GIFT or A TEEATISE ON ALGBBEA , MACMILLAN AND CO., Limited LONDON • BOMBAY • CALCUTTA MELBOURNE THE MACMILLAN COMPANY NEW TOEK • BOSTON • CHICAGO DALLAS -SAN MIANCISCO THE MACMILLAN CO. OF CANADA. Ltd. TOEONTO A TEEATISE ON ALGEBEA BY CHARLES SMITH, MA. MABTEB OF SIDNEY SUSSEX COLLEGE, OAMBRIDQEL MACMILLAN AND CO., LIMITED ST MARTIN'S STREET, LONDON 1913 COPYRIGHT First Edition, 1888. Second Edition, 1890. Third Edition, with additions, 1892. Fourth Edition, with additions, 1893. Fifth Edition, 1896. Reprinted 1898, 1900^ 1903, 1905.1908, 1910,1913 I'-n '^"^ ,.»' PREFACE TO THE FIEST EDITION. The following work is designed for the use of the higher classes of Schools and the junior students in the Universities. Although the book is complete in itself, in the sense that it begins at the beginning, it is expected that students who use it will have previously read some more elementary work on Algebra : the simpler parts of the subject are therefore treated somewhat briefly. I have ventured to make one important change from the usual order adopted in English text-books on Algebra, namely by considering some of the tests of the conver- gency of infinite series before makings any use of such series : this change will, I feel sure, be generally approved. The order in which the different chapters of the book may be read is, however, to a great extent optional. A knowledge of the elementary properties of Deter- minants is of great and increasing practical utility ; and I have therefore introduced a short discussion of their fundamental properties, founded on the Treatises of Dostor and Muir. No pains have been spared to ensure variety and inte- rest in the examples. With this end in view, hundreds of examination papers have been consulted ; including, with 383520 VI PREFACE. very few exceptions, every paper which has been set in Cambridge for many years past. Amongst the examples will also be found many interesting theorems which have been taken from the different Mathematical Journals. I am indebted to many friends for their kindness in looking over the proof-sheets, for help in the verification of the examples, and for valuable suggestions. My especial thanks are due to the following members of Sidney Sussex College : Mr S. R. Wilson, M.A., Mr J. Edwards, M.A., Mr S. L. Loney, M.A., and Mr J. Owen, B.A. CHARLES SMITH. Cambbidge, December 12th, 1887. PREFACE TO THE THIRD EDITION. A Chapter on Theory of Equations has been added, which it is hoped will increase the value of the book. CONTENTS. Definitions CHAPTER I. PAGE 1 CHAPTER 11. Fundamental Laws, Negative quantities 9 Addition of terms 10 Subtraction of terms 11 Multiplication of monomial expressions 14 Law of Signs 15 The factors of a product may be taken in any order . . , 16 Fundamental Index Law 18 Division of monomial expressions 19 Multinomial expressions 22 Commutative Law, Distributive Law and Associative Law . . 23 CHAPTER IIL Addition. Subtraction. Brackets. Addition of any multinomial expressions 26 Subtraction of multinomial expressions 27 Brackets 28 Examples 1 29 VIU CONTENTS. CHAPTER IV. Multiplication. page Product of two multinomial expressions 32 • Detached coefficients 36 Square of a multinomial expression 39 Continued products 40 Examples IL . 42 CHAPTER V. Division. Division by a multinomial expression ..... .46 Extended definition of division 60 Examples lU. 51 CHAPTER VI. Factors. Monomial factors • 53 • Factors found by comparing with known identities . , . 54 Factors of quadratic expressions found by inspection ... 65 Examples rV 66 Factors of general quadratic expression 57 Factors found by rearrangement and grouping of terms . , . 60 Examples V. 64 Divisibility of a:" ± a~ by a; ± a 65 ' Remainder Theorem 66 f An expression of the nth degree cannot vanish for more than n values of x, unless it vanishes for all values of a; . . .69 - Cyclical order 71 Symmetrical expressions 71 Factors found by use of Remainder Theorem 72 Examples VI 73 CHAPTER VII. Highest Common Factor. Lowest Common Multiple. Monomial common factors 76 Multinomial common factors 77 Examples Vn 84 . Lowest common multiple „ ... 84 ^ Examples Vni 86 CONTENTS. IX CHAPTER VIII. Fractions. PAGE A fraction is not altered by multiplying its numerator and denomi- nator by the same quantity 88 Eeduction of fractions to a common denominator .... 90 Addition and subtraction of fractions 91 Multiplication and division of fractions 93 Important theorems concerning fractions formed from given fractions 96 Examples IX. .99 CHAPTER IX. Equations. One Unknown Quantity. General principles applicable to all equations 106 Simple equations 107 Special forms of simple equations 108 The problem of solving an equation the same as the problem of finding the factors of an expression 110 Quadratic equations 110 Discussion of roots of a quadratic equation 113 Zero and infinite roots . . 114 Equations not integral 116 Irrational equations 119 A quadratic equation can only have two roots 121 Relations between the roots and the coefl&cients of a quadratic equation 123 Relations between the roots and the coefficients of any equation . 124 Equations with given roots 125 Discussion of possible values of a trinomial expression . . , 127 Examples X 130 Equations of higher degree than the second 135 Equations of the same form as quadratic equations . . . 135 Reciprocal equations . 137 Roots found by inspection 138 Binomial equations 139 Cube roots of unity 140 Examples XI. 141 X CONTENTS. CHAPTER X. Simultaneous Equations. PAGE Equations of the first degree with two unknown quantities , . 145 Discussion of solution 149 Equations of the first degree with three unknown quantities . . 150 Method of undetermined multipliers 151 Equations with more than three unknown quantities . . . 154 Examples XII. 155 Simultaneous equations of the second degree 157 Examples XIII 163 * Equations with more than two unknown quantities . . . 165 Examples XIV 170 CHAPTER X*. • Graphical representation of functions, and approximate solution of equations 172 a CHAPTER XI. Problems. . Problems not always satisfied by the solutions of the corresponding equations 173 Examples XV 177 'CHAPTER XII. Miscellaneous Theorems and Examples. Examples of elimination 182 Equations with restrictions on the values of the letters . . . 186 Identities deduced from the factors of a^ + b^ + c^- Babe . . . 187 Examples XVI 190 CHAPTER XIII. Powers and Roots. Fractional and Negative Indices. • Index Laws 200 Roots of arithmetical numbers 202 Surds obey the Fundamental Laws of Algebra .... 203 Fractional and negative indices 205 ♦ Rationalizing factors 210 Examples XVII 212 A-' CONTENTS. -^ XI CHAPTER XIV. Surds. Imaginary and Complex Quantities. PAGE Properties of Surds 214-216 If a + ^b = c + ^d, where a and c are rational and ^b and ^d are irrational, then a = c and 6 = d 217 If either of two conjugate quadratic surds is a factor of a rational expression, so also is the other 217 Square root of a + /^6 218 Examples XVIII 220 Imaginary and complex quantities 221 Complex quantities obey the Fundamental Laws of Algebra . . 223 Definition and properties of the modulus of a complex quantity 224-226 If either of two conjugate complex quantities is a factor of a real expression, so also is the other 226 CHAPTER XV. Square and Cube Roots. Square roots found by inspection 229 Square root of any algebraical expression 230 Square root found by equating coefficients 231 Extended definition of Square root 232 When any number of terms of a square root have been found, as many more terms can be found by ordinary division . . 233 When n figures of a square root have been found by the ordinary method, n - 2 more figures can always be found by division . 234 Cube root 235 Method of finding the nth root of any algebraical expression . . 238 Examples XIX 238 CHAPTER XVI. Ratio and Proportion. Ratio. Compound ratio. Duplicate ratio 241 A ratio is made more nearly equal to unity by adding the same positive quantity to each of its terms 242 Incommensurable numbers 243 Proportion 244 Continued proportion. Mean proportional . . , , . 245 Geometrical and algebraical definitions compared .... 246 XU CONTENTS. PAGE Variation 248 Indeterminate Forms 251 Examples XX 253 CHAPTER XVII. The Progressions. Arithmetical progression 255 Geometrical progression 260 Harmonical progression 265 Examples XXI 267 CHAPTER XVIII. Systems of Numeration. ;• Expression of any integer in any scale of notation .... 271 Radix fractions . 273 The difference between any number (expressed in the scale r) and the sum of its digits is divisible by r - 1 275 Rule for casting out the nines 276 Examples XXII 277 CHAPTER XIX. Permutations and Combinations. Permutations of different things 280 Permutations all together of things which are not all different . 281 Combinations 283 ifir = nPn-r 285 Greatest value of „C^ 286 (!frfl/)<^n = x^n + a;<^n-l-l/^l+---+yC'„ 287 Vandermonde's Theorem [see also p. 310] 288 Homogeneous products [see also p. 352] 289 Examples XXni 294 CHAPTER XX. The Binomial Theorem. i Proof of the binomial theorem for a positive integral exponent . 297 > Proof by induction 298 Greatest term 301 Examples XXIV 803 -• Properties of the coefficients of a binomial expansion . . . 805 CONTENTS. Xlll PAGB Continued product of n binomial factors of the form x-\-a, x + h, &g. 308 Vandermonde's Theorem [see also p. 288] 310 The multinomial theorem 311 Examples XXV 314 CHAPTER XXI. ■ CONVERGBNCT AND DIVERGENCY OF SERIES. Convergency and divergency of series, all of whose terms have the same sign . ......... 318-326 Series whose terms are alternately positive and negative . . . 326 » Application to the Binomial, the Exponential and the Logarithmic Series 327-329 Product of convergent series 329 If 2a^*■=2&^^ for all values of x for which the series are con- vergent, then af=bf 331 Examples XXVL . 332 CHAPTER XXII. Thb Binomial Theorem. Any Index. ^ Proof of the theorem 886 / Euler's proof 836 Greatest term 340 Examples XXVII 841 Sum of the first r + 1 coeJBficients of Uq + a^x + a^x^ + 343 « Binomial Series 344^ * Expansion of multinomials 348(^ Combinations and Permutations with repetitions .... 350 Homogeneous products [see also p. 289] 852 Examples XXVHI .856 CHAPTER XXIII. Partial Fractions. Indeterminate Coefficients. i Decomposition into partial fractions 362 * Case of imaginary factors 365 * Case of equal factors 366 » Indeterminate coefficients , 368 Examples XXIX 369 XIV CONTENTS. CHAPTER XXIV. n Exponential Theorem. Logarithms. Logarithmic Series. PAGE The Exponential Theorem 373 Examples XXX 878 Properties of logarithms 380 The logarithmic series .... . . ; . 381 Cauchy's theorem 383 Series for calculating logarithms 383 Examples XXXI . . .386 Common logarithms 388 Compound interest and annuities 391 Examples XXXII 393 CHAPTER XXV. Summation of Series. Sum of series found by expressing u^ in the form v„ - v„_i . . 395 Sum of series whose general term is {(a+ n~i . 6) {a + nb)...{a+ n + r -2.1)} . . 397 Sum of series whose general term is !/{(«+ n^ . 6) {a + nb)...{a+ n + r -2 . b)} . . 400 Sum of squares and sum of cubes of the first n numbers . . 403 Sumofr + 2'-+. ..+«'• 404 Piles of shot 405 Figurate Numbers . 406 Polygonal Numbers 407 Examples XXXIII 407 - . . , ,t . a{a + x){a + 2x)...{a+n-l . x) ... Sum of series whose general term is -^ '-^ — ^^ = 410 6(6 + x)(6 + 2x)...(6+n-l . x) Zflpa;!', where ap=J^j)»-+^r-iP''-i + ... + ^o ^^^ Series whose law is not given 414 Method of Differences 415 Recurring Series 417 Convergency of Infinite products 423 Conditions for convergency of 1 ± m + A— ^- =*= V^ 2 3 "*"*•• 2u^ and Sa*Ht^„ both convergent or both divergent .... 426 CONTENTS. XV PAGE »(-^') Sm„ convergent or divergent according as the limit of ^1 428 Examples XXXIV 429 CHAPTER XXVI. Inequalities. Elementary Principles . . 435 Product of any given number of positive quantities, whose sum is given, is greatest when the quantities are equal . . . 436 The arithmetic mean of any number of positive quantities is greater than their geometric mean 437 The sum of any given number of positive quantities, whose product is given, is least when the quantities are all equal . • . 438 If m, a, /S,... are positive and m = a + /3 + ..., then ai"*+...+a„»*^<+... + a,'^ ai^+... + a/ ■> n Sa« - — > \ — j- , unless a; is a positive fraction .... 442 Examples XXXV 443 CHAPTER XXVII. Continued Fractions. Convergents of a + - - ... are alternately less and greater than the continued fraction 447 Law of formation of successive convergents 447 Reduction of any rational fraction to a continued fraction . 449 Properties of convergents 460 Examples XXXVI 454 General convergent 456 Periodic continued fractions 458 Oonvergency of continued fractions 460 Reduction of quadratic surds to continued fractions . . 465 Series expressed as continued fractions 470 Examples XXXVH 473 XVI CONTENTS. CHAPTER XXVIII. Theory of Numbers PAOB The Sieve of Eratosthenes 479 Properties of primes 480-482 Highest power of a prime contained in [» 433 The product of any n consecutive integers is divisible by |n . . 483 Fermat's Theorem [see also p. 493] ^85 Number of divisors of a given number '^^" Number of positive integers less than a given number and prime to it [see also p. 495] . . 487 Forms of square numbers 489 Examples XXXVIII .490 Congruences 492 Wilson's Theorem 494 Extension of Fermat's Theorem 497 Lagrange's Theorem 498 Eeduction of Fractions to Circulating Decimals .... 499 Examples XXXIX 601 CHAPTER XXIX. Indeterminate Equations. Integral solutions of ax^by=:c can always be found if a and & are prime to one another 504 General solution of as- by =c, having given one solution . . 504 General solution of ax + by = c, having given one solution . . 505 Number of positive integral solutions of ax + 6y=c . . . . 506 Integral solutions oi aa: + by + cz = d, a'x + b'y + c'z = d' . . . 508 Examples XL 610 CHAPTER XXX. Probability. Definition of probability 512 Exclusive Events 514 Independent Events 515 Dependent Events 616 '♦ CONTENTS. XVU PAQB Probability of an event happening r times in n trials . , .. 518 Expectation 620 Inverse Probability 621 Probability of Testimony 523 Examples XLL .526 CHAPTEE XXXI. Determinants. Definition and properties of determinants .... 530-542 Multiplication of determinants 543 Simultaneous Equations of the First Degree 545 Elimination 547 Sylvester's method of Elimination 548 Examples XLII 549 CHAPTEE XXXII. Theory of Equations. '' Every equation of the nth degree has n roots 553 « Eelations between the roots and the coefficients of an equation . 554 Sum of the mth powers of the roots 556 Symmetrical functions of roots 557 Transformation of Equations 559 Reciprocal Equations 561 Examples Xliin 664 ^ Imaginary and quadratic surd roots occur in pairs . . . 565 'Roots common to two equations 567 Roots connected by any given relation 567 Commensurable Roots 568 Examples XJjIV 569 Derived Functions 571 Equal Roots 572 Continuity of a rational integral function . . . . . 574 If /(a) and/(/S) are of contrary signs, a real root of /(a;)=0 lies between a and /S 575 The Discriminating Cubic 576 Eolle'B Theorem _^. . . 577 xvm CONTENTS. PAOE X Deicartes' Rules of Signs 578 Examples XL V 680 * Cubic Equations 582 * Biquadratic Equations 583 Sturm's Theorem 585 « Synthetic Division 590 y. Horner's Method of approximating to the real roots of any equation 593 Examples XL VI 596 Miscellaneous Examples 599 Answers to the Examples ....•••! gig CHAPTER I Definitions. 1. Algebra, like Arithmetic, is a science which treats of numbers. In Arithmetic numbers are represented by figures which have determinate values. In Algebra the letters of the alphabet are used to represent numbers, and each letter can stand for any number whatever, except that in any connected series of operations each letter must through- out be supposed to represent the same number. Since the letters employed in Algebra represent any numbers whatever, the results arrived at must be equally true of all numbers. 2. The numbers treated of may be either whole numbers or fractions. All concrete quantities such as values, lengths, areas, periods of time, &c., with which we have to do in Algebra, must be measured by the number of times each contains some unit of its own kind. Thus we have lengths of 4, |, 5J, the unit being an inch, a yard, a mile, or any other fixed length. It is only these numbers with which we are concerned, and our symbols of quantity, whether figures or letters, always represent numbers. On this account the word quantity is often used instead of nwmber, 3. The sign + , which is read ' plus,' is placed before a number to indicate that it is to be added to what has gone S.A. 1 2 DEFINITIONS. before. Thus 6 + 3 means that 3 is to be added to 6 ; 6 + 3 + 2 means that 3 is to be added to 6 and then 2 added to the result. So also a + 6 means that the number which is represented by b is to be added to the number which is represented by a ; or, expressed more briefly, it means that b is to be added to a ; again a + b + c means that b IS to be added to a and then c added to the result 4. The sign — , which is read 'minus/ is placed before a number to indicate that it is to be subtracted from what has gone before. Thus a—b means that b is to be subtracted from a; a — b — c means that b is to be subtracted from a, and then c subtracted from the result; and a — b + c means that b is to be subtracted from a, and then c added to the result. Thus in additions and subtractions the order of the operations is from left to right. 6. The sign x, which is read 'into/ is placed between two numbers to indicate that the first number is to be multiplied by the second. Thus axb means that a is to be multiplied by b; also axbxc means that a is to be multiplied by b, and the result multiplied by c. The sign x is however generally omitted between two letters, or between a figure and a letter, and the letters are placed consecutively. Thus ab means the same as axb, and 5ab the same as 5 x a x 6. The sign of multiplication cannot be omitted between figures : 63 for example does not stand for 6x3 but for sixty-three, as in Arithmetic. Sometimes the x is replaced by a point, ^ which is placed on the line, to distinguish it from the decimal point which is placed above the line. Thus axbxc, a.b.c and abc all mean the same, namely that a is to be multiplied by b and the result multiplied by c. 6. The sign -r-, which is read 'divided by' or *by/ is placed between two numbers to indicate that the fiist DEFINITIONS. 3 number, called the dividend, is to be divided by the second number, called the divisor. Thus a-r-h means that a is to be divided by h ; also a-i-h-r-c means that a is to be divided by 6, and the result divided by c; and a -7- 6 X c means that a is to be divided by h and the result multiplied by c. Thus in multiplications and divisions the order of the operations is from left to right. 7. When two or more numbers are multiplied together the result is called the continued product, or simply the product; and each number is called a factor of the product. When the factors are considered as divided into two sets, each is called the co-efficient, that is the co-factor of the other. Thus in 3a6a?, 3 is the coefficient of abx, 3a is the coefficient of hxy and 3a6 is the coefficient of x. When one of the factors of a product is a number expressed in figures, it is called the numerical coefficient of the product of the other factors. 8. When a product consists of the same factor repeated any number of times it is called a power of that factor. Thus aa is called the second power of a, aaa is called the third power of a, aaau is called the fourth power of a, and so on. Sometimes a is called the first power of a. Special names are also given to aa and to aaa ; they are called respectively the square and the cube of a. 9. Instead of writing aa, aaa, &c., a more convenient notation is adopted as follows : a" is used instead of aa, a' is used instead of aaa, and a** is used instead of aaaa the factor a being taken n times; the small figure placed above and to the right of a shewing the number of times the factor a is to be taken. So also a^h^ is written instead of aaabh, and similarly in other cases. The small figure, or iSiter, placed above a symbol to 1—2 4 - DEFINITIONS. indicate the number of times that symbol is to be taken as a factor is called the index or the exponent Thus a" means that the factor a is to be taken n times, or that the nth power of a is to be taken, and n is called the index. When the factor a is only to be taken once, we do not write it a\ but simply a. 10. A number which when squared is equal to any number a is called a square root of a, and is represented by the symbol ^a, or more often by »/a: thus 2 is ^^4, since 2^ = 4. A number which when cubed is equal to any number a is called a cube root of a, and is represented by the symbol j^a : thus 3 is ^27, since 8" = 27. In general, a number which when raised to the nth power, where n is any whole number, is equal to a, is called an nth. root of a, and is represented by the symbol ^a. The sign V was originally the initial letter of the word radix. It is often called the radical sign. 11. A root which cannot be obtained exactly is called a surd, or an irrational quantity: thus ^/7 and 4/4 are surds. The approximate value of a surd, for example of ^7, can be found, to any degree of accuracy which may be desired, by the ordinary arithmetical process ; but we are not required to find these approximate values in Algebra: for us \/7 is simply that quantity which when squared will become 7. 12. A collection of algebraical symbols, that is of letters, figures, and signs, is called an algebraical expression. The parts of an algebraical expression which are con<- nected by the signs 4- or — are called the terms. Thus 2a — Sbx + ocy^ is an algebraical expression con- taining the three terms 2a, — Sfa?, and + ^cy\ DEFINITIONS. 5 13. When two terms only differ in their numerical coefficients they are called like terms. Thus a and 3a are like terms ; also ba%^G and 3a^6'c are like terms. 14. An expression- which contains only one term is called a monomial expression, and expressions which contain two or more terms are called multinomial expres- sions; expressions which contain two terms, and those which contain three terms are, however, generally called binomial and trinomial expressions respectively. Thus Sab^c is a monomial, a^-{- 36^ is a binomial, and cw?* + 6a; + c is a trinomial expression. 15. The sign =, which is read 'equals,' or 'is equal to,' is placed between two algebraical expressions to denote that they are equal to one another. The sign > indicates that the number which precedes the sign is greater than that which follows it. Thus a>b means that a is greater than 6. The sign < indicates that the number which precedes the sign is less than that which follows it. Thus a-{b^-S(^)d, (iii) a»-6»-2(a-& + c)», (iv) a{b^-c^) + b{c^-d^) + d(a^-c% (v) 3(a + 6)2(c + d)-2(6 + c)2{a + d), , / ., 2a2 262 20" 2cP and (vi) z , , -\ r . b + c c + a b + d a + b Am. 9, 18, 3, 11, 21, 3. 4. Find the values of Va« - h-\ Joab + c, J{h^c^ + h^<^) 9.ndi ^o? + W^AV\ when . a = 5, 6 = 4, c = 3. ylns. 3, 13, 60, 5. 5. Shew that a^-b"^ and (a + 6) {a -6) are equal to one another (i) when a = 2, b = l\ (ii) when a = 6, 6 = 3; and (iii) when a =12, 6 = 5. 6. Shew that the expressions a8-63, (a-6)(a2 + a6+62), (a- 6)8 + 3a6(a-6), and (a + 6)8 - 3a6 {a + b)- 263 are all equal to one another (i) when a = 3, 6 = 2; (ii) when a = 5, 6 = 1; and (ii) when a = 6, 6 •■=3. CHAPTER II. Fundamental Laws. 17. We have said that all concrete quantities must be measured by the number of times each contains some unit of its own kind. Now a sum of money may be either a receipt or a payment, it may be either a gain or a loss ; motion along a given straight line may be in either of two opposite directions ; time may be either before or after some particular epoch ; and so in very many other cases. Thus many concrete magnitudes are capable of existing in two diametrically opposite states : the question then arises whether these magnitudes can be conveniently distin- guished from one another by special signs. 18. Now whatever kind of quantity we are consider- ing + 4 will stand for what increases that quantity by 4 units, and — 4 will stand for whatever decreases the quantity by 4 units. If we are calculating the amount of a man's property (estimated in pounds), + 4 will stand for whatever increases his property by £4, that is + 4 stands for £4 that he possesses, or that is owing to him ; so also — 4 will stand for whatever decreases his property by £4, that is, — 4 will stand for £4 that he owes. If, on the other hand, we are calculating the amount of a man's debts, + 4 will stand for whatever increases his FUNDAMENTAL LAWS. 9 debts, that is, + 4 will now stand for a debt of £4 ; so also — 4 will now stand for whatever decreases his debts, that is, - 4 will stand for £4 that he has, or that is owing to him. If we are considering the amount of a man's gains, + 4 will stand for what increases his total gain, that is, + 4 will stand for a gain of 4 ; so also — 4 will stand for what decreases his total gain, that is, — 4 will stand- for a loss of 4. If however we are calculating the amount of a man's losses, + 4 will stand for a loss of 4, and — 4 will stand for a gain of 4. Again, if the magnitude to be increased or diminished is the distance from any particular place, measured in any particular direction, + 4 will stand for a distance of 4 units in that direction, and — 4 will stand for a distance of 4 units in the opposite direction. 19/ From the last article it will be seen that it is not necessary to invent any new signs to distinguish between quantities of directly opposite kinds, for this can be done by means of the old signs + and — . The signs + and — are therefore used in Algebra with two entirely different meanings. In addition to their original meaning as signs of the operations of addition and subtraction respectively, they are also used as marks of distinction between magnitudes of diametrically opposite kinds. The signs + and — are sometimes called signs of affection when they are thus used to indicate a quality of the quantities before whose symbols they are placed. The sign +, as a sign of affection, is frequently omitted ; and when neither the + nor the ~ sign is prefixed to a term the + sign is to be understood. 20. A quantity to which the sign + is prefixed is called a positive quantity, and a quantity to which the sign — is prefixed is called a negative quantity. The signs + and — are called respectively the positive and negative signs. 10 FUNDAMENTAL LAWS. Note. Although there are many signs used in algebra, the name sign is often used to denote the two signs + and — exclusively. Thus, when the sign of a quantity is spoken of, it means the + or — sign which is prefixed to it; and when we are directed to change the signs of an expression, it means that we are to change the + or — before every term into — or + respectively. 21. The magnitude of a quantity considered inde- pendently of its quality, or of its sign, is called its absolute magnitude. Thus a rise of 4 feet and a fall of 4 feet are equal in absolute magnitude ; so also + 4 and — 4 are equal in absolute magnitude, whatever the unit may be. Addition. 22. The process of finding the result when two or more quantities are taken together is called addition, and the result is called the sum. Since a positive quantity produces an increase, and a negative quantity produces a decrease, to add a positive quantity we must add its absolute value, and to add a negative quantity we must subtract its absolute value. Thus, when we add -f 4 to + 6, we get + 6 + 4 ; and when we add — 4 to + 10, we get + 10 — 4. Hence + 6 + (+ 4) = + 6 + 4, and + 10 + (- 4) = + 10 - 4. So also, when we add + 6 to + a, we get + a + 6 ; and when we add — 6 to + a, we get + a - 6. Hence + o + (+ 6) = + a + ^>, and + a + (— 6) = + a — 6. We therefore have the following rule for the addition of any term : to add any term affix it to the expression to which it is to he added, with its sign unchanged. When numerical values are given to a and to 6, the numerical values of a + 6 and a — b can be found ; but ADDITION OF TERMS. 11 until it is known what numbers a and b stand for, no further step can be taken, and the process is considered to be algebraically complete. 23. When b is greater than a, the arithmetical operation denoted by a — 6 is impossible. For example, if a = 3 and 6 = 5, a — b will be 3 — 5, and we cannot take 5 from 3. But to subtract 5 is the same as to subtract 3 and 2 in succession, so that 3-5 = 3-3-2 = 0-2 = - 2. We then consider that —2 is 2 which is to be sub- tracted from some other algebraical expression, or that — 2 is two units of the kind opposite to that represented by 2 ; and if — 2 is a final result, the latter is the only view that can be taken. In some particular cases the quantities under con- sideration may be such that a negative result is without meaning ; for instance, if we have to find the population of a town from certain given conditions ; in this case the occurrence of a negative result would shew that the given conditions could not be satisfied, and so also in this case would the occurrence of a fractional result. Subtraction. 24. Since subtraction is the inverse operation to that of addition, to subtract a positive quantity produces a decreasey and to subtract a negative quantity produces an increase. Hence to subtract a positive quantity we must subtract its absolute value, and to subtract a negative quantity we must add its absolute value. Thus, to subtract + 4 from +10, we must decrease the amount by 4 ; we then get +10 — 4. Also to subtract — 4 from + 6, we must increase the amount by 4 ; we then get +6 + 4. Hence +10-(+4) = + 10-4 = + 6, and + 6-(-4) = + 6 + 4 = + 10. 12 FUNDAMENTAL LAWS. So also, in all cases a — (+ 6) = a — 6, and a — (—h) = a-\-b. We therefore have the following rule for the subtraction of any term : — to subtract any term affix it to the expression from which it is to be subtracted but with its sign changed. 25. We have hitherto supposed that the letters used to represent quantities were restricted to positive values ; it would however be very inconvenient to retain this restriction. In what follows therefore it must always be understood, unless the contrary is expressly stated, that each letter may have any positive or negative value. Since any letter may stand for either a positive or for a negative quantity, a term preceded by the sign + is not necessarily a positive quantity in reality ; such terms are however still called positive terms, because they are so in appearance; and the terms preceded by the sign -- are similarly called negative terms. 26. On the supposition that b was a positive quantity, it was proved in Articles 22 and 24, that a-\-(+b) = a-\-b.. (i) a + (-6) = a-6 (ii) a — (+ 6) = a — 6 (iii) and a — (-6) = a + 6 (iv), We have now to prove that the above laws being true for all positive values of 6 must be true also for negative values. Let b be negative and equal to — c, where c is any positive quantity ; then + 6==4.(-c) = — c from (ii), and -6 = — ( — c) = + c from (iv). Hence, putting — c for + 6, and + c for — 6 in (i), (ii). .(A). SUBTRACTION OF TERMS. 13 (iii), (iv), it follows that these relations are true for all negative values of 6, provided a 4- (— c) = a — c, a + {+c) =a + c, a — (— c) = a 4- c, and a — (+ c) = a — c, are true for all positive values of c ; and this we know to be the case. Kence the laws expressed in (A) are true for all values of 6. 27. Def. The difference between any two quantities a and b is the result obtained by subtracting the second from the first. The algebraical difference may therefore not be thq same as the arithmetical difference, which is the result obtained by subtracting the less from the greater. The symbol a • 6 is sometimes used to denote the arithmetical difference of a and h. ♦ Def. One quantity a is said to be greater than another quantity b when the algebraical difference a — b is positive. From the definition it is easy to see that in the series 1, 2, 3, 4, &c., each number is greater than the one before it ; and that, in the series — 1, — 2, --3, — 4, &c., each number is less than the one before it. Thus 7, 5, 0, — 5, — 7 are in descending order of magnitude. EXAMPLES. Ex. 1. Find the sum of (i) 5 and - 4, (ii) - 5 and 4, (iii) 5,-3 and - 6 and (iv) - 3, 4, - 6 and 6. Am. 1, - 1, - 4, 0. Ex. 2. Subtract (i) 3 from - 4, (ii) - 4 from 3, and (iii) - a from -b. Ans. -1,1, -b + a. Ex. 3. A barometer fell '01 inches one day, it rose -015 inches on the next day, and fell again '01 inches on the third day. How much higher was it at the end than at the beginning? Am. - '005 inches. Ex. 4. A thermometer which stood at 10 degrees centigrade, fell 20 degrees when it was put into a freezing mixture: what was the final reading? ^ Am. -IC. 14 FUNDAMENTAL LAWS. Ex. 5. Find the value of a-b + e and of -a + b-c, when a=l, 6=:-2andc = 3. 47w. 6, -G, Ex. 6. Find the value of -a+b-c when a=l, 6= -2, 0= -1; also when a=s-2, 6=-l, crr-3. Am. -2,4. Ex. 7. Find the value of a - ( - 6) + ( - c) when a= -3, 6= -2, c=-l, Ans. -^. Ex. 8. Find the value of - a + ( - 6) - ( - c) when a= -2, 6= -3, c= -6. Ant. 0. Ex. 9. Find the value of -{~a) + b-{-c) when a= -1, 6= -2, c= -3. Am. -&. Multiplication. 28. In Arithmetic, multiplication is first defined to be the taking one number as many times as there are units in another. Thus, to multiply 5 by 4 is to take as many fives as there are units in four. As soon, however, as fractional numbers are considered, it is found necessary to modify somewhat the meaning of multiplication, for by the original definition we can only multiply by whole numbers. The following is therefore taken as the definition of multiplication : " To multiply one number by a second is to do to the first what is done to unity to obtain the second." Thus 4 is 1 + 14-1+ 1.; .-. 5 X 4 is 5 + 5 + 5 + 5. Again, to multiply ^ by J, we must do to f what is done to unity to obtain J; that is, we must divide f into four equal parts and take three of those parts. Each of the parts into which ^ is to be divided will be fr-—. , and ^ ^ 7x4 by taking three of these parts we get ^ — 7 . Thus s x ? _5_x_3 "7x4* r MULTIPLICATION OF MONOMIAL EXPRESSIONS. 15 So also, (-5)x4 = (-5) + (-6) + (-5) + (-5) =_5_5_5_5 = -20. With the above definition, multipHcation by a negative quantity presents no difficulty. For example, to multiply 4 by — 6. Since to subtract 5 by one subtraction is the same as to subtract 5 units successively. -5 = -l-l-l-l-l; .*. 4x( — 5) = — 4— 4 — 4 — 4 — 4 = -20. Again, to multiply — 5 by — 4. Since -4 = -l-l-l-l; ,.(_5)x(-4) = -(-5)-(-6)-(-5)- (-5) = + 5 + 5 + 5 + 5 [Art. 26] = + 20. We can proceed in a similar manner for any other numbers, whether integral or fractional, positive or nega- tive. Hence we have the following rule : To find the product of any two quantities, multiply their absolute values, and prefix the sign + if both factors be positive or both negative, and the sign — if one factor be positive and the other negative. Thus we have ( + a)x( + ^) = + a6 (i) ^ (~a) x( + b) = -ab (ii) ( + a)x(-6) = -a6 (iii) ( -a) X (-6) = + a&., (iv) j The rule by which the sign of the product is determined IS called the Law of Signs. This law is sometimes enunciated briefly as follows : Like signs give +, and unliht signs give — . .(B). 16 FUNDAMENTAL LAWS. 29. The factors of a product may be taken in any order. It is proved in Arithmetic that when one number, whether integral or fractional, is multiplied by a second, the result is the same as when the second is multi- plied by the first. The proof is as follows : when the numbers are integers, a and h suppose, write down a series of rows of dots, putting a dots in each row; and take h rows, writing the dots under one another as in the following scheme : a m a row * ***** #****♦ 4( « 4t * ^ iK I h rows. Then the whole number of the dots is a repeated h times, that is a x 6. Now consider the columns instead of the rows : there are ' clearly h dots in each column, and there are a columns ; thus the whole number of dots is h repeated a ^times, that is 6 x a. Hence, when a and h are integerSy ah = ha. When the numbers are fractions, for example f and f , 5 3 5x3 we prove as in Art. 28 that 17 x t = k — 7 • And, by the , rr,. 5x33x5, 5335 above proof for integers, = — ^ = 7 — =; hence i? >< 7 = 7 ^ ir- Hence we have ah = ha, for all positive values of a and 6; and the proposition being true for any positive values of a and 6, it must be true for all values, whether positive or negative ; for from the preceding Article the absolute value of the product is independent of the signs, and the sign of the product is independent of the order of the factors. Hence for all values of a and h we have ah — ha (i). If in the above scheme we put c in place of each of the MULTIPLICATION OF MONOMIAL EXPRESSIONS. 17 dots; the whole number of the c's will be ab; also the number of c's in the first row will be a, and this is repeated b times. Hence, when a and b are integers, c repeated ab times gives the same result as c repeated a times and this repeated b times. So that to multiply by any two whole numbers in succession gives the same result as to multiply at once by their product; and the proposition can, as before, be then proved to be true without restriction to whole numbers or to positive values. Thus, for all values of a, b and c, we have axbxc = ax(be) (ii). By continued application of (i) and (ii) it is easy to shew that the factors of a product may be taken in any order, however many factors there may be. Thus abo^cab = cba, &c (C). 30. Since the factors of a product may be taken in any order, we are able to simplify many products. For example: 3ax4a = 3x4xaxa = 12a^ (- 3a) X (- 46) = + 3(x X 46 = 3 X 4 X tt X & = 12ab, (aby = abxab = axaxbxb = d^b^, (V2a)' = ^/2a x V2a = y/2x^2xaa = 2a'. Although the order of the factors in a product is indifferent, a factor expressed in figures is always put first, and the letters are usually arranged in alphabetical order. 31. Since a^ = aa, and a' — aaa; we have So also and In the above examples we see that the index of the product of two powers of the same letter is equal to the sum of the indices of the factors. We can prove in the following S.A. 2 18 FUNDAMENTAL LAWS. manner that this is true whenever the indices are positive integers : since by definition a'" = aaaa ... to m factors, and a" = aaaa ... to n factors ; .*. a'" X a" = {aaa ... to m factors) x {aaa ... to n factors) = aaa ... to {m + n) factors, = a*"^, by definition; hence a'"xa" = a"'^ (D). The law expressed in (D) is called the Index Law. 32. Since (- a) x {~a)=-^- a' = (+ a) (+ a) [Art. 28], it follows conversely that the square root of a* is either H- a or — a: this is written s/d'' = ± a, the double sign being read * plus or minus.' Thus there are two square roots of any algebraical quantity, which are equal in absolute magnitude but opposite in sign. EXAMPLES. 1. Multiply 2a by -46, a^ by -aSand -2a86by -3a&». Am. -8ab, -a"^, 6a*b*. 2. Multiply - 2xy^ by - 3yh, Saxh/ by - Ba^xy^, and Sa^c^x by 12ab^cx^. Ans. Qxyh, - IBa^x'^y'^, SQa^b^c^x*. 3. Multiply la'^bh^ by - Sa^^c^, and - 2ab'^xY by - Aa^b^x^i/. Ans. -21a76«c», Sa^ftSx^yS. 4. Find the values of ( - a)\ ( - a)^, ( - a)* and ( - a)\ Ans. a^, -a', a*, -a^. 6. Find the values of ( - ab)\ {a%y and ( - Sab^c^)*. Ans. a%\ a8&4, -21a^b^c*. 6. Shew that the successive powers of a negative quantity are alternately positive and negative. 7. Find the cubes of 2a»6, - Zdb^c^, and - 2a%xh)'K Am. 8a«63, - 27a36«c» and - Qa^h^x^y'^. DIVISION OF MONOMIAL EXPRESSIONS. 19 8. Find the value of {-afxi-b)^ of ( - 2a62)3 x ( - Sa^ft)', and of {-Sabc)^x{2a%)K Am, -a2&3, 216a96», 72a»b^c\ 9. Find the value of 3abc~2a^bc^ + Ac*y when a — 2, &=-!, and c=-2. Am. 12. 10. Find the value of 2a^bc-Sb^cd+4,c^da - 5cPab, when a=-l, 6= -2, c= -3 and d= -4. 4m. -148. Division. 33. Division is the inverse operation to that of multi- plication; so that to divide a by 6 is to find a quantity c such that cxb = a. Since division is the inverse of multiplication and multiplications can be performed in any order [Art. 29], it follows that successive divisions can be performed in any order. Thus a-^6-r-c=a-^c-^6. It also follows from Art. 29 that to divide by two quantities in succession gives the same result as to divide at once by their product. Thus a-r-b -r-c = a-7- (be), which is usually written a -r- be. Not only may a succession of divisions be performed in any order, but divisions and multiplications together may be performed in any order. For example axb-^c = a-i-cxb. For a = a-r-oxc; .\ a xb — a-i-cx cxb = a-i-cxbxc'f [by Art. 29] therefore, dividing each by c, we have a X b-r- c = a^c xb. Hence we get the same result whether we divide the product of a and b by c, or divide a by c and then multiply by 6, or divide 6 by c and then multiply by a. 34. The operation of division is often indicated by placiDg the dividend over the divisor with a line between 2—2 20 FUNDAMENTAL LAWS. them : thus j- means a-i-b. Sometimes a/b is written for r . When a-r-b is written in the fractional form 7- , a is b called the numerator, and b the denominator. Since - = 1 -r c, c It -xc=l^cxc = l. c Also ax-xc = ax(-xc)=axl=a. c \c J Therefore, dividing by c, 1 a X - = a -f- c, c so that to divide by any quantity c is the same as to multiply by the quantity -. c Hence a xb -7-c = a-r- c xb^ can be written, ax6x-=ax-x6, c c in which form it is seen to be included in Art. 29 (C). 35. Since a? x a^ = a^, and aJ xa^ =■ a*®; we have con- versely a^ ^a* = a^ and a^" -^ a^ = a'. And, in general, when m and n are any positive integers and m > w, we have for by Art. 31 a"'~" X a** = a*". Hence if one power of any quantity be divided by a lower power of the same quantity, the index of the quotient is equal to the difference of the indices of the dividend and the divisor. DIVISION OF MONOMIAL EXPRESSIONS. 21 and * a'b'o'^a'b'c' = a'b\ 36. We have proved in Art. 28 that ax (—b) = — ab; ,\ (— ab) -5- (~ 6) = a, and (— a6) -r- a = — 6 ; we have also proved that (-a)(-6)= + a6 = (+a)(+6); .\ (+ ab) ~ (- a) = -- 6, and (+ ab) h- (+ a) = + 6. Hence if the signs of the dividend and divisor are alike, the sign of the quotient is + ; and if the signs of the dividend and divisor are unlike, the sign of the quo- tient is — ; we therefore have the same Law of Signs in division as in multiplication. Thus -a'b'^ab' = -a'b\ and - 2a'bG' ^ - Sa^bc" = | ac\ . EXAMPLES. 1. Divide 10a by - 2a, Ba^b^ by - 2a6», and - la'^b^* by - Sa^b^c\ AriM. -6, -^a, -a-^6c^. 2.' Divide - 2a^b'^c^ by 4^a^bc^, - Qx'^y* by Sx^y, and - ba^^x'^y^ by -2a6%V- Am. -^a^b% -2xhj\ |aa;V- 3. Multiply - 2a36c^ by - BaWc^ and divide the result by 8a=*6«c«. Ana. -aV^e. 4 37. The fundamental laws of Algebra, so far as monomial expressions are concerned, are those which were 22 FUNDAMENTAL LAWS. marked A, B, C, D in the preceding articles, and which are collected below : + (+a) = + a + (-a) = -a - (- a) = + a (+ a) (+ 6) = + ab (+ a) (- 6) = - ah (- a) (+ 6) = - ab (- a) (- 6) = 4- ri6 a6c = cba = ca6 = &c. (A), (B), •(C), .(D). It should be remarked that the laws expressed in (A), (B), (C) have been proved to be true for all values of a and 6; but both m and n are supposed in (D) to be positive integers. Multinomial Expressions. 38. We now proceed to the consideration of multi- nomial expressions. We first observe that any multinomial expression can be put in the form a + h + c-\- &c., where a, b, c, &c. may be any quantities, positive or nega- tive. For example, the expression Sx*y — ^xy* — 7ccyz, which by (A) is the same as Safy + (- ^ccy^) + (— 7a:yz), takes the required form if we put a for Sx^y, b for — ^xy^, and c for (— Ifxyz). It therefore follows that in order to prove any theorem to be true for any algebraical expression, it is only necessary MULTINOMIAL EXPRESSIONS. 23 to prove it for the expression a + b + c-i- &c., where a,b,c, &c. are supposed to have any values, positive or negative. 39. It follows at once from the meaning of addition that the sum of two or more algebraical quantities is the same in whatever order they are added. For example, to find how much a man is worth, we can take the different items of property, considering debts as negative, in any order. Thus a + 6 + c = c + a + 6 = 6 + c + a = &c (E). The laws [C] and [E] are together called the Com- mutative Law, which may be enunciated in the following form : Additions or Multiplications may be made in any order. 40. Since additions may be made in any order, we have a + (b-{-c + d+ ...) = (6 + c + cZ+ ...) + a (from E) = b-\-c-\-d+ ... + a = a + b+c + d-\- ... (from E). Hence, to add any algebraical expression as a whole is the same as to add its terms in succession. Since the expression -\-a — b-{-c — d may be written in the form + a + (— ^) + c + (— c?), we have -\-{+a-b + c-d}=+{+a+(-b)+c-h(-d)} = -]-a-\-{-b)-\-c + (-d). Whfen we say that we can add the terms of an expres- sion in succession, it must be borne in mind that the term^ include the prefixed signs. 41. Since subtraction and addition are inverse opera- tions, it follows from the preceding that to subtract an expression as a whole is the same as to subtract the terms in succession. Thus a — {b + c-\-d+...) = a — b — o — d—.,. 24 FUNDAMENTAL LAWS. 42. If c be any positive integer, a and h having any values whatever, then (a + 6) c = (ct + 6) + (a + 6) + (a + 5) + ... repeated c times = a-\-h + a-\-h + a + h+ .., [Art. 40] = a + a + a + . . . repeated c times + 6 + 6 + ^ + . . . repeated c times = ac + ho. Hence, when c is a positive integer, we have (a + 6)c= ac + ho (F). Since division is the inverse of multiplication, it follows that when d is any positive integer And hence {a-\-h) X c ^ d = [(a -\-h) X c] ^ d = (ac + hc) -T- d^ac^d+bc-i-dj c c c that is (a + 6) X -, = a X -, + 6 X 1 . ^ d d d Thus the law expressed in (F) is true for all positive values of c ; and being true for any positive value of c, it must also be true for any negative value. For, if (a + 6) c = ac + 6c, then {a -\-h) {— c) = — {a-\-h) c^ — dc — ho = a(-c)+6(-c). Hence for all values of a, h and c we have {a-\-h)c = ac-\-hc (F). Thus the product of the sum of any two algebraical quantities by a third is the sum of the products obtained by multiplying the quantities separately by the third. The above is generally called the Distributive Law. MULTINOMIAL EXPRESSIONS. 25 43. Since (a + 6) -r- c = (a + 6) x - c c we see that the quotient obtained by dividing the sum of any two algebraical quantities by a third is the sum of the quotients obtained by dividing the quantities separately by the third. 44. From Art. 40 it follows that a + 6 + c + c2 + e + . .. = (a + 6) + c + (d + e) + ... = a + (6 + c + c?) + e + ... = &c., so that the terms of an expression may be grouped in any manner. Again, from Art. 29, it follows that ahcde ... = a (be) (de) ... =a (bed) e ... = &c., so that the factors of a product may be grouped in any manner. These two results are called the Associative Law. 45. We have now considered all the fundamental laws of Algebra, and in the succeeding chapters we have only to develope the consequences of these laws. CHAPTER III. Addition. Subtraction. Brackets. Addition. 46. We have already seen that any term is added by writing it down, with its sign unchanged, after the expres- sion to which it is to be added ; and we have also seen that to add any expression as a whole gives the same result as to add all its terms in succession. We therefore have the following rule : — to add two or more algebraical expressions, write down all the terms in succession with their signs unchanged. Thus the sum of a — 26 + 3c and — 4c2 — 5e + 6/ is a-26 + 3c-4d- 5e + 6/. 47. If some of the terms which are to be added are 'like' terms, the result can, and must, be simplified before the process is considered to be complete. Now two 'like' terms which have the same sign are added by taking the arithmetical sum of their numerical coefficients with the common sign, and affixing the com- mon letters. For example, to add 2a and 5a in succession gives the same result, whatever a may be, as to add 7a; that is, +2a + 5a= +7a. Also, to subtract %i and 5a in succession gives the same result as to subtract 7a; that is, - 2a - 5a = - 7a, ADDITION. SUBTRACTION. 27 Also two *like' terms whose signs are dififerent are added by taking the arithmetical difference of their numerical coefficients with the sign of the greater, and affixing the common letters. For example, + 6a - 3a = + 2a + 3a - 3a = + 2a, also +8a-6a= +3a-3a-2a= -2a. Thus, when there are several *like' terms some of which are positive and some negative, they can all be reduced to one term. Ex. 1. Add 2a + 56 to a - 66. The sum is a-66 + 2a + 66 = a + 2a-66 + 56 = 3a-6. Ex.2. Add 3a2-5a6 + 76', - 4a2 - 2a6 + 36% and 2aa + 6a6-86s. The sum is 3a' - 5a6 + 762 _ 4^2 _ 2ab + 36' + 2a' + 5a6 - 86«. The terms Sa^, - 4a', and + 2a' can be combined mentally; and we have a'. Similarly we have - 2a6 and + 26'. Thus the required sum is a' - 2a6 + 26'. The beginner will find it desirable to put like terms under one another. Subtraction. 48. We have already seen that any term may be sub- tracted by writing it down, with its sign changed, after the expression from which it is to be subtracted ; and we have also seen that to subtract any expression as a whole gives the same result as to subtract its terms in succession. We therefore have the following rule : To subtract any alge- braical expression, write down its terms in succession with all the signs changed. Thus, if a— 2b-}- 3c be subtracted from 2a — Sb — 4.l>s. 6«=- 105a76«. 54. Product of a multinomial expression and a monomial. It was proved in Art. 42 that the product of the sum of any two algebraical quantities by a third is equal to the sum of the products obtained by multiplying the two quantities separately by the third. 32 MULTIPLICATION. Thus {x + 2/) z = xz-\-yz (i). Since (i) is true for all values of x, 3/ and z, it will be true when we put {a + h) in place of x ; hence [{a + h)-\-y]z=^{a + h)z+ yz = az + bz + yz. .*. (a-^b + y)z = az + bz + yz. And similarly (a-\-b-\- c+ d+ ,..) z = az -\-bz + cz -\-dz -h ,.,, however many terms there may be in the expression a + b-\-c + d+ ... Thus the product of any multinomial expression by a monomial is the sum, of the products obtained by multiplying the separate terms of the multinomial expression by the monomial. 55. Product of two multinomial expressions. We now consider the most general case of multiplication, namely the multiplication of any two multinomial ex- pressions. We have to find (a + 6 + c + ...)x(a; + y + ^:+...); and, from Art. 38, this includes all possible cases. Put M for x + y -h z-h ...] then, by the last article, we have (a-^b + c+ ...)M=aM-\-bM-\-cM-\-... = Ma + Mb+ Mc-{- ... ==(x+y + z+ ...)a + (x-^y + z+ ...)b + (x-^y + z-{-...)c-^... = ax + ay -{-az-\- ...-\-bx-^by + bz-\- ...-^cx + cy -\-cz-^ ... Hence (a -\-b-i- c-^ ...) {x-{-y + z -\- ...) = a^-\-ay -\-az-\- ... -\-bx+by + bz+ ...+cx-{-cy-\-cz + ... MULTIPLICATION. 33 Thus, the product of any two algebraical expressions is equal to the sum of the products obtained by multiplying every term, of the one by every term of the other. For example {a + b) {c + d) = ac + ad + be -\- hd ) also (3a + 56) (2a -+86) = (3a) (2a) + (3a) (36) + (56) (2a) + (56) (36) = 6a' + 9a6 + 10a6 + 156' = M + 19a6 + 156'. Again, to find (a —h){c — d), we first write this in the form [a + (- 6)} {c +(— d)}, and we then have for the product ac + a (- rf) + (- 6) c + (- 6) (- d) — ac — ad — bc + bd. In the rule given above for the multiplication of two algebraical expressions it must be borne in mind that the terms include the prefixed signs. 56. The following are important examples : — I. (a + 6)* = (a -H 6) (a + 6) = aa + a6 + 6a + 66 ; .-. (a + 6)* = a'' + 2a6 + 6^ ^"^ Thus, the square of the sum of any two quantities is equal to the sum of their squares plus twice their product. II. (a-6)*=(a-6)(a-6)=aa + a(-6)4-(-6)a + (-6)(-6) = a*-a6-a6 + 6^ ... (a-6)^ = a*-2a6 + 6l Thus, the square of the difference of any two quantities is equal to the sum of their squares minus twice their product. III. (a + 6)(a- 6) = aa+a(-6) + 6a + 6(--6) = a'' - a6 4- a6 - 6' ; /. (a + 6)(a-6) = a*-6l Thus, the product of the sum and difference of any two quantities is equal to the difference of their squares. S.A. 3 84 MULTIPLICATION. 57. It is usual to exhibit the process of multiplication in the following convenient form : a' + 2ab - b' a'-2ab -\-b' a' + 2a'b - a'b' - 2a;'b - 4^M' + 2a¥ d'h' + 2idf - -6* a* - Wh' + 4a6^ - -b\ The multiplier is placed under the multiplicand and a line is drawn. The successive terms of the multiplicand, namely a^, + 2a6, and — 6', are multiplied by a^ the first term on the left of the multiplier, and the products a*, + 2a^b and — aJ^b"^ which are thus obtained are put in a horizontal row. The terms of the multiplicand are then multiplied by —2ab, the second term of the multiplier, and the products thus obtained are put in another hori- zontal row, the terms being so placed that 'like' terms are under one another. And similarly for all the other terms of the multiplier. The final result is then obtained by adding the rows of partial products ; and this final sum can be readily written down, since the different sets of * like ' terms are in vertical columns. The following are examples of multiplication arranged as above described : a + 6 a + 6 a+6 a-b a^ + ab -ab- ■b^ a^ + ab + b* a-b a^-\-ab a/^ + a% + ab* - a:^b -ab^-b» a'* + 2a6 + 62 a^ 6' a3 -63 a-\-b-\-c a + b + c. lix' + xy-2y^ d^ + ab + oc + ah + ac + fe2+ be + bc + c^ + 3arV- x-y- + 2xy^ a" + 2a 6 + 2ac + b'^ + 2bc + c^ ^x* - arV + 4.ri/3-V 58. If in an expression consisting of several terms which contain dift'erent powers of the same letter, the MULTIPLICATION. 85 temi which contains the highest power of that letter be put first OD the left, the term which contains the next highest power be put next, and so on ; the terms, if any, which do not contain the letter being put last ; then the whole expression is said to be arranged according to descending powers of that letter. Thus the expression a^ + d'h-^ah'^ + h^ is arranged according to descending powers of a. In like manner we say that the expression is arranged according to ascending powers of h. 59. Although it is not necessary to arrange the terms either of the multiplicand or of the multiplier in any particular order, it will be found convenient to arrange both expressions according to descending or both according to ascending powers of the same letter: some trouble in the arrangement of the different sets of *like' terms in vertical columns will thus be avoided. 60. Definitions. A term which is the product of n letters is said to be of n dimensions, or of the nth degree. Thus Sabc is of three dimensions, or of the third degree ; and 5a^¥c, that is 5aaabbc, is of six dimensions, or of the sixth degree. Thus the degree of a term is found by taking the sum of the indices of its factors. The degree of an expression is the degree of that term of it which is of highest dimensions. In estimating the degree of a term, or of an expression, we sometimes take into account only a particular letter, or particular letters : thus aa;^ + bx-\-cis of the second degree in X, and is often called a quadratic expression in x ; also ax^y + bxy + cx^ is of the third degree in x and y, and is often called a cubic expression in x and y. An expression, or a term, which does not contain x is said to be of no degree in x, or to be independent of x. When all the terms of an expression are of the same dimensions, the expression is said to be homogeneous. Thus a' + ZaJ^b — 56' is a homogeneous expression, every 3—2 36 MULTIPLICATION. term being of the third degree ; also aa^ + hxy + cy* is a homogeneous expression of the second degree in a; and y. 61. Product of homogeneous expressions. The product of any two homogeneous expressions must be homogeneous ; for the different terms of the product are obtained by multiplying any term of the multiplicand by any term of the multiplier, and the number of dimensions in the product of any two monomials is clearly the sum of the number of dimensions in the separate quantities; hence if all the terms of the multiplicand are of the same degree, as also all the terms of the multiplier, it follows that all the terms of the product are of the same degree ; and it also follows that the degree of the product is the sum of the degrees of the factors. The fact that two expressions which are to be multi- plied are homogeneous should in all cases be noticed ; and if the product obtained is not homogeneous, it is clear that there is an error. 62. It is of importance to notice that, in the product of two algebraical expressions, the term which is of highest degree in a particular letter is the product of the terms in the factors which are of highest degree in that letter, and the term of lowest degree is the product of the terms which are of lowest degree in the factors: thus there is only one term of highest degree and one term of lowest degree. 63. Detached Coefficients. When two expressions are both arranged according to descending, or to ascending, powers of some letter, much of the labour of multiplication can be saved by writing down the coefficients only. Thus, to multiply Zx^ — a; + 2 by 3a;'^ + 2a? - 2, we write 3-1+2 3 + 2-2 9-3 + 6 6-2+4 -6+2-4 9+3-2+6-4 MULTIPLICATION. 37 The highest power of x in the product is clearly a;*, and the rest follow in order. Hence the required product is 9a?* + 3ic' - 2x^ + 6a; - 4. When some of the powers are absent their places must be supplied by O's. Thus, to multiply a?* — 2a;* + a; - 3 by a;* + a;^ - a; - 3, we write 1+0-2+1-3 1+1+0-1-3 1+0-2+1-3 1+0-2+1-3 _l_0+2-l+3 -3-0+6-3+9 l+l_2-2-5-l+5+0+9 Hence the product is x^^-x' - 2a;« - 2af - bx"- - a;" + Sa;* + 9. This is generally called the method of detached coefficients. 64. We now return to the three important cases of multiplication considered in Art. 56, namely, (a + hy = a' + 2ah + If (i), (a-6)* = a*-2a6 + 6^ (ii), {a + h){a-h) = a^-h^ (iii). A general result expressed by means of symbols ia called Q. formula. Since the laws from which the above formulae were deduced were proved to be true for all algebraical quantities whatever, we may substitute for a and for h any other algebraical quantities, or algebraical expressions, and the results will still hold good. 38 MULTIPLICATION. We give some examples of results obtained by substi- tution. Put —b in. the place of b in (i) ; we then have [a + (- b)Y =a' + 2a (- b) + (- b)\ that is (a - by = a' - 2ab + b\ Thus (ii) is seen to be really included in (i). Put V2 in the place of b in (iii) ; we then have (a + V2) (a- V2) =«*- (V2)* = a' - 2. [We here, howevei, assume that all the fundamental laws are true for surds: this will be considered in a subsequent chapter.] Put 6 + c in the place of b in (i); we then have la+ (b -\- c)Y = a' + 2a (b + c) -{-{b +cy ; .-. (a + b-\~cy = a^ + 2ab + 2ac + b' + 26c + c* (iv). Now put — c for c in (iv), and we have {a + b + (- c)Y = a^ + 2ab+2a (- c)-\-b' + 26 (- c) + (- c)*; .-. (a 4- 6-c)'= a' 4-2a6 - 2ac + 6" - 26c+ c*. Put 6 + c in the place of 6 in (iii); we then have {a+(6 + c)}{a-(6 + c)}=a*-(6 + c)' = a''-(6*+26c + c*); /. (a + 6 + c) (a - 6 - c) = a'' - 6* - 26c - c\ The following are additional examples of products which can be written down at once. (a2 + 263) (aa - 2b-^) = (a^)^ - (262)2 = ^4 _ 454. {a^ + jBb^){a^ - ^362) = (a2)a _ (^363)2 = ^4 - 3?A {a-b + c)(a + b-e)={a-{b-c)}{a + {b-c)}=aJ'~{b-c)K (a» + ab + 62) (a« -ab + b^) = {{a^+ 6«) + ab}{(a^ + b^) - ab} = (a» + 62)2 - (a6)2 =a* + a^b^ + 6*. = (ar« +x)2 - (a!2 + l)2 = x« + 2a;^ + .'c2 - {x*+2x^ ^l) = x<^ + x*-x^-l. MtTLTIPLlCATlOlf. S9 65. Square of a multinomial expression. We have found in the preceding Article, and also by direct multiplication in Art. 57, the sqaare of the sum of three algebraical quantities; and the square of the sum of more than three quantities can be obtained by the same methods. The square of any multinomial expression can however best be found in the following manner. We have to find {a->rh + c + d+ ...)(a + 6 + c + fZ+ ...)• Now we know that the product of any two algebraical expressions is equal to the sum of the partial products obtained by multiplying every term of one expression by every term of the other. If we multiply the term a of the multiplicand by the term a of the multiplier, we obtain the term a* of the product: we similarly obtain the terms h'\ c^, &c. We can multiply any term, say h, of the multiplicand by any different term, say dy of the multiplier; and we thus obtain the term hd of the product. But we also obtain the term hd by multiplying the term d of the multiplicand by the term h of the multiplier, and the term hd can be obtained in no other way, so that every such term as 6d, in which the letters are different, occurs twice in the product. The required product is therefore the sum of the squares of all the quantities a, 6, c, &c. together with twice the product of every pair. Thus, the square of the sum of any numher of algebrai- cal qvxintities is equal to the sum of their squares together with twice the product of every pair. For example, to find (a + & + cf. The squares of the separate terms are a^, h^, (?. The products of the different pairs of terms are ah, ac and be. Hence {a + b + c)'^=a^ + b' + c^ + 2al) + 2ac + 2bc. Similarly, {a + 2h- 3c)2=a2+ (26)2+ { - Scf+2a (26) + 2a{ - 3c) + 2 {2b){- 3c) = a2 + 462 ^ 9^2 + ^ab - 6ac - 126c. 40 MULTIPLICATION. ^ And (a-6 + c-d)2 = a2+(-6)2 + c2 + (-d)2 + 2a(-6) + 2ac + 2a(-d) + 2(-6)c4-2(-6)(-d) + 2c(-d) = a2 + 62 + c2 + d2_2a6-h2a« - 2ad - 2bc + 2bd - 2cd. After a little practice the intermediate steps should be omitted and the final result written down at once. To ensure taking twice the product of every pair it is best to take twice the product of each term and of every term which follows it. 66. Continued Products. The continued product of several algebraical expressions is obtained by finding the product of any two of the expressions, and then multiplying this product by a third expression, and so on. For example, to find {x + a){x + b) {x+c), we have x + a x + b z^ + ax + bx + i x^ + ia + b)x + ab x + c a;3 + (a + b) x'^ + abx + cx^ + {a + b)cx + abc x^ + {a + b+c)x''^ + {ab + ac-bc)x + ahe In the above all the terms which contain the same powers of x are collected together : it is frequently necessary to arrange expressions in this way. Again, to find (x^ + aP'f {x + af {x - af. The factors can be taken in any order ; hence the required product = [(x - a) (x + a) {V? + a?)f = [{x^ - a^) [x^ + a^)f = {x^ - a^)^ =x»- 2a^v* + a*. 67. We have proved in Art. 55 that the product of any two multinomial expressions is the sum of all the partial products obtained by multiplying any term of one expression by any term of the other. To find the continued product of three expressions we must therefore multiply each of the terms in the product of the first two expressions by each of the terms in the third; hence the continued product is the sum of all the partial products which can be obtained by multiplying together any term of the first, any term of the second, and any term of the third. MULTIPLICATION. 41 And similarly, the continued product of any number of expressions is the sum of all the partial products which can be obtained by multiplying together any term of the first, any term of the second, any term of the third, &c. For example, if we take a letter from each of the three factors of {a + h) {a + h) {a-h b), and multiply the three together, we shall obtain a term of the continued product; and if we do this in every possible way we shall obtain all the terms of the continued product. Now we can take a every time, and we can do tl\js in only one way; hence a^ is a term of the continued product. We can take a twice and h once, and this can be done in three ways, for the b can be taken from either of the three binomial factors; hence we have Sa^b. We can take a once and b twice, and we can do this also in three ways; hence we have Sab^. Finally, we can take b every time, and this can be done in only one way; hence we have ¥. Thus the continued product is a» + Sa'b + Sab' + 6', that is (a + bf = a' + Sa'b + Sab' + b\ The continued product (x + a) (a; + b) {x + c) can simi- larly be written down at once. For we can take x every time: we thus get x'. We can take two x's and either a or ^ or c: we thus have x^a, x'^b and a?V We can take one x and any two of a, b, c: we thus have xaby xac, and xbc. Finally, if we take no x% we have the term aba, 42 MULTIPLICATION. Thus, arranging the result according to powers of x, we have {x + a) [x + h) {x -\- c) = x^ -\- x' {a + h + c) + X {ah + ac-\- be) -\-abc. 68. Powers of a binomial. We have already found the square and the cube of a binomial expression; and higher powers can be obtained in succession by actual multiplication. The method of detached coefficients should be used to shorten the work. The following should be remembered: {a + by = a' + 2ab + ¥, (a + by = a« + Sa'b + 3a6' + b\ and (a + by = a* + 4>a'b + QaV + ^ab' + b\ To find any power, higher than the fourth, of a binomial expression a formula called the Binomial Theorem should be employed: this theorem will be considered in a subse- quent chapter. EXAMPLES IL 1. Multiply 2x-a by x — 2a. 2. Multiply 3a; - J by ^x- 3. 3. Multiply x^ + X+1 by x-l. 4. Multiply a^-xi/ + 7/' by x + y, 5. Multiply 1 + 05 + a:* + ic' by x-\. 6. Multiply aj* + y^y + ^y^ + xy^ + y* by y -x. 7. Multiply a;' - a; + 2 by a;' + a; - 2. 8. Multiply 1 + aa; + a*x^ by 1 — aa; + a^x*, 9. Multiply a;* + a;' + 1 by x* - x'' ^- 1. EXAMPLES. 43 10. Multiply 3£c» -xy-^ %f by Sy* - ajy + 2aj*. 11. Multiply a;' - 5a;' + 1 by 2a;« + 5a; + 1. 12. Multiply 2a;'' - t)x^y + y' by / + ^V + 2x^ 13. Multiply 3a»- 2a^6 + ZaV" - 36* by 2a*+ 5a^6-4a6»+6*. 14. Multiply 2aV - 3aVy + 5/ by aV + 4aa:s^^ - 2/. 15. Multiply 2a - 3a» + da" - 7a' by 1 - 2a' + 'oa''. 16. Multiply a* — ah — ac + b^ — he •\- c* by a + 6 + c. 17. Multiply a? ■¥ y^ ■>*- ^ - y% - zx — xy by a; + y + «. 18. Multiply 4a'+ 96* + c' + 36c + 2ca - 6a6 by 2a + 36 - c. 19. Multiply together a;*+ 1, 3;"+ 1 and a;'— 1. 20. Multiply together a;* + 16^, a;'+ 4^*, a; + 2y and x - 2y. 21. Multiply together (x - yY, (x + y)' and (a;* + yy. 22. Multiply together (x' + 1)^ (a; + 1)' and (x - l)^ 23. Multiply together a;'-a;+l, a;'+a;+l and a;* - a;' + 1 . 24. Multiply together a' - 2a6 + 46', a* + 2a6 + 46' and 25. Find the squares of (i) a + 26 - 3c, (ii) a* — a6 + 6', (iii) 6c + ca + a6, (iv) 1 - 2a; + Sx^, and (v) a;* + a' + a; + 1. 26. Find the cubes of (i) a + 6 + c, (2) 2a - 36 - 2c and (iii) 1 + a; + jc". 27. Simplify (x + y + zy-(-x + y+zy + {x-y + 2)'- (x + y- zf. 28. Shew that (x-¥ y) {x + z) -x'' = {y ^ z) {y + x) -f = {z ■¥ x) (z + y) - z'. 29. Shew that (y + z)' +{z + xy + (x + yy~x'-y'~s^={x-ry + zy. 44 EXAMPLES. 30. Simplify {x (x + a) - a (x - a)]{x {x - a) - a {x + a)}. 31. Shew that {y-zy+{z-xy+{x-yy = 3(y-z)(z-x){x-y). 32. Shew that a' + 6' = (a + by - dab (a + b\ and that a* + 6* = (a + by - 4ab (a + by + 2a'b\ 33. Shew that {x" + xy + y'y - ixy (x' + y') = (af-xy+ y'y. 34. Shew that {y + zy+(z + xy + (x + yy + 2(x + y){x + z)+2(y + z)(y + x) + 2(z + x)(z + y) = 4: {x + y + zf. 35. Shew that {a' + b') (c» + d')^{ac + bdy +{ad- bey. 36. Shew that, if ic = a + c?, 2/ = ^ + ^, and z = c + d; then will of ■\-'i^ + z^ - yz^zx-xy = a* -\-b'^ + c* -be- ca- ab. 37. Shew that, if a; = 6 + c, y = c + a, and z = a + b; then will of ■^y' + T^ — yz — zx-xy = a' + b' + c'—bc — ca-ab. 38. Shew that 2{a-b){a-c) + 2(b-c){b-a) + 2(c-a){c-b) = {b-cy + (c-ay+(a-by. 39. Shew that (a^ + y" + z') (a' + b' + c') -{ax + by + czy = (bz — cyy + (ex- azy + (ay - bx)'. 40. Shew that, if a; = a* - 5c, y = b*- ca, z = c' — ab; then will ax + by + cz = (x ■^- y + z) (a + b -h c)j and be (x^ - yz) = ca (y' — zx) = ah (z* - xy), 41. Find the value of {x - ay + {x-by + (x-cy-Z{x- a) {x -b){x- c) when Zx = a + b-^c. 42. Shew that (a» + 6" + cV = (6" + cy + (ab 4- ac)' -{-(ab-aey + a* = (6c + ca + aby + (a* - bey + (6' - ca)» + (c» - a5)«. 43. Shew that (sc' + xy + y') (a' + ab + b') = (ax - byy + (ax- by) (ay + bx + by) + (ay + bx + by)'. EXAMPLES. 45 44. Shew that l+a' + b' + c' + b'c' + cV + a'b" + a'b'c' = (1 ~ 6c — ca - aby + (a + b + c~ obey. 45. Shew that (a« + 6« + c« + dy = (a' + b'- c«- dy+ 4:{ac + bdy+ 4:(ad - bey. 46. Shew that (i) {a + 2y-4:{a + iy+6a'-4:{a-iy+{a-2y=0. (ii) (a + 2) (6 + 2) - 4(a + 1) (6 + 1) + Gab - 4 (a - 1) (6 - 1) + (a - 2) (6 - 2) = 0. 47. Shew that (i) (a + 2)»- 4 (« + 1)«+ 6a«- 4 (a- 1)^+ (« - 2)« = 0. (ii) (a + 2)(6 + 2)(c + 2)- 4 (a + l)(b + l)(c + l) + 6abc -4(a-l)(5-l)(c-l) + (cj-2)(6-2)(c-2) = 0. 48. Shew that {a + b + cy + (b + c - a) (c + a - b) {a + b - c) = ia' {b + c) + 46' (c + a) + 4c' (a + 6) + 4a6c. 49. Shew that x{x-y + z){x + y-z)+y{x + y-z){-x+y-{-z) + z{- x+y-\-z){x -y + z) + (-x + y+z){x-y + z)(x + y-z) = 4:xyz. 50. Multiply a' + b' + c^ + d' -bc-ca-ab-ad — bd-cd by a + b -^c + d. 51. Shew that (x'+x+l)(af'-x+l)(x*-x'+l){x'-x'+l)...{x''-x'''" + l) = x* + a- -♦- 1. CHAPTER V. Division. 69. Division by a monomial expression. We have already considered the division of one monomial expression by another. We have also seen (Art. 43) that the quotient obtained by dividing the sum of two alge- braical quantities by a third is the sum of the quotients obtained by dividing the quantities separately by the third; and we can shew by the method of Art. 54 that when any multinomial expression is divided by a monomial the quotient is the sum of the quotients obtained by dividing the separate terms of the multinomial expression by that monomial. Thus {pi^x — Sax) -i-ax = a^x -^ ax — Sax -^ cw; = a — 3. And (12a;' - bax" - 2a*x) h- 3.r = 12a;' -r 3a; - bax" -h 3a; - 2a*a; -^ 3a; = 4a;'^ - f aa; — |a*. 70. Division by a multinomial expression. We have now to consider the most general case of division, namely the division of one multinomial expression by another. Since division is the inverse of multiplication, what we have to do is to find the algebraical expression which, when multiplied by the divisor, will produce the dividend. Both dividend and divisor are first arranged according DIVISION. 47 to descending powers of some common letter, a suppose ; and the quotient also is considered to be so arranged. Then (Art. 62) the first term of the dividend will be the product of the first term of the divisor and the first term of the quotient ; and therefore the first term of the quotient will he found by dividing the first term of the dividend hy the first term of the divisor. If we now multiply the whole divisor by the first term of the quotient so obtained, and subtract the product from the dividend, the remainder must be the product of the divisor by the sum of all the other terms of the quotient; and, this remainder being also arranged according to descending powers of a, the second term of the quotient will be found as before by dividing the first term of the remainder by the first term of the divisor. If we now multiply the whole divisor by the second term of the quotient and subtract the product from the remainder, it is clear that the tldrd and other terms of the quotient can be found in succession in a similar manner. For example, to divide Sa" + ^a% - 4^a¥ + b^ by 2a 4- b. The arrangement is the same as in Arithmetic. 2a + 6 ) 8a« + 8a'6 + ^ab' + ¥ ( 4a' + 2ab + 6» 8a' + 4a'6 4a=-'6 + 4a6' + b^ 4>a'b + 2ab' 2ab* + 6" 2ab' + 6' The first term of the quotient is 8a' -*- 2a = 4a'. Multiply the divisor by 4a* and subtract the product from the dividend : we then have the remainder 40/^6 + 4'ab'^ + 6^ The second term of the quotient is 4a''fe -r- 2a = 2ab. Multiply the divisor by 2a6, and subtract the product from the remainder: we thus get the second remainder 2a6' 4- b\ The third term of the quotient is 2a¥ -i-2a = b\ Multiply the divisor by 6^ and subtract the product from 48 DIVISION. ^ 2ab'^-\-b', and there is no remainder. Since there is no remainder after the last subtraction, the dividend must be equal to the sum of the different quantities which have been subtracted from it; but we have subtracted in suc- cession the divisor multiplied by 4a^ by + 2ah, and by + b^ ; we have therefore subtracted altogether the divisor multiplied by 4a' + 2ab + b^. And, since the divisor mul- tiplied by 4a^ f 2ab + 6* is equal to the dividend, the required quotient is 4a* + 2ab + b^. The dividend and divisor may be arranged according to ascending instead of according to descending powers of the common letter, as in the last example considered with reference to the letter b ; but the dividend and the divisor must both be arranged in the same way. 71. The following are additional examples: Ex. 1. Divide a* - a^b + 2a^b^ - ab» + b* by a" + bK a^ + b^)a*-a^b + 2a^b^~ab^ + b^(a^-ab + b^ -a"^& + a26a -aW^b*' - d^b - ab^ + a262 +b^ + a^b^ +b* Ex. 2. Divide a* + a^b^ + b* by a^ - ab + b\ a^~ab + b^)a* +a^h^ +b*(a* + ab+b^ + a?b + &4 In this example the terms of the dividend were placed apart, in order that ' Uke ' terms might be placed under one another without altering the order of the terms in descending powers of a. The subtractions can be easily performed without placing *like' terms under one another; but the arrangement of the terms according to descending (or ascending) powers of the chosen letter shoald never be departed from. Ex.3. Divide a3 + [;3^^_3^5j; by a + 6+c. DIVISION. 49 a+b + c)a^- Sabc -i- b^ + c^ ( a^ - ab - ae+b'^ - be + e^ 'a3 + a26 + a2c ^ -a26- -a26- -ab'' -3a6c+63 + c3 -abc . - a^c + a62 - -a2c -2a6c + 63 -abc -ac + c» 2 + a62- + a62 - abc + ac"^ + 63 + C3 + 63+ 62c - abc + ac2 -abc -62c + c8 -62c-6c» + ac^ + ac^ + 6c2 + c3 + 6C2 + C3 Where, as in the above example, more than two letters are involved, it is not sufficient to arrange the terms according to '' descending powers of a; but b also is given the precedence over c. y By using brackets, the above process may be shortened. Thus o+6 + c)a3-3a6c+63 + c3(a2-a(6 + c) + (62-6c + c2) ^a^ + a^(b + c) ^ -a2(6 + c)-3a6c + 63 + cS -o='(6 + c)-a(6 + c)8 a 62-6c + c2) + 63 + c' a(62 -6c+c2) + 6^ + c» 72. The method of detached coefficients may often be employed in Division with great advantage. For example, to divide 2a;'-7aj"+5a;* + 3a;»-3a;* + 4^-4 by 2x^ - Sx^ -h x - 2, we write — 2-3 + 1-2)2-7 + 5 + 3-3 + 4-4(1-2-1+2 2-3+1-2 -4+4+5-3+4- -4+6-2+4 -4 -2 + 7- -2 + 3- -7 + 4- -1+2 -4 4- 4- -6 + 2- -6 + 2- -4 -4 The first term of the quotient is a/^ and the other powers follow in order : thus the quotient is x^-2aP-x^2. S. JL 4 50 DIVISION. 73. Extended definition of Division. In the process of division as described in Art. 70, it is clear that the remainder after the first subtraction must be of lower degree in a than the dividend ; and also that every re- mainder must be of lower degree than the preceding remainder. Hence by proceeding far enough we must come to a stage where there is no remainder, or else where there is a remainder such that the highest power of a in it is less than the highest power of a in the divisor, and in this latter case the division cannot be exactly per- formed. It is convenient to extend the definition of division to the following : To divide A by B is to find an algebraical expression G such that B x C is either equal to A, or differs from A by an expression which is of lower degree, in some particular letter, than the divisor B, For example, if we divide a^ -h Sab + W by a + b, we have a + 6 ) a' -f 3a6 + 46' ( a + 26 a^ + ab 2ab + 46'' 2ab + 2 6' + 26* Thus (a* + Sab + 46^^) -4- (a + 6) = a + 26, with remainder 26=^ ; that is a' + Sab + 46'"^ = (a + 6) (a + 26) + 26*. We have also, by arranging the dividend and divisor diiferently, 6 + a ) 46' + 3a6 + a" ( 46 - a W + 4(i6 — ab-V(J^ — ab — a^ + 2a'^ Hence a change in the order of the dividend and divisor leads to a result of a different form. This is, how- ever, what might be expected considering that in the first DIVISION. 51 case we find what the divisor must be multiplied by in order to agree with the dividend so far as certain terms which contain a are concerned, and in the second we find what the divisor must be multiplied by in order to agree with the dividend so far as certain terms which contain h are concerned. When therefore we have to divide one expression by another, both expressions being arranged in the same way, it must be understood that this arrangement is to be adhered to. 74. Def. A relation of equality which is true for all values of the letters it contains, is called an identity. The following identities can easily be verified, and should be remembered : {a? + 2gw? -\- a^) -^ {x -\- a) = X -\- a, (x^ — 2ax -\- a^) -7- (x — a) = X — a. (a?' — a*) -7- (a? + a) = a? + a. (x^ + a') -7- (a? T a) = ^ ± aa; 4- a\ (x* - a*) -^ (a; + a) = ic" ± ax^ + a^x ± a\ {x* + aV -\- a*) -^ {x^ T ax -\- a^) = a^ ± ax -^ d\ (a;'+ 2/' + 2^ — ^xyz) -^{x-\-y-\-z)^ x^^- y'^+ i^ — yz — zx - xy. EXAMPLES IIL oJ* - 92/* by (K + Zy. a;*- 161/* by a;* - V- 27a;»+6V by ^y + ^x. 2>x^ - ixy - 42/' by 2y - x, 1 - 5a^ -I- 4a;* by 1 - x. a^ ~ 5xy* + 4y* by x-y. 4—2 X. JL^l VICIC 2. Divide 3. Divide 4, Divide 6. Divide 6. Divide 62 EXAMPLES. 7. Divide 1 - Cic^ + 5£C« by l-2x + a^. 8. Divide m' - 6mn'^ + 5n' by m* — 2mn + ri*. 9. Divide l-7xf^ + Qa^ by (1 - x)'. 10. Divide 1 - a^ by l-x\ 11. Divide 1 + a; - 8a;' + 19a;» - 16ar* by l-\-3x- doc". 12. Divide 4 - 9ic' + 1 2a;« - 4£c^ by 2 + 3x- 2x\ 13. Divide 4ic* - 9afy' + 6xy^ - y* by 2x' + dxy - y'. 14. Divide a;^ - 3x V 3a; + y' - 1 by x-\-y-l. 15. Divide a:" + x^y + x^y^ + x^y^ + xy^ + ?/^ by x^ + xy + y*. 16. Divide a" - bx*y + 7a;V - a;*/ - ^xy* + 22/"^ by a;' - 3a;V + 3a;y* - y^. 17. Divide a' - 26' - 6c' + a6 - ac + 76c by a - 6 + 2c. 18. Divide a' + 26' - 3c' + 6c + 2ac + 3a6 by a + 6 - c. 19. Divide 6a* + 46' - a»6 + 13a6« + 2a'6' by 2a« + 46' - 3a6. 20. Divide x* + y* - z' + 2xY ■¥ 2z' - I by «" + y' - ^^ + 1. 21. Divide a" - 3a'6 + 3a6^ - 6' - c' by a-6-c. 22. Divide a' + 86' - c' + 6a6c by a + 2b-c. 23. Divide a* + 86* + 27c' - 18a6c by a' + 46' + 9c' - 66c - 3ca - 2a6. 24. Divide 27a" - 86» - 27c» - 54a6c by 3a - 26 - 3c. 25. Divide acx^ + (ad -he) a?- (ac + hd) x + bc by oa; - 6. 26. Divide 2aV-2(6-c)(36-4c)2/' + a6a;2/ by aa;+2(6-c)y. 27. Divide 9a«6' - 1 2a*6 + 36» + 2a«6' + 4a» - 1 la6* by 36' + 4a» - 2a6'. 28. Di\T^de x* + y' hj x + y; and from the result im^e down the quotient of (a; + y)' + a' by x + y + z. 29. Divide af — y* by x — y; and hence lorite down the quotient of {x + y)' — 8«' by x + y — 2z. CHAPTER VL Factors. 75. Definitions. An algebraical expression which does not contain any letter in the denominator of any term is said to be an integral expression : thus ^a^b — \h^ is an integral expression. An expression is said to be integral with respect to any particular letter, when that letter does not occur in the denominator of any term : thus — I — -, is integral with '' a a-\-o respect to x. An expression is said to be rational when none of its terms contain square or other roots. 76. In the present chapter we shall shew how factors of algebraical expressions can be found in certain simple cases. We shall only consider rational and integral expres- sions ; and by the factors of an expression will be meant the rational and integral expressions, or the expressions which are rational and integral in some particular letter, which exactly divide it. 77. Monomial Factors. When some letter is common to all the terms of an expression, each term, and therefore the whole expression, is divisible by that letter. Thus 2ax + x^=x(2a + x), ax + a^x^ =ax{l + ax) , and 2a»6'a: + 3a%hf = a%'^ {2ax + Bby). Such monomial factors, if there be any, are obvious on inspection. 64 FACTORS. 78. Factors found by comparing with known identities. Sometimes an algebraical expression is of the same form as some known result of multiplication: in this case factors can be written down at once. Thus, from the known identity a^-b^ = {a + b)(a-b), we have a2 - 462 = a2 - (26)2 = (a + 26) (a - 26), a2-2 = a2-(^2)2=(a + V2)(a-^2), a* - 166'' = (a2)a - (462)2 = (a^ + 462) (^2 _ 462) = (a2 + 462)(a + 26)(a-26), and a3 - 9a62 = a{a^- 962) = a (a + 36) (a - 36). Again, from the identity a3 + 63 = (a + 6)(a2-a6 + 62), we have a3 + 863 = ^3 + (26)3 = (a + 26) { a2 - a (26) + (26)'} = (a + 26)(a2-2a6 + 462), 8a3 + 276«= (2a)3 + (362)8 = (2a + 362) | (2a)2 - (2a) (362) + (352)2 j = (2a + 362) (4^2 - 6a62 + 96-») , and a^ + x^ = (a3)3 + {x^f = (a^ + x^) (a^ - a^a^ + a;«) = {a + x){a^-ax + x^) {a^ - a^a^ + x% And, from the identity a3-63=(a-6)(a2 + a6 + 62), we have a363 _ ^a^3= fab - ^xy\ (^a%^ + ^ abxy + \ xY^ . The following are additional examples of the same principle : (i) (a + 6)2-(c + d)2={(a + 6) + (c + d)}{(a+6)-(c + d)} = (a + 6 + c H- d) (a + 6 - c - d). (ii) 4a262 - (a2 + 6^ - c2)2 = {2a6 + (a2 + 62 - c2) } { 2a6 - (a2 + 62 - c2) } ; and, since 2a6 + a2 + 62-c2 = (a + 6)2-c2=(a + 6 + c)(a + 6-c), and 2a6-a8-62 + c2=c2-(a-6)2 = (c + a-6)(c-a + 6), we have finally 4a262-(a2 + 62-c2)2=(a + 6 + c)(6 + c-a)(c + a-6)(a + 6-c). FACTOKS. 55 (iii) (a + 26)3 -(2a + 6)3 = {(a + 26)-{2a + 6)}{(a + 26)2 + (a + 26)(2a + 6) + (2a + 6)a} = (6-a)(7a2+13a6 + 762). 79. Factors of x^+px + q found by inspection. From the identity {x + a) (a? + 6) = «* + (a + 6) a? + ab, it follows conversely that expressions of the form 0!^ +px-{-q can sometimes, if not always, be expressed as the product of two factors of the form x + a, x-hb. We shall presently give a method by which two factors of x^-^px + q of the form x + a and x + b can always be found ; but whenever a and b are rational, the factors can be more easily found by inspection. For, if {x -ha) (x + b), that is of -\- (a ■]- b) X -\- ab, is the same as x^ +px-h q, we must have a-\-b=p and ab = q. Hence a and b are such that their sum is p, and their product is q. For example, to find the factors of x^+7x + 12. The factors will be jc + a and x + b, where a + 6 = 7 and ab = 12. Hence we must find two numbers whose product is 12 and whose sum is 7: pairs of numbers whose product is 12 are 12 and 1, 6 and 2, and 4 and 3; and the sum of the last pair is 7. Hence a;^ + 7a; + 12 = (x+ 4) {x + 3). Again, to find the factors of x^-7x + 10. We have to find two numbers whose product is 10, and whose sum is -7. Since the product is +10, the two numbers are both positive or both negative; and since the sum is - 7, they must both be negative. The pairs of negative numbers whose product is 10 are - 10 and - 1, and - 5 and -2; and the sum of the last pair is -7. Hence a;2-7a; + 10 = {x-5){x-2). Again, to find the factors of a;2 + 3a;-18. We have to find two numbers whose product is - 18 and whose sum is 3. The pairs of numbers whose product is - 18 are - 18 and 1,-9 and 2,-6 and 3, - 3 and 6,-2 and 9 and - 1 and 18 ; and the sum of 6 and - 3 is 3. Hence ai^+3x-18 = {x + &){x- 3). It should be noticed that if the factors oi x^-}-px-{-q be x-{- a and x + b, the factors of x^ -{-pxy + qy^ will hQ x + ay and x-\-by; also the factors of {x -{- yY -\- p [x + y) z -h q? will he x-\-y-{-az and x-\-y + bz. 56 FACTORS. Hence from the above we have x^ + 3xy^ -lSy* = {x + ey^) (x - 3y\ {a + by^-7{a + b)x+10x^={a + b-5x){a + b-2x), = {x+2){x-2){x + l){x-l). EXAMPLES IV. Find the factors of the following expressions : 1. a*-16b\ 2. 16x*-8Wb\ 3. l6-{3a~2b)\ 4. 4.y'-{2z-xy. 5. 20aV-45aa;2/». 6. 36aV-4aV2/*. 7. (3a'-by-(a'-3by. 8. (5a^ - 36y - (3a« - 56^ 9. (5af + 2x-Sy-{x'-2x-3y. 10. (3x' -ix- 2y - (3x' +4:x- 2)'. 11. 32a'b'-U\ 12. (a«-.2&c)»-86V. 13. a'-2a~8, U a;+12-aj«. 15. 1-Ux-Q3a^. 16. 8a-4a^-4. 17. a'b-ia'b' + 3ab'. 18. a'b + 5a%' + ia'b'. 19. (6 + c)^-6a(6 + c) + 6a«. 20. 9(a + 6)*-6(a + ^»)(c + c/) + (c + c/)». 21. a:* - 29a;^ + 100. 22. lOOx* - 29.xy + y\ 23. x'~8xyz'+16y*z\ 24. 9a' - lOa'b' + a'b\ 25. x'-2ax-6« + 2a6. 26. x' + 2xy - a' - 2a/}/. 27. 4 («6 + cdy - (a' + b'^c'- dj. 28. i(xy- aby - {x' + y'-a'- by. FACTORS. 57 80. Factors of general quadratic expression. We proceed to shew how to find the factors of any ex- pression of the second degree in a particular letter, x suppose. The most general quadratic expression [Art. 60] in x is ax^ -\-bx + c, where a, b and c do not contain x. The problem before us is to find two factors which are rational and integral with respect to x, and are therefore each of the first degree in x, but which are not necessarily, and not generally, rational and integral with respect to arithmetical numbers or to any other letters which may be involved in the expression. The method of finding the factors of ax^ + bx-^c con- sists in changing it into an equivalent expression which is the difference of two squares. We first note that since ai^ + 2ax + a^ is a perfect square, in order to complete the trinomial square of which x^ and 2ax are the first two terms, we must add the square of a, that is, we must add the square of half the coefficient ofx. For example, a^+Bx ia made a perfect square, namely [ a; + ^ j , by the addition of ( ^ j ; also x^~px is made a perfect square, namely (« - 1) , by the addition of (^- 1^ = ~- . 81. To find the factors of ax^ -\- bic-\-c, aaf-\-bx + c==alx'-\--x+-]. \ a a) Now a?-^ - a; is made a perfect square, namely [a? + ~ ] , by the addition of f ^ ] =7-1- And, by adding and sub- tracting j^-j to the expression within brackets, we have 58 FACTORS. ^ Hence as the difference of any two squares is equal to the product of their sum and difference, we have h = ^^^+2a Thus the required factors have been found*. Ex. 1. To find the factors of a^ +4« + 3. «2 + 4a; + 3 = x2 + 4a; + 4-4 + 3 = (x + 2)2-l = (a; + 2 + l)(x + 2-l) = ix + 3){x + l). Ex. 2. To find the factors of x^-5x + S. ..-5..3=.2.,.,(_|y.(.|)V3=(.-|y-^ =(^-i+\/T)(^-|-\/T)- Ex. 3. To find the factors of 3ar» - 4a; + 1. s.-...i=s(.»-|..|)=3{.-|..(iy-(l)%i| Ex. 4. To find the factors of x^ + 2a4; - &' - 2ah. iti+2ax -¥ -2ab=^x^ + 2ax + a^ - a^ -b^ -2ab = {x + ay - (a + b)* = {x + a + {a + b)} {x + a- {a + b)} = {x + 2a + b) {x - b). • It will be proved later on [see Art. 91] that an expression containing X can be resolved into only one set of factors of the first degree in «. FACTORS. 59 82. Instead of working out every example from the beginning we may use the formula and we should then only have to substitute for a, h and c their values in the particular case under consideration. Thus to find the factors of 3a;3 - 4a; + 1. Here a = 3, 6 = - 4, c = 1. „ /62 - 4ac /16 - 12 /I 1 ,^ . . Hence ^ —4-3- = y - gg— = \/ 9 = 3 ' *^® expression is therefore equivalent to 3 (*-q + q)(^-q - a)= ^ (^~'q) (*~^)' 83. We have from Art 81 ac\ 7)2 __ Ann Now, for particular values of a, 6, c, — ^Ta — may be positive, zero, or negative. I. Let — 7-2 — be positive. Then the two factors of cwj* + 6a; + c will be rational or irrational according as — 7-2 — is or is not a perfect square. II. Let — T~2— be zero. Then Hence aa^ + 6iP + c is a perfect square in x, if 6^ — 4ac = 0. III. Let — T~T~ b® negative. Then no positive or negative quantity can be found whose square will be equal to — TTT" j ^or all squares, whether of positive or nega- tive quantities, are positive. 60 FACTORS. Elxpressions of the form V— a, where a is positive, are called imaginary, and positive or negative quantities are distinguished from them by being called real. We shall consider imaginary quantities at length in a subsequent chapter : for our present purpose it is sufficient to observe that they obey all the fundamental laws of Algebra ; and this being the case, the formula of Art. 81 will hold good when b^ — 4ia^ is negative. Note. For some purposes for which the factors of expressions are required, the only useful factors are those which are altogether rational: on this account irrational and imaginary factors are often not shewn. Thus, for example, the factorisation of x^ — S is for many purposes complete in the form (os — 2) (as^ + 2a; + 4) *, the imaginary factors of a;^ + 2ic + 4, namely a;+l + V^ and a^ + l-V^, not being shewn. 84. We have in Art. 81 shewn how to resolve any expression of the second degree in a particular letter into two factors (real or imaginary) of the first degree in that letter. It should be noted that the factors of the most general expression of the third degree, or of the fourth degree, can be found, although the methods are beyond the range of this book ; expressions of higher degree than the fourth cannot however, except in a few special cases, be resolved into factors. 85. Factors found by re-arrangement and grouping of terms. The factors of many expressions can be found by a suitable re-arrangement and grouping of the terms. For example = {l + ax){l + x){l-x); * The reason of thia will appear from Art. 179 and Art. 193, FACTORS. 61 or we may write the expression in the form and the factors 1 - x^, 1 + ax are now obvious. For the best arrangement or grouping no general rule can be given : the following cases are however of frequent occun'ence and of great importance. I. When one of the letters occurs only in the first power, the factors often become obvious when the expres- sion is arranged according to powers of that letter. Ex. 1. To find the factors of ab + be + cd + da. Arranged according to powers of a we have a{b + d) + bc + cd, which is at once seen to he a{b + d) + c {b + d) = {a + c) {b + d). Ex.2. To find the factors of 3tP + {a + b + c)x + ab + ac. The expre88ion = a {x + b + c)+x^ + bx + cx = {a + x){x + b + c). Ex. 3. To find the factors ofax^ + x+a + 1. asfi + x + a + l = a{a^ + l) + x + l = {x + l){a{xi^-x + l) + li. Ex. 4. To find the factors of a^ + 2ab - 2ac - Bb^ + 26c. The given expression is of the first degree in c ; we therefore write it in the form a^ + 2ab - '6b^ -2c {a- b) = {a-b){a + 3b)-2c{a-b) = {a-b)(a + Bb-2c). II. When the expression is of the second degree with respect to any one of the letters; factors, which are rational and integral in that letter, can be found as in Art. 81. Ex. 1. Find the factors of a^ + 36^ - cH 26c - 4a6. Arranging according to powers of a, we have ' a«-4a6 + 36'-c» + 26c=a2-4 = fa + -^y - ^ {462 _ 46c + c» + 1262 _ 206c + 8c2} = (a + 36-2c)(a-6 + c). Ex. 4. Find the factors o{ x* + x^-2ax + l- a?. Arranging according to powers of a, we have - {a^ + 2ax -1- x^ - x^} ^ - {a? + 2ax^x^ -l-2x^ - X*} = -{(a + x)a-(l + a:2)2}=-(a + a; + l + x2)(a + a;-l-a:2). III. When the expression contains only two powers of a particular letter and one of those powers is the square of the other, the method of Art. 81 is applicable. Ex. 1. To find the factors of x^ _ I0a;2 + 9. a:* - 10x3 + 9 =a:4 - 10a;2 + 25 - 25 + 9 = (a:» - 5)2 - 16 = (a;2-5 + 4)(iB2-5-4) = (xa-9)(ar»-l) = (a; + 3)(a;-3)(x + l)(x-l), or thus:— «* _ i0a;2 + 9 = (a:' + 3)» - IGx' = («» + 3 + 4a;) (x2 + 3 - 4x) = (x + 3) (x+ 1) (x - 3) (x - 1). Ex. 2. To find the factors of x" + x^ + 1. Two real quadratic factors can be found as follows : X4 + X2+I=(x2 + 1)2-X2=(x2+1 + X)(x3+1-X). Ex. 3. To find the factors of x« - 28x» + 27. x«-28x3 + 27 = x«-28x3+143-142 + 27 = (x3-14)»-13» = (x»-l)(x3-27) = (x-l)(x-3)(x2 + x + l)(x2 + 3x + 9). In this case, and also in Ex. 1, two factors can be seen by inspection, as in Art. 79. FACTORS. 03 Ex. 4. To find the factors of a* + b* + c*- 2b^c^ - 2c«a* - 2a^l^. Arranging according to powers of a, we have a4 - 2a2 (62 + gS) + 54 + ^4 _ 252^2 = a* - 2a2 (&2 4. c2) 4. (^2 + c2)2 _ (52 + c2)2 + 64 + c4 _ 262^2 ^ = {a2 - (62 + c2)}2 _ 46 V = (a^ - 62 - c^ - 26c) (a^ - 62 - 0^+ 26c) = {a2-(6 + c)2}{a2-(6-c)2} = (a + 6 + c) (a - 6 - c) (a - 6 + c) (a + 6 - c) . IV. Two factors of aP^ + bP + c, where P is any expression which contains x, can always be found by the method of Art. 81 ; for we have aP^ + bP + c Ex. 1. To find the factors of [x^ + xf + 4 (a;^ + x) - 12. Since P2 + 4P- 12 = (P-2) (P + 6), the given expression = (a^ + a; - 2) (x* + a; + 6) = (a; + 2)(a;-l)(a;2 + flc + 6), the factors of ot^ + aj + G being imaginary [see Art. 83, Note]. Ex. 2. To find the factors of (aj2 + a; + 4)^ + 8x (x2 + a: + 4) + 15x2. The given expression = { (x2 + x + 4) + 3x} { (x2 + x + 4) + 5x} = (x2 + 4x + 4)(x2 + 6x+ 4) = (x + 2)2(x2 4 6x+4). Ex. 8. To find the factors of 2(x2 + 6x + l)2 + 5(x2 + 6x + l)(^=» + l) + 2(x2+l)». Since 2P2+ 5PQ + 2Q2=(P + 2(2)(2P+ g), the given expression = {(x2 + 6x+l) + 2(xHl)}{2(x2 + 6x+l) + xHl} = (3x2 + 6x + 3) (3x2 + 12x + 3) = 9(x+l)2(x2 + 4x + l). Ex. 4. To find the factors of (x» + x + 1) (x^ + x + 2) - 12. The given expression = {x^ + x)2 + 3 (x2 + x) - 10 = (x2 + x-2)(x2 + x + 5) = (x + 2)(x-l)(a;2 + «+5). . 64 EXAMPLES. EXAMPLES V. Find the factors of the following expressions i 1. 03* + a^ -x — a. 2. ac — bd — ad+ be. 3. ac' + bd'-ad'-bc\ 4. acx^ + (be + ad) xy + bd'tf, 5. acx^ + bcaf + adx + bd. 6. {a + 6)' + (a + c)« - (c + (i)^ - (6 + J)*. 7. a' + a'b-ab'-b\ 8. a* - a»6 _ ab"" + b\ 9. a'b'-a'-b'+l. 10. afi/'-x'z'-y'z' + z\ 11. a;VV-a;'«--/;s + l. 12. oj* + a;^ + xz"^ + y^. 13. x(x + z) -y{y + z), 14. a;*-7a:^-18. 15. a;* - 23x' + 1. 16. x*-Uxy-i-y\ 17. x' + aj' + l. 18. x*-2{a' + b')x'+(a'-b')\ 19. aj* - 4a;y 2» + 4:y*z\ 20. ic*- 2 (a + 6) a; - a5 (a - 2) (b ^ 2). 21. a:* + 6a^ + CKC + a6. 22. {l+yy-2a^{l+y')^x*{l-yY 23. af-y'- 32» - 2a» + 4y». FACTORS. 66 24. 2^/* - 6xy + 2x^ -ay-ax- a^ 25. a' - 3b' - 3c' + 106c - 2ca - 2ah. 26. 2a» - 7a6 - 226« - 5a + 356 - 3. 27. l + (6-a»)aJ*-a6£c^ 28. 1 - 2aa5 - (c - a') a?" + aca;°. 29. a«(6 -c) + h\c-a) + c\a - h). 30. 6*c + 6c' + c'a + ca' + a*6 + a6* + 2a6c. 31. a'6 - a6' + a*c - ac' - 2abc + b'c + be'. 32. x\a -\-l)-xy(x-y) {a-b)+j/'{b + 1). 33. ax {/ + 6') + by (6a;' + a'y). 34. 2a;' - ^a;*!/ -a^z + 2xy' + 2xyz - y'z. 35. xyz (a;" + / + «") - yV - «V - ajy. 36. {a;'+a;)»-14(a;' + a;) + 24. 37. (a;' + 4a; + 8)' + 3a; (a;' + 4a; + 8) + 2a;'. 38. (a;+l)(a; + 2)(a;+3)(a; + 4)-24. 39. (a;+ 1) (a;+ 3) (a;+ 5) {x + 7) + 15. 40. 4(a; + 5) (a; + 6) (a; + 10) (a; + 12) - 3a!«. 86. Theorem. The expression of" — a" is divisible by x — a, for all positive integral values of n. It is known that x — a, x^ — a^ and x^ — a^ are all divisible hy x — a. We have a;" - a" = a;" - aa;""' 4- ax*"'^ - a" = a?"-' (a; - a) + a (a?""^ - a""*). Now if x — a divides a;""* — a**"^ it will also divide a;*"* (a? - a) + a (a;""* - a""*), that is, it will divide a;" - a\ Hence, if x — a divides a;"'* — a""* it will also divide X — a . S.A. 5 66 FACTORS. But we know that x — a divides of — a^;it will therefore also divide cc" — a*. And, since x — a divides x* — a* it will also divide x'^ — a*. And so on indefinitely. Hence a;** — a" is divisible by x — a, when n is any positive integer. 87. Since a;" + a" = a;" - a" + Sa" it follows from the last Article that when a?" + a** is divided by a; — a the remainder is 2a**, so that a?" + a" is never divisible by a; — a. If we change a into — a^x—a becomes x — {—a) = x-\-a\ also a?" — a** becomes a?" — (— a)**, and a?" — (— a)" is a;" + a** or a;" — a** according as n is odd or even. Hence, when n is odd a?" + a" is divisible by a? 4- a, and when n is even a;" — a" is divisible by a; + a. Thus, n being any positive integer, x-^ a divides a?" - a** always, x — a „ a;" + a" never, a? + a „ a?" — a" when w is even, and x + a „ a;" + a" when n is odd. The above results may be written so as to shew the quotients: thus " = a;--' + a;"-^ a + a;""' a'^ + + «-», = a;""' - a?""* a + a;""' a' - ± a""', x-{- a the upper or lower signs being taken on each side of the second formula according as ri is odd or even« 88. Theorem. If any regional and integral expres- sion which contains x vanish when a is put for x, then will x — a be a factor of the expression. or -a" X — a x"" + a" FACTORS. 67 Let the expression, arranged according to powers of x, be cw?" + 6a?"~' + ca?**"^ + Then, by supposition, aa'*-i-6a""' + ca''~'+ =0. Hence aa?" + 6a;""* + ca;""' + = a«" + 6^""' -i- ca?"^ + - (aa" + 6a"-' + ca""* + ...) = a (a;" - a") + 6 (^"~* - a""*) + c (^""^ - a""') + But, by the last Article, a?" - a", aj""' - a"-\ a;""'' - a♦*-^ &c. are all divisible by x — a. Hence also cm?" + 60?"-* + co;""' + is divisible by X —OL. The proposition may also be proved in the following manner. Divide the expression aaj"+6a;**~' + ca;""'' + by a? — a, continuing the process until the remainder, if there be any remainder, does not contain x\ and let Q be the quotient and K the remainder. Then, by the nature of division, aa;" + 6a;"-* + ca;"-='+ =Q(a;-a) + -B, and this relation is true for all values of x. Now since H does not contain a;, no change will be made in R by changing the value of x : put then x = (x, and we have aa" +6a"-* + ca"-'^+ =Q(a-a) + R==R Hence, if any expression rational and integral in x be divided by x~a, the remainder is equal to the result obtained by putting a in the place of x in the expression. It therefore follows that the necessary and sufficient condition that an expression rational and integral in x may be exactly divisible by x — a is that the expression should vanish when a is substituted for x. 6—2 68 FACTORS. Ex. 1. Find the remainder when 3^ - 4lX^ + 2 is divided by a; - 2. The remainder = 2" - 4 . 22 + 2 = - 6. Ex. 2. Find the remainder when x^ - 2a^x + a' is divided hy x-a. The remainder is a* - 2a* + a^=0, so that x^ - 2a^x + a^ is divisible hj x-a. Ex. 3. Shew by substitution that «-l, x-5, x + 2 and x+4: are factors of x* - 23x2 _ 18a; + 40. Ex. 4. Shew by substitution that a - 6 is a factor of a^{b-c) + b^{c-a) + c^{a-b). Put a=b and the expression becomes a^ {a - c) + a^ {c - a), which is clearly zero : this proves that a - 6 is a factor. Ex. 6. Shew that a is a factor of (a + 6 + c)8-(-a + 6 + c)3-(a-6 + c)»-(a + 6-c)». 89. We have proved that w — a is a factor of the expression oaf* + hx"*'^ + cx**~^ + , provided that the expression vanishes when a is put for x. If the division were actually performed it is clear that the first term of the quotient, which is the term of the highest degree in x, would be cw?""\ Hence the given ex- pression is equivalent to (a? - a) (ow;""'' + &c ). Now suppose that the given expression also vanishes when x=l3; then the product of ic — a and 0^""^ + will vanish when x = 0; and since x— a does not vanish when x= ^, it follows that aa?"~* 4- must vanish when a?=/S. Hence x — ^ is a factor of ax^~^ + &c.; and, if the division were performed, it is clear that the first term of the quotient would be cw;""*. Hence the original expression is equivalent to (a;_a)(a;-/3)(aa;*-*-f-&c ). Similarly, if the original expression vanishes also for the values 7, B, &c. of x, it must be equivalent to (x -(i){x- /3) (x -y)(x- 8) (OA-"-*- + &c ), FACTORS. 69 where r is equal to the number of the factors x — a, X - py &c. If therefore the given expression vanishes for n values a, y8, 7, &c. there will be n factors such as a? — a, and the remaining factor, oa?""*" + &c. will reduce to a; and hence the given expression is equivalent to a{x — a)(w — 13) {x — 7) OOR If any of the factors x — a, x — P, ... occur more than once in aa?" + 6a;"~^ + . . . , it can similarly be proved that the expression is equivalent to a(x — ay (x — iSf . . . , the factors x — a, x — ^, ... occurring respectively ^, ^, •.. times, and p -{-q-\- ... = n. 90. Theorem. An expression of the nth degree in x cannot vanish for more than n values of x. For if the expression Gw;" + 6a;""^ + ca;""* + vanishes for the n values a, ^^y , it must be equivalent to a(x — a)(x-~0)(x — y) If now we substitute any value, k suppose, different from each of the values a, ^, 7, &c.; then, since no one of the factors k — a, k — ^, &c. is zero, their continued product cannot be zero, and therefore the given expression cannot vanish for the value x = k, except a itself is zero. But, if a is zero, the original expression reduces to fta?""^ + ca;""* + , and is of the (n — iy degree; and hence as before it can only vanish for n — 1 values of x, except b is zero. And so on. Thus an expression of the nth degree in x cannot vanish for more than n values of x, except the coefficients of all the powers of x are zero; and when all these coefficients are zero, the expression will clearly vanish for all values of a?. 70 FACTORS. 91. Theorem. If two expressions of the nth degree in X he equal to one another for more than n values of x, they will be equal for all valves of x. If the two expressions of the nth degree in x cw;" + 6a?"-' + ca;"-'' + , and px"" + qaf"'^ + rx*'~^-{- , be equal to one another for more than n values of x, it follows that their difference, namely the expression {a - j9) a;" + (6 - q) a?""^ + (c-r) x^'^ + , will vanish for more than n values of x. Hence, by Art. 90, the coefficients of all the different powers of x must be zero. Thus a-_p = 0, 6-^ = 0, c-r = 0, &c. that is, a — p,h—q,c = r, &c. Hence, if two expressions of the nth degree in x are equal to one another for more than n values of x, the coefficient of any power of x in one expression is equal to the coefficient of the same power of x in the other expression. When any two expressions, which have a limited number of terms, are equal to one another for all values of the letters involved, the above condition is clearly satisfied, for the number of values mast be greater than the index of the highest power of any contained letter. Hence when any two expressions, which have a limited number of terms, are equal to one another for all values of the letters involved in them, we may equate the coefficients of the different powers of any letter. 92. Theorem. A rational integral expression con- taining X can he resolved into only one set of factors of the first degree in x. For, if it be possible, let the expression ax"* + hx^^ + . .. be equivalent to a{x— ay (x — j3y..., and also to a (^ - f)' {x — ly)" ... FACTORS. 71 Put OP = 01 in both expressions; then a (a - ^y (a — t?)*". . . must vanish, and therefore one at least of the quantities f , 17, ... must be equal to a. Let f = a ; remove one factor x — a from both expressions, and proceed as before. We thus prove that every factor of one expression occurs to as high a power in the other expression; the two ex- pressions must therefore be identical. 93. Cyclical order. It is of importance for the student to attend to the way in which expressions are usually arranged. Consider, for example, the arrange- ment of the expression bc-\-ca + ah. The term which does not contain the letter a is put first, and the other terms can be obtained in succession by a cyclical change of the letters, that is by changing a into 6, h into c and c into a. In the expression a^ (b — c)-\- ¥ {c—a)-\- & {a — h) the same arrangement is observed; for by making a cyclical change in the letters of a^ (b — c) we obtain ¥ (c — a), and another cyclical change will give c' (a — b). So also the second and third factors of (b — c){c — a) (a — b) are obtained from the first by cyclical changes. 94. Symmetrical expressions. An expression which is unaltered by interchanging any pair of the letters which it contains is said to be a symmetrical expression. Thus a-\-b + c, bc-\- ca-\-ab, a' + 6^ + c^ — Sabc are symmetrical expressions. Expressions which are unaltered by a cyclical change of the letters involved in them are called cyclically sym- metrical expressions. For example, the expression (b — c){c — a) (a — b) is a cyclically symmetrical expression since it is unaltered by changing a into b, b into c, and c into a. It is clear that the product, or the quotient, of two symmetrical expressions is symmetrical, for if neither of two expressions is altered by an interchange of two letters their product, or their quotient, cannot be altered by such interchange. 72 FACTORS. It is also clear that the product, or the quotient, of two cyclically symmetrical expressions is cyclically sym- metrical. Ex. 1. Find the factors of a^ (6 - c) + 6^ {c-a)->r c^ (a - 6). If we put 6=s:c in the expression a2(6-c) + &2(c-a) + c3(a-6) -. (i) it is easy to see that the result is zero. Hence &-c is a factor of (i), and we can prove in a similar manner that c-a and a — & are factors. Now (i) is an expression of the third degree; it can therefore only have three factors. Hence (i) is equal to L(})-c){e~a){a-h) (ii), where L is some number, which is always the same for all values of a, by c. By comparing the coefficients [See Art. 91] of a' in (i) and (ii) we see that L=-l. We can also find L by giving particular values to a, 6 and c. Thus, let a = 0, & = 1, c = 2; then (i) is equal to -2, and (ii) is equal to 2L, and hence as before L = - 1. Ex. 2. Find the factors of a^ (6 - c) + 6« (c - a) + c^ (a - 6). As in the preceding example, (6-c), {c-a) and (a -6) are all factors of a8(6-c) + J3(c-a) + c»(a-6) (i). Now the given expression is of the fourth degree ; hence, besides the three factors already found, there must be one other factor of the first degree, and this factor must be symmetrical in a, 6, c, it must therefore be a + & + c. Hence the given expression must be equal to L (6 - c) ip-a) (a-b) (a + b + c) (ii), where L is a number. By comparing the coefficients of a' in (i) and in (ii) we see that L = - 1 ; hence a*(6 - c) + 63(c - a) +c8(a - 6) = - (6 - c) {c-a) {a - b) {a + b + c). We can also find L by giving particular values to a, b, and c. Thus, let a = 0, & = 1, c=2; then (i) is equal to -6 and (ii) is equal to 6L, so that L = - 1. We may also proceed as follows: Arrange the expression according to powers of a ; thus a^{b-c)-a (63 - c^) + be {b'- - c% FACTORS. 73 It is now obvious that 6 - c is a factor, and we have (b-c) {a^-a{b^ + bc + c^) + bc{b + c)} = (& - c) { 62 (c - a) + & (c2 - ac) + a3 - ac2 } = (6 - c) (c - a) {&2 + bc-a^-ac}=-{b- c) (c - a) (a - 6) (a + 6 + c). Ex. 3. Find the factors of bh^ {b-e) + c^o? {(i-d)-\- a^V^ (a - &). By putting 6=c in the expression V^c^{b-c) + c^a^{c-a) + a%^{a-b) (i), it is easy to see that the result is zero; hence h-c is a factor of (i). So also c-a and a - 6 are factors. The given expression being of the fifth degree, there must be, besides the three factors b-c, c-a, a-b, another factor of the second degree ; also, since this factor must be symmetrical in a, b, c, it must be of the form L [a^ + b^ + c^) + M {be + ca + ab). Thus (i) is equal to {b-c){c-a){a-b){La^ + Lh^ + Lc^ + Mbc + Mca + Mab}...{n). Equating coefficients of a* in (i) and in (ii) we see that L = 0; and then equating coefficients of bh^ we see that M=-l. Hence (i) is equal to -(b-c) {c- a) {a - 6) {be + ca + ab). We may also proceed as follows. Arranging according to powers of a, the factor b-c which does not contain a becomes obvious ; then, arranging according to powers of 6, the factor c-a which does not contain 6 becomes obvious; and so on. Thus 6V(6 - c) - a2(68 _ c3) + a3 (52 _ c2) = (&-c){62c2-a3(62 + 6c + c2) + a3(& + c)} = (6 - c) {62 (c2 - a2) + a26 {a-c) + a^c (a-c)} = {b-c){c-a){b^{c + a)-a^-a^c} = {b-c){c- a) {{62 - a2) c + 62a - a26} ss - (6 -c) (c-a) (a -6) (6c + ca + a6). EXAMPLES VI. Find the factors of the following expressions i 1. {y-zy + (z-xy + (x-yy. 2. {y-zy + (z^xy + (x-yy, 3. a' (b' - c') + h' (c' - a') + c' (a' - h'), 4. a{b-cy-\-h{c-ay + c(a-h)\ 74 EXAMPLES. 5. a(b- cY + h{c-ay + c{a- h)\ 6. he (6 - c) + ca (c - a) + ah {a - h). 7. 6V (h-c)-\- ea"" {c~a)-¥ a'6^ (a - 6). 8. a* (6 - c) + 6* (c - a) + c* (a - 6). 9. a''(6-c) + 6»((;-a) + c'(a-6). 10. (a + 6 + c)» - (6 + c - a)» - (c 4- a - 6)» - (a + 6 - c)^ 11. (a + 6 + c)'-(6 + c-a)'-(c + a-6)*-(a + 6-c)*. 12. a (6 + c - a)' + 6 (c + a - 6)' + c (a + 6 - c)' + (6 + c - a) (c + a - 6) (a + 5 - c). 13. a* (6 + c - a) + 5' (c + a - 6) + c^ (a + 6 - c) - (6 + c — a) (c + a - 6) (a + 6 - c). 14. (6 + c - a) (c + a - 6) (a + 6 - c) + a (a - 6 + c) (a 4- 6 - c) + 6 (a + 6 - c) ( - a + 6 + c) + c (- a + 6 + c) (a - 6 + c). 15. (6 - c) (a - 6 + c) (a + 5 - c) + (c - a) (a + 6 - c) (-a + 6 + c) + (a - 6) (- a + 6 + c) (a - 6 + c). 16. (a; + 2^ + is)' - £c' - / - z^. 17. (aj + 2/ + 2;)'' - a;* - g/** - «\ 18. (6-c)(6 + c)' + (c-a)(c + a)»+(a-6)(a + 5)'. 19. (6-c)(6 + c)» + (c-a)(c + a)' + (a-^»)(a + 6)^ 20. (6-c)(ft + c)*+(c-a)(c + a)^+(a-6)(a + 6)*. 21. a^ + 6^ + c" + 5a6c - a{a-h) {a- c) - h {h - c) {h - a) — c{c — a){G — h). 22. a» (a + 6) (a + c) {h-c) + h' (h + c){h + a) {c - a) + c'(c + a) (c + h){a- b). 23. (y + z) (z + x) (x + y)+ xyz. 24. a*(6 + c)' + 6'(c4-a)» + c*(a + 6)' + a6c(a + 6 + c) + {a^ + 6' + c") (6c + ca + a6). 25. (a; + 2/ + »)* - (y + »)* - (a + a;)* - (a; + y)* + a;* -1- y* + 2;*. 26. a« (6 + c - 2a) + 6' (c + a - 26) + c" (a + 6 - 2c) + 2 (c'-a^)(c- 6) + 2 (a'-6')(a-c) + 2 {¥-c'){b - a). EXAMPLES. 75 27. {h + c-a-dY{h-c)(a-d) + {c-^a-h-dy{c-a){b-d) + {a + b-c-dy{a-b)(c- d). 28. Shew that 12 {ix + y + «)«- - (y + »)'" - (« + xf"" -{x + yy + x"' + y'" + z'"] is divisible by {x + y + zy-{y + zY -{z-k- x)*- {x + y)* + a* + y* + z\ 29. Shew that a«(6 + c-a)»+6»(c + a-6)* + c*(a + 6-c)»+a6c(a' + 6»+c») + (a V 6' + c' ^ 6c - ca - a6) (6 + c - a) (c + a ~ 6) (a + 6 - c) = 2a6c (Jc H- ca + a6). 30. Shew that (6-c)« + (c-a)' + (a-6)«-9(6-cy(c-a)'(a-6)' = 2 {a-hf{a-cy + 2 (6 - c)« (6 -a)^+ 2 (c - a)" (c - 6)«. 31. Shew that {h-^ cf + {c + af -^ {a + hf + (a + dy + {h + dy -\- {c+ dy = Z(a+h + c-^d){a' + h' + c' + d^). 32. Reduce to its simplest form 4 (a' + aft + hy - (a - 6)' (a + 26)^ (2a + 6)^ 33. Shew that «* (J« + c« - a^)^ 4- 6* (c» + a» - 6^)» + c* (a'' + 6' - c'')* is divisible by a* + 6* + c* - 26^c» - 2c'a« - 2a'h\ 34. Resolve into quadratic factors 4 {cc^ (a^ - h') + aft (c^ - d')]' + {(a' - 6^) (c' - 1^») - 4a6c^}». 35. Shew that (^^-^)(l+xy) (1 -^xz) + (^'-a;^) (l + y«)(l + yx) + {x'-y') (1 +!2;a;)(l+«2/) = (y-z) (z-x) (x-y) {xyz +x+y+z). 36. Find the factors of a'(b^c)(c-d){d-b)-b'{c-d)(d-a)(a-c) + c\d-a){a-b){b-d)-d^(a-b)(b-c)(c-a). 37. Find the factors of b'c'd' (b -c){c- d) (d-b)- c'd'a' (c -d){d- a) {a -c) + dWb'(d- a)(a-b)(b-d) - a'b'c' {a - b) {b - c) (c - a). CHAPTER VII. Highest Common Factors. Lowest Common Multiples. 95. A Common Factor of two or more integral alge- braical expressions is an integral expression which will exactly divide each of them. The Highest Common Factor of two or more integral expressions is the integral expression of highest dimensions which will exactly divide each of them. It is usual to write h.c.f. instead of Highest Common Factor. 96. The highest common factor of monomial expressions. The highest common factor of two or more monomial expressions can be found by inspection. Thus, to find the highest common factor of a^h^c and a^ft^'. The highest power of a which will divide both expressions is a'; the highest power of & is 6^. and the highest power of c is c; and no other letters are common. Hence the h.c.f. is a^h^c. Again, to find the highest common factor of a^6*c^, a%^ and a%c^. The highest power of a which will divide all three expressions is a^ ; the highest power of & which will divide them all is 6 ; and c will not divide all the expressions. Hence the h.o.f. is a%. From the above examples it will be seen that the H.C.F. of two or more monomial expressions is the product of each letter which is common to all the expressions taken to the lowest 'power in which it occurs. HIGHEST COMMON FACTORS. 77 97. Highest common factor of multinomial expressions whose factors are known. When the factors of two or more multinomial expressions are known, their h.c.f. can be at once written down, as in the pre- ceding case. The h.c.f. will be the product of each factor which is common to all the expressioTis taken to the lowest power in which it occurs. Thus, to find the h.c.f. of (a: -2)3 (a; -1)2 (a; -3) and (a; - 2)* (« - 1) (« - 3)». It is clear that both expressions are divisible by {x - 2)^, but by no higher power of «- 2. Also both expressions are divisible by a; - 1, but by no higher power of a; - 1 ; and both expressions are divisible by a; -3, but by no higher power of a; -3. Hence the h.o.f. is {x-2,f{x-l){x~^). Again, the h.c.f. of a%'^{a-}>Y{a + hf and a%'^{a-h){a-\-hY is In the following examples the factors can be seen by inspection, and hence the H.C.F. can be written down. Ex. 1. Find the h.c.f. of a^}^-a%* and a^lfi + a^l^. Am. a262(a + 6). Ex. 2. Find the h.o.f. of a«6=»-4a4&* and a%'^-l%a^h^. Am. a262(a3_452). Ex. 3. Find the h.o.f. of a»+3a»6 + 2a6» and a*+6a*6 + 8a262. Am. a (a + 26). 98. Although we cannot, in general, find the factors of a multinomial expression of higher degree than the second [Art. 84], there is no difficulty in finding the highest common factor of any two multinomial expressions. The process is analogous to that used in arithmetic to find the greatest common measure of two numbers. If the expressions have monomial factors, they can be seen by inspection; and the highest common factors of these monomial factors can also be seen by inspection: we have therefore only to find the multinomial expression of highest dimensions which is common to the two given expressions. 78 HIGHEST COMMON FACTORS. Let A and B stand for the two expressions, which are supposed to be arranged according to descending powers of some common letter, and let A be of not higher degree than B in that letter. Divide B by A, and let the quotient be Q and the remainder R ; then B = AQ + R; .-. B==B-AQ. Now an expression is exactly divisible by any other if each of its terms is so divisible; and therefore B is divisible by every common factor of A and R, and R is divisible by every common factor of A and B. Hence the common factors of A and B are exactly the same as the common factors of A and R; and therefore the h.c.f. of A and R is the H.C.F. required. Now divide A by R, and let the remainder be >Si ; then the H.C.F. of jK and S will similarly be the same as the H.C.F. of A and R, and will therefore be the H.C.F. re- quired. And, if this process be continued to any extent, the H.C.F. of cmy divisor and the corresponding dividend will always be the H.C.F. required. If at any stage there is no remainder, the divisor must be a factor of the corresponding dividend, and that divisor is clearly the h.c.f. of itself and the corresponding divi- dend. It must therefore be the H.C.F. required. It should be remarked that by the nature of division the remainders are successively of lower and lower dimen- sions ; and hence, unless the division leaves no remainder at some stage, we must at last come to a remainder which does not contain the common letter, in which case the given expressions have no H.C.F. containing that letter. Since the process we are considering is only to be used to find the highest common multinomial factor, it is clear that any of the expressions which occur may be divided or multiplied by any monomial expression without destroying the validity of the process; for the multinomial factors will not be affected by such multiplication or division. HIGHEST COMMON FACTORS. 79 Thus the h.c.f. of two expressions can be found by the following Rule : — Arrange the two expressions according to descending powers of some common letter, and divide the expression which is of the highest degree in the common letter by the other [if both expressions are of the same degree it is immaterial which is used as the divisor). Take the remainder y if any, after the first division for a new divisor, and the former divisor as dividend ; and continue the process until there is no remainder. The last divisor will be the h.c.f. required. The process is not used for finding common monomial factors, these can be seen by inspection ; and any divisor, dividend, or remainder which occurs may be multiplied or divided by any monomial ex- pression. Ex. 1. Find the h.o.f. of »» + a;^ - 2 and a^-{-2x^- 3. x^ + aP-2\x^ + 2x'-S(l J x^+ x\-2^ ' x^-x ^ 2 f T.- — V.\ x^ + x-2 x^ -1 a;- -05 x-1 x-1 Thus the h.c.f. is x- 1. The work would be shortened by noticing that the factors of the first remainder, namely x'^ - 1, are x-1 and a; + 1. And by means of Art. 88 it is at once seen that x-1 is, and that a; + l is not, a factor of a* + a;*- 2, Ex. 2. Find the h.o.f. of x^ + 4:x'^y-8xy^ + 24:y^ and x^-x^ + 8x^y^-8xy*. The second expression is divisible by x, which is clearly not a common factor: we have therefore to find the h.o.f. of the first expression and x*-x^y + 8xy^-8y*. st^ + 'kc'^-8xy^ + 2iy^\x^- x^y + 8xy^ - 8y* Ix-by f x^ + ^ahf - 8x'^y'^ + 2^xy^ ^ - 5x^y + 8xh/ - 16an/3 - »y* - bx^y - 2(ixhj'^ + A Qxy^ - 120y* 28«2i/2-66xj/3 + 112t^* 80 HIGHEST COMMON FACTORS. The remainder =28y^{x^-2xy + 4y^): the factor 28^^18 rejected and x^ - 2xy + iy^ is used as the new divisor. s^-2xy + Ay^\x^ + 'hPy-8xy^ + 24y»(z + 6y I x^ - 2x'^y + 4an/2 ^ Qxh/-12xy^+2^y^ Qx^y-12xy ^ + 24:y^ Hence «* - 2xy + iy^ is the H.o.r. required. Ex. 3. Find the h.o.f. of 2x^ + 9x^ + lix + S and 3x*+ 16a^ + 6«3 + 10a? + 2. To avoid the inconvenience of fractions, the second expression is multiplied by 2: this cannot introduce any additional common factors. The process is generally written down in the following form : 2a;4+ 9a;3+ Ux + 3 \ 3a:* + 15a;3 + 5*2 + lOx + 2 /2 6x^ + S0x^ + 10x^ + 20x + 4:(S 6a;^ + 27a:8 +42x4-9 ^ '6x^ + 10x^-22x-5\2x* + 9x^+Ux + 3 6a;^ + 27a;3 + 42a; + 9 ( 2x 6x^ + 2(h^-4.ix^-10x^ 7x^+Ux^ + 52x + 9 3 21x3 + 132a^» + 156a; + 27/7 21x8+ 70a;2-154a;-35 V 62 |62a;^ + 31Qa: + 62 ar^+ 5x+ 1 x'' + 5x + l\Sci^ + 10x^-22x-5(Sx-5 hx^ + 15 x^+ Sx V - 5a:2-25x-6 - 5a;a-25a;-5 Thus x' + 6a; + 1 is the h.o.f. required. Detached coefficients should generally be used [Art. 63]. 99. The labour of finding the h.c.f. of two expres- sions is frequently lessened by a modification of the pro- cess, the principle of which depends on the following Theorem : — The common factor of highest degree in a particular letter, x suppose, of any two expressions A and B is the same as the H.C.F. of pA + qB and rA + sB, HIGHEST COMMON FACTORS. 81 where p, q, r, s are any quantities positive or negative which do not contain x. To prove this, it is in the first place clear that any common factor of A and B is also a factor of pA -f qB and oirA-^sB. So also, any common factor of pA 4- qB and rA + sB is also a factor of s{pA-]-qB) — q{rA+sB), that is, of {sp — qr) A. Hence, as {sp — qr) does not contain x, any common factor of pA +qB and rA -\-sB must be a factor af A, provided only that p, q, r, s are not so related that sp — qr = 0. Similarly any common factor of pA + qB and rA + sB is also a factor of r {pA + qB)—p {rA + sB)^ that is of {rq —ps) B, and therefore of B. Since every common factor of A and 5 is a factor of pA + qB and of rA + sB, and every common factor of pA + qB and rA + sj5 is a factor of A and of B, it follows that the H.c.F. of A and B is the same as the h.c.f. of pA + qB and rA + sB. Ex. To find the h. c.p. of 2ic* + a:* - 6a;2 - 2a; + 3 and 2a;* - 3a;3 + 2as - 3. We have, by subtraction, 4aJ_6x'»-4a; + 6 (I); and, by addition, 4x4-2a;»-6a;«=2a:»(2x2-a;-3) (II). The required h.c.f. is the h.c.f. of (I) and (II), and therefore of (I) and 2ar»-a;-3 (in). Multiply (III) by 2 and add (I), and we have another expression, namely 4a;»-2x2_6a; = 2a;(2a;2-a;-3) (IV), such that the h.c.f. of (III) and (IV) is the h.c.f. required. But the H.C.F. of (III) and (IV) is obviously 2x^-x-'^. 100. U R, S be the successive remainders in the process of finding the H.C.F. of the two expressions A and B by the method of Art. 98 ; then, as we have seen, every common factor of A and 5 is a factor of R, and therefore a common factor of J. and R. Similarly every common factor of A and ii^ is a common factor of R and S. And so on ; so s. A 6 82 HIGHEST COMMON FACTORS. that every common factor of A and 5 is a factor of every remainder, and therefore must be a factor of the H.c.F. Hence every common multinomial factor of two ex- pressions is a factor of their highest common multinomial factor ; and this is obviously true also of monomial factors. Therefore every common factor of two expressions is a factor of their H.c.F. It can be shewn that every remainder, in the process of finding the H.c.F. of two expressions A and 5 by the method of Art. 98, is expressible in the form FA + GB where F and G are rational and integral in x. For, if Qi, Qa, Qjj ••• Qn be the successive quotients, and i^i, R^, R^, ...Rn be the successive remainders, the last of which, Rn, is the H.c.F. of A and B ii A and B have any common factor, but is independent of a; if J. and B have no common factor containing x ; then we have R,^B -A. Q,, R2 = A — Ri' Qa, Rs = Ri — -Ba . Qa, Mn— Mn—i lin—\' ^n* Now i^i is clearly of the required form {F being — Q^ and G being 1), and substituting for jRi in the second equation it will be at once seen that R^ is of the required form. Also jRfc is of the required form provided that both Rjc_^ and Rje^ are of that form ; for, if i^;fc_i = LA + MB and Rk-^ = L'A^M'B\ then Rh = Rh-a - Bk-i Qtc = {L' -Qjc.L)A + {M'-Qi.M)B, and the expressions L — Q]c > L and M' — Qj^ . M are both integral, since by supposition L, M, L', M' and Q^ are all integral expressions. Hence Rn = FA + GB, where F and G are integral expressions. If now A and B have no common factor in x, then Rn does not contain x. And, dividing throughout by Rn, since F/Rn and G/Rn are integral in x, we have where P and Q are integral in x. i fflGHEST COMMON FACTORS. 83 Thus, if A and B he any two integral expressions con- taining X, hut which have no common factor containing a?, two other expressions^ P and Q, can he found hoth integral in X, and such that PA + QB = 1. Ex. FindPand Q when .4= a:»- 3*2+1 and B=a;« + 2a; + 2. Am. P=-l(8a; + 6). Q=i(8x''-35i;+39). 101. The H.C.F. of three or more multinomial expres* sions can be found as follows. Let the expressions be -4, 5, (7, Z),.... Find Q the H.C.F. of A and B. Then, since the required H.c.F. will be a common factor of A and B, it will be a factor of Q : we have there- fore to find the H.c.F. of 0,0,1).... Hence we first find the H.c.F. of two of the given ex- pressions, and then find the H.c.F. of this result and of the third expression ; and so on. Note. The highest common factor of algebraical ex- pressions is sometimes, but very inappropriately, called their greatest common measure (G.C.M.). If one expression is of higher dimensions than another, in a particular letter, we have no reason to suppose that it is numerically greater : for example, a' is not necessarily greater than a; in fact, if a is positive and less than unity, a' is less than a. It should also be noticed that if we give particular numerical values to the letters involved in any two ex- pressions and in their H.C.F., the numerical value of the H.C.F. is by no means necessarily the G.C.M. of the values of the expressions. This is not the case even when the given expressions are integral for the particular values chosen. For example, the h.cf. of l^aj'^^- 15a;-f-l and 22a;' + 23a; + 1 will be found to be a;+l; but if we suppose X to be |, the numerical values of the expressions will be 12 and 18, which have 6 for G.C.M., whereas the numerical value of the H.O.F. will be |. 6—2 84 LOWEST COMMON MULTIPLE. EXAMPLES VIL Find the h. c. f. of 1. a" - 5ab + ib' and a" - 5a'b + ib'. 2. 2af-5x + 2 and 12a;'- 8a;'- 3a; + 2. 3. 2x*-doi^y' + y* and 2x''-3xy + y\ 4. 2a;" + 3a;*y - y' and 4:X^ + xi/'-y\ 5. a;*- V+122/«-9«' and a;* + 2a» - 42^ + 8^2 - 3«*. 6. 20a* - 3a'b + 6* and 64a* - Sab' + 5b\ 7. a» - a'b + ab' + 14&» and 4a» + Za'b - 9ab' + 2b\ 8. 2a;* + a;»-9a;' + 8a;-2 and 2a;*- 7a;" + 1 la;"- 8a; + 2. 9. 1 la;* - 9aa;» - aV - a* and 13a;* - lOaa;" - 2aV - a\ 10. a;* + a;»-9a;»-3a;+18 and a;' + 6a;' - 49a; + 42. 11. a;* - 2a;" + 5a;' --4a; + 3 and 2a;* - a;" + 60;" + 2a; + 3. 12. a;* + 3a;' + 6a; + 35 and a;* + 2a:" - 5a;' + 26a; + 21. Lowest Common Multiple. 102. Definitions. A Common Multiple of two or more integral expressions, is an expression which is exactly divisible by each of them. The Lowest Common Multiple of two or more integral expressions, is the expression of lowest dimensions which is exactly divisible by each of them. Instead of Lowest Common Multiple it is usual to write L.C.M. LOWEST COMMON MULTIPLE. 85 103. When the factors of expressions are known, their L.C.M. can be at once written down. Consider, for example, the expressions ' a'h^x-afioo-hy and ah'ix-a)' {x-h). It is clear that any common multiple must contain a* as a factor ; it must also contain 6^ (x — of and {x — hf. Any common multiple must therefore have a'6* {x — of (x — by as a factor ; and the common multiple which has no un- necessary factors, that is to say the lowest common multiple, must therefore be a^6* (x — a)^ (x — by. From the above example it will be seen that the L.C.M. of two or more expressions which are expressed as the product of factors of the first degree, is obtained by taking every different factor which occurs in the expressions to the highest power which it has in any one of therri. Ex. 1. Find the l.o.m. of ^xh/z, 27sc?yh^ and SxyV. Ans. 54a;^yV. Ex. 2. Find the l.o.m. of Gab^ {a + bf and ia^b [a^ - 62). Ans. 12a^b^a + by{a-b). Ex. 8. Find the l.o.m. of 2axy {x - y)\ Sax^ {x^ - y^) and 4y^ {x + y)^. Ans. 12aa;V{^^-2/T- Ex. 4. Find the l.o.m. of 3c^-Bx + 2, x^-5x + 6 and x^-4:x+ 3. Am. {x-l){x-2){x-S). 104. When the factors of the expressions whose L.C. M. is required cannot be seen by inspection, their H.C.F. must be found by the method of Art. 98. Thus, to find the l.o.m. of a;3 + a;2-2 and x^ + 2x?^3. The H.o.F. will be found to be « - 1 ; and x» + x^-2 = {x-l){x^ + 2x + 2), ««+2a;a-3 = («-l)(«3 + 3a; + 3). Then, since x2 + 2a; + 2 and x^ + 3x + S have no common factor, the required l.o.m. is {x ~ 1) {x^ + 2a5 + 2) (x^ + 3a; + 3). 86 LOWEST COMMON MULTIPLE. 105. Let A and B stand for any two integral ex- pressions, and let H stand for their H.C.F., and L for their L.G.M. Let a and h be the quotients when A and B respec- tively are divided by ^; so that A=H .a and B = H.b. Since H is the highest common factor of A and B, a and b can have no common factors. Hence the L.C.M. of A and B must he H x axb. Thus L = H.a.b. Hence L — Ra x -jf = A Xjj. (i); also L xH = Hax Hh^A x B (ii). From (i) we see that the L.C.M. of any two expressions is found by dividing on£ of the expressions by their H.C.F., and multiplying the quotient by the other expression. From (ii) we see that the product of any two expressions is equal to the product of their H. c. F. and L. c. M. EXAMPLES YIIL Find the l. c. m. of 1. 6a^ - 5aa; - 6a' and 4aj' - 2aa? - 9a'. 2. 4a* -5ah + b' and 3a' - Sa'b + ab' - h\ 3. 3aj» -Ux' + 23aj - 21 and 6a3» + x^- Ux + 21. 4. x*-l Ix' + 49 and 7x* - 40ic' + 75a;" - 40a; + 7. 6. aj' + 6a;' + lla; + 6 and a;* + a;"-4a;'-4a;. 6. a;*-a;» + 8a;-8anda;* + 4a;«-8a;« + 24a;. 7. 8a' - 18a6", 8a» + Sa'b - Qab' and 4a' - 8a6 + 36' EXAMPLES. 87 8. a^-7x + 12, 3x'-Qx-9 and 2x'-6x'-Sx, 9. 8a;' + 27, 16a;* + 36a;' + 81 and 6a;' -5x-6. 10. x' - 6xy + 92/', x^-xy- 6y' and 3a;' - 1 2^. 11. of — 7ocy + 122/', a;' - 6xy + Sy^ and a;' - 5xy + Qy* 12. Shew that, if aa;' + 6a; + c and a'a;' + 6'a; + c' have a com- mon factor of the form x +f, then will (ac' - a'cY = (6c' - 6'c) (ah' - a'h), 13. Shew that, if ax^ + haf + Gx + d and a'x^ + b'x" + c'a; + d' have a common quadratic factor in x, then will ba' — b'a cal — c'a da' — d'a ad — a'd bd' - b'd cd' — c'd ' 14. Find the , condition that ax^ + bx + c and a'x^ + b'x + c' may have a common factor of the form x -¥f. 15. If g^,g^i g^ are the highest common factors, and ?j, ?,, l^ the lowest common multiples of the three quantities a, 6, c taken in pairs; prove that g^g.29]'}}z= {phcf. 16. If -4, jB, (7 be any three algebraical expressions, and {BG\ (GA), (AB) and (ABC) be respectively the highest common factors of B and C, C and A, A and B, and A, B and G j then the l.c.m. of A, B and C will be A.B.G. (ABG) ^ {{BG) . (GA) . (^^)}. CHAPTER VIIL Fractions. 106. When the operation of division is indicated by placing the dividend over the divisor with a horizontal line between them, the quotient is called an algebraical fraction, the dividend and the divisor being called respec- tively the mmierator and the denominator of the fraction. Thus T means a-r-h. Since, by definition, ^ = a -i- 6, it follows that 7- x 6 = a* 107. Theorem. The value of a fraction is not altered by multiplying its numerator and denominator by the same quantity. We have to prove that a _am b~b^' for all values of a, b and m. a Let ^ = 7- ; then a xb = Txb== a, by definition. FEACTIONS. Hence a; xh x m = a x m; ,\ X X (hm) = am. [Art. 29, (ii).] Divide by hm, and we have ,, V am X — am -^ (6m) = 7— . 108. Since the value of a fraction is not altered by multiplying both the numerator and the denominator by the same quantity, it follows conversely that the value of a fraction is not altered by dividing both the numerator and the denominator by the same quantity. Hence a fraction may be simplified by the rejection of any factor which is common to its numerator and ax denominator. For example, the fraction tj- takes the a^ simpler form ^ , when the factor a?, which is common to its numerator and denominator, is rejected. When the numerator and denominator of a fraction have no common factors, the fraction is said to be in its lowest terms. To reduce a fraction to its lowest terms we must divide its numerator and denominator by their h.c.f. ; for we thus obtain an equivalent fraction whose numerator and denominator have no common factors. Ex. 1. Eeduoe -^f ^ to its lowest terms. Qarxy The H.c.y. of the numerator and denominator is 3axy ; and Saxhf _ 3axhf-r-Saxy _ x_ Qa^xy ~ 6a''^xy-7-daxy "" 2a * Ex.2. SlmpUfy5Ll|S±12i;. ^ *' x^-8xy + 12y^ x^ - 7xy + lOy^ _ jx - 2y) {x - 5y) __ x-5y x^-8xy + 12y^~ {x-2y){x-6y) x-6y' 90 FRACTIONS. Ex.3. Simplify ^^. x^-ax x{x- a) a^-x^~ {a- x) {a + x)' Since a; - a= - (a - a;), if we divide the numerator and denominator by a - a?, we have the equivalent fraction ^^ : and if we divide the a + x numerator and denominator by a; - a, we have — , r . By the -(a+x) '' Law of Signs in Division = -— : = , and the last form a-\-x -{a + x) a + x is the one in which the result is usually left. Ex.4. Sunpbfy jj^;^^^^^^^^. The H.c.F. will be found to be x^-Zx + 1; and, dividing the numerator and denominator by x^-ix + l, we have the equivalent ^^^^^^"^ i^T5^T3- 109. Reduction of fractions to a common de- nominator. Since the value of a fraction is unaltered by multiplying its numerator and denominator by the same quantity, any number of fractions can be reduced to equivalent fractions all of which have the same de- nominator. The process is as follows. First find the L.C.M. of all the denominators ; then divide the L. C. M. by the denomi- nator of one of the fractions, and multiply the numerator and denominator of that fraction by the quotient; and deal in a similar manner with all the other fractions : we thus obtain new fractions equal to the given fractions re- spectively and all of which have the same denominator. For example, to reduce a h and ^y{x + y)' xy^{x-y) oc^yHx^-y'')* to a common denominator. The L.o.M. of the denominators is ahi^{x^-y^). Dividing this L.C.M. by c(^y{x+y), xy^{x-y) and xhf^{x^-y^), we have the FRACTIONS. 91 quotients y^{x-y), x^{x+y) and xy respectively. Hence the required fractions are a _ axy^{x-y) _ ay^{x-y) x^y{x + y) x^y{x + y)xy^{x-y) a^^{x^-y^)* b _ bxix^{x+ y ) _ bx^{x + y) -y) X x^{x + y) ~ a^y^{x^- cxxy _ cxy xY i^^ - y^) ^y^ {^ -y^)^xy x^ (x^ - y^) ' It is not necessary to take the lowest common multiple of the denominators, for any common multiple would answer the purpose j but by using the l.o.m. there is some saving of labour. 110. Addition of fractions. The sum (or differ- ence) of two fractions which have the same denominator is a fraction whose numerator is the sum (or difference) of their numerators, and which has the common denominator. This follows from Art. 43. When two fractions have not the same denominator, they must first be reduced to equivalent fractions which have the same denominator : their sum, or difference, will then be found by taking the sum, or difference, of their numerators, retaining the common denominator. When more than two fractions are to be added, or when there are several fractions some of which are to be added and the others subtracted, the process is precisely the same. The fractions must first be reduced to a common denominator, and then the numerators of the reduced fractions are added or subtracted as may be required. Ex. 1. Find the value of — - + — r . a+b a-b The L.c.M. of the denominators is (a + b) {a-b); and 1 1 a-b a+b a + b "^ o^ ^ (a + b) (a-b) "*" {a-b) (a + b) {a-b) + {a + h) _ 2a a^-b^ ~a»-b^' a ab Ex. 2. Find the value of r + f-^ s . a-b b^-a^ FRACTIONS. The L.O.M. of the denominators is a'- 6^; and we have a{a + b) -ah a{a-\-h)-db _ a^ + Ex. 3. Simphfy + — — + „ , „ + -^j-— . . ^ a-x a + x a^ + ar a* + x* La this case it is not desirable to reduce all the fractions to a oommon denominator at once : the work is simplified by proceeding as under: a a _a{a + x)+a{a-x) _ 2a^ a-x a + x~ a^-x^ "a^-x^* , 2a^ 2a^ _ 2a^{a^ + x^)+2a^{a^-x^) _ 4a< "^®° a^-x^'^a^ + x^~ a*-«* ~a*-c^' .^ „ 4a* 4a* 4a* (a* + aj*) + 4a* (a* - a;*) 8a^ and finally ^j-^ + ^-^ = — ^ ^,-^ 1 = ^3--, . [The above would be shortened by observing that, except for the factor 2, the second addition only diifers from the first by having a^ and aP instead of a and x respectively; and hence the result of the addition can be written down at once. So also the result of the third addition can be written down from the first or second.] Ex. 4. Simplify J_ - -1- + -1- - -i- . *^ 05-3 x-1 x + 1 x + S Here again it is best not to reduce all the fractions to a common denominator at once : much labour is often saved by a judicious arrangement and grouping of the terms. and then The L.o.M. of the denominators is (6-c)(c-a)(a-6) [See Art. 93]. Hence we have a^(c-b) + b^{a-c) + c^{b-a) {b-c){c-a){a-b) Now we naturally test, by the method of Art. 88, whether either of the factors of the denominator is a factor of the numerator : we are thus led to find that the numerator is the same as the denomi- nator, so that the given expression is equal to unity. 1 «-8 1 x + 3 {x + S)-{x-3)_ x'-9 ' 6 9' 3 x-1 ^^^- -S{x+1) + S{x- x'^-l :i) = -6 x^-1' x''-9-^x^- 6 6( -1" 'x^-l)-6{x^-9) {x^-9){x^-l) -(x3 48 -9){x^- 1)' StiTYTnlifiLr a2 &2 u. c2 FRACTIONS. 93 Ex. 6. Simplify a' 6^ c^ (o - 6) (a - c) (x + a) "*" ( & - c) (6 - a) (a; + 6) (c - a) (c - 6) (a; + c) * The L.c.M. of the denominators is {b-c){c-a){a-h){x + a) [x + l) {x + c). The expression is therefore equal to the fraction whose denominator is this L.c.M. , and whose numerator is a2(c-6)(a; + &)(a; + c) + 62(a-c)(a; + c) {x + a)+c^{h-a) {x-\-a) {x + h). Arranging the numerator according to powers of «, the coefficient of «« is a?{c - 6) +6'{a - c)-\-c^{b - a)= (6 -c) (c -a) {a- b). The coefficient of x is a^ (c« - b^) + fe^ (a» - c^) + c^ {b^ - a») = 0. The term which does not contain x is abc{a{c-b) + b{a-c)+c{b-a)}=0. Hence the numerator is x^{b-c){c-a){a-b), and therefore the given expression xHb-c){c-a){a-b) x^ ''(b-c){c-a){a-b){x + a){x + b){x + c) {x + a) (x + h) {x + c)' 111. Multiplication of fractions. We have now to shew how to multiply algebraical fractions. n c Let the fractions be y and -, , a Let ^=6'...)*-+^+v+~. = a;".X6i«6/ V--» since a + y9 + 7+ ... =?i.* Hence any term of ^ = a?" x corresponding term of B ; .'. sum of all the terms of -4 = a;" x sum of all the terms of B, that is A ^x"" .B\ which proves the theorem. Theorem III. If the denominators of the fractions j^ J ~, J-*, he all positive, then will the fraction ,^ — ," . / — '-L1111 J)0 greater than the least and less than O1 + O2 + O3 + ^ the greatest of the fractions 1^ , if t <^o. = a;; Let r be the ejreatest of the fractions, and let ~ then r^ *^6n will each fraction be equal to o a J{a^-2ac + 2c^) Put - = -=«; ^{a^-2ac + 2c^) _ ^{b^x^-2bxdx + 2d^x^) _ ,_ s/{b^ - 2bd + 2d'') ~ J{b^-2bd+ 2d-) - V^ - ^• [This is a simple case of Theorem II., Art. 113.] Ex. 8. Shew that, if ^^^ = ^^+^ = ^^±^ , then wUl box cay abz cay _ ~ al + bm+cn al-bm + cn al + btn-cn EXAMPLES. 99 Each of the given equal fractions ^ -a{cy + bz) + b{az + cx) + c{bx + ay) 2bcx -al+bm + cn ~ -al + bm+cn and similarly =-7-^- = —^^^1— . cU-bm + cn al + bm-cn EXAMPLES IX Simplify the following fractions : 3Qa'bc'x'9jz' • ^' a'c*xy ' 3 «'-8«5 + 7 6' 7xY-Sa fy' + l ' a'-3ah-28b'' ' 28a;y + 3a;y - 1 ' • (a:« + 2/^)(a;-2/)- * {x'' - y') {x* - y*) ' • a;' + 2a;'+2ic' + 2a;+l' a* - 2a;' - a'' - 2a; + 1 2ic' + 5x'y + xy' - Sy' 3x' + 3x'y - ixV - xy^ + 2/* * 54a;' -27a;* -3a;' -4 36ar' + 3a;«+3a;-2 * {a + b){{a + by-c'} Wc'-(a'-b'-cy' 10. 11. 12. x^W-z')+y'(^-x')-^z'^(:^-y') x'{y-z) + y'{z-x) + z' (x-y) 13. ^'(.y-g) + y'(g-a;) + g'(«^-y) {y + zY + {z+ xy +(x + yY ,^ a{b-c){c-d) -c(d-a) (a- b) b{c-d)(d-a)-d{a-b){b-cy 7—2 100 EXAMPLES. a^ (2/ - «) + / (« - a) + «' (a? - y) 2a 26 a* + 6' 16. — , 4- — -T + 17. a + h ' a — b ' h* — a*' Z-x Z + x 1 - 16a; 1 - 3a; 1 + 3a; Ga;* - 1 * 18. ^ y i^-vT x+2y 2y-x af - 4^ _ - a; - 2a a; + 2a 8aa; 19. ^ - ^ + x+2a 2a — X of - 4a* ' o« 1 3 3 1 20. s ; + a; + 2 a; + 4 x+ 6 x + S 21. J ?- + -^ L_, a;+a a; + 3a x + 5a x + 7a' or> 1 4 6 4 1 22. ;r + + 23. 24. 25. x — 2a x-a x x + a x + 2a 1 2 1 a;* - 5xy + Qy' a;* — 4:xy + 33/' a;* - 3xy + 2^/* a 6 c (a -b) (a- c) (b -c){b- a) (c - a) (c - 6) * a' 6» c' (a _ 6) (a - c) (6 - c) (6 - a) (c - a) (c - 6) ' (l+a6)(l+ac) {l + bc)(l+ba) (l+ca)(l+cb) ^^' (a-b){a-c) "^ {b-c){b-a) "^ (c-a)(c-6) ' 6c (a + c?) ca (6 + d) ah(c + d) (a-6)(a-c)"^ (6-c)(6-a)"^ (^a)(c-6)* 28 ^-y^ 1 y"-^^ , ^-^ {x + y)(x + z) (y+z){y + x) (z + x){z + y)' EXAM?L>iS. 101 (y-x){z-x) (g-y)(^-y) {x-2y-\-z){x + y- 2z) (x + y-2z){-2x + y + z) (-2x + y + z)(x-2y + z)' x + a a; + 5 x + c ^{x + a){x + b){x + c) X — a x — h X — c {x — a)(x-b)(x — c) X X X ^x^+ (be + ca + ab) x x-a x — b x — c {x - a) {x -b) {x - c) g' 6' c' •• {a-b) (a-c)'^ (b- c) (6- a) "^ (c-a) (c-b)' 32 ^' «L_ {a -b)(a-c) (b- c) (b-a)+(c- a) (c-b)' 33. ^« (« + ^)(« + c) ^ ^, (64-c)(6 + a) ^ ^, (c + a)(c + 5)^ (a — 6) (a - c) (b — c)(b — a) (c — a) (c — 6) * \J c/ \c a/ Va bj 85. & c/ \c aj \a b) 1 1 {a — b + c) {a -¥ b - c) (a + 6 - c) (- a + 6 + c) 1 (- a + 6 + c) (a - 6 + c) * ^ h-c c-a a-b a'-ib-c)' b*-{c-ay c'-(a-b)' 87. Shew that 16 + 1^:^ . ^^ - 2 ?;z4i'= 16 (^y, {x-a x + a x+a) \x-aj 88. Shew that a+6 a-b a*x + ¥y _ aV-6y oa; + 62/ ax -by aV + 6*y' aV - ^y ' 102 EXAMPLES. 89. Shew that ^^^ (a-b)(a-c)(l+ax)'^ {b-c){b-a)(i+bx) c' 1 (ii) (iii) (c -a)(c- b) (1 + ex) (1 + ax) (1 + 6ic) (1 + ex) a b (a-b){a-c)(l+ax) (b-c) (b-a) (I +bx) c —X (c -a)(c- b) (1 + ex) (1 +ax)(l+ bx) (1 + ex) ' 1 1 (a -b){a- e) (1 +ax) (b- e) (b - a) (1 + bx) 1 x' {e-a)(c-b)(l + ex) (i +ax){l +bx){l +cx)' 40. Simplify {a+p)(a + q) _^ (b+p){h + q) ^ {c+p){c + q) {a-b){a-c){x + a) (b -c) (b - a) (x + b) {e-a){e-b){x+ey 41. Simplify a(b + e — a) b(c + a — b) c(a + b — e) (a-b) (a -c) ■*" {b-e) (b-a)'^ (e-a) (e-b)' 42. Simplify (a-b + c)(a + b-e) {a + b -c) (-a + b + e) {a-b)la-e) "^ (b-c) (b-a) {-a + b + e){a-b + c) "*• {e-a)(c-b) • 43. Simplify a{b + c) b{e + a) c{a + b) b+e-a c+a-b a+b—c EXAMPLES. 103 44. Shew that |m + — ) + (n + -) -»- (mn + — ) \ m/ \ w/ \ mn/ - (m + — ) (n + - ) (mn + — ) = 4. \ mj \ nj \ mnj 45. Shew that 46. Shew that h — c c — a a — h (b — c)(c — a){a-b) l+bc l+ca l-hab (1 + be) (1 + ca) (1 + ab) 47. Simplify 48. Shew that, if y + z _ z + x _ x + y b — c c—a a— b* then will each fraction be equal to J of + y' + !i^ j{(b-cy + {c-ay+{a-by} X Ob 49. Shew that, if - = r i tl^en will y b x' + a^ y' + b' (x + yY + (a + 5)" x + a y + b x + y + a + b 50. Shew that, if a; _ y _ z b+c-a c+a-b a+b~e* then will (b - c) x + {c - a) y + (a -b) z = 0. 104 EXAMPLES. 61. Shew that, if '!lz£y^'JL^^ 0-c c-a and c be not zero, then will each equal — — 7- , cit — V and a{y — z) + b(z-x) + c{x-y) = 0. 52. Shew that {a-h){a-c){a-d)'^ {h-c){h-d){h-a)'^ {c-'d){c-a)(c-h) {d-a){d-h){d-'c) 63. Shew that a: a' a •¥}>■¥ c -^d. + .. + < is equal to zero if r be less than n-\^ to 1 if r = /i - 1, and to ttj + a^ + . . . 4- a^ if r = r*. 54. Shew that x-tty (x-a^)(x-a^) (x-a^{x-a^{x-a^ 1 + a; — «! {x — a^ (x-a,) (a:-aj...(a;-aj («-«,) (a; - aj...(x-a,) 55. Shew that 6 + C + G?+ ... +^ + ^ 6 c + 7 7T-7 ; ^ + ... a(a+6+c+...4-^4-^) a{a + h) {a + h){a + h +c) I (a+6 + ... + A;) (a + 6 + ... + ^ + ^ CHAPTER IX. Equations. One Unknown Quantity. 114. A STATEMENT of the equality of two algebraical expressions is called an equation; and the two equal expressions are called the members, or sides, of the equation. When the equality is true for all values of the letters involved the equation is, as we have already said, called an identity, the name equation being reserved for those cases in which the equality is only true for certain particular values of the letters involved. For the sake of distinction, a quantity which is sup- posed to be known, but which is not expressed by any particular arithmetical number, is usually represented by one of the first letters of the alphabet, a, h, c, &c., and a quantity which is unknown, and which is to be found, is usually represented by one of the last letters of the alphabet x, y, z, &c. 115. We shall in the present chapter only consider equations which contain one unknown quantity. To solve an equation is to find the value or values of the unknown quantity for which the equation is true ; and these values of the unknown quantity are said to satisfy the equation, and are called the roots of the equation. Two equations are said to be equivalent when they have the same roots. 106 EQUATIONS. ONE UNKNOWN QUANTITY. An equation which contains only one unknown quantity, X suppose, and which is rational and integral in x, is said to be of the first degree when x occurs only in the first power ; it is said to be of the second degree when of is the highest power of a; which occurs; and so on. Equations of the first, second and third degrees are however generally called simple, quadratic and cui^ic equa- tions respectively. 116. In the solution of equations frequent use is made of the following principles. I. An equation is equivalent to that formed by adding the same quantity to both its members. For it is clear that A +m = B -\-m when, and only when, A=^ B. II. Any term may be transformed from one side of an equation to the other, provided its sign be changed. Let the equation be a + h — c—p — q + r. Add —p-\-q~r to both sides ; then a-^h — c—p + q — r—p — q + r — p + q^r, that is, a-\-h — c—p + q — r — 0. We thus have an equation equivalent to the given equation, but with the terms p, —q,-\-r changed in sign and transposed. By means of transposition all the terms of any equa- tion may be written on one side of the sign of equality and zero on the other side. III. An equation is equivalent to that formed by multiplying (or dividing) each of its members by the same quantity which is not equal to zero. For, ii A=B, it is clear that mA = mB. Conversely, if mA = mB, that i%m{A-B)= 0, it follows that A-B^O, since m is twt zero. Hence mA — mB when, and only when, A=B. EQUATIONS. ONE UNKNOWN QUANTITY. 107 The case of division requires no separate examination, for to divide by m is the same as to multiply by — . 117. Simple Equations. The method of solving simple equations will be seen from the following examples. Ex. 1. Solve the equation 13a; - 7 = 6a; + 9. Transpose the terms 5a; and - 7; then 13a; - 5a;= 7 + 9. That is 8a; = 16. Divide both sides by 8, the coefficient of x; then a; =2. Zx 2x Ex. 2. Solve the equation ~r - 2 = — + 5. 4 o We may get rid of fractions by multiplying both members by 20, the least common multiple of the denominators; we then have 15a; -40 = 8x4-100, or transposing 15a; - 8a; = 100 + 40 ; .-. 7a; =140. Divide by 7, the coefficient of x; then a; = 20. Ex. 3. Solve the equation a{x-a) = 2ab ~b{x- 6). Eemoving the brackets, we have ax-a^ = 2ab-bx + b\ or transposing ax + bx=2ab + b^ + a^, that is a; (a + 6) = (a + 6)2. Divide by a + 6, the coefficient of x; then x=- ~=:a + b. a + b From the above it will be seen that the different steps in the process of solving a simple equation are as follows. First clear the equation of fractions, and perform the algebraical operations which are indicated. Then trans- pose all the terms which contain the unknown quantity to one side of the equation, and all the other terms to the other side. Next combine all the terms which contain the unknown quantity into one term, and divide by the 108 SIMPLE EQUATIONS. coeflRcient of the unknown quantity : this gives the re- quired root. 118. Special Case9. Every simple equation is re- ducible to the form cw? + 6 = 0, the solution of which is x = — . The following are special cases. I. If 6 = 0, the equation reduces to cw? = ; whence a? = 0. II. If 6 = and also a = 0, the equation is clearly satisfied for all values of x. III. Ifa = and 64=0. Suppose that while h remains constant, a takes in succession the values :r^ , r^, -j-Trg,.--; then will a; take in succession the values — 106, — 10^6, — 10^6.... Thus as a becomes continually smaller and smaller, x will become continually greater and greater in absolute magnitude; moreover, by making a sufficiently small, x will become greater than any assignable quantity; for example, in order that the absolute value of x may be greater than lO'"*" it is only necessary to give to a an absolute value less than j^ . This is expressed by saying that, in the limit, when a becomes zero, the root of the equation oo? + 6 = is infinite. The symbol for infinity is oo . EXAMPLES. Solve the equations 1. l(x-2)-|(a;-3) + ^(a;-4) = 4. Am. «=12. 2. g(a;-3)-^(a;-8) + g(x-6) = 0. Ans. a: = 0. EQUATIONS. ONE UNKNOWN QUANTITY. 109 8. a{x-a) = b{x-b). Ans. x = a + b. 4. {x + a){x + b)-{x-a){x-b) = {a + b)K Am, x=-^{a + b). 6. a{2x-a) + b{2x-b) = 2ab. Ans. x=^{a + b). 6. {a^ + x)(b^ + x)-{ab + x)K Ans. x=0. 7. 3{a; + 3)3 + 5(a; + 6)2 = 8(x + 8)a. Ans. x=-&. 8. {x + a)*-{x-ay-8aK^ + 8a'^ = 0. Ans. x=-a. 9. {x-lf + K^ + {x + l)^ = Sx{x'^-l). Am. a: = 0. 10. ix+a)»+{x + bf+{x + c)^=S{x + a){x + b)(x + c). Am. x = --{a + b + c). 119. Equations expressed in Factors. It is clear that a product is zero when one of its factors is zero ; and it is also clear that a product cannot be zero unless one of its factors* is zero. Thus {x — 2) (a; — 3) is zero when a? — 2 is zero, or when a? — 3 is zero, and in no other case. Hence the equation (a;-2)(a:-3)x=0, is satisfied if a; — 2 = 0, or if a? — 3 = ; that is, if a; = 2, or if a? = 3, and in no other case. The roots of the equation are therefore 2 and 3. Again, the continued product {x — a)(x — h){x — c)... is zero when a; — a is zero, or when a; — 6 is zero, or when a; — c is zero, &c. ; and the continued product is not zero except one of the factors a) — a,x — b,x — c, &c. is zero. Hence the equation {x — a) {x — h) {x — c) . . . = is equivalent to the system of equations a? ~ a = 0, a; — 6 = 0, a? — c = 0, &c. From the above it will be apparent that the solution 110 QUADRATIC EQUATIONS. of an equation of any degree can be written down at once, provided the equation is given in the form of a product of factors of the first degree equated to zero. Now all the terms of any equation can be transposed to one side, so that any equation can be written with all its terms on one side of the sign of equality and zero on the other side. It follows therefore that the problem of solving an equation of any degree is the same as the problem of finding the factors of an expression of the same degree. Ex.1. Solve the equation x^-5x = %. Transposing, we have «'' - 5a; - 6 = 0, that is (a;-6)(x+l)=0; .-. x-G = 0, or a; + 1 = 0. Hence x=6, or a;=-l. Ex.2. Solve the equation 3^-x^=^(jx. Transposing, yue have x^-x^- or a;- -^ — =0, 2a 2a 2a 2a that is a; - - = 0, or a; - a = 0. a Thus the roots are a and - . a Note. In most cases the factors can be written down at once, as in Art. 79, without completing the square; and much labour is thereby S2^ved. EQUATIONS. ONE UNKNOWN QUANTITY. 113 EXAMPLES. Find the roots of the following equations : 1. 9a;2- 24a: + 16=0. -• 1- 2. 6(a:2 + 4)=4(x2-h9). Ans. >t4. 3. 3a;2=8a; + 3. Ans. 3, -i. 4. 16x2 + 16a: + 3 = 0. ... -i, -?. 6. »»+(a-a!)2=(a-2x)«. Am. 0, a. 6. a!3+(a-2a:)2=(a-3a:)2. Ans. 0, |. 7. x^ + x = a^ + a. uins. a, -a-1. 8. x^ + 2ax = b- + 2ab. ^7W. 6, -2a-b. 9. (a:-a)2 + (a:-6)2=(a-6)S. Am. a, b. 10. (a-a:)» + (a:-6)3 = (a-fc)3. Am. a, b. 11. {b~c)a^+{c-a)x+{a-b): = 0. Am. 1, 7 • b-c 12. (x-a + 2b)^-{x-2a+b)^= :(a + 6)». Am. a -2b, 2a -b. 121. Discussion of roots of a quadratic equation. In the precediDg article we found that the quadratic equa- tion aa^ + 6a? + c = had two roots, namely 2 and — X . / — T-^ 2a V 4a* Since * / — ^""a — is real or imaginary according as b^ — 4iac is positive or negative, it follows that the roots of aa^ + bx + c = are real or imaginary according as 6* — 'itac is positive or negative. The roots are clearly rational or irrational according as b^ — 4iac is or is not a perfect square. It should be remarked also that both roots are rational or both irrational, and that both roots are real or both imaginary. s. A. 8 114 SPECIAL FORMS OF QUADRATIC EQUATIONS. If 6* — 4ac = 0, both roots reduce to — h- , and are thus Za equal to one another. In this case we do not say that the equation has only one root, but that it has two equal roots. It is clear that the roots will be unequal unless 6* — 4ac = 0. Hence in order that the two roots of the equation aa^ + 6a; + c = may be equal, it is necessary and sufficient that b^ = 4ac. When 6' = 4ac, the expression ax* -\-bx-\-c is a perfect square in x, as we have already seen. [Art. 83.] 122. Special Forms. We will now consider some special forms of quadratic equations, in which one or more of the coefficients vanish. I. If c = 0, the equation reduces to oar* + bw — 0, or X (ax + 6) = 0, the roots of which are and — . a II. If c = and also 6 = 0, the equation reduces to ax^ = 0, both roots of which are zero. III. If 6 = 0, the equation reduces to aa^ + c = 0, the roots of which are + , / — . The roots are therefore ~ V a equal and opposite when 6 = 0, that is when the coefficient of X is zero. IV. If a, 6 and c are all zero, the equation is clearly satisfied for all values of x. V. If a and 6 be zero but c not zero, put x = - in the equation ax^ + bx + c = 0; EQUATIONS. ONE UNKNOWN QUANTITY. 115 then we have, after multiplying by i/*, cy^ + 6y + a = 0. Now from I. and II. one root of this quadratic in y is zero if a = 0, and both roots are zero if a = and also 6 = 0. But since a; = -, a; is infinity [Art. 118] when y is zero. Thus one root of ax^ + 6ic + c = is infinite if a = ; also both roots are infinite if a = and also 6 = 0. Thus the quadratic equation (a - a!) a;2 + (& - 6') « +c - c'=0 has one root infinite, if a = a' ; it has two roots infinite, if a=^a' and also 6 = 6'; and the equation is satisfied for all values of x, if a=a', 6=6' and e=c'. Again, the equation a(a: + 6) (a: + c) + 6 (x + c) {a; + a)=c (a; + a) (a; + 6), is a quadratic equation for all values of c except only when c = a + 6, in which case the coefficient of x^ in the quadratic equation is zero. When c — a + h we may still however consider that the equation is a quadratic equation, but with one of its roots infinite. Note. It is however to be remarked that since in- finite roots are not often of practical importance in Algebra, they are generally neglected unless specially required. 123. Zero and infinite roots of any equation. The most general form of the equation of the nth degree is aa;"+ hx*"'^ +...+ ka) + l=^0 (i). If ^ = 0, the equation may be written X (aa;"~* + 6a;"~' +. . .+ A;) = 0, one root of which is clearly zero. Similarly two roots will be zero if i^ = and also k = 0; and so on, if more of the coefficients from the end vanish. 8—2 116 EQUATIONS NOT INTEGRAL. Put a; = - ; then we have, after multiplying by y", a-¥hy + + %""' + /y" = 0. From the above, one root of the equation in y will be zero when a = 0; and two roots will be zero if a = and also 6 = 0. But when v = 0, x = - = co . ^ y Thus one root of (i) is infinite when a = 0, and two roots are infinite when a and 6 are both zero ; and so on, if more of the coefficients from the beginning vanish. 124. Equations not integral. When an equation is not integral, the first step to be taken is to reduce it to an equivalent integral equation. An equation will be reduced to an integral form by multiplying by any common multiple of the denominators of the fractions which it contains, but the legitimacy of this multiplication requires examination. For if we multiply both sides of an integral equation by an ex- pression which contains the unknown quantity, the new equation will not only be satisfied by all the values of the unknown quantity which satisfy the original equation, but also by those values which make the expression by which we have multiplied vanish. Thus if each member of the equation A=^B, be multiplied by P, the resulting equa- tion PA = PB, or P {A — B)= 0, will have the same roots as the equation A—B = together with the roots of the equation P = 0. When however an equation contains fractions in whose denominators the unknown quantity occurs, the equation may be multiplied by the lowest common multiple of the denominators without introducing any additional roots, for we cannot divide both sides of the resulting equation by any one of the factors of the L.C.M. without reintroducing fractions, which shews that there are no roots of the re- sulting equation which correspond to the factors of the expression by which we multiply. EQUATIONS. ONE UNKNOWN QUANTITY. 117 3 2x Ex. 1. Solve the equation = + = 6. x-5 x-B Multiply by (as- 5) (x- 3), the u c. m. of the denominators; then we have 3(a;-3) + 2a;(a;-5) = 5(a:-5)(a5-3); /. 3x2 -33ar + 84 = 0. Whence x=4 or x=7. Ex. 2. Solve the equation —= — - + 2 + =0. x^-1 x-1 Multiply by x^-1, the l.o.m. of the denominators; then we have x»-3x + 2(xa-l)+x + l=0, which reduces to 3x»-2x-l=0, that is (3x+l)(a;-l)=0. Thus the roots appear to be - J and 1 ; the latter root is however due to the multiplication by a^- 1. x"-3x 1 x^-Sx + x + 1 (x-l)2 x-1 Since V3T + ^= x^-1 =Wl=i^^i' the equation is equivalent to ^ + 2=0, which has only one root, namely x= - J. From the above example it will be seen that when an equation has been made integral by multiplication, some of the roots of the resulting equation may have to be rejected. Ex. 3. Solve the equation : X x-9 _ »+! . »-8 x-2'^x^~x^'^x^Q' In this case it is best not to multiply at once by the l. o. m. of the denominators of the fractions; much labour is often saved by a judicious arrangement and grouping of the terms. By transposition we have X x+1 x-9 ag~8 _Q 08-2 x-1 x-7 x-G" The first two terms {x-2){x-iy 118 EQUATIONS NOT INTEGRAL. -2 and the other terms = x-7){x-6y Hence the equation is equivalent to 2 2 = 0. {x-2){x-l) (*-7)(x-6) Now multiply by the l.o.m. of the denominators; then 2 (x-l) {x -&) -2 {x - 2) {x -1)=0, which reduces to 20a; -80=0; Or thus : — The equation is equivalent to x-2 ^ ^x-7 ^ -x-1 ^ ^x-6 *» *v, . • 2 2 2 2 that IS _^_-_^ = _^-_^; - 10 - 10 " {x-2){x-7)~{x-l){x-Q)* from which we find as before that a;=4. Ex. 4. Solve the equation : a b e _rt x + a x + b x + c~ We have —^-1 + — ^.-1 +-- 1 = 0; x+a x+b x+c X X X n :. — + -— i + ——=0. X + a x + b x + c Hence a;=0 (i), or else + -—7 + — — =0. x + a x + h x + c Multiply by the l. 0. m. ; then {x + b){x + c) + {x + e){x + a) + {x + a){x + b)^% that is Zx'^ + 2x{a + b + c) + bc + ca + ab = 0^ EQUATIONS. ONE UNKNOWN QUANTITY. 119 the roots of which are -^{{a + b + c)^J{a^+b^ + c^~bc-ea-ab)} (ii). Thus there are three roots given by (i) and (ii). Ex. 6. Solve the equation: b+c , e+a ^ a bc-x ca-x ab -X a + b + c X n is equivalent to 6+c a c + a bc-x x^ ca-x b - - + X a + b ab-x c x' =a Taking the terms in pairs we have {a + b +c)x-abc {a + b + c)x-ahc (a + b + c) x-dbe_^ X {be -x) x {ca -x) x{ab-x) ~ ' Hence (a + 6 + c)a:-a6c = 1, 1 1 1 X {be -x) X {ca — x) X {ab - x) abe = XL From I. we have x= , a + b + e From n. we have on multiplication by the l. o. m. {ca - x) {ab -x) + {ab - x) {be -x) + {be - x) {ca - x) = 0, that is 8052 - 2x {be + ca + ab) + abc {a + b + c) = 0, whence x=i{bc + ea + ab^ Jb'^c"^ + e^a^ + a^b^ -abc {a + b + c)), 125. Irrational Equations. An irrational equa- tion is one in which square or other roots of expressions containing the unknown quantity occur. In order to rationalize an equation it is first written with one of the irrational terms standing by itself on one side of the sign of equality: both sides are then raised to the lowest power necessary to rationalize the isolated term ; and the process is repeated as often as may be necessary. 120 IRRATIONAL EQUATIONS. Ex. 1. Solve the equation Jx + A+ Jx + 10 - 2 Jx + Tl = 0. We have JxTi + Jx + 20 = 2jx + 11. Square both members : then 2X + 24 + 2 Jx + ljx + 20 = 4:{x + n), which is equivalent to Jx + l Jx + 20= X + 10. Square both members : then (a; + 4)(a; + 20) = (a; + 10)*, whence x=5. Ex.2. Solve the equation j2x + Q-2jx + 5=2. Square both members: then 2a; + 8 + 4(x + 5)-4 J2x + Q Jx + 5=^4^\ :. 3a: + 12 = 2 J2x + S Jx + 5, Square both members : then 9x2 + 72a; + 144 = 4 (2a; + 8) (x + 5)> /. a;a=16, whence a;=4 or x= -4. Ex. 3. Solve the equation s/ax+a+ Jbx + p+^cx+y=sO» We have Jax + a + Jbx + p= - Jcx + y. Square both members : then we have after transposition (a + 6-c)x + o + /3-7= - 2 J ax + a ^Jbx +p. Squaring again, we have {{a + b-c)x + a + p-y}^=4:{ax + a){bx + p)t that is x^{a^ + b^ + c^-2bc-2ca-2ab) + 2x(aa + bp + cy-by-ep-ca-ay-ap-ba) + a2 + /32 + 72 - 2/37 - 27a - 2a/3 = 0. Thus the given equation is equivalent to a quadratic equation. It should be observed that it is quite immaterial what sign is put before a radical in the above examples; for there are two square roots of every algebraical expression and we have no symbol EQUATIONS. ONE UNKNOWN QUANTITY. 121 whi ch r epresents o ne on ly to the exclusion of the other ; so that + Jx +i and - Jx + l are alike equivalent to :^Jx + l\ also X + Jx + 1 has the same two values as xJ=iJx+l. 126. By squaring both members of the ratioDal equa- tion A=B, we obtain the equation A^ = B^; and the equation A^ = B\ or A^ — B^== 0, is not only satisfied when J. — 5 = 0, but also when A-\-B — 0, Hence an equation is not in general equivalent to that obtained by squaring both its members; for the latter equation has the same roots as the original equation together with other roots which are not roots of the original equation. Additional roots are not however always introduced by squaring both sides of an irrational equation. For example, the equation a? + 1 = V^+13 is really two equations since the radical may have either of two values ; and by squaring both members we obtain the equation (a?+ ly = a; + 13, which is equivalent to the two. [See Art. 152.] 127. A quadratic equation can only have two roots. We have already proved that an expression of the nth degree in x cannot vanish for more than n values of a?, unless it vanishes for all values of x. This shews that an equation of the nih. degree cannot have more than n roots, and in particular that a quadratic equation cannot have more than two roots. The following is another proof that a quadratic equa- tion can only have two roots. We have to prove that a^ -\-hx + c cannot vanish for a, /8, 7 three unequal values of x. That is we have to prove that aa^ -{-hoL +c = (i), a/3*-K6/5-hc = (ii), and 07^-1-67+0 = (iii), cannot be simultaneously true, unless a, 6, c are all zero. 122 A QUADRATIC EQUATION HAS ONLY TWO ROOTS. From (i) and (ii) we have by subtraction that is (a-/3) {a(a + ^) + b} = 0. But a — yS + O; hence a(a + ^) + b = (iv). Similarly, since ^ — 7 4= 0, we have from (ii) and (iii) a(/S+7) + 6 = (v). From (iv) and (v) we have by subtraction a(a — y) = (vi). Now (vi) cannot be true unless a = 0, for a — y^O. Also when a = 0, it follows from (iv) that 6 = 0, and then from (i) that c = 0. Thus the quadratic equation aa^ -\-bx-\-c = cannot have more than two different roots, unless a = 6 = c = ; and when a, 6, c are all zero it is clear that the equation aa^ -{■bx-\-c = will be satisfied for all values of a;, that is to say the equation is an identity. Ex. 1. Solve the equation a^) ^^t [ + ^ n rk r=«*. (a - 6) (a - c) (6 - c) (6 - a) The equation is clearly satisfied by fl;=a, and also by 05=6; hence a, h are roots of the equation, and these are the only roots of the quadratic equation. [The equation is not an identity , for it is not satisfied by a5 = c.] Ex. 2. Solve the equation ni (^-fe)(a?-c) . .a {^-c){x-a) 3 {x-a){x--b) {a~h){a-cy {h-c){h-ay (c-a)(c-6) The equation is satisfied by as = a, by a; = 6, or by a; = c. Hence, as it is only of the gecond degree in as, it must be an identity, Ex. 3. Solve the equation (x-6)(x-c ) {x-c){x-a ) i,x-a)[x-h) _ (a-6)(o-c)"*" {h-c){h-ay (c-a)(c-6)~ * EQUATIONS. ONE UNKNOWN QUANTITY. 123 The equation is satisfied hj x=a,hj x=b, and hy x=c; and the equation is not an identity, since the coefficient of x^ is not zero. Hence the roots of the cubic are a, b, c. Ex. 4. Shew that, if (a - a)^x + {a-^)^y + {a- y)^z = (a - 5)3, (6-o)3x + (6-/S)2y + (6-7)2« = (&-5)3, {c-a)^x+{c-p)^y + {c-y)^z = {e-d)\ then will (d-a)^x + {d-^Yy + {d~y)^z=={d-8)\ where d has any value whatever. The equation {X-a)*x + (X-p)^y + {X-y)H=.{X-8)\ is a quadratic equation in X, and it has the three roots a, 6, c. It is therefore satisfied when any other quantity d is put for X. 128. Relations between the roots and the coeffi- cients of a quadratic equation. If we put a and 13 for the roots of the equation aaf + bx-{-o=0,yfe have 2a / 5'~4ac V 4a'^ ' and '^^-^.-V""^^"'' By addition we have «+/3 = -^ (i). By multiplication we have b* h^-4tac c ,..>. ''^ = W—W=a (")• The formulae (i) and (ii) giving the sum and the product of the roots of a quadratic equation in terms of the coefficients are very important. 124 RELATIONS BETWEEN ROOTS AND COEFFICIENTS. 129. Relations between the roots and the co- efficients of any equation. By the following method relations between the roots and the coefficients of an equation of any degree may be obtained. We have seen that if the expression of the nth degree in X aic* 4- hx"^^ + ca;'*'* + doT'^ + . . . , vanish for the n values a? =*a, a? = ^, a; = 7, &c., then will cw;" + 6x--' + ca?"-^ + o^a;""' + ... = a (a; - a) (a; - /S) (a; - 7)... We have therefore only to find* the continued product (a; — a) (ic — /8) (a? — 7) and equate the coefficients of the corresponding powers of sc on the two sides of the last equation. For example, if a, yS, 7 be the roots of the cubic equa- tion aaf + hx^ + coj + (Z = 0, we have aa^ + hx^ -^ ex -{- d = a {x — a) {x — /8) {x — 7) = a {«' - (a + /3 -f 7) a^ + (^y + yoL+a^) x-ajSy], Hence, equating coefficients, we have a+^+7=-^, /87 + 7a + a/3= - , Of a/37 = -- ^ It should be remarked that the sum of the roots of any equation will be zero provided that the term one degree lower than the highest is absent*. We may make use of the above to prove certain identical rela- tions between three quantities whose sum is zero. For a, b, c will be the roots of the cubic a^ +px + 2=0, provided that a + 6 + c = 0, and that p and q satisfy the relations * See Art. 437. EQUATIONS. ONE UNKNOWN QUANTITY. 125 be,+ ca + ab=p (i), abc = - q (ii). Tlien, since a + b + c = (iii), we have a^ + b^ + c^=(a + b + c)^-2{bc + ca + ab) = -2i> (iv). Also, since a, b, c are roots of x^+px + q=Of a^+pa + q = \ lP+pb + q = I (v). c^+pc + q = ) From (v) by addition a^ + h^ + c^=-Sq (vi). Multiply the equations (v) in order by a'*-^, 6"-3, c"-^, and add ; then a« + 6« + c" +i) (a»-a + 6""* + c»»-2) + q (a^-s + fe**-* + c"-^) = 0. Hence we have in succession a* + b^ + c* = 2p\ a^ + ¥ + c'^=5pq, a« + 6« + c« = 3g2-2j)3, o^ + 67 + c7=-7p2gr. Hence also a°+&'^ + cg _ a2 + 6g + c» a» + b^ + c* 6 ~ 2 • 3 • a7 + b'^ + cf a^ + h^ + c^ a^ + b^ + c^ = 2. 2*6' a» + b^ + c^ a* + b* + c* 3 • 4 • [See also Art. 308, Ex. 2.] 130. Equations with given roots. Although we cannot in all cases find the roots of a given equation, it is very easy to solve the converse problem, namely the problem of finding an equation which has given roots. For example, to find the equation whose roots are 4 and 6. We want to find an equation which is satisfied when a; =4, or when x — 5; that is when a; - 4 = 0, or when a; - 5 = ; and in no other cases. The equation required must be (a?-4)(a;-5) = 0, that is, aj«-9a+20=0, 126 EQUATIONS WITH GIVEN ROOTS. for this is an equation which is a true statement when af-4=0, or when a; - 5=0, and in no other case*. Again, to find the equation whose roots are 2, 3, and - 4. We have to find an equation which is satisfied when a; - 2 = 0, or when a; - 3 = 0, or when a; + 4 = 0, and in no other case. The equation must therefore be {x - 2) {x - S) [x + 4) = 0, that is x^-x^-Ux + 2i = 0. Ex. 1. If a, /3 are the roots of the equation cLX^ + bx + c=iO, find the equation whose roots are - and - . P « The required equation is that is «3_a;'L±^ + l=,0. Now, by Art. 128, we have b c ..a^^^^ = ';-2l; Hence the required equation is „ b"- 2ac a; + l = 0. Ex. 2. If a, j9, 7 be the roots of the equation fla:* + &x^ + ca; + d=0, find the equation whose roots are /S7, 7a, a/3. The required equation is (a;-^7)(ar-7a)(x-a/3)=xO, that is «»-a:2(/37 + 7a + a/3)+xa/J7(a + ^ + 7)-a^/SV = 0. * The equation a;^ - 9ar + 20 = is certainly an equation with the proposed and with no other roots ; but to prove that it is the only equa- tion with the proposed and with no other roots, it must be assumed that every equation has a root. If, for example, the equation a:* + 7a;'-2 = had no roots, then (x - 4)(a; - 6) [x^ + lx^ - 2) = would also be an equation with the proposed roots and with no others. The proposition that every equation has a root is by no means easy to prove; the proof is given in works on the Theory of Equations. EQUATIONS. ONE UNKNOWN QUANTITY. 127 Now, by Art. 129, we have h a + p + y=--, and 0'^y= — • Hence the required equation is a?- -x^ + x-^ - -0=0, or 0^3^ - acT? + hdx- cP = 0. 131. Changes in value of a trinomial expression. The expression act? + hx + c will alter in value as the value of X is changed ; but, by giving to x any real value between — oo and + oo , we cannot make the expression ax^ -\-hx-\-c assume any value we please. We can find the possible values of ax^ + hx-\-c, for real values of x, as follows. In order that the expressicm ax^ -\-hx-\-c may be equal to X for some real value of x, it is necessary and sufficient that the roots of the equation ax^ -\-hx-\-c = \ ^< be real, the condition for which is 6'-4a(c-X)>0, that is 6*-4ac + 4aX>0 (i). I. If 6' - 4ac be positive, the condition (i) is satisfied for all positive values of 4aX, and also for all negative values of 4aX which are not greater than 6' — 4ac. Thus, when h^ — ^ac is positive, aa? + hx ■{■ c can, by giving a suitable value to x, be made equal to any quantity of the same sign as a, or to any quantity not absolutely greater than — 7 and whose sign is opposite to that of a. II. If 6' — 4ac be negative, the condition (i) can only be satisfied when 4aX is positive and not less than 4ac — 6'. 128 CHANGES IN VALUE OF A TRINOMIAL EXPRESSION. Thus, when 6* — 4ac is negative, aa;* -\-bx + c must al- ways have the same sign as a, and its absolute magnitude can never be less than — -. . 4a III. If h* — ^ac be zero, the condition (i) is satisfied for all positive values of aX. It follows from the above that the expression aa^+bx+c will keep its sign unchanged, whatever real value be given to X, provided that 6* — iac be negative or zero, that is provided that the roots of the equation aa^ -\- bx -\- c =^ he imaginary or equal, and also that the expression can be made to change its sign when the roots of do^ -{-bx+ c—0 are real and unequal. We give another proof of this proposition. If the equation ax^ + bx-\- c=0 have real roots, a, yS suppose, then ax^ -^bx-i- c = a(x — a)(x - jS). Now (x — OL){x — /3) is positive when x has any real value greater than both a and /8, or less than both a and jS : but ix — a) (x — jS) is negative when x has any real value intermediate to a and y§. Thus for real values of x the expression ax*+bx-\-c has always the same sign as a except for values of x which lie between the roots of the corresponding equation ax^ + 6a; + c = 0. 132. We can also prove that the expression aa^-\-bx-^c will or will not change sign for different values of x accord- ing as 6^ — 4ac is positive or negative, as follows. ax^ + bx-\- c r + 2-aj-— 4-^J I. Let 6* — 4ttc be positive. The whole expression within square brackets will clearly be negative when a; = — ^ ; also, when x is very EQUATIONS. ONE UNKNOWN QUANTITY. 129 great, (^ + 2" J '^^^^ ^^ greater than — 7-3— , and there- fore the whole expression within square brackets will be positive. Thus when 6''—4ac is positive the expression aa;*+5a7+c can be made to change its sign by giving suitable real values to x. II. Let 6' — 4ac be negative (or zero). Since lx+ ^\ is positive for all real values of sc, and 2~8 — ^^ ^^^^ positive (or zero), the whole expression within square brackets must be always positive. Thus when h^ — 4iaG is negative or zero, the expression GUK* + bx-\-c will always have the same sign as a. 133. It follows from Article 131 or 132 that if an expression of the second degree in x can be made to change its sign by giving real values to x, then must the roots of the corresponding equation be real. Consider, for example, the expression a'(x -^){x-y)-\- b' (x-y) (a;- a) + c' (x - a) (x-fi), where the quantities are all real, and a, /3, 7 are supposed to be in order of magnitude. The expression is clearly positive if x = a, and is negative if x = ^. Hence the expression can be made to change its sign, and therefore the roots of the equation a'(x-^)(x-y)+b\x-y){x-OL)-\-c\x-a)(x-^)^0 are real for all real values of a, 6, c, a, A 7. Ex. 1. Shew that {x -l){x- 3) {x - 4) (a? - 6) + 10 is positive for aU real values of x. Taking the first and last factors together, and also the other two, the given expression becomes (x2 - 7a; + 6) (ar» - 7x + 12) + 10 = {a;a-7a;)3+18(«2-7a;) + 82 = {(x^-7x) + 9y + l, which is clearly always positive for real values of x. S.A. ' 9 130 EXAMPLES. Ex. 2. Shew that, by giving an appropriate real value to a:, TTT-s — 7^ T can be made to assume any real value. 12a;2 + 8x + l ^""^ 12x2 + 8x + l-^' then «2 (4 _ i2\) + (36 - 8X) a; + 9 - X =0. Now in order that x may be real it is necessary and sufl&cient that (36 - 8X)» - 4 (4 - 12X) (9 - X) > 0, or that X2-8X + 72>0, or (X-4)a + 56>0, which is clearly true for all real values of X. Thus we can find real values of x corresponding to any real value whatever of X. x^ — Sx + 4 Ex. 3, Shew that -s — ^ 1 ca^i never be greater than 7 nor less «-' + dx + 4 than - for real values of x. ^* a;^ + 3x + 4 =^' then x2(l-X)-3x(l + X) + 4(l-X)=0. In order that x may be real it is necessary and sufl&cient that 9(1 + X)2-16(1-X)3>0. that is -7X3 + 50X-7>0, or _(7X-l)(X-7)>0. Hence 7X - 1 and X - 7 must be of different signs, and therefore X must lie between - and 7, which proves the proposition. EXAMPLES X Solve the follovring equations: 1. (x-a + 2by-(x-2a-i-by = (a + b)*. 2. (c + a-2b)af + (a + b-2c)x + {b + c-2a)=0, a (x-ay {x+by a+x b+x b+x a+x 5. EQUATIONS. ONE UNKNOWN QUANTITY. 131 ax-hb _cx + d a + bx c + dx' a-x 1-bx 1 -ax h-x ' 7. r- = a^-k-2x . x+\ a;+l Q . X* x' 5x 8. X+l + — r = i + x'-l a; + l ic«-l' x-0 X- 6 x + 6 x+ 8 10 2^5 3 _4^ a;+8 a;+9 a:+15 """aj + S* 11. J-.. ' ' 2x-3 x-2 3a;+2* 12. ?r| + ?i:* + ^-f=3. «-o x-c x-a . x + a x + b x + c „ 13. + + = 3. a-x o—x c-x 14. ^±f + ^* + ?±5^3 X-a x-b x-c 15. -!£Z1+!^1 = 4^^7 x+1 x+2 x-1 ,„ X 2 X 3 ,„ *-rc* w — a x+0 X — o 18. 19. x + a h a; — a «- a x+6 a;-6 X- 6 x + a x + 6" x-l x + 1^ X- x + -4 4 a;-2 a;-3 ~x + 2 * a;+3 1 - = 1 af+a + - 1 a:-a + - "T '_' x + b 9—2 132 EXAMPLES. 20. 111^ 21. a + b a + c 2{a + b + c) x + h x + c ~ a; + 6 +c 22. a + c h + c a + 6 + 2c x + % x + 2a x + a + b ' 23. x-b x — a 2(a-b) x-a X— b x-a-b' - , (x + a){x + b) _(x + c) (x + d) x+ a+ b x+c+d 25. x+a x+d x+b x+o 26. x-a x-b b a a x-a x-b 27. a-b b - c c-a ^ 1 + r -f- = 0. x+a-b x+b-c x+c-a 28. 1 2 3 ^ I + 2x 2 + 3x S + 4:X 4 + 5£c "' 9.fl {x -a){x- b) {x 4- a) (a; + b) (x - ma) {x - mb) (x + ma) {x + mb) 30. ^2^T9 - 7^^ = V^TT. 31. ^/(x-l){x-2) + ^/(x-S){x-i)=:^2. 32. V7a;-5 + V4£c- 1 = V7a; - 4 + V4a;- 2. 33. tja' -x+ ^/b' + x = a + b. 34. tja-x + *jb-x^^a + b-2x. 35. ^/a-bx + >Jc-dx = ^/a + c-{b + d)x. 36. ^aaj + 6* + v 6a; + a' = a - 6. EQUATIONS. ONE UNKNOWN QUANTITY. 133 37. Ja + x+ Jb+x = J a + b + 2x, 38. Ja-x + Jb + x = J2a + 26. 39. J{a + x)(x + b) + J(a-x)(x^) = 2 yosc. 40. Ja{a + b + x)- Ja(a + b-x) = x. 41. ^a;^ + aa; + 6» - Vaj' - oo; + 6' = 2a. 42. Js^T^x + a' + Jac^-ax + a* = J2a'-2b'. 43. Jax~^ + ^ccc + 6 = ^oaj + ft + ^ca? - 6. 44. Jx {a + b - x) + Ja (b + X - a) + Jb (a + X - b) = 0. 45. /Jx + a + Jx + b + Jx + c = 0. 46. JabJa + V+s^ = Ja(a + b) (6 - a;) + ^6 (a + 6) (a - a;). ' 47. Jx'~b'-(^ + Jx'-c'-a' + Jaf-a!'-b'=^x. 48. J'^r^+ J¥^^+ sl7'^^= Ja' + b' + c'-af. 49. For what values of a; is ^14 - (3a; - 2) (a; - 1) real. x'+ 34a; — 71 50. Shew that — = — =- can have no real value between a;' + 2a;-7 6 and 9. 51. Shew that, if aj be real -5 — ^ r can never be less a;" + 2a; + 1 than —J. 52. What values are possible for —j ^ , x being real. 53. Find the greatest and least real values of x and y which satisfy the equation a^ + y' = 6x-8y. 54. Find the greatest and least real values of x and y when aj" + 4y»-8a;-16y-4 = 0. 55. When x and y are taken so as to satisfy the equation (a;* + y")" = 2a" (x" - y^\ find the greatest possible value of y. 134! EQUATIONS OF HIGHER DEGREE THAN THE SECOND. 56. Shew that if the roots of the equation a? (b' + b") + 2x {ah + a'h') + a'-^a" = be real, they will be equal. 57. If the roots of the equation aaf + 6a; + c = be in the ratio m : n, then will mnlf = (m + ny ac. 58. If aa3* + 2hx + c = and aV + 2h'x + c' = have one and only one root in common, prove that 6' — ac and 6" — a'c' must both be perfect squares. 59. If ajj, X be the roots of the equation oa;' + 6a; + c = 0, .. x^ x^ find the equation whose roots are (i) x^ and x^^ (ii) -^ and — ^' ajj ajj (iii) h + ax^ and h + ckc,. 60. If x^ , a;, be the roots of ax^ + 6a5 + c = 0, find in terms of a, 6, c the values of ajj' (bx^ + c) + a;/ (ftaj^ + c), and aj^^ (Jaj^ + c)' + aj^^ (62;^ + cf. 61. Shew that, if x^ , a;, be the roots of x^ + ma; + m^ + a = 0, then will ai^* + a;ja;, + a;/ + a = 0. 62. If a?! , a5j be the roots of {a? + 1) (a* + 1) = max (ax - 1), then will {x' +1) (aj^* + 1) = majja;^ (ajia;^ - 1). 63. If aj,, x^ be the roots of the equation A (x' + m') + Amx + ^mV = 0, then will A {x^ + x^) + Ax^x^ + -Ba;j'a;/ = 0. 64. Prove that, if x be real, 2 (a — a;) {x + >/a;* + 6*) cannot exceed a' + 6^ «. ^. , , , .,, , ^ 2a;' -4a;' 4- 9a;'' -4a; + 2 65. Find the least possible value of -7—% — fva » for real values of x* EQUATIONS. ONE UNKNOWN QUANTITY. 135 Equations of higher degree than the second. 134. We now consider some special forms of equations of higher degree than the second, the solution of the most general forms of such equations being beyond our range. 135. Equations of the same form as quadratic equations. The equation ax*' + 6a;^ + c = can be solved in exactly the same way as the quadratic equation aoc^ + 6^? + c = ; we therefore have . ^ b J¥- 4>ao 2a- 2a Hence x /{b Jb'-4>ac \ V I 2a- 2a j Thus there are four real or imaginary roots. Similarly, whenever an equation only contains the unknown quantity in two terms one of which is the square of the other, the equation can be reduced to two alternative equations : for, whatever P may be, aP'+6P+c = is equivalent to P — — ^r- -^ — ^r . 2a - 2a Ex. 1. To solve x* - IQx^ + 9=0. We have {x^-^){x^-l)=Q\ .\ x^=9, giving x= ±3; or else ar*= 1, giving a;= ± 1. Thus there are four roots, namely +1, - 1, +3, -3. 136 EQUATIONS OF HIGHER DEGREE THAN THE SECOND. Ex. 2. To solve {x^ + x)^ + ^ {x^ + x) - 12 = 0. The equation may be written {x^ + x + 6) {x^ + x-2)=z 0. Hence x^ + x+&=0^ or x^+x-2=0. The roots of a;^ + a; + 6 = are - ^ ± --J^2Z, The roots of ic2 + a?-2 = are 1 and -2. Thus the roots are 1, -2, --.^-J-IX Ex. 3. (a;2 + 2)2 + 8a; {f + 2) + 15a;2 = 0. The equation is equivalent to (a;2 + 2 + 6a;)(a3 + 2 + 3a;) = 0. The roots of ar» + 3a;+2=0 are -1 and -2. Therootsof a;2 + 5a; + 2 = are -s=»= ^^. 5 \ Thus the equation has the four roots -1, -2, -h^'^o v17. Ex.4. To solve aa;* + &a5 + c + p Ayaa;2 + 6a; + c + g = 0. Put y=ijax^ + hx + c\ then 2/'' + Pl/ + 3 = 0, whence we obtain two values of y, o and j3 suppose. We then have ax^ + fea; + c = a^, or aa;* + 6a; + c = jS^, and the four roots of the last two quadratic equations are the roots required. Ex.6. To solve 2a;^-4a;+3^/«'-2a; + 6 = 15. The equation may be written 2(x2-2a; + 6) + 3^(a;2-2a; + 6)-27=0. Put y = ;^(x2 - 2a! + 6) ; then we have 1y^ + 3y - 27 = 0, 9 whence y=3, ory=--. Hence «2-2a; + 6 = 9, giving aj=3 or -1; or else a;"-2»+6=-j-, giving a;=l±s n/61. Thus the roots are 8 ; - 1 ; 1 ± x ^61. EQUATIONS. ONE UNKNOWN QUANTITY, 137 9 Ex. 6. To solve {x + a){x + 2a) {x + 3a) {x + 4a) = r^ a*. Taking together the first and last of the factors on the left, and also the second and third, the equation becomes of the form we are now considering. We have {x^ + 5ax + 4a2) {x^ + 5ax + 6a^) = ^ a\ Hence (x» + 5ax)^ + lOa^ {x^ + 5ax) + 24a* = ^ a\ .*. !e^ + 5ax= --ra\ or else x^+5ax= - -ra\ 4 4 Hence «+5a = 0, or «+ a==b-^10. Thus the roots are ~o*» ''^^^9. '^^^ ' 136. Reciprocal Equations. A reciprocal equa- tion is one in which the coefficients are the same whether read in order backwards or forwards ; or in which all the coefficients when read in order backwards differ in sign from the coefficients read in order forwards. Thus ax^-\-baf-hba) + a = 0, ax* + hx^ +cx^-\-hx + a=Oy and ax^ + hx*' -\- cx^ — ca^ — hx — a =^ are reciprocal equations. [See also Art. 442.] Ex. 1. To solve ax^-\-b^ + 'bx + a-=^ 0. We have o (a;'+l) + 6a;{a; + l)=0, that is (x + l){a{3^-x + l) + hx)=Q. Hence as=s - 1, or else aai?-\-(h-a)x + a—0» Ex. 2. To solve ax^ + 6a:' + cx^ + 6x + a = 0. Divide by x^ ; then we have (x«+i)+6(.+i)+c=a Now put x + ~=y\ then a;2+_=y2_2. Hence a(y2-2) + &y + c=0. 138 BOOTS FOUND BY INSPECTION. Let the two roots of the quadratic in y he a and /S;. then the roots of the original equation will be the four roots of the two equations 1 J 1 o ajH — =a and x + -=fi. X X ^ Ex.3. To solve ax'^ + bx* + cx^ - cx^ - bx - a=zO. We have a{x^ -l) + bx {x^-l) + cx^ (a;- 1) = 0, that is (x-l){a{x* + x» + x^ + x + l) + bx{x^ + x + l) + cx^} = 0. Hence a;=l, or else cuK^+{b + a)a^ + {a + b + c)x^ + {b + a)x + a=0. ^ The last equation is a reciprocal equation of the fourth degree and is solved as in Ex. 2. 137. Roots found by inspection. When one root of an equation can be found by inspection, the degree of the equation can be lowered by means of the theorem of Art. 88. Ex. 1. Solve the equation x{x-l){x-2) = a(a-l){a-2). One root of the equation is clearly a. Hence a; - a is a factor of x{x-l){x-2)-a{a-l){a-2), and it will be found that x{x-l){x-2)-a{a-l){a-2) = {x~a){x^-{3-a)x+{a-l){a-2)}. Hence one root of the equation is a, and the others are given by x'^-{S-a)x + {a-l){a-2) = 0. Ex. 2. Solve the equation 0:3 + 2x3- 11a; + 6 = 0. Here we have to try to guess a root of the equation, and in order to do this we take advantage of the following principle : — K «= dL- be a root of the equation ax'^+bx''-^+ ... + k=0, where a, 6, ... fc are integers and - is in its lowest terms, then a will be a factor of k and /3 a factor of a. As a particular case, if there are any rational roots of x^+ ... + k = 0, they will be of the form x=±a, where a is a factor of k. In the example before us the only possible rational roots are ±1, ±2, ±3, and ±6. It will be found that a; =2 satisfies the equation, and we have (a; - 2) (a;2 + 4a; - 3) = a;3 + 2a;2 - 1 la; + 6. Hence the other roots of the equation are given by x^ + ^x-S = 0, and are therefore - 2 ± Jl. EQUATIONS. ONE UNKNOWN QUANTITY. 139 Ex. 3. Solve Since x=a and x = b both satisfy the equation, (x-a) (x-b) will divide (a - x)^ + {x~ b)* - (a - 6)^, and as the quotient will be of the second degree, the equation formed by equating it to zero can be solved. We may however proceed as follows. The equation may be written {a-x)*+{x-b)*={{a-x) + {x-b)y = {a-x)* + ^{a-x)^(x-b) + 6(a-x)^{x-b)^ + 4(a-x){x-b)» + {x-b)*; .'.2{a-x){x-b){2{a-x)^ + S{a-x){x-b) + 2{x-b)^} = 0. Thus the required roots are a, 6 and the roots of the quadratic aP-x(a + b) + 2a^- Sab + 2b^ = 0. Ex. 4. Solve the equation A^-b){x-c) {x-c){x-a) . Ax-a){x-b) _ {a-b){a-cy {b-c){b-a)'^ {c-a){c-b)~ The equation is clearly satisfied by a:=a, by x=b, and by a;=c. Also, since the coefficient of x^ is zero, the sum of the roots is zero. [Art. 129.] Hence the remaining root must be -a-b-c. Thus the roots are a, &, c, - (a+6 + c). 138. Binomial Equations. The general form of a binomial equation la oT ±k=0. The following are some of the cases of binomial equations which can be solved by methods already given — for the general case De Moivre's theorem in Trigonometry must be employed. Ex.1. To solve a?- 1=0. Since a?-l = {x-l)(x^+x+l), we have a;- 1 = 0; or else as' + a; + 1 = 0, the roots of which are _1 x/^3 Hence there are three roots of the equation a^=l; that is there are three cube roots of unity, and these roots are 1. -5 + 3*/^3ana -i-i^/^3. 2 ' 2 140 CUBE BOOTS OF UNITY. Ex. 2. To solve a;4-l=0. Since 3e*-l = {x-l){x+l) {x + J~^) {x - V -^). tt»e /owr fourth roots of unity are 1, -1, V^ and -J^, Ex.3. To solve x»-l=0. 3cf^-l = {x-l){x* + gi? + x^ + x + l). Hence x=l; or else x* + x^+x^ + x + l = 0. The latter equation is a reciprocal equation. Divide by x^^ and we have «3+-„+a5+- + l=0. x^ X Put x+-=y; then a;2 + -^=y2-2; Hence x+ — _ , a; 2 that is a;a_a. -^^V5 _^^^Q^ Hence x = Jll + n/^ ± ^ ^-10-2^5, or a;= ~-'-~">^^ dbi V-IO + V^. or x=l. Ex. 4. To solve x* + 1 = 0. a;4 + l = {a;a+l)»-2.T2 = (a;!» + l-v'2a;)(a;» + l+V2a;). Hence «*=F^2a; + l = 0; •••*--^;^ • 139. Cube roots of unity. In the preceding article we found that the three cube roots of unity are 1, H-i+s/~3), H-i-y-3)- An imaginary cube root of unity is generally repre- sented by o); or, when it is necessary to distinguish EQUATIONS. ONE UNKNOWN QUANTITY. 141 between the two imaginary roots, one is called oy^, and the other G),, so that 1, co^ and ©^ are the three roots of the equation a?" — 1 = 0. Taking the above values, we have 1 +«, + 0), = 1 + H- 1 +y^) + i(- 1 -s/^^) = 0, also co.co, = i(- 1 + s/=^) (- 1 - 7=^) = 1. These relations follow at once from Art. 129 ; for the sum of the three roots of a?* — 1 = is zero, and the product is 1. Again CO,' = i(- 1 + J-^f = K"" 1 - J^) = «,> and < = i(- 1 - J^r = K- 1 + ^/^) = «i> so that co,=co^ and co^=(o,. These relations follow at once from CD, ft). = 1 and ft) * = ft) ° = 1. Thus if we square either of the imaginary cube roots of unity we obtain the other. Hence if (o be either of the imaginary cube roots of unity, the three roots are 1, co and g)*. We know that a9 + b^ + c^-Sabc = {a + b + c){a^ + b^ + c^-bc-ca-ab). Hence a+b + cia & factor of a^ + 6^ + c* - 3abc, and this is the case for all values of a, 6, c. Hence a + (w&) + U^c) is a factor of a^ + {(abf + {uPcf - 3a (w6) (w^c), that is of a^ + li^ + i^-^abc, and a+uP'b + wc can similarly be shewn to be a factor. Hence a^ + h^ + (^ -Zabc = {a + b-\-c) {a + ub + (a^c){a + orb + (ac). I EXAMPLES XI. Solve the following examples ; 2. a;*'4-7aV-8a' = 0. 142 EXAMPLES. 3. a;«-7aV-8a'' = 0. d ^ a^ + 1 _ 5 5 a;' + 2 x^ + 4x+l _5 af+4:x + l'^ a^ + 2 ~2' 6. {af + x+l){x' + x + 2) = l2, 7. (aj« + 7aj + 5)' - 3x' - 2\x = 19. 8. JU-1x^x'=x'+1x-\, 4 9. 6v/x''-2ic + 6 = 21+2ic-ic«. 10. {a-\){\+x + xJ = {a+l){\+x' + x*). 11. (a; + 1) (a; + 2) (a; + 3) (ic + 4) = 24. 12. {x + a) (a; + 3a) (a; + 5a) (aj + 7a) = 384a*. 13. {x - 3a) (aj - a) (a; + 2a) (a; + 4a) = 2376a\ 14. {x + 2){x + 3) (a; + 8) (a; + 12) = 4a;". 15. 2a;«- 3a;-21 = 2aj ^a;^ - 3a; + 4. 16. a;' - 2 (a + 6) a;* + a^ + 2a6 + 6« = 0. 17. X* - 2a;V - 2x'h' + aV 6* - 2a'6' = 0. 18. 4a;*-4a;"-7a;*-4a; + 4 = 0. 19. 9a;* - 24a;» - 2a;' - 24a; + 9 = 0. 20. a;' + l=0. 21. a;«-l=0. 22. 3a;'-14a;« + 20a;-8 = 0. 23. a;* - 15a;' + 10a; +24 = 0. 24. a;* + 7a;' - 7a; - 1 = 0. 25. (x^ay (b-cy + (x-by (c-ay + (a; - c)» (a-6)» = 0. 26. a;(aj-\)(a;-2) = 9.8.7. EQUATIONS. ONE UNKNOWN QUANTITY. 143 27. x{x-l){x-2){x-3) = 9.8.7.6. 28. («-«)»+ {b-xy= {a + b-2xY. 29. (a -- xy + {b-xy = {a + b- 2xy. 30. (a-xy + {b-xy = (a+b-2xy. 31. ^^a^^ + ^F^ = 'Ja + b- 2x. 32. i/a- x + tjb-x = ^a + b- 2x. 33. (a - «)*-+ (a - 6)* = (a - 6)\ 31 Ua-x-\-^x-b = Ua-b. 35. J^a - aj + ^a? - 6 = ^a - b. 36. «* + («- a?)* = 6*. 37. (a; + ay + (a; + 6)' = 17 (a - 6)*. 38. *Jx + 7a^= 4/&. 39. a6a; (a; + a + 6)^ - (aa; + 6a; + a6)' = 0. 40. obex {x + a+b + cy - (xbc + xca + xab + abc)' = 0. (a-xy + (x-by _ a* + b* ' (a+b-2xy ^{a + by 42. x* + b(a + b)x''+{ab-2)b'o(f-(a + b)b^x + b' = 0. 43. {of + by = 2ax' + 2a¥x - aV. 44. (x + b + c) (x + c + a) {x+ a+b) + abc = 0. AIT ^ ^ C n r\ 45. T + + — I +3=0. o+c—x c+a—x a+o—x *°- (x-ay-{b-cy ^ (x-by-{c-ay "*" (a;-c)^-(a-6)« ~ * (a; + a) (x + b) (x - a) (x-b) _(x + c) (x + d) {x -a) {x- b) (x + a) (x + b)~ (x- c) {x - d) {x +c){x + dy CHAPTER X Simultaneous EQUATioNa 140. A SINGLE equation which contains two or more unknown quantities can be satisfied by an indefinite number of values of the unknown quantities. For we can give any values whatever to all but one of the unknown quantities, and we shall then have an equation to deter- mine the remaining unknown quantity. If there are two equations containing two unknown quantities (or as many equations as there are unknown quantities), each equation taken by itself can be satisfied in an indefinite number of ways, but this is not the case when both (or all) the equations are to be satisfied by the same values of the unknown quantities. Two or more equations which are to be satisfied by the same values of the unknown quantities contained in them are called a system of simultaneous equations. The degree of an equation which contains the unknown quantities a?, y, 2^. . . is the degree of that term which is of the highest dimensions in a?, y, ^. . . . Thus the equations ax + a^y + a^z = a*, wy-\-(io-\-y + z = Ot a^-^y'^ + i^ — Sxyz — 0, are of the first, second and third degrees respectively, SIMULTANEOUS EQUATIONS OP THE FIRST DEGREE, 145 141. Equations of the First Degree. We proceed to consider equations of the first degree, beginning with those which contain only two unknown quantities, x and y. Every equation of the first degree in x, y^ Zy.,, can by transformation be reduced to the form ax-\-hy ■\' cz-\- ,,,—k, where a^h, c, ...k are supposed to represent known quan- tities. Note. When there are several equations of the same type it is convenient and usual to employ the same letters in all, but with marks of distinction for the difi'erent equations. Thus we use a, 6, c. . . for one equation ; a\ h\ d . . . for a second ; cb' , 11 \ c"... for a third ; and so on. Or we use ttj, \y Cj for one equation ; a,, 6,, Cj for a second ; and so on. Hence two equations containing x and y are in their most general forms ax+hy — c, and a'x + h'y = o\ and similarly in other cases. 142. Equations with two unknown quantities. Suppose that we have the two equations ax+hy = Ct and cbx + h'y = c'. Multiply both members of the first equation by 6', the coefficient of y in the second ; and multiply both members of the second equation by Z>, the coefficient of y in the first. We thus obtain the equivalent system ah'x + hh'y = cb\ a'hx + hh'y — c'h. S.A. 10 146 SIMULTANEOUS EQUATIONS OF THE FIRST DEGREE. Hence, by subtraction, we have {ah' — ah)x = ch' — ch ; cb' — ch whence ah'— a'h Substitute this value of a? in the first of the given equations; then ch' - c'h , a —n 77 + oy = c, ah —ah ^ , _ c {ah' — a'h) — a (ch' — c'h) ••• ^y- W^a'h ' , ac' — a'c whence y = -tt 7? • ^ ah'- a'h The value of y may be found independently of oc by multiplying the first equation by a and the second by a ; we thus obtain the equivalent system a'ax + a'hy = a'c, a ax + ah'y = ac'. Hence, by subtraction, we have {a'h — ah') y = a'c — ac' ; _ a'c — ac' •'• ^'aT-^'' which is equal to the value of y obtained by substitution. Note. It is important to notice that when the va] ue either of x or of y is obtained, the value of the other can be written down. For a and a' have the same relation to x that h and b' have to 2/; we may therefore change x into y provided that we at the same time change a into h, b into a, a into b', and h' into a. Thus from ch' — c'h , ca' — c'a X = —n 7-, we have y= 7—7 — p- . ah -ah ^ ha -ha SIMULTANEOUS EQUATIONS OF THE FIRST DEGREE. 147 It will be seen from the above that in order to solve two simultaneous equations of the first degree, we first deduce from the given equations a third equation which contains only one of the unknown quantities ; and the unknown quantity which is absent is said to have been eliminated. 143. From the last article it will be seen that the values of x and y which satisfy the equations ax-\-hy — c, and a'x + h'y = c', can be expressed in the form _ y _ -1 he' — h'o ~ ca' — da ah' — ah So also, from the equations GWJ + 6y + c = 0, and a'x + h'y + c = 0, we have _ y _ he —h'c ca'^c'a ah'— ah It is important that the student should be able to quote these formulae. Ex. 1. Solve the equations 3ar + 2y = 13, and 7aJ + 3t/=27. 2, .27- -3. . 13 ■" 13. 7-27. 3 3.3-7.2' that is i5~l0~5* and y-'h- 10—2 148 SIMULTANEOUS EQUATIONS OF THE FIRST DEGREE. Ex. 2. Solve the equations X y * ^ and = - 7. A X y J These may be considered as two simultaneous equations of the first degree with - and - as unknown quantities. X y We therefore have 1 1 X 11 -1 S(-7)-(-6)2 2.2-(-7)4"4(-5)-2.3* 1. 1 that IS — — i = -^ = — : '^ - 11 32 26 ' 1 11 26 •••i=-26'°'*=-n- ., 1 32 13 ^«« r26'^' 2^ = 16- Ex. 3. Solve the equations x-y=a-b, ax-6y = 2(a2-62). We have X y - -2(a^-h^) + b{a-h) a(a~6)-2(a2-Z>2) _6 + a' , . * _ y _ ■'■ *^**^ 62 + a6-2a2"~262-a6-a2~6^' 6a + a&-2a» , ^ .♦. «= r = & + 2a; — a , 2&2-a&-a» and « = — r = a+26. — a Instead of referring to the general formulae of Art. 143, as we have done in the above examples, the unknown quantities may be elimi- nated in turn, as in Art. 142 ; and this latter method is frequently the simpler of the two. Thus in this last example we have at once, by multiplying the first equation by a and then subtracting the second, (6-a)y = a(a-6)-2(a2-Z>2); -a2-a6 + 262 * b-a SIMULTANEOUS EQUATIONS OF THE FIRST DEGREE. 149 Then x = (a+2b)+a-bi ,\ x = 2a + b. 144 Discussion of solution of two simultaneous equations of the first degree. We have seen that the values of x and y which satisfy the equations aw+by=c (i), and a'x + b'y = c' (ii), are given by (aJ/ - a'h) x = cb' — ch (iii), (ba — h'a) y = ca' — ca (iv). Thus there is a single finite value of x, and a single finite value of y, provided that ah' — a'h 4= 0. If ah' — a'h = 0, x wiU be infinite [see Art. 118] unless ch' — c'h — O; and, if aV — a'h and ch' — o'h are both zero, any value of x will satisfy equation (iii). So also, y will be infinite if ah' — a'h = 0, unless ca'—c'a is also zero, in which case any value of y will satisfy equation (iv). If ab' — a'h = 0, then — = t/ ; and if ah' — a'6 = and a also ch' — c'h = 0, then — = -t-.= -,, a b c When equations cannot be satisfied by finite values of the unknown quantities, they are often said to be incon- sistent. Thus the equations ax + hy—c and a'x + h'y — d are inconsistent if — = t7 1 unless each fraction is equal to c -7 , in which case the equations are indeterminate. In fact c when -7 = T7 = — , it is clear that by multiplying the terms of equation (i) by — we shall obtain equation (ii), so that a the two given equations are equivalent to one only. 150 SIMULTANEOUS EQUATIONS OF THE FIRST DEGREE. We have hitherto supposed that a, a', b, V were none of them zero. It will not be necessary to discuss every possible case : consider, for example, the case in which a and a' are both zero. /» When a and a' are both zero, we have from (i) y — t y c' and from (ii) y—jj- These results are inconsistent with one another unless r = n • (f Hence, if a=a' = 0, and t=t,, the equations (i) and (ii) are satisfied by making y = j-, and by giving to x any finite value whatever. If however r =+ r/ , the equations hy = c and h'y = c' cannot both be satisfied, unless they are looked upon as the limiting forms of the equations (ix+hy=c and a'x+h'y=c\ in which a and a' are indefinitely small and ultimately zero. But from (iii) we see that when a and a' diminish without limit, x must increase without limit, cb' — c'b not being zero. Thus, in the equations (i) and (ii), when a and a' diminish without limit, and cb' 4= c% the value of x must be infinite. Equations with three unknown quantities. 14)5. To solve the three equations : ax + by + cz = d (i), a'x -^ b'y -{■ c'z = d' (ii), a"w-\-b"y + c"z^d'' (iii). Method of successive elimination. Multiply the first equation by c', and the second by c ; then we have ac'x + bc'y + cc'z = dc'. SIMULTANEOUS EQUATIONS OF THE FIRST DEGREE. l^X and a ex + b'cy + &cz = d'c ; therefore, by subtraction, (ac — a'c) X + (6c' — 6'c) y — dc* — d'c (iv). Again, by multiplying the first equation by c" and the third by c and subtracting, we have {ac" - a"c) X + {W - Vc) y = do" - d"c (v). We now have the two equations (iv) and (v) from which to determine the unknown quantities x and y. Using the general formulae of Art. 143, we have {he' - h'c) {do - d"c) + {dc - d'c) (be" - h"c) X — {ac' - ac) {be' - b"e) - {be - b'c) {ac" - a"e) ' Method of undetermined multipliers. Multiply the equations (i) and (ii) by X and fi, and add to (iii); then we have the equation X {\a + fjui' + a") + y{\b + fib' + b") + ^ (Xc 4- fic' + e") = {\d + tid' + d"\ which is true for all values of X and //,. Now let X and /x be so chosen that the co-efficients of y and z may both be zero, ^, Xd + fid'+d" then X = .——--7-; — r, , Xa -V fia + a where X and //. are found from Xb+fih' + b"=0, and Xc + fic' + c'' = ; /^ 1 '• b'c"-b"c' b"c-be" bo -b'c' Hence _ d {b'c" - b"c) + d' {h"c - be") + d" (be' - b'c ) ^ a {b'c" - b"e') + a [b"c - be") + a" {he - b'c) ' [The numerator and the denominator of the first value of X, which was obtained by eliminating z and y in succes- 152 SIMULTANEOUS EQUATIONS OF THE FIRST DEGREE. sion, can both be divided by c ; and the two values of x will then be seen to agree.] Having found the value of x by either of the above methods, the values of y and z can be written down. For the value of y will be obtained from that of x by interchanging a and h, a' and 6', and a" and 6". The value of y can also be obtained from that of a; by a cyclical change [see Art. 93] of the letters a,h,c\ a, h', c ; and a"y h'\ c"; and a second cyclical change will give the value of z. It should be remarked that the denominators of the values of x^ y and z are the same, and that there is a single finite value of each of the unknown quantities unless this denominator is zero. Ex. 1. Solve the equations : a; + 22/ + 3« = 6... (i), 2x + 4y+ z = 7 (ii), dx + 2y + 9z = 14: (iii). Multiply (ii) by 3, and subtract (i) ; then 5a; + 102/ = 16 (iv). Again multiply (i) by 3, and subtract (iii) ; then 4y = 4 (v). From (v) we have y = 1 ; then, knowing y, we have from (iv) x = l; and, knowing x and y, we have from (i) ^ = 1. Thus x=y = z = l. Ex 2. Solve the equations : x + y + z=l (i), ax + by + cz = d (ii), a^x + bhf + cH = d'^ (iii). Multiply (i) by c and subtract (ii) ; then {c - a)x+ {c -b)y =c - d (iv). Again multiply (i) by c' and subtract (iii) ; then {c^-a^)x+{c^-b^)y=c^-d'^ (v). Now multiply (iv) by c+6 and subtract (v); then (c - a) (6 - a)x={c -d){b-d); {b-d){c-d) ^' ''-(b-a)(c-a)' SIMULTANEOUS EQUATIONS OF THE FIRST DEGREE. 153 The values of y and z may now be written down : they are _ {c-d){a-d) _ {a-d){b-d) y~{c-b){a-b)' '~{a-c){b-c)' Instead of going through the process of elimination, we may at once quote the general formulsie. Thus _ {bc^ - b^c) + d{b^- c^) + d^{c- b) "~{bc^- b^c) + a (62 _ c^) + a^{c- b) _ {b-c){-bc + d(b + c)-dj^} ~ {h-c){-bc + a{b + c)-a^) = ;,~ \,~ I t as above. (6 -a){c-a)* Ex. 3. Solve the equations : x + y+z=a + b + c (i), ax + by + cz = bc + ca+ab (ii), hcx + cay + abz = Sabc (iii). We have (a + b + c) {ab^ - ac^) + {be + ca + ab) {ca - ab) + Sabc (c - &) *~ ab^- ac^ + a{ca - ab) + bc{c- b) a(b-c){ {b + c){a + b + c)-bc-ca-ab- Sbc} ~ {& -c){ab + ac-a^- be] a{b-cy ~ "" (a - 6) (a - c) * The values of y and z can now be written down : they are _ 6 (c - aY c{a-bY '^~ (6-c)(6-a)* *~ {c-a){c-by Ex. 4. Solve the equations : a;+ay + a2« + a' = (i), a; + 6y+ 622 + 63=0 (ii), a; + ct/ + c^«+ 0^ = (iii). The equations may be solved as in the preceding examples, or as follows. It is clear that a, 6, c are the three roots of the following cubic inX X8 + z\2+t/\ + a;=0. Hence from Art. 129, we have at once 2= _(a + 6 + c), y = 6c+ca+a6, and «= -dbc. 154 SIMULTANEOUS EQUATIONS OF THE FIRST DEGREE. 146. Equations with more than three unknown quantities. We shall return to the consideration of simultaneous equations of the first degree in the Chapter on Determinants, and shall then shew how the solution of any number of such equations can be at once written down. The method of successive elimination or the method of undetermined multipliers can however be extended to the case when there are more than three unknown quan- tities. For example, to solve the equations aco +bi/ -{-cz -\-dw =e (i), a'x +b'^ +c'2 +d'w =e' (ii), a"x -\-h"y -^-g'z +d"w =e" (iii), a"(c + h"'y + c'"z + d"'w = e"' (iv). Multiply (i) by X, (ii) by jjl, (iii) by v, and add the products to (iv). Then we have X (a\ + a> + a''v + a'") + y(h\ + h'fi + h"v + h'") -\- z(c\ + c> + g"v + d") + w{dX + d'fi + d"v + d'") = eX + e'/x + e"v + e" ( v). Now choose X, //,, i/ so as to make the coefficients of y, z and w in the last equation zero ; then ex+ea-^ev + e , .. X= , , ; , JT-, 777 (vi), where X, fi, v are to be found from the equations 6X + h'lJL + h"v + h'" = 0] cX + c> +c'V +c'" = ol (vii). d\ + d'ix^-d"v + d"' = Q\ Hence we have to solve (vii) by Art. 145 and then substitute the values of X, /a, and v in (vi) ; this will give the value of x\ and the values of the other unknown quantities can then be found by cyclical changes of the letters, a, 6, c, c?, &c. SIMULTANEOUS EQUATIONS OF THE FIRST DEGREE. 155 EXAMPLES XXL Solve the following equations. 1 ^ ^ -^ 2 X Zy 1 6 10 2* — + - = 10. X y y 2' 4 3 ^ 6 S -^ +5=-+- = 10. X y X y y 3 5. oa? + 6y = 2a6, 6. a; + a^/ + a^ = 0, hx-ay = h*- a?, x + by + b' = 0. 7. a; + 2/= 2a, 8. (5 + c) aj + (6-c) 2/ = 2a6, (a — 6) 03 = (a + 6) 2/. (c + a)x + (c-a)y = 2ac. 9. bx + (iy = 2ab, a^x + b*y = a^ + 6'. 10. (a + b)x + by = ax + (b + a)y = a' -b\ 11. x + y-^z=lf 12. x + y + z = lf 2x+3y + z = i, X y . 4x + 9y + z=16. 2 4"^ ■"' 6 3 « , 13. x + 2y + Sz = Sx + y + 2z = 2x + 3y + z = 6, 14. y + »=2a, 15. ^ + 2; - a; = 2cr/, « + a; = 26, « + oj - 2/ = 26, x + y=2o, x + y-z = 2c. 166 EXAMPLES. 16. 2/ + «-3a3=2a, 17. aa; + 6y + c«=l, x + y-Zz = 2c. cx + ay + bz = l, 18 y + g-^ ^ g + g-y _ x ^y-z _ b + c c + a a + b ~ 19. x + y + z^'Of 20. x + y + Z'^a + b + c, ax + by + cz= ly bx + cy + az = bc+ca+abf a'x + b'y + c^z = a + b + c cx + ay+bz = bc + ca+ab. 21 x + y + z = a + b + Cf bx + cy + az = a' + b' + c*, cas + ay + 6« = a' + 6' + c'. 22. a; + 2/ + » = 0, (6 + c) a; + (c + a) y + (a + 5) « = (5 - c) (c - a) (a - 6), bcx + cay + abz = 0. 23. ax + by + cz = ay 24. x — ay+ a'z — a' = 0, 6a; + cy + «« = 6, a? - 61/ + 6*« - 6' = 0, cx + a/y + bz = c, x— cy + c'z— c^ =0. 25. OKC + 63/ + c» = m, a'a; + b^y + 0*2; = m', a'a; + 5V + c^z = m\ 26. ax + cy + bz = a'+2bGy cx + by + az = b' + 2caj bx + ay + cz = c^ + 2ab. 27. x + y + z = 2a + 2b +2c, aoi + by + cz = 2bc + 2ca + 2ahf (b-c)x + {c — a)y + (a-b)z = 0» 28. ax + by +cz = a + b + c, a'x + b'y + c*z={a + b + c)', bcx + cay + abz = 0. SIMULTANEOUS EQUATIONS OF THE SECOND DEGEEE. 157 29. x + y + z^l + m + n, Ix + my + nz = mn + nl + hn^ {m-n)x+{n~l)y + (l-m)z = 0. 30. Ix + ny + mz = nx + my + lz = mx +ly + nz = l^ + m^ + n^- Slmn. 31. l^x + m^'y + n'z = Imx + rtmy + nlz = nlx + Imy + mnz ^l+m + n. 32. * 4- ^ +- "^ 1 a + a a + (3 a + y * ^ ^ y \ ^ 1 b+a h+p h+y ' " + \+ ' 1. C+a c+p c+y 33. 2/ + » + t(; = a, » + 10 + a; = 6, 10 + a; + y = c, oj + y + » = d 34. X + ay -[■ a'z + a^w + a* = 0, a + fty + 6*» + 6^2^ + 6* = 0, ' X + cy -^ c^z + — irrrr^ \ » -V 2{b + c-a) , . ., 1 /{a + b- c){b + c-a) and similarly y = ± ^ 2{c + a-i ) ' /{b + e~a){c + a-b) and '=^^'-~2^^rb^:c) — ' Ex. 3. Solve the eqaations : x^+2yz=a ^), y^ + 2zx^a (ii), «a-|.2x2/ = 6 ^ (iii). By addition {x+y + z)^ = 2a + & ; /. x + y + z- Jtzj2a + b (iv). From (i) and (ii) by subtraction (.T-y)(a: + y-22) = 0. Hence x=y (v), or else « + t/ - 2«=0 (vi). I. Ii x=yy we have from (ii) and (iii) by subtraction «^ + x'-2xz=:6-a; .*. z-x=^^b-a,... (vii). Hence, from (iv), (v) and (vii), M=^{±j2a + b±2jb-a}. SIMULTANEOUS EQUATIONS OF THE SECOND DEGREE. 16Y U. When x + y-2z-~0, we have from (iv) and «+t/=±g»y2a + 6. Also, from (ii), y^ + x{x + y)=a, which with the previous equation gives and Ex. 4. Solve the equations : We have bh + chf = xyz (i), c'x 4- a'z = xy« (ii), and ahf + h^x=xyz (iiij Multiply (i) by - a', (ii) by b^, and (iii) by c', and add ; then 2b^c^x =(-«*■' + 6^ + c^) xyz. Hence x-d, 2&2c3 or else y, = ___^^-__^ . If jf =0, y and « must also be zero. Hence x=y=«=0; 2J»c2 or else y^: 6a + c3-a2' and similarly ^^'" c» + a«-&-^ ' 2o2&2 and ^=a» + 62-c3* The solution then proceeds as in Ex. 2, Ex. 6. Solve the equations : x^-yz = a. xy=c. 168 SIMULTANEOUS EQUATIONS OF THE SECOND DEGREE. We have (a? - yz)* - (y' - zx) (z* -xy) = a?- 6c, that is x{a^ + y^ + i^-3xyz) = a^-bc. Hence, from the last equation and the two similar onen^ X _ y _ z a^-bc~'b'^- ca c^-ab' Hence each fraction is equal to V (a^ - &c)'» - (6^ - ca) {c^ -ab) V («' + **+ c* " 3«&<^) ' Ex. 6. Solve the equations : x + y + z=a + h + e...,^ «.^ (i), a;2 + 2/2 + ^2^aa + 62 + c2 (ii), - + | + - = 3 (iii). a b c It is obvious that x = a, y = 6, z=c will satisfy the equations: put then a:=a + \, y=6 + /it, « = c + i', and we have after reduction \ + ^ + i» = (iv), ^f-^» <^)- 2(a\ + 6AH-cv)+X2 + Ai'' + i'*=0 (vi). From (iv) and (v) X _ A* __ p a{b^) " b{c~a) ~ cjd-bj ' whence from (vi) X 2{b-c){c-a){a-b) **' a(6-c)~a2(6-c)2 + 62(c-a)a + c2(a-6)a' Hence «=a, y=b, z=ci or else g-g _ y-b _ «-c 2 (6 - c) (c - a) (a - 6) a{b^'c) ~ b{c^) ~ c{a-b) " a^ (6 - cf + h''\c-a)^ + c'{a- bf ' Ex. 7. Solve the equations : a; + y+«= 6, yz + zx + xy = ll^ xyz= 6. This is an example of a system of three symmetrical equations. Such equations can generally be easily solved by making use of the relations of Art. 129. Thus in the present instance it is clear that X, y, z are the three roots of the cubic equation X»-6X2 + llX-6=0 SIMULTANEOUS EQUATIONS OF THE SECOND DEGREE. 169 The roots of the cubic are 1, 2, 3. Hence x=l, y = 2, z—d; or j; = l, y = 3, z=2; or «=2, y = 3, « = 1; &o. Ex. 8. Solve the equations : x + y +z=:a i.i (i), UUUi (ii), X y z a ^ " yz+zx + xy= -c^ (iii). This again is a system of symmetrical equations, and two of the relations of Art. 129 are already given ; we have therefore only to find the third. We have from (ii), yz+zx + xy _ 1^ xyz ~ a * .'. xyz=: -ac* (iv). Then, from (i), (iii) and (iv), we see that x, y, z are the roots of the cubic X^ - a\^ - c»X + ac^ = 0, that is X*(X-a)-c2(\-a)=0; .-. X=a, or X= ±c. Thus x=a, y=c, z=-~c; &g. Ex. 9. Solve the equations : n:^{y-z)=za^(b-e)f y^z-x) = b^c-a), zHx-y)=::c^{a-b), By addition x^{y-z)+y*{z-x) + z^{x-y)=a\b-c) + b^e-a) + c^(a~b), that is (y -z){z- x) {x-y) = {b- c) (c -a) (a- b). By multiplication arV V {y-z){z-x){x-y) = a'Wc^ (6 -c){c- a) (a - 6) ; .-. xh/H'^-a^V^c^. Hence xyz = abc (i), or xyzss ~abe (ii). Again a' (6 - c) y + b^ (c - a)x=xPy {y -z) + xy^ (« - «) =xyz{y-x) (iii). Hence, if xyz = abc, we have from (iii) {b^{c-a) + abe}x + {a*{b-c)-abc}y=Of that is bx {be + ca~ ab) - ay {be + ca - a&) = ; /. - = r> aJ^d therefore each = - . a b e ITO EXAMPLES. Thus, when xyz=abc, we have - =s ^ = - . a b c Hence each is equal to a /^ = V^* Thus - = ! = -=!. or ^=f = i = l. aw' ftw* cw* If xyz = - a6c, we have from (iii) - (6c -ca^ab) = ~^ (ca -ab-^ he). Hence also each =- (aJ - 6c - ca) =s/{ - (6c - CO - o6) {ca ~ab- be) (ab-bc- ca)}, EXAMPLES XIV. Solve the following equations : 1. 3. yz^a', 2. x{x + y^z) = a\ zx = h\ y{x + y-¥z) = b% xy = c'. z{x + y + z) = c^. — yz + zx + xy - = a. 4. yz = a(y + z), yz-zx^- xy = = 6, zx = b(z + x), yz + zx — xy = = c. xy = c(x + y). yz=^by + czy 6. x^ + 2yz=l2, zx = cz + ax, y'+2zx = l2, xy = (ix + by. z* + 2xy=\2. {y + z){x + y + z)^ a, 8. {y + 6) (» + c) = a% (z + x) {x + y + z) = by (z + c) (a? + a) = b', (oj + y) (flc + jr + z) = c. {x + a) (y + b) = C*. SIMULTANEOUS EQUATIONS OF THE SECOND DEGREE. 171 9. x" - (y - zy = a'y 10. x{y + z-x) = af y'-(z- xf = 6", y(z + x-y)=^b, z' - {x - yY = c^. z(x + y-z) = c. 21. 11. y-i-z z + x x + y a b c = 2xyz. 12. 3^ + z z+x x+y b ~ c x' ~a' + / + «• + b' + c*' a 13. yz = --a + y + z. 14. yz = a ',{y + z) + «, zx = = b + z + x, zx=a ^{z + x) + A xy-- = c + x + y. xy=a '{x + y) + y. 15. yz- zx- -r = cy + bz, -g^^az + cx^ 16. x + y' ~3' xy- -h* = bx + ay. y + z' z + x' ■^=4. 17. x + y + z= 6, 18. x + y + z = 15, af-^ .^ + z^=U, a;' + y' + «' = 495, xyz= 6. xyz = 105. 19. x + y + z^ 9, 20. x+y + z=^ 10, «* + 2/'-f«"= 41, yz + zx + ocy= ZZy x* + if + z*^ 189. (y+«)(«+aj)(a;+y) = 294. yz _ zx xy a;' + y* + g * bz + cy~ ex + az ay + bx a' + b* + o'' ? + |+? = l, 23. ax^^*'-, X b c ' z y X b z ^ I z X -+- + -=1, oy = - + -, aye ^ X z X V e ^ X y -+|+- = 1. cz = -+^, a z y X l'?^2 EXAMPLES. x' + y'-z{x + y) = c\ 25. x' + yz-a'' = y' + zx-b* ^z% a^ - c' = ? (a:» + y» + z\ 26. a (a; + y + «) - (y« 4- »* + y«) = a, y {x ■¥ y -k- %) - (z^ ¥ a? -^r zx) ^ 6, « (a; + 2^ + 2) - (a;' + y' + xy) = c. 27. a; + 2/ + « = « + & + c, a:« + y« + ;s^ = a» + 6« + c', (6 - c) a; + (c - a) y + (a - ^») 2; = 0. 28. (a; + y)(a; + «) = aa;, 29. x^-yz = aai, {y-^z){y + x) = hy, f-zx^hy, {z + x) {z ■\- y) = cz. z'-xy^cz. .30. a3' + a(2a; + y+«)=y» + 6(22/+« + a;) = «« + c(22 + a: + y) s= (a; + y + »)'. 31. 2/' + 2/0 + «* = a*, »* + ;sa; + a^ = 6% a^ + xy + y' = c*. 32. a»a; + 6"«/ + c*2!=0, (6-cy^(c-«)» , (et-5)' _Q aa; 6y C2; * 1 1 1111 ^2 ;sa; a^ a 6 « CHAPTER X* Graphical Representation of Functions. I. Co-ordinates. Let two straight lines XOX\ YOY' be drawn in a plane at right angles to one another, and through any point P in the plane let the straight lines PM, PN be drawn parallel to OX, OY respectively to meet OY, OX respectively in M and N, Then it is easily seen that the position of the point P in the plane XOY can be found when the lines ON, OM are given. The lines ON and OM, or ON and NP, which thus define the position of any point P with reference to the fixed lines OX, Y are called the co-ordinates of the point P with reference to the axes OX, Y. The point of inter- section of the axes is called the origin. The co-ordinate which is measured along the axis OX IS denoted by x, and that measured along the axis OF is denoted by y. The point for which x — a and y = 6 is called the point (a, 6). . When the co-ordinates of difterent points are given, the positions of the points with reference to the axes can be marked in a diagram. This is called plotting the points. In order to determine the position of a point we must know not merely the lengths of the lines ON, OM but also the directions in which they are drawn. Now, if a length measured in one direction be taken as positive, the same length measured in the opposite direction must be taken !:;:;:::!:::::s;:;::::;:;;:::s!::::h:::::u::;::::::k;::::::;::::::: as negative. Thus, for example, if OJS'' is positive, NO is negative, and conversely. OX and 7 are always taken as the positive directions, and OX', OY' are therefore drawn in negative directions. By using paper ruled with two sets of equidistant parallel lines, as in the figure, the position of points whose co-ordinates are given can be readily plotted. Thus the points P, Q, R, S in the figure are the points (12, 8), (- 9, 10), (- 9, - 5), (15, - 6) respectively. II. The Graph of a Function. If any function of o! (that is, any algebraical expression which contains x) be .taken, and different values of x be set off along the line vjrxfcxi.A . jkjxjii X n. X iv^j OX, and the corresponding values of the function be set off in a perpendicular 'direction, the curve through the series of points so obtained is called the Graph of the Function. For example, to find the graph of 2w + 6. Put y = 2.x + 6. Then, if a; = 0, y = 6; if x = l,y = S; if a7 = 2, y = 10; and so on. Now, if the points (0, 6), (1, 8), (2, 10) be plotted on squared paper, it will be found that these three points lie on a straight line; and it will also be found that the points corresponding to any other values of x, for example the points (— 1, 4), and (- 2, 2), will lie on the line through the first two points. Thus the graph of y = 2^7 + 6 is a straight line. It will similarly be found that the graphs of y = 6a; - 11, y = ^ — lcc, 2y — — ^x — 7 are straight lines. [The above are special cases of the following general theorem : — The graph of y = ax-\-h, where a and h are any given numbers, is a straight line. 17 2d GRAPHICAL REPRESENTATION OF FUNCTIONS. -H To prove this, it is only necessary to point out that it follows from the properties of sirhilar triangles that the graph of y = ax is a, straight line through the origin ; and that, if we increase the y co-ordinates of points on y — ax by the same quantity 6, we know from Geometry that we shall obtain points on a straight line parallel to that de- termined by 2/ = ax.'\ An equation of the form y = mx 4- n, or ax-\-by-\-c = 0, where m, n, a, b, c are supposed to be known constants, is called a Linear Equation. In order to draw the graph of a linear equation it is only necessary to find two pairs of corresponding values of X and y, for a straight line is determined when any two points on it are known; and pairs which are arithmetically easy to obtain and are not near together should be chosen. Ex. 1. Draw the graphs of y = Sx + 7 and 3y = 4x + 6, and find their point of intersection. Ex. 2. Solve by graphical constructions the simultaneous equations y = 2x-16, Sy = 7~4:x. Ex. 3. Draw the graphs of Bx + 2y = 12 and 6x + Ay = S&, and shew that they do not intersect at a finite distance from the origin. Ex. 4. Plot the two lines given by 5x + 2y = 10, 10a5 + 4t/ = 15, and shew that the two lines are parallel. Ex. 6. Plot the two lines given by 5x + y = ll, 2x-Sy = lS, and find their point of intersection. Ex. 6. Draw the graph of 3a; -4^ = 6, and hence find solutions of the equation in positive integers. Ex. 7. Find positive integral values of x and y which satisfy the equation 2x-{-y = ll. III. To find the graph of By = a;*. [5y is used because x^ increases very rapidly as x is increased.] GRAPHICAL REPRESENTATION OF FUNCTIONS. 172 6 The following are correspondin| y values of c c and y- X 1 2 3 4 5 -1 -2 ... y •2 ^8 1-8 3-2 5 •2 •8 ... Mark these points on squared paper, and draw a curve through them. It will easily be seen that y is the same for any two values of oo which differ only in sign, so that the graph is symmetrical about the axis of y. Ex. 1. Find the graphs of 5i/= -x^, and 5t/= - x' + 5. Ex. 2. Find the graphs of y = Jdx and y = VSa; + 1. IV. To find the graph of lOy = 2x^ - Sx + 5. Corresponding values of x and y are : X 1 2 3 4 5 -1 ... y •5 -1 -•3 -•1 •5 1-5 1-5 ... Since y is positive when x = and is negative when a? = 1, the graph cuts the axis of x between and 1*. See Art. 454 and Art. 455. 3. A. 12 172/ GRAPHICAL REPRESENTATION OF FUNCTIONS. To find more exactly where the graph cuts the axis of X (which will give the value of the roots of the equation 2a;^ — 8d? + 5 = 0), we find in succession the following series of corresponding values : X •5 •6 •7 •8 y •15 •09 •038 -012 N.B. Decimals should always be used. 1 Isiii :::Ei 111: 1 III • ■ 1 iil in Ill 1 M? :::jsii|[||S:||isjisjiassg L;:is!;::aa:a;:Sa::aH;siai s 1 He" 1 111' I:: !■■■ 11 !» 1 JlllllhisilKl II |[n|gi:g| mm ill [:r:ss::::| ■■■■■I 11 ■■■■■■ 1 III 1 1 1 1 1111:111:111 ■■sssRs:::! mill ill Hence oo is between '7 and '8 and very nearly divides the interval in the ratio 38 : 12 [for the curve from x=-7 to a? = '8 is nearly a straight line]. Thus the root is '776 nearly, and the other root can be found in a similar manner to be 3-224 ; these agree with the values deter- mined directly from the equation, namely 2 + JV6 = 2 + l-2247.... y. The graph of a quadratic function will shew roughly its least or greatest value. Thus, in the above figure, the least value of 2af-8x-^5 appears to be — 10 x "3, and it has this least value when x = 2. GRAPHICAL REPRESENTATION OF FUNCTIONS. 172 ^r It should be noticed that ^x^ — 8^ + 5 has its least value for that value of x which is midway between the roots of 2x'^ — ^x + b = 0, or midway between any two values for which 2x^ — 8a; + 5 has the same value. This is true for all quadratic functions. [These results can be obtained at once by Algebra, as we have already shewn. For 2a;2 _ 8a3 + 5 = 2 (a? — 2)^ — 3 ; and, since 2 (a; — 2)^ is positive for all real values of x, it is obvious that — 3 is the least value of 2ar* — 8a? + 5 and that a; is 2 when the expression is least.] We can also find the value of y for which the roots of 2a;2 — 8a; + 5 = 2/ ^^^ equal. [The condition for equal roots of 2a;2-8a;+5-2/ = is 8^ = 4 x 2 x (5 - 3/), whence 3/ = 3.] If these results are obtained from the graph they should always be checked by Algebra ; for, unless the values happen to be integral, the graph will generally give only a rough approximation. VI. Approximations to the roots of a quadratic equa- tion may also be obtained in the following manner. Consider, for example, the equation 2a?^ — 8a7 + = 0, or a?' — 4a; + 2'5 = 0, whose roots have just been found. \. Y > \ ^\ ^.^ ^ ^ q -^ ^ Draw the graphs of lOy = o^, and IO3/ - 4a; + 2-5 = 0, as in the figure. Then where these two graphs intersect we shall have a^ = lOy = 4a? - 2-5, so that a^* - 4a; + 2*5 = 0. 12—2 172 A GRAPHICAL REPRESENTATION OF FUNCTIONS. Hence the roots of the equation are the two values of X at the points of intersection of the graphs, and these are seen to be approximately "7 and 3*3. N.B. If the graph of 10y = a^ has been drawn on squared paper .very accurately and on a large scale, the second method would enable any one to find the approxi- mate roots of a quadratic equation without great waste of time. For any quadratic equation can be reduced to the form x'^-\-ax-\-h = 0, and the roots required are the values of X where the graph of lOy = ar^ is cut by the straight line 10y + ax-\-h = 0\ and having found two points on the straight line, a ruler could be placed along it and the values of x read off without actually drawing the graph of the straight line, so that the graph of l(dy = a^ could be used over and over again. Except for this advantage, this second method is inferior to the former, for in the first method the roots can be found to any required degree of accuracy, by successive approximations, as shewn in IV. Graphical methods are, however, after all only methods of getting rough approximations by those who know no A Igehra, EXAMPLES XIV a. 1. Calculate the values of y for a5 = 0, 0'5, 1, 1*5, 2, 2*5, 3, (i) when 3/ = 2*5 + Q'lx ; and (ii) when y = 5 - OSa;. Draw the graph of y in each case using the same lines of reference for the two graphs. Find by calculation, the values of X and y which will satisfy the two equations simultaneously, and check the result by considering the point of intersection of the two graphs, 2. Calculate the values of {x—l){x—3) for the following values of a; : - 1, 0, 1, 2, 3, 4, 5. Plot these values on squared GRAPHICAL REPRESENTATION OF FUNCTIONS. 172 i paper and draw through them a curve to shew how the expression varies for diflferent values of x. Find from your curve the values of x when the expression is equal to 5, and verify the result by calculating the roots of the equation (a; — 1) (aj— 3) = 6 to two places of decimals. 3. Draw the graph of 103^ = (a;- 4) (a;— 6) between the values x = —2 and aj = 9, and use the graph to determine when (x - 4) (a; — 6) is equal to 5. Verify by calculation. 4. Draw the graph of 10y = a?-Sx between the values a; = — 4 and x = 5, and use the graph to determine when x^ - Sx is equal to 5. Shew that x^-Sx has its least value when x is midway between the values just found, and find the least value. 6. Draw the graph of 102/= 10 + 4a;— a;^ between the values x = -S and a; = 7, and use the graph to find the roots of the equation 10 + 4a; - ar^ = 0, and also to find the greatest value of 10 + 4a; - ar^. Verify the results by calculation. 6. Draw the graph of i/=:lS-Qx-x^, and shew that y is greatest for the value of x midway between the two values for which y = 2. 7. Calculate the values of ar'-3*4a; + 4 for the values 0, 0-5, 1, 1*5, 2, 2-5, 3 of x. Plot these values on squared paper, and draw a smooth curve to shew how the expression varies for diflferent values of x. From your diagram, determine (i) the least value of the expression, (ii) the value of x for which the expression is least. 8. Draw the graph of 3/ = a;* — 6a; + 1, and hence shew that the smaller root of the equation ar^ - 6a; + 1 = is approxi- mately 0-17. 9. Draw the graph of y=a^-4:X + 2, and find the smallest value of y for real values of x. Find also the values of x for which 2/ = 7 and for which y = 23. 10. Draw the graph of y = 15 + 3x-x\ For what values of X will X and y be equal ? 172 k GRAPHICAL REPRESENTATION OF FUNCTIONS. 11. Draw on the same scale and with the same lines of reference the graphs of the following functions from a; = to x = 5 : (i) y = x-2, (ii) y = {x-2){x-3). Find two values of x, each of which gives the same value of y in both graphs. 12. Plot the graphs of 10y = a^a.ndl0y-5x + 2 = 0, and find their points of intersection, and so find approximately the roots of s^ — 5x + 2 = 0. Compare the result with the alge- braical solution. 13. Plot the graphs ol y = x^ and of y = 5a3 - 5, and hence write down the approximate roots of the equation a?—bx+b=Q. Verify the results by Algebra. 14. Plot (i) the graph of y = x^ — Zx-{-\ -5, and (ii) the graphs of \Oy = x^ and lOy - 3a; + 1 -5 = 0. Find the roots of ar^-3aj-i-l'5 = 0by means of the graphs and compare the results. 15. Find where the graph of y = a;^ - 3a; + 4 is cut by the graph of y = mx^ (i) when m = 2, (ii) when m = 3, and (iii) when m = — 7. Find also the value of m in order that the two points may coincide. 16. Find where the graph of y = a;^ — 4a; + 9 is cut by the graph of y = Qx. Find also the values of m if the graph of y = x^ — 4cX + 9 is touched by the graph oi y = mx. 17. Trace the graph in x and y given by the equations x = f—\,y = ^ + t—\ from the values < = — 3 to < = 3. Find from the graph the value of x when y = — ^, and verify by Algebra. 18. Trace the graph in x and y given by the equations x = f-\.t-¥l, y = t^ + 2t + 2 from the values t = -Z to < = 3. Find by means of the graph the least values of x and y, and verify by Algebra. GRAPHICAL REPRESENTATION OF FUNCTIONS. 1721 Sa? VII. To find the graph of 2/ = ^^ _ ^^^ ^^ _^ ^^ Corresponding values of x and y are : X •25 •5 •75 1 1-6 2 5 00 y •8 -2 -5-14 00 3-6 2 •625 When X is ^er^/ little less than 1, 3/ is negative and very great, when a; = l, t/ is infinite, and when x is very little greater than 1, 3/ is positive and very great. Thus as a? passes through the value 1, y becomes infinite and changes sign. So also when x passes through the value —1,2/ becomes infinite and changes sign. [It is important to notice that, in general, a function changes sign whenever it passes through the value zero or infinity; this is not, however, the case when the factor which by itself vanishing makes the function vanish if it is in the numerator, or become infinite if it is in the de- nominator, is of an even power. For example, the function vanishes but does not change sign as x passes X through the value 0; also the function 7 -:- becomes infinite but does not change sign as x passes through the value 1.] By changing x into —x, it will be seen that y is changed into — y. Hence the required graph is as in the figure. 172 m GfiAPHICAL REPRESENTATION OF FUNCTIONS. I GRAPHICAL REPRESENTATION OF FUNCTIONS. 17 2 n Ex. 1. Draw the graph of y ■■ {x-2){x-S) {x-l){x-^) Corresponding values of x and y are x\ \ -5 \ 1 1-5 2 2-5 1 3 3-5 4 1 5 I 6 00 y \1'5\ 2-14 1 00 -•5| -1| -2 1 -•6 -00 |0 •11 1 1 -•6 00 |l-6 1-2 1 1-3 1 1-2 1 1-11 1 1 The graph outs the axis of x when x = 2 and when x = 3, it changes from + oo to - oo as a; passes through the value 1, and it changes from - oo to + oo as a; passes through the value 4. Thus the graph is as shewn below. It will be seen that y does not in this case, as in the preceding, pass through all values as x increases from - oo to + oo . The actual limits to the possible values of y can be found roughly from the graph, or in the following manner. We have {x-2){x-S) ^ y {x-\){x-^y This quadratic will give the two values of x which correspond to any assigned value of y. [For example, the two values of x when t/ = 2 are the roots of -a;2 + 5a;-2 = 0, which will be found to be '44 and 4-66 nearly.] 172 GRAPHICAL REPRESENTATION OF FUNCTIONS. Now the values of x corresponding to any possible assigned value of y must be real, the condition for which is that 25 (1 - y )2 - 4 (1 - y ) (6 - 4t/ ) > 0, [Art. 121 ue, {9y-l){y-l)>0. From which it follows that no real values of x correspond to values of y between ^ and 1, but that real values of x correspond to any value of y which is less than ^ or greater than 1. When y = ^ it will be found that both values of x are 2-5 ; and when y = l both values of x are infinite. EXAMPLES XIV 6. 1. Plot the graph of y = x + - for the following values of x: -1, -2, -5, 1, 1-5, 2, 4, 10, 20. Find from the graph the least value of y, and verify by calculation. 1 9 2. Plot the graph of y = jx + - {or the following values oix: 1, 1-5, 2, 3,5, 8, 10, 15, 20. Find from the graph the least value of y, and verify by calculation. 1 2 3. Plot the graph of y = -^x + -ior values of x between 0-3 and 10. Also find, from the graph, the value of m for which the 1 2 quadratic ■^x+ - = m has equal roots. X 3 4. Plot the graph oi y = -= + - from x='6 to a; = 5. o X Find from the graph the value of m for which the line y^m cuts the graph in coincident points. GRAPHICAL REPRESENTATION OF FUNCTIONS. Il2p a^ — x+1 5. Draw the graph of 2/= o^ -, > and shew from the graph that y is not greater than 3 nor less than J. Verify by calculation. 6. Draw the graph of y = - ^g, and shew that the (03-1) least value of 2^ is -J. 7. Draw the graphs of i)cy = 10 and x + y = 8, and hence write down the solutions of the simultaneous equations. Verify by Algebra. x — 1 8. Draw the graphs oi y = =- and 2y=l — x, and write X "r L down the solutions of the simultaneous equations. VIII. Approximate solutions of equations with numerical coefficients. Approximate solutions of numerical equations can be found in the following manner. Consider, for example, the equation a;' - 3a; + 1 = 0. Put y=x^-3x + l', then corresponding values of x and y are a; |0| 1 I 2 I 3 I -1| -2| -3 1 2/|l| -1|3|19| 3 I -1| -17 1 Now, since y does not become infinite for any finite value of x, the graph must cut the axis of x between - 2 and - 1, between and 1, and again between 1 and 2. Thus the graph of y = a;^ - 3a;+ 1 is as in the figure below (p. 172 q). To find closer approximations to the points where y = 0, that is to the required roots, we have ^1 •3 1 •4 1 1-5 1 1-6 1 -1^9 1 -1-8 y\ •127 1 - -136 1 - -125 1 •296 1 -•159 1 •57 If the points are all plotted and a curve be drawn through them freehand we shall obtain approximations to the roots near enough for most practical purposes. If the points (-3, -127) and (•4, - '1S&) are joined by a straight Une, this line will cut the axis between -3 and -4: in the ratio -127 to 127 •136, and for this point y = -3 + -lx .,^- , .,--. = •348. The actual Jl^ I + loo 172 5' GRAPHICAL REPRESENTATION OF FUNCTIONS. value of this root found by Homer's Method [see Chapter xxxn. p. 592] is 'Si?...; but, from the slope of the graph in this neighbour- hood, it will be seen that the actual root is slightly less than that given by the straight line. We find in a similar manner that 1'529 and - 1-878 are close approximations to the other two roots ; and here again we see from the graph that 1-629 is leas than the actual value, which will be found by Horner's Method to be 1-632...; also -1-878 is greater than the true value, namely -1-879..,. Turning values. It appears from the graph that the value of x'-Sx + l increases from x= -d to x= -1, and that it then begins to diminish; also that it goes on diminishing until a;= -1, after which x^-dx + 1 increases continuously. These points are called the turning points of the function a^-3a; + l. It will also be seen that at the turning points, namely when x=-l or x=+l, the tangent line to the graph is parallel to the axis XOX\ The position of the turning points can always be determined, roughly at any rate, from the graph. GRAPHICAL REPRESENTATION OF FUNCTIONS. 172 r To determine the turning values by Algebra, we have to find the values of k for which the line y = k outs y=x^-Sx + lin coincident points. Hence we have to find the values of k for which two of the roots of x^-Sx + l = k are equal, and this method is applicable whatever may be the degree of the function. To find the condition that an equation may have equal roots see Art. 453. The roots of the above cubic could be found, as in VI., by drawing the graphs of y=x^ and y -3x + l = 0, when the values of x at the intersections of the two graphs would give the required roots. Unless, however, the graph of y = x^ is drawn very accurately and on a very large scale the results will not be of much value. EXAMPLES XIV c. 1. Find approximate values of the real roots of the following equations : (i) £c'»-4aj + 2 = 0. (ii) ar^-ll£c-5 = a (iii) o(^-Sx-l=0. (iv) a?-x+3 = 0. (v) a^-3x'' + 2x-l = 0. (vi) a^-3x' + 4:x-l=0. (vii) x'^-6o(^ + 7x^+6x-7 = 0. (viii) ar^ - 5a; + 1 = 0. 2. Draw the graph of y = a^—12x + 2 from a; = -4toa; = + 4. Find the values of x for which the value of y reaches a turning point, and obtain the corresponding values of y (i) by measurement, (ii) by calculation. 3. Draw the graph of y = a^ — 5x^ +3x + 2. Find the approximate values of the roots of the equation a^-5x' + 3x+2 = 0, and find the values of x for which oc^ - 5x^ +3x + 2 reaches a turning value. 172 5 GRAPHICAL REPRESENTATION OF FUNCTIONS. 4. Draw the graph of y = a^-3x, and find from the graph the values of m for which two of the roots of the equation a:^ — 3x + m = are equal. 5. Draw the graph oi y = 2oc^ — 3x^ + 4, and find from the graph the values of m for which two of the roots of the equation 2a^ - 3aj2 + m = are equal. 6. Draw the graph of 20?/= a? (a;'- 36), from x=-S to a;=8. Find from the curve for what approximate values of x the value of y reaches a turning point, and obtain the correspond- ing values of y (i) by measurement, (ii) by calculation. IX. In Scientific investigations when a series of cor- responding values of two different quantities have been obtained by experiment or observation, a series of points are plotted so that the co-ordinates of each point represent corresponding values of the two different quantities. It is then sometimes possible to determine the law connecting the magnitudes of the two different quantities. For example, if the points lie very nearly on a straight line, it may be taken as probable that the points would all lie exactly on a straight line if the observations or experi- ments were quite free from errors, which can however never be actually realised; moreover, by drawing the straight line which lies above some of the points and below others but w^hich on the whole appears to pass most evenly amongst the points, we should reduce the effect due to errors of observation to a minimum. Curves of this kind which shew at a glance the con- nection between two variable quantities are very often employed even when there is no expectation of finding any law connecting the two quantities; for example, curves are given in the newspapers to shew the time variation in the height of the barometer or of the ther- mometer. GRAPHICAL REPRESENTATION OF FUNCTIONS. 1*1 2 t X. If the two quantities whose corresponding vahies are to be plotted are both of the same kind, the scales on which they are represented should generally be the same. If, however, the quantities are of different kinds there can be no connection between the scales on which they are represented — for example, one centimetre along the axis of X may represent a pressure of one pound, and one millimetre along the axis of y may represent one cubic foot, or any other number of cubic feet which may be most convenient for the diagram. It should be noticed that when a number of corre- sponding values of two different quantities have been found, and these results are plotted and a curve drawn freehand through them, the best approximation to corresponding values intermediate to those given can be found from the curve so drawn. It should also be noticed that the slope of the graph at any point gives us a means of comparing the rates of increase of the two quantities at that point. EXAMPLES XlYd, 1. Find the values of x and y which satisfy the equations y = x + 2, y =^bx + 3. Draw the graphs of the equations on squared paper and indicate the values of x and y which satisfy both equations, (i) when 6 = 0-7; (ii) when 6 = 0-8; (iii) when 6 = 0-9. Is it possible to find values of x and y which satisfy both equations when 6 = 1? 2. For a certain book it costs the publisher £100 to prepare the type and 2s. to print and bind each copy. Find an expression for the total cost in pounds of x copies. Also make a diagram on the scale of 1 inch to 1000 copies and 1 inch to £100, to shew the total cost of any number of copies up to 5000. Read off the cost of 2500 copies, and the number of copies costing £52 o. 17 2 U GRAPHICAL REPRESENTATION OF FUNCTIONS. 3. The mean temperature on the first day of each month shewed on an average of 50 years the following values : January 37° May 50° September 69° February 38 June 57 October 54 March 40 July 73 November 46 April 45 August 62 December 41 Represent these variations by means of a curve, neglecting the difference of length of different months. 4. A manufacturer has priced certain lathes ; the largest sells at £175, and the smallest at £40. He wishes to increase his prices so that the largest will sell at £200 and the smallest at £50. Assuming that the new price (P) and the old price (Q) are connected by the relation Q = a + bP, find the values of a and b ; and, to the nearest pound, the new prices of lathes originally valued at £130, £125. 10s., and at £76. 5. From the following numbers, obtained on determining the solubility of saltpetre in water, plot a curve on squared paper : Temperature Grams of saltpetre dissolved ] per 100 grams of water j State the inferences you draw from the curve, and also find from it the approximate weight of saltpetre which 100 grams of water would dissolve at 35°. 6. The following are the areas of cross sections of a body at right angles to its straight axis : 0° 10° 20° 30° 40° 50' 13 21 31 45 64 86 A in square inches 250 292 310 273 215 180 135 120 X inches from one end 22 41 70 84 102 130 145 Plot A and x on squared paper. What is the probable cross section at a; = 50 and at aj = 90 ? GRAPHICAL REPEESENTATION OF FUNCTIONS. 172 V 7. The following pairs of corresponding values of two quantities x and y were measured : . 1 1-8 2-8 3-9 6-1 6-0 y •223 •327 •625 •730 •910 1^095 It is known that the quantities are connected by a law of the form y = ax + b, but the given values are subject to errors of observation. Plot the values of x and y on squared paper and find the most likely values of a and h. 8. It is thought that the following quantities (in which there are probably errors of observation) are connected by a law like y = ae^. Test if this is so, and find the most probable values of a and b. X 2-30 3 •ID ^•00 4-92 6-91 7-20 y 33-0 39-1 50^3 67-2 85-6 125-0 9. The following table gives corresponding values of two quantities x and y : X 10-16 12-26 14-70 20-80 24-54 28-83 y 37-36 81-34 26-43 19 08 16-33 14-04 Try whether x and y are connected by a law of the form yx^ = c ; and, if so, determine as nearly as you can the values of n and c. What is the value of x when 2/ = 17-53 1 10. Draw the graph of y = -^ — -. . Shew that y cannot Xi — Tt lie between "25 and 1. (x — ^)(x — 6) 11. Draw the graph of y = ) ' ) j^ , and shew that (JB - 2) (03 - 4) ' the line y=c will cut the graph in two points for all values of c. S. A. 13 172 W GRAPHICAL REPRESENTATION OF FUNCTIONS. 12. Draw the graph oi y = -, =4-^^ Jr , and find for what values of c the line y = c will cut the graph in coincident points. (£c-4) (a; -7) 13. Draw the graph oi y= -. — ^^ — =--r^ , and shew that between a; = 4 and aj = 7 the greatest value of 3/ is |-. CHAPTER XL Problems. 152. We shall in the present chapter consider a class of questions called problems. In a problem the magni- tudes of certain quantities, some of which are known and others unknown, are connected by given relations; and the values of the unknown quantities have to be found by means of these relations. In order to solve a problem, the relations between the magnitudes of the known and unknown quantities must be expressed by means of algebraical symbols: we thus obtain equations the solution of which gives the required values of the unknown quantities. It often happens that by solving the equations which are the algebraical statements of the relations between the magnitudes of the known and unknown quantities, we obtain results which do not all satisfy the conditions of the problem. The reason of this is that in a problem there may be restrictions, expressed or implied, on the numbers concerned, which restrictions cannot be retained in the equations. For example, in a problem which refers to a number of men, it is clear that this number must be integral^ but this condition cannot be expressed in the equations. Thus there are three steps in the solution of a problem. We first find the equations which are the algebraical expressions of the relations between the magnitudes of the 13—2 174 PROBLEMS. known and unknown quantities; we then find the values of the unknown quantities which satisfy these equations ; and finally we examine whether any or all of the values we have found violate any conditions which are expressed or implied in the problem, but which are not contained in the equations. The necessity of this final examination will be seen from some of the following examples of problems. Ex. 1. A has £5 and B has ten shillings. How much must A give to B in order that he may have just four times as much as JS ? Let X be the number of shillings that A gives to B. Then A will have 100 - x shillings, and B will have 10 + as shillings. But, by the question, A now has four times as much as B, Hence we have the equation 100-a;=4(10 + a;); :. x=12. Thus A must give 12 shiUings to B. It should be remembered that x must always stand for & nurnber. It is also of importance to notice that all concrete quantities of the same kind must be expressed in terms of the same unit. Ex. 2. One man and two boys can do in 12 days a piece of work which would be done in 6 days by 3 men and 1 boy. How long would it take one man to do it ? Let ae=the number of days in which one man would do the whole, and let y=the number of days in which one boy would do the whole. Then a man does - th of the whole in a day ; and a boy does - th X y of the whole in a day. By the question one man and two boys do ^^jth of the whole in a day. Hence we have 1 2_J^ «"*"y""l2* We have also, since 8 men and 1 boy do ^th of the whole in a day, 3^1 1 - + -=» J. X y ^ Whence a; =20. Thus one man would do the whole work in 20 days. PROBLEMS. 175 Ex. 8. In a certain family eleven times the number of the children is greater by 12 than twice the square of the number. How many children are there ? Let X be the number of children ; then we have the equation 11a; = 2*2 + 12, or 2ar»-lLB+12=0, that is (2a!-3)(a?-4)=0. Hence «=4, or a:=f . The value x=\ satisfies the equation, but it must be rejected, since it does not satisfy all the conditions of the problem, for the number of children must be a whole number. Thus there are 4 children. . Ex. 4. Eleven times the number of yards in the length of a rod is greater by 12 than twice the square of the number. How long is the rod? This leads to the same equation as Ex. 3; but in this case we cannot reject the fractional result. Thus the length of the rod may be 4 yards, or it may be a yard and a half. Ex. 5. A number of two digits is equal to three times the product of the digits, and the digit in the ten's place is less by 2 than the digit in the unit's place. Find the number. Let X be the digit in the ten's place ; then a; + 2 will be the digit in the unit's place. The number is therefore equal to 10a; + (a! + 2). Hence, by the question, 10x + (a? + 2)=3a;(a; + 2); .-. 3«3-6a;-2=0, or (a;-2)(3x + l)=0. Hence «=2, or x— -\. Now the digits of a number must be positive integers not greater than nine; hence the value x= -^ must be rejected. The digit in the ten's place must therefore be 2, and the digit in the unit's place must be 4. Hence 24 is the required number. Ex. 6. A number of two digits is equal to three times the sum of the digits. Find the number. Let X be the digit in the ten's place, and y the digit in the unit's place; then the number will be equal to IQx+y. Hence, by the question, lQx + y=^{x + y)\ /. lx=2y. 176 PROBLEMS. Since x and y must both be positive integers not greater than 9, it follows that x must be 2 and y must be 7. Thus the required number is 27. Ex. 7. The sum of a certain number and its square root is 90. What is the number ? Let z be the number ; then we have the equation x+^x = 90; ,\ {x-90y=x, or «»- 181a; + 8100 = 0, that is {x - 81) (a; - 100) = 0. Hence ar=81, or a;=100. If, in the question, the square root means only the arithmetical square root, 81 is the only number which satisfies the conditions. If, however, ' its square root ' is taken to mean * one of its square roots,' both 81 and 100 are admissible. Ex. 8. The sum of the ages of a father and his son is 100 years ; also one-tenth of the product of their ages, in years, exceeds the father's age by 180. How old are they ? Let the father be x years old; then the son will be 100 - x years old. Hence, by the question, ^a;{100-a;) = a; + 180; .-. ar^- 90a; +1800 = 0, that is (x-60)(a;-30)=0. Hence a; =60, or a; =30. If the father is 60, the son will be 100-60=40. If the father is 30, the son wUl be 100-30=70, which is impossible, since the son cannot be older than the father. Hence the father must be 60 and the son 40 years old. Ex. 9. A man buys pigs, geese and ducks. If each of the geese had cost a shilling less, one pig would have been worth as many geese as each goose is actually worth shillings. A goose is worth as much as two ducks, and fourteen ducks are worth seven shillings more than a i)ig. Find the price of a pig, a goose, and a duck respectively. Let a;=the price in shillings of a pig, y= „ >i „ ,* goose, and«= „ „ „ „ duck. Then, by the question, a pig is worth y times (y - 1) shillings ; .-. x=y{y-l) «. (i). PROBLEMS. 177 Since a goose is worth 2 ducks, ••• J/=2z (ii). And, since 14 ducks are worth 7 shillings more than a pig, Uz = 7+x (iii). From (i) and (ii) we have the values of x and z in terms of y ; and, substituting these values in (iii), we have 7y = 7 + y(y-l), or y2-8y + 7=0; .-. y = 7, or y=l. If y=7, a; = 42 from (i), and z = ^ from (ii). If y = l, «=0 from (i), and 2=^ from (ii). These values are how- ever inadmissible, since pigs cannot be bought for nothing. Hence a pig cost 42«., a goose 7«.,.and a duck 3s. 6d, EXAMPLES XV. 1. Divide 60 into two parts, such that twice one part is equal to three times the other. 2. A has £6 less than jB, G has as much as A and JB together, and A, By C have £50 between them. How much has each % 3. One man is 70 and another is 45 years of age ; when was the first twice as old as the second *? 4. How much are eggs a score, if a rise of 25 per cent, in the price would make a difference of 40 in the number which could be bought for a sovereign 1 5. A bag contains 50 coins which are worth XI 4 altogether. A certain number of the coins are sovereigns, there are three times as many half-sovereigns, and the rest are shillings. Find the number of each. 6. A can do a piece of work in 20 days, which B can do in 12 days. A begins the work, but after a time B takes his place, and the whole work is finished in 14 days from the beginning. How long did A work ? 178 EXAMPLES. 7. A man buys a certain number of eggs at two a penny, four times as many at 5d. a dozen, five times as many at Sd. a score, and sells them at Ss. Sd. a hundred, gaining by the transaction 3^. Qd. How many eggs did he buy] 8. A bill of £63. 5s. was paid in sovereigns and half-crowns, and the number of coins used was 100; how many sovereigns were paid 1 9. A man walking from a town A to another B at the rate of 4 miles an hour, starts one hour before a coach which goes 12 miles an hour, and is picked up by the coach. On arriving at B he observes that his coach journey lasted two hours. Find the distance from A to £. 10. Two passengers have altogether 600 lbs. of luggage and are charged for the excess above the weight allowed 3s. 4(i. and lis. 8d. respectively. If all the luggage had belonged to one person he would have been charged £1. How much luggage is each passenger allowed free of charge? 11. A piece of work can be done by A and ^ in 4 days, by A and (7 in 6 days, and by B and C in 12 days : find in what time it would be done by ^, ^ and C working together. 12. A father's age is equal to those of his three children together. In 9 years it will amount to those of the two eldest, in 3 years after that to those of the eldest and youngest, and in 3 years after that to those of the two youngest. Find their present ages. 13. A 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 at two- thirds his previous pace they would have met in 4 hours. Find the distance between the towns. 14. A traveller walks a certain distance : if he had gone half a mile an hour faster, he would have walked it in |^ of the time : if he had gone half a mile an hour slower he would have been 2^ hours longer on the road. Find the distance. EXAMPLES. 179 15. Divide 243 into three parts such that one-half of the first, one-third of the second, and one-fourth of the third part, shall all be equal to one another. 16. A sum of money consisting of pounds and shillings would be reduced to one-eighteenth of its original value if the pounds were shillings, and the shillings pence. Shew that its value would be increased in the ratio of 15 to 2 if the pounds were five-pound notes, and the shillings pounds. 17. £1000 is divided between A, B, C and B. B gets half as much as A^ the excess of Cs share over Z)'s share is equal to one-third of ^'s share, and if B's share were increased by £100 he would have as much as C and D have between them ; find how much each gets. 18. Find two numbers, one of which is three-fifths of the other, so that the difference of their squares may be equal to 16. 19. Find two numbers expressed by the same two digits in different orders whose sum is equal to the square of the sum of the two digits, and whose difference is equal to five times the square of the smaller digit. 20. A man rode one-third of a journey at 10 miles per hour, one-third more at 9 miles per hour, and the rest at 8 miles per hour. If he had ridden half the journey at 10 miles per hour and the other half at 8 miles per hour, he would have been half a minute longer on the journey. What distance did he ride 1 21. Two bicyclists start at 12 o'clock, one from Cambridge to Stortford and back, and the other from Stortford to Cambridge and back. They meet at 3 o'clock for the second time, and they are then 9 miles from Cambridge. The distance from Cambridge to Stortford is 27 miles. When and where did they meet for the first time ? 28. Divide £1015 among A^ B, C so that B may receive £5 less than A, and G as many times B's share as there are shillings in ^'s share. 180 EXAMPLES. 23. On a certain road the telegraph posts are at equal distances, and the number per mile is such that if there were one less in each mile the interval between the posts would be increased by 2^ yards. Find the number of posts in a mile. 24. The sum of two numbers multiplied by the greater is 144, and their difference multiplied by the less is 14 : find them. 25. A and B start simultaneously from two towns and meet after five hours; if A had travelled one mile per hour faster and B had started one hour sooner, or if ^ had travelled one mile per hour slower and A had started one hour later, they would in either case have met at the same spot they actually met at. What was the distance between the towns 1 26. A battalion of soldiers, when formed into a solid square, present sixteen men fewer in the front than they do when formed in a hollow square four deep. Required the number of men. 27. A number of two digits is equal to seven times the sum of the digits; shew that if the digits be reversed, the number thus formed will be equal to four times the sum of the digits. 28. A sets out to walk to a town 7 miles off, and B starts 20 minutes afterwards to follow him. When B has overtaken A he immediately turns back, and reaches the place from which he started at the same instant that A reaches his destination. Supposing B to have walked at the rate of 4 miles an hour : find ^'s rate. 29. A starts to bicycle from Cambridge to London, and B at the same time from London to Cambridge, and they travel uniformly : A reaches London 4 hours, and B reaches Cambridge 1 hour, after they have met on the road. How long did B take to perform the journey 1 30. A number consists of 3 digits whose sum is 10. The middle digit is equal to the sum of the other two ; and the number will be increased by 99 if its digits be reversed. Find the number. EXAMPLES. 181 31. Two vessels contain each a mixture of wine and water. In the first vessel the quantity of wine is to the quantity of water as 1 : 3, and in the second as 3:5. What quantity must be taken from each in order to form a third mixture, which shall contain 6 gallons of wine and 9 gallons of water ? 32. Supposing that it is now between 10 and 11 o'clock, and that 6 minutes hence the minute hand of a watch will be exactly opposite to the place where the hour hand was 3 minutes ago : find the tima 33. -4, B and C start from Cambridge, at 3, 4 and 6 o'clock respectively to walk, drive and ride respectively to London. C overtakes ^ at 7 o'clock, and G overtakes A 4 J miles further on at half-past seven. When and where will B overtake A % 34. A train 60 yards long passed another train 72 yards long, which was travelling in the same direction on a parallel line of rails, in 12 seconds. Had the slower train been travelling half as fast again, it would have been passed in 24 seconds. Find the rates at which the trains were travelling. 35. 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 less than A does. How much does A give to each person ? 36. Three vessels ply between the same two ports. The first sails half a mile per hour faster than the second, and makes the passage in an hour and a half less. The second sails three-quarters of a mile per hour faster than the third and makes the passage in 2 J hours less. What is the distance between the ports ? 37. Two persons J., B walk from P to ^ and back. A starts 1 hour after /i, overtakes him 2 miles from Q, meets him 32 minutes afterwards, and arrives at F when -5 is 4 miles ofi". Find the distance from P to ^. CHAPTER XIL Miscellaneous Theorems and Examples. 163. Elimination. When more equations are given than are necessary to determine the values of the un- known quantities, the constants in the equations must be connected by one or more relations, and it is often of importance to determine these relations. Since the relations required are not to contain any of the unknown quantities, what we have to do is to eliminate all the unknown quantities from the given system. The following are some examples of Elimination : Ex. 1. Eliminate x from the equations ax + b=0, a'x + 6' = 0. From the first equation we have x= -- , and from the second equation we have «= — j. Hence we must have - = -., or ba'-b'a=0; which is the required result. Ex. 2. Eliminate x and y from the equations ax + by + c=0, a'x + b'y + c' = 0, ELIMINATION. 183 From the first two equations we have [Art. 143] a? _ y _ 1 he' - b'c ~ ca' - c'a ~ aV - a'h ' These values of x and y must satisfy the third equation ; hence ah - ah ah -ah or o" {be' - h'c) + h" {ca' - c'a) + c" (ah' - a'h) = 0, the required result. The general case of the elimination of n - 1 unknown quantities from n equations of the first degree will be considered in the Chapter on Determinants. Ex. 3. Eliminate x from the equations a'a;2 + 6'x + c'=0. As in Art. 143, we have a^ X 1 he' - b'c ca' - c'a ah' - a'h Hence (he' - b'c) [ah' - a'h) = (ca' - c'a)', the required result. It should be remarked that the above condition is also the condition that the two expressions aa^ + hx + c and a'x^ + h'x + c' may have a common factor of the form as -a; for if the expressions have a common factor of the form x-a they must both vanish for the same value of as. Ex. 4. Eliminate x from the equations ax''i-bx + c=Ot a'afi+b'x + c'=0. As in Ex. 3, we have a^ X 1 . he' - b'e ea' - c'a ah' - ah b&-b'c __( ca'-e'a \»^ '' ab'-a'h~\ab'-a'b) ' .-. (6c' - b'e) {ab' - a'h)^ = {ca' - c'a)^, the required relation. 184 ELIMINATION. Ex. 6. Eliminate x from the equations ax^ + bx + c=0 (i), a'x^ + b'x^ + c'x+d'=0 (ii). Multiply (i) by a'x, (ii) by a, and subtract; then, {ab'-ba')x^ + {ac'-ca')x + ad'=0 (iii). We can now eliminate x from (i) and (iii) as in Ex. 3. Ex. 6. Eliminate a;, y, z from the equations x + y + z = a (i), a:2 + t/2 + 22^&2 ^jjj^ X^ + y3 + z3^C^ (iii)^ xyz = (p (iv). From (i) and (ii) we have 2yz + 2zx + 2xy = a^-b^. From (iii) and (iv) we have x^ + y» + z3-Bxyz = c^-3d^, ie. {x + y+z){x^ + y'^ + z^-yz-zx-xy)=c^-Sd:i, Hence a{b^-^{a^-b^)}:=c^-Sd^.^ .'. a8 + 2c3-6d3-3a&2=0, the required result. Ex. 7. Eliminate x, y, z from the equations x^{y+z) = a^ (i)^ 2/2(2+a;) = 6a ^^^^ z^{x + y) = ^-2xa/ + af^){y^-2yy' + y'^) + xh/^-2xa^yy' + x'Y=0, that is {x - x')^(y - y'Y + (ary' - «V)' = 0. Hence xy'-x'y—Q and (x-x'){y-y')=^Q. From the second relation x=x' or y=y'; and either of these combined with the first relation shews that both x=x' and y=y'. IDENTITIES. 187 Ex.3. If a^ + a^ + a^^Jf =jf)3, V + V + V + = 3'» and Orp^^t a^^+ a^h^+ =^g, the quantities being all real; then will ^1 = ^ = ^3^ &C.=^. \ Oj &8 q Multiply the equations in order by 2^, p^ and - 2pg respectively, and add ; we then have (2«i-2>6i)' + (2«»-P&2)*+(3«8-i'&8)'+ =0- Hence goj -!p\ = = qa^ -1>&3= 2^3 - ^^3= &c. Therefore ei=? = «» = ^= &o. 156. We have already proved that a" + 6' + c" - 3a6c = (a + 6 + c) (a' + 6* + c'' - 5c - ca - a6) = i(a + 6 + c) {(6 - c)'' + (c - af^ (a - hf] — {a-\-h-\-c){a + (oh + co^c) (a + ay^b + coc), where « is either of the cube roots of unity. [See Art. 139.] From the above many other identities can be found. Ex.1. (6 + c)» + (c + a)8+(a + 6)8-3(6 + c)(c + a)(a + 6) = 2(a3 + 63 + c»-3a6c). Left side = ^^ {& + c + c + a + a + 6} {(c + a-a+ 6)^+ two similar terms} = (a + 6 + c){(6-c)2+(c-a)2 + (a-6)2} = 2(a3 + 63 + c8-3a6c). Ex.2. (6 + c-a)»+(c + a-6)3 + {a + 6-c)8 -3(6 + c-a)(c + a-6)(a + &-c) = 4(a34-&3 + c'-3a6c). Left side = ^ (a + 6 + c) {(26 - 2c)^ + two similar terms} =4(a3 + 6» + c»-3a6c). Ex. 3. (a;2 - y^)^ + (ya - za;)» + {z-* - xy^ -3{x^- yz) {y^ - zx) {z^ - xy) = {x^ + y^ + z^-3xyzy. Left side =s ^{x^ + yi + z^-yz-zx-xy[{y^-zx-i?-xyy + two similar terms] ^i{x^ + y^ + z^-yz-zx-xy){x + y + z)^[iy~z)^ + two similar terms] = (x + y + z)^ {x^ + y^ + z^-yz - zx- xy)^ = (x^ + y^ + z^- 3xyz)'^. S. A. 14 188 EXAMPLES. Ex. 4. Shew that (iB« + y3 + 28 - 3(tyz) (a» +h^+c^ - 3abc) can be expressed in the form X^+T^ + Z^-HXYZ. We have {x + y + z){a + b + c)=:{ax + hy + cz) + {hx + cy + az) + {ex + ay + 6«), {x + (ay + ux^z) (a + co^ft + wc) = {ax + hy + cz) + u^ {hx + cy + az) + + ajy + ar» + TO(a; + y) = (i). So also, since y^z^ «* + 2/2 + y^ + wi (« + y)=0 (ii). From (i) and (ii) we have by subtraction a^ - «2 + J/ (« - 2) + i» (a; - «) = 0. Hence, as x^t^wQ have aj+y + «+m=0 (iii). Substitute - (a; + y + 2) for »n in (i) ; and we have x^+ xy + y^ ~ {x+y){x -{-y ■\- z) = 0\ :. yz + zx + xy=0 (iv). Then y^ + z^ + m{y^ + z^)=y^+z^-{y^+z^){x + y+z) from (iii) = -{y^x+z^+y^g+z^) = -y {xy+yz)-z {yz + zx) =2xyz from (iv), 14—2 190 EXAMPLES. Ex. 8. Shew that, if a+b + c + d=0, then will a* + b* + c* + d*=2 {ab - cd)^ + 2{ac- bdf + 2 {ad- be)' + Adbed. We have to prove that Sa4=2Sa26a-8a6cd. Since a + 6 + c + d=0; we have, hy squaring and transposing, a2 + 6a + c2 + d2=-2(6c + ca + a& + ad + 6d + cd). Hence by squaring Sa* + 22a26a=4(S&c)a. Now (26c)2 = 262^2 + Qabcd + 2bcd {b + c + d) + 2cda {c + d-\-a) + 2dab (d + a + b) + 2abc {a + b + c) = ^b^c^ + 6abcd - 8abcd. Hence 2a* + 2^a%^ = 42a262 _ Sabcd ; .-. 2a*=22a262_8a6cd. Ex.4. Prove that, if ax + by + cz=0, and - + - + - = 0, then will z y z ' ax^ + by^ + cz^= - (a + b + c){y + z){z + x){x + y). From the given relations we have, as in Art. 143, a _ b c y -- ~ t _ ? ~f y * z y X z y X Hence [Art. 113] each fraction is equal to aa^ + by^ + cz^ a + b + c \z yj " \x zj \y xj z y X z y X Hence ax^ + 62/8 + cg8 ^ a^ ^y^^ _ ^8) ^ yi (^.a _ ^.2) + g4 (^a _ ^8) o + & + C x{y^-z'') + y{z^~ x^) + 2 (a^» - y^) -{y^-z^){z^-x^){x^-y^) *" {y-z)(z-x){x-y) = -{y+z){z+x){x + y). EXAMPLES XVL 1. Shew that, if = a, — ^ = 6 and = c ; then will . + , — - + , = 1. l+a 1+6 1 +c EXAMPLES. 191 2. Shew that, if ax + b^ = and csc^ + dxy + eif = 0, then will (j?e + 6*c = ahd. 3. Eliminate a;, y, z from the equations y — z z-x , ic-y ^ — =a, =6, - = c, y + z z + x x + y 4. Eliminate x, y, « from the equations y X z y J x z (SB « y x z y 11 1 5. If a; + - = 1 and y + - = 1 ; prove that « + -=!, 6. Eliminate a; from the equations a + c— — dxj x a — c = — bx, X I 7. Eliminate a, y, z from the equations ^ x'~yz = a, y^-zx = b, z' -xy=Cy ax + by + cz = d, 8. Prove that the equations x + y + z = a, aJ' + y' + ^ = b% a^ + ^ + z^-Sxyz = c\ do not give any roots, but simply a relation between a, b and c. 9. Shew that, if bz + cy = ex + az= ay + bXf and x^ + y^ + s^ - 2yz - 2zx - 2xy = ; then will a =fc 6 ± c = 0. 10. Shew that, if - +| +- = 1 and - +- +- =0; then I ' a 6 c X y z ' ' .„ a;^ 2/' z* - will -T + TT + ^ = !• a^ b cr 11. Ifa; + - = i/ + - = » + -; then xWz^ =1, or x = y = z. y ^ z X 192 EXAMPLES. 12. Shew that, if x = cy + bZf y = az + cx and z = bx + ay ; then g* y* a* l_a»-l_6«-l_c»' 13. Shew that, if af = y' + z'+ 2ayz, j/* = »' + sc* + 2bzx and «* = «:' + 2/' + 2ca;^; then of _ y' _ z' 1 _ a« - 1 _ 6» ~ 1 - c' • 14. Shew that, if x, y, » be unequal, and a + bz a + bx , a + by y = X > ^2! = -T- and 03 = /- , c+a2; c + dx c + ay then will aci + 6c + 6' + c* = 0. 15. Eliminate a;, y, « from the equations af yz . y* zx 2* an/ yz or zx y' xy z' 16. Eliminate a?, y, z from the equations baf + Ix + c = 0, cy'+my + a = Of as^ + nz + b = 0, xyz = \. 17. Eliminate x, y, « from the equations 't^ ■\- s? = ayz^ s^ + af — bzXf af + y' = cxyf xyz not being zero. 18. Eliminate (i) a;, y, « and (ii) a, 6, c from the equations o--\-c-=a. c- + a- = b. and a-+ b-=e, z y X z y X 19. Eliminate a;, y, e from the equations ax + yz = bCf by + zx = cay cz-\-xy = dbf and xyz = a6c. 20. Eliminate a;, y, « from the equations a? — xy-xz_y*-yz-yx^z'-zx-zy a ~ b " c * and ax + by + cz = 0. EXAMPLES. 193 21. From the equations d?yz = a* (y + zf^ hhx = p' (z + x)", c^xy = y'(x + yY, deduce the relation abc a' b' c' . ^ — ■ -I- 4. — _ 4 a)8y a» ^ ^« y' 22. Prove that, if y' + z' + yz = a*y «* + sja; 4- a;' = 6", a? + xy +y'=^ c* and yz + zx + xy ==0 'y then will a ± 6 ± c = 0. 23. Prove that, if - + =-+- = r , then will a b c (a+ b + c) 1 1 1 _ 1 where n is any positive integer. 24. Shew that, if J . -.8 rfl _« . 7.8 _t = 1, 1. 26c 2ca 2ab then (6 + c - a) (c + a - 6) (a + 6 - c) = 0, and \ 2bc J "^ V 2ca ) '^\ 2ab J 25. If aV + 6y + cV = 0, and — a" = — 6' = c": a; 2/ » prove that aV + 6y + cV = 0, and a V + 6y + c V = a V + 6*2/' + cV. 26. if « - ^ = y -.-^ =z / , and x, y, 2? be unequal; XT y z then each member of the equations is equal tox + y + z—a. (z - xY 27. If x, y, z be unequal, and if 2a-3y = ^ ^ and 2a-3z= ^^~^^\ then will 2a-3x= ^^~ ^^* , and x + y + z = a. 194 EXAMPLES. 28. If a; + -^ — 5 » ^ Jio* altered in value by inter- changing x and 2/, it will not be altered by interchanging x and «, and it will vanish if x + y + z=l, the letters being all unequal. 29. If X, y, z be unequal, and y" + «* + m (2/ + «)=«* + ic^ + w (« + «) = »• + y + m (35 + y), then each will equal 2xyz. 30. If X, y, z be unequal, and y* + z^ + myz = 2^ + cc' + mzx = af + y' -{• mocyy then each will equal J (a?" + y* + ^). 31. If X, y be unequal, and if (i^Ui-f)" = {^V^^Z^ , X y then will each equal ^^ — . 32. Shew that, if a, 6, c, c? be all real quantities not zero, and [a^ + Jf) (c* + d^) = 4:abcd : then will a = ^h and c='^d. 33. If a, 6, c, aj be all real quantities, and c b then -= - =x. a 34. Shew that, if (ai^ + y^ + z") (a' + b' + c') = (ax + by + cz)^, th en xja = y/b = z/c. 35. Prove the following : (i) If 2 {a' + b') = (a + b)% then a = h. (ii) If 3(a» + 6* + c') = (a + 6^-c)^ thena=ft=c. (iii) If 4(a» + 6' + c' + c^«) = (a + 6 + c + (i)«, then a = 6 = c = c?. and (iv) If 7i(a' + 6' + c'+ ) = (a + 6 + c+ )', then a = b = c = , n being the number of the letters. EXAMPLES. 195 36. Prove that, if a, b, c, d be all real and positive, and a^ + 6* + c* + c?* = 4a6c£i; then will a = h = c=^d. 37. If 5 {n - 1) a' + 2a; {a^ - a J + a/ + 2a/ + 203* + . . . + 2a;.i + aj = 2 {a^a^ + a^a^ + + a^_,aj for real values of «, a^, a,, ..., a,; then will a, — «, = CTg — a, = ... = a„ — a„_j = x, Yerify the following identities : 38. aJ" (6 + c) + 6' {c + a)-\- c^ {a + h) + ahc (a + h+c) = {a^ + 5" + c') (6c + ca + ai). 39. (6 + c-a-cZ)*(6-c)(a-ci) + (c+a-6-c^)*(c-a)(6-fQ + (a + 6 - c — c?)^ (a - 6) (c - c?) ^ 1 6 (6 - c) (c - a) (a - ^)) (c/ - a) {d -h){d- c). 40. 8 (a + 6 + c)»- (6 + c)» - (c + a)» - (a+ 6)» = 3 (2a + 6 + c) (a + 26 + c) (a + 6 + 2c). 41. (a+h + c-^d)'-{h + c-^dY-(c + d-\-af-{d-¥a-¥ h)' - (a + 6 + c)» + (6 + c)' + (c + ay + (a + 6)' + (a + c?)' + (6 + df + (c + <^)' -a'-b'-c'-d' = 60abcd (a + b + c + d). 42. (a + 6 + c)' a6c - (be + ca + abf = abc (a' + 6' + c') -(6V + cV + aV). 43. (a' + 6 V c^)-^ + 2 (6c + ca + a6)' - 3 (a« + 6'' + c") (6c + ca + abf - (a^ + 6" + c" - 3a6c)». 44. (ca - b") (ab - c') + (a6 - c') (bo - a') + (6c - a') (ca - b') = (6c 4- ca + a6) (6c + ca + a6 - a^ - 6^ - c*). 196 EXAMPLES. 45. 2 (c" + ca + a') (a' + ab + b') - (6» + 6c + c')" + 2(a" + a6 + 6')(6"+6c+c')-(c« + ca+ay + 2(6" + 6c + c')(c' + ca + a')-(a" + a6 + 6y = 3 (6c + ca + a6)'. 46. Shew that (3a-6-c)»+(36-c-a)«+(3c-a-6)' -3(3a-6-c)(36-c-a)(3c-a-6)= 16 (a' + 6« + c'-3a6c). 47. Shew that (rwi - 6 - c)* + (w6 - c - a)* + (ric - a - 6)* — 3 {na — b — c) (nb — c — a) (no — a-b) = (w + Vf in - 2) (a» + 6« + c'' - 3a&c). 48. Shew that (aj* + 2yzy + {y*+ 2zxy + («» + 2xyy -Z(a?+2yz) {y^ -\-2zx) {z' + 2xy) = (x"" + y^ + z' - Sxyz)'. 49. Shew that (by + azy + (bz + aa;)* + (Jo? + ay)' -3(by+az) (bz + aa;) (6a; + ay) = (a»+ 6') (a» + ^ + »*- 3a;2/2). 60. Shew that, if 1 + co + o>' = 0, then [(6 - c) (aj - a) + 0) (c - a) (a; - 6) + a)'(a - 5) (a; - c)]' + [(6 - c) (a; ^ a) + co' (c - a) (a; - 6) + w (a - 6) (a?- c)]' = 27 (6 - c) (c - a) (a - b) (x -a)(x- b) (x-c). 51. Shew that the product of any number of factors, each of which is the sum of two squares, can be expressed as the sum of two squares. 52. Verify the identity (a' + 6" + c* + d^) (p' + q' + 7^+ «') = (ap + bq + cr + d^y + (aq -bp + c8 — dry+ (or —b8-cp+ dqy •\- (as + br — cq - dpy. Hence shew that the product of any number of factors, each of which is the sum of four squares, can be expressed as the sum of four squares. EXAMPLES. 197 53. Shew that {a?-\-xy -^^) (a' + ab + b') can be expressed in the form X' + XY+ Y\ 54. Shew that {a^ + pxy + qy') (a' + pa^ + qb') can be ex- pressed in the form X* +pXY + qY\ 55. Shew that, it 2s=a + b + e, (i) a(8-b){8-c) + b{8-c){s 'a) + c{8-a){8- b) + 2 {8-a) {8-b){s-c) = abc (ii) («-a)» + («^6)" + (»-c)» + 3a6c = «». (iii) (b + c) 8 {s - a) •¥ a {8 -b) {8 - c) - 2s6c = (c + a) 8 (» - 6) + 6 (s - c) (« - a) — 2sca = (a + 6) « (« - c) + c (« - a) (« - fe) - 2sdb. (iv) a(6-c) («-«)■+ 6 (c-a)(«-6)" + c(a-6)(«-c)'=0. (v) 8 (« - 6) (« - c) + « (« - c) (s - a) + s (« - a) (« - 6) -(«-»)(«- 6) (« - c) = a6c. (vi) («-«)' (8-6)' («-c)" + «»(«-6)'(s-c)" + «« («-c)» («-»)« + «»(«- a)* («-6)» + «(«-a)(8-6)(s-c)(a» + 6« + c«) = a''6V. 56. Shew that, if 2s = a + 6 + c + c?, 4(6c + a^)*-(6» + c*-a»-c^»)« = 16(s-a)(s-6)(8-c)(«-c;). Shew also that a (« - 6) (s - c)(» - c?) + 6 (8 - c) (s - c?) (« - a) + c(«-fl?) («-a) («-6) + c?(s-a)(«-6)(«-c) + 2(s-a)(s-6)(s-c)(s-c?) — 8 {bed + cc?a + c^ai + abc) = — 2abcd. 57. Shew that, ifa4-6 + c + c? = 0, then flki (a + c?)' + 6c (a - c?)" + a6 (a + 6)' + cc? (a - 6)* + ac (a + c) V 6c? {a - c)' + 4a6cc? = 0, 198 EXAMPLES. 58. Shew that, if {a + 6) (6 + c) (c + d) (d + a) ^(a + b + c + d) (bed + cda + dab + ahc) ; then ac = bd. 59. Shew that, if a + b + c = and x + y + z = 0, then 4:{ax + by + cz)' -3{ax+by + cz) (a" + b' + c'){af + y' + ^) - 2 (6 - c) (c - a) (a - b) {y -z){z~ x) (x-y) = diabcxyz. 60. Shew that, ii a + b + c = ; then (i) 2{a' + b' + c'') = 7abc(a* + b* + c*). (ii) 6{a' + b' + c') = 7{a' + b' + c')(a^+b* + c'), (iii) a' + 6« + c« = 3a^6V + l{a' + b' + c'f. (iv) 25(a^ + 6^ + c0(a' + 6« + c«) = 21(a'' + 6» + c7. 61. If a + b + c + d=0, prove that (a* + 6' + c** + d^y = 9 (6cc? + cda + dab + abc)' = 9 (be — ad) (ca — bd) {ah - cd). 62. Shew that, if a + 6 + c = 0, then /b-c c — a a — b\/a h c \ . \ a c J\o — c c—a a—bj 63. Prove that, if 1 \-\-l-¥ln 1+m + m^ l+w + nm =1, A ? ml 1 1+Z+^w 1+m + m? 1+n+nm * and none of the denominators be zero, then will l = in = n. 64. Shew that a + (1 - a) 6 + (1 - a) (1 - 6) c + (1 - a) (1 - 6) (1 - c) i + ... = 1 - (1 - a)(l - b) (1 -c)(l - d) EXAMPLES. 199 65. Shew that - = 1 + 2 (1 - a) + 3 (1 - a) (1 - 2a) + ... + {n{l-a){l-2a)...{l-n- la)] + h(l-a)(l-2a)..,(l-na)}. (Jb 66. Shew that a" + a"-' (1 - a") + a"-' (1 - a") (1 - a"-') + ... + {a (1 - a-) (1 - a"-^) ... (1 - a')} + {(1 - a") (1 - «"">) ... (1 - a)} = 1. 67. Shew that, if n be any positive integer, l-a- (l-a')(l-a'-) (1 -a") (!-»- ) (l-""") , ^ (l-a-)(l-a-)...(l-«) ^ l-a" 68. Prove that, if a + b + c-\-d=Of x + y + z + u = 0, and ax + by + cz + du = 0; then 2 (a^c + b*y + c*z + d*u) = (a'x + 6V + c'2; + c? V) (a* + 6^ + c' + d^. 69. Prove that, if n be any positive integer, 1111 111 1 ( 2^4 2n n + l n-i-2 2n 70. Prove that, if u V u + a v — h u+a' v-b' /' then /2 (a6' - a'bf = aa'66' (a - a') (6 - 6'). CHAPTER XIIL Powers and Roots. Fractional and Negative Indices. 159. The process by which the powers of quantities are obtained is often called involution; and the inverse process, namely that by which the roots of quantities are obtained, is called evolution. We proceed to consider some cases of involution and of evolution. 160. Index Laws. We have proved in Art. 31, that when m and n are any positive integers, a'^xar = ar** (i). This result is called the Index Law. From the Index Law, we have arxarxcf= oT^ X(f = oT^^, and so on, however many factors *here may be. Hence a"* x a" x a** x . . . = a"**""^ (ii). Thus the index of the product of any number of powers of the same quantity is the sum of the indices of the factors. Also, a"* X a*" X a"* X ... to n factors ni+n»+»i+ to n terms POWERS. 201 Hence {ory = ar'' (iii). Thus, to raise any power of a quantity to any other power, its original index must he multiplied by the index of the power to which it is to be raised. Again, to find (a6)**. (a6)*" = ah xab X ahx to m factors, by definition, = (axaxa to m factors) x (bxb xb to m factors), by the Commutative Law, = a*" X 6"*, by definition. Hence (aby = a** x h^. Similarly (a6c...)"* = a*" xb^'xc^x (iv). Thus, the mth power of a product is the product of the mth powers of its factors. The most general case of a monomial expression is a'6V Now (a"W. .,...)" = {oTT Q>T {cT from (iv) = a"'"6'^c'"* from (iii). Hence (a^h^c' )*" = a'^b'^'c"^. . . . . .(v). Thus any power of an expression is obtained by taking each of its factors to a power whose index is the product of its original index and the index of the power to which the whole expression is to be raised. As a particular case 161. It follows from the Law of Signs that all powers of a positive quantity are positive, but that successive powers of a negative quantity are alternately positive and negative. For we have 202 BOOTS OP ARITHMETICAL NUMBERS. (-a)* = (-a) (~a) = 4-a', (-a)^ = (-a)^(-a) = (+a'0(-ti) = -a«, (-a)^ = (-a)«(-a)=(-a»)(-a)= + a^ and so on. Thus (-ar = 4-a^ and (-a)^«*^=-a^"-^\ Hence all even powers, whether of positive or of negative quantities, are positive; and all odd powers of any quantity have the same sign as the original quantity. 162. Roots of Arithmetical numbers. The approximate value of the square or of any other root of an arithmetical number can always be found: this we proceed to prove. It will be seen that the process described would be an extremely laborious one; we are not however here concerned with the actual calculation of surds. Consider, for example, i^Q% First write down the squares of the numbers 1, 2, 3, &c. until one is found which is greater than 62 : it will then be seen that 7* is less and 8* is greater than 62. Now write down the squares of the numbers 7'1, 7*2, 7*3, ..., 7'9 : it will then be seen that (7"8)'* is less, and (7*9)^ greater than 62. Now write down the squares of 7-81, 7-82, ..., 7-89 : it will then be seen that (7 '83)^ is less, and (7 84)* greater than 62. By continuing this process, we get at every stage two numbers such that 62 is intermediate between their squares, and such that their difference becomes smaller and smaller at every successive stage; moreover, this difference can, by sufficiently continuing the process, be made less than any assigned quantity however small. Thus, although we can never find any number whose square is exactly equal to 62, we can find two numbers whose squares are the one greater and the other less than 62, and whose difference is less than any assigned quantity however small. The limiting value of these two numbers, SURDS OBEY FUNDAMENTAL LAWS. 203 when the process is continued indefinitely, is called the square root of 62. The process above described for finding a square root can clearly be applied to find any other root. Thus an nth root of any integral or fractional number can always be found. 163. Surds obey the Fundamental Laws of Algebra. The fundamental laws of Algebra were proved for integral or fractional values of the letters ; and it can be proved that they are also true for surds. Consider, for example, the Commutative Law. We have to prove that We can find whole numbers or fractions x, y and p, q such that ic > 7a > 2/, and p>"^h> q; and the difference between a; and y, and also the difference between p and q, can be made less than any assigned quantity however small. Hence xxp> ^ax1Jh>y xq, and pxx>'^h X J^a>qxy. But, since x, y, p, q are integral or fractional numbers, we know that x x p=p xx, and y xq= q xy; also the difference between px and qy can be made less than any assigned quantity however small. It therefore follows that ^a x 76 and "^6 x 7a, which are both always intermediate to xp and yq, must be equal. Thus the Commutative Law holds for Surds, and the other laws can be proved in a similar manner. 164. We already know that there are two square roots, and three cube roots of every quantity ; and we may remark that there are always n nth roots. Thus there is S. A. 15 204 PROPERTIES OF ROOTS. an important diflference between powers and roots; for there is only one nth power, but there is more than one nth root. 165. We have proved in Art. 160 that the mth power of a product is the product of the mth powers of its factors; and, since surds obey the fundamental laws of Algebra, the proposition holds good when all or any of the factors are irrational. Hence Also {Jab...y = ah...y by definition. Hence ^a x <\Jb... must be equal to one of the square roots oi ah... . We can write this \/a ^Jh... = Jab... , meaning thereby that the continued product of either of the square roots of a, either of the square roots of b, &c. is equal to on^ or other of the square roots of ab ... Similarly we have, tuith a corresponding limitation, , :Ja "/a- ^a76... = 7a6..., and ^K^ sj I ' Also J a"" = X^a*™^, for their npih powers are both equal to a*"^ Again, since the nth power of a monomial expression is obtained by multiplying the index of each of its factors by n, it follows conversely that an nth root of a monomial expression is obtained by dividing the index of each of its factors by n, provided the division can be performed. Thus one value of \/a* is a*, one value of Ja^ 6' c* is a^ b^ c, and one value of Ja^ 6"^ c"^ is a" b^ c^. EXTENSION OF MEANING OF INDEX. 205 Fractional and Negative Indices. 166. We have hitherto supposed that an index was always a positive integer ; and this is necessarily the case so long as we retain the definition of Art. 9 ; for, with that definition, such expressions as a and a"^ have no meaning whatever. We might extend the meaning of an index by assign- ing meanings to a** when n is frax^tional and negative. It is, however, essential that algebraical symbols should always obey the same laws whatever their values may be ; we therefore do not begin by assigning any meaning to a** when n is not a positive integer, but we first impose the restriction that the meaning of a" must in all cases be such that the fundamental index law, namely shall always be true ; and it will be found that the above restriction is of itself sufficient to define the meaning of a** in all cases, so that there is no further freedom of choice. For example, to find the meaning of a . Since the meaning is to be consistent with the Index Law, we must have Thus a* must be such that its square is a, that is a* must be ^a. Again, to find the meaning of a"^. By the index law a' 1 a^ X a^ = a ^"^" = a* ; therefore a ' = -^ = - . ' a^ a Thus a~^ must be - . ^ 15-2 206 FRACTIONAL AND NEGATIVE INDICES. 167. We now proceed to consider the most general cases. 1 I. To find the meaning of a**, where n is any positive integer. By the index law, 111 a** X a" X a" X to n factors -+-+-+ to n terms - = a** *» « =a» = a' = a. Hence a" must be such that its nth power is a, that is II. To find the meaning of a", where m and n are any positive integers. By the index law, a^ X a« X to 71 factors = a- "^» *°*»*«"°« = ^«^ «^^m Hence a^= H^. We have also 11 1 ^1 , * * »» " - . r i i:+ — f" to «» terms — a^ xa'^x to m factors = a** » = a**. 22 1 Hence a" = (a")"*. Thus we may consider that a* is an wth root of the mth power of a, or that it is the mth power of an nth root of a ; which we express by With the above meaning of a" it follows from Art. 165 that a" = a"^ FRACTIONAL AND NEGATIVE INDICES. 207 Note. It should be remarked that it is not strictly true that ^(O = (l/aY except with a limitation corre- sponding to that of Art. 165, or unless by the nth root of a quantity is meant only the arithmetical root. For example, }/{a^) has two values, namely +a^ whereas ( ^aY has only the value + a^. III. To find the meaning of a^. By the index law a' X a"'=^a'^ = a'^y :, a' = a''' -^ a"^ = 1. Thus a° = 1, whatever a may be. lY. To find the meaning of a~^, where m has any positive value. By the index law, a- X a"* = a-^'^ = a' ; and a° = 1, by III. Hence a~^ = -=, , and a"* = -^^^ . a a 168. We have in the preceding Article found that in order that the fundamental index law, a*" x a" = a*"*", may always be obeyed, a*" must have a definite meaning when n has any given positive or negative value. We have now to shew that, with the meanings thus obtained, a'" X a" = a"'^, (»"»)•» = a"*", and {aby = a"6*, are true for all values of m and n. When these have been proved, the final result of Art. 160 is easily seen to be true in all cases. 208 INDEX LAWS. I. To prove that oT x a"" = oT^'' , for all values of m and n. We already know that this is true when m and n are positive integers. Let m and w be any positive fractions - T and - respectively. Then oTx aJ" = a* X a'= ^aFx *Ja\ by definition = '^oT X V^= '4/^^ [Art. 165] ps+rq = a ?* , ' by definition P+r tn+n = a9 « =a' Thus the proposition is true for all positive values of m and n. To shew that it is true also for negative values, it is necessary and sufficient to prove that a"~ X a"* = a"*""", and a™ x a"" = a"*"" where m and n are positive. Now a-^xa-" = --x- = -^,- =a-"-«. U/ U/ lib And, if m — 7i be positive, a*""* X a* = a*", and a" x a"" x a" = a* ; therefore a"*"" = a*" x a"". Hence, if m — w be negative, -^^ x — = -^p;;, , that is, a*" X a"* = a'""". Hence a"* x a" = a"^, for aZ^ vaZwes of m and n. Cor. Since a*""" xa'' = a'^ for all values of m and n, it follows that a"" -^ a"" = a"*"". INDEX LAWS. 209 II. To prove that (a'")" = o^""*? ^^ all values of m and n. First, let n be a positive integer, m having any value whatever. f Then {a^'Y = oT x a"' x a"^ x to n factors, r -^Qm + m+m+ to n terms bv I. = ft"*". I Next, let 71 be a positive fraction - , where p and q are positive integers. Then (0'* = K)*=^{(aT}, = V(0> since p is an integer, mp Finally, let n be negative, and equal to —p. Then {arr = {aT = (~lp = ^ = "^^ = """■ Hence for all values of m and n we have III. To prove that (a6)" = a"6^ for all values of n. We have proved in Art. 160 that (afe)" = a"6", where w is a positive integer. And, whatever m may be, provided that ^ is a positive integer, we have {oTiry = a'^lr x arlr x ... to ^ factors __ Q7)i+m+,.. to q terms ^ J^m+m+ ... to q terms Let n be a positive fraction -, where p and g are positive integers. Then 210 RATIONALIZING FACTORS. p (ahy^(ahy^X/(aby = ;^{a''b^), since _p is a positive integer. Also (a"6")' = a*"^b*^, since g is a positive integer. Hence a"6" = ^(a^'b'') = (ab)\ Thus (aby = a"6**, for all positive values of n. Finally, if n be negative, and equal to — tw, we have V / V / (a 6) a 6 Ex. (i) Simplify a^ xa~K Ex. (ii). Simplify aH^ x a^&». ah^ xah^o = a^ + h^ + %=. aW' = a^fc*. Ex. (iii). SimpUfy (a-^b^) ' ^. Ex. (iv). Simplify J {a " ^ 6»c " t) 4- v^(a^ ft^c-i). 169. Rationalizing Factors. It is sometimes re- quired to find an expression which when multiplied by a given irrational expression will give a rational product. The following are examples of rationalizing factors. Since {a-\-Jh){a-Jh) = a^-h^ it follows that a^Jh is made rational by multiplying by a^FsJh. So also ajh =t c ^d is made rational by multiplying hy a Jh^c Jd, Again from the known identity 262ca + 2c2a3 + ^a?})^ _ a* - 6* - c* = (a + 6 + c){-a + 6 + c)(a-6 + c)(a + &-c), RATIONALIZING FACTORS. 211 it follows that the rationalizing factor of The rationalizing factor of Jp + ijq+ ^Jr may also be found as follows, {Jp + Jq + s/r)iJp+s/q~Jr)=p + q-r+2jM, and {p + q -r+2jpq) (p + 5 _r - 2 Jpq) = (p + g - r)^ - ^pq. Thus the required rationalizing factor is {>Jp-\rJq-slr){p + q-r-2^pq), which is the same as before. Again, from the identity the rationalizing factor of a + 6^ is seen to be a^ - db^ + h^. 170. To find the rationalizing factor of any binomial. P r Let the expression to be rationalized be gw?'± hy'. P r Put X^ax", and Y=hy\ and let n be the L.C.M. of q and s. Then it is easily seen that X" and F" are both rational. Hence, from the identities (Z+y) {Z"-^-Z«-'^F+...+ (- ir^F-^} =:Z" + (- ir^F" and (X- F)(X'^-^ + X'^-'F+ + F"-^) = Z''- F", the rationalizing factors of X 4- F and X — F are seen to be respectively X"-'-X"-^F+ + (-l)"-^F""^ and X'*-^ + X"-^F+ + F"^ 212 EXAMPLES. Ex. To find a factor which will rationalize xs _ ay^ . Here X=x^, Y=ay^, n=6. The factor required is therefore a;^ + ax^y^ + a^x"y ^ + a^x^y^ + a*x^y^ + a^y^^ . EXAMPLES XVIL L Simplify a^b^ x oT^b'^. 2. Simplify at X a-S X (a«)-i X — . 8. Simplify (aJ-V)* X (a»6"c-";^. 4. Simplify (a;c-oja-6 ^ (a;«-&)6-c X (a;5^)«-« • 5. Multiply s^ + x^y^ + y^ by x^-y^. 6. Multiply a;' + 1 + a;-« by a:' - 1 + a;"*. 7. Multiply a;^ + yt + »3 _ 2^i ;si _ ;g7£ci - a;7y J by a;^ + y^ + ;s^. 8. Divide a5*-2 + ic~* by x^-x~^, 9. Divide a^-x by a^^-a^. 10. Divide x^-xy^ + xy-y^ by x^-y\ 11. Shew that a;* - 4a;^ -*- 2x^ + 4a; - 4a;^ + a;^ = (a;* - 2x^ 4- x^)\ 12. Multiply 4a;' - 5aj - 4 - 7a;"' + 6a;~' by 3a; - 4 + 2«-» and divide the product by 3a;- 10 + lOa;"' - 4a;~'. EXAMPLES. 13. Divide x-x-'^2{x^''X-^) + 2(x^-x-^) by x^-x-*, 14. _,. ... ax~'+a-'x+2 Simplify , _, _i X . a^x s + a ^x^ - 1 15. 16. Shew that X x^ 1 1 1 „ — . + , -x^ + 2. 213 5^-1 X^^ +1 X^ -1 X^+l 17. Shew that (2x + y-') {2y + a;"') = {2xiyi + aj-^y-^/. 18. Shew that a' + b'-g-'-b-' {a-a-')(b-b-') ^ ^ 19. Shew that, if x^ + y^ + z^=Oj then (x + y + zy = 27xyz, 20. Find factors which will rationalize the following expressions : (i) a^ + 0, (ii) a^x^ + y^, (iii) a-\-bar + cx^^ and (iv) x^ + y^ + z^. 21. Shew that, if (l-x')Hy-z) + {l-y')^{z-x) + (l-z')h(x-y) = 0, and X, y, z are all unequal, then CHAPTER XIT. Surds. Imaginary and Complex Quantities. 171. Definitions. A surd is a root of an arithmetical number which can only be found approximately. An algebraical expression such as sja is also often called a surd, although a may have such a value that ^a is not in reality a surd. Surds are said to be of the same order when the same root is required to be taken. Thus V2 and V^ are called surds of the second order, or quadratic surds ; also i/4f is a surd of the third order, or a cubic surd ; and l/a is a surd of the nth. order. Two surds are said to be similar when they can be reduced so as to have the same irrational factors. Thus V8 and \/18 are similar surds, for they are equivalent to 2\/2 and 3\/2 respectively. The rules for operations with surds follow at once from the principles established in the previous chapter. Note. It should be remarked that when a root symbol is placed before an arithmetical number it denotes only the arithmetical root, but when the root symbol is placed before an algebraical expression it denotes one of the roots. Thus ^/a has two values but f^2 is only supposed to denote the arithmetical root, unless it is written ± \/2. SURDS. 215 172. Any rational quantity can be written in the form of a surd. For example, 2 = ^4 = ^8 = ^2", and a = ^a' = ^a^ = J^a^ Also, since V^ x ^6 = y/ab [Art. 165], we have 2^2 = ^4x^2== \/(4 x 2) = ^S, 5 J/3 = ^5' X ^3 = ^(5^ X 3) = ^375, and aZ^= "Ja" x J^/ab = ^(a" x ab) = ^a^. Conversely, we have V18 = V(9 x 2) = V9 x V2 = 3^2, and ^135 + ^^40 = ^(3^x5) +^(2^x5) = 3^5 + 2^5 = 5 75. 173. Any two surds can be reduced to surds of the same order. For if the surds be ^a and *l/b, we have ^a = '7a'", and 76 = "76" [Art. 165]. Ex. Which is the greater, ^14 or 4/6? The surds must be reduced to equivalent surds of the same order. Now 4/14^:4/142=4/196, and 4^6 = 4/63=4^216. Hence, as ^216 is greater than 4^196, 4^6 must be greater than 4^14. Thus we can determine which is the greater of two surds without finding either of them. 174. The product of two surds of the same order can be written down at once, for we have yaxyb= J^ab. Hence, in order to find the product of any number of surds, the surds are first reduced to surds of the same order ; their product is then given by the formula ;yax :y6x :^c,.,= ;/abc.,. Ex. 1. Multiply ;^5 by 4/2. n/5 X 4/2 = 4/53x 4'22=4/(68x 22) = 4/500. Ex. 2. Multiply 3^5 by 24/2. 3^5 X 24/2 = 3 x2x<^5x 4/2 = 6x4/53x4/22=64/500. 216 MULTIPLICATION OF SURDS. Ex. 3. Multiply ^/2 by 4^2. V2 X ^2 = 4/2» X ^23=^25^22 =4/32. Or thus: ^2x^/2=2^ x 2i = 2Hi = 2^ = 4/i:5. ^ Ex. 4. Multiply J2 + ^S by V3 + v/5. (^/3 + ^/2)(V3^-^/5)=^3 X V3 + ^/2 X V3 + v/3 X ^^5 + ^2 X 5 = 3+VG + Vl5 + x/10. Ex. 5. Divide 4/4 by ^8. < /4a '32 4'4^^8 = 4/4=^4'8'=^g=^4 175. The determination of the approximate value of an expression containing surds is an arithmetical rather than an algebraical problem ; but an expression containing surds must always be reduced to the form most suitable for arithmetical calculation. For this reason when surds occur in the denominators of fractions, the denominators must be rationalized. [See Art. 169.] The following examples will illustrate the process: 2^_ 2x^/5 2 v/5~^5x^5 5 J5. 3 _ 3(^5 + 1) _3(^,^,j. ^/o-l (^5-l)(^5 + l) 4 = I(n/3~1)(n/5-1). 1 + V3 + V5 + V15 (l + ^3)(l + ;^5) 8 176. The product and the quotient of two similar quadratic surds are both rational. This is obvious; for any two similar quadratic surds can be reduced to the forms a^b and C\/6- Conversely, if the product of the quadratic surds V« and ^/b is rational and equal to x, we have x = \lax \/b', therefore x\Jb = \/a x \/b x \/b = b\la, which shews that the surds are similar. So also, if ^a -J- >Jb is rational, the surds must be similar. SURDS. 217 177. The following theorem is important. Theorem. If a + V^ = ^ + Vy? where a and x are rational, and sjh and xjy are irrational; then will a — x^ and b = y. For we have a — a? + V^ == ^/y. Square both sides ; then, after transformation, we have 2(a — x)^/h — y-b -(a- xf. Hence, unless the coefficient of s/h is zero, we must have an irrational quantity equal to a rational one, which is impossible. The coefficient of sjh in the last equation must there- fore be zero, so that a = x. And when a = x, the given relation shews that i^h — isjy, and therefore h = y. As a particular case of the above, Aja^h-\- sICy unless 6 = and a — c. Hence si a + sjc can only be rational when it is zero. Ex. 1. Shew that Ja+^h + ^c^i^O, unless the surds are all similar. For we should have Ja + ^b = -^c; and therefore a + b + 2 Ja^h = c. Hence Jajh is rational, which shews [Art. 176], that Ja and ^6 are similar surds. 178. The expressions a-\-i^h and a — sjh are said to be conjugate quadratic surd expressions. It is clear that the sum and the product of two conju- gate quadratic surd expressions are both rational. Conversely, if the sum and the product of the expres- sions a + V^ ai^d c + /s/d are both rational, then a = c and i^b -\- Ajd=0, so that the two expressions are conjugate. For a + c + \/6 + V^ can only be rational when ^/b + ^/d is zero. [Art. 177.] And, when ^/d^-^b, the product (a + s/b)(c+ ^d) = ac -\- {c — a) %/b — b, which cannot be rational unless c = a. 179. In the expression CM?" + bx""'' + c^""" + + k, where a, b, c, k are all rational, let a-^-sJ^ be substi- 218 SURDS. tuted for x\ and let P be the sum of all the rational terms in the result and Q s/^ the sum of all the irrational terms. Then the given expression becomes P + Q s/fi. Since P and Q are rational, they contain only squares and higher even powers of VA and hence P and Q will not be changed by changing the sign of VA Therefore when a — Vy^ is substituted for x in the given expression the result will be P - Q VyS. If now the given expression vanish when a + s/^ is substituted for a?, we have P + OViS = 0. Hence, as P and Q are rational and tj^ is irrational, we must have both P = and Q = ; and therefore P-Qv'^ = o. Therefore if the given expression vanish when a + ^JP is substituted for x it will also vanish when a — V/^ is sub- stituted for X. Hence [Art. 88], if a; — a — V/S be a factor of the given expression, x — a-\- V/3 will also be a factor. Thus, if a rational and integral expression he divisible by either of two conjugate quadratic surd expressions it will also be divisible by the other, 180. The square root of a binomial expression which is the sum of a rational quantity and a quadratic surd can sometimes be found in a simple form. The pro- cess is as follows. To find sJCa + \Jh), where »Jb is a surd. Let \/{a + s/b) = *^x + hjy. Square both sides ; then a-\-slb = x-\-y-\- ^Jxy. Now, since sjb is a surd, we can [Art. 177] equate the rational and irrational terms on the different sides of the last equation; hence x + y — a, and ^xy — 6, SURDS. W 219 Hence x and y are the roots of th 4 and these roots are J (a + V(a" - &)} and Hg-J/ (^' - &)}. equation Thus V(aW6)VM"^^H7^=^^- It is clear that, unless \/(ot^ ~ ^) is rational, the right side of the last equation is less suitable for calculation than the left. Thus the above process fails entirely unless a'' — 6 is a square number; and as this condition will not often be satisfied, the process has 4ot much practical utility. It should be remarked that if x and y are really rational, they can generally be written down by inspection. Ex.1. Find ^(6 + 2^5). Let Mj{& + ^fJ5) = ^x + Jy. Then, by squaring, we have 6 + 2/^5 ssx + y + 2ijxy' Hence, equating the rational and irrational parts, 05+1/ = 6 and xy = 5. Whence obviously x = l and y = 5. Thus ^(6 + 2V5) = l + ^/5. Ex.2. FindV(28-5J12). Let ;^(28 - 5 J—a^ = ai-j it is therefore only necessary to use one imaginary expression, namely J—1, 185. Let XOX\ YOY' be two rectangular axes. [See Chapter X*.] Then, any positive or negative real quantity oo, is represented geometrically by choosing any fixed length for unit and laying off a length OM=x units, measured from the fixed point in the direction OX or its opposite according as x is positive or negative ; and we may con- sider that the quantity x is represented either by the position of the point M, or by the straight line OM. 223* COMPLEX QUANTITIES. Again, any purely imaginary quantity iy, is represented by laying off a length ON — y units, measured from in the direction F or its opposite according as y is positive or negative; and we may consider that the imaginary quantity iy is represented by the point iV, or by the length OK Y N P ."^^ X' M X In order to represent the complex quantity (c + iy, complete the rectangle MONP, then we shall consider that the point P, as also the straight line OP, represents 00 -^-iy. Let r denote the absolute length of OP, and ^ the angle XOP. [The angle AOB in Trigonometry always means the angle described by a line revolving from OA to 05, the angle being positive when the revolution is in a counter-clockwise direction.] Then a; = r cos ^, y = r sin 6,x-\- iy = r (cos 6 -{-% sin d\ Also r = isj{x' + 2/') and 6 = tan-^ ^ . X Thus the complex quantity x + iy can be obtained by taking a length equal to r units along OX, where r is the positive quantity n/(«^ + y^)t and then turning through an angle tan-^ yfx. COMPLEX QUANTITIES. 223** Definitions. The positive quantity r = sJ{a^-\-y^) is called the modulus and the angle 6 is called the argument of the complex quantity x-\-iy. The addition of complex quantities. If two complex quantities x^-^-iyx, x^ + iy^ are represented by OPy OQ, and OR is the diagonal of the parallelogram POQR, it is easily seen that the projection of OR on OX is Xi + X2 and the projection on OF is 3/1 + 3/2 ; from which it follows that OR represents the sum of the complex quantities represented by OP and OQ. Since OR is less than the sum of OP and OQ unless OP and OQ coincide in direction, it follows that the modulus of the sum of two complex quantities is less than the sum of their moduli, unless their arguments are equal. Thus the sum of two complex quantities is obtained geometrically hy adding the straight lines which represent them according to the parallelogram law. The multiplication of complex quantities. By the definition of multiplication, in order to multiply the complex quantity represented by OP by the complex quantity represented by OQ, we have to do to OP what was done to the unit to obtain OQ, that is to say we must multiply OP by the modulus of OQ and then turn OP through an angle equal to the argument of OQ. Thus the product of two complex quantities is the product of their moduli and the argument is the sum of their argu- ments. With the above geometrical representation of complex quantities, due to Argand, it will be seen that they obey the fundamental laws of Algebra. Some of the results of the following Articles follow at once from this geometrical interpretation. 224 CONJUGATE COMPLEX EXPRESSIONS. 186. If a + bi = 0, where a and h are real, we have o = _ l)i. But a real quantity cannot be equal to an im- aginary one, unless they are both zero. Hence, if a + bi = 0, we have both a = and 6 = 0. Note. In future, when an expression is written in the form a + bi, it will always be understood that a and 6 are both real. 187. If a-{-bi^c-^di, we have a — c + (b — d)i = 0; and hence, from Art. 186, a — c—0 and b — d = 0. Thus, two complex expressions cannot be equal to one another, unless the real and imaginary parts are separately equal. 188. The expressions a + bi and a — bi are said to be conjugate complex expressions. The sum of the two conjugate complex expressions a-\-bi and a — 6i is a + a + (6 — 6)^ = 2a; also their pro- duct is oa + ahi — abi — 6V = a^ + 6^ Hence the sum and the product of two conjugate complex expressions are both real. Conversely, if the sum and the product of two complex expressions are both real, the expressions must be con- jugate. For let the expressions be a + bi and c + di. The sum is a H- 6i + c + a5i = a H- c + {b-\-d)iy which cannot be real unless 6 + d = 0. Again, (a + bi) (c + di) = ac+bci+adi 4- bdi^ = ac — bd-^ (be + ad)i, which cannot be real unless bc + ad = 0. Now, if 6 4- c? = and also bc-\- ad = 0, we have b (c — a) = ; whence a = c or 6 = 0. If 6 = 0, c^ is also zero, and both expres- sions are real ; and, if 6 =|= 0, we have a = c, which with b — — d, shews that the expressions are conjugate. 189. The modulus of the complex quantity a + i6, namely the positive value of the square root of \/(a' + b^) MODULUS OF A COMPLEX EXPRESSION. 225 is written mod (a + bi). Thus mod (a + bi) = + \/a^ + b\ It is clear that two conjugate complex expressions have the same modulus ; also, since (a + bi) (a — bi) = d^ + }^ [Art. 188], the modulus of either of two conjugate complex expressions is equal to the positive square root of their product. Since a and h are both real, a* -f })^ will be zero if, and cannot be zero unless, a and 6 are both zero. Thus the modulus of a complex expression vanishes if the expression vanishes, and conversely the expression will vanish if the modulus vanishes. If in mod (a + bi) = + >J a^ + b^ we put 6 = 0, we have mod a = + s/ a*, so that the modulus of a real quantity is its absolute value. 190. The product of a + bi and c + di is ac + bci + ddi + bdi^ — ac — bd-\- (be + ad) i. Hence the modulus of the product of a + bi and c + di is y/{(ac - bdy + {be + ady] = V{(a' + 6'0 (c' + d')} = ^/(i(l' + b')x^/{c'-hd'). Thus the modulus of the product of two complex expressions is equal to the product of their moduli The proposition can easily be extended to the case of the product of more than two complex expressions ; and, since the modulus of a real quantity is its absolute value, we have the following Theorem. The modulus of the product of any number of quantities whether real or complex, is equal to the product of their moduli. 191. Since the modulus of the product of two com- plex expressions is equal to the product of their moduli, it follows conversely that the modulus of the quotient of two expressions is the quotient of their moduli. This may also be proved directly as follows : 226 MODULUS OF A PRODUCT. / 7 -x / 7 .\ a + bi c — (a + bi) ^ (c + dz) = —— p X c + di c — di _ac + bd + (bo — ad) % ~ TTd' • Hence mod j^'l = ^{(-o^WHho-adf} [c + di) c^ + dr _ is/\^_+^ _ mod(a + 6i) " V{c' + d'] ~ mod (c + c^i) * 192. It is obvious that in order that the product of any number of real factors may vanish, it is necessary and sufficient that one of the factors should be zero, a,nd, by means of the theorem of Art. 190, the proposition can be proved to be true when all or any of the factors are complex quantities. For, since the modulus of a product of any number of factors is equal to the product of their moduli, and since the moduli are all real, it follows that the modulus of a product cannot vanish unless the modulus of one of its factors vanishes. Now if the product of any number of factors vanishes its modulus must vanish [Art. 189] ; therefore the modu- lus of one of the factors must vanish, and therefore that factor must itself vanish. Conversely, if one of the factors vanishes, its modulus will vanish; and therefore the modulus of the product and hence the product itself must vanish. 193. In the expression where a,b,c,...k are all real, let a + /3i be substituted for a?, and let F be the sum of all real terms in the result, and Qi the sum of all the imaginary terms. Then the given expression becomes P + Qi. Since P and Q are both real, they can contain only MODULUS OF A PRODUCT. ^ 227 squares and higher even powers of i, and hence P and Q will not be changed by changing the sign of i. Therefore when a — ^i is substituted for a; in the given expression the result will be P — Qi. If now the given expression vanishes when a + ^i is substituted for a?, we have F + Qi = 0. Hence, as P and Q are real, we must have both P = and Q = 0, and therefore P—Qi= 0. Hence if the given expression vanishes when a + jBi is substituted for x, it will also vanish when a — fii is substituted for x. Therefore [Art. 88] if x — a-jSi is a factor of the given expression, x — a-\-^i will also be a factor. Thus, if any expression rational and integral in x, and with all its coefficients real, he divisible by either of two conjugate complex expressions it will also he divisible by the other. CHAPTER XY. Square and Cube Roots. 194. We have already shewn how to find the square of a given algebraical expression; and we have now to shew how to perform the inverse operation, namely that of finding an expression whose square will be identically equal to a given algebraical expression. It will be seen that our knowledge of the mode of formation of squares will enable us in many cases to write down by inspection the square root of a given expression. 195. From the identity a' ± 2ah + 6^ = (a ± b)\ we see that when a trinomial expression consists of the sum of the squares of any two quantities plus (or minus) twice their product, it is equal to the square of their sum (or difference). Hence, to write down the square root of a trinomial ex- pression which is a perfect square, arrange the expression according to descending powers of some letter ; the square root of the whole expression will then be found by taking the square roots of the extreme terms with the same or with different signs according as the sign of the middle term is positive or negative. Thus, to find the square root of - 4a*'-12a*6''4-96*. SQUARE ROOT. -^^ 229 The square roots of the extreme terms are ± 2a* and + 86^ Hence, the middle term being negative, the re- quired square root is ± (2a* — 36^). Note. In future only one of the two square roots of an expression will be given, namely that one for which the sign of the first term is positive: to find the other root all the signs must be changed. 196. When an expression which contains only two different powers of a particular letter is arranged accord- ing to ascending or descending powers of that letter, it will only consist of three terms. For example, the expression a^ + 6' + c^ + 26c 4- 2ca 4 2ah when arranged according to powers of a is the trinomial a' + 2a (6 + c) + (¥ + c^ + 26c). It follows therefore from the preceding article that however many terms there may be in an expression which is a perfect square, the square root can be written down hy inspection, provided that the expression contains only two different powers of some particular letter. Ex. 1. To find the square root of a^ + b^ + c^ + 2bc + 2ca + 2db. Arranged according to powers of a, we have a2 + 2a(6 + c) + (6 + c)2, thati8{a+(& + c)}«. Hence the required square root is a + 6 + c. Ex. 2. To find the square root of 4tx^ + V + 16^* + 12a;2t/2 _ iQxh^ - 24.y'^z\ The given expression is 4«4 + 4x2 (3^2 _ 4^3) + 9^4 _ 242/2^2 + 1824, that is, {2a;2)a + 2 (2x2) (3j/2 - 4^2) + (3i/2 - 4^2)3^ which is { 2x2 + ( 3^2 _ 4^2) j 2. Hence the required square root is 2x^ + 3y2 - 4^*. n 230 SQUARE ROOT. Ex. 3. To find the square root of a? + lahx + (62 + 2ttc) «« + 26ca;» + c^x*. Arrange according to powers of a ; we then have a2 + 2a (ftx + cx2) + 62a;2 + 2&cjk3 + c^S that is, a2 + 2a (6x + ct?) + (6a; + cx^)\ Hence the required square root is a+6x + ca;2. Ex. 4. To find the square root of «« - 2a;'' + 3x4 + 2a;3 (i/ - 1) + a;2 (1 - 2y) + 2x?/ + 1/«. The expression only contains y^ and y ; we therefore arrange it according to powers of y, and have ya + 2?/ (a;3 _ a;2 + x) + a;6 - 2a;5 + Sx* - 2a;' + x^. Now, if the expression is a complete square at all, the last of the three terms must be the square of half the coefficient of y ; and it is easy to verify that (aH» - a;2 + a;)2 = a;8 - 2a;6 + 3 J.4 _ 2aa + x«. Hence the required square root is y-\-x^-Q^-\-x. 197. To find the square root of any algebraical ex- pression. Suppose that we have to find the square root of (J. 4- -B)*, where A stands for any number of terms of the root, and B for the rest; the terms in A and B being arranged accord- ing to descending (or ascending) powers of some letter, so that every term in A is of higher {or lower) degree m thai letter than any term of B. Also suppose that the terms in A are known, and that we have to find the terms in B. Subtracting A^ from {A + B)^, we have the remainder {^A^B)B. Now from the mode of arrangement it follows that the term of the highest (or lowest) degree in the remainder is twice the product of the first term in A and the first term mB. Hence, to obtain the next term of the required root, that SQUAEE ROOT. 231 is, to obtain the highest (or lowest) term of B, we subtract from the whole expression the square of that part of the root which is already found, and divide the highest {or lowest) term of the remainder by twice the first term of the root. The first term of the root is clearly the square root of the first term of the given expression; and, when we have found the first term of the root, the second and other terms of the root can be obtained in succession by the above process. For example, to find the square root of «« - 4a;5 + 6iB* - 8x^ + 9x^ - 4a; + 4. The process is written as follows : x» - 4^^ + 6x^ -8a^ + 9x^ -ix + i ( x^ -2x^ + x-2 {x^)^ = x^ ^ {x^ - 2x2)2 =^«j-4^«j+4^ {x^ - 2a;2 + a;)2 = x^- 4a;g + (ix^ - 4a;3 + x^ {x^ -2a^ + x -2)^ = x^ - ^+Qx* -8x^ + 9x^ - 4x + i We first take the square root of the first term of the given expression, which must be arranged according to ascending or de- scending powers of some letter: we thus obtain a^, the first term of the required root. Now subtract the square of x^ from the given expression, and divide the first term of the remainder, namely -ix^, by 2x^: we thus obtain - 2x^, the second term of the root. Now subtract the square of a:*-2a;2 from the given expression, and divide the first term of the remainder, namely 2x'^, by 2a^ ; we thus obtain x, the third term of the root. Now subtract the square of a^ - 2a;2 + x from the given expression, and divide the first term of the remainder, namely - 4a;*, by 2a^^ : we thus obtain - 2, the fourth term of the root. Subtract the square of a^-2x^ + x-2 from the given expression and there is no remainder. Hence x^ - 2^2 + a; - 2 is the required square root. The squares of a^, x^-2x'\ &c. are placed under the given expression, like terms being placed in the same column, so that in every case the first term of the remainder is obvious. 198. The square root of an algebraical expression may also be obtained by means of the theorem of Art 91. Take for example the case just considered. 232 SQUARE ROOT. The required root will be ax^ + ba^ -^ ex + d, provided that the given expression is equal to (ax^ + bx^ -\- cx + df^ that is equal to d'x^ + 2a6a;' + (2ac + 6') a?* + 2 {ad + be) a? + {%d + c') x^ + 2cdx + d\ Hence, equating the coefficients of corresponding powers of x in the last expression and in the expression whose root is required, we have tt'^^l; 2a6 = -4; 2ac + 6''=6; 2ac^ + 26c = -8; . 26c^ + c' = 9; 2ccZ = -4; (f = 4. The first four of these equations are sufficient to determine the values of a, 6, c, d; these values are (taking only the positive value of a), a = 1, 6 = — 2, c = 1, (i = — 2. The last three equations will be satisfied by the values of a, 6, c, d found from the first four, provided the given expression is a perfect square, which is really the case. Thus the required square root is a?' — 2a;^ + x — 2. 199. Extended Definition of Square Root. The definition of the Square Root of an algebraical expression may be extended so as to include the case of an expression which is not a perfect square. For, although an expres- sion may not be a perfect square, we can find, by the methods of Art. 197 or Art. 198, a second expression whose square is equal to the given expression so far as certain terms are concerned. Thus the square root of x^ + 2x may be said to be x-\-l, (x + iy being equal to x'^-\-2x so far as the terms which contain x are concerned. Again, the square root of 1 + a? may be said to be X X OC/ l-f-orl+^— — , the square of the former differing from 1+ X by — , and the square of the latter differing by 4! SQUARE ROOT. 238 OR — Jd?' + ^^a?*. Thus, provided x is small, 1 + s is an approximation to the sq uare root of 1 4- a;, and 1 + 5 — -q is a closer approximation, and by continuing the process we can approximate as closely as we please to the square root of 1 -f a? ; this however is by no means the case when X is not a small quantity. 200. When any number of terms of a square root have been obtained as many more can be found by ordinary division. For suppose the expression whose square root is to be found is the square of (aja?"+ a,aj"-* + . . . + a^""^') + (ar+i^""' + • • • + a«X""^') + -K. The coefficients a^, a^,,..a^ can be found by equating the coefficients of the first 2r powers of x in the square of the above to the coefficients of the corresponding powers of X in the given expression. The square of the above expression is ia/" + a,a;""' + . . . + a^"^')' + ^ (ai^?" + . . . + a^"^') + 2R (a^^.x'^^ + . . . + a^x''-^'^') + R']. Now, since the highest power of a; in -B is a?**'*^, the highest power of x in the expression within square brackets is a;'^"-^. Hence the expression within square brackets will not affect any of the terms from which a^, a,, ...a^ are deter- mined, for the first 2r terms of the given expression ex- tend from ar"* to oT-^K It therefore follows that, if the square of the sum of the first r terms of the root be subtracted from the given S. A. 17 234 SQUARE ROOT. expression, and the remainder be divided by twice the sum of the first r terms, the quotient will give the next r terms of the root. 201. When n figures of a square root of a number have been found by the ordinary method, n — 1 more figures can be fownd by division, provided that the number is a perfect square of '2,n—l figures ; if however this be not the case, there may be an error in the last figure. Let N be the given number, which is the perfect square of a number containing 2n — 1 figures, and let p be the number formed by the first n figures followed by n—1 zeros, and let q be the number formed by the remaining w — 1 figures. Then ^N=p + q; .-. (N-f)/2p = q + qy2p. Now 2p > lO*'"'. Hence qy2p must be a fraction ; whence it follows that if p^ be subtracted from N and the remainder be divided by 2p, the integral part of the quotient will be q. Next, let ^N contain m figures, where m is greater than 2n — l. Let p be the number formed by the first n figures of the root followed by m — n zeros, let q be the number formed by the next n — 1 figures followed by m — 2n + 1 zeros, and let r be the number formed by the m — 2n-\-l remaining figures. Then N^^ip + q-^rY; .'. (N-p')l2p - ^ = (5^ + r* + 2qr)l2p + n Now 10"* >p ^ lO'^'S 10""" >q^ 10"-"-*, and 10'""^**-''>rH:lO"-^ CUBE ROOT. 235 whence it follows that {(f + r^ + 2qr)/2p is less than 1 f\m-2n+l Hence (q" + r'^ + 2qr)/2p + r is less than 2 x i0'"-2»+\ but it is not necessarily less than 10"*"*"'*'\ Hence {N-p')l2p may differ from q by more than lO*"-^"^^; it must however differ by less than 2 x lO*""^""*"^; so that the n — 1 first figures of the quotient (iV^ — p*)/2p are either the n—1 figures of q or differ only in the last figure, and in that case by 1 in excess. Cube Root. 202. From the identity (a + by = a' + Sa'h + Sah' + h\ we see that the cube of a binomial expression has four terms, and that when the cube is arranged according to ascending or descending powers of some letter, the cube roots of its extreme terms are the terms of the original binomial. Hence the cube root of any perfect cube which has only four terms can be written down by inspection, for we have only to arrange the expression according to powers of some letter and then take the cube roots of its extreme terms. For example, if 27a'' - 54a''6 + 36a*62 _ SaSftS is a perfect cube its cube root must be Sa^ - 2ah ; and by forming the cube of Sa^ - 2ab it is seen that the given expression is really a perfect cube. When an expression which contains only three different powers of a particular letter is arranged according to powers of that letter, there will be only four terms. It therefore follows that however many terms there may be in an expression which is a perfect cube, the cube root can be written down by inspection, provided that the 17—2 236 CUBE ROOT. expression contains only three different powers of some particular letter. For example, to find the cube root of a3 + 6» + c» + 3a2& + Sa^c + Sab^ + Sac^ + 6ahc + 3h^c + 36c». Arranged according to powers of a, we have a8 + 3a2 (6 + c) + Sa {b^ + c^ + 2bc) + b^ + c^ + Sb^c + 36c», that is, 08 -f- 3a» (6 + c) + 3a (6 + c)2 + (6 + c)«. Hence the required root is a + b + c. 203. To find the cube root of any algebraical expression. Suppose we have to find the cube root of {A + Ef, where A stands for any number of terms of the root, and B for the rest; the terms in A and B being arranged according to descending (or ascending) powers of some letter, so that every term of A is of higher (or lower) degree in that letter than any term of B. Also suppose the terms in A are known, and that we have to find the terms in B. Subtracting A^ from {A + B)^, we have the remainder {^A^ + ^AB + B')B. Now from the mode of arrangement it follows that the term of the highest (or lowest) degree in the remainder is 3 X square of the first term oi A x first term of B. Hence to obtain the next term of the required root, that is, to obtain the highest (or lowest) term of B we subtract from the whole expression the cube of that part of the root which is already found and divide the highest {or lowest) term of the remainder by three times the square of the first term of the root This gives a method of finding the successive terms of the root after the first; and the first term of the root is clearly the cube root of the first term of the given expression. CUBE BOOT. 237 For example, to find the cube root of The process is written as follows : a;6 _ Q^y + 21x*y^ - 44a;V + 63a;2t/4 _ 543,^^3 + 271/6 (a;2)»=x« (a;2 - 2xy + 3y 2)3 = a;6 _ Gar^y + 21a;^2 _ 44a;3p + 63a;'V4 _ 543.^3 4. 27y», Having arranged the given expression according to descending powers of x, we take the cube root of tlie first term : we thus obtain x^t the first term of the required root. We then subtract the cube of x^ from the given expression, and divide the first term of the remainder, namely - Qxh/, by 3 x (052)2 . we thus obtain - 2xy, the second term of the root. We then subtract the cube of 052 - 2xy from the given expression, and divide the first term of the remainder by 3 x (a;2)2 ; this will give the third term of the root. Note. The above rule for .finding the cube root of an algebraical expression is rarely, if ever, necessary. In actual practice cube roots are found as follows. Take the case just considered ; the first and last terms of the root are x^ and Sy^, the cube roots of the first and last terms of the given expression ; also the second term of the root will be found by dividing the second term of the given expression by 3 x (aff, so that the second term of the root is — 2a;y. Hence, if the given expression is really a perfect cube, it must be (a?" — 2xy + Syy, and it is easy to verify that (x^ — 2xy + Zy^ is equal to the given expression. Again, to find the cube root of x' - Qx'y + 15*Y - 29^y + 51a;y - eOxY + 64«y -6SxY + 2'7xy'-27y\ If the given expression is really a perfect cu be the first and last terms of the root must be y^a;' and \/— 27y^ respectively, that is x^ and — 3y^. 238 EXAMPLES. The second term of the root must be — Qx^y -r- 3 (a;')* = — 2a^y ; and the term next to the last must be 27^' - 3 (-Sj/-)' = + «;/. Hence the given expression, if a cube at all, must be (a;* — 2x^y + xy^ — 3^")' ; and by expanding (a^ - ^y + xy^ — Z^y it will be found that the given expression is really a perfect cube. 204. From the identity [see Art. 253] (a + 6)" = a** + n(f~^}> + terms of lower degree in a, it is easy to shew, as in Articles 197 and 203, that the n*^ root of any algebraical expression can be found by the following Rule. Arrange the expression according to descending or ascending powers of some letter, and take the n^^ root of the first term : this gives the first term of the root. Also, having found any number of terms of tiue root, subtract from the given expression the n*^ power of that part of the root which is already found, and divide the first term of the remainder by n times the (n — iy^ power of the first term of the root : this gives the next term of the root. EXAMPLES XIX. Write down the square roots of the folio wing expressions i 1. 4a;'"'-12a;y + V- 2. aj» + 9xy»-6a;y. 3. a' + 46* 4- 9c* + 1 26c - Qca - ^ah. 1 25a* + 96* + 4c* + 1 26V - 20c*a'' - 30a»6'. EXAMPLES. 239 Find the square roots of 5. x' + 2x' + 3x* + ^x' + 3x' + 2x+l. O 6. ^x* - 8a^y + 4032^' + y*. 7. 49 + 112aj^ + 70^' + 64a;* + 80a;» + 25x'. 8. a;* - 2x^ + 5x'-6x + 8- Qx'' + dx'' - 2a;-" + a;-\ y" 25a;* y 5a; 10. a;* - 4a;^ + 2a; + 4a;^ + a;* . 11. x^ - 4a;* + 4a; + 2a;^ - 4a;^ + x^. 12. a;^ - 2a"'^a;'^ + 2a^a;^ + a'^a;^* - 2a^a;^ + a\ Find the cube roots of 13. a;« - 24a;' + 192a; -512. 14. x' - Sx'y + 6xy - 7a;y + 6xY - 3xy' + y\ 15. 1 - 9a;' + 33a;* - 63a;« + 66a;'' ~ Sea;^" + 8a;''. 16. Find the square root of 2a' (6 + c)' + 26' (c + a)' + 2c' (a + 6)' + 4a6c (a 4- 6 + c). 17. Find the square root of aj* (a;* + 2/' + «') + 2/'«' + 2a; (2/ + z) (yz - a;'). 18. Find the square root of (a - 6)* - 2 (a' + 6') {a -hf^2 (a* + 6*). 19. Shew that {x + a) (a; + 2a) (a: 4 3a) {x + 4a) + a* is a perfect square. 20. Prove that x* + px^ + ja;' f ra; + 8 is a square, if jp'« = r* and ^^ - ipq + 8r = 0. 240 EXAMPLES. 21. Find the vaiues of ^, ^ and C in order that 4a;*'-24a;'* + Ax'' + Bvi" + Cx'-40x + 25 may be a perfect square. 22. Sliew that, if aoif + bx* + cx + d he a perfect cube, then 6" = 3acandc'=36d 23. Find the conditions that oaf + by' + cz' + 2fyz + 2gzx + 2hxy may be the square of an expression which is rational in a;, y and z. 24. Shew that if (a -X)x' + {b- X) y" -I- (c - X) z' + 2fyz + 2gzx + 2hxy be the square of an expression which is rational in. Xy y and z, then will ' gh . hf_ fg f 9 h 25. Shew that when the first r terms of the cube root of an algebraical expression are known, r more terms can be found by ordinary division. 26. When n + 2 figures of the cube root of a number have been obtained by the ordinary method, n more can be obtained by ordinary division, provided the number is a perfect cube of 2?/ + 2 figures. 27. Shew that, if n + 2 figures whose numerical value is a have been found of a positive root of the equation 0^ + qx — T = Oj q being supposed positive, then the result of dividing r-qa-a^ by Za' + q )vill give at least ?* - 1 more figures correctly. \ CHAPTER XVL Ratio. Prcportion. 205. Definitions. The relative magnitude of two quantities, measured by the number of times the one contains the other, is called their ratio. Concrete quantities of different kinds can have no ratio to one another : we cannot, for example, compare with respect to magnitude miles and tons, or shillings and weeks. The ratio of a to 6 is expressed by the notation a : h; and a is called the first term, and b the second term, of the ratio. Sometimes the first and second terms of a ratio are called respectively the antecedent and the consequent It is clear that a ratio is greater, equal or less than unity according as its first term is greater, equal or less than the second. A ratio which is greater than unity is sometimes called a ratio of greater inequality, and a ratio which is less than unity is similarly called a ratio of less inequality. The ratio of the product of the first terms of any number of ratios to the product of their second terms, is called the ratio compounded of the given ratios. Thus ae:hd\& the ratio compounded of the two ratios a : h and c : d. The ratio a^ : ¥ is sometimes called the duplicate ratio of a : 6 ; so also a^ : 6°, and ^/a : sjh are called respectively the triplicate, and the sub- duplicate ratio of a : 6. 242 RATIO. 206. Magnitudes must always be expressed by means of numbers^ and the number of times which one number contains another is found by dividing the one by the other. Thus ratios can be expressed d^ fractions. The principal properties of fractions and therefore of ratios have already been considered in Chapter viii. Thus, a ratio is unaltered in value by multiplying each of its terms by the same number. [Art. 107.] Different ratios can be compared by reducing to a common denominator the fractions which express their values. [Art. 109.] The theorems of Art. 113 are also true for ratios. The following theorem is of importance : 207. Theorem. Any ratio is made more nearly equal to unity by adding the same positive quantity to each of its terms. By adding a; to each term of the ratio a : b, the ratio a-^x : b + x is obtained. T^T a - a — b , a-ho) - a — b Now T — 1 = —7 — , and j—, 1 = t— — , b b b + x 6 + aj and it is clear that the absolute value of v is less than b -¥x t that of , , for the numerators are the same and the denominator of the former is the larger : this proves the proposition. When X is very great, the fraction r is very small ; Snd q , which is the difference between f and 1, b-\-x b+x ^an be made less than any assignable difference by taking sufficiently great. RA.TIO. 243 This is expressed by saying that the limiting value of r , when x is infinite, is unity. Now two quantities, whether finite or not, are equal to one another when their ratio is unity. Thus a-\-x and h-\-x are equal to one another when x is infinite, a being supposed not equal to h. [See Art. 118.] 208. Since any ratio is made more nearly equal to unity by the addition of the same quantity to each of its terms, it follows that a ratio is diminished or increased by such addition according as it was originally greater or less than unity. This proposition is sometimes enunciated : A ratio of greater inequality is diminished and a ratio of less iiiequality is increased by the addition of the same quantity to each of its terms. 209. Incommensurable numbers. The ratio of two quantities cannot always be expressed by the ratio of two whole numbers ; for example, the ratio of a diagonal to a side of a square cannot be so expressed, for this ratio is V2 : 1, and we cannot find any fraction which is exactly equal to V2. Magnitudes whose ratio cannot be exactly expressed by the ratio of two whole numbers, are said to be in- commensurable. Although the ratio of two incommensurable numbers cannot be found exactly, the ratio can be found to any degree of approximation which may be desired ; and the different theorems which have been proved with respect to ratios can, by the method of Art. 163, be proved to be true for the ratios of incommensurable numbers. 244 proportion. Proportion. t 210. Four quantities are said to be proportional when the ratio of the first to the second is equal to the ratio of the third to the fourth. Thus a, b, c, d are proportional, if a : h = c : d. This is sometimes expressed by the notation a : h :: c : d, which is read " a is to 6 as c is to d.'* The first and fourth of four quantities in proportion, are sometimes called the extremes, and the second and third of the quantities are called the means. 211. If the four quantities a, b, c, d are proportional, we have by definition, a __c b~d' Multiply each of these equals by bd ; then ad = be. Thus the product of the extremes is equal to the product of the means. Conversely, if ad — be, then , a, b, c, d will be propor- tional. For, if ad-- = be, then ad W be ~bd' . ^ ••r ~d' that is a •.b = c :d. PROPORTION. 245 Hence also, the four relations ( a : b = c : d, a '. c =h '. d, h : a = d : c, and b : d = a : c, are all true, provided that ad = be. Hence the four proportions are all true when any one of them is true. Ex. 11 a : b=c : dj then will a + b : a-b==c + d : c-d. This has already been proved in Art. 113: it may also be proved as follows : a+b : a-b=c + d : c-d, if {a + b){c-d) = {a-b)(c + d), that is, if ac-bd + bc-ad = ac-bd-bc + adi or, if be =04. But be is equal to ad, since a : b = e : d. 212, Quantities are said to be in continued proportion when the ratios of the first to the second, of the second to the third, of the third to the fourth, &c., are all equal. Thus a, b, c, d, &c. are in continued proportion if a:b = b:c=c:d — &c., that is, if -=- = _ = &c. bed If a : b = b : c, then b is called the mean proportional between a and c; also c is called the third proportional to a and b. If a, b, be in continued proportion, we have a b ', b^ = ac, or b = Ji ac 246 PROPORTION, Thus the mean proportional between two given quantities is the square root of their product. a b b b Also j- X - = - X - 6 c »c that IS c n Thus, if three quantities are in continued proportion, the ratio of the first to the third is the duplicate ratio of the first to the second. 213. The definition of proportion given in Euclid is as follows: Four quantities are proportionals, when if any equimultiples whatever be taken of the first and the third, and also any equimultiples whatever of the second and the fourth, the multiple of the third is always greater than, equal to or less than the multiple of the fourth, according as the multiple of the first is greater than, equal to or less than the multiple of the second. If the four quantities a, b, c, d satisfy the algebraical test of proportionality, we have t=-j', therefore for all „ J ma mc values 01 m and n, -^ =—3 . no nd > > Hence mc = nd, according as ma = nb. Thus a, b, c, d < < satisfy also Euclid's test of proportionality. Next, suppose that a, 6, c, d satisfy Euclid's definition of proportion. If a and b are commensurable, so that a: b = m: n, where m and n are whole numbers ; then a m V = — ; .*. na = 7uo. b n ' PROPORTION. 24'7 ition ■ > > J = md according as na = mh, < < no — md; ^ c m a " d~ n~b' Hence Thus a, h, c, d satisfy the algebraical definition. If a and b are incommensurable we cannot find two whole numbers m and n such that a :h — m: n. But, if we take any multiple na of a, this must lie between two consecutive multiples, say mh and (m + 1) 6 of 6, so that na > mh and na<{m-\-\)h. Hence by the definition, vo>md and 7ic<(m + l)d Hence both y and ^ lie between — and . a n n Thus the difference between y and ^ is less than - ; and h d n as this is the case however great n may be, -j must be equal to Y , for their difference can be made less than any assignable difference by sufficiently increasing n. Ex. 1. For what value of x will the ratio 1 + x : 12 -fa; be equal to the ratio 5 : 6? Am. 18. Ex. 2. If 6a;' + 62/2 = l3a;y, what is the ratio of x to y? Am. 2 : 3 or 3 : 2. Ex. 3. What is the least integer which when added to both terras of the ratio 5 : 9 will make a ratio greater than 7 : 10? Ans. 6. Ex. 4. Find x in order that x + l : a; + 6 may be the duplicate ratio of 8:6. ^m. |. 248 VARIATION. Ex. 6. Shew that, if a : 6 : : c : d, then (i) a^ + ab + b^ : c^ + cd + d^ :: a^-ab + h^ :c^-cd + cP, (ii) a + 6 : c + d : : ^(2a2 _ 363) . j^2c^ _ 3^2^. (iii) aa+62 + c2 + da ; {a+b)*+{c + d)^ :: (a+c)«+(& + d)2 : (a + 6 + c + d)«. [See Art. 113.] Ex. 6. Ii a : b :: c : d, then will ab + cd be a mean proportional between a^ + c^ and b^ + dK Variation. 214. One magnitude is said to vary as another when the two are so related that the ratio of any two values of the one is equal to the ratio of the corresponding values of the other. Thus, if ctj, a^ be any two measures of one of the quantities, and b^, b^ be the corresponding measures of the other, we have — = r'i and therefore 7^ = 7^ . Hence the measures of corresponding values of the two magnitudes are in a constant ratio. The symbol cc is used for the words varies as: thus AocB is read 'A varies as B \ If a oc 6, the ratio a : 6 is constant ; and if we put m for this constant ratio, we have 5- = m; .*. a = mh. To find the constant m in any case it is only necessary to know one set of corresponding values of a and b. a 15 For example, if a oc 6, and a is 16 when 6 is 6, we have - =m=-v- ; A a =36. VARIATION. 249 215. Definitions. One quantity is said to vary in- versely as another when the first varies as the reciprocal of the second. Thus a varies inversely as h if the ratio a : r is constant, and therefore ab = m. One quantity is said to vary as two others jointly when the first varies as the product of the other two. Thus a varies as h and c jointly if a « be, that is if a = mbc, where m is a. constant. One quantity is said to vary directly as a second and inversely as a third when the ratio of the first to the product of the second and the reciprocal of the third is constant. Thus a is said to vary directly as b and inversely as c, if a : 6 X - is constant, that is, if a = m- , where m is a c c constant. In all the different cases of variation defined above, the constant will be determined when any one set of corresponding values is given. For example, if a varies jointly as 6 and c ; and if a is 6 when b is 4 and c is 3, we have a=mbc, and 6=mx4x3. Hence m=jr , and therefore a=-bc, a It 216. Theorem. If a depends only on b and c, and if a varies as b when c is constant, and varies as c when b is constant; then, when both b and c vary, a will vary as be. Let a, b, c; a', b' , c and a'\ b\ d be three sets of corresponding values. s. A. 18 260 VARIATION. Then, since c is the same in the first and second, we have —, — Tf (i)- a! h And, since V is the same in the second and third, we have — =- (ii). a G Hence from (i) and (ii), — , = tt-, , which proves the proposition. The following are examples of the above proposition. The cost [C] of a quantity of meat varies as the price [P] per pound if the weight [IF] is constant, and the cost varies as the weight if the price per pound is constant. Hence, when both the weight and the price per pound change, the cost vaiies as the product of the weight and the price. Thus, if C QC P, when W is constant, and GolW, when P is constant ; then G oc PW, when both P and W change. Again, the area of a triangle varies as the base when the height Ib constant; the area also varies as the height when the base is constant; hence, when both the height and the base change, the area will vary as the base and height jointly. Again, the pressure of a gas varies as the density when the temperature is constant; the pressure also varies as the absolute temperature when the density is constant ; hence when both density and temperature change, the pressure will vary as the product of the density and absolute temperature. Ex. 1. The area of a circle varies as the square of its radius, and the area of a circle whose radius is 10 feet is 314-159 square feet. What is the area of a circle whose radius is 12 feet ? Ans, 452-38896 feet. Ex. 2. The volume of a sphere varies as the cube of its radius, and the volume of a sphere whose radius is 1 foot is 4-188 cubic feet. What is the volume of a sphere of one yard radius? Am. 113-076 feet. Ex. 3. The distance through which a heavy body falls from rest varies as the square of the time it falls ; also a body falls 64 feet in 2 seconds. How far does a body fall in 6 seconds? Am. 676 feet. Ex. 4. The volume of a gas varies as the absolute temperature and inversely as the pressure; also when the pressure is 15 and the temperature 260 the volume is 200 cubic inches. What will the volume be when the pressure becomes 18 and the temperature 390? Am. 250 inches. INDETEKMINATE FOEMS. 251 Ex. 5. The distance of the offing at sea varies as the square root of the height of the eye above the sea level, and the distance is 3 miles when the height is 6 feet : find the distance when the height is 72 yards. Am. 18 miles. Indeterminate Forms. 217. A ratio or fraction sometimes assumes an in- determinate form for some value or values of a contained letter. Thus, when x=0 both the numerator and the denominator of the fraction -^—— vanish, and the fraction assumes for this value of X the indeterminate form - ; and this is also the case when x=l. Again, when «=qo both the numerator and the denominator of the above fraction become infinitely great, and the fraction assumes the indeterminate form — . 00 We proceed to shew how to find the limiting values of fractions which assume these indeterminate forms. x^ —1 Consider, for example, the fraction -g — -, which as- sumes the form ^ when x=l. Now ^~1_ (^-l)(a^+l) . x'-i'ico-lXx' + x+l)' and, provided x — 1 is not really zero, we may divide the numerator and denominator by a? — 1 without altering the value of the fraction, and we can do this however small x—1 may he. Q^ \ X -\-\ Hence, when a? - 1 is very small, -g — zr = -5 =• , X -~ L X "TT X "X" 1. and the limiting value of the latter fraction, as x ap- 2 preaches indefinitely near to 1, is at once seen to be ^ . 18— 2' 252 INDETERMINATE FORMS. Hence, as x approaches indefinitely near to 1, the x' —1 . . 2 fraction -g — =- approaches indefinitely near to the value ^. a^ — 1 2 This is expressed by the notation X,„, ^^^ — « . x^ — bx + 6 Ex. 1. Find the limiting value of -s — r^j ^^ when x—2, x^- 10a; + 16 It follows from Art. 88 that « - 2 is a common factor of the numerator and denominator. a;2_53; + 6 _ (a;-2)(a;-3) _ a;-3_;_l ^*-« a;a-10a; + 16~ *"« (a;-2) (aj-S)"-^*"* a;-8~6' Ex. 2. Find the limiting value of ^—^ — —- when a;=0 and when « = ao . a;» + 2a; _ a;(a; + 2) _ a; + 2 2 *"•*' 2a;2 + 3x~ *"* a:(2a; + 3j~ *"* 2a; + 3~3* 2 3 since - and - are both zero when x is infinite. X X Ex. 3. Find the limiting value of the ratio l + 2a; : 2 + 3ar when x increases without limit. ^" 2+8.-^«- 7(17!) '■" r:|=»' Ex. 4. Find the limiting value of ^ , „ ^,^ when x becomes 5a^ - 40 indefinitely great. y 2a;a + 100a; + 600 ,. ^ V^ x ^ x^ ) (-:-?) _r 2a;2 2 _ EXAMPLES. 253 EXAMPLES XX 1. Shew that, if a + 6, 6 -»- c, c + a are in continued propor- tion, then b + c ; c + a — c — a i a — h. 2. Shew that, ifa;:a = y:6 = «:c, then a^ y' »* _ (a; + 2/ + 2;)* 3. Shew that, ii {a ->fh •\- c ■¥ d)(a -h - c ■{- d)={a -h + c - d) (a+b-c-d), then a, 6, c, t , - are in arithmetical progression. Thus, if quantities are in harmonical progression, their reciprocals are in arithmetical progression. 231. Harmonic Mean. If a, 6, c be in harmonical progression, - , r, - will be in arithmetical progression. TT 2 11 Hence r = ~ + - > a c , 2ac .*. = a + c Thus the harmonic mean of two quantities is twice their product divided by their sum. If we put A, G, H for the arithmetic, the geometric, and the harmonic means respectively of any two quantities a and b, we have A=i(a + bX G^Jab,S = ^^; Thus the geometric mean of any two quantities is also the geometric mean of their arithmetic and harmonic means. 232. Theorem. The arithmetic mean of two unequal positive quantities is greater than their geometric mean. If a, b be the two positive quantities we have to shew that \{a + h)> Jab, or i (^a - s/bf > 0. EXAMPLES. 267 Now {i\Ja — hjhf is always positive, and therefore greater than zero, unless a = 6. Since the arithmetic mean of two positive quantities is greater than their geometric mean, it follows from Art, 231 that the geometric mean is greater than the harmonic. 233. To insert n harmonic means between any two quantities a and h. Insert n arithmetic means between - and r , and the a reciprocals of these will be the required harmonic means. The arithmetic means are 1 1 /I 1\ 1 _2_/l 1\ o Hence, by simplifying these terms and inverting them, the required harmonic means will be found to be {n + 1) ah {n + 1) ah (n + 1) ah nb + a '(7i-l)6 + 2a* ' b + na 234. It is of importance to notice that no formula can be found which will give the sum of any number of terms in harmonical progression. EXAMPLES XXI. 1. Shew that, if a, 6, c be in a. p., then will a'{h + c), 6" (c + a), c' (a + b) be in A. p. 2. Find four numbers in A. P. such that the sum of their squares shall be 120, and that the product of the first and last shall be less than the product of the other two by 8. 3. If a, 6, c be in a. p., and 6, c, c? be in h.p., then will a: b — c'.d. 4. Find three numbers in g.p. such that their sum is 14, and the sum of their squares 84. 19—2 \ 268 EXAMPLES. \ 5. If a, 6, c be in arithmetical progression, and x be the g^metric mean of a and 6, and y be the geometric mean of 6 and c j then will a;', 6', y" be in arithmetical progi^ession. o. Shew that, if a, 6, c be in harmonical progression, then will 7 , , and — -j , be also in harmonical o-\-c-a c + a-b a + b-c progression. 7. Shew that, if a, 6, c, d be in harmonical progression, then will 3{b-a){d-c) = (c-b){d-a). 8. Shew that, if a, 6, c be in harmonical progression, 2 1 1 6 6 — a 6 — c* 9. Shew that, if a, 6, c be in h.p., then will b+a b+c ^ I + I = 2. b~ a b — c 10. If a, 6, c be in a. p., 6, c, o? in o. p., and c, J, e in ii. p. ; then will a, c, e be in o. p. 11. If a, by c be in h. p., then will a-^, ^,c-^bein g.p. 12. If a, 6, c are in h.p., then a^ a — c, a~b are in ii. P., and also c, c — a, c - 6 are in n. p. 13. If X, CTj, o,, 2/ be in a. p., «, <7j, ^„ 2/ in g.p., and ^» ^i> ^8> y ^^ H. p., then hjh^ h^ + h^' 14. The sum of the first, second, and third terms of a o. P. is to the sum of the third, fourth and fifth terms as 1 : 4, and the seventh term is 384. Find the series. EXAMPLES. 269 15. If a,, ttj, ^3, , a, be in harmonical progression, prove that a^a^ + a^a^ + a^a^ + + a^.i^^ = (n - 1) a^a^ . 16. If a, 05, J/, 6 be in arithmetical progression, and a, w, Vy h be in harmonical progression, then xv= yu = ab. 17. Three numbers are in arithmetic progression, and the product of the extremes is 5 times the mean ; also the sum of the two largest is 8 times the least. Find the numbers. 18. If = Y » ^» 1 r- be in A. p. : then a, ,- » <^ will be 1- ah I -be h in H.p. 19. If a, 6, c be in a. p., and a*, &*, c* be in h.p., prove that - ^ , 6, c are in G. p., or else a — h = c. 20. If aj be any term of the arithmetical progression and y be the corresponding term of the harmonical progression whose first two terms are a, 6, then will x — a : y~a ::h : y. 21. Shew that, if a be the arithmetic mean between h and c, and h be the geometric mean between a and c, then will c be the harmonic mean between a and h. 22. The series of natural numbers is divided into groups as follows: 1; 2, 3; 4, 5, 6 j 7, 8, 9, 10; and so on. Prove that the sum of the numbers in the B^ group is \k {k* + 1). 23. An A. p. and an h. p. have each the first term a, the same last term ?, and the same number of terms n ; prove that the product of the (r + 1)'^ term of the one series and the (n — ry^ term of the other is independent of r. 24. Terms equidistant from a given term of an A. p. are multiplied together ; shew that the differences of the successive teims of the series so formed are in a. p. 25. Shew that, if S^y /S,,, S^^ be the sum of n terms, of 2n terms, and of 3/1 terms respectively of any q.p., then will 26. If a, 6, c be all positive and either in A. p., in g.p., or in H. p., and n be any positive integer, then a" + c" > 26". 270 EXAMPLES 27. If P, Q, R be respectively the p^\ q^'^, and r^ terms (i) of an A. p., (ii) of a g. p., and (iii) of an h. p., then will (i) P(q-T)+Q{T-p) + R{p-q)^0, (ii) P»-'. ^'-'.i?'-' = l, (iii) QR (q-r) + RF (r -p) + PQ (j»-?) =0. 28. Shew that, if a,, aj* %> > », ^e in h.p., then a «, «, a, + a3+ ... +a,' a, + a.+ . ..«/ will be in h. p. '«!+«.+ •••+«.-, 29. Shew that, if «j, a,, a, , a^ be all real, and if (a^ + a^+ + «\-i)K* + V+ +0 = (a,a, + a^a, + + a^^.aj, then will a^, a,, ^g, be in g.p. 30. Shew that any even square, (27i)", is equal to the sum of n terms of one series of integers in a. p., and that any odd square, (2n + 1)^, is equal to the sum of n terms of another a. p. increased by unity. 31. Prove that any positive integral power (except the first) of any positive integer, p^ is the sum of p consecutive terms of the series 1, 3, 5, 7, &c. ; and find the first of the p terms when the sum is p". 32. If an a. p. and a g. p. have the same first term and the same second term, every other term of the a. p. will be less than the corresponding term of the g.p., the terms being all positive. CHAPTER XYIIL Systems of Numeration. 235. In arithmetic any number whatever is repre- sented by one or more of the ten symbols 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, called figures or digits, by means of the convention that every figure placed to the left of another represents ten times as much as if it were in the place of that other. The cipher, 0, which stands for nothing, is necessary because one or more of the denominations, units, tens, hundreds, &c., may be wanting. The above mode of representing numbers is called the common scale of notation, and 10 is said to be the radix or base. 236. Instead of ten any other number might be used as the base of a System of Numeration, that is of a system by which numbers are named according to some definite plan, and of the corresponding Scale of Notation, that is of a system by which numbers are represented by a few signs according to some definite plan ; and to express a number, N, in the scale whose radix is r, is to write the number in the form d^d^d^d^, where each of the digits d^,d^,d^,d^ is less than r, and where d^ stands for d^ units, cZj stands for d^ x r, d, for d^ x r^, and so on. Thus N:=d^ + d^r + d/^ Note. Throughout this chapter each letter stands for a positive integer, unless the contrary is stated. 272 SYSTEMS OF NUMERATION. 237. Theorem. Any positive integer can he expressed in any scale of notation, and this can he done in only one way. For divide N by r, and let Q^ be the quotient and d^ the remainder. Then N = d,-\-rxQ^, Now divide Q^ by r, and let Q, be the quotient and d^ the remainder. Then Q, = c?, + rQ, ; therefore N = d^ + rd^ + r*Q, . By proceeding in this way we must sooner or later come to a quotient, Qn = c?„, which is less than r, when the process is completed, and we have N==d^ + rd^-{- r% + r'cZ, + ^X. so that the number would in the scale of r be written dn d^d^d^d^. Each of the digits d^, d^, d^, is less than r, and any one or more of them, except the last, d^ , may be zero. Since at every stage of the above process there is only one quotient and one remainder the transformation is unique. The given number N may itself be expressed either in the common or in any other scale of notation. Ex. 1. Express 2157 in the scale of 6. The quotients and remainders of the successive divisions by 6 are as under : 6 1 2157 6 1 359 remainder 3=d^ 6j69 6=ii 6 [9 6=^3 1 3=d, Thus 2167 when expressed in the scale of 6 is 13553. Ex. 2. Change 13553 from the scale of 6 to the scale of 8. We have the following successive divisions by 8, remembering that since 13553 is in the scale of 6 each figure is $ix times what it would be if it were moved one place to the left, so that to begin with we have to divide 1x6 + 3, and so on. SYSTEMS OF NUMERATION. 273 8 [13553 8 [1125 remainder 5 8[63 6 4 1 Hence the number required is 4155. Ex. 3. Change 4165 from the scale of 8 to the scale of 10^ Proceeding as before, we have 10 1 4165 10 1 327 remainder 7 10[25 5 2 1 Thus 2157 is the number required. Or thus : Since 4155 = 4 x 88+1x82 + 6x8 + 5= {(4x8 + 1)8 + 5)8 + 5, the required result may be obtained as follows : — Multiply 4 by 8 and add 1 ; multiply this result by 8 and add 5 • then multiply again by 8 and add 5. Ex. 4. Express 3166 in the scale of 12. [Represent ten by t, and eleven by e.] Am. I9et. Ex. 5. Express ^ in the scale of 4. Ans. -— . Ex. 6. In what scale is 4950 written 20301? Ans. 7. 238. Radix Fractions. Radix fractions in any scale correspond to decimal fractions in the ordinary scale, so that 'abc... stands for - + -5 + -s + r r r To shew that any given fraction may be expressed by a series of radix fractions in any proposed scale. Let F be the given fraction ; and suppose that, when expressed by radix fractions in the scale of r, we have F^-abc ^^ + 4 + ^ + , r r* r^ where each of a, 6, c is a positive integer (including zero) less than r. 274 SYSTEMS OF NUMERATION. Multiply by r; then TT be ^xr = a + - + 3- + r r Hence a must be equal to the integral part, and & c - + -^ + must be equal to the fractional part of Fr, (If Fr be less than 1, a is zero.) Let F^ be the fractional part of Fr; then K=l+i+ r r Multiply by r; then Fxr=6+-+ Hence 6 must be equal to the integral part of F^r. Thus a, b, c, can be found in succession. Ex. 1. Express jr- by a series of radix fractions in the scale of 6. Hence '012 is the required result. Ex. 2. Express = by a series of radix fractions in the scale of 3. 13 *i 9 2 fi ^x3=0+^; 2x3=l+f; |x3=0+?; Hence -610212 is the required result. Ex. 3. Change 324-26 from the scale 8 to the scale 6. The integral and fractional parts must be considered separately. SYSTEMS OF NUMERATION. 275 6|324 •26 6 1 43 remainder 2 6 ....„, 6 6 2-04 6 0-30 6 2-20 6 1-40 3-00 Thus the required result is 552-20213. Ex. 4. In the scale of 8 express '16315 as a vulgar fraction. 2^= -16315; .♦. 822^=16-315; A 8W=16315-§lS; 16315 - 16 ^ 16315 - 16 _ 16277 •• ^- 8«-82 77700 "77700* Ex. 5. In the scale of 7 express *23i as a vulgar fraction. ^"^•330- Ex. 6. CSiange 314-23 from the scale of 5 to the scale of 7. Am. 150-3564. 239. Theorem. The difference between any nvmber and the sum of its digits is divisible by r—1, where r is the radix of the scale in which the number is expressed. Let N be the number, S the sum of the digits, and let d^t d^yd^ be the digits. Then N= d^ + rd^ + r^d^ + + rH^, and ^ = (Z^ + cZj + d,+ + d^. :, N-8^{r^l)d,-^{r'-r)d, + + (r«-l)rf^. Now each of the terms on the right is divisible by r-1 [Art. 86]. Hence iV— ^ is divisible by r— 1. Since N — S is divisible by r — 1, iV and S must leave the same remainder when divided by r—1. 276 SYSTEMS OF NUMERATION. Ex. 1. The difference of any two numbers expressed by the same digits is divisible by r- 1. For the sum of the digits is the same for both ; and since N-.-S and N^-S are both divisible by r-1, it follows that Ni-N^ is divisible by r - 1. Ex. 2. Shew that in the ordinary scale a number is divisible by 9 if the sum of its digits be divisible by 9, and by 3 if the sum of its digits is divisible by 3. N-S IB & multiple of 9 ; hence, if 5 be a multiple of 9, so also is N ; and, if S be a multiple of 3, so also is N. Ex. 3. Shew that any number is divisible by r+ 1 if the difference between the sum of the odd and the sum of the even digits is divisible by r + 1. Let N=dQ + djr + d^r^ + d.^r^+ , and D = dff-d^ + d^-dg+ Then N - D = di{r+l) + d^{r^ -l) + d3{r^+l) + Each of the terms on the right is divisible by r + 1 [Art. 87J; .*. N-D is divisible by r + 1. Hence if D is divisible by r + 1 so also ibN. Ex. 4. If Nj^ and N^ be any two whole numbers, and if the remainders left after dividing the sum of the digits in Ni, N^ and in NixN^ by 9 be rij, n, and p respectively ; then will Tijn^ be equal to p, or differ from phy & multiple of 9. For A\ = ni+a multiple of 9, andA^^ = nj4-a multiple of 9; therefore N, x N^ = Wj x n, + a multiple of 9. Hence n^n^ + a multiple of 9 is equal to p+& multiple of 9. If the above is applied in any case of multiplication, and it is found that n^n^ does not equal p, or differ from it by a multiple of 9, there must be some error in the process of multiplication. This gives a method of testing the accuracy of multiplication; the test is not however a complete one, for although it is certain that there must be an error if n^ x Wj does not equal p, or differ from it by a multiple of 9, there may be errors when the condition is satisfied, provided that the errors neutralize one another so far as the sum of the digits in the product is concerned. This is called the *'Rule for casting out tbe nines." Ex. 5. A number of three digits in the scale of 7 has the same digits in reversed order when it is expressed in the scale of 9: find the number. Let a, 6, c be the digits ; then we have 49a + 76 + c = 81c + 96 + a, where a, b, c are positive integers less than 7- Hence 40c + 6 = 24a. EXAMPLES. 277 Now 40c and 24a are both divisible by 8; therefore b must be divisible by 8. But b is less than 7: it must therefore be zero. And since 6 is zero, we have 6c=3a, which can only be satisfied when c=3 and a— 5. Thus the number required is 503. Ex, 6. A number consisting of three digits is doubled by reversing the digits ; prove that the same will hold for the number formed by the first and last digits, and also that such a number can be found in only one scale of notation out of every three. Let the number be abc in the scale of r. Then we have {abc) x 2 = cba. Since cba is greater than abc, c must be greater than a. Hence we must have the following equations : 2c = a + r (i), 2& + l = 6 + r (ii), 2a + l=e (iii). From (i) and (iii) we see that the number represented by ca is double that represented by ac. Also 4a + 2=2c=a+r; .-. r-2 = 3a. Hence, as a is an integer, r - 2 must be a multiple of 3, so that the number must be in one of the scales 2, 6, 8, 11, &c., the numbers corresponding to these scales being Oil, 143, 275, 3t7, Ac. EXAMPLES XXII. 1. Find the number which has the same two digits when expressed in the scales of 7 and 9. 2. In any given scale write down the greatest and the least number which has a given number of digits. 3. A number of six digits is formed by writing down any three digits and then repeating them in the same order ; shew that the number is divisible by 1001. 4. Of the weights 1, 2, 4, 8, &c. lbs., which must be taken to weigh 1027 lbs. 1 278 EXAMPLES. 5. Shew that the number represented in any scale by 144 is a square number. 6. Shew that the numbers represented in any scale by 121, 12321, and 1234321 are perfect squares. 7. Find a number of two digits, which are transposed by the addition of 18 to the number, or by converting it into the septenary scale. 8. A number is denoted by 4*446 in the quinary scale, and by 4-54 in a certain other scale. What is the radix of that other scale ? 9. If S be the sum of the digits of a number iT, and 2Q be the sum of the digits of '2J}1, the number being expressed in the ordinaiy scale, shew that ^S'- ^ is a multiple of 9. 10. If a whole number be expressed in a scale whose radix is odd, the sum of the digits will be even if the number be even, and odd if the number be odd. 11. Prove that, in any scale of notation, the difference of the square of any number of three digits and the square of the number formed by reversing the digits is divisible by r* — 1. 12. Prove that, in any scale of notation, the difference of the square of any number and the square of the number formed by reversing the digits is divisible by r* — 1, 13. A number of three digits in the scale of 7, when expressed in the scale of 11 has the same digits in reversed order : find the number. 14. Prove that all the numbers which are expressed in the scales of 6 and 9 by using the same digits, whether in the same order or in a different order, will leave the same remainder when divided by 4. 15. There is a certain number which is expressed by 6 digits in the scale of 3, and by the last three of those digits in the scale of 12. Find the number. EXAMPLES. 279 16. Find a number of four digits in the scale of 8 which when doubled will have the same digits in reverse order. 17. The digits of a number of three digits are in a. p. The number when divided by the sum of its digits gives a quotient 15; and when 396 is added to the number, the sum has the same digits in inverted order. Find the number. 18. Find the digits a, 6, c in order that the number 13a645c may be divisible by 792. 19. Prove that there is only one scale of notation in which the number represented by 1155 is divisible by that represented by 12, and find that scale. 20. Find a number of four digits in the ordinary scale which will have its digits reversed in order by multiplying by 9, 21. In the scale of notation whose radix is r, shew that the number (r* — l)(r''— 1) when divided by r— 1 will give a quotient with the same digits in the reverse order. 22. Shew that, in any scale of notation, ^-l^,= -(,n3...(r-3)ir-l), ■ the circulating period consisting of all the figures in order except r — 2 which is passed over. For example, in the ordinary scale, ^ = 012345679. 23. There is a number of six digits such that when the extreme left-hand digit is transposed to the extreme right-hand, the rest being unaltered, the number is increased three-fold. Prove that the left-hand digit must be either 1 or 2, and find the number in either case. 24. Find a number of three digits, the last two of which are alike, such that when multiplied by a certain number it still consists of three digits, the first two of which are alike and the same as the former repeated ones, and the third is the same aa the multiplier. CHAPTER XIX Permutations and Combinations. 240. Definition. The different ways in which r things can be taken from n things, regard being had to the order of selection or arrangement, are called the per- mutations of the n things r at a time. Thus two permutations will be different unless they contain the same objects arranged in the same order. For example, suppose we have four objects, represented by the letters a, 6, c, cZ; the permutations two at a time are ah, hay ac, ca, ad, da, he, ch, hd, dh, cd, and dc. The number of permutations of n different things taken r at a time is denoted by the symbol ^P^. 241. To find the numher of permutations of n different things taken r ata time. Let the different things be represented by the letters a, h, c, It is obvious that there are n permutations of the n things when taken one at a time, so that ^P^ = n. Now in the permutations of the n letters r together, the number of permutations in which a particular letter occurs first in order is equal to the number of permuta- tions of the remaining w — 1 letters r — 1 at a time. This is true for each one of the n letters, and therefore PERMUTATIONS. 281 Since the above relation is true for all values of n and r, we have in succession n— 2 r-2* ..,P,., = («-l)x„.,P, But ' "_«P. = (n-r + l). Multiply all the corresponding members of the above equalities, and cancel all the common factors; we then have ,P, = n (w - l)(7i - 2) (n - r + 1). If all the n things are to be taken, r is equal to n, and we have ^P^ = n{n-l)(n- 2). 3 .2.1. Definitions. The product w (n - 1) (r? - 2) . . . 2 . 1 is denoted by the symbol [w or by n! The symbols [n and n ! are read * factorial n.' The continued product of the r quantities n, n — 1, w — 2, (n — r4-l), n not being necessarily an integer in this case, is denoted by n^. Thus w, = w (n — V){n~ 2). Hence we have JP^ = \n, and ^P^ = n^. 242. To j^rwi ^Ae number of permutations of n things taken all together, when the things are not all different. Let the n things be represented by letters; and sup- pose p of them to be a's, q of them to be 6's, r of them to be c's, and so on. Let P be the required number of per- mutations. s. A. 20 282 PERMUTATIONS. If in any one of the actual permutations we suppose that the as are all changed into p letters different from each other and from all the rest; then, by changing only the arrangement of these p new letters, we should, instead of a single permutation, have \p different permutations. Hence, if the a's were all changed into p letters different from each other and from all the rest, the 6's, c's, &c. being unaltered, there would be P x Ip permutations. Similarly, if in any one of these new permutations we suppose that the 6's are all changed into q letters different from each other and from all the rest, we should obtain \q permutations by changing the order of these q new letters. Hence the whole number of permutations would now he P x\px \q. By proceeding in this way we see that if all the letters were changed so that no two were alike, the total number of permutations would be Pxlp x l^'x Ir... But the number of permutations all together of n different things is jw. Hence P x |^ x l^' x lr...= In; ~ _ In " Ex.1. Ex.2. Ex.3. Ex.4. Ex. 6. Ex. 6. Find the number of permutations of all the letters of each of the words acacia^ hannah, success and mississippi. Am. 60, 90, 420, 34650. Ex. 7. In how many ways may a party of 8 take their places at a round table; and in how many ways can 8 different beads be strung on a necklace ? Ans. [7, i [7. Ex. 8. In how many ways may a party of 4 ladies and 4 gentlemen be arranged at a round table, the ladies and gentlemen being placed alternately? Am. 144, ' \i\q\r--' Find ePg, jP, and jPj, Alls. 120, 120, 5040. Shew that 10^4=7^7- If,,P5 = 12x„P3, find». Ans. 7. n i^Pg = 100 x^Pa, find n. Ans. 13. If2„P,=2x„P., findw. Ans. 8. COMBINATIONS. 283 Ex. 9. The number of permutations of n things all together in which r specified things are to be in an assigned order though not necessarily consecutive is (n/ [r. Ex. 10. The number of ways in which n books can be arranged on a shelf so that two particular books shall not be together is (n - 2) \ n- 1. Ex. 11. Find the number of permutations of n things r together, when each thing can be repeated any number of times. Here any one of the n things can be put in the first place ; and, however the first place is filled, any one of the n things can be put in the second place ; and so on. Hence the number required =nxnx »x ,..=71''. Combinations. 243. Definition. The dilBferent ways in which a selection of r things can be made from n things, without regard to the order of selection or arrangement, are called the comhinations of the n things r at a time. Thus the different combinations of the letters a, h, c, d three at a time are abc^ ahd, acd and bed. The number of combinations of n different things r at a time is denoted by the symbol ^G^. 244. To find the number of combinations ofn different things taken r at a time. Let the different things be represented by the letters a, b, c, ... Now in the combinations of the n letters r together the number in which a particular letter occurs is equal to the number of combinations of the remaining n — 1 letters r — 1 at a time. Hence in the whole number of combina- tions r together every letter occurs ^_fi^_^ times, and therefore the total number of letters is w x ^_fi^i ; but, since there are r letters in each combination, the total number of the letters must be r x ^G^. 20—2 284 COMBINATIONS. Hence rx jO^^nx ^S'r^v Since the above relation is true for all values of n and of r, we have in succession Also -_^,C^, = w — r + 1. Hence, by multiplying corresponding members of the above equations and cancelling the common factors, we have |r x^(7^=w(7i-l)(n-2) (n-r-f 1), that is c^^ nin-l)(n-2)...{n-^r + l) -^n ^ By multiplying the numerator and denominator of the fraction on the right by In — r, we have n (w — 1) (n — 2). . .(n — r + 1) X In — r fi _ \r \n- .(ii). Ir j^— '^ By comparing the above result with that obtained in Art. 242, it will be seen that ^P^ = ^G^ x jr. The relation ^P^ = jO^x [r can however be at once obtained from the consideration that every combination of r different things would give rise to [r permutations, if the order of the letters were altered in every possible way. Note. In order that the formula (ii) may be true when r — n, we must suppose that |0=1, since JJ^ = 1. We should also obtain the result |0 = 1 by putting n = 1 in the relation \a = n \n— 1. COMBINATIONS. 285 245. Theorem. The number of combinations of n different things r together is equal to the number of the combinations n — r together. The proposition follows at once from the fact that whenever r things are taken out of n things, n — r must be left, and if every set of r things differs in some par- ticular from every other, the corresponding set oi n — r things will also differ in some particular from every other ; and therefore the number of different ways of taking r things must be just the same as the number of different ways of leaving or taking n — r things. The result can also be obtained from the formula (ii) of the last Article. \r \n — r \n— r \r It should be remarked that the first method of proof holds good when the n things are not all different, to which case the formulae of Art. 244 are not applicable. Ex. 1. Find ^^G^, ^Cg and joCiy. Ans. 210, 220, 1140. Ex.2. U„C7,=,Cn,find„C'i,. Am. 15S. Ex. 3. Find n, having given that n^8 = n^«' ■^^' H* Ex. 4. Find n, having given that 3 x »C4 = 5 x n-iCg. Ans. 10. Ex.5. Find w, having given that ^(74 = 210. Am. 10. Ex. 6. Find ?i and r having given that „P,=272 and ^(7^=136. Ans. w=17, r=2. Ex. 7. Find n and r having given ^C^^ : „C^ : ^G^+i :; 2 : 3 : 4. Am. 71 = 34, r=14. Ex. 8. How many words each containing 3 consonants and 2 vowels can be formed from 6 consonants and 4 vowels ? The consonants can be chosen in gC,=20 ways; the vowels can be chosen in ^C^ = & ways ; hence 20 x 6 different sets of letters can be chosen, and each of these sets can be arranged in 5P5=120 ways. Hence the required number is 20 x 6 x 120. 286 COMBINATIONS. Ex. 9. How many different sums can be formed with a sovereign, a half-sovereign, a crown, a half-crown, a ehilling and a sixpence ? Number required = gCj + jC, -f- jCg + ^G^, + gCg -h jC7g = 63. Ex. 10. Shew that, in the combinations of 2n different things n together, the number of combinations in which a particular thing occurs is equal to the number in which it does not occur. Ex. 11. Shew that, in the combinations of 4n different things n together, the number of combinations in which a particular thing occurs is equal to one-third of the number in which it does not occur. Ex. 12. Out of a party of 4 ladies and 3 gentlemen one game at lawn- tennis is to be arranged, each side consisting of one lady and one gentleman. In how many ways can this be done ? Ans. 36. Ex. 13. The figures 1, 2, 3, 4, 5 are written down in every possible order : how many of the numbers so formed will be greater than 23000? Ana. 90. Ex. 14. At an election there are four candidates and three members to be elected, and an elector may vote for any number of candidates not greater than the number to be elected. Li how many ways may an Sector vote? ^ (^ ^ ^ C ^ Ans. 14. 246. Greatest value of rnOf> To find the greatest value of ^C^ for a given value of n. I Ci ^ . /() cfc From the formulae of Art. 244 we have i^' (^>^-. j} /^~'^) n-r-Vl r^.j) ,a = „a.iX- ^;— . ,^^ ^ Hence J3^ = „(7^_i, according as n — r + 1 = r j that is, according as r = J (w + 1). > Thus the number of combination of n things r together increases with r so long as r is less than J (n -f 1). n If then n be even, JO^ is greatest when r = ^. If n be odd, JO r%nGr.y as r^i(n-f 1), and ,(7,= n^T-i when r = ^ (w + 1). Thus, when n is odd, «^i{»-i) ~ n^i{n+i) ^^^ these are the greatest values of JO^. COMBINATIONS. 287 For example, if w=10, „(7y is greatest when r=5. Also if n=ll, ,Cr is the same for the values 5 and 6 of r, and ^Cf ^^ greater for these values than for any other value. 247. To p'ove that „G, + „0,., = ^,(7,. If the total number of combinations of (w + 1) things r together be divided into two groups according as they do or do not contain a certain particular letter, it is clear that the number of the combinations which do not contain the letter is the number of combinations r together of the remaining n things, and the number of the combinations which do contain the letter is the number of ways in which r — 1 of the remaining n things can be taken. Thus The above result can also be proved as follows : From Art. 244 we have _^ n(n-l)...(n-r + l) n(n- l)...(n-r + 2) * -"^" '■-' 1.2 r ^ 1.2 (r-1) n(n-l)...(n-r-\-2) , ^ , '^ 1.2...... r Hn-r-^l+r} _ (w4-l)n(n-l)...(r?-r + 2) _ ^ ~ 1.2 r '*+'•" Ex. To prove that „4.,Py=„P,, + r.„P^_i. A particular thing is absent in ^P^ of the permutations of the (n + 1) things and occurs in JPr-i', also when it does occur it can be in either of r places. Hence 248. Theorem. To prove that, if x and y he any two positive integers such that x-\-y=^m, then will ^0. = .0, + .0... . ,0. + ,G,_ . ,0, + ... + .C, . ,6V. + .G,. Suppose that m letters a, 6,..., p, q,..., are divided into two groups a, 6,..., and p, q,..., there being x and y letters respectively in these groups. Then the whole number of sets of n out of the m things will clearly be 288 COMBINATIONS. equal to the sum of the number of sets formed by taking n out of the first group and none out of the second, n—\ out of the first group and one out of the second, n — 2 out of the first group and 2 out of the second, and so on. Now n can be chosen from the first group in JJ^ ways. Also n — 1 can be chosen from the first group in JJ^^ ways, and any one of these sets of w — 1 things can be taken with any one of the ^G^ sets of 1 from the second group, so that the number of sets formed by taking n—\ from the first group and 1 from the second is J^^_^ x JJ^. Similarly, the number of sets formed by taking n — 2 from the first group and 2 from the second is J0^_^ x yC^. And, in general, the number of sets formed by taking n—r from tha first group and r from the second is gpn-r ^ w^r' Hence we have li X OX y be less than n some of the terms on the right will vanish; for JJ^ = if r > w. 249. Vandermonde's Theorem. From the last Article, if a?, y and n be any positive integers such that a; + y is greater than w, we have, since JO^ = .— , \n |n"^ |yi-l - |1 "^ |n-2 '|2 "*" ••• \n — r \r_ \n Multiply each side of the last equation by |w, and we have n (w — 1) {x + y\ = x^ + nx^_^y^ + \ ^ "^^^^^ "^ *•• + |7[^^«-2/.+ +yv HOMOGENEOUS PRODUCTS. 289 The above has been proved on the supposition that x and y are positive integers such that a; 4- y is greater than w; and by means of the theorem of Art. 91, the proposi- tion can be proved to be true for all values of x and y. For the two expressions which are to be proved identical are only of the nth degree in x and y. But, if y has any particular integral value greater than n, the equation is known to be true for any positive integral value of Xy and thus for more than n values; and hence it must be true for that value of y and any value whatever of x. Hence the proposition is true for any particular value whatever of x, and for more than n values of y ; it must therefore be true for all values of x and for all values of y. This proves Yandermonde's theorem, namely : — If n he any positive integer, and x, y have any values whatever ; then will n(n — X) [r \ n — r Homogeneous Products. 250. The number of different products each of r letters which can be made from n letters, when each letter may be repeated any number of times, is denoted by the symbol JS^. For example, the homogeneous products of two dimensions formed by the three letters a, 6, c aie oH^, b^, c^, he, ca, ab. To find JI^, Since in each of the r-dimensional products of n things there are r letters, the total number of letters in all the products will be ^H^ x r ; and, as each of the n letters occurs the same number of times, it follows that the 290 HOMOGENEOUS PRODUCTS. number of times any particular letter, a suppose, occurs is Now consider all the terms which contain a at least once. If any one of these terms be divided by a the quotient will be of r — 1 dimensions ; and, when all the terms which contain a are so divided, we shall obtain without repetition all the possible homogeneous products of the n letters of r — 1 dimensions. Now the homogeneous products of r — 1 dimensions are in number „H^_j^ ; and, by the above, the number of a's they contain is r — 1 X „H^^. Hence, taking into account the a which is a factor of each of the „J5^i terms, the total number of as which occur in all the r-dimensional products of the n letters is ,Sr.. + '^ X A.„ that is '^±1^ „ff,... Hence equating the two expressions for the number of a's, we have r rr n-^r — 1 „ n n « r i» n^r = X n^r-1' r Since the above relation is true for all values of n and r, we have in succession jr _ ^+r-2 rr n-"r-l "~ •• — 1 ^ B-'-'r-a* T r _ yi + r—S rr Also ^ITj is obviously equal to n. EXAMPLES 291 Hence, by multiplying and cancelling common factors, we have -w^-iA^-fO jl^ = n(n + l).^ (n + r-1 ) ,g^^ ^j^^ j^^^ 293]. Ex. 1. Find the number of combinations three at a time of the letters a, by c, d when the letters may be repeated. Ans. 20, Ex. 2. Find the number of different combinations six at a time which can be formed from 6 a's, 6 6'b, 6 c's and 6 d's. Ans. 84. Ex.3. Shew that «ffr=n-A + n^Vi. and deduce that Ex. 4. Shew that n^r = n-l^r + n-l^r-1 + «-l^r-S + + n-Jh + !• 251. Many theorems relating to permutations and combinations are best proved by means of the binomial theorem : examples will be found in subsequent chapters. [See Art. 292.] Ex. 1. Find the number of ways in which mn different things can be divided among n persons so that each may have m of them. The number of ways in which the first set of m things can be given is mn^m » ^^^t whatever set is given to the first, the second set can be given in mn-m^m ways ; so also, whatever sets are given to the first and second, the third set can be given in mn-nn^m ways; and so on. Hence the required number is nm^tn ^ in(n-l) ^m ^ m ""• Ex. 8. Shew that n straight lines, no two of which are parallel and no three of which meet in a point, divide a plane into ^ (/i + 1) + 1 parts. The nth straight line is cut by each of the other n- 1 lines ; and hence it is divided into n portions. Now there are two parts of the plane on the two sides of each of these portions of the nth line which would become one part if the nth line were away. Hence the plane is divided by n lines into n more parts than it is divided by n - 1 lines. Hence, if F {x) be put for the number of parts into which the plane is divided by x straight lines, we have JP(n)=F(»-l)+n. Similarly l^(n-l)=F(n-2) + (n-l), and F(2)=F(l) + 2. But obviously JP(1)=2. Hence ^(n)=2 + 2 + 3 + 4+ +n = l + ^n(n + l). Ex. 4. Suppose n things to be given in a certain order of succession. Shew that the number of ways of taking a set of three things out of these, with the condition that no set shall contain any two things which were originally contiguous to each other is ^ (n - 2) (n - 3) (n - 4). Shew also that if the n given things are arranged cyclically, so that the nth is taken to be contiguous to the first, the number of sets is reduced to \n (n - 4) (n - 5). EXAMPLES. 293 Consider the second case first. Let the different things be represented by the letters a, 6, c, Suppose that a is taken first. Then, if either of the two letters next but one to a be taken second, any one of n - 6 letters can be taken for the third of the set. If, however, the second letter is not next but one to a, but in either of the n - 5 other possible places, there would be a choice of n - (5 places for the third letter of the set. Hence the total number of ways of taking 3 letters in order a being first is 2 (n - 5) + (n - 5) (n - 6), that is (n - 4) (n - 5). There is the same number when any one of the other letters is taken first ; hence, as the order in which the three letters in a set are taken is indifferent, the total number of sets is ^n (n - 4) (n - 6). In order to obtain the first case from the second, we have only to suppose that a and I are no longer contiguous. Hence the number in the first case is the same as that in the second with the addition of those sets which contain a and 2, and there are n-4 of these. Hence the number in the first case is in(n-4)(n-5) + (3i-4) = i(n-2)(n-3)(n-4). Ex. 6. There are n letters and n directed envelopes : in how many ways could aU the letters be put into the wrong envelopes ? Let the letters be denoted by the letters a, 6, c... and the corre- sponding envelopes by a', b', c', Let F (n) be the required number of ways. Then a can be put into any one of the n-1 envelopes 6', c', Suppose a is put into k'; then k may be put into a', in which case there will heF{n- 2) ways of putting all the others wrong. Also if a is put into fc', the number of ways of disposing of the letters so that k is not put in a', 6 not in 6', &c. is JP'(n - 1). Hence the number of ways of satisfying the conditions when a is put into fc' is F(n-l) + F{n-2). The same is true when a is put into any other of the envelopes 6', c',... Hence we have F{n) = {n-l){F(n-l) + F{ri-2)}; F (n) -nF(n- 1)= -{F{n-1)- (n- 1) F(n- 2)}. Similarly F(n-l)-{n-l) F {n-2)= - {F {n-2) - {n-2) F{n-d)} F (3) -3F{2)=-{F (2) - 2F (1) }. But obviously F (2) = 1 and F (1) = ; F{n)-nFin~l) = {-l)\ F{n) F{n~l) _ 1_ \n |n-l "^ ' '\n' 294 EXAMPLES. ^ and 12"" ir-^"^^ |2" Hence, by addition, -("'=41-^34—1?} the number required. SXAMPLES XXin. 1. In how many different ways may twenty different things be divided among five persons so that each may have four 1 ^ A crew of an eight-oar has to be chosen out of eleven men, five of whom can row on the stroke side only, four on the bow-side only, and the remaining two on either side. How many different selections can be made ? 3. There are three candidates for a certain office and twelve electors. In how many different ways is it possible for them all to vote ; and in how many of these ways will the votes be equally divided between the candidates ? 4. Shew that ^C^^ : ^JO^ is equal to " l". 3.5 (4n-l) {1.3.5 (2n-l)}«* 5. Find the number of significant numbers which can be formed by using any number of the digits 0, 1, 2, 3, 4, but using each not more than once in each number. 6. Shew that in the permutations of n things r together, the number of permutations in which p particular things occur 7. There are n points in a plane, no three of which are in the same straight line ; find the number of straight lines formed by joining them. 8. There are n points in a plane, of which no three are on a straight line except m which are all on the same straight line. Find the number of straight lines formed by joining the points. EXAMPLES. 295 ' 9. There are n points in a plane, of which no three are on a straight line except m which are all on a straight lina Find the number of triangles formed by joining the points. " 10. Shew that the number of different rt-sided polygons formed by n straight lines in a plane, no three of which meet in a point, is J j 71— 1. 11. There are n points in a plane which are joined in all possible ways by indefinite straight lines, and no two of these joining lines are parallel and no three of them meet in a point. Find the number of points of intersection, exclusive of the n given points. 12. Through each of the angular points of a triangle m straight lines are drawn, and no two of the 3m lines are parallel ; also no three, one from each angular point, meet in a point. Find the number of points of intersection. 13. The streets of a city are arranged like the lines of a chess-board. There are m streets running north and south, and n east and west. Find the number of ways in which a man can travel from the N.W. comer to the S.E. comer, going the shortest possible distance. 14. How many triangles are there whose angular points are at the angular points of a given polygon of n sides but none of whose sides are sides of the polygon ? 15. Shew that 2n persons may be seated at two round tables, n persons being seated at each, in ■^ different ways. 16. A parallelogram is cut by two sets of m lines parallel to its sides : shew that the number of parallelograms thus formed is J (w + l)'(m + 2)'. 17. Find the number of ways in which p positive signs and n negative signs may be placed in a row so that no two negative signs shall be together. 18. Shew that the number of ways of putting m things in n + \ places, there being no restriction as to the number in each place, is (m -hn) l/mln' 296 EXAMPLES. 19. Shew that 2n things can be divided into groups of n \2n pairs in ^^ ways. 20. Find the number of ways in which mn things can be divided into m sets each of n things. 21. Shew that n planes through the centre of a sphere, no three of which pass through the same diameter, will divide the surface of the sphere into n' -n+2 parts. 22. Shew that the number of parts into which an infinite plane is divided by m + n straight lines, m of which pass through one point and the remaining n through another, is mn + 2m + 2n—l, provided no two of the lines be parallel or coincident. 23. Find the number of parts into which a sphere is divided hj m + n planes through its centre, m of which pass through one diameter and the remaining n through another, no plane passing through both these diameters. 24. Find the number of parts into which a sphere is divided by a + b + c+ ... planes through the centre, a of the planes passing through one given diameter, b through a second, c through a third, and so on; and no plane passing through more than one of these given diameters. 25. Shew that n planes, no four of which meet in a point, divide infinite space into J (n* +671 + 6) different regions. 26. Prove that if each of m points in one straight line be joined to each of n points in another, by straight lines termin- ated by the points ; then, excluding the given points, the lines will intersect Jmw (m—l)(n—l) times. 27. No four of n points lying in a plane are on the same circle. Through every three of the points a circle is drawn, and no three of the circles have a common point other than one of the original n points. .Shew that the circles intersect in if^ (n - I) (n - 2) {n - S) {n - ^) {2n - \) points besides the original n points, assuming that every circle intersects every other circle in two points. CHAPTER XX. The Binomial Theorem. 252. We have already [Art. 67] proved that the con- tinued product of any number of algebraical expressions is the sum of all the partial products which can be obtained by multiplying any term of the first, any term of the second, any term of the third, &c. 258. Binomial Theorem. Suppose that we have n factors each of which is a + 6. If we take a letter from each of the factors of (a-f 6)(a + &)(a-f 6) and multiply them all together, we shall obtain a term of the continued product; and if we do this in every possible way we shall obtain all the terras of the continued pro- duct. [Art. 67.] Now we can take the letter a every time, and this can be done in only one way; hence a" is a term of the product. The letter b can be taken once, and a the remaining (w — 1) times, and the number of ways in which one b can be taken is the number of ways of taking 1 out of n things, so that the number is „(7, : hence we have .C,.a'-'b. s- A. , „ 21 298 BINOMIAL THEOREM. Again, the letter h can be taken twice, and a the remaining (n — 2) times, and the number of ways in which two 6's can be taken is the number of ways of taking 2 out of n things, so that the number is ^G^ : hence we have And, in general, h can be taken r times (where r is any positive integer not greater than n) and a the re- maining n — r times, and the number of ways in which r 6's can be taken is the number of ways of taking r out of n things, so that the number is ^G^: hence we have Thus (a + b) (a -\- b) (a -{- b) to n factors = a" + „a, . a"-*6 +nC, . a"-^6» + +„a. a""'^>'+ ... the last term being jOy^'^b'', i.e. 6". Hence, when n is any positive integer, we have (a + by = a" + .a^ . a"-*6 + ,(7, . a"-»6^ + ... The above formula is called the Binomial TJieorem. If we substitute the known values [see Art. 244] of „0j, ^Cj, „0g,... in the series on the right, we obtain the form in which the theorem is usually given, namely (a '{■ by = a" + 7ia"-' b + ""^^"^^ a"^b' + . . . J. . '' + .•• + 6". Now if we assume that the theorem is true when the index is n, and multiply by another factor a + 6, we have, when like terms of the product are collected, (a + by^' = a"-^^ + (1 + „a,) a"6 + (A + „a,) a"-^ ^'^ + . . . Now i + ^a, = n-n = „,,a„ xri- and, in general, .a-. + .a = «.a[Art.247]. Hence (a + 6r^ = a"-^^ + „,,C, . a% + „^,a, . a^^ft'' + • . . Thus i/^the theorem be true for any value of n, it will be true for the next greater value. Now the theorem is obviously true when n = 1. Hence it must be true when n — 2; and being true when n = 2, it must be true when 7i = 3; and so on indefinitely. The theorem is therefore true for all positive integral values of 71. Ex.1. Expand (a + 6)4. We have = a* + ia^b + 6tt362 + Aab^ + 6*. 21—2 300 GENERAL TERM. Ex.2. Expand (2a; -i/)3. Put 2x for a, and - y for 6 in the general formula : then (2a:-y)3=:(2a;)8 + 3 (2x)M-y) + f^-| (2x) (- j,)» + ^ (-y)» Ex.3. Expand (a -6)". Change the sign of 6 in the general formula; then we have 1. « l5 ,+ Fi^ «"-"(-*)'•+ +(-")' 1 .J In +^-ir|7i^^"-^&''+ +(-i)"ft*. 255. General term. By the preceding articles we see that any term of the expansion of (a + by by the Binomial Theorem will be found by giving a suitable value to r in In \r \n — r On this account the above is called the general term of the series. It should be noticed that the term is the (r + l)th term from the beginning. [See Note Art 244.] 256. Coefficients of terms equidistant respec- tively firom the beginning and the end are equal. In the expansion of (a + by by the Binomial Theorem, the (r + l)th term from the beginning and the (r + l)th term from the end are respectively nC^.a'-^b^ and A-. • «^^""^- But „a=«C^n^- [Art. 245.J GREATEST TERM. 801 Hence, in the expansion of (a + 6)", the coeficients of any two terms equidistant respectively from the beginning and the end are equal. This result follows, however, at once from the fact that (a + 6)", and therefore also its expansion, would be unaltered by an interchange of the letters a and h ; and hence the co- efficient of a'^'^V must be equal to the coefficient of ft""*" a*". 257. If, in the formula of Art. 253, we put a = 1 and 6 = a;, we have (l+a?)" = l + ^ + ''t'^^^ ^' + - + r-r^ — ar + ... + «?«. ^ ^ 1.2 r 71 — r This is the most simple form of the Binomial Theorem, and the one which is generally employed. The above form includes all possible cases: if, for example, we want to find {a + hy by means of it, we have = a" + 7ia"-^6 -f "^^ a^-^h' + ... 258. Greatest term of a binomial expansion. In the expansion of (1 + a?)", the {r + l)th term is formed n ~~ r -^- 1 from the rth by multiplying by w. T^T- n — r + l /n + l ,\ , n-\- 1 JNow X — — Ijoo, and clearly diminishes as r increases : hence cc diminishes - r as r is increased. If x be less than 1 for any 302 GREATEST COEFFICIENT. value of r, the (r + l)th term will be less than the rth. In order therefore that the rth term of the expansion may be the greatest we must have n—r+1 ^ ,n—r— 1+1 a; < 1, and = x>l. r r — 1 Hence r > -^ — , and r < i — \- 1. a?+ 1 a?+ 1 The absolute values of the terms in the expansion of (1 + (cY will not be altered by changing the sign of x ; and hence the rth term of (1 — a?)" will also be greatest in absolute magnitude if r > ^ — -f— , and r < ^^ ^ + 1. x-h 1 x + l If r = ~- , then x—1: and hence there x+l r * is no one term which is the greatest, but the rth and r+ l|th terms are equal, and these are greater than any of the other terms. Since (a + x)' f-x)*»i8 (n+1) the rth term of (o+x)*» is the greatest when r>- and < +1, a a Ex. 1. Find the greatest term in the expansion of (l + a?)*, when 21 21 The rth term is the greatest, if r>— and r <.!+-=-, Hence o 5 the fifth term is the greatest. Ex. 2. Find the greatest term in the expansion of {1+xy, when 6 The rth term is the greatest, if r > 5 and < 6. Thus there is no one term which is the greatest, bnt the 5th and 6th terms of the EXAMPLES. 303 expansion are equal to one another and greater than any of -the other terms. Ex. 3. Find the greatest term in the expansion of (10 + Sx)^'^ when a; = 4. Am. The ninth term. The greatest coefficient of a binomial expansion can be found in a similar manner. For in the expansion of (1 + a?)" the coefficient of the (r + l)th term is formed from 71 — T* -|- 1 that of the rth by multiplying by ± . Hence the rth coefficient will be the greatest in absolute magnitude, if < 1 and > 1. r r — 1 That is if r>'^ and < 1 + ^. Hence when n is even, the coefficient of the rth term is the greatest when r = ^ + 1 ; and when n is odd, the coefficients of the — q— th and — ^ — th terms are equal to one another and are greater than any of the other terms. For example, in {l + x)^ the coefficient of the 11th term is the greatest; and in (1 + a;)" the coefficients of the 6th and 7th terms are greater than any of the others. EXAMPLES XXIY. Write out the following expansions : 1. (x + ay. 2. {2a -xy. 3. (l-xj. 4. (2a -Say. 5. (2af-3y. 6. {x' - 2yy. 7. Find the third term of {x - Sy)*". 8. Find the fifth term of (3aj - 4)''". 9. Find the twenty-first term of (2 - x)^, 10 Find the fortieth term of {x - y)**. 'n 304 EXAMPLES. 11. Find the middle term of (1 +xf, 12. Fiml the middle terms of (1 4- a;)". 13. Find the general term of {x - 3y)". 14. Find the general term of (of + y^)". 15. Write down the first three terms and the last three terms of(3ic- 22/)''. 16. Find the term of (1 + xy^ which has the greatest coefficient. 17. Find the two terms of (1 + aj)'* which have the greatest coefficients. 18. Shew that the coefficient of x" in the expansion of (1 + xy is double the coefficient of x" in the expansion of (1 + xy''-\ 19. Shew that the middle term of (1 +a;)^'* is [n 20. Employ the binomial theorem to find 99*, 51* and 999^ 21. Shew that the coefficient of x' in the expansion of \n is ("jy \ i(n + r)\i{n-r) ' 22. Find the middle term of (a; - - j . 23. The coefficients of the 5th, 6th and 7th terms of the expansion of (1 + a;)* are in arithmetical progression : find n. 24. For what value of n are the coefficients of the second, third and fourth terms of the expansion of (1 +a;)" in arith- metical progression 1 25. If a be the sum of the odd terms and b the sum of the even terms of the expansion of (1 + aj)", shew that il-af)''=a'-b'. PROPERTIES OF THE COEFFICIENTS. 805 259. Properties of the coefficients of a binomial expansion. It will be convenient to write the Binomial Theorem in the form (l+a?)" = c„ + Cja; + C2a^ + ...+cX+ ... c.a?" (i), where, as we have seen, c^ = c„ = 1 ; c^ = c„_i = n ; and, m general, n = c„_ = ; — r= — . -' ° "^ \r \n — r I. Put a; = 1 in (i) ; then 2'* = c„ + c, + c,+ ... + c„. Thus the sum of the coefficients in the expansion of {l+xyisr, II. Put a; = — 1 in (i) ; then (l-ir = c,-c, + c,- + (~irc,; ••.0 = (Co + c, + c,+ ...)-(c,+ C3 + c, + ...). Thus the sum of the coefficients of the odd terms of a binomial expansion is equal to the swm of the coefficients of the even term^. III. Since c^ = c„_^, we have (1 +xy = c^-{-c^x-\-c^a^ + ... + cX+ ... + cjv\ and (1 + xy = c„ + c„_^x + c„^^ + . . . c^_X + . . . + c^^x''. The coefficient of of in the product of the two series on the right is equal to Co^+C,^ + c/+ +C Hence [Art. 91] c,' + c,^ -^ ... +c,' + + cj' is equal to the coefficient of x^ in (1 + xy x (1 + a?)", that 1271 is in (1 + xf" ' and this coefficient is -'=^ • 306 PROPERTIES OF THE COEFFICIENTS. Hence the smn of the squares of the coeflScients in the expansion of (1 +a?)' is ~t- • \n\n IV. As in III, we have (1 + xf = Co + c^x + c^a^ + ... + c„a;", and (1 - xf = c, - c^^,x + c^_,x^ + ... + (- Vfc^"". The coefficient of o^ in the product of the two st^rios on the right is equal to . (-l)"(c.'-c.» + c,= - + (-l)V}. The coefficient of a;^ in (1 + xy x (1 — x^, that is in - 1^ (1 - s^y, is zero if n be odd, and is equal to (— 1)'. rr^v- if n be even. Hence c^ - c^'' + c,* - . . . + (- 1)X' is zero n or (— l)^/i!/(^7i!)', according as 7i is odd or even. Ex. 1. Shew that We have Ci + 2c2 + 3c8+...+nc,, „n(n-l) n(n-l)(n-2) I" . =njH-(n-l) + j-2 + +]r-lln-.r+ + ^/ (r-l|n-r r:i(l + l)»»-l = n2'»-i. Ex.2. Shew that Co-^c, + ^c,- + (-!)«_£._ = _i_^. We have c©- g^i + s *-'» " *^°' "= ■•■ ~ 2 ** '^'S 172 ~ PROPERTIES OF THE COEFFICIENTS. 307 1 La.1 (^+1)« ■ (r^ + l)n(n-l) J - ^ ^ (l fnlin ^''^^^'' (n+l)n(n-l) ./.m-Hil J^ ?_ (1 _ l)n+l __J_ n+1 n + 1^ ' n+1 Ex. 3. Shew that, if n be any positive integer, o__£l_ -.^... + (-!)" In X x + 1 x+2 ' " ^ ' x+n x(x + l)..^{x+n) Assume that X a; + l^x + 2 ^^ ' x + n « (« + l). ..(« + »)' for all t;alM<« of «, and for any particular value of n. Change x into x + 1 ; then « + l a; + 2^a: + 3 ^^ ^ x + » + l !« (a? + l)(a; + 2)...(a; + n + l)* Hence, by subtraction, X X + 1 ^ X + 2 ~ "^^ ^^ x + r "^ 1 |W + 1 ^^ ' x+n + 1 x(x+l) (x+n + 1)' But «C^+nCf-i=n4-i^r. for all values of r [Art. 247 j. Hence we have i _ w+l^l , n+lA _ +(_l)n+l n±l^»+l 1^ x~ X+1 x + 2 ^ ' x+n+1 X {x + 1) ...{x + n+1)' Hence if the theorem be true for any particular value of n it will be true for the next greater value. But the theorem is obviously true for all values of x when n=l: it is therefore true for all positive integral values of n, [See also Art. 297, Ex. 3.] By giving particular values to x we obtain relations between Cj, Cj , &c. For example : Put x = l; then we have £o £i 4. ^ _ _ _L l"'2"^3 ~n + l' 308 PROPIGRTIES OF THE COEFFICIENTS. - ^=^-"^M- =rx3;^,- Ex. 4. Shew that Coa-Ci(a-l) + Ca(a-2)-C8(a-3) + + (-l)nc„(a-n)=0, and that Coa3-Ci(a-l)2 + Ca(a-2)2-c,(a-3)3+ + (-l)»'c„{a-n)2=0. We have from 11., if w be any positive integer, i-„+!L(±il)- "("-iH«-2) ^ ^,_,,.^„ ,,,_ Hence, if n > 1 1_(„_1,+ (lzi)|zl> _ + (_i).-.=o (ii). Multiply (i) by o and (ii) by n and add ; then a-n(a-l) + ^^^^a-2)- + (_i)n(a-n) = (iii), where n is > 1. Change a into a - 1 and n into n - 1 in (iii) ; then, n feeing > 2, we have a-l-(w-l)(a-2) + + (-l)'*-i(a-n)=0., (iv). Now multiply (iii) by a and (iv) by n and add; then aa-n(a-l)a + -^-Ji^y(a-2)2- + ( - l)«(a-n)2 = 0, provided n is greater than 2. By proceeding in this way we may prove that aP-n{a-l)P + ^^'!^~^\ a-2)P- + (- l)'»(a-7i)P=0, provided that p is any positive integer less than n. [See also Art. 305.] 260. Continued product of n binomial factors of the form x + a, x-}-b, x-\-c, &c. It will be convenient to use the following notation : /S, is written for a+6 + c + ..., the sum of all the letters taken one at a time. 8^ is written for ab -\- ac •}- . . ,, the sum of all the products which can be obtained by PRODUCT OF BINOMIAL FACTORS. 809 taking the letters two at a time. And, in general, S^ is written for the sum of all the products which can be obtained by taking the letters r at a time. Now, if we take a letter from each of the binomial factors of (x-¥a){x + h){x-\-c){x-{-d) , and multiply them all together, we shall obtain a term of the continued product ; and, if we do this in every possible way, we shall obtain all the terms of the con- tinued product. We can take x every time, and this can be done in only one way; hence a?" is a term of the continued product. We can take any one of the letters a, b, c..., and x from all the remaining n—1 binomial factors ; we thus have the terms ax^'\ bx*"'^, ca?'*"^ &c., and on the whole S^.x""-'. Again, we can take any two of the letters a, b, c..., and X from all the remaining w — 2 binomial factors; we thus have the terms abx'''^, acx*''^ &c., and on the whole S^ . of*-*. And, in general, we can take any r of the letters a, 6, c..., and x from all the remaining n — r binomial factors ; and w^e thus have S^ . a7""^ Hence (x-{- a) (x -\- b) (x + c) = x''-\-S^.ar-''-\-S^.x''-'+... + S,.x*'-' + ... the last term being abed , the product of all the letters a, 6, c, d, &c. By changing the signs of a, 6, c, &c., the signs of /Sj, /Sfg, S^, &c. will be changed, but the signs of S^y S^, 8^, &c. will be unaltered. Hence we have {x — a) (x — b) (x - c) ^x''--S,.x'-'+S,.x^-'-,..-^('-iyS,.ar-\..-{-(''iyabcd,,. 310 vandermonde's theorem. 261. Vandermonde's Theorem. The following proof of VandermoiKie's Theorem is due to Professor Cayley*. [See also Art. 249.] We have to prove that if n be any positive integer, and a and b have any values whatever ; then will / 7 x J n(n — l) J (a + bX = a, + na,_J)^ -f -\—2^ »„., 6, -I- . . . 6^ + ... +6.. ir |7i — r " ' *^ * Assume the theorem to be true for any particular value of n. Multiply the left side by a + 6 — w ; it will then become (a + 6)^1. Multiply the successive terms of the series on the right also by a + 6 - w but arranged as follows : — for the first term {(a — n) + 6} ; for the second {(a — n + 1) + (6 - 1)} ; and for the rth {(a - w + r - 1) + (6 - r + 1)}. We shall then have (a + bX,,:^a^{{a-n) + b\+A.a„_,b,{(a-n + l) + (b-l)] + nC^,.a«-A{(«-^ + 2) + (6-2)} + ... + nOr-x . ttn-^+X^'.-l {(a - M + T - 1) + (6 - r + 1)} + nCr . a,. A {(a-n-^r)-\-(b- r)} + ...+6Ja + (6-n)}. Now a„ {{a - n) + b] = a,^, 4- a„ 6,, „G, . a^^A ((a - n + 1) + (6 - 1)) = ^C\ (a, b, + a,_,6,), „a-i . a„-,.+i^.-i {(a - n + r - 1) + (6 - r + 1)} nCr . a„-r^r {(tt - ?i + r) + (6 - v)} = ^C, (a^.^-ift, + a,_,6^x) • Messenger of Mathematics, Vol. y. MULTINOMIAL THEOREM. 311 Hence (a + b)^, = a„+i + (1 + ^C^) a,,\ + . . . since „0^i + JJ^ = n+iC'r- Thus, t/ the theorem be true for any particular value of n, it will also be true for the next greater value. But it is obviously true when w = 1 ; it must therefore be true when 71 = 2 ; and so on indefinitely. Thus the theorem is true for all positive integral values of n. 262. The Multinomial Theorem. The expansion of the nth power of the multinomial expression a + 6 + c + ... can be found by means of the Binomial Theorem. For the general term in the expansion of (a + 6 + c + (i + ...)", that is of {a +(6 + c + d+ ...)}", by the Binomial Theorem is .vu f "^ \n ' ^ Ir |w — r Similarly the general term in the expansion of by the Binomial Theorem is \n — r , 5>(c + d + ...)—-. The general term in the expansion of (c + cZ + . . .)" "^^ by the Binomial Theorem is !"-'•- \t\n — r-s — t c'(d +...)"-"--'. Hence the general term in the expansion of is \r\n--r \8 \n — r-8 \t\n — r-s—t 312 MULTINOMIAL THEOREM. ^ that is j—Tu — a'b'c' ..., \r\s_\t... where each of r, 5, ^ . . . is zero or a positive integer, and r4-5 + ^ + ...=7i. The above result can however be at once obtained by the method of Art. 263, as follows. We know [Art. 67] that the continued product (a + 6 + c+...)(a4-6 + c+...)(a + 6 + c + ...)--- is the sum of all the different partial products which, can be obtained by multiplying any term from the first multi- nomial factor, any term from the second, any term from the third, &c. The term a*" 6* c* . . . will therefore be obtained by taking a from any r of the n factors, which can be done in „G^ different ways; then taking h from any s of the remaining n — r factors, which can be done in „_,.(7, different ways ; then taking c from any t of the remaining n — r — s factors, which can be done in „_^,C7^ different ways; and so on. Hence the total number of ways in which the term a*" 6* c*. . . will be obtained, which is the coefficient of the term in the required expansion, must be that is nOr X .-A X , ,-,-.(?, X .... 1" . n — r \n- -r — s _ t • |rl£|£...- \r n — r \s \n-r—s \t \n- -r-s-t"" Hence the is general term in the expansion of {a + b + c+.. ••)' l£ii- EXAMPLES. 313 Ex. 1. Find the coe£Qcient of abc in the expansion of (a + & + c)". 13 The required coefficient = = 6. Ex. 2. Find the coefficients of a^lP, bcd^ and abed in the expansion of {a + b + e + d)\ We have the terms Thus the required coefficients are 6, 12 and 24 respectively. 263. By the previous Article, the general term of the expansion of {a -\- bx -i- car^ -\- daf^ + )*• is In ^ a' {hxj {cx'J {dxy , r \s \t\u.. that is ,-T-h a''?>V(i" ^.«+2*-»^«--. \r\s \t\u... Hence to find the coefficient of any particular power of «, say of af^y in the expansion, we must find all the different sets of positive integral values of r, s, ^,... which satisfy the equations 5 + 2« + 3w+ =a, r + s + ^ + w+ =w. The required coefficient will then be the sum of the coefficients corresponding to each set of values. Ex. 1. Find the coefficient of «* in the expansion of (1 + 2a: + ^x^)\ 1^ The general term is -j — j^^ 2*3'x*+2*, and the terms required are til 11 those for which « + 2« = 5 and r+« + t=4. Since each of the quantities r, s and t must be zero or a positive integer, the only possible sets of values are t = 2, «:=!, r=l and 14 t=l, s = 3, r=0, the corresponding coefficients being p . .y, .2.3^ |l \i\z s. A. 22 314 EXAMPLES. li and . 23 . 3, that is 216 and 96 respectively. Hence the required coefficient is 312. In simple cases the result can be readily obtained by aotnal expansion. We have (1 + 2a; + 3x^)*= 1 + 4 (2a; + 3cc2) + 6 (2a; + 3x^)^ + 4 (2a; + 3a;2)8 + (2x + Sx^. Only the last two terms will contain x'^ and the coefficients of ar^ in these terms will be found to be 216 and 96 respectively, so that the required coefficient is 312. Ex. 2. Find the coefficient of x* in the expansion of (1 + a; + ar')^. Am. 6. Ex. 3. Find the coefficient of x^ in the expansion of (1 + a; + aP)*. Am. 16, Ex. 4. Find the coefficient of a;* in the expansion of (2 + a; - x^)^. Am. Ex. 5. Find the coefficient of a;^" in the expansion of (7 + x + x^ + x^ + x* + x^)». Am. 39. Ex. 6. Find the coefficient of the middle term of the expansion of (l+a; + a;- + x3 + a;«)». Ans. 38L EXAMPLES XXV. 1. Prove that c„ - 2c, + 3c, - + (- 1)- {n +l)c^ = 0. 2. Prove that c,-2c, + 3c3- 4-(-l)--'nc,-0. 3. Prove that c„+2c, + 3c, 4- + (w + l)c, = 2"-'(n+2). 4. Prove that c, + 2C3 + 3c, + + (n- 1) c, = 1 + (n- 2) 2'-\ 5. Prove that c„ + 3c, + 5c, + + (2n + 1) c„ = (n + 1) 2". 6. Prove that 3c, + 7c,+ IIC3+ + {in-l)c^ = l + (2n-l)2\ EXAMPLES. 316 7. Prove that 1^2 3 ri+1 7*+l 8. Prove that ^0 ^2 ^4 ^'s 2* 13 5 7 n+l 9. Prove that c. c- c. 4-^4- _ 2"-l 2 "4 "6 " ~ n+l" 10. Prove that ^ ^ ^ c, _ 1 + »i S"-"^ . 2"^ 3 "^ 4"^ "^n + 2~ (ri + l)(w + 2)* 11. Prove that ^ - § + % - + (- If -^ -i= - + - + + - . 12 3 ^ ' w 1 2 n 12. Prove that 1"4^7 ^^ ^^ 3n+l~ 1.4.7. ..(3n + l)- 13. Pjove that 14. Prove that, if (1 + a;)" = c„ + c,a; + c,£c* + ,. +c„a;", then n(l + ic)"-* =c, + 2Cj£c + 3030;' + +wc^a;""*, and {1 + (ti + 1) £c} (1 + a;)""' = c„ + 2Cja; + + (r« + 1) c^as". Hence prove that, |2n-l ' ' * " |n- 1 \n-\ (n+2) |2yi,-l and c; + 2c/+3c;+ + (^ + l)C= [^y^.l^TP "' 22—2 316 EXAMPLES. 15. Shew, by expanding {(1 +«)"— 1}"*, where m and w are positive integers, that -c, . A - .c. . ,.c„ + „c. . ..c, - = (- 1)— »-. 16. Prove that, if t* > 3, (i) a_n(a-l)4-^?^^^)(«-2)-. + (- 1)" (a_r*) = 0. (ii) db-nia-l) (6- 1)+ -&*^-)(a-2) (6-2)- * + (-l)"(a-w)(6-n) = 0. (iii) a6c-n(a-l)(6-l)(c-l) + ^?^^^(a-2)(6-2)(c-2) + (-l)"(a-w)(6-n)(c-7i)=0. 17. Shew that, if there be a middle term in a binomial expansion, its coefficient will be even. 18. Shew that the coefficient of x" in the wth power of «" + (a + 6) aj + a6 is a' + ^C^'a^-'b + ^C^'a^-'b' + + b\ 19. If n be a positive integer and F^ denote the product of all the coefficients in the expansion of (1 + £c)", shew that F^ [n • 20. Shew that 1 . 2 2L Shew that, if tz be a positive integer, - 1 +x n{n-l) 1 + 2jb \+nx 1.2 {l + nxy - ^(^-l)(^ -2) 1 + 3a; ^ 1.2.3 (IT^)'"^ 22. Shew that (a + 6 + c + c? + e)* = Sa» + 55a'6 + 1 0^a%* + 202a'6c 4- 302a'6'c + ^0%a%cd + 1 mahcde. EXAMPLES. 317 23. If (1 + re + xy = «„ + «i^ + ^s^** + n{n-\) (-l)'!^ ^ prove that a^-na^_,+ ^^ ^ a,.,- "^ [r |n -r """" * unless r is a multiple of 3. 24. Shew that, in the expansion of (1 + a? + £c'+ + a;'^)'*, where w is a positive integer, the coefficients of terms equi- distant from the beginning and the end are equal. 25. If a,,, cb^,a^i be the coefficients in the expansion of (1 + ic + a?Y in ascending powers of x, prove that a/ — a/ + a*— + a^^' = a^, and that 26. If (1 + a; + a?*)" = a,, + ^i^ + V* "*" ^3^' + • • •> prove that a^a^-a^a^ + a/i^- = 0. 27. Shew that, in the expansion of (aj + a, + ctg + ... + a^)*, where n is a whole number less than r, the coefficient of any term in which none of the quantities a^j a,, 1, a. b and x bemg positive and b > a\. *• rb + x b + x ' ' *^ ■• The latter series is convergent, and therefore also the given series. To ensure the convergency of the first series it is not necessary that all its terms should be less than the corresponding terms of the second series, it will be sufficient if all the terms except 'a, finite number of them 322 CONVERGENCY AND DIVERGENCY. be less than the corresponding terms of the second, for the sum of a finite number of terms of any series must be finite. 4 43 48 44 46 4s Ex. Shew that the series l + T2"*"]3 + j4+i5+i6+l7 + - i^ con- vergent. From the sixth term onwards, each term is less than the corre- 4s 46 spending term of the series gjp + -^^-j^ + .... Hence the series beginning at the sixth term is convergent, and therefore the whole . series is convergent. X 270. Theorem Hi If the ratio of the corresponding wrms of two series be always finite, the series will both be convergent or both divergent. Let the series be respectively U=u^-^u^ + u^-\- , and F = Vj + Vj H- 1;^ -f Then, since the quantities are all positive, -^^ must lie between the greatest and least of the fractions — [Art. 113]. Hence 27 : F is finite. It therefore follows that if U b finite so also is F, and if CT is infinite so also is F. For example, the two series ^r-^ + ^— + + i 2.3 3.4 (n+l)(n + 2) ' *" and T + + + - + ... are both convergent or both divergent. For the ratio of the rth terms, namely -, — ttt ;cr : - is equal to •' (r+l)(r + 2) r ^ , . .,,, xr , which is > 1 and < 8 for all values of r. Now we have (r + l)(r + 2) already proved that the second series is divergent : the first series is therefore also divergent. 271. Theorem III. A series is convergent if after any particular term, the ratio of each term to the preceding is always less than some fixed quantity which is itself less than wiity. CONVERGENCY AND DIVERGENCY. 323 Let the ratio of each term after the r^ to the preceding term be less than k, where A; < 1. Then, since ^ < A;, ^^<7c, , we have ^r + ^r+i + Wr+» + < U^ (1 + k + Jc" + ) < 3 — ^ , since k is less than 1. 1 — K Hence the sum of the series beginning at the r^ term is finite, and the sum of any finite number of terms is finite ; therefore the whole series must be convergent. 272. Theorem IV. A series is divergent if, after any particular term, the ratio of each term to the preceding is either equal to unity or greater than unity. First, let all the terms after the r^ be equal to u^; then w^^j + ii^^g + . . . + u^^.^ = nu^, and nu^ can be made greater than any finite quantity by sufficiently increasing n. The series must therefore be divergent Next, let the ratio of each term, after the r^^ to the preceding term be greater than 1. Then w^^j > u^, u^ > w^^^ > u^, &c. Hence u^_^_^-^ u^^^+ ,,, -\-u„_^^> nu/, the series must therefore be divergent. 1 2 22 2» 2*»-i Ex. 1. In the series T+o + ^ + :r + + + » *^6 'a^o 12 3 4 n * -^— = i , which is greater than 1 ; the series is therefore w„ n + 1' divergent. X. 2. In J that is ( 1 + - I X. Now, if x be leas than 1, and any fixed quantity k jtween x and 1, the test ratio will be the first which makes (n + 1)^ Ex. 2. In the series l*+22a? + 3%2+ the test ratio is ^ — ^x, 7L* be chosen between x and 1, the test ratio will be less than k for all terms after the first which makes 324 CONVERGENCY AND DIVERGENCY. Hence the series is convergent if a; < 1. If x = 1 the series is 1^ + 2^ + 3^ + which is obviously divergent, and if x > 1 the series is greater than l' + 22-f3^+ Thus the series l^ + 2,^x + '6^a^+ is divergent except when x is less than unity. 273. When a series is such that after a finite number of terms the ratio -^^ is always less than unity but becomes indefinitely nearly equal to unity as n is in- definitely increased, the test contained in Theorem III. fails to give any result ; and in this case, which is a very common one, it is often difiicult to determine whether a series is convergent or divergent. For example, in the series 1» + 2* ■*■ 3* "^ 4* "^ the ratio u^ (n + 1)* (-jy Hence, if A; be positive, the test ratio is less than unity, but becomes more and more nearly equal to unity as w is increased. We cannot therefore determine from Theorem III. whether the series in question is convergent or divergent. I 274. To shew that the series y* + o» + q* + ••• ** ^o^' L Ji o V vergent when k is greater than unity, and is divergent when \J k is equal to unity or less than unity. First, let k be greater than unity. Since each term of the series is less than the pre- ceding term, we have the following relations : 1 1 2 CONVERGENCT AND DIVERGENCY. 325 ^- 1+1+1 5* ^ 6* ^ 7* 4 1 1 + .. 1 2" 2n* + (2" + 1)* 1 (2»«_i)« Hence the whole series is less than 1 2 p + 2* + ^ |.^ 2" + 2^ + •••» t is, less than ^^ ^ + ^ :Y,+ 1 -23(-^ i. + + 2S (*-l) + 1 ' 2*-' 2'**- 2«-i + 2 "*" "^ 2" 4- But this latter series is a geometrical progression whose common ratio, -^^ , is less than unity, since k>l. Hence the given series is convergent. Next, let k = l; then we can group the series as follows : 1 1 ri 11 n 1 1 11 i-^2+U-'4j + b'"6 + 7 + 8j + + ['2«-^ + 1 therefore, as each group of terms in brackets is greater than ^, the given series taken to 2" terms is greater than l4-i + i + i + taken to w + 1 terms, that is, greater than 1 + |w, which increases indefinitely with n. Ill Hence t + o + q + ^^ divergent. X Ji O Lastly, let k be less than unity; then each term of the series ^4--^+ is greater than the corresponding term of the divergent series t + h + ; the series is therefore divergent when A; < 1. CONVERGENCY AND DIVERGENCY. ^ 275. The convergency or divergency of many series can be determined by means of Theorems I. and II., using the series of the last Article as a standard series. The method will be seen from the following examples. , . . 2n Ex. 1. Is the series whose general term is -3^ convergent or divergent ? Since -^> - , if n> 1, it foUows that S -^^->S^. But si On is divergent; therefore S -g — j is also divergent. 71 + 2 Ex. 2. Is the series whose general term is -5 — - convergent or divergent ? Now4±i<'^ ivergent when x= - 1, provided n > 0. [See Ait. 338. J ■ 1 ; and it is con^ CONVERGENCY AND DIVERGENCY. 329 III. The Logarithmic Series. In the logarithmic series, namely the ratio u _^Ju is = — x ll ] ; and hence n+l/ « ^+1 \^ 71+1/ u^^.Ju^ will be numerically less than unity provided x is numerically less than unity. The logarithmic series is therefore convergent when x has any value between i - 1 and + 1. J If a;=l, the series becomes 1— ^ + J— ..., which is V convergent by Theorem V. r^ If a? = — 1, the series becomes — (1 + ^ 4- ^ + . . .)> which is known to be divergent. [Art. 274.] 279. The condition for the convergency of the product of an infiuite number of factors, and also some other ^theorems in convergency, will be proved in a subsequent Chapter. [See Art. 337.] The two important theorems J which follow cannot however be deferred. J 280. If the two series U = Uq-\-u^x + u^x^ -\- + w„a;'*+ ..., and V = v^ + v^x + v^x^ -{■ +v„^" + ..., be both convergent, and the third series + +(V« + ^iVi+ +^„'yo)^"+ — be formed, in which the coefficient of any power of x is the same as in the product of the two first series; then P will be a convergent series equal to UxV, provided (1) that the series U and V have all their terms positive, or (2) that the series U and V would not lose their con- vergency if the signs were all made positive*. • This Article, and in fact the whole of this Chapter, is taken with slight modifications from Cauchy's Analyse Algehrique. s. A. 23 330 CONVERGENCY AND DIVERGENCY. First, suppose that all the terms in U and V are positive. Then U^ x V.^^ = -^271 + terms containing ^r^+^ and higher powers of x. Hence U^ x V^, > Pm- Also P,„ = i^n X F„ + other terms. Hence P^^ >U^xy^. Hence P^^ is intermediate to U^ x F„ and U^ x F^^. Now, the series U and F being convergent, U^ and ^. both approach indefinitely near to U, also Fj„ and F„ both approach indefinitely near to F, when n is indefinitely increased. Hence ^j„ x F,„ and U^x V„, and therefore also Pj^ which is intermediate to them, will in the limit be equal to U x V. Hence, when all the terms are positive, P=:Ux V. Next, let the signs in the two series be not all positive, and let U' and V be the series obtained by making all the signs positive in U and F; and let P' be the series formed from W and F' in the same way as P is formed from f7and F. Then U^„ x F,„ — P,„ cannot be numerically greater than U'^ X V\„ — P\^ , for the terms in the latter expres- sion are the same as those in the former but with all the signs positive. Now, provided the series U and F do not lose their convergency when the signs of all the terms are made positive, it follows from the first case that U'^ x V\^ — P'^, and therefore also U^x V^ — P^ which is not numerically greater, must diminish indefinitely when n is increased without limit. Hence the limit of P^ is equal to the limit of U^^ X Fj^ ; so that P must be a convergent series equal to the product of U and F. If the series U and F are convergent, but are such that they would lose their convergency by making the signs of all the terms positive, the series P may or may not be convergent; and, when P is not convergent, the relation UxV = P does not hold good, for P has no definite value and cannot therefore be equal to U x V, CONVERGENCY AND DIVERGENCY. 331 although the coefficient of any particular power of x in the series P is always equal to the coefficient of the same power of X in the product of the series U and F*. 281t_Jf the two series a^-^- a^x + a^x^ + a^x^ -k- , and h^ + \x -f h^x"" + \x^ + , be equal to one another for all values of x for which they are convergent ; then will a^ = 6^, a^ = h^^ a^ = b^, &c. For if the series are both convergent, their difference will be convergent Hence ao-^o + (ai-^)^ + K-^2)«^+ =0 (i), for all values of x for which the series is convergent. The last series is clearly convergent when a; = ; and putting a; = we have a„ — Z>„ = 0. Hence a„ = 6^, . We now have X {a^ - 6j +(a, - h^)x-h(% - h^)x^ + } = (ii). Now for any value x^ for which the series in (i) is convergent, a^ — b^ + (% — b^) ^i + is equal to a finite limit, Xj suppose. Hence (ii) may be written x^ {a^ — 6^ + a?^ Z J = ; and, since this is true for all values of x^, however small, it follows that ttj — 6j, must be numerically indefinitely small compared with L^ ; that is, a^ — 6^ must be zero. It can now be proved in a similar manner that a, — 6^ = 0, a, - J, = 0, &c. Hence if two series which contain x be equal to one another for all values of x for which the series are conver- gent, we may equate the coefficients of the same powers of x in the two series. The particular case of two series which have a finite number of terms was proved in Art. 91. * It can be proved that P is convergent if either Z7 or F is absolutely convergent. See Chrystal's Algebray Part ii., p. 127. 23—2 332 EXAMPLES. EXAMPLES XXYI. ■^ Determine whether the following series are convergent or divergent : 1 1 1 1 '" 1 . 2"*' 3. 4 "^5. 6 ■^••* "^(2^+1) (2/1 + 2) **■••* 1 1 1 + • a(a + 6) (a + 26)(a + 36) (a + 46) (a + 56) * ? ijJ^ 3' ^- ^ 3. 4...(n + 2) ^' 4"^4.6"^4. 6. 8'^-'^4. 6...(2n + 2) "*" "• 4 ? ?J_? 3-5.7 3.5. 7...(2n + l) 4 ■*■ 4. 7 "^4. 7. 10 ^•••■^4. 7. 10...(3«+ 1) ^ " 5 1 Ll_^ 1.3.5 1.3. 5...(2n-l) 3"^3.6'^3.6.9"^*""^ 3.6.9...3/* "*" '" « 1 1 1 1 05 a;+ 1 a;+ 2 a; + 3 7 JL 1 1 o 1 1 1 8. ^ + :; -„ + 3 ■ ••! l+£c l+aj' l+os' ^ 1 X X* OJ* 1+a; 1+ar 1 + a;* 1+ic^ in 1 1 1 1 1 + a; l + 2ic^ l+3ic'' 1 + nx J- 1 X of x^ r72'^273'^3:^"^-'^(n+l)(n + 2)'^'- 12. 1 - ^-- + ^' _... + (- 1)- -^1- + .„ 1+a l + 2a ^ ' \+7ui EXAMPLES. 333 _- , 1 + 2 1 + 3 1+n ^3- ^^rT2»^rTy'^-^iT^«-*-- 2'-P 3' -2' n'-{n-iy ^*- ^■^2»+P"^3» + 2''^'-'^n« + (n-l/"*' __ m m* m" 15. + 7T-+- x+m a; +2/72, sc+Sm 16. ^ + + 05+1 x + m x + m* (l+a)(l+6) (2+a)(2 + 6) (n + a)(n + 5) 1.2.3 2.3.4 ■^w(/i+l)(n+2)"^ no 1 2 3 19. 20. 1+72 1 + 2^3 1 + 3^4 • l+nV^n _N^ + _V3_^_^^+ +^^ + 2 + ^2^3+ 3^4+74 -^w + Vn^- 21. (^2-l) + (J5-2)+...+(7;?Tl-n) + .„ a; 1* ■ 3* ■ 5* "^ "^ «„ 2 4a; 6a;" 2na:" 23- 2^T^T0^-^n'+l^ 24. 3 1 1 , 2n-5 „ , 4 2 12 w'-5w 25. Shew that the series 111 1 V-x ' V-x' 3«-a;' - ' n'-x is convergent for all values of a;, except only when x is the square of an integer. 26. %(-J—--l-\rrx\ 27. ^v} {J{n -l)-2J{n-2) + J(n- 3)} a;\ CHAPTER XXII. The Binomial Theorem. Any Index. 282. It was proved in Chapter xx. that, when n is any positive integer, (l + Wy:r.l+rUV + '^-^^K'-^ We now proceed to prove that the above formula is true for all values of w, provided that the series on the right is convergent. When n is a positive integer the above series stops, as we have already seen, at the (w -f l)th term; but when n is not a positive integer the series is endless, for no one of the factors ti, w — 1, n — 2, &c. can in this case be zero. It should be noticed that the general term of the binomial series, namely — ^ ^-^ ~ — ^ (^> Ir cannot be written in the shortened form ■, — ~ — a?'' unless \r \n — r n is a positive integer; we may however employ the notation of Art. 241, and write the series in the form ^ [2 [3 [r BINOMIAL THEOREM. ANY INDEX. 335 283. Proof of the Binomial Theorem. Represent, for shortness, any series of the form l + ppa? + y^a;'^ + ... [1 |£ + .— ^ «;'■+... by /(m). Thus* 11 /W-l+g- + J'^+ ^^,r^ /(^)^1+|^^ + |^^+ + %^+ , and Now the coefficient of cf in the product /(m) xf(n) is + „ . . + ^' ' - + ... + — ^^^r- + ... + ^r+ ••• + l^'~« |» '^r-«^«+'" +^rr |r |r-l |1 |r-2 |2 |r-g |g "• \r' that is 1 f k Ir ( r — » 15 And, by Vandermonde's Theorem [Art. 249 or 261], this coefficient is equal to ^ — i ?, which is the coefficient [r of af in/(m + n). Thus the coefficient of any power of x in /(m + n) is equal to the coefficient of the same power of x in the product f{in)xf{n)\ also the series /(m), f{n) and /(m + n) are convergent, for all values of m and w, when a; is numerically less than unity [Art. 278]. It therefore follows from Art. 280 that /(m) x/(7i)=/(m + n) (a), for all values of m and n, provided that x is numerically less than unity. BINOMIAL THEOREM. ANY INDEX. Now it is obvious that/(0) = 1, and that /(I) = (1 + a;); "we also know that if r be a positive integer f{r) = (1 + xj. Hence, by continued application of (a), we have f{m) xf{n) xf(p) X ... =f(m + n) xf(j)) x ... =/(m + n+^ + ...). T Now letm = n=|?=... = -, where r and 8 are positive integers; then taking s factors, we have {/(.-)}'=/6")=/» But, since r is a positive integer, /(r) is (1 + a?/; .•.{/0}' = (1 + .)'; This proves the Binomial Theorem for a positive fractional exponent: the theorem is therefore true for any positive index. And, assuming that the binomial theorem is true for any positive index, it can be proved to be true also for any negative index. For, from (a), /(-»)x/(«)=/(-« + «)=/(0). Hence, as/(0) = 1, we have /(- n) = -jj-r =77- — r-„ , since n is positive, fW (1 + a?) =(i+^r. Hence (1 + a;)"** —f(— n\ which proves the theorem for any negative index. 284. EuleiOs Proof. Euler's proof of the Binomial Theorem is as follows. euler's proof. 337 Represent, for shortness, any series of the form ^ , ^ m(m-l) ^ ^ . m(m-l)...(m — r+1) ^ 1.2 k* by/(m): thus |2^ + - + P /(m)=l+m^ + ^^a;» + ... + !^^aj- + (i), f(n)^l+7i^ + p^« + ... + ^^ ^^ + (ii), and, Now, if the series on the right of (i) and (ii) be multi- plied, and the product be arranged according to ascending powers of x, the result must involve m and n in the same way whatever their values may be. But, when m and n -^ are positive integers, we know that f(m) is (1 + x)"*, and > that/(w) is (1 4- a;)**, and the product /(m) xf(n) is there- .^s fore (1 + a?)"*'^, which again, as m + w is a positive integer, V is /(m + n). Hence when m and n are positive integers the product f(m) xf(n) is / (m H-n); and, as the form of ^ the product is the same for all values of m and n it follows that f(m)xf(n)=f(m + n) (a), for all values of m and n provided /(m) and f{n) are absolutely convergent. [Art. 280.] From this point the proof is the same as in Art. 288. Ex.1. Expand (l + a;)-i. Put n= - 1 in the above formula; then we have =zl-x-\-x^-sc^+ + (-1)'*t'-+ a:^ + .- 338 BINOMIAL THEOREM. ANY INDEX. This example illustrates the necessity of some limitation in the value of x; for we know [Art. 229] that l~x + x^- is not equal to =— — unless x is between - 1 and + 1. 1 +x Ex.2. Expand (l-a;)-2. We have (l-,)--l + (-2, (-., + t|LM)(.,),^(^2H-8^, ^_^^. = l + 2x+3x^+^9+ + {r+l)x''+ is clear that the result can] example, we should have 1 = 1 + 2. 2 + 3.22 + 4.2' + Here again it is clear that the result cannot be true for all values of a: ; if «=2, for example, we should have which is absurd. Ex.3. Expand (l + a;)i aeing TT-^ ^^ that is ( - l)r-i Lll^-iSr-S) ' 2.4.6...2r ^ +••— We have (1+a;) the general term being Hence (l+.)i=l4.-^A^ .. + _1^^^ Ex. 4. Expand (l-a;)"J. weh»ve (i-«)-i=i+{-j)(-.)+i::^v:L)(-^,H . + — \ (-^)''+ Hence a-.)-i=,,i.,i.3^,, .L|.5^^^),., BINOMIAL THEOREM. ANY INDEX. 389 Ex. 6. Expand {a* - Sa^x)'^ according to ascending powers of x. (a3-3a=.)i={a3(l-^)}*=„.(l-^)* ^ ^ ^""•y-'"-^' (i)v ]• After the second, all the signs are positive ; for in the general term there are r - 2 + r, that is an even number, of negative factors. that is 285. The (r + l)th term of the expansion of (1 + a?)" 7 he rth by multiplying by is obtained from the rth by multiplying by w, by ( — 1 H ) ^. Now — 1 H is always negative if n + 1 is negative ; and, whatever n + 1 may be, 71 + 1 — 1 H will be negative for all terms after the first for which r > w + 1. Hence, if x be positive, the ratio of the r + 1th and rth terms will be always negative when r > n + 1. The terms of the expansion of (1 + xY will therefore be alternately positive and negative after r terms, where r is the first positive integer greater than n + 1. If X be negative, the ratio of the (r+l)th and rth terms will be always positive when r > n + 1. The terms of the expansion of (1 — a?)* will therefore be all of the same sign as the ?^h term, where r is the first positive integer greater than n 4- 1 ; and, as a particular case, all the terms of the expansion of (1 — a?)" are positive when n is negative. 340 GREATEST TERM. For example, all the terms in the expansion of (1 -«)* are of the same sign as the rth, where r is the integer next greater than | + 1, . so that r is 3. Also, after the ninth, the terms of the expansion of 1 (l + a;)"*^ are alternately positive and negative. M 286. Greatest Term. In the expansion of (1 + xy by the binomial theorem, we know that the ratio of the 71 — 7* -f- X (r + l)th term to the rth is ± a?, that is T a?(l ); we also know that x must be numeri- cally less than 1, unless w is a positive integer. First suppose that n -\-\ is negative, and equal to — m. Then the absolute value of the ratio of the (r + l)th term to the rth term ia a il ■{ — J . Hence the rth term is =(r + l)th term according as x (l ■{ — ) = 1; that is, ,. > mx ^, ^ . > — (1 +n)a7 according as r = ^ , that is = — ^z^ ^— , Hence, if — ^j — be an integer, r suppose, the rth X —" X term will be equal to the (r+l)th term, and these will ('1-4- Yl) X be greater than any other terms. But, if — ^^j — be not an integer, the r*th term will be the greatest when r IS the integer next above — ^j — , 1 —X Next, suppose that n + 1 is positive, and let k be the integer next greater than n + 1. Then, if r be equal or 71 -f- 1 greater than k, 1 will be negative and less than unity; hence, as x must be less than unity, each term after the Mh will be less than the one before it, and therefore the greatest term must precede the A;th. And since, for values of r less than ti -f 1, '^ 1 will be GREATEST TERM. 341 positive ; the rth term will be = (r + l)th according as I — 1 J a? = 1 ; that is, according as r = ^^^ ~ . Hence, if ^^-:j — be an integer, r suppose, the rth term will be equal to the (r + l)th, and these will be greater than any other terms. But, if ^^- — be not an integer, the rth term will be the greatest when r is the , , (w -f 1) ^ integer next above ^^ — Tt" • Ex, 1. Find the greatest term in the expansion of (1 -«)"«, when x=^. Here n + 1 is negative, and \ '"'/^ _ i_^ _ 4^ Hence the fourth and fifth terms are equal to one another, and are greater than any other terms. Ex. 2. Find when the expansion of (l-x)" a" begins to converge, if 3 ^=4. Here n + 1 is negative, and — ^- — - ^ — 5 = 22i. Hence the 1 — x ^ convergence begins after the 23rd term. Ex. 3. Find the greatest term in the expansion of {a + x)^, when 4a;=3a. Since (a + x)'^ — a^ {l + -\ , the greatest term required is the term corresponding to the greatest term in ( 1 + - J . Now X / x\ 21 3 7 9 (n + l)--s-(l + -j=— .2^2 = -; hence r must be the integer next 9 greater than - , so that the 5th term is the greatest. EXAMPLES XXVII. 1. Find the general term in the expansion of each of the following expressions by the binomial theorem. (i) 0--X)-', (ii) (\-x)-', (iu) il-x)-'. 34(2 EXAMPLES. (iy) (l+a^)-^, (v) (l+x)i (yi) (l+x% (vii) (l-5a;)-i (viii) (l-5a:)*, (ix) (1-a;)-?, (x) (2a 4- 3a;)-*, (xi) (a* - 2ax)^, and (xii) (4-7a;)f 2. Find the first negative term in the expansion (i) of (1 + |a;)^, and (ii) of (1 + ^x)^. 3. Find the greatest term in the expansion of (1 + x)~^' when a; = J. 4. Find the greatest term in the expansion of (1 -fa;)"' when 05 = j. 5. After what term will the expansion of (l—x) « begin to converge, when x = ^1 6. Shew that the coefficients of the first 19 terms in the 19 — 21a; expansion of -p. rg- are all positive, and that the greatest of them is 100. 7. If ftj, a^y a^, a^ be any four coefficients of consecutive terms of an expanded binomial, prove that 8. Find the general term in the expansion by the binomial theorem of each of the following expressions according to ascending powers of x : ,.. a .... a + x ..... /a + xy (iv) {a + x)^ (a - a;)-i, (v) (a + x)' (a - a)"", and (vi) (a-xy(a + x)~^. 9. Shew that the coefficient of x'" in the expansion of {l+xy{l-x')-'h2n. 10. Shew that the coefficient of x' in the expansion of (1 + 2a;)* (1 -a;)-'' is 27 (n- 1), n^3. SUM OF COEFFICIENTS. 343 287. Sum of coefficients. The sum of the first r + 1 coefficients of the expansion of (1 — a?)* can be ob- tained as follows. We have also (1 -a;)~' = l-f-a? + a^+...+a^+... From [Art. 281] the coefficient of af in the product of the two series is equal to the coefficient of «*" in (1 — xy X (1 — a?)"*, that is in (1 — «)""*; hence we have = coefficient of x' in (1 - a:)""' = (- IY^^^t^ , LI Similarly, if (j> (x) = a„ + a^x + a^a^ + . . . + aX + • • •, the sum a„ + ttj 4- . . . + G^r will be the coefficient of a?*" in ^^ . JL ~~ X Thus, to find the sum of the first r + 1 coefficients in the expansion of (f) (x), we have only to find the coefficient of a?*" in the expansion of ^^ ^ . Ex. 1. Find the sum of the first r ooefBcientB in the expansion of (l-x)-». Ans. ir(r+l){r + 2). The sum required is the coefficient of x^^ in (1 - x)-*. Ex. 2. Find the sum of n terms of tha series 1.2.3 + 2.3.4 + 3.4.5 + Since(l-ar)-'»=;i-4-^[l-2.3 + 2.3.4a; + 3.4.5x'+ ]; the sum required = 6 x sum of the first n coefficients in the expansion of (1 - :r)-*= 6 X coefficient of .t»-i in (1 -ar)-»=jTO (w + 1) (n + 2) (n + 3). Ex. 3. Find the sum of the first n+r coefficients in the expansion of (l+x)« {l-xy- 344 BINOMIAL SERIES. The sum required = coefficient of x^*^~^ in the expansion of j^±^. Now(H-x)«=(2-l^r=2»-n.2»-i(l-a:) ^ n^-1) 2^_j ^j _ ^^a ^ higher powers of (1 - x). „ (1 + x)' 2" n2^i » (n - 1) 2»-» , -41 expression of the (n - 3)th degree. The coefficients of a;«+^i in (l-x)-*, (l-a;)-» and (l-ar)-i re- spectively are ^{n+r) (n+r+l), n+r, and 1; hence the coefficient of x«+-iini?^3i8 (1 - x)^ 2*-i (n + r) (w + r + 1) - 2«-i n (n + r) + 2«-3n (to - 1). Ex. 4. Find the sum of n terms of the series x + »+ ^^2 + 1.2.3 "^ .in*. (2n-l)!/n! (n-l)l. 288. Binomial Series. Series which are derived from the expansion of (1 + a;)" by giving particular values to X and n are of frequent occurrence: it is therefore of importance to be able to determine at once when a given series is a binomial series. The case in which the index is a positive integer needs no remark. When the index is a negative integer, we have n{n + l)...(n + r-l) ^ ^ and it should be carefully noticed that this expansion can be written in the form ... + {(r + l)..,(r + n - 1)) a?" + ...]. BINOMIAL SERIES. 345 When the index is fractional, —p/q suppose, we have _ j?(j) + g)(i) + 2g) /a;y ^ ^^^ Here we notice that (i) there is an additional factor both in the numerator and in the denominator for every successive term, (ii) the successive factors of the numerator are in an A. P. whose common difference is the denominator of the index, (iii) the successive factors of the denominator are 1, 2, 3, 4, &c., or multiples of these. Bearing in mind the above laws, there will be no difficulty in determining the expression which will pro- duce a given binomial series. Ex. 1. Find the sum of the series 1 1.3 1.3.6 ..^.. s + O-^sTeTg-*- ^^^^^^7- Writing the series in the form i ^4.1l2 i 1.3.6 j. ^ \\/l [2*3a"*" |3 •33'^ ~ ' we see from (A) that it is obtained from the expansion of (1 - a;)"* X 1 by giving to x the value found from 5 = 5. 2 3 1 3 (2\~» 1 2 2*2 /2\^ ^-3) =^+2-3 + r:2(3) + = 1+^' *^^^^^°^« -g = v/3-l. Ex. 2. Find the sum of the series . 2 2.6 2.6.8 ^ . ^ .. '-^6 + 67i2 + 6n2Tr8+ tomfimty. Writing the series in the form ■•■ [1 * 6 "*" ! 2 62 "^ |3 '68"^ we see from (A) that it is obtained from the expansion of (l-at)"* s. A. 24 346 BINOMIAL SERIES. X 1 by giving to x the value 5 = « . Hence the sum required is o o {^-T-^- 3 37 3 7 11 Ex. 3. Find the sum of the series jg + ^3' 24 + i s' 24 30 "^ '° infinity. In this case the factors of the denominator, although multiples of 1, 2, 3, 4, &c., do not begin at the beginning. Additional factors must therefore be introduced in the denominator, and corresponding additional factors in the numerator. We then have (-5)(-l)3 1 (-6)(-l)3.7 l' |3 63"^ j4 6*"^ Now the terms of this latter series are terms of (A), if 5=4, 2>= -5, andj = g. We can therefore find the required sum, as follows : A_4\i_ 5 1 6.1 1 5.1.3 1 5.1.3.7 1 \ 6J ~ l'& [2 6a''" |8 e*"*" I4 6*"^ _ 5 _5_ _6_rj^ 3.7 3.7.11 "I ~ 6"^6.12'^6.12 Ll8"^ 18.24"^ 18.24.30"^ J' /l\l ,555-, -[y =^-6 + 72 + 72^- Whence fif = i{84/27 -17}. Q Q fi *l i5 7 Ex. 4. Find the sum of the series -r + -^ + . ' ' + to infinity. 4 4.8 4.8.12 [--(-!)"*]• Am. 2^2-1. Ex. 5. Find the sum to infinity of the series J— -111 1-3.5 28 |3 2* [4"^ 2»|6 ~ [From (1 + 1)^]. Am. ^-- ^2. Ex. 6. Shew that 1 + 1^1^ + ^^ + 14^6^ "> ^ ... ,22.6 2.5.8 2. 5.8. 11 ^ ■ ~ -^ fimty = l + g + J-3-2+ ---j-.-_ +6_ia^j^8^g^ + to mfimty. [s..(i-|)-*=(i-^)-^. THEOREMS OBTAINED BY EQUATING COEFFICIENTS. 347 289. We know from Art. 281 that if any expression containing x be expanded in two different convergent series arranged according to ascending powers of x, the coefficients of like powers of x in the two series will be equal. By means of this very important principle many theorems can be proved. Ex. 1. Shew that, if n he any positive integer, Wehave (l-^).=l-„^+2j2^) x>-ii2f4<^^+ n(n-l)...(n-iiri) ^ ' 1.2...n Also, provided a? > 1, we have {^-r- , .^1 . w(M + l) 1 . n(n + l)(n + 2) 1 , ^ + "^■^"1:2-^+ 1.2.3 i5 + yt(n-t-l)...(n + n-l) 1 "•■ 1.2...n x""^ is equal to the coefficient of ar® in (1 - x)» x ( 1 - - j , that is equal to the coefficient of «<* in ( - l)*»a;«, which is zero. [See also Art. 251, Ex. 3.] Ex. 2. Find the sum of . -.. 1 / t.1-3 .-. 1.3.5... (2n-l) („+l)+n.^ + (n-l)2^ + + 1. _____L___Z. [Equate coefficients of x» in (1 - x)~^ x (1 - x)-^ and in (1 - x)"^.] .7 ...(271 + 2.4...2/1 ^™. 5 •I.-J2«+3) Ex.8. Shew that l-3« + '^-l"^''-''' -;..... = (-l)'- J. . A We have :; = = = « = = 73 : . Hence (l + a;){l -xs + ajs... + ( -irx^"+ ...} = 1 + a- (1 - «) + «2 (1 - a;)2 + „ . + a:3n+i (1 _ ^)3n+i .^. _ The coefficient of x^"^^^ on the left is ( - 1)". 24—2 348 EXPANSION OF MULTINOMIALS. The terms on the right which give x'^+i are aja^H-i (1 _ a.)»i.+i + a;8'»(l - x)^+7^-'^ (1 -ar)3»-i + ...; and hence the coefficient of ac^^^ will be found to be ,_ (3n-l)(3n-2) (3w - 2) (3n - 3) (3n - 4) . l_8n+ j-^ J-2--3 +..... 290. Expansion of Multinomials. Any multi- nomial expression can be expanded by means of the binomial theorem. Since (p -{- qx -\- nc* -\- . . .)" may be written in the form p*ll-{--x+ - x^ -\- ...] , it is only necessary to consider \ p P J expressions in which the first term is unity. Now in the expansion of {1 + aa;4- ta;" + ca:* + ...}", that is of (1 +(cM7 + 6a;* + ca?'+ ...)}*, by the binomial theorem, the general term is —5^ ^^ .^ — ^ ^^ (ax + ha? + ex* +...); also in the expansion of (aaj + fta?" + ca;' + ...)'', r being a positive integer, the general term is by Art. 262 r— T^^l a**6^c^...a;*+2^+»v+-, where each of a, /3, 7,... is zero or a positive integer, and a + /8 + 7+...=r. Hence the general term of the expansion of the multinomial is n(n-l)(n-^2)...(n-r + l) ^.^^.....^..^.a,..., |a [^ |7... To find the coefficient of any particular power of a?, say of of, we must therefore find all the different sets of positive integral values (including zero) of a, ^, 7,... which satisfy the equation a + 2^ + 87 + ... = A;; the cor- responding value of r is then given byr=a + ^ + 7+..., and the corresponding coefficient is found by substituting EXPANSION OF MULTINOMIALS. 349 in the formula for the general term. The required coeiBQ- cient will then be the sum of the coefficients corresponding to each set of values of a, /3, 7 Ex. 1. Find the coefficient of afi m{l-x + 2x^- Bx^)-2' The values of a, /S, 7 which satisfy a + 2/8 + 87 =6 will be found to be 0, 1, 1; 2, 0, 1; 1, 2, 0; 3, 1, 0; and 6, 0, 0. The cor- responding values of r will be 2, 3, 3, 4 and 5 respectively; and the corresponding coefficients will be L|iJi(.)>(-3).. HK_|)(J)<-in-3)-. lili illi (-D(-l)(-i)(-i)(-: and V ./ ^ ./ V .^ V ./ V 2/ (.ij». ,, , . 9 45 16 35 , 63 thatis--. ^, J. -jg and — . 31 Hence the required coefficient is — ^ . 256 291. From the above example it will be seen that the process of finding even the first six terms in the expansion of a multinomial is very laborious; in many cases, however, the work can be much shortened, as in the following examples. Ex. 2. Find the coefficient of x^^ in the expansion of (l + a;+x2 + ar» + a;*)-3. We have {l+x + x^ + x^ + x*)-^= f f^)'^ = ^^ ~ ^^^ ^^ ~ ^'^^~' = (l-2x + a:a)(l + 2x» + 3xi« + 4a;"+...). Hence the coefficient required is zero. Ex. 3. Find the coefficient of «» in the expansion of {l+x+x^+a^)"\ We have {l + x+x' + x^)-^=: ___i— _= ll^ ' 1+X + X^ + X^ 1-X-* ^(l-x){l + x*+x^+...+x*r+...). 360 COMBINATIONS WITH HEPETITIONS. Hence the coefficient of x*^ is 1, the coefficient of x*^^ is - 1, the coefficient of x*^'^^ is zero, and the coefficient of oc^^'^^ is zero. Thus the coefficient of a;" is I when n is of the form 4r, it is - 1 when n is of the form 4r + 1, and it is zero when n is of either of the form8 4r + 2 or 4r + 3. Ex. 4. Find the coefficient of «♦" in the expansion of (l + 2x + 3«a + 4a;3+ to infinity)". Since l + 2x + Sx^+ = (l-a;)"^ the required expansion is that of (1 - x)~^^; the coefficient of x^ is therefore 2?i (2/1 + 1). ..(2n + r-l) \L 292. Combinations with repetitions. The number of combinations of n things a together of which p are of one kind, q oi a. second, r* of a third, and so on, can be found in the following manner. Let the different things be represented by the letters a,b, c,...'y and consider the continued product (l+cw;+aV+...+a'a;*')(l4-6ic+...+6V)(l+ca;+...4-cV)... It is clear that all the terms in the continued product are of the same degree in the letters a, 6, c,... as in a;; it is also clear that the coefficient of of- is the sum of all the different ways of taking a of the letters a, h, c,... with the restriction that there are to be not more than p a'a, not more than q 6*s, &c. ; so that the coefficient of of- in the continued product gives the actual combinations required. Hence the number of the combinations will be given by putting a = 6 = c=... = l. Thus the number of the com- binations of the n things a together is the coefficient of a^ in (1 +a; + ar' + ... + a;**) (l+a?+...+ a;') (1+^+ •••+«?")... Permutations. The number of permutations of the n things a together being represented by P^, it is easily seen that (, , aj , a;* x^) (^ x x^ x") li P P P PERMUTATIONS. 351 For la X the coefficients ofaf^ in aV . aV] {'-5-f--f) is the sum of all possible terms of the form 6V . 6W ^ lii m .., a'S"*,. for which Z + m + ... = a, and the number of permutations a together formed by taking I of the a's, m of the 6's, &c. is 111 m Ex. 1. Find the number of combinations 7 together of 5 a's, 4 6's and 2 c's. The number required is the coefficient of x"^ in {l + a; + ...-f o^) (l + a;+ ...«*) (1 + a: + x2), that is in {I - x<^) {1 - x^) {1 - x^) {1 - x)-K Eejecting terms of higher than the seventh degree in the continued product of the first three factors, we have (1 - a;8 - aH^ - a;«) (1 + 3x + 6x2 + lOx* + 15a:* + 21ar» + 28x8 + 36*7 +...) ; and the coefficient of x' is 36 - 15 - 6 - 3= 12. Ex. 2. Find the total number of ways in which a selection can be made from n things of which p are alike of one kind, q alike of a second kind, and so on. The total number of the combinations is the sum of the coef- ficients of a;\ x^..., x« in (l + a; + ... +xP) (l+x + ...-1-a;^)... ; and this sum is obtained by putting x=l in the product and subtracting 1 for the coefficient of afi. Hence the required number is {p + l){q + l)...-l. The above result can, however, be obtained at once from the consideration that there are^+1 ways of selecting from the a's, namely by taking 0, or 1, or 2,... or j? of them; and, when this is done, there are g + 1 ways of selecting from the 6's; and so on. Hence the total number of ways, excluding the case in which no letter at all is selected, is {p + l){q + 1)... - 1. [Whitworth's Choice and Chance, Prop, xm.] 352 HOMOGENEOUS PRODUCTS. Ex. 3. A candidate is examined in three papers to each of which m marks are assigned as a maximum. His total in the three papers is 2m; shew that there are -(m+l)(m + 2) ways in which this may occur. The number of ways is the coefficient of x^^ in (l + a5 + a:2+ ...x"*)', that is in (1 - x»"+i)3 (1 - x)-8 = (l - 3a;»»+i+ ...) x -{1.2 + 2. 3a:+...+w(nn-l)a;"*-i + ...+ (2m + l){2m + 2) x^+ ...), Hence the number required = i{(2m+l)(2m+2)-3»n(m + l)}=i(m+l)(m+2). Ex. 4. Shew that the number of permutations four at a time which can be made of n groups of things of which each consists of three things like one another but unlike all the rest is n* - n. The number required is equal to J4 x the coefficient of a;* in 293. Homogeneous Products. We have already [Art. 250] found the number of homogeneous products of r dimensions which can be formed with n letters, where each letter may be repeated any number of times. We now give another method of obtaining the result. Suppose the letters to be a, 6, c,... ; then if the continued product (1 +aa; + aV + aV -f ...) x (1 + 6a; + 6V + 6V + ...) X (1 + ca; + cV + cV + ...)... be formed, the coefficient of oc^ will clearly be of r dimen- sions in the letters a, 6, c,..., and will be the sum of all the possible ways of taking r of the letters*. Hence the number of the products each of r dimensions will be given by putting a=6=c=... = l in the continued product Thus the number required is the coefficient of of in (1 + a? + a;* + . . .)", that is in (1 — a;)"*. Hence n{n + l)...{n + r'-l) |n + r — 1 M, t n-1 This result can be expressed in the form ^11^ = n+r-fir- * An expression for the sum of the homogeneous products will be found in Art. 300, Ex. 4. EXAMPLES. 353 I f Cor. The number of terms in the expansion of (a, + a, + a,+ ,.. + aJ is ^—^^. 294. We shall conclude this chapter by solving the following examples. Ex. 1. Find ,^14, by the binomial theorem, to six places of decimals. =4 {1 - -0625 - -001953 - -0001220 - 0000095 - -0000010} =3-741667. Ex. 2. Shew that, when x is small, (l-3x)-f + (l-4«)-i ,3 . ^, ^--j — ^^ ^ = 1 + jr a; approximately. Since x is small, its square and higher powers may be rejected ; and when all powers of x except the first are neglected the given expansion becomes equal to , 1 o , 1 . 2 + 2« 1 + x l + g.3x + l + ^.4x (l + |a;)(l + *r = ^l + 5x)(l z) = l + ^x. Ex. 3. Shew that the integral part of (s/3 + 1)^+^ is (^3 + l)2«-i-i - (^3 - l)2«+i. Since ^3-1 is a proper fraction, (v'3-l)''"+i must also be a proper fraction. It therefore foUows that if ( ^3 + 1)^+^ - (x/3 - 1 )2«+i be an integer, it must be the integral part of (/v^3 + l)2«+i. Now (V3 + l)^+^-(x/3-l)2«+i = {3V3 + (2n + l)3* + ^-?^^t^3»-V3+ + (2n + l)^3+l} -{3»V3-(2n + l)3»+ /3 + 1)^**"^^ - (iy/3 - l)2»+i is an integer, and is moreover an even integer. By the following method it can be proved that (x/3 + l)2»+i - (,^3 - l)2»+i is an integer divisible by 2~+i. Represent (^3 + l)2«+i - (,^3 - 1)^+^ by I^^. Then Ji=2; and it will be found that 28=20, and also that (x/3 + l)9 + (x/3-l)2=8. Hence 8l2«+i= {(x/3 + 1)=^+^ - {s/3 - l)^+n {(V3 + 1)' + {v/3 - 1)'} = (x/3 + 1)2"+3 _ (^3 - l)a«+8 + 4 {(^3 + l)2«-i - ( J3 - l)2«-i} ; •*• ■f2n+3 = ^^2n4-l~^^2n-l {^)- It follows from the last relation that I^n+s will be an integer if ■Tan+i aiid I^-i are integers. Now we know that I^ and /, are integers ; hence by induction lin-^i is always an integer. The relation (A) also shews that I^^^ will be divisible by 2"+* provided Ijjn+i is divisible by 2»+i and I^n-i by 2'*. Now we know that Ii is divisible by 2^ and Ij by 2^; hence Ij must be divisible by 2^ ; and it will then follow that ly must be divisible by 2* ; and so on, so that I^n^i is always divisible by 2**+i. Ex. 4. To shew that, if n be any positive integer, a"-n(a + 6r + "^^\a + 26)*- = (-6)"ln. Put ^—r— for X in the identity proved in Art. 259, Ex. 3 ; then, after reduction, we have l^! ^_0o c^_ {y + a){y + a+b) ...{y + a+nb) y + a y+a+b Now expand the expressions on the two sides in powers of - . ln&* |n6» Left side = j l j ,. = -—rr + higher negative powers of y. Eight .iae = ^(l + ?)--... + ,-ir|(l^"±^y+ , hence the coeflS*oient of -j^^ on the right is (-l)*[Coa*-Ci(a + &)*+ + (_i)rc^(a + r6)*+...]. Hence S (- l)»*Cy(a + r&)* is zero if k»-l) ,... 2«(2«-l)(2»-2)(2»-3) ' "~M~ El! •• |2n [47* "** jn [»"|2w [2»* 22. . Shew that n(n+ l)...(yi + m~l) n{n + 'i)...{n + m-4) fi — — n(n-l) n{n+l)..,(n + m — 7) 0, 1.2 |m-6 if m > 2n, and = 1 if w = 2w. 23. Find the coeflGlcient of x" in (l+a;)(l+a;»)(l+a;*)(l+a;»)... 24. Shew that, if a; be a proper fraction, 25. In how many ways can 12 pennies be distributed among 6 children so tliat each may receive one at kast, and none more than three 1 EXAMPLES. 359 26. There are n things of which p are alike and the rest unlike ; prove that the total number of combinations that can be formed of them is (;? + 1) 2"~'' - 1. 27. Shew that the number of ways in which n like things can be allotted to r different persons, blank lots being admis- sible, is ,^,_,(7,_,. 28. Shew that the number of combinations n together of 2n things, n of which are alike and the rest are all different, is 2". 29. The number of combinations n together of 3n things, of which n are alike and the rest all different, is 2'"-' + |2n-l /|n |?^-l. 30. A man goes in for an examination in which there are four papers with a maximum of m marks for each paper ; shew that the number of ways of getting half marks on the whole is 31. Find the coefficient of a;* in (1 - 2r- 2x')^. 32. Find the coefficients of a:' in the expansions of (l+a; + jc' + as' + xy and (I +x + x' + x^ +x* + x^, 33. In a shooting competition a man can score 5, 4, 3, 2, 1 or points for each shot. Find the number of different ways in which he can score 30 in 7 shots. 34. In how many ways can 20 be thrown with 4 dice, each of which has six faces marked 1, 2, 3, 4, 5, 6 respectively? 35. Find the coefficient of oj' in the expansion, according to ascending powers of a;, of (4a' + 6ax + 9a^)~\ 36. Shew that the coefficient of x^"" in the expansion of l+x . g, T -,Tb is 2w+ 1. {l + x + a^y 37. Shew that the coefficient of oj' in the expansion of (1 + 2a; + 3a;*+ ...)' is J (r + 1) (r + 2) (r + 3). -J\ 360 EXAMPLES. 38. Find the coefficient of a;" in the expansion of {1 . 2 + 2 . 3a; 4- 3 . 4ic» + ... to infinity}'. 39. Find the coefficient of t^ in the expansion of (1.2 + 2. 3. 2a; + 3. 4. 2 V + + (w + 1) (« + 2) 2-0^ + ... to infinity)". 40. Shew that the coefficient of v^ in the expansion of (1 +a;+ 2a;« + 3a;' + ...)* is Jr (r» + 11). 41. Shew that if p-q be small compared with p or q^ then will l'p_ ^ (n+l);? + (n-l)g ^ q (n-l)p + {n+l)q 42. If (6 76 + 14)""-'' = iV^, and F be its fractional part; then will IirF= 20'"-^'. 43. If (3 ^3 + 5)^"'+' = I+F, where I is an integer and F a proper fraction, then will F{I+F) = 2*'*\ 44. Shew that the integer next greater than (3 + ^7)"** is divisible by 2'""*'*. 45. If w be a positive integer, the integer next greater than (3 + ^5)"* is divisible by 2"*. 46. Shew that the general term in the expansion of l+x+y + xy 1 + X + y |m + n— 2 \m-\ 47. Shew that the coefficient of a;'' in the expansion of ^ (r'-l')(r'-2')(r'-3') ^. ^ _ | ^ EXAMPLES. 361 48. Shew that 1.2.n + 3.4l^M + 5.6^*i!4^^ + {2n-S)(2n-2).n + {2n-l)2n.l= 2 V. 49. Shew that the coefficient of x"*'"^ in the expansion of 50. Shew that the coefficient of ic"*''"* in the expansion of |i^|is(-ir(.-2„)2'-. 51. Shew that 7i" - w (w - 2)" +— ^ — ^ (w - 4)" - ... to n + 1 terms = 2.4.6. 8...2n. 52. Shew that a--^^ -n(a + 6)"^' + !L^L^) (« + 26)"*^ - ... = J |n+l (2a + nb) {- b)\ 53. If three consecutive coefficients in the expansion of any power of a binomial be in arithmetical progression, prove that the index, when rational, must be of the form q* — 2, where ^ is an integer. 54. Shew that the sum of the squares of the coefficients in the expansion of (1 +aj + a;*)", where n is a positive integer, is tl 2w-2r' 55. Shew that, if n is any positive integer, n(n-l) n(n-l)(n-2){n-S) 2(2r+l)'*" 2.4(2r + l)(2r + 3) '^ '" -9» r(r + l)(r + 2)...(r + n- l) 2rr2r+l)(2r+2)...(2r + n-l)* s. A. 26 CHAPTER XXIII. Partial Fractions. Indeterminate Coefficients. 295. In Chapter viii. it was shewn how to express as a single fraction the algebraic sum of any number of given fractions. It is often necessary to perform the converse operation, namely that of finding a number of fractions, called partial fractions, whose denominators are of lower dimensions than the denominator of a given fraction and whose algebraic sum is equal to the given fraction. 296. We may always suppose that the numerator of any fraction which is to be expressed in partial fractions is of lower dimensions in some chosen letter than the denominator. For, if this be not the case to begin with, the numerator can be divided by the denominator until the remainder is of lower dimensions : the given fraction will then be expressed as the sum of an integral expression and a fraction whose numerator is of lower dimensions than its denominator. 297. Any fraction whose denominator is expressed as the product of a number of different factors of the first degree can be reduced to a series of partial fractions whose denominators are those factors of the first degree. For let the denominator be the product of the n factors x — a, x — b, x — c,...; and let the numerator be represented by F{x), where F(x) is any expression which is not higher than the (n — l)th degree in x. PARTIAL FRACTIONS. 363 We have to find values of A, B^ 6',... which are independent of x and which will make i^(^) _ A B G ^ {x — a){x — h){x — c)... X — a x — b x—c ' or, multiplying hy (x — a)(x — b)(x — c) , F(x) = A(x-b){x-'C) +B(x-a)(x-c) + G(x-a)(x'-b) .....(i). In order that (i) may be an identity it is necessary and sufficient that the coefficients of like powers of x on the two sides should be equal. Now F(x) is of the (n— l)th degree at most, and the terms on the right of (i) are all of the (n — l)th degree ; hence, by equating the coefficients of x^, x^,... a?""* on the two sides of (i), we have n equations which are sufficient to determine the n quan- tities A, B, Cf The values of A, B, G,... can however be obtained separately in the following manner. Since (i) is to be true for all values of x, it must be true when x = a; and, putting 37 = a, we have F(a) = A {a — b)(a — c) ; and therefore A = F(a)/(a — b)(a — c) Similarly we have B = F{b)/(b - a) (6 - c). . . ; and so for G, D,.. .. We have thus found values of A, B, G,... which make the relation (i) true for the n values a, b, c, . . . of x ; and as the expressions on the two sides of (i) are of not higher degree than the (w — l)th, it follows [Art. 91] that the relation (i) is true for all values of x. Thus Fja^) ^^ Fja) 1 (x — a) (x — b) (x — c). ,. (a — 6)(a — c). . . a? — a * In the above it was assumed that all the factors of the denominator were known and were all different. The general theorem is the following: — 25—2 363* PARTIAL FRACTIONS. If .^ })Q a/fiy fraction in which iV, P, Q are rational and integral functions of x, and N of lower degree than PQ; then, provided P and Q are prime to one another in x, two other functions A and B, rational and integral in w, can be found such that PQ-P'^Q- For, since P and Q are prune to one another, two integral functions in oo, G and D suppose, can always be found such that GQ-\-I)P = 1, [Art. 100] CN DJSr N ^^^••- -p-^-Q^PQ (°^>- Now let CN/P = L-\-AIP, where L is an integral expression in oc and A is of lower dimensions than P in a; ; and similarly let DN/Q = if + B/Q. Then, since N is of lower dimensions than PQ, it follows from the identity (a) that i/ + ilf=0, and that PQ'P'^Q ^^^• From (l3) it immediately follows that, if a, y8, 7, ... are all prime to one another, we can always find integral functions A, B, G, ... of lower dimensions in x than a, y8, 7, ... respectively such that N A B G + -7^+-+.... x + 2 Consider, for example, the fraction x^ + 2x + 1 ~x + S)x^ + 2x + l{x-l a;2 + 2a;-3 PARTIAL FRACTIONS. 363* Then x + S = x^ + 3x + 4:- {x^ + 2x+l), 4 = a;2 + 2x+l-(x + 3)(a:-l) = x^ + 2x + l-{x^ + 3x + 4~{x^ + 2x + l)}{x-l) =:{x^ + 2x + l)x-{x^+3x + 4,){x-l). Hence 4t {x +2) = (x^ + 2x + l)x {x + 2) - {x^ + Bx + i) {x-1) ix + 2); . x + 2 X (a; + 2) _ (a;-l)(a; + 2) *'• {x + 1)2 (x2 + Bx + 4) ~ 4^2 + 3a. + 4) 4 (^2 + 2a; + 1) Mi__^±i4_iJi__^_±i_l 4[ a;2 + 3a; + 4j 4] a;'^ + 2a; + lj x+B x+4 4(a;2 + 2a;+l) 4(a;2 + 3x + 4)* Again, for the fraction ^^^^^^^^^^^ . ,3 a;2 + 6a; + 9|a:^ + 6a;2+12a; + 81a5 9 x^ + Gx^+ 9x 3.'r + 10|9a:2 + 54a; + 81| 3a; + 8 9x2 + 54a; + 80 Thu8 3a; + 8 = (x + 2)3-a;(a; + 3)2, and l = 9(a; + 3)2-(3a; + 8)(3a; + 10) = 9 (x + 3)2 - { (x + 2)3 - a; (a; + 3)2} (3a; + 10) = (3a;2 + 10a; + 9) (X + 3)2 - (3a; + 10) {x + 2)^. Hence a^ a;2(3ar'+10x + 9) _ x2(3a;+10) (a; +2)3 (a; + 3)2- (a; + 2)3 (a; + 3)2 ^ ^. 21x2 + 72a; + 64 „ . 21x + 72 = ^^-« + — (^T2P ^^ + ^-l^?3F _ 21x2 + 72x+64 21X + 72 (x + 2)3 (x + 3)2 • Ex. 1. Resolve -. -r-, — — - into partial fractions. (x-l)(x-2) Assume 3x + 7 _ ^ . ^ . ^"'""^^ (x-l)(x-2) =.rri + ^32 ' then 3x + 7 = ^(x-2) + fi(x-l). 364 PARTIAL FRACTIONS. In this identity put x = l; then 10 = -^. Now put x=2; then 18 =B. Sx + 7 13 10 Thus {x-l)(x-2) x-2 x-1' Ex. 2. Resolve ; ^)v^ — '^rr- — I i^*^ partial fractions. {x-a)(x-b){x-c) (t-.J (o-a)(a-t) ^ J_^B^ {x-a){x-b){x-c) x-a x-b x-c* then (h-c){c-a){a-b)=A{x-b){x-c)-\-B{x-c){xr-a) + G{x-a)(x-b). Putting a; = a, we have (b - c){e - a){a -b) = A{a -b){a - c)\ there- fore A=c-b\ and the values of B and G can be written down from symmetry. Thus ('-0(;-')(«-'') ^ «^ ^ gjif ^ ^ng . Ex. 3. Resolve —. rr-7 z- -. r into partial fractions. Assume '+ —i:+^+ -/-+...+ a;(a? + l)(a + 2) ... (a5 + w) x as + 1 a? + r "■ a; + n Then, we have l = Af,{{x + l){x + 2)...{x-\-n)}-\-A^{x{x + 2){x + Z)...{x->rn))-if...+ il^{a;(a; + l)...(« + r-l)(a; + r+l)...(a; + n)}+...+^^{a;(« + l)...(a? + n-l)}. If we put a;=0, all the terms on the right will vanish except the first, and we shall have 1 = ^5X In, so that ^o=l/jw. To find the general term, put x= - r ; we then have l=^^{(-r){-r+l)...(-l)(l)(2)...(n-r)}, that is 1= ( - 1)*-^, |£ jn - r; hence ^,= ( - l)*'/[r |w-r. Hence the required result is «(«+l)...(«+n) \n\x -'^^ ^^ \r\n-r x+r'-^^ ^x + n\' [See Art. 259. Ex. 3.] (a-b){a-e) x-a PARTIAL FRACTIONS. 865 Ex. 6. Resolve -. , , , „ ^ into partial fractions. The factors of x" + 2a; + 5 are the complex expressions a; + 1 + 2i and a; + 1 - 2i, where t is written for \/ - 1. x^+lb A B C Assume ;; rrrii — 7i 5\= r + (x-l)(a;2 + 2a; + 5)-a;-l a5+l + 2i a; + l-2i ' .% «3 + 15=.i(a;+l + 2i)(a;+l-2i) + B(«-l)(a; + l-2t) + C{x-l){x + l + 2i). Put a;=l; then 16=8^4, so that 4 = 2. Puta;=-l-2t, then (l + 2i)2 + 15 = J5 (-2-2i) (-4i), that is 12 + 4i = B ( - 8 + 8i) ; therefore JB = - ^^. . Change the sign of i in the value of B, and we have C= - Thus 2 + 2i * ar«+15 2 3 + i 1 3 (ar-l)(a;a + 2x + 5) x-\ 2-2ia? + l + 2» 2 + 2i a? + l 298. We have in the last example resolved the given fraction into three partial fi-actions whose denominators are all of the first degree, two of the factors of the denomi- nator being imaginary. Although it is for most purposes necessary to do this, the reduction into partial fractions, of a fraction whose denominator has imaginary factors, is often left in a more incomplete state. Take, for example, the fraction just considered, and assume fl^ + 15 A Bx + G [It is to be noticed that we must now assume for the numerator of the second fraction an expression containing X but of lower degree than the denominator.] Then 0^-^15=: A{x' + 2x + 5) + (Bx-{-G)(x-l). Putting a; = 1, we have 16 = 8^, so that A = 2. Put il = 2 in the above identity; then after transposi- tion -x'-4^x + 5 = (Bx + C)(x-l); or, dividing by a? — 1, Bx-\'C = — x — 5. a^-\-lo 2 x + 6 Thus {x-l)(x'-\-2x-^5) a?-l a;'+2a? + 6* 366 PARTIAL FRACTIONS. 299. We have shewn in Art. 2!J7 that a fraction can always be resolved into partial fractions the denominators of which are prime to one another. The process thus indicated would, however, be very tedious. When factors are repeated the following method may be used. Ex. 1. Express , ^, 5-. in partial fractions. We may assume that 2a; + 5 A B _^ . ^ (a;-l)»(x-3)-(x-l)3"^(x-l)a"*" (x-l)'^ x-S' or, clearing from fractions, 2x + 5 = A{x-S) + B{x~l){x-3) + C{x-l)Hx-3) + D{x-l)». By equating the coefficients of x*, x^, x^, x^ on the two sides of the last equation, we shall have four equations to determine the four quantities A, B, C, D, so that the assumption made is a legitimate one. The actual values of A, B, C, D are not however generally best found from the equations obtained by equating the coefficients of the different powers of x. In the present case, the following method is more expeditious. Put x-l=y; then we have 2 + 2y + 5=:A(y-2)+By{y-2) + Cy^y-2) + Dy\ Now equate coefficients of y^, y^, y^, y^, and we have 7=^ -2Ai 2 = A-2B; 0=5-2(7; andO = D + (7. XKTU .« 7 o 11 ^ 11 , ^ 11 Whence A = -^, B= - j-, C= -— &ndD=— . „ 2x + 5 11 7 11 11 Hence {x-l)^x-S)''8{x-B) 2(a;-l)» 4(a:-l)> 8(aj-l)* (l + a;)" Ex. 2. Express the fractional part of ^ — ~-= in partial fractions. (1 — Zx) Assume (l + a;)« A B C . . , (T^2i)»^ (Tr2^+ (1^12^+ (132^) + ^ ^^*'^'^^ expression. Then (l + x)*=:A+B{l-2x) + C{l- 2x)* + (1 - 2« )» X integral expression. Now put 1 - 2a?=y ; then {H-x)«=(?-|y=i-J3'»-«3H-iy+^i^3»-2y2^. terms con. taining higher powers of y). PARTIAL FRACTIONS. 367 Also right Bide— A + By + Cy^ + y^ x integral expression in y. Hence, equating coefficients of j/", y^, y^, we have 300. The following examples will illustrate the use of partial fractions. Ex. 1. Find the coefficient of a** in the expansion of j— ^ ^ according to ascending powers of x. 1 3 2 We have l-5a; + 6a;» 1-3j: l-2a; = 3{l + 3x + (3.T)«+... + (3a:)*+...} -2{H-2x + (2x)»+...+(2a;)«+...}. Hence the required coefficient is S^+i - 2"+i. (1 + x\^ Ex. 2. Find the coefficient of a;""*^ in the expansion of -^ — --., . ^ (1 - 2xf From Ex. 2, Art. 299, we have (1+a;)" _ S" 1 _ n3»^i 1^ _ n (w - 1) 3*-^ 1 (1 - 2a;)3 ~ 2» (l-2a;)» 2'* (T^ip "*" 2«+i 1 - 2a; + an integral expression of the (n - 3)th degree. Whence the re- quired result. Ex. 3. Shew that the sum of all the homogeneous products of n dimensions of the three letters a, 6, c is equal to g*-^ (c - 6) + b"-^'' {a-c) + c'*+'^ (6 - o) (& -c){e- a) (a - 6) The sum of all the homogeneous products of n dimensions is the coefficient of x* in the product (l + aa; + a2a;a+...)(l + 6a; + 62a;2+...)(l + ca; + c2a;2+...)[See Art.293]; that is in yz r-^; — r-m ; . which will be found to be equal to (1 - oar) (1 - bx) (1 -ex) a? 1_ &» 1 c« 1 (a-h){a-e)l-ax (6 - c) (6 - a) 1- 6a; (c - a) (c - 6) 1 - ca; ' and the coefficients of a;" in the expansions of these partial fractions is easily seen to be ^n+a jn+a cn-t-2 + ,T .-.-i V + {a-h){a-c) (6-c)(6-a) {c-a){c-b) which equals (& - c) (c - a) (a - 6) 368 INDETERMINATE COEFFICIENTS. Ex. 4. To find the sum of all the homogeneous products of n dimensions which can be formed &om the r letters aifa^^a^^ , a^. As in the previous example, the sum required will be the co- efl&cient of x* in -; r-jz r-r^ r , which will be found a.r-1 I to be equivalent to 2-; r-* ; — -:; . Hence the required sum is 2) («i-«a)(«i-«8) ••(oi-ar) 301. Indeterminate coefflcients. We shall con- clude this Chapter by giving two examples to illustrate a method, called the method of indeterminate coefficients, which depends upon the theorems established in Articles 91 and 281. Ex. 1. Find the coefficient of a:' in the expansion, according to ascending powers of x, of {l + cx){l-\-c^x){l + c^x)...{l + c^x). The continued product is of the nth degree in a; ; we may therefore assume that (l + cx){l + c^x)...{l + c'*x) = AQ + AiX + A^^+... + Arxr+... + A^x*, where A^, A^, A^,... do not contain x. Now change x into ex ; then, since Aq, A^, A^, &g. do not contain X, we have (l+c*x){l + c^x)...(l + c'^+^x) = Aq + AiCX + A^c^x^ +... + ArC^air + . . . + A^c*x'^. Hence (1 + c"+ia;) (.lo + ili«+^j^+ ... + ^^+ ... + J„x«) = (1 + ca;) (ilo + Aycx + A^c'^x^ + . „ + Ar^x^ + . . . + ^^c^x"). Now equate the coefficients of «»" on the two sides of the last identity, and we have Ar + c«+M^i = AfCf + A^^cT ; •• ^'•^ c*--! ^«-i=^ c«--l ^*-^ ^*^- By continued application of (a) we have cr cr-l «a , (C*-*^^ - 1) (C"-*^-- 1)...(C*-^ - 1) {C- - 1 ) •' •••' •'' (c'-l)(c-»-l)...(c^-l)(c-l) ^»' iri.M.i.(c"-l)(c»-l-l)...(c'»-'^l-l) , . . ^ . , , INDETERMINATE COEFFICIENTS. 369 Ix. 2. To find the sum of the series l^ + 2^ + S^ + . . . + n^. Let V + 2^ + S^ +.,. +71^= A jn + A^rv' + Agn^ (a) for some particular value of w, where A^, A^, A^ do not contain n. The relation (a) will be true for n + 1 as well as for n, provided ia + 2«4.32+...+«3 + (n + l)»=^i(w + l) + ^,{n+l)2 + ^3(n + l)3; or, subtracting (a), provided (n+l)2=^i + (2n + l)^ + (3n» + 3w + l)^3. Now the last relation will be true for all values of n if we give to Aj, A^, Ag the values which satisfy the equations found by equating the coefficients of n', n} and n®, namely, the equations 3^8= 1» 3^g + 24j=2, and A^ + A^ + Aj^ = l, from which we obtain 6^i=2^j= 3^3=1. Hence, if the relation V+2^+...+n^=-n + -n^+-'n?, be true o Z o for any value of n, it will be true for the next greater value. But it is obviously true when n = l; it will therefore be true when n= 2; and, being true when n=2, it must be true when n=3j and so on indefinitely. The sum of the cubes, or of any other integral powers, of the first n integers can be found in a similar manner. [See also Ajt. 321.] EXAMPLES XXIX. Resolve into partial fractions : Sx ^ x + l 1. -=- X ^ af+ 1 (2-xy(i+x)' ^' (x + 2){af + iy 9. . /-"/\... 10. x(x+iy' af + x+l a^-4x' + x + 6' l + Jx-af {l+Sxy{l-10xy 5-9x (af+l)(x-iy' {l^3xy(l-^x)' {X- {X- -l)'(x-2)(x' l + 2x + 1) 370 EXAMPLES. • (a;^ + l)(a;-2)(aj + 3)* a;^(a;+ 2)^ («+!)• «« (a; + 2^ (a; - 1) 17. Find the coefficient of x" in the expansion of X+ 4: x' + 5x+6' 18. Find the coefficient of aj" in the expansion of a;-2 (x+2)(x-iy' 19. Shew that the coefficient of x"*~^ in the expansion of x + 5 . , 1 IS 1 - (aj'-l)(aj + 2) 2'"' 20. Find the sum of the n first coefficients in the expansion . 3 -2a; l-2a;-3ar«' 21. Find the sum of the n first coefficients in the expansion » 2-5a; ° (l-5a;)(l~3a;)(l-2x)' 22. Find the coefficient of a;* in the expansion of y^ ^ . Find also the sum of the n first coefficients. 23. Shew that the coefficient of a?"*' in the expansion of [l±gis(-2)'(r-2«+l). 24. Shew that EXAMPLES. 371 {x-a^)(x-a^)...(x-aj ar' 1 = a; + a. +«-+ ... + a +S . .. . ' * (ax-a.)K-a3)...a;-a^ 25. Shew that the coefficient of 2;""* in the expansion of {(1 -z)(l- cz) (1 - c'z) (1 - c'^z)}-' is (1 _ C-) (1 - 0"-) (1 - c--)/(l - c) (1 - c«) (1 - O. 26. Prove that a{h-c){hc-aa'){ar-a"^) b{c-a){ ca-hh'){ir -b"^) a -a' h-h' c(a-b) (ab - cc') (c*" - c"") c- c' ^_ -i_ (5 _ c) (c -a) (a- b) {be - aa') {ca - bb') {ab - cc) H^_^, where aa' = bb' = cc\ ajid //„_, is the sum of the homogeneous products of a, 6, c, a', 6', c' of w — 3 dimensions. 27. Shew that the product of any r consecutive terms of the series 1 - c, 1 - c*, 1 — c',. • • is divisible by the first r of them. 28. Shew that, if c be numerically less than unity, (1 + ex) (1 + 0*05) (1 + c^x) ... to infinity -1 ^ c' c^("+^> .^ -^ + l_c^"^(l-c)(l-c«)^'*" •*•'*' (l-c)...(l -c")^"^- 29. Shew that, if c be numerically less than unity, (1 + ex) (1 + c*a5) (1 + c^x) ... to infinity _, c c* ^ c^ 3 ~ "^ rr7^^(l-c»)(l-c^) "^ (1 -c«) (1 -c*) (1 -c«) "^ ■*" - 30. Shew that, if c be less than unity, = l + r + (l-»)(l-ca;)(l-c^a;)... l-c (l-cXl-c") ^ (l-c)(l-c-)(l-c- )^- t^^^^^-3 372 EXAMPLES. 31. Shew that, if c be less than unity, ( I + cx)(l + c'x) (I + c'^x) ... ^ l+c (1 + c){l+ c*) ^ '{l-x){l-cx)(l-c'x)... ~ l-c^'^(l-c){l-c')'^ ' [Gauss.] 32. Shew that the coefficient of x"" in the expansion of (I + cx)(l +c'x){l + c'x) ... {l-cx)(l-c'x){l-c'x)... (l + l)(l+c)...(l+0 "^ (l-c)(l-c')...(l-cO' c being less than unity. 33. Shew that -x I -ax l-a'x l-a^x _ 1 x a^ a? l-y l-oy 1- ay 1 — ay 34. Shew that x 2x' 3x' ^x* l-x^\-a?'"\-a^^\-x'''^"* X x' a? + T-. ST. +Tq i:^+ ... (I-.:)' (1 -«:»)• -(l-a:») 35. Shew that Lambert's series, namely. x l-x'"l-x'^\-x'^l^x*'^"' is equivalent to \+x . 1 + a^ , 1 + cc* r^, , J»i +«*= a + aTr -„+... rClausen.) \-x l-oj' 1-x' •■ ' CHAPTER XXIV. Exponential Theorem. Logarithms. Logarithmic Series: J 302. The Ex ponential Theorem. If 1/n be nu- merically less than unity, (1 +-j can be expanded by the Binomial Theorem ; and we have \ nj n 1.2 nr ruv (nx — 1) {nx — 2) 1 nx(na;—l)...(nx — r-{-l) 1 .2.3 n ^ n which may be written - -^ xlx—-] x\x I (a; ) V^n) =1+^ + — r-Tr-+ .-^r^. + 1.2 ' 1.2.3 / 1\ / r-l\ X [x ]..Ax . V nj \ n ; . Putting^a; = 1, we have l + l+-7^r- + 2 ' |3 p +■■■■ 374 EXPONENTIAL THEOREM. xiix! ) mix )( =- / (1 =- ) , that is less than - . But an integi w + l/Vw + l/ n ^ cannot be less than 1/n ; it therefore follows that e cannot be equal to the commensurable number 7^/71. 304. The following proof of the Exponential Theorem is due to Prof. Hill*. It will be seen that it only assumes the truth of the Binomial Theorem for a positive integral exponent. 771 771/*" Let /(tti) denote the series 1+7^-4-7^+ + t— +.... Thus /(7n)= 1+771 + ^' + + ~ + \r * Proceedings of the Cambridge Philosophical Societi/, Vol. v. p. 415. Substantially the same proof is however given in Cauchy's Analyse A Ige- brique. s. A. 26 376 EXPONENTIAL THEOREM. /(n)sl+» + ^ + + ^ + and/(m+n) = 1 +(m + n)+<^i+i^V ... + 2- + ... =2. 16. If log, •:! 1 ^ be expanded in a series of positive 1 — a? ^ au ■T" as 1 3 integral powers of a?, the coefficient of a;" will be - or - accord- n n ing as n is odd or even. EXAMPLES. 387 17. Shew that the coefficient of a;'' in the expansion of c"' — 1 1 :j -, is r-{V + 2' + 3"+ +»-}. Hence find the sum of n terms of the series 1* + 2' + 3' + ..., and also of 1' + 2* + 3' + ... 18. Shew that, if a^ be the coefficient of aj*" in the ex- pansion of e*', then 1 (V 2- 3' ) Hence shew that -13 Q8 OS and that 1* 2* 3* ,. 19. Shew that r 71 n{n-l) n(n-l)(n-2) 1 i 1' r.2* P. 2^3" J -1 .rml^. ^^-^^X^^^) (7.-Hl)(n+2)(n + 3) 20. Shew that the sum of n terms of the series j + h + o + • • •> beginning at the (n+ l)th, becomes equal to log, 2 when n is increased without limit. 21. Shew that log, (1 + n) < y + 2 + . . . + - < 1 + log, (l + n). 22. Prove the following : — (i) (x + yy -x'-y' = Ixy (x ^y){a^ + xy + y")', (ii) {x + yY' -x''-y'' = l Ixy (x + y) (of + xy + y') {(x^ + xy + yy + xy(x + yy}, (iii) {x + yy^ - a;'' - y'' = 1 "^xy {x + y) {x* + xy + y^ {{x' + xy + yy + ^Qt^y" {x + yy\. — 3r^2r 388 COMMON LOGARITHMS. 23. Shew that a^" + y'" + (x + y^ «x n.ii w(w-3)(w-4)(n-5) ,_, , - Sp" + n (r* - 2)p'-y + -i^ ^^3 ^ ^^ ^ p" V . ^ ( y^-r-l)...(yi-3r+l) "*■ 3.4...2r ^ ^ ■^•••' where p = a^ + xy + y' and g' = ojy (a; + y). 24. Shew that, (i) if n be any uneven integer, (b — c)" + (c - a)" + (a - by will be divisible by (b - c)* + (c - a)« + (a - 6)'; (ii) if 7i be of the form 6w*l, it will be also divisible by (6 — c)' + (c ~ a)' + (a- 6)' ; and (iii) if w be of the form Qm+l it will be divisible by (6 — c)* + (c - a)* + (a - 6)*. # Common LoGARrrHMS. 311. In what follows the logarithms must always be supposed to be common logarithms, and the base, 10, need not be written. If two numbers have the same figures, and therefore differ only in the position of the decimal point, the one must be the product of the other and some integral power of 10, and hence from Art. 307, II. the logarithms of the numbers will differ by an integer. Thus log 421-5 = log 4'215 + log 100 - 2 + log 4-215. Again, knowing that log 2 = '30103, we have log '02 = log (2 -r- 100) = log 2 - log 100 = -30103 - 2. On account of the above property, common logarithms are always written with the decimal part positive. Thus log -02 is not written in the form - 1-69897 but 2-30103, the minus sign referring only to the integral portion of the logarithm and being written above the figure to which it refers. Definition. When a logarithm is so 'written that its decimal part is positive, the decimal part of the logarithm is called the mantissa and the integral part the character- istic. CHARACTERISTICS FOUND BY INSPECTION. 389 312. The characteristic of the logarithm of any nnmher can he written down by inspection. For, if the number be greater than 1, acd n be the number of figures in its integral part, the number is clearly less than 10" but not less than 10""'. Hence its logarithm is between n and n — 1: the logarithm is therefore equal to n — 1 + a decimal. Thus the characteristic of the logarithm of any number greater than unity is one less than the number of figures in its integral part. Next, let the number be less than unity. Express the number as a decimal, and let n be the number of ciphers before its first significant figure. Then the number is greater than 10"""' and less than 10"". Hence, as the decimal part of the logarithm must be positive, the logarithm of the number will be — (n + 1) + a decimal fraction, the characteristic being — (n + 1). Thus, if a number less than um^ity he expressed as a decimal, the characteristic of its logarithm is negative and one more than the nwmber of ciphers before the first signifi- cant figure. For example, the characteristic of the logarithm of 3571*4 is 3, and that of -00036714 is 4. Conversely, if we know the characteristic of the logarithm of any number whose digits form a certain sequence of figures we know at once where to place the decimal point. For example, knowing that the logarithm of a number whose digits form the sequence 35714 is 3*55283, we know that the number must be 3571 '4. 313. Tables are published which give the logarithms of all numbers from 1 to 99999 calculated to seven places of decimals : these are called ' seven-figure * logarithms. For many purposes it is however sufficient to use five- figure logarithms. 890 USE OF TABLES OF LOGARITHMS. , In all Tables of logarithms the mantissae only are given, for the characteristics can always, as we have seen, be written down by inspection. In making use of Tables of logarithms we have, I. to find the logarithm of a given number, and II. to find the number which has a given logarithm. I. To find the logarithm of a given number. If the number have no more than five significant figures, its logarithm will be given in the tables. But, if the number have more significant figures than are given in the tables, use must be made of the principle that when the difference of two numbers is small compared with either of them, the difference of the numbers is ap- proximately proportional to the difference of their loga- rithms. This follows at once from Art. 308, for log,o {N + x)~ log,, N = log,, (l + 1=) = /^ H. (l + 1") ^^ XN^^W^'") ^^N approximately, when -^ small, jM being the modulus 1/log^ 10. An example will shew how the above principle, called the Principle of Proportional Differences, is utilised. Ex. To find the logarithm of 357-247. We find from the tables that log 3-5724= -5529601, and log 3-5725 = -6529722; and the difference of these logarithms is -0000121. Now the difference between 3*57247 and 3-5724 is ^ths. of the difference between 3-5724 and 3-5725 ; and hence if we add ^ths, of •0000121 to the logarithm of 3-5724 we shall obtain the approximate logarithm of 3-57247. Now /^f/ia. of -0000121 is -00000847, which is nearer to '0000085 than to -0000084. Hence the nearest approxi- mation we can find to the logarithm of 3-67247 is -5529601 + -0000085 = •5529686. The characteristic of the logarithm of 357-247 is obviousiy 2, and therefore the logarithm required is 2*6629686. II. To find the number which has a given logarithm. For example, let the given logarithm be 4 •5529652. We find from the tables that log 3-5724 = -5529601 and that log 3-5725= -5529722, the mantissa of the given logarithm falling is I COMPOUND INTEREST AND ANNUITIES. 391 between these two. Now the difference between •5529601 and the 51 given logarithm is r^ of the difference between the logarithms of 3-5724 and 3-6725.; and hence, by the principle of proportional differences, the number whose logarithm is -5529652 is 3-6724 + ,^ X ■0001 = 3-6724+ •00004=3-57244. [The approximation could only be relied upon for one figure.] Thus -5529662 = log 3-57244, and therefore 4 -5529652= log -000357244. Compound Interest and Annuities. 314. The approximate calculation of Compound In- terest for a long period, and also of the value of an annuity, can be readily made by means of logarithms. All problems of this kind depend upon the three fol- lowing : — [The student is supposed to be acquainted with the arithmetical treatment of these subjects.] I. To find the amount of a given sum at compound interest, in a given number of years and at a given rate per cent per annum. Let P denote the principal, n the number of years, lOOr the rate per cent, per annum, and A the required amount. Then the interest of P for one year will be Pr, and therefore the amount of principal and interest at the end of the first year will be P (1 -\- r). This last sum is the capital on which interest is to be paid for the second year ; and therefore the amount at the end of the second year will be {P (1 + r)} (1 + r) = P (1+ rf. Similarly the amount at the end of n years will be P (1 -f r)". Thus ul = P (1 + r)" ; and hence log A = log P + n log (1 -f r). If the interest is paid, and capitalised, half yearly, it can be easily seen that the amount will be P ( 1 + ^ 1 . s. A, .27 392 ANNTTITIES. Ex. Find the amount of £350 in 25 years at 5 per cent, per annum. Here P = 350, r=^ and n = 25; /. log 4 = log 350 + 25 log (l + fQQJ = log 350 + 25 (log 105 - log 100). From the tables we find that log 350 = 2-5440680 and log 105 = 2-0211893; hence log J:= 3-0738005. Whence it is found from the tables that ^ = £1185-22. II. To find the present value of a sum of money which is to be paid at the end of a given time. Let A be the sum payable at the end of n years, and let P be its present worth, the interest on money being supposed to be lOOr per cent, per annum. Then the amount of P in w years at lOOr per cent, per annum must be just equal to A. Hence from I. P = ^ (1 + r)'". III. To find the present value of an annuity of £A payable at the end of each of n successive years. If the interest on money be supposed to be lOOr per cent, per annum ; then from II. The present value of the first payment is J. (1 -H r)~* second -4(1 + r)~* ?ith A(l-\-ry. Hence the present value of the whole is Ex. Find the present value of an annuity of £30 to be paid for 20 years, reckoning interest at 4 per cent, Here^=80,„=20,r=A = ^l_. EXAMPLES. 393 Hence the present value = 30x25Jl-(2c) \ • Now log (II y" = 20{log 25 - log 26} = 20 {1-3979400 - 1 -4149733} =20 (--0170333)= - •340666=1-659334 =log -456389, from the Tables. Hence the value required = 30 x 25 x (1 - -456389) = £407-7... EXAMPLES XXXIL The following logarithms are given log 1-02 = •0086002 log 1-6386 = -2144730 log 1-025 = •0107239 log 1-6387 = •2144995 log 1-033 = •0141003 log 1-7292 = •2378452 log 1-04 = •0170333 log 1-7349 = •2392744 log 1-05 = •0211893 log 2 •3010300 log 1-06 = •0253059 log 2^0829 = •3186684 log M467 = •0594498 log 3 -4771213 log M468 = •0594877 log 3-0832 = •4890017 log 1-2258 = •0884196 log 4-4230 = •6457169 log 1-2620 - •1010594 log 5-1 •7075702 log 1-4816 - -1707310 log 5-577 = •7464006 log 1-4817 ^ •1707603 log 6-3862 = -8052425 log 7-4297 = •8709713 log 7-4298 = •8709771 1. Find SyiOS. 2. Find jy51. 3. Find the amount of .£100 in 50 years at 5 per cent per annum. 4. Shew that money will more than double itself in 15 years at 5 per cent, per annum, and in 18 years at 4 per cent, per annum. ^ 27—2 394 EXAMPLES. 5. Find the amount of £500 in 10 years, interest at 4 per cent, being paid half yearly. 6. The number of births in a certain country every year is 85 per 1000 and the number of deaths 52 per 1000 of the population at the beginning of every year: shew that the popu- lation will be more than doubled in 22 years. 7. A man invests £30 a year in a Savings Bank which pays 2J per cent, per annum on all deposits. What will be the total amount at the end of 20 years? 8. What sum should be paid for an annuity of .£100 a year to be paid for 40 years, money being supposed to be worth 4 per cent, per annum ? 9. A corporation borrows £30000 which is to be repaid by 30 equal yearly payments. How much will have to be paid each year, money being supposed to be worth 4 per cent, per annum? 10. A house which is really worth £70 a year is let on a lease for 40 years at a rent of £10 a year, the lease being re- newable at the end of every 14 years on payment of a fine. Calculate the amount of the fine, reclconing interest at 6 per cent. CHAPTER XXV. Summation of Series. 815. We have already considered some important classes of series, namely the Progressions [Chapter xvii], Binomial series [Art. 288], and Exponential and Logarith- mic series [Chapter xxiv]. In the present chapter some other important types of series will be considered. 316. The nth term of a series will be denoted by u^, and the sum of n terms by 8^. When the series is con- vergent its sum to infinity will be denoted by S^. 317. No general method can be given by which the summation of series can be effected ; but . in a great number of cases the result can be obtained by expressing the general term of the series, i*„, as the difference of two expressions one of which involves w — 1 in the same manner as the other involves n. For example, in the series a)(ic + a) (a? + a) {x + 2a) (x -t- 2a) {x + 3a) a the nth term, namely == , is equal to {x -\-n — l.a) {x-\-na) 7 =^- . Hence the series may be written x -k- {n — \) a X •\- na '' 396 SUMMATION OF SERIES. yx" x-{-a) \d?+a x-\-2a/ \x+2a a? -f Sa/ + J ; -V [ ' and it is now obvious that all \x+{n — l)a a? -f- na) the terms cancel except the first and last ; hence o„ = ; = —7 — ; r . X OS + na a;{£c + na) Ex. 1. Find the sum of n terms of the series Ans. 1 = . n+1 1.2 "^2. 3 ■^3.4"*' '^n{n+l)^ Ex. 2. Find the sum of n terms of the series Ex. 3. Find the sum to infinity of the series _i- J_ Jl_ 1 3|_l + 4l2 + 5[3'^ '^(n+2)\^'^ Here w„= , ^ - -. ^ . Ans. rr. " \ n + l \ n + 2 2 Ex. 4. Find the sum to infinity of the series 3 5 _J__ 2n+l 12 . 28 ■*" 23 . 32 "^ 32 . 42 "•" ■*"^(n + !)«"*" Ex. 5. Find the sum of n terms of the series 1 2 3 n 1.3 "•'1.3. 5 "^1.3. 5. 7"*" '^1.3.5...(2n+l)' [^''"=1.3. 5... ^7^1) " 1.3.6...(2rt-l)(2n + l)J * ^^- H^"l.3.5.^ 2n + 1 )}- SUMMATION OF SERIES. 397 Ex. 6. Sum to infinity the series 2 13 14 1 n+1 1 1. 3 3 ' 8. 6 • 33 ' 6. 7 • 3» ' ' (2n-l){2n+i) 3« Tsincc "■♦■^ ^M M L ° (2n-l)(2n + l)"4 V2n-1 271 + 1^ ' 11 1 i"! "~2n-13«-^ 2n + 13'*J' Ex. 7. Find the sum to infinity of the series 1111 28_il43-l '6»-l '8^-1 + ^r..\. Ex. 8. Find the sum of n terms of the series X x^ x^ (l-x)(l-x2) (l-x^)(,l-x') ' {l-x'){l-x') ■■■■ ... 1 1 {l-xf (l-a:)(l-a;~+i) 318. To find the sum ofn terms of the series {a (a + h) ,. .{a + V^. h)] + {{a + h) (a + 26). ..(a + rh)] + ... + {(a + w - 1 . 6) (a +n6) ... (a + n + r - 2 . 6)} + . .. In the above series (i) each term contains r factors, (ii) the factors of any term are in arithmetical progression, and (iii) the first factors of the successive terms form the same A.P. as the successive factors of the first term. Consider the series which is formed according to the same law but with one factor added at the end of every term, and let v^ be the nth term o f this new series, so that t;^={(a + w-1.6)(a + n6)...(a+n + r-l.&)1. Then v^ - v^_^ = {(a + n-1 .b)(a + nb) ...(a + n + r-1 .b)} -{(a + n-2.6)(a + 7i-1.6)...(a + n + r-2.6)} = {(a-^n-l.b)...(a + n + r-2.h)}{{a-hn+r-l.b) -{a + n-2.b)} = (r+l)6{(a+?i-l .6)...(a + 7H-r-2. 6)}. 398 SUMMATION OF SERIES. Hence v„ — v^_^ = (r + 1) 6 x u^. Changing n into w — 1 we have in succession v, - Vi = (r + 1) 6 X w,. Also Vj — v^ = (r + 1) 6 X Wj, where v^ is the term preceding v^ which is formed accord- ing to the same law, that is Vq= {(a — b)a(a + b)...(a + r — ib)], so that v^ is obtained by putting n = in the expression for v^. Hence by addition ^n-'^o^(r+l)bS^; r.S^ = {v^-v,)l{r-^l)b. Ex. 1. Sum the series 1.2 + 2.3 + 3.4+ +w(w + l). Here M„ = n(n+1), v„= n(n+l)(7t+2), Vq = 0. 1. 2, r = 2, and 6=1. Hence Sn=^n{n + l){n+2). Or, by using the above method without quoting the result, which is preferable in very simple cases, we have n(n+l)=^{w(w+l)(n+2)-(n-l)n(n + l)}. («-l)n=i{(n-l)n(n+l)-(«-2)(M-l)«}, 1.2 = ^{1. 2.3-0. 1.2}. Hence S^=-w(w + l)(n+2). Ex. 2. Sum the series 1 .2. 3 + 2.3.4+ +w(n+l) (n + 2). Ans. -n(n + l) (n + 2) (u + 3). SUMMATION OF SERIES. 399 Ex. 3. Sum the series 1.2.3.4 + 2.3.4.5 + +n{n + l){n + 2) (n + B). Ans. vn(w + l)(n + 2)(n + 3)(n+4). Ex. 4. Find the sum of n terms of the series 3.5.7 + 6.7.9 + 7.9.11 + Here w„=(27i + l)(2w + 3)(2n + 5), v„=(2n + l) (2ri + 3) (2n + 5) (2n+7), ^0 = 1.3.5.7, r=3, and 6 = 2. Hence /Sf„=^{{2w + 1) (271 + 3) (2n + 5) (271 + 7) -1.3.5.7}. Many series which are not of the requisite form can be expressed as the algebraic sum of a number of series which are all of the required form ; and the sum of the given series can then be written down. The following are examples. Ex. 5. Find the sum of n terms of the series 1.3 + 2.4 + 3.5 + Here M„=ro(n + 2)=72 (n+l)+w. The sum of the series 1.2 + 2. 3+...+n(n + l) is -J-{n(w + l)(n + 2)-0.1.2}, and the sum of the series 1 + 2+ ... +n is ^{n(w+l)-0 . 1}. Hence the required sum i8-n{n + l){n + 2) + --n{n + 1). Ex. 6. Find the sum of the series 2.3. 1 + 3.4. 4 + 4. 5.7+ + Cu + 1) (n + 2) (37i-2). Here M„=(n+1) (n+2) (3n-2) = 3n(n+l) (71+2) -2 (71 + 1) (w + 2). /. S„=| {71(71 + 1) (71 + 2) (71 + 3)^0. 1 .2.3} -|{(7i+l)(7i + 2)(n + 3)-1.2.3} =i (971 - 8) (n + 1) (71 + 2) (71 + 3) + 4. 400 SUMMATION OF SERIES. 319. To find the sv/m of n terms of the series whose general term is l/{(a + 71 - 1 . 6) (a + n6) (a + n + 1 . 6)...(a + n + r - 2 . 6)j. Consider the series which is formed according to the same law but with one factor taken away from the beginning of each term, and let v^ be the nth term of this second series, so that v^—\l{{a-\- nh) ...{a-\-n-\-r — 2.h)]. Then 1 [{a + nh) ... {a ^- n -\- r -2 .h)] 1 {(a + 71 - 1 . 6) (a + n6) . . . (a + ri + r - 3 . 6)} {{a + 71- 1 . 6). . .(a+ 7i+r-2 . h)] -(a+7i + r-2.6)}; Changing n into ti — 1 we have in succession v^^—v^ — — {r—l)h X u.^. Also v^ — VQ=^ — (r- 1)6 xu^, where v^ is the term which precedes v^ and which is formed according to the same law, that is t;, = 1 / {a (a + 6). . .(a + ^'2 . 6)j. Hence, by addition, v„-v,==^-(r-l)hxS^; -Sn = (Vo-vJI{r-l)b, Ex. 1. Sum the series tt— „ + ^—. + ... + 2.3 ^3.4 •"^(n + l)(n + 2)* SUMMATION OF SERIES. 401 Hence S^=j-j j^ " n + 2l " 2 " ^T2 * Ex. 2. Sum the series j-^^ + 2:^5^ --^ n{n+l){n + 2){n + Z) to w terms and to infinity. „ _ 1 1 liere «n-„(^^ij(^ + 2)(»+"3)» *'»-(n+l)(n+2) (n+3) ' Hence ^«=37i jf^Ts " (n + 1) (n+^)F+3)|' and S^ = ^. 3 1.2.3 18 Ex. 3. Sum the series x-^^n + s-T7-^r?+ ••• + 3.7.11 7.11.16 (4n-l){4w + 3)(4n + 7) Ans. fif„=g |377-(4„+3)(4^ + 7)[ • Many series which are not of the above form can be expressed as the algebraic sum of a number of series which are all of the required form; and the sum of the series can then be written down. The following are , examples. Ex. 4. Sum the series ^-q + 2~i + 3~6'^ *- Here 1 • n+1 1 . 1 »""n(n + 2) w(n+l)(w + 2) (w + l){n + 2) ' w(n + l)(n + 2) The series whose general terms are - — ., q. and (n+l)(n + 2) n(n+l)(n + 2) are of the required form. Hence the sum of the given series is given hy _/l_JLA . l/_i L ^=! 2n + 3 ^«~V2 n+2j'*"2Vl.2 (n+l)(w + 2); 4 2(n + l)(n + 2)* 402 SUMMATION OF SERIES. r. . o .^ . 1 1 1 Ex, 6. Sum the senes ., » ^ + 77— j — ^ + ... + —. — -^rr-. — — jv . 1.3.6 2.4.6 n{n + 2){n + 4) _ 1 ^ (w + l)(n+3) "«~n(7H-2)(n + 4)~n(n + l)(n + 2)(n + 3)(n + 4) n(w + 4) + 3 "■n(n+l)(n + 2)(w + 3)(» + 4) = ^^ + ? (n + l)(n+2)(n+3)^n(n+l)(n + 2)(n + 3)(n + 4)' Hence g =liJ____J__i .?J_J __i 1 " 2]2.3 (n+^)(n+3)j "^4 (1.2.3.4 (n+l){n + 2)(n + 3)(n+ 4)f * 320. The sum of series of the kind just considered may be obtained by means of partial fractions. The method will be seen from the following example. To find the sum of the series r— ^ + jr—r + 77—?+ ... + — ; . 1.3 2.4 3.5 n(n + 2) that ^ = 5 and S=-i. a 2 Hence 2w» = . * n n+2 We have therefore the following series of equations : ^=1-5- '^-l-V M-g = ^"»--;i^2-s- ^"-=s:^i-j7i' '«'^2"«=i-r^- Hence, by addition, '^«-i"*'2 irri"^rr2' the other terms all cancelling. 3 4 2(n + l)(n + 2)' Hence S.=| - 2n + 3 321. 2^0 /nc? ^Ae si^m 0/ the rth powers of the first n whole numbers. We will first consider the two simplest SUM OF SQUARES AND CUBES. 403 Case I. To find the sum of 1' 4- 2' + 3' + ... + n\ Here u^ = n^ = n {n -^-1) — n. Hence, by Art. 318, "^ S,^\n{n + l){n + 2)-\n{n-\-l) = gn(n + l)(27i + l). Case II. To find the sum of 1» + 2" + 3» 4- . . . + n\ Here t/-^ = w' = ri (w + 1 ) (w + 2) - Sti' - 2n = n (n + 1) (w + 2) - 3/1 {n + 1) + w. Hence, by Art. 318, /S„ = ^n(7i + l)(7i + 2)(7H-3)-|7i(7i + l)(^4-2) = j7i(n + l){(7i + 2)(7H-3)-4(7i+2) + 2} Since l + 2 + ...+n = 2^(^ + l), the above result shews that r + 2'' + ...+7i» = (l + 2 + ... + ny, so that the swm of the cubes of the first n whole numbers is equal to tlie square of the sum of the numbers. The sum of the cubes of the first n integers can also be easily found by means of the identity 4w2 = {n (n + 1)}^ - {(»- 1) «}'. For we have in succession 47j.^ = {7t(w + l)}2-{(n-l)n}2, 4{«-l)8={(w-l)n}3-{(n-2)(n-l)}2, 4. 23 = (2, 3)»- (1.2)2, and 4. 18 = (1.2)3 -(0.1)8. Hence, by addition, ^Sn=n^ {n + l)\ 404 SUM OF POWERS OF INTEGERS. Case III. To find the sum of 1" + 2*^ + 8' + . . . + n^ The sum for any particular value of r can be found by the same method as that adopted for the values 2 and 3. For example, the sum of the fourth powers can be written down as soon as n* is expressed in the form n' = w (71 + 1) (71 + 2) (71 + 3) - 6n(n+l){n+2) 4- Tti (n + 1) - 71. By means of the Binomial Theorem a formula can be found which gives the sum of the rth powers in terms of the sum of powers lower than the rth ; and this formula can be used for finding the sum of the 2nd, 3rd, 4th, &c. powers in succession. The formula has however the great disadvantage that in order to find by means of it the sum of the rth powers, it is necessary to know the sums of all the powers lower than the rth. By the Binomial Theorem, we have in succession (71+ ir^ = tT' + (r + 1) 7?: +^^li^7i'-^ + ... + 1, (7ir=(7i-ir+(r+i)(7i-ir+(^:±^(7i-ir +...+1, g^, ^ 2*^* + (7- + 1) 2'- + ^—^- 2*-^ + ... + 1, 2'-^ = l*^* + (r + 1) r + ^-^^li^ 1-^ + ... + 1. Hence, by addition, we have (n + l)**^^ — (ti + 1) = (r + 1 ) ^/ + (l+^ ^, -^ + . . . + (^ + 1 ) s:, where SJ" is written for the sum of n terms of the series r + 2- + 3''+... PILES OF SHOT. 405 We can in a similar manner find a formula for summing the rth powers of any series of quantities a, a + 6, a + 26, ... in arith- metical progression. The result is (a + n6)'^i-a'-^i-w6-*-i = (r + l)&V + ^~^^- b^S^'--^ +..'. + {r+1) b^S„\ where /S„'-=a»- + (a + 6)*'+ ... + (a + 7^U6)^ 322. Piles of Shot. To find the number of spherical balls in a pyramidal heap, when the base is (I) ai^equilateral triangle, (II) a square, and (III) a rectangle. I. In a pile of this kind the balls which rest on the ground form an equilateral triangle, and upon this first layer a number of balls are placed forming another equi- lateral triangle having one ball fewer in each side than in the side of the base ; and so on ; a single ball being at the top. If n be the number of balls in each side of the base, the total number in the base will be 71 + (n - 1) + (w - 2) +. . .+ 2 + 1, that is ^n (n + 1). The whole number of the balls in the pile will therefore be ^ {w (71 + 1) + (w - 1) n +. . .+ 1 . 2}, that is Jn (w + 1) (n + 2). II. In this case the balls in any layer form a square with one ball fewer in each side than in the layer next below. Hence if n be the number of balls in each side of the lowest layer, n^ will be the number of balls in the base, and therefore the whole number of the balls will be 71* + (71 - I)'^ + (71 - 2)^^ +. . .+ 1^ that is jTi (71 + 1) {2n + 1). III. In this case the balls in any layer form a rectangle with one ball fewer in each side than in the layer next below. Hence if n and m be the number of balls in the sides of the lowest layer, nm will be the number of balls in the base and therefore the whole number of the balls will be, n being greater than m, nm + (71-1) (m- l) + ( n - 2) (tti - 2) + ... (n-m + 1) 1 —{n — m + m)m + {n — m + 'm—l){m — l)-\-...{n — m+ 1)1 406 FIGURATE NUMBERS. = (n-m){m + (m-l)+...+ l} + m'* + (m-l)»+...+ l' = ^(7i-m)m(m + l) + Jm(m+l)(2m + l) S = Jm (m + 1) (3m -m + 1). Ex. 1. How many balls are contained in 8 layers of an unfinished triangular pile, the number in one side of the base being 12 ? If the pile were completed it would contain ^ . 12 . 13 . 14 balls; ♦ 1 and there are ^ . 4 . 5 . 6 missing from the complete pile; hence the D required number is ^ (12 . 13 . 14 - 4 . 5 . 6). Ex. 2. How many balls are contained in 10 layers of an incomplete pile of balls whose base is a rectangle with 20 and 25 balls in its sides ? The number=Sn(n + 5) from n=ll to w=20. An*. 3260. 323. Pigurate numbers. Series of numbers which are such that the nth term of any series is the sum of the first n terms of the preceding series, all the numbers of the first series being unity, are called orders of figurate numbers. Thus the different orders of figurate numbers are : — First order, 1, 1, 1, 1, 1, Second order, 1, 2, 3, 4, 5, Third order, 1, 3, 6, 10, 15, It follows from the definition that the nth term of the second order of figurate numbers is n; the nth term of the third order will therefore be (1 4- 2 + 3 + . . . + n), that is \n (n + 1); the nth term oi the fourth order will therefore be i {n(n + l) + (n- l)n+...+ 1.2}, that is ''^'''^^^^^''^^^ the nth term of the fifth order will therefore be 2^{n(n + l)(n + 2) + (n-l)n(n + l)+...+ 1.2.3j, that POLYGONAL NUMBERS. 407 is 2 n {n -^ 1) (n -{- 2) (n + Sy, and so on, the nth term of the rth order being 71 (w + 1) (n + 2)...(n + r-2) 324. Polygonal numbers. Consider the arithmetical progressions whose first two terms are respectively 1, 1 ; 1, 2 ; 1, 3 ; 1, 4 ; and so on. Then the series formed by taking 1, 2, 3,..., n of the terms of these different arith- metical progressions, namely the series 1, 2, 3, , n, 1, 8, 6, , in(n + l), 1, 4, 9, , < 1, 6, 12, , n-\-^n(n-l), 1, r, Sr-S,..., n-\-in(n-l){r-2), ... are called series of linear, triangular, square, pentagonal^,, r-gonal numbers. The sum of n terms of a series of r-gonal numbers can be written down at once, for the sum of n terms of the series whose general term is n -\- \n {n — V) {r — 2) is \n{n+l) + i{n-\)n{n + \){r-2) [Art. 318]. EXAMPLES XXXIII. Find the sum of n terms of each of the following series, and find also the sum to infinity when the series is convergent. 1. 4.7.10 + 7.10.13 + 10. 13.16 + ... 1 1 1 3.7.11 "*"7.11.16 ■*"11.15.19"*'"* 3. 1.3.4 + 2.4.5 + 3. 5.6 + ... 4. 1.5 + 3.7 + 5.9 + 7. 11 + ... s. A. 28 408 EXAMPLES. 6. 1.2. 3 + 2. 3. 5 + 3. 4. 7 + 4. 5.9 + ... 6. 1.2*+2.3« + 3.4» + 4.5«+... 7. 1.3V3.5* + 5.7=' + 7.9=»+... 1 1 1 1 ^' 1 . 3 . 7 "^ 3 . 5 . 9 "^ 5 . 7 . 11 "^ 7 . 9 . 13 ■*■ **• 1 1 1 1 1.3.4'^2.4.5"^3.5.6'^4.6.7'**'*" 10. 4 5 6 7 1.2.3 2.3.4 3.4.5 4.5.6 ,,12 3 4 11. ,— 7C— = + A ^ ^ + 1.3.5 3.5.7 5.7.9 7.9.11 3_ 4 5 6 1.2.4"^2.3.6"*"3.4.6'^4.5.7'^" .oil 1 1 13. - + r ;: + 11+21+2+31+2+3+4 ,, 1« P + 2» V + 2' + 3' l" + 2^+3" + 4* 14. ^ + -2-^ ^3—^ 4 + -. 15. l.P + 2(P + 2») + 3(r+22 + 3^) + 4(P + 2* + 3^ + 4=')+... 16. a'+{a + by + {a + 2by+.„ 17. a^+(a + bY+{a + 2by+„. 18. l'+3» + 5» + 7" + ... 19. l« + 5« + 9« + 13^+... 20. Shew that l«_2»+3»-4«+... + (2n + l)* = (n+l)(2n+l). 21. Shew that 1" - 2« + 3« - 4« + . . . - (2n)^ = -n {2n + 1). 22. Shew that l»_2«+3»-4»+...+(2n+l)»=4n« + 9n» + 67j+l. 23. Find the sum of the series l.w+2(n-l) + 3(w-2) + ...+n.l. EXAMPLES. 409 24. Find the sum of the series n.n+(n-l)(n + 'l) -k- (w- 2) (n + 2) + ... + 2 (271 - 2) + 1 . {2n - 1). 25. Find the sum of n terms of the series ab + ia- 1) (b-l) + (a- 2) (6 - 2) + ... 26. Prove that, ifS;=r + 2'+ ... n"; then wiU (i) 6S; = QS^'xS;-SJ'. (ii) ^;+^;=2(^;)v 27. Find the sum of the following series to n terms: 1 o O (i) __. 2 + — - 2' + -— 2^ + ^^ 2.3 3.4 4.5 _3_1 ^J^ 5 1 ^^'^ 1. 22^2. 32^ ■^3.42^"^- ..... 4 /2\ 6 /2V 6 /2V ("^) 172(3)^273(3) ^3-4(3)^- (^^) r:2r3 (7) ^ 2-7374 (7) ^ 37175 (7) ^ - / X 9 /3\ 10 /3\> 11 /3V (") 1727^ (4) ^ 2-7374 (4) ^ 37175 (4) ^ - . ., 15 /6\ 16 /6V 17 /6Y ("^) 17273 (7) ^ 2-7374 (7) ^ 3—475 (7) ^ - 28. Shew that the sum of all the products of the first n natural numbers two together is ^ (w — 1) w (n + 1) (3n + 2). 29. Shew that the sum of all the products of the first n natural numbers three together is -j-^in - 2) (n — 1) n' (n + 1)*. 30. Shew that the sum of the products of every pair of the squares of the first n whole numbers is ^«K-l)(4»'-l)(5«+6). 28 — 2 410 SUMMATION OF SERIES. 325. To find the sum of n terms of the series a a(a-\-a)) a (a -\- x) (a -i- 2a?) b "^ JJb + xj ■*■ 6 (6 + a?) (6 + 2^) ■*■ • * * a{a + x)...(a-\-n—lx) b(b + x)...{h-^n-lx) In the above series there is an additional factor both in the numerator and in the denominator for every succes- sive term, and the successive factors of the numeiator and denominator form two arithmetical progressions with the same common difference. Consider the series formed according to the same law but with an additional factor in the numerator, and let v^ be the general term of this second series, so that _ a (a + x). . .(a -{• n — 1 x) (a -\- nx) 6 (6 +«?)...(& + ^r^ a;) Then _ « (o^ + g^)" '(c^ + n - 1 a?) (g + 7ia?) a(a^x)...{a-\-n-lx) b{b + x)...{b-\-^^^x) a{a-\-x)...{a-\-n — lx){, , ^, . J b{h+x)...{b + n-lx)\: ^ So also v^j - Vn-s = '^n^t X (a + a; - 6) v^- v^ = u^x {a -\- X -b). [a^-x) . . Wj X (a +ar-6) + bu^. Also ..-^(« + ^) SUMMATION OF SERIES. 411 Hence S^x(a + w — b) = v^ — a; •~a + a;-6|6(6 + a?)...(6 + n^a;) J' The sum of n terms of the series a a{a — w) a(a — x){a — 2a;) b~'b(b + x)'^ b(b + x)(b-{-2xj'^'" in which the successive factors of the numerator and denominator form two arithmetical progressions whose common differences are equal in magnitude but of opposite sign, can be found by changing the sign of a in the previous result : the sum can, however, be obtained inde- pendently by the same method. Thus a __ 1 Va a (a — x)! h ~ a+b-x [l "^ b J 1 ra(a- x) a (a— x) (a — 2xy\ ^a-\-b-x I b "^ b{bTxj J a(a — x) b(b + x) C_ i)"-i (^((^-^)-"(ci'-n-lx) 6(6 + ^)...(6 + ^r^a;) _ ._ . „_, 1 f g (a — x)...(a — n—l x) a + b-x [b{b + x)...{b+n^2x) a(a — x)...(a — nx) ~| b(b + x)...(b-i-n-lx)j' Hence S ^ ["i _ /_ pn (^ - ^) (^ - 2^)- • (Q^ - nx) l « a+6-^L b{b-^x).,.(b + n-lx) y Ex. 1. To find the sum of n terms of the series - + ^ + hJ^ + 3 3.6 3.6.9 412 SUMMATION OF SERIES. We have 2_l/2.5 2\ 2. 51/ 2. 5. 8 2 . 5\ 8~2l^ 3 ij' 3.6~2V 3.6 3 )* 2.5.8...(3n-l) _ 1 / 2.5.8...(3w + 2) _ 2. 5. 8...(3n-l) | 3.6.9...3n ~2 t 3.6.9...3n 3 .6. 9...(3w-3)j ' ■ „ 1 j2.6.8...(3n + 2) 2) [This particular series is a binomial series, the successive terms being the coefficients of x, x^, &c., in the expansion of (l-x)"*. Hence [Art. 287] 1 + /S«=sum of the first (n + 1) coefficients in the expansion of (1 -x)~^= coefficient of «« in {l-x)~^ x {l-x)-\ that is in (l-x)'^]. 2 2 6 2 6 10 Ex. 2. Find the sum of n terms of the series - + ^^ + 5-^ _'■,., +... o 3.7 3.7.11 2| 6.10...(4n + 2) 1 "^^^ 3 i3.7...(4n-l) 7* Ex. 3. Find the sum of n terms of the series - m TO (m - 1) m (to - 1) (to - 2) ~T'*"~T72 1.2.3 "*"■•• Am. ( i)»-i ("^-l)(^-2)...(m-7i + l) ^ 326. The sum of n + 1 terms of the series where a„ is any integral expression of the rth degree in n, can be found in the following manner. S^ = cto + (i^a)+a^af + ... + ajc'', (1 ~ajr^= 1 - (r+ l)a; + (!:il^^^- ... + (_ l)^^a^r^\ Hence ^„ x (1 - w)"^' = a, + {a, - (r + 1) a,} a; +... SUMMATION OF SERIES. 413 Now aj, is by supposition an integral expression of the rth degree in p ; hence a, = Ay + A^_,f-'^ + A^_,p^ + . . . + ^0. where A^, A^^,..., A^ do not contain p. Also, by Art. 305, the sum of the series p» _ (^ + 1) (^ _ 1)» -f (!l±^(p - 2)*- . . . to (r+ 2) terms, is zero for all integral values of k less than r -f 1. Hence a,-(r+ 1) a^., + ^^3^- a^- . . . to (r + 2) terms is zero for all values of ^. All the terms of the product S^ x (1 —x)"^^ will there- fore vanish except those near the beginning, or the end, for which the series %—{r+V)aj^^-\-,.. is not continued for (r + 2) terms, that is all the terms of the product will vanish except the first r + 1 terms and the last r + 1 terms. Heoce S^ X (l-a:y'=a,-h{a,-(r + 1) ^o} ^ +— whence the value of S^ is found, Ex. 1. Find the sum of the series l + 2ar + 3a;2 + 4x»+ + (n + l)a;'»' -S^i = l + 2x + 3x2 + 4a;3+ + {n+l)x\ {l-x)^=l-2x + x^; .'. {l-xr^S^^=l + x^+^{n-2{n + l)} + {n + l)x^+\ [all the other terms vanishing on account of the identity k-2{k-l) + {k-2)=0] «1 - (w + 2) a:"+i + (n + 1) a;"-^; *• ^"+1 {l~x)^ {l-xf 414 SERIES WHOSE LAW IS NOT GIVEN. Ex. 2. Find the sum of n + 1 terms of the series V + 2^x + S^x^+ + {n+lfx\ 5f^i = 18 + 23x + 3%2+ + (n + l)3a;», {1-x)*=1-4jc + 6x^-4x» + x*', /. Sf^iX(l-a;)4=l + (23-4)a; + (38-4.23 + 6.18)a;a + (43-4. 33 + 6. 2»-4.1»)a:» + {-4(n + l)3 + 6n8-4(n-l)' + (n-2)3}a:''+i + {6 (n + 1)» - 4n3 + (n - 1)3} «"+=' + {-4(n + l)»+n3}a;~+3 + (n + l)3a;»+*. [The other terms all vanishing, since k^-4{k-l)* + 6{k-2)^-^{k-Sf + {k-4f = identically.] Hence /S„+i = [1 + 4a; + a;^ _ (n» + 6n^ + 12n + 8) a;"+i + (3n3 + 15n^ + 21n + 6) x»+2 - (3n3 + 12n2 + 12n + 4) a;«-*-3 + (n + l)3a:«+*]/(l-a;)^ When X is numerically less than 1, the series is convergent, and the sum of the series continued to infinity is {l + 4x + x^)l{l- a;)*. 327. Series whose law is not given. We have hitherto considered series in which the general term was given, or in which the law of the series was obvious on inspection. We proceed to consider cases in which the law of the series is not given. With reference to series in which the law is not given, but only a certain number of the terms of the series, it is of importance to remark that in no case can the actual law of the series be really deter- mined : all that can be done is to find the simplest law the few terms which are given will obey. There are for instance an indefinite number of series whose first few terms are given by a; + a?* H-a^ + ..., the simplest of all the series being the geometrical progression whose nth term is a?" : another series which has the given terms is that formed by the expansion of r ^ , METHOD OF DIFFERENCES. 415 which agrees with the geometrical progression except at every 10th term. Note. In what follows it must be understood that hy the law of a series is meant the simplest law which satisfies the given conditions. Method of Differences. 328. If in any arithmetical series a, + a2 + a,+...+ a„, each term be taken from the succeeding term, a new series is formed, namely the series (a, - a,) + (a, - aj +...+ («„- a„_,) +. . ., which is called the first order of differences. If the new series be operated upon in the same way, the series obtained is called the second order of differ- ences. And so forth. Thus, for the series 2, 7, 15, 26, 40, ... , the first order of differences is 5, 8, 11, 14, ..., and the second order of differences is 3, 3, 3,... 329. When the law of a series is not given, it can often be found by forming the series of successive orders of differences ; if the law of one of these orders of differences can be seen by inspection, the law of the preceding order of differences can often be found, and then the law of the next preceding order of differences, and so on until the law of the series itself is obtained. The method will be seen from the following examples. Ex. 1. Find the nth term of the series 1 + 6 + 23 + 58 + 117 + 206+ The first order of differences is 5 + 17 + 35 + 59 + 89 + . .. .„ „ second „ „ „ 12 + 18 + 24 + 30+ „ third „ „ „ 6 + 6 + 6 + ....^ 416 METHOD OF DIFFERENCES. The second order of differences is clearly an arithmetical progres- sion whose nth term is 6 (n + 1). Hence, if t?„ be the nth term of the first order of differences, we have in succession «'n-«'n-l = 6»; Vn-1-Vn-2=6(W-1); ] V^-Vj^ = 6 .2, Also »i=6 . 1-1. Hence, by addition, t;„=6(l + 2 + +n)-l = 3n(n + l)-l. Then again, we have in succession u^ - w„_i = v„_i = 3 (n - 1) n - 1 ; «„_i-w„_a=3(n-2)(n-l)-l; ..,; M2-Wi = 3 . 1. 2-1. Also 1*^ = 1. Hence m„=3 {(n-l)n+ + 1 .2} -n + 2 = (n- 1) n(TO + l) -w + 2. Ex. 2. Find the nth term and the sum of n terms of the series 6 + 9 + 14 + 23 + 40+ The first order of differences is 3 + 5 + 9 + 17+ „ second,, „ ,,2 + 4 + 8+ Hence the second order of differences is a geometrical progression, the (n - l)th term being 2»-i. Hence, if r„ be the nth term of the first order of differences, we have in succession ^«-^^n-l = 2~-^ i'„-i-Vn-2 = 2"-^ , v^-v, = 2\ Also Vi = 3. Hence, by addition, v„ = (2 + 2^ + + 2»-i) + 3 = 2" + 1. Then again, we have in succession u^-u^-i=v^_^= 2^-^ + 1, **n-l-Wn-a=2"~^ + l , W3-Wi = 21+1. Al80Mi = 6. Hence M,»=(2»-i+...+2) + n+5=2«+n + 3. The sum of n terms of the series can now be written down : for the sum of n terms of the series whose general term is 2" + n + 3 is (2 + 2a+...+2«) + {n + (n-l)+... + l} + 3n = 2«+i-2 + in(n+l) + 3n. Note. By the method adopted in the preceding examples the nth term of a series can always be found provided the terms of one of its orders of differences are all the same, or are in geometrical progression. 330. It is of importance to notice that when the nth term of a series is an integral expression of the rth degree in n, all the terms of the rth order of differences will be the same. RECURRING SERIES. 417 For, if w„ = Ay + A.^{!nr^ + . . . + J.^ , where A^, A^_^,, , . do not contain n, the nth term of the first order of differ- ences will be [4,(«+l)'+4^,(» + ir+ ...} - {^X + ^^.n-' + ...}, which only contains n to the (r — l)th degree. Similarly the wth term of the second order of differ- ences will be of the (r — 2)th degree in n ; and so on, the nth term of the rth order of differences being of the {r — r)th degree in n, so that the nth term of the rth order of differences will not contain n, and therefore all the terms of that order of differences will be the same. When therefore it is found that all the terms of the rth order of differences are the same, we may at once assume that u^ = A^ + A^^^ + ... + A^y and find the values of A^y A^_^y ..., ^o ^7 comparing the actual terms of the series with the values obtained by putting n = 1, n = 2, &c. in the assumed value of u^. This method will not however give the value of u^ in a convenient form for finding the sum of the series ; for, if r be greater than 3, the sum of n terms of the series whose general term is Ay ■\- A^^^-\- ... cannot be found [see Art. 321] without a troublesome transformation which will in fact reduce u^ to the form in which it is obtained by the method of the preceding Article. A much better method would be to assume that u^ = A^ (n\ + A^^ (^)r-i + •••> and then to find A^, A^^,...jiA^ as above. t; Recurring Series. 331. Definitions. When r + 1 successive terms of the series a^ -f a^x + a^x^ + + a^x^ +. . . are connected by a relation of the form a^ a?" + px (a^_^ a?""^) + qa^ (a^_^ a?""^) + ... = 0, the series is called a recurring series of the rth order, and 1 •\- px-\- qx^ + ... is called its scale of relation. The relation does not hold good unless there are r terms before the nth^ so that the relation only holds good after the first r terms of the series. 418 RECURRING SERIES. For example, the series 1 + 2a? + 4a;* + 8.z;' + is a recurring series of the first order, the scale of relation being 1 — 2x. Again, it will be found that the series 1 4- 3a; + 5a^ + 7a;' + 9a;* + is a recurring series of the second order, the scale of relation being 1 — 2a; + a;*. \ 332. To find the sum of n terms of a given recurring Let the series be a„ -H a^x 4- + (X„a;**-|-..., and let the scale of relation be 1 + pa; 4- qa^. [This assumes that the recurring series is of the second order, but the method is perfectly general]. Then .-. S^ (1 + pa; + qx^) = a, + (a^ + pa^ x-\-{a^+ pa^ + qa^) o? + ... + (»„+ V^.-X + ^«n J ^" + ( i>«« + ^«»-i) ^"^' + ^«« ^"^ = a, + (Oi + pa,) X + (^a„ + ^a„_,) a;"""' + ^a„ a;""^, since all the other terms vanish in virtue of the relation ay^ a;* + px {a^_y a;*"^) + qo^ {a^_^ a?*"') = 0, which is by sup- position true for all values of h greater than 1. Hence ^ ^ «o + (o^i + pa^ ^ + (^o^n + go^n-i) a?**^' + qa^x''^ " 1 + pa? + g'a;* If the given series be a convergent series, the nth term will be indefinitely small when n is increased without limit; and the sum of the series continued to infinity will in this case be given by ^ ^ ao4-(a^+pa o)a? l+pa; + ga;' The expression % . ^ — ^-^ is therefore such that if it ^ 1 ^-px 4- qof can be expanded in a convergent series proceeding accord- ing to ascending powers of a;, the coefficient of a?" in its expansion will be the same as in the recurring series. RECURRING SERIES. 419 On this account the expression -^ — ^-^^ — - — %— is ^ 1 +px + qar called the generating fwnction of the series. 333. A recurring series of the rth order is determined when the first 2r terms are given. For let the series be a^ + a^x-\-a^a;^ -h +«„«;'*+ Then, the series being a recurring series of the rth order, if we assume that the unknown scale of relation is 1 -h2)^x+p^x^-\-,..+p^x'', we have by definition the follow- ing equations ^r^l+i'l^r + P,a^,+ ...+Prao =0, C^r« + Pr^r^i +P,(J'r + • • • + PA = 0, =0, We have therefore r equations which are suiSficient to determine the r unknown quantities p^, p^, "-yPr '^^ ^^® scale of relation ; and when the scale of relation is deter- mined the series can be continued term by term, for a^^^ is given by the equation aj^, + p^a^ + . . . + p^a^ = ; and when ajj^i is found, a^^^ can be found in a similar manner ; and so on. The series is similarly determined when any 2r con- secutive terms are given. 334. From Art. 305 we know that if ^ < r + 1, to r + 2 terms = 0, for all values of k. 420 RECURRING SERIES. This shews that the series l"- + I'x + 3V +. . .4- (w + 1/a;" +. . . is a recurring series whose scale of relation is (1 — xf^^. It also shews that the series a^ + a^x + a^o^ +. . .+ a„a?" +. . . is a recurring series whose scale of relation is (1 — xY"^ whenever a^ is a rational and integral expression of the rth degree in n. 335. In order to find the sum of any number of terms of a recurring series by the method of Art. 332, it is neces- sary to know the general term of the series; we must therefore shew how to obtain the general term of a recurring series when the first few terms are given. By Art. 333 the scale of relation of a recurring series of the rth order can be found when the 2r first terms are given ; and, having found the scale of relation, the genera- ting function is at once given by the formula of Art. 332. Now, provided the scale of relation can be expressed in factors of the first degree, the generating function can be expressed as a series of partial fractions of the form A A or of the form 7^ rv, and the coefiScient of any 1-ax {l-axf* ^ power of X in the expansion of the generating function can be at once written down by the binomial theorem; and thus the general term of the series is found. When the value of x is such that the given recurring series is not convergent, the generating function will not be equal to the given series continued to infinity nor can it be expanded in a series of ascending powers of x ; but, taking as an example the generating function in Art. 332, the expression - ^ _J * — ^^{^ can always be expanded in ascending powers of y, if y be taken sufficiently small, and RECURRING SERIES. ' 421 the coefficients of y® and y* in this expansion will clearly be a^ and a^ respectively and all succeeding terms will obey the law a^ +paj,_j^ + qa^_^ = 0, and hence all the coeffi- cients of the expansion will be the same as the corre- sponding coefficients in the given series. We may there- fore in all cases, whether the series is convergent or not, find the general term of a recurring series by writing down the expansion of its generating function in ascending powers of w on the supposition that a? is sufficiently small. Ex. 1. Find the nth term of the reourrmg series 3 + 4a: + 6ir* + lOx' + . . . In an example of this kind, in which the order of the recurring series is not given, it must always be understood that what is wanted is the recurring series of the lowest possible order whose first few terms agree with the given series. In the present example there is a sufficient number of terms given to determine a recurring series of • the second order, but an indefinite number of recurring series of the third, or of any higher order than the second, could be found whose first four terms were the same as those of the given series. [See Art. 327.] Assuming then that the scale of relation is 1 +px + qx\ we have the equations 6 + 4^) + 3g = 0, and 10 + 6^ + 4g = 0, whence ^ = - 3 and 2 = 2. Hence the scale of relation is 1 - 3a; + 2x2. The generating function is therefore 3 + (4-9)x_ 3-5a; _ 2 1 1 - 3a: + 2a;a "" 1 - 3x +2a:3~ l-a?"*" 1 - 2x s=2{l+x+...+a;«-i} + {l + 2a;+...+2»-ia;«-i + ...}. Hence the general term of the series is (2 + 2"~^) x**"!. The sum of n terms can now be found by the method of Art. 332; the sum can however be written down at once, for the sum of n terms of the series 2 (1 + x + a;^ + . . .) is 2 (1 - x") / (1 - x) and the sum of n terms of the series 1 + 2a; + 4a;''* + ... is (1 - 2**a;'») / (1 - 2x). We may remark that the given series is convergent provided a; < ^. Ex. 2. Find the nth term and the sum of n terms of the series 1 + 3 + 7 + 13 + 21 + 31 + .... Consider the series 1 + 3a; + 7a;» + 13a;3 + 21a;* + 313;^* + ... Then, assuming that the series is a recurring series, and also that a sufficient number of terms are given to determine the recurring series completely, it follows that the series is of the third order. Let then the scale of relation be l+px + qx^ + rx^) we then have the following equations to find j?, g, r : 422 RECURRING SERIES. lS + 7p + Sq + r = 0, 21 + r6p + 7q + 3r=0, and n + 21p + lSq + 7r=0, whence p= -3, q = S and r= -1, so that the scale of relation is 1 - 3x + Bx"^ - x'. The generating function is now found to be 1 + x^ _ 2 2 1_ {l-x)^~{l-xf (l-a:)2 1-x Hence the general term of the series l + Sx + 7x^+.'.. isa;"-i{n(n + l)-27H-l}=(n2-n + l)a'» ^ Thus the general term of the given series is w^- w+ 1. Having found the general term of the series the sum of the first n terms can be written down, for the sum of n terms of the Series whose wth term is n (w - 1) + 1 is - (n - 1) n (n + 1) + n. o Ex. 3. Find the nth term of the series 2 + 2 + 8 + 20 + Considered as a recurring series of the lowest possible order, the generating function of 2 + 2x + 8a;2 + 20x8+.., will be found to be 2 -2a: l-2a;-2a;2' Now the factors of 1 - 2a; - 2x^ are irrational, and therefore the nth term of the series, considered as a recurring series of the second order, will be a complicated expression containing radicals. On the other hand, by the method of Art. 329, we should be led to conclude that the nth term of the series was (Sn^ - 9n + 8) x^~^, which by Art. 334 is a recurring series of the third order. As we have already remarked, the actual law of a series cannot be determined from any finite number of its terms, and the above is a case in which it would be difficult to decide as to what is the simplest law that the few terms given obey, for the recurring series of the lowest order which has the given terms for its first four terms is not the recurring series which gives the simplest expression for the nth term. CONVERGENCY AND DIVERGENCY. 3n6. We shall now investigate certain theorems in convergency which were not considered in Chapter XXI. CONVERGENCY. , 423 337. Con vergency of infinite products. A product composed of an infinite number of factors cannot be con- vergent unless the factors tend to unity as their limit ; for otherwise the addition of a factor would always make a finite change in the continued product, and there could be no definite quantity to which the product approached without limit as the number of factors was indefinitely increased. It is therefore only necessary to consider infinite pro- ducts of the form n (1 + «,) = (1 + «.) (1 + u,) (1 + u,). . .(1 + «.).... .where u^ becomes indefinitely small as n is indefinitely [increased; and the convergency or divergency of such products is determined by the following theorem. Theorem. The infinite product II (1 + u^), in which ill the factors are greater than unity, is convergent or livergent according as the infinite series Xu^ is convergent or divergent. Since e* > 1 -f- a;, for all positive values of x, it follows that (1 + Wj) (1 + u,) (1 + M,). . . < e"' . e»« . e**» . . . < e«, +«,+«,+... Hence, if %u^ be convergent, 11 (1 + u^) will also be convergent Again, (1 + wj (1 + w^) > 1 + m, + w^, (H-w,)(l+w,)(l+W8)>(l+^i+^2)(l+^)>l + ^i+^+^8> and so on, so that Hence, if Xu^ be divergent, H (1 + u^) will also be divergent. ■n , mi. XV ^a{a + l)(a + 2)...(a + n-l) . . „ .. , Ex. 1. To shew that y^v — '\. ' ), :r[, is infinite or zero, when b(h + l){b + 2)...{b + n-l) n is indefinitely increaeed, according as a is greater or less than 6. s. A. 29 424 CONVERGENCY. For, if a > 6, the expression may be written in the form (-^)(-l^:) (-^i) which is greater than ! + („_,) |l+_l-+^_i-^ + }• But T + -, — T + i — s + ... is a divergent series [Art. 274] : the given ex- 0+1 0+2 pression is therefore infinite when n is infinite, a being greater than 6. If 6>a: then as before, — ) -J4 „! is infinite; and a(a+l)(a + 2) ^- , a(a + l)(a + 2) ,, therefore -^4-, — rrr^ — ?:r must be zero. 6(& + l)(6 + 2) Ex. 2. Determine whether the series a a{a + x) a (a + x) {a + 2x) 6"*" 6(6+a;)"^ b(b+x){b + 2x)'^ is convergent or divergent. From Art. 325, we have _ a I {a + x){a + 2x)...{a + nx) ] ''~a + x-b\i,(^b + a:}{b + 2x)...{b + n^l.x) \' Now by Ex. 1, ^^ — — £zu_i_ is mfimte or zero according b{b + x)...{b + n-lx) as a + X ^ b. Hence the given series is convergent, and its sum is then ~ , 0— a — X if 6 > a + a;. Also the series is divergent \ib<.a + x. Also if 6=a + a;, the series becomes r-+, V-, — jr- which is b h + x b + 2x known to be divergent [Art. 274]. 338. The Binomial Series. We have already proved that the binomial series, namely - , , m(m-l) „ m(m-l)(m-2) - . l + m^+ 1.2 ^ + 1.2 3 "''+•" is convergent or divergent, for all values of m, according as a; is numerically less or greater than unity. If flj = 1, the series becomes 1 I m I ^(^-^) I m(m-l)(m-2) CONVERGENCY. 425 Now we know that the terms of this series are alternately positive and negative after the rth term, where r is the first positive integer greater than m + 1. More- over the ratio u^^Ju^ is numerically less or greater than unity according as m + 1 is positive or negative. The series will therefore, from theorem V. Chapter XXI. be convergent when m + 1 is positive provided the nth. term decreases without limit as n is increased without limit. .. ^1 1.2 n Now ± — u^ (— m) (1 — m). ..{n — l—m) u^ m\ l—mj\ 2 — mJ \ n — l — mj Now, if m 4- 1 be positive and less than r, the product of the factors from the rth onwards is greater than ^ ^ [r — m r + 1— m J and the product of the preceding factors is finite. Hence, when n is increased without limit, 1/u^ is in- finitely great, and therefore u^ indefinitely small, provided 1 + m be positive. Thus the binomial series is convergent if ic = l, pro- vided m > — 1. If a? = — 1, the series becomes 1 m I ^(^-1) m(m-l)(m-2) The sum of n terms of the above series is easily found to be [see Art. 287 or Art. 325] (1 - m) (2 - m) (3 - m).,.(n - 1 -- m) 1.2.3...(w-l) The sum of n terms of the series is therefore pEx. 1, Art. 337], zero or infinite, when n is infinite, according as m is positive or negative. Thus the binomial series is convergent when a? = — 1, provided m is positive, 29--2 426 CONVERGENCY. 339. Cauchy's Theorem. If the series u^ + u^-^ u^ ... +u^+ ... have all its terms positive, and if each term e less than the preceding, then the series will be convergent divergent according as the series Wj + au^ + a^Uas + ... + a"Uan-\- ... 15 convergent or divergent, a being any positive integer. For, since each term is less than the preceding, we have the following series of relations Wi + w, + . . . + Wa < a^ < (a - 1) ^^ + ^, w^i -\-u^^...-\-Ua^< (a' -a)u,<(a-l)au^, m Uan+i + Uan+2 + . . .+ Won+i < (a""^^ - a") Uan < (a - 1) aJ'Uan, Hence, by addition, >Si< (a — 1) S + u^ (I), where S and S stand for the sum of the first and second series respectively. Again, we have since a is «w« a {u^^ + w^^2 +. . .+ tt^a) > a{a^ — a) Ua^ > G^UcHy a (Ua^-i+i + Wa-i+2 +. . .+ Ua") > a (a"" - a""') Ua» > a'^Uan. Hence aS>X —v^ (II). From I and II it follows that if S is finite so also is S, and that if S is infinite so also is S. Ex. To shew that the series —r, rz is converffent if fe be greater n (log n)* ^ ^ than unity, and divergent if k be equal or less than unity. By Cauohy's theorem the series will be convergent or divergent a" according as the series whose general term is -^j: — ^ is convergent or divergent. CONVERGENCY. 427 a* (log a**)*~ n* (log a)* ~ (log a)* n* * it therefore follows from Art. 274 that the given series is convergent if fc > 1 and divergent if fc :f 1. 340. We shall conclude with the two following tests of convergency which are sometimes of use, referring the student to Boole's Finite Differences and Bertrand's Differ- ential Calculus for further information on the subject. 1 '341. Theorem, A series is convergent when, from \and after any particular term, the ratio of each term to tJie preceding is less than the corresponding ratio in a known convergent series whose terms are all positive. For let the series, beginning at the term in question, be 27 = 1^, + w, -f W3+...+t<„ + ..., I and the known convergent series, beginning at the same term, be F = Vj-|- V, + Vg+ ... + v„ + .... U V Then, since -^* < -^* for all values of r, we have u, V, F= V, + v, . -^ -I- v, -' -^ + V, -* -' - + ... u^ u. u. u. u^ u^ Hence as F is convergent, U must also be convergent. I The given series is therefore convergent, for the sum I of the finite number of terms preceding the first term of I U must be finite. ; We can prove similarly that if, from and after any [ particular term, u^^ : u^ > v^^, : v^, and all the terms of Xu^ have the same sign ; then 1u^ will be divergent if Xv^ be divergent i 428 CONVERGENCY. 342. Theorem. A series, all of whose terms are posi- tive, is convergent or divergent according as the limit of nil ^^ J is greater or less than unity. For let the limit of w ^1 - ^'] be a. Consider the series ^ — 5 = Xv^ : then / v^^A _ Un-^iy — n^\ _ fin^ + lower powers of n \ v^J ] (n-\-iy ) n^ + lower powers of n Hence the limit of 71 [ 1 —^ ) , when n is infinitely great, is /8. First supposie a > 1, and let yS be chosen between a and 1. Then since the limit of n(l ^M is greater than the limit of n [ 1 ^M , there must be some finite value of n from and after which the former is constantly greater than the latter. But when nfl- ^^] >n(l- '"^'\ , we have -^* > -^^ . Hence, by the previous theorem, '^u^ will be conver- gent if 2v^ be convergent; but Xv^ is convergent since Similarly, if a be < 1, and y9 be taken between a and 1, we can prove that 2m^ is divergent if Xv„ is divergent, and the latter series is known to be divergent when y9 < 1. If the limit of w (l ^M be unity the test fails. EXAMPLES. 429 „ , , „ .a a{a + l) a(a + l){a + 2) „ . B.. 1. Is the senes -+ ji^« + ^^^^^21^.»+... convergent or divergent ? Here ''^^rr J^a;, the limit of which is x. Hence, either from M„ b+n the beginning or after a finite number of terms, — "*"^ ^ 1 according as a; ^ 1. Hence the series is divergent if x > 1, and convergent if a; < 1. If a;=l, the limit of -^- is unity. But {^-"^M^-m- I the limit of which is 6 - a. Thus, if x = l, the series is convergent when 6-a>l and divergent when 6 - a < 1. "When 6=a+ 1, the series becomes a a a 6"*" 6 + 1 "^6 + 2"^ * which is divergent. [These are the results arrived at in Ex. 2, Art. 337.] EXAMPLES XXXIY. 1. Find the sum of each of the following series to n terms, and when possible to infinity : — ■ 4 4.7 4.7.10 ,.., 2 2.5 2.5.8 (^^) 4^4T7^4TnO^-- 3 S^ 3.5.7 ^''^^ 8"^8.10"^8.10.12'^-* 11 11.13 11.13.15 ^'""^ 14 "^14. 16 ^14. 16. 18"*" •••• 2. Find, by the method of differences, the nth term and the sum of n terms of the following series : — (i) 2 + 2 + 8 + 20 + 38+.... v/(ii) 7 + 14+19 + 22 + 23 + 22 + .... (iii) 1 + 4 + 11 + 26 + 57 + 120+.... 430 s, EXAMPLES. yfv) 1 + + 1 + 8 + 29 + 80 + 193+.... ^) 1 + 5+15 + 35 + 70+126+.... XAji) 1 + 2 + 29 + 130 + 377 + 866 + 1717 + .... 3. Find the generating function of each of the following series on the supposition that it is a determinate recurring serieB : — V(i) 2 + 4a;+14ar» + 52a;»+.... (ii) l + 3a;+lla;' + 43a^+.... (iii) l + 6a; + 40a;»+288a3'+.... ; (iv) 1 + (B + 2a5* + 7a;« + 14^' + 35a;» + .... J (v) 1" + l^x + 3 V + 4*a^ + 5 V + 6 V + . . . . 4. Find the nth term, and the sum of n terms of the following recurring series : — %(i) 2 + 6 + 14 + 30+ .... (ii) 2-6 + 29-89+.... (iii) 1 + 2 + 7 + 20+.... 5. Find the nth term of the series 1, 3, 4, 7, 1 and jo + n > 0. 13. Shew that, if m be greater than 1, 1 1.2 1.2.3 1+ T + rn+l (m+l)(7» + 2) (m+ 1) (m + 2) (7At + 3) + ... to infinity m m— 1 14. Shew that _1 n-1 • (n-l)(n-2) 1 m + 1 (m+l)(m + 2) (w + l).(m + 2) (m + 3) '** m + n' if m + w be positive, or if w be a positive integer. 15. Shew that, if n be any positive integer, n n(n — l) n{n- 1) (n - 2) ;rri ~ (n+l)(n+2) "^ (rM^T)(n + 2)(n + 3y'*' "• yt(7i- l) (n-2)... 2.1_ 1 "^ {n+\){n + 2).,:2n ~2' 432 EXAMPLES. 16. Shew that, if m be a positive integer, 2n + 1 m(m-l) (2n + l) (2n + 3) _ ^""^27*1^^ 1.2 (2n + 2)(2n + 4) "* 1.3.5...(2m-l) ~ (2w+ 2) (27i + 4) ... (2w + 2m) * 17. Shew that, if m, n and m — n + l are positive integers; then m n(n-l) m(m-l) 1 +w m-n + 1 1.2 (m-n+ l)(wi-ri + 2) n(n-l)(n-2) m(m-l)(m-2) "*" 1.2.3 (m-n+l){m-n+2){m~n + 3) ^ , ,,, (m+ l)(m+2)...(m + m) + ...to (n+1) terms = 7 "^ , , / ^ ^ ^, ^ . ^ -v. ^ ' (m-7i+ l)(m-fH- 2) ... (7»-w + m) 18. Shew that, if m + 1 > 0, then 1-1 I m(m- l) l w(w-l)(m-2) 2 S''*'^^ 172"~"5 1.2.3 ■^■•* 1 (m+l)(m+2)* 19. Shew that, if P^ be the sum of the products r together of the first n even numbers, and Q^ be the sum of the products r together of the first n odd numbers; then will 1+P,+P,+ +P^=1.3.5...(2n+1), and l + Q^ + ^^+ + <^^ = 2.4.6..,2n. 20. Prove that [a + {a+ l)+(a+2) + ... +(a + n)}{a" + (a+ l) + (a + 2) + ... ... + (a + n)} = a^ + (aH-l)' + ... ^-(a + 7l)^ 21. Shew that the series l-g" (l-a")(l-a"-^) (1 -a")(l -a^-^)(l -g --^) 1-a^ (l-a)(l-a«) (1 - a)(l -a*)(l -a«) ^ '" is zero when n is an odd integer, and is equal to (1 -a) (1 -a') ... (1 — a"~^) when w is an even integer [Gauss]. EXAJtfPLES. 433 22. Find the sum of the series n n— 1 n— 2 + 77-71—7 + -^r—r-P + ... + 1.2. 32. 3. 43. 4. 5 w(w+l)(n + 2)' n^ « , . ^ ., 2(B» 3a:» 4£C* 23. Sum to infinity =— x - ^ - + 3-5- •••• 24. Sum, when convergent, the series X a? x" 172 "^ 273 "*■••• "^ ^^^) "^ ••• • 25. Sum to infinity the series 1 . 2 . 3 + 3 . 4 . 5a; + 5 . 6 . 7a;' + 7 . 8 . 90;" + ..., X being less than unity. 26. Shew that, if n is a positive integer l-3n^^"f-^)-^"(^Vt^f-^)^... = 2(-ir. 1 . iB 1.2.0 27. Shew that, if a^, a^, a,,.-, be all positive, and if ttj + a, + ttj + . . . be divergent, then + .. a^ + 1 (a^ + l)(a,+l) (a, + l)(aj+ l)(a,+ l) is convergent and equal to unity. 28. Shew that the series _i_ Z!_ ^ ^^^ is convergent if «> 1, and is divergent if a? :|> 1, 29. Shew that, if the series u^+u^ + u^+ ... +Un+ ... be divergent, the series Wj w,+w, *" Wj + i^, + ... +^^,_^ will also be divergent. 434 EXAMPLES. 30. For what values of x has the infinite product (1 + a) (1 + ax) (1 + ax") (1 + ax^) ... a finite value? 31. Prove that, if v^ is always finite and greater than unity but approaches unity without limit as n increases indefinitely, the two infinite products ViV^v^v^...y v^v^v*v^ ... are either both finite or both infinite. 32. Test the convergency of the following series : — ijifff' 1 2« 3^ n" ^,..., 2 2.4 2.4.6 2.4.6...2ri + .... 3.5.7...(2w-f l)(2n + 2) \ _2__ 1.3 1.3.5 ^^""^ 2.3'^2.4.5^2.4.6.7'*'*" 1.3.5...(2w-l ) ■^2.4.6...2w(2w+l)'^""* /v^ a;g a(a+ 1)^(^+1) , r(r+i) a(a+l)(a-h2)^()8+l)(/? + 2) 1.2.3.y(y+l)(y + 2) CHAPTER XXVI. INEQUALITIES. 343. We have already proved [Art. 232] the theorem that the arithmetic mean of any two positive quantities is greater than their geometric mean. We now proceed to consider other theorems of this nature, which are called Inequalities. Note. Throughout the present chapter every letter is supposed to denote a real positive quantity. 344. The following elementary principles of inequal- ities can be easily demonstrated : I. If a > 6 ; then a + x>h + x, and a—x>h — x. II. If a > 6 ; then —a< — h. III. If a > 6 ; then ma > mb, and — ma < — mb. IV. Ua>b,a'>b\a''>b\&c.; then a + a'-|-a"+... > 6 + 6'+ 6"+ ..., and aa'a". . . > bb'b". . .. V. If a > 6 ; then a"* > &"', and a"^ < Z)"^. Ex. 1. Prove that a» + 6» > a% + ah^. We have to prove that a» - a% - ab^ + 63 > 0, or that (a' - 6«) (a - &) > 0, which must be true since both factors are positive or both negative according as a is greater or less than b. 436 INEQUALITIES. Ex. 2. Prove that a"» + a~^ > a" + a"", if m >• n. We have to prove that (a"* - a") (1 - a~**~*) > 0, which must be the case since both factors are positive or both negative according as a is greater or less than 1. Ex. 3. Prove that (P + m» + n^) {V + wi'» + n'^) > {IV + mm! + nn'f. It is easily seen that (Z» + TO» + n2) (ra + m'a + n'») - {IV + mm' + nn')« = {mn' - m'nf+(nV - n'l)^ + {Im' - Vm)^. Now the last expression can never be negative, and can only be zero when mn' - m'n, nV - n'l and Im' - I'm are all separately zero, the conditions for which are ^ = — ^ = -7 . V m n' Hence (P + m^ + w*) (Z'^ + m'^ + n") > (Zr + mm' + nw')', except when lll'—mjm'=nln'f in which case the inequality becomes an equality. 345. Theorem I. The product of two positive quanti- ties, whose sum is given, is greatest when the two factors are equal to one another.. For let 2a be the given sum, and let a+ x and a — x be the two factors. Then the product of the two quanti- ties is a* — x^, which is clearly greatest when x is zero, in which case each factor is half the given sum. The above theorem is really the same as that of Art. 232 ; for from Art. 232 we have {^\ ("- J^) > ab. 346. Theorem II. The product of any number of positive quantities, whose sum is given, is greatest when the quantities are all equal. For, suppose that any two of the factors, a and h, are unequal. Theu, keeping all the other factors unchanged, take i(a + 6) and ^(a + b) instead of a and b : we thus, without altering the sum of all the factors, increase their continued product since ^(a+b) x ^(a + 6)> ab, except when a = b. Hence, so long as any two of the factors are unequal, the continued product can be increased without altering the sum ; and therefore all the factors must be equal to one another when their continued product has its greatest possible value. INEQUALITIES. 437 .. a )■• Thus, unless the n quantities a, b, c, ... are all equal, and therefore a + h-{-c + d+ ... > :^(ahcd ...). By extending the meaning of the terms arithmetic mean and geometric m^an, the last result may be enunci- ated as follows : — Theorem III. The arithmetic mean of any number of positive quantities is greater than their geometric mean. Ex. 1. Shew that a» + 6' + c' > Sabc. _. ^3 j_ J[)8 _L /jS "We have > ^{a* . 6' . c*) >• dbc. Ex.2. Shew that '^ + ^+-' + +^>n. ag a, a^ a^ Wehavei(^^ + ^^+ + <)^ ::/ h.'^.jA :> ::/!. Ex. 3. Find the greatest value of (a -x){b- y) {ex + dy), where a, 6, c are known positive quantities and a-x, b-y are also positive. The expression is greatest when {ac - ex) {bd - dy) {ex + dy) is greatest, and this is the case, since the sum of the factors is now con- stant, when ac-cx = bd-dy = cx + dy. Whence the greatest value is found to be {ac + bd)^l27cd. Ex. 4. Find when x^y^z^ has its greatest value, for different values of a, y and z subject to the condition that x + y + z is constant. Let P^x'^y^z'^; then P rx' 'x\a (yy (z\y _??? y y y t ^ ^ ~a' a'a ^*/S'^ 7* 7*7 The sum of the factors in the last product is constant, since there are a factors each - , S fiotors each ^ , and y factors each - , and therefore the sum of all the factors ia x + y + z. 438 INEQUALITIES. Hence, from Theorem II, (-■) (V) (-j^bas its greatest value when all the factors are equal, that is when - = | = - . « P 7 It is clear that P is greatest when Pja'^py'^ is greatest, since o, /3, 7 are constant ; hence P is greatest when xla = ylp=zly. In the above it was assumed that a, /3, y were integers ; if this be not the case, let n be the least common multiple of the denomina- tors of a, /S, y. Then x'xfz^ will have its greatest value when x^^-y^^z^"^ has its greatest value, which by the above, since na, n/3 and ny are all integers, will be when — = ^L = — that is when ' ^ na n^ ny Thus, whether a, /3, y are integral or not, x^y^z^ is greatest for values oix,y and 2; such that x + y + zia constant, when xja = yj^ = z/7. 347. Theorem IV. The sv/m of any number of positive quantities, whose product is given, is least when the quantities are all equal. First suppose that there are two quantities denoted by a and 6. Then, if a and h are unequal, {s/a ~ \/l>f > 0, and there- fore a + b> Jab + Jab. Hence the sum of any two unequal quantities a, b is greater than the sum of the two equal quantities Jab, Jab which have the same product. Next suppose that there are more than two quantities. Let a, by any two of the quantities, be unequal. Then, keeping all the others unchanged, take J ah and Job instead of a and b ; we thus, without altering the product of all the quantities, diminish their sum since Jab + Jab r ; then, imless a^ — ai = as = Sac, a: + ar+... + a- ^^^ n greater than n n We have to prove that or that (n - 1) « + a,"» + ...)> 2 (a, V + a^'^-XO, or that 2 (a,*" 4- < - a;a,'"-' - a^^-^a^) > 0, every letter being taken with each of the (n — 1) other letters. Now which is positive since a^ — a^ and a^'""'" — a,*""** are both positive or both negative according as a^ is greater or less than a^. Hence t (o^"* + a,*" - aX""" " ^r'^sO > 0, which proves the proposition. By repeated application of the above we have ta^ tal tal_ Xay n n ' n ' n '"* where a, jS, y, ... are positive quantities such that a + )S + 7+ =m. Ex. 1. Shew that 3 {a* + P + c^) > (a + b + c) (a^ + b^hc^), Ex. 2. Shew that a« + ft'' + c» > abc (a^ +b^ + c^). From Theorem V, > r . b*^ > o o o aa + 52 + c2 > Q . abc, from Theorem III. s. A. 30 440 INEQUALITIES. . 349. Theorem VI.* To prove that, if ay b,Cy.,. and a, ^,y, ... be all positive, then V a + b + c + .., J ^ ' First, let a,b,c, ... be integers. Take a things each a, b things each 0, and so on. Then, by Theorem III, (a + a+ ... to a terms) + (y6! + yS + ... to b terms) + ... >«*VKi^' }, that is "^:;f^t.:' >°^v{«y...}. If a,b,c,... be not integral, let m be the least common multiple of the denominators of a, 6, c, ... ; then ma, mb, mc, . . . are all integers, and we have maa + mb0-\-... ^ ^^^^^ |a"'«^'»^ I ma -\- mb -\- . . . '^ ^ '"*' Hence i^^j >--^ w Cor. I. Put letters a, b, . . . ; a = -,/3 = r,...5 and let t a then • • [a + 6 + ...j ^a"6'...' Cor. II. Substitute in (A) a** for a, 6" for 6, ... ; also substitute a"'"' for a, 6*""^ for ^, ..., where m > r. a'XVt::] >Kfe"-}"-' [B]. * See a paper by Mr L. J. Eogers in the Messenger of Mathematics, Vol xvu. INEQUALITIES. 44)1 Again, substitute a*", If, ... for a, b, ... respectively, and a*"*", l/'", ... for a, /8 respectively, where ^< r. Then a' + 6*+...! K + 6'' + ...^ Hence, as m — r and r — ^ are both positive, we have from [B] and [C] Hence, provided m>r >t, {a'"+6'"+...px{a'"+6*'+...}*-^x{a'+6*+...)'""'>l....[D]. The following are particular cases of [D]. Put t = 0'y then, since a® + 6°+ ... = w, we have provided m>r f'^J>{'~M" f^^- Again, put ^ = 0, m = 1 ; then since m>r>t, r must be a proper fraction. Hence, if r he a proper fraction, r-±^f>°^^- m. Again, put < = 0, r = 1 ; then m > 1. Hence, ifm>l we have r-±^-r- [°i- Now put m = 1, r = 0, then < is negative. Hence, pro- vided t be negative, {a + b+ ...)-* X rT' X (a* + b* + ...)>1 ; • -'4-6^ + ... /a + 6+,^y ^^^^ 30—2 442 INEQUALITIES. From [F], [G] and [H] we see that a' + b'-^... > r a + 6 + ... r n < L '^ J according as x is not or is a proper fraction. 350. We shall conclude this chapter by solving the following examples. [See also Art. 133.] Ex.1. Shew that, if « = ai + aa+... + a^, 1 H... + > — =-, unless flh=aa=a...=a-. Unless ai=aa=...=a^, we have 1/ < $ s \ -/ «* nV-Oi s-a^ '" s-aj^ y (s-a^){s-a^...{8-aj* ^,t%)±(fZ^±,::±(lZ£^>yj(._^)(.-,,)...(._aJ}. By multiplication, since (« - Oj) + (« - aj) + . .. + (» - a J = n* - «, we have n^ \»-«i 8-a^ '" 8-a^J Ex. 2. Shew that, if a + 6 + c + d = 3s, and a, 6, c, d, a-a, «-6, <-c, » - d are all positive ; then will abed > 81 (s - a) (s - 6) (s - c) (s - d). For 84/{(«-6)(»-c)(»-d)}<{(»-6) + («-c) + (»-d)}^^{3r'yyz^)', #« -t-^ I-* V ^a^yV^^ Vx+y+z/ ^ If «, y, t be not integral let m be the least common multiple of theii- denominators ; then mx, my and mz are integral, and we have by the first case — ?. ) > (ffw:)"« imy)^y {mzY>^\ \ mx + my + mz J \ / \ sf/ \ / * EXAMPLES. 443 that is I ^ ) X »n"»(»^+«) > (ar^yi'z*)"* x m»»»(*-^^'+') j \ x+y+z J \ x+y + z J ^ The Theorem can in a similar manner be proved to be true for any number of quantities. EXAMPLES XXXY. Prove the following inequalities, all the letters being supposed to represent positive quantities : — 1. y^z' + z'a:^ + (C*y^ -^^ xyz (x + y + z). 2. («,« + a/ + a3«+ )(6^« + 6.- + V+ ) ^^ ) (^+ ^'^ ^+ ) _ fx y z\ /a h c\ . _ \a cj \x y zj ^ 6. (a + 6 + c) (a' + 6* + c«) (a;j + a;j+ +£cj^>a. 444 EXAMPLES. 14. Prove that, if cCj, x^, x^, , x^ be each greater than a, and be such that (x^-a){x^-a) (x^ -a) = b% the least value of x^x^x^...x^ will be (a + 6)*, a and b being positive. (a + b)xy . ax + by 15. Shew that ^ ^ > -/ . ay + 0X a + o no 2 2 2^9 6+c c+a a+6 a + 6 + c* 3 3 3 3 . 16 17. 5 J + ; + -J r + i 6 >c; then / a + c y / 6 + c y \a — cj\b-cj* 19. If x' = i/' + s^f then will a;" ^ y" + z" according as n ^ 2. 20. Shew that (a6cc?y+*^^^ lies between the greatest and 1111 least of a\ 6*, c', c?* . 21. Shew that l+x + x' ■¥ ^x" <^(2n+l) x\ 22. If n be a positive integer, and a > 1 ; then a > a^--l a-r 23. (m + l)(m + 2)(m + 3) {m+ 27i-l)^(7n + n)'^-\ 24. Shew that, if all the factors are positive: — abc ^ (b + c ~ a) (c + a — b) (a + b — c). 25. abed ^ (b -i- c + d-2a) (c + d + a- 2b) (d + a + b - 2c) (a + b + c — 2d). 26. a,a.a,...a, ^ o+c—a c+a—b a+b—c a+b+c EXAMPLES. 445 28. Shew that, unless a = h = c, (b-€y{b + c-a) + {c-ay(c + a-b) + (a-by{a + b-c)^0, 29. Shew that, if a, 6, c be unequal positive quantities, then a'(a-b){a-c) + b'{b-c){b-a) + c'{c-a){c~b)^0. 30. Shew that px'''" + qx'~^ + ra^"' > jo + ^ + r, unless a; = 1, or p = q = r. 31. Shew that c, a a c, < , / o"" TT' 2. 4. 6...2n v 2n + 1 on CI. XT. X 3.7. 11. ..(471-1) / 3 32. Shew that _ ^ ,» — )-. =^ < ^ / -; = . 5.9. 13. ..(471+1) \/ in + Z 33. Find the greatest value of a3*^«^, for different values of X, 2/, and z subject to the condition that aa* + by^ + cz^ = c?. 34. Prove that, if w > 2, (|n)' > w". 35. Shew that, if n be positive, (l+£c)"(l+a;")>2'-'V. 36. In a geometrical progression of an odd number of terms, the arithmetic mean of the odd terms is greater than the arithmetic mean of the even terms. 37. Prove that, if an arithmetical and a geometrical pro- gression have the same first term, the same last t^erm, and the same number of terms; then the sum of the series in A. p. will be greater than the sum of the series in G. p. 38. Shew that, if P^ denote the arithmetic mean of all those quantities each of which is the geometric mean of r out of n given positive quantities; then /*,, P,, ..., F^ are in descending order of magnitude. 39. Shew that, if 8 = a-\-b ■¥ c -^ ..., /« - ay ( 8 - b \^ ( 8 - c y /«\* \n-\) \n-\) V^^^/ "*"^W' n being the number of the unequal positive quantities a, 6, c, .... 40. Shew that, if n be any positive integer, — (-^r(r)'C-i-0""--t^)'©' ^ CHAPTER XXYir. Continued Fractions. 851. Any expression of the form a±b e± &c. is called a continued fraction. Continued fractions are generally written for con- venience in the form ^ h d f ' c± e± g± 852. The fraction obtained by stopping at any stage is called a convergent of the continued fraction. Thus a and a ± - , that is =- and — =- , are respectively the first and second convereents of the continued fraction a + — — - . . . ~o± e± The rth convergent of any continued fraction will be denoted by — . The fractions a, , -, &c. will be called the first, c e second, third, &c. elements of the continued fraction. CONTINUED FRACTIONS. 447 853. In a continued fraction of the forvi a-\ — . . . , where a, 6, c, ,-i [fromi.]. x| Similarly q^^ = b^q^ + a„g„.,. Thus the law will hold good for the (n+l)th con- vergent if it holds good for the nth convergent. But we know that the law holds good for the third convergent ; it must therefore hold good for all subsequent ones. Cor. I. In the fraction a, -i — — ... , ^--^^ • a2 + as + Pn=f^nPn-x + p„., and q, = a^q,.^ + q„. 2* CONTINUED FEACTIONS. 449 CoK. 11. In the fraction 7^ ^ r" •.•! Pn =- KPn-t- ^nPn-, ^nd ^„ = b^q^_^ - a^q^_^. Ex. By means of the law connecting successive convergents to a continued fraction, find the fifth convergent of each of the following fractions : 1111 . 11111 ^^f ^ + 1 + 2 4- 3 + 4* ^"^ 1 + 1 + 4 + 1 + 4 + '" ,..., 12345 ,.,^2222 (^) 2 + 3 + 4 + 6 + 6- <^^) 3+g_^g^-^-^... 12345 /N^^?^! ^^^ 1 + 1 + 1 + 1 + 1* t'^^^ 4 + 3 + 2 + 1 + 2 + *" ,..,22222 , .... 11111 ^") 3-3-3-3-3-- <^")l_i_I_4-l-- 356. The convergents to continued fractions of the form a-\-T - t . • • • > where a, b,c,d,,,, are all positive i- c +a + integers, have certain properties on account of which such fractions have special utility: these properties we proceed to consider. We first however shew that any rational fraction can be reduced to a continued fraction of this type with a finite number of elements. For let — be the given fraction ; then, if m be greater than n, divide m by w and let a be the quotient and p the remainder, so that — = a + - . Now divide n hy p n n and let b be the quotient and q the remainder; then v 1 1 ^=-=s . Now divide p hy a and let c be the ^ !^ 6 + i r p quotient and r the remainder : then - = - = — . By P P c+^ ^ 9. 450 CONTINUED FRACTIONS. proceeding in this way, we find — in the required form, , m , p , 1 11 namely —= a-\-- = a -{■ — ^— = a + - - .... ^ w ,£ 6+c-f- Since the numbers jp, g', r, ... become necessarily smaller at every stage, it is obvious that one of them will sooner or later become unity, unless there is an exact division at some earlier stage, so that the process must terminate after a finite number of divisions. It should be noticed that the process above described is exactly the same as that for finding the G.C.M. of m and n, the numbers a, h, c, ... being the successive quo- tients On this account the numbers a, b, c &c. in the continued fi-action a + r - ••• are often called the + c + first, second, third, &c. partial quotients. It is easy to see that the continued fractions, found as above, for — and — ^ , where k is any integer, will be the same. 491 Ex. Convert ^Hiii *^d 3 •14159 into continued fractions, and find in 71 355 each case the fourth convergent. Aru. ^fpj » rfo • 357. Properties of Convergents. Let the continued 11 v fraction be a, H — . — . ... , and let ^ denote the nth > a,+ a.+ q^ convergent i L From Art. 355 we have Pn _&i= «n Pn-1 +Pn-, Pn-r ^ Pn-An-X - JPn-xg>.-« . CONTINUED FRA.CTIONS. 451 So also in succession But p^q^ -p^q^ = (a,a, + 1) - a^a^ = 1. Hence jp„?^i -i)„.,g„ = (-l)" (i). Hence also ^_£t:» = t:_J_ (ii). Cor. In the continued fraction — — H... , which is less than unity, we have PnQ .-P .q =(-1)""' and & - ^» = tiZL . II. Every common measure of p^ and q^ must also be a measure of Pn "" P-' ^-^ (-1)" Now \ is less than 1, and q^ is greater than g^.^ ; hence F -.^ is less than F-^^ . % 9.n-X Thus any convergent is nearer' to the continued fraction than the immediately preceding convergent, and therefore nearer than any preceding convergent. lY. If any fraction, - suppose, be nearer to a continued «/ fraction than the nth convergent, then - must from III. be also nearer than the (n--l)th convergent; and, as the continued fraction itself lies between the nth and the (w — l)th convergents [Art. 353], it follows that - must if also lie between these convergents. Hence &=^ ~ - must be <-^* ^ 2? ; q^ (p„_, y ~ q^.,x). Hence, as all the quantities are integral, y must be greater than q^ . CONTINUED FRACTIONS. 453 Thus every fraction which is nearer to a continued fraction than any particular convergent must have a greater denominator than that convergent, V. We have seen in III. that where \ is a positive quantity less than unity. Hence F'-^'^> ^-^ c ; also ^-.^i<_l_. Thus any convergent to a continiced fraction differs from the fraction itself by a quantity which lies between -j-j- and -f-rr——T\ , where d, and cZ, are respectively the d^a^ d^ (cfj + a ) denominators of the convergent in question and the need succeeding convergent. Ex. 1. Shew that, if "Prlq,. be the rth convergent to the continued fraction a, + — — — , then wiU t-^<.. 1 1 For we have jp„ = a^„_i + p„_a , Pn-l = (^n-lPn-ii+Pn-i* = Pi=a^Pi + l, andpi = ai. Hence -^=:a„+^^^'= a_ + = Ph-1 " Pn-l " Pn-l 2>n-3 1 «a„+ ' ' -a ^ 111 454 EXAMPLES. It can be proved in a similar manner that ^„ 1 1 1 Ex. 2, To shew that — ^ = pr ^ ^ ... to n quotients, where n is n + 1 2 - 2 - 2 - a positive integer. We have n 1 n-1 1 2_ 1 n n-1 2 Hence =- ^ - ^ .«... to n quotients. Ex. 3. Shew that, if Prl^r ^^ *^® ^^ convergent o' 7 t r .«...; then wDl 1)^1 =05,. EXAMPLES XXXYL 1. Shew that, if ^ , ^ , ^ be three successive converffents ^1 9, gz to any continued fraction with unit numerators, then will 2. Shew that, if ^ be the nth convergent of -7-' r' t^ qn "^ 6, +6,4-6, + ...; then wm p,5'„_,-;7^_,g', = (-l)-X«. «„• V 3. Two graduated rulers have their zero points coincident, and the 100th graduation of one coincides exactly with the 63rd of the other: shew that the 27th and the 17th more nearly coincide than any other two graduations. / EXAMPLES. 455 4. Shew that, if a^, CTj j , a^ be in harmonical progression; then will — ^=- - ... - -s. a,_, 2-2- -2-aj 5. Shew that ^111 ,111 * na^ + na^ + na^ + ^ ^ n\ + % + n\ + * 6. Shew that, if Pe - ^ - ... ~, a +b +c + +^+1' and b + c +d + h then will P(a+Q+l) = -a + Q. 7. Find the value of n w — 1 n — 2 2 1 11 n+w-1 +W-24- *** +2 + 1 +2' 8. Shew that, whether n be even or odd, t t t t 1 —4 — 1 — 4- ^. ^ 2n to w quotients = =■ . n+\ 9. Prove that the ascending continued fraction — — -^ ... is equal to -i + - * + — ?— + ... 10. If p^ be the numerator of the wth convergent to the fraction -^ ^ =^ .... shew that a linear relation connects every successive four of the series jt>,', p/^ Pg',...; and find what the relation is. 11. If pjq, be the rth convergent of - 5- - ^ —j shew that p^^^^ =p^^ + 55^,^, and that g^^, = «/>,„ + {ab + 1)^^. S. A. 31 456 EXAMPLES. 12. If pjq^ be the rth convergent of the continued fraction - T - - T - ..-, shew that », ., = 6», + (6c + l)^, . a+6+c+a + 6+c+ ' '^■••+» ^»" ^ ''^8- 13. If jt?^/^, be the rth convergent oi ~ j =- - ..., shew that ;?,„ ^-^^.i - 1/',, ;?,„_, = - oTir. 14. Shew that, if — be the nth convergent to the continued fraction 15. Shew that, if ^= a + \ - ...\ 1; then wiU q 6+C+ A; + Z' 11 11 1^1 1 1 l^J^ ?-<-^ + ***+c+6+a i+A + ***+c+6 "JOS'' p 16. Shew that, if -pr be converted into a continued fraction, P the first quotient being a, and the convergent preceding -= being -; then, if — be converted into a continued fraction, the last convergent will be (F -aQ)/{p-aq). 17. Shew that, if - and —. be any two consecutive conver- gents of a continued fraction a;, then will ^ J a;* according as qn=4i>„-i + (2n-3)(2n-l)i)„_a. The above relation may be written i)«-(2n+l)i)„_i= -(2w-3) {i)n-i-(2n-l)l>„_3}. Changing n into w - 1 we have in succession Pn-i - (^ - l)l>n-2= - (2» - 5) {j)„_a - (2n - S)^^.,}, l>8-7P2=-3{l>2-5i>i}. But, by inspection, ^=1, Pa=4; /. p^-5pi= -1, Hence p^ - {2n + 1) p^_, = ( - 1)*"! (2to - 3) (2n - 5) ... 3 . 1. Then again Pn J>n-1 _ (-1)"-^ 1.3...(2w + l)" 1.3... (2/1-1) (2w + l)(2w-l)* Pi Pi _(-l)^ 1.3.6 1.3~ 3.5 ' *°^ 173 =173- _ 11 ( — 1)""^ °®''°® 1.3.6.."(2n+l) = r73~376"^ + (27i + l) (2n-l)- Since the denominators of convergents are formed according to the same law as the numerators, we have from the above ^„-(2n + l)g«_i=(-l)»-23.6...(2w-3){ga-5gr,} = 0, since gi=3 and q2=15. 31—2 468 GENERAL CONVERGENT. Hence 1» gn-i _ _ ga _ gl _-i. (2n + l)(2w-l)...3.1~(2»-l)...3.1 5.3.1 3.1" ' .-. g^=1.3...(2»-l)(27i + l). Hence p^lQn* *^® '"'^^ convergent required, is 1 ^ ,+ ,_„+. <-!)""' 1.3 8.5 (2»-l)(2n + l)" Ex. 2. To find the nth convergent of the continued fraction 12 3 4 1 + 2 + 3 + 4+ The necessary transformations are given in Ex. 5, Art. 2 11 ( - l)«-i It wiU be found that p^= |2 " |3 "*" "*■ u^+i • 11 ( - D* and that gr^=i-+-l __... + ^ > ;SSH |2 • |3 •*"••• ' \n±l If n be supposed infinitely great 359. Periodic continued fractions. When the elements of a continued fraction continually recur in the same order, the fraction is said to be a periodic continued fraction ; and a periodic continued fraction is said to be simple or mixed according as the recurrence begins at the beginning or not. rm. 111111 Thus a + - - - - - - .•• b+c +a+b+c +a+ is a simple, and - r r r '^ ^ a + b + b + b + is a mixed periodic continued fraction. GENERAL CONVERGENT. 459 360. To find the nth convergent of a periodic continued fraction with one recurring element. h h h Let the fraction be a + - . - - ... Then, for all c-\-c + c-\- convergents after the second, we have jp„ = cp^^.^ + hp„_^ where h and c are constants, that is, are the same for all values of n. Now, if Wj + u^x + u^off* + . . . +uj3cr^ + ... be the recurring series formed by the expansion of :j r- ^, the suc- cessive coefficients after the second are connected by the law u^ = cw^_i + &^,_a- Hence, if A and B are so chosen that tl^=p^ and u^^p^y then will u^=Pn for all values of n. The necessary values of A and B are respectively p^ and p^ — cp^, that is a and h. Hence the numerator of the nth convergent to the continued fraction a + - - ... is the coefficient of a;""* in c + c + J.X. • n a + hx the expansion of = j-^ . Similarly the denominator of the nth convergent is the coefficient of x*~^ in the expansion of -* ^^' r^- , 1. ~~ ex ~~ OSCT that is of 1 — cic — hx* ' Ex. 1. Find the nth convergent of the continned fraction 3 3 3 '^2 + 2 + 2 + '"' The numerator of the nth convergent is the coefl&cient of ar^* in tiie expansion of j^z^^TsS^, that is of ^^^j-g^^ - ^^^ . Hence J>„=^ {3"+ (-!)»}. Also 3,4= coefficient of a?""^ in the expansion of 8 ^ 1 l-2x-Bx^ 4(l-3x) 4(l + x) 460 GENERAL CONVERGENT. Hence tfi»=7{3'*- (-1)"}. Thus the nth convergent is 2 — — - — —• Ex. 2. Find the nth convergent of the continued fraction a c a e^ b+d+b+d+ "" We have p^ = dp^_j^ + cp^.^. Hence, eliminating ^2n-i ^-nd j?2n-3» ^e ^^.ve i)a„ - (a + c + 6d) p2«-a + ««i'2n-4 = 0- Since the last result is symmetrical in a and c, and also in h and d, it follows that Pin-i-ia + c + hd) p2„_8 + acp2„-6 = 0. Hence the relation i>n- (« + c + 6d)i>„_3+acp„_4=0 holds good for aU values of n. Hence p^ will be the coefficient of x^~^ in the expansion of A + Bx + Gx^ + Dx^ 1 - (a + c + 6d) ar^ + ocar* ' provided the values oi A, B, G, D are so chosen that the result holds good for the first four convergents. It wiU thus be found that p^ ib the coefficient of x""* in the expansion of a+adx-acx^ 1 - (a + c + bd) X* + acx^ * It will similarly be found that g„ is the coefficient of ar""* in the expansion of b + {bd + c)x- aca^ l-{a+e + bd)x^+(icx*' 361. Convergency of continued fl-actlonB. When a continued fraction has an infinite number of elements it is of importance to determine whether it is convergent or not. When an expression can be found for the nth con- CONVERGENCY OF CONTINUED FRACTIONS. 461 vergent, the rules already investigated can be employed; the nth convergent cannot however be often found. In the continued fraction y-^ j-* r^ ... it is easy to K + ^ + ^8 + shew, as in Art. 357, that 5n ^«-l ^n-l = * + T • . • T . Q' A; + +6 TT v 7 . 1 111 Hence, if y = f + 7- ... - ^ t .... . Py±Q we have V^prp-q) :,fP' + y{q-P)-Q^O (ii). The roots of (ii) are obviously of different signs, and the positive root is the value of the continued fraction ^k-\-"' -Ya + l + "" From (i) and (ii) we see that the positive root of (ii) is equal in magnitude to the reciprocal of the negative root of (i); and therefore the reciprocal of the negative root of (i)is_(r+i^..._^l_^i_^...) The positive roots of (i) and (ii) are both greater than unity, as may be seen by inspection; the negative root of (i) must therefore be less than unity. The fraction - ^ ... y - ... requires no special 464 PERIODIC CONTINUED FRACTIONS, examination, for we have only to change x into - , and y into - ; thus - r . ••• . r . t . ~ . ••• is equal to the positive root of PV— {Q —P) a; — Q = 0, and the negative root is —\l-\- I 1 11 Hence, as before, one root of the quadratic equation in w is greater and the other is less than unity. 364. Theorem. Every mixed periodic continued ^ \fraction, which has more than one non-periodic element, is a ^slroot of a quadratic equation with rational coefficients whose roots are both of the same sign. Let the fraction be 1 111 1111 '^"^"^6 + - + A; + a + ^ + - +y^ + i;+a + y8 + -' and let 1 1111 ^ 13+ +/i + v + a + y8 + A' A Let -^7 and -^ be the two last convergents of the non- periodic part ; then yA + A' ... F P Let ^ and ^ be the last two convergents of the first period of y ; then y P + F .... '-WTQ' ^^^>- The elimination of y from (i) and (ii) will clearly lead to a quadratic equation in x with rational coefficients. Now, if the positive root of (ii) be substituted in (i) we REDUCTION OF QUADRATIC SURD. 465 shall clearly obtain a positive value of a?, and this will be the actual value of the given continued fraction. Also, from the preceding article, the negative value of -is — -^1; + - ... -[: and, if this value be substituted in (i), we have we have to shew that this is positive, li k>v the result is obvious; if A; > -y &c. Hence t^N—ay \/N—a^, s/N—a^, &c., and therefore also N—a^, N — a^, iV— ttg^ &c., are all positive. That is r^, r^y r^, &c. are ail positive. Again, since b is the integer neoct helow , it follows that ^J^+acr2, for then yjNa; hence the only possible values of a^ are 1, 2,..., a. Then, from the relation a„4-a„+i = Aj.r„, where A; is a positive integer, it follows that r„ cannot be greater than 2a. Hence the expression ^ cannot have more than 2a X a different values ; and therefore after 2a* quotients, at most, there must be a recurrence. 368. Theorem. Any quadratic surd can he reduced to a 'periodic continued fraction with one non-recurring element, the last recurring quotient being twice the quotient which does not recur; also the quotients of the recurring period, exclusive of the last, are the same when read hack- wards or forwards. Let Js/I\fhe the quadratic surd. Then, from the preceding Article, we know that ^/N is equal to a periodic continued fraction. We also know that any periodic continued fraction is equal to one of the roots of a quadratic equation with rational coefficients ; and the only quadratic equation in x with rational coefficients of which ^/ N is one root is the equation a;* — iV= 0. Now the roots of af — N = are hoth greater than unity in absolute magnitude, and the roots are of different signs : it therefore follows from Articles 363 and 364 that the continued fraction which is equal to ^/N must be a mixed recurring continued fraction with one non-recurring element. Hence we have , 11 1111 ^^-''-^h^c+'" +h + k+l+b + "'' /AT 11 1111 470 SERIES EXPRESSED AS CONTINUED FRACTIONS. XT 1 1 1111 . ^, Now T .-.•••• r . 7~ . 7 . r . ... is the positive root of a quadratic equation with rational coefficients; and as this positive root is ^jN—a, the negative root must be —tjN—a. Hence, Art. 363, we have __i_ = l 1 1 1 + 1 + 1 ViV + a l+k + h+'" +c h 1 + '"* iliT 7 1 1 111 ^ k+h+ +c+6+^+ TI 7.11 111 Hence i — a + y t ••• - t t = 4-11 111 whence it is easy to see that l — a = a,k = b,h = c, .... Series expressed as Continued Fractions. 369. To shew that any series can be expressed as a continued fraction. Let the series be u^-^u^ + u^ + u^-\- ,..+u^+ (i). Then the sum of n terms of the series (i) is equal to the nth convergent of the continued fraction 1 - v^ + u^ — u^ + u^ — u^-\-u^— '" — w^i + w^— "*^ ^' This can be proved by induction, as follows. Assume that the sum of the first n terms of (i) is equal to the nth convergent of (ii). Another term of the series is taken into account by changing u^ into u„ 4- u^+ ; and, by changing u„ into u^ + u^^, — **"'"* will become SERIES EXPRESSED AS CONTINUED FRACTIONS. 471 **~^^ " r^ which is easily seen to be equal to u^..u„ u^,u^, ^j^^^ ^^^ ^^^ of n + 1 terms of (i) will be equal to the (?i+l)th convergent of (ii) provided the sum of n terms of (i) is equal to the nth convergeut of (ii). But it is easily seen that the theorem is true when 7i is 1 or 2 or 3 : it is therefore true for all values of n. Thus u^ + u^-\-u^-\-u^-\- .,,to n terms W, ^5 ^l^» ^8^4 J. X- ^ PAT = -i ^ — — *-^ — *^^ ... to n quotients... [A J. 1 —u^-\-u^ — u^ + u^ — u^ + u^— ^ It can be proved in a precisely similar manner that u^—u^ + u^ — u^-{- to n terms = -i — ?— — ^-^ — ^-^ ... to n quotients ... [B|. The formula [B] can also be deduced from [A] by olianging the signs of the alternate terms. 370. The following cases are of special interest : ^ + tV + 111 + ... to w terms = r - rr^ - a¥^ t - *« ^ quotients... [C], all the upper signs, or all the lower signs, being taken. And — ± — I — + — h...to7i terms «i ^2 ctg ^4 1 3 3 = — _ — i — _ — ? — _ ... to 71 quotients . . . [D], «! + a, ± a^ + ftg ± ttj + all the upper signs, or all the lower signs, being taken. These can be proved by induction as in the preceding Article. s. A. 32 472 SERIES EXPRESSED AS CONTINUED FRACTIONS. Thus to prove [C]. It is obvious by inspection that the theorem is true when n=2. Assume then that [C] is true for any particular value of n: then, to include another term of the series 7^ must be changed into P^ =t: f »^w4-i ^ ^^^ therefore _~=1^ will become which can easily be seen to be equal to J*~^ ** n n+i — ^ Thus, if [C] be tKue for any value of n, it wiU be true for the next greater value ; hence as [C] is true when n=2, it is true for all values of n. The following are particular cases of [C]. Oi ± a^a^ + a^a^a^ ± a^a^a^a^ + 1 +I±a3+ 1 ±a,+ l ±a, + ^ h and 2 + Jl +-J_ + _J_ + ... ai + a2±l + a,±l + a,±l+ *'""'- ^' Ex. 1. Shew that 1 1» 32 5« * . « .* V TT -TT -^ ... to mfinity. 1+2+2+2+ ^ = --- + ---+... to infinity. [Brouncker.] 13 5 7 Put ai=l, aa=3» o»=5> &o. in [D], Ex. 2. Shew that 1 IS 22 39 1 4. r+ 1 + r+ - *° i°fi°i*y=log-2. [Euler.] Put ai=l, aj=2, 03=3, &c. in [D]. • The formula [A] is due to Euler ; [C] is given by Dr Glaisher in the Proceedings of the London Mathematical Society^ Vol. v. EXAMPLES. 473 Ex. 3. Find the value of :j- - - - j ... to infinity. 1+1+2+3+4+ From [F] we see that 1 + 1 + 2 + 3+ *^^^^^^*y 13 3 Ex. 4. Find the nth convergent of 5 - - .... o + 2 + J + From [F] we have 1 1 1 _1 ? ? 8 3.3"^3.3.3 •••"3 + 2 + 2+ •••• Hence the nth convergent required =2 -/l-(-._j L, Ex.6. Shew that 1 + ^ _!_. JL JL _^^r^ 1 -r+2-r+3-r+4- _ - r , r.r r .r.r r .r.r .r ' =^+i + O + 1:2:3 + ir2TF.-4+ - 1 . *• »■ 2r 3r . _„ = l + l_,-T2-r-T3-r-?l--»^^°^[^l- EXAMPLES XXXYII. 1. Find the continued fractions equivalent to the follovring quadratic surds : (1) V17, (2) JUO, (3) V33, (4) ^43, (5) J{a'+l\ (6) VK + 2«). 2. Shew that JI^ = a + — - — - . . ., where a has any value whatever, and b = N—a'. 3. Find the value of « l-^3+2+3+2 + -*^"^^"^*y- 82—2 474 EXAMPLES. /..v ,11111 ^ . . .^ /.-. 111111 ^ • . ., (ill) 7^ o T K T F ••• *<^ infinity. ^^2+3 + 4 + 5+4+5+ •^ 4. Shew that 7 + ^rr TT ...to infinity = 5 (1 +75 ^r ... to infinity). 14 + 14+ '' \ 2 +2 + •" 5. Shew that \a +b +c + d +a + '" J \ c+b+a+d+"'J _b + d+hcd ~ a + c + acb ' 6. Shew that, ifa5 = y + 7r- -r- ... to infinity, then "^2^ + 2^ + 7. Shew that, if a; = - 7- - r ... to infinity, a + b +a +b + •" 2^=i + i + i + i + -*"^^"^*y' ""^ *=i+i+i + i+-*^^^"^*y' then will a; (y« - «') + 21/ («« - a') + 3« (ar» - y') = 0. 8. Shew that, if n be any positive integer, n«_p n»_2« w«-3* 3 + 5 + 7 + 9. Shew that l+a» + aV... + a'" ^ 1 ]_ 1 a + a' + a*+... + a'"-^~ a 1 1 ... a + a + — a a to n quotients. EXAMPLES. 475 10. Shew that, if a c a e , c a e a b+d+b+d+ ^ d+b+d+b+ ' then bx-dy = a — c. 11. Shew that the ratio of 1111 ,^1111 l+6+a+l+ l+a+b+l+ is 1 + a : 1+6. 12. Shew that the nth convergent of 14 2 2 . 2--1 ...IS 3_3_3_3_-- -"2'' + r 13. Shew that the wth convergent of ^""2 + 2 + 2+-'' (l + ^2)"-(l-V2)- • 12 2 2 14. Shew that the nth convergent of ^ ^r 77 tt i — o — 6 — o is 2- - 1. 15. Shew that the wth convergent of 1 ab ab . a* -b* ... IS -z-rz T-TT. a-^b —a + b —a + b — 16. Find the nth convergent of the continued fraction 2 3 8 r'-l 1_5_7_ - _2r+l-'"' 17. In the series of fractions — , — , »-3) (n-l)l Z = I + r- =- = — = ... = — . 1-n + l- n+l - n + l - - w + 1 86. Shew that (l +5 (1+3(1+^,)... _ as a (a; + a) a' (a? + a') a— x + a+a' —x+a' +a' ^ 37. Shew that - ^ . ^ r ... to infinity 5= ... to infinity, w-l+2n-l+3w-l+ -^ CHAPTER XXVIIL Theory of Numbers. 371. Throughout the present chapter the word number will always denote a positive whole number ; also the word divide will be used in its primitive meaning of division without remainder. The symbol M(p) will often be used instead of * a multiple of p.' Definitions. A number which can only be divided by itself and unity is called a prime number, or a prime. A number which admits of other divisors besides itself and unity is called a composite number. Two numbers which cannot both be divided by any number, except unity, are said to be prime to one another, and each is said to be prime to the other. 372. The Sieve of Eratosthenes. The different prime numbers can be found in order by the following method, called the Sieve of Eratosthenes. Write down in order the natural numbers from 1 to any extent that may be required : thus 1, 2, 3, 4, 5, 6, 7, 11, 12, 13, 14, 15, 16, 17, 21, 22, 23, 24, 25, 26, 2*7, 81, 32, 33, 34, 35, 36, 37, Now take the first prime number, 2, and over every second number from 2 place a dot : we thus mark all 8. S, 10 18, 19, 20 28, 29. 30 38, 39, 40 &c. 480 THEORY OF NUMBERS. multiples of 2. Then, leaving 3 unmarked, place a dot over every third number from 3 : we thus mark all mul- tiples of 3. The number next to 3 which is unmarked is 5 ; and leaving 6 unmarked, place a dot over every fifth number from 5 : we thus mark all multiples of 5. And so for multiples of 7, &c. Having done this, all the numbers which are left unmarked are primes, for no one of them is divisible by any number smaller than itself, except unity. It should be here remarked that if a composite number be expressed as the product of two factors, one of these must be less and the other greater than the square root of the number, unless the number is a perfect square, in which case each of the factors may be equal to the square root. Hence every composite number is divisible by a prime not greater than its square root. On this account it is, for example, only necessary to reject as above mul- tiples of the primes 2, 3, 5 and 7 in order to obtain the primes less than 121, for every composite number less than t 121 is divisible by a prime less than 11. \ 373. Theorem. The number of primes is infinite. ^ For, if the number of primes be not infinite, there must be one particular prime which is greater than all others. Let then p be the greatest of all the prime num- bers. Then \p will be divisible by p and by every prime less than p. Hence 1^+1 will not be divisible by p or by any smaller prime ; therefore [^ + 1 is either divisible by a prime greater than p, or it is itself a prime greater than p. Thus there cannot be a greatest prime number ; and therefore the number of primes must be infinite. Ex. Find n consecutive numbers none of which are primes. The numbers are given by jn + 1 + r, where r is any one of the ^ numbers 2, 3, ..., (n + 1). ^74. Theorenii No rational integral algebraical formula can represent prime numbers only. THEORY OF NUMBERS. 481 For, if possible, let the expression a±ba)±ca^± dx*±,., represent a prime number for any integral value of as, and for some particular constant integral values of a, 6, c, ... . Give to a? any value, m suppose, such that the whole expression is equal to p, where p is neither zero nor unity ; then p = a±hm± cm^ ± ... . Now give to x any value m-\-np, where n is any positive integer; then the whole expression will be a±h(m + np) ± c (m 4- np)^ + . . . = a ± 6m + crn^ ± ... + M(p)==p-^M(p). Thus an indefinite number of values can be given to x for each of which the expression a±bx±ca^ ± ... is not a prime. In connexion with the above theorem, the following formnlae are noteworthy : — (i) x^+x+ 41, which is prime if a; < 40. [Euler.] ^) x^ + x + 17, which is prime if « < 16. [Barlow.] (iii) 2a;^ + 29, which is prime if a; < 29. [Barlow.] 375. The student is already acquainted from Arith- metic with many properties of factors of numbers : these all depend upon the following fundamental ' Theorem : — If a number divide a product of two factors, and be prime to one of the factors, it will divide the other. For, let ab be divisible by x, and let a be prime to x. Reduce - to a continued fraction, and let - be the con- X q vergent which immediately precedes - ; then [Art. 357, L] X qa—px=±l; .'. qab—pxb — ±b. Now qab is, by supposi- tion, divisible by x) and therefore qab—pxb must be divisible by x, that is b must be divisible by w. From the above theorem the following can easily be deduced : — "i way. 482/ THEORY OF NUMBERS. 9| ^I. If a prime number divide the product of several facials it must divide one at least of the factors. yll. If a prime number divide a" it will divide a. /IIL If a be prime to each of a, /3, 7, ... it will be prime to the product a^y. . . . f iV. If a be prime to 6, a* will be prime to 6*". ^. If a number be divisible by several primes separately it will be divisible by the product of them alL 376. Theorem. Every composite number can be re- sotvedrtnto prime factors ; and this can be done in only one For, if iV be not a prime number, it can be divided by some number, a suppose, which is neither iV nor 1: thus N=ah. Again, if a and b be not primes, we have a = cxc?, 6 = e x/, and therefore N—cdef. Proceeding in this way, since the factors diminish at every stage, we must at last come to numbers all of which are primes. Thus N can be expressed in the form aX)8x7xSx..., where a, y8, 7, S, . . . are all primes but are not necessarily all different, so that N may be expressed in the form a'/3*7*"..., where a, ^, 7,... are the different prime factors ofiV^. Next, to shew that there is only one way in which a number can be resolved into prime factors. Suppose that N = ahcd... ^ where a, b, c, d,... are all primes but are not necessarily all different ; suppose also that N = a/878. . ., where a, y8, 7, S. . . are also primes. Then we have abed... = a^yB Hence a divides a^yB...; and therefore, as all the letters represent prime numbers, a must be the same as one of the factors of ay37S Let a — a; then we have bed... —fiyB..., from which it follows that b must be equal to one or other of y3, 7, 3, . . . ; and so on. Hence the prime factors a, 6, c,... must be the same as the prime factors a, )S, 7, . . . . Ex. Express 29646, 13689 and 90508 in terms of their prime factors. Am. 6 . 72 . lia, 8* . 132 and 2^ . 11» . 17. THEORY OF NUMBERS. 483 377. To find the highest power of a prime number contained in Iw. Let /(-) denote the integral part of-; and let a be any prime number. Then the factors in hi which will be divisible by a are a, 2a, 3a, ...,/(- j . a. Thus / ( - j factors of \n will be divisible by a. Similarly 11-%) factors will be divisible by a*. And so on. Hence the whole number of times the prime number a is contained in In is /(-) +J^(— ) +-^(~8) +••♦. Ex. 1. Find the highest powers of 2 and 7 contained in (50. H.e z(f)-., .(-)=x. x(-)=e. x(-)=a. I ( -g j = 1. Hence 2*^ is the required highest power of 2. I Again, I ( — J = 7, I [ ^ j = 1. Hence 7* is the required highest ^ -^ power of 7. jUiij 2. Find the highest powers of 3 and 5 which will divide [80. I Am, 33«, 6". — jj"*^-^ 8. Find the highest power of 7 which will divide [1000. Am. 7i«*. ^ 378. Theorem. The product of any r consecutive \ numbers is divisible by \r. h\ Let n be the first of the r consecutive numbers ; then we have to shew that "(^ + l)('» + 2) (n + r-l) ir n + r — 1 or I , , is an integer. \r n-1 ^ In + r — 1 The theorem follows at once from the fact that ' ' v — , ^71 — 1 is n+r-fir> ^^^ ^^^ uumber of combinations oi n-\- r^l 484 THEORY OP NUMBERS. things r together must be a whole number for all values of n and of r. The theorem can also be proved at once from first principles by means of Art. 377. For it is obvious that / ( '^"^^""^ ) < I (^) + -^ Q » /( a — )^^\ — 2-) + -^(-^), and so on. Hence from Art. 377 it follows that the number of times any prime number is contained in \n-\-r — 1 can never be less, although it may be greater, than the number of times the same prime number is contained in \n — l x |r. Thus every prime number which occurs in In— 1 x Ir, occurs to at least as high a power in |w + r — 1, which proves that \n-{-r — l is divisible by |n — 1 x jr. In It can be proved in a similar manner that r-^ — is an integer, where a + ^-\-y-\-..,—n. 379. If n he a jpHme number the coefficient of every term in the expansion of (a + by except the first and last terms is divisible by n. For, excluding the first and last terms, any coefl&cient , n(n— l)...(n — r + 1) , is given by -^ —r- , where r is any integer between and n. Now, by the preceding Article, — ^^ ^^ ^"^ is an integer; and, as n is a prime number greater than r, n must be prime to r; and therefore -^^ — — ^^ ■ — - must be an integer. Hence every coefficient, except the first and last, is divisible by n. Similarly, if n be a prime number, the coefficient of THEORY OF NUMBERS. 485 every term in the expansion of (a + 6 + c+...)* which contains more than one of the letters, is divisible by n. For the coefficient of any term which contains more \n than one of the letters is of the form Y—rkr 1 where a+y8+7+... = n. Now . '-7 is an integer; and, as w is a prime greater than any of the letters a, /8, 7,..., n must be prime to lal^ly... ; and therefore the coefficient of every term which contains more than one letter is divisible by n. - -Ex: 1. Shew that n (n + 1) (2n + 1) is a multiple of 6. Ex. 2. Shew that, if n be odd, (w=» + 3) {n^ +7) = M (32). Ex. 3. Shew that/if n be odd, n* + 4w' + 11 = Jf (16). Ex. 4. Shew that 1 + T^^+i = M{8). ^ Ex. 5. Shew that 19*» - 1 = Jlf (360). \ £z. 6. Shew that, if n be a prime number greater than 3, \ n (w2 - 1) (n« -4:) = M (360). * H 380. FermaVs Theorem. If n be a prime number^ Dand m any number prims to n ; then rrC^ — 1 will be Invisible by n. We know that when n is a prime number, the coeffi- cient of every term in the expansion of (a, + a, +...+ a„)", which contains more than one of the letters, is divisible by n. Now there are m terms each of which contains only one letter and the Coefficient of each of these terms is 1. Hence, putting a, = a, =...= 1, we have mr==m-^ M(n) ; .-. m (w"~' - 1) = M(n). Hence, if m be prime to n, m""* — 1 will be a multiple of 7i. Ex. 1. Shew that, if n be a prime number, ^n-i + 2«-i + 3^-1 + . . . + (w - l)-*-! + 1 = M{n). 2. Shew that, if a and b are both prime to the prime number n ; then will a**"^ - 6"-^ be a multiple of n. Ex. 3. Shew that n^-n=M (30). 486 THEORY OF NUMBERS. Ex.4. Shew that nf-n=M (42) . Ex. 5. Shew that a?" - y " = M{1S65) , if a? and y are prime to 1365. ...^-Ex. 6. Shew that, if m and n are primes ; then m^-i + ri'nr-i -i = M (mn) . Ex. 7. Shew that, if m, n and p are all primes ; then (np)"*-i + (pwi-)»-i + {mnf-^ - 1 =ilf {mnp). ^„,„.^x. 8. Shew that the 4th power of any number is of the form 5m or 6w+l. Ex. 9. Shew that the 12th power of any number is of the form 13m or 13m + 1. Ex. 10. Shew that the 8th power of any number is of the form 17m \or 17m ± 1. 381. To find the number of divisors of a given number. Let the given number, N, expressed in prime factors, be a'6V Then it is clear that N is divisible by every term of the continued product Hence the number of divisors of N, including N and (^ + l)(y + l)(^ + l) Ex. 1. The number of divisors of 600, that is of 2» . 3 . 5^, is (3 + l)(l + l)(2 + l) = 24. Ex. 2. Find the sum of the divisors of a given number. The given number being N=a''bvc*...y the sum required is easily seen to be (1 - o^i) (1 - fty+i) (1 - c*^^^)... (l-a)(l-6)(l-c)... • Ex, 3. Find the number of divisors of 1000, 3600 and 14553. Ans. 16, 45, 24. Ex. 4. Shew that 6, 28 and 496 are perfect numbers. [A perfect number is one which is equal to the sum of all its divisors, not considering the number itself as a divisor.] Ex. 6. Find the least number which has 6 divisors. Am. 12. Ex. 6. Find the least number which has 15 divisors. Ans. 144. Ex. 7. Find the least number which has 20 divisors. Am. 240. Ex. 8. Find the least numbers by which 4725 must be multiplied in order that the product may be (i) a square, and (ii) a cube. Am. 21, 245. THEORY OF NUMBERS. 487 382. To find the number of pairs of factors, prime to each other, of a given number. Let the given number he N= a'h^c*... ; then, if one of two factors prime to each other contains a, the other does not ; and so for all the other different prime factors. Hence the factors in question are the different terms in the product (1 + a") (1 + 6") (1 +c")..., the number of them being 2*, where n is the number of different prime factors of N. The number of different pairs of factors \prime to each other is therefore 2""^ in which result N and 1 are considered as one pair. \ \ 383. To find the number of positive integers which are less than a given number and prime to it. Let the given number be iV= a*6V..., where a, b, c,... are the different prime factors of N. The terms of the series 1, 2, 3,..., iV which are divisible by a are a, 2a, 3a,..., — a: and therefore there are — N numbers which are divisible by a. So also there are -j- N . . . iV . . . numbers divisible by b, j- divisible by be, -r- divisible by abc, and so on. We will now shew that every integer which is less than iV and not prime to iV is counted once and once only in the series 2^_2 - + S— -S— + . (a) a ab abc abed Suppose an integer is divisible by only one prime factor of N, a suppose; then that integer is counted once in iV (a), namely as one of the — numbers which are divisible by a. INext suppose an integer is divisible by r of the prime (actors a, b, c,... , then that integer will be counted r s. A.. 33 488 THEORY OF NUMBERS. iV r(r — l). N times in S — , it will be counted v o times in 2 -i; , a i . z ao it will be counted — ^ — = — ^r^ times m 2, -7- , and so 1 . Z . 6 aoc on. Hence the whole number of times an integer divisible by r of the prime factors is counted, is r(r-l) r(r-l)(r-2) r(r-l)...l r ^p-2-+ f--273 ...+ (-1) ^ = 1 - (1 - ly = 1. Thus every number not prime to N is counted once in (a); and therefore the number of positive integers less than iV and not prime to iV is given by (a) ; provided however that unity is considered to be prime to N. Hence the number of positive integers less than iV and prime to j/V is a ao aoc { a ao aoc ) Ex. 1. Find the number of integers less than 100 and prime to it. Since 100 = 2^ . 5^, the number required is 100 (i-i)(i-i)- 1=39. Ex. 2. Find the number of integers less than 1575 and prime to it. Am. 719. Ex. 3. Shew that the number of integers, including unity, which are less than N[N>2] and prime to N is even, and that half N these numbers are less than -^ . For if a be prime to N so also is N-a; and if a > - , then N-a THEORY OF NUMBERS. 489 384. Forms of square numbers. Some of the different possible and impossible forms of square numbers will be seen from the following examples : — Ex. 1. Shew that every square is of the form 3ot or 3w + 1. For every number is of the form Bm or 3m ± 1. Hence every square is of the form 9m or 3m + 1. Ex. 2. Shew that every square is of the form 5m or 57n ± 1. For every number is of the form 5m, 5m J= 1 or 6m ± 2 ; and there- fore every square is of the form 5m, 5m +1 or 5m + 4. Ex, 3. Shew that, if a? + 6*=c^ where a, b, c are integers ; then will abc be a multiple of 60. First, every square is of the form 3m or 3m+l; and therefore the sum of two squares neither of which is a multiple of 3 is of the form 3m + 2 which cannot be a square. Hence either a or b must be a multiple of 3. Again, every square is of the form 5m or 5m =t 1. The sum of two squares neither of which is a multiple of 5 is therefore of one of the forms 5m, or 5m ± 2. Now no square can be of the form 5m ± 2 ; and if a square be of the form 5m, its root must be a multiple of 5. Hence, if ab is not a multiple of 5, c will be a multiple of 5. Thus, in any ease, abc is a multiple of 5. Lastly, since every number is of the form 4to, 4m + 1, 4m + 2 or 4m + 3, every square is of the form 16m, 8m +1, 16m + 4. Now a and 6 cannot both be odd, for the sum of their squares would then be of the form 8m + 2 which cannot be a square. Also, if one is even and the other odd, the even number must be divisible by 4, for the sum of two squares of the forms 8m + 1 and 16m + 4 respectively is of the form 8m + 5 which cannot be a square. It therefore follows that ab must be a multiple of 4. Thus abc is divisible by 3, by 6 and by 4; hence, as 3, 4 and 6 are prime to one another, a6<; = ilf (60). Ex. 4. Shew that every cube is of the form 7to or 7m ± 1. Shew also that every cube is of the form 9m or 9m ± 1. Ex. 5. Shew that every fourth power is of the form 5m or 5m + 1. Ex. 6. Shew that no square number ends with 2, 3, 7 or 8. Ex. 7. Shew that, if a square terminate with an odd digit, the last figure but one will be even. Ex. 8. Shew that the last digit of any number is the same as the last digit of its (4n + l)th power. Ex. 9. Shew that the product of four consecutive numbers cannot be a square. 33—2 490 THEORY OF NUMBERS. EXAMPLES XXXVIII. 1. Shew that the difference of the squarep of any two prime numbers greater than 3 is divisible by 24 2. Shew that, if w be a prime greater than 3, n K- l)(nV- 4)(n^ - 9) = Jf (2' . 3'. 5 . 7). 3. Shew that, if ti be any odd number, {n + 2m)- - (n + 2m) = J/ (24). 4. Shew that «*"+' - a*''*' = J/(30). 5. Shew that, if ^—a' = x and (a + 1)' - iV = y, where ^ and y are positive; then I^—xyiaa, square. y 6. How many numbers are there less than 1000 which are not divisible by 2, 3 or 5? 7. P, Qj i?, p, q, r are integers, and p, q^ r are prime P O R to one another; prove that, if — h — + — be an integer, then — , — and — will all be integers. p q r 8. Shew that 284 and 220 are two 'amicable' numbers, that is two numbers such that each is equal to the sum of the divisors of the other. 9. Shew that, if 2"-l be a prime number, then 2"~'(2''-l) will be a 'perfect' number, that is a number which is equal to the sum of its divisors. 10. Find all the integral values of x less than 20 which make a;'* - 1 divisible by 680. 11. Shew that no number the sum of whose digits is 15 can be either a perfect square or a perfect cube. 12. Shew that every square can be expressed as the differ- ence between two squares. THEORY OF NUMBERS. 491 13. Find a general formula for all the numbers which when divided by 7, 8, 9 will leave remainders 1, 2, 3 respectively; and shew that 498 is the least of them. 14. If n be a prime number, and iV prime to n, shew that ]^n-n_i ^M{n'), and that ir»' -«"■'_ 1 = M {n'). 15. Shew that, if n be a prime number and iV be prime to n, then wiU iV^'+=^-+^"-« =*= 1 = if (w*). 16. Shew that, if p be a prime number, and (1 + xy~'= 1 + a^x + a^af + ajpc'' + .,,; then a^ + 2, a, -3, ^8 + 4, J 386. Theorem. Ifa^^b(nwd.x),anda^ = h^(mod.iic); then will a^ + a^ = b^ + b^ (mod. x), and a^a^ = bp^ {mod. x). For let ttj = m^x + r^ , and a, = m^x + r, ; then, by sup- position, b^ = n^a? 4- r-j and 6^ = "^2-^ + ^2* Hence a^ + a^ — {b^ + 6^ = (m^ + 7?ijj - n, - ti^) a; ; .-. {a^ + a,) - (6j + 6,) = (mod. a;), or «! + a, = 6j + 62 (mod. a;). Again, it is easily seen that a^a^ — bfi^ = a multiple of X, and therefore a^a^ = bf>^ (mod. ic). The proposition will clearly hold good for any number of congruences to the same modulus. 387. Congruences have many properties analogous to equations. For example, if the congruence Aa^-\-Bx^G=0 (mod. p\ wherein A,B,G have constant integral values, be satisfied by the three values a, b, c of x, which are such that a — b is unity or prime to p, and so for every other pair, then the congruence will hold good for all integral values of a?, and A.ByG will all be multiples of p. For we have Aa' + Ba+G=0 {modi. p\ and Ab' -\-Bb + G = (mod. p) ; THEORY OF NUMBERS. 493 .*. by subtraction (a -h){A(a + h) + B}=0 (mod. p). Hence, as a — 6 is unity or prime to p, we have A(a + b) + B = (mod. p). Similarly, A (a + c)-\-B=0 (mod. p). Hence, by subtraction, A {b — c) = (mod. p). Therefore ^ = (mod. p\ since b — o is unity or prime to^. Then, since A = (mod. p), it follows that B=0 (mod. p), and then that C = (mod. p). Then, since A, B, G are all multiples of p, it follows that Ax^ + Bx + (7 is also a multiple of p for all integral values of x. We can prove in a similar manner the general theorem, namely : — If a congruence of the nth degree in x be satisfied by more than n values of x, which are such that the difference between any two is unity or is prime to the modulus, then the congruence will be satisfied for all integral values of x, and the coefficients of all the different powers of x will be multiples of the modulus. 388. Theorem. If a and b are prime to one another, the numbers a, 2a, Sa,..., (b — l)a will all leave different remainders when divided by b. For suppose that ra and sa leave the same remainder when divided by b. Then ra- sa = M(b); but if b divide (r — s) a, and be prime to a, it must divide r — s, which is impossible if r and s are both less than b. Hence the remainders obtained by dividing a, 2a, ... , (6 — 1) a by 6 are all dififerent; and since there are 6—1 of these remainders, they must be the numbers 1, 2, , (6 — 1) in some order or other. 494 THEORY OF NUMBERS. If a be not prime to 6 the remainders obtained by dividing a, 2a, 8a, ..., (6 - 1) a by 6 will not be all different. For let A be a common factor of a and 6, and let a=ka atid 6 = /c/S. Then it is easily seen that (r + /3) a and ra will leave the same remainder when divided by b, and {r + ^)a and ra are both included in the series a, 2a, ..., (6- 1) a provided r + /3 < 6 - 1. Cor. If a be prime to 6, and n be any integer what- ever, the remainders obtained by dividing n, n -\-a, \ n + 2a, ...,n + (b — l)a by b will all be different, and will \ therefore be the numbers 0, 1, 2, ..., (6 — 1). ^ 389. Fermat'8 Theorem. From the result of the preceding article, Fermat's theorem can be easily deduced. For, if a and b are prime to each other, the numbers a, 2a, ..., (b — l)a will leave, in some order or other, the re- mainders 1, 2, . . ., (6 — 1), when divided by b. Hence we have a. 2a . 3a... (6-1 . a) = 1 . 2 . 3. ..(6 - 1) (mod. 6). that is |6-1 (oT' - 1) = (mod. 6). Now, if 6 be a prime number, 1 6 — 1 will be prime to 6; \and we have a*^* — 1 = (mod. ~5X which is Fermat's theorem. 390. Wilson's Theorem. If n be a prime number, 1 -I- 1 71 — 1 will be divisible by n. If a be any number less than the prime number M, a will be prime to n, and hence, from Art. 388, the remainders obtained by dividing a, 2a, ..., (n — 1) a by n will be the numbers 1, 2, ..., (w — 1); hence one and only one of the remainders will be unity. Let then a6 be the multiple of a which gives rise to the remainder 1; then, if 6 were equal to a, we should have a* = 1 + M(n), or (a + 1) (a — 1) = M{n), and this can only be the case, since n is a prime, if a = l or a = n — 1. Hence the numbers 2, 3, . . . (n - 3), {n — 2) can be taken in pairs in such a way that the product of each pair, and therefore the product of all the pairs, is of the form M{n) + 1. Thus 2 . 3 . 4...(n - 2) = M{n) 4- 1 ; .'. In - 1 = M{n) X (n - 1) + ^i - 1. THEORY OF NUMBERS. 495 Hence |ri-l + 1 = M(n). Wilson's theorem may also be proved as follows : — From Art. 305, we have („-!)■-_(„- l)(n - 2)-' + ("-^) (^"- ^) („-3)--... (- i)-^ ("-i)(»-^)-^ r- = |„ _ 1. Now by Fermat's theorem (ri - 1)" ^ = 1 + M(n), Hence we have = |yi-l, that is (1 - 1)"-* - (- ir'-{-M(n) = \n-l ; hence, as w-l is even, I n — 1 + 1 = M(n). Wilson's theorem is important on account of its express- ing a distinctive property of prime numbers; for 1 + in— 1 is not divisible by n unless n is a prime. For if any number less than n divide n it will divide I n — 1 and therefore cannot divide In — 1 + 1. 391. Theorem. If the number of integers less than ahy number n and prime to n he denoted by (f> (n) ; then, if a, 6, c,... are prime to each other, (f>{ahc...) = {cb) X (j>(b) X (j>{c) , provided that unity is considered to he prime to any greater number. First take the case of two numbers a, h and their product ah. Arrange the ah numbers as under : 1 , 2 , 3 , , k , a a + 1 , a + 2, a + 3, , a + k, 2a 2a+l, 2a + 2, 2a+3, , 2a+A;, 3a (6-ija+i, (6-i)a+2,**(6-iya+3, *.*.., (6-i)a+^ 496 THEORY OP NUMBERS. Then it is clear that all the integers in the kth vertical column will or will not be prime to a according as A; is or is not prime to a. Hence there are (b), so that (j) (ab) =^ (a) x (b). The proposition can at once be extended, for we have (abc,..) = (f>{axbc...) = (f>ia) x <^(6c...) = 4,{a)(h).4,(c)... 392. The number of integers less than a given number and prime to it can be found by means of the theorem in the preceding article. For let the number be ^ = a*6^c^..., where a, 6, c,... are the different prime factors of N. To find the number of integers less than a* and prime to it, (unity being considered as one of these numbers) we must subtract a*~^ from a*; for the numbers a, 2a, 3a,..., a*~^ . a are not prime to a, and these are the only numbers which are not prime to a ; thus ^ (a") = a« - a*-i = a« (l - - j THEORY OF NUMBERS. 497 Similarly <^ (6^ = 6^ ^1 - g) , (c"^) = cy(l-^Y &c. But, by the preceding article, H..- ♦m.jr(i-!)(i-J)(i-?)..., where a,b,c,... are the different prime factors of iV, and unity is considered to be prime to a, 6, c, &c. 393. The following is an extension of Fermat's Theorem : — If a and m are two numbers prime to one another, and (m) the number of integers, including unity, which are less than m and prime to m; then a*^'^ — 1 = {m,od. m). Let the

(m) remainders are all different, and are all prime to m, they must be, in some order or other, the <^(m) numbers 1, a, /3, 7... Hence a.aoL. ayS a (m - 1) = 1 . a . /8 . 7. . . (m — 1) (mod. m) ; .-. {«*("»)- 1} 1 . a .^...(m- 1)= (mod. m). Hence as 1. a.y8...(m--l) is prime to m, we have a<^(»»)_l = 0(mod. m). If m be a prime number, <^ (m) = m — 1, and we have Fermat's Theorem. 498 THEORY OF NUMBERS. 394. Lagrange's Theorem. Ifip he a prime number, the sum of all the 'products r together of the numbers 1, 2, 3 ..., p — 1, is divisible by p, r being any integer not greater than p — 2. Consider the identity Change x into x—1; then (x'-2){x-S),..(x--p)=^(x-iy-'-8,{x-iy-'+.., Hence (x -p) {x^-' -S.x^'-hS^x^' +...+(- 1)'^' 8,_;\ =(^-i){(^-ir^-^,(^-ir^+... + (-ir^^,.j. Equate the coeiBficients of the different powers of x in the above identity ; and we have * 1.2 ' 2 . (h) or one of its sub-multiples, let <^ (6) — ka-\-ry where r < a. . Then 10*^ - 1 = 10** . lO*" - 1 = {M (b) + 1}* . lO*- - 1 = if (6) + lO*- - 1 ; .-. 10- -1 = if (6), which is impossible since r < a, and a is the lowest power of 10 which is equal to if (6) 4- 1. Hence, ifhhe the factor of the denominator of afraclian which is prime ^o 10, the number of recurring figures in the equivalent decimal is either cj> (6) or one of its sub-multiples. 396. We shall conclude this chapter by considering the following examples : — Ex. 1. Shew that B^+^ - 8w - 9 is a multiple of 64. We have 32n+9_8„_9 = (i + 8)'»+i-8n-9 = l + (n+l)8 + Jlf(82)-8n-9=. Ex. 2. Shew that '6^ - S2ri> + 24n - 1 = (mod. 612). Let u„ = 32n_32w» + 24n-l; then w^i = 32»»+« - 32 (n + 1)' + 24 (« + 1) - 1. Hence u„+i - 9m„ = 25Qn^ - 256n = 266n (n - 1) = M (512), sinoe n (» - 1) is divisible by 2. / EXAMPLES. 501 And since Uj^i-9u^ = (mod. 612), it follows that tt„4.i = (mod. 512) provided m„ = (mod. 512). The theorem is therefore true for all values of n provided it is true for « = 1, which is the case since Mi = 0. Ex. 3. Shew that no prime factor of «*+ 1 can be of the form 4m - 1. Every pritne number, except 2, is of the form 2ft + 1. Let then 2/c + l be a prime factor of n^ + 1. Then n is prime to 2/c + l, and therefore by Fermat's theorem n^* = JI (2ft + 1) + 1. But, by supposition, n^ + l = M (2ft + 1) ; n2*={lf(2ft + l)-l}* = M(2ft + l) + (-l)*. Since n^ = M [2k + 1) + 1 and ii^ = Mi2k + l) + {-l)^ it follows that ft must be even, and therefore every prime factor of w'' + 1 is of the form 4/71 + 1, and therefore no prime factor can be of the form 4m-l. Since the product of any number of factors of the form 4wi + 1 is of the same form, it follows that every odd divisor of «' + 1 is of the form 4w + 1. Ex. 4. Shew that every whole number is a divisor of a series of nines followed by zeros. Divide the successive powers of 10 by the number, n suppose, then there can only be n different remainders including zero, and hence any particular remainder must recur. Let then 10^ and 10^' leave the same remainder when divided by n ; then 10* - 10" is divisible by w and is of the required form. EXAMPLES XXXIX 1. Prove the following : — (i) 2'"**-9n'^ + 3n-2 = if(54). (ii) 5^"-^' + n' - 5n» + 4w - 5 = if (120). (iii) 4=^-^* + 3"-^« = 0(mod. 13). (iv) 3*»-^' + 2 . 4*"+^ = (mod. 17). 2. Shew that, if a be a prime number, and b be prime to a; then IV, 2«6', , ^^Y fe» will give different re- mainders when divided by a. 3. Shew that, if 47i + 1 be a prime number, it will be a factor of { j 2nY + 1 ; and that, if 4n - 1 be a prime, it will be a factorof{|2^-l}'-l. 4. Shew that, if n be a prime number, and r be less than n; then will Ir-l ln-r + (-l)''-' = J[/(w). 502 EXAMPLES. 6. Shew that, if m and n are prime to one another, every odd divisor of m' + n^ is of the form 4A; + 1. 6. Shew that ts+oS + o;: + 7s + •••*<> in6nity (■4.r(>-«"('-r where 2, 3, 5,... are the prime numbers in order. 7. Shew that the arithmetic mean of all numbers less than n and prime to it (including unity) is ^n. 8. Shew that, if N be any number, and a, 6, c, ... be its different prime factors ; then the sum of all the numbers less than N and prime to iV is — (1 — j (1-t) (1~") •••> *^^ the sum of the squares of all such numbers is fl(,-!)(.-!)....f„-.„.-«... 9. If <;|!> (m) denote the number of integers less than m and prime to it; and if c?j, d^, c?^,... be the different divisors of n] then will %<^{d) = n. 10. Shew that, if a fraction ^ , where h is prime and prime to 10, be reduced to a decimal, and if the number of figures in the recurring period is even ; then the sum of the first half of the figures added to the last half will consist wholly of niues. 11. If - be converted to a circulating decimal with p -\ figures in its recurring period, shew that p must be prime and that the recurring period being multiplied by 2, 3, (jP - 1) will reproduce its own digits in the same order. 12. Shew that, if -p has a circulating period of jo figures, ^ of q figures, and p <^^ ** figures,..., and if P, Q, JR,... are prime, then y — will have a circulating period of n figures, where n is the l.o. m. of p, q^ r.... CHAPTER XXTX. Indeterminate Equations. 397. We have already seen that a single equation with more than one unknown quantity, or n equations with more than n unknown quantities, can be satisfied in an indefinite number of ways, provided there is no restric- tion on the values which the unknown quantities may have. If, however, the values of the unknown quantities are subject to any restriction, n equations may suffice to determine the values of more than n unknown quantities. We shall in the present chapter consider some cases of equations in which the unknown quantities are restricted to integral values. 398. It is clear that every equation of the first degree with two unknown quantities x and y can be reduced to one or other of the forms ax-\-hy=±c, ax — hy=±c, where a, b, c are positive integers. By changing x into — x and y into —y, ax^-hy — c will become ax + by = — c, and ax — by = c will become — a^-\-by=C'y hence in order to shew how to find integral solutions of any equation of the first degree in x and y, it is only necessary to consider the two types ax -i-by — c and ax—by—c. Now, it is evident that the equation ax ±by — c cannotl be satisfied by integral values of x and y, if a and b havej any common factor which is not also a factor of c ; and, ifj a, b and c have any common factor, the equation can be s. A. 34 i 504 INDETERMINATE EQUATIONS. divided throughout by that factor. In what follows it will therefore be supposed that a and h are prime to one another. 899. To shew that integral values can always he found which will satisfy the equation aw ±by = c, provided a and h are prime to one another. Let T be reduced to a continued fraction, and let - be the convergent immediately preceding -r. Then, from Art. 357, aq—ph=±l; /. a{±cq)-b(±cp) = c (i), and ci(±cq) + 6(+ cp) = c (ii). Hence it follows from (i) that either x = cqy y = cp or a; = — eg, y = — cp is a solution of the equation ax — by = c; and from (ii) that either x = cq, y — — cp or x = — cq, y = cp ia a solution of the equation ax + hy = c. Hence at least one set of integral values of x and y can always be found which will satisfy the equation ax ±by=c. The above investigation fails when a or 6 is unity. But the equation ax ±y — c is obviously satisfied by the values x — a, ±y = c — acL, where a is any integer. So also x±by = cis satisfied by the values x = c^ b/3, y = ^, where yS is any integer. Hence the equation ax±by = c always admits of at least one set of integral values. 400. Having given one set of integral values which satisfy the equation aw — by = Cy to find all other possible integral solutions. Let x — a, y = j3 be one solution of the equation ax — by — c'j then a/x-b^ = c. Hence, by subtraction, a(x-a)-b(2/-^) = 0. INDETERMINATE EQUATIONS. 505 Now since a divides a{x— a), it must also divide b(y — fi); a must therefore be a factor of y — ft since it is prime to b. Let then y — ff = ma, where m is any integer ; then a(x—a) = mba, and therefore ic=a-\- mb. Hence, if a; = a, y — ^ Ke one solution in integers of the equation ax — by = Cy all other solutions are given by x = a + mb, y = ^ + ma, where m is any integer. It is clear from the above that there are an indefinite number of sets of integral values which satisfy the equation ax—by — c, provided there is one such set ; and, from the preceding article, we know that there is one set of integral values. It is also clear that, whether a and ^ are positive or not, an indefinite number of values can be given to m which will make a + mb and fi + ma both positive. Hence there are an infinite number oi positive integral solutions of the equation ax — by = c. 401. Having given one set of integral values which satisfy the equation a^ -\-by = c, to find all other possible integral solutions. Let x^a^ y =fi he one integral solution of the equation ax + by =c; then a(x+ b^ = c. Hence, by sub- traction, a(x — a)-\-b(y — ^) = 0. Now, since a divides a(x— a), it must also divide b(y —ff); a must therefore be a factor oiy — ^, since it is prime to b. Let then y —13 = ma, where m is any integer ; then a(x — a)= — b{y-^) = — mab ; and therefore x = a — mb. Hence, ii x= a, y = fi he one solution in integers of the equation ax — by = c, all other integral solutions are given by a; = a — mb, y — ^-\- ma, where m is any integer. 506 INDETERMINATE EQUATIONS. From the above, together with Art. 399, it follows that there are an indefinite number of sets of integral values which satisfy the equation ax + hy = c. The number of positive integral solutions of the equation is, however, limited in number. 402. To find the number of positive integral solutions of the equation ax-\-hy — c. We have proved in Art. 399, that the equation ax + hy = c is satisfied by the values x = cq, y = — cp, or by the values x — — cq,y = cp, where p/q is the penultimate convergent to a/b. First suppose that x = cq, y — — cp satisfy the equa- tion ; then all other integral -values which satisfy the equation are given by x—cq—m\ y — — cp-\-ma (i), where m is any integer. From (i) it is clear that in order that x and y ma/both be positive, and not zero, m must be a positive integer, and that the greatest permissible value of m is J^(-r) and its least value /( — ) + l, so that the number of different values of m is -^(■^J ~ ^ { ] • Hence, as one set of values of x and y corresponds to each value of m, the number of solutions is/f-^j — I (-^j . Letf = /.+/. andf = /,./, the.£ = «(l«^^ = f-f =/.-/.+/.-/.. Hence / (^^^) is 7. - /, or fj — /, — 1 according as f is not or is less than f . Thus the number of solutions is ll-jj + l or/f-yj INDETERMINATE EQUATIONS. 507 according as the fractional part of -^ is or is not less than the fractional part of — . It can be shewn in a similar manner that if x= — cq, y = cp satisfy the equation, the number of solu- tions in positive integers is 7 ( -y j + 1 or 7 f -^ J according as the fractional part of — is or is not less than the frac- tional part of -^ . Ex. 1. Find the positive integral values of x and y which satisfy the equation 'Jx-lSy = 2Q. 7 111 We have 7^5 = 7 7 ? . the penultimate convergent is therefore 13 1 + 1 + i. Then 7.2-13. 1 = 1; .-. 7 (2 x 26) - 13 (26) = 26. 2 Hence one solution is a;=52, 2/ = 26; the general solution is there- fore «= 52 + 13wi, y = 26 + 7m. [In this case the solution x=0, y— -2 can be seen by inspection; and hence the general solution is a; = 13ot, y= -2 + 7m, which is easily seen to agree with the previous result.] Ex. 2. Find the positive integral values of x and y which satisfy the equation 7x + 10^ = 280. 7 111 2 Here tr = t s o » *^® penultimate convergent being - . Then 10 1 + 2 + o 7 . 3 - 10 . 2=1; .-. 7 (3 . 280) + 10 ( - 2 . 280) = 280. Hence x ~ 840, y = - 660 is one solution in integers. The general solution in integers is therefore a; = 840 -10m, y= -660 + 7m; and, in order that x aud y may be positive m 4- 84 and m -^ 80. Thus the only values are a; = 40, y = 0; x = SO, y = 7; a; = 20, 2/ = 14; « = 10, 2/ = 21; x = 0,y=28. Ex. 3. Find the number of solutions in positive integers of the equation 3x + 5y = 1306. 3 111 Here c=t , t , 5, whence 3 .2-5. 1 = 1; o X + 1 + J .-. 3 . (2 X 1306) + 5 ( - 1306) = 1306. Hence the general solution is a; =2612- 5m, y= 3m- 1306. For positive values of x and y we must have m>436 and m:^522. Hence the number of solutions is 522 - 435=87. 508 INDETERMINATE EQUATIONS. 403. Integral solutions of the two equations cus -\- by + cz = d, a'x + h'y + cz = dl can be obtained as follows. Eliminate one of the variables, z suppose; we then have the equation {ad — a'c) X + (be' — b'c) y=^dc' — d'c (i), and this equation has integral solutions provided ac' — a'o and be' — b'c are prime to one another, or will become prime to one another after division by any common factor which is also a factor of dc' — d'c. Hence from (i) we obtain, as in the preceding articles, the general solution x = a-\- (be' - b'c) n, y = ^ — (ac — a'c) n, where a; = a, y =^ is any integral solution, and n is any integer. Now substitute these values of x and y in either of the original equations : we then obtain an equation of the form Az + Bn— G, from which we can obtain integral solutions of the form z = y-\- Bm, n = 8 — Am, provided A and B are prime to one another, or will become so after division by any common factor which is also a factor of (7. Ex. Find integral Bolutions of the simultaneous equations 5x + 7y + 2zz=24, 3ar-y-4z = 4. Eliminating «, we have 13a; + 131/ = 62, or x + y = 4:. Whence x=2 + n,y = 2-n. Then 5 (2 + n) + 7 (2-n) + 22 = 24, thatis«-n=0. Hence the general solution isa?=2 + n, y = 2-n, z — n. If X, y and z are to be positive, the only solutions are a;=4, y = 0, «=2; x=S, y = l, z = l; &nd x=2, y = 2, z = 0; and, if zero values are excluded, there is only one solution, namely a;=3, y = l, «=1. 404. The following are examples of some other forms of indeterminate equations. Other cases will be found in Barlow's Theory of Numbers. Ex. 1. Find the positive integral solutions (excluding zero values) of the equation 3x + 2y + 8z=40. It is clear that z cannot be greater than 4, if zero and negative values of x and y are inadmissible. INDETERMINATE EQUATIONS. 509 Hence we have the following equations : «=4, Sx + 2y= 8; 2 = 3, 3a; + 2y = 16; z = 2, 3a; + 2y = 24; « = 1, 3a; + 2y = 32. And it will be found that all the solutions required are 2, 1, 4, 4, 2, 3; 2, 5, 3; 6, 3, 2; 4, 6, 2; 2, 9, 2; 10, 1, 1; 8, 4, 1; 6, 7, 1; 4, 10, 1; and 2, 13, 1. Ex. 2. Find the positive integral solutions of the equation 6x^-lSxy + &y^=16. We have (3a; - 2y) {2x - 3y) = 16 ; hence, as x and y are integers, 3x ~ 2y must be an integer, and must therefore be a factor of 16. Thus one or other of the following simultaneous equations must hold good 3x-2y=^16, 2x-3y=± 1 (i) ; 3a;-2y=± 8, 2x-3y=± 2 (ii) ; Sx-2y==L 4, 2x-3y=± 4 (iii); 8a;-2y=± 2, 2x-3y=± 8 (iv); 3a;-2y=± 1, 2a;-3y=±16 (v). Whence we find that 5x must be ± (48 - 2), =t (24 - 4), ± (12 - 8), i (6 - 16) or dt (3 - 32). Hence the only integral values of x are 4 and 2, the corresponding values of y being 2 and 4. Ex. 3. Solve in positive integers the equation 3x^ + 7xy-2x-5y-35 = 0, We have y {7x - 5) + 3a^»- 2aj- 35=0; 3a;a-2a;-35 ^ ••• y+ — n s — =0; * 7a; -5 * .-. 49y + 21a:+l-^H = o. Hence „-— > °i"st be an integer, and therefore 7a; - 5 must be a factor of 1710. Whence it will be found that the only positive integral solutions are x = 2, y = 3 and a;=:l, ^ = 17. 510 EXAMPLES. EXAMPLES XL. L Find all the positive integral solutions of the equations : (1) 7a; +152/ = 59. (2) Sx + Uy = 13S. (3) 7a;+9y=100. (4) 15a; + 712/ = 10653. 2. Find the number of positive integral solutions of 2x + Sy= 133 and oi7x + Uy = 2312. 3. Find the general integral solutions of the equations (1) 7a;- 133/= 16. (2) 9a;-ll2/ = 4. (3) 119a;- 1052/ = 217. W 49a;- 692/= 100. 4. Find the positive integral solutions (excluding zero) of the equations (1) 2x+Sy + 7z=23. (2) 7a; + 42/ +182!= 109. (3) 5a; + 2/ + 7» = 39, (4) 3a; + 2^/ + 3« = 250, 2a; + 42/ + 9« = 63. 9x- ^y + 5z= 170. 5. Solve in positive integers (excluding zero) the equa- tions : (i) 2a;2/-3a; + 22/=1329. (ii) a^-xy+2x-3y=ll. (iii) 2a;" + 5a;2/-122/' = 28. (iv) 2x^ - xy - y' + 2x + 7y = 84. 6. Shew that integral values of x, y and z which satisfy the equation ax + by -hcz^dj form three arithmetical progressions. 7. Divide 316 into two parts so that one part may be divisible by 13 and the other by 11. 8. In how many ways can £1, 6». Qd. be paid with half-crowns and florins? 9. In how i?iany ways can ^£100 be made up of guineas and crowns 1 EXAMPLES. 511 10. In how many ways can a man who has only 8 crown pieces pay 11 shillings to another who has only florins? 11. Find the greatest and least sums of money which can be paid in eight ways and no more with half-crowns and florins, both sorts of coins being used. 12. Find all the different sums of money which can be paid in three ways and no more with four-penny pieces and three- penny pieces, both sorts of coins being used. 13. Find all the numbers of two digits which are multiples of the product of their digits. 14. Two different numbers each of two digits, and which end with the same digit, are such that when divided by 9 the quotient of each is the remainder of the other. Find all the sets of numbers which satisfy the conditions. 15. A man's age in 1887 was equal to the sum of the digits in the year of his birth : how old was he 1 16. Shew that, if (1 -JB«i)(l-a3^«)...(l -(C««) * then the number of solutions in positive integers (including zero) of the equation a^x^ + a^x^ -I- ... -I- ajic^ ^m^ is A^, a^jU^j ..., a^ being all integers. The number of solutions of the equations x+2y = n is i{27i + 3 + (-in. At an entertainment the prices of admission were Is., 2s. and £5, and the total receipts ,£1000; shew that there are 1006201 ways in which the audience might have been made up. 17. The money paid for admission to a concert was £300, the prices of admission being 55., 3s. and Is.; shew that the number of ways in which the audience may have been made up is 1201801. CHAPTER XXX. Probability. 405. The following is generally given as tlie defini- tions of probability or chance: — Definition. If an event can happen in a ways and fail in b ways, and all these ways are equally likely to occur, then the probability of its happening is r and the pro- bability of its failing is -^ . To make the above definition complete it is necessary to explain what is meant by * equally likely.' Events are said to be equally likely when we have no reason to expect any one rather than any other. For example, if we have to draw a ball from a bag which we know contains unknown numbers of black and white balls, and none of any other colour, we have just as much reason to expect a black ball as a white ; the drawing of a black ball and of a white one are thus equally likely. Hence, as either a black ball or a white ball must be chosen, the probability of drawing either is J, for there are two equally likely cases, in one of which the event happens and in the other it fails. Again, if we have to draw a ball from a bag which we know contains only black, white and red balls, but in unknown proportions, we have just as much reason to expect one colour as to expect either of the others, so that the drawing of a black, of a white and of a red ball PROBABILITY. 513 are all equally likely ; and hence the probability of draw- ing any particular colour is J, for there are three equally likely cases, and any particular colour is drawn in one case and is not drawn in the other two cases. Another meaning may however be given to ' equally likely ;' for events may be said to be equally likely when they occur equally often, in the long run. For example, if a coin be tossed up, we may know that in a very great number of trials, although the number of * heads ' is by no means necessarily the same as the number of ' tails,' yet the ratio of these numbers becomes more and more nearly equal to unity as the number of trials is increased, and that the ratio of the number of heads to the number of tails will differ from unity by a very small fraction when the number of trials is very great; and this is what is meant by saying that heads and tails occur equally often in the long run. Now, if each of the a ways in which an event can happen and each of the 6 ways in which it can fail occur equally often, in the long run, it follows that the event happens, in the long run, a times and fails 6 times out of every a + 6 cases. We may therefore say, consistently with the former definition, that the prohahility of an event is the ratio of the number of times in which the event occurs, in the long run, to the sum of the number of times in which events of that description occur and in which they fail to occur. Thus, if it be known that, in the long run, out of every 41 children born, there are 21 boys and 20 girls, the probability-of any 21 particular birth being that of a boy is ^r • Again, if one of two players at any game win, in the long run, 6 games out of every 8, the probability of his winning any particular . 6 game is - . We may remark that, in the great majority of cases, including all the cases of practical utility, such as the data used by Assurance Companies, the only way in which pro- bability can be estimated is by the last method, namely, by finding the ratio of the actual number of times the event 514 PROBABILITY. occurs, in a large number of cases, to the whole number of times in which it occurs and in which it fails. 406. If an event is certain it will occur without fail in every case : its probability is therefore unity. It follows at once from the definition of probability that if p be the probability that any event should occur, 1 — ^ will be the probability of its failing to occur. When the probability of the happening of an event is to the probability of its failure as a is to 6, the odds are said to be a to 6 for the event, or 6 to a against it, according as a is greater or less than h. 407. Exclusive events. Events are said to be mutually exclusive when the supposition that any one takes place is incompatible with the supposition that any other takes place. When different events are mutually exclusive the chance that one or other of the different events occurs is the sum of the chances of the separate events. It will be sufficient to consider three events. Let the respective probabilities of the three events, expressed as fractions with the same denominator, be ^^ ^ and -» d' d ^"""^ d' Then, out of d equally likely ways, the three events can happen in a,, a^ and a, ways respectively. Hence, as the events never concur, one or other of them will happen in a^ + a, + a^ out of d equally likely ways. Hence the probability of one or other of the three events happening is °- + y°Sthatis| + |^ + |^ This proves the proposition for three mutually ex- clusive events; and any other case can be proved in a similar manner. PROBABILITY. 515 Ex. 1. Find the chance of throwing 3 with an ordinary six-faced die. Since any one face is as likely to be exposed as any other face, there is one favourable and five unfavourable cases which are all equally likely ; hence the required pr oba bility is ^ . Ex. 2. Find the chance of throwing an odd number with an ordinary die. Am. -. Ex. 3. Find the chance of drawing a red ball from a bag which con- tains 6 white and 7 red balls. Here any one ball is as likely to be drawn as any other ; thus there are 7 favourable and 5 unfavourable cases which are all equally 7 likely ; the required probability is therefore — . ^' Ex. 4. Two balls are to be drawn from a bag containing 6 red and 7 white balls ; find the chance that they will both be white. Here any one pair of balls is as hkely to be drawn as any other pair. The total number of pairs is 13^2 > ^^^ *^6 number of pairs which are both white iq ^G^: the required chance is therefore 7.6 / 12 . 11 _ 7 1.2 / 1.2"~22' Ex. 6. Shew that the odds are 7 to 3 against drawing 2 red balls from a bag containing 3 red and 2 white balls. Ex. 6. Three balls are to be drawn from a bag containing 2 black, 2 white and 2 red balls ; shew that the odds are 3 to 2 against drawing a ball of each colour, and 4 to 1 against drawing 2 white balls. Ex. 7. A party of w persons take their seats at random at a round table : shew that it is n - 3 to 2 against two specified persons sitting together. 408. Independent Events. The probability that two independent events should both happen is the product of the separate probabilities of their happening. Suppose that the first event can happen in a^ and fail in \ equally likely ways; and suppose that the second event can happen in a, and fail in 63 equally likely ways. Then each of the a^-\-b^ cases may be associated with each of the eta + 6, cases to make {(ii-\-h^{a^-\-b^ compound cases which are all equally likely; and in a^a^ of these compound cases both events happen. Hence the proba- 516 PROBABILITY. bilitv that both events happen is , ~-^ — — ^. , that IS — ^-7- X — V » which proves the proposition. ai + 61 aj+62' ^ ^ ^ Thus the probability of the concurrence of two inde- pendent events whose respective probabilities are jp^ and f^ Cor. If ^ and 'p^ be the probabilities of two inde- pendent events, the chance that they will both fail is (1 — ^i)(l — i?a), the chance that the first happens and the second fails is 'p^ (1 — jo,), and the chance that the second happens and the first fails is (1 — Pi)Pi. It can be shewn in a similar manner that, if Pi, p^^p^,... be the probabilities of any number of independent events, then the probability that they all happen will be jp, .^j .^s- • •> and that they all fail (1 —^1) (1 —pd 0- ~Pi)' • •> <^c. 409. Dependent Events. If two events are not independent, but the probability of the second is different when the first happens from what it is when the first fails, the reasoning of the previous article will still hold good provided that p^ is the probability that the second event happens when the first is known to have happened. Thus if pi be the probability of any event, and jo, the probability of any other event on the supposition that the first has happened ; then the probability that both events will happen in the order specified will be PiXp,. And similarly for any number of dependent events. Ex. 1. Find the probability of throwing two heads with two throws of a coin. The probability of throwing heads is 5 for each throw ; hence the required probability is, by Art. 408, s >< 5 = 7 • 2 2 4 Ex. 2. Find the proteability of throwing one 6 at least in six throws with a die. ,.^__ PROBABILITY. 517 The probability of not throwing 6 is ^ in each throw. Hence the o probabiUty of not throwing a 6 in six throws is, by Art. 408, ( « ) « and therefore the probability of throwing one six at least is Ex. 3- Find the chance of drawing 2 white balls in succession from a bag containing 6 red and 7 white balls, the balls drawn not being re- placed. 7 The chance of drawing a white ball the first time is =-^ ; and, having drawn a white ball the first time, there will be 5 red and 6 white balls left, and therefore the chance of drawing a white ball the second time will be jrr . Hence, from Art. 409, the chance of 7 6 7 drawing two white balls in succession will ^ ts >< tt = on • [Compare Ex. 4, Art. 407.] Ex. 4. There are two bags, one of which contains 5 red and 7 white balls and the other 3 red and 12 white balls, and a ball is to be drawn from one or other of the two bags ; find the chance of drawing a red ball. The chance of choosing the first bag is ^ , and if the first bag be chosen the chance of drawing a red ball from it is t^; ; hence the chance of drawing a red ball from the first bag is o '^ 12 ~ 24 * Similarly the chance of drawing a red ball from the second bag is 13 1 2 X r^ s= z^ . Hence, as these events are mutually exclusive, the , ... 5 1 37 chance required is — + — = — . Ex. 5. In two bags there are to be put altogether 2 red and 10 white balls, neither bag being empty. How must the balls be divided so as to give to a person who draws one ball from either bag, (1) the least chance and (2) the greatest chance of drawing a red ball. [The least chance is when one bag contains only one white ball, and the greatest chance is when one bag contains only one red ball, 1 fi the chances being — r and :pr respectively.] 518 PROBABILITY. 410. When the probability of the happening of an event in one trial is known, the probability of its happen- ing exactly once, twice, three times, &c. in n trials can be at ODce written down. For, if p be the probability of the happening of the event, the probability of its failing is 1 —p — q. Hence, from Art. 408, the probability of its happening r times and failing n — r times in any specified order is p'^q'"'''. But the whole number of ways in which the event could happen r times exactly in n trials is „C^, and these ways are all equally probable and are mutually exclusive. Hence the probability of the event happening r times exactly in n trials is „C^p''g**"''. Thus, if (^ + qY be expanded by the binomial theorem, the successive terms will be the probability of the happen- ing of the event exactly n times, n — 1 times, n — 2 times, &c. in n trials. Cor. I. To find the most probable number of successes and failures in n trials it is only necessary to find the greatest term in the expansion of (p + qy. Cor. II. The probability of the event happening at least r times in n trials is Ex. 1. Find the chance of throwing 10 with 4 dice. The whole number of different throws is 6*, for any one of six numbers can be exposed on each die; also the number of ways of throwing 10 is the coefficient of x^^ in {x + x^+ ...+x^)\ for this co- efficient gives the number of ways in which 10 can be made up by the addition of four of the numbers 1, 2, ..., 6, repetitions being allowed. Now the coefficient of x^° in {x + x^+ ...+x^)*', that is in x^ [ ) , is easily found to be 80. Hence the required chance is 80 5 6.6.6.6 «1* Ex. 2. Find the ohance of throwing 8 with two dice. Ans. — , 36 PROBABILITY. 519 Ex. 3. Find the chance of throwmg 10 with two dice. Am, ^h • 5 108' Ex. 5. A and B each throws a die; shew that it is 7 : 5 that ^'s throw is not greater than B's. Ex. 6- A and B each throw with two dice : find the chance that their throws are equal. - 73 '^'"- 648- Ex. 7. A and B have equal chances of winning a single game at tennis : find the chance of A winning the 'set' (1) when A has won 5 games and B has won 4, (2) when A has won 6 games and B has won 3, and (3) when A has won 4 games and B has won 2. Am. (l)|,(2)|,(3)g. Ex. 8. A and B have equal chances of winning a single game; and A wants 2 games and B wants 3 games to win a match : shew that it is 11 to 5 that A wins the match. Ex. 9. A and B have equal chances of winning a single game ; and A wants n games and B wants n+1 games to win a match : shew that ^, ,, , , 1.3.5. ..(2n-l), , 1.3.5...(2n-l) theoddson^ aore 1^ 2.4.6...2n *^ ^- 2.4.6...2n ' 3 Ex. 10. A*s chance of winning a single game against B ia-=: find the 5 chance of his winning at least 2 games out of 8. ^"•- 125- 2 Ex. 11. A'b chance of winning a single game against B is - : find the o chance of his winning at least 3 games out of 6. , 192 v ^"*-243- Ex. 12. What is the chance of throwing at least 2 sixes in 6 throws with a die? ^ 12281 ^^^- 4-665-6- Ex. 13. A coin is tossed five times in succession : shew that it is an even chance that three consecutive throws will be the same. Ex. 14. Three men toss in succession for a prize which is to be given to the first who gets 'heads'. Find their respective chances. A 4 2 1 7 7' 7 S. A, 35 520 PROBABILITY. 411. The value of a given chance of obtaining a given sum of money is called the expectation. If is the chance of obtaining a sum of money M, a + o a then the expectation is if x— — 7 . For if E be the expectation in one trial, E(a-[-h) will be the expectation in a + b trials. But the chance being =■ , the sum M will, on the average, be won a times in every a -\-b trials ; and hence the expectation in a-\-b trials is Ma. Hence E{a-\-b) = Ma; therefore E=^Mx a + b Thus the expectation is the sum which may be won multiplied by the chance of winning it. Ex. 1. A bag contains 5 white balls and 7 black ones. Find the expectation of a man who is allowed to draw a ball from the bag and who is to receive one shilling if he draws a black ball, and a crown if he draws a white one. 7 The chance of drawing a black ball is — ; and therefore the expectation from drawing a black ball is 7d. The chance of drawing 5 a white ball is — ; and therefore the expectation from drawing a white ball is 2s. Id. Hence, as these events are exclusive, the whole expectation is 2s. 8d, Ex. 2. A purse contains 2 sovereigns, 3 half-crowns and 7 shillings. What should be paid for permission to draw (1) one coin and (2) two coins ? Am. (1) 4«. 6^^. (2) 9s. Id. Ex. 3. Two persons toss a shilling alternately on condition that the first who gets 'heads' wins the shilling: find their expectations. Ans. 8d., id. Ex. 4. Two persons throw a die alternately, and the first who throws 6 is to receive 11 shillings : find their expectations. Ana. 6s., 5s. PKOBABILITY. 521 412. Inverse Probability. When it is known that an event has happened and that it must have followed from some one of a certain number of causes, the deter- mination of the probabilities of the different possible causes is said to be a problem of inverse probability. For example, it may be known that a black ball was drawn from one or other of two bags, one of which was known to contain 2 black and 7 white balls and the other 5 black and 4 white balls ; and it may be required to determine the probability that the ball was drawn from the first bag. Now, if we suppose a great number, 2N, of drawings to be made, there will in the long run be N from each bag. But in N drawings 2 from the first bag there are, on the average, ^ N which give a black ball ; and in N drawings from the second bag there are ^ N which 2 give a black ball. Hence, in the long run, ^N out of a total of 2 5 ^N+-N black balls are due to drawings from the first bag; thus the probability that the ball was drawn from the first bag is 2 _ /2 „ 5„\ ., . . 2 iV^-j-Qi^+^J^y thatis We now proceed to the general proposition : — Let Pj, Pg,..., P„ 6e the probabilities of the eodstence ofn causes, which are mutually exclusive and are such that a certain event must have followed from one of them; and let i>i, Pi, "-yPn ^^ ^^^ respective probabilities that when one of the causes P^, P^, ..., P^ eodsts it will be followed by the event in question; then on any occasion when the event is known to have occurred the probability of the rih cause is Let a great number iV of trials be made ; then the first cause will exist in JV.P, cases, and the event will follow in N .P^.p^ cases. So also the second cause exists and the event follows \u N .P^.p^ cases ; and so on. Hence the event is due to the rih. cause in N .P^.p^ 35—2 522 PKOB ABILITY. cases out of a total of iV' (P^^^ + P^Pa + • • • + -P«P«) 5 *^e P V probability of the rth cause is therefore y^- Having found the probability of the existence of each of the different causes, the probability that the event would occur on a second trial can be at once found. For let P/ be the probability of the existence of the rth cause ; then p^ is the probability that the event will happen when the rth cause exists ; and therefore P/ . p^ is the probability that the event will happen from the rth cause. Hence, as the causes are mutually exclusive, the probability that the event would happen on a second trial is Ex. 1. There are 8 bags which are known to contain 2 white and 3 black, 4 white and 1 black, and 3 white and 7 black balls respectively. A ball was drawn at random from one of the bags and found to be a black ball. Find the chance that it was drawn from the bag con- taining the most black balls. 1 3 17 Here Py^=P^=^P^=^. Also jpi=g, p^:=-^ and i>3=io- Hence the required probability is Ex. 2. From a bag which is known to contain 4 balls each of which is just as hkely to be black as white, a ball is drawn at random and found to be white. Find the chance that the bag contained 3 white and 1 black balls. The bag may have contained (1) 4 white, (2) 3 white and 1 black, (3) 2 white and 2 black, (4) 1 white and 3 black, and (5) 4 black ; and 14 6 4 1 the chances of these are respectively ^r^ , ^ , -^ , — ; and — ^ . lb io Id lb lb Art. 410. Also the chances of drawing a white baU in these 3 11 different cases will be 1, - , - , j and respectively. 1 3 7 ■ 10 7 3* 5^3" 117 10 5 "•" 3 • 10 Hence the required probability = 4 8 16-4 3 1 8 4 16 1 4 8 16 "*■ i * 16 "^ 2 • 16 "*" 4 * 16 PROBABILITY. 523 413. Probability of testimony. The method of dealing with questions relating to the credibility of wit- nesses will be seen from the following examples : Ex. 1. A ball has been drawn at random from a bag containing 99 black balls and 1 white ball; and a man whose statements are accurate 9 times out of 10 asserts that the white ball was drawn. Find the chance that the white ball was really drawn. The probability that the white ball will really be drawn in any case is =^r^ , and therefore the probability that the man will truly assert 1 9 that the white ball is drawn is r^ x =^ . 99 The probability that the white ball will not be drawn is -rrpr , and 100 therefore the probability that the man will falsely assert that the 99 1 white ball is drawn is =^ x -^ , Hence as in Art. 412 the required probability is 100 ^10 1 1 9 99 1 12 100 ^ 10 "^ 100 ^ 10 Ex. 2. From a bag containing 100 tickets numbered 1, 2, ..., 100 respectively, a ticket has been drawn at random ; and a witness, whose statements are accurate 9 times out of 10, asserts that a particular ticket has been drawn. Find the chance that this ticket was really drawn. In 1000 iV^ trials the ticket in question will be drawn ION times; and the witness will correctly assert that it has been drawn 9 JV times. The ticket will not be drawn in 990 N cases, and the witness will make a wrong assertion in 99 -N" of these cases ; but there are 99 ways of making a wrong assertion and these may all be supposed to be equally likely; hence the witness will wrongly assert that the particular ticket has been drawn in N cases. Hence the required 9 probability is r^ , so that the probability is in this case equal to the probability of the witness speaking the truth. Ex. 3. A speaks the truth three times out of four, and B five times out of six ; and they agree in stating that a white ball has been drawn from a bag whicL was known to contain 1 white and 9 black balls. Find the chance that the white ball was really drawn. The probability that the white ball will be drawn in any case is 624 PROBABILITY. — r , and therefore the probability that A and B will agree in truly jl^ 3 5 10^4^6* The probability that a black ball will really be drawn in any 9 case is ^tx ; and therefore the probability that A and B will agree in 9 11 falsely asserting that a white ball is drawn is r^ x ^ x ^ . Hence, as in Art. 412, the required probability is J^ 3 5 10 ^ 4 ^ 6 5 1 £ 5 9^ 1 1 8* 10 ^4^6"^ 10 ^4^6 Ex. 4. A speaks truth three times out of four, and B five times out of six ; and they agree in stating that a white ball has been drawn from a bag which was known to contain 10 balls all of different colours, white being one. What is the chance that a white ball was reaUy drawn? The probability that the white ball will really be drawn in any case is jr- , and therefore the probability that A and B wiU agree in l ft K "l truly asserting that the white ball is drawn is=7rX:rX;; = =^. 10 4 D Id The probability that the white ball wiU not be drawn in any case 9 1 is jtr . The probability that A will make a wrong statement is j ; hence, as there are nine ways of making a wrong statement which, may all be supposed to be equally Ukely, the chance that A will wrongly assert that a white ball is drawn is ^r x ;r. Therefore the 4 9 chance that A and B will agree in falsely asserting that a white ball' is drawn is ^ _1_ 11 10^4x9^6x9~2160* ifi mE Hence the required probability is , = — — . 1 1 136 16"*" 2160 Ex. 5. It is 3 to 1 that A speaks truth, 4 to 1 tbat B does and 6 to 1 that C does : find the probability that an event really took place which A and B assert to have happened and which G denies ; the event being, independently of tiiis evidence, as likely to have happened as not. Am. |. PROBABILITY. 525 414. We shall conclude this chapter by considering the following examples, referring the reader who wishes for fuller information on the subject of Probabilities to the article in the Encyclopaedia Britannica, and to Tod- hunter's History of the Mathematical Theory of Proba- bility. Ex. 1. A bag contains n balls, and all numbers of white balls from to n are equally likely ; find the chance that r white balls in succes- sion will be drawn, the balls not being replaced. The chance that the bag contains s white balls is — ^ ; and the n + l chance that r balls in succession will be drawn from a bag contain- 1- n i. T-- t- x.-i. • s (s-l)...(»-r+l) ing n balls of which « are white is —j =4 — ) \ • n(w-l)...(n-r + l) Hence the chance required is 1 f w(7i-l)...(n-r+l) (w-1) (w- 2). ..(n-r) n+l\n{n-l)...{n-r + l) n{n-l)...{n-r + l) ■ r(r-l)...l \ '" n(n-l)...(n-r+l)j * Now {1.2...r} + {2.3...(r+l)} + ... + {(n-r+l)...(n-l)n} ^(n-r+l)(n-r + 2)...n(n + l) ^^ g^g r+1 Hence the required chance is — -. , which is independent of the whole number of balls in the bag. If it be known that r white balls in succession have been drawn, the probability of the next drawing giving a white ball can be at once found from the preceding result. For in a great number N^ of cases, there will be r white balls in N N succession in — — y cases, and r+1 white balls in succession in — - cases. Hence the required chance is — ^ -i = — - , ^ r + 2 r + 1 r + 2 Ex. 2. Two men A and B, who have a and b counters respectively to begin with, play a match consisting of separate games, none of which can be drawn, and the winner of a game receives a counter from the loser. Find their respective chances of winning the match, which is supposed to be continued until one of the players has no more counters, the odds being p : q that A wins any particular game. 526 PROBABILITY. Let -4'b chance of ultimate success when he has » counters be w^. Then A'b chance of winning the next game is — — — , and his chance of ultimate success will then be w»4.i; also -4*8 chance of losing the next came is — ^ , and his chance of ultimate success will then be Hence «„ = -^ WnrW ■! r" ^n-i » P + 9. P + 2 .'. pu^i-{p + q)Un + qu^-i=0, from which it follows that w„ will be the coefficient of «•* in the expansion of — ; — r — ; , p-{p + q) x + qar provided A and B be properly chosen. Now ; r-^ J can be expressed in the form + ; p-{p + q)x + qx^ P-2^ 1-^ and hence the coefficient of a;** is D + — ( - J . Thus w„=D + -(-j , where G and D have to be determined. But it is obvious that ^'s chance of winning is zero if he has no counters and unity if he has a + b, so that W(, = and ^^+6=1 ; hence 0=Z>+-, and l=D + -l-] , whence the values of C and D P P \jpj are found, and we have -{-(I)"} / HT)- Hence A'b chance of winning the game is Similarly JS's chance of winning the game is I-©} /{-(in- EXAMPLES XLI. 1. A and B throw alternately with two dice, and a prize is to be won by the one who first throws 8. Find their respective chances of winning if A throws first. EXAMPLES. 527 2. -4, B and G throw alternately with three dice, and a prize is to be won by the one who first throws 6. Find their respective chances of winning if they throw in the order Aj B, C, 3. Three white balls and five black are placed in a bag, and three men draw a ball in succession (the balls drawn not beiQg replaced) until a white ball is drawn : shew that their respective chances are as 27 : 18 : 11. 4. What is the most likely number of sixes in 50 throws of a die 1 5. Shew that with two dice the chance of throwing more than 7 is equal to the chance of throwing less than 7. 6. In a bag there are three tickets numbered 1, 2, 3. A ticket is drawn at random and put back; and this is done four times : shew that it is 41 to 40 that the sum of the numbers drawn is even. 7. From a bag containing 100 tickets numbered 1, 2, 3, ...100, two tickets are drawn at random; shew that it is 50 to 49 that the sum of the numbers on the tickets will be odd. 8. There are n tickets in a bag numbered 1, 2, ..., w. A man draws two tickets together at random, and is to receive a number of shillings equal to the product of the numbers he draws : find the value of his expectation. 9. An event is known to have happened n times in n years : shew that the chance that it did not happen in a particular year is f 1 — j . 10. If p things be distributed at random among p persons ; shew that the chance that one at least of the persons will be void is 1- L^ . 11. A writes a letter to B and does not get an answer; assuming that one letter in rn is lost in passing through the post, shew that the chance that B received the letter is vn ~~ 1 =- , it being considered certain that B would have answered 2m -I the letter if he had received it. 528 EXAMPLES. 12. From a bag containing 3 sovereigns and 3 shillings, four coins are drawn at random and placed in a purse; two coins are then drawn out of the purse and found to be both sovereigns. Shew that the value of the expectation of the remaining coins in the purse is 11». Qd. 13. From a bag containing 4 sovereigns and 4 shillings, four coins are drawn at random and placed in a purse; two coins are then drawn out of the purse and found to be both sovereigns. Shew that the probable value of the coins left in the bag is 29 J shillings. 14. If three points are taken at random on a circle the chance of their lying on the same semi-circle ia J. 15. A rod is broken at random into three pieces : find the chance that no one of the pieces is greater than the sum of the other two. 16. A rod is broken at random into four pieces : find the chance that no one of the pieces is greater than the sum of the other three. 17. Three of the sides of a regular polygon of 4w sides are chosen at random; prove that the chance that they being produced will form an acute-angled triangle which will contain ^, . . (n-l)(n-2)_ the polygon IS ^^—3^^-^^-^. 18. Out of m persons who are sitting in a circle three are selected at random; prove that the chance that no two of .1 . (m - 4) (m - 5) those selected are sitting next one another is 7 :— ^rr . {m — i) ym — ^) 19. If m odd integers and n even integers be written down at random, shew that the chance that no two odd numbers are \n \n+l adjacent to one another is ; *=f =- , m being i> w + 1. \m + n\n — m+i 20. If 771 things are distributed amongst a men and b women, shew that the chance that the number of things ■ AX. .1. ^ ' AA • l{b-^ar-(b-ar received by the group of men is odd, is -x ^^ — --; . EXAMPLES. 629 21. The sum of two whole numbers is 100; find the chance that their product is greater than 1000. 22. The sum of two positive quantities is given; prove that it is an even chance that their product will not be less than three-fourths of their greatest product; prove also that the chance of their product being less than one-half their greatest product is 1 — j^. 23. Two men A and B have a and h counters respectively, and they play a match consisting of separate games, none of which can be drawn, and the winner of a game receives a counter from the loser. The two players have an equal chance of winning any single game, and the match is continued until one of the players has no more counters. Shew that J.'s chance of winning the match is ^ • a + o 24. An urn contains a number of balls which are known to be either white or black, and all numbers are equally likely. If the result oi p + q drawings (the balls not being replaced) is to give p white and q black balls, shew that the chance that the ,1 next drawing will give a black ball is -^ k • ° ® p + q + 'i 25. Two sides play at a game in which the total number of points that can be scored is 2m + 1 ; and the chances of any point being scored by one side or the other are as 2m +1—0! to 2m + 1 — y, where x and y are the points already scored by the respective sides. Shew that the chance that the side which scores the first point will just win the game ia ( 2m! 2m+ iy {mXfm-^ 11 4m4-ir CHAPTER XXXL Determinants. 415. If there are nine quantities arranged in a square as under : «. a« % \ h K Cj 0. ^8 then all the possible products of the quantities three to- gether, subject to the condition that of the three quantities in each product one and only one is taken from each of the rows and one and only one from each of the columns, will be ap^c^y afi^c^, afi^c^, aj)^c^, afi^c^, and aj)^^^. Let now these products be considered to be positive or negative according as there is an even pr an odd number of inversions of the natural order in the suffixes ; then the algebraic sum of all the products will be «! Vs - ^J>fPi + «Aci - tta^iCs + %K<^^ - ^8 Vi W ; for there are no inversions in afi^c^, there is one inversion in ajb^c^ since 3 precedes 2, there are two inversions in ajb^c^ since 2 and 3 both precede 1, there is one inversion in ajfejCg since 2 precedes 1, there are two inversions in ajb^c^ since 3 precedes both 1 and 2, and there are three invei'sions in o,J)^c^ since 3 precedes both 1 and 2 and 2 precedes 1. DETERMINANTS. 631 The expression (A) is called the determinant of the nine quantities a^, a,, &c., which are called its elements; and the products afi^c^, cififpii <^c. are called the terms of the determinant. 416. Definition. If there are n^ quantities arranged in a square as under : a^ % ^8 ••• ... a, K K 63 ... ... 6. ^1 0, Os ..• ... c. m. m. ^3 •• ... m, the members of the same row being distinguished by the same letter, and the members of the same column by the same suffix ; and if all the possible products of the quan- tities 71 at a time are taken subject to the condition that of the n quantities in each product one and only one is taken from every row and one and only one from every column, and if the sign of each product is considered to be positive or negative according as there is an even or an odd number of inversions of the natural order in the suffixes ; then the algebraic sum of all the products so formed is called the determinant of the n^ quantities or elements. To denote that the ti* quantities are to be operated upon in the manner above described, they are enclosed by two lines, as in the above scheme. The diagonal through the left-hand top comer is called the principal diagonal; and the product of the n elements ftj, 63, Cg, , m„ which lie along it, is caMed the principal term of the determinant. All the other terms can be formed in order from the principal term by taking the letters in their alphabetical order and permuting the suffixes in every possible way: on this account a determinant is sometimes represented by enclosing its principal term in brackets ; thus the above determinant would be written [aj)^c^...m^], the 632 DETERMINANTS. determinant is also often represented by the notation 2(±a,6,c,...mJ. When only one determinant is considered it is generally denoted by the symbol A. A determinant is said to be of the nth order when there are n elements iii each of its rows or columns, and therefore also n elements in each of its terma 417. Since there are as many terms in a determinant of the nth order as there are permutations of the n suffixes, it follows that there are \n terms in a determinant of the nth order. There are, for example, six terms in a deter- minant of the third order. 418. The law by which the sign of any term of a determinant is found is equivalent to the following : Take the elements in order frma the successive rows beginning at the first ; then the sign of any term is positive or negative according as there is an even or an odd number of inversimis in the order of the colwmns from which the elements are taken. We will now shew that the words row and column may be interchanged in the above law. To prove this, consider any product, for example, ^A^i^i^eJ's ^^^ ^^ equivalent Cj/jfegS^a^eg, where in the first form the letters follow the alphabetical order and in the second form the numbers follow the natural order. We have to shew that the number of inversions in the suffixes in the first form is the same as the number of inversions of the alphabetical order in the second form. This follows immediately from the fact that if, in the first form, any suffix follow r suffixes greater than itself; then, in the second form, the letter corresponding to that suffix must precede r letters earlier than itself in alphabetical order. Thus, in the example, 2 follows four suffixes greater than itself in afi^c^d^e^f, and / precedes four letters earlier than itself in c^fb^d^a^e^. DETERMINANTS. 533 Since the words rows and columns are interchangeable in the law which determines the sign of any term, we have the following Theorem. A determinant is unaltered by changing its rows into columns and its columns into rows. For example a„ 6, k k Ex. 1. Count the number of inversions in 2314, 3142 and 4231. Am. 2, 3, 6. Ex. 2. Count the number of inversions in 4132, 35142 and 531264. Ans. 4, 6, 7. Ex. 3. What are the signs of the terms bfg, cdh and ceg in the determinant a b c d e f g h k [The order of the columns is 231, 312 and 321.] Ans. +, +, -. Ex. 4. What are the signs of the terms bgiq, celn and dfkm in the determinant a b c d 1 e f g h i j k I [The order of the columns is 2314, 3142 and 4231.] Ans. +, ' 419. Theorem I. If in any term of a determinant any two suffixes be interchanged, another term of the determinant will be obtained whose sign is opposite to that of the original term. Let P .ha. kfi be any term of a determinant, P being the product of all the elements except ha and k^ ; then, by interchanging a and yS we have P.h^.ka. Now since P . ha. kp is a term of the determinant, P can contain no element from the rows of ^'s and k's and no element from 534 DETERMINANTS. the a or ^ columns ; and this is a sufficient condition that Phpka should also be a term of the determinant. We have now to shew that the two terms have different signs. First suppose that two consecutive suffixes are inter- changed. Consider the term AhJc^B where A denotes the product of all the elements which precede h^ and B the product of all the elements which follow k^. By interchanging a and fi we have Ah^kaB, which we have already found is a term of the determinant. Now the number of inversions in the two terms must be the same so far as the suffixes contained in J., or in B, are concerned, whether compared with one another or with a and ^ ; but there must be an inversion in one or other of a/3 and y8a but not in both. Hence the numbers of the inversions in the two terms differ by unity, and therefore the signs of the terms must be different. Now suppose that two non-consecutive suffixes are interchanged ; and let there be r elements between the two whose suffixes, a and fi suppose, are to be interchanged. Then a will be brought into the place of ^ by r+1 in- terchanges of consecutive suffixes, and /3 can then be brought into the original place occupied by a by r interchanges of consecutive suffixes ; and therefore the interchange of a and ff can be made by means of 2r + 1, that is by an odd number, of interchanges of successive suffixes. But, by the first case, each such interchange gives rise to a loss or gain of one inversion ; and hence there must on the whole be a loss or gain of an odd number of inversions : the sign of the new term will therefore be different from the sign of the original term. 420. Theorem II. A determinant is unaltered in absolute value, but is changed in sign, by the interchange of any two columns or any two rows. Suppose that in any determinant the rows in which the letters h and k occur are interchanged. Then, if DETERMINANTS. 635 A .ha.B .kp .G he any term of the original determinant, the term of the new determinant formed by the elements which occur in the same places as before will be AJcaBhpG; and these two terms must have the same sign in the two de- terminants. Now by Art. 419 we know that A .ka.B .hp,G is a term of the original determinant and that its sign is different from that of A . ha . B . kp . G. Hence any term of the new determinant is also a term of the original determinant but the sign of the term is different : the two determinants must therefore be equal in absolute magni- tude but different in sign. The proposition being true for rows is, from Art. 418, true also for columns. For example A certain number of the terms of A will contain a^ let the sum of all these terms be a^.A^. Similarly let the sum of all the terms which contain a^, a^ and a^, be respectively cr, . A^, a, . A^ and a^ . A^. Then, since no term can contain more than one of the letters a^, a,, a,, a^ we have A = a^A\ + a^^ + asA^ + a^A^ (i). Now, since no term of A which contains Oj can contain any element from the column or the row through a^, it follows that every term of A which contains a^ is the product of Oj and some term of Aa^; conversely the product of Oi and any term, T, of Aa^ will be a term, of A, and the sign of the term a^.T of A will be the same as the sign of the term T of A^^, for there is no change in the number of 36—2 538 DETERMINANTS. inversions. Hence the sum of all the terms of A which contain a^ is Oj. A^,. So also, every term of A which contains a, is the product of fla and some term of A^,, and the product of a, and any term, T, of A^, will be a term of A, but there is one more inversion in the term ci^.T oi A than there is in the term T of A^^, since 2 precedes 1. Hence the sum of all the terms in A which contain a^ is — a, . A^^. Similarly the sum of all the terms of A which contain ttj are a, . A^j,; and the sum of all the terms which con- tain a^ are — a^ . A at' Hence A = a, . A a^ - a, . A^ + a, . A«, - a, 'a*' .(ii). By means of Articles 419 and 420, we can shew in a similar manner that A = - 6i Aft, + 62 Aj, - 6, Aj, + 6, As^ = «! A«^ - 61 Aj^ + Cj Ae, - (^1 Ad, = &c. Cor. By comparing (i) and (ii) we see that the co- factors of the elements a^, a^, &c., are equal in absolute magnitude to the minors of the same elements. 425. We have in the previous article considered the case of a determinant of the fourth order ; the reasoning is however perfectly general, so that if A be a determinant of the nth. order having a,, a,,..., a^ for the elements of its first row or column; then will A = a,.A«.-a,A«. + ... + (-ir^a„A„,. So also A = (-irMA;,.A,,-A;,.A,.4-... + (-irA:„AU. Where \,k^y...yk^ are the elements of the rth row. For example % ^2 Oj =«i h h -a. 6i 6, + c^ai + c^^i + Csyiy Ojaa + ajj/Sa + aaYa, b^a^ + b^^ + b^y^, CiO^ + c^Pi + c^y^, which is by Art. 429 equivalent to the product of — 1 , that is 1, and 1-10 0-1 OiOi + OajSj + 0,71 , ftitti + 62J81 + 6371 > CiOi + Csj/^i + CgTi aiOa + a^jSa + OaTa' b^a^+b^^ + hyz* ^lOj, + Cjj^j + C372 aia3 + a2/33 + a,73, &ia8 + ^2/^8 + ^873. Ci^s + Ca/Sg + 0373 Hence the required product is the determinant last written. Ex. 1. Multiply X y z z X y y z X The required product is by a c b h a e e b a X Y Z Z X Y Y Z X Y= ay + bz+ ex, and Z=az + bx + cy. where X=ax+by + cZf Since X y z * X y y z X ■a^+y^+z^- Bxyz, and the other determinants are of the same form, we see that the product of any two expressions of the form a^ + y^ + z^-Sxyz can be expressed in the same form. [See Art. 156, Ex. 4.] Ex. 2. Shew that | 26c - a\ b^ Form the product of 2ac - b\ , -6, e -a, b -c, a 62 2a6 - c2 and :(a»+62+c«-3a6c)3. Ex. 3. Shew that ^1 ^1 Ci ^a J'a ^i ■^8 -^3 ^8 -a, 6 c -c, a b -6, c a 61 Cj 2^ where ^j, -Sj, &0. 6, ca «8 *8 Cj are the oo-factors of Oj , 6^ , &o. in the expansion of the determinant ra,6ac,]. For DETERMINANTS 5 ^1 ^1 ^1 . Oj Oa aj = [ajfeaCj] ^2 -^2 C'a 6i &a 63 [(hP^zl ^8 ^» <73 «i c» c» [oiVs] 545 and Hence [A^B^C^ . [ajb^c^] = [a^h^c^f. 431. The notation a. cto a. a. 6. 6. 6. b. X a o V is employed to denote the system of four determinants obtained by omitting any one of the columns. 432. We conclude with the following important appli- cations of determinants. Simultaneous Equations of the First degnree. The solution of any number of simultaneous equations of the first degree can be at once obtained by means of the foregoing properties of determinants. First take the case of the three equations a^x + b^y + c^z — k^, a^ + b^ + c^ = A?,, a,x+b^ + c^^k^. Multiply the equations in order by A^, A^, A^, where A^, A^,A^ are the co-factors of a^^a^, a^ respectively in the determinant Then we have by addition {a,A, + a^, + a,A;) x + (Mi + KA + ^s^s) V 54j6 DETERMINANTS. that is [a^ 6, c^] x = \k^ \ cj, for from Art. 426 the coefficients of y and z are zero. Similarly we obtain [»! K ^3] y = K ^2 cj, and [«! 62 C3] <2 = [ttj 62 A^g]. Now consider n equations of the form ^1^1 + M« + ^1^8 + ^1^4 + ^h' As before, multiply the equations in order by A^, -4.,, ^3, &c. the co-factors respectively of a,, a^, a^, &c. in the determinant [a^ b^ c,...] ; then we have by addition (a^A^ + a,A^ + %A^ +...)«?= Mi + M* + Ms +• • •» the coefficients of y, z, &c. being all zero by Art. 426. Hence So also Ex. 1. Solve the equations 2x + 4y + « =7, The values of a;, y, 2 are respectively &C. 6 2 3 16 3 12 6 7 4 1 2 7 1 2 4 7 14 2 9 3 14 9 and 3 2 14 12 3 12 3 12 3 2 4 1 2 4 1 2 4 1 8 2 9 3 2 9 3291 and it will be found that each determinant is «=2/=«=l. Ex. 2. Solve the equations X +y +z +w +k =0, ax +by +CZ +dw +k^ = 0, a»x + 6% + ch + dhp + k*=0. 20. so that We have DETERMINANTS. 1 Ilk -5- 1 1 1 1 b c d k^ a 6 c d 62 c» d? F a2 62 C2 d2 63 c3 d3 A:* a3 6» C8 d3 547 _ ;b(c-d)(d-6)(6-c)(fc-6)(fc-c)(A;-cg) ^ "" {c-d)(d-b)(b-c)ia-b)ia-c){a-d) ' _ fc(A;-6)(fe-c)(fe-d) •'• '^"" {a-b)(a-c){a-d) ' and the values of y, z and «o can be written down from that of x. 433. Elimination. To find the condition that the three equations a^x + \y + Cj = 0, (i^ + \y + Cg = 0, a^x + 63^ + c, = 0, may be simultaneously true. Multiply the equations in order by (7j, (7,, C^, the co-factors of Cj, c,, Cg respectively in the determinfint Then by addition we have »1 &. C| a. K ^2 «8 h C3 that is, from Art. 426, a, 6, a„ = 0. which is the required condition. The three homogeneous equations ajX + biy + Cj^z = a^ + b2y + c^ = a^ + h^ + c^z=0 are obviously satisfied by the values x=y=z=0. If however x, y, z are not all zero, it follows from the above that the condition [ajbcf^]=0 must hold good. 548 DETERMINANTS. ^ It can be shewn in a similar manner that the condition that n equations of the form a^x + b^y -^ ... + k^ = 0, with (n — 1) unknown quantities, may be simultaneously true is 434 Sylvester's method of Elimination. This is a method by which x can be eliminated from any two rational and integral equations in x. The method will be understood from the following examples. Ex. 1. Eliminate x from the equations ax^ + 6a5 + c =0 and px^ + qx + r=0. From the given equations we have ax^ + bx^+cx =0, ax^ + bx + c=:0, px^ + qx^ + rx =0, Aud px^ + qx + r=0. Now we may consider the different powers of x as so many different unknown quantities ; and the result of eliminating x\ x^ and x from the four last equations is by Art. 433 a b e =0. a b c p q r p q r [This result is equivalent to that obtained in Art, 153, Ex. 3.] Ex. 2. Eliminate x from the equations ax^ + bx^ + cx + d=0 and px^ + qx + r=0. From the given equations we have ax^ + bx' + cx^ + dx =0, ax^ + bx^ + cx + d=0, px^ + qx^ + rx^ =0, px^ + qx^-^rx =0, px^ + qx + r=0. Eliminating x*, a^, x\ x from the five last equations as if the different powers of have the condition were so many different unknown quantities, we d = 0. EXAMPLES. 649 EXAMPLES XLn. 1. Shew that 2. Shew that 3. Shew that Shew that 6. Shew that 6. Shew that 7. Shew that o. Shew that e. Shew that lO. Shew that 6>+c2 ab c^ + a^ cb ac be a^ + b^ =4a2&2c«. 1 a a?-bc 1 6 b^-ca 1 c c^-ab =0. b + e c + a a + b b' + c' c' + a' a' + b' b" + c" c" + a" a" + b" = 2 a h c a' b' c' a" b" c" • a + b + 2e c b + e a 6 c + 2a 6 a c + a + 26 = 2(a + 6 + c) \ a^+ab ba (& + c)» c^ 6» -be ca + a^ ab + a^ (a + 6)2 ca be {6 + c)3 b^ c^ a I a ( b c C c b c be 63 62 + 6c c2 {e+a)* a« 6c + 62 -ca a6 + 62 ea (6+c)2 ab a2 (c + a)e c« » c I 6 I a ', ac + c* oc c2 62 a2 (a + 6)'^ 6c + c2 ca + c2 -a6 6c ab (c + a)2 :4a262c2. = 2(6c + ca + a6)'. = (6c + ca + a6p. = 2a6c(a + 6 + c)3. a"" 62 (a + 6)2 Oil 1 c2 1 C2 1 2a6c(a + 6 + c)», 62 a* (a + 6 + c){-a + 6 + c)(-6 + c + a) 11. Prove that 1 62 62 c2 a2 «3 n aa aa 6^3 C7 cy 6/3 ;-c + a + 6). 6^ C7 cy b^ aa aa 650 EXAMPLES. 13. 12. Shew that j 1 1 1 il 1+a 1 11 1 1+& |l 1 1 Shew that 1+a 111 11+6 1 1 1 1 1+c 1 1 1 1 1+d Shew that abed 14. 1 1 1 1 + c sabc* ..aicd(l + l + l + Ul) b c a d e dab e b a = (a + 6 + c + d)(a + 6-c-d)(a + c-6-d)(a+d-6-c). 15. Shew that 16. Shew that 17. Shew that 18. Shew that 19. Shew that 20. Shew that 1+a? 1 1 1 a -h -c -d 2 3 4 2 + x 3 4 2 3 + a; 4 3 4 + a; c d -d c a -b b a 2 h a d -c =:a;»(a; + 10). (a2 + 62+c2 + d2)a. a^ a^ + bcd 63 6' + cda c2 c* + da6 = a(6-a) a t-by-cz ay + bx cx + az ^-(a-b)*. ay + bx by-cz- ax bz + cy cx + az bz+ey cz -ax -by : (a2 + 6» + c2) {a;2 + y a + «a) (ax + 6y + cz) . EXAMPLES. 551 ai. Shew that aa a2-(6-c)2 be 62 62_(c-a)» ca c2 c2-(a-6)' a6 22. Shew that {6-c)2 (a -6)2 (a-c)2 (6-a)» (c-a)2 (6-c)2 (c_a)a (c-6)a (a -6)3 23. Shew that, if any determinant vanishes, the minors of any one row will be proportional to the minors of any other row. = {b-c) (c -a) (a-b) {a + b + c) (a^ + b^ + c^). = -2(a2 + 62 + c2-6c-ca-a6)3. 24. 25. 26. Shew that a2+l ab ac ad ba 62 + 1 be bd ca cb c3 + l cd da db dc d^ + 1 Shew that 1 1 1 = ( 67-C/3 1 a^ + a^ ab + a^ ac + ay 1 ab+ap b^ + ^ bc + py 1 ac + ay bc + ^y c^ + y^ Shew that the determinants ■ a^+b^ + c^ + d^+1. = (67 - Cj8 + ca - ay + a^- 6a)2. y X X y are all zero. 27. Shew that 28. Shew that \ c -6 -c \ a b -a \ x^-yz y^-zx z^-xy = X y z «2 -xy x^-yz y^-zx y z X y^-zx z^-xy x^-yz z X y a^ + X^ ab + \c ac-\b ab-\c 62 + X2 6c + Xa ac + X6 6c-Xa c^ + \^ = X3{XHa2 + 62 + c2)3. 29. Shew that X y a b d c w z x+w y+z a+d b+c x-w y -z a-d b-c S. A. 37 CHAPTER XXXII. Theory of Equations. 435. Any algebraical expression which contains x is called a function of x, and is denoted for brevity by / {x\ F (x), (x), or some similar symbol. The most general rational and iotegral expression [Art. 76] of the nth degree in x may be written aoX"' + Oiic"-^ + ajpz;"-* + . . . + a„, where cto* <^i> ctav do not contain x. Since all the terms of any equation can be transposed to one side, every equation of the nth. degree in x can be written in the form ttoo?" + Oia?**-^ + aao;"-* 4- . . . 4- a„ = 0, where n is any integer, and the coefficients a©, (h, (h--- do not contain x. Now any equation in x is equivalent to that obtained by dividing every one of its terms by any quantity which does not contam x ; and, if we divide the left side of the above equation by ao, the coefficient of a;**, we shall obtain the equation of the nth degree in its simplest form, namely a?" 'hpix''-^ -f ^2^**-^ + ...+;?„ = 0, where pi, p^, ps,... do not contain x, but are otherwise unrestricted. THEORY OF EQUATIONS. 553 436. If we assume the fundamental theorem* that every equation has a root real or imaginary, it is easy to prove that an equation of the wth degree has n roots. For suppose the equation to be / {x) = 0, where Since /(/r) = has a root, a^ suppose, we have /(oi) = 0, and therefore [Art. 88] f {x) must be divisible by x — a^, so that f (x) — (x — Oi) ^ (x\ where 0(a?) is an integral function of x and of the (n ~ l)th degree. Similarly, since the equation (x) = has a root, Oj suppose, we have (p (x) = (x — a^ yjr (x), where 'xjr (x) is an integral function of x of the (n — 2)th degree. Hence f(x) =:=(x- aj)(x-ai)± (x). ^ Proceeding in this way we shall find n factors o{ f(x) of the form x — ai, and we have finally /(x) = (x- ai) (x - aa). . .(x - a„)i It is now clear that Oi, a^,..., an are roots of the equation f{x) — 0', also no other value of x will make f{x) vanish, so that the equation can only have these n roots. In the above the quantities a^, a,, as,." need not be all different from one another ; but if the factors x- Oy, x — a^fX — Ot, &c. be repeated p, q, r, &c. times respectively inf(x), we must have f(x) = {x- a^y {x - a^y {x - a^f. . ., where p-\-q-{-r-\-...—n. The equation/(a;) = has in this case p roots each Oi, q roots each a^, &c., the whole number of roots being p-\-q^-r-\- ... =n. • Proofs of this fnndamental proposition have been given by Cauohy, Clifford and others: the proofs are, however, long and difficult. 37—2 554 THEORY OF EQUATIONS. 437. Relations between the roots and the coeffi- cients of an equation. We have seen that if Oj, Oj, a„... be the roots of the equation/ (a?) = ; then Hence [Art. 260] x^ + p^a^-"^ +_pjpr"-» +...+;)„ = a;«- ^1 . x^-^ + S^.x-^-^ - ...+(- If Sn, where Sr is the sum of all the products of Oi, a^, a^,... taken r together. Equating the coefficients of the different powers of x on the two sides of the above identity, we have 438. By means of the relations obtained in Art. 437, which give the values of certain symmetrical functions of the roots of an equation in terms of its coefficients, the values of many other symmetrical functions of the roots can be easily obtained without knowing the roots themselves. The following are simple examples : Ex. 1. If a, b, c be the roots of the equation a^+pa^ + qx + r^O, find the value of (i) Sa^ and (ii) Za^b\ "We have a + b + c=: -p^ be + ca + ab = q and abc = - r. Hence a* + b^ + c^={a + b + c)^-2{bc + ca + ab)=p^-2g. Also, 262ca = (&c + ca + a6)» - 2abc {a + b + c) = q^- 2pr. Ex. 2. If a, &, c,... be the roots of x^+PiX^~^+p^*~^+ ...+p^=0, find the values of Sa' and Ha^. We know that 2a = -p^, i:ab=p^ and Sa6c= - p^. Now(Sa)a=(a + 6 + c+...)2=Sa2 + 2Sa6[Art. 65]; 2a2 = (2a)'' - 22a6 =p^^ -2p^. Again 2a2 . 2a = 2a3 + 2a2&, and I>a% = 2a6 . 2o - 32a6c. THEORY OF EQUATIONS. 555 [For in 2a6 . Sa there can only be terms of the types a^b and abc ; of these the term a^b will occur once, bnt the term abc will occur three times, for we can take either a or 6 or c from Sa and multiply by 6c, ca or ab respectively from 2a6. Thus 25a6 . 2o= 2a'6 + SSaZ/c] Hence 2a»= Sa3 . 2a - 2a6 . 2a + 32a&c = {2)i» - 22)3) ( -pj) -p^ ( -^j) - Spj. 439. Theorem. If there are any n quantities Oi, Oj, Os, &c., and m be any positive integer not greater than n ; then will + Xoi . SoiCta. . .amr-i ± 'fri . Saifta. . .a^. The following relations hold good : ...[A]. To prove the first relation it is only necessary to notice that the product Xa^ . Xoi"^^ can only give rise to terms of the types Oi"* and Oa^^^^ch; also every term of either type will occur, and no term can occur more than once. Thus 2ai . Soi"^* = Soi"* + Soi^^'aa. The other relations, except the last, will be seen to be true in a similar manner. Also, the product Xoia^. . .a„^i . Soi can only give rise to terms of the types Oi^a^cis" -dm-i and Oitta-.-a^; the first of these terms can only occur "once, namely as OiO^ai. . .afn_i X tti ; the second term will, however, occur m times, for we get the term by taking any one of the m factors it contains from 2ai and multiplying this by the proper term of SctiOa-.-cw-i. Hence 556 THEORY OF EQUATIONS. From the relations [A], we have at once Sai"* = Soi*^^ . Xoi ~ Soi"*-'' . Xoifl, + Soi^^'SaiaaO, — ... ± m.taia^...am [B]. If now Oi, a,, a„ &c. be the n roots of the equation we know that Xai = — pi, Saiaa=«^a> Saia^ai — —pty &c. Hence, by substituting in [B] and transposing we have + pm.m^0 [0]. The formula [C] gives the sum of the mth powers of the roots of an equation of the Tith degree [m if n] in terms of the coefficients and the sums of lower powers of the roots. [See also Art. 471.] The sum of the mth powers of the roots of an equation can therefore be obtained from the formulae Xoi' +i?iSai» + piXon. + 3p, = 0, Xtti* + PiSoi* + p^Xa^^ + Ps • 2)ai + 47)4 = 0, If we eliminate ^Oj* and Soi from the first three equations we have Pi Pi Sjps + Soi* 1 ^1 2^a 1 Pr = Soi' + p^ Pi 3^8 1 pi 2p, 1 p. = 0. To find toy"^ we must eliminate Xai"»-\ Xal"*-^..., ^Oj from the first m equations, and we have THEORY OF EQUATIONS. 657 ^1 i>3 Pz"'Pm-i m.pm -^^Oi'" 1 pi Pi.,.prr^^ {m-l)ptr,^, 1 pi...prn-., (m-2)^^2 l...pm-A {m-S)prf^s 1 Pi 0. The coefficient of Xa^^ is a determinant of which all the elements on one side of its principal diagonal are zeros, the elements along the principal diagonal being all equal to 1 ; the determinant is therefore equal to 1. Hence Soi"* is equal to an integral function oi pi, p^, &c. If m be greater than n the relation corresponding to [C] can be very easily obtained. For, since Oi, ctj,... are roots off(x) = 0, we have n equations of the type < + jPiOi"-! + p^a^""-^ + . . . + pn = 0. Multiply by Oj*"'"^, a^*"'"^,... respectively; then we shall have n equations of the type Ch^ + ^i^i"^' + P^Oa^^ + . . . + pnOa"^ = 0. Hence, by addition, we have + j[)n5:ai"^^ = 0...[D]. By means of the relations [C] and [D], which were first given by Newton, it is easily seen that the sum of the mth powers of the roots of any equation can be expressed as a rational and integral function of the coefficients, m being any positive integer. 440. Any rational and integral symmetrical function of the roots of an equation can be expressed in terms of the coefficients by means of the relations Xax = —px, Soiaj|=j9a, ^a,ia^i — — pt, &c. Consider the symmetric functions of the third degree. 568 THEORY OF EQUATIONS. It is easily seen that Thus we have three equations to determine Soi*, Soi^Oj and XoiaaOa, and these are the only symmetrical functions of the third degree. Similarly each of the products pi\ Pi*Pi, PiPi, p^ and p^ can be expressed in terms of symmetric functions of the fourth degree, and there will be as many such equations as there are symmetric /mictions of the fourth degree. The same will hold good with respect to symmetric functions of any other degree. The sum of the suffixes of the p's will in all cases be equal to the degree of the symmetrical function. 441. A rational and integral symmetrical function of the roots of an equation can also be expressed in terms of sums of powers of the roots, and thence by Newton's Theorem in terms of the coefficients of the equation. The method will be seen from the following examples. Ex. 1. Express "La^a^ in terms of 2a/, Soj* and Sai»^+«. 'La^=a^^ + a^^-\-aj^-^ Sai«=ai« + a3«+a,«+ .'. Za^P . 2ai« = Zaj*^ + 'La^^a^^. Thus 2ajPa3 = \ {^La^^f - \ Zoj-^p. Ex. 2. Express ^a^^a^n{' in terms of the sumv of powers of the separate roots. 2a/=/Sp = ai«' + a/ + ajP+ 2ai«=/Sfg = aj« + a,«+tt8«+ 2ai'' = 5^=ai'- + a/+a3'-+ THEORY OF EQUATIONS. 669 Hence Sp.S^.Sr= ^Oj^^^ + Sa/+«fl/ Hence, from Ex. 1, "Zor^ac^a^ = Sp.Sq.Sf~ Sp.^^ . Sf — Sp^ . Sq — Sq^ . Sp + 2Sp+^4^ . The above will only hold good when jp, q, r are all different. If p = g =: r we shall have .-. 2a/a/a3P = i {Sp^-SS^p . Sp + 2>Ssp}. Transformation of Equations. 442. We now consider some cases in which an equa- tion is to be found such that its roots are connected with the roots of a given equation in some specified manner. I. To find an equation whose roots are those of a given equation with contrary signs. If the given equation be / (x) = 0, the required equa- tion will be f(—y) = 0. For, if a be any root of the given equation so that /(a) = 0, then — a will be a root of /(-y) = o. Thus if the given equation be Pi/v*" + jhx"^"^ -|-j[>aa;"-« -I- -\-pn = 0, the required equation will be i>*(- y)** +Pi i-yT-'' +i>a (- 2/)""' + +i?n = o, or ^o2/" -i>i3^' +^^**-^ - -I- (- lYpn = 0. II. To find an equation whose roots are those of a given equation each multiplied hy a given quantity. Let f(x) = be the given equation, and let c be the quantity by which each of its roots is to be multiplied. Let 2/ = CO?, or ^ = x; then /(-) = is the equation 660 THEORY OF EQUATIONS. required. For, if a be any root of / (x) = 0, so that / (a) = 0, ac will be a root of/ (^^ == 0. Thus, if the given equation be the required equation will be ^»(f)"+^'(!r+^'(fr+ +^''="' or j3oy** + Picy^~^ + PiC^f^ + 4- ^nC" = 0. The above transformation is useful for getting rid of fractional coefficients. Ex. Find the equation whose roots are the roots of x^-ia^ + ^x + ^^O each multiplied by c. The required equation is y3 _ ^y a + ^2y + ^^c» = 0. We can now choose c so that all the coefficients may be integers ; the smallest possible value of c is easily seen to be 6. III. To find an equation whose roots are those of a given equation each diminished by the same given quantity. Let / (a?) = be the given equation, and let c be the quantity by which each of its roots is to be diminished. Let y = x — c, or aj = y+c; then / (y + c) = will be the equation required. For, if a be any root of / (x) = 0, so that / (a) = 0, a — c will be a root of /(y -f c) = 0. An expeditious method of finding /(y + c) will be given later on. [Art. 472.] The chief use of above transformation is in finding approximate solutions of numerical equations ; it can also be used to obtain fi*om any given equation another equa- tion in which a particular term is absent. Ex. Find the equation whose roots are those of ar'- 3x'-9x + 5 = each diminished by c, and fuQd what c must be in order that in THEORY OF EQUATIONS. 661 the transformed equation (i) the sum of the roots, and (ii) the sum of the products two together of the roots, may be zero. The equation required is /(y+c)=0, that is (y + c)»-3(y + c)»-9(y + c) + 6=0, or y»+(3c-3)y» + (3c«-6c-9)y + c»-3c2-9c + 5=0. The sum of the roots will be zero if the ooefl&cient of y' be zero ; that is, if c = l. The sum of the products two together of the roots will be zero if the ooeflGioient of y be zeroj that is, if c'- 2c -3=0, or (c-3)(c + l)=0. IV. To find an equation whose roots are the reciprocals of the roots of a given equation. Let f{x) — be the given equation. Then the equa- tion /( -j = is satisfied by the reciprocal of any value of a which satisfies the original equation. This transformation enables us to find the sum of any negative power of the roots . of the equation f(x) = 0, for we have only to find the sum of the corresponding positive power of the roots of the equation/ ( - j = 0. 443. A reciprocal equation is one in which the reciprocal of any root is also a root. To find the conditions that an equation may be a reciprocal equation. Let the equation be p^ -\- PiOf-^ + p^""-* -^ -\-p„ = 0. Then the equation whose roots are the reciprocals of the roots of the given equation is P' S)"+ ^' ©""+ P' ©"""+ •••••• ^P" = 0- or, multiplying by a^, P9-^Pia!+pap[f+ +i3fta;" = 0. 562 THEORY OF EQUATIONS. The equation last written must be the same as the original equation, the ratio of corresponding coefficients must therefore be the same throughout. Thus Pn Pnr-i Pn^i Po' From the first and last we have p^^^p^^ so that />n= ±i?o> whence it follows that the coefficients are the same when read backwards as forwards, or else that all the coefficients read in order backwards differ in sign only from the coefficients read in order forwards. These two forms of reciprocal equations are often said to be of the first and of the second class respectively. 444. The following important properties of reciprocal equations can easily be proved. I. A reciprocal equation of the first class and of odd degree has one root equal to - 1. II. A reciprocal equation of the second class and of odd degree has one root equal to + 1. ni. A reciprocal equation of the second class and of even degree has the two roots ± 1. [These follow at once from Art. 87.] IV. After rejecting the factor corresponding to the roots given in I, II, III, we are in all cases left with a reciprocal equation of the first class and of even degree. V. The problem of solving a reciprocal equation of the first class and of even degree can, by means of the substitution x-\-x~^=y^ be reduced to that of solving an equation of half the dimensions. For the equation may be written ao (a;'* + 1) + Oi («*»-^ + a;) + . . . = 0. Divide by «" ; then a© (^* + «"•*) + , c^ + a\ a? + 6^ (vi) bc-a^,ca-b\ ab-c\ 9. If a, 6, c, d be the roots of the equation x^ +px^ + qx^ + rx + s=0, find the equation whose roots are (i) b + c + d,&o. (ii) b + c + d-2a, &q. (iii) 6« + c» + d2-a3, &c. 10. Find the equation whose roots are the cubes of the roots of the equation a:* +px^ + qx + r = 0. 446. In any equation with real coefficients imaginary roots occur in pairs. For, if a + 6V^ be a root of/ (x) = 0,x-a-b V -1 will be a factor of f(oo), and therefore [Art. 193] x^ a-\-h v^^ will also be a factor, whence it follows that a — 6 V — 1 is also a root of / (a?) = 0. Corresponding to the two roots a±h V — 1 off{x) = 0, / (a?) will have the real quadratic faxitor [{x — a)^ + b^]. 566 THEORY OF EQUATIONS. 447. In any equation with raiional coefficients quad- ratic surd roots occur in pairs. For, if a + '^b he a. root of f(x)==0, sjh being irra- tional, X — a — isjh will be a factor of / (a?), and therefore [Art. I79]a7 — a-hV^ will also be a factor of/ {x), whence it follows that a — >^h will also be a root of/ {x) = 0. Corresponding to the roots a± ^h, f (x) will have the rational quadratic factor [{x — a)^ — b}. Ex. 1. Solve the equation x* - 2x^ - 22a:2 + 62a! - 15 = 0, having given that one root is 2 + JS. Since both 2 + /^3 and 2-^JS are roots of the equation, {x-2-^B){x-2 + s/Sh that is x^-4:X + l, must be a factor of the left-hand member of the equation. Thus we have {x^ - 4x + 1) {x^ + 2x- 16) =0. Whence the roots required are 2JoJS and the roots of a;3 + 2x-15 = 0. Ex. 2. Solve the equation 2x^ - ISx^ + 46a; - 42 = 0, having given that one root is 3 + -^ - 5. Since 3 ± ^7 - 5 are roots of the equation {x-d-J~^){x~S+J~^) must be a factor of the left-hand member of the equation, which may be written {(x-3)2 + 6}{2a;-3) = 0. I 3 Whence the roots required are 3 ± a/ - 5, ^ • Ex. 3. Solve the equation a;« - 4a;5 - llx* + 4:0a^ + llx^ - 4a; - 1 = 0, having given that one root is s/^+^S. If fja + jb be a root of any equation with rational coefficients, ^a and ^Jb not being similar surds, then zk^a^Jb will all four be roots. Hence in the present case {x-J2-s/S){x-s/2 + s/^){x + ^2-jS){x + J2 + ^S), that is x4-10a;2 + l will be a factor of f{x). The equation may therefore be written (a;* - 10x3 + 1) (a;2 - 4a; - 1) = 0, whence the roots are ±(^2±^3, 2 ±^5. THEORY OF EQUATIONS. 667 Ex.4. Solve ar^-ar^- 9x2- 14a; + 8 = 0, having given that one root is -1 + ^3. «+l- v^3 is a factor of f{x); and therefore, as J (x) is rational, the rational expression of lowest degree of which a; + l-^3 is a factor, namely the expression (a; + l)*-3, must be a factor of f{x). Thus we have {(a; + l)3-3}(a;-4) = 0. Thus the roots are 4, -1+4/3, -1 + 0,4/3, -l + «2^3, where w is an imaginary cube root of unity. 448. Roots common to two equations. If the two equations f(jc) = and (f>{a)) = have one or more roots in common, / (a?) and (x) must have a common factor, which will be found by the process of Art. 98. Ex. Find the common roots of the equations ar»-3a:2- 10a; + 24 = and a;» - 6a;2 - 40ar + 192 = 0. The H. 0. F. of the left-hand members will be found to be a; - 4. Hence a; =4 gives the common root. 449. When it is known that two roots of an equation are connected by any given relation, these roots can be found. Ex. 1. Solve the cubic x' - Bx^ - 10a; + 24 = 0, having given that one root is double another. Let a and 6 be the two roots and let a = 26. Then, since a is a root of the given equation a8-3a«-10a + 24=0 (i). Also, since 6 is a root, (iy-(iy--(f)-^-«- or a3-6a5»- 40a +192 = (ii). The factor common to the left-band members of (i) and (ii) will be found to be a -4. Thus a = 4 and 6 = 2; the remaining root of the cubic is then easily found to be - 3. Ex. 2. Solve the cubic 2a;2- 15a;2 + 37a; -30=0, having given that the roots are in a. p. The sum of the roots is equal to three times the mean root, s. A. 38 568 THEORY OF EQUATIONS. a suppose. Thus 3a = — , whence a = ^. Divide /(x) by the factor 2x - 6, and the remaining roots are given by x^-5x + 6. Hence the roots are 2, - , 3. In the general case suppose that a and h are roots of the equation /(a?) = connected by the relation 6 = (a) can both be found. Ex. (i). Find the condition that the roots of x^+px^ + qx + r=zO may be (i) in Arithmetic Progression, (ii) in Geometrical Progression. Let a, h, e be the roots in order of magnitude. (i) a + 6 + c = 36; .'. 6= -|. Hence, as & is a root, we have (-iy-(-iy-*(-i)-=^ whence 2p» - 9pg + 27r = 0. (ii) a6c = 6»; .-. 6=4/^7. Hence, as 6 is a root, we have -r+p (- r)^ + q (- r)^ + r=0, whence pH=q^. 450. Commensurable roots. When the coefficients of an equation are all rational the commensurable roots can easily be found. It is at once seen that an equation with integral coefficients and with unity for the coefficient of the first term cannot have o. fractional root. For if r be a root of f{x) = 0, ^ being a fraction in its lowest terms, we have s)"+^-(Fr+ +P"=^- Multiply by 6^^; then all the terms will be integral except the first which will be fractional [for a is prime to THEORY OF EQUATIONS. 669 b and therefore a" is also prime to 6], and this is im- possible. Now, from Art. 442, ii., any equation can be trans- formed into another with integral coefficients and with unity for the coefficient of its first term ; hence, from the above, we have only to find integral roots. Now it is clear that if a be an integral root of /(a?) = 0, so that X — aia a. factor off {x), a must be a factor of the term which is independent of x. Thus if we apply the test of Art. 88 to all the factors of />„ we shall discover all the integral roots. Ex. Find the commensurable roots of ar* - 27a;' + 42j; + 8 = 0. Here the commensurable roots, if any, are factors of 8. Hence we have only to test whether any of the numbers ±8, ±4, ±2, ±1 are roots. It will be found that 4 and 2 are roots. Having found two roots the equation can be completely solved ; for we have {« - 2) (a; - 4) (x* + 6x + 1) = 0. Hence the roots of the equation are 2, 4, - 3±2;^2. EXAMPLES XLIV. 1. Solve the equation a:* + 2x8 - ig^a - 22ar + 7=0, having given that 2 + /^3 is one root. 2. Solve the equation 3a:*-23a;2 + 72x-70=0, having given that 3 + sf-^ is one root. a. One root of the equation 3aH^ - 4x* - 42a:» + 56x» + 27« - 36 = is ^2 - ijb, find the remaining roots. 4. One root of the equation 2a:«-3a;' + 6a;* + 6a;'- 27a; + 81 = is J2 + J- 1. Find the remaining roots. 5. Find the biquadratic equation with rational coefficients one root of which is J^->Jb. 6. Find the biquadratic equation with rational coefficients one root of which is J2 + J^^. 7. Shew that a:^ - 2a;2 - 2a; + 1 = and a:* - 7a;2 + 1 = Q have two roots in common. 8. Solve the equation ar*-4a:3 + llx'-14aj + 10=0 of which two roots are of the form a+pj^l and a + 2§J^. 38—2 570 THEORY OF EQUATIONS. 0. Find the condition that the roots of x^ +px^ + qx + r = may be in Barmonical Progression. 10. Find the conditions that the roots of x*+px^ + qx^+rx + 8 = may be in a. p. 11. Find the roots of the equation a?' - 3x* - 13x + 15 = 0, having given that the roots are in a. p. 1 2 . Solve the equation X* + 2a:» - 2 Ix^ - 22a; + 40 = 0, having given that the roots are in a. p. 13. Find the commensurable roots of (i) x»-7ar^ + 17a; -15 = 0, (ii) x^-x^-lSx^+Ux-4S=0, (iii) Ba^ - 26x^ + 34a; -12 = 0. 14. Solve the equation 4a;3- 32x^-3;+ 8=0, having given that the sum of two roots is zero. 15. Solve the equation a;* + 4a;'- 5x2-8a; + 6=0, having given that the sum of two roots is zero. 16. Find the condition that the sum of two roots of the equation as*+px^ + qx^ + rx + 8=:0 may be equal to zero. 17. Solve the equation a;* - 79a; + 210 = 0, having given that two of the roots are connected by the relation a =2/3 + 1. 18. Solve the equation 3a;* -32a;» + 33a; + 108=0, having given that one root is the square of another. 10. Shew that, if the roots of the equation -.iW(n — 1) _„ - a;* + np x*^-^ + — V-i,- g «""'* + ...= 0, 1 . A be in a. p., they will be obtained from -p + r -[ ^ ^ ^ l by giving to r the values 1, 3, 5,... when n is even, and the values 2, 4, 6,... when n is odd. ao. Find the condition that the four roots a, /3, 7, 8 of the equation x*+px^ + qx^ + rx + $ = may be connected (i) by the relation aj8=75, and (ii) by the relation ap + y8=0. 21. Shew that, if four of the roots of the equation ax* + bx^ + cx'^ + dx + e = 0, be connected by the relation a + /S=7 + 5, then will 4a6c-6'-8a'd=0. 22. If a, 6, Cy... be the roots of the equation X^ + J3lX"^l +^2^'»-2 +...+Pn = 0, prove that (l-a^){l-b^){l-c^)... = A*+B^+C^-BABG, where ^=i>n+i?n-3+— » ■B=P„-i+J>n-4+—l and C=p„_2+jp„_6 + .« THEORY OF EQUATIONS. 671 451. Derived functions. Let f{x) = p^-^-\-p^af'-^+p^'^-^+ +p»; then, a x + hhe put for x, we have f{x + h)=p,{x-\^ hY +i?i (a? + hy-"^ +i?a (a? + hy-^ + . . . -^pn. If now (x + hy, (x-\-hy-\ &c. be expanded by the Bi- nomial Theorem, and the result arranged according to powers of h, we shall have f(x + h) =f(x) + h {npox'^-^ + {n-l) p^x + {n-1)p^^-^+,.....+Pr^,} + higher powers of h. This expansion is usually written in the form n—% [The reader who is acquainted with the Differential Calculus will see that the expansion of / (a; + A) in powers of h is an example of Taylor s Theorem.] It will be seen at once that /' {x) is obtained by multi- plying every term of / {x) by the index of the power of x it contains and then diminishing that index by unity. It will also be easily seen that /" (x) can be obtained from /' {x) in a similar manner, and so for f" (x), &c. in succession. We shall however in what follows only be concerned with /' (x). Def. The function /' (x) is called the first derived function off{x), the function /" (a?) is called the second derived function off(x), and so on. Thus if / (x) =p^ +i>ia;3 ^p^^ +jp^ +p^, /' (x) = 4poaJ» + ^p^x^ + 2p^ +i?3 , /" (x) = 12p^x-^ + Qp^x + 2p^ , 672 THEORY OF EQUATIONS. 452. Theorem. If (so) he any rational and integral function of x and f {x) he its first derived function, then wiU f'i^.)=m^m^fSEL^'. •^ ^ ^ x — Oi x — a^ x — a^ where Oi, aj, a,, are the n roots, real or imaginary, of the equation f(x) = 0. We know that / (a?) = i)o (a; - Oi) (a? - aa) (a? - tts) Hence f(x-\-h) =po (x — ai-\-h)(x — a^ + h)(x-a3-\- h) The coefficient of h in the expression on the right is by Art. 260 equal to p^ x (sum of all the products w — 1 together of the n quantities x — ai, x—a^, , x — a^. But f(x + h)=f(x) + hf (x) 4- higher powers of h. Hence f'(x) = poX(sxim of all the products n — 1 together of the n quantities x — ai,x—a^, , x — an)- Hence /'(^) = /M+/W+ •'^^ x — a^ X — a^ In the above the quantities Oi, a^, , «« need not be all different from one another ; but if ai occur r times, and a^ occur s times, &c., we shall have /'(^)=^Z(?) +*/(?)+ •^ x — ai aj — Oa 453. Equal Roots. We have seen in the preceding Article that if ai, aj, , a„ be the n roots of the equa- tion / (x) = 0, so that / (a?) = po(x — a^) (x — a^) (x — an)] then will f'(x)=pQx(s\im of all the products n—1 together of the n quantities x — a^,x — a^, ,x — a„). Now, if any root, for example Oi, is not repeated, so that the factor x — a^ occurs only once in f (x), then the factor X — ai will be left out of one of the terms of /' (x) but will occur in all the others ; whence it follows that /'(a;) is not divisible hy x — a^. Thus a root of f{x) = which is not repeated is not a root of f (x) = 0. THEORY OF EQUATIONS. 573 If, however, r roots of the equation f(ixi)=^0 are equal to Oi, the factor a? — Oj will occur r times in f (oc), and therefore as — Oi will occur at least r — 1 times in every term of/' (x), for every term of/' (x) is formed from / (x) by omitting one of its factors. Hence a root of f{x) = which is repeated r times is also a root of f {x) = re- peated r—1 times. We can therefore find whether the equation f(x) = has any equal roots, by finding the H. c. F. of / (x) and /' (x) ; and if / (x) be divided by this H. c. F. the quotient when equated to zero will be an equation whose roots are the different roots of / (x) = 0, but with each root occur- ring only once. Ex. 1. Find the equal roots of the equation «* - 5x3 - 9a;a + 81a; - 108 = 0. Here f{x)^x*- 5x^ - 9x» + 81a; - 108, /' (x) = 4ar» - 15a;2 - 18« + 81. The H.O.F. of /(«) and/' (x) will be found to be x^-Qx + 9, that i8(x-3)». Since {x - 3)^ is a factor of /' (x), (x - 3)" will be a factor of / (x), and it will be found that / (x) = (x - 3)' (x + 4). Thus the roots of the given equation are 3, 3, 3, - 4. Ex. 2. Shew that in any cubic equation a multiple root must be commensurable. This follows from Art. 445 and 446, and from the fact that a cubic equation can only have three roots. Ex. 3. Solve the equation x» - 15x3 + i0a;3 + 60x - 72 = by testing for equal roots. '. Here / (x) = x« - 15x» + lOx^ + 60x - 72 ; /' (x) = 5x4 - 45x2 + 20x + 60. ; It will be found that the h.c.p. of /(x) and/' (x) is x3-x2-8x + 12. If now we divide /(x) by x3-x^-8x + 12 the quotient will be ! x'+x-6, and the roots of x'^ + x- 6 = are 2 and -3. Thus the given equation has only two different roots, namely 2 and - 3 ; and it will be found that /(x) = (x - 2)3 (x + 3)^. Thus the roots of/(x) = are 2, 2, 2, -3, -3. 574 THEORY OF EQUATIONS. 454. Continuity of any ra4:lonal and integral function of x. Let po^" + p^x^~^ + _p2^"~2 + -\-pn be any rational and integral function of x arranged according to descend- ing powers of x. Then each term will be finite provided x is finite ; and therefore, as the number of the terms is finite, the sum of them all will be finite for any finite value of x. It can be easily proved that the first (or any other term) can be made to exceed the sum of all the terms which follow it by giving to a; a value sufficiently great ; and also that the last (or any other term) can be made to exceed the sum of all the terms which precede it by giving to a? a value sufficiently small. For let k be the greatest of the coefficients; then l>ix'^i + ... +p^ "" fc {x«-i + ... 1) ^ kx^ ^ k ^' "■>' Now ^ (a; - 1) can be made as great as we please by sufficiently K increasing x. We can prove in a similar manner that Pnl{Pn-i^+ "• +PoX*^) can be made as great as we please by sufficiently diminishing x. Now suppose that x is changed into x-\-h\ then we shall have f(x + h) -fix) = hfix) + ^f"{^) + >1 where the coefficients /'(a?), f"{oc), &c. of the different powers of h are finite quantities. Then by the above, the first term on the right (or if this term vanishes for any particular value of x, then the first term on the right which does not vanish for that value) will exceed the sum of all the terms which follow it, provided h be taken small enough. But the first term will itself become indefinitely small when h is indefinitely small. Therefore /{x-^-h)—/ (x) can be made as small as we please by taking h sufficiently small. This shews that as X changes from any value a to another value b THEORY OF EQUATIONS. 575 f {x) will change gradually and without any interruption from f (a) to fib), so that f {x) must pass once at least through every value intermediate to / (a) and / (6). It must be noticed that it is not proved that f {x) always increases or always diminishes from / (a) to / (6), it may be sometimes increasing and sometimes diminish- ing as a; is changed from a to 6; what has been proved is that there is no sudden change in the value of / {x). 455. Theorem. If f (a) and /(yS) have contrary signs one root at least of the equation f (x) = must lie be- tween a and /3. For since /(a;) changes continuously from /(a) to/(/3), it must pass once at least through any value intermediate to /(a) and/()S); it therefore follows that for at least one value of X intermediate to a and ^ it must pass through the value zero, which is intermediate to / (a) and / (yS) since /(a) and /(/S) are of contrary sign. Thus the equation f(x) = is satisfied by at least one value of x which lies between a and /3. For example, if /(a:) = jr3- 4a; + 2, then /(1) = -1 and / (2) = 2. Hence one root of the equation x' - 4a: + 2=0 lies between 1 and 2. 456. Theorem. An equation of an odd degree has at least one real root Let the equation hef(x) = 0, where f(x)=^x^+^ -f jhx"^ + +p^+i. Then /(+x) is positive, f(0) = p^+^, and /(-oo)is negative. Thus there must in all cases be one real root, which is positive or negative according as ^271+1 is negative or positive. 457. Theorem. An equation of even degree, the coefficient of whose first term is unity and whose last term is negative, has at least two real roots which are of con- trary signs. 676 THEORY OF EQUATIONS. Let (c^ + p^aP^-^ + + ;)!m = be the equation, pan being negative. Then /(+ oo ) is positive, /(O) =|)an, and /(— oo ) is positive. Hence, as ^an is negative, there must be one real root at least between + oo and 0, and also one at least between and — 00 . 458. The following is a very important example. To prove that if a, &, c, /, ^, h he all real the roots of the equation {x-a){x-h){x-c)-f^{x-a)-g''{x-h)-h''{x-c)-2fgh=0, will always he real. We may suppose without loss of generality that a->h>'C, Write the equation in the form {x-a){{x^h){x-c)-P]-{9''{x-h)-^h^{x-e) + 2fgh}=Q. By substituting + oo , 6, c, - oo respectively for x in (x-h){x-4)-f\ we see that the roots of the equation (x - 6) (« - e) -P=0 are always real ; and if a and /S be these roots, where a>/9, then a>6>c>/S. Now substitute + oo , a, /S, -co for x in the left-hand member of the cubic equation, and we shall have respectively the following results + <». - {gsl^-b + hja - c}9, +{gJh-§+hJc-^}\ - oo. Hence there is one root of the cubic between + od and a, one root between a and /9, and one root between /3 and - oo . If, however, a=/3 the above proof fails ; but if o=/3, then {x -h){x- c) -f\ must be a perfect square, whence it follows that b=c and/=:0. The cubic equation in this case becomes (a? - o) (x - &)« - (5f2+ Zi3) (x - 6) =0, the roots of which are at once seen to be all real. If a be a root of the cubic equation itself, there will be another real root less than ^. Hence aU the roots of the cubic must be real, for the equation cannot have one imaginary root. The cubic equation considered above is of great importance in Sohd Geometry, and is called the Discriminating Cubic. 459. Theorem. Iff(a) and f(0) are of contrary signs, then an odd number of roots of f {x) — lie between a and ^ ; also iff (a) and f (fi) are of the same sign, then THEORY OF EQUATIONS. 577 no roots or an even number of roots of /(x)^0 lie between a and yS. Let a, by c, , k be all the roots of the equation f(is) = which lie between a and /S ; then / (at) = (sp — a) (w — b) (x — c) (a? — k)(^ {x\ where (x) is the product of quadratic factors (correspond- ing to pairs of imaginary roots) which can never change sign, and of real factors which do not change sign while x lies between a and yS. Then /(a) = (a-a)(a-6)(a-c) (a-k)(a\ and /(/S)=:()8-a)(ye-6)(;5-c) {^-k){P). Now, supposing a > /3 all the factors a — a,a — bf , a — k are positive; and all the factors ^ — a, ^ — b, , fi — k are negative ; also (a) and <^ (/3) have the same sign. Therefore if/ (a) and/(/3) have contrary signs there must be an odd number of the roots a, 6, c, , k. Also, if / (a) and / (/5) have the same sign there must be no such roots or an evert number of them. 460. Rollers Theorem. A real root of the equation f'(x) — lies between every adjacent two of the real roots of the equation f{x) = 0. Let the real roots of f(x) = 0, arranged in descending order of magnitude, be a, 6, c, . . ., k. Then f(x) = (x-a)(x'- b). . ,(x -k)(x) Now all the terms on the right except the first contain the factor x — a, and that term is Hence f{a) = (a-b) (a - c)...(a-A;)<^ (a). So f{b)=Q>-a){b-cy..Q>-k){b\ f{c)^{o-a){c-b)...{c-k)(c\ Now (c), &;c. have all the same sign. Hence as a > 6 > c. . ., the signs of /' (a), /' (6), f (c), &c. are alternately positive and negative. Hence there is at least one root of /' {x) — between a and 6, one root between b and c, &c. 461. Descartes' Rule of Signs. In any equation f(x) — the number of real positive roots cannot exceed the number of changes in the signs of the coefficients of the terms in f(x), and the numiber of real negative roots cannot exceed the number of changes in the signs of the coeffixyients 0/ /(.-'»)■ We shall first shew that if any pol3rDomial be multi- plied by a factor x — a, where a is positive, there will be at least one more change in the product than in the original polynomial. Suppose that the signs of any polynomial succeed each other in the order +H \--\ h— , in which there are five changes of sign. Then writing only the signs which occur we shall have + + - + + + - + - + + - + + + - Now we cannot write down the second partial product for we do not know that all the possible terms in the THEORY OF EQUATIONS. 579 polynomial are present ; but whenever there is a change of sign in the first partial product it is clear that if there is in the second row any term of the same degree in x, so that it would be put under this term which has the changed sign, it must arise from the multiplication of the next preceding term so that the two terms would have the same sign. Thus whenever there is a change of sign in the first partial product that sign will be retained in the addition of the two lines of partial products. The number of changes of sign, exclusive of the additional one which must be added at the end, cannot therefore be diminished. Hence the product of any poljrnomial by the factor x — a will contain at least one more change of sign than there are in the original polynomial. If then we suppose the product of all the factors corresponding to negative and imaginary roots to be first formed, one more change of sign at least is introduced by multipl3dng by the factor corresponding to each positive root. Therefore the equation f(x) = cannot have more positive roots than there are changes of sign in the coefficients of the terms in f{x). The second part of the theorem follows at once from the first, for the positive roots of /(— a;) = are the nega- tive roots of f{x) = 0. The above proof may be made clearer by taking as a definite example the multiplication of «' + 2x^ - ar* + 4a;3 + 3a; _ l by «-l. The signs of the two lines of partial products will be + + - + + - - - + - ~ + + - + - + In the third line the only signs written down are those under the changes in the first line, which changes are all retained in the final product. Hence no matter what has occurred in the intervals the number of changes (exclusive of the one at the end) cannot be diminished. 462. Descartes' Rule of Signs only gives a superior limit to the number of real roots of an equation, but does 680 THEORY OF EQUATIONS. ^ not determine the actual number of real roots. The number of the real roots of any equation with numerical coefficients can be found by means of Sturm's Theorem. Before considering Sturm's Theorem we shall shew how to find algebraical solutions of cubic and biquadratic (quartic) equations in their most general forms. Abel has proved that an algebraical solution, that is a solution by radicals, of a general equation of higher degree than the fourth cannot be found, although particular forms of such equations can be solved, for example any reciprocal equation of the fifth degree can always be solved. EXAMPLES XLV. 1. Solve the following equations each of which has equal roots : (i) 4a;2^12a;3-15a;-4=0, (ii) aj*-6a;3 + 13a;»-24x + 36=0, (iii) 16a;4-24a;» + 16x~3 = 0, (iv) 2x* - 23x3 + 84a;3 _ 80a; - 64 = 0. 2. Find the condition that the equation ax^ + 3bx^ + Scx+d=0 may have two equal roots. 3. Shew that, if the equation ax^ + 36a;' + 3ca; + d= have two equal roots, they are each equal to 1 be -ad 2 ac-b^* 4. Shew that the roots of the equation x-a' x-b' x-c' x-k' are all real. 6. Shew that all the roots of the equation a' 62 c» , + - + +...=w + n=a; z-a «- /S X- 7 are real. THEORY OF EQUATIONS. 581 0. It a^y a^y a^, ...f a^^he in descending order of magnitude, and if b be positive, prove that the roots of the equation will all be real, and find their positions. 7. Prove that if a, &, c, d be unequal positive quantities, the roots of the equation X X z , ^ + r+ — +x + d=0 x~a x-b x-e will all be real ; and that, if the roots be a, /S, y, 8, then will (a-a)(a-i8)(a-7)(a-5)'^(6-a)(6-/S)(6-7)(&-5) C8 = 0. ^(c-a)(c-^)(c-7)(c-5) a. Form the equation whose roots are the values of pu + jw"^, where w is a fifth root of unity, and shew that tlie equation is :fi-5pqx^ + Bp^q^x -p^-q^—0, O. If o, /S, 7, 3 be the roots of the equation a;^ + 4^3?' + Cgx' + 4rx + s = form the equations whose roots are (i) a/S + 75, a7 + jS5, a5 + /S7. (ii) (a+/3)(7 + 5), (a + 7)(|3 + 5), (a + 5)(/3 + 7). 10. If a, /S, 7, 5 be the roots of the equation it^ + 4pa;8 + 6ga;' + 4ra: + « = form the equation whose roots are (a + /S_7-5)a, (a-/S + 7-5)2, (a - /S -- 7 + 5)'. 11. If Oj, a,, a^ be the roots of shew that Za^a^ — '2p-^pg - 'p^p^ + J^aPs • 682 THEOEY OP EQUATIONS. CtJBic Equations. 463. The most general form of a cubic equation is a? + aa? + 6a; + c = 0. We have however seen [Art. 442, iii.] that by in- creasing each root by ^ , the equation will take the simpler form ^'" + pa? + g = 0. We shall therefore suppose that the equation has already been reduced to this simplified form. 464. To solve the cubic equation a^ -\-px-\-q = 0. The solution is at once obtained by comparing the equation with a? - Sabx + a» + 6» = 0, i.e. (a; 4- a + 6) (a? + wa + (o^b) (x + (o^a -\- mb) = 0, where o) is an imaginary cube root of unity [Art. 139]. Thus the roots required are — a — b, — coa — oy^b, — a>^a — ooh, where a and b have to be determined from the equations p = ^Sab, q=^a* + b\ Whence a* and 6' are given by |iV(M?))- 465. The foregoing solution is a slight modification of that called Garden's solution. It is a complete algebraical solution of the equation and the values found for x would satisfy the given equation identically. If, however, nu- merical values be given to p and q, the numerical values of a and b cannot be found when j; + 9^ is negative, for we THEORY OF EQUATIONS. 583 cannot reduce an expression of the form (3 + 5\/ — l)i for example, to the form a + /SV — 1. Thus when p and q are numerical quantities such that ^ + 9I7 is negative, Garden's solution altogether fails to give a numerical result. This case is called the 'irreducible case/ and we shall see further on [Art. 467, Ex. 3] that when p^ Q* , . ^ + ^ is negative all the roots of the cubic are real. It should also be noted that in any case the approxi- mate values of the real roots of a cubic can be obtained much more easily by Horner's general process [Art. 475] than by Garden's solution. Ex. Solve the cubic equation x* + 4x - 6 = 0. Comparing with «» - Sabx + a' + 6* = 0, we have -3a6=4 and a' + 6*=-6, whence a and b are given by {-I4./2793}*. The approximate values of a and 6 can therefore be found, and then the roots are - a - 6, -wa- 0)^6, - w^a - w6. In this example the solution can be obtained in a very simple manner. For, using the test given in Art. 449 for commensurable roots, we are led to find that 1 is a commensurable root, and writing the equation in the form {x-1) (a;^ + a; + 6)=0, the roots are at once seen to be 1, -i(liV^^). Biquadratic Equations. 466. Several methods of solution of a biquadratic equation have been given. In all of them the solution is shewn to follow from the solution of a cubic equation. IThe simplest method of solution is that due to Ferrai-i. § s. A. 39 584 THEORY OF EQUATIONS. To solve the equation a^ + poc^ + qa^ ■\-rx-\- s=0. Ferrari's Solution. Add {ax-\-^y to both sides of the equation; then a^^-px''^{q-\-a')a?-^{r-\-2a^)x-\-s-¥^ = {oLX-\-^)\ Now the left-hand member will be a perfect square, namely f a?* + ^ a? + X j , provided 2X + ^ = g + a», ;)\ = r-l-2a/9 and X» = 5 + ^^ Eliminating a and ^S, we have a cubic equation to determine \, namely 4 (X» -.s){^X^^-q\- {p\ - ry = 0. One root of this cubic equation is always real, and if this root be found the values of a and yS are determined. We then have (x'-^^x-^xj = (ax + l3)\ whence ai'+^x -{-\ ±(ouc + ^) = 0, where a, /8 and \ are known. Thus the biquadratic equation can be completely solved. Ex. Solve the equation Add (cur + /S)* to both sides ; then ar* + 6x3 + (14 + a2) a;« + (22 + 2a^) a; + 6 + /Sa = (aa; + /3)2. The left-hand member is the square of a;^ + 3a; + X provider? 9 + 2\=14 + a2, 6X = 22 + 2aj3 and \^=5 + ^\ Whence (X'-S) (2\- 6)- (3\- 11)2 = 0; X8-7X» + 28X-48 = 0. The real root of the oubio is 3. THEORY OF EQUATIONS. 585 Then, taking \=3, we have a^=l, 2aj8= -4, /S2=4. Hence {x^ + 3a; + 3)' = (a; - 2)», whence we obtain the roots - 2 ± «y3, - 1 ±2^-1. Sturm's Theorem. 467. Sturm's Theorem. Let /(a;) = be an equa- tion cleared of equal roots, and let /i (x) be the first derived function of f (x). Let the process of finding the highest common factor of / (x) and /i (x) be performed with the modification that the sign of every remainder is changed before using it as a divisor, and let the operation be con- tinued until a remainder is arrived at which does not contain x (this will always happen since f(x) = has no equal roots and therefore f(x) and fi (x) have no common measure in a?), and change also the sign of this last remainder. Let /a (a?), fs(x\..., fm(^) be the series of modified remainders so obtained, of which the last, fm (x), does not contain x. Then the number of real roots of the equation f(x) = between a and yS, [/8 > a] is equal to the excess of the num- ber of changes of sign in the series f(x), f (x) , f (x), . . . , /m (^) when x = a over the nu/mber of changes of sign when x = 0. For, let qi, g',,..., qm-i be the successive quotients; then we have the series of identities f,(x) = qsfi(x)-f^(x), /m-a (a?) = qm-ifm^i ((c) -fm («?)• Now (i) it is clear that no two consecutive functions n vanish for the same value of x, for in that case all the iceeding functions, including fmi^c), would vanish for ,hat value of (x) ; and, (ii) it is also clear that when any 89— S 686 THEORY OF EQUATIONS. one of the functions except /(a;) vanishes, the two adjacent functions will have contrary signs. It follows from (i) and (ii) that so long as the increasing value of X does not make f{x) itself vanish, that is unless we pass through a real root of the equation f{x) = 0, there can he no alteration in the number of changes of sign in the series of Sturm's functions ; for no function will change sign unless it passes through a zero value, and when this is the case for any function, since the two adjacent func- tions have opposite signs, there must be one and only one change in the group of three. Next suppose that a is a real root of the equation /(a7) = 0. Then /(a + X) =/(a) + X/' (a) + &c. ; and as f{a) = 0, the sign of the series on the right will, if \ be very small, be the same as the sign of + >f' {a). Hence, however small \ may be, the sign of f{a — X) must be opposite to that of /' (a), and the sign of f(a + X) must be the same as the sign of /' (a). Thus as X increases through a real root of the equation f(x) = 0, the series of Sturm's functions will lose one change of sign. Since we have proved that as x increases the series of Sturm's functions never lose or gain a change of sign except when x passes through a real root of the equation f(x) — 0, in which case one change of sign is always lost, it follows that the excess of the number of changes of sign when x = a over the number of changes when x = ^ must be equal to the number of real roots of the equation which lie between a and ff. To find the total number of real roots of an equation we must substitute — oo and + oo in Sturm's functions ; then the excess of the number of changes of sign in the series in the former case over that in the latter will give the whole number of real roots, Ex. 1. Find the number of the real roots of the equation Here f{x)=x* + 4^-^x- 13, /i(x) = 4(a;3 + 3x2-l). THEORY OF EQUATIONS. 687 N.B. We may clearly multiply or divide by positive numerical quantities as in the ordinary process for finding h.c.f. It will be found thai f^{x)=x^ + x + 4., /g(a;) = 2.r + 3, f,{x)=-19. Substitute -oo , 0, +qo in the above functions, and the series of signs will be + - + --; -- + + -; + + + + -. Thus there is one real root between - oo and 0, and one real root between and + oo . Ex. 2. Find the number and the position of the real roots of the equation a:" - 5a; + 1 = 0. Here f{x)=x^-6x+lf and f^{x) = 5(^-1). It will be found that Mx) = ^x-1, /,(a;)=+255. The following are the series of signs corresponding to the values of X written in the same line -2, - + - + -1, + - + 0, + - - + 1, - + + 2, + + + + Hence there is one real negative root between - 2 and - 1, one positive root between and 1 and another between 1 and 2, the remaining two roots being imaginary. Ex. 3. Find the condition that all the roots of the equation afi+px+q=0 may be real. f{x)=x^+px + q, f^(x) = 3x^+p. The other functions will be found to be f^{x)=-2px-^3q, Mx)=-{27q^ + ip^). 588 THEORY OF EQUATIONS. n The signs for - oo and + oo are -, +, +2p, -(27g« + 4p»), and +, +, -2p, - {27 q^ + ip^). In order that the roots may be all real, it is necessary and sufficient that there shall be three changes of sign in the first line and none in the second, the conditions for which are that p and 27g2^4^3 must both be negative, the second of which implies the first. 468. Although Sturm's Theorem completely solves the problem of determining the number and the position of the real roots of an equation, it is often a very laborious process. In some cases the position of the real roots can be determined without difficulty by actual substitution; and sometimes the necessity for using Sturm's Theorem can be obviated by some special device. Ex. 1. Find the number and position of the real roots of the equation a^_41a;2 + 40a. + 126 = 0. Substitute in f{x) the values 1, 2, 3, 4, 5, 6 in succession, and the signs will be +,+,-, -,-,+. Hence there is one root (at least) between 2 and 3, and one (at least) between 5 and 6 ; but by Descartes' Rule of Signs there cannot be more than two positive roots. Hence there are two positive roots which lie between 2 and 3 and between 6 and 6 respectively. We can find in a similar manner that there are two negative roots which lie between - 1 and - 2 and between - 6 and - 7 respectively. Ex. 2. Find the number and position of the real roots of the equation x*-14a;» + 16a; + 9=0. In this case wo should easily find the two negative roots which lie between and - 1 and between - 4 and - 6 respectively. The positive roots would, however, probably escape notice (unless Sturm's Theorem were used) as they both lie between 2 and 3 ; it will in fact be found that/ (2) is +,/(2i) is -, and/ (3) is +. Ex. 3. Find in any manner the number and position of the real roots of the equation a:« - 6a:» - 7*^ + 8a; + 20 = 0. By Descartes' Rule of Signs we see by inspection that there cannot be more than two positive roots and there cannot be m<>re than two negative roots. Now /(I) is +, /(2) is -; thus one real root lies between 1 and 2. Since / (co ) is + , there must be another positive root which is . easily found to lie between 5 and 6. THEORY OF EQUATIONS. 589 Change x into - x, then the negative roots of the given equation are positive roots of a:« + 5x»-7a;2-8aj + 20 = 0. Now f{x) must clearly be positive for all positive values between and 1 ; and if a; > 1, / (a;) >6x*- 15x2 + 20, which is always positive since 4x6x20-15«>0. Hence there can be no real negative roots. 469. Equation of DifTerences. If /(a?) = be any equation of which a and /3 are any two roots, and if y = OL — P, we shall obtain an equation in y whose roots are the differences of the roots of the given equation by eliminating /3 from the equations /(y + /8) = and/(y8) = 0. Since a — yS and y8 — a will both be roots of this equation, it follows that the equation in y will only contain even powers of y ; and it is easily seen that the equation in y^ has as many positive roots as there are pairs of real roots of the equation f{x) = 0, and that the equation in y'^ has all its roots real and positive if the roots of the equation f{x) = are all real. In the case of the cubic a^-{-px + q = the equation in y^ can be found more easily as under. To find the equation whose roots are the differences of the roots of the cubic «* +px + q = 0. Let y^ — {a-^)\ where a, /3, y are the roots of tne cubic. Then y«=(a + /S)2-4a/3 = r' + ^j .-. y«-3/ay+4g=0. Also 7*+i>7 + 3 = 0. Hence y=Sql{y^+p)\ .-. 27q^ + Spq{y^+p)^ + q{y^+p)^ = 0, y« + 6py* + 9pY + 4i>' -^ 27q^ = 0, which is the equation required. By Descartes' Rule of Signs the above cubic in y^ cannot have three positive roots unless p and Ap^ + 27q^ are both negative Also 590 THEORY OF EQUATIONS. ^ if ip^ + 27q^ be negative the equation in y^ will have three positive roots which are separated by +ao, -2p, -p, 0. Thus a necessary and sufficient condition that the cubic a^ +px + q = may have all its roots real is that 42)» + 27g' may be negative. 470. We shall conclude by shewing how to find the approximate values of the real roots of any equation. This can be done in various ways; we shall, however, only give Horner's method. We must first give the explan- ations of the separate processes which are employed. 471. Synthetic Division. Suppose that when /(x) = a^/c'^ + Oia?**-! + a^'^-^ + . . . + On is divided hy x — X the quotient is Q = 6oa;"-^ + M«-» + 62^"-' + . . . + hn-i, and that the remainder is R, where R does not contain x Then f(x) = Qx(a!-X) + E. But Q X (a? — X) + jR is at once seen to be 60^" + (61 - X60) x""-' + (b, - \h,) x^-^ + . . . Equating coefficients of the dififerent powers of x mf(x) and in the expression last written, we have bo = ao, hi — \bo = (ii, bfi — Xbi = a»i, br^i - \bn^ = a„_i , R — Xbn-^ = an. From the above relations it will be seen that the values of 60, 61, 62, (fee. can be obtained at once by the process indicated below : Oq Oi a^ as an-i an Xbff Xbi X63 Xbn-i Xbn-i bo bi 6, 63 bn-i R First 60 = «o; multiply &<, by X and add to Oi, the sum ii THEORY OF EQUATIONS. 591 61; multiply 61 by \ and add to a^, the sum is 63; proceed in this way to the end. Ex. Find the quotient and the remainder when is divided by a; - 2. l_6 + 2 + 15 + 04- 7 2-8-12 + 6 + 12 1-4-6+ 3 + 6 + 19 Thus the required quotient is x4-4a;»-6x2 + 3a; + 6, the remainder being 19. The -above process is called the method of Synthetic Division. The method can easily be extended to the case when the divisor is a multinomial expression, but this extension is not needed for our present purposes. 472. The actual values of &„, \, 63, &o. in terms of a^, a^, a^, &o. and X can be at once written down ; they are &o = ^o» 6i=ai + Xao> 62=^ + ^% + ^X» and i2 = a^+Xa„_i+...=/(X). Thus ^^=aoa;'»-i + (ai + Xao)a;»-»+... From the above we can obtain the formula of Art. 439. For, if a, 6, c, ... be the roots of the equation /(a;) =0; then x-a x-b = {agX*-^ + («i + ««o) a;"~" + (oa + aoi + a^do) a;*"* + ... } + { a^«-i + («! + 6ao) ar^-a + (aj + feoi + b^a^) x^-^ + . . . } + ... = na^x^^^ + (nttj + ao2a) a:**"' + (noa + ajSa + a^Za^) ar"-' + . . . But /' {x) = naoxn-i + (n - 1) ajX^-^ + (n - 2) aaa;»-^ + .„ 592 THEORY OF EQUATIONS. Equating the coefficients of like powers in the two expansions, we have naQ=naQj (n - 1) Oj = noj + ttflSa, (n - 2) a, = na^ + aj2a + aoZa*, Whence the required result follows at once. 473. We have already seen [Art. 442, ill.] that in order to dimmish each of the roots of the equation /(ai) = by \, we have only to substitute y + X for a; in /(a?). Let the equation whose roots are those of aoa;« + Oja;'*-^ + aaa;'*-^ + . . . + an = 0, each diminished by \, be boy"" + &i2/"-^ + %"-' + . . . + 6„ = 0. Then, since y — x — \ the last equation is equivalent to h, {x - \Y + h^{x- xy^^ + . . . + fe„-i (a; - X) + 6n = 0. The equation last written must be identical with f(x) — 0. Hence we have identically f{x) = h,{x-Xf + \ {x - X)*^! + ... + 6n-i («-X) + 6n. From the form of the right-hand member of the above identity, it follows that if we divide f{x) by a; — \, and then divide the quotient by {x — \), and so on, the suc- cessive remainders will be the quantities 6n> &n-i»"»» ^i> ^o- Ex. 1. Find the equation whose roots are those of a;*-2a58 + 3x-6=0, each diminished by 2. Using the method of Art. 469 to perform the successive divisions, the whole operation is indicated below, the successive remainders being printed in black type. 1_2+ 0+ 3-5 2 6 1 3 + 1 2 4 8 1 2 4 11 2 8 1 4 2 12 1 6 THEORY OF EQUATIONS. 593 The first division gives the quotient «*+3 with remainder 1; the second division gives the quotient 05^^ + 20; + 4 with remainder 11; the third division gives the quotient x + A with remainder 12, and the last division gives the quotient 1, and remainder 6, Ex. 2. Find the equation whose roots are those of x»-x^-x + A=0, each increased by 3. The divisor is here a; + 3, and the work is as under. 1- i_ 1+ 4 i_ i_ 1 + 4 - 3 + 12-33 i_ 4 + 111^29 " ^ + ^^ 1-10 1- 7 + 32 - 3 1-10 Thus the transformed equation is ar>-10a;2 + 32a; -29 = 0. We shall in future write the operation as on the right, the multi- plication and addition being performed mentally, and the result only being written down. 474. In order to multiply all the roots of the equation a^ + Oia?**-^ + a^''*^ + ... -f On = by ten, we must substitute :^ for a? in its left-hand member. If we then multiply throughout by 10**, the transformed equation will be Ooa?** + lOoia?**-! + Wa^"^ + ... + lO'^a* = 0. Thus in an equation with numerical coefficients the roots will be multiplied by 10 by affixing one nought to the coefficient of «"~^, two noughts to the coefficient of x^~\ and so on. For example, the equation whose roots are those of X*- 2*34. 5a; + 8=0, each multiplied by ten, is X* - 20a:» + 5000a? + 80000 = 0. 475. Homer's method of approximating to the real roots of equations with numerical coefficients. Having found (by trial or by Sturm's Theorem) two consecutive integers between which a real positive root of 694 THEORY OF EQUATIONS. the given equation must lie, the first step is to diminish all the roots of the given equation by the smaller of those integers. Then, by supposition, the transformed equation will have a root between and 1. We now multiply all the roots of the last equation by 10 by the process of Art. 472, so that this new equation has by supposition a root between and 10; now find by trial between what two integers less than 10 the root lies, and diminish the roots of the equation by the smaller of these integers. Then again multiply the roots by 10, and continue the process until the required degree of accuracy is attained. After the roots of the given equation have been di- minished by the integral part of the required root, the roots are multiplied by 10 in order to avoid decimals in the work, the next integral root found must therefore originally have been so many tenths. After again multi- plying the roots by ten, the next integral root found corresponds to hundredths in the original equation; and so on. By the above process it is clear that we are con- tinually approximating to the root sought; care must, however, be taken that we do not pass beyond the root, which would be shewn by the change in sign of the constant term. The negative roots can be found approximately in a similar manner after changing x into — x. Ex. 1. Find to two places of decimals, the positive root of the equation «»-3a:-4=0. There can only be one positive root, and by trial this must lie between 2 and 3. First diminish the roots by 2, and the transformed equation will be found to be «*+6a;2 + 9a;-2 = 0. Multiply the roots by 10 and we have the equation «« + 60a;* + 900ar- 2000 = 0, which will be found to have a root between 1 and 2. Diminish the roots of this last equation by 1, and the transformed equation will be a;" + 630;''+ 1023a; -1039 = 0. Multiply the roots of this equation by 10, and the resulting equation will be found to have a root between 9 and 10. Diminish the roots of the last equation by 9, and the resulting equation is a;3 + 657a;8 + 113883a; -66641 = 0, which could be used to obtain a more accurate result. THEORY OF EQUATIONS. 595 The work is written as under, lines being drawn to indicate the completion of each stage of the process. -3 -1 -4 ( 2-19.„ 2 H -2000 4 1 60 900 " 961 -1039000 61 62 102300 108051 - 66541 630 639 648 113883 657 Ex. 2. Find the cube root of 30. We have to find the positive root of the equation x* - 30 : proceed as under 10 -30 ( 3-107 :0. We 8 9 - 3000 6 1 2700 90 2791 1 91 28830000 92 - 209000000 9300 28895149 9307 1 28960347 6733957 9321 It will be seen that after two or three multiplications of the roots by 10, the numbers in the two last columns will become very much greater than in the others; a contracted process can then be em- ployed, namely, instead of afl&xing one, two, three, &o. zeros to the coefficients after the first in order from left to right, we may cut off one, two, three, &c. figures from the coefficients after the first in order from right to left. Proceeding in this way with the above example after the stage at which it was left, we have 935^1 28960347 2896220 2896^ai$ 6733957(3-1072325 941517 72597 14669 189 The first of the new figures is 2 ; and after finding 2, the numbers standing in the columns will be 93, 2896406, - 941517, the original first column having previously disappeared. We then cut off one figure from the second column and both from the first ; we then have only to divide 941517 by 289640, cutting oS one figure from the divisor 696 THEORY OF EQUATIONS. at each successive stage, as in the ordinary method of contracted division. 476. Zmasinary root*. The nnmerioal values of the imaginary roots of an equation can theoretically be obtained in the following manner, but the work would except in very simple cases be very laborious. Ex. Find the numerical values of the imaginary roots of the equation «» + 3a;-l=0. Put a + 1/3 for « in /{«), and equate the real and imaginary ex- pressions separately to zero; then we shall have a» + 3a-l-3a/32=0 and 3a»/3-/S» + 3/3=0. Rejecting the factor /S=0, which corresponds to a real root of the given equation, we have by eliminating /3 the equation 8a» + 6an 1=0. Now a must be a real root of the equation last written, and this real root will be found to be - '16109 Then /S«=3(a«+1), whence /5 is found to be 1-75438.... Thus the required roots are - -16109... ±1-75438...V^^ EXAMPLES XL VI. 1. Solve the following equations : (i) «»-12a;+66=0. (ii) «»-9a;+28=0. (ui) a^- 48a; -620=0. (iv) «»- 21a; -344 = 0. (v) a;'-2a;+5=0. (vi) a;» - 6a? - 11 = 0. 2. Solve the following equations : (i) ar^-f 2x3 + 14a; +15=0. (iij a:*- 12a; -5=0. (iii) a;*-12a;3 + 24a; + 140=0. (iv) 4a;* + 4a;»-7a;a-4a;-12=0. 3. Apply Sturm's Theorem to find the number and position of the real roots of the following equations : (i) «»-3«+6=0. (ii) a;»-a;»-33x + 61=0. (iii) 2a;<-a;»-10a; + 8=0. (iv) a;*-14a;3 + 16a; + 9=0. (v) a;*-7a-a + 3a;-20=0. 4. Find all Sturm's functions for the equation a? + 3px^ + 3qx + r = 0, and hence shew that, if p^-t^-4a^-16=0. (ii) a»-8a;-40 = 0. (v) a:4-14ar» + 16x + 9 = 0. (iii) x^-Qx^ + 9x-S=0. (vi) {r»-2 = 0. O. Find the number and position of the real roots of (i) X* + 2a;3 _ 23x3 - 24a; + 144 = 0, and (ii) x*- 26x3+ 48a? +9=0. 10. Prove that the equation x« - 7x3 + 15x3 + 3x- 4=0 cannot have more than 4 real roots ; prove also that these roots must lie between 1 and -1. 1 1. Solve the equation 2x«-7x* + 6x«-llx2 + 4x + 6=0, having given that the real roots are commensurable. 12. Find the equation whose roots are the square of the roots of x» - 3px^ - 3 (1 - i?) X + 1 = 0, hence shew that the given equation has three real roots for all real values of p. 18. Prove that the equation x»-3i)x3-3(l-^)x + l=0 has three real roots for all real values of p. Prove also that, if these roots be o, )S, y then |8(l-7)=7(l-a) = a(l-/S) = l, or /3(l-a)=7(l-/3) = a(l-7) = l. 14. Shew that the equation whose roots are the sum of pairs of roots of the quintic x° +px + g = is xW - 3px« - llgxB - 4p2x3 + 4^gx -q^=0. 16. Prove that the equation X* + 4ax» + Sa^x^ + 4ax + 1 = has no real roots unless and that the equation has two real roots if a* is between these limits. 598 THEORY OF EQUATIONS. 16. Prove that, if a be the root of the equation a?* + oa?* - 6a;2 - oac + 1 = 0, , . 1 + a 80 also IB ;, . 1-a Prove also that the other two roots are 1 , a-1 — and -. a a-t-1 17. Prove that, if a, /S, 7, ... be the roots of then Sa2/9^7 = - p^p^ + Bp^Pi - ^Ps • 18. Shew that, if Oj, a2,...be the roots of x"^ - 5pa^ + 5p^x -q = 0, then Zaj*a^'ai^a^ + Sq^ + 60()p» = 0. MISCELLANEOUS EXAMPLES. 1. Find the factors of the expressions : (i) a^ {b - c)(c + a - h){a + b - c) + b^ {c - a)(a + b - c) (h + e-a) ■^-(^ (a-b){b + c-a){c + a-b). (ii) abed (a* + b^ + c^ + (P)^ b^c'd^ - c^6?d? - d^a'b^ - a*bh\ (iii) 2 (a» + 6* + c') + a'b + a'c + b^'c + b^a + c»a + c% - Zabc. (iv)a*(6+c)+6*(c + a)4-c''(a + 5) + a»(6 + c)^+6»(c+a)» + c* (a + 6)' + 2abc (be + ca + ah). 2. If a' - a» = 6=* - y8* = c« - 7^ prove that 6y - c/3 ca - ay aB - ba . _£ ^ + _^ — ^ + -i:: = 0. a — a — p C-y 3. Shew that, il yz ■\- zx + oyy = a', then 111 jf. 4- yz {a^ + a^) zx (a* + y*) xy (a' + z^) 2a* xyz V{(a* + £c») (a« + y=) (a» + «»)} ' 4. Shew that, if 2/2; + «a; + ay = 0, then (y + 2)» (« + a)* (a + y)« + 2ar»2/V = «* (2/ + zf + y* (» + x)' + 2;* (a; + g/)* 5. Prove that, \if{x), any rational integral function of a;, be divided by (a: -a){x- 6), the remainder will be a— 6 a— 6 6. Prove that 2 {(6 - c)* + (c - «z)* + (a - 6)*} is a perfect square. 7. Find the square root of 3{(6-c)« + (c-a)« + (a-6)«-2(a« + 6»+c«-6c-ca-a6)»}. 8. Shew that S. A. 40 1^ 600 MISCELLANEOUS EXAMPLES. 9. Shew that, if a, 6, c be positive quantities and not all equal to one another, be (a -b){a-c) + ca {b — c)(b-a) + ab (c -a){e- b) will be positive, 10. Shew that, if a + 6 + c = 0, then will a{a-bf{a-cY + b(b-cf(b-ay + c{c-ay{c^by + 27 abc (be + ca+ ab) = 0. 11. Shew that, ifa+5 + c + c? = 0, (i) (Sa»)» = 2^a' + Sabcd, (ii) ^' = -^' . ^ . Shew also that %a^^'^^ is divisible by 2a', if n be a positive integer. 12. Shew that, ii a + b +c + d = = a^ -^-b^ -i- c* + (P, then a^^b' + c^ + d' = ^(a' + b* + c' + (P)\ 13. Shew that, ifa + 6 + c + c?+e +/= 0, and a* + b* + (^-^ and (he' - h'c) (ah' - a'h) - (ca! - c'af > 0. 25. Find the condition that the solution of the equations 2a (1 - a;) + (6 + c) as = 26 (1 - y) + (c + a) y, 2c? (1 - «) + (e +/) aj= 2e (1 -y) + (/+ (^) y may be indeterminate. 26. Having given that a + y + « = 0, and a? f s» ^ b — c e — a a — b prove that 2 (6 - c) (6 + c - lafla? = 0. 27. Prove that, if x + y +« = = a'^ic + 6'y + c*«, then (a + 6 + c) (a'a; + h^y + c's;) = (he + ca + a6) (6ca; + cay + a6«). 28. Solve the equations : (i) j27?-\ +Joc*-3x-2=:^j2x' + 2x + 3-i-Joe'-x + 2. ^ ' \a; + a/ \x + hj \x + c) + 2 ^"-^4^4^^.1 = 0. (oj + a) (a; + o) (a; + c) MISCELLANEOUS EXAMPLES. 603 (iv) x + y + z=Oj ax + hy + cz = 0, aV + by + c V - 3 (6 - c) (c - a) (a - 6). (v) 2yz = 2/ + «, (vi) a; (a + 2/ + ») + «/« = «^ 2205 = z + x, y(x + y + z)-^zx = h\ 2xy = x-i-y. z (x + y + z) -^ xy = c^. (vii) o^{x' + ^) = y''{y'+4) = z^z''i-4)=bxyz. (viii) ay« + by + cz = hzx + cz + ax = 0072/ + ax + by = a + b + c. (ix) a; - y = 2, rr^ - yt^ = 3, ic«'' - ytcr^ = 5, and 0^2;* — yid^ == 9. (x) x + y + z =0f a b e = 0. (c - a) oj (a - b) y {b-c)z * b—e e-a a-b 111 + + = - + -+ -. X y z a b c . .. cy + bz az + ex bx + ay (xi) ^ = = ^ = x + y + z, ^ ' y + z z + x x + y ^ (xii) yz + bcx 4- aby + caz = 0, zx + ahx + cay + 6c« = 0, xy + caa; + bey + afta; = 0. 29. Shew that, if X y z ^ a; + a 6+a c + a ay* and + r-^— + =1: a + y 6 + y c +y then will a; V « 1 tti^Y a 6 c aba 604 MISCELLANEOUS EXAMPLES. 30. Prove that the three equations a h a h' f. — a-T7y + — + — = 0, a h " X y b c b' c' f. c y z c a c' a' f. -tZ ^3?+— +— =0, c a z X are inconsistent unless a + b + G = 0. 31. Eliminate x, y, z from the equations p = ax + cy + bzj q = cx + by + aZy r=bx + ay + cZj (x^ + 1^ + z^ — yz — zx — xy -0. 32. Determine those pairs of positive integers whose product is twelve times their sum. 33. Prove that, if be the three solutions of the equations ^ + ^ + ^ + ci'Xyz = 0, Ix-^ my + na? = 0, then x^x^x^ + y^y^ys + z^z^^ = 0. 34. Shew that, if ic + 2/+«=j»i, yz + zx+ocy=Piy xyz=psy and if a?-\-yz = a, y' + zx=bf s^ + xy = c; then a+b + c = p*—p^, bc + ca + ab-p^p^-2p^^-p^, and abc = p^p^ - ^PxPiPi + Pi + ^Pz- 35. Shew that, if Xx = x^-\- x^ + x^ + ... ad inf., x^ = x^+'x^ -k- x^ + ... ad inf., then nx^ = nx^ + n'^a;,,' + n%' + . . . ad in£ MISCELLANEOUS EXAMPLES. 605 36. Find four numbers a;, y, z, u such that x-l, y-1, z+1, u+1 are in arithmetical progression, as are also x', y^, 7? and ic", «*, w*. 37. Shew that any number can be expressed in the form where /7i, jOj, ... are positive integers, and^i<2, P2<3, and so on. Express 999 in this way. 38. Find N so that the arithmetic, geometric and harmonic means of the first and last terms of the series 25, 26, 27, iV-2, N-\, N, may be terms of the series. 39. The result of multiplying a whole number of three digits in the scale of r by (r - 1 ) is to interchange the first and last digits and to increase the middle digit by the difierence between the first and last. Find the number. 40. Shew that the number of permutations of n things r together when two of the things are excluded from having definite positions is (n»-3n + 3)„_3P,_3. 41. Shew that, if n be a positive integer, the series (2n-l) (2n-l)(2n-2) "TT"^ |2 - + / nn-> (2.^-l)(2n -2)...(n+l) is equal to ( - !)»-» (2w - 2) (2n - 3) ... (w + 1) nj \n-l . 42. Sum the series 111 1 ^ 1 12 \n-2 [2 |n-3 [3 •*• ^ [1. \ n-l ' 43. By means of the identity {\-x('i-x)\-i = (\-x)-\ 606 MISCELLANEOUS EXAMPLES. prove that |2n |2n-2 |2n-4 ... = 2». |w |w |1 |n-l |n-2 |2 |n-2 | n-4 44. Shew that, if a, 6, c, c? be the coefficients of any four consecutive terms in the expansion of (1 + a;)**, then {a + h) {c» - hd) = {c + d) (6» - ac). 45. Prove that the coefficient of of in the expansion of (l+a;)"(l-a;)-» IS + r-l ^ [^ + ^-2 _^ n(n-l) ^^_,h + ^-3 |n-l |r ""'^ [w-2[r^ 1.2 |n-3 |r ^ + (-l)"-in. 2. 46. Sum the series (1> + 1) |l + (2»+l) |2 + (3»+l) [3+...+(n» + l)[w. 47. Shew that, if ?* be any positive integer greater than 2, l.„_!^I(.-l). (-^H-^) (n-2) -( "-l)("-^)("-^ )(n-3).... to » terms is equal to zero. 48. Shew that, if j-^i-^ = 1 +:P.x + p.r' + ... +p^ + ... . then iz^ . ^ ^ ^ } ^ = 1 + p^^x +p^^a? + ... , oj being such that both series are convergent. 49. Prove that, if oj < 1, X a? a? (T3^» "*• (1 - xy ^ (r=^)» ^ •*• X 2rr» 3a^ 1-03^ 1-fic^ 1-ar MISCELLANEOUS EXAMPLES. 607 50. Prove that k + (l-a)(l-6)...(l k)} a{2-a) b(2-b) k{2-k) ~ (1 -ay "^ (l-ay (1-6)'"*" ••• ■^(l-a)^(l -b)\..{l-kf ' 51. Find the value of the infinite series 1 . 4 1.4.7.10 1. 4 . 7 .10.13. 16 "^ 4. 8"*" 4.8. 12. 16 "^4. 8. 12.16.20. 24 ■*■"•• 52. Sum the series 32 5a fji l* + rT + 7-^+T^+-«« ^ infinity. 11 ll 11 53. Find the nth term of the series 2 + £c + 4ar'+ 19a^4-70£c* + 229a^+ ... , it being assumed that there is a linear relation between every four consecutive coefficients. 54. Find the nth term of the recurring series 1, 2, 3, 5, 8, 13, &c., in which each term, after the second, is the sum of the two preceding terms. Shew that in this series the number of terms which have the same number of digits is always either 4 or 5. 55. Find the sum to infinity of the series 3 6 7 66. Prove that, if n be a positive integer, 1 n 1 n.n-1 1 ^ r , v5 + — 5 — TT- 7 HTi — ... to n + 1 terms (x+iy I {x + 2y 1.2 (x + sy / 1 n 1 n.n-1 _1 \f ^ ^ ' \^+ 1 ~ 1 * a; + 2 "^ 1.2 ' x+3~"') \xTl "*" .r + + 2^ -1-.). x + nj 608 MISCELLANEOUS EXAMPLES. 57. If |2n y = x + a^ + 2a:^+... + , V~ t ^*^+"- ad inf., ^ w \n + l prove that y^ — y + x = Of and deduce that 3 \2n \n—l In + 2 68. If n^r denote the sum of all the homogeneous pro- ducts of r dimensions of the n letters a^b, c^ . . . ; shew that the sum of all the homogeneous products of r dimensions wherein no letter is raised to so high a power as the mth, will be 59. Shew that the sum of all the homogeneous products of a, 6, c of all dimensions from to w is „»+8 5»+8 ^n+8 (a_6)(a-.c)(a_l) "*" {h-c){h-a){h-\) "^ {c-a){c-h){c-\) 1 "(a-l)(6-l)(c-l)- 60. A man addresses n envelopes and writes n cheques in payment of n bills; shew that the number of ways of en- closing within each envelope one bill and one cheque in such a manner that in no instance are the enclosures completely correct, is , r , (n-l)I («-2)! , ,,^1!) 61. In a plane n circles are drawn so that each circle inter- sects all the others, and no three meet in a point. Prove that the plane is divided into n^ — n + 2 parts. MISCELLANEOUS EXAMPLES. 609 62. Shew that n" _ n (n-iy n(n - 1) (w-2)* x + n la5 + w-l 1.2 x + n — 2 xr~'\n (x+l)(x+2)...(x+ny 63. Test the convergency of the following series : ... 2 3 n /•x 2 3 n (") 2^1 + 3»— l^-^r.^:^ ■'•••• ,..., 2»-2 + l - n^-n+1 „ (^") ^■^2^r2Mri«^-^n^3^?:n"-' — 2'. 4» 2' . 4» . 5» . 7' (!▼) 38 , 33 + 3» 3» 68 08+ ••• 2». 4». 5». 7»...(3n--l)»(3n + l)» ■*■ 3^ 3». 6» . 6»...(3n)»(37i)» ^ (v) (l.log|-l) + (21og|-l) + ... / , 2w + l -\ ■^r^"^2^r=-i-V+- f 64. Find the condition that the series whose rth term is (m + nY {2m — n) (3m - 2??) ... (rm — r-1 n) {m — ny (2m + n) (3m + 2n) . . . {rm + r — 1 n) may be convergent. 65. Shew that the limit when n is infinite of r r(m+l)(m+2)(m+3)...(m + n))" . _ 1' 1.2.3...n -\ ''^' 66. Shew that 1111 2V(n"+l) = 2n + - -L ± -L ... , ^^ ' w + 4n+w+4n + 610 MISCELLANEOUS EXAMPLES. 67. If , , . - a 1 a* 1 a* X 2! cc(a;+l) 3! a;(a + l)(a; + 2) ' ., .„ a{x) a;+a; + l + a! + 2 + 68. If Pnjqn and Pn-i/qn-i ^6 the last and last but one convergents to 11 11 a + b + ... + k+ I * prove that 11 1111 1 "^ ^ Pnqn+PnPn-l a+b + ... + k+ I + a + b + ... + k + I qn+Pn.9n-i 69. Shew that the wth convergent to the continued fraction 12 3 ji^ . Vi2;__ 2 - 3 - 4 - ... - n + i - .!. ^^ 1 + ^7* \r_ ' 70. Find the nth convergent to the continued fraction 1 i A -^' 2-5- 10-...-7i« + l-.... 71. Prove that the nth convergent to the continued fraction Oj + i— ttg+l— aj + l- is cTnfio-n + 1 ), where o-„ = Oi + a^a^ + a^a^a^ + . . . to n terms. 72. Prove that the continued fraction 1 a a^{l--a) a^{l-a^) 1 I -a I -a l-a" »(l+a)(l + a»)(l+a»).... MISCELLANEOUS EXAMPLES. 611 73. Prove that a being greater than 1, the nth. convergent to the continued fraction 1 (o-l)(2a-l ) (2a-l)(3a-l) (^a-\){na-\) 2a- 1+ 2a 4- " 2a + "* 2a + is equal to .(a - 1) (2a- 1) (2a - 1) (3a - 1) {na - 1) (w + 1 a - 1) -l)(w + la^l)i 74. The difference of the first convergents of the square roots of two consecutive odd integers expressed as continued fractions is 1, and the difference of their third convergents is 497 oQ^piK i fiiid the integers. 75. Shew that, in the ascending continued fraction 6i+ 63+ 6,+ 64 -f ttj a^ a^ a^ * the n convergent Pnln, shew that if X be positive 1 +x + 3s' + ... + x^-^> 1 +x + x^ + ... + x^-'^ m < n * according as as ^ 1. 84. Shew that, if a be any positive quantity and p > q^ aP-\ a'^-l p q unless p > > q. 85. Prove that, if x be any positive quantity, a;~^ > a;"* > 1 + x -x^. 88. Prove that, if all the letters denote positive quantities, m + n/ and deduce the minimum value oi x + y when a;™y" = a. 87. If a, 5, c be positive, prove that the least value which can take for positive values of a, y, » occurs when a; =. ctj y = 6, a = c. 88. Shew that, if a, 6, c, c?, a, j8, y, 8 be positive quantities such that a>a, h'> p, c>y, c?>S, then ^abcd + 8a)Sy8 > (a + a) (6 + yS) (c + y) (c? + S). 89. If Oj, ttj, a,, ..., a„; Jj, 63, 6,, ..., h^; Ci> ^2) c,, ..., c^ be three sets of positive quantities which in each set are arranged in descending order of magnitude ; then will OiftiCj + a^b^c^ + as^jCj + . . . Soj 56i ^Cj n n ' n ' w ' 614 MISCELLANEOUS EXAMPLES. 90. Prove that, if a, 6, c be positive quantities, and (b + cf (c + a)** (a + 6)* < -^5 (a -»- 6 + c)[- a+ft-fe 91. Prove that, if a is any prime number greater than 19, then a^' — 1 is a multiple of 9576. 92. Prove that, if the radix of a scale of notation be the product of the different prime numbers m+ 1, n+ l,jt? + l, the (mnp + l)th power of any integer will end in the same digit as the integer itself. 93. Shew that the number of solutions in positive integers, zero included, of the equation x + 2y + 3z = Qn is 3w«+3n + l. 94. Shew that, if n be any positive integer, (n+l)(n+2){n + 3)...(n + n) will be a multiple of 2". 95. Prove that, if g be a prime number and p prime to q, then I!? will, when reduced to its lowest terms, have for its deno- minator a power of q. 96. Shew that, if p be any prime number, the sum of the products p-2 together of the numbers 1, 2, 3, i i?-l will be divisible by p*. 97. Shew that, if p be any prime number, the sum of the rth powers of the numbers 1, 2, 3, , p- 1 will be divisible by Pf if r be any positive integer ^ p—2. MISCELLANEOUS EXAMPLES 615 98. Shew that, if jo be prime, [2£-2|£|^ is divisible by j9*. 99. If a white balls and h black balls be placed in a bag, and be withdrawn one at a time, without replacing them ; shew that the probability that all the black balls will have been drawn in the process of drawing a balls is [a + 6 \a — h * a being not less than h. 100. A bag contains 10 coins, and it is known that 2 of these are sovereigns. Two coins are drawn out and neither is a sovereign ; shew that the probability there were only two sovereigns in the bag is one-third. 101. A bag contains any number of balls, which are equally likely to be white or black ; one is drawn and found to be white. Shew that the chance of drawing another white one, the first ball not being replaced, is two-thirds. 102. Prove that, if a, ft y be the roots of the cubic then the equation whose roots are (a-^)(a-y), (/?-y)(^-a), (y-a)(y~^), is Q^ + ^H^ - 27 {G'' + 4Zf 2) = 0. 103. If Sf denote the sum. of the rth powers of the roots of the equation a"* 4- oar* + 6 = 0, shew that, if w > 6, S^_x = 0. 104. Two roots of the equation are of the form a + t^, y8 + la. Find all the roots of the equation. S. A. 41 616 MISCELLANEOUS EXAMPLES. 105. In the equation a^n+a + (2 - a) x^^' +{b-a+l)a^-\-(a+ 2nb) x + a + (271- 1)6 = 0, prove that two of the roots are equal, and that of the rest the sum one, two, three, &c. together are in arithmetical pro- gression. 106. If n be an even integer, and p^, jpi, ^, ...be all positive, and if H be the greatest of the ratios Pi Pz Po Pi Pn-l Pn-i Pn-l and K be the least of the ratios Pi Pa Pi' Pz' then will all the real roots of the equation lie between H and iT, and all the roots of the equation will be imaginary unless H be greater than K. 107. Shew that 1 1 1 1 1 + a 1 1 1 1 + a to n rows and columns 108. Shew that 1+a 1 1 1 1+a 1 1 1 1+a to n rows and columns 109. Prove that 1111 a h c d V c' d' aa' hh' cc' dd a {a-b)(c-d){a'-c')(b'-d') '{a-c){h-d){a'-h'){c'-d:). MISCELLANEOUS EXAMPLES 110. Shew that a' + 6" + c", bc + ca + ab, hc+ca + ah bc + ca + aby a' + 6* + c^, bc + ca + ab be + ca+ ab, bc + ca-\- ab, a' + 6' + c* 111. Solve the equation ar*- a\ x^-b\ a^-c» (x-ay, (x-b)\ (x-cf (x+ay, (x+by, (x-^cy 112. Shew that 6V, he, 6 + c = <^a\ ca, c -^-a a^b\ obi a+b 113. Simp lify 617 = (a» + 6» + c»-3a6c)». = 0. 1, 0, p, 1, 6 + c, be, {p-^b'^){p + c^) 1, c + a, ca, (p + c") (p + a^) 1, a+b, ab, (p + a*){p + b^) 114. Shew that 6a(6« + c') -cV bc{a^ + P) caib^ + c") eb{c* + a^) -a^b^ 115. Shew that b' + c' + l, c'+l, b'+l, h + c <^ + l, c' + a'+l, a'+l, c + a 6»+l, a»+l, a« + 6* + l, a+6 6 + c. c + a, a + 6, 3 (6V + cV + a«62)8 (Sc + ca 4- a6)^ 41—2 ANSWERS TO THE EXAMPLES. EXAMPLES I. 1. is. a. -2«-62/-4». a. 6 6^6 4. a*. 6. -a:y-4y«. e. 2m*+2mn+2n^ 7. 3a3+26* + c»+a6-4ac + fcc. 8. a«6 + 106». 9. -2a+56-4c. iO. 13^ 8 11. ««-«-9. 12. -5a*+Sa^b-Sab»+5b* 13. 2a;3-7a;2/ + 7y^ 14. -bc+Aca + iab. 16. - 3aa+ 2&a _ 3c»+ 6c + ca + a6. 10. x-y. 17. ~5y-3z. 18. ~2x + 2y. 19. & + (2. 20. y. ai. 4a. aa. -8x + dy. as. -4n+4m. 34. 20. as. -20. EXAMPLES n. 1. 2x^-5ax + 2a?. a. x*~x+i. 8. a^-1. 4. «* + y*. 6. «*-l. 6. y'^-x'^. 7. sc*-a;2 + 4a;-4. 8. 1 + aV + a<«*. 9. a;8 + a;4 + i, lO. 6x* - 5a;8y + 14a: V - ^a:y» + 61/*. 11. 2««-10aH' + 6a;*-22«8-5»a+6a; + l. ANSWERS TO THE EXAMPLES. 619 12. 4e8 - lOx'^y + 10a;^3 _ 21a;32/3 _ 5^:22,4 ^ s^^s + ^e, 13. 6a8 + lla56-16a463^20a36S_29a264 + 15a65_36«. 14. 2a«x8 - 3a«a:5i/2 ^ Qa*xHf* - lla^xh/'^ + ^^x^ + ^Oaxy^^ - lOy^*. 16. 2a - 3a2 + a^ + Ga* - Ba^ - ISa* + 44a' - 42a8. 10. o» + &» + c»-3a/;c. 17. «» + y» + «3_3a;i/2. 18. 8a3 + 2763-c»+18a6c. 19. x^-\. 20. «8-256i/8. 21. a:«-2a;V + y^- 22. xi2- 3x8 + 3x^-1. 28. «8 + 2a;6+8«4 + 2a;Hl. 24. a8 + 8a862 + 48a464 + l28a«6« + 25668. 25. (i) a2 + 463 + 9c2 + 4a6-6ac-12&c, (ii) a* - 2a36 + 3a26a _ 2a6» + 6*, (iii) h^e^ + e^a^ + a?h^ + 2a26c + 2a62<; + 2a&c». (iv) l-4a; + 10ara-12a;3 + 9a;*. (v) ««+2a;6 + 3x* + 4a:3 + 3a;3 + 2a; + l. 26. (i) a» + 6» + c3 + 3 (a^ft + aft* + a^c + flc' + &«c + 6c«) + 6a&c. (ii) 8a3 - 2763 - ^ - SGa^ft + 54a6a - 24a2c + 24ac2 - 546ac -366c>+72a6c. (iii) l + 3«+6aj»+7«»+6«*+3a;5 + a;«. 27. 8a;«. 80. X* - 2a3a; - 2a«3 - a*. 41. 0. «0. Sa'-32a6c. EXAMPLES in. 1. x-Zy. 2. ««+4y2. 8. 9a^ - 12a;y + 16t/*. 4. -3a;-2y. 6. l+« + a;''* + x»-4a;*. 6. ar* + x3t/ + xV + *2/'-4y*. 7. l + 2a: + 3«'» + 4a;3+5a;*. 8. m* + 2m3n+3m2n* + 4mn» + 5n*. O. l + 2x + 3x2 + 4x3 + 6a*+6*«. 10. H-x2 + x* + x«. 11. l-2x + 3x3. 12. 2-3a5+2x2. 18. 2ac2-3xy+y». 620 ANSWERS TO THE EXAMPLES. 14. x^-xy-2x + y^ + y + l, 16. x^-2xy-2y\ 18. a + 2b + Sc. ao. a^ + y^ + z^-l. 22. cfi+^1^ + c^-2dh + ac + 2bc. 24. 9a» + 46» + 9c2 - 66c + 9co + 6ad. 26. c«" + da;-c. 26. 2ax-{Sb-Ac)y. 28. a;2-a;y + 2/», (a; + y)2-« (ar + y)+2». 29. x^ + xy + y\ {x + y)^+2z{x+y) + 4z\ 16. x3 + y». 17. a + 26 -3c. 10. 3a«+4a6 + 62. 21. a^-2ab + ac + b^-bc + c'. 23. a + 26 + 3c. 27. a2-3a6 + 62. EXAMPLES IV. {2x - 3a&) {2x + Bab) {ix^ + 9a^^. {2y + 2z-x) {2y-2z + x). l&{a-b){a + b){a^ + b^). I. (a-2&)(a + 26)(a2 + 462). 2. a. (4 + 3a-2&)(4-3a+26). 4. 6. 5ax (2aa; + Sy) {2ax ~ By). 6. 7. 8(a-6)(a+6)(a*+62). q. 9. 24x(x-l) (« + !)«. 11. 46»(2a-62)(4a2 + 2a&2 + 64). 18. (a -4) (a + 2). 16. (l-21a;)(l + 3x). 17. oft (a -6) (a -36). 19. (6 + c-a)(6 + c-5a). 21. {x^2){x + 2){x-5){x + 5y 28. (a:»- 4^22-2)8= (a; + 2^2)2 (x-2yz)a. 24. a« (a + 6) (a -6) (3a +6) (3a -6). 26. (a:-6)(« + 6-2a). 26. («-a)(a; + 2y + a). 27. (a + 6 + c-fl!) (a + 6-c + d) (a-6 + c + d)( -a + 6 + c-i-(i). 28. {x + y + a + b){x+y-a-b){x-y + a-b)(-x + y + a-b). lO. 16«(2-3ar»). 12. (aa-46c)(a4-2a26c + 462c2). 14. (4- a;) (3+ a;). 16. -4 (a -1)3. 18. a26(a + 6)(a+46). 20. (3a + 36-c-d)». 22. (5x - y) {5x + y){2x- y) (2a; + y). ANSWERS TO THE EXAMPLES. 621 EXAMPLES V. 1. {x + l){x-l){x + a). 2. (a + 6)(c-^. 3. {a-b){c + d){c-d), 4. {ax + by) (cx + dy). 5. {ax + b){cx^ + d). 6. 2{a-d) {a + b + c + d). 7. {a + b){a-b)(a^ + ab + b^). 8. (a-6)«(a« + a6 + &2). 9. (a + l)(a-l)(6 + l)(6-l). lO. {x + z){x-z)iy + z) {y -z). 11. (a;22;_l){y2;j_l), 12. {x + y){x + z){x^-xz + z^). 13. (a;-y)(a; + y + «). 14. (ar + 3) (a;-3) (a:2 + 2). 1 6. {x^ + 5a; + 1) (a;^ -5x + 1). 1 6. (x^ + 4:xy + y'^) [x"^ - 4:xy + y^). 17. (a;3 + a: + l)(a;»-a; + l)(a:4-aa + l). 18. {x + a-\-b){x-a- b) {x-\-a - b) {x - a + &). 19. {x^-2yh^)\ 20. {x-2b + ab){x-2a-db). 21. (a; + 6)(ar + a). 22. (l-x^){l-\.y + x{l-y)}{l + y-x{l-y)}. 23. (a; + y-3«)(x-y + «). 24. {piy -x->ra) [y -2x-a). 26. (a-36 + c)(a + 6-3c). 26. (2a- ll& + l)(a + 2&-3). 27. (l-aa;)(l + ax + &a?*). 28. (1 - oa) (1 - oa; - ca;-). 20. ~{b-c){c-a){a-b). 30. (6 + c) (c + a) (a + 6). 31. (a-6)(a-c)(6 + c). 32. {x^-xy + y^){x{a+l)+y{b + l)), 33. (a;?/ + db) {ay^ + b'^x) . 34. (2a; -z){x- y)\ 36. {x'^-yz){y^-zx){z^-Ty). 36. (a;+4)(a; + 2)(a;-l)(a;-3). 37. (a; + 4) (a; + 2) (a;2 + 5a;+8). 38. x(;c + 5)(a;2 + 5a; + 10). 30. (a; + 2) (a; + 6) (a:^ + 8x + 10). 40. (x + 8) (2a; + 15) (2x2 + 35a; + 120). 522 ANSWERS TO THE EXAMPLES. EXAMPLES VI. 1. 3{y-z){z-x){x-y). 2. 5 {y -z) {z - x) (x-y) {x^ + y^+z^-yz- ^x - xy). 3. {b + c){b-c){c + a){e-a){a + b){a-b). 4. {b-e){c-a){a-b){a + b + c). 6. {b-c){c-a){a-b){a^ + b^ + c^+b^c + bc^ + c^a + ca^ + a''b + ab'^ - 9a6c). 6. - (6 - c) (c - a) (a - 6). 7. - (& - c) (c - a) (a - 6) \b\^ + c^a^ + a^ft^ + a&c (a + & + c)]. 8. -(6-c)(c-o)(a-6){a2 + 62 + ca + &c + ca + a6). 9. _(6_c)(c-a)(a-&)(a» + &» + c8 + &'»c + 6c2 + c2tt + ca3 + a26 + a&3 + a6c). lO. 24a&c. 11. 80a6c(a2 + 63 + c»). la. 4aic. 18. 2a&c. 14. 4a6c. 15. -4(6-c) (c-a) {a-6). 16. 3(2/ + «)(« + a;)(a; + y). 17. 5 (y + «) (z + a;) (« + J/) (a^ + 2/'' + 2'' + |/2 + 2;a; + a;t/). 18. -(6-c)(c-a)(a-6). 19. -2 (6-c) (c-a) (a-6) (a + 6 + c). 20. - (6 - c) (c - a) (a - 6) (Sa" + 363 ^ ScS + gftc + 5ca + 5a6). 21. (6 + c)(c+a)(a + 6). 22. -(6-c) (c-a) (a-6) (a + 6 + c)=^, 23. {x + y+z){yz+zx + xy). 24. (6 + c) (c + a) (a + 6) (a + 6 + c). 25. 12x1/2 (x + y + «). as- -3 (6-c) (c-a) (a -6). 27. 16 (6-c) (c - a) (a-6) (d -a) (d- 6) (d- c). 82. 27a26Ma + fe)'. S*" (aH62)2(c2 + d2)^ 86. (6-c)(c-o)(a-6)(a-d)(6-d)(c-d). 37. -(6-c)(c-a)(a-6)(a-d)(6-d)(c-d)(6cd + cda + d«6 + a6c). ANSWERS TO THE EXAMPLES. 623 EXAMPLES Vn. 1. a-6. 2. 2«-l. a. x^-yK 4. 2x-y, 6. x-2y + Sz. 6. 4aa-3a6 + &«. 7. a+2b. 8. 2x^-Sx + l. 9. ^-a. lO. x^+x- 6. 11. x^-x + B. 12. x^-Bx + 7, EXAMPLES VIIL 1. 12x^ + 2ax^ - ia^x^ - 21a^x - 18a*. 2. (4a-6)(a-&)(3aa + 62). 3. (a;»-2a; + 7) (6a;8 + a;2-44a; + 21). 4. («« + 6a; + 7) (7ar» - 40a;3 + Ihx"^ - 40x + 7). «. «(«+!) (a; + 2)(«-2)(x + 3). 6. «(«-l)(« + 2)(« + 6){a:»-2a; + 4). 7. 2a (2a -6) (2a -36) (2a + 36). 8. &r (a? + 1) (ar - 3) (a; - 4). 0. (3a; + 2)(8a:3+27)(8«»-27). lO. 3 (x - 3y)a («» - 42/2). 11. («-2y)(x-8y)(a?~4y). 14. (ac'-o'c)*=(6a'-6'a)2(6'c-60. 66*c*y'i 2. 4. tfjll., e. 4xV+l 2a;-l 0:2 + 1* o 2x + 3y 8x3-y2- — 6a:« + 3a;2-l EXAMPLES IX 3. «-^ a + 46' x^ + xy + y^ x'^-xy+y^' 8. .^ -i)V 3x + l' lO. 9x3 -3x-2 624 AJ^SWERS TO THE EXAMPLES. "• ^a-h + cH-a + b + cy "• yz + zx-^xy. 18. ■'h{y-z)(z-x){x-y), 14. ^-^. 15 yz+zx + xy o^ + ft' x + y + z ' ' a^-b^' 17. .-4^. 18^ ^^y' l-9a;3 ar3-42/ 2x + 4a 48 18. ^^7— . 20, 81 2a " {x + 2){x + 4:){x + 6){x + B)' 48a» _^ 24a* (a; + a)(a; + 3a)(a; + 5a)(x + 7a)* " x {x^ - a^x^ -la^' 83. 0. 24. 0. 85. 1. 86. -1. 87. <2. 28. a 89. 1. 30. 2. 81. a + 6 + c. 38. a^ + b^ + c^ + bc + ca + ab. 33. (a+6 + c)2. 84. a + b+e. 35. . 88. 0. (-a + & + c)(a-6+c) (a+ft-c) ■'°V^*-aV ^®- ^a^x^-b'y^' 2a6c(a + 6 + c) *'• (-a + 6 + c){a-6 + c){a + &-c)' *''• 2(x + t/ + .). EXAMPLES X. * o- ^ o». _ 1 6 + c-2a « A 2a5 1. 2a - 6, a - 26. 8. 1, — . 8. 0, t , c + a-26 ' h-a 4. a -26, 6 -2a. 6. ±1. 6. ±1. 7. 1, -3. 8. 1. 8. 0, ±5^2. lO. 6, -ei. ANSWERS TO THE EXAMPLES. 625 60 a^c + b^a + c'^b - 3abc 11. r^r. 12. 29 a^ + b^ + c^-bc-ca~ab 18. 0,-s{a+b + c^y/{a*+b*+c!'-bc-ea-ab)}. 14. [6c + ca + ab ± ^{6V + c^a^ + a%^ - aJc (a + 6 + c)}]-^ (a + 6 + c). 16. 6,-|. 16. ±V6. 17. ±v/«^ ^J-ab. 18. 0, --. 20. {a + 6 + c±V(a' + J^ + c»-6c-(ja-a6)}. 21. **' -6+7- 22. -2(a + 6 + c). 28 a» + 6a 24. cd(a + 6)-a6{c + d!) a + b ' ab-cd 25. ab-cd e+d-a-b' 26. 0.-'.^'- 27. Bib-e)ie- {b-c)^+{c-a a)(a-i %■ - 0, i(-19±V-3). 29. ^ ^Jmab. do. 8, -6. 81. 2, 3. 82. 1. 33. 0, a2-&a. 34. a, ft. 86. a e 6'5- 36. 0,4 {a + b). 37. -a, -&. 88. i(a-6). 39. ±7^. 40. 0, ±2^ab. o , i^ 42. /b*-a* «1. i=^3a»-3&^ 43. 0. 1. ., . . <«a^+(6«-3a6c)aj + c»=r0. (ii) a^cx^ + xb(ir^-Bac) + ac^ = 0. (iii) x^-bx+ac = 0. EXAMPLES XL 1. ±2, JkJ^. 2. a, aw, aw^, -2a, -2aw, -2aci»'. 3. -a, —aw, -aw', 2a, 2aw, 2aw^ 4. 1, j(l±7^ri6). 5. 0,1,3,-8. 6. 1, -2, ^(-1±V^19). 7. -1, -6, li-7^3^5). ®- §•"¥• i(-7=^^N/2). ». 8. -1, l=t2Vi9. lO. i(-l±N/^3), i(a:fc^i^^:4). ANSWERS TO THE EXAMPLES. 627 11. 0, -6, i(-5±V-15). 12. a, -9a, -4a±aV^Il6. 13. 7a, -8a, |(-1±V^^I67J. 14. -4, -6, ^(-16± ^129). 21 16. 3, -T^. 16. ±^(a + 6). 17. dba±6. 18. 2.^, 1(-3±^A^). 19. 3,i, |(-liV^). 20. 3' 3 1.. ,.. ,, „ 1 ~1, ^[l + v/5±^/{ + 2V6-10}], |[1-n/5±V(-2n/5-10)]. ai. ±1, ±^31; ii_±yr|. aa. 2, 2, I 23. -1, 2, 8, -4. 24. ±1, -(-7±3J6). 26. a, 6, c. 26. 9, -8±^-47. 27. 9, -6, i(8±^-215). 23. a, 6, ^(a + 6). 29. a, 6; l{a + b)^~{a-b)J^. 30. a, 6, i(a + 6). i(a + 6)±^(a^6)V"r3. 81. a, 6, 2(a + &). 32. a, 6, |{a + J±l(a-&) ^"1681. 83. a, 5, ^{a + b±{a-b)J^}. 34. a, 6. 36. a, 6, and roots of (a - a;) (a? - &) = 16 (a - 1>)'. v^ a . / - 3a2 87. a-26, 6-2a, -^ {a + 6±(a-6) V" 15}. 38. B.ootaof X [a- z) = (jb:i=/i^ ^ + ^\\ 628 ANSWERS TO THE EXAMPLES. ««, (^^ &' /— r ^ be ca ab o a ^ a o e 42. |{-6=t^(62 + 4)|, ||_„±^(^a + 4)}. 43. | {a±V(«*-46»)}. 44. -(a + 6 + c), -.|(a + 6 + c)±i^(Sa2-226c). 2 1 46. a + 6 + c, -(a+ 6 + 0)^5^(20" -7S6c). 46. a, ft, c. o o 47. ft =fc /J cd(a + 6) + o&(c + d)[ / | a6 (c + d) -cd(a + 6) ) * V t ^6+7+d r VI c+d-a-h r EXAMPLES Xn. 18 8 1. a; = l, y=-l. a. a;=y, y=^. 2 3. a;=3, y = 6. 4. «=:g, y=3. 6. « = &, y=ra. 6. « = o6, y=-a-6. 7. a;=a + 6, y=a-6. 8. aj=y = a. 9. a; = a, y = 6. lO. x = a{a-b), y = b{a-b). 11. a;=-3, y = 3, « = 1. 12. « = 2'y = 3»* = 6' 13. a:=y=i; = l. 14. a;=6 + c-a, y=:c + a-b, z=:a+b-e, 16. »=6 + c, y = c + a, z = a + b. 10. a;=-^(2a + 6 + c), y= --(a + 2& + c), «= -^ (a + 6 + 2c). 17. z = y = z= a + b + e 18. x = ^{2a + b + c), j/ = -(a + 2& + c), 2 = 5(0 + 6 + 20). 10. «= ANSWERS TO THE EXAMPLES. b 629 20. X 21. X 22. X: 23. X: 24. X. 26. 2;= 27. a;r {a-6)(a-c)'^ (6-c)(6-a)' '"(c-a) (c-6)- = a, y = 6, z=c. = -a + 6 + c, y = a-6 + c, z = a + b-e, = a{b-c), y = b{c-a), z = c{a-b). = 1, y = 0, « = 0. = a6c, y=6c + ca + a&, «=« + ?> + «. _m {m - b) {m - c) a(a-b){a- c) &o. b + c, y=c + a, z=:a + b. 26. x=a, y = b, z=e, 28. ar-J^J'^ + ^ + c) (a-6)(a-c) , &o. 29. :r 80. a; = 31. Ix: 32. a; = 34. x = z~ 10 = (a + a)(a + 6)(a + c) . 1 aftcd, y = - (6cd + cda + dab + abc), bc + ca + ab + ad + bd + cd, -{a + b + c + d). EXAMPLES Xin. 1. 12, 11. 3. ±3, ±1; ±4 2. 1, 1, 1 1 15' 15' Vs' ^ivs' 4. ^2, ±3; =..^^3. :^^^3. 6. 12, 7; -7, -12. ^ 2ab 2ab b^a ' a+6* 6. a, 6; 2a-6, 2&-a. 8. ±fx/^N^, ±^VW6"^ 630 ANSWERS TO THE EXAMPLES. _ a , b O. ±-, =fc&; ±a, ±-. lO. / 5»-a» / &« - a8 11. ±7, =p6; ±5, T=7. 13. 64, 8; 8, 64. 12. 9, 4; 4, 9. 14. -6±s/30, 6=1=^/30. 16. i(ld=V-ll), ^(l=F^/-ll); 2. -1; -1,2. 7 14 3' 3 * 19. 2a, 26 J -a, -&, ai. &, a. 16. 1, 1; 2=fc^7, 2tx/7. 17. 2,4; 18. 2, 1; 1,2. ao. i,2. aa. 6,6; -|(ldbV6). -|(1=fn/5). / ±a6 ^ / ±a& 24. ±i Vl + «^ =^jT+a\ as. 8, 4; 2,4. ae. 4, 2; 2, 4; 3±V-13, Stn/^H^ 1111 2' 6' 6' 2 a9. l,^; -1,-2. 81. haij,~. a8. 3, -6; - 1 2 3' 3" 80. 0, 0; a+b b-a 2a6 ' 2ab ' 8a. 6,a; 6, ^; y.a; ^/a^fi, 4/^. 1. EXAMPLES XIV. he ca db be ca ah a. -„ = 2/_« a ' 6 ' c 1 aa-&a-c3"" s/(aa + 6^ + c^) * ANSWEES TO THE EXAMPLES. 631 {b + e) = y{c + a) = z{a + b)=± ^ i^{b + c){c + a) (a + b)^ , 4. 0,0,0; 9. X 2abc 2abc 2abc 6. 0, 0, 0; -bc + ca + ab^ bc-ca + ab* bc + ca-ab' 26c 2ac 2ab b+c-a* c+a-b* a+b-e' 6. x=y = z= J=2. » * y _ « _^ 1 b + c-a c + a-b a + b-e ,J{2a + 2b + 2c)* » .6c t. ^ca , ab 8. -ai— , -6±--, -c±— . a c o. ? y ' -jL 1 a2(62 + c2) 62(c3 + a3)~c2(a2 + 62)- 2a6(j' lo. ^ _ y _ ' a(-a + 6 + c) 6(a-6 + c) c{a + b-c) ^{{-a + b + c){a-b+c){a + b-c)} b + e-a c + a-b a+b-c 1 ^ {{b + c-a) {c+a-b) {a + b-c)} j^2 a; ^ y „ g 2{a^ + b^ + c^) b+e-a c + a-b a + b-c {b + c-a)^ + {c + a-b)^+{a + b~c)^' z = y = z = 0» 13. .=1. /■(»±iH£±i),&„. V a+1 le 1 2 3- ^ ^ 2 16. 1,2, 3,-, ^, -. 17. 1, 2, 3. 18. 3, 5. 7. 19. 0, 4, 6. 20. 3, 3, 4 21. 0, 0, 0; ^, t, £. ' ' 2 2' 2 s. A. 42 632 ANSWERS TO THE EXAMPLES. 22. -a, 6, c; a, -&, c; a, 6, -c; |(lix/-7), |{1±n/^7), ^®' *-^7{(^^ + c)(a + 6-c)}V*^- ^** ^ ~ v/{2a6 + 26« + 2c« - ea^feV) * 26. {6 + c)2-(c + a)(a + &) {c + a)2- (a + 6) (6 + c) « 1 (a + 6)3-(6 + c)(c + a) 2 ^(3a&c - a» - 6» - c^) 27. a, 6, c; ^(2& + 2c-a), 5(2c + 2a-6), K(2a + 26-c). 28. a;= =tj-7- (6c- ca + a6) (6c + ca-a&), &c. -^AAA 3 a a b S, b c c 3 80. 0, 0. 0; -^a, ^^ 2,' g' "3^' 2' 2* 2' "2"^ _(-a + 6 + c), 2(a-6 + c), ^C^ + ^-c). _ a^x b^y c^z ,, 32. T — = — ^ = -.=V(-a6c), 6 -c c-a a-6 ax _ by _ cz _ h-c~ c-a~ a- b~ \/ bc + ca + ab' _ I — abc ~ \/ 6c + ca + < EXAMPLES XV. I. 20, 30. 2. A £10, £ £15, (7 £25. 8. 20 years ago. 4. 28. 6. 6, 15, 30. 6. 5 days. 7. 1800. 8. 58. 9. 30 miles. 10. 120 lbs. 11. 4 days. 12. 36,9,12,15. ANSWERS TO THE EXAMPLES. 633 13. 48 miles. 14. 15 miles. 16. 54, 81, 108. 17. A £450, B £225, G £237. 10s., D £87. 10«. 18. rb5. 19. 38,83. 20. 18 miles. 21. At 1 o'clock, 15 miles from Cambridge. 22. A £10, J5 £5, G £1000, 23. 25. 24. 9, 7; 8^/2, v'2. 25. 50 miles. 26. 576. 28. 3 miles an hoar. 29. 3 hours. 80. 253. 31. 2 gals, from the first, and 12 gals, from the second. 32. 15 minutes past 10. 33. 9 o'clock, 30 miles from Cambridge. 34. 45 and 22^ miles an hour. 35. £3. 86. 450 miles. 87. 30 miles. EXAMPLES XVI. 8. a + & + c + a&c = 0. 4. (6 + c-a) (c + a-6) (a + 6-c)=8. 8. a» + 2c3_3a5a_o. 16, p + 7n« + n»-Zmn-4 = 0. 16. aZ2 + 6m'' + cn3 + Z7?m=4a&c. 17. a2 + &2 + c2-a6c-4 = 0. 18. a3 + 63 + c«-5a6c = 0, y3z3 + «V + a;V + «V«'^=0. 19. S6»c8 = 5a262c2. 20. a3 + &^ + c'-&c(6 + c)-ca(c + a)-a6{a + 6) = 0. EXAMPLES XVIL 1. aUi 2. 1. 3. ^4-» 4. 1. 5. x-y, 6. a:* + l + ar-*. 7. a; + t/+«-3a;^?/^A 8. x^-x~^' 9. a^ + a"'nra;T + aT^a;^+at*'^a;^ + x^. lO. a? + y. 12. 4a;H 3a; + 2 - 3x-i. 13. a;^ + 2a;i + 1 + 2a;-i + a;"^. 42—2 634 ANSWERS TO THE EXAMPLES. 14. a^ x~^ + as x~^ + oT^ x^ + a~^ x^ . 16. x^y~^ - x'Sy~^ + x^y~^ -l + x~^y^ - x~^y^ + x-^y^ . 20. (i) a^-a^b^ + Jb^-ab* + aib^-b^. (ii) a » x'* — a^ x^' y^ + a^a;* y — a^ x^ y^ + a^ x* y^ — w* , (iii) a^ + b^x^ + c^x^-bcx-cax^-abx^. (iv) [x'^ +y's + z^ -y^ z''^ - z^x'^ - x^y^ {{x + y + z)* + 3x^y^ z^ {x + y + z)} +9x^y^ z^. EXAMPLES XVm. 1. 2-^3. 2. 6-^15. 8. ~>J2. 4. 52. 5. 0. e. 14i. 7. V3 + V2^>y30 ^^ V30 + 2V3-3V2 _ O. i(V21+^/10-^/14-^15). lO. 1(^6+^/10-^/21-^35). »• 2^*^ '^ '^ '" V*";- ,, 4^^4 + 24/2 + 4 11. 3 12. 34^3 + 1. 14 4/12-4/4 16. 7-2^13. 17. 1+^/5 + ^7. 18. V3-1. 13. 4/2-1. 16. 5-^3. 10. 2-^3. 20. 1. 24. 1 + ^2+^3. 25. V3+x/2+V6. 26. 2+J^-^5-J^, EXAMPLES XIX. 1. 2aH»-32/'. 2. a?*-3a;V- «« a -26 -3c. 4. 5a3 - 36-' - 2c«. 6. x^ + a^^ + a^ + l. 6. 2x*-2xy^-y*. 7. 7 + 8;ra + 5a:3, 8. a;3-x + 2-a;-i + a;-2. 9. —-2-^. y 5x ANSWERS TO THE EXAMPLES. 635 lO. x^-2x^-x^. 11. x^-2x^ + x^. 12. a~^x^-x^-a^. 13. x-8. 14. x^-xy+y\ 16. l-3a;2 + 2iB*. 16. 2{bc + ca+ab). 17. x^-x{y + z)-yz. 18. a^ + ft^. ai. 4 = 20,5 = 68, C=-U; or 4 = 52, £=-68, C=76. 23. af=gh, hg = hf^ and ch=fg. EXAMPLES XX. 6. x = 3. 6. a:6:c=2:3:4. 17. (i) |,1. (ii) 0,1. (iii) 00,0. 18. ^^, -j=. EXAMPLES XXI. a. 2, 4, 6, 8; -2, -4, -6, -8. 4. 2,4,8. 14. 6, ±12, 24, &o. 17. 3, 9, 15. EXAMPLES XXn. 1. 31. a. (r-l)(r-l) ,1000.... 4. lib., 21b., 10241b. 7. 46. 8. 6. 13. 502 or 361. 16. 288, 289 or 290. 16. 2775 or 2525. 17. 135. 18. a=8, 6=0, c=6. 19. 7. ao. 1089. as. 142857, 285714. 34. 166, 199. EXAMPLES XXIIL I. [20/{|4}». a. 145. 3. 3i»; [12 /([£)'. 6. 260. 7. ^n(n-l). 8. in(n-l) -im(m-l) + l. 9. g{n(7i-l){n-2)-m(m-l)(m-2)}. 636 ANSWERS TO THE EXAMPLES. 11. gn(n-l)(n-2)(n-3). 12. SmK 13. '_^ . -^' I \p + l \mn 14. ^w(w-4)(w-5). 17. , , ., . 20. , -;— ,^ 23. 2(mn+m + n-l). 24. 2Sa+ 2Sa6 - 2 (n - 1), where n is the number of given diameters. EXAMPLES XXIV. 1 . x'^ + baa^ + 10a V + lOa^x^ + &a*x + a». 2 . 32a8 - 80a% + 80a V _ 40a«xS + lOao^ - «». 3. 1 - 6ar» + Ibx*' - 20a;« + 15a:« - 6«io + a^^. 4. 16a4-96a» + 216a«-216a7 + 81a». a. 16a;8-96«« + 216a;4-216a;2 + 81. 6. a;i0-10a;V + 40a;V-80A^ + 80a;22/i3-32yi». 120 7. 405a;V- 8- TTi=T 3^«4*a;i». O. 9242r'», |16^ 142 121 !21 In In 13. (-irr-^= — 3''a;»-'-y'-. 14. — ^ ^in-r [r p- r 15. {3x)i» - 30 (3a;)i^ + 420 (3a;)i3ya. . . - Ubx^ {2yy* + 45a; (2y)i* - {2y)i». 16. 924a;«. 17. 64354;^ 6435ir8. 22. ( - 1)** [2tc / [n [w. 23. 7 or 14. 24. 7. EXAMPLES XXVI. 1. Convergent. 2. Convergent. 3. Convergent. 4. Convergent. 6. Convergent. 6. Divergent. 7. Divergent. 8. Convergent if x > 1 ; Divergent if a; i> 1. 9. Divergent if x = 1 ; Convergent if a: 4= 1. lO. Convergent if a;> 1 ; Divergent if a; 4* 1. ANSWERS TO THE EXAMPLES. 637 11. Convergent if x :^ 1 ; Divergent if a: > 1. 12. Convergent if a: 4* 1 ; Divergent if x > 1. 13. Divergent. 14. Divergent. 15. Convergent if m < 1 ; Divergent if w 1. If x = 1, then Convergent if A; > 1 and Divergent if & ::f 1. 23. Convergent if a; < 1 ; Divergent if a; -f: 1. 24. Convergent if a; i^ 1 ; Divergent if a; > 1. 26. Convergent if a; < 1, Divergent if a; > 1. If a; = l, then Convergent if m < i and Divergent if m -^ ^. 27. Convergent if a; < 1, Divergent if a; > 1. If a;=l, then Convergent if A < i and Divergent if /c 4^ i. EXAMPLES XXVn. 1. (i)(r + l).rr (ii)Hr + l)(r + 2)xr. (^,) (^ + 1) (r+^(r + n-l) ^^ _ 2.5.8...(3r-l) _ 2.1.4...(3r-5) ^'"^^^ ^^ 3.6,9...3r *• ^^^ ^ ^^ 3.6.9...3r ^- , .. , ,,_5.2.1.4.7...(3r-8) ^ , ... 8. 8 . 13...{5r-2) , 2.3.8.13...(5r-7) .. . g (g+i>) (g + 2p)...(g +f3I. j,) ^^^^ 1.2.3...r • ^ ' ^j7 *• / x g»8.1.1.3 .5...(2r-7) , . , . (xi) i ' x^a^, r>3. , ... 2.6. 12.. .(7r- 9) ,,o 2. (i) The ninth term, (ii) the eighth term. 3. The 39th term. 4. The first and second terms. 638 ANSWERS TO THE EXAMPLES. 6. After the 12th term. 8. (i) ^-4^ l"'}^''~^K ^-^''x^. o • o . y . . . or (ii) 2a-*'x*". (iii) 4ra-^x^. (iv) ' ' . ""' ' a-^x^ when n is ^ ' ^'2.4. 6...» 1.3.6...(»-2) „ „ , . ^, even, ^r — ^ — j; — 7 =-; a-"x" when n is odd. 2.4. 6,..(n-l) (v) (2r2 + 2r + l)a-'-ix'-. (vi) ( - 1)*" 16 (r - 1) aS-'-a;*'. EXAMPLES XXVin. J. (i) 2. (ii) ^|. (iii) 44/4. (iv) J27-2. (V) 1. (vi) 4/4. (vii) ^^. (viii) |. ,,. (_!).,. (n-2)(.-3)...(n-r) ^ ^^ ^ ^^ ^^^ \ r — L 81. -245/8. 82. 246,792. 88. 462. 84. 35. 36. Coefficient of x^^ is 33*"2-8»-3o-3'^3, of x^^^ is - 3=^^i 2-3''-3 a-^» and of x=^'^^ is 0. 38. ^(n + l)(n + 2)(w+3)(n + 4)(n + 6). 39. 2«+'' [3n + r-l J |r_ |37t - 1 . EXAMPLES XXIX. 18 3 _4 8_ • 5(x + 6) 5(a; + l)' "• ar-3 x-2' 33 5 1 • 4(a: + 3) 8(x + 6) 8(a; + l)* 1 2 2 1 1_ x~(x + l)^' ®- (x-2)2 a;-2''"x + l* 1 7 13 „ 1 4a5-8 + -7-/ ST' 7- ^-7 .-K^ + 12(a;+l) 3(a;-2)^4(a;-3)' 5 (a; + 2) ^ 6 (a:»"+l)' ANSWERS TO THE EXAMPLES. 639 1 1 1 « 1 . 1 l-10x^3(l + 3a;) 3{1 + Sx)^' ' 2 (x^ + l) ^ 2 (a;-!)^' 3 21 21 7 2(l-3a;)8 8(l-3a;)3 32(1 -3a;) 32(H-x) 1^ J. 1_ 2 4 a; + 2 ^^' a;3+I''*«-2 a; + 3' (« - 2)8 "•" 5 (a; - 2) ■*" 5 (a:^ + 1) ' 2 11 11a; -4 ^®* 6(x-l)3'*"25(a;-l) 25(a;2 + 4)* 3 11 a; + 2 14. 6{a;-2) 2(a;-l)2 2(a;-l) 10(a;2 + l)' l^J- 1_ 17 1 3 "' 8a;3 16a; a; + l "^ 16 (a; + 2) "'' (a; + 2)2'^ 4 (x + 2)3* 1 1 11 1 1 3_ "• 4(« + 2)»"'"6(a; + 2)2'^144(a; + 2)''"9(a;-l) Sa;^ 16aj' 4 / l\«+i 1 17. (-l)"{2-"-3-~-i}. 18. -(^-^j -9(3n + 7). 20. ^{3«~l)-5{(-l)«-l}. 21. l{9 + 5«-W-2.3"+«-2'*+<}. 22. (n3 + 7Ji + 8)2«-8; ^ (n» + 9»8 + 14n) 2»-*. EXAMPLES XXXII. 1. 1-262. 2. 1-48169. 8. £1146-74. 6. £742. 19«. 6d. 7. £785. 10». 8. £1979. 68. 6d. O. £1736 nearly. lO. £122-58. EXAMPI^ES XXXIII. I. i{(3n + l)(3n + 4)(3n + 7)(3n + 10)-1.4.7.10}. 8 |3.7' {4n + 3)(4n + 7)) * " 168* 3. in(n + l)(3w2 + 23n + 46). 4. |to(w+1) (n4-2)-3n. 640 ANSWERS TO THE EXAMPLES. 6. in(7H-l)2(n + 2). 6. j^n(n + l) (n + 2) (3» + 6). 7. 2-4(2n-l)(2»+l){2n + 3){6n + 7)+g. 21 6n + ll Q -^L ®- 180 12 (2n + l)"(2n + 3) {2n+6) ' * ~ 180 ' _5 3n + 5 _^ 36 6{w + l)(n + 2)(n + 3)' *"36* 4 2(n + l)(n + 2)' "" 4 • 8 8(2w + l)(2n + 8)' «~8* 29 6na + 27n-h29 _29 36 6(n + l)(n + 2)(n + 3)' ^^'"se* "• ^'^~=^' ^• = ^- "• ^(« + l)(« + 2)(4n + 3)-|. 16. ji^n{n+l)(n+2)(8n» + lln + l). 16. na3 + n(n-l)a6 + g(w-l)n(2ro-l)6>. 17. na^ + ln {n - 1) a^ft + i (n- 1) to (2n - 1) afts + ^na (n - 1)»6«. 18. i«(4na-l). 10. in(16n2-12n-l). o o 23. in(n + l)(n + 2). 24. ^n(n + l)(4n -1). 26. na6-iTO(n-l)(a + 6)+iTO(n-l)(2n-l). 27. (i) ^;-i. (ii) 1-j^^ ("^) 2-n + l (3) • ^^") 4 - 2(n^l)(n + 2 ) (tJ /\ B 3 (By 6 /6\" ^^^ 2 (n + l)(n + 2)V4; ' ^""'^ "* (n + 1) (n + 2) ^7^ * ANSWERS TO THE EXAMPLES. 641 EXAMPLES XXXIV. 4.7.10...(3n + 4) 2. 5. 8... (3n+2) *• ^^' 2.5.8...(3» + 2) ^' ^"^ 4.7.10...(3n + l) ^' ,...,, 5.7...(2n + 3) , /. X 1, f, 13.15...(2w + ll)) ,, 2. (i) 2 + 3(71-1) (w-2); 2n + n(n-l) (n-2). (ii) 7n-(n-l)(n-2); ! n (n + l)-in(n- 1) (n-2). (iii) 2"+i-n-2; 2«+2-l_^(n + 2) (n + 3). (iv) 2»+i-n(n+l)-n; 2»+2-4-|n (w + 1) (n + 2) -in(n+l). (v) ^n(n + l)(n + 2)(n + 8); y|Qn(n+l) (n + 2) (« + 3) (w + 4). (vi) (n-2)(n-l)w(n + l) + (n-l)w-n + 2; i(n-2)(n-l)n(» + l)(n + 2)+i(n-l)»(n + l)-in(n + l)+2n. ^ ... 2-4x .... l-2« ^.... l-6a: l-4x + x^' ^ ' l-5ar + 4«a' ^ ' l-12a; + 32a^» ^^ 16-14x-35a;«-42ar8* ^"^^ (l-a;)«* 4. (i) 2»+i-2; 2«+2-2n-4. (ii) ^{3»+ll(-4)-}4 + ^-|^^-4)-. (iii) i {3'» - ( - 1)«} ; g {3«+i - 3} when n is even, and i {3»+i - 1} when n is odd. 2« •• ii{(l + N/5)" + (l-N/5)'*}. 6 a=l, 6=4, c = l, d=0. 2-3a;-a;* 1 9. • (l-«)»(l-ac)* "• (l-a;)2(l-2a;)* a6c...Z "• ^n^y - K^4)'°«'^->-J-i 642 ANSWERS TO THE EXAMPLES. a*. l-(l-^)log(l-.,. «. «-^^. 80. . a + /3, Divergent if 7 < a + /3. EXAMPLES XXXVL n + 1 n + 2' 10. b^.^\ - {b^a^ + & V«-l) P\-l - «»&«-! iKK-l + «n) J»'n-« EXAMPLES XXXVn. *• W ^ + 8 + ^"^ 11 + 1 + 4 + 1 + 22 + ,..., ^1111 M 6+1+2+1+10+ .111111111 1^ (IV) t>+i^i + 3^i + 5 + j + 3 + l + l + 12 + 8. (i) ^|. (ii) J(4 + x/37). (ill) ^(28-^30) 16. i(n2 + 3n). 26. «~i. EXAMPLES XXXVin. 6. 266. 10. 3, 7, 9, 11, 13, 19. X8. 604»-6. EXAMPLES XL. 1. (i) x=2, y=3. (ii) x=l, y = 10; x=U, y = 2. (iii) a; = 4, 2/ = 8; a; = 13, y = l. (iv) 696, 3; 626, 18; 554, 33; ; 57, 138. ANSWERS TO THE EXAMPLES. 643 a. 22, 30. a. (i) a?=4 + 13m, y = l + 7m. (ii; a; = 9 + llm, y = 7 + 9m. (iii) x=15m-7, y = 17m-lO. (iv) a;=644-69m, i/ = 44 + 49m. 4. (i) 3, 1, 2; 6, 2, 1; 2, 4, 1. (ii) 1, 21, 1; 5, 14, 1; 9, 7, 1; 3, 13, 2; 7, 6, 2; 5, 5, 3; 3, 4, 4; 1, 12, 3; 1, 3, 5. (iii) 2, 8, 3. (iv) 8, 38, 50; 19, 44, 36 ; 30, 50, 20; 41, 56, 5. 5. (i) 1326, 2; 441, 3; 101, 8; 77, 10; 33, 21; 25, 27; 5, 112; 1, 333. (ii) 6, 3. (iii) 8, 6. (iv) 6, 1; 13, 14. 7. 195, 121; 62, 264. 8. 8. ©. 20. lO. 3. 11. £3. 14s. 6d., £4. 5s. 6d. 12. 2s. Id., 2«. lOd., 2«. lid., Bs. Id., 3s. 2d., 3s. 3d., 3s. 4d., 3s. 5d., St. 6d., 8s. 6d., 8s. 9d. and 4s. 18. 11,12,15,24,36. 14. 15, 55; 26, 65; 35, 75. 15. 21. EXAMPLES XLI. 86 31 11664 11124 10609 • 67' 67* • 33397' 33397' 33397* *' 117 8. 3n2 + 6n + 2 pence. 15. -:. 16. 5. 21. 5. 4 ^ (7 EXAMPLES XLIII. 1. (i) q. (ii) 27g. (iii) - 2p. (iv) - 3q. (v) 2pK (vi) Sq. (vii) -2i>2. (viii) Sp^. (ix) -p^. (x) piq. (xi) i?/2g. (xii) -2>2/(8gr2+^3). 2. (i) 0. (ii) -Sp. (iii) -4g. 3. (i) 8i)3-16p^ + 64r. (ii) {q^-4:pqr + 8r^)lT^. (iii) {q^-p^r)lr*. 4. (i) 28, -24. (ii) 44, -252. 5. {i)pi'-2p,. {ii) Bp^p^-p^^-Sps. (iii) (Pn-l'-2Pn-2Pn)/Pn'. • (iv) 2'i+2'n-i(2p2-V)/2>n- (V) (2'i' + 32^3 -3piPo)p«_i/i)„ + 2^2 -i)!^. (vi) {BpiP^-pj^ - Bp^) (jp„-i3 - 2p^_^p^)lp^^+p^. 644 ANSWERS TO THE EXAMPLES. e. ar3-10a;3 + 31a; -31 = 0. 7. 6. 8. (i) x'-qx'^+prx-^r^=0, (ii) s(^ + 2px^ + {p^ + q)y -r+pq = 0. {iu).x^^{r-pq)+x^{Sr-2pq+p'^) + x(Sr-pq) + r=0. ^' (iv) a^-2qx^+{q^+pr)x + r^-pqr = 0. 2r (v) Eliminate x between given equation and y = {p + xy^-i — . X (vi) y3 - {Sq - jp«) y^ + (Sq^ - qp^) y + rp^-q^ = 0. 9. (i) Substitute - {y +p) for x. (ii) Substitute -^{y +p) for x. (iii) Substitute ^ {p^ -^q-y) for x\ lO. (y + r)3 + q^y +pY - ^pqy (y + r) = 0. EXAMPLES XLIV. 1. 2±JS, -3±J2. a. |,3±V^. o 6. x^-16x^ + 4. = 0. 6. a;* + 2ar» + 25 = 0. 8. l±v^^l±2V^. 9. 2g^-9pqr + 21r*=0. 10. 2)3 - 4pg + 8r = and {p^ + 4$) (36g - IV) - 1600s. = 0. 11. 6,1, -3. 12. 4, 1, -2, -5. 13. (i) 3. (ii) +4, -4. (iii) |. 14. 8, ±^. 16. ±J2, -2±V7. 16- r3-pgr+i)2g^o. 17. 3, 7, -10. 18. 3, 9, -|. 20. (i) r2- 2)25=0. (ii) 2)2s+r2=42«. EXAMPLES XLV. 1. (1) -i, -|,4. (ii) 3,3, ±2 V^i. (iii) i, |. i. -|. (iv) 4, 4, 4, - i . 2. (6c - ad)2 = 4 (ft^ _ ^c) (c' - bd). 9. (i) y3-6^t/2 + 4(4py_s)y_8(22)2«_3grs + 2r2). (ii) {y - 6q)^ + 6g (t/ - 6q)^ + 4 (42)r - «) (y - Qq) + 8 (2^)% - Sqs + 2r») =0. lO. (y - 162)2 + 24^)3 - 24g {y - 16p^ + 24g)2 + 64 (ipr -s)(y- IQp- + 24g) - 512 {2ph - Sqs + 2r'^) = 0. ANSWERS TO THE EXAMPLES. 645 EXAMPLES XL VI. 1. (i) -5, -w-4w2, ~u^-^(a,i.e. -6, 0=^0'^"^* (ii) -4, -w-3a;», -w^-Sw. (iii) 10, 2a; + 8w2, 2w3 + 8w. (iv) 8, a) + 7&>2, w3 + 7a,. (v) -2094..., - 1-703. ..w- -391. ..w^ - 1-703. ..w»--391...w. (vi) 30913, 2-1699w+-9214w2, 2-1699w2+-9214w. a. (i) -1, -3, l±2i. (ii) 1±V2, -1=*= 2 v'"^ (iii) 3±V^ -3±V^rT. (iv) -2, |, i(-l±7^0[5). 3. (i) One real root between - 3 and - 2. (ii) One between -7 and -6, one between 1 and 2, and one between 6 and 6. (iii) One between and 1, and one between 1 and 2. (iv) Two between 2 and 3, one between and - 1, and one between - 4 and - 5. (v) One between 2 and 3, and one between - 3 and - 4. 8. (i) 1.-3569, 1-6920. (ii) 4-189L (iii) -4679, 1-6527, 3-8793. (iv) 2-2317. (v) 2-1622, 2-4142. (vi) 1-1487. 9. (i) 3, 3, -4, -4. (ii) 3, 3, -3±V8. 11. 1.3, -g, ±V^2. MISCELLANEOUS EXAMPLES. I. (i) -{h-e){c-a){a-b){2a)\ (ii) iab-cd){ac-dh){ad-bc). (iii) Sa (2Sa2 - S6c). (iv) (6 + c) (c + a) (a + 6) Sa^. 6. {(6-c)» + (c-a)« + (a-6)a}». 7. 3 (6-c) (c-a)(a-6). 21. (6+c-a)(c + a-6)(a + 6-c)=0, (y + z-x){z + x-y){x + y-t)=0. 28. (i) -2, |(-3±V5). (ii) a, 6, i{a+6=fc(a-6)^-3}. (iii) 0, ±,^6c±>^ca±/^a6 (ail signs being positive or only one positive). (iv) A =^L = * =^. (V) 0,0, 0; 1, 1, 1. ^ ' b-c c-a a-b ^abc (vi) 2abc a; = =t {c^-i + a'ft^ - V^c^), &c. (vii) 0,0,0; ±4, ±4, ±4; ±1, ±1, ±1. [Signs in ambiguities must be taken so that xyz may be positive.] . .... - - - a + b + c a + b + c a + b-^c 646 ANSWERS TO THE EXAMPLES. (ix) 1, -1, 1, 2; 1, -1, 2, 1. (x) a{b-c),b{e-a),c{a-h)i a-b b-e c-a \a-b b-e c-a) l\a^ b"^ cj' (xi) 0,0,0;0,^^>,a^;&c. O — C C — (xii) 0, 0, 0; a'^-bc, b^-ca, d^-db. 31. 'p'^ + q^ + 7^-qr-rp-'pq=^0. 32. 13, 156; 14, 84; 15, 60; 16, 48; 18, 36; 20, 30; 21, 28; 24, 24. 36. 49, 225, 1225. 37. Digits 1, 0, r-1. 38. 7,13,17,23; -\, 1, -\, \. 39. l+|^+2|3_+|4_+2(5+|6. 42. (2«-2)/n! 46. n(n + l)! 61. sl^+K/Tj' ea. l + 6<. es. 3»-i-3n + 4. 64. {(V5 + ir+i + ( - 1)" (v/5 - l)«+i}/ 2«+i . ^5. 2, « 1 66. 3log.2--. 83. (i) 2). (ii) 0. (iii) If a>l, D. If a:f 1, C. (iv) D. . (v) C. 64. m