IN MEMORIAM FLORIAN CAJORI Digitized by tine Internet Archive in 2008 witii funding from IVIicrosoft Corporation littp://www.arcliive.org/details/collegealgebraOOtaylricH COLLEGE ALGEBRA BY J. M. TAYLOR. A.M., LL.D., n PROFESSOR OF MATHEMATICS IN COLGATE UNIVERSITY SIXTH EDITION ALLYN AND BACON Boston antj Cf)icago Copyright, 1889, By Allyn and Bacon. Berwick & Smith, Norwood, Mass., U.i PREFACE THIS work originated in the author's desire for a course in Algebra suited to the needs of his own pupils. The increasing claims of new sciences to a place in the college curriculum render necessary a careful selection of matter and the most direct methods in the old. The author's aim has been to present each subject as concisely as a clear and rigorous treatment would allow. The First Part embraces an outline of those fun- damental principles of the science that are usually required for admission to a college or scientific school. The subjects of Equivalent Equations and Equivalent Systems of Equations are presented more fully than others. Until these subjects are more scientifically understood by the average student, it will be found profitable to review at least this por- tion of the First Part. In the Second Part a full discussion of the Theory of Limits followed by one of its most important ap- plications, Differentiation, leads to clear and concise IV PREFACE. proofs of the Binomial Theorem, Logarithmic Series, and Exponential Series, as particular cases of Mac- laurin's Formula. It also affords the student an easy introduction to the concepts and methods of the higher mathematics. Each chapter is as nearly as possible complete in itself, so that the order of their succession can be varied at the discretion of the teacher; and it is recommended that Summation of Series, Continued Fractions, and the sections marked by an asterisk be reserved for a second reading. In writing these pages the author has consulted especially the works of Laurent, Bertrand, Serret, Chrystal, Hall and Knight, Todhunter, and Burnside and Panton. From these sources many of the prob- lems and examples have been obtained. J. M. TAYLOR. Hamilton, N. Y., 1889. PREFACE TO THIRD EDITION. In this edition a number of changes have been made in both definitions and demonstrations. In the Second Part, derivatives, but not differentials, are employed. Two chapters have been added ; one on Determinants, the other on the Graphic Solution of Equations and of Systems of Equations. j. M. TAYLOR. JiAMILTON, N, Y., 1895. CONTENTS, FIRST PART. CHARTER I. Pagb Definitions and Notation 1-9 CHARTER n. Fundamental Operations 10-21 CHAPTER HI. Fractions 22-24 CHAPTER IV. Theory of Exponents 25-32 CHAPTER V. Factoring . 33 Highest Common Divisor 38 Lowest Common Multiple 41 VI CONTENTS. CHAPTER VI. Page Involution, Evolution 42 Surds and Imaginaries 46 CHAPTER VII. Equations S^ll Equivalent Equations ^'j Linear Equations 63 Quadratic and Higher Equations 65 CHAPTER VIII. Systems of Equations 78-91 Equivalent Systems 79 Methods of Elimination 80 Systems of Quadratic Equations 85 CHAPTER IX. Indeterminate Equations and Systems .... 92 Discussion of Problems 98 Inequalities loi CHAPTER X. Ratio, Proportion, and Variation .... .104-114 CHAPTER XI. The Progressions 115-121 CONTENTS. Vll SECOND PART. CHAPTER XII. Page Functions and Theory of Limits 122-133 Functions and Functional Notation 123 Theory of Limits .125 Vanishing Fractions 132 Incommensurable Exponents 132 CHAPTER XIII. Derivatives I34-U7 Derivatives I34 Illustration of A («:r2) 136 Rules for finding Derivatives ........ 137 Successive Derivatives MS Continuity 146 CHAPTER XIV. Development of Functions in Series .... 148-170 Development by Division I49 Principles of Undetermined Coefficients 150 Resolution of Fractions into Partial Fractions ... 154 Reversion of Series I59 Maclaurin's Formula 161 The Binomial Theorem 163 viii CONTENTS. CHAPTER XV. Page CONVERGENCY AND SUMMATION OF SERIES . . . I7I-I92 Convergency of Series 171 Recurring Series 177 Method of Differences 182 Interpolation 188 CHAPTER XVI. Logarithms 193-214 General Principles 193 Common Logarithms 197 Exponential Equations 203 The Logarithmic Series 205 The Exponential Series 211 CHAPTER XVn. Compound Interest and Annuities 215-223 CHAPTER XVHI. Permutations and Combinations 224-236 CHAPTER XIX. Probability , 237-253 Single Events 237 Compoufid Events 241 CONTENTS. IX CHAPTER XX. Page Continued Fractions 254-265 Conversion of a Fraction into a Continued Fraction . 255 Convergents 256 Periodic Continued Fractions 263 CHAPl^ER XXI. Theory of Equations 266-317 Reduction to the Form F(x) = 266 Divisibility of /^(.r) 267 Horner's Method of Synthetic Division 268 Number of Roots 272 Relations between Coeff cients and Roots 274 Imaginary Roots 276 Integral Roots 280 Limits of Roots 281 Equal Roots 284 Change of Sign of /'(r) 286 Sturm's Theorem 288 Transformation of Equations 295 Horner's Method of Solving Numerical Equations . 300 Reciprocal Equations 307 Binomial Equations 310 Cubic Equations 312 Biquadratic Equations 315 CONTENTS. CHAPTER XXII. Page Determinants 318-346 Determinants of the Second Order 318 Determinants of the Third Order 322 Determinants of the «th Order 328 Properties of Determinants 330 Minors and Co-factors 336 Expression of A in Co-factors 337 Eliminants 340 Multiplication of Determinants 343 CHAPTER XXIII. Graphic Solution of Equations and of Systems 347-363 Co-ordinates of a Point 348 Graphic Solution of Indeterminate Equations . . . 349 Graphic Solution of Systems of Equations .... 353 Properties of -^(:r) and of /^(.r) = illustrated by the Graph of /^ (;r) 357 Geometric Representat'on of Imaginary and Complex Numbers - . 360 ALGEBRA FIRST PART. CHAPTER I. DEFINITIONS AND NOTATION. 1. Quantity is anything that can be increased, di- minished, or measured; as any portion of time or space, any distance, force, or weight. 2. To measure a quantity is to find how many times it contains some other quantity of the same kind taken as a unit, or standard of comparison. Thus, to measure a distance, we find how many times it con- tains some other distance taken as a unit. To measure a por- tion of time, we find how many times it contains some other portion of time taken as a unit. 3. By counting the units in a quantity, we gain the idea of ' how many ' ; that is, of Arithmetical Number. If, in the measure of any quantity, we omit the unit of measure, we obtain an aritJunctical number. It may be a whole number or a fraction. Thus by omitting the units ft., lb., hr., in 6 ft., 3 lbs., and Yi hr., we obtain the whole numbers 6 and 3, and the fraction Yi, 2 ALGEBRA. 4. Positive and Negative Quantities. Two quanti- ties of the same kind are opposite in qiuxlity, if when united, any amount of the one annuls or destroys an equal amount of the other. Of two opposites one is said to be Positive in quality, and the other Negative. Thus, credits and debits are opposites, since equal amounts of the two destroy each other. If we call credits positive, debits will be negative. Two forces acting along the same line in opposite directions are opposites ; if we call one positive, the other is negative. 5. Algebraic Number. The sign +, read positive, and — , read i:eg t've, are used with numbers, or their symbols, to denote their quality, or the quality of the quantities which they represent. Thus, if we call credit positive, + $5 denotes $5 of credit, and — $4 denotes $4 of debt. If + 8 in. de- notes 8 in. to the right, —9 in. denotes 9 in. to the left. By omitting the particular units $ and in., in + $Sy ~ ^4» + S i"-» ~ 9 ^"•' ^v^ obtain the algebraic numbers + 5, — 4, + 8, — 9. + 5 is read ' positive 5,' — 4 is read ' negative 4.' Each of these numbers has not only an arithmetical value^ but also the quality of one of two opposites ; hence An Algebraic Number is one that has both an arithmetical value and the quality of one of two opposites. Two algebraic numbers that are equal in arith- metical value but opposite in quality destroy each other when added. DEFINITIONS AND NOTATION. 3 The element of quality in algebraic number doubles the range of number. Thus, the integers of arithmetic make up the simple scries, o, I, 2, 3, 4, 5» 6, 7, ..., ^; (i) while the integers of algebra make up the double series, -:»,.. .,-4, -3,-2, -I, ±o, +1, +2, +3, +4, • • •,+ ^- (2) An algebraic number is said to be i7tcreased by adding a positive number, and decreased by adding a negative number. If in series (2) we add + i to any number, we ob- tain the next right-hand number. Thus, if to + 3 we add -t- I, we obtain -I-4; if to —4 we add 4- i> we obtain — 3 ; and so on. That is, positive numbers increase from zero, while negative numbers decrease from zero. Hence positive numbers are algebraically the greater, the greater their arithmetical values; while negative numbers are algebraically the less, the greater their arithmetical values. All numbers are quantities, and the term quantity is often used to denote number. 6. Symbols of Number. Arithmetical numbers are usually denoted by figures. Algebraic numbers are denoted by letters, or by figures with the signs 4: and — prefixed to denote their quality. A letter 4 ALGEBRA. usually represents both the arithmetical value and the quality of an algebraic number. Thus a may denote +5, — 5, — 8, + 17, or any other algebraic number. When no sign is written before a symbol of number, the sign + is understood. Known Numbers, or those whose values are known, or supposed to be known, are denoted by figures, or the first letters of the alphabet, as ^, b, c, a', b\ c\ a I, b^, c^, Uftknoivn Numbers, or those whose values are to be found, are usually denoted by the last letters of the alphabet, as, x, j, z, x', j/', z\ x^, y^, z^. Quantities represented by letters are called literal; those represented by figures are called numerical, 7. Signs of Operation. The signs, + (read plus), — (read minus'), X (read multiplied by), -^ (read divided by), are used in algebra to denote algebraic addition, subtraction, multiplication, and division, respectively. The use of the signs + and — to indi- cate operations must be carefully distinguished from their use to denote quality. In the literal notation, multiplication is usually denoted by writing the mul- tiplier after the multiplicand. Thus, a b — a X b. Sometimes a period is used; thus, 4 • 5 == 4 X 5- Algebraic division is often denoted by a vinculum ; thus -^ — a -^ b. b 8. Signs of Relation and Abbreviation. The sign of equality is —. The sign of identity is =. The sign DEFINITIONS AND NOTATION. 5 of inequality is > or < , the opening being toward the greater quantity. The signs of aggregation are the parentheses ( ), the brackets [ ], the brace { }, the vinculum ', and the bar |. They are used to indicate that two or more parts of an expression are to be taken as a whole. Thus, to indicate the product of c — f/ multi- plied by Xy we may write (c — d) x, \c— d\ x, {c — d\ x, c — dx, or —d\ . The sign .*. is read heJtce, or therefore; the sign •.* is read sijice^ or because. The sign of continuation is three or more dots ... , or dashes — , either of which is read and so on. 9. The result obtained by multiplying together two or more numbers is called a Product. Each of the numbers which multiplied together form a product, is called a Factor of the product. 10. A Power of a number is the product obtained by taking that number a certain number of times as a factor. If ;/ is a positive integer, a" denotes aaaa .,.\.o n factors, or the «th power of a. In a!\ n denotes the number of equal factors in the power, or the Degree of the power, and is called an Exponent. 11. A Root of a quantity is one of the equal factors into which it may be resolved. The wth root of a is denoted by '"{/a. In ^a, m denotes the number of equal factors into which 6 ALGEBRA. ^ is to be resolved, and is called the Index of the root. The sign ^/~ (a modification of r, the first letter of the word radix) denotes a root. If no in- dex is written, 2 is understood. 12. Any combination of algebraic symbols which represents a number is called an Algebraic Expression. 13. When an algebraic expression consists of two or more parts connected by the signs + or — , each part is called a Term. Thus, the expression ^^ + {c — X) y -\- hz^ \ c ^ d consists of four terms. A Monomial is an algebraic expression of one term; a Polynomial is one of two or more terms. A poly- nomial of two terms is called a Binomial; one of three terms a Trinomial. 14. The Degree of a term is the number of its lite- ral factors. But we often speak of the degree of a term with regard to any one of its letters. Thus, %cP'b'^x^, which is of the ninth degree, is of the second degree in a, the third in b, and the fourth in X. The degree of a polynomial is that of the term of the highest degree. An expression is homogeneous when all its terms are of the same degree. A Liu'^ar expression rs one of the first degree; a Quadratic expression is one of the second degree. DEFINITIONS AND NOTATION. 7 15. Any algebraic expression that depends upon any number, as x, for its value is said to be a Function of X. Thus, 5 x"^ is a function of x ; ^ x^ -{- a^ — y x is a function of both x and a; but if we wish to con- sider it especially with reference to x, we may call it a function of ;r simply. A Rational Integral Function of X is one that can be put in the form Ax" + Bx''-'^ + Cx"-^ + ... + F, in which n is a whole number, and A,B, ..., /^denote any expressions not containing x. Thus, ax^— 4x^ — dx + c and x^ — \x are rational integral functions of x of the third degree. 16. The Reciprocal of a number is one divided by that number. 17. If a term be resolved into two factors, ei- ther is the Coefficient of the other. The coeffi- cient may be either ninnerical or literal. Thus, in ^abc^, 4 is the coefficient of abc^, 4a of bc^, and 4 a d of c^. When no numerical coefficient is written, i is understood; thus, a = (^+ 1^ a, and -a = (-l}a. 18. Like or Similar Terms are such as differ only in their coefficients. Thus, 4a bc^ and loabc^ are like terms ; 6cP'l^y^ and 4ay^y^ are like, if we regard*6rt2 and 4^ as their coefficients, but unlike if 6 and 4 be taken as their coefficients. 8 ALGEBRA. 19. A Theorem is a proposition to be proved. 20. A Problem is something to be done. 21. To solve a problem is to do what is required. 22. An Axiom is a self-evident truth. The axioms most frequently used in Algebra are the following: 1. Numbers which are equal to the same number or to equal numbers are equal to each other. 2. If the same number or equal numbers be added to, or subtracted from, equal numbers, the results will be equal. 3. If equal numbers be multiplied by the same number or equal numbers, the products will be equal. 4. If equal numbers be divided by the same num- ber, except zero, or by equal numbers, the quotients will be equal. 5. Like powers or like roots of equal numbers are equal. 23. Identical Expressions. Equal expressions that contain only figures, or expressions that are equal for all values of their letters, are called Identical Expressions. Thus, 4 + 6 and 5X2 are identical expressions; so also are {a ^b) {a- b) and a' - b\ 24. An Equality is a statement that two expressions represent the same number. The two expressions are called the Members of the equality. DEFINITIONS AND NOTATION. 9 25. Identities and Equations. Equalities are of two kinds, identities and equations. The statement that two identical expressions are equal is called an Identity. In writing identities, the siy^n =, read ' is identical with,' is often used instead of the sign =. Thus the equality 5 + 7 = 4X3 is an identity ; so also is aP- — x^ =■ {a ■\- x) (a —x), since it holds true for all values of a and X. To indicate that these equalities are identities, they may be written $ + 7= 4^3 and a^ — x^= (a + x) {a — x). If two expressions are not identical, and one or both of them contains a letter or letters, the state- ment that they are equal is called an Equation. Thus the equalities 3,1- — 6 = 0, 7^ = 2^ + 5, and 2j — 4:r = 6, are equations. The first holds true for x= 2, and the second ior a= I. The equation 2j — 4jr=6 holds true for^ = 2JI- + 3; hence if :r = i , j = 5 ; if ^- = 2, j = 7 ; and so on. 26. Algebra is that branch of mathematics which treats of the equation, its nature, the methods of solv- ing it, and its use as an instrument for mathematical investigation. The history of algebra is the history of the equation. The notation of algebra, including symbols of operation, rel ition, abbreviation, and quantity, was invented to secure conciseness, clearness, and facility in the statement, transformation, and solu- tion of equations. The number of algebra, which has quality as well as arithmetical value, was conceived in the effort to interpret results obtained as solutions of equations. Hence the study of the nature and laws of algebraic number and of the methods of combining, factoring, and transforming algebraic expressions should be pursued as auxiliary to the study of the equation. This will lend interest and profit to what might otherwise be regarded as dull and useless. 10 ALGEBRA. CHAPTER II. FUNDAMENTAL OPERATIONS. 27. Addition is the operation of finding the result when two or more numbers are united into one. The result, which must ahvays be expressed in the sim- plest form, is called the Sum. 28. Subtraction is the operation of taking from one number, called the Minuend, another number, called the Subtrahend. The result, which must be expressed in the simplest form, is called the Remainder. The subtrahend and the remainder are evidently the two parts of the minuend; hence, since the whole is equal to the sum of all its parts, we have minuend = subtrahend + remainder. 29. To multiply one number by another is to treat the first, called the Multiplicand, in the same way that we would treat i to obtain the second, called the Multiplier. Thus, 3=1 + 1 + I ; .-.4x3 = 4 + 4 + 4. Again, ^=i-+3 X 2; .-. 9X ^ = 9 + 3 X 2. 30. Having given a product and one factor. Division is the operation of finding the other factor. The given product is called the Dividend ; the given fac- tor, the Divisor; and the required factor, the Quotient. From their definitions, the divisor and quotient are evidently the two factors of the dividend ; hence, FUNDAMENTAL OPERATIONS. II since any number is equal to the product of its factors, we have quotient X divisor = dividend. (i) Let D denote the dividend, and d the divisor ; then the expression D -^ d will denote the quotient, and by (i) we shall have {D-.d)xd = D, (2) 31. Law of Order of Terms. Numbers to be added may be arraiiged in any order ; that is, a^b = b^ra. (A) For let there be any two quantities of the same kind, one containing a units and the other b units. Now if we put the second quantity with the first, the measure of the resulting quantity will be ^ + <5 units ; and if we put the first quantity with the second, the measure of the resulting quantity will be ^ + ^ units. It is self- evident that these two resulting quantities will be equal; hence their measures will be equal; .' . a ^ b = b ^ a. A similar proof would apply to an expression of any number of terms. 32. Law of Grouping of Terms. Numbers to be added may be grouped in any manner ; that is, a^^b^c^a^ib^c), (B) For by the laiv of order we have a-\-b^ c=b ^c-\- a =^ {b -^ c) ^ a = a ^ {b ^- c). A similar proof would apply in any other case. 12 ALGEBRA. 33. Law of Quality in Products. Two like signs give + ; tzvo unlike signs give — . By the definition of multiplication we have + 3-0+(+i) + (+i)4-(+i). .-. (+4)x(+3)-+(+4)+(+4) + (-f4)-+i2, (i) and (-4) X (+3) = + (-4) + (-4) + (-4) =- 12. (2) Again, -3=0-(+i)-(+i)-(+i). ••• (+4)X(-3)=-(+4)-(+4)-(+4)=-i2, (3) and (-4)x(-3) — (-4)-(-4)-(-4)-4-i2. (4) From (i ) and (4) it follows that two factors like in quality give a positive product ; and from (2) and (3) it follows that two factors opposite in quality give a negative product. Thus the product ab \^ positive or negative according as a and b are like or unlike in quality. 34. Cor. I. Any product containing an odd number of 7iegative factors will be negative ; all other products will be positive. Hence, changing the quality of an even number of factors will not affect the product; but changing the quality of an odd number of factors will change the quality of the product. 35. Cor. 2. The quality of any term is changed by changing the quality of any one of its factors, or by multiplying it by — \. The quality of any expression is changed by changing the sign before each of its terms. FUNDAMENTAL OPERATIONS. 13 36. Law of Order of Factors. Factors may be ar- ranged in any order ; that is, ab = ba. (A') For from arithmetic we know that any change in the order of factors will not change the arithmetical value of their product; and from the law of quality, any change in the order of factors will not change the quality of their product. Thus, (+ 4) (- 3) (- 5) = (- 5) (+ 4) (- 3) = (- 3) (- 5) (+ 4). 37. Law of Grouping of Factors. Factors may be grouped in any manner : that is, abc = a{bc), (B') For by the law of order we have abc — bca. = {bc) a = a (b c). Since a (bcd)= bc(da), a product is multiplied by any number a by multiplying one of its factors by a. Note. The laws of order and grouping are often called the commutative and associative laws of addition and multiplication. 38. Equimultiples of two or more expressions are the products obtained by multiplying each of them by the same expression. Thus, A m and B jn are equimultiples of A and B. 39. If Am = Bm (i) and m is not zeroy then A = B. (2) For dividing each member of identity (i) by m, we obtain identity (2). 14 ALGEBRA. 40. Distributive Law. The product of two expres- sions is equal to the sum of the products obtained by multiplying each term of either expression by the other^ and conversely. That is, {a-\-b-\-c-\- ,..) x^ax^bx-\-cx-\-.,,, (C) Let m and n denote any positive integers^ and a and b any numbers whatever; then we have {a -\- b) m^i{a -{- b) -^ {a ^ b) ^ . .. io m terms ^^am + bm. (i) Again, {a + b) {ni -^ n)^ a {m -=r fi) -\- b{m-^ n). (2) For multiplying each member of (2) by n, we obtain the identity (i); hence, by § 39, (2) is an identity. Hence, {a + b)z^az + bz, (3) in which z is any positive number. Again, {a J^b){-z)=a{-z)-\-b (- z), (4) For changing the quality of the members of (4), we obtain identity (3) ; hence, by § 39, (4) is an identity. The same reasoning would apply to any polynomial as well as to ^ + ^/ hence {C) is proved for all values oi X. By this law similar terms are united into one; thus 3^;r — y bx-\-/^cx= (■^a — yd + 4c)x. 41. Law of Exponents. Let 7n and n be any positive integers, then by definition we have a"' a" = (a a a ... m factors) {a a a ...ton factors) = aaa ...m +n factors = a"'+"- FUNDAMENTAL OPERATIONS. 1$ 42. From the laws (A), (B), (C) of §§ 31, 32, 40, we have the following Rale for Addition : Write the expressions iinder each other ^ so that like terms shall be in the same coltmm ; then add the col- ufuns separately. 43. Rule for Subtraction. To subtract one algebraic expression from another, add to the minuend the sub- trahend with its quality chafigcd. For let 5 denote the subtrahend, and R the re- mainder; then {R + S) will denote the minuend, and — 5 the subtrahend with its quality changed. But Hence, parentheses preceded by the sign — may he removed if the sign before each of the included terms be changed fro7n + to — or from — to +. Thus, a c — {fn — 2 c n ^ "i^ a x) — a c — m ^r ic n — },a X. In arithmetic addition implies increase, and subtraction de- crease ; but in algebra addition may cause decrease, and sub- traction increase. To solve any problem of subtraction in algebra, we first reduce it to one of addition. 44. Since 3^ — 4^ = 3^— (4-4)<^ = 3^ + (— 4)<^> we evidently may regard the sign connecting two terms as a sign either of quality or of operation. In general formulas, it is of advantage to regard the sign + or — as a sign of operation ; but in most other cases it is better to regard the sign written before any term as the sign of its numerical coeffi- cient, the sign of addition being understood between each two consecutive terms. l6 ALGEBRA. 45. From the commutative, associative, and dis- tributive laws of multiplication we have the three following rules: 1. To multiply monomials together, multiply to- gether tJieir numerical coefficients , observing the law of sights ; after this result write the product of the literal factor Sy observing the laiv of exponents, 2. To multiply a polynomial by a monomial, mul- tiply each term of the polynomial by the mono- mial^ and add the results. In applying the law of signs, each term must be considered as having the sign which pre- cedes it. 3. To multiply one polynomial by another, mul- tiply the multiplicand by each term of the multiplier y and add the results thus obtained. Let the student state in words the following im- portant theorems: {a-^cy = a^\2ac^c^. (0 {a — cY-a^—iac^c^. • (2) (a^c){a.-c)=^a^-c\ (3) 46. Law of Quality of Quotient. Like signs in dividend and divisor give + in the quotient; unlike signs give — . Let d = divisor, ^ = quotient ; then g d= dividend. By § 33 i^ ^ ^^^ Q^ have like signs, q must be +; while if d and q d have unlike signs, q must be — . FUNDAMENTAL OPERATIONS. 1 7 Hence changing the quality of both dividend and divisor does not affect the quotient ; but changing the quahty of either the dividend or the divisor changes the quahty of the quotient. 47. AX - = ^^ (I) m m For muhiplying each member of (i) by W we obtain A = A. By (i), ±_£= (^ad)c- =al^~' §39. e e e That is, any product may be divided by any number by dividing one of its factors by that number. 48. Let D = the dividend, d = the divisor, and ^ = the quotient; then D = dq. Hence, by § § 23, 40, 47, we have m£> = d(mq)y or -=d^\ (i) mm JD={md)^, or D = ~(mq)', (2) m m ^ "^ and ml?= (m d) q, or - = [^ a. (t\ m \mj ^^^ From equations (i), (2), and (3), respectively, it follows that, (i.) Multiplying or dividing the divideJid by any quantity multiplies or divides the quotient by the same quantity. 1 8 ALGEBRA. (il.) Multiplying or dividi7ig the divisor by any quantity divides or multiplies the quotient by the same quantity. (iii.) Multiplying or dividing both dividend and divisor by the same quantity does not affect the quotient. 49. Law of Exponents. If m and ;/ are positive integers, and ;;/ > Uy a"' aaaa ...to m factors a" aaaa... to ;/ factors == aa a ... to m — n factors § 48. = a—\ (i) 50. Corollary. If in (i) ;;^ = n, the first mem- ber is evidently i, and the second is a^; hence a^= i. That is, any quantity with zero as an exponent equals unity. 51. The Distributive Law. A quotient equals the sum of the quotients of the parts of the dividend di- vided by the divisor. For ^jLfpA^(a^c~d)\ §47- b b FUNDAMENTAL OPERATIONS. I9 52. To divide one monomial by another. (i.) If all the literal factors of the divisor appear in the dividend, divide the numerical coeffi- cient of the dividend by that of the divisor, observing the law of signs ; then divide the literal parts, observing the law of expojietits (§§47.48). (ii.) If all the literal factors of the divisor do not appear in the dividend, cancel all the factors common to both the dividend and divisor (§§47.48). 53. From the distributive law we have the follow- ing two rules: (i.) To divide a polynomial by a monomial, di- vide each term of the polynomial by the monomial and add the results. (ii.) To divide one polynomial by another, ar- range both dividend and divisor according to the powers of some letter. Find the first term of the quotient by dividing the first term of the dividend by the first term of the divisor. Multiply the divisor by the term thus foimd, and subtract the product from the dividend. Treat this remainder as a neiv dividend aiid repeat the process until there is no remaiftder, or until a remainder is found ivhich will not contain the divisor. Write the remainder over the divisor as a part of the qtwtie?tt. 20 ALGEBRA. The several products and the remainder are the parts into which the process has separated the divi- dend ; and the quotient found is made up of the quo- tients of these parts divided separately by the divisor. Hence by the distributive law it is the quotient required. 54. Detached CoeflBcients. If two polynomials in- volve but one letter, or are homogeneous and involve but two letters, much labor is saved in finding their product or quotient by writing simply their coeffi- cients. The coefficient of any missing term is zero, and must be written in order with the others, (i) Multiply ^x^ + 2x^ -S by ^ir^ + 3 — 5 x. 3+2+0-8 I- 5+ 3 3+ 2+ o -8 — 15 — 10 —0 + 40 + 9 +6+ 0-24 3 - 13 - I -2 + 40-24 Hence the product is ;^ x^ — 1$ x^ — x^ — 2 x^ + 40 x — 24. (2) Divide 2 x^ - S x + x^ + 12 - y x^ by x^ + 2 - ^ x. 1 + 2— 7— 8+i2|i — 3 + 2 ■-3+ 2 ,+5+6 + 5- 9- 8 + 5- 15 + 10 + 6- 18 + 12 + 6- 18 + 12 Hence the quotient is jr^ + 5 :ir + 6. FUNDAMENTAL OPERATIONS. 21 EXERCISE I. 1. Find by multiplication the value of (x + jY, {x +/)*, (x + yy% (x + y)% {x + y)\ {X + 7)«. Verify by division or multiplication, and fix in mind, the following identities : 2. If « is any positive whole number, x — y If ^ = I, this identity becomes, 3- If // is any positive even number, x -\- y "^ "^ 4. If;/ is any positive odd number, = ;t:"-^ — ^"-2^ + a'"-V ^y-24-y-i. ^' +/'_,,_, ^_2 x -\- y 5. Show that x" + v" is not exactly divisible by ^ + j; or X — y, when /i is any even whole mmiber. 22 ALGEBRA. CHAPTER III. FRACTIONS. 55. An Algebraic Fraction is the indicated quotient of one number divided by another. The dividend is called the Numerator, and the divisor the Denominator of the fraction. The numerator and denominator of a fraction are called its terms. In fractions division is denoted by the vinculum. Thus, , denotes the quotient oi a — b divided c ■\- a by c + d. Here the vinculum between the terms serves as a sign both of aggregation and of division. 56. Law of Signs. The law of signs in fractions is the same as that in division. The sign before a fraction is the sign of its coefficient Thus, -Zl^ := (_ i)IL_^ ^ ^. §§ 34, 46. b 57. An Entire or Integral Number is one which has no fractional part. A Mixed Number is one which has both an entire and a fractional part. 58. The terms Simple Fraction, Complex Fraction^ Compound Fraction, and Common Denominator are defined in algebra as in arithmetic. FRACTIONS. 23 59. The Lowest Common Denominator of two or more literal fractipns is the expression of lowest de- gree that is exactly divisible by the denominators of each of the fractions. 60. To reduce a fraction to an equivalent entire or mixed number, perform, in whole or in party the indicated operation of division. 61. To reduce a mixed number to an equivalent fraction, vinltiply the entire part by the denominatory to the product add the numerator^ and nnder the sum write the denominator (^^^ 22, 51). 62. To reduce a fraction to its lowest terms, can- cel all the factors common to both the Jiumenitor and the denominator (§ 48). 63. To reduce fractions to a common denominator, multiply both terms of each fraction by the denomina- tors of all the other fractions. Or^ find the lowest common denominator of the givcji fractions. Then multiply both terms of each fraction by the quotient of the lowest common denominator divided by the denom- inator of that fraction. This operation will not change the value of the fraction (§ 48), and the resulting fractions will evi- dently have a common denominator. 64. To add or subtract fractions, reduce them to a commofi denominator, add or subtract their numera- 24 ALGEBRA. tors^ and place the sum or remainder over the common denominator (§§ 48, 51). 65. To multiply a fraction by an entire number, multiply the numerator or divide the denominator by the entire number (§ 48). 66. To divide a fraction by an entire number, divide the numerator or multiply the denominator hy the entire number (§ 48). "• |x^=i^X^ §§^3,65,66. a c Yd §65, Hence the product of two or more fractions equals the prodtict of their numerators divided by the product of their denomitiators. 6a \^'^^^c §§48,65. bat be Hence the quotient of one fraction divided by an- other equals the pr^oduct of the first multiplied by the second invei'ted. _ i 69. Corollary. Since - -^ ^ = -. the reciprocal I b a ^ of a fraction equals the fraction inverted. THEORY OF EXPONENTS. 2$ CHAPTER IV. THEORY OF EXPONENTS. 70. U a, b, in, 11, denote any numbers, the five laws of exponents may be expressed as follows : a"' X a" = a'" + ". (i) a" (2) {ary = a"'\ (3) {abY = a'"b'\ (4) \b) b- (s) These laws hold for any exponents, whether they be integral, fractional, positive, or negative. 71. To prove the five laivs, when the exponents m and n are positive integers. (i.) Law (i) is proved in § 41. (ii.) Law (2) is proved in § 49, when w > « or w = «. (iii.) {a'")" = a"" a'" ...to u factors § 10. __ ^,« + ,« + ... to « terms La^ ^ j ^ = a"'". 26 ALGEBRA. (iv.) iaby =^ab'ab-ab...\.om factors § lo. — {aa..,X.o tn factors) {b b ...iom factors) = ^-^-. §§ 38, 39- ^"^KV = J 7/^ -to ^factors §10. aa a-" io m factors bb b -"io ?n factors §67. 72. A Positive Fractional Exponent denotes a root of a power. The denominator indicates the root, and r the numerator the power; that is, a"" — ^oT. 73. Let r and s be any positive whole numbers, and let \'a — c^ or a — c^ \ then {'S/^^Y = ^^ and a^ = (O' = ^"^ = (f 0'. ... ^^ = c''= ('{/ay. r Hence a' denotes either v^a' or its equal (v^a)^ 74. Negative Exponents. If we assume law (2), § 70, to hold when m — o, we have or That is, a~ " denotes the reciprocal of a". 75. To prove the five laws, when the exponents m and n are positive fractions. THEORY OF EXPONENTS. 2/ (i.) Let /, q, r, s, denote any positive integers ; then by § 73 we have 09 a? = («^ «''... to / factors) (a' a^ .--to r factors), and a^ •'" = (a'^ a'^ -- - to /> factors) (a' a^ -"to r factors). /! r l^- 9 n' = /7^ •'- Cf 4.. I a* a" = a p_ (ii.) -^^^^'x _ a' a' t -L >*_♦' (iii.) \a'0 = a^ ' a^ -" to r factors ^ + -^ 4- ... to r terms = a" ^ = «^. ... {a'y={a'^y §§23,72. = [U')'t § 72. Now one of the s equal factors of one of the q equal fac- tors of any number is evidently one of the q s equal factors of that number ; that is, (iv.) (^1^)^=^^"^ = [(f b' -a'b' ... to s factors j" § § ^Z, 39. I I 28 ALGEBRA. r r = a" b\ (v.) Let | = r, or a = b c; § 73' then g^'^ and a' = {bey = b' c% or ~ = c' (I)--? If p.r rp Corollary. By (i), ^^ = GO" = a^'. 76. To prove the five laws, when the exponents arc negative. Let h and k be any positive numbers. (i.) .-«- = -^X^ §74. §§67, 71. /j + /fe « = ^-<^' + ^) = ^-^-^ 74. ^"•^ ^^ = ^-^ = ^-'-' §48. THEORY OF EXPONENTS. 29 (iii.) («-'■)-*= I ^(^y § 74. = ■:;.=«-. §§71,68. <'^-> (''")-^=(.V §74. I II -*^-*. <-) .©"'=-(1)' _ . «* _ ^* «-* <^-* _ «-* ~ ^ ^ ^ ~ ^ * ^^ ' /^-'^' ~ ^' * Note. The introduction of fractional and negative expo- nents is evidently not necessary ; but they supply us with a new notation of very great convenience. 77. If we use the term power to signify what is in- dicated by any exponent, the five laws of exponents may be stated as follows : (i.) The product of the vnth and the nth power of any number equals the (m + n)/// power of that number, (ii.) The quotient of the vath power of any number divided by its nth power equals the (m — n)/// power of that number. (iii.) The v\th power of the vs\th powei" of any number equals the m vith pozver of that number. 30 ALGEBRA. (iv.) The mth power of the product of any niiniher of factors equals the product of the vnth powers of those factors. Corollary. The rth root of the product of two or more factors equals the product of their ^th roots. (v.) The vnth power of the quotient of any two quan- tities equals the quotient of their mth powers. Corollary. The rth root of the quotient of any two numbers equals the quotient of their rth roots. 78. To affect a monomial product with a given exponent, multiply the exponent of each factor by the given exponent. This rule follows from laws (3) and (4). Thus, {^a'b-^c'^)^ = 4^ {J)"^ (/^ - ^)^ {c^^ ^_ a"" b^ ar^ _ _b^ Hence a factor 7nay be changed from one term of a fraction to the other if the sign of its expojient be changed. EXERCISE 2. I. Multiply 3 a^ b^ c"^ by 2 a^ b* c^ ; 7 ^2 ^ -«^ - 1 by 6 a^ x'" ~ "7^". THEORY OF EXPONENTS. 3 1 2. Perform the operations indicated by the exponents in each of the following expressions : \2 a"^ x~ ^y^J ; (125 Jx-^^ ■ (8 a^ b^ c' d-^) ~ ^ ; (64 «- ^ a:~ ^)5 ; 3. In each of the following expressions introduce the co- efficient within the parentheses: 8 {a^ — x"^)- ; a^ {a -\- a^xy ; x^{i- x^)^ ; x^ (a - xj ; x'' (x'' - ay) ~ ^. (8)1 = (8)3 '1 = (8^)^ = 4! .-. 8 (^2 - X^)i = 4^ («2 _ ^2)1 = (4 «2 - 4 ;r2)t. {•-if i-^i? (-3 4. Simplify — ~r^ — ; — ; - ,2\| / ,.vA! (x^±r\^ (£!±2^ (x^+y^)^ ^ \ "*" xy _ [ x' ) _ _(£:)!__ -i-^ (x^-hy^)^ 5. Remove a monomial factor from within the parentheses in each of the following expressions : 3 (d'^ — a^ b'^) "2 ; 2 {()a''b- i^ab^)^', f(2 7^^^^-54««/^*)^. 32 ALGEBRA. 7. Square the following binomials: b'' x~'"^ ~ a^ x" ; 8. Free of negative exponents a-^ b^ , Sx~^y~i . g x~^y~^ 2-'^ 3 :r X — I x^-\- y r " ^ a 23 y x^ — y^ 6 3 2 / ;<; 10. Simplify ^ I :v I + ^ I 2 jr'^ I + - I +^4- X I — X a + b a— b a^ + ^ a'^x -\- ^ ax^ -}- x^ ^.^ c -\- d c — d x^ — y^ c — d c •{■ d x^ + xy + y'^ 2. Simplify (^ + 1^ + ?) - f^+li' - ^-). , Simplify (^ + i).(5-i + ^). FACTORING. 33 CHAPTER V. FACTORING, HIGHEST COMMON DIVISOR, LOWEST COMMON MULTIPLE. 80. Factoring is the operation of finding the fac- tors of a given product. It is the converse of multipHcation. 81. When each term of an expression contains the same factor, the expression is divisible by that factor. Thus, x'^y + xy^ + ^ax - zdx = x (xy -i- y^ -\- 4a ~ 36). Also, ac(ac + d) + d(ac + d) = {ac + d)(ac + d). Binomials. 82. Whatever be the values of m and n, That is, t/ie diffcrmce between any two quantities is equal to the product of the sum and difference of their square roots. Thus, a^ - x^- (a* + ^'*) (a* - x^). 34 ALGEBRA. 83. From the examples of Exercise i, page 21, we have, (i.) x'' ~ y" is divisible by x — y \{ n is any whole number; and the ;/ terms in the quotient are all positive. (ii.) x'' — y" is divisible by x + y \( n is even; and the n terms in the quotient are alter- nately positive and negative. Thus, c^-b^^ («3 )' _ (^i)* = («3 _|. ^i) (^ _ (^ ^5 + ^¥ 4 - ^5 ). (iii.) x"" +y' is divisible by x + y \i n is odd; and the n terms in the quotient are alternately positive and negative. Thus, ^fio + j5 = (;t-2 + y) {x'^ - x^y + :r ^^ - -^V^ + JJ'*) • (iv.) ;r"+j/" is not divisible hy x ^ y or x — y when ?2 is even. 84. For any value of ;/ we have, For 71 = I, (i) becomes a:* + y = (a;2 + / + ^jj; /y/a) (a;^ +y^ — xy ^fi). factoring. 35 Trinomials. 85. x^ ± 2 ax -h a^ = (x ± ay. That is, if tivo tenns of a trinomial are positive, and the third is ± twice their square roots ^ the trinomial equals the square of the sum or difference of the two square roots. 86. x^ + {a + b)x -^ ab = {x -\- a) {x -[- b) ; hence x"^ -\- c x -\- d = {x + a) {x + ^), if a -\- b = c and ab = d, (i) Equations (i) can always be solved for a and b by the method of § i66; hence a trinomial of the form x'^ -\- ex ■{■ d can be resolved into two linear factors in X. Equations (i) however may often be solved by inspection. Example. Factor x^y^ — 1 1 x^y^ + 30. ^6^,4 _ 1 1 x^y2 + 30 = {x^y^y^ - 1 1 (:r3y) + 30; hence a + b = — 11, and « ^ = 30 ; therefore ^z=— 5, b = — 6; whence x^y*' - 1 1 x^y^ + 30 = (x^y'^ - 5) (x^y"^ - 6). 87. nx^ + ex + d= — ' ^^ ^—^ Now (^nxy -f- c(ux) + nd can be factored by § 86. Hence any trinomial of the form nx"^ + ex -\- d can be resolved into two linear factors in x. ^6 ALGEBRA. Example. Factor 15 Jt:^ — 7 jc — 2. 15 (i^x ~ 10) (15 X -\- 2) / \ /■ , \ 88. When, by increasing one of its terms, a trino- mial can be made a perfect square, it can be factored by § 82. Example. Factor x* — ^ a^ x"^ + g a^. x^- Za^^'^ + ga^ = ^^ + 6a^x^ + ga^~ga^x^ = {x^+ 3 a'^f - (3 a^ xf = (;r2 + 3 «* + 3 a'^x){x'^ + 3 «* - 3 a'^x). 89. A polynomial of four or more terms may often be factored by properly arranging its terms, and applying the foregoing principles. 1. cx"^ — cy'^ — ax'^ + ay'^ = c (x'^ —j") — ^ (-^^ —y^) = (c-a) (x-y) (x+y). 3. x*-x'^~g-2a^x^+a^+6x = (x^ - a'^f - {x - 3)2 = {x^-a^+x- 3){x^- a'^-x + 3). EXERCISE 3. Resolve into their simplest factors : 1. a^c^-\-acd-\-abc-\-bd. 2. a'-y^ - Py x"" - a" dy'' + ^' dx\ FACTORING. 37 3. 10^^ + 30 x^ y — 8 x)p- — 24^. 4. {a^bf-i. II. 2X'-<,xy ^ 2>y''' 5. d^b^ — T^abc — \0(^. 12. 12 jc^ — 23 ;c^ + 5^. 6. 98 — 7 jc — j(;-. 13. 9 x^ + 24xy + i6y\ 7. ^'' + I. 14. ^^ + i6jc2 + 256. 8. ^« - I. 15. 81 a' + 9 ^' b' + b\ 9. 7+10^ + 3:1:^ 16. 2 + 7^— ^S-^^- io. 6 ;c^ + 7 :x: — 3. 17-3 ^' + 41 ^ + 26. 18. 31^ — 35 — 6^=^. 19. a^ ^rb'^-c^- d^ + 2 ^2/^2 _ 2 ^2^2^ 20. I — «^ jc^ — ^^_y- + 2 rtt <^ :i'_y. 21. ^^ JC — ^'"^JC + «^7 — ^^j. 22. Resolve A:^_y — jc^ j^^ — x^ y- -\- x y^ into four factors. 23. Resolve c^ — 64 a^ — ^z^ + 64 into six factors. 24. Resolve a, {a b ^ c df - {a^ + ^,2 _ ^ _ ^2y -^^^^ four factors. 25. Write out the following quotients : {x^-yh - {^^-yh> (x^ + J) -^ {x^ + a^). 38 ALGEBRA. HIGHEST COMMON DIVISOR. 90. A Common Divisor of two or more expressions is an expression that divides each of them exactly. Two expressions are prime to each other if they have no common factor other than unity. 91. The Highest Common Divisor of two or more algebraic expressions is the expression of highest degree that will divide each of them 'exactly. The abbreviation H. C. D. is often used for the words highest common divisor. 92. When the given expressions can be resolved into their simple factors, or such as are prime to each other, their H. C. D. is obtained by taking the product of all their common factors, each being raised to the lowest power in which it occurs in any of the expressions. Thus, the H. C. D. of 6 (:r-i) (;r+2)3 and 3 (;r-i)2 (:r + 2)2 {x - 3) is zipc- i)(;r+ 2)2, 93. When the given expressions cannot be resolved into their factors, the method of finding their H. C. D. is based on the following theorem : If A ^ B Q + R, then the H. C. D. of A and B is the same as the H. CD. of B and R, Since R — A — B Q, by §81 every factor com- mon to A and B divides R ; hence every factor HIGHEST COMMON DIVISOR. 39 common to A and B is common to B and R. Con- versely, since A — B Q + R every factor common to B and R divides A ; hence every factor common to B and R is common to A and B. Hence the H. C. D. of i>' and R is the H. C. D. of A and /?. 94. To find the //. C D. of two algebraic quantities. Let A and ^ denote any two rational integral func- tions of X, whose H. C. D. is required, the degree of B not being greater than that oi A. Divide A by B aitd let the quotient be Q^ and the remainder R^. Divide B by /?, and let the quotient be Q2 and the remaiftdcr R^,. Divide R^ by R^ and let the quotient be (2.., ^'^^ ^^^^ remainder R^. Continue this process until the remainder is zerOy or does not contaift x. If the last remainder is zero, the last divi- sor is the H. C. D. ; if the last rejnaiitdcr is not zero^ there is no H. C. D. From the process above described, it follows that A = B Q,^ R^, B = R,Q, + R,, ^1= ^2 a + ^3. Now by § 93 the pairs of expressions, A and B, B and 7?,, R, and R^, ..., R ,,-2 and R,,_^, all have the same H. C. D. 40 ALGEBRA. (i.) If R„ = 0, R„_2 = i?,,_i g,. Hence the H. C. D. o{ R^_, and 7?„_„ or R,^_, a, is y?„_i. Hence ie«_i is the H. C. D. oi A and ^. (ii.) If R„ is not zero, the H. C. D. of ^ and B is the H. C. D. of R,,_^ and R, (§ 93)- But, since R^ does not contain x, Rn-\ and R„ have no common factor in x. Hence A and B have no common divisor. 95. Corollary. To avoid fractions, and to other- wise simpHfy the work in finding the H. C. D., it is important to note that at any stage of the process, (i.) We may multiply either the dividend or the divisor by any quantity that is not a factor of the other. (ii.) We may remove from either the dividend or the divisor any factor that is not common to both. (iii.) We may remove from both the dividend and the divisor any common factor, provided it is reserved as a factor of the H. C. D. 96. To find the H. C. D. of three expressions, A, B, C, find the H. C. D. of A and B, and then find the H. C. D. of this result and C. This last H. C. D. will be the H. C. D. of A, B, and C, LOWEST COMMON MULTIPLE. 4I LOWEST COMMON MULTIPLE. 97. A Common Multiple of two or more expressions is an expression that is exactly divisible by each of them. The Lowest Common Multiple (abbreviated L. C. M.) of two or more expressions is the expression of low- est degree that is exactly divisible by each of them. 98. Hence when two or more expressions can be resolved into their factors the L. C. M. of these ex- pressions is the product of their factors, each being raised to the highest power in which it occurs in any of the expressions. 99. To find the L. C. M. of two expressions, as A and B, when they cannot be factored, divide A by the H. CD. of A and D and multiply the quotient by B. For the L. C. M. of A and B must evidently con- tain all the factors of ^, and in addition all the factors of -^ not common to A and B; hence the rule. EXERCISE 4. Find the H. C. D. of the following expressions : 1. x^ ■{■ 2x^ — Zx — id, x^ -\- 2iX' — 2>x — 2/^. 2. 2 x^ — 2 x^ -\- X' -\- -^ X — d, 4 ^* — 2 ^^ + 3 ^ — 9. 3. 4^^+I4^^+20Ar3+70^^ %X^^-2Zx^'—'^0(^—\2X^^^(iX^, Find the L. C. M. of the following expressions : 4. x^ -^ a x^ ■\- a^ X ■\- a*, x* -{• a^ x^ + a^. 5. x^ — <) x^ -f 26 ^ — 24, x^ — 12 x^ ■\- ^'J X — 60. 42 ALGEBRA. CHAPTER VI. INVOLUTION, EVOLUTION, SURDS, IMAGINARIES. 100. Involution is the operation of finding a power of a number. Evolution is the operation of finding a root of a number. For the involution and evolution of monomials see § 78, of binomials see § 275. 101. A root is said to be even or odd according as its index is even or odd. By the law of signs it follows that, (i.) Any odd root of a quantity has the same sign as the quantity itself. (ii.) Any even root of a positive quantity may be either positive or negative. In this chapter only positive even roots are considered. (iii.) Any even root of a negative quantity is not found in the series of algebraic numbers thus far considered. An even root of a negative number is called an Imaginary number. For sake of distinction all other numbers are called ReaL EVOLUTION. 43 102. To find the square root of any ntnnber. The rule is given by the formula, {a + hf = a"" -\- {2 a + b) b, in which a represents the first term of the root or the part of the root already found ; b the next term of the root ; 2 a the trial divisor in obtaining b; and 2 a -]r b the true divisor. In finding the square root of any polynomial, as 4 a:^ 4- py + 13 x^y'^ — 6 xy^ — 4 x^y, its terms should be arranged according to the descending powers of some letter, and the work may be arranged as below : I 2x^-xy+'iy^ 4x*- 4x^y + 1 3 x^y^ - 6xy^ + 9J/* 4^ 2a + b = 4x^ — xy {2a + b)b = -4jr8j/+l3^V — 4x^y + x^y^ 2a + b= 4x^ - 2xy + 3 j/2 {2a + b)b = 1 2 x^y^ — 6 xy^ + gy* I2.r2y -6xy^-\-gy* At first a=2x^ and b = —xy; then a = 2x^ — xy and b = 3/^ The root is placed above the number for conven- ience. In extracting the square root of any number expressed in the decimal notation, we first divide it into periods of two figures each, beginning with units' place. We then proceed essentially as with the polynomial above, bearing in mind that a denotes tens with reference to b. 44 ALGEBRA. 103. To find the cube root of any number. The rule is given by the formula, {a ^by = a^^{zd'^iab\ b'')b, in which a represents the first term of the root or the part of the root already found ; b the next term of the root; ^ (f the trial divisor in obtaining b; and 3 ^^'-^ + 3 ^^^ + <^^ the true divisor. In finding the cube root of any polynomial, as 8 ;ir^ — 36 ;r^ + 66 ;i:4 + l — 63 ;r3 — 9 ;ir + 33 ;t^, its terms should be arranged according to the descend- ing powers of some letter, and the work may be arranged as below: I 2;i-2 — 3;r+ I a" — ,2 8 :r6 - 36 ;jr5 + 66 Ji'i - 63 -r^ + 33 ;i-2 - 9 ;r + I (3«'^ + 3^^ + ^2)^ = ■ ^6x^ + 66 z^ — 6s :*^ 36;r^+54:r^ — 27jr3 3^2— 12X^~S^X^+27X^ Sab + d^= 6x^—gx+ I {3a' + 3ab+b^)b = I2x^ — 26x^ + 22x-~gx+i 1 2 jr* — 36 jr^ + 33 :i'2 — 9 jf + I At first a = 2x^ and b = —^xj then a = 2x^ — ^x and In finding the cube root of any number expressed in the decimal notation, we first divide it into periods of three figures each, beginning with units' place. We then proceed essentially as with the polynomial above, a denoting tens with reference to b. EVOLUTION. 45 104. In finding the fourth root of any number, we may obtain the square root of its square root, or follow the rule given by the formula. The rule for finding the fifth root is given by the formula, (^ + ^)6 = ^« + (5^^+ voa^b ^ 10^-^2^ 5^^' + b')b. The sixth root may be obtained by finding the cube root of its square root, or by using the formula, In like manner we may obtain any root of a quantity. EXERCISE 5. Find the square root of 1. 25 ^* — 30^;^'* + 49 ^^^'^ — 24^^^ + 16 a*. 2. () x'^ — \2 X* -\- 22 X^ -{■ X^ -\- \2 X -\- /^, 3. 384524.01. 4. 0.24373969. Find the cube root of 5. I — 6 X + 2 1 ^'-^ — 44 ^^ + 63 Jt:* — 54 ^^ + 2*1 x^. 6. 2d,x^y^-\-()(iX^ y^ — (iX^y-{-x^—()(ixy^-\-(i/i^y^—t^(iX^y^. 7. 3241792. 8. 191. 102976. 9. Find the fifth root of 32 a:^ — 80.%* 4- 8o.;c^ — 40Jt;^ + lo.:*: — i. 10. Find the fourth root of 16^* — 96 ^^jt: + 216 a^x'^ — 216 «^^ + 81 .;c*. 46 ALGEBRA. SURDS. 105. If the root of a quantity cannot be exactly obtained, its indicated root is called a Surd or Irra- tional Quantity. All quantities which are not surds are called rational quantities. The order of a surd is indicated by the index of the root. Thus, V« and "ija are respectively surds of the second and ?/th orders. The surds of most frequent occurrence are those of the second order; they are often called quadratic surds. 106. Surds of different orders may be transformed into others of the same order. The order may be any common multiple of the orders of the given surds ; but usually it is most convenient to choose the L. C. M. Thus, ^a -(h = (^- /yV, and ^J^^fi =b^ ^ ^¥. 107. A surd is in its simplest form when the smallest possible entire quantity is under the radi- cal sign. Surds are said to be Like when they have, or can be so reduced as to have, the same irrational factor ; otherwise they are said to be Unlike. Thus, 2/y/5 and ^\/s ^^^ I'l^^ surds, so also are /\/i8 and \/^. 108. In adding or subtracting surds reduce them to their simplest form by the principles of § 70, and combine those that are like. SURDS. 47 109. The product or quotient of surds of the same order may be obtained by the laws of exponents (§ 70). If they are of different orders they may be reduced to the same order. Thus, x^a X ^\/^=^^aKi = xda^'^J^ = xb ^^*7«. 110. When two binomial quadratic surds differ only in the sign of a surd term, they are said to be Conjugate. Thus, ^Ja + ^b is conjugate to ^/a — ^, or — ^/a + ^/b. The product of two conjugate surds is evidently rational. 111. The quotient of one surd by another may be found by expressing the quotient as a fraction, and then multiplying both terms of the fraction by such a factor as will render the denominator rational. This process is called rationalizing the denominator. The cases that most frequently occur are the three following: I. When the denominator is a monomial surd, as V^j, the rationalizing factor is evidently y~^ . II. When the denominator is a binomial quad- ratic surd, as V^ + V^, the rationalizing factor is its conjugate, ^/a — Vi or — V^ + V^. 48 ALGEBRA. 111. When the denominator is of the form "s/a + "s/b + V^, first multiply both terms of the fraction by ^/a + ^Jb — "s/c; the denominator thus becomes (V^+ V^)2-(V?)2, or (^+^-^ + 2 V^. Then multiply both terms of the fraction by {a ^ b — c) — 2 "s/ab ; and the denominator becomes the rational quantity (a -\- b — cy^ — /i^ab. 112. To find a factor that will rationalize aiiy given binomial surd, as v a ± 'V^b. Let n be the L. C. M. of r and s ; then {^\/dy and ('v/^)" are both rational and so also is their sum or difference. There are three cases I I I. When the given surd is a'' — b' ; then by § 83 II l/rl n —X «— 21 «— 32 «— 1 a' — ^ = the rationalizing factor. I z II. When the given surd is a'' + b% and n is even/ then by § 83 ( ^) ' I/O" "— ^ >'-2 i. "—3 1 "— ^ — X ~ V ^ ^ ' -^^ ^^+^'' ^ ^' dr + <^'' = the rationalizing factor. SURDS. 49 j_ 1 III. When the given surd is a'' + lf% and u is odi^ ; then by § 83, a"- + d' = the rationalizing factor. In each case the rational product is the numerator of the fraction in the first member of the identity. Example. Find the factor that will rationalize Vs + V5. V3 + V5 ____ ___ = 9V3-9\^5+3V3v^25-i5+5V3'\?^5-5v^2S. EXERCISE 6. Find the value of 1. (\/~2 + Vi - Vs) (a/^ + V3 + Vs)- 2. (4 + 3 A/2) -^ (5 - 3 V2)- 3- 17 -^ (3 V7 + 2 a/3)- 4. (2 a/3 + 7 V2) ^ (5 V3 — 4 A/2). a/3 + \/2 . 7 + 4 V3 2 — V3 a/3 — V2 6 gV^+_^ . 8V3-6a/5 5 + V^ ' 5V3-3V5 50 ALGEBRA. Rationalize the denominator of 7. 3_+ a/6 _ 5 V3 — 2 V12 — V32 + V50 Vi + ^^— a/i — ^'' Vi + ^"^ + a/i — ^■'^ a/2 V^3 12. V2 + V3 - a/s Vz + A^9 a/io + Vs — \/3 a/2 ^3 V3 + Vio — V5 A/3 + a/2 • V^3 - I a/8 + \/4 II. ^-^^^ 14. — z ^• V 3 + I V8 — V4 113. 77^^ square root of a rational quantity cannot be partly rational and partly a quadratic surd. If possible let ^fa — ;/ + \/m, then a ^^ 71^ -\- m -\- 211 ^m ; J— a — n^ — m ,'. Wm — , 2 71 which is impossible, since a surd cannot equal a rational quantity. 114. /;/ any equality containing rational quantities and quadratic surds, the rational parts in the two members are equal, and also the irrational paints. SURDS. 51 Suppose a + ^/~b — x -\- \fy. If possible let a = x ^- m; then ^ + w+ ^Jl^ X ■\- Vj/ or \/y — m -\- V^, which is impossible by § 113. Hence a = x^ and therefore V^ = Vj* 115. To find the square root of ^ ± 2 Vh. Since Vx + y ± 2 ^Jxy = ^/x ± \fy; therefore ' y a ± 2 \fb — V-^ ± V>'» (i) if :jc + ^ = «, and :»: j' — b, (2) Solving equations (2) as simultaneous (§ 166), and substituting the results in (i), we obtain Equations (2) may often be solved by inspection. Example. Find the square root of 13 + 2 Vso* Here x -\- y — 13 and xy — yy\ X — \o and y ~ Z- H^"^^ V13 + 2 S'Jo - V"^ + V3- 52 ALGEBRA. EXERCISE 7. Find by inspection the square root of 1. 7 — 2 Vio. 4- i8 — SVs- 7- 19 + 8^/3- 2. 5 + 2 V6. 5. 47-4 '^2>Z' 8. II + 4 V6. 3. 8 — 2^7' 6. 15— 4Vi4- 9- 29 + 6^/22. IMAGINARY QUANTITIES. 116. Imaginary quantities frequently occur in mathematical investigations, and their use leads to valuable results. By the methods of Trigonometry, any imaginary expression may readily be reduced to the form of a quadratic imaginary expression. We give below some of the laws of combination of quadratic imaginaries. 117, By the definition of a square root we have V-^i X ^/— I = — I. . • . ^/a V— I X '\/a V— i = — ^ / that is, (V^ V^y = —a^ {^^^af. By this principle any quadratic imaginary term may evidently be reduced to the form c V— i- Thus, /y/— II «2 — ^i I ^2 ^_ J _ ^ y'l I ^. IMAGINARY QUANTITIES. 53 118. To add or subtract quadratic imaginaries, reduce each imaginary term to the form c V— I, and then proceed as in the case of other surds. Thus, V-4 + V--9 = 2 V^. + 3 V-^ = 5 a/-^- 119. To find the successive powers of V— i . ... ^^—^Y =.(_i)(vcr7) =-V=T; ... (V^^ =(-i)(V=^)^ = +i; .-. (V=T)*''=(+ i)" =+i, in which n is any positive integer. Hence, in general, 120. An expression containing both real and im- aginary terms is called an Imaginary or Complex ex- pression. The general typical form of a quadratic imaginary expression is a + d V— i. U a = 0, this becomes d V— i. 121. Two imaginary expressions are said to be Conjugate when they differ only in the sign of the imaginary part. Thus, a — l> \/— I is conjugate to a + d ^J^'x, 54 ALGEBRA. 122. The Sinn and product of two conjugate imagi- nary expressiofts are both real. For {a -\- b ^— \) -\- {a — b V^^) = 2 ^ and {a-^ b V^) {a - b V^ ■= d" - {- b"-) = 0"+ b'' The positive square root of the product a'^ + b"^ is called the Modulus of each of the conjugate expres- sions, a ^- b V— I and a — b V— i. 123. If two imaginary expressions are equal, the real parts must be equal and also the imagijiary parts. For let a -\- b V—^ =z c ^ d V^^ ; then a — c = (d — b) V— i. Hence (^ - d' = - {^ - b)\ which is evidently impossible, except a ^= c and b = d. 124. Corollary. U a -\- b V— i == 0, a = 0, and ^ = 0. 125. To multiply or divide one imaginary expres- sion by another, reduce them each to the typical form, then proceed as in the multiplication or divi- sion of any other surds, obtaining the product or the quotient of the imaginary factors by § 1 19. Thus, /y/— a X \/— b = A^a ^- i x ^/b ^— i = a/^ ^~b ( v^)' = - V^- IMAGINARY QUANTITIES. 55 Remark. The student should carefully note that the product of the square roots of two negative num- bers is not equal to the square root of their product Thus, -y/— 2 X y'— 8 does not equal '\/i6. 126. When the divisor is imaginary the quotient may be found by expressing it as a fraction and then rationalizing the denominator. Thus, I ^ S + g/y/JV-i ^i 2_V3 /— - 3-2V"3~ 9+12 7+ 21 V • EXERCISE 8. Perform the following indicated operations : I. 4 V— 3X2 V— 2. 2. (V2 + V— 2) (v^ — V— 2). 3. (2 \/=^)^ 4. (2 a/- 3 + 3 V^) (4 V^ - S V^)' 5. V— 16-^ V'-4- 6. (3 V^ - 5 V^ (3 V^ + 5 V^). 7. (i + V^) ^ (I - V=^). 8. (4 + V^^) H- (2 - v'^. 3 V^ri — 2 V— 5 rt— V— ^ 9- 7= 7= • 10. i^ j^' 3 V— 2 + 2 V— 5 ^ + V — ^ 1 1 . What is the modulus of 3 + 2 a/— 3 ? Of 5 — 3 V^ ? 56 ALGEBRA. CHAPTER VII. EQUATIONS. 127. An Equation is an equality that is true only for certain values, or sets of values, of its unknown quantities. Any such value, or set of values, is called a Solution of the equation. Equations are classified according to the number of their unknown quantities; thus, we have equations of one unknown quantity ; of two unknown quantities ; of three un- known quantities; and so on. For example, the equality 5 or = 15 is an equation of one un- known quantity x ; its single solution is x — 2>' Again, {x — 5) (;r — 4) = is an equation ; its two solutions are evidently X — ^ and X — \. The equality j/ = 2 :r + 3 is an equation of two unknown quantities :ir and y ; one of its solutions is ;ir = i, y — S\ another is ;ir = 2, / = 7 ; another is ;r = 3, j/ = 9 ; and so on for an unhmited number of solutions. 128. An equation is said to be Numerical or Literal. according as its known quantities are represented by figures only, or wholly or in part by letters. 129. When an equation contains only rational in- tegral functions of its unknown quantities, its Degree is that of the term of highest degree in the unknown EQUATIONS. 57 quantities. Thus, the equations x^ + x^ -{- 4 = and xy^ + x}' = 5 are each of the third degree. A Linear equation is one of the first degree. A Quadratic equation is one of the second degree. A Cubic equation is one of the third degree. A Biqiiadratic equation is one of the fourth degree. Equations above the second degree are called Higher Equations. Equivalent Equations. 130. Two equations involving the same unknown quantities are said to be Equivalent when they have the same solutions; that is, when the solutions of either include all the solutions of the other. Thus, ^x-\oa = ^x — 4a and 2x = 6a are equivalent equations ; for the only solution of either is jr = 3 rt. A single equation may be equivalent to two or more other equations. Thus, (3jr-6rt) (;r2-9^2) =0 (l) is equivalent to the two equations 3 ;r — 6 « = 0, (2) and ;i-2 - 9 (2) or x'^-6x+S = 0. Hence (x - 4) (x - 2) = 0. (3) Now the only solution that could be introduced by multiplying (i) by ;r — I is ;ir = I. But the solutions of (3) are x= 4 and x= 2 '^ hence the solution x= i was not introduced. * The reason for this exception to § 135 is that in this case the solutions of ;// = do not make both A and B finite. Thus, the solution of .r — i = 0, or ;r= i, does not render the first member of (i) finite. EQUATIONS. 6 1 To avoid introducing new solutions in clearing an equation of fractions : (i.) Those which have a common denominator should be combined. (ii.) Any factor common to the numerator and denominator of any fraction should be cancelled. (iii.) When multiplying by a multiple of the de- nominators always use the L. C. M. Example. Solve i = 6. (0 X — I I — X Transposing and combining, we have I — = - 6. X — I .*. I — (;r + l) = - 6, or ;r = 6. But if we first clear (i) of fractions, we obtain X— I— x'^ = — I— 6x+6, or (x-6)(x-i) = 0, of which the roots are 6 and i. But as ;r= i does not satisfy (i), the root i was introduced in clearing of fractions. 137. Corollary 2. In solving an equation of the form m A = m B, or m {A — B) = 0, we should write its two equivalent equations, 7Tt = 0, and ^ — ^ = 0, and solve each. 62 ALGEBRA. Thus, the equation ;i'3 — i = maybe written in the form (x- i)(x^ + x+ i) = 0, (I) which is equivalent to the two equations ;ir - I = and x^ i- x + i = 0, the solutions of which are readily found. 138. If both members of an equation be raised to the same integral pozuer, in general, new solutions will be introduced. Let the equation be A = B. (i) Squaring both members of (i) we obtain ^2 ^ B\ or A^-B^ = 0, which can be written in the form {A -B){A-\-B) = 0. (2) Now (2) is equivalent to the two equations ^ - ^ = and ^ + ^ = 0. Therefore the solutions of ^ + ^ = were intro- duced by squaring both members of (i). Hence if in solving an equation, we raise both its members to any power, we must reject those solu- tions of the resulting equation which do not satisfy the given equation. Example i. Let the equatioi ibe X — 4- V4- - X — Squaring (i), 4- - x = x^- -8;r+ i6, or ;r2 - 7 r + 12 = 0. - Hence ( x-A){x~ 3) = 0. 63 (I) (2) Now the solutions of (2) are evidently x = 4 and ;r = 3, of which x= 3 does not satisfy (i). Hence the solution x = 3 was introduced by squaring (i). Example 2. Let the equation be jy-2 = x. (I) Squaring (i), ( j - 2)2 = x'', or (/ - 2)2 - x^ = 0. Hence (jy - 2 - x) (y - 2 + x) = 0. (2) Now (2) is equivalent to the two equations J — 2 — r = and ^ — 2 -}- jr = 0. Hence the solutions of j^ — 2 + x = were introduced by squaring (i). Linear Equations of one Unknown Quantity. 139. Any solution of an equation of one unknown quantity is called a Root of the equation. 140. By the preceding principles any Linear equa- tion of one unknown quantity can be reduced to an equivalent equation of the form ax =: c. ( I ) Dividing both members of (i) by ^, we obtain X = c-^ a. Hence a linear equation has one, and only one root. 64 ALGEBRA. EXERCISE 9. Solve the following equations ; that is, find their roots I. h + ^ = 0. 7 3 21 4 (^ + 2) ejx-y) 2. = 12. 3 7 7 - 5:y ^ II - i5 >r I + ^ I + 3:^ * 3^—1 4J\? — 2_i 2:r — I ^x — I 6 4 (-^+3) 8jr+37 7-^—29 5- 18 5^—12 30 4- 6 ^ 60 4- 8 a: 48 6. , 1- ; =14-1 ~ . ^4-1 X + 3 X -\- I Reducing the first two fractions to a mixed form, we have ^ 24 „ 36 48 X + I ^+3 X + I 36 24 32 — or — ■ — = — ; — , etc. ;r + 3 X + 1' ^ X + -^ x+ i ^+5 X — 6 :r — 4 X — 15 X -\- ^ X — y X — 5 X — 16 .5 ^ — .4 2 X — .1 9. V-^ — 32 = 16 — -y/x. lO. EQUATIONS. 65 5^ — 9 Vs^- 3 Vs^ + 3 11. ^/x — Vx — Vi — .T = I. In example 10, cancel the factor common to the terms of the first fraction. Multiplying both members by y^S^+S would introduce the root of the equation ^5^+ ^ = 0. In exam- ple II, neither of the two roots obtained will satisfy the given equation ; which therefore has no root, and is impossible. ax — I \/a X — I 12. — __ =: 4 + Vax -{■ I 2 13- sVx — 4 ^ 1 5 + A/ 9-y 2 + ^/x 40 + ^/x M- . . F + V« — X •{■ \^a \^a — X — ^a X 15. V-^— Vat — 8 V.^-8 Quadratic Equations of one Unknown Quantity. 141. By the preceding principles any quadratic equation in x can be reduced to an equivalent one of the form ax^ ^- bx \- c=^. (1) In this equation a cannot be zero, for then the equation would cease to be a quadratic. 66 ALGEBRA. U d or c is zero, equation (i) assumes the form a x^ -\- c = 0, or a x^ + d X = 0. (2) Either of equations (2) is said to be Incomplete. The first is called a Pure quadratic. If <^ = ^ = 0, (i) becomes ax'^ = ; .*. ;i; = ± 0. 142. To solve the pure quadratic a x^ + c = 0. Solving the equation for jr^, we obtain x^ ^ — {c -h- a), c ± V- T a The two values of x will be real or imaginary, ac- cording as c and a have unlike or like signs. Hence a pure quadratic has two roots, arithmetically equal zvith opposite signs ; both are real, or both are imaginary. 143. To solve the incomplete quadratic ax^ + b x = 0. This equation may be put in the form x{ax -Y b)=0. . (i) Now (i) is equivalent to the two equations ^r = and a X + b =^0j whose roots are and — (b -^ a), respectively. 144. To solve the complete quadratic ax^ + bx + c = 0. (i) EQUATIONS. 6^ We first transform equation (i) so that its first member shall be a perfect square. To do this we transpose r, then multiply both members by 4 a, and finally add f^ to both members. We thus obtain /^a^ x^ -\- ^ab X -^ b"^ ~ b'^ — a^ac, ■ or (2 ax + by- = b^ — ^ac. 2ax ■\- b = ± \b'^ — A,ac, 2.a Hence, to solve a complete quadratic, traitsform the equation so that its first member shall be a perfect square^ and then proceed as above ; or put the equation in the form of {i), and then apply formula (2). 145. Sum and Product of Roots. Representing the roots of ax'^ + bx + c = by a and /3, we have _ — b + \lb^ — 4ac , K — » \v 2a Adding (i) and (2), we find the sum -+^ = -y (3) Multiplying (i) by (2), we find the product ap = '-. (4) 68 ALGEBRA. Dividing both members o^ ax^ -{- dx -{- c = by {7, we obtain ^^4.*a-4-£ = 0. (5) a a From (3) and (4) it follows that if a quadratic be put in the form of (5), (i.) The Sinn of its roots is equal to the coefficient of X with its sign cJianged. (ii.) The product of its roots is equal to tJie known term. For example, the sum of the roots of the equation 3:^^ + T x + 12 = is —J, and their product is 4. 146. Corollary i. If the roots a and ^ are arith- metically equal and opposite in sign, the equation is a pure quadratic. For \i -b^ a^a-\- ^ = ^y b^^. 147. Corollary 2. If the roots are reciprocals of each other, a = c ; and conversely, if a — c^ the roots are reciprocals. » For \{ c ^ a- a/3= I, a = c ; and conversely, if a = c, a l3 = 1. 148. Character of Roots. From the values of a and in (i) and (2) of § 145, it evidently follows that, (i.) If b'^ — 4ac ^ 0, the roots are real and un- equal. EQUATIONS. 69 (ii.) If b"^ — A^ac = 0, the roots are real and equal, (iii.) If ^2 — 4^^ < 0, the roots are conjugate im- aginaries. (iv.) 1( d^ — 4ac \s a. perfect square, the roots are rational; otherwise they are conjugate surds. Thus, the roots of 3 jr^ — 24 :tr + 36 = are real and unequal ; for here d^ -4ac= (;- 24)2 - 4 X 3 X 36 > 0. Again, the roots of 3 Jir^ — 12 ^ + 135 = are conjugate im- aginaries ; for here 6^-4ac= (-12)2-4 X 3 X 135 < 0. 149. Number of Roots. Since - = — (a + fi) and c ^ - = a/3, we may write the general equation a a in the form a:^ — (a + jS) at + a y8 = 0, (i) or {x-a){x-fi) = ^. (2) Now a and /? are evidently the only values of x that will satisfy (2). Hence every quadratic equation has two, and only two roots. From either (i) or (2) we see that a quadratic equation may be formed of which the roots shall be any two given quantities. ;0 ALGEBRA. Thus, if the roots are 5 and - 3, by (2) the equation is (x- s) (x+ 3) = 0, or x-^ ~2x~ is = 0. If the roots are 2 ± -v/— 3, their sum is 4, and their product is 7, hence by (i) the equation is x'^- 4X+ 7 = 0. 150. Resolution into Factors. Any quadratic ex- pression of the form ax'^ + dx -{- c can be factored by finding the roots, a and /3, of the equation. a x'^ + l^ X + c = 0. For a x^ + If X + c = a Ix'^ + ~ X -{- -) \ a a) = a{x — a) {x — p). § 149. Example. Factor 2 x^ — 14^ + 36. The roots of the equation 2 x'^ — 14 r + 36 = are I + i V^^ and I - i V-235 hence 2x'^ -\^x-\-'>,(i^2{x-\-\ \/-23) (:r - J + ^ V-^iO 151. Solution by a Formula. Any complete quad- ratic may be reduced to the form x^+J>x + ^ = 0. (i) Solving (i), we obtain f. (2) 2 V 4 EQUATIONS. 71 Formula (2) affords the following simple rule for writing out the two roots of a quadratic equation in the form of (i): The roots equal one half the coefficient of x with its sign changed, increased and diminished by the square root of the square of one half the coefficient of x dimin- ished by the known term. Thus, to solve ;r2 + 3 :ir + 1 1 = 0, we have 152. Solution by Factoring. In solving equations the student should always utilize the principles of factoring and equivalent equations. Example. Solve x^ = \, (i) Transposing and factoring, (i) may be written in the form (:r-i)(:r2 + :r+i)(r+ i) (y-* - ;r + i) = 0. (2) The roots of (2) are i, — i, and those of the two equations :r* + :r + I = and :r2 - :r + I = 0, which are readily solved. EXERCISE 10. By § 145 determine the sum and the product of the roots of each of the four following equations. By § 148 find the character of the roots of each. Then solve each by § 151. 1. 5^:2 — 6:^ — 8 = 0. 3. 2x — x^^^, 2. ^2 4- II = 7 a:. 4. 5 jc'^ = 17:^; — 10. 72 ALGEBRA. Form the equations whose roots are 5. 3, -8. m n 6. §, f ^* n'~ m' 7- 3 ± Vs- ^ + ^ ^ _ 10 o ,/^ a — b a -^ b 8. 2 ± V— 3- Solve the following equations : II. ^ = -. 14- 2;c— 7 ^ — 3 7-^ — 5 2:^—13 :r + 4 ^V" — 2 , c /I ^ ^2. r— T + r-— : = 6J. 15. ^ — 4 X — 3 Jt: — 2 a: ^ + 6 -2:^— I 6 ^ X -\- a X — 2 a 13. ; = I . 16. ^~-^- , =1. /^x -\- 1 X ■\- "] X — 2 a X -\- a 3 T I 17. 2(^'^-i) 4(^+0 8 „ JC+I ^— I 2X— I 18. — [- ^ X + 2 X — 2 X — I 'YAx -\- 2 4 — 's/x 19. z^ = = • 4 + ^x ^Jx a+ 2b a" 4 ^^ 20. = -^ a — 2 b {a — 2 b) X X 21. a^ x"^ — 2 a^ X -\- a"^ — 1 ==0, 22. ^ a^ X = {a^ — b"^ -\- xy. EQUATIONS. 73 23. I — 4 Vx — a/?-^ + 2 = 0. (i) Transposing yji x ■\- 7. and squaring, we obtain I - 8 y'^ + 16 r = 7 ;r + 2, or 9 jr - I = 8 V-^. (2) Squaring (2), 81 :ir'^ - 18 jr + i = 64 jr. .-. 8i;ir2-82;r+ I =0. .-. (r- I) (81 ^-0 = 0. (3) The roots of (3) are evidently i and ^. Hence, if (i) has any root, it must be i or ^y. But neither of these roots satisfies (i) ; hence (i) has no root, or is impossible. It should be noted, however, that if we use both the positive and negative values of ^x and ^Jl x -^ 2, we obtain in addition to (i) the three equations I - 4 /y/;ir + V7 •** + 2 = 0, (4) + 4 V-^ - V7 ^ + 2 = 0, (5) I + 4 y:r + y; ;r + 2 = 0. (6) Multiplying together the first members of (i), (4), (5), and (6), we obtain the first member of (3) ; hence equation (3) is equivalent to the four equations (i), (4), (5), (6). Now i is a root of (4), and ^\ is a root of (5), but neither is a root of (6) ; hence (6) is impossible. Equation (3) could be obtained from (4), (5), or (6) in the same way it was from (i). 24. 2 V 4 + \/2 ;«•* + JC* = ^ 4- 4. 25. X \/6 + at"'' = I + x^. 74 ALGEBRA. 26. ^ - V-^ + I ^ _5_ 27. + X + V2 — JV'-^ -^ ~ \/2 — X^ 2 28. JC + Vi + ^^ = Vl + 5 (3:^— i) 2 /- I + 5 V ^ yx ^ — 82(jc + 8)_3jc+ 10 3,. _i^ + 4 _ 3^ 5 — ;;=: 6. Muln'ply both members by x, thus introducing the root zero; but this must not be included among the roots of the given equation. EQUATIONS. 77 4. X* — 2 ^^ + X = 380. 5. X* — 4 x^ + S x'^ — 8 X = 21. 6. 4x*+^x = 4x^ + :^:^. 7. :v + 16 — 7 V^ + 16=10 — 4 \x+ 16. 8. 2X^— 2X+2^/2X■^— Tx+6 = sx — 6. 9. ;c2 — ^ 4- 5 \/2 -^^ - 5 -^ + 6 = 2 (3 -^ + 33)- 10. x^ + 4x^= 12. J y 14. :v" + 6 = 5 X". 11. a:* = 81. I I I . , I ^ 15. 3 ^' - ;c^ — 2 = 0. 13. 6x^ = 7x^ -2X~K 16. i+8a* + 9 V^« = 0. 17. 3 :«:^ — 7 + 3 V3^^^-i6T+2i = 16 X. 18. 8 + 9 a/(3-^ - i) (:c - 2) = 3 :c2 - 7 9. ;<;2 + 3 — ^2 ^^ — 3 :t + 2 = I (;c + i). a:^ \ ^/ 9 21. a/^ + 12 + V^ + 12 = 6. 20 yS ALGEBRA- CHAPTER VIII. SYSTEMS OF EQUATIONS. 154. A single equation involving two or more un- known quantities admits of an infinite number of solutions. Thus, of _y = 2 ;r + 3, one solution is x — \^y — ^\ another is jjT = 2, jj/ =r 7 ; another is ;ir = 3, j = 9 ; and so on. In fact, whatever value is given to x^ y has a corresponding value. Oi y — \x— I, one solution is :r = i, j = 3 ; another is ;ir=2, / = 7 ; another is x —% y — 11; and so on. Both equations have the solution ;r = 2, y = y, which is therefore a solution of the two equations. 155. Equations which are to be satisfied by the same set or sets of values of their unknown quantities are said to be Simultaneous. Simultaneous equations which express different relations between the unknown quantities are said to be Independent. Of two or more independent equations, no one can be obtained from one or more of the others. Thus, of the simultaneous equations (i), (2), and (3), any two are independent, since no one can be obtained from an- other. But the three are not independent, for any one of them can readily be obtained from the other two. SYSTEMS OF EQUATIONS. x-2y + 3z = 2, (0 2X-3jy+ 2=1, (2) 3^-5jy + A2 = 3- (3) 79 Thus, by adding (i) and (2), we obtain (3) ; and by subtract- ing (2) from (3), we obtain (i). 156. A System of equations is a group of two or more independent simultaneous equations. 157. A Solution of a system of equations is any- set of values of the unknown quantities which will satisfy each of the equations. 158. Two systems of equations are Equivalent when they have the same solutions; that is, when every solution of either system is a solution of the other. 159. If each equation of a system contains only one unknown quantity, the system is solved by previous methods. But if each equation contains two or more unknown quantities, we must combine the equations of the system so as to obtain finally an equivalent system in which each equation con- tains only one unknown quantity. This process is called Elimination, and is dependent on the following principles. 160. T/ie equation obtained by addi^ig or subtracting' any two equations of a system may be substituted for either one of I hem. • 8o ALGEBRA. Let A = A^ and B = B' be any two equations in X, f, Zy . . . ; then the systems {a) and {h) are equivalent. vy ' A±B = A' ±B'j^'^ A = A'],_^ A = A' B = B' For it is evident that systems (^) and (^) are each satisfied by any set of values of x, y, z, .. ., that will render A equal to A^ and B to B\ and neither is sat- isfied by any other set. Hence systems {a) and {U) are equivalent. Either of the equations of the given system may evidently be multiplied by any known quantity be- fore they are added or subtracted. 161. Elimination by Addition or Subtraction. This method is based on the principle of § i6o. We will illustrate it by two examples. Example i. Solve Multiplying (i) by 7, Multiplying (2) by 8, Adding (3) and (4), or Similarly, we obtain By § 160, (5) and (6) may be, substituted for (i) and (2), respectively; hence the solution of system {ii) is given in {b). Zx^ 8j= 25, (2) J I2;ir— 7 J = 22. 2i;r+56_>/= 175. (3) Oi6x — s6y = 176. (4) li7;ir=35T, y = 2. (6) J SYSTEMS OF EQUATIONS. 8 1 m Example 2. Solve _ + ^-. (0 1 nV n' — + - = c', (2) ntn' nn' , ^ ^ Multiplying (i) by «', -j- +— - = en'. (3) m' n nn' Multiplying (2) by «, -— + — - = ^ «. (4) K^) Subtracting (4) from (3), {in n' - in' n)- = e n' - d n^ m n' — m'n ^ m'n-mn' \ ^ ^ Similarly, we obtain y - ^^^,_^,^ -J By § 160 the solution of system {a) is given in {b). 162. If (a) be any system of equatiojis in which A does not contain x, afid (b) ^ system obtained from system (a) /^/ substituting A / By § 160, (5) and (6) may be substituted for (i) and (2), respectively ; that is, the solution of {a) is given in (5) and (6). Since (i) and (2) are general equations of the first degree be- tween X and/, (5) and (6) may be used as general formulas for solving any system of simple equations in x and j'. Hence, a system of two linear equations in x and y has in general one, and only one, solution. 3- m + ^ = I, X m = I. 4- 5 X + '- = y 30, 9 X _5 _ y ■■ 2. 84 ALGEBRA. EXERCISE 12. Solve the following systems of linear equations : 1. 6x + 4y= 236, SX+ i5y = S73' 2. ax + l^y = a\ bx + ay = b"^. 5. (^ _ ^)^ + (^ + i,)y = 2 (a'- b'), ax — by = a^ + b^. 6. 1 =:m + n, 9. - H ^- - = 36, ^ 1 ^ 2 > 2 X y 7. 2x+:^y + 42 = 2g, 3:^ + 2 V + 52: = 32, 4-^ + 3 J' + 22:= 25. 8. «JJ -\- b X =: Cf ex + a z = bf b z ■\- cy = a. I I z 28, 3y I 2 Z 20. 10. 32 ■]-^U = 33 J 7^ — 2 z + 3^^ =:' 17. 4J — 2 z + z^ = II, 4y -ZU 4- 2 z^ = 9, sy -3^ — 2U = 8. SYSTEMS OF EQUATIONS. 85 Systems of Quadratic Equations. 166. A system consisting of one simple and one quadratic equation has ifi general two, and only twOy solutions. This theorem will become evident from the solu- tion of the following example: Example. Solve 8;r-4j = — 12, (i)^ 3:1:2 + 2^-^ = 48. (2)]' Solving (i) for J, j/ = 2 ^ + 3. (3) From (2) and (3), :r2 + 2 ;i- = 3. (4) From (4), x= i, or - 3. (5) From (3) and (5), J = 5» or - 3. (6) From § 162 the systems (rt), {b)^ and (^) are equivalent; hence the two solutions of {a) are given in (<:). 167. A system of two complete quadratic equa- tions m X and y has in general four solutions. Any such system can evidently be reduced to the form x^ -\-bxy-\- cjy^ + d x + ey +/= 0, (i) 1 ;r2 + b'xy + c' y'^ + d' x + €> y +/' = 0. (2)} Subtracting (2) from (i), and solving the resulting equation for Xj we obtain (t^' ~d)y + ti' -d ' ^^^ Substituting in (i) the value of x in (3), we shall evidently obtain a general equation of the fourth degree in y. This S6 ALGEBRA. equation will give four, and only four, values for y (see exam- ple of § 153). For each value of j in (3), x has one, and only one, value. Hence a system of two complete quadratics has in general yi7//r solutions. If, however, c = c' and l^ = b' , (3) will be a linear equation in x and j, and therefore system {a) will be equivalent to one consisting of a linear and a quadratic equation ; hence in this case the system has but two solutions. 168. By § 167 the solution of a system of two complete quadratics involves in general the solution of a complete biquadratic, of which we have not yet obtained the general solution. But many systems of incomplete quadratics can be solved by the methods of quadratics. We shall next consider some of the most useful methods of solving systems involving incomplete equations of the second and higher degrees. 169. When the system is of the form X ± y — a^ x^ ^ y'^=^a\ ^^ x^ \ y"^ = a\ xy = bl xy = b] x+y = b\ it may be solved symmetrically by finding the values o( X + f and X — f. Example i. Solve x—y 4, (01., >o. (2) J xy =: 60. From(i), x'^-2xy+y''= 16. (3) Multiplying (2) by 4, 4 xy = 240. (4) Adding (3) and (4), x^ + 2xy +y^= 256, or ^+y = ± 16. (5) SYSTEMS OF EQUATIONS. 87 Now (i) and (5) are equivalent to the two systems ^-J= 4, 1 x-y = 4, 1 x+j/^iS;] x+j/ = —i6.j Whence x=io,] ;r = — 6, 1 y= 6;J >/ = -io.J The values in (c) evidently satisfy system (a). Equations (2) and (5) would form a system having other solutions than those of system (a), introduced by squaring (i). Example 2. Solve x^-\-y^ = 6s, (i) xy = 28. (2) Multiplying (2) by 2, then adding ^nd subtracting, we have x'^+ 2Xf 4-j2= 121, and x^-2x_y+y^= 9; and ■^— / = ± 3- Now (d) is equivalent to the four systems x-\-jy=ii, ]x+_y=ii, , _ - i . - , , . x-y: } = 11, ^ x+_y=ii, "1 x-\-jr=-u,') ;r+J=-ll,l snce = 7,1 ;r = 4, 1 x = - 4, ]^ x = -7,^ =4; J j = 7; J y = -7;j y = -A' } Whence X y By § 160 systems {a) and {b) are equivalent, so also are sys- tems {c) and (^). Any system of equations of the form x'^ ±p xy +j2 = ^2 :r ± jK = ^, in which / is any numerical quantity, can evidently be reduced to the first form given above. 88 ALGEBRA. 170. Systems in which all the unknown terms are of the second degree may be solved as below: Example. Solve x^ -^ xy -\- 2j2 = ^, (i) | 2x^ — xy +y^ = i6. (2) j ^^^ Multiply (i) by 4, 4x^ + ^xy + 8y^ = 176. (3) Multiply (2) by 11, 22 x^ — 11 xy -j- iiy^ = 1 76. (4) Subtract (3) from (4), i8x^ -isxy -]- 3^2 = 0. Factor, {y - sx) (y - 2x) = 0. (5) Equations (i) and (5) form a system equivalent to system (a) or to the two systems (d) and (c), x^ + xy + 2/2 = 44, I x^ + xy + 2y^ = 44, } y-3x=0; P ^ y-2x = 0', ) ^^^ which are readily solved. Their solutions are x= V2, — V2, 2, -2. ^ = 3^2, -3^, 4, -4- The above method is sometimes applicable to other systems than those of the class considered above. ^6y. i2)\^' Example. Solve x"^ - 2y^ = 4/, 2,x^ + xy — 2_y2 Multiply (i) by 4, 4:r2 - 8j2 = i6j/. (3) Subtract (2) from (3), x'^ — xy — 6y^ = 0. Factor, (x + 2y) (x - 3 j) = 0. (4) Equations (4) and (i) form a system that is equivalent to (a) or to the two systems (d) and (c), x^-2y^ = 4y, I ^2 _ 2^2 ^4^, I which are readily solved. Their solutions are ^=0, -4, 0, -v^. .y = o, 2, 0, f SYSTEMS OF EQUATIONS. 89 EXERCISE 13. Solve the following systems of equations : 19. I. x+y= 51, 5- ^->' = 3» xy=s^^' ;i:2_3^_j;+/z3 — 2. x—y= 18, 6. :^^-^J+/ = 76, xy= 1075. X +y= i^. 3. ^-y= 4, 7- x^+xy+y' = 67, x' + y'= 106. :r2— .r>'+/ = 39• 4. ^'+y=i78, 8. ^^— 2JCJ— j^ = I, x-^y= 16. X +y= 2. 9. 3^^*^— 2/ + 5^— 2j= 28, x+y+ 4= 0. 10. :r2— 3^>'+y 30.2— a:>'+3y =: 13. 16. ^^y^s y X 2 X +y = 6. 17. II. 3^' -5/ = 28, 3^7-4/= 8. 12. jr^ 4- 3 jc J ^ 54, xy + 4/= 115- '3- ^V+ 5 0:^ = 84, ^+j= 8. 14. x^+y^-3 = 3^y, 2x''-6+f = 0. I X I y = I ■ — J 3 h^ I ? = 5. 9* 18. x'^—2xy—y'^— 31, ^x''^-\-2xy — y^ =^ loi. 3i^>'-3a-2-5/=45. ^2_^^_^^y^ g^ 90 ALGEBRA. 171. When X and y are symmetrically involved in two simultaneous equations, the system can fre- quently be solved by putting x — v ■\- w and y ■=^ V — w. Example. Solve ;r^ + i/'*= 82, (i)"l x-y = 2. (2) J Put ^=:^^-^^^ (2)^ and y = V — TV. (4) From (2), (3), and (4), w = 1. (5) From (I), (3), (4), and (5), or z/ = ± 2, or ± /y/— ID. (6) From (3), (5), and (6), x=3,-i,i± ^/^o. ] , ^ >{c) From (4), (5), and (6), j = i, _ 3, - i ± y_ 10. J 172. I7i general, system (a) is equivalent to the two systems (b) and (c). AB = A'B',) ^ A = A',) , -^5 = 0,) By § 162, system (a) is equivalent to the system AB = A'B,l ^^ {A-A')B B = A'B,l ^^ {A-A^)B = 0, I = B\ S B = B\ S which is equivalent to both (d) and (c). The first equation in system (<^) is obtained by dividing the first equation in (^) by the second. If ^ and B^ cannot each be zero, system (<:) is im- possible, and system {d) is equivalent to system (<^). SYSTEMS OF EQUATIONS. 91 This principle is often useful in solving systems involving higher equations. Example. Solve»r» + ;j^y+y = 737I> (Ol x^^x_y-\-jy^= 63. Dividing (i) by (2), xHxy+y^= 117. (3) Adding (2) to (3), A-2+yz=9o. ^^^\(d) Subtract (2) from (3), 2xy= 54. (5) J Hence x = = +9. -9>+3, -3,1, , = +3, -3, +9, -9- J Here A" = 63, and the system, ^ = 0, ^' = 0, is impossible ; hence equations (2) and (3) form a system equivalent to (a). Therefore all the solutions of (a) are given in (c). EXERCISE 14. Solve the following systems of equations : 1. A« + y = 637, 7. A« -/ = 56, x+y= 13. :t- + ATj + y- = 28. 2. x^ -\- y = 126, x^ — xy + _y^ = 21. 3. x^ 4- J^^^*^ + J''* = 2128, x^ _j_ ^^^ -j- ^2 __ ^^ 4. .V + ^ — V^ = 7' x^ -\r y^ \ xy^ 133. 5. a: + >' = 4, ^4 -f / == 82. 6. f-V-^''^?, ,2. ^±^ + ^:-:^=.s, j^' .r 2 x — y X + y 2 3 _ -^^ + ^^ = 20. 8. •^ J (-^ + ^) = 30» ^*+/ = 35- 9- jc«-y = 127, x^y — j^y-^ = 42. 10. 5^v2- 5/ = ^+j, 3^^'-3/ = -^-7- II. a--* +y = 272, x-y= 2. 92 ALGEBKA. CHAPTER IX. INDETERMINATE EQUATIONS AND SYSTEMS, DIS- CUSSION OF PROBLEMS, INEQUALITIES. 173. An Impossible Equation or System of Equations is one that has no finite solution. Such an equation or system involves some absurdity. Two equations are said to be Inconsistent when they express relations between the unknown quanti ties that cannot coexist. Any system that contains inconsistent equations or embraces more independent equations than unknown quantities is impossible. Thus, |:r + |^-5 = ^:r + ^'^;jr+8is an impossible equa- tion ; for attempting to solve it, we obtain the absurdity = 312. C ax ^- by — c^ ( i ) Thesystem iy^^^ ^ ^^^ ^ ^^^ (,) is impossible ; for attempting to solve it, we obtain the absurdity 3 = 5- Equations (i) and (2) are evidently inconsistent. r ;r + r= 9. (0 The system -| 2 ;r + j = 13, (2) [x-h 2y= 16, (3) is impossible, for solving (i) and (2), we obtain x = 4, y = $; substituting these values in (3), we obtain not an identity, but the absurdity, 14 = 16. Similarly the solution of any other two of these three equations will not satisfy the third. INDETERMINATE EQUATIONS AND SYSTEMS. 93 Again, if in the system ax — by — c, (2) J ^ "^ we divide equation (i) by (2), we obtain ax^by-c. (3) Now (2) and (3) form a system equivalent to system {a) ; hence {a) has but one solution. But by § 166 such a system as {a) has in general two solutions. System ('/) is called a defective system.* Nearly all the systems in Exercise 14 are defective. 174. An Indeterminate Equation or System of Equa- tions is one that admits of an infinite number of solu- tions. Hence a single equation containing two or more unknown quantities is indeterminate (§ 154)- Again, any system of equations that contains more unknown quantities than mdepejidcnt equations is indeterminate. Thus, the system , , _ . a' X + b'y -\- c'z = 0,} is indeterminate ; for solving the system for j and 2 we may give to X any value, and find the corresponding values oty and z. * An impossible equation or system of equations is, in general, but the limiting case of a more general equation or system, the solutions of which in the limit become infinity. Thus, the equation a x — b becomes impossible only when a = 0, and then its root b -^ a becomes ^ -^ 0, or infinity. It will be seen in § 176 that a system of linear equations becomes impossible only for a certain relation between the coefiicients of its equations, which renders both x and y infinite. Again, the general system ax — {b ■\- e)y ^^ Cy ' becomes the defective system {a), only when e — Q. 94 ALGEBRA. ( 3 ^ — 4y — 9 1 The system < „ f is indeterminate, for its equa- tions are not independent but equivalent. Again, the system 2x+3y- z= IS, (i) 3x-y + 2z=8, (2) 5 ^ + 2 J + ^ = 23, (3) is indeterminate. No two of its equations are equivalent, but any one of them can be obtained from the other two ; thus, by adding (i) and (2) we obtain (3). Hence the system contains but two independent equations. 175. Sometimes it is required to find the positive integral solutions of an indeterminate equation or system. The following examples will illustrate the simplest general method of finding such solutions. (i) Solve 7^+ I2_y =■ 220 in positive integers. Dividing by 7, the smaller coefficient, expressing improper fractions as mixed numbers, and combining, we obtain 5^ — 3 ^+^^ + ^^=31. (I) Since x and y are integers, 31 — :ir — y is an integer ; hence the fraction in (i), or any integral multiple of it, equals an integer. Multiplying this fraction by such a number as to make the coefficient oi y divisible by the denominator with remainder I, which in this case is 3, we have Hence IS J' — 9 >' — 2 ■ — _ 2j/ - I + -an mteger. y — 2 —- — _ an mteger =/, suppose. .-. y = 7p + 2. (2) [) and (2), ;r=28 — 12/. (3) INDETERMINATE EQUATIONS AND SYSTEMS. 95 Since x and y are positive integers, from (2) it follows that / > — I, and from (3), that ^ < 3 ; hence i^ = 0, I, 2. (4) From (2), (3), and (4), we obtain the three solutions x= 28, 16, 4; J = 2, 9, 16; (2) Solve in positive integers the system or x^y-^z^ 43, 0) Io:ir+ 5J + 2-2r = 229. (2) Eliminating z, 8 :r + 3 j = 143, IX— 2 y + 2x+ — — = 47- (3) AX — A. X — I .'. ^—- — = X- \ -{ — = an mteger. .♦. = an integer = ^, suppose. .-. x=2>p-\-i. (4) From (3) and (4), ^ = 45 - 8/. (5) From (I), (4), and (5), z^sp-Z- (6) From (6),/ > ; and from (5),/ < 6; hence P=i, 2,3,4, 5. Whence x= 4, 7, 10, 13, 16; ^ = 37,29,21, 13, 5; z= 2, 7, 12, 17, 22. Thus, the system has five positive integral solutions. (3) Show that ax -{- by = c has no integral solutions, if a and b have a common factor not a divisor of c. 96 ALGEBRA. Let a = m d, b = n d, c not containing dj then md X + n dy — c, or mx + ny = c -^ d. (i) Now r -^ and its reverse < ; the opening being toward the greater quantity. U a — d is positive, a> b ; if ^ — ^ is negative, a < b' The expression a > b > c indicates that b is less than a but greater than c. The expression a^ b indicates that a is either equal to or greater than b. 179. The following principles, used in transform- ing inequalities, will upon a little reflection becoftie evident: (i.) An inequality will still hold after both mem- bers have been Increased or diminished by the same quantity; Multiplied or divided by the same positive quantity; Raised to any odd power, or to any power if both members are essentially positive. (ii.) The sign of an inequality must be reversed after both members have been Multiplied or divided by the same negative quantity ; Raised to the same even power, if both mem- bers are negative. I02 ALGEBRA. (iii.) If the same root be extracted of both mem- bers of an inequahty, the sign must be reversed only when negative even roots are compared. 180. In estabhshing the relation of inequahty be- tween two symmetrical expressions, the following principle is very useful. If a a7id b are uneqtial and real, a^ + b^ > 2 a b. For {a - by > 0, , or a''-2ab-^b^>(}; (i) hence d^ ^ b'^y 2 ab, (2) (i) Prove that the arithmetical mean between two unequal quantities is greater than the geometrical mean. If in (2) we put d^ — x and ^^ —y_^ ^e obtain — , x-Vy . — X -^y •> 7. ^xy, or — ^ > ^^xy. (2) Show that a^^b^> d^b^ab\\ia-\-b>^. From (i), a^-ab + b'^> ab. Multiplying hy a + b, a^ + b^ > d^ b + a b\ (3) Show that the fraction ^ ^ / 4. ^ \ ... ^^ < the greatest and > the least of the fractions ^, ^, •••, ^, all the denominators being of the same quality. Suppose that -/ is the least and ^ the greatest of these fractions, and that the denominators are all positive. INEQUALITIES. IO3 Then ^ = J, .-. ^i-^iX^i; T- ^ -7 J • • "-2 ^ ^2 X V- > Adding, ^^ + ^3 + • • • + ^« >(/^i + ^2 + • • • + /^;,) -^^ . ^1 +y^2 + ••• + ^« <^i' Similarly we may prove that ^1 + ^2 + H g>t ^ a^ In like manner the principle may be proved when all the denominators are negative. EXERCISE 16. 1. Show that the sum of any fraction and its reciprocal is numerically greater than 2. The letters denoting unequal positive numbers, show that 2. a^ + d'^ -\- c^> ad + ac+ dc; m^ + i > m"^ + m. 3. a^ + /^' +^ > K^'^ + «^^ + «V + «r^ + ^V + dc^). a + d 2 ab a b i i 4- ~7~>^T^' ^ + ;r^ ^'b'^a' 5. If a^ -\- b"^ -\- c^ = I, and m^ + n^ + r^ = i, show that am -\- b n + cr ^ ^ — f , and I — ^ x < S — 2 x, show that the values of x lie between 1 1 and -^/. 104 ALGEBRA. CHAPTER X. RATIO, PROPORTION, VARIATION. 181. The Ratio of one abstract quantity to another is the quotient of the first divided by the second. When the quotient a -^ b or t is spoken of as a ratio, it is often written a : b, and read '* a is to h;" a is called the Antecedent and b the Consequent of the ratio. 182. By § 48 the value of a ratio is not changed by multiplying or dividing both its antecedent and consequent by the same quantity. Two ratios may be compared by reducing them as fractions to a common denominator. 183. When two or more ratios are multiplied to- gether they are said to be compounded. Thus the ratio compounded of the three ratios 2:3, a : d, and b : e'^ is 2 a b : ^ de\ The ratio a^ : ^^ compounded of the two identical ratios a : b and ^ : ^, is called the Duplicate ratio of a : b. Similarly a^ : b^ is called the Triplicate ratio of a : b. Also a"^ : b'^ is called the Subduplicate ratio of ^ : ^. RATIO. 105 184. The ratio of two positive quantities is called a ratio of greater or less inequality according as the antecedent is greater or less than the consequent. 185. A ratio of greater inequality is diminished^ and a ratio of less inequality is increased, by adding the same positive quantity to both its terms. Let Uy b, and x be any positive quantities; then a + X \ b ^ X a '. b, according as a> or <,b. aa-\-x_x(a — b) ,. Now the second member of (i) is evidently positive or negative according as ^ > or < ^. TT ^ + -^ ^ Hence 7— — < or > -, accordmg as ^ > or < b. In like manner it may be proved that a ratio of greater inequality is increased, and a ratio of less ine- quality is diminished, by stib trading the same qna?ttity from both its terms. XS6. By§68, |:£Hff Hence, unless surds are involved, the ratio of two fractions can be expressed as a ratio of two integers. If the ratio of any two quantities can be expressed exactly by the ratio of two integers, the quantities are said to be Commensurable ; otherwise they are said to be Incommensurable. I06 ALGEBRA. 187. Ratio of Concrete Quantities. If A and B be two concrete quantities of the same kind, whose nu- merical measures in terms of the same unit are a and b, then the ratio of ^ to ^ is the ratio of a to b. If A and B are incommensurable, that is, cannot be exactly expressed in terms of the same unit, we can ahvays find two integers whose ratio differs from that oi A \.o B by as Httle as we please. For divide B into n equal parts ; let ^ be one of these parts, so that B — n^. Suppose ^ is con- tained in A more than m times and less than m + i times ; then it is axiomatic that A\ B > 7n (i : n P diud < (w + i)P:nP; that IS, the ratio A : B lies between — and . Hence the ratio of ^ to ^ differs from that of m to n by less than i -^ ;/, which by increasing n can be made as small as we please. The ratio oi A \.o B is the fixed value to which the ratio of m to n approaches indefinitely near when n is increased without limit. PROPORTION. 188. Four quantities, a, by c, d, are said to be pro- portional if the ratio a \ b \s equal to the ratio c : d. The proportion is written a : b = c : d, a : b : : e : d, or — = - , a and is usually read " a is to h as z is to d." PROPORTION. 107 The first term and the last are called the ExtrenieSy and the other two the Means of the proportion. 189. Continued Proportion. The quantities a, by r, d, . . . , are said to be /;/ continued proportio7i if a\ b — b '. c — c \ d — '" (i) In (i), ^ is said to be a mean proportional between a and r, and c a thii'd proportional to a and b. Also b and c are said to be two ineaji proportionals between a and d, and so on. 190. If four quantities are in proportion, the product of the extremes is equal to the product of the means. If T = -, » then by § 23 a d = c b. o a 191. Corollary. \i a\ b = b\ c, then I^ = ac, or b = Va c. 192. Conversely, if tRe product of two quantities equals the product of tivo other quantities, two of them may be made the extremes and the other two the means of a proportion. \i a d = c b, then by dividing both members hy b d we obtain a \ b — c\ d. 193. If four quantities are in proportion, they are in proportion by I08 ALGEBRA. (i.) Inversion : 7/" a : b = c : d, the7i b : a = d : c. (ii.) Alternation : 7/" a : b = c : d, then a : c = b : d. (iii.) Composition : //"a : b = c : d, theji a + b : b = c + d : d. (iv.) Division : If a : b = c : d, then a — b : b = c - d : d. (v.) Composition and Divisiori: If a : b = c : d, then a + b:a — b = c + d:c — d. These propositions and those of §§ 194 and 195 are easily proved by the properties of fractions or by §§ 190 and 192. 194. 7^ a : b = c : d, and e : f = g : h, then a e : b f = c g : d h. 195. //■ a : b = c : d, then (i.) ma:mb = nc:nd; (ii.) ma:nb = mc:nd; (iii.) a" : b" = c" : d", n being any exponent. 196. If we have a- series of equal ratios, the sum of the antecedents is to the sum of the consequents as any one antecedent is to its consequent. T- ^ ace L«t ^=;/=7 = •••'"'• then a = bry c= dr, e =fr, ... PROPORTION. 109 Adding these equations, we obtain « + ^ + ^+ ••• = (^ + ^ + /+---)^. a + c -i- e + •" _ _^_ Remark. The method of proof used above might be employed in § § 193, 194, 195. The proof in the next article further exhibits the directness and sim- plicity of this method. ^^ a c , ma ■\- nb m c -\- n d 197. If 7 = ,, then — , r — — — \ 3 • b d pa -\- gb pc -\- q d _ a c . ma-\-nb mbr + nb mr-\-n Let 1 = ~j = ^; then - — r— r = , t — ; — r = t — ; — > b d pa-\-qb bpr-\-qb pr-\-q tnc •\- nd mdr -\- n d m r ■{• n pc-\-qd pdr + qd p r -\- q ma -\- nb _mc -\- nd pa -\- qb pc ^ qd 198. Proportion of Concrete Quantities. If ^, B^ be two concrete quantities of the same kind, whose ratio is a : b, and C, D, be two other concrete quantities of the same kind (but not necessarily of the same kind as A and B)^ whose ratio is ^ : ^; then A.B^C'.D, when a\ b — c\ d. The last proportion can be transformed according to the theory of proportion given above, and the re- sult interpreted with respect to A, B, C, D. and no ALGEBRA. VARIATION. 199. If the relation between y and z is expressed by a single equation, then y and z have an infinite number of sets of values (§ 154), and are called variables. There are an infinite number of ways in which one variable, y, may depend upon another, -z. For exam- ple, we may have y = az, y=:az + c,y = az'^ + bz + c, and so on. In this chapter we shall consider only the simplest relation, y = az, in which z denotes any variable, and a is a constant ; that is, a has the same value for all values of j^ and z. 200. \{ y =^ a z, the ratio oi y \o z is the constant a. The expression j/ oc 2; denotes that the ratio of y \.o z is some constant, and is read '' y varies as zT The symbol oc is called the Sign of Variation, Hence, if y — a z, y 01 z, and conversely. 1( y cc z, or y = a z, then any set of values of y and z are proportional to any other set ; for the ratio of each set is the constant a. Hence _^ oc ^ is often read ''y is proportional to ^." I X 201. If in § 200, z = - , XV, -, X -\- 7/, we have "^ XV (i.) If Y = a. ~ , then y oc -, and conversely. y oz I -^ ;r is often read '' y varies inversely as x' VARIATION. 1 1 1 (ii.) If y = axv, t/ien y oc x v, and conversely, y oi xv\^ often read '' y varies as x and v jointly ^ X X (iii.) y/" y = a -, then y cc - , and co?iversely. y cc X -^ V is often read ''y varies directly as x and inversely as v!' (iv.) i^ y = a (x + v), then y oc x + v, and conversely. 202. The simplest method of treating variations is to convert them into equatiofis. Of the six following propositions we give the proof of the first, and leave the proof of the others as an exercise for the student. (i.) If \x oc y, and y oc x, then u oc x. By § 200, ti — ay, y = b x {a and b being con- stants), .*. ti = a b X, or n ^ x. (ii.) //■ u X X, and y oc x, then u ± y oc x, and u y X x^. (iii.) If n cfi -K, and z x y, then u z x x y. (iv.) Ifw X X y, //^^// X X u ^ y, and y x u -H x. (v.) If \i cc -x., then z u x z x. (vi.) If \i cc -K, then u" x x". 203. //■ u X x w/?^;^ y /i" constant, and u x y when x w" constant y then u x x y when both x ^«rtf y ^r^ 112 ALGEBRA. The variation of u depends upon that of both x and y. Let the variations of x and y take place successively ; and when x is changed to x^^ let u be changed to ?/ ; then since u c^ x when y is constant, by § 200 we have Next let y be changed to ^j, and in consequence let 71 be changed from u' to u^ ; then, since u oz y when X is constant, u^ __y (2) Multiplying (i) by (2), hence by § 200 u oc jirjj', This proposition is illustrated by the dependence of the amount of work done, upon the number of men, and the length of time. Thus, Work oc time (number of men constant). Work X number of men (time constant). Work X time X number of men (when both vary). The proposition given above can easily be extended to the case in which the variation of ii depends upon that of more than two variables. Moreover the varia- tions may be either direct or inverse. Example. The time of a railway journey varies directly as the distance, and inversely as the velocity; the velocity varies directly as the square root of the quantity of coal used per mile. VARIATION. 1 1 3 and inversely as the number of cars in the train. In a journey of 25 miles in half an hour, with 18 cars, 10 cwt. of coal is re- quired ; how much coal will be consumed in a journey of 21 miles in 28 minutes with 16 cars ? Let / = the time in hours, d= the distance in miles, V = the velocity in miles per hour, n = the number of cars, ^ = the quantity of coal in cwt. Then /x -, and v ccYl. ^ n nd nd ,'. /oc— =, or / = ^ — — . (i) Now ^ = 10 when d= 25, / = i, and n = 18 ; hence from (i) 1 18x25 y\/To . -_ y\/To nd 2 ^s/Vo 36 X 25 36 X 25 ^q Hence when ^=21, t — \% and n— 16, we have 28 _ /y/io X 16 X 21 _ ^~ 25x36^^ ' *"' ^~ ^' Hence the quantity of coal consumed is 6| cwt. EXERCISE 17. 1 . \i a \ b — c \ d^ and b ;x = d:y^ prove that a : x =: c. y, 2. li a : b =^ b '. c^ prove that a : c ^^^ a^ : b'^ =^ b^ : c^. 3. \i a \ b ■= b '. c =^ c \ d^ prove that a : d ^=^ a^ -. b^ z= ... Let r =. a -^b ; then a =^ b r, b r= c r, c — dr. .'. abc—bcdr^j .-. a -^ d = r^ — a^ -^ b^ = ,., 114 ALGEBRA. li a : b = c \ d, prove that 4. a^c+ ac'':b''d+ b d'^ = {a -\- cf : {b + df. 5. a — c. b — d= ^a^ + r : \/b' + d\ 6. V^M^: V^'' + ^'=\/^^+~:\/bd+~. 7. If «, ^, c, d, be any four numbers ; find what quantity must be added to each to make them proportional. 8. If y varies as x, and y — 8 when x = 1^ ; fin*: y when a: = 10. 9. If y varies inversely as x, and y = J when ^ = 3 ; find ^ when jc = 2^. 10. If ti varies directly as the square root of x, and in- versely as the cube of y, and if z/ ~ 3 when x = 256 and y = 2 ; find jjc when 2/ = 24 and y = \. 11. If 2/ varies as x and_y jointly, while x varies as -s:^, and y varies inversely as u ; show that ti varies as z. 12. The pressure of wind on a plane surface varies jointly as the area of the surface, and the square of the wind's veloc- ity. The pressure on a square foot is i lb. when the wind is moving at the rate of 15 miles per hour. Find the velocity of the wind when the pressure on a square yard is 16 lbs. THE PROGRESSIONS. II5 CHAPTER XI. THE PROGRESSIONS. 204. An Arithmetical Progression is a series of quantities in which each, after the first, equals the preceding plus a common difference. The common difference may be positive or nega- tive. The quantities are called the terms of the progression. 205. Let d denote the common difference, a the first term, and / the «th, or last term. Then by definition the 2d term = a -\- d, the 3d term — a -\- 2 d, and the «th term — a -\r {11 — i) d. Hence i = a -\- {n — i) d. (i) Let ^ denote the sum of the terms ; then S = a + {a -\- d) + (a ^ 2 d) -{- ... + {I - d) ^ I, S= I^ {I -d) -^{i - 2d) ■\- ... + {a-^ d) -]- a. Adding these two equations, we obtain 2S=n{a-\- I), 1 6 ALGEBRA. Hence S =^ {a + i). (2) From (i) and (2), S=~{2a + (fz-i)d}. (3) If any three of the five quantities, a, I, d, n, 5, be given, the other two may be found by the formulas given above. 206. The m terms lying between any two terms of an arithmetical progression (A. P.) are called the m Arithmetical Means between the two terms. 207. To insert m arithmetical means between a and b. Calling a the first term, b will evidently be the {in + 2)th term; hence from (i) of § 205, b = a -\- (m -\- i)d. J b — a .-. d= , Hence the required terms are , b — a 2 (b ~ a) ^ m (b — a) a+ — , a -\ ^ — ■ -, ..., a -^ m + I m -\- 1 m + 1 208. Corollary. If ;;^= i, the arithmetical mean b — a a -\- b is ^ H ^- , or I -f I 2 THE PROGRESSIONS. II7 209. If any two terms of an A. P. are given, the progression can be entirely determined ; for the data furnish two simultaneous equations between the first term and the common difference. Example. The 54lh and 4th terms of an A. P. are — 61 and 64; find the 27th term. Here — 61 = the 54th term = /z + 53 ^y and 64 = the 4th term = a + 3 ^. Hence ^= — -|, ^ = 71^. . •. 27th term = a + 26^=6^. 210. A Geometrical Progression (G. P.) is a series of quantities in which each, after the first, equals the preceding multiplied by a constant factcr. The con- stant factor is called the ratio of the progression. 211. Let r denote the ratio, a the first term, / the «th, or last term ; then the 2d term = ^r, the 3d term = a r^. and the ;/th term = a r"-\ Hence /=ar*'-\ (i) Let 5 denote the sum of the terms ; then S^za + ar+ar'^ + ar^-] + ^r""^ = a(i + r+r^ + ... + r"-i) Hence 3=^-^^::^^:^. (2) /'—-I Il8 ALGEBRA. 1( r < I, formula (2) is usually written S='Lii-=^. (3) From (3), S = I — r a ar^ 1 — r Now \i r < I, then the greater the value of n, the a r" smaller the value of f'y and consequently of . Hence, if n be increased without limit, the sum of the progression will approach indefinitely near to a 1 — r That is, the limit of the sum of an infinite number of terms of a decreasing G. P. is , or more briefly, the sum to infinity is I — r 212. The m terms lying between two terms of a G. P. are called the m Geometrical Means between the two terms. 213. To insert m geometrical means between a and b. Calling a the first term, b will be the i^ + 2)th term; hence by (i) of §211. =0 1 (i) Hence the required terms 'are ar, ar'^y "- ar*", in which r has the value in (i). THE PROGRESSIONS. II9 214. Corollary. l(m = ijr=y-y, and there- fore ar= ^/a b ; hence the geometrical mean between a and b is the mean proportional between a atid b. 215. A series of quantities is in Harmonic Progres- sion (H. P.) when their reciprocals are in A. P. Thus, the two series of quantities, i»i, \^\^ •••, and 4, -4, -|, ..., are each in H. P., for their reciprocals, I, 3. S» 7. •••, and i -i -|, ..., are in A. P.' We cannot find any general formula for the sum of any number of terms of an H. P. Problems in H. P. are generally solved by inverting the terms and mak- ing use of the properties of the resulting A. P. (i) Continue to 3 terms each way the series 2, 3, 6. The reciprocals ^, ^, ^ are in A. P. ; .-. d=—\. .'. TheA. P. is I, 1,1,^,^,^,0,-^,-}. .-. The H. P. is I, f, f, 2, 3, 6, CO , -6, -3. (2) Insert 4 harmonic means between 2 and 12. The 4 arithmetical means between \ and ^-^ are ^^tj, }, J, J ; hence the harmonic means required are 2f, 3, 4, 6. 216. If // be the harmonic mean between a and by then by § 215, I a T I 1' are in A. P. I I I I ' * ~H~ a ~b 7^' 2 a I b' or . i^ 2 a a-\- b ■b 1 20 ALGEBRA. 217. Corollary. If A and G be respectively the arithmetical and the geometrical mean between a and by then (§ § 208, 214, 216) A — , G — \a b, H — a ■\- b ^^ a -{- b 2 ab , ^„ .-. A X 11=. — — X — —7 =ab= G\ 2 a -{• Hence A.G=G:B; That is, the geometrical mean between two num- bers is also the geometrical mean between the arith- metical and harmonic means of the numbers. EXERCISE 18. 1. Sum 2, 2>\) 4ij •••> to 20 terms. 2. Sum f, f, i^g? '"^ to ^9 terms. 3. Sum a — 2ib, 2a — ^by 3 ^ — 7 <^, • • • , to 40 terms. 4. Sum 2 a — b, ^a — ^b, 6 a — ^ b, • • • , to « terms, 5. Insert 17 arithmetical means between 3^ and — 41^^. 6. The sum of 15 terms of an A. P. is 600, and the com- mon difference is 5 ; find the first term. 7. How many terms of the series 9, 12, 15, •••, must be taken to make 306 ? 8. Sum i, ^, f, •••, to 7 terms. 9. Sum I, VSj 3j •••> to 12 terms. THE PROGRESSIONS. 121 10. Insert 5 geometrical means between 3f and 40^. 11. Sum to infinity f, — i, |, ••• 12. Sum to infinity 3, Vs^ i, ••• 13. The 5th and the 2d term of a G. P. are respectively 81 and 24; find the series. 14. The sum of a G. P. is 728, the common ratio 3, and the last term 486 ; find the first term. 15. The sum of a G. P. is 889, the first term 7, and the last term 448 ; find the common ratio. 16. Find the 4th term in the series 2, 2^, 3 J, ... 17. Insert four harmonic means between § and -j^^. 18. Find the two numbers between which 12 and 9^ are respectively the geometrical and the harmonic mean. 19. If a body falling to the earth descends 16^ feet the first second, 3 times as far the next, 5 times as far the third, and so on ; how far will it fall during the /th second ? How far will it foil in / seconds? 20. A ball falls from the height of 100 feet, and at every fall rebounds one fourth the distance ; find the distance passed through by the ball before it comes to rest. 21. According to the law of fall given in Example 19, how long will it be before the ball in Example 20 comes to rest? Ans. ^% A/579 = 7-4805 seconds. 122 ALGEBRA. SECOND PART, CHAPTER XII. FUNCTIONS AND THEORY OF LIMITS. 218. A Variable is a quantity that is, or is sup- posed to be, changing in value. Variables are usu- ally represented by the final letters of the alphabet, as X, y, 2. The time since any past event is a variable. The length of a line while it is being traced by a moving point, is a variable. li X represents any variable, x^, 3 x^, and 2 x^ — 4:1; will denote variables also. A Constant is a quantity whose value is, or is sup- posed to be, fixed. Constants are usually represented by figures or by the first letters of the alphabet. Particular values of variables are constants, and they are often denoted by the last letters with accents, as x'y y, x", y'K The time between any two given dates is a constant, as is also the distance between two fixed points. Figures denote absolute, and letters denote arbitrary constants. 219. An Independent Variable is one whose value does not depend upon any other variable. FUNCTIONS. 123 A Dependent Variable is one whose value depends upon one or more other variables. A dependent variable is called a Function of the variable or varia- bles upon which it depends. If the radius of the base of a cylinder is an independent va- riable, and the aUitude is always four times the radius, then the altitude is a function of the radius, and the surface and volume are different functions of both the radius and the altitude. The expressions ax^, x* — ex, a"^, represent functions of ;ir. If in any equation between x and j/, we regard x as an inde- pendent variable, then j/ is a function of x. Thus, if y = 2x''^ -{- X — 6, and x increases ; then when ^=-4, -3, -2, -I, 0, I, 2, 3, 4, ... jy= 22, 9, 0, -s, -6, -3, 4, 15, 30, ... Here while x increases from — 4 to — i, j decreases from 22 to — 5 ; while x increases from — i to i , y first decreases and then increases ; and while x increases from i to 4, j/ increases from — 3 to 30. 220. Functional Notation. The symbol /(x), read " function of x" is used to denote any function of x. When several dififerent functions of x occur in the same discussion, we employ other symbols, as /' (,i'), F(x)^ (;r), which are read "/" prime function of x" " large F function of x," '' function of :r," respectively. The symbols f(a), /(2), /(r), f(c + ^), repre- sent the values of /(;r) for x = a, 2, 2, c + d, respec- tively. Thus, if /(x) =x^ + x, /(a) -a^-j-a, f{2) = 10, and /(^ + ^) = ^ + 3 ^^^+ 3 ^^^'^ + ci^ + c + d 124 ALGEBRA. Since /(;ir) denotes any function of x,y=f{x) represents any equation involving x and j/, when solved for y, . • 221. The symbol ^, read "factorial ?2," denotes the product of the first n whole numbers ; that is, 1^= I • 2 . 3 . 4 n. Thus, [3=1.2.3 = 6; 14 = 1-2.3.4 = 24. EXERCISE 19. In the first three examples the student should carefully note \iQ'N f {x) changes as x increases. 1. /(.r) = 5 a:^ - 3 ^ + 2 ; find /(-3),/(- 2),/(_ i), /(O), /(i), /(2),/(3)> /(4), /(5), /(6). 2. f{x) = 4.x^ — x*+ 2X— IT, find /(— 2), /(— i), /(O), /(I), /(2), /(3). /(4), /(5), /(6)- 3- fix) =x' + x' + 2; find /(- 4), /(- 3), /(- 2), /(- i), /(- 0-3), /(- 0.2), /(O), /(.), /(2). 4- ^(x) =x^+ sx; find /"(^j + 2), i^(a:i + «), 7^(5 x^), 5- -/^ (•^) =^^ + 4^ + 3; find -^ (3 -^i). -^ (*i + '^). ^(^1-5), fie-, a). 6. fix) = (« + ^)'" ; find /(O), /(i), /(2), fib), fiz). 7- /W = 2 + Vs+-")^. Now however small k may be, each one of the n variables, Vi, V2, Vs, ••• , can be made less than k -^ n ; therefore their sum can be made less than k ; hence lt(z;i + 2^2 + ^'3 + •••) = 0. Hence It (^+7 + 2+ ...) == ^ + /^ + ^ 4- ••• 230. The limit of the variable product of two or more variables is the product of their limits; that is, if lt(x) = a, and It (y) = b, then \t(xy) =ab = \t(x) \t (y). Let Vi = a — X and 7'2 = b —y; then x = a — Vj, y = b — v^y and X y = a b — a V2 — b 7^1 + z^i z'2 • .*. It (xy) =\\.{ab — av^— It {b v^ + It {v^ v^ § 229. = ab^\i{x) It {y). In like manner the theorem is proved for n variables. THEORY OF LIMITS. 1 29 231. The limit of the variable quotient of two varia- bles is the quotient of their limits ; that isy \\.{x-ry) = \\.{x)-^\\.{y). Let z=^x-^y, Qxx^^yz; then \\.{z) = \i{x-^y), and lt(^)=:lt(>'0)=lt(j)lt(2). (i) Hence It (^) = It (^) -- It (/). (2) Whence It {x ^ y) ^ It {x) H- It {y). Remark. The demonstration given above fails, and the theorem is not true, when It(^), or the Hmit of the divisor, is zero; for then we cannot divide (i) by lt(j) to obtain (2). 232. Whc7i the product, quotient, or sum of two or more variables is equal to a constant, the product, quo- tient, or sum of their limits, is equal to the same constant, (i.) Let xy = m ; then xy z = mz. .-. \\.(x)\\.{)^\t{z) = m\l(z). .'. It (x) It (>') = m. (ii.) Let X -^ y = m ; then x=7ny. .'. It (.v) = m It (y), or It (x) -4- It (y) = m. (iii.) Let x+y+z + '" = m; then y + z -{-'•' = m — X. .-. It (y) + It (z) + ... = m — \t (x). .-. \t(x) + \t(y) + \t(z) + ... = m. I30 ALGEBRA. 233. If a is finite, and ;r = 0, then the fraction - X will numerically increase without limit. \{x increases without Hmit, then - = 0. X A variable that increases without limit is called an Infinite. The value of an infinite is denoted by the symbol oo , and ;t: — x is read '' x increases without limit," or '' x is infinite." With this notation the two statements made above may be written as follows : If jc = 0, then - r= DO ; X if ^ = 00, then - = 0. Example i. Find It ^^_. — -ttq.^ if ;»r=». *ExAMPLE 2. If ;r= 0, and « > 0, then It (^^) = i. Let a^ \, X positive, and k a positive number as small as we please ; then, as i -^ ;f = », ,'. 1 + k > a'' > I, or It (^^) = I. The proof is similar for a < i. For x negative we have ^z^ = (i -f a)"", in which — x is positive. In this example x is commensurable, and only the positive real values of «* are considered. THEORY OF LIMITS. 131 EXERCISE 20. 1. Prove It (at") = [lt(A:)]^ Lt (x'') =\t(x - X '" to n factors) = It (x) . It (^) ... to « factors = [it(^)]-. 2. Prove It U~'/ = [It (or)]". Let x*" = z; then xf" = z", etc. 3. Prove lt(^y = /;/ -H [It (x)^. If It (x) = a, It ( j) = ^, It (2) = c, and It (z;) = 0, find 4. Lt (^^y-j + axz). 5. Lt (a;2 ^? -{■ mx z^ + nxzv). ,i_ xy ■\- nx^ ■\- my f x"^ y^ ■\- ms^ -\- nzv\ \ xy ■\- nofi ■\- my v J Find the limit of each of the following expressions, (i.) when ;«: = 0, (ii.) when ;c = ao : 8. 2^* + 4 mx^ -\- nx^ -\- px -\- q' 10, (2 ^ - 3) (3 - 5 ^) 70:'^— 6x + 4 ''^* 2;c^— I ■ 2 x^ I. + 9 (3 + 2 ;c«) (a: -5) (4^= -9) (I + ^) I — ^' . I — X 132 ALGEBRA. 234. Vanishing Fractions. If in the fraction ir^ — ^ — we put X — a, it will assume the inde- terminate form - . A fraction which assumes this form for any particular value of x\ as a^ is called a Vanishing Fraction for X =^ a. To find the value of such a fraction for ;r= «, we find its limit when x = a. The limit of ^ o when ;ir = ^ is often x^ — a^ limit \x^ -\- ax — 2 a'^l written . o 2 • X = a I x^ ~ a^ \ ^ „. , limit \x^ + a X — 2a^'\ Example. Find . — —^ ^ . X = a \_ x^ — a^ J Here the limit of the divisor, jr^ — a^^ is zero, and we cannot apply § 231. But as long as x is not absolutely equal to a, we may divide both terms of the fraction by x — aj hence x"^ ^ a X — ia^ X -\- 1 a x^-a'^ x + a limit [ x"^-^ ax-2 ^^ _ limit p + 2 fz l _ 3 x= a\ x^ — a^ \~ x = a\ X -{- a \ 2 *235. Incommensurable Exponents. If di ts positive and m is incomtnejisurab ^e^ the positive real value of a"" is the limit of the positive real value of a" when X ~ m. Let X and jj/ be commensurable, and let X < m < y, \i{x) = m = It {y). Then, if ^ > i, we assume as axiomatic that a""' < a^, or «"" — a"" <, a^ — a". THEORY OF LIMITS. 133 But a^ — a'' = (f{a^-'' — i) = 0. Example 2, § 233. /, oT — It (t?') when a > \ and x = m. The proof is similar when a < i. This proof applies also when m is commensurable. Example. Prove the laws in § 70, when m and n are in- commensurable, a and b being positive. Let X and J denote any two commensurable fractions whose limits are m and n respectively ; then It («* a>) = It (^*+>) = «'" + ", also It {a^ay) = It («0 • It (^-») = a'^a\ Similarly the other laws are proved. Remark. Except when the limit of a divisor is zero, the limit of each function considered in this chapter is the result obtained by substituting for the variables their respective limits. EXERCISE 21. Find Limit r^r" — il X = iix — I y Limit [x^ -}■ 1*1 ^ = — I Ix^ — I J * Limit X = a (x - a)^ Limit [x* — ^*1 4. . — . x^=alx — aj Limit rx^ — a^l X = al X — a 1 \/a — \/x Limit X — a I ^a — X . In Example 6 rationalize the numerator. 134 ALGEBRA. CHAPTER XIII. DERIVATIVES. 236. The amount of any change (increase or decrease) in the value of a variable is called an Increment. If a variable is increasing, its increment IS positive ; if it is decreasing, its increment is neg- ative. An increment of a variable is denoted by writing the letter A before it ; thus A x, Ay, A z, denote the increments o{ x,y, z, respectively. Hence if x^ denote any value of ;r, x^ -\- A x denotes a subsequent value of x. \{ y is a function of x, and y = y^ when x — x^ ; then y =y + Ay^ when X = x^ + A X. 237. The Derivative of a function is the limit of the ratio of the increment of the function to the increment of the variable as the increment of the variable approaches zero as its limit. If ^ is a function of x, the derivative of y with respect to x is often denoted by D^y. Hence by definition, we have limit t^-^-y w DERIVATIVES. 135 Example i. Given j/ = ax^ + cx-{-l>, to find Z>^^. Let x^ and y be any two corresponding values of x and jy then y = ax'^ + cx' + d. (i) When X = x' + A X, y =/ + Ajy hence we have y + A_y = a(x' + Axy + c(x+ Ax) +d = ax'^ + 2ax'Ax + aAx^ + ex* + CAX+ d. (2) Subtracting (i) from (2), we obtain Af= 2ax'Ax+aAx^ + cAx, Dividing by A x, we have — ^ = 2ax* + aAx + c, Ax == 2ax' + c. Hence, as x' is any value of x^ we have in general, £>xy -D^{ax'^ ^cx-\-b) = 2ax-Vc. In case J/ is a linear function of x, as when j = ^;r, by the method above, or by inspection, we find that Ay -^ Ax =. a,2i constant. Here the ratio of A^ to A,r, being a constant, cannot approach a limit as A;ir = 0. In this case the deriva- tive o{ y is the constant ratio of Aj^ to A;r; that is D^y = D,(ax) = a, l{ a= I, we have D^^y = D^x — i. Thus, Z)^ (2 :r) = 2 ; D^{tix) = n\ D^{^cx) — -c\ D^x=i, 136 ALGEBRA. 238. The de7'ivative of a function is positive ornega- tive according as the function increases or decreases^ when its variable increases ; and conversely. For \i y increases when x increases, Ay and Ax have hke signs; hence, from [i] of § 237, i>^_;/ is positive. \i y decreases when x increases, Ay and Ax have unHke signs; hence Dj,y is negative; and conversely. EXERCISE 22 Find the derivative of _>', if \. y ■= x"-. 2. y = a x^ -\- b, 3. y — c x^ — a X, 4' y = x\ 5. y=zcx^ — ax\ Note. To the expression x"" — cx^ -\- 4.x. 4 6. y 7. y = ax 8. y = cx' — bx\ 9. y =^ a -^ X. 10. y = X -^ (a — x) limit Lax] Ax = {) LA:rJ^^^ cannot apply the principle of § 231, for the limit of the divisor, Ax, is zero. 239. Geometric Illustration of a Derivative, ceive a variable right triangle with constant angles as gene- rated by the perpendicular BC moving uniformly to the right. Let X denote the variable base, 2 ax its altitude, and y its area; then, by geometry, y = ax^. Let A B he any value of ;ir, and let Ax = B H; then t^y = zxediBHNC. a Con- DERIVATIVES. 137 Let MS join the middle points oi B H and C N; then, by geometry, 'QiQ3iBHNC=BHx MS. Ay zx^z.BHNC BHxMS . , „ ,, '-'-£= BH - BH =^^' W As A ,r z^ 0, MS =B C ; hence, from (i) we have limit rM'l limit \ms\ = BC; ,\ D^y — BC=2ax, §237. or D^ {a x^) = za x. 240. The sign of the operation of finding the de- rivative of a function with respect to x is D^. Thus, D^ in D^{x^) indicates the operation of finding the derivative of ,r^, while the whole expression Z)^ (x^) denotes the derivative of x^. We could obtain the derivative of any function by the method of § 237 ; but in practice it is more expe- dient to use the following general principles: 241. Tke derivative of equal functions of the same variable are equal. \{u 2CCiAy are equal functions of ;r, we are to prove that D, u = D,y, A« A^ If u=y,^u = ^y', '•^^ = ^ 138 ALGEBRA. Tj. limit r^'^l liii^it [-^yl n Hence, ^ U^^ 1^ . §227. .-. Z>^u =■ D^y. § 237. 242. The derivative of the product of a constant and a variable is the product of the constant and the deriva- tive of the variable. We are to prove that D^ {ay) — a D^y^ in which y is some function oi x. Let u— ay, and let x denote any value of x, and y and t/ the corresponding values of j^ and u respec- tively; then a = ay', (i) When x — x'-^Ax, then yz=iy' -\- Ay^ and u — u'-\-Au\ hence u' -\- Au — a (/ + Ay) = a/ + a Ay, (2) Subtracting (i) from (2), we have Au ■= a Ay. , Au _ Ay Ax~ Ax Ajv=OLA^J Ajt: = OLA::cJ limit r^ J^ R ^^Q = « • __^ . § 228. Aa:= OLa^vJ .-. D,u = aD,y. § 237. Hence, as x^ is any value of ;t-, we have in general D^ u = D^ (ay) — a D^y, Thus, A (3 ^ ^;r8) = 3 « 3 . Z7^ {x^y DERIVATIVES. 1 39 243. The derivative of a constant is zero. If ^ is a constant, A(a; = ; .'. — = 0, .*. D^a = ^. Ax 244. The derivative of a polynomial is the algebraic sum of the derivatives of its several terms. We are to prove that D^{v ^ y -\- z -\- a^ = D^v -f D^y + D^z, in which v, y, and z are functions of ;r. Let n — v +y + z + a, and let y represent any value of ;r, and v\y, z, and t/ the corresponding values of V, y, z, and «, respectively ; then u' = v' +y + z' + a. (i) When X = x^ + Ax, then v = v' + Av, y=y + Ay, z = si + Az, and u= u^ -\- Au; hence u' + Au = v' + ^v -{-/ + ^y + z' + Az + a. (2) Subtracting (i) from (2), we have Au = Av + Ay -\- ^z. . Au _Av Ay A z ' ' Ax~ Ax Ax Ax . limit [Aw]^ limit [^ , M' , ^1 §227 '*A:^ = 0LajcJ A^ = 0LA^ A^ Ax\ __ limit r^^l , limit TAjH limit ["A si I40 ALGEBRA. Hence, as x! is any value of x, we have in general = 3a'D (x^) -2n(x^) + sa. 245. T/ie derivative of the product of two variables is the first into the derivative of the second^ plus the second into the derivative of the first. We are to prove that D^ {y z) —y D^2 -\- zD^y, in which _y and s are functions of ;ir. Let u—yz, and let x represent any value of ;r, and y, 2^, and u' the corresponding values of y, z^ and Uy respectively; then u'^y'^. (i) When x = x' ^ ^x^ then ^ = / + A j, 2 = s' + A 2:, and u^u' -^ ti.u\ hence //+ A ?/=(/+ ^y) (/+ A ^)=//+y A2 + /Aj;4- A0 Aj. (2) Subtracting (i) from (2), we obtain A«=yA0 + '2;'A_y + A^r A J. . A u AJt: -^ A^ ^ ^ A^ limit A j; lit TA^/l limit r ,A^ (^, ^^^Aj-l = ^^"^^^ [y-l+ ^^"^^' r(. + A.)^i^l A.t=OL AjcJ A.t=OL^ A^J =y ^i"^it r^^l+ ^i^^t [z + Asl . ^'"^'^ F^l •^ Ax=OLA:rJ A^ = OL J A:^ = OLa;<;J /. D^u =y'D,z + 2' Dxy^ DERIVATIVES. I4I Hence, as x' is any value oi Xy we have in general D,u = D, {yz) =yD,z ^zD.y. 246. The derivative of the product of any number of variables is the sum of the products of the derivative of each into all the rest. We are to prove that D^ {vy z)—yzD^v-Vvz D^y + vyD^z, Let u = vy; then Dx {vyz) = A («^) § 241. = zD,u + uZ>xZ §245. = zDj, {vy) + vy £>xZ = yzD^v -\- vzJDj^y -\- vyZ>^z. §245. In a similar manner the theorem may be demon- strated for any number of variables. 247. The derivative of a fraction is the denominator into the derivative of the numerator ^ minus the numc- 7'ator into the derivative of the denominator^ divided by the square of the denominator. We are to prove that A \-]= J! — ^' ^^ which y and z are functions of x. Let « = -^ ; then uz=y. z .*• u D^z ■\- zD^u = D^y. §245. 142 ALGEBRA. D^y — uD^z _ n^y-^D^z .'.n^u=. 248. By §247, A (^l) _ zD,y-yD,z "" z^ z D^a — aD^z ^. § 243. Hence, the derivative of a fraction with a constant numerator is minus the numerator into the deriva- tive of the denominator divided by the square of the denominator. 249. By§.43.A@ = A(i.)=>., = ^. 250. The derivative of a variable base affected with any constant exponent is the product of the exp07ient into the base with its exponent diminished by one, into the derivative of the base. (i) If the exponent is a positive integer, ^s m\ then D^ {f) — D^iz-z-z -" \.o m factors) = 2"*-' D^z + z""'^ £>^z + '•' to m terms = mz'''~^ Z>^z. DERIVATIVES. 1 43 1ft (2) If the exponent Is any positive fraction, as — ; let u — z"\ then «" — z"\ .'. n ti""-^ D, u = m z'"-' D,z, .*. D^u = - — — D^z= D^ z m (3) If the exponent is any negative quantity, as — ;/ ; then^-"= -. . (0 By § 248, we obtain from (i) — — nz-"-^D,z. Thus,Z?.(;r8) = 3;i-2; Z?, (:rt) = f x? ; Z? (x-0 = -2^-3. 251. By§250,Z>.(V'<^) = A-(^) = J^~^A^=J^- Hence, /^^ derivative of the square root of a variable is the derivative of the variable divided by twice the square root of the variable. 252. The general symbol for the derivative oifix) is/' (.r) ; that is A [/W] =/' {x). 144 ALGEBRA. EXERCISE 23. Find the derivative of, I. X^ + SX + 2X^ D^{x^ + 8^ + 2x') = I^.(x')+DX8x)+jD^(2x') § 244. = 3x^ + 8 + 4x. §250. 2. /(x) = 3 ax^ — ^nx — 8 m. fix) ~D,{3ax''-^7ix-8ni) = A(3 «^') - ^.(5 «^) - A(8 m) ~ 6 ax — 5 «. f(x)= loax-gPx'' — 4adx\ 5. y = ax2+^x^-\rc. 6. /(x) = (d + ax^)i. 7' y=(i + 2X'') (i + ^ x^). In Example 8 D,y = (^ + ^)^-(^+^^)-(^ + a^)DAr+^) ^ b - a^ 10. y^{a^x)^Jl^^. II. /(^) = _1^. DERIVATIVES. I4S 13- /(*) = v/f 14- /W = 5 • '9- /W = (i-*' (1 + ^) 15. ^ = — ^=z=7^ • 20. f{x) = ^ax + ^/c^x^, 16. /(^) = ^4x~J* ^^- y = 2/^. ^ X -\- X In Example 21, Z);t(j^) = D^iip x),t\.z. 22. rt^_y^ -f ^2 ^2 _ ^2 ^2^ 23. 2 .v_>'^ — ay^ = x^. 24. Prove the theorems of §§ 246, 247 by the general method used in the previous articles. 25. Assuming the binomial theorem, prove the theorem of § 250 by the general method. 253. Successive Derivatives. Since f'{x)y the de- rivative of /(a'), is in general another function of jr, it can be differentiated the same asy(.i'). The deriva- tive oi fix) is called the Second Derivative of the original function /(jr), and is denoted hyf'ix). The derivative c>{ f'\x) is called the Third Derivative oi f\x), and is represented hy f"'{x) ; and so on. f"{x) 146 ALGEBRA. represents the nth. derivative of/(;r), or the derivative Thus, if /(x) = x^, then /' (x) = 4 x», f" (;f ) = 1 2 x\ f" {x) = 24 X, /'M^) = 24, /^W^O. /'W> /"W* /'"(-^). •••, /"W are called the 5?/^- cessive Derivatives of /(:i'). EXERCISE 24. Find the successive derivatives oi f{pc)^ when 1. /(^) = ^^ + 2 :r2 + ;r + 7- 3- /W = (^ + ^)^ 2. fix) =,cx^ ^ ax" ^b. 4. /(x) = (^ + xy. 5. /(^) = ^o+ A-^ + ^2^'+ -43-^'+ -^4-^'+ ^5^'+ ^,^T«. 254. Continuity. A variable is Continuous, or varies contimwiLsly, when in passing from one value to an- other it passes successively through all intermediate values. Otherwise it is disconti^mous. A Continuous function is one that varies continu- ously, when its variable is continuous. Hence ^ is a continuous function of x, if for each real finite value of ;ir, y is real, finite, and determinate, and \{ A y = when A X = 0. The time since any past event is a continuous variable, as is also the length of a line while being traced by a moving point. The velocity acquired by a falling body, and the distance fallen, are continuous functions of the time. DERIVATIVES. 1 47 The area and the altitude of the triangle in § 239 are contin- uous functions of the base. The number of sides of a regular polygon inscribed in a circle, when indefinitely increased, is a discontinuous variable, as is also the perimeter or the area of the polygon. For each of these variables passes from one value to another without passing through all intermediate values. In general i -^ ;r is a continuous function of .ry but when x increases and passes through zero, i ^ x leaps from — » to + ao; hence i -f- x is discontinuous for x—^. 255. Any rational integral function of x is con- tinuous. Let « be a positive integer, and y = A^x" + A^x'—"^ H h A„_^x + A,„ then for each real finite value of x^ y has one real finite value, and only one. Again, if by the method in Example i of § 237, we obtain Ay in terms of Ax, Ay will be found equal to A X multiplied by a finite quantity ; hence J j/ = when Ax = 0. Therefore y, or A^x" + A^x"''^ + *,. + A„, is a continuous function of x. Thus, if J/ = x^ + 2x'^ + X, (i) Aj = (3^^+ 3^'^^ + t:^^''^ + 4^ + 2 Ax + i) A;t-. (2) From (i),/has one finite real value for each finite real value of .1', and from (2), Ay = when A :r = 0. Hence x^ + 2x^ + x is a continuous function of x. 148 ALGEBRA. CHAPTER XIV. DEVELOPMENT OF FUNCTIONS IN SERIES. 256. A Series is an expression in which the suc- cessive terms are formed by some regular law. A Finite series is one of which the number of terms is limited. An Infinite series is one of which the num- ber of terms is unlimited. 257. An infinite series is said to be Convergent when the sum of the first n terms approaches a limit as n is increased indefinitely ; and the limit is called the Sum of the infinite series. If the sum of the first n terms of an infinite series does not approach a defi- nite limit when ;^ = », the series is Divergent^ and has no sum. Thus, the infinite geometrical series I + - + - +o + --- + -^ + --- 248 2"-l is convergent; for by §211 the sum of its first n terms ap- proaches 1-^(1— ^), or 2, as its limit, when ;/ = x. The series i + i + i + i+---is divergent, since the sum of its first ti terms increases indefinitely with ;/. The series i — i + i — i+---is divergent; for the sum of its first n terms does not approach a limit, but alternates be- tween I and according as n is odd or even. DEVELOPMENT OF FUNCTIONS IN SERIES. 1 49 258. To Develop a function is to find a series the sum of which is equal to the function. Hence the development of a function is either difi7iite or a co7i- vergent infinite series. Thus, the development of the function {a + xy is the finite series a* + ^a^x + 6a^x^ + ^ax^ + x*. 259. Development of Functions by Division. I — X** Example i. Develop by division. Dividing i — jr" by i — ;r, we obtain I -X** I —X = l + x + x^ + x^+...x'*-K (i) If n is finite, the series in identity (i) is finite and is the de- velopment of the function for all values of x. Example 2. Develop by division. I — X Dividing i by i — x, we obtain I I —X = i+x + x^ + x^ + ... + x»-'^ + ..., (l) in which x"-"^ is the «th or general term of the series. The series in (i) is infinite, and hence it must be convergent to be the development of the function. If X is numerically less than i, the series is evidently a de- creasing geometrical series, of which the first term is i, and the ratio xy hence by § 211 the sum of ;/ terms approaches I — X as a limit when « = ». 150 ALGEBRA. If X is numerically greater than i, the sum increases indefi- nitely with nj hence the series is divergent, and is not the de- velopment of the function. Thus, for or = 2, the series becomes I + 2 + 4 + 8+ 16 + ..., while the function equals — i. \i x= I, the sum of the first n terms is n, and therefore the series is divergent. If :ir = — I, the series becomes the divergent series I - I + I - I + I -I + ... Hence the series in (i) is the development of only for values of ;r between — i and + i. Example 3. Develop — — by division. I + X Dividing ;r by i + A", we obtain -J^ = x-x^ + x^-x^ + x^ + (-i)«-i;f'' + ... I + X Here the series is evidently divergent for all values of x except those between — i and + i. For values of x between — I and + I the series is a decreasing geometrical progression of which the first term is x and the ratio is — xj hence the sum of n terms approaches ■ as its limit, when ?i = 00. Principles of Undetermined Coefficients. 260. Undetermined Coefficients are assumed coeffi- cients whose values, not known at the outset, are to be determined in the course of the demonstration of a theorem or the solution of a problem. DEVELOPMENT OF FUNCTIONS IN SERIES. 15I 261. If ^0 = Bo, Ai = Bj, A2 = B2, Ag = B3, ..., theft Ao + A,x + A2x2+ . . . =Bo + B,x + B2x2+ . . .;(i) that is, if in tJie two members of an equality the coeffi- cients of the like powers of y. are identical^ the equality is an identity. For by hypothesis we have the identities Aq = Bq, Ai X = Bi X, Azx'^ = B2x\ • • . Adding these identities, we obtain the identity (i). 262. Conversely, if A, + A,x + A^x^ + ... = ^0 + ^1^ + ^2^' +•••, (i) then Ao = Bo, Ai = B^ . » -; that isy in an identity, the coefficiejits of the like powers of the variable in the two members are identical. For Aq and ^0 ^^e respectively the limits of the equal varying members of (i), as ;r = ; hence A, = B,. (2) Subtracting (2) from (i), and then dividing by ;ir, we obtain A^ + A^x + A^x" + ... = ^1 + B^x + B^x^ + ... (3) l( X = 0, from (3) we obtain A,=B,. In like manner we may prove A2 = B2, A3 = Bs, . . . 152 ALGEBRA. 263. Development of Functions by Undetermined Coefficients. Example i. Develop I -x^ Clearing (i) of fractions, and for convenience writing the coefficients of the like powers of x in vertical columns, we obtain x^^A^^A^ x + A^ x'+A^ x' + Ai + A + A, + A, --^o -^1 -A, X^+... (2) In the first member of (2), the coefficient of any power of x that does not appear is zero. Equating the coefficients of the like powers of x in the two members of (2), we obtain ^0=1,^0 + ^1=0, ^2 + ^i-^o=-i» ^3 + ^2-^1 = 0/ y^4 + y^3-^2 =r 0, ... , A„ + An-l-An_2 = ^'. Solving the system of equations (3), we obtain A, , A^ — I, Aq _ — 2, A^ ,A„ = A„_2-A„_i. As = 3, Substituting these values o( Aq, A^, . > . , in (i), we have I -X' I +x = 1 -x + x' 2x^ + :^x* + (3) (4) (5) which is an identity for such values of x as render the series convergent. The values of Aq, A^, A^, "-, A„, given in (4), render (2) an identity; for they satisfy equations (3), and therefore render the coefficients of like powers of x in (2) identical. Now if DEVELOPMENT OF FUNCTIONS IN SERIES. 1 53 (2) is an identity, then (i), or (5), also is an identity for such values of ;ras render the series convergent (§ 259). The law of coefficients of the series in (5) is A„^A„_^-A„_^. (6) By (6) the series can be readily extended to any number of terms. Thus, ^6 = /?8 - /^4= - 2 - 3 = - 5, ^6 = ^4-^6 = 3 + 5 = 8, . . . I Example 2. Develop :^-x^ Assume ^A^x-'^^A^ x-^ + A^ + A^x + x^-x^-x* Clearing (i) of fractions, we have 1= Aq + A^ -A, x + A.^ -A, -A. x^ + A, -A, -A, x^ + A, -A, -Ao x* + (0 (2) Equating the coefficients of like powers of x in (2), we have Aq — i» A I — A, 0-0, A^-Aj^-A^ = 0, A^-A^-A^ = 0^] A^-A^~A., = 0,...,A„- A„_i - A„_, = 0.]}^^ '. ^0= I, ^1= I, ^2 = 2, ^8 = 3, ^4=5»-" (4) Here the taw of coefficients is A„ = A„_i 4- A„^2. Substituting in (i) the values in (4), we obtain _^_i^-- =^-2 + ^-1 + 2 + 3^+5^24...., (5) which is an identity for such values of x as render the series convergent. [Let the student give the proof.] Note. The form assumed for the series must in each case be such that when the equality is cleared of fractions, no power of X will appear in the first member which is not also found in 1 54 ALGEBRA. the second. For otherwise, the system of equations obtained by equating the coefficients of the hke powers of ;r will be im- possible. For example, let us assume x^ — x^ — x^ ^ Clearing of fractions, and equating the coefficients of x^, we obtain the absurdity i = 0, which shows that we have assumed an impossible form for the development. By the laws of expo- nents in division we know that the first term of the series will contain x~^ \ hence we assume the form in (i). EXERCISE 25. Develop the following functions by the principle of unde- termined coefficients, and verify the results by division : 1 -\- X x^ -{■ x^ -\- -i ^' y V 4- T : — z 2. I + 2 X — 2>^^ I — 3 ^2 X^ -\- 2 Jf ^ + t 6. I + ^ + X' I + X 2 x:^ + 3 x^ 1+^^ -^ i~ X -Y x^ x^ -{■ 2>^^ Resolution of Fractions into Partial Fractions. 264. In elementary Algebra a group of fractions connected by the signs + and — are often united into a single fraction whose denominator is the lowest common denominator of the given fractions (§ 64). DEVELOPMENT OF FUNCTIONS IN SERIES. 1 55 The converse problem of separating a rational frac- tion into a group of simpler, ox partial, real fractions frequently occurs. The denominators of these par- tial fractions must evidently be the real factors of the denominator of the given fraction. These real factors may be I. Linear and unequal. II. Linear and some of them equal. III. Quadratic and unequal. IV. Quadratic and some of them equal. To present the subject as clearly as possible, we shall consider these cases separately. 265. Case I. To a lijiear factor of the dejtomina- tor, as X — a, there corresponds a partial fraction of the form • X — a Example. Resolve ^—^ — into partial fractions. x^ + X* — 2X . 2X+ 3 A , B , C . . Assume -— — ^- — - = — + + — ■ — . (i) X {x — I) (x -j- 2) X X— I x+ 2 Clearing (i) of fractions, we have 2X+ 3 = A (x-i) (x+2) + B(x+ 2)x + C(x- i)x (2) = (A + B+ C)x^+ (A -^2B- C)x-2A. (3) Equating the coefficients of like powers of :r in (3), we have A+B+C=0, A + 2B-C=2, -2A = 3. • (4) 156 ALGEBRA. Solving equations (4), we find A = -l£ = i, C=-i. (5) Substituting these values in (i), we obtain --^^:±i_ = -J- + _5 ^ (6) x^ + x^ — 2X 2x 3(;^r— l) 6(;f+2)* The values o{ A, B, and C given in (5), render (3) and there- fore (i) an identity; for they satisfy (4), and therefore render the coefficients of like powers of x in (3) identical ; hence (6) is an identity. If we assume that (2) is an identity, the values of Ay B. and C may be obtained as follows : Making x = 0, (2) becomes $ = — 2 A ; .♦. A = — ^. Making x= i, (2) becomes 5 = 3 ^y .-, B= f. Making x = — 2, (2) becomes — i = 6 Cy .-. C = — ^. 266. Case II. To r e^iml linear factors of the de- nominator as (x — b)', there corresponds a series of r partial fractions of the form {x - by ^ (jc - by-' ^ ^ x-b Example. Resolve 7 t^t — \ — ^ into partial fractions. i^x — i)^ (;r + I) ^ Assume T A B C ^ + :;73T + :;ri-r- (0 {x _ 1)2 (;f 4. I) (;r - 1)2 ' jr - I ' ;ir Clearing (i) of fractions, we have \=A{x^-\)-^B{x-\){x-V\)-\-C{x~\y- = iB + C) x^ + {A - 2 C) X + A - B + C. (2) DEVELOPMENT OF FUNCTIONS IN SERIES. 1 5/ Equating the coefficients of like powers of x, we have B+C=0, A-2C=0, A-B-i-C^i. (3) Hence, A = :^, B = - \, C = \. (4) Substituting these values in (i), we have I I I (5) (x-iy{x+ I)- 2(;r-i)« 4(1--^) 4(-^+i) Equality (5) is an identity; for the vahies o( A, B, and C given in (4) satisfy (3), and hence render (2) and therefore (i) an identity. 267. Case III. To anjy quadratic factor of the de- nomhiator, as x^ + p x + q, there corresponds a par- . r . . . r . Ax + B tial fraction of the form -^—, n •^ -' -^ x^4-px + q x^ Example. Resolve -^— — ^Z^ into partial fractions. Assume £2 Ax^B C D (jr2+2)(;r+ i)(;i'- I) ~ jt* + 2 "^ :r + i "^ ;r - i* ^^^ Clearing (i) of fractions, we obtain ;i-2 = (^;r + ^)(jr2-i)+C(;f2+2)(;r-i)+Z^(;r2+2)(;r+i) = {A^C-VDy^{p-C^B)x'^^(:2.C-\-iD-A)x^2D-zC-B. (2) Equating coefficients of like powers of ;r in (2), we have (3) 2C+2Z>-/?=0, 2Z>-2C-i? = 0. ' ^^^ Whence, /^ = 0, ^ = f, C= - ^ Z? = J. (4) Substituting these values in (i), we have X"- _ Z I T j^+2;r-2-3(:r2+2) 6(;ir+ I) +6(;ir- I)* ^^^ (5) is an identity for the same reason as that given above. 158 ALGEBRA. 268. Case IV. To x equal quadratic factors of the denominator, as (x^ + p x + q)'^, tJiere corresponds r partial fractions of the form Ax + B Cx + r> Lx-\-M {x^^px-\-qY {x'^ -\- p X ^ qy-^ x^-\-px-^q In any example under this case, by clearing the assumed equation of fractions and equating the co- efficients of like powers of ;r, we would, as in the first three cases, evidently obtain as many simple equa- tions as there are undetermined quantities, and the values of A, B, C, ..-, M, thus determined would make the assumed equality an identity. 269. In what precedes, the numerator is supposed to be of a lower degree than the denominator. If this is not the case, the fraction can be sepa- rated by division into an entire part and a frac- tion whose numerator is of a lower degree than its denom'nator. For example : x^ _ 5 ^' - 4 X^ -\- 1 X'^ - X ~ 2.^ ^ ^'^ X^+2X^-X-2' A 5 ^'-4 ^ ^6 _ I „_i ^^^ X^-h2X^~X-2~ 3(x-^2) 2 (;f + I) "^ 6 (^ - I) X* , t6 I I • ;r3 + 2;r2-:r-2-'^ ^"^3(r+2) 2(.r+i) 6(j--i) DEVELOPMENT OF FUNCTIONS IN SERIES. 1 59 EXERCISE 26. Resolve the following fractions into partial fractions x^-2 3 ^^ - 7 a: + 6 7- (^-i)a X— X" X^ I « X" -V 2X :r + 3 23 ;c — 1 1 ^ '^ ^* i^* + "x - 2 * ^* (2 ^ - i) (9 - ^O ^ +3^+2^ ' i8jc^+ 12 .y — 3 4* (r=- 2 :r) (i - ^^) * '''• (3^+2)« 2 JC''— 12^** — SjC+ 2 4 2 — \^X 5- :^^-5^^ + 4 "• (^+'0^-4)* 6. . :9__. X^ -\- X — \ x^ + x-i Ax-\-B Cx + D Assume -^-T^:^ =^ (P^T^^ + ^^^^TV * 2:r*— iiJf+5 x^ — 2 X -\- ^ ^3- (^2_^2^_3)(^_3)- H- -^2-^77)2- * 270. Reversion of Series. Given y^=zax-{-bx'^-\-cx^-{--"f (l) the series being convergent, to express x in terms of y ; that is, to revert the series. Assume a: = ^^ + ^/ + C/ + ••• (2) i6o ALGEBRA. Substituting in (2) the value ofj/ given in (i), x^ -\- Ac + 2Bab x^ + (3) Equating the coefficients of Hke powers o{ x in (3), Aa=^i, Ab -\- B a^^Q, Ac+ 2Bab + Ca^ = 0, ... b „ 2b'' Hence ^ — -, B = , C^ ac or a" Substituting these values in (2), we obtain a c I b ^^2b''- f- which is the result sought. If the series to be reverted be of the form y ^ aQ-\- ax ■\- bx^ -{■ cx^ + •••, we express x in terms of j^ — a^. Example. Revert the series y — 2 + 2X — x'^ — X^+2X^-\---. From (i), 2 = 2X — X^ — X^+2X^ + " Assume x= A {y - 2) + B {y - 2y + C {y - 2y + Substituting in (3) the value oi y — 2 given in (2), x= 2 A X — A x^-A X^+2A + 4B . -4B -3B + 8C -12C + 16D (0 (2) (3) DEVELOPMENT OF FUNCTIONS IN SERIES. l6l Equaling coefficients of like powers of ;ir, we obtain 2A = i, 4B-A=0, SC-4B-A=0, ] i6Z>- i2C-3^ + 2^ = 0, ...J Hence, A=\, B = \, C = \, Z^ = A*-- Substituting in (3), we have EXERCISE 27. Revert the following series : 1. y = X -\- x^ -\- x^ -\- •'» 2. y = X — 2x^ -{■ $x^ — •" 3. y = x-ix^+ ^X^ 4. j=i +x-hi^+h^' + ih^'+ '•' 5. y = x + sx^ + s^'^ + 7^* + "' 6. y=2X + s^^ + 4^^ + 5^''-^ Maclaurin's Formula. 271. Mac/a?/rm's Formula is a formula for develop- ing a function of a single variable into a series of terms arranged according to the ascending powers of that variable, with constant coefficients. l62 ALGEBRA. 272. To deduce Maclaurin s Formula. We are to find the values o{ A^, A^, A^, ..., when /(;r) can be developed in the form /{x) =A, + A,x + A^x' + A,x' + A,x' + ... (i) in which Aq, A^, A^, ••., are constants, the series being finite, or infinite and convergent. Finding the successive derivatives of (i), we obtain /i(x) = A, + 2A,x + sA,x''+ 4A,x^ + ... (2) /" {x):=2A, + 2.s A,x + 3 .4A,x' + ... (3) /"'{x) = 2.sA,+ 2.S'4A,x + ... ^ (4) /-(^) = 2.3.4^4 + - (5) Let x=0', then from equations (i) to (5), we obtain /(O) = A„ /> (0) = A, , /" (0) = 2A„ /"'(0) = \3A„ /-(0) = [4^., ... Solving these equations for ^0. ^1 > ^2, •••, we have /" (0) A„ =/(0), A, =/' (0), A, = 11 ' A _ /"'(0) . _/"(0) . . _ /'-'(Q) Substituting these values in (i), we obtain f{x)^A^)^f\^)x\f\^f^^^ (6) which is the required formula. DEVELOPMENT OF FUNCTIONS IN SERIES. 1 63 This formula, though bearing the name of Mac- laurin, was first discovered by James Stirling in the early part of the last century. The Binomial Theorem, Logarithmic Series, Ex- ponential Series, and many other formulas, are but particular cases of this more general formula. Binomial Theorem. 273. The Binomial Theorem is a formula by which a binomial with any exponent may be expanded in a series. Its general demonstration was first given by Sir Isaac Newton. It was considered one of the finest of his discoveries, and was engraved on his tomb. 274. To deduce the Binomial Theorem, To do this we develop {a + jr)'" by the formula f{x) =/(0)+/'(0).r+/"(0)^+...+/"-H^)£^V... (i) Here f{x) = (^ + .r)- ; .-. /(O) = a"^, fix) = m{a^ or)'" -^ ; .-. /'(O) = m rz'""'. f"(x) = m{m-i){a + x)—^; .'. f"{0) = m{m-i)a"^-\ f"(,\) = m {m — 1) {m — 2) {a + x)'"-^ ; ' .-. /'<'{{)) =m(m-i) (m - 2) a"'-\ /"-i (x) = m (m — j)...(m~n+ 2) (a + x)"" -« ^ 1 ; ••• /'"' (0) = m (m - i) ... (m - « + 2) a'"-- + \ 1 64 ALGEBRA. Substituting these values in formula (i), we have 7nim~\) „ „ m(m—i)bn~2) LI lA + . . . 4- — «"* - " + 1 ;»;" - 1 + . . . , in which the last term is the /^th, or general, term of the formula. Example. Find the 6th term in the expansion of (.r^— ^^)~^. Here n = 6, a = x^^ x — —b'^, and w = — | ; hence ;;/ — « + 2 = — \^. Substituting these values in the «th term of the for- mula, we obtain 6thterm = ^^lHzlK=lK^V)(z^(,.)-|-5(_,i)^ 1.2.3.4.5 ' ^ ^ ^ "^ ^ 729 275. By an inspection of the Binomial Theorem we discover the following laws of exponents and coeffi- cients, which are very useful in its applications: (i.) The exponent of a in the first term of the series is the same as that of the binomial, and it decreases by unity in each succeedifig tei^m. (ii.) The exponent of x is unity in the second term, and increases by unity in each succeeding term. (iii.) The coefficient of the first term is imity, and that of the second is the expottent of the binomial. DEVELOPMENT OF FUNCTIONS IN SERIES. 1 65 (iv.) If i?t any term the coefficie^it be multiplied by the exponent of a, a7id this product be divided by the exponent of x increased by unity, the result will be the coefficietit of the next term. Example i. Expand (^+3^)*- Here a-c, x=3y, m = 4; = c* + 12 ^V + 54 ^V + 108 cy^ + Siy*. (2) In the series in (i) the exponent of c in the 5th term is 0; hence by (iv.) the 6th term is 0, and therefore the expansion consists of 5 terms. Example 2. Expand («2 _ c'^) "i or [«2 + (_ c^)y^. Applying the laws given above, noting that here a — n\ x = — c% and m = —^, we have = n~^ + ^n~^c^ + ifr'^c* + ^{n~^c^ + ". (2) By(iv.) the coefficient of the 3d term in (i) is (— ^) (— |) -T- 2, or f ; that of the 4th term is f (- f ) -^ 3, or — |*, etc. In (i) the 2d term has two negative factors; the 3d, two; the 4th, four, etc.; hence the signs of all the terms in (2) are -f . This development could be obtained by substituting ;/2, — f2, and — \, respectively, for a, x, and w, in the formula, but the process would be longer. In the series in (i), the exponent of w'^ cannot be in any term ; hence no term can have as a factor of its coefficient, and thus vanish. Therefore the expansion is an infinite series, and equals the function only when convergent. 1 66 ALGEBRA. 276. When m is a positive whole nnmber, the bino- mial series is finite and consists ^ m + i terms ; when m is fractional or 7iegative, the series is infinite. For when m is a positive whole number, the expo- nent of a in the {^n + i)th term is 0; hence by (iv.) of § 275 the {7n + 2) th term and all succeeding terms are 0. Therefore the series consists oi m ^- i terms. But when m is fractional or negative, the exponent of a cannot be in any term ; hence no term can have as a factor, and the series is infinite. Thus, the expansion of {x^yy^ is a finite series of 14 terms; while the expansion of (:r 4-j)'^ or of {x ^ y)-"^ is an infinite series. 277. When m is a positive zvhole nnmher, the coefii- cients of terms equidistant from the begimiing and e7td of the expansion ^ (a + x)'" are equal. For m{in—\) „ „ W {a+ xy = a"" + m a"^-^ x+^^ — a»'-^x^+... + T^x'», (i) {x+ ay ^x'^ + m x*"-^ a + ^^ — ' x'^'-'^aP'^ '"'^f^^"'' ^^^ Now the series in (2) has the same terms as the series in (i), but in reverse order; whence the proposition. Hence, in expanding any positive power of a binomial, after we have computed the coeflficients of the first half of the series, the remain- ing coefficients are known to be those already found written in reverse order. II 12 DEVELOPMENT OF FUNCTIONS IN SERIES. 1 6/ EXERCISE 28. 1. If m is a positive integer, what is the sign of the even terms in {a — xY ? Why ? 2. Write out the expansion of (i + a:)'". Expand ID. {\-\-df, 13. {a~^-2b''c^. 14. (i — AT^y. 24. (9 + 2 jc)^. 25. (4^-8^)-*. 26. (r^«-2 _^2^-§)-i I 27. 28. 6- 4- 5- (^ + by. 6. (2- ixy. 7- {r-^ -**r. 8. {r^- 3«-^r. 9. (r 2x) • Expand to five terms 15- (3 + x>)i. 16. (8 + 12 «)^. 17- {y + .)-». 18. + a:^)-^ 19. G- 3-)*. 20. G- 3-)-4. 21. («^ + .i)?. 22. (- d^)-K 23- (/i- -c-i)-i. a/x" -f I ai- -i-h a '^- (,^-f_^,v-^^)^ 1 68 ALGEBRA. Find 30. The 4th term of (x — 5)^^. 31. The loth term of (i — 2 x)'^. ( by 32. The 5th term of \2 a J . II. The 7th term of f — ^^-j . 34. The 6th term of (^^ - c'x')-^. 35. The 5th term of (^"^ + e~^)~^, 36. The 7th term of {a^ — b-^~^^. 37. The 6thterm of (^~^ — dt^^*)"^ 278. To find the ratio of the (n ■\-\)th term to the nth. Substituting n -\- \ for n in the ?/th term of the bi- nomial theorem, we obtain as the (// + i)th term, m{m- i)(m-2)...im-n+ i) ^^_„^« Dividing this by the nth. term, we obtain (m — n + i\x fm + I \ x , >. ^— )-, or --^^ I -, (i) n J a \ 71 ) a as the ratio sought; that is, (i) is the quantity by which we multiply the n\\\ term to obtain the next term. DEVELOPMENT OF FUNCTIONS IN SERIES. 169 This ratio affords the following simple proof of the principle in § 276: When m is a positive integer, this ratio is evidently- zero, for it — ni-\- I ; hence the (;// + 2)th term and all the succeeding terms are zero, and therefore the series consists of ;« + i terms. But when ;« is fractional or negative, no value of n (ft must be integral) will make the ratio zero ; hence no term can become zero, and the series is infinite. 279. Any root of a number may be found approx- imately by the Binomial Theorem. Example. Find the approximate 5th root of 248. ^^ = (243 + 5)^ = (3' + 5)* (I 22.3 \ = 3(1+ 0.0041 152 - 0.0000338 + 0.0000004 - • • •) = 3.0122454, which is correct to at least six places of decimals. 280. Expressions which contain more than two terms may be expanded by the Binomial Theorem. Example. Find the expansion of {x^ + 2 or — i)*. Regarding -zx — \ as a single term, we have [;r2 + (2:r- i)P = (.1-2)8 + 3(:r2)2(2;r- i) + 3a-2(2;t—i)2+(2;r- 1)8 = x^ + 6x^ + ()x^—^x^—^x'^+6x—i. I/O ALGEBRA. EXERCISE 29. Expand and write the «th term of I. (i_^)-i. 2. {i-x)-\ 3. {i-x)-\ Find to five places of decimals the value of 4- V'ai- 6. v^^. 8. ^^2400. Find the expansion of 10. (1 + 2^ — ^^2)4. II. {^x'^— 2ax -^ :^a'^Y. Find the /zth term of the expansion of 4 14. 3(1 — 2^) (2 — ^)^ 5 _ 3:^2 + ^—2 ^^' 3(2-^)* '^' (^-2)2(1-2:^)' By§266,-l£^±-£ll^ = _ ^ +-5 -1- •^ ^ '(;r-2)2(r-2;r) 3(i-2;r) 3(2-;r) {2-xf Hence, by Examples 12, 13, and 14, the «th term is V 3 3 2" 2"-V I + llJC+28^=^ '' (i — Jt:^) (i — 2a:) Expand to four terms in ascending powers of x (2 + ;,) (i _ :r) • ^- (x - I) (^2 + i) * CONVERGENCY OF SERIES. I71 CHAPTER XV. CONVERGENCY AND SUMMATION OF SERIES. 281. An infinite series is divergent, if the nth term does not approach zero as its limit when « = x. For if the nth term does not approach zero when ;/ = x, the sum of n terms cannot approach a hmit. Thus, the series -_:^_^4_5^._ ^ 'i±_L ^ . . . js diver- 1234 « gent ; for the «th term approaches unity and not zero as its limit. A series may be divergent even though the «th term approaches zero as its limit when 7i = x. Example. Show that the harmonic series I + - + -H [-...-f_ + ... is di versrent. 234 n If after the first two, the terms of this series be taken in groups of two, four, eight, sixteen, etc., we have Each parenthetical expression is evidently greater than ^. Regarding these as single terms of series (i), the sum of 7n terms is greater than ^m. But w increases indefinitely with n. Hence the series is divergent, although its «th term = 0, when 172 ALGEBRA. 282. The following three important principles are almost self-evident : (i.) An infinite series of positive terms is conver- gent, if the sum of its first n terms is always less than some finite quantity, however large n may be. For as the sum of 71 terms must always increase with fly but cannot exceed a finite value, it must ap- proach some finite limit. (ii.) If a series in which all the terms are positive is convergent, then the series is convergent when some or all of the terms are negative, (iii.) If, after removing a finite number of its terms, a series is convergent, the entire series is convergent; if divergent, the entire series is divergent. For the sum of this finite num- ber of terms is finite. 283. An infinite series ift which tJie terms are alter- nately positive and negative is convergent, if its teiiris decrease numerically, and the limit of its nth terin is zero. Let the terms of the series be denoted by n^, — n^, u.^y . . . y and their sum by s ; then s — u^ — u^Ar u^ — u^-\- u^ . ± u„ :f ... (i) Since u„ = when ;/ = x, the sum of the series is evidently the same whether we take an even or an odd number of terms. CONVERGENCY OF SERIES. 1 73 Now (i) may be written in the form s = ii^ — («2 — u^) — (//^ — u^ (2) or s = (//, - //2 ) + ('^3 - '0 + ('^5 - ^O + • • • (3) Since «i > //^ > «3 > ••• > ^«> the expressions u^ — «2> ^2 — ^'sj ^3 — "4> • • • 2ire all positive. Hence from (2) we know that s < u^; therefore by (i.) of § 282 the series in (3) is convergent. lyi Thus, the series ^ humrvM^ ^ I I I I I \ ^ V' I 1 1 ±-T..« IS convergent. 2^3 4 5 « If we put this series in the forms -e-^e-;)---'(-i)*(i-^ we see that its sum is less than i and greater than ^. 284. Ah infinite series is convergent if the ratio of each term to the preccdirig (erni is less tJian some fixed quantity that is itself numerically less than unity. Let all the terms be positive ; then f = Wj + //g + //g + ^'4 H = „.f, + '^ + «» + «. + ...) \ U^ «1 «i / (rd- 174 ALGEBRA. Let ^ be a fixed quantity less than i, but greater than any of the ratios — > - ^ — > • • • ; then from (i) 5 < U,{y + ^ + ^2 _^ ^3 _^ . . .)^ or •$■ < 2^1 T > a finite quantity. § 259, Ex. 2. Hence by (i.) and (ii.) of § 282 the series is conver- gent whether its terms are all positive or some or all negative. 285. An infinite series is divergent if the ratio of each term to the preceding term is numerically equal to or greater than unity. For if this ratio is unity, or greater than unity, the /2th term cannot approach zero as its limit, and the series is divergent by § 281. 286. In the application of the tests of § § 284, 285, it is convenient to find -^^' ; let this limit be n — y^V u^ J denoted by r. If r < I, the series is convergent. § 284. If r > I, the series is divergent. § 285. \i r — I, and u^^^~ u^ > i, the series is divergent by § 285 ; if r = T, and u„^^ ~- u„ < i, the test of § 284 fails, and other tests must be applied. CONVERGENCY OF SERIES. 1 75 Example i. For what values of;iris the logarithmic series x^ x^ X* X" jr" + i convergent? Here ''-* M = '™'' [f ' _ ,U = _;.. Hence, if :r < i numerically, the series is convergent. If jr > I numerically, the series is divergent. If X = I, the series is convergent by § 283. If jr = — I, the series becomes — (i + ^ + J + •••)» and is divergent by Example of § 281. Hence (i) is convergent for jr = i, or .r > — i and < -f I. Example 2. When the binomial series is infinite, for what values of x is it convergent? Here '™" [^^'1 = '™" r(^^'_Afl = ^- §278. Hence, ii x rt numerically, the series is divergent. U x = a numerically, the test of § 284 fails. Thus, the theorem will develop (8 + 2)2, but not (2 + 8)2, Hence when m is fractional or negative, the binomial theo- rem will give the development of (a + x)"* or (x 4- «)'", ac- cording as ;r < or > a numerically. li x = a numerically, {a + xy^ becomes (2^)"' or 0'", and the formula is not needed. Example 3. For what values of x is the series p+^ + ^ + ^.+ ••• + -. + •••convergent? 1 76 ALGEBRA. (i.) If ;ir> I, the first term is i; the sum of the next two 2 terms is less than — ; the sum of the next four terms 4 is less than—; the sum of the next eight terms is 8 less than ■^; and so on. Hence the sum of the series 2 A R is less than that of i-i -l^-l— _|_..., 2^ ^ 4^^ ^ 8-^ ^ ' which is a geometrical progression whose common ratio, 2 -^ 2*', is less than i ; hence the series is convergent. (ii.) If ;ir = I, the series is the harmonic series, and is diver- gent by Example of § 281. (iii.) If JT < I, each term is greater than in case (ii.), and therefore the series is divergent. EXERCISE 30. Determine which of the following series is convergent and which divergent : 2' 3' 4' i ^ 11 5- ■^+- + - + - + - 234 SUMMATION OF SERIES. 1 7/ «- + ^^^ + ^» + 1.2 2-3 3-4 8. + + + + 1-2 2-3 3-4 4.5 X X^ X^ X* 2^ 3* « SUMMATION OF SERIES. 287. The Summation of a series is the process o< finding an expression for the sum of its first ;/ terms. Formulas for the sum of the first ;/ terms of an A. P. and of a G. P. were obtained in Chapter XI. We proceed to deduce formulas for the sum of other series. Recurring Series. 288. When the «th term of the series t^l + «2 + //g + 7/^ + 1- u„_, + u„ is connected with the m preceding terms by a rela- tion of the form the series is called a Recurring Series of the mth 178 ALGEBRA. order. The multipliers pi, p^, • • ',p^ remain unchanged throughout the series. In the G. P. i + 2;ir + ^x'^ + ^x^ + • • •, Un — ^^' ^h—l', hence this series is a recurring series of ihtjirst order, in which 2;r is the multiplier. In the series i -\- 2 x -\- Z x'^ + 28 r3 + ioo;ir4 + ... (i) we have Un — Z^-'^n-i-\'^^'^''^n-2' (2) Thus 2>x'^ = ^x.2x+2x^'i. and 28 ;r8 = 3 ;ir . 8 y-^ + 2 ^"^ • 2 :r. Hence series (i) is a recurring series of the second order in which 3 X and 2 jr^ are the /w^ multipliers. The series i + 3 + 7 + i3 + 2i + 3iH is a recurring series of the third order, in which the three multipliers are 3, —3, and I. Thus 31 = 3 X 21 - 3 X 13 + 7- 289. If we have given the m muhipliers of a recur- ring series of the mth. order, any term can be found, if we know the m preceding terms. Thus, to find the 6th term of series (i) in § 288, we have Mq — '^x- loox^ + 2x'^ ■2^x^ = 356;lr^ To find the 7th term of the last series in § 288, we have 7/y = 3 X31 -3 X 21 + 13 = 43. 290. To find the midtipliers of a recurring series. (i.) If the series is of the first order, let px be the multipHer, then u^, =Pi u^y or/i = u^-^ Uy SUMMATION OF SERIES. 1/9 (ii.) If the series is of the second order, let/i and /a be the multipliers ; then and «/4=A^3+A^2- From these two equations the values of A and p^ may be found when the first four terms of the series are known. (iii.) If the series is of the third order, let /i, A* and /a be the multipliers ; then from any six consecutive terms we can obtain three equations which will determine the values of A' A» and py If the series is of the wth order, we must have given 2 m consecutive terms to find the m multipliers. Example. Find the multipliers of the recurring series Let the multipliers be/j and /a ; then to obtain /j and/21 we have the equations 13 ;r2 = 5 xpx + 2/2 and 35 ^8 = 13 x"^ p^ + 5 xpi, /?2» A» ••• Dr-> denote respectively the first terms of the successive series of differences; then Z>i = u^ — u^; .-. Ui = 7/1 -f A • Z>2 = «3 — 2 «2 -r- ^1 ; •*• ^h — u^+ 2 D^-\- D^. A = ^4— 3^^3 + 3 «2 - ^1 ; •'• ^4= ^1 + 3 A + 3 A + A • A = «5 — 4 ?^4 + 6 7/3 — 4 u.-^ + ?/i ; .-. u, = //, + 4 Z?, + 6Z>, + 4 A + A- The reader will notice that the coefficients in the value of H^ are those in the expansion of (^ + xy ; a similar relation evidently holds between ?/g and {a + xYy 21^ and (a + ,i')^, etc. ; hence ,/ -,. ^(^ i\n ,(^- 0(«-2) r. , («-i)(«-2)(; y-3) Example. Find the loth term of the series I, 2, 6, 15, 31, 56, ... Here the successive series of differences are : 1st differences, i, 4, 9, 16, 25, ... 2d differences, 3, 5, 7, 9, ••• 3d differences, 2, 2, 2, ... 4th differences, 0, 0, ... Hence ^/j =1, « = 10, T^j =1, A = 3, T^g = 2, Z>4 = 0. Substituting these values in formula (i), we obtain «j^j = I -}- 9 + 108 + 168 = 286. 1 84 ALGEBRA. 295. To find the sum of n terms of the series ?/j, ?/.,, u^, u^, ?/,, u^, ... (i) Assume the new series 0, U^ , U^ + //2 J ^ti + Z^2 + ^3 » ^\ + ^^2 + ^3 + «4 » • • • (2) Now the sum of n terms of series (i) is evidently equal to the {11 + i)th term of series (2). Moreover, the series of first differences of series (2) is series (i) ; hence the second differences of series (2) are the first differences of series (i); the third differ- ences of series (2) are the second differences of series (i); and so on. Hence we may obtain the (;/ + i)th term of series (2), or S„ of series (i), by putting in (i) of § 294 Making these substitutions, we have Example. Find the sum of n terms of the series t2 o2 -j2 a2 c2 ... «2 1st differences, 3, 5, 7, 9, ••• 2d differences, 2, 2, 2, ... 3d differences, 0, 0, ... Hence u-^ =1, D^ = 3, D^ =2, Z^g = 0. Substituting these values in the formula, we obtain = J « (2 «2 + 3 « + I) = J K (« + l) (2 « + I). SUMMATION OF SERIES. 1 85 EXERCISE 32. 1. Find the 7th term of the series 3, 5, 8, 12, 17, ... Ans. 30. 2. Find the 15th term of the series 3, 7, 14, 25, 41, ... 3. Find the 7th term of the series 286, 205, 141, 92, 56, ... 4. Find the 9th term of the series 194, 191, 174, 146, no, ... 5. Find the «th term of the series i, 3, 6, 10, 15, 21, ... Find the sum of each of the following series : 6. I, 3, 5. 7, 9> ••, 2;/- I. 7. 2, 4, 6, 8, ..., 2n, 8. I^3^5^7^ -.(^^-i)'. 9. 2^4^6^8^...,(2«)^ 10. m + I, 2 (;;/ + 2), s(m + 3), ..., n(m + n). Ans. S„ = ^ n (n + 1) (3 /// + 2 // + i). 11. Find the number of balls that can be placed in an equilateral triangle with « on a side ; that is, find the sum of the series i, 2, 3, 4, 5, ..., n. Ans. ^n (n + i). 12. Obtain the series whose nth term is ^n(n + i), and find the sum of n terms. Ans. J // (// + i) {n + 2). 13. Show that x«+ 2« + 3^ + 4' + ••• + ^' = (i + 2 + 3 + 4 + ... + n)\ 1 86 ALGEBRA. 296. Application to Piles of Balls. An interesting application of the preceding theory is that of find- ing the number of cannon-balls in the triangular and square pyramids, and rectangular piles, in which they are placed in arsenals and navy-yards. Triangular Piles. When the pile is in the form of a regular triangular pyramid, the top course con- tains one ball, the second course contains three balls, and the n\\\ course from the top is a triangle of balls with ;^ on a side, and therefore contains \ n (n + i) balls (Example ii of Exercise 32). Hence the whole number of balls in a triangular pyramid hav- ing n balls on a side of its bottom course is the sum of the series I, 3, 6, 10, T5, 21, ..., ^n(n + i). Hence by Example 12 of Exercise 32, S,, = ^n(n+ i){n+2). (i) Square Piles. When the base of the pile is a square having n balls on a side, the top course con- tains one ball, the second course 2^ balls, the third course 3^ balls, and the ;^th course 71^ balls. Hence the number of balls in the pile is the sum of the series . Hence by Example of § 295 Sn = i « (« f I) (2 ;^ + i). * (2) SUMMATION OF SERIES. 1 8/ Rectangular Piles. When the base of the pile is a rectangle having n balls on one side and /« + « on the other, the top course will be a single row of w + i balls; the second course will contain 2 {in + 2) balls; the third course 3 (;;/ + 3) balls; and the bottom course n (m + «) balls. Hence the number of balls in the pile is the sum of the series m -{■ 1, 2(m -\- 2), 2>{^ -^ 3)> •••> n(m-\r n). Hence by Example 10 of Exercise 32, Sn = ln {n +1) (3 fn + 2 « + i). (3) If we put m = 0, (3) becomes identical with (2), as it should ; for when m = 0, the pile is a square pyramid. Incomplete Piles. If the pile is incomplete, find the number of balls in the pile supposed complete, then find the number in the part that is lacking, and sub- tract the last number from the first. EXERCISE 33. 1. Find the number of balls in a triangular pile of 12 courses. How many balls in the lowest course? How many in one of the faces? 2. If from a triangular pile of 20 courses, 8 courses be removed from the top, how many balls will be left? 1 88 ALGEBRA. 3. If from a triangular pile of h courses, c courses be removed from the top, how many balls will be left? 4. How many balls in a square pile of 25 courses? How many balls in each face ? 5. How many balls in a square pile having 256 balls in its lowest course ? 6. Find the number of balls in the lower 12 courses of a square pile having 20 balls on each side of its lowest course. 7. The top course of an incomplete triangular pile con- tains 2 1 balls, and the lowest course has 20 balls on a side. How many balls in the pile ? 8. Find the number of balls in an oblong pile whose low- est course is 52 balls in length and 21 in breadth. If 11 courses were removed from the top of this pile, how many balls would be left? 9. Find the number of balls in an incomplete oblong pile whose top course is 10 balls by 30, and whose bottom course is 45 balls in length. 10. Find the number of balls in a rectangular pile which has II balls in the top row and 875 in the bottom course. 297. Interpolation is the process of introducing be- tween the terms of a series intermediate terms which conform to the law of the series. It is used in find- ing terms intermediate between those given in mathe- matical tables, but its most extensive application is in Astronomy. SUMMATION OF SERIES. 189 The formula for interpolation is that for finding the nth. term of the series by the inetJiod of differences. Thus to find the term equidistant from the ist and 2d terms of a series we put ;/ = \\ in (i) of § 294; to find the term equidistant from the 2d and 3d terms we put n — 2|. Example i. Given log 97= 1.9868, log 98= 1.9912, log 99 = 1-9956; find log 97.32. Series, 1.9868, 1.9912, 1.9956. 1st differences, 0.0044, 0.0044. 2d differences, 0. Hence u^ = 1.9868, Z>i = 0.0044, D^ = 0, u = 1.32. •*• log 97-32 = 1.9868 + 0.32 X 0.0044 = 1.9882. Example 2. Given -v^is = 3-55^89, ^47 = 3.60882, ^49 = 365930, v^iT = 3-70843 ; find ^48. Here «i = 3.55689, Z>i = 0.05 193, /?, = -0.00145, Z>j = 0.0001, « - I = |. Hence ^48 = 3 55689 + I (0.05193) -f I (- 0.00145) - ^\ (o.oooi) = 3-63424. EXERCISE 34. 1. Given Vs = 2.23607, V6 = 2.44949, Vj = 2.64575, VS = 2.82843 ; find Vs^, a/6^. 2. Given the length of a degree of longitude in latitude 41° ==45.28 miles; in latitude 42° = 44.59 miles; in lati- I90 ALGEBRA. tude 43° = 43-88 miles; in latitude 44° = 43- 16 miles. Find the length of a degree of longitude in latitude 42° 30'. Ans. 44.24 miles. 3. If the amount of ^ i at 7 per cent compound interest for 2 years is ;^i.i45, for 3 years ^1.225, for 4 years ^1.311, and for 5 years ^1.403, what is the amount for 4 years and 9 months ? for 3 years and 6 months ? 298. The summation of some series is readily ef- fected by writing the series as the difference of two other series. Example i. Sum the series 1.2 2.3 3.4 4.5 «(«+!) TV ^ — ' • ' _ ^ ^ . I . 2 ~ 2' 2.3 2 3 ' *** Writing the positive and negative terms separately, and de- noting the sum of n terms of the given series by S^ we have [^ \2 3 4 n/ n+ I ^ If the series is infinite, (i) becomes 6'oo = i« Example 2. Sum the series + H 2 + «»» 1.4 2.5 3.6 Here the «th term is evidently -- — ; — : • SUMMATION OF SERIES. 19I Now --L_ = '(l--^); §265. S„=' 3I _fI + ... + M ^ ' L_ I ^ViL ^^ -^ '-). 3\ 6 A/ + I « + 2 // + a^' Hence 6'» = {\. Example 3. Sum the series i 4- 5 + i + ^ + ••• Multiplying and dividing by 2, we have .'. »S*»= 2 . Example i. 2 2« «+ I EXERCISE 35. Find the «th term, the sum of n terms, and the sum of all the terms in each of the following series : 1-3 3-5 5-7 4,4.4, 4 , 2. 1 1 + ••• 1.5 5.9 9 . 13 13-17 192 ALGEBRA. + -^ + ^. + . 3-4 4-5 5-6 ' +-^ + 2.7 7.12 12-17 6. ;^ ' S 6 . 12 9 . 16 7. The series of which the nth term is 8. -^ + -^— + (3'^+2) (3« + 8) I • 4 4 • 7 7 • 10 9. Sum « terms of the series i, 2*, 3*, 4*, ... 10. Show that the number of balls in a square pile is one- fourth the number of balls in a triangular pile of double the number of courses. 11. If the number of balls in a triangular pile is to the number of balls in a square pile of double the number of courses as 13 to 175, find the number of balls in each pile. 12. The number of balls in a triangular pile is greater by 150 than half the number of balls in a square pile, the num- ber of courses in each being the same. Find the number of balls in the lowest course of the triangular pile. 13. If from a complete square pile of ;^ courses a triangu- lar pile of the same number of courses be formed, show that the remaining balls will be just sufficient to form another tri- angular pile, and find the number of its courses. LOGARITHMS. 1 93 CHAPTER XVI. LOGARITHMS. 299. The Logarithm of a number is the exponent by which a fixed number, called the base, must be affected in order to equal the given number. That is, if ^^ — N, X v=> the logarithm of N to the base Uy which is written jt = log,. tV". Thus, since 32 = 9, 2 =log3 9. Since 2^=16, 4 = log^^ ^^• Since 10^=10, lo^^ 100, lo' = 1000, . . . , the positive numbers i, 2, 3, . , . , are respectively the logarithms of 10, 100, 1000, . . ., to the base 10. To the base 10 the logarithms of all numbers between i and 10, 10 and 100, 100 and 1000, ..., are incommensurable. Since 3"^^ = ^ -2 = Iog3f Since 10-1 = 0.1, 10-2 = 0.01, 10-' = 0.001, .. ., the negative numbers —1, —2, —3, ..., are respectively the logarithms of o.i, o.oi, o.ooi, ..., to the base 10. 300. Any positive number except I may evidently be taken as the base of logarithms. The logarithms of all positive numbers to any given base constitute a System of Logarithms. 194 ALGEBRA. In any system, the logarithms of most numbers are incommensurable. Before discussing the two systems commonly used, we shall prove some general propositions that are true for any system. 301. The logarithm of i is 0. For tf° = I ; .-. log^ i = 0. 302. The logarithm of the base itself is i . For a^ =: a ; .-. log^^;— i. 303. The logarithm of a product equals the sum of the logarithms of its factors. Let log« M= X, log,, N=y; then J/= a\ N^a\ § 299. Therefore MN=a'---^-\ Hence log« {MN) =x +y = log, M + log^ JV. Similarly, log.(i^iV^0= log.^+ log.^+ log^Q; and so on, for any number of factors. 304. The logarithm of a quotient equals the loga- rithm of the dividend fninus that of the divisor. Let M=cf, N=a''; then M-^ N =za''-''. Hence loga {M -^ N) = x — y — \og^ M — log« N. LOGARITHMS. 195 305. The logarithm of a positive number affected with ariy exponent equals the logarithm of the number multiplied by the expofient. Let M^a''; then, whatever be the value of/, Hence log, {M^) —px—p log., M. 306. By § 305, the logarithm of any power of a number equals the logarithm of the number multi- plied by the exponent of the power ; and the loga- rithm of any root of a number equals the logarithm of the number divided by the index of the root. 307. From the principles proved above, we see that by the use of logarithms the operations of multi- plication and division may be replaced by those of addition and subtraction, and the operations of in- volution and evolution by those of multiplication and division. Example. Express log«-^^!— - in terms of log<, b, log^ z^ \ogaX> Loga ^^ = log, ^t - log, (^2 ^f ) § 304. = loga di - (log,, Z^ + loga X^) § 303. = |l0g„ 6-2\QgaZ-^ loga X. § 305. 196 ALGEBRA. 308. If a > 1, and ^* = iV; then ii JV > I, X Is positive; if ^< I, X is negative; if JV= -x), X = yo ; if j\r=o, x = ~yo. That is, if the base is greater than unity ^ (i.) The logarithm is positive or negative according as the number is greater or less tJian nnity. (ii.) The logarithm of an infinite is infinite ; and the logarithm of an infinitesimal is a nega- tive infinite, or, as it is often stated, the logarithm of zero is negative infinity, EXERCISE 36. 1. Find log, 16; log, 64; loggSi; log, ^V ; log^^; logs^V; Io&ttV; ^^^Z-o^hz) logs 125. 2. If 10 is the base, between what integral numbers does the logarithm of any number between i and 10 He? Of any number between 10 and 100? Of any number between 100 and 1000? Of any number between o. i and i? Of any number between o.oi and 0.1 ? Of any number between 0.00 1 and 0.01 ? In the next ten examples express log« y in terms of log« by log, c, log^ X, and log, z. 3. y=.z^b^, 5. y^^^z^x'^. 4. J — y^2 ^ ^^^ 6. y = "s/z^x . ^zb~^ , LOGARITHMS. 1 97 g. x^ \ bz^ '.: y^ : xb^ , 10. x~^ '. c^y^ \\ ^ '. x^bi. z^ J b_2^ V^ z^ ^/x^ 12. 2" Common Logarithms. 309. Although there may be any number of sys- tems of logarithms, there are in general use only two, the Natural and the Common. The Natural system, called also the Napierian^ from Baron Napier, is used for analytical purposes only ; its base is 2.71828. The Common system is the system used in practical computations; its base is 10. It was introduced in 161 5 by Briggs, a contemporary of Napier. ' Both Napierian and common logarithms are writ- ten decimally. Hereafter when no base is written, the base 10 is understood. 310. From the equation 10* = A^ it is evident that the common logarithms of most numbers consist of an integral part and a fractional part. 198 ALGEBRA. For example, 2146 > lo^ and < 10* ; .♦. log 2146 = 3 + a decimal fraction. Again, 0.04 > iq-^ and < iq-^ ; .'. log 0.04 = — 2 + a decimal. The integral part of a logarithm is called the Characteristic, and the decimal part the Mantissa. For convenience in the use of common logarithms, mantissas are always made positive. Hence the logarithm of any number less than unity consists of a negative characteristic and a positive mantissa. 311. The characteristic of the common logarithm of any number can be determined by one of the two following simple rules : (i.) If the number is greater than unity, the charac- teristic is positive a7id numerically one less than the number of digits iji its integral part. For a number with one digit in its integral part lies between 10^ and lO^; a number with two digits in its integral part lies between 10^ and 10^; and so on. Hence if N denote a number that has n digits in its integral part, then N lies between 10" ~' and 10"; that is, jy -- J Q (m — 1) + a fraction. .*. logiV^= {71 — i) + a mantissa. Thus, log 2178.24 = 3 + a mantissa; log 3872416 = 6 + a mantissa. LOGARITHMS. I99 (ii.) If the mimber is less tha7t tmityy the character- istic is negative and numerically one greater than the number of ciphers immediately after the decimal point. For a decimal with no cipher immediately after the decimal point lies between 10 "^ and 10^; thus, 0.327 lies between o.i and i ; a decimal with one ci- pher immediately after the decimal point lies between 10 ~ 2 and io~^; thus, 0.0217 Hes between 0.0 1 and 0.1 ; and so on. Hence if Z^ denote a decimal with n ciphers immediately after the decimal point, then D lies between 10 -^''-^^^ and io~"; that is, ^___ jQ — («+l) + a fraction. .'. log /? = — (« + i) + a mantissa. Thus log 0.003217 = — 3 + a mantissa ; log 0.000081 =: — 5 + a mantissa. The converse of rules (i.) and (ii.^ may be stated as follows : (i.) If the characteristic of a logarithm is + n, there are n 4- i integral places in the corresponding ntimber. (ii.) If the characteristic is — n, there are n — i ciphers immediately to the right of the decimal point ift the number. 312. I^g {Ny,\o^'')=^\ogN ±n. § 303. Hence if « is a whole number, log N and log (A^ X 10-") have the same mantissa. Therefore if 200 ALGEBRA. a number be multiplied or divided by an exact power of lo, the mantissa of its logarithm will not be changed. That is, the common logarithms of all numbers that have the same sequence of significant digits have the same mantissa. Thus, the logarithms of 21.78, 2178, and 0.002178 have the same mantissa. 313. The method of calculating logarithms will be explained in §§ 319, 322. The common logarithms of all integers from i to 200000 have been computed and tabulated. In most tables they are given to seven places of decimals ; but in abridged tables they are often given to only four or five places. Common logarithms have two great practical advantages : (i.) Characteristics are known by § 311, so that only mantissas arc tabulated. (ii.) Mantissas are determined by the sequence of digits (§ 312), so that the mantissas of inte- gers only are tabulated. When the characteristic is negative, the minus sign is written over the characteristic, to indicate that the characteristic alone is negative, and not the whole expression. Thus 3.845098, the logarithm of 0.007, is equivalent to — 3 + 845098, and must be distinguished from —3.845098, in whicli both the integral and decimal part are negative. LOGARITHMS. 201 To transform a negative logarithm, as —3 26782, so that the mantissa shall be positive, we subtract i from the characteristic and add i to the mantissa. Thus — 3.26782 = — 4 4- (i — 0.26782) = 4.73218. To divide 3.78542 by 5, we proceed thus : i (3.78542) -H- 5 + 2.78542) = 1.55708. 314. For logarithinic tables and directions in their use, the student is referred to works on Trig- onometry. For use in this and the next chapter we give below the common logarithms of prime numbers from i to 100. No. Logarithms. No. 29 Logarithms. No. Logarithms. 2 0.3010300 1.4623980 61 ,1.7853298 3 O.4771213 31 I.49I3617 67 1.8260748 7 0.8450980 37 1. 56820 1 7 71 I. 8512583 II I. 041 3927 41 I. 6127839 73 I.S633229 13 I.I 139434 43 1.6334685 79 I.F97627I 17 1.2304489 47 1.6720979 83 1. 9190781 19 1.2787536 53 1.7242759 89 1.9493000 23 1. 3617278 59 1.7708520 97 1. 98677 1 7 Log 5 = log (10 -^ 2) =: log 10 — log 2 0.30103 = 0.69897. In like manner the logarithms of all integers between i and 100 can be obtained from those given in the table above. 202 ALGEBRA. The utility of logarithms in facilitating numerical computations is illustrated by the following example. .2 8 Example. Find the value of 33 x 0.9^ ^ 0.494, given log 2.87686 = 0.458919. log (3^ X 0.92 - 0.49^) = f log 3 + 2 log ^\ - f log ^%\ = flog3 + 2(log32-i)-|(log72-2) = I log 3 + 4 log 3 - 2 - 1 log 7 + f = ¥log3-|log7-i = 2.3265661 — 1.267647 — 0.5 = 0.458919 = log 2.8y686 ; .-. 3^ X 0.92 H- 0.49^ = 2.87686. EXERCISE 37. 1. Given log 2659 = 3.424718; find log 26.59, log 0.2659, log 265900, log 0.0002659. 2. Given log 2389 = 3.378216; find the number whose logarithm is 1.378216, 0.378216, 2.378216, 5-378216, 3.378216, 4.378216. Find the common logarithm of 8. 1.05. 13- 3. 84. Vo-oio5« 4. 0.128. 9. 0.0183. 10. 0.02134. 14. 15. 86^ 5. 0.0125. V3S-^ 27. 6. 1.44, II. ^^42^ 16. 4|. 7. 1.06. 12. V374- 17- 25^. LOGARITHMS. 20$ l8. .2I01T 19. 0.015^ ^ V2 ^' ^X2Tx^ ' 20. o-ooiS"^. 3,. 0.63^. 24. v5^i^. ,6. (---^-7)'. .JUL '^^^ (0.002-^3)3 22. (14^15)^. ^ ^^ 27. Find the seventh root of 0.00324, given log 4409.2388 = 3-644363- 28. Find the eleventh root of 39-2-^, given log 19.48445 = 1.2896883. 29. Find the product of 3 7- 203, 3.7203, 0.0037203, and 372030; given log 372.03 = 2.570578, and log 191.5631 = 2.282312. 315. Exponential Equations. An exponential equa- tion is one in which the unknown quantity appears in an exponent. Thus 2"" = 5, b^'' ■\- If — c, and x* = 10 are exponential equations. Exponential equa- tions are solved by the aid of logarithms. Example i. Solve 32-^43*= 54 ^r 2^+1, Taking the logarithms of both members, we have 2 ;i' log 3 + 3 X log 22 = 4 jf log 5 + (jir + I) log 2 ; .'. (2 log 3 + 6 log 2 - 4log 5 - log 2) ;r = log 2, ^log2 or x = 2log3 + slog2-4logs 0.301030 0.336488 0.894-}-. 204 ALGEBRA. Example 2. Find the logarithm of 32/^/4 to the base 2y'2. Let X = log 32y^4 to base 2^2, or 22 ; then (2^)'' = 32^4 = 2*^ X 25 ■^ 2 Hence "^ ;ir log 2 = $ log 2 + log 2 ; 2 5 27 3 18 .*. x= — -^ ^ = — = 3.6. 525-^ Example 3. Solve 32-^ — 14 x 3"" + 45 = 0. The equation may be written in the form (3" - 9) (3" - 5) - 0, which is equivalent to the two equations r = 9 and 3- = 5. From y — g, X = 2] and from y = 5, ^^ logs ^ 0698970 ^ ^5 log 3 0.477121 EXERCISE 38. Solve the following literal equations : 2. ^2x ^8^ _ ^5^ 4. ^ = -^f, 16. a'''b''y = m\ x^ =y\ a^- f'y = m^\ Logarithmic and Exponential Series. 316. The Derivative of log^^. Let j> = nz, (i) 71 being an arbitrary constant, and y and js functions of X. Then log^^ = log,, ft + log,, z ; .-. A(log.>')-Z>,(log.0). (2) Dividing the derivatives of the members of (i) by (i), we obtain ^ = ^^ (3) Dividing (2) by (3) we obtain, D, (I0&7) : ^ = ^. (log. ^) : ^- (4) y z 206 ALGEBRA. It is evident that the equal ratios in (4) are constant for any particular value of z. Let m denote their con- stant value when z = z ; then D, (log,^) = m ^, (5) when y =. n^. But as n is an arbitrary constant, 71 ^ denotes any number; hence (5) holds true for all values of y, m being a constant. The constant m is called the Modulus of the system of logarithms whose base is a. Hence, the derivative of the logarithm of a variable is equal to the modulus of the system into the derivative of the variable divided by the variable. EXERCISE 39. Find the derivative of 1. J = log, (i + ^). 4- /W = (log«^)^ 2. 7= log. {x^ + oc^)' 5- /W = ioga-^^- 3. ^ = log, (^2 + ^ + b). 6. fix) = X log, X, 'J. y = log„ Vi—x^=i log« (i — x^). S. y = log, (x^ + x)2. 10. y = log, (x^ — cx^)t , X Vi + -^ 9. y = \og,--==z. II. j; = log,— =.r. yi + x^ yi — X 317. To deduce the Logarithmic Series. To do this, we develop log, (i ■\- x) by Maclaurin's formula, /'(.v) = i+x' f>\x) = m (i+xf' /'"(*) = 2 m (i+A)" f\x) = |3«» (i+^r' LOGARITHMS. 207 /W -/(O) 4-/'(0) Y + /" (0) ^' +/'"(0) ^' +/'M0) ^ + . . . Here /(^) - log, (i + x\ .-. /(O) - ; .. ///(O) =_;;,. . /iv(0)=_i3^, Substituting these values in the formula, we have lofc(i+^) = «(x-^%^-^V^-...), (A) which is the general logarithmic series, 318. Napierian System. The system of logarithms whose modulus is unity is called the Napierian or Natural system. The symbol for the Napierian base is e. If in (A) of § 317 we put m = i and a = e, we have ^2 ^,3 „4 „5 log, (I + a:) = ^ -- + --- + --... , (B) 2345 which is the Napierian logarithmic series. 208 ALGEBRA. By Example i of § 286 the series in (B) is con> vergent only for values of x between — i and + i ; hence formula (B) cannot be used to compute the Napierian logarithm of any number greater than 2. 319. To obtain a formula for computing a table of ISlapierian logarithms. Putting — ;!; for ;ir in (B) of § 3 18, we have (0 log (l+^)-l0g,(l-.T) = 2(^ + - + ^ + -+...).(2) Let ^ = TT-r-;; (3) log. (I -:.)=- — X — x" 2 x^ 3 ^4 4 x' ' 5 Subtracting (i) from (B), we have then 22+1' I + ^ z -^ 1 1 — X .'. log, (i + x) - log, (i - ^) = log, (2: + i) - log, z. (4) Substituting in (2) the values in (3) and (4), we obtain l0g,(^+l)=l0g.^+2(^ + ^^j^+|y^^, + ...).(C) Since, in (3),^< i for ^>0, the series in (C) is con- vergent for all positive values of ^; hence log, {2 + 1) can be readily computed when log, z is known. LOGARITHMS. 20g Example. Compute to six places of decimals logv2, log^3, log, 4, log, 5, log, lo. Putting ^ = I in (C), we obtain, since log^ i = 0, Summing six terms of this series, we find log, 2 = 0.693147. Putting z = 2'in (C), we have = 1. 0986 1 2. Log, 4 = 2 log, 2 = 1.386294. Putting 2- = 4 in (C), we obtain = 1.6094379. Log, 10 = log, 5 + log, 2 = 2 302585. In this way the Napierian logarithms of all positive numbers can be computed. The larger the number the more rapidly convergent is the series. 320. Value of m. Dividing (A) by (B), we have log,, (i +x) .. in which i + x lies between and 2. Let N be any number, and let V = locr. JV, or A/'= a^ : 2IO ALGEBRA. then \og,JV=y . log, a = log^iV^. log, a. Hence log^JV I , . LetiV^ = I + ;r; then from (i) and (2), we have «« = ,-7:^- (3) That is, t/2e modulus of any system of logarithms is equal to the reciprocal of the Napierian logarithm of its base, 321. From (2) and (3) of § 320 we have log,, N=^ 7)1 log, N. That is, the logarithm of a number in a7ty system is equal to the Napieria7i logaritlnn of the same number multiplied by the modulus of that sysiem. 322. Value of M. If in (3) of § 320, M denote the value oim when a — 10, we obtain Jf = ;— = IT = 0-4342Q4. log, 10 2.302585 ^^^ ^^ That is, the modulus of the Common System to six places of decimals is 0.434294. Hence to obtain common logarithms from Na- pierian, multiply the Napierian by 0.434294; to ob- tain Napierian from common, multiply the common by 2.302585, or log, 10. LOGARITHMS. 2 1 1 Multiplying both members of (C) by M, we obtain which is a formula for computing common logarithms. Multiplying both numbers of (C) by ;//, we obtain a general formula for computing logarithms to any base a, 323. An Exponential Function is one in which the variable enters the exponent, as ^^X^ o!"^'"* 324. To find the derivative of a^ Let y =^ a' ) then log^^ = zlog^a, (i) Hence -^ = log, a - Z>^z, y or D^y = A («~~) = a'' log,« • A z- That is, t/ie derivative of an exponeiitial function with a constant base is equal to the fu7iction itself into the Napierian logarithm of the base into the derivative of the exponent. 325. To develop a*, or deduce the Expojiential Series, Here f{x) = (f, .-. /(O) = i ; fix) = or \og,a, .-. /HO) = \og,a; f\x) = cf (log.«)^ .-. f\Q) = Qog^ay ; f"(x) = a^ {\og,a)\ ,'.f"{0) = (log,«)« ; 212 ALGEBRA. Substituting these values in Maclaurin's formula, we have a'=i + (Xog^a) X + {Xog^af ^ + {\og,aff- + ... , (i) Li 11 which is the exponential series, 326. Value of e^ . Putting ^ = ^ in (i) of § 325, we have ^•2 ^8 ^4 327. Value of e. Putting x=\ in (i) of § 326, we obtain ^= 1 + I + P + i- + ^ + ••• = 2.718281. That is, the Napierian base — 2.7 1828 1 +.* EXERCISE 40. 1. Find to five places log^ 6, log^ 7, log^ 8, log^ 9, log^ 1 1. 2. Find to five places the moduli of the systems whose bases are 2, 3, 4, 5, 8, 9, 12. 3. By § 321, prove that the logarithms of the same num- ber in different systems are proportional to the moduH of those systems. * In the " Proceedings of the Royal Society of London," Vol. XXVII., Prof. J. C. Adams has given the values of e, M, log, 2, log, 3, log^ 5, to more than 260 places of decimals. LOGARITHMS. 21 3 4. By the formula of § 322 compute log 2, log 65, log 131, log 3; log 82, log 244. 5. Obtain the formula, log. (. + I) = log.. + . m (^^ + ^^^l^^y +•••)• 6. Obtain the following formulas : log.(^+ '^"^''^^""^"^2 + ^8 W log.^ - log.(0 - = ^ + 2-72 + ^8 + ••• (2) log, (. + I) - log, (._ 1) = 2 (i + ^'^ + ^ + ...). (3) These formulas are convergent for > i, and may be used in computing logarithms. To obtain (i), in (B) substitute (i —z) ioxx; to obtain (2), substitute ( — i -H 2) for ^. 7. Obtain the formulas corresponding to (i), (2), and (3) of Example 6, for common logarithms : for any system. 4/- lOy- 8. Show that log -Y^ —_ = i log 5 - f log 2 - § log 3. V 18 . V2 9. Show that log y 729 y 9~~^X 27 ^ = log3. 10. Find the logarithms ofi/^s-^ —=, \/^~^ > to base a. 11. Find the number of digits in 3^^ X 2^ 214 ALGEBRA. AiNioo 12. Show that I — 1 is greater than I oo. 13. Find how many ciphers there are between the deci- mal point and the first significant digit in (^)^'^°^ 14. Calculate to six decimal places the value of m X I25\2 X32 J ' ;iven log 9076.226 = 3-95 79< ^53- Solve 15. 2^+-^ =6-^, 16. 3^— > = 4-^ s^'^sx 2^ + \ 22.-1 = 33.-^ 17. If log {x^ y^) = a, and log(^ -J- 7) = d, find log x and log J. o o, , hmit 18. Show that m = unit / xK*" f x\*^ X 7n(m—\)lx\'^ V tn) tn p \7nj [2 [3 limit / :r\'' I +-) rz i+;r + — + l:^ + ••• = ^• limit ( x\^ x^ x^ 19. If « is positive, the positive real value of a'' is a con- tinuous function of x. For ^'^ has one positive real value for each value of x, and A(«^) = a^ (a^ — i) = when A^ = 0. COMPOUND INTEREST. 21 5 CHAPTER XVII. COMPOUND INTEREST AND ANNUITIES. 328. To find the interest I and ainoimt M of a given sunt P /;/ n years at r per centj compoimd interest. (i.) When interest is payable anmially. Let R^= the amount of $ i in i year; then R = i + r, and the amount of P at the end of the first year is PR; and since this is the principal for the second year, the amount at the end of the second year is PR X R, or P R^. For hke reason the amount at the end of the third year is P R^, and so on; hence the amount in n years is P R" ; that is M= PR", or P{i + ry, (i) Hence I=P(R"-\). (2) (ii.) When the interest is payable q times a year. If the interest is payable semi-annually, then the interest of $1 for i a year is \r ; hence the amount of P in \ a year is P(i + 2 r); the amount of P in one year is P (i + J r)^ ; the amount of P in ;/ years is /* (i + | r)^". 2l6 ALGEBRA. That is, M=F{Y-\-\rf\ (3) Similarly, if the interest is payable quarterly, M=P{Y^\ry\ (4) Hence, if the interest is payable q times a year, G+^r (5) M^ P 9 In this case the interest is said to be "converted into principal " q times a year. Example. Find the time in which a sum of njoney will double itself at 10 per cent compound interest, interest payable semi-annually. Here i ■\-\r ~ 1.05. Let P = $i; then M = $2. Substituting these values in (3), we obtain 2= (i.05)2«; .*. log 2.— in ' log 1.05 ; loof 2 2 (log 5 + log 3 + log 7 - 2) o 30^03 ■~ 2 (0.69897 + 0.477 1 21 3 -I- 0.845098 — 2) = 7-103. Therefore the time is 7.103 years. 329. When the time contains a fraction of a year^ it is usual to allow simple interest for the fraction of the year. Thus the amount of P in 11 -\ years is m COMPOUND INTEREST. 21/ PR^ + PR'^ -, or PR" (i + -V When interest is payable oftener than once a year there is a difference between the nominal annual rate and the true anmtal rate. Thus, if interest is payable semi-annually at the nominal annual rate r, the amount of ;^l in one year is (i + \ r)^, ox \ -\- r-\-\ r^, so that the true annual rate is r + J r^. Thus, if the nominal annual rate is 4 per cent, and interest is payable semi-annually, the true annual rate is 4.04 per cent. 330. Present Value and Discount. Let P denote the present value of the sum J/ due in ;/ years, at the rate r ; then evidently, in ;/ years, at the rate r, P will amount to M ; hence M^PR'\ or P=MR-''. Let D be the discount ; then £> = M-P=M{i -R-"). EXERCISE 41. 1. Write out the logarithmic equations for finding each of the four quantities M, R, P, n. 2. In what time, at 5 per cent compound interest, will $100 amount to $1000? 3. Find the time in which a sum will double itself at 4 per cent compound interest. 21 8 ALGEBRA. 4. Find in how many years $1000 will become $2500 at 10 per cent compound interest. 5. Find the present value of $10,000 due 8 years hence at 5 per cent compound interest; given log 67683.94 = 4.8304856. 6. Find the amount of $1 at 5 per cent compound interest in a century; given log 13 15 =1=3.11893. 7. Show that money will increase more than seventeen- thousand-fold in a century at 10 per cent compound inter- est, interest payable semi-annually; given log 17213.13 = 4.23786. 8. Find what sum of money at 6 per cent compound interest will amount to $1000 in 12 years; given log 49697 = 4.6963292, log 106 = 2.0253059. 9. Find the amount of a cent in 200 years at 6 per cent compound interest; given log 115. 128 = 2.06118. 10. The present value of $672 due in a certain time is $126 ; if compound interest at 4^ per cent be allowed, find the time. ANNUITIES. 331. An Annuity is a fixed sum of money that is payable once a year, or at more frequent regular intervals, under certain stated conditions. An Aft- miity Certain is one payable for a fixed number of years. A Life Annuity is one payable during the ANNUITIES. 219 lifetime of a person. A Perpetual Annuity, or Per- petuity, is one that is to continue forever, as, for in- stance, the rent of a freehold estate. A Deferred Ajifiuity is one that does not begin until after a certain number of years. 332. To find the amount of an annuity left unpaid for a given number of year's, allowing compound in- terest. Let A be the annuity, n the number of years, R the amount of one dollar in one year, M the required amount. Then evidently the sum due at the end of the 1st year = A, 2d year = A R •\- A. 3d year ^ A R"" + A R -^ A. ni\i yedix = A R'-'' + A R"-"" + ... + A R + A _ A (R' - i) '~ R- I " That is, M=^(R"-i). (i) Example i. Find the amount of an annuity of $100 in 20 years, allowing compound interest at 4.^ per cent; given log 1.045 = 0.0 191 163, log 24.117 = 1.382326. r ^ ^ 0.045 By logarithms 1 .04520 = 2.41 1 7 ; 0.045 *^-J -5' 22(3 ALGEBRA. Example 2. Find what sum must be set aside annually that it may amount to $50,000 in 10 years at 6 per cent compound interest ; given log 17.9085 = 1.253059. Solving (i) for A we obtain Mr $ 50,000 X 0.06 By logarithms 1.0610=1.79085; $ 3000 333. To find tJie present value of an annuity to con- tinue for a given number of years y allowing compound interest. Let P denote the present value ; then the amount o{ P in n years will equal the amount of the annuity in the same time ; that is, PR" = ^{R"-^)i (i) '.■.P=-^{x-R-). (2) 334. Perpetuity. If the annuity be perpetual, then n = -Xi, R'"" == 0, and (2) of § 333 becomes r 335. Deferred Annuity. If the annuity commences after / years, and continues 7t years thereafter, then the present value will evidently be the difference ANNUITIES. ' 221 between the present value of an annuity to continue n ^r p years and one to continue/ years; that is, P=-^{R-^-Rr''-% (i) „ A R'-z. 336. If the annuity be perpetual after/ years, then R~"-^= 0, and (i) of § 335 becomes F=- R-K r 337. Solving (i) of § 333 for ^, we obtain Fr 7?" A = R" which gives the value of the annuity in terms of the present value, the time, and the rate per cent. 338. A Freehold Estate is an estate which yields a perpetual annuity, called rent; hence the value of the estate is the present value of a perpetuity equal to the rent. Example i. Find the present value of an annual pension of $200 for 10 years at 5 per cent interest ; given log6. 13917 = 0.78S107. A $ 200 P=-{i-R--) = - (I ^1.05-10). r^ / 0.05 ^ ^ '' By logarithms 1.05- ^^ =: 0.613917 ; .-. F = $4000 X 0.386083 = $ 1544.33. 222 ALGEBRA. Example 2. The rent of a freehold estate is $350 a year. Find the value of the estate, the rate of interest being 5 per cent. A $350 F = — = ■ — ^ = $ 7000. r 0.05 ^ ' Example 3. Find the present value of an annuity of % 1400 to begin in 8 years and to continue 12 years, at 8 per cent in- terest; given log 25.1818 rr 14010868, log 466.1 = 2.668478. ^ A R" - \ $1400 1.0812-1 ^ = 7 "^ 1F+} = "^.^ ^ ~7:o8"^o- = $5700.09. Example 4. Find what annuity $ 5000 will give for 6 years when money is worth 6 per cent; given log 14.185 = 1. 15 18344. ^ Pr/e« 1.066 A = j^,^ _ ^ = % 5000 X 0.06 X ^^Q^6_i - $ 1016.84. EXERCISE 42. 1. If A leaves B $1000 a year to accumulate for 3 years at 4 per cent compound interest, find what amount B should receive; given log 112.4864 = 2.05 11 062. 2. Find the present value of the legacy in Example i ; given log 888.9955 — 2.9488998. 3. Find the present value, at 5 per cent, of an estate of $1000 a year, (i) to be entered on immediately, (2) after 3 years; given log 17376.75 — 4.2374621. 4. A freehold estate worth $120 a year is sold for $4000 ; find the rate of interest. ANNUITIES. 223 5. A man borrows $5000 at 4 per cent compound inter- est; if the principal and interest are to be repaid by 10 equal annual instalments, find the amount of each instal- ment; given log 675560 = 5.829666. 6. A man has a capital of $20,000, for which he receives interest at 5 per cent; if he spends $1800 every year, show that he will be ruined before the end of the 1 7th year. 7. When the rate of interest is 4 i)er cent, find what sum must be paid now to receive a freehold estate of $400 a year 10 years hence ; given log 6.75560 = 0.829666. 8. The rent of a freehold estate of $882 per annum, deferred for two years, is to be sold ; find its present value at 5 per cent compound interest. 9. The rent of a freehold estate, deferred for 6 years, is bought for $20,000 ; find what rent the purchaser should receive, reckoning compound interest at 5 per cent; given log 1.340096 = 0.1271358. 10. Find the present worth of a perpetual annuity of $ 10 payable at the end of the first year, $ 20 at the end of the second, $ 30 at the end of the third, and so on, increas- ing $ 10 each year, interest being taken at 5 per cent per annum. 24 ALGEBRA. CHAPTER XVIII. PERMUTATIONS AND COMBINATIONS. 339. Fundamental Principle. If one thing can be done in m ways, and {after it has been done in ajty one of these ways) a second thing can be done in n ways ; then the two things can be done in m X n ways. After the first thing has been done in any one way, the second thing can be done in n different ways ; hence there are ;/ ways of doing the two things for each of the m ways of doing the first ; therefore in all there are m n ways of doing the two things. Ihis principle is readily extended to the case in which there are three or more things, each of which can be done in a given number of ways. Example i. If there are ii steamers plying between New York and Havana, in how many ways ran a man go from New York to Havana and return by a different steamer ? He can make the first passage in ii ways, with each of which he has the choice of lo ways of returning ; hence he can make the two journeys in ii x lo, or no, ways. ^ Example 2. In how many ways can 3 prizes be given to a class of 10 boys, without giving more than one to the same boy ? PERMUTATIONS AND COMBINATIONS. 225 The first prize can be given in lo w.iys ; with each of which the second prize can be given in 9 ways ; hence the first two prizes can be given in 10 x 9 ways. With each of these ways tlie third prize can be given in 8 ways; hence the three prizes can be given in 10 x 9 x 8, or 720, ways. 340. Each of the different groups of r things which can be made of ;/ things is called a Combination. The Permutations of any number of things are the different orders in which they can be arranged, taking a certain number at a time. Thus of the four letters a, b, c, d, taken two at a time, there are six combinations ; namely, ab^ ac^ ad^ be, bd, cd. Each of these groups can be arranged in two different orders; hence of the four letters a.b, c, d, taken two at a time there are tv.elve permutations ; namely, ab, ac, ad, be, bd, ed, ba, e a, da, eb, db, de. Of a group of three letters, 7is abe, when taken all at a time, there are six permutations ; namely, a be, aeb, be a, bae. cab, eba. The symbol "P^ will be used to denote the number of permutations of 71 things taken r at a time. Thus, ^P^, ^P^, ®/*4, denote respectively the number of per- mutations of 9 things taken 2, 3, 4, at a time. Similarly "C,. will be used to denote the number of combinations of ;/ things taken r at a time. 226 ALGEBRA. 341. To find the number of permutations of n dis- similar things taken v at a time. The number required is the same as the number of ways of fining r places with ;/ things. Now, the first place can be filled by any one of the n things, and after this has been filled in any one of these n ways, the second place can evidently be filled in {n -- i) ways; hence with n things two places can be filled in n (it — i ) ways ; that is, -F^ = n(7i-i). (i) After the first two places have been filled in any one of these n{n — i) ways, the third place can be filled in (ji — 2) ways ; hence three places can be filled in ;/ (n — i) (ji — 2) ways ; that is, "Pg — n{n— i) {11 — 2). (2) For like reason we have «/>, z=n{n- i) {11 -2){n~i)\ (3) and so on. From (i), (2), (3), ..., we see that in "P^ there are r factors, of which the rth is ;^ — r + i ; hence ^P^ ^n {n— i) {n — 2) ... {?i — r -\- i). (A) 342. Value of "P„. U r — n, (A) of § 341 becomes "^. = \n. (B) That is, the number of permutations of n things t.:ken all at a time is \n. PERMUTATIONS AND COiMBINATIONS. 22/ 343. Circular Permutations. When n different let- ters are arranged in a circle, any one of their per- mutations can without change be revolved so that any letter, as a, shall have a given position. Hence we may regard a as having the same position in all the permutations. Now -the number of the possible arrangements of the remaining ;/ — i letters in the other positions is \n— I . Hence, the number of the Circular Permutations of n things is \n—\ . EXERCISE 43. 1. A cabinet-maker has 12 patterns of chairs and 7 pat- terns of tables. In how many ways can he make a chair and a table? Ans. 84. 2. There are 9 candidates for a classical, 8 for a mathe- matical, and 5 for a natural-science scholarship. In how many ways can the scholarships be awarded? 3. In how many ways can 2 prizes be awarded to a class of 10 boys, if both prizes may be given to the same boy? 4. Find the number of the permutations of the letters in the word numbers. How many of these begin with fi and end with s ? 5. If no digit occur more than once in the same number, how many different numbers can be represented by the 9 digits, taken 2 at a time ? 3 at a time ? 4 at a time ? 228 ALGEBRA. 6. How many changes can be rung with 5 bells out of 8? How many with the whole peal ? The first number = ^P^ = 6720. 7. How many changes can be rung with 6 bells, the same bell always being last? 8. In' how many ways may a host and 6 guests be seated at a table in a row? In how many ways if the host must have Mr. Jones on his right and Mr. Smith on his left? In how many ways if the host must sit between Mr. Smith and Mr. Jones? 9. In how many ways may 15 books be arranged on a shelf, the places of 2 being fixed? 10. Given "F^ = 12 . '^7^2 ; find n. 11. Given n : ""F^ : : i : 20; find n, 12. In how many different orders may a party of 6 be seated at a round table? 13. In how many different orders may 10 persons form a ring? 14. In how many different orders may a host and 8 guests sit at a round table, provided the host has Mr. A at his right and Mr. B at his left? 15. Given "F^ : " + ^^3 : : 5 : 12, to find n. 16. Given 'F^ : ^"F^ : : 13 : 2, to find n. PERMUTATIONS AND COMBINATIONS. 229 344. To find the number of combinations of n dis- similar things taken r at a time. By § 342 there are \r permutations of any com- bination of r things; hence we have = n (n — 1) (« — 2) ... (« — r + i). Hence -C = "<"-')("- g"" ^^ -"+'> . (C) 345. Corollary i. Multiplying the numerator and denominator of the fraction in (C) by \n — r , we obtain _ n{n-i) {n-2) ... {n-r+i) \n - r [r \n-r or «C=i— pl= — . (D) Formula (C) should be used when a numerical result is required. In applying this formula, it is useful to note that the suffix r in the symbol "C^ denotes the number of the factors in both the nu- merator and denominator of the formula. Formula (D) gives the simplest algebraic expression for "C^. 346. Corollary 2. Substituting n — r for r in (D) we obtain "C,._r= , '- , . (l) \n — r \r ^ From (D) and (i), "C = "C.-.. (E) 230 ALGEBRA. The relation in (E) follows also from the con- sideration that for each group of r things that is selected, there is left a corresponding group of ;/ — r things. This relation often enables us to abridge arithmetical work. Thus, ^^^13 = 15^2 = ^^^^==105. EXERCISE 44. 1. How many combinations can be made of 9 things taken 4 at a time ? taken 6 at a time ? taken 7 at a time ? The last number = ^Q = ^Cg = 36. 2. How many combinations can be made of 11 things taken 4 at a time? taken 7 at a time? 3. Out of 10 persons 4 are to be chosen by lot. In how many ways can this be done ? In all the ways, how often would any one person be chosen? 4. From 14 books in how many ways can a selection of 5 be made, (i) when one specified book is always included, (2) when one specified book is always excluded? 5. On how many days might a person having 15 friends invite a different party of 10? of 12? 6. Given ^"C^ = 15, to find 71. 7. Given " + ^C; = 9 X ''C^, to find 71, 8. In a certain district there are 4 representatives to be elected, and there are 7 candidates. How many different tickets can be made up ? PERMUTATIONS AND COMBINATIONS. 23 I 9. Of 8 chemical elements that will unite with one another, how many ternary compounds can be formed ? How many binary ? 10. On a table are 6 Latin, 7 Greek, and 8 German books. In how many different ways may 2 books from different languages be chosen ? In how many ways may 3 ? The first number =6x7 + 6x8 + 7x8 = 146. 11. In how many ways can 10 gentlemen and 10 ladies arrange themselves in couples? 12. How many different arrangements of 6 letters can be made of the 26 letters of the alphabet, 2 of the 5 vowels being in every arrangement? 13. How many different straight lines can be drawn through any 15 points, no 3 of which lie in the same straight line? 14. In a town council there are 25 councillors and 10 aldermen ; how many committees can be formed, each consisting of 5 councillors and 3 aldermen? 15. Find the sum of the products of the numbers 3, — 2, 4» ~ 5» i» (0 taken 2 at a time, (2) taken 3 at a time, (3) taken 4 at a time. 16. Find the sum of the products of the numbers i, 3, 5, 2, (i) taken 2 at a time, (2) taken 3 at a time. 17. Find the number of combinations of 55 things taken 45 at a time. 18. If ^"Ca : "Cj = 44 : 3 ; find n. 232 ALGEBRA. 19. If "C,2 = "Q ; find "C,. ; find ^''C^, 20. Ill a library there are 20 Latin and 6 Greek books; in how many ways can a group of 5 consisting of 3 Latin and 2 Greek books be placed on a shelf ? 21. From 3 capitals, 5 other consonants, and 4 other vowels, how many permutations can be made, each begin- ning with a capital and containing in addition 3 consonants and 2 vowels? 22. Ifi«C = ''C+2; find'-Q. 23. From 7 Englishmen and 4 Americans a committee of 6 is to be formed ; in how many ways can this be done when the committee contains, (i) exactly 2 Americans, (2) at least 2 Americans? 24. Of 7 consonants and 4 vowels, how many permutations can be made, each containing 3 consonants and 2 vowels ? 25. How many different arrangements can be made of the letters in the word courage, so that the consonants may occupy even places? 347. //"N de7iote the mimber of permutations of x\ things taken all at a time, of which r things are alike, s others alike, and t others alike ; then N= \L\L\L Suppose that in any one of the A^ permutations we replaced the r like things by r dissimilar things; then, PERMUTATIONS AND COMBINATIONS. 233 from this single permutation, without changing in it the position of any one of the other n — r things, we could form V new permutations. Hence from the ^original permutations we could obtain NW permu- tations, in each of which s things would be alike and t others alike. Similarly, if the s like things were replaced by s dissimilar things, the number of permutations would be iVIr !.r, each having / things alike. Finally, if the t like things were replaced by t dissimilar things we should obtain N\r\s \t_ permutations, in which all the things would be dissimilar. But the number of permutations of « dissimilar things taken all at a time is \n. Hence iV^[r [f; [£ = |«. \n Therefore N 348. To find the number of ivays in which m 4- n things caji be divided into two groups containing re- spectively m and n things. The number required is evidently the same as the number of combinations of m + n things taken m at a time ; for every time a group of i7t things is selected a group of ;/ things is left. Hence the required number = ■ ■ § 345. 234 ALGEBRA. 349. By § 348 the number of ways in which m + n + / things can be divided into two groups containing respectively m and n -\- p things is \m + ^i -\- p \ni \n +/ Again, the number of ways in which each group of n -\- p things can be divided into two groups con- taining respectively n and / things is V^ + P \n\p_ Hence the number of ways in which in \ n ■\- p things can be divided into three groups containing respectively ;;/, ;/, and p things is \m ■\- n ■\- p \n + p \m -\- n ■\- p \m^\Ti -\- p (^ \P ' \m\71\p This reasoning can be extended to any number of groups. *350. The sum of all the combinations that can be made of n things, taken i , 2, . . . , w at a time^ is 2" — i . By the binomial theorem we have / X n{f^ — i) ^ n(n — 1)(«— 2) , , . (i+;v)«=:i + ^^^+-h ^-x^^^ f^ ^a:H - (i) \^ [3 In (i) the coefficients of x, x^, x^, ..., x"' are evi- dently the values of "Q, "C2, "Q, ..., "C„; hence (i) may be written (i+xy=i + "C,x + "C2x''-{-"Qx^+'^' +"C„x\ (2) PERMUTATIONS AND COMBINATIONS. 235 Putting X = I, and transposing i, we obtain 2" _ I =. "C, + "C, + "Q + ... + "C«, which proves the proposition. *351. "Cr is greatest wJien r=^n^rr = Kn ± i), according as n is even or odd. I n Evidently "C> or \ — ^= — » is greatest when \r\n — r ^ \r \n — r ■- ' is least. Since \a -\- i [^ — 'i is obtained from \a \a by mul- tiplying hy a -\- I and dividing by a, it follows that Irt 1^ < 1^ + I \a~ \ < |« + 2 1^ — 2 < ... Hence when n is even, \r \n — r is least, and there- fore "Cr is greatest, when r= 71 — r, or r = | ;/. Again |^ \fi -\- i = | /^ + i |^ and | /^+ I [^ < 1^4- 2 | /^ - i < |/^ + 3 | /^-2 < ... Hence when n is odd, Ir |« — r is least, and there- fore "Cr is greatest, when r = n — r ± i,orr=|(« ± I). EXERCISE 45. 1. How many different arrangements can be made of the letters of the word commencement ? Of the 12 letters, 2 are ^'s, 3 are m^s, 3 are ')" ? i^ + yf'' ? PROBABILITY. 23/ CHAPTER XIX. PROBABILITY. 352. If an event may happen in a ways and fail in b ways, and each of these ways is equally likely, the a Probability, or the Chance, of its happening is — — :, b ^"^^ and the probjbi'tty of its failing is -. Hence to find the probability of an event, divide the Jiimibcr of cases that favor it by the whole Jiumber of cases for and against it. For example, if in a lottery there are 5 prizes and 22 blanks, the probability that a person holding i ticket will win a prize is ^, and the probability of his not winning is |^. Example i. From a bag containing 8 white, 7 black, and 5 red balls, one ball is drawn. Find the chance, (i) that it is white, (2) that it is black or red. In all there are 20 ways of drawing a ball ; of these 20 ways 8 are favorable to drawing a white ball, and 12 to drawing a black or a red ball; hence the chance of the ball being white is -^Q, or |, and that of its being black or red is f . Example 2. From a bag containing 7 white and 4 red balls, 3 balls are drawn at random. Find the chance of these being all white. The whole number of ways in which 3 balls can be drawn is "Cg; and the number of ways of drawing 3 white balls is 'Cg; therefore, of drawing 3 white balls , , 'Q 7-6.5 7 the chance = r^ - — — = — • "Cg II. 10. 9 33 238 ALGEBRA. 353. Unit of Probability. The sum of the proba- . . . a b . . bihties and is unity; hence if the prob- a ^ b a + b ^ ^ abihty that an event will happen is/, the probability that it will fail is i — /. If b is zero, the event is certain to happen, and its probability is unity ; hence certainty is the tuiit of pi'obability. Instead of saying that the probability of an event a is 7, we sometimes say that the odds are a /^ b in a -\- b favor of the event, or h to a. agaijist it. Example i. Find the chance of throwing at least one ace in a single throw with three dice. Here it is simpler to first find the chance of not throwing an ace. Each die can be thrown in five ways so as not to give an ace ; hence the three can be thrown in 5^, or 125, ways that will exclude aces (§ 339). The total number of ways of throwing 3 dice is 6^, or 216. Hence the chance of not throwing one or more aces is 125 -^ 216 ; so that the chance of throwing at least one ace is i - \\% or ^^Ve (§ 353)- Here the odds against the event are 125 to 91. Example 2. A has 3 shares in a lottery in which there are 4 prizes and 7 blanks; B has i share in a lottery in which there is I prize and 10 blanks ; show that A's chance of success is to B's as 26 is to 3. A can get all blanks in "^Cg, or 35, ways ; he can draw 3 tick- ets in "Cg, or 165, ways ; hence A's chance of failure — /g^ = jg- Therefore A's chance of success = i — /j = f |. B's chance of success is evidently ^^\ .'. A's chance : B's chance = ff : i\ = 26 : 3. PROBABILITY. 239 Or to find A's chance we may reason thus : A may draw 3 prizes in ^Cg , or 4, ways ; he may draw 2 prizes and i blank in *C^ x 7, or 42, ways; he may draw i prize and 2 blanks in 4 x 'Q, or 84, ways ; hence A can succeed in 4 + 42 + 84, or 130, ways. Therefore A's chance of success = y^ = |f. EXERCISE 46. 1. From the vessel on which Mr. A took passage one person has been lost overboard. There were 60 passengers and 30 in the crew. Find, (i) the chance that Mr. A is safe, (2) the chance that all the passengers are safe, (3) the probability that a passenger is lost. 2. There are 15 persons sitting around a table ; find the probability that any 2 given persons sit together. Wherever one of the 2 persons sits, the other may occupy any one of 14 places, of which 2 will put the 2 persons to- gether. 3. According to the Carlisle Table of Mortality, it appears that out of 6335 persons living at the age of 14 years, only 6047 reach the age of 2 1 years. Find the probability that a child aged 14 years will reach the age of 21 years. Find the chance that he will not reach it. 4. From a bag containing 4 red and 6 black balls, 2 balls are drawn; find the chance, (i) that both are red, (2) that both are black, (3) that one is red and the other black. 5. From a bag containing 4 white, 5 black, and 6 red balls, 3 balls are drawn ; find the probability that (i) all are white, (2) all black, (3) all red, (4) 2 black and i red, (5) i white and 2 black. Ans. ^f^, ^, ^\, H» /t- 240 ALGEBRA. 6. When two coins are thrown, find the chance that the result will be, (i) both heads, (2) both tails, (3) head and tail. 7. When two dice are thrown, what is the probability of throwing, (i) a 5 and 6, (2) two 6's? 8. ;^rom a committee of 7 Republicans and 6 Democrats, a sub-committee of 3 is chosen by lot. What is the proba- bility that it will be composed of 2 Republicans and i Democrat? , 9. From a committee of 8 Democrats, 7 Republicans, and 3 Independents, a sub-committee of 4 is chosen by lot. Find the chance that it will consist, (i) of 2 Democrats and 2 Republicans, (2) of i Democrat, 2 Republicans, and i Independent, (3) of 4 Democrats. 10. In a single throw with two dice, show that the chance of throwing 5 is l- ; of throwing 6 is /^. IT. One of two events must happen, and the chance of the first is two thirds that of the second ; find the odds in favor of the second. 12. In a bag are 4 white and 6 black balls; find the chance that, out of 5 drawn, 2, and 2 only, shall be white. 13. In Example 12 show that the chance of 2 at least being white is %\. 14. Out of TOO mutineers, a general orders two men, cho- sen by lot, to be shot ; the real leaders of the mutiny being TO, find the chance that, (i) one of the leaders will be taken, (2) two of them. PROBABILITY. 24 1 15. A has 3 shares in a lottery containing 3 prizes and 9 blanks ; B has 2 shares in a lottery containing 2 prizes and 6 blanks ; compare their chances of success. 16. There are 4 half-dollars and 3 quarter-dollars placed at random in a line ; prove that the chance of the extreme coins being both quarter-dollars is f. In the case of in half- dollars and 11 quarter-dollars, show that the chance is n{n-\) {m -\- n){m -\- n — i) 17. There are three works, one consisting of 3 volumes, another of 4, and tlie third of i volume. They are placed on a shelf at random ; prove that the odds against the vol- umes of the same works being all together are 137 to 3. 18. A man wants a particular span of horses from a stud of 8. His groom brings him 5 horses taken at random. What is the chance that both horses of the span are among them ? 19. Of the three events A, B, C, one must, and only one can, occur ; A can occur in a ways, B in b ways, and C in ^ ways, all the ways being equally likely ; find the chance of each event. Compound Events. 354, Thus far we have considered only single events. The concurrence of two or more events is sometimes called a Compound event. Events are said to be Dependent or Independent^ according as the happening (or failing) of one event does or does not affect the occurrence of the other. 242 ALGEBRA. 355. The probability that two independeitt events will both happen is equal to the product of their sepa- rate probabilities. Suppose that the first event may happen in a ways and fail in b ways, all these cases being equally likely ; and suppose that the second event may happen in a^ ways and fail in b^ ways, all these cases being equally likely. Each of the a -{■ b cases may be associated with each of the a^ + b' cases to form {a + b) {ci + U) compound cases, all equally likely to occur. In a a' of these compound cases both events hap- pen, in by of them both fail, in ab^ of them the first happens and the second fails, and in a^ b of them the first fails and the second happens. Hence 7 ;^v^i 77T === the chance that both events happen ; {a + b) {a} + ^0 bb' = the chance that both events fail ; (a + b) (af + b') a b^ f the chance that the first happens and =1 {a + b) (a' + b') \ the second fails ; a' b { the chance that the first fails and the + ^') 1 (a + b) (a' + b^) \ second happens. 356. The probability that any number of independent events will all happen is equal to the product of their separate probabilities. Let ply p2, and /a be the respective probabilities of three independent events. The probability of the concurrence of the first and second events is pi p2\ PROBABILITY. 243 the probability of the concurrence of the first two events and the third is {pipi) pz, ox p^p,p^\ and so on for any number of events. 357. By § 356, if / is the chance that an event will happen in one trial, the chance of its happening each time in 71 trials is/". 358. The chance that all three of the events in §356 will fail is (I -A) (I -/,) (I -/a). Hence the chance that some one at least of them will happen is i — (i — /i) (i — A) (» — Pz)- The chance that the first two will happen and the third fail is AA(i -/a). Example i. Find the chance of throwing an ace in the first only of 2 successive throws with a single die. The chance of throwing an ace in the first throw = \. The chance of not throwing an ace in the second throw = |. Hence the chance of the compound event = J x ^ = ^. Example 2. From a bag containing 6 white and 9 black balls, 2 drawings are made, each of 3 balls, the balls first drawn being replaced before the second trial ; find the chance that the first drawing will give 3 white, and the second 3 black balls. The number of ways of drawing 3 balls = ^^q . " " " " 3white = 6<^;; 3 black = 9(73. Hence, of drawing 3 white balls at first trial the chance = ..^ = ' ^ ' ^ = ~- . i^Cg 15. 14. 13 91' 65* 4 I2_ 48 5915 244 ALGEBRA. and, of drawing 3 black balls at second trial the chance = -p-^ = —^ — ^-^— i-^Cg 15. 14. 13 Hence the chance of the compound event: Example 3. If the odds are 1 1 to 9 against a person A, who is now 38 years old, living till he is 68, and 4 to 3 against a person B, now 43, living till he is 73 ; find the chance that one at least of these persons will be alive 30 years hence. The chance that A will die within 30 years = il ; the chance that B will die within 30 years = -f ; hence the chance that both will die = ^l X ^ = II ; therefore the chance that both will ?wl die, that is, that one at least will be alive, = i — -i| == ||. Example 4. In how many trials will the probability of throwing an ace with a single die amount to |? Let X = the number of trials. By § 357, the chance of failing to throw an ace each time in x trials is (l)"- Hence the chance of throwing an ace once at least in x trials is 1 — (|)^ ; .-. i-(l)^ = ior(i-)- = ^; • X = ^"g3 ^ 0-4771 213 ^ g^^^ log 6 — log 5 0.0791813 Hence in 6 trials the chance of success is a little less than |, and in 7 trials it is greater than |. 359. Dependent Events. A slight modification of the meaning of ^ and <^' in § 355 enables us to estimate the chance of the concurrence of two dependent events. Thus, if after the first event has happened, a' denote the number of ways in which the second can follow, and U the number of ways in which it will not fol- low; then the number of cases in which the two PROBABILITY. 245 events will both happen is aa\ and the chance of , . . ^ ^' their concurrence is rr^—. 77-. {a + /;) {a' + b') Hence if/ is the chance of the first of two depen- dent events, and /' the chance that the second will follow, the chance of .their concurrence is //'. Example. From a bag containing 6 white and 9 black balls, two drawings are made, each of 3 balls, the balls first drawn not being replaced before the second trial ; find the chance that the first drawing will be 3 white and the second 3 black balls. At the first trial, 3 balls may be drawn in ^^Cg ways ; and 3 white balls may be drawn in ^C^ ways ; hence the chance !^ = ^-5-4 ^ 4_ 'C3 15.14.13 91- After 3 white balls have been drawn the bag contains 3 white and 9 black balls ; therefore, at the second trial, 3 balls may be drawn in ^^Q ways ; and 3 black balls may be drawn in ®6"g ways ; hence, of drawing 3 black balls at the second trial, , , »C. 9-8.7 21 the chance = ttt^ = = — . i^Cg 12 . II . 10 ss Hence the chance of the compound event = ^|- x |^ = -jSos- The student should compare this result with that of Example 2 in § 358. EXERCISE 47. 1. Show that the chance of throwing an ace in each of two successive throws with a single die is ^jr. 2. Show that the chance of throwing an ace with a single die in two trials is \^. of 3 white balls at the first trial = j. 246 ALGEBRA. 3. A traveller has 5 railroad connections to make in order to reach his destination on time. The chances are 3 to i in favor of each connection. What is the probability of his making them all? 4. Mr. A takes passage on a ship for London. The probability that the ship will encounter a gale is -^. The probability that she will spring a leak in a gale is ^. In case of a leak, the probability that the engines will be able to pump her out is f. If they fail, the probability that the compartments will keep her afloat is £. If she sinks, it is an even chance that any one passenger will be saved by the boats. What is the probability that Mr. A will be lost at sea on the voyage? 5. In how many trials will the chance of throwing an ace with a single die amount to ^ ? Ans. In 4 trials the chance is a little greater than ^. 6. The odds against A's solving a certain problem are i to 2, and the odds in favor of B's solving the same problem are 3 to 4 ; find the chance that the problem will be solved if they both try. 7. The chance that a man will die within ten years is ^, that his wife will die is ^, and that his son will die is ^ ; find the chance that at the end of ten years, (i) all will be living, (2) all will be dead, (3) one at least will be living, (4) husband living, but wife and son dead, (5) wife living, but husband and son dead, (6) husband and wife living, but son dead. PROBABILITY. 247 8. A bag contains 2 white balls and 4 black ones. Five persons, A, B, C, D, E, in alphabetical order each draw one ball and keep it. The first one who draws a white ball is to receive a prize. Show that their respective chances of winning are as 5 : 4 : 3 : 2 : i. A's chance of winning the prize is easily obtained. That B may win, A must fail. Hence to find B's chance, we find, (i) the chance that A fails, (2) the chance that if A fails B will win. We then- take the product of these chances. That C may win, (i) A must fail, (2) B must fail, (3) C must draw a white ball. Hence C's chance of winning is the product of the chances of these three events; and so on. 9. A and B have one throw each of a coin. If A throws head, he is to receive a prize ; if A fails and B. throws head, he is to receive the prize. If A and B both fail, C receives the prize. Find the chance of each man winning the prize. 10. From a bag containing 5 white and 8 black balls two drawings are made, each of 3 balls, the balls not being replaced before the second drawing ; find the chance that the first drawing will give 3 white and the second 3 black balls. 11. In three throws with a pair of dice, find the chance of throwing doublets at least once. 12. Find the chance of throwing 6 with a single die at least once in 5 trials. 13. The odds against a certain event are 6 to 3, and the odds in favor of another event independent of the former are 7 to 5 ; find the chance that one at least of the events will happen. 248 ALGEBRA. 14. A bag contains 17 counters marked with the numbers T to 1 7. A counter is drawn and replaced ; a second draw- ing is then made ; find the chance that the first number drawn is even and the second odd. * 360. If a7i evefit can happen m two or more different ways, which are mutually exclusive, the cha^tce that it will happen is the sum of the chances of its happening in these different zvays. When these different ways are all equally probable, the proposition is merely a repetition of the definition of probability. When they are not equally probable, the proposition is often regarded as self-evident from that definition. It may, however, be proved as follows : Let -r and ~ be respectively the chances of the happening of an event in two ways that are mutually exclusive. Then out of b^ hi cases there are a^ b-i cases in which the event may happen the first way, and a^ bi cases in which the event may happen the second way; and these ways are mutually exclusive. There- fore out of bi b^ cases, ai b^ + a^ b^ cases are favorable to the event; hence the chance that the event will happen in one of these two ways is a, be. -f- a^ b, a, a^ ■ \ , - , or T^ + X • bi b^ b^ b^ Similar reasoning will apply to any number of exclusive ways in which an event may happen. PROBABILITY. 249 Hence if an event can happen in n ways which are mutually exclusive, and if /i, /.,, /s, ••., /„ are the probabilities that the event will happen in these dif- ferent ways respectively, the probability that it will happen in some one of these ways is/i+/2+/3 + ...+A. Example i. Find the chance of throwing at least .8 in a single throw with two dice. 8 can be thrown in 5 ways, .*. the chance of throwing 8 = ^ 9 can be thrown in 4 ways, .*. the chance of throwing 9 = ^ 10 can be thrown in 3 ways, .*. the chance of throwing 10 = j\ ir can be thrown in 2 ways, ••. the chance of throwing 11 = 3!^ 12 can be thrown in i way, .-. the chance of throwing 12 = ^^. These ways being mutually exclusive, and 8 at least being thrown in each case, the required chance = A+8^ + A + A + ^ = tV Example 2. One purse contains 2 dollars and 4 half-dollars, a second 3 dollars and 5 half-dollars, a third 4 dollars and 2 half-dollars. If a coin is taken from one of these purses selected at random, find the chance that it is a dollar. The chance of selecting the ist purse = \\ the chance of then drawing a dollar = | = ^; .-. the chance of drawing a dollar from ist purse = ^ • ^ = ^. Similarly, the chance of drawing a dollar from 2d purse = \\ and the chance of drawing a dollar from 3d purse = f . .-. The required chance = \ + \ + ^ = \\- It is very important to note that when, as in the two ex- amples given above, the probability of an event is the sum of the probabilities of two or more separate events, these separate (vents must be mutually exclusive^ 2SO ALGEBRA. * 361. If p denote the chance of an evefit happening in one trial, and q == i _ p ; then the chance of its hap- pening r times exactly in n trials is "Cr p' q"~'. For if we select any particular set of r trials out of the whole number n, the chance that the event will happen in every one of these r trials and fail in the rest is/''/'"'' (§§ 355, 357); and as in the n trials there are "Cr sets of r trials, which are mutually ex- clusive and equally applicable, the chance that the event will happen r times exactly in n trials is * 362. The chance that an eveiit will happen at least r times in n trials is /« + ''C^r-^q + "C.p'^-^g'' + .-. + "C/-^''-^ or the sum of the first n — r + i terms of the expajision ^/(p + q)"- For an event happens at least r times in n trials, if it happens n times, or ;/ — i times, or n — 2 times, ..., orrtimes; and if in "Crp''q"~''wQ put r equal to n, 71 — I, 71 — 2, .»., r, in succession, and add the results, remembering that "C-r = "C we obtain the expression given above. Example. In 5 throws with a single die, find, (i) the chance of throwing exactly 3 aces, (2) the chance of throwing at least 3 aces. PROBABILITY. 2$ I Here p — \^ ^ = f , « = 5, ^ = 3 ; hence the chance of throwing exactly 3 aces = ^C^ (1)^ (|)2 = Yy^L. The chance of throwing at least 3 aces is the sum of « — r + i, or 3, terms of the expansion of {\ -\- \)^^ or the chance = {\f + 5 (i)Ht; f 10 i\y (t)^ = ^A- *363. Expectation. If / be a person's chance of winning a sum of money M, then Mp is called his expectation, or the value of his hope. The phrase probable value is often applied to things in the same way that expectation is to persons. Example. A and B take turns in throwing a die, and he who first throws a 6 wins a stake of $ 22. If A throws first, find their respective expectations. In his first throw, A's chance is i ; in his second throw, it is {\y^ X ^ ; in his third, it is {\y x \ ; and so on. Hence A's chance = H^ + (1)^ + (6)* + ••• 1- Similarly, B's chance = I • i { i + (1)'-^ + (|)* + ..•}• Hence A's chance is to B's as 6 is to 5 ; or their respective chances are ^^^ and ^^. Therefore their expectations are $ 12 and % 10 respectively. Note. The theory of probability has its most important applications in Insurance and the calculation of Probable Error in physical investigations. It is also applied to testimony and causes. But the limits of this treatise exclude further consider- ation either of the theory or its. applications. For a fuller treatment the student may consult Hall and Knight's Higher Algebra, Todhunter's Algebra, Whitworth's "Choice and Chance," and the articles Annuities, Insurance, and Proba- bility in the "Encyclopaedia Britannica." A complete account of the origin and development of the subject is given in Tod- hunter's " History of the Theory of Probability from the time of Pascal to that of Laplace." 252 ALGEBRA. EXERCISE 48. 1. Find the chance of throwing 9 at least in a single throw with two dice. 2. One compartment of a purse contains 3 half-dollars and 2 dollars, and the other 2 dollars and i half-dollar. A coin is taken out of the purse ; show that the chance of its being a dollar is y^g. 3. If 8 coins are tossed, find the chance, (1) that there will be exactly 3 heads, (2) that there will be at least 3 heads. 4. If on an average i vessel in every 10 is wrecked, find the chance that out of 5 vessels expected, (i) exactly 4 will arrive safely, (2) 4 at least will arrive safely. 5. If 3 out of 5 business men fail, find the chance that out of 7 business men, (i) exactly 5 will fail, (2) 5 at least will fail. 6. Two persons, A and B, engage in a game in which A's skill is to B's as 3 to 4 ; find A's chance of winning at least 3 games out of 5. 7. If A's chance of winning a single game against B is f, find the chance, (i) of his winning exactly 3 games out of 4, (2) of his winning at least 3 games out of 4. 8. A person is allowed to draw two coins from a bag containing 4 dollars and 4 dimes; find the value of his expectation. 9. From a bag containing 6 dollars, 4 half-dollars, and 2 dimes, a person draws out 3 coins at random ; find the value of his expectation. PROBABILITY. 253 10. Two persons toss a dollar alternately, on condition that the first who gets "heads" wins the dollar; find the expectation of each. 11. Find the worth of a lottery-ticket in a lottery of 100 tickets, having 4 prizes of $ 100, 10 of $ 50, and 20 of $ 5. 12. Three persons, A, B, and C, take turns in throwing a die, and he who first throws a 5 wins a prize of ^ 182 ; show that their respective expectations are $ 72, $60, and $50. 13. A has 3 shares in a lottery in which there are 3 prizes and 6 blanks ; B has i share in a lottery in which there is i prize and 2 blanks. Compare their chances of success. 14. Show that the chance of throwing more than 15 in one throw with three dice is j^-g. 15. Compare the chances of throwing 4 with one die, 8 with two dice, and 12 with three dice. 16. There are three events A, B, C, one of which must, and only one can, happen. The odds are 8 to 3 against A, 5 to 2 against B ; find the odds against C. 17. A and B throw with two dice ; if A throws 9, find B's chance of throwing a higher numbsr. 1 8. The letters in the word Vermont are placed at random in a row ; find the chance that any two given letters, as the two vowels, are togetlier. 254 ALGEBRA. CHAPTER XX. CONTINUED FRACTIONS. 364. An expression of the general form b a + d is called a Continued Fraction. We shall consider only the simple form «i + «2 + ^+..., 3 a^ in which a^y a^, a^, ... are positive integers. This is often written in the more compact form III a H — . . . The quantities a^, ^2» ^3» • • • ^^e called quotients, or partial quotients. A continued fraction is said to be terminating or non- terminating, according as the number of the quotients a^, a^, a^, ... is limited or unlimited. Any terminating continued fraction can evidently be reduced to an ordinary fraction by sim- plifying the fractions in succession, beginning from CONTINUED FRACTIONS. 255 the lowest. Hence any terminating continued frac- tion is a commensurable quantity. I a^ a^ (^^1^2+ + ^1 Thus, rt^i + — = ^1 + a^a^^- I a^a^-\- I 365. To convert a given fraction into a continued fraction. ffi Let — be the given fraction ; divide m by ;/, and let ^1 be the integral quotient and r^ the remainder; /// r, I then ~ — a^-\ — = a^ -\ — Divide n by r^ with quotient ^2 and remainder ^2, then - = «3 + -^ = ^2H Divide rx by ^2 with quotient a^ and remainder ^3 ; and so on. TJ W I II Hence — = ^. -| , or a, -\- — ; . . . i^ .,+ .3+ «3 + --- \int< ;/, Ui = 0, and a^ is obtained by dividing n by m. The above process is evidently the same as that of finding the G. C. D. of in and n ; therefore if m and ft are commensurable, we must at length obtain as the remainder, and the process terminates. 256 ALGEBRA. Hence any fraction whose terms are commensur- able can be converted into a terminating continued fraction. Example i. Reduce — — to a continued fraction. Here the quotients are 3, 5, 7, 9 ; 1051 _ III 329 -3 + ^ ^ 9' Example 2. Reduce j~ to a continued fraction. Here the quotients are 0, 3, 5, 7, 9 ; 329 I I I I 1051 -3+ 5+ 7^ 9' 366. Convergents. The fractions obtained by stop- ping at the first, second, third, ..., quotients of a continued fraction, as <7 I ' II -i, «i + — , ^i + — T -, ••• I «2 ^2 + ^3 or when reduced to the common form, a^ a^a^-\- i a^ {a^ a^-\- i) + a^ 1 * «2 ' ^3 ^2 + I ' * " are called respectively the Ji7'st, secondy thirds ..., convergents. 367. The successive convergents are alternately less and greater than the continued fraction. The first convergent, a^ , is too small, since the part ... is omitted. The second convergent, «2 + «8 + CONTINUED FRACTIONS. 257 a^-i , is too great, for the denominator ^2 is too ^2 II. small. The third convergent, a^ -\ , is too J ^2 + ^8 small, for a^ -\ is too great (^3 being too small) ; and so on. 368. To establish the law of formation of the sue- ' cessive convergents. If we consider the first three convergents, a, ^ a,a^+ I ^ a, (^, a, + i) + ^^i ^ g ^66. we see that the numerator of the third convergent may be obtained by multiplying the numerator of the second convergent by the third quotient, and adding the numerator of the first convergent; also that the denominator may be formed in a similar manner from the denominators of the first two con- vergents. We proceed to show that this law holds for all subsequent convergents. Let the numerators of the successive convergents be denoted by/,, /._,» A^ •••> and the denominators b\ (7p <72' ^8» ••• Assume that the law holds for the ;/th convergent ; then A = ^«A-i+A-2, (i) and ^„ = a„^„_i + ^„_2' (2) 258 ALGEBRA. The (;/ + i)th convergent evidently differs from the nth. only in having a, •] ^ - in the place of a„; ^n + I hence ^ an+l {Cinpn-X ^Pn-^ ^-pn-1 an+x{cinqn-\ + ^„ - 2) + Qn-X ^n + lPn + Pn~\ , , x , , x "■n + 1 y « i^ y « - 1 Hence the law holds for the {n + i)th convergent, if it holds for the ;/th. But it does hold for the third ; hence it holds for the fourth ; and so on. Therefore it holds universally after the second. The method of proof employed in this article is known as Mathematical Induction. Example i. Calculate the successive convergents of I I I I I ^"'"BT r+ T+ II + 2" Here ^1, a^, ^3, ^4, ^5^ ^6J are 2, 6, I, I, II, 2. Hence ^^ = 9i 2 I ' A_2 , 1-L3. A I X 13 + 2 15 1x6+1 ~ 7 ' A IX 15 + 13 28 Qz ^4 1x7 + 6 13' A 11x28+15 323 A 2 x 323 + 28 674 ^6 II X 13 + 7 "" 150' ^6 2X150+13 313 CONTINUED FRACTIONS. 259 Example 2. Find the successive convergents of I I I I I I I 2T 2T 3+ i~+ 4T 2T 6* Here a^, a^, n^, a^, ^5, ^5, ^7, a^, are 0, 2, 2, 3, i, 4, 2, 6. wpnr. 9 i ^ _! 9 _43 95 ^ nence j, 3, ^, 17, 22' 105' 232' 1497* are the successive convergents. EXERCISE 49. Compute the successive convergents to I I I I II I I I I I ^' 7+7+6 + ^10* I 1 I I I I Express each of the following fractions as a continued fraction, and find its convergents : Reduce to a continued fraction, and find the fourth con- vergent to, each of the following numbers : 12. 0.37. 13. 1. 139. 14. 0.3029. 15. 4.316. Write 0.37 as a common fraction, ^^\ then proceed as above. 260 ALGEBRA. 369. The difference betiveen any tivo consecutive con- vcr gents is iDiity divided by the product of their de- nofninaiors ; that is^ Vu ^H + 1 ^n^>i+l The law holds for the first two convergents, since ^1 ai a^ -\- I I ^2 (I) Assume that the law holds for — — ^ and — , so that qn-x qn Pnqn-l--qnPn-l= Ij (2) then by § 368 Pn qn+1 '^Pn+l qn = Pn (^«+l ^« + ^..-l) ^ qn {^n+lPn + Pn-l) ' = Pnqn-l-qnPn-l = I, by (2). Hence, if the law holds for one pair of consecutive convergents, it holds for the next pair. But by (i), the law does hold for \hQ first pair; therefore it holds for the second pair ; and so on. Therefore it is universally true. 370. Any convergent/„-r ^„ is in its lowest terms. For if /„ and q„ had a common factor, it would also be a factor o{ p,,q„j^^ ^ p„^^q„, or unity; which is impossible. * The expression x ~j denotes "the difference between ;r and ^." CONTINUED FRACTIONS. 26 1 371. Let X denote the value of any continued frac- tion ; then, by § 367, x Hes between any two consecu- tive convergents; hence ,'. x^^<-^— <-^ . §§369,368. qn qnqn + l (^«)^« + l P T That is, — differs from x by less than qn {q.y',~ P>,q..+\ _ I qn + 1 q„^. i{A q„ + 1+ q,i) q,, + 1 (^ $^„ + 1 + q,^ ' Now ^ > I, and q„ < ^„ + ,; hence the difference between the (« + i)th convergent and x is less than 262 ALGEBRA. the difference between the ;/th convergent and x ; that is, any convergent is nearer to the continued fraction than the next preceding convergent, and therefore than any preceding convergent. From this property and that of § 367, it follows that The convergent s of an odd oi^der continually increase ^ but are always less than the continued fraction. The convergents of an even order continually decrease^ but are always greater than the continued fraction. Example. Find the successive convergents to 3. 141 59. Here the quotients are 3, 7, 15, i, 25, i, 7, •..; hence the convergents are f , ^, ||f , f f |, • • • If the 4th convergent, which is greater than 3. 141 59, be taken as its value, the error will be less than i ^ 25 (113)^ and there- fore less than i -^ 25 (loo)^ or 0.000004. The convergents above will be recognized as the approxi- mate values of tt, or the ratio of the circumference of a circle to its diameter. This example illustrates the use of the properties of continued fractions in approximating to the values of incom- mensurable ratios or those represented by large numbers. 373. Any convergent approaches more nearly the value of the continued fraction, x, than any other fraction whose denominator is less than that of the convergent. f p^^ For let the fraction - be nearer to x than — ; then s q„ is it nearer to x than the (;/ — i )th convergent (§ 372) ; and since x lies between the ;/th and the CONTINUED FRACTIONS. 263 (;;_i)th convergent, r -^ s does also; hence we have s q„-\ qn qn-\ q»qn-i s , . .'. rq„_i^sp„_i <-. (I) Now the first member of(i) is an integer; hence r pn s > q„; that is, if - is nearer x than is — , J > q„. 374. Periodic Continued Fractions. A continued fraction in which the quotients recur is called a periodic^ or recurring^ continued fraction. Any quadratic surd can be expressed as a periodic continued fraction. We give the following example to illustrate this principle, and to exhibit the use of the properties of continued fractions in approximating to the value of a quadratic surd. Example. Convert /y/TJ into a continued fraction, and find its convergents. Since 3 is the greatest integer in ^JTs, we write * A/ I V15 + 3 (I) _Vl5±3^i + \/i5ii3^,+ (2) 6 '6 V^+3' VTI+3 ^6^Vri-3^^^ 6_^ The last fraction in (3) is the same as that in (i); hence after this the quotients i, 6, will recur. 264 ALGEBRA. I I I I Hence v^=3 + — gr; 7+ 6 The quotient in each of the identities, (i), (2), (3), is the greatest integer in the value of its first member. The nu- merator of each fraction is rationahzed so that the inverted fraction will have a rational denominator. To find the convergents, we have the quotients 3, I, 6, I, 6, I, 6, ...; .3 4 2 7 .31. i\3 2 4JL ... 1> U T ) 8 ' 55 ' 63 » are the first six convergents. The error in taking the sixth convergent as the value of /v/iy is less than 1-^-6 (63)'^, and therefore less than 0.0C005. 375. Every periodic continued fraction is equal to one of the surd roots of a quadratic equation with rational coefficients. The following example will illustrate this general truth. I T I I , Example. Express i + — -— — ^ — • • • as a surd. Let X denote the value of the continued fraction ; then I I I I ~2 + 3+ 2 + 3 + I I ~2 + 3 + (^- 0' ix'^A-ix-T- = 0. The continued fraction, being positive, is equal to the positive root of this equation, or \ {^^J ^S ~ i)- CONTINUED FRACTIONS. 265 EXERCISE 50. Reduce to a continued fraction, and find the sixth con- vergent to, each of the following surds : I. Vl' 3- V6- 5- Vm- 7- 3^5- 2. V^. 4. \/T^. 6. V22' 8. 4V10. 9. In each of the above examples the difference between the surd and the sixth convergent is less than what? 10. Find the first convergent to I I I I I 1 + 3 + 5 + 7 + 9 + 11 + which differs from it by less than 0.000 1. 11. Find the first convergent to VToi that differs from it by less than 0.0000004. 12. Given that a metre is equal to 1.0936 yards, show that the error in taking 222 yards as equivalent to 203 metres will be less than 0.000005. 13. A kilometre is very nearly equal to 0.62138 miles ; show that the error in taking 103 kilometres as equivalent to 64 miles will be less than 0.000025. 14. Find the first six convergents to the ratio of a diago- nal to a side of a square. The difference between each of the six convergents and the true ratio is less than what? 15. Express 3 + ^ A: ^ • • • as a surd. 16. Express — ; : ... as a surd. I + 3 + I + 3 + 266 ALGEBRA. CHAPTER XXI. THEORY OF EQUATIONS. 376. The General Equation. Let n be any positive integer, and/i,/2.A, •••>/«> be any rational known quantities; then the equation X'' +A-^"~' +A-^""' + ••• +A-i-^ + A = (A) will be the general type of a rational integral equa- tion of the ?/th degree. In this chapter we shall let F{x) = x''+ A -^'^-^ + A ^"~' + • • • + A and write equation (A) briefly F (x^ = 0.* 377. A Root of the equation F{x) = is any value of ;r, real or imaginary, that causes the function, F{x)y to vanish. 378. Reduction to the form F(^x) = 0. In general, any equation in x having rational coefficients can be transformed into an equation of the form F(x') = 0. The following example will illustrate the general truth. * What properties of the equation F{x)=0 belong also to the equation formed by putting any rational integral function of x equal to zero, the reader will readily discover. THEORY OF EQUATIONS. 267 I — x^ x—^ 4- 3 Example. Reduce — — — = — r to the form oiF{x) — 0. i + -^ X^+2. Clearing the given equation of fractions we obtain x^ — x^ + 2 — 2 x^ = x~'^ + ^ + I + Z^' Multiplying by x to free of negative exponents, we obtain X^ — X^ -2X^ = I + 2X+ :iX^. (l) To transform (i) into another equation with integral expo- nents, put x=jy\ 6 being the L. C M. of the denominators of the fractional exponents of x. We thus obtain y _^i8 _ 2jio = I + 2j/« + 3>/i2, or ys + 3^2 4. 2 jlO _y -I- 2_;/6 + I = 0, (2) which is in the required form. The roots of (i) and (2) hold the relation x=j^^. EXERCISE 51. Reduce the following equations to the form I^(x) = : 2 L I. 3^.- + ^x- — 1 = 1. 2. x^- - I I I ^-2 I + J -3-* 3- Vi -X^: 4. V2 X — ■zx^ — X = \/l — X. 379. Divisibility of F(x). If F (x) is divided b), X — a, the remainder will be F (a). 268 ALGEBRA. Divide Fix) hy x — a until a remainder is obtained that does not involve x. Let /\ (;r) denote the quotient, and R the remainder; then F{x) = (x-~a) F, {x) + F. Since R does not involve x, it is the same for all values of x. Putting ;r = ^, v^e obtain F{a) = X ^1 W + ^ = ^, the remainder. //* F (a) = 0, F (x) is divisible by x — a ; ajid con- versely. Example. If n is even, show that x" — ^" is divisible by Since n is even, and F{x) m x" — b^; .-. Fi-b)^b^-b^ = 0. Hence x» — b" is divisible by ;ir — (— b), or x + b. 380. If 3. is a root of the equation F (x) = 0, that is, if F (a) = 0, theft F (x) is divisible by x — a (§ 379). Conversely, if F (x) is divisible by x — a, the 71 F {a) = 0, that is, a is a root of the eqtmtion F (x) = 0. 381. Horner's Method of Synthetic Division. Let it be required to divide Ax^-\- Bx'^+ Cx^ D hy x-a. In the usual method given below, for convenience we write the divisor to the right of the dividend and the quotient below it. + *{Aa^ + Ba + C) THEORY OF EQUATIONS. 269 Ax^+Bx^ +Cx +D *Ax^- Aax^ (Aa + B)x^ *(Aa + B)x^ -{Aa'^ + Ba)x iAa^-\-Ba+C)x * (A a'^ + B a ^ C) X - (A a^ -{- B a^+ C a) Aa^ + Ba^+Ca + D Here the remainder, A a^ -{- B a^ -{- Ca + D, is the value of the dividend, A x^ + B x^ + C x -{-£>, ior x = a, which aliorcis a second proof of § 379. In the shorter or synthetic method, we write the coefficients of the dividend with a at their right as below : ^B +C +B \a_ + Aa +Aa^-{-Ba +Aa^-]-Ba^-\-Ca Aa + B Aa^ + Ba + C Aa^ + Ba'^+Ca + D Multiplying A by a, writing the product under B^ and adding, we obtain Aa + B. Multiplying this sum by «, writing the product under C, and addin^"~^ + ••• + A = (^ — ^1) («^ — ^2) ••• {^ — O- Multiplying together the factors of the second member, and equating the coefficients of like powers of X (§ 263), we obtain the theorem. Thus, when « = 2, we have x^ +PiX +p^=x^ — (a^ + a^) X + a^ a^ ; r. —A = ^h + ^2> A = ^1^2* THEORY OF EQUATIONS. 275 When n = 3, we have = x^— (a^ + ^2 + ^3) ^^ + (^1 ^2 + «i ^3 + ^2 ^3) -^ — ^1 ^2 ^3 -' ••• —A = ^1+^2+^3? A = «1^2+^1^3+^2«3> — A = ^1^2^3- From the laws of multiplication it is evident that the same relation holds when ;/ = 4, 5, 6, ... If the term in x"~^ is wanting, the sum of the roots is 0, and if the known term is wanting, at least one root is 0. Thus, in the equation x^ -\- 6 x"^ — 11 x — 6 = 0, the sum of the roots is 0; tiie sum of their products taken two at a time is 6; the sum of their products taken three at a time is 11 ; and their product is — 6. Note that - A = (~0A' A = (- ^fPv • • • Note. The coefficients in any equ ition are functions of the roots; and conversely, the roots are functions of the coeffi- cients. The roots of a literal quadratic equation have been ex- pressed in terms of the coefficients (§ 144). The roots of a literal cubic or biquadratic equation may also be expressed in terms of the coefficients, as will be shown in §§ 421, 423. But the roots of a literal equation of the fifth or higher degree cannot be so expressed, as was proved by Abel in 1825. EXERCISE 53. By § 3S6, form the equations whose roots are given below, and verify each equation by § 387 : I- I, - 3, - 5- 4- - §, 3 ± V^, 5- 2. I, V2, — a/2. 5. I, 2, V'3- 3. I, ± A/3, ± V5. 6. 3, - 4, V^^. 2/6 ALGEBRA. 7. i, I ± V3, I ± V5- 9- V3» V^-2- 8. ± V- I, 3 ± V-2, 2. Note. In each of the above examples, the student sTiould note that the coefficients of the equation obtained are a// rational whenever the surd or imaginary roots occur in conju- gate pairs. 10. Solve 4x^ — 24.x^ -{- 2^x -{- 18 = 0, having given that its roots are in arithmetical progression. Reduce the equation to the form ^{x) = 0, and denote its roots by a — d, a, and a -{- b ; then by § 387 3 a := 6, 3 «2 _ ^2 ^ .2_3^ a (^2 _ ^2>) ^ _ 9. (i) Hence ^ = 2, and b = ± ^\ therefore the roots are — \, 2, |. The values of a and b must satisfy all three of the equations in (I). 11. Solve 4 ;c^ + 16 ^^ — 9 ^ — 36 = 0, the sum of two of the roots being zero. 12. Solve ^x^ -{- 20 jc^ — 23 a: + 6 = 0, two of the roots being equal. 13. Solve 3 ..^ — 26 Ji:^ + 52 ^ — 24 = 0, the roots being in geometrical progression. 388. Imaginary Roots /;/ the equation F (x) = 0, imaginary roots oeeur in conjugate pairs ; that is, if a + hV— lisa root of F (x) = 0, then a — b V — I is also a root. THEORY OF EQUATIONS. 2/7 If <2 + 3 V — T be substituted for x \\\ F(x), all its terms will be real except those containing odd powers of ^ V — I, which will be imaginary. Representing the sum of all the real terms by A, and the sum of all the imaginary terms by ^ V — i, we have F{a + b^-l)=A + B^/-l = 0. (i) Now F(a — b V — i) will evidently differ from F{a -\- b V — i) only in the signs of the terms containing the odd powers of b V — I ; that is, in the sign of ^ V — I ; hence F{a-b^/ - i) =A-B\^^^, From (i) by § 124, A = 0, and ^ = 0; hence F{a - b V^) = 0. Example. Onerootof ji-8 — 4:ir2 + 4;ir — 3 =0 is K^+a/"^)' find the others. Since ^(i + /y/— 3) is one root, ^i — \/^~3) is a second root. The sum of tliese two roots is i, and 1)y § 387 the sum of all three roots is 4; hence the third root must be 3. Here F{x^ ^ (.r - 3) (^ - ^ - i V^) (^ - * + ^ \/=l) = (^-3)[(^-|)*+|] = (^-3)(^^-^+i); that is, the real factors of ;t-8 — 4 ;i-2 + 4 ;r — 3 are ^ — 3 and 389. By § 388 an equation of an odd degree must have at least one real root; while an equation of an even degree may not have any real root. 278 ALGEBRA. 390. Real Factors of F{x). Since {x~a — b^/~ i) (x — a -\- b ^/ — i') = (^x — a^ + b'^, the imaginary factors oi F {£) occur in conjugate pairs whose pro- ducts are of the form {x — a)'^ + b'^. Hence, F (x) cafi be resolved hito real linear or quadratic factors in x. 391. To transform an equation into another whose roots shall be some multiple of those of the first. If in the equation ^« +A^"~' + A^""' +Pz^"~' + ••• + A = 0, (i) we put X = x^^ ay and multiply by a'\ we obtain a:/'+ A ^ ^i - '+ A ^'•^i" ~ '+ A ^^^1 "'+••• +/« a'' = 0. (2) Since x^ = ax, the roots of (2) are a times those of (I). Hence, to effect the required transformation, mtil- tiply the second term of F (x) = ^ the givett factor, the third by its square, and so on. Any missing power of x must be written with zero as its coefficient before the rule is applied. The chief use of this transformation is to clear an equation of fractional coefficients. Example. Transform the equation x^ — ^x^^ -\- \x — ^^ = into another with integral coefficients. THEORY OF EQUATIONS. 279 Multiplying the second term by a, the third by a^^ the fourth by a^-, we obtain x^-lax'^ + \a'^x- {^a^ = 0. (i) By inspection we discover that 4 is the least value oi a that will render the coefficients of (i) integral. Putting « = 4, we have ;f3 - 10 ;r2 + 28 :ir - 12 = (2; as the equation required. The roots of (2) each divided by 4 are the roots of the givei equation. EXERCISE 54. 1. One root of ^^— 6x' -\- 57.^—196 = is i— 4V— 3: find the others. 2. One root of a;^ — 6 ;c + 9 = is | (3 + V — 3) ; find the others. 3. Two roots oi x^ — x^ -^ x^ — x^ + X — 1 = are — V^^ and ^ (i + V — 3) ; find the others. 4. One root o{ x^ — 2x^ ■{■ 2x— i=0 \s ^{i -\- -v/ — 3) ; find the real factors oi x^ — 2 x^ -\- 2 x — i. Find the real factors of F {x) in the Examples from i to 3. 5. Prove that, if ^t + V^ is a root of F{x) = 0, a — ^/b is a root also. (See proof in § 388.) Transform the following equations into others whose co- efficients shall be whole numbers, that of .r" being unity : 6. x^^%x-l = ^. 8. ^« _ 3 .,;2 + ^ ^ _ 2 ^ 0. 9. X^ — \ X^ — I ^2 + O.I JC + y^ly^ = 0. 28o ALGEBRA. 392. A Commensurable real root is one that can be exactly expressed as a whole number or a rational fraction. An Incommensurable real root is one whose exact expression involves surds. Thus, of the equation (r — 5) (^ — ^) (;r — yy/2) (x + y^i) = 0, 5 and ^ are commensurable roots, and y/2 and — y/2 imom- mensurable. 393. Integral Roots. If the coefficients ^ F (x) are all wJiole numbers^ any commensurable real rcot of F(x) = is a whole number and an exact division of p,,. s Suppose-, a rational fraction in its lowest terms, to be a root of F{x) = ^\ then, by substitution, we have -n +A ^33 +A^-;73T, + ••• +A = 0. (.) Multiplying by /" ', and transposing, we have ^' = -(A«f""'+A^^""' + ---+A/""')- (2) Now (2) is impossible, for its first member is a fraction in its lowest terms, and its second member is a whole number. Hence, as a rational fraction cannot be a root, any commensurable root must be a w^hole number. Next, let a be an integral root oi F (x) = 0. Substituting a for x, transposing /„, and dividing by a, we have . THEORY OF EQUATIONS. 28 1 The first member of (3) is "integral; hence the quotient/,, 4- « is a whole number. Thus, any commensurable root of x-^ — 6 x^ -\- 10 jr — 8 = must be ±1, ±2, ± 4, or ± 8 ; for these are the only exact divisors of — 8. 394. The Limits of the Roots of an equation are any two numbers between which the roots lie. The limits of the real roots may be found as follows: Superior Limit. In evaluating F{fi) in Example i of § 381, the sums are all positive, and they evi~ dently would all be greater for x > 6. Hence F{x) can vanish only for x < 6\ and therefore all the real roots of F{x) — ^ are less than 6. Hence, if in computing the value of Y (c), c being positive, all the sums are positive, the real roots of F (x) = are all less than c. Inferior Limit. In evaluating F{—6) in Example 2 of § 38r, the sums are alternately — and +, and they evidently would all be greater numerically for X < — 6. Therefore all the real roots of F {x) = {) are greater than — 6. Hence, if in computijig the value ^F(b), b being negative, the sums are alternately — and +, all the real roots' of ¥ (x) = are greater than b. Therefore, if its coefficients are alternately + and — , F {x^ = cannot have any negative roots. 282 ALGEBRA. Example. Solve x* *+ 2 x^ — i^ x^ — 14 x + 24 = 0. In evaluating /^(4), the sums are all + ; and in evaluating 7^{— 5), the sums are alternately — and + ; hence the real roots of I^{x) = lie between — 5 and 4. By § 393, the commensurable roots are integral factors of 24. Hence any commensurable root must be ± i, + 2, ± 3, or — 4. The work of determining which of these numbers are roots may be arranged as below : I +2 - 13 - 14 + 24 l_r^ + 1 4-3 - 10 - 24 I -}- 3 - 10 - 24 |-2 — 2—2+24 + 1 - 12 Hence J^(x) is divisible by x — i, the quotient x^ + ^ x^ — 10 jr — 24 is divisible by :r + 2, and the depressed equation is X^+ X- 12 = 0, of which the roots are evidently 3 and —4. Therefore the required roots are i, — 2, 3, —4. EXERCISE 55. 1. Show that the real roots of :v:^ — 2 .:v — 50 = lie between — 2 and 4. 2. Show that any commensurable real root of x* — ^x^ — 1S^ ~ loooo = is ± I, ± 2, ± 4, ± 5, ± 8, or 10. 3. Show that the real roots of :v^ + 2 ^* + 3 :t^ + 4 x^ + 5 jc — 54321 = lie between — 2 and 9, and that any commensurable real root must be ± i or 3. 4. Any commensurable root ofx^ — 15^^^+ io;t:+ 24 = must be one of what numbers ? 5. Find the roots of the equation in Example 4. THEORY OF EQUATIONS. 283 Solve each of the following equations, and verify the roots of each by § 387 : 6. x^ — 4 X* — 16 .T^ 4- 1 12 jc^ — 208 X + 128 — 0. 7. ^* — 4 a:^ — 8 ^ + 32 =: 0. 8. x"" — zx^ ^ X ^ 2^^, 9. x^ — d x"^ -\- \\ X — 6 = 0. 10. jc'' — 9 ;c^ + 1 7 x'^ 4- 27 Ji — 60 = 0. 11. x^ — d x"^ -\- \o X — 8 = 0. 12. x"^ — 6 x^ + 24^:1; — 16 = 0. 13. ^^ — 3 x^ — 9 jt-^ + 2 1 .r^ — 10 jc + 24 = 0. 14. x"^ — x^ — 39 x"^ + 24 a: + 180 = 0. 15. ^» + 5 -^"^ — 9 -^ — 45 = 0- 16. ^* — 3 jc* — 14 Ji:^ + 48 .r — 32 = 0. 17. x^ -{- x^ — 14^^ — 14 a* + 49 .V* + 49 ^'^ — 36 a: - 36 = 0. 18. ^«+5 ^^— 81 a;*— 85 JtH 964^2^ 780 ;t:— 1584 = 0. 19. ^' — 8 a:^ + 13^ — 6 = 0. 20. x^ -\- 2 x^ — 23 jc — 60 = 0. 21. ^^ — 45 :r^ — 40 j«r 4- 84 = 0. 22. x^'-'jx' + iix'- Tx'^ i4x''-28x-\- 40 = 0. 284 ALGEBRA. Solve the following equations by first transforming them into others whose commensurable roots are whole numbers : 23- ^^ — t ^'' — ih ^ + /ff = 0- 24. 8 Jt^ — 2 ..- — 4 ^ '+ 1=0. 26. gx^ — (^x^i-^x^ — ^x+^ = 0, 27. S x^ — 26 o;"^ + 1 1 -^ + ro = 0. 28. X* — 6 x^ + g^ x^ — ;^ X -{- 47} = 0. 395. Equal Roots. Suppose the equation F(x) = has r roots equal to a, and let J^(x) = {x — aycfi{x); (i) then F' (x) = r {x - (i)'-'' cIj(x) + (x — aY cfj' (x) . (2) From (i) and (2) it is evident that {x — ay~' is a common factor of i^(;r) and F' (;r). Hence if F (;l) = has r roots equal to a, (x — ay~ ^ will be a factor of the H.C.D. of F (^x) and F' {x). Any linear factor will occur once more in F (x) than in the H.C.D. o{F{x) and F' (x). Example i . Solve x* — 1 1 jr^ + 44 ;i-2 — 76 ^r + 48 = (i) Here F (x) ~ x^ - i\ x^ -^ 44 x"^ - 76 x + 4S -, .-. F'(x) = 4 ;r3 - 33 x^ +S8x- 76. By the method of §94 we find the H.C.D. of F(x) and F'{x) to be ,r — 2 ; hence two roots of (i) are 2 each. % § 387, the sum of the other two roots is 7, and their pro- duct 12 ; hence the other two roots are 4 and 3. THEORY OF EQUATIONS. 285 Example 2. Solve x' + S^ + dx^^dx^- i5;i-3-3;i'2-|- 8:r+.4 = 0. (i) Here the H. C. D. of F{^x) and F{x) is x^ -\- -i^x^ ^- x'^ — 3 -r — 2. (2) The H. C. D. of function (2) and //s derivative is x + i ; hence (x + i)-^ is a factor oi (2). liy factoring we obtain x^ + Zx^ + x'2-3x-2 ^{x-^ 1)2 (x -f- 2) (^ - I). (3) Hence three roots of (i) are — i each, two — 2 each, and two I each. EXERCISE 56. Solve the following equations, each having equal roots : 1. X*— i4x^ + 61 x^ — 84 ^ + 36 = 0. 2. x^ — 'J x"^ ■\- 16 X — 12 = 0. 3. a:* — 24 ^^ + 64 ;t: — 48 = 0. 4. AT* — 1 1 a:^ + i8 a: — 8 = 0. 5. X* + 13 ;f» + Z2> ^^ + 31 -^ + 10 = 0. 6. x^ — 2 x^ + 3 ^^8 — 7 .t2 + 8 x — 3 = 0. 7. ;c* — 12 :r^ + 50 A-2 _ 84 .r + 49 = 0. 8. A* + 3 a^ - 6 :^* - 6 :i8 + 9 ^' + 3 ^ - 4 = 0. 9. Show that the equation ^^ + 3 ffx +(9 = will have two equal roots, when 4 H^ + (9^ = 0. 10. If 4 r =/2^ find the roots of x* — / jc'^ + r = 0. 286 ALGEBRA. 396. If only two of the roots of a higher numeri- cal equation are incommensurable or imaginary, the commensurable real roots may be found by the methods already given, and the equation depressed to a quadratic, from which the other two roots are readily obtained. When a higher numerical equation contains no commensurable real root, or when the depressed equation is above the second degree, the following principle is useful in determining the number and situation of the real roots. 397. Change of Sign of F(x). 7/" F(b) and F(c) have unlike signs, an odd number of roots o/F (x) — lies between b and c. If X changes continuously, then F{x) will pass from one value to another by passing through all in- termediate values (§ 255). Therefore to change its sign, F{x) must pass through zero ; * for zero lies between any two numbers of opposite signs. Hence \{ F{b) and F{c) have opposite signs, F{x) must vanish, or equal 0, for one value, or an odd number of values, o{ x between b and c. If F{b) and F {c) have like signs, then we know simply that either no root, or an even number of roots, of F (x) = lies between b and c. * A function may change its sign by passing through infinity (§ 254) ; but evidently F{x) or any other integral function of x can- not become infinite for a finite value of x. THEORY OF EQUATIONS. 28/ Example. Find the situation of the real roots of By § 394 we find that all the real roots lie between — 2 and 6. Herei^(-2)=-4, /^(O) = + 8, 7^(4) = - 16, /r(_i) = + 9, yr(,) = _i, /^(5) = + 3- ' Since F(— 2) and F(~ i) have unlike signs, at least one root of F(x) = lies between - 2 and - i. For like reason a second root lies between and i, and a third between 4 and 5. Hence the roots are - (i- +), 0- +, and 4. +. 398. Every equation of an odd degree has at least one real root whose sign is opposite to that of the knoivn term p„. \iF(x) is of an odd degree, then ir(_ x) = - X, /^(O) =/,„ 7r( + ^) ^ + ^^. Hence, if /„ is positive, one root of F{x) — lies between and — x (§ 397) ; and if /„ is negative, one root lies between and + x. 399. Every equation of an even degree in which p„ is negative has at least one positive and one Jtegative real root. Here F{-^)=: + x, 7^(0) is — , 7^(+ x) rn + x. Hence one root of FCx) = lies between and — yo, and another between and + x. 288 ALGEBRA. EXERCISE 57. Find the first figure of each real root of 1. x^ + x^—2x—i=0. 6. x^—4x^ - 6x = ~S, 2. .:r^— 3 J\;^— 4:^+11 =0. 7. ^'*— 4^^ — 3 Jt: = — 27. 3. :v^— 4^'— 3 x+ 23 = 0. 8. ^^ + :r — 500 = 0. A. X^ — 2X — ^ = 0. 9. ::c^+ 10 a:^+ 5^ = 260. 5. 20;(;^— 24;^;^+ 3 = 0. 10. :t:^+ 3 ^^ + 5 ^ = 178. II. :v^ — 11727^ + 40385 = 0. Sturm's Theorem. *400. The object of Sturm's theorem is to deter- mine the number and situation of the real roots of any numerical equation. Though perfect in theory, Sturm's theorem is laborious in its application. Hence, when possible, the situation of roots is more usually determined by the method of § 397. *401. Sturm's Functions. Let F {x) = be any equation from which the equal roots have been re- moved, and let F' (x) denote the first derivative of F{x). Treat F{x) and F {x^ as in finding their H. C. D., with this modification, that the sign of each remainder be changed before it is used as a divisor, THEORY OF EQUATIONS. 289 and that no other change of sign be allowed. Con- tinue the operation until a remainder is obtained which does not contain x, and change the sign of that also. Let F^ {x), h\ {x), . . ., F^ (,r^) denote the several remainders with their signs changed; then F{x^, P{x\ F,{x), F^{x), ..., /s,, Ot^)are called St linn s Functions. F(x) is the primitive function, and P {x)y F^{x)y ..., F,„ {x^^) are the auxiliary functions. We use ^ in- stead of Ji' in F„^ {^)y since F,^ (,r^) does not contain x. Example. Given x^ — ;^x^ — 4x -\- 13 = 0; find Sturm's functions. Here F (x) ^ x^ - 3 x^ - 4 x + 13 ; .-. P(x) = 3x'^-6x-4. Dividing; F(x) by P(x), first multiplying the former by 3 to avoid fractions, we find that the first remainder of a lower degree than the divisor is — I4;r + 35. Changing the sign of this remainder and rejecting the positive factor 7, we have F,(x)^2x-5. Proceeding in like manner with 3 or^ — 6ar — 4 and 2 x — ^, we find the next remainder to be — i ; hence F^^x^) :i: + i. If an equation has equal roots, the process of find- ing Sturm's functions will discover them, and then we can proceed with the depressed equation. *402. A Variation of sign is said to occur when two successive terms of a series have unlike signs ; and a Permanence, when they have like signs. 290 ALGEBRA. Thus, if the signs of a series of quantities are + -I h + — t- , there are four variations and three permanences of sign. Again, in the Example of § 401, we have J^ (x) = x^ - 3 x^ - 4x + IS, Fi(x) = 2x- 5, F\x) = 3x^~6x-4, F^{x^) = + i. When F{x) F' {x) F^{a') F^{x^) X = + — — +2 variations. x=z 3 -\- + + +0 variations. * 403. Sturm's Theorem. 7/" F (x) = has no equal roots and b be substituted for yi in Sturm s functions, and the number of variations noted, and then a greater number c be substituted for x and the 7iumber of varia- tions noted ; the first number of variations less the second equals fhe number of real roots of F (x) = tJiat lie between b and c. (i.) Since each of Sturm's functions is an integral function, to change its sign a Sturmian function must vanish (§ 255). (ii.) Two consecutive functions cannot vanish for the same value of x. For if Fi {x) and Fz (x) both vanished when x — a, each would contain the factor x — a. Hence, by §§94 and 401, F{x) and F' {x) would have the common factor X — a ; whence, by § 395, F{x) = would have equal roots, which is contrary to the hypothesis. (iii.) When any auxiliary function vanishes, the two adjacent functions have opposite signs. THEORY OF EQUATIONS. 29I Let the several quotients obtained in the process of finding Sturm's functions be represented by q^y q^^ ^3, ...; then by principles of division we have F'{x)=F,{x)q,-F,{x), F,{x)=F,(x)q,-F,{x\ Hence, if any auxiliary function, as R, (;r), vanishes when X — a, from the third identity we have F,{a)=-F,{a). (iv.) The number of variations of sign of Sturm's functions is not affected by a change of sign of any of the auxiliary functions. Suppose ^^2 C-^) to change its sign when x — a ; then, by (i.) and (ii.), neither Fx{x) nor F^^x) can change its sign when x = a. Hence F^ {x) and Fz {x) will have the same signs immediately 2Xx.^x x — a that they had immediately before, and by (iii.) these signs will be unlike. Now, whichever sign be put between two unlike signs, there is one and only one variation. Hence the change of sign of F,^ {x) does not affect the number of variations of sign. The same holds true of any other auxiliary function except F„, {x^), which, being constant, cannot change its sign. (v.) If X increases, there is a loss of one, and only one, variation of sign of Sturm's functions when F {x) vanishes. 292 ALGEBRA. When F(x) vanishes, F\x) is + or — . li F' (x) is +, F{x') by § 238 is increasing when it vanishes, and therefore must change its sign from — to +. Hence, immediately before FQv) vanishes, we have the variation 1-, and immediately afterward the permanence + +. \{ F' (x) is — , F(x) is decreasing when it vanishes, and therefore must change its sign from + to •— . Hence, when Fi^x') vanishes, the variation + — be- comes the permanence — — . Whence there is a loss of one variation of Sturm's functions when F(x) vanishes, and only then. Therefore the number of variations lost while x increases from b to c \s equal to the number of roots of F(x^ = that lie between b and c. Example i. Determine the number and situation of the real roots of the equation x^ — 2> x'^ — ^x -\- 13 = 0. By § 394, all the real roots lie between — 3 and 4. Here F{x') :=3 jir^ — 3 jr^ — 4 ;ir + 13, F^ (.r) = 2 ;r — 5, F' {x) Ez 3 ^2 _ 6^- -4, F^{x^) ^ + I. Beginning at ;ir = — 3, we find the following table of results : F,{x) F,{x) — +3 variations. — +2 variations. — +2 variations. + +0 variations. Hence there is one negative root between — 3 and — 2 (§ 403), and two positive roots between 2 and 3. To separate the two positive roots, we substitute in the Sturmian functions some When F{x) F\x) x=-z - + X=~2 + + :r= 2 + - x= 3 + + THEORY OF EQUATIONS. 293 value of X between 2 and 3, as 2.5. When x= 2.5, the suc- cession of signs is +, which gives but one variation, whether has the sign + or — ; hence one positive root lies betweeti 2 and 2.5, and the other between 2.5 and 3. Example 2. Find the number and situation of the real roots of 2x^ — 13^-2+ 10 X ~ 19 = 0. Sturm's theorem may be applied to an equation in this form, since there is nothing in its demonstration that requires the coefficient of x" to be unity. By § 394, the real roots lie between — 4 and -f 3. Here F{x)^2x* — I'^x^ + \ox ~ i<), F'{x)^2(/[x^-iZx+s\ F^ix)^i2,x''-iSx + z^' Since, by § 148, the roots of F^{x) ~ 13 ;i'2 - 15 jt- + 38 = are imaginary, F-^{x) cannot change its sign for any real value of xj hence there can be no loss of variations beyond F^ (:r), and it is unnecessary to obtain F^{x) and F^{x'^). When in F(,x) F'{x) -^iW :r=-4 + - + 2 variations. x=-^ + - + 2 variations. X = —.2. — - + I variation. X= 2 - + + I variation. x= 3 + + + variations. Hence there are two real roots, one of which is — (2. +) and the other 2. +. Example 3. Find the number of the real roots of the cubic x^ + 2>^^-+G = 0. (I) Here F{x) ^ x^ ^- ^ Hx ^ G, F^ {x) ~~2 Hx - G, F' (A) : 3 {x^ + H), F., {x) = - (6^2 + 4 ^3). 294 ALGEBRA. If G^ + 4l/^>0, H may be either + or — ; so that When F{x) F' {x) F^{x) F^{x) X = — T> — + ± — 2 variations. :f = +jo4- + T — I variation. Hence when 6^^ + 4 H^ \s positive, only ojte root is real. If 6^-^ + 4 //3 < 0, evidently // is — ; so that we have .r = — 30— + — +3 variations. ;jr=:+x+ + + + variations. Hence when G^ + 4//^ is negative, all ///r\ if ^ = 0, the root b ^ a — b ^ ^, or infinity. Again, the roots, in § 145, of the quadratic equation ax'^-\-bx^-c = ^, by rationalizing the numerators, may be put in the forms, ic ic b-^b^-^ac -b+ ^b'^ -4ac THEORY OF EQUATIONS. 297 Now if ^ = 0, then a-:^ — c-^b and ^ = >d ; if « = 0, one root is finite, and the other is infinity. If ^ = and ^ = also, then a = do and /3 = x ; if « = ^ = 0, both roots are infiiiity. 408. To transform an equation into another whose roots shall be those of the first diminisJied by a given quantity. If in the equation Fix) = X" +A^"~' + A^"~' + ••• + A-i^-+A = 0, (i) we put X — x^ ■\- h, we obtain F{x, -\-h) = {x, + hy + A ('^i + ^)"-' + ... + A-i(-^i + /0+A-0. (2) As X = Xy -\- hy or x^=x — //, the roots of (2) equal those of (i) diminished by h, h being either positive or negative. Hence, to eflfect the required transformation, sub- stitute X, + h for X, expandy and reduce to the form of F (x) = 0. 409. Computation of the Coefficients of F(jc^ + //). Since F{x^ + //) may be reduced to the form of F{x), put F{x^ + h) = Xj" + qiXi"-^ + q^x{'-'^+ ... + $^„-i^i + q„. Substituting x — h for Xi, we obtain F{x) = (x-h)" + q,{x-/iy'-^ -i- ...-{- q„_,(x--/i) + q^. 298 ALGEBRA. Dividing F {x) by x — //, we obtain (^^ _ /,)«-! + g,{x- hy-^ + ..• + qn.-i (i) as the quotient, and q,, as the remainder. Dividing the quotient (i) hy x — h, we obtain as the quotient, and q„_^ as the remainder. The next remainder will be qn-i\ the next qn-z\ and so on to q^. The last quotient will be the coefficient of ;tr". Hence, if F (x) be divided successively by x — h, the successive remainders and the last quotient will be the coefficients o/F (x^ + h) in reverse order. Example. Transform the equation x^ — ^x^ — 2x+s = ^ into another whose roots shall be less by 3. The work of dividing /^(x) successively by ;ir — 3 to compute the coefficients of i^(^i + 3) may be arranged as below : I -3 — 2 + 5 |3 + 3 + -6 I - 2 -^ = q + 3 + 9 --^2 I + 3 + 7 = + 3 I + 6 = ^1 Hence the transformed equation is F(xi + 3) Xj^ + 6xi^ + 7x^-1= 0. THEORY OF EQUATIONS. 299 410. Equation Lacking any Term. If the binomials in equation (2) of § 408 be expanded, the coefficient of or/'"' will evidently be ;/// +A; hence if we put n h -\- py = 0, or // = — /i -^ ;/, the transformed equa- tion will lack the term in x"~\ In like manner an equation can be transformed into another which shall lack any specified term. Example. Transform the equation x^ — 6x'^-\-^x-\-$ — into another lacking the term in xK Here /^ = — 6, ;/ = 3 ; .'. h = — p^ -^ n = 2.. Transforming the given equation into another of which the roots are less by 2, and writing x for x^, we obtain :i-8-8jr-3 = 0, an equation which lacks the term in x-. ■ EXERCISE 59. Transform each of the following equations into another having the same roots with opposite signs : 1. x^ + a:^ — a-2 — 5 a: + 7 = 0. 2. ;c^ - 7 a:^ — 5 ;t:2 + 8 = 0. 3. :j;^ — 6 ^* — 7 ^^^ + 5 ^ = 3. /[. x' — T X* ■\- ^x"^ — ^ X -{- 2 = 0, 300 ALGEBRA. Find the equation whose roots are the reciprocals of those of each of the following equations : 5. x^ — 'jx^ — 4x+2=0. 7. x^ — x^ + ^x'^+ S = 0. 6. x^—8x^ + T = 0. 8. x^ — x^ +'jx^ + g = 0. Transform each of the following equations into another whose roots shall be less by the number placed opposite the equation : 9. ^^ — 3 ^^ — 6 = 0. 5. . 10. x^ — 2 x'^ -\- ^ x^ -{- 4. X — y = 0. 4. 11.:^* — 2 JC^ + 3 Jt:^ + 5 -^ + 7 = 0. — 2. 12. X* — 18 ::t:^ — ^2 x^ -\- I'j X -\- ig = 0. 5. 13- 5 ^* + 28 ^^ + 51 j\;^ + 32 jc — I = 0. — 2. Transform each of the following equations into another which shall lack the term in x''^ : 14. x^ — ^ x^ + S ^ + 4 = ^' 15. x^ — 6x''+ 8x— 2 = 0. 16. x^ + 6 x^ — J X — 2 = 0, I 7. j:!::^ — 9 ^^ + 12 a: + 19 — 0. 411. Horner's Method of Solving Numerical Equa- tions. By this method any real root is obtained, after Its situation has been determined. The main THEORY OF EQUATIONS. 3OI principle involved is the successive diminution of the roots of the given equation by known quantities, as explained in § 408. Thus, suppose that one root of /^(x) =: is found to lie between 40 and 50; to find this root we transform the equation yr (i-) =: into another whose roots shall be less by 40, and "•^'^i" ^(4o + r,) = 0, (i) of which the positive root sought is less than 10. If this root is found to lie between 6 and 7, we transform equation (i) into another whose roots shall be less by 6, and obtain ^ j-^^ ^ (g _^ ^^>j-| = /r (46 + :,,) = 0, (2) of which the positive root sought is less than i. If this root lies between 05 and 0.6, we transform equation (2) into another whose roots shall be less by o 5, and obtain 7^(46.5 + ;ro) = 0, (3) of which the positive root sought is less than o.i. First, suppose this root of (3) to be o 03, we then have -^(46-53) - ; that is, one root of F(x) = is 46.53. Next, suppose this root of (3) to lie between 0.03 and o 04 ; then it follows that one root of F{.x) = lies between 46.53 and 46 54. By transforming equation (3) into another whose roots shall be less by 0.03, the thousandths figure of the root can be found ; and so on. By repeating these transformations we can evidently obtain a root exactly, or may approximate to any root as nearly as we please. 412. One of the practical advantages of Horner's method is that the first figure of the root of any 302 ALGEBRA. transformed equation, after the first, is in general correctly obtained by dividing the last coefficient with its sign changed by the preceding coefficient, which is therefore often called the trial divisor. The figure obtained in this way from the first transformed equation is likely to be too large. Example. Find the root of the equation that lies between 3 and 4. We first transform equation (a) into another whose roots shall be less by 3. The work is given below. -3 -4 + 11 [3. + 3 + j2 -4 - 1 + 3 + 9 + 3 + 5 + 3 + 6 Thus F(3) = — I, and the first transformed equation is ^(3 + ;ri) = :ri3 + 6:ri2 + 5 ;ri - I = 0, (l) of which the root sought is positive and less than i. Since x^<, \,x^ < x^ < x^. Hence it is probable that the Jirst figure of this root of (i) will be correctly given by the quadratic equation THEORY OF EQUATIONS. 303 We next diminish the roots of (i) by o.i. + 6.0 + 5-00 — 1.000 |o.i + O.I + 0.61 + 0.561 + 6.1 + 5.61 - 0.439 + I + 0.62 + 6.2 + 6.23 + 0.1 + 6.3 Thus /^(3.i) = — 0.439, ^"^ ^^6 second transformed equa- tion is F(3. i + x^)- x^ + 6.3 x^ + 6.23 ^2 - 0.439 = 0, (2) of which the root sought is positive and less than o.i. Since x,^^ < o. i, x^ and x.^ are each much smaller than .i^. It is probable therefore that the first figure of this root of (2) will be correctly given by the simple equation 6.23 x^ — o 439 = ; ''. X2 = 0.06 +. We next diminish the roots of (2) by 0.06. 1 +6.30 +6.2300 —0439000 1 0.06 + 0.06 +0.3816 +0.396696 + 6.36 +6.6116 -0.042304 + 006 + 0.3852 + 6.42 + 6.9968 + o 06 + 6.48 Thus 7^(3.16) is — , and the third transformed equation is /^(3. 16 + ^a) = V + 6.48 V + 6.9968 ^3 - 0.042304 = 0, (3) of which the root sought is positive and less than o.oi. 304 ALGEBRA. Dividing 0.042304 by 6.9968 we find the next figure of the root to be o 006. Diminishing the roots of (3) by 0.006 will give the next transformed equation, which will furnish the next figure of the root ; and so on. But, since x^ < o.oi, and the coefficient of x^^ is less than that of Tg, it is probable that the first Iwo figures of x^ will be correctly given by the simple equation Xq = 0.042304 -^ 6.9968 = 0.0060 -f. Hence 3.1660 is the required root of (a) to four places of decimals. Observe that the known term of any transformed equation is the value of J'ix) for the part of the root thus far found. 413. As seen above, the known term of any trans- formed equation is the value of F{x) for the part of the root thus far found ; hence the known term must have the same sign in all the transformed equations. If any figure of the root is taken too large, the known term in the next equation will have the wrong sign. If a figure is taken too small, the root of the next equation will evidently be of the same order of units. Hence, eack figure of the root is correct if the next transformed equation has a known term of the same sign as that of the preceding equatio7i and a root of a lower order of units. Example. Find the root oi x^ — 'i,x'^ — ^x -\- \\ =0 that lies between i and 2. We give below the work of the successive transformations written together in the usunl form. The broken lines mark the THEORY OF EQUATIONS. 305 conclusion of each transformation, and the figures in black-letter are the coefficients of the successive transformed equations. 1. -3 -4 I - 2 4- II 1 1.782 - 6 — 2 i -6 — I 5.000 - 4-557 I -7.00 0.49 0.443000 — 0.428448 0.0 0.7 -6.51 0.98 0.014552000 — 0.010340232 0.7 0.7 1.4 0.7 - 5.5300 0.1744 -5.3556 0.1808 0.004211768 2.10 0.08 - 5.174800 4684 2.18 0.08 — 5.1701 16 4688 2.26 0.08 - 5.165428 2.340 2 2.342 2 2.344 2 2.346 Here we find that the second figure of the root is correctly given by dividing 5 by 7; the third by dividing 0.443 by 5.53 ; the fourth by dividing 0.014552 by 5.1748 ; and so on. 306 ALGEBRA. Since x^<. o.ooi, and the coefficient of x^^ is much less than that of x^, it is probable that the first three figures of x^ are correctly given by x^ = 0.00421 176S -^ 5.165428 = 0.000815. Hence i. 78281 5 is the required root to six places of decimals. How many figures of the root will in this way be correctly given by the last transformed equation may be inferred from the value of its root and the relative values of its leading coefficients. 414. Negative Roots. To find a negative root, it is simplest to change the sign of the roots (§ 405), obtain the corresponding positive root, and change its sign. Thus to find the root of .r^ — 3 ;ir2 — 4 ;r + 11 = that lies be- tween — I and — 2, we obtain the positive root of the equation x^ -\- 2 x^ - 4 X — II = § 405. that lies between i and 2, and change its sign. It is evident that Horner's method is directly ap- plicable to an equation in which the coefficient of x" is not unity. Note. For a fuller discussion of Horner's method, for its application to cases where roots are very nearly equal, and for contractions of the work, see Burnside and Panton's or Tod- hunter's "Theory of Equations." EXERCISE 60. Find to five places of decimals the root of the equation 1. x^ + X — 500 = 0, that is 7 +. 2. x^ — 2 X — ^ = 0, that is 2 +. 3. x^ — sx^ + Sx— I =0, that is +. THEORY OF EQUATIONS. 307 4. x^ + 2 X- — ^ X — g = 0, that is i +. 5. 2 x^ + $x -^ go = 0, that is 3 +. 6. x^ -[■ x^ + Tox -{- 300 — 0, that is — (3 +). 7. 5 ^^ - 6 a:^ 4- 3-^ + 85 = 0, that is - (2 +)• 8. sx' -\- x^-{- 4x^+ s^- 375 = 0, that is 3 +. 9. a* — 80 Jt' + 1998 a:^ — 14937 X + 5000 = 0, that lies between 30 and 33 ; between ;^;^ and 40. Find the positive root of each of the equations : 10. 2x» — 85 jc^ — 85:^-87 = 0. 11. 4:<;3— 13.V2 — 3i.r — 275 = 0. 12. 20 a:^ — 121^^— 121. r — 141 =0. 13. Solve a;^ — 315 Ji'"^ — 19684^:1; + 2977260 = 0. Reciprocal Equations. 415. A Reciprocaly or Recurring, equation is one that remains unaltered when x is changed into its reciprocal; that is, when the coefficients are written in reverse order (§ 406). Hence the reciprocal of any root of a reciprocal equation is also a root. Therefore, if the equation is of an odd degree, one root is its own reciprocal ; hence one root of a reciprocal equation of an odd degree is + i or — i. 308 ALGEBRA. Example. Given 5 as one root, to solve the equation 5;ir5 — 51 jr^ + i6o;r3 — l6o;if2 + 51 ;»; — 5 = 0. (i^ Since (i) is a reciprocal equation, and one root is 5, a second root is \. Since (i) is of an odd degree, one root is + i or — i. By inspection we see that + i is a root. The depressed equation is x^- 4X+ 1=0, I from which x = 2 + x/ 3 or 2 — a/ 3, which equals ,~ 2 + V3 A reciprocal equation of an odd degree can be depressed to one of an even degree; for one of its roots is always known. 416. Let^'^+A^"- '+/2^"-'+---+A-i^+A=^. (0 and A^"+A-i^""'+A-2^"~'+--- + A-^"+i = 0, (2) be equivalent equations; that is, let (i) be a re- ciprocal equation. Dividing (2) by A to put it in the form of (i), and then equating their last terms, we have A= I -^A' ••• A = ± I- Reciprocal equations are divided into two classes, according as A is + i or — i. Ftrs^ Class. \{ pn = i ; then, from (i) and (2), P\—Pn-\, A=A-2V*--^ that is, the coefficients of terms equidista7it from the ends of F (x) are equal. THEORV OF EQUATIONS. 309 Second Class. If /„ = — i ; then, from (i) and (2), P\ = — A-ij A =— A-2J -"'f that is, t/ie coefficients of terms equidistant from the ends of F (;r) are equal numerically but opposite in sign. If in this class n is even, say « = 2 m ; then p,^ — —pmy or p,„ = 0) that is, the middle term is wanting. 417. Standard Form. Any equation of the second class of even degree can evidently be written in the form ^" - I -\-p,x{x--^ - i) + ... = 0. (i) Since n is even, F(x) in (i) is divisible by ;r2— i ; hence two roots of (i) are ± i. The depressed equation will evidently be a recip- rocal equation of the first class of even degree, which is called the standard for7n of reciprocal equations. Hence, afty reciprocal equation is in the standard form or can be depressed to that form. 418 Any reciprocal equation of the standard form can be reduced to one of half its degree. The fol- lowing example will illustrate this truth. Example. Solve ;r* — 5 ;r8 + 6.r2 — 5 :r + i = 0. • Dividing by (;f*)2, or x^^ we obtain 310 ALGEBRA. Since x^ + -^= Ix + ~] — 2, from (i) we have ;ir + - = 4 or I ; .- I ± a/— 3 ... ;r=2± V3, ^ EXERCISE 61. Solve the following equations : 1. x'^ + x^ + x^ + x'^ + X + 1 =0, 2. X* — ^ x^ -\- ^ X — I = 0. 3. X* — lojc^ + 26:^:^ — lojt; + I = 0. ^. x^ — ^ x"^ + g x^ — c) X- -{- ^ X — I =^ 0. 5. x^ + 2 x^ — ;^ x^ — ;^ x"^ + 2 X + I = 0, 6. ^^ — ^^ 4- ^^ — -^^ + ^ — I = 0. 7. 6 Jt:" + 5 ^-^ — 38 Jt^ + 5 .r + 6 = 0. 8. ^» — /7^2 + /^jt:— I =0. 419. Binomial Equations. The two general forms of Binomial Equations are x^ — d = 0, (i) and x"-{-d = 0, {2) in which d is any positive number. THEORY OF EQUATIONS. 3H By § 405, when n is odd, the roots of (2) are the roots of (^i) with their signs chajiged. The n roots of either (i) or (2) are unequal; for .r" q: b and its derivative «^"~' have no C D. From (i), ;r = "^Z b ; from (2), x = y— b ; that is, each of the n unequal roots of (i) or (2) is an «th root of + ^ or — ^. Hence, anj/ number has n tmequal nth roots. By § 391 » the n roots of ;r" — i = multiplied by y a are equal to the n roots of X" — a = 0. Hence, «// //;^ nth roots of any number may be obtained by multiplying any one of them by the nth roots of unity. If n is even, ;r" — i = has two real roots, ± i. If « is even, x" ■\- i = has no real root; for ^J —\ is imaginary when 71 is even. If n is odd, ;ir" — i = has one real root, + I ; and ;r" + I = has one real root, — i. 420. The Cube Roots of Unity. The roots of A:« - I = (X - l) (^2 ^ AT + i) = 0, were found to be I, - ^ + \ V"^, -\-\ \/^. If ft) denote either of the imaginary roots, by actually squaring, the other is found to be d?. Hence the three cube roots of + i are i, ft), and ft)^. 312 ALGEBRA. Therefore by § 419 the three roots of ^"'^ + i = 0, or the three cube roots of — i, are — i, — w, and — w^. Example. Find the five fifih roots of 32 and — 32. The equation ;r5 = i is equivalent to ;ir — i = and x^ + x^ + x^ + x+ 1 =0. (i) 2 From (I), ^^+i) + ^;ir + -|,j = i; ... ■. + 1 = ^14^5. X 2 Solving ;ir — I = and the two quadratics in (2), and multi- plying each root by /y^32, or 2, we find the five fifth roots of 32 to be 2, - I + Vs ± V- 10 - 2 V5 - I - V5 ± V- 10 + 2 V5 ■ 2 ' 2 These roots with their signs changed are the roots of - 32. *421. Solution of Cubic Equations. By §410 the general cubic equation can be transformed into another of the simpler form x^+ 7,Ifx+ = 0. (i) To solve this equation, assume xz=r^ -{- s^ ; (2) ,-. x^ — 3r^s^x-'(r + s) = 0. (3) THEORY OF EQUATIONS. 313 Comparing coefficients in (i) and (3), we have r^s^ = -Jjr,r + s = -G, (4) Solving these equations, we obtain r=},{-G+VG' + 4^% (5) Substituting in (2) the value of s^ obtained from the first of equations (4), we have ^ = rh + ^, (6) the value of r being given in (5). If ^ r denote any one of the cube roots of r, by §419, rS" will have the three values, v^r, w v^r, w^ Vr, and the three roots of (i) will be 3 -If 3/- , -If , 3/- , -B- If in (6) we replace r by s, the values of ;i: will not be changed ; for, by (2) and the relation t^ s^ = ~ H, the terms are then simply interchanged. Moreover, the other solution of equations (4) would evidently repeat these values of ;r. Note. The above solution is generally known as Cardan's Solution, as it was first published by him, in 1545. Cardan obtained it from Tartaglia ; but it was originally due to Scip'o Ferreo. about 1505. See historical note at th^ end of Burnside and Panton's "Theory of Equations," 314 ALGEBRA. * 422. Application to Numerical Equations. When a numerical cubic has a pair of imaginary or equal roots, by Example 3 of § 403, 6^^ _|_ ^ //3 -> ^j. _ q. hence r in (5) of § 421 is real, and therefore the roots may be computed by the formula (6). When, however, the roots of a cubic are all real and unequal, by Example 3 of § 403, G^ -\- ^H^ < 0; whence r is imaginary, and the roots involve the cube root of a complex number. Hence, as there is no general arithmetical method of extracting the cube root of a complex number, the formula is useless for purposes of arithmetical calculation. In this case, however, the roots may be computed by methods involving Trigonometry. When the real root of a cubic has been found by (6) or (2) of § 421 it is simpler to find the other two roots from the depressed equation. Example. Solve ;ir^ — 15 ^ — 126 =: 0. (i) Put x=r^ + s^', .'. x^-^r^s^ X- (r + s) = 0] (2) r^ s^ = 5, r + s = 126; r^ = 5, s^ = I ; x= r^ + s^ = S + T =6, Hence the depressed equation is x^ + 6x+ 21 =0, and the three roots are 6, —3 + 2 /y/^^ — 3 — 2 v^'s* THEORY OF EQUATIONS. 315 EXERCISE 62. 1. Find the 6 sixth roots of 729 ; of— 729. 2. Find the 8 eighth roots of 256 ; of— 256. Solve the following equations : 3. x^—iSx = ^^. 6. ^' + 21 X = — 342. 4. :x:3 + 63^ = 316. 7. .T» + 3.t'^ + 9.r = 13. 5. x^ -\- 'J 2 X — J J 20. 8. x^ — 6 a- + 3.r = 18. 9. x^ — 6x- + J^x = 10. 10. x^ — 15 .^^ — 33 'V = — 847. * 423. Solution of Biquadratic Equations. Any bi- quadratic equation can be put in the form X* + 2px^ + qx"^ -\- 2rx ^ s-= 0. (i) Adding {ax + ^)^ to both members, we obtain a:* ^ 2px^ -\- {q + a-) x^ + 2 (r + ^7 /») x + s + l,' = (ax + l,y, (2) Assume X* + 2p x^ + (q + a^) x^ + 2 (r + a li) X + s + P = (x'+/>x + ^y, (3) Equating coefficients (§ 262), we have P^+ 2k = q + a\ />k = r + ad, J^^ = s + P. (4) 3l6 ALGEBRA. Eliminating a and b from (4), we have or 2k^ — q k"^ -{- 2 {p r — s) k — p'^ s -^ q s — r'^ — ^, From this cubic, find a real value of k (§ 389). The values of a and b are then known from (4). Subtracting (2) from (3), we have {x^-^rpx -^ kf-{ax + bf = 0, which is equivalent to the two quadratic equations x^ + (p — a) X + {k — Z*) = 0, and ^2 ^ (/ + «) ^ + (^ + ^) = 0, of which the roots are readily obtained. Note. The solution given above is that of Ferrari, a pupil of Cardan. This and those of Descartes, Simpson, Euler, and others, all depend upon the solution of a cubic by Cardan's method, and will of course fail when that fails. For a full discussion of Reciprocal, Cubic, and Biquadratic Equations consult Burnside and Panton's "Theory of Equations." Example. Solve x^ — 6 x^ ^ \2 x'^ — 14 ;ir + 3 = 0. Adding (ax -\- by- to both members, we obtain r4-6jr3+(i2 + «2);t-2+2(^^-7)^ + ^^ + 3 = («-^+'^)^. (0 Since ^ = — 3, assume or* — 6 ;jr3 + (12 + «2) ^2 _f_ 2 (« 3 _ 7) ;r + ^-^ + 3 = (^'-3^ + >C02; (2) .♦. 12 + ^2^9+ 2/^, ab-T=-zk, ^2 + 3^^2. ... y&3_6^2+ 18k -20 = 0. THEORY OF EQUATIONS. 317 Whence ^ = 2; hence d^ = \^ b"^ = i^ ab = i. (3) From (i), (2), and (3), we obtain (;r2 - 3 ;r + 2)2 ~ {x + 1)2 z= 0, which is equivalent to the two equations ;r2 - 4;ir + I = 0, ;i-2 -2 ^ + 3 = ; « .'. x=2±^/]„ I ± V^-^. EXERCISE 63. Solve the following equations : 1. x^ ^ ^x^ -\- ()x^^2>x— 10 = 0, 2. x^ — 3 r^ — 42 jc — 40 = 0. 3. X^ + 2X^— 7^:^ — 8^+ 12=0. 4. x^ — 2>^'^ — ^^ — 2 = 0. 5. x^ — 1 4 ^' + 59 :*r2 — 5o X — 36 = 0. t 6. x^—2x^— 12 x"^ + io.r + 3 = 0. 7. :«:* — 2 .;c^ — 5 *•- + 10 a: — 3 = 0. 3i§ ALGEBRA. CHAPTER XXII. DETERMINANTS. 424. Determinants of the Second Order. The square array «2 h (') is called a determiitant of the second order. The quantities a\, a^, bu b^ are called Elements. A horizontal line of elements, as a^, bu is called a Row ; and a vertical line, as ^i, a^, a. Column. The Order of a determinant is determined by the number of elements in a row or column. The downward diagonal, a^ b^, is called the Princi- pal Diagonal ; and the upward diagonal, a& , 4- ^-y x^- c b X +y ^'M xy ^ f 5. Write as a determinant ax — by; as -\-bk; ax -\- by; axb + cdy; x^ -\- y^; c— s ; a + b; ax. Show by expanding that ^1 «1 = 0; ai h «2 a2 di b. = 0. I. That is, if the corresponding elements of two col- umns or tivo rows of a determinant are alike ^ the deter^ninant equals zero. 7- a^x C\ =. X a-i Cx = a a^, and (5) (6) whose elements are the coefficients of the unknowns in system (a) written in their order, is called the Determinant of the System. Hence, the value of either unknown is a fraction whose denominator is the determinant of the system ^ and zvhose mimerator is the deterntinant of the system with the k7iown terms substituted for the coefficients of the unknown whose value V5 sought. Note. It was the study of systems of linear equations that suggested the idea of determinants and their notation. The next few pages are designed to illustrate the beauty and utility of de- terminants in the soludon of such systems. Example. Solve the system zx — 7^ + Writing the given system in the form of system {a), we liave 2^-7 = 9. ) 3^-7j= 19. f 4; y^ =_I. f 19 = 3 a-.') 9 — 1 19 -7 2 -I 3 -7 2 9 3 19 2 — I 3 -7 322 ALGEBRA. EXERCISE 65. Solve the system 1. sx-\-6y= 17, 6x +sy = 16. 2. Sx— y^S4, X + 8y = 53. 3. ^x='jy-2j,l 2JX = gy + 75. i 12 a: = 9_>' 5- >^^+ ^^J'^i 6. :v + 7 = rt; + ^, j ax — dy = P — d^. \ 7. ;«: — ^ = jj; _ ^^ « ^ — 2 ^" == /^_); — 2 ^ 8. bx— b^ =ay, ) ax — by — a^. \ 9. a^x ■\- b^y = c^, a^x + b^y = ^. 10. ax + by — a^ + z^^, ,'o>' — 3-^ bx-\-ay = 2ab. II. (^ + /^) Jt: — (rt! — Z') _)/ = 3 rt; ^, (rt; + /;) J* — (rt; — /^) j:t: = rtt <^. Note. The student should now solve the preceding exam- ples mentally^ without writing out the values of x and y in the determinant form. 426. Determinants of the Third Order. The ordinary form of a determinant of the third order is that in (i), h , h ^2 (l) ^72 \h '^2 (2) and its expansion, or development, is ^1 K ^3 + ^2 ^3 ^1 + ^3 '^l ^^2 — ^3 <^2 <^i — ^2 '^i '^3 — ^1 ^2, ^2- Thus, the expansion of a determinant of the third order is the sum of the three products obtained from the principal diagonal and lines parallel thereto, as DETERMINANTS. 323 indicated in form (2), minus the sum of the three products obtained from the secondary diagonal and lines parallel thereto. Each product includes three elements, one, and only one, from each column and each row. The term a^ ^2. ^3 is called the Principal Term. In the above notation the number of the column to which a given element belongs is indicated by the letter, and that of the row by the subscript number; thus, the element b.^ belongs to the second column and the third row. According to this notation we have X —m s y n t Z — O 71 ^z: xnu — y OS — z JH t — z ns ■{- y mu -\- xo t. 2 - 1 —4 3-35 = - 5 + 24 - 24 -}- 12 -I- 20 - 12 = 15. Evaluate EXERCISE 66. I 2 3 4- 4 — I — 2 7- X y z 2 3 4 3 V w u 3 4 5 3 -7 4 t r s 4 5 2 5- I — I I 8. ^1 a^ a^ — I 2 3 3 I -4 K K b„ 6 -^ 5 2 -3 -5 ^1 C.2 c^ I —I I 6. I \ \ 9- a ~-2a b 4 -3 \ \ \ 3* — c Ad 3 2 - -5 \ \ \ 2C : 5^ -4^ 324 ALGEBRA. lo. Write as a determinant each of the following expres- sions : (i) ams — dmx — els — atr -\- cix + dlr ; (ii) aks — rka + sma + r^ I — s^l — amr ; (iii) r^ — k'^ -\- 2srx — 2sxk, Show by expansion that II. 12. 13- ^2 ^2 = 0, a. ^ ^1 cii ^1 ^1 ^2 h ^2 = 0. ma-^ b. Cl — in ma^ Ih Ci ma^ K 'z «i + h^ ^2 + bo ^2 «2 ^2 ^2 + ^1 ^1 ^2 ^2 ^1 + '^l ^2 + ^2 Note. Examples ii, 12, 13 prove for determinants of the third order the principles I., II., III. of Exercise 64. «3 + ^3 ^3 427. A determinant whose elements are expressed by letters with stibscripts is often denoted by writing between bars its principal term, from which all its other terms are known. Thus, the identities in Exam- ples 12 and 13 of Exercise 66 may be written briefly as follows : \max b^ c^\ = m\ai b^ c^\, and I ^1 + /^i 4 I = I ^i ^1 4 1 + I ^1 ^2 ^3 DETERMINANTS. 325 428. To solve by determinants the system ax X + /?iy + CiZ = mi, a^x ^ b.^y + C2Z = m^, a^x + b.^y + CzZ = m^, {a) In §425 the first determinant of equation (3) or (5) may be obtained from the determinant of the sys- tem by writing the known terms in place of the coefficients of the unknown whose value is sought; and the second may be obtained from the first by putting for the known terms their values as given in (i) and (2). Writing equation (4) below according to this law, we find the value of x as follows ; wi bi ^1 = aix -\- biy ■ ■{- CiZ bi Cx m^ b^ ^2 a,x + b^y ■\- c^z b^ c^ (4) nt^ ^8 ^3 H^-^b^y^c^z ^3 ^3 = ax X b^ Cx + b^y bx Ci + c^z bi Cx a^x b^ c^ biy b^ Ci c^z bi ^2 a^x b^ ^3 bzy b^ ^3 ^3^ h ^3 = ^ki bi c^\^-y\bi b^ rgl + ^ki ^2 ^al = x\ai b^ c^\ x = \mx b^ ^sl-^l^i ^2 h Similarly we obtain and y = \a^ m^ c.\~\ax b^ c^\^ z = \ax b^ W3 \-^\ax b^ c^\ . 326 ALGEBRA. Hence, in a system of three linear equations, the value of each unknown may be written out according to the rule in § 425. Example. Solve the system x = 2 —2 5 3 - -2 3 - 3 .r - 2 J H- 2- = 2 2 X + 3/ - 2" = 5 X + y -\- z — 6 y ■\ 3 2 I 2 5 -I I 6 I 3 — 2 I 2 3 I I I I = 2, ^ = 3. EXERCISE 67. Solve the system X 2 X 3^ 2j|; + 32:1=: 2, 3j>; + 2 =: I, ^ +22 = 9. 2. ^+J^'+2:=:I,^ 2^+3^ + 2:= 4, |- 4 ji; + 9j/ + 2: — 16. J 3. ^+ 2j>; — 32= 6, \ 2^ + 4_y— 72: = 9, |- 3^- y — 5^ = 8. J :r + 27 + 2 <3r = II 2 JC + y + 2: = 7 3-^ + 4y + 2 = 14 5. X + sy + 4^= 14, Ji: 4- 27 + 2; = 7, 2^+ J/ +20= 2. 6. «:v + dy = ij ^ ^j; +r2: = I, |- ^2: + ax = I. -^ 7. v rtts -{- ex = c a, y b X -\- ay=^ ab. ) 8. x—ly= 6,\ y -\z^ 8, > z — \x=^ 10. ^ DETERMINANTS. 327 In examples 9 and 10, consider the reciprocals of x,y, z as the unknowns. 9- I < I 1 ^ c X y z 10. I + 4=0, X y Z X 429. Assuming that determinants of the fourth order have the properties I., II., III. of Exercise 64, we may apply the method of § 428 to solve the fol- lowing system : aix + l>iy + c^z -\- d-iW — m^y «2 •^" + ^2 >' + ^t^ -\- d^w = m2, a^x ■\- b^y + qz ■]- d^w = m^, a^x + b^y ■\- c^z -\- d^w = vi^. -A ( (3)1 (4) («) Wi b^ ri //, ^ ^1 ^ + <^i JJ' + ^1 2: + ^1 «^ ^1 ^1 dx W2 ^2 ^2 ^2 a^x -\- b^y -\- c^z ■{- d^w ^2 ^2 ^4 ^3 ^3 ^3 ^; «8^ + ^3^ + ^8 2 + ^^3 ^^ ^3 ^3 ^3 W4 ^4 ^4 ^4 rt;^ X + ^4 JV + ^4^ + ^4 ^ ^4 ^4 ^4 = . x\ax b^ c^ d^\ ,'. x = \ w, /^2 ^3 < I -^ I ^1 h ^3 '^rj . (5) Finding the values of j, 5-, and w is left as an exercise for the student. The work should be written out in full, applying only one principle at a time. The expansion of each of the determinants in (5) contains twenty-four terms. Similarly the solution of a system of five linear equations gives rise to determinants of the fifth order. 328 ALGEBRA. The expansion of a determinant of this order contains one hun- dred and twenty terms. Thus we see that, by the determinant method of notation, systems of linear equations are readily solved, and large expressions are written in concise forms. Determinants of the htu Order. 430. Inversions of Order. The number of inver- sions in a series of integers is the number of times a larger number precedes a smaller. In the determi- nant (i) of § 426, the letters and the subscripts in the principal term, ^1 ^2 c^, are in the natural order. If the letters, or the subscripts, are taken in any other order, there will be one or more inversions of order. Thus, in the series 2, 4, i, 3, there are three inversions; 2 pre- cedes I, and 4 precedes both i and 3. In the series ij 6, 5, 8, 2, 4, 9, (i) 3> 7» i» 6, 8, 5, 2, 4, 9. (2) Neglecting the relations of 5 and 8 to each other, the inversions of one series are evidently the same as those of the other. Interchanging 5 and 8 in series (i), giving series (2), will increase, and interchanging DETERMINANTS. 329 5 and 8 in series (2), giving series (i), will decrease, the number of inversions in the series by one. 432. A product of eleinc7its is said to be even or odd according as the total number of inversions of both its letters and its subscript numbers is even or odd. Thus, the product aiC<2,b^ is even, while the product b^c^ai is odd. 433. The character of a product as even or odd is not cJiaiiged by changing the order of its elements. For by the interchange of any two adjacent ele- ments in a product, as a^ c^ b^ d^, the number of inver- sions of the letters is changed by one (§ 431), so also is the number of inversions of the subscripts; hence the total number of inversions in the product is changed by an even number, that is, by 2 or 0. Hence an even (or odd) product will remain even (or odd), whatever change be made in the order of its elements. Thus, the product d^ a.^ c^ bi is even, if a^ b^ c^ d^ is even. 434. Hence the simplest method to determine whether a product is even or odd is to arrange its elements in the natural order of the letters, and then count the number of inversions of the subscripts ; or we might arrange the elements in the natural order of the subscripts^ and theft count the number of inversiojis of the letters. Thus, the product b^ d^ ^i ^^2 is even, since a^ b^ Ci d^ or Ci a.^ b-i d^ is even. 330 ALGEBRA. EXERCISE 68. Show that the product 1. <7o b^ Ci J 2 X -'- + y 4 z : 2.9, J 5_ X y 10 •4 = Z { 2 + 1^ -_ y ^ 8 X = 14.9.) 451. Eliminants. When the number of equations in a system is greater by one than the number of un- knowns, the system is impossible, that is, its eqiiatious are inconsistent, unless the relation between the co- efficients is such as to make one equation of the system depend upon the others. DETERMINANTS. 341 Let us consider the following system of three linear equations involving only two unknowns. aix + b^y + nix = 0, (i) \ a^x -{- d.^y + ;//2 = 0, (2) > {a) a^x + b^y + m. = 0. (3)) Multiplying equations (i), (2), (3), respectively by the co-factors of ;«i, m2, m^, in the determinant 1^1 b-i m^y and adding the resulting equations, we obtain "1 J^3 a^x -\r biy + mi ai bi ^9 bo a^x -\- b^y + W2 a^x ■\- b^y -\r nn a^ b^ .*. I a^ b^ Ph I = 0. (4) If the equations in {a) are consistent, the relation in (4) must hold true ; conversely, if (4) is satisfied, the equations in {a) are consistent. Note that | a^ b^ m^ \ is the determinant of the coeffi- cients and the known terms in system {a). Example. Test the consistency of the equations X ^- y + 2z = g,' x+ y~ 2 = 0, 7.x- y+ 2 = 3, X — 2y + 22 = l._ Here the determinant of the coefficients and the known terms is found to equal zero ; hence the equations are consistent. Equation (4) is called the Eliminant or Resultant of system (a), because it is the result obtained by elimmating the unknowns from its equations. i■ (a) 2x-3y- 2 = 0, (3)) indeterminate? If so, find the ratios of x,y, z. DETERMINANTS. 343 The determinant of the system equals zero ; hence the system is indeterminate. Solving (i) and (2) for x and _y, we obtain 453. Multiplication of Determinants. Let A = o o o o I o X2 (0 then ^1 b, ■— I — aA Xi jFi X2 y-i 1 ^1 «2 X ^1 71 ^1 Jh ^2 yi — I x^ yi x^ y% (2) Increasing each element of the first column in (i) by a^ times the corresponding element of the third column plus a^ times the corresponding element of the fourth column ; in like manner increasing each element of the second column by b\ times that of the third //;/^ b^_ times that of the fourth ; we then have A = — I — I cixx^ + ciiyx bxx ^ h^yx Xx yi ^,^2+ ^2^2 \ x^ + hyi ^2 y^ ax Xx + a^yx hx xx + b^yx ^1^2 + ^2j^'2 bx X2 + ^2 y^ (3) Equating the values of A in (2) and (3), we have xxyx X ^1 ^2 = bxb,\ (txXx + a^yx bxXx-^b^yx ax x-i + a^y^ bx x^ + b^ y^ (4) 344 ALGEBRA. Each column of elements in the product may be obtained as follows : Multiply the elements of the first row, Xi, yi, by ai, a2 respectively, and take their sum ; then multiply the elemefits of the seeottd row, Xg^ y.,, by ai, a^ respectively, and take their sum ; these sums are the elements of the first column in the product. The elements of the second column in the product are found in a similar manner, by using the elements of the second row of the multiplier instead of those of the first. By letting A a\ bx «2 b, ^3 ^3 ^1 —I ^■2 I O o o - - I o o o — I Xi yi Zl ^2 i'2 22 •^8 yz Zz we may in like manner prove that ^1 yi Zl X ^2 J'2 Z-2 ^8 J'3 Zz ^1 <^2 bi b^ Ci C2 ai Xi -\- a^y^ + a^ Zi bi x^ + ^2 Ji + ^3 Zi c^ x^ + <^2 J^i + ^3 -^i ax X.2 +a2y2 + ^3 2^2 ^i -"^2 + <^2 J2 + bs z.^ c^ x^ -f c^y^. + c^ z^ ^1 ^3 + ^2^3+^3 Zz bi xz + biyz + b^ 03 ^1 ^3 + c^y^ + ^3 2^3 (5) The same method of proof will apply to a deter- minant of any order. Hence the rule given above will enable us to obtain one form of the product of two determinants. DETERMINANTS. 345 Thus we have 3 4 X 4 3 = 2 3 5 4 232 X 2 I 2!E| 322 I 2 3 I 3 I 2 2 4 12+12 15 + 16 j — 24 31 8+9 10 + 12 I 17 22 4 + 3 + 4 2 + 6 + 6 4 + 6 + 8 6 + 2 + 4 3 + 4 + 6 6 + 4 + 8 2 + 3 + 2 1+6 + 3 2 + 6 + 4 = 0.) EXERCISE 70. Find the ratios of the unknowns in the system I. — 4x+y + z = 0, i 2. 2 X + jy — 2Z X— 2^ + 2 = 0. ) y + 4.Z — 4W jr— 5J'+ 2 + 2W 3. U aix + l>iy + Ciz = a2X -\- dzy + C2Z = a^x + b^y + ^sZ = a^x -{- If^y + c^z ; find the relation that must exist between the coefficients. Put each trinomial equal to — « ; then the required relation will be 1 rtj <^2^8 ^1 = 0- This relation expresses the condition that the resulting system shall be indeterminate. 4. Eliminate x from px^ + gx-\-r=0, (i) aindax^ -{- dx^ -\- ex -{- d=0. (2) From (i), px'^ + gx + r=0. (a) From (2), ax^ + bx'^^-cx^d- 0. id) . Multiply (2) by ,r, ax^ + b x^ + cx'^ + dx =0. Multiply (i) by Jir, px^ -\- q x"^ + rx =0. Multiply (i) by x^ p x^ ■}- qx^ + rx"^ = 0. These five equations involve the four unknowns x^, x^, x'^f x; hence, by § 451, their eliminant is P Q r a b c d a b c d P ^ r P i r = 0. (3) 346 ALGEBRA. Equations (i) and (2) being consistent, equation (3) is their eliminant ; or, to test the consistency of (i) and (2), ascertain whether or not (3) is satisfied. EHminate x from the consistent equations 5. ax^ -\- bx + c = 0, \ 6. ax^-\-bx!^-{-cx-\-d = ^, x^ \ qx ■\- r ^^.) p x^ ■\- qx^ + rx-{- J- =0. ) 7. ax'^ -^ bx^ -\- cx^ -\- dx -\- e ^=^, = 0.f p x^ -^ qx ■\- r Test the consistency of the equations 8. 2x^ — 2>x'-\- 6a: = 0, x'^ — 2 X — 3 =0. 9. dx^ — 3 x'^y — xy^ — 1 2 j^ = 0, ^ 4^^ —Zxy-^ 3 j/2 — 0. 3 Divide the first equation by x^ and the second by x^^ then regard the different powers of j -f at as the unknowns. 10. Find the product of II. Find the product of I 2 I 7 2 15 7 1 2 2 I and Note that 12. Find the product of and GRAPHIC SOLUTIONS. 347 CHAPTER XXIII. GRAPHIC SOLUTION OF EQUATIONS AND SYSTEMS. 454. Let XX' and Y Y' be two fixed straight lines at right angles to each other at O. Let OX and 6^ F be positive directions; then OX' and O Y' will be negative directions. The lines XX' and Y Y' divide their plane into four equal parts called Quadrants, which are numbered as follows: The First Quadrant is X Yy the Second YOX', the T/iird X' O F. and the ^'' Fourt/i Y'OX. The lines XX' and YY', are called Axes of Reference, and their intersection O the Origin. P>om P, any point in the plane of the axes, draw PJf parallel to Y Y' ; then the position of P will be determined when we know both the leiigths and the directions of the lines 6^ J/ and MP, Fig. I. 348 ALGEBRA. The line O M \s called the Abscissa of the point P; and MP is called the Ordinate of P. The abscissa and ordinate together are called the Co-ordinates of P. Thus, O A and A P' are the co-ordinates oi F' \ the abscissa, O A^ is negative, and the ordinate, A /", is positive. O C, the abscissa of F'", is positive, and C F"\ its ordinate, is negative. An abscissa is usually denoted by the letter x^ and an ordinate by j. The axis XX' is called the Axis of Abscissas, or Axis of x; and Y V, the Axis of Ordi- nates, or Axis of y. The point whose co-ordinates are x and j/ is written (x,j/). Thus (2, —3) denotes the point of which the abscissa is 2, and the ordinate — 3. We use a system of co-ordinates analo- gous to that explained above whenever we locate a city by- giving its latitude and longitude. In this case, the equator is one axis, and the assumed, meridian the other. Example. Construct the point (-2, 3) ; (-3, — 4). In Fig. I, lay off (9 /^ = —2, and on A F' parallel to V V lay off /? /" = +3 ; then is P' the point (—2, 3). To construct (— 3, —4), lay off (9^ = —3, and on B F" par- allel to YV lay off B F" = -4; then is F" the point (-3,-4). EXERCISE 71. 1. Construct the point (2, 3); (4,7); (3,-5); (-2,-3); (4,-2); (-5,-3); (-2, 5). 2. In which quadrant is the point (^', j^^), when .x and _y are both positive ? both negative ? x negative and y positive ? X positive and y negative ? GRAPHIC SOLUTIONS. 349 3. In which quadrant may (x, j) be, when x is positive? X negative ? y positive ? y negative ? 4. In what Hne is (x^ o) ? (o, y) ? Where is (o, o) ? (4, o) ? (-3,0)? (0,2)? (0,-5)? 455. To solve an indeterminate equation graphically is to draw the line or lines which include all the points, and only those, whose co-ordinates satisfy the equation. The line or lines is called the Graph of the equation. 456. To draw the graph of an indeterminate equa- tion, we obtain a number of its solutions, then con- struct a sufficient number of the corresponding points to determine the form of the graph, and through these points trace a continuous curve. Example i. Solve y^x^-x-^y graphically. If we put ;r = —3, —2, ... , we obtain when x- —3, —2, — i, o, i, i, 2, 3, 4,... j)/= 6, o, -4, -6, -6^, -6, -4, o, 6,... Drawing the axes XX' and Y V in Fig. 2, and assuming O i as the linear unit, we construct the points (—3, 6), (-2, o), (-1,-4),... Tracing a continuous curve through these points, we obtain the curve A a B 2.^ the graph of the given equation. As X increases from 4, j or :r2 — ;r — 6 continues positive and increases; hence there is an infinite branch of the locus in the first quadrant. As x decreases from — 3, j continues posi- tive and increases; hence there is an infinite branch in the second quadrant. 350 ALGEBRA. Example 2. Solve j = x^ — 2x graphically. When x= —2, — i, —0.8, o, 0.8, i, 2,... y= -4, I, I.I, o, -I.I, -I, 4, .. . Locating these points as in Fig. 3, and tracing a continuous curve through them, we obtain the curve MNP Q as the graph oiy = x^ ~ 2 X. Here evidently one infinite branch is in the first, and the other in the third quadrant. Fig. 2. Whenever there is any doubt about the form of a graph be- tween any two determined points, intermediate points should be located. Example 3. Construct the graph of/ 3 :r2 + 4. When :r= -f, y = —6.1 The 5:raph is given in Fig. 4 I, -1, o, \, I, 2, 3, . . . o, 3.1, 4, 3.4, 2, o, 4, . . . GRAPHIC SOLUTIONS. 351 Example 4. Solve y = x^-\-x^ — 'ix'^—x-\-2 graphically. When x= -|, -2, - 1, -i, _^, o, \, i, |, j= 9.2, o, —1.6, o, 1.7, 2, 09, o, 2.2. The graph is given in V\g. 5. Of the infinite number of real solutions of an inde- terminate equation each is represented geometrically by the co-ordinates of some point in its graph ; hence the graph of an indeterminate equation represents geometrically all its infinite number of real solutions. EXERCIS: Solve graphically the equation y ^= x""— 2 X y 7^-+ 10. y — X^—:^X^ + y — x^ -\- 4X + 352 ALGEBRA. 5. Construct two points of the graph of the linear equa- tion y = 2 X -\- ;^, and prove that the unlimited straight line through them is the graph of this equation. Similarly the graph of any linear equation can be con- structed ; hence t/ie graph of any linear equatiofi in x atid y is a straight line. 6. Construct the graph oi y — ^iX— 2) of2_y = — 4^+ i; of 37 + 5 -^'+2=0; of:v = 3; ofa: = — 5; of 7 = 2 ; of y = -^• The equation/ — b may be written in the form y — ox -\- b^ in which, for any value q>{ x^y = b; hence the graph of_7 = ^ is a line parallel to the axis of x. 7. Show that the graph of the equation j*; ^ + _>'^ — 5- is a circle whose centre is at the origin, and whose radius is 5. Evidently the graph of any equation of the form x^ +y^ = r^ is a circle whose centre is at the origin and whose radius is r. 8. Construct the graph of ^^ + y = 9 ; oi x^ -{■ y"^ = 16 \ of 2^^ + 2jv^ = 8. 9. Construct the graph of 4 jr^ + 9 ^ = 36. Herej = ±| Vg - x\ Evidently —3 is the least real value of x that will render y real ; hence no part of the graph can lie to the left of the line :r = — 3. For like reason no part of the graph can lie to the right of jr = 3. When;r=:— 3, —2.5, —2, — i, o, i, 2, 3, j = ±o, ±1.1, ±1.5, ±1-9' ±2, ±1.9, ±1.5, ±0. The graph is the ellipse ANBS (Fig. 9, page 356), the semi-axes being 3 and 2. GRAPHIC SOLUTIONS. 353 lo. Construct the graph oiy^—4x — x^ /' Herey=-xy . ^, The graph is the curve H P P' A (Fig. 8, page 355). It is evident that when x — v>,y approaches -x\ hence, in either quadrant, the infinite branch approaches indefinitely near to, but cannot reach the line/ — -x, or D B. 1 1. Show that the graphs oi y = 2x -\- i and _y = 2 a: + 3 intercept equal segments on lines parallel to Y V\ and are therefore parallel. Show also that y = x and y = x + c are parallel, and that each makes an angle of 45° with the axis of X. 12. Construct the graph ofy= 4 jc; of 4^— 97^=36; of (i + x^) y = X. a'x+c'. (2)) ^ ^ 457. Graphic Solution of Systems of Equations. Example i. Solve the system / / = Let the graph of (i) be the straight line MB, and that of (2) the line R P; then the single solu- ^M tion of system {ii) is ^ the co-ordinates of the common point P. This illustrates the general truth that a system of linear equa- tions has one and only one solution. By measuring the co-ordinates O A and A P, the numerical solution of the system may be obtained approximately. 23 Fig. 6. 354 ALGEBRA. (a) li a = a' and <: = ^', the graphs of (i) and (2) will evidently co- incide ; that is, the graphs of equivalent equations coincide. \i a — a\ and c = not c', the graphs of (i) and (2) will be parallel ; for they will intercept equal segments of the value c — c\ on all lines parallel to Y Y' j that is, the graphs of in- consistent linear equations are parallel. As the lines PR and P M approach parallelism, their inter- section P recedes to an infinite distance from the origin. When they become parallel, their intersection is lost at infinity, which illustrates the infinite solution of an impossible linear system (Example of § 176). Example 2. Solve the system x^ -\- y'^ = 25, (i) y-x = c. (2) / If O A =5, the graph of (i) is the circle P' P'" P, and, if c — ^., the graph of (2) is the straight line M Nj hence the two solutions of system {a) are the co-ordinates of the two points, P and P'. By measurement we find the two so- lutions to be ;r = — V-,j/ = -2/. For ^ = 5 |/2 or " _ 5 |/2, the graph Fig. 7. of (2) is the tan- gent N' M' or N'' M'\ and the two solutions of the system are equal. For c < 5 \/2 and > — 5 \/2, the graph of (2) lies between N' M' and N" M" ^ and the two solutions of the system are real and unequal. / ^ ^ — ^^ /^ A /!\ N^ ^1 / ^ \ v' X A X- \ k ■^^ 'o y^'" X GRAPHIC SOLUTIONS. 355 = 4-^-^, (I) = ax i- c. (2) [(«) For c > 5 I '2 or < 542, the graph of (2) does not cut the circle, and both solutions of the system are imaginary. Example 3. Solve the system y' y The graph of (i) is the curve' H P O F' A, of which the infinite branches approach indefinitely near to the graph of ^ = — x, Qx D B (Example 10 of Exercise 72). \ia— I, and c= 0, the graph of (2) is the line M N^ and the three solu- tions of system {a) are the co-ordinates of the points P, (9, and P' . If ^ = — I, and ^ = 0, the graph of (2) is DB, and system {a) is defective; only one solution is finite, the other two being infinite. \i a — — \ and r = 2, the graph of (2) is A' A', and two solutions of {a) are finite, the third being infinite. If c were increased from 2, the two finite solutions would approach equality, become equal, and then become imaginary. If in equation (i) we put v = — x. we obtain - ;i-8 = 4 r - .r8, or o .r« + o.r^ - 4 :r = 0, which illustrates § 407, since the abscissas of both P and P' become infinite, when the line M N is revolved clockwise about O to the position of D B. Example 4. Solve the system ^x^ ^ gy^ = 36, (i) X^ i- j2 — ^2^ The graph of (i) is the ellipse A NBS, in which O A = 2, and ON— 2. Fig. 8. 356 ALGEBRA. If r = |, the graph of (2) will be the circle P F P" , and all four solutions of the system will be real and unequal. If r = 3, the circle will bj tangent to the ellipse at A and B; hence two solutions of the system will be ;ir = 3, j = 0, and the other two jr = — 3, j/ = 0. If r =^ 2, the circle will be tangent to the ellipse at A^and 6^. If r < 2 or > 3, the two graphs will have no common points, and all the solutions of the system will be imaginary. P^^_ _r_^ i pHn^^ ^f^'" ^—-'^y^ ^^ Y Fig. 9. li r = ^, by clearing (2) of fractions, and then subtracting it from (i) we obtain 5^/^ = 11, of which the graph is the parallel lines P P' and P'" F' . These lines cut either the ellipse or the circle in all points common to both of these curves, and only in these points. This illustrates the equivalency of system {a) to the system 4X^ + gy^ = 36 > x^ + /^ = V'' I 458. A system of equations involving x a7id y, one of the nth degree and the other of the ist, has n and onfy n solntions. GRAPHIC SOLUTIONS. 35/ For substituting the value ofjj/, as obtained from the Hnear equation, in the equation of the ;/th degree, we obtain in general an equation of the ;^th degree in X. This equation gives ;/ values for x, each of which gives one value for j/ in the linear equation. 459. A system of equations involving x and y, one of the mth degree and the other of the nth, has in gen- eral mn solutions. The general proof of this theorem is too long and difficult to be given here. The theorem is very evident, however, when one of the equations can be resolved into equivalent linear equations- For example, the system (x-2y- I) (x+y-2) (^ + 3/) = 0, j ^^^ is equivalent to the three systems, X* — dy* = axy \ x^ — by^ = axy\ x^ — by^ = axy} X— 2y = I )' X -\- y = 2 ) :ir+37=0 > each of which by § 458 has four solutions; hence the given system (a) has 4 x 3, or 12, solutions. 460. By its ordinates, the graph of j^== ^ (^) rep- resents graphically the continuous series of values of F {x) corresponding to a continuous series of values of .r; hence the graph oiy = F(^x^ is often called the graph of F (:r). The graph of /^(;ir) clearly illustrates the following properties of F (x} and of the equation F (x) = 0. 358 ALGEBRA. (i.) The abscissa of any point in which the graph of F {x) cuts or touches the axis of x is one of the unequal or equal real roots of the equation F (x) = 0. Hence the real roots of F {x) = may be obtained approximately by measuring the abscissas of the points in which the graph of F (x) cuts the axis of ;ir. At a point of tangency the graph is properly said to meet the axis of ;ir in two coincident poirits. Thus from the graph in Fig. 2 we learn that one root of the equation :r'^ — ;f — 6 = is — 2, and the other is 3. In Fig. 3, the graph crosses the axis of x between ;r = — 2 and X — ~ \\ hence one root of x^ — 7.x :=^ lies between - 2 and — I ; a second root is zero, and the third lies between i and 2. In Fig 4, the graph cuts the axis of ;ir at ;jr = — 1, and touches it at ;r=: 2; hence one root of ^ir^ — 3 jr^ + 4 = is — i, and the other two are 2 each. (ii.) The grc ph of F {x) renders evident the theorem of § 397. For it is clear that the portion of a continuous curve between any two points must cut the axis of X an odd number of times when these points are on opposite sides of that axis ; and an even number of times, or not at all, when the points are on the same side of that axis. Thus, in Fig. 3, the graph cuts XX' an odd number of times between M and /V, or M and Q, and an even number of times between M and P. GRAPHIC SOLUTIONS. 359 In Fig. 5, the graph cuts XX' an odd number of times be- tween M and N, or N and /4, and an even number of times between M and R, R and A, ox M and A. (ill.) The graph of F {x) also illustrates the fact that equal roots form the connecting link between real and imaginary roots, and that imaginary roots occur in pairs. Thus, by slightly diminishing the absolute term 4 of the function ;r8 — 3:^2 -f 4, its graph in Fig. 4 would be moved downward, and would then cut the axis of x in three points ; by slightly increasing the term 4, the graph would bs moved up- ward, and would then cut the axis of x in but one point. This illustrates the fact that the two equal real roots of the equation ^^ - 3 :f 2 + 4 = would become unequal real roots or imaginary, according as the absolute term 4 were diminished or increased. By increasing the absolute term of F{x^ (Example 4, § 456) by 2, all the roots oi F{x) — would become imaginary. (iv.) The graph of F (a-) illustrates §§ 398 and 399. For when F (,r) is of an odd degree, one infinite branch of its graph will be in the first quadrant, and the other in the third; hence the graph will cut the axis of X in at least one point. When F {x^ is of an even degree, one infinite branch of its graph will be in the first quadrant, and the other in the second. Now if pn is negative, the graph will cut the axis of jj/ below the origin; hence it will cut the axis oi x in at least two points, one to -^the right and the other to the left of the origin. 36o ALGEBRA. EXERCISE 73. Construct the graph of F (x) and locate the real roots of F ix) — ^ in each of the following examples : \. F {x)^o(^ ^ X —2. ^. F {x)^x^ — ix^—\x-^\\. 2. F{x)^x^-2x-^. 5. >^(jt:) = ^3-4^-6^+8. 3. F{x)^x'^ — ^x^^\o. 6. ^(.t) = :^*— 4^^ — 3X+23. 7. i^ (^) = ^^ -h 2 :r^ — 3 ^-^ — 4 -^ + 4- 8. F(x) = a;^ + 4^^— 14 a;^ — 17^ — 6. 461. Geometric Representation of Imaginary and Complex Numbers. A line whose value includes both its length and direction is called a Directed Line, or a Vector. We pro- ceed to show that any algebraic num- ber, real, imagi- nary, or complex, can be represented by some vector. Let vector O A represent + i ; then OA' = -i. But (-M)x(-i) = -i; hence — i as a fac- tor reverses the yq.c- ior A, or turns it through 180°. Therefore + y'-T, or— \/^^(that is, one of the two equal factors of— i) 15 i \ ,-! + 1 1 "3 B' Fig. 10. GRAPHIC SOLUTIONS. 36 1 will revolve the vector O A through 90°. Suppose V— I to revolve the vector OA counter-clockwise; then — V— I vvill revolve it clockwise : hence O B = + V^i and O B' = - V^T. For brevity, the symbol V— i is generally denoted by t. Hence t as a factor revolves a vector through 90° counter-clockwise, and therefore — t revolves a vector 90° clockwise. Hence we have (i) ii= {+1) ll:=-I, Hi— (-i- i) //• i — (— i) /, or — /, iiii—{-\- 1) ii- it = (— i) // = -f i. (ii) ai = ia) that is, (-h \)ai— (-h i) ia, (i) For multiplying the unit by a and then revolving it through 90° gives the same result as first revolving the unit through 90° and then multiplying it by a. ai bi= ab ' ii, a ii • b, or //' • a b, (2) For multiplying the vector ai hy b and then re- volving it through 90° gives the same result as revolving the vector ^^ through 90° twice in succes- sion, as multiplying the vector aii (or — a) by b, or as multiplying the vector ii (or — i) hy ab. That is, the commutative and associative laws of multiplication hold true for imaginary factors. Next let us consider the complex number a -\- b i, where a and b are both real. 362 ALGEBRA. Let a — \ ^2, and b B ^-''' T ^~--^ P> ! XP / \ 1 / / \ 1 / \ / \ 1 / / \ ' / / \. ! / \ / / a'I \ 1 / \ 1 / \ M' /0\ M / 1 \ / 1 \ \ \ / 1 \ / \ / 1 \. / \ / [ \ / \ / P"X I /?'" ^--~, i .-"" \ \/2. Lay o^ O M=\ /^2, and on MP drawn perpendicular to OA lay off J/P = I ^2 ; then M — + i V2, M P = \ V2 • ^"• V'2 B' Fig. II. + \^2'i ^ M + MP= OP. (i) The vector OP equals the sum of the vectors O M and MP; for trans- ference from O to M followed by transference from M to P gives the same result as transference from directly to P. For like reason OP"'r=^ \^2-\^2i, (2) OP^ =-\^2-V\^2i, (3) OP" ^-h^/2-\ ^2i. (4) In like manner any complex number may be represented by some vector. In Fig. 10, the vectors O B and OB' represent the two square roots of — i ; for either multiplied by itself gives — I. In Fig. II, by geometry we know that OP, O P\ OP", OP"' are each a unit in length. Hence J ^/2 GRAPHIC SOLUTIONS. 363 (i + /) as a factor revolves the unit O A through 45°; whence [^ ^2 (i + 0? == ^^' = - I- (S) As a factor \^^2 (i — i) revolves OA through — 45° i whence l\ ^2{i - i)]"^ = A' = -i. (6) As a factor -- ^ /^2 (i — i) revolves OA through 135°; whence [- ^ ^2 (i — t)f = A' = - i. (7) As a factor — ^ ^2 {i + i) revolves OA through -135°; whence [- \ ,^2 {\ -\- i)f = O A' = -i. (8) Hence OP, OP', OP", OP'" represent the four fourth roots of — i. In like manner we could represent geometrically the six sixth roots of — i ; and so on. SCIENCE. 47 Physics for Uniuersity Students. By Professor Henry S. Carhart, University of Michigan. Parti. Mechanics, Sound, and Light. With 154 Illustrations. i2mo, cloth, 330 pages. Price, ^1.50. Part II. Heat, Electricity, and Magnetism. With 224 Illustrations. i2mo, cloth, 446 pages. Price, $1.30. THESE volumes, the outgrowth of long experience in teach- ing, offer a full course in University Physics. In preparing the work, the author has kept constantly in view the actual needs of the class-room. The result is a fresh, practical text-book, and not a cyclopaedia of physics. Particular attention has been given to the arrangement of topics, so as to secure a natural and logical sequence. In many demonstrations the method of the Calculus is used without its formal symbols ; and, in general, mathematics is called into ser- vice, not for its own sake, but wholly for the purpose of establish- ing the relations of physical quantities. It is believed that the work will be helpful to teachers who adopt the prevailing method of a combination of lectures and text-book instruction. As it is intended to supplement, not super- sede, the teacher, it leaves ample scope for the personal equation in instruction. Professor W. LeConte Stevens, Rensselaer Polytechnic Institute, Troy, N. Y. : After an examination of Carhart's University Physics, I have unhesitat- ingly decided to use it with my next class. The book is admirably arranged, clearly expressed, and bears the unmistakable mark of the work of a successful teacher. Professor Florian Cajori, Colorado College : The strong features of his Uni- versity Physics appear to me to be conciseness and accuracy of statement, the emphasis laid on the more important topics by the exclusion of minor details, the embodiment of recent researches whenever possible. Professor A. A. Atkinson, OAio University, Athens, O. : I am very much pleased with the book. The important principles of physics and the essentials of energy are so well set forth for the student for which the book is designed, that it at once commends itself to the teacher. Professor Sarah F. Whiting, Wellesley College, Mass. : I am using it with one of my classes, and find that it is well adapted to supplement lectures and to put the student in possession of salient points. 48 SCIENCE. The Elements of Physics. By Professor HENRY S. Carhart, University of Michigan, and H. N. Chute, Ann Arbor High School. i2mo, cloth, 392 pages. Price, ^1.20. THIS is the freshest, clearest, and most practical manual on the subject. Facts have been presented before theories. The experiments are simple, requiring inexpensive apparatus, and are such as will be easily understood and remembered. Every experiment, definition, and statement is the result of practical experience in teaching classes of various grades. The illustrations are numerous, and for the most part new, many having been photographed from the actual apparatus set up for the purpose. Simple problems have been freely introduced, in the belief that in this way a pupil best grasps the application of a principle. The basis of the whole book is the introductory statement that physics is the science of matter and energy, and that noth- ing can be learned of the physical world save by observation and experience, or by mathematical deductions from data so obtained. The authors believe that immature students cannot profitably be set to rediscover the laws of Nature at the beginning of their study of physics, but that they must first have a clearly defined idea of what they are doing, an outfit of principles and data to guide them, and a good degree of skill in conducting an investigation. WiJliam H. Runyon, Armour Institute, Chicago : Carhart and Chute's text- book in Physics has been used in the Scientific Academy of Armour Institute during the past year, and will be retained next year. It has been found concise and scientific. We believe it to be the best book on the market for elementary work in the class-room. W. C. Peckham, Adelphi Academy, Brooklyn, N. Y. : Carhart and Chute's Physics on the whole impresses me as the best book for a beginner to use in getting his first view of the general principles of the whole subject. Professor A. L. Kimball, Amherst College : As a text-book to be used in high school classes, I do not know of any that is superior to it. Professor C. T. Brackett, Princeton University: I have examined this work with care and with pleasure, for it presents the fundamental prin- ciples of physics with exactness and with clearness. Professor George F. Barker, University of Pennsylvania : The book is an excellent one ; the best of its grade in the market. SCIENCE. 49 Electrical Measurements, By Professor Henry S. Carhart and Asst. Professor G. W. Patter- son, University of Michigan. i2mo, cloth, 344 pages. Price, $2.00. IN this book are presented a graded series of experiments for the use of classes in electrical measurements. Quantitative experiments only have been introduced, and these have been selected with the object of illustrating general methods rather than applications to specific departments of technical work. The several chapters have been introduced in what the authors believe to be the order of their difificulty involved. Explana- tions or demonstrations of the principles involved have been given, as well as descriptions of the methods employed. The Electrical Engineer, New York : We can recommend this book very highly to all teachers in elementary laboratory work. The Electrical Journal, Chicago : This is a very well-arranged text-book and an excellent laboratoi7 guide. Exercises in Physical Measurement By Louis W. Austin, Ph.D., and Charles B. Thwing, Ph.D., University of Wisconsin. i2mo, cloth, 198 pages. Price, ^1.50. THIS book puts in compact and convenient form such direc- tions for work and such data as are required by a student in his first year in the physical laboratory. The exercises in Part I. are essentially those included in the Practicum of the best German universities. They are exclu- sively quantitative, and the apparatus required is inexpensive. Part II. contains such suggestions regarding computations and important physical manipulations as will make unnecessary the purchase of a second laboratory manual. Part III. contains in tabular form such data as will be needed by the student in making computations and verifying results. Professor Sarah F. Whiting, Wellesley College : It comprises very nearly the list of exercises which I have found practical in a first-year college course in Physics. I note that while the directions are brief, skill is shown in seizing the very points which need to be emphasized. The Introduction with Part II. gives a very clear presentation of the essential things in Measurements, and of the treatment of errors. 60 SCIENCE. Principles of Chemical Philosoph y. By JosiAH Parsons Cooke, late Professor of Chemistry, Harvard University. Revised Edition. 8vo, cloth, 634 pages. Price, ^3.50. THE object of this book is to present the philosophy of chem- istry in such a form that it can be made with profit the subject of college recitations. Part I. of the book contains a statement of the general laws and theories of chemistry, together with so much of the principles of molecular physics as are con- stantly applied to chemical investigations. Part II. presents the scheme of the chemical elements, and is to be studied in con- nection with experimental lectures or laboratory work. Elements of Chemical Physics. By JosiAH Parsons Cooke. 8vo, cloth, 751 pages. Price, $4.50. THIS volume furnishes a full development of the principles of chemical phenomena. It has been prepared on a strictly inductive method and any student with an elementary knowledge of mathematics will be able easily to follow the course of reasoning. Each chapter is followed by a large number of problems. Chemical Tables. By Stephen P. Sharples. izmo, cloth, 199 pages. Price, ^2.00. Logarithmic and Other Mathematical Tables. By William J. Hussey, Professor of Astronomy in the Leland Stan- ford Junior University, California. 8vo, cloth, 148 pages. Price, ^i.oo. IN compiling this book the needs of computers and of students have been kept in view. Auxiliary tables of proportional parts accompany the logarithmic portions of the book, and all needed helps are given for facilitating interpolation. Various mechanical devices make this work specially easy to consult ; and the large, clear, open page enables one readily to find the numbers sought. SCIENCE. 51 Anatomy, Physiology, and Hygiene. " A Manual for the Use of Colleges, Schools, and General Readers. By Jerome Walker, M.D. i2mo, cloth, 427 pages. Price, ^1.20. THIS book was prepared with special reference to the require- ments of high and normal schools, academies, and colleges, and is believed to be a fair exponent of the present condition of the science. Throughout its pages lessons of moderation are taught in connection with the use of each part of the body. The subjects of food, and of the relations of the skin to the various parts of the body and to health, are more thoroughly treated than is ordinarily the case. All the important facts are so fully explained, illustrated, and logically connected, that they can be easily understood and remembered. Dry statements are avoided, and the mind is not overloaded with a mass of technical material of little value to the ordinary student. The size of type and the color of paper have been adopted in accordance with the advice of Dr. C. R. Agnew, the well-known oculist. Other eminent specialists have carefully reviewed the chapters on the Nervous System, Sight, Hearing, the Voice, and Emergencies, so that it may justly be claimed that these impor- tant subjects are more adequately treated than in any other school Physiology. The treatment of the subject of alcohol and narcotics is in conformity with the views of the leading physicians and physiol- ogists of to-day. The Nation, New York : Dr. Jerome Walker's Anatomy, Physiology, and Hygiene appears an almost faultless treatise for colleges, schools, and general readers. Careful study has not revealed a serious blemish ; its tone is good, its style is pleasant, and its statements are unimpeachable. We cordially commend it as a trustworthy book to all seeking information about the body, and how to preserve its integrity. Journal of the American Medical Association: For the purposes for which it is written, it is the most interesting and fairest exponent of present physiological and hygienic knowledge that has ever appeared. It should be used in every school, and should be a member of every family, — more especially of those in which there are young people. It is a pleasure to read and review such an excellent book. SCIENCE. The Elements of Chemistry. By Professor Paul C. Freer, University of Michigan. i2mo, cloth, 294 pages. Price, ^1,00. IN the preparation of this book an attempt has been made to give prominence to what is essential in the science of Chemistry, and to make the pupil famiHar with the general aspect of chemical changes, rather than to state as many facts as possible. To this end only a few of the most important elements and compounds have been introduced ; and the work, both in the text and in the laboratory appendix, has been made quantitative. Chemical equations have been sparingly used, because they are apt to give the pupils false notions of the processes they attempt to record. Considerable space has been given tb physi- cal chemistry, and a constant effort has been made to present chemistry as an exact science. The apparatus required to perform successfully the experi- ments suggested will not be found expensive, the most costly being such as will form part of the permanent equipment of a laboratory, and if properly handled will not need to be replaced during a long term of years. Professor Charles Baskerville, University of North Carolina: It is the most excellent book of the character which has ever come to my notice. It is clear, scientific, and thoroughly up to date. Professor E. E. Slosson, University of Wyoming-: Freer's Elementary Chemistry gives the most completely experimental and logical proof of the fundamental laws of chemistry for beginners that I have seen. It teaches chemistry as a rational, and not merely as a descriptive, science. Willard R. Pyle, Media Academy, Media, Pa. : It emphasizes the nature of chemical changes, connects facts, and leads the student to think. In these respects it is superior to any elementary chemistry I know of. Lewis B. Avery, High School, Redlands, Cat. : The book has proved far better than I had dared to hope for in adopting it. The rational mode of treatment and the remarkable perspicuity of the work encourage sound scientific thought and genuine interest among pupils such as I have not, in ten years' experience with the best of the books now most in use, been able to obtain. SCIENCE. 53 Descn'ptiue Inorganic General Chemistry. A text-book for colleges, by Professor Paul C. Freer, University of Michigan. Revised Edition. 8vo, cloth, 559 pages. Price, $3.00. IT aims to give a systematic course of Chemistry by stating certain initial principles, and connecting logically all the resultant phenomena. In this way the science of Chemistry appears, not as a series of disconnected facts, but as a harmo- nious and consistent whole. The relationship of members of the same family of elements is made conspicuous, and resemblances between the different families are pointed out. The connection between reactions is dwelt upon, and where possible they are referred to certain prin- ciples which result from the nature of the component elements. The frequent use of tables and of comparative summaries les- sens the work of memorizing and affords facilities for rapid refer- ence to the usual constants, such as specific gravity, melting and boiling points, etc. These tables clearly show the relationship between the various elements and compounds, as well as the data which are necessary to emphasize this relationship. They also exhibit the structural connection between existing compounds. Some descriptive portions of the work, which especially refer to technical subjects, have been revised by men who are actively engaged in those branches. In the Laboratory Appendix will be found a list of experiments, with descriptive matter, which materially aid in the comprehension of the text. Professor Walter S. Haines, I^usA Medical College, Chicago : The work is worthy of the highest praise. The typography is excellent, the arrange- ment of the subjects admirable, the explanations full and clear, and facts and theories are brought down to the latest date. All things considered, I regard it as the best work on inorganic chemistry for somewhat advanced general students of the science with which I am acquainted. Professor J. H. Long, Northwestern University, Evanston, III. : I have looked it over very carefully, as at first sight I was much pleased with both style and arrangement. Subsequent examination confirms the first opinion that we have here an excellent and a very useful text-book. It is a book which students can read with profit, as it is clear, systematic, and modern. 56 MA THE MA TICS. An Academic Algebra. By Professor J. M. TAYLOR, Colgate University, Hamilton, N.Y. i6mo, cloth, 348 pages. Price, ^i.oo. THIS book is adapted to beginners of any age and covers sufficient ground for admission to any American college or university. In it the fundamental laws of number, the literal notation, and the method of solving and using the simpler forms of equations, are made familiar before the idea of alge- braic number is introduced. The theory of equivalent equa- tions and systems of equations is fully and clearly presented. Factoring is made fundamental in the study and solution of equations. Fractions, ratios, and exponents are concisely and scientifically treated, and the theory of limits is briefly and clearly presented. Professor C. H. Judson, Furman University, Greenville, S.C.: I regard this and his college treatise as among the very best books on the subject, and shall take pleasure in commending the Academic Algebra to the schools of this State. Professor E. P. Thompson, Miami University, Oxford, O. : The book is compact, well printed, presenting just the subjects needed in preparation for college, and in just about the right proportion, and simply presented. I like the treatment of the theory of limit?, and think the student should be introduced early to it. I am more pleased with the book the more I examine it. E. P. Sisson, Colgate Academy, Hatnilton, N. V. : It has the spirit of the best modern thought on mathematics. The book is conspicuously meri- torious : First, In the clear distinction made between arithmetical and algebraic number, which lies at the foundation of an understanding of Algebra. Second, The introduction at the very first of the equation as an instrument of mathematical investigation. By this instrument many of the demonstrations of the theorems which follow have a conciseness and clearness which could not otherwise be obtained. Third, Dr. Taylor's presentation of the doctrine of equivalency is clear and rigid. Fourth, The treatment of the subject of factoring is concise, comprehensive, and logical. ... I am using it with satisfaction in my own classes. Arthur G. Hall, University of Michigan: Its clear, concise, and logical presentation renders it well adapted to high school classes. It is alto- gether the best text-book for secondary schools that I have seen. The Critic, June, rSgs : On the whole the book is the best elementary Algebra, written by an American, that has come to our notice. MA THEM A TICS. 57 A College Algebra. By Professor J. M. TAYLOR, Colgate University, Hamilton, N.Y. i6mo, cloth, 326 pages. Price, ^1.50. A VIGOROUS and scientific method characterizes this book. In it equations and systems of equations are treated as such, and not as equalities simply. A strong feature is the clearness and conciseness in the state- ment and proof of general principles, which are always followed by illustrative examples. Only a few examples are contained in the First Part, which is designed for reference or review. The Second Part contains numerous and well selected examples. Differentiation, and the subjects usually treated in university algebras, are brought within such limits that they can be success- fully pursued in the time allowed in classical courses. Each chapter is as nearly as possible complete in itself, so that the order of their succession can be varied at the discre'.ion of the teachers. Professor W. P. Durfee, Hobart College, Geneva, N, Y. : It seems to me a logical and modern treatment of the subject. I have no hesitation in pro- nouncing it, in my judgment, the best text-book on algebra published in this country. Professor George C. Edwards, University of California: It certainly is a most excellent book, and is to be commended for its consistent conciseness and clearness, together with the excellent quality of the mechanical work and material used. Professor Thomas E. Boj'ce, Middlehury College, Vt. : I have examined with considerable care and interest Taylor's College Algebra, and can say that I am much pleased with it. I like the author's concise presentation of the subject, and the compact form of the work. Professor H. M. Perkins, Ohio Wesleyan University: I think it is an excellent work, both as to the selection of subjects, and the clear and concise method of treatment. S. J. Brown, Formerly of University of Wisconsin : I am free to say that it is an ideal work for elementary college classes. I like particularly the introduction into pure algebra, elementary problems in Calculus, and ana- lytical growth. Of course, no book can replace the clear-sighted teacher ; for him, however, it is full of suggestion. 46 SCIENCE. Primary Batteries. By Professor HENRY S. Carhart, University of Michigan. Sixty- seven Illustrations. i2mo, cloth, 202 pages. Price, ^1.50. THIS is the only book on this subject in English, except a translation. It is a thoroughly scientific and systematic account of the construction, operation, and theory of all the best batteries. An entire chapter is devoted to a description of stand- ards of electromotive force for electrical measurements. An ac- count of battery tests, with results expressed graphically, occupies forty pages of this book. The battery as a device for the transformation of energy is kept constantly in view from first to last ; and the final chapter on Thermal Relations concludes with the method of calculating electromotive force from thermal data. Professor John Trowbridge, Harvard University : I have found it of the greatest use, and it seems to me to supply a much needed want in the literature of the subject. Professor Eli W. Blake, Brown University : The book is very opportune, as putting on record, in clear and concise form, what is well worth know- ing, but not always easily gotten. Professor George F. Barker, University of Pennsylvania : I have read it with a great deal of interest, and congratulate you upon the admirable way in which you have put the facts concerning this subject. The latter portion of the book will be especially valuable for students, and I shall t)e glad to avail myself of it for that purpose. Professor John E, Davies, University of Wisconsin : I am so much pleased with it that I have asked all the electrical students to provide themselves with a copy of it. . . . I have assured them that if it is small in size, it is, nevertheless, very solid, and they will do well to study and work over it very carefully. . . . I find it invaluable. Albert L. Amer, formerly of Iowa State University : I am using your work on Primary Batteries, and find it the best guide to practical results in my laboratory work of anything that I have yet secured. It is a book we long have needed, and it is one of that kind which is not exhausted at a first reading. Professor Alex. 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