he originally thought of is one more than the (?i + l)th part of the final result. 3. Prove that the sum of the squares of two quantities is equal to twice the sum of the squares of half their sum and half their difference. 4. Express the following statement in algebraical symbols. " The difference of the cubes of any two numbers divided by the difference of the numbers is equal to the sum of the squares of the two numbers together with their product." Prove that the statement is true. 5. From the difference of the squares of two numbers subtract the square of their difference, and shew that the result is a multiple of the smaller of the given numbers. 6. From the difference of the cubes of two quantities subtract the cube of their difference, and shew that the result is a multiple of both the given quantities. MISCELLANEOUS PROPOSITIONS. IG' SUBSTITUTIONS. 173. The following examples may be treated as illustrations of substitution, though many of them can be proved (often more elegantly) by other processes. Ex. 1. Shew that., if .r - 3/ = 1, then hf- - if-)- = .v' -f + xij. Since A'-y=l, .*. x=y-\r\. • Hence (.r2 - y2)2 = ^^ _ ^)2 (.^ + yf Also x''-fj^:,:y = {y-\-\f-f'\y(^y^\) = ^f-^^y-^\. These results are the same, .*. {pfi — y^Y = .'if'-y^ + xy. Ex. 2. Find the value of a--h--{a-by\ when « + 2/> = 13 and 2a + b = S2. The equations a + 26 = 13 and 2a + 6 = 32 are simultaneous equations. They can be solved by the methods given in the last chapter, and it will be found that a = l7, b= - 2. Hence a^-b-^-{a-by- = l7'--{-2f-{l9f Ex. 3. Shew that, if x=pq, y = qr, z = 7's, and .v-\-y + z=ps, then {x-^yY + {y + zy = {x + z)\ The relation (x+yy- + {i/ + zy- = {x + 2f is true, if .r'- + 2xy +y^--\-y^-\- 2yz +z^= x^ + 2xz + z^. Transpose all the terms to the left-hand side, .-. it is true if %2 + 2xy + 2yz - 2xz = 0, that is, if y (2/ + ^' + 2) - xz = 0, that is, if iqr) (ps) - (pq) {rs) = 0, which is the case. Ex. 4. Shev: that, if x^ = x— 2, i/ien x'^= -x+6. We have x^ = x- 2. .-. x*={x-2y 168 MISCELLANEOUS PROPOSITIONS. = -3.r + 2. .'. .r-^=.^<^x^^ = .r (-3.^ + 2) = -3.r2 + 2A\ But x'^ = x-% .'. :c^= -3(.r-2) + 2.^^= -cc-\-Q, which was to be proved. Ex. 5. If x-\-y + z=0, prove that X {x'^ - yz) -\-y ( ?/2 - zx) +z{z^- xy) = 0. Multiplying out, we have X {.v^ - yz) + y (y2 - zx) + z{z''-- xy) =.ofi+y'^-\-z^ — Zxyz = {x+y-\-z){x''--\-y'^ + z'^-xy-xz-yz) [Art. 108.] .-. =0-x{x''-+y''--\-z''--xy-.vz-yz) =0. EXAMPLES. XIV. B. 1. Find the value of {a — c) {a + c) — {a + cf, when 3« + 2c = 45 and 3c + 2a = 15. 2. If a + 6 = 1 , prove that {a^- - h'^)'- = a^ + h^ - ah. 3. Shew that, if ^2 = ^ _l_ 1 ^ then x^ = hx + 3. 4. Shew that the expression x {y^ - z^) -\-y (z^ — x^) + z (x^ - y^) is not changed by adding the same quantity to x, to y, and to z. 5. If a+T=l, and if «6c+l=0, then will h + - = l. c 6. If a + 6 + c = 0, prove that a^ {b-c) + ¥ {c-a) + c^{a-h) = 0' 7. Shew that, if a + 6 + c = 0, then ah {a + b) + hc {b + c) + ca (c + a) + 3abc = 0. 8. Ux+y + z = 0, prove that (^-3 + y^-^ z^)^ = 2no?y\^. 9. Prove that, if s = a + & + c, then {as + 6c) (65 + ca) (cs + ah) = (6 + c)^ (c + af- {a + 6)2. 10. Shew that, if -^ + -^ = - , then 1 + - = -. a-b a-c a b c a MISCELLANEOUS PROPOSITIONS 169 ELIMINATION. 174. We solved two simultaneous equations of the first degree between two unknown quantities by eliminating one of them — that is, we combined our equations in such a way as to get rid of one of the quantities involved [see Art. 160]. In general we can eliminate one quantity between two equations, two quantities between three equations, three quantities between four equations, and so on ; in each case ob- taining one equation as the result. The following are a few simple examples. Ex. 1, Eliminate y between the equations y — x-\-\ = and y'^-\-^x^ = Z. The first equation gives y = x — \. Substituting this value for y in the second equation wherever y occurs there, we have .'. 3.^2-2^-2 = 0, an equation in which y does not occur, and which is therefore the required result. Note. Wherever we are given two equations involving a certain quantity (such as y in the above example), and one of these equations is of the first degree in it, we can always eliminate the quantity by finding its value from the equation which is of the first degree in it, and substituting this value in the other equation. Ex. 2. Eliminate x between the equations ax'^-\-by''''-^c — ^ aiid bx''--ay''-+d=0. The second equation gives bx'^- = ay^ - d. ay^ - d • • ■* =— r- Substitute this value of x^ in the first equation, «.^^'- -^ + 6?/2 + c = 0. b Simplifying, this reduces to d^y^-ad+b^y^ + bc = 0. .'. (a^ + b-)y^-ad+bc = 0, an equation which is independent of .r, and is therefore the required result. 170 MISCELLANEOUS PROPOSITIONS. EXAMPLES. XIV. C. Eliminate x between the equations given in examples 1, 2, 3. 1. ±v + Zy-\=0 and Zx-2y + l=0. 2. ax-hy = and x^-\-xy-{-y'^ = ah. 3. x^a-l = and x''--cv^ + l = 0. Eliminate y between the equations given in examples 4, 5, 6. 4. y-x = and 2.rH S?/^ + 4 = 0. 5. y = 'mx-\-c and y'^ = 'iax. 6. - + | = 1 and -,+f-! = l. 7. Eliminate y and ^ between the equations 2.r4-3y-^ = 3^v-2y + 2s = 4.u+3/-5,: = 3. SYMMETRY. 175. Symmetrical Expressions. If an expres- sion involving certain letters be such that its value is unaltered when two of the letters are interchanged, it is said to be symmietrical with respect to them. Thus each of the expressions a + h and ^a^ + Zab + 2b'^ is sym- metrical with respect to a and h ; for, if a and h be interchanged, the values of the expressions are unaltered. If an expression involving certain letters be such that its value is unaltered when any two of those letters are interchanged, it is said to be symmetrical with respect to all of them. Thus each of the expressions ahc and a^ + h^ + c^ + d^ + Zahc is symmetrical with respect to «, 6, and c ; but the second of them is not synnnetrical with respect to a, 6, c, and d ; since if a and d be interchanged, its value is altered. Similarly, the expression ia-b){b-c) ' {b-c){G-a) {c-a){a-b) is symmetrical with respect to a, b, and c. 176. Cyclical interchanges. The value of the expression at the end of the last paragraph was found in Art. 150, Ex. 4 (p. 120). As written in Art. 175, the three terms which compose it are so arranged that MISCELLANEOUS PROPOSITIONS. 171 the second term is obtainable from the first by writing h for a, c for h and a for c ; the third term is obtainable from the second by the same rule ; and if we apply the same rule to the third term, we get back to the first term. Letters so arranged are said to be taken in cyclical order. One advantage of writing the above expression in this way is that from one term each of the other terms can be written down by symmetry. And always, if the letters of an expression, such as a, 6, c, ... , be arranged round the circumference of a circle, then to make a cyclical change in the expression we replace every letter by the letter immediately in front of it. 177. The form in which an expression can be written is a matter of great importance, and the student will find that his power of successfully applying analysis to the problems he has to solve will often depend to a large extent on his power of arranging his symbols in a symmetrical form. 178. Attention to the symmetry of expressions will moreover frequently save the student from mis- takes in his work. For example, if in the product the coefficient of .r^ were said to be a-\-h + 2c, it is obvious (by inspection) that there must be an error; for the given expression is symmetrical with respect to a, b, and c, and therefore the coefficient of every power of x in the product must be so. 179. Examples of the following kind are not un- common in the applications of algebra. A rational integral algebraic expression of two dimensions in X and y is symmetrical and homogeneous. Its value is 4, wlien x=\ andy=\', and its value is 1, when .r=0 and y=l. Find the expression. Since the expression is homogeneous and of two dimensions 172 MISCELLANEOUS PROPOSITIONS. in X and y, every term in it must be of one of the forms ax'^, hxy, and cy^. Let u stand for the expression, .-. u=ax'^ + hxy-\-cy'^, where a, h, c are numerical coefficients which we have to find. Again, the expression is symmetrical in x and y ; that is, if x and y be interchanged, no change is made in the expression. This requires that a shall be equal to c. Hence we must have u = ax^ + hxy + ay^. Now we are also given that if x = \ and y = l, then ?^ = 4, • •• A = a-\-h + a (i). Also, if ^ = and y = l, then il= 1, .-. 1=0 + 0+^ (ii). The equations (i) and (ii) give, when solved, a = 1 and 6 = 2. Substituting these values, we obtain ^i = x^-\-'^xy+y'^, which is the required expression. It will be observed that we first wrote down a homogeneous expression, next we made it symmetrical, and lastly we deter- mined the coefficients that were then unknown by means of the given relations. EXAMPLES. XIV. D. 1. Which of the following exj)ressions are symmetrical, and with respect to which letters are they symmetrical ? (i) abx + aby''"; (iii) a^ + ^^ + Sc^; (ii) a2 + 62-Hc2; (iv) {a-h)c^-\-{h- c) a^ + {c-a)h\ 2. Can a-[-2h + c be the coefficient of x^ in the product {x +b - c) {x + c - a) {x +a - b) 1 Can a^ + b^-\- c^ be the coefficient of x^ in this product ? 3. "Write down by cyclical interchanges of a, b, and c the quantities corresponding to (i) b-c; (ii) 62_c'2; (iii) a(b-c); (iv) {a-b){b + c). 4. Find a rational integral homogeneous expression of the first degree in x and y, which is equal to 3 if .^7= 1 and y= 1, and is equal to 4 if .r = 2 and y = l. 5. Find a rational integral homogeneous and symmetrical expression of the first degree in x and y, which is equal to 6 if x = l and y = l. MISCELLANEOUS PROPOSITIONS. 173 COMPARISON OF UNEQUAL QUANTITIES. 180. Comparison of unequal quantities. The comparison of the magnitudes of unequal quantities depends on the following propositions. An inequality is unaltered in character (i) if equal quantities be added to each side of it, or (ii) if equal quantities be subtracted from each side of it, or (iii) if each side be multiplied or divided by a positive number ; but (iv) an inequality is reversed in character if each side be multiplied or divided by any negative number. To prove these propositions, consider the definition of an inequality. A quantity a is said to be greater than a quantity 6, that is, a>h, if a — 6 be positive [Art. 48]. But, if a — 6 be positive, then {a±x) — (h± x) is positive, and therefore a±x>h ±x, that is a ±x \^ greater than h±x. Similarly, \i ah, then a — 6 is positive. Therefore, if m be a positive number, m{a — b) is positive, and there- fore ma > mh : but, if m be a negative number, then m{a — h) is negative, and therefore ma < mh. This proves (iii) and (iv). As a particular case of (iv), we have the result that an inequality is reversed in character if the sign of each side he changed (that is, if each side be multiplied by - 1). 181. To compare two or more unequal fractions, we must express them in an equivalent form having a common positive denominator. To do this, we must find the L.c.M. of their denominators and make this L.C.M. the denominator of each fraction. 174 MISCELLANEOUS PROPOSITIONS. The following examples will illustrate the use of Arts. 180, 181. Other examples will be found in Art. 296, which the student may here consult. Eob\ 1. Arrange the fractions H, 2, ^-^ in order of magnitude. The L. c. M. of their denominators is 24 ; .•. we have I -^^f, and f = A-f. Hence i<2 4rh is the greater, a and b being positive quantities. Ine fraction -7 is > or <- , , a + 26 a + b { a-\-3b){a + b) Ja + 2by^_ {a + 2b){a + b) ^ ^(a + 6)(«+-26)' that is, as (a + 36)(a + 6) is > or <(rt + 26)-, that is, as a^^^ab^-W- is > or or < 46-. But 362 is < 46^. a + 36 . rt + 26 * ■ a + 26 ^'"^ "^ a-\-b ' Ex. 3. Determine 'which is the qreater of the fractious = and ^ — , , ivhere a is any positive niwiber and b is a positive niunber less than 7. rvu f 4.- Q + a . _ 6-6 the fraction - is > or < ,= — , , l-\-a i -b ,. (6 + a)(7-6). ^(6-6)(7 + a) according as ^^- ^^y^^^^) ^ > «^' < (7^^57(7 +a) ' that is, as (6 + a) (7 - 6) is > or < (6 - 6) (7 + «), [if 7>6, Art. 180 (iv). that is, as 42 + 7a - 66 - ah is > or < 42 - 76 + Q>a - ab, or, transposing, as 7a — 6a + 76 - 66 is > or < 0, that is, as a + 6 is > or < 0. But, if both a and 6 be x)ositive, a + 6 is > 0. 6+a . 6-6 •'■r+a''> 7-6- M ISC ELLA NEO US PROPOSITIONS. 1 75 Ex. 4. Shew (hat, if a and h he unequal quantities, then the sum of a?" and 6- loill he greater tlian 2ab. Wc sec that a^ + b'^>2ah, if a^-2ah + h~>0, that is, if {a-hy>0. Now a is not equal to h, and the square of any quantity (whether positive or negative) is a positive quantity and there- fore greater than zero ; .-. {a-bY>0. Hence a^-\-h^>2ab. Note. If a = h, then a-h^O, .-.(«- b)- = 0, . • . a'^ + h^ = 2ah. Wc may tliereforc say that, in all cases, a^ + ?/^ <|: 2ah. Ex. 5. Shew that, if x^ = ar--\-l)\ and i/- = c--\-d'\ then xy will he greater than ad+bc, provided that ac is not equal to bd, and x and y are of the same sign. Wc see that xy >ad+ he, if x'f>{ad-^hcf, that is, if (a2 + 62) (cHc^^) >{ad + hef, that is, if a'-^c- + aH'^ + 6 V + bW > dhP -f 2ahcd + IS^e^ or transposing, \id^c'^—2abcd-\-b'H'^>0, that is, if {ac-bdf>0. Now ac is not equal to bd, and therefore {ac - hd) is not equal to 0. Also the square of any quantity (whether positive or negative) is a positive quantity and therefore greater than zero, .-. {ac-bdf>0. .'. xy>ad+bc. EXAMPLES. XIV. E. 1. If X is positive, which is the greater of the fractions '-— -^ ,X+4: and -— ? x + 7 1^ 4- 5 5 2. For what values of x is ;— -- gi-eatcr than - I >■ I jj ^ + 2 3. For what values of x is ' --, greater than — -_ ? X + 4: X+0 4. Shew that -7 is never greater than „ ( -r, -H yv ah 2 \a- /r 176 MISCELLANEOUS PROPOSITIONS. MISCELLANEOUS EXAMPLES. XIV. F. 1. Shew that half the difference between the cubes of two consecutive odd integers is greater by unity than three times the square of the intermediate even integer. 2. A number is equal to the product of two factors ; if one of these factors be increased by 1 and the other diminished by 2, the product is increased by 1. Prove that, if the factors be again increased by 1 and diminished by 2 respectively, the product will be less by 2 than the original number. 3. Prove the truth of the following statement. Take any two proper fractions such that their sum is unity, subtract the square of the smaller from the square of the greater, and add unity to the remainder ; the result will always be equal to twice the greater of the two fractions. 4. At an election there are two candidates, A and B, of whom one is to be chosen. A has a majority of those who vote by proxy and also of those who vote in person, and it is observed that if a certain number of those who voted for him in person had voted by proxy and thus trebled his majority by proxies only, then his majority by voters in person would be \ of what it was before. Shew that his whole majority is fom' times his majority by proxies only. 5. A common conjuring trick is to ask a boy among the audience to throw two dice, or to select at random from a box a domino on each half of which is a number. The boy is then told to recollect the two numbers thus obtained, to choose either of them, to multiply it by 5, to add 7 to the result, to double this result, and lastly to add to this the other number. On men- tioning the final number thus obtained, the conjurer knows the two numbers originally chosen. How is this done 1 6. Shew that, if ^ - 3y = 2^* + ?/ - 1 5 = 1 , then x^ + y^ = 4x^ - 3. 7. Find the value of 25^2 + (a + 46)2, when 3a + 26 = 7 and a-6 = 4. *8. If ^^ = 1 + V2, prove that x{x-\)=x+\. 9. U {x + l f = X, find the value of 1 L^'^ + 8^-2 + 8^- - 2, 119 119 10. U1+ -— = -^, shew that - + ^ = -. a a- c a-b a o c 11. Shew that the expression x^ + ff + z^~^z-zx-x}/ is not changed by adding the same quantity to x, to y, and to z. MISCELLANEOUS PROPOSITION'S. 177 12. If (a + -V = 3, prove that cv^+\ = 0. 13. If « + ^> -I- c = 0, prove that (i) a?-[^h'^^c^ + ^{a^-h){h + c){c + a) = 0•, ('') W+d^-a^ '^ c'+a^-b-' "^ a2 + ^2_^.2 = 0- *14. Write down the most general possible rational integral algebraical expression of the second degree in .v and ?/, (i) which is homogeneous in .v and i/, (ii) which is both homogeneous and symmetrical in x and y. Also, write down a rational integral algebraical expression which is of a degree not higher than the second and which is symmetrical. *15. A homogeneous expression of two dimensions is sym- metrical in .r, ?/, and z. Its value is 9 when .v=^ = z=l, and its value is 36 when .v=l, 2/ = 2, z = 3. Find it. 16. Eliminate :c between the equations 2x^ — 7/^ = 3i/'^-/v^ = 2. *17. Eliminate x, ?/, and z between the equations CLV -\-hy-\-cz = bx-\-cy-\-az = ex + ay ■\-hz = 0. ^2_^,2 a^ — .r^ 18. Of the fractions —^ — —^ and — ; — '—. , which is the greater ? *19. If A'2+ 3/2 = ^2, then will x^+y^^22. If a {by + cz- ax) = b{cz + ax- by) — c {ax + by- cz), and if a + b + c = 0, shew that x+y + z = 0. *23. Prove that if a + 6 + c + o? = 0, then a^ + P + c^ + d^ + S{a + b){b + c){c-^a) = 0, and {a + b){a + c){a + d) = {b + c){b + d){b + a) = {c-\-d)(c + a){c + b) = {d+a){d+b){d+c)=-{a + b){b-\-c){c + a). *24. Shew that, if a, b, c, d be positive quantities of which the sum of any three is greater than the fourth, and if ( c + c^)2-(a-6 )2 {d^a f-{b-cf \a-\-c^-df-b'^ "^ {a+b + df~c^ ~ ' then a-\-b^-c + d. B. A. 12 EXAMINATION PAPERS AND QUESTIONS. [The first two of the folloiving papers {A and B) contain those questions on Elementary Algebra on which Junior Students in two of the recent Ca7nhridge Local Examinations ivere expected to satisfy the Examiners — but the two questions in each paper which are marked luith a * are concerned with subjects treated in Chapters XV, XVI. The next two papers (C and D) contain all those questions on Elementary Algebra on which Candidates in two of the recent examinations for Higher Certificates of the Oxford and Cambridge Schools Examination Board were required to pass. These papers are followed by groups of questions taken from papers set in recent years under the authority of the same Examination Board to boys in the lower forms of various Public Schools.] Paper A. 1. Simplify {b — c){c + a) — {c-a){a + b)-a(a-{-b-c). x + y x — y 2(x^ — y^) 2. Simplify o"^o+ TT-^3- 4 , 2 f r-A' 3. Divide {x"- - y'^f - ^ {x"^ + ?/2) + 1 by .^2 - (3/ + 1 f. 4. Resolve ^x^ + 13^-6 into factors. 5. Resolve ab (c^ + d^) + cd {ci^ + b^) into factors. *6. Extract the square root of a^ + ^xy-\-x^- 4.^5 - 1 2.^4 - 8.1-3 _,_ 4,^2 + 1 e.r + 1 6. 7. Solve the equation {x - 3a) (3.1- -a)- {x - 2a) {2x - a) = {x - af. *8. Solve the equation 30 3-.r_ x-\-% 5 "*'' • 9 Prove that, if a2 + a + 1 = 0, and if 0^ = 1, then x^-\ = {x-\) {x-a){x-a^). 10. The Eiffel Tower is 580 ft. higher than the spire of Salisbury Cathedral, while the number of inches in the height of the spire is 4520 greater than the number of yards in the height of the Tower. Find the height of each building. EXAMINATION PAPERS. 179 Paper B. 1. If a = 3, 6 = 4, c = 5, d=Q, find the numerical values of 2 \/^M^+\/a^ + 63 + c3 _ ^^^ {^aj^h){ c+ d)-{b + c) (d+a) d-c + h — a ' '' ah—hc-\-cd — da 2. Divide ^7- 13^- 30 by x'^-2x + 3. 3. If x'^-{-7a; + c be exactly divisible by .r + 4, what is the value of c ? 4. State and prove the rule for finding the Least Common Multiple of two algebraical expressions. Find the l.c.m. of 9x^-x-2 and 3x^- lOx^-lx-^. *5. Find the relation between a, b, c in order that ax^ + bx + c may be a perfect square. Extract the square root of 9^-6 -12^^ + 22.^ + .v^ + 1 2^ + 4. 6. Simplify L^4-^ + -LJL^. 7. Simplify jl —^-i- -^1 jl + -4l - -^A . 8. Solve the equations (i) i[^-^.^-i(^-^'')}] = 53; 7.y-ll 3.77-2 ^2.v + 5 ^ ^^^^ 4^-7 12^77-1 x+2 ' (iii) lly-^=10, 11.2;-101y = 110. 9. A boy spent one-third of his money in cakes, one-fourth in apples, one-fifth in oranges and one-sixth in nuts, and has 1^0?. left: how much had he? 10. In reducing a quantity of ore, it is passed through three processes which remove respectively — th, - th and - th of whatever is subjected to them. If the weight left be 120 lbs., and the weight lost in the third process he 30 lbs., 40 lbs., or 60 lbs. according to the different orders in which the processes can be performed, what was the original weight ? 12—2 180 EXAMINATION PAPERS. Paper G. 1 . Eemove the brackets from a^ — [(6 - c)^ - {c^ -{a — hY] ]. 2. Find the vahie of .i-^ -\-{p-Z)x^ + {q- 3p) x - 2q, when x = 3. 3. Divide (/ - hh/ + by - ¥) {f + by + by + ¥) by y* - b\ 4. In finding the h.c.f. of two given expressions, can we reject either altogether or temporarily a factor occurring in both expressions ? Justify your answer. Find the h. c. f. of x'^-Sx'^+x^-4x^ + l2x-4 and ^x^-Sx^ + Zx"- -3x-{-l. 5. Simplify ^-- - —^ + (i) b a^-b ' «26-62 ^ ^- Aab ^ , , , T „.r-2a .r + 2a IQab 6. If .r = -, , find the value of — -7^. ^ - . ,9- . 7. Solve the equations : 2.1^4-1 4Q2-3.y_ 471 -6.r ^ ~29 12 2 ' 8. Two persons, a certain distance apart, setting out at the same time, are together in 25 minutes if they walk in the same direction, but they meet in 15 minutes if they walk in opposite directions. Compare their rates of walking. Paper D. 1. If a = 4, & = 5, and c = 3, find the value of ^{5 (62 _ c2) - a^] + ^3{a(a2-c2)-l}. 2. Express .r^ — 8.^; - 84 in factors. 3. Find the factors of (2.i- + 3)^ - {x - 3)2. 4. Divide a? Q) - c)-\-b^ {c- a)-\-c^ {a -b) by rt + 6 + c, and find the factors of the quotient. 5. Shew that if a quantity x divide A and B exactly, it will also divide mA + iiB. Find the h.c.f. of G.v* - 2.^^ + 9.^2 + 9^. _ 4 ^nd ar4 + 80.r2_9 What value of x will make both these expressions vanish ? EXAMINATION PAPERS AND QUESTIONS. 181 6. Define a fi-caction, and hence ])rove that t = — r . b mb Reduce the following fractions to their simplest forms : ... x^-bx^ + A (ii) Zx + 2 ' {b-cf + {c~af-\-{a-bf {a-b) {a- c) + {b - c) {b - a) + {c - a) {c - b)' 7. Solve the equations : ,., x + 2 x-2 „ W 1^3 + ^6 = ^' ,.., 2x . 3^-1 5^^-11 ^ {a + b)x'-{a-b)i/ = 3ab,] ^ ' {a-\-b)y-{a-b)x= ab.f 8. Find the diflerence of the squares of the highest and lowest of any three consecutive numbers in terms of the middle number. Examination Questions. 1. Remove the brackets from the expression - 3 [{a + b) - {(2a- 36) -(5a + 76- 16c) - ( - 13a+ 2b-3c-bd)}l and find its value when a = l, 6 = 2, c = 3, d=4. 2. Find the highest common factor of x^-\-\ and .v* - x^ — 2. 3. Find the lowest common nniltiple of ^2_5.i- + 6, x^--^x-\-3, and x^-3x + 2. 4. Reduce the following fractions to their simplest forms : ... bx^-Ax-l ,.., x'^ + x-6 2xi - 3^2^ 1 ' ' ' x^- 13^H 36 ■ 5. At an election where there are three candidates A, Z), C, and two persons to be chosen, A obtained 486 votes, B 461, C 457. Every elector voted for two candidates. How many voted for B and C ; how many for C and A ; and how many for A and^? 6. Prove that, if a^+pa'^ + qa-\-r=0, then x^+px'- + qx + '> a multiple of x - a. Find the factors of x^ + 8x'^- 79^7+70. 182 EXAMINATION QUESTIONS. 7. Prove the rule for finding the highest common factor of two algebraical expressions : and state the circumstances under which a factor may be introduced or omitted at any stage of the process. Find the h.c.p. of a'^-h'^ and 27a3 + 16a26 + 30a62 + 4l63. 8. Add together the fractions aH2^>c _6M-2ca_ cH2a6 {c-a){a-by {a-h){h-cy {b-c){c-a)' 9. Solve the equations : ,., ^ + 8 x-14: ,.., x-2 j; + 2 18 x-4: x-2 ' a; + l x—l x^—l' 10. A vessel can be filled by three pipes, L, M, N. If 31 and ^y run together, it is filled in 35 minutes ; if N and L, in 28 minutes ; if L and 31, in 20 minutes. In what time will it be filled if all run together? 11. A boy is a years old, his mother was b years old when he was born, his father is half as old again as his mother was c years ago. Find the joresent ages of his father and mother. 12. Prove that every common multiple of any number of algebraical expressions is also a multiple of their lowest common multiple. Find the highest common factor and lowest common multijjle of a^^-Sab-lOb"-, a2 + 2a6- 3562, and a^-8ab+l5b^. 13. Simplify the expressions 2a - 36 36 2-\-x 3-x 6-a; , 2a-66 2a + t: ; and x + d x-4: x + 2' 2a-Sb 36 + 2a 2a -6b 14. A merchant added to a cask of wine ten per cent, of water. If he were now to add nine gallons more water, the mixture would contain fifty per cent, of water. How many gallons of wine had he at first] 15. If |lZi_+_li:£ +^3/^0, prove that two of the 1+3/2 l+zx l-\-xi/ ' ^ quantities x, y, z are equal to one another. 16. Find the continued product of EXAMINATION QUESTIONS. 183 1 7. Find the h. c. p. of x^ - x and .^"* - 2.^2 + 3.); - 2 ; and also the G. c. jr. of their numerical values when .r = 2. Account for the fact that this G. c. m. is not the same as the numerical value of their h.c.f. when x = % 18. Subtract —^ — =-z ^r^r from .^•2 + ll.r+30 .^2 + 8^+ 15' 19. Solve the following systems of equations: (i) lLv-17y=17.f-28j/-|-2 = 28; (ii) {ci-\-h) x-ay^a?, {cfi^-h'^)x-ahy = a?. 20. A farmer bought a certain number of sheep for £30. If he made a profit of 20 per cent, (on those sold) by selling all but five of them for £27, find how many he bought. 21. Simplify (^ - -^Vf^ - ^ ) • ^ ^ \x-\-y x-yJl\x'^+y'^ x^-y'^j (a + 2.r)2 22. What must be the value of x in order that + 70c/^ + 3.2,-2 may be equal to 1^- when a is equal to 67? 23. Solve the equations : ,., 48-:^' 352-20.r 5.^^-28 3 « -¥- + —¥— = -5- -4'^-' (ii) -4.2;- -OU- + •002.^■ = 11-7 --OOlOx 24. Solve the simultaneous equations ax -{-cy -hz = a^, cx + by—az = 6^, bx -ay + cz = c^, 25. .4 can walk forwards foiu* times as fast as he can back- wards, and undertakes to walk a certain distance {\ of it back- wards) in a certain time. But, the ground being bad, he finds that his rate per hour backwards is \ mile less than he had reckoned, and that to win his wager he must walk forwards two miles an hour faster. What is his usual rate per hour forwards % 26. Find the factors of 20a24.21a6 - 276^. 27. Shew that the following system of equations is indeter- minate : 7a;+y- 52 = 24, bx-Zy^1z = \\, 4?/ - 7^ + 2.^ = 13. 28. An egg-dealer bought a certain number of eggs at \s. Ad. per score, and five times the number at 6s. 3d. per hundred. He sold the whole at lOd per dozen, gaining £1. 7s. by the transaction. How many eggs did he buy ^ CHAPTER XV. EVOLUTION (square ROOTS AND CUBE ROOTS). 182. We have already explained [Art. 23] that the w*'' i'oot of an expression is a quantity such that its 71^^ power is equal to the given expression ; and that if the given expression be denoted by X, its ?i*^' root is represented by a/X. A quantity which has an exact 7i*'' root is called a 2)erfect 7i"' potver. In particular, a quantity which has an exact square root is called a perfect square, and a quantity which has an exact cube root is called a i^erfect cube. When no exact quantity can be found which when raised to the ti"^ power is equal to a, then the ?i"' root of a is called an irrational quantity. An expression which involves no irrational quantity is said to be rational. 183. The process of finding a root of a given expression is called evolution. The most direct test whether a certain quantity (which we may denote by y) is the ?i"' root of a given expression (which we may denote by X) is that the n^^'' power of y is equal to X. That is, y^N\\\ = yX, provided y'' = X: and if 7/'^= A", then y = yX. Of course (VZf=Z. EVOLUTION. 185 *184. Since rt"-^ = (±«)^ it follows tlint Ja'^ = ±a ; that is, cither 4- a or - a is the square root of a. Thus, strictly speaking, there are two square roots of a quantity denoted by a^. Again, we shall see presently that there are three roots of the equation x^ = a^^ and strictly speaking, there are therefore three cube roots of a quantity denoted by a^. The student will find later that similarly the number of n^^ roots of a given quantity is ?i, but some of these are imaginary, and since we here confine ourselves [see Art. 119] to real values of the quantities considered, we shall make no attempt to find more than one root out of all the ^i"* roots of the given quantity. We shall moreover adopt the usual notation and speak of any ?i"' root as the ti*"" root, though it would be more accurate to say an n^^ root. 185. Roots of Simple Quantities. If we want the /i"' root of a power of a simple quantity, and if the index of the power to which that quantity is raised happen to be a multiple of n (say, ^ times 7i), then the root required will be the quantity raised to the j)^^ power : in other words, we must divide the index of the power to which the quantity is raised by n. For, if y be the 'n}^ root of" a quantity like a"^', we have y = ya^P. :. yn = ce^^ = {aPy\ [Art. 7G] 7/ = aP. For example, the cube root of a^ is a raised to tlie power of 2, (that is, 6 divided by 3), which is expressed in symbols thus. Similarly, ^la'^^ = ce\ 4/.^io = ^-2; and ;y.v2i = A-7. 186. Roots of Products. The w^'' root of a product of any factors is the j^^^oduct of the ?i^'" roots of the different factors. Suppose there are two factors, a and h. We want then to shew that {/{ah) is equal to {/a ,{/b. 186 EVOLUTION. Now {l/a . ^hT = (yaf . {'{/hf [Art. 75] = a.h [Art. 183] = ah. ^Similarly, ^{abc) = {/a. y/b .y/c. Ex. 1. v^(rt46'2) = ^/a4 . ^62 _ a%. Ex. 3. ^(a263) = ^a2 ^ ^^3 == ^^2 ^ 6. EXAMPLES. XV. A. Find the values of the following quantities. 1. sjx^^. 2. J{a%% 3. V|^. 4. ^^. 5. Vf . 6. V49^ 7. ^/|S. 8. .V^. 9. 4^ -^3. 10. \^27a'^6^. 11. ^-8^-j. 12.^ 13. ^/^y". 14. ^32^^^. 15. y^- 243^^*1,. 16. !y^ 187. Square Root of a Compound Expression. The general method for obtaining the square root of a compound expression is somewhat more complicated. We shall first confine ourselves to the comparatively simple cases of expressions containing only three terms, and of expressions which are of the second degree in some quantity ; and after discussing them shall consider the general case of any compound expression. We shall assume that the expressions considered are perfect squares. We shall also suppose that every expression has been reduced to its simplest form, and arranged in descending powers of some letter. 640^4 27c^^' 6^". EXTRACTION OF SQUARE ROOTS. 187 188. Square Root of a Trinomial. First, consider the case of a trinomial expression, that is, an expression consisting of only three terms. Suppose it to be arranged in descending powers of (say) :v. Let us suppose that its square root is an expression like A ±B, where the quantities, A, B, are arranged in descending powers of .r. Then the given expression must be the same as {A + By, that is, as A'^±2xiB + B^, the terms of which are arranged in de- scending powers of a\ If then the trinomial be a perfect square of a quantity like A ±B, and if it be arranged in descending powers of x, then its first term will be A'\ and its last term will be B'^. Hence the square root of the first term will be equal to A, and the square root of the last term will be equal to B ; and their sum or difference will be the square root required — the sum or difference being taken according as to whether the sign of the middle term is positive or negative. The rule applies only to those cases where the given trinomial is a perfect square. Hence (unless we know this to be the case) the correctness of the result must be tested by forming its square and comparing it with the given trinomial. Example. Find the square root of 9.^^ — 6x^a^ + a\ Here we take the square root of the first term, which is Sx^ ; and then the square root of the last term, which is a^ ; also, since the middle term is negative, we take their difference. Hence the required square root is Zx"^ - a^. The correctness of the result can be tested by forming its square, which will be found to be equal to the given quantity. The reader will remember that either + (3^'^ — d^) or -i^x^ — d') may be taken as the square root required [Art. 184]. EXAMPLES. XV. B. Find the square roots of the following expressions. 1. a%2_2a6^ + 62^ 2. 121^2_374^+289. 3. If^'rlyz^lz^ 4. ^a^a^-\-\^a^a^ - Ma^a^a^a^, 5. 64-8j9 + ijo2. 6. blOt^mn + milhn?' + 225^2,^2^ 7. 1 1^-2 9a;2 3y "^ 4?/^ ' 8. 1 ^2 1 188 EVOLUTION. 189. Square Root of a Quadratic Expression. A quad- ratic expression is one which contains only first and second powers of some quantity, sucli as x. AVe can therefore arrange the terms in three groups, namely, those that involve x^, those that involve x, and those that are independent of x. When thus written as a trinomial, the preceding rule applies ; but since it is sometimes difficult to write down by inspection the square root of one of these groups of terms, we can (if we prefer) proceed by the following method. Let us compare the given expression with the square of an expression like A + B, where we will suppose that A involves X but that B is independent of x. If A + B be the required square root, then the given expres- sion must be identically equal to (^4 + Bf, that is, to ^4- -f 2.45 + B-, the terms of which are arranged in descending powers of x. Hence, if A be taken equal to the square root of that part of the given expression which involves x'^, then the terms in the given expression which involve x to the first power must be 'iAB. Thus, the remaining part of the square root (namely, B) will be obtained by dividing by 2.4 those terms in the given expres- sion which involve the first power of x. [It may be noticed in passing that the square root of d^ + 6- is not a-\-h\ since the square oi a + h is d^-{-b'^ + 2ah.'\ Ex. 1. Find the square root of x"- -f a2 + 52 ^ 2a.2; - 2hx - 2ah. Arranging the expression in descending powers of x^ we have x'^ + 2{a-h)x + d^-^ah-\-lfi. Hence the first term in the square root is the square root of .^•2, that is, x. If the expression be a perfect square, then the rest of the square root is the quotient of 2 {a — h) x by 2c^', that is, {a - h). The remaining terms in the given expression, namely a^ — '2ah + b^^ are the square oi a — b. Hence the required square root is x + a-b. Note. If the square oi a-b had not been equal to the group of terms in the given expression which were independent of .r, the given quantity would not have had an exact square root, Ex. 2. Find the square root of a^ + a%2 _ 2aH^ - 2ax^ -I- 2x + 2a^ + 2a + x^ + 3a^-\-l. EXTRACTION OF SQUARE ROOTS. 189 Arranging the given expression in powers of x and a, we have The square root of the terms involving x^- is {a-\)x. This then is the first term of the required square root. If the given expression be a perfect cube, then the quotient of —'2,{a^-\)x by 2{a — \)x will give the remaining terms of the root: this quotient is —{a^ + a-\-l). The square of this last quantity is equal to the terms in the given expression which are independent of x^ namely. Hence the required square root is {a-\)x-{a?-\-a^-\). Note. If the square of -{a?-\-a-\-\) had not been equal to the group of terms in the original expression which were in- dependent of x^ the given quantity would not have had an exact square root. EXAMPLES. XV. C. Find the square roots of the following expressions. 1. A^H2aA^ + 26^'-l-rt2 + 2a6 + R 2. a2^'2 + 2ahx - 2ax + &2 _ 26 + 1 . 3. a\v^ + Mhx + 62 _ 2hx + .r2 - ^ax'^. 4. aKv"- - 4a^2 ^_ 4^,2 _ 2t«6^ + 4.hx + 2ax - Ax + h'^ - 26 + 1 . 5. a2 + 962 + 2 5c2 - 306c + 1 Oca -6ab. - 2 «'\ ^^ , 2a.^' ab , 6. -^ + ^ + I + -3--T-^^^'- 190. Square Root of a Multinomial. The preceding rule can be extended so as to enable us to find the square root of any multinomial expression. We shall, as before, assume that the expression is a perfect square, and that it is arranged in descending powers of some letter, such as x. Let us suppose that its square root is an expression like A + B -\- C -\- ... , where the line of dots after C signifies that there may 190 EVOLUTION. be a number of similar terms, and where the successive terms, ^, B,C..., are arranged in descending powers oix. The given expression is equal to (A + B + C + . . .f. Denote B + G+...hj P. Then (A+B + c+,..y = {A+py = A''-h2AP + P' = A' + 2A(B + G + ...) + (B+C+...)\ The term of the highest dimensions in this last ex- pression is A"^, and since this last expression is the same as the given expression, the term of the highest dimen- sions in the given expression must be equal to A^. Hence the first term of the required square root, that is, A, is the square root of the term of highest dimen- sions in the given expression. Now, since A, B, G ... are arranged in descending powers of oo, therefore 2AB is of higher dimensions than any of the other terms in 2A(B -}-G + ...). Again, B^ is the term of highest dimensions in the expansion of (jB+ G+ ...y, but 2AB is of higher dimensions than B^, because A is of higher dimensions than B. There- fore (except for A^) the term 2AB is of higher dimensions than any other term in the square of (A + B + G -\- ...). Hence, if we subtract A^ from the given expression, the terms of the highest dimensions in the difference must be equal to 2AB : if therefore these terms be divided by 2A, we obtain B, which is the second term in the required square root. We now know A and B. By similar reasoning to that given above, we see that if we subtract (A -{-By from the given expression, the terms of the highest dimensions left will be equal to 2AG : if therefore these terms be divided by 2A, we obtain G, which is the third term in the required square root. Proceeding in this way, every term in the required square root can be successively obtained. EXTRACTION OF SQUARE ROOTS. 191 191. Process of extracting a square root. The process of extractiug the square root of a given expression is usually arranged in three columns, as shewn in the following example, where the square root of is determined. The first term of the square root is the square root of xf\ that is, x^. This is written to the right of the given expression in the column III. Subtracting the square of this from the given expression, we obtain 'iaf' + ^x'^ — 2jc^ + 5.^•- — 2^- 4- 1 . Twice the part of the root already determined is then written in the column I. to the left of the expression. By the above rule, we have now to divide the first term of the remainder, which is written in column II., by 2xK The quotient, namely -f 2.^^^ is then written both in the column I. and the column III. Thus I. 11. III. I afi + 4x^ + 2x^-2x^ + 5x^-2x+l {x^ + 2x'^ ^6 2.273 + 2.^^ I 4:a;^ + 2x^-2a;^ + 5x^-2x+l We now have to subtract (x^+2x^Y from the given expression, that is, we have to subtract x^ + 2x^ (2.v^ + 2x^). We have already subtracted x^\ and w^e have thus only to subtract from the re- mainder the product of 2x^ and the expression in the column I. Writing this product below the expression in the column II., and subtracting, we obtain - 2.^^ — 2x^ + ^x*- - 2x + 1. Twice the part of the root already determined is then written in the same line as this and in the column I. Dividing the expression in the column II. by 2.r3, we obtain — .r as the next term of the square root. This is then written both in the colunni I. and the colunm III. Thus I. II. III. x^ -I- ^ar' + 2x^ - 2x^ + 5.^2 _ 2.r + 1 {x^-\-2x''-- x 2.^-3 + 2.^2 2x^ -h 4.^2 - X ^x^ + 2a^ - 2.^3 + 5.^2 - 2x-\- 1 4^^ + 4.z?4 2x^ - 2x^ + 5.^2 - 2.?; + 1 We have now to subtract (.r^ + 2.v;2 - xY from the given expression. This is equivalent to subtracting from the ex- pression in the last line of the column II. the product of the last term in the column III. (namely, —x) and the expression in 192 EVOLUTION. the last line of I. Writing this below the expression in IL, and subtracting, we obtain 2^ + 4x2- 2^'+ 1. Twice the part of the root already determined is then written in the same line as this and in the column I. Dividing the expression in the column 1 1, by 2^3^ ^e obtain + 1 as the next term in the square root. This is then written both in the column I. and the column III. Pro- ceeding as above, we have now to multiply the quantity in the last line of the column I. by the last term in the column III., and subtract it from the expression in the last line of II. There is no remainder. Hence the original quantity is a perfect square, and its square root is exhibited as follows : afi -f 2.^2 — X +\. The whole process is 2.^3 ^2.r 2.^3 + 4.^2 _,^, 2.r3 + 4^2_2^-+l x^ -f 4.aP + 2.^?4 _ 2.r3 -|- 5.^2 _ 2^ + 1 ( .^ + 2.^2 - .r + 1 Ax^ -\- ^x'^ - 2^3 + 5^.2 _ 2;^; + 1 4.r5 + 4:X^ -^x^-^x^ + bx'^-'Ux+l -2x^-4:X^+ x^ 2x^ + 4:X^-2x-\-l 2.r3 + 4.r2-2.r+l 192. The following is another example, and exhibits the process for finding the square root of Ax^ - Aax^ - a^x^ + a^x + ^a\ Ax^ - Aax^ - a^x''- + a^x + ^a^ ( 2.^2 _ ^^^ _ 1 ^^2 4^ 4x2- 4^-2 - 2ax — ^a^ 4ax^+ a^x^ -2a^v^ + a?x+^ The process exemplified in this and the last article is analo- gous to the process used in arithmetic for finding the square root of a number. 193. Square Roots by Inspection. If however the given expression be of a degree not higher than the sixth, and i/" we know that it is a perfect square, then we can generally write down its square root by inspection, as illustrated by the following method of obtaining the two square roots which are given in Arts. 191,192. EXTRACTION OF SQUARE ROOTS. 193 194. The first term in the square root of ^.6 _,. 4^^ + 2^-4 _ 2,r3 + 5^.2 _ 2^' + 1 is the square root of x^, that is, x^. The next term is obtained by dividing 4^ by twice the term already obtained, that is, by 2.1-3 ; hence it is ^x^. Similarly, if we write the given expression in ascending powers of x^ that is, write it backwards, as 1 - 2^- + 5^-2 _ 2^ + 2x^ + 4^-5 + .^•^ the first term of its square root, as now written, will be the square root of 1, which is 1. This therefore is the last term of the square root of the expression as originally written. Since however the square root of a^ is either a or —a, we cannot tell whether, if we begin the square root with the terms x^ + 2x'\ we shall end with -f 1 or - 1. It is better therefore to leave it ambiguous, and take the last term as ±1. The term before this is got by dividing -2x by twice the term just obtained, that is, by ± 2 ; hence it is + x. Thus four of the terms of the required square root are in order x"^, 2./;-, +Jt', and ±1. Moreover, since the required square root is of the third degree, there can be no other terms. Thus, the required square root is one of the forms x^-\-^x'^±{-x + \) (a). Inspection, or at the longest a trial by squaring, will deter- mine which sign in the ambiguity must be taken. In this case, if we form the square of (a), we find that the coefiicient of a-* is 4 + 2, and since the coefficient of o:^ in the given expression is 2, we must take the upper sign in order to make these coefficients the same. Thus the required square root is A-3 + 2.^2 -./;-!- 1. 195. Similarly, the first term in the square root of 4ar* — 'iax^ - a2^'2 ^ ^3^. + 1 rt* is the square root of 4^'*, that is, 'Hx'^. The next term is the quotient of —^ax^ by 2 (2^2), hence it is -ax. The last term is similarly the square root of ^a'*, that is, ±\a^. The term before that is the quotient of a^x by 2(±^«2); hence it is ±(jlv. But we have already found that it is - ax, hence we must take the lower sign in the ambiguities. Thus the required square root is 2x--ax — \a'^. B. A. " 13 194 EVOLUTION. *196. Square Roots of Expressions which are not perfect squares. If we apply the method of Art. 191 to a multinomial which is not a perfect square, we obtain a series of terms whose square is approximately equal to the given expression. Example. Find three terms {in descending powers of x) of the square root of x'^ + ax+b. x^-\-ax + b x^ 2. + I ax + b a2 ax + -- 4 -.-:.-t"^ (, + 2 + ta^ 2x + ... Hence the required terms are x + --\ — - — . will find that the next term in the square root is - The student 4a6 - a^ 16^2 • EXAMPLES ON SQUARE ROOTS. XV. D. Find the square roots of the following expressions numbered 1 to 15. 1. 3. 5. 7. 9. 10. 11. .r* + 4.^3-8^ + 4. 4x^-12x^ + 46:^ 36^-1 - 36.t?3 + 1 7^2 _ 4^ + A 2. x^- 8x^ + 1 0^-2 4. 24^' 4- 9. 54.r-H81. 4. 4.7^-4x^ + 3x^-x + l. 6. 4a4 + 9(l-2a) + 3a2(7-4a). 4a4 + 9(l+2a) + 3a2(7 + 4a). 8. 4{x-l){x^-l) + 9x^. (2^+l)(2.y + 3)(2.i- + 5)(2^;+V) + 16. 9.^6 _ 12^4 + SOx^ + 4jf- - 20^ + 25. 9^-0 - 1 2^^ + 22^4 +x'^-\-\ 2x + 4. 12. 9^ - 12.^3^ + 10^2^2 _ 4^^y 4.^4^ 13. 9x^ - 24.^;3^ + 40.^%2 _ 32.^^3 _,. 1 g/. 14. 4x^ + 26^-3 - (4a - ^b) bx"^ - ab\v + a2^2. 15 . (a2 + 62)2 + (c2 + (^2)2 j_ (^ 4. J)2 (^ _ ^/)2 + (^^ _ ^,) 2((, 4. ^^2 ^ g^^CC^. EXTRACTION OF CUBE ROOTS. 195 16. Shew that, if .r = ^'^^^ , then {a - cxf^{x'- - 1) (62 - d^) is a perfect square. 17. Prove that the product of any four consecutive even integers increased by 16 is a perfect square. *18. The first two terms of a certain perfect square are 49.1-^ - 28.r^, and the last two terms are 6.i' + f. Find the expres- sion, and its square root. *19. Extract the square root of rt2m^^2»i + iOco^2m-2,;2,i + i «2 - 6ci»* + i.r» - 1 -f 25c2a2»i-4^^2n + 2 _ 30ca"^ - i.y« + 9 —^ . *197. Cube Roots of Multinomials. The method for finding the cube root of a multinomial expression is analogous to that given above for finding the square root. We shall not here discuss the general method, which is strictly analogous to that explained in Arts. 190, 191 ; and shall only briefly indicate some methods for finding the cube root by inspection in a few simple cases. The following articles apply only to expressions which are perfect cubes. *198. First, consider the case of an expression which is a perfect cube and consists of four terms. Suppose it arranged in descending powers of some letter. We know that {A±Bf = A-^±ZA^B + ^AB'-±B\ If then the given expression be a perfect cube of a quantity like A±B, its first term will be A^, and its last term will be ±B^. Hence the cube root of its first term will be A , and the cube root of its last term will be ±B. The algebraic sum of these quanti- ties is the required cube root. There is only one real cube root of any given real quantity, and thus there is no ambiguity. Ex. 1. Find the cube root of a/;3 - 36a^-2 + oAci^x - ^la\ The first term is the cube root of Sx^, that is, 2x. The last term is the cube root of — STa^, that is, —3a. Hence the required cube root is 2.r-3a. 13—2 196 EVOLUTION. The cube of this is equal to the given expression. If this had not been the case, the given quantity would not have had an exact cube root. Ex. 2. Find the cube root of \'2,bofi - ^^-ax^ ■\-^-^d^x'^ -\a^. The cube root is 4/(125^'^) - ^/^a^, that is, bx"^ — -^a. The cube of this is equal to the given expression. ■^199. Xext., consider a nmdtinomial of a degree not higher than nine. In this case, we may proceed as in Art. 194. Arrange the expression whose cube root is required in descending powers of some letter, such as x. Let us compare it with the expansion of (.4+Z? + (7+...)3, where J, ^, C... are arranged in descending powers of x. Then the two terms of highest dimensions in (^+^ + (7+...)^ are A^ and 3^2^. Thus, if the required cube root be ^+^+C+..., the cube root of the first term in the given expression will be J, and the quotient of the second term by 3^2 will be B. Similarly, the cube root of the last term in the given expres- sion will be the last term of the required cube root, which we will denote for the moment by R\ and the quotient of the last term but one by ZH'^ will be the last term but one of the required cube root. Thus the two first terms and the two last terms of the required cube root can be written down by inspection. This rule will enable us to write down the cube root of any compound expression whose degree, say in x.^ is not higher than nine, provided it is a perfect cube, since the only possible terms in the cube root are terms involving x^^ x'^, x, and an absolute term. Ex. Find the cube root of 8x^ - 36x^ -f- 66^7 _ 87^6 ^ iqq^ _ 37^ + QU^ - 42x^ +12^-8. The first term is the cube root of 8x^, i.e. is 2x^. The next term is the quotient of - 36.^8 by 3 {2x^y, i.e. is - 3.^2. The last term is the cube root of - 8, i.e. is - 2. The term before the last is the quotient of 12.^ by 3 (-2)2, i.e. is x. We thus have the terms involving x^, x^, x, and the term independent of x. There can be no other terms. Therefore the cube root is 2.^•3-3^•2 + ^■-2. EXTRACTION OF CUBE ROOTS. 197 *200. Lastly, consider the case of any multinomial. To find the cube root of any multinomial, we can either use a method analogous to that given in Arts. 190, 191, or we can write down the two first and the two last terms of the cube root, determined as in Art. 199, and insert between them terms involving the intermediate powers of x with unknown coefiicients. By cubing this expression, and comparing it with the given one, we can at once obtain these unknown coefiicients. Ex. Find the cube root of x^ - 3^11 ■{■ 3jt'io + 2^9 - ^x^ + 9.^'' - ^x^ + 9.ri - 2.i;3 _ ^£. + 3.^, _ 1 . The two first terms are x^ and -x^ [Art. 199]. The two last terms are similarly x and -1. The only other term which the cube root can contain is one involving x'^: suppose that its coefficient is h. Therefore the required cube root is x^-x^ + hx'^-^x-l. The cube of this is This must be the same as the given expression. Comparing the coefficients of x^^, we see that 3 + 2A = 3. .-. h = 0. Hence the required cube root is x^-x'^ + x-l. *EXAMPLES ON CUBE ROOTS. XV. E. Find the cube roots of the following expressions. *1. a3_|.6a2_^12a + 8. *2. 8/ -I- 60^+1502/2 + 125. *3. 27 - 135a^ + 225.^2 - 125^3. *4. a^ - Qa^ + ISa* - 20^^ + 15«2 - 6a -h 1. *5. Sxf^ - 12.^-5 + 18.r4 - 13.r3 + 9.^2 - 3.r+ 1. *6. 8.^6 - 2Qaf> + 114^- 207.^^ + 285.^2 - 225.i- + 125. *7. 27/ - b^ay^ + QZa^y^ - UaY + 21ay - 6ay + a^. *8. (a + 26)3 + (2, _ 2c)3 + 3 (« + 26) (6 - 2c) (a + 36 - 2c). *9. a3m _ Q^2m + l^n + X 2a"^ + 2^y2n _ 8a3^n, CHAPTER XVI. QUADRATIC EQUATIONS. 201. We have already defined [Art. 93] a quad- ratic equation involving only one variable, say x, as an equation in which no power of the symbol representing the unknown quantity is involved except the first and second, namely x and x-. Thus, 2^2 = 3, 3^2^2.^ = 0, and cuf- + hx + c = are quadratic equations. A quadratic equation in which the term involving the first power of the unknown quantity is absent is sometimes called a pure quadratic. Other quadratic equations are called adfected quadratics. We proceed now to consider the solution of a quadratic equation involving only one variable. 202. Any term of an equation can be transposed from one side of the equation to the other side [Art. 95]. It is therefore possible to move all the terms of an equation to one side, and the equation will then take the form that the algebraical sum of those terms is equal to zero. Thus ax'^ + hx + c = (i) is the general form of a quadratic equation, all the terms having been brought to the left-hand side. If we divide by a so that the equation takes the form h c «2 + -^+_ = (ii) a a ^ ^ [or, what comes to the same thing, if a = \ in (i)] it is said to be expressed in its simplest form. QUADRATIC EQUATIONS. 199 When written in the above form, the term which does not involve x (or the sum of such terms) is called the absolute term of the equation. c Thus the absolute term in (i) is c, and in (ii) is - . 203. First method of solution. Resolution into Factors. This method of solution depends on resolving the left-hand side of (i) or (ii) into factors. This can always be effected by Arts. 115 — 117, but if the factors be not obvious by inspection, it will be less trouble to find the roots by the process of completing the square, as hereafter explained, than to use the results of the articles above mentioned. We shall therefore here confine ourselves to cases where the factors are obvious. This method depends on the following self-evident proposition! 204. If the product of two quantities he zero, one of them must he zero. Let P and Q be the two quantities, then by hypo- thesis PQ = 0. Now, if neither P nor Q be zero, their product cannot be zero, which is contrary to the hypothesis. Hence either P ov Q must be zero. Conversely, if either P or Q be zero, their product is zero (unless the other be infinite). 205. Thus, if the product {:v-l)(x-2) be zero, then either ^-1=0, or :?;-2 = 0; that is, either .r=l, ov x = 2. If therefore we have the quadratic equation {x-l) (.r- 2) = 0, either (^-1) must be zero, or {^v-2) must be zero. That is, either ^-1=0, or .r— 2 = 0. That is, A' = l, or ^ = 2. That is, the roots are 1 and 2. Hence, if we had to solve the equation x^-3x+2 = 0, then since we can express it in the form {.v-l){x-2)^0, we are able to write down the roots at once. 200 QUADRATIC EQUATIONS. 206. Conversely, if we want to form an equation whose roots are 1 and 2, we have merely to reverse the process. We have x = \ or x=2. .-. x-i=0 or ^-2 = 0. .-. (^-l)(^-2) = 0. 207. In the following examples the resolution into factors can be performed by inspection. In every case we (i) simplify the equation, (ii) take all the terms to one side of the equation, (iii) resolve the resulting expression into factors. It follows from Art. 204 that, if we remove a factor involving x, or if we divide each side of the equation by such a factor, we shall obtain one root of the equation by equating that factor to zero. The student must carefully bear this in mind ,when he is simplifying any equation which he is trying to solve. Ex. 1. Solve the equation x^ = a^. This is equivalent to x'^ -a^ = Q. .-. {x-a){x-\-a) = 0. .'. x-a = 0, or x-\-a = 0. .'. x = a, or x= - a. Hence the roots are ±a, that is, are a and —a. We can therefore solve an equation of this form by taking the square root of each side of the equation, and prefixing to one square root the sign +. Ex. 2. Solve the equation x'^=ax. This equation is x'^ — ax = 0. .-. x{x-a) = 0. .'. x = 0, or x = a. Hence the roots are and a. Ex.3. Solve the equation x'^-\-bx + Q = 0. The equation is (.^ + 2) (^ + 3) = 0. .'. .^; + 2 = 0, or ^ + 3 = 0. .*. x= -2, or x= -3. Hence the roots are - 2 and - 3. QUADRATIC EQUATIONS. 201 Ex. 4. Solve the equation x^ — x—Q = 0. The equation is {x - 3) (.^ + 2) = 0. .-. .17-3 = 0, or ^^ + 2 = 0. .'. ^ = 3, or x= -2. Hence the roots are - 2 and 3. Ex. 5. Solve the equation ahx"^ — x {a^ + 1>^) + ah = 0. The equation is {ax - h) {hx — a) = 0. . • . a.r - 6 = 0, or hx—a = 0. .-. x = b/a, or x^ajh. Hence the roots are hja and a/6. EXAMPLES. XVI. A. Solve the following equations by resolution into factors. 1. .^'2-3J^;=0. 2. a^2 _|_ 5^^ — 0. 3. x'^ = 2. 4. .r2 = a2 4.2a6 + 6=^. 5. .172 + 307 + 2=0. 6. ;r2 + 3a.i^ + 2a2=0 7. o;2-llo- = 60. 8. .r2 + 9.r + 20 = 0. 9. x'^-1x = 20. 10. .^2_3^_4 = 0. 11. 072 + 707=60. 12. 7^2 _^ 6^ _ 1=0. 13. 2/ + 33/ + l=0. 14. ^•2 + 10 = 13(07 + 6). 15. (o.'-l)(o7-2) = 5(.^'-3) + 2. 16. ^2 + (a + 6)^+a6 = 0. 17. y-y=2. ^ 2(.r2 + l)=-5^. 19. 3^2_9^^ + 20a2=o. ®) 6(2.r2-l)=-^. 21. (2.r-3)(3.r + 4) = 39. r 3.r2 + 2«.v-65a2 = 0. 23. .^^2_(«_5),^.= ■(c_a)(c-6). ^3) (m-m)-^ 25. (.17 + 5)2-9 (07 + 5) (2.i7 + l) + 20(2o7 + l)2 = 0. 26) (6 + c).i72-(c + a).i7 + a-6 = 0. 208. Second method of solution. Completing the square. Where the factors are not obvious, we might find them by the method given in Arts. 115 — 117; but it is generally better to proceed directly hy completing the squai^e, as is illustrated by the following solution of the equation ^-'-138 =-17^. 202 QUADRATIC EQUATIONS. First. Transpose the absolute term to the right-hand side and all the terms involving x or x'^ to the left-hand side, .-. ^2 + 17.^ = 138. Second. Divide by the coefficient of x'^. In this case it is 1, and the equation is not thereby altered. Third. Add to each side the square of half the coefficient of X, that is, (YO^ .-. .i;2 + 17.^^-l-(V)- = 138-h(-V)2. Fourth. Collect the numbers on the right-hand side into a single term, 552 -f 289 Fifth. Express each side as a perfect square. The left-hand side is written in such a form that it is a perfect square, viz. (.r-j-Js^)"'^, hence its square root is .27-f y . The square root of the right-hand side must also be found. " ^^^lere it is a number, we can do this by arithmetic; if letters or algebraical symbols be involved, the result will generally be obvious, but if not, we must use the method described in the last chapter. In this case the square root of -^^ is ^■^. Hence the equation is equivalent to (^+¥)'=(¥)- Sixth. Take the square root of each side of the equation. [Art. 207, Ex. 1, p. 200] ••• x + ^J=±^. If we take the upper sign in the ambiguity ± , then ••• ^=-¥-¥ = 6. If we take the lower sign, then x + \}-= -^. = - iz. Hence the roots of the equation are 6 and - 23. Note. If we had seen that the factors of .r^ 4- 17^-- 138 were X -\- 23 and x — 6, we should have obtained the answer at once ; and generally, we only make use of the above method of com- pleting the square when we cannot see the factors. V = ±¥. X- =¥+-¥ = =¥^ = 4, x- = ¥-¥= =f= 4. QUADRATIC EQUATIONS. 203 209. The following are additional examples. Ex. 1 . Solve the equa t ion x""- - 6x + 3 = Ix -2x^-1. (i) Transposing, we have 3x^ — \3x= — 4. (ii) Divide by 3, . • . .^2 - ^>-x= - ^ . (iii) Complete the square : that is, add (^ ^Y to each side, ... ^.i^^ + iy^=.f,ii^, (iv) Simplify, =-i + W _ 121 — '36" • (v) Express each side as a perfect square, (vi) Take the square root of each side of the equation, . ' . X ■ Hence the roots are and Ex. 2. Solve the equation x''- + 6^ - 1 00 = 0. Transpose, .•. x'^-hQx =100. Complete the square, . • . x^ + 6x + (3)2 = 100 + (3)2 = 109. Take the square root of each side, .'. x + 3= ±\/i09. Hence the roots are x= - 3 + Vl09, and x= — 3- \/l09. We cannot find the exact square root of 109; and in such cases it is usual to leave surds in the result. But we can, if it be deemed desirable, take the square root of 109 to as many places of decimals as we like, and thus approximate to the roots. In this case \/l09 = 10-4403.... Hence the roots are 0,'= -3 + 10-4403...= 7-4403..., and x'= -3- 10-4403... = -13 4403.... *Ex. 3. Solve the equation x^-\-2x + 2 = 0. Following the above order of procedure, we have x^- + 2x =-2. ... .^2 + 2.r+l=-2 + l=-l. .-. (.^+1)=±\/^. That is, the roots are - 1 + \/ - 1 and - 1 - V— 1, both of which are imaginary [Art. 119]. 204 QUADRATIC EQUATIONS. Ex. 4. Solve the equation ax^ -\-hx-\-c = 0. This is the general form of a quadratic equation, and there- fore includes all the preceding examples. Transpose the absolute term .'. ax'^ + hx=—c. Divide by «, .-. x'^ + - x= — . a a Complete the square, .-. ^2 + ^ ^+ (^^ ^ . 4ac. They are equal if b^ = 4ac. They are imaginary if ¥ < 4ac. The student will notice that one root exceeds — r— by as 2a -^ much as the other falls short of it. *211. Third method of solution. Method by Substitution. The following is another method of solving a quadratic equation, but the student will not find it so convenient in practice as the methods already given. It is known as the method by substitution. QUADRATIC EQUATIONS. 205 Suppose the equation to be ax^ -\-hx->rC = 0. Let us put x=y + h, where h is some quantity which we will ultimately fix as may be most convenient to us, ••• a{ij^-hf-\-h{y + h) + c = 0, that is, ay2 + ga Ay + aJi^ -\-hy + hh-\-c = 0, that is, «/ ^y (^2aji + 6) + aJi^ + hh + c=0. Now we may give any value that we like to h. Let us choose it so that 2ah + 6 = 0; that is, take h= - - - . 2 a The equation last written will then become «^^ + 4a-2a+^ = ^' that IS, y2 = __^^_^ the roots of which are y = + ^^-^ — . But ^,;=:y + /,=y___. Hence the values of .r, which are the required roots, are + V (^^-4«c) _ _^ - 2a 2a' which agree with the results given in Art. 210. EXAMPLES. XVI. B. [Additi07ial examples loill be found in the collection at the end of the chapter., and numbered XVI. C] Solve the following equations. 1. x^ + 6x = 55. 2. 2jf- + 3x=2. 3. 2^2 _ 3^. = 2. 4. 5^-2_i7^,+ 14=0. 5. 4^2 + 4^_3^0. 6. 4:^2-13^ + 3 = 0. 7. 1 5.^2 -|-34r + 15 = 0. 8. 15jt'2+;^_6 = 0. 9. 13.^2 _ 90^^, _ 7^0. 10. 4.1-2- lU'-3 = 0. 11. 14a'2-13^; + 3 = 0. 12. 5a'2-12.^; = 9. 206 QUADRATIC EQUATIONS. 13. 3^^2 + 7^' -76 = 0. 14. 17.^^H19J^^- 1848 = 0. 15. 2U-2-2.r = 3. 16. 4i'2_65.r + 126 = 0. 17. 4^2_8,^. + 3 = 0. 18. Q{x^-l)=Z{±v-^). 19. Zx'^ + l=--f-x. 20. 2.v2 + 697^' = 349. 21. (3^ -4) (4^ -3) = 10. 22. {x -h){x + h)=--a^^h'^. 23. (^-19)Gr-21) = 8. 24. {x-2f + {x-Zf = {x^-Qf. 25. 2aa;2 + 6^r=2a-6. 26. oj^-\-2{a-h) x + h'^ = ^ah. 27. a (^^2 + 1) = ^^ (^2 + 1). 28. 2a^2_4^+2a = a2^-. 29. ahx'^-i^a^-^x^-l. 30. (a^-^a) (^.2_ i) = 4^^,^^ 31. 5-.^'{x•-3(3.r-5)} + 2(2.r+l)2 = 0. 32. .^'2 _ (^ _ ^) ,y + ((^ _ 5 + c) c = 2c.«^ + ah. 33. (p-5').t;2-(^ + 2').i' + 2g = 0. 34. 3.^2 - 6a*' -126 = 4*-- 96* - 8a. 212. The equations may contain fractions, involv- ing X in the denominators. In such cases we must multiply throughout by the L.c.M. of the denominators, and simplify the resulting equations, before we can apply the above rules. We must however be careful to multiply only by the L.C.M., as we may otherwise introduce a factor which will apparently give us a root of the given equation, but which will not really satisfy it. This is illustrated in Example 2 which is worked out on the next page. Should the numerator of any fraction be of equal or of higher dimensions than the denominator, it may be convenient to begin by dividing it by the denominator [Art. 142]. This is ilhis- trated in Example 3 on page 208. But although this may simj)lify the work, it is not necessary. We may add that the simplification of equations involv- ing fractions, and the combination of the fractions involved, frequently afford opportunity for ingenuity in arranging the work so as to enable the solution to be obtained in a few lines ; but no precise rules can be laid down, and nothing but practice QUADRATIC EQUATIONS. 207 combined w'th a certain natural aptitude will give the requisite skill. All the examples at the end of this chapter are soluble by direct analysis, and do not necessarily require the use of special devices. 213. The following examples illustrate these re- marks. 5 4 3 Ex. 1. Solve the equation — -r = — - . ^ x-l x+l x-\-l The L.c.M. of the denominators is (^' - 1) {x + 1) (^ + 7). Multi- plying throughout by this, we have bi^c + l){x + l)-A:{x-\){x + l) = ^{x-\){x + \). .-. 5(.y2 + 8.r + 7)-4(^2 + 6^-7) = 3(.i-2_i). .-. 2x^-l6x- 66=^0. .'. ^•2_ai'-33 = 0. .-. (a;-ll)(^+3) = 0. .'. .27 = 11, or x= —3. Hence the roots are 11 and —3. K,j. 2. Solve the eq,cation -,-^±L_ + _|^ _ _L_ = o. The L.c.M. of the denominators is (^' + 2) {x - 1) (.^+ 1). Multi- plying throughout by this, we have {x+\f + {x-\f-{x + 2) = 0. .'. 2x'^-x = 0. .'. .r(2^-l)=0. .-. x = 0, or 2.r-l=0. .*. x=0, or x = ^. Hence the roots are and -1. [Now if we had multiplied the given equation by the product of the first two denominators, instead of by the l.c.m. of the three denominators, we should have obtained as the resulting equation (x-^l){.v^ + 3x + 2) + {x-l)(x'-+x-2)-{x'- + 4x + 4) = 0, which reduces to 2x^-{-3x'^-2x=0. If we divide by x, which is a factor of the left-hand side, we get .r = and 2.r2 -|- 3j; - 2 = 0. The latter equation may be written (2.r- l)(.r-h2) = 0, and its roots are therefore x = ^ and x= -2; and we might think that 0, ^, and - 2 were all roots of the given equation. But the root 208 QUADRATIC EQUATIONS. x= —2 arises from the factor x + 2 in 2a;^ + 3^' — 2 ; and this factor was introduced by our having multiplied every term in the original equation by x + 2 in addition to the l.c.m. of the denominators. The resulting expression of course had ^ + 2 as a factor, but ^- + 2 was no factor of ^ the expression which was given as equal to zero. We may say generally that no factor of an expression by which the given equation is multiplied will be a root of the equation. Hence we must be careful to multiply the given equation only by the l.c.m. of the denominators, and we shall then avoid the introduction of any unnecessary factor.] ^2 -|- 1 x^ — 2 Ex. 3. Solve the equation -\ = 2x. X — i X — A Since the numerators of the two fractions on the left-hand side are of a higher order than their respective denominators, it will be better to begin by dividing each numerator by the cor- responding denominator. The equation then becomes •••2-+3+.:;f^-i+;?42=2... 2 2 .-. 3 + -^-f-^ = 0. x-\ x — 2 The L.C.M. of the denominators is (.:*;- l)(.^'-2). Multiplying throughout by this, we obtain 3 {x- \){x - 2) -f 2 (^' - 2) + 2 {^x - 1) =0. .-. 3x^-5x = 0. .-. x{3x-5) = 0. .-. x = 0, or 3.1- -5=0. Hence the roots are and |. *ET . e 7 .1 *' ^ + ^ « b 6-c *Ex. 4. /Solve the equation — ; — ; — = — —j —, . ^ x-\-a + c x + a x + o x + o-c Since the numerator of each of these fractions is the same as the part of the denominator which is independent of x, we had better begin by writing the equation in the form (i _ _^\ _ fi _ _^^ = A _ _-L_v /'i ^ \ . \ x-\-a + cJ \ x + aj \ x + bj \ x + b-cj QUADRATIC EQUATIONS. 209 x + a + c x-\-a x+b x-^b — c' ^ is a factor of every term, and therefore [Art, 206] x = m one root. The other root is determined by the equation 1 _1__ 1_ 1 x + a + c x + a x + b x + b-c' This equation is equivalent to -{x-\-a)-\-{x+a + c) _ -{x+b-c)-\-{x+b) {x -\- a) {x -\- a + c) ~ {x + b){x-\-b-c) c c ' ' {x-\-a){x-\-a-\-c) {x + b){x-{-b-c)' Divide by c ; and multiply up, .'. {x + b){x->rb-c) = {x + a){x + a + c). .'. X"-\-x{2b-c) + b^-bc = x'^ + x{2a + c)-\-a^ + ac. .'. x(2b-2c-2a) = a^-\-ac-b'^ + bc. .'. -2x{a-b + c) = (a + b){a-b + c). .•. -2x = a + b. .: x= -^{a + b). Hence the roots are and -^{a + b). MISCELLANEOUS EXAMPLES. XVI. C. 1. Shew that the definition of the roots of an equation is satisfied by the statement that 1 and 2 are roots of the equatic^n .r2- 3.^ + 2 = 0. 2. Determine whether 1, |, -|, or any of them, are roots of the equation 3 (4r'^ + l) = j;(7A^ + 8). 3. Determine whether -fl, - 1, or either of them, is a root of the equation {2x^ + 1 ) \/2jf+^x+\ = x^ - 99.r - 100. 4. Form equations of which the roots are respectively (i) 1 and 2 ; (ii) - 3 and - 5 ; (iii) and 4 ; (iv) a and 6. Solve the following equations. 5. x'^-4.x = 5. 6. 6x^-x-l=0. 7. 5.r2-26.r+5 = 0. 8. 1 3a-2 _ i y.r = 66. 9. 17.r2-14'P = 40. 10. 27^-2 _ 24^-16 = 0. B. A. U 210 QUADRATIC EQUATIONS. 11. 15.^^2 + 7^=2. 12. Tt{x^-\) = nx. 13. 5:^2 + 6^-104 = 0. 14. 20.372-:?;- 1=0. 15. 15.2;2 + 7^ = 74. 16. 12.^2_95^_8 = o. 17. 12ji;2_^_6 = 0. 18. 6.272-11.27-35=0. 19. 15.272 + 207-77 = 0. 20. 5^2 + 4^_i=,o. 21. 4.272-17.27+4 = 0. 22. 18.272-11.27-24 = 0. 23. 28072-13^^-6 = 0. 24. (3o7-ll)(2o7- 10)=072-9. 25. 5(07-9)2-(207-ll)2 + (307-21)2 = 53. 26. (2o7+l)(3.27-7) + (2o7-5)2 = (o: + 2)(2o7-5). 27. (07-2)2 + (07+5)2=(07 + 7)2. 28. 6(207-7)2 -(07-13)2 + 507-15=0. 29. .r (207-4 (507-3)} = 2(307-2)2. 30. (307+1)2-3(30^+1) (o7 + 5) + 2(o.'+5)2=0. 31. (07-l)(o:-2)(07-3) = (o72-2)(o7 + 25). 32. ao72 + 6o7=co?. 33. a6(o7+l)o7+a2=(a2_^262)o7 + 462. 34. 072-2ao7-26o7+4a6 = 0. 35. ao72 + 26o7 = a-26. 36. -^. + -^ = 2. 37. ^^ + ^=a + 6. x-b x-a a + .v b-\-x 38. 1207-5=21. 39. 5^-^ + ^3 = 1. 40. 4- +^+1=2^. 41. ^? + i = |. 07+1^ 07 ^ ^' 07 + 7 O- 6 42. i(o7+f) = i?. 43. — ? =-'— '^ '^ Qx *'^- 307-2 5(07-1) 20' 44 ^ 5 07+1 07 + 2 **• 3^'-5-2^n-^- *^- ^+2 + .rT3 = ^r^- 46 7o7-ll 3o7-2 ^ 2o7 + 5 ^Z^, _i 3o7-l • 407-7 ■''12.27-1" 07+2 • ■*'• 4 ■*"o7-5~ 4 • • ^^■^^^"^' *^' "^"^Ih"*" (07 + 6) (07 + 4) '"^* . 07-1 ^-2 3_ 207 + 3 3o7 + 2_ 07-1 ^"' 0.-2 ^~i + 2-^- ^^- 2":^:::3 + 3^:r^-^^TT' 52 307+2 307-2 ^ 4o72 + 12o7+2 1 1 ^ 9 . QUADRATIC EQUATIONS. 211 p.. , , 2 7 , 4 ^ 3 '^(a;-7)(^-3) ^-3" • ^--3 (^-3)(^'-5)* ^-3 24 _^+2 2^' 3.r-l 5^-11 ^ 58. 3- ^ ^ ^ .r+2 (.i--f2)(2.r-3) 4r-6' ^ x-l _ _2 x-Q x-\ 2 ~.i.--6 ^" ' x-\ x-^ x-Z + . - O » , ■ =0- • x^-hx^% x'^-^-\-Z x'^-Zx-^\ 1 1 x__ _ ^ 1 _ 1 16(u.--2)"^ 16(^+2) 87x2 + 4)~2(^-l)"^2(.r + l) .i-" 1 1 ^ _ _J^ 1 _ J_ ° 4(^-l)'^4(x-+l) 2(:i'2+l)~8(.';-2)"^8(.r + 2) 4^-' fio 1 I -^-4 ^zi__=o 7(^-3)Gr-2)"^(.r-l)(:r-3) {x-\){x-'l) ' «>• 1 2 3 ^^ -^ , ^ :?^i^jr2=^- 65. -^ =1. ^ 66. — ^- ^+a+l=0. 2+ 5 X a x-Z 67. '^ + ^^=iii. 68. 't^\-±=o. x—oa — x^ x^ + ab x 69. -^+,4 =^. 70. ^ + A=1- a + ^ b + x ab x-a x — b 71. ^ I ^ _^. 72. l-+A-=^^ 2^ — 5a 2^ — a a * * ^ - a .^ — 6 ah ' x + h a_x^+ab x x + b c b + c a + b b bx x + a x a + c c 75 _J__ = i + l + i 76 «.^ + ^ bx + a ^ {a + b){x + 2) x+a + b X a b' ' cx + b cx + a cx+a + b 14—2 CHAPTER XVII. SIMULTANEOUS EQUATIONS, OF WHICH AT LEAST ONE IS OF A DEGREE HIGHER THAN THE FIRST. 214. We now proceed to consider the solution of two or more simultaneous equations, one or more of them being quadratics, or at least of a degree higher than the first. We shall begin by treating the case where there are only two equations involving two unknown quantities, and one of the equations is of the first degree. We shall next discuss the solution of two simultaneous equations involving two unknown quantities, where both the equations are quadratics. The theory of simultaneous equations of a higher order and not reducible to one of these two cases lies out- side the limits of this work. 215. Solution of Two Simultaneous Equa- tions, one being a Simple Equation and the other a Quadratic Equation. We commence with the case of two equations involving only two un- known quantities, where one of the equations is of the first degree. We can always solve such a system by the following rule. First. From the simple equation, find the value of one of the unknown quantities (say x) in terms of the other unknown quantity (say y) and the known quantities. .'SIMULTANEOUS EQUATIONS. 213 Second. Substitute this value of x in the quadratic equation. The result will be a quadratic equation involving only y and known quantities. Third. The solution of this quadratic will give two values of y. Fourth. To each of the values of y so determined will correspond a certain value of x, which can be determined from the value of x in terms of y which was originally obtained from the simple equation. 216. The following examples illustrate the method. Ex. 1. Solve the simultaneous equations j 2.i' + 3y = l (a), |3.r2_,y^ + 2?/2 = l6 (6). (i) It is usually best to solve the simple equation for that variable whose numerical coefficient is the smaller. In this case, therefore, we solve it for .v. From (a), we have 2x=l-3y, (ii) Substitute this value of x in (6), Multiply by 4, collect like terms, and simplify, .-. 4l3/2_20y-61 = 0. (iii) This quadratic must now^ be solved. Resolving into factors, we have (y+l)(41y-61) = 0. [These factors ought to be obvious by Art. 114, since the only positive integral factors of 41 are 41 and 1, and of 61 are 61 and 1. If however the factors be not obvious, then the quad- ratic in y must be solved by the method of Art. 208.] .-. y + l=0, or 41y-61=0. .-. 3/=-l, or y = £i- (iv) We have now to find the values of x which correspond to these values of y. "We have .r = |(l — 3y). 214 SIMULTANEOUS EQUATIONS, If y=-l, .-. ^=i(l-3y) = ^(l + 3) = 2. Ify=fi, .-. ^=Hl-3y)=Hl-3.f})=-|}. Hence the roots of the original system are ^=2, y=-l', and x=-l\, y = i{. Ex. 2. Solve the simultaneous equations (a h ^ - + - = 2 1, b+p=2 (")• Here, we treat - and - as the unknown quantities. X y From(i), - = 2--. X y 1 / 6\2 62 Substitute this value of - in (ii), .'. 2-- ) + -5 = 2. X ^ ' \ yj y2 Simplifying, .-. y2_26j/ + 62^0. .-. {y-hf = 0. Hence the two roots of y are equal to one another, and each is equal to h. We have now to find the values of x which correspond to these values of y. We have - = 2 — . •^ X y .-. if v = 6, - = 2-- = 2-y = 2-l = l. •^ ' X y b .*. ^' = a. Hence the roots of the original equations are x = a\ , x = a^ Ex. 3. /So^ve the simultaneous equations - ^ + ^=2 X y ax-hy = a^-h'^ (ii). a2 _ 52 _^ }yy a Now (i) may be written ay-\-bx = ±xy. 2 + *=2 (i), X y ^ '' From (ii), we have x= ~ (iii). SIMULTANEOUS EQUATIONS. 215 Substitute in this latter equation, the value of x given in (iii), ■''^y^^ — ^=2 — ^y- This, on simplification, reduces to 26/+y(a''^-362)-6(a2-62) = (iv). .-. (y-6)(2&3/ + a2_52) = o. [If these factors be not obvious to the student, he must solve equation (iv) by one of the methods given in the last chapter, when he will obtain the roots given in the next Hne.] a2_52 These values must now be successively substituted in (iii), and we shall thus obtain the corresponding values of x. First, if y = b, ,. by (iii), ^■ = «'^*!±i^ = ?!^^!±i' = a. Second, if y= —^— , .-. by (m), a-= = — - — . Hence the roots of the original equations are a2-62 «2_52 x = a,y = b;^ndx=-^^, y=--^^. Ex. 4. Solve the equations x-7/ = d, x^ — 7/^ = 6. Although the method by which the three preceding examples were solved is always applicable, yet the work may sometimes be slightly facilitated by noticing whether the terms involving X and i/ in the given simple equation form a factor of part of the quadratic equation. In this case, the second equation may be written {x-y){x + 7/) = 6. Dividing each side by the corresponding side of the first equation, we obtain {x -2/)(x+ i/) ^ 6 x-y 3* .-. x+y = 2. We have now the simultaneous equations x-i/ = 3, x + i/ = 2, the roots of which are x=f, y= —^. The equations given in this example might have been solved in the same way as the equations in Examples 1, 2, and 3 above. 216 SIMULTANEOUS EQUATIONS. 1. EXAMPLES. XVII. A. Solve the following systems of equations. V+y = 6. 2. 1^2+3/2 = 4]^ 1^+3/ =9. r9x-2_ 43^2 = 576^ [23/ - 3^ = - 12. 4. 25.^2 _ 93^2 = 675^ 33/ + 5^=45. (x +y =8, V +3/2 = 50. 6. ■^2 + 2^'3/ = 5, ^+3/ = 3. r x-y = 3, ".^2 + ^3/ -1-3/2 = 93. 8. V +3/2 = 37. ■3^ + 63/ = 21, \33/ + 6.r = 4^3/. 10. r.r2+/= 10a2+ 1062- I2a6, V +3/ =2 (a + 6). r^(y + 7)+3/G'«; + 5) = 5, 73/ + 4.r = l. 12. |^•3+3/3 = 351, V +y =9. p+y= ft, 14. (^+y = 6. (6^-a3/ = a2-62, |52(^_«) + «2(3/„5) = 0, 7. 9. 11. 13. 15. •j-^_.y!_^_^ 16. <_i 1 i__Q (a2 62 62 a2 • |_2; - 6 3/ - a a — 6 217. Solution of Two Simultaneous Quad- ratic Equations. The solution of any two simul- taneous quadratic equations involving two unknown quantities is not always possible by the methods which are explained in this book. But any simultaneous equations of the form ax^ + hxy + cy"^ = d Aa^ + Bxy+Cy = D\ (where all the terms involving the unknown quantities are of the second degree) can always be solved by the following method. SIMULTANEOUS EQUATIONS. 217 Dividing each side of the first equation by the corresponding side of the second equation, we have Ax' + Bxy + Cy- D' Now put y = vx, and substitute vx for y wherever y occurs, pi?{a-\- hv-\- CV-) _ d •'• p^'{A+Bv + Gv')~D' This equation reduces to a quadratic in v. Solving this, we obtain two values of v ; let us denote them by a and /3. (i) If y = a, .-. y — OLx. Put y=OLX in the first of the given equations; we can from the resulting equation obtain two roots of x, to each of which will correspond a value of y, which may be obtained from the equation y = ax. (ii) If V = /3, we can similarly obtain two values of X ; and, thence, the two corresponding values of y. Hence there are altogether four pairs of roots. 218. The following examples illustrate the method. Ex. 1. Solve the simultaneous equations a-2+3.*-y=10, Vy + 4y2 = 6. Dividing, we have x^ + ^xy _\0_b xy-\-4:y^~ 6 ~3' Let u = vx. . • . —^, — :7r = - . This reduces to 3 (I + St?) = 5 (v + 4y2). .-. 20y2-4y-3 = 0. 218 SIMULTANEOUS EQUATIONS (i) li v=\, .-. y = \x. Substitute this value in the first of the given equations, .-. ^2 = 4 .-. x=±2. If ^=+2, ,-.y=\x=\ (i). If ^=-2, .-. y = \x=-l (ii). (ii) If v= — ^, .-. y= —^x. Substitute this value in the first of the given equations, .-. ^2__9_^2 = io. .-. ^=±10. If ^=+10, .-. y=-^x=-Z (iii). If ^=-10, .-. y=--^x = Z (iv). Hence the four pairs of roots are .r = 2) x=-2'\ ^ = 10 1 ,^^=-10') y=l]^ 3,= -l|' .,= -3}' ^^% = 3 }• [The student, in writing down the roots of simultaneous equations, must be careful to arrange them in their proper pairs : thus, in this example, ^ = 10, y = - 3, satisfy both the given equations. Values of x and y taken from different brackets will not in general satisfy the given equations: thus x = \0, y = 3, satisfy neither of the given equations.] Alternative method. We can sometimes arrive at the result more simply by first forming from the two given equations a new one, of which both sides are perfect squares, but this method is only applicable in particular cases. Thus, in the above case where the given equations are x^ + Zxy = lO, xy + 4y^ = Q, we obtain by addition x^ + 4xy + 4:y^=l6, that is, (^+2y)2 = 16. .-. x+2y=±4. (i) If x + 2y = 4:, we have the simultaneous equations x + 2y = 4:, x'^ + Sxy=lO. SIMULTANEOUS EQUATIONS. 219 The roots of these can be found by the method of Art. 215, and are .(i). (ii) If x + 2y= — 4, we have the simultaneous equations x-\-2y=- -4, x^ + Zxy^lO. The roots of these can be found by the method of Art. 215, and are A and . \ (n). y=-l) y= 3 J Ex. 2. Solve the simultaneous equations ^2+^^ = 21, 3/2_|_^ry = 28. If we adopt the first method of solution [Art. 217], we have x'^ + xy _2l _3 y^+xy~28~'i' Let y = vx, /. „; ^ . — r = - . [The quantity 1 + v is a factor of both the numerator and the denominator of the fraction on the left-hand side. For the reason given in Art. 213, Ex. 2, it does not lead to a root of the quadratic] Dividing the numerator and the denominator of the fraction on the left-hand side by l + v, we have 1_3 V~ 4:' .-. v = ^. But y = vx, hence 3/ = f •^• Substitute this value of y in the first of the given equations, .-. x^ + ^x'^ = 2l. .'. 3.^2 -f- 4^2 = 63. .-. x^ = 9. .-. x= ±3. Bvity = ix, .-. y=±4. Hence the roots are x= ±3, y = ± 4. 220 SIMULTANEOUS EQUATIONS. The alternative method of solution explained in Ex. 1, is applicable to the equations given in this example. If we adopt it, we obtain, by the addition of the given equations, x^ + 2xy +3/^ = 49. .'. x+y= ±7. If x+y = 1, this, combined with the first of the given equations, gives, by the method of Art. 215, x= 3, y = 4. If X -ry = — 7, this, similarly, gives ^ = — 3, y = - 4. ^Ex. 3. Solve the equations x'^=y-\-\% y^=x-\-\2. These equations are of such a form that they can be solved by a process analogous to the second method used in the last two examples. Subtracting, we obtain x^—y^=y — x. .'. (x-y){x+y)=-{x-y). ... {x-y){x+y + l)=0. .'. x — y = 0, or x+y+l=0. (i) U x-y = 0, .-. y = x. Substitute this value of y in the first of the given equations, .-. .^2 = ^^,4.12. .-. {x-4){x + 3) = 0. .'. x = 4:, or x= -3. But y = x, .'. the corresponding values of y are 4 and - 3. Hence ^ = 4, y=4 ; and ^= - 3, y = - 3 are two sets of roots. (ii) If ^+y+l=0, .-. y=-(l+.v). Substitute this value of y in the first of the given equations, .-. x^+x-U=0, the roots of which are x = ^~ — . 2 The corresponding values of y are y= -^ — . These form two more sets of roots. Thus altogether there are 4 sets of roots, namely, a; = 4) x=-3) ^=H- 1+^45)1 ^^ = ^(-1-^45)] y = 4/' y=-3/' y = H-l-x/45);'^"^y = H-l + V45)J ' SIMULTANEOUS EQUATIONS. 221 219. We may sometimes combine the equations so as to get rid of all the terms of the second degree and thus obtain an equation of the first degree. The case will then be reduced to one similar to that discussed in Arts. 215, 216. Ex. 1. Solve the simultaneous equations (^' + I)(y4-2) = 101 xy^ 3J • Subtracting, we obtain 2.r+y + 2 = 7. .-. y = b-±i\ Substitute this value of y in the second of the given equations, .-. ^(5 -2^) = 3. ... 2^2_5^4.3 = 0. .-. (^-l)(2^-3) = 0. .'. .r=l, or x = ^. But y = 5-2cr, .'. (i) if ^ = 1, 3/ = 3; and (ii) if ^'=f, 3/ = 2. Hence the roots of the given equation are x=\, 3/ = 3; and .^• = f, y = 2. Ex. 2. Solve the equations axy = c {hx + ay)^ bxy = c {ax — by). We have, by division, ^ = ^ . •^ '6 ax -by Multijply up, . •. a {ax -by) = b {bx + ay). .'. {a^-b^)x=2aby. •■••'■ = 2 52^652' (')• Substitute this value of x in the first equation, ^ a^b , /2«62 which on reduction gives (a2 + 62)c ^ = ^' "^^=-2^6-- The corresponding values of x will, by substitution in (i), be found to be x=0, and x= o d • 222 SIMULTANEOUS EQUATIONS. EXAMPLES. XVII. B. Solve the following systems of equations. 7y- 23^ = 200/ • hx-\-ay = a^ + + /+/ = 27r • 32/2 + ^ = ir x^ — xy m X 7. 9. 11. 13. 15. 17. *19. 21. x^-hxy=l\ y'^^-Zxy x^ — xy+y^ = 31 x^-\-xy+y'^ = l) 3x'^-7xy + 4y^ = 5] 4^2 _7^y + 3^2=2/ x^+xy-y^ = 29\ xy + 2y2 == 7J * 2x^-3xy=U\ 3y^-x^ + l= OJ • x^-\-4xy = Sb\ 2xy-iey'^= Ij* x^-y^ = 8x-16\ 3xy = 6x + 7y}' x'^+xy = ' = 104| = 15S' = 50\ = 7;- - xy = 4\ -7xy = 4j' 32/-2x^ = in x^ + 4y'^ = l0j' 2g x^-2xy=-2) y^-\-5xy = ll J ' xy-y 25 ^^'■ ^^- «./2. TTo "r =^ — 62 a^ } 6. 8. 10. 12. 14. 16. 18. 20. 22. 24. 26. 28. 30. .r2 + 2^.y = 39l ^y 4-2^2 = 65/ * ^2 + 2^y = 32l 2y2 + ^y=16j ■ .^2 — 3^3/= -5"! 14 /• ?/2 4- 5jn/ : 2^2 _ 9^y4. 9y2= 5| 4x'^-\Qixy^-\\y'^ = ^^ 3^-2 + 5^^ = 221 Il^j/-3y2=i9j • 4^2 + ^j/ = 3^3/ +3/2 = 18 ^2 + ^y=10' y^-xy= 3 ;i^2 + 2.^ = 39| 23/2-3.^3/= 5J 3.^2 -5.^3/ + 23/2 = 14 2x'2- 5^3/ + 33/2= 6 4x2 + 3^.^=10| 33/2'+'4^y = 20j * x'^ — xy+y'^ = l \ x^ + x^ }- i+3/4 = i33J- ^2 + 3^3/= 7| ^3/ + 33/2 = 14j • ^2 _,_ ^2^2 = jo2 + 2pq + 2^2| xy=p + q } SIMULTANEOUS EQUATIONS. 223 '^y=p-9 )' ' f+^- .V7/ = 2a^ + 2h^ + 5ab) ' ' xy-y- 35. 2.r2 + ^3/=/-2^-4 = 12. 36. ^2 _ 2^^7^ = 23/24.^^2^ + 4 = 16. (.r + 2)(y + 3) = 15| ^3/^2^ = 211 ^3/= 2j* ^'y + 3y=50j ' 37. 9.r+83/ + 7.r3/ = 0| 41. --^ = -^\ 42, ^-y = 3 -J (^-3)2 + (y-3)2 = 34| (3.t--y)(33/-.r) = 21 | .ry-3Gr + ?/)= 6J ' 3^^ (3.r-2?/) = 49-3/2j (2.r-y)(23/-^) = 101 (.^ + 2y)(2.r+y) = 20 1 (.;+3/)(^-y)= 7/' *'• 4.^^G^+3/) = 16-3/2/ • 48. - + ^ = 5 ] *49. v^ = -^ \ 0+3/ a + ^ V. ^V^ = - 1 3/^ 2 V. ^2 + 3y2 = 28j 33/2 = 28j a^+63/=(:r+3/)2j ^gQ 2.^2 + 3.^;y = 271 ^^ (^ + 5)2 + (y + 5)2 = 851 xy+y'^= 4J' • ^3/4.5 (^+^) = 17/' 52. ;p^ 2 = 3, -^ h — =9. 53. A' =a-h = y. Ax^ y'- 4^'' xy y^ y x ^ 7(.r2 + l) ^ 5 (a;y + 1) ^^+1 ^. .t;3 +3^3 = 4681^ ^ • 10(a-3/ + 1) 7(3/2+1) 3/+I' ^^- .r^= 35J • *56 ^^^y + ^^y = 1801 .^4-3/^ = 4071 ^°- ^+3/3 = 189/ • ^'- ^2^ + .r3/2=308j • *58. a^2 4. ^jpy 4. cyl — Ij;2 4. ^^^ 4. ^^2 _ ^^ 4. J 4. ^_ *59. :i?2 4.y2=7, y^+zx=7, z'^ + xy =11. *60. ^ + 2W = 444, yz+ux=156, zx+yu = l80, xyzu = bl84', where x>y>z>u. (Person's Problem.) CHAPTER XVIIL . PROBLEMS LEADING TO QUADRATIC EQUATIONS. 220. We proceed now to the consideration of problems in which the given relations can be ex- pressed by means of quadratic equations. We shall first confine ourselves to problems whose solution de- pends on that of a single quadratic equation, and sliall then proceed to problems which require the solution of simultaneous equations. Here, as in the problems leading to simple equations which have been previously considered, the formation of the equations is a mere translation from ordinary language into algebraical language. The solution of the resulting equation or equations can be effected by the processes explained in the two preceding chapters. 221. The following examples are typical of some of the more common problems in which the relations can be expressed by means of quadratic equations. We may add that though there are always two roots of a quadratic equation, yet the student will sometimes find that it is only one of these roots that will satisfy the problem under discussion. The other root will generally be found to satisfy some analogous problem, the algebraical expression of which leads to the same quadratic equation as that formed from the given problem. Ex. 1. Divide sixteen into two parts such that their product shall he six times their difference. Let X be one part. Therefore the other part is \Q — x PROBLEMS. 225 Now the product of the two parts is, by the question, six times their difierciice, .-. a;{lQ-x) = Q{x-{\Q-x)] (i), that is, \Qx-x'^ = Qx-^Q-\- Qx. .-. (^-12)(^ + 8) = 0. .'. .^■=12, or x= -8. The latter root is not appHcable to this problem ; therefore one part is 12, and the other is 16 — 12, that is, is 4. Note. In forming the equation (i), we subtracted (16-^-) from X to get their difference. Had we subtracted x from \Q-x to get their difference, the equation corresponding to (i) would have been x{lQ-x) = Q{{\Q-x)-x], which would have led to the equation A'- -28.^ + 96 = 0, of which the roots are 4 and 24. The latter root is inapplicable to this problem, since it is greater than 16. Hence the answer would have been, that one part was 4 and the other part was 16 - 4, that is, 12. This is the same answer as before. Ex. 2. The price of photographs u raised 35. per dozen, and mtstomers consequently receive 7 less than before for a guinea. What loere the prices charged ? Suppose that before the price was raised, the charge was x shillings a dozen ; 12 .'. for l5. a man got — photographs, . for 21, 1^^ • • lor /i6. „ ,, „ „ After the price was raised to {x + 3) shillings a dozen, 12 X 21 then for 2l5. a man got photographs. X -{- 6 This, by the question, was 7 less than before, 12x21 _ 1 2x21 X x-\-3 B. A. 15 226 PROBLEMS LEADING TO Divide each term by 7, and multiply up, .-. 36(^ + 3) = 36a;+x'(^ + 3). .-. ^2 + 3^_ 108 = 0. .-. (^-9)(^+12) = 0. .'. ^=9, or x= — 12. The latter solution is obviously inapplicable to this problem, hence the answer is 9 shillings a dozen. The student can verify that at this price the given relation would be true. {^Note. We can find what is indicated by the negative root of the resulting quadratic equation by observing that if the given relation had been that when the price was lowered 35. a dozen the number received was 7 7nore than before, we should have obtained the equation 12 X 21 _ 12x21 X x-Z ' which reduces to the equation x'^ — Zx— 108 = 0, of which the roots are 12 and —9 ; hence in this case the price would have been 12s. a dozen. The root — 12 in the original problem corresponds to this solution of this problem.] Ex. 3. A boat's creio can row at the rate of 9 miles an hour in still water. What is the speed of the current of a river, if it take them 2| hours to row 18 tniles, of which 9 7niles are up stream and 9 miles down stream'? Suppose that the river is flowing at the rate of x miles an hour. The crew can row at the rate of 9 miles an hour in still water. .-. the crew row „ „ (9-.^') „ „ upstream, and „ „ „ ,, „ (9+.r) „ „ down stream. 9 Therefore to row 9 miles up stream takes ^ hours [Art. 101], 9 — X and „ „ down „ ^-^-^ „ The sum of these is stated in the question to be 2|^ hours, ... _l_ + ^_2.=9 QUADRATIC EQUATIONS. 227 Divide by 9, and multiply up, .-. 4(9 + ^) + 4(9-.r) = (9-^-)(9 + '^')- .'. 36 + 4^+36-4^=81-^2^ .-. JJ7=+3. The negative root is obviously inadmissible. Hence the answer is 3 miles an hoiir. ^Ex. 4. The perimeter of a rectangular field is 3| times its diagonal, and the length exceeds the breadth by 70 yards. What is its area? Suppose that the breadth of the field is x yards, .'. the length of the field is ^ + 70 yards ; .*. the perimeter is 2^ + 2 (^ + 70) yards, and the diagonal is sf{^^ + {^-^^^f] yards, and the required area is ^(^+70) sq. yds. By the relation given in the question, we have 2a; + 2 (^ + 70) = f |V{-^2 + (^ + 70)2) . .-. 13 (2^ + 70) = 17V{2^H140ji; + 4900}. Squaring both sides, we obtain 169 (4^2 + 280^' + 4900) = 289 (2.i;2 + 1 40.^^ + 4900). Simplifying, this reduces to ^^2 + 70^^ = 6000. Completing the square, .-. .t'2 + 70^' + (35)2 = 6000 + (35)2 = 7225. .'. ^ + 35= ±85. .". ^'=50, or -120. The latter root, being negative, is inadmissible, .*. the breadth is 50 yards, and the length is 120 yards. Hence the area is 50 x 120 square yards, that is, 6000 square yards. 15—2 228 PROBLEMS LEADING TO 222. We proceed now to give a few examples of problems which lead to simultaneous equations, of which one or more is of a degree higher than the first. Ex. 1. Find two 7iumhers whose sum is 34 and product is 145. Let a; and y be the numbers, ••. .^+3/ = 34 (i), and £by=145 (ii). We have therefore two equations involving two unknown quantities. From (i) we have ^ = 34-y. Substitute this value of x in (ii), .-. (34-y)y = 145. .-. y'^- 34?/ 4- 145 = 0. .-. (;y-5)(y-29) = 0. [If these factors be not obvious, tiie equation must be solved by completing the square, and it will be found that the roots are the same as those given in the next line.] .-. y = 5, or ?/ = 29. Ify = 5, .-.^'=34-^ = 34-5 = 29. If y = 29, .-. .517= 34 -y = 34 -29 = 5. Hence the required numbers are 5 and 29. Ex. 2. Find a numbet- of tivo digits such that, if it he divided hy the product of its digits the quotient is 2, and if ^1 he added to the numher the order of the digits is reversed. Let X be the tens' digit, and y be the units' digit, .'. the number is \Ox-\-y. We are given that —=2 (i). xy ^ ^ The number formed by reversing the order of the digits is lOy+x. Hence, by the relation given in the question, I0x+y + 27 = l0y + x .(ii). QUADRATIC EQUATIONS. 229 We have therefore two equations involving two unknown quantities. The equation (i) may be written \0x-\-y = 2xy (iii). The equation (ii) reduces to A'-y + S^O. .-. y=x + Z. Substitute this vahie of y in (iii), . • . 1 0.r + (*' + 3) = 2x {x + 3). Simplify, .-. ^x'^-bx-Z = 0. .-, (2^+l)(.^^-3) = 0. .'. x= -^, or ^ = 3. To satisfy our problem x must be an integer, hence the only root suitable for our purpose is x = 3. But y = X + 3, and therefore if x = 3, y = x + 3^3 + 3 = 6. Hence the required number, which is lOx + y, is 36. The student can satisfy himself that this number satisfies the given conditions. Kv. 3. The 7iumher of apples ivhich can be obtained for a shilling is two more than the number of pears which can be obtained for a shilling^ and the price of seven pears exceeds that of seven apples by a penny ; find the prices charged for apples and pears respectively. Suppose that each apple costs x pence, and that each pear costs y pence. 12 The number of apples obtained for a shilling is — , 12 1/ and „ of pears „ „ „ 12 12 Hence, by the question, — = ^ — 1-2, X y which reduces to Qy = Qx + xy (i). Also the price of 7 apples is Ix pence, and „ „ 7 pears is 7y „ but seven pears cost one penny more than six apples. .-. ly = ^x+\ (ii). We have therefore two equations involving two unknown quantities. From (ii), we obtain y = x-\-\. 230 PROBLEMS LEADING TO Substitute this value of y in (i), .'. 7x'^-\-x-6 = 0. .-. (7^-6)(^ + l)=0. .-. x=^, or X— -1. Now X cannot be negative, .'. x = ^; also, if .r=f , .-. y = x + l = ^+} = l. Therefore the pears cost a penny each, and the apples cost f of a penny each, that is, the apples were sold at the rate of seven for sixpence. Ex. 4. FiTid the sides of a rectangle, lohose area is unaltered if its length he increased hy 3 inches while its breadth is diminished hy 2 inches, and whose area is diminished hy one quarter if its length he increased hy 9 inches while its hreadth is diminished hy 5 inches. Suppose that one side is x inches long, and that the next side is y inches long, .'. the area is xy square inches. Had the length been {x-\-^) in, and the breadth (y-2) in., the area would have been {x-\-Z){y-'2) sq. in. Hence, by the question, xy={x + Z){y -2), which reduces to 3y — 2^-6 = (i). Had the length been (^+9) in. and the breadth (?/-5)in. the area would have been (^ + 9)(?/-5) sq. in. Hence, by the question, ^xy = {x-\-^) (y-5), which reduces to xy + 36y - 20.r — 180 = (ii). We have therefore two equations involving two unknown quantities. From (i) we obtain ^=|y - 3. Substitute this value of x in (ii), •■• (|y - 3) 3/ + 3% - 20 (|j/ - 3) - 180 = 0, which reduces to ^/^ + 2y - 80 = 0. .-. (y-HlO)(^-8) = 0. .-. 3/= -10, or 3/ = 8. Now 2/ cannot be negative, .*. 3/ = 8; also, if3/ = 8, .-. .r = fy-3 = |8-3 = 9. Hence one side is 9 inches long, and the other is 8 inches long. QUADRATIC EQUATIONS. 231 Ex. 5. Two places^ A and B^ are 168 miles apart, and trains leave A for B and B for A simultaneously. They pass each other at the end of 1 hr. 52 min., and the first reaches B half-an-hour before the second reaches A. Find the speed of each train. Suppose that the train from A to B travels at the rate of x miles an hour; and that the train from B to A travels at the rate of y miles an hour. At the end of 1 hr. 52 min. the first train has gone Y^'^y.x miles, and „ „ „ „ second „ „ i|^xy „ . The sum of these is the whole distance of 168 miles, .-. l|§x.r + lf§xy = 168, which reduces to x+y^QO (i). The train from A sroes 168 miles in hours, ° X ' A 7? 168 and „ B „ „ „ — - ,, but the train from A travels 168 miles in \ hoiu* less than that from B, 168 _ 168 _ 1 •'• X ~Y 2' which reduces to 336y = 336.r - ^y (ii). "We have therefore two equations involving two unknown quantities. From (i) we have x='dQ-y. Suljstitute this value of x in (ii), .-. 336y = 336(90-y)-(90-3/)y. .-. 2/2 _762y= -30240. Completing the square, .-. /- 762^/ + (381)2 = (381)2 -30240 = 114921. .-. y- 381 =±339. .-. ?/ = 720, or y = 42. If 3/ = 720, .-. .^' = 90-y = 90-720= -630. Ify = 42, .-. .r=90-y = 90-42 = 48. Both X and ?/ must be positive, hence the only solution suitable for the problem is .^^ = 48, ?/ = 42, Therefore the train from A to B travels at the rate of 48 miles an hour, and the train from B to A travels at the rate of 42 miles an hour. 232 PROBLEMS LEADING TO EXAMPLES. XVIII. 1. Divide 18 into two parts whose product may be 12 times their difference. 2. The product of two consecutive odd numbers is greater than the sum of the numbers by 79. Find the numbers. 3. A distributes £180 in equal sums amongst a certain number of people. B distributes the same sum, but gives to each person £6 more than A, and gives to 40 persons fewer than A does. How much does A give to each person ? 4. On a certain road the number of telegraph posts per mile is such that, if there were 3 less in each mile, the interval between the posts would be increased by 9f yards. Find the interval between two consecutive posts. 5. A rectangular room is 14 feet longer than it is broad, and 7 feet broader than it is high. If 1 foot were added to its height, 2 feet added to its breadth, and 6 feet taken from the length of the room, its cubical content would be unaltered. Find its dimensions. 6. The area of the floor of a certain room is 224 square feet; it is found that, by leaving a margin of 18 inches all round, a saving of 12 yards of carpet, which is three-quarters of a yard wide, is effected : what are the dimensions of the room ? 7. A colonel draws up his whole regiment in two solid squares; he then increases the side of the smaller square by 6 men, and finds that he must diminish that of the larger by 4 ; he next increases the side of the (original) large square by 6 men, and finds that he has one man over. How many men are there in the regiment? 8. The men in a regiment can be formed into a solid square, and also into a hollow square 4 deep, the number of men in the front in the latter formation exceeding the number of men in the front in the former formation by 25. Find the number of men in the regiment. 9. The men in a regiment can be arranged in a hollow square 4 deep ; if the number of men be increased by 129 they can be arranged in a solid square having on each side 10 men less than were on each outer side of the hollow square. Find how many men were in the regiment at first. QUADRATIC EQUATIONS. 233 10. The men iu a regiment can be arranged in a column twice as deep as its breadth ; if the number be diminished by 206, the men can be arranged in a hollow square three deep having the same number of men in each outer side of the square as there were in the depth of the column ; how many men were there at first in the regiment ? 11. The area of an oblong room is 221 square feet ; its perimeter is 60 feet. Find its length and breadth. 12. T"wo vessels, one of which sails faster than the other by 2 miles an hour, start together upon voyages of 1152 and 720 miles respectively ; the slower vessel reaches its destination one day before the other : how many miles per hour did the faster vessel sail? 13. A man travels 39 miles on a bicycle ; if his wheel had made 13 more revolutions per minute he would have covered the distance in 10 minutes less time, and if the circumference had been 1 foot greater, and had made the same number of revolu- tions as it did, he would have gone 3j miles farther. What time did he take 1 14. A man travels 24 miles on a bicycle ; if his wheel had made 8 more revolutions per minute he would have covered the distance in 10 minutes less time, and if the circumference had been 1 foot greater and had made the same number of revo- lutions as it did he would have gone 2 miles farther. What time did he take? 15. Two steamers ply between the same two ports a distance of 420 miles. One travels half-a-mile per hour faster than the other, and is two hours less on the journey. At what rates do they go ? 16. Two bricklayers can together build a wall in 18 days. Find how long it would take each of them to build it by himself if it be known that one takes 15 days longer than the other. 17. A farmer bought a number of oxen for £240, and after losing 3, sold the remainder for £8 more a head than they had cost him, and gained £59 on the transaction : what number did he buy? 18. A merchant lays out £16. 3.s. in buying silk. If it had cost Qd. less per yard, he would have got 8 yards more for his money. What was the cost per yard ? 19. A man bought a number of articles for £20, if each had cost 2s. less, he would have got 10 more for the same money : how many did he buy ? 234 PROBLEMS LEADING TO 20. The number of pence which a dozen apples cost is less by 2 than twice the number of apples which can be bought for Is. How many apples can be bought for 8s. ? 21. The number of eggs which can be bought for one shilling is equal to the number of pence which 27 eggs cost. How many eggs can be bought for one shilling ? 22. What are eggs a dozen, when if two more were given for a shilling, the price would be lowered a shilling a dozen ? 23. What is the price of oranges per dozen, when if two less were given in a shilling's worth the price would be raised one penny per dozen 1 24. A woman spends five shillings in eggs ; if she had paid threepence a dozen less for them, she would have got a dozen more for the same money. How many did she buy 1 25. The number of fives-balls which can be bought for a pound is equal to the number of shillings in the cost of 125 of them. How many can be bought for a pound ? 26. A man bought a number of articles for £21. If he had got 6 more for the same money they would have cost Is. Zd. less apiece : how many did he buy ? 27. A number of articles were sold for £6. 5s., and the seller by charging for each Qd. more than the cost price received for all but 10 as much as all had cost : how many were there ? 28. A number of apples were bought for 5s. 10c?. Half of the apples cost Id. a dozen more than the others, and 12- more of the cheaper than of the dearer were obtained for Is. How many apples were there ? 29. A man bought a certain number of shares in a compaiiy for £375 : if he had waited a few days until each share had fallen £6. 5s. in value, he might have bought five more for the same money. How many shares did he buy 1 30. What two numbers are those whose sum is 47 and product 312? 31. Find a fraction such that the denominator exceeds the square of half the numerator by unity, and the product of the sum and difference of the numerator and denominator is 225. 32^ A number consists of 2 digits of which that in the units' place is the greater ; the difference between their squares is equal to the number, and if they be inverted the number thus formed is 7 times their sum. Find the number. QUADRATIC EQUATIONS. 235 33. A number of two digits is equal to double the product of its digits and also equal to four times their sum. Find it. 34. A certain number exceeds the product of its digits by 52 and exceeds twice the sum of its digits by 53 : find the number. 35. Find two numbers expressed by the same two digits in difterent orders, whose sum is equal to the square of the sum of the digits and whose difierence is equal to the cube of the difference of the digits. 36. Find a number of three digits in which the digits in the hundreds' and tens' places each exceed that in the units' place by 1, and which is such that if it be divided by the digit in the units' place the quotient has 1 in the hundreds' place, and the other two digits are equal to half the digit in the hundreds' place in the original number. *37. Find two numbers such that their product is 48, and the difference of their squares is to the sum of their cubes as 13 to 217. *38. Divide the number 26 into three parts such that the sum of their squares may equal 300, and the square of the middle part may be half the sum of the squares of the greatest and least parts. 39. Find the sides of a rectangle whose area is unaltered, if its length be increased by 4 feet while its breadth is diminished by 3 feet, and which loses one-third of its area, if its length be increased by 16 feet while its breadth is diminished by 10 feet. 40. The area of a certain rectangle is equal to the area of a square whose side is three inches longer than one of the sides of the rectangle. If the breadth of the rectangle be decreased by one inch and its length increased by two inches the area is unaltered. Find the lengths of the sides of the rectangle. *41. The diagonal of a rectangular field is 182 yards, and its perimeter is 476 yards. What is its area ? 42. It costs twice as much per square yard to stain the floor of a room as to paper the walls, but it is found that the total sum spent on the floor is equal to that spent on the walls ; moreover, if the room had been 2 yards shorter and 1 yard broader, the total sum spent on the floor would also have been the same as that spent on the walls. Given that 8 sq. yards of the walls are occupied by doors and windows and that the room is 12 feet high, find its other dimensions. 236 PROBLEMS LEADING TO 43. A rectangular grass-plot, the lengths of whose sides are as 3 to 4, is surrounded by a gravel walk of uniform width, the outer boundary of which is also rectangular. The area of the walk is to that of the grass-plot as 2 to 3, and the length of a diagonal of the grass-plot is 100 feet : find the width of the walk, and the dimensions of the grass-plot. 44. Two rectangular fields were supposed each to contain 4 acres, but on measurement it was found that one contained 140 sq. yards more, the other 160 sq. yards less than that area, and that one was 10 yards longer and also 10 yards narrower than the other : find the dimensions of the fields. 45. Sixty-four square yards of the wall surface of a room which is 12 feet high are occupied by doors and windows, and the rest of the wall is papered at one shilling a sq. yd. The whole of the floor is also stained at a shilling a sq. yd. It is found that equal amounts of money are spent on paper and on staining, and that this would also be the case if the room were one yard shorter and two yards broader. Find the length and breadth of the room. 46. Two elevens A and B play a cricket-match. The total of ^'s first innings is the square of the difference of the totals of 5's two innings, and the total of .4's second innings is one-fourth the sum of the totals of ^'s two innings ; A scored 215 more in their first innings than in their second, and lost the match by three runs. What were the respective scores 1 47. A, B, C are candidates at an election. A polls a number of votes equal to the square of B^s majority over C, and C polls a number equal to the square of J's majority over B : A polls 27 more than C. Find what each polls. 48. Two trains run, without stopping, over the same 36 miles of rail. One of them travels 15 miles an hour faster than the other and accomplishes the distance in 12 minutes less. Find the speed of the two trains. 49. A starts to walk from P to Q: B starts 1 h. 40 m. later to drive from P to Q. B overtakes A at 10 miles from P, and picking him up brings him to ^ in 5 hours from the time at which A started, which is 2 h. 30 m, less than A would have occupied in walking the whole distance. Find the rate at which A walks and B drives. 50. A man bought a certain number of articles of equal value for £75. By selling them at £1. 165. each, he gained as much as ten of them cost him. How many did he buy ? QUADRATIC EQUATIONS. 237 51. A boy intends to spend a certain sum of money in la^Yn-tennis balls. There are two prices, each of the cheaper ones being Zd. less than each of the better ones. He finds he can buy as many of the better kind as he gives pence for each ; but he can get 4 more of the cheaper ones. How much has he to spend, and what are the prices of the balls ? 52. A person has ten guineas to spend in books of equal value ; if, however, he has any of them bound, they will cost six shillings a volume extra. He reckons that if he gets half of them bound he will possess nine more volumes than he would were they all bound: what is the cost apiece of the books unbound ? 53. A girl worked two square pieces of worsted work of the same kind, one of w^hich was an inch longer than the other ; the first took 12^ skeins, the second 18 skeins. What was the length of the side of each square ? 54. A sum of £19,950, if invested in the 4 per cents., would give an annual income of £8. 85. more than if it were invested in the 3 per cents. : if however each stock were 1 per cent, higher in price, the former would give £9. 175. l\d. more. What is the price of each stock 1 ■*^55. John and Hodge met together at market, John had bought sheep, and Hodge pigs and geese. Whilst taking a friendly pot together they agreed to an exchange of goods, viz. John to give his sheep for Hodge's pigs and geese. The value of each sheep was the same as that of a pig and goose, and that of each goose was two shillings. Now the number of the pigs and geese together exceeded the number of sheep by 16 ; and the number of geese exceeded that of the pigs by 10. What was the number of each, the pigs and geese together being worth five pounds? {The Ladies Diary ^ 1756.) CHAPTER XIX. EQUATIONS REDUCIBLE TO A QUADRATIC FORM. 223. It is customary to exclude from elementary algebra any discussion as to the general solution of equations of a degree higher than the second, but to include a. few equations which are immediately re- ducible to a quadratic form. We may remark that the solution of an algebraical equation can always be reduced to the solution of one which is (i) rational [Art. 182] and (ii) integral [p. 98]. Rational integral algebraical equations are classified according to the number of variables involved and their degree, and a rational integral algebraical equation involving one variable can always be solved, if it be of a degree not higher than four. 224. Quadratics. The following examples are typical of the more common equations which are quadratics in a power of x, and are at once reducible to a quadratic form. Bx. 1. Solve the equation x^-4,x'^-\-Z = 0. This is a quadratic in x^. Resolving into factors, (.^'2 - 3) {x"^ - 1 ) = 0. .-. ^2_3 = o, or .^'2-l=0. .'. .r=+^3, or x=±\. Hence there are four roots, namely, + ^3, ±1. EQUATIONS REDUCIBLE TO QUADRA TICS. 239 We may also begin the solution by putting x^^y. The given equation then becomes the roots of which are y = 3 and ?/ = l. ,*. ^2 = 3, or x^ = \. .-. ^= + ^3, or x=±\. Ex. 2. Solve the equation .^"*' + 8a;^4-12=:0. This is a quadratic in .r^, .-. (^^ + 6)(^3 + 2) = 0. .-. ^3 + 6 = or .^-■^ + 2 = 0. .-. x^=-% or .r'5=-2. .-. .y = 4/(-6) or x=^{-2). As a matter of fact there are three cube roots of — 6, and also three cube roots of — 2, but we shall not here concern ourselves with finding them. Thus there are six roots of the given equation. Similarly, if the given equation be of the degree 7i, there will always be n roots. AVe might also have begun by putting x^=y^ and then have proceeded as in the last example. Note. Any equation of the form is a quadratic in a-**, or is reducible to a quadratic by putting x^ = y. The solution of this quadratic gives two values of x"^. The ii^""" roots of these values give the roots of the given equation. "^ EXAMPLES. XIX. A. Solve the following equations. ^ 1. :r4- 13^2 + 36 = 0. 2. 3*-^ -44^-2 + 121=0. •^^ 16^-4 -40a2a;2 + 9a4 = o. 4. ^ - T^-"^ - 8 = 0. . ^2 + 16 25 > _^ ^2-62 ^ _ '• 25 "^^^2 + 16 ■ -5^ a2 '^a;2-62-^* 240 EQUATIONS REDUCIBLE TO 225. The method of solving another similar class of equations is illustrated by the following example. Ex. Solve the equation {x^ + 2^-3)2 + 3 (.v^ + 2^- + 2) - 1 3 = 0. V\xix'^^2x=y, ' .-. (3/-3)2 + 3(y + 2)-13 = 0. ... f-Si/-\-2 = 0. .-. (y-2)(y-l) = 0. •'• ^ = 2, or 7/=l. (i) If j/-=2, .-. .^2 + 2^^ = 2, - the roots of which are — 1 ± >/3. (ii) If.y=l, .'. .^2 + 2^=1, the roots of which are - 1 ± ^2. Hence the four roots of the given equation are -l±v/3, -l±s/2. If we put x'^ + 2x in a bracket, and treat it like a separate symbol, we need not introduce 7/ ; but the beginner will j^robably find it easier to use a subsidiary symbol as in the above example. JVote. Any equation like p{ax^-\- hx + cf-\-q {ax'^ ■\-hx-\-d) + r = 0, can be solved in a similar way. Because, if we put ax'^ + hx=y^ we obtain a quadratic in y. Suppose the roots are h and k. Then, (i) if y = h, .*. ax^ + bx=h, which gives two roots of x. And, (ii) if y=k, .*. ax'^-\-bx = k, which gives two more roots of x. Hence there are altogether four roots of x. EXAMPLES. XIX. B. Solve the following equations. 1. {x'^-4:X+b){x'^-4:X + 2) + 2 = 0. 2. (.^2 + ^_l)2_4(^.2_|_^,_l)_5^0. 3. (.^2 4.3^_i)2_i2(^2 + 3^._2) + i5 = o. 4. 12.y2_9^4.34. ^]^ =77. A QUADRATIC FORM. 241 226. Equations involving Surds. Equations involving surds may frequently be reduced to quadratic equations. Examples of these are given later in Chapter XXIV. (pp. 298 —302). 227. One root known. If one root of an equa- tion be known or be obvious, we can reduce the degree of the equation by unity. If the given equation be of the third degree, we can thus reduce it to a quadratic. E.v. 1. One root of the equation x^ + \Ox'^-x=lO is 1. Find all the roots. Since ^=1 satisfies the equation, therefore .v^ + lOx^ — x—\0 vanishes when x is put equal to 1 [Art. 90], and therefore is divisible by x - 1 [Art. 120]. Dividing x^ + 10.^2 - ^^ - 10 by x - 1, we obtain .1-2 +1U-+ 10 = 0. .-. (.r+10)(^+l) = 0. .-. x=-\0^ and a'=--l. Hence the other roots are - 10 and — 1 ; and therefore all the roots are 1, -10, —1. Note. Before dividing by .^' — 1, we have to take all the terms to one side of the equation. Ex. 2. One root of the equation x^ — a^ is a. Find the other roots. Since xi^-a^ vanishes when x=a [Art. 90], .*. .v^-a"^ is exactly divisible hj x—a, .'. ^-a3 = o can be written {x - a) {x^ -{-ax-\- a^) = 0. One root is given by x — a — 0. The other roots are given by x^ + ax + a'^ = 0, and .*. by Art. 210 are x = ^{-a±'\/ — 3a^). Hence there are three cube roots of a^, namely a, ^ ( — a + V — 3a2)j and ^ ( - a - V — Sa^) ; for each of these when cubed is equal to a^. This was what was asserted in Art. 184. Of these three cube roots, two are imagi- nary, and only one is real. B. A. 16 242 EQUATIONS REDUCIBLE TO EXAMPLES. XIX. C. One root of each of the following equations is printed in square brackets by its side. Find the other roots. 1. .i73-|-4r2 + .^-6 = 0, [-3]. 2. 5.^3- 13^2 _ 13^^ + 21 =0, [1]. 3. Qx^ - 6 1 ax"- - 1 2 1a\v - GOa^ = 0, [ - «]. *4. .r3 + F = 0, [-/[•]. 228. Reciprocal Equations. An equation in x is said to be reciprocal if, when - is substituted for cc X, and the resulting equation simplified, the form of the equation is unaltered. An equation of the form ax^ + hx^ + cx^ ±^hx + a = is reciprocal, since, if - be substituted for x and the resulting equation be simplified, its form is unaltered. An equation of this form can be solved by dividing by a-^, and putting 1 X "^ An equation of the form ax^±Kifi + cx'^+hx + a = can be solved by dividing by x^, and putting 1 X — = y. X ^ If — be substituted for x in this equation and the resulting X equation be simplified, its form is unaltered. Such equations are analogous to reciprocal equations. A QUADRATIC FORM. 243 229. The following are examples of reciprocal equations. Re. 1 . Solve the equation .r* _ Zx^ 4. 4-r2 - 3.r +1=0. Divide by .^2 .-. x''--Zx-\-4.-2--\-\ = Q. (..^4)-3(.+l) + 4 = 0. .'. the given equation becomes (^2_2)_3y + 4 = 0. .-. 3/2_3^^2=0. .-. (y-2)(y-l) = 0. .-. 3/ = 2, or 3/ = l. (i) If ^ = 2, .-. ^ + ^ = 2. the roots of which are 1 and 1. (ii) If y = l, .-. ^^ + ^=1. .-. x^+\=x, the roots of which are ^ (1 + V - 3). Hence the four roots of the given equation are 1, 1, ^(1 ± V - 3). Ex. 2. Solve the equation "— —^ + 2 .^ = 3. X -Y L X -j- 1 Let ^^=y, .-. the given equation becomes 3/ + 2l = 3. .-. 3/2_33^ + 2 = o, the roots of which are y = 2 and 3/ = !. 16—2 244 EQUATIONS REDUCIBLE TO QUADRATICS. ... jp2_2^_i=o. .-. ^(^-l)=0. .•. ^=0, or .r = l. Hence the four roots of the given equation are 1 ± ^2, 0, and 1. EXAMPLES. XIX. D. Solve the following reciprocal equations. 1. 2^+^-6^-+A- + 2 = 0. 2. 6.r4 + 35^ + 62^H35^' + 6 = 0. 3. 2^-9^-3_|.i4_^2_9^, + 2 = 0. 4. 6^ - 25.r3 + 1 2.r2 + 25.^• + 6 = 0. *5. 2.^■4 - bx^ + 6:^2 - 5.^^ + 2 = 0. CHAPTER XX. THE THEORY OF QUADRATIC EQUATIONS. 230. We discussed, in Chapter XVI., the means of solving a given quadratic equation, and we also there explained [Art. 206] how we could form a quadratic equation which has given numbers for its roots. We shall now proceed to investigate some of the general properties of the roots of any quadratic equation, and shall conclude the chapter by shewing how the form of the solution of a quadratic equation sometimes enables us to determine the greatest and least values which a given expression can assume. 231. We will begin by reminding the student [see Art. 210] that the roots of the equation ax'^i-hx + c = 0, -b-\- Jb^-^c are oc = ^ Hence the roots are 7^eal if ¥ > ^ac, „ „ „ equal if ¥ = 4iaCy „ „ „ imaginary if 6^ < 4ac. Moreover, if ¥ — 4ac be a perfect square, then we can take the square root of it, and the roots of the given equation will be rational [Art. 182]; but if ¥ — 4ac be not a perfect square, then the roots of the given equation will involve an irrational quantity. 246 THEORY OF QUADRATIC EQUATIONS. 232. Sum and Product of the Roots of a Quadratic Equation. The following is an important proposition. The sum of the roots of the quadratic equation ax^ + 6^' 4- c = is equal to , and the pro- c duct of the roots is equal to - . If we solve the equation, we find that the roots are h Vffi-4ac) ^" 2a'^ 2a and x = b_ _ \/{h~ - 4ac) 2a 2a b The sum of these is — tr ^ , that is 2a 2a The product of the roots ^ 2a'^ 2a } \ 2a 2a J ¥ W L 2a J [Art. 62, Ex. 3. ¥ ¥ - ^ac 4a2 4a2 4ac 4>a' c a ' Note. The sum of the roots of the equation x^-\-px + q = is —p, and their product is + q. THEORY OF QUADRATIC EQUATIONS. 247 Hence, if we are given that the sum of two quan- tities is h and that their product is k, we know that the two quantities must be the roots of the equation x^ — hx + ^^ = 0, Ed\ 1. Find the sum and the product of the roots of the equa- tion x'^ + 1x + 6 = 0. The sum is - 7, and the product is 6. If the equcation be solved, the roots will be found to be — 6 and - 1, which agrees with the above statement. Ex. 2. If a and /3 be the roots of the eqitation '2.x^ - 5.^r + 3 = 0, find the value of a? + /3^. We have a+/3=|, and aj3 = f ; .-. (a + /3)3 = (|)3, that is, a3 + ^3 + 3a/3 (a + /3) = ^K ... „3 + ^3 + 3.|.| = l|5. EXAMPLES. XX. A. Write down the sum and the product of the roots of the following equations, numbered 1 to 6. 1. ^2_3^_|.4:=o. 3. ax'^-bc.v-\-a^ = 0. 4. 2^2_3 = 6(jf-l). 5. 3.rG^-2) = 2(l-3.^'). 6. {px- q) {qx -p) = {x -p) {x - q). 7. If a and jS be the roots of the equation 3^^ _ 7^. _^ 4 = o, find the values of aH^^ and {a-^f. 8. If a and /3 be the roots of the equation px'^-\-qx-]rp = 0, what is the value (i) of a-/3, and (ii) of a^-^^1 9. If a and ^ be the roots of the equation x"^ -px-\-q = 0, find the values of a2 + ^2^ and (a + 5) (^ + „) ' 248 THEORY OF QUADRATIC EQUATIONS. 233. Resolution into Factors. If h and k he the roots of the equation ax^ -\-hx-\- c = 0, then will aaf -\- hx -\- c = a (x — h) (x — k). We have h -^ k = -- , smd hk = -; [Art. 232] a a' ^ ,". h = — a{h-\-k), and c = ahk. Substitute these values of h and c in ax'^ + 6^ + c, .*. ax^ -hbx + c = ax^ — a{h-\-k) x+ ahk = a[x^—Qi-^ k) X + hk] = a{x — h) (x — k). 234. Hence, if we want to resolve the expression ax'^ + hx+ c into factors, we can proceed thus. Put the given expression equal to zero, and solve the resulting equation. If the roots be h and k, then the required factors will be a {x — h) (x — k). Ex. 1. Fiyid the factors of 42x'^ - 5x - 63. If we equate this to zero, we obtain the equation 42^2 _ 5^^ _ 63^0. The student, on solving this, will find that the roots are f and -J. .-. 42^2_5^^_63_42(^^_|)(^^_^l) = (7^-9) (6^^ + 7). Ex. 2. Find the factors of the expression (ah + 1) (.x'2 + 1) + (a + 6) (x2 - 1) - (a2 _|. ^,2 _ 2) ^. If we equate this to zero, and arrange it in descending powers of X, we obtain the equation {ab + a + b+l)x^-(a^ + b^-2)x-\-ab-a-b + l=0. The student, on solving this by one of the methods given in Chapter XVI., will find that the roots are and t—— • a+1 b+l THEORY OF QUADRATIC EQUATIONi^. 249 Hence the given expression is equal to which reduces to {(a + 1) x -{b- 1)} {{b-\-l)x-{a- 1)}. EXAMPLES. XX. B. Find the factors of the following expressions. 1. 9.^2-24^+16. 2. l6a;^+l6a; + 3. 3. {:c-l){x-2)-6. 4. a;^--2ax + a^-b^. 5. .r2 - a2 _ 2ca; - 6^ + 2ab + cK 6. .r2-2(a + 6)a.--a6(a-2)(6 + 2). 7. (a- — 6) (.j; — c) 4- (.r — c) (.a; — a) -f {x - a) {x — b). *S. yz{y — z)-{-zx{z — x)-\-xy{x — y). 235. If one root of a quadratic equation be obvious, the other root can be obtained after division. For if /i be a root of the equation, ob- tained by equating a quadratic expression to zero, the expression vanishes when x is put equal to h [Art. 90], therefore the expression is exactly divisible hy x — h [Art. 120]. The quotient will be an expression of the first degree in x, which gives a simple equation for the other root. This is the same principle which we used in Art. 227 to reduce the degree of an equation by unity when one root was known. Example. Solve the equation - + - = - + v • a X a It is obvious that this equation is satisfied by x^b, .'. b is one root of it. On reduction and simplification, the equation becomes bx'^-x{a'^ + b'^)-\-a% = 0. Since the left-hand side is known to vanish when .37 = 6, it must be exactly divisible hj x-b. Dividing hj x—b, we obtain bx - a2 = 0. Hence the other root is -r- . b 250 THEORY OF QUADRATIC EQUATIONS. EXAMPLES. XX. 0. Solve the following quadratic equations, numbered 1 to 3. , a-\-h a + h ^ 1,1 1. \-ax= Vac. 2. x-a = 6-a~, . X c x-a b-a 3. (*'-l)(^-2) = (a-l)(a-2). 4. Find for what other value of x, the expression has the same value as it has when x = a. 236. There cannot be more than two unequal roots of a Quadratic Equation. First proof. For suppose, if possible, that there are three roots of the equation ax"^ + 6« + c = 0, no two of them being equal. Denote them by h, k, I. Since h and k are roots we have [Art. 233] ax^ -\-hx + c = a(x~h) (x — k). But since Z is a root, it satisfies the equation [Art. 90] ... a(l-li){l-k) = 0. Therefore one of the factors of this last expression must vanish. [Art. 204.] But a cannot be zero, and if either l — h or l — k vanish, then I is not different from h and k. That is, two of the assumed roots must be equal, which is contrary to the hypothesis. Thus there cannot be three unequal roots. Second proof. This proposition may also be proved thus. Since h and k are roots of the equation ax'^-\-bx + c = 0, we have ah'^ + bh-^c = and ak^-\-bk-^c = 0. Subtract, .'. a {/i^-k^) + b{h-k)=0. .-. a{h-k){h + k) + b{h-k) = 0. .'. {h-Jc){a{h+k) + b] = 0. Now h — k'i^ not equal to 0, therefore the other factor of the product must vanish [Art. 204] ; .-. a{h-\-l-)-\-b = Q (i). THEORY OF QUADRATIC EQUATIONS. 251 Similarly, since h and I are roots of ax^-^hx-\-c = 0, we have the equations ali--\-hh + c = and aP-hl + c = 0; and from these, in the same way as above, we obtain a{h + l) + b = (ii). Subtracting (ii) from (i), we have a{k-l) = 0. And since a is not equal to 0, therefore k-l = 0, that is, two of the roots must be equal, contrary to the hypothesis. JVote. The proposition that there cannot be more than one root of an equation of the first degree can be proved in a similar way. 237. The use of the propositions in Art. 232 enables us to form equations whose roots are con- nected in a given manner with the roots of a given quadratic equation. Ex. 1. If a and 13 be the roots of .v'^+px-\-q = 0, form the equation lohose roots are - and - . a ^ The required equation is that is, {ax - 1) (/3.r - 1 ) = 0, that is, a/3.);2 -(a + ^)x + l= 0. Now by Art. 232, a + ^=-p and a/3 = q. Hence the required equation is qx^+px +1=0. Ex. 2. If a and /3 he the roots of the equation x'^ + 4.v + 2 = 0, find the equation lohose roots are - and - . The required equation is [•^'-^) ( -v — ) = 0, that is, x'^-(^ + ^yv+\ = 0; 2 _|_ Q2 that is, x^ ^— .r + 1 = 0. op 252 THEORY OF QUADRATIC EQUATIONS. Now, in this Ccase, a+^^-A and 0/3 = 2, [Art. 232. .-. a2-|-/32 = (a + ^)2-2a^ = (-4)2-2x2 = 12. Hence the required equation is ^2- ^^'+1=0, that is, x'^-Qx^-\ = 0. MISCELLANEOUS EXAMPLES. XX. D. \The following miscellaneous examples on the subject of this chapter are to he solved hy the itse of the foregoing articles, and not hy finding the roots of the quadratic equations which are given."] 1. If a and /3 be the roots of Ax'^ = Zx- 1, find the value of ^2^a2- 2. Shew that, if a and ^ be roots of the equation x^-px-\-q=0^ then g^ , ^^ _ 3. Shew that, if a and ^ be the roots of x'^-cx-\-d=Oj then (a2 + (^)2- 02+^)2 = ^3 (a-/3). 4. Prove that the difference of the roots of x'^-\-px + q = is equal to the difference of the roots of x'^-\-Zpx + '2p'^-^q = 0. 5. If a and ^ be the roots of the equation x^-2px-\-q = 0, prove that a^ + /S^ = 8p3 _ g^^^ 6. If the roots of x^^-px + q = 0, and those of x^-\-qx-^p = differ by the same quantity, then ^-j-2' + 4 = 0. 7. Find the value of c in terms of a and h, in order that the sum of the roots of the equation x'^ + ax + h = Q may be equal to the difference of the roots of the equation x^-\-cx + {a-\-c)h = 0. 8. Shew that one root of the equation ax'^ + hx + c=0 wiU be the reciprocal of the other root, if a = c. 9. If a and ^ be the roots of the equation 2^2 — s.^r -j- 3 = 0, find the equation whose roots are - and - . THEORY OF QUADRATIC EQUATIONS. 253 10. If a and fi be the roots of the equation px'^-\-qx + r = Oj form the equation whose roots are — , - - . 11. Prove that the roots of x^+2mx-ir^n = are 2a and 2/3, where a and fi are the roots of x^-\-mx-{-n = 0. 12. If a and ^ be the roots of the equation lx^ = 'ix-\-\^ find 2 /32 the values of a^ + /S^ and of - H — . P a 13. If the roots of the equation ax'^-\-x-\-h = be equal to a and 6, what are their numerical values? 14. If the roots of the equation px'^ + qx-\-\=0 be equal to - and - , what are their numerical values ? P 9. 15. If a and /3 be the roots of the quadratic x'^-{-rx-\-s = 0, shew that the roots of 4x'2 + 2r.r+s = are \a and \^. 16. Shew that, if a and ^ be the roots of x'^-hx-{-c = 0^ then (a2-c)2-(/32-c)2 = 6(a-^)3. 17. Find the quadratic which has equal roots, each being equal to the sum of the roots of the equation Zx'^-\-bx-\-\=0. 18. If a and /S be the roots of px'^-itqx + r = 0, shew that the equation pqx'^-\-{pr-\-q^) x-\-qr = has roots — — _ and a-f ^. 19. Prove that, if a and /3 be the roots of px'^-\-qx-\-r = 0^ then qrx"^ -\-{pr-\- q^) x +pq = has roots — -^ and - + s • 20. If a and /3 be the roots of the equation x^-px-\-q = 0, then will | + ^=^^-iS.V2. j3^ a^ 2"^ 2 21. Find the relation that must exist between the quantities a, 6, c, so that the equation ax'^-\-bx + c = may have one of its roots double of the other. 22. Prove that, if a + 6 + c = 0, then each pair of the equations .r^-f a^-|-6c = 0, x'^-\-hx-\-ca = 0, and x'^ + cx-\-ah^O will have a common root. 254 THEORY OF QUADRATIC EQUATIONS. 23. If the roots of ax^ — hx+c = be the reciprocals of the roots of a^x^—hiX + Ci = 0^ prove that ab^ = bci, and aai=cci. 24. Prove that, if ax^ + bx-\-c = and ax^ + b'x-c^O have a common root, then their other roots will be equal in magnitude but opposite in sign. *25. If ax^ + bx + c = 0, cx'^ + kx + a = have a common root, prove that h must be a root of the equation *26. The expressions x'^-\-^x-\-b and x'^-\-Vlx-^'ib have a common factor. What numerical values can b have ? *27. How can you tell, without solving, whether the roots, supposed real, of a quadratic equation are positive or negative? Prove that the positive root of .r^- 8.^-8 = is greater than 8. *28. The numerical term in a quadratic equation of the form x'^ -^ px -\- q — is misprinted 18 instead of 8, and a student in consequence finds the roots to be 3 and 6. What were they meant to be? *29. Two boys attempt to solve a quadi'atic equation. After reducing it to the form x'^-\-'px-\-q=^^ one of them has a mistake only in the absolute term, and finds the roots to be 1 and 7 ; the other has a mistake only in the coefficient of .r, and finds the roots to be — 1 and — 12. Find the roots of the correct equation. *288. Application to Maxima and Minima. The maximum or minimum values of expressions in- volving a variable quantity (say oo) to a power not higher than the second can often be found by equating the expression to y, and solving for x. The method is explained and illustrated in the following examples. Ex. 1. Find the least value which the expression x^ — Qx + \0 can have for any real value of x. Let x'^-Qx+\0=y. .'. ^2_6^+10-3/ = 0. .-. .r=3±x/y-l. Now, if X be real, y — l cannot be negative, .'. y is not less than 1. Hence the smallest possible value of y is 1. THEORY OF QUADRATIC EQUATIONS. 255 Ex. 2. Divide a line into two parts so that the rectangle con- tained hy them shall be a maximum. Let the length of the line be a; that is, suppose that it contains a units of length. Let .v be the length of one part into which it is divided, .*. a-.r is the length of the other part. Thus X {a - x) is to be a maximum. Let x{a- x)^y. .'. x'^-ax-^-y^^^. a + sJ a^ — Aai Now, if X be real, a^-^y cannot be negative. Therefore the greatest possible value of 4y is a^\ that is, \d^ is the greatest possible value of y. If yz=\a?-^ then x = \a\ that is, the line must be bisected. Ex. 3. Find the greatest and the least values which the ex- pression -^ can have^ where x is any real quantity. J ^ x'^ + x + l Let -^ -^ = y. .-. .'v''- + x-\-l=y {x'^ - x + l). Solve for x^ _ -(l+:y)±\/(l+y)^-4(l-y)^ 2(l-y) Now X is real, .*. (l -\- y)^ - 4 {1 - yY must be positive, that is, {(1 +y) + 2 (l-y)} {(1 +y) - 2 (1 -y)} is positive, that is, {3 -y} {^3/ - 1} is positive. Hence the factors 3 - y and 3y — 1 must be of the same sign. Now, if y>3, the first of these factors is negative and the second positive; also, if 3j/, went to market to buy sheep. A bought 8 sheep more than B, and B bought 16 sheep more than C. The number of sheep bought by C and D was 72. The sum of the squares of the number of sheep that each person bought was a minimum. Find what number each person bought. CHAPTER XXI. INDETERMINATE EQUATIONS. \^The student ivho is reading the sicbject fo?' the first time may omit this chapter.] *239. A single equation between two or more un- known quantities, such as ax -\-by = c, can be satisfied by an infinite number of values of the unknown quantities. For, if there be two unknown quantities, any number or value may be given to one of the unknown quantities, and we shall then get an equa- tion to determine the corresponding value of the other unknown quantity. Such equations are called indeterminate. More generally we may say that a system of simultaneous equations where the number of equations is the same as the number of variables is usually determinate^ that is, the equations enable us to determine a finite number of values of the unknown quantities which satisfy all the equations. But if the number of equations given be less than the number of unknown quantities contained in them, the system is indeterminate. *240. It may however happen that in a numerical indeterminate equation the number of roots which are whole numbers or integers is determinate. The general method of obtaining such roots lies beyond the scope of this work, but it may be illustrated by one or two easy examples. B. A. 17 258 INDETERMINATE EQUATIONS. *Ex. 1. Fhid all the positive integral solutions of the equation We begin bj dividing every term by the coefficient of x or the coefficient of y, taking the smaller of the two. In this case, therefore, we shall divide by 3, ••• .^'-h§^+y = 7 + l. We now take all the integral terms to one side of the equation, and the fractional terms to the other, •••• •^+y-7=:i-|A- (i). Now X and y are integers, .*. x-\-y — l is an integer, .'. ^ — %x must also be an integer. Let us denote it by jt?, ••• t'-|^^"=P, .-. \-%v = '^p. This is another indeterminate equation between x and p, but of a simpler form than that from which we started. Con- tinuing the above process, we must now divide by 2, and then transpose the integral terms to one side of the equation and the fractional terms to the other side. We thus obtain .^■+^ = l-lp (ii). Now X and p are integers, .'. ^-\p must be an integer. Let us denote it by q, ■'■ k-\P = 1- .-. l-jo = 2^ (iii). This is another indeterminate equation between p and q, but as the coefficient of one of the variables, namely jo, is now unity we need not continue the process any further. From (iii), we have p = \-'2,q. Substitute this value of jo in (ii), and we obtain ^' = 32'- 1. Substitute this value of x in (i), and we obtain y = 9-5^. The integral solutions of the equation are all included in the values x = '^q-\, and y = 9-5q, where q may be any integer whatever. It is obvious that, if q be an integer, the corresponding values of x and y will also be integers ; and it is easy to verify that whatever value q may have, these values of x and y satisfy the given equation. Now in this case x and y are not only to be integers, but are also to be positive. To make y positive, q must be less than 2 ; to make x positive, q must be positive. Hence the only INDETERMINATE EQUATION>S. 259 viiluG which (J can have is 1. The only positive integral roots are thus given by putting 5' = !, and are .17 = 2, ^=--4, *Ea;. 2. Eiiid all the positive integral solutions of the eqication We have (.^' +7/) {x -y) = 63. Since x and y are positive integers, .*. x-\-y dA\diX-y are integers. Also, x-\-y is positive and greater than x—y. Now the only positive integral factors of 63 are 63 and 1, 21 and 3, 9 and 7. Hence the only solutions are given by x-y^ Ij' x-y= 3j ' x-y = l] These give x = 32 and v/ = 3 1 ; .i- = 1 2 and y = 9 ; .^^ = 8 and y = 1 . *Ex. 3. Find in how many ways a debt of 3s. Sd. can be paid in sixpences and francs — the value of a franc being taken as lOd. Let X be the number of francs, and y the number of sixpences, which are used in paying the debt : the value of these is equal to {I0x + 6y) pence. This, by the question, is equal to 35. 80?. or 44o?. .-. 10^ + 6y = 44. .-. 5.^ + 33^ = 22. But by Ex. 1, there is only one solution of this equation in posi- tive integers ; namely, x = 2 and y=4. Therefore there is only one way of paying the debt, namely, by paying two francs and four sixpences. ^EXAMPLES. XXI. Solve the following equations in positive integers. 1. 2x + 3y = 9. 2. 3x-{-7y=58. 3. 13.^-'+2y = 119. Find the general solution of the following equations, and also the least positive integral values of x and y which satisfy each of them. 4. Sx-2y = l0. 5. 7x-9y = 2d. 6. 13.v-17y = 9. 7. In how many ways can a sum of 15s. 6d. be paid in threepenny pieces and fourpenny pieces ? 8. In how many ways can five pounds be paid in dollars (worth 4s. each), and francs (25 of which are worth £1) ? 9. Find in how many ways the sum of £4. 15s. 6d. can be paid in half-guineas and half-crowns. 17—2 260 INDETERMINATE EQUATIONS. 10. The sum of two numbers is 35, and their highest com- mon factor (or greatest common measure) is 7. Find all the positive integers which these numbers may be. 11. A boy had between 150 and 180 nuts. He divided them into parcels of four, and found there was one over. Then he divided them up into parcels of five, and found that none were left over. How many nuts had he ? 12. Three Arab jugglers, travelling to Mecca, with a per- forming monkey, purchased a basket of dates. The basket con- tained enough dates to divide equally among the three and one over. During the night one juggler awoke, and, after giving one date to the monkey, secretly ate one-third of the remainder. Afterwards, each of the others in turn awoke, and, after giving one date to the monkey, secretly ate one-third of the remaining dates. In the morning, enough dates remained to divide equally among the three, with again one over. What is the least number of dates which must have been purchased to make this possible ? 13. A owes B 4s. 8c/. A has only half-crowns, and B has only fourpenny pieces. How can A most easily pay B ? 14. Shew that the equation 4:X—Qy = \l cannot be satisfied by integi'al values of x and y. 15. Divide 25 into two parts, such that one of them is divi- sible by 3 and the other by 2. 16. The sum and the product of two integers are together equal to 41. What are the two integers ? 17. A person bought an exact number of shares in a certain company at £24. \0s. per share, spending between £200 and £300. Some time after, he bought another exact number at £4. 10s. He then sold out the whole for £400, which was the money he gave for them. Find how many shares he bought on each occasion. 18. "There came three Dutchmen of my Acquaintance to see me, being but lately married ; they brought their Wives with them. The Men's Names were Hendrick, Claas, and Cornelius; the Women's Geertruij, Catrijn, and Anna: But I forget the Name of each Man's Wife. They told me they had been at Market to buy Hogs ; each Person bought as many Hogs as they gave Shillings for each Hog; Hendrick bought twenty-three Hogs more than Catrijn ; and Claas bought eleven more than Geertruij ; likewise, each Man laid out three Guineas more than his Wife : I desire to know the Name of each Man's Wife." {The Woman's Almanccck, 1739.) EXAMINATION QUESTIONS. 1. Find the square root of 144.^4 _ 600.r3 + 913.^.2 _ gOO.r + 144 ; and prove that the square root is divisible by 3.v - 4. 2. Solve the equations : 3.^-2 2^-5 ^'^ 2.1^-3 .r-1 ~ ' (ii) 2^-3y=l, .?;(.r-2y)+?/(3/- 2.r) + 26 = 0; (iii) .r2-3/2 + 2.v-43/-f5 = 0, .^^ + 2.r +?/=!. 3. Shew that the sum of the squares of the roots of the equation x^ + rx + s = is equal to r^ - 2s. If a, /3 be the roots of the equation ax^ -{-hv + a -\-b = 0, prove that(l-a2)(l-^2)_4„^. 4. A man buys a number of yards of cloth for £20. He sells three-quarters of them at 3s. per yard, and the rest at two- fifths its cost price, when he finds that he has neither gained nor lost : how many yards did he buy ? 5. A rectangular plot of grass is surrounded by a gravel walk of equal area 6 feet wide. The diagonal of the rectangle formed by the outer boundary of the walk is 58 feet : find the dimensions of the grass-plot. 262 EXAMINATION QUESTIONS. 6. Find the square root of \Qa^- ■8a^- ma^ + a ^2 + 4a + 4 7. Solve the equations : (i) 3^2_ 16^+16^0; ^ ah ^ y X i 1 + 64 8. Shew that there cannot be more than two unequal roots of a quadratic equation. Form the equation whose roots are the reciprocals of those of the equation (^ + 5)2-4^=160. 9. A starts to walk from Wimbledon to London, and B from London to Wimbledon at the same time. As they pass one another, A diminishes his pace one mile an hour, and B in- creases his one mile an hour. Each arrives at his destination at the same time. Find how much faster one started than the other. 10. The length of a rectangular room is twice its height, and its height is two-thirds of its width. If 5 feet were taken oft' its length, 5 feet added to its breadth, and 5 feet added to its height, its cubical contents would be increased by 1500 cubic feet. Find its dimensions. 11. The united ages of a man and his wife are six times the united ages of their children. Two years ago their united ages were ten times the united ages of their children, and six years hence their united ages will be three times the united ages of the children. How many children have they ? 12. Find the cube root of 64a<5 - 48a5 - 84^^ + 47^3 + 42^2 - 1 2a - 8. 13. Find the factors of 4 (.^H 1 ) - 20.'^ (.^2 ^ i ) ^ 33^^.2^ 14. Solve the equations : (i) 11.^2 _ 13^^ = 18; (ii) ^2 + 2^^ = 3, .r2-3?/2 = 6. 15. A bag contains 180 gold and silver coins of the value altogether of £60. Each gold coin is worth as many pence as there are silver coins, and each silver coin as many pence as there are gold coins. How many are there of each kind ? EXAMINATION QUESTIONS. 263 16. In a cricket-match, the score of the captain is three times tliat of the next best player, and the scores of these two together make up three-fifths of the total of the innings. The average of the rest of the eleven is seven rims, and nine runs are scored for extras. Find the captain's score and the total of the innings. 17. If X=-pqx-{-{py + qz){p-\-q\ J^= - pqy + {pz -f qa:) {p + q), Z= -pqz + {px + qy){pJrq\ prove that XH J'2 _^^2 ^ (p2 ^^,^ + ^2)2 (.^,2 +y2 + ,2^ 18. Solve the equations : ^ ' a+x b+x (ii) 2x -f y = .^2 _ 2.r -f 6/ + 4y = 5. 19. Shew that the product of the roots of the equation x^-\-px + q = is q. Find the equation whose roots are the reciprocals of those of x^+px + q = 0. 20. From a square field is taken as much ground as is required to form a road of uniform breadth skirting the sides of the field. The ground so taken is ^\ of the whole field. The total length of the road measured along the middle is 700 yards. Find the breadth of the road. 21. Find the square root of 9a2 _|. 4^2 _^ ^2 + 25(^2 _ Uab - 10c^+ 4hc - 6ca -t- 20da - 20bd. 22. Solve the equations : (i) x^+2 x-2 _ 2.r + 5^ ^ .r^"*'^+2~ x" ' ., a—bx^ ax^—b _ a-b ^"^ a + 6^2 + ^^2:^^^,-^^qr6' (iii) .r2 -f .ry = 22, y'^+yx= —\^. 23. If in a number of 3 digits, the sum of the tens' and hundreds' digits be 3 times the sum of the units' and tens' digits, prove that the number is divisible by 7. 24. The sum of the cubes of three consecutive integers is 33 times the middle integer : find the numbers. 264 EXAMINATION QUESTIONS. 25. A man buys two kinds of cloth, black and brown, the brown cloth costing Qd. a yard more than the black. He pays £5. 55. for the black cloth, and £4. \0s. for the brown, but obtains 3 yards more of the former cloth than of the latter. Find the cost of a yard of each kind of cloth, and the number of yards he bought. 26. Find the square root of (3.^2 _ 5)2 _ 4^. (3.^.2 + 5) + 64^2^ 27. Solve the equations : . x + l .r-l_2.t-+l| (ii) x-y=\, afi—y^=\^. 28. Find the least value which the expression x^ - 4.r + 7 can have for real values of .^'. *29. A and B are two stations on a railway. A fast train leaves A for B at the same time that a slow train leaves B for A . They arrive at their destination three-quarters of an hour and an hour and twenty minutes respectively after they passed each other. Find how long each took for the journey. *30. On two different railways the stations on each are at equal distances. A train on one runs at the rate of 20 miles an hour, and a train on the other runs at the rate of 24 miles an hour, while both trains lose two minutes at every station. The arrival at every station on the first line coincides with the arrival at every fourth station on the second line. Find the distances between the stations, supposing that neither train can run more than 20 miles without stopping, and that on each railway the distances between the stations are an exact number of miles. CHAPTER XXII. FRACTIONAL AND NEGATIVE INDICES. 241. We have hitherto supposed that the indices which have been employed are positive integers ; and, in fact, the definition of x''\ given in Art. 22 (namely, the product of n quantities, each equal to x), is un- intelligible unless n is a positive integer. We shall now proceed to see whether we cannot frame some other definition 6f ^*\ which shall be in- telligible whatever n may be ; but, of course, the definition must be so framed that, when n is a positive integer, it shall agree with that already given. 242. The method we shall adopt is strictly analogous to that used in Art. 40, when we were seeking to extend our definitions of addition and subtraction to inchide the case of negative quantities. In that case, we first proved that certain formulae were true so long as a was greater than h, and, next, by assuming those formulae to be true whatever numbers were represented by a and 6, we were able to extend om* ideas of addition and subtraction to include negative as well as positive quantities. The process by which, in Art. 55, we obtained the meaning to be assigned to the product of two negative quantities was similar. 243. We have already proved some propositions about indices, and in particular have shewn [Art. 72] that 266 FRACTIONAL AND NEGATIVE INDICES. when m and ii are positive integers. We shall now assume this result to be true, whatever m and n may be, and from it shall deduce a meaning to be assigned to x'^, when n is either negative or fractional ; and this meaning we shall take as our definition of oc^^. 244. The only other propositions about indices which we have proved are [see Arts. 76, 86], that, if m and n be positive integers, then if m > ?z, «^ -f- x"- = x'''-''' ; and (^'^)^^ = x^'^. These results may be regarded as true for all values of m and n, provided we can shew that they are satis- fied by the values which we are going to assign to ^"^ and x'^ when m and n are fractional or negative. This we shall do in Art. 246. 245. We shall begin then by assuming the formula x^x^ = x^'^^, and shall thence deduce in succession the meanings to be assigned to a quantity raised to the power of (i) a positive fractional index, (ii) a zero index, (iii) any negative index (integral or fractional). p Theorem (i). To shew that x^ is equal to the q^^' root of x^, where p and q are any positive integers. Whatever numbers m and n may be, we have and similarly, x^ x x'"" x x'' x ...= ^^+'»+"+- (a). p p p p Hence {xi)i = xi x x^ x xi x ... (q factors) [Art. 22 n n p - + -+- + . ..(pterins) n , . = ^999 [by (a) = XP. FRACTIONAL AND NEGATIVE INDICES. 2G7 Take the 5^'' root of each side of this equality, V. .'. x^ is equal to the cf^^ root of /r^. Theorem (ii). To shew that a^ is equal to unity. Whatever numbers m and /i may be, we have Let m = 0, :. x"" X x"^ = x'K Divide both sides by x'^, :, x^ = \. Theorem (iii). To shew that x~^ is equal to the reciprocal of x'\ luhatever number n may he. Whatever number m and n may be, we have Let m = — n, .'. x~^^ x x''^ = a;~'^+*^ = *•« = 1. 1^ Divide both sides hy x''^, .'. x ^^ = Thus, whatever n maybe, positive or negative, inte- gral or fractional, we have found a meaning to be given to x\ 246. Next, we have to shew that this meaning is consistent with the results of the propositions collected in Art. 244. Whatever numbers m and n may be, we have, by Art. 245, ^•« X"" X'^ ,,,111 1 and —7 = .-r'" X - = — - X X" X" X"^ X'^ ^n-m 268 FRACTIONAL AND NEGATIVE INDICES. Thus the first two propositions quoted in Art. 244 are consistent with the extended meaning now given to .^•*'* and x'\ We have next to shew that, whatever numbers m and n may be, we always have {x'^Y = ^'"" '■> ^-nd here we shall consider in succession the cases where n is a positive quantity, zero, or a negative quantity. (i) If m be any number, and 7i =p/q, where ^ and q are positive integers, then we have to shew that p mp This, by Art. 245 (i), means that ^{x'^y = ^^, a result which is true. [Art. 76.] (ii) If m be any number, and n = 0, then we have to shew that {x"'f = x"'''\ This, by Art. 245 (ii), means that 1 = 1, a result which is obviously true. (iii) If 111 be any number, and n = — s, where s is a positive quantity, then we have to shew that (x'")-' = x-'"\ This, by Art. 245 (iii), means that 7 — r- = — , a result which by (i) of this section is always true. 247. Index Laws. All the above results are included in the statements that, whatever numbers m and n may be, and (a?^**)" = x'""". These results are known as the index laws. FRACTIONAL AND NEGATIVE INDICES. 269 248. The following examples illustrate the appli- cation of the index laws to the simplification of algebraic expressions. Ex. 1. Simplify ai' + ^ixai>^i^ai-P. The given expression = a^ + ^a x a?J + « x a^ " « _ ^3/) + 39_ Ex. 2. Simplify tlie fraction Ll^___l_i^ L, This fraction = ' ) «4.»4 r..\ir (^^p + q + ry. .f^np + 2nq) + {nq + 2nr) + {nr + 2np) ~ rf,n{p + q + r) j3np + 3nq + 3nr j>np + nq + nr __ j.2np + 2nq + 2nr EXAMPLES ON THE INDEX LAWS. XXII. A. Simplify the following quantities. ' a'i-P * ' x'^.x*. xf a"^-". a»^-3«.a'"-^'' (^)2 {pi^f {off ' Qn-m Qn-3m Qn-bm' X*^" ^ ■*" " X^'''^^' {x'''-^Y.{x^'-'y.{x'^^y (a^-«xa9-»-)2 ' {x<'7^x'y ' (a2)^x(a-«)2 • ^n-vi ^ Qn-2m ^ ^fti-Sm h^V . qV^ + ^V /l)\V(^-x+l) 1 _3 _ i_ ^ /~^ J 7 9. a^xa ^xax v^a^xfti^x V a^ x(tfc ^)6, 270 FRACTIONAL AND NEGATIVE INDICES. 249. The following examples illustrate the use of the index laws in questions involving the application of algebraic processes — multiplication, division, evolu- tion, &c. — to quantities raised to fractional or negative indices. The work will usually be facilitated if all the expressions be arranged in descending powers of some letter (see Art. 65). 11 11 Ex. 1. Multiply x-Zx^y'^+y by ±v^-by^. Following the usual process, we have 1 1 1 1 3 11 ^x^- Qxy^+ %x^y 113 - fixy'^ + 1 bx^y — by"^ 3 1 1 3 Add . • . 2.^-2-1 \xy^ + Vlx^y - by^ 3 113 11 Ex. 2. Divide 2.^•2- I7.ry2+17.,A/- 15/^ by 2.^2-15/1 Following the usual process, we have 3 1 2.f2_i7W^ 3 1 '2x'^— Ibxy^ 113 11 3 11 2a;2- 15?/2 ) 2.f2_ 17.^/^^- 17^1-2^- 15y2 ( x-x^f^y 1 1 - 2^3/2 +17a;2y - 2.ry2-f I5^2y 1 3 Ixhj-lby'^ 1 3 '2.x^y— \by^ 1 1 Thus the required quotient is x-x^y^+y. FRACTIONAL AND NEGATIVE INDICES. 271 Ex. 3. Find the square root of 5 48 S3 43 3 4^5 - 1 2a263 + 963 4. 1 6^2^,4 _ 246V + 1 ^c\ Following the usual process, we have 54 53 8 4 3 SS 4 3 4a5 - 1 2a263 4- 16a V + 9^,3 _ 246¥ + 1 6c'^ ( 2a^ - 3b^ -f 4c* 4a5 6 4 4a^'-363 543 4«2_663 + 4c4 54 53 8 12a^"6H16a¥ + 963 54 8 12a263 +963 53 43 3 16«¥ -2463c4 + 16c2 53 43 3 16a¥ -2463c4 + 16c2 543 Hence the required square root is 2a^-Zb^-{-4c^. Ex. 4. Find a root of the equation 3^ + ^ + 9^ = 6804. We have 3x3^ + (32)^ = 6804, that is, 32^+3.3^ = 6804. This is a quadratic equation in 3^. Let 3^=y, .-. y2_|.3^^6804. Completing the square, .*. y^-\-Zy-{-{fp = mOi: + l = '^^. Take the square root of each side, .'. y + f = ±-|-. .-. ^^. _3+i|i = 81, or j/=-f-i|^=-84. Now if X be real, 3-*= must be positive, hence the negative value of y is inadmissible, .-. 3^ = 81. But 81 = 3-*, .-. 3^ = 3^ which is satisfied by x=A. Therefore 4 is a root of the given equation. Ex. 5. Find a solution of the simultaneous equations 2M2'=128, \ These equations can be written 2^222/ = 2^ " that is, 2* + 2;/ = 2', b'^y-^ = b. 25^' 522/ 2^222/ = 2^ ^ = 5; 272 FRACTIONAL AND NEGATIVE INDICES. These are obviously satisfied by Adding, we get 4?/ = 8, .-. y = % Subtracting, we get 2^=6, .•. x = Z. Hence a solution of the given equations is a-- = 3, ^ = 2. *2.50. The use of the index laws also enables us, in some cases, to extract roots of expressions of an order higher than the second or third. For example, the fourth root of an expression can be found by extracting the square root of the square root of it. For suppose that we desire to find the fourth root of X. Then we have Example. Find the fourth root of 6561^4 - 43740a36 - bm^a^c + lOQBSOa^fta + 29160a266' + 1944aV - 121500a63 - 48600a62c - 6480a&c2 - 288ac3 + 506256^ + 270006% + 540062c2 + 4806c3 + lQc\ Extracting the square root of this expression by any of the methods given in Chapter XV., we find that it is 81a2 - 270«6 - 36ac + 22562 -f 606c 4- 4c=^. The square root of this latter expression is 9a-156— 2c', which is therefore the required fourth root. *251. Similarly, the sixth root of an expression can be found by extracting the cube root of its square root. For suppose that we want to find the sixth root of X, Then we have ^x = x^ = {x^f- = ll\7T]. The same method shews that we might also obtain the sixth root of an expression by extracting the square root of its cube root. 1 n - in FRACTIOXAL AND NEGATIVE INDICES. 273 MISCELLANEOUS EXAMPLES ON INDICES. XXII. B. 1 1 1. Simplify ^—^i — > ^^^^ fi^^d its value when a = 2, & = 3, c = 432. Simplify the following expressions, numbered 2 to 9. 14 _1 1 2. (4/a7) X (4/a9) X a'^^a^. 3. (^/a^) x {^la^) x a"^-fa2i. . (X/aY^^ r>C/a\»^ rj * W«J ' W«J ' V a'^ "" V «^ "" V a- • 8. a4U+6;-|-a3U+5,'. 9. j(^^] ^[.^'»-«..r2'']'»l 1111 10. Prove that a-h = {d'- h^) {a^ + h'-). 1 12 11 2 11. Prove that a-h = {a^- h^) {a? + a^h^ + h^). 1 12 11 2 12. Prove that a + 6 = (a^ + 63) («3 _ f^363> 63). 1 1111 1 13. Prove that a-h = (a* _ 6*) (a* + 6*) (a^ 4- 62). 3 3 1 14. Multiply 3.(,'2 - 5.r + 4 by 2^-2 _ ,^2 _ 4. 11 1 111 15. Multiply 6-i + 2a26"2_c-f«2 by c + rt2_2rt262 4_6-i. Ill 111 16. Multiply (.i-^-y*)2+3/2 by (.t-4 4-/)2+y^. 17. Find the contmued product of 1111 11 11 {x-\-a + a^x\ (a2+.r2), (a_a2^;t^2^_^.)^ and {,v^-aPj. 1 11 18. Divide by cfi - 2b- - 6*2 the expression 3 1 1 1 1 1111 1 33 «2 _ «62 - ac^ - 3a26 + a^c - 2cMc^- b'^c + 36c2 + 262 _ ^2. 19. Divide (.r - 5^-2 + 6) {x - 5.r2 - 1 4) by .r - 10.1-2 _^ 2I . B. A. 18 274 FBACTIONAL AND NEGATIVE INDICES. Ill 20. The product of two quantities is x + B>y-2'lz-\-\d>a:^yh'^^ and one of them is a^ + 2i/^-3z^ : find the other. 21. Simplify ^('^•^')'^:i^. ^ ^ x-\-y 1 1 22. bmiphfy ]1 ^ + __^lj_j_l. 23. Simplify 111 1 1 1 1~1 11 1~1 11 !• Extract the square root of the expressions numbered 24 to 27. 24. .r2 + -^,-4^- — + 10. 4 2 1 25. 4r2 - 1 2.^3 + 28.r + 9.v3 - 42.y3 + 49 5 1 11 2 1 111 26. X - 4.^•%e _j_ 6.^^2c2 + ^a^ki^ - 1 2.t'%^c2 ^ 9^^ 27. rt« + 6-c + 4a56-i + 26-5a + 2«^6-2-36-%2_ 6^,3^,-3^ ■^28. Extract the fourth root of 8Lr4 - 216.r3 + 216.^2 - 96.i' + 16. *29. Extract the sixth root of .^'^ x^y bx^y^ bx^y^ bx^y'^ xy'^ y^ 64 "^ T6 "^ ~48"" "^ "54~ "^ Tos" "^ "sT "*■ 729 * Find solutions of the following equations, numbered 30 to 33. 30. 4^ = 2^ + ^ 31. 9(9^-i + 3) = 28x3*. 2 222 22 22 22 32. («3 -f 1 ) (.^•3 _ ^,3) - (Z,3 4. 1) (^^^- _ „3) = (rt3;,3 4. 1 ) (^^3 _ ^,3). 33. (.r»*' + « ' 4- .r - 2«»») («"»' + «2 - a - 2'nn) -f (^"i'' + '*2 - X - 2'»») («»«' + «' + «- 2"»») = 0. 34. Prove that .r* ^^ = (^ V^)« is satisfied by x = 2|. 35. Solve the simultaneous equations 27^ = 9^, 8P/3^ = 243. Qc a 1 XI • ,^ ^. 2^+3^'= 171 Jb. oolve the simultaneous equations ^ ^., „ ^, ^>. 2* "^ - — 3^ "^ * = 5J CHAPTER XXIII. LOGARITHMS. [The results of this chapter are not required in the immediately following chapters^ and the discussion of logarithms may be deferred for the present if the student desh^e itJ] 252. Logarithm. The logarithm of a number to a given base is the index of the power to which the base must be raised to be equal to the number. Thus, if tt* = n, then x is called the logarithm of n to the base a. This is written as x=\ogan. The equations a* = n and x = log^ n express the same relation between a, x, and n ; and this relation may be written in whichever of these forms is the more convenient. 253. We can approximately calculate the log- arithm of a number to any given base in the manner described later [Arts. 414, 416] ; but in some cases, the exact value can be at once obtained from the definition. The first two of the following examples are important. Ex. 1. The logarithm of 1 is zero, whatever the base may be. For let a be the base, and let log^l =.t\ Therefore, by the definition, a^= 1. This equation is satisfied by .r = 0. [Art. 245, (ii). 18—2 276 LOGARITHMS. Ex. 2. The logarithm of any number to that number as base is unity. For let a be the base, and let \ogaa = x. Therefore, by the definition, a^=a. This equation is satisfied by x=\. [Art. 22. Ex. 3. Find the value of log^\. Let log4^ = .r. • Ax — L that is, 22^ = 2-1. This is satisfied by 2x= - 1, .•. x= -\. Ex. 4. Find the value of Zo^3,-(27)~^. Let x = \og^r^{21)-K 1 .-. (93)x=(33)-l^ ?x This is satisfied by g.v=3 -3, .'. x= -f^. EXAMPLES. XXIII. A. Determine the values of the following logarithms. 1. logiolOO. 2. logioo 10. 3. log^ 64. 4. log^rSl. 5. log-ooi 100. 6. log^oo'OOl. 7. \og,, .2. 8. log.oilOOO. 9. logV7 28. 10. log, 4. 11. logiooo -0001. .2. log^2 32. 13. log^5-008.' 14. logs 125. .5. logl44T7V8- 16. log256 32. 17. l0g343^49. .8. log.3-027. 19. log3V3(243)"l *20. lOgaO. 254. We now proceed to discuss some of the ele- mentary properties of logarithms : these are at once deducible from the two fundamental propositions re- lating to indices, namely, x"^x'^ = x'*^^''^ and {cc^^y^ = x'^'^ [Art. 247]. We shall suppose that the logarithms are calculated to any number, a, as base. LOGARITHMS. 277 255. Logarithm of a Product. The logarithm of a product is equal to the sum of the logarithms of its factors. Let m and n be factors of the product. Let m = a^, .' . x = \oga f^^ ] and let n = a^, •'- y= loga 7i. .-. mn = a"" . ay = a'^+y. .-. \oga{mn)=x + y = logam-\-\oga,n. Similarly, if m, n, p,... be factors of the product, let m = a^, n = a^, p = a^, &c., then mnp. . . = a^a^a^ ... = a^+y+z+-\ .'. loga {mnp ...)=x-^y + z + ... = log^m + log,,?i + log^p + ... . 256. Logarithm of a Quotient. Tlie logarithm of a quotient is equal to the logarithm of the dividend diminished by the logarithm of the divisor. Let m be the dividend, and let n be the divisor. Let m = a^, .'. oo = logam; and let ?i = a^, •'• V^ loga ^^• 771 _ a^ n a^ •*• ^^^^©^'^-^ = loga 'in - log„ n. 257. Logarithm of a Power. The logarithm of a potver of a number is equal to tlie product of the index of the power and the logarithm of the number. Let 771 be the number, and let y be the power to which it is raised ; y may be either an integer or a fraction. ,x = a^'-y. 278 LOGARITHMS. Let m — a^, .'. ir = loga??L = 2/loga?/i. 258. It follows from the result of the last article that logarithms can be used to extract the roots of numbers. For suppose that N is some given number, and that we want to find the n"' root of it. Let ^N^x. 1 .'. -\ogaN-=^\oga,x. [Art. 257. Hence, if the logarithm of iV to any base a be known, then we can deduce the logarithm of x to that base ; and hence, by means of certain tables which are published, we can find the value of x. In numerical calculations, the number ten is usually taken for the base [Art. 262]. 259. It is on the results of Articles 255, 256, 257, that the use of logarithms largely depends ; and in order to apply those propositions, it is necessary to express the number, whose logarithm is required, in the form of a product or a quotient. This is illustrated by the following examples. Ex. 1. Having given log 2 =-'30103 and log 3 = ' All 1^2,^ find the logarithms of (i) 5 ; (ii) 225 ; and (iii) '003 — the base in all cases being ten. We are given the logarithms of 2 and 3, and we know by Art. 253, Ex. 2, that the logarithm of 10 (to the base 10) is unity. Hence, by Art. 257, we know the logarithms of any powers of 2, 3, and 10. Therefore, by Art. 255, we know the logarithms of any product of powers of 2, 3, and 10. We shall LOGARITHMS. 279 therefore in each case begin by expressing the number, whose logarithm is required, in factors which are powers of 2, 3, and 10 only. (i) Wehave5 = i^'^ .-. Iog5 = logl0-log2 [Art. 256. = 1 - -30103 = •69897. (ii) Dividing 225 by 2, 3, and 5, as often as we can, we find that 225 = 32x52. .-. log 225 = log (32x52) = log 32 + log 52 = 2log3 + 21og5 = 2 log 3 + 2 log -1^- = 2 log 3 + 2 (log 10 - log 2) = 2(-47712) + 2(l--30103) = 2-35218. (iii) In the same way, -003 = 9^0=^0 = 3"^ x 10 -2. .-. log (-003) = log (3- 1x10-2) = log (3-1) + log (10 -2) = - log 3 - 2 log 10 = -•47712-2 = -2-47712. Ex. 2. Having given log 4 = -6020600 and log 36 = 1 -5563025 ; find log 9 and log -15 — the base in all cases being ten. We must first resolve 4 and 36 into the simplest possible factors. We have 4 = 22, .., 2 log 2 = log 4 = -6020600. Also, 36 = 22 X 32, . • . 2 log 2 + 2 log 3 = log 36 = 1 -5563025. Solving these two equations for log 2 and log 3, we obtain log 2 = -3010300, log 3 = '4771213. We now know log 2 and log 3. Hence, if we can express the given numbers as products of powers of 2, 3, and 10 (or 5), we can obtain their logarithms as in the last example. Now, 9 = 32. .-. log9 = 2log3 = 2 (-4771213) = -9542426. 280 LOGARITHMS. Again, •1 "S- 15 100 3 '20 3 X.O - 2x10" log (-15) = =log3- -logs i - log 10 = •4771213- •3010300 - 1 = - -8239087 EXAMPLES. XXIII. B. [/?^ the first four examples, a knowledge of the numerical values of log 2, log 3, log 7, log 11 is assumed. These values are log 2= •3010300, %3= -4771213, logl= -8450980, % 11 = 1-0413927. All the logarithms in the following set of examples are calculated to the base ten.'\ 1. Having given the numerical value of log 2 (see above) ; find the logarithms of (i) rk; (ii) 3-125; (iii) v^^^025; (iv) {10-24)2. 2. Given the numerical values of log 2 and log 3 (see above) ; find the logarithms of the following numbers. (i) 15; (ii) 1944; (iii) 4-5; (iv) 2400; (v) 75; (vi) -0045; (vii) '0036; (viii) '003; (ix) If; (x) ^072; (xi) -75; (xii) 7-29; (xiii) -00125. 3. Having given the values of log 2 and log 7 (see above); 1 _i 1 find the logarithms of (1-75)^; (24-5) 3; (-0056)2. 4. Having given the numerical values of log 2, log 7, and 210 log 11 (see above); find the logarithms of 1-4; -0154; and wgYjg- 5. Given log 2 = -3010300 and log 6 = -7781513; find the logarithms (to the base 10) of 15, and -0025. 6. Given log 21 = 1*3222193 and log 49 = 1-6901961 ; find log 3. 7. Given that log 27 - 1-4313638 and log 5 = 0-6989700; find log 135, and log 3. LOGARITHMS. 281 8. If log 125 = 2-0969100 and log 49=r6901961 ; find the 3 logarithm of (SS)*. 9. Calculate log 14, it being known that log 392=. 2-5932861 and log 1715 = 3-2342641. 10. Given log 72 = 1-857332 and log 45 = 1-653212 ; find log 30, and log -0135. 11. Having given log 242 = a, log 80 = 6, log45 = c; find log 66, log 3993, and log 36 in terms of a, b, c. 260. To find the relation hetiueen the logarithms of the same number to different bases. Let m be the number, a and h the bases. Let loga m = X, and log^ m = y. .'. 711 — a^, m = by. .-. a''=by. Therefore a = b'^, that is, l=log,a. :. y = x\ogia. ■. logb m = \oga m X Xogi, a (i)- Similarly, h = ay, that is, ^ = log.6. .-. x = y\ogab. .-. Xoga m = logi m X log„ b (ii). X ?/ Also, we have log^ h x log;, a = - x - = 1. y ^ Thus Iog„ b and log^ a are reciprocals each of the other. Example. Having given logiQ 2 = -3010300, log^Q 3 = -4771213 ; Jind log ^27 correct to four places of decimals. It is better to work examples like these from first principles than to quote formulae. 282 LOGARITHMS. Let ^ = logs 27. .-. 5^ = 27 = 33. .-. ^* log 5 = 3 log 3. Now log 5 = log V^- = 1 - log 2 =: 1 - -3010300 = -6989700. .-. ^(-6989700) = 3 (-4771213). 3 (-4771213) •6989700 2-04782. We are asked to determine x correct to four places of decimals. To obtain the last decimal figvu'e, it is necessary to find X to five places of decimals, in order to see whether x is more nearly equal to 2*0478 or 2*0479. As the fifth decimal figure is less than 5, the answer will be 2-0478. EXAMPLES. XXIIL C. 1. Having given log^o 2 = '3010300, log^o 3 = -4771213; find, correct to three places of decimals, (i) log3 6; (ii) logg5; (iii) log5 3. 2. Having given logj^ 2 = -3010300, log^o 7 = -8450980 ; find logy 4, and logg 70. 3. Given log^o 3 = -4771213, log^o 7 = '8450980 ; find log3V7, and log^7 3. 261. Common Logarithms. In practical nu- merical calculations, all logarithms are calculated to the base 10, and the symbol shewing the base to which they are calculated is usually omitted. Logarithms to the base 10 are called common logarithms. The only logarithms considered in the rest of this chapter are common logarithms, and the base 10 need not be inserted. Tables of the logarithms of the numbers from 1 to 100,000 have been calculated. That is, values of ^' which satisfy the equations 10^= 1, 10^ = 2,. . . have been found. The method of finding the values of these logarithms will be explained later [Art. 416], but it may be stated that the exact roots of these equations cannot in general be found, though the numerical values of the roots can LOGARITHMS. 283 be obtained as accurately as is desired. Most tables give the results to at least five places of decimals. 262. Advantages of taking ten as the base. The advantage of using 10 as the base arises from the consideration that log 10 =1 [Art. 253, Ex. 2. log 100 = log (10-) = 2 log 10 = 2 [Art. 257. log 1000 = log (10=^) = 3 log 10 - 3, and, universally, log (10") = n log 10 = n. Hence log (A^ x 10") = logiV+log(10'0 = logiV+ n, N and log — — = log A"— log (10") = log A"— n. Thus, if the logarithm of any number, such as A", be known, we can immediately determine the logarithm of the product or quotient of that number by any power of 10. Thus, if we know that log 2 = 0"30103, we have at once log 20 = log (2x10)= log 2 + log 10 =1-30103. Similarly, log "02 = log (2/ 100)= log 2 -log 100= -2 + -30103. The latter example illustrates the advantage of keeping the decimal part of the logarithm positive, since then the logarithm of all multiples or quotients of the number l)y powers of 10 will hav^e the same decimal figures, and will only difler in their integral parts. 263. Mantissa. Characteristic. When a lo^- arithm is written so that it is the algebraic sum of a positive decimal fraction and a certain integer (whether positive or negative), the positive decimal part is called the mantissa, and the integral part is called the characteristic. If the characteristic be negative, it is usual to write the negative sign above the number. 284 LOGARITHMS. Thus 2-70516 stands for -2 + 0-70516; while - 2-70516 would signify - 2 - 0-70516. The latter number = -3 + (l -0-70516)= -3 + -29484 = 3-29484. 264. Characteristics can be determined by inspection. In the common system of logarithms the characteristic of the logarithm of any number can he determined hy inspection. For suppose the number to be greater than unity, and to lie between 10''' and 10'^+^; then its logarithm must be greater than n and less than ?i + 1 ; hence the characteristic of the logarithm is n. Next, suppose the number to be less than unity, and to lie between :j-^ and Ycvn+i ' ^^^* ^^^ between 10~'* and 10~<"+^' ; then its logarithm will be some negative quantity between —n and — (?i + l); hence, if we agi-ee that the mantissa shall always be positive, the characteristic will be — (?i + 1). 265. Since the characteristic of the logarithm of a number can be written down by inspection, it is suffi- cient to give in the tables the mantissa only. Example. Find the value of log 2173. We find in the tables opposite to 2173 the number 3370597. The characteristic is 3. Hence the required logarithm is 3-3370597. 266. Conversely, if we know the characteristic of the logarithm of a number, we know the number of digits in the integral part of the number. If therefore the arrangement of figures in the number be given, we can tell where the decimal point must be. Example. Find the number whose logarithm is 2-4560774. Opposite to 4560774 in the tables is the number 28581. Therefore log 2-8581 = -4560774. .-. log -028581 =2'-4560774. LOGARITHMS. 285 EXAMPLES. XXIII. D. 1. Add together 8-2152630, 6-3212579, and 3-3725700. 2. Subtract 3-1527943 from 2-4984732. 3. Divide 2-5188142 by 6, and 8-5502766 by 7. 4. Given that log^o 95-882 = 1-981 7371 ; write down the logarithms of 9588200, 9588*2, and -0095882. 5. Given that log^^ 6-4145 = -8071628 ; find the numbers whose logarithms are 3-8071628 and 3-8071628. 6. Find the characteristic of log^ 350, of log^ '065, and of the logarithm of 500 to the base 3. 7. Find the characteristics of the following logarithms, log, 21 ; logii iV ; log3 95 ; log.^, 29 ; log^^ "0003 ; log, 63; log^ 16829. a Find the logarithms to the base 10 of -001 1 °^\ and of •0001 1^*^^- 267. Uses of Logarithmic Tables. The chief purposes for which we want tables of logarithms are (i) to find the logarithm of a given number, and (ii) from a given logarithm to find the number of which it is the logarithm. If the number or the logarithm be given in the table, this can be done at once. It will be convenient to postpone until Chapter XXXII. the explanation of how a table of logarithms is used to find numbers or logarithms which are not expressly given in- the tables [see Arts. 418 — 422] ; but if the student will assume the result of Art. 418, he may read here the examples which are worked out in Arts. 419—422. 268. A few miscellaneous examples on logarithms are here added. E:v. 1. Having given log^^ 3 = '477121 3, find how many digits there are in 3^^. Let .^ = 3100. .-. log .r = 100 log 3 = 47-71213. .'. the characteristic of log^r is 47. .•. there are 48 digits in x. 286 LOGARITHMS. Ex. 2. Fi7id^ correct to three places of decimals^ values of x and y which satisfy the equations 2*^3^= 1 =3* + 1 2^-^ ; having given % 2 = -3010300, % 3= -4771213. Take the logarithms (to the base ten) of each side of the two given equations. .-. X log 2+3/ log 3 = log 1=0, [Art. 253, Ex. 1. and (A' + l)log3 + (?/-l)log2 = logl=0. These are two simple equations between x and y. Solving them, we find _ log 3 log 2 '^~ ~log2 + log3' ^~log2 + log3* Substituting for log 2 and log 3 their numerical values, we obtain •4771213 / /81513 •3010300 __^^ ^ = 778151-3 = '''''- The value of x is nearer - -613 than - -614, and that of y is nearer -386 than -385. Hence the answer is .r= — -613, y = -386. Ex. 3. Find how long it loill he hefore a sum of money 'put out at compound interest at the rate of 3 per cent, per ayimim^ payable annually, has doubled itself: it being given that log 2 = -30103, and % 103-2-0128372. Let n be the number of years required, and P the sum originally put out at interest. At the end of the first year, the amount is P+x^ P=P (1*03). Call this amount P^ . Then, at the end of the second year, P^ has amounted to Pi(l-03), that is, to P (1-03)1 Similarly, at the end of the 7i"' year, the total amount will be P(l-03)". This, by the question, is 2/'. .-. P(r03)« = 2P. .-. (l-03)" = 2. .-. w log (1-03) = log 2. -30103 „^ ^^ , •*• '^ = -0128372 = ^^'^^' ^''^"^^"^^- It will therefore take a little less than 23^ years before a sum of money doubles itself under these conditions. LOGARITHMS. 287 MISCELLANEOUS EXAMPLES. XXIII. E. [In the first nineteen examples the values of the following logarithms are supposed to he given, log^Q 2= -3010300, log^^ 3= -4771213, log^^ 7= -8450980, %io 11 = 1-0413927. All the logarithms in the folloiving examples are supposed to he taken to the hase 10, unless the contrary is stated.^ Find the logarithms of the following numbers, numbered 1 to 9. I. -032. 2. -36. 3. 720. 4. log 72-9. 5. 7-5. 3 6. -00045. 7. 4/-00012. 8. (29-7)"^ 9. ^'002835. 10. Find logyVS, and logv2 7. II. Determine which is greater, (i) -01 or(f)9; (ii) -1 or (D^'^. 12. Determine which is the greater {'^Y^ or 100. 13. Find how many digits there are in 5^^ and (54)^*^, and in the integral part of my^. 14. How many ciphers are there between the decimal point and the first significant figure in (-6)^^"? 15. If the number of births in a year be jh^ of the popula- tion at the beginning of the year, and the number of deaths ^j^ ; find in what time the population will be doubled. 16. In what time will £100 amount to £500 at 5 per cent, per animm compound interest? 17. In how many years will a sum of money double itself at compound interest, interest being i)ayable yearly at the rate of 10 per cent, per annum ? 18. Solve the equations 31 - x - y 4?/ == 3^ - Si, 22^ - 1 = i . 19. Given logj2 36 = a, find logj2 48. 20. Given log 24 = 1-3802112 and log 36 = 1-5563025; find log 8-64, log 1-5. 21. Given log,o 864 = 2-9365137, log,,, 486 = 2-6702459; find 288 LOGARITHMS. 22. Find the numbers whose logarithms to the base 64 are 2, 23. What is the smallest number of logarithms (to the base 10) required in order to calculate the logarithms (to the base 10) of each of the following numbers ? 64 ; 125 ; 50 ; 30 ; f ; -6 ; -375. Find the characteristics of the logarithms of these numbers to the base 2. 24. If log (1^) + log (6-|) = 1, what is the base? 25. If there be 27647 digits in the integral part of (1.89)100000^ find log 1890. 26. Given that the integral part of (3-981)iooooo contains sixty thousand digits ; find log^o 39810, correct to five places of decimals. 27. Shew that the square root of 372*86 is very nearly ten times its ninth root; having given log 37286 = 4*5714293. 28. Assuming that the sixth and seventh powers of 7 are 117649 and 793543 respectively; prove that the first digit in the mantissa of log^^ 7 is 8. 29. How many positive integers are there whose logarithms to the base 2 have 5 for a characteristic? 30. Find x and y, if (a.r)>°g« = (cj/)'°sc and c'^s^=:a'°-^. 31. Prove that log f + log — -f log ^ = 0. ^hc ° ca ° ab 32. Find the value of 7 log2 1 1 + 5 logg |f + 3 logg |^. 33. Find the sum of the logarithms (to the base 10) of the roots of the equation x^ - 14.27+100 = 0. 34. Express {a'^-\-h'^f-{a^-¥f-{a^-\-h'^-c^f in a form fitted for logarithmic calculation. *35. If log2a a = .r, and logg^ 2a =3/, shew that 2^ - ^2' = 3^^ " *2'. *36. Prove that, if the logarithm of ^« to the base ofi be equal to the logarithm of .^'^'' to the base y^\ then each of them is equal *37. If loga b = log,, c, then will each be equal to log„ b + log,, c. b a *38. A man borrows £500 from a money-lender. The bill is renewed every half-year with an increase of 12 per cent. What time will elapse before it reaches £5000 ? [log 112 = 2 -0492 1 8.] CHAPTER XXIV. SURDS. [T/ie results of this chapter are not required in the immediately following chapters, and the discussion of surds may he deferred foi' the present, if the student desire itJ] 269. It may be convenient if we repeat here that the root of a quantity is called a surd. The n*'^ root of a quantity X is denoted by '{/X, and is such a quantity that its n^^ power is X ; it is called a surd of the n>^^ order. If there be no exact n^^ root of X, the surd is said to be irrational [Arts. 23, 182]. 270. Surds which have the same irrational factor are called like surds. Thus, 2^3, |x/3, and -3v/3 are like surds, because the irrational factor, JZ, is common to each of them. The sum of like surds can be combined into a single term in the same way as any other like quantities. Thus, 2V3 + iV3-3v/3 = (2 + ^-3)V3=-^V3. Surds which have not the same irrational factor are said to be unlike. Thus, 2>/3-3V2 is the algebraical sum of two unlike surds, and cannot be simplified further. 271. We shall consider in this chapter some of the more simple propositions about surds. We shall begin B. A. 19 290 SURDS. by discussing surds formed by the roots of simple expressions ; next, we shall treat of compound expres- sions involving surds; and lastly, of the solution of equations which involve surds. SIMPLE EXPRESSIONS INVOLVING SURDS. 272. The extension of the meaning of indices, which is given in Chapter XXII., enables us to write the root of a quantity either as a surd or as the quantity raised to a fractional power. The latter method of expression enables us to reduce two or more expressions to surds of the same order. 273. Any rational quantity can he expressed as a surd of any required order. For, if a be any quantity, and n any positive integer, 1 a = {/a"- = {a'^Y which is a surd of the n^^ order. 1 Thus, to express 2 as a quadratic surd, we have 2 = J A = 4?. 1 Similarly, 2 can be written as a surd of the third order, 2 = 4^8 = S^. 274. The product of a rational quantity and a sur^d can be expressed as a surd. For, let a be the rational quantity, and y/h the surd. Then, a^h = {/a"" xyb = VaFb, which is a surd of the n^^ order. This proposition may also be proved thus. ax6" = (a")"x6" = (a"6)". For example, 7 y,f5 = \/V^^ = ^/245. SURDS. 291 275. If the quantity under the root-sign of a surd can he resolved into factors, tlie surd, can be expressed as the product of surds. For we proved, in Art. 186, that y/ab = ya x -^yb. Hence, if one of these factors, such as a, be the n^^ power of any quantity, we can resolve the given surd into the product of a rational factor and an irrational factor. For example, ^x'^y = ^x'^ x^2/ = x^y, s/'Vdx^y-^ = \/49^V^ X iJy=*locy^y^ ^a^ - slxy'^ = Xs!x-y>Jx = {x-y) ^x^ sJa^-)rfJxh/=x,^fx-\-XsJy=x{^x + Jy). 276. Where we have a fraction with a surd (or a product of surds) in the denominator, it is usually con- venient to make the denominator rational. Thus, if a surd like Va occur in the denominator, we multiply numerator and denominator by \/a*^~\ and thus make the denominator rational and equal to a. For example, A_1^-2V?_ ,9 V2~V2V2~ 2 ~^''' 2^3 _ 2^/3x^/5 _ 2^15 v/5 ~ Ubf - 5 • 277. A sw^d of the n'* order can be expressed as a surd of the mn^* order, luhere m and n are positive integers. This proposition is proved by a method analogous to that given in Art. 273. For, if {/a be the given surd, we have 19—2 292 SURDS. This proposition may also be proved thus. For example, (i) Ja = ^fa\ (ii) ^b = ^bK (ill) V(4^^^) = 2.r^/.v=2.r4/.t^2. 3 3 3 13 (iv) V(4'^'V) = 2^x// = 2;^ X (3/2)2 = 2.1^1/4 = 2x^f. 278. J.?i7/ ^-z/zo s?^?"c^5 can be expressed as surds of the same order. tn p For, if rt" and If be two surds, we can express each of them as a surd whose order is the L. c. M. of n and q. For, a" = a"«'=v/a"S p pn and }f = h^-=^Vh'\ The above surds are each of the order nq. But the process depends on reducing the fractions — and - to a common denominator. Therefore if n and q have a common factor, the order of the resulting surds will be the L.c.M. of n and q, and not nq. Example. Which is the greater ^'4 or »JZ ? Keduce the surds to equivalent surds of the same order ; in this case, the order will be the l. c. m. of 3 and 2, that is, will be 6. 1 1 1 Then, 4/4 = 4^ = (42)6 = (16)«, 111 and V3 = 32=(33)6=(27)«. 1 1 But, since 27>16, .-. (27)6>(16)«. Hence, V3>v/4. 279. The product or the quotient of any two surds can he tuHtten as a surd. For, by the last article, we can express the two surds as surds of the same order. Also, we know that the SUHDS. 293 product or quotient of two surds of the same order can be written as a surd, for ya X y 6 = Vab, and ^a ^ ^6 = \/a/h. Thus, m p ituj fni 2_ EXAMPLES. XXIV. A. 1. Express 3 as a surd of the second order, a surd of the third order, and a surd of the fourth order. Express the following quantities, numbered 2 to 6, as surds. 2. a^h. 3. 3x/2. 4. 2^3. 5. x'^Jy. 6. x^y"^ J^. Express each of the following fractions, numbered 7 to 11, as a fraction having a rational denominator. 7 ^ 8 ^ J5 slah x^^y 7- 72- ^- V3- ^^ 77- ^^' jjvc' y^^' 12. Prove that (i) 3^'^"^= 27^; (n) {^W^={^^I^T'^' \Z, Is ^84 greater or less than 4^28 ? 14. Of the following quantities, which is the greatest and which the least? ^35, ^¥214, 4/1290. COMPOUND EXPRESSIONS INVOLVING SURDS. 280. The rules for the multiplication or the division of expressions by quantities like a + \Jh are the same as those for multiplication or division by rational quantities. If we have to divide one such expression by another, or if the work involve the multiplication of surds of different orders, it is usually convenient to 294 SURDS. express all the surds with fractional indices ; but simple cases of multiplication may be treated directly. Ex. 1. Multiply a-\-Jx by b + ^x. We have a -\-^x h -\-lfx 2. Find the have ah + hjx a^x+x Ex. We ah-{-{a+h)sJx + x square of 5-3^/-^. 5- 3Jx 5- Zjx 2b-\bsfx -Ibsjx + ^x 25-30x/^+9^ Ex. 3, Find the product of ^ X — sj y (J^nd >Jx-\-Jy. We have iA-B){A + B) = A^-£^, EXAMPLES. XXIV. B. 1. Find the product of 2 — d^Jx and 4 -bsjx. 2. Find the square of 7 - 2 <^a. 3. Divide 6-2a-Ja by 3-2sJa. 4. Divide y/x^-UsJx-\-6 by ^'•^-3. 5. Divide ^-81 by ^x-\-3. 6. Prove that, if ^^ = 2 + ^2, then {x-l){x-2)= x. 281. Where we are deaUng with a compound expression in- volving surds, it is generally desirable to begin by writing every term and the whole expression so that the denominator is rational. Where the denominator is a surd or a product of surds, this can be efifected by the rules given in Art. 276. If the denominator be a compound expression, the process is somewhat more complicated, but it will be sufficient here to remark that a fraction, whose denominator is of the form a + Jb, will be rationalized, if both its numerator and its denominator be SURDS. 295 multiplied by a - Jb, since the denominator will then take the form a^-b. Similarly, a fraction, whose denominator is of the form a - Jb, will be rationalized, if both its numerator and its denominator be multiplied by a-\-Jb. TW Z _ X{a-Jb) _X{a-Jb)^ ' a+^b~{a + yjb){a-^b)~ a^-b ' X _ X{a+Jb) _ X{a + sJb) a-Jb~ {a-sfb){a+Jb)~ a^-b ' For example, 5-V2 ^ (5-v/2)(3 + V2) ^ 15 + 2^2-2 ^13 + 2v/2 3-V2 (3-V2)(3 + x/2)~ 9-2 " 7 ' 282. The following is an important proposition. If X -\- \/i/ = a + \/h, where x and a are rational quantities^ and \ly and f^/h are irrational surds, then will X = a, and y = h. We have x-\- \fy = a-\- \Jh. :. X — a-\- sjy= ^b. Squaring both sides of this equation, .'. (x - aY -{- y -h 2 (x - a) \/y = b. .'. 2 {x — a) ^/y = b — (x — ay — y. Therefore, unless (x — a) is equal to zero, we have a multiple of an irrational quantity equal to a rational quantity. This is impossible, and therefore x — a = Oy that is, X = a. Again, if x = a, the given relation re- duces to ^y = \/b, and therefore y = b. Note. We have assumed that the square root of y is not rational, and our proof requires that s/b and ^^/y shall both be irrational sm-ds. Thus, from the relation 2-\-fj4 = 3 + Ji where v/4 and ^^1 are not irrational surds, we could not infer that 2 = 3 and 4 = 1. 283. We have just shewn that, if x + ^/y = a-\- sjb, then x = a and ^Jy = \/b. It therefore follows that X — sly = a — i^b. 296 SURDS. Expressions like x-V sly and x — s/y are said to be conjugate. Thus if two expressions be equal, their conjugates are also equal. 284. The square root of an expression like a-\-h s/c can sometimes he found as the sum of two quadratic surds. Suppose \/(a + b \/c) = s/x + sjy. Square both sides, .•. a-\-h \Jc — x -^ y -\-^ Jxy. Hence, by Art. 282, a = x-Yy, and h \/c = 2 Jxy. These are two equations from which we can obtain x and y : and, since x + y = a and xy = i¥c, it follows [Art. 232] that X and y are the two roots of the equation z^ — az -h l¥c = 0. These roots are a-{-s/{a^-¥c , a-s/(a^-¥c) 2 ~~2 • Therefore the required square root, namely, s/x + s/y, is ^/(a' - If'c)] v^r-i^^f^'lVr 2 This expression is however more complicated than the original expression unless a^ — h^c is a perfect square. In any particular case where a^ — hrc is a perfect square, the square root reduces to the sum of two quadratic surds. The square root of a — h\/c can be similarly deter- mined. Example. Find the square root of 27- lOv/2. Let V(27 -10^/2) = s,lx - sjy. S^Note. It is convenient to assume that ^Jx and sjy have opposite signs when the two given numbers (in this case, 27 and — 10 ^^2) are of opposite signs ; and that they have the same sign when the two given numbers are of the same sign.] Square both sides, . • . 27 - 10 ^2 = ^' +y - 2 ^J{xy), .-. 27 = .v+y, and 10V2 = 2V('i*y), [Art. 282. SURDS. 297 that is, y = 27-A', and xy = bO. .'. a; (27-^) = 50. .-. (.^-25)(^;-2) = 0. But 3/ = 27-.f. Hence, if .^ = 25, 3/ = 2; and if a- = 2, ?/ = 25. The first solution is alone applicable to this problem. Therefore \/27 - 10^2 = 5-^2. If we had tried to find the square root of 27 + 10^2, we should have assumed it to be equal to Ijx + Jy, where Jx and Jy would have to be taken of the same sign. In this case, we should have obtained the same quadratic equation for x as that written above ; and the answer woul4 be 5 + ^/2. MISCELLANEOUS EXAMPLES ON SURDS. XXIV. C Simplify the following expressions, numbered 1 to 5. _i ^____ f\/3-ll 2 sjy^t-yt^x ^slx^^y'^-y rys-ii ^ Vs+iJ \lx^ + y ^ + y X — \/x^ - y^ a-\ld^-h'^ \Ja?-\-¥-h 1 +2^2+^-^ + 2a- ^ ( sjx 1 - >Jx \ ^ / sjx _ l-s!x \ 6. Shew that, if ^ = 3 - ^3, then x'^ + ^ = 24. 1 1 7. If A- = 2 + V3, find the value of ^^^ - ^^ ^ . 2 112 11 8. Multiply A-^ + A-^y^ 4-^3 by ofi-y^. 2 2 9. Divide x'^-\-y'^ by 0(^+y^. 10. Divide 2.1-3-6^ + 5 by .r 4/2 + ^/4 -fl. 11. Divide (.r* + ?/) ^3 + ^.^2^2 ^ ^.y {ofl -\-y^) (^6 + x/2) by a;HA-j/v^2+yV3. 298 SURDS. Find the square roots of the expressions, numbered 12 to 15. 12. 30 + 12^6. 14. 107-42^2. 13. 7 + 2 V6. 15. 5 + V2_ Find the square roots of the expressions, numbered 16 to 20. 16. ^x^ - 6.^72 + 24^2 + 1 _ _?_ + 1? , sjx X 17. f +^+3-2 /f -2^/2. 18. ^+'Jf^\ 19. a+V(a2 + 26c-62_c2). ^20. 4 + v/5 + x/(17-4v/15). 21. Simplify \/30 + 10V5. ^2. Simplify ^3!-^+-^--^^. *23. Simplify— ?±£=+-^::,^L^. m Simplify f-±^-.-^^+^-:^^+^5^^.iy. EQUATIONS INVOLVING SURDS. 285. It not unfrequently happens that an equa- tion proposed for solution contains surds in which the unknown quantity appears under the radical sign, and in such a case we must get rid of the surd before we can solve the equation. The usual method of effecting this is (after sim- plifying the equation as far as possible) to transpose one radical to one side of the equation, and to transpose all the other quantities to the other side. By then squaring both sides (or raising them to a suitable power) we get rid of that radical. Repeating the process again, we can get rid of another radical. Con- tinuing the process, we finally obtain an equation SURDS. 299 which is rational and integral in the unknown quantity ; and this equation must then be solved by the methods already described. We may add that the roots of the equation thus obtained may contain surds which involve nothing but numbers or known quantities [see Art. 105, Ex. 4], though such siu"ds should always be simphfied as far as possible. It may also be well to warn the beginner to be careful that the square of that side of the equation which contains a radical and another term is written down correctly. It is a not un- common mistake to write the square of a-\-Jb as a^ + h^ whereas it is a^-\-'2asJb-\-h. In a similar way, beginners sometimes think 1 that {x^+y^Y is equal to x-\-y\ this is not the case, for we have already shewn that {x^-\-2xy-\-y^Y is equal to ±{x-\-y). 286. The method is illustrated by the following examples. Ex. 1. Solve the equation y/^-4 = 0. Here ^Jx = 4:. Squaring both sides, .*. ^ = 16. Ex. 2. Solve the eqitation 2 sjx - 1 + sJA:X + 5 = 9. Transpose one radical to one side, and all the other terms to the other side, .-. 2\/.r-l=9-\/4^+5. Square both sides, .-. 4(a;-l) = 81-18\/4.^T5-f4jp-|-5. Transpose the radical to one side of the equation, and all the other terms to the other side ; collect like terms, and simplify, .-. 18\/4^ + 5 = 90. .-. V4a' + 5 = 5. .'. 4r+5 = 25. .-. x=b. The two examples just given led to simple equations. The two following examples lead to quadratic equa- tions. 300 SUEDS. Ex. 3. Solve the equation 2 \l2x +1-3 V^ -3 = 3 Transposing, 3 s/x-"^ = 2 \/2x +1-3. Squaring both sides, . • . 9 (^ - 3) = 4 (2a,' + 1) - 12 \l2x-\-\ + 9. Transpose the radical to one side of the equation, and all the other terms to the other side, and simplify, .-. 12\/2^ + l=40-x Squaring both sides, / . 144 (2.r + 1) = 1600 - mx-\-x\ .-. ^2 _368,r + 1456 = 0, the roots of which are 4 and 364. If now we proceed to verify this solution, we find that x = A satisfies the given equation, since on putting .v=4 in the equation it becomes 2 ^'9- 3^1 = 3, which is clearly true. If however we put .r = 364 in the equation, it becomes 2^729-3^/361 = 3, that is, 2x27-3x19 = 3, or 54-57 = 3, which is clearly not true. Hence, only one of the roots lohich loe ham found satisfies the original equation. It is only hy trial that ive can find which of the tioo roots is the one we require. [The explanation of this paradox is that we shall obtain the same quadratic equation for x from another equation involving surds, and the root 364 satisfies this other equation. In fact, if we solve the equation 2 \/2^+l -Z^]x-Z= - 3, we have 3V.r-3 = 2\/2.r + l+3. .-. 9(:^;-3) = 4(2.r;+l) + 12V2.v+l+9. .-. -12V2^^-|-l=40-.t^, and squaring both sides of this last equation, we obtain the same quadratic as before, whose roots are 4 and 364. The root 364 satisfies the equation above considered, but the root 4 does not satisfy it. The equation 2 sj'^x + 1+3 a/-^' -3 = 3 also leads to the same quadratic equation for x ; but neither of the roots of the quadratic will satisfy this equation. This equation has no rational root.] Ex. 4. Solve the equation ^x + \/x + « - \/x + 6 = 0. We shall follow the above procedm'e without specifically indicating each step. SURDS. 301 We have ^x=\lx + 6 - ^ x + a. .'. x={x + b) -2\/{x + b) {x-\-a) + x + a. .-. 2\/{x+b){x+a)=x+b + a. .-. 4{x+b){x+a) = {x + a+bf. .-. 3x^+2{a + b)x-{a-bf = 0. .', .r=J{-(a + 6) + V4(a2-a6 + 62)}. We cannot determine which of these two roots satisfies the given equation unless we know the numerical values of «, or of b, or some relation between them. If radicals appear in the denominator of a fraction^ the equation must be simplified so as to get rid of them. Ex. 5. Solve the equation '\/x-a + \/x= six- a Multiplying up, we have x — a-V sjx X s] X — a — a. .'. ^xxs/x — a = 2a — x. .'. x{x — a) — 4^2 _ 4.ax + x^. .'. 3ax=4a^. .'. x = ^a. Ex. 6. Solve the equation Jx-\-Z = - 7- . ^ 4 — .Jx Multiplying up, we have ( V.^-' + 3) (4 - ^/.r) = 6. .-. ^x-ir\2-x = Q. .'. y/x — x — 6. .'. x = (x-G)\ .-. ^2 _ 13^ + 36=0. .-. (.^-4)(^-9) = 0. Of these roots, x = 9 alone satisfies the original equation. In some cases, it is convenient to introduce a sub- sidiary symbol in the same way as we did in the ex- amples worked out on pp. 239, 240, 242, 248. This is illustrated by the following examples. 302 SURDS. Ex. 7. Solve the equation x^-\x-{- ^/2.^^^ - 3:r + 5 = .r + 1 5. Simplifying the equation, it becomes 2a^2_3^^2V2^2- 3:^^ + 5 = 30. Let 2x^-Zx—y^ then the equation becomes 3/+2\/3/T5 = 30. .-. 2\/y + 5 = 30-3/ (i) .-. 4 (2/ + 5) = 900 -60?/ +^2, .-. 3/2_64^=_880. .-. /-64y + (32)2 = (32)2 -880 = 144. .-. ?/-32=±12. .-.^ = 32 + 12 = 44, or y = 32-12 = 20. Of these two roots, the latter alone satisfies the equation (i) ; therefore y = 20. But 3/ = 2;r2_3^^ .-. 2jr2_3^^20. .*. x^-^x—\0. .-. ^'2_|^+(|)2=10 + (|)2 = i6_9 16 • .'. .'^|=±^/. .^;=|+^ = 4, or^^ = |-^=-f. Hence the required roots of the given equation are 4 and - f . Ex. 8. Of a swarm of bees clustered on a tree, the sqzcare root of half their 7iumher flew away. Eight-ninths of the original number then departed, leaving but two behind. How many loere there at first? (Bhaskara's Bija Ganita, circ. 1160.) Let X be the number of bees. Then we have, by the question, \/^^ + f^ + 2 = .r. .'. \/\x=x-^x-'2. = \x-'2,. Square both sides, .-. ■|;r = (^;r- 2)2. Simplify, .-. 2x^-\bZx-\-MS = 0, of which the roots are 72 and |. The latter root is not applicable to the problem. Hence there were 72 bees. SURDS. 303 EQUATIONS INVOLVING SURDS. XXIV. D. Solve the following equations, numbered 1 to 46. 1. ^x'-^ + x=lO. 2. ^Ja-\-x-it-\/a-x = ^h. 3. V^T2 + \/^ + 9 = 7. 4. \/7^-V3^ + l = l. 5. V8^ + 5-2\/2^-l = l. 6. V.^ + 19 + \/^r + 3 = 8. 7. .r + V«-^'-a^=3a. 8. ^x=lO- \/bx+Q. 9. \/.<^ + V« + -^ = -7^. 10. 2\/i + 2^'2 = 5. \/x 1 11 11. x^ + {a-\-xf = 2ax~\ 12. V.r + 3 + \/2:^' - 1 = 9. 13. V7^- + 1-a/3^+4 = 1. 14. \/l0.r-l = 3 + \/3^n. 15. V4x + 5-2V^^ = l. 16. \/^+\/5.t^ + l=5. 17. \/3+:y + V7-^ = \/4 + 2.r. 18. \/4 + a- + \/6^ = V6 + 2.r. 19. \/^+5 + 2\/^^+l = V3:r+7. 20. V-^+28 + v9a;-28 = 4\/2^-14. 21. \/7.r-6 - \/4^Ti = \/3^-17. 22. \/l4^ + 9 + 2\/^l + \/3a.-+l = 0. 23. Vl2:«?-3 + \/A' + 2 + V7a?-13 = 0. 24. \/2^ + 7 + \/3^-18 = \/7.r + l. 25. \/.^ + 3 - 2 V^^ + l = \fbx + ^. 26. Vs^^+l - Vi^+5 + \/^"- 4 = 0. 27. sj^x + 1 - V^^+T = \/3^. 28. aV^2 + «2=^2_a2^ 29. -7=^— + 31. a/.'p - 0? i\lx - c m\/c^ -x^-n{c-\-x) ma- s/c^-x^+n{c+x) rna+Tih 32. x^ + {x - 2^ (.r - 3) + s/^x"^ - 5.r + 6 = 6. 33. 2 (2.r - 3) (.-r - 4) - \J2x'^ - ll.r + 1 5 = 60. 304 SURDS. 34. 5^2 + 1 lo? - 12 \/(.r + 4) (5^ - 9) = 36. 35. ^■\-4.\JZx'^-lx-{-Z = Zx{x-\)-[-Q. 36. .r2 + \/''^2 + 3^. + 5 = 7_3.^. 37. y+4a„-2>^«^-"=0. 38. 4.^2 + V4;^'2_io^'+l = \0x + 1. 39. ^2_3^_3^^_3^_ 10 = 118. 40. X \/x^ + 1 2 + .r \/a2 +6 = 3. *41. 2.r \/^Ha2 + 2.^ ^x^+¥ = a2 - 52, *42. :^'2 -j- a2 _f. ^^4 ^^4 = 2a V.r'-^ -f V^^ + a^. * ■ • " J* * x-y=l I X-7/ = 7 x-^x^-2e-=0] 3.^-2\/^ + 9 = 01 ^5- 3V3/-2\/-^-7=OJ ' * 5\/-^-3\/y-3 = oJ ' *47. Explain the fact that the vahie of x obtained by solving the equation '\/x + 8-\-'\/x~\=l does not appear to satisfy the equation. CHAPTER XXV. RATIO AND PROPORTION. 287. Ratio. The relation which two quantities bear to one another in magnitude may be regarded in two ways. We may consider how much one of them is greater than the other : we call this their difference. Or, we may consider how many times one of them contains the other : we call this their ratio. 288. The quantities in the definition of ratio must be of the same kind. Thus, we can compare a length with a length, or an area with another area, or a sum of money with another sum of money, but we cannot compare inches with shillings, or acres with weeks. Moreover, the quantities to be compared must be expressed as multiples of the same unit. Every quantity, as we have already remarked, is measured by the number of times it con- tains a certain unit of its own kind. Thus, if we take a mile as our unit of length, then any length will be measured by the number of miles it contains. If, for example, a certain length be equal to a j mile, the numerical measure is j when the unit of length is a mile. The same distance might have been ex- pressed as 440 yards, in which case a yard is the unit of length, and 440 is the numerical measure of the magnitude. To compare quantities, we must express each as a multiple of the same unit, and we shall then only have to compare their numerical measures. Thus, whether the quantities in the definition of ratio given in Art. 287 be abstract or concrete, their ratio will be measured by the number of times which one number contains another number. B. A. 20 306 RATIO AND PROPORTION. 289. Notation. The ratio of a to h is written a : h, which is read as a to h. The quantities a and h are called the terms of the ratio ; of these, a is called the first teiiyi or antecedent, and h is called the second term or consequent. A ratio is said to be a ratio of greater inequality if the first term be greater than the second term ; a ratio is said to be a ratio of less inequality if the first term be less than the second term. 290. Definitions. The following terms are used: The duplicate ratio of a : b is the ratio a^ : b^. The triplicate ratio of = or < the ratio c : d, as the fraction r i« > = or < the fraction -i , b a , , , . ad . hd that IS, as 7-^ is > = or < r- , hd he that is {p'rovided hd is positive) as ad\^ > = or <6c. Art. 180 (iv). 20—2 308 RATIO AND PROPORTION. 296. Since ratios are measured by fractions, many of the following examples may be considered as examples of the properties either of fractions or of ratios. Ex. 1. What number must be added to each term of the ratio 7 : 10 ^o 7nake it equal to the ratio 2 i 31 Let JO denote the required number. Therefore, by the question, 7 + x _2 10T^~2' Multiply up, . • . 3 (7 +.r) = 2 (10 -\-x). .-. 21+3^ = 20+2x .-. ^=-1. Hence the number required is — 1. The interpretation of this result is that unity must be subtracted from each term of the given ratio to make it equal to the ratio 2 : 3. Bx. 2. Of the ratios 2{j;-\-lf+l : 3.^2 + 6a- -f 5 aiid 2 : 3, ivhich is the greater? rp, ,., 2(^+1)2+1 . 3 The quantity 3^2^6a. + 5 ^« > ^^- <2 according as 3 {2 (.^--f 1)2 + 1} is > or <2{3x^ + Qx-\-b\ [Art. 295. that is, as 6.r2-fl2jt + 9 is > or or <10. But 9 is < 10, ...|(f±l)!±li3<^. 3x^-\-Qx-\-b 3 Ex. 3. Determine whether the ratio a+x : b-{-x (formed from a given ratio a : b by adding a positive quantity x to each term of it) is greater or less than the given ratio a : b. rrii ,-, a-\-x . a The quantity -. is > or < r '' b-^x b as 6(a+^) is > or or or < ax^ that is, (since x is positive) as 6 is > or < a. RATIO AND PROPORTION. 309 Hence a ratio of greater inequality is decreased by adding any positive quantity to each term of it. For in this case, 6a, and therefore a-\-x : 6 + .r is >a : 6; that is, the new ratio is greater than the given one. [For example, 1 + 1 :2 + l>l :2.] Ex. 4. Determine whether the ratio a — x : b-x {formed frojii a given ratio a : h by subtracting a positive quantity x from each term of it) is greater or less than the given ratio a : b. The quantity t is > or < 7- ^ *' b-x b as b {a - x) \^ > OY or < a6 — ax, that is, as — bx is > or < — ax. Transposing, that is, as ax is > or < bx, [Art. 180 (iv). that is, as a is > or < b. Hence a ratio of greater inequality is increased by subtracting any positive quantity from each term of it. For in this case, a > 6, and therefore a — x : b — x>a : b ; that is, the new ratio is greater than the given one. [For example, 3 — 1 : 2 — 1>3 : 2.] Similarly, a ratio of less inequality is decreased by subtracting any positive quantity from each term of it. For in this case, at f). -r», a a -i- c + e a PC CL Next, let -p=y, :. -^>y, and 7- > y. ,-. e=fy, c> dy. a>hy. Add, . *. a-}-c + e>by + dy +fy, that is, a + c + e > 2/ (^ + f^ +/)• -r,, e a + c + e e But y = —;, :. ^ — >>-F. Similarly, if we have any number of unequal positive ratios a : 6, c : 0?, e : /, ^ : A, . . . , and we form a new ratio a + c^e-\-g+ ... : 6 + c?+/+A + ..., whose first term is the sum of the antecedents of these ratios and whose second term is the sum of their consequents, then its value will be intermediate in magnitude between the greatest and the least of the given ratios. EXAMPLES ON PROPORTION. XXV. B. 1. Shew that, if a : 6 : : c : c?, then (i) wia + 6 : mc + d :: pa + b :pc + d. (ii) ma -\- Tib : ma-nh :: mc-\-ml : mc-nd. (iii) a2 + 52 . c^j^^i2 .. (^a-\-hf : {c-\-df. 2. If a : h = c : d, prove that {a^-\-c^){h''--\rd'^) = {ah^-cdf. 3. lia:h = c'.d, prove that a2 + 6'■^ + c2 + c^^ {a + hf^{c-\-df, (« + c)2 + (6 + c?)^ and {a-\-h-\-c-\-df are in proportion. 4. Shew that, if c :d :: x -.y, then cd : xy :: c'-^ + c^^ : x'^->ry'^. 5. Shew that, if a : 6 :: c : c^, and a be the greatest of these four quantities, then d will be the least. 6. Prove that, \i x : y :: a : h, then x^ : a^ :: x^-{-y^ : a^ + i^. 13. 316 RATIO AND PROPORTION. 7. li a \ b :: c : d :: e : f^ prove that each of these ratios is , ^ ma-\-nc + pe , . , ab + cd+ef Shew that, if a : b = c : d=e : f, then the results, numbered 8 to 13, will be true. 8. a + 3c + 2e :a-e ::b-\-3d-\-2f :b-f. 9. a + 4c+'Se : a + c :: fe + 4o?+3/ : b + d. 10. a3 + c3-ace : ab^ + cd^-adf :: (a-e)^ : (6-/*)2. ^ \b + 2d+Zf) ~bd + df' 12. (a2e2 _^. c4) (5/+ ^2)2 = (Z,2/-2 ^ ^4) (^^g ^ c2)2. pa^ + qc^ _ (qc + re){la + me) pb^+qd'^ ~ {qd+rf){lb^mf) ' 14. If the ratios a : :r, 6 : y, c : ^ be all equal, prove that pbc + qca -\- rab, pyz-\-qzx-\-rxif^ pa?-\-qb^ + rc^^ and px^-\-qy^-\-rz^ are in proportion. 15. If ^' : 5 :: ?/ : 8, find the ratio of x-\-b to ,y + 8. 16. If x — Zy \ y — 2x : : 3 : 2, find the value of the ratio x^ — xy : x'^ + 2y2. 17. Shew that, if .r - ^ :y-z = x^ \ y"^, then x + z : ?/ + 5=- + 2 : ^ + 2. J/ A' 18. The first and fourth terms of a proportion are 5 and 54, and the sum of the mean terms is 39. Find the mean terms. 19. Having given a-\-c : b = c : a = a : c-b, determine the ratios a : b : c. 20. Shew that, if 2a +36, 2a -36, 2c + 3^, and 2c -3d be in proportion, so also are a, b, c, d. 21. Shew that, if a + b — c : c + d+a = a - c : 2d, then b : a-c = a + c-d : 2d. 22. If a^ + (^ : ab + cd :: ab+cd : b'^ + d'\ prove that a : b :: c : d. 304. Continued Proportion. Quantities are said to be in continued proportion when the ratio of the first of them to the second, the ratio of the second to the third, the ratio of the third to the fourth, and so on, are equal. RATIO AND PROPORTION. 317 Thus a, b, c, d, ... arc in continued propoi*tion, if a : b = b : c = c : 0?=..., ,,,..- a b c that IS, if - = - = - = .... ' bed If a, 6, c be in continued proportion, a is called the first term of the proportion, b is called the mean propor- tional between a and c, and c is either called the third proportional to a and 6 or is called the third term of the proportion. 305. The mean proportion between two given num- bers is the square root of their product. For suppose that a, b, c are in continued proportion. Then a _b b~c' .-. b^ = ac. b = Jac. 306. The following examples illustrate the methods of treating questions concerning quantities in continued proportion. Ex. 1. If three qicantities be in continued proportion, the ratio of the first to the third is equal to the duplicate ratio of the first to the secotul. Let a, 6, c be the quantities. We want to prove that a '. c = a^ : b^. We have y = - . b c T i. a b Let T = -^% • • - = oc. b ' c From the last of these relations, b = ex. From the first of these relations, a = bx = cxxx = cx'^. Thus each term of the proportion is expressed in terms of c (the last term of the proportion) and of x. In a continued 318 RATIO AND PROPORTION. proportion, this is always possible. Now take each of the ratios which it is desired to prove equal, and wherever a or 6 appears substitute the values just found. Thus c c Also Therefore a a? Ex. 2. Prove that, if a, b, c, d be in continued proportion, then a-d : b-c = bd{b + c + d) : c^. We have t ==- = -/ • bed T . a b , c Let -j=-^\ .'. -=x, and -j=x. b ' c d From the last of these relations, c = dx. From the second „ „ b = cx = {dx)x = dx^. From the first „ „ a=bx= {dx'^) x = dx^. a-d dx^-d d{x^—l) c^-^+^+l Hence Al- bd{b+ c+d) _dx^ d(dx^ + dx + d) _ x^--\-x+l c^ d^x^ X c dx^-dx dx{x-V dx^d (dx^ + dx -\- d^ a-d bd{b+c+d) ' ' b - c (^ EXAMPLES ON CONTINUED PROPORTION. XXV. C. 1. H a -b : b-c :: b : c, shew that a, 6, a are in continued proportion, 2. If a, b, c, d be in continued proportion, shew that {a-bf : {b-cf = a : d. 3. Find the mean proportional between 3^ and 1^. 4. Find the mean proportional between x"^ — ^ and z^ — -^. 5. Shew that, if ,r — y be a mean proportional between y and y-\-z— 2x, then x will be a mean proportional between y and z. RATIO AND PROPORTION. 319 6. Find a quantity, such that when it is subtracted from each of the quantities a, 6, c, the remainders are in continued proportion. 7. Find a quantity, such that when it is added to each of the three quantities a + 6, 6 + c, c-\- «, the sums are in continued proportion. 8. If h be the mean proportional between a and c, and if the same quantity x be added to a, b, c, determine which is the greater ratio, a+x : b + x or b + .v : c + x. 9. If the mean proportional between a and d be equal to that between b and c, shew that a : b :: c : d. 10. li a : b :: c : d, prove that the mean proportional between b and c is a mean proportional between the mean proportional between a and b and that between c and d 11. If a + b, b + c, c-\-a be in continued proportion, prove that 6 + c, c + a, c — a, a-b are proportionals. 12. The third proportional to two numbers is 48, and the mean proportional between them is 6. Find the numbers. *13. If a : b = b : c = c : d= , prove that each of these ratios is equal to a2 + 62 + c2+ J XT X « a3 + 63 + c3 + . and that ~, = ab + bc-\-cd-\- ' d b^ + c^ + (P-{-. MISCELLANEOUS EXAMPLES. XXV. D. 1. If 5.>; - 4j/ : 3.r- 2y = 4 : 1, find the ratio of x to ^. 2. Find the new ratio formed by subtracting the quantity — , — - — fi'om each term of the ratio a : b. pb — a 3. If -1 : B be the duplicate ratio of A -\- x : B + x, prove that jf' = AB. 4. Find what quantity must be added to each term of the duplicate ratio of a : 6, in order that the new ratio thus formed may be equal to that of a : b. 5. If X — 4i/ : 7/ — Sx = 3 : 2, find the value of the ratio x'-'-xy^-y^ : x'^-^xy-\-'ip-. 6. If the ratio a : 6 be compounded with a ratio of greater inequality, prove that the resulting ratio is greater than the ratio of a \b. 320 RATIO AND PROPORTION. 7. What is the least integer which must be added to the terms of the ratio 9 : 23 so as to make the new ratio so formed greater than the ratio 8 : 13? 8. Find a number which when subtracted from each term of the ratio 9 : 13 will make the new ratio so formed equal to 13 :9. 9. A is 32 years old, B is 5 years old; what is the least number of years after which the ratio of their ages will be less than 3 : U 10. The terms of a ratio are seven and three ; what number must be added to each term in order that the value of the ratio so formed may be half that of the original ratio 1 11. If a and x be positive quantities, shew that the ratio a^-x^ : a^ + x^ is greater or less than the ratio a^ — ofi : a^ + x^ according as the ratio a : x is one of less or greater inequality. 12. The age of the eldest of three children is equal to the sum of the ages of the other two, the ages of these two being in the ratio 2 to 3 : in ten years time, the age of the eldest will be five years more than half the sum of the ages of the other two. Find their present ages. 13. The ages of a man's three sons are as 1 : 2 : 3. In 12 years time, his age will be equal to the sum of the ages of the three sons ; and in 14 years more, his age will be equal to the sum of the ages of the two elder. What are their present ages ? 14. If ax-\-by : a^-\-h^ = a^y -b^x : a^b^{a—b)y find x : y. 15. If a : b :: c : d, shew that a^-b^ : c^ -cP :\ ab : cd. 16. Shew that, if a : b = c : d, then If a : b = c : d=e : f, prove the relations numbered 17 to 21. 17. pa-\-qc + re : pb + qd+rf=e :f. 18. ae:c^ = bf '.d\ -Q a?x-\-c^y-{-e^z _ cex-\-eay-\-acz ^^' b'^x+d'^y+Pz ~ dfx-^-fby + bdz' 20. {a+ef : {b+ff :: c ^J'^f^^ : ds/b'^-dK 21. a:b :: y/imW + yi'^c^-p'^e^) : s]{inW\nM'^ -p^P)- RATIO AND PROPORTION. 321 22. If «! : fti = a2 : ^2=^3 • ^3) prove that C16263 + C26361 H- C36162 ~ 6^2 ' (ii) \/«i^i + \/a2^2 + Vtts^s = \/(«i + a2 + %) (^1 + ^2 + ^3)- 23. If X : a=y : b = z : c, then will each of these ratios be equal riyz-\-mzx-{-nz^ Z6c+mca+nc2 24. Prove that, if ^ : ??i=jo : q=r : s, then each of these ,. , ,, /lmpq + lh—\lmpQr ratios IS equal to . / ^f- — 5 ,- ^^. . 25. Shew that, if a : 6 : : c \ d :: e : f :: g : h ::.,., then 26. Shew that, if ,— = f^ = ... =t^ , then each fraction will be eaual to p^iV^+ V2" + >.-+^n<"l^ 27. Prove that, if a, 6, c, 0? be in continued proportion, then (b-cY + {c-af + {d-b)^ = {a-d)\ 28. Shew that, if 3x + 2u, 3x-2u, 3y + 2v, 3y-2v be in proportion, so also are ^, y, w, v. 29. If y : A' + y = ^ + 2-y : y + 0-A?=^-f y + s : 2^ + y+ 2^, find the ratios x \y \ z. 30. If a (y + 0) = 6 (2 4- ^) = c (.r 4-y), prove that .r — y : c{a — h) =y - s : a (5 - c) = s — .r : h{c — a). 31. Prove that, ify — 2;:a::2 — :r:5:: x—y : c, then a (y — 2)2 + 6 (2 — ^)2 + c (:?; — y)2 = a(2-:^;)(.r-y) + 6(^-y)(y-2)+c(y-2)(2-a:). 32. Shew that, if ^ : y : z=a-\-'2b + c : 2a + 6-c : 4a-46 + ^f' 37. If a, 6, c be in continued proportion, then will J{a^-b^) . a . . sj{a-c) . ^a a ' sj{a^ + b^) " \/a * s/{a + c) ' *38. Give the algebraical and geometrical definitions of pro- portion : and deduce the latter from the former. 39. What number must be added to each of the numbers 3, 4, 13, 16 that the sums may form a proportion? *40. Find two numbers such that their difference, the differ- ence of their squares, and the difference of their cubes, are in the ratio of 1 : 9 : 61. 41. The third proportional to two numbers is 162, and the mean proportional between them is 6. Find the numbers. 42. Two numbers, each less than 100 and having the same digits, are to one another as 5 : 6 ; what are the numbers ? 43. Divide £1230 among three persons, so that if their shares be diminished by £5, £10, and £15 respectively, the remainders shall be in the ratio 3:4:5. 44. £3000 is to be divided among A, B, and C : if each received £1000 more than he actually does, the sums received would be proportional to the numbers 4, 3, 2. Find what each receives. 45. A vessel is half full of a mixture of wine and water. If filled up with water the quantity of water bears to that of wine a ratio ten times what it would be were the vessel filled up with wine. Determine the original quantities of wine and water. 46. Two cisterns, connected by a pipe, contain 25 and 24|^ gallons of water respectively. Find how much water must be allowed to flow out of one cistern into the other, so that the quantity of water in the first cistern may be to that in the second in the ratio of 5 to 6. 47. A rectangular court has a grass-plot in its centre, sur- rounded by a gravel walk of unifoi-m width. If the area of the grass-plot be half the area of the court, and the length of the grass-plot be equal to the width of the court, find the ratio of the width of the grass-plot to that of the gravel walk. CHAPTER XXVI. VARIATION. 307. Variation. When two quantities are so re- lated that the ratio of their numerical measures is always constant, each quantity is said to vary directly as the other. We shall see later [Arts, 312, 314] that there are other kinds of variation besides direct variation ; but if one quantity is said to vary as another, it is understood that the former varies directly as the latter. sc Thus X varies directly as y (or varies as y), if - be constant. For example, the distance traversed by a man walking at a uniform pace varies directly as the time during which he walks. If he walk at a pace of three miles an hour for one hour, he will walk three miles ; if he walk for two hours, he will walk six miles : and so on : the ratio of the numerical measures of the distance walked and the time occupied being always the same. 308. Notation. The symbol x is used as an abbreviation for the words varies directly as. Thus X cc y, is read as x varies directly as y, or sometimes (for brevity) as x varies as y. X 309. If a? X V, we have - = m, where m is some . , y \ constant quantity, that is, a quantity which contains neither x nor y. Therefore x = my. The beginner, when he is dealing with questions involving variation, will generally find it convenient 21 2 324 VARIATION. to introduce a symbol, like m, to denote this con- stant ratio of the variable quantities. If the particular value of X corresponding to any particular value of y be known, then the value of m can be at once deter- mined. Ex. 1. If xccy and if 3/ = 3 whe^i x = 2, fi7id the relation betiveen x and y. "We have x=my, where m is a constant. But if x = % y = 3, .-. 2 = 3m. .-. ??t = |. .-. x = y. Ex. 2. If (.r-f 1)2x3/2, and if y = '^ when x=% find the relation between x and y. We have {x + Vf = my^^ where m is a constant. li x = % y = 2, .-. (2 + l)2 = w23. .-. 9 = 8m. .-. m = f. ••• (^ + l)' = fy'. 310. When one quantity varies directly as another, any increase or decrease in the one causes a proportional increase or decrease in the other. For \et XQC y\ and suppose that when x is increased to a? + x', then y is increased to y + y' . X -. , X -p X .'. - = m, and also 7 = 1^1, y y + y where m is the constant value of the ratio of the nu- merical measures of the quantities. .'. x = my, and x-\-x' == m (y + y'). Subtract, .'. x =my'. X X .'. —, = m = -. y y That is, the ratio of x' to y is the same as that of x to y. Or, in other words, the increase of x bears the same ratio to the corresponding increase of 3/ as ^ does to y. VARIATION. 325 EXAMPLES ON DIRECT VARIATION. XXVI. A. ^) If X vary as y, and if ^ = 7 when y = 4, find the value of X when y = l. 2. If X vary as the square of y, and if y = 2 when x = 9, find the equation between x and y. 3. If a a 6, and if a = 2 when 6 = 3, find the vahie of ab when a- 6 = 1. J ^ 4. If y2 X .^2- 1^ and if y = l, ^ = v/5 be simultaneous values of y and x, find the value of x when y = 2. 5. Shew that, if a^- b^ vary directly as c% and if c = 2 when a = b and 6 = 3, then 6 is a mean proportional between a — 2c and a + 2c. ^^6. If xccy + Zy and 2x^', and if x = 2 when y = 4, find the value of y when x = \. ir. If 2 vary as (^; + a)(y + 6), and be equal to {a + bf when 0;=^ and y = a\ shew that when x = a + 2b and y = 2a + 6, then 2 = 4(a+6)2. 8. If X cca + b, accy, and 6 x -, and if ^=18 when y = l, and x — 2d^ when y = 2, find x when y = 7. 9. If a X 6, and axe, shew that 6c x a^. 311. Inverse Variation. One quantity is said to vary inversely as another quantity when the first varies directly as the reciprocal of the second. Thus 0) varies inversely as y if a; oc - , that is, if a) = m- , where m is some constant quantity. y Also, since if a; = m- , then y =m - , it follows that ' y "^ X if X vary inversely as y, then y varies inversely as x. Again, if x = m- , then xy = in, hence the product of two quantities, one of which varies inversely as the other, is always the same. 326 VARIATION. 312. As an example of inverse variation, consider the case of a man walking at a uniform rate. Then the time required to walk a given distance varies inversely as the rate at which he walks. In fact, if s be the distance traversed, t be the time occupied in traversing it, and v be the velocity with which it is traversed, we have s = vt. Therefore t=-. Hence, if s be constant, t varies inversely as v. Again, suppose that the length of the base of a triangle is a feet, and that its altitude is h feet. Then we know that the area of the triangle is \ab square feet. Hence, if we have a number of triangles whose bases and altitudes are different but such that the total area of each triangle is the same, then we have a6 = constant = m (say), . • . a = 7n-j-, that is, a varies inversely as h. Similarly, b = m- , and therefore h varies inversely as a. EXAMPLES ON INVERSE VARIATION. XXVI. B. \\,i If X vary inversely as y, and if 3/ = 3 when x—h^ find the value of x when y = 5. 2. If X vary inversely as y, and if y = 3 when ^'=10, find the value of x when y = 8. ('3.) If a^ vary inversely as 6^, and if 6 = 3 when a = 2, find the relation between a and h. fC* A man, walking at a uniform pace, walks from one town to another. He can cover the distance in 4 hours, when he walks at the rate of 3 miles an hour. How long will he take, when he walks at the rate of 3^ miles an hour ? 5k If X vary as p-Vq^ p vary as y, and q vary inversely as y, and if ^ = 18 when 3/ = !, and ^=19^ when y=2, find x when y=ll. 6i Shew that, if x vary inversely as yzj{y -z), and be equal to 5 when y = 7 and z=% then xyz = \'i{y -z). '"^\ 1 1 ■ 7.j Prove that, if - + - vary inversely as .r+y, then x'^-\-y^ X y varies as xy. VARIATION. 327 313. Joint Variation. Ona (\\i2,nt\iy varies jointly as two (or more) quantities when it varies as their pro- duct. Thus X varies jointly as y and z when x oc ifz. Hence x = myz, where m is some constant quantity, that is, a quantity independent of x, y, and z. 314. A quantity x is said to vary directly as y and inversely as z when it varies jointly as y and the reciprocal of z. 1 . y In this case, .r a y-, that is, x=m-. where m is some constant quantity. 315. The following is an important proposition. If X vary as y tvhen z is kept unchanged, and x vary as z when y is kept unchanged, then x will vary jointly as y and z when both y and z are changed. Let u denote the value of the ratio x : yz. Then the proposition will be true, if we prove that u is a, constant quantity, that is, one which involves neither X nor y nor z. Since ~ = u, :. x — uyz. yz ^ Now, by hypothesis, if z be kept unchanged, x oc y. Therefore uz cannot involve x or y. [Art. 309.] Thus, u cannot involve x or y. Similarly, u cannot involve x or z. Hence u is con- stant, and therefore the proposition is true. 316. For example, consider the case of a man walking for t hours at the uniform rate of v miles an hour. If v be constant, the space traversed varies as ^. If ^ be constant, the space traversed varies as v. Hence, if both v and t vary, then the space traversed varies as the product of v and t. In fact, we know that s=vt, and therefore in this case the constant value of the ratio of the measures of this space and this product is unity. Again, the area of a triangle varies as the length of its base if its altitude be kept constant (Euc. vi. 1) ; and a 328 VARIATION. similar proof shews that it varies as its altitude if its base be kept constant. Hence, if both its altitude and its base be changed, the measure of its area varies as the product of the measiu-es of its base and altitude. In fact, if the lengths of the base and altitude of a triangle be respectively a feet and b feet, we know that the area contains \ah square feet. In this case, the constant value of the ratio of the measures of this area and this product is ^. *317. Similarly, physicists have shewn by experiment that the volume of a quantity of gas varies inversely as the pressure to which it is subjected if the temperature be kept constant (Boyle and Mariotte's Law); and that the volume varies as the absolute temperature if the pressure be kept constant (Dalton and Gay- Lussac's Law). Hence, if both the pressure and temperature vary, the volume of the gas varies directly as the pressure and inversely as the absolute temperature. Again, it has been shewn by experiment that the space described by a body falling from rest varies as the square of the time occupied if the accelerating effect of gravity be constant (which at any given place is the case) ; and varies as the accele- rating effect of gravity (at different places) if the time of fall be constant. Hence, if both the time and the accelerating effect of gravity vary, the space varies as the product of the measures of the accelerating effect of gravity and of the square of the time. EXAMPLES ON JOINT VARIATION. XXVL C. f 1. ' If X vary directly as 3/^ and inversely as 2, and if when y = 2 and 2=3 then ^ = 1, find the value of x when y = 3 and 2 = 2. 2. If X vary inversely as y^ and directly as 2, and if when ^ = 4 and y = 2 then s = 4, find the value of x when 2 = 2 and >'%] If A vary directly as the square root of B and inversely as the cube of (7, and if ^ = 3 when ^ = 256 and (7=2, find B when A = 24 and (7=|. 4. If x^ vary jointly as 3/ — 2 and y + z, and if x=2 when 3/ = 5 and z = 3, shew that 2 is a mean proportional between y — 2x and y + 2x. 5. Shew that the number of years in which the simple interest on a sum of money will accumulate to a certain fixed amount varies inversely as the sum of money put out to interest and also inversely as the rate per cent, of the interest. VARIATION. 329 318. A few miscellaneous examples on questions involving variation are here appended. Ex. 1. If X vary directly as y aiid also directly as z^ shew that y will vary directly as z. Since x i cub. in., „ „ „ „ 10 in. = m X 10^ = lOOOw cub. in. .-. the vol. of the single sphere = (216?n + 512m + 1000??i) cub. in. = 17287?! cub. in. Let the radius of this sphere be x inches, therefore its volume is mofi cubic inches. Hence, by the question, m.r3 = 1728m. .-. .r=^1728 = 12. Hence the required radius is 12 inches. Ex. 5. The price of a passengers ticket on a certain French raihoay is proportional to the distance he travels; he is allowed 25 kilogrammes of luggage free, hut on every kilogramme heyond this amount he is charged a sum proportional to the distance he travels. If a journey of 200 miles loith 50 kilogrammes of luggage cost 25 francs, and a journey of 150 miles loith 35 kilogrammes cost 16^ francs, what will a joxmiey of 100 miles with 100 kilo- grammes of luggage cost? Suppose that the passenger is charged x francs for each mile that he travels, and is charged y francs per mile for each kilo- gramme of luggage over the 25 kilogrammes which are carried free. VARIATION. 331 Then, if s miles be the distance travelled and w kilogrammes the total weight of luggage, we are told that the price of the ticket is sx francs, and that (if ^^ > 25) the sum charged for the luggage is sy (w - 25) francs. Hence the total sum charged is {sx + sy [to — 25)} francs. If 5 = 200 and z^ = 50, the charge is 25 francs, .-. 25 = 200 {x + 2by} (i). If 5 = 150 and ?(; = 35, the charge is 16^ francs, .-. 16^ = 150{:r + 10y} (ii). Solving (i) and (ii), we obtain x = ^q, y = Tnoo- Therefore, if s=100 and ?(;=100, the charge = {100a.-+100y(100-25)} francs, = 17^ francs. EXAMPLES ON VARIATION. XXVI. D. 1. If z vary as x when y is constant, and vary as y^ when x is constant, how does z vary when neither x nor y is constant ? 2. If X vary inversely as yz, and y vary directly as z'^, shew that z varies inversely as i^lx. 3. If X vary directly as y, and y vary inversely as 2^, shew that z varies inversely as ^x. 4. If y2 cc ^2_ 1^ and if y = \, x^sJ'H be simultaneous values of y and ^, find the value of x when y = 2. 5. Prove that, if x x y, and x ^ z^ then will x oc sjij/z). 6. Given that x varies directly as (y + -) , and that ^ = 202 when y = 10, find x when y = 5. 7. If vary inversely as x-y, prove that ^2_|_y2 varies X y as xy. 8. If X vary directly as the square of y and inversely as the cube root of z, and if .r=2 when y = 4 and 2 = 8, find y when x — Z and 2 = 27. 9. Prove that, if x vary as the sum of the reciprocals of y and 2, and be equal to 3 when y = 1 and 2 = 2, then xyz = 2(^ + z). 332 VARIATION. 10. If X vary as the sum of the squares of two quantities, y and ^, whose product is constant, find the value of x when y = 2, it being given that ^ = 3, when y = 3 and 2 = 3. 11. li X vary as the sum of the cubes of two quantities, y and 2, whose sum is constant, find the value of x when 3/ = 2, it being given that ^ = 3, when 3/ = 3 and ^ = 3. 12. If A vary directly as P, inversely as §, and directly as R; and if when P=bc, Q = ca, R = ab, A = abc; find ^ when F=a% q = h\ R = c\ 13. li z vary as (^+a)(y + &), and be equal to ah{a-\-h) when ;^- = and y = 0, prove that z={a + l)f when ^'=6 and 3/ = a. 14. If ax-\-hy-\-\ =0, where a and 5 are constants and x and y are variable, and if the values of x be 2 and - 9 when the values of y are 1 and —4 respectively, what is the value of X when y is zero? 15. Given that w varies as ^+y, and that y varies as ^^^ and that 2 = 2, when the values of w, x, and y are 26, 1, and 12 respectively ; express w in terms of x and z. 16. Shew that, if x vary directly Sisy-z and inversely as ys;, and be equal to 5 when y = 7 and z=2, then ^y2 = 14(y-s). 17. If a mixture of gold and silver, of which three-quarters of the weight is gold, be worth £49, what will be the value of a mixture of equal weight, of which a half is gold, the value of gold being 16 times that of silver? 18. If a mixture of gold and silver, of whose weight seven- eighths is gold, be worth £15, what will be the value of a mixture of equal weight, of whose weight five- eighths is gold, the value of gold being 16 times that of silver ? 19. It being given that the arc of a circle varies as the length of the radius and also as the angle the arc subtends at the centre, find the length of an arc of a circle of radius 20 feet, subtending a certain angle at the centre, when the length of an arc of a circle of radius 4 feet, subtending three times the former angle at the centre, is 9 feet. 20. Given that the area of a sector of a circle varies as the product of the radius and the arc on which the sector stands, find the area of the sector of a circle, of given radius, standing on an arc 15 feet long, when the area of the sector of a circle of 5 times the radius, standing on an arc 12 feet long, is 100 square feet. VARIATION. 333 21. The volume of a right circular cone varies jointly as its height and the square of the radius of its base. If the volume of a cone 14 ft. high with a base whose radius is 3 ft. be 132 cubic feet, find that of a cone twice as high, standing on a base whose radius is half as large as the other one. 22. The volume of a cylinder varies jointly as its height and the square of the radius of its base. Shew that, if the heights of three cylinders of equal volume be in continued pro- portion, so also are the radii of their bases. 23. The volume of a sphere varies as the cube of its radius. If three spheres of radii 9, 12, 15 inches be melted and formed into a single sphere, find its radius. 24. If the quantity of water which flows through a pipe in a given time vary as the square of the diameter of the pipe, and if two vessels whose contents are as 16 : 7 be filled by two pipes in 7 and 4 minutes respectively, compare the diameters of the pipes. 25. The value of a diamond varies as the square of its weight, and the value of a ruby varies as the cube of its weight. If the values of a ruby and a diamond, each weighing a carats, be equal, compare the values of a ruby and a diamond weighing respectively h and c carats. 26. It is found that the quantity of work done by a man in an hour varies directly as his pay per hour and inversely as the square root of the number of hours he works per day. He can finish a piece of work in six days, when working 9 hours a day at \s. per hour. How many days will he take to finish the same piece of work, when working 16 hours a day at \s. Qd. per hour ? 27. The price of a passenger's ticket on a certain French railway varies as the distance he travels ; he is allowed 30 kilo- grammes of luggage free, but on every kilogramme beyond that amount he is charged a sum proportional to the distance he travels. If a journey of 200 miles with 55 kilogrammes of luggage cost 25 francs, and a journey of 150 miles with 40 kilogrammes cost 16^ francs, what will a journey of 200 miles with 105 kilogrammes of luggage cost ? 28. The amount of fuel consumed in a slow-combustion stove varies as the square of the diameter of the stove when the time for which it is kept burning is constant, and varies as the time for which it burns when the diameter of the stove is constant. A stove 10 inches in diameter can be used for 21 days at a cost of 3s. 6r{n-\)d, we have ^ = 2001, a = 2, ^=7-2 = 5, .-. 2001 = 2 + (n-l)5, that is, 5n = 2004. This would require n to be a fraction, which is impossible. Hence 2001 is not a term of the given series. ARITHMETICAL PROGRESSIONS. 337 [Since ?i = 400i, the given number will lie between the 400"" and 401" terms. The 400"' term of the series will be found to be 1997, and the 401" term to be 2002.] Ex. 3. The 9"* term of an arithmetical progression is zero, and the 21" term is -36. Find the series. We shall know the series if we know the first term and the common difference, since in that case we can write down as many terms of it as we like. Let a be the first term of the series and d the common differ- ence. Then, using the formula l = a-\-{n—\)dy we have 1=0 if )^ = 9, = a-\- Sd\ and Z=-36 if ?i = 21, -36 = a4-20(iJ' These are two simple equations for a and d. Subtracting, we have - 36 = 20o? -M=\ 2d. .-. d=-3. Bxit a-\-8d=0, .-. a= -8c^=24. Thus the series is 24, 21, 18, 15,.... EXAMPLES. XXVII. A. 1. Which of the following series are in a. p. ? (i) 1, -2, -4, -7,...; (ii) -4,0,4,8,...; (iii) 11, -10, -31, -51,.... 2. Write down the first five terms of an arithmetical series, whose first term is 6 and whose tenth term is 20. 3. Write down the 17"* terms of the following series, each of which is in a. p. (i) 4,7,10,13,...; (ii) 3, -1, -5, -9,...; (iii) {a + bf, a^ + b% {a-h)\..,. 4. Is 1000 a term of the series (ii), given in question 1 1 5. Find the n^^ term of an a. p., having given that a +6 is the fii'st term and a — h the common difference of the series. 6. Find the arithmetical series whose lO"' term is -100 and whose 48*'' term is 128. R A. 22 338 ARITHMETICAL PROGRESSION'S. 7. Find the IS"' term of an arithmetical progression whose 8"" and IS'*" terms are respectively 17 and 25. 8. Find the 18"' term of an arithmetical progression whose 6"" and IS"" terms are respectively 22 and 43. 9. The 5"" term of an arithmetical progression is 64, and the lO"' term is 729 : find the Z'^ and the 7"" terms. 10. Every term of a series of numbers in a. P. is multiplied by the same quantity. Is the new series so formed in a. p. ] 323. Sum of n terms of an A.P. The sum of n terms of an arithmetical progression, whose first term is a and common difference is d, is ^n {2a + (n — 1) d}. Let s be the required sum, and let I be the n^^ term. Writing down the terms of the series, we have s = a + {a + d) + (a -{- 2d) + ... + (I - 2d) + (I - d) + I. If we write the series in the reverse order, we have s = l^(l-d) + {l-2d)-h ... +(a-h2d) + {a-\-d) + a. Adding these series together, we get n terms as follows : 2s = {a + l)-[-{a-\-l) + {a-\-l)-\-...-h{a-{-l)-\-{a+l)+{a-\-l), that is, 2s = n{a + I). ••• « = |(« + (i). But l = a-\-(ji-l)d, n :. s=^{a + a + (n-l)d] = '^{2a + {,i-l)d} (ii). 324. Note. If the first and last terms of the series had been given, the formula (i) would have given the sum. If any three of the four quantities a, 1, 7i, s be given, the equation (i) will serve to determine the fourth quantity. ARITHMETICAL PROGRESSIONS. 339 Similarly, if any three of the quantities a, d, n, s be given, the equation (ii) will serve to determine the fourth quantity. E.v. 1. Fmd the sum of 21 terms of the series 17, 11, 5, — 1,.... This is an arithmetical progression, whose first term is 17 and whose common difierence = 1 1 - 17 = —6. Hence, in the above formula, a = 17, o?= -6, 7^=21, ••• s=^{2a + {n-\)d} =^{34 + (20)(-6)} = 21(17-60} = - 903. Ex. 2. Fhid the suin of n terrns of the series 1 + 3 + 5 + 7 + .. . This is an arithmetical progression, whose first term is 1 and whose common difierence = 3 — 1 = 2. Hence, in the above formula, a = l, o?=2, .-. 5 = |{2a + (n-l)c^} = |{2 + (7i-l)2} Ex. 3. The 4"" term of an arithmetical progression is 8, and the 11"* term is 22. Find the sum of 9 terms. Let a be the first term, and d the common difference. Then, using the formula l = a-\-{n-l)d^ we have, if n = 4, ^ = 8, .-. 8 = a+ M\ and if n = ll, Z = 22, .-. 22 = a + lOd]' Solving these two equations, we find a = 2, o?=2. We want the sum of 9 terms of this series, . • . n = 9. .-. s = |{2a + (7i-l)c^} =1(4 + 8.2} = 90. 22—2 340 ARITHMETICAL PROGRESSIONS. Ex. 4. How many terms of the series 1^ + 3 + 4^ + 6+... must he taken in order that their sum may he 99 ? We use the equation s=- {^a-\-{n-\)d]. Here s = 99, a = l^, d=\\^ and n is required, .-. 99=|{3 + (.i-l)xf} _n 371 + 3 .'. ?i2 + ^_ 132 = 0. .-. {n + 12){n-ll) = 0. .*. n= —12, or 71 = 11. To make the solution intelligible, n must be a positive integer. Hence the root — 12 is inapplicable to this problem, and the answer is 11. [Note. The interpretation of the answer ?^= - 12 is that if, beginning with the first term, we count 12 terms backwards, then the sum of those terms will be —99. J EXAMPLES. XXVII. B. Sum the series of numbers (each series being in arithmetical progression) given in examples 1 to 17. 1. 8, 12, 16, 20,... to 20 terras. 2. 9, 26, 43,... to 90 terms. 3. 8, 27, 46,... to 100 terms. 4. 32 + 32^ + 32| + ... to 19 terms. 5. 3 + 2^ + If + . . . to 10 terms. 6. 3 + 3^ + 3| + . . . to 15 terms. 7. 2 + 2j+2^+... to 12 terms. 8. | + li + 2 + ... to 10 terms. 9. - 1 + ^ + 1 + ... to 29 terms. 10- ¥> - 1 >-¥>••• to 9 terms. 11. 13-1, 11-4, 9-7,... to 15 terms. 12. 1-6 + 2-4 + 3-2... to 6 terms. ARITHMETICAL PROGRESSIONS. 341 13. 16-6 + 14 + 11-4... to 10 terms. 14. 3'5 + 4-l+4-7+5-3 + ... to 12 terms. 15. 2-5 + 3-2 + 3-9 + 4-6 4-... to 12 terms. ,^ x+l x+2 x+S 16. — ^ + — ^ + — ^ + ... to a terms. a a a 12 3 17. n — , 3n — , 5n — .... to ?i terms. ji n n 18. The first term of an arithmetical progression is 38, and the fourth term is 86. Find the sum of the first 12 terms. 19. The first term of an arithmetical progression is 36, and the fourth term is 90. Find the sum of the first 10 terms. 20. The 20* term of an a. p. is 15, and the 30"' term is 20. Find the sum of the first 25 terms. 21. Find the sum of 24 terms of an arithmetical progression, whose 13"" term is 25 and whose 19''' term is 37. 22. In an arithmetical series, consisting of 15 terms, the sum of the last five terms exceeds the sum of the first five terms by 100. Find the common difierence. 23. If the sum of n terms of an arithmetical progression whose first term is 11 and common difference is 3, be equal to 377, find the value of n. 24. The sum of ?i terms of the series 3, 7, 11,... is 820. Find n. 25. If n be so chosen that the sum of the fii'st n odd numbers is 900 ; find n. 26. If the sum of n terms of an arithmetical progression be n^, and the common difference be 2, find the last term. 27. If the common difference of an a. p. be double the first term, prove that the sum of n terms of the series varies as n^. 325. Arithmetic Mean. When three quantities are in arithmetical progression, the middle one is called the arithmetic mean or average value of the other two. The letters A.M. are often used as an abbreviation for the words Aiithmetic Mean. 342 ARITHMETICAL PROGRESSIONS. 326. The arithmetic mean of two quantities is half their sum. For let a, x, h be three quantities in arithmetic progression. .*. X — a = h — x. .'. 2x = a + b. .'. x = i(a-hb). 327. Arithmetic Means. When any number of quantities are in arithmetical progression, the terms intermediate between the two extreme ones are called the arithmetic means (or arithmetical means) of the two extreme terms. 328. To insert p arithmetic means between two quantities a and b. We have two quantities, a and b. We want to insert p quantities between them, so that the (^ + 2) terms so formed may be in a. p. The first of these p + 2 terms is a ; the last of them is b. Let d be the common difference of this progression. Then the (^ + 2)"' term must he a + {p + 1) d. .'. b = a-{-{p + l)d. , b - a Hence the required series is b-a ^ b—a b—a , a, a-\ -. a + 2 ;-, , a+p -, b. And the arithmetic means are . b-a , _ b-a . b-a ''+i^>«+2?^ '«+^iH:i' that is, are ^*, (^-I'^-^ft «+£&_ p + \ p + \ + 1 Ex. 1. Insert three arithmetic mea7is between 4 and 1. We want to insert 3 terms between 4 and 1, so that the 5 terms so formed shall be in a. p. Let d be the common difference of the required series. Therefore 1 is to be the S"* term of an a. p. whose first term is 4 and whose common difference is d. ARITHMETICAL PROGRESSIONS. 343 Using the formula ^ = a + (?i - 1 ) c?, we have l = 4 + 4c^, .'. the series is 4, 3|, 2^, 1|, 1 ; and the required means are 3|, 2^, 1|. Ex\ 2. jPi>Ki ^Ae sti7)i of the p arithmetic means inserted hetiveen a and h. The sum of an a. p. of n terms, whose first term is a and last term is I, is \n{a-^l) [Art. 323 (i)]. Therefore the sum of the {p + 2) terms, whose first term is a and last term is h, is ^(p + 2)(a + 6). Hence the sum of the arithmetic means is \{p + ^){a+h)^a-h = \p{a-\-h). EXAMPLES. XXVII. C. 1. What is the arithmetic mean between 4 and 8? 2. Insert three arithmetic means between (i) 4 and 8 ; (ii) 1 and 9 ; (iii) 3 and - 4. 3. Insert seven arithmetic means between \ and 7. 4. Insert two arithmetic means between a and h. Insert also two arithmetic means between the reciprocals of the means just found. 5. Insert four arithmetic means between ofi and x~^. 6. Insert five arithmetic means between {a—by and (a + 6)2. 7. Prove that the sum of ?i terms of an arithmetic progression is equal to n times the arithmetic mean between the first and Ti"* term. 8. Find the sum of the arithmetic means inserted between each two consecutive terms of an arithmetical progression of which the first term is a and the last I. 9. A man training for a mile race runs the distance every day for 24 days, his time improving at a uniform rate. On the first day he takes 8 minutes, on the last 4§ minutes. What is his average time? 10. The square of the arithmetic mean of three numbers in arithmetical progression is less by 1 than the arithmetic mean of the squares of the extremes, and the cube of the mean is less by 18 than the arithmetic mean of the cubes of the extremes. Find the numbers. 344 ARITHMETICAL PROGRESSIONS. 329. The student will notice that all the propo- sitions about arithmetical progressions are deduced from the three formulae, I = a-\-{n — \)d, s = \n{a + I), s =- \n[2a -^ {n -1) d], where a is the first term of the series, d the common difference, I the ?i"^ term, and s the sum of n terms. MISCELLANEOUS EXAMPLES. XXVn. D. [The following miscellaneous examples are on arithmetical progressions. Other examples on arithTnetical progressions will he found on pp. 377 — 379.] 1. Sum the following series, each of which is in a. p. (i) 7 + 32 + 57 + ... to 20 terms; (ii) l + 4 + l + ...to 12 terms; (iii) {a - 36) + (2a - bh) + (3a - 76) + . . . to 40 terms ; (iv) (7i-l) + (n-2) + (7i-3) + ... to n terms. 2. How many strokes are struck in a day by a clock that tells the hours, but not the quarters or halves ? 3. Prove that, if every alternate term of an arithmetical progression be struck out, the remaining terms will form an arithmetical progression. If the terms that remain be , — — , ..., supply n ^ n ^ n ' ^^ -^ the terms that have been removed. 4. Shew that, if any odd number of quantities be in a. p., the first, the middle, and the last of them will be in a. p. 5. The 4"' term of an a. p. is 15, and the 20"* term is 2.3^; find the sum of the first 20 terms. 6. The sum of the first three terms of an arithmetical pro- gression is 9, and the sum of the three subsequent terms is 27. Find the series. 7. Shew that the sum of the first 5 terms of the series 11 + 9 + 7 + .. . is equal to the siun of the first 7 terms. Explain this. ARITHMETICAL PROGRESSIONS. 345 8. Find the arithmetical progression whose sum to 5 terms is 45 and whose second term is 7. 9. How many terms of the series 29 + 27 + 25 + 23 + . . . must be taken so that their sum may amount to 200? 10. The sum of n terms of the a. p. 2, 5, 8,... is 950 ; find n. 11. The sum of n terms of the series 3, 6, 9,. . . is 975 ; find n. 12. Find the number of terms of the arithmetical progression 18, 15, 12,,.., which must be taken so that their sum is 45 ; and explain the double answer. 13. Find the last term in the series 201, 204, 207,..., when the sum of all the terms is 8217. 14. The sum of a certain number of terms of an a. p. is 36, and the first and last of these terms are 1 and 11 respectively. Find the number of terms and the common difference of the series. 15. If the sum of the first n terms of an a. p. be ^^n^, where n is any positive integer ; find the r"* term. 16. If the ?•"' term of an a. p. be 32r-16, where r is any positive integer ; find the sum of the first n terms, 17. Shew that, if unity be added to the sum of any number of terms of the series 8, 16, 24,.., the result will be the square of an odd number. 18. An arithmetical progression contains 27i + 1 terms. Shew that the sum of the odd terms : the sum of even terms = 7i + 1 : n. 19. Find the sum of 2m terms of an arithmetical progression, of which the two middle terms are a-h and a-\-b. 20. How many terms of the series 19 + 17 + 15 + 13 + ... must be taken to amount to 100? 21. Shew that the sum of an arithmetical progression, whose first term is a, whose last term is z, and whose common difference is 6, is z^ — aP- z + a "^f "*" ~2" • 22. Shew that, in every arithmetic progression in which the first term is equal to the common difference, the {p-qf" term is equal to the difference between the jo"' and 5"* terms. 23. If each of two arithmetical progi'essions be composed of 2w + l terms, and if their middle terms be equal, shew that their sums are also equal. 24. Find the sum of all the numbers between 200 and 300 which are divisible by 3. 346 ARITHMETICAL PROGRESSIONS. 25. A man stands by a heap of 100 stones. How far must he walk, carrying one stone at a time, to place the stones sepa- rately, at intervals of 20 feet apart, in a straight line having one end where the heap is? 26. The sum of n terms of an a. p., whose first two terms are 43 and 45, is equal to the sum of 2n terms of another a. p., whose first two terms are 45 and 43. Find the value of n. 27. The first term in a series of numbers in a. p. is ?i, the middle term w, and the sum of the series is m'^-mn+m. Write down the last three terms of the series. 28. Shew that, if -, , f — , ^— be in arithmetical pro- a-0 b-c b-c gression, then a, b, c will be also in arithmetical progression. 29. The sum of four numbers, which are in arithmetical pro- gression, is 16; and the product of the second and third of these numbers exceeds by 8 the product of the first and fourth. Find the numbers. 30. There are four numbers in arithmetical progression, such that the sum of twice the square of the first and the square of the last is equal to the square of the sum of the first and second, and the square of the last exceeds three times the square of the first by 1. Find them. 31. Divide 15 into 5 parts which are in arithmetical pro- gression, and such that the sum of the squares of the least and greatest of them is less than the sum of the squares of the other three parts by 3. 32. Find three numbers in the ratio of 3 : 6 : 10, such that, if each be increased by 1, the square roots of the sums are in arithmetical progression. 33. The common difference of an a. p. is 2, and the square roots of the first, second, and fourth terms are in a. p. Find the series. 34. Write down the n^^ term and the last term of the series 1 + 4 -h 7 + . . . continued to 27i + 1 terms ; and shew that, if S^ and >S'2 be the sums of the first n and the last n terms respectively, then S2-Si=6S, where S is the sum of the first ?i. natural numbers 1, 2, 3, &c. 35. Prove that any even square, {2nY, is equal to the sum of n tenns of one series of integers in arithmetical progression, and that any odd square, (2n + l)^ is equal to the sum of n terms of another such series increased by 1. ARITHMETICAL PROGRESSIONS. 347 36. The 11^^ terms of the two series 2+3H-5| + ... and 187 + 184| + 18U + ... are the same. Find n. 37. The first terms of two arithmetical series are a and 2a, and their common differences are respectively d and \d. Prove that, if 4a be a multiple of o?, it will be possible to find a number n such that the sum of n terms of each series is the same. 38. Ten balls are placed on the ground in a line at equal distances apart. A boy, standing in a line with them and 12 feet from the nearest ball, places a basket on the ground, fetches each ball in succession, and places it in the basket. When he has returned with the last ball, he finds he has walked a quarter of a mile. How far apart were the balls placed ? 39. A man starts to explore an unknown country carrying provisions for 10 days. He can walk 15 miles a day when carry- ing provisions for 10 days, and he can go an extra mile a day for each day's provisions he gets rid of. What distance will he have walked by the time he has just exhausted his provisions? 40. The cost of boring an Artesian well 500 feet deep is 25, So?, for the first foot, and a halfpenny in addition for each succeeding foot. What is the whole cost ? 41. A man, who is training for a race, runs on the first day a number of miles equal to one-sixth part of the number of days he is in training, and each succeeding day runs three miles farther than he did the day before. Altogether he runs 51 miles. How many days is he in training? 42. Three squares of ground, the lengths of whose sides are in A. p., are paved with square tiles of equal size. In the two smaller squares together there are 45 tiles more than in the largest square; and, if there were 9 tiles more in the middle square, the numbers of tiles in the three squares would be in A. p. How many tiles are there in each square? 43. A man walks a distance of 21 miles in 5 hrs. 37 min., starting at the rate of 3| miles per hour, and increasing his rate by a certain quantity after completing one-third of the distance, and again by the same quantity after completing two-thirds of the distance. What is his final rate ? 44. A and B start to walk to a place 96 miles off and back again, B always walking at the rate of two miles a day faster than A. A starts at the rate of 10 miles a day and daily in- creases his rate by two miles. Determine when and where he will meet B comingr back. CHAPTER XXVIII. GEOMETRICAL PROGRESSIONS. 330. Geometrical Progression. A series of numbers is said to be in geometrical progression (or to form a geometrical progression) when the ratio of any term in the series to the one immediately preceding it is always the same. The letters G. P. are often used as an abbreviation for the words Geometrical Progression. Each number is called a tenn of the series. The constant ratio of any term to the one imme- diately before it is called the common ratio of the progression. 331. Condition for a G.P. The condition that the numbers represented by the letters a, 6, c, d ... shall be in geometrical progression is that h _c _d _ a h c Thus a, b,c... are in continued proportion [Art. 304]. 332. Ecach of the following series is an example of a geo- metrical progression, 1, 2, 4, 8, (the common ratio being 2), 1, I, I, I, (the common ratio being ^), 2) -ij +T8-> ~To8j (the common ratio being -^), a, ar, ar^, m-^, (the common ratio being r). Where all the terms of a series, which are given, form part of a geometrical progression, it is usual to assume that the whole of the series is in geometrical progression. GEOMETRICAL PROGRESSIONS. 349 333. Expression for the n^^' term of a G.P. The ?i*'^ term of a geometrical progression, whose first term is a and common ratio is r, is ar''^~'^. For the second term is ar. The third term is obtained from the second by multiplying it by r, hence it is ar'^. Similarly, each successive term is obtained by multi- plying the term before it by 7\ Thus, the fourth term is ar^, the fifth term is ar^, and so on ; the index of r in any term being 1 less than the number of the term. Hence the n^^ term consists of the product of a and ?'"~^ ; and, if we denote the n}'^^ term by Z, we have I = ar'^-\ Ex. 1. Find the Q^ term of the series 1, 2, 4, .... This is a geometrical progression. The first term (denoted above by a) is 1, the common ratio r = f = 2. Hence the 6"* term =ar"~i = 1x25 = 32. Ex. 2. The fourth term of a geometrical progression is 4 and the seventh term is -^. Find the series. We shall know the series if we know the first term and the common ratio, since in that case we can write down as many terms of it as we like. Let a be the first term of the progression, and r the common ratio. Then using the formula l = ar^~'^^ we have 4=ar3'l Dividing, we obtain ^3 = — 2=_i^ .-..= -4. Substitute this value of r in the equation 4 = a7^, .-. 4 = a(-i)3, .'. a=-32. Hence the series is -32, 16, -8, 4, -2, 1, -I,.... 350 GEOMETRICAL PROGRESSIONS. EXAMPLES. XXVIII. A. 1. Which of the following series are in g. p. 1 (i) i, 1, f, ^,.-.; (ii) 1, -1, 1, -1,...; (iii) 2, 4, 8, 12,...; (iv) ^, x^y, xy\ y\,.., 2. Write down the sixth terms of those series given in question 1 which are in g. p. 3. Find the 6"' term of the series 3, 6, 12, ... . 4. What is the eighth term of the geometrical progression whose first and second terms are respectively 2 and - 3 % 5. What is the sixth term of a geometrical progression whose first and second terms are 3 and - 4? 6. The 2°'' term of a geometric progression is 21, and the 3"^ term is 147. Find the 1'' and 5* terms. 7. The 2°'^ term of a geometric progression is 15, and the 3"^ term is 75. Find the 6"" term. 8. Find the geometrical progression whose fifth and ninth terms are respectively 1458 and 118098. 9. The sum of the second and fourth terms of a geometrical progression is 20, and the difference of the first and fifth terms is 30. Find the series. 10. Shew that, if the terms of an A. P. be written down and any quantity be raised successively to those powers, the results will be in g. p. 334. Sum of n terms of a G.P. The sum of n terms of a geometrical progression, whose first term is a and common ratio is r, is a =- . r— 1 Let s be the required sum. Then s = a +ar +ar^ -{■ ... + ar^^~^ + ar"~^ (i). Multiply by r, .'. sr = ar + ar"^ + a?-^ + . . . + ar'^~'^ + ar''^ (ii). Subtract (ii) from (i). .'. s — sr = a — ar^\ GEOMETRICAL PROGRESSIONS. 351 since all the other terms on the right-hand side cancel. .-. s{l-r)=o.{l-7'''). 1-r' .'. s = a 335. If any three of the four quantities a, r, n, and s be given, this equation will determine the fourth quantity; but, if a, n, and s be given, and if n>2, it will not generally be possible to solve the resulting equation for r. If I be the last term, we have I = a7^^^~\ _ ar^^ — a _rl — a which gives a different form for s. Ex. 1. Find the sum of 8 terms of the series 1, 2, 4,.... This is a geometrical progression whose first term is 1 and common ratio = 2. Hence, in the formula of Art. 334, a= 1, r=2, ?i = 8, _ ^"~1 ~ r-\ 28-1 2-1 256-1 = 255. 1 Ex. 2. Find the sum of n terms of the series a^, ab, b'\.... This is a geometrical progression whose first term is a^ and whose common ratio = — = —r = - . Hence the formula a-^ ao a r»-l s=a 7- gives s=a^ a b" - a"" '^b-a)' 352 GEOMETRICAL PROGRESSIONS. Ex. 3. It is said that chess loas invented hy Sessa for the amusement of an Indian rajah, narned Sheran, who rewarded the inventor by promising to place 1 grain of loheat on the 1" square of a chess-hoard, 2 grains on the 2""^, 4 grains on the 3'^, and so on, doubling the number for each successive square on the board. Find the number of grains which Sessa should have received. The number of grains on the successive squares form a g. p. whose first term is 1 and common ratio is 2. There are 64 squares, .•.in the above notation a = l, r=2, 7i = 64. Therefore the total number of grains is given by r"-l s=a ;- r — \ = ^^2-1 = 264-1. This number has been calculated, and is equal to 18,446,744,073,709,551,615. The quantity of wheat required to meet Sessa's claim is far greater than all the wheat in the world. Ex. 4. How many terms of the series 3, 9, 27,.,. must be taken in order that their sum may be equal to 120? ^»_ 1 Using the formula s = a — r-, we have a =3, r=3, s = 120, .-. 120 = 3 1^^^ =f (3- -1), .-. 40x2 = 3'^-l, .-. 3"=81. Now 81 = 92 = 34, .-. 71=4. Hence 4 terms of the series must be taken. *Ex. 5. How many terms of the series 1, \, j, \,... mu^t be taken in order that their sum may be equal to 21 j'n _ 1 Using the formula s = a — -y, we have a=l, r = ^, and s=2, =2{i-(in GEOMETRICAL PROGRESSIONS. 353 .-. 1 = 1 -(i)«, ••• (i)''=o. N \^ab, that is, that a + b- 2\/ab > 0, that is, that {\/a — \/by > 0. But the square of any quantity (whether positive or negative) is itself a positive quantity, and therefore greater than zero. Hence {\/a — \/by is > 0. .-. i(a + b)>^/ab. Note, If we have two equal quantities, a and 6, then their arithmetic mean = ^{a-\-h) = ^ (2a) = a ; and their geometric mean = ^Jab = \/«'^ — a. Thus, in this case, the arithmetic mean is equal to the geometric mean. We may therefore say that the arithmetic mean of any two quaiUities is not less than their geometric mean, 343. Geometric Means. When any number of quantities are in geometrical progression, the terms intermediate between the two extreme ones are called the geometric means (or geometrical means) of the two extreme terms. 360 GEOMETRICAL PROGRESSIONS. 344. To insert p geometric means between two quantities a and b. We have two quantities, a and b. We want to insert p quan- tities between them so that the p + 2 terms so formed may be in G.P. The first of these p + 2 terms is a ; the last of them is b. Let r be the common ratio of this progression. Then the {p + 2y^ term must be ar^ + i. a •■•'•=V; Hence the required series is of which the terms intermediate between a and b are the required geometric means. Ex. 1. Insert tioo geometric means between 1 awc^ 27. We want to insert 2 terms between 1 and 27, so that the 4 terms so formed shall be in g. p. Let r be the common ratio of the required geometric progression. Therefore 27 is to be the fourth term of a G. p. whose first term is 1 and whose common ratio is r, .-. 27 = lxr*-i. .-. 27 = '/-3. .-. r=3, .•. the series is 1, 3, 9, 27; and the required means are 3 and 9. Ex. 2. Find the sum of the p geometric means inserted between a and b. Let r be the common ratio of the progression, of which a is the first term and b is the (jo + 2)"' term. GEOMETRICAL PROGRESSIONS. 361 The sum of the p means is rP— 1 ar 4- ar"^ + . . . + ar^^ = ar r—\ ^•P + 1 _ ^' r-1 h — r a = a =- r-1 h-ar r-l And we know, by Art. 344, that r^^X^hja, hence the sum can be expressed in terms of p, «, and h. EXAMPLES. XXVIII. C. 1. Find the geometric mean between (i) 2 and 8; (ii) {x + af ^ndi {x - a)\ 2. Insert three geometric means between (i) 1 and 16; (ii) (*• + «)- and {x-cif; (iii) 1 and 2. 3. Insert two geometric means between .r*'" and ?/^". 4. Insert six geometric means between lOJ and V^. 5. Insert eight geometrical means between 512 and 19683. 6. Find the geometrical mean of 4^2 -12^' + 9 and 9^'2 + 12.i' + 4. 7. Prove that, in a geometrical series containing an odd number of terms, the middle term is a geometric mean between the first and last terms. 8. Determine the ratio of two numbers, when the ratio of their arithmetic mean to their geometric mean is as 13 to 5. 9. The arithmetic mean between two numbers is 39, and the geometric mean between them is 15. Find the numbers. 10. If one geometrical mean, G^ and two arithmetical means, p and q, be inserted between two given quantities, shew that G^^ = {2p-q)m-p). 362 GEOMETRICAL PROGRESSIONS. 11. If one arithmetical mean, A, and two geometrical means, p and 2', be inserted between two given quantities, shew that p^j^q^ = 2Apq. *12. Between each consecutive pair of terms of a series of numbers in g.p., m arithmetic means are taken. Find the sum of all the means thus obtained. 345. The student will notice that all the pro- positions about geometrical progressions are deduced from the formulae I = ar''-\ r^ - 1 where a is the first term of the series, r the common ratio, I the 7i^^ term, and s the sum of n terms. It has been shewn [Art. 336] that it follows from the latter of these formulae that the limit of the sum of an infinite number of quantities in geometrical progression, whose common ratio, r, is numerically less than unity, is a MISCELLANEOUS EXAMPLES. XXVIII. D. [The folloioing miscellaneous examples are on geometrical progressions. Other examples on geometrical progressions will he found on pp. 377—379.] 1. Sum the following series, each of which is in g. p. (i) 14 + 42 + 126 + ... to 8 terms; (ii) 864 + 1296 + 1 944 + ... to 6 terms; (iii) 8 + 6 + 4| + ... to infinity; (iv) 2^/5-5^6 + 15^5-... to 8 terms. 2. Find, to four places of decimals, the sum to infinity of •nil the series 1 + 73 + o + ---- GEOMETRICAL PROGRESSIONS. 363 3. Can any of the following geometrical series be summed, the number of terms in each being infinite? If so, find their respective sums. (i) l-f+f-...; (ii) l-§+f-...; (iii) •9 + -81 + -729 + ...; (iv) -1 + -5 +2-5 + ...; (V) (V2 + 1) + 1 + (V2-1) + .... 4. Sum the following series : (i) x{x+y)+x^ {x^ + y2) -j- ^3 (^^ ^ y3) 4. ^ ^ ^ ton terms ; (ii) (^ + a) + (.r2 4-2a) + (^ + 3a)-}-... to n terms. 5. Find the sum of n terms of the series whose r"' term is ( — a)'", r being any positive integer. 6. The first term of a geometrical progression exceeds the second term by 1, and the sum to infinity is 100 : find the series. *7. Find the sum of n terms of a series in which each term is half the sum of all that precede it, and in which the first term is a. 8. The alternate terms of any G. p. form a series in G. p. 9. If an odd number of quantities be in g.p., then will the first, the middle, and the last of them be in g. p. 10. Shew that, if every fourth term of a geometric series be picked out, these terms will themselves form a geometric series. 11. Prove that, if each term of a G. p. be squared, these terms will also form a g. p. 12. Shew that the logarithms of a series of numbers in geometrical progression will themselves be in arithmetical pro- gression. 13. Prove that, if an odd number {n) of consecutive terms of a geometrical progression be multii)lied together, the product will be the n^^ power of the middle term of the progression. 14. Each term in a certain infinite geometric progression is equal to the sum of all that succeed it. Find the common ratio of the progression. 15. Prove that, if >S' be the sum to infinity of a g.p. whose first term is a, then the common ratio of the series is 1 — ^ . 16. An odd number of consecutive terms in a certain geo- metrical progression is taken ; the middle term is 3, and the continued product of all the terms is 243. How many terms are taken ? 364 GEOMETRICAL PROGRESSIONS. 17. If the ratio of a g. p. be not less than 2, shew that any term is greater than the sum of all that precede it. 18. Three quantities are in g. p. The first exceeds the sum of the other two by unity; and the excess of the first over the second is greater than the excess of the second over the third by unity. Find the numbers. 19. The sum of four integers in geometrical progression is 255, and the fourth exceeds the second by 180 : find the integers. 20. Find three numbers in g. p., such that if they be increased by 4, 7, 1 respectively, the sums form a G.P., whose common ratio is less by 1 than that of the original series. 21. From three numbers, which are in geometrical pro- gression, three others, which are also in g. p., are subtracted. Prove that, if the remainders be in g.p., all these series have the same common ratio. 22. The sum of three quantities in geometrical progression is ^, and the sum of their squares ^^^. What are the quantities ? 23. The continued product of three numbers in geometrical progression is 64, and the sum of the products of them in pairs is 84 ; find the numbers. 24. If X, y, a be in arithmetical progression and if .^\ ?/, 6 be in geometrical progression, shew that .r, x—y^ h-a are in geo- metrical progression. *25. Shew that, if s^, ^2, s.^... be an infinite series of sums of infinite geometrical progressions, whose common ratios are the same, and whose first terms are respectively the terms of the series s^ , then .^^ + ^3 + . . . is greater or less than s^ according as the common ratio is greater or less than one-half. CHAPTER XXIX. HARMONIC AND OTHER SERIES. *346. We are constantly concerned in mathe- matics with series of numbers, of which the successive numbers (or terms) are formed according to various rules. Arithmetical and geometrical progressions are instances of such series. A few series whose terms are formed according to other rules are reducible to arithmetical or geometrical progressions, and we shall deal in this chapter with some of the more simple of such series. We shall discuss in succession (i) Harmonical Progressions [Arts. 347 — 356], (ii) Series whose terms are the squares or cubes of numbers which are in arithmetical progression [Arts. 357 — 361], and (iii) Series whose terms are the products of corresponding terms of a series of quantities in a. p. and a series of quantities iuG.P. [Arts. 362—365]. HARMONICAL PROGRESSIONS. 347. Harmonic Progression. A series of numbers is said to be in hminonical progression (or harmonic pi'ogression, or to form a harmonical pro- gression), when their reciprocals are in arithmetical progression. The letters H.P. are often used as an abbreviation for the words Harmonical Progression. 3G6 HARMONICAL PROGRESSIONS. 348. Condition for an H.P. The condition that the series of numbers denoted by the letters a, b, c,d... shall be in harmonical progression is that 1111 a' b' c' d'" shall be in arithmetical progression ; that is, that 1_1_1_1_1_1_ b a c b d c For example, 3, 4, and 6 are in harmonic progression because 3 J J, i are in arithmetical progression, the common diflference being —^^. 349. Fundamental Property of three numbers in H.P. If three quantities be in harmo7iic progression, the ratio of the difference between the first and second of them to the difference between the second and third of them is equal to the ratio of the first of them to the third of them. This property is sometimes taken as the definition of a harmonic progression. Let a, b, c, be in harmonic progression ; .-. y-- = --J. [Art. 348 a c b Multiplying throughout by the l.c.m. of the denomi- nators, .*. ac — bc = ab — ac, that is, c(a-b) = a{b- c). .'. a — b : b — c = a : c. *350. The n"; term of a H.P. To find the yi''^ term of a harmonic progression, whose first and second terms are given. Let a and b be the two first terms, and let x be the required h"" term. IIARMONICAL PROGRESSIONS. 3G7 Therefore - i« the ?i"' tcnii of an arithmetical pi-ogression whose two first terms are - and ^ . a .'. the common difference of this A.p. = y = — , . b a ab Hence, by the formula l = a-\-{ii—\)d{A.i't. 322], the required term is given by 1 1 , ^.a-b - = -->r{n-l) — r- X a ab ^ b + {n-\){a -b) ~ ab ' ab b + {7i-l){a-b)' *351. It is impossible to express the sum of a number of terms of a harmonical progression by an algebraical formula of a concise form similar to the corresponding formulae in a. p. and G. p. *352. Harmonic Mean. When three quantities are in harmonic progression, the middle one is called the harmonic mean of the other two. The letters H.M. are often used as an abbreviation for the words Harmonic Mean. *353. The harmonic mean between a and b is 2abl{a + b). Let the harmonic mean between a and b be .v. Therefore a, x, b are in harmonical progression. Ill '• a' - , -r are m a. p. x' b 1_1 1 1 X- a ~b x' 2 a b .'. X 2ab ~a + b' *354. The geometric mean of two quantities is also the geometric mean of their arithmetic and harmonic means. Let a and b be two quantities ; and let A be their aritlimetic mean, G their geometric mean, and // their harmonic mean, 368 HARMONICAL PROGRESSIONS. . a + 6 ^ y-y „ 2a6 ... ^ = -^, G = ^ah, H=^^^. 2 a + 6 ' which is the condition that G may be the G. m. of A and H [Art. 341]. Since .4^ = G-\ we have A:G=G,H. But yl <^ G, [Art. 342], .-. GJfiH^ that is, the geometric mean of any two quantities is not less than their harmonic mean. This property can be proved directly by a process similar to that given in Art. 342. *355. Harmonic Means. When any number of quantities are in harmonic progression, the terms inter- mediate between the two extreme terms are called the harmonic means of the two extreme terms. *356. To insert p harmonic means between a and b. We have two quantities, a and b. We want to insert p quan- tities between them, so that the p-h2 terms so formed may be in H.P. Therefore the reciprocals of these p + 2 quantities will be in a. p. We want, therefore, to form an arithmetical progression, con- taining jy -f 2 terms, of which the first term shall be - , and of which the last term shall be r • The series so formed can be con- b structed by the method given in Art. 328, and will be found to be 1 ( p + i)b + {a-b) {p+i )b-{-2{a- b) 1 a' {p+l)ab ' ' {p + l)ab ' ' *"' b' The reciprocals of these terms will be in harmonical progression. Hence the required harmonic means are { p + l)ab (p-\-l)ab (p + l)ab {p + l)b + {a-by {p + l)b + 2{a-by ••" {p + l)b + p{a-by Ex, 1. Insert tivo harmonic means between 2 and ^. We want to find 4 terms in an a. p. of which the first term is \ and the fourth term is 2. Let d be the common difference of this a. p. Then using the formula l = a-\-{n-\)d [Art. 322], we have .'. d=l. IIARMONICAL PROGRESSIONS. 369 Hence the a. p. is \, 1, f, 2. The corresponding h.p. is 2, 1, §, \\ and the required harmonic means are 1 and §. Ej\ 2. Sheio that^ if a^ b, c be in G. p., then a + b^ 2b, 6 + c will be in H. p. Since «, b, c are in g.p., we have ac = b^. Now the quantities a-\-b, 2b, b + c will be in H. p., ... 111.. it J, zrr , , bemA.p., a + b' 26' b+c ' ^, ^ . .„ 11 11 that 18, if ~7 i- = ,- -7 , ' 26 a + b b + c 26' which reduces to v = , - . a+o b+c This is true, if {a - 6) (6 + c) = (6 - c) {a + b), that is, if ac-b^=b^ — ac, that is, if 2(ac-62) = 0, which is true. .'. a + 6, 26, 6 + c are in h.p. Fa;. 3. The sum of three mimbei's in h.p. is \\, and their con- tinued prod'iict is 36. Fi7id the numbers. Let the numbers be x, y, z. Since they are in H. p., .*. -, -, -are in a. p. ^ ' .r' y' z .•J-l = i-l (i). y X z y The sum of the numbers is 11, .*. x+y + z=^\\ (ii). The product of the numbers is 36, .*. xyz = ^Q (iii). We have therefore three equations to determine three un- known quantities. Equation (i), on simplification, reduces to 2xz=y{x+z) (iv). Now by (ii), x+z—ll-y; and by (iii), xz = — . Substituting these values of x+z and xz in (iv), we have 2j=y{U-y), .-. y^-lly- + 12 = 0. B. A. 24 370 HARMONIGAL PROGRESSIONS. This is a cubic equation, and we have not discussed the rule which enables us to solve such an equation. This particular equation can however be written in the form that is, /(y-3) = 8(3/-3)(y + 3). Hence one root is given by 3/ - 3 = 0, [The other roots are given by 3/"^=8(y + 3), hence they are not integers, and therefore they are not applicable to this problem.] Thus the required root is y = 3. If in equations (ii) and (iii) we put y = 3, we obtain x-\-z = S^ xz = l^. From the two latter equations, we obtain .r = 6 and z=2, or x = 2 and z = Q. Hence the required numbers are 2, 3, 6. ^EXAMPLES. XXIX. A. 1. What is the fourth term of a h.p. of which the second term is \ and the fifth term is ^^1 2. Find the sum of four terms of a h. p. of which the first term is 1 and the third term is i. 3. Write down the fourth, fifth, and sixth terms of the h. p. of which the first term is 1 and the second term is 2. 4. Find the harmonic mean of (i) 1 and 4 ; (ii) 2 and 5 ; (iii) a and — . 5. Insert three harmonic means between (i) 1 and ^j\ (ii) 17 and 1 ; (iii) | and |; (iv) a + 6 and h. 6. Shew that, if a, 2^, 6 be in h. p., then x-a : x-b = a^ : b^. 7. Shew that, if a, b, c be in geometric progression, and if p, q be the arithmetic means between a, b and 6, c respectively then b will be the harmonic mean between p and q. HARMONIC AND OTHER SERIES. 371 SERIES EACH OF WHOSE TERMS IS A POWER OF THE SUCCESSIVE TERMS OF A SERIES OF NUMBERS IN A.P. *357. We proceed next to the consideration of series each of whose terms is the square (or the cube) of the successive terms of an arithmetical progression. We shall begin by considering the special case of the deter- mination of the sum of the squares of the numbers 1, 2, 3, 4.... These numbers, when written in this order, are sometimes called the natural mimhers. To this case, we shall reduce one or two other series. We shall then determine in a similar vyay the sum of the cubes of the natural numbers. *358. The sum of the squares of the first n natural numbers is equal to ^n (n-\- l)(2/i-|- 1). Fii'st proof Let >S'„ denote the required sum, .-. >Sf,,= l■^ + 22 + 3^+... + ?l^ It is easy to verify that (?i -fiy - 7i3 = 3/i2 4- 3?H- 1. Write n -1 for n on both sides of this identity, .-. n^ - (n -iy = 3 (n - 1)- + 3 (n - 1) + 1. Similarly, (n-iy-(n-2y=S(n-2y-\-S(n-2)-^l. Continuing this process, we finally get the identities 3^-2^ = 3.22+3.2+1, 2'-V = S. 1-' + 3.1 + 1. Now, add all the right-hand sides of these n identities together, and also all the left-hand sides. The sum of the left-hand sides reduces to (n+iy—l'\ since all the other terms cancel one another. Therefore (n + 1)=^ - 1 = 3 (P + 2-^ + ... + n') + 3 (1 + 2 +...+n) + ?i. 24—2 372 HARMONIC AND OTHER SERIES. But 1 + 2 + . . . + w is an A.P., and its sum = ^n (?i + 1) ; .-. (n + 1)3 _ 1 = 3>Sf,, + fn {n + 1) + n. _ {n + lf-l-^n{n+\)-n •■• '^^~ 3 • Simplifying the right-hand side, and resolving the result into factors, we obtain _ 7^(?^ + l) (271 + 1) n This result is sometimes written '2m^ = ^n{n-\-\) {2n-\-\)y n 1 where 2m^ stands for the words "the sum of all quantities like m^, for integral values of m from ??2 = 1 to m = 7i" *359. Second proof. This is an important series, and we shall give another proof of the result. Suppose that we knew or had guessed the value of the sum, and merely wanted to verify the result. We could effect this in the following manner, which is an illustration of what is known as mathematical induction. We assume that we know or suspect that Sn=l'-h2'+...+7i' = i?i (71 + 1) (2n + 1) ... (i). Add (n + 1)" to each side, .-. 12 + 22 +. . .+ ?i2 + (n + 1)2 = i?i (71 + l)(2n + l)+(n + 1)^ = (/l + l){l7l(27l + l) + (7l + l)l, which on simplification =^{n + l)(n+ 2) (2/i + 3). Now this result is exactly what we obtain from (i), if in it we write n + 1 for 7i. Hence, if the formula (i) enable us to find the sum of the squares of the first n natural numbers, it will also enable us to find the sum of the squares of the first n + 1 natural numbers. But, if 71 = 1, the formula (i) is true, since 1^ = ^.1.2.3; hence, it is true for the case of n = 2. But, since it HARMONIC AND OTHER SERIES. 373 is true if n = 2, :. it is true if n = 3. Again, since it is now known to be true when ?i = 3, .•. it is true when n = 4. Continuing this process, we see that the formula (i) is true for any positive integral value of n. Ex. 1 . Find the sum of n terms of the series 1 .2-1-2.3 + 3.4 + .... Let Sn denote the required sum, .-. ^S^^- 1.2 + 2. 3 + 3. 4 + ... + 7K^i+l) = l(l + l) + 2(2 + l) + 3(3 + l) + ...+?^0i + l) = (12 + 22 + 32+. ..+w2) + (l + 2 + 3 + . ..?i) = ^'^(n + l){2n,.l) ^ .U^ ^^^^^^ 3^3^ 323^ = i?i(?i + l)(?i + 2). Alternative proof This result is sometimes proved in another way. We have 1.2 = K1.2.3) 2.3 = ^2.3.4-1.2.3) 3.4 = 1(3.4.5-2.3.4) n{n-^\) = l{n{n + \){n + 2)-{n-l){n + \)]. Add, .-. 1 . 2 + 2. 3 + ... + ?i(?i + l) = ^?i(?i + l)(7i+2), since all the other terms on the right-hand sides cancel one another. Ex. 2. Find the sum of the squares of n terms of an arith- metical progression. Let the series in x. p. be a, S'„ = a2 + (« + c^)2 + (a + 2c^)2 + ...+ {a + (7i-l)(ff = a2 + {«2 + 2cw^+(^2| + /^2_^2.2a€^ + (2c^)2} + ... + {a2 + 2 (u - 1) «c/+ (ri - l)2c^2|. Collect like terms, .-. ,s;=/ia2 + 2«^{l+2 + ... + (H-l)}+c^2.fi2^22+... + (n-l)2}. 374 HARMONIC AND OTHER SERIES. But i + 2 + ... + (w-l) =-|(H-l)n, [Art.323 and i24.22 + ... + (^_l)2 = i(^i_l),i(27i-l), [Art. 353 .-. S,,=:na?-{-n{n-\)ad+ln{n-\){^n-l)(P. *360. The sum of the cubes of the first n natural numbers is equal to \n'^ (n + 1)-. Let Sn denote the required sum, . • . >S',i = 1^ + 2^ + . . . + 71'. Now {n+iy-n*=4n^-{-Qn^-\-4:7i-\-l. Write n—l for n in this identity, .-. n'^-{n-iy = 4:(n-lf + Q{n-iy + 4.{7i-l) + l. Similarly, (n-l)4-(7i-2)* = 4(?i-2)3 + 6(w-2)2 + 4(?i-2) + l. Continuing this process, we finally obtain the identities 34-24 = 4.23 + 6.22+4.2+1, 24-14 = 4.13 + 6.12 + 4.1 + 1. Adding these results, we have (h + 1)4-14=4/S'„ + 6(12 + 22+...+h2) + 4(1+2+...+w) + h, = 4.Sr, + 6.ln{n + l){'2n + l) + 4.^n{7i + l) + n. Transposing Sn to one side of the equality, and all the other terms to the other side, and simplifying, we finally obtain S,,=^7i^n + iy\ This result can also be proved by induction in the same way as the proof given in Art. 359. The result of this article gives us the theorem that the sum of the cubes of the first n natural numbers is equal to the square of the sum of the first n natural numbers. For ]3 + 23 + ... + n3 = iw2(n + l)2, and 1+2 + ... + ?? =^n{7i + l), .'. 13 + 23 + ... + 7^3^(1+2+. ..+7l)2. HARMONIC AND OTHER SERIES. 375 Example. Find the sum of n terms of the series 1.2.3 + 2.3.4+3.4.5 + .... Let Sn denote the required sum. The w"' term is n {n-\-\) {n + 2), which may be written n^ + 3n^ + 2n. The other terms can be obtained from this by successively writing for n the numbers 1,2,3,.... Hence ,S'^ = (13 + 3. 124.2. 1)4.(23 + 3. 22 + 2. 2) + . .. + (?i3 + 3.?i2 + 2?0 = (13 + 23+...+9i3) + 3(l2422+...+9l2) + 2(l+2 + ...+?0 =^7i^7i + lf + 3.^n{7i + l){2n+l)-h2.in(7i + l). [Arts. 360, 358, 323. Simplifying the right-hand side, we obtain S^ = i7i{n+l) (71 + 2) (71 + 3). *.361. By a process similar to that used in Arts. 358, 360, we can find the sum of the fourth (or higher) powers of quantities in A.P. EXAMPLES. XXIX. B. 1. The sum of the squares of the first ?i natural numbers is equal to 20??. Find 7i. 2. Shew that the sum of the squares of the first 71 odd numbers is equal to ^n (4?i2 _ i). 3. Shew that the difference between 71 and the sum of the squares of any ?i odd numbers is a multiple of 8. 4. Sum to 71 terms the series 1.3 + 2.4 + 3.5 + ...* 5. Find the sum of m terms of the series 1 .2m + 2.(2wi-l) + 3.(2wi-2) + .... 6. Find the sum of 15 terms of a series whose w"' term is {2n— 1) (.3?i + l), where 71 is any positive integer. 7. Shew that the sum of the cubes of the first 71 odd numbers is equal to ^2 (2^2 - 1 ). 8. If ?i be an even number, prove that the sum of the series I + 22 + 3 + 4. + ... to „ terms is »(« + 4)(2>»+l) ^ LA 376 HARMONIC AND OTHER SERIES. SERIES WHOSE TERMS ARE THE PRODUCTS OF CORRE- SPONDING TERMS OF AN A.P. AND A G.P. *362. We proceed next to determine the sum of the terms of a series of which any term, such as the ?i*^ is the product of the n^^ term of an A.P., and the n^^^ term of a G.P. *363. To find the sum of n terms of the series 1 + 2^' + 3a'2 + 4*-3 + ... . Let S denote the required sum, .-. S=l + 2x +Zx^ + ... + {n-\)x'<'-'^^nx''-\ .'. Sx=x-\-'2.x'^ + Zafi-\-...-\-{n-\)x''-^^'Hx''. Subtract, .'. S-Sx=\-^x + x'^-\-...+x''-'^-nx'^, that is, = lzfL_^^^.n [Art. 334 \-x ^ *364. The following is the general case. To find the sum of n terms of the series a, [a + b] r, [a + 26} r", {a + 36} i-^, .... Let S denote the required sum, .-. S=a +{a + b}r + .,.+ {a + (7i-2)b}r''-^ + {a + {7i-l)b}r''-\ .-. Sr = ar+{a+b}r^- + ...+ {a + {n-2)b}r''-^ + {a + {n-l)b}r'\ Subtract, and simpUfy the result on the right-hand side, .-. S - Sr = a + br + br^-\- ... + br^'-'^- {a + {7i-l)b}r'\ .-. S{l-r) = a + br^~^^'^^ ^ -{a + {n-l)b}r\ ^ ^^ a.[a + (^n-l)b}r^ ^^ l-r^-i 1-r (l-r)2 ' HARMONIC AND OTHER SERIES. 377 *365. Note. In Art. 363, we formed Sx from ,S', and by .sub- traction obtained S{\-x)', thus we multiplied S by (1 -.r), and reduced the question to summing a geometrical progression. In a similar way, if ,S'=l2-|-2% + 3%2 + ... + (?i + l)%2, the multiplication of S by (1-^')-^ reduces the determination of S to the summation of a geometrical progression. *MISCELLANEOUS EXAMPLES. XXIX. C. [7%e following examples are on arithmetical and geometrical progression, as well as on the subject-matter of this chapter. 1. Determine whether the following series are in arithmetical or in geometrical progression, and sum each of them to six terms. (i) 18 + 15 + 12 + ...; (iv) § + !+# + ...; (ii) 3 + 41 + 5^ + ...; (v) l-48-2-22 + 3'33- ... ; (iii) 12 + 4 + 1.1 + ; (vi) l'3-3-l -7*5- .... 2. Sum the following series to n terms : (i) (2w-l) + (2?i-3) + (27i-5) + ...; (ii) {x + a) + {x^ + 2a)^-{j,f> + ba)^...; (iii) a6 + 2a62 + 3a63+.... 3. The first and eleventh term of a series are a^ and a~^. Find the sum of 11 terms of the series, (i) on the supposition that it is in a.p., (ii) on the supposition that it is in g.p. 4. Find the difference between the sums of the series n n-\ n-2 ,, c^ ^ •. - + H h ... (to 2?i terms), n n n ^ and + -r , r., + : —7. + . . . (to mtinity). 5. The sum of 10 terms of an a.p. is 100, and the second term is zero. Find the fii'st term. 6. Prove that, if a, 6, c be in arithmetical progression and a, 6 - a, c - a be in geometrical progression, then a = ^h = \c. 378 HARMONIC AND OTHER SERIES. 7. If 1, X, y be in arithmetical progression, and 1, y, x be in geometrical progression, find x and y. 8. The arithmetic mean between two numbers is 25, and the geometric mean between them is 15. Find the numbers. 9. Find three numbers in the ratio of 6 : 11 : 20, such that if each be increased by 1 they become in g.p. 10. Shew that in every geometric progression in which the common ratio is 5, the arithmetic mean between the 2"*^ and 4'*" terms is 13 times the 2"*^ term. 11. If xy, y^j z^ be in arithmetical progression, shew that y, ^, 2y - X are in geometrical progression. 12. If a, 2>, G be in g.p., and if x be the a.m. between a and b, and y the a.m. between b and c, prove that - + - = 2. 13. The first term of a geometrical series is a, the sum to infinity is pa. Find the oi^^ term. 14. The sum of an infinite geometrical progression is 4^, and the sum of the first two terms is 2|. Find the series. 15. The first terms of an arithmetical and of a geometrical progression are equal to 8, the second terms are equal, and the third term of the latter series exceeds the third term of the former by 2. Find the two arithmetical and geometrical pro- gressions which satisfy these conditions. 16. Insert between 6 and 16 two numbers such that the first three terms of the series so formed may be in a. p. and the last three terms in g.p. 17. If X, y, a be in a.p., and x, y, b in g.p., shew that, whether the series proceed in ascending or descending order of magnitude (provided only that x a,ndy are real positive quantities) b must be greater than a. Find the values of x and y, if a = 21 and 6 = 25. 18. If a, b, c be in arithmetical progression, and A, G, be the arithmetic and geometric means between a and 6, and A', G', be the arithmetic and geometric means between b and c, prove that A-'-G-^ = A'-'-G'l 19. The sum of three numbers in g.p. is 14, and the sum of their reciprocals is | : find the numbers. HARMONIC AND OTHER SERIES. 379 20. The sum of 37i terms of a g.p., whose first term is a and common ratio is r, is equal to the sum of n terms of another G.p. series, whose first term is h and common ratio is i-^. Prove that h is equal to the sum of the first three terms of the first g.p. 21. Shew that, if rt, 6, e be three numbers in g.p., and if p and j(/ be respectively the a.m. and h.m. between a and 6, and a and q' the a.m. and h.m. between h and c, then p, p\ q, q' will be proportionals. 22. Three numbers are in h.p. ; if 4 be taken from each, they are in g.p. ; if 1 be added to the middle one, they are in A.P.: find them. ■^23. An arithmetical, a geometrical, and a harmonical pro- gression have the same first and second terms ; and the third terms of the three series are x, y, z respectively. Shew that e-^) ■^ryz + zx-xyhjx-\-y-z, and divide {pq-\-rsf-{ps->t-qrf by {p-r){q-s). 2. Keduce to its lowest terms ;:— ; — - — ^ . 8x^-Ux^-9 3. Simplify the expressions : 3^-2-5 4y-{-5 1 ^^^ ^-1 x^+x+l'^x^' .... f X x+1] ( X , x+l] 4. Solve the equations : ,., 3(6-5^?;) 63.1; 3.^ 36 ^^ 5 ^50 2 125 ' (ii) 4.27 + 5 .^2+ 6.^+10 (yy-= 5. Extract the square root of 1 a*;* - 24.^3 + 25.^2 - 1 2x + 4. 6. A man can walk from A to B and back in a certain time at the rate of 4 miles an hour. If he walk at the rate of 3 miles an hour from A to B, and at the rate of 5 miles an hour from B to .1, he requires 10 minutes longer for the double journey. What is the distance AB^, EXAMINATION PAPERS. 381 7. Prove that, when m and n are positive integers, a"* X a** = «'»' + ". 1 How is a meaning given to a'^ ? 8. Prove that, if a : 6 = c : o?, h {a + h - c - d) = {a-\-h) {h - d). What number must be added to each of the numbers, 3, 5, 7, 10 that the sums may be in proportion ? 9. Solve the quadratic equations : (i) 2.^2 -2L^^ + 40 = 0; 1 5 _? ^^^' Qx-ba'^Qx-a~ a' 10. Sum the series : (i) 1296 + 864 + 576 + ... to 7 terms, and to infinity; (ii) 1296 + 1080 + 864 + ... to 12 terms. Paper B. 1. Solve the equations : (i) {x-l){x-2){x-Z)^-(x-4.){x-b){x-Q) = {x-2){x-'^){x-^) + {x-Z){x-^){x-b); 3^+5 x + Z x + \ _ x+Z b {x + 2) ^"^ a^^^+l"^ 6 "'"^~~a;+l'^"'l2~' (iii) ^^{x-]ry)=x+\, l{ij-x) = 2x-l. 2. If the difference between the roots of the equation x'^ + {a-\-b)x-\-0^ = O is the same as that between the roots of x^-^cx+{a + b)c = 0^ prove that a + 6 is equal either to c or to - 5c. 3. Solve the equations : . .^ + 4 3x- + 10_2a; + 3 ^ ^^^ 2^ + 3 "^ ~~2ar " x-l ' (ii) \/]~^x + \/2 (1 + x) = VG - 2.r ; ^ ^ V+'^'y + '»— 205. 382 EXAMINATION PAPERS. 4. li p :q '.r :\ a + 2b-{-2c : b + 2c + 2a : c + 2a + 2b, prove that a : b : c :: 2q-\-2r-^p : 2r + 2p-3q : 2/> + 2q — 3r. 5. If X vary as the sum of two quantities, one of which varies directly as y and the other inversely as y'^, and if .r = 37 when 3/ = l, and jt' = ll when y = 2, find the value of .a? when 3/ = 3. 6. Find two numbers whose sum is to their difference as 9 : 2, and whose product exceeds the difference of their squares by 5. 7. A starts to walk from P to ^ at 10 a.m. ; 5 starts to walk from (^ to P at 10.24 a.m. They meet 6 miles from Q. B stops 1 hour at P, and A stops 2 hrs. 54 min. at Q ; retiu-ning they meet midway between P and Q at 6.54 p.m. Find the distance from P to Q. 8. The areas of two rectangles are as 9 : 10 ; the greater side of the less : the less side of the greater as 3:2; the diagonal of the less is equal to the greater side of the greater, and the difference of their diagonals is 2 feet. Find their sides. 9. Find the ?i"' term of a geometrical progression whose first two terms are a and b. If a and b be the first two terms of an arithmetical progression and also of a geometrical progression, and if the ratio of the third term of the former to the third term of the latter be 5 : 9, find the ratio of their sixth terms. 10. Sum the following series : (i) 25 + 20 + 15 + ... to 8 terms; (ii) 25 + 20+16 + ... to 8 terms; (iii) 1^-1+ 4-... to infinity. 11. Insert four arithmetical means between a and b. If the square of the arithmetical mean between two quantities be increased by the square of half their difference, the sum is the arithmetical mean between the squares of the two quantities. 12. The difference of the first and second terms of a geo- metrical progression is 8, and the sum of the second and third terms is 12. Find the series. EXAMINATION QUESTIONS. 383 Examination Questions. 1. Explain how iiieaiiings are assigned to a^ and a~^. Divide .1-4^"^ 2 + .?;'V by x^y~~^-\^-x'^y\ 2. Prove that, if a, 6, c, o? be proportionals, then a? h^ (^ d^ , , a ct c 3. Find the sum of n terms of an arithmetical progression, having given the first term and the common difference. Find the sums of the series : (i) 16 + 24 + 32 + ... to 7 terms; (ii) 16 + 24+36 + ... to 7 terms; (iii) 36 + 24 + 16 + ... to infinity. 4. Find the G. m. of a and h. The A.M. between two numbers is 1. Shew that their h.m. is equal to the square of their g. m. 5. The velocity of a train varies directly as the square root of the quantity of coal used per mile and inversely as the number of carriages in the train ; and the train is supposed to travel with uniform velocity. In a journey of 25 miles in half-an-hour with 18 carriages, 10 cwt. of coal are used. How much coal will be consumed in a journey of 15 miles in 20 minutes with 20 carriages? 6. Prove that, if the equations x^ + hx + ca = 0, x^ + ex ■\-ah = have a common root, their other roots will .satisfy the equation .1''^ + ax + bc = 0. 7. State the index laws. Explain what is meant by 3 2 a ^ and a^. 2 3 1 Simplify the expression (^^.y^) '^x~^y\ 384 EXAMINATION QUESTIONS. 8. If <2 : 6 = c : 0?, shew that 3a-2c : Zh-2d=2a-Zc : 26-3d If = — = ^ , then will each fraction be equal y+zcx—azay — ox to yjx unless 6+c=0. 9. Find the ?i* term of a geometrical progression whose first two terms are a and h. The S"* term of a geometrical progression is 64, and the 11"" term is 729. Find the 7"" term. 10. Two clerks were awarded pensions, the amount of which was proportional to the square root of the number of years they had served. One had served 9 years longer than the other, and received a pension greater by £50. If the length of service of the first had exceeded that of the second by 4| years only, their pensions would have been in the ratio 9 : 8. What were the amounts of their respective pensions ? 11. Define a continued proportion. If a : h = h : c = c : d, shew that ad^=c^. 12. If X vary directly as y and inversely as z^, and if x=l when 2/ = 2 and z = S, find the value of x when y = S and z = 2. 13. Find the sum of (i) 16-I-19 + 22 + ... to 20terms; (ii) 16 + 15 + 14 + ... to 30 terms; (iii) 16-12 + 9+.. .to infinity. 14. Divide 76 into three parts in g.p. such that the sum of the first and third is to the second in the ratio of 13 : 6. 15. A carrier charges 3d. each for all parcels not exceeding a certain weight ; and on heavier parcels he makes an additional charge for every pound above that weight. The charge for a parcel of 14 lbs. is Is., and the charge for a parcel of 12 lbs. is twice that for a parcel of 7 lbs. What is the scale of charges? EXAMINATION QUESTIONS. 385 1« ^imi.lifv N_^(f±^) x/(-^'-^)"l V(^-^') 1 7. Shew that f (^/3 + 1)"^ - 2 (^2 - 1)^ = ^(59 + 24 v/6). 18. Find the sum of an inj&nite number of terms in g.p., the first term and common ratio being given. Determine the limits between which the ratio must lie in order that the formula may be true. Find the sum of 1 - '1 + -01 - ... to six terms and to infinity. The first term of a geometrical progression exceeds the second term by 2, and the sum to infinity is 50 : find the series. 19. Find a\ y, and z where a, x, h are in a. p. ; a,y,h are in G. P. ; and a, z, h are in h. p. 20. If G be the g. m. of two quantities A and B, shew that the arithmetic and harmonic means of A and G and the arith- metic and harmonic means of G and B are in proportion. B. A. 25 CHAPTER XXX. PERMUTATIONS AND COMBINATIONS. 366. Permutations. Combinations. The dif- ferent groups which can be formed, each consisting of a certain definite number of things selected from a given collection of such things, are called permutations when the order in which the things are arranged in each group is taken into account ; and are called co7n- hinations when the order in which the things are arranged in each group is not taken into account. For example, suppose that we have three things, denoted by rt, h, c respectively, and we form all possible groups of them taken two at a time. Then there are six permutations, namely, the groups ab, he, ca^ ha, ch, ac; the groups ah and ha being regarded as different, since the order of arrangement is taken into account. On the other hand, there are only three combinations, namely, the groups ah, he, ca, since arrangements like ah and ha are the same combination. 367. Notation. The number of permutations of n things taken ?• at a time is commonly denoted either by the symbol „P,., or by the symbol '^P,., or by the symbol P (n, r). It follows from the definition that r cannot be greater than n. The number of combinations of n things taken r at a time is commonly denoted either by the symbol „(7^, or by the symbol "C,., or by the symbol C (n, r). It follows from the definition that r cannot be greater than n. PERMUTATIONS. 387 Using this notation, the results of the illustration given in the last article would be written ^P.^ = 6, and gC^ = 3. 368. It is obvious from the definitions that nP\ — n, and ,fii = )i, since ii things can be taken one at a time in n separate ways. 309. Number of permutations of n things taken r together. TJie number of pevDiutations of n different things taken r at a time is given by the formula nPr = n{n-\) (?i-r+l). Let us denote the things by the letters a, b, c, .... Suppose that we knew all the different permutations of n things taken ?- — 1 at a time ; the number of these is denoted by the symbol „P,._i. Then, by pre- fixing to any one of these permutations any one of the n — r + 1 letters which it does not contain, we obtain one of the ^Dermutations of n things taken r at a time. Repeating this process on each of the permutations in n-iPfj we obtain all the permutations in ^Pr- Now every permutation in nPr-\ gives rise to n — r-{-\ of the permutations in nPr, .'. nPr=(n-r-\-l)nPr-i- This relation is true for all values of r which are not greater than n. Therefore, writing for /• successively r—l,r — 2, , we have nPr-i = (n-r + 2)nPr-2, nPr-2 = {n-r-}-S)nPr-3, also nPi = n. [Art. 368. The product of all the right-hand members of these equalities must be equal to the product of all the left- hand members. Cancelling the factors common to both sides, we have ,,P,. = (,t-r + l)0i-r + 2) (71-1)/?. 25—2 388 PERMUTATIONS. Eeversing the order of the factors on the right-hand side, we have ,P, = 7i(n-l)(ti-2) Oi-r + 1). 370. If in the result of the last article we put r = n, we obtain the theorem that the number of jyermii- tations ofn tilings taken all together is n{n-l){n-2) 2.1. 371. Factorials. The product of the first n natural numbers (that is, of n consecutive integers beginning with unity and ending with n) is called factorial n. Factorial n is denoted either by the symbol \n, or by the symbol n ! In foreign books, it is sometimes denoted by the symbol F (n +1). 372. The factorial notation enables us to express the results of Articles 369, 370 in a more concise form. We have nPr — n(n — 1) . . . (n — r + 1). Multiply and divide the right-hand side by \ n-r. _ {7i{n-l).. .(n-r +l)]{(n-r)(7i-r-l)..,2A] •*• "^' ~~ (71 - r) (n - r - 1) ... 2 . 1 n — r Similarly, «P« = n (n - 1). . .2 . 1 = \n. 373. The product of r consecutive positive integers, of which n is the greatest, is denoted either by the symbol [?i],., or by (n),., or by ?i,.. Thus, the result of Art. 369 would be written either as nPr = [H-^ or as „P,. = (?i)r, or as nPr = '^r- 374. Note. The method of proof given in Art. 369 is sug- gested by the form of the answer. For, if the result be true for all values of r less than ?i, then we have ,,P, = n{n-l){n-2)...(n-r+2)i7i-r + l), and ,,Pr-i = n{7i-l){n-2)...{n-r+2). PERMUTATIONS. 389 Hence ,iPy = (?i-r+l),iP^_i, which is the relation we commenced by proving. We might, in a similar way, have proved the relation which would have given us a different form of the proof. Ex. 1. How many dif event numbers can he formed hy using 3 out of the 9 digits 1, 2, 3, 4, 5, 6, 7, 8 and 9? "We have 9 different things, and we are to take them 3 at a time, .*. the required number = 9 Pg =9.8.7 = 504. Ex. 2. Iloir many of the numbers formed as in Ex. 1 lie between 300 a7w?400? Each number contains three digits. Hence the digit in the hundreds' place must be a 3. There remain 8 different digits, and any pair of these can be put in the tens' and units' places. .*. the required number =§^2 — ^ .7 = 56. Ex. 3. Four men hire a four-oared rowing boat. In how many ways can they be arranged as a crew? They are all to row at the same time, .*. we want the number of pernuitations of 4 men taken all together. .-. the required result = 4P4= |4 = 4 . 3 . 2 . 1 = 24. 375. To find the number of permutations of n things taken all together, when the things are not all different. Let the n things be denoted by the letters a, h, — Suppose that p of them are alike, all being " a "s ; q of them are alike, all being " h "s ; r of them are alike, all being " c "s ; and so on. Let P be the required number of permutations. Each of the permutations contains all the letters, and therefore contains p " a "s. If, in any permutation, the ''a"s were replaced by J9 new letters (quite distinct both from one another and from the letters already 390 PERMUTATIONS. used), then by changing the arrangement of these new letters amongst themselves, and keeping the other letters unaltered in position, we should get \p different per- mutations. If this were done to each of the P per- mutations, we should get altogether P x |_p different permutations. Similarly, if in any one of these P x \p permuta- tions, all the " b "s were replaced by q new letters, then by permuting these new symbols amongst one another, and keeping the other letters unaltered in position, we should get \q different permutations. If this were done to each of the P x \p permutations, we should get altogether P x \p x \q ditferent permutations. Similarly, if we replaced all the letters of which any one was like any other by new symbols distinct both from one another and from all the others used, then we should get altogether P x ^ x '5' x |?* ... different permutations. But the case is now reduced to finding the number of permutations of ?i things, which are all different, taken all together ; and the number of these is n. [Art. 370.] n. .'.P\p\q\r.. In ' p = U ^ \p\q_\r.., Ex. 1. Find the mimher of permutations of the letters in the word ALGEBRA taken all together. Here there are 7 letters, of which 2 are alike ; that is, n = 7, p = 2, in the above formula. 17 .'. the required number =y^ = 2520. PERMUTA TIONS. 39 1 Ex. 2. Find the number of permutations of the letters in the u-ord COMBINATION taken all together. Here there are 11 letters, 2 of them are "o"s, 2 of them are "i"s, and 2 of them are "74 "s. Ill .-. the required number =-J=-^ = 4989600. EXAMPLES. XXX. A. 1. Write down the values of r,Po', 7P5; 4P4; oA- 2. If nP^^WO, what is the numerical value of n1 3. In how many different ways can the letters of the word v:oman be arranged (i) taken all together, and (ii) bxken three at a time? 4. How many different arrangements can be made of the letters of the following words, in each case all the letters of the word being used in every arrangement? (i) school; (ii) number; (iii) feeler; (iv) cricket. 5. Find the number of permutations of the letters in the words fiddle-de-dee and Mississippi, in each case all the letters being used in every permutation. 6. If the number of permutations of 12 things taken r together be 42 times the number taken r—2 together, find r. 7. In how many ways can six boys stand in a line to receive an electric shock, two only being willing to stand at the extremities of the line? 8. With three consonants and three vowels, how many words of six letters can be formed, each word beginning with a consonant and ending with a vowel ? 9. If the number of permutations which contain a par- ticular thing be equal to the numl)er which do not contain it, prove that n must be even, and r must be equal to ^n. 392 COMBINATIONS. 376. Number of Combinations of n things taken r together. The numher of combinations of n different things taken r at a time is given by the formula P _ ^ 0^ - 1) (^^ - 2). . .(?i - r + 1) First Proof Each of the combinations of n things taken r at a time contains r different things : and by arranging these r things in every possible order, it would give rise to r permutations. Hence the number of permutations of n things taken r at a time is n^r x b'- n^r n (n - 1). . .(71 - r +1). [Art. 369. n(n— l)...(n — r +1) Second Proof The above proof reduces the deter- mination of nCr to that of nPy We now proceed to give an independent proof Let us denote the things by the letters a, b, c,.... Then those combinations of them (taken r at a time) in which a occurs can be obtained by first writing down all possible combinations of the ?i - 1 letters b, c,... taken r — 1 at a time, and then prefixing a to them. Therefore the number of combinations in which a occurs will be n~iCr-i. Similarly, the number of combinations in which b occurs will be n-fir-i ' and so also, the number of com- binations in which each of the other letters occurs will be n-iCr-i' Now, if we collect these n combinations of n — 1 letters taken r — 1 at a time together, the number of them will be n x n-i^^r-i ; and in this collection we shall get all possible combinations of ?i things taken r together. But in this collection, every combination will be COMBINATIONS. 393 repeated r times; because, if the combination happen to contain the r letters hkl..., it will occur once with h in the first place, once with k in the first place, and so on; and since the order of arrangement in a combination is immaterial, these will all be the same combination. Hence the number of arrangements in this collection is also ?' X nC)-. . , ?i X ,^_iO^._i ^ T X n^v . . n^.j. = — X n—i^r—i' This relation is true for all values of n and r (pro- vided of course that n and 7^ are positive integers, and that r is not greater than n). Therefore, writing n — 1 for n, and ?' — 1 for r, we have n - "^^ X r n — 2 Similarly, n-oCr-2 = ^73^ ^ *^-3^'-3- Finally, n-r+oC. = ^ — |i- X n-r+iC\ . Also, n-r+iCi =n-r-\-l. [Art. 368. The product of all the right-hand members of these equalities must be equal to the product of all the left- hand members. Cancelling the factors common to both sides, we have n n—1 n — 2 ?? — r + 2 n — r-{-l tiGr — r>-l>-2" 2 n (n — 1) ... (n — r + 1) r If we multiply both the numerator and the denomi nator by \n — r, we have 394 COMBINATIONS. n (n - 1) ... (n - ?' + 1) (n-r)(n- r -1) ...2 .1 n^r — r \n — r \n that is, ,,C. = -, — r= — , which is sometimes a more con- \r\n — r venient form for JJy than that given in the enunciation of the proposition. Ex. 1. Find the number of diferejit whist parties which can be made out of six players. We want the number of possible sets of 4 people which can be made out of 6 people. .'. the required number = g(74 =: ' 9 ' q '/i ~^^- Ex. 2. El ho\o many loays may a cricket eleven^ of whom tioo at least must be boiolers, be formed from 14 players, it being known that only 4 of the players can bowl? There are two groups of players, namely, 4 bowlers and 10 other players. The eleven may contain only 2 bowlers, and 9 others selected out of the remaining 10 players. Now from the 4 bowlers we can select 2 in 462 ways, that is, in 6 ways ; and from the 10 non-bowlers we can select 9 in jqCq ways, that is, in 10 ways. Any of these 6 pairs of bowlers can be taken with any of the 10 possible sets of 9 non- bowlers which can be selected out of the group of non-bowlers. Thus, altogether there are 4^2 ^ 10^'9> that is, 6 X 10 possible elevens, each of which would contain two bowlers only. Similarly, the number of ways of forming an eleven which would contain 3 bowlers and 8 players selected out of the group of non -bowlers is 4^3 ^ 10^85 since the number of sets of 3 bowlers which can be formed out of the group of 4 bowlers is 4C3, and the number of sets of 8 non- bowlers which can be formed out of the group of 10 non-bowlers is iQ^g, and any of these sets of 3 bowlers can be taken with any of these sets of 8 non-bowlers to make an eleven. Lastly, the number of ways of forming an eleven which would contain 4 bowlers and 7 non-bowlers will be 4^4 ^ 10^ 7' COMBINATIONS. 395 Hence, the total number of possible elevens is 4^2 ^ 10^9 + 4^3 ^ 10^8 + 4^4 ^ 10^7' , ^ . 4 . 3 4 10 . 9 , 10 . 9 . 8 *^^^'^^' 1.2'1' 1T2+^- 1.XT' which is equal to 6. 10 + 4.45 + 120 = 60+180 + 120 = 360. Thus, there are 360 possible elevens. EXAMPLES. XXX. B. 1. Find the values of ^Cq\ ^C^\ 47^45- 2. If ,,(72-21, find 71. 3. If ,A- 2 = 10, find n. 4. If ioC';, = 45, find.r. 5. How many parties of 10 men can be formed from a com- pany of 20 men ? 6. How many parties of 10 men can be formed from a com- pany of 20 men (i) so as always to include a particular man, (ii) so as always to exclude a particular man ? 7. A boat's crew of 4 men has to be selected from 6 men, of whom two can only row on the stroke-side and two can only row on the bow-side. How many crews can be selected, no account being taken of the way in which the crew arrange them- selves ? 377. The number of combinations of n things taken r* together is equal to tlie number of combinations of n things taken n — r together. For we have In In I ?? nC,. =rV= — > and nCa-r = r \7i — r n — r n — (n — r) n — r \r • • n^r — n^n- The result of this proposition is otherwise obvious ; since for every combination of r things, which is formed out of the n given things, there must be left a collection of the remaining 7i — r things, which constitutes a combination of the 71 things taken n-r tosrether. 396 COMBINATIONS. . 378. To shew that ^^V + wC'y-i = n+iOr- By Art. 376, we have nC,. + nO,—i _n (n — I) . . .{n-r+2)(n-7^-\-l) 7i (n — 1) . . . (n-r -\- 2) ~ 1.2...(r-l)r "^ 1.2...(r-ir )i{n—l)...(n — r-\-2)^n — r+l 1.2... (r-1) ( r _n (n - 1) . . . (n - r + 2) n + 1 ~ 1.2. ..(r-1) * r _ (n + l)n(n-l)...(n-r + 2) This result, like that of the last article, can be obtained by considering the meaning of the symbols. For suppose the n-\-l things to be denoted by the letters a and n other letters b, c Then the number of combinations of these n + l things taken r at a time will be equal to the number of combinations of the n things b, c, . . . taken r together phis the number of combinations of these 7i things taken r-1 together when a is prefixed to each of them ; that is, „ + iC,. = ,i^r + nPr - 1 • *379. The proposition given in the last article can be extended by repeated applications of it, thus : n + 2^r = n + l^J- "r n + 1^ r - 1 = {n^r + n^r - 1) + [n^r - 1 + n^'r - 2} = n^r + 2^6,. _ 1 + „6,. _ 2 . Similarly, „ + gC,. = „(7^ + 3„(7,. _ ^ + 3„C;. _ 2 + nOr - 3 • Proceeding in this way, we can prove the following theorem, by mathematical induction, The (Z- + iy^ term is — ^ ^—^ n^r-k ', and all the terms (after the first) can be deduced from the (Z- + l)*^ terra by putting k successively equal to 1, 2,...m. COM BIN A TIONS. 397 This result can also be written in the form since, by Art. 376, wi = mCi, — ^ — 5— = H^Cg, «&;c. It will be noticed that ?>i, n, r are positive integers, and r must be less than m and less than n. 380. To yi?i(i what value of r makes nCr greatest, n being a given number. We have C - »("- 1) ■■■(«->■ + 2)0/i - r + 1) we have ,.0,- l.2...(r-l)r ' • _ »(n-l). ..( » -»■ + 2) -""•^ "^'- - r:27:7(,,^^i) — • _ M-r+l Therefore we can get n^r from n6V_i by multiplying the latter by the fraction ^ . Hence, if this frac- tion be greater than unity, ^Or will be greater than n^r-i- p -^ n so long as n—v-\-\>r, that is, so long as y^ + 1 > 2?', that is, so long as r<\ (n + 1 ). If n be even, ^CV will be greatest when /• = in. If /i be odd, then ^ (n + 1) is an integer, and if ?' = J (n + 1) the fraction ^ is equal to unity, and therefore this value of r makes ,iCV = ,fir-i '■> hence, if n be odd then when ?■ = -J- (n + 1) or r = ^ (n + 1) - 1 = J (n - 1), the number of combinations of n things taken r at a time is greater than for any other value of r. 398 PERMUTATIONS AND COMBINATIONS. 381. The product of r consecutive positive integers is divisible by I r. The number of combinations of n things taken r together is necessarily an integer. Hence, we have the theorem that n{n-l){n-2) ...{n-r+l) \r is an integer, that is, the product of r consecutive positive integers is always divisible by jr. Another proof will be given later [Art. 437, p. 455, Ex. 5,]. MISCELLANEOUS EXAMPLES. XXX. 0. 1. How many of the permutations of the six letters a, 6, c, d, e!,/ taken four at a time contain a, and how many begin with a1 2. How many of the permutations of the six letters a, b, c, dj e, f taken four at a time contain (i) the letters ab in that order, (ii) the letters a and b next to one another ? 3. Prove that the number of permutations of 2?i things taken ?i together is 2" . 1 . 3 . 5. ..(2/1- 1). 4. Out of 5 consonants and 3 vowels, how many words can be formed, each consisting of 3 consonants and 2 vowels ? 5. Out of 30 oranges at a halfpenny each, how many selec- tions can be made in buying a shilling's worth ? 6. Two landing parties of 50 men each are to be formed. The only men available are four English officers, fom' English non-commissioned officers, and 100 native troops. Each party is to contain two officers and two non-commissioned officers. How many different possible parties can be made up ? 7. How many diflerent numbers can be formed out of the digits 1, 2, 2, 3, 3, 3, 4, all the digits being used in each number? 8. In how many ways can 15 be thrown with four dice ? 9. How many words, each of seven letters, can be formed from three vowels and four consonants, such that no two con- sonants are next to one another ? 10. If the number of permutations of n things 3 together be equal to 6 times the number of combinations of them 4 together, hnd the value of u. PERMUTATIONS AND COMBINATIONS. 399 11. If „_i(7, : „C, : n + iCr = 6 : 9 : 13, find n and r. 12. What value should (on the usual conventions as to the meanings of algebraic symbols) be assigned to JJ^ 1 *13. Shew that „Cp = 2 {j^C',. x n-nfip- X 14. There are n railway signal posts at a junction, and the arm of each signal can be moved into three positions : how many different signals can be made ? 15. In a railway carriage, holding 5 people on each side, there are 10 people, 4 of whom refuse to travel with their backs to the engine, and 3 of whom insist on doing so. In how many ways can they be arranged ? 16. If the ratio of the number of combinations of 2;i things taken /«.- 1 at a time to the number of combinations of 2 {n- 1) things taken w at a time be 132 : 35, tind n. 17. If the number of combinations of n things taken /• to- gether be equal to the number of combinations of n things taken 2r together, and if the number of combinations of n things taken /•+1 together be ecpial to V- times the number of combinations of n things taken r - 1 together, tind n and r. 18. In how many ways may a man vote at an election where every voter gives six votes which he may distribute as he pleases among three candidates ? *19. In how many ways can eight counters be arranged in four groups, each group containing two counters ? *20. In the list of those who passed a certain examination, 21 candidates were placed in the first class, 27 in the second, and 17 in the third. In how many ways could a staff" of 6 masters be selected containing 2 from each class I *21. How many trios can be formed by taking 1 girl and 2 boys from n girls and 2?i boys ; and in how many ways can they be seated at n tables so that no two boys sit next each other ? *22. In how many ways can two bishops be placed upon a chess-board (i) upon squares of the same colour ; (ii) upon squares of different colours ? *23. Find the sum of all the integral numbers consisting of 5 figm-es, which can be formed by the digits 1, 2, 3, 4, 5, 6, 7, 8, 9, no digit being used more than once in any number. *2^. A boat-club consists of 15 members. Find in how many ways a crew of 9 can be chosen, (i) so as always to include a particular man, (ii) so as always to exclude him. 400 PERMUTATIONS AND COMBINATIONS. *25. There are 20 stations on a railway, find the number of kinds of tickets required so that a person may travel from any one station to any other. *26. Find how many diflerent sums can be made with the following coins : a penny, a sixpence, a shilling, a half-crown, a crown, and a sovereign. *27. In a college of 100 men there are 4 coxswains ; shew that 14(96 there are 7^777^-^4 ways in which four crews, each consisting of a coxswain and 8 men, may be chosen from the members of the college. *28. How many changes can be rung with five out of eight bells, and how many with the whole peal ? *29. In how many ways can a whist table be made up out of six married couples and four single persons, so as not to include any husband and wife together ? *30. In how many ways can 5 people be arranged at a round table ? *31. A party of 4 ladies and 4 gentlemen contains two mar- ried couples. How many difierent arrangements of the party can be made at a round table, every lady sitting between two gentlemen, neither of whom is her husband ? *32. In how many ways can a pack of 52 cards be divided into 4 sets, each containing 1 3 cards ? *33. With 1 red, 1 white, 1 blue, and 3 black balls, in how many ways can 4 of them be arranged in a row ] *34. I have four black balls (exactly alike), and also one red, one white, one green, and one blue ball. In how many ways can I make up a row of four balls, no two rows being alike ? *35. A polygon is formed by joining n points in a plane. Find the number of straight lines, not sides of the polygon, which can be drawn joining any two angular points. *36. On each of n given straight lines m points are taken. No other straight line can be drawn through any three of these mn points. How many triangles can be formed by joining the points 1 *37. Find the number of combinations of the letters in the word annunciation taken 4 at a time. CHAPTER XXXI. THE BINOMIAL THEOREM. 382. We have found by actual multiplication (see pp. 38, 42), expressions for the square aud the cube of a binomial, such as a-\-h. The object of the Binomial Theorem is to find an expression for the n^^' power of a binomial. We shall first confine ourselves to the case where n is a positive integer, and shall then discuss the extension to cases where n is either a positive fraction or any negative quantity. THE BINOMIAL THEOREM FOR A POSITIVE INTEGRAL EXPONENT. 383. To sheiv that, if n he a positive integer, then (a + by = a'' + na''-'b + ^^3^-^- ^''~'^' + • • • ^ n(n-l)...(n-r + l) ^,_,^, ^ _^ ^^^^^„_, ^ ^,, 1 . 2 ...r It is evident that this result includes such results as we have already proved. If, for examj)le, we put 7i = 2, we obtain (a + bf = a^ + 2ab + b^. If we put n = 3, we obtain {a + bY = a^ + 3aVj + 3ah'^ + P. B. A. 26 402 THE BINOMIAL THEOREM. The term ^(^- l)'--(^-^+l) a^-rh-'' is the (r + 1)'*" term. It 1 . 2...r is called the general term, because any term may be obtained from it by giving r the proper value — the second term corre- sponding to r = 1, the third term to r= 2, the fourth term to r = 3, and so on. The result that we want to prove may be written in the form (a -f- 6)" = a" + „(7i . a'^-iJ + «(72 . a"-262 + . . . + „C;. . a'^-^6'- + . . . + 6", which is more convenient for our purpose. 384. First Proof {Positive Integral Exponent). Every term in the product of 7i factors, each equal to a + 6, must contain one letter taken from every factor, hence it is of n dimensions. The terms of the pro- duct consist therefore of certain multiples of a'\ a^~'^h, ..., 6^ The term involving a'^ arises from the product of an " a " taken from each factor. Hence, a*^ is a term in the product. The terms involving a'^~% arise from the product of a " 6 " taken from one factor and n — 1 "a"s, one being taken from each of the remaining n — 1 factors. But the number of ways in which a *'6" can be selected from n factors is the number of combinations of n things taken one at a time, that is, 7i. Therefore there will be altogether n terms in the product, each equal to a^~%. Hence, 7ia^~^h will be a term in the product. A similar argument applies to every term in the product. The terms involving a^~'^Jf arise from the product of r " 6"s (one being taken from each of r of the factors) and of n — r"a "s (one being taken from each of the remaining factors). But the number of ways in which r "6"s can be selected from n factors is nCr- Hence, ^0^ . a*^-*'6^ is a term in the product. Hence, the expansion of (a + by is (a + by = a*^ -f nC\ . a''-'b + • . . + nC^ . a^~**^^ -f . . . + 6". POSITIVE INTEGRAL EXPONENT. 403 385. Second Proof {Positive Integral Exponent). The proof of the binomial theorem given above is the one which would naturally occur to any one investi- gating the subject for the first time. But if the result be known or guessed, then the following proof, by mathematical induction, is shorter and easier. Let us assume that the theorem is true for n factors, that is, {a + hy = a^ + nOi . ««"16 + . . . + nOr. «"-''^>'" + ••. + n^l • f^?>""l + 6^ . .(i). Multiply both sides by a + 6, = a'* + 1 + „(7i . a"6 + „(72 . a^'-^b'^ -f . . . + „(7, . a" -'• + 16'- + . . . + «Z>« + a»6 + „(7i . a''-i62 + ^C^ . a'^-^ + . . . + „C,. . a^-'V + 1 + . . . + 6» + 1 + {nCr + nGr-i)a^-^^'b^'+...+{l+nOi)ab^'+b^^\ But nOr + nGr-i = n + i(^r [Art. 378]; heuce recollectiug that 1 =„Co, we have (« + 6)" + i = a" + i+,i + i(7i . rt"6 + ,i + i(72. a^^-^^H... -f^ + i(7,..a" + i-»-6'-+...+„ + i(7i.rt6'* + 6'* + i (ii). Now the formula (ii) is exactly the same as the formula (i), except that n + l is written for n wherever it appears in (i). Hence, if the formula (i) be true for the exponent w, then (ii) shews that it is also true when the exponent is 7i + l. Now (i) is obviously true when h = 1, therefore it must be true when n = 2. But since it is true when ?i = 2, therefore it must be true when ?i = 3. Continuing this line of reasoning, we see that it is true for any positive integral value of n. 386. If, in the formula given in Art. 383, we put a = 1, 6 = a?, we have n ( n 1 1 (l+d?)" = 1 + nx + ^.^ ^ ^ X' + ... n(n — 1) ... (n— r + l) 1 . 2 .... r 26—2 404 THE BINOMIAL THEOREM. This result includes the expansion of (rt + 6)'*, for («-|-6)"=--a" jl + -r [ a I .2 a^ J Since the expansion of (1+^)" given above includes the ex- pansion of (« + 6)", it is often taken as the standard form. If we change the sign of x in the expansion of (1 + .Ty\ we have (1 _ a;)n = 1 _ n^ + n (71-^1) ^, _^ _ ^ _^ ( _ ^^y^ the last term being + x^^ or — a''\ according as n is even or odd. Ex. 1. Write dovm the expayision of {a-\-hy'. By the formula in Art. 383, we have = a5 + 5«45 _^ 10^362 -f 10^253 + 5rt64 + js, Ex. 2. TFnVe c^02^9i ^Ae expansion of {Zx — 2y)\ In the general formula, Art. 383, put 3.^ for a and - 2y for 6, .-. (3.r-23/)^ = (3^)4 + 4 (3.,)3(-2y) + i^ {^^f{-yf = 8U'^ - 216.'i;3^ + 2\QxY - 96.^3 J^lQyi, Ex. 3. i^mo? ^Ae coefficient of x^ in the expansioii of (1 - 2.r)^. The term involving x^' in the expansion of (1 4-^'^)" is n{n-\){n-2)...{n-r-\-\) ,. .-. the term involving x"^ in the expansion of (1 - 'ixf is 9.8. 7. ..3, . ,_ POSITIVE INTEGRAL EXPONENT. 405 9 8 7 3 Hence, the coefficient of x^ = , ' ^-^^ ( - 2)' 1.2. 3. ..7^ 1.2 = - 4608. Ex. 4. Find the 12"' term in the expansion of {x-^yY^. The (r+l)"" term in the expansion of (a + 6)" i« n{n-l)...{n-r + l) ^„_^.^,. |r .•. the required term = —^ — ^ x^ ( - ^yY^ 1 . ^...11 387. It is usual to add the following additional propositions connected with the expansion of a binomial when raised to a positive integral power. 388. To find the greatest coefficient in the ex- pansion o/ (1 -j- xy^. The coefficient of the {r + If^ term is ,^0,.. If n be even, the greatest value of this is when r = \n [Art. 380]. Hence, in this case, the coefficient of the {\n + \y^ term is greater than that of any others. If n be odd, the values of „(7^ and nGr-\ ai"e equal when r = \{n+\), and these values are greater than for any other value of r. Hence, in this case, the co- efficients of the i{n-\- If^ and ^ {n + 3)"' terms are equal, and either is greater than the coefficient of any other term. 389. To find the greatest term in the expansion of {\+xy\ The (r+iy^ term is nOr-o)''. The r^^ term is nCr-i . x'^~'^. Therefore we can obtain the (r + 1)*^ term from the z*^'' term by multiplying the latter by the 406 THE BINOMIAL THEOREM. fraction X = X = l]x. This multiplier decreases as r increases, but it is greater /ji y -f- 1 than unity so long as ^ > 1, that is, so long as {n — r-{-l)x> r, that is, so long as (?i + 1) ^ > r + rx, ^■u 4. • 1 {n-Vl)x that IS, so long as r < ^ ^ . If this be not an integer, and if we take r equal to the integer next below it; then the (r + l)*^ term is the greatest. But if ^^-^j — be an integer, then for this value of r the fraction above written is equal to unity, and the ?^'^^ and (r + 1)"' terms are equal, and each of them is greater than any other term. 390. The expansion of (1 + a;)" is sometimes writ- ten as (1 + Xj' = Co + CiX + CoA-^ + . . . + CX + . . . + CnOC'\ and we shall use this notation in enunciating the fol- lowing propositions. Arts. 391—395. 391. The coefficients of the ter^nis in the expansion of (1 + xj^ luhich are equidistant from the two ends are equal. This proposition follows from the fact that Co = Cn ^^ J- J Ci = Cn—\ = "n, \n and Cr = \ — r= — = Cn-r- [Art. 3 / 7. I r n —r ^ POSITIVE INTEGRAL EXPONENT. 407 Note. This is equally true of the expansion of (a + ft)", pro- vided that the numerical coefficients of a and h are equal ; but if a and h have unequal numerical coefficients, it may not l)e true. Thus, in the expansion of {l-lxf^ the coefficients of powers of x which are equidistant from the two ends are not equal. 392. The sum of the coefficients of all the terms in the expansion of (1 + a?)"- is 2^. For (1 -\- xy = Co + Ci*' + . . . + CrX'' + . . . -t- CnOO'K Puta;=l, .-. 2" = Co + Ci +...+C,. +... + Ch. 393. The sum of the coefficients of the odd terms in the expansion of (1 + xj^ is equal to the sum of the coefficients of the even terms. For (1 + xy = Co + Ci« + . . . + CrX>' + . . . + c>,x'\ Put ^ = - 1 .-. (1 - 1)" = Co - Ci + C, - Cg + . . . . .-. = (Co + Co + C4 + . . .) - (Ci 4- Cg + . . . ). 394. The coefficients of x in the expansion of {l+x)"- are the various combinations of n things taken different numbers together. Hence, numerous theorems connected with the com- binations of things can be proved by the binomial theorem, and such proofs are often more simple than those suggested by the last chapter. Thus, the theorems just proved may all be read as theorems in combinations. For example, the result of Art. 392 is equivalent to a statement that the sum of all possible combinations of 7i things taken any number at a time is 2" - 1 , for the required sum = c^ + c, -|- . . . -|- c,„ and Cq = 1 . [See p. 440, Ex. 7.] 395. Numerous theorems concerning the binomial coefficients can be obtained, either directly by making use of their known values, or in some cases by using the binomial expansion. Example. FiTid the value of c^ -f q^ + Cg^ -f . . . -I- c^. We have ( l-f ^)" = Cq -{- c^x + c^'^- -I- . . . + c„.^''*, x'^ Multiplying the two series together, we see that the quantity whose value we want consists of the constant term in the product, 408 THE BINOMIAL THEOREM. .'. CQ^->rc^ + c^+ ...+ c„- = constant term in ( 1 + .^•)" ( 1 + - = coefficient of x"" in (1+^)" x (1 +0--)" |2n 896. Multinomial Theorem. We can expand the ?i"' power of a multinomial by repeated applications of the binomial theorem. The process will be sufficiently illustrated by considering the expansion of the n^'^ power of a trinomial, such as a + h + c. {a + h + cY=={a-\-{h + c)Y \n = a»-j-*ia'^-i(6 + c) + ... + --J= — a»-/-(6 + cy + ... + (6 + c)». \r \n — r ^ Now every power of the binomial {h + c) which occurs on the right-hand side can be expanded by the binomial theorem. If, for example, we want the coefficient of a"-~^'¥'Pc^\ we see that it must arise from the term in the original expansion which involves a^~^. The coefficient of b^'-i'cP in the expansion of {b + cY is Ir Hence the coefficient of a"'~''b^'-Pcf> is \P\^-P' \r hi - ■, that is. \p\r-p In-rlr-p'p' Hence the coefficient of a^'b^&' (where p + q-\-r = n) is \p_\q_\7^' 397. The expansion of an expression like {a + b + c + d+,,.y^ can be formed in a similar manner; but the process is tedious, and it is preferable (if possible) to express the multinomial a + b + c + d+ ... in some simple form, which frequently enables us to reduce the expansion to that of a binomial. EXAMPLES. XXXI. A. Write down the expansion of the following binomials. 1. (l + ^>^ 2. {.v-2)-. 3. {x-yf. 4. (2a + 6)11. 5. C^x-yf. 6. (2a + ^6)^. 7. (i^•+;ly)«. 8. {l-\a'y. 9. (.^-^>'. POSITIVE INTEGRAL EXPONENT. 409 10. Find the coefficient of jc in the expansion of {.v^ + — I . 11. Write down the vahie of {x-\-\f + {x-\ f. 12. Write down the 6th term in the expansion of {x — x^Y''. . 13. Find the coefficient of o^ in the expansion of (^ + 2)^. 14. Find the coefficient of y^ in the expansion of (2^ - 3?/)''. 15. Find the coefficient of a^h^ in the expansion of {ha — fhy^. 16. Write out the expansion of {'^.a — x)'' in full. 17. Find the middle term in the expansion of (1 +.r)^'\ 18. Find the 7"> term of {a - ^xf^. 19. What is the coefficient of x"^ in (3.v - ^f 'I 20. Find the term involving x^^ in the expansion of {ax - 3*-^)^. Find the greatest coefficients in the expansions of 21. {l+x)\ 22. {l-xfK 23. {x + afK 24. (l-|-2*')^. 25. Find the greatest terms in the expansions of (i) (1+^)^ when .*-10; (ii) {x + \y^, when ^'=4; (iii) (.r-f2)i', when x = Z. 26. Find the greatest coefficient in the expansion of {\-^xY^{\-x). 27. Employ the binomial theorem to find the values of (101)'', and of (99)5. 28. The third, fourth, and fifth terms of a binomial series are 20412, 22680, 15120 ; find the expansion. *29. If n be a positive integer, find the value of 1 . zi 3?i(3/i-3)(37i-6) ,,, ,„ , 172:3 — '* ^^ + -^"^ ^ + • • •• 30. Prove that (1 -xY = {\ JrxY - 2nx (1 +.r)-3 + M?|:Z^^,2 (i +^,.)n-.2 _ ..,. ! 31. Write down the first four terms in the expansion of (1 +2./;-|-.i-2)" in powers of x. 32. Find the expansion of (1 +.v- x^Y in powers of ;v. 33. Find the coefficient of a^h'^c"- in {a + h + cf. 34. Find the middle term of the expansion of (1 - ±v+x^y\ 410 TEE BINOMIAL THEOREM. 35. What is the sum of all the coefficients in the expansion of (2a-- 1)"? 36. Find the sum of the numerical coefficients of (3a - 2^)^. 37. If ( 1 + xY = Co + c-^x + c^'^ + . . . + c^x'\ find the value of (i) Co-Ci + C2-... + (-l)X; (ii) CoC^ + CiC„_i + ...+c„Co. 38. If n be a positive integer, prove that the middle coefficient of (l+^)2« is equal to the sum of the squares of the coefficients of (1 +^)". *39. If Cq, e^, Cg,... be the coefficients of the powers of x in the expansion of (l+^)»* ; and if (7o, C^, Cav be the coefficients of the powers of x in the expansion of (1 -^-xY'^ ; prove that *40. Prove that the sum of the squares of the first, third, &c., coefficients in the expansion of {a + 6)2" differs from the sum of the squares of the second, fourth, &c., by — -^ — ; — J" . *41. Shew that, if ^j;. be the coefficient of ^ in the expansion of (1+A')'S then *42. Employ the identity (1 + 2^ + x^Y = (1 +xY'', to prove that on , ^ (^ - 1) 0.-2 , ^ (^^ - 1) (^ - 2) in - 3) ,_, _ \^ n being a positive integer. THE BINOMIAL THEOREM FOR FRACTIONAL AND NEGATIVE EXPONENTS. 398. We have already shewn in Chapter XXII. that a definite meaning can be assigned to (l+^r)" when n is fractional or negative. We now proceed to discuss the possibility of finding an expansion for such an expression in positive powers of x, and we shall find that, if xhe numerically less than unity, then /I . \n 1 . . ^^0^ — 1) o n(n—l)..Xn—7^+l) 1.2 L . z ...r FRACTIONAL AND NEGATIVE EXPONENTS. 411 Note. The coefficients of x in the expansion of (l^-^)**, namely, the quantities n{n-l) ?i(?i- l)(?i-2) ^' ''' 1.2 ' 1T2T3 ' - are known as the binomial theorem. 399. The formula has been shewn [Art. 386] to be true if n be a positive integer. That, under certain limitations, it is true, whatever be the value of 11^ is rendered probable by the fact that when we test the result in a few simple cases we find that under certain conditions it is true. If we put 71= - 1, the right-hand side reduces to 1 4-^+^^ + ... while the left-hand side becomes (1 — ^)~i, which is equivalent to . But if we divide 1 by 1 - x. we find that the successive \-x J •> terms in the quotient are \-\-x-'rx'^-\- ..., which agrees with (i). This illustration is particularly valuable because it leads to a series which we have already discussed at length [Art. 339] ; and we have shewn that the series can only be represented by the expression , when x is numerically less than unity. Thus, the expansion of {\ — x)~^ in a series in positive integral powers of X by means of the formula (i) is only permissible provided X is numerically less than unity. 400. In a similar manner, if we put ?i = ^, the left-hand side of (i) is equivalent to \J\ -\-x. If now we proceed to extract the square root of \-\-x by the process given in Art. 196, we shall find (however far we go) that the successive terms in the square root are exactly the terms of the right-hand side of (i). But, if X be numerically greater than unity, the remainder (which represents the difierence between the root so far as we have ex- tracted it and the given expression \/l+x becomes larger and larger the further we proceed, and thus, the further we go, the less is the series on the right-hand side of (i) an accurate repre- sentation of the expression on the left-hand side. On the other hand, if x be numerically less than unity, the remainder becomes smaller and smaller, and the further we go, the more accurately does the series on the right-hand side of (i) represent the ex- pression on the left-hand side. 412 THE BINOMIAL THEOREM. It can be proved that, if n be not a positive integer, then a limitation similar to that given in this and the last article is always necessary ; and if n be not a positive integer, then the formula (i) is only true when x is numerically less than unity. *401. A rigorous proof of the binomial theorem for any value of the exponent cannot be given without introducing considerations beyond the limits of this book, but the following sketch of one method of estab- lishing it, which was suggested by Euler, is worthy of careful study. Let/(7?) stand for the series on the right-hand side of(i), .•./(n) = l-F7za;+ \^^ ^ x'^ + Then if n be a positive integer, we know [Art. 386] that f{n) = (1 -h xy'\ also, we have/(0) = 1- Writing m for n, we have f{m) = \+mx-^-A^—^x'^- Similarly, f{m -f 70 = 1 ^ {m -h n) x -f '^ '-^-^ ^ x''+.... Now Vandermonde proved that the coefficient of x^ in the product of f(m) and f(n) is equal to the coefficient of x^ in f{m 4- n) ; and Cauchy proved that, in that case, the equation f(m)xf{n)=f(m + n) is true for all values of m and n, provided the series represented by these symbols are convergent. Lastly, Cauchy proved that these series are convergent, pro- vided X is numerically less than unity. We shall not here jjrove these results, but the student will see that if they be granted it follows from them that, if x be less than unity, f(m)xf(n)=f(m+n). FBACTIONAL AND NEGATIVE EXPONENTS. 413 [Eulcr arrived at this result in another way, which is an ilhistration of the principle of the permanence of equi cedent forms ^ and is an extension of the method used in Arts. 40, 55 to obtain meanings to be assigned to the sum, difference, and product of negative quantities. First, he shewed that the equation /(«i) xf{n)=f{m-\-n) was true when m and n are any positive integers, since then it was equivalent to the equation (1 ^-.^^)'" x (1 -f A'f = (l +^^)'» + ^ Next, he asserted that the product of the series /(m) and the series f{7i) must be of the same form, whatever be the values of m and n. (This however is only true of series containing an infinite number of terms if they be convergent ; that is, in this case, if .r)"^" ^ ^^ 1.2.3...M 2»^ ^^ * _f j^,„-, 1.3...(2»t-3) /^Y' ^ ^ 1.2.3...n V2/ (0 1.3...(2??-3) /^^Y 1.2.3...W 1.3...(27i-3) 2.4.6...2?i ■' ■ 416 THE BINOMIAL THEOREM. To justify the use of the factor (2?i-3) in the numerator, we must suppose that n is greater than 1. E.v. 4. Find the general term in the expansion of {\—x) . ( -i)(-i-i)(-i-2)-..(-f 1.2.3...?i (-i)(-i)(-f)...(-^) ThegeneraIterm^(---Mri-l)(-i-^)---(-i-^ + l)(_.,)n 1.2.3.... (-1^^" ^ ^^ 1.2.3...9i 2-^ ^^ -^ _ 1.3.5...(2.-1) 1 ^ ^ 1.2.3...^ 2«'^ _ 1.3.5...(2r.-l) 2.4.6...2?i 13 ^.^. 5. TfA^>^ is the first term m ^/?e expansion o/(l+i^)4, ?«?/?05e coefiicient is negative ? The (^i+ l)th term in the expression = ^ ' ^'"^"^ ^^—^(U')" 1 . 2...?i ^'^ ^ This will first be negative, as soon as J^^-'^ + l is negative, that is, 13- 4?i + 4 is to be negative, .-. 4?^>17, .-. n = b. Hence the sixth term (that is, the term involving x^) is the first whose coefficient is negative. u Ex. 6. Find the greatest term in the expansion of (l+.r)^, x = l. The (r-f 1)**^ term is obtained from the r'*' by multiplying it by ¥-^+1 X. Hence, as long as this multiplier is greater than unity, each suc- cessive term is greater than the term before it. This multiplier decreases as r increases, but it is greater than unity, so long as -^ # > 1, FRACTIONAL AND NEGATIVE EXPONENTS. 417 that is, so long as 2 {\^- - ;-)>3/', that is, so long as 5r<^^. .-. if r=l, then this multiplier is >1, but if ?•>!, the multiplier is <1. Therefore the second term is the greatest. 404. We can sometimes sum a given series of numbers by shewing that it consists of the successive terms in the expansion of some binomial. This is suf- ficiently illustrated by the two following examples. *Ex. 1. Find the sum of the series ^ -^ ^ + 1 tl • ^ + rlil • ^+ - ^" ^■^>*'^^- If the series can be expressed as the expansion of a binomial, we can sum it. Comparing the given series with the expansion of (l+A-)", namely l+^Lf4--Aj — -x--\-...^ we must have n^' = l, , n{n-\) ,3.41 and^y^-— .-. The solution of these equations gives x=—^^ n— —^. These values make the fourth terms in the two series the same. Hence we conclude that the given series is the expansion of (1 - 4^)"^. Its value, therefore, is (i-i)-^=(§)-^=(f)^=¥. With practice, it is generally possible to write down the answer to examples such as this one by inspection. *Ex. 2. Find the sum of the series Compare the given series with the following expansion P( Pi :i±a;/^=l + (-|)(±^-)+ ^\ J ' {±xY -Pf^PA{_P__^\ ^-'-Hr2^-'-^(±^^)^-- -'■^T^^ 1.2 y ^ 1.2.3 ' \jj +••• B. A. 27 418 THE BINOMIAL THEOREM. The given series can be written in the form 1 V9 j ''" 1 . 2 V9/ ^ 1.2.3 W '^•'* _p These terms are the same as those in the expansion of (1 ±^) « X 2 if jo = 4, 2' = 7 — 4 = 3, and - = 5, that is, x=\. Hence we conclude that the given series is the expansion of (1-|) ^ except that the first term in the expansion is missing. There- fore the sum of the given series _4 _4 4 = (1-|) 3-l = (^) 3-1=33-1=4/81-1. 405. Application to approximations. The fol- lowing typical examples illustrate one of the most important applications of the binomial theorem, namely, the determination of the approximate values of given quantities. Ex. 1. Find the approximate value of ^ , ivhen x is a quantity so small that its square may he neglected. We have -^ =\->rX-\-x'^ + ... l — x For, since we may neglect the square of x, therefore x must be less than unity, and the series is convergent. Moreover, the cubes and higher powers of x must be smaller than x% and therefore may also be neglected. [For example, if x = ^q, •^"""roo' "^ ~Tooo> approximately. Ex. 3. Find approximately the value of the cube root 0/ 1001. 1 Here v^iooi = (10'H1)3 1 -M^-i^'f. -1) 1 ^ 2 10«^ ^] = 10 {1 + 0-0003 - 0-ooooooi + .. .} = 10 {1-0003332...} = 10003332..., which gives the result correct to six places of decimals, since the terms neglected cannot affect these figures. 406. The three following examples deal with ques- tions involving the expansion of multinomials. Ex. 1. Find the first three terins in the expansion of 1 (1+2^-^2)2 in ascending powers of x. We have {1 + {'2.x - x'^)Y = \+^{'2.x- x^) + ^^f "^^ {2x - x^ + . . . We only want the terms involving x and x^, therefore we need not go further in the expansion, since {2x — x'Y must involve x^ : similarly, we may neglect x^ and higher powers of x in the ex- pansion of {2x — x^)'^. .'. {1 4-2x - A-2}^= 1 +1 (2x-x^)-l{Ax'^+...) ==l+x-^x'^-^x^+... = l+x-x'^ + ... 27—2 420 THE BINOMIAL THEOREM. Ex. 2. Find the coejlcient of x^^ in the expansion of {1-x+x^-x^+x^)-^ in ascending poivers of x. The expression in the bracket is a g.p., = (l+^)-2 (1+^)2 = (l-2.rH 3.^10-.. .)(l + 2^'+^2). The only term in this product which can give x^^ is the product of Zx^^ and 2^. Therefore the coefficient of x^^ is 6. Ex. 3. A candidate is examined in two papers, the full marks of each paper being n. In how many ways can he get a total of n marks F In each paper he can get 0, 1, 2,... or n marks. These are the coefficients of the powers of x in x^ + x^ + x^ + ...+xf^. Now in the product {l+x + x^ + ... +x'') {l+x+x'^+...+ X''), the coefficient of x'^ will be the sum of the product of the co- efficients of x^ and x^\ the product of the coefficients of x^ and ^•'^~i, &c. Each of these products corresponds to one way of obtaining a total of ?^, marks ; namely, in the first paper and 71 in the second, 1 in the first paper and n-l in the second, and so on. Hence, the number of ways in which the candidate can get a total of n marks is the coefficient of x''^ in this product. But (1 _ ^.n + 1\ 2 = (l_.^H + l)2(l_^)-2 = {l-2^" + l+^2n + 2}|l+2A' + 3A'H...+(H+l).t'"+...}. Hence the coefficient of ^-^ is ?i+ 1. Therefore the candidate can get 7i marks in n + l ways. This particular example could be solved more simply, but the above work is inserted as an illustration of the method of treating such problems. iVbie. Similarly, the number of ways in which the number n can be formed as the sum of k of the numbers a, b, c... is the co- efficient of *•"■ in (^■« + .y^ + .f« + ...)*. HOMOGENEOUS PRODUCTS. 421 *407. Homogeneous Products. The number of homogeneous powers and products of r dimensions which can he formed from n letters a, b, c ..., ivhere each letter may be repeated any number of times, is n{n-\-\) ...{n + r-1) \r This number is denoted by the symbol nHr^ Consider the continued product , (I a- /-, ^ ^' A ^ c c^ \ 1 + -+ -. + ..., 1+ -+-, + . ..1 + - + -2 -f ....... X OCT J \ X X^ J \ X X^ } where there are altogether n factors, and each factor is an infinite series. Each factor is homogeneous and of no dimensions. Therefore the product is homo- geneous and of no dimensions [Art. 74]. Hence, the coefficient of — in the product must be a homogeneous X function of r dimensions in a, 6, c ..., and clearly it will be the sum of every power and product of r dimensions in these letters. Now we only want the niimher of such products, and if we put a = 6 = c=... = l, we make each term in the coefficient of — equal to unity, and therefore in X this case the coefficient of — will be rjly. Therefore, x^ nHr is the coefficient of — in the product iH h-4-. .. )(l + - + - + . ..).--^ factors. XX" J \ XX- 1 . /, 1 1 - m 1 + - + - X^ \ X x^ .'. n^r = coefficient of — in (1 + - + :;^+ • 422 THE BINOMIAL THEOREM. Let - = y, .'. nHr = coefficient of ?/^ in (1 + y + 2/' + . . .)" X = „ „ in (1-2/)-^' _n{n + 1) . . . {n -\- r — 1) |r ~ [p. 415, Ex. 2. iVo^e. This result can be proved directly by a method analo- gous to that given in Art. 369. For, from the given result we have (see Art. 374) TT _ n + r-\ JI-" r— • n^ 7- - 1 ) and this relation can be proved directly, and thence the value of „H. can be determined. EXAMPLES. XXXI. B. [It iinay he assumed that the series involved in the following examples are considered only for such values as will make them convergent.l Write down the first five terms in the expansions of the following binomials, numbered 1 to 12. 2 5 2. (1 + ^)3. 3. (1-^)1 i _^ 5. {\-^xf. 6. {l+ax) «. n 1 8. {\-2x)~^. 9. {x-x'^)\ 11 ^ 12 ^^-±^ Express, in their simplest forms, the (r + 1)"' terms in the expansions of the following binomials, numbered 13 to 21. 1 3 1 13. (l-2^)"2. 14. (l-4.r)"2. 15. {l-x)~^K 1 3 _U 16. (l-?i^)«. 17. (1 + 2^)2. 18. (l-|.r)"3. 1. {l-x){ 4. (1+2^) "2. 7. 7 {a-bxf. 10. 1 FRACTIONAL AND NEGATIVE EXPONENTS. 423 22. Determine which is the first term with a negative co- 9 p efficient in the expansions of (i) (1 +f ^)^ ; (ii) (1 +^)«. 23. Prove that, if n be a positive fraction, all the terms in (l-.v)» after a certain term will have the same sign. What is this sign if ?i = V-? Write down the first two terms which have this sign. 24. Find the greatest term in the expansion of (l+.^')'\ when X = ? and n= —A, 25. Find the greatest term in the ex^jansion of (l+.r)~", when o:=h and >? = 7. 26. Find the greatest term in the expansion of (1-.^^)-", when ^' = 1 and n = ^. „„ n.! • , 2 3.4,4.5.6., 5.6.7.8, 27. The series l+-.^ + — ..2 + ^-_^^..3+_____ ,4 + ... is a binomial expansion, the index being fractional ; find the expression which is thus expanded. 28. Find the sum of 1 + x .^^ + 7~—, x^ + ^' / a ^^' + • • • 2 2.4 2.4.6 29. Findthesumofl + l + i^^ + i^^:|^ + ... *30. Find the sum of 2.5.7.11 2.5. 7.11.14 2 . 5...(37i + 8) "*■ 70.4 "*" 70|2_.42 '^•••"*" 70,^. 4'^ ^"' 31. Prove that o /o i_q/1 1 1-1 1-1.3 1.1.3.5 \ 2V^-l-3|2 + 22^-23-3"^ 241^ ~~^5j^ + -|- „ _ 1.3.5...(2r-3) ^, ^ ^^' ^''^ 2.4.6...2r >P^Q^e^^^^ l-pi-/?2-...-i3n = 2C^i+l)p„ + i. 33. Find the square root of 101, correct to five places of decimals. 34. Find ^'2^bb^ correct to 3 places of decimals. 35. Find .^999, correct to 6 places of decimals. 36. Expand sj N'^+x in ascending powers of .2?, as far as the third power of x. 424 THE BINOMIAL THEOREM. 37. If X be so small that its square can be neglected, prove ., , \/9 + 2^ + j8-3^ , , 13 that „ ^ -=! + --— .r. ^27 -6^ + .^32 + 10^ 360 38. Find the value of V^±£±v(l^' ^l,en x=^, cor- rect to 3 decimal places. 1 39. If jo and q be the ?i"' terms of the expansions of (1 -^) 2 and (1 -.r) ^ respectively, shew that q={^n-V)'p. 40. Prove that, if jt?^ and 9-^ represent the coefficients of .r'' in the expansions of (1+^)" and (l+^)~", then Vn -^Vn - 1^1 +Pn - 2?2 + • • • +i^l9'n - 1 + ^n = 0. 41. Prove that the sum of the coefficients of the first 7* terms 1 in the expansion of (1-^) ** bears to the coefficient of the ?'"' term the ratio of \-\-n{r — \) to 1. 42. Shew that, if jt?, q., s be respectively the product, quotient, and sum of two quantities, q being less than unity, then p^-- r 2 4 „ 4.5 , 4.5.6 , l 43. If ( 1 + .^)" = Co -f c^x + c^x"- + . . . + c„.^'" +..., find the values of (i) Co-2ci + 3c2-... + (-l)»(w + l)c„. (ii) (Co-C2 + C4-...)2 + (Ci-C3 + C5-...)2. 44. Find the coefficients of x^'^ and of .r3» + 2 j^ ^j^jg expan- sion of y; \-x \-\-x-\-x'^' 45. Find the coefficient of ^r^n + i jj^ ^i^^ expansion of 46. Find the coefficient of x^^ in the expansion of 3 (l-2.r-f4ji;2_ 8.-^3 +...)" 2. 47. Find the coefficient of x^^ in the expansion of {\-x^ 2 A^2 _ 2.^3 + 4^.4 _ 4^5 + 8.r« -...)- 1. 48. Prove that the coefficient of x^ in ^]r'^^}l is 2?i2 - 6w + 1. (1 - x)^ FRACTIONAL AXD NEGATIVE EXPONENTS. 425 *49. If n be a positive integer, and (b Lv. 7i) = — 71 — - H ^ — z~ .... ^^ ' ^ X j; + l 1 . 2 .v+2 ' find a relation connecting (f) (.r, 7i) and (:i? + 1, ?i - 1) ; and thence hi x—1 shew that d) U\ n) = , . ^ ^ ' ^ |^ + ?^ *50. Prove that the sum of the homogeneous products of ?i dimensions of three things a, 6, c is a'^ + ^(6-c) + 6»-^2(c-a)-fc» + 2(f^_5) a-^ (6 - c) + 6--^ (c - a) + c'-^ (a - 6) *51. Shew that the sum of all homogeneous products of a, by c of all dimensions from to ?i is ^n + 3 Jn + 3 ^n + 3 + ., TTt sTi ^,X + (a-6)(a-c)(a-l) ' {b-a){b-c){b-l) ' (c-a)(c-6)(c- 1) 1 {a-l){b-l){c-iy *52. Prove that the coefficient of .r"~i in the expansion of {(1 - x) (1 -- ax^){l - a2^)(l - a\v)} -i in ascending powers of x is (l-a")(l-a*^ + i)(l-a» + 2) (l-a)(l-a2)(i_a3) • *53. A man enters for an examination in which there are four papers with a maximum of 7n marks for each paper. Shew that the number of ways of getting half marks on the whole is ^(m+l)(2m2-|-4m4-3). *54. In a certain examination there are 3 papers for each of which 200 marks are obtainable as a maximum ; but if less than 50 marks are obtained in a paper, they are not counted towards the aggregate. Shew that there are 1326 ways in which a can- didate can just get a total of 200 marks. CHAPTER XXXII. THE EXPONENTIAL THEOREM. 408. Definition of e. The series . , 11 1 . ■ n '. l + l+|-x + |-^ + ... + p + ..., to infinity, is denoted by the letter e. This series contains an infinite number of terms, but it is convergent [Art. 338], since (i) it is greater than 2 and (ii) it is less than 1 + 1+- + ^^+,,. + — + ..., that is, it is less than 1 + :j — j , that is, it is less than 3. Its value lies therefore between 2 and 3, and (by actual calculation) is found to be equal to 2 '71 8... We shall now shew that e is an incommensurable number, and therefore its numerical value is a non-terminating and non-repeating decimal. 409. The number denoted hy e is incommensurable. For, if possible, let e be commensurable ; and suppose that e = — , where m and n are positive integers. Multiplying both sides by \n, we have an integer = the sum of a number of integers 111 + -T-r + r-TTTT— TTTx + n+\ ' (?i-f l)(7i-|-2) {n+\){n + 2){ii+^) "• ^^" .T^+ (.-H)V + 2) + (.+ l)(n + 2)(.-f3) +- ^^^ ^^^^^*^) THE EXPONENTIAL THEOREM. 427 is obviously greater than — -^ , and it is less than This latter series is a geometrical progression of which the com- mon ratio is less than miity, and its sum is - . [p. 355, Ex. 2] Thus — -4-7 TTi ?^ + ... lies between — — and -, and is ?i + l (?H-l)(w-|-2) n + l n' therefore a proper fraction. But a proper fraction cannot be equal to the difference between two integers. Therefore e cannot be commensurable. 410. The Exponential Theorem. To shew that /yi2 /v>3 rpT e^=\j^x^-r^^-~^...^-y+...to infinity. I_ l_ \L [The following demonstration is not altogether satisfactory ; but, since it is usual to give it in ele- mentary text-books, it is inserted here. It is assumed that n and x are real quantities.] We know that {('n)T=(-*r «■ We shall now expand each side of this identity by the binomial theorem. We shall suppose n to be greater than unity; and therefore, since - is less than unity, this expansion will always be possible. The right-hand side, when expanded, becomes , 1\"^ , 1 nxCnx-l) 1 nJ n 1.2 n^ nx {nx — 1) (nx — 2) j^ x\x 1 x[x \[x—- T , , V n) \ nJ \ ih , = l4-^+-y-2- + ^-273 + ... 428 THE EXPONENTIAL THEOREM. Putting ^ = 1, we obtain ^ + »j -^ + ^ + T^+ 1.2.3 — + • Hence, the relation (a) may be written i_i (i-i)(i-?) n . V nj V nj 1.2.3 ^ + ^+172+ 1.2.3 +- = 1 +^+ , ^ + 1.2 ' 1.2.3 This is true for all values of n greater than unity, and we may therefore make n as great as we like. Take n infinitely great, then - will be infinitely small and therefore ultimately zero ; then it can be shewn that the above relation will become f, , 1 1 Y ^ x" a? f + ^-^^ + ^-^-j =l+^+^ + r^+- that is, e^=i + -x, + ^ + ^4.... \A \± Note. It can be proved that this series is convergent for all values of x. N.B. The quantity e^ is sometimes denoted by the symbol exp (x). 411. If a be any number, then loill a- = l +^fo^,a + ^^^+ (i^i^ + ... Let the logarithm of a to the base e be h. .'. loge a = h; that is, a = e^. .-. a^=e^^=e^i°^'«= l+^log,a + (^^^|li^' + ...[Art. 410]. THE EXPONENTIAL THEOREM. 429 412. The Logarithmic Series. To sheiu that, if X he numerically less than unittj, then loge{l+x)^x---^^-...->r{- ly-H- ... (^0 infinity). We have a^=l + n log, a + l^'i^l^l' + . . . [Art. 411], If where a and n are any numbers whatever. Let us take a = 1 + X, .-. (1 + xj' = 1 + n log, (1 + ^O + A {^^ log, (1 + ^)p + . . . 1^ .'.^- ^^'^f"^ = l0ge(l+^-) + in{l0g,(l+^)i-+... We now proceed to find another expression for the left-hand side of this equality. If x be numerically less than unity, we can expand (1 + xj^ by the binomial theorem. Hence we have (l4.^)n_i - n(n—\)„ l-\-nx->f -^"^ x^^ ... 1 . Z - 1 11 -I . , (n-l)(?i-2) , -" + T72^^+ 1.2.3 -^'^+- Equatmg the two series to which ^— — — ^^ is n equal, we have This is true for all values of n. Put n = 0. Every term, after the first, on the left-hand side vanishes. Therefore log, {l+x) = x-\- ^^ x^ + ^" ^l}'^ x^ ^ .., a;2 x^ 430 THE EXPONENTIAL THEOREM. The student should insert the general term in the expansion of (1 -^-xY, and satisfy himself that it reduces to the general term in the expansion of loge(l +x), which is given in the enunciation of the proposition. EXAMPLES. XXXII. A. 1. Find the sum of ^— ^ - j^— 3 + ^ "gT^ - ... (to infinity). 2. Find the sum of 1 -i + i-i + ...(to infinity). 3. Find the sum of 4. Shew that 5. Shew that \ + — + ^ +^ + ..- = ^6. Ii_ l_ l_ 6. Shew, by taking logarithms, that, if jj>y> z^ then (S)' (.-3)..-+...^o. THE EXPONENTIAL THEOREM. 431 41 eS. Natural or Napierian Logarithms. Logarithms calculated to the base e are called natural logarithiiis or Napierian logarithms — the latter name being derived from that of John Napier of Merchistoun, to whom the invention of logarithms is due. In theoretical investigations, it is usual to assume that all logarithms are calculated to the base e, unless the contrary is expressly stated ; and the symbol shewing to what base the logarithms are calculated is frequently omitted. 414. Calculation of Napierian Logarithms. The logarithmic series enables us to calculate the numerical values of the logarithms of the successive natural numbers, namely, of 1, 2, 3, &c. We have, if x be less than unity, loge(l+^) = ^--|^2 + i^,3_^,^ [Art. 412 and \og^{\-x)= -x-\x'^-\ct^- ... \ + V ••• loge ^^ = loge(l+^)-log,(l-^0 i — X 2{x + lx^-\-\afi-\-.. Put ' = — , that is, let x= . Since x is numerically less than unity, this is permissible if m and n be positive integers. •*• ^"s«^=M^M^"^n^'^^^y +5V^M^J +...|...(a). (i) Tofindloggl. This we know is zero. [Art. 253, Ex. 1.] (ii) To find loge2. In (a), put m = 2 and 7i = l, .-. loge2 = 2{HMF + ia)^+...} = 0-693... (iii) To find logg3. In (o), put wi = 3 and n = 2, .-. l0ge3-l0ge2 = 2{HHi)' + HF+...}- .-. loge3- 0-693... =0-405... .-. log,3 = 1-098... 432 THE EXPONENTIAL THEOREM. The method is general. From the logarithm of any number, we can get the logarithm of the next greater number ; and thus a table of the Napierian logarithms of all the natural numbers can be calculated. The logarithm of any fraction or decimal can be deduced by the aid of Art. 256. 415. Common or Briggian Logarithms. Logarithms calculated to the base 10 are called covfi- mon logarithms or Briggian logarithms [Art. 261] — the latter name being derived from that of Henry Briggs, to whom their introduction is due. In numerical calculations — esi^ecially where approximations are required — it is usual to assume that all logarithms are cal- culated to the base 10, unless the contrary is expressly stated ; and the symbol shewing the base to which the logarithms are calculated is frequently omitted. 416. Calculation of Common Logarithms. The result of Article 260 enables us to deduce the numerical value of the logarithm of any number cal- culated to the base ten from the value of its logarithm calculated to the base e. In the result (i) of Art. 260, put m=x ; and divide each side by log^a, .-. log..^-|^^. Put a = 10, and b=e. logio^' = logelO We have shewn how to calculate the logarithm of any number to the base e ; and therefore we know the values of loggX and logelO. Hence, if we divide the natural logarithm of any number by the value of logJO, we shall have the value of the common logarithm of the number. The fraction , — , by which the natural logarithm of any number must be multiplied in order to produce the common logarithm of the same number, is called the modulus. Its value is 0-43429... THE EXPONENTIAL THEOREM. 433 417. Tables of Logarithms. The numerical values of the common logarithms of the natural numbers from 1 to 100,000 have been calculated, in some cases to fourteen places of decimals ; and their values to seven (or five) places are published in a tabular form. As an illustration, a table of the logarithms of the first 100 natiu-al numbers, calculated to 5 places of decimals, is here added. No. Log. No. Log. NO. Log. No. Log. No. Log. 1 00000 21 32222 41 61278 61 78533 81 90849 ! 2 30103 22 34242 42 62325 62 79239 82 91381 3 47712 23 36173 43 63347 63 79934 83 91908 4 60206 24 38021 44 64345 64 80618 84 92428 6 69897 25 39794 45 65321 65 81291 85 92942 6 77815 26 41497 46 66276 66 81954 86 93450 7 84510 27 43136 47 67210 67 82607 87 93952 : 8 90309 28 44716 48 68124 68 83251 88 94448 i 9 95424 29 46240 49 69020 69 83885 89 94939 : 10 00000 30 47712 50 69897 70 84510 90 95424 1 11 04139 31 49136 51 70757 71 85126 91 95904 12 07918 32 50515 52 71600 72 85733 92 96379 13 11394 33 51851 53 72428 73 86332 93 96848 \ 14 14613 34 53148 54 73239 74 86923 94 97313 i 15 17609 35 54407 55 74036 75 87506 95 97772 16 20412 36 55630 56 74819 76 88081 96 98227 17 23045 37 56820 57 75587 77 88649 97 98677 i 18 25527 38 57978 58 76343 78 89209 98 99123 i 19 27875 39 59106 59 77085 79 89763 99 99564 ! 20 30103 40 60206 60 77815 80 90309 100 00000 It will be observed that the characteristics of the logarithms are omitted from the table, and only the mantissas are printed [see Art. 265]. The characteristic of the common logarithm of any number can be written down by inspection [Art. 264]. The result of multiply- ing or dividing the number by any power of ten will change the characteristic of the logarithm, but will not affect the mantissa. B. A. 28 434 THE EXPONENTIAL THEOREM. We however constantly meet with numbers which lie between two numbers whose logarithms are known; and to determine the logarithms of such numbers we want the theorem given in the next article, which is known as the rule of proportional parts. Conversely, we frequently meet with a logarithm, opposite to which in the tables no number is placed, though other logarithms very nearly equal to it are the logarithms of known numbers; and to find the number of which it is the logarithm we require the rule of proportional parts. 418. Rule of Proportional Parts. The numerical value of loQa N being given, to find approximately the value of log a {N + x) where x is small compared with N. We have N+x loga {N+ X) - \0ga N= l0g« —^ =log.(l+|) 1 X where u = , . Hence, if the squares and higher powers of -^ ^ loge a -'>' may be neglected in comparison with ^, we have, approximately, \0ga{N+X)-\0gaN=fJL^. If the tables give us the logarithms of the successive natural numbers, we may suppose N to be an integer and ^ to be a proper fraction. From the tables, we know the values of log^i^^ and loga {N+ 1). Hence, we have log« {N+x) - \ogaN=fj,~, and log„ (iY + 1 ) - log„ N =im^^. . loga{N+x)--\ogaN^ '■ log„('^+~l)-log,iV^ *• If loga {N+ x) be given, the equation last written determines x; \i X be given, it determines log^, (xV+^O- THE EXPONENTIAL THEOREM. 435 419. The rule of proportional parts enables us to find approxi- mately the values of the logarithms of numbers intermediate between numbers whose logarithms are known : and conversely, from a given logarithm to find the number of which it is a logarithm. To take a simple illustration, we will suppose that we have a table like that printed on page 433, which gives us the logarithms of numbers of only two digits. Suppose then that log 92 = 1 "96379, and log 93 = 1 '96848, are given in the tables, and we want to find log 92-7. By Art. 418, we have log 92-7 - log 92 _ 7 _ ^ k)g93-log92 ~ 1 ~ To* .'. log 92-7 -log 92 = ^^ (log 93 -log 92). .'. log 92-7 = log 92 -|-^7_ (log 93 _ log 92) = 1 -96379 + y^ (-00469) = 1 -96707, if we retain only 5 pla.ces of decimals. Thus the value of log 92*7 is known approximately. Conversely, if we are told that 1-96707 is the logarithm of some number, we find from the tables that the number is between 92 and 93. We therefore take it to be 92 +x We then have the proportion _ log ^92+^) -log 92 ■^~ "log 93 -log 92 ' ,, ^ . 1-96707-1-96379 -00328 ^^^ *^"' ^^' '"= 1-96848-1-96379 = ^00469 = '^^^'" Thus the required number is (very approximately) 92-7. 420. We proceed now to take two examples, one of each kind, which will sufficiently illustrate the rule. 421. To find the logarithm of a number not given in the tables: for example, to find the logarithm of 575-056. In the table, opposite the numbers 57505 and 57506 will be found their logarithms, namely, 7597056 and 7597132— the cha- racteristics being omitted. .-. log 575-06 = 2-7597132, and log 575-05 = 2-7597056. 28—2 436 THE EXPONENTIAL THEOREM. The logarithm of the given number lies between these numbers ; suppose it to be 2-7597056 + x The rule of proportional parts shews us that the difference between the numbers is proportional to the difference between the logarithms, .-. 575-056 -575-05 : 575-06 - 575-05 = x : 2-7597132-2-7597056, that is, 6 : 10 = ^ : -0000076, . • . x = -^q{ -0000076) = -00000456. Since we are only keeping 7 places of decimals, we write this •0000046, because this is nearer the truth than -0000045. Hence log 575-056 = 2-7597056 + -0000046 = 2-7597102. 422. To find the number which has a given logarithm {not printed in the tables): for example^ to find the number whose log- arithm uT-0211972. In the table, opposite the logarithms 0211893 and 0212307 will be found the numbers 10500 and 10501, .-. log -10500 = 1-0211893, .-. log -10501 = 1-0212307. The required number is between -10500 and '10501. Suppose it to be -10500 + j^. Then the rule gives us .r : -10501 - -10500=1-0211972- 1-0211893 :T-0212307-T-0211893, that is ^ : -00001 = -0000079 : -0000414, •■• ^'=4^1: (•00001) = "0000019, if we keep only 6 places of decimals, = -000002. Hence l-0211972 = log(-]0500 + -000002) = log -105002. 423. Examples similar to those given in Chapter XXIII. can be worked out to a close degree of approxi- mation by the aid of the rule of proportional parts. Example. Find approximately the fifth root of 2, having given %2 = -30103, % 1-1482 = -060175, % 1-1483 = -060554. 1 Let ^=25, .-. log^=i log 2 = 1 (-30103) = -060206. From the given logarithms, we see that x lies between 1-1482 and 1-1483. Hence, by the rule of proportional parts, we have = l-1482+^V(-00001) = l-14821, approximately THE EXPONENTIAL THEOREM. 437 EXAMPLES. XXXII. B. \^All the logarithms i)i the folloiving examples are calculated to the base ten.'] 1. Given log 11111=4-0457531, and log 11112 = 4-0457922; find log 11-1112, and the number whose logarithm is 2-0457777. 2. Given log 8-6223 =-9356231, and log 8-6224 = '9356282; calculate log 86-22384. 3. Given log 3-1156 = -4935417, and log 3-1 157 = '4935556; find log -03115625. 4. Given log 1-3287 = -1234269, and log 1-3288 = -1234596 ; find log -00132874. 5. Calculate log -0361356 ; having given log 3-6135 = -5579281, and log 3-6136 = -5579401. 6. Find log -172306; having given log 1-723 = -2362853, and log 1-7231 = -2363105. 7. Find log 13-664357, and find the number whose logarithm is 2-1356053 ; having given log 13664 = 4-1355779, and log 13665 = 4-1356096. 8. Employ logarithms to divide 32-0576 by -69665, having given log 3-2057 = -5059229, log 4-6016 = -6629089, log 3-2058 = -5059364, log 4-6017 = -6629183. log 6-9665 = -8430146. 9. Find the value of (1-25)'* to six places of decimals, having given log 2 = -30103, log 2-4414 = '3876389, log 2-4415 = -3876567. 10. Find (1-467799)' ; having given log 14677 = 4-1666373, and log 14678 = 4-1666669. 11. Shew that the cube root of 19307 is very nearly ten times its tenth root, having given log 19307 = 4-2857290. 12. Given log 1-7783 = -2500050, and log 1-7782 = -2499806; find the value of (17-7828)^ 438 THE EXPONENTIAL THEOREM. 13. Find (13-89492)8 ; having given log 1-3894 = '1428273, and log 1-3895 = -1428586. 14. Find the fifth root of 5-4; having given log 3 = '4771213, log 5 = -6989700, log 14011 = 4-1464691, log 14012 = 4-1465001. 15. Find the fifth root of 6'4;_ having given log 2 = '30103, log -14495 = 1-1612182, log -14496 = 1-1612482. 16. Find the eleventh root of (39-2)2 . having given log 2 = -3010300, log 7 = '8450980, log 19484 = 4-2896781, log 19485 = 4-2897004. 17. Extract the seventh root of 3115455 ; having given log 3-1 154 = -4935138, log 3-1155 = -4935278, log 8-4653 = -9276424, log 8-4654 = -9276475. 1 18. Find the value of 2i28 3^ 10 - ^^ ; having given log 2 = -3010300, log 3 = -4771213, log 4-239 = -627263, log 4'24 = -627366. 19. Prove that log 7+ log 11+ log 13 is approximately equal to 3. To how many decimal places is this true? EXAMINATION PAPERS. 439 EXAMINATION PAPERS. The first of the following papers vxts set to the Senior Students at one of the recent Cambridge Local Examinations. The next paper {B) was set recently in the Cambridge Higher Local Ex- aminations. The next two papers (C) and {D) are two of those set recently to candidates for admission into the Military Academy, Woolwich. Paper A. 1. Prove that 2. If a, /3 be the roots of the equation ax^-\-bx-\-c = 0, find the value of a^ + jS- ; and form the equation whose roots are a + - and /3 + -. 13 a 3. If ab + bc + ca=0, prove that (i) (a + 6 + c)2 = a2 + 62+c2. (ii) {a + b-\-c)^ = a^ + P + c^-3abc; (iii) {a + b + cy = a'^-\-b^+c*-4abc (a + b+c). 4. Solve the equations : (i) ax -\-by = mxy =px + qy ; (ii) .{b- c) x^-h{c-a) x + {a- b)=0 ; (iii) ^/^+3-^/5^-7 = ^/2; x^ +3xy + 2y = l2) ^ ' xy + 43/2+ x= 8J • 5. The distance from A to B is 20 miles by one road and 24 miles by another. A bicyclist goes from J. to ^ by one road and returns by the other ; but in returning, travels 2 miles an hour slower than in going. He finds that, if he goes by the longer and returns by the shorter road, he takes 6 minutes less time to travel than if he went in the reverse order. What is his speed in going? 6. Insert n arithmetic means between a and 6, and find the sum of these means. If the 3^^ the {p + 2)"" and the {Zpf' terms of an arithmetical progression be in geometrical progression, prove that the (/>- 2)"" term of the arithmetical progression is double the first term. 440 EXAMINATION PAPERS. 7. Find the number of combinations of n different things taken r together. Prove that the whole number of ways in which a selection of one or more things can be made out of n different things is 2"— 1. li p of the things are alike, prove that the whole number of ways in which such a selection can be made is (jo + 1) 2"~^ - 1. 8. Prove the Binomial Theorem for a positive index. If (1 + xY = (7o + C^x + (72^2 + . . . + c^x-^, prove that ^« 2 "^3 •••■^ n+\ -^^V 9. Define a logarithm, and prove Find, correct to two places of decimals, values of x and y which satisfy the equations (2-5)^=1000, (•25)2'= 1000 ; having given log 2 = -3010300 ; and prove that i - i = i . o: y 6 Paper B. 1 . Prove that ah = ha, and {a-\-h)c = ac + he, where a, 6, and c are positive integers. Shew that {a + h-\-c-\-df-\-^{a-\-h-c-d){a-\-c-h-d){a + d-h-c) = 4(a3 + 63 + c3 + t;3) + 12a6c^(^i4-r + - + -,V \a c dj 2. Solve the simultaneous equations : (i) y-\-z-ax=p, z+x-hy = q, x->ry-cz = r; (ii) x'^-\-ay=y'^ + ax = h'^. 3. Find the relations between the roots and the coefficients of a quadratic equation. If the length of a field were diminished and its breadth in- creased by 12 yards, it would be square. If its length were in- creased and its breadth diminished by 12 yards, its area would be 15049 square yards. Determine the area of the field. EXAMINATION PAPERS. 441 4. Define an arithmetical progression and a geometrical progression; and shew that, if a^, «2, a^ etc. be a g. p., then logaj, logttgj log% 6tc. is an a. p. If P, Q, R be the /?"*, q^^^ and r"' terms of a g.p., shew that and deduce the relation between three given terms of an a. p. 5. Find from first principles the number of combinations of n things taken r together. In how many diflferent ways can m shillings and m + 7i florins be given to m' boys and m' -\-n' girls, one coin to each, where 6. Under what limitations does the Binomial Theorem hold when the Index is not a positive integer \ Determine the greatest term in M - — - j , where m and n are integers and m is less than n. 7. Find the cube root of 100 to 3 places of decimals. Paper C. 1. Find the value of x^ -bofi-l2x'^- \Zx - 7, when x=-\{\ + ^J^^). 2. Find the factors of (a + 6 + c)2 - aH 6^ - c\ and of xr^-y'^- {x-yf. 3. Keduce to its lowest terms 2^-^ + 17.^3 + 30^^8.2; -5 x^ + Aa^- \M^ - 29x - 10 ' 4. Simplify . f)X^ — bxy — ^y^ 1 bx^ + ^xy — 1 2y2 ^ ^^^ l4:x'^-2Zxy + Zy'^ ~ Zbc^+nxy + Qy'^ ' /••x c , 6 . a ah{h — c){c — a) ac{a—b){b — c) hc{a-h){c-a)' 5. Apply the process for extracting the square root to find m and w, when x^ + ajfi + mx'^ + ct; + ?i is a complete square. 6. If the roots of the equation be equal, shew ihsit p^ = Aq. 442 EXAMINATION PAPERS. 7. Solve the equations : ^^' ^'+10 "^^ + 4 .r + 7' (ii) sJ^x + Q-'sJx-l =2 ; 111 ^ , +-4 -=^'^. a-o o-a 8. Find the value of 3 3 (4 + \/l5y + (4-\/l5y 3 3* (6 + V35)'^-(6-\/35)^ 9. The metal of a solid sphere, radius r, is made into a hollow sphere, whose internal radius is r ; required its thickness. [The volume of a sphere of radius r may be taken as |f r^.] 10. Two rectangular lawns have the same area, which is given and is equal to a?; but the perimeter of the one is one- fourth longer than that of the other, which is a square. Find the dimensions of the lawns. 11. Shew that, if the sum of the first ^ terms in an a. p. be equal to zero, then the sum of the next q terms is -^ , ^ . p — i 12. Is the coefficient of x'^' in the expansion of (1 -x)~'^ equal to the number of combinations of {n + r) things taken r together? If not, amend the proposition, and prove it as amended. Paper D. 1. Simplify the expression {(a + 6)3_c3}|(a + 6+c)} 2. Shew that x"^ -\-y^ + z^- Zxyz is divisible exactly by x +y + z ; and hence, or otherwise, shew that (6_c)3 + (c-a)3 + (a-6)3 = 3(6-c)(c-a)(a-6). 3. Find the highest common factor of 16^-^ + 36x^^ + 81, and 8^3 ^ 27 ; and the lowest common multiple of 8.:i^ + 27, 1 6^* + 36a,-2 + 81, and 6^2 _ 5^^ _ 6. 4. Extract the square root of 4^2a-2 + 16^-2a2_i2--24-+25. a X 5. Determine what relation must hold between a, 6, and c, in order that the roots of the equation ax'^-\-hx-\-G = ^ may be real and different. EXAMINATION PAPERS. 443 If the roots of the equation (a^ + 62) x"^ -'2.h{a + c)x + h'^ ^-€^=0 be real, prove that a, 6, c are in g.p., and that oc is their common ratio. 6. Solve the equations : (ii) -^+^.-2 = 0; ^ ' x-a x-b (iii) X +y + 2=19, 1 , having given that y is a mean pro- x^ + 7/'^ + z^ = l 33J portional to x and z. 7. Two casks, A and B, are filled with two kinds of sherry ; in cask ^-1 they are mixed in the ratio 2 : 7, and in cask B they are mixed in the ratio 1 : 5. What quantity must be taken from each cask to form a mixture which shall consist of 2 gallons of the first kind of sherry and 9 gallons of the second kind of sherry ? 8. Prove the formula for the sum of ?i terms of an arithmetic progression, whose first and last terms are given. If Sj, Sgj %» &c., be the sums of m arithmetic series, each to n terms, the first terms being 1, 2, 3, &c., respectively, and the differences 1, 3, 5, &c., respectively, shew that ^1 + ^2 + ^3 + • • • +Sni=l'>nn {ran + 1). 9. The number of combinations of n letters, taken 5 together, in which a, 6, c occur is 21. Find the number of combinations of the ?i letters, taken 6 together, in which «, b, c, d occur. 10. Write down the two middle terms in the expansion of -\ — ) : and shew that, if x be a small fraction, then X ^^^ ^ ^^- ^ is very nearly equal to 1 - %x. l-\-x-{-\/l+x 1 1. Prove that the logarithm of the quotient of two numbers is the difference of the logarithms of the numbers. If the logarithms of a, 6, c be respectively p, q, r, prove that ««-»•. b^'-P . cP-« = l. Prove that log ^f - 2 log f + log ^^ = log 2. 12. If the number of persons born in any year be equal to j^^ of the whole population at the beginning of the year, and the number who die be equal to ^y^' of it, find in how many years the population will be doubled. [In this question, the student is supposed to have access to tables of logarithms.] ^CHAPTER XXXIII. PROPERTIES OF NUMBERS. *424. The properties of numbers, and the forms of numbers, which are subject to certain conditions, are treated better in works connected with higher arith- metic than in connection with algebra. Some of the more elementary propositions about numbers are how- ever usually inserted in text-books on algebra, and a few of these propositions are given in this chapter, which may be regarded as an appendix to the present work. SCALES OF NOTATION. *425. Denary and other Scales. In arithmetic, the ordinary method of expressing an integral number by figures is to write down a succession of digits ; each digit represents the product of that digit and a power of ten, and the number represents the sum of these products. Thus every digit has a local value. For example, 2017 signifies (2 x 103) + (0x 102) + (1 x 10) + 7 ; that is, the 2 represents 2 thousands, i.e. the product of 2 and 10^, the represents hundreds, i.e. the product of and 10'^; the 1 represents 1 ten, i.e. the product of 1 and 10, and the 7 represents 7 units. Similarly, a decimal fraction is the sum of a number of fractions whose denominators are powers of ten. Thus 0-317 stands for 1 + ^, + ^^3 . PROPERTIES OF NUMBERS. 445 This mode of representing numbers is called the common scale of notation, or the scale to the base ten, or the denary scale of notation; and 10 is said to be the base or radix of the common scale. We shall now prove that, in a similar way, a number can be expressed in a scale whose radix (instead of being 10) is any positive integer greater than unity. We shall confine ourselves to scales whose radices are positive integers. *426. Whole numbers. A given whole number can he expressed as a whole number in any other scale of notation. By a given whole number we mean a number expressed in words, or else expressed by digits in some assigned scale. If no scale be mentioned, the common scale is supposed to be intended. Let N denote the given whole number, and let r be the radix of the scale in which it is to be expressed. We have to shew that N can be written in the form N=p,,r'' +Pn-\ r""-^ + '" -^P-f^ +Pxr +i?o 5 where each of the numbers ^q, p^, ..., pn is a positive integer less than r, or is zero. Following the analogy of the common system of numeration, the numbers jDq, p^, ..., jt?n are called the digits of the number in the scale r. Divide N by r, then the quotient, say Q^ , is pnr'' ~ ^ +Pn ~ 1^*" ~ H . . . +p.f +i>i , and the remainder is jo^. Thus p^^ is determined. Next, divide Q^ by r, then the remainder is p^. Thus p^ is determined. Continuing this process (and taking ?•"■ as the highest power of r which is contained in T), we obtain in succession the digits Pdi Ply •••> Pn as the remainders of these divisions. *427. The following examples illustrate Art. 426. Ex. 1. Express 2176 in the septenary scale., that is, the scale radix is 7. The rule given in Art. 426 requires us to divide by 7, and repeat the operation on each successive quotient. Thus 446 PROPERTIES OF NUMBERS. 7 12176 7 1 310 with remainder 6 7"[44 „ „ 2 6 „ „ 2. Hence 2176 = (6 x 7^) + (2 x 72) + (2 x 7) + 6 = 6226, when expressed in the scale 7. Ex. 2. Transform 6226 from the septenary scale to the com- mon scale. This is the converse of Ex. 1. We have to divide 6226 by 10. Our first step in dividing 6226 by 10 is to find how often 10 will divide into 6. Since 6 is less than 10, we cannot divide 6 by 10. Next, we have to divide ^"l, i.e. (6x7) + 2, by 10; this gives 4 as quotient with remainder 4. Following the usual process, we divide 42, i.e. (4x7) + 2, by 10; this gives 3 as quotient, with remainder 0, Next, we divide 6, i.e. (Ox 7) + 6, by 10; this gives as quotient, with remainder 6. Hence the quotient obtained by dividing 6226 by 10 is 430 with remainder 6. Next, we have to divide 430 by 10. Continuing this process, the whole work is as follows : 10 1 6226 10 1 430 with remainder 6 10"]30 „ „ 7 2 „ ,, 1. Hence 6226 in the septenary scale = 2176 in the common scale. Ex. 3. Change 241 8 /rom the nndenary scale {radix 11) to the duodenary scale {radix 12). We shall want symbols for ten and eleven, since they may now be digits. Denote them by t and e respectively. Following the same process as in Ex. 2, we have 12 1 2418 12 |21^ with remainder 9 12^ „ „ e 1 „ „ 9 Hence 2418 in the undenary scale = 19e9 in the duodenary scale. Ex. 4. Determine which of a series of weights lib., 2 lbs., 4 lbs. and 8 lbs. must be used in a balance to weigh 13 lbs. — not more than one weight of each kind being taken. This is equivalent to requiring us to express 13 in the binary scale (radix 2). PROPERTIES OF NUMBERS. 447 The above process gives 13 ill the denary sctale = 1101 in the binary scale, that is, 13 = (lx23) + (lx2-0 + (Ox2)+l = 8 + 4 + l. Hence the weights required are those marked 8 lbs., 4 lbs., 1 lb. Ex. 5. A weight of 11 lbs. is placed in one scale-pari of a balance. Shew hoio to place a series of weights 1 Ib.^ Slbs., 3'^ lbs., 3^ lbs., 3* lbs. in the scale-pans so as to make the beam of the balance even — not more than one weight of each kind being taken. If we express 77 in the ternary scale (radix 3), we find that 77 = (2 X 3'^) + (2 X 32) -1- (1 X 3) -f 2. But since we have only one weight of each kind, each of the digits by which we express it in the scale of 3 must be unity or zero, though as we can put a weight in either scale we can make the digit positive or negative. If therefore we have a 2 as a remainder in our division, we must write it as 3 — 1. Hence we have 3|77 3 j'26 with remainder -1 3][9 J) )? -1 313 •>■) 55 1 )) 55 .*. 77 = (1 x34)-(lx 3)- We must therefore put the 3* lbs. weight in one scale-pan, and in the other scale-pan we must place the weights 1 lb. and 3 lbs. in addition to the given weight of 77 lbs. *428. Vulgar Fractions. A vulgar fraction in one scale can be expressed as a vulgar fraction in any other scale by expressing the given numerator and the given denominator in the new scale. *429. Radix Fractions. The term radix fraction is used to denote a fraction, expressed in the scale r, in a manner analogous to that in which a decimal fraction is expressed in the denary scale. Thus, just as the decimal fraction . abc... stands for 10^102^103^'"' 448 PROPERTIES OF NUMBERS. so the radix fraction . ahc. . . in the scale r stands for a h c where a, b, c,... are positive integers, each being less than the radix of the scale. *430. A given radios fraction can he expressed as a radix fraction in any other scale of notation. Let F denote the given fraction, and let r be the radix of the scale in which it is to be expressed. We have to shew that F can be written in the form r ?'2 r^ where. each of the numbers q-^, q.y, q^, .,As st. positive integer or zero, and is less than r. Multiply F by r, then q^ is the integral part of the pro- duct, and J^ + ^+... is the fractional part, say Fj^. Thus q^ is determined. Next, multiply F^ by r, then the integral part of the product is ^2- Thus q^ is determined. Continuing this process, we obtain in succession the numbers qi, ^21 ••• ^^ ^^^ integral parts of these products. If part of a number be integral and part fractional, the parts must be treated separately : that is, the integral part must be treated by the method of Art. 426, and the (radix) fractional part by the method of Art. 430. Ex. 1. Express ^ as a radix fraction in the binary scale, that is, in the scale whose radix is 2. We have I X 2 = 1 + ^ ; hence the 1'' figure is 1, 1x2 = + 1; „ „ 2"^ „ 0, and the figures now recur. Hence | in the denary scale = 0'i6 in the binary scale. In other words, 2 10 1 ,^ ■ . -^ ^ 3 = 2 + 2'2 + 2"3 + ^4 + - (to infinity). PROPERTIES OF NUMBERS. 449 Ex. 2. E.vpress 0'261 as a radix fraction in the scale of 5. We have to multiply the given number by 5 ; take the fractional part of the product, and multiply it by 5 ; and so on. The process is as follows 0-261 5 1'305 thus the Y^ figure in the radix fraction is 1, 5 1'525 „ 2"' „ „ ,, 1, 5 2"^625 „ 3"^ „ „ „ 2, 5 3*125 ,, 4 „ „ „ 3, 5 0^5 „ 5*'' „ „ „ 0, and the last two figures recur. Hence 0*261 in the denary scale = 0*11236 in the scale of 5. *431. The difference between any number in a scale of radix r and the sum of its digits is divisible by r — 1. Let N be the number, and 8 the sum of the digits ; and S=_po+i5i +...+pn. The right-hand side vanishes if r = l, therefore it is divisible by 7' — 1 [Art. 120]. Hence, N — S is divisible by r— 1. For example, in the denary scale, the difference between any number and the sum of its digits is divisible by 9. Now any multiple of 9 is divisible by 9, therefore the sum of its digits is also divisible by 9. *432. The rule for casting out the nines, which is given in some text-books on arithmetic as a check on the accuracy of multipli- cation, aflfords another illustration of the use of scales of notation. Suppose that the product of two whole numbers A and B is found to be P. Divide the sum of the digits in A by 9, and let B. A. 29 450 PROPERTIES OF NUMBERS. a be the remainder. Similarly, let h and p be the remainders when the sums of the digits in B and P are respectively divided by 9. Then the rule for casting out the nines is that the differ- ence between ah and jt? will be either zero or a multiple of 9. If we use the symbol J/ (9) to signify any multiple of 9, we have A=M{Q) + a; B = M{9) + b; P=i/(9)+j9. But P=AB, .-. M(9)+p = {M{9) + a}{M{9) + b} = M{9) + ab. .-. ab-p = M{9). Therefore, if ab -p be neither zero nor a multiple of 9, there must be a mistake in the work. But it does not follow that when the condition is satisfied the result must be right ; and in fact, if the number found for the product differ from the correct result by a multiple of 9, the rule will not serve to detect an error. *433. A number of m digits is necessarily less than IQm Hence, its square is necessarily less than 10^'" : that is, its square cannot contain more than 2m digits. Similarly, its cube cannot contain more than Sin digits ; and so on. *434. The last proposition can be used to determine some of the digits of the square root of a number which is a perfect square, by the following rule. If the square root of a given number {which is a perfect square) contain in all (2?i + l) digits, and if the first (n+1) of these digits have been obtained by the usual arithmetical process, the remaining n digits can be obtained by division. Let N represent the given number. Let a be the part of the root already obtained, which consists of n-\-\ digits followed by n ciphers; and let x represent the part of the root which remains to be found. We have ^N=a-\-x. .'. N=a^ + '2.ax+x\ N-a^_ ^ Now X contains n digits, .'. x"^ cannot contain more than 2w x^ digits [Art. 433]. But a contains (271+1) digits. Therefore — is a proper fraction. PROPERTIES OF NUMBERS. 451 Therefore, if we subtract ci^ from N^ and divide the result by 2a, then the integral part of the quotient will be x. For example, the first four digits in the square root of 12088868379025 are 3476. Since there are 14 digits in the given number, there will be altogether 7 digits in the square root. Hence the remaining three digits are the integral part of 1 2088868379025 - (3476000)^ 2 X 3476000 which will be found to be 905. Therefore the required square root is 3476000 + 905 = 3476905. *EXAMPLES. XXXIII. A. [^The numbers in the following examples are expressed in the common scale, unless the contrary is stated.] 1. Express 725 in the scale 6. 2. Express 1171 in the undenary scale. 3. Change 1234 from the scale 5 to the common scale. 4. Change 111000111 from the scale 2 to the scale 12. 5. Express |f as a vulgar fraction in the scale 7. 6. Change "42 from the common scale to the scale 9. 7. Change 14*23 from the scale 6 to the scale 5. 8. Express I as a radix fraction in the scale 6. 9. A radix fraction in the scale 3 is -110201. Express it as a vulgar fraction in the scale 7. 10. In what scale is 712 expressed as 871 ; and in what scale as 598? 11. Multiply together 1461 and 6253 in the scale 7. 12. Divide -1000000 by 10000 in the scale 2. 13. Extract the square root of 25400544 in the senary scale. 14. Extract the square root of 1022121 in the scale 3. 15. Extract the square root of 769 in the scale 12. 16. Shew that the difference between the square of any number and the square of the number obtained by reversing the digits is divisible by r'^ - 1 . 17. Shew that, if the radix of the scale be odd, the difterence between a number and the sum of its digits must be even. 29—2 452 PROPERTIES OF NUMBERS. 18. Prove that a number is divisible respectively by 7 or 11 or 13, when the number expressed by the last three digits differs from the number expressed by the digits before the last three, by a quantity which is either zero, or a multiple of 7 or 11 or 13. 19. If, in a number, the difference between the sum of the digits in the units', hundreds', ten-thousands', &c. places, and the sum of the digits in the tens', thousands', hundred-thousands', &c. places, be zero or a multiple of 11, prove that the number is divisible by 11. 20. Prove that, if the digits of a number iV^, expressed in the denary scale, reckoning from the units' place, be o^oj <^n •••<^n) then 10»(«„ + 2a„_i + ... + 2"«o)-#is divisible by 19. ■'*"21. Shew that the figure nine cannot occur in the decimal part of a fraction whose denominator, when expressed as a vulgar fraction, is less than ten. In the scale whose radix is r, what figures, if any, cannot occur in the radix part of a fraction whose denominator, when expressed as a vulgar fraction, is less than r ? 22. How many numbers can be formed with 4 digits in the scale of r, no digit being used twice in the same number ? What will be the sum of these numbers ? 23. If any number be multiplied by a number which exceeds the radix by unity, shew that the result may be obtained by adding each digit to the digit to the left of it, beginning from the right, and carrying unity to the next pair when the sum of any pair is not less than the radix. Multiply in this way by 11 the number 1243201 in the scale of 5. 24. Enunciate and prove the proposition analogous to that given in Art. 434, about the square root of a number which contains in all 2n digits. 25. The highest digits in the square roots of the following numbers, which are perfect squares, are given. Find the re- maining digits. (i) ^236144689 = 153...; (ii) V142091.3025 = ,376...; (iii) ^285970396644= 5.347...; (iv) ^48303584206084 = 6950 ... . 26. If the cube root of a given number, which is a jDerfect cube, contain 2n digits, and if the first {n-\-\) of these digits have been obtained in any way, shew how the remaining {n-\) of them can be obtained by division. PROPERTIES OF NUMBERS. 453 PROPERTIES OF NUMBERS. *435. We have already seen [Art. 172] that we can make use of algebraical notation to illustrate some of the more obvious properties of numbers. We shall here explain the notation commonly used in the subject, and add a few examples to those given in Art. 172. *436. Notation of the theory of numbers. Either of the following systems of notation is employed. When a number, denoted by N, is an exact multiple of a number n, the relation is expressed by the symbols iV" = il/(?i), which is read as N is equal to a multiple of n. Similarly, N = M{n) + m signifies that N is equal to the sum of a multiple of n and of m ; that is, that if N be divided by n, the remainder is m. If two numbers, X and Y, when divided by n, leave the same remainder, they are said to be congruent to the modulus n. This relation is expressed by the symbols X=F(mod. n), which is called a congruence \ and is read as X is congruent to Y to the modulus n. It is evident that X — F is exactly divisible by ?i, and there- fore we have X — Y=0 (mod. n). *437. We shall conjfine ourselves to working out a few examples. We shall, for brevity, use the word number as meaning a positive integer expressed in the common scale of notation. Ej:. 1. Shew that a number^ \ohich is a perfect square, must be of one of the forms bn or 5?i + l, n being a positive integer. Every number, iV, can be written in one of the forms 5m, 5m — 1, bm — % 5»i + l, or 5m + 2. If ^V=5m, .-. i\^2 = 25m-' = J/(5) = 5;i (say). If iV=5m±l, .-. iV^"'^ = i/(5) + l = 5^1+1 (say). If iV^=5m±2, .-. iV2 = J/(5) + 4 = J/(5)-f5-l=J^(5)-l = bn- 1 (say). All of these forms are included in bn or 5/1. + 1. 454 PROPERTIES OF NUMBERS. Ex. 2. Shew that a number^ which is both a square and a cube, must be of one of the forms ^n ov In + l. Every number, N, can be written in one of the forms 7wi±a, where a = 0, or a=l, a = 2, or a = 3. Hence, a square is of the form (7m±a)2, which can be written M{l)-\-a^. If a = 0, then a2 = 0; if a = l, then a2 = l; if a = 2, then 0^ = 4; if a = 3, then 0^ = 9 = 7 + 2. Hence, a square number must be of one of the forms In, or 7w + l, or ln-\-2, or 7?z + 4. Similarly, a cube must be of the form (7wi + a)^, which can be written M{i)±a^. If a=0, then a^ = 0\ if a = l, then a^^l; if a = 2, then a3=:8 = 7-f-l; if a = 3, then a3 = 27 = 28-l. Hence, a cube number must be of one of the forms In, or 7?i+l, or ln — \. Therefore a number which is both a square and a cube can only be of one of the forms In or 7?i+ 1, Ex. 3. Shew that a number, which exceeds any odd 'power of 7 by unity, is a midtiple of 8. The number is of the form 72"'^i + l, where n is some positive integer. Let S'Aez^' that the product of any n consecutive integers is divisible by factorial n. We deduced this result in Art. 381 from the fact that the number of combinations of n things taken r at a time was necessarily an integer. We now proceed to give a direct proof of it. We shall first assume that the product of any n - 1 consecu- tive integers is divisible by 1.2. ^...{n- 1) ; and shew that, if this be the case, then the product of any n consecutive in- tegers is divisible by 1 . 2 . 3...(7i- 1)^. But one integer is always divisible by 1, therefore, the product of 2 consecutive integers is divisible by 1 . 2 ; hence, the product of 3 consecutive integers is divisible by 1 . 2 . 3 ; and continuing the process, the proposition will be proved. We assume, then, that the product of any n — 1 consecutive integers is divisible by 1 . 2 . 3...(?i- 1). Let {r) stand for the product of n consecutive integers from r upwards, that is, (^(r) = r(r-M)(r-|-2)...(r-i-)i-2)(/'-F^i-l), .-. <^(r-hl) = (r-l-l)(r-H2)(r-f3)...(r-|-?i-l)(r-f?0, .-. (^(/'-|-l)-0(r) = (r4-l)(r-l-2)...(r-h7^-l)[(r-|-7i)-/-] = n{r + l){r + 2)...{r-\-n-\). But (r+l)(/*-|-2)...(/- + ?i-l) is the product of n-\ consecutive integers, and is therefore a multiple of 1 . 2 . 3...(?t - 1), .-. {r) be a multiple of 1 . 2 . 3...n, so also is <^ {r-{-\). But (^(1) = 1 . 2...?i, therefore the proposition is true of 0(2) ; hence, it is true of (3) ; and continuing the process, it is true of 0(r). Thus, if it be true for any n-l consecutive integers, it is true for any n consecutive integers, and therefore, as explained above, it is true generally. 456 PROPERTIES OF NUMBERS. *EXAMPLES. XXXIII. B. 1. Shew that a number, which is a perfect cube, must be of the form *ln or ln±\. 2. Shew that a number, which is a perfect square, cannot be of the form 3>i - 1. 3. Shew that the fourth power of a number is of the form bn or bn-\-\; also, that it must be of the form In, or Tti + I, or 7w + 2, or 7?i + 4. 4. Shew that the twentieth power of a number is of the form 25 w or 25?i+l. 5. Shew that the sixth power of a number is of the form In or 7^4-1. 6. Shew that S^™ + 2 _ S;^ - 9 = i/(64). 7. Shewthat22'^ + i-9>i2 + 3n-2=J/(54). 8. Shew that S^" + 1 - \An - 8 = if (49). 9. Shewthat?i(yi2-l) = J/(6). 1. 7. 2. 14. 3. 8. ^. 15. 3. 4. 2. 5. -h- 12. i. 19. 1. 26. 12. 13. 18. 20. 0. 27. 9. ANSWERS TO THE EXAMPLES. Chapter I. Definitions and Notation. I. A. Pages 9— 10. l.^ahx. 2. (i)i;(ii)i. 3. 7;nu. merical. 4. 23a. 5. y\ literal. 6. 1; numerical. 9. 2. 10. §. 11. h 12. 4. 13. h 17. 2. 18. -£6. 8s. 19. -7 miles. I. B. Page 12. 1. lOS. 2. 50. 3. ■ 6. 2. 7. 4. 8. 32. 9. 5. 10. 5. 11. 5. 14. 54. 15. 18. 16. 1458. 17. 1. 18. 0. 21. 384. 22. 0. 23. 4. 24. 24. 25. t- 28. 75. 29. 3. 30. 22. 31. 4. 32. 3. 33. 49. I. C. Page 14. 1. 2. 2. 0. 3. 0. 4. 4. 5. 4. 6. 4. 7. 1. 8. I. 9. h 10. 4/4. 11. t. 12. |. 13. (i) 10; (ii) 7. I. D. Page 17. 1. Seven is the coefficient of a^; and three is the index, shewing the power to which a is raised. 2. (i) ^^ stands for XXX XX', (ii) 3a; stands for the product of 3 and x. If x = l, then .t3=1 and 3.r = 3, .'. 3x>.t3. If .^ = 2, then x^ = B, and 3a; = 6, .-. ;r3>3a;. 3. 3?^%. 4. 3 + Z; + ?^ + o + i/. 5. 3. 6. 5. 7. 3. 8. 4. 9. 1. 10. n. 11. (i) 1; (ii) 4; (iii) 1; (iv) 3; (v) 1 ; (vi) n. 12. 1. 13. 3. 14. n. 15. No. 16. 180. 17. (i) 2; (ii) 7. 18. (i)137; (ii)3f. 19. (i)-V-; (ii)l- Chapter II. Addition and Subtraction. II. A. Page 20. 1. 4Z> + c. 2. -2a-26 + ^c + 5«. 3. 3.t2-8^2. 4. 0. 5. ii3- + JT'^g2 + 2r2-r. 6. 0. 7. v/6. 458 ANSWERS TO THE EXAMPLES. II. B. Pages 27— 28. i. i. 2.-1- 3. -^-y- 4. -2a;2 + 32/2 + 22. 5_ _(jz, 6, _5. 7, _9. 8. «-&• 9. ^-y- 10. He lost 30s., or he gained - 30s. H. -a-x-^y. 12.2a;. \Z. \a + 2x + y. 14.2?/. 15. -2a + 56-. r. IQ. x'^ + h'' ox -{x'^ + h'^). 17. fl6-6c. 18. 3a + 3& + 3c; 33. 19. 263-36%. 20. -2?/-z. 21. 6 + d. 22. -5a. 23. Ifa + A^ + ZiTC. 24. -^a;2 + |?/2-2:r. 25. - ^ - T^m + ln. 26. 6 - a. If a be greater than 6, the answer will indicate that it is (a - h) years since the man was 6 years old. 27. -2 miles. 28. 6- a- 29. - &• ZO. x^-{a- + h% 31. x'^-if-2xy. 32. a;-!/. 33. a-h. 34, . 22. Chapter III. Multiplication. III. A. Page 31. l. %ahxy. 2. ^ahcd 3. A^a%HK 4. ^'6%^. 5. 2464a262c2d2. 5. ahcdefx^\ 7. a6cj;2|/V. 8. a%^xhfz''. 9. a353^^2. 10. ^VZ%%^ 11. -7a6a;2. 12. -7a6a;2. I3. -a262c2a:3. 14. - U^rri^nxY. 15. -48aia2%33a>5. 16. -\ha^y-^^. III. B. Pages 32— 33. l. 2x'^ + ^xy + xz. 2. -laHn-lahmn-laciv^. 3. Pnyh - Imnxyz - In^xy^. 4. -i-&- 5. 0. 6. 20:c - 4y ; a binomial ; coefficient of x is 20. 7. - 100a- + 100a; a binomial ; coefficient of x is - 100. 8. - 26 + 48c - lO.r ; a trinomial ; coefficient of x is - 10. 9. -19a; + 12?/; a binomial; coefficient of a; is -19. 10. 7a; -6?/; a binomial; coefficient of x is 7. 11. -80a;2+112a; + 16. 12. a2-a6 + 262. 13. xy + ay, x + a. III. C. Page 39. 1. (i) abxy; (ii) -abxy; (iii) x'^. 2. xy-Sx + 2y-6. 3. a;2-3a; + 2. 4. a;^- aa; + 6a;-a6. 5. 7a;2-50a; + 7. 6. 49a2-m2. 7. .t2 _ 8^2^^ + ^2. 8. a6a;2-a3a;-63x + a262. Q. x^-aK 10. a^x"^ - abxy - 2bh/. 11. 6a6i3 _ 962Z»i2 _ 4^anhn + Gaftm^. 12. 12abxY - Sa'^xh - 9bh/z + 6abxyzK III. D. Page 41. 1. 6a;^ + 2a;=^-15a;2 + 13a; + 6. 2. 21a;4-50a:3_33^2 + 5o.r + 21. 3. a;3 + a3. 4. x^-a^ 5. a;'* + a2a;2 + a^ 6. a=^ + 6=* + c^ - 3a6c. ANSWERS TO THE EXAMPLES. 459 III. E. Page 42. i, .x.4_49. 2.6400-9 = 6391. 3. .r'4-27. 4. 4.r--r21. ^. {c -hf- d'= - a' + h''- -^bc + c^. 6. ii''-h\ 7. ir-{a + lf. III. F. Page 45. i. x^^. 2. - ild^x^''''. 3. I2a'^^%'^'^ij^. 4. .i--"-l. 5. a;2" + ax" + 6a;" + a&. 6. a:^'^- 2x-"-a;2 + l. 7. 2x-'" - .f'"+" - 6x2" + 7x"» + 7a;" + 3. 8. a2«&» + feSngU + g2u^n _ ^n^2n _ 5ng2n _ gn^2n. 9. (/■-«//-" -a"&" (a " + ?>").r"+ (a '^ + ^»)x3«-x4". 10. a;iooiooi, III. G. Pages 46— 47. 1.375. 2.9. ^. Aad + ^hc. 4. a- + Z;-. 6. (i) a- + c-; (ii) 8a6. 7. 4a6 + 4cd. 8. -r2x^-30x- + 114x-180. 9. a^-^x^y + 2&x}f-2hf. 10. a**- 729x6. 11. l-|x + \ix2-^x3 + fx4-^x^ 12. .r' - 4rt2j;3 + 3a^x - a^. 13. ^r^ - oax^ + lOaS.r^ - l^dix"^ + Ua^x - ^aP. 14. .r« + 3.c5y - 3.rV - llx^j/^ + Qxh/ + 12x2ar/3g. IV. B. Page 52. l. a + 2&. 2.hax-^b-^. 3. -Al-' + Smy. 4. -3«x3-^?^x2 + 7,cx-38. 5. |x2 + |x + |4. IV. C. Pages 56—57. 1. x-l6. 2. x2 + 25x + 8. 3. 2x3+5x2 + |x + ^. 4. x2-ax + a2^ 5. x2 + ax + a'-. 6. x'^ + y^ + z^- xhf — x-^z - xy- - xz^ - y'^z - yz"^ + 2x^2 . 7. x5 + 2x-« + x=^-4x2-llx-10. 8. 4x2 + 5xy + 10?/2. 9. 2x-'-4x3 + 3x2-2x + l. 10. a:--ax + a2. 11. (a + 3)x + (a -3) y. 12. 6a& + 66c. 13. ^V (-r*^ - 12x5 + 60x^-160x=^ + 240x2 -192x + 64); fx3-|ic2 + fx-l. 14. bx'^ + cx-f. IV. D. Page 58. 1. lla^^x^^. 2. - 17a/3x2. 3. 9a». 4. x"-l. 5. x'^ + x-1. 460 ANSWERS TO THE EXAMPLES. IV. E. Pages 58—59. 1. l + 2x- + 3x2 + 4^3. 2.x^-x{y-l) + y^ + y + l. 3. (a + &)2- (a + fo) c + c^. 4. 3x2-4a; + 5. 5. -60a; + 17. 6. (a + 6)3-c(a + &)2 + c2(a + &)-c3- 7. a;4-3a;3|/ + 6a;V-9a;?/2 + V- 8. 2a;2 + 9.r?/ - 4i/2. 9. 2^2 - 4rry + 5?/2. 10. -a2-2a6 + 262. H. a;6 + 2xy-3xV- 6^V + 2^V + 4a^I/^° + l/^■^• 12. 6a: + a?/ + 1. 13. Sx^ - 4x2?/ + ^^y2 + 2i/3. 14. x'-[h + 1) x + 3. 15. l+5x + 10x2 + 20x3 + 40x*. 16. 2a2 + 3a6 + 2Z>2. 17. .t2 - x?/ + /. 18. 2a + 36. 19. 2/2-2?/. 20. if -hx/-hhj + h\ 21. x2 + (a + 2)x + 3. 22. x6 - 3x5 + 9x4 - 27x3 + 81x2 -243x + 729. 23. a2_2a6-262. 24. x + 1. 25. a^ - 63 - c3 - 3a6c. 26. 7x2 _ 2xy + Sxz + 4?/2 _ 27/z + 22. 27. x3i-2x2 + 3x + 4. 28. x^ + x^y-x'Y + xy^ + y^' 30. a^{h + c){c + d)[d + h) + a{hc + cd + dhf + hcd{hc + cd + dh). 31. 12. Chapter V. Simple Equations. V. Pages 66—68. 1. (i) Yes; (ii) No; (iii) Yes [see Art. 90]. 2. See Art. 90. 3. 1- 4. 6§. 5. 3^. 6. 3. 7. I- 8. 20. 9. 6f. 10. 20/27. 11. f. 12. 1. 13. 2. 14. 9. 15. 20. 16. 7. 17. 12. 18. 1. 19. 72. 20. 7. 21. 13. 22. 1. 23. 10. 24. 7. 25. 12. 26. H- 27. 5- 28. 22. 29. 12^. c 2(a-6j 37.1. 38.2. 39. Iff. 40.-1. 41.-1. 42.1. 43. 44. a + 6. 45. ah. 46. 6. 47. 1. 48. 2070/2079. Chapter VI. Problems. VI. Pages 77—80. 1. 28, 18. 2. a/{m+l), ma/(m + l). 3. lOSJ, 131§. 4. 48, 216. 5. 72. 6. 360. 7. 10. 8. 12. 9. 28, 800. 10. £49. 11. 200 cavalry, 600 artillery, 1800 foot. 12. 5. 13. 3tV 14. 147 miles. 15. 15 miles and 18 miles. 16. £14. 8s. 17. 54, 55. 18. 9, 8. 19. 48, 24. 20. 18. 21. 88 yards. 22. £15. 23. 180 tickets; £50. 24.45. 25. 17, 22, 51. 26. 600; namely, 300 in the gallery, 200 in the pit, and 100 in the stalls. 27. 40 miles an hour. 28. 26. 29. 48s. 30. £9600. 31. 3^ days. 32. 20 shillings, 4 half sovereigns, 4 sovereigns. 33. 3s. 34. £1250. 35. A, 12s.; B, 6s. 36. 4h21'^r\. 37. 9t^7 past 10, and 21x\ minutes past 10. AirSWUBS TO THE EXAMPLES. 461 Examination Questions. Pages 81—84. 1. (i) 8; (ii) -2. 2. 10a-2c. 3. xij-^2x^2ij'~zx; y + 2-z. 4. 4. 5. 7. 6. (i) 190; (ii) 34^. 7. (i) 2a + 4& ; (ii) - 91a. - 56/;. 8. 3^5 + llx^y - 12.rV _ 19,^2^:$ + ^if. 9. ^ + 6. 10.' ^ tad 135s., £ had 90.s. H. 8xif + x'^-6i/ + xhj. 12. (i) -2a + 46-4c; (ii) lOOx-24; (iii) 1. 13. da'^-a^b^ + Aab^ - 4¥; if a=:l and 6 = 2, the given expressions become respectively 9 and -3 and their product becomes -27. 14. 7. 15. 35 miles. 16. 7; -1; 4. 17. (i) 2a -3; (ii)a;-3; (iii) 0. 18. 2a^ + 5a^b- 5ab^-2¥; four. 19. (i)lOa; (ii) -2|. 20. 528. 21. 2a2 + 5a6 + 3&2; Sa^-id^b-GaHj'^-Anb-^ -b*; a"^ -b'- + 2bc - c\ 22. 3&c; a2 + aa: + a;2; Sx + 2y-z. 24. 3. 25. 15 days. 26. (i) x^ + 2x^ + 2x^ + Sx-2', (ii) - a3 - 63 - c3 - 2a6c + a^i + a62 + a^c + ac^ + b'^c + bc^. 27. (.T2 + 15a: + 50)-^(10 + .T) = a; + 5; if a: = 2, 84^12 = 2 + 5. 28. 2x3 _ 3^2 ^. .^ _ 2 ; rt-* + 2a36 + 6a-62 + 2a63 + b\ 29. (i) 1 ; (ii) 3. 30. 22 miles. 31. (i) ac; (ii) - 100a; + 124a; -100. 32. a^- 646-1; a3 + 3a26 + 9a62 + 2763. 33. 2a2-5a6 + 262. 35. 8 miles an hour. 36. 36. 37. x^ - 2x^ + x-^ - 2a; + 1. 40. £4680; £4720. Chapter VII. Factors. VILA. Page 86. I. x{a + b). 2. 5.r(l-4a:). 3. «(a-36). 4. 3Zm(m-30. 5.xy{x + y + l). Q, p {llp^- -2pq -Sq^). 7. 4a;?/22(22/ + 3a;-42'^). 8. {a-b){x-y). 9. (a + b){c + d). 10. {x-S){x + l). II, {a+p)(x + y+z). 12. {ay-l)(y + l). VII. B. Page 88. 1. (« - ii) (a + ii). 2. (9 - 12) (9 + 12) = - 3 . 21 = - 63. 3. {-ry - H) {xy + 11). 4. (1-86) (1 + 86). 5. {x-l){x + l). 6. {x-^/o){x + ^/5). 7. (2-a-)(2 + a-). 8. (a;-2) (a: + 2). 9. (3- 2a) (3 + 2a). 10. (llZ-5;n)(llZ + 5/H). 11. (9iJ -a6) (9iJ + a6). 12. a^ (b- x) (b + x). 13. {ab-xy){ab + xy). 14. (a; - 12/m) (a: + 12Zm). 15. a;2(a;-12w)(.r + 12w). 16. (a - 6) (a + 6) (a^ + 62). 17. (2a;-3a)(2a; + 3a)(4x2+9a2). 18. (2a62c3- 3a:2) (2a62c3 + 3.r2). 462 ANSWERS TO THE EXAMPLES. \^. {x + y-ll){x + y + ll). 20. (a-^-3c)(a-6 + 3c). 2\. ^xy. 22. (a2 + 62_a)(a2 + 62 + a,). 23. [a? -h"^ -h)[a'^-h'^ + h). 24. 4 (a + 6) c. 2b. -l{x-y){x + y). 26. (3a -26) (26 -a). 27. 3a(a-2). 28. {«- (•'c + ?/)} {« + (^ + l/)}. 29. {l-n){l-\-2m + n){P + 2lm + 2m'' + 2mn + n''-). 30. 5(3a:-l)(.r + 3)(a;2 + l). VII. C. Page 89. 1. (^ + 1)-. 2. {x-^f. 3. (2.r-3y)2. 4. (.r2-4)-. 5. (a-p)-. 6) {« + &-2(rt-6)P=(a-36)2. VII.D. Page90. 1. (•^-l)(.i^' + .'r + l). 2. (a + l)(a2-a+l). 3. (2aa;-l){4a%2 + 2aa; + l). 4. (.-c + i/) (T.r^- 13.ry + 7i/2). 5. (.'^ + 1^. 6. (1-?/)^. 7. (.T + rt + llfr^ + fl^ + i-a^-a-fj.T^). VII. E. Page 90. 1. (l-a)(l + a)(l+a2). 2. (ar - 2) (a: + 2) (x^ + 4). 3. 8 {x + ?/) (2.i; + 3?/) (o.t^ + 14.t?/ + 10//2). 4. «(a + 2)(a2 + 2a + 2). 5. (.V - 1)'. 6. (a2 + rt6 + 62 + ft_& + i)(a2_fl& + t2 + 3^_3j + 3), VII. F. Page 91. 1. ^x'^-^xy^-^y'^. 2. a(13a2-10a6 + 262). 3. Ix + y+z. 4. 4(7x2 + 4?/2 + 722 + 8.r// + 8?/5; + 22.'r). 5. 7a2 + 1362 + 21c2 + 19a6 + 336c + 24ca. 6. {a + h){c-d). VII. G. Page 94. 1. {x + %){x + ^). 2. (a-3)(a-4). 3. (3 + 2/) (4 + 2/). 4. (a: -8) (a;- 11). 5. («-7)2. 6. (2/ - 7) (2/ - 12). 7. (a - 26) [a - 36). 8. (xy - 2) (a:?/ - 27). 9. (1-2Z)(1-11Z). 10. (Il.r2 + l)(2a;2 + l). H. (a -86) (a- 116). 12. («' - 66') {a' - 116'). 13. {v - 2) [ly - 9). 14. (1 - Uz) (1 + 6,-). 15. (.T + 2)(.r-l). 16. (a-2)(a + l). 17. {x-10){x + 1). 18. (2/-7)(y + 6). 19. (a + 76) (a -46). 20. (1 -8m) {l + 3m). 21. (a; -12) (^ + 3). 22. (6 + 7) (6-6). 23. (.r^- 15) (.r2 + 6). 24. (a + ll)(a-10). 25. (x-7)(.r + 6). 26. {x + Sij) {x-2y). 27. {y-ll){y + ^). 28. (.r-17) (.r + 5). 29. (ai-19)(ai + 3). 30. («6-ll)(a6 + 22). VII. H. Page 97. I. {x-kf-{lf={x-2){x + l). 2. {y-'^Y-{\f={y-^){y-^)- 3. {n-i2f-i-^={n-i^){n-5). 4. (a+i)^ - (¥)''=(« - 6) (a + 7). 5. {x - ^)^ - (^)2= {x - 13) {x - 8). 6. (1 - 662) _ (ii&)2^ (1 _ 17^,^ (1 + 5t), ANSWER.^ TO THE EXAMPLES. 463 7. (a - V6)'^ - (-^f^6)- = (a - 13i)(a + 2t). 8. [x^ + ^)'' - 3'-^= (x^ + l){x^ + 7). 9. ( V-a)' - (^« + bf = {7a - h) (14a + b). 10. 10{(.T+y)2-(e)2} = (10a:-l){:r + 8). 11. ll{(a + U^)'-(|-|'>)'}=(ll«-2i>){a + 7&). 12. 14 {(y-M^)'-(W-} = (7y-2^)(27/-32). 13. {a:-V(-l)}{^' + v/(-l)}. 14. {a:-2-V(-l)}{^ 15. {^ + t-ix/(-3)}{x + |+iV(-3)}. -2 + V(-l)}- VII.I. Pages99— 100. 1. (i) Yes; (^ + l)(a: + 2). (ii) Yes; x^ + x-o. (iii) No. 2. (i)Yes; (j- + 1) (a; + 3). (ii) No. (iii) Yes; .t2-2x + 6. 3. (i) (x-l) (x + l) {x-2). (ii) 0- (x - 1) (a; - 6). (iii) {x - 1) (x - 2) (x + a). (iv)(x-2)(x + 2)(x + 5). (v)(x+l)2(x-2)(x + 3). VII. J. Page 100. 1. x(x-6)(x + 6). 2. 3x(x + 6). 3. (x-2/)(x2-5xy + 7?/2). 4. (27x3- 7f) (27x3 + }f) = (3x - ?/) (3x + y){9x'^ + 3x7j+y^) {9x^ - Sxij + y"^). 5. Q{a-h){a + h){a^ + h% 6. (a- 1) (a + 1) (ft - 1) (& + 1). 7. (J/ -!)(?/ + 1)2. 8. 2/(x-y)(2x2 + x?/ + 2/2). 9. (3a + 2c) (3a + 2b - 2c). 10. (a -y){a + z) (a^ -az + z^). 11. (a + &)(a-6 + 4). 12. (2x + 3a) (3x-a- 1). 13. (a + &-4c)(a-6 + 4c). 14. (a + 6) (a + 6 + 2c). 15. (3a - 4c) (3a + 2b + 4c). 16. (9x + 8) (8x - 9). 17. (-a + 5c)(3a + 46 + c). 18. (x- 1) (x+l) (x + 3) (x + 5). 19. (x-l-V2)(x-l + ^2)(x+l-V2)(x + l + V2). 20. {a + b){b + c){c + a). 21. a''-abj2^^ + b^; a^ + b^={a? - ab J2 + b''){a^ + ab j2 + b''). 22. x^ + a^x* + a^; .r^ - 2ax5 + 8a3x3 - 32a5x + 64a«. 23. (x + 2/-2)2(y + 2-x)2(2 + x-i/)2. 25. 2a&c(a + t + c). 26. {a-b){b-c){c-a). 27. a\a-b){a-2b). Chapter VIII. Highest Common Factors. VIII. A. Page 102. 1. abc\ 2. ^xYz"". 3. 2;>Vr. VIII. B. Page 103. I. x-y. 2. a. 3. •^+a. ^.x-2. 5. x-l. 6. 2x + l. 7. 3a6(a-i). VIII. C. Page 103. \. a + b. 2.a{a-b). 3.^ + 1. 464 AJ^SWFRS TO THE EXAMPLES. VIII. D. Pages 107— 109. 1. 4^-5. 2.a-l. 3. a + 2. 4. ^ + 3. 5. 2j:-1. 6. J^-9. 7. 2x4-3. 8. 3x2 + ar-3. 9. a;2-l. 10.^^-1. 11. 2:r2 + 3. 12. a;2 + 3x-4. 13. x"--x-l. 14. 2.r-l. 15. ^'-2. 16. x-2. 17. a;2-3a; + 4. 18. x^-Sx-4:. 19. a;2 + 3^ + i. 20. 2.r2-4a; + 3. 21. 2^2 -3a: + 4. 22. 2^2 + 3^ + 1. 23. x--2.r + l. 24. x'^ + Sx + 1. 25. .r2-2a; + 4. 26. x^-lxy + 9i/. 27. .t2 - o.r?/ + 7?/-. 28. 3a; -1. 29. x'^-a'^. 30. 4 (a: -a). 31. &{x + a). 32. .t2 + .r?/ + 3i/2. 33. x + 1. 34. l+a:3-.T4. 35. 2.r + l. 36. ay + bx. 37. -3. 38. 6; x-2. 39. 2rt + 76; 75. 40. x-l. 41. -1- Chapter IX. Lowest Common Multiples. IX. A. Page 111. 1. 6x5. 2. I080a^bc^xyz. 3. lOSai^oSV-^V- 4. {x'^-y^)^- 5. (a: + 2?/)2(x-2?/)3. 6. 72(.r-^)2(x3 + ?/3). IX. B. Pages 112— 113. 1. (.r'^ + i) (x^-i). 2. x(3x-y)(x3 + ?/3). 3_ {x-l){6x^-5x-&). 4. x^ (a;9 + 1) (x6 - .tS i-x^- x^ + x''-x + l). d. {x-l)^x+ 1). 6. {x - 8){x^ - 7x + 9) (x^ -lOx + ll). 7. («-4)(3.r-2)(3x2 + 2x + l). 8. (x + l)2(a;2-3.T + 5)(a:2 + 3x + 5). 9. (a; - 1) (a: + 6) (x^ + 8). 10. {a;2 + a; + l)(2a;3-14.T3 + 26a;-30). H. (x^ + l) {x^-8x + S). 12. {a:2 - 2/2) (a;3 - ^3). 13^ (.^2 + 2x + 3) (3.t2 + X + 2) (2x2 + 3x + 1). 14. (a;2 - 1) (a;2 - 4). 15. (9x2 _ 4) (4^2 _ 9). 16. x^y {X - y) (a;2 - y-^) (x^ - y^). VJ , xY (^' - ?/') (-^^ - V')- 18. a;?/ (x - ?/) (x3 - 3x2?/ + 5x1/2 _ 6^3). ^g ^3^ 2ax2 - a^x - 2a^. 20. x2-12x + 35. Chapter X. Fractions. X A Parens 1 — ^ 2 ^^^^' 3 ''^^^' X. A. rage 118. i- ,. + 5- ^- .r2+8?/2- "^^ x2 + v- 4fl(a2 + l) x2 (5x - 1) x3 + 2x2-11x4-6 *• 3a2 4-l ' 2.1- -1 • ^' 2x2 4-4x-7 * x2 - 5xy 4- 2y'' 2X2 _ 3^^ ^ y 3a2-a6 4-fe2 ?• isffl^- 8. (.-l)(.-l)=..-.-..l. 2a2 4-62 10. 1. ANSWERS TO THE EXAMPLES. 465 X. B. Pages 121— 122. 1. ^^^-j^- 2. -2 2x-a x + ^a 6 3. ~T - ' 4. — ; — . 5. x + a x-\-a (a;2-l)(a; + 3) 2(a:-6) „ 2(& + 2/) Q 5a6{3a2 + 462) O. -? Sw ^r; 7w r \ • *• ~~i • O' x-2)(x-3){.T-4){a;-5) '• 6-?/ "* a-'-lGft^ / /(3x' + 8jy + 36y2) .r-5 3:'^-.f + l 2{x^-m/) ' •••"• (a; - 3) (x - 4) • •'■•^- a-^- ox + 6 12.0. 13. \. U.''<'?±^». 15. ^-* x^-l' '■^' a^ - l)-" ' ^^' {x - 2) (x - 3) 16.^,^,. 17. , o,. '.w .. . 18. ' 19 ^ 20 ^ 21 22 ^<^ + ^'^'^ 23. 0. 24. 0. 25. 0. 26. l. 27. — . 28. tn + n. 29. X. C. Page 124. 1. i. 2. ^^ri:^y 3. a2 + a6 + 62 1 ^ 2xw- ^ ?/2 4. . 5. — ^' 6. — . X + ?/ a; + 3?/ xz X.D. Page 126. 1. ?^^ *-5J?p- 5. a + 6 + c. 6.32/3. X. E. Pages 128—131. 1. o. 7--ll^- ^-^'y- 9- 25^1- 10- -■ 11- 1- 12.0. 19.-3. 20.0. 21.0. 22.0. 23."^,*^. 24.1. 25.0. 2Q.lp±3^^. 27. '^^ 2. a;-4 a^ a; + 4" • x + a * 7. 2:r 3y 7^-^+4l-- 4. Each=^ + ^. a h ^■^.- (2a + 3)(oa + 6) * '^ • x{l + x^) ' 28. -^^Iz^,- 29. a^ + b\ 30. b* + c^. 31. —. 32 """^ a6 {a - b)'^ ' ' ' xy ' (1 + ^^) B. A. 30 466 ANSJVEMS TO THE EXAMPLES. 33. a + h. 34.""-". x + y 35. l-a; + .T2 ^^ a2-62 X ' 2^' a^6 • '''■n- 38. 1. 39. a- -a + 1 ^a"^l * 40.1. 41.^'. 42.1 1^ -^ 44. 0. 45. 1. 46.1. 47. :^^ x + y x-y 48. x:'-xy + ,f-l. Chapter XI. Simple Equations continued. XI. Pages 135— 138. 1. i- 2. f 3. -J- 4.33/16. 5.7. 6. -*• 7.2- 8.9. 9.2, 10. -J. 11. a + i- 12.6. i3_(2a-!iia^6)^ 14. i(d-o). 15. i (a-). 16.^ + "- 17.„^?|;^.- 18. f. 19.« + 6 + o. 20.11. 21. 4. 22. - -^"'^^"-^-"-^-^-^^^ . 23. 388 acres. a + o + c 24. 16. 25. £570. 6s. M. 26. 7 miles. 27. 14 miles. 28. ^ miles. 29. ^ in 4'« 37^; jB in 4" 35^ 30. Rate of boat is 11 times rate of stream. 31. 2^ miles. 32. 35 7o above cost price. 33. 6 "/(). 34. 4500. 35. 8 times as much spirit as water. 36.136. 37.50. 38. S^l'i'^and 5'^42t\™; 15i\minutes. 39. 1tt9 seconds past 12. 40. 3/^ hours; 2^\ hours. 41. l/:j minutes to 12. 42. 8f o'clock. 43. ^^ . pb {mil - mp + ain + an) ma{)i-p) Chapter XII. Simultaneous Simple Equations. [The numbers placed first and second in the answer are respectively the values of x and y tvhich satisfy the equation. For example, the solution of Ex. 1 is a; = lll, y = ll.] XII. Pages 148—151. 1. Hi; n. 2. 9; 8. 3. 91; 19. 4. f; f- 5. 3; 10. 6. f; |. 7. 2; 3. 8. 39; 2. 9. -iV;i- 10. 5; 3. 11. 16|; 10|. 12. f;-i. 13. a; b. ^^•h-'- 15. i;^. 16.^; ^\ 17. A; xV 18. 3; 4. 19. 3; -1. 20. 6; 1. 21. 7; -5. i2. h h 23. I ; i- ANSWERS TO THE EXAMPLES 467 24. 3; 0. 25. 12; 3. 26. 13; 17. 27. 2; 3. 28. 3; 4. 29. 4; 3. 30. 4; -5. 31. 112; 168. 32. 36; 40. 33. 3; 4. 34. -1/37; -2/37. 35. t^V; i- 36. 15; 16. 37. -12;-14§. 38. 12; 16. 39. 12; 14. 40. 15; 6. 41. 144; 216. 42. 40; 30. 43. 308; 21. 44. 15; 18. 45. 12; 24. 46. 14; 18. 47. 10; 12. 48. t'tt; tV 49. l\ 3. 50. 12; 8. 51. 5;2i. 52. i; f. 53. i; f 54. |; i- 55. ^; i- ,-« , , ^„ h a __ ah ah 56. i; J. 57. --;--. 58.— ,;-„-^. __ h' + C'-a^ cr + a^-h- ^/n «, « , 59. -^^— ; —2^7—- 60. 2&-a;2a-6. 61. i {7a + 86); ^ (8a + 76). 62. -1;1. 63. a + c; & + c. 64. a + 6; a-&. 65. 12; -2. 66. 3; 3. 67. 16; -4. 68. 14; 17. 69. i; -17. 70. 192/65; 64/35. 71. -{a + &); a + 6. 72. 4; 5; 6. 73. 9; 11; 8. d-{ah-hc + ca) ^ dr {he - ca + ab) d? [ca-ah + hc) 74 2a6c ' 2a&c 2a6c Chapter XIII. Problems leading to Simple Equations. XIII. Pages 158—164. 1. 36; 35. 3. 41; 14. 4. 73. 5. 31; 13. 6. 31/43. 7. h 8. i 9. 9; 7; 5. 10. £37. 10s.; £30. 11. £38; £32. 12. 12s.; 6s. 13. Is. Mr, 2s. 6d.; 2s. 2d. 14. 135s.; 90s. 15. 24 yds.; 16 yds. 16. 120 lbs. ; £16. 10s. 17. 30 years ; Ih, years ; 6 years. 18. 12 years. 19. 17 years, 14 years, 12 years, 9 years ago. 20. 23 + a; | (23 + &). 21. p(a-c)/(6-c); p {h- a)l{b -c). 22. £2400; £900. 23. £450; £225; £237. 10s.; £87. 10s. 24. 50 and 30 miles an hour. 25. 359; 241; 128. 26. £20 for an ox; £2. 10s. for a sheep. 27. 10 of each. 28. £150, 125 qrs. of wheat, 200 qrs. of barley. 29. £9975; £95, £105. 30. 4; 10; 2; 12. 31. 108. 32. 15§; 13^; 18|. 33. 1881. 34. 6770. 35. 12 and 15 miles. 36. 6 minutes. 37. ^ in 5 minutes ; B in 5 min. 20 sees. 38. ^ in 24 days, B in 32 days, and G in 48 days ; iu the ratio 7:3:2. 39. 48 miles. 40. 44 sovereigns, 208 half-crowns, 600 shillings. 41. A has 6s., B has 10s., C has 16s. 42. £800 at 3%, £1200 at 40/0- 43. 3 pence, 6 halfpence, 2 three-penny pieces. 30—2 468 A.VSWUBS TO THE EXAMPLES. 44. 3 florins, 6 sixpences, 4 half-crowns. 45. 2s. 6d. 46. 12 quarts. 47. 160 acres arable, 60 acres pasture. 48. A has £3000, B has £5000. 49. £2800 at 4o/(,, £1200 at 7%. 50. £2320 and £2350. 51. 40 men; 72 days. 52. 13000 town subscribers, 14000 country subscribers. 53. 9 gallons; 18 gallons. 54. 3 miles an hour, 4 miles an hour, 5 miles an hour. 55. 55 seconds; 11 seconds. 56. 12 miles. 57. £88; £118. 58. 9 yards by 4 yards. 59. The quick, which anives at 10'* 6*" : the slow steamer arrives at 10^ 15°^. Chapter XIV. Miscellaneous Propositions. XIV. A. Page 166. XIV. B. Page 168. 1. 216. XIV. C. Page 170. 1. 13?/ -5 = 0. 2. i/{a'' + ah + h^) = a%. 3. fl-l = 0. 4. 5x2 + 4 = 0. 5. m-x- + 2 (mc -2a)x + c'^ = 0. 6. x''-ax = 0. 17. 37.r-39 = 0. XIV. D. Page 172. 1. (i) Symmetrical to a and Z; ; (ii) Symmetrical to a, h, c; (iii) Not symmetrical; (iv) Symmetrical to a, h, c. 2. No (see Art. 178): no (see Art. 74). 3. (i) c-a, a-b; (ii) c^ - a~, a^-b"^; (iii) b {c-a), c{a-b); (iv) (h -c){c + a), (c -a){a + b). 4. x + 2y. 5. 3{x + i/). XIV. E. Page 175. 1. '— y. 2. Any positive value of X, and any negative value of x less than - 7. 3. If x> -I, and if -4>a;> -5. XIV. F. Pages 176— 177. 1. (2;i + l)2-(2;i-l)3 = 4(2n). 5. Subtract 14 ; the result is a number of two digits, which are the numbers thought of. 7.226. Q. 1. 14:. {^) ax'^ + bxij + cy^; (ii) ax^ + bxy + a?/- ; a {x- + ij-) + bxy +c {x + 7j) + d. 15. x^ + if + z^ + 2{xy + yz+zx). 16. 5y^-Q = 0. 17. a3 + 63 + c3 - Sabc = 0. 18. The first, if x'^>a^; the second, if x^ < a"^. ANSWERS TO THE EXAMPLES 469 Examination Papers and Questions. Paper A. Page 178. l.ah-ac-c'. 2.4^. 3. .r--(j/-l)-. 4. (8.r-3)(.r + 2). b. (ac + hd) {ad + hc). 6, .r4 + .-r3_2x--4. 7. 0. 8. % -7. 10. 984 feet, 404 feet. Paper B. Page 179. 1. 8; -i 2. .r^ + 2.r4 + .T^ - 4.r2 - ll.r - 10. 3. 12. 4. The h. c. f. is S.r^ + 2.t + 1 ; the L. c. M. is (9x=* -x-2){x-4). ^. h^ = 4.ac', 3^3 - 2x'^ + 3a; + 2. 6. frf- 7. 1. 8. (i)144; (ii)3; -1; (iii) .r = lll. 7/ = ll. 9. 2s. M. 10. 300 lbs. Paper C. Page 180. 1. 2a&-262+26c. 2. 0. 3. {ll^+h^?- 4. a-2-3«+l. 5. -^. 6. 0. ft "h O 7. (i)72; (ii).T = 5, ?/ = 4. 8. 4 : 1. Paper D. Page 180. 1. 7. 2. (.r - 14) (^ + 6). 3. 3x(j; + 6). 4. -(a-Z>)(6-c)(c-a). 5. 3a;- 1; a; = J. 6. (i) a;2 + 3a; + 2; (ii) 2. 7. (i) 4§; (ii) IJ; (iii) a; = i(2a + 6), y = l{2a-h). 8. 4 times. Examination Questions. Pages 181 — 183. 1. 27a-396 + 57c + 15fZ; 180. 2. a' + l. 3. (.r- 1) (x- 2)(a;-3). 4- <■' '^£h ■' <«) (^T2^ • 5. 216; 241 ; 245. 6. See Art. 120; (.r - 1) (a; - 5) (x + 14). 7. a + b. 8. -3. 9. (i)3; (ii) 3. 10. 17^ minutes. H. a + h; ^{a + b-c). 12. H.c.F. = a-5b; h.c.:si. = {a - 5b) {a - Sb) {a + 2b) {a + 7b). .^ a;3 + 9x2 + 3a;-166 , ,. ,. .. i^ 1 « 13. (^r4)(:^2H^T5J' ^- 14- ^^^^^1^^^- 16. -s-««. 17.(.-i)(.-.4-l). 18. ^^^^3^^J,^^^^^ -^^. 19. (i) a; = 18, y = lO; (ii) x = a, y = b. 20. 20. 21. ^^-^ . 22. -i- 23. (i)96/5; (ii) 3000/101. 24. x = a,y = b,z = c. 25. 4 miles an hour. 26. (4rt - db) {5a+9b). 28. 4320. 470 ANSWERS TO THE EXAMPLES. Chapter XV. Evolution. XV. A. Page 186. 1. x^- 2. ah\ 3. ^a^ 4. t . b 5. ^. 6. l^V- 7. ^^. 8. .r. 9. -■■>;• 10. 3a%=. 11. -f. 12.^|-^ 13. xV. 14.2.I,'. 15. -|. 16. a&*'. XV. B. Page 187. l.ax-h. 2. Ha: -17. 3. i2/ + i^. 1 a; 4. Sfliaj- 4^304. 5. 8-^^. 6. 19Zw + 15Zw. 7. 3- - o" * XV. C. Page 189. \. x + a + h. 2. ax+h-l. 3. (a-l)a; + 6. 4. (« - 2) a:- (6- 1). 5. a-36 + 5c. Q, x + la-lh. XV. D. Pages 194— 195. \. x'' + 'ix-2. 2.^2-4a:-3. 3. 2a;2-3a: + 9. 4. 2.r2-a; + i. 5. 6.T2_3a; + |. 6. 2a2-3a + 3. 7. 2a2 + 3a + 3. 8. 2x2-.t + 2. 9. 4a;2+16x + ll. 10. ^x^-2x + 5. 11. 3x3 -2a;2 + 3x4-2. 12. 3^2 _ 2a:i/ + ?/2. 13. Zx^-^xy + ^y^. 14. 2a;-^ + i6x-a&. 15. a^+h'^-Vc'^ + d^. 18. 49x4- 28x3- 17x2 + 6x + f; 7x2-2x-f. 19. fl"*x" + 5ca^-2x"+i - 3fl/x. XV. E. Page 197. 1. «+2. 2. 2^2+5. 3. 3-5x. 4. a2-2a + l. 5. 2x2 -x + 1. g^ 2.r2-3.i) + 5. 7. 3.?/2- 2fl?y + a2. 8. a + 36 -2c. 9. a"*-2ax". Chapter XVI. Quadratic Equations. XVI. A. Page 201. 1. 0,3. 2.0,-^. 3. ±3. 4. ±(a + 6). 5. -1, -2. 6. -a, -2a. 7. 15, -4. 8. -4, -5. 9. 10, -3. 10. 4, -1. 11. -12,5. 12. -l,f. 13. -1, -*. 14. 17, -4. 15. 3,5. 16. -a, -b. 17.2, -1. 18. -2, -i. AJ^SWEBS TO THE EXAMPLES. 471 19. orf, 4rt. 20. -1, S- 21. -2t, 3. 22. -5a, -V- a. 23. a-c,c- b. 24. -1, -2i. 25. 0,|. 26. l,^^^ XVI. B . Pages 205— 206. 1.-11,5. 2.-2,^. 3. 2, -h 4. 2, If. 5. *, -li. 6. 3, i. 7. -f, -^ 8. -§, h 9. 7, -iV 10.-1,3. 11. i, f 12.-1,3. 13. 4, -6i 14. -11, 9}f. 15. h -h 16. 14, 2i. 17. i, |. 18. h h 19. h t- 20. i -349. 21. 2, J.. 22. ±a. 23. 17, 23. 24. -1, 23. 25. -1, ^^. 26. ^ &-2a. 27. «,^ oq 2 a on 1 1 on '' + ^ ^''' 31. 1, tV 32. a + c, c-b. 33. 1, -^. 34. 1^, 2a -3i. XVI. C. Pages 209—211. 2. All of them are roots. 3. -lis a root. 4. (i) .r2_3a:4-2 = 0; (ii) .r2 + 8a; + 15 = ; {Hi) x^-4x = 0; {i\) x'^--{a + b)x + ab = 0. 5.5,-1. 6.^,-^. 7. 5,i. 8. 3, -l^v 9. 2, -l,v 10. h -|. 11. -§, i. 12. -T^T, V-. 13. 4, -5i. 14. -i, i- 15.2,-37/15. 16. -1^?, 8. 17. -§, |. 18. i, -f. 19. V, -1- 20. -1, i. 21, h 4. 22. -I, f. 23. -h I- 24. V-, 7. 25. 6, Hi. 26. h 2. 27. - 2, 10. 28. 5, 22/23. 29. h h 30. 2, 9. 31. -1, 44/31. 32. 0, ^\ 33. ^, "-^' . a a 34. 2a, 2?>. 35. -1, "L::^. 36. « + &, i(a + 6). 37.0,-—. a a + o 38. -i, 2. 39.2,31/5. 40.-3,2. 41.3,14. 42. -¥, i- 43.^,2. 44. --V-, 2. 45. -¥, 1. 46. -i, 3. 47.-3,6. 48.1,3. 49. -4, -v- 50. 0, |. 51.0,1. 52.1,5. 53. -¥, 4. 54. 8, 9. 55. 4, 8. 56. 4, 7. 57. f, 4. 58. -23/12, 2. 59. 2, 5. 60. ±2. 61. ±4. 62. =ti. 63.8. 64. f. 65.3. 66. -1, a'^ + a. 67. i (3a + 26), 4 (2a + 36). a2 &2 a2 + 62 68. -^, -. 69. 0, --^^' 70. a + 6±V(«--«i + Z>'). 71. t«, 3a. 72. a + 6, M 73. a, ^^^""^^ a + o a 74 c Jl^^^J^) 75 -a -& 76 a'-^ ^' -c{a' + h^ ) ^^''^ ab + bc-ca- ^^' ''' ^- ^^- ^' c(a2 + 62_«e-6c)- 472 ANSWERS TO THE EXAMPLES. Chapter XVII. Simultaneous Quadratic Equations. [The pairs of roots of each system of equations are separated by a semicolon. For example, the roots of the equations 1 in XVII. A. are x = 5, y = l; and x=l, y = C).] XVII. A. Page 216. I. 5, l; l, 5. 2. 5, 4; 4, 5. 3.10,9. 4.6,5. 5. 7,1; 1,7. 6. 1,2; 5, -2. 7. 7,4; -4, -7. 8. 6,1; 1,6. 9. 3,2; 1-, ^. 10. Sb-a,da-b; 3a-b,Sb-a. 11. 2,-1; 15/8, -13/14. 12. 7, 2; 2, 7. 13. 5/12, 3/16; 3/16, 5/12. 14. 4, 2; -6, 12. 1R ^ ??! ift ^ ^ a {2b -a) b{2a-b) ^^' 6 ' a ■ ■^°' 6 ' a ' & ' a ' XVII. B. Pages 222—223. 1. 6, 3; 3, 6 ±^, =Fi. 8. ±4, ±2. 9. ±2, ±1; ±1, ±2. 10. ±1, ±2 ±1, ±|. 11. ±1, ±2. 12. ±4, ±1; ±13^(5/68), i 10^(5/68) 13. ±5, ±1; ±13, =f7. 14. ±2, ±1; i-V", ±¥-. 15. ±2, ^1 ±7, ±4. 16. ±1, ±3; ±1, t12. 17. ±5, ±i; ±7^1, ±W| 18. ±2, ±3; ±5^1, Tv/i- 19. 7, 3; |, -f ; 1(13^7^=^167) ^(11^7-167). 20. ±3, ±5; ±V-, t^ 21. ±8, ±5 ^ISJh ±3^1. 22. ±3, ±1. 23. ±1, ±7; ±7, ±1 24. ±1, ±2. 25. ±2, ±2; ±1, zpi. 26. 2, 3; 3, 2 27. ±3, =F^; ±23VuV, ±¥V7V 28. ±l, ±2 29. ±2, ±1; ±i ±JJ^. 30. ^iP + q), ±1; ±?, ±^^±^ 31. ±(.-,),±l; ±,, ±^-^^ 32.^^^, ^J^;^y 33. ±(2a+6), ±(a + 26); ±(a + 2&), ±{2a + b). 34. 3,4; 5,2 35. ±3, t2; ±^2, =t4;^2. 36. ±2, ^3; ±4^2, ±^/2. 37. 1, 2 I, t._ 38. 7^; I, -V-. 39. 0, 0; 4, -1. 40. 0, 0; 2, -2 ±7^^; 4±Vr2: 41. 4, 1; |, --V-. 42. h i^\ ¥» f 43. 4, 5; -i, -i. 44. 8, 6; 6, 8; -2, 0; 0, -2 45. ±3, d=2. 46. ±4, ±3; ±K/(-l), ±lv/(-l) 47. ±1, ±2. 48. ±4, ±2; ±2VA, ±4^3^. ANSWERS TO THE EXAMPLES. 473 «• u^' U^y «o- *^' ^'-^ *V<-«' -«^"-«- 51. 1, 2; 2,1; -11, -12; -12, -11. 52. ±i, =f1. 53. «,&;«,&• 54. -2, -3. 55. 5, 7; 7, 5. 56. 5, 4; 4, 5. 57. 7, 4; 4, 7. -Q __,i _^_JL-± /S g + fe+c ) Oii.x-y- i; ^^_^_^_^_ y )^(^-2 + ^2 + c2_ftc_ca-a6)i • 59. .T= ±2, ?/= ±1, 2= ±3; x= ±1, ?/= ±2, z= ±3; or x = y, which gives another (but irrational) solution. 60. x = M, y = lS, z=6,u = 2\ and .r = 36, y = 12, z = 4:, « = 3. Chapter XVIII. Problems leading to Quadratic Equations. XVIII. Pages 232—237. 1. 12, 6. 2. 9, 11. 3. £3. 4. 75i yards. 5. 30 feet long, 16 feet broad, 9 feet high. 6. 16 feet by 14 feet. 7. 2210. 8. 784. 9. 496. 10. 578. 11. 17 feet long, 13 feet broad. 12. 8 or 12 miles an hour. 13. 2 hours. 14. 2 hours. 15. 10 miles per hour ; lOi miles per hour. 16. 30 days, 45 days. 17. 16. 18. 4s. 9d. 19. 40. 20. 72. 21. 18. 22. 3.S. 23. 8^?. 24. 48. 25. 50. 26. 42. 27. 50. 28. 240. 29. 15. 30. 39 and 8. 31. tV 32. 48. 33. 36. 34. 73. 35. 74 and 47. 36. 665. 37. 16 and 3. 38. 2, 10, 14. 39. 16 feet and 15 feet. 40. 16 inches and 9 inches. 41. 11760 sq. yds. 42. 30 feet and 18 feet. 43. 10 feet; 80 feet by 60 feet. 44. 150 yards by 130 yards, and 160 yards by 120 yards. 45. 9 yds. and 8 yds., or 8 yds. and 6 yds. 46. A, 324 and 109; B, 227 and 209. 47. A, 196; B, 183; C, 169. 48. 45 and 60 miles an hour respectively. 49. A at 31 miles per hour, B at 7^ miles per hour. 50. 50. 51. 12s.; 9d. and Is. 52. 4s. 53. 5 inches, 6 inches. 54. £125, £95. 55. 10 sheep worth 10s. each, 8 pigs worth 8s. each, 18 geese worth 2s. each. Chapter XIX. Equations reducible to Quadratics. XIX. A. Page 239. 1. ±3, ±2. 2. ±v/il, ±v/-V-. 3. ±K ±t«- 4. 4/8 = 2, 4/(-l)=-l. 5. ^/(-a±6). 6. ±v/3, ±x/-V-. 7. ±3. 8. ±x/(a--^ + i-). XIX. B. Page 240. 1.1,2,2,3. 2.0,-1,2,-3. 3. 1, -4, 2, -5. 4. 2, -f, ^V(9± 7v/33). 474 ANSWERS TO THE EXAMPLES. XIX. C. Page 242. 1. -2, l. 2. 3, -t. 3. -^a, 12«. 4. 4A;{l±V(-3)}. XIX. D. Page 244. 1. i,i,-i, -2. 2.-^,-2,-^,-3. 3. 1, 1, 2, 1. 4. - i 2, - 1, 3. 5. 1, 1, HI ±x/( - 15) ^ Chapter XX. The Theory of Quadratic Eciuations. XX. A. Page 247. 1. 3; 4. 2. 2; f. 3. &c/«; «'• 4.3;|. 5.0; -I. 6.^-^:^^^;0. 7.-V,|. 8. (i) ^^ (ii) ^t^ii^EM. 9. .-2,; ,.2.^. XX. B. Page 249. 1. (3.r-4)2. 2. (4.r + l)(4.r + 3). 3. (.T - 4) {x + 1). 4. {x-a + h)[x-a~ h). ^. {x + a-h-c){x-a + h-c). 6. (^-2a-a6) (a;- 26 + tt6). 7. |{3a;-a-6-c + V(a2 + 52 + ^2 _ j^ _ ^^ _ ^^^ | { 3a; - a - 6 - c - ,^(a2 + 62 4. c2 _ jg _ g^ _ ^5j I ^ 8. (.r-?/)(a;-z)(j/-^). XX. C. Page 250. 1- ^' ^- 2. &, « + ^ . 3. a, 3-«. 4. --. XX. D. Pages 252—254. 1. -f 7. c = a + 4&, orc=-a. 9. 6a;2-13.T + 6 = 0. 10. ra;2- ga;+2) = 0. 12. if;-|^ 13. a=±l, &==f2. 14. i?=-2, (/ = 1. 17. 9.r2 + 30.r + 25 = 0. 21. 262 = 9ac. 26. 0, 9. 28. 1, 8. 29. 2, 6. XX. E. Page 256. 3. -i- 4.5. 5. -i- 6.9. 7. Each is ^a. 8. ^ bought 32, J5 bought 24, C bought 8, D bought 64. [If X be the number bought by C, then x- - 16.t is a minimum; hence, a; = 8.] Chapter XXI. Indeterminate Equations. XXI. Pages259— 260. 1. 3, l; 0,3. 2. 17, l; 10,4; 3,7. 3. 1, 53; 3, 40; 5, 27; 7, 14; 9, 1. 4. x = 2t, y = St-5; 4, 1. 5. .r = 9f-l, y = 7t-A; 8, 3. 6. .r = 17f + 2, y = lBt + l; 2, 1. Aiys]vi:j^s TO the examples. 475 7. 15. 8. 26 or 24, according as payment in one kind of coin alone is or is not reckoned as permissible. 9. 2. 10. 7 and 28; 14 and 21. H. 165. 12. 79. 13. A gives 2 half-crowns and receives 4rf. 15. 3 and 22; 9 and 16; 15 and 10; 21 and 4. 16. 1 and 20; 2 and 13; 5 and 6. 17. 11, 29. 18. Hendrick and Anna, Claas and Catrijn, Cornelius and Geertruij, were respectively man and wife. Examination Questions. Examination Questions. Pages 261 — 264. 1. 12a;2-25.r + 12 = (3x-4)(4x-3). 2. (i) 4, ^; (ii) 5, 3; -AJ, -ss. (iii) 0, 1; -2, -5; -1±3V(-1), -2=fV(-1). 4. 160. 5. 28 feet by 30 feet. 6. 4a3 - a - 2. 7. (i) *, 4; (ii) ±3, ±J(-3); (iiij a/&, 6/a; hja, ajb. 8. 135x2 - 6.r - 1 = 0. 9. 1 mile an hour. 10. 20 feet long, 15 feet broad, 10 feet high. 11.3. 12. 4a2-a-2. 13. (2.r- l)2(.r-2)2. 14. (i)-A,2; (ii) ±3, =f1; ±^(-3), =p^/(-3). 15. 60 gold coins, 120 silver coins. 16. 81; 180. 18. (i) 0, -2a6/(a + &); (ii) 3,-1; -V-. f • 19. qx-+px + 1 = 0. 20. 25 yards. 21. 3fl -26-c + 5d. 22. (i) -1,4; (ii) ±1, ±1; (iii) x=±ll, y=^9. 24. 2,3,4. 25. 15 yards of black at 7s. a yard; 12 yards of brown at 75. Qd. a yard. 26. Sx'^-'Ix + b. 27. (i) - i, 3; (ii) x = S, y = 2; x=-2, y= -S. 28. 3. 29. 1 hour 45 minutes, 2 hours 20 minutes. 30. 12 miles, 3 miles. Chapter XXII. Fractional and Negative Indices. XXII. A. Page 269. 1. a^p+^a. 2. l. 3. a^^m-m. 4. 1. 5. 1. 6. «-'«"^'". 7. a9»-9'^ 8. {hc)v. 9. a. 1 _1 l_ XXII. B. Pages 273—274. 1. a ft^^^^. 1 2. a\ 3. fl. 4. 1. 5. 1. 6. a^. 7. (1^. 8. rt-^*24, 9^ .-j.2(7n+n). 53 1 14. 6a;3-10a;2-3a;2 + x2+20.r-4a;2-16. 1_1 11 13 1 _] 15. b-^ + 2a^h 2c + 2fl262c + 2a2?r2_c2 + 2a2&-i + 2a6'2 1 1 1 -2a26~2_3fl_ 2ab^. 11 1 _ 16. x + ^. 17. x^-n-: 18. a + a262_;,_&2c2 + (.. 19. x-4:. 21. 1 1 • x^ + y^ 25. 2a;- -3.r3 + 7. 476 ANSWERS TO THE EXAMPLES. 2 2 2 11 11 11 20. a;3 + 4?/3 + 9«3 _ 2.'c»?/3 + 6?/V3 + ^xH^ . 22. — i- 23.0. 24. a;-2 + -. l-.r2 1 11 1 26. .r2-2o%3 + 3c2, 27. a3 + 2a26-i-rt6-2-6-3. 28. 3.r-2. 29. 4a; + i?/. 30. h. 31. 0, 3. 32. ah. 33. «-^. 35. 1,1- 36. .r = 3, ^1=2. Chapter XXIII. Logarithms. XXIII. A. Page 276. 1. 2. 2. h 3. 3. 4. f 5. -|. 6. -f. 7. i. 8. -f. 9.2. 10.-2. 11. -|. 12.10. 13.-6. 14.3. 15. -f. 16. f. 17. tV 18.3. 19. - -^' 20. An indefinitely large negative quantity. XXIII. B. Pages 280—281. 1. (i) -2-1072100; (ii) -4948500; (iii) --53402; (iv) 2-52575. 2. (i) 1-1760913; (ii) 3-2886965; (iii) -6532125; (iv) 3-3802112; (v) 1-8750613; (vi) - 3 + -6532125 = - 2-3467875 ; (vii) - 3 + -5563025 = - 2-4436975 ; (viii) -3 + -4771213= -2-5228787; (ix) -2552725; (x) - 2 + -8573325 = - 1-1126675 ; (xi) - 1 + 8750613 = - -1249387 ; (xii) -8627275; (xiii) - 3 + -0969100= -2-9030900. 3. -121519; - -4630553; -1-125906. 4. -1461280; - 2 + -1875207= - 1-8124793; -9-8025455. 5.1-1760913; -4 + 1-3979400= - 2-6020600. 6. '4771213. 7. 2-1303338; -4771213. 8. 1-158051. 9. 1'146128. 10. 1-4771213; -2 + -1303339= -1-8696661. 11. i(3a + 2& + 3c-5); i (9a- 26 + 3c - 1); 6 + C-2. XXIII. C. Page 282. 1. (i) 1-631; (ii) -898; (iii) -683. 2. -712; 6-129. 3. '886; 1-129. XXIII. D. Page 285. 1. 5-9090909. 2. 1-3456789. 3. 1-7531357; 2-9357538. 4. 6-9817371; 3-9817371; 3-9817371. 5. 6414-5; -0064145. 6. 3; 3; 5. 7. 2; 2; 4; 0; i"; 5; 5. 8. --003 = 1-997; --0004 = 1-9996. XXIII. E. Pages 287 288. 1.2-5051500. 2.1-5563026. ANS]VER^ TO THE EXAMPLEiS. 477 3. 2-8o73325. 4. 1-8627275. 5. -8750613. 6. 46532125. 7. 2-6930604. 8. 1-1163460. 9. 2-7262766. 10. -178; 5-615. 11. (f)'>-01; (*)i«>-l. 12. (#i)i«^>100. 13. 18; 87; 53. 14. 22. 15. About 83^ years. 16. 33 years nearly. 17. 7 years. 18. x = 8293039/7043652, y = 4771213/7043652. 19. 3 -a. 20. -9365137; -1760913. 21. 2 8115750. 22. 4096; 32768; 16384 ; 8192. 23. log^o 2 and log^o 3. If logjo 2 = « and log^^ 3 = &, then the logarithms of the given numbers are 6a; 3(1 -a); 2 -a; 1 + 6; 3a -1; a-6; 6- 3a. The characteristics of their logarithms (to the base 2) are 6 ; 6; 5; 4; - 1 ; - 1; -2. 24. 10. 25. 3-27646. 26. 4-59999. 29. 32. 30. .r = l/a, y = \\c. 32. 1. 33. 2. 34. (a + ^> + c)(a + 6-c)(a-6 + c)(-a + Z; + c). 38. 10 years. Chapter XXIV. Surds. XXIV. A. Page 293. 1. v/9; 4/27; 4/8I. 2. i/^ 3. v/18. 4. v/12. 5. sl^j. 6. sl~xhr\ 7. ^. 8. ^'^. 9. ^1^. 10. ^. 11. &. 13. Less. 14. x/35< 4/214 < 4/1290. XXIV. B. Page 294. 1.8- 22^^ + 15.r. 2. 49 - 28^" + 4a. 3 1 1 3. 2 + ^rt . 4. a; + '6^x - 2. 5 . a;^ - 3^-2 + 9a;"^ - 27. XXIV. C. Pages 297—298. 1. ^^^ 2. C • a X" 1 1 o ^^ ^ l-a;^ + 2a;"^ ^ 1 •3. ^.. 4. ^ ^,. 5. 2— . . 7. 2(1-^3). l-ar4 + 3x2 4 2 2 4 2 1 4 2 •8. x-y, 9. a;5-xV + ?/"a. 10. 2V- (2 + 2"^) u; + (2JJ-2JJ+ 1). \\,x^sj^^xy^2-^xjK 12.2^3 + 3^2. 13. l+v/6. 14.7^/2-3. 21. 5+^5. 22. 0. 23. ^/3. 24. v/,- + *+^|- 478 ANSWERS TO THE EXAMPLES. XXIV. D. Pages 303— 304. 1.8. 2. W(4«&-&')- 3. 7. 4. i. 5. 1. 6. 6. 7. 2a. 8. 2. 9. ia. 10. I or 4. 11. f «. 12. 13. 13. ¥• 14. 5. 15. 5. 16. h 17. 6. 18.5. 19.-1. 20.17. 21.6. 22. None : the roots of the rationalized equation are and 8, but neither satisfies the given equation. 23. None: the roots of the rationalized equation are -2 and 7; but neither satisfies the given equation. 24. 9. 25. -|- 26.5. 27.0,3. 28. ±av/3. 29. (c + d)-, i(c + d)2. 30.25. 31. ^-^^^^. 32. h 2. 33. -I, 7. 34. -4, 5, -^, f. 35. 2, 3, -§, 1. 36. -4, 1. 37. -2a± V4a^ + 2a. 38. 0, |. 39.-11,14. 40.4. 41.,-^^^!^^. 42.0. 43. a; = 16, ?/ = 9 ; a = 8, ?/ = l. 44. a; = 16, ?/ = 9. 45. a;=169, 7/ = 121. 46. sjx= -d, ^y= -^', Jx = ^,Jy = U. Chapter XXV. Ratio and Proportion. 1 1 XXV. A. Pages 309—311. 1. 3 : 5. 2. (i) -t'- : if\ 113 3 11 (ii) x^ : if; (iii) x^ : y\ 3. (i) 1; (ii) 16 : 3; (iii) a^ : fcs. 4. (i) 28 : 33; (ii) U : 13. 5. In both cases, 4-a; : 3 -.r<4 : 3. 6. 5 : 2. 7. 19 : 3. 8. 25 : 4. 9. 15. 10. 15. 11. 4 : 7. 16- ^2 /^ 7 , T,o • 17. 8, 22. 18. 7 years (actually to 7^ years). 19. .45, 30. 20. 12, 21; 24, 42. 21. 7, 13. 22. 136. 23. 4 : 7. 24. £242, £300, £358. 25. 3, 12. XXV. B. Pages 315—316. 15. 5:8. 16. 9 : 209. 18. 9, 30. 19. 2:3:4. 1^ xz * 6. — ^. 7. r-, ^ . 8. The former. a + c-2b b + c-2a 12. 3, 12. XXV. C. Pages 318—319. 3. I 4. x^ XXV. D. Pages 319—322. 1. 4 : 7. 2. « : pb. 4. ab. 5. 1 : 3. 7. 14. 8. 22. 9. 9. 10. 21. 12. 10, 6, 4. 13. Ages of sons are 2, 4, 6; age of man is 36. 14. a" : b^. AJVSWEBS TO THE EXA3fPLES. 479 29. 2:3:4. 39. 2. 40. 5, 4. 41. 2, 18. 42. 45, 54. 43. i^305, £410, £515. 44. A receives £1666. 13s. U., B receives £1000, C receives £333. 6s. 8(/. 45. 1 : 2, or 2 : 1. 46. 2i gallons. 47. 4 : 1. Chapter XXVI. Variation. XXVI. A. Page 325. 1. 12^. 2. 4x = 9//-. 3. 6. 4. s/17. 6. 2. 8. 78. XXVI. B. Page 326. 1. 3. 2. 3^. 3. d'b''=i2. 4. Bf hours. 5. 78. XXVI. C. Page 328. 1. H- 2. 128. 3. 4. XXVI. D. Pages 331— 334. \.z^.xif. 4. Vn. 6. 104. 8. 6. 10. ^t- 11. 4. 12. ^. 14. -h 15. t(; = 2(x + 3^2). 17. £34. 18. £llih- 19. 15 feet. 20. 25 square feet. 21. 66 cubic feet. 23. 18 inches. 24. 8 : 7. 25. V^ : ac-. 26. 3 days. 27. 35 francs. 28. 12s. 29. 32 : 25. 30. 26§ miles. 31. £30. 32. The 4'^, 5^ and 6*^ days; 6 days. 33. 20 trucks. 34 g{^^y(g-^0-«g^(/-/^)} beh{c-f) Chapter XXVII. Arithmetical Progressions. XXVII. A. Pages 337—338. 1. (ii) Is in a.p. 2. 6, 7f, 9^ 10§, 12|. 3. (i) 52; (ii) -61; (iii) a^- -SOab + b'^. 4. Yes, the 252"'. 5. na -{n-2)b. Q, - 154, - 148, - 142, &c. 7. 31. 8. 58. 9. -202, 330. 10. Yes. XXVII. B. Pages 340—341. 1. 920. 2. 68895. 3. 94850. 4. 665. 5. 0. 6. 80. 7. 40*. 8. 36$. Q 290. 10. -303|. 11.18. 12.21-6. 13.49. 14." 81-6. 15.76-2. 16. x + M« + l)- 17. i(2/i=*-n-l). 18.1512. 19. 1170. 20. 287^. 21. 576. 22. 2. 23 13. 24.20. 25.30. 26. n^ + 11-1. XXVII. C. Page 343. 1. 6. 2. (i) 5, 6, 7; (ii) 3, 5, 7; (iii) 1^, -^, -2|. 3. i, n, 2s, 3f, 4a, 42, 6f. . 2a + b a + 2b 4a + 5& 5a + 4b 3*3' {a + 2b)(2a + b)' (a + 2b) {2a + b) ' 480 ANSWERS TO THE EXAMPLES. 5. i(4:r3 + a:-2), ^ ^3a- + 2a;-2), \{2x^ + ^x-% i(ar^+4x-2). 6. a? + h"~%ah, a? + li^-%db, a^ + b^ a2 + 6'^ + fa6, a? + h- + ^ah. 8. i(« -!)(« + 0- 9. 6i minutes. 10.5,6,7. XXVII. D. Pages 344—347. 1. (i) 4890; (ii) 25i; (iii) 820a - imOh ; (iv) i» {n - 1). 2. 156. 3. ^^^ , ^-^^ , &c. 5. 369tV 6. 1, 3, 5, 7... 8. 5, 7, 9, 11... 9. 10 or 20. 10. 25. 11. 25. 12. 10 or 3. 13. 297. 14. 6; 2. 15. 32 (2;-- 1). 16. 16^2. 19. 2/«a. 20. 10. 24. 8217 (see Ex. 13). 25. 37i miles. 26. 10. 27. 2wi-?i-4, 2m-7i-2, 2m-». 29. 1, 3, 5, 7. 30. 4, 5, 6, 7. 31. 1, 2, 3, 4, 5. 32. 24, 48, 80. 33. i, 2^, 4^, &c. 34. 3n-2; 6n + l. 36. 41. 37. 4a = («-l)d. 38. 12 feet. 39. 195 miles. 40. £326. lis. 3d. 41. 6 days. 42. 12, 15, 18. 43. 4 miles an hour. 44. In 6 days ; at 90 miles from the starting place. Chapter XXVIII. Geometrical Progressions. XXVIII. A. Page 350. 1, (i), ("), and (iv) are in g.p. 2. if; -1; r/-^'. 3. 96. 4. -2187/64. 5. -1024/81. 6. 3; 7203. 7. 9375. 8. 18, ±54, 162, ±486, &c. 9. 2, 4, 8, 16, &c.; or -32, 16, -8, 4, &c. XXVIII. B. Pages 357—358. 1. 665. 2. 296/735. 3. 16383/9604. 4. -1261/384. 5. -8425/8748. 6. -25862/6613488. 7. 910. 8. 1093711/890625. 9. 416|. 10. 281i. H. 64. 12.101 13. 2^. 14.38-4. 15. 35f|. 16. 211(^/3 -V2)/81. 17. 781. 18. 3741, or -53|. 19. 748^, or -1491- 20. 2. 21. ±2. 22. 3, I, T%, &c.; or 1, |, ^^, &c. 23. 765 or -255. 24. 4^^. 25. 3; 4. XXVIII. C. Pages 361— 362. 1. (i)±4; (ii)±(a;2-a'-^). 13 3 1 2. (i) 2, 4, 8; (ii) ^{x-af{x + af, x'^-a^, ^{x-af{x + df; 113 (iii) 1\ 22, 2^. 3. x^^if^, .T2'^«/4n. 4^ 2^, 9^ 3^ 2, |, f . 5. 768, 1152, 1728, 2592, 3888, 5832, 8748, 13122. 6. ± (6a;2 - 5.r - 6). 8. 25 : 1. 9. 3, 75. 12. a»i(r»-i- 1) (;-+l)/(r- 1), where a is the first term and r is the common ratio of the given g.p. AiVSWENS TO THE EXAMPLES. 181 XXVIII. D. Pages 362 -364. 1. (i) 45920; (ii) 17955; (iii)32; (iv) -3893^/5(v'30-2)/8. 2. 2-366. 3. (i) is not a g.p.; (ii) -6; (iii) 9; (iv) the common ratio is greater than 1; (v) ^ (4 + 3y^2), A :. •^^-(•^•-"-1) , ■rj/(.ry-l), ,,,., a;(x"-l) ."n(n + l)^ *• ^ .r--l ^ .ry-'l ' ^''^ a; - 1 "*" 2 "'• 5. -^"x"^' ^- "^^' ^' ^'■^' *^* ^' <")"~^«- 14. i. 16. 5. 18. 4, 2, 1. 19. 3, 12, 48, 192. 20. 5, 20, 80. 22. ^, -1, |. 23. 1,4,16. Chapter XXIX. Harmonic and other Series. XXIX. A. Page 370. 1. i^. 2.^1 3. 2. -1, 4. (i) If; (ii) 2f ; (iii) j^3. 5. (i) i, i, iV; (ii) ¥, V-, H; riii^ 5 5 5. /i^. 46(a + &) 2b(a + 6) 4Ma + 6) XXIX. B. Page 375. 1.7. 4.i«(M + i)(2« + 7). 5. |/u(m + l)(2»i + l). 6. 7305. XXIX. C. Pages 377—379. 1. (i) in a.p., 63; (ii) in A.P., 36|; (iii) in g.p., 17-H; (iv) in g.p., 13||; (v) in g.p., 1 _ r2'i -6-15125; (vi) in a.p., -58-2. 2. i}) n^\ (ii) x y--^ + n^a ; ("'»'"' {(T59-"i?-4 • 3. (i) V(a= + „-=); (ii)^^,. 4.1-1 = 0. 5.-20/7. 7. x = ^, i/= -1. 8.5,45. a in- 1V*~1 9. 24, 44, 80. 13. ^^^„./ . 14. I, 1, f,...; or -V-» -4, -V.-- 15. Either the a.p. 8, 12, 16, ..., and the g.p. 8, 12, 18...; or the a.p. 8, 4, 0, ..., and the g.p. 8, 4, 2,... . 16. 9 and 12; or 1 and -4. 17. ,r = 9, // = 15; or .t = 49, ?/ = 35. 19. 2, 4, 8. 22. 6, 8, 12. 26. 9, 7, 5, 3. 29. ^rr^^ - (1 - xY 30. J«(n + l)(3H2 + 23;n-46). B. A. 31 482 ANSWERS TO THE EXAMPLES. Examination Papers and Questions. Paper A. Page 380. I. x^ + i/- z^ + 'dxiiz-, {p^r){q+s). 4. (i) li; (ii) -i; (iii) .r^308 and ?/ = 39. 5. 4.r2-3x+2. 6. 5 miles. 8. 5. 9. (i) 2J or 8; (ii) ^a or a. 10. (i) 3660i, 3888; (li) 32U8'>/312.5. Paper B. Pages 381—382. 1. (i)3i; (ii)3; (iii) .1 = 9, y -111. 3. (i) -1 or 6; (ii) tV or 1; (iii) x = l and ij = ll, or x= --%^- and ij=-^-. 5. 7. (S. ±11, ±7. 7. 15 miles; .i and jB respectively walk at the rates of 3f and 3 miles an hour. 8. 12 feet by 9 feet, and 15 feet by 8 feet. 9. ft'^-Va""-; H : 243, or -3125 : 243. 10. (i) 60; (ii) 1690981/15625; (iii) 25/36. H. i(4a + 6), i(3a + 2&), i(2a + 36), i(a + 46). 12. Either 16, 8, 4,..., or 2, -6, 18,.... Examination Questions. Pages 383 — 385. 1 _i 1 _i _i 1 _i 1 1. x^y ^ + x\j ^ + x y + .r V. 3. (i) 280; (ii) 514|; (iii) 108. 5. 6cwt. 7. a '=l--^/a->, a=^ld'- x'^y*. Q, 144. 10. £250, £200. 12. 27/8. 13. (i) 890; (ii) 45; (iii) 9f 14. 16, 24, 36. 15. 3(/. up to 5 lbs., and Id. for every additional lb. iR 2a 999,999 10 ....^ ,. ^ 16- ^/{x + ay 18. (Hi i^oo^ooq; ji- (m) 10, 8, .... 19. x = ^{a + b), y=Jc7b, ~^-^^^^; 21?/ -a). Chapter XXX. Permutations and Combinations. XXX. A. Page 391. 1. 6; 2520; 24; 6. 2. 11- 3. (i) 120; (ii) 60. 4. (i) 360; (ii) 720; (iii) 120; (iv) 2520. 5. 69300; 34650. 6. 7. 7. 48. 8. 216. XXX. B. Page 395. 1. 7; 70; 1081. 2.7. 3.5. 4. 2 or 8. 5. 184756. 6. (i) .o^^,; (ii) i,C,,. 7. 19. AXS]VERS TO TUK EXAMPLES. 483 XXX. C. Pages 398—400. 1. 240; fio. 2. (i) 30; (ii) 72. 4. BOO. 5. rmllr^. 6. ;^<) x ,o,/"r.o- 7. 420. 8. Coefficient of .t^'- in ;.r + .(•- + .^' + .r-^ + .r■■• + .r'';^ that is, in{.r (1 -u;«)/(l -a;)i-* = 140. 9. 144. 10. 7. 11. ;i = 12, r = i. 12. 1. 14. H". 15. 43200. 16. 6. 17. n = 15, r-5. 18. 28. 19. 105. 20. ,7C2X2^C,Xi.a, = 10024560. 21. u.o„Co = (2/i-l);i2; |2nx|>i/2". 22. (i) 496; (ii) 1024. 23. (11111) (3150) (4). 24. (UiA; (ii)i4<^2 + 42240a%" + 42240rt' i-* + 29508rj«/;5 + 14784rt5i6 + 5280a4// + 1320a'''63 + 220a%^ + 22ah^^ + h'K 5. 243.r5 - 810x^y + 1080x3?/2 - 720.1-2?/^ + 240.r//4 - 32//^ 6. left-* + V-a36 + |ft262 ^ ^S^„^3 ^ ^YJ4. 7. a;6/64 + xhjIW + 5xV/48 + 5.t"\;y3/54 + oxh/l'^-OS + .r.vVSl + ?/fi/729. 8. 1 - 7a2/2 + 21a4/4 - 35a6/8 + 35a8/16 - 21aio/32 + 7ai2^64 - r/i4/128. 9. x^<> - %x^ + V--^' - ^-v' + ^.^2 - ^\^. 10. 10^9. 11. 2(.T8 + 28a;6 + 70x4 + 28a;2+l). 12. - 53130.r'^ 13. 10. 14. - 35 . 2-1 . S^x^= - 15120a;'*. 15. - 55/288. 16. 128a'' - 448a6a; + 672aV _ 560a%=^ + 280a-V - 84a2r5 + Uax^ - x'^. 17. [2n.T"/( ?0^- 18. 59136ft''a:fi. 19. -2268. 20. - 945«4. 21. 3"^ term; coefficient =6. 22. H*^' and 12*^ terms. 23. 6"' and 7*'' terms; each coefficient = 402. 24. 7"' term; coefficient = 5376. 25. (i) 10*'' term; (ii) 3'"'i term; (iii) 8"' term. 26. coefficient of. r-' = 90. 27. (10 + 1)' =107213535210701; (10-1)" = 9509900499. 28. (3 + 2)^. 29. (l + .x^)". 31. (l + a-)J-*=l + 14a; + 91a;2 + 364a;^+... 32. l+4a; + 2a:2-8a;«-5.T'' + 8.T5 + 2.r6-4a-7 + x«. 33. 90. 34. (-l)\l^,^^ 35. 1. 36. (3a-2)5. 37. 0)0; (ii) Tj^^,. 484 AN'SWUES TO THE EXAMPLES. XXXI. B. Pages 422—425. I. l-ir-ir2-j-Vjc3-^f^a;4. 2. 1 + 2.r/3 - a;2/9 + 4x3/81 - 7.t-'/243. 3.1- 5a;/4 + 5.r2/32 + 5.r3/128 + 35x^2048. 4. 1 - 3.r + ^x^ - ^-x^ + ^^x\ 5.1- 2.T/3 + .r^/lS + .t3/162 + 5x*IBS8S. 6. l-lax + P^pa^x^-P-^P±g^±^a^x^ 4\ q'^ 7 W-iA^l 4."^^^ o 7.4.1 ?>^ . 7.4.1.2 ?>-^ , '• " V 3«^'^32.2!a2''' 3^3! r,^*^ 34.4! a^"" r. S ( . 11 , 88 .„ „ 616, „ „ 4004 ,, A 8. 2 ( l + -^4.r+g2 42^^+-i^43.r3 + __-- 4%M . 9. /'(l + i.r + t.r2+J^.T3+_«^.T4). 10. l + i.r2+fr4 + ^t-.T6 + ^W^ II. 1+4x3 + 1.^6 + ^V'P9 + _?^5_^12. 12, l + 5a^+J_5.T.2^^5^.^3__5_,54 13 l-8-S-;P^-l),.. 14. 3^1^^) 2'.^ 7'! r! _ ( n+l)(2n + l )...(m-n + l)^,. (n- l)(2n- l)...(m-7i- 1 ) 15. ^j^^ ^ • lb. ^., ^^^ (,_l). 3.1.1.3.5...(2.-5, ^.. ^3^ "■l^;-(f^«) .n 21. „H,..r-"--''. 22. (i) 5*^; (ii) if ^ be negative, then the 2"'i; if ^ be positive and be >n and <» + l, then the (;j + 3)'''. 23. If « = -Vi the sign is +; 17.14.11.8.5.2.1 , ^ 17. 14. ..2.1 „ 3^77! ^'^ ""^ —w^ir- ^- 1 24. The S**'. 25. The e^^^ and T\ 26. The 1^^ and 2"'^. 27. (1 - 4.r) ". 1 _i _n u 28. (l-a^)'". 29. (l-4)"'=x^2. 30. (1-|)"'=4'. 33. 10-04987. 34. 3-995. 35. 9-996666. 36. A>^. - ^.3 + jg^,^. 38. '983. 43. (i)0; (ii) 2". 44. Coefficient of .r'^" = ^'^'^_- j^^^ ~ ^ ^ ; coefficient of .t3i+2 = o. 45. -1- 46.429/256. 47. -1- 49. f-^', ^0 = {^, n - 1) - (.r + 1, 7? - 1), ANSWEIx'S TO TllK EXAMPLES. 485 Chapter XXXII. The Exponential Theorem. XXXII. A. Page 430. 1. '-'• 2. log 2. 3. log 4. XXXII. B. Pages 437—438. 1. 10457609; 111116. 2. 1-9356274. 3. 2-4935452. 4. 31234400. 5. 2-5579353. 6. 1-2363004. 7. 1-355892; 136-6486. 8. 460168. 9. 2141406. 10. 14-677976. 12. 1-778278. 13. 1-389495. 14. 1-401131. 15. 1-4495593. 16. 1-948446. 17. 8465369. 18. 4239-01. 19. To 3 places. Examination Papers. Paper A. Page 439. 2. ncx'^^h{a-\-c)x + {a-vcf={). 4. (i) .r = -^^,, y = ^^,; (ii) 1, ^— ; (iii) *; (iv) .r = 8 and // = - 2, or .r = 2 and y = l. 5. 10 miles an hour. 6. See Art. 328, and p. 343, Ex. 2. 7. See Art. 394. 9. .T = 7-54, ?/=-4-98. Paper B. Page 440. ^ ' «. + 6 + c-afoc + 2 (ii) .r = 7/ = i(-a± s!a^^^h\ or .r = ^(a± ^4fc2-3a2) and y = \{a^ V46^^^3a2). 3. 15481 square yards. 4. If -P, Q-, -R be the i)***, q^^, and r'^^ terms of an A.P., then P(g-r) + Q(r-^) + iJOj- 5) = 0. 6. The(r + l)t'> term, where r is the integer just less than (n - l)/(wt - 1). 7. 4-642. Paper C. Page 441. 1. o. 2. 2(a4-t)(6 + c); 5.r.v [X - y) (.x^ - xy + y-). 3. -^^i^Tg^ g - 4. (D 4^;^, : ^, ; (n) - ^y^- 5. TO = .|a2 + 2c/«; n = c^ja^. 7. (i) -20; (ii) 5; (iii) a, h. 13 11^/7 13^/5 11^7 10. The sides of one arch are equal to o ; of the other are 2a and ^a. 12. The given coefficient = „_^,._iCy. Paper D. Pages 442— 443. 1.1 3. 4^2 6r + 9; (3.r + 2)(64.i'^-729). 4. 2.iYr ^ - 3 + 4^/.t-'. 5. ^2>4ac. 6. (i) ItrV". (ii) a-\-h, h{a + h)\ (iii) x = 9, y/ = 6, ^ = 4, or .i=4, y = ^, x = 9. 486 ANSWERS TO THE EXAMPLES. 7. 3 gallons from the first and 8 gallons from the second. 9. 15. |2w + l j 2w+l I 10. , "i , X, -. '-. 12. Between 114 and 115 years. Chapter XXXIII. Properties of Numbers. XXXIII. A. Pages 451—452. 1. 3205. 2. 975. 3. 194. 4. Sle. 5. 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Gardner 2/- Lamb Tales from Shakespeare Flather lie Macaulay Lord Clive Innes 1/6 Warren Plastings ,, 1/6 William Pitt and Earl of Chatham „ 2/6 t z John Bunyan ,, 1/- t „ John Milton Lays and other Poems Flather 1/6 1/6 Mayor A Sketchof Ancient Philosophy from Thales to Cicero 3/6 )) Handbook of English Metre W- More History of King Richard IH Utopia Lumby 3/6 Milton Arcades and Comus Verity 3/- ,, Ode on the Nativity, L'AUe ■\ 2/6 gro, 11 Penseroso & Lycidas^ " t „ Comus & Lycidas J5 ± )) Samson Agonistes >J 2j6 )) Sonnets >) 1/6 Paradise Lost, six parts )> 2/- each Pope Essay on Criticism West ^1- THE PITT PRESS SERIES, ETC. ENGLISH cotUimied. Author Work Editor Price Scott Marmion Masterman 2/6 91 Lady of the Lake )> 2/6 Lay of the last iSIinstrel Flather ^\- )) Legend of Montrose Simpson 2/6 > ) Lord of the Isles Flather i\- Old Mortality Nicklin 2/6 .[ Kenilworth Flather 2/6 Shakespeare A Midsummer-Night's Dream Verity 16 1) Twelfth Night M 1/6 »» Julius Caesar 1/6 «• The Tempest ,, 1/6 King Lear ,. 1/6 ,, Merchant of Venice > J 1/6 «) King Richard II >) 1/6 As You Like It J> 1/6 5> King Henry V >» 1/6 Macbeth >) 1/6 Shakespeare & Hetcher Two Noble Kinsmen Skeat 3/6 Sidney An Apologia for Poetrie Shuckburgh 3/- WaUace Outlines of the Philosophy of Aristotle 4/6 West Elements of English Grammar 2/6 >} English Grammar for Beginners i/- )) Key to English Grammars 3/6 net Carlos Short History of British India •/- MiU Elementary Commercial Geography 1/6 Bartholomew Atlas of Commercial Geography 3/- Robinson Church Catechism Explained 2/. Jackson The Prayer Book Explained. Part I 2/6 »» ,, Part II In pj-eparation MATHEMATICS, Ball Elementary Algebra 4/6 tBlythe Geometrical Drawing Part I 2/6 Part II ^Z- Euclid Books I— VI, XI, XII Taylor 5/- ,, Books I— VI jj 4/- ^, Books I— IV ^^ 3/- Also separately )) Books I, & li; III, & IV; V, & VI ; XI, & XII [/6 ^a^A *« Solutions to Exercises in Taylor's Euclid W. W. Taylor 10/6 And separately „ Solutions to Bks I — IV 1* 6/- „ Solutions to Books vi. XI 6/- THE PITT PRESS SERIES, ETC, MATHEMATICS conthmed. Author Work Hobson& Jessop Elementary Plane Trigonometry Editor Loney tSanderson Smith, C. Hale, G. Elements of Statics and Dynamics Part I. Elements of Statics „ II. Elements of Dynamics Elements of Hydrostatics Solutions to Examples, Hydrostatics Solutions of Examples, Statics and Dynamics Mechanics and Hydrostatics Geometry for Young Beginners Arithmetic for Schools, with or without answers Part I. Chapters I — viii. Elementary, with or without answers Part II. 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