GIFT ©r •v'l C e^GMEERlNG LIBRARY A \ r^' MESSRS MACMILLAN AND CO.'S PUBLICATIONS. WORKS BY THE SAME AUTHOR. ELEMENTARY ALGEBRA. Second Edition. Globe 8vo. /\s. 6d. SOLUTIONS OF THE EXAMPLES IN CHARLES SMITH'S ELEMENTARY ALGEBRA. By A. G. Crack- NELL, B.A., late Scholar of Sidney Sussex College, Cambridge ; Mathematical Master of Sunbury Military College. Cr. 8vo. los. 6d. A TREATISE ON ALGEBRA. Second Edition. Cr. 8vo. ys. 6d, 4 SOLUTIONS OF THE EXAMPLES IN A TREATISE ON ALGEBRA. Cr. Svo. loj. 6d. AN ELEMENTARY TREATISE ON CONIC SECTIONS. Cr. Svo. js. 6d. SOLUTIONS OF THE EXAMPLES IN AN ELEMENTARY TREATISE ON CONIC SECTIONS. Cr. Svo. los. 6d. AN ELEMENTARY TREATISE ON SOLID GEOMETRY. Cr. Svo. 9^-. 6d. MACMILLAN AND CO. LONDON. A TEEATISE ON ALGEBEA ^^m' "^ TEEATISE ON ALGEBRA livr CHARLES SMITH, M.A. MASTER OF SLDNBY SUSSEX COLLEGE, CAMBRIDGE. FOURTH ED in ox Hontion: MACMILLAN AND CO. AND NEW YORK. [The night of Translation is reserved.'] t J « c First Edition, 1888. Second Edition, 1890. Third Edition, with additions, 1892. Fourth Edition, with additions, 1893. GIR ^^/^. S^^:?- tNGIN. UBRARY PREFACE TO THE FIRST 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 making 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 M166388 VI PREFACE. very few exceptions, every paper winch 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. E. Wilson, M.A., Mr J. Edwards, M.A., Mr S. L. Loney, M.A., and Mr J. Owen, BA. 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 II. Fu2Jdame:ntal Laws. Nef;ative quantities Addition of terms Subtraction of terms .... Multiplication of monomial expressions . Law of Signs The factors of a product may be taken in any order Fundamental Index Law .... Division of monomial expressions . Multinomial expressions .... Commutative Law, Distributive Law and Associative Law 9 10 11 14 15 16 18 19 22 23 CHAPTER TIL Addition. Subtraction. Brackets. Addition of any multinomial expressions 2G Subtraction of multinomial expressions 27 Brackets 28 Examples 1 29 Vlll CONTENTS. CHAPTER ly. Multiplication. Product of two multinomial expressions . Detaclied coefficients Square of a multinomial expression Continued products . , Examples II. . PAGE 32 36 39 40 42 CHAPTER V. Division. Division by a multinomial expression Extended definition of division Examples III. CHAPTER VI. Factors. Monomial factors . Factors found by comparing with known identities . . Factors of quadratic expressions found by inspection Examples IV Factors of general quadratic expression .,..,. Factors found by rearrangement and grouping of terms . , . Examples V Divisibility of a;"^ ^ a" by a; ± a . Remainder Theorem An expression of the 7ith degree cannot vanish for more than n values of x, unless it vanishes for all values of a; Cyclical order Symmetrical expressions ......... Factors found by use of Remainder Theorem Examples VI CHAPTER VII. Highest Common Factor. Lowest Common Multiple. Monomial common factors Multinomial common factors 46 50 51 53 54 55 56 57 60 64 65 66 69 71 71 72 73 Examples VII. Lowest common multiple . Examples VIII. 76 77 84 84 86 CONTENTS. IX CHAPTER VIIT. Fractions. A fraction is not altered "by multiplying its numerator and denomi nator by the same quantity Reduction of fractions to a common denominator ... Addition and subtraction of fractions Multiplication and division of fractions Important theorems concerning fractions formed from given fractions Examples IX. PAGE 88 90 91 93 96 99 CHAPTER IX. Equations. One Unknown Quantity. General principles applicable to all equations Simple equations Special forms of simple equations The problem of solving an equation the same as the problem of finding the factors of an expression . Quadratic equations Discussion of roots of a quadratic equation Zero and infinite roots . . . . . Equations not integral ...... Irrational equations A quadratic equation can only have two roots . Relations between the roots and the coefficients of a equation . Relations between the roots and the coefficients of any equation Equations with given roots ..... Discussion of possible values of a trinomial expression Examples X. . . Equations of higher degree than the second Equations of the same form as quadratic equations Reciprocal equations Roots found by inspection Binomial equations . Cube roots of unity . Examples XI. . quadratic 106 107 108 110 110 113 lU 116 119 121 123 124 125 127 130 135 135 137 138 139 110 141 CONTENTS. CHAPTER X. Simultaneous Equations. Equations of the first degree with two unknown quantities Discussion of solution Equations of the first degree with three unknown quantities Method of undetermined multipliers Equations with more than three unknown quantities Examples XII. Simultaneous equations of the second degree . Examples XIII Equations with more than two unknown quantities Examples XIV. PAGE 145 149 150 151 154 155 157 163 165 170 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^ 4-^,3 ^c3_3(j;,c . , . 187 Examples XVI 190 CHAPTER XIII. Powers and Roots. Fractional and Negative Indices. Index Laws Boots of arithmetical numbers Surds obey the Fundamental Laws of Algebra Fractional and negative indices Rationalizing factors .... Examples XVII 200 202 203 205 210 212 •-if>- CONTENTS. XI CHAPTER XIV. Surds. Imaginary and Complex Quantities. PAGE rroperties of Surds 214-216 If a + fjb = c + fjd, where a and c ar3 rational and ^Jb and ,Jd are irrational, then a = c and i = 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 229 230 . 231 . 232 been found, as Square root of any algebraical expression Square root found by equating coefficients Extended definition of Square root . When any number of terms of a square root have 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 xii CONTENTS. PAGE Variation 248 Indeterminate Forms . . . . . . • • .251 Examples XX 253 CHAPTER XVII. The Progressions. Arithmetical progression 255 Geometrical progression . 260 Harmonical j)rogression ......... 265 Examples XXI 267 CHAPTER XVIII. Systems op Numeration. Expression of any integer in any scale of notation .... 271 Eadix fractions . 273 The difference between any number (expressed in the scale r) and the sum of its digits is divisible by r - 1 275 Eule 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 differen;; . 281 Combinations 283 ifir = ifin-r 285 Greatest value oi" ,^(.'^ . . 286 (x+l/)^n = a;^n + a;^n-l •y^l+ ••• +S^C'„ ....... 287 Vandermonde's Theorem [see also p. 310j 288 Homogeneous products [see also p. 352] 289 Examples XXIII 294 CHAPTER XX. The Binomial Theorem. Proof of the binomial theorem for a positive integral exponent . 297 Proof by indnction .298 Greatest term 301 Examples XXIV 303 Properties of the coefficients of a binomial expansion . . . 305 CONTENTS. XIU PAGE Continued product of n binomial factors of the form x + a, x-\-h, Sec. 308 Vandermonde's Theorem [see also p. 268] 310 The multinomial theorem 311 Examples XXV 314 CHAPTER XXI. COXVERGENCY AND DIVERGENCY OF SERIES. Convergeucy and divergency of series, all of whose terms have the same sign . ......... 318-32ft Series whose terms are alternately positive and negative . . . 326 Ajiplication to the Binomial, the Exponential and the Logarithmic Series 327-329 Product of convergent series 329 If 2a^'' = S&^T'', for all values of x for which the series are con- vergent, then aj. = b^ 331 Examples XXYI 332 CHAPTER XXII. The Binomial Theorem. Any Index. Proof of the theorem 535 Euler's proof 336 Greatest term 340 Examples XXVII 341 Sum of the first r + 1 cocfiicients of flo + aia; + flr2^2_|. 34.3 Binomial Series 344 Expansion of multinomials ........ 348 Combinations and Permutations with repetitions .... 350 Homogeneous products [see also p. 280] 352 Examples XXVIII 355 CHAPTER XXIII. Partial Fractions. Indeterminate Coefficients. 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. 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 385 Common logarithms 388 Compound interest and annuities , . 391 Examples XXXII 393 CHAPTER XXV. Summation op Series. Sum of series found by expressing ii^ in the form v„ - r^^j . . 395 Sum of series whose general term is {{a+ n-1 . b) {a + nb)...{a+ n-\-r -2 .())} . . 397 Sum of series whose general term is ll{{a+n^. h){a + nh)...{a+n + r-2.b)} . . 400 Sum of squares and sum of cubes of the first n numbers . - i03 Sum of 1'- + 2'-+...+ w*- 404 Piles of shot 405 Figurate Numbers 406 Polygonal Numbers 407 Examples XXXIII 407 c( r • 1 li • aUi' + a:){a + 2x)...{a+'n-l . x) .„ Sum of series whose general term is — i— — — — ^^ : 41U b {b + x) {b + 2x) ... (h + n - 1 . x) 'ZopxJ', where ap = Jrj9'' + ^r-i2^''~^ + -" + -^0 Series whose law is not given . Method of Differences Recurring Series Convergency of Infinite products 412 414 415 417 423 ^ ,. . <. ^, mim-1) . vi\m-l)i.)n-2) ,„, Conditions for convergency of l±m+ :, — tt- =•= ^ — o~5 + •-• ^^^ St<„ and ^a''^a „ both convergent or both divergent .... 42G CONTENTS. XV 2w- convergent or divergent according as the limit of n (-^) Examples XXXIV. PAQK 428 429 CHAPTER XXYL 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 prodnct is given, is least when the quantities are all equal . . . 438 If m, a, B,... are positive and m = a + j3 + ..., then m^ ^nm a/+... + a„'* ai^ + ...+a/ 439 n n n > ^ — 1- , unless a; is a positive fraction .... 442 n \n \ Examples XXXV 443 CHAPTER XXVII. Continued Fractions. Convergents of a + - are alternately less and greater than the continued fraction Law of formation of successive convergents Reduction of any rational fraction to a continued fraction Properties of convergents Examples XXXVI. . General convergent . Periodic continued fractions Convergency of continued fractions Reduction of quadratic surds to continued fractions Series expressed as continued fractions . Examples XXXVIT 447 447 449 450 454 456 458 460 465 470 473 XVI CONTENTS. CHAPTER XXVIII. Theory of Numbers PAGE The Sieve of Eratosthenes 479 Properties of primes , . 480-482 Highest power of a prime contained in |?i 483 The product of any a consecutive integers is divisible by |n . . 483 Fermat's Theorem [see also p. 493] ^^^ 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 XXXVni 490 Congruences ........... 492 Wilson's Theorem 494 Extension of Fermat's Theorem 497 Lagrange's Theorem . . . 498 Reduction of Fractions to Circulating Decimals .... 499 Examples XXXIX 501 CHAPTER XXIX. Indeterminate Equations. Integral solutions of ax Jcby=iC can always be found if a and h are prime to one another 504 General solution of ax -hy = c, having given one solution . . 504 General solution of aa; + &?/ = c, having given one solution . . 505 Number of positive integral solutions of ax + 6y = c . . . . 506 Integral p.o\utions oi ax + by + cz = d, a' x + b'y + c' z = d' . . . 508 Examples XL 510 CHAPTER XXX. Probability. Definition of probability 512 Exclusive Events 514 Independent Events 515 Dependent Events 51G CONTENTS. XVU PAGE Probability of au event happening r times in n trials . . . 518 Expectation 520 Inverse Probability .521 Probability of Testimony 523 Examples XLI 52G CHAPTER XXXI. Determinants. Definition and properties of determinants Multiplication of determinants Simultaneous Equations of the First Degree Elimination Sylvester's method of Elimination . Examples XLII 530-542 . 543 . 545 . 547 . 548 . 549 CHAPTEE XXXII. Theory of Equations. Every equation of the nth. degree has n roots . . . , Relations between the roots and the coefficients of an equation Sum of the mth powers of the roots Sj'mmetrical functions of roots Transformation of Equations Reciprocal Equations Examples XLIII Imaginary and quadratic surd roots occur in pairs Eoots common to two equations Roots connected by any given relation Commensurable Roota Examples XLIV Derived Functions .... Equal Roots Continuity of a rational integral function If /(a) and/(j3) are of contrary signs, a real between a and /3 . . . The Discriminating Cubic RoUe's Theorem .... root 0f/(.T) lies 553 554 556 557 559 561 564 565 567 567 568 569 571 572 574 575 576 577 XVIU CONTENTS. PAGE Descartes' Rules of Signs 578 Examples XLV. .......... 580 Cubic Equations 582 Biquadratic Equations - 583 Sturm's Theorem 585 Synthetic Division 590 Horner's Method of approximating to the real roots of any equation 593 Examples XL VI 596 Miscellaneous Examples , . o , 599 Answers to the Examples ♦.,,,.,, 618 CHAPTER I. Definitions. 1. Algebra, like Arithmetic, is a science which treats of numbers. lu 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, f, b\, 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, wliether figures or letters, always represent numbers. On this account the word quantity is often used instead of number. 8. The sign + , which is read 'plus,' is placed before a number to indicate that it is to be added to what has gone y. A. 1 :^ , 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 + b 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 + 6 + 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 — 6 + 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. 5. The sign x, which is read 'into/ is placed between tAvo numbers to indicate that the first number is to be multiplied by the second. Thus axb means that a is to be multiplied by 6; also axb x c 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 sls 5 x a x b. 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 axb xc, 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 -^, which is read 'divided by' or *by/ is placed between two numbers to indicate that the first DEFINITIONS. 6 number, called the dividend, is to be divided by the second number, called the divisor. Thus a-r-b means that a is to be divided by b ; also a-^b-r-c means that a is to be divided by b, and the result divided by c ; and a-i-b X c means that a is to be divided by b 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 Sabx, 3 is the coefficient of aba), Sa is the coefficient of bx, and Sab is the coefficient of o). 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, aaaa 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 aV is written instead of aaabb, and similarly in other cases. The small figure, or letter, 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 o}, 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 ^a : thus 3 is ^27, since S'^ = 27. In general, a number which when raised to the ^th jDOwer, 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 a/ 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 >v/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 + or — are called the terms. Thus 2a — 3^^ + ocy^ is an algebraical expression con- taining the three terms 2a, — Sbx, and + 5cy'^. 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 5a^6^c and otc^b^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 midtinomial expres- sions ; expressions which contain two terms, and those which contain three terms are, however, generally called binomial and trinomial expressions respectively. Thus 3«6^c is a monomial, a^+ 36^ is a binomial, and acd'' -^-hx + o is a trinomial expression. 15. The sign =, w^hich 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>h means that a is greater than h. The sign < indicates that the number which precedes the sign is less than that which follows it. Thus 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 — h 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 b = o, 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 = S-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 decrease, 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 + G + 4. Hence + 10-(+4) = + 10-4 = + 6, and + G-(-4) = + G + 4 = + lU. 12 FUNDAMENTAL LAWS. So also, in all cases a — (+h) = a — h, and a — (—h) = a + h. 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 he 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 p)ositive quantity, it was proved in Articles 22 and 24, that a + (+6) = a + 6 (i) \ a-{-(—b) = a — b (ii) a — (+b) = a — b (iii) and a — (—b) = a + b (iv), We have now to prove that the above laws being true for all positive values of b must be true also for negative values. Let b be negative and equal to — c, where c is any positive quantity ; then + 6 = -|-( — c) = — c from (Hy, and —b = — ( — c) = + c from (iv). Hence, putting — c for + b, and + c for — 6 in (i), (ii), • • • • m • •\ jr\. f* SUBTRACTION OF TERMS. 13 (iii), (iv), it follows that these relations are true for all negative values of b, provided a + {— c) = a — c, a + (+ c) =a + c, a — (— c) =a-\- c, and a — {-\-c)=a — c, are true for all positive values of c ; and this we know to be the case. Hence the laws expressed in (A) are true for all values of 6. 27. Def. The difference between any two quantities a and h is the result obtained by subtracting the second from the^r5^. The algebraical difference may therefore not be the 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 h when the algebraical difference a — 6 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 -6and(iv) -3,4, - 6 and 5. Ans.l, -1, -4,0. Ex. 2. Subtract (i) 3 from - 4, (ii) - 4 from 3, and (iii) - a from -h. Ans. -7, 7, -h + 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? Ans. - '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? Ans. -10. A71S. 6, - -6, Ans. -2, 4. Ans. - 4. Ans. 0. Ans. - 6. 14 FUNDAMENTAL LAWS. Ex.5. Find the value of a-b + c and of -a + h~c, when a=l, 6 = - 2 and c = 3. Ex. 6. Eind the value of -a + b-c when a=l, 6= -2, c= - 1; also when a= -2, 6= -1, c= -3. Ex. 7. Find the value of a -(-&) + (- c) when a=-3, 6=-2, c=-l. Ex. 8. Find the value of - a + (-&)-(- c) when a= —2, 6= -3, c= —5. Ex. 9. Find the value of - ( - a) + & - ( - c) when a= -1, 6= -2, c= -3. Multiplication. 28. In Arithmetic, multiplication is first defined to be tlie 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 luliole 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 + 1 + 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 |; that is, we must divide f into four equal parts and take three of those parts. Each of 5 the parts into which f is to be divided will be ^ — - , and 5x3 53 by taking three of these parts we get »^— r- Thus s x j 5x3 ~7x4* MULTIPLICATION OF MONOMIAL EXPRESSIONS. 15 So also, ( - 5) X 4 = ( - 5) -f ( - 5) + (- 5) + ( - 5) = — 5 — 5 — .") — 5 = -'20. With the above definition, nmltipHcation by a negative quantity presents no difficulty. For example, to multiply 4 by — 5. Since to subtract 5 by one subtraction is the same as to subtract 5 units successively, _5=_1_1_1_1_1; .-. 4x(-5) = — 4-4-4 — 4 — 4 = -20. Again, to multiply — 5 by — 4. Since -4 = -l-l-l-l; .,(_5)x(-4) = -(-5)-(-5)-(-5)-(-5) = + 5 + 5 + 5 + 5 [Art. 2G] = + 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 tivo 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( + Z>) = + «6 (i) ^ (-a) X { + b) = -ab (ii) ( + a) x{-b) = -ah (iii) (-a) X (-6) = + a6 (iv) 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 unlike 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 h rows. Then the whole number of the dots is a repeated h times, that is axh. 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 integers, ah = ha. When the numbers are fractions, for example ^ and f , 5 3 5x3 we prove as in Art. 28 that 1^X7= if — r • And, by the ^^., 5x3 3x5, 5 33 5 above prooi tor mtegers, ^ — r = -. — ^ ; hence ^ x - = 7 x =-. ^ ^7x44x7 7447 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 h, 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 cs will be ah; also the number of c's in the first row will be a, and this is repeated b times. Hence, when a and h are integers, c repeated ah times gives the same result as c repeated a times and this repeated h 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 a xbx c = a X (he) (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 ahc ^ cab = eha, &c (C). 30. Since the fiictors of a product may be taken in any order, we are able to simplify many products. For example : 3ax4a = 3x4xaxa = 12a^ (- 3a) X (- 46) = + 3a X 46 = 3 X 4 X a X Z) = 12a6, {ahy = ahxah = axaxbxh = a'^h'\ {i^2af = \/2a x V2a = a/2 x V2 x aa = 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 d^ = aa, and a' = aaa\ we have So also a^ X a?= aax aaa = a^ = a^"^^ 14 T 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 - a? and - 2a?h by - 3a&3. Ans. -Sab, -a^, Qa^b'^. 2. Multiply - 2xy^ by - Sij^z, Sax^y by - Ba^xy'^, and 3a%c^x by 12ab^cx^. Ans. Qxy'^z, - 15a^x^y^, 36a^6^c%^. 3. Multiply la'^b^c^ by - Sa^bh\ and - 2ab'^xY by - ^a^^xY- Ans. -21a'^b»c^, 8a%^x^y^. 4. Find the values of ( - a)^, ( - a)'^, ( - a)^ and ( - a)^ Ans. a?, -a^, a*, -a^. 5. Find the values of ( - a6)2, {aPb^ and ( - Sab^c^)^. Ans. a%\ a»b\ -21a^b^c^. 6. Shew that the successive powers of a negative quantity are alternately positive and negative. 7. Find the cubes of 2a%, - ^ah'^c'^, and - 2a"bxhj\ Ans. QaPb^, - 21a%^c^ and - Sa%^x^y^. DIVISION OF MONOMIAL EXPRESSIONS. 19 8. Find the value of (-a)-x(-t)3, of {-2abn'^x(-3a-b)^, and of ( - Subc)- X {2a'^byK Ans. -a~b\ 2l(ja^b^, 72a^b^c^. 9. Find the vahie of Sabc - 2a-bc^ + ic\ when a = 2, &=-!, and c=-2. Alls. 12. 10. Find the value of 2a'bc-Sb'^cd+ ic\la - 5drab, when a= -1, b= -2, c= -3 and d= -4. Ans. -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-f-6-i-c = a-=-c-H6. 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-^6-^c = a-^ (be), which is usually written a -^ 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 a>n,we have for by Art. 31 a'" - a'* = a'"-". 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 Hence a^"^ -^ a^b = a^ x ¥ -^ a^ -7- b = ct' -^ a" x 6" -i-b = d'b, aud a'bY - a'b'c* = a'b\ 36. We have proved in Ai't. 28 that ax (— b) = — ahj .'. (— ah) H- (— 6) = a, and (— ab) -^a= — b', we have also proved that (-a) i-b) = + ab = (+ a) i+b) ; .'. (+ ab) ^ (—a) = — b, and (+ ab) -r (+ 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' = - d'b\ and - 2a'bc' -=- - Sa'bc" = | ac\ o EXAMPLES. 1. Divide 10a by - 2a, Sa-b^ by - 2ab^, and - la^bh^ by - Sa^^b'^c^-. 3 7 Ajis. -5, —-a,-a^bc^. 2. Divide -2a'>b'^c^ by ia^c"^, -Qx^y* by Sx^y, and -5a-b*x'^y^ by - 2ab*xY- Alls. - - a?b^c, - 2x-?/3, - ax^y^. 3. Multiply - 2oi?bc' by - ^aWc^ and divide the result by Sa^i^c^. Ans. -aWc. 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 wLich are collected below : + (+ a) = + a -\-{—a) = — a — (+ a-) = — a — {—a) = + a {+a){-b) {-a)(+h) {-a){-h) — ah — ah •{■ah ahc = cha = cah = &c. a"a'' = a ni+n (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 h ; 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 + J + c + &c., where a, h, c, &c. may be any quantities, positive or nega- tive. For example, the expression Sx^y — ^xy"^ — *Jxyz, which by (A) is the same as ox^y + {—^xy'^) + {—Ixyz), takes the required form if we put a for Kx^y, h for — f.^i/Vand c for (— ^xyz). 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 + 6 + c + &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-\-b + c = c + a + h = b + 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 he made in any order. 40. Since additions may be made in any order, we have a + (& + c + cZ+...) = (^ + c + cZ+...) + a (from E) = a-\-h-\- 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 — h -\- c — d may be written in the form -\- a -[- {— h) + c + {— d), we have + {+ a - 6 + c - c?} = + {+ a + ( - 6) + c + (- cZ)} = + a + (- 6) + c + (- d). When we say that we can add the terms of an expres- sion in succession, it must be borne in mind that the terms include the j^refixed 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— (6 + c + cZ+...) = a— 6 — c — cZ— ... 24 FUNDAMENTAL LAWS. 42. If c be any positive integer, a and h having any values whatever, then (a + 6) c = (a + 6) + (a + 6) + (a + &) + ... repeated c times = a + 6 + a + & + a + & + ... [Art. 40] = a + a + a + . . . repeated c times 4- 6 + 6 + ^ + . . . repeated c times = ac + ho. Hence, when c is a positive integer, we have {a + h)G= ao + ho (F). Since division is the inverse of multiplication, it follows that when d is any positive integer {a -\-h) -r- d — a ^ d -{- h -r- d. And hence (a + h) X c -^ d = {{a -\-h) X c] ^ d = {ac-[-hc)-^ d = ac ^d+bc-^ d, that is (a + 6)x-^ = ax-j + Z)X-i. 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 + he, then (a + 6) (— c) = - (a + 6) c = — ac - 6c = 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. MULTINOMIA.L EXPRESSIONS. 25 43. Since (a + 6) ^ c = (a + 6) x - c 1 7 1 = a X - + X - = a-^c + o-^c, 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 + cZ + 6 + . . . = (a + 6) -f c + ((Z + e) + ... = a + (b + c-\-d)-\-e-^... = &c., so that the terms of an expression may be grouped in any manner. Again, from Art. 29, it follows that abode ... = 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. Subteaction. Brackets. Addition. 46. We have already seen that any term is added hy 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 algehraical expressions, write down all the terms in succession with their signs unchanged. Thus the sum of a — 26 + So and — 4x^ — 4aj + 2 from 4tx^ — 5x- 7. 12. Subtract 5a' - 3a'b + ia'b' from 56' - 3a6' + 4a=6l 13. What is the difference between - 3a;' — 5xy + 43/' and -5x' + 2x2/-3f'i 14. What must be added to 26c — 3ca-4a6 in order that the sum may be 6c + ca ? 15. What must be added to 3a' -26' + 3c' in order that the sum may be 6c + ca + a6? 16. Simplify 3x-{2y + (5x -3x + y)]. 17. Simplify x-[3y + {3z-x- 2y] + 2x']. 18. Simplify y -2x-[z-x-y-x + z). 19. Simplify a-[a-6-{a-6 + c-a-6 + c- (/}]. 20. Simplify 2x - [3x -9y- {2x -3y-{x + 5y)}]. 21. Simplify a - [3a + c - {4a - (36 - c) + 36} - 2a]. 22. Subtract x - {dy - z) from y — {2x - z - y}. 23. Subtract 2m — (3m - 2?^ — m) from 2n — (3?^ - 2m — n). 24. Find the value of {a-(b- c)Y + {6 - (c - a)}' + {c-{a- 6)}' when a = -l, 6 = — 2, c=-3. 25. Find the value of {a' - (6 - c)'} - {6' - (c - a)'} - {c' - (a - 6)'} when a = 1, 6 = 2, c - - 3. CHAPTER IV. Multiplication. 5.?. Product of monomial expressions. The multiplication of monomial expressions was considered in Chapter II., and the results arrived at were: (i) The factors of a product may be taken in any order. (ii) The sign of the product of two quantities is + when both the factors are positive or both negative ; and the sign of the product is — when one factor is positive and the other negative. (iii) The index of the product of any two powers of the same quantity is the sum of the indices of the factors. From (i), (ii) and (iii) we can find the continued product of any number of monomial expressions. Thus ( - 2a'bc^) x ( - Sa%"c) = + 2a'^bc^ x Sa-^b^ from (ii), = 2 X B X a- . a^ . b . b^ . c^ . c, from (i), = 6a56^c*, from (iii). Again, ( - Sa^b) (- 5ab^) (- la'^b^) = { + Sa-b . 5ab^ \ ( - la^b^-) = -3 . 5 . 7 . a2 . a . a^ . 6 . 63 . 62^ - lOoa'b^. 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 -\-y) z = xz -\-yz (i). Since (i) is true for all values of ^, 3/ and z, it will be true when we put (a + h) in place of x ; hence [{a -{-h) + y] z = {a + 1) z + yz — az + hz + yz. ,'. {a + h + y)z = az-\-hz -\- yz. And similarly (a + b + c+ d + .,.) z = az -{■bz + cz + dz+ ..,, 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 midtiplying the separate terms of the multinomial exp7'ession 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 + b + c + ...) x(x-{-y + z-h ...); and, from Art. 38, this includes all possible cases. Put M for x-\-y + z-\- ...\ then, by the last article, we have {a-\b-\-c+ ...)M=aM+bM+cM+... = Ma + Mb-\- Mc-V ... = {x -{■ y + z + . . .) a -\- {x -\- y + z -{■ . . .)b + (^ + 2/+^+...)c+ ... — ax-^ay-\-az-\-...-\-bx-{-by+bz-{-...-\-cx-\-cy-{- cz+ . . . Hence {a -[-b + c -{-...) (x + y + z -\- ■.-) = ax -\- ay + az + . . . -\- bx + by -^ bz -{- . . . + ex + cy + cz + . . . MULTIPLICATION. 33 Thus, the product of any two algebraical expressions is equal to the sum of the products obtained by niultiplying every term of the one by every term of the other. For example (a + b)(c-\-d) = ac + ad + be -^ bd ; also (3a + 5b) (2a + Sb) = (3a) (2a) + (3a) (Sb) + (5b) (2a) + (5b) (Sb) = Qd' + 9ab + lOab + 15b' = Qa^ + 19ab + 1561 Again, to find (a -b)(c — d), we first write this in the form [a + (- b)} {c +(— d)], and we then have for the product ac + a(-d)-\-(-b)c-\- (- b) (- d) = ao — ad — bc + bd. In the rule given above for the multiplication of two algebraical expressions it must be borne in mind that the ter^ms include the prefixed signs. 56. The following are important examples : — I. (a + by =(a-\-b)(a-{-b) = aa-\-ab + ba-\-bb; .-. (a-{-by = a' + 2ab + b\ Thus, the square of the sum of any two quantities is equal to the sum of their squares plies twice their product. II. (a - by =(a-b)(a-b)=aa-\-a(-b) + (- b) a -h{-b)(-b) = d'-ab-ab-{-¥; ... (a-by = a'-2ab + b\ Thus, the square of the difference of any tiuo quantities is equal to the sum of their squares minus twice their product. III. (a + 6) (a - 6) = aa + a (- b) -{-ba + b (- b) = a^ — ab + ab- h^ ; .-. (a + b)(a-b) = d'-b\ Thus, the product of the sum and difference of any tiuo quantities is equal to the difference of their squares. S. A. 3 34 MULTIPLICATION. 57. It is usual to exhibit the process of multiplication in the following convenient form : a"- + 2a6 - 6' a'^-2a6 +6^ - 2a'b - 4>a'b' + 2ah' a'b' + 2ab' - 6' a' - 4a^6^ + 4>a¥ - b\ The multiplier is placed under the multiplicand and a line is drawn. The successive terms of the multiplicand, namely a^ + 2ab, and — b^, are multiplied by a^, the first term on the left of the multiplier, and the products a*, + 2a^b and — a^¥ 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 + b a + b a^ + ab + h^ a + b a-b a-b a^ + ab d^ + ab d-^ + a-b + ab^ + ab + b^ -ab-b^ - d^b - ab"^ - b- a + b + c Sx^-xy + 2y' a + b + c Sx^ + xy-2y^ a- + ab + ac 9x^-dx^y + 6x^y^ + ab +b^+ be +3x^y- x^y^ + 2xy^ +ac + bc + c^ -Qx'h/ + 2xy^-h/ d^ + 2ab + 2ac + b^ + 2bc + c^ dx'^ - x^y^ + 4.xy^ - 4y'^ 58. If in an expression consisting of several terms which contain different powers of the same letter, the MULTIPLICATION. 35 term which contains the highest power of that letter be put first on 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' + a'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. CO. Definitions. A term which is the product of n letters is said to be of n dimensions, or of the nth degree. Thus 3a6c 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 liighest 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 ax"^ -\-bx-\-c is 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 ?/, 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 Xy 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' + »3a'^6 — 56^ is a homogeneous expression, every 3—2 36 MULTIPLICATION. term being of the third degree ; also ac^ + 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 sxiwj 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^ — x + ^hy *^x^ + 2x-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 9x* + Sx"" - 2^^^ +Qx-4s. When some of the powers are absent their places must be supplied by O's. Thus, to multiply x* -2x^ + x-S by x* -\-x^ -x- 3, we write 1+0-2+1-3 1+1+0-1-3 1+0-2+1-3 1+0-2+1-3 -1-0+2-1+3 -3-0 + 6-3 + 9 l+l_2-2-5-l+5+0+0 Hence the product is x' + x'- 2x' - 2x' - 5x' -x' + Bx' + 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' + 2ab + b' (i), (a-by = a'-2ab-\-h' (ii), (a + 6) (a - 6) = a' - 6^ (iii). A general result expressed by means of symbols is called Si 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 b 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 — 6 in the place of b in (i) ; we then have {a + (- b)Y =a'-^2a (- b) + (- 6)^ that is (a - by = a'- 2ab + ¥. Thus (ii) is seen to be really included in (i). Put \/2 in the place of b in (iii) ; we then have (a + V2) {a-^/2)=a'-{^/2f=a' - 2. [We here, however, 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 {a-\- (b + c)Y = a' + 2a (b + c) + {b + cy ; .\(a-^b + cy = a' + 2ab-i-2ac + ¥ + 2bc + c' (iv). Now put — c for c in (iv), and we have {a + b+.(-c)Y = a' + 2ab+2ai-c) + b' + 2bi-c)-\-{-cy; :. (a + b -cy = a' + 2ab -2ac + b'' - 2bc+ c\ Put 6 + c in the place of b in (iii); we then have [a + {b+c)}{a-{b + c)}=^a'-{b + cy = a'-{b'+2bc + cy, :,{a + b + c)(a-b-c) = a'-¥- 2bc - c\ The following are additional examples of products which can be written down at once. (a2 + 262) (^2 _ 21)-) = {rt2)2 _ (262)2 = a* - 46-1. (a2 + ^362) (a2 - ^36^) = (a2)2 - (^36^)2 = a* - 36^. (a-6 + c)(a + 6-c)={a-(6-c)}{a + (6-c)}=a2-(6-c)2. (a2 + a6 + 62)(a2-a6 + 62) = {(a2 + 62) + a6}{(a2 + 62)-a6} = {a^ + 62)2 - (a6)2 = a* + a262 + b\ {x^ + x^ + x+l){x^-x'^ + x-l) = {{a^ + x) + {x^ + l)}{{x^ + x)-{x^+l)} = (x3 + a;)2 - (aj2 + l)^ = x^ + 2x* + x^ - {x^ + 2x'^ + l) = x^ + x*-x^-l. MULTIPLICATION. 39 65. Square of a multinomial expression. We have found in the preceding Article, and also by direct multiplication in Art. 57, the square of the sum of tliree 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 + h + c-{-d+ ...){a + h + G-\-d+ ...). 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 oJ^ of the product: we similarly obtain the terms 6^ c^, &c. We can multiply any term, say h, of the multiplicand by any different term, say d, 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 hd, 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 algehrai- cal quantities is equal to the sum of their squares together with twice the product of every pair. For example, to find {a + h + cf. The squares of the separate terms are a?, IP, c^. The products of the different pairs of terms are ah, ac and be. Hence {a + b + c)' = a' + b- + c- + 2ab + 2ac + 2bc. Similarly, {a + 2b - 3t)2 = a2+ (26)2 + ( - 3c)'^+2a{2b) + 2a( - Be) + 2 {2b) (- Sc) = a2 + 4^2 + 9c2 + 4ai - Gac - 12bc. 40 MULTIPLICATION. And {a-b + c-d)^ = a^+{-b)'^ + c" + {-d)^ + 2a{-h) + 2ac + 2a{-d) + 2{-b)c + 2{-b){-d) + 2c{-d) = a^ + b^ + c^ + d^-2ab + 2ac - 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 x^ + ax -Vbx-\-ab iC + C a;^+ [a-\-b)x'--\-abx + CT? + (a + 6) ca; + aba o;^ + (a + 6 + c) x^ + {ab + ac - 6c) a; + abc 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^ + c^f [x + af {x - a)\ The factors can be taken in any order ; hence the required product = [{x -a)(x + a) (x2 + a2)]2 = [(x^ - a^) {x^ + a^)f = {x* - ay=x^ - 2a'^X^ + 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 + b) {a + 6), 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 this 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 6 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 Sa¥. 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 + Sah' + b\ that is (a + bf = a' + Sa'b + Sa¥ 4- b\ The continued product (oc + a) {x + b) (x + c) can simi- larlv 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 6 or c: we thus have x'^a, x^b and x'^c. We can take one x and any two of a, b, c: we thus have xab, xac, and xbc. Finally, if we take no x'a, we have the term abc. 42 MULTIPLICATION. Thus, arranging the result according to powers of x, we have {x + a) {w + b) {oc -{- c) = x^ + x^ {a -\- b -\- c) -\- OS (ab -\-ac-\- be) 4- 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 = 0" + 2ab + b'\ (a + by = a' + Sa'b + Sab' + b\ and (a 4- by = a' -\- 4a'6 + Qa'b' + 4^a¥ + 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 II. 1. Multiply 2x-a by x - 2a. 2. Multiply 3£c-i by ^cc- 3. 3. Multiply x^ + X + 1 by x-l. 4. Multiply x^-xy + y^ by x + y. 5. Multiply \-¥x+v? -vv? by x-\. 6. Multiply x^ + x^y + ^y^ + xy^ + y" by y-x, 7. Multiply x^ -x^2 by x^ + x-2. 8. Multiply 1 + ace + a^x" by 1 - ax + a V. 9. Multiply X* + x^ +\ by x* - x" + \. EXAMPLES. 43 10. Multiply 3a;'' -xy + 2if by 3?/' -xy + 2m\ 11. Multiply x^ -bx^ +1 by 2x^ + 5ic + 1. 12. Multiply 2a;'' - bx'y + y" by f + f>xy^ + 1x\ 13. Multiply 3a'-2a'6 + 3a6'-36'by "la? ■¥ ba:h - iah^ ■Yh\ 14. Multiply 2aV - 3aV2/' + 5/ by aV + 4a.xy - 2/. 15. Multiply 2a - 3a' + ba? - 7a' by 1 - 2a' + 6a\ 16. Multiply a^ — ab- ac + b'' -be + c^ by a + 6 + c. 17. Multiply x^ +y' + ii^ - yz — zx — xy by aj + 2/ + 2;. 18. Multiply 4a' + 96' + c' + 36c + 2ca - Gab by 2a + 36 - c. 19. Multiply together 03*+ 1, a;'+l and a;'-l. 20. Multiply together x* + lGy'*, a;'+ iy', x + 2y and x - 2y. 21. Multiply together {x - y)\ (x + yf and (a;' + ?/')'. 22. Multiply together (x' + 1)^ {x + 1)^ and {x - l)^ 23. Multiply together a;' - a; + 1, x^ +x+ 1 and a;* - a;' + 1 . 24. Multiply together a'-2a6 + 46', a' + 2a6 + 46' and a'-4a'6'+166\ 25. Find the squares of (i) a + 26 - 3c, (ii) a^ - ab + b', (iii) 6c + ca + a6, (iv) 1 - 2a; + 3a;', and (v) x^ + x^ + x+ 1. 26. Find the cubes of (i) a + 6 + c, (2) 2a - 36 - 2c and (iii) 1 +a; + a;'. 27. Simplify (x + y + zY-{~x + y + zy + {x-y + zf- (x + y- zy. 28. Shew that (a; + y) {x -f- z) - .^•' = (y + z)(y + x)~y' = (z + x) {z + y) - z\ 29. Shew that (y + zY +(z + xf +{x + yf -x'-y'- z' ^{x + y + z)\ 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^ + b^={a + by - Sab {a + b), and that a*+b*=(a + by - iab {a + by + 2a'b\ 33. Shew that {x^ + xy + fy - ixy (x^ + y') = {x' -xy+ yy. 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 x = a + dj y = b + d, and z = c + d ', then will of + y^ + z^ - yz^zx — xy = a^ + b^ + c^ -be — ca— ab. 37. Shew that, if a; = & + Cj y = c + a, and z = a + b; then will x^ ^-y^ + z^ — yz — zx — xy = a^ +b^ + G^—bc — ca- ab. 38. Shew that 2(a-6)(a-c) + 2(6-c)(6-a) + 2(c-a)(c-6) = {b-cy + {c-ay+{a-by. 39. Shew that {x" + y^ + z') {a' + b' + c') - {ax + by + czy = (bz — cyy + (ex — azy + (ay — bxy. 40. Shew that, if x = a^—bc, y = b^ — ca, z = c^ — ab; then will ax + by + ez = {x + y + z)(a + b+e), and be (x^ — yz) = ca {jf — zx) = ah {z^ — xy). 41. Find the value of {x - ay ■\-{x- by + {x-ey -S{x- a) {x -b){x- e) when Zx = a-\-b + c. 42. Shew that {d' + b^ + cy = {b^ + cy + (ab + aey + (ab - aey + a" = (be + ea + aby + (a' - bey + (b' - cay + (c' - dby. 43. Shew that (x^ + xy + y^) (a^ ■\-ab + b^) = (ax - byy + (ax - by) (ay + bx + by) + (ay + bx + by)". EXAMPLES. 45 44 . Sliew that l-\-a' + b' + c' + b'c' + c'a' + a'h' + a'b'c' — (I — be — ca — ah)' + {a + b + c — abcy. 45. Shew that (a' + 6» + c' + dy = (a- + b'- c"- dy'+ 4{ac + bdy+ 4{ad - be)'. 46. Shew that (i) {a + 2y - 4{a + ly + (ja' - 4:{a- ly + {a-2)' = 0. (ii) (a+2)(b + 2)-i{a + l){b+l) + Gab -4{a-l){b-l) + (a-2){b-2)^0. 47. Shew that (i) {a + 2y-i{a + l y + Ga' - 4 (« - 1)^ + (a - 2)-'' = 0. (ii) {a + 2){b + 2){c + 2)-4:(a + l){b + 1) (c + 1) + Qabc -4:(a-l){b-l){G-l) + {a-2){b-2){c-2) = 0. 48. Shew that (a + b + cy + {b + c-a) (c + a -b) (a + b-c) = W (b + c) + 46^ (c + a) + 4c^ (a + J) + 4a6c. 49. Shew that x(x-'i/ + z)(x + y-z) + y(x + 7/-z)(-x + y + z) + z{- x+y + z)(x - y + z) + (- X + y + z) {x - y + z) (x + y - z) = 4a;2/2?. 50. Multiply a' + b' + c' + d^-bG-ca-ab - ad-bd-cd by a + 6 + c + cZ. 51. Shew that (x'+x+l){otf^x+l)(x*-x'+l){x'-x'+l)...{x'^-ar'"\\) ol + l o" -1 = x' +x- +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 (a^x — Sax) -~ax = c^x -r- ax — Sax -^ax — a — S. And (12^' - hax'^ - 2a^ ^ dx = 12x' -i- Sx - 5ax' -r Sx 2a^x ■r-Sx = 4!X^ — ^ax — lo^. 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 (piotient 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 be found by dividing the first term of the dividend by 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 beinof also arrancjed according to descending powers of a, the second term of the quotient tvill 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 third and other terms of the quotient can be found in succession in a similar manner. For example, to divide 8a' + Sa^b — 4a6'^ + W by 2a + b. The arrangement is the same as in Arithmetic. 2a + 6 ) 8ct' + Mb + 4a6' + b^ ( 4a' + 2a?> + 6' 8a^ + 4a'6 4a=^6 + 4a6' + 6' 4a'^6 + 2a6' 2a6'^ + 6^ 2a6^ + 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 4a'6 + 4a6''' + 61 The second term of the quotient is 4a'6 -4- 2a = 2a6. Multiply the divisor by 2a6, and subtract the product from the remainder : we thus get the second remainder 2a6' + 61 The third term of the quotient is 2a6' -h 2a = 6'. Multiply the divisor by U\ and subtract the product from 48 DIVISION. 2ah^ + }fy 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 4- 2a6, and by + h^ ; we have therefore subtracted altogether the divisor multiplied by 4a^ + 2(x6 + 61 And, since the divisor mul- tiplied by 4a^ 4- 2a6 -I- h^ is equal to the dividend, the required quotient is M + 2ah + 6^ 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 h ; 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% + 2a262 _ ah^ + 6* by a^ + &2. a2 4- 62 ) a4 _ ^35 + 2a2i2 - a&3 + &" (a^ - a& + 63 -a^ + a^h^ -a63 + 64 - a^h - aW + a262 +64 Ex. 2. Divide a^ + a^fts + 54 by a2 - a6 + 62. a2-a6 + 62)a4 +a262 +h^(a^-\-ab + h^ + a^b +64 + a36-a262 + a5g + a%'^-ab^ + b* + a262-a63 + 64 In this example the terms of the dividend were placed apart, in order that ' like ' 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 should never be departed from. Ex.3. Divide a^ + b^ + c^ - Sabc by a + b + c. DIVISION. 41) a + b + c)a^- 'dabc + b'^ + c'^ (a^-ab- ac + b'^ -bc + e^ a^ + a-b + a^c - a^b - d^c - 3abc +b'^ + c^ - a?b - ab^ - abc - arc + aft" _ 2abc + b^ + c^ — (I'C - abc — ac- + ab^ -abc + ac^ +b^ + c'' - abc + ac"^ - b'^c + c'^ - abc — b^c - bc^ + ac^ + be- + c^ + ac^+bc'^ + c;^ Where, as in the above example, more than two letters are involved, it is not sufficient to arrange the terms according to descending povv^ers of a; but b also is given the precedence over c. By using brackets, the above process may be shortened. Thus a + b + c)a^-Sabc+b^ + c^(a^-a{b + c) + {b^-bc + c") ^a^ + a-{h + c) ^ -d'{b + c)-Sabc + b^-i-c^ ^a^{b + c)-a{b + c)^ a{b^-bc + c^) + b^ + c^ a{ b^-bc+c^) + b3 + c^ 72. The method of detached coefficients may often be employed in Division with great advantage. For example, to divide 2x' -7x' -}- 5x' + Sa)' -Sx'' + 4!X-4i by 2^' - 3^' + *• - 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 x^ and the otlicr powers follow in order : thus the quotient is x'-2x''-x-h2. S. A. 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 w^e 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 xG is either equal to A, or differs from A by an expression which is of lower degree, in some l^articular letter, than the divisor B. For example, if we divide a^ + 8a6 + W by a 4- &, we have a + 6 ) aH 3a5 4- 46' ( a + 26 a'+ ab 2a6 + W 2ab + 2b' Thus {a? + Zab + 46') 4- (a + 6) = a + 26, with remainder 26' ; that is a' 4- 3a6 + 46' = {a-\-b){a + 26) + 26'. We have also, by arranging the dividend and divisor differently, 6 + a ) 46' + 3a6 4- a' ( 46 - a 46' 4- 4a6 — a6 4- a' — a6 — a' 4- 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 : {x^-¥2ax + a^)-r {x^ - 2ax + a^) -^- {x^-a")--- (x' + a') - {x' - a*) - [x ■\- a) = X -{- a. (x — a) = X — a. [x±a)=xT a. {x T ct) =x^ ±ax-\- a^. {x^ a) =x^ ± ax^ + a^x ± aj\ {x*' + aV + a*) -^ {x^ T dec -^^ a^) — x^ ± ax + a^. (a;*+ 2/' + / — ^xyz) ^ [x -\- y -\- z) = x^-\- y^-\- ^ — yz — zx~ xy. EXAMPLES III. 1. Divide ar* - 9?/* by a; + Sy. 2. Divide a;* - 16^ by a' - iy\ 3. Divide 11^ + 6 V by 4?/ + 3aj. 4. Divide 3ar - ixy - iy^ by 2y - x, 6. Divide I - 5od^ + 4x'^ by 1 - x. 6. Divide a^ - 5xy^ + 4:y^ by x-y. 4—2 52 EXAMPLES. 7. Divide 8. Divide 9. Divide 10. Divide 11. Divide 12. Divide 13. Divide 14. Divide 15. Divide 16. Divide 1 - 6x^^ + 5x^ by 1 - 2a; + x\ m^ — 6mn^ + 6n^ by m^ — 2m7i + 7^^. 1 - 7£c« + Gx" by (1 - xy. 1 - a^ by 1 -x^. l+aj-8a;'+19a;'-15a;^ by l + 3ic-5a:^ 4 - 9x' + 12a:' - 4aj' by 2 + 3x- 2x\ 4a;" - 9a;y + 6 V - 2/' by 2a;'' + 3a;?/ - y^ x^-Sx^ + 3x + y^ -I by a; + 2/ - 1. a;^ + x^y + x^y^ + x^y^ + xy"^ + 'if by x^ -\-xy ^y"^ X' - 5a;V + 7a;'2/^ - x^y^ - ixy" + 2y^ by a;^ - 3a;^v/ + 3a;^^ - v/^ 17. Divide a' - 26' - 6c' + ab-ac + 7bG by a - 6 + 2c. 18. Divide a' h- 26' - 3c' + 6c + 2ac + 3^ 25. a;* - 2aa; - 6' + 2a6. 26. a;' + 2xy - a' - 2ayr-- 27. 4 (a6 + C6?)' - (a' + 6' - c' - cZ')'. 28. 4 (a;2/ - ab)' - (a;' + 2/' - a' - U)\ FACTORS. 57 80. Factors of general quadratic expression. We proceed to shew how to iiiid the factors of any ex- pressiou of tlie second degree in a particular letter, x suppose. The most general quadratic expression [Art. 60] in x is ax^ -\-hx-\- c, where a, h 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^ + hx-\-c con- sists in changing it into an equivalent expression which is the difference of two squares. We first note that since x^ + 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 of x. For example, x^-\-5x is made a perfect square, namely ( a; + ^ ] , by the addition of ( - J ; also x^-'px is made a perfect square, namely / a; - - j , by the addition of [ - -^ ) = x * 81. To find the factors of cuic^ -\-hx-\-c. h c ax^ -\-hx + c = a[x'^ -\- - x + - \ a aj Now x^-{--x\s> made a perfect square, namely ( ^ + — J , by the addition of ( — ) = — . And, by adding and sub- tracting - 2 ^ the expression within brackets, we have 58 FACTORS. alos + -a)-\- 7-^ — 1—7^ + - = a ■{\x + 2 J \y 4a'^ Jr Hence as the difference of any two squares is equal to the product of their sum and difference, we have ax^ + bx-\-G Thus the required factors have been found*. Ex. 1. To find the factors of x^ + 4a; + 3. a;2 + 4a; + 3=a;2 + 4a; + 4-4 + 3 = (a; + 2)2-l = (a; + 2 + l)(a; + 2-l) = (x + 3){a;+l). Ex. 2. To find the factors of x'^-Bx + 3. =(^-i + \/T)(^-i-\/T)- Ex. 3. To find the factors of 3a;2 - 4a; + 1. =a|(.-|y-i}=3(.-|4)(.-|-i) = s(.-i)(.-x). Ex. 4. To find the factors of x^ + 2aa; - &2 _ 2a&. jB2 + 2aa;-62-2a&=»2 + 2aa; + a2_a2-62_2a& = (x + a)2-(a + &)2 = {a;+a+(a + 6)}{a! + a-(a + 6)} = (a; + 2a + 6)(a;-6). * 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 x. FACTORS. 59 82. Instead of working out every example from the beginning we may use the formula ax^ -{-bx + c b /h^ — ^tac) { b Jb'^ — ^ac a ix 4- -^ + f + 2V ■/ and we should then only have to substitute for a, b and c their values in the particular case under consideration. Thus to find the factors of 3x2-4a; + l. Here a = 3, 6 = -4, c=l. „ /62 - 4ac /16 - 12 /I 1 ,, . . ^^°°" V""^^""" \/~36~= V 9 = 3' *^^ expression is therefore equivalent to3(x-^+-|(a;-^--j = 3(a; — ^j (x-1). 83. We have from Art. 81 ax^ -\-bx + c ( b /b'-^ac\f b /b^-4!ac\ =n"'"2s''v~i^;r-'2^-\/^^J- Now, for particular values of a, b, c, — j-r^ — may be positive, zero, or negative. I. Let — 2~~2 — b® i^ositive. Then the two factors of dx^ + bx-{- G will be rational or irrational according as ¥ — 4ac . . ^ ^ ^ — y-2 — IS or IS not a perfect square. II. Let — -r-^ — be zei^o. Then Hence cui? + 6a; + c is a perfect square in x, if 6^ — 4ac = 0. y^ 4(zc III. Let — 2n^ — be negative. Then no positive or ^a negative quantity can be found whose square will be equal to — T~2~ ; for all squares, whether of positive or nega- tive quantities, are positive. 60 FACTORS. Expressions 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 If — 4ac 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 a?^ — 8 is for many purposes complete in the form {cc — 2) {cc^ +2^ + 4) *, the imaginary factors of x^ -H 2^ + 4, namely « + 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 groupiug of the terms. For example ' \-\-ax-x^- ax^ = 1 + ax - a:^ (1 + ax) = (1 + ax) (1 - x^) = {l + aa;)(l + a;){l-a;); * The reason of this will appear from Art. 179 and Art. 193. FACTORS. Gl or wo may write the expression in the form l-x^ + ax-ax^=l -x^ + ax(l -x-), and the factors 1-x^, 1 + ax are now obvious. For the best arrangement or grouping no general rule can be given : the following oases are however of frequent occurrence 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 + bc + cd + da. Arranged according to powers of a we have a{b + d) + hc + cdy which is at once seen to be a{b + d) + c {b + d) = {a + c) {b + d). Ex. 2. To find the factors of x^ + {a + b + c)x + ab + ac. The expression = a (a; + 6 + c) + aj^ + &a; + ca; = (a + jc) (x + & + c). Ex. 3. To find the factors of ax"^ + a; + a + 1. ax^ + x + a + \ = a{x* + V)-\-x + l = {x + l){a{x'^-x + l) + \]. Ex. 4. To find the factors of a? + 2db - lac - 3^2 + 2?;c. The given expression is of the first degree in c ; we therefore write it in the form a^ + 2a& - 362 _ 2c (a - 6) = (a - 6) (a + 36) - 2c (a - 6) = (a - 6) (a + 36 - 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^ - c^ + 26c - 4fl6. Arranging according to powers of a, we have a»-4a6 + 362-c2 + 26c = a2-4a6 + 462-462 + 362-c2 + 26c = (a-26)2-(6-c)2={(a-26) + (6-c)}{(a-26)-(6-c)} = (a-6-c)(a-36 + c). Ex. 2. Find the factors of a^ - ja _ ^s + ^^2 _ 2 {ad - be). The expression = a2 - 2ad - 6= - c^ + d'^+2bc = a2-2arZ + d2_63-c2 + 26c = (a-d)2-(6-c)2 ss{a d + b-c)(a-d-b + c). 62 FACTORS. Ex. 3. Find the factors of a^ + 2ah -ac- Zlf- + 56c - 2c\ The expression = a'' + a(26-c)-3&2 + 56c-2c2 = a? + a(2h-c)+ (^!!^y-(^^J-Sb^ + 5bc-2c^^ = fa + -^) -- {462 _ 46c + C2 + 12&2- 206c + 8c2} = (a + 36-2c)(a-6 + c). Ex. 4. Find the factors of a;* + a;^ - 2aa; + 1 - a^. Arranging according to powers of a, we have - {a2 + 2aa; - 1 - x2 - a;4} = _ {a? + 2aa; + a;2 - 1 - 2a;2 - a;^} = -{(a + a;)2-(l + a;2)2}=_(a + a;+l + a;2)(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 a^ - 10a;2 + 9. jc* - lOar^ + 9 = a;4 - 10a;2 + 25 - 25 + 9 = {a;2 - 5)2 - 16 = {x2-5 + 4)(a;2-5-4) = (a;2-9)(x2-l) = (a; + 3)(a;-3)(a; + l)(x-l), or thus:— a;4-10a;2 + 9 = (a;2 + 3)2-16a;2 = (a* + 3 + 4a;) (a;2 + 3 - 4x) = (a; + 3) (a; + 1) (a; - 3) (a; - 1). Ex. 2. To find the factors of a;" + a;2 + 1. Two real quadratic factors can be found as follows : «4+x2^1^ (3,2 + 1)2 _ a;2= (a.2 + 1 ^ -P) {a;2+ 1 - a;). Ex. 3. To find the factors of x^ - 2'da? + 27. a;6 - 28a;3 + 27 = a;6 - 28a:3 + 142 - 142 + 27 = (x3 - 14)2 - 132 = (a;3 _ 1) (a^ ^ 27) = (« - 1) (x - 3) (a;2 + a; + 1) (a;2 + 3a: + 9). In this ease, 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^- 2/j'c"' - Ic^o? - 1a"\i'. Arranging according to powers of a, we have a-» - 2a2 (62 + c-) + ft"* + c* - Ih-c^ = a^- 2a2(62 + c2) + (62 + c2j2 _ {]fi ^ c^f J^h^ Jf d^ - llt^c^ ~ {a-^ - {62 + c2) }2 - 462c2 = (a2 - [,2 _ ^2 _ 2/;c) {a2 - ^2 _ c-' + 2hc) = {a2-(6 + c)2}{a2-(6-c)2} = (a + 6 + c) (a - 6 - c) (a - Z> + c) (a + ?j - c) . IV. Two factors of aV"^ -^-hF -\r c, where P is any expression which contains x, can always be found by the method of Art. 81 ; for we have Ex. 1. To find the factors of (a;2 + a;)2 + 4 (a;^ + a;) - 12. Since P2 + 4p_ i2=:(p_2) (P + 6), the given expression = (x2 + a; - 2) (x^ + a; + 6) = (x + 2)(a:-l)(a:2 + a; + 6), the factors of x2 + a; + 6 being imaginary [see Art. 83, Note]. Ex. 2. To find the factors of (x2 + a; + 4)2 + 8x (x^ + a; + 4) + 15x2. The given expression ={(x2 + a; + 4) + 3x}{(x2+x + 4) + 5x} = (.-r2 + 4x + 4)(x2 + 6x + 4) = (x+2)2(x2+6x+4). Ex. 3. To find the factors of 2 (x2+ 6x + 1)2 + 5 (x2 + 6x + 1) (x2 + i) + 2(x2+ 1)2. Since 2P2+ 5PQ + 2g2=(p + 2g)(2P+ g), the given expression = {{x2 + Gx+l) + 2(x- + l)}{2(x2 + 6x + l) + x2 + l} = (3x2 + 6x + 3)(3x2+12x + 3) = 9(x + l)2(x2 + 4x+l). Ex.4. To find the factors of (x^ + a; + 1) (x^ + a; + 2) - 12. The given expression = (x2 + x)2 + 3 (x2 + x) - 10 = (x2 + x-2)(x2 + x + r)) = (x + 2)(x-l)(.T3 + a;+5). 64 EXAMPLES. EXAMPLES V. Find the factors of the following expressions : 1. oi? + a^ — X — a. 2. ac — bd — ad -\- be. 3. ac^ + bd^ — ad^ — bc^. 4. acx^ + (be + ad) xy + bdif. 5. ac^x^ + bcdi? + adx + bd. 6. (a + 6)^ + (a + cy - (c + c^)' - (6 + 1/)^ 7. a* + a^6 - aW - b\ 8. a* - a% - ab^ + b\ 9. a'^'^-a^-J^^l. 10. a;y - x\' - yh' + z\ 11 . a:'2/^2;'' - cc^;s - 2/^2; + 1 . 12. x^ + x^y + fljs;^ + yz^. 13. aj(a; + 2;) -2/(2/ + ^;), 14. a;^-7a:'-18. 15. x^ - '2dx' + 1. 16. x*-l4:xy + y\ 17. («« + aj'+l. 18. x*-2 (a' + b') x' + {a' - b'f, 19. X* - 4:xyz' + 4:y*z\ 20. x'-2{a + b)x-'ab (a -2)(b + 2). 21. x^ + bx^ + ax-'r ab. 22. (1 + yf - 2x' (1 + /) + x'{\- y)\ 23. a;' - 2/' - S^r" - 2a;2; + iyz. FACTORS. 05 24. 2v/^ - bxy + 2x' -ay-cvx- a\ 25. a' - Zh' - 3c» + 106c - 2ca - 'lah. 26. 2a^ - lah - 22b' - 5a + 356 - X 27. l+(6-a-)a;-'-«6a;"'. 28. l-2ax-{c-a')x' + acx\ 29. a^(6 - c) + 6^(c - «) + c\a - b). 30. 6-c + 6c' + c^'a + ca- + a'6 + ab^ + 2a6c. 31. a'6 - ab^ + a'c - ac" - 2((6c + 6*c + 6c'. 32. x\a ^\)~xij (x -y){a- b) + if{b + 1). 33. ax (if + 6') + by {bx' + a'y). 34. 2x^ - ix'y - xh + 2x1/ + 2xyz - y'z. 35. xyz {x^ + 2/' + z^) - y^z^ - «V - x'y^. 36. {x'-^xy-U{x' + x) + 2\. 37. {x" + 4a; + 8)' + 2>x (x' + ix + 8) + 2x^. 38. (a; +l)(x + 2) (a; + 3) (a; + 4) - 24. 39. (a; + 1) (a; + 3) (a; + 5) (aj + 7) + 1 5. 40. 4(a; + 5) (a; + 6) (x + 10) (a; + 12) - 3x\ 86. Theorem. The expression x"* — a" is divisible by x — a,for all positive integi^al values ofii. It is known that x — a, x^ — u'} and x^ — a' are all divisible by a? — a. We have a;" - a" = a;" - a^-"'' + ao;""' - ft" = a;"-' (a; - a) + a (a-""' - a"''). Now if x — a divides a;""^ — a""^ it will alsd divide a""^ (a; - a) + a (./;""' - a""*), that is, it will divide a;" - a". Hence, if x - a divides a;""' - a""' it will also divide a,-" - a". S. A. 5 6G ^ FACTORS. But we know that x — a divides os^ — a^; it will therefore also divide a?* — a*. And, since x — a divides x^ — a* it will also divide x^ — a^ And so on indefinitely. Hence x" — a" is divisible by x — a, when n is any positive integer. 87. Since ii;" + a" = ^" - a~ + 2a" it follows from the last Article that when x^ + a" is divided by a; — a the remainder is 2a'*, so that a;" 4- a" is nevei^ divisible hj x — a. If we change a into — a, x—a becomes x — {—a) = x + a', also a?" — a"" becomes «" — (— a)'\ and x^ — (— a)" is i»" + a" or ^" — a*" according as n is odd or even. Hence, when n is odd a;" + a" is divisible by ^ + a, and when n is even x^ — a" is divisible hy x -\- a. Thus, n being any positive integer, x — a divides a;" — a^ always, x — a ■ „ x'' + a" never, X -]- a „ ^'" — a" when w is even, and X -\- a „ ^" + a** when n is odd. The above results may be written so as to shew the quotients: thus ^" ~ ^^" = ^"-' + ^"-'^ a + x'^-'a' + + a"-\ x—a X -^ a = x""-' - x""-' a + x""-' a" - ± a""\ the upper or lower signs being taken on each side of the second formula according as n is odd or even. 88. Theorem. If any rational and integral expres- sion which contains x vanish when a is put for x, then will x — a he a factor of the expression. FACTORS. 67 Let the expression, arranged according to powers of x, be ax^' + hx''"' +cx""' + Then, by supposition, act" + 6a"-' + ca"-'+ =0. Hence ax^ -f hx''"' + cx''~- + = ax'' + 6^""' + c^""" + - (eta" + hoT-^ + ca""^ + . . .) = a (a-" - a") + h (a-"-* - a""0 + c (^""^ -«"■')+ Bat, by the last Article, a;"- a", a?""' - a^"', a-" - - a""', &c. are all divisible by a; — a. Hence also ax^ + hx^~^ + cr""^ + is divisible by X —OL. The proposition may also be proved in the following manner. Divide the expression ax"" +hx''~'^ + cx''"^ -\- hy x— a, continuing the process until the remainder, if there be any remainder, does not contain x\ and let Q be the quotient and R the remainder. Then, by the nature of division, ao;" + 6^"-' + c^"-' + =Q {so- a) + R, and this relation is true for all values of x. Now since R does not contain x, no change will be made in R by changing the value of x : put then x = a, and we have aa" + ^a"-^ + ca"-' + =Q{a-a)-hR = R. Hence, if any expression rational and integral in x he divided by x — a, the remainder is equal to the result obtained by putting a in the j^kwe 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 fur x. 5—2 68 FACTORS. Ex. 1. Find the remainder when x^ - Ax^ + 2 is divided by x - 2. The remainder = 23 - 4 . 22 + 2 = - 6. Ex.2. Find the remainder when x^ - 2a^x + a^ is divided by x - a. The remainder is a^ - 2a^ + a^ = 0, so that x^ - 2a^x + a? is divisible by a; - a. Ex. 3. Shew by substitution that x-1, a; -5, x-\-2 and a; + 4 are factors of x^ - 23^2 - 18a; + 40. Ex. 4. Shew by substitution that a - 6 is a factor of a3(&_c) + &3(c_a) + c3(a-6). Put a = h and the expression becomes a^{a-c)+a^{c-a), which is clearly zero : this proves that a - 6 is a factor. Ex. 5. Shew that a is a factor of (a + 6 + c)3-(-a + & + c)3-(a-& + c)3-(a + &-c)3. 89. We have proved that x — a is a factor of the expression a^" + 6a;""^ + c«""^ + , 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 aa;"~\ Hence the given ex- pression is equivalent to (a; - a) (aaj'^*' + &c ). Now suppose that the given expression also vanishes when x = ^\ then the product of a; — a and ax""'^ + will vanish when a? = /3; and since x— ol does not vanish when x= (3, it follows that a^"~^ + must vanish when x = l3. Hence x — ^ is a factor of a^''~^ + &c.; and, if the division were performed, it is clear that the first term of the quotient would be a«"~^. Hence the original expression is equivalent to {x -OL){x~ 13) (^ax''-' + &C ). Similarly, if the original expression vanishes also for the values y, B, &c. of x, it must be equivalent to {x-a) {x - /3) (x-y) (x - 8) (a^'^"'' + &c ), FACTORS. GO where r is equal to the number of tlic factors x-% X - ^, &c. If therefore the given expressiim vanishes for n values a, yQ, ry, &c. there will be n factors such as x—a, and the remaining factor, ow;""'' + &c. will reduce to a; and lieiicu the given expression is equivalent to a(x — a) (a: — /3) {x — y] Cor. If any of the factors x — a, x — jB, ... occur more than once in a^r" + 6a;"~^ + . . . , it can similarly be proved that the expression is equivalent to a{x — a)" [x — jSf ... , the factors x — a, x — 13, ... occurring respectively j9, 7, ... times, and p -{■ (i-\- ... = n. 90. Theorem. An expression of the nth dejree in x cannot vanish for more than n values of x. For if the expression ow;" + Zjo;""' + c^"~' + vanishes for the n values a, /3, 7 , it must be equivalent to a {x — a) {x — 13) (x — j) If now we substitute any value, k suppose, different from each of the values a, /3, 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 6a;""' + c^""' + , and is of the (n-lf degree; and hence as before it can only vanish for ?i - 1 values of x, except h is zero. And so on. Thus an expression of the 71th 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 coelHcienta are zero, the expression will clearly vanish for all values of X. 70 FACTORS. 91. Theorem, If tivo 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 values of x. If the two expressions of the nth degree in x cw;" + 6^""' + c^""' + , and px"" + qx""-^ + r«"-'+ , be equal to one another for more than n values of x, it follows that their difference, namely the expression {a - p) x"" -\- (h - q) x""-^ + {c-r) x''-' -{■ , will vanish for more than w values of x. Hence, by Art. 90, the coefficients of all the different powers of x must be zero. Thus a-p=-0, b-q = 0, c-r = 0, &c. that is, a = p, b = 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 must 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 be resolved into only one set of factors of the first degree in x. ^-^ For, if it be possible, let the expression ax"" + bx"'^ + ... be equivalent to a {x - ay (x-^y ..., and also to a {x - f / {x - ?;)'" . . . FACTollS. 71 Put X = a in both expressious ; tluni (i (a - f )* (a — 77)"'. . . must vanish, and therefore one at least of the (luiiiitllius f , 77, ... 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 tlie student to attend to the way in which cx])re.s.sions arc usually arranged. Consider, for example, the arrange- ment of the expression hc + ca + ab. 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, b into c and c into a. In the expression d^ (b —c)-\- h^ (c — a) + c^ {a — b) 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 {h — c)(c- a) {a — h) are obtained from the first by cyclical changes. 94. Symmetrical expressions. An expression which is unaltered by interchanging any pair of the letters wliicii it contains is said to be a symmetrical expression. Thus a + 6 + c, bc-\- ca-\- ab, a^ + b^ -\-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 cjuotient, of two symmetrical expressions is symmetrical, for if neither ot two expressions is altered by an interchange of two letters their product, or their (piotient, cannot be altercil by sucli 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) + b^ {c~a) + c^ {a - &). If we put b = c in the expression a^b - c) + h^c - a) +c^{a - h) (i) it is easy to see that the result is zero. Hence 6 - c is a factor of (i), and we can prove in a similar manner that c-a and a — h are factors. Now (i) is an expression of the third degree; it can therefore only have three factors. Hence (i) is equal to L{b-c){c-a){a-h) (ii), where L is some number, which is always the same for all values of a, b, 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, b and c. Thus, let a = 0, 6 = 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^{b -c) + b^{c- a) + c^{a - b). As in the preceding example, {b-c), (c-a) and (a-b) are all factors of a^ {b - c) + h^ {c - a) + c^ {a - b) (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 Jirst degree, and this factor must be symmetrical in a, b, c, it must therefore be a + & + c. Hence the given expression must be equal to L{b-c) (c-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 Z/= - 1; hence a^{b-c) + b^{c-a)+c^{a-b)= -{b-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, 6 = 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 a3 {b-c) -a (63 _ c3) + be (6^ - c^). FACTORS. 7;{ It is now obvious that 6 - c is a factor, and wo have [b-c) {a^-a{b'- + bc + c^) + bc(b + c)} = {b-c){b-{c-a) + h{c-- ac) + a^- ac^ } = {b-c) {c-a) {b- + bc-a^-ac}= - (b-c) {c - a) {a-b) {a + b + c). Ex. 3. Find the factors of b^c^ {b - c) + c-a- {c-a) + a-b'^ {a - b). By putting & = c in the expression b'-C'{b-c) + c^i^{c-a) + a%''{a-b) (i), it is easy to see that the result is zero; hence b-c is a factor of (i). So also c-a and a-b are factors. The given expression being of the Jifth degree, there must l)c, 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- + UP- + Lc"^ + Mbc + Mca + 3/a6} . . . (ii). Equating coefficients of a* in (i) and in (ii) we see that 7. = 0; and then equating coefficients of b^c'' we see that M = -\. Hence (i) is equal to - (b - c) (c - a) (a - b) [be + ca + ab). We may also proceed as follows. Ai-ranging according to powers of a, the factor b-c which does not contain a becomes obvious ; then, arranging according to powers of b, the factor c-a which does not contain 6 becomes obvious; and so on. Thus = {b-c) {&2c2 _ a2 (6- + bc + c"^) + a=* (6 + c) } = (6 - c) { t2 (c2 - a2) +a-b{a-c)+ a2c (a - c) } = {b-c){c- a) {b'^{c + o)- a-b - a^c } ~(b-c){c-o) {{b^-a^)c + b^a- a^-b} = - (6 - c) (c - a) (a - b) [be + ca + ab). EXAMPLES VI. Find the factors of the following expressions: 1. (y - zy + {z- xf + {x- yy. 2. {y-zy-\-{z^xy + (x-yy. 3. a* (b' - c') + h* (c' - a') + c' (a» - h'). 4. a{b- cy + b(c- ay + c{a- b)\ 74 EXAMPLES. 5. a{h- cf + h(G-aY + c{a- h)\ 6. he {h —c) + ca {c — a) + ah {a — b). 7. b'c^ {b-c) + c'a' {c - a) + a'b' {a - b). 8. a" {b-c) + b" {c-a) + c" {a - b). 9. a' {b-c) + b^ {c-a) + c' {a - b). 10. {a + b + cy - (b + c- ay - (c + a- by - {a + b - c)^ 11. {a + b + cy - {b + G - ay - (c + a - by - {a + b - cy. 12. a(b + c-ay + b{c + a-by + c{a + b- cy + (b + c - a) {c + a - b) {a + b - c). 13. a^ {b + c-a) + b^(G + a-b) + c^ (a + b - c) - {b + c — a) (c + a ~ b) (a + b - c). 14. {b + c- a) (c + a- b) (a + b-c) + a(a-b + c){a + b -c) + b (a + b — c) { — a + b + c) -\- c {- a +'b + c) {a - b + c). 15. (b — c){a — b + c)(a + b-c) + {c-a){a + b -c){-a + b + c) + (a-b) {-a + b + c){a-b + c). 16. (x-i-y + zy -x^-y^-z\ 17. (x + y + zy-x'-y^- z\ 18. (b-c) {b + Gy + (c-a) (c+ ay + (a-b) (a + by. 19. {b-c)(b + cy + {c-a){c + ay+{a-b}{a+by. 20. (b-c) (b + cy+ (G-a) {c + ay + (a-b){a + by. 21. a^ + b^ + c^+ 5abc - a {a -b) {a - c) - b {b - c) {b - a) — G(c — a) (c — b). 22. a' (a + b) {a + c) (b -c) -i-b' (b + c) {b +a) {G-a) + 0^^ (c + a) (c + b){a- b). 23. {y + z) {z + x) (x + y) + xyz. 24. a^ (6 + Gy + 6^ (c + ay + c^ (a + by + abG(a + b + c) + (a^ + b^ + c^) (be + ca + ah). 25. (x + y ^- zy - (y + z)* -(z + xy - {x + yY + x'^ + y* + z*. 26. a'(b + c- 2a) + b' (c + a- 2b) +c'(a + b- 2c) + 2 (c'-a')(c -b) + 2 (a' - b')(a-c) + 2 {b'-c')(b - a). EXAMPLES. 76 27. {b + c-a- dy {h - c){a - d) f (c + a-h- d)' {r,~a){h-d) + {a + h-c- dy {a - 0) (c - d). 28. Shew that I2{{x + y + zY" - (y + zY" -{z + xy -(x + i/Y" + af + if t c''} is divisible by (x-\-y + zy-(y + zy -{z + a:)*- (x + y)* + x* + y* + ;:*. 29. Shew that a« (6 + c - a)' + 6' (c + a - 6)' + c'{a + b-cY+ abc {a' + bU c') + (a^ + 6^ + c^ — 6c — ca — a^) (6 + c - «) (c + a — 6) (a + 6 - c) = 2abc [be ■^ca + ab). 30. Shew that {b - cY + (c - ay + (a-bY - ^ {b - cY {c - aY (a- by = 2 (a-by (a- cY + 2 (b - cY (b -ay + 2{c-ay{c-by. 31. Shew that (/, + cY + (c + a)' + {a + by + {a + dy + (6 + dy +{c + dy = 3{a+b + c+d){a' + b' + c' + d'). 32. Reduce to its simplest form 4 {a' + ab + ¥y -{a- by (a + 2bY {2a + by. 33. Shew that a* {b' + c'- aj + V {c' + a'- b'Y + c* (a' + 6' - c=)' is divisible by a* + b' + c'- 2b"-c' - 2c'a' - 2a'b\ 34. Resolve into quadratic factors 4 {cd (a^ - b') + ab {c' - d"-)Y + {(«' - ^'') {c' - d') - iabcdy. 35. Shew that {y'-z'){\+xy) (I +xz) + {z^-x') (l + 2/c;)(l + yx) + {x'-f){l+zx){\+zy) = {y-z){z-x){x-y){xyz^x^y^z\ 36. Find the factors of a'{b-c){G-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'dW (c - (/) ((/ - a) {a - c) + dW-b' (d - a) {a - b) (6 -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%^c and a'^h^c^. The highest power of a which will divide both expressions is a' ; the highest power of 6 is 6^ ; and the highest power of c is c ; and no other letters are common. Hence the h.c.f. is a%^c. Again, to find the highest common factor of a%'^c^, arh^ and a^hc^. The highest power of a which will divide all three expressions is a^ ; the highest power of h which will divide them all is h ; and c will not divide all the expressions. Hence the h.c.f. is a^h. 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 exjyression^ taken to the lowest power in wliich it occurs. HIGHEST COMMON FACTORS. 77 97. Highest common factor of multinomial expressions whose factors are known. WIkmi tlie factors of two or moro multiiKJiiiial exprossiuiis uro 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 eackfartor lohich is conwion to all the expressioiis taken to the lowest power in which it occurs. Thus, to find the h.c.f. of {x-2)^x-l)-{x-d) and (a;-2)2(x- 1) (x-.3)». It is clear that both expressions are divisible by (x - 2}^, but by no higher power of x-2. Also both expressions arc divisible by x - 1, but by no higher power of x-1; and both expresbiuns are divisible by x-S, but by no higher power of x-3. Hence the H.c.ir. is (x-2)2(x-l)(x-3). Again, the h.c.f. of a'-P {a - b)'- {a + b)^ and a^b^ {a - b) {a + b)^ is a262(a-6)(a + 6)2. 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*b--a%^ and a*b^ + a%*. Am. arb'(a-irb). Ex. 2. Find the h.c.f. of a^U'-^w'b^ and a^'-KSa^b^. Aiui. a-b-ia^-ib-). Ex. 3. Find the n.c.r. of a^ + da'b + 2ab' and a* + Qa% + 8a%\ Ans. a{a+2b). 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 oi any two multinomial cxpression.s. The process is analogous to that u.sed in arithmetic to find the fi-reatest 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 J. and ^ 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 B, and B 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 J. and B] and therefore the H.c.F. of A and B is the H.c.F. required. Now divide A by B, and let the remainder be S ; then the H.C.F. of B and S will similarly be the same as the H.c.F. of A and B, and will therefore be the H.C.F. re- quired. And, if this process be continued to any extent, the H.C.F. of any 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 multinjomial factors will not be affected by such multiplication or division. HIGHEST COMMON FACTORS. 7lj 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 hy the other {if both expressions are of the same degree it is immaterial which is used as the divisor). Take the remainder, 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 he the H.C.F. required. The process is nut used for finding cornmon monomial factors, these can he seen hy inspection ; and any divisor, dividend, or remainder which occurs may he multiplied or divided hy any monomial ex- pi^ession. Ex. 1. Fiud the h.c.f. of a;^ + a;2 _ 2 aud x^ + 2x^ - 3. x^ + x--2\x^ + 2x-'-S(l X'^-X x^ + x-2 x^ -1 x-l\x'^-l/x + l ' X -x^ 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'^-l, are x-1 and x + 1. And by means of Art. 88 it is at once seen that x-1 is, and that x + 1 Ib not, a factor of x^ + x^ - 2. Ex. 2. Find the h.c.f. of x3 + 4x2j/-8x?/ + 24y3 and x^ -x^y + 8xY-8xy*. The second expression is divisible by x, which is clearly not a common factor: we have therefore to find the u.c.f. of the first expression and x^ - x^y + 8xy' - 8y*. ar» + 4x-»-8xi/''' + 2-lyMx<- x^y + Bx;/' - 8//< Ix-by -I i> ^ ;^ + 4a ;3y-8x V + 24.rj/»V -5x3i/ + 8x=v'-l<'-'-i/'-''^"* - iix^ y - 2Q xY +A^^l' -^-'V* 28x=^y- - 6«)xy* + 1 1 2y* 80 HIGHEST COMMON FACTORS. The remainder = 28y^ {x^ - 2xy + Ay^) : the factor 2%y^ is rejected and x^ - 2xy + Ay^ is used as the new divisor. x^ - 2xy + 4y^ \x^ + Ax^y - 8xy^ + 2iy^ / a; + % ^ x"^ - 2x-y + Axy"^ ^ &x^y-12xy^ + 24.y^ Hence a;2-2xy + 4?/2 is the h.c.f. required. Ex. 3. Find the h.c.f. of 2x4 + 9a;3 + 14a; + 3 and dx^ + 15a;=« + 6x^ + lOx + 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 : 2x4+ 9x3+ 14a; + 3 \ 3a;4 + is^^a + ^x^ + lOx + 2 /2 6x4 + 30a;3 + i0a;2 + 20x + 4 / 3 6x^ + 27x3 +42x + 9^ 3x3 + 10x2 - 22x - 5 \ 2x4 + 9x3 + 14a; + 3 6x^ + 27x3 + 42x + 9 / 2x 6x4 + 20x3- 44x2 -lOx^ 7x3+44x2 + 52x + 9 3 21x3 + 132x2 + 156x + 27/7 21x3+ 70x2 - 154x - 35 V 62| 62x2 + 310x + C2 x2 + 5x + 1 x2 + 5x + l\3x3 + 10x2-22x-5/3x-5 / 3x3 + 15x2+ 3^ \ ^~5x2 - 25x - 5 - 5x2-25x-5 Thus x2 + 5x + 1 is the h.c.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, lITClfKST COMMON FACTORS. Si where p, y, r, s are any quantities positive or negative wkick do not contain x. To prove this, it is in tlie first place clear that am/ common factor of A and B is also a factor of pA -i-f/li and of rA + sB. So also, any common factor of p^ + qB and rA + sB is also a factor of s(pA -\-qB) — q{7'A -{-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 of ^, 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 4- qB) —p {rA + sB), that is of (i^q —ps) B, and therefore of B. Since every common factor of A and B is a factor of 2)A + qB and of rA + sB, and every common factor of pA + qB and rA + s5 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.CF. of pA + qB and rA + sB. Ex. To find the h.c.p. of 2x* + s(^- Gx^ - 2x + 3 and 2x* - 3x^ + 2x- 3. We have, by subtraction, 4a:3_6x2-4x + 6 (T) ; and, by addition, 4x'»-2a;S-6x2=2x5(2x'-a;-3) (IT). The required h.c.f. is the h.c.f. of (I) and (II), and therefore of (I) and 2x2-a;-3 (IH)- Multiply (III) by 2 and add (I), and we have unolher expression, namely ,^^. Ax^-2x'^-Qx = 2x{2x'-x--;i) (IV), such that the h.c.f. of (III) and jIV) is the h.c.f. required. But the H.C.F. of (in) and (IV) is obviously 2x'^-x- 3. 100. 1( B, S be the successive remainders in tho process of finding the H.C.F. of the two expressions .4 and B by the method of Art. 98 ; then, as we have seen, every common factor of ^ and B is a factor of 7^. and therefore a common factor o^ A an.l B. Similarly every common factor of A and R is a common factor of R and S. AnxY + y\ 4. 2a;'' + ^x^'y - / and 4a:' + xy^ - y'. 5. x" - 4?/" + 1 2yz - dz" and a:' + 2a;^ - 4^/= + Syz - 3z\ 6. 20a' - 3a'b + b' and 64a' - Sab' + 5b\ 7. a' - a''^ + ab' + 146' and 4a' + 3a'6 - 9a6' + 26'. 8. 2a;' + a;'-9a;' + 8a;-2 and 2a;* - 7a;' + 11a;'- 8a; + 2. 9. 1 la;' - 9aa;' - a^x' - a' and 1 3a;' - 1 Oaa;' - 2a'a;' - a\ 10. a;' + a;' - 9a;' - 3a; + 18 and x' + 6a;' - 49a: + 42. 11. a;'-2a;' + 5a;'-4a; + 3 and 2a;' - a;' + 6a;' + 2a; + 3. 12. a;' + 3a;' + 6a; + 35 and a;' + 2a;' - 5x' + 26a; + 21. Lowest Common Multiple. 102. Definitions. A Common Multijjle 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-ay{x-hy and ah\x-ay {x-h). It is clear that any common multiple must contain a' as a factor ; it must also contain h^, (x — a)* and (x — 6)^ Any common multiple must therefore have a^b* {x — a)* {x — bf 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^b* (x — ay (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 eveiy different factor which occurs in the expressions to the higJiest power which it has in any one of them. Ex. 1. Find the l.c.m. of dxhjz, 21x^ifz- and &xy-^. Ans. 54a;^V. Ex. 2. Find the l.c.m. of 6ab'^{a + bf and ^a'^b{a^-b-). Ans. 12a^b^a + b)^{a-b). Ex. 3. Find the l.c.m. of 2axy {x - ijY, Saa:^ [x^ - y"^) and Ay"^ {x + y)-. Am. 12aa;2j/2(a;2_ 2/2)2. Ex. 4. Find the l.c.m. of a;^ - 3a; + 2, a:^ - 5a; + 6 and a;2 - 4a; + 3. Ans. (a;-l)(a;-2){a;-3). 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.c.m. of x^-^x'^-2 and x^ + 2x2 - 3. The H.c.F. will be found to be a; - 1 ; and a;3 + a;2-2 = (a:-l)(x2 + 2x + 2), a;3 + 2a;2 - 3 = (x - 1) (x2 + 3a; + 3). Then, since x2 4-2x + 2 and x'' + 3x + 3 have no common factor, the required l.c.m. is (x - 1) (x- + 2x + 2) (x^ + 3x + 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.C.M. Let a and h be the quotients when A and B respec- tively are divided by if ; so that A = H .a and B = H .b. Since H is the highest common factor of A and B, a and h can have no common factors. Hence the L.C.M. of A and B must he K x a x b. Thus L = H .a .b. Hence L = Ha x -yt = A x-fr (i) ; also L X H = Ha x Hb = A x B (ii). From (i) we see that the L.C.M. of any two expressions is found by dividing one of the expi^essions by their h.c.f., and multiplying the quotieiit by the other expression. From (ii) we see that the product of any ttvo expressions is equal to the product of their h.c.f. and L.C.M. EXAMPLES YIIL Find the L.C.M. of 1. (Sx^ - bax - 6a^ and 4x^ - 2ax' - 9<2^ 2. 4:a^ - 5ab + b' and 3a' - 3a'b + ab' - b\ 3. 3a;' - 13a;' + 22>x - 21 and 6a;' + x^ - iix + 21. 4. x' -l\x' + 49 and 1x* - 40a;' + 75a;' - 40a; + 7^ 5. a;' + 6a;' + 11a; + 6 and a;"* + a;' -4a;' -4a;. 6. a;* - a;' + 8a; - 8 and a;* + 4a;' - 8a;' + 24a;. 7. 8a' - 18a6', 8a' + 8a'6 - Qa¥ and 4a' - 8a6 + W. EXAMPLES. 87 8. x' - 7a; + 1 2, 3x' - 6a; - 9 and 2a;' - 6a;- - 8x. 9. 8a;' + 27, 1 6x* + 36x' + 81 and 6a;' - 5a; - 6. 10. x' - 6xy + 97/-, x'-xy- 6p' and 3a;' - 1 27/^ 11. a;- — 7a;^ + 1 2?/", x^ — Gxy + Sy^ and a;' — 5xy + Oy'. 12. Shew that, if ax^ + bx + c and a'x^ + h'x + c' have a com- mon factor of the form x +f, then will {ac — a'cY = {he — b'c) {ah' — ah). 13. Sliew that, if ax^ + hx^ + cx + d and a'x^ + h'x' + c'x + d' have a common quadratic factor in a;, then will b(i — h'a ca - c'a da — d'a ad' — a'd hd' — h'd cd' — c'd ' 14. Find the condition that ax^ + bx + c and a'x^ + h'x + c' may have a common factor of the form x +/. 15. If ^j, <7„, ^3 are the highest common factors, and l^, Z,, l^ the lowest common multiples of the three quantities a, 6, c taken in pairs; prove that g^O.oJJ'J'i^ {ahc)~. 16. If A, B, G be any three algebraical expressions, and {BC), {CA), {AB) and {ABC) be respectively the highest common factors of B and G, G and A, A and B, and A, B and C; then tlie L.c. m. of A, B and G will be A.B.G. {ABC) ^ [{BG) . (GA) . (AB)]. CHAPTER VIII. 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 numerator and the denominator of the fraction. Thus Y means a-^h. Since, by definition, y- = a -^ 6, it follows that y xb = a, 107. Theorem. The value of a fraction is not altered hy multiplying its numerator and denominator by tJie same quantity. We have to prove that a _ am b bm ' for all values of a, b and 7n. r^ Let X= rr-\ then a)xb = jxb = a, by definition. FRACTIONS. 89 Hence x x h x 711 = a x m; .'. X X (bm) = am. [Art. 29, (ii),] Divide by hm, and we have OS = am -7- (bi)i) — ^ 01 am tia 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 d^x denominator. For example, the fraction jr- takes the cc a' simpler form ^ , when the factor x, 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. Reduce ,. „ to its lowest terms. ba-xy The H.C.F. of the numerator and denominator is Saxy ; and ^axhf _ 3ax^y-~Saxy _ x (ja^xy ba^xy -~ daxy la ' ■coo- vr a;2 - Txy + 10(/2 x^ - Ixy + 10y2 _ (x-2y)(z-5;v ) _ x- By x2^8xy+12y2 - (x-2y){x-^y) ~x'-Gy' 90 FRACTIONS. Ex. 3. Simplify '-j x^ - ax a-- X- x^-ax X (x - a) a^ - 7? (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 x~ a, we have - -> .- . 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. Simpbfy ^j^^3- -^-^^23^^ . The H.c.F. will be found to be a;^- 3x4-7; and, dividing the numerator and denominator by a;^-3x + 7, we have the equivalent fraction —z — = , a;2 + oa; + 3 109. 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 a^y (x + y)' xf{x- y) x-y' (x- -y-)* to a common denominator. The L.c.M. of the denominators is x^y'^{x^-y'^). Dividing this L.c.M. by x^y{x + y), xi/{x-y) and a;-xcZ = YX-, x6xfZ b a a J c J a by the Commutative Law. ct c But, by definition, j x b = a, and -.xd = c; .'. a;xbxd = axc'j ac bd 94 FRACTIONS. Thus the product of any two fractions is another fraction whose numerator is the product of their numerators, and whose denominator is the product of their denominators. The continued product of any number of fractions is found by the same rule. For ace ac e ace b d f hd f hdf and similarly, however many fractions there may be. Hence f a\^ a a aa oJ^ , . , /a\" a" a as. ijet a c 112. Division of fractions. Let r and -, be any d "^ two fractions ; and let x = b ' d' rri-. caeca Then icx-7 = Y-4-^x,= ^; d b d d b c d _a d d c c Tx ad Hence £c = 7- x — , b c c d cd ^ smce -; X - = -y- = 1. d c dc Thus to divide by any fraction -^ is the same as to midtiply by the reciprocal fraction - . c As particular cases of multiplication and divisionTwe have a _ a c _ac ^ X n ^ ^ X - == y , FRACTIONS. 95 , a _a c _a l_o^ b \ b c DC Note. It should be Doticed that the rules for the multiplication and division of algebraical fractions are simply rules concerning the order in which certain opera- tions of multiplication and divibion may be performed, and have really been proved in Art. 33. Thus r X j = (a-^b) x{c^d) = a-^ b X c -T-d = a X c^b -^d = (ao) -r (bd) = y-. . T* -I- /9 T ^ f1 Ex. 1. Simplify -^ ^ x x^-^-a? x-a _ {x^ + a^){x-a) s^^^^ ^ {x + af ~ \x^-w'){x + aY __{x^-ax + a^) {x + a) (x - (j) _ x^ - ox + n^ "~ {x-a){x + a)^ ~ (x + a)2 Ex. 2. Simplify 1 1 o X If xy y - X y"^ - x" _y - X x'-xp' _ .r?/ 1 y -x'' xy xhf xij y^-x^ x + y X- y a+x a- X T o o- ,.. fi-x a-\-x Ex. 3. Sunplify ■ . a+x a -X a — X a + x a + x a-x _(a + x) {a + x) - {a - x) (a- x) _ iax a-x a + x a^ - x'^ a'^ -x"^^ a + x a-x _{a+x) {a + x) + (a- x) {a- x) _2a^ + 2x- and ■ 1 — r, — Q ;;— • a- X a+x a^-x^ a* - x^ 96 FRACTIONS. Hence the given fraction is equal to Aax , 2a^ + 2x^ _ Aax a'^-x^ _ 2ax 113. The following theorems (the second of which includes the first) are of importance : Theorem I. If the fractions ^ , j^ , ^% dx. he all equal to one another, then will each fraction be equal to pa^ + qa^ + ra^-]- p\-\'qb., + rb^+ Let each of the equal fractions be equal to x. Then, since -r" = x, a,=b,x\ .*. pa^ = pb^ X, so also qa^ = qb^ x, ra^ = rb^ x, Hence, by addition, pa^ + qa^ + ra^-\- = (pb^+qb^ + rb^+ ...)x; . pa,-hqa, + ra^ + _ ^ _ ^^i _ ^.^ pK + ^K + '^h-^ K Theorem II. If the fractions j^ > if > ^^ ^'C- ^^ ^^^ equal to one another, then luill each fraction be equal to ^-^ , where A ts any homogeneous expression of the nth degree in a^, a^, a^, &c. and B is the same homogeneous expression with b^ in the place of a^, b^ in the place of ttg, &c Let each of the equal fractions be equal to x, so that »! = b^x, a^ = b^jjo, a^ = b^x, &c. t'RACTlONS. 97 Let \a^ af a^... be any term oi A\ then Xb^'^kf ^3'-" will be the corresponding term of B ; and since the ex- pressions A and B are homogeneous and of the ?ith degree, a -{- ^ + y -{-... =n. Now Xci^-* a/ a.y ...=\ {h,xY {\xY Q)^x)y. . . = x'' .Xb.'^b/by..., since a + ^ + y+...=n* Hence any term of A = x" x corresponding term of B ; .'. sum of all the terms of ^ = x" x sum of all the terms of B, that is A =x'' .B; . VA wliich proves the theorem. Theorem III. If the denominators of the fractions j^ , ~, —, be all positive, then luill the fraction j^ — J- ^ — '^^^^ he qreater than the least and less than h-\-h + K + the greatest of the fractions y , ^ , ct'C. Let j^ be the greatest of the fractions, and let -p = a; ; then j^ then will each fraction be equal to b a J{a'^-2ac + 2c^) ^{b-^-2bd + 2d'^)' T, , a c >J{a^-2aG + 2c^) _ J{b'^x'^-2bxdx + 2d^x^) _ , ^_ ^{b''-2bd + 2d^)~ J{b^-2bd + 2dP) -V^'-^- [This is a simple case of Theorem II., Art. 113.] .— ^ n o 01- ii i. -P cw + 62 az + cx bx + ay ^. ... Ex. 3. Shew that, if -^ — = = — ^^— ^ , then will I m n bcx cay abz - al + bm + en al-bm + en al + bm - en EXAMPLES. 00 Each of the given equal fractions - a (r// + hz) + b {az + ex) + c ( t.r + ay) 2bcx — al+ bm + en - al + bin + en , . ., , 2cay 2abz and similarly = -,-— 7 = —, — j . al- bm + en al + bm - en EXAMPLES IX. Simplify the following fractions : 3. 9. 10. 11. 12. 13. 11 'dija'bc'x'i/z' ' ' a'c'xy ' a'-Sab^i^7b' IxYj-^^y' +\ a'-'3ab~-28b' ' 28a;y + Sx'/- 1 * (x' - f) {x + y ) jx' - f) {X - y ) ^- {x-'^f){x-yy ■ {x^-f){x^-yr 2a;' + 3a;'-l ^ x'-x^-x+\ x' + 'Ix' + lx'+'lx+l ' x'-2x^-x^-2x+l' 2x^ + 5x^y + xy^ - 3y^ dx' + 3x'y - 4:xY' - xy' + y^ ' 5ix'-27x*-Zx'-4: 36a;' + 3a;' + 3a; - 2 ' {a + b){{a + by-c' } 46V - {a' -K-- cy ' X' {y-z) + y' (z-x) + z^ {x-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 100 EXAMPLES. x' (1/ - z') + / {z' - x') + z^ (x' - y') 15. x^ (2/ - ^) + y^ {z — x) + ^ {x — y) 2a 26 a' + ¥ 16. V + ^ + 17, 18. a + b a-b b^ -a^' 3 — 03 3 + £c 1-1 6a; ^ y__ _ {^ - y Y x+ 2y 2y — x x^ — iy' ' -^ x — 2a x + 2a Sax 19. ^ — ^ + 20. x+ 2a 2a — X x^ - ia^ ' 13 3 1 x + 2 x + 4: x + Q 03 + 8* 21. — V + x + a x + 3a x + 6a x+7a' 22. • ^ + + x — 2a x — a x x + a x + 2a' 1 2 1 23. — s = ;;:; — 5 o ~; 7: — a + x^ — 5xy + 6y^ x^ — 4:xy + 3y^ x^ — 3xy + 2y^ ' nA a b G 24, -, TT-, ^v + n ^7^ :■ + (a ~b) [a- c) {b - c) (b -a) (c - a) (c-b)' 25. ; z—. T + -n ^^. T + {a — b) (a - c) (b - c) {b -a) {c -a) (c-b)' (l+ab){l + ac) (l + 6c)(l +ba) {l+ca){l+ch) (a -b) (a- c) (b -c) {b — a) (c — a){c — b) -_ be (a + d) ca (6 + d) ab (c + d) ~ {a — b) (a- c) (b -c) (b — a) (c - a) (c-b)' 28. «^'-3/^ ^ y^-zx ^ z^-xy {x + y) {x + z) {y+ z) {y + x) (z + x) (z + y) ' 29. EXAMPLES. 'lO'] (y-cc)(^-a;) ^ (g-y)(a;-2/) {x-2y + z){x + y- 'Iz) (x + y- 2z) (- 2a; + y + ;;) ^ {z-x) { z-y ) ^ (- 2ic + ?/ + 2) (cc - 2?/ + ;^) ' a; + a x \-h x + c ^{x + a) {x + h) [x -\- c) X — a X — b X — G {x — a)(x-b)(x — c) X X X o ^^ + (^c + ca + ah) x x — a X — b x — c {x — a) {x -b) (x — c) a^ ¥ c' 31. , T^, r+r; TT? 7 + 30. {a - b) (a- c) (6 -c) (b- a) {c - a)ic-b)' «* ^' c* 32. TT-, C + 33. {a -b){a- c) (b -c) {b-a) + {c- a) (c-b)' ^2 (^ + b) (« + c) ^ j2 (6 + c) (6 + a) ^ ^, (c + a)(c + Z >) ^ (a - 6) (rt - c) (6 — c) (b -a) (c - a) (c-b)' a' 34. \3 c/ \G a) \a b) .^ c. 35. ^— ^ A , ,+ (a - 6 + c) (a + 6 - c) (a + 6 - c) (- a + 6 + c) 1 + (- a + 6 + c) (rr - & + c) * 6-c c — a a — h Ob. — s^ Ti rT> T" i"5 7 To I fr-(6-c)^ 6''-(c-a)= c'-(a-^)^ 37. Shew that '^ "' /aj* + a^2 1G+ 4 + 2-^ -\ -IG (^ ta; - « a; + a a; + a') \x ^x - a 38. Shew that ax + 6?/ a - 6 a^x + bSj _ a*x^ - b\f ax - by a^x^ + b'y' a*x* - b*y* ' 102 .. EXAMPLES. 39. Shew that (i) (a -h)(a- c) (1 + ax) {b -c){h- a) (1 + hx) c 1 (c -a) {c- b) (1 + ex) (1 + ax) (1 + bx) (1 + ex) ,..v a b (a -b) {a- c) (1 + ax*) (6 - c) {b - a) (1 + 6a;) c —a; + (c - a) (c - 6) (1 + ex) (1 + aa;) (1 + bx) (1 + c.tj) ("i) /„ M /„ ^^ n , „^^ + ^ (a -b){a- e) (1 + aa;) (6 - c) (6 - a) (1 + bx) 12 x + (c - a) (c - 6) (1 + caj) (1 + ax) (1 + bx) (1 + cic) 40. Simplify {a+p)(a + q) {b + p){b + q) (c + p) {c + q) (a- 6) (a- c)(a; + <:t) (6 - c) (6 - a) (ic + 6) (c— a)(c-6)(a; + c)' 41. Simplify a(b + c — a) b (c + a — b) c(a + b — c) + ~n ^ — 71 c "f" (a -b) (a- e) {b -c) (b - a) (c — a) (c — b)' 42. Simplify (a-b + e) (a + b- e) (a + b - c) {- a + b + c) (a - b) {a -e) (b - c) {b - a) (-a + b + c)(a-b + c) {c-a){c-bX 43. Simplify a{b + c) b (c + a) c{a + b) b + c — a c + a — b a + b — c' EXAMPLES. 10:3 44. Shew that (m + —) + ln+ -] + Imn + — | \ on J \ nj \ 11 in J - im + —] {n + -\ ( inn + — ) = 4. \ m/ \ 11/ \ mnj 45. Shew that 46. Shew that h — G c — a a — h (h — c){G — a) (a-b) l+bc 1+ca I + ab {I + be) {I + ca) {I + ab)' 47. Simplify (y^ + zx + xy) (- + -+-)- xyz ( — + - 4- - ) . ^ ^ \x y zj \x- y- z'J 48. Shew that, if 2/ + « %-\-x x + y b — c c — a a— 6' then will each fraction be equal to Jx' + y' + z■■ J{{b-cy+{c-aY+{a-by\ 49. Shew that, if - = - , then will y b x" + a' y^ + b^ _(x + yf + {a + by x + a y + b x + y + a + b 50. Shew that, if X _ y _ z b + c-a c + a- b a+ b — c^ then will {b- c)x + (c- a)y + (a — b)z = 0. 104 EXAMPLES. 51. Shew that, if -7 = , o — c c— a and c be not zero, then will each equal ^ , ^ a — o and a (2/ - «) + 6 (« - a;) + c (a; - 2/) = 0. 52. Shew that a^ 6' c* + 71 r-77 TT^l r + {a-b){a-c){a-d) {h- c){b-d) (h- a) {c-d) {G-a){c-h) d^ {d — a) [d — b){d — c) 53. Shew that -a + b + c-\-d. + : < K-»2)(«. -^3) K-^0 K-«i)(«2-«3) K-«J + ...+ is equal to zero if r be less than -^i - 1, to 1 if r = 7i — 1, and to a, + a„ + . . . + a ii r = n. 12 n 54. Shew that x — a^ ipG-a^) {x — a^ [x -a^ (x — a^ {x — a^ ... + a a;""' £c" (a?-aj) {x-a^)...{x-a^) (a^-«i) (a; - a2)---(^ -««) 55. Shew that b-\-c + d+ ... + /;+ ^6 c + -; rr-7 ? r + . . . a{a+b + c+ ..,+k + l) a{a + b) {a + b){a + b+c) I ... + {a+b + ... + k) {a + b + ... +k + l) 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 rrieinhers, 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, b, 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. lOG 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 x^ is the highest power oi x which occurs; and so on. Equations of the first, second and third degrees are however generally called simple^ quadratic and cubic 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 -hm = 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-\-b — c=p — q + r. Add —p + q — r to both sides ; then a + b — G—p + q — r=p — q-\-r — p-\-q — r, that is, a + b — c—p + q — r = 0. We thus have an equation equivalent to the given equation, but with the terms ^, —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, it A=B, it is clear that mA = mB. Conversely, if mA = mB, that is m (A—B) = 0, it follows that A—B = 0, since m is not zero. Hence mA — mB when, and only when, A=B. EQUATIONS. ONE UNKNOWN QUANTITY. 1 07 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 13x-7 = 5x + 9. Transpose the terms 5x and - 7; then 13a; - 5x=7 + 9. That is 8a; = 16. Divide both sides by 8, the coefficient of x; then x=2. Ex. 2. Solve the equation — -2 = — +5. 4 5 We may get rid of fractions by multiplying both members by 20, the least common multiple of the denominators; we then have 15a; -40 = 8a; + 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) = 2ah -b{x-h), Removing the brackets, we have ax - a'^ = 2ab -hx + h^, or transposing ax + bx = 2ah + b^ + a^f that is x[a + b) = {a + by^. Divide by a + &, the coefficient of x; then x= r^ = a + &. a + o From the above it wall 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. coefficient of the unknown quantity: this gives the re- quired root. 118. Special Cases. Every simple equation is re- ducible to the form aoo + b = 0, the solution of which is X — — . The following are special cases. Qj I. If 6 = 0, the equation reduces to ax = 0\ whence oc = 0. II. If 6 = and also a = 0, the equation is clearly satisfied for all values of x. III. If a = 0, and 6+0. Suppose that while h remains constant, a takes in 111 succession the values zr^, r-^, -^3,...; 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 4. This is expressed by saying that, in the limit, when a becomes zero, the root of the equation a^ + 6 = is infinite. The symbol for infinity is 00 . EXAMPLES. Solve the equations 1. |(a;-2)-|(a;-3) + |(ic-4) = 4. Ans. a;=12. 2. i(a;-3)-|(a;-8) + ^(a;-5) = 0. Ans. .r = 0. EQUATIONS. ONE UNKNOWN QUANTITY. lOf) 3. a{x-a) = h{x-h). Ans. x = a + b. 4. {x + a) {x + b)- (x-a) {x-b) = {a + b)'^, Ans. x = ~ (a + b). 5. a(2x-a) + t (2x-6) = 2«&. A.ns. a5=-(a + 6). 6. (a2 + x)(62 + .T) = {aZ> + a;)-. Ans. a;=0. 7. 3(x + 3)2 + 5(x + 5)2=8(x + 8)2. Ans. a;=-6. 8. (x- + (0-'-(x-a)4-8aa:3 + 8a''=:0. ^/is. x=-a. 9. (x-l)3 + cc3 + (a; + l)3 = 3x(x2-l).' ^tw. a; = 0. 10. (a; + af + {x + ^f + (a; + c)3= 3 (a; + a) {x + 6) (a; + c). -47is. a;=-- (a + & + c). 3 ' 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;-8) = 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 (a; — a) (a; — 6) (a; — c)... is zero when a; — a is zero, or when a; — 6 is zero, or when a; — is zero, &c. ; and the continued product is not zero except one of the factors x — a,x — h,x — c, &c. is zero. Hence the equation (a; — a) (a; — 6) (a7 — c) . . . = is equivalent to the system of equations a; — a = 0, x — b = 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 a;- -5a: = 6. Transposing, we have a;" - 5a; - 6 = 0, that is (a;-6)(a; + l)=:0; .'. a; -6 = 0, or a; + 1 = 0. Hence cc=6, or a; = -l. Ex.2. Solve the equation x^-x^ = &x. Transposing, we have a;^ - a;^ - 6a; = 0, that is a;(cc-3){cc + 2) = 0; .*. a; = 0, or a; = 3, or a; = -2. 120. Quadratic Equations. When all the terms of a quadratic equation are transposed to one side it must be of the form ax^ -\-bx-\-c = 0, where a, b, c are supposed to represent known quantities. We have already [Art. 80] shewn how to resolve a quadratic expression into factors : the same method will therefore enable us to find the roots of a quadratic equation. Hence to solve the quadratic equation ax^ + bx-\- c = 0, we proceed as follows. Divide by a, the coefficient of x^; the equation then becomes x^+ - x-{-- =0. a a EQI'ATIONS. ONE JJNKNOVVN QUANTITY. Ill Now add and subtract the square of half the coefficient of X, that is the square of i - . Then we have a x'-\-- x-\-[^] -U- +-=0, a \2aJ \2.aj a "••'i' ("IJ-|y(.S-3F=«^ that is or Hence ^ + 2^ + ^-4^^ =(>^ Thus there are ^'wo roots of the quadratic equation, namely 2a - V ^d' ' Ex.1. Solve x2-13x + 42 = 0. We have a;2_ 13^+ /^IV _ ^^\%42 = 0, tliat is \^~~9)~A.~^' or 13 1 _ la 1 ^ x--2- + 2 = 0, or X--- 2 = 0. .•. ;r = 6, or x = 7. 112 QUADRATIC, EQUATIONS. Ex.2. Solve 3a;3_i0a; + 6 = 0. Dividing by 3, we have Hence ^.-_.+ (^_) . (^gj +2 = 0, that is (^~q) ~Q~^' or x ••^-3+ 3=^'^^ ^-3-3=^- Hence a; = -(5-^7), or a; = ^(5 + ^7). Ex.3. Solve a{x'^ + l) = x{a^ + l). Divide by a and transpose ; then a Hence ^^-^^^^ ("^J - ^^1=0. a2 + l a?-l ^ a^ + 1 a^-l . 2a 2a 2a 2a that is X — = 0, or x-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 saved. EQUATIONS. ONE UNKNOWN QUANTITY. ] 1 f} EXAMPLES. Find tbo roots of the following equations: 1. 9j;--24.r + 16 = 0. 3. 3a;2 = 8a; + 3. 4. lGx- + 16x1-3 = 0. 6. x'^ + {a-x)^={a-2x)-. 6. x- + {a-'2x)-^={a-3x)\ 7. x'^ + x = a^ + a. 8. X' + 2ax = b'-\-2ab. 9. {x-ay^ + {x-bf^{a-b)^. 10. (a-a;)3 + (a;-6)3 = („-^)3. 11. {b-c)x"+{c-a)x + {a-b) = 0. 12. (a; - a + 2/>f - (.c - 2a + 6)^ = (a + b)^. 121. Discussion of roots of a quadratic equation. In the preceding article we found that the quadratic equa- tion ax^ -i-bx + c = had two roots, namely b /b'-^ac , b /6'-4ac - ^ + . / — 7-T- and - yins. 4 3* Jns. ±4. A US. 3, - 1 3* Ans 1 4' " 3 Am. 0, a. ii7l5. 0, 2* Ans. a, - a- -1. Ans. 6, -2a- -b. m ^ns. a, b. .4rw. a. b. A 1 ^~ il/lS. 1, - — - b c 7JS. a - -2^, 2tt -b. 2a V 4a' 2a V 'ia' * o,. /b' — 4ac . , . . T Since . / — 7—2 — IS real or imaginary accordmg as b^ — 4ac is positive or negative, it follows that the roots of aa^ -\-bx + c = are real or imaginary according as H^ — 4ac is positive or negative. The roots are clearly rational or irrational according as 6' — 4ac is or is not a perfect square. It should be remarked also that both roots are rational or botJi irrational, and that both roots are real or both imaginary. S. A. 8 114 SPECIAL FORMS OF QUADRATIC EQUATIONS. If 6^ — 4 X--1 x-1 x^-l X--1 x + l the equation is equivalent to ^ + 2 = 0, x+ X 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 x + 1 x-8 + -r,= T + x-2 x-1 x-1 x-6 In this case it is best not to multiply at once by tlie l. c. 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-n a5-8_-. X- 2 X- 1 x-1 X- G 2 The first two terms = -. ttt-, ^^ , (x - 2) (x - 1) 118 EQUATIONS NOT INTEGRAL. -2 and the other terms = {x -l){x- 0) Hence the equation is equivalent to 2 2 = 0. {x-2){x-l) {x-7){x-Q) Now multiply by the l. cm. of the denominators; then 2 {x -1) {x -6) -2 {x -2) {x -1)=0, which reduces to 20a; -80 = 0; Or thus : — The equation is equivalent to X ^ x-d - x+1 ^ x-S x-2 x-1 x-1 a;-6 2 2 2 2 that is x-2 x-1 x-1 x-G' -10 -10 " {x-2) {x-1) (a; - 1) (a; - G) ' from which we find as before that a; = 4. Ex. 4. Solve the equation : a b c n + 7 + -—=3. x-\-a x-^-l) x + c Wehave -^-1 +4:^-1 +7^-1=0? x-Va x + h x' + c X X X f. + -—-7 +--— = 0. " x + a x + h x + c Hence cb = 0)' 1 1 1 n or else 1 -7 H — = "• x-\-a x + h x + c Multiply by the l.c.m.; then {x + h){x + c) + {x + c){x + a) + {x + a){x + h) = 0, that is 3a;2 + 2a;(a + 6 + c) + &c + ca + a6 = 0, EQUATIONS. ONE UNKNOWN QUANTITY. I I |J tho roots of which are -\{{a + b + c)±J(a^ + b^ + C'-bc-ca-ab)} (li). Thus there are three roots given by (i) and (ii). Ex. 5. Solve the equation : b + c c + a a + b a + h + c H T be -X ca- X ab — X x The equation is equivalent to b + c a c + a b a + b c 1 — - — 1 — — =: 0. bc — x X ca-x X ab-x x Taking the terms in pairs we have {a + b + c)x-abc {a + b + c)x-abc {a + b + c) x-abc _^ X (be -x) X {ca -x) X {ab - x) Hence {a + b + c)x-abc = 1, 1 1 1 n or — ry- . +—. ; + — n \ = ^ ^^• x{bc-x) x{ca — x) x{ab-x) From I. we have x a + b + c From n. we have on multiplication by the l. cm. {ca - x) {ab -x) + {ab - x) {be - x) + {be - x) {ca - x) = 0, that is 3a;2 - 2x {be + ca + ab) + abc (a + 6 + c) = 0, whence x= J {bc + ca + ab=i=Jb-c^+cW + a:^b'^ -abe {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 ^ya: + 4 + ^/.T + 20-2 ^^0; + 11 = 0. We have Jx + ^ + Jx + 20 = 2 ^a; + ll. Square both members : theu 2a; + 24 + 2 J^+l Jx + 1Q = i [x + 11), which is equivalent to Jx + l 7^+20 = a; +10. Square both members : then (a; + 4)(a; + 20) = {a; + l0)2, whence a; = 5. Ex. 2. Solve the equation J'lx + 8-2 Jx + 5 = 2. Square both members: then 2a; + 8 + 4{x + 5)-4 J2x-^Q Jx + 6 = i\ :. 3a; + 12 = 2 J2xTQ JocT^. Square both members : then 9a;2 + 72a; + 144 = 4 (2x + 8) (x + 5) J /. a;2 = 16, whence a; = 4 or a;=-4. Ex. 3. Solve the equation sjax + a + Jhx + /? + J ex + 7 = 0. We have J ax + a + Jhx + /3 = - J ex + 7. Square both members: then we have after transposition (a + 6-c)a; + a + j3-7= -2 Jax + a Jbx + p. Squaring again, we have {{a + b - c) X + a + ^ - y}^ = 4: {ax + a) {bx + ^), that is a;2 (a^ + b^ + c^ - 2bc - 2ca - 2ab) + 2a; {aa + b^ + cy-by-c^ - ca - ay- a^- ba) + a2 + ^2 + 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 exami)les; for there are two square roots of every algebraical expressi'' n and we have no symbol EQUATIONS. ONE UNKNOWN QUANTITY. 121 which represents one only to the exclusion of the other; so that + Jx + 1 and -^x + 1 are alike equivalent to ^Jx + l; also x-v Jx + 1 has the same two values as x^Jx+1. 12G. By squaring both members of the rational equa- tion A=B. we obtain the equation A^ = B'^\ and the equation A^ = B'\ or A^ — B^ = 0, is not only satisfied when A — B = 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 x + l= sja)+ 13 is really two equations since the radical may have either of two values ; and by squaring both members we obtain the equation (x + ly = cc + IS, 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 ?ith degree in x cannot vanish for more than n values of w, unless it vanishes for all values of x. This shews that an equation of the ?ith 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 ax^ + hx + c cannot vanish for a, /3, 7 three unequal values of x. That is we have to prove that ao^ -i-boL ^-c = (i), al3' + hl3 + c = (ii), and ay^ + ^Y + c = (iii), cannot be simultaneously true, unless a, b, c are all zero. 122 A QUADRATIC EQUATION HAS ONLY TWO ROOTS. From (i) and (ii) we have by subtraction that is (oL-l3){a(oL + l3)+b] = 0. But (2 — ^=1=0; hence a(a + l3) + b = (iv). Similarly, since jS — y^O, we have from (ii) and (iii) a(0 + j) + b = O (v). From (iv) and (v) we have by subtraction a(a-j) = (vi). Now (vi) cannot be true unless a=0, for a — ry =)= 0. Also when a = 0, it follows from (iv) that 6 = 0, and then from (i) that c = 0. Thus the quadratic equation aaf + bx + c = cannot have more than two different roots, unless a = 6 = c = ; and when a, b, c are all zero it is clear that the equation aaf + bx + c = will be satisfied for all values of x, that is to say the equation is an identity. Ex. 1. Solve the equation a^ ^^^\^^^ + b^ i^~^jff~^^^ =x^ {a-b){a- c) (b - c) {b - a) The equation is clearly satisfied hy x = a, and also hy x=b; hence a, b are roots of the equation, and these ai'o the only roots of the quadratic equation. [The equation is not an identity, for it is not satisfied by x=c.] Ex, 2. Solve the equation ^2 { x-b){x-c) _^ ^2 ( x-c){x-a) _^ ^2 (3;-a)(--r-?^) ^^, (a -b) (a- c) {b -c){b- a) [c - a) {c - b) . The equation is satisfied by a; = a, by a; = &, or by a; = c. Hence, as it is only of the second degree in x, it must be an identity. Ex. 3. Solve the equation „3 i^-b){x-c ) {x-c){x-a) (x^-a) {x-b) _ {a-b){a-c)'^ {b-c){b-a)'^ (c-a){c-b)~ ' EQUATIONS. ONE UNKNOWN QUANTITY. 123 The equation is satisfied by x = a, by x = b, and by x = c ; and the equation is not an identity, since the coellicient of x^ is not zero. Ucnce the roots of the cubic are a, b, c. Ex. -4. Shew that, if (a - a)2x + {a- ^fy + [a - y^z = {a- S)\ (h - aYx + {h - ^)'-y ■\-{b-yfz = {h-bY, {c-afx+{c-^Yy + {c-yYz = {c-5)\ then will {d-afx+{d-^fy + {d-yfz = {d-b)\ where d has any value whatever. The equation {X-afx + {X-fir-y + {X-yfz = {X-5Y is a quadratic equation in X, and it has the three roots a, b, 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 /3 for the roots of the equation ax^ + bx + c=0, we have "~ 2a "^V 4a^ ' and /3 = — 5 , / "j-.r~ • 2a V ^^^ By addition we have a+0 = ~^ (i). a By multiplication we have 1/ h'^ — 4ac c r-\ '■"^ = ^'—i^'- = a ^">- The formulae (i) and (ii) giving the sum and the product of the roots of a quadratic e([uation in terms of the coefiicieuts 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 aa?" + ha;'"''- + cx''^'' + dx""'^ + . . . , vanish for the n values x^a, x = ^^ x = (a"-'' + &""'' + c"-2) + q {rt"-3 + ?>"-3 + c«-3) = 0. Hence we have in succession a* + b^ + c^=2p^, a^ + b^ + c'^=5pq, a6 + 66 + c6 = 3(/2_2^^ a'' + 67 + c7=-7^2^. Hence also a^ + b'^ + c ^_ a'^ + b^ + c^ a^ + b^ + c^ 5 ~ 2 • 3 ' a^ + b "^ + c7 _ a^+b^ + c^ a5 + i5 + c^ ' 7 " 2 ~ • 5 * _ a^ + b^ + c^ a^ + b^ + c* -2. ^ . ^ . [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 5. We want to find an equation which is satisfied when x = 4, or when x = 5; that is when .t - 4 = 0, or when x - 5 = ; and in no other cases. The equation required must be {x-4){x-5)=0, that is, a;2-9a; + 20 = 0, 126 EQUATIONS WITH GIVEN ROOTS. for this is an equation which is a true statement when a; -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 (a; - 2) (a; - 3) (a; + 4) = 0, that is a;3-a;2-14x + 24 = 0. Ex. 1. If a, ]3 are the roots of the equation aa;^ + 6a; + c = 0, find the equation whose roots are — and - . p a The required equation is that is x'^-x":^^ + l = 0. ap Now, by Art 128, we have a a &2 c a^ a ,a2+^_/&2 c\ c_&2_2ac aj3 ya"- a J a ac Hence the required equation is x^ a; + 1 = 0. ac Ex. 2. If a, /3, 7 be the roots of the equation ax^ + hx^ + cx + d=Oy find the equation whose roots are ^y, 7a, a^S. The required equation is (a:-/S7)(a;-7a)(.r-a/3) = 0, that is x^- X- {py + 7a + a/3) + xa/37 (a + j3 + 7) - a-/3273 = 0. * The equation x'^ — 9a; + 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 x^ + 7x^ -2 = had no roots, then {x - 4:){x - 5) (a-^ + 7a;2 - 2) = would also be an equation with the proposed roots and with no others. The proiiosition 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 + ]8 + 7= --, and a/37= — , a Hence the required equation is a a-* a-^ or a^^^ - acaP + bdx -d^ — 0. 131. Changes in value of a trinomial expression. The expression auf -\-hx ■\- q will alter in value as the value of X is clianged ; but, by giving to x any real value between — oo and + oo , we cannot make the expression oic^ -\-hx-\-c assume any value we please. We can find the possible values of a'o^ -{-hx-\- c, for real values of x, as follows. In order that the expression ax^ + hx-\-c may be equal to \ for some real value of x, it is necessary and sufficient that the roots of the equation ax^ ■\-hx + c = \ be real, the coudition for which is 6' - 4a (c - X) > 0, that is y^— 4ac + -iaA.> (i). I. If ¥ — ^ac 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 \^ — 4!ac. Thus, when 6^ — 4ac is positive, ax"^ + bx i- 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 — ^~ and whose sign is opposite to that of a. IT. If I)^ — 4ac be negative, the condition (i) can only be satisfied when 4aX is pcjsitive and not less than 4rtc — h*. 128 CHANGES IN VALUE OF A TRINOMIAL EXPRESSION. Thus, when 6^ — 4>ac is negative, ax^ +bx -{- c must al- ways have the same sign as a, and its absolute magnitude can never be less than — . 4a TIT. If b^ — 4 0, or that X2-8X + 72>0, or (X-4)2 + 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^ — 3x + 4c Ex. 3, Shew that ^ j can never be greater than 7 nor less than - for real values of x. Put =X' a;2 + 3a; + 4 ' then a;2(l-X)-3a;(l + X) + 4(l-X) = 0. In order that x may be real it is necessary and sufficient that 9{1 + X)2-16(1-X)2>0, that is _7X2 + 50x_7>o, or -(7X-l)(X-7)>0. Hence 7X - 1 and X - 7 must be of different signs, and therefore \ must lie between - and 7, which proves the proposition. EXAMPLES X. Solve the following equations: 1. (x-a + 2by-(x-2a + bf = (a + by. 2. {c + a-2b)x^ + (a + b-2c)x + {b + c- 2a) = 0. 3 '' '' {x - ay (x + by ' A a + x b + x ,1 + X a + X 5. 6. EQUATIONS. ONE UNKNOWN QUANTITY. 131 ax+ b CX + d a + bx c + dx ' a — x _1 -bx 1 —ax b -X ' r7 3a;-4 , . 7 7. T- = cc + 2^c - a;+ 1 x+l' , X* x'^ 5x- i 8. a; + 1 + — — r = If + 3 1 . x' - 1 x + l x^-l 9. « — -^ + ^ + ^ + 5 = 0. X- b X- b X + b x+ 8 in 2 6 3 4 10. ^+ 7^= z— + 19. a; +8 x+d x + 15 x + 6' n 2 1 6 11. 7^ X + 2a;-3 x-2 3x + 2' _- x — a x~b X — c 12. r + + = 3. a; — o a; — c x—a __ a; + a a; + 6 a; + c „ 13. + ^^ + 3. a- x b-x G — x .. x + a X + b x + c 14. + . + =: 3. x — a X — b x — c T ^ 2a; - 1 3a; - 1 , x-7 15. ^+ ^-=44._ — . x+ 1 a; + 2 a;-l ,« a; 2 a; 3 16. -+-=-+-. J a; 3 a; __ x+a x—a x+b x—b 17. 4- + + = 0. x — a x + a x—b x + b ,o x—\ 33 — 4 a; — 2 a;— 3 18. T + i = -A + x + l x + i x + 2 a;+3* 1 1 1 ~ 1 • x+a\ ; x—a + x + b X ~ b 0—2 132 EXAMPLES. 20. -1-4-— l-+^-i-=0. oa — x 6o — x oG — X 21. + = — ?^ — L . x + o x + c x + b +c nn Oj+ c b + c a -i-b + 2g 23. 29 X b x — a_2{c(j-b) x — a x — b x — a — b' nA {x + a){x + b) _{x-\- c) (x + d) x+ a + b x + c + d 25 (^{c-d ) d(a-b) _b(c - d) c(a-b) x + a x + d x + b x + c nn x — a X — b b a 26. —^r- + = + b a x — a X — h ' 27. -^^+ '>'" ^ "-" .0. x+a—0 x+o—G x+c— a 28 — 1___J_4._J__ ^___0 • 1 + 2ic 2 + 3a; 3 + 4cc 4 + 5x (x - a) {x- b) _ {x + a){x + b) {x — ma) {x — mh) {x + ma) {x + 7nb) ' 30. J2x+^ - J^^ = V^TT. 31. '^(x-l){x-2) + ^/{x-3){x-^) = J2. 32. ^703 - 5 + ^/4:x - 1 = \/7x - 4 + V4a;-2. 33. ^/a^ _ aj + \/6^ + x = a + b. 34. ^a-x + ^b-x = \/a + b-2x. 35. ^Ja -bx + \/c-dx = \la + g— {b + d)x. 36. >^ax +b^ + \/bx + a^ = a-b. l\..r EQUATIONS. ONE UNKNOWN QUANTITY. 1 .']3 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-b) = '2, Jax. 40. J a (a + b + x) — J a (a + b-x) = x. 41. Jx- + ax + b^ - Jx^ -ax + b" =^2a. 42. Jx' + ax + a^ + Jx^ -ax-¥c^ = JlaJ" - 2b\ 43. J ax ~b + J ex + b = J ax + b + J ex - b. 44. Jx {a + b -x) + J a (b + x - a) + Jb (a -h x - b) = 0. 45. Jx + a + Jx + b + Jx + c = 0. -^ 46. Jab (a + b + x) = J a {a + b) (b- x) + Jb {a + b) (a- x). 47. Jx^ - b' - c^ + Jx^ -G^ - a^ + Jx^ - a^ - b' = x. 48. J'^^^+ J¥^^+ J7^^^ J a' + b' + G'- x\ 49. For what values of cc is ^ 14 - (Sec -2){x-\) real, x^ + 34ic-71 50. Shew that — 3 — —- can have no real value between x" + 2x-l 5 and 9. 51. Shew that, if x be real — ; — — =- can never be less a^ + 2a; + 1 than — ^. rjf^ aj + 1 52. What values are possible for — , , x beinc: real. ^ x- + x+\ ° 53. Find the greatest and least real values of x and y which satisfy the equation x^ + if = ^x- 8v/. 54. Find the greatest and least real values of x and y when 0;" + 4?/' -8a; -16?/- 4 = 0. 55. When x and y are taken so as to satisfy the equation {x^ + yY= 20^ {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 x^ {jf + 5^2) + 2^ {ah + a'b') + a' + a" = be real, they will be equal. 57. If the roots of the equation ax^ + bx + c = be in the ratio m : n, then will nmb^ = (m + ny ac. 58. If ax^ + 2bx + c = and a'x^ + Ib'x + c' = have one and only one root in common, prove that b^ — ac and b'^ — a'c' must both be perfect squares. 59. If x^, X be the roots of the equation ax^ + hx + c = 0^ 2 2 X X find the equation whose roots are (i) x^^ and x^^, (ii) — ^ and ^' (iii) b + ax^ and b + ax^. 60. If fljj , x^ be the roots of ax^ + 5aj + c = 0, find in terms of a, b, G the values of cCj^ {bx^ + c) + x^ (bx^ + c), and x^ (bx^ + c)^ + a;/ (5ajj + c)^ 61. Shew that, if a;, , cc^ be the roots of x^ + mx + m^ + a = 0, then will x^^ + x^x^ + x^^ + a=0. 62. If a?! , x^ be the roots of (a;* + 1) (a^ + 1) = ^*<^^ (^^ — !)> then will (aJ,^ + 1) (a?/ + 1) = 'wzaija!^ (ajia;^ - 1). 63. If x^, ajg be the roots of the equation A (x^ + m^) + Amx + Bnfx^ = 0, then will A {x^ + a?/) + ^ai^aj^ + Bx^x^ = 0. 64. Prove that, if x be real, 2(a — a;) (aj + Jx' + b^) cannot exceed a^ + b^. 65. Find the least possible value oi ^-s — tt:; — , ^ {x + \y 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^ + ^o;^ + c = can be solved in exactly the same way as tlie quadratic equation ax^ + 6.^ + c = ; we therefore have 2 h . Jh' - 4ac X = h • 2a - 2a Hence .= , ^j -^^ , ^«} . Thus there are /ow-r 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, aPH6P + c = is equivalent to P = — -^ + '^^—^ — — • ^ 2a - 2a Ex.1. To solve a;-'- 10.t"^ + 9 = 0. We have {x^-d){x^-l)=0\ .'. x^=9, giving x= ±3; or else a;^=l, giving a; = ±1. Thus thnre arc four roots, namely +1, - 1 , 4-3, - 3. 136 EQUATIONS OF HIGHER DEGREE THAN THE SECOND. Ex. 2. To solve {x^ + x)^ + 4: {x^ + a;) - 12 = 0. The equation may be written {x^ + x + Q) {x^ + x - 2) = 0. Hence x^ + x+& = 0, or x^ + x-2 = 0. The roots of a;^ + .r + 6 = are - - ± ^J^23. The roots of a;2 + a;-2 = are 1 and -2. Thus the roots are 1, - 2, - p =*= nv ~ 23. Ex.3. . (a;2 + 2f + 8a; {x^ + 2) + 15a;2 = 0. The equation is equivalent to (a;2 + 2 + 5a;) (a;2 + 2 + 3a;) = 0. The roots of a;2f3a;+2 = are - 1 and -2. K. I'\n The roots of a;2 + 5a; + 2 = are - jj ± ^'^^ . Thus the equation has the four roots -1,-2, - - i - ^/l?. Ex.4. To solve ax'^ + hx^-c + y >Jax^ + hx + c + q = 0. Put y=Jax^ + bx + c; then y^+py+q=o, whence we obtain two values oi y, a and /3 suppose. We then have ax^ + bx + c = a% or ax^ + bx + c=^f and the four roots of the last two quadratic equations are the roots required. Ex.5. To solve 2a;2 - 4a; + 3 ^a;^ - 2a; + 6 = 15. The equation may be written 2 (a;2 - 2a; + 6) + 3^(^:2 - 2a; + 6) - 27 = 0. Put y = V(^^ - 2a: + 6) ; then we have 2if + Sy-21 = 0, whence 2/ = 3, or y= --. Hence x^-2x + 6 = 9, giving a; = 3 or -1; ~^ or else a;2-2x+6 = -j-, giving a; = l±- ^^/gT. Thus the roots are 3 ; - 1 ; 1 ± 5 J HI. a EQUATIONS. ONE UNKNOWN QUANTITY. 137 9 Ex. C. To solve (x + a){x + 2a) {x + 3a) {x + 4a) = - 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 {a;2 + 5ax + 4a-) (x^ + 5ax + Ga^) = ~ aK lo Hence {x^ + 5ax)'^ + lOa^ {x^ + 5ax) + lia^ = ^ a\ .'. x'^ + 5ax= --—a"^, or else x^ + 5ax=. — ^a^. 4 4 5 5 a Hence a; + -a = 0, or a;+ •a= ±-,^10. 5 5 a Thus the roots are ~ o ^» ~ o ^ ^ o V^^ • loC. 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 forv^^ards. Thus ax^ -^hx^ -\-hx-\-a = 0, ax*' + hx^ + cx^ -\-hx + a=0, and ax^ + hx^ + cx^ — cx"^ — bx — a = are reciprocal equations. [See also Art. 442.] Ex.1. To solve ax^ + bx^ + bx + a=:0. We have a{x^ + l) + bx{x + l) = 0, that is {x + l){a{x'^-x + l) + bx}=0. Hence 05= -1, or else ax- + {b-a)x + a = 0. Ex. 2. To solve ax* + bx^ + cx^ + bx + a = 0. Divide by x^ ; then we have Now put then Hence 1 x + - = X ■y; x^ -y- -2. a 0/-2) + by- 4-c = :0. 138 ROOTS FOUND BY INSPECTION. Let the two roots of the quadratic in ?/ be a and /3; then the roots of the original equation will be the four roots of the two equations 1 ^ 1 ^ x + -=a and x + -=8. X X Ex. 3. To solve ax^ + hx^ + cx^ - cx^ -bx~a=0. We have a {x^ -l) + bx {x^ - 1) + cx^ {x-l) = 0, that is {x - 1) {a{x^ + x^ + x^ + x + 1) + bx{x^ + X + 1) + cx^} = 0. Hence 03=1, or else ax^ + {b + a) x^ + {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 re - 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'^-{S-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 x^ + 2x^-nx + & = 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 : — ■ If a;= ±- be a root of the equation ax^^ + bx"-'^+ ... + k = 0, where a, b,...k are integers and - is in its lowest terms, then a will be a P factor of k and ^ 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 x=2 satisfies the equation, and we have {x -2) {x'^ + Ax -d)=x^ + 2x- -llx + 6. Hence the other roots of the equation are given by a;2 + 4r-3=:0, and are therefore - 2 ±^7. EQUATIONS. ONE UNKNOWN QUANTITY. 139 Ex. 3. Solve Since x = a and x = b both satisfy the equation, {x - a) (x- 6) will divide (a - a;)* + (x - 6)* - (a - b)"*, 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-hy={{a-x) + {x-b)}^ = {a- x)^ + ^{a- xfix - b) + 6{a - xy- {x -b)^ + 4{a-x){x-by+{x-by; .•.2{a-x){x-b){2{a-x)^ + 3{a-x){x-b) + 2{x-by}=0. Thus the required roots are a, b and the roots of the quadratic a?-x{a + b) + 2a^-Bab + 2b^ = 0. Ex. 4. Solve the equation ^^ {x-b){x-c) ^ ^4 {x -c){x- a) _^ ^^ {x -a){x- b) ^ ^^^ (a -b){a- c) {b -c){h- a) (c -a){c-b) The equation is clearly satisfied by a; = a, byaj = 6, and by a; = c. Also, since the coefficient of ^ 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, 6, c, - (a + 6 + c). 138. Binomial Equations, The general form of a binomial equation is x"^ ± 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^-l==0. Since aP-l = {x-l){x'^ + x + l), we have a; - 1 = ; or else x^ + x + l = 0, the roots of which are Hence there are three roots of the equation x^=l; that is there are three cube roots of unity, and these roots are 140 CUBE ROOTS OF UNITY. Ex.2. To solve a;4-l = 0. Since x'^-l = {x-l){x+l) {x + J -1) {x- J -1), the four fourth roots of unity are 1, -1, iJ-1 and -/J-1. Ex.3. To solve a;5-l = 0. x^-l = {x-l) (x'^ + x'^ + x^ + x + 1). Hence x = l; or else ic^+x^ + x^+x + l = 0. The latter equation is a reciprocal equation. Divide by x^, and we have x^+-+x + - + l = 0. X" X Put x + -=y; X then x'^ + -^ = y^-2; 2 • ... ?/=--. ± Hence a; + - = ^^ , ic 2 that is g^-a; „ +1 = 0. Hence x = —^^ ± ^ ^-10-2^5, or ^^ -1-^5 ^^^_;^Q^2V5, or a; = l. Ex.4. To solve a;4 + l = 0. a;4 + 1 = (x^ + 1)2 - 2a;2 = (a;2 + 1 - ^2a;) (a;^ + 1 + ^2a;). Hence x^ =f yv/2^ + 1 = 0; _±1±V^ •'• "'" ^/2 139. Cube roots of unity. In the preceding article we found that the three cube roots of unity are 1, 4(-1+n/^3), li-l-J'^). An imaginary cube root of unity is generally repre- sented by ft) ; or, when it is necessary to distinguish EQUATIONS. ONE UNKNOWN QUANTITY. 141 between the two imaginary roots, one is called co^, and the other 0)^, so that 1, co, ami co^ are the three roots of the equation x^ —1 = 0. Taking the above values, we have 1 + to. + 0,, = 1+ J (- 1 + v/^) + i(- 1 -n/^) = 0, also ft)j«, = J(_ 1 + yiTs) (- 1 - 7^) = 1. These relations follow at once from Art. 129 ; for the sum of the three roots of .1;^ — 1 = is zero, and the product is 1. Again a>;' = K" 1 + J^^T = K" 1 - J^) = co,, and o)./ = i(- 1 - J-Sy = K- 1 + J^) = ^,> so that co^^ = co^ and (o^ = (d^. These relations follow at once from &)jft)2 = 1 and (o^ = ci)/ = 1. Thus if lue square eitlier of the imaginary cube roots of unity we obtain the other. Hence if co be either of the imaginary cube roots of unity, the three roots are 1, w and ai\ We know that a3 + 63 + c3 _ 3a6c = [a + h + c){a^-\- b- + c^-bc-ca- ab). Hence a + b + c is a factor of a^ + b^ + c^- 3abc, and this is the case for all values of a, b, c. Hence a + {uh) + {wh) is a factor of a^ + (ub)^ + (w-c)^ - 3a {wb) (urc), that is of d^ + b'-^-\- c^ - Babe ; and a + urb + lac can similarly be shewn to be a factor. Hence a^ + t'^ + c^- 3a6c = (a + 6 + c) (a+ w& + a/^c){a + w-& + wc). EXAMPLES XI. Solve the following examples : 1. x' - 'Ix^ -8 = 0. 2. x^ + 7«V - 8a' = 0. k 142 EXAMPLES. 3. x^ - 7aV - 8a' = 0. X cc* + 1 5 4. —^ — r + 5. x'+l X 2 • a;'' + 2 a;^ + 4a; + 1 5 a;''+4aj + l a;' + 2 2 6. (a;Va;+l)(a;' + aj + 2) = 12. 7. (a;^ + 7aj+5)^-3a;'-21aj = 19. 8. \/l6-7a;-aj' = a3'+7£C-^. 9. 6 s/a;' - 2a; + 6 = 21 + 2a; - a;^ 10. (a-l)(l +a; + a;7 = (a+l)(l+a;' + a;''). 11. (a; + 1) (a; + 2) (a; + 3) (a; + 4) = 24. 12. (a; + a){x + 3a) {x + 5a) {x + 7a) = 384a\ 13. (a; - 3a) (a; -a){x+ 2a) (x + 4a) = 2376a\ 14. (a; + 2) (a; + 3) (a; + 8) (a; + 12) = 4a;^ 15. 2a;'-3a;-21 = 2x^/a;'-3a; + 4. 16. a;^ - 2 (a + 6) a;' + a^' + 2a6 + b' = 0. 17. X* - 2a;V - 2x'b' + a' + b^- 2a'b' = 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;-l = 0. 25. (x - ay {b - of + {x- by (c - ay + {x- cy (a - by = 0. 26. a;(a;-l)(a;-2) = 9.8.7. EQUATIONS. ONE UNKNOWN QUANTITY. 143 27. x{x-l){x-2)(x-3) = 0.8.7.(j. 28. {a - xy +{b-xy^{a + b- 2x)\ 29. (a - xY + {b- xy = {a + h- 2xy. 30. (a - xy + {h- xy = {a+b- 2xy. 31. ^a-x + ^b-x = l/a -\-b-2x. 32. ^a - X + Jb — x = J a + b — 2x. 33. {a - xy + (re - h)' = {a- by. 34. J a — x + Jx - b = J a - b. 35. Ja -x+ ^x -b = ^a — b. 36. x' + {a-xy = b\ 37. {x + ay + {x + by=l1 {a - by. 38. i/x + ^a-x= ^b. 39. abx {x + a + by - (ax + bx + aby = 0. 40. abcx (x + a+b + cy — (xbc + xca + xab + abcy = 0. (a - xy + {x- by _ a' + b' ' ■ (a + b-2xy ~{a + by' 42. x* + b{a + b) x^ + {ab - 2) b'x' -(a + b) b'x + b' = 0. 43. (x' + by = 2ax' + 2ab'x - aV. 44. (x + b + c) (x + c + a) (x+ a + b) + abc = 0. 45. 1 + + 7 +3=0. + c — X c + a — x a + — X 4fi (^-< {x-by jx-cy (x-ay-{b-cy {x-by-(c-ay "^ {x-cy-{a-by * {x + a) {x + b) {x - a) {x - b) _(x + c) (x + d) (x - a) (x -b) {x + a) (x + b) {x- c) (a: - d) (x -c) (x- d) [x +c){x-¥dy CHAPTER X. Simultaneous Equations. 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 x, y, z... is the degree of that term which is of the highest dimensions in x^ y, z Thus the equations ax + d^y + a^z — a*, xy + x + y + z^O, a^ + y'' +z^ - Sxyz = 0, are of the first, second and third degrees respectively. SIMULTANEOUS EQUATIONS OF 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, z,... can by transformation be reduced to the form ax -{-by ■¥ cz + ... =k, where a, 6, 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 different equations. Thus we use a, h, c... for one equation; a\ b', c ... for a second ; a", 6", g' ... for a third ; and so on. Or we use ttj, 6j, Cj for one equation; a^^b.^, Cg 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 + b'y = c\ and similarly in other cases. 142. Equations with two unknown quantities. Suppose that we have the two equations ax +by = c, and ax-\-b'y = c'. Multiply both members of the first equation by b\ the coefficient of y in the second ; and multiply both members of the second equation by h, the coefficient of y in the first. We thus obtain the equivalent system ah'x + bb'y = cb\ a'bx + bb'y = c'b. s. A. 10 146 SIMULTANEOUS EQUATIONS OF THE FIRST DEGREE. Hence, by subtraction, we have {ah' — a'h) x = ch' — c'h ; ch' — c'h whence x = ah'— a'h ' Substitute this value of x in the first of the given equations; then ch' — c'h ^ a — TT jj + hy = c, ah —ab "^ J _ c (ah' — a'h) — a (ch' — c'h) •*• ^^- W^al) ' , ac — a'c whence y = —77 77 . ^ ah — ah The value of y may be found independently of x by multiplying the first equation by a and the second by a ; we thus obtain the equivalent system a'ax + ahy = a'c, a' ax + ah'y = ac. Hence, by subtraction, we have {a'h — ah') y = a'G — ac' ; _ ac — ac' '*• y^'a'h'^^^" which is equal to the value of y obtained by substitution. Note. It is important to notice that when the value 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 h' have to 2/ ; we may therefore change x into y provided that we at the same time change a into 6, b into a, a' into h\ and h' into a'. Thus from ch' — c'h , ca — c'a X = —n 7-7 we have 7/= y-, — r^ . ab —ab ^ ba — ba SIMULTANEOUS EQUATIONS OF THE FIRST DECREE. 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 + b'y = c\ can bo expressed in the form X y _ -1 he — h'e ~ ca — oa cb' — ah So also, from the equations ax -\-hy + c = 0, and ax + h'y + c' = 0, we have ^ 2/ he —h'c ea—e'a ah'— ah It is important that the student should be able to quote these formulae. Ex, 1. Solve the equations 3a; + 2y = 13, and 7a; + 3^=27. X 2/ _ -_L_ We have that is 2. 27 -3. 13 "13.7- 27. 3 3.3-7.2 ^ _ 2/ _ 1. 15 ~ To ~ 5 ' .'• X = -_' ^ 0| and y = — = 2. 10—2 148 SIMULTANEOUS EQUATIONS OF THE FIRST DEGREE. Ex. 2. Solve the equations X y and = - 7. X y 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 y -1 3(- -7)-(-6)2 2.2-(-7)4~4(-5)-2.3' that is 3^ 1 X y 1 _ - 11 " 32 ~ 26 ' 1 11 26 •• X- 26' °^ ^- 11- Also 1 32 13 ^ = 26'^^2/ = i6. Ex. 3. Solve the equations x-y = a-'b, ax - by = 2 {a'' -h^). We have X y ^i -2{a^-b'^) + b{a-b) a{a-b)-2{a^-U^} -b + a' that IS _ ;» _ . b-^ + ab-2a^ ^U^-ab-a- b-a' &2 + a&-2a2 , „ .*. x= = 6 + 2a; b-a 2&2_a&-a3 and y z = a + 2&. b-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, {b - a)y = a{a -b) -2 [a^ -b")', -a^-ab + 2b'^ .*. y = i — = a + 2b. b-a V:. S1MULTA:sE0US equations of TiJi: Flli^ST DEGREE. 140 Then x = (a + 26)+a-6; 144:. Discussion of solution of two simultaneous equations of the first degree. We have seen that the Viilues of j; aud y which satisty the equations a.v+bij=c (i), and a'x + b'i/ = c (^ii), ai'e given by (ah' — ah)x = ch' — c'b (iii), {ba — h'a) y = ca' — c'a (\\). Thus there is a siuole finite value of x, aud a siuirle finite vaUie of y, provided that ab' — a'b^O. If ab —ab = 0, x will be injinite [see Art. 11 S] unless cb' — c'h = 0; and, if ah' — ah and ch' — c'b are both zero, any value oi x will satisfy equation (iii). So also, y will be infinite if ab' — a'b = 0, unless ca —ca is also zero, in which case any value of y will satisfy equation (iv). If ab' — a'b = 0, then — = v, ; and if ah' — a'b = and a also ch' — c'b = 0, then — = — =-. a c When equations cannot be satisfied hy fnite values of the unknown quantities, they axe often Siiid to be incon- sistent. Thus the equations ax -\-by=c and a'x + b'y = c are inconsistent if — = t? , unless each fraction is equal to a o — , in which case the equations are indeterminate. In fact c when -7 = T-' = — y it is clear that by multiplvincr the terms a b c "^ r . o of equation (i) by — we shall obtain equation (ii), so that the two given equations iu:o equivalent to one only. 150 SIMULTAI^EOUS EQUATIONS OF THE FIRST DEGREE. We have hitherto supposed that a, a, h, 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 = y , g' and from (ii) y-jj. These results are inconsistent with c c' one another unless t= r,- c - c' Hence, if a= a' = 0, and j =j,, the equations (i) and (ii) are satisfied by making y =^ r , and by giving to x any finite value whatever. c c' If however r =f= tv > the equations hy = c and h'y = c' cannot both be satisfied, unless they are looked upon as the limiting forms of the equations ax+by=c and ax+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'h not being zero. Thus, in the equations (i) and (ii), when a and a diminish without limit, and cb' =}= cb, the value of x must be infinite. Equations with three unknown quantities. 145. To solve the three equations : ax + by -\-cz = d (i), a'x-^b'y-\-c'z = d' (ii), a"x + Vy + c"z = d" .(iii). Method of successive elimination. Multiply the first equation by d , and the second by c ; then we have ac X f bey + cc'z = dc, X SIMULTANEOUS EQUATIONS OF TUE FIRST DEGREE. 151 and a ex + h'cij + c'cz = dc\ therefore, by subtractioD, {ac — a'c) X + ihc' — b'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 + {he" - h"c) y = dc" - 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 - (be' - b'c) (do" - d"c) + {do - d'c) (be" - h"c) {ac — a'c) {be" — b"e) — (be — b'c) {ac" — a'c) Method of undetermined multipliers. Multiply the equations (i) and (ii) by \ and /jl, and add to (iii); then we have the equation X {\a + ua + a") -\-y(kb + fib' + h") +z{\e + \xd + c") = (Xd + fid' + d"), which is true for all values of X and fi. Now let \ and /x be so chosen that the co-efficients of y and 2^ may both be zero, \d + ad' + d" then ^=>", ^, — '/ » \a-\- fia -{-a where \ and /a are found from Xb+fxb' + b"=0, and Xc + /ac' + c" = ; . ^ _ /^ _ _1 • • b'e" - b"c b"c - be" bo - b'c ' Hence d {b'c" - b"c') + d' {b"c - be") + d" {be' - b'c ) ^ - a {b'c" - b"c) + a [b"c - be) + a {be - b'e) ' [The numerator and the denominator of the first value uf a;, which was obtained by eliminating z and ij in succcs- 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 b, a' and h', and a" and b". The value of y can also be obtained from that of a? by a cyclical change [see Art. 93] of the letters a, b,c\ a, b', c ; and a\ b", 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 : x + 2y + ^z = ^ (i), 2a; + 4?/+ z = l (ii), 3x + 2?/ + 92 = 14 (iii). Multiply (ii) by 3, and subtract (i) ; then 5a: + 102/ = 15 (iv). Again multiply (i) by 3, and subtract (iii); then 42/ = 4 (v). From (v) we have ?/ = 1 ; then, knowing y, we have from (iv) a; = 1 ; and, knowing x and y, we have from (i) z = l. Thus x=y = z=.l. Ex. 2. Solve the equations : x + y + z=\ (i), ax + by + cz = d (ii), a^x + h^y + ch = (P (iii). Multiply (i) by c and subtract (ii) ; then {c - a) x+ {c -h)y =c - d (iv). Again multiply (i) by c^ and subtract (iii) ; then {c^-a^)x + {c^-h'^)y=c^-cP (v). Now multiply (iv) by c + & and subtract (v) ; then {c-a) {b - a)x={c - d) {b - d); • ^Jb-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 : tlicy are {c-d){a-d) ^ {a-d){b-d) ^ (c - 6) (a - 6) ' {a-c){b-c)' Instond of going througli the process of elimination, wc may at once iiuote the general formulae. Thus _ {bc^-- hh) + d{b'^-c^) + d'^{c- h) ^~(bc- - Uh) + a[b'^- c^) + a^{c- b) _ {b-c){-hc + d{b + c)-d''] ~ {h-c){-bc + a{b + c)-a-\ {b -d){c-d) = )-j f7 f , as above. (b - a) (c - a) Ex. 3. Solve the equations : x + y + z = a+b + c (i), ax + by + cz = bc + ca + ab (ii), hex + cay + dbz = Zahc (iii). We have _{a + b + c) (at^ - ac^) + {be + ca + ab) [ca - ab) + Sabc (c - b) ~ ab' - acP" -Vaica- ab) + bc{c- b) _a{b-c){{b + c){a + b + c)-bc-ca-ab-Sbc} ~ (b-c) {ab + ac-a^- be] _ a{b- c)3 ~ [a -b) (a-c)' The values of y and z can now be written down : they aro h{c- a)2 _ _ c (g - by- y~ {b-c){b-a)* ^~ (c -a){c-by Ex. 4. Solve the equations : x+ay-\-a-z-\-a^ = (i), x + by+h"z + h'^ = Q (ii), X •\- cy + c^z + c^ = (iii). The equations may be solved as in the preceding examples, or as follows. It is clear that a, b, c are the three roots of the following cubic in X \^ + z\^ + 7j\ + x = 0. Uence from Art. 129, we have at once z= -{a + b + c), y = bc + ca + ab, and .T= - abc. 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 ax +hy ■\- cz '-\-dw =e (i), ax +h'y ■\-dz -\-d\u = e' (ii), a"x + h"y + c"z + d"w =e".... (iii), a"x + h"'y + c"'z + d"'w = e" (iv). Multiply (i) by X, (ii) by //,, (iii) by v, and add the products to (iv). Then we have X {aX -f a> + a"v + a") + y {h\-\- h'fi + h"v + 6'") 4 ^ (cX + c> + c"v + g") + w {d\ + d!ix + dl'v + d!") = eX + e'yL6 + GV + 6'''' (v). Now choose X, jx, v so as to make the coefficients of y^ z and 11} in the last equation zero ; then /// eK + e Uj 4- e'v + e , .. X= ■ jT—, 777 (vi), a\ + afjb + a v + a where X, fju, v are to be found from the equations cX + c'lM+c'v+c" = ol (vii). d\ + d'fjL + d''v + d'" = oj Hence we have to solve (vii) by Art. 145 and then substitute the values of X, fi, 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, b, c, d, &c. SIMULTANEOUS Et^UATIONS OF THE FlUST DEGREE. 155 EXAMPLES XII. Solve the following equations. X K>y \ 18 8 -„ -r:-i-^ = o- —+-=10. 5 10 2 ^ y 3. a; + -=-, 4. +5=-+-= 10. 2/2 X y X y y 3 5. ax + hy = 2ab, 6. x + ay + a" = 0, hx-ay = b'- a'. x + by + b'^ = 0. 7. X + 7/ = 2a, 8. (5 + c) X + (/j -c)y = 2ab, (a - b) X = (a + h) y. {c + a) x + (c- a) y = 2ac. 9. bx + ay = 2ab, o?x + b^y = a^ + W. 10. (a + 6) a; + 6?/ = aa; + (5 + a) 2/ = ^&^ - ^^ 11. x + y + z=l, 12. a: + 2/ + ;r = l, 2a;+32/ + c; = 4, ? ■ 2/ , 4.^1 4.x + 0y + z=lG. 2 i"" "' '' 5 3 ;3 , 3-^n^-r^- 13. a; + 2?/ + 32; = 3a.- + 7/ + 2;^ = 2.x + 32/ + ;:: = 6. 14. y + z=2a, 15. y + z-x = 2a, z + x-2b, z + x — y=2b, x + y=2c. x + y - z = 2c. 156 EXAMPLES. 16. y + z-3x=2ay 17. ax + bj/ + cz=l, z + x-3y = 2b, bx + cy + az=lf x + y -3z = 2c. ex + ay + bz=l. iQ y+z—x z+x~y x+y-z lO. - f = = r = 1. + c c + a a + 19. x + y + z = 0, 20. x + y + z = a + b + c, ax + by+cz=l, bx + cy + az — bc+ca+aby a^x + ¥y + c^z = a + b + c. cx + ay +bz = bG + ca + ab. 21 x + y + z = a + b + Cj bx + cy + az = a^ + b^ + c^, cx + ay + bz — a^ + b'^ + c^ 22. x + y+ z = 0, . {b + c)x + (c + a)y + {a + b)z=^{b-c) (c- a) {a - b), bcx + cay + abz = 0. 23. ax + by + cz = a, 24. x - ay + a^z - a^ =■ 0, bx+ cy +az = bf x—by + ¥z - 6^ = 0, cx + ay + bz = c. x— cy + c^z — c^ = 0. 25. ax + by + cz = m, a^x + b^y + c^z = m^, a^x + b^y + &z = m\ 26. ax + cy + bz = a^ + 2bcy cx + by + az = b^ + 2ca, bx + ay + cz = c'^ + 2ab. 27. x + y + z = 2a + 2b +2c, ax + by + cz = 2bc + 2ca + 2ab, (b — c)x + (c — a)y + (a — h)z-—Q> /^ 28. ax + by + CZ =a + b + c, a'x + ¥y + c'^z - (^a + b + c)^, bcx + cay + ahz = 0. SIMULTANEOUS EQUATIONS OF THE SECOND DEGREE. 157 29. X + y + z = I + 7/1 + n, Ix + my + 7iz —■ mn + nl + Im^ {in — n)x+{n-l)y + (l-m)z = 0. 30. Ix + ny + mz = nx-\-my + lz = mx+ly + nz = P + m^ + n^- Zlmn. 31. I'x + m^y + Tt'z = Imx + mny + nh = nix + Imy + mnz = I + m + n. 32. +—->,+ =1, a + a a + (d a + y X y z -, 6+a b + /3 b + y ' X y 2 -, + -^, + =1. 33. 2/ + » + t« = a, « + 2« + a; = 6, e« + a; + 2/ = c, x + y-\-z = d. 34. x + ay + a^z + a'w + a^ = 0, x + by + b'z 4- 6^w; + 6* = 0, £c + cy + c^« + c^z« + c* = 0, x+ dy + d^z + cZ^w; + cZ* = 0. Simultaneous Equations of the Second Degree. 147. We now proceed to consider simultaneous equa- tions, one at least of which is of the second or of higher degree. We first take the case of two equations containing two unknown (luantities, one of the equations being of the first ,:i docrree and the other of the second. 158 SIMULTANEOUS EQUATIONS OF THE SECOND DEGREE. For example, to solve the equation b : Bx + 2y = 7, From the first equation we have X- g . Substitute this value of x in the second equation ; we then have whence y"^ + 14?/ + 13 = 0, that is {y + lS){y + l) = 0; .'. y= -1, 01 y= -13. Uy=-1, xJ-^=3; and if ?/=-13, a; = 11. Thus x = S, y=-l; or a; = ll, y=-13. From the above example it will be seen that to solve two equations of which one is of the first degree, and the other of the second degree, we proceed as follows : — From the equation of the first degree find the value of one of the unknown quantities in terms of the other un- known quantity and the known quantities, and substitute this value in the equation of the second degree ; one of the unknown quantities is thus eliminated, and a quadratic equation is obtained the roots of which are the values of the unknown quantity which is retained. The most general forms of two equations such as we are now considering: are *& Ix + m?/ + 71 = 0, aa^ + hxy + cy'^ -\- dx + ey +/= 0. From the first equation we have my + n x= J-— SIMULTANEOUS EQUATIONS OF THE SECOND DEGREE. 150 Hence on substitution in the second equation we have to determine y from the quadratic equation a {my + iif — Ihy (my + n) + cPy^ - dl (my -\-n)+ ery +/Z' = 0. Having found the two values of y, the corresponding values of a) are found by substitution in the first equation. 148. It should be remarked that we cannot solve any two equations which are both of the second degree ; for the elimination of one of the unknown quantities will in general lead to an equation of the fourth degree, from which the remaining unknown quantity would have to be found ; and we cannot solve an equation of higher degree than the second, except in very special cases. For example, to solve the equations ax^ + hx+ c = y, a)^-\-y^=d. Substitute ax^ + bx-\-c for 3/ in the second equation, and we have x"^ + {ax^ + bx + cf = d, which is an equation of the fourth degree which cannot be solved by any methods given in the previous chapter. 149. There is one important class of equations with two unknown quantities which can always be solved, namely, equations in which all the terms which contain the unknown quantities are of the second degree. The most general forms of two such equations are ax^ + hxy + cy^ = d and a'x^ + h'xy + cy^ = d'. Multiply the first equation by d', and the second by d and subtract ; we then have {ad' - ad) x"" + {hd' - b'd) xy + {cd' - cd) if = 0. 160 SIMULTANEOUS EQUATIONS OF THE SECOND DEGREE. The factors of the above equation can be found either by inspection, or as in Art. 81 ; we therefore have two equations of the form Ix + my = either of which com- bined with the first of the given equations will give, as in Art. 147, two pairs of values of x and y. Ex. 5. To solve the equations : y^-xy = 15 (i), x^ + xy = 14: (ii). We have 14 {y^ - xy) = 15 {x^ + xy) ; .-. 15a;2 + 29a:2/-14i/2z=0, that is (5a; - 2y) (3a; ■¥Ty) = 0. Hence 5x-2y = 0, or else 3a; + 7y = 0. If 5x-2y = 0, we have from (i) whence ?/ = ± 5. Hence also a; =±2. If 3x + ly = 0, we have from (i) whence 2/ = ± — - , and then x= ^ -j^ . 7 3 Thus a;=±2, 2/=±5; or x=d=-~, 2/==f^. 150. The following examples will shew how to deal with some other cases of simultaneous equations with two unknown quantities ; but no general rules can be given. Ex.1. To solve x-y=2, ^~-- xy = 15. Square the members of the first equation, and add four times the second ; then (a; + 2/)2 = 64. SIMULTANEOUS EQUATIONS OF THE SECOND DEGREE. IGl Hence ir + j/=±8, which with x-y = 2, gives x = 5 or -3, and ?/ = 3 or -5. Thus x=5, y = 'd\ orx=-S,y=-5. Ex.2. To solve X' + xy + y''- = a^ (i), «* + a;y- + 2^ = 6^ (ii). Divide the members of the second equation by the corresponding members of the first ; then x^-xy + y^ = ^ (iii). From (i) and (iii) by subtraction we have 2xy = a^- -J (iv). From (i) and (iv) From (iii) and (iv) we have x^-2xy + y^= -,^\ •• ^-y==^\/ 2ar- ^^'^- Finally, from (v) and (vi) we have and y=2i=^\/~^^^v ~2^^r Ex.3. To solve x^-2tf = iy, Bx^ + xy-2y-^ = lC)y. Multiply the first equation by 4, and subtract the second; then x^-xy- 6y^ = 0, that is {x + 2>f)(x-Sy) = 0i .'. x-\-2y^0, or else x-Sy = 0. S. A. 11 1G2 SIMULTANEOUS EQUATIONS OF THE SECOND DEGREE. If x + 2i/ = 0, the first equation gives 4y2_2y2 = 4y. .-. y = or y = 2, whence x = or a; = - 4. If x-3y = 0, the first equation gives %2-2i/2 = 4y; 4 .-. y = or y=-r whence 12 a; = or a; = — . Thus x = 0, 2/ = 0; a;=-4, 2/ = 2; or 12 4 Ex.4. To solve x'^ + y^ = {x + y + l)\ x^ + y''={x-y + 2)\ By subtraction we have {x + y + lf-{x-y + 2f = 0, that is (2a; + 3)(2^-l) = 0. Hence 2a; + 3 = 0, or 2y-l=0. If 2a; + 3 = 0, we have whence y= -2. If 2?/ - 1 = 0, we have whence .'c= -" 3* 3 Thus ^=~2' y-"-^' or ^=-g, 2/ = 2- Ex.5. To solve x + y = 2bf x* + y^ = 2a^. Put ae = ?;+r; then, from the first equation, y = h~Z, SIMULTANEOUS EQUATIONS OF THE SECOND DEGREE. 103 Honce (b + z)* + {b -zy = 2a\ wheuce after reduction .-. z'^=-Sb'±JSh* + a*; .-. z = ±^{ - 31'^ ± JSbU^'^*} . Thus x = h±J{-Sb^:i^j8b^ + a-'}, and y = b^^{-Sb^J^j8b* + a-^]. EXAMPLES XIII. Solve tlie following equations : — 1. x + y ^x^-y^ = 23. 2. x^-4y' + x+3y = 2x-y=l. 3. x' + xy=l2, 4. af + 2y^=22, xy-2y' = l. ^y'-xy- x' = 1 7. 5. x-y = 6y G. X + y = a + b, 115^ _a_ b y X 84:' X + b y + a ' 7. a(x + y) = b (x-y) = xy. 8. 1 1 1 a ' — 2 7 X xy a 1 1 1 3/* yx~ b^' 9. —^ +—2=10, x' y' xy 10. x + y = 2a, 11. a;Va;y + 2/' = 4251. 12. a;' -\ xy + if = 133, 13. a; + y = 72, x + Vi^'2/ + y = ^9. 11 2 164 EXAMPLES. 14. -+-=2, 15. x-\-y=l, X y ' ^ ' X y 16. x^ -^y^ + 3xy - 4:(x + y) + 3 = 0, xy + 2{x + y)-5 = 0. 17. x'' + xy + x=14:, 18. x^ + y^ = 9, y^ + xy + y = 28. a^^xy + y" = 3. 19. x(y — b)=y(x-a)=i2ab. 1 . 20. a; + - = l, 2/ 1 , 2/ + ^ = 4. 22. ^ + 2^=12, 2/ aJ 1 1 1 X y 3 ^ 2/ 26. a; + 2/ = 6, 27. a? + 2/ =80:2/, (a;' + y') (x' + y') = U4:0. x' + y'= iOx'y". 28. x'-xy = Sx+3, 29. -^—^- = 3, ^ * \~xy * xy -y^ = Sy-Q. ^ — 2/ 1 1 + xy 3 * 21. ax + by= 2ab, 2/ aJ 23. — hxy = a f ^ + xy = b\ x 25. 2/' ,. «^ + 2/ + '^ =14» SIMULTANEOUS EQUATIONS OF THE SECOND DEGllEE. 1G5 30. x-y = a{x'-y% 31. ?-h| = ^+^, x + y = b{x'- if). X* if _ 6* a 9 2 12 2 12* a a nn X a y h X y 32. - + - =^^ +- =- +'^-, ax b y y X 151. Equations with more than two unknown quantities. No general rules can be given for the solu- tion of simultaneous equations of the second degree with more than two unknown quantities: all that can be done is to solve some typical examples. Ex. 1. Solve the equations : {x + y){x + z) = a'^ (i), {y + z){y + x) = h'^ (ii), (2 + a;)(z + i/) = c2 (iii). Multiply (ii) and (iii) and divide by (i) ; 7 2 2 then (2/ + ^)^=-;t5 Similarly we have he ... .-. 2/ + 2= ± — (iv). z^x=^- (v), - ah , .. and x + y= =b— (vi). Also from the original equations it is clear that the signs must all be positive or all be negative. Add (v) and (vi) and subtract (iv) from the sum ; then /ca ah hc\ \b c a J c^a^ + a'^b^-h\'^ 2a 6c ^^^^'^^ y=^ — 2^6c — ' &V + c2a2-a-'ta and 2=± fi-T • 2aoc 166 SIMULTANEOUS EQUATIONS OF THE SECOND DEGREE. Ex. 2. Solve the equations : xiy+z)=a (i), y{z + x)=b (ii), z{x + y)=c (iii). We have y{z + x)+z{x+y)-x{y + z) = b + c-a, that is 2yz = 6 + c - a. Similarly 2zx =c + a-bt and 2xy—a + b-c, Hence (2^y) (2^3^) ^ {a + b-c){c + a-b) . 2yz b + c-a (b + c-a) Hence ^^^ Al±^zRi^±^, \ 2{b + c-a) -, . ., 1 . /{a + b-c){b + c-a) and similarly 2/ = i ^ 2{c + a-b) ' /(b + c-a) (c + a-b) and z= :i= . / iTT — . , > . \/ 2{a + b-c) Ex. 3. Solve the equations : x'^+2yz=a (i), y^ + 2zx = a (ii), z^+2xy = b (iii). By addition {x + y + z)^ = 2a + b; .*. x + y + z= ±j2a + b (iv). From (i) and (ii) by subtraction {x-y){x + y-2z) = 0. Hence x=y (v), or else x + y-2z — (vi). I. If x=y, we have from (ii) and (iii) by subtraction z^ + x^-2xz — b-a; .: z~x=ziz fjb-a (vii). Hence, from (iv), (v) and (vii), x = y = -{^ sj2a + b^ sjb-a}^ 3 1 3 z = -{^j2a+b^2jb-a]. SIMULTANEOUS EQUATIONS OF THE SECOND DEGREE. 767 II. Wbcu x + y-2z — 0, we have from (iv) and « + ?/= ±H \/2a + &- Also, from (ii), y^ + x{x + i/)=ay which with the previous equation gives /a-b 1 /- r /a-b 1 / and 2/ = =F w — g— =*= 3 \/2a + b. Ex. 4. Solve the equations : bh + c^i/ = c^x + o?z = o?y + b'^x = xyz. We have h^z + chf = xyz (i), c^x + aH = xyz (ii) , and a^y + b^x = xyz (iii). Multiply (i) by - a^, (ii) by b^, and (iii) by c-, and add ; then 2b\^-x ={-a' + b^ + c-) xyz. Hence x=0, 2Z>2c2 or else yz = — ., , , .. - — s • If a; = 0, y and 2 must also be zero. Hence x=y = z = 0; 2bh^ or else ^^ = £.27^23^2 ' 2c2a2 and similarly zx= ^ 2 _ / j » 2a262 and ^2/ = a2 + 62_,2- The solution then proceeds as in Ex. 2. Ex. 5. Solve the equations : X2 -y2: = a, J/- - zx = b, 22 -xy = c. 1G8 SIMULTANEOUS EQUATIONS OF THE SECOND DEGREE. We have (x^ - yz)^ - (?/^ - 2-t) iz^ - xy) = 0^- be, that is X (.r- + y-^ + z^- ?>xyz) = a^- be. Hence, from the last equation and the two similar ones, ^ - y = ^ a? -be W" - ca c^ — ah ' Hence each fraction is equal to / X^-XjZ _ I 1 V (a2-6c)2-(&2_ca)(c-^-a&) ~ "^ V (a^ + fcS + c3 - 3a&c) ' Ex. G. Solve the equations : x-\-y-\-z=a-\'b^e (i), a;- + 2/2 + 2;2 = a2 + 62 + c2 (ii), X y z ^ ..... - + r + -=3 ni). a b c It is obvious that x = o, y — h, z=c will satisfy the equations: put then x = a + X, 3^ = & + /*, z = e-\-v, and we have after reduction X + At + i/ = (iv), - + j + -=0 (v), a b c 2(aX+&;w + cj')+X2 + /i2 + ^2 = (vi). From (iv) and (v) X _ )K _ I» a{b - c)~ b{e - a)~ c {a -h)* whence from (vi) X=/i=i' = 0, X _ 2(&-c)(c-a)(a-&) **' a{h-c) ~ a?{b- cY + 62 (c - a)2 + c'(a- bf ' Hence x=a, y = b, z=c\ or else x~a __ y-b _ z-c _ 2{b-c){c-o) (a-b) a{b^) ~ b{c - a) ~ c]^^ ~ a'^{b-cy + b'^{c-af + c^{a-bf ' Ex. 7. Solve the equations : x + y+z= 6, yz + zx + xy = lly 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, 2 are the three roots of the cubic equation X3-6X2 + llX-6 = SIMULTANEOUS EQUATIONS OF THE SECOND DEGREE. 1G9 Tlie roots of the cubic are 1, 2, 3. Hence x = l, y = 2, z = 3; or x=l, y = 3, z = 2; or x = 2, y = 3, z = l\ &c. Ex. 8. Solve the equations : X + y + z = a (i), 1111 ,.., - + - + - = - (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^\ + ac^ = 0, that is \^\-a)-c^X-a)=0; .•. X = a, or \= ic. Thus x=a, y=c, z = — c\ &c. Ex. 9. Solve the equations : a;2(i/-2) = a2(&-c), y"^ {z - x) = h^{c - a)i z^ {x - y) = c^{a - b). By addition a;2 {y-z)+ y"^ {z-x) + z"^ {x-y)= a" (6 - c) + 6^ {c-a)-^ c^ {a-b), that is {y -z){z- x) {x-y) = {b- c) (c - a) {a - b). By multiplication x-y V {y -z){z-x){x-y) = a'b\^- {b - c) {c-a){a-b)] .'. xY'^'^ = a^bh\ Hence xyz = abc (i), or xyz= -abc (ii). Again a^{b-c)y + b- (c - a)x = x-y {y-z) + xxf {z - x) = xyz{y-x) (iii). Hence, if xyz = abc, we have from (iii) {b^{c - a) + abc] x + {a'^{b - c) - abc} y = 0, that is bx {be + ca- ab) - ay {be + ea- ab) — ; ,: - = ; , and therefore each = - . a b 170 EXAMPLES. Thus, when xyz=abc, we have - = ^ = - , ^ a b c Hence each is equal to . / -^ = ,^1. Thus - = ^ = -=1, or — = /- = — = !, a c ao} b(jj cu) ^ _ y _ ^ _■, If a:?/2 = - abc, we have from (iii) - {bc-ca — ab)=j (ca-ab-bc). Hence also each =-{ab-bc- ca) ~i^{ - {be - ca - ah) {ca-ab-bc) {ab-bc-ca)}. EXAMPLES XIV. Solve the following equations : 1. yz = a^, 2. x{x + 9/ + z) = a^, zx = b^j y (x + y + z) = b^, xy^c'. z{x + y + z) = c^ 3. -yz + zx + xy = a, 4. yz = a(y + z), yz-zx + xy = b, zx = b{z + x), yz + zx — xy = c. xy = c{x + y). 5. yz = by + cz, 6. x^ + 2yz ==12, zx = cz + ax, y^ + 2zx = 12, xy = ax + by. z^ + 2xy =12. 7. {y + z){x + y + z) = a, 8. {y + b) (z + c) = a% {z + x) (x + y + z) = b, {z + c){x + a) = b% (x + y)(x + y -\-z) = c. {x + a) (y + b)^ c'. SIMULTANEOUS EQUATIONS OF THE SECOND DEGREE. 17 1 9. x'-{y-zy = 0^, 10. x{y + z-x) = ay y'-{z- xf = 6', y(z + x-y) = by z* -{x- yY =c*. z(x + y-z) = c. 11. y + z z + X x + y = 2xyz. a b c 12. y + z z + x x + y x' + 2/' + z' a b G a + 6^ + c'' 13. yz = a + y+z, zx = b + z + Xy xy = c + x + y. 14. yz = a(y + z) + ay zx'= a (z + x) + fty xy = a{x + y) + y. 15. yz -P = cy + bz, zx — g^ = az + CXf xy-h^ = bx + ay. 16. -I 3 x + y =2' _i 7 2/ + ^ =3, « + »"' = 4. 17. x + y + z= Q, 18. x + y + z= I5y x' + y' + z'=Uy x^ + y' + z'=^95, xyz= 6. xyz = 105. 19. x + y + z= 9, 20. x + y + z= 10, x^ + y^ + z'= 41, yz + zx + xy= 33, a« + 2/'4-»' = 189. {y+z){z+x){x+y) = '2d4:. 21. 22. yz zx xy ay + bx x' + 2/' + ^' bz + cy ex + az a" + 6^ + c'- a y z . -+^+- = 1, X c 23, » aic = - + - , z y X b z ^ aye , 2; a; •^ a; « X y e ^ - + f- + - = 1. a z 172 EXAMPLES. 24. 7/ + z^ -x(y + z)- a*, z^ + x^ -y{z + x) = h^y x^ + y^ --z(x + y) = c^. 25. x^ + yz-a^ = y^ + zx- h* =2^+ xy - c^ = ^ (x- + y'^ + z"). 26. xix + y + z)- (2/^ + z^ + yz) = a, y {x + y + z) - {z^ + x^ + zx) = h, z{x + y + z)- {x^ + y^ + xy) = c. 27. x + y + z = a + b + c, x^ + y^ + z^ = 0^ + b^ + c^, (b -c) x + {c- a)y + (a-b) z = 0. 28. {x + y) (x + z) = ax, 29. x^ — yz = ax, (y + z){y + x) = by, y^-zx = by, (z + x) (z +y) = cz. z^ -xy = cz. 30. x^ ■¥ a {1x + y ^ z) = y'^ ■¥ b {'iy + z + x) = z^ + G {2z + X -\- y) = {x + y + zy. 31. y^ + yz + z^ = a^, z^ + zx + x'^ = b^f x^ + xy + y^ = c^. 32. a^x + b'y + c'z = 0, {b-cr {c-af {a-h)l_ 111111 — +— + — =-+- + -. yz zx xy a G CHAPTER XL Pkoulems. 152. We shall in the present chapter consirler a class of questions called problems. In a problem the magni- tudes of certain quantities, some of which are known and others unknowm, 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 knoAvn 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 maej-nitudes of the 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 £ ? Let X be the number of shillings that A gives to B. Then A will have 100 - x shillings, and B will have 10 + x shillings. But, by the question, A now has four times as much as B. Hence we have the equation 100-a; = 4(10 + a;); /. a; = 12. Thus A must give 12 shillings to B. It should be remembered that x must always stand for a numher. 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 ic = 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 xV^h of the whole in a day. Hence we have 1 2_J^ x'^y~ 12' We have also, since 3 men and 1 boy do ^th of the whole in a day, 3 11 X y Q Whence a; = 20. Thus one man would do the whole work in 20 days. PROBLEMS. 175 Ex. 3. In a certain family eleven times the nnmbor 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 Ux = 2x^ + 12, or 2a;2- 11a; + 12 = 0, that is (2a:-3){a;-4) = 0. Hence x = i, or a; = f . The value re = f 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 x + 2 will be the digit in the unit's place. The number is therefore equal to 10a; + (a; + 2). Hence, by the question, 10x + (x + 2) = 3a;(x + 2); .-. Bx'^-5x-2 = 0, or (a;-2)(3x + l)=0. Hence x = 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 requu-ed 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 IOj; + y. Hence, by the question, 10x + y = S{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 reqr.ued number is 27. Ex. 7. The sum of a certain number and its square root is 90. "What is the number ? Let X be the number ; then we have the equation a; + v/a; = 90 ; .-. {x-iece 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 hj A, B 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 w^alks a certain distance : if he had gone half a mile an hour faster, he would have walked it in 4 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, oue-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 pound.s 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 D. B gets half as much as A, the excess of C's share over Z)'s share is equal to one-third of ^'s share, and if ^'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 journe}^ 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 ? 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 ? 22. Divide £1015 among A^ B, C so that B may receive £5 less than A, and C as many times ^'s share as there are shillings in -4's share. 12—2 180 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 m.eet after five hours ; if A had travelled one mile per hour faster and B had started one hour sooner, or if B 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 J.'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 ? 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 5 gallons of wine and 9 gallons of water 1 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 time. 33. Aj B and C start from Cambridge, at 3, 4 and 5 o'clock respectively to walk, drive and ride respectively to London. C overtakes ^ at 7 o'clock, and C overtakes A 4| 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 X6 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^ hours less. What is the distance between the ports ? 37. Two persons A, B walk from P to ^ and back. A starts 1 hour after B, overtakes him 2 miles from Q, meets him 32 minutes afterwards, and arrives at P when ^ is 4 miles off. Find the distance from P to Q. CHAPTER XIL Miscellaneous Theorems and Examples. 153. 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 tlie equations ax + b = 0, a'x + 6' = 0. From the first equation we have x= — , and from the second a equation we have x=. — r, a h 7»' Hence we must have -.= —,, or ha' -Va—Qi which is the a a required result. Ex. 2. Eliminate x and y from the equations ax + hy + c = % afx + h'y + c' — O, a"x + b"y + c' = 0. ELIMINATION. 183 From the first two equations we have [Art. 143] X _ y _ 1 he' - b'c ca' - c'a ah' - a'^ * These values of x and y must satisfy the third equation; hence ,,hc' -h'c ,,,ca'-c'a ,, . a"-r-, 17 + ^ -T' Tj+C' = Q, ab - ah ah - ab or a" {be' - h'c) + h"{ca' - c'a) + c"{ab' - a'b)=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 ax' + bx + c = 0, a'x^ + b'x + c' = 0. As in Art. 143, we have x"^ X 1 he' - h'e ca' - c'a ah' - a'b ' Hence {he' - h'e) [a h' - a'b) = {ca' - c'a)\ the required result. It should he remarked that the above condition is also the condition that the two expressions ax- + bx + c and a'x^ + b'x + c' may have a common factor of the form x-a; for if the expressions have a common factor of the form x-a they must both vanish for the same value of x. Ex. 4. Eliminate x from the equations aa; ' + hx + c = 0, a'x^ + b'x + c' = 0. As in Ex. 3, we have a^ X 1 he' - b'c ca' - c'a ah' - a'b ' be' -b'c _ /c a'-c'a y ^ .-. {he' - h'c) {ab' - a'b)- = {ca' - c' v^ft > y, and 'p>''l]h> q\ and the difference between x and y, and also the difference between 'p and q, can be made less than any assigned quantity however small. Hence x xp> ^a x "'Jh >y x q, and px x>'"^b x '^a > q x y. But, since x, y, p, q are integral or fractional numbers, we know that x x p=p x x, and y x q = q xy; also the difference between j^^ and qy can be made less than any assigned quantity however small. It therefore follows that ^a x'^b and X^6 x ^a, which are both always intermediate to xj) 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 204 PROPERTIES OF ROOTS. an important difference 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 mih. 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 {\/a X ^b.,.y = {\Jaf X {\/hf ... =a6.... Also {Jab...y = ah..., by definition. .*. {^/a X ^/b...y= {Jab...y. Hence fs/a x ^b... must be equal to one of the square roots of ab... . We can write this i^a\/b... = 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 one or other of the square roots oi ab ... Similarly we have, with a corresponding limitation, 'J a _ "/a- :ja :^b... = Jab..., and ^~\/^- Also V «"* = V ot'^^j for their npth 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 91, 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 v^a* is a^, one value of l/a^ b^ c^ is a^ b^ c, and one value of Jja''"- b""^ c''y 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 a 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 fractional 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 he 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 a' X a = a ^ = a^ = a. Thus a^ must be such that its square is a, that is a'^ must be s/a. Again, to find the meaning of a"\ By the index law a~^ X a^ = a"^"^^ = d^ ; therefore a~^ = ^ = a a Thus a ^ must bo - . a 206 FRACTIONAL AND NEGATIVE INDICES. 167. We now proceed to consider the most general cases. 1 I. To find the meaning of (jb\ where n is any positive integer. By the index law, 111 a'* X a" X a" X to n factors -+-+-+ tow terms - = a^ *^ ** =a'^ = a^ = (2. 1 Hence a" must be such that its ?ith power is a, that is 1 a" = ^a. II. To find the meaning of a!\ where 77i and n are any positive integers. By the index law, mm m , ni ^ ^ . m — — , r- , TT +;r '^ to n terms - x n „, a^ X a^ X ton factors = a'^ *» =a^ = a"'. m Hencea" = 7a^\ We have also 11 1,1,,. m - - - -H — h to OT terms fV^ xct^ X to m factors = a^ ** = a^\ Hence a^= {a'T- in Thus we may consider that cC' is an ??th root of the mth power of a, or that it is the in\h power of an Tith root of a ; which we express by rn With the above meaning of if it follows from Art. 165 that a" = ct'^. FRACTIONAL AND NEGATIVE INDICES. 207 Note. It should be remarked that it is not strictly true that ^(O^^C/^)'" except with a limitation corre- sponding to that of Art. 165, or unless by the nth root of a quantity is meant only the aritlimetical root. For example, v^(a'') has two values, namely ±d\ whereas ( ^a)* has only the value + «^ III. To find the meaning of a°. By the index law a*'xa"' = a°"^ = a'"; .'. 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" = I, by III. Hence a~^' = —, , and a"' = -^, . a a 1G8. 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 an} given positive or negative value. We have now to shew that, ivith the meanings thus obtained, a'" X a" = a"*-'", (a"')" = a"'", and (ah)" = a"6", are true for all values of in and n. When these have been proved, the final result of Art. IGO is easily seen to be true in all cases. 208 INDEX LAWS. I. To prove that a"' x a" = a'""*"", for all values of m and n. We already know that this is true when m and n are positive integers. Let m and n be any positive fractions - and - respectively. Then p r orxa'' = a''x a' = 'Ja^ x 'Ja\ by definition = ':J^' X '^ct^= ''sl^F^' [Art. 165] = a 5« ^ by definition Thus the proposition is true for all positive values of 771 and n. To shew that it is true also for negative values, it is necessary and sufficient to prove that a""* X «-" = a""*"", and a*" x a" = a"^ where m and n are positive. Now a-" X a-» = i, X ;^ = -L. = a"'""". Cti ct ct And, if m — w be positive, a'"-** X a** = a'", and a"" x a"" x a** = a*" ; therefore a'"~" = a'" x o~". Hence, if m — n be negative, -^^ x — a ■•' a- a"'"' ' .m-n that is, a'" x a"" = a" Hence a"* x d^ = a'"+", for all values of m and n. Cor. Since a'""" x a** = a"' for all values of m and n, it follows that a"' -^ a" = a"'"". INDEX LAWS. 209 II. To prove that (a'")" = «"'", f'or all values of m and n. First, let n be a positive integer, m having any value whatever. Then (a'")" = a" x a"* x a"' x to n factors, __ Qni. + m + 7n+ to n terms |-)y J^ = a*"". Next, let n be a positive fraction - , where j^ ^^(^ + 3 - r + 2 sj]^i, and ip + q-r + 2jpq) {p + q-r-2 J^) ={p + q- r)' - 4pq. Thus the required rationalizing factor is Up + \/(7 - s/r) {p + q-r-2 Jpq), which is the same as before. Again, from the identity {a + h){a'-ab + h^) = a^ + P, the rationalizing factor of a + b^ is seen to be a^-ab^ +1'^, 170. To find the 7xitionalizing factor of any binomial. P r Let the expression to be rationalized be ax''± hij\ P r Put X=ax\ and Y=hy\ and let n be the L.CM. of q and 5. Then it is easily seen that Jl" and F" are both rational. Hence, from the identities (Z+F) {Z"-^ -Z"-'^F+...+ (- 1)«-^F"-^} =Z'* + (- \y-'Y" and (X- F)(X"-^ + X"-^F+ + F"-0 = X"- Y\ the rationalizing factors of X -\- Y and X — Y are seen to be respectively X"'^ - Z"-'^ F + + (- 1)"-^ Y"-\ and X"-^ + Z"-^F+ + F"-^ 14—2 212 EXAMPLES. Ex. To find a factor which will rationalize 2 5 Here X=x^ , Y=ay^, n=6. The factor required is therefore 10 85 n„5 46 210 2'^ x^ + ax^y^ + a^-xhj ^ + a^x'^ y'^ + a^x'^y "S' + ahf «*" . EXAMPLES XVII. 1. Simplify a^b^ x a~^h~^. 2. Simplify a^ x a i x (a^) ^ x j — • 3. Simplify (a6-V)^ x {a^^c'^f. &+C 1 c+g 1 « + & Jl_ 4. Simplify (a;C-«)'*-?' x (a;«-^)^-c x (a;^"^)^^^ • 0. Multiply x^ + x^'if + y^ by x^-y^. 6. Multi})ly £c- + 1 + a;"^ by x^ -\+ x~^. 7. Multiply 2 2 2 iiil 111 1 ii a;^ + 2/^ + ;^^ - 2/^ ^^^ - 5:=^ ^^ — x'^y^ by cc'-' + ?/" + ;s^^\ 8. Divide £C« — 2 + aj~^ by x^ - a;~ "\ 9. Divide a- — x by a^ir — jc^. 10. Divide or - xy^ + x'^y — y^ by x'^ — y'^. 11. Shew that x^ - ^x^ + 2a;^ + 4aj - 4a;^ + a;^ = {x^ - 2x^ + cc-^. 12. Multiply 4a;=-5ic-4:-7£c-' + 6a;-' by 3a;-4 + 2a;-i and divide the product by 3a;— 10 + 10a;~* — ix~'\ EXAMPLES. 21:1 13. Divide x-x~^ -2 {x^ — x~'^) + 2(x'^' — x~'^) by X'^-x~*. ax~^ + a~^x+ 2 14. Simplify — j ^ (I'x y + a "a;" — 1 7. 14 1 2 15. Divide -T* + S- ^y — + "^ • y~^' xs ?/5 a;^ 16. Shew that a; a^ 1 1 ^ o + -^ — - =x^ + 2. x^ -I x^ +1 a;-' - 1 a;^ + 1 17. Shew that (2a; + 2/"') {2y + x~^) = (2a;%^ + a;~%"^)'. 18. Shew that a^ + h'-g-'-b-' {a-a-'){b-h -') _ ^ a'b^-a~-b~'^ ab + a~^b~^ 19. Shew that, if a:^ + 2/ + ^"^ = 0, then {x + y + zf = 21xyz. 20. Find factors which will rationalize the following; expressions : (i) (j^ + h^-^ (ii) a^x^-^y^^ (iii) a + 6a;' + cd^^ and (iv) a;* 4- 2/'* + ^ • 21. Shew that, if (1 - x^)^ (2/ - 2;) + (1 - 2/^)^3 (;.- a;) + (1 - z')^ ix-y)^ 0, and .r, 2/, ^ iJ-J'C all unequal, then (l-./)(l-v/)(l-;:^) = (l-a:/pr. CHAPTER XIV. Surds. Imaginary and Complex Quantities. 171. Definitions. A surd is a root of aa arithmetical number which can only be found approximately. An algebraical expression such as ^ja is also often called a surd, although a may have such a value that ^Ja 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 a/2 and \/6 are called surds of the second order, or quadratic surds ; also ^4 is a surd of the third order, or a cubic surd ; and ^a is a surd of the 71 th order. Two surds are said to be similar when they can be reduced so as to have the same irrational factors. Thus V8 and VI 8 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 ^Ja has two values but V^ is only supposed to denote the arithmetical root, unless it is written + ^2. SURDS. 215 17- Any rational quantity can be written in the form of a surd. For example, and a = ^d' = ^a^ = ^ a". Also, since *^a x ^b = \/ah [Art. 165], we have 2^2 = ^4x^2 - V(4 x 2) = a/S, 5^/3 = ^5^ X ^3 = ^(5^ X 3) = ^375, and Lr:,1Ib = lya" x :Jah = ':J{cC' x ah) = :Jar%. Conversely, we have ^18 = >v/(9 x 2) = V^ x -v/2 = 3\/2, and ^135 + ^4:0 = ^/(3^ X 5) + ^(2^ x 5) = 3^5 + 2^5 = 5^/5. 173. Any two surds can be reduced to surds of the same order. For if the surds be ^a and '^/b, we have ^a = ":j/a"*, and 76 = ":y6" [Art. 165]. Ex. Whicli is the gi-eater, 4/14 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 = 4/68 = 4/216. Hence, as 4/216 is greater than ,^^196, 4/6 must be greater than /^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 ya xyb = 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 ::jax ybx 7c...= 7a6c... Ex. 1. Multiply Jo by 4/2. ^/5 X ^2 = J6^ X 4/2-* = 4/(53 X 22) ^ ^500. Ex. 2. Multiply 3J5 by 24/2. 3,^/5 X 24/2 = 3 X 2 X V5 X 4/2 = X 4/53 X 4/2== O4/5OO, 216 MULTIPLICATION OF SURDS. Ex. 3. Multiply ^2 by ^2. ^2x^2=4/23x^22=^2^^^22=4/32. Or thus: ^^2 x 4/2 = 2^ x 23 = 22 + 3 = 2^=4/25. Ex. 4. Multiply V2 + x/3 by v/3 + J5. (x/3+V2){x/3 + ^5)=^3x^3 + V2xV3 + V3x;^5 + V2x5 = 3 + ^6 + V15 + v/10. Ex. 5. Divide 4/4 by 4/8. 4/4^4/8 = 4/42^4/83=^1=^1^. 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 "x/5xV5 = >■ 3 3i U'o + 1) _ s/o- ■1"(n/5- ■l)(v/5 + l) 1 1 1 + ^/3 + V5 W15 - {1+V3) (1 + v/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\/b. Conversely, if the product of the quadratic surds f^a and \/b is rational and equal to x, we have x = ^Ja x ^Jh ; therefore x\/h = \/ax \/b x \/b = b\Ja, which shews that the surds are similar. So also, if \/a -h ^b is rational, the surds must be similar. SURDS. 217 177. The following theorem is important. Theorem. If a-\- \/b = x + \Jy, where a and x are rational, and ^/b and ^/y are irrational ; then will a = x, and h = y. For we have a — x -\- tjh = \Jy. Square both side.s ; then, after transformation, we have 2(a — x)^b = y — h —(a — xf. Hence, unless the coefficient of \Jb is zero, we must have an irrational quantity equal to a rational one, which is impossible. The coefficient of i^b in the last equation must there- fore be zero, so that a = x. And when a = x, the given relation shews that ^/b= \ly, and therefore b = y. As a particular case of the above, si a 4= 6 + sjc, unless 6 = and a = c. Hence sja + \Jc can only be rational when it is zero. Ex. 1. Shew that ^/a + y^y^> + ;^c^:0, unless the surds are all similar. For we should have fja + tjb = -ijc ; and therefore a + b + 2Ja^b = c. Hence ^Jajh is rational, which shews [Ait. 170], that Ja and Jh are similar surds. 178. The expressions a + \/b and a — \/b 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-\- \/b and c + ^/d are both rational, then a = c and V^ + ^/d = 0, so that the two expressions are conjugate. For a + c -\- \/b -\- sjd can only be rational when sjb + s/d is zero. [Art. 177.] And, when \Jd = — ^Jb, the product {a + isjb) (c + /^d) = ac + {c — a) *s/b — b, which cannot be rational unless c = a. 171). In the expression ax"" -h bx""-"^ ^ cx""^ + + k, where a, b, c, k are all rational, let a -f ^(B be substi- 218 SURDS. tilted for x; and let P be the sum of all the rational terms in the result and Q \//3 the sum of all the irrational terms. Then the given expression becomes P + Q\//3. Since P and Q are rational, they contain only squares and higher even powers of \/P, and hence P and Q will not be changed by changing the sign of \l^. Therefore when a — \//3 is substituted for x in the given expression the result will be P — Q V/3. If now the given expression vanish when a + ^JP is substituted for x, we have P + (3ViS = 0. Hence, as P and Q are rational and \//3 is irrational, we must have both P = and Q = ; and therefore Therefore if the given expression vanish when a + t\/^ is substituted for x it will also vanish when a — Vy5 is sub- stituted for X. Hence [Art. 88], if a? — a — \//3 be a factor of the given expression, x — a-{- aJ(3 will also be a factor. Thus, if a rational and integral expression he divisible hy 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 ^J{a + ^Jb), where i^b is a surd. Let VC^ + V^) = V^ + V^. Square both sides ; then a + ^Jb = x-\-y-\^ 2jxy. ^^ Now, since ^Jb is a surd, we can [Art. 177] equate the rational and irrational terms on the different sides of the last equation; hence x -h y = a, and 4fxy = 6, SURDS. 219 Hence x and y arc the roots of the equation X — ax + - = 0, and these roots are h {a + ^J{ct' - h)] and -J {a - V(«' - ^)]- Th us ^J((l + V^>) = ^ -^ ^ + y^ ^2 ^ • It is clear that, unless isj{ci^ — h) 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 not 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 V(6 + 2^5). Let;^(6 + 2/^yo) = y/a; + ;^y. Then, by squaring, we have 6 + 2^5 = x + y + 2Ajxy- Hence, equating the rational and irrational parts, x+y = Q and xy = 6. Whence obviously .-c = 1 and y = 5. Thus Ex.2. Find V(28- 5^12). Let Ji2S-5^12)=Jx-^y. Then, as before, ixy = 25xl2, or xy = 15 &iidx + y = 2S; whence a; = 25 and 7/ = 3. Thus ,^(28-5 >yi2) = 5-mJS. [If we had taken x = S and ?/ = 25 we should have had the negative root, namely ^S - 5.] Ex.3. Find V(18 + 12^3). In this case ^{a^ - b) is irrational and therefore the required root cannot be expressed in the form sJx-\-iJy where x and ?/ arc rational. The root can however be expressed in the form ^x + ^y\ for V(18 + 12V3) = v/W3 (12 + 0^3)} =4/3 X ^(12 + 0^3) = 4/3 x(3 + V3) = 4/243 + 4/27. Ex.4. Find V(10 + 2^6 + 2^10 + 2^15). Assume ^(10 + 2^6 + 2^10 + 2^15) = ^x + Jy + Jz ; then 10 + 2;^/G + 2 ^IIQ ■\-2^\^ = x -iry + z + 2 ^ xy + 2 J xz + 2 J yz. We have now to find, if pvssihlc, rational values of x, y, z such that xy = 6, 0:2 = 10, 7/2 = 15 and x + y + z = 10. The first three equations are satisfied by the values x = 2, 7/ = 3, 2=5, and these values satisfy x + y + z=10. Hence ^/(lO + 2^/r. + 2^/10 + 2 ^/15) = ^/2 + ^3 + %/•"'• 220 IMAGINARY AND COMPLEX QUANTITIES. Ex. 6. Prove that, if ^(a + ^Jb) = x + Jy; then will ^{a - ijh) = x- ^y. We have a + ijb = {x + JyY = x^ + Zxy + Jy {^x"^ ■\-y). Hence, equating the rational and irrational parts, we have a=x^ + ^xy, and s/b = ^y{3x' + y). Hence a-^Jb = x^' + Sxy-^y{3xz + y)', •■• sj{a-sjh)=x-^y. EXAMPLES XVIII. Simplify the following : 1 s/3-1 2^5 V3 + 1' ■"• v/5 + 73- 3. 1 1 V8 + v/3 ' ^8-^3* 4. (2-73)- + (2 + V3r- 5. 3 V2 4 V3 ^/3 + V6 VG + ^2 "^ ^ v/6 2 + ^3- fi (7-2 75) (5 + ^7) (31 + 13 ^5) u. (6 -2^7) (3 + ^5) (11 4 -4V7) • 7 1 8 1 q V2 + 73 + V5- 1 ■ V3 + ^5-v'2- t7. J\0 + JU-^-Jlb + J'2l • 10 1 JLK). ^6 + ^^21-^10-^35 • 11. 1 1 ;/2-l ' ^2 + 1* TO ^ 5 13 1 11 ^ XQ. 1 + ^72 + 4/4 • ^ ^2-;/6 + vi8' lAlAUlNAUY AN1> COMPLEX QUANTITIKS. 221 15. V(101-2SV13). 16. 7(28-5 712). 17. V{ll + 2(l + 7r,)(l-f77)}. 18. 7{6-4 73+v/(lG-8 73)}. 7(3 + 2 72) -72 19. \/{d7 - bG J?j). 20. 21. J2-J{3-'2J2) ' 22. 23 72 + 7(7-2 710)- 73 + 72 73 - 7: 72 + 7(2 + 73) 72-7(2+73)* 7(5 + 2 76) - 7(5 - 2 76) 7(5 + 2 76) + 7(5 - 2 76; • 21 7(6 + 272 + 273 + 276}. 25. 7{11 +672 + 473 + 276}. 26. 7{17 + 4 72 - 4 73 - 4 76 - 4 v^5 - 2 710 + 2 730}. 27. Shew that 1 1 2 7(12-7140) 7(8-760) 7(10+784) 28. Shew that 1 3 4 = 0. 7(11 - 2 730) 7(7 - 2 710) 7(8 + 4 73) = 0. Imaginary and Complex Quantities. 181. We have already seen that in order that the formula obtained in Art. 81 for the factors of a quadratic expression may be applicable to all cases, it is necessary to consider expressions of the form J — a, where a is 222 IMAGINARY AND COMPLEX QUANTITIES. positive, and to assume that such expressions obey all the fundamental laws of algebra. Since all squares, whether of positive or of negative quantities, are positive, it follows that J— a cannot represent any positive or negative quantity; it is on this account called an imaginary quantity. Also expressions of the form a + hJ—1 where a and h are real, are called complex quantities. 182. The question now arises whether the meanings of the symbols of algebra can be so extended as to include these imaginary quantities. It is clear that nothing would be gained, and that very much would be lost, by extending the meanings of the symbols, except it be possible to do this consistently with all the fundamental laws remaining true. Now we have not to determine all the possible systems of meanings which might be assigned to algebraical symbols, both to the symbols which have hitherto been regarded as symbols of quantity and to the symbols of operation, subject only to the restriction that the funda- mental laws should be satisfied in appearance whatever the symbols may mean: our problem is the much simpler and more definite one of finding a meaning for the imaginary expression J— a which is consistent with the truth of all the fundamental laws. 183. We already know that — 1 is an operation which performed upon any quantity changes it into a magnitude of a diametrically opposite kind. And, if we suppose that ^/— 1 obeys the law expressed by 1 x J — 1 x J— 1 = — 1, it follows that J—1 must be an operation which when repeated is equivalent to a reversal. Now any species of magnitude whatever can be re- presented by lengths set otf along a straight line; and, when a magnitude is so represented, we may consider the COMPLEX QUANTITIES. 223 openition J— I to be a revolution through a right angle, lur a repetition of the process will turn the line in the same direction through a second right angle, and the line will then be directly opposite to its original direction. Hence, when magnitudes are represented by lengths measured along a straight line, we see thatV— 1, regarded as a symbol of operation, has a perfectly definite meaning. The symbol J —1 is generally for shortness denoted by i, and the operation denoted by i is considered to be a revolution through a right angle counter-clockwise, —i denoting revolution through a right angle in the opposite direction. 184. It is clear that to take a units of length and then rotate through a right angle counter-clockwise gives the same result as to rotate the unit throuorh a ricjht anofle counter-clockwise and then multiply by a. Thus ai = ia. Again, to multiply ai by hi is to do to ai what is done to the unit to obtain hi, that is to say we must multiply by h and then rotate through a right angle; we thus obtain ah units rotated through two right angles, so that ai X hi= — ah = ahii. From the above we see that the symbol i is commuta- tive with other symbols in a product. Since {ai) x {ai) = aaii = a^ {—\) = — a^, it follows that J—a^ — ai\ it is therefore only necessary to use one imaginary expression, namely J —1. 185. With the above definition of V— 1 or i, namely that it represents the operation of turning through a right angle counter-clockwise, magnitudes being represented by lengths measured along a straight line, the truth of the fundamental laws of algebra for imaginary and complex expressions can be proved. Some simple cases have been considered in the previous Article: for a full discussion see De Morgan's Douhle Algehra; see also Clifford's Common Sense of the Exact Sciences, Chapter iv. §§ 12 and 13, and Hobson's Trigonometry, Chapter Xiii. 224 CONJUGATE COMPLEX EXPRESSIONS. 186. If a + hi = 0, where a and h are real, we have a = — bi. But a real quantity cannot be equal to an im- aginary one, unless they are both zero. Hence, if a 4- hi = 0, we have both a = and h = 0. Note. In future, when an expression is written in the form a + hi, it will always be understood that a and b are both real. 187. If a + hi = c-\- di, we have a~ c-\'(h — d)i = 0; and hence, from Art. 186, a — c = and h — d=0. Thus, two complex expressions cannot he equal to one another, unless the real and imaginai^y parts are separately equal. 188. The expressions a + hi and a — hi are said to be conjugate complex expressions. The sum of the two conjugate complex expressions a + hi and a — hi is a + a + (h — h)i=2a'j also their pro- duct is aa + ahi — ahi — 6V = a^ + h'\ Hence the sum and the product of two conjugate complex expressions are hoth 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 s6 + hi and c + di. The sum is a + 6* + c + c?^ = a + c + (6 + c?) ^, which cannot be real unless h -\- d = 0. Again, {a + hi) (c 4- di) = ac+hci+adi + hdi^ = ac — hd-\- (be + ad)% which cannot be real unless hc-h ad=^ 0. Now, if 6 + cZ = and also hc + ad = 0, we have h (c — a) = ; whence a = c or 6 = 0. If 6 = 0, c? is also zero, and both expres- sions are real ; and, if 6 4= 0, we have a = c, which with b = — d, shews that the expressions are conjugate. 189. Definition. The positive value of the square root of a'"^ + U^ is called the modulus of the complex MODULUS OF A COMPLEX EXPRESSION. 225 ([iiaiitity a-\-hi, and is written mod (a + 6i). Thus mod (a + hi) = + Jd' + b\ It is clear that two conjugate complex expressions have the same modulus ; also, since (a + bi) (a — hi) = d^ + h^ [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^ + h'^ will be zero if, and cannot be zero unless, a and b 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" + 6^ we put 6 = 0, we have mod a = -\- sA?, so that the modulus of a real quantity is its absolute value. 190. The product of a 4- bi and c + di is ac + bci + adi + bdi" = ac — bd + (bo + ad) i. Hence the modulus of the product of a + bi and c ^-d^ is sj[{ac - bdf + {be + adj] = V((a' + b') (c' + d')} = V(a'^ + b') X ^/(c^ + d'). 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 : s. A. 15 226 MODULUS OF A PRODUCT. , , .. , 7 .. a-\-hi c — di c-\-di c — di _ac+hd + (be — ad) i ~ J+d' • Hence mod l^l = ^{(^o^bdJHbc-adn _ V{ci' + fe'l _ mod (a + bi) ~~ Aj{c' + d^} ~ mod (c + di) ' 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, and, 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 exj^ression where a,b,c,...k are all real, let a + ^i be substituted for x^ and let P be the sum of all real terms in the result, and Qi the sum of all the imaginary terms. Then the given expression becomes P4^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 — jSi is substituted for x in the given expression the result will be P — Qi. If now the given expression vanishes when (x + /3i is substituted for x, 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 + /3i is substituted for x, it will also vanish when a — (3i is substituted for x. Therefore [Art. 88] if x — a- ^i is a factor of the given expression, x — a.-^ /Si will also be a factor. Thus, if any expression rational and integral in x, and with all its coefficients real, be divisible by either of tivo conjugate complex expressions it will also be divisible by the other. lb— 2 CHAPTER XY. Square and Cube Eoots. 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' ± 2ab + b' = (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 4!a^ - 12a'b' ~\- 9h\ SQUARE ROOT. 229 The square roots of the extreme terms are ± 2(X* and + 861 Hence, the middle term being negative, the re- quired square root is ± (2(X* — 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 thi^ee terms. For example, the expression a^ -\- b^ -\- c^ -{■ 2hc + 2ca + 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 a2 + 62 + c2 + 26c + lea + 2ab. Arranged according to powers of a, we have a2 + 2a(6 + c) + (& + c)2, thatis {a+(6 + c)}^ Hence the required square root is a + b + c. Ex. 2. To find the square root of 4x* + 9y^ + ICiz* + Ux-y"^ - 16xh^ - 2hrz\ The given expression is 4x* + 4x3 (3^2 _ 4^2) ^ 2y* - 2hjh^ + 162*, that is, (2x2)3 + 2 (2x2) {Sy^ - 4^2) + (37/ - iz-)'', which is {2x2 + (3?/ -4^2)}^. Hence the required square root is 2x2 ^ 3^2 _ 42'', 230 SQUARE HOOT. Ex. 3. To find the square root of a- + 2abx + (b^ + 2ac)x^ + 2hcx^ + c^x*. Arrange according to powers of a ; we then have a^ + 2aibx + ex-) + b'^x^ + 2bcx^ + c%\ that is, a^ + 2a{bx + cx^) + [bx + cx'^)\ Hence the required square root is a + bx + ex". Ex. 4. To find the square root of ^6 _ 2a;5 + 3x^4- 2a;3 {y _ 1) +a;2(l - 2y) + 2xij + y^. The expression only contains y^ and y ; we therefore arrange it according to powers of y, and have y^ + 2y {x^ -x^ + x) + x^- 2x^ + Sx^ - 2x^ + x'^. Now, if the expression is a complete square at all, the last of the three terms must be the square of half the coefl&cient of y ; and it is easy to verify that (aj3 -x^ + xf =x^- 2x^ + 3a;4 - 2a? + x\ Hence the required square root is y+x^-x^ + x. 197. To find the square root of any algebraical ex- pression. Suppose that we have to find the square root of (J. + Bf, where A stands for any number of terms of the root, and B for the rest; the terms mA 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 that 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 + BY, we have the remainder {2A + 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 in 5. Hence, to obtain the next term of the required root, that SQUARE ROOT. 231 is, to obtain the highest (or lowest) term of B, we svhtract from the whole expression the square of that j^ctrt of the root ivhich is already found, and divide the highest (or loiuest) term of the remainder hy 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 The process is written as follows : a;6 _ 4x5 + 6a;4 - 8x3 + 9a;2 - 4.t + 4 f a;3 - 2x2 + x - 2 (x3 - 2a;2)2 = x^-ix^ + ^x^ {x^ - 2a;2 + x)- = x^- 4a;^ + (3.r^ - Ax^ + x"^ (x^ - 2x2 + a; - 2)2 = a;6 _ 4,x^ + 4 j,4 _ q^a ^ 9-2 _ 4a; + 4 We first take the square root of the first term of the given expression, which must he arranged according to ascending or de- scending paicers of some letter: we thus obtain x^, the Jirst term ot 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 - 2.r-, the second term of the root. Now subtract the square of x^~2x^ from the given expression, and divide the first term of the remainder, namely 2x'*, by 2x'^: we thus obtain x, the third term of the root. Now subtract the square of x^- 2.x2 + a; from the given expression, and divide the first term of the remainder, namely -4x*, by 2x^: we thus obtain - 2, the fourth term of the root. Subtract the square of x"^-2x-4-x-2 from the given expression and there is no remainder. Hence x' - 2x- + x - 2 is the required square root. The squares of x^, 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 ao^ + ho^ -\-cx-\-d, provided that the given expression is equal to {ax^ + ha? -^ cx-\- df^ that is equal to a^x'' + laM + (2ac + ¥) x*-\-2 (ad + he) x"" -f (2bd + 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 a'=l; 2a6 = -4; 2ctc + 6'=6; 2ad + 2hc = -S; 2hd-hc' = 9; 2cd = -4<; d'^4i. The first four of these equations are sufficient to determine the values of a, h, c, d; these values are (taking only the positive value of a), a = 1, 6 = — 2, c = 1, d = — 2. The last three equations will be satisfied by the values of a, h, 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 x^ — 2x^ + 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+1, (^ + 1)'^ 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 X l-}--orl+-— — , the square of the former differing from 1 + X by — , and the square of the latter differing by SQUARE ROOT. 233 — g^"^ + (j?^'"*- Thus, provided x is small, l+o i^ an X x^ approximation to the square root of \-\- x, and 1 + „ — -5- 2 8 is a closer approximation, and by continuing the process we can approximate as closely as we please to the square root of \-\- x\ this however is by no means the case when X is not a small quantity. 200. ^Yhen 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 {a^x" + a^x""-' + . . . + aX~"^') + (a^^.x''-' + . . . + a^X''*^') + ^• 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 giveu expression. The square of the above expression is (a^x"" + a^x''-' + . . . + a^-^'f + 2 (a^ + . . . + a^^''^^ + LK-..^""" + • • • + ^^X"""')' + 2i^ (a,^" + . . . + aX"''') + 2R (a,^,^"-^ + . . . + a^x"-"^') + R']. Now, since the highest power of a; in J? is a;"~% the highest power of x in the expression within square brackets is x'''-'''. Hence the expression within square brackets will not affect any of the terms from which a^, a^, •••^g^ are deter- mined, for the first 2r terms of the given expression ex- tend from a;"* to x^"-""*'. It therefore follows that if the square of the sum of the first r terms of the root be subtracted from the given 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 nimiher have been found by the ordinary method, n — \ more figures can he found by division, provided that the number is a "perfect square of 2n—\ 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 ^n — 1 figures, and let _p be the number formed by the first n figures followed by n — \ zeros, and let q be the number formed by the remaining n — 1 figures. Then sJN=p-\-q\ .-. (N-py2p = q-\-(f/2p. Now 2p <(: 2 .10'""' and q > 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 V-^ contain m figures, where m is greater than 2n — 1. 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 — 27z + 1 zeros, and let r be the number formed by the m — 2n + 1 remaining figures. Then J^=(p + q-^ry; .-. (JSr - p')l2p -q = (q'' + r'' + 2qr)/2p + r. Now 10"* > p < 10"'"\ /-^ 10"'"">^<10"'""~\ and X0"'"'""''>r<10' \iii—'in CUBE ROOT. 235 ■whence it follows that {(f -{-r^ •\-2qr)l2p is less than 10"'"'""''. Hence {q~ + r + 2(p')/2p + r is less than 2 x 10"'-'"'', but it is not necessarily less than 10'""'^'''^\ Hence (N-]f)l2p may differ from q by more than 10"*-^"+'; it must however differ by less than 2 x 10"'"'"''^ so that the n — 1 first figures of the quotient (iV — p^)/2jj 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 + Sab' + b\ 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^-5ia^b + ZGa'^h'^-Sa^b^ is a perfect cube its cube root must be 3a--2a6; and by forming the cube of 3a^-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 bi/ inspection, provided that the 236 CUBE ROOT. expression contains only three different powers of some particular letter. For example, to find the cube root of a3 + 63 + c3 + Sa^b + 3a^c + dah^ + dac^ + Qahc + Sb'^c + Sbc\ Arranged according to pov/ers of a, we have a3 + 3a2 {b + c) + Sa {b^ + c^ + 2bc) + b^ + c^ + Sb^c + 3bc^, that is, a3 + 3a^ {b + c) + 3a (& + c)^ +{b + cf. Hence the required root is a + 6 + c 203. To find the cube root of any algebraical expression. Suppose we have to find the cube root of {A + By, 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 + By, we have the remainder {SA'-\-SAB + 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 of ^ 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 ROOT. 237 For example, to find the cube root of The process is written as follows : .T« - 6x5y + 21x^y'- - Uxh/ + CiSx-y"^ - 5ixi/ + lly^ {p(? - 2xyf = x^- ^xhj + 12.T*?/2 - 8a;3|/3 (x- - 2xy + 3?/-)"- = x^ - 6x^y + 21afy^^4&y^ + OSx"Y ~ 54x2/3 + 27y\ Having arranged the given expression according to descending powers of x, we take the cube root of the fii'st 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 lirst term of the remainder, namely -6x^y, by Sx{x^f: we thus obtain - 2xy, the second term of the root. We then subtract the cube of x"^ - 2xy from the given expression, and divide the first term of the remainder by 3 x [x-)" : 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 3t/^, 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 {ixf'Y, so that the second term of the root is — 2^y. Hence, if the given expression is really a perfect cube, it must be {x^ — ^xy + ^y^, and it is easy to verify that {x^ — 2xy + Sy'^y is equal to the given expression. Again, to find the cube root of x' - Qx'y + 15a-'?/' - 29.'i^y + 5UY - ^OxSf + 64a;y -63Ay + 27A/-27/. If the given expression is really a perfect cu be the first and last terms of the root must be ^x^ and iZ—^ly^ respectively, that is x^ and — 3_y '. 238 EXAMPLES. The second term of the root must be — Qx^y -=- 3 (x^y = — 2x^y ; and the term next to the last must be 21 xf -- 3 (- ^y'Y = + xy\ Hence the given expression, if a cube at all, must be (x^ — 2x^y + xy^ — Sy^y ; and by expanding (x^ — 2x'^y + wy"^ — Sy^ it will be found that the given expression is really a perfect cube. 204. From the identity [see Art. 253] (a + by = a" + na"~^b + terms of lower degree in a, it is easy to shew, as in Articles 197 and 203, that the ?i*^ 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 the 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 — 1)*^ poiuer of the first term of the root : this gives tJw next term of the root. EXAMPLES XIX. Write down the square roots of the follow Id g expressions : 1. 4:x''-l2xY + ^y'' 2. x^ + 9xV'' - 6a;y. '^ 3. a' + ih"" + 9c' + 1 25c - ^ca -iah. 4. 25a" + 96* + 4cV 1 26V - 20c'«i- - 30^ V. EXAMPLES. 239 Find the square roots of 5. x' + 2x' + 3x* + ix^ + 3x' + 2x + 1. 6. ix* - 8x-V' + ^xif + y\ 7. 49 + 112a;' + 70.r' + 64a;* + 80a;' + 2.3a;^ 8. x" - 2.x' + 5a;' - 6a; + 8 - 6a;-' + Sa;"" - 2a;-=' + a;"*. 25a;' y" ^r.^ .^V ,^ 10. a;^ - 4a;* + 2a; + 4a;^ + a;^. 11. a;^ - 4a;^ + 4a; + 2a;^ - 4a;^ + a;^. 12. a;° - 2a; "^a: ^ + 2a^a;"^ + « ^x" - 2d^x'' + a\ Find the cube roots of 13. a;'-24a;' + 192a;-512. 14. a;« - 3a;V + ^x'y' - IxSf + 6a;y - ^xy' + y\ 15. 1 - 9a;' + 33a;* - 63a;« + 66a;« - 36a;''' + 8a;'^ 16. Find the square root of 2a' (6 + c)' + 26' (c + a)' + 2c' (a + hf + 4ak (a + 6 + c). 17. Find the square root of a;' (a;' + 2/' + «^) + 2/'^^ + 2a; (y + 2;) {yz - x"). 18. Find the square root of {a -by -2 (a' + 6') (a -by + 2 {a' + 6 '). 19. Shew that (x + a) (x + 2a) {x + 3a) (x + 4a) + a* is a perfect square. 20. Prove that x* + 2)x^ -\-qx- + rx + s is a square, if ^/^ = r' and /^'^ - 47?5' + 8r = 0. 240 EXAMPLES. 21. Find the values oi A^ B and C in order that Ax' - 24a;' + Ax" + Bx^ + Cx' - 40a; + 25 may be a perfect square. 22. Shew that, if cvjf + hx^ + cx + d be a perfect cube, then b^ = Sac and c^ = 36(i. 23. Find the conditions that ax^ + h}f + c^ + '^fyz + 2^;2;a; + 2/ia:2/ may be the square of an expression which is rational in a;, y and %. 24. Shew that if (a - A.) a;^ + (6 - A) t/^ + (c - X) 2;^ + 2/?/5; + 2^;2;a; + "XTixy be the square of an expression which is rational in cc, ?/ and z, then will qh , 7?/ /or - J g ^^ 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 9^ + 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 29^ + 2 figures. 27. Shew that, if n + 2 figures whose numerical value is a have been found of a positive root of the equation a?-\-qx — r = 0, q being supposed positive, then the result of dividing r-qa-a? by Zo?-\-q will give at least n-\ more figures correctly. _ CHAPTER XYI. Eatio. Proportion. 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 : 6 ; and a is called the first term, and h 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 ac'.hdiQ the ratio compounded of the two ratios a : h and c : d. The ratio a^ : h^ is sometimes called the duplicate ratio of a : 6 ; so also a^ : b^, and ^/a : ^Jb are called respectively the triplicate, and the sid)- duplicate ratio of a : 6. s.A. 16 242 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 3iS 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. 118 are also true for ratios. The following theorem is of importance : 207. Theorem. Any ratio is made more nearly equal to unity by addhig the same positive quantity to each of its terms. By adding x to each term of the ratio a : b, tlie ratio a + ^ : 6 + a? is obtained. ,, a ^ a — b , a + cG ^ a — b Now T - 1 = -1—- , and j— 1 = ,-- — , b b + as b + X and it is clear that the absolute value of j is less than b -\- X that of — ^i — , for the numerators are the same and the 6 denominator of the former is the larger: this proves the proposition. When X is very great, the fraction y is very small ; "T" X and , , which is the difference between j and 1, b+x b+x can be made less than any assignable difference by taking X sufficiently great. RATIO. 243 This is expressed by saying that the limitivr/ value of J- — - , 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 aiid a ratio of less inequality 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 \/2 : 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 ditierent theorems which have been proved with respect to ratios can, by the method of Art. 163, Ix^ proved to be true for the ratios of incommensurable numbers. 16—2 244 proportion. Proportion. 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 : b :: c : d, which is read " a is to 6 as c is to dJ* 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, 6, c, d are proportional, we have by definition, a _G 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 that is ad = = be, then ad be bd' bd J a ''' b'~ c ~d' a : b = = c : d. pRoroiiTioN. 245 Hence also, tlie four relations a : h = c : d, a : c = b : d, b : a = d : Ci and b : d = a : Cy are all true, provided that ad = be. Hence the four proportions are all true when any one of them is true. Ex. If a : 6 = c : d, then will a+b : a- b = c + d : c-d. This has already been proved in Ai't. 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 + ad; or, if be = ad. But be is equal to ad, since a : b = c : 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, 6, c be in continued proportion, we have a _b b~~c' .". y^ = ac, or 6 = Jac. 246 PROPORTION. Thus the mean proportional between two given quantities is the square root of their product. a h h h Also ^ X - = - X - , c c c , . ^-^-^ c c 6 Thus, if three quantities are in continued proportion, tlie 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 a c test of proportionality, we have j =-i'i therefore for all , „ , ma mc values 01 m and w, -y = — ^ . no na > > Hence mc = nd, according as ma = nh. Thus a, b, c, d < < satisfy also Euclid's test of proportionality. Next, suppose that a, b, 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 7 T = — ; .'. na= mo. b n PROPORTION. 247 But by definition > > nc = md according as na = lub. < < Hence nc = 7nd; c _m _a d n b' • • Thus a, h, c, d satisfy the algebraical definition. If a and h are incommensurable we cannot find two whole numbers m and n such that a : b = m : n. But, if we take any multiple na of a, this must lie between two consecutive multiples, say mb and (m + 1) 6 of 6, so that na > mb and na < (m + 1) b. Hence by the definition, nc > md and nc < (m + l)d. Hence both y and -, lie between — and . a n n a c 1 Thus the difference between -j and -^ is less than - ; and b d n as this is the case however great ?i may be, -j must be equal to J- , 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 7 + .t : 12 + a; be equal to the ratio 5:6? Arts. 18. Ex. 2. If Qx'^ + Qtj^ = 13xy, what is the ratio of x to y? Ans. 2 : 3 or 3 :.2. Ex. 3. What is the least integer which when added to both terms of the ratio 5 : 9 will make a ratio greater than 7 : 10? An^. 5. Ex. 4. Find x in order that x + 1 -. x + O may be the duplicate ratio of 29 3 : 5. Ans. -- . lb 248 VARIATION. Ex. 5. Shew that, ii a :h :: c : d, then (i) a^ + ab + b"^ : c^- + cd + d^ :: a^-ab + b^ : c^-cd + dP. (ii) a + b:c + d:: ^{2a^ - Sb^) : ^(2c2 - Bd^). (iii) a^+b^ + c'^ + d^ : (a + 6)2 + (c + d)2 :: (a + c)2+(& + (Z)2 : {a + b + c + d)^. [See Art. 113.] Ex. 6. li a I b :: c : d, then will ab + cd be a mean proportional between a^ + c- and i^ + d^. Yartation. 214. One magnitude is said to vaiy as anofher when the two are so related that the ratio of any two vahies of the one is equal to the ratio of the corresponding values of the other. Thus, if a^, a^ be any two measures of one of the quantities, and b^, h^ be the corresponding measures of the other, we have — = 7"^ ; and therefore -^-^ = -^^ . a, \ b, b^ 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 A ccB is read 'A varies as B \ If a cc 6, the ratio a : 6 is constant ; and if we put m for this constant ratio, we have T- — 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 ccb, and a is 15 when b is 5, we have - = m = — ; o ,*. a = Sb. VAT^TATION. 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 : y- 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 b and c jointly if a oc he, that is if a = mhc, 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 h 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 b and c ; and if a is 6 when b is 4 and c is 3, we have a = mbc, and 6 = mx4x3, Hence in = -, and therefore a — -be. 216. Theorem. If a depends only on h and c, and if a varies as b when c is constant, and varies as c ivhen h is constant; then, when both b and c vary, a luill vary as be. Let a, h, c; a', b', c and a", b\ c be three sets of corresponding values. 250 VARIATION. Then, since c is the same in the first and second, we have — / = T7 (i)- a And, since h' is the same in the second and third, we ■t 0/ G /• • \ nave — r/ = — (n). a c Hence from (i) and (ii), -r, = tt-/ , ^ ^ a be 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 [W] 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 varies as the product of the weight and the price. Thus, if C cc P, when W is constant, and G (X. W, when P is constant ; then G cc PW, when both P and W change. Again, the area of a triangle varies as the base when the height is 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 7 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. Whatisthe volume of a sphere of one yard radius? Ans. 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? Ans. 57& 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 2G0 the volume is 200 cubic inches. What will the volume be when the pressure becomes 18 and the temperature 390? Ans. 250 inches. INDETERMINATE FORMS. 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 : tiud the distance when the height is 72 yards. Ans. 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 = both the numerator and the denominator of the fraction -;; vanish, and the fraction assumes for this value of x"^- X X the indeterminate form - ; and this is also the case when x = l. Again, when a: = oo 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. 0)^ — 1 Consider, for example, the fraction — — z-, which as- sumes the form - when x=l. Now — ^ ^ ^ ^'-1 {x-l)(cc' + a)+l)' and, provided x — 1 is not ideally 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—\ may he. Hence, when a; — 1 is very small, —s — t = —i ? > and the limiting value of the latter fraction, as x ap- 2 proaches indefinitely near to 1, is at once seen to be ^ . 252 INDETERMINATE FORMS. Hence, as x approaches indefinitely near to 1, the 0^—1 . . 2 fraction -^ — =- approaches indefinitely near to the value ^ . This is expressed by the notation X^=i 3 — =- = 75 . x^ - 5x + 6 Ex. 1. Find the limiting value of -^ — -— —7 when x = 2, aj-^-lOoj + lo It follows from Art. 88 that x-2 is a common factor of the numerator and denominator. x^-5x + 6 {x-2){x-S) _ a;-3_l '^«=2 a;2-10a; + 16"*^=2 (a;-2)(a;-8)~ *=2 x-S^Q' x^ + 2x Ex. 2. Find the limiting value of ^r—, — r— when a; = and when 2,x-' + ox x- = 00 . •t-ar-^o x'^ + 2x 2x^ + dx~ = ^a; = x{x + 2) ^(2a; + 3)~ *=<> a; + 2 2 2a; + 3~3* ■^x = « x'^ + 2x 2:r2 + 3a;" = Lx^^ <-!) , j; 1 .= f2 + ?\ ^^ '" ~r2' \ XJ X 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 + 3x when x increases without limit. l + 2^_ LiU-fi-r "'"•'g 2 *=" 2 + 3a;~^=" /, 2\~^=" ^ 2~3' a;(3 + -) 3 + - Ex. 4. Fmd the limiting value of =-r — j-r when x becomes oic-^ - 40 indefinitely great. 2a;2 + 1003; + 500 _ '^^^ 5a;3-40 — ^a!=« ^ /„ 40 - /^ 100 500\ V X x^ ) x^ I o 40\ -L ?^'-L A-0 EXAMPLES. ^53 EXAMPLES XX. 1. Shew that, U a + b, 6 4- c, c + a are in continued propor- tion, then b + c : c + a = c — a : a — b. 2. Shew that, ii x : a = y : b = z : c, then x^ y^ z^ _(x + y ■\- zf -. + ^2 + c'^\a + b + cY' 3. Shew that, \i (a + b + c + d){a -b - c + d) = {a -b ■¥ c - d) {a+b — c — d), then a, 6, c, d are propprtionals. 4. Shew that, if b^ + c^ = a^, then a -\-b + c : c + a — b = a + b~c: b + c — a. 5. What number must be subtracted from each of the numbers 7, 10, 19, 31 in order that the remainders may be in proportion 1 6. Find a : b : c, having given 6 a + c-b a + b + G a + b b+c — a 2a + 6 + 2c* 7. If » _ y _ * b+c—a c+a—b a+b—c^ shew that (a + b + c) {yz + zx + xy) = (x + y + z) (ax + by + cz). 8. If a(y + z) = b(z + x) = c(x+ y), prove that y — z z — x oc — y a{b-c) b (c— a) c(a -b)' 9. Shew that the ratio ^,»i + ^2«2 + ^3«3 + ' h^, + KK^KK^ is intermediate to the greatest and least of the ratios a^: b^, «a : b^, (kc, the quantities being all positive. 10. If a : b :: c : d, then a^- + h'" + c"' + d"' _ 11. Show that, if (a + b) (b + c) (c + d) (d + a) = (a + b + c + d) (bed + cda + dab + abc), then a : b :: d : c. 254 EXAMPLES. 12. If (hcd + cda + dab + ahcY — abed {a + b + c + dy = Oy then it will be possible to arrange a, b, c, d so as to be propor- tionals. X y z 13. Shew that, if then a + '2b + c a — c a — 26 + c' a b c x + 2y + z x — z x — 2y + z 14. Shew that, if ax^ + by^ + cz"^ + 2fyz ^ 2gzx + 2hxy = and X + y + z = are only satisfied by one set of ratios x : y : z, then be -f^ + ca - g^ + ab - A" + 2 (gh — af) + 2{h/-bg) + 2(fg-ch)=^0. 15. Shew that, if a b c then p{px — qy — rz) q (qy — rz — px) r{rz — px — qy)* p q r a (ax — by — cz) b {by — cz — ax) c (cz — ax— by) 16. Shew that, if ab = cd, then either of them is equal to (a + c) (a + d) (b + c) (b + d)l{a + 6 + c + d)\ Also, \i a+b = c + d, then either of them is equal to abed I - + 7- + - + -J /(«& + ed). \a c dj I ^ 17. Find the limiting values of the following fractions when x—2y and when x = go . ,. 03^ — 707+ 10 x^ — 4:X + 4: ..... x"+6x—lQ aj"-9aj+14' ^ ^ x'-6x + 6' ^ ^ a;'-~ 12a;+ 16* 18. Find the limiting values of the following when x = a, ^^ iJa-^x' ^"^ J(:^-ar) ' CHAPTER XVII. Arithmetical, Geometrical, and Harmon ical Progression. 218. Series. A succession of quantities the members of which are formed in order according to some definite law is called a series. Thus 1, 2, 3, 4, , in which each term exceeds the preceding by unity, is a series. So also 3, 6, 12, 24, , in which each term is double the preceding, is a series. We shall in the present Chapter consider some very simple cases of series, and shall return to the subject in a subsequent Chapter. Arithmetical Progression. 219. Definition. A series of quantities is said to be in Arithmetical Progression when the difference between any term and the preceding one is the same throughout the series. Thus, a, b, c, d, &c. are in Arithmetical Progression [a. P.] lib — a~c — b = d — c = &c. The difference between each term of an A. p. and the preceding term is called the common difference. 256 ARITHMETICAL PROGRESSION. The following are examples of Arithmetical progressions ; — 1, 3, 5, 7, &c. 3, -1, -5, -9, &c. a, a + 26, a + ib, &G. In the first series the common difference is 2, in the second it is - 4, and in the last it is 26. 220. If the first term of an arithmetical progressioD be a, and the common difference d; then, by definition, the 2nd term will he a -{- d, „ 3rd „ „ a + 2d, „ 4th „ „ a + Sdy and so on, the coefiicient of d being always less by unity than the number giving the position of the term in the series. Hence the nth. term will be a + (n — l)d. We can therefore write down any term of an A. P. when the first term and the common difference are given. For example, in the a. p. whose first term is 5, and whose common difference is 4, the 10th term is 5 + (10 -1)4 = 41, and the 30th term is 5 + 29 x 4 = 121. 221. An arithmetical progression is determined when any two of its terms are given. For, suppose we know that the mth term is a, and that the nth term is jS. Let a be the first term, and d the common difference; then the ??ith term will be a + (m — 1) d, and the nth term will he a + (n — l) d. Hence a + (in — 1) cZ = a, and a-\-{n —l)d = l3. "~^ Thus we have two equations of the first degree to determine a and d in terms of the known quantities m, n, a and /S. ARITHMETICAL PllOGRESSION. 257 Ex. Find the 10th term of the a. p. whose 7th term is 15 and whose 21st term is 22. If a be the lirst term, and d be the common difference, we have a + Gd = 15, and a + 20rf = 22. 1 9 Hence d = ^, a = 12. The 10th term is therefore 12 + -==1GL 2 2 ^ 222. When three quantities are in arithmetical pro- gression, the middle one is called the Arithmetic Mean of the other two. If a, h, c are in A. P., we have, by definition, b — a = c — b; and therefore 6 = |(a 4- c). Thus the arithmetic mean of two given quantities is half their sum. When any number of quantities are in arithmetical progression all the intermediate terms may be called arithmetic means of the two extreme terms. Between any two given quantities any number of arith- metic means may be inserted. Let a and b be the two given quantities, and let n be the number of terms to be inserted. Then b will be the n + 2th term of the A. P. whose first term is a. Hence, if d be the common difierence, b = a-\~{n-\-l)d; and therefore d = ^ . n + 1 Then the series is b — a cx^ — a „ a, a 4- -— -, , a + 2 — — , &c., n + 1 71 + 1 the required arithmetic means being b - a ^b — a b — a a H , a + 2 , a 4- ?i r , 71+1' 71 + 1 7i+l* na + b {n-\)a+'lb (7i-2)a + .3Z^ a-\-nh °^ ¥+T' ^T+i ■' mH: ' '^T' S.A. 17 258 ARITHMETICAL PROGRESSION. 223. To find the sum of any number of terms of an arithmetical progression. Let a be the first term and d the common difference. Let n be the number of the terms whose sum is required, and let I be the last of them. Then, since I is the nth term, we have 1 = a-\-(n — l)d (i). Hence, if S be the required sum, S=a-{-(a-hd)-{-(a+2d) + -{-(l-2d) + (I -d) + l. Now write the series in the reverse order ; then S = l-^{l-d)-\-(l-2d)+ +{a-{-2d) + (a + d) + a. Hence, by addition of corresponding terms, we have 2S = {a-{-l) + {a + l) + {a + l)-\- to ti terms = n{a-\- 1); .-. S='^(a+l) (ii), or, from (i), >Sf = |{2a + (7i-l)cZl (iii). From the formulae (i), (ii), (iii) the value of all the quantities a, d, n, I, S can be found when any three are given. Ex. 1. Find the sum of 20 terms of the arithmetical progression 3 + 6 + 9 + &C. Here a = 3, d = 3, w=20; .-. S = ^{6 + 19x3} = 630. Ex. 2. Shew that the sum of any number of consecutive odd numbers, begmning with unity, is a square number. The series of odd numbers is /^^ 1+3+5+ Here a = l, d = 2; hence the sum of w terms is given by S = ^{2a + {n-l)d} = -{2 + {n-l)2}=n\ ARITHMETICAL PROGRESSION. 259 Ex. 3. How many terms of the series 1 + 6 + 9+ must be taken iu order that the sum may be 190? WehaveS' = -{2a + (n-l)d}, where /S=190, a=l, d = 4. Hence n is to be found from the quadratic equation 190 = |{2 + 4(n-l)}, or 2n2-w- 190 = 0, that is (n-10)(2n + 19) = 0. 19 Hence n=10. The value n= -— is to be rejected for n must necessarily be & positive integer*. Ex. 4. How many terms of the series 5 + 7 + 9+ must be taken in order that the sum may be 480? Here we have 480=|{10 + {n-l)2}; /. 7i2 + 4n- 480 = 0, or (n-20)(n + 24) = 0. Hence n must be 20, for the value n= - 24 must be rejected as a negative number of terms is altogether meaningless*. Ex. 5. What is the 14th term of the a. p. whose 5th term is 11 and whose 9th term is 7? Ans. 2. Ex. 6. What is the 2nd term of the a. p. whose 4th term is b and whose 7th term is 3a + 46 ? Ans. - 2a - h. Ex. 7. Which term of the series 5, 8, 11, &c. is 320? Am. The 106th. Ex. 8. Shew that, if the same quantity be added to every term of an A. p., the sums will be in a. p. Ex. 9. Shew that, if every term of an a. p. be multiplied by the same quantity, the products will be in a. p. * The inadmissible value is a root of the equation to which the problem leads, but it is not a solution of the problem. [See Chapter xi.] It should be remarlced that a negative value of n cannot mean a number of terms reckoned backwards. 17—2 260 GEOMETRICAL PROGRESSION. Ex. 10. Shew that, if between eveiy two consecutive terms of an A. p., a fixed number of arithmetic means be inserted, the whole will form an arithmetical progression. Ex. 11. Find the sum of the following series : (i) 2^ + 4| + 6|+ to 23 terms. (ii) 2'''6~6~ to 12 terms. (iii) {a + 9b) + (a + lb) + {a + 5b) + to 10 terms. ,, , w-1 n-2 71-3 , . , (iv) 1 1 h ton terms. ^ ' n n n Ans. (i) 621, (ii) - 16, (iii) 10a, (iv)|(?i-l). Ex. 12. The 7th term of an a. p. is 15, and the 21st term is 8; find the sum of the first 13 terms. Ans. 195. Ex. 13. Find the sum of 21 terms of an a. p. whose 11th term is 20. Ans. 420. Ex. 14. Shew that, if any odd number of quantities are in a. p., the first, the middle and the last are in a. p. Ex. 15. Shew that, if unity be added to the sum of any number of terms of the series 8, 16, 24, &c., the result will be the square of an odd number. Ex. 16. How many terms of the series 15 + 11 + 7 + must be taken in order that the sum may be 35? Ans. 5. Ex. 17. The sum of 5 terms of an a. p. is - 5, and the 6th term is -13; what is the common difference? Ans. -4. Ex. 18. Find the sum of all the numbers between 200 and 400 which are divisible by 7. Ans. 8729. Ex. 19. If a series of terms in a. p. be collected into groups of n terms, and the terms in each group be added together, the results form an A. p. whose common difference is to the original common difference as w2 : 1. Geometrical Progression. 224. Definition. A series of quantities is said to be in Geometrical Progression when the ratio of any term to the preceding one is the same throughout the series. GEOMETIIICAL PllOGRESSlON. 261 Thus a, h, c, d, &c. are in Geometrical Progression (g.p.) if - = - = - = &c. a c Tlie ratio of each term of a geometrical progression to the preceding term is called the common ratio. The following are examples of geometrical progressions : 1, 3, 9, 27, &c. 4, -2, 1, -^, &c. a, a^, a^, a7, &g. In the first series the common ratio is 3, in the second series it is - ^, and in the third series it is a^. 225. If the first term of a G.P. be a, and the common ratio r; then, by definition, the 2nd term will be ar, „ 3nl „ „ a7'\ „ 4th „ „ ar\ and so on, the index of r being always less by unity than the number giving the position of the term in the series. Hence the nth term will be ar"*'^. We can therefore write down any term of a G.P. when the first term and the common ratio are given. For example, in the g.p. whose first term is 2, and whose common ratio is 3, the 6th term is 2 x 3^, and the 20th term is 2 x 3^'->. 226. A Geometrical Progression is determined when a7iy two of its terms are given. For, suppose we know that the mth term is a, and that the nth term is /3. Let a be the first term, and r the common ratio ; then the 7?ith term will be a?'"'"S and the 7ith term will be ar""-'. 262 GEOMETRICAL PROGRESSION. B.ence ar'"-' = a, ar'"'' = ^ ; and .-. ?^"=^. l-n 1-m Hence r = a'''-^/9'^-"*, and therefore a = a"^-«/3"-"*. Ex. Find the first term of the g.p. whose 3rd term is 18 and whose 5th term is 40^. If a be the first term, and r the common ratio, we have aj-2 = 18, ar^=~; /. r2 = |. 4 Hence a=18x-=8. y Thus the series is 8, 12, 18, &c. 227. When three quantities are in G.P., the middle one is called the Geometric Mean of the other two. If a, 6, c are in G.P., we have by definition - = y ; .•. = + Jac. a ~ ^ Thus the geometric mean of two given quantities is a square root of their product. When any number of quantities are in geometrical progression all the intermediate terms may be called geometric means of the two extreme terms. Between two given quantities any number of geometric means may he inserted. For let a and h be the two given quantities, and let n be the number of means to be inserted. Then h will be the (n + 2)th term of a G.P. of which a is the first term. Hence, if r be the common ratio, we have b^ar""^^; /. r= . -. V a a Hence the required means are ar, ar^y jft^", n 1 w-1 2 1 n that is, a'*+i6^^+i, a^+^t^+i, a"+i6^^+i. GEOMETRICAL PROGRESSION. 2()3 228. To find the sum of any nuviher of terms in geometrical progression. Let a be the first term, and r the common ratio. Let n be the number of the terms whose sum is required, and let I be the last of them. Then, since I is the nth term, we have I = af"^. Hence, if S be the required sum, /S'=a + ar+ar'+ + ar""\ Multiply by r ; then ^r = ar -\- ar^ + ar^ + + a?'""^ + ar\ Hence, by subtraction, S-Sr = a- ar"" ; .*. o = a — . 1—r Ex. 1. Find tlie sum of 10 terms of the series 3, 6, 12, etc. Here - a = 3, r=2, n = 10. 1 -2^0 Hence iS = 3-— — =3 (2i0- 1) = 30C9. X — ^ 220. From the preceding article we have 1 — r" a ar"" S = a \—r 1—r 1 — r' Now when r is a proper fraction, whether positive or nega- tive, the absolute value of r" will decrease as n increases ; moreover the value of r" can be made as small as we please by sufficiently increasing the value of n. Hence, when r is numerically less than unity, the sum of the series can be made to differ from - — by as small a quantity as we please by taking a sufficient number of terms. 264 GEOMETRICAL PROGRESSION. Thus the sum of an infinite number of terms of the geometrical progression a + ar -\- ar^ -\- , in which r is numerically less than unity, is ^j . Ex, 1. Find the sum of an infinite number of terms of the series 9-6+4- 2 Here a=9, r= --. XT c « 9 27 Hence o = =~ — = ' , r,s = -r- 1-r ^ f 2\ 6 (-:) Ex. 2. Find the geometrical progression whose sum to infinity is 4J, and whose second term is - 2. Let a be the first term, and r be the common ratio. a 9 Then we have ar= - 2, and z = - . 1-r 2 Whence 9r2-9r-4 = 0. -rx 14 Hence r = - ^ , or r= - . o o If r=-l,a=^ = 6; 2 . and the series is 6, - 2, - , &c. 4 The value r = - is inadmissible, for r must be numerically less than o unity. Ex. 3. The 3rd term of a g.p. is 2, and the 6th term is -|; what is the 10th term? Ans. -i^. Ex. 4, Insert two geometric means between 8 and - 1, and three means between 2 and 18. Ans. - 4, 2 ; ±2 ,^3, 6, ± 6 ,^3. Ex. 5. Shew that if all the terms of a g.p. be multiplied by the same quantity, the products will be in g.p. /^^ Ex. 6. Shew that the reciprocals of the terms of a g.p. are also in g.p. Ex. 7. Shew that, if between every two consecutive terms of a g.p., a fixed number of geometric means be inserted, the whole will form a geometrical progi'ession. HARMONICAL PllUGRESSION. 2G5 Ex. 8. Find the sum of the following series; (i) l2 + 9 + Gf+... to 20 terms. 2 4 (ii) 1 -- + -+... to 6 terms. (iii) 4 + -8 + '1Gh-... to infinity. ^rt.. (i) 48 |l - (^lyi , (ii)l|?, (iii)5. Ex. 9. Shew that the continued product of any number of quantities n in geometrical progression is equal to (gl)^, where n is the number of the quantities and g, I are the greatest and least of them. Ex. 10. Shew that the product of any odd number of terms of a g.p. will be equal to the 7ith power of the middle term, n being the number of the terms. Ex. 11. The sum of the first 10 terms of a certain g.p. is equal to 244 times the sum of the first 5 terms. What is the common ratio ? Ans. 3. Ex. 12. If the common ratio of a g.p. be less than |, shew that each term will be greater than the sum of all that follow it. Harmonical Progeession. 230. Definition. A series of quantities is said to be in Harmonical Progression when the difference between the first and the second of any three consecutive- terms is to the difference between the second and the third as the first is to tlie third. Thus a, h, c, d &c., are in Harmonical Progression [h. p.], if a — h : h — c : : a : c, h — c : c — d :: h : d, and so on. If a, h, c be in harmonical progression, we have by definition a — h : b — c :: a : c; ,', c {a — b) = a(b — c). 266 HARMONICAL PROGRESSION. Hence, dividing by ahc, we have 1_1_1_1 bach which shews that - , j i - are in arithmetical progression. Thus, if quantities are in harmonical progression, their reciprocals are in arithmetical progression. 231. Harmonic Mean. If a, h, c be in harmonical progression, - , ^ , - will be in arithmetical progression. TT 2 11 Hence r ~ - + - > a c , 2aG .*. b = 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 6, we have A=^{a + b\ G = Jab, H=-'_^~^; .\A.H=G\ Thus the geometric mean of any two quantities is also the geometric mean of their arithmetic and harmonic mea7is. 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 + b)> Jab, or 1 (Va - ^/by > 0. EXAMPLES. 2G7 Now (^^a — ^/by is always positive, and therefore greater than zero, unless a = b. 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 6. Insert ?i arithmetic means between - and ^ , and the a reciprocals of these will be the required harmonic means. The arithmetic means are a n + l\b aj a n-\-l\b aj Hence, by simplifying these terms and inverting them, the required harmonic means will be found to be (72 + 1) ab (n + 1) ab (n + 1) ab nb + a ' (n — l)b + '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, h, c be in a. p., then will a' (6 + c), 6* (c + a), c' {a + h) be in A. p. 2. Find four numbers in A. P. such that the sum of their squares sliall 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: h = c \ d. 4. Find three numbers in o. p. such that their sum is 14, and the sum of their squares 84. 268 EXAMPLES. 5. If a, b, G be in aritlimetical progression, and x be the geometric mean of a and b, and 9/ be the geometric mean of b and c ; then will x^, 6^, y^ be in arithmetical progression. 6. Shew that, if a, b, c be in harmonical progression, then will J— , r and z be also in harmonical o+c-a c+a—b a+b—c j)rogression. 7. Shew that, if a, 5, c, d be in harmonical progression, then will 2>(b-a){d-c) = {G-b){d-a). 8. Shew that, if a, 6, c be in harmonical progression, 2 1 1 + b b — a b — c' 9. Shew that, if a, b, c be in h.p., then will b + a b + c b — a b — G 2. 10. If a, b, c be in a. p., b, c, d in g. p., and c, d, e in h. p. ; then will a, c, e be in g. p. 11. If a, b, c be in h. p., then will a — -, ^ , c — ^ be in g.p. 12. If a, 6, c are in h.p., then a, a — c, a -6 are in h.p., and also c, c - a, c -b are in h. p. 13. If a?, ^j, a^) y be in a. p., £c, g^^ g^, y in g.p., and ic, h^, h^, y in h.p., then 14. The sum of the first, second, and third terms of a G. 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,, a^, a.^, , a ^ be in harmonical progression, prove that a^a^ + a^a^ + aji^ + + <*„-i'^„ = («- — !) f^i»„« 16. If a, X, y, b be in arithmetical progression, and a, u, V, b be in harnionical 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 } ^j 1 T- be in A. P. : then a, v » c will be l-ao I - be ' '6 in H. p. 19. If a, b, c be in a. p., and a^, b'^, c^ be in h. P., prove that — - , 6, c are in G. p., or else a = b = c. 20. If X be any term of the arithmetical progression and y be the corresponding term of the harmonical progression whose first two terms aie a, 6, then will x — a \ y — a wb \ y. 21. Shew that, if a be the arithmetic mean between b and c, and b be the geometric mean between a and c, then will c be the harmonic mean between a and b. 22. The series of natural numbers is divided into groups as follows: 1; 2, 3; 4, 5, 6; 7, 8, 9, 10; and so on. Prove that the sum of the numbers in the W^ group is \h (k^ + 1). 23. An A. p. and an h.p. have each the first term a, the same last term I, and the same number of terms n ; prove that the product of the (r+1)*^ term of the one series and the {n — o-y^ 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 difierences of the successive terms of the series so formed are in a.p. 25. Shew that, if «S'^, /S^^, S^^ be the sum of n terms, of 2n terms, and of 3n terms respectively of any g.p., then will 26. If a, b, 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" > '2b". 270 EXAMPLES 27. If P, Q, R be respectively the j)^^, ?*^, 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{r-p) + B{p-q) = 0, (ii) P«-\ Q'-^ . i2^-» = l, (iii) QE (q -r) + JRP (r -p) + PQ {p - q) = 0. 28. Shew that, if a^, a^, a^, , «^ be in h. p., then a. a a 1 2 n a„ + a.+ ... + a^ a.+a.+ ... a^^ ^ a^+a+ ... +a , 23 nl3 n 12 i»~l will be in h. p. 29. Shew that, if a^, a^, a^ , a^ be all real, and if « + <+ +<-i)« + <+ +0 = {a^a^ + a^a^+ +«„-i«X then will a^, a^, a^, be in g. p. 30. Shew that any even square, (27^)^, is equal to the sum of n terms of one series of integers in a. p., and that any odd square, {'2n f 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, />, is the sum of /) consecutive terms of the series 1, 3, 5, 7, tfec. ; and find the first of the p terms when the sum is ^j'. 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 XVIII. 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^, Si 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. 18—2 276 SYSTEMS OF NUMERATION. Ex. 1. The diffeieuce of any two numbers expressed by tbie same digits is divisible by r - 1. For the sum of the digits is the same for both ; and since N-y - S and N^-S are both divisible by r-1, it follows that Nj^-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 is Sb multiple of 9 ; hence, if S 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 + dir + d2r'^ + d^r'^+ , and D = dQ-dj + d2-dg+ Then N-D = dj^{r + l)+d2{r^--l) + d^{i^ + l) + Each of the terms on the right is divisible by r + 1 [Art. 87 j; .*. N-D is divisible by r + 1. Hence if D is divisible by r + 1 so also is 2^. Ex. 4. If N-, and J^g ^^ ^^7 t'^o whole numbers, and if the remainders left after dividing the sum of the digits in N-^, N^ and in N-y x N^ by 9 be Wj, Wg and p respectively ; then will n^n^ be equal to p, or differ from p hj a. multiple of 9. For i^i = Wi+a multiple of 9, and N=n2 + B, multiple of 9 ; therefore N-i^xN^^n^x n^ + a multiple of 9. Hence iijii^ + a multiple of 9 is equal to p+a, multiple of 9. If the above is applied in any case of multiplication, and it is found that w^Wg 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 n^ does not equal ^, 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 the 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, by c be the digits ; then we have 49a + 76 + c = 81c + 9& + a, where a, 6, c are positive integers less than 7> Hence 40c + & = 24a. EXAMPLES. 277 Now 40c and 24*/. are both divisible by 8; therefore h must be divisible by 8. But b is less than 7: it must therefore be zero. And since b 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), 26 + 1 = 6 + r (ii), 2a + 1 = (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, 5, 8, 11, &c., the numbers corresponding to these scales being Oil, 143, 275, Stl, 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, tfcc. lbs., which must be takeu to weigh 1027 lbs.] 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 -440 in the quinary scale, and by 4*54 in a certain other scale. What is the radix of that other scale 1 9. If S be the sum of the digits of a number iV, and 2Q be the sum of the digits of 21^, the number being expressed in the ordinary scale, shew that jS~ Q 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, Avhen 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 5 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. 270 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, b, c in order that the number 13a64:5c 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 dibits 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 (7'^ — 1) (;•" — 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, ^j^, = -0l23...(,-3)(r-l), the circulating period consisting of all the figures in order except 7* — 2 which is passed over. For exjyiiple, 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 as the multiplier. CHAPTER XIX. Permutations and Combtnations. 240. Definition. The different ways in whicli 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, b, c, d; the permutations two at a time are ab, ba, ac, ca, ad, da, be, cb, bd, db, 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 number- of permutations of n different tilings taken r at a time. Let the different things be represented by the letters a, by 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 ^ — I letters i — 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 \alues of n and ?', we have in succession „^A-, = (" - 1) X ..-A-.. But „.,^,P, = (» - r + 1). Multiply all the corresponding members of the above equalities, and cancel all the common factors; we then have ^P^ = n(n-l){n-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 n (?i — 1) (r? - 2) . . . 2 . 1 is denoted by the symbol [?i or by ?i! The symbols |_?i and n ! are read ' factorial n.' The continued product of the r quantities r?, n — 1, n — 2, (n — r+1), n not being necessarily an integer in this case, is denoted by n^. Thus n^ = n{n — \){n — 2). Hence we have „P„ = I ^ , and P^ = n. n n I ' n r r 242. To find the 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 2^ 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. 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 etters. 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 r... of permutations would be P x p x I ^ x But the number of permutations all together of n different things is I n. Hence P x |_p x q x I • • • n; P n \pW Ex.1. Ex. 2. Ex. 3. Ex. 4. Ex. 5. Find fiP,, .P, and ^P. 7"- 7' Shew that xqP^^^P-j. If„P5 = 12x„P3, finclw. If2„P3=100x„P2, findn. Ans. 120, 120, 5040. Ans. 7. Ans. 13. Ans. 8. Ex. 6. Find the number of permutations of all the letters of each of the words acaciUy hannali, success and mississippi. Ans. 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, ^ [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? Ans. 144. COMBINATIONS. 283 Ex. 9. The number of pciTautations of n things all together in wliich 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 7i books can be arranged on a shelf so that two particular books shall not be together is {?i - 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 rec[uired = nxnxnx ... — u''. COxMBINATIONS. 243. Definition. The different 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 combinations of the n things r at a time. Thus the different combinations of the letters a, h, c, d three at a time are ahc, abd, 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 „_iC'^_i times, and therefore the total number of letters is ?i x „_fi^_^ ; but, since there are r letters in each combination, the total number of the letters must be r x C_. 284 COMBINATIONS. Hence rx^G^ = nx ^.fir-x* Since the above relation is true for all values of n and of r, we have in succession (^ - 1) X, .-.(?.-. = (»-l)x„-A-., (>• - 2) X ._,C,., = (« - 2) X a^ n-3 r-3' 2 X _^,(7, = (71 - r 4- 2) X „i,^,C^. Also ^_^.^^a^ = ^_r+l. Hence, by multiplying corresponding members of th( above equations and cancelling the common factors, we have \r X J0^=n{n- l){n-^) (?i-r + 1), thatis„a = -^ / ^ '- = - (i). By multiplying the numerator and denominator of the fraction on the right by In — r, we have n{n-l){n- 2). . .{n - r + 1) x fflr — n — r n — r \n (ii)- Ir \n — r By comparing the above result with that obtained in Art. 242, it will be seen that P, = (7, x in The relation ^P^ = ,,Cv 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 „(7„ = 1. We should also obtain the result |0 = 1 by putting n = 1 in the relation \n = n \ n— 1. COMBINATIONS. 285 245. Theorem. The numhdr of combinations of n differ eat tilings r together is equal to the number of i'ne 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. \n \n For _(7= — ^ — .and„a = "- n r ^1^ «> 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^, y^C^ and aoCi?- ^ns. 210, 220, 1140. Ex.2. If „C8 = „(7i2,findnCi6- ^rw. 153. Ex. 3. Find n, having given that ^Cg = ^Cj . Ans. 11. Ex. 4. Find n, having given that 3 x„C4 = 5 x^.^Cj. Ans, 10. Ex.6, Find 71, having given that ,j (74 = 210. Ans. 10. Ex. 6. Find n and r having given that „P,.= 272 and „C,. = 136. Am. n = n, r = 2. Ex. 7. Find n and r having given „C^-i : „C^ : n^r+i "• 2 : 3 : 4. Ans. n=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 gCj=20 ways; the vowels can be chosen in 4(72 = ways; hence 20 x different sets of letters can be chosen, and each of these sets can be arranged in 5/^5=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 shilling and a sixpence ? Number required = 6<^i + 6^2 + 6^'3 + 6^4 + 6^6 + 6^6 = 6^- 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 An 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 msmj of the numbers so formed wiU be greater than 23000? Ans. 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. In how many ways may an elector vote? Ans. 14. 246. Greatest value of „G^. To find the greatest value of ^C^ for a given value of n. From the formulae of Art. 244 we have n-r + 1 Hence ^p^ = „C,._j, according asTi — r + l=r; that is, according as r= ^(n+1). Thus the number of combination of n things r together increases with r so long as r is less than | (n + 1). n If then n be even, „(7^ is greatest when ^ = h • If n be odd, „0, > JJ,_, as r > i (7t + 1), and ,fl, = JJ,_^ when r = J (n + 1). Thus, when n is odd, n^^i(n-i) "^ fPh(n+\) ^^^ thcse are the greatest values of „(7^. COMBINATIONS. 2 7i. 249. Vandermonde's Theorem. From the last Article, if a>, y and n be any positive integers such that ic + 2/ is greater than w, we have, since ^fl^ = i~ > I n-r J r i i Jn • • • "P . I ~r • • • I I • n— r \r_ \n Multiply each side of the last equation by \ny and we have , . n(n — l) /^ (^ + y)n = ^„ + ^^n-l2/i + \ 2 "^"-^^^ "^ ■ " \n + , — = — X y + +V,.. HOMOGENEOUS rilOJ)UC'JS. 289 The above has been proved ou the supposition that x and y are positive integers such that x ■\-y '\$> greater than n; 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 whicli 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 X, and thus for more than n values ; and hence it must be true for that value of y and any value luhatever of x. Hence the proposition is true for any particular value whatever of x, and for more than n values of 2/ ; it must therefore be true for all values of x and for all values of y. This proves Vandermonde's theorem, namely : — If n he any positive integer, and x, y have a.iy values whatever; then will 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 „H^. For example, the homogeneous products of two dimensions formed by the three letters a, b, c are a^, t-, c'-', be, ca, ah. To find. II r- 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 S. A. 19 290 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 „-ff^_i ; and, by the above, the number of a's they contain is T ~~ 1 X JH^^. Hence, taking into account the a which is a factor of each of the „^,._i terms, the total number of a's which occur in all the ?^-dimensional products of the n letters is nHr-l + -^ X nHr-u that IS ^^— ^H,_,. Hence equating the two expressions for the number of a's, we have r jj. _ n + r — l ^ Since the above relation is true for all values of n and r, we have in succession If — ^ + ^~^ ff Also JH^ is obviously equal to n. EXAMPLES. 291 Hence, by multiplying and cancelling coniruon factors, we have ^ «('' + l)^ (n + r-l ) ^,^^ ^^^ 1.2 r u — r Ex. 1. Find the number of combinations three at a time of the letters fl, b, 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's, G c's and 6 d's. Ans. 84. Ex.3. Shew that rJIr=n-l^r + Jh-l^ and deduce that n"r — n "r-l + n-l"-r-\ + n-2"^r-l + + l^^r-l • Ex. 4. Shew that 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 jnn^m 5 ^^^^j whatever set is given to the first, the second set can be given in fnn-m^m "^^-js ; so also, whatever sets are given to the first and second, the thiid set can be given in mn-zm^m '^^ays; and so on. Hence the required number is mn^m ^ min—l) ^m ^ m(n-2) ^m ^ ^ 2m^w» -^ m^m \mn \m(n- 1) lm(;i-2) j2m \m \m \m{ii- 1) jm \ m(7i-2) \m \ m (n-'d) [w^ | m \m {mn Ex. 2. Prove that Since „(;,.„//, = " (^L:ili^--i!Llijd) . nini-l) {n-,r-l) n r n r ^ j^ _ 7i'^{7i'-l-) {n-^-r-P) P. 2-^ r* ' 19—2 292 EXAMPLES, we have to prove that _ n^ n2 (n2 - 12) _ n" {n^ - 1^) {n^ - 2^) V'^ 12.22 12.22.32 "^ -,.„. ^r(7l^-l2)...(n2-n-l2) _ "^^ ^ 12.22...712 71^ — 12 Now the first two terms = - 12 ' (n2-12)(7l2-22) — 12.22 {n2-12)(n2-22)(n2. -0) .' three = + lOUl = 12 02 32 and so on. Hence the sum of all the terms on the left _r .. Jn^-l'){n'-2^) (n2-n2) _ ~^ ' 12.22 7i2 Ex. 3. Shew that n straight lines, no two of which are parallel and no three of which meet in a point, divide a plane into ^11 {n + 1) + 1 parts. The 7ith 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 ?ith line which would become one part if the ?ith 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 F{n)=F{n-l)+n. Similarly F {7i-l) = F {n-2) + (n - 1), and F{2) = F{l) + 2. But obviously F (1) = 2. Hence 2^(n) = 2 + 2 + 3 + 4+ -^-n = l + hn{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 Urst. Let the different things be represented by the letters a, b, c, k, I. Suppose that a is taken first. Then, if either of the two letters next but one to a be taken second, any one of ri - 5 letters can be taken for the third of the set. If, however, the second letter is not nest but one to a, but in either of the n-5 other possible places, there would be a choice of n - 6 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 (?i-5) + (n-5) (/i-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) (?i - 5). 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 I, and there are n - 4 of these. Hence the number in the first case is ^n {n- i) {n- 5) + {n- 4:) = l [n -2) {n-d){n-i). Ex. 5. There are n letters and n directed envelopes : in how many ways could all the letters be put into the wrong envelopes ? Let the letters be denoted by the letters a, h, c... and the corre- Bjponding envelopes by a', h', c\ Let F {n) be the required number of ways. Then a can be put into any one of the ti- 1 envelopes h', c\ Suppose a is put into A;' ; then U may be put into a', in which case there will be i^(n-2) ways of putting all the others wrong. Also if a is put into /c', the number of ways of disposing of the letters so that fc is not put in a', h not in 6', &c. is 2*^(71-1). Hence the number of ways of satisfying the conditions when a is put into A:' is F (n-V)-^F [n-2). The same is true when a is put into any other of the envelopes 6', c', . • • Hence we have J?'(7i) = (n-l){F(n-l) + i^(n-2)}; P{n)-nF{n-\)= - { J'' (n- 1) - (w- l)F(7i- 2)}. Similarly F (n - 1) - (n - 1) i*' (?i - 2) = - {F {n -2)-{n-2)F^n-o)\ F(3)-3i^(2)=-{F(2)-2i''(l)}. But obviously F (2) = 1 and F (1) = ; F(^n)-nF{n-V) = {-\Y\ „ Fin) i<'(7i-l) , ,^ 1 Hence -^ -'- ^^ r-^= - 1)" . — . 294 EXAMPLES. Similarly -j^ - -^^= ( - l)-i ^^. . F{2) F(l) . ..gl Hence, by addition, _^ — ^ ^W-IZi||2 |3+|4 •••+ 1^ / the number required. EXAMPLES XXIII. 1. In how many different ways may twenty different things be divided among five persons so that each may have four ? 2. 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„ : „ C is equal to 1.3. 5 (4n-l) {1.3. 5 {2n-l)f' 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 tilings 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 tliree are on a stiaight line except m which are all on a straight line. Find the number of triangles formed by joining the points. 10. Shew that the number of different w-sided polygons formed by n straight lines in a plane, no three of which meet in a point, is ^ jw — 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 2>m 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. corner to the S.E. corner, 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 \2n 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 ^ (m + iy{m + 2f. ■ 17. Find the number of ways in which 2^ 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 + 1 places, there being no restriction as to the number in each place, is (m + n) l/ml n\ 296 EXAMPLES. 19. Shew that 2)i 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. '•o^ 21. Shew that n planes through the centre of a sphere, no three of wliich pass through the same diameter, will divide the surface of the sphere into 71^ — 71 + 2 parts. 22. Shew that the number of parts into which an infinite plane is divided hy m + n straight lines, m of which pass through one point and the remaining n through another, is 7nn + 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 hj7n + 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 + h + c-\- ... planes through the centre, a of the planes passing through one given diameter, h 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 ^ (n^ + 5?^ + 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 J?mi (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 y^n (n—l) {n— 2) (n — 3) {n - 4) {2n - 1 ) 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. 253. Binomial Theorem. Suppose that we have n factors each of which is a + h. If we take a letter from each of the factors of (a -}- 6) (a + 6) (a 4- 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 terms 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 h can be taken once, and a the remaining (n — 1) times, and the number of ways in which one h can be taken is the number of ways of taking 1 out of n things, so that the number is „0j: hence we have .C,.a»-'6. 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 (7„ : hence we have And, in general, b 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 -f 6) (a + 6) (a + 6) to n factors ^a'' + ^G, . a'^-'b + ,a^. a^'-W +......+ ^G,. a^'-'b'- -h ... the last term being G a''~"6'*, i.e. b\ Hence, when n is any positive integer, we have The above formula is called the Binomial Theorem. If we substitute the known values [see Art. 244] of „(7j, ^Cjj, jjCg,... in the series on the right, we obtain the form in which the theorem is usually given, namely {a + bf = a" + na''-' b + !i^i:^) a'^'^b' + . . . \r \n — r The series on the right is called the expansion of {a + 6)". 254. Proof by Induction. The Binomial Theorem may also be proved by induction, as follows. BINOMIAL THEOREM. 299 We have to prove that, when n is any positive integer, \n ■i--r-r= — a""- 6'-+... + 6", \r \n —r or that (a + 6)" = a" + „(7,a"-^ 6 + „CX"'i' + . . . + .a^-'-^'- + . . . + Z>". 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 + hy^' = a-^ + (1 + ^fi^) a% + ^(7, + „(7,) cr^ W^... Now l^^fl^ = l + n = ^^,fi,, «^i + «^ - ^ + ^ 2 "" 1 2 — ~ «+i 2' and, in general, „6;., + „C, = „,,a, [Art. 217]. Hence Thus zy 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 rz = 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 n. Ex.1. Expand (a + Z>)^. We have / v.j 1 ^ ^, 4.3 „,„ 4.3.2 ,„ 4.3.2.1, (a + !,)-a« + 4a3i,+ j^2 '='''-+1:23 "''■'+1-2X4''* ^ni a^ + 4a^6 + Qa^b- + Anlfi + 6*. 800 GENERAL TERM. Ex.2. Expand (2a; - ?/)3. Put 2x for a, and - ?/ for 6 in the general formula : then {2x -yY= {2xf + 3 {2xf ( - 2/) + 1;| {2a;) {-yf^ |^ ( - y^ = 8x^-12x'^y + 6xy^-7f. Ex. 3. Expand {a - Z>)". Change the sign of b in the general formula ; then we have (a - &)« = a« + na»-i ( - 1)+^^^-^ a'^-^ {-h)"+ J. . a \n n 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 \n -r-h — «""'■^'■• \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 from the beginning and the end are equal. In the expansion of (a + 6)" by the Binomial Theorem, the (7' + l)th term from the beginning and the (r + i)th term from the end are respectively „a,.a"-'7>'- and „0_ . «'-6'^"'' But ' „a-na-.. [Art. 245.] GREATEST TERM. 301 Hence, in the expansion of (a + 6)", the coefficients of any two terms equidistant respectively from the beginning and the end are equal. TJiis result follows, however, at once from the fact that (a 4- by, and therefore also its expansion, would be unaltered by an interchange of the letters a and b ; and hence the co- etficient of a"~*'6'' must be equal to the coefficient of b"'^ a^. 257. If, in the formula of Art. 253, we put a = 1 and b = X, we have /. Nn -. n(n — l)^ W „ „ (1 -\-xy = l+nx-}- \ ^ a.-+... + -^— ^ of + ... +x^. 1.2 r n — 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 + 6)" by means of it, we have «.H-i)-.(a(l4g)-..-(l + ^)- = a" + na''-'h + ""^f"^^ a^'^P + . . . 1 . 2 258. G-reatest term of a binomial expansion. In the expansion of (1 + xY, the (?- + l)th term is formed from the rth by multiplying by x. Now X = ( 1 ) ^, and clearly diminishes as r increases ; hence x diminishes r ji ^« _j_ ]^ as r is increased. If x be less than 1 fur any 302 GREATEST COEFFICIENT. value of r, the {r + l)th term will be less than the rtb. In order therefore that the rth term of the expansion may be the greatest we must have n — r + 1 ^ ,w — r— i + 1 a; < 1, and ^ > 1. r ?'— 1 Hence r > — , and r < i — 1- 1. x+1 x^-l The absolute values of the terms in the expansion of (1 + «)" will not be altered by changing the sign of x ; and hence the rth term of (1 — xy will also be greatest in absolute magnitude if r> (n+l) X , (?? + 1) ^' ^ . -, , and r < '—^ + 1. If r = ^^ 7— , then x=\', and hence there 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 + a;)" = a»»( 1 + - j , tlie rth term of (a + xY is the greatest when (n + 1)- (n + 1)- r> and < +1. + 1 - + 1 a a Ex. 1. Find the greatest term in the expansion of (1 + a;)-'', when 1 21 21 The rth term is the greatest, if r> — and r 5 and < G. Thus there is no one term which is the greatest, but the 5th and 6th terms of the EXAMPLES. 308 expansion arc equal to one another and greater than any of tlie other terms. Ex. 3. Find the greatest term in the expansion of (10 + 3x)i5 when x = 4. Ans. The ninth term. The greatest coefficient of a binomial expansion can be found in a similar manner. For in the expansion of (1 ± xy the coefficient of the (r + l)th term is formed from 71 — 7* -|- 1 that of the rth by multiplying by ± ; . Hence the rth coefficient will be the greatest in absolute magnitude, . ,. n - ?• + 1 T n-r -l + l if < 1 and > 1. r r — 1 That IS if r > — r— and < 1 H ^r— . 2 2 Hence when n is even, the coefficient of the rth term is the greatest when 7- = - + 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 + a:)2o the coefficient of the 11th term is the greatest; and in (l + a;)^! the coefficients of the 6th and 7th terms are greater than any of the others. EXAMPLES XXIV. Write out the following expansions : 1. (rc + a)\ 2. (•2a-£c)^ 3. {\-x-)\ 4. (2a- 3^7. 5. (2a;'-3)\ 6. ix' - 2y^) 7. Find the third term of (x > 3?/)'°. 8. Find the fifth term of (3a; - 4)=^*'. 9. Find the twenty-first term of (2-a;)^'. 10 Find the fortieth term of [x — y) 3\5 V42 304 EXAMPLES. 11. Find the middle term of (1 +xY. 12. Find the middle terms of (1 + x)'\ 13. Find the general term of {x — ^y)". 14. Find the general term of (x^ + yy. 15. Write down the first three terms and the last three terms of (3x — 2yy\ 16. Find the term of (1 + xy^ which has the greatest coefficient. 17. Find the two terms of (1 +xy^ which have the greatest coefficients. 18. Shew that the coefficient of x" in the expansion of (1 + xY" is double the coefficient of re" in the expansion of (i + xy-K 19. Shew that the middle term of (1 + a;)"" is 1.3.5 ...(2n-l)^„^„ I ^ X , [n 20. Employ the binomial theorem to find 99*, 51* and 999^ 21. Shew that the coefficient of x'' in the expansion of (^^x) 1\" \^ is I ijn + r) i{'i-r) ' 22. Find the middle term of (-S" 23. The coefficients of the 5th, 6th and 7th terms of the expansion of (1 +x)" 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 +x)" in arith- metical progression 1 ~^ 25. If a hti the sum of the odd terms and b the sum of the even terms of the expansion of (1 + xY, shew that {l-xy = a'-b\ PROPERTIES OF THE COEFFICIENTS. .^05 259. Properties of the coefficients of a binomial expansion. It will be convenient to write the Binomial Theorem in the form (1 + ^y = Co + c^x + c^x^ + . . . + cX + • • • c,,-^" (i), where, as we have seen, c,, = c„ = 1 ; c^ = c„.i = n; 'n n —r and, in greneral, c, — c„_, = — I. Put x=l in (i) ; then 2'* = Co + Cj + C2+ ... + c„. Thus the sum of the coefficients in the expansion of (1 + xY is 2\ II. Put A' = — 1 in (i) ; then (i-iy = c,-c,-\-c,- + (-!)" c„; .-. = (Co + C.3 + C4 +...)- (Cj + C3 + Cg + .. .). Thus the sum of the coefficients of the odd terms of a binomial expansion is equal to the sum of the coefficients of the even terms. III. Since c^ = c,^_^, we have (1 + xY = Co + c^x + c.,a;' + . . . + c,./ + . . . + c^x'\ and (1 -f- xY = c„ + c„_^x + c„.^'^ + . . . c„_X + • • • + ^o-^" The coefficient of x" in the product of the two series on the right is ecjual to Co+o' + c^'-h +c„l Hence [Art. 91] c^' + c^"' + . . . + c/ + + c,^ is equal to the coefficient of x" in (1 + x)" x (1 +xy, that \'2u is in (1 +x)'": and this coefficient is — ^- • S. A. 20 306 PROPERTIES OF THE COEFFICIENTS. Hence the sum of the squares of the coefficients in the expansion of (1 + xX is -==~ . \n\n IV. As in III, we have (1 -f xf = Co + c^x + c^sc^ + . . . + cX> ■ and (1 — xy = c^- c^.^x + c„_2«' +...+(— 1)%^^ The coefficient of x"* in the product of the two series on the right is equal to (- 1)" {c/ - C.= + 0/ - +(-l)Vl- The coefficient of x"" in (1 + xy x (1 — x^, that is in (1 — xy\ is zero if n be odd, and is equal to (— l)^ n~|jr In ieru ii n ue uuu, ciiiu is tjqucii ijU ^ — x^ . py LI. if 71 be even. Hence c/ - c^^ + c/ - . . . + (- lyc^^ is zero or (— iyn\/(^n\y, according as n is odd or even. Ex. 1. Shew that Cj + 2c2 + Scg + . . . + rcy + . . . + 7ic„ = ?z2"-i. We have Cj + 2c2 + Scg + . . . + wc„ w .n(n-l) ^n(n-l)(n-2) ._ = n + 2 \ ^ +3 ^ '\ + + r— ^ + ...+11 1.2 1.2.3 |£ \ n-r =Ji + („.i) + (ii^H^2) ^^^^^ ^4 ( ' 1.2 \r- 1 \ n-r j = n(l + l)«-i = «2"-i. Ex.2. ShesYthat Co-^c^ + ^c^- + (-ir^ = ^l- We have Co-~c^ + -C2-&G.=l--n+- —,^ ore. A I" n + 1 PROPERTIES OF THE COEFFICIENTS. 307 {n + l)n (n + l)n(n-l) | = 4,-4-,|l-0^ + l) + ^^-^^^^l:^ + + (-lH- 7i + 1 7i + 1 I 1.2 1 . z . d ) _ _i L- (1 _ l)n+l — ^ n+1 n+1 n+1' Ex. 3. Shew that, if n be any positive integer, rc x + 1 x+2 x + n x {x + l)...{x + n) Assume that ^ r' r n |w x~ x+1 X + 2 a + n a; (a: + l)...(a; + ;0' for all values of x, and for any particular value of n. Change x into x + 1; then X + 1 x + 2 x + '6 ^ ' x + n + 1 \!L {x + 1) {x + 2). ..{x + n + 1)' Hence, by subtraction, X X + 1 X + 2 x + r 1 lw + l ^ ' x + n + 1 x{x + l) {x + n + 1)' Eut n^r + nC'r-i = n+iCr> ^or all values of r [Art. 247]. Hence we have ^ r f c \n+l _ _ "+1^1 _L. »±1_2_ I ( pn+l n+l^n+ l ' X x + 1 x + 2 x+n+1 x{x + l) ...{x + n+iy Hence if the theorem be true for any particular value of n it will be true for the next gi'eater 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 Cf^^c-^, &c. For example : Put x = l\ then we have t-'o _ ^1 I ^2 _ _ _^_ 1 2 "^"3 ~7t + l' 20—2 308 PROPERTIES OF THE COEFFICIENTS. 1 c c p 2" In Put .=1 ; then 'I - f + I - = OCTTIj • Ex. 4. Shew that Cfltt - Ci (a - 1) + Cg (a - 2) - Cg (a - 3) + + ( - l)«c^ (a - ?i) = 0, and that Coa2-Ci(a-l)2 + c2(a-2)2-c3(a-3)24- + (-!)« c„ (a- n)2=0. We have from II., if n be any positive integer, i-„+_L_J_^_^i__!+ + (-i)«=o (.). Hence, if n > 1 i-(-i)+ '"-^i'.^^' - + (-i)»-=o (ii). Multiply (i) by a and (ii) by n and add ; then a_w(a-l) + ^i^^(a-2)- + (_ i)u(a_,i) = (iii), where 7i is > 1. Change a into a - 1 and n into n - 1 in (iii) ; then, n being > 2, we have a_l-(7i-l)(a-2) + + (-!)«-! (a -w) = (iv). Now multiply (iii) by a and (iv) by ?i and add; then a^-n{a-l)^ + -^-~^^{a-2y-- + ( - l)"(a-w)2=0, J. . ^ provided n is greater than 2. By proceeding in this way we may prove that av-n{a-l)p + '"^^^^^{a-2)p- + (- 1)« (a-«)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-\-h, x -\- c, &c. It will be conveuient to use the following notation : 8^ is written for a4 6 + c+..., the sum of all the letters taken one at a time. >S.^ is written for ah + ac+ ..., the sum of all the products which can be obtained by PRODUCT OF BINOMIAL FACTORS. 309 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 x"^ is a term of the continued product. We can take any one of the letters a, 6, c..., and x from all the remaining n—1 binomial factors ; we thus have the terms aa;""\ 6a;""\ c^"'\ &c., and on the whole S^ . x''-\ Again, we can take any tivo of the letters a, h, c..., and X from all the remaining n — 2 binomial factors ; we thus have the terms ahx''~'^, aca^'"^, &c., and on the whole 6;.a;"-^ 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^ .a;""''. Hence {x ■\- a) (x -\-h) {x -V c) the last term being ahcd , the product of all the letters a, 6, c, d, &c. By changing the signs of a, 6, c, &c., the signs of iSfj, ^Sg, /Sg.. &c. will be changed, but the signs of ^S*^, S^, /Sg, &c. will be unaltered. Hence we have {x — a){x — })){x — c) = a;"- >Sf,.a;'*-'+ >S;.a;"--- ...+ (- l)'■>S;.a;"-^.. + (- l)"a6cc^... 310 vandermonde's theorem. 261. Vandermonde's Theorem. The following proof of Vandermonde'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 h have any values whatever ; then will {a + b\ = a„ + na^__J)^ + ^^^ ^ a^^J)^ -f . . . n r m — r Assume the theorem to be true for any particular value of n. Multiply the left side hy a-{-h—n\ it will then become {a + 6X+1. Multiply the successive terms of the series on the right also by a + h —n but arranged as follows : — for the first term {{a — n) + h]', for the second {{a — n + l) + (h - 1)} ; and for the rth [(a — n-\-r — l)-\-(h — r + 1)}. We shall then have {a + h\,, = a„ {{a - ti) + 6} + ^fi\ . a,_,\ {(a-n + l) + (b -1)} + nG,.a„_,h^{{a-n + 2) + (h-2)} + ... + nCr-l • Cin-r^^K-l {(a - Tl + T - 1) + (b - V + 1)} + „a . a^_X {(a-n + r) + (b- r)} + ... +b^{a + {b-n)}. Now a^ {(a -n)-\-b}= a,^+, + a„ 6, , „(7, . a„.j6, {(a - 71 + 1) + (6 - 1)1 = rflx (^n ^x + ^n-1^2)» n(^r-l • a«-r+l^r-l [(ci - U + T - V) + Q) - V + 1)} JJ^ . a„_,6^ {(a - ?i + r) + (6 - r)) = „(7, (a„_,+i6^ + (in-rK+i) * Messenger of Mathematics, Vol. v. MULTINOMIAL THEOREM. 311 Hence (a + b\^, = a,,^^ + (1 + „C,) a,fi^ + . . . • • • + LGr-l + u(^r) (^n+l-A + . . . + 6,+ = ^n+1 + n+fil^J\ + • • • + „+iC^r<^,H-l-r^r + • • • + ^.+1> since „(7^_i + ,.CV = „+i6V. Thus, i/" the theorem be true for auy particular value of n, it will also be true for the next greater value. But it is obviously true when n = 1 ; it must therefore be true when ?i = 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 + c^ + ...)", that is of {a +(6 + c +cZ+ ...)]", by the Binomial Theorem is In Similarly the general term in the expansion of by the Binomial Theorem is \n — r s n—r—s The general term in the expansion of (c + (^ + . . .)" '' 'by the Binomial Theorem is In — ■? — s & (d + . . .y-'-'-'. t n — r — s — t Hence the general term in the expansion of is In In — r In — r — 5 - — X i— rin— r 5 X n — r — s \t n — r — s—t . , , Ct c • • •• 312 MULTINOMTAL THEOREM. \n that is ■; — r~- — a** h' c* where each of r, s, t ... is zero or a positive integer, and r-{-s-\-t + ... =71. The above result can however be at once obtained by the method of Art. 253, as follows. We know [Art. 67] that the continued product (a + 6 + c+ ...) (a + 6 + c-f- ...)(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^'b' c* ... will therefore be obtained by taking a from any r of the n factors, which can be done in „(7^ different ways; then taking b 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 „_^,C^ 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 n \n—r \n — r — s \n X " X 7— ' X ... = |V \n — r \s\n-r—s \^\n—r—s-i '" |rls|i...* Hence the general term in the expansion of (a + Z> + c + . . .)" is n \r s\t... EXAMPLES. 313 Ex. 1. Find the coefficient of ahc in the expansion of {a + b + c)'^. The required coefficient = ~ ■ = 6. Ex. 2. Find the coefficients of a^b'^, bcd^ and abed in the expansion of (a + b + c + d)*. We have the terms "^^^' I, n i.T ^^^^ ^^^ ,., ,., .■■ ,., abed. [2^ '|1^[2 Ullllll Thus the required coefficients are 6, 12 and 24 respectively. 263. By the previous Article, the general term of the expansion of (a + 6« -f cx^ + doc^ + )" is n a"- {bxy (caij {dx'f , Ir Is l^lw... In that is r^-b a"6VcZ" a;*-'''+'" Hence to find the coefficient of any particular power of x, say of cc^, in the expansion, we must find all the different sets of positive integral values of r, s, t,... which satisfy the equations s-\-2t + Su+ = (x, 7'-\-s + t + u+ = n. Tiie required coefficient will then be the sum of the coefficients corresponding to each set of values. Ex. 1. Find the coefficient of x^ in the expansion of (1 + 2x + '6x^)*. The general term is -; — : — ;— 2«3'a;'+2< and the terms required are those for which s + 2t = 5 and r + s + 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, s = l, r=l and |4 t = l, s = 'd, r=0, the corresponding coefficients being t- . .^ -2.3^ li \± l£ 814 EXA3IPLES. li and . 23 . 3, that is 216 and 96 respectively. Hence the '_ i_ lA required coefficient is 312. In simple cases the result can be readily obtained by actual expansion. We have {l + 2x + 3x2)4= 1 + 4 (2a; + Sx'^) + 6 (2a; + 3a;2)2 + 4 {2x ■{- 3x^)^ + {2x + Sx"^)-. Only the last two terms will contain x^ and the coefficients of x^ 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 + x + x^)^. Ans. 6. Ex. 3. Find the coefficient of x^ in the expansion of {1 + x + x^)^. Ans. 16. Ex. 4. Find the coefficient of x^ in the expansion of (2 + a; - x^)^. Am. Ex. 5. Find the coefficient of x^^ in the expansion of {7 + x + x^ + x^ + x^ + x^)^. Ans. 39. Ex. 6. Find the coefficient of the middle term of the expansion of {l + x + x^ + x^ + x"^)^. Ans. 381. EXAMPLES XXY. 1. Prove that c„-2c,4-3c,-......+(-l)''(^4-l)c„ = 0. 2. Prove that c,-2c, + 3c3- + (-iy-^nc^=^0. 3. Prove that c„+2cj + 3c,+ + (rj + l)c„ = 2'-^(7i + 2). 4. Prove that c„ + 2c^ + dc^ + +(w-l)c„=l + (n-2)2"-\ 5. Prove that c„ + 3c, + Sc^ 4- + (2n + l)c^ = {n + 1) 2". 6. Prove that 3c, + lc^-¥ IIC3+ + (47^ - 1) c„ = 1 + (2?i- 1) 2". EXAAIPLES. 315 7. Prove that ^ + ^^ + '^ + 12 3 c 2 + — " = 1+1 -1 n+\ n+ i 8. Prove that 1 3 9. Prove that c, c. c. c^ 2" 5 7 ^y ^^ ^s — + — + — + 2 4 6 10. Prove that — + — + — + 2 3^4 11. Prove that + n + 1 * 2"-l n+ 1 ' 1 + 7^2"-^^ n + 2 (?i + l)(ri + 2)' 2 3 ^ -4 +I3_ , - .„ , C 11 1 ^ ' n 1 2 7i 12. Prove that ^0 _ ^ + ^ _ 1 4 7 13. Prove that + (- 1)" 3"|7i CoCr + CxC,^.+ 3?2+l 1.4. 7. ..(371 + 1)* \2n +C G = - , ""' " \n + r \n-r 14. Prove that, if (1 +xy = c^ + c^x + c^x''+ +c„a;% then n(l+ a;)""* = c, + 2c„aj + Scjc" + +nG x"~\ and {1 + (n+ l)a;} (1 +xY-' = c^ + 2cx + + {n+ l)c^x\ Hence prove that, |2n-l c'+2c'+'dcj+ +nc' = n—1 \n—\ and c/ + 2Cj^ + 3c/ + + (n+l)c^' {n + 2) I 2?^ - 1 [?!_ I rt — 1 316 EXAMPLES. 15. Shew, by expanding {(I + x)" - l]"", where m and n are positive integers, that 16. Prove that, if ^ > 3, (i) a-n{a-l) + '^^'^~J'^ (a-2)- +{-!)" {a-n) = 0. (ii) ab-n{a-l){b-l) + ''^^^^^-^{a~2){b-2)- + {-!)" {a-n){b-n) = 0. (iii) abc-n(a-l){b-l){c-l) + '^^!'~^\ a-2){b-2){c-2) + (- 1)" (a-n) (b-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 nth power of x^ + {a + b)x + ab is a"+ G:a"-'b+ C'a"-'b'+ +b\ 19. If n be a positive integer and P^ denote the product of all the coefficients in the expansion of (1 + x)", shew that •P. ui ■ 20. Shew that Q.-xy = {l+x)"-2nx{l-\-x)"-'+ ^"'^^"'~^^ x-(l+xy '- 1 . Z 21. Shew that, if n be a positive integer, , 1 +a; n(n—l) 1 + 2a; 1+nx 1.2 {i + nxy _ n{n-l)(n-2) I + 3x _^ 1.2.3 (JTVixy'^ ~ ^ 22. Shew that {a + b + c + d + ey = ^a' + bta'b + I0%o?b' + lO^a^bo + ^O^a'b'c + QO^a'bcd + 1 lOcibcde. EXAMPLES. 317 23. If (I + x+ x^ = ^0 + ^1^ + ^s^' "•" ,, , n(n-l) (-irk _ prove that a^ - 7ia^_, + — — — a,_, - -i- , ^^_^. a„ = 0, unless r is a multiple of 3. 24. Shew that, in the expansion of (1 +x + x^+ + a;'')", where n is a positive integer, the coefficients of terms equi- distant from the beginning and the end are equal. 25. If a^, a^, a^, be the coefficients in the expansion of (1 + a; + x^y in ascending powers of x, prove that a^ — a^ -k-a^ - + a^^ = a^, and that 26. If (1 + a; 4- x^)" = % + ci^x + a^^ + a^ + . . ., prove that 27. Shew that, in the expansion of {ci^ + a^ + a^ + ... + aj", where /i is a whole number less than r, the coefficient of any term in which none of the quantities a^, a^j ^c- appears more than once is n\ 28. Shew that, if the quantities (1 +ic), (l + cc + .x-'), , (1 + a? + a;^ + +aj") be multiplied together, the coefficients of terms equidistant from the beginning and end will be equal ; and that the sum of all the odd coefficients will be equal to the sum of all the even, each being \{^i + 1) ! 29. Shew that the coefficient of a;" in the expansion of (1 fa: 4- a;-)" is lUn^-X)^ n(n-l)(n-'2){u- 3) n{n-l) {u - •2){7i - 3)(n - 4)(n - 5) 30. Shew that 18 can be made up of 8 odd numbers in 792 different ways, where repetitions are allowed and the order of addition is taken into account. CHAPTER XXL CONVERGENCY AND DIVERGENCY OF SERIES, 264. A series is a succession of quantities which are formed in order according to some definite law. When a series terminates after a certain number of terms it is said to be a finite series, and when there is an endless succession of terms the series is said to be infinite. We have already found that when the common ratio of a geometrical progression is numerically less than unity the sum of n terms will not increase indefinitely, but that the sum will become more and more nearly equal to a fixed finite quantity as n is increased without limit. Thus the sum of an infinite series is not in all cases infinitely great. When the sum of the first n terms of a series tends to a finite limit S, so that the sum can, by sufficiently increasing n, be made to differ from S by less than any assignable quantity, however small, the series is said to be convergent, and S is called its sum. Thus l + J + :| + g + ... is a convergent series whose sum is 2. r^ When the sum of the first n terms of a series increases numerically without limit as n is increased indefinitely, the series is said to be divergent. Thus 1 + 2 + 3 -1- 4 + . . . is a divergent series. CONVERGENOY AND DIVERGENCY. 310 \\^lien the sum of the n first terms of a series does not increase indefinitely as n is increased without limit, and yet does not approach to any determinate limit, the series is neither convergent nor divergent. Such a series is some- times called an indeterminate or a neut7^al series, or the series is said to oscillate. For example, the series 1 — 1 + 1 — 1 + ... is an oscilla- tory series, for the sum of n terms is 1 or according as n is odd or even. It is clear that a series whose terms are all of the same sign cannot be indeterminate, but must either be conver- gent or divergent. For unless the sum of n terms increases without limit as n is increased without limit, there must be some finite limit which the sum can never exceed, but to which it approaches indefinitely near. 265. If each term of a series be finite, and all the terms have the same sign, the series must be divergent. For, if each term be not less than a, the sum of n terms will be not less than na, and no. can be made greater than any finite quantity, however large, by sufficiently increasing n. 266. The successive terms of a series will be denoted by u^yU^yii^,... ; and, since it is impossible to write down all the terms of an infinite series, it is necessary to know how to express the general term, u^, in terms of n. The sum of the n first terms will be denoted by U^ ; and the sum of the whole series, supposed convergent, in which case alone it has a sum, will be denoted by U. Thus U =u^ + u^-\-u^+ ... +u^-[- w„+i + . . ., and U^^ = u^-{-u,^ + u^-\- .,. -\- u^. 267. In order that the series u^, u,^, u^, u^, ,w,^, ^n+i' ^^- ^^y ^^ convergent it is by definition necessary and sufficient that the sum 320 CONVERGENCY AND DIVERGENCY. should converge indefinitely to some finite limit U sls n h indefinitely increased. Hence U , U ^,, U _^^, &c. ... must differ from U, and therefore from one another, by quantities which diminish indefinitely as n is increased without limit. Now ^„+i- ^„ = ^„-.i> U- U,=u^,+u,, + u^ n '^n+l ' "^11+2 ' ""rt+S ' Hence, in order that a series may be convergent, the (n + l)th term must decrease indefinitely as n is increased indefinitely, and also the sum of any number of terms beginning at the (7i+l)th must become less than any assignable quantity, however small, when n is indefinitely increased. For example, the series t + h + o + '-- + -+-" cannot be con- 12 3 n vergent, although the ?ith term diminishes indefinitely as n is increased indefinitely; for the sum of n terms beginning at the (n + l)th is -H -+ ... + 2^- , which is greater than -r- x n, that is, greater n + 1 n + 2 2/1 ° "^ii than ^ . 268. We shall for the present consider series in which all the terms have the same sign ; and as it is clear that the convergency or divergency of such a series does not depend on whether the signs are all positive or all negative, we shall consider all the signs to be positive. The convergency or divergency of series can generally be determined by means of the following theorems. '^^ 269. Theorem I. A series is convergent if all its terms are less than the correspondiiig terms of a second series which is known to he convergent. CONVERGENCY AND DIVERGENCY. 321 Let the two series be respectively TJ=u,-^u^ + u^-\- and F='yj+V2+Vg+ Then, since u^ < v^ for all values of r, it follows that U is less than V. Hence, as V is finite, U must also be finite : this proves the theorem, for a series must be convergent when its sum is finite and all the terms have the same sign. It can be proved in a similar manner that a series is divergent if all its terms are greater than the corresponding terms of a divergent series. Ex. (i). To shew that the series - + -— - + , ^ , + -——_—-+... is 1 1.2 1.2.6 1.2.3.4 convergent. The terms of the series are less than the terms of the series 111 1 T + J— 2 + ■. 2 2 "^ logo + •••' ^^^ *^^^ latter series is a geo- metrical progression whose common ratio is -, which is therefore 2 known to be a convergent series. The given series must therefore also be convergent. Ex. (ii). Shew that the series (a + x) {a + x){2a + x) (a + x) {2a + x){3a + x) (b + x)'^ {b + x){2b + x) '^ {b + x){2b+x){-6b + x) "^ is convergent if a, h and x are all positive, and a<.b. The terms of the given series are less than the corresponding ... a + x ia + xf ^ {a + x)'^ terms of the series r—— + j^ — '- + }— — '- + . . . , b + x {b + xy {b + xf [since — < :; if r > 1, a, 6 and x being 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 o, finite number of them S. A. 21 322 CONVEKGENCY 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. Ex. Shew that the series I + ttt+itt + tt +77; + 77^ +-n^ + ... is con- 4 42 43 44 45 46 |2 "^ 13 "^ J4 "^ |5 "•■ [6 '^n_ vergent. From the sixth term onwards, each term is less than the corre- 45 46 spending term of the series r-r^ + ^yj^ + .... Hence the series beginning at the sixth term is convergent, and therefore the whole series is convergent. 270. Theorem Hi If the i^atio of the corresponding terms of two series he ahuays finite, the series will both be convergent or both divergent. Let the series be respectively U=u^ + u^-\-u^-{- , and V = v^ + v^+v^^ Then, since the quantities are all positive, -^^ must lie between the greatest and least of the fractions — [Art. 118]. Hence tT : F is finite. It therefore follows that if U is finite so also is F, and if U is infinite so also is V. 1 .1, ^ . 8 16 8» For example, the two series ^r— ^ + — — 4- + 2.3 3.4 (u+l)(n + 2) ••• and T + 7; + H ^ '•- ^'^^e both convergent or both divergent. 8r 1 For the ratio of the rth terms, namely -, —-, -r- : - is equal to ^ (r + l)(r + 2) r ^ ; , 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 titan some fixed quantity which is itself less than unity. CONVERGENCY AND DIVERGENCY. 323 Let the ratio of each term after the r^^ to the preceding term be less than k, where k <1. Then, since "^-^ u^, u^_^ > u^_^_^ > u^, &c. Hence it^^^ + ?^^^2 + • • • + ^n+r > ^^r j *li® series must therefore be divergent. 1 2 2^ 2^ 2"~^ Ex. 1. In the series t + k + -:7 + t- + + + > t^^ ^^^^^ 12 3 4 n — "-^ = 5 , which is greater than 1 ; the series is therefore u^ n + 1 divergent. in + 1)2 Ex.2. In the series 1^ + 2-x + 3-a;2 + , the test ratio is ^ — 5-^ a;, that is ( 1 + - ) X. Now, if x be less than 1, and any fixed quantity A; be chosen between x and 1, the test ratio will be less than k for all terms after the first which makes (l + ^)v'x^/^^. 21—2 324 CONVERGENCY AND DIVERGENCY. Hence the series is convergent if a; < 1. If a; = 1 the series is 1^ + 2"^ + 3^ + which is obviously divergent, and if a; > 1 the series is greater than 1^ + 22 + 32+ Thus the series 12 + 22a; + 32a;2+ is divergent except when x is less than unity. 273. When a series is such that after a finite number 77 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 difficult to determine whether a series is convergent or divergent. For example, in the series 1111 ^A; + 2^ "*" 3* "^ 4^ "^ the ratio ^n+l _ ^* ^n (n + iy /i + iy Hence, if Jc be positive, the test ratio is less than unity, but becomes more and more nearly equal to unity as n is increased. We cannot therefore determine from Theorem III. whether the series in question is convergent or divergent. 274. To shew that the series ^ + ^^ + ^ + ... is con- i 1 i 1* "^ 2* "^ 8 vergent when k is greater than unity, and is divergent when 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 : 111 2* "^ S"-' ^ 2* ' CONVEllGENCY AND DIVERGENCY. 825 •" K* + A* "■" 17* ^ Ak » 5 6* ' 7' 1 and 2"* + (2"+ 1)'"^ ••••'•••• + ^2""'' - 1) Hence the whole series is less than 2" i. 1 1 1 1* "^ 2* "^ 4' "^ 8"= "*" 2" that is, less than 1 i "I" «t-l "f" 92 (A; -1) "T" 23(A:-1) "l" "T" Qri(ft-l) "f- •• • • But this latter series is a geometrical progression whose common ratio, — ^ , is less than unity, since A; > 1. Hence the given series is convergent. Next, let h = \\ then we can group the series as follows : 1+2 + 1 1 3 + 4 + + 2"-^ + 1 1 1 1 r 5+6+7+8 + )n-l + 2 + + . + 2'* + therefore, as each group of terms in brackets is greater than \, the given series taken to 2" terms is greater than 1+^ + 4 + ^- + taken to ti + 1 terms, that is, greater than 1 + \n, which increases indefinitely with n. XT 111 Hence- + 2 + 3 + is divergent. Lastly, let k be less than unity; then each term of the 11. series -i-+-Q* + is greater than the corresponding term of the divergent series - + - + ; the series is therefore divergent when k < 1. 326 CONVERGENCY AND DIVERGENCY. 275. The convergency or divergency of many series can be determined by means of Theorems I. and IL, 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 —^ — = convergent or divergent ? Since —5 — - > - , if w > 1, it follows that S — rA- > S - . But S - n^ + 1 n 7r + l w n 2m is divergent ; therefore S -^ — r- is also divergent. 7r + 1 71 + 2 Ex. 2. Is the series whose general term is — — - convergent or 71*^+ 1 divergent ? .^ w + 2 w + 2 3/1 3 ^ „w + 2 _^ 1 T.^^1 Now -^ — r- < — 3- < -^ < — , . Hence S -^ — =• < 3S -^ . But S -5 n^ + 1 n-* 7i"* 71- n^ + 1 71^ 9i2 71 + 2 is convergent [Art. 274]; therefore S-^ — r- is also convergent. 276. We have hitherto supposed that the terms of the series whose convergency or divergency was to be deter- mined were all of the same sign. When, however, some terms are positive and others negative, we first see whether the series which would be obtained by making all the signs positive is convergent; and, if this is the case, it follows that the given series is also convergent ; for a con- vergent series, all of whose terms are positive, would clearly remain convergent when the signs of some of its terms were changed. If, however, the series obtained by making all the signs positive is a divergent series it does not necessarily follow that the given series is divergent. For example, it will be proved in the next Article that the series j — i + i—i+... is convergent, although the series T + i + 3+i + --- is divergent. A series which would be convergent if all the terms had the same sign is called an absolutely convergent series. 277. Many series whose terms are alternately positive and negative are at once seen to be convergent by means of CONVERGENCY AND DIVERGENCY. • 327 Theorem V. A series is convergent token its terms are alternately positive and negative, provided each term is less than the preceding, and that the terms decrease without limit in absolute magnitude. Let the series be ^1 - '"'2 + ^^3 - ^ + • • • ± % + ^^n,i ± ^i„+2 + By writiDg the series in the forms Wj - 1*2 + (lis ~ '^0 + (^^5 - '^e) + "•> and u^ — {u^-u^)-(u^-%)- , we see that, since each term is less than the preceding, the sum of the series must be intermediate to u^ — u^ and u^ ; and hence the sum of the series infinite. It is also similarly clear that the absolute value oi U — U^ is intermediate to the absolute values of u^+i—iin+2 ^^^ "^^n+i* ^'^^ therefore U— U^ becomes indefinitely small when n is increased without limit. The series must therefore be convergent. For example, the series T-9 + 0-7+ i^ convergent, since the terms are alternately positive and negative and decrease without 2 3 4 5 limit. The series t-o + q-t+ is not however a convergent i. Z O 4: series although its sum is a finite quantity between - and 2, for the rith term, namely , does not diminish indefinitely as n is 74 indefinitely increased. 278. We will now apply the preceding tests of con- vergency to three series of very great importance. I. The Binomial Series. In the binomial series, namely m (m — 1) o 1 .2 m (m — 1) ... (m — M + 1) „ \n 328 CONVERGENCY AND DIVERGENCY. the number of terms is finite when m is a positive integer ; but when m is not a positive integer no one of the factors m, m— 1, m — 2, &c. can be zero, and hence the series must be endless. To determine the convergency of the series when m is not a positive integer we must consider the ratio TVT '^«+i m — n + 1 (^ m + l\ ^i,,., : u^. JNow — ^^^ = x =^ — x \\ . *^ u^ n \ n J Hence, for all values of n greater than m + 1, w,,^^, and u^ have different signs when x is positive, and have the same sign when x is negative. Moreover, as n is in- creased, the absolute value of u^_^Ju^ becomes more and more nearly equal to x. If therefore x be numerically less than unity, the ratio u^+Ju^ will, either from the beginning, or after a finite number of terms, be numeri- cally less than unity. Hence by Art. 271 the series formed by adding the absolute values of the successive terms will be convergent, and therefore also the series itself must be convergent, whether its terms have all the same sign or are alternately positive and negative. Thus the binomial series is convergent, if x is numeri- cally less than unity*. II. The Exponential Series. In tl series, namely x'^ x^ c ^+^ + -2 + |3+ + ■ rf n'^'"' the ratio u.Jit is xln. Hence the ratio u _^Ju is nu- n+V n I ^ n+1/ n merically less than unity for all terms after the first for which n is numerically greater than x. The series is therefore convergent for all values of x. * The series is also convergent when x — 1, provided 7i > - 1; and it is convergent when .1;= - 1, provided n > 0. [See Art. 838.] CONVERGENCY AND DIVERGENCY. 329 III. The Logarithmic Series. In the logarithmic series, namely ^-2 + 3- -(-1) « + ■••' the ratio u^^Ju is ^ = — a; ( 1 ] ; and hence 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 — 1 and + 1. If x=l, the series becomes 1— -l^ + J— ..., which is convergent by Theorem V. If a; = — 1, the series becomes — (1 +4" + J+ ...)> which is known to be divergent. [Art. 274.] 279. The condition for the convergency of the product of an infinite number of factors, and also some other theorems in convergency, will be proved in a subsequent chapter. [See Art. 337.] The two important theorems which follow Ccinnot however be deferred. 280. If the two series TJ = u^ + u^x + u^x^ + + UJJC'' + . . . , and V = Vq-^v^x + v^x^ + + v"+ ..., be both convergent, and the third series 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 U x V, provided (1) that the series U and V have all their terms positive, or (2) that the series U and V would uot 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. 330 CONVERGENCY AND DIVERGENCY. First, suppose that all the terms in TJ and V are positive. Then U^^ x V^^ = P^^ + terms containing x^" and higher powers of x. Hence U^^ x V^^ > P^ Also P^„ = CT X F, + other terms. Hence P^„ >U^xV^ 2» Hence P is intermediate to C/^ x F„ and CT" x V Now, the series C/'and F being convergent, TI^^ and £/"„ both approach indefinitely near to U, also Fg^ and F„ both approach indefinitely near to F, when n is indefinitely increased. Hence U^^ x Fg^ and U^x F„, and therefore also Pj^ which is intermediate to them, will in the limit be equal to U xV. Hence, when all the terms are positive, F=UxV. 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 U' and V in the same way as P is formed from U and F. Then U^^ x V^^ — 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^^y and therefore also JJ^n 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*. 281. If the two series a^ + a^x + a^x^ + a^x^ + , and hQ-\-h^x-\-\x'^ -\-h^x^ + , be equal to one another for all values of x for which they are convergent ; then will a^ = 60, a^ = 6^, a^ = b^, Sac. For if the series are both convergent, their difference will be convergent. Hence for all values of x for which the series is convergent. The last series is clearly convergent when x = ; and putting a; = we have a^^ — b^^ 0. Hence a^ = h^. We now have x{a,-\^{a^^-h.^x+{a^-\)x'+ } = (ii). Now for any value x^ for which the series in (i) is convergent, ftg ~ ^2 + (^3 "~ ^3) ^1 + ^^ equal to a finite limit, Xj suppose. Hence (ii) may be written x^ [a^ —h^ + x^ ij = ; and, since this is true for all values of x^, however small, it follows that a^ — h^, must be numericnlly indefinitely small compared with L^\ that is, cb^—h^ must be zero. It can now be proved in a similar manner that a.^ — 63 = 0, «3 - ^ = ^^ ^c. Hence if two series which contain x he 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 ilie 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 cither V ox K is absolutely convergent. See Chry.stal's Algebra, Tart 11., p. I'll. 332 EXAMPLES. EXAMPLES XXVI. Determine whether the following series are convergent or divergent : 1 1 1 1 1 . 2"^ 3. 4 "^5. 6'^•••"^(2^^+l)(2?^ + 2)■^•" 2 1 1 1 • a{a + b)'^ {a + 26) {a + Zh) '^ (a + 46) (a + 56) "^ * * ' 3 3 . 4 3.4.5 3 . 4...(n + 2) 4'*'4T6'^4. 6. 8'*"'""^4T6...(2n+2)'*""' 4 5 1:_^ 3.5.7 3.5. 7...(2n + l) 4"^T77 "^4. 7. 10"^"-'^4. 7. 10...(3ri+l)"^'** 3 "^ 3 . 6 "^ 3 . 6 . 9 ^ "• "^ ~3 . 6 . 9. ..3m "^ "'° _ 1 1 1 1 X x+ 1 £c+ 2 a; + 3 7 1, 1 , 1 , 1+fl? l + 2a; l+3a; 1 1 1 \ +x \ -\-x^ \+x _ X X X X \+X \+X^ \+x^ \+x 1 1 1 ^L •^"* YV^^\'^%^^'^T^x''^"'^\^nx-^"' Irv* rv^ /y»^ 1 . 2 ■*" 2T3 "^ 3T1 ^ '" "^ (n+ 1) (n + 2) ■"' •" 12. l-^^4- ^ ^^ -. .. + (-!)'■ -^— + ... 1+a l + 2a ^ ^1+na EXAMPLES. 333 1 + 2 1 + 3 1 + n 13. 1 + :.— ^,, + :;— „, + . . . + ^i— 2 + ••• 2'-V 3' -2^ n'-{n-\ y 2^+P''^FT2^"^ -"'^rr + {n-lf'^"' 15. + -— + ^— +... x + m x+ "lin X + o\n 1 m m^ 16. =^ + + =+... aj + 1 x + 'in X + m (l + a)(l+b) (2 + a)(2 + 6) (^ + a) (r^ + 5) 1.2.3 2.3.4 n{n+l){n+2) 1 2 3 n ^^- l + J^-^ 1 + 2 J3^ 1 + 3^4:-^ -^l+nj^^^l^'" ^''- 2 + ^2'*"3+ 3^4 + V^^-'^w + >"^'" 2^- K^-i2)40-73)-^-;;^C-i)^- 21. (V2-l) + (^5-2)+...+(77i^+l-n) + .„ 1 a; a^' a;" 2 4a; 6a;' 2na;" ^^' 2"^T"^T0"^ •••■^7^'Tl"'••• o. 3 1 1 „ 2n-6 „_, 24. 7+-a + 3-^a;-+ +-r — —x '+... 4 2 12 nr — bn 25. Shew that the series 111 1 i" "2^2 •f' ^Q ■+■ ••• '2 ••• 1^-a; 2'-a; 3'-a; ?^--a; is convergent for all values of x, except only when x is the square of an integer. 27. 2-/ {J(H - 1 ) - 2 V('i - 2) + J(n - 3)1 a". CHAPTER XXII. The Binomial Theorem. Any Index. 282. It was proved in Chapter xx. that, when n is any positive integer, n(n—l)...(n — r + l) ^ r We now proceed to prove that the above formula is true for all values of n, 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 {n + l)th term; but when 71 is not a positive integer the series is endless, for no one of the factors n, n — \,n — 2, &c. can in this case be zero. It should be noticed that the general term of the ,. ., . 1 ^(?i— l)(w — 2)...(w — r + 1) _ binomial series, namely — ^^ -^-^ -^ ^^ x\ cannot be written in the shortened form ; — ]= of 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 1 +?i^+!|a;' + ^^'+... + p^a;'" + 12 .3 |r BINOMIAL THEOREM. ANY INDEX. 885 283. Proof of the Binomial Theorem. Represent, for shortness, any series of the form 1 + 77^^ + 7^^^+'" + —'• «;'■+... by/(m)- Thus '0)'' + and r + — a;'^ + 7' f(m + n)=l+^-— — -a; + ^ ,^ ^" ar^ + ... -f ^H ^^' ... I 1 |_z |r Now the coefficient of x^ in the product /(??i) xf(n) is m. T ^-^ + ... -\- :zl\l r — 2 2 \i that is Y- ]m^+ ... + \L 7' — 5 5 *^ ^r-*^* + . . . + n. And, by Yandermonde's Theorem [Art. 249 or 261], this coefficient is equal to ^^ — r— '^> which is the coefficient |r of a;*" in/ (7?? + n). Thus the coefficient of any power of x in f(m + n) is equal to the coefficient of the same power of x in the product f(ffi)^f(n); also the series /(w), f(n) and f{m-\- n) are convergent, for all values of m and ti, when X is numerically less than unity [Art. 278]. It therefore follows from Art. 280 that f{m) X /(/i) =/(7M + 7i) (a), for aZZ values of m and ?i, provided that x is numerically less than unity. 386 BINOMIAL THEOREM. ANY INDEX. Now it is obvious that /(O) = 1, and that/(l) = (1 + a?) ; we also know that if r be 2^ positive integer f(r) = (1 +xy'. Hence, by continued application of (a), we have /(m) x/(w) xf(p) X ... =f(m + n) xf{p) x ... =f(m + n +P+ ...)• Now let m = n=p= .., = -, where r and s are positive integers; then taking s factors, we have But, since r is a positive integer, /(r) is (1 +«)''; ••• {/0f=(i + -/; .•.(!+«.)" =/0. 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), f(-n)xfin)=f(-n+n)=/iO). Hence, as/(0) = 1, we have 1 1 /(- ^) = j^ = Q ^ xn , since n is positive, Hence (1 + xy =/(— n), which proves the theorem for any negative index. 284. Euler^s Proof. Euler's proof of the Binomial Theorem is as follows. * euler's proof. 337 Represent, for shortness, any series of the form l+7nx+ \ ^ ^ x^-\- ... +—^ ^—-^ J-la;''+ ... i. . ^ \r by f{m) : thus f(m) = l+mcc + '^x'-\-... + '^x'+ (i), f(n)=l-h7ia; + ^x' + ... + '^^ x' + (ii), and, /(m + n) = l ■i-(m-\-n)x + ^ — ^^^^2^"'+ ... +^— r^^'^a;'^+ ... p \r Now, if the series on the right of (i) and (ii) be multi- plied, and the product be arranged according to ascending powers of so, 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 /(7?2) is (1 +xy\ and that/(n) is (1 +a;)'*, and the product /(r/i) 'xf{n) is there- fore (1 -f-a?)"'"^", which again, as m -f- ti is a positive integer, is/(m + ?i). Hence when m and n are positive integers the product /{tti) xf(n) is f(m + n); and, as the form of the product is the same for all values of m and n it follows that f(m) xf(7i)=f(m+n) (a), for all values of m and n provided f(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;)-^ Put n= - 1 in the above formula; then we have (-l)(-2) (-r)^. ... + ' — — — r ^ — ■ ^ + [r = l-x + x^-x^+ +(-l)'-x'-+ s. A. 22 338 BINOMIAL THEOREM. ANY INDEX. This example illustrates the necessity of some limitation in the value of x; for we know [Art. 229] that 1-x + x^- is not equal to :; unless x is between - 1 and + 1. 1+x Ex. 2. Expand (1 - x)-\ We have (i-.)-=n-(-2)(-.,.<^f^(-.)^.l^Mz5Uzl)(_.)3 , (-2)(-8).^.^...(-rTl) (.^,,^ = l + '2x + Sx^ + 4cX^+ + {r + l)x''+ Here again it is clear that the result cannot be true for all values of x; if re = 2, for example, we should have 1 = 1 + 2.2 + 3.22 + 4.23+......, which is absurd. Ex. 3. Expand {l + x)i. ^^1+1 Jij)...J(:s)v'^) We have (l + a;)^=l + ^ a; + ^^- ^ ■T- + " ^ ^~'^ \^ ~' .rH the general term being K-^)(-i) g-^) .. ,, . . / ,.,. ,1.3. 5.. .(2; -3) ^ ^ , that IS ( - 1)' -1 —^ x^. Hence (1 + a;)2 = l + _ a; - — ^ a;^ + ^ f^ x^+ 1.3.5...(2.-3) ^ ' 2.4.6...2r Ex. 4. Expand (1 - a;) "a. .(JKJbi:^;.,,.. Hence ,- ,-h 1 , -, 1.3 „ 1.3.5 (2r-l) ,. All the terms are positive, for in the general term there are 2r negative factors. 6 BINOMIAL THEOREM. ANY INDEX. 339 Ex. 5. Expand (a^ - Bd^x)^ accordiug to ascendiug powers of x. (a3-3.=.)l={.3(x-|^)}L.(..|)' - Jilit_(hd(.|)v ] . r, 5 X 5.2 fxY 5.2.1 /x\3 5.2.1.4.7...(3r-8) /xy "1 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. 285. The (r + l)th term of the expansion of (1 +a;)" is obtained from the rth by multiplying by — x, that is by ( — 1 H ) ^- Now — 1 -\ is always negative if ?i + 1 is negative ; and, whatever n-\-l may be, ?i + 1 — 1 H will be nefjative for all terms after the first for r * which r> n-\-l. 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 -\-l. If X be negative, the ratio of the (r + l)th and rth terms will be always positive when r>n+l. The terms of the expansion of (1—^)" will therefore be all of the same sign as the r*th term, where r is the first positive integer greater than n + 1 ; and, as a particular case, all the terms of the expansion of (1 — xY are positive when n is negative. 22—2 340 GREATEST TERM. For example, all the terms in the expansion of {1-x)^ are of the same sign as the rth, where r is the integer next greater than | + 1, Bo that r is 3. Also, after the ninth, the terms of the expansion of (1 + x)^ are alternately positive and negative. 286. Greatest Term. In the expansion of (1 + cvy by the binomial theorem, we know that the ratio of the (r + l)th term to the rth is + x, that is + « ( 1 J ; we also know that x must be numeri- cally less than 1, unless w is a positive integer. First suppose that n + 1 is negative, and equal to — m. Then the absolute value of the ratio of the (r + l)th / 77l\ term to the rth term is a; ( 1 + — J . Hence the rth term is ^ (r + l)th term according as i»(l+— ) = 1; that is, ,. > mx ., . . >~(1 +n)x accordmg as r = ^ , that is = — ^, — . ° ' ° < 1+a? 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 ^ -^ l-\- X integer, the rth term will be the greatest wlien r is the ill/ -\- \) X integer next above ^^ = — . X ~p JL Ex. 1. Find the greatest term in the expansion of {l — x)~^, when 8 Tx 1 • i.- J -{l + n)x ixf . _ ^. x=-. Here n + 1 is negative, and — — =f — ^ = 4. Hence the 9 l-xl-| fourth and fifth terms are equal to one another, and are greater than any other terms. Ex. 2. Find when the expansion of (1-a;)" 2' begins to converge, if 3 Here n + 1 is negative, and — \, ^— = '^ , ^ = 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. 19 1 9 X 3j\ 2 Since (a + x)'^ = a~2~ M +_ j , the greatest term required is the term corresponding to the greatest term in ( 1 + - j . Now , ,,a; /^ a;\ 21 3 7 9 , . u .^. • . (n + 1)--^ 1 + - ]=-—.- — -- = -] hence r must be the integer next a \ a/ 2 4 4 2 9 greater than - , so that the 6th 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) (1-.T)-; (ii) (l-a•)-^ (iii) {i-x)-\ 342 EXAMPLES. (iv) (l+x)-'^, (v) (1+^)-^, (vi) {l+x)% (vii) (1— 5a;)~6, (viii) (1 — oxy, (ix) (l-x)~pj (x) {2a+3x)~i, (xi) (a^ - 2ax)^, send (xii) (4 - 7x)^. 2. Find the first negative term in the expansion (i) of (1 + ^xp-, and (ii) of (1 + ^xf^'. -12 3. Find the greatest term in the expansion of {1 + x) ' when x = ^. 4. Find the greatest term in the expansion of (1 — ^x)' when £C = f . 122 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 - 2lx expansion of -jz r^ are all positive, and that the greatest [l —X) of them is 100. 7. If a^, a^, a^, a^ be any four coefficients of consecutive terms of an expanded binomial, prove that ^1 , ^3 _ ^^2 a^ +a a + a^ a. + a.^ 12 3 4 2 3 8. Find the general term in the expansion by the binomial theorem of each of the following expressions according to ascending powers of x: ,.v a .... a + x ..... /a + a;"\2 (iv) (a + x)^ [a - oc)~^^ (v) {a + xf (a - a;)"', and (vi) (a — xY (a + x)~^. 9. Shew that the coefficient of x^" in the expansion of (1 + xy (I - x')-' is 2n. 10. Shew that the coefficient of x" in the expansion of (1 + 2xy (1 - x)-' is 27 (^-1), n ji 3. SUM OF. COEFFICIENTS. 343 287. Sum of coefficients. The sum of the first r + 1 coefficients of the expansion of (1 — xf can be ob- tained as follows. We have (1 _,,)" = l_!|.^+J|^^-...+(_iyp:^'-+..., also (1 — c7')~* = 1 + a; 4- .?/ + . . . + ^'" + . . . From [Art. 281] the coefficient of x"" in the product of the two series is equal to the coefficient of x"" in (1 — xY X (1 — ^)~\ that is in (1 — «)'*~^; hence we have |_1 |2_ '"^^ ^ \r_ = coefficient of x' in (1 - xY'' = (- IV ^^""^)- . Similarly, if cf) (x) = %-\- a^x + a^x"^ + . . . + a^.x'' + . . . , the (b (x) sum a„ + a, + . . . 4- a- will be the coefficient of x" in J^ , ^ 1 — a; Thus, to find the sum of the first ?' + l coefficients in the expansion of (f) (x), we have only to find the coefficient of X m the expansion oi .T ^ . Ex. 1. Find the sum of the first r coefficicuts in the expansion of {l-x)-\ Ans. lr{r+l){r + 2). The sum required is the coefficient of x^~''- in {1-x)-*. Ex. 2. Find the sum of n terms of the series 1.2.3 + 2.3.4 + 3.4.5 + Since (1- x)-*=—^r-- [1 . 2 . 3 + 2 . 3 . 4x + 3 . 4 . SxH ] ; the sum required =6 x sum of the first n coefficients in the expansion of (l-x)-'=Gx coefficient of rr«-i in {1 - x)-^ = -n{n+l) {n + 2) {n + S). Ex. 3. Find the sum of the first n + r coefficients in the expansion of {l+x)2 (1 - .r)--^ • 344 BINOMIAL SERIES. The sum required = coefficient of x^'^^~^ in the expansion of |^_ 3 . Now(l + a;)"=(2- l-a;)»=2"-?i. 2"-i(l -ic) + !l-^!--i) 2«-2 (1 - a:)2 + higher powers of (1 - x). ^ (1 + a;)" 2~ n2"-i n(n-l) 2*^-3 ^ Hence 7- ^^ = ^ ^ " 7i :?+ 1 — ^ + ^^ integral (1 - xy (1 - rK)3 (1 - a;)-* 1-x expression of the {n - 3)th degree. The coefficients of a;"+^-i in (l-x)-^, {1-x)-^ and {1-x)-^ re- spectively are ^{n + r){n + r + l),n + r, and 1; hence the coefficient of (1 - xf 2«-i (n + r) (n +r+ 1) - 2"-in (?H-r) + 2'»-3n (w - 1). Ex. 4. Find the sum of n terms of the series ^"^^""^ 1.2 •*" ^172.3^+ ^7w. {2w-l)l/n! (n-l)l. 288. Binomial Series. Series which are derived from the expansion of (1 + xy 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 a /I \-n 1 . . ^i (^ + 1) 2 \ (1-cc) "" = 1+7100+—^ — '-^af+... j ^ n(n + l)...(n + r-l) • • • ~i j X ~\~ • • «j and it should be carefully noticed that this expansion can be written in the form . . . + { (r + 1 ) . . . (r + 71 - 1 ) } a?*" + . . . ] . BINOMIAL SERIES. 345 When the index is fractional, —pjq suppose, wo have _f(^+^£+2£) /^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. i'iud the sum of the series 11. 3 1.3.5 ^ • a : 3 + + 37679 + to infinity. Writing the series in the form |1^'3+ [2_'32+ j3 '33 + ~^' we see from (A) that it is obtained from the expansion of {\ - x)~^ X 1 by giving to x the value found from 5 = 5. 1 3 (2\ ~2 12 2 '2 /2\2 ^-h) =^ + 2-3 + — (b) + =^+^' *^^^^^^°^^ 5 = ^/3-1. Ex. 2. Find the sum of the series ,2 2.5 2.5.8 . ■ r '^ ^+6 + 6712 + 67108+ toinfmity. Writing the series in the form + ]r*6+ [2 62 + ~"[3~*G3 + ' we see from (A) that it is obtained from the expansion of (l-a;)"^ 346 BINOMIAL SERIES. by giving to x the value 5 = 5-. Hence the sum required is o o 2 Hr=^'- _ 5 5 5 r3 . 3.7 . 3.7.11 ~ 6 "^ 6T12 "^ 67l2 6 Ex. 3. Find the sum of the series ^^ + ^+ J^l^+ to 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 (-5)(-l)3.7 1 |3 63"^ |4 6^ Now the terms of this latter series are terms of (A), if q—4:,p=-5, and 7 = -r . 4 6 We can therefore find the required sum, as follows : / 4\f _ 5 1 SJL^ 5.1.3 1 5.1.3.7 1 V 6J ~ 1'6"^ 12^62''" |3 63"^ 14 G^"*" 5 r_3^ 3.7 3.7.11 -| , "^6712 LlS"^ 18724 ■^18.24.30"^ J' /l\t ,5 5 5 „ Whence 5f = ^ {8 4/27 -17}. 3 3 5 3.57 Ex. 4. Find the sum of the series - + — ^- + . ' ' + to infinity. 4 4.8 4.8.12 fFrom/'l-^') n. Ans, 2^2-1. Ex. 5. Find the sum to infinity of the series ; 23 [3 24^"^ 25 15 23 2 [From (1 + 1)^J. ^^^- 24 ~ 3^^' T. P au ^i-i-. 1 1-4 1.4.7 1.4. 7. 10, , . Ex.6. Shew that l + j + ;p—+ . „ .,„ + ; „ ..^ ^^ + to in- 4 4.8 4 . 8 . 12 4 . 8 . 12 . lb ^., ,2 2.5 2.5.8 2. 5.8. 11 ^ - x- -^ ^^^^y"^ + 6 + 67r2 + 6:i2:r8 + 6.12.18.24 + tomfimty. [S.ce(l-|)-=(l-i)-j. TJIEOKKMS OlVrAlNKD JiY Ec^UATING COEKFKMENTS. 347 289. We know from Art. 281 that if any expression containing x be expanded in two different convergent scries arranged according to ascending powers of x^ the coefficients of like powers of x in the two series will be ecjual. By means of this very important principle many theorems can be proved. Ex. 1. Shew that, if n be any positive integer, 12 + — 12,22 12.22.32 + -'^• We have (1 -x)»=l-nj;+ V ' x- ^— — ^^ -x^-v + ( - 1)** -^^ — 1-- 9^; =^ • Also, provided a; > 1, we have ( 1 1\'" , .„! . n(/i + l) 1 . n(» + l)(n + 2) 1 , ^'x) =^ + ^i+-T:2-r2+ 1.2.3 x^^ w(?t+l)... (n + n- l) \^ Hence l-n=.'i^^. ,,.,,.^l^^lz^:M^^Zm is equal to the coefficient of a;® in (1 - x)** x ( 1 - - J , that is equal to the coefficient of x° in (-l)^c", which is zero. [See also Art. 251, Ex. 3.] Ex. 2. Find the sum of 1 , ,vl-3 , 1.3.5.. (2/1-1) (^ + l) + n.- + {n-l)2^ + + 1> 2.4.6...2n -- [Equate coefficients of x" in (l-x)"2x (l-a:)-2 and in (l-x)"'^.] 5 .7...(2n + 3) '2.4...2n • ' ^ X ^J. — Ex.3. Shew that l-3n + ^-^^i:^^i^^^^^- = (-1)". We have r, = , — :, = :j -p. ^> . l + X"* l-aj + x^ l-x(l-x) Hence (1 + a;){l -arVx"... + ( - irx'"+ ... } = l + x(l-x) + x2(l-x)2+...+x-''«+i(l-.r)^'»+> + ... The coefficient of x3"+i on the left is ( - 1)". 348 EXPANSION OF MULTINOMIALS. The terms on the right which give a;2"+^ are aj3«+i (1 - a;)»'»+i + ir3« (1 - a;)^" + a;3n-i (i _ ^fn-i + . . . ; and hence the coefficient of a^^+^ will be found to be (37i-l)(3n-2) (3n - 2) (3» - 3) (3n - 4) 1-Sn + 1.2 1.2.3 290. Expansion of Multinomials. Any multi- nomial expression, can be expanded by means of the binomial theorem. Since (p + qx + ro)^ + ...)" may be written in the form p" ( 1 + - ^ -}- - i3?^ + . . . ) , it is only necessary to consider expressions in which the first term is unity. Now in the expansion of [1 + ax -{- hx^ + cx^ + ...}", that is of {1 + {dx + hx^ + cx^ + ...)}", by the binomial theorem, the general term is — ^ ^-^ / — ^ ^ (ax -\- hx^ + cx^ + . . .Y \ also in the expansion of (ax-\-hx^ -{■ cx^ + ...yy r being a positive integer, the general term is by Art. 262 a 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 \a ^ |7^... To find the coefficient of any particular power of x, say of a^, we must therefore find all the different sets of positive integral values (including zero) of a, ^, 7,... which satisfy the equation a + 2/3 + 87 + ... = A;; the cor- responding value of r is then given by r=a + ^ + 7+ ..., and the corresponding coefficient is found by substituting EXPANSION OF MULTINOMIALS. •'^49 iu the formula for the general term. The required coeffi- cient will then be the sum of the coefficients corresponding to each set of values of a, /S, y Ex. 1. Find the coefficient of a;« in (1 - x + 2x2 - Sx^)-h' The values of a, /3, y which satisfy a + 2/3 + 87 = 5 will be found to be 0, 1, 1; 2, 0, 1; 1, 2, 0; 3, 1, 0; and 5, 0, 0. The cor- responding values of r will be 2, 3, 3, 4 and 5 respectively; and the corresponding coefficients will be LlILlllro^w qNi V" 2)V"2)V"2) , ,,,, ,,. [TjT (2) (-3), ^ (-1)( 3)' {-1){-1){-1) _ (-^)(--2)(-2)(-2) ^^ NKiMLIKJ,.,,., ^, ^ . 9 45 15 35 , G3 thatis-^. jg, 4, -j^ and — . 31 Hence the required coefficient is ;r-- . 25t> 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: + a;2 + a;3 + a;4)-2. We have {l-\-x-{-x^ + x^ + xY^=Q^^\''^ ={l-xf{l-x^)-'^ = (1 - 2a; + a;2) (l + 2r5 + 3^10 + 4.C15 +._,). Hence the coefficient required is zero. Ex. 3. Find the coefficient of a;» in the expansion of (l + a! + a;- + .r^)-'. We have (l + a; + a:2 + a;3)-i= , —^ ■,= \—^. ' l + X + X' + X^ 1 -X* 350 COMBINATIONS WITH REPETITIONS. Hence the coefficient of a;^** is 1, the coefficient of a;*''+i is - 1, the coefficient of a;'*'"^'^ is zero, and the coefficient of x^''^^ is zero. Thus the coefficient of x^ is 1 when n is of the form 4?-, it is - 1 when n is of the form 4r + 1, and it is zero when n is of either of the forms 4r + 2 or 4?- + 3. Ex. 4. Find the coefficient of x^ in the expansion of (1 + 2a; + 3x2 + 4a;3 + to infinity)**. Since 1 + 2a; + 3x2+ _^j^_-j.^-2^ tj^e required expansion is that of (1 - a;)~2"; the coefficient of x^ is therefore 2H( 2u + l)...(2M + r-l) E 292. Combinations with repetitions. The number of combinations of n things a. together of which p are of one kind, ^ of 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, . . . ; and consider the continued product (l+a^+aV+...+a^a^)(l+6ic+...+6V)(l+c«+...+cV)... It is clear that all the terms in the continued product are of the same degree in the letters a, b, c,... sls in x; it is also clear that the coefficient of of- is the sum of all the different ways of taking a of the letters a, b, c,... with the restriction that there are to be not more than p as, not more than q b's, &c. ; so that the coefficient of a?* in the continued product gives the actual combinations required. Hence the number of the combinations will be given by putting a = b = c=... = 1. Thus the number of the com- binations of the n things a together is the coefficient of of- in Permutations. The number of permutations of the n things a together being represented by P^, it is easily seen that ^^ f, X x^ x'^\ f, X x^ x''] p p p .1 2 \n PEKMUTATIONS. J^51 For a X the coefficients of a;'' in , ^ hx 6 V is the sum of all possible terms of the form la a}}f\... IZ jm ... for which I -\-m -\- ... =a, and the number of permutations a together formed by taking I of the a's, m of the 6's, kc. is \l\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 a:^) (l+a:+ ... a;'»)(l + a; + a;'^), that is in {1 - x^) (1 - x^) {1 - x^) {1 - x)-^ . Ilejecting terms of higher than the seventh degree in the continued product of the first three factors, we have (1 _ x3 _ a;5 - a;6) (1 + 3a; + 6x2 + 10^3 + 15^4 ^. 21x5 + 28a;« + 36x7 +...) ; and the coefficient of a;^ 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 ahke of a second kind, and so on. The total number of the combinations is the sum of the coef- ficients of a;\ «-,..., x'^ in (l + a; + ... +x^) [l-\-x+ ... +x^)... ; and this sum is obtained by putting a;=l in the product and subtracting 1 for the coefficient of aP. Hence the required number is (2> + l)((/ + l)...-l. The above result can, however, be obtained at once from the consideration that there arep + l ways of selecting from the a's, namely by taking 0, or 1, or 2,... or p of them; and, when this is done, there are 5 + I ways of selecting from the i's; and so on. Hence the total number of ways, excluding the case in whicb no letter at all is selected, is (p + 1) (i4-l ~ '^-^2»-l (^)* It follows from the last relation that l2)i+3 ^'^^^ ^^ ^^ integer if i2n+i and lon-i ^^Q integers. Now we know that Jj and /g are integers ; hence by induction Jan^i i^ always an integer. The relation (A) also shews that l2n+3 will be divisible by 2"+2 provided 1271+1 i^ divisible by 2"'+i and /2H-1 by 2". Now we know that Ii is divisible by 2^ and Jg by 2^ ; hence I5 must be divisible by 2^; and it will then follow that I^ must be divisible by 2^ ; and so on, so that i2n+i is always divisible by 2'^+^. Ex. 4. To shew that, if n be any positive integer, a^-n{a + 6)» + ^^ ^l^ ~ ^^ (a + 2&)» - = (^-h)n\n. Put — — for X in the identity proved in Art. 259, Ex. 3 ; then, after reduction, we have "O ^1 {y + a,){7/ + a + b) ...{2j + a + nb) y + a y + a + b ... + {-l)r ^+... y + a + rh Now expand the expressions on the two sides in powers of - . y Left side = ■■'■ ■'— ■■; — « . , = ^xr + higher negative ,»«(,..)...(,,«_±!^) y- powers of y. Eight siae = |(l + ?)-'-... + (-irJ-'(l + «_±'y + ..,..., hence the coefficient of -z-r, on the right is (-l)^-[Coa*-Ci(a + &)^+ + (_l)rc^ (a + ?•&)*+...]. Hence ^ { - lycj. {a + rh)^ is zero if k"j«.if k = n. EXAMPLES. :Soo EXAMPLES XXVIII. 1. Find the sum to inlinity of each of the following series : 11 Lll^ 1.3.5 3" (0 ■^1^2'"^ 12 2'* "^ |3 2^' ■*■••• ^"^ 2 2"^ 2.4 2- 2.4.6 2"""^ ••• 41 4.71 4.7.10 1 ("^) ^^114^7^4^^ A ^ ~' • • • [2 4' 3 4" 3^ 3.5.7 j. 5.7.9 ^'''' 3.6'*'3.6.9"^3.6 . 9Ti2'^'" (A _A. 3-4 3.4.5 3.4.5.6 ^'^^ 2 . 4 "^ 2 . 4 . 6 "*■ 2 . 4 . 6 . 8 "^ 2 . 4 . 6 . 8 . 10 "^ '■■ ,., , 2 2.5 2.5.8 ("^ ^^6-'6n2-*-67r27r8+- (..> -, O O.O O.O.I ""^ ^-4^r:8-4:87r2-^- . .... j4^ 4. 12 4.12.20 ^''"'^ 18*^18.27 "^18. 27.30"^ ••• ... - 2 2.5 2.5.8 ^'""^ ^"^9"^9TT8"^9.18.27"^- , , 1 1.3 1.3.5 9.18 9.18.27 9.18.27.36 "' , . 1 1 . 3 1.3. 5 ^""'^ 2.4.6"^2.4.6.8"^2.4.6.8.10'*""' (xn) ~ ^ -^^ 7.28.49 ^^'^^ 72 "^72.96 '*'72.96. 120'^'" 2. Shew that , a n(n+\)/ a Y 1 + '* E + — 1 V- ( 7 ) + . • . b ?i(?i+ 1) / 6 >^- _ ^** 2:J— 2 l+n f + \ o ( - • + ••• a + 1.2 \a + bj 356 EXAMPLES. 3. Shew that ^ ' y 1 + X 1 . 2 \1 + ,T/ 71 (n + l)(/^ + 2) /I -a^Y "i 4. Shew that, if x be greater than - ^, £C _ aj 1 / ic Y 1 . 3 / cc Y T)~TT^'^2 VIT^; ■*"271vr+^/ 1.3.5 / a; Y , "^2.4.6 \y^x) ^ 5. Shew that (1 -xy^{\-\-xY-%ix{\^xY-' + ?^^^^a;^(l+a,f'' -=-... 6. Shew that a — x nin-\-V) /a — x\^ \+n + -^i — ^ I + a + x 1.2 \a + xJ fa + xV" 7. Shew that (1 + xY" = (1 + x)" + nx (1 4- xy-' + '' ^f ^,^^ o;^ (1 + xf-' 1 . 2 8. Shew that, if a < 6, 97 9 / 7X4 fa" 4 rt^ 4.5 a" 4.5.6 a^ ) ai=(« + J) |__j.^-, + _-_j-^^. + |. 9. Shew that Tz + a:; {71 ■\- 2x) {n ~ \) [n + 2)x) {71 - \) {n - 2) __ ~ r+^ ^ [2(1+ £c)-" |3(l+r«)2 ■^'■■^ ' 10. Shew that, if the numerical value of y be less than one- third of that of X, — . ^ ^n{7i+\) f 2y\- n{n-\-\)(n^'2) / '2y ^ ^ ^^^_2^^.^(^/J^Y x + y 1.2 \x + yj 1.2.3 \x + yy a; - 2/ 1.2 \x-yy ^ 2ij lb in- 1) / '2y \- EXAMPLES. 357 11. Find the value of n{n— I) n{n-l){n-2) n — to r terms. 12. Shew that, if n be a positive integer, 7r{n-l) 'nr{vr-V){n-2) + ... (1(2 + |2^ + ... = 0. n 13. Shew that, if n be a positive integer, n(7r~V) 7i{n'-V}{n'-2') [1 |2 "^ [2~]3 ~ +(-ir "^"^:^l-.^f"'^^ ^--(-ir'- t r + 1 14. Shew that if ?i be a positive integer <|; 4 l_4^ + _____ ___ ___ +..._u. 15. Shew that l.7i{n+\) + 2{)i-l)7i + 3{n-2){n-l)+ ... +n. 1 .2 1 = Y^ ?i (?i + 1) (?i + 2) (m + 3). 16. Prove that n 1 . n (h + 1 ) 4- ;f . (71 - 1) 71 + ''^'.'X^^ (n - 2) (/i - 1 ) n(7t + l)0i + 2) , ox/ nx « |2^+1 ^ 1.2 3 ^ ^-3)(n-2).....2-L 71+2* 17. Shew that, if ;,, = i4^|;:g^), 358 EXAMPLES. no -Ti. 1.3.5...(2r- 1) - 5.7...(2r + 3) 1^- ^* P^ - 2.4.6...2r ^ ^^^ ^^ = 2.4...2r ' P^'""^ that p^ +Pr-^^,^Pr-2 • • • — ^i 1.2 m — 6 if 'lit > 27Z,, and = 1 if m = 2n. 23. Find the coefficient of x" in {\ +x) {\ +x') {I +x') {\ + x^).., 24. Shew that, if a; be a proper fraction, {\-x)(l-x'){l-x'){l-x')...^ (^ + a;) (1 + x') {l+x")... 25. In how many ways can 12 pennies be distributed among 6 children so that each may receive one at least, and none more than three ? EXAMPLES. 350 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 {p+\) 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 ,, ,C ,. ' n+r— 1 r— 1 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 toofether of Zn things, of which n are alike and the rest all different, is 2"'-'-^\2n-li\n \n-\. 30. A man goes in for an examination in which there are four papers with a maximum of wi marks for each paper; shew that the number of ways of getting half marks on the whole is |(m+l)(2m'+4/7^ + 3). 31. Find the coefficient of a;* in (1 - '2x - 2x')'^. 32. Find the coefficients of x^ in the expansions of (1 + ic + cc^ + aj' + x^y and (1 + cc + a;^ + aj^ + a;* + x^y. 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 a;'" in the expansion, according to ascending ])owers of x, of {^d^ + 6ax + 9a;^)~'. 36. Shew that the coefficient of x^"" in the expansiou of 1 +a; . o T -„ IS 2m + 1. »3\3 (1 + a; 4- x") 37. Shew that the coefficient of a;"" in the expansion of (1 + 2a; + 3a;'+ ...f is I (r + 1) (r + 2) (r + 3). 860 EXAMPLES. 38. Find the coefficient of cc" in tlie expansion of {1 . 2 + 2 . 3a; + 3 . 4x' + ... to infinity}''. 39. Find the coefficient of £c'" in the expansion of (1.2 + 2.3.2X + 3.4. 2-V+ + (n + 1) (n + 2) 2''x" + ... to infinity)". 40. Shew that the coefficient of x" in the expansion of {1 +x + 2x' + dx^ + ...y is }r (r^ + 11). 41. Shew that if p-q he small compared with p or 5', then will V 9 (n-l) p + {n+ 1) q 42. If (6 JQ + Uf'-'^JSr, and F be its fractional part; then will iVi^=20'"+\ 43. If (3 Jd + bf-' = I+F, where / 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" )m + l 45. If m be a positive integer, the integer next greater than (3 + J5)'" is divisible by 2"\ 46. Shew that the general term in the expansion of 1 +X + y + x]/ 1 + x + 1/ \m + n— 2 ia (- 1)"'+" -i=__- x-y. m - 1 n-1 47. Shew that the coefficient of x' in the expansion of X . r r^-r (7- - P) (r= - 2^) . is 9M 1 + — — — c + ^ -^ "' {1-xy-cx [ 3 [6 ^ (r--r)(r--2-^)(r-30 ^3, EXAMPLES. 3G1 48. Sliew that 1 . 2 . „ + 3 . 4 ^ii;i:^> + 5 . 6 '-i-(»fA)i^ + ... + (2n-3)(2n-2).7i + {2n-l)2n.l^- 2 V. 49. Shew that the coefficient of a;"^''"' in the expansion of (l±4-]is2'-'(n+2r). (1 - xY 50. Shew that the coefficient of a;"'^''"' in the expansion of |i^);is(-ir(,-2»)2'-.. 51. Shew that ?i" - w {71 - 2)" + —\-^ {n-^Y- ... to n+1 terms J. . ^ = 2.4.6. 8...2n. 52. Shew that a"+> - 71 (a + 6)"^' + '"(.''"^^ (a + 26)"-^^ - .. . = i|n+l (2a + ?i6) (- 6)". 53. If three consecutive coefficients in the exjjansion of any power of a binomial be in arithmetical progression, prove that the index, when rational, must be of the form q" — 2, where q is an integer. 54. Shew that the sum of the squares of the coefficients in the expansion of (1 +x + x^)", where ?i is a positive integer, is \2n ^» \r\r \27i-2r' 55. Shew that, if n is any positive integer, n(n-l) 7i(n-l){7i-2){n-3) 2(2r+l)^ 2.4(2r4-l)(2r + 3) ^*" ^ 2" '^ ^*^— ?i^^' + 2)...(r + ?t-l) 2r72r+l)(2r + 2)...(2r + n-l)* CHAPTER XXTII. 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 00 — a, x — h, 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)ih. degree in x. PARTIAL FRACTIONS. 363 We have to find values of A, B, 6',... which are iudependent of w and which will make F(x) ABC {x — a)(x — b)(x — c)...~' X — a x — h x—c or, multiplying by {x — a){x — h){x — c) , F{x) ~A{x-h){x-c) +B{x-a){x-c) ^-G{x-a){x-h) (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 a?^ x^,... x""'^ on the two sides of (i), we have n equations which are sufficient to determine the n quan- tities A, B, G, 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 x = 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 C, 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 (n — l)th, it follows [Art. 91] that the relation (i) is true for all values of x. Thus F(x) ^ ^ F(a) 1 (x — a)(x — b) {x — c). . . ^ {a — b){a — c). .. x — a' Sx + 7 Ex, 1. Resolve . .,,, into partial fractions. {x-l){x-2) . Sx + 7 A B Assume ; -— — = -\ ; (x-l){x-2) a; - 1 re - 2 ' then dx + 7=A{x-2) + B{x-l). 364 PARTIAL FRACTIONS. In this identity put x = l; then 10 = -A. Now put x = 2 ; then 1S = B. Thus ^^ + 7 - 13 10 {x-l){x-2) x-2 x-1' Ex. 2. Kesolve ,- {■; ,4? iiito partial fractions. (oj-a) (a;- o)(a;-c) Let (&-c)(c-a)(a-& )^ ^_^ B ^ C (a; — a) (a; - 6) (a; - c) x-a x-b x-c * then (6 -c){c-a){a-b)=A{x-b){x-c) + B{x- c) {x - a) + G(x-a){x~ b). Putting x = a, we have {b - c) [c - 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 (^-^)(^-^)(^-^) ^ ^_Z^ + ^f + ^-^ (cT - a) (a; - &) (a; - c) x-a x-b x- c ' Ex. 3. Kesolve — ; tt-t ^rr ; r into partial fractions, x{x+l){x + 2)... (x + n) ^ Assume a;(a; + l) (x + 2) ... (a; + ?i)~ .t a; + l "* x + r '" x + n' Then, we have l = Af^{{x + l){x + 2)...{x + n)}+A^{x{x + 2){x + S)...{x + n)} + ...+ Aj.{x{x + l)...{x + r-l){x + r + l)...{x + n)}+ ...+An{x{x + l)...{x + n-l)}. If we put x = 0, all the terms on the right will vanish except the first, and we shall have 1 = AqX \n, so that ^o = l/|n. To find the general term, put x=-r; we then have l = ^,{(-r)(-r + l)...(- 1)(1){2).. .(n-r)}, that is 1 = (- 1)M^ |r jw-r; hence Aj,= {-iy/\r \n-r. Hence the required result is x{x + l)...{x + n) [nix ' ^ ' \r\n-r x + i' x + n) [See Art. 259. Ex. 3.] ^.^ Ex. 4. Express - — , , \, , , in partial fractions. {x-a){x- b) {x-c) ^ Ans. S {a—b)[a-c) x-a ' PARTIAL FRACTIONS. 365 X'+ 15 Ex. 5. Resolve -. , . , .> — r — - iuto partial fractions. {x-l){x^ + 2x + 5) ^ The factors of a;- + 2j: + 5 are the complex expressions a: + l + 2i and a; + 1 - 2i, where i is written for ij -1. J- + 15 A B C Assume ; .,,,., — ^ -^ = H :. — —. + {x-l){x- + 2x + o)— x-1 x+l + 2i x + l-2i* .: X'-{-l5 = A{x + l + 2i){x + l-2i)+B{x-l)(x + l- 2i) + C{x-l){x + l + 2i). Put a; = l; then 16 = 8^, so that A = 2. Puta;=-l-2i, then (l + 2i)2 + 15 = B (-2- 2t) (- 4i), that is S + i 12 + 4i = J5 ( - 8 + 8i) ; therefore B=- ^r—^. . Change the sign of i 3 - i in the value of B, and we have C= - Thus 2 + 2r a;'-^ + 15 2 S + i 1 3-i {x-l){x- + 2x + o) x-l 2-2ix + l + 2i 2 + 2i x + l-2i 298. "We Lave in the last example resolved the given fraction into three partial fractions 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 w^hose denominator has imaginary factors, is often left in a more incomplete state. Take, for example, the fraction just considered, and assume ^' + 15 A Bx + G + {x-\) {fc' + 2.C + 5) x-l x" + 2x+d' [It is to be noticed that w^e must now assume for the numerator of the second fraction an expression containing ^ X but of lower degree than the denominator.] llhQnx^ + lb = A{x'' + 2x + b) + {Bx + G){x- 1). Putting a; = 1, we have 16 = 8/1, so that A = 2. Put ^ = 2 in the above identity; then after transposi- tion -x'^-^x-\-b = {Bx + (7) (^ - 1) ; or, dividing by x—\, Bx -\-C = — x — 5. x' + Id 2 x-\-o Thus {x-l){x'+2x + 5) x-l x'+2x+5' 366 PARTIAL FRACTIONS. 299. We have hitherto supposed that the factors of the denominator of the fraction which is to be expressed in partial fractions, were all different from one another. The method of procedure when this is not the case will be seen from the following examples. 2a; + 5 Ex. 1. Express , r-rir, tt. in partial fractions. {x-iy{x-S) We may assume that 2x + 5 A B € D {x-l)^x-B)-{x-lY' (x-l)2' {x-1) x-S' or, clearing from fractions, 2x + 5=A{x-S) + B{x-l){x-B) + G [x - ly'ix - 3) + D {x-lf. 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, G, 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) + Gy^iy -2) + DtjK Now equate coefficients of t/^, y'^, y^, y^, and we have 7= -2A ; 2 = A-2B; = B-2G', andO = D + C. Whence ^ = -- , 5= - — , (7= -— and D = — . J 4 o o „ 2x + 5 11 7 11 11 Hence {x-lf{x-S) 8{x-S) 2(a:-l)3 4{x-l)^ 8(.t-1)* (1 + xV^ Ex. 2. Express the fractional part of -^ — ^^ in partial fractions. Assume \ (1 + a;)** A B G . . , . (1^^= (11^+ (r::2^+ 0^2^) + ^^ '""^'^'^^ expression. Then {l + x)^=A + B (1 -2x) + C{l- 2a;)3+ (1 -2x)^x integral expression. Now put l-2x = y; then (l + rr)« = (|- |y= L{sn_nsn-iy+'l^lz3sn-2y2^texms con- taining higher powers oi y). PARTIAL FRACTIONS. 367 Also right B'\de = A+By+C7/ + y^ x integral expression m y. Hence, equating coefficients of ?/", y^, y^, we have 3»» w3»-^ 71 (w- 1)3'*-'^ 300. The followicg examples will illustrate the use of i^artial fractions. Ex. 1. Find the coefficient of a*'* in the expansion of ^ — z s according to ascending powers of x. w 1 1 3 2 = 3{l + 3a; + (3x)2+... + (3a:)«+...} -2{l + 2a: + (2a;)2+... + (2a;)«+...}. Hence the required coefficient is 3"+^ - 2""*"^. (1+x)^ Ex. 2. Find the coefficient of .t"+'' in the expansion of 7; — — \„ . (1 - 2x)^ From Ex. 2, Art. 299, we have (l + a-)"_3« 1 w3«-i 1 7i(n-l)3"-2 1 (l-2a;)3 2« (l-2a;)3 2» (l-2a;)2' 2«+i 1-2.T + 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 a"+2 {c-b) + b''+~ (a - c) + c"+2 (& _ g) {b-c) {c -a){a-b) The sum of all the homogeneous products of n dimensions is the coefficient of x^ in the product {1 + ax + a^x^ +...) (1 + bx + b^x^ +...){! + ex + c^x'+...)[iiee Art. 293]; that is in 7:; r-rz — ; — r-r; c > which will be found to be equal to {1 - ax) (1 - bx) (l - ex) a2 1 ^2 1 c2 1 + 71 rr-. : ,— ,- + {(i-b) {a-c) l-ax {b-c) (b-a) 1-bx {c-a){c-b) 1-ex' and the coefficients of x" in the expansions of these partial fractiona is easily seen to be a«+2 jn+a c"+2 -t + (a -b){a-c) {b- c) {b-a) (c- a) (c - 6) ' which equals a'^{c-b) + t»+2Ja - c) + c"+^6 - a) {b-c){c-a){a-b) 368 INDETERMINATE COEFFICIENTS. Ex. 4. To find the sum of all the homogeneous products of n dimensions which can be formed from the r letters aiya^^a^, , a^. As in the previous example, the sum required will be the co- efficient of a;" in -j r-^^j r— j r , which will be found (1 - a^x) (1 - a^x) (1 - a^x) to be equivalent to S-; p: r — — . (aj — tta) (% — ag) . . . 1 — a-^x an+r-l Hence the required sum is 2 (ai-a2)(ai-a3)...(ai-a^)' 301. Indeterminate coefficients. 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 {1 + cx) {1 + c^x) (l + c%)...(l + c"a;). The continued product is of the ?ith degree in a; ; we may therefore assume that [l-V ex) {l + c'^x)..\l-\- c'^x) = Aq + A-^x + A^^x"' + .. . + ArX'ii' + ...■\- A^^x'^, where Aq, A-^, A^,... do not contain x. Now change x into ex ; then, since ^o» -^i* -^2' ^^' ^'^ ^<^^ contain X, we have (1 + c^x) (1 + c^x) . . . (1 + C^+^iC) = ^0 + ^1^^ + ^2^^^^ + • • • Hence (l + c'^+ia;) (^0 + ^1^ + ^^^+ ••• +'<^»^''+ •••+^n^") = (1 + co;) (^0 + ^i<^^ + -^ac^^ + . . . + A^c'^x'^ + . . . + A^e'^x^). Now equate the coefficients of x'^ on the two sides of the last identity, and we have Ar + c'^+My-i = ArCi- + Ay_^c1' ; ^ _ c^+i-c^ ^ _ ^ c^->-+i - 1 ^ .*. A^— r_i A^_-^ — C Qf^-X r-i (<*)• By continued application of (a) we have "•"■'^ •'' •••''•^ (C'--1)(C'-1-1)...(C2-1)(C-1) "' ft"' - 1^ (c'^~'^ - 1) (c'^-r+1 _ 1 \ = -'^'"" ' (C-- 1) (.^- -1)' .(0 - 1) ' ■ '"' ^' '- ""'-'""-'^ ^- INDETERISIINATE C0P:FF1CI KNTS. [i{j\) Ex.2. To liiul the sum of the series P + 2^ + 3- + . . . + n-. Let l- + 2'- + 32+...+n3 = .4i7i + J3n2 + ^3n3 (a) for some particular vahie of 7?, where J^ , A^, A^ do not contain n. The relation (a) will be true for rt + 1 as well as for n, provided 13 + 23 + 32+. .. + n2 + {n + 1)2 = ^ J„ + l) + J3(» + ip + j3(,i + !):»; or, subtracting (a), provided (u + 1)2 = ^1 + (2n + l).l2+ (3/(2 + 3n + l)./3. Now the last relation will be true for all imlites of n if we give to Jj, ^2' -^3 *^6 values which satisfy the equations found by equating the coefficients of 71^, n^ and /i'', namely, the equations 3.43 = 1, 3^3 + 2^2 = 2, and ^3 + ^.^ + ^1 = 1, from which we obtain GAi = 2A2 = SA2 = l. Hence, if the relation ^ + 2'^+ ... + n^=-^ n+ -n^ +- n^, be tiue b J o for any value of 77, it will be true for the next greater value. But it is obviously true when /7 = 1; it will therefore be true when ti = 2 ; and, being true when ?t = 2, it must be true when n = 3; 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 Art. 321.] EXAMPLES XXIX. Resolve into j)artial fractions : 1. -3-4^-^. 2. 3. . .-.-^..r. ^. 4. x^ + 7x + 6' X {x+ l){x+'6){x + 5)' S-x {'2-xY{l+x)' x'-3 {x + 2){x'+l)' x^-x+l 8. {x'-^l){x-\f' ^^' (l-;Uf (1+a;)' x+ 1 x' - 5,>j + G * af+l x{x+iy' x' + X + 1 x^ - ix- +X + G* 1 + 7x-i v' (l4.3.r)'(l- Kb;) 5 -9a; {X- -2y{x ^ + 1)* x'- -x+ 1 (X- -Ifix - 2) {x' + 1) 1 + : 2x 370 EXAMPLES. 1 1 Qx^ + x-1 x^+2 {x'+l)(x-2){x + 3j' 1 Q X + X _ . '^' {x-iy{x' + 4:) • ■^*' 15 l + 2a; 17. Find the coefficient of x" in the expansion of £C + 4 x' + 5x + 6' 18. Find the coefficient of cc" in the expansion of x-2 {x + 2){x-iy' 19. Shew that the coefficient of x^"~^ in the expansion of x + 5 1 {x'-l){x + 2) 20. Find the sum of the ti first coefficients in the expansion 3 -2a; l-2a;-3a;^* 21. Find the sum of the n first coefficients in the expansion „ 2 — 5a; ^ (l-5.x)(l-3a;)(l-2a;)* 22. Find the coefficient of x" in the expansion of j= ^ . Find also the sum of the n first coefficients. ^ 23. Shew that the coefficient of x"'^" in the expansion of [|^is(-2)'(,.-2„.l). EXAMPLES. y7i 24. Shew that x"^' {x-a^){x-a„) ... {x-aj a = x + a, + a.^ + ... + a+2t 25. Shew that the coefficient of z"~^ in the expansion of {(1 - ^) (1 - cz) (1 - c'z) (1 - c'z)}-' is (1 _ e") (1 - c"-) (1 - c"-0/(l - ^) (1 - c') (1 - c^). 26. Prove that a{b-c) (be - aa') (a'" - a^") 6(c- a) (ca - hh') (6"' - b"") a — a b — b' c{a-b) (ab - cc') (c" - c"") ■^ c-c' . (^ - c) (c — a) (a — b) (be - aa) {ca - bb') {ab — cc) JI^_^t ahc where aa! - bb' = cc ., and U^_^ is the sum of the homogeneous products of a, 5, c, «', b\ c' of m — 3 dimensions. 27. Shew that the product of any r consecutive terms of the series 1 - c, 1 — c^, 1 — cV • • is divisible by the first r of them. 28. Shew that, if c be numerically less than unity, (1 + ex) (1 + c^x) (1 + c^x) ... to infinity 29. Shew that, if c be numerically less than unity, (1 -^ cx){\ + c^x) (1 + c^x) ... to infinity " -^ ^ nr7 -"^ "^ (i-c^)(i-c'') ^' "^ (i_c^')(i-c'')(i-c'') ""'■*"••• 30. Shew that, if c be less than unity. 12 . a; x = 1 +r + (l-a;)(l-cx)(l-ra;)... 1-c (l-c)(l-c-) 24—2 ^ + 372 EXAMPLES. 31. Shew that, if c be less than unity (1+CX)(1+C''X)(1+C^X)... _ l+G (1 4-c)(l +C^ {l-x){l-cx)(l- c'x)... ~ "^ n^ "^ ■*■ (1 - c) (1 - r) ■'^ [Gauss.] 32. Shew that the coefficient of a?'' in the expansion of (1 + cx)(\ +c^x){\ + c^x)... (1 — cx)(l — c^x) (1 —c^x) ... (14-1)(1+C)...(1+C-) (l-c)(l-c^)...(l-c'-) ' c being less than unity. 33. Shew that + , + . o + . -^ 3 + 1— £c 1— ax 1— a^ic 1 — a" ic \ X ai" a? + :; +^; 5- + 1-2/ 1 —ay 1 — a^2/ 1 — a^y 34. Shew that X 2x' 3ic" ix^ + -. s +•= s +:: -.+ . 1-X I— of l — X^ 1-X^ X X^ X^ l"/1 OvO"!"/-! 3\2* {i-xy {i-xy {i-xy 35. Shew that Lambert's series, namely, X x^ x^ X* + -^ s 4- :; :: + ,. 4 + ... I ~X I -x' 1-03^ 1-03 is equivalent to 1. + X . L + X o L + X rr^ii 1 X + X 5 + X r. + . . . Clausen. i—x l — x' l—x "- i- CHAPTER XXIV. Exponential Theorem. Logarithms. Logarithmic Series. 302. The Exponential Theorem. If Ijn be nu- merically less than unity, ( 1 + - J can be expanded by the Binomial Theorem ; and we have ('-j= 1 nx {nx — I) 1 n 1.2 if = l-\-nx + — \-^ — ^ —^ + nx{nx — l){nx — T) 1 nx(nx—l)...{nx — r-\-V) 1 which may be written (^nj =n- + -o-+ TO -^••• / 1\ / r-l\ X [x \...\x V nj \ n J ^ Putting a; = 1, we have I/: 374 EXPONENTIAL THEOREM. 1\"^" \i\ 1 (I \ 1 ' I \ I -S('- / 1\ / 1\/ 2^ X [x x\x , \ nj \ n/ \ Uj 1+''^ + ' TZ + io ' + »•■ 2 "^ 13 L = u + i+— — - + 71 V n) V 72/ V, 3 + ...} . The above relation is true for all values of n however great, and therefore when n is infinite ; but when n is infinite, Ijn is zero, and the relation becomes* X X /- ^ . X ^+ +l^+-=(l+l + ^^ l + ^ + Tl^+ + - + ... = (1 +1+ IS +.-. + —+-..1. 11 1 Denoting the series 1 + 1+t^+ — +... + j~ + ...hje, we have the Exponential Theorem, namely 2 T 2 Ir It should be remarked that the above series for e" is convergent for all values of x [Art. 278]. 303. The quantity e is of very great importance in mathematics. It is obvious that it is greater than 2 and it is clearly less than 1 + 1 + 2"^ + 2"' + 2'^ + . .., and therefore less than 3. Its actual value can be found to be 2*71828.... * This requires more careful examination not only to find the limit of each term, but also because the limit of a sum is not necessarily equal to the sum of the limits of its terms unless the number of the terms is finite. This examination is however omitted here for the investigation in Art. 304 is preferable. EXPONENTIAL THEOREM. ^7') To prove that e is an incoimnemurahle number. If possible, let e = m/n, where m and n are integers ; then we sliould have m , , 1 11 1 1 -=1+1 + -+ +- + -^ + , 71^+1 , o + --- n + 1 |n + 2 n + 3 Multiply both sides by \n; then all the terms will become integral except 11 1 + : — r^^w — rr-x + t — . on/ - . ^s / — r^ + ••. 71+1 ' {n+2)(7i + l) (7H- 8) (/I + 2) (71 + 1) Hence 11 1 + 7 — — TV7 — — TT+z :;t^ — r-7^r^ — — t-x + ... n + 1 (?^ + 2)(?i+l) (?i+ 3)(7i+2)(7i + l) must be equal to an integer; but this sum is less than 111 -\- -, TT^ + T TY3 + --'' ^^^ therefore less than 71 + 1 (Vl + 1)"^ (71 + 1) z- / (1 =- 1 , that is less than - . But an intefi^er n + 1 I \ n + lj n ^ cannot be less than 1/n ; it therefore follows that e cannot be equal to the commensurable number mjn, 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. 8 r fft ni 'Letf(vi) denote the series l+m+;r-+ + . +.... 3 r Thus/(w)- 1+771 + -^+ + 17 + ' * Proceedings of the Cambridge Philosophical Society, Vol. v. p. 41;"). Substantially the same proof is however given in Cauchy's Analyse Alge- brique. 87 G EXPONENTIAL THEOKEM. n^ n' /(70sl+7l + -y- + +-^ + 2 l.s p and f{m+n) = 1 + (m + 7i)+ ^ ^ ^ +.., +^ — ^ + . . LP Now the coefficient of mV in / (771) xf(n) is and in /(7?i + n) the term wiV can only occur in r \s r + s ' ?' + s 1 and its coefficient will therefore be , , that is \r \s - ' " r + s r s Hence, as the series /(m), f(n) and f(m + n) are convergent for all values of m and n, and the coeffi- cient of any term in^n* is the same iny(??i) xfiji) as in f{in + n), it follows from Art. 280 that f{m)-xf{n)=f{m + n) (1) for all values of m and n. Now let a? be a positive integer; then from (i) we have /(I) x/(l) x/(l) + to CO factors, =/(l + 1 + 1 + to^ terms), ••• {/(1))'=/W (ii)- Next let 00 he a, positive fraction - , where p and q are positive integers. Then from (i) Ki)F=-^(f+f + f + to , terms) =/0.) = i/(l)j'',from(ii); '^ •••/g) = {/(!)}". Hence, for ^iW. positive values of a.', {/(l)J'' =/(^')- EXPONENTIAL THEOREM. .S77 Lastly, let x be liegative, and equal to — y, so that y is positive ; then/(- //) x/(.y) =/(0) from (i) ; but/(0) = 1, therefore /(- y) = Ijfiy). Hence /(•^') =f{- y) ^fJJ) = \f{\)\' ' """'"^^ ^ '^ positive, = {/(i)r={/a)r Hence, whatever a; may be, i/(i)r =/w- But /(l) = H--i + i + |+ se, 2 r OC X '7 therefore e"' =/(-^') = l+TT+9 + + r,+ 305. To shew that ,," _ n {n - ly + ^-^^^^ (72 - 2)" - ... = |?2. J. . i^ '~~~ We have from Art. 304 Also, by the binomial theorem, : ■,^ n(n — l) Now the coefficient of a;*" in ( •'-■ + 775+ 70"+ ••• ) ^^ y-cro, if r is less than n, and is 1 if r = n. Also the coefficient of .t*" in c'"-nfc<"""'^ + " :^~ ^ g'»-2)«- ... is jij„.-„(n-lK+'i<(^^i»(«-2)'-...j. Hence, equating the coeflicieuts of x^ in the expansions of the two expressions for (e' - 1)", we have j^|„»_„(„-i,» + !ii!L-i)(„-2)»-...[=i. 878 EXPONENTIAL THEOREM. The above theorem may be generalised as follows : We have d { (a - h)^x^ i ' (gax _ ghxyi=gni>x (g(a-J)a; _ l)n _ gnbx j (a - &) X + ^ r-^ + ... [ X" Hence, equating coefficients of a;" in the two expressions for -i((na)"-n(n^.a + 6)'*+ ^^^^ (w^ . a + 2&)« -...[ = (a- 6)». I 71 I A. m a J If we put na = x and b-a=y, the last result becomes t;"-n(a; + y)"+ !; ^ (a; + 2y) - ... = (-!)« .?/" In.. We have also, if X; be any positive integer less than n^ x^'-n{x + y)^+ ^ ^ (a; + 2?/)^- ... to w+1 terms=0. The following particular cases are of impoitance, k being less than n. p-yt2^ + ^^(^~^) 3^-... to 71 + 1 terms =0, and ni^-n {m- 1)*+ — ^ -^ (m- 2)''= - ... to w + 1 terms = 0. EXAMPLES XXX. / x\^ Ex. 1. Shew that the limit when n is infinite of 1 1 + - | is e^. Ex. 2. Shew that the limit when 71 is infinite of ( 1 + - J is c& . Ex. 3. Shew that n«+i-n(n-l)"+i + ^^^?^^(n-2)«+i- ... =ln\n + l, " Ex. 4. Shew that «~+2 _ „ (^ _ i)n+2 + ^^^^ (w - 2)"+a - ... = ^ (37i + 1) |w + 2). LOGARITHMS. 370 Ex. 5. Shew that 2 4 6 Ex.6. Shew that e-i = -j-5-+ T— + 7^ +... li II l_Z Ex. 7. Shew that 3 1+2 1+2+3 1+2+3+4 -e_l+ ^ + ^ + ^^ +... Ex. 8. Shew that /_ 1 1 1 \2 , /, 1 1 1 \2 Ex. 9. Shew that 1 1 e-i = + + A + 1.3 1.2. 3.5 '•• 1.2.3...{2;i-l)(2/t + l) Ex. 10. Shew that e-1 e + 1 (1 1 1 •) (1 1 1 ) Ex. 11. Shew that e^ + l L 1 1 1 ) (, 1 1 1 Ex. 12. Shew that the coefficient of a;" in the expansion of '• f • (l + 2x) (l + 2:r)-^ (l + 2a;)3 1 + — j^ + — j2— + -jj- + ... 18 2''e !n Logarithms. 306. Definition. The index of the power to which one number must be raised to produce a second number is called the logarithm of the second number with respect to the first as base. Thus, if tt^ = y, then x is called the logarithm of y to the base a, and this is expressed by the notation x = log„ y. 380 PROPERTIES OF LOGARITHMS. We proceed to investigate the fundamental properties of logarithms, and to shew how logarithms can be found, and how they can be employed to shorten certain approxi- mate calculations. 307. Properties of Logarithms. The following are the fundamental properties of logarithms. I. Since a° = l, for all values of a, it follows that loga 1=0. Thus the logarithm of 1 is 0, luhatever the base may he. II. If log,, x = a, log„ y = l3, log„ z = y, ... then x = a°-, y = a^, z = a'^, ...\ .'. xyz ... =a'^ .a^ .ay ... = a''+^+y+- .-. log„(^2/^...) = a+/3 + 7+ ... = loga ^ + log. y + log„ z + ... Thus the logarithm of a product is the sum of the logarithms of its factors. III. If log„ x = a, and log^ y = (3', then a; = a", y = a^, and .*. ^-hy = a*~^; .-. log„ {x-ry) = a-^ = log^ X - log„ y. Thus the logarithm of a quotient is the algebraic differ- ence of the logarithms of the dividend and the divisor, IV. If ^ = a* ; then ^™ = a'"*, for all values of m. Hence log,, ^"' = ma = m log„ x. Thus the logarithm of any 'power of a number is the product of the logarithm of that number by the index of the 2:)ower. V. Let \og^ x — a, and logj, x = /3 ; then x = a'^ = b^; and hence a = b'^, and a^ = b. LOGARITHMIC SERIES. .SSI Therefore = logr a and -^ = loW>. a ^ y3 ° Hence log„ 6 x loiT;, a = -:: x ^ = 1. pa Also /3 = a log^ a, that is logj, a; = log^ x . log,,a. Hence the logarithm of any number to the base b will be found by multiplying the logarithm of that number to the base a by the constant multiplier log^ a. 308. The logarithmic series. Let a = e\ so that /j = log, a; then ^-^ = e'^'-= e^^iogaa^ Hence from Art. 304, we have Iz |r Put a = l + y; then we have (1 + 2/)' = 1 + ^ log, (1 + y) + ji {^ log, (1 + y)j--' + . .. Now, provided y be numerically less than unity, (1 -{-yY can be expanded by the binomial theorem ; we then have _ , x{x-l) 2, ,x(x—l)(x-2)...(x-r+V) , 1+0;?/+ '^ ^ ^ 7/^+...+-^ ^^^ j-^^-^ 2^^y +... = l+^log„(l+3/)+-i{^log,(l + 7/)}^4-... The series on the riorht is convergent for all values of x and y, and the series on the left is convergent for all values of X provided y is numerically less than unity. Hence, for such values of 3/, we may equate the coefficients of x on the two sides of the equation. We thus obtain log.(l+y) = y-^ + |'- + (-ir'^+... This is called the logarithmic series. 382 LOGARITHMIC SERIES. Ex. 1. To express a^ + b^ in terms of powers of ah and a + b. From the identity (1 - ax) (1 - 6a;) = 1 - (a + b) x + abx^ = l-sx+px% where s is put for a + b and p for ab, we have loge (1 - ax) + loge (1 - 6a;) = loge (1 - sx +px"). Hence lax + -^ + -^+ j + {bx+ 2~ "^ ~3" ^ J = i^xis-px) + ^^^^\^-^^^ + [ . Equate the coefficients of x'^ on the two sides of the last equation. [This is allowable since the series can clearly be made convergent by taking x sufficiently small.] Then the coefficient of x^ on the left is -(a^+h^). On the right we have to pick out the coefficient of a" n ' from the terms (beginning at the highest in which it can appear) /».» <,.n-l /v.n-2 (s -pxY^ + — - (s -px)^-'^+ (s -px)^~^+. n n-1 71-2 the coefficient of x'^ is therefore Hence we have a^+b^={a + by' - nab {a + 6)"-^ + ''}l^1z3 a%^ {a + 6)'^"^ JL • ^ w(w-4)(n-5) „-„ . , J... g , 1 — ^9 -aW {a + by' " + ... JL • ^ • o ... + (-l).,^(^-^-l)<^^7-^)-('^-^'- + ^)a^6>-(a + &r-^''+ Ex. 2. Shew that, if a + 6 + c = ; then will 10 {a7 + 67 + c7) = 7 (a2 + 6^ + c^) (a^ + 6^ + c^). Put -p for 6c + ca + a6, and g for abc ; then we have the identity {l-ax){l-bx){l-cx) = l~px^-qx^. ^^ Now take logarithms, and equate the coefficients of the different powers of x in the two expansions. This gives - (a** + 6*" + c*") in terms of p and g, and the required result follows at once. [See also Art. 129.] cauchy's theorem. 383 Ex, 3. To express a'^ + i^ + c** in terms of abc and bc + ca + ab, when a + b + c = 0. Put -p = bc + ca + ab, and q = abc\ then we have the identity (1 - ax) (1 - bx) (1 - ex) = 1 -px^ - qx^. Hence, by taking logarithms, and equating the coefficients of Hke powers of x, we have - (a* +&" + c") = coefficient of a;" in 'L- x^'^ip + qx)*, n If- ^^ ^ ' ' which gives the required result. The terms in S - .t^*" {p + qx)'' which contain .t^"»±i are — - a;-«»» ip + qx)^ + x'^^^+'^ (p + qx)-'^+^ + ~ a;4'»+4 ( » + ga;)-'"+2 + . . . + ^ r a;6^-2 (_p + (/a;)3'"-i + n— a;*^ (p + ga;p^ Now by inspection we sec that the coefficient of x^^~^ in each of the above terms in which it occurs contains pq as a factor ; and also that the coefficient of a;*>"^+i in each of the terms in which it occurs contains p^q as a factor. Hence, when a + b + c = 0, a^+b'^ + c'^ is algebraically divisible by abc {be + ca + ab) when n is of the form 6m -1, and a**+&" + c** is algebraically divisible by abe {bc + ca + ab)^ when n is of the form 6m + 1. If we put c= -{a + b), bc + ca + ab becomes -{a^ + nb + b^), and we have Cauchy's Theorem, namely that a"+?>"- (a + 6)'' is divisible by ab{a + b) (a^ + ab + b-) when ii is of the form 6m -1, and by ab {a + b) {a^ + ab + b^)^ when n is of the form 6m + 1. [Sec paper.s on Cauchy's Theorem by Mr J. W. L. Glaisher and Mr T. Muir in the Quarterhj Journal, Yo\. xvi., and in the Messenger of Mathematics, Vol. vm.] 309. In order to diminish the labour of finding the approximate vahie of the logarithm of any number, more rapidly converging series are obtained from the funda- mental logarithmic series. Changing the sign of y in the logarithmic series we have log«(l+2/) = 2/-f + |-| + (0, 2 3 4 log.(l-2/) = -:y-|-|-|- (ii)- 384^ LOGARITHMIC SERIES. Hence log, ^-— ^ = log, (l + y) ~ log, (1 - y) %J = 2(2/+^ + ^"+ ) (iii). Put — for ^ — ■ , and therefore for ?/ ; then n ^ — y m -\-n "^ log,- = 2[ — ■ — ^ + - ■\--[ +...k..(iv). w (m + ^ o \m + nj 5 Vwi + ?i/ J ^ ^ We are now able to calculate logarithms to base e without much labour. For example : — Put m = 2, n = 1, in formula (iv) ; then from which it is easy to obtain the value log, 2 = •693147... Having found log, 2, we have from (iv) log.3-log,2 = 2|^ + |.^3+^.^+...} = -405465. Hence log, 3 = -693147 + -405465 = 1-09861. Proceeding in this way, the logarithm to base e of any number can be found to any requisite degree of approximation. 310. Logarithms to base e are called Napierian or natural logarithms. The logarithms used in all theoretical investigations are Napierian logarithms; but when approximate numeri- cal calculations are made by means of logarithms, the logarithms used are always those to base 10, for reasons which will shortly appear: on this account logarithms to base 10 are called Common logarithms. We have shewn how logarithms to base e can be found; and having found logarithms to base e, the logarithms to base 10 are obtained by multiplying by the constant factor log^/, or by l/log,10. [Art. 307, V.] This constant factor is called the Modulus: its value is '43429... EXAMl'J.ES. 385 EXAMPLES XXXI. 1. Shew that log (.« + 7b) = log x + log ( 1 + - J + log (1 + ) +log( 1 + — — ) + +log(l + = ) 2. Shew that log, 7l2 = l + (^-+-)- + (^- + ^^)-,+ 'J4)i^(^4)i^-^ toinfimty. 11 11 11 to 3. Shew that log, ^10 ={1 + 39 + 5 92 +79^ + iDfrnityj + j^+ 39-3+595+ 797+ toinfinity|. 1 Shewthatlog,2-J = ^3+-3-i-^ + ^+ to infinity. 5. Shew that ^-|-^+_J— +j,-?--^+ to iu- finity ^ 3 log, 2-1. 6. Shewthatlog,^^ = 2{^+l-Ap {2x- ly 1 1 '^5{2x-ly^ ni xi . 1 ^ - 1 1 aj' - 1 1 r/.-^ - 1 7. Shew that log, x = + - r— „ + - ^^-^ + ^^ a; + 1 2 (a; + 1)- 3 (a; + 1)^ 8. Shew that a + x 2ax 1 / 2ax V' 1 / 2ax V s. A. 25 386 EXAMPLES. 9. Shew that 5 H'^'a+s- j = !--«?- 3 (r^^'j+sirr^j- 10. Shew that {log,(l+x)P = 2{i.'-J(14)x'4g44).'-...}. 11. Shew that, if logg (l+rr + jc^) be expanded in powers 12 of a;, the coefficient of x" is either - or , and distinguish the cases. 12. If logg [1 — x + x^) be expanded in ascending powers of X in the form a^x + a^x' + (X^o;^ + , then will a^ + ag + ^a + • • • = I loge 2. X -f* Q^ -f- 9^"' 13. Expand logg ~ ^ in ascending powers of a;. X "~ y> "T" yl/ 14. Shew that 1 x x^ a? — I 1 — — — Y h ... n n{n-v\) ?i (?^ + 1) (?i + 2) n (ri + 1) (w + 2) (n + 3) =-fi-ir H/ iX/ *Aj + 77^7 ^-^7-7 :^ + (?^+l) [2(^yi + 2) [3(m + 3) 15. From the identity 2 log (1 — .r) = log (1 — 2a; + a;"), prove that2'-».2-' + "("-^> 2-'- "/ + y' and 5' = a;?/ (x + y). 24. Shew that, (i) if n be any uneven integer, {h — c)" + {c-ay + {a-by will be divisible by (b-cf + {c-ay + (a-bY; (ii) if n he of the form 6m ±1, it will be also divisible by (6 — cy + (c - ay + (a- by ; and (iii) if n be of the form 6m + 1 it will be divisible by (6 — cy + (c - ay -\- {a - by. Common Logarithms. 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 -^ 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. niARACTKRISTTCS FOUND BY INSPECTION. ,S89 312. Tlie characteristic of the logarithm of any numher can he ivritten down by inspection. For, if the number be greater than 1, and n be the number of figures in its inteoral part, the number is clearly less than 10" but not less^han 10"~\ Hence its logarithm is between n and w — 1 : the logarithm is therefore equal to ti — 1 + a decimal. Thus the characteristic of tlie logarithm of any numher greater than unity is one less than tlie member 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 — (?i+ 1) + a decimal fraction, the characteristic being ~{n-\- 1). Thus, if a numher less than unity he expressed as a decimal, the characteristic of its logarithm is negative and one more thari the numher of ciphers before the first signifi- cant figure. For example, the characteristic of the logarithm of 3571'4 is 3, and that ot -00035714 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 "i. 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 lo^jarithms. 'o"''-' '^o' 890 USE OF TABLES OF LOGARITHMS. In all Tables of logarithms tlie mautissae 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 nuinher. 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 logio {N + oc) - log^„ N = log^ . . 1 = ytA itt approximately, when -^- is small, jjb being the modulus l/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 = •5529722; 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 ^^ths. of -0000121 is -00000847, which is nearer to -0000085 than to -0000084. Hence the nearest approxi- mation we can find to the logarithm of 3-57247 is '5529601 + -0000085 = •5529686. The characteristic of the logarithm of 357 "247 is obviously 2, and therefore the logarithm required is 2^5529686. ~^ 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 COMPOUND INTEREST AND ANNUITIES. 891 between these two. Now the dififeronce between '5.529601 and the given logarithm is ^— of the difference between the logarithms of 3*5724 and 3*5725 ; and hence, by the principle of proportional differences, the number whose logarithm is -5529052 is 3-5724+^ X •0001 = 3-5724+ •00004 = 3-57244. A. aj A. [The approximation could only be relied upon for one figure.] Thus -5529652 = 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.] T. 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 + r)\ Similarly the amount at the end of n years will be P (1 -f ?')". Thus ^ =: 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 . 392 ANNTTITIES. Ex. Find the amount of £350 in 25 years at 5 per cent, per annum. 5 Here P = 350, r=:^ and 9i = 25; . /. log ^ = log 350 + 25 log ^1 + ^^ = log 350 + 25 (log 105 - log 100). From the tables we find that log 350 = 2-5440680 and log 105 = 2-0211893; hence log 4 = 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 he 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 n years at lOOr per cent, per annum must be just equal to A. Hence from I. P-=A{1+ ?•)"". 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 ^ (I + r)~^ second -4(1 + r)~^ nth J[(l+7')-". Hence the present value of the whole is 1 . I 1 ] A{^ 1 [1+r (l+rf '" ' (l+ryj r [ 0-+ry\' Ex. Eind the present value of an annuity of £30 to be paid for 20 years, reckoning interest at 4 per cent. Here ^ = 30, 7^ = 20, r = ~ = ^^^. 100 2o EXAMPT.ES. 393 Hence the present value = .^0 x 25 il - ( ..,. I \ . ^j =20{log25-log26} = 20 {1-3979400 - 1 -4149733} = 20 (--0170333)= - •340606 = 1-659334 = log -450389, from the Tables. Heuce the value requii-ed = 30x25 x (1- •456389) = £407 7 .. EXAMPLES XXXII. The following logarithms are given log 1-02 = ■0086002 log 1-6386 = •2144730 log 1025 = •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 logl-OG = -0253059 log 2-0829 = •3186684 log 1-1467 = •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 = •170731U log 5-577 = •7464006 log 1-4817 = •1707603 log 6^3862 = •8052425 log 7-4297 = •8709713 log 7-4298 = •8709771 1. Find ^105. 2. Find iy51. 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 16 years at 5 per cent, per annum, and in 18 years at 4 per cent, per annum. 394 EXAMPLES. 5. Find tlie 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 2| 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 1 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, reckoning interest at 6 per cent. CHAPTER XXV. Summation of Series. 315. We have already considered some important classes of series, namely the Progressions [Chapter xvil], 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 S^. 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 tlie general term of the series, u,^, as the difference of two expressions one of which involves n — 1 in the same manner as the other involves n. For example, in the series a a a x{x -^ a) {x + a) {x + 2a) (x + 2a) (x -\- 3a) the nth term, namely r= , is equal to {x + n— \.a) {x-\-na) Hence the series may be written X -\- {n — \)a X -\- na 396 SUMMATION OF SERIES. \x x + aj \x-\-a x-\-2a/ \x + 2a x-{-Sa. + \ 7 T^; -(■ 3 a,nd it is now obvious that all [x -\-{n — l)a X -\r na] the terms cancel except the first and last ; , ^11 na hence o„ = X X -\- na X (x-\- na) ' Ex. 1. Find the sum of n terms of the series 111 1 1 1- — + A 1- 1.2 2.3 3.4^ ^ n{>i + l) Ex. 2. Find the sum of n terms of the series 12 3 n [2 "^]3"^TI "^ -••••• '^Ini-l"^ Ans. 1 71+1* Here v,, = i -3- . Ans. 1 - — — n+1 \n + L '' \n Ex. 3. Find the sum to infinity of the series 111 1 , 3 11 4|2 5|3 (71 + 2) lii Here w„ = , ^ ^ . Ans. ~ . *7l" r? + 1 7? + 2 ' ' 2 ' a. Ex. 4. Find the sum to infinity of the series 3 5 7 2n + l 12 . 22 + 22 . 3^ "*" 32. 42"^ '*"7i2 {n + iy ' Here w„ = -o - , , i,o • ^^« 1 • "■ 72,2 (71+1)- Ex. 5. Find the sum of n terms of the series 12 3 n _ + , .. . + :r— o . r, + + 1.31.3.5 1.3.5.7 1.3.5...(2?i + l) U, 1 I __, 1 L ''~1 . 3 . 5...(2u - 1) 1.3. 5. ..(2/1 - 1) (271 + 1)J • SUMMATION OF SERIKS. .'U)7 Ex. 0. Sum to infinity the series 1.3 3"^ 3.5 • 33"^ 5. 7 • 3^'"^ "''(2/t- l)(2n+ 1) 3'*"^ [ „. 71+1 _1 / 3_ _ 1 \ ^^^^ (2»-l)(2;i + l)~4 V2/i-l 2n + l)' ^"'""2^r:i3^i"2;TTl3^J' '*''"* 4* Ex. 7. Find the sum to infinity of the series 1 111 1 Ex. 8. Find the sum of n terms of the series .•2\ "*" /I ~,'2\ /I ».:<\ ■" /I „s\ /I -.4\ "^ Ans. X x^ x^ (l-a:)(l-a;2) "^ (l-a;^) (l-.x"') "^ (1 -a;^) (1 -a;'*) 1 1 (l-a;)2 (l-a;)(l-a;"+i)* 318. To find the sum of n terms of the series [a{a + l) ., . (a + r-1 . /;)} + {(« 4- 6) (a + 2Z^) . . . (a 4- rh)] + . . . + {(a + ?i - 1 . 6) (a + 7?6) ... {a-\- n + r — 2 .h)] + ... 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 accordinsj to the same law but with one factor added at the end of every term, and let v^ be the nth term of this new series, so that ^„ = {(<^ + 11— 1 .h) {a + nh) ... {a + n + r — 1 . h)\. Then v^ — v^_^ = {(a + 71 — 1 . 6) (a + nh) ...(a + n -\-r — l . b)} - {(a + 7^-2 . h) (a + ti^ . 6) . . . (a + n + ?• - 2 . 6)} = {(a + 71 - 1 ./;)... (a + 7t + r-2.6)}{(a+7i+r-l .h) - (a + ?i - 2 . 6)} = (?• + 1) ^ [(a + 71-1 . 6). . .(a + ?i + 7' - 2 . 6)}. 398 SUMMATION OF SERIES. Hence v^ — v^^_^ = {r -\-\)h xu^. Changing n into ?i — 1 we have in succession V , —V _^ = ir + V)h xu , n-l n-2 \ / n-\ v^ — v^ = {r + l)hx u^. Also v^ — v^ = {r-\-l)h X u^, where v^ is the term preceding v^ which is formed accord- ing to the same law, that is Vq = {(a — 6)a(a + 6)...(a + r — i 6)}, so that v^ is obtained by putting ti = in the expression for v^. Hence by addition Ex. 1. Sum the series 1.2 + 2.3 + 3. 4+ +?i{w + l). Here m„=w(w + 1), v„= n{n + l){n+2), ?;o = 0.1.2, r = 2, and 6 = 1. Hence S^=~n{n+\) {n-\-2). o Or, by using the above method without quoting the result, which is preferable in very simple cases, we have n(7i + l)=-{?l(?l + l)(?l + 2)-(7Z-l)7l(?l + l)}, (n - 1) n=\{{n - 1) n (n + 1) - (n - 2) (n - 1) n}, 1.2 = ^{1.2.3-0.1.2}. Hence /S„=-7i (w + 1) (?i + 2). ^-^ Ex. 2. Sum the series 1 .2. 3 + 2.3.4+ +n(7i + l) (;i + 2). Ans. - n (n + 1) (?i + 2) (?i + 3). SUMMATION OF SERIES. 399 Ex. 3. Sum the aeries 1.2.3.4 + 2.3.4.5 + +7i(?i + l) (n + 2) (n + 3). Ans. -n(n + l)(n + 2) (n + 3)(n + 4). Ex. 4. rind the sum of n terms of the series 3. 5.7 + 5.7.9 + 7.9.11 + Here «„={2n + l)(2n + 3)(2/i + 5), r„=(2n + l) (2/t + 3) (2n + 5) (2n + 7), t;„=1.3.5,7, r = 3, and 6 = 2. HenceSf„ = ^{(2?j + l)(2n + 3)(2« + 5)(2n + 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 u^=n {n + 2) = n (n + l) + n. The sum of the series 1 . 2 + 2 . 3 + . . . + n (?i + 1) is J{n(n+l)(n + 2)-0. 1.2}, and the sum of the series 1 + 2+ ... +7i is - {n{n + l) -0.1}. Hence the required sum is - ?i (n + 1) (n + 2) + ^ n (>i + 1). Ex. 6. Find the sum of the series 2. 3. 1 + 3. 4. 4 + 4. 5.7 + + (« + 1) (n + 2) (3/4- 2). Here !/«=(« + 1) (n + 2) (3;t - 2) = 3n (n + 1) (n + 2) - 2 (» + 1) (» + 2). •■• 5„ = 7{n(n + l)(n + 2)(Ti + 3)-0.1. 2.3} -|{(u + l)(ri + 2)(n + 3)-1.2.3} = y^2(^"-^)(" + l)(^^ + 2)(n + 3)+4. 400 SUMMATION OF SERIES. 319. To find the sum of n terms of the series luhose general term is ll{{a + n — 1 . h) {a -\- no) {a + ??. + 1 . 6)...(a + ?^ + r- 2 . h)}. Consider the series which is formed accordinor to the same law but with one factor taken aiuay from the beginning of each term, and let v^^ be the wth term of this second series, so that v^=l j {{a-\- nh) . . . (a + ti + r — 2 . 6)}. Then 1 [{a+nh) ...{a-\-n + r-2 . h)] 1 [{a + n — 1 .h) {a -\- nh). ..{a-\-n + r —"S . h)] [{a-\-n — 1.6) {{a + n-l .h).. .(a+ ti+r-2 . 6)} — {a+n-]-r— 2 . Z))j ; .•• '?;n-Vt = -(^-l)^x Un- changing n into ?^ — 1 we have in succession v^ — V j^ = — {r — 1) h X u,y Also v^ — Vf^ = — (r— l)b xu^, where v^ is the term which precedes v^ and which is formed according to the same law, that is v^=l / {a(a + b). ..{a-i- r-2 . b)}. Hence, by addition, v,^-v, = -(r-l)bxS^; r^ .■.S,={v,-vJI{r-l)b. 1 1 1 Ex. 1. Sum the series jr— _ + 7: — ■. + ... + 2.S^ 3 . 4:^ '" '^ {n + 1) {n + 2)' SUMMATION OF SERIES. 401 XT e 1 jl M ^ 1 Hence 0,. = :; — r -^k - c^r =7i ?; • *' 1 . 1 (2 71 + 2J 2 ?i + 2 Ex. 2. Sum the series - — - — -— + --— — — _ + ... h 1.2.3.4 2.3.4.5 ^71 (/i+l){?i + 2)(n + 3) to 71 terms and to infinity. xlere u = 1) ^ ■ • " 7e(;( + l)(7i + 2)(7i + 3)' » (» + l) (7i + 2) (n + 3) ' 1 1.2.3 1 *'o=i o o> ^^-i. and 6 = 1 Hence 8^ = " 3.1 (1.2. 3 (?i + l)( A ,.111 ^''^ ^- = 3 -172: 3 = 18 _1_ I 71 + 2) (71 + 3) j* Ex. 3, Sum the series ^ „ ^^ H — — -- + . . . -4 . 3.7.11 7.11.15 (4u-l) (471 + 3) (4// + 7) ^ns. Sf/ is written for the sum of n terms of the series 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 + b, a + 21), ... in arith- metical progression. The result is {a + nb)^^-a'^^-nb^^ = {r+l)hS^^ + ^^t-^lT b^Sj-'^+ ... + {r + l) h^S,^\ X.J where S^** sa*- + (a + 6)'- + . . . + (a + u - Iby. 322. Piles of Shot. To find the number of S2'>herical balls in a pi/ramidal heap, when tlie base is (I) an 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 iayer 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 + (w - 1) + (?i - 2) +. . .+ 2 + 1, that is \n (n + 1). The whole number of the balls in the pile will therefore be ^ [n {n + 1) + {n - 1) n +. . .+ 1 . 2}, that is Jn (n + 1) (n -f 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 7i' + (n - ly + (n - 2y +. . .+ 1', that is -^n {n + 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, ?i?/t will be the number of balls in the base and therefore the whole number of the balls will be, n being greater than '/n, nm + (n - 1) {m - 1) + (?i - 2) {m - 2) + . . . (vt - m + 1)1 = {n — nL + m) ni -\- {ii — m + TYi — l)(v;t — l) + ...(;<. — 7/t + 1) 1 2 406 FIGURATE NUMBERS. = (w - m) {m + (m - 1) +. . .+ 1} + m^ + (771 - 1)' +. . .+ 1 = ^ (71 — m) m (m + 1) + ^m (m + 1) (277i + 1) = -Jm (m + 1) (371 — 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; and there are 77 . 4 . 5 . 6 missing from the complete pile; hence the 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 = 2/1 (n + 5) from n=ll to 7z = 20. Ans. 3260. 323. Figurate 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 Tith term of the third order will therefore be (1 + 2 + 3 + ... + tz), that is \n (n + 1); the nth term oi the fourth order will therefore be ^ {n{n + 1) -{- {n - 1) n +...-h 1.2}, that is — ^ —~ ; the nth term of the fifth order will therefore be --- {n (71+1) (n + 2) + (n -l)n(n + l)+...+ l .2 . 3}, that POLYGONAL NUMBERS. 407 is n {n + 1) (n + 2) (/? + 3); and so on, the ntii term of the rth order being 7i(7i + l)(?i+ 2)...(n-\-r- 2) r- 1 824. Polygonal numbers. Consider the arithmetical progressions whose first two terms are respectively 1, 1 ; 1, 2 ; 1, 8; 1, 4; and so on. Then the series formed by taking 1, 2, 8,..., ?i of the terms of these different arith- metical progressions, namely the series 1, 2, 8, , n, 1, 3, 6, , ^n(n+l), 1, 4, 9, , 7^^ 1, 5, 12, , 7i-\-^n(n-l), 1, r, 3?' — 8, . . ., n + ^n (n — 1) (r — 2), ... are called series of linear, triangular, squat^e, 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 ?i + Jn (n — 1) (?' — 2) is in (n + 1) + i (w - 1) 71 (?i + 1) (r - 2) [Art. 818]. EXAMPLES XXXIII. Fiud the sum of n terms of each of the following series, and tiad also the sum to iutinity when the series is convergent. 1. 4.7.10 + 7.10.13 + 10. 13. 16+... 2 1_ 1 1 3.7.11 "^7. 11.15 "^ll. 15.19"****' 3. 1.3.4 + 2.4.5 + 3. 5.0 + .. . 4. 1.5 + 3.7 + 5.9 + 7. 11 + ... 408 EXAMPLES. 5. 1.2. 3 + 2. 3. 5 + 3. 4. 7 + 4. 5.9+... 6. 1.2^^ + 2.3^ + 3.4^ + 4.5^+... 7. 1.3^ + 3.5^ + 5.72 + 7.92+... 1 1 1 1 1.3.7 "^3.5.9 "^5.7.11 "*" 7. 9. 13"^"' J 1 1 1 17374 "^2.4.5 "^3.5. 6 ■^4.6.7"^'" ,^4567 1.2.3 2.3.4 3.4.5 4.5.6 n 1 2 3 4 1.3.5 "^3.5.7"^ 5.7.9 "^7.9. 11 ^•■' nn 3 4 5 6 1 9. . J J 1 u 1.2.42.3.53.4.64.5.7 1 1 1 1 ^^' T'^rr2'^l + 2 + 3 + l + 2 + 3 + 4"^-- ,, P' P + 22 P + 2' + 3^ r + 2^+3^ + 4' 14. y-+-2-^ 3 + 4 +••• 15. 1 . P + 2(P + 2') + 3(r+22+ 3') + 4(l-^ + 2^ + 3^ + 4^)+ ... 16. a'+{a + by + {a + 2bY-\-... 17. a^ + {a + by + {a + 2by+... 18. r+32 + 5'^ + 7'+... 19. P + 5'+9'+13^+... 20. Shew that p _ 2« + 3^ - 4' + . . . + (2n + 1)' = (71 + 1) (2';i + 1). 21. Shew that P - 2' + 3' - 4"^ . . . - (2/*)' = - t^ (29^ + 1). 22. Shew that !» _ 2^ + 3' - 4' + . . . + (2?z + 1)' = 4:01' + 9?^- + 6?i + 1. 23. Find the sum of the series 1 . n+ 2 («- 1} + 3 {n-2) + ... +n. 1. EXAMPLES. 409 24. Find the sum of the series n . n + (n - 1) (n + \) + (n-2)(nf 2) + ... + 2(2n-2) + l . (2>t-l). 25. Find the sum of n terms of the series ab + (a- 1) (6 - 1) + (a - 2) (6 - 2) + ... 26. Prove that, if SJ =V-\-2' + ... n"; then wiU (i) 5^ * = 6*S' ^ X .S' ^ - ^ =. \ / n n n n 27. Find the sum of the following series to n terms : _3_1 _4_ 1_ 5 1 ^^'^ 1.2 2 "^2.3 2^' "^374 2"^"^ •■• ..... 4 /2\ 5 /2y 6 /2y (^") rrrs (7 j ^ 2-7371 (7) ^ 3:^-5 (7) ^ - / x 9 /3\ 10 /3V 11 /3V 1 . 2 . 3 V4y 2 . 3 . 4 V4/ 3 . 4 . 5 V4. ^ .. 15 /6\ 16 /6y 17 /GV ^^•^) 1:^3(7)^27374(7) -^3:4:5(7) ^- 28. Shew that the sum of all the products of the first n natural numbers two together is — (n — 1) 11 (n + 1) (3n f 2). 29. Shew that the sum of all the products of the first n natural numbers three together is — (vi - 2) (n — 1) n' (n + l)^ 30. Shew that the sum of the products of every pair of the squares of the first n whole numbers is g~nK-l)(4«'-l)(5n+G). 410 SUMMATION OF SERIES. 325. To find the sum of n terms of the series a a{a + x) a{a + x){a + 2x) h^ h{h + x)^ h{h + x){h + 2x)'^"' a(a + x)...(a + n— Ix) + — ^ ' — ^^ ■ -^4-... h (h -\- x). . .(h -\- n — 1 x) 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 numerator 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 V = a(a + x). ..(a-{-7i — Ix) {a + nx) b {b + x). . .{b + n — 1 x) Then ^n - -^n-, = _ a (a -[- x). . .(a + n — 1 x) (a -{- nx) h(J) -{■x)...{b -\- n— 1 x) a{a + x)...{a + n — 1 x) b(b + x)...(b + n—2x) a(a + x)...(ct+ '^ — ^ x) b(b +x)...(b + n — So also v„_j - v^_^ = u^^x(a + x-b) =-^ ](a + n^) — {b+n — l x) \ ; lx)[ J v^ — v^ = u^x{a + x — b), * 1 (a-{- x) , . Also v^=a - — -. — - = {a + X) u^ = u^ X (a ■{•x — b)-\- bu^. SUMMATION OF SERIES. 411 Hence S,^ x (a -\- x — b) = v^^ — a ; .'. S = a (a + x)...(a + nx) a + X - b [b (b + x) . . .(b + n - 1 x) The sum of n terms of the series a a(a — x) a(a — x) (a — 2x) b ~ b {b + x) '^ bjb + x)(b + 2x) ~ • • • 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 b a-\-b — x a{a — x) _ 1 b (b + x) a-\-b — X a a {a — x) l"^ b a {a- x) a {a— x) (a — 2jj) b "^ b{b-\-x) / -. Y,_i « (a — ^)...(a — 71 — 1 a;) = (- 1)" -I a(a — x)...(a — n— I x) _b(b-\- x).. .(b + ?i — 2 x) a (a — x). . .{a — nx) + b(b + x)...(b-\-n-l x)_ Hence 6' = a a-\- b — X i-(-i) ^(a — x) (a — 2x).. .(a — nx)' b{b -hx)...{b + n—1 x) 2 '^5 '^5 8 Ex. 1. To lind the sum of n terms of the series - + *''' + — — '— + 33. 63. 6. y 412 SUMMATION OF SERIES. We have 2 1/2.5 2\ 2.5 1/2.5.8 2.5 3 2V3 iy'3.6 2V3.6 2.5.8...(37?-l) _ 1 ( 2.5. 8. ..(371 + 2) _ 2. 5 . 8...(3n- 1) ] 3. 6. 9. ..3/1 ~2 I 3.6.9...3n 3 . 6 . 9...(3/i-3)j' ^ „ 1 f2.5.8...(3?i + 2) 2 Hence o« = ;r ' ** 2 [ 3. 6.9...3ri 1 [This particular series is a binomial series, the successive terms being _2 the coefficients of x, x^, &c., in the expansion of {l-x) ^. Hence [Art. 287] l + >S„ = 8um of the first (n + 1) coefficients in the expansion of (1 - a;)~^= coefficient of x"" in (1 - x)~^ x (1 - a;)-i, that is in (1 - x)~^. Ex. 2. Find the sum of n terms of the series - + „-^- + " * 3 3.7 3. 7.11 2 (6. 10...(47i + 2) 3 [3. 7. ..(471 Ex. 3. Find the sum of n terms of the series ^ VI m{vi — l) VI {rn — 1) [m — 2) I ■*" ~T72 1.2.3 "*■ •'• ^-4- Am ( l^n-l (^»-l)(^^^-2)-(m-7^ + l) ^"^^ ^ ^^ 1.2...(n-l) (n-1) 32G. The sum of ti + 1 terms of the series a^ + a^x + a^x^ + . . . + cijc^, where a„ is any integral expression of the ?'th degree in n, can be found in the following manner. - /S^„ = ttg + a^x + a,jjc^ + . . . + a,^a;", (1 - xy^' = 1 - (r + 1) ^> + (r^L^^: ^.-^ _ . . . + (_ ly^^x^K Hence S^ x (1 - ^)'^' = «„ + \^i - 0' + 1) "ol *'+••• + K - 0' + 1) «i>-i + i~2 ^>--^ "•••] '^■'' + ••• SUMMATION OF SERIES. 41.3 T^ow a^ is by supposition an integral expression of the rth degree in p ; hence where A^, j4, ._,,..., J.^ do not contain p. Also, by Art. 305, the sum of the series / - (r + 1) ( p - 1)^= + t±}lr {p-2y-...to(r+2) terms, is zero for all integral values of k less than r + 1. Hence (r + l)r ap-(r+ l)ap_, + ^ ^ - ^g- ... to (r + 2) terms is zero for all values of _p. All the terms of the product >S^„ x (1 —a;)"^^ will there- fore vanish except those near the beginning, or the end, for which the series a^— (r + 1) ap_j +... 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. Hence >Sf„ X {l-xy^' = a^ + {a^-(r 4- 1) a,] x+... + l«r-x - (^ + 1) «r-, + ...+(- '^)\r + 1) aj a;*- + ...+(-ir^a„^"^'-^ whence the value of >S' is found. n Ex. 1. Fiud the sum of the series l + 2x + 3x2 + 4x3+ +(n + l)a;^ ^n+i = l + 2-c + 3x2 + 4a;3+ ^{n + l)x\ {l-x)- = \-2x + x~\ .-. (l-a:)-'S,»+i=l + a;«+i{n-2(n + l)} + (n + l)rc«+^ [all the other terms vanishing on account of the identity h-2{k-l) + {k-2)=0'] = 1 - (n + 2) a;"+i + (7t + 1) a;«+2 J 'n+1 (l-.r)2 iX-xf 414 SERIES WHOSE LAW IS NOT GIVEN. Ex. 2. Find the sum oin + 1 terms of the series l3 + 23a; + 33^2+ + {n + lfx\ Sn+i = 1^ + 2^« + 33a;2 + + {7i + lfx\ {l-x)^ = l-ix + 6x^-4:X^ + x^; ,\ /Sr„+iX(l-.'c)^ = l + (23-4)a; + (33-4.23 + 6.13).'i;3 + (43-4. 33 + 6.23-4. 13) a;3 + I _ 4 (II + 1)3 + 67i3 - 4 (n - 1)3 + (n - 2)3} a;«+i + {6 (w + 1)3 - 4w3 + (n - 1)3} a;»*+3 + |_4(;i + l)3 + „3|a;«+3 + (n + l)3a;"+^ [The other terras all vanishing, since F - 4 (/c - 1)3 + 6 (/c - 2)3 - 4 (A; - 3)3 + (/c - 4)3 = identically.] BenGeS,,+^ = [l + 4x + x^-{n^ + Qn^+12ii + 8)x''+^ + (3^3 + 157i2 + 21w + 6) a;"+2 - (37i3 + 12n2 + 12?i + 4) .r''+3 + {n + lfx^+^]l{l-x)'^. When X is numerically less than 1, the series is convergent, and the Bum of the series continued to infinity is (1 + 4a; + a;-)/(l -a;)-^. 827. 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 x + oc^ -}-£c^ -h ..., the simplest of all the series being the geometrical progression whose nth term is of : another series which has the given terms is that formed by the expansion of ^ , METHOD OF DIFFERENCES. 415 which agrees with the geometrical progression except at every 10th term. Note. In what follows it must be understood that by the law of a series is meant the simplest law which satisfies the given conditions. Method of Differences. 328. If in any arithmetical series each term be taken from the succeeding term, a new series is formed, namely the series which is called the first or^der 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 0, 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 rzth term of the series 1 + 6 + 23 + 58 + 117 + 20c + The first order of differences is 5 + 17 + 35 + 59 + 89 + „ second „ „ „ 12 + 18 + 24 + 30+ „ third „ ,t M G + 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 v^ be the nth term of the first order of differences, we have in succession Also Vi = Q.l-l. Hence, by addition, v^=Q{l + 2+ +n)-l = Sn{n + l)-l. Then again, we have in succession w„ - w^.i = t'„_i = 3 (w - 1) n - 1 ; w„_i-M„_2 = 3(n-2)(7i-l)-l; ...; ii2-u-^ = S .1.2-1. Also 2t^ = l. Hence m„ = 3 {{?i-l)w+ + 1 .2} -n + 2 = {n-l)n{n + l) -n + 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 (7i-l)th term being 2""^. Hence, if v„ be the nth term of the first order of differences, we have in succession ^«-^n-i = 2«-i, v^_,-v^_^ = 2^-\ , v,-v, = 2\ AlsoVi=3. Hence, by addition, u„= (2 + 22+ + 2^^-!) + 3 = 2^* + 1. Then again, we have in succession w,i- Wn-i=^n-i=2"~^ + 1' «n-l - «n-2 = 2""^ + 1, » ^h - "l = 2^ + 1. Also U^ = 6. Hence M„ = (2«-i+...+2) + n + 5 = 2» + 7i + 8. 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'* + ?i + 3 is (2 + 22+ ...+2") + {7i+(n-l) + ... + l} + 3?i = 2«'+i-2 + -?i(n+ l) + 37i. 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 deofree in n, all the terms of the rth order of differences will be the same. RECURRING SERIES. 417 For, if u„ = Ay -{- A ^_{rf~^ + ... + A^, where A^, A^_^,... do not contain n, the Tith term of the first order of differ- ences will be {A^(n + iy-i-A^_^{n + ir+ ...] - {^y ^^^.y- + ...), which only contains n to the (r — l)th degree. Similarly the nth. term of the second order of differ- ences will be of the (?• — 2)th degree in n; and so on, the ??th term of the ?'th order of differences being of the (r — ?')th degree in n, so that the nth. term of the 7'th 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 ?'th order of differences are the same, we may at once assume that u^ = A^ + A^_ji''~^ + . . . + A,^, and find the values of ^^, Ar_^, ...,-4(j by comparing the actual terms of the series with the values obtained by putting n = 1, w = 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 An'' + A^_^nr^-i- ... cannot be found [see Art. 321] without a troublesome transformation which will in fact reduce w„ 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^_^ Wr-i + • • •; ^^^ then to find A^, A^_^,,.., A^ as above. Recurring Series. 331. Definitions. When ?- + 1 successive terms of the series a^ -f a^x + a^x" + + a„.^'" +• • • arc connected by a relation of the form a^ a;" -}- j^^ (o^„_i ^"'0 + 9'^^ (^n-2^"""^) + ...= 0, the series is called a recurring series of the ?-th 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. s. A. '21 418 RECURRING SERIES. For example, the series 1 + 2^ + 4 — is therefore such that if it ^ 1 -^-px + qx can be expanded in a convergent series proceeding accord- ing to ascending powers of x, the coefficient of x^ in its expansion will be the same as in the recurring series. RECURRING SERIKS. 411) On tins account the expression -^ — - — „" is ^ 1 -^-px + qx called the generating function of the series. 333. A recurring series of the xth order is determined when the first 2r terms are given. For let the series be a^ + a^x + a^x- + + a^x"" 4- Then, the series being a recurring series of the rth order, if we assume that the unknown scale of relation is 1 -\-p^x-\-p^x^ -\-...-\-p^x'', we have by definition the follow- ing equations Ctr+2 + Ih^r^l ■^V-P'r + • • • + Vr^X = ^' =0, We have therefore r equations which are sufficient to determine the r unknown quantities p^, p^, •", Pr iii ^^^ scale of relation; and when the scale of relation is deter- mined the series can be continued term by term, for dj^j is given by the equation a^^^ + Pi^> + • • • + Pr<^r = ^ > ^^^ when tto^^i is found, Ogr+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 ?• + 2 terms = 0, for all values of k. 27—2 420 KECURRING SERIES. This shews that the series is a recurring series whose scale of relation is (1 — x) It also shews that the series r+l ^2 a^ + a^x + a^x +. . .+ a^x +. . . is a recurring series whose scale of relation is (1 — xj^'^ whenever a^ is a rational and integral expression of the ?^th deofree in n. 'O' 835. In order to find the sum of any number of terms of a recurring series by the method of Art. 3.32, it is neces- sary to know the general term of the series ; w^e 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-- rv., and the coefficient of any \ — olx (1 — axy "^ 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 ^ ^ — ^~t^ cai; always be expanded in ascending powers of y, if y be taken sufficiently small, and RECURRING SERIES. 421 the coefficients of 'if and y^ in this expansion will clearly be a^ and a^ respectively and all succeeding terms will obey the law cbk+P^^'k-\~^^^k-i = *^> ^^^ hence all the coefti- 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 x on the supposition that x is sufficiently small. Ex. 1. Find the nth term of the recurring series 3 + 4a; + Qx^ + lOa;^ + . . . 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 tidrd, 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. 827.] Assuming then that the scale of relation is \+px-\-qx'^, we have the equations 6 + 4^) + 'iq = 0, and 10 + G^ + 4g = 0, whence ^ = - 3 and q = 2. Hence the scale of relation is 1 - 3a; + 2x\ The generating function is therefore 3 + (4-9)a; _ 3 -5a; 2_ 1 1 - 3a; + 2x-2 ~ 1 - 3a; + 2x-2 ~ 1 - a; "^ ITo^ = 2 {l+a;+ ... +a;«-i} + {1 + 2a;+ ... + 2»-ia;~-i + ...}. Hence the general term of the series is (2 + 2'*~i) a;"~^ 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{l + x + x^+ ...) is 2 (1 - a;") / (1 - a;) and the sum of 71 terms of the series 1 + 2a; + 4a;- + ... is (1 - 2»»a;") /(I - 2a;). We may remark that the given series is convergent provided a; < i. Ex. 2. Find the nth term and the sum of 7i terms of the series 1 + 3 + 7 + 13 + 21 + 31+.... Consider the series l + 3a; + 7a;2+13a;3 + 21x4 + 31a;'+ ... Then, assuviiytj 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 tliird order. Let then the scale of relation be l+2ix + qx- + rx^; we then have the following equations to find p, q, r : 422 RECURRING SERIES. lS + 7p + dq + r = 0, 21 + ldp + 7q + Sr = 0, and Sl + 21p + lSq + lr = 0, whence p=-3, g = 3 and r=-l, so that the scale of relation is l-dx + Sx^ - x^. The generating function is now found to be 1 + x^ ^ _2 2__ J_ (1 - xf ~{1- xf (1-x)^ '^l-x' Hence the general term of the series l + Sx + 7x^+... is x'^-^{n{n+l)-2n+l} = {)i^-n+l) .x'^-i. Thus the general term of the given series is n^-n + 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 ?ith term is ?i (n - 1) + 1 is - {n - 1) n {n + 1) + w. o Ex. 3. Find the nth term of the series 2 + 2 + 8 + 20+ Considered as a recurring series of tlie lowest possible order, the generating function of 2 + 2.r + 8a;2 + 20a;3+... will be found to be 2 -2a; 1-20; -2a;2' Now the factors of 1 - 2x - 2x'^ are irrational, and therefore the 7ith 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 conchide that the 7ith 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. 336. We shall now investigate certain theorems in convergency which were not considered in Chapter XXI. CONVERGENCY. 42.*) o37. Convergency 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(i+t^,) = (i + zO(i + ?g(i + i^3)...(i + ?0..., 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 11 (1 + u^), in whicli all the factors are greater than unity, is convergent or divergent according as the infinite series Xu^ is convergent or divergent. Since e* > 1 + x, for all positive values of x, it follows that (1 + Wj) (1 + u,^ (1 + u^. . . < e«' . e«2 . e«3 . . . < qu,+u^+u^+... Hence, if Xu^ be convergent, FI (1 + ?/,.) will also be convergent. Again, (1 + wj (1 + u^^ > 1 -f zt^ + u^, (1 + u^ (1 + u^) (1 +u.) > (1 +2*, +w..) (1 + u^) > 1 + 2t,+W2+^3, and so on, so that n (1 + tO > 1 + 2i/,. Hence, if Sz/y be divergent, H (1 + u^ will also be diveroent. 'o T. 1 m 1 XI . cr(a + l)(a + 2)...(a + 7i-l) . . ^ ., , Ex. 1. To shew that -rr; — ttt; — \- — V, r>i is infinite or zero, when b{b-k-\){h + 2)...[b + n-l) n is indefinitely increased, according as a is greater or less than h. 424 CONVEEGENCY. For, if a > &, the expression may be written in the form (^-^')(i-^0 (^-fI^O which is greater than i+(a-(,)|i+^^+^-l-^+ |. But - + = — :7 + 7 — 5+ ... is a divergent series [Art. 274] : the given ex- pression is therefore infinite when n is infinite, a being greater than b. If 6 > a ; then as before, — } -f-^ -^ is infinite ; and ' a{a+l) (a + 2) therefore ^-4i — tt-ti — tJ——— must be zero. b{b + l){b + 2) Ex. 2. Determine whether the series a a{a + x) ^ a (a + x) {a + 2x) ^ T I" b b{b + x) b{b+x){b + 2x) is convergent or divergent. Prom Art. 325, we have a I {a + x){a + 2x) ...(a + nx) S^= f {a + x){a + 2x)...{a + nx) ^ \bib + x)(b + 2x)..Ab + n'-^l.x) I « a + x-b [b{b + x){b + 2x)...{b + n-l.x) -^ u -n 1 {a + x){a + 2x)...{a + nx) . . „ ., ,. Now by Ex. 1, ^ — — -^ is infinite or zero according 6 {b + x)...{b + n-lx) &s a + X b. Hence the given series is convergent, and its sum is then b- a-x* if b> a + x. Also the series is divergent if & < a + a;. • Also if b = a + x, the series becomes - + , \- - — -- which is b b + x b + 2x known to be divergent [Art. 274]. 338. The Binomial Series. We have already proved that the binomial series, namely m(m — 1) o m(m — l)(m — 2) , l-\-mx+ ^^ ^ ^ x^ + — ^ — 123 ' ^ + ••• is convergent or divergent, for all values of m, according as X is numerically less or greater than unity. If d? = 1, the series becomes 1 . .„ , m( m-l) m(m-l)(7?i-2) , CONVERGENCY. 425 Now we know that the terms of this series are alternately positive and negative after the ?"th term, w]\ere r is the first positive iutegcr greater than m 4- 1. More- over the ratio u^^^/u^ 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 + — = w^ (— in) (1 — m)...(n — 1 — ni) 1 1 /^ 1 + ni\ (^ l + m\ (^ 1 + m ± — = - 1 + . -1+1^ ... 1 + u in\ 1 — an) \ 2 — m \ n — \ — mJ Now, if m + 1 be positive and less than r, the product of the factors from the rth onwards is greater than (l+m)] + — — + ...[; [r — Tti 7* + 1 — ?7t j and the product of the preceding factors is finite. Hence, when n is increased without limit, l/w„ is in- finitely great, and therefore u^ indefinitely small, provided 1 4- m be positive. Thus the binomial series is convergent if x = l, pro- vided 7?l > — 1. If a; = — 1, the series becomes m (m — 1) m (?7i — 1) (m — 2) l-m+-y-2 j^^3 +... 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...(?i-l) • The sum of n terms of the series is therefore [Ex. 1, Art. 337], zero or infinite, when n is infinite, according as m is positive or negative. Thus the binomial series is convergent when x = —\, provided m is positive. 426 CONVERGENCY. 339. Cauchy^s Theorem. If the series u^-\-u^-\- u^ + . . . + u^^-\- ... have all its terms positive, and if each term he less than the preceding, then the series will he convergent or divergent according as the series u-^ + au^ + a^Ua-i + • . . + a''Uan+ ...is convergent or divergent, a heing any positive integer. For, since each term is less than the preceding, we have the following series of relations Wj + ^2 + . . . + Ma < au^ <{a~-l)u^-\- Wj, w„+i + Wa+2 ...■\-Ua".<{pj' - a) u^ <(a-l) au^, Wa«+1 + ^a»+2 + • • •+ ^a"+i < (^"^^ ~ ^") '^a» < {a — 1) a^'lCan. Hence, by addition, S<(a — l)X-\-Ui (I), where S and X stand for the sum of the first and second series respectively. Again, we have since a is au^ « (^Wi + ^^.+2 +• • •+ ^as) > a (a^ - a) Ua^ > ahia^., Hence aS>S —u^ (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 X. Ex. To shew that the series -— ^ rjr is convergent if k be^ greater n (log 71)^ than unity, and divergent if k be equal or less than unity. By Cauchy's theorem the series will be convergent or divergent according as the series whose general term is ~— -^ is convergent or divergent. CONVERGENCY. 427 Xow Z ''** =1^ ^ ^ = ^- S — • "^ a*» (log a")* ~^?t* (loga)* (log a)* 7i* ' it tliereforo follows from Art. 274 that tlie given series is convergent if k> 1 and divergent if A; :|> 1. 840. 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. 341. Theorem. A series is convergent when, from and after any particular term, the ratio of each term to the preceding is less than the corresponding ratio in a known convergent series whose terms are all j^ositive. For let the series, beginning at the term in question, be U = ?/', + l/g 4- tig + ... + t(,, + . . . , and the known convergent series, beginning at the same term, be V = v^-{-v^ + v^-^ ... + v,^ + .... U V Then, since — ^^ < -^^ for all values of r, we have u V- r ^ F= -y + V -^ + V. -' -' + V -*-'-' + ... > ?;, + V, - + V, - -' + V, -* -^ --' + ... > '(w, + w--f-?/ +w,+...)> ' u. U^ ^12 3 4 'Wj Hence as V is convergent, U must also be convergent. Tlie given series is therefore convergent, for the sum of the finite number of terms preceding the first term of U must be finite. We can prove similarly that if, from and after any particular term, u^_^_^ \u^ ^"^r+i -^r' ^^^^ ^^^ *^^® terms of %u^ liave the same si2:n ; then iz/ will be divercient if 5)t;, be divergent. 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 n ( 1 - -''+M be a. Consider the series S -^ = ^v„ ; then / v^^A _ Un -\-iy — n^) _ ^n^ + lower powers of n \ v^ ) \ {n-\- ly J n^ + lower powers of n Hence the limit of nil ^^ j , when n is infinitely great, is /3. First suppose a>l, and let /3 be chosen between a and 1. Then since the linfiit of nil "^ j is greater than the limit oi nil ^^M , there must be some finite value of n from and after which the former is constantly greater than the latter. \ n But when nil - ^^^^]>nil - -^^^ , 'n we have -^^ > -^^' Hence, by the previous theorem, Sw„ will be conver- gent if 2v„ be convergent; but Xv^ is convergent since yS>l. Similarly, if a be < 1, and ^ be taken between a and 1, we can prove that l^u^ is divergent if %v^ is divergent, and the latter series is known to be divergent when /3 < 1. If the limit of ti ( 1 ^M be unity the test fails. EXAMPLES. 429 _ , T ii . a a{a + l) a(a + l){a+2) „ Ex. 1. Is the series - + _-^_ .: + ^i^_^^i^_^ ^,2 + ... convergent or divergent ? Here -**— = , x, the limit of which is x. Hence, either from ?/„ b + n ' the beginning or after a finite number of terms, — ^-^ ^ 1 according as a; ^ 1. Hence the series is divergent if a; > 1, and convergent if a; < 1. If x=l, the limit of -^^^ is unity. But Un n (-'t')="(^-:-f:)' the limit of which is b -a. Thus, if .r = l, the series is convergent when b-a>l and divergent when b-a <.l. When b = a + l, the series becomes a a a b'^bTl'^b + 2'^ ' which is divergent. [These are the results arrived at in Ex, 2, Ai't. 337.] EXAMPLES XXXIY. 1. Find the sum of each of the following series to n terms, and when possible to infinity : — 4 4^ 4.7.10 ,.. 2 2^ 2.5.8 ^"^ 4"^4.7'*'4.7.10'*'"*' ^"'^ 8 "*'8TT0'*'8. 10. 12'^'"' n 11.13 11. 13.15 ^^^^ 14 "^14.16 "*" UTTgTTS "^ " • • 2. Find, by the method of difTercnces, the nth term and the sum of n terms of the following series : — (i) 2 + 2 + 8 + 20 + 38+.... (ii) 7 + 14+19 + 22 + 23 + 22 + .... (iii) 1+1+11 +2G + 57 + 120+.... 430 EXAMrLES. (iv) 1 + + 1 + 8 + 29 + 80 + 193 + .... (v) 1 + 5 + 15 + 35 + 70 + 126 + .... (vi) 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 series : — (i) 2 + 4:x+Ux'' + 52x^+.... (ii) 1 + 3x + llx' + 4:3x^ + .... (iii) 1 + 6a; + 40a;' + 288a;' + .... (iv) 1 +a; + 2a;'+7a;^ + 14a;''+ 35x'+ .... (v) r + 2^x + 3 V + i'x' + 5 V + 6'a;' + . . . . 4. Find the nth term, and the sum of qi terms of the following recurring series: — (i) 2 + 6 + 14 + 30+.... (ii) 2-5 + 29-89+.... (iii) 1 + 2 + 7 + 20+.... 5. Find the nth term of the series 1, 3, 4, 7, &c. ; where, after the second, each term is formed by adding the two preceding terms. 6. Determine a, b, c, d so that the coeflicieut of x" in ,, . J, a + hx + cx^ + dx^ . , ^,„ the expansion or jz -^ may be [n + 1) . (1 —X) 7. Shew that the series 1''+ 2''a; + 3V + 4V + ... is the a„ + a^x+ ... + ajif also that a^=0; and that ar_g = a _j. expansion oi an expression or the lorm — — ^ —^ — - — j shew 8. Find the sum to infinity of the recurring series 2 + 5a; + 9a;' + 15a;' + 25a;'' + 43a;' + ... supposed convergent, it being given that the scale of relation is of the form 1 + px + qx^ + ra;^ Shew that the (n + l)th term of the sei-ies is (2" + 2n+ l)x". EXAMPLES. 431 9. Find the sum to infinity of the series 1 + 4a; + 1 la;' + 26x' + 57x* + 120a;* + X being less than ^. 10. Find the sum of n terms of the series X x(x + a) x(x + a) (x + b) 1 +- + -^—f — - + —" P -+ . a ao abc 11. Shew that i ' a ah + : r-, ?T + -, r-7 7^^ X + ... x+ a [x + a) {x + b) (x + a) {x+ b) (x + c) abc.k 1 abc.kl {x -t a) {x + b) . . .{x + k){x + I) x x{x + a) {x + h) . . . {x + I)' 12. Shew that 1 \+n ' (l+n)(l+2«) + 1 n 7^ — : + p + n (p + n){p + 2n) {p + n) (p + 2n) (p + ^n) + ... to infinity = p-i' provided that p>l and p + n> 0. 13. Shew that, if m be greater than 1, 1 1.2 1.2.3 1 + ^ + -: TT-, ^T + 7/t+l (m+l)(m + 2) (m+ 1) (m + 2) (m + 3) + ... to infinity = ^ . -^ m-1 14. Shew that _1 n-1 (n--l)(n-2) 1 7/i + l (m+l)(m + 2) (?7i + 1) (m + 2) (m + 3) '" Qu+n^ if m + n be positive, or if n be a positive integer. 15. Shew that, if ii be any positive integer, n n(n—l) n(n-l){n — 2) + 7 ^r^ 7>.^~~, ;,v + • • • n + l (n + l)(;i+2) (n+l)(n + 2)(n + 3) n{n- 1 ) (n- 2). ..2.1 _ 1 '^ {oi+\f{^'2)T772n "2' 432 EXAMPLES. 16. Shew that, if tti be a positive integer, 2n+l m(m-l)(2n + l)(2n + 3 ) '27rf2'^ 1.2 (2n + 2) (27^ + 4) •'• 1.3.5...(277i-l) (2n + 2) (2n + 4:) ... {2n + 2m)' 17. Shew that, if ?n, 7i and m — n + 1 are positive integers; then _ m ?2(n— 1) m(m— 1) m — n + 1 1.2 (tti — n + 1 ) (tti — ?i + 2) n (oi — 1) (71 - 2) m (m — 1 ) (m — 2) 1.2.3 (m — 7^ + 1) (m — n + 2) (??i — n + 3) + ...to(«+l)terir,s = , {m^lHm+2) (m^m) _ ^ ' {m — n+l){ni — n+2)...{m — '}i + m) 18. Shew that, if m + 1 > 0, then 1 1 lm(m— 1) 1 m(^m — l)(m—2) 2~3'^'^4 172 5 17273 '^ '" (m + 1 ) (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 l+F^+F^+ +P =1.3.5. ..(2n+l), and l + Q^ + Q_^+ + Q^^2 . 4: . Q ... 2n, 20. Prove that {a + {a+l) + {a + 2) + ... + (a + n)] {a^ + {a + l) + (a + 2) -1- ... ... + (a + n)} = a^ + (a + ly + ... + (ci + rif. 21. Shew that the series l-g" (1 - aP) (1 - g"-') (1 - oT) (1 - or- ') (1 - g"-^) 1-a "*■ {l-a){\~a') ~ (1 - a) (1 - ay(l - a^) '^ '" is zero when n is an odd integer, and is equal to (1 —a) (1 -a^) ... (1 — a"~') when oi is an even integer [Gauss]. EXAMPLES. 433 22. Find the sum of the seiies n n—\ n—2 + ^ .. . + .. . ^ + ... + 1.2.3 2.3.4 3.4.5 " n(w+l)(?i + 2)' 2X^ 3a;3 4/;g< 23. Sum to infinity :; — ^s - ^^—, + .^^- .... 1.3 2.4 3.5 24. Sum, when convergent, the series o n XX-' X + 7^~K + . . . + —7 r; + ■ 1.2 2.3 n{n+\) 25. Sum to infinity the series 1.2.3 + 3.4. 5a; + 5.G.7ic' + 7.8.9a;'+..., X being less tlian unity. • 26. Shew that, if n is a positive integer 1 o, 3«(3.»-3) 3«(3»-4)(3»-5 ) _ i_rf„+ —^ ^-^-^ +...-.(-1). 27. Shew that, if «j, a^? '^a'--* be all positive, and if ttj + rtg + ttg + ... be divergent, then a a^ a. ' + -, r^/ TT + -, Tx~r- ^-^m r-v + ... a,+ l (a, + l)(6*^+l) (a, + l)(rT,2+l)(«3+l) is convergent and equal to unity. 28. Shew that the series r+x "^ Orn + j "1" im +jr + •• • + / 1 \i/i * , + • •« is convergent if re > 1, and is divergent if x ;:)> 1. 29. Shew that, if the series u^+u,^ + u^+ ... +v^^+ ... be divergent, the series u„ v., u -'■ + ■'— + ... + " 1- ... will also be divergent. s. A. 2S 434 EXAMPLES. 30. For what values of x has the infinite product (1 +a)(l + 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^..., v^v^v^v^ ... are either both finite or both infinite. 32. Test the convergency of the following series : — 1 ?! ^ n" (1) 2^ + 3-8 + 44 + ••• + (^+ i)n+i + •••• W 1+ 2/23+ 3/34+ ••• + ^23 ^^s^-^-^^yT.-^ l-'\ Jl^ 2.4 2.4.6 2.4. 6 ...2n + .... 3.5.7...(2n^-l)(27^ + 2) X _1_ 1-3 1. 3.5 ^"-^^ 2.3"*"2.4,5"^2.4.6.7"^"* _1^^3_^^5^2ri-2)_ "*" 2.4.6...27i(27zTr) "*" •*' * a/3 a(a+l)^(;g+l) '^ ^ l.y"" 1.2.y(y+l) "^ - a(a+l)(a + 2)/?(^ +l)( /?+2) 1.2.3.y(y+l)(y + 2) CHAPTER XXVI. INEQUALITIES. 843. 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>'b -^ x, and a — x>h — x. II. If a > 6 ; then — a < — h. III. If a > 6 ; then ma > mb, and — ma < — mh. IV. Ua>h,a'> h\ a" > b'\ &c. ; then a + a' 4 a" +. . . > 6 + 6' + 6" + . . ., and aaa"... >hb'b".... V. If a > 6 ; then a"* > b"\ and a~"' < b' \->n Ex. 1. Prove that a^ + h^> a% + ab^. We have to prove that aS - a% - ah- + 6^ > 0, or that {a^ - 6«) (a - 6) > 0, which must be true since both factors are positive or both negative according as a is greater or less than b. 28—2 436 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^ + m'^ + n'^) > {IV + mm' + im'f. It is easily seen that ip + m2 + 7i2) (^2 + m'2 + to'2) - {IV + mm' + wn')2 = {mn' - m'nf + {nV - n'lf- + (Zm' - Vm)^. Now the last expression can never be negative, and can only be zero when mn' - m'n, nV - n'l and Im' - Vm are all separately zero, the conditions for which are — = — r = -- . V 111 n Hence {P + m'^ + n^){V^ + m''^ + n''^)> {IV -^-mm' + nn')^t except when HV = mlm' = njn', 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 la 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 f —^- \{ -^\> ab. 846. Theorem II. Tlie 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. Then, keeping all the other factors unchanged, take \{a + h) and ■J(a + 6) instead of a and h : 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 Tlius, unless the n quantities a, b, c, ... are all equal, 'a + 6 + c + (/ + . / # r -^^ I I —*■- r • • I I -A— 1 ahcd . . . < 1 , , V n j and therefore a-Yh + c-\-d-\-... n > :^(abcJ ...). By extending the meaning of the terms arithmetic meaii and geomettHc mean, the last result may be enunci- ated as follows : — Theorem III. The arithmetic mean of any mimber of positive quantities is greater than their geometric mean. Ex. 1. Shew that a' + 63 + c^ > 3a6c. We have > ^{a^ . b^ . c^) > abc. Ex.2. Shew that ^ + ^ + ^^+ +^>n. ttg ttg a^ «! Wehave^(^^ + ^^+ +^A > ;y fli.l^..M >;/!. Ex. 3. Find the greatest vakie of {a-x) {b-y) {cx + dy), where a, b, c ai'e 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-ex = bd - dy = cx + dy. Whence tlie greatest vaUie is found to be {ac + bdy^j'27cd, Ex. 4. Find when x^y z^ has its greatest vahie, for different values of X, y and z subject to the condition that x + ?/ + 2 is constant. Let F=.x^y z^ ; then xV^v'W 'W '\y) XXX y y V z z z a a*a /3 /3 ^ y 7*7 The sum of the factors in the last product is constant, since there are a factors each - , 8 factors each ^ , and 7 factors each - , and a "^ fi 7 therefore the sum of all the factors is x + y + z. 438 INEQUALITIES. Hence, from Theorem II, {-■) (V) (-) lias its greatest value X y z when all the factors are equal, that is when - = ^ = - . a ^ 7 It is clear that P is greatest when Pfa^'^^y* is greatest, since a, /3, 7 are constant ; hence P is greatest when xla = y I j3=zjy. In the above it was assumed that a, /3, 7 were integers ; if this be not the case, let n be the least common multiple of the denomina- tors of a, )8, 7. Then x'^y^z^ will have its greatest value when ^nayti^^ny j^^^ -^^ greatest value, which by the above, since Tia, n^ V "it z and ny are all integers, will be when — = —- = — , that is when ' ^ no. n^ ny X _y _z a~^~y' Thus, whether a, /3, 7 are integral or not, x^'y^z' is greatest for values of a;, y and z such that a; + 2/ + 2 is constant, when x\a. = r///3 =2/7. 347. Theorem IV. TAe sum 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 b. Then, if a and b are unequal, (a/o^ ^ ^bf > 0, and there- fore a -hb> 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, 6, any two of the quantities, be unequal. Then, keeping all the others unchanged, take Jab and Jab instead of a and b : we thus, without altering the product of all the quantities, diminish their sum since Jab 4- Jab < a-f 6. Hence, so long as any two of the quantities are unequal, their sum can be diminished without altering their product'; and therefore all the quantities must be equal to one another when their sum has its least possible value. INEQUALITIES. 439 348. Theorem V. //"m and r be j^ositive, and m > r; then — wiU be greater than a, +a., + ... + a ^ a^ + a, +... + a„ n n We have to prove that n (a;* + a;" + ...)> (a/ + a/ + . . .) (a/"-'' + a;"--- +...), or that {n - 1) (a,"* + a,*" + ...)> :S (a/a^"" + «;""•<)» or that every letter being taken with each of the (/i — 1) other letters. Now a ™ 4- a,*" - «>;'-•• - a;'-^a; = («; - <) (a,'""'" - <-^), 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 £ (a^* + a.,"» - aX'"" " ^r^^aO > 0, which proves the proposition. By repeated application of the above we have Sa/" ta^^ Sa/ 2a v 1 v^ 1 I 1_ n n n n where a, /3, 7, ... are positive quantities such that a + /3 + 7+ =?H. Ex. 1. Shew that 3 (a» + ?^ + c^) > (a + 6 + c) (a" + 6* '- c^). Ex. 2. Shew that a« + i;5 + c^ > a 6c (a^ + 62 + c^) . From Theorem Y, > . ( ^ — j . > " - . a6c, from Theorem III. o 440 INEQUALITIES. 349. Theorem VI.* To prove that, if a, b, c, ... and a, 13, y, ... be all positive, then \ a-\-b + c + ... J ' First, let a, 6, c, ... be integers. Take a things each a, b things each /3, and so on. Then, by Theorem III, (a + a + ... to a terms) + (/3 + /3 + . . . to b terms) + . . . a + b + ... ^u+M-^J^a^. j, that is (l^-^hl^+-" ^ a^^.^ . a^„^ ^ ^ I If a,b, c, ... be not integral, let m be the least common m ultiple of the denominators of a, b, c, . . . ; then ma, mb, 7UG, . . . are all integers, and we have 'inaCl + WP/j + • • ■ ma+mD+.../ f/v"'« /P'"^ 1 ma + mb + ... ^ '• ""■'* Hence j^rijr^^j— | >«^ ...[A]. 1 1 Cor. I. Put a = -, ^ =r , ..., and let there be n of the Cv o letters a,b, ... \ then f 71 1 ""''■'- 1 > [a + 6 + ...j a%\..' X r. {a4y+:.j >K*"-)"- [B]. * See a paper by Mr L. J. Rogers in the Messenger of Mathematics^ Vol. XVII. • • INEQUALITIES. 441 Again, substitute a*", Jf, ... for a, b, ... respectively, and a'"^ t'"*", ... for a, /3 respectively, where tr >t, {a'» + 6'"+...px(ft'-+6'-+...r"'x{ft'+6'+...)'"-^>l....[D]. The following are particular cases of [D]. Put i = ; then, since a° + 6"+ ... = 7i, we have provided 7)1 >r -^i 1 ^1 n 1 [^J- Again, put ^ = 0, m = 1 ; then since m> r> t, r must be a proper fraction. Hence, if r he a proper fraction, fft + 6 + ...r ft*" + 6''+... r,,. Again, put ^ = 0, r = 1 ; then m > 1. Hence, if m > 1 loe have ft"' + //" + ... [ ft+6 + n I 71 Now put m = l, r = 0, then ^ is negative. Hence, piv- vided t he negative, (a + 6 + . . .)"* X n'~^ X (ft' + 6' +...)> ; ft' +6'+... /ft + 6 + ...V pTn n \ n J >\-^^'z^l [G]. 442 INEQUALITIES. From [F], [G] and [H] we see that g"' + ^'^ + . • ■ > n < "(X + 6 + n according as a) 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 s = ai + a2+... + an, " s s n^ + +... + ^> Z-, unless ai=a2= ••• = «/ s-a^ s-a^ s-a^ n-1 Unless ai = a2=...=a„, we have !/_£_ 8 S \ "/ S» n\s-a^ s-a^ '" s-a^J V (s-ai)(s-a2)...(s-aj ' ,^a (^-''^' + »-''^' + - + (^-''») >y{(.-a,)(«-«,)...(«-aJ}. By multiplication, since (s - Uj) + (s-a^) + ... + {s-cin) = ns - s, we have n-1 / s « » \ -, — 2- + + •.. + >1. n^ \ s - «i s-a^ s- a^j Ex. 2. Shew that, ifa + & + c + cZ=3s; then will aicd > 81 (s - a) (s - 6) (s - c) (s - d). For 34/{(s-6) (s-c) (s-d)}<{(s-6) + (s-c) + (s-(Z)} x^yVz', unless x — y = z. \ x + y-vz J First suppose that a;, y and « are integral; then by Theorem III. (3; + j: + ... to a; terms) + (?/ + y + •■■ to y terms) + (^ + ^ + ... to ^ terms) x + y+z > ='-^+^[x''yyz^) ; .*. I — I > x^yVz'. -'—, \x+y+zj If a;, ?/, z be not integral let m be the least common multiple of their denominators ; then mx, my and mz are integral, and we have by the first case EXAMPLES. 443 /x24-7/3 + 22\m {x+v+z) that is -T-'_L_ ) X ?n»n(x+v+2) > ix^'yVz^)^ x m"» (*+''+'»: .*. — > x^xiyz*. \ x+y+z J The Theorem can iu a similar mamier 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^ + 2; V + x^y'^ abc (a + 6 + c). 9. o* + 6' + c' <^ a6c (a + 6 + c). 10. a' + 6* + c' + c/5 (x', + x'a + + ^ „)' > a. 444 EXAMPLES. 14. Prove that, if x^, x^, x^, , x^ be eacli greater than a, and be such that {x^ — a){x^ — a) (x^^ — a) = 6", the least value oi x^x^x^...x^ will be (a + 6)^, a and h being positive. . , (a + h)xy . ax + by 15. Shew that ^^ ^ > / . ay -^ ox a + n^ 2 2 2-^9 b+c c+a a+b a + b + c' nr, 3 3 3 3 ^ 16 6 + c + c/ c + c^ + a d + a + b a + b + c a + b + c + d' 18. Shew that if a > 6 > c; then \C& — C/ \o — C/ 19. If x^ = y^ + z^, then will x"^ y" + z" according as ?i ^ 2. 20. Shew that (abcd)'"^^'^^' lies between the greatest and X 1 1 1 least of a^, b\ c', d* . 21. Shew that \+x + x^+ +x'" <^ (2n + l)a;". 22. If 7* be a positive integer, and a>\] then ^'"■^^ + 1 a n —^„ — =— > a"'— i a—1 ' 2n 23. (tn + 1) {m + 2) (7/1 + 3) {m + 2n- 1) < (m + jz)-"-'. 24. abc <^(b + c-a){G + a-b)(a + b- c). 25. abed <^ {b + c + d-2a){c + d + a-2b){d + a + b-2c) (a + b + c — 2d). 26. a^a^a^...a^^ {n-lf (s-a^) (s-a^)...{s-ajy where {n—l)s = a^ + a^+ +a„. 27. If a, 6, c be unequal positive quantities and such that the sum of any two is greater than the third, then 1 _J. _1_ 9 b + c — a c + a — b a + b — c a + b + c' EXAMPLES. 445 28. Shew that, unless a = b = c, (^b-ay(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 2>x''~'' + qx'"'' + raf' > ]j + q + r, unless £c=l, or p = q = r. 31. Shew that 1 ■ 3- 5. (2»-l) ^ / 1 _ 2. 4. 6...2n V 2>i+ 1 on oi- XI . 3.7.11...(4ri-l) / 3 32. Shew that , , ., ^ — )-. .i < / . 5.9. 13...(4rn-l) V 4?i + 3 33. Find the greatest value of x^t/z^, for different values of X, y, and z subject to the condition that ax"^ + hif + c;^ = d. 34. Prove that, if n > 2, [\nf > n". 35. Shew that, if n be positive, (l+a;)"(l+a;")>2''-'V. 36. In a geometrical progression of an odd number of terms, the arithmetic moan 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 term, 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 F^ 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, , P,,..., F^ are in descending order of magnitude. 39. Shew that, if s = a+b + c -^ ..., ( 8 -aV" /s -by /s - ex" /s' ^i) {^ U-3i) •■•<' n being the number of the unequal positive quantities a^b.c, ..., 40. Shew that, if n be any positive integer, 2-'>(«.Ir(~)■r--^^...C,^yC)'• CHAPTER XXYII. Continued Fractions. 351. Any expression of the form a ±b c -\- d 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 7 7 second convergents of the continued fraction a± c ± e ± The rth convergent of any continued fraction will be denoted by — . b d The fractions a, , ~ , &c. will be called the first, c e second, third, &c. elements of the continued fraction. CONTINUED FRACTIONS. 417 3 o 3 . In a continued fr actio n ofth e form a 4- where a, h, c, dx. are all positive, the convergents are alternately less and greater than the fraction itself. For the first convergent is too small because the part ... is omitted; the second convergent, a + -, is too great because the denominator is really greater than c; then again, the third is too small, because c + - is greater than c H ...; and so on. e-\- 354. In order to find any convergent to a continued fraction, the most natural method is to begin at the bottom, as in Arithmetic : thus ^ ^^ ^ = ^1 = ^hK + ^x^z If only one convergent has to be found, this method answers the purpose ; but there would be a great waste of labour in so finding a succession of convergents, for in finding any one convergent no use could be made of the previous results: the successive convergents to a continued fraction are, however, connected by a simple law which we proceed to prove. 355. To prove the law of formation of tJie successive canvergents to the continued fraction , '^'l ^2 ^'3 a-\- - - — 6, + 6, + 63 + The first three convergents will be found to be a ab^ + g, ^^^ abfi^ + cta^ + ^^A 1 ' 6, bA + «2 448 CONTINUED FKACTIONS. Now the third convergent can be written in the form (ab^ + a^) b^ + (a) a^ b,.b^ + l.a^ from which it appears that its numerator is the sum of the numerators of the two preceding convergents rmdt'qdied respectively by the denominator and numerator of the last element which is taken into account; and a similar law holds for the denominator. We will now shew by induction that all the convergents after the second are formed according to the a,bove law provided there is no cancelling at any stage. For, assume that the law holds up to the nth con- vergent, for which the last element is a^_Jbn_^, and let pjqr denote therth convergent; then by supposition Then the {n + l)th convergent will be obtained by changmg -^^ mto , "'* 7^ , that is into , — ^=^-^ — . _n— 1 n— 1 n n—\ n n Hence in (i) we must put a„_i^„ for a^_^ and b^_^b^-\-a^ for b^_^; we then have i^«+l = (K-rK+^n)Pn-l + ^n-APn.2 = K (K-xPn-X + <^n-lPn-2) + «"i^«-l = KPn-^^nPn-l [from i.]. Similarly q^^^ = b^q^ + a^q^^_^. 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, H — — ... , ' «2 + ^3 + Pn = «n P.-l + Pn-, ^^"d q^ = «//„_, + q„. 2* CONTINUED FRACTIONS. 449 Cor. TI. In the fraction y-i -:-? , ^ .... ^1 - K -K- Pn = KPu-1 ~ ««Pn-2 ^nd cj^ = 6„g„.^ - a,,^^.,. Ex. By means of the law conuecting successive convergents to a continued fraction, find the fifth convergent of each of the following fractions : ,.^ ,1111 ,.., 11111 (^) l + I + 24-S + r ^"^ 4 + I + 1 + 14-4 + - r-, 12345 ,. ,_2222 ^ 2+3+4+5+6 ^ ' 5+5+5+5+ 12345 I \ i 1 "i }l \ ^^^ l + l + l + l+l* y^^^ 4 + 8 + 2 + 1 + 2 + - ,..'22222 , .... 11111 <"") 3-3-3-3-3-- ^'^^^)i_4-l-4-r-- 856. The convergents to continued fractions of the 111 form a + V - -7 .... where a, b,c,d,... are all positive 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* 71, divide m by ?i and let a be the quotient and p the remainder, so that — = a + - . Now divide n hy p and let b be the quotient and q the remainder ; then •^ = -= . Now divide p by g and let c be the n n , a - 6 + - P P quotient and r the remainder; then - = - = — ^— . By n ]) p r S.A 2i) 450 CONTINUED FRACTIONS. proceeding in this way, we find — in the required form, , m p 1 11 namely — =a+- = a H =a+ y - .... '' n n 1 Q 0+C + 6 + - P Since the numbers p, q, 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, b, c, ... being the successive quo- tients. On this account the numbers a, b, c &c. in the continued fraction a + r - -•• are often called the 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 saine. 491 Ex. Convert j^^. and 3"14159 into continued fractions, and find in 71 355 each case the fourth convergent. Ans. - — , — . 177 113 357. Properties of Convergents. Let the continued 11 p fraction be a, H — — . . . , and let — denote the ni\\ convergent. I. From Art. 355 we have ^ Pn _iV, ^ a. Pn-t +Pn-2 Pn-^ ^ Pu-.ln-i - Pn-iQ: '-2 . CONTINUED FKACTIONS. 451 So also in succession Ps^h-p,»3»-,-p.-,?„ = (-ir and £-"-^^. = (^'. Hn jfn-1 jLtil)i-l II. Every common measure of J9„ and q„ must also be a measure of Pnqn-\~ Pn-iqn^ ^^^^ is, from I., a measure of + 1. Hence p^ and q^ can have no common measure. Thus all convergents are in their lowest terms. III. Ifi^=a, + - - ... - ...; then i^ will be obtained from the nih convergent by putting — in the place of — . Hence F =• ''" ^ ; =P^-^^'^^-^ , where X is written instead of ... , fo that \ is some j)ositive quantity less than unity. 452 CONTINUED FRACTIONS. Hence n — — — ^;^ — — / , >> \ (- 1)"-' X qniqn + Mn-iY Also F -^"-^ = ^" "^ ^^""^ ^""^ = ^ Pn 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 P ^ Pn-i ^ where X is a positive quantity less than unity. Hence i^ ~ ^-=1 > also i^~^i<_J_ . Thus any convergent to a continued fraction differs from the fraction itself by a quaiitity which lies between 1 1 -y-T and T , J -J-. , where d^ and d^ are respectively the denominators of the convergent in question and the next succeeding convergent. Ex. 1. Shew that, if Prl^lr ^^ the rth convergent to the continued fraction a, H — - — , then \Yill ■'■ " — - d J . Vn-l " «n-l+ +a2 + a/ For we have Vn = ^nPn-i + Vn-2 . Pn-l = ^n-l2^n-2 +i^u-3 » ^2 = "2^1 + 1. and ^jj-^j. nence J^ = a^+^'''^=a^ + — =a^- Vn-l Pn-l Pn-l „ ,Pa-i Pn-i Pn-2 1 1 p. = «» + fln-l + «n-2 + + «3 + P2 1 111 = + 454 EXAMPLES. It can be proved in a similar manner that Ex. 2, To shew that — i =- - - ... to n quotients, where n is n+1 2-2-2- ^ a positive integer. We have n _ 1 n-1 _ 1 2 _ 1 o_^^-l »i 2-"— ^ 2- n n—1 2 Hence =- - ;r r- to 7i quotients. 7i + 1 = 2-2-2- ^ Ex. 3. Shew that, if PrlQr ^e the rth convergent of v t r ; then will pn+i = ^Qn • n '~hA.n ~ hn 4-'n * "^^^^^ -Pn+1 — «7ii» EXAMPLES XXXXl, p p p 1. Shew that, if ^ , — , — be three successive converijents qx q, qz to any continued fraction with unit numerators, then will 2. Shew that, if » , ,, , , C ^, ^o <^« — be the rith. convercjent or -7-' ^ ^ v-^ then will P„q„.,-p,-,q,={-iy~'a^a„ a.. "a* n 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^ a.,, , a ^ be in harmonical progression; .1 .,, a 11 1 f^, then will — ^ = - ^ ... - ^ . a, 2-2- -2- a, 5. Shew that 111 ,111 ?ia, + — — — ... =n{a^ + —„ — — -^5 — ... }. * 'ua^ + na^ + na^ + ' ^ n a^ + a^ + n^a^ + 6. Shew that, if P= - ^ "^ ^ and Q = T - J ••• t; then will P {a + Q + \)=^a+ Q. 7. Find the value of n 7i-\ n-2 2 11 7i+w— 1+71-2 + **' +2 + 1+2* 8. Shew that, whether 71 be even or odd, T t t t ' '1—4 — 1—4 — to n quotients = ;- . ^ 72.+ 1 9. Prove that the ascending continued fraction — — — ... is equal to — + ^ H 1- ... 10. If ;?„ be the numerator of the 7ith convergent to the fraction ^ -,^ =^ .... shew that a linear relation connects ^I + ^2 + ^ + every successive four of the series p^^, p^, p^,...', and find what the relation is. 11. If p,lq, be the rth convergent of - + ^ + ^ + ^ + ---' .shew that p,„^^=^p,„ + bq^„, and that q.^^^^ - a;?.,,, + (ab + l)^',,.. 456 EXAMPLES. 12. If pjq, be the rth convergent of the continued fraction - r - - r - ..., shew that p^ ..^ = 6)t?, +(6c+l)(7„. 13. If via, be the Hh convergent of - - - - .... ^'•/^' *=> 1+1+1+14.' shew that p^^ g^,,., - 5'2„;?2„_, = -a"h\ P 14. Shew that, if — be the nth convergent to the continued 2/1 fraction - zr- - - .... then (7., =w ,, and b(f„ ^,=ap„ + cthp^ ... 15. Shew that, if-=a+^ - ...^ -; then will 11 111^11 l'i_J_ . p 16. Shew that, if -^ be converted into a continued fraction, the first quotient being a, and the convergent preceding -^ being - ; then, if — be converted into a continued fraction, the last (1 q convergent will be {P - ciQ)/(p- aq). 17. Shew that, if - and —. be any two consecutive conver- ' q q' ^ gents of a continued fraction ic, then will ^-^ ^ x^ according as q^q" ^ 358. To find the nth convergent of a continued fraction. We have in Art. 355 found a law connecting three suc- cessive convergents to a continued fraction, so that the GENERAL CONVERGENT. 457 convergcnts can always be determined in succession. In some cases an expression can be found for any convergent which does not involve the preceding convergents : the method of procedure will be seen Irom the following examples. Ex. 1. To find the nth convergent of the continued fraction 1 1^ 3^ 5^ 3+4+4 + 4 +•••• Here the nth element is -^ , and therefore 4 i'« = 4p„-i + (2/1 - 3) {2n - l).?r„_2. The above relation may be written p^-{2n + l)p^_,= -{2n-S) { Pn-i - {^n - 1) p ,_.r . Changing n into w- 1 we have in succession Pn-i - {2'i - l)2'n-2= - (2^1 - 5) {|J„_2 - (2n - 3)p„_o}, But, by inspection, ^=1, p^^^', .: P2-^Pi= -'^' Hence ;>„ - (2n + 1) p„_j = ( - l)n-i (2n - 3) (2n - 5) . . . 3 . 1. Then again Pn 7>n-i ^ L- 1)""' _ 1.3...(2;i + l) 1.3...(2»-1) (2n + l)(2n-l)' P2 a.=(-_i)i 1.3.5 1.3 3.5* Ilcncc P- - ^ -L+ +— Izll!:i_ 1.3.5...(2n + l)~1.3 3.5^ ^ ('Zn + l) {2>i -I)' Since the denominators of convergents are formed according to the same law as the numerators, we have from the above (7„-(2n + l)g„_i = (-l)»-23.6...(2u-3){g3-5(/,}=0, pince 2i = 3 and q2=15. 458 GENERAL CONVERGExXT. Hence ^ ^ gn-l _ _ g2 =ll ^\. (2?i + l)(2w-l)...3. 1 (2u-l)...3.1 5.3.1 3.1 ' .-. g„=l. 3. ..(271-1) (2^ + 1). Hence p^jq^, the nth convergent required, is J^_ ]_ ( - l)»-i 173 3.5"^ '''(2/i-l)(2» + l)' Ex. 2. To find the ?ith convergent of the continued fraction 12 3 4 1 + 2 + 3 + 4+ "*' The necessary transformations are given in Ex. 5, Art. 251, 1 1 (-l)'^-i It will be found that »„ = tt; - 77. + + —, r— • " |2 |d \n + 1 11 ( - 1)** andthat ^^^^^ is a and h. Hence the numerator of the n\j\\ convergent to the continued fraction a + - - ... is the coefficient of a;""* in c -V c-\- ^, . ^ a^hx the expansion oi z. ,—3 . JL "~" CX ~~ OX Similarly the denominator of the ?ith convergent is the coefficient of a;""^ in the expansion of —^ — ^^ r^i- , \—cx — ox' 1 that is of 1 — cx — hx* ' Ex. 1. Find the nth. convergent of the continued fraction 3 3 3 The numerator of the 7ith convergent is the coefficient of x""^ in the expansion of jA^^^,, that is of ^-^^^ - Y^2x' Hence Pn = | {3"+(- W- Also (7„ = coefricifnt of x^~^ in the expansion of 1 _ 3 1 1 - 2a; - 3^2 - 4 (1 - 3ar) "*■ 4 (1 + x) ' 460 GENERAL CONVERGENT. Hence 3n = J{3''- (- 1^}- Thus the nth convergent is 2 ^ — j- — —^ , Ex. 2. Find the nth convergent of the continued fraction a c a c • h + d + b +d+ "" Weliave P2n = dl>2n-l + (^P2n-2i Pin-l — KP2n-2 + ^Pin-S » P2n-2 — "p2n-3 + ^P2n-4 • Hence, eliminating ^2»i-i ^^^ Pzn-s^ "^^® have P2n - («^ + C + bd) 2)2ft-2 + <^(^P2n-4 = ^• Since the last result is symmetrical in a and c, and also in & and d, it follows that P 2n-l ~{a + C + M) pon-z + acp.^n 5 = 0. Hence the relation Pn-{a + c + hd) Pn-2 + «'7i'n-4 = holds good for all values of n. Hence p^ will be the coefficient of x^~^ in the expansion of A+Bx+Gx^- + Dx^ l-{a + c + bd)x'^ + acx^ ' provided the values of ^ , I?, C, D are so chosen thit the result holds good for the first four convergeuts. It will thus be found that p^ is the coefficient of a;"~i in the expansion of a + adx - acx^ l-(a + c + &d)a;2+ acx'^' It will similarly be found that ^ g'„-i>^ (^„-i^/„-2 + ««-i^«-3) > f^qn-2' n n— 1 fi— 1 Hence < ""^"^-•f-^ ^ . Whence u„ < ^ "^ * Todhunter's Algebra, Art. 783. 462 PERIODIC CONTINUED FRACTIONS. But (1 + ky~^ increases indefinitely with n, since k is finite; hence w„ decreases without limit as n is increased without limit. We have therefore the following Theorem. The infinite continued f/'action j^ -^ ,- ...An which all the letters represent positive quantities, is convergent if the ratio ^n^n-i • ^n '^^ alwaijs greater than some fixed finite quantity. It should be remarked that any infinite continued 11 fraction of the form a + 7- - .... in which a, h, c,... 6 4- c+ ' ' ^ ' are positive integers, is convergent. 862. In the following five Articles the continued fractions M'ill all be supposed to be of the form .11 I, 7 •.• • a + ^ - . . ., where a, 0, c,... are positive integers. This form of continued fraction possesses two great advantages, for we know that every convergent is in its lowest terms, and we can also see by inspection, within narrow limits, the difference between any convergent and the true value. *363. Theorem. Every simple periodic continued fraction is a root of a quadratic equation with rational coefficients whose roots are of contrary signs, one root being greater and the other less than unity. Also the reciprocal of the negative root is equal in magnitude to the continued fraction which has the same quotients in inverse order. Let the fraction be 11 1111 6 + C+ +^•+/+a+6+ -- P' P Let jy and yr be the last two convergents of the first * Articles 363, 304, and 368 are taken from a paper by Gerono, Nouvelles Annates de Mathematiques, t. i. PEKIODIC CONTINUED FRACTIONS. 463 period. Then _ /rP 4 P' :,x^Q + x{Q' -P)-F = (i). The roots of (i) are obviously of different signs, and the positive root is the value of the continued fraction. Now, from Art. 357, Ex. 1, P _ 1 11 and ^ = ^ + - ... _ . XT .. , 1 111 Hence, \[ y = l-\-- ... - - - ... we have , = ^; .■.fP' + y{Q'-P)-Q = (ii). The roots of (ii) are obviously of different signs, and the positive root is the value of the continued fraction 1 11 k+ + a + Z + 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 (1) IS — U + ;- ... - -y ... 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. 11 11 The fraction - , ... -r ... rtciu ires no special a+6+ +/+a+ ^ ^ 464 PEllIODIG CONTINUED FKACTIONS. examination, for we have only to change x into - , and 11 into -: thus - 7 ... 7- 7 - ... is equal to the positive root of PV— (Q'—P) x — Q = 0, and the negative . • [7 1 11) root IS— U + Y ... 7- -> . Hence, as before, one root of the quadratic equation in o) is greater and the other is less than unity. 364. Theorem. Every mixed periodic continued fraction, which has more than one non-penodic element, is a root of a quadratic equation with rational coefficients whose roots are both of the same sign. Let the fraction be ''~'''^b-{-"' + k + a + ^ + "' ■^fM + v+a + ^ + "" and let _ 1 1111 ^ 13+ +yu, + i/ + a + /3 + A' A Let -^ and -^ be the two last convergents of the non- periodic part ; then ^~ yB + E ^'^■ P' P Let Yy and -^ be the last two convergents of the first period of y \ then yp + p n... y-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 HEDUCTIOX OF QUADRATIC SURD. 4G5 shall clearly obtain a positive value of x, 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 _ ^1 11 6+ + k — v — [JL -{- we have to shew that this is positive. If A:> z/ the result 1 1 is obvious ; if A; < z^, -, - ... is nes^ative but is less k—v — iJb+ ° than 1, and therefore x is positive provided one element at least precedes k ; also k cannot be equal to v, for in that case the periodic part would really begin witli k and not with a. Hence both values of x are positive in all cases. Reduction of Quadratic Surds to Continued Fractions. 365. It is clear that a quadratic surd cannot be equal to a continued fraction with a finite number of elements ; for every such continued fraction can be reduced to an ordinary fraction whose numerator and denominator are commensurable. It will be shewn that a quadratic surd can be reduced to a periodic continued fraction of the form a + r ,-,•.., where a, h, c, ... are positive integers. The process will be seen from the following example. Ex. To reduce JQ to a continued fraction. The. integer next below y/8 is 2; and we have /8-2+ /8 o-o,( VQ-^)(N/Q + ^ )-g, 4 _ 1 ^8-2 + ^8-2-.+ ^8 + 2 -^^^/8T2-^ + 78T2' 4 /8 + 2 The integer next below "^—r — is 1 ; and we have S. A. 3U 406 REDUCTION OF A QUADRATIC SURD N^:^2_ V8:^2_ __4___ J^ The integer next below ,^8 + 2 is 4 j and we have The steps now recur, so that /8-9 + 1 1 1 1 Thus Av/8 is equal to a periodic continued fraction with one non- periodic element, which is half the last quotient of the recurring portion; and it will be proved later on that this law holds good for every quadratic surd. 366. We now proceed to shew how to convert any quadratic surd into a continued fraction. Let ViV be any quadratic surd, and let a be the integer next below sJN\ then N — a^ 1 ^JN=a + ^JN — a = a-\- —^ = a + -r^ , where r, = N—a^. ^ Since ^N— a is positive and less than 1, it follows that is greater than 1. Let then b be the integer next below ; then \AZVj-a_, Vi\^ - (b7\ - a) where a„ = hi\ — a and r„ = N-al' T ^i Then, as before, ^ is greater than unity ; and if r TO A CONTINUED FRACTION. 467 c be the integer next belotu — '^ , we have Vi\^ + ti^ _ ^ ^ ^/JS' - (cr.^ - a,) — G r ~ r r ' % 2 ^ r. wherea3=o...-a.and.3 = "'-''-^ 3 ' 2 The process can be continued in this way to any extent that may be desired. Thus JN =a + r - ? 867. To shew that any quadratic surd is equal to a re- curring continued fraction. It is first necessary to prove that the quantities which, in the preceding Article, are called a, a^, a^,..., r^, 7\^, ^\,--- are all positive integers. It is known that iV is a positive integer, and that a, h, c, d, ... are all positive integers. Wc have the following relations : r, = N-a\ (i) ttg = bt\ — a , r,?'2 = iV — «./ (ii) 0's = c^\-%> V3 = ^-< Oil) «4 = ^'^^'s - «3 > ^s^\ = lY - a/ (i v) jind so on. Now it is obvious from (i) that r^ is an integer. From (ii) we have r^ = ^^ — ' = 1 + 2ab — h'r^ , since N—a^ = r^. Thus a.^ = bi\ — a,, and r ^ = 1 + 2ah — 6Vj ; wlience it follows that rtj and r^ are integers, since ?\ is an integer. 30—2 468 REDUCTION OF A QUADRATIC SURD From (iii) we have similarly a^ = cr^ - a^ and Tj = r ^ + 2a^c - c\ ; whence it follows that a^ and r^ are integers, since a^ and 7*2 are integers. Then again, from (iv) we have a^ = dr^ - ^3 and r^ = 7\ + 2a^d - d\', whence i"t follows that a^ and r^ are integers, since a^ and i\ are integers. And so on; so that a„ and r^ are integral for all values of 71. We have now to prove that a^ and r,, are positive for all values of n. We know that a, b, c, &c. are the positive integers nea^t below ^/N, ^^^±^, ^/J^ + a, ^ ^^^ ^^^^^ ^N-a, T r 12 \/N-a^, y/N—a^, &c., and therefore also N — d\ N ~ a^\ N -a^, &c., are all positive. That is r^, i\, r^, &c. are all positive. Again, since h is the integer next below ^ , it follows that \/N'-\-acr^, for then sjN a; hence the only possible values of a„ are 1, 2,..., a. Then, from the relation a„ + ct„+i = A;. r„, where ^ is a positive integer, it follows that 1\ cannot be greater than 2a. Hence the expression '' cannot have more than 2a X a different values ; and therefore after ^a^ quotients, at most, there must be a recurrence. 868. 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 back- wards or forwards. Let isj N hQ the quadratic surd. Then, from the preceding Article, we know that \J 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 sj N \^ one root is the equation x^ — N =0. Now the roots of x^ — N = are both 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 V^ must be a mixed recurring continued fraction luith 07ie non-recurring element. Hence we have /Ar_ 11 1 1 1 1 /v 11 1111 470 SERIES EXPRESSED AS CONTINUED FRACTIONS. AT 1 1 1111 . ,, JNow T - ... r 7- T 7- ... IS tne positive root of a quadratic equation with rational coefficients; and as this positive root is \jN — a, the negative root must be — ^N —a. Hence, Art. 363, we have 1 111 111 - + T + T ^jN + a I +k^h+"' + c^ b^ 1 + '"' ,, _, 1 1 111 ...^/i\+a_^ + __^_^ ••• + C+6 + I + '- W 7^11 111 Hence l — a + -r 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 caji be expressed as a continued fraction. Let the series be u^ + u^ + u„ + u^-^ ... -\-n^-\- (i). Then the sum of ?i terms of the series (i) is equal to the nth convergent of the continued fraction 1 - ^1 4- t^a — ^2 + ^3 — 'Wg + i^4 — '** — U^_^ + '^n—" This can be proved by induction, as follows. Assume that the sum of the first n terms of (i) is equal to the Tith convergent of (ii). Another term of the series is taken into account by changing u^ into u^ + u^^ ; and, by changing ii^ into ?^, + w„.i, — ""'' " will become SERIES EXPRESSED AS CONTINUED FRACTIONS. 471 , wliich is easily seen to be equal to w„-2 (u„ + u„^,) JW^^ JVA+L, Thus the sum 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 convergent of (ii). But it is easily seen that the theorem is true when n is 1 or 2 or 3 : it is therefore true for all values of n. Thus i^j + Wj + Ug + w^ + . . . to n terms = rr — — ^-^ — ~- ... to 71 quotients... A. 1 - 2^, + ^2 — ^^2 + "^^8 ~ ^3 + ^4 ~ It can be proved in a precisely similar manner that u^—u^ + u^ — u^+ to n terms = -^ — ^ — — ^—^— — ^-^- ... to 71 quotients ... [Bl. 1 -\- u^- u^ -{■ u^ - % + %—'t'^4 + The formula [B] can also be deduced from [A] by changing the signs of the alternate terms. 370. The following cases are of special interest: 7- ± tV + -rrV ± • • • to n terms a b^a^ b^^ _ . . . to n quotients. . . [C], all the upper signs, or all the lower signs, being taken. A J 1111 , , And — + — I — + — f-...to7i terms a, ^2 ^3 «4 12 2 — ' "^ _... to 71 quotients ... [D], all the upper signs, or all the lower signs, being taken. These can be proved by induction as in the preceding Article. 472 SERIES EXPRESSED AS CONTINXTED FRACTIONS. Thus to prove [C]. It is obvious by inspection that the theorem is true when w = 2. Assume then that [C] is true for any particular value of n ; then, to include another term of the series -~ must be changed into ^ =t , ^\^+^ , and therefore , ""-^ " wiU become i± which can easily be seen to be equal to -'LJ-Jk — n n+i — ^ Thus, if [C] be true for any value of n, it will 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]. ttj ± a^a^ + a^ac^a^i ± a^^a^a^a^ + 1 4- 1 ± a.3 + 1 + ^3 + 1 + a^ + 111 1 and — ± — H ± f- . . . , _ 1 «! «2 ^3 [E], [F]. «! + ftg ± 1 + (Xg + 1 + Ex.5. Shew that 1 + ^ -^ — ^ — ^ ... = c''. l-r + 2-r + 3-7' + 4- ^ . r r.r r .r . r r .r , r .r ^1^1. 2^1. 2. 3^1. 2. 3. 4^*" -, f ^ 2r 3r ^ ^^, = l + r -^ 5 7 ....from [CI. 1 - r + 2 - r + 3 -r+4 - ' ^ ^ EXAMPLES XXXVII. 1. Find the continued fractions equivalent to the following quadratic surds : (1) J\1, (2) VUO, (3) V33, (4) 743, (5) 7(fr + l), (6) V(a^ + 2«). 2. Shew that JIi= (i + jr- -^ • • ., where a has any value ACl> + Act + whatever, and b = N -a^. 3. Find the value of W ^^3+2+3+2+ •••^"^^^^'^^'^- 474 EXAMPLES. /•••^ 111111 . • fi •, (ill) - 77 - -= -J ^ ... to mnmty. ^ ^ 2+3 + 4 + 5+4+5+ -^ 4. Shew that 7 + TT TT • • • t« infinity = 5(1+^ - ...to infinity). 14 + 14 + -^ V 2 + 2 + *^' «+ "/\ c +b + a +d + '" J 5. Shew that 1111 ^a + b+c+d + b + d+bcd ~ a + c + acb ' 6. Shew that, ii x = y + -^r- -— ... to infinity, then ^ 22/ + 22/+ ^' 7. Shew that, if a; = - -z- - r ... to innnity, a +b + a +b + "^ 1111 f • fi f V = -K- 777- r^- ?:t- ... to mnnity, ^ 2a + 2b +2a + 2b + ^' and « = 7^ 77T 77- TTT ... to infinity I 3a + 36 + 3« + 36 + '' ' then will a; (y^ - z') + 2^/ (2;' - x') + 3^ (aj^ - y"") = 0. 8. Shew that, if n be any positiye integer, n^-V n'-2' n^-3^ 3 + 5 + 7 + 9. Shew that l+a^ + a^+...+a'" 1 1 1 a + a^ + a'+ ... + a^"-^ a 1 1 ,,, a -I a-\ a a to ti quotients. EXAMPLES. 475 10. Shew that, if a c a c , c a c a x = T -J Y -. ... and V = -. j- -, ,- b + a + b + d + '^ d+b+d+b + then bx - dy = a — c. 11. Shew that the ratio of 1111 ,,1111 a + T T - T ... to6 + r- — - - l+6+a+l+ l+rt+6+l+ is 1 + a : 1 + 6, 12. Shew that the nth convergent of ••> 14 2 2 .2 n ■X - - ... IS 3-3 -3 -3-*" 2" + r 13. Shew that the nth convergent of 9^1 i 1 . (i+v2r--(i-v2)-' ""2 +2 + 2 + •••''' (1 + ^2)" -(1-72)" • 12 2 2 14. Shew that the nth convergent of t o o o 1 — o — o — o is2"-l. 15. Shew that the nth convergent of \ ah ab . a"-b" IS n + l l,, + \ ' a + b—a-i-b—a + b— a"^^ - b 16. Find the nth convergent of the continued fraction 2 3 8 r'-l 1 _5_7_ ••• -27+1 _•••• 17. In the series of fractions - , — j *fcc., where the law of formation is p, = <7r-i» 5'r = (^^ ~ l)7^r-i "^ ■^^'^-w P^'^^e that the limit of- when r is infinitely great is n+\- '' 1 n 2n . ■ a 'J. to mnnity. n~l +2n — 1 + 3n—l + CHAPTER XXVIII. Theory of Numbers. 371. Throughout the present chapter the word numher 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 numbe?^ 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, 8, 9, 10 11, • • 13, ii, 15, 16, 17, 18, 19, • • 20 • • 21, 22, 23, • • 24, 25, 26, 27, • • 28, 29, • • • 30 31, 3*2, 33, 34, • • 35, 36, 37, 38, 39, ■io (fee Now take the first prime number, 2, and over every second number from 2 place a dot : we thus mark all 480 THEORY OF NUMBERS. multiples of 2. Then, leaving 3 unmarked, place a dot over every third number from 8 : we thus mark all mul- tiples of 3. The number next to 3 which is unmarked is 5 ; and leaving 5 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 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^ and by every prime less than p. Hence 1^ + 1 will not be divisible by p or by any smaller prime ; therefore I j:) + 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 \n + 1 + r, where r is any one of the numbers 2, 3, ..., (?? + ]). 374. Theorem. No rational integral algebraical formula can represent prime numbers only. THEORY OF NUMBERS. 481 For, if possible, let the expression a ± hx ± cc(^± dx^±.., represent a prime number for any integral value of a;, and for some particular constant integral values of a, b, c, ... . Give to X any value, m suppose, such that the whole expression is equal to p, where p is neither zero nor unity ; then p = a ± bm ± cm^ ± ... . Now give to x any value m + np, where n is any positive integer ; then the whole expression will be a ±b {in + np) + c (in + np)^ ± ... = a± bm + cm^ ± ... + M(p)=p-\-M(p). Thus an indefinite number of values can be given to x for each of which the expression a ± bx ± cx^ ± ... is not a prime. In connexion with the above theorem, the following formulae are noteworthy : — (i) X'-\-x->r 41, which is prime if a; < 40. [Euler.] ^li) x^ + x-\- 17, which is prime if a; < 16. [Barlow.] (iii) 2a;2 + 29, which is prime if a; < 29. [Barlow.] 875. 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 p)rime to one of the factors, it will divide the other. For, let ab be divisible by x, and let a be prime to x. a r) Reduce - to a continued fraction, and let - be the con- X <1 vergent which immediately precedes - ; then [Art. 357, I.] qa —px = + 1 ; .'. qab—pxb = ±b. Now qab is, by supposi- tion, divisible hy p; and therefore qab—pxb must be divisible by ^, that is b must be divisible by p. From the above theorem the following can easily be deduced : — s. A. 3 1 482 THEORY OF KUMBERS. I. If a prime number divide the product of several factors it must divide one at least of the factors. II. If a prime number divide a" it will divide a. III. If a be prime to each of a, fi, y, ... it will be prime to the product afiy. . . . IV. If a be prime to 6, a" will be prime to 6*". V. 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 he re- solved into prime factors ; and this can be done in only one way. For, if N be not a prime number, it can be divided by some number, a suppose, which is neither N nor I thus N= ah. Again, if a and b be not primes, we have a = cxd, b = 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..., Avhere a, y8, 7, S, . . . are all primes but are not necessarily all different, so that N may be expressed in the form oJ^fi^rf..., where a, /3, 7,... are the different prime factors of N. Next, to shew that there is only one way in which a number can be resolved into prime factors. Suppose that N =abcd... y where a, &, c, d,... are all primes but are not necessarily all different ; suppose also that N = u/3yS. . . , where a, /3, 7, S. . . are also primes. Then we have abed... = a^yB... . Hence a divides aj3yS...; and therefore, as all the letters represent prime numbers, a must be the same as one of the factors of afiyB — Let a = a] then we have bed... —jSyS..., from which it follows that b must be equal to one or other of /S, 7, S,... ; and so on. Hence the prime factors a, h, 0,... must be the same as the prime factors a, yS, 7, . . . . Ex. Express 29645, 13689 and 90508 in terms of their prime factors. Ans. 5.TK IP, 34 . 13- uud 2'-^ . 11^ . 17. THEORY OF NUMBERS. 483 377. To find the Inghest power of a prime number contained in \n. - j denote the integral part of - ; and let a be any prime number. Then the factors in In which will be divisible by a are a, 2a, 3a, . . . , / 1 - ) . a. Thus I\-\ factors of \n will be divisible by a. Similarly I\~A 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 +/(-3) + Ex. 1. Find the highest powers of 2 and 7 contained in |50. I ( ^ j = 1. Hence 2'*7 is the required highest power of 2. Again, l[~\ = 1, l[_A=\. Hence 7^ is the required highest power of 7. Ex. 2. Find the highest powers of 3 and 5 which will divide |80, Am. 3^«, 519. Ex. 3. Find the highest power of 7 which will divide [1000. Am. 7ifr*. 378. Theorem. The product of any r consecutive numbers is divisible by I r. Let n be the first of the r consecutive numbers ; then we have to shew that — ^ ^— ^ , II \n-\-r — 1 or 1 , ■ - , is an integer. \r |n-l ° In + r— 1 The theorem follows at once from the fact that ' ,- -. vr\n — \ is „+r-i^r» ^^^ ^^^^ number of combinations of n + ?^^^1 31—2 484? THEORY OF 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 / (^'^1^) <^ / {^^\ + ^ Q , / y- ^ — J <(: / ( — 2- j + li-^) , 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 — lx\r. Thus every prime number which occurs in w— 1 x |r, occurs to at least as high a power in I ?i + r — 1 , which proves that \n-\-r — 1 is divisible by I ?i — 1 x \r. n It can be proved in a similar manner that -, ,^ is an integer, where a + fi -{- 'y+ ... =n. 379. If n be a prime number the coefficient of every term in the expansion of {a + 6)" except the first and last terTYis is divisible by n. For, excluding the first and last terms, any coefficient , n(n—l)...(n — r + l) , IS given by — -^—,- -, where r is any integer between and n. ■KT T 1 T A • 1 n(n—l)...(n — r-{-l) JNow, by the preceding Article, — ^^ > is an integer; and, as ti is a prime number greater than r, n .1 • X . IIP (n-l)(n-2)...(n-r-\-l) 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. 4cS5 every term in the expansion of (a + 6 + c + ...)" ^vliich contains more than one of the letters, is divisible by n. For the coefficient of any term which contains more than one of the letters is of the form - ,-— , where In . . ~ a +/3+7 + ...= 7i. Now I , '"- is an intetier; and, as n is a prime greater than any of the letters a, /3, 7, ..., 71 must be prime to |al^|7...; and therefore the coefficient of every term which contains more than one letter is divisible by n. Ex. 1. Shew that 7i{n + l) (2n + l) is a multiple of 6, Ex. 2. Shew that, if n be odd, (n^ + 3) {11^ + 7) = M (32). Ex. 3. Shew that, if n be odd, n* + i)i^ + 11 = M (16). Ex.4. Shewthat l + 7-«+^ = iJ/(8). Ex. 5. Shew that lO^" -1 = 31 (360). Ex. 6. Shew that, if n be a prime number greater than 3, «(n2-l)(n2-4) = J/(360). 380. Fermat's Theorem. If n he a prime number, and m any number prime to n; then nC'~^ — 1 will be divisible by n. We know that when n is a prime number, the coeffi- cient of every term in the expansion of {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^=.,.= l, we have 771" = m-\- M{n)', :. m (m""' - 1) = M {n). Hence, it" m be prime to ?i, m"~' — 1 will be a multi2)le of n. Ex. 1. Shew that, if n be a prime number, in-l + on-l + S"-! + . . . + (71 - l)*-! + 1 = il/ (n) . Ex. 2. Shew that, if a and b are both prime to the prime number n; then will a**"^ - h^ ^ be a multiple of /«. Ex. 3. Shew tliat n' -71 = 31 (30). 486 THEORY OF NUMBERS. Ex. 4. Shew that w^ - n = ill (42) . Ex. 6. Shew that x^^ -y^^ = M (1365) , if x and y are prime to 1365. Ex. 6. Shew that, if m and n are primes ; then 7?i"-i + n'"-! -1 = M {vm) . Ex. 7. Shew that, if m, n and p are all primes ; then [ii^yn-i + (p^)n-i + (7W7i)P-i - 1 = Jl (mnp). Ex. 8. Shew that the 4fch power of any number is of the form. 5m or 6m + 1. Ex. 9. Shew that the 12th power of any number is of the form lB?n or 13m + 1. Ex. 10. Shew that the 8th power of any number is of the form 17m or 17?/i=tl. 381. To find the number of divisors of a given number. Let the given number, iV, expressed in prime factors, be a^'^V Then it is clear that iV" is divisible by every term of the continued product (l + a+a'+...+ aO(l + 6 + 6'+...+ 60(l + c + c'+...+c*)... Hence the number of divisors of I^, including N and 1, is {x + l){y + \){z + 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'^b^c^... , the sum required is easily seen to be (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. 5. Find the least number which has 6 divisors. Ans. 12. Ex. 6. Find the least number which has 15 divisors. Ans. 144. Ex. 7. Find the least number which has 20 divisors. Ans. 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. Ans. 21, 245. "^»^Pjk. THEORY OF NUMBERS. 487 382. To find the number of imirs of factors, prime to each other, of a given number. Let the given number be iV= a*6V... ; then, if one of two factors prime to each other contains ft, the other docs 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 + b^) (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 ofj^ositive integers which are less than a given number and prime to it. Let the given number be JV= a'^^V..., where a, 6, c,... are the different prime factors of N. The terms of the series 1, 2, 3,..., iV which are divisible N N by a are a. 2a, 3a,..., — a; and therefore there are — •^ a a N numbers which are divisible by a. So also there are -r- N . . . N . . . numbers divisible by b, y- divisible by be, -j- divisible by abc, and so on. We will now shew that every integer which is less than N and not prime to N is counted once and once only in the series 2:^-2- + X— -:E— + (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 N (a), namely as one of the — numbers which are divisible by a. Next suppose an integer is divisible by r of the prime factors a, 6, c,..., then that integer will be counted r 488 THEORY OF NUMBERS. i\^ . . r (r — 1) N times in IS — , it will be counted -\—^— times in S -^ , a 1.2 ab' it will be counted ^ — = — ^-^t; times in 2V-7- , and so L . z . 6 abc on. Hence the whole number of times an integer divisible by r of the prime factors is counted, is _ ^0^-1) , r (r - 1) (r - 2) _ , r(r^- 1). . .1 1.2 "^ 1.2.3 •••"^'^ ^ = 1-(1-1X-1. Thus every number not prime to K is counted once in (c/.); 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 he prime to N. Hence the number of positive integers less than N and prime to N is N N N a ao abc ( a ab abc =^(i-D(i-S(^-S f^^*-^«"j- Ex. 1. Find the number of integers less than 100 and prime to it. Since 100 = 2^ . 5^, the number required is Ex. 2. Find the number of integers less than 1575 and prime to it. Ans. 719. Ex. 3. Shew that the number of integers, including unity, which are less than iV[2V^>2] and prime to N is even, and that half N '-'"^ these numbers are less than -^ . ' N For if a be prime to N so also is N -a; and if a > „- , then [ N -a < ^ . THEORY OF NUMIJERS. 41^9 884. 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 3m or Sm + 1. For every number is of the form 3m or 3wi ± 1. Hence every square is of the form 9w or 3m + 1. Ex. 2. Shew that every square is of the form om or 5m ± 1. For every number is of the form 5m, 5m ± 1 or 5m ± 2 ; and there- fore every square is of the form 5m, 5m +1 or 5m + 4. Ex. 3. Shew that, if a^ + b^=c^y where a, 6, 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 h must he a multiple of S. Again, every square is of the form 5m, or 5m ± 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 case, abc is a multiple of 5. Lastly, since every number is of the form 4m, 4m +1, 4m + 2 or 4m + 3, every square is of the form 16m, 8m + 1, 16jn + 4. Now a and b 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 Sm + 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 5 and by 4; hence, as 3, 4 and 5 are prime to one another, abc = M {60). Ex. 4. Shew that every cube is of the form 7m 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 (4/1 + l)th power. Ex. 9. Shew that the product of four consecutive numbers cannot be a square. 490 THEORY OF NUMBERS. EXAMPLES XXXVIII. 1. Shew that the difference of the square? of any two prime numbers greater than 3 is divisible by 24 2. Shew that, if 7^ be a prime greater than 3, n{n' -l)(n' - 4:){n' - 9) = M{2\3\ 6 ,7). 3. Shew that, if n be any odd number, (n + 2my - (n + 2m) = J/ (24). 4. Shew that a'""^^ - a'"^' = i/(30). 5. Shew that, if N-a^ = x and {a^Vf-N=y^ where x and y are positive; then N -xy is a square. 6. How many numbers are there less than 1000 which are not divisible by 2, 3 or 5] 7. P, ^, U^ p, q, r are integers, and p, q, r are prime to one another ; prove that, if — h h — be an integer, then ' r p q r — , — 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"-! 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. W- -r-^ THf:ORY OF NUMBERS. 491 13. Find a general formula for all the numbers which when divided by 7, 8, 9 will leave renuiinders 1, 2, 3 respectively; and sliew that 498 is the least of them. 14. If n be a prime number, and N prime to n, shew that iV«'-«-l = iJ/(n'), and that iV^»'-«'" - 1 = M {ny 15. Shew that, if n be a prime number and N be prime to n, then will iyr^+^-'+^-^i 1 =i/(n^). 16. Shew that, if /> be a prime number, and (1 +xf~^= 1 + a^x + a^x- + a^x^ + ...; then a^ + 2, a^- 3, a^ + 4, &c. will be multiples of J9. 17. Shew that if three prime numbers be in A. P. their common difference will be a multiple of 6, unless 3 be one of the primes. 12a 126 18. Shew that , ^—r ' — -^ is an inteft-er. la |6 a + 6 t)^ 19. Shew that -. — ,, is an integer. [n+ 1 [n \nr 20. Shew that r-Trin is an integer. [n{[r} 21. Each of two numbers is the sum of n squares ; shew that the product of the two numbers can be expressed as the sum of|?i(n— 1) + 1 squares. 22. Shew that a' + If cannot be divisible by 3, unless both a and b are divisible by 3; shew also that the same result holds good for the numbers 7 and 11. 23. Shew that, if a' + 6- = c-', then ah{a'-b') will be a multiple of 84. 24. Shew that no rational values of a, 6, c, d can be found which will satisfy either of the relations a" + 6^ = 3 (c^ + d^)^ a' + b' = 7 {c' + d') ora' + b'=n {c' + d'). 25. Shew that, if a' + c' = 2b% then a'-b' = M (24). 492 theory of numbers. Congruences. 385. Definition. If two numbers a and h leave the same remainder when divided by a third number c, they are said to be congruent with respect to the modulus c; and this is expressed by the notation a = b (mod. c), which is called a congruence. For example, 21 = 1 (mod. 10), and (a + 1)^=1 (mod. a). The congruence a=6 (mod. c) shews that a — h is a multiple of c, which can be expressed by a — 6 = (mod. c). 386. Theorem. Ifa^ = b^(mod.x),anda^ = b^(mod.a;); then will a^-\-a^ = b^ + b^ (mod. x), and a^a^ = bj)^ {mod. so). For let a^ = m^x + r^, and a^ = m^cc + r^ ; then, by sup- position, 6j = n^o) + r J and b^ = n.^sc -f r^. Hence a^-\-a^ — (b^ + b^) = (m^ + m^ — n^- n^ x ; .-. {a^ + a^ - (b^ + b^) = (mod. x), or a^ + 0-2 = 6j + b^ (mod. x). Again, it is easily seen that a^a^ — bfi^ = a multiple of X, and therefore a^a^ = bjb^ (mod. x). 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 Ax"" + 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 x, and A, B, G will all be multiples of p. For we have Aa^ + Ba+ G=0 (mod. p\ and Ab' -}- Bb + G = (mod. p); THEORY OF NUMBERS. 403 .'. 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 + h) + B = (mod. J)). Similarly, A (a-\- c) + B=0 (mod. p). Hence, by subtraction, A (b — c) = (mod. p). Therefore ^ = (mod. p), since 6 — c is unity or prime to^. Then, since A = (mod. p), it follows that B=0 (mod. p), and then that (7=0 (mod. p). Then, since A, B, G are all multiples of p, it follows that Ax"^ + Bx + C 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 he satisfied by more than n values of x, which are such that the difference between any tivo 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, 3a,..., (6 — 1) a will all leave different remainders when divided by b. For suppose that ?'a and sa leave the same remainder when divided by b. Then ?'a — 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, ... , (b — l)ahyb are all different; and since there are 6 —1 of these remainders, they must be the numbers 1, 2, , (b — l) in some order or other. 494 THEORY OF NUMBERS. If a be not prime to 6 the remainders obtained by dividing a, 2a, 3a, ..., {b -1) ahy b will not be all different. For let fc be a common factor of a and b, and let a—ka and 6 = /fj8. Then it is easily seen that (r + /3) a and ra will leave the same remainder when divided by b, and (r + /3) a and ra are both included in the sreies a, 2a, ..., (6 - 1) a provided r + /3 < 6 - 1. Cor. If a be prime to h, and 7i be any integer what- ever, the remainders obtained by dividing n, n -^ a, n + 2a, . . . , n + Q) — 1) a by 6 will all be different, and will therefore be the numbers 0, 1, 2, ... , (6 — 1). 389. Fermat^s Theorem. From the result of the preceding article, Fermat's theorem can be easily deduced. For, if a and h are prime to each other, the numbers a, 2(1, ..., (6 — 1) a will leave, in some order or other, the re- mainders 1,2,.. ., (h—1), when divided by h. Hence we have a. 2a . 3a... {h — l.a) = l .2 .3. ..(6 — 1) (mod. b). that is 6 - 1 (a'-' - 1) = (mod. b). Now, if 6 be a prime number, 16—1 will be prime to b; and we have a^~^ — 1 = (mod. 6), which is Fermat's theorem. 390. Wilson's Theorem. If n be a j^rime number, 1+71—1 will be divisible by n. If a be any number less than the prime number n, a will be prime to n, and hence, from Art. 388, the remainders obtained by dividing a, 2a, ...,(n — l) a by n will be the numbers 1, 2, ..., (n — 1); hence one and only one of the remainders will be unity. Let then ab be the multiple of a which gives rise to the remainder 1; then, if b 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 ?i is a prime, if a = 1 or a = n — l. 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. . .(?i - 2) = M(n) -f 1 ; /. n-l = M(n) X (n-l) + n-l. THEORY OF NUMBERS. 495 Hence Itz— 1 + 1 = M(n). Wilson's theorem may also be proved as follows : — From Art. 305, we have (n-iy-'- (« - 1 ) (n - 2)"- + ("-J)(^'-2) („ _ 3)"-' _ . . . + (- iy.-^(" - 1) (" - 2) - 2 i„-, ^ , _ 1 ^ ^ \n — 2 ' Now bv Fermat's theorem (?i - 1)""' = 1 + il/(n), (n - 2y-' = i + M(n), &c. Hence we have l-(/?-l) + ^-^^^^|-^-... + (-ir^(n-l) + il/(n) = ln-l, that is (1 - 1)""' - (- ir'+M(n) = \n-l; hence, as n-1 is even, I ?i — 1 -f- 1 = il/(?i). Wilson's theorem is important on account of its express- ing a distinctive property of prime numbers; for 1 + n—1 is not divisible by n unless w is a prime. For if any number less than n divide n it will divide In — 1 and therefore cannot divide \n — 1 +1. 391. Theorem. If the number of integers less than any number n and prime to n be denoted by (p (n) ; then, if a, 6, c, . . . a7^e prime to each other, (f>{abc...) = (f)(a) X (^(6) x (b). The proposition can at once be extended, for we have (\){abc...)= ^(axbc...)= (f) (a) x (j)(bc...) = (j) (a)

(c . . .) = (/)(a).^(6).(/)(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 iV=a*5^cY..., 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 (f) (a'') = a''- a«-i = a'^ ( 1 j . TUEOilY OF NUMBERS. 41)7 Similarly <^ (¥) = bf" (l - '^Y cj) (cv) = cy(l- ^^ , &c. But, by the preceding article, ^'•^''(^-D-^^-o: Hence ,^ W = i\^(i - 1) (l _ 1) (i _ 1)..., where a, 6, c,... are the different prime factors of N, 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 (f) (w) the number of integers, including unity, which are less than ni and prime to m; then a*^(™^ — 1 = {mod. m). Let the (m) integers less than m and prime to m be denoted by 1, a, /3, 7,... , (m — 1). Then the products a . 1, aoL,a^,ay,..., a{m—l) must all leave different remainders when divided by m, for if any two, ra and sa suppose, left the same remainder, (r — s)a would be a multiple of m, which is impossible since a is prime to m and r — 5 is less tlian m. Moreover the remainders must all be prime to m, since the two factors of any one of the products are both prime to m ; and therefore as the cp (m) remainders are all dift'erent, and are all prime to m, they must be, in some order or other, the n) -1 = (mod. m). If m be a prime number, (j) (m) = m—l, and we have Fermat's Theorem. s. A. 32 498 THEORY OF NUMBERS. 394. Lagrange's Theorem. Ifi^ 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 + ...+(-ir>^,_,. Change x into oc — l; then (x-2)(x-S)...{a)-p) = (x-iy-'-S,{x-iy-'-^-... Hence (x -p) [x^-' - >Sf, x^-' + ^^ ^-^ + . . . + (- 1)^"' >Sf^_J - (^ - 1) [{x - ly-' -8,{x- ly-' + ... + (- 1)^-^ >s;.j. Equate the coefficients of the different powers of x in the above identity ; and we have ^ ^ ~ 1.2 * a _ p(p-l)(p-^) . o (p-l)(p-2) "^'^^ r:¥73 '^ '' 1.2 . c. p(p-l)(j>-2)(p-3) (;,-l)(p-2)09-3) '^3"" 1.2.3.4 ^* 1.2.3 (^-2)(i;-3) 2 • TO ' .,, 9^ q ._ j>(p-l)-..2 (y-l)(p-2)...2 ^i> -^ ^^-2- 1.2... 09-1) ■^'^»- 1.2. ..(29 -2) (j )-2)...2 3^ ^'^^•l.2...(p-3)"^ ••• '^'^^-'Al.2- Since jo is a prime the first term in each right-hand member is divisible by_p ; whence it follows from the first equation that S^ is a multiple of p, and then that S^ is a multiple of p, and so on to ^S^^..^. THEORY OF NUMBKRS. 490 Lagrange's Theorem may also be deduced from the Theorem of Art. 387, assuming that Fermat's Theorem is known. For the conc-Tuence (x- 1) (x - 2)...(oo-p + 1) -^""^ + 1=0 (mod. ;;), is of the (n — 2)th degree in x, and by Fermat's Theorem it is satisfied by the ^ — 1 values 1, 2, ..., ^ — 1, which are sucli that the difiference between any pair is either unity or is prime to p. Hence, by Art. 387 it is true for all integral values of x, and the coefficients of all the different powers oi X are multiples of p. It should be remarked that Wilson's Theorem follows at once by putting x = 0. 395. Reduction of fractions to circulating decimals. It is obvious that a fraction whose denominator con- tains only the factors 2 and 5 can be reduced to a ter- minating decimal, for 2''5' ~ 10^+* ' If, however, the denominator contains any factor which is prime to 10, the fraction can only be reduced to a cir- culating decimal. Let the fraction in its lowest terms be ^r— — r , where 2^5^ . b b is prime to 10. Let this fraction be equivalent to a cir- culating decimal with a recurring and fi non-recurring figures. Then a _ a. 5". 2' _ JST ^ 2^1m;~ 10"*". b ~10^(10'^-1)'' /. 10^.6.A^ = a.5^.2MO^(10'^-l). Hence, as b is prime to a and to 10, 10'^ — 1 = M (b), and OL is the lowest power of \0 for which tJiis is true, for 32—2 500 THEORY OF NUMBERS. otherwise the fraction could be expressed as a circulating decimal with fewer than a recurring figures. It should be noticed that the number of recurring figures in the circulating decimal depends only on h and is not affected by the presence of 2^5^^ in the denominator, for the number is a, where a is the lowest power of 10 which is equal to if (6) + 1. We will now prove that a is either equal to (f> (h) or to one of its sub-multiples. By the extension of Format's Theorem [Art. 393] we have IQHb) _ I = M (h). We have also 10* - 1 = M(b). Hence, if a be not cj) (b) or one of its sub-multiples, let (j) (h) — k'x-\- r, where r < a. Then 10*^^) - 1 = 10«^ . lO*" - 1 = {M(b) + 1}* . 10*- - 1 = 31 (b) + 10'- - 1 ; 10'--l=l/(6), which is impossible since r < a, and a is the lowest power of 10 which is equal'to M(b) + 1. Hence, if b be the factor of the denominator of a fraction which is prime ^o 10, the number of recurring figures in the equivalent decimal is either (p (6) or one of its sub-multiples. 396. We shall conclude this chapter by considering the following examples : — Ex. 1. Shew that 32«+2 - 8n - 9 is a multiple of 64. We have 32n+2 _ 8w _ 9 = (1 + 8)"+i - 87i - 9 = 1 + (?i + 1) 8 + 31 {S^) - 8n - 9 = 3I{S% Ex. 2. Shew that 3-» - S2ii^ + 24n - 1 = (mod. 512). ^^ Let 7t^ = 32'*-32?i2 + 24?i-l; then M„+i = 32"+2 - 32 (n + 1)2 + 24 {n + 1 ) - 1. Hence ii„+i-9!f„ = 256n2-256?i = 256w (/i- l) = il/(512), since n (n - 1) is divisible by 2. EXAMPLES. 501 And since v„+i-9i/„ = (mod. 512), it follows that t/„+i = (mod. 612) provided J/„ = (mod. 512). The theorem is therefore true for all values of n provided it is true for n = l, which is the case since m^ = 0. Ex. 3. Shew that no prime factor of n^+l can be of the form 4m- 1. Every prime number, except 2, is of the form 2k + 1. Let then 2/:4-l be a prime factor of 7i'^ + l. Then n is prime to 2^ + 1, and therefore by Fermat's theorem n^'' = 3I (2A; + 1) + 1. But, by supposition, 7i^ + l = M {2k + l); n'''={M{2k + l) -l}^ = M{2k + l) + {- 1)*. Since 7l^ = M {2k + l) + l and 7i^ = 31 {2k + 1) + { - 1)'^ it follows that k must be even, and therefore every prime factor of n'^ + 1 is of the form 4?n + l, and therefore no prime factor can be of the form 47n.- 1. Since the product of any number of factors of the form 4m + 1 is of the same form, it follows that every odd divisor of n^+l is of the form 4w + l. 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 n and is of the required form. EXAMPLES XXXIX. 1. Prove the following : — (i) 2-""-'^ - 97i2 + 3n-2 = Jf (54). (ii) 6'"*' + n' - 5n^ + ^n - 5 = M (120). (iii) 4-"-'' + 3"-^'' = 0(mod. 13). (iv) 3^'-^' + 2.4^"^^E0(mod. 17). 2. Shew that, if a be a prime number, and b be prime to a; then Vb^, 2%^, , [ —^~) b' will give different re- mainders when divided by a. 3. Shew that, if 4?i + 1 be a prime number, it will be a factor of { 2n}* + 1 ; and that, if in — 1 be a prime, it will be a factor of { 12^-1 ^-1. 4. Shew that, if n be a prime number, and i' be less than 7i; then will |r-l |n-r+ (- 1)'"' = J/ (u). 502 EXAMPLES. 5. Shew that, if m and n are prime to one another, every odd divisor of m^ + n^ is of the form 4/^ + 1. 6. Shew that pi+2S + 3n + ^ + ---to infinity -i}-ff{\-WHr 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, h, c, ... be its diflerent prime factors ; then the sum of all the numbers less than iV and prime to iVis— (1 )(l— -j (l— -j..., and the sum of the squares of all such numbers is 9. If 435 and 7/i:^522. lleuce the number of solutions is 522-435 = 87. 508 INDETERMINATE EQUATIONS. 403. Integral solutions of the two equations ax -\- hy -\- cz = d, ax + Vy + c'z = 3! can be obtained as follows. Eliminate one of the variables, z suppose ; we then have the equation {ad — a'c) X + (be' — h'c) y-dc' — d'c (i), and this equation has integral solutions provided ac — a!G and hd — h'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 dd — d!c. Hence from (i) we obtain, as in the preceding articles, the general solution x = a-Y (bd — h'c) n, y = /3 — (ad — a'd) n, where x = ol, y = /B 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 = S — 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 G. Ex. Find integral solutions of the simultaneous equations 5x + 7y + 2z = 2i, 3x-y-4:Z = 4. Eliminating z, we have 13a; + 13?/ = 52, or x + y = 4:. Whence x=2 + n,y = 2-n. Then 5 (2 + w) + 7 {2-n) + 2z = 24,t}iatisz-n = 0. Hence the general solution is x = 2 + n, y = 2-n, z = n. If X, y and z are to he positive, the only solutions are 05=4, y — 0, z=2; x=S, y = l, z = l; and x=2, y = 2, z = 0', and, if zero values are excluded, there is only one solution, namely x=d, y = l, z=.l. 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 3a; + 2?/ + 8^=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 wo have the following equations : z = 4, Sx + 2y= 8; 2 = 3, Sx + 2y = 16] « = 2, Sx + 2y = 24t\ i = l, Sx + 2y = S2. 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; G, 7, 1; 4, 10, 1; and 2, 13, 1. Ex. 2. Find the positive integral solutions of the equation 6x2-13a;?/ + 6?/2=16. We have (3a;-2i/) {2a;-3?/) = 16; hence, as x and y are integers, Sx - 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 3a:-2?/=±16, 2x~Zy=^l (i) ; Zx-2y=^ 8, 2a;-3i/=± 2 (ii) ; Sx-2y=± 4, 2x-Sy=± 4 (iii); Sx-2y=^ 2, 2x-Sy=± 8 (iv); Sx-2y=± 1, 2x-3y=±16 (v). Whence we find that 5x must be ± (48 - 2), ± (24 - 4), ± (12 - 8), ± (6 - 16) or =t (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 Sx^ + 7xy-2x-5y-S5 = 0. We have 2/ (7a; - 5) + 3^2 _ 2a; - 35 = ; 3x2 - 2a; - 35 ^ ^ ^ a; -245 ,, .-. 4'Jy + 21^+l-^ = 0. Hence _ — ^ must be an integer, and therefore 7a; -5 must be a (X — factor of 1710. Whence it will be found that the only positive integral solutions are a; = 2, y = S and a; = l, y = 17. 510 EXAMPLES. EXAMPLES XL. 1. Find all the positive integral solutions of the equations: (1) 7a; +15?/ = 59. (2) So; + 132/ = 138. (3) 7x+9y=^100. (4) 15a; + 71^/ = 10653. 2. Find the number of positive integral solutions of 2x + Sij= 133 and of 7ic + 11?/ = 2312. 3. Find the general integral solutions of the equations (1) 7x-lSy=15. (2) 9a;- 11?/ = 4. (3) 119a'-105y = 217. (4) 49aj- 69?/= 100. 4. Find the positive integral solutions (excluding zero) of the equations (1) 2x+3y + 7z=23, (2) 7a; + 4?/+ 18;^ - 109. (3) 5x + y + 7z = 39, (4) 3x a- 2y + 3z ^ 250, 2x + 4:y + dz = 63. 9a; - 4?/ + 5«= 170. 5. Solve in positive integers (excluding zero) the equa- tions : (i) 2a;?/-3x + 2?/=1329. (ii) x^ — xy + 2x - 3y = 11 . (iii) 2x' + 5xy-12y' = 2S. (iv) 2a;' -xy-y^ + 2x + 7y^ 84. 6. Shew that integral values of x, y and z which satisfy the equation ax + by -i- cz = d, form three arithmetical progressions. 7. Divide 316 into two parts so that one jDart may be divisible by 13 and the other by 11. 8. In how many ways can ^1. 6s. Qd. be paid with half-crowns and florins 1 9. In how many ways can i^lOO be made up of guineas and crowns ] EXAMrLP:s. 511 10. In liow iiijiny ways can a man wlio has only 8 crown pieces pay 11 shillings to another who has only tiorins? 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 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 = l+A^x+ ... +A^x" -h ...f (1 - iK^i) ( 1 - x^^i) ... (1 - a;«") then the number of solutions in positive integers (including zero) of the equation a^x^ + a^x^ + ... + ajx^ = m, is A^^ ^^t^ a^, . . ., a beinsr all intesers. The number of solutions of the equations x + 2f/ = n is i{27i+3-h(-l)"}. At an enteitainment the prices of admission were Is., 25. and £5, and the total receipts £1000; shew that there are 1005201 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 5s., 3s. and Is.; shew that the number of ways in which the audience may have been made uyy is 1201801. CHAPTER XXX. Probability. 405. The following is generally given ag the 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 y and the pro- bability of its failing is =- . •^ * a + b 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. olS 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 ru7i. Now, if each of the a ways in which an event can happen and each of the b 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 b times out of every a + b cases. We may therefore say, consistently with the former definition, that the probability 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 — . Again, if one of two players at any game win, in the long run, 5 games out of every 8, the probability of his winning any particular . 5 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 findiuiu^ the ratio of the actual number of times the event s. A. 33 514 PROBAEILITY. 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—2^ '^vi^^ ^^ ^^^ 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 h, 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 5 ^ nud -^ da a 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 ^l±%±*s,thatis^ + ^» + ^^ '^ d d d d This proves the proposition for three mutually ex- clusive events; and any other case can be proved in a sirailar manner. PKOBABTLITY. 515 Ex. 1. Fiud 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 probability is ^ . Ex. 2. Find the chance of throwing an odd number with an ordinary die. Ans. -. Ex. 3. Find the chance of drawing a red ball from a bag which con- tains 5 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 5 red and 7 white balls; find the chance that they will both be white. Here any one pair of balls is as likely to be drawn as any other pair. The total number of pairs is jo^s' ^-^^ *^® number of pairs which are both white is 7 Cg : 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 n 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 prohah'dity that two independent events should both happen is the pi^oduct of the separate probabilities of their happening. Suppose that the first event can happen in a^ and fail in 61 equally likely ways; and suppose that the second event can happen in a^ and fail in b„ equally likely ways. Then each of the «i + 6i cases may be assjciated with each of the a^ + b^ cases to make (a^ + t^) (^2 + 62) compound cases which are all equally likely; and in a^a^ of these compound cases both events happen. Hence the proba- 516 PROBABILITY. bility that both events happen is -. =-^-7^ =— ^ , that is — ^- X — V J which proves the proposition, aj + Oj a2+^2 Thus the probability of the concurrence of two inde- pendent events whose respective probabilities are 'p^ and "p^ is^i xpj. Cor. If p^ and p^ be the probabilities of two inde- pendent events, the chance that they will both fail is (1 — J9i)(l — P2X ^^ chance that the first happens and the second fails is j?i (1 —p^, and the chance that the second happens and the first fails is (1 — p^p^. It can be shewn in a similar manner that, if Pi, p^yp^,... be the probabilities of any number of independent events, then the probability that they all happen will hep^.p^.p^..., and that they all fail (1 —^1) (1 —p. 2) 0- —Ps)---, <^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 p^ 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 p^xp^. 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 - for each throw ; hence the required probability is, by Ai-t. 408, -x- = -. ^ ^ 'X Ex. 2. Find the probability of throwing one 6 at least in six throws with a die. PROBABILITY. 517 5 The probability of not throwing 6 is ^ in each throw. Hence the o probability 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 5 red and 7 white balls, the balls drawn not being re- placed. 7 The chance of drawing a white ball the first time is r-^ ; 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 c the second time will be — . Hence, from Art. 409, the chance of 7 fi 7 drawing two white balls in succession will ^^ to ^ Tl — oo' i.a XJ. ^ij [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 5 chosen the chance of drawing a red ball from it is — ; hence the 15 5 chance of drawing a red ball from the first bag is „ x to = oT • Similarly the chance of drawing a red ball from the second bag is 13 1 :^ X — ; = — - . Hence, as these events are mutually exclusive, the 2 lo 10 , . , . 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, the chances being — and — 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 ouce written down. For, if p be the probability of the happening of the event, the probability of its failing is l—p = q. Hence, from Art. 408, the probability of its happening r times and failing n — r times in any specified order is p^(f^. 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^q'"'''. Thus, a (p-{- 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 + q^. Cor. II, The probability of the event happening at least r times in n trials is „-, n(n— 1) „_„ „ p^ i.^ \r n—r K + ™. />- V + -^^f^V"¥ + • ■ • + rT^=— i'-'i 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 a;!" 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~«i* Ex. 2. Find the chance of throwing 8 with two dice. Ans. — 36 PROBABILITY. 519 Ex. 3. Find the chanco of throwing 10 with two dice. Ans. -, 5 Ex. 4. Find the chance of throwing 15 with three dice. Ans. j-r^. Ex. 5. A and B each throws a die; shew that it is 7 : 5 that A's throw is not greater than U'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 5 games and B has won 3, and (3) when A has won 4 games and B has won 2. Ans. (1)|.(2)§,(3)^. 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 7i + l games to win a match: shew that ^ ^^ , , 1.3.5...(2;i-l) , . 1.3.5...(2n-l) the odds on A are 1 + ^^ . ^ ' to 1 o a a. — o:: — • 2 . 4 . 6 . . . 2n 2 . 4 . D . . . 271 3 Ex. 10. ^'s chance of winning a single game against B is -: find the u chance of his winning at least 2 games out of 3. Ans. ^. 2 Ex. 11. ^'s chance of winning a single game against B is - : find the chance of his winning at least 3 games out of 5. 192 Ans. — . Ex. 12. ^Yhat is the chance of throwing at least 2 sixes in 6 throws with a die? 12281 '^'^- 46656* 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. Ans. „ , „ , - . Ill 520 PROBABILITY. 411. The value of a given chance of obtaining a given sum of money is called the expectation. ft If y is the chance of obtaining a sum of money M, a + b then the expectation is ifx 7. ^ tt + 6 For if E be the expectation in one trial, E(a + h) will be the expectation in a + b trials. But the chance being 7 , the sum M will, on the average, be won a times in a + b' . . 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 Id. 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 ? Ans. (1) 4s. 6^d. (2) 9*'. 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., 4id. Ex. 4. Two persons throw a die alternately, and the first who throws 6 is to receive 11 shillings : find their expectations. Am. 6s. 1 5s. PROBABILITY. 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 5 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 -2V + -2^ black balls are due to drawings from the first bag; thus the probability that the ball was drawn from the first bag is We now proceed to the general proposition : — Let P, , Pg,. . ., P,, he the ])rohabilities of the existence of n causes, which are mutually exclusive and are such that a certain event must have followed fi^om one of them ; and let Pj, jt?2, •••,Pn ^^ th^. respective probabilities that when one of the causes P^, P^, ..., P„ eocists it will be followed by the event in question; then on any occasion when the event is knoiun to have occurred tJie probability of the rth cause is PrPr - (P, l\ + P,P, + • . • + i'dK)- Let a great number N of trials be made ; then the first cause will exist in N .P^ cases, and the event will follow in N .P^ .p, cases. So also the second cause exists and the event follows in N . P,^. p.^ cases ; and so on. Hence the event is due to the rth cause in N .P^.p^ 522 PROBABILITY. cases out of a total of N {P^p^ + P^^p^ + • • • + PnP^ 5 ^^^^ P p probability of the rth cause is therefore ^ ^ '" . 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 3 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. Pind the chance that it was drawn from the bag con- taining the most black balls. 1 3 17 Here P^=P^=P^=-. Also p^=-^, p.^ = ^ and B^jq- Hence the required probability is Ex. 2. From a bag which is known to contain 4 balls each of which is just as likely 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 — ; , zrp, , s^ , v^ and r-s . lb lb lo 16 lb Art. 410. Also the chances of drawing a white ball in these 3 11 —^ different cases will be 1, ^ > « . t and respectively. ^^ ^ ^ ^ Hence the required probability = -^ i ■ =-. l6 "^ i • 16 ■*" 2 • l6 "*" 4 • 16 1 7 3' 10 7 13 111 3* 5"^3' 5'^ 3' 7 "15 io 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 -— ^ , and therefore the probability that the man will falsely assert that the 99 1 white ball is drawn is r— - x ^n • Hence as in Art. 412 the required probability is 1 ^ 100 ^10 1 1 9 99 1 12 100 ^ lO ■*" 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 is/' trials the ticket in question will be drawn 10 JV times; and the witness will correctly assert that it has been drawn 9N times. The ticket will not be drawn in 990 lY 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 ^7: , 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 which 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 524 PROBABILITY. — , and therefore the probability that A and B -will agree in truly 13 5 asserting that a white ball is drawn is rpr x - x - . The probability that a black ball will really be drawn in any 9 case is ^k ; and therefore the probability that A and B will agree in 9 11 falsely asserting that a white ball is drawn is r^ x j x ^ . Hence, as in Art. 412, the required probability is 13 5 10 "^ 4 ^ 6 5 13 5 9 11 i0''4''6"^10''i''6 8 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 really drawn? The probability that the white ball will really be drawn in any case is — , and therefore the probability that A and B will agree in 13 5 1 truly asserting that the white ball is drawn isv7:X-7X- = -^. 10 4 6 Id The probability that the white ball will not be drawn in any case 9 . .1 is ztt: . The probability that A will make a wrong statement is - ; hence, as there are nine ways of making a wrong statement which may all be supposed to be equally likely, the chance that A will wrongly assert that a white ball is drawn is -7 x ;: • Therefore the 4 9 chance that A and B will agree in falsely asserting that a white ball is drawn is 9 111 X - — - X Hence the required probability is 10 4x9 6x9 2160 16 135 "136 16 "^ 2160 Ex. 5. It is 3 to 1 that A speaks truth, 4 to 1 that B does and 6 to 1 that G 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 this evidence, as likely to have happened as not. Ans. |. 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 71+ 1 chance that r balls in succession will be drawn from a bag coiitain- u ^■t e ^ • 1 v,-*. ■ S {s -1) ...{s -r+l) ing 71 balls of which s are white is —i -^ — ] . ?i{n-l)...{n-r + l) Hence the chance required is 1 J n(w-l)...(n-r + l) (n- 1) (n-2) ...(n-r) n + l\n{n-l) ,..{n-r+l) n {n- I) ...(n-r + l) r(r-l)...l ) 71 {71- 1) ...(/i-r-t-l)j * Now {1.2...r} + {2.3...(r+l)} + ... + {(7i-r+l)...(7i-l)n} _(n-r + l) {n-r + 2)...7i{7i+l) r+l , by Art. 318. Hence the required chance is — ^ , which is independent of the r+l 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 y succession in — ^ cases, and r + l white balls in succession in -^, r+l r+2 N N r + l cases. Hence the required chance is ~ = — - . ^ r + 2 r + l 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 ^'s chance of ultimate success when he has n counters be J are found, and we have Hence ^'s chance of winning the game is ( \PJ 1 / I \2^ Similarly JB's chance of winning the game is 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. A, B and C throw alternately with three dice, and a prize is to be won by the one who first throws G. Find their respective chances of winning if they throw in the order A, 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 being 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, ..., 7i. A man draws two tickets together at random, and is to receive a number of shilliugs 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 [ 1 — ] . 10. li p things be distributed at random among p persons ; shew that the chance that one at least of the persons will be void IS Li_ . f 11. A writes a letter to B and does not get an answer ; assuming that one letter in iii is lost in passing through the post, shew that the chance that B received the letter is TtX ~~ 1 q- , it being considered certain that B would have answered 2m— 1 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 lis. Gd 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 29J- shillings. 14. If three points are taken at random on a circle the chance of their lying on the same semi-circle is |. 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 in sides are chosen at random; prove that the chance that they being produced will form an acute-angled triangle which will contain the polyajon is , , 7-'-- — -~ . ^ "^^ (in- I) (in -2) 18. Out of m persons who are sitting in a circle three are selected at random; prove that the chance that no two of those selected are sitting next one another is -. =^, ^ . 19. If m odd integers and n even integers be written down at random, shew that the chance that no two odd numbers are In l7i+ 1 adjacent to one another is ' — =-, m being '^ n+1. m + n n — m+ 1 20. If m things are distributed amongst a men and b women, shew that the chance that the number of things received by the group or men is odd, is 7^^ V™ — • EXAMPLES. 529 21. The sum of two whole luiiiibers 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-fourt])s of their greatest product ; prove also that the chance of their product being less than one-half their • -. 1 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 winninijr the match is y . ° a + h 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 q+ 1 next drawing will give a black ball is p) + q-\- 1 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 — a; to 2m + 1 — ?/, where x and y are the points already scored by the respective sides. Shew that the chance that the side which scores the hrst point will just win the game is {2m\ 2m +11)2 (m\f m+ 11 hii-^lV S. A. 3^! CHAPTER XXXI. Determinants. 415. If there are nine quantities arranged in a square as under : ^1 «2 »3 h K ^3 ^1 ^2 ^3 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 (^h%> «1^3C2' ^^2^1' ^2^C3' ^3^C2' ^^^ ^J^'Pl' Let now these products be considered to be positive or negative according as there is an even or an odd number of inmi'sions of the natural order in the suffixes ; then the algebraic sum of all the products will be afif^ - afi^c^ + afi^c^ - afi^c^ + afi^c^ - aj),c^ (A) ; for there are no inversions in ctfi^c^, there is one inversion in (i^b^c^ since 3 precedes 2, there are two inversions in aj)^c^ since 2 and 3 both precede 1, there is one inversion in a.^^jCg since 2 precedes 1, there are two inversions in ajb^c^ since 3 precedes both 1 and 2, and there are three inversions in ajb^c^ since 3 precedes both 1 and 2 and 2 precedes 1. DETERMINANTS. 531 The expression (A) is called the determinant of the nine quantities a^, a.^, &c., which are called its elements; and the products aj)^c^, <^Pz%^ ^'C. are called the terms of the determinant. 416. Definition. If there are 71^^ quantities arranged in a square as under : a. «2 «a a, m, m„ iii„ ni 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 n 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 corner is called tlie j)rincipal diagonal; and the product of the n elements a,, h^, Cg, ,7?i„ which lie along it, is Cdi\\^^lih.Q 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 he written \(ip.f.^-'-t^i^, the 34—2 532 DETERMINANTS. determinant is also often represented by the notation S(±a,6,C3...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 in each of its rows or columns, and therefore also n elements in each of its terms. 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 from 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 inversions in the order of the columns 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, cipf^d^e^f and its equivalent cj'jy^d^a^e^, 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 earher than itself in alphabetical order. Thus, in the example, 2 follows lour suffixes greater than itself in apf^d^ej^, and / precedes four letters earlier than itself in c, fjjxi.aj}^. ly 2 3 4 5 6 DETERMINANTS. 533 Since the words rows and columns are interchangeable in the law which determines the sign of any term, we iiave the following Theorem. A determinant is unaltered hy changing its TOWS into columns and its columns into rows. r example a, K 0, = a^ «2 «3 a. h\ ^2 K K ^^3 ^3 K ^3 ^1 ^2 % Ex. 1. Count the number of inversions in 2314, 3142 and 4231. Am. 2, 8, 5. Ex. 2. Count the number of inversions in 4132, 35142 and 531264. Ans. 4, 6, 7. Ex. 3. Wliat are the signs of the terms bfg, cdh and ccg in the determinant a h 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 abed e f g h i j k I m n p q [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 he obtained whose sign is oj^tj^osite 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 ^ we have P .h^. ka. Now since P .ha. kfi is a term of the determinant, P can contain no element from the rows of A's and k's and no element from 534 DETERMINANTS. tlie a or yS columns ; and this is a sufficient condition that Fhpka 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 Ahak^B where A denotes the product of all the elements which precede ha and B the product of all the elements which follow k^. By interchanging a and /S we have Ah^k^B, 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 ^, or in B, are concerned, whether compared with one another or with a and (B\ but there must be an inversion in one or other of a/3 and /3a 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 P suppose, are to be interchanged. Then a will be brought into the place of /3 by r-f-l in- terchanges of consecutive suffixes,and /3 can then be brought into the original place occupied b}'^ a by ?• interchanges of consecutive suffixes; and therefore the interchange of a and /3 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, hut 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. o.3o A . Iia ' B . kfi .0 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 AkaBh^C ; 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. ^or example «i «s «a =r — «1 «3 ^•2 — «1 «3 «2 ^ 62 ^3 ^ h h. <^1 C3 ^2 Cj C2 C3 Cl C3 Co h ^3 h 421. Theorem III. A determinant, in which two rows or two columns are identical, is equal to zero. When two rows (or two columns) are identical, the determinant is unaltered either in sign or magnitude by the interchange of these two rows (or columns). But, by Theorem II, the interchange of any two rows (or columns) of a determinant changes its sign. Thus the determinant is not altered in value by changing its sign : its value must therefore be zero. Ex. 1. Find the value of 1 1 a a^ h 62 It is obvious that two rows woukl become identical, and therefore the determinant woirtd vanish, if a = 6. Hence A must be equal to an expression whicli has a - 6 as a factor. Similarly h-c and c — a must be factors of A. But A is by inspection seen to be of the third decree in a, h, c ; hence A = I/ (6 -c) (c-a) (a-ft), where L is numerical. The principal term of A is hc^ and this is the only term which gives bc"^, and the coefificient of hc^ mL{h- c) (c -a){a- h) is L ; therefore L = 1. Thus A = (/> - c) (c - a) (a - h). 586 DETERMINANTS. Ex. 2, Find the value of 1 a a^ a^ 1 6 fe2 ^3 1 c c^ c^ 1 d d? d? A ns. - (6 - c) (c - a) {a - h){a- d) [h -d){c~ d). Ex. 3. Find the value of 1 h V^ h^ 1 1 d d2 d^ Ans. -{h-c){c~ a) [a -h) {a- d) {h - d) {c - d) (a + h + c + d). 422. Theorem IV. If all the elements of one row 07^ of one column of a determinant be multiplied by the same quantity, the whole determinant will be midtiplied by that quantity. For every term of the determinant contains one element and only one from each column and from each row ; and it therefore follows that if all the terms of one row or of one column be multiplied by the same quantity, every term of the determinant, and therefore the sum of all the terms, will be multiplied by that quantity. Cor. From the above, together with Theorem III, it follows that if tryo rows or two columns of a determinant only differ by a constant factor, the determinant must vanish. For example mui mbi mc^ WOg wftg ''^2 pog P&3 pc^ Also ■ vmp a. a„ \ K a. ma na mb nb mc nc 1 1 1 3 ' vm . ^2 a a b b c c 1 1 1 ma.2 nb^ pc^ ma^ n&3 pc^ = 0. 423. Minor determinants. When any number of columns and the same number of rows of a determinant are suppressed, the determinant formed by the remaining elements is called a minor determinant. DETERMINANTS. 537 A minor determinant is said to be of the first order, or to be a first minor, when one column and one row are suppressed ; it is said to be of the second order, or to be a second minor, when two columns and two rows are sup- pressed ; and so on. The determinant obtained by suppressing the line and the column through any particular element is called the minor of that element, and will be denoted by A^. where x is the element in question. are first minors of Thus "i K > «i ^1 and h ^2 ^2 h ^3 C3 ^3 ^3 ^j h '•] , and are A^ , A^^ and A^^ respect a.y h.2 ^2 «3 ^3 ^3 424. Development of determinants. determinant of the fourth order Consider the A = a. 6„ b„ a. d, d^ d, d^ A certain number of the terms of A will contain a 1 3 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 a^ . A^, a^ . A^ and a^. A^. Then, since no term can contain more than one of the letters Oj, a^, a^, a^ we have A = a^A J + a^2 + a^A^ + a^A^ (i). Now, since no term of A which contains a^ can contain any element from the column or the row through a^, it follows that every term of A which contains «, is the product of ttj and some term of A^,; conversely the product of a, and any term, T, of A^^ 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 538 DETERMINANTS. inversions. Hence the sum of all the terms of A which contain (Xj is a^ . A^^ . So also, every term of A which contains a^ is the product of ^2 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 a^ . T of 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 ag is — ttg . A^^. Similarly the sum of all the terms of A which contain ag are a^ . ^a^\ and the sum of all the terms which con- tain a^ are — a^ . A^^. Hence A = aj. A^^-a^.A^^ + ttg. A«3-a4. A^^ (ii). By means of Articles 419 and 420, we can shew in a similar manner that A = - 6i Afi^ + &2 Afi^ - 63 A63 + 64^64 = «! A^^ - \ Afi^ + Ci A^^ - d^ Adi = &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 ?ith order having a^, a^,..., a„ for the elements of its first row or column; then will A = rti . A^,^ - a^ A«^ + . . . + ( - 1)""^ a,, A^^. So also A=(-ir{*, Where k^, k^,... For example 61 /;, k„ are the elements of the rth row. «3 = Ciag + Cg/Ss + CgYs, which is by Art. 429 equivalent to the product of — i — 1 , that is 1, and 1-10 0-1 «!«! + «2^l + «37l . &1«1 + ^2/^1 + ^37l , ClO-l + Cg/?! + CgTi «l«2 + «2i^2 + «372> h^2+^2^2+hy-2> ^102 + 02)82+6372 aia3 + 03/33 + a373, Vs + ^3 + ^373 . <^ia3 + Ms + ^373 Hence the required product is the determinant last written. ::. 1. Multiply X y z by a c h z X y b a c y z X c b a The required product is X Y Z Z X Y Y Z X Y=:ay + bz+cx, and Z=az + hx + cy. Avhere X = ax + by + cZf Since = x^ + y^ + z^ -bxyz, and the other determinants X y z z X y y z X are of the same form, we see that the product of any two expressions of the form x^ + y'^ + z^-3xyz can be expressed in the same form. [See Art. 156, Ex. 4.] Ex. 2. Shew that 2bc-a^, h', 2ac - b-, Form the product of Ex. ii. Shew that a^ a, -b, c, -a, b, - c, A, B, C\ ^2 ^2 jBq Co C b a b^ 2ab - c- and = {a^+b^+c^-'6abc)^ -a, b a c c b a c, 2, where Ai, Bi, &Q. are the co-factors of a^, &i , &c. in the expansion of the determinant [ajb.^c^]. i DETERMINANTS. 545 For j J, L'l (7o A^ £, &i 62 [^itg'^g] [a^lK,c^] • [fli&./al since and Heuce 431. A ja^ -^-A^an + A gOg = B161 + BJ)^ + I'g&g = CjCj + CoCo + C3C3 = [ai^/g], ^i&i + yl2^2 + ^3^3 = &c. = [Art. 4JG]. [^i^oCg] . [aiV3] = [«i^V3p- The notation a.. a„ a. is emplo3'ed 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 degree. The solution of any number of simultaneous equations of the first deo^ree can be at once obtained bv means of the foregoing properties of determinants. First take the case of the three equations a^x + \y + c^z = 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 a, a.. Then we have by addition (a^A^ + a^A^ + a,A^) x -f (6,.4, -f h^A ,^ + b,A^) y S. A. 35 546 DETERMINANTS. that is [a^ b^ cj x = [k^ h^ c^, for from Art. 426 the coefficients of y and z are zero. Similarly we obtain [»i K C3] y = [a, h^ C3], and K K C3] ^ = [«i ^2 ^3]- Now consider n equations of the form a^x^ + h^x^ + c,a?3 + c?,^;^ + = k^. As before, multiply the equations in order by A^, A^y A^, &c. the co-factors respectively of a^, a^, a^, &c. in the determinant [a^ b^ Cg...] ; then we have by addition {a^A^ + a^A^ + a^A^ +...)x= k^A^ + k^A^ + k^A^ +. . ., the coefficients of y, z, &c. being all zero by Art. 426. ■] Hence X _ ft K ^3- K b,c^...y So also ■2 ^3- 2 ^3- 2 ^3- •] •]' Ex. 1. Solve the equations 2x + 4:y + z =7, Sx + 2y + 9z = 14:. The values ot x, 7j, z are respectively &C. 6 7 14 2 4 2 3 1 9 1 2 3 2 4 2 3 1 9 1 6 3 2 7 1 3 14 9 1 2 3 2 4 1 3 2 9 and 1 6 2 4 7 3 2 14 1 2 3 2 4 1 3 2 9 and it will be found that each determinant is -20, so that x = y=z = l. ^_^ Ex. 2. Solve the equations X +y +z +%o +k =0, ax +by +CZ +div +k'^ = 0, a^x + hhj + cH + d?w + k^ = 0, a^x + b^y + c'^z + 2. 6. Shew that [ [b + cf 7. Shew that Shew that 9. Shew that lO. Shew that &2 -be ca + a^ c- &C + 62 -ca 62 a^ ab + a- ab + b- (a + 6)"^ ca {6 + c)2 ab ca be (b + cf 62 a-" (c + a)2 a b c a c 6 6 c a c b a (a + 6)2 bc + c^ ca + c2 -ab be ab {c + a)2 a2 62 (a + 6)2 1 1 1 c2 = 2{be + ea + ab)\ = {bc + ea + ab)^. = 2abc (a + 6 + c^. = 2abc (a + 6 + c')3. 1 6- a2 1 6= a2 = -(a + 6 + c)(-a + 6 + c)(-6 + c + a)(-c + a + 6). 11. Prove that a2 62 o 7" /S2 62 o 7' /3^ aa 6^ cy aa C7 6j8 6^ cy aa C7 6i3 aa EXAMPLES. 13. 12. Shew that 1 1 1 1 Shew that 1 + a 1 1 1 + 6 1 1 1 1 14. Shew that 1 1 + a 1 1 1 1 1 + c 1 c d a b a b b a c d d c 1 1 1 + b 1 1 1 1 1 + d d c b a 1 1 1 1 + c =:abc. — abed I 1 + 1 1 1 1\ - + 7 + - + ;, ) a b c dj = {a + b + c + d) {a + b-c-d){a + c-b-d) (a + d-b-c). 15. Shew that 1+x 1 1 1 2 2 + x 2 2 3 3 3 + x 3 4 4 4 4 + ic = x^{x + W). 16. Shew that a b c -b a — ~c d a -d -c b d d c -b a = (a2 + &-^ + c2 + d-)3. 17. Shew that 18. Shew that 19. Shew that 1 1 1 1 a a a a a a b a a b c d a b a a b b b a 62 c2 d^ a a 6 a 6 a a a = 0. a^ + bed 6^ + cda c^ + dab d^ + abc a =a{b-a)^, a a b 6 ! = -(«- b)K a 6 6 20. Shew that ax — by — cz ay + bx ex + az ay + bx by -cz — ax bz + cy cx + az bz + cy cz - ax - by = {a^ + 62 + c2) (x2 + ?/2 + ^2) (^ax + by + cz) . EXAMPLES. 551 21. Shew that a- a--(6-c)2 be ca ab = (6-c)(c-a)(a-6)(a + 6 + c)(a2+62 + c2). 62 62_(c_a)2 c2 c2-(a-&)2 22. Shew that (6 - c)2 (a - 6)2 (a - c)2 (6 - a)2 (c - a)2 (6 - c)2 (c-rt)2 (c-6)2 (a -6)2 23. Shew that, if any determinant vanishes, the minors of any one row will be proportional to the minors of any other row. = - 2(n2 + h^ + c-- be -ca- ab)^ 24. 25. Shew that a2 + l ab ac ad ba 62+1 be bd ca cb c2 + l cd da db de fZ2+l Shew that 1 1 1 1 ={ by-cB = a2+62 + c2+(Z2+l. 26. 1 a2 + a2 ab-\-a§ ac -^^ ay 1 a6 + a/3 62 + ^2 he + ^y 1 ac + ay be + ^y c^ + y^ Shew that the determinants X X y y z z a a a 6 6 6 e () (• c are all zero. 27. Shew that 28. \ -c b 29. Shew that c -b X a -a X Shew that X y a b d c 10 z x^ -yz ?/2 - zx z^ - xy z^ - xy x^- yz y^ - zx 2/2 - zx ^2 _ xy x^ - yz a2 + X2 ab + Xc a6-Xc 62 + X2 ac4-\6 6c -Xa X y z y z X z X y z c /. y w d a X x + w a + d ac-\b bc + \a c2 + X2 = X3(X2 + a2- y + z b + c • X -IV y - z a- d b-c CHAPTER XXXIL Theory of Equations. 435. Any algebraical expression which contains x is called a function of x, and is denoted for brevity by f (cc), F {x), (f) (x), or some similar symbol. The most general rational and integral expression [Art. 75] of the nib. degree in x may be written where ao, a^, a.2,... 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 aoX"^ + aiX'^~^ + ttao;""- + . . . + a„ = 0, where n is any integer, and the coefficients a^ ai, a^... 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 contain x ; and, if we divide the left side of the above equation by a^, the coefficient of x^, we shall obtain the equation of the nth degree in its simplest form, namely x^ +p^x"'-'- ■\-jp^x'^-^ + ... +i)n = 0, where p-^, 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 ?ith degree has n roots. For suppose the equation to be / (x) = 0, where Since f{x) = has a root, a^ suppose, we have f{a-^ = 0, and therefore [Art. 88] f {x) must be divisible by x — a^, so that f{x) = {x — a^^{x), where (f){x) is an integral function of x and of the {n — l)th degree. Similarly, since the equation (j> (x) = has a root, ag suppose, we have (f) (x) = (x — a^ -v/r (^•), where i/r {x) is an integral function of x of the {n — 2)th degree. Hence f{x) = {x— ai) (x — ao) ^jr (x). Proceeding in this way we shall find n factors of f(x) of the form x — a^, and we have finally f{x) = {x — tti) (x — a^. ..{x — ttn). It is now clear that Oj, a.,,..., an are roots of the equation f(x) = ; 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^, a-i,... need not be all different from one another; but if the factors x- a^, X — a^, x — as, &c. be repeated p, q, r, &c. times respectively in f{x), we must have f{x) = (x — ai)P (x — a2y{x — O3)'*. .., where p + g + r + . . • = n. The equation /(a;) = has in this case p roots each a^, (J roots each a^, &c., the whole number of roots being p-{- q + r-^ ... =n. * Proofs of this fundamental proposition have been given by Cauchy, Clififord and others: the proofs are, however, long and dillicult. 554 THEORY OF EQUATIONS. 437. Relations betTveen the roots and the coeffi- cients of an equation. We have seen that if ai, a^, (is,... be the roots of the equation f (x) — ; then f(pc) = {x — ai) {x — a^...{x— «,^). Hence [Art. 260] where 8r is the sum of all the products of a^, a^, cis,... taken r together. Equating the coefficients of the different powers of x on the two sides of the above identity, we have >S'i = - 2h> S2 =P2,-- -Sr = (- iypr,...Sn = (- Vf Pn- 438. By means of the relations obtained in Art. 437, which give the vahies 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. li a,h, c be the roots of the equation x^ +px^ + qx-\-r = 0, find the value of (i) Sa^ and (ii) Sa^ft-. We have a-{-h^c= -py hc + ca + ab — q and abc— -r. Hence a^ + b^ + c^ = {a + b + c)^-2{bc + ca + ab)=p^-2q. Also, SftV = {be + ca + dbf - 2abc {a + b + c) = q'^- 2pr. Ex. 2. If a, b, c,... be the roots of x'^+p-^x''^'^ +2^2^^'^+ ...+Pn = 0, find the values of Sa^ and 2a^ We know that Sa= -p^, Sa6=p2 ^^^ Sa6c= - Jh- Now (2a)2=(a + & + c + ...)2 = Sa2 + 2Sa6[Art. 05]; Sa2 = (Sa)2 - 2Sa6 =pi^ - 2p^ . Again Sa^ . Sa = Sa^ + Srt2Z>, and 'La?b = 'Zab . Sa - BZobc. THEORY OF EQUATIONS. 555 [For ill Zah .Zit there can only be terms of the types a-b and a 6c ; of these the term a-b will occur once, but the term abc will occur three times, for we can take either n or 6 or c from Sa and multiply by 6c, ca or ab respectively from 2a6. Thus 2a6 . Sa = 2a-6 + 32a6c.] Hence 2a3 = Sa2 . 2a - Sa6 . Sa + 32c6c = {p^^ - 2p2) (-Pi)- p.2 {-Pi)- ^P^- 439. Theorem. If there are any 71 quantities a^, «,_,, as, &c., and 7/t be any positive integer not greater than n'y then will 2a/" = Sa^-^ . 2ai - Soi"^"' . Xa,a. + ta,"^-^ . Xa^aM, - ... + X^i . ^01^2. . .ttm-i ± ^?i • -aia2. . .Clm- The following relations hold good : la,'"' = loaK"'' - la,-^-' aa, \ la^'^-'a^ = ta^a, . Sa^^-^ - Xa,'^-'' a^a^, la^'^-^a.^as = Xa^a.^a^ . ta,'"-'^ - loy^-^a^a^a^, f • • • [ A]- la^a^a^. • -Oin-i = la^a^. . .am-i • 2<^i — mla^a^. ..am ) To prove the first relation it is only necessary to notice that the product la, . Sa,^~'^ can only give rise to terms of the types a,^ and ai"^~^a2; also every term of either type will occur, and no term can occur more than once. Thus la, . 2ar"' = lar + S^r"^ a,. The other relations, except the last, will be seen to be true in a similar manner. Also, the product la^a^. . .am-i - l n\ in terms of the coefficients and the sums of lower powers of the roots. [See also Art. 471.] The sum of the wth powers of the roots of an equation can therefore be obtained from the formulae Sgi+^i =0, 2^1^ ^r 'p^o^ -^ 'P^dy + 3j^3 = 0, If we eliminate '^a-^ and Soj from the first three equations we have Pi P2 ^Pi + ta^^ 1 ^1 2^2 1 pi = %a,'-h Pi P2 3^3 1 pi 2^2 1 p. -0. To find Soi"* we must eliminate Xai*"-i, Xai'*'-^..., toi from the first m equations, and we have THEORY OF EQUATIONS. 00 i Pi P-- P3- • • Pm—1 m.prn + ^Cii 1 pl P2- • " Pjn—2 {m-l)pm-i 1 Pl' • • Pm—3 {m-2)2},n-2 • • 1. • • Pm—A (m - S)2),n-3 = 0. 0.. pl The coefficient of ^a^^ 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 Xoj^ is equal to an integral function oi p^, p^, &c. If m be greater than n the relation corresponding to [C] can be very easily obtained. For, since Oj, a^,... are roots oif{x) = 0, we have n equations of the type a n +i^iai""' -f p^di'"' + . . . +pn = 0. Multiply by a^~^, a^~^,... respectively; then we shall have n equations of the type a m + pia,"^-^ + _p,a/«'-2 + . . . + pna^'^-'' = 0. Hence, by addition, we have ta^"^ + pitch"^-^ + p^^a^"^-' + . . . + pnta,'^-'' = 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 he expressed as a rational and integral function of tJie 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 Stti = — pl, la^aa =Pi, '^a^a^as = — p^i, &c. Consider the symmetric functions of the third degree. 558 THEORY OF EQUATIONS. It is easily seen that Thus we have three equations to determine 2a/, Sa^aa and l^a^aMz, and these are the only symmetrical functions of the third degree. Similarly each of the products p^^, pip^, 2^^Ps> P-^ ^^^^ pi can be expressed in terms of symmetric functions of the fourth degree, and there will be as many such equations as tJiere are sy in metric functions 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 ps 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 Sa/ag^ in terms of Sa^P, Sa^^ and SajP+a. 'La^P=a^ + a^v + a^->r Sa^^ = Oj^ + fla^ +03'^+ Thus Sa/a2« = Sa^P . Saj^ - ^ciyP+'i. If, however, p = q\ then we have Thus Sa/a./ = J {'^a^f - i '^a{-P. Ex. 2. Express 'Za^a^a^ in terms of the sums of powers of the separate roots. 2a/ = 5'p = a/ + OTo" + «3^ + Sai« = Sq = a^i + a.Ji +03^+ Za/=S^=a{' + a./ + a./+ i THEORY OF EQUATIONS. 550 Hence S^.S^. S^ = 2aiP+«+'- + >:«/-t-«cr/ Hence, from Ex. 1, The above will onl}- hold good when p, q, r are all different. If p = q = r we shall have /. Zflj^'aaPagP = ^ { V - 3S2p • Sp + 2S^p}. 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 ivhose roots are those of a given equation with contrary signs. If the given equation be / {x) = 0, the required equa- tion will be / (— y) = 0. For, if a be any root of the given equation so that / (a) = 0, then — a will be a root of Thus if the given equation be PoX"" + }\x^-^ + p.x^-- -t- + ;9„ = 0, the required equation w^ill be Po (- yY +Pi (~ yy-' + p. (- yT-"~ + + i^n = 0, or i)o2/" - Piy""-' + p-.y''-' - + (- 1 )'^ iJ>« = 0. II. To find an equation luhose roots are tJiose of a given equation each multiplied by a given qiiantity. Let f{x) = bo the given equation, and let c be the quantity by which each of its roots is to be multiplied. Let y = ex, or - — x\ theny(-) = is the equation 560 THEORY OF EQUATIONS. required. For, if a be any root of / {x) — 0, so that / (a) = 0, ao will be a root of/ {p\ = 0. Thus, if the given equation be the required equation will be or ^0^/^ + PiGy'^~^ + p.2d^y'^~^ + + jJ^iC^ = 0. The above transformation is useful for getting rid of fractional coefficients. Ex. Find the equation whose roots are the roots of each multiplied by c. The required equation is 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 f(x)= 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 x = y -\- c\ then / (y + c) = will be the equation required. For, if a be any root of / {x) = 0, so that f (a) = 0, a — c will be a root oif{y + 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 from any given equation another equa- tion in which a particular term is absent. Ex. Find the equation whose roots are those of x^ - Sx^ - 9a: + 5 = each diminished by c, and find what c must be in order that in THEORY OF EQUATIONS. 501 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 f{y + c) = 0, that is (7/ + c)3-3(y + c)2-9(i/ + c) + 5 = 0, or if+{Sc-'d)y^+{Sc^-6c-9)y + c^-Sc--dc + 5 = 0. The sum of the roots will be zero if the coefficient of j/^ be zero ; that is, if c=:l. The sum of the products two together of the roots will be zero if tlie coefficient of y be zero; 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 / {x) = be the given equation. Then the equa- tion / f - J = is satisfied by the reciprocal of any value of X 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 i^o^"" + PiX^'"- + ;j2«"-2 4- -f Pn = 0. Then the equation whose roots are the reciprocals of the roots of the given equation is or, multiplying by x^, po + Pi^ + P'lX- + + PnX'' = 0. s. A. 36 562 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 Pn-i Pn-2 Po' From the first and last we have Pn=po^, so that Pn= ±Po> 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 recij)rocal equations can easily be proved. I. A reciprocal equation of tlie 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. III. 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 a^ {x-'' + 1) + ai (.r^-i + a;) + . . . = 0. Divide by x^ ; then Uq (a;" + .T-«) + «! (.x-»*-i + .f-"+i) + . . . = 0. Now, if x + a;~^ = y, x^ + x~- = ?/- - 2 ; and, from the general relation rf.n + a;-n _ (a-n-i + a;-«+i) (x + rc-1) - (.T«-2 + a;-«+^) , it follows by induction that .r'^ + x""'^ can be expressed as a rational and integral expression of the nih. degree in y. THEORY OF EQUATIONS. 5G3 Ex. Solve the equation 6x^ - 25a;5 + Slx^ - 31a;2 + 25.t -6 = 0. As in III, the expression on the left has the factor x' - 1 corre- sponding to the roots ± 1. Thus we liave G (.c« - 1) - 25.C (.r^ - 1) + 31a-2 {x- - 1) = 0. Hence the required roots are ± 1 and the roots of Gx-* - 25.1-3 + 37.r- - 25.t; + 6 = 0. Divide by x-; then 6(.r2 + i^ _ 25 ('a: + ^^ +37 = 0. Put x + -=ij; .: .r2 + -=«2-2. Hence C,ij--2oy + 25 = 0i 5 5 15 1 From X + - = - , we have x = 2 or - . a; 2 2 From a; + - = ^r , we have a; = 77 (5 ± \/ - 11). a; o 6 Thus the required roots are ± 1, 2, - , - (5 ± \'' - 11). 445. The method of dealing with other cases of trans- formatioQ will be seen from the following examples. Ex. 1. If a, b, c be the roots of the equation x^ +})x^ + qx + r = 0, find the equation whose roots are be, ca, ab. Clbc T T Since be = — = - , if we put v— - , the three values of y cor- a a " X responding to the values a, b, c of x will be 6c, ca, ab. Hence the V equation required will be obtained by substituting - for x in the given equation, so that the required equation is or r^ + pry i-qij^ + y^ = 0. Ex. 2. Find the equation whose roots are the squares of the roots of the equation x^+2)-t- + qx + r = 0. We have x {x^ + q)= - (px"^ + ?•) ; x^x^ + q)^=(px'^ + r)'. Now put y = x^, and we have the required equation, namely 36—2 564 THEORY OF EQUATIONS. Ex. 3. If a, b, c be the roots of x^+px^ + qx + r=0, find the equation whose roots are a{b + c), b {c + a), c {a + b). a {b + c) = a {-p-a) ; &c. Hence, if we put y = x{-p-x), y will have the values required provided x is restricted to the three values a, b, c; that is provided X satisfies the equation x^ +px^ + qx + r — 0. Thus if we eliminate x between the given equation and the equa- tion x^+px + y = 0, we shall get the required equation in y. Multiply the second equation by x and subtract ; then (y -q) x = r. Now substitxate for x in the second equation, and we obtain the equation required, namely r^-+pr{y-q) + y{y-q)^ = 0. EXAMPLES XLIII. 1. If aj, 02, Oo be the roots of the equation x^+px + q = 0, find the values of (i) {a2 + a^){as + aj){a^ + a2). (ii) {aci + a^-2a{}{a^ + a^-2a.,){a■^^ + a2-2as). (iii) 2ai2. (iv) Sa^s. (v) :Ea^\ (vi) Saj^a- (vii) Sfli^a- (viii) S (a./ - a^a^) {a^ - a-^a.-y). (ix) {a-^ - a^a^) {a^ - a^a^) {a^^ - Uia^). 1 , .. „ 1 , ... ^ 1 V ^J (x) S . (xi) S . (xii) ^2 + ^3 «2 + '^3-% a^'^ + a^Og 2. Find the sum (i) of the squares, (ii) of the cubes and (iii) of the fourth powers of the roots of the equation x'^+px + q = 0. 3. If a, b, c be the roots of the equation x^+px' + qx + r = 0, find the values of (i) (& + c - 3a) (c + a - 36) {a + b- 3c). ,ii) (l+i.l\(iH.l-^)(i+l 1). ^ \b c a J \c a b J \a b c) 4. Find the sum of the squares and the sum of the cubes of the roots of the equations (i) a;3-14.r + 8 = 0. (ii) a;^ - 22a;2 + 84.r - 49 = 0. THEORY OF EQUATIONS. 5G5 6. If a, b, r,... be the roots of the equation find the values of (i) 2a3. (ii) Sa3. (iii) 2 \. (iv) S"^'. (V) Sf. (vi) sg. 6. Find the equatiou each of whose roots exceeds by 2 a root of the equation 7. Find the equation whose roots are those of the equation each multiplied by c, and find the least value of c in order that the resulting equation may have integral coefficients with unity for the coef- ficient of the highest power. 8. If a, b, c be the roots of the equation x'^ + px"^ + qx +r=0, find the equation whose roots are (i) be, ca, ab, (ii) b + c, c + a. a + b. (iii) , , , -. ^ ' ^ ' ^'b+cc+aa+b (iv) a{b + c), b{c + a), c{a + b), (v) b^ + c^, c^ + a\ a" + 6^. (vi) be - a^, ca - b^, ab - c^. 9. If a, b, c, d be the roots of the equation x^+px^ + qx^ + rx + s = 0, find the equation whose roots are (ij b + c + d,&c. (ii) b + c + d-2a, &g. (iii) b^ + c^ + d^-a^,&c. 10. Find the equation whose roots are the cubes of the roots of the equation x^ +px'^ + qx + ?• = 0. 446. In any equation with real coefficients imaginary roots occur in pairs. For, if a + ?>V — 1 be a root of/ {x) = 0,iC — a — b"^ — 1 will be a factor of f(x), and therefore [Art. 193] X — a-\-by— 1 will also be a factor, whence it follows that a — h V - 1 is also a root of / (x) = 0. Corresponding to the two roots a + 6 V — 1 off(x) = 0, f{x) will have the real quadratic factor [{.c — ay + 6^J. 566 THEOKY OF EQUATIONS. 447. Ill any equation with rational coefficients quad- ratic surd roots occur in pairs. For, if a + V^ be a root of / (os) = 0, \/h being irra- tional, X — a — \lh will be a factor of / {x), and therefore [Art. 179] X — a-^ sih will also be a factor oif{x), whence it follows that a— \/b will also be a root off (x) = 0. Corresponding to the roots a ± \/h, f {x) will have the rational quadratic factor {{x — af — h]. Ex. 1. Solve the equation a;* -2x3-22a;2 + 62a:- 15 = 0, having given that one root is 2 + ,^^. Since both 2 + J^ and 2-^3 are roots of the equation, (a; -2 -^3) (.-c- 2 + ^3), that is a;^-4x + l, must be a factor of the left-hand member of the equation. Thus we have (a;2 _ 4a; + 1) (.X.2 + 2.r - 15) = 0. Whence the roots required are 2 ±^3 and the roots of a;2 + 2x-15 = 0. Ex. 2, Solve the equation 2a;=^ - loa;^ + 46a; - 42 = 0, having given that one root is 3 + >/ - 5. Since 3 ± ,^ - 5 are roots of the equation (a;-3-V-5)(a;-3+V-5) must be a factor of the left-hand member of the equation, which may be written {(.'C-3)2 + 5}(2a;-3) = 0. Whence the roots required are 3^/^-5, ^ . Ex. 3. Solve the equation a;^-4a;5- ll.T4 + 40a;3+ lla;2- 4a;- 1 = 0, having given that one root is ^2 + ^3. If ^a + s^fb be a root of any equation with rational coefficients, f^a and ^Jb not being similar surds, then ±^a±^6 will all four be roots. Hence in the present case (a; - V2 - sfB) (a; - ^2 + ^3) (a; + ^2 - J3) {x + ^2 + ^3), that is a;*-10a;2 + l will be a factor of /(a;). The equation may therefore be written (a;* - 10a;2 + 1) (a;2 - 4a; - 1) = 0, whence the roots are ±^2 ±^3, 2±y^o. THEORY OF EQUATIONS. 567 Ex.4. Solve x*-x3-9;r2-14.c + 8 = 0, having given that one root is - 1 + ^3. x+ 1-4^3 is a factor of /(.t) ; and therefore, as f{x) is rational, the rational expression of lowest degree of which a: + l-;y/3 is a factor, namely the expression (a; + 1)^-3, must be a factor of /(x). Thus we have Thus the roots are 4, -1 + 4/3, -I + C04/3, -1 + 0,24/3, where w is an imaginary cube root of unity. 448. Roots common to two equations. If the two equations / (x) = and (f){x) = have one or more roots in common, / {oc) and cf) {x) must have a common factor, which will be found by the process of Art. 98. Ex. Find the common roots of the equations a;3_ 3x2 -10a; + 24 = ^nd x^ - G-r^ - 40a; + 192 = 0. The H. c. F. of the left-hand members will be found to be a; - 4. Hence x = 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 a;^ - Sx^ - lOx + 24 = 0, having given that one root is double another. Let a and h be the two roots and let a = 26. Then, since a is a root of the given equation rt3-3a3-10a + 24 = (i). Also, since 6 is a root, or a3-6a3-40a + 192 = (ii). The factor common to the left-hand members of (i) and (ii) will be found to be a -4. Thus a =4 and h = 2\ the remaining root of the cubic is then easily found to be - 3, Ex. 2. Solve the cubic 2a;=^ - ISx^ + 37.r - 30 = 0, having given that the roots are in a. p. The sum of the roots is equal to three times the mean root, 568 THEORY OF EQUATIONS. 15 5 a suppose. Thus 3a = — , whence a = -. Divide /(.-r) by the factor 2.r - 5, and the remaining roots are given by x^-5x + 6. 5 Hence the roots are 2, - , 3. In the general case suppose that a and h are roots of the equation y(^) = connected by the relation b = (f>(a). Then f (oj) = and / {<^ (^)} = have a common root, namely a ; and this common root can be found as in Art. 448. Thus a and cj) (a) can both be found. Ex. (i). Find the condition that the roots of x^'+px^ + qx + r = may be (i) in Arithmetic Progression, (ii) in Geometrical Progression. Let a, b, c be the roots in order of magnitude, (i) a+b + c = Sh; .'. 6= -|. Hence, as 6 is a root, we have (-iy-(- -ly- Hh- = 0, whence 2^- 9pq + 27 r=0. (ii) ahc- = &3. ... b=^-r. Hence, as b is a root, we have whence ' -r+p{-i r)3 + r = 0, 4.50. 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 s, fractional root. For if ^ be a root of f{x)= 0, j- being a fraction in its lowest terms, we have faV'' , faV'-' ^ \b) ^^'[bj +"''"-+Pn = 0. 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. 5G9 h 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 oif(x) = 0, so that ^r — a is a factor of/* {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 pn we shall discover all the integral roots. Ex. Find the commensurable roots of ar* - 27a;2 + 42a; + 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 (a; - 2) (a; - 4) (.t2 + 6a; + 1) = 0. Hence the roots of the equation are 2, 4, - 3 ± 2J2, EXAMPLES XLIV. 1. Solve the equation x* + '2x^ - l^x^ - 22a; + 7 = 0, having given that 2^-J'6 is one root. 2. Solve the equation B-c^ - 23a;- + 72a;- 70 = 0, having given that ?>-\-J- 6 is one root. 3. One root of the equation 3a;5 - 4a;^ - 42a;3 + bQ>x- + 27a; - 36 = is J2 - Jo, find the remaining roots. 4. One root of the equation 2a;« - 3a;5 + 5a;'* + 6a;3- 27a; + 81 = is J2 + J- 1. Find the remaining roots. 6. Find the biquadratic equation with rational coefficients one root of which is J^ - ^5. 6. Find the biquadratic equation with rational coefficients one root of which is J2 + J- 3. 7. Shew that a;^ - 2a;2 - 2a; + 1 = and ar^ - 7j;- + 1 = have two roots in common. 8. Solve the equation a;^- 4a;3 + llx'^- 14a; + 10 = of which two roots are of the form a + ^J- 1 and a + 2/3>^- 1. 570 THEORY OF EQUATIONS. 9. Find the condition that the roots of x^+px'^ + qx + r = may be in Harmonical Progression. 10. Find the conditions that the roots of x^+px^ + qx^+rx + s = may be in a. p. 1 1. Find the roots of the equation x^ - 3x^ - ISx + 15 = 0, having given that the roots are in a. p. 12. Solve the equation a;4 + 2x=^-21a;2- 22a; + 40 = 0, having given that the roots are in a. p. 13. Find the commensurable roots of (i) x^-lx'^ + nx-15 = 0, (ii) x*-x^-lSx^ + iex-^S = 0, (iii) dx^-2Qx^ + Six-U = 0. 14. Solve the equation 4:X^-S2x^-x + 8 = 0, having given that the sum of two roots is zero. 15. Solve the equation x^ + ix^ - 5x'^~8x + Q = 0, having given that the sum of two roots is zero. 16. Find the condition that the sum of two roots of the equation X* +px^ + qx"^ + rx + s = may be equal to zero. 17. Solve the equation x^ - 79a; + 210 = 0, having given that two of the roots are connected by the relation a = 2^ + l. 18. Solve the equation Sx^ - 32a;2 + 33a; + 108 = 0, having given that one root is the square of another. 19. Shew that, if the roots of the equation a;" + np a;^-^ + ^ ^^ ~ ^ -^ q x""-^ +...=0, be in a. p., they will be obtained from -p + r \ ■, Y by giving to r the values 1, 3, 5,... when n is even, and the values 2, 4, 6,... when n is odd. 20. Find the condition that the four roots a, /3, y, S of the equation x^ + p>x^ + qx"^ + rx + s = may be connected (i) by the relation al3 = yd, and (ii) by the relation a^ + 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 + ^=y + d, then will iabc - b^ - 8aH = 0. 22. If «, b, c,... be the roots of the equation ' x''+p^x^-^+p^x'^-^+... +Pn=0, prove that (1 - a^) (1 - U^) (1 - c^)... = A^ + B-^+C^-SABCy where A=zPn+Pn-3+ -■•y ^=Pn-l+Pn-A+ —t and C =Pn-2 + Pn-b + • • • THEORY OF EQUATIONS. 571 451. Derived ftinctions. Let / (.r) = poX"" + ^i^'"-i + p.^""-^ + +pn, ; then, if ^ + // be put for cc, we have /(x + h)=po{x-\- hY' +p,(x + hy-' +p,{x + hy-- + . . . +p,,. If now (x + hy\ (x + /?)"~\ &c. be expanded by the Bi- nomial Theorem, and the result arranged according to powers of h, we shall have —a f(x-\- h) =f(x) + h {np,.T''-^ + (?i - 1) jh^''^ + (n - 2) p^"--' + + p,^i} + higher powers of //. This expansion is usually written in the form [The reader who is acquainted with the Differential Calculus will see that the expansion of / (^ + h) in powers of h is an example of Taylor's Theorem.] It will be seen at once that /' («•) is obtained by multi- plying every term of f (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 f (x) can be obtained from /' (x) in a similar manner, and so for /"' (x), &c. in succession. We shall however in what follows only be concerned with f (x). Def. The function f (x) is called the first derived function off{x), the function /"(a--) is called the second derived function of/(j.), and so on. Thus if f{x)=Po^*+Pi^^+Pi^^+P3^+PA* /' (x) = 4pox3 + Sp^x^ + 2p.-^ +P3 , f"{x) = 12poX^-^6p,x + 2p^, 572 THEORY OF EQUATIONS. 452. Theorem. If {x) he any rational and integral function of x and f {x) he its first derived function, then will f'(x)=^^^-^^-^ +.^-A_Z +^^_w_^_ ^ X — 05^1^ X — (X2 ^ ^3 where Oj, a^, a^, are the n roots, real or imaginary, of the equation f{x) = 0. We know that f{x)=;pQ{x-a^{x-a^{x-a^ Hence f(x + h) =po (^ — «! + h) (a? — tta + h) (^ — (Xg + h) The coefficient of h in the expression on the right is by Art. 260 equal to po x (sum of all the products n — 1 together of the n quantities x — ai, x—a^, , x — a^. But f{x + h)—f{x)-\- hf {x) + higher powers of h. Hence /'(^) = Po x (sum of all the products n — \ together of the n quantities x — a'^,x — a^, ^x — a^). Hence f'(i,) = l^+l^+ In the above the quantities ai, ag, , «n need not be all different from one another ; but if ai occur r times, and ttg occur s times, &c., we shall have y'(^) = »£(£) +«/(?) + X tt]^ X ~~ Cv2 453. Equal Roots. We have seen in the preceding Article that if a^, a^, , a^ be the n roots of the equa- tion / (x) = 0, so that / (x) = po{x — cii) (x-a^) (x — an)] then will /' (x) = po x (sum of all the products n—1 together of the n quantities x — ai,x — a2 ,x — an). Now, if any root, for example aj, is not repeated, so that the factor x — ai occurs only once in / (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 f'(x) is not divisible hy x — Oa. Thus a root off(x) = tuhich is not repeated is not a root of f {x) = 0. THEORY OF EQUATIONS. 573 If, however, r roots of the equation / (^) = are equal to fli, the factor x — a-^ will occur r times in / (x), and therefore x — aj will occur at least i — 1 times in every term of/' (x), for every term of/' (x) is formed from f (x) by omitting one of its factors. Hence a root of f (x) = which is repeated r times is also a root of /' (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 f (x) be divided by this H. c. F. the quotient when equated to zero will be an equation whose rcots are the different roots of / (x) = 0, but with each root occur- ring only once. Ex. 1. Find the equal roots of the equation x^ - 5x^ - 9a;2 + 81a; - 108 = 0. Here /(a;) = a;4-5a;=^-9a;2 + 81a;- 108, /' {x) = 4a;3 - 15a;2 - 18a; + 81. The H. c.F. of f {x) and/' (x) will be found to be a;^- 6a; + 9, that is {x - 3)2. Since {x - 3)^ is a factor of /' (a:), {x - 3)^ will be a factor of /(a;), and it will be found that / (a;) = (a;- 3)^ (a; + 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 cxCbic equation can only have three roots. Ex. 3. Solve the equation x^ - 15ar^ + lOa;^ + 60a; - 72 = by testing for equal roots. Here /(a;) =a;5- 15x3 + 10.T2 + 60a;- 72; /' (a;) = 5a;4 - 45a;2 + 20a; + 60. It will be found that the h.c.f. of /(a;) and/' (a;) is a;3 - a;2 - 8a; + 12. If now we divide /(.r) by .r^ - a;2-8a;+12 the quotient will be a;'+a;-6, and the roots of x- + a;- 6 = are 2 and -3. Thus the given equation has only two different roots, namely 2 and -3; and it will be found that/(a;) = (a;-2)3 (a; + 3)-. Thus the roots of/(a;) = U are 2, 2, 2, -3, -3. 574 THEORY OF EQUATIONS. 454. Continuity of any rational and integral Amotion of x. Let po^" + pi^^~^ + ii.,x^~'^ + ■\-'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 oi 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 value sufficiently small. For let k be the greatest of the coefficients; then p^x^-^ -h ... +Pn k{x'^~^ + ...1) kx^ k Now -J {x - 1) can be made as great as we please by sufficiently increasing x. We can prove in a similar manner that PnliPn-i^+ ••• +Po^'^) 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 /(« + /0 -/ W = ¥'(*) + !/"(*■) + , where the coefficients f'{x), fix), &c. of the different powers of In 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) — f {x) can he made as small as we please by taking h sufficiently small. This shews that as X changes from any value a to another value h i^ THEORY OF EQUATIONS. 575 f {x) will change gradually and without any interruption from f {a) to f ih), so that f {x) must pass once at least through every value intermediate to /(a) and f {h). It must be noticed that it is not proved that f {x) always increases or always diminishes from f {a) to /(6), it may be sometimes increasing and sometimes diminish- ing as X 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 f(^) have contrary signs one root at least of the equation f {x) = must lie be- tween a and ^. For since /(^) changes continuously from y (a) tof(^), it must pass once at least through any value intermediate to /' (a) and f {{3) ; it therefore follows that for at least one value of X intermediate to a and /S it must pass through the value zero, which is intermediate to f (a) and f (/3) since /(a) and /(/3) are of contrary sign. Thus the equation /(a?) = is satisfied by at least one value of x which lies between a and /^. For example, if /(j-) = x3- 4x + 2, then /(1) = -1 and f(2)=2. Hence one root of the equation a;^- 4a; + 2 = lies between 1 and 2. 456. Theorem. A71 equation of an odd degree has at least one real root. Let the equation \)Qf{x)= 0, where / {x) = 57-"+^ + ^ia;2« + + p.n+1. Then /(+x) is positive, /(0) = _p2«+i. and /(-:o)is negative. Thus there must in all cases be one real root, which is positive or negative according as p.>n+i is negative or positive. 457. Theorem. An equation of even degree, the coefficient of luhose first term is unity and whose last term is negative, has at least two real roots which are of con- trary signs. 576 THEORY OF EQUATIONS. Let ^2n _[_ ^^^71-1 _l_ 4-^2^=0 be the equation, pon being negative. Then /(+ go ) is positive, /(O) =pm> and /(— oo ) is positive. Hence, as p2n 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, b, c, f, g, h be all real the roots of the equation {x -a){x- b) {x - c) -/2 {x-a)- cf {x -b)- li^ {x-c)- 2fgh = 0, will always be real. We may suppose without loss of generality that a>b> c. Write the equation in the form {x - a) {{x -b){x- c) -/2} - {^2 {x-b) + h^ {x-c) + 2fgh} =0. By substituting + oo , b, c, - oo respectively for x in {x-b){x-c)-f\ we see that the roots of the equation (x -b) {x- c) ~f^=0 are always real ; and if a and j8 be these roots, where a>^, then a>b>c>^. Now substitute + oo , a, /3, - oo for x in the left-hand member of the cubic equation, and we shall have respectively the following results + 00, -{gs/a-b + hj'^^c}\ + {g.sjb - p + hjc - 13]^ - oo . Hence there is one root of the cubic between + oo and a, one root between a and j8, and one root between /3 and - oo . If, however, a = j8 the above proof fails ; but if a = /3, then {x -b) {x- c) -/2, must be a perfect square, whence it follows that b = c and/=0. The cubic equation in this case becomes (x - a) {x - 6)2 - (92 + h^) (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 /3. Hence all 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 Solid Geometry, and is called the Discriminating Cubic. 459. Theorem, //"/(a) and f {(3) are of contrary signs, theri an odd number of roots of f {x) = lie between a and /3 ; also iffipC) and /(/3) are of the same sign, then THEORY OF EQUATIONS. 577 no roots or an even number of roots of f (.x-) = lie between a and /3. Let a, b, c, , k be all the roots of the equation f(oo) = which lie between a and yS ; then f (x) = (x — a) {w — b)(x — c) {x — 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 -6)(«-c) (a -ZO (/)(«), and fW) = (0 - «)(/3 -b){^- c) {^-k) (/3). Now, supposing a > /3 all the factors a — a, a — 6, , a — k are positive ; and all the factors jS — a, jS — b, , fi — k are negative ; also (a) and (^ {(B) have the same sign. Therefore if/ (a) and/(/3) have contrary signs there must be an odd number of the roots a, b, c, , k. Also, if /(a) and f(l3) have the same sign there must be no such roots or an even number of them. 460. RoUe's 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, b, c, ..., k. Then f{x) = (x — a)(x — b). ..(x — k)(f) (x), where cj) (x) is the product of real quadratic factors corre- sponding to pairs of imaginary roots and these quadratic expressions kc ep their signs unchanged for all real values of a;. Then f{x -\-\) = (x - a -\-X) (x - b + X). . .(x - k + X) X [<^ (j;) -I- \(f)' (x) + higher powers of X] [See Art. 452.] s. A. 37 578 THEORY OF EQUATIONS. ■'^ X — a X— X — fc 9 {x) Now all the terms on the right except the first contain the factor x — a, and that term is (x — h){x — c). ..(x — k)(j) (x). Hence /' (a) = {a-b)(a — c). . .(a — k) (f> (a). So /' (b) =(b-a){b- c). ..{b-k)^ (6), f (c) = (c - a) (c - 6). . .(c - ^0 (c), Now (f){a), (j){b), (c), &c. have all the same sign. Hence as a>b>c..., the signs of /' (a), /' (6), f (c), &lc. are alternately positive and negative. Hence there is at least one root of /' (x) = between a and b, 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 number of real negative roots cannot exceed the number of changes in the signs of the coefficients of A- a:). We shall first shew that if any polynomial be multi- plied by a factor x — a, where a is positive, there will be at least oner more change in the product than in the original polynomial. Suppose that the signs of any polynomial succeed each other in the order +-I \- -\ H— , in which there are five changes of sign. Then writing only the signs which occur we shall have + + - + + + - r^ + -- + + - + +- + - 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 pol}'Tiomial 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 polynomial by the factor x — a will contain at least one more chauge 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 multiplying by the factor corresponding to each positive root. Therefore the equation f{po) = 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 /(— x) = 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 x^ + 2x^ - x^ + ix^ + 3x-l by x-1. 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 37—2 580 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 degi^ee can always be solved. EXAMPLES XLV. 1. Solve the following equations each of which has equal roots : (i) 4x2-12a;2-15a;-4 = 0, (ii) x^-Qx^ + ldx^-2^x + S6 = 0, (iii) 16.t4 - 24a;2 + 16a; - 3 = 0, (iv) 2.r4 - 23a;3 + 84a;2 - 80a; - 64 = 0. 2. Find the condition that the equation ax^ + Sbx^ + Scx + d = may have two equal roots. 3. Shew that, if the equation ax^ + dbx'^ + 'dcx + d = have two equal roots, they are each equal to 1 be -ad 4. Shew that the roots of the equation a2 &2 c2 , , Ti^ ^ - x-a' x-b' x-c' "' x- k are all real. 6. Shew that all the roots of the equation a2 62 c2 „ - - + - + +...=m + )v'x x-a X- p X- y are real. THEORY OF EQUATIONS. 581 6. If aj, a^, flg, ..., Con be in descending order of magnitude, and if 6 be positive, prove that the roots of the equation {x - flj) {x-a^) ...{x- ajn-i) + h{x- a^) {x - aj ...{x-a^) = will all be real, and find their positions. 7. Prove that if a, 6, c, d be unequal positive quantities, the roots of the equation 3> %C *C -w r^ + — r+ +x + d = x-a x-h x-c will all be real ; and that, if the roots be a, p, y, d, then will + (a-a)(a-/3){a-7)(a-5) {b-a){b- p) [b-y] [b- 8) c2 + {c-a){c-p)(c-y){c-8) ^' 8. Form the equation whose roots are the values of pu + qo}~^j where 0) is a fifth root of unity, and shew that the equation is x^ - 5pqx^ + 5p^q-x -p^ -q^ = 0, 9. If a, /3, 7, 5 be the roots of the equation X* + 4:px^ + &qx^ + ^rx + s = form the equations whose roots are (i) ap + yS, a7 + /35, ad + ^y. (ii) (a + i8) (7+5), (a + 7)(/3 + 5), (a + 5)(^ + 7). 10. If a, /3, 7, 5 be the roots of the equation a;"* + 4pa;^ + Ggx^ + Arx + s = form the equation whose roots are (a + p-y-d)^ (a-3 + 7-5)-, (a-^-7 + 5)2. 11. If Oj, ttj, flj be the roots of x^+p^x'^ + p^x +p^ = 0, shew that ^ai^a^'^'^PiPz' PiP-f +lhP3* 582 THEORY OF EQUATIONS. Cubic Equations. 463. The most general form of a cubic equation is x^ + ax^ + 6^ + c = 0. We have however seen [Art. 442, iii.] that by in- creasing each root by - , the equation will take the simpler o form a;' + pa? + g = 0. We shall therefore suppose that the equation has already been reduced to this simplified form. 464. To solve the cubic equation x^ ■\- px + q = 0. The solution is at once obtained by comparing the equation with x^ - Sahx + a^ + ¥ = 0, i.e. (x + a -h b) (x + (oa -i- co^b) (x + cohi -\- cob) = 0, where co is an imaginary cube root of unity [Art. 139]. Thus the roots required are — a — b, —03a — co^b, — co'^a — cob, where a and b have to be determined from the equations p = — Sab, q = a^ + b\ Whence a^ and b^ are given by 1^ 4- /(t 4. ^' [2-y \4> 27, 465. The foregoing solution is a slight modification of that called Cardenas 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 ^ + {^ is negative, for we THEORY OF EQUATIONS. 583 cannot reduce an expression of the form (S + 5\/ — 1)\ for example, to the form ol-\- p^J — 1. Thus when p and q are numerical quantities such that ^ + |ir 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 "^ + ^,^ 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 i^eal 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^ + 4.r - 5 = 0. Comparing with x^ - Sabx + a^ + b^ = 0, we have -3a6 = 4 and a'^ + b^= -5, whence a and b are given by {-|4.m93}*. The approximate values of a and b can therefore be found, and then the roots are — a-b, — wa — a] is equal to the excess of the num- ber of changes of sign in the series f(x), f (x) , f (x), ..., fm (x) ivhen x = OL over the number of changes of sign when x = ^. For, let ^1, g'2v, qm-i be the successive quotients; then we have the series of identities / («) = 9'i/i (^) -/a (^)> /i W = 5^2/2 (^) -fz {x\ • • • ' • • • Now (i) it is clear that no two consecutive functions can vanish for the same value of x, for in that case all the succeeding functions, including fmix), would vanish for that value of {x) ; and, (ii) it is also clear that when any 586 THEORY OF EQUATIONS. one of the functions except /(^) 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 /(^) = 0. Then /(a + \)=/(a) + X/'(a) + &c.; and as f{a) = 0, the sign of the series on the right will, if X be very small, be the same as the sign of + Xf {a). Hence, however small X may be, the sign of f{a — \) must be opposite to that of /'(a), and the sign oi f(a + X) must be the same as the sign of /' (a). Thus as X increases through a real root of the equation f(^oo) = 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 = (3 must be equal to the number of real roots of the equation which lie between a and /3. 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 /(a;) = a;4 + 4x3-4a;- 13, /i(a;) = 4(a;3 + 3x2-l). THEORY OF EQUATIONS. 587 N.B. "We may clearly imiltiply or divide by positive numerical quantities as in the ordinary process for finding H.c.r. It will be found that f^{x) = 2x + S, f,{x)=-ld. Substitute -oo , 0, +00 in the above functions, and the series of signs will be + - + --; -- + + -; + + + + -. Thus there is one real root between - 00 and 0, and one real root between and + 00 . Ex. 2. Find the number and the position of the real roots of the equation a;' -5.1- + 1 = 0. Here /(j;) = a;5- 5.T + 1, and f^{x) = 5{x^-l). It will be found that /a (a;) = 4^-1, fs{x)= +255. The following are the series of signs corresponding to the values of X written in the same line -co, — + — + -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 a^+px + q = may be real. f{x)=x^+px + q, f^{x) = ^x^+p. The other functions will be found to be /2(a:)= -2px~3q, Mx)=-{27q^ + -ip^). 588 THEORY OF EQUATIONS. The signs for - oo and + oo are + , +, -2p, -{27g2 + 4p3). and 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 27q^ + 4:p^ must both be negative, the second of which implies the first. 468. Although Sturm's Theorem completely solves the problem of de terra ining 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;4-41a;2 + 40x+ 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 5 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 In this case we should easily find the two negative roots which lie between and - 1 and between - 4 and - 5 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 +,/(2^)is -, and/ (3) is +. Ex. 3. Find in any manner the number and position of the real roots of the equation a;6 _ 5a;5 _ 7^2 + 8.T + 20 = 0. /^ By Descartes' Rule of Signs we see by inspection that there cannot be more than two positive roots and there cannot be viore than two negative roots. Now /(I) is +, /(2) is -; thus one real root lies between 1 and 2. Since / (oo ) is + , there must be another positive root which is easily found to lie between 5 and G. THEORY OF EQUATIONS. 589 Change x into - x, then the negative roots of the given equation are positive roots of a;8 + 5x5 - 7a:3 - 8a; + 20 = 0. Now f(x) must clearly be positive for all positive values between and 1 ; a*id if .r > 1, /(a;)> 6x^-15x2 + 20, which is always positive since 4x6x20-152>0. Hence there can be no real negative roots. -"o*- 469. Equation of Differences. If /(^) = be any equation of which a and /3 are any two roots, and if y = a — /3, 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 f(y + 0) = O and /(/3) = 0. Since a — $ and /3 — a wdll 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 /(«) = 0, and that the equation in y^ has all its roots real and positive if the roots of the equation /{a:) = are all real. In the case of the cubic a?* +_/;a; + 5 = 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 x^+px + q = 0. Let 2/2 = (a-/3)2, where a, ^, y are the roots of the cubic. Then y^={a + ^f-'ia^ = y- + '^-^; •'■ y^-ify + 4q = 0. Also y^+py + q = 0. Hence 7 = 3g/(?/2+/j) ; .-. 21q^+dpq{y^+pf + q{l/+pr=:0, i. e. i/« + (Jpy* + \)pY + 4^^ + 279- = 0, which is the equation required. By Descartes' Rule of Signs the above cubic in j/^ cannot have three positive roots unless p and 4^^ + 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 +00, -2p, -p, 0. Thus a necessary and sufficient condition that the cubic cc^+px + q = may have all its roots real is that 4^^ + 27 q^ 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 f{x) = aooc'' + (h'^'^-'- + a-^x''-^ + ...+«„ is divided by ^ — X the quotient is Q = h.x^-'- + \oc^-^ + h^''-^ + . . . + hn-i, and that the remainder is R, where R does not contain x. Then f{x) ^Qx{x-\) + R. But Q X (^ — X) + jR is at once seen to be h^'' + (61 - X60) x'^-"- + (62 - X61) x''-^ + . .. + (^n-l - ^K-2) X-\- R — \bn-i. Equating coefficients of the different powers of x iiif(x) and in the expression last written, we have bo = ao, bi — \bo = ai, 62 — A6i = a2, On—i A0ji_2 = Clfi—\. } -^ A,0ji—2 ^^ ^71* From the above relations it will be seen that the values of bo, bi, 62, &c. can be obtained at once by the process indicated below : ^-^ CIq di CI2 Cls ^n— 1 ^n Xbo Xbi X?>o X6,i_2 'Xb-n-i bo h h 63 bn-1 R First 60 = Go; multiply 60 by \ and add to Gj, the sum is THEORY OF EQUATIONS. 591 hi', multiply h^ by X and add to cfo, the sum is h/, proceed in this way to the end. Ex. Fimi the quotient and the remainder when is divided by x-2. 1-6 + 2 + 15 + 0+ 7 2-8-12 + 6 + 1 2 1-4-6+ 3 + 6 + 19 Thus the required quotient is x*-ix^-Qx^- + 3x + Q, 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 ftg, ft^, feg) *c. in terms of Qq, a^, a.^, &c. and \ can be at once written down ; they are bQ = aQ, &i = ai + Xa0, b^^a^ + Xui + X'^aQ, h = ^:i + ^^2 + '^"^i + ^^'^o » > and J2 = a„ + \a„_i+...=/(X). Thus ^j'=ao^'»-i + (ai + Xao)a;"-2+... From the above we can obtain the formula of Art. 439. For, if a, b, c, ... be the roots of the equation /(a;) = ; then •^ ^ ' X - a x-b = {ayX"-! + (ai + aao) x^-^ + {a^ + aaj + u-Oq) x^-^ + . . . } + I ttox"-! + (aj + bao) a;«-2 + (a^ + fta^ + b'^Uo) a:"-^ + . . . } + ... = naoX^~^ + {na^ + a^Za) x^~^ + (7ia^ + ajSa + a(,2a=) x"-' + . . . But /'(x) = /tapX'»-i + {« -i)ajX"-a+(n-2)a2X«-* + ... 592 THEORY OF EQUATIONS. Equating the coefficients of like powers in the two expansions, we have naQ = na^ , (n - 1) a^ = na-^ + aoSa, (w - 2) ttg = na^ + a{Za + a^Sa-, Whence the required result follows at once. 473. We have already seen [Art. 442, iii.] that in order to diminish each of the roots of the equation f{x) = by X,, we have only to substitute y + \iovx mf{x). Let the equation whose roots are those of affic^ + a^x^~^ + tts^'""^ + . . . + a^ = 0, each diminished by \, be 6o2/" + 6a2/"-^ + h,if-^ + . . . + 6n = 0. Then, since y = x — \, the last equation is equivalent to bo (x - \y + 6i (ic - xy-"- + ... + bn-i {x-X) + bn = 0. The equation last written must be identical with/(^) = 0. Hence we have identically f(cc) =.bo{x- X)" + b^(x- \)^-i + . . . 4- bn-i (a? - X) + 6^. From the form of the right-hand member of the above identity, it follows that if we divide /(«) by x — \ and then divide the quotient by (x — \), and so on, the suc- cessive remainders will be the quantities 6^, 6^_i,..., bi, b^. Ex. 1. Find the equation whose roots are those of a;4-2a;3 + 3a;-5 = 0, each diminished hy 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. 598 The first division gives the quotient a;3+3 with remainder 1; the second division gives the quotient x^ + 2x + A with remainder 11 ; the third division gives the quotient x + -i with remainder 12, and the last division gives the quotient 1, and remainder 6. Ex. 2. Find the equation whose roots are those of each increased by 3. The divisor is here x + S, and the work is as under, 1- 1-1+4 i_ i_ 1 + 4 - 3 + 12-33 1- 4 + 11-29 - 3 + 21 1- 4 + 11 -29 1-7 +32 1-10 1- 7 + 32 - 3 1-10 Thus the transformed equation is a^-10x^ + 32x-2d = 0. We shall in future write the operation as on the ripht, 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 aooo''' + Oi^^-i + a 0. 25. Find the condition that the solution of the equations 2a (1 — a;) + (6 + c) ic = 26 (1 -y) + (c +a)y, 2d{l-x) + (e+f)x--^2e{l-y) + {f-,d)y may be indeterminate. 26. Having given that x + y + z = 0, and x^ ifi ^ ^ b — c c — a a -b prove that 2 (6 - c) (6 + c - ^aflx" = 0. 27. Prove that, if x + y +2; = = a}x + b'^y + c^z, then (a + 6 + c) (a?x + 6^y/ + c^^;) = (6c + ca + ab) {bcx + cay + a62;). 28. Solve the equations : (i) j2x?-\ + Jx'-^x-'i = j2a^ + 2x + 3 + Jx'-x + 2. (a-xy+(b-xy (ii) / ^2 — 71 ^ =(a- by. ^ ^ (a- xy + {b-xY ^ '■ (iii) f^%f*^*y+f^ ^ ' \x-v a) \x + bj \x+ cj ^ ^ {x-a){x-b)(x-c) ^_Q (x + a) {x + b)[x + c) I MISCELLANEOUS EXAMPLES. 003 (iv) X + y + z - 0, ax + b^ + cz = 0, aV + by + cV - 3 (6 -- c) (c -a) {a- h). (v) 2yz = y -^ z, (vi) x{x+ y -^z) + yz = a\ 2zx = z + X, y {x + y + z) + zx = b'^j ' '2xy = x + y. z (x + y + z) + xy = c^. (vii) or (x^ + 4) = 2/- (?/- + 4) = z^ {z- + 4) = 5xyz. (viii) aj/2; + by + cz = bzx + cz + ax = cxy + ax + by = a + b + c. (ix) x — y = 2, xz — yw - 3, xz^ — yw^ = 5, and xz^ — ytv^ = 9. (x) x + y + z -Oj a h C r^ + 7—- + -7 r-- = 0, (c-a) X {a — b)y {b — c) z b — c c — a a — b 1 1 1 + + = -+-+-. X y z a c , ., cy + bz az + ex bx + cui (xi) — = = = x->ry ^z, ^ ' y + z z + x ^ + y (xii) yz + box + aby + caz — 0, zx -!- abx + cay + bcz = 0, xy + cax + bey + abz = 0. 29. Shew that, if X y z ^ X + a b + a c + a X y z ^ 1 X y z , and + Y^^— + = 1 ; a + y 6 + y C4-y then will _+£+_ = !+ ^Jl. 604 MISCELLANEOUS EXAMPLES. 30. Prove that the three equations a h a h' ^ h G h' c ^ y ^ G y z ' c a c a' ^ c a z X are inconsistent unless a + h + g = 0. 31. Eliminate Xj y, z from the equations p = ax + cy + hz, q = GX + by + az, r — bx + ay + cz, a^ + y^ + 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 x^ + y^ + z^ + axyz = 0, lx + my + nz = Of then x^ x^ x^ + y^ y^ yz + ^^^^0% = 0. 34. Shew that, if x + y+z=pi, yz + zx+xy=p2, xyz=p^, and if x^ + yz = a, y^ + zx=b, z^ + xy = g; then a+b + G = p^—p2, bc + ca + ab=p^p2—^Pi2h~ pii and abG = p^p^ - ^p^P2Pz + P^ + 8^3^. 35. Shew that, if ^-^ x-^ = x^ + x^ ■¥ x^ + ... ad inf., x^ — x^ + x^ + x^ ■¥ ... ad inf., then nx,^ = nx^ + ii?x^ + ti^x^ + . . . ad inf. MISCELLANEOUS EXAMPLES. 605 36. Find four numbers .r, y, z, u such that x — \, y-l, z+\, u+\ are in arithmetical progression, as are also x^, v/-, z^ and .x-^, z^y n". 37. Shew that any number can be expressed in the form whore jOj, p.,, ... are positive integers, and jji < 2, ])2<'^j ^^^^^ so on. Express 999 in this way. 38. Find iVso that the arithmetic, geometric and harmonic means of the first and last terms of the series 25, 26, 27, iV^-2, iV^- 1, 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)„_2 Pr-2' 41. Shew that, if n be a positive integer, the series (2» - 1) Qn-l)(2n-2) _ ' [I ^ \2 , ( i..-(2»-l) (2»-2)...(»+l ) ^ ' 'n — 1 is equal to ( - 1)"-^ {2n - 2) {2)1 - 3) ... (n + 1) n/ |7i-l . 42. Sum the series 111 1 71-1 II \n-2 |2 |n-3|3 '■• |l|n-l 43. By means of the identity {l-x{2-x)]-^ = (\ -x)-\ 606 MISCELLANEOUS EXAMPLES. prove that \'2n |2n-2 \2n-4: \n^ ~ [f Jn-T|n-2 "^ |2 |n-2 |n^ "^ '" " ^' 44. Shew that, if a, b, c, d be the coefficients of any four consecutive terms in the expansion of (1 +x)% then {a + h) (c^ - bd) = {G + d) {b^ - ac). 45. Prove that the coefficient of x^ in the expansion' of (l+cc)^(l-a;)-" is «« h"*"^-! \7i + r-2 nOi-l) k + r-3 91-1 Ir * In- 2 J' 1.2 In-'S \r + (-lf-i?i.2. 46. Sum the series (P + 1) |2+(2^ + l) [2 + (3=+l) _3+ ... +(n' + l) [n. 47. Shew that, if 7i be any positive integer greater than 2, n-l , {n~\)(n- 2) . Li ' to 71 terms is equal to zero. 48. Shew that, if :; o= 1 +2J,X + p.x'^+ ... +p,.X^ + ... , I- ax — axr ^ 1 f - I > ? then 1 -o^ a; 1 2 , , 2 J + «a; 1 - (cr + 2x) x + a~x' i ^ i ^ a. being such that both series are convergent. 49. Prove that, if a; < 1, X x^ a^ (l-xf (l-xj {l-xj ~ + -. -. + ■^ -, + X 2x'' 3a;3 X' l-x' \-a? [ MISCELLANEOUS EXAMPLES. 607 50. Prove that a b c "^ rr^ "^ (i-a){i-b) "^ {\ - a) {I - hjiT-c) "^ ••• k (l-a){l-b)...{l-k) J a (2 -a) 6(2-6) k{2- k) ~ "^ "(T-"^ "^ (1 -af (l-6)2"^--"^(l-«)^(l-6)^..(l-^f • 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 Q2 53 ^2 1^ + -T-+-Fr+T^+--- to infinity. II ll Iji 53. Find the ?ith term of the series 2 + x + ix^+ 19a;=' + 70a;^ + 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, (fcc, 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 5 7 —•7 + 1.2.4 3. 4. 65. 6. 8 56. Prove that, if n be a positive integer, 1 n 1 71 . n - I 1 1 xo -1 • / -.xo -I i — ?r~ 7 svo — ... to n + 1 terms {x+\y 1 {x+2y 1.2 (x + sy / \ n 1 n.n-11 \[ ^ ^ '' \x +1 " I • ^+2 "^ "T72~ ;^T3 "") W-Tl "*■ ^"2 ^ •*• •f X + 71 J 608 MISCELLANEOUS EXAMPLES. 57. If \2n y =:X + x^ + 2x^+ ... + -j — = — r-£c^+^+ ... ad inf., ^ In \n + 1 prove that y'^ — y + x=-0, and deduce that 3 2n ^ \n-l n + 2 58. If n^Ir denote the sum of all the homogeneous pro- ducts of r dimensions of the n letters a, h, 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 7?ith, will be 59. Shew that the sum of all the homogeneous products of a, b, c of all dimensions from to n is ^ra+3 ^n+3 f,n+3 + (a-b){a-c){a-l) {b -c) {b - a) (b -1) (c-a){c-b){c-l) 1 (a-l)(6-l)(c-l)- 60. A man addresses n envelopes and writes n cheques in payment of 7i bills; shoAv 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 I 1! 2! ••• + V ^^ nlj \^ n 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 tiie plane is divided into n"^ —n + 2 parts. MISCELLANEOUS EXAMPLES. 609 62. Shew that x+n \x+n-\ 1.2 x+n-i {x+l){x+2)...(x+7ty 63. Test the convergency of the following series : 2 3 n (ill) a+Tii — Ki — =-a^+ ... + -T— 5 — ^ «"+.... ^ / 2^-2^+1 w*-n^+l 2^ 4^ 2^ . 4^ 5^ 7^ 2^ 4^^. 53. 7^..(37^-l)^(3?l + lf "^ 3^ 3^ 63 . 6^.. (3H)^(3n)3 ' (v) (l.log?-l)+(^2 1og|-l) + ... ^r^"^2;rr-i-V + - 64. Find the condition that the series whose 9'th term is (m + nY (2m — ?i) (3m - 2n) . . . {rm — ?• - 1 n) (m — 7i)'' (2?^ + 7i) {'dm + 2/6) ... (r/u + r — 1 11) may be convergent. 65. Sliew that the limit when ii is infinite of r (m+l)(w + 2)(m + 3)...(m + n) l" . \ 1.2.3...n J '^ ^• 66. Shew that 1111 r + s. A. 89 ^ ^ 71 + 4?i + ri + 4?? GIO MISCELLANEOUS EXAMPLES. 67. If a 1 a^ J^ (^ then will a; 2 ! c«(jc+l) 3 ! r« (a^ + 1) (^ + 2) a<^{x+l) _a a a x(f){x) ~ x+ x + 1 + x + 2 + " 68. If Pn/qn and Pn-i/qn-i be the last and last but one convergents to 11 11 a + b + ... + k+ I * prove that 11 111^ 1 '^ ^ Pnqn+Pn-iqn-l a+b+... + k+l+a + b + ... + k + l q^^ + 2Mn-i 69. Shew that the nth convergent to the continued fraction 12 3 n . ^i^k IS 2_ 3- 4- ... -n + 1-... 1 + Sj"|r* 70. Find the nth convergent to the continued fraction 14 9 n 2 2-5- 10- ...-n^ + l-.... 71. Prove that the nth convergent to the continued fraction CL-t Ctn tvq ^1+1 — «2+l— a3 + l — is (Tnjio'n + l)j where o-,„ = % + a^a^ + a^a.^a^ + . .. to n terms. 72. Prove that the continued fraction 1 a a? (1 - a) a? {\ -a?) I-l- 1 - 1 -••• ^ \ — a \ —a \—a^ = (1 +a)(l + a2)(l+a=^).... .j^mia. MISCELLANEOUS EXAMPLES. Gil 73. Prove that a being greater than 1, the nth. convergent to the continued fraction 1 (a-\){2a- l) (2a- 1) (3^-1) (^a-\) {na-\) 2a- 1+ 2a + 2a + "* 2a + is equal to ^^^~ -^ [{a - 1) (2a - 1) ~ (2a - 1) (3« * 1) "^ •** . — tiir^ |. (na- 1) (tz + 1 a- 1)J 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 3855 ^r^ ; find the integers. 75. Shew that, in the ascending continued fraction 6i+ h.2+ bs+ b^ + ttj aa a^ a^ the n convergent p^ /q^ is given by the laws Hence shew that the value of the fraction 1+ 2+ 3 + 4 + 2 3 4 5 - ^^^^^*- is unity. 76. Prove that the continued fraction 1 X x+l 2(a; + 2) (n-l){x + n+l) 1 — o:+l~x+3— x + 5— " x + 2n — I is equal to {x + I) {x+ 2) ... (x + n)l [n. 77. If shew that 39—2 2/= a, 1 1 1 + — — ... - , a^ + rt3 + + ((,. CI 2 MISCELLANEOUS EXAMPLES. 78. Prove that, if p^jqr be the rth convergent to the con- tinued fraction r+ 1 + ••• + 1 ' and }>,!lqr be the rth convergent to the continued fraction 1+ 1 + ••• + 1 ' then qn=P'n-l + S'n-l) ^n-1 = Q n-\ and i?n-l= ^l?n-2- 79. Shew that the value of the continued fraction m^ im+lf (m+2)2 , . . . — ^j^^ j/- -^^ -!~ ... ad mt. IS m. 2m + 1 - 2m + 3 - zm + 5 — 80. Shew that, if n be any positive integer, N'^ can be expressed as a non-recurring continued fraction with unit numerators, and in the particular case when iV = 2, 7^ = 3, prove that 4 5 29 3' 4' 23 are convergents to it. 81. Find the greatest value of (a — x) (h — y) (c — z) (ax + hy + cz), where a, b, c are known positive quantities, and a — Xj — j^ c — z are also positive. 82. If a, h, c be three positive quantities such that the sum of any two is greater than the third, and if ax + by + cx= 0, prove that ayz + hzx + cxy is always negative for real values of x. MISCELLANEOUS EXAMPLES. 613 83. If in and n be positive integers and m > n, shew that if X be positive 1 + x+ x^ + ... + .r"*~^ > 1 +x + x^ + ... + a;"~^ m < n ' according as oj 1. 84. Shew that, if a be any positive quantity and p > q, > , P 9. unless ^? > > g. 85. Prove that, if x be any positive quantity, x~^ > cc~* > \-\- X -x^. 86. Prove that, if all the letters denote positive quantities, r^m . ^y'i j> ( ^ ) . 77i"i . n"", \m + nj and deduce the minimum value o/L x-\- y when x'^y'^ = a. 87. If a, b, c be positive, prove that the least value which (x + y + zY^^^'' can take for positive values of x^ y, % occurs when x = a^ y = h, z = c. 88. Shew that, if a, b, c, d, a, (3, y, 8 be positive quantities such that a > a, b> /3, c> y, d> 8, then Sabcd + Sa/SyS >{a + a){b + jS) {c + y) (d + 8). 89. If cfj, a^, ag, ..., a„; b^, b^, 63, ..., 6„ ; Cj, c^, Cg, ..., c^ be three sets of positive quantities which in each set are arranged in descending order of magnitude ; then will a^biCi + a^bnC^ + ((sb^c^ + ... 2 5, S^m^x = 0. 104. Two roots of the equation x'-^x^+ 18a;2~30cc + 25--0 are of the form a + i^, ^ -i- ta. Find all the roots of the equation. 616 MISCELLANEOUS EXAMPLES. 105. In the equation r^n+2 4. (2 - a) x""^' + {b-a+l) x''^ +{a+ 2nb) x + a + {2n - 1) 5 = 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 71 be an even integer, and p^, p^, p^ positive, and if H be the greatest of the ratios be all Pi Vz Pn-X and K be the least of the ratios P2 Ih P n J > • • •) J Pi Pz Pn-l then will all the real roots of the equation 2)qX^ - 2h^'^~^ + p<2,'^^~^ - ... +pn=0 lie between ff and K, and all the roots of the equation will be imaginary unless // be greater than K. 107. Shew that 1 1 1 1 1 + a 1 108. Shew that 1+a 1 1 1+a 1 1 1 1 1 + a 1 1 1+a to n rows and columns to n rows and columns 109. 1 a a' aa Prove that 1 1 1 bed U c' (V bb' cc dd' = (a -b)(c- d) (a' - c) (b' - d') -ia-c){b- d) {a' - b') (c' - d'). MISCELLANEOUS EXAMPLES. 617 110. Shew that a^ + b^ + c^, be + ca + ab, be +ca + ah be + ca + ah, a- + b^ + c^, be + ca + ah be + ca + ah, be + ca + ab, a^ + b"^ + c? 111. Solve the equation {x — ay, (x — by, (x — cy (x + ay, (x + by, {x + cy 112. Shew that = (a' + P+e^- 3ahcy = 0. b\^, be, b + c c%^, ea, e + a a~h% ah, a+ b 0. 113. Simplify 1 , 0, p, 1, h + c, be, (p + 6^) (p + C-) 1 , c + a, ea, (p + c^) (p + a^) 1, a + b, ah, (p + a"^) (j? + b^) 114. Shew that ha (6^ + c") - c V be {a" + Jy") ca{h^ + c^) ch{c^ + a^) -aW 115. Shew that = (i V + c V + a^bf. 6^ + c^ + l, c^+1, c^ + l, c' + a^+l, 6'-+l, a'+l, 6 + c, c + a, 6' + l, 6 + c a^+ 1, c + a a'^ + fe^ + 1, a+ 6 a + 6, 3 = (6c + crt + ahy. ANSWERS TO THE EXAMPLES. exampj.es I. 1. 4a;. 2. -2x-6y-4:Z. 5 5 5 6. 2«i2 + 2OTn + 2/i2. 7. 3a2 + 262 + c2 + a&-4ac + &c. 8. a26 + 1063. 9. -2a + 56 -4c. 1 O Q lO. -a--jrb + -c. 11. x^-x-9. D O 12. -SaHSa^ft-SaSS + S^j*^ 13. 2x^-7xy + 7y^. 14. -&c + 4ca + 4a&. 15. -3a^ + 2b'^-Sc'^ + bc + ca + ab. 16. a;-?/. 17. -5y-Sz. 18. -2a; + 2?/. 19. & + fZ. 20. y. 21. 4a. 22. -3a; + 3y. 23. -4n + 4m. 24. 20. 25. -20. EXAIMPLES n. QO 1. 2a;2-5aa; + 2a2. 2. x'^-^x + 1. 3. a;S-l. 4. a:3 + ?/. 5. a;^-!. 6. y^-x\ 7. a;4-x2 + 4a;-4. 8. 1 + a-x'^ + a'^x^. 9. a;8 + a;4 + l. lO. Ga;* - ^x^y + 14a;2?/2 - Sxi/^ + 62/**. 11. 2x6 - 10a;5 ^^ 5x* - 22a;3 - 5a;2 + 5a; + 1. ANSWERS TO THE EXAMPLES. 619 12. 4x6 _ lOx^y + lOx-tj/2 - 21.t3,/3 _ 5.X-7/4 + 5xif + ?/«. 13. Ga6 + llrt'^6 - IGa^i^ ^ 20a3;>3 _ 29a264 + iSq^s _ 3^6. 14. 2(i<5x6 - Sa^xhj- + Sa'^x^y-^ - lla^x^y^ + Qa'-x'Y + '^Oaxi/^ - lOif-. 15. 2a - 3a2 + a^ + Ga-* - oa^ - 18a« + 4'la7 - 42a9. 16. a3 + 63 + c3-3a^c. 17. a^^ + T/S + ^s. 3^^_2.^ 18. 8a3 + 2763-c3 + 18a6c. 19. a;8-l. 20. a;8-25G7/8. 21. a;8 -2x^ + 2/^- 22. a;i2-3a;8 + 3a;-'-l. 23. x^ + 2x^-\-^x^ + 2x'^^l. 24. a8 + 8a'562 + ^Qa'^b^ + 128a2&6 + 2^1^. 25. (i) a- + 462 + 9c2 + 4a&-6ac-12&c, (ii) a* - 2a36 + 3a262 _ 2ah^ + 6S (iii) 62^2 + c2a2 + a2i2 + 2a%c + 2a &2c + 2a6c2. (iv) l-4j; + 10a;2-12a;3 + 9x4. (v) a;H2x5 + 3a;4 + 4x^ + 3a;2 + 2.r + l. 26. (i) a3 + 63 + c3 + 3(a2& + a62^.a2c + ac2 + 62c + 6c-) + Ga&c. (ii) 8a^ - 276^ - Sc^ - 36a^6 + 54a&2 _ 24a2c + 24ac2 - 5462c -366c2 + 72a6c. (iii) 1 + 3a; + 0x2 + 7x3 ^6^4+ 3^5 + a;6. 27. Qxz. 30. a;4 - 2a3x - 2tta;'* - a*. 41. 0. 60. 2a3-32a6c. EXAMPLES III. 1. x-'^y. 2. a;2 + 4z/2. 3. 9j;2 - 12x?/ + IG?/^. 4. -Sx-2y. 5. l + a: + a;"-' + a;3-4a;'. 6. x* + x^y + x-y' + xif-'ky^. 7. l + 2a; + 3a;2 + 4x3 + 6a;4. 8. m* + 2m^n + 3m'^7i'^ + 4tTnn^ + 5n*. 9. l + 2x + 3x2 + 4x3 + 5x'* + Gx°. 10. l + x2 + a;4 + x«. 11. l-2x + 3x2. 12. 2-3x + 2x2. 13. 2jr-3xy + y\ 620 ANSWERS TO THE EXAMPLES. 14. x--xy-2x + y^ + y + l. 15. x^ + ij'^. 16. x2-2.T7/-22/2. 17. a + 2b-3c. 18. a + 2b + Bc. 19. Sa^ + 4:ab + b^. 20. x2 + t/2 + 22_i. 21. a^-2ab + ac + b'^-bc + c^. 22. a2 + 4i2 + c2-2a6 + ac + 26c. 23. a + 26 + 3c. 24. 9a2 + 462 + 9c2-66c + 9ca+6a&. 25. ca;2 + fZa;-c. 26. 2ax-{Sb-'ic)y. 27. a^-Sat + fi^. 28. x'-xy + y"^, {x + y)^-z{x + y) + z^. 29. x^ + xy + y% {x + y)^+2z{x + y) + 4.z'^. EXAMPLES lY. 1. {a-2h){a + 2b)(a^ + Ab'^). 2. (2a;-3a6) (2a; + 3a6)(4x2 + 9a2J2). 3. (4 + 3a-2&)(4-3a+26). 4. (2?/ + 22-a;) (22/-22 + x). 6. 5ax {2ax + By) {2ax - By). 6. 4:a'^x^ (Bx^ + y^) {Bx^ - y^). 7. 8(a-6)(a + &)(a2+62). a. IQ (a-b) {a + b){a^ + b^). 9. 24a; (a; -1) (a; + 1)2. lO. 16a; (2 - 3a;2). 11. 4¥{2a-b^~){4:a^+2ab- + b*). 12. (a2-46c)(a4-2a2ic + 462c2). 13. (a-4)(a + 2). 14. (4-a;) (3 + a;). 15. (l-21a;)(l + 3a;). 16. -4 (a -1)2. 17. ab{a-b){a~Bh). 18. a^b [a+b) {a + 4b). 19. {b + c-a){b + c-5a). 20. {Ba + Bb-c-df. 21. (a;-2)(x + 2)(a;-5){a; + 5). 22. (5x-?/) (5a; + ?/) (2a;-i/) (2a; + i/). 23. (a;2 - h/z-)^= {x + 2yzy^ {x - 2yzf. 24. a?{a + h)[a-h){Ba + b){Ba-b). 25. [x-h){x + b~2a). 26. (a;-a) (a; + 2?/ + a). 27. {a + h + c~d) {a + b-c + d) {a-b + c + d) [- a + b + c-rd). 28. (a; + t/ + a + 6) (a; + ?/-a-i>) (a;-?/ + a-6)(-a; + y + a-6). ANSWERS TO THE EXAMPLES. 621 EXAMPLES V. 1. {x + l){x-l){x + a). 2. {a + b){c-d). 3. {a-b){c + d){c~d). 4. {ax + by) (cx + dij). 5. {ax + b){cx^ + d). 6. 2 (a-d) [a + b + c + d). 7. {a + b){a-b){a^- + ab + b-). 8. {a - b)^ {a"^ + ab + b^) . 9. {a + l){a-l){b + l){b-l). lO. (x + z) (x-z){y + z) (y-z). 11. (a;2z-l)(y2^-l). 12. {x + y){x + z){x^-xz + z^). 13. (a:-?/)(j; + ?/ + -s). 14. {a; + 3) (a;-3) (a;2 + 2). 16. {x^ + 5x + l){x^-5x + l). 16. (a;2+4a;2/ + 2/-) {a;2-4r?/+?/2), 17. {x^ + x + l){x^-x + l){x^-x'^ + l). 18. (x + a + 6) (a;-a-6) (.T + a- 6) (a;-a+&). 19. (x^ - 2?/2z2)2, 20. {x-2b + ab)(x-'^a-ab). 21. (x+6)(x2 + a). 22. {l-x^){l + y + xil-y)}{l + y-x{l-y)}. 23. (a; + t/-32) (a;-y + 2). 24. {2y -x + a) (y -2x- a). 25. (a-36 + c)(a + &-3c). • 26. (2a- 11& + 1) (a + 26 - 3). 27. {l-ax){l + ax + bx^). 28. (1 - ax) (1 -oos-cic"^). 29. -(6-c)(c-a)(a-6). 30. (& + c) (c + a) (a + 6). 31. {a-b) {a-c){b + c). 32. (x2 - a;?/ + 1/2) {a;(a+l) +2/(6 + 1)}. 33. (j;?/ + a6) (a?/2 + 62j;). 34. (2x - z) {x - i/)^. 36. (x2-y2)(?/2-2a:)(22_a:^). 36. {x + 4:){x + 2){x-l){x-'d). 37. (j; + 4) (a; + 2) (a;2 + 5aj + 8). 38. a;(x + 5)(a;- + 5x + 10). 39. {x + 2) (x + ij) {x- + 8x+10). 40. (x + 8)(2x + 15)(2x2 + 35x + 120). (J22 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-zx- xy). 3. {h + c){h-c){c + a){c-a){a + b){a-h). 4. {h-c){c-a){a-'b){a+h + c). 6. {h-c){c-a){a-h){a? + h^ + c^ + l}'^c + 'bc'^ + c^a + ca^ + a?h + db^ - 9a&c). 6. - (6 -c) (c-a) (a-6). 7. -(6-c)(c-a)(a-Z>)[Z>%2 + c2a2+a262 + a6c(a + & + c)]. 8. -(Z>-c) (c-a)(a-&)(a2 + &2^g2 + ji,g + g^^a^)^ 9. -{h-c){c-a){a-'b){a^ + h^ + c^ + h\-vhc'^ + c^a + ca? + a%-\-ah'^ + ahc). lO. 24rt6c* 11. 80a&c(a2 + 62 + c2). 12. 4a&c. 13. 2a&c. 14. Aahc. 15. -4(&-c) (c-a) (a-6). 16. Z{y + z){zJrx){x + y). 17. 5 (t/ + 2) (2 + a;) (a; + y) (x^ + y^ + z^ + yz + zx + xy). 18. -(&-c)(c-a){a-&). 19. -2 (6 -c) (c-a) (a- 6) (a + 6 + c). 20. -(6-c)(c-a)(a-&)(3a2 + 362 + 3c2 + 56c + 5ca + 5a&). 21. (6 + c)(c+a)(a + 6). 22. _ (&-c) (c- a) (a- &) (a + 6 + c)2. 23. {x + y + z){yz + zx + xy). 24. (& + c) (c + a) (a + &) (a+Z> + c). 25. lJxyz{x + y + z). 26. - 3 (6 - c) (c- a) (a-&). 27. 16(6-c){c-a)(a-&)(rf-a)(d-&)(d-c). 32. 27a262(a + 6)2. 34. (a2 + &2)2 (^2 + ^2)2. 36. {b-c){c-a)(a-b){a-d){b-d){c-d). ^~" 37. - (6 - c) (c - a) (a - b) {a - d) (6 -d){c- d) {bed + ccZa + dab + a&c). ANSWERS TO THE EXAMPLES. 623 EXAMPLES VII. 1. a-b. a. 2a;- 1. 3. x--rj^. 4. 2x~y. 6. x-2y + Sz. 6. 4:a-'-Bab + h\ 7. a + 2b. 8. 2x^-Sx + l. 9. x-a. lO. X' + x-Q. 11. a-2-a; + 3. 12. x'^-3x + 7. EXAMPLES Vni. 1. 12a;* + 2ax^ - ia-x- - 27a^x - 18a\ 2. (4a-6)(a-&)(3a2 + 62). 3. (a;2-2a; + 7) (Ga;3 + a;2-44a; + 21). 4. (a;2 + 5a; + 7) (Ta;^ - 40a;3 + 7oa;2 _ 40a; + 7). 5. a; (a; + 1) (a; + 2) (a; -2) (a; + 3). 6. a;(a;-l)(a; + 2)(a; + 6)(a;2-2a;-f4). 7. 2a (2a - 6) (2a -3&) {2a + 36). 8. 6a; (a; + 1) (a;- 3) (.r- 4). 9. (3a; + 2)(8a;3 +27)(8a;3-27). 10. 3 (a; - 32/;2 (a;2 - 4?/2). 11. {x-2y){x-Sy){x-iy). 14. {ac' - a'c)^ = {ba' - b'a)' {b'c - he'). EXAMPLES IX. bb-chjH'^ dab-cH^z*' a-h Qa^x'^ ' ' !/* ' ' a + 46 * _ ,, _ x'^ + xy + y- x^-xy + y^ a;2j/=-l 4a;-i/2+l' 2x-l x^+l- 2x + 3y x^-xy + y^ x'^ + y- 7. ^^^ — f. 8. a;2-3a; + l* 9a;3 - 3a; - 2 ®' 3x-^2- ^®- 6a;3 + 3a;2-l 624 ANSWERS TO THE EXAMPLES. 11. , ; r- , , , . 12. yz + zx + xy. {a -b + c}(-a + b + c) " 13. -h{y-z)(z-x){x-y). 14. 1^. yz + zx + xif a^ + h^ xo. x + y + z 17. 1 1-9x3* 19. 2x + ia ~x-2a' 18. 20. a:y + y^ x^ - 4?/2 * 48 (a: + 2)(a; + 4)(a; + «3)(a; + 8}' 48a3 ^ 24a4 21. , T-, ■:r-^, i^— ^ =-T . 22. (a; + rt)(d; + 3a)(x + oa)(a; + 7a)* ' .r (.^2 _ a2) (a;2 _ 4a2) * 23. 0. 24. 0. 25. 1. 26. -1. 27. d. 28. 0. 29. 1. 30. 2. 31. a-'rh-'rc. 32. a2 + &2 + c2+&c + ca + a&. 33. (a + 6 + c)2. 34. a + 6+c. OK a+6+c (-a + 6 + c)(a-6 + c){a + 6-c)* 37. 16f^:y. 38. 4^?fl^^\ 40 (x-p)(a;-g) >., o ^^ A ^a 2a6c(a + 6 + c) EXAMPLES X. 1. 2a-6,a-26. 2. 1. ^±^. 3. 0, ^ . c + a-2b b-a 4. a -26, 6 -2a. 6. il. 6. il. 7. 1, -3. 8. 1. 0- 0, ±5^2. lO. 6, -81. ANSWERS TO THE EXAMPLES. 625 50 a^c + h^a + c-b - Sabe ^^' 29' ^^' a^ + U^ + c^-bc-ca-ab' 13. 0,-{a + b + c^^{a^ + b^ + c'^-bc-ca-ab)}. o 14. [hc + ca + abJ=^{b^c^ + c^a' + a%'^-abc{a + b + c)]]^{a + b + c). 5 15. 6, -J. 16. ±V6. 17. ±^ab, ^J-ab. 18. 0, - - . //?) A 19. ^ x/G-0- 20. {a + 6 + c±;^(a2 + 62 + c3-6c-ca-a6)}. &2 + c2 21. '^' 6 + c- 23. a2 + 62 2.S. a&-c^ 22. -2(a + & + c). cd (a + 6) -a6 {c + d) 24. — -, -r. 26. 0, a+&, c+a-a-o a6-cd a + b [b~cY+[c-aY + {a-bY 2b -v / 20. 0, i^jTIoft. 30. 8, -5. 31. 2, 3. 32. 1. 33. 0, a2_62^ 34. a, &. 35. ?, ^. 36. 0, 4(a + 6). 37. - a, - 6. d 38. -iCa-ft). 30. ±*ya^- *®- 0' ^2 Jab. 2 , /b*-a* 41. ±-v/3a2-36^ 42. ± >/ ^^232^2' 43. 0. 44. ±a±6. 45. -- {a + ft + c ±2 ^/(a3 + 62 + ^2 _ 6c -ca -«&)}. a6(a + &) Sab (a + b ) ^h^c^+fa^ + aPb^ *®" a2 + a& + &2' a2 + a6 + 62- ' 2abc S. A. 40 (J26 ANSWERS TO THE EXAMPLES. 48. ^.J^J{2a-b^c^{a^ + b'^ + c^)-h*c^-c*a^-a^b-} 2aoc ^{{bc + ca + ab) {-bc + ca + ab) {bc-ca + ab) (bc + ca-ab)}. 2abc 4 49, Values between 3 and -k. 62. Values between 3 and - . o 63. X lies between - 2 and 8, and y between - 9 and 1, 64. X between - 2 and 10, and y between - 1 and 5. a 56. -. 69. (i) a^x^+{b^-Sabc)x + c^ = 0. (ii) a^cx^ + xb{b^-Sac) + ac^ = 0. (iii) x-^-bx+ ac = 0. 60. (i) -2-. (ii) ^-2-. 66. ~. EXAMPLES XL 1. ±2, ±7^. 2. a, aw, aw", -2a, -2aw, -2aw*. 3. - a, - aw, - aw-, 2a, 2aw, 2aw^. 4. 1,^(1 ±7'=^). 5. 0. 1, 3, -8. 6. 1. -2, 2(-l±V^^). 7. -1, -6, i{-7±3V5), 1 15 ][ °" 2' "T' 2^"'^^^^/^)- ®- 3,-1,1^2^19. lO. -(-l±7-3), i(aiVa'-4). 1 ANSWERS TO THE EXAMPLES. 627 II. 0, -5, -{-5J=J -15). 12. a, -Ua, -4a±a^^o[5. 13. 7a, -8a, |(-1±^-167). 14. -4, -6, -(-15± ^129). 21 15. 3, -j^. 16. ±^(a + 6). 17. ±a±6. 18. 2,^, 1(-3±V^. 19. 3, i, 1(-1±7T8). 20. -1, ^[l + V5±V{+2s/5-10}], |[l-V5±v/(-2v/5-10)]. ai. =^1. =^V^; =^^=^^-^ 22. 2,2.^. 23. -1,2,3,-4. 24. ±1, ^(-7±3;V5). 25. a, &, c. 26. 9, -3±^-47. 27. 9, -6, i (3 ±7^^215). 28. a, 6, ^(a + b). 29. a, 6; - (a + 6)±— (a- 6) ^^. 30. a, b, -{a + h), -[a + h)±-{a-b) J^. 31. a, 6, ^{a + b). 32. a, 6, - {a + 6±— (a-6) 7^63}. 33. a, b, -{a + b±{a-b)J~^}. 34. a, 6. 35. a, 6, aud roots of (a - x) (x -- 1) = 16 (a - 6)". 37. a-2i, 6 -2a, -- {a + 6=t (a - &) ^/-15}. 38. Hoots of ar (a - a-) = f v'i; -fc a/ ^ + - j . 40—2 628 ANSWERS TO THE EXAMPLES. ^3 53 ])c ca ab ^^- h^ a"-^^- ^''- a' y T- , a^ + h^ 2ab 41. 0,a + b, ^, r. ' ' a + b a + b 42. ^{-tdbV(Z>^ + 4)}. |{-a±V(a2 + 4)}. 43. I {a ^ J {a^ - Ab^}} . 44. -{a + b + c), -~(a + 6 + c)±^^/(Sa"-226c). 2 1 46. a + 6 + c, - (a + 6 + c)±5,^{2a2-7S&c). 46. a, b, c. 47. ^ ^ /] cd{a + b) + ab{c + d) ] / \ ab {c + d) -cd{a + b) ] V i a + 6 + c + (Z j' V t c + d-a-6 |* EXAMPLES XIL 11 « 18 8 1. x = l, 2/=-l. 2. a;=y, 2/ = g. 2 3. a;=3, y = 6. 4. a;=-, y = 3. 5. x = b, y=a. 6. x = ab, y= -a-b, 7. a; = a + Z>, y = a-b. 8. x = y = a. 9. x = a, y = b. 10. a; = a {a-6), ?/ = &(a-6). Ill 11. a;= -3, 2/ = 3, 2; = 1. 12. a; = - ,?/ = -,« = - . 13. x=y = z = l. 14. a; = 6 + c-a, y = c + a-b, z=:a + b-c. 15. x = Z> + c, y = c + a, z = a + b. 16. x=--(2a + 6 + c), 2/=-^(a + 2& + c), ;2= - 1 (a + & + 2c). -- 1 17. x = y = z = , . -^ a + b + c 18. x = -{2a + b + c), y = ^{a + 2b + c), z = ^{a + b + 2c). ANSWERS TO THE EXAMPLES. 629 a b ^®- ^=n^ — n-n — T^y v=iu — vttt — i» ^= (rt -h){a-cy •' (& - c) (& - a) ' (c - a) (c - ^j ' 20. x = a, y = b, z = c. 21. a;=-a + ft + c, y = a-b + c, z = a-{-b-c. 22. a; = a(6-c), ]) = b (c-a), z = c{a-h). 23. a; = l, y = 0, ^; = 0. 24. x = abc, y = bc + ca + ab, z=a + b + c. ^ m (m - 1) (/?i - c) 25. a; = — ^ -— -, &Q. 26. x = a, i/ = b, z=c. a (a-b) [a- c) ' j ' -,_ 7 ■, ^ a(a + b + c) „ 27. x = b + c, y = c + a, z = a + b. 28. x= , ^ ,^ , ^, - 16 1 2 3- A S 2 16. i, J, d, ^^, g, -. 17. 1, 2, 3. 18. 3, 5, 7. 19. 0. 4, 5. 20. 3, 3.4. 21. 0, 0, 0; |, |. ^. 032 ANSWERS TO THE EXAMPLES. 32. -a, b, c; a, -h, c; a, h, -c; |(l±^-7), -^{1±J-1), c 4 1(1=^ V-7). ''^- '^-^{(«-6 + c)(a + &-c)}'^''- 26. " - y (b + c)--{c-ra){a + b) (c + a)^- {a + b){b + c) z 1 = d= ~(a + 6)3-{6 + c)(c + a) 1^{^abc-d^ -b^-c^)* 27. a, 6, c; -(26 + 2c-a), -(2c + 2a-6), :^(2a + 2&-c). o o o 28. x=^j-^{^c-ca^-ab){\ic-\-ca-ab), ho,. 29. a, 0, 0; 0, b, 0; 0, 0, c; ^-^ '-^^, — |. 3aoc - a-* - b-^ — c^ 30. 0, 0, 0; -^a, -, -; -, --6,-; -. -, --c; 2(-a + 6 + c), ^{a-b + c), ^{a + b-c). a^x bhj cH 32. = = -=J{- abc\ b-c c-a a-b ^ ^ ' ax by cz I - abc ax _ by _ cz _ / -c c-a~ a~b~ \/ bc + ca + ab' EXAMPLES XV. 1. 20, 30. 2. A £10, B £15, G £25. 3. 20 years ago. *. 2«. 6. 5, 15, 30. 6. 5 days. 7. 1800. 8. 58. 9. 30 miles. lO. 120 lbs. 11. 4 days. 12. 36,9,12,15. ANSWERS TO THE EXAMPLES. 633 13. 48 miles. 14. 15 miles. 15. 54, 81, 108. 17. A £450, B £225, G £237. lO.-., D £87. 10^. 18. ±5. 19. 38, 88. 20. 18 miles. 21. At 1 o'clock, 15 miles from Cambridge. 22. A £10, B £5, C £1000. 23. 25. 24. 9, 7;8^2, ;^2. 25. 50miles. 26. 576. 28. 3 miles an hour. 29. 3 hours. 30. 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. 36. 450 miles. 37. 30 miles. EXAMPLES XYI. 3. a + h + c + abc = 0. 4. {b + c-a) (c + a-h) {a + b-c)=S. c^ a^ 8. a^ + 2c^-Bab^ = 0. 15. P + m? + ii^-lmn-4: = 0. 16. aP + bm^ + cnP + lmn=4:abc. 17. a'^ + b- + c^-abc-^i: = 0. 18. a3 + &3 + c3_5a6c = 0, yh^ + z^x^ + xhf + xhjh-^O. 19. 263c3=5a262c2. 20. a^ + b^ + c^- be (6 + c) - ca {c + a)- ab {a + b) = 0. a^b"^' 2. 1 EXAMPLES XVIL 1. b^' 5. x-y. 6. x* + l + x-*. 7. x + y+z-Sx^y^z^. 8. x'^-x~i' 9. aT7 + aT^a;^ + a^^^a;^ + aT 1 ; Divergent if x 4- 1. lO. Convergent if x > 1 ; Divergent if x 4* 1. ANSWERS TO THE EXAMPLES. 637 11. Convergent if a; 4" 1 ; Divergent if a- > 1. 12. Convergent if a; :|» 1 ; Divergent if a; > 1. 13. Divergent. 14. Divergent. 15. Convergent if 7/i< 1 ; Divergent if m ^ 1. 16. Convergent if m < 1 ; Divergent if m <^ 1. 17. Divergent. 18. Divergent. 19. Divergent. 20. Divergent. 21. Divergent. 22. Convergent if x 1. If a; = 1, then Convergent if A; > 1 and Divergent if /i :f 1. 23. Convergent if a; < 1 ; Divergent if a; «|; 1. 24. Convergent if a; :j- 1; Divergent if a; > 1. 26. Convergent if a; < 1, Divergent if a: > 1. If a; = l, then Convergent if 7?t < ^ and Divergent if m -j: ^. 27. Convergent if a; < 1, Divergent if a; > 1. If a; = l, then Convergent if A < ^ and Divergent i( k ^^. EXAMPLES XXVII. , .^ , ,, 5.2. 1.4.7. ..(3r-8) ^ , ... 3 . 8 . 13...(5r ^x^ . ...^ 2. 3. 8. 13.. .(5r- 7) ^, ...,. g (g+i^) (g + 2i>)-(g + r- 1 . f ) ,, (VIU) z pr— 5 X^. (IX) _-— JJ\ ^ ' 1.2.3...r P la , ., 5.3.1.1.3.5...(2r-7) ^ . , „ (xi) ■ ^^ ■ x^a^'', r > 3. . ... 2.5.12...(7r-9) ^,a ("") - 4.8.12:..4r " ''^- 2. (i) The ninth term, (ii) the eighth term. 3. The 3yth term. 4. 'J'lio first and .second terms. C^S ANSWERS TO THE EXAMPLES. ,., 1 .4.7...{3r-2) i_2r ,, 6. After the 12th term. 8. (x) 3 g "9 — sT^ 13 5 (n-1) (ii) 2a-^x\ (iii) 4ra-^x'^. (iv) ' ' '"^ a-'^x^ when n is ^ ' ^ ' ^'2.4. 6...ri 1. 3.5...(n-2) „ „ , -1, even, ,, , ,, — r- a-^a;** when n is odd. 2 .4. 0...(7i~ 1) (v) (2r2 + 2r + l)a-'-ix'-. (vi) ( - 1)'' 16 (r - 1) aS-'-.i'-. EXAMPLES XXVIII. 1. (i) 2. (ii) y/|. (iii) 44/4. (iv) V27-2. (V) 1. (vi) 4/4. (vii) ^A. (viii) 1. 3/9 10 1 1 (i^) /y/i- (^) y-SV^l^- (^) 24* ^^"^ 37_ 245 . ,. . (7i-2)(n-3)...(n-r) « , 11. (-l)*--!^ '- { — 5^ '-. 23. 1. 25. 141. \ ' \ r-l 31. -245/8. 32. 246,792. 33. 462. 34. 35. 35. Coefficient of x^'* is S^*" 2-a'-2 a-3J-2^ of x^^+^ is - 3=^'+i 2-3'--3 a-S'-a and of x^^'^ is 0. 38. ^ (« + 1) ('I + 2) (?i + 3) (71 + 4) [n + 5). 39. 2»-t-'' |3n + r-l ^ |r_ i3/i-l . EXAMPLES XXIX. 1 ^Q _ 3 _4_ 3 • 5(a: + 6) 5(a; + l)* ^- a;-3~a;-2' 3 5__ _ 1 4(a; + 3) 8(a; + 5) 8(a;+l)* 4 1 2 _2 1_ 1_ a; {x + lf *• (x-2)-' a:-2'*'a; + l* 6 _i Z__ + _^ « 1 4a;- 8 r2(a;+l) 3(rc-2)'^4(x-3)* ^* 5 (a: + 2) + 5 (.^2+ 1) ' 3. ANSWERS TO THE EXAMPLES. 639 1 1 1 1 1 1-lOa: 3(l + 3a;) 3(H-3x)2* 2 (a;^ + l) ^ 2 (x- 1/-^ * 8 21 21 7 lO. ^TT-. — rr-T^ + TT^. — 77-r. + 7^K^. — ^^ + 2 (1 - 3.r)' 8 (1 - 3x)2 ^ 32 (1 - Sx) 32 {1 + x)' 1 ^ 1_ 2 4 _5 + 2 ^*' a;2 + l'^a:-2 x + 3' {x-2f'^ 5 {x-2)'^ 5 {x" + l}' 2 11 11a; -4 13. — TTo + 14. 5 (x - 1)2 ^ 25 (.-r - 1) 25(a;2 + 4)* 3 11 a; + 2 5(a;-2) 2 (a; -1)2 2(a;-l) 10(a;^ + l)* 1 1 1 17 1 3 15- cT-o-Tir-- — ^ + .^... . o> + ... . .-..■> + 8^2 IGx a; + l 16(a; + 2)^(a; + 2j2^4(a; + 2)3* 1 1 11 113 4(x + 2)3^6(a; + 2)2^144(a; + 2)^9(a;-l) Sa;-^ lliic* 4 / IX^+i 1 17. (-l)-{2--3-"-i}. 18. -(^--j -^(3n + 7). 20. ^(3'*-l)-g{(-l)'^-l}. 21. i{0 + 5"+2-2.3"+2-2"+4}. 22. (7i2 + 7n + 8)2'»-3; \{n^ + 9n" + Un)2^-\ o EXAMPLES XXXII. 1. 1-262. 2. 1-48169. 8. £1146-74. 5. £742. 19s. Gd. 7. £785. 10s. 8. £1979. 5s. 6d. 0. £1735 nearly. lO. £122-58. EXAMPLES XXXIII. ( (9,'n -U 1 \ r^« -J- ±\ 12 1- /5{(3n + l)(3n + 4)(3n + 7)(37i + 10)-1.4.7.10}, ■ 8t3.7'{4n + 3)(4H + 7jj' * 168* 3. --n(n + l)(3n2 + 23n + 46). 4. i w (« + l) (n-f-2)-3ri. G40 ANSWERS TO THE EXAMPLES. 6. ~n{n + lf{n + 2). 6. —n{7i + l){n + 2){Bn + 5). 7. i^(2n-l)(2ri+l)(2n + 3)(6^i + 7) + |. °' 180 12(2?i + l)(2n + 3)(2n + 5)' "~180' 5 3n + 5 . q _ ^ ®' 36"6(n + l)(n + 2){n + 3)' *~36* 5 2n + 5 _5 4 2(n + l)(?i + 2)' ^^o^S' 1 4:71 + 3 1 ^^ 8 8(2n + l)(27i + 3)'^~8' 29 6n2 + 27w + 29 _29 ^^' 36 6(n + l)(n + 2)(7i + 3)' ^~36* . ^®- '^'*"r^' '^«"^- "• 4{'^ + l)(^i + 2)(in + 3)-|. 15. ^n(n+l)(7i + 2)(8;i2 + llw + l). 16. 7ta2 + w (n- 1) a6 + jT (ti- 1) ti (27t - 1) b\ 17. na^ + fn {n-1) a'-b + hn-l) n (2/i - 1) afcs + l 2 ^^^ _ 1)2^,3, 18. -7i(47i2-l). 19. ^n{lQn^-12n-l), 23. -7i(7i + l)(ri + 2). 24. ^ti (71 + 1) (471 - 1). 25. 7ia& --n{7i-l){a + b)+^n{n- 1) (27i - 1). .. 2«+i 1 n + 2 ^ '' (7i+l)2«* ^-- ^ ^ 71 + 1 VS; ^'''^ 4 2 (71 + 1) (71 + 2)1,7 j M ^ ^ /3\" 6 /G\" (n + l)(n + 2) V4; • V^ (;i + l)(7i + 2) V7 ANSWERS TO THE EXAMPLES. G41 EXAMrLES XXXIV. 4.7.10-(3» + 4) 2^8.^(3n + 2)_ *• ^'^ 2.5.8...{3n + 2) ^* ^"^ 4 . 7 . 10... (3n + l) ^• ,...,, 5.7...(2n + 3) ^ ^^11i1 13 . 15...(2» + 11 )) .. 2. (i) 2 + 3(/i-l)(w-2); 2n + w(?i-l)(7i-2). (ii) 7n-(n-l)(«-2); ^ n (/i+l) -^n (n-1) (w-2). (iii) 2«+i-n-2; 2"+2- 1 -i (7i + 2) (w + 3). (iv) 2"+i-n{n + l)-n; 2"+2- 4--n (n+1) (7i + 2) --n(n + l). (v) ~n{n + l){n + 2){n + S); — n (n+ 1) (w + 2) (n + 3) (7i + 4). (vi) (n - 2) (n - 1) 71 (n + 1) + (n - 1) w - n + 2 ; hn-2)(n-l)n{n+l){n + 2)+-{n-l)7i{n + l)--n{7i + l)+ 2n. 2-4a; „.. 1 - 2a; l-6g ®- ^'^ 1-^ + x'' ^"^ l-5x + 4a;-' ^'"^ 1 - 12a; + 32.r;2 * 15 + a;-19a;2 . . 1 + x ^^^^ 15-14a;-35x2-42a:3' ^^^ (l-a:)^' 4. (i) 2"+i-2; 2''+2_2n-4. (ii) 1 {3«+ U (- 4)»-n ; i; + 5;^ -i-^ (--!)». (iii) ^ {3" - ( - 1)«} ; i {3«+i - 3} when n is even, and - {3"+i - 1} 4 o o when 71 is odd. e- |,{(l + v/'5)"+(l-\/'^r}. 6 a = l, h = i, c = h d = 0. 2 - 3x - a;2 ^ 1 8. — — -— . 0. (1 - x)-^ (1 - 2x) * ■ (1 - a)-* (1 - 2j;) • lO. (^ + q)(^ + ^) - (^ + ^ ahc.l r?-Vn 1/ l\i ,, X a: 1 S. A. 41 642 ANSWERS TO THE EXAMPLES. 24. l-(^l--jlog(l-^). 25. —j ^_^^, . 30. x a + /3, Divergent if 7 < a + /3. EXAMPLES XXXVI. n + 1 7. n + 2* 10. hn-ip'n - (^»«n + KK-l) P\-l - ^nK -i iKK-l + «n) P'^n-a EXAMPLES XXXVII. 1. (i) 4 + J (ii) 11 + ^ ^ - - B+ ^ ' 1 + 4 + 1 + 22 + ..... „ 1 1 1 1 M ^+1 + 2 + 1+10+ (iv) + 1 1 1 i 1 i i 1 1 1 1 + 1 + 3 + 1 + 5 + 1 + 3 + 1 + 1 + 12 + ^ (•) V^^ (") ^(4 + v/37). (iii) ^(28-^30). le. J(n= + 3n). 26. 6"^ EXAMPLES XXXVIII. ®- 733. 10. 3, 7, 9, 11, 13, 19. 13. 504w-6. EXAMPLES XL. 1. (i) x = 2, y = 3. (ii) .^. = 1^ ^^^10; .t = 14, y = 2. (iii) .t:^4, 7/ = 8; .t = 13, y = l. (iv) 696, 3; 025, 18; 554, 33; ; 57, 138. A^sbVVEKS TO TUE EXAMPLES. 643 2. 22, 30. 3. (i) x = i + lSm, y = l + 7in. (ii; x = d + lhn, y = 7 + dm. (iii) a; = 15wi-7, ?/ = 17m-10. (iv) a; = 64 + 69//i, ?/ = 44 + 49m. 4. (i) 3, 1, 2; 5, 2, 1; 2, 4, 1. (ii) 1, 21, 1; 5, 14, 1; 9, 7, 1; 3, 13, 2; 7, 6, 2; 5, o, 3; 3, 4, 4; 1, 12, 3; 1, 3, 0. (iii) 2, 8, 8. (iv) 8, 38, 50; 19, 44, 35 ; 30, 50, 20; 41, 56, 5. 5. (i) 1325, 2; 441, 3; 101, 8; 77, 10; 33, 21; 25, 27; 5, 112; 1, 333. (ii) 5,8. (iii) 8,5. (iv) 6, 1; 13, 14. 7. 195, 121; 52, 264. 8. 3. 9. 20. lO, 3. 11. £3. 14s. 6tL, £4. 5s. (jd. 12. 2s. 7^., 2s. lOd., 2s. lid., 3s. Id., 3s. 2d., 3s. Sd., 3s. id., 3s. 5d., 3s. Qd., 3s. 8d., 3s. 9d. and 4s. 13. 11,12,15,24,36. 14. 15, 55; 25, 65; 35, 75. 15. 21. EXAMPLES XLL 36 31 11664 11124 10600 *• 67' 67' ^' 33397' 33397' 33397* ^' 117 8. 3n2 + 5n + 2 pence. 15. 2* 1®- o* 2^- q EXAMPLES XLin. 1. (i) q. (ii) 27^. (iii) - 2p. (iv) - 3q. (v) 2p\ (vi) 3q, (vii) - 2p''. (viii) 3p^. (ix) -p\ (x) 2)/?. (xi) pl2q. (xii) _2,2/(872+;,3). 2. (i) 0. (ii) -3p. (iii) -4?. 3. (i) 3p3-16/>7 + 64r. (ii) (7' - 4p7r + 8r2)/r3. (iii) {q^-p^r)lr^. 4. (i) 28, -24. (ii) 44, -252. 5. (i) J)i2 - 2p^. (ii) 3piP2 -i'l^ - 3P:i • (^ii) (2'n-r - '^Pn-lVn)iPn"' (iv) lh-^Pn-ii^P-2-Pi')IPn- (V) (i>i' + 3;93-3;),p.,);5„_i/iJn + 2;5,-jPi2. (vi) (3iJi2>2 -2?i3 - SiJ^) (2?„-i- - 2i?„_2P„)/p«=+2'i. 04-4 ANSWERS TO THE EXAMPLES. 6. x*-10x- + 'Mx-Sl = 0. 7. 6. 8. (i) x^- qx'+prx-r"=0. (ii) x^ + 2px'+{p^ + q)y-r+pq = 0. (iii) x'^ (r -pq) + x- (3r - 2pq +p^) + x{dr -pq) + r=0. (iv) a-' - 25X- + (q- +pr) x + r^- pqr = 0. 2r (v) Eliminate a; between given equation and y = (p + x)'^-\ . (vi) 7/3 _ (3^ -2)2) ?/+ (332 - qp^^) y + rp^ - gS^O. 9. (i) Substitute- (y +2?) for a;. (ii) Substitute - 5 (2/ +i>) for x. (iii) Substitute - {p^ -^Q-y) for x"-'. 10. {1/ + r)3 + q;^y +p^y^ - dpqy {1/ + r) = 0. EXAMPLES XLIV. 1. 2±^/3, -3±^2. 2. 5,3±Ay^- 3- ^, ±s/2±V5. 4. ±^2±v'-l> ^(1^-7). 6. a:^-16.c- + 4 = 0. 6. 0^ + 2x^ + 25 = 0. 8. 1±V^, 1±2 7-T. 9. 223_9p5r + 27r2 = 0. 10. 7>-'-4;)9 + 8r=0 and (252 + 42)(36g-llp2)_iG00s.=0. 11. 5, 1, -3. 12. 4, 1, -2, -5. 13. (i) 3. (ii) +4, -4. (iii) ^. 14. 8, ±|. 15. i^2, -2 ±^7. 16. r2-p(/r+iy-5 = 0. 17. 3, 7, - 10. 18. 3, 9, -i. o 20. (i) r2-p-'^— 0. (ii) p~s + r^=^s. EXAMPLES XLV. 1. (i)-^,-^^,4. (ii)3,3, i2V3T. (iii)l, 1, |, -|.^ (iv) 4, 4, 4, --. 2. {hc-adf = 4.{b'^-ac){c^-hd). 9. (i) y'^-(jqy- + ^{ipr-s)y-%{2ph-dq!i + 2r^). (ii) (j/ - 6<7)"« + 6g (// - 6g)2 + 4 {^pr - s) (?/ - Gg) + 8 (22A- - Sgs + 2r'^) = 0. 10. {y- \(\p'^ + 245)3 - 245 [y - IG^y- + 2AqY + G4 (4i;r - s) (y - Up- + 24 j) - 512 {2p^>i - dqs + 2r-) = 0. ANSWERS TO THE EXAMPLES. 045 EXAMPLES XL VI. 1. (i) -5, -b}-Acci^, - ur-4^u), i.e. -5, x^:^\'-'^- (ii) _ 4, _ 0, _ 30,2^ _ o;2 _ 3a,. (iii) 10, 2cj + 8w-, 2a,2 + Sw. (iv) 8, w + 7w-, up' + Tw. (v) -2-094..., -l-703...w--391...a;2, - 1-703... w^ - -391... w. (vi) 30913, 2-1699w + -9214a)^ 2-1699w-2 + -92140;. 2. (i) -1, -3, l±2i. (ii) li^2, -1± 2V^1^ (iii) 3iV^, -3±V3T. (iv) -2, |, i(-l±V-i5). 3. (i) One real root between -3 and -2. (ii) One between - 7 and - 6, one between 1 and 2, and one between 5 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) 4-1891. (iii) '4679, 1-0527, 3-8793. (iv) 2-2317. (v) 2-1622, 2-4142. (vi) 1-1487. 9. (i) 3, 3, -4, -4. (ii) 3, 3, -3i^8. 11. 1, 3, -5, ^J~^. MISCELLANEOUS EXAMPLES. 1. (i) -(6-c)(c-a)(a-6) (2a)2. (ii) {(xb-cd){ac-dh){ad-hc). (iii) 2a(2Sa2-2Z>c). (iv) \h^c){c^a){a^h)^(j?. 6. {(6-c)2 + (c-a)2 + (a-6)2}2. 7. 3 (6-c) (c-a) (a-&). 21. (6 + c-a)(c + a- J) (a + Z>-c) = 0, {\j^z-x){z-vx-]j)(x-\-y-z) — Q. 28. (i) -2, l(-3±Jo). (ii) a, 6, ^{a + Z;±(a-6)V-3}. (iii) 0, ±^njc^Jca:i^^ab (all signs being positive or only one positive). (iv) / = '-^ =-?_ = -3^. (v) 0,0, 0; 1,1,1. b-c c-a a-o ^abc (vi) 2abc x = ± {c^a^ + a^b"^ - b^-c-), etc. (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 (vni 1,1,1; -, , , , ,. ^ a- b-c b-c -a c-a-b (jm; answers to the examples. (ix) 1, -1, 1, 2; 1, -1, 2, 1. (x) a{b-c),bic-a),c{a-b}; X y _ z _ [b-c c-a a -b\ // 1 1 , A cr-b~ b-c~ c-(i~ \a-b b-c c-aji\a b cj' r. ^ ^ (i-lb-a) c-a (XI) 0,0,0; 0, -^', «^:7-^; &0. (xii) 0, 0, 0; a'^-bc, b'--ca, c^-ab. 31. p^ + q- + r--qr-r2J-2Jq = 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, ^3; ~g' o' ~5' o ' 39. l+|2 + 2j3+|4 + 2 j5_+|6. 42. (2'*-2)//i! 46. 7i{ii + l)l 51. ^/.i+ V|. 52. l + 5e. 63. 3»'-i-3/i + 4. 54. { {Jo + l^'+i + ( - 1)" (v/5 - l)»+i}/2»+i . J5. 55. 3log.2--. 63. (i) D. (ii) C. (iii) If a>l, D. Ifa:|>l, C. (iv) D. (v) C. 64. m