THOMSON'S NEW >;>*ATHFMATI'"\\ T . SFJME !. THE COLLEGIATE ALGEBRA ADAPTED TO COLLEGES AND UNIVERSITIES. BY JAMES B. THOMSON, LL. D., AUTHOR OF NEW MATHEMATICAL SERIES, AND ELIHIT T. QUIMBY, A.M., LATE PROFESSOR OF MATHEMATICS IN DARTMOUTH COLLEGE. OF NEW YORK: Clark & Matnaed, Publishers 5 Barclay Street. chicago: 46 madison street. 1SS0. T TinyMSOF^'llTilEiiTICAL SEKIES. I. A Graded Series of Arithmetics, in three Boohs, viz. .- New Illustrated Table Book, or Juvenile Arithmetic. With oral and slate exercises. (For beginners.) 128 pp. New Rudiments of Arithmetic. Combining Mental with Written Arithmetic. (For Intermediate Classes.) 224 pp. New Practical Arithmetic. Adapted to a complete business education. (For Grammar Departments.) 384 pp. II. Independent Boohs. Key to New Practical Arithmetic. Containing many valuable sug- gestions. (For teachers only.) 1G8 pp. New Mental Arithmetic. Containing the Simple and Compound Tables. (For Primary Schools.) 144 pp. Complete Intellectual Arithmetic. Specially adapted to Classes in Grammar Schools and Academies. 168 pp. III. Supplementary Course. New Practical Algebra. Adapted to Higb Schools and Academies. 312 pp. Key to New Practical Algebra. With full solutions. (For teachers only.) 224 pp. New Collegiate Algebra. Adapted to Colleges and Universities. By Thomson & Quimby. 346 pp. Complete Higher Arithmetic. (In preparation.) *** Each book of the Series is compleU in itself. Copyright, 1879, 1880, by J. B Thomson and E. T. Qrmr.v. Electrotyped by Smith & McDougal, S2 Beekman Street, New York. P REFA CE. SOON after the publication of the " New Practical Alg bra/' the author was urgently requested by sever; mathematical professors to prepare a higher work on the sam general plan, adapted to the wants of Colleges and Univers ties. In compliance with these requests, the present treat i; was undertaken and is now presented to the public. To facilitate its preparation he was fortunate in securing the co-operation of Prof. E. T. Quimby, of Dartmouth College, a gentleman of more than twenty-five years experience in teaching mathematics. The work presents a full discussion of all the subjects usually contained in the most complete text-books in use, as the Demonstration of the Binomial Formula, the Computation of Logarithms, Theory of Equations, Sturm's Theorem, Inde- terminate Coefficients, Series, Infinitesimal Analysis, Horner's Method of Approximation, Loci of Equations, Exponential Equations, etc. A few subjects, as Probabilities, etc., thought to be of less importance have been thrown into an Appendix. While the most approved authors have been freely consulted and their methods carefully compared, the plan and execution of the work are the results of long personal experience in the class-room. The arrangement is systematic, each subject appearing in its natural order, and the dependence of the principles upon each other is shown by frequent references. i v P B B 1-" ACE. The Examples are numerous, and have been prepared with a view both to illustrate the principles under discussion, and to stimulate thoughl on the part of the student. Great pain- have been taken to make the rules and defini- tions clear and concise, and the demonstrations simple, rigorous, and logical. The subject has been brought down to the present time, and the best of the improved methods of teaching the various topics have been adopted, to the exclusion of such as are obsolete. Originality of mailer is not to be expected in a book of this kind : but it will be found that several subjects have been treated in a manner more or less original. The reader is referred t<> the Articles on the use of the Directive Signs, or Factors of Direction, the treatment of Imaginary Quantities, Logarithms, Scries, etc. The work is designed to meet a want which has been felt to a greater or less extent, but which heretofore has not been supplied in a satisfactory manner. It is earnestly commended to the attention of instructors ami students, with the hope that it will be found to contain enough that is new and useful to satisfy, in a measure, the views of those who believe in progress, and who have desired some departures from the beaten track. In conclusion, the authors would avail themselves of the opportunity to express their obligations to their friends who have favored them with many valuable suggestions upon the subji Brooklyn, X. Y., ./ y, 1S79. CONTENTS. PAGE Introduction, 9 Definitions, 9-15 Functions of Quantities, 10 Axioms, . 15 Notation, . 16 Symbols of Quantity, 16 Symbols of Operation, 17 The Distinction between Factors and Terms, . . 18 Symbols of Eelation, 20 Symbols of Abbreviation, ...... 20 Algebraic Expressions, 21 Positive and Negative Quantities, . . . . 22 Directive Signs, 23 The Directive Force of the Signs + and — , . . 25 Rule for the Directive Signs, 26 Exercises in Notation, 27 Algebraic Addition, 29 General Eule, 30 Adding Similar Polynomials, 32 Algebraic Subtraction, 32 General Rule, t>2> Use of Parentheses, 34 Multiplication, 36 Multiplication of Monomials, 37 Signs in Multiplication, . ... 38 Multiplication of Polynomials, 39 Multiplication- by Detached Coefficients, . . . 40 Division, ......... 4 1 Dividing a Monomial by a Monomial, . . . . 42 vi CONTENTS. PACK Analogy between Coefficient and Exponent, . . 43 Signa in Division, 45 Dividing a Polynomial by a Monomial, ... 46 Dividing a Polynomial by a Polynomial, . . - 47 Dividing by Detached Coefficients, .... 50 S3 nthetic Division. . 51 Factoring, - 53 Theorems for Factoring Binomials, . . . .53 Finding the Binomial or Polynomial Factors of a Polynomial,. 55 Greatest Common Divisor, 58 Least Common Multiple, 6r Fractions, 63 Signs of Fractions, 64 Reduction of Fractions, 65 Addition and Subtraction of Fractions, ... 69 Multiplication of Fractions, 70 Division of Fractions, 72 General Rule, 74 A Finite Quantity divided by Zero, as a -4- o, . . 75 An Infinitesimal divided by a Finite Quantity, as o-j-a, 75 Equations of the First Degree, ... 77 Conditional Equations, . . 78 Reduction *4' Equations, 7S Methods of Reducing Equations of the First Degree, . 79 Proof of Reduction of Equations, .... 81 Solution of Problems, 83 Discussion of Problems, 85 Problem of the Couriers, 86 Sim ultaneous Equations of the Fii'st Degree, 91 Elimination, 92 Three Unknown Quantities, 98 Powers and Hoots, 101 Radical and Rational Quantities defined, . . . 102 \ Surd or Irrational Quantity 102 Powers of Monomials. 103 Towers of Binomials, Ic >4 CONTENTS. vii PAGE Binomial Formula, 104 Involution of Polynomials, . . . . - io 5 Multinomial Theorem, 105 Evolution of Polynomials, 107 Signs of Boots, 109 Imaginary Quantities, 113 Calculus of Radicals, 120 Multiplication of Radicals, 121 Division of Radicals, . . . . . . .122 Former Notation of Imaginary Quantities, . . . 1 24 Reduction of Radicals, 126 Radical Equations, 131 Equations of the Second Degree, . . . 132 A Complete Equation, 133 Completing the Square, 134 Higher Equations solved by Quadratics, . . .140 Problem of the Lights, . . . . . .143 Simultaneous Equations of the Second Degree, . - '45 Inequations, . . . . . .151 Reduction of Inequations, 152 Ratio and Proportion, 154 Theorems, 155-160 Permutations and Combinations, . 163 Formulas, 164 Infinitesimal Analysis, 166 Notation, . .168 Differential Coefficient, .169 Differentiation, .... ... 170 Indeterminate Coefficients, . . .176 Development of Functions, 177 Decomposition of Fractions, . 180 Demonstration of the Binomial TJieorem, . 184 Logarithms, . . . . .187 Briggs' System, . . . . . . . . 191 Table of Logarithms, 193 To Multiply and Divide by Logarithms, . . 195, 196 Computation of Logarithms, .... 200 CONTENTS. FACJ Scries, . 207 Interpolation <>! T{' Roots, . 247 Signs of Root-. .250 Limits of Roots, 252 Limiting Equation, . . . . . . 254 Equal Roots, 255 Commensurable Roots 258 Incommensurable Roots, 263 Sturm's Theorem, 264 Horner's Method of Approximation 267 Recurring Equations, 27.) Binomial Equations 277 Exponential Equations, 279 Appendix, 280 Probabilities 280 ( !ardan's Formula, 282 I I jcartes' Formula, 286 ContiDUi '1 Fractions . 288 M cellaneous Problems, 298 Formulas, 303 No 305 ALGEBRA. INTKODUCTIOlSr. Art. 1. Mathematics is the science of quantity. 2. Quantity is anything which can be measured; as, distance, space, time, etc. 3. The Measure of a Quantity is the number of times it contains another quantity of the same kind, called the unit of measure ; or, it is the ratio of the quantity to the unit of measure. Hence, 4. Number is the measure of quantity. 5. A quantity is measured mechanically by applying the unit of measure directly to the quantity, and counting the number of times it is applied. Thus, the application of the yard-stick to measure the length of a piece of cloth, and of the surveyor's chain to measure distances, are examples of the mechanical measurement of quantities. 6. When the unit is not contained in the quantity to be measured an integral number of times, this unit may be divided into any number of equal parts, and one of these parts taken as a unit. In this way, a fraction or a mixed number may express the measure of a quantity. Thus, we may have a piece of cloth 5^ or 5 J yards long 1 . The exact measure of a quantity cannot be found with a unit which cannot be divided into such a number of equal parts that one of these parts shall be contained an integral number of times in the quantity to be measured. 7. Commensurable Quantities are those that cnn be measured mth the same unit. 1° .*. : : .■ ••• .....XN^b.od-ijction. 8. Incommensurable' Quantities are those that cannot be measured with the same unit. Thus, the sick of a square and its diagonal are distances that mnnotf fa /„,,/.*///•,l" the other. 9. A Single Quantity is called commensurable or incommensurable according as it can or cannot he measured with the unit we are using. 10. Quantities which are of different hinds or natures, as time and distance, cannot be compared one with the other; for, the magnitude or extent of one is wholly unlike that of the other, so thai one cannot he made the unit of measure for the other. 11. Any quantity may he made the unit of meas- ure for quantities of its own hind, but for the purposes of truth and other mutual uses, the unit must he generally known ami accepted ; hence the necessity of units established by law as a national standard. Note. — In common language, we speak of measuring apile of wood, or a piece of lifnd, but, strictly speaking, that which we reuUy meas- ure is the space occupied by the wood, and the area of the land. So also we measurt the It ngth, the /reight, the density, the elasticity, etc., of mati rial bodies, but not the bodies themselves. 12. Quantities as they appear in nature have certain defi- nite nla! ions to each other, which make them mutually dependent, and from which the measure of one may be found when the measures of others arc known. Mathematics investigates these relations and determines the measures of quantities indirectly, or without direct measurement. 13. Quantities thus mutually dependent arc called Functions of each other. For example, 14. In all motion, three (piantities are involved, viz.: Time. Distance, and Velocity, These (piantities are -<> related that neither can change without changing one or both of the others. If the velocity increase, the distance will increase or the time decrease, or both these results may follow. INTRODUCTION. 11 Notes. — i. In common language, we say the time depends on the distance and velocity ; the distance on the time and velocity; and the velocity on the time and distance. 2. In mathematical language, the time is a function of the distance. and velocity ; the distance is a function of the time and velocity ; and the velocity is & function, of the ft'm« and distance. 15. The question, ''What function is one quantity oi others ? " refers to the manner in which the latter quantities must be combined or treated to give the measure of the former; as, "What function is the distance of the time and velocity?" The answer to which would be, " TJte product"; that is, the distance is the product of the time by the velocity. 16. In like manner, let the pupil answer the following questions : r. What function of the side of a square is its area ? 2. What function of the radius of a circle is its diameter? 3. What function of the diameter of a circle is its circum- ference ? Its area ? 4. What function of the principal, rate per cent, and time, is the interest on a note ? The amount ? 5. What function of the itumber of pounds and the price per pound is the cost of an article ? 6. What function of its sides is the area of a rectangle ? 17. A Pvojwsit ion is something proposed for demon- stration, or solution. 18. A TJieovem is a proposition /or demonstration. 19. A Problem is a proposition for solution. Note. — A theorem affirms, " This is true," and requires demon- stration. A problem inquires, " What is true ? " and requires solution. 20. An A.riom is a self-evident theorem. 21. A Postulate is a self-evident problem.. Note. — A truth is called self-evident when it commands the instant assent of one who is acquainted with the subject to which it relates, and cannot be made plainer by any proof, 12 INTRODUCTION. 22. A Demonstration is an arrangement of defi nitions, ((.minis, and postulates, by which the truth of a theorem is established. Note.— A direct demonstration proves that a theorem is t vue, by assuming the truth of certain definitions and axioms, and from these premises deducing other truths, till we arrive at the one which is to be established. An indirect demonstration proves that a theorem is not untrue, by proving that the supposition of its contrary involves an absurdity. 23. A Solution is an arrangement of axioms and postu- lates by which the answer to & problem is determined. 24. The Hypothesis of a proposition is: ist. Iii a theorem; — The conditions on which the theorem is affirmed. 2d. In a problem ; — TJie data from which the required truth is to be determined. 25. A Corollary is an inference from a preceding demonstration or solution. 26. An Equation is aw expression of equality between two quantities. The equation is used to express in algebraic language the relations between quantities which are functions of each other. 27. Known Quantities are those from which other quantities are to be determined. Arbitrary values may there- to it be assigned to them at pleasure. 28. Unknown Quantities are those whose values are to be determined from their relations to other quantities. They are regarded d& functions oi known quantities and cannot therefore have arbitrary values assigned to them. 29. Problems are of two kinds: [st Those which require some geometrical or mechanical Iruction : as, To construct a triangle from three given INTRODUCTION. 13 2d. Those which require the measure of a quantity from its relations to other quantities; as, To hud the base of a right-angled triangle from the other sides. 30. The Solution of a problem of the second kind is made up of three distinct parts or steps: 1st. Finding the equations which express the relations between the quantities involved. 2d. Finding from these equations ivliat function the unknown quantity is of the known quantities. 3d. Substituting in this function the numbers representing the known quantities. 31. The first of these steps requires a knowledge of that branch of mathematics or physics to which the problem belongs; as, Geometry, Mechanics, etc. The second is the province of Algebra. The third belongs exclusively to Arithmetic. 32. For illustration take the following problem : A rope 50 feet long attached to the top of a vertical pole, reaches the ground 40 feet from the foot of the pole, on a horizontal plane; how high is the pole? First step : By geometry we learn that the square of the length of the rope equals the sum of the squares of the length of the pole and the distance on the ground. This rela- tion expressed by an equation is a? = b°~ + z 2 , in which the letters a, b, and x have been put for the length of the rope, distance on the ground, and height of the pole, respectively. Second step .' By algebra this equation is reduced to the form x = 's/a 2 — b~, which shows how the known quantities must be combined to produce the unknown; in other words, what function the unknown is, of the knoini. 1-4 I N I BO I) UC Tlo.N. Third Step : By arithmetic the numbers 50 and 40 are substituted for a ami b : thus, x = Vs oi — 4° 2 — 3°> Ans. 33. The equation a' 2 = b 2 + x 2 cannot be true unless the quantities a, b, and x are mutually dependent; thai is, unless eac// is a function of the o^fer £k>o. The equation therefore Implies that a; is a function of a and £, but does not state explicitly what function. In such an equation a; is said to be an Implicit Function of a and J. The equation , 1 x = Va> - V 2 states explicitly what function x is of a and b, and it is therefore called an Explicit Function. 34. This change from an implicit to an explicit function is called " reducing the equation." The explicit function resulting from the reduction of the equation is called a Formula, and is the expression in algebraic language of an arithmetical rule. The above formula for finding the perpendicular of a right-angled triangle when the hypothenuse and base are given, being translated into common language, becomes the I'm 1 In wing. I!i 1.1;. — Subtract the square of the base from the square of the hypothenuse, and take the square root of the difference. From the preceding illustrations Ave have the following ilelinii ions : 35. Algebra is the science of the Equation. (Art. 34.) [ts objeel is the reduction of equations, by which formulas for arithmetical computations are obtained. 36. Arithmetic is the Science of Numbers. Its objeel is the substitution of numbers in algebraic formulas; or, what is equivalenl to this, the combination of numbers in accordance with rules furnished by Algebra, INTRODUCTION. 15 37. The Reduction of an Equation consists in such transformations as will make the unknown quantity an explicit function of the known quantities. 38. The Reduction of Equations is based on the following AXIOMS. i°. Equal quantities equally affected remain equal. 2°. Equal quantities unequally affected become unequal. 3°. Unequal quantities equally affected remain unequal. 4°. Quantities equal to the same quantity are equal to each other. 5°. Quantities differing equally, in both magnitude and direction, from the same quantity are equal to each other. 6°. TJie whole is greater than its part, and is equal to the sum of all its parts. CHAPTER I . NOTATION. 39. Notation in Algebra is the method of expressing quantities, their relations, and combinations, by general sym- bols. The algebraic symbols differ from the Arabic figures, or numerical measures, in this respect; the latter represent specific quantities, the former general quantities. Note. — This notation constitutes what is called the algebraic language. It is necessarily general, since its object is to furnish general formulas for arlthm, Ural compvtations. 40. The Symbols used may be classified as follows: ist, Symbols of Quantity ; 2d, of Operation; 3d, of Relation; 4th, of Abbreviation. SYMBOLS OF QUANTITY. 41. Quantities ore commonly represented by the letters of the alphabet Any letter may be used to repre- -. ni any quantity, and the same letter may represent different quantities, subject to one limitation; the same letter must always stand for the same quantity throughout the same discussion. 42. For uniformity and convenience, the following order should be observed : 1st. It is customary to employ the "first loiters of the alpha- be! to represenl known quantities, as, a, b, c, etc.; and the last for unknown quantities, as, x, y, :. etc. 2d. Initial letters are frequently used: as, r or 7? for nut ins ; c for circumference j s for sum, etc. NOT ATION. 17 3d. Different Quantities of the same kind may be represented, in the same problem, by the same letter, with accents or subscript figures to distinguish the different quantities. Thus, when a problem involves the radii of several circles, we may use ?■', r", /"'", etc. (read, "r prime," "r second," "r third," etc.), or r lf r.,, r & , etc. (read, " /• sab one," " ;• sub two," etc.). 4th. The Greek letters are commonly used to represent angles, but sometimes other quantities. Thus, the Greek .i is used for the ratio of the circumference of a circle to its diameter. 5 th. The symbol a> (the figure 8 placed horizontally) represents infinity, or a quantity greater than any assign- able quantity. 6th. The symbol o (zero) represents an infinitesimal quan- tity, or a quantity less than any assignable quantity. 43. Quantities when expressed by numbers are called numerical; when expressed by letters, they are called literal quantities, SYMBOLS OF OPERATION. 44. The Fundamental Operations in Algebra are Addition, Subtraction, Multiplication, Di- vision, Involution, and Evolution. 45. Addition and Subtraction are expressed as in Arithmetic by the signs + and — ; as, a + b — c, read, " a plus b minus c." Quantities connected by the signs + and — are called Terms. 46. When the same term is to be added or subtracted more than once, a number is placed before it to show how many times it is to be used. Thus, a + b + b— c— c— c may be written a 4- 26— 3c. A number thus used to show how many times a quantity is taken as a term is called a Coefficient. IS N O T A T I X . 47. A Coefficient may be integral or fractional, the latter showing what />"/•/ of a quantity is taken as a term; us 3a I «/'. It '" :IV a ' 6u ' j e numerical, or literal, or both. Thus, in the expression 211b, 2a may be regarded as indicating how ma \y times b is taken : or 2 how many times ab is taken, or b how many times 2a is taken. Note. — The word coefficient, however, usually refers to the numeri- cal factor of a term. Hence, generally, 48. A Coefficient is the factor or- factors of a term indi- cating the number of times the rest of the term is taken, or, what equal terms are taken. When no coefficient is expressed, 1 is always understood. 49. The double sign ± is used when a quantity may be either added or subtracted, and is read, "plus or minus." Thns, a ± b means that the conditions of the problem will be satisfied either by adding or subtracting b. 50. A Posit ire Quantity is one whose sign is +. 51. A Negative Quantity is one whose sign is — . Note. — The signs + and — in Algebra have a more general mean- ing than merely addition and subtraction, which will be explained in the proper place (Arts. 81-94). MULTIPLICATION AND DIVISION. 52. Quantities used as multipliers or divisors are called Wactors, in distinction from Terms, which are added or subtracted. 53. Multiplication is expressed: 1st. By the usual sign, x ; as, 2 xaxbxc. 2d. By the period ; as, 2-4-6 = 2x4x6. 3d. By writing the factors one after the other without any sign : as, 2a.be = 2 x a x b x c. \.'ii.— It is customary to write numerical factors first, and literal (acton after, in alphabetical order, as, ytbex. 54. Division \- expressed : ist. By the usual sign, -:- : as. a -t- b. NOTATION. 19 2d. By writing the divisor under the dividend in the form of a fraction ; as, -r = a-i-b. 3d. By a colon : ; as, a : b = a -j- b. 4th. By a negative exponent; as, ab~ x = a-i-b, as seen below. 55. When the same factor is used more than once, either as a multiplier or divisor, a figure called an Exponent is placed a little above and to the right of it, to show how many times it is used. aaci cfi Thus, instead of aaabb we write « 3 6 2 , and for --- we write r- , or bb ¥ a 3 6- 2 (Art. 54, 4th). INVOLUTION AND EVOLUTION. 56. Involution is the multiplication of equal factors. 57. Evolution is the process by which a cpiantity is separated into equal factors. 58. A Power is the product of any number of the ecpial factors of a quantity, and is expressed by an exponent ; as, a% a 3 , a*, etc., read, " a second power " or " a square," " a third power" or "a cube," "a two-thirds power," etc. Note. — In reading poioers, do not omit tlie word "power," reading " a fourth," " a third," etc., for that means a"" , a'" , etc. 59. A Root is one of the equal factors of a quantity. It is therefore a power whose exponent is a fraction with 1 for a numerator ; as, a?, a?, which may be read, " the square root of «," " the cube root of a," or " a one-half power," "a one-third power." 60. The Denominator of a fractional exponent shows the number of equal factors into which the quantity is sepa- rated. 61. The Numerator shows lioiv many of those factors are taken. Hence, 62. An Exponent shows what equal factors are taken. 20 NOTATION. 63. The Radical Sign, V , is often used to express . a figure being written over it to indicate what root is taken. Thus, y/a = a*; tya = a* ; y'a'-' = a*. When no figure is written over the sign, 2 is understood ; as, -y/a = y/a = a?. Notk. — The figure placed over the radical sign is called the Index of the root, because it denotes the name of the root. SYMBOLS OF RELATION. 64. The Sign of Equality is = ; as, a = t + c, read, "a equals b + c." 65. Inequality is expressed by two lines forming an acute angle, and opening towards the greater quantity; as, a > b, or a < b, read, "a is greater than b" or "a is less than b." 66. The Sign of Variation is oc . It shows that the quantities between which it is placed have a constant rat in. Thus, x x //, read, " x varies as y," means that however x may change its value, y also changes, so that the quotient x-i-y does not cliange. SYMBOLS OF ABBREVIATION. 67. The symbol .*. is used for the word therefore, and V for becan 68. The Vinculum, horizontal, ~ , or vertical, the Parenthesis, ( ), Brackets, [ ], and Braces, { are used to connect several quantities with the same coefficient, exponent, or sign. 'I'h us, \[(a b — c)° — x + y + z~\ — (x — a)\ -, indicates ist. Thai the quantity a + b— c is to be squared. 2d. That the quantity x \-y+s is to be squared. 3d. That the second of these squares is to be subtracted from the first, and the difference raised to the third power. .jtli. Thai the quantity (a*— a) is to be subtracted from this cube, and (he square rool of the difference taken. DOTATION". 21 It is sometimes convenient to use the vinculum vertically ; as, ax which is the same as (a + b— c)x. + ° — c 69. In writing a series of terms, or factors, the expression is often abbreviated by omitting a part of the quantities, where they can be easily supplied, and indicating the omission by a succession of dots or short dashes. Thus, i + 2 4- 3 .... 8 ; meaning the sum of the numbers i, 2, 3, to 8, inclusive. The product of the numbers 1, 2, 3, etc., to any given number may- be expressed in this manner (1.2-3.... 10), but it is usually still further abbreviated by writing the last factor ; thus, |io —i -2- 3 .... 10. This is read, "factorial 10," meaning the product of the natural numbers from 1 to 10 inclusive. So \n (read " factorial n") = 1 . 2 • 3 . . . . n. ALGEBRAIC EXPRESSIONS. 70. An Algebraic Expression is any quantity ex- pressed in algebraic language; as, 3a, 5a — 7 b, etc. 71. Tbe Terms of an algebraic expression are the quan- tities which are connected by the signs + and — . Thus, in a + b there are two terms; in x + y xs — a there are three, y x 2 being a single term. For, quantities connected by the signs x or -5- do not constitute separate terms. 72. A Monomial is an algebraic expression containing only one term ; as, a, tab, etc. 73. A Binomial has two terms ; as, a + I. 74. A Trinomial has three terms ; as, a + b + c. 75. A Polynomial has tkvee or more terms ; as, ax + by — z — x. Note. — A binomial is -wwaetrfnes- carted a polynomial. 22 NOTATION. 76. The "Degree of a term is the number of its literal factors. As the number of these factors is indicated by the exponents, the degree of a term will be the sum of the exponents of Us literal factors. Thus, 2til>, 211-, and 3«.r are of the second degree, and a-b, ah 1 *, and ¥ are of the third degree. 77. A Homogeneous Polynomial has all its terms of the same degree. Thus, ab- + rt 2 6 + b 3 is homogeneous, but a 2 &- — 2ab + b- is not homo- geneous. 78. Like or Similar Terms are those containing the same powers of the same letters ; as, d~x and zdrx. 79. Unlike or Dissimilar Terms contain different Utters, or the same letters with different exponents. Thus, 2a-x and 2oa?, 2d- and 2a-x, a' 2 x- and «-.*%, etc., are dissimilar terms. 80. The Reciprocal of a quantity is unity divided by tli at quantity. Thus, a and - are reciprocals of each other. POSITIVE AND NEGATIVE QUANTITIES. 81. The signs + and — are used to indicate addition and subtraction ; but if we inquire why certain quantities in the solution of a problem are added and others subtracted, we shall find that it depends on the direction of the quantities. This will he best illustrated by a few examples. ist. A man walks several distances, some north and some south. II"\\ far north of his starting-point does he stop? To answer this, we add n<>rtlt distances and subtract south, because the former increast , while the latter decrease the result, I low far south of his starting-point does he stop? Here we add south distances and subtract north, for the same reason NOTATION. 23 2d. A man has bills payable and bills receivable. What is their net value to him ? Add bills receivable and subtract bills payable. They have opposite directions; one represents cash coming to him and the other going from him. What is the net amount of debt these bills represent ? Add bills payable and subtract bills receivable, the question having- been reversed. 3d. What is the value of my bank account ? Add deposits, subtract drafts, because the former come to me, the latter go from me ; opposite directions as before. We might also give illustrations involving time past and future, suggestive of direction backwards and forwards, as well as other varieties of oppositeness, not in the nature of the quantity, but in its relation to the problem; all of which, without severe stretch of the imagination, may be called opposite directions. 82. These opposite directions determine whether a quan- tity shall be added or subtracted ; that is, they determine the direction in which the quantity is to be used ; for addition and subtraction, by which a quantity is put in or taken out as a term, may be considered as opposite directions. 83. It will be observed that a quantity having a particular direction is not always to be added, nor is one having the oppo- site direction always to be subtracted ; but when the conditions require one to be added, the other must be subtracted. 84. These opposite directions are called Positive and Negative f that direction which tends to increase the quan- tity sought being usually called positive, and the opposite direction negative, though either direction may be assumed as positive at pleasure. 85. Since the signs -f- and — represent the direction of a quantity, they may be called Directive Siyiis, or Fac- tors of Direction. 24 NOTATION. 86. In the use of these signs it is necessary to consider the following PRINCIPLES. i°. A quantity may be considered without reference to its direction. It has then no sign, and is neither positive nor negative. Thus, the answers to the questions, How far? How long? How many ? etc , are of tins nature. Bu1 when a question includes the idea of direction, as, How far north? How Ion.;- before? How long after? How much did you receiti .' etc., the answer must be either + or — . 2°. Questions involving the idea of direction may always be reversed. Thus, How in north? How far south ? How much did you receive? How much did yon give? etc. 3°. The nature of problems is such, that tie quantities involved must //(/re one of two opposite directions. When a problem asks, "How far north or south?" other directions have nothing to do with the answer. So of time past or future. Mils payable and bills receivable, etc... the only possible direct inns are two, and these are opposite. NOTE. — Nothing but some impossible condition in a problem will introduce into the solution a quantity out of the line of positivt and nega- tie , and wben by reason of an impossible condition such a quantity is introduced, it is called an impossibl of lino, jinn ry quantity, and its direction is expressed by a method to be explained hereafter (Art. 292.) 4°. There is no direction which is naturally positive, but it is cum 1 unary to consider the direct inn named in a problem as positive, unless the opposite be made so by special assumption. 87. In operations upon quantities having directive signs, it is no! necessary to consider the nature o£ the quantities, nor bhe hind of oppositeness which gave rise to these signs, whether of distance or time, or of debt and credit ; for when once the equations arc formed, the signs are used in accordance with the arbitrary meaning assigned to them ; chat is. in accordance with the following definitions of the directive signs. NOTATION. 25 DIRECTIVE SIGNS. 88. The sign -\- indicates the positive direction, but has no power to control the direction of a quantity in the presence of the sign — . 89. The sign — reverses the direction of a quantity. 90. In the application of these definitions we observe : ist. The sign + having no power as a factor of direction, may be omitted, except when its omission would lead to some misunderstanding; jnst as the factor i is omitted because it has no power as a factor of magnitude or value. 2d. Two or more + signs mean no more than one, as two or more i's as factors have no more effect than one such factor. 3d. When a quantity has several signs, some of which are + and some — , the direction of the quantity will depend wholly on the negative signs. 4th. When a quantity has several minus signs, each of these signs will inverse its direction. For illustration : Let the minute-hand of a clock, when pointing to the hour XII, represent the positive direction of a quantity, or its direction with no written sign. A single minus sign will reverse that direction, turning the hand backwards* till-it jwints to VI. Another minus sign will continue this revolution, and bring the hand back to XII, or the positive direction. Thus we see that — a is negative, a is positive, and a is negative, etc. These may therefore be written — a, + a, — a, etc., or if we wish to show how many minus signs are used in giving the direction, we may use an exponent for this factor of direction, as well as for factors of magnitude, and write, — a, — 2 a, — 3 a, — i a, etc. * We say backward* (meaning opposite to the natural motion of the hands of the clock), because it is customary to consider revolution in this direction as pod/ire, and in the opposite direction negative. Taking away a — sign from a quantity would reverse it in the negative direction. There is nothing, however, to forbid reversing this supposition and making positive revolution agree with the natural motion of the hands. 2 26 NOTATION. 91. From these illustrations we have the following Rule.— An even power of — is positive, an odd power negative. Or, An even number of minus signs gives + , and an odd a a in her — . 92. Apply this rule to the following EXAM PLES. i. What is the sign of -\ \- a? 2. What is the sign of — 2 1 2 a? 3. What is the sign of a and what of b in the following expression : \- (a — b) ? 4. What is the sign of — ± + a? Ans. z f. 5. What is the sign of ±a? Ans. + . 6. What is the sign of ± — =F « ? Ans. -f- . Note. — When there are several double signs, the upper signs are generally taken together, and also the lower. Thus, in the last exam- ple, the sign of a will be either + or + , each of which will give +. 93. What signs should be given to the following answers ? 7. How far north did you go ? Ans. 10 miles. 8. How much did you pay? Ans. 5 dollars. 9. How much older are you than John ? Ans. 5 years younger. 10. How much older than Henry? Atis. 3 years. 11. How many books have you ? Ans. 10. 12. How many hooks did you take from the table? Ans. 5. 13. How old are you ? Ans. 50 years. 14. How far is it to New York? A?is. 100 miles. Some of the above answers have no sign; which are they, ami why ? NOTATION. 27 94. The force of a directive sign is limited to the term immediately following, unless several terms are connected by a parenthesis or vinculum. Thus, in —a?b + ab' 2 , the sign — affects only a%; but if we write — (a 2 6 + ah 2 ), the whole quantity is negative. 95. In like manner, the sign — before a fraction ; as, , affects the whole fraction, and if in the course of a 2 solution the denominator be removed, it must not be written — a — b, but — (a — b), or — a + b. EXERCISES IN NOTATION. 96. To Translate an Algebraic Statement from Common into Algebraic Language. i. The product of the sum and difference of any two quan- tities is equal to the difference of their squares. Solution. — In this translation, it is necessary first to assume letters to represent the quantities. Let a and b be the letters. Then the statement becomes (a + b)(a — b) = a- — b-, Ana. Hence, the Eule. — For the words, substitute the letters and signs which indicate the relations of the quantities and the operations to be performed. Translate the following into algebraic language : 2. The square of the sum of any two quantities is equal to the sum of their squares increased by twice their product. 3. The square of the difference of any two quantities is equal to the sum of their squares decreased by twice their product. 4. The square of the sum of any two quantities added to the square of their difference is equal to twice the sum of their squares. 5. The difference of the squares of two quantities divided by the difference of the quantities equals the sum of the quantities. 28 NOTATION. 6. The difference of the square roots of two quantities divided by the sum of their fourth roots equals the difference of i licit- fourth roots. 7. The square of the difference of two quantities subtracted from the square of their sum is equal to four times their product. 97. To Translate Algebraic into Common Language. 1. Translate (x + y) 2 — (x — y) 2 = Ley into common language. Solution. — The square of the sum of any two quantities, diminished tiy the square of their difference, is equal to 4 times their product Hence, the Rule. — For the letters representing quantities and the sigm indicating the given relations and operations, substitute words. Translate into common language the following 2. (a + b) 3 = a 3 -f- yi 2 b + yib 2 + b 3 . 3 . 1 (« + i) + 1 {a — h) =a . 4- i (a + b) - i (a - b) = b. 98. If a = 3, b = 2, c = |, m = a + b, n = a — b, what are the numerical values of the following expressions: l(a-b)(a + b) 5 ' > negative quantify is the *an@ a* adding an equal positive one. 4°. The sum of the difference and subtrahend is equal to the minuend. SUBTRACTION. 33 113. The only difference between Subtraction and Addition in Algebra is that when we propose to subtract a quantity, we give that quantity another negative sign, and therefore change all its signs. Hence the following GENERAL RULE. 114. Change the signs of quantities to be subtracted, and unite the terms as in addition. (Art. 107.) Notes. — 1. For convenience, similar terms may be written under each other, as in addition. (Art. 109.) 2. When a result is to be found from several quantities by adding some and subtracting others, the operation may be made one by the above rule. 3. In finding the difference between two quantities, it is immaterial which is made the subtrahend. The result will be the same in either case, except its sign. Thus, the difference between 7 and 4 is 7 — 4 = 3, or 4 — 7 = — 3. When, therefore, a problem requires only the difference, without regard to sign, either quantity may be made the subtrahend. EXAMPLES. 1. From a 2 + 2ax + x 2 take a 2 — 2ax + x 2 . 2. From $x 2 — 2^ + 5 take x 2 — 2 + x. 3. From $ab + b 2 take — 2ab + b 2 . 4. From a 2 + 2ax -+- x 2 take ax — a 2 + x 2 . 5. From a (x + y) — b (x + y) take (x + y) — b (x—y). 6. From 3\/# 2 + « 2 take 5V% 2 + « 2 . 7. From a + b — c + d take a — b + c + d. Simplify the following: 8. 3 ^A_ [(« _|_ l& — tf) _ (zx 3 + zbx 2 + a)]. 9. 3 (ax 2 — ab 2 ) — 3a (x 2 — b 2 ). 10. (nix)? + m*z* — 2\ / 'mx. 34 SUBTRACTION. ii. 2db — ib~ + ib (a + b). 12. a + b — c — //> -f » — (a — & + c + m + n). 13. 2«i — /*« — («b + & 2 ). USE OF PARENTHESES. 115. It will be observed in the above examples that the use of Ptirrntlieses is 1 st. To bring several terms under the influence of the sign -. 2d. To connect several terms to the same coefficient. 3d. To subject several factors to the same exponent. 116. These parentheses may be removed in the several cases : 1st. By applying the sign — to each term separately, or, what is the same thing, by changing the sign of each term. 2d. By connecting the coefficient with each term. 3d. By applying the exponent to each factor. 117. The parenthesis or vinculum, when used to save the repetition of some sign, coefficient, or exponent, ca^i generally be removed without changing the value of the expression: By applying the. sign, coefficient, or exponent to each of the quantities separately to which it belongs. 118. In doing this, it must be remembered that coefficients belong to terms, and exponents to factors. We nausl not therefore apply an exponent to the several terms nor a coef- ficient to the several factors of the quantity within the parenthesis. Thus, the expression 2 (fib + b") does not equal 2a- 26 + 26 s (applying the 2 to both factors a and 6), but 211b + 2b' 2 . So (a — /')- is not the same as a a — ft 4 . « Note. — The parenthesis is here used to indicate that a — b is used as a single factor, and it cannot be removed in the manner explained above. SUBTRACTION- 35 119. Remove the signs of abbreviation from the follow- ing, and reduce to the simplest form : i. 2 \a - [b + (c + zx) — (y — 2)]}. 2. 3 [a — (b + c) + 2 (to — i) — (to + i)]. 3. (a + b — c) — (b — a + c). a\x — a I x + 2a\x. — c — c\ + b 5. ^abc' — 2\/abc + {2abc)^. 6. a [a — a (b — c)] — (a 2 + « 2 Z>) + a?c. 7. Vab + d*b$ — yxif — 2 (a% fyb 2 + 2a;?/ 2 — aW). 8. x — s \[d$+ tab — (ab)i] _ \ab + a*j£ - (^p)" 3 -x)]\, 9. (a + 5 — c) V% + # — (a + * + c) (jb + y)i 10. (a + 2«^to + 5) — (a — 2\/ab + b). 11. (a — 2«2#2 4- j) _2 (^ _j_ 2 \/aVb ± #)• 12. ?w (a 4- b) — wi (« — &) + 2m (J — a). 120. In the present chapter we have considered quantities as Terms. It now remains to treat of them as Factors, in connection with the following subjects, viz. : Multiplication, Division, Factoring, Greatest Common Divisors, Least Common Multiples, Fractions, Powers, and Roots. 7v CHAPTER III. MULTIPLICATION. 121. Multiplication is the process by which the sum of any number of equal terms is found without performing the addition. The Multiplicand is the quantity multiplied. The Multiplier is the number by which it is multiplied. The Product is the result of multiplication. The multiplier and multiplicand are called Factors. 122. To illustrate the difference between multiplication and addition, take the following example : What is the sum of four distances, each equal to 5 feet? We may find the answer to this by saying mentally, 5 and 5 are io, and 5 are 15, and 5 are 20; or we may say. 4 times 5 are 20. The men- tal work of the first is that of successive additions ; of the second it consists simply in giving the required sum from memory, knowing the /, mi and number of times it is used. In addition, if one does not remember what numbers and 5 make, he may begin at 5 and count, adding one at a time, and thus find the sum. In multiplication, the result can be given only by memory, which associates certain products with certain factors, and if the memory fail, the answer can be found only by addition. In the example above, when the 20 is considered as made up by the addition of four lives, it is called a sum : but when it is found by con- sidering the numbers 5 and 4, and giving from memory the result without adding, it is called >t /induct. The two operations ;,re expressed thus : 1st. 5 + 5-1-5 + 5 = 20. 2d. 5x4 = 2D. In the first, the 5's are terms and the 20 is their sum. In the second, 5 and 4 are factors and 20 is their product, (Art. 11.) MULTIPLICATION. 37 Note. — It is important that the distinction between terms and fac- tors should be kept in mind : ist. Terms are quantities united by algebraic addition. 2d. Factors are quantities one of which shows how many times the other is used as a term. 123. A product formed by two factors may be multiplied by a third factor, and this product by a fourth, and so on indefinitely. A product may therefore contain any number of factors. Hence the definition : 124. M/ultiplication is the process of combining fac- tors into a product, PRINCIPLES. 125. i°. Tlie multiplier must be considered an abstract quantity. 2°. The product is of the same nature as the multiplicand ; for, repeating a quantity does not alter its nature. 3°. The product of two or more factors is the same in whatever order they are multiplied. 126. Multiplication may be considered under two cases : ist. Multiplication of monomials. 2d. Multiplication of polynomials. CASE I. 127. Multiplication of Monomials. By algebraic notation (Art. 53), we have for the multipli- cation of monomials the following Rule. — To the product of the numerical coefficients annex the literal factors, giving each an exponent equal to the sum of all its exponents in the several factors. Note. — This rule applies to any number of monomial factors. 128 The Signs of the Factors in multiplication are to be treated as factors of direction, and must enter the product precisely as the other factors. (Art. 91.) Hence, 38 MULTIPLICATION. 129. For the Siyus in Multiplication we have the following IiUle. — An even power of — is positive} an odd power negative. Or, An err// number of negative factors gives a positive product, an odd number a negative. 130. When there are only two factors, this gives the common Eule. — Like signs give + ; unlike, — . EXAMPLES. Find the product of the following : i. 2«b 2 .c x 3« 2 £ 3 £ 2 . Ans. 6a s b 5 z?. 2. yfixy x zaxy % x sa 3 x- x \-^yn x .r"// 1 ". 1 c . a; 72,3 w m:! x #" ( w— 2) f/wj (m— l)^ 16. 2rt 2 .' - x — 3SC 5 x — rt.r 2 x a5. 17. rt5 2 x ± rtV x — frr 2 x rt//r. 18. -\/a£ x — (>"Ir- X (— rt.r) 2 x 2rt#. 19. $fyax X — 2\/rt.r x {—a '■•'"-') X rt.'C 2 . 20. (— rt.r 3 ) (— ""'./) (,/V 2 ) y/ax. MULTIPLICATION. 39 CASE II. 131. Multiplication of Polynomials. The Multiplication of Polynomials is performed by the following Rule. — Multiply each term of the multiplicand by each term of the multiplier, and add the products. Notes. — i. This does not differ in principle from the method of multiplying numbers, where each figure is multiplied separately and the products added. The multiplier may be a monomial, 2. For convenience in adding the partial products, like terms should be placed under each other. 3. The multiplication of polynomials may be indicated by inclosing each factor in a parenthesis, and writing one after the other. Thus, {a + b + c) (a + b + c ) is equivalent to (a + b + c) x (a + b + c). 21. Multiply x 2 — 2ax -f- a 2 by a + x. Solution. — We write the multiplier under the multiplicand, and proceed thus : OPERATION. x 2 — iax + a? a + x Multiplying by a, ax' 1 — 2f'-d 6. 4a 3 bc 2 d~ l — 6a*b 2 cd. ga 5 b VV 7 * 3 r H >' 8. i$cPlfar*y -^ sa'WxPy- 1 . DIVISION". 45 145. When the signs of the dividend and divisor are considered, they must be treated as the other factors, in accordance with the principles already established. (Art. 128.) That is, The quotient will have the sign — , with an exponent equal to its exponent in the dividend minus its exponent in the divisor. Take the following example : 9. Given — ^ — 7-fj j^r) ; ; , to find the — 2 («) (— b") (— c*) (/r) quotient. Solution. — Cancelling or removing the factors of this divisor from the dividend, we have 2d 2 b 3 d i . In the same way, we may cancel factors of direction, remembering that the sign + , like the factor i, has no power. We have here five minus signs in the dividend and three in the divisor, or, what is the same thing, — 5 and — 3 , leaving — 2 , or + for the quotient. It will he readily seen that the same result would be given by adding the exponents of — , for whenever the difference of the exponents is odd, the sum will be odd, and when the difference is even, the sum will be even, hence we may use the same rule for the sign of the quo- tient in division as for the product in multiplication. (Art. 129.) Hence, the Eule. — I. An even number of negative signs gives a posi- tive quotient, an odd number a negative. II. If there be but two signs : Like signs give + ; unlike signs, — . Note. — If it be asked why these signs are thus treated in multipli- cation and division, the answer is, that such use of them meets the wants of mathematical analysis, as will be abundantly illustrated in the solution of problems. Divide the following: 10. — 8a m b n -=- 2db. 11. 6« 5 Z> 4 -\ 3a b 2 . 12. — $a m+n b m ~ n -. a n b m . 46 DIVISION". 13. _8o l b 2 -T- 2«- 2 ^- 4 . 14. 2^- 2 i- 4 -. Sa^b~ 2 . 1,1 1,1 15. y^b* -. tt*o*. 16 . -a?{-ab) + $(<£$). 17. — " by ab. Divide a T + b' bv a /> . division. 47 CASE III. 147. To Divide a Polynomial by a Polynomial. i. Divide sub 2 -f yi 2 b + b' A + a 5 by a + b. Analysis. — ist. Since the dividend is the product of the divisor and quotient, that term of the dividend which has the highest power of a must have been produced by the multiplication of those terms in the divisor and quotient which contain the highest powers of a. If, there- fore, we divide that term, of the dividend containing the highest power of a by the corresponding- term of the divisor, we shall find one term of the quotient. 2d. Multiplying the divisor by this term of the quotient and sub- tracting the product from the dividend, we shall have a remainder to be divided as before. 3d. Whenever this remainder becomes zero, the division will be complete. 4th. The division may be stopped at any time and the quotient be completed by adding the remainder over the divisor in the form of a fraction to indicate the uncompleted division. It will be more convenient in dividing to arrange the dividend and divisor in the order of the ascending or descending powers of the same letter. OPERATION. Dividend, a 3 + ^aPb + ytb- + b 3 | a + b Divisor. ist product, a 3 + (Pb a 2 + 2ah _,. 52 Quotient, 2a 2 6 + 306 s + b 3 ist remainder. 2d product, 2a-b + 2«6 2 + ab' 2 + b 3 2d remainder. 3d product, + ob 1 * -f- b 3 148. \Ye have from the above illustration the following Rule. — I. Arrange the terms of both dividend and divisor in accordance ivith the ascending or descending powers of the same letter. II. Divide the first term of the dividend by the first term of the divisor, and write the result for the first term of the quotient. III. Multiply the divisor by this term of the quotient, and subtract the product from the dividend. 48 DIVISION". IV. Divide this remainder as before, and so on till the division is complete or a remainder is found which has no term di risible by lite first term of the divisor. V. Write t lie final remainder, if any, over the divisor in the form of a fraction and add it to the quotient. EXAMPLES. 2. Divide a 5 —.satb + loaW — io« 2 £ 3 + ydS — b 5 by a? — 2ab + b\ OPERATION. a 5 — 5^ 4 5 + io« 3 & 5 — ioa 2 6 3 +5 4 — £> 5 a- — inb + b- Divisor. a 5 -2afib+ aW a 3 -3«-& + 3a#-6 3 , Quot. ist rem., — 3 2 + 2a¥ — b* — 2a 3 b + « 2 6 2 +0 +0 Quotient, a- — 2ab + 6 2 Analysis. — 1st. Divide a 4 by a 2 and set down the quotient a 2 . 2d. Multiply the several terms of the divisor, except the first, by this quotient, and set the products 2a z b and — « 2 & 2 as above. 3d. Add the second column and set the sum —2a 3 b below. 4th. Divide this by the first term of the divisor, and proceed as before. 149. By this method, the dividend and divisor must be arranged in accordance with the ascending or descending powers of each letter ; that is, tbe exponent of each letter must either increase or decrease by unity from left to right in both dividend and divisor. Terms may be inserted with o for coefficients, if by so doing the exponents may be made to form an increasing or decreasing series. It will also be observed that the signs of the divisor, except the first, have been changed, which enables us to add the products instead of subtracting. The product obtained by multiplying the first term of the divisor is not written, since it always cancels a term of the dividend. The other products are written each under a similar term of the dividend, and in line with that term of the divisor from which it was obtained. Let the student work in this way such examples as are suited to this process, found in Art. 148. 3 50 DIVISION DIVISION BY DETACHED COEFFICIENTS. 150. The .work of division may often be abbreviated by dropping the literal factors and replacing them in the cpuotient. This can only be done when the dividend and divisor can be arranged according to the ascending or descending powers of each letter, as in multiplication. (Art. 132.) 1. Divide a 4 — b 4 by a — b. Tliis may be written a 4 + oa y o + oa2& 2 + oab 3 — ¥, in which the exponents of a decrease and those of b increase by unity in the successive terms. Writing the coefficients only, and dividing, we have, 1+0 + + 0— 1 ' 1 — 1 Coef. of Divisor. 1 — 1 1 + 1 + 1 + 1 Coef. of Quot. + 1+0 1 — 1 1 + o 1 — 1 1 — 1 1 — 1 It is evident by dividing a 4 by a that the first term of the quotient will be a 3 , and we may then write n 3 + a"b + ab 2 + ft 3 , Ans. EXAMPLES. Divide by detached coefficients : 2« 3 — 6a 2 x 4- 6ax 2 — 2X 3 by 2a — 2X. cfi — b 3 by a 2 + ab + &■ a A — x* 4- 20.2? — (fix* by ax + a? — x 2 . x 4 — 4-r 3 + Gx 2 — 42 + i by x — 1 . x 5 + x 4 !/ — 5>' 3 # 2 + 6-ry 4 + 2 if by x 2 4- ^xy + y 2 . 1 — 4b 4- ioZr — 16b 3 + 17& 4 — 12b 5 by 1 — 2b + 3J 2 . 2u k + wdhc + 20(( 2 x 2 4- ^rtrc 3 + 2.C 4 by a 2 + 3«.r 4- 2Z 2 . DIVISION. 51 SYNTHETIC DIVISION. 151. The operation may be still further shortened, when the first coefficient of the divisor is unity, by using detached coefficients, in the manner of Art. 148, Ex. 18. 5. Divide « 4 — 4a 8 * + 6aW — ^a¥ + ¥ by « 2 — 20b + b\ OPERATION. I + 2 — I i— 4 + 6 — 4 + 1 + 2 — 4 + 2 — 1 + 2 — 1 1 — 2 + 1 .•. Quotient = a? — iab + b 2 . Note. — Observe that the coefficient of each remainder becomes the coefficient of a term of the quotient. This method is called Synthetic Division* In like manner divide the following: 6. a 6 + 2a 3 b 3 + b 6 by « 2 — ab + P. 7. x 3 + x 2 y — xy 2 — y s by x — y. 8. a 5 -+- 5« 4 £ + \oa % x 2 + locfcfl+saxl+a? by a 2 -\-2ax + x 2 . The method of Synthetic Division is especially convenient in case of a binomial divisor. Thus, 9. Divide x 5 — 2X* + 53? + 3a; 2 — 122; — 28 by x — 2. OPERATION. I + 2 I — 2 + 5 + 3 — 12 — 28 + 2 + + 10 + 26 + 28 I + O + 5 + *3 + 14 .*. a? 1 + 00? + 5a: 2 + 132+ 14 = a? 1 + 5a; 2 + 132+ 14 = Quotient. Or, transferring the divisor to the right of dividend, and omitting the first term, an arrangement frequently more convenient, we shall have 1 — 2 + 5+ 3 — 12 — 28 I 2 + 2 + 0+10 + 26 + 28 1+0 + 5 + 13 + 14 And x* + 5X 2 + 132: + 14 = Quotient. * It was first proposed by W. G. Horner, of England. (See p. 307, Note 6.) 52 DIVISION. 10. Divide x 5 — 32 by x — 2. I+O + O + O+ O — 32 I 2 + 2 + 4 + 8 + 16 + 32 1 + 2 + 4 + 8 + 16 x 1 + 2a; 3 + 4a: 2 + 8x + 16, Ans. 11. Divide a 4 — 2a; 2 + 32; — 7 by x + 3. 1+0-2+ 3- 7 I - 3 -3 + 9-21 + 54 1 - 3 + 7 - 18 + 47 cr 3 — 3.C 2 + 7^ — 18 h — — , Ans. x + 3 Note. — In this case we have the remainder 47, and therefore the division is incomplete. Divide in like manner: 1 2. a; 6 — 3a 4 + 2cc 2 by 2 — 1. 13. x 5 + jx 4 — 3Z 2 + -jx — 5 by x — 3. 14. a 4 — 2a; 3 + 82; — 16 by a* — 2 ; also by k + 2. 15. x 5 — 5-r 3 -f- 2a; 2 — 7 by a; + 1. 16. x 7 — 5a; 5 + 2x 2 — 1 by x + 3. 17. z 6 — 4a; 5 + Ox 4 — Ox 2 + 42; — 1 by z— 1 and by x+ 1. 18. .r 5 — 7a; 3 + 6x — 5 by x + 2 and by a; — 3. 19. .r 3 — 1 by x — 1 and by x + 2. 20. .r 7 — 1 by a: — 1 and by x + 1. 21. a n + a; 12 by a* + a 4 and by a 3 + a^. 22. a 40 + y w by JB 2 + if and by a' 5 + if. 23. a- 14 + 1 by x 2 + 1 and by x" + 1. 24. x w + a 36 by a: 6 + a« and by a; 2 + « 4 . 25. 2? + y« by a- 3 + ?/' ; also, « 5 + b w by a + 6 2 . 26. a- 5 — 6x* + 5a- 3 — 7a: 2 — 4X + 9 by x— 3 and a- -+3. 2 7- -'' 7 + 3'' 3 — 2a 4 — 5_/' 3 + 33 + r by a; + 1 and a-— r. 28. a; 6 + 4.f 5 — 7.i 4 + 2ar 3 — 7a: 2 + 4.t+i6 by x+i and x— 1. CHAPTER V. FACTORING. 152. Factoring is the process of separating a quantity into factors. Unity having no power as a factor, will not be considered as a factor in the following discussions. A Prime Factor is one which does not contain other factors. Factors are prime to each other when they have no common factor. 153. The operation of factoring is performed either by inspection, in which we employ our previous knowledge of the forms of products ; or by trial, in which we determine by division whether one quantity is a factor of another, and at the same time, discover the other factor. Note. — Finding the equal factors of a quantity is the work of Evolution, which will be treated in its place. We are at present con- cerned especially with unequal factors, though equal factors may often be found by the same methods. 154. The Factors of a Monomial are too easily discovered to need explanation, since they are all indicated by the exponents. The same may be said of the monomial factors of a polynomial, since they must be factors of each term of the polynomial. Thus, a"o is a factor of ia z o + yiW — 5« 3 6 3 , and we may write a"b(2a + 3& 2 — 5a6 2 ). 155. The Factoring of Binomials will be facili- tated by the following theorems : Theorem I. — The product of the sum and difference of any two quantities is equal to the difference of their squares. Demonstration, (a + b)(a — b) = a i — b\ 54 FACTORING. Theorem II. — The difference of any two quantities is a factor of the difference of any like positive integral powers of the same quantities. Demonstration. — Let a and 6 represent the quantities, and a 11 and b n the like powers, in which n is a positive integer. We are to prove that a' 1 — b n is divisible by a — b. Dividing, «" ~ b" | a — b a n _ a d-i i ^ n _ l We have the remainder, a n ~ l b — b" One of the factors of the remainder is the difference of like powers of the same quantities, whose exponent is one less than in the dividend. If this factor be divisible by a — b, the dividend is also divisible by it. We have therefore proved that if the Theorem be true for one value of n, it is also true for a value one unit greater. For example, if it be true when 71=2, it will also be true when 11=3, and for the same reason when n = 4, 5, 6, etc. But we know that a — b is a factor of a- — b' 2 ; that is, the theorem is true when n = 2. Hence, it is universally true. Cor. i. — The sum of two quantities is a factor of the differ- ence of any like even positive integral powers of the same. That is, a- n — 6 2 " is divisible by a + b. For, a 2 " - b in = (a 2 )" - {¥)", which by the Theorem is divisible by a- — b' 2 = (a + b) (a — b). Cor. 2. — TJie difference of any tivo powers whose exponents have a common factor, may be separated into factors one of which is a binomial. For, a mn — &'"■ = (a"') n — (b T ) n , which has a m — b T as a factor. Theorem III. — The sum of any two quantities is a factor of the sum of any like odd positive integral powers of the same. That is, a 2 " H + b- n + l is divisible by a + b. For, dividing, «"" M + I I a + b We have the remainder, — a' 2 "b + b-"+ l = (a 2n — b-") (— b). But a 2 " — b' 2n has been Bhown to be divisible by a 4-6; hence, frtn+i j a a i so divisible by a + b. FACTOEING. 55 Cor. — Tlie sum of any two powers each of whose exponents has the same odd factor, may be separated into factors one of which is a binomial. That is, a^ n + 1 '> m + &0+ 1 )-- has a binomial factor. For this may be written, (a'") 2n+I + (b r f+\ which has the factor a"' + b r . For example, a 6 + b* = (a-) s + (b 3 f = (a' 2 + b 3 )(a i -aW + a 6 ). It is evident that m and r may be equal ; as, a 6 + b'' — (a 2 ) 3 + (b' 2 ) 3 , which has the factor a 2 + ¥. They may also be negative ; as, a~ b — b~ 10 = (a -1 ) 5 — (& -2 ) 5 » which has the factor a -1 — 6~ 2 . Factor the folio wins: : I. a i - V. 2. a 3 — b\ 3- a 3 + b 3 . 4- a~* - b~K 5- fl-3 _ J-3. 6. a~ 3 — b\ 7- a 6 — b~ 6 . 8. a 6 + ¥. 9- a 12 + b 15 . IO. a 12 - b vi . 1 1. x 6 + # 9 . 12. a; -6 — y~ 9 . 13- x G + y~ 9 . ^ws. (a 2 + V)(a — b) (a + b). Ans. (a - J) (a 2 + ad + £ 2 ). Ans. (a + J) (a 2 - ab + J 3 ). 14. z 10 + # 15 . T 5- a; 12 + I/ 21 . 16. or 12 — ?/ 9 . 17. 2-14 _|_ yU 18. X U - I. 19. I 4/ ff 10 . • 20. I — a" 10 . 21. a 72 — £ 96 . 22. jplOO + yM 156. To find the binomial or polynomial factors of a polynomial, the forms of products must be observed. The following theorem will aid in this inspection : Theorem IV. — The square of a polynomial is equal to the sum of the squares of its terms plus ttvice the sum of the products of the terms taken two and two. Demonstration-, (a + b + c) 9 = a' 2 + 20b + lac + b' 2 + 2bc + c 2 , as is shown by performing the multiplication, and the process is such as to make it evident that the same would apply to a polynomial of any num- ber of terms. Hence, 56 FACTORING. Cor. i. — Tlte square of (he sum of two quantities equals the sum of their squares plus twice their product ; and the square of the difference equals the sum of their squares minus twice their product. Thai is, (a + bf = a? + 2 ab + J 2 , and (a — b) 2 = d l — 2ab + b 2 . Cor. 2. — If two polynomials having like terms, but ruith unlike signs be multiplied together, the product icill be the same as the square of one of the polynomials ; except, ist. The squares of the terms having unlike signs in the two polynomials will be negative. 2d. The double products formed by those terms which have like signs in one polynomial and unlike in the other will not be found in the result ; and the double products formed by terms having like signs in each polynomial, but unlike in the two, will be negative. 157. From this theorem and its corollaries factor the following examples: 23. 4a 2 — gb 2 + c 2 + 4ac. Analysis. — The terms which are perfect squares suggest the terms of the factors sought. These give 2a, 36, and c. The only double product found is 40c ; hence, 2a and c have either like or unlike signs in both factors, but 2a and 3?;, and 3ft and c have unlike signs in one and like signs in the other factor. This requires for the factors, 2a — 36 + c and 2a + 36 + C 24. 4a 2 — gb 2 + 6bc — c 2 . 25. 4« 2 — 4«c — gb 2 + c 2 . 26. a 2 + 2ab + b 2 + ac + be. Observe that the first three terms are the square of a + b, which is also a factor of the last two terms. 27. 4 + 46 + b 2 + 2f + be. 28. a 2 4- Ga 4- 8. Separate into terms thus : a s + 4^ -1- 4 4- 2a + 4. FACTORING. 57 29. (fix* + 2ttX? + a 4 — a 4 4- 2(fix — x % . 30. 4a 2 — 4a, v — 24a; 2 . 3 1 . qcfix 2 — 4CIX 3 — 4a 4 4- 402% 4- ^xhf — ahf — 2ay 3 — y* 4- a 4 . 32. «6 — ay 4- 5a 4- as — ay 4- ?/ 2 4- 2:2: — by — yz. 33- « 2 + 3«^ + 2 ^ 2 - 158. By multiplying (x +_a x ) (« + a 2 ) (» + O, and collecting the terms containing the like powers of x, we shall find, 1st. The highest power of x will be x n , and its coefficient will be 1. 2d. The coefficient of x n ~ 1 will be a l 4- a 2 4- « 3 . . . . « n . 3d. The coefficient of a; will be a x a 2 . . . . #». Let the student illustrate this by multiplying five or six such factors. 159. From this we may often discover the binomial factors of a polynomial function of a single letter. 34. Required to factor x 2 — 2X — 15. Solution. — The factors of this will have numerical terms whose product is — 15, and whose sum is — 2, and we find that 3 and — 5 fulfill these conditions. The factors, therefore, are x + 3 and x — 5. 35. Factor yfi — ga~ — 12a 4- 36. Solution.— Dividing hy 3 to make the coefficient of a 3 unity, we have, of' — 3a 2 — 4(1 + 12. The product of the numerical terms of the factors is 12, their sum is — 3, and there are three of these factors. The only three factors whose product is + 12 and sum — 3 are + 2,-2, and — 3. These give a + 2, a — 2, and a — 3, which with the 3 already taken out, are the factors sought. Find the factors of the following : 36. a: 2 — 2X — 35. 39. 4.? • 2 + 32X 4- 60. 37- x* 4- x — 20. 40. 5a ■ 2 4- 152 — 140. 38. x 2 — gx 4- 20. 41. x z — x 2 — 142; 4- 2 4« 42. x 3 — 8a; 2 4- 112: 4- 20. 43- 5^ — 35* 3 + 85.1' 2 — 853 4- 30. 44. 2ttX 3 4- 2«.t' 2 — 7,2aX 4- 40a. 45- x 5 4- a 4 — 5x z — 5a- 2 4- 4X + 4. CHAPTER VI. DIVISORS AND MULTIPLES. 160. The Factors of a quantity are sometimes called its Divisors, since it may be divided by any one of them. 161. Commensurable Quantities have a common divisor; as, ac and ax, both of which have the divisor or facto?' a. 162. Incommensurable Quantities have no com- mon divisor ; as, abc and :ryz. Such quantities cannot be measured with the same unit ; hence the name. (Art. 8.) GREATEST COMMON DIVISOR. 163. The Greatest Common Divisor of two or more quantities is that common divisor which contains the greatest number of factors. 164. If the prime factors of the quantities can be discovered by inspection, the f/,c.d. may be found by combining all the common factors into a product, using each as many times as it is found in every one of the quantities. 165. When these factors cannot be thus discovered, we may employ a process based on the following Theorem.— A factor common to two quantities is a factor of tlw remainder resulting from the division of one of the quantities by tin' <>// 3 — i6r 2 - 8.r + 16 + 32:'-' — 12 Rejecting the factor 3 from the remainder, and dividing divisor by remainder, x 3 — 2X 2 — x + 2 x- — 4 ^_ 4 X x — 2 — 2X- + 2>X + 2 - 23? + 8 33- — 6 Rejecting the factor 3, we have the remainder x — 2, which will of course divide x- — 4. (Art. 155.) Therefore the f/. c. d. is ± (x — 2) x, or ± (a? — 2x). 2. 4-f 3 — 6.r' 2 — 4X + 3 and 2.C 3 -j- .r 2 -j- x — 1. The operation may be put in the following convenient form. OPERATION. 1st dividend, 4X 3 — 6x' 2 — 47- + 3 4.r 3 + 2x- + 2x — 2 2d divisor, 2d quotient, 3d dividend, 4th divisor, - 8 r- - 6a 4 5 - S.j: 2 - 8a; 2 + 6x 36a; + X 5 16 - 21 ) - 42.r + 21 2a; - 1 ± (2X — X + 4 1) = q. c, (I. A us. 2X i + X 2 + X 1st divisor. 2 1st quotient. 8.T 3 + 4a; 2 + 4a; — 4 2d dividend. Sx 3 + 6x* — 53; — 2.r2 + gx — 4 3d divisor. 4 3d quotient. — 2.C 2 + 9a 1 — 4 4th dividend. — 23! 2 + x + 8x — 4 8.r - 4 MULTIPLES. Gl 3. «« 4 + 2a 2 « 3 + a 3 ^ 2 — ax 2 — 2« 2 a; — a 3 and a 6 — 2a 4 a: 2 4-a 2 ar*. 4. x 4 4- io.c 3 4- 24a; 2 — 10a; — 25 and 4a; 4 — 212: 3 + 5. 5. 4« 2 ^ 3 + a 3 b + 2aW + 2a¥ + « 2 & 2 — b 4 and a 3 Z> — aW — a& 3 + 6*. 6. (a 4 — £ 4 ) aa; and (a 3 + 6 3 ) 5a;. 7. (a 3 — Z» 3 ) (a — x) and (a 2 — J 2 ) (a 4- a;). 8. 3a; 5 + 7a; 3 — 5a: 2 4- 3 and 3.T 3 — 2a; 2 — 1. 9. o 2 — $az + 4a; 2 and a 3 — a 2 x 4- 3«a; 2 — 32 s . 10. x 4 — 6x 3 4- 13X 2 — \2x 4- 4 and a^ — 4a; 2 4- 52; — 2. 169. The g. c. d. of more than two quantities may be found by finding the g. c. d. of two, and then of this g. c. d, and a third, and so on till all are used. Find the g. c. d, of the following : 11. x 2 — x — 6 ; x 2 4- 4X 4- 4 ; a; 2 — 4. 12. 3.T 2 4-6a; + 3; 6a; 2 — 30a; — 36; g:i?-{-2'jz-\- 18; 12a; 2 — 12. 13. a; 3 4- 4a: 2 4- 6x 4- 9 ; x 3 4- x 2 — 2a; 4- 1 2 ; x 2 — % — 12. 14. x 4 — x 3 — 4X 2 4-i6a; — 24; x 3 — 5a; 2 4- 8x — 4; x 2 — 2a; — 8.. 15. x i — 8a; 2 4- 16 ; x 3 4- 2a; 2 — 4a; — 8 ; x 3 — 2X 2 —4X 4- 8. LEAST COMMON MULTIPLE. 170. A Multiple of a quantity is the prochict of that quantity by any factor. Hence, It is any quantity which is divisible by the given quantity. A Common Multiple is a multiple of several quantities. The Least Common Multiple is the quantity which contains no factors except those which are necessary to make it a multiple of the several quantities. The I. c. m. will therefore contain every factor found in the given quantities, and each factor will be found in the I. c. in. as many times as it is found in any one of the given quantities. Hence, • 62 MULTIPLES. 171. To find the I.e. in. of several quantities we have this Eule. — I. Separate the quantities into their prime factors. II. Give each of these factors an exponent equal to the largest exponent it has in any of the given quantities. III. The product A ^S. 3 3 quired. Hence, the Rule. — Multiply the integral part by the denominator ; to the product add the numerator, and place the sum over the denominator . Note. — An entire quantity may be reduced to the form of a fraction by making 1 its denominator. Thus, a = - • Reduce the following to the fractional form : 2. ax 2 5. a — x — - — — ■-• a a — x x — a 2 b 2 , 7 a 2 + b 2 3. ab + r 6. a + b —r- ab a + b (x—1) 2 „ „ a 4 — 1 4. ic + 1 + - — — ■ — • 7. a 3 + ^ + a; + 1 — • CASE IV. 196. To Reduce a Fraction to any required Denominator. By Art. 191, 3 , we have the Rule. — Multiply both terms by the factor which will give the fraction the required denominator. Notes. — 1. An entire quantity may be reduced to a fraction having' a given denominator, by the same rule, by first writing under it the denominator i. 2. When the required denominator is not a multiple of the given denominator, the result will be a complex fraction. 68 REDUCTION OF FRACTIONS. EXAMPLES. tt 2 -\- 2 i. Reduce - — ■ to a fraction whose denominator is c a 2 c — c. 2. Reduce — n to a fraction whose denominator is a 2 + i a 6 + i. 2. Reduce - - to a fraction whose denominator is « 4 — i. a — i X 4- C 4. Reduce - — - to a fraction whose denominator is x — 7 X 2 — 2X — 35. <. Reduce -= to a fraction whose denominator is D a 2 — 1 a 3 — 1. 6. Reduce x to a fraction whose denominator is x + 1 x 2 — 1. CASE V. 197. To Reduce Fractions to a Common Denominator. By Case IV we can reduce fractions to any required denominator; but to avoid complex fractions that denomi- nator must be a multiple of the given denominators. To express the fractions in the lowest terms, it must be the least common multiple. We have then the following Rule. — I. Find a common multiple of the denominators for the common denominator, the least common multiple being preferred. II. Multiply loth, terms of each fraction by that factor which will give it the required denominator. 198. When the least common multiple is taken as a common denominator, it is called the Least Common Denominator, ADDITION AND SUBTRACTION. (i!) Eeduce the following fractions to the least common denominator : 2. 3- 4- 5- 6. x + y' i JC 4 — i' a + i 2a 2 + 2 4 x 2 - 1 ' a; .T* - I ' .r + y X 2 a 4 27* x* — y 2 ' "^ 2 3 x*+ 1' z 3 + 1 a — 1 a 2 — 1 4« 3 — 4 ' 8a 6 — 8 x — 2 2' x + 2 a 2 + x — a; 2 — £ — 2 X 2 -f I x 2 — 1 x A + 4Z 2 + 3 ' a 4 + 2£ 2 — 3 x — y x 2 + y 2 x A — y*' x 2 + y 3 ' a; 2 — ?/ 3 ADDITION AND SUBTRACTION OF FRACTIONS. 199. According to Art. 113, fractions may be added or subtracted by the following Kule. — Reduce the fractions to a common denominator, and add or subtract their numerators, writing the result over the common denominator. 2ax 3a 1. Add — =- and - — 30 2X Solution. — Reduced to a c. d., the fractions become — -- and %r- , tax 6bx which, by adding tbe numerators, give — ~~2r~ > Ans. 2. From -— take - — ■ x y Solution. — Reducing to a c. d., we have - — - and - — ; and xy .11/ "jaby $cdx _ yaby — scdx . xy xy xy 70 MULTIPLICATION OF FRACTIONS. Perform the operations indicated in the following: 3- 2 + c) b c a abc x — 2 x — 4 x — 6 x — 8 ?» (./• 4- a) n (x 4- b) a- 4- b x 4- « X 4- S 2 —12 Z. 2 3 a; — 4 ca; 4- 14 1 3X 4 - 5 — 6x + 3 12 7a: 4- 14 17 — 3a; 4a- 4- 2 3 ?o — a; 6a; — 8 4a; — 4 xx — x + - = x — 2 * + 4, 2 7 5 5 zs + 4 _ 6a: 4- 7 7 a — 13 , 3 9 6a; 4- 3 ' (x — b\ (x + b\ x 3 / v a; h - x a 4- 1 a — 1 c a; 2 4- a; a 2 — # 2 a — b a + b x — 2 , .301 „ .r — 2 h — — = .ooia- 4- .6 — 5 -5 -05 THE FIRST DEGREE. 83 SOLUTION OF PROBLEMS. 234. The Solution of a Problem, which requires the finding of an unknown quantity from its relations to known quantities, consists of three distinct steps. (Art. 30.) 1st. The conditions of the problem must be expressed hy a conditional equation. 2d. That equation must be reduced from an implicit to an explicit function, called a formula. 3d. The numerical values of the known quantities must be substituted in that formula. 235. The first of these steps does not belong to Algebra ; but as the practical value of Algebra cannot be illustrated except by the solution of problems, it is important to become familiar with the translation of the conditions of a problem into an equation. The problems, however, which may properly engage the attention of the student are those relating to subjects with which he is supposed to be familiar. 236. This analysis of the process of solving a problem gives the following Rule. — I. Represent the quantities involved hy proper letters, in accordance with the usage of the algebraic language. (Arts. 41-69.) II. With these letters express the conditions of the problem by a conditional equation. III. Reduce this equation. (Arts. 224-227.) IV. To apply the solution to a special case, substitute the numerical values given in the special case, in the formula obtained. 84 EQUATION SOF 237. The following problems will illustrate the rules: Problem i. Find the number which being divided by two given numbers will give quotients differing by a given number. Solution. — ist. Let x = the number to be found. m and n = the given divisors. d = the difference of quotients. Having assumed this notation, we are prepared to express the con- ditions of the problem by an equation. This equation will be x x — ii m n ' in which x is an implicit function of m, n, and d. 2d. The second step in the solution is the reduction of this equation, by which x becomes an explicit function of the given quantities. This reduction gives _ mnd n — m 3d. The third step consists in applying this explicit function or formula to any given case of the problem. 2. Find a quantity whose fifth part exceeds its sixth part by 5- Here m — 5, n — 6, and d = 5 ; and x = — = mo, Ans. 6-5 238. The third step of the solution, which is arithmetical. may be performed before the second by substituting in the equation, before it is reduced, the numbers belonging to the special case. Thus, - - - = 5. 5 6 5 ' Reducing, x = 150, Ans. 239. The advantage of reducing the equation before the arithmetical substitutions are made is evident from the fact that a formula or rule is obtained by which the arithmetical part of the solution may be performed for any special case of the problem. m + x n + x m — x = rn THE FIRST DEGREE. 85 DISCUSSION OF FORMULAS. 240. A problem is said to be Generalized when its conditions are stated in general terms and reduced to -a formula. The Discussion of a Formula consists in applying it to such special cases of the problem as will show the differ- ent forms the result may take. 3. Find the time when the ages of two persons, A and B, will have a given ratio, the present age of each being given. Let x = the time required, r = the given ratio, m = A's age, and n = B's age. Then, by the conditions, and r — 1 To discuss this formula : 1st. Suppose that A is now 30 years old and B 20. How long before A will be twice as old as B ? In this case, r = 2, m = 30, n = 20. 30 — 40 — 10 .-. x = — = = — 10. 2 — 1 +1 This means that the event occurred 10 years ago. 2d. Again, suppose A is 30 years old and B 30. How long before A will be 3 times as old as B ? Here m = 30, n = 30, r = 3. r, , -, ,. 30 — 90 — 60 substituting-, x = — = = — 30. 3-i +2 That is, 30 years ago, when the age of each was zero, A might be said to be 3 times as old as B. 0x3 = 0. 3d. Again, let A's age be 30 and B's 30. When will the ratio between their ages be 1 ? 30 — 30 o Bv the formula, x = — = - , 1 — 1 o which, by (Art. 209), is indeterminate, and means that any time future or past will satisfy the conditions of the problem, since their ages are now equal (or their ratio is 1), and have been and will continue equal. 86 EQUATIONS OF 4th. Again, let A's age be 30, and B's 15. How long before A will be twice as old as B ? Substituting, x = — — = = o. 2 — 1 +1 That is, in zero time, or now, their ages are in that ratio. 5 tli. Once more, let A's age be 30 and B's age 20. How long before their ages will be equal. By the formula, _ 30 — 20 _ + 10 _ 1 — 1 o That is, only at the end of an infinite time ; in other words, they will never be of equal age. PROBLEM OF THE COURIERS. 4. Two couriers, A and B, are travelling to tbe' cast, the former m and the latter n miles per hour. At noon, A passes a given point 0, and B is a miles in advance of A. How long after noon and how far from will they be together? Let t — the required time, and d = the required distance. Then mt — nt = a, and t — • m — a ant ~~ m — n To discuss this result, we make the following suppositions: 1st. Let a, m, and n be positive, and m > n. .". t and d are positive, and the time of meeting is after noon, and the place east of O. t> — and d > a. m 2d. Lei a, m, and n be positive, and in < v. .-. t and d are negative, and the time is before noon, and the place west of 0. 3d. Let a = o, and m % n (read, " m greater or less than n"). r. t = o and d = o. The time of meeting is noon and the place at 0. THE FIRST DEGREE. 87 4th. Let m = o, and a and n be positive. :. t = and d = o. The time is before noon and the place as n before, but for a different reason. By the former supposition, A passed I O at noou ; by the latter, he remains at all the time. 5th. Let n = o, and a and m be positive. .'. t = — and d = a. B now remains at a miles east of 0, but the m time is, as it should be, — hours after noon. m 6th. Let m = n, and a be positive. .*. t = oo and d = 00 . Their rate of travelling being the same, it will require an infinite time and distance for A to overtake B. 7th. Let m = n, and a = o. .-. t = - , and d = - • Hence, o o They are together all the time and everywhere; as they should be, being together at noon and travelling at the same rate. 8th. Let a and m be positive, and n negative. .•. t and d are positive, and the result is similar to 1st, but t < — m and dy _ 4 + iy 5 From which, y = 3, as above. 94 SIMULTANEOUS EQUATIONS CASE III. 250. For Elimination by Substitution we have the following Kule. — Find from one of the equations the value of the unknown quantity to be eliminated in terms of the other unknown and known quantities, and substitute that value for this unknown quantity in the other equation. BY SUBSTITUTION. Finding x from (i), c _ 13 - 3V . 2 (3)" Substituting in (2), x 3 — 3V 5 2 31 V = 4- (4)" Giving as before, y = 3- CASE IV. 251. For Elimination by Division we have the Kule. — I. Clear the equations effractions and transpose all the terms of each to one member. II. Proceed as if to find the greatest common divisor of the polynomials thus found, and when a remainder is obtained from which one of the unknown quantities has disappeared, put this remainder equal to zero for the equation sought. BT DIVISION. Transposing (1), ix + 3?/ — 13 = 0. (3)'" Transposing 12), 5.7' — 21/ — 4 = 0. (4)'" Dividing (4)'" x 2 by (3)"' ioz - 4?/ — 8 1 2* + 33/ — 13 iox + isy — 65 5 — 19^ + 57 . . . Remainder. Putting remainder = 0, - 193/ + 57 = * = l x - 6. as -f by = h, x + y — d. 7- ax + bx = y~, x = hy. 8. 4X 2// 7~y = 4 ' 6x = gy. 9- x + y — 24, 3 4 x + sy = Ii6 - 10. \x + toy = 124, 2£ + 9^/ = I24. 11. x — 2ij = a, 2x + Sy = b. 12. ax + by — c = o, «'# + 6'?/ — c' = 0. *3- ^^ — 3. ? y = to^/' 4» + iy = //• 14. y = ax -\- b, y = ax + &'. 15- y — a = 2 (x — b), y — b = 2 (x — a). 16. x + VJ _ 7l 2 - 7 *' 4« + 5?/ _ s 4 17- ft + 3// _ c 3 _ 3' £ — 3JJ d 3 "3 18. 2* + y 4 =4, 3* — 3y 6 - Io a- + 1 1 19. _ 2/ 3 a; 1 y + 1 • 96 SIMULTANEOUS EQUATIONS PROBLEMS. i. A number consists of 3 digits. The middle digit is the sum of the other two, the first "is twice the lust, and inverting the order of the digits gives a number 33 more than half the first number. What is the number? 2. A person spends 50 cents for apples and oranges, buy- ing oranges at 5 cents and apples for 2 cents apiece. If he had taken half as many of each and paid 6 cents apiece for oranges and one cent for apples, they would have cost him 23 cents. How many of each did he buy ? 3. What fraction is that whose numerator being increased by 5 and the denominator decreased by 2, equals 1 ; but whose denominator being increased by 5 and the numerator decreased by 4, becomes | ? 4. Three pipes discharge into the same cistern. The first and second will fill it in -\ hours, the second and third in 12 hours, and the first and third in 8^2 hours. In what time will each pipe fill the cistern ? 5. A certain sum of money at interest amounted to $550 in 10 months, and to 8560 in 12 months. What was the sum and the rate per cent ? 6. Two persons, A and B, can together reap a field of grain in 10 days. They work together 6 days, when A is left to finish the work, which he does in 10 days more. In v. hat time can each reap the field ? 7. A and B engage to do a piece of work in 12 days, but after a time, finding themselves unable to accomplish it, C was called in to help them, and the work was finished in time. The rate of working of each was such that A could do the work alone in f the time required for B to do it. and C could do it with A in f of the time in wdiich he could do it with B, and the three together could do it in 9 days. What part of the work did each one do ? How long did C work ? 8. A banker has two kinds of money. It takes a pieces of one and b pieces of the other to make a dollar. If c pieces be given for a dollar, how manv of each will be used ? OF THE FIRST DEGREE. 97 9. A, B, and C lunch together. A furnishes 3 loaves and B 2 loaves and a basket of fruit, the whole cost of which was 50 cents ; but 0, having no provisions, agrees to pay for his share 25 cents in money, when it is found that only 2 cents of this will belong to A. What was the cost of the loaves and what of the fruit ? 10. What fraction is that, whose numerator being doubled and denominator increased by 7, the value becomes f ; but the denominator being doubled and the numerator increased by 2, the value becomes f ? 11. A merchant has two kinds of wine. If he mixes a gallons of the first with b gallons of the second the mixture is worth c dollars a gallon ; but if he mixes m gallons of the first with n gallons of the second the mixture is worth p dol- lars a gallon. What is the price of each kind of wine ? 12. What two fractions have their sum if, and the sum of their numerators equal to the sum of their denominators ? 13. A loaned 6500 in two separate sums, the less at 2% more than the other. If the per cent on the greater be increased and that of the less diminished by 1, the whole interest will be increased 25$ ; but if the per cent on the greater be so increased without changing the other, the interest will be increased 2>2>i%- What were the sums and the rate per cent of each ? 14. What is the fraction which becomes | when 1 is added to the numerator, and } when 1 is added to the denominator ? 15. Two men commence business at the same time, A hav- ing $1000 more capital than B. At the end of a year A had lost an amount equal to B's capital and B had gained, the same amount, when A's capital is found to be 1 2 000 more than B's. What was the capital of each ? 16. Two men buy a farm in company for $2000, each put- ting in all the money he had and giving a mortgage for the balance. >{f A should pay the mortgage, he would then have invested §200 more than twice as much as B. If B should pay the mortgage, he would have invested $200 more than A. How much cash did each put in, and how much was the mortgage ? 08 SIMULTANEOUS EQUATIONS THREE OR MORE UNKNOWN QUANTITIES. 252. When there are three or more unknown quantities, and a like number of equations, the reduction is made by the following Eule. — I. Eliminate the same unknown quantity from different pairs of the given equations, thus forming a set of equations independent of this unknown quantity, and one less in number than the given equations. II. From these, in like manner, eliminate another unknown quantity, and so continue till an equation is found with hut our unknown quantity. III. From this find the value of that unknown quantity, and. substitute it in a previous equation to find the value of another unknown quantity. Substitute these two to find a third, and so on till all are found. The following example will illustrate this process: i. Given x + 2ij — $w + % = 4 (1) 2X — y + 2w — sz = 1 (2) 5 X — ZV — w — 2Z = 11 (3) ix + 41/ — 5W + 6z = — 9 (4) to find the value of x, y, w, and z. 3x + by — qic + 32 — 12 (5) 5* + sy - I" = 13 (6) (7) (8) (9) (10) (") (12) (13) 443 + gy = - 62 (14) 44' 1 - 88.y = 1 32 (15) 97y = - 104 (16) y = — 2 (I) X 3. (2) + (5), (I) X (3) + (7), (I) X 6, (9)" (4), (8)- (6), (6) x 13. (10) x 7, (12).- - (13). (II) - < 22, (14)" - (15), (16)- - 97, 2X + 4y — bw + 23 = 8 "jx + y — "jw = 19 bx + i2y — 1S10 + bz = 24 3.r + 8y — 1310 = 33 2X — \y = 6 65.r + 657/ — giw = 169 2\x + 567/ — 917c - 231 OF THE FIRST DEGREE. 99 Substituting in (n), 2X + 8 = 6 And £C = — X Substituting in (8), — 7 — 2 — 7«o = 19 And w = — 4 Subst. in (1), —1 — 4 + 12 + 3 = 4 And 2 = — 3 .-. x = — 1, y — — 2, w = — 4, and 2 = — 3, ^.ns. 2» + 3# — 42 = 8, 6. 5X— iy+ z-\- u = 2, Z% — 42/ + 2z = 3, ix— sy— $z+2u = 2, 4X — 21) — 32 — 5. w— 2\j = 2, ax + by + cz = m, x + 5l/— 2Z = 2. bx + cy -{- az = n, 1 £ 1 ex- + ay + fo: == r. x y ~ a' x + y = 12, I , T _ I y-* = 3, y z y z + u= 7, 1 1 _ 1 W + £ = 8. Z £ C x y z __ 8. aa 4- Jy + « = o, 2 ~ 3 4 ~ «'« + #';/ + c'2 = o, » _ y , « _ 23 «".T + Z»"?/ + C"^ = O. 3 4 2 12 9. «# 4- bx — cy = m, n. % , y - = -• a v + by — ex = 4 2 3 — 3 10. jb^2 = a (xy + y« — xz) = d (xy + xz — yz) = 'c (xz -\- yz — xy). PROBLEMS. i. A number consists of 4 digits. The first is half the second; the third, twice the second plus the first ; the fourth, the sum of the second and third; and the sum of the digits is 15. What is the number? 2. Find three numbers, such that \ the sum of the first and second shall be 50, \ of the second and third shall be 65, and I of the first and third shall be 55. 3. The average age of A, B, and C is a. The average age of A and B is b, and of B and C is c. What are their ages ? 100 SIMULTANEOUS EQUATION'S. 4. Divide the number 150 into three parts, such that \ the first shall be \ of the second, and \ of the second shall be \ of the third. 5. A person has 2 horses and 2 saddles, all of which are worth £265. The poorer horse and better saddle are worth 85 less than the better horse and poorer saddle, while the better horse and better saddle are worth 845 more than the poorer horse and poorer saddle, and the horses are worth 5 times as much as the saddles. "What is the value of each horse and saddle ? 6. Three brothers, A, B, and C, bought a farm for -Si 000, A's money with \ of B's and \ of C's would pay for it ; so also would B's money with -^ of C's and ^ of A's, or C's with I of B's and \ of A's. How much money had each ? 7. A, B, and C bought a farm for a dollars. A's money with — of B's and - of C's, or B's money with — = of A's and m n m — of C's, or C's money with — 77 of A's and —77 of B's will pay iv lit ih for the farm. How much money had each ? 8. A and B can do a certain piece of work in a days : B and C in b days; C and D in c days ; and A and C in cl days. In what time can each do the work alone, and how long would they be in doing it, working all together? 9. Three men began trade at the same time. A had $2000 more than twice as much capital as B, and C had 8500 less than A and B together. The first year A gained as much as B's original capital, B gained as much as A's capital, and C gained as mnch as A's and B's capital together, when each had an ecpial sum. How much had each at first? 10. A doctor visits a patient, and when half way home he is overtaken by a messenger, and called to return 3^ miles to visit a second patient ; and again, when half way home, he is called to return to visit a third patient, 3 miles farther away than the first. On his return to bis office, he finds he has driven 20 miles. How far did each patient live from his office? CHAPTER X. POWERS AND ROOTS. 253. A JPowev is the product of any number of the equal factors of a quantity. 254. Powers are expressed by exponents, which show what equal factors are taken. Hence, A quantity with any exponent is a power. 255. The quantity itself is called the Base of the power. 256. The word power is used with reference to the effect of a quantity as a Factor, while its effect as a Term is called its value or magnitude. Thus, if I of the factors of a are required, it is written a* ; if two- thirds of a as a term, it is written \a. 257. The operation of finding any part of the factors of a quantity is similar to that of finding any part of its terms. For example, f of the terms of 27 are found by separating 27 into three equal terms (27-7-3 = 9), and combining two of them (9 + 9 = 18, or 9x2 = 18). So f of the factors of 27 are found by separating 27 into three equal factors (271 = 3), and combining two of them (3x3=: 9). 258. Evolution is the process by which a quantity is separated into any number of equal factors. 259. Involution is the process of finding the product of equal factors. 102 POWERS A X D ROOTS. .... XorE3. — i. If the exponent of the required power have i for its denominator (that is, if the exponent be integral), the factor to be found by evolution will be the base, and the work of finding the required power will be wholly involution. 2. If the exponent have x for a numerator, the work will be wholly evolution. 3. If both numerator and denominator be 1 (that is, if the exponent be unity), the power is already found in the quantity itself ; that is, 260. The First Power is the quantity itself, or base. 261. As a coefficient shows by its denominator the number of equal terms into which a quantity is to be separated, and by its numerator the number of these terms that are to be taken, so an exponent shows by its denominator the number of equal factors into which a quantity is to be separated, and by its numerator how many of these factors are to be taken. 262. A Hoot is one of the equal factors of a quantity; as, a*, a*, a?, etc.. which may be read, "a one-half power, ] a one-third power, ," etc.; or, "the square {or second) root of a, flu' cube {or third) root of a, the fourth root of a,"" etc. (Art. 58, 59.) 263. Quantities with fractional exponents are called Bad leal Qua tit ities. They may be expressed by the radical sign, the numerator of the exponent remaining as an exponent of the base, and the denominator being placed over the sign, except when it is two (2), in which case it is omitted. Thus, a* = <\/a ; a% = tya- ; a* — \/a 3 , etc. 264. A Power which can be expressed without a fractional exponent, as a?, is called Rational, and may be written a 2 . When it cannot be so expressed, it is called Irrational, or Sard ; as, aK a*, etc. Thus, 8^, 4*, and 8' are rational ; but S*, $, and g 3 are irrational. POWERS OF MONOMIALS. 103 265. To make the use of exponents familiar to the student, let the teacher ask such questions as the following: i. What is the value of 16&? 16^? i6* ? 2. What is the value of 1 6* ? 16* ? i6°? 3. What is the value of 16- 1 ? i6~ 2 ? 16 2 ? i6~$ ? 16-*? 1 6-°? 4. What is the value of (16- 1 ) 2 ? (16- 1 )- 1 ? (16 !)- 2 ? 325? 32-5 p e tc, etc. POWERS OF MONOMIALS. 266. Any power of a quantity may be expressed by giving it the exponent of the required power; as, (8a 3 6 2 )t, (cM + 2 £ 2 )f. In the case of the monomial, the parenthesis may be removed by applying the exponent to each factor separately. Thus, (8a 3 6-)i = 8* (a z f (6 2 ) 1 = 4« 2 &*. (Arts. 117, 118.) Hence, To find any power of a monomial we have the following Eule. — Multiply the exponent of each factor of the mono- mial by the exponent of the required power. Notes. — 1. It will be observed that the numerical factor or coefficient must be included, and that when this becomes rational the required power may be taken, as in the example above. 2. This rule applies to all powers of a monomial, integral or frac- tional, and includes the work of evolution and involution. 5. Find the m th power of (— a s b 2 ). 6. Find the third power of (x — y)*. 267. When the quantity whose power is taken is regarded as having a directive sign, the same rule applies to the sign. Thus, (— aW) s , which equals — * a?bk We have already seen the force of the signs with integral exponents. Their meaning with fractional exponents will be considered hereafter. 104 BINOMIAL FORMULA POWERS OF BINOMIALS. 268. Any power of a binomial may be found by the Binomial Formula, given below, in which a and x represent the terms of the binomial and n the exponent of the power, which may be integral or fractional, positive or negative. Note. — We are not yet prepared to demonstrate this formula, but the student may verify it for any integral powers of (a + x) by actual multiplication, and by committing it to memory it can be used with equal facility before or after demonstration. BINOMIAL FORMULA.* (a + z) n = a n + na n ~ x x ^ s— — '- a n ~ 2 x 2 x ' 1-2 n (n — i) (n — 2) H — — *-* -a"- 3 *?, etc. 1-2-3 269. An inspection of the formula will show that, I. The exponents of the leading letter (a), beginning with the exponent of the power (n), decrease by unity in the succes- sive terms. II. The exponents of the follovring letter (x), beginning with o, increase in like manner by unity. III. The sum of the exponents in any term equals the exponent of the required power. IV. Hie coefficient of any term {after the first, whose coefficient is 1) is the product of the coefficient of tin- preceding term by the exponent of the leading letter in that term, divided by the exponent of the following letter in the term itself V. The number of terms in the series /rill be n + i when the exponent of the poiver is integral and positive, and infinite in all other cases. VI. lite m th or general term is 11 (n - 1) (n - 2) . .. . (w-ffi + a^ ^ 1 - 2 • 3 • 4 . . . . (m — 1 ) Note. — This formula translated into common language is called the Binomial Theorem. * This formula was discovered by Sii Isaac Newton. (See p 30c. Note 1.) INVOLUTION". 105 270. Although the terms of the formula are all +, it must be remembered that as a, x, and n may oue or more of them be negative, the sign of each term must be determined by the general Rule of Signs. (Art. 91.) 271. Signs. — I. If both terms of the binomial are positive, and the exponent integral and positive, the terms of the series will all be positive. II. If both terms of the binomial are positive and the exponent negative {whether integral or fractional), the terms of the series will be alternately positive and negative. III. If both terms of the binomial are positive, and the exponent is a positive fraction, the terms of the series ivill be positive until the term whose number is the next integer greater than (n -f- 2). This term will be negative, and the following terms will be alternately positive and negative. IV. If the second term of the binomial be negative {the first being positive), the alternate terms, beginning with the second, will have signs opposite those given in the cases above. Note. — Let the student verify each of the statements in Arts. 269 and 271 by an examination of the formula. INVOLUTION OF POLYNOMIALS. 272. In finding the powers of polynomials, we must per- form the involution and evolution separately. The square of a polynomial may be written by Theorem IV, Art. 156. 273. The cube of a polynomial may be written by the following Rule. — Tlie cube of a polynomial is equal to the sum of the cubes of its several terms, plus three times the products of the square of each term by each of the other terms, plus six limes the products of the terms taken three at a time. Note. — Higher integral powers of polynomials can be found by actual multiplication. 274. A Multinomial Theorem, by which any power of a polynomial may be written in the same manner as powers of a binomial by the Binomial Formula, is sometimes given, but it is of little use in ordinary mathematical operations. 106 INVOLUTION. EXAMPLES. Develop the following by the Binomial Formula : i. (« + J). = rf + 5^ + 5^* + L4J^ |£ |3 |4 1 5 = ft 5 + 5« 4 i + io«% 2 + loarb 3 + s«i 4 + J 5 , Jms. 2. (ft — #) 3 = ft 3 — y&b + 3«i 2 — b s , A us. 3 . (a + 5)4 = a* + ia-U + iiipil ar*P + etc. r = a* + — ; i i + etc., ^««. 4. (ft + b — c + ft 7 ) 2 = ft 2 -+- 2a5 — 2ac + 2ad + b-—2bc -\- zbd + (? — zed + ft 12 , -<4rcs. 5. (ft + #) 2 and (a — xf. 6. (2« + J) 2 and (ft — 2b) 2 . 7. (a + by and (a — by. 8. (.r + y) 6 and (x — y)"'. 9. (2ft + 2#) 5 and (3ft — 3&) 3 . 10. (« + a;) -1 and (a — x)~ z . 11. (ft -f «)* and (ft — x)z. 12. (ft + c)* and (ft + c)~?. 13. (ft — c)~^ and (ft — c)*. 14. (ax -f- J^)* and (ax — by)~~. Expand the following by Theorem IV and Rale, Art. 156 : 15. (ft + zb — 4c) 2 . 16. (2ft — 3ft.r + lr — xy + zf. 17. (ax + by — 32 + 5) 2 . 18. (ft + b — c + df. 19. (zx — 3!/ + s) 3 - 20. (ax + by + z -\- m — ay. EVOLUTION. 107 EVOLUTION OF POLYNOMIALS. 275. To find any Hoot of a Polynomial, the mono- mial factors should be removed and their roots taken as factors of the required root. Let P represent any polynomial whose n th root is required, and from which the monomial factors have been removed, and let x represent one term and y the algebraic sum of the remaining terms of the root. Then P = (x + yy = x" + nx"-* y + etc., in which y may be a polynomial. (Art. 268.) Now if none of the literal factors of x are found in any term of y, the root sought may be found at once by taking the n' h root of that term of P represented by x n for one term of the root, and dividing all the terms represented by nx"~ l y by nx"" 1 for the rest of the root. But, as some of the factors of x may be found in one or more terms of y, {x+y)" may have similar terms, which by uniting will prevent finding by inspection the value of nx :i ~ s y. And yet, since there will be at least one term of y which does not contain x (the monomial factors having been removed), we shall always be able to find at least two terms of the root, which being raised to the n lh power and subtracted from P, will give a remainder from which other terms of nx a ~ x y can be found. A repetition of this process will give the whole root. 276. This process may be expressed by the following Eule. — I. Find by inspection those terms of P which are the 11 th powers of the terms of the root, and take the roots of these for terms of the root. II. Representing these terms by x, x x , x 2 , etc., find the terms representing nx n ' l y, nx x n ~ x y, etc., and divide them respectively by nx n ~ 1 , nx x n " x , etc., for other terms of the root. III. Raise the part of the root so found to the n th poiver, subtract it from P, and find in the remainder other terms of the farm nx n ~ l y, and divide as before. Notes. — 1. The terms to be found by Part I of this rule will be those containing the highest powers of the different literal factors ; or, if there are two terms that contain higher powers of any letter than any others, the one containing the least number of other literal factors will be the term required. 2. The term.; to be found in Part II will be those containing the rie^t lower powers Qi tl) - letters, 108 EVOLUTION. 277. By making n = 2, we have the rule for taking the square root, and n = 3 gives the rule for the cube roof. 278. Any Hoot whose index is the product of two or more factors may be found by taking successively the roots indicated by those factors. Thus, the sixth root is the cube root of the square root. EXAMPLES. 1. Find the cube root of 8a 6 — s6a 5 b + 33a 2 b i + 66a 4 b 2 — 630:^ — gab 5 + W. Solution. — By Part I of rule we find that 8er 6 and ¥ must be cubes of terms of the root, and we get from them 2a- + b* as a part of the root. By Part II we take the term containing the next lower power of a, and divide it by n(2a 2 ) a ~ 1 = 3 (2a 2 ) 2 = 12a 4 (u in this case being 3). This gives — 36a 5 6 -f- 12a 4 = — 306. Also, taking the term containing the next lower power of b and dividing, we have — gab 5 -¥■ 3& 4 = — 30b. By each of these last two steps we find — 3«& as another term of the root. This does not mean that — yib is found twice in the root, for the same term of the root will frequently be found more than once by the method above. By cubing the root, 2a' 2 + b' 2 — ytb, we shall find the given poly- nomial. 2. Find the cube root of 8»% — 362;%* -f 122^* — 27.1- + 542%* 4- 272:%* + ifl + 6.r%* _ gxky — 36./ ; 7/ ; '. Solution. — From Part I of rule we have 2.rfy T — 3a:* + y* as terms of the root, and Part II gives no other terms. This root cubed gives the polynomial. 3. Find the square root of a 2 — 2a?x — 2a z x + ^a z x 2 —2ac + a 2 x 2 — 2a s .c 3 + 2acx-\-ah? — 2ah?-\-2a z cx+ofa£ — 2a 2 cx 2 -\-c 2 . Solution. — By Part I of the rule, ± c is one term of the root. By Part II, dividing the terms containing c first power (viz., — 2nd + 2acx + 2(i'cx—2a*cx' : ) by ±2c gives the otherterms, to ± ax, ;. ± (c — a + ax + a i x — «' 2 .r°) = the root required. EVOLUTION-. 109 EXAMPLES. Find the square roots of i i i. a + 2a~x^ -f x. i i 2. a — 2a~x~ + x. 3. a %n ± 2a n x n + a* 1 . 4. a 4 -f 4« 3 .f + 6« 2 .c 2 -f- 4«a; 3 + a^. 5. a 6 + 6« 5 £ + i5« 4 ^ + 2ort 3 ^ 3 + i5«V + 6ax 5 + x\ Find also the cube root of the last polynomial and of the following : 6. a 6 + 3« 4 .£ 2 + 3« 2 2 4 + x 6 . 7. ^ + | a + | a J + 1 8. « 3 — 3« 2 a; + 3ffi^' 2 — ic 3 . Find the fifth root of 9. a 5 — 5a 4 + 10a 3 — ioa 2 -f- 5a — 1. 279. The square root of numerical binomials of the form a + myb may often be found by separating the rational term into parts, to give it the form x 2 ± zxy + y 2 . 10. Find the square root of 7 + 4V3. Solution. 7 + 4\/3=4 + 4V / 3 + 3 = (2 + V^) 2 - ■"• (7 + 4^/3)' — 2 + y^, Ans. 11. Find the square root of 11 — 6V2. 12. Find the square root of 41 ± 12 V 5. 13. Find the square root of 33 ± 20V2. SIGNS OF ROOTS. 280. A Power of a quantity, by definition, is that quan- tity affected by any exponent whatever; while a Hoot is the quantity affected by a fractional exponent whose numer- ator is 1. Note. — We use the word root in the present discussion to distin- guish such powers, and not, as it is frequently used, for fractional powers in general, 110 EVOLUTION". 281. In considering what sign shall be given to fractional powers, we first find what are the proper signs of roots; and as all fractional powers are integral powers of roots, we may find their signs hy Art. 91. For example, . 282. The sign of a root is found by the general rule, giving the sign of the quantity whose root is taken the expo- nent of the root. (Art. 90.) Thus, ( _ «2)i _ _i a . ( + a 2)i- = + h a , etc. If we have a positive quantity (a), it may he written -f- a, — 2 a, — 4 a, — 6 «, etc.; or, for uniformity, we may write, — ° a, * — 2 a, — 4 a, — 6 a, etc.; (1) using any even power of — to express the positive direction. So, also, if a be negative, it may be written — a, — 3 a, — 5 a, — 7 «, etc. (2) 283. If now we take any root of + a, we may use for + a any of the expressions in (1) ; or if we take a root of — a, we may use any of the expressions in (2). This will give for the n th root of -f- «> — n a n , — n a n , — * a n , — n a n , etc. ; (3) and for the n th root of — a, 11 < 1 — nan, —"a". 5 1 — n a n , 7 1 etc., (4) n being any integral number. * A^ o°, which has no power as a factor, equals 1, ho +, which hae no pov factor of direct iou, may be represented by — " , EVOLUTION. Ill 284. From this we see that we may have different signs for the same root, depending on the sign we use to express the direction of the quantity whose root is taken. These are called the different roots of a quantity, the only difference however being in the sign. 285. To find the number of these roots, examine first the signs of the roots of a positive quantity. The first sign in (3) is _» — _o _ _f_ > Hence, One root, of whatever degree, of a positive quantity is positive. 286. The other exponents of the signs in (3) will be frac- tions until we come to one whose numerator is divisible by n. If n be an even number, we shall have the numerators, giving a root whose sign is n — « = — l = — . Hence, One of the even roots of a positive quantity is negative. If n be odd, the second, and if even, the third integral exponent of — will be — = 2, and we shall have the root whose sign is — 2 = + , which is the same as the first root. Thee of n, in The exponents of the sign after — or 2 will be, for all values 2 4 2 + - , 2 + - , etc. n n 287. As it will not affect the signs to drop from the exponent an even number, they may be written ~ , — , etc., n n showing that the series will repeat itself from that point, and the number of different roots, or more properly of different signs for the root, will be n. 112 EVOLUTION. 288. If we now examine the signs of the roots of a nega- tive quantity as given in (4), we shall find the first integral exponent, when n is an odd number, to be n giving the sign — 1 =—. Hence, One of the odd roots of a negative quantity is negative. 289. When n is an even number, we shall find in (4) no integral exponents of the sign ; but whatever be the value of n, we shall find the exponent 271 4- I I — = 2 + -, 11 n after which we shall get 2 + ^, 2 + ^, 2+ 1 -, etc.; n n 11 and as we may omit the 2, the exponents repeat themselves, and we have as before the number of roots equal to n. Hence, in general, 290. Every quantity has as many roots as there are units in the degree of the root. This may be illustrated as follows : 1. (+ 4)* = (_» 4 )* = (- 4 4)^ = (- 6 4 )", etc., which may be written + 2, — 2, — 2 2, — 3 2, etc., the last two being the same as the first two, and we have only the two roots + 2 and — 2. 2. (_ 4 )i = (_s 4 )i = (_5 4 )i = {-U)\ etc.; or i _l 2, —-2, — * 2, etc.; which give only —^2 and —*2, the others being the same as these. IMAGINARY QUANTITIES. 113 3 . (+ 8)3 = (— 2 8)3 = (— 4 8)^ = (— 6 8)3, which become + 2, — » 2, — i 2, — 2 2, and we get three cube roots of + 8. 4. (_S)i ■ = (- 3 8)i = (- 5 S)i = (- 7 8)^; or, — ia, —2, -^2, —^2 = (— *a), and we have also three cube roots of — 8. In the same way the student may find 4 fourth roots of 16, and 5 fifth roots of 32, etc. 291. We have taken rational quantities for illustration, but the same would of course be true for irrational quantities. We have found that the signs of some of these roots are + and some — , while most of them can only be expressed by the sign — with a fractional exponent. IMAGINARY QUANTITIES. 292. The force of the sign — , with any integral exponent, has already been explained. (Arts. 90, 91.) It now remains to consider what interpretation shall be given it when affected by a fractional exponent. Let AB, Fig. 1, be a line whose length is 2, and whose direction is positive, reckoned from A to B. Then AC, lying in the opposite direction, will = — 2, and these lines will represent the sq. roots of + 4. Since the sign — , in reversing the line AB, c turns it in the direction indicated by the arrow (Art. go), when it has ex- pended J its power, the line will have the direction AD, which will therefore be expressed by —\ 2. In like manner, the -I) power of 4- 2 1U I M A G 1 :N' A It Y (iL'AXTITIES. this sign (— ') brings the line to AE. These lines represent the square roots of — 4. In like manner, all fractional powers of — indicate directions out of the line + and — , which are found by taking such part of a reversal as the exponent of the power indicates. 293. The directions of the cube roots of a positive quantity are represented by the lines AB, AD, and AE, Fig. II. Fig.il. If these lines be taken 2 units in length, they will represent the cube roots of + 8 both in magnitude and direction. So also AF, AC, and AG represent the cube roots of — 8. 294. A Ileal Quantity is one whose sign is either + or — . 295. An Imaginary Quantity is one whose sign is — , with some fractional exponent. • Note. — In the solution of problems, quantities are always considered as lying in a certain line in one of two opposite directions, called positive and negative. (Art. 86, 3 . > Any quantity not in that line is regarded as unreal, and is therefore called imaginary. Such a quantity in a result indicates the introduction of some impossible condition. IMAGINAKY QUANTITIES. 115 296. From (Arts. 281-289) we deduce the following PRINCIPLES. i°. Of the even roots of a positive quantity, tivo are real, one + and the other — . 2 . Of the odd roots of a positive quantity, one only is real, and its sign is + . 3 . Of the odd roots of a negative quantity, one only is real and Us sign is — . 4 . All the even roots of a negative quantity are imaginary. 297. In mathematical computations, the real roots of quantities are used in all cases, when there are such ; hence the only case which necessitates the use of imaginary quantities is that which requires the even root of a negative quantity. For this reason, an imaginary quantity is often defined as the even root of a negative quantity. 298. In Fig. II, if the lines DE and FGT be drawn perpen- dicular to BO, then AJST = +i; MD = (-3)*5 AM=-i; NO =-(-3)*; NF = -M = (- 3)*; ME = - (- 3 )l Note. — The student who understands Trigonometry may verify these values. AN + NF = 1 + V^3; AN + NG = 1 — V^3 ; AM + MD = - 1 + V^3 ; AM + ME = - 1 - V~~2,. For all purposes of mathematical calculation, AN + NF = AF ; AM + MD = AD ; AN + NG = AG ; AM + ME = AE. 116 IMAGINAKY QUANTITIES. 299. llence we have the following equations: — *2 = I + V- 3; (0 _§ 2 = t _ V— 3; ( 2 ) -»2 = -I + V— 3; (3) _# 2 = .— I — V— 3- (4) The first two are the imaginary cube roots of — 8 and the last two of + 8. Notes. — 1. It is evident that travelling over the distance AF from A is equivalent to travelling over AN and NF successively, so far as the result is concerned. 2. But it is not so evident that the second members of these equa- tions may be used for the first members in mathematical computations. The student may, however, verify the statement. For example, 1. Add the 1st and 4th, and we have, = 0. 2. Add the 2d and 3d, and " " o = o. 3. Cube the 1st or 2d, and " " — 8 = — 8. 4. Cube the 3d or 4th, and " " 8 = 8. Observe in adding that — * 2 and — » 2 lie in opposite directions, and being equal in distance, their sum is zero. 300. The forms in the second members of these equations are always used instead of those in the first members. The advantage of this will be readily seen in the fact that the second form has its imaginary part always a square roof, which will lie along a line at right angles to BC, in one of two opposite directions. Such quantities may therefore be added and subtracted like real quantities. Thus, we may add - 1 + V-~3 and — 1 — y — 3 Sum, — 2 which is represented by the line AC But we cannot by any algebraic process reduce — ' 2 and — * 2 to one term. IMAGINARY QUANTITIES. 117 Notes. — i. We may, however, see that by a different method of addition (not algebraic), the sum of — •> 2 and — j 2 is — 2 ; for if we go from A to D, and then from D a distance and direction equal to — * 2, that is, from D to C, we shall reach the same point C to which the line AC or — 2 extends. 2. To find the imaginary roots of a number in this binomial form requires a knowledge of Trigonometry. The student who is not acquainted with the fundamental principles of that subject will not fully understand Arts. 301-305. It will, however, be for his advantage to read them. 301. The process of finding imaginary roots is the same as finding the base and ■perpendicular of a right-angled triangle, when the liypothenuse and angles are given. The liypothenuse is the real root of the quantity, obtained in the nsnal way by evolution, and the angle at the base is found by the sign of the root. 302. Since the sign — represents an angle of 180 , we have for these angles, for the roots of positive quantities, o° 360 2-360° 3-360° n ' n ' n ' n and for the roots of negative quantities, 180° viSo° 5-iSo° 7-180° .. , , , ^ , , etc. (Art. 283.) Note. — Any one of these expressions which represents a multiple of 180 gives a real root. The rest are imaginary. Whatever be the value of n, the angles less than i8o° will correspond to those greater than i8o°. That is, if we have an angle equal to 180° — (3°, there will be another. iSo c -f- (3°. Hence the roots will always be found in pairs of the form a ± V^b. 303. The two roots having these relations to each other are called Conjugate Hoots. The formula for these, by Trigonometry, will be r (cos ± V '— sin 2 0), in which r is the real root of the quantity, and the angle indicated by the sign. etc.; 118 IMAGINARY QUANTITIES. EXAMPLES. i. Find the imaginary values of 32^. Here n = 5, = 72 and 144% and r = By the formula the imaginary roots are 2 (cos 72 ± ^/— sin 2 72"), and 2 (cos 144 ° ± ^ — sin- 144 ). Introducing approximate values for sine and cosine of 72° and 144' we have 2 (.309 ± -V/-.951 2 ), and 2 (— .809 ± yf— .588 s ). 2. Find the imaginary values of (— 32)^. Here «. = 5, r = 2, and = 36" and 108 . Hence the imaginary roots are 2 (cos 36. ± \/— sin- 36°), and 2 (cos 108° ± y/— sin 2 108°). Or, substituting values of sine and cosine of 36" and 108°, 2 (.809 ± -v/^Tsss 2 ), and 2(— .309 ± y/ — .951 2 ). Note. — Hence it appears that the imaginary fifth roo:s of 32 and of — 32 are the same, except in the sign of the real term. 3. Find the imaginary values of (729) 1 ' and ( — 729)*. For the first, n = 6, r = 3, 6 = 6o° and 120°. The values are therefore 3 (cos 6o° ± y/ — sin 2 6o°), and 3 (cos 120 ± y/ — sin- 120 ). Or 3 (£ ± hV^jl and 3 (- 1 ± */v/~3). For the second, n = 6, r = 3, = 30 , 90°, and 150 . The values are 3(iV3 ± i\/ :i 2)' 3(°±\/-*)> and 3(-iV3 ± *A/-2) The second pair of values reduce to ± — * 3 = — * 3 or — * 3. IMAGINARY QUANTITIES. 119 304. From the preceding we have a direct method of find- ing any power of an imaginary expression of the form a ± V— b. Since a is the base and \/b the perpendicular of a right-angled triangle whose liypothenuse, with a proper sign, is equivalent to the binomial imaginary, we may operate by the following rule, in which v represents the liypothenuse and m the proper exponent of — to give the quantity the right direction. Eule. — I. Change the expression to the form — m r. (Art. 305.) II. Raise the resulting monomial to the required power. (Art. 266.) III. Restore the form a ± V— b. (Arts. 302, 303.) 305. To make the change from a ± V — b to — m r \vt have by Trigonometry, r = \/a 2 + b; m = -7T-03 in which 6 =z cos -1 -• 1 80 r The following examples will illustrate the process: 1. Find the cube root of — 4 + 4 V— 3- SOLUTION. m = ——- 180 II. III. — » 2 = 1. r = \/d 1 + b = ^16 + 48 = 8 ; _ coa-' (— I) _ 120° _ 2 - 180° ~ 180° — 3" -4 + 4 \/—3 = - l 8. (- S 8)» = -3.2 2 (cos 40° + y^sin 2 40 ) = 2 (.766 + y / — 643?), Ans. This gives one oikthe cube roots of —4 + 4 /y/— 3. The other twc will be found by using for —3 8, its equal —5 8, or — " 8. (Art. 290.) 2. Find the 5th root of 243 (£ — \ \/— 3). 3. Find one pair of the imaginary values of 64*. 4. Find the cube roots of — 125. 120 RADICALS. CALCULUS OF RADICALS. 306. AH mathematical operations upon radicals are per- formed by the same rules as like operations upon rational quantities. The student need only become familiar with the use of the signs and symbols of the algebraic language, and with the principles involved in the fundamental mathematical operations, and then apply them alike to rational and radical, to real and imaginary quantities. Any attempt to make a difference in their application to real and imaginary quantities, is liable to result in confusion. One principle only in addition to those already given requires attention, viz., Quantities wl/ose signs are neither the same nor oppo- site, cannot be united in one term. Thus, 2>a aud — = 5a are neither the same nor opposite in direction, and therefore cannot be united in the same term. They can be added only by writing them with their signs ; thus, 3« — * yi, or, as it is frequently written, 3a + 5«V — r > in which \/— 1 has no force except as & factor of direction, 1 having no power as a factor. Those, however, who prefer this form, can use -y/— 1 for y'- and — \/~ 1 for — \/- or or 1 It is wholly immaterial which form is used, if the student does not allow the presence of the 1 to give him the impres- sion that something besides a direct ire sign is intended. 307. For Addition and Subtraction of Radicals we have the following Rule. — Change the signs of terms to be subtracted, a id unite similar real terms in one; also similar imaginary terms, and connect dissimilar terms by their signs. CALCULUS OF RADICALS. 121 Note. — Changing signs means reversing the direction by applying a minus sign to the term. Thus, changing the sign —I makes it — i ; that te, + y/~ becomes — y/ — ■ i. Simplify ab — V— a + 2\/ — a + sab. Ans. ^ab + V — a. 2. Subtract 2a + 3V — b from 7a — 5 V — b. Ajis. $a — 8a/— #• 3. Subtract — *8 from — * 12. Changing sign of subtrahend, it becomes — * 8 and therefore lies in the same direction as — ^ 12. Adding, after changing the sign, gives — a 20, J./2.S. 308. For Multiplication of Radicals we have the following Eule. — I. ^o tfZie product of the numerical factors annex the literal factors each with an exponent equal to the sum of its exponents in the several factors. (Art. 127.) II. Give the product the sign — , with an exponent found by adding the exponents of — in the several factors, and subtracting from the turn the greatest even number. (Art. 127.) 4. Multiply a + y/ — b by a — y/ —b. OPERATION. a + y/— b a — V '— 6 a 2 + ay/— 6 — a\/— b + b a 2 + b, Ans Note. — a- is positive, because every power of + is + , or because in both factors the sign — has zero for an exponent. The product of a x (4- y/—b) has the sign — - or + y/ — , the sum of the exponents of — being \. Also, a x (— \/a~x — 4V — ax + 2\/ — ax. 8. Vf< 2 — x* 4- 2 Va 2 — x* - 5 (fl 2 — a*)* + 4 (tf 2 — * 2 )*- * This agrees with the treneral use of + and — , for if a positive exp i« nt indicates revolution in one direction, a negativt expont nt Bhould indicate revolution in the oppJb = -\/ab ; = V b V— J ; \/a V '— 1 \/a - _ _ — = -*-=. Ans. y/b -\/— 1 ^/b 2. Divide v^ by V — J. Solution. >y/a = — -y/a x — 1 = — \/a \/— 1 <\/— 1 ; y/— b = \/b \/— 1; — 's/a -y/— 1 \/— 1 _ — \/a <\/— _£ _ ^/b *J — 1 yft = -| Ans. 3. Divide 9 v 7 — 8 by 3 v 7 — 4- 4. Divide 4 V — 3 by 2 y — 9. 5. Divide 4V — « 2 by 2a V — 2. 6. Divide 4 -f- v 7 — 2 by 2 — v — 2. 126 REDUCTION. REDUCTION OF RADICALS. 314. Radical expressions may frequently be simplified: ist. By removing or introducing rational factors. 2d. By reducing radicals to like indices. 3d. By rationalizing one term of a radical fraction. 315. To Remove Rational Factors. Rule. — Separate the quantity under the radical sign into rational and irrational factors x and taking the required root of the rational factors, write them outside the radical sign. 1. Given (2^a 3 bx 2 )^ + (gaPbx 2 )* — (i6a z bx 2 )?, to simplify the expression. OPERATION. (25a 3 &E 2 )* = (25« 2 a; 9 . atifi = {2$a?x°f (ab)* = sax{ab)^ ; (ga s bx^ = (g^x 2 ) 1 (ab)i = ?,ax {abf (i6a 3 bx-)-- = 4ax(ab)?. .'. (25rt 3 6x 2 )' + (ga^bx 2 )* - (i6a s bxrf = 5ax(ab)* + 3ax(ab)^ — ^ax{db)^ =■ <\ax(ab)\ Ans. 2. Simplify (a z x 2 — a 2 x^. OPERATION. (a 8 x 2 — a 2 z 3 )* = (a 2 :c 2 / 3 (a — x)* = ax (a — x)^, Ans. 3. Simplify \/8i. OPEKATION. ^/8i = ^27 x 3 = ^27 ^3 ; .-. ^81 = 3 y3, Ans. 4. Simplify (aWx 2 — a 5 tfx)*. OPERATION. (aW-'-rt^x)' ~ (a z b 3 )'(a i b-x i -a'-b\r)' - ab(a i b°-x' i -a°b i x) } >, Ans. RADICALS. 127 316. To Place a Rational Factor under a Radical Sign or Fractional Exponent. Rule. — Involve the rational factor to a power indicated by the reciprocal of the fractional exponent, and combine it with the radical factors, if there be any. 5. Eeduce a 2 bc to a radical of the second degree. OPERATION. a?bc - (tfbcf = [(tfbcf]* = (aW)", Ans. In this case there were no radical factors with which to combine the rational factors. 6. Eeduce ab (ax — z 2 )? entirely to a radical form. OPERATION. ab = (a' 2 6 2 )l .-. ab (ax — a; 2 )* = [a 2 6 2 [ax — x")]-- = (aWx — a 2 &V)«, Ans. 317. To Reduce Radicals to Like Indices. Rule. — Eeduce their exponents to a common denominator. 7. Reduce a* xa§ x eft to the simplest form. OPERATION. a* = a*, ffl* = a ", a" = a*. .-. a*a*a* = a 1 *' = a 3 . 318. To Rationalize one Term of a Fraction. Rule. — Multiply both numerator and denominator by that factor which will render the required term rational. 8. Rationalize the denominator of vf. OPERATION. V* = Vt - V* x 2 - 3 V 2 > Am - 128 RADICALS. 9. Kationalize the denominator of V" + Operation. — Multiply both terms of the fraction by ^a—x, which will give a — x a — x (a + g)* + {a - x)$ < (a + x)? — (a — x)* 10. nationalize the denominator of OPERATION. Multiply by the numerator, and we have [(a + ,r)* -t- (a — .?■) ']- _ a + x + 2 (a + x)* (a — xfi + a . — x (a + x) — {a, — x) 2X a + (a- — x-)' , Ans. EXAMPLES. Simplify the following: 1. a 56 — ] a ■ — 3 (c — b) + 2 [c — a — 2b — c »(]'• 2. 1 a/« 4 — b A 4a \ 4 ' a 2 _ tf ' T ' I + X f-51 1 — S ' I — X I + \/ 1 -f- X I — \A + # /a 2 .r + 2rt.T 2 + a?U /a 2 r — 2ax~ + a^\i \ a 2 — 2Cix -\- x* / \ a 2 + 2fu - + .t 2 / 5- (5 + 2V3) (5 — 2V3). V2 + I2v6 — 4V10 6. 4Vs RADICALS. 129 4 — V— 2 — 2^/2 (i ± v~3) 3 + (- 1 ± V^y, a a + a + Va? — b 2 a— Va? — & (a + x )i + (a — z)* (a + x)^ — {a — .r)£ (a + &)* — (a — a)i (a + a)* + (a — x)* — sV^l + ioV—3 + 5^—3, IOV — 2 — sV^o 5V— 2 2-\/ — 2A/^ " X — — X 3V — b Va JZT. y io 5 x \fy— 3-^3^— 3°- 13- 4V — 2 X —3' 2\/x 14. 15- 2(1 — y^) 1 2 (i + V^)- (i + ^) 2 -(i-^) 2 h (I+ *)* _ (1 -*)*}■--. r + n (I +^-(i-#p- 2 r 1 + 16. T 7- (i + .T 2 )* 2 Vz + Vi + x 2 VaW — a 2 x 3 + V« 5 ^ 4 — a 4 ^ 5 + V(a — aO*. 130 1 8. 19. RADICALS. x + a x — a x — a x + a x — a x + a x + a x — a x — a x -f a (cfi — x 5 — s^x + sax* + ioa 3 x 2 — iocfa?ft. (a 3 — $a 2 b — i2abx + 6a 2 x + 120a; 2 + yib 2 + S.r 3 — 12^ + 6«z— &»)*. ac 22. 23- 24. 25. 26. V— «c x — V^c X — ' a {a + 5)- 1 + fl (a — &) -1 a (a — J)" 1 — 6 (a + &)-*' 121 ° 1 1 S 1 ^ 11 Av X a^b^tr' x a m i"c* x 0"&V. W+w m+n m+n m+n m a n b m c mn x a m b n c". a + V— ^ « — V— x X — V— » be. A, 3,1 1 7 1 Rationalize the denominators of a x 2 2 7- — • 3°- 3 1 - 29 I Vb « + y- b 3 32. 1 X* + r X 2 1 — y* 1 33- rr 3 + y* 34- —- $. — a* x$ — y? Note. — For the last nine examples see Art. 155, and examples 26-30, 35- — i- a~- + y~~* page 123. RADICALS. 131 RADICAL EQUATIONS. 319." Equations containing Radicals may often be reduced to simple equations by the following Rule. — Involve both members to a power indicated by the reciprocal of the radical exponent. Notes. — i. Before involving the quantities, it is generally best to clear of fractions, and transpose the terms, so that one member of the equation shall contain but one radical term. 2. In reducing such equations, the student should remember that any of the methods of reduction or rationalization of radicals may be employed, in accordance with Axiom i. (Art. 38.) 1. Given (12 + xp = 2 -+- Vx, to find x. OPERATION. Squaring, 12 + ;r = 4 + 4\/% + x. Transposing and uniting terms, 4yx = ^- • Dividing by 4, and squaring, x = 4, Ans. 2. (2X + 3)^04 = 7. 3. V'X — 16 = 8 — Vx. 4. V4« + x = 2 (b +. a;) 5 — Vx. Vx + 2ft Vx -f- 4ft Vx + b Vx + 3/y 6 3^—i __ t [ V3X — 1 V3X + 1 2 -V nix — V m Vx -f m Vex — Vc Vx + c yC 8. (5 + x)* + ^5^' = VZ. 9 . V^ + -{t* _ V x - X \ - 3 ( _ lp j . CHAPTER XI. EQUATIONS OF THE SECOND DEGREE. 320. Equations of the Second Degree are called Quadratics, and may all be reduced to the general form Ax* + Bx + = o, (i) in which A is positive, and B and C are either positive or negative. 321. y! cannot be o, for the first term would then disap- pear, and the equation be of the first degree. If B = o, the second term disappears, but the equation is still of the second degree. If C = o, the third term disappears, and the equation, though still of the second degree, may be changed to one of the first degree by simply dividing by x. 322. Hence we have but two cases of equations of the second degree, represented by Ax 2 + C = o, which is called the incomplete equation of the second degree, or the pure quadratic; and Ax* + Bx + = o, called the complete equation of the second degree or the affected quadratic. THE SECOND DEGREE. 133 323. A Complete liquation is one in which the series of powers of the unknown quantity is complete, from the highest to the lowest; as, ax z — bx 2 + ex + d = o, in which the exponents of the powers of x form an unbroken series, from 3 to o. 324. An Incomplete Equation is one in which one or more terms of this series are wauting ; as, ax 3 + bx + c = o, in which the term containing x 2 is wanting. 325. Dividing equation (i) by A, we have „ B C , , B C Substituting for -j and -r, 2a and b, x* + tax + b = o, (3) 1 a form to which every quadratic may be reduced, in which a and b represent any quantities whatever, integral or fractional, positive or negative, and a may be zero. 326. To reduce equation (3), if b = a 2 , we may take the square root of both members (Art. 156, Cor. 1), which will give x + a = ±0, and x = — a ± o. If # ^ « 2 , we may always add to b such a quantity as will make it equal to a 2 . This quantity will be a 2 — b. (Art. 1 1 2, 4 .) But to preserve the equality, the same quantity must be added to both members (Art. 38, Ax. 1), which will give x 2 -f- 2ax -\- a 2 =z a 2 — b. (4) 134 EQUATIONS OF 327. This is called Completing the Square, by which is meant Making /he first member a Perfect Square. 328. It will be observed that the addition of a? — b to both members is the same as transposing b and adding a 2 to both members. Hence, To make the first member of a quadratic a perfect square, we have this Rule. — Transpose the absolute term, and add to both members the square of half the coefficient of x. 329. Taking the square root of equation (4), x 4- a = ± V a % — b, and x = — a ± V« 2 — 0. (A) If a = o, the equation is incomplete, and (A) becomes x= ± V— 0. (B) As equation (A) is general, applying to all equations of the second degree, complete and incomplete, aits discussion will furnish all the principles relating to the roots of such equa- tions. These may be enumerated as follows : i°. The equation has two roofs (Art. 232), — a 4- Va~ — b and — a — \/a? — b. 2 . The sum if the roofs equals — 2a. or the coefficient of j first power with its sign changed. Hence, 3 . The sign of the numerically greater root will be unlike the sign of the second term or 2/u: (Art. 106.) 4 Q . The product of the roots equals b, or the coefficient of x zero power. Hence, 5°. The roots will hare lil-e siims when b is positive, and ■unlike signs when b is negative. (Art. 130.) 6°. The rooh will be equal when b = a 2 , and numerically equal with opposite sigi n. Then both values of x will be positive, one less and one greater than d ; that is, one point will be between A and B, nearer to B, and the other beyond B. This is evidently as it should be, since the point sought must be nearer the less light. In like manner, if m < n, the point between A and B comes nearer to A, and the other point without is on the side of A. 144 EQUATIONS OF 2d. Let m = n. This gives the two values x — Id and x = co . The first of these is evidently correct, since if the lights are equal, the point of equal illumination ought to be equidistant from each. The second value ( co ) does not so readily appear true ; but when we consider that ca ± d = co (Art. 412), we find this point also equidistant from the lights. Another question might arise here, whether the co be + or — , and if + why not also — . This is answered by observing that while the lights differ in intensity, the point without is on the side of the less light, and as the less light becomes more nearly equal to the greater, this point recedes until, when the difference between the lights is infini- tesimal, its distance is infinite, but still on the side of the light which is infinitesimally less. 3d. Let d = o, and in ^ n. Then both values of x are o. The lights are then at the same point, and no place except this point will be equally illuminated. At this point (by the theory that gives u — - - 1, if X = o, the illumination is infinite. This supposes the light to come from a mathematical point, which has no dimensions, a thing which never occurs, since the source of light is always some portion of matter having dimensions Practically, therefore, no such conditions can be fulfilled; but it furnishes a good illustration of the general character of mathematical analysis, which does not stop with the possibilities of physical conditions, but gives the results which would follow from given laws, if the physical conditions could be realized. 4th. Let n be negative. Then the values of x are imaginary, showing that with such condi- tions there is no point of equal illumination. In (3d), if the conditions could be fulfilled, the result would be real, and the problem admits of a definite answer. In (4th), even if the conditions could be fulfilled, there would be no such point as the one sought, and the result is therefore imaginary. To fulfill the conditions of n being negative, it would require that B should diffuse in all direc- tions some light-absorbing vapor, by which the light from A should be partially neutralized, the law of its diffusion being the same as that of the diffusion of light. But even thin it is evident there would be no point equally illuminated by A and B. If m and n are both negative, the result becomes real. THE SECOND DEGREE. 145 5th. Let d = o and m = n. Then x = o and -• This last value is indeterminate (Art. 210), and o represents any quantity whatever, which is evidently a correct result, since equal lights situated at the same point should illuminate equally all points at whatever distance. SIMULTANEOUS EQUATIONS OF THE SECOND DEGREE. 342. The general equation of the second degree between two unknown quantities is ay 2 + bxy + ex 2 + dy + ex + / = o. This equation, combined for elimination with another of like form, would produce an equation of the fourth degree, but combined with the general equation of the first degree, dy -f b'x + c' = o, gives an equation of the second degree. Hence, 343. Simultaneous equations with hvo unknoivn quan- tities, one of the second and one of the first degree, can always be reduced as quadratics. 344. There are also certain classes of simultaneous equa- tions with two unknown quantities which may be reduced as quadratics, when both are of the second degree. These are, 1st. Homogeneous Equations. 2d. Symmetrical Equations. 345. Hoinof/c neons Equations contain the same number of unknown factors in each term, except the absolute term. Such equations can always be reduced by substituting for one of the unknown quantities the product of the other by a new unknown factor. 146 EQUATIONS OF 346. Symmetrical Equations have the unknown quantities similarly involved ; as, x 2 + xy + y 2 = 19. These can usually be reduced by substituting for x and y respectively the sum and difference of two other unknown quantities. 347. The following is an example of a quadratic and a simple equation: j x 2 + y 2 = 20 (x — y = 2 (0 (2) From (2), x = 2 + y (3) Substituting in (1), (2 + yf + y' 2 = 20 (4) Or, y 2 + 2y = 8 y = — 1 ± \/s + 1 = — r ± 3 y = 2 or — 4, ) Substituting in (2), x = 4 or — 2. \ Arts. 348. The following illustrates the reduction of homogeneous equations : -i X 2 + xy = 10 (1) wy + y 1 = x 5 (*) Put x = zy Then z-y* + zy- = 10 (3) Zy* + y* = 15 ,(4) From (3), y Z- + 2 (5) From (4), « 2 = " 2 + I (6) Equating, IO 15 2 2 + 2 2+1 Hence, 3 = % or — 1 Substituting | for 2 in (5) or (6), we have, y - ± 3 rr = ± 2 Substituting — 1 for 2 in (5) or (6) y = ± 00 a; = =? <» we hav e THE SECOND DEGREE. 147 Since the combination of these equations of the second degree give an equation of the fourth degree, there should be four roots. Two of them in this case are oo . Substituting + oo for x, and — oo for y, we have, CO 2 — OO 2 = IO OO 2 — OO- = 15 results which are consistent, as will be seen, Art. 412. 349. As an example of Symmetrical Equations, we have, ( z 8 + V 1 = 13 (1) 3 ' 1 xy = 6 (2) Multiplying (2) by 2, and adding it to and subtracting it from (1), X 2 + 2xy + y 2 = 25 x 2 — 2xy + if = 1 x + y = ± 5 x - y = ± 1 Adding, 2x = ± 6 Subtracting, 2y = ± 4 x = ± 3 ) , \ Ans. y — ± 2 S 350. Symmetrical Equations with two unknown quantities may frequently be reduced, when one is of the first degree and the other of a degree higher than the second. 3 4- ■ ; X + y = 8 (I) . x 9 + y z = 152 (») Let X = « + V y — u — v Then x + y = 211 — 8 And u = A Substituting in (2), (4 + vf + (4 — -y) 3 = 152 Developing and reducing, 241; 2 = 24 v = ± 1 a; = M + « = 4±i = 5 or 3 y = u — !) = 4ti = 3 or 5 148 EQUATIONS OF Ileduce the following by these or other methods, as shall be found feasible : 5. x — y — 4, 17. x 3 y 2 + x 2 y 3 = — 4, X s — y 3 = 124. x 2 y + xy 2 = 2. 6. x + y = 9, 18. Vz + V# = 6, z 2 + */ 2 = 53- x + y = 20. 7. x + */ = a, i 9 . Vx + Vy = 4, & + y* = h. x % + yl = 28 . 8. a: + ?/ =5, 20. £B — # = 1, ^ + f = 97- a 4 — y* = 15. 9. 23 2 +■ xy =14, 21. x 2 + 3.r# — y 2 = 36, 2?/ 2 — ## = 12. 3^ + 21J = I 6. 10. II. x 3 y — y = 21, a; 2 ?/ — .ry = 6. z 2 +2Z# + iy + 3.T=73, 22. *4_y y + x x + xy + y = 14- 2/ 2 + 3!/ + « =44- 2 3- s 2 + xy = 40, •■ 12. 13- #2y2 _|_ ^^ = 12, 2- + y = I. x —y — 2, x 3 — y 3 = 56. 24. 2 5- 3»y— 2 «/ 2 = 27. x 2 + 2.r?/ 4- y 2 = 81, a; 2 — 22/y + ?/ 2 = 9. x 2 — gy 2 = 16, # 3 — ?/ 2 = 24. 14. a- 2 - if = 8, 26. a; 2 (x — y) = 4, T 27. x 2 (2x + $y) = 28. xy 2 + 2$ = 24, v^ 2 + y 2 = ~' 15. x + y x — y 10 x — y x + y — 3 ' 28. xj/ 2 + x =40. x 2 + y 2 =13, a 2 + y 2 = 5. zxy — x — y — 7. 16. 1 1 1 x y~ a' 29. x + y __ 7 x — y ' 3' x 2 + y 2 = Z>. x—if= 1. 3°- a; + y 2X a? — x 2 y 3. V z + y y 3 - x 2 y ~ 5' y 2 - -x 2 = 5- THE SECOND DEGREE. 149 PROBLEMS. i. Divide a into two parts, whose product shall equal b. Let x and y represent the parts. Then, by the conditions, x + y - a (i) xy = b (2) From (1), y — a — x Substituting in (2), (a — x) x = b x* — ax + b = o x = \a± y /a? The two parts are ^a + JU b A 2 . and ia — y b, a? from which it appears that if b> — , the values are imaginary. Hence, Cor. — The product of two quantities cannot be greater than the square of half their sum. Or, The jjroduct of the tioo parts of a given quantity is greatest when those parts are equal. 2. Find three numbers, the difference of whose differences is 8, their sum is 41, and the sum of their squares 699. 3. Find three numbers, the difference of whose differences is 5, their sum is 44, and their continued product is 1950. 4. The fore-wheel of a carriage makes 6 revolutions more than the hind-wheel in going 120 yards, but if the circumfer- ence of each wheel be increased one yard, it will make 4 revolutions more than the hind-wheel in the same distance. "What is the circumference of each wheel ? 5. What number being divided by the product of its two digits gives the quotient 2, and if 27 be added to the number, the digits will be inverted ? 150 EQUATIONS OF THE SECOND DEGREE. 6. The difference between the hypothenuse and base of a right-angled triangle is 6, and the difference between the hypothenuse and perpendicular is 3. What are the sides ? 7. A and B put out at interest different sums amounting to $200. B's rate of interest was \ c / more than A's. At the end of 5 years, B's accumulated simple interest was $4 less than the double of A's. At the end of 10 years, A's principal and interest was f of B's. What was each sum and rate per cent? 8. Two partners, A and B, gained $140 in trade. A's money was 3 months in trade, and his gain was £60 less than his stock, and B's money, which was $50 more than A's, was in trade 5 months. What was each man's stock? 9. Find two numbers, the difference of whose squares is m % , and which being multiplied respectively by a and b, the difference of the products is n 2 . 10. Divide a and b each into two parts, such that the product of one part of a by one part of b shall be m, and the product of the remaining parts n. 11. What is the side of a cube which contains as many units of volume as there are linear units in its diagonal ? 12. Find two numbers whose sum, product, and sum of their squares shall be equal to each other. 13. Find two numbers whose sum, product, and difference of their squares are equal to each other. 14. Find two numbers whose product equals the difference of their squares, and the sum of their squares equals the difference of their cubes. 15. The product of the sum and difference of two numbers is 8, and the product of the sum of their squares and the difference of their squares is 80. What are the numbers ? 16. A and B bought 600 acres of laud for $600, each pay- ing $300. In dividing, A took the best land and paid 75 cents an acre more than B. How T much land did each get and at Avhat price ? CHAPTER XII. INEOUATIONS. 351. An Inequation is an expression of inequality between two quantities ; as, a > b (read, " a is greater than #") ; x < y (read, " x is less than y "). The quantity on the left of the sign is called the First Member, and the one on the right the Second Member of the inequation. 352. Algebraically, a negative quantity is said to be less than zero (Art. 104) ; and of two negative quantities, that which is numerically greater is algebraically less. Therefore, If a — b > o, a > b, and if a — b < o, a < b. 353. In the transformation of inequations, it is necessary to observe when the sign of inequality will be reversed. When this sign is reversed, the tendency of the inequation is said to be changed. Thus, a > b and c> d, are inequations of the same tendency, and and a > b and c < d are of opposite tendency. 354. The tendency of an inequation is not changed, 1. By any like operation upon both members, except changing their signs. 2. By adding or multiplying by the corresponding members of an inequation of the same tendency: Provided that in multiplying, the signs of the members be not changed. 152 INEQUATIONS. 3. By subtracting or dividing by the corresponding members of an inequation of opposite tendency : Provided that in dividing, the signs of the members be not changed. 355. The tendency of an inequation is changed, By changing the signs of both members. (Art. 104.) For, 2 < 3, but — 2 > — 3. 356. The tendency of an inequation becomes doubtful, 1. By subtracting or dividing by the corresponding mem- bers of an inequation of the same tendency. 2. By adding or multiplying by the corresponding members of an inequation of opposite tendency. For it is evident that if a < b and e d Thus, (1.) 5 < 15 and 3<6. Subtracting, 2 < 9. Dividing, 5 <--- 5 3 ^- 1- (2.) 5 < 15 and 2 < 12 Subtracting, 3 = 3- (3) 5 < 15 and i<3- Dividing, 5 = 5- (4.) 5 < 6 and K 4 . Dividing, 5>l- Subtracting, 4>2. 357. The Red}(ction of an Inequation consists in so transforming it that the unknown quantity may stand alone as one member, while the other member contains only known quantities, the value of which is a limit to the value of the unknown quantity in one direction. If two inequations containing the same unknown quantity can be reduced with opposite tendencies, limits in both directions will be found. INEQUATIONS. 153 EXAMPLES. X X i. Given — (- 3 > ^ 4- 2, to find a limit for x. 4 o Solution. — Clearing of fractions, 6a; + 72 > 421 + 48. Transposing, 2X > — 24. Dividing by 2, a; > — 12, Ans. 2. Given ■} _ ]■ to find limits for z. 2£ + 4 > 16 — 2X Solution. — Transposing and uniting terms, 2a; < 10 and 4.?' > 12. x < 5 and x > 3, ^4ws. 3. Given 32; + 72; — 30 > 10, to find a limit for x. 4. Given x 4- |# — |x > 4, to find a limit for x. ax , _ a 2 5. Given 6. Given 7. Given ^2 4- bx — a# > — 5 5 bx b 2 — — ax 4- ab < — I 7 7 to find limits for x. $x — 4< x + 6) to find an integral value 53 + 7 > 3 X + l 3) of &• £(a; + 2) + !£ < i (^—4)+3 ) to find an inte- ^ (a; 4- 2 ) 4- J 2; > - 2 - (z 4- 1 ) + i j gral val ue of x. 8. A certain integral number, doubled and diminisbed by 7, is greater tban 29 ; and 3 times the number diminished by 5 is less than double the number increased by 16. What is the number ? 9. A boy sold a number of apples, such that triple the number increased by 2 exceeds double tbe number increased by 61 ; and 5 times the number diminished by 70 is less than 4 times the number diminished by 9. How many did he sell ? 10. The sum of two whole numbers is 25. If the greater be divided by the less, the quotient will be greater than § ; and if the less be divided by the greater, the quotient will be greater than |. What are the numbers ? CHAPTER XIII. RATIO AND PROPORTION. 358. A Ratio is the quotient arising from the division of one quantity by another. The sign of division commonly used to express ratio is the colon (:), as a : b, which means the ratio of a to b. But a -=- h and 7 express the same thing. 359. A Proportion is an equality of ratios, or an a c equation each of whose members is a ratio ; as ■=■ = -, or b a a : b = c : d. Notes. — i. The double colon (: :) is frequently used as the sign of equality in a proportion, but without good reason. 2. It is also questionable whether it were not better to express ratio in ike form of a fraction, instead of using a special sign of division. 360. The First Term of a ratio is called the Ante- cedent; the Second Term the Consequent. 361. The First and. Last Terms of a proportion are called the Extremes ; the Second and Third the Means. 362. When the same quantity is used for both ///ran*, it is called a Mean Proportional between the other two: and the last term a Third Proportional to the other two. In this case the three quantities are said to be in Continued Proportion ; as, a : b = b : c. 363. A Compound Ratio is the product of two or a c in more ratios ; as, T x -, x — • 7 o a n RATIO AND PROPORTION. 155 364. A Compound Proportion is one in which there is a compound ratio. Theorem I. 365. In any proportion, the product of the extremes equals the product of the means. Demonstration. — Let a :b = c : d, a c 0r r. = *■ b d Clearing of fractions, ad = be. Cor. i. — If three terms are in continued proportion, the square of the mean is equal to the product of the extremes. For, if a : b = b : c, .'. b 2 = ac. Cor. 2. — A mean proportional between tivo quantities is the square root of their product. For, if b 2 = ac, .'. b — ^/ac. Cor. 3. — Tlie product of the extremes divided by one mean equals the other mean, and the product of the means divided by one extreme equals the other extreme. For, if ad = be, .'. a — — , and d = — . d a ad ad Also = — , and c = — . c b Theorem II. 366. If the product of two quantities be equal to the pro- duct of two other quantities, the factors of either product may be made the means with the factors of the other product for the extremes of a proportion. Dem. — Let ad = be ; Dividing by bd, - = - ; Or, a : & = c : (?. 156 RATIO AND PROPORTION". Cor. i. — If the factors of one product are the same, that factor is a mean proportional between the other tico. Cor. 2. — Tli e means or extremes may change places ; for the order in which the factors arc t alien is not material. If ■ a :b = c : d, Then a : c = b : d. Tliis change is called Alternation, Theorem III. 367. A proportion will remain true if loth of its ratios be inverted. Dem. — If two quantities are equal, their reciprocals will be equal. That is, if % = e -, o a Then * = * a c This is called proportion by Inversion, Theorem IV. 368. The sum or difference of the terms of the first ratio is to either the antecedent or consequent of that ratio, as the sum or difference of the terms of the second ratio is to the antecedent or consequent of the second ratio. Dem.- -Let a : b = c : d, Or, a c b~d' Then a c b ±1 = d ±X ' Or, a ±b c ± d ~b~ ~d~' Again, b _d a ~~ c ' b d I ± - = i ± - a c RATIO AND PROPORTION. 157 a ± b c ± d a c Hence, a ±b :b = c ± d d; And a ±b : a — c ± d c. This is called proportion by Composition or Divi- sion ; the former when the sum, the latter when the difference is used. Cor. i. — If four quantities are in proportion, the sum of the first and second is to the sum of the third and fourth as the difference of the first and second is to the difference of the third and fourth. For by Alternation, a + b : c + d = a : c, And a — b : c — d — a : c. But a : c = b : d; Therefore, a + b: c + d = a — b : c — d; Also a + b : a — b = c + d : c — d; hence Cor. 2. — The sum of the first and second is to their differ- ence as the siim of the third and fourth is to their difference. Cor. 3. — In any number of equal ratios, the sum of the antecedents is to the sum of the consequents as any one antecedent is to its consequent. For if - = - = i = r; b d f r ' a + c + e + &c. a c ■ • iT7-^ * i — = r = r = &c. (Art. 200.) b + d + J + &c. b v ' 369. These changes may all be expressed as follows : If four quantities are in proportion, they will be in proportion by Alternation, Inversion, Composi- tion, or Division. 158 RATIO AND PBOPDRTION, Theorem V. 370. If four quantities are in proportion, the proportion will he true if both antecedents, both consequents, both terms of either ratio, or ail the terms be multiplied by the same quantity. Let the student prove each of these by the properties of ratios. Theorem VI. 371. If both antecedents or both consequents be increased or diminished by adding or subtracting quantities having the same ratio as the antecedents or consequents, the results will be in proportion with the antecedents or consequents, or with each other. Dem. —Let a : b = c : d ; Then a c - ± m = -±m; b d Or a ±mb c ± md b d Also b d - ± n = - ±n; a c Or b ± na d± nc II c •"• a ± a ib : e ± md = l ~ na Theorem VII. d ±nc. 372. A proportion is not destroyed by affecting each term by the same integral or fractional exponent. Let the student furnish the proof. Note. — The ratio of the squares of two quantities is called the Duplicate ll<(tio, and the ratio of the cubes the Triplicate llatio. Also the ratio of the square roots \b called the Sub-duplicate, and of cube roots the Sub-triplicate ratio. RATIO AND PROPOETION. 159 Theorem VIII. 373. Tlie products of the corresponding terms of any num- ber of proportions are proportional. This is only the multiplication of several equations, member by member, As, a _ c b ~ d' m _ x _ n _ y' am _ ex bn ~ dy 374. When two quantities have the same ratio as the reciprocals of two other quantities, they are said to be Reciprocally Proportional ; as, a : = - : -=• c d This may be written a : b — d : e, in which the terms of the second ratio are inverted ; hence they are also said to be inversely proportional. This has no meaning unless a, b, c, and d are so related that c in some way belongs to a, and d to b. Otherwise the order d : c would be no more an inverted order for the second ratio than c : d. ILLUSTRATION. Suppose two men travel at different rates the same dis- tance. Then the times of travelling would he different. Let r and r' be the rates, t and f the corresponding times. We shall then have r:r' = l^ = f:t. The terms of the second ratio must here be taken in inverse order, or the reciprocals must be used. If they travel at the same rate but different times and distances, we have, putting d and d' for distances, t :t' - d: d'. In this case tlie quantities are said to be Directly Proportion &l ; in the former they are Reciprocally or Inversely Proportional, ICO RATIO AXD PROPORTION". PROBLEMS. i. Find a third proportional to 25 and 50. 2. The last three terms of a proportion are 36, 24, and 16. What is the first term ? 3. Find a mean proportional between 5 and 45 ? 4. If a men working b hours a day can complete a certain work in c days, in how many days can a men working V hours a day do it ? 5. Two wagons with their loads have their weights in the ratio of 4 to 5. Parts of their loads in the proportion of 6 to 7 being removed, they weigh in the ratio of 2 to 3, and the sum of their weights is then 10 tons. What were the weights at first ? 6. A starts to travel from Cto D, and 3 hours afterwards B starts from D towards C, travelling 2 miles an hour more than A. When they meet the distances they have travelled are in the ratio of 13 to 15. If A had travelled 5 hours less and B had gone 2 miles an hour faster, they would have been in the ratio of 2 to 3. How many miles did each go, and how long did each travel before they met ? 7. Find 3 numbers such that if 6 be added to the first and second the sums will be in the ratio 2 to 3, and if 5 be added to the first and third the sums will be in the ratio of 7 to 11, but if 36 be subtracted from the second and third, the remain- ders will be as 6 to 7. 8. Find the two numbers whose sum is to the less as 5 to 2. and whose difference multiplied by the difference of their squares is 135. 9. A certain number has 3 digits. The first is to the third as 16 to 6, the third to the second as 1 to 2, and the sum of the digits is 17. What is the number? 10. Find three numbers whose sum is to the first as 6 to 1, to the second as 3 to 1, and to the third as 2 to 1. 11. Find two numbers the ratio of whose difference to their sum is m, and the ratio of the difference of their squares to the sum of their squares is n. VARIATION. 161 VARIATION. 375. The 'Relations of Quantities are sometimes expressed by saying that one varies directly or inversely as the other, or as the square or cube, or some other function of the other. 376. The Sign ,of Variation ( oc ) is used to express these relations ; as, x oc y, which is read " x varies as y," and means that x and y are such functions of each other that their ratio remains the same, and that whatever changes may take place in the quantities themselves, the increase or decrease of one must always be proportional to the increase or decrease of the other. 377. The same thing may be expressed in other ways. Let m represent a constant quantity ; that is, a quantity which in the same discussion does not change its value, and we may write x oc y ; x = my ; or - = m, each of which expresses the same relation. 378. If x t and x 2 represent two different values of x, and y x and y^ the corresponding values of y, then the same thing is also expressed by the proportion x x x 2 = = l/i y*> OT Xi _ x$ - I 1 . y* 379. If X oc i y then x - = m i x - y = m ~y or * x i ^y = m, or xy = m. In this case x varies reciprocally as y, while in x oc y, x is said to vary directly as y. 162 VARIATION. 380. From x oc - we have the proportion ii Xi '. x% = : 5 2/1 y* in which x and y are said to be reciprocally proportional, or x x :x. 2 = y 2 : ?/, ; in which they are inversely proportional. (Art. 374.) EXERCISES. 381. Form the equations and proportions which are implied in the following : 1. x oc y 2 . x = my 3 ; — = m ; x x : x a = y x 9 : y., 2 , Ans. 2. x oc — 2 - 4. x oc y 2 + ^ 3 . T 3. x oc tf + y. 5. x oc 2/ + r 6. If a; oc y, and a; = 3 when y = 5, what is the value of y when k = 15 '? When a; = 7 ? What of x when 3/ = 20 ? When y = 8 ? 7. If a; oc ?/ 2 , and a; = 1 when y = 5, what is the value of a; when _?/ = 7 ? 3? 2? o? 8. If z oc mi- + y, and 2 = 3 when a; = 1 and y = 2, and z = 5 when a; = 2 and y = 3? what is the value of m ? 9. If x 2 oc y 3 , and a; = 2 when y = 3, what is the value of y in terms of x ? 10. If y oc v 4- w, r oc x, and w oc -; and, if y = 4 when a: = 1, and 7/ = 5 when x = 2, what is the value of y in terms of a;. n. If a body falls 192 inches the first second and the dis- tance it falls varies as the square of the time, how far will it fall in 10 seconds ? In how many seconds will it fall 400 ft. ? C H APTER XIV. PERMUTATIONS AND COMBINATIONS. 382. Permutations are the different orders in which things can be placed ; as, ab and ba. Note. — Observe that " things " does not mean quantities. Letters or figures which represent quantities may be the things whose permuta- tions are considered, but they are regarded merely as so many objects which may be arranged in different orders. 383. Combinations are the different groups that can be formed of any number of things, taking a given number at a time. Thus, from the letters abc we can form three groups if we take two at a time ; viz., ab, ae, and be. The order in which the individuals are placed in the group is not considered, ab and ba being but one combination. 384. The object of the theory of permutations and com- binations is to determine the number of orders in which things can be placed, and the number of groups that can be formed. 385. The two problems may be thus stated : I. To find the number of permutations of n things taken m at a time. II. To find the number of combinations of n things taken m at a time. Note. — For convenience we adopt the notation, P for permutations and G for combinations, and to indicate the number of things and the number taken at a time, write a subscript fraction whose denominator shall be the whole number of things and the numerator the number 164 PERMUTATIONS AND COMBINATIONS. taken at a time. Thus, P 5 = number of permutations of S things taken 3 at a time ; and C± — number of combinations of S things taken 3 at a time. 386. To find a general expression for P m and C m we have, n n evidently, P\ = n, and if we take 2 at a time P 2 = n(n — 1); ;i n for each of the n permutations formed by taking one at a time, may be placed before each of the (n — 1) remaining things. By the same reasoning we shall have P 3 = n (n — 1) (n — 2), and Pn — n (n — 1) (n — 2) . . . (n — m + 1), (1) P n = n (n — 1) (n — 2) . . . 1 = \n. (2) n 387. Every combination of w things, taken mata time, may have P m = |m permutations. Therefore, \m x C m = P m n n ■L m - n (n —i)....(« — ;?? 4- 1) or C/ m = — - = -■ (3) 388. If the things whose permutations are to be found are not all dissimilar, the number of permutations will be less, being evidently divided by the number of permutations which could be made with the identical things if they were unlike That is, if p of the things are alike and q others are alike, etc., then \n P l = Q^lptc7 (4) PERMUTATIONS AND COMBINATIONS. 165 EXAMPLES. i. How many permutations can be made with the 9 digits taken all at a time ? That is, P 9 = what ? 2. Pr, = what? 4. (7 4 = what? 3. <% = what ? 5. Show that C» = CU, . 6. How many products can be formed of six factors taken two at a time ? 7. What is the number of products that can be formed from the 9 digits, taken 3 at a time ? 8. How many more products can be formed of 50 numbers taken 40 at a time, than taken 1 o at a time ? 9. P„ -i- C s = what ? n n 10. C & — C 9 = what? 11. If P 5 = 120C3, what is the value of w ? 12. Find the value of m that will make Cm. the greatest possible. 2 " 13. Find the value of m that will make C m the greatest possible. 2 " +1 14. How many permutations (P„) can be made with the " Ll 2 letters of the word Permutations'? Ans. -. — 15. How many permutations can be made with the letters of the word Ecclesiastical ? 16. How many permutations can be made with the letters of the word Divisibility ? CHAPTER XV. INFINITESIMAL ANALYSIS. 389. In the preceding chapters, quantities have been distinguished as hnoivn and unknown. The problems consid- ered have involved quantities to which arbitrary values could be assigned, and others whose values were to be found from these. In these problems, changes in the values of quantities have been made by adding or subtracting finite differences. 390. But there is a large class of problems, in which quan- tities must be conceived as passing from one value to another by a process of growth. A quantity changing in this way from one value to another, passes through all intermediate values ; as, when a point moves from one position to another, it passes through all intermediate positions ; or as time in passing from one hour to another, passes through every instant of intermediate time. This leads to the conception of quantities as constant and variable. 391. A Variable is a quantity conceived as changing from one value to another in such manner as to pass through all intermediate values. 392. A Constant is a quantity whose value remains the same during the same discussion. Constants are of two kinds, absolute and arbitrary. 393. An Absolute Constant is a quantity expressed by a number whose value never changes; as, 4. 10, etc. INFINITESIMAL ANALYSIS. 107 394. An Arbitrary Constant is represented by one or more letters, to which values may be arbitrarily assigned, but which remain the same throughout the same discussion. 395. Consecutive Values of a variable are those values between which there are no intermediate values. 396. The Differential of a variable is the difference between two of its consecutive values. Note — It is evidently impossible to conceive of this difference ; for any conceivable difference would imply intermediate values ; but the existence of strictly consecutive values is a necessity from the manner in which a variable changes its value. 397. A Function (in the m6nitesimal analysis) is a quantity or algebraic expression whose value depends on one or more variables. A function is therefore a variable quantity, having its consecutive values corresponding to the consecutive values of the variables on which it depends. 398. The Differential of a function is the difference between two of its consecutive values. 399. We may suppose the infinitesimal increments or differentials by which a variable passes from one value to another to be equal to each other ; that is, we may assume the differential of a variable to be constant, since there is nothing to forbid such a supposition, but the differential of the func- tion which depends on the differential of the variable will not usually be constant. Hence, 400. The Differential of a Function may be defined as the infinitesimal change in the function produced by a change in the variable from one value to its consecutive value. 401. Differentiation is the process of finding a gen- eral expression for the difference between any two consecutive values of a function ; in other words, of finding a general expression for the differential of a function. 168 INFINITESIMAL ANALYSIS. 402. The variable whose increments are arbitrarily assumed is called the Independent Variable. (Art. 399.) 403. The function whose increments depend on the incre- ments of the independent variable is called the Dependent Variable. Thus, in the expression x°- — x, if we assume that x increases by a constant increment, x will be the independent and x- — x the dependt id variable, or, making u — x- — x, u is the dependent variable. 404. Infinitesimal quantities, expressed in terms of any finite unit, are all o. (Art. 42, 6th.) In order, therefore, to express the relations of infinitesimals to each other, fome infinitesimal unit must be employed. It need be no objection to the use of such a unit, that the conception of its magnitude is impossible, for mathematics is concerned only with the ratios of quantities, and never with their absolute magnitude. It is of no advantage whatever to the mathematician to know the magnitude of the unit employed. We may therefore assume as the unit of measure for differentials the Differential of the Independent Variable. (Art. 399.) 405. To Differentiate a Function will then be. to find the differential of the function in terms of the differen- tial of the variable on which the function depends. NOTATION. 406. The Differential of a Variable is expressed by writing before it the letter d; thus, dx, dy, dz, etc., to be read, "differential of .r," "differential of//," etc. Note. — The d in these expressions is not a factor, but only an abbreviation of the word differential, and must be so read. The same abbreviation placed before a function indicates the differential of the function; as, d(x 2 ), d(a?—x), are read and mean, "the differential of aP/V'the differential of a: 2 — x." In all such cases, the function whose differential is expressed must be enclosed in a parenthesis. Thus, d {.)■-) is the differential of x 2 ; but da? is the square of the differential of x. INFINITESIMAL ANALYSIS. 1G9 407. When any function of a variable is under discussion to avoid its repetition, we write the variable in parentheses with/; 0, or ip before it; as, f(x), f(ij), etc., read, "function of x," "function of y," etc. Different functions arc expressed by fi (x), f(x), = o; etc. A = 1; B = — 1 ; C=i; B = — 1 ; etc. Note. — The values of these coefficients enable us to determine the law of the series, and to write any required number of terms, as follows : 1 = 1— x + x- — x 3 + x* — x s + etc. I + X 178 INDETERMINATE COEFFICIENTS, 2. Develop (a — xp. OPERATION. Assume (a — a;)* = A + Bx + C'X ! + Dx 3 + Ex 4 + etc. (i) Squaring, a — x = A* + lABx + lAC x- + 2AD j x 3 + zAE\ x 4 + etc. + B- + 2BC I + 2BD j + C- .'. Art. 419, Cor., A 2 = a ; A = a} ; 2AB = — 1 : B = 1 2a 1 1 5 128a 7 etc. etc. 2AG + B* = o; G=- 2AD + 2BG = o ; , D = - 2AE + 2BD + C* = o ; E = - Substituting in (1), ,1 1 x a; 2 x 3 sx 4 (a — x)* = a* — — k - - — - , etc. 2a 3 8a 11 16a 3 128a 5 422. From these solutions we have for the development of a function of a single variable the following Rule. — I. Assume the function equal to a series with indeterminate coefficients, containing all the fmvers of the variable which the development requires. II. Free this equation from fraction* and from parentheses which include different powers of the variable, and make the coefficients of like powers of the variable in the two members equal to each other ; or transpose all the terms to one member, and make the several coefficients equal to zero. III. From the equal inns /has formed find the values of the coefficients and substitute them in the assumed series. Notes. — 1. The form of the function must determine what powers of the variable to assume. One thing should always be observed, viz. . The assumed equation should give no absurd result when x — o. Thus, if iu equation (1), Example 1, x = o, A = I, a result involving no INDETERMINATE COEFFICIENTS. 179 absurdity; but if the series were Ax + Bxi + Cx z , etc., x = o would give i = o, an absurdity. 2. If any power or powers of the variable belonging to the develop- ment are omitted from the assumed series, it will be shown by some such absurdity in the course of the solution, if it does not appear by making x = o. 3. If powers of the variable not found in the development be assumed, their coefficients will be found to be zero, and the function will be correctly developed. It appears, therefore, that it is only necessary to make sure of including all necessary powers, since it does not vitiate the development to include those that are unnecessary. Let the student illustrate this by assuming for the function above, = A + Ex* + Ci A + Dx* + etc. 1 + x Also, = Ax-' 2 + Bx- 1 + C + Dx + Ex 11 + etc. Develop the following : 3- 1 + X 4- 1 X -\- X 2 5- (,- - l)i 6. ax — x 2 X s — X* 7. (a + x)-k 8. a (x + a) 2 x 1 — 1 9- X 3 + X 2 — 2X + l 10. X + I I — X 2 11. (!-*)» 12. (a -f- x)~\ X 2 — 2X + 1 1 o- X s — X 2 14. a (a — x)~\ 15- (x - l)l 16. 1 — yx x — \/x 17- X + \/% X 2 — X* + X* 18. 1 X — X » (1 + ^) 3 19. I 1 " I — £« 1 X* X 3 (1 4- x*) (1 — X?) Note. — Observe what powers of x will be found iu the development of the last five examples. 180 DECOMPOSITION DECOMPOSITION OF FRACTIONS. 423. A Rational Fraction, a function of a single variable, whose denominator lias rational factors, may be separated into two or more fractions, whose sum shall be equal to the given fraction. This is called decomposing the fraction, and the several fractions are called, partial fractions. 424. The fraction to be decomposed is understood to have its numerator of a lower degree than its denominator ; other- wise it would give by division one or more integral terms. 425. The Decomposition of a Fraction is per- formed by the following Bulk — I. Assume the given fraction equal to the sum of several fractions with indeterminate numerators, and whose denominators include all the denominators possible for the partial fractions. II. Clear the equation of fractions, and collect like powers of the variable in one term. III. Equate flic coefficients of these like powers, and from the equations thus formed determine the values of the numerators. IV. Substitute these values in the assumed fr act inns. 426. The manner in which the numerators and denomi- nators of the partial fractions are assumed needs special notica By the rule, we are to include in the denominators all denominators possible to the partial fractions. To ascertain what these will be, it should be observed that, since the given fraction is the sum of the partial fractions, each denominator must be a factor of the given denominator. (Art. 197.) Tiie denominator of the given fraction being a rational function of a single variable, the prime rational factors will have one of the following forms. x, x ± a, or x 2 ± ax -\- b. (Art. 549, Cor. 2.) OF FRACTIONS. 181 Any one or more of these forms may be found with any exponent, so that x n , (x ± a) n , and (z 2 ± ax + b) n , will represent all the different factors of the given denom- inator. ■** 427. Considering the first of these, x n , we see that it must be one of the partial denominators, otherwise it would not be a factor of the denominator of the sum. (Art. 197.) So also x n ~ x may be one denominator, and in like manner a? 1 ' 2 , x n ~ 3 , .... x, are all possible denominators. Hence they must all be used, and from x n we shall have the partial denominators x, x 2 , x 3 , . . . . x 11 . In like manner, (x ± a) n will give the denominators x±a, {x±af, .... (z±a) n , and (x 2 + ax + b) n will give x*±ax + b, (x* ±ax + b) % , .... {x i ±ax + b) n . 428. To determine what numerator to assume for each denominator, consider first the proper form of the numerator for x n . This numerator must be independent of x, for if it (Ax 4- B\ as — 1, it would form a fraction capable of further decomposition, and the partial fractions would not be the simplest possible. The assumed fraction therefore must be of the form — ■ x n The same may be said of all the fractions with monomial denominators. 429. The numerator also for (x — a) n must be independ- ent of x for the same reason, and therefore all the fractions having denominators of that form will have numerators of the zero degree. 182 DECOMPOSITION 430. The denominators of the form (x~ ± ax + b) n are quadratic factors, whose binomial factors are imaginary, and therefore cannot be further resolved. Their numerators may therefore contain the first as well as the zero power of x, and must be in the form Ax + B. This will give for the different forms of the partial fractions, A B Cx + D and &' (x ± a) n ' (x 2 ± ax + b) n In the third form, a may be zero, reducing it to Cx + D (.r2 + by The process will be made plainer by the solution of a few i. Decompose -= EXAMPLES. x 5 — 3Z 3 + i X Z ( X — 2) 3 (a* + 2) 2 (2? — 2X + 2)2 Solution. — Assuming the partial fraction as above indicated, ; .5_ 3>C 3 +I ABC D E + -i + —- + r~-^ + X* (X—2) 3 (X- + 2y (x' 2 —2X+2) i X 0?' 2 X—2 (X—2)' ! (X—2, 3 Fx+G Hx + I Kx + L Mx + N + -5—7- + 7-S—C, + .. _,. - + .{'-' + 2 (.r + 2)- .!■ — 235 + 2 (,C 2 — 2.r+2) 2 Note. — This example is given for tire purpose of including all possi- ble forms of partial fractions, and the student should compare it with the preceding explanations. The complete solution would occupy too much space to be conveniently printed. The student may complete it for his own practice and satisfaction. _. .T- — 2X + 2 2. Decompose -= x 3 + 2.1 -2 — x — 2 Solution. — The factors of the denominator are x+i, x—i, and x+2. Assume x" — 2.r + 2 A B G X 3 + 2J.' 2 — X — 2 _ X + I X — X X+2 OF FRACTIONS Clearing of fractions and uniting terms, 183 X 2 — 2X + 2 = — 2A + 2B + A I x + A I & + 3 B\ + b c \ + u .-. - 2^ + 25 - C = 2. J. + 3# = - 2. A + 5 + G = i. From which, -A = — s, •*» — * » i io a;-' — 2* + 2 _ 5 , . + —- — — - • ••• * + *=*=*- ~ 2(x + D + 6(* - i) 3.<* + ») 3- Decompose ^T^ZT^) 4 . Decompose -—-^-^ 5 . Decompose prZl^+T) (a? + s + 4) /^4 _t_ ^;2 _1_ J 6. Decompose ^^T^TTt?' a; 3 — 3a; -f 3 7. Decompose -— — ^^—^^ «# — a 2 8. Decompose -nr-^T+V 1 9. Decompose . __ ^ 10. Decompose ^ r ^ + 7^ 1 11. Decompose & + ^ _ bx _ a b 2X 5 + 3iC 4 _ 7^3 _|_ 9a ;2 _ (,x + 4 12. Decompose (^T^)^- t) CHAPTER XVII. DEMONSTRATION OF THE BINOMIAL THEOREM. 431. The Binomial Formula has already been given, and the student is familiar with its use. It only remains to give the proof, which he was not prepared to understand at an earlier stage of his progress. 432. Let it be required to develop (a + x) n into a series of ascending powers of x, n being any number whatever, positive or negative, integral or fractional. (a + .r)" — a h 1 1 + - 1 . Put - = g, a Then n"(i + *) = a n (i + z) n , and the development of (i + z) n will give the required series when the value of z is restored aud the series multiplied by a". Assume, (i + z)« = A + Bz + Cz* + Bz 3 + Ez* + etc., (i) in which A, B, C, etc. , are indeterminate coefficients, whose values are to be found. Performing- the successive differentiations of (i) and dividing each by dz, we have n (i +2)»-i = B + 2Cz + ?,Dz* + \Ez 3 + $Fz 4 + etc. (2) n (1, - 1) (1 +g)"-2 = 2C + 2 . 3DZ + 3 • 4Ez i + 4 ■ 5F2 3 + etc. (3) n(n — 1)(» — 2)(i+2)»- 3 = 2 -3D + 2. 3.4^3 + 3-4-5F2 2 + etc. (4) n(n— i)(n-2)(n-3)(i+z)"- i = 2 . 3 . 4E + 2 • 3 4. 5^2 + etc. (5) n(n—i){n—2)(n — 3) (n - 4) {1 + z)»- 5 = 2.3.4- 5-^+ etc. (6) BINOMIAL THEOREM. 185 Making z = o, we have from these equations A = i ; B — n; G _ n(n—i) I) I • 2 _ 71 (ft — I) (71 — 2) I- 2-3 £" — n(n— i )( n — 2) (ft — 3) _ 1.2.3.4 ^, _ 7^ (ft - I) (ft — 2) (ft — 3) (ft - 4) I.2-3-4-5 from which we readily determine the law of the coefficients, and may write the series, , \ n{n—i)„ n(n — i)(?i— 2) , (1 + z)» - i + nz + — y s 2 + — — -— ' z 3 + etc. 1-2 1-2-3 Restoring the value of a and multiplying by a n , we have the formula as given in Art. 268, . . ft (ft — 1) „ „ 71 (ft — 1) (ft — 2) „ , (a + x)» = a n + na n ~ x x + — '-a»- 2 x 2 + -± - '■ a*-SaP 1-2 1-23 7t(ft — I)(» — 2) (ft — 3) . . H 1 -- ao-W + etc. 1-2-3-4 433. The m th term of the series is n(» -i)(»- 2) (» -m+J) an _ m+la;m _ 1 I • 2 • 3 • 4 .... (7ft — I) The (7ft + i) th term is ft (ft - 1) (ft - 2) . . . . (ft - 7ft + 2) (ft - 7ft + 1) a)l _ ma;m> 1-2-34 ■ ■ • • (7ft — I) 7ft Dividing the (/ft + i) th term by the m' h term gives n — m + 1 x In +1 \ x _i x _ / ft + I \ a ~ \ 7ft / a This is the variable factor by which any term may be multiplied to produce the next, 7ft representing the number of the term multiplied. 186 BINOMIAL THEOBEM. EXAMPLES i. Expand by the formula, 2. Expand by the formula, 3. Expand by the formula, 4. Expand by the formula, 5. Expand by the formula, 6. Expand by the formula, 7. Expand by the formula, 8. Expand by the formula, 9. Expand by the formula, 10. Expand by the formula, 11. Expand by the formula, 12. Expand by the formula, 13. Expand by the formula, 14. Expand by the formula, 15. Expand by the formula, 16. Expand by the formula, 17. Expand by the formula, 18. Find the 11 th term of the development of (ax + b)K 19. Find the m th term of the development of (m — x)"\ 20. Find the nP term of the development of (a — x) - a + b) 5 . a - b)i a — b)- 3 . a + b)~\ x - y)~k 1 (x -y? (m + n)l 1 (m — 7i)i 1 m + n)* x — 2 p. X + 2) 3 . X + 2)i ax + by) 5 , ax + ty)h ax + %) _? S a — 7 .r) 2 . 3« + 4ar)i Note. — The student will observe that any function of .?• which can be developed into a series of ascending powers of .;•, may be developed by the same method. The general formula for this is called, from its originator, McLaurin's formula. (See p. 306, Note.) CHAPTER XVIII. LOGARITHMS. 434. The Logarithm of a number is the measure of its factors. In other words, it is the exponent which shows how many times the number contains a given number as a factor. 435. Heretofore, when we have spoken of the measure of a quantity, the measure of its terms has been meant. If we wish to find the measure of a line 27 feet long, with the yard as a unit, we may do it in either of three ways : 1st. Subtract 3 feet (the length of the unit) from 27, and again from the remainder, and continue so to do until nothing is left. The number of subtractions will be the measure of the terms of 27. 2d. Add 3 + 3 + 3, etc., until the sum is 27, and the number of times 3 is used will be the same measure. 3d. Divide 27 by 3, and the quotient will be the measure. We may represent the three processes as follows : 1st. 27-3 — 3 — 3 — 3-3 — 3 — 3-3 — 3= o. 2d. 3 + 3 + 3 + 3 + 3 + 3 + 3 + 3 + 3 = 27. 3d. 27 -J- 3 = 9. Therefore 9 is the measure of 27 as a term when 3 is taken as the unit term ; that is, 3 taken 9 times as a term equals 27. We express this by using 9 as a coefficient, thus 9 ■ 3, or if we let y represent the yard, Then, gy = 27 ft. 436. In a similar manner we may measure the fac- tors of a number. The measure of the terms of a number is the measure of its effect when added or subtracted, while 188 LOGARITHMS. the measure of its factors measures its effect when used as a multiplier or divisor. To find this measure we must assume a unit factor, as in measuring the terms we assumed a unit term. Take the same number 27 to measure its factors, and assume 3 as the unit factor. As before, we may find the required measure in three ways : 1st. Take the factor 3 from 27, and again from the remaining factors, and so on until no factor remains. The number of times we can remove the factor 3 from 27 will be the measure required. 2d. Use 3 as a multiplier until the product equals 27 ; and the num. ber of times 3 is used will be the measure. 3d. By a process to be explained hereafter, this measure can be found. The first two processes we may represent thus : 1st. [ (27 h- 3) h- 3] ■*■ 3 = 1. 2d. 3 x 3 x 3 — 27. In the first case we have taken the factor 3 from 27 three times and 1 only is left, which is a factor of no power, and occupies the same place with reference to factors that zero does to terms. (Art. 152.) In the second case we find that 3 used three times as a factor equals 27. Both give us 3 as the measure of the factors of 27 with 3 as the unit factor. This is expressed by using an exponent thus, 3 3 = 27. 437. The measure of the /floors of a quantity is expressed by an Exponent; the measure of terms by a Coefficient. 438. When in any mathematical computation only one unit term is employed, the symbol for that term is omitted, and the coefficient or measure only is written. Thus, the surveyor who measures all his distances with the rod as a unit, does not write 10 r, 15 r, 20 r, etc., but simply 10, 15, 20, etc. Yet, in his computations he bears in mind the fact that the symbol for rod has been omitted, for when he multiplies two of these measures together, as 15 x 10, he calls it 150 ?•'-, or 150 square rods. 2' — 2 2 6 = 64 2° = I 2 2 = 4 2 1 = 128 2-'= i 2 3 = 8 2 8 = 256 2" 2 = J 2 4 = 16 2 9 = 512 o-3 — X — 8 2 ; ' = 32 2 10 = IO24 r>— 4 1 2 — T6" LOGARITHMS. 189 439. In a similar manner, computations in which numbers are used as factors may be abbreviated by measuring the factors of each with the same unit factor, and omitting that factor, writing only the exponents which express the measures of the factors. Exponents used in this way, as we have seen by the definition, are called Logarithms. (Art. 434.) Thus, if we adopt 2 as the unit factor, we have 2 as the measure of the factors of 4, 3 as the measure of the factors of 8, and 4 of 16, aud so on. That is, we have : 2- 5 = A 2~ 6 = «V 2 — T2¥ c— 6 _ 1 2 — 256 o—9 — 1 2 — 51? Note. — The abbreviation " log." is used to express logarithms. Thus, log. 16 indicates the logarithm of 16, and is read "logarithm of 16." 440. The factor adopted as the unit factor is called the Base. This may be any number except 1. That 1 cannot be used as the unit factor in measuring the factors of other quantities is evident, since 1 as a factor has no power, and it would require an infinite number of such factors to produce any number greater than 1. For the same reason we cannot use zero for the unit of measure for terms. 441. In measuring the factors of numbers in this manner, the signs of the numbers are not and cannot be considered. We measure only the factors of magnitude or value, and not the factors of direction. Hence the base or unit of measure for these factors is a number without a sign. 442. But the factors measured may be used as multipliers or divisors, and the measure should indicate this. As the measure of terms is either + or — , according as the terms are used in addition or subtraction, so the measure of factors (logarithms) are either + or — , according as the factors are used as multipliers or divisors. 190 LOGARITHMS, For example, if we measure the factors of 8 with 2 as a unit factor, the measure is +3 ; but if the 8 be used as a divisor, making it -J-, the measure is —3. This will be further illustrated by referring to Art. 439. We have from that, when 2 is the base, log. 2 = 1 log. 1 — o "4 = 2 " £ = -1 "8 = 3 - i = -2 " 16 = 4 « 1 = _ 3 1 — -4 If now we make 2~' (= I) the base, we have, log. 2 = — I log. 1 = O " 4 = -2 « 1 I " 8 = -3 •< 1 — 2 " 16 = —4 " 1 8 — 3, etc., etc If we take the base 5, we have, log. 1=0 log. I = " 5 = i " A - 5 — — 1 " 25 = 2 •' 1 — ^5 — — 2 " 125 = 3 « 1 3 If 5- 1 (= 1) be taken as the base, then log. 1 = log. I = " 5 = -1 " I = 1 " 25 = —2 " 1 =: 2 " 125 = -3 " jh = 3 443. We see that log. 1 will evidently be o for all bases, since 1 as a factor has no power, and its measure with any unit factor must be o. So also the logarithm of the base will always be 1, since the measure of any quantity with itself as a unit must be 1. 444. The logarithms given above are all integral, because we have selected such numbers as contained the base as a factor an integral number of times. Intermediate numbers have intermediate logarithms. LOGARITHMS 191 This may be illustrated by taking 16 as a base. We shall then have, log:, i = o Og. I = " 2 = 1 4 = •25 " 4 = 1 = •5 " 8 = 3 = •75 " 16 = I " 32 = 5 = 1-25 " 64 = 3 = i-5 '< 1 = 1 i = - •25 << 1 = 1 5 = — ■5 "■ 1 8 = 3 = — •75 16 = — I " 1 32 = 5 3" = — I •25 << 1 6? = 3 2 = — I 5 The logarithms of numbers between these are incommensurable, and can be expressed only by approximation. 445. The logarithms of numbers with a given base are called a system of logarithms.* The system in common use has 10 for a base, and from its originator is called Briggs' System. In this system, log. 1=0 10 = 1 100 = 2 1000 = 3 The logarithms of numbers between and 10 100 .1 .01 10 100 1000 .1 .01 .001 log. .1 = -1 " .01 = —2 " .001 = -3 " .0001 = —4 s o + a decimal. 1 + 2 + — 1 + " — 2 + -3 + 446. The integral part of a logarithm is called its CJiftv- acteristic, and the decimal part its Mantissa. It is customary in expressing negative logarithms to make the mantissa positive, as indicated above. Thus, if we have the logarithm —2.754526 (the whole being nega- tive), it may be expressed thus, - 3 + -245474, or, as it is commonly written, 3.245474, in which the 3 only is negative, the sign — being placed over it to indi- cate this. * The method of computing by logarithms was invented by Lord Napier. (See p. 306, Notes 3 and 4.) 192 L0GAKIT1IMS. 447. It will readily appear that the characteristic of the logarithm of a number in the common system may be known by the following Eule. — The number of places from the first significant figure of a number to units' place, counting the latter, is equal to the characteristic of the common logarithm of thai number. When counted to the right, the characteristic is positive; when counted to the left, negative. Note. — The reason fortius rule is found in the fact that multiplying or dividing a number by 10 moves the decimal point one place to the right or left, and at the same time increases or diminishes its logarithm one unit. Hence, moving the decimal point of a number to the right or left does not affect the mantissa of its logarithm. 448. What are the characteristics of the logarithms of the following numbers: I. 2785.62. Ans. 3. 2. 5437L Ans. 4. 3- .0075. Ans. — 3. 4- 1.075. 5- 16.0005. 6. .0000589. 10. 5.89. 7- 0.00589. 11. 589. 8. 0.0589. 12. 58900. 9- 0.5S9. 13. 5890000 449. When the number has integral figures, The characteristic of its logarithm is one less than the number of integral places, and is positive. 450. When the number is entirely decimal. The characteristic of its logarithm is one more than the number of ciphers between the decimal point and the first significant figure, and is negative. LOGARITHMS. 193 TABLES OF LOGARITHMS. 451. A Table of Logarithms is one which contains the logarithms of all numbers between given limits. 452. The Table found on the following pages gives the mantissas of common logarithms to five decimal places for all numbers from i to iooo, inclusive. The characteristics are omitted, and must be supplied by inspection. (Arts. 447-450.) Notes. — 1. The first decimal figure in column is often the same for several successive numbers, but is printed only once, and is under- stood to belong to each of the blank places below it. 2. The character ( ♦ ) shows that the figure belonging to the place it occupies has change! from 9 to o, and through the rest of this line the first figure of the mantissa stands in the nest line below. 453. To Find the Logarithm of any Number from I to 10. Rule. — Look for the given number in the first line of the table ; its logarithm will be found directly below it. 1. Find the logarithm of 7. Ans. 0.84510. 2. Find the logarithm of 9. Ans. 0.95424. 454. To Find the Logarithm of any Number from 10 to 1000, inclusive. Rule. — Look in the column marked A 7 " for the first tivo figures of the given number, and for the third at the head of one of the other columns. Under this third figure, and opposite the first two, ivill be found the last four decimal figures of the logarithm. The first one is found in the column marked 0. To this decimal prefix the proper characteristic. (Art. 447.) Note. — If the number has 4 or more figures, find the logarithm of the first three figures and add to it the product of the remaininrr figures considered as a decimal, by the tabular difference (from column D) oppo- site the logarithm of the first three figures. 194 LOGARITHMS. 3. Find the logarithm of 108. Ans. 2.03342. 4. Find the logarithm of 176. Ans. 2.24551. 5. Find the logarithm of 1999. Ans. 3.30085. 455. To Find the Logarithm of a Decimal Fraction. Rule. — Take out the logarithm of a whole number consist- ing of the same figures, and prefix to it the proper negative characteristic. (Art. 450.) Note. — If the number consist of an integer and a decimal, find the logarithm in the same manner as if all the figures were integers, and prefix the characteristic which belongs to the integral part. (Art. 449.) 6. What is the log. of 0.95 ? Ans. 1.97772. 7. What is the log. of 0.0125 ? Ans. 2.09691. 8. What is the log. of 0.0075 '• Ans. 3.87506. 9. What is the log. of 16.45 ? Ans. 1.21616. 10. What is the log. of 185.3 • Ans. 2.26787. 456. To Find the Number belonging to a given Logarithm. Rule. — I. Find in the table the mantissa next less than the mantissa of the given logarithm, and the corresponding number will be the first three figures of the required number. II. Subtract the mantissa found in the table from the mantissa of the given logarithm, and divide the remainder by the corresponding tabular difference, for the remaining figures of the required number. III. Place the decimal point of this number as required by the characteristic of the given logarithm. (Art. 447.) Note. — If the characteristic of a logarithm be negative, the number belonging to it is a fraction, and as many ciphers must be prefixed to the number found in the table, as there are itnits in the characteristic leas 1. (Art. 450.) LOGARITHMS. 195 ii. What number belongs to 2. 17231 ? Ans. 148.7. 12. What number belongs to 1. 25261 ? Ans. 17.89. 13. What number belongs to 3.27715 ? Ans. 1893. 14. What number belongs to 2.30963 P Ans. 204. 15. What number belongs to 4.29797 ? Ans. 19858.29. 16. What number belongs to 1. 14488? A ns. 0.1396. 17. What number belongs to 2.29136 ? Ans. 0.01956. 18. What number belongs to 3.30928 ? Ans. 0.002038. 457. Computations in which numbers are used as factors may be made by logarithms upon the following principles: i°. TJie sum of the logarithms of two numbers is equal to the logarithm of their product. Let a and c denote any two numbers, m and n their logarithms, and 6 the base. Then bm = a And b n = c Multiplying, &'«+" = etc. 2 . TJie logarithm of the dividend diminished by the logarithm of the divisor is equal to the logarithm of the quotient. Let a and c denote any two numbers, m and n their logarithms, and b the base. Then b" 1 = a And b" = c Dividing, 6m—" = a -~ c. 458. To Multiply Numbers by their Logarithms. Eule. — Add the logarithms of the factors ; the sum will be the logarithm of the product. (Art. 457, i°.) Notes. — 1. If one or more of the characteristics be negative, make them positive by adding 10 to each, and reject as many io's from the sum. • 2. The sign of the product or quotient is determined as when multi- plication and division are performed in tbe usual manner. (Art. 129.) 196 LOGARITHMS. i. Required the product of 35 by 23. Solution. — The log. of 35 = 1.54407 " " 23 = 1. 36173 Adding, 2.90580. The corresponding number is 805, Ans. 2. What is the product of 109.3 by 14.17 ? 3. What is the product of — 1.465 by —1.347 ? 4. What is the product of .074 by —1500? 459. To Divide by Logarithms. Rule. — From the logarithm of the dividend subtract the logarithm of the divisor ; the difference will be. the logarithm of the quotient. (Art, 457, 2 .) Note. — If either or both characteristics are negative, add 10 to each and the result will not be affected. 5. Required the quotient of 120 by 15. Solution. — The log. of 120 = 2.07918 " " 15 = 1. 17609 " " " quotient = 0.90309. Ans. 8. 6. What is the quotient of 12.48 by 0.16 ? 7. What is the quotient of .045 by 1.20 ? 8. What is the quotient of 1.381 by .096 ? 9. Divide — 128 by — 47. 10. Divide — 186 by — 0.064. 11. Divide — 0.156 by —0.86. 12. Divide — 0.194 by 0.042. 460. To Involve a Number by Logarithms. Rule. — Multiply the logarithm of the number by the exponent of the required power. Notr. — 1. This rule depends upon the principle that logarithms are the exponents of powers, and a power is involved by multiplying its exponent into the exponent of the required power. 2. Let the student remember that power includes also root. (Art. 253.) LOGARITHMS. 197 13. What is the cube of 1.246 ? Solution. — The log. of 1.246 is 0.09551 Index of the required power i3 3 Logarithm of power is 0.28653. Ans. 1.93435. 14. What is the fourth power of . 135 ? 15. What is the tenth power of 1.42 ? 16. What is the twenty-fifth power of 1.234 ? 17. What is the square root of 1.69 ? Note. — Log. 1.69 must he multiplied by 4. or what is the same thing, divided by 2. 18. What is the cube root of 143.2 ? 19. What is the sixth root of 1.62 ? 20. What is the eighth root of 1549 ? 2i. What is the tenth root of 1876 ? 461. If the characteristic, of the logarithm be negative, and caunot be divided by the index of the required root without a remainder, add to it such a negative number as will make it exactly divisible by the divisor, and prefix an equal positive number to the decimal part of the logarithm. 22. It is required to find the cube root of .0164. Solution. — The log of .0164 is 2.21484. Preparing the log., 3)3+ 1.2 1484 140494. Ans. 0.25406 + . 23. What is the sixth root of .001624 ? 24. What is the seventh root of .01449 ? 25. What is the eighth root of .0001236 ? 198 LOGARITHMS TABLE O F COMMON LOGARITHMS. N. 1 .00000 2 3 4 5 6 7 8 9 D. .3oio3 •477 '2 .60206 .69897 .778.5 .845io .90309 .95424 10 .ooooo 0432 0860 1284 1703 2119 253 1 2938 3342 3743 416 11 4i3 9 4532 4922 53o8 5690 6070 6446 6819 7188 7555 379 12 7918 8279 8636 8991 9342 9691 ♦037 ♦38o ♦721 1059 349 i3 .11394 1727 2057 2385 2710 ! 3o33 3354 3672 3 9 88 43oi 322 14 46i3 4922 5229 5534 5836 6137 6435 6732 7026 7319 3oi i5 7609 7898 8184 8469 8752 9o33 9312 9590 9866 ♦ 140 281 16 .20412 o683 0952 1219 1484 1748 201 1 2272 253 1 2789 264 n 3o45 33oo 3553 38o5 4o55 43o4 455 1 4797 5o42 5285 249 18 5527 5768 6007 6245 6482 6717 6951 7184 7416 7646 235 19 7875 8io3 833o 8556 8780 9003 9226 9447 9667 9885 223 20 .3oio3 0320 o535 0750 0963 1 175 1387 1 597 1806 2oi5 212 21 2222 2428 2634 2838 3o4i 3244 3445 3646 3846 4044 203 22 4242 4439 4635 483o 5o25 52i8 341 1 56o3 5793 7658 5984 ig3 23 6i 7 3 636 1 6549 6736 6922 7107 -291 7473 7840 1 85 24 8021 8202 8382 856i 8739 8917 9094 9270 9445 9620 .78 25 9794 9967 ♦ 140 ♦3 1 2 ♦483 ♦654 ♦824 ♦993 1162 i33o •7i 26 }.4i497 1664 i83o 1996 2160 2325 2488 265i 28i3 2975 1 65 27 3i36 3297 3457 36i6 3 77 5 3 9 33 ,'"/! 4248 4404 456o 1 58 28 4716 4871 5o25 5'79 5332 5485 5637 : 5939 6090 1 53 29 6240 638g 6538 6687 6835 6982 7129 7276 7422 7 56 7 i47 3o •477 ' 2 7 85 7 8001 8i44 8287 843o 8572 8714 8855 8996 i43 3i 91 36 9276 941 5 9554 9693 9 83 1 9969 ♦ 106 ♦ 243 ♦379 i38 32 .5o5i5 o65i 0786 0920 io55 1188 l322 1455 1 587 1720 i33 33 i85i i 9 83 2114 2244 2375 25o4 2634 2763 2892 3020 i3o 34 3 148 3275 34o3 3529 3656 3782 3908 4o33 41 58 4283 126 35 4407 453 1 4654 4777 4900 5o23 5 1 45 5267 5388 55o9 123 36 563o 5751 5871 5 99 i 61 10 6229 6348 6467 6585 6703 119 37 6820 6937 7024 7'7i 7287 74o3 7519 7634 7749 7864 116 38 7978 8o 9 3 8206 8320 8433 8546 865 9 8771 8883 8995 n3 39 9106 9218 9 32 9 9439 955o 9660 977° 9879 9988 ♦097 1 10 4o .60206 o3i4 0423 o53i o638 0746 o853 0959 1066 1172 108 4i 1278 i38 i ', 1490 i5g5 1700 i8o5 1909 2014 2118 2221 io5 42 2325 2428 253 1 2634 2737 283 9 2941 3o43 3 144 3246 102 43 3347 3448 3548 3649 3749 3849 3949 4048 ,< r 4246 100 44 4345 4444 4542 4640 4738 4933 5o3i 5 128 5225 98 45 5321 5418 55 14 56io 5706 58oi 58o6 5992 6 > s - 6181 9 5 46 6276 6370 6464 6558 6652 6745 683 9 6g32 7117 93 8 7210 7302 7394 7486 7578 8485 7669 7761 - ■ 7943 8o34 V 8124 82i5 83o5 83q5 85 7 4 8664 8 7 53 8842 8o3i 89 49 9020 9108 9'97 9285 9 3 7 3 946i 1 9548 6 9 636 9723 9810 88 N. 1 2 3 4 5 7 8 9 D. L G A K I T II _M S . 199 N. 1 2 3 4 5 6 7 8 9 ♦672 D. 5o .69897 9984 ♦070 ♦ 1 57 ♦ 243 ♦329 ♦4i5 ♦5oi ♦586 86 5i .70757 0842 0927 1012 1096 1181 1265 1 349 1433 l5l 7 85 52 1600 1684 1767 i85o 1933 2016 2099 2181 2263 2346 83 53 2428 2 509 2591 2673 2754 2835 2916 2997 3078 3 159 81 54 3239 3320 3400 3480 356o 3640 3719 3 799 3878 3 9 57 80 55 .74o36 41 1 5 4194 42 7 3 435 1 4429 4507 4586 4663 4741 78 56 4819 4896 4974 5o5i 5i28 52o5 5282 5358 5435 55u 77 57 5587 5664 5740 58i5 58 9 i 5967 6042 6118 6193 6268 76 58 6343 6418 6492 6567 6641 6716 6790 6864 6g38 7012 75 5 9 7o85 7i5 9 7232 73o5 7379 7432 7525 7597 7670 7743 73 6o .778i5 7887 7960 8o32 8104 8176 8247 83i 9 83go 8462 72 6i 8533 8604 86 7 5 8746 8817 8888 8 9 58 9029 9099 9169 7' 62 9239 9309 9 3 79 9449 9 5l 9 9 588 9 65 7 9727 9796 9 865 69 63 9934 ♦oo3 ♦072 ♦ 140 ♦ 209 ♦ 277 ♦346 ♦4i4 ♦482 ♦55o 68 64 .80618 0686 0754 0821 0889 0956 1023 1090 11 58 1224 67 65 1291 i358 1425 1491 1 558 1624 1690 1757 1823 1889 66 66 1934 2020 2086 2l5j 2217 2282 2347 24i3 2478 2543 65 67 2607 2672 2737 2802 2866 2930 2995 3o59 3i23 3187 64 68 32DI 33t5 3378 3442 35o6 3569 3632 36 9 6 3759 3822 63 69 3885 3 9 48 401 1 4073 4i36 4198 4261 4323 4386 4448 63 7° .845io 4572 4634 4696 4757 4819 4880 4942 5oo3 5o65 62 7i 5i 26 5i8 7 5248 5309 5370 543 1 5491 5552 56i2 5673 61 72 5733 5 79 4 5854 5914 5 97 4 6o34 6094 6i53 62i3 6273 60 73 6332 6392 645i 65io 6370 6629 6688 6747 6806 6864 5 9 74 6923 6982 7040 7099 7 i5 7 7216 7274 7332 7 3 9 o 7448 5 2 75 75o6 7364 7622 7680 7737 7795 7852 7910 7067 8024 58 76 8081 8i38 8196 8252 83o 9 8366 8423 8480 8536 85o3 57 77 8649 8705 8762 8818 8874 8930 8986 9042 9098 9134 56 78 9209 9265 9321 9376 9432 9487 9542 9 5 97 g653 9708 55 79 9763 9818 9873 9927 9982 ♦037 ♦091 ♦ 146 ♦200 ♦ 255 55 80 .90309 o363 0417 0472 o526 o58o o634 0687 0741 0795 54 81 0849 0902 0936 1009 1062 1 1 16 1 169 1222 1275 i328 54 82 i38i 1434 1487 1 540 i5 9 3 i645 1698 i 7 5i i8o3 i855 32 83 1908 i960 2012 2o65 21 17 2169 2221 2273 2324 2376 52 84 2428 2480 253i 2583 2634 2686 2737 2788 2840 2891 52 85 2942 2993 3o44 3095 3i46 3197 3247 3298 3349 3399 5i 86 345o 3 5oo 355i 3 60 1 365 1 3702 3752 38o2 3852 3go2 5i 87 3952 4002 4o52 4101 41 5i 4201 425o 43oo 435o 4399 30 88 4448 4498 4547 4596 4645 4694 4743 4792 4841 4890 49 89 4939 4988 5o36 5o85 5i34 5i82 523i 5279 5328 53 7 6 48 90 .95424 5472 5321 5569 56i 7 5665 5 7 i3 5 7 6i 5809 5856 48 9 1 5904 5932 6000 6047 6095 6142 6190 6237 6284 6332 47 92 6379 6426 6473 6320 656 7 6614 6661 6708 6755 6802 47 9 3 6848 6893 6942 6988 7o35 7081 7128 7174 7220 7267 46 94 73 1 3 7 35 9 7403 745i 7497 7543 7 58 9 7635 7681 7727 46 9 5 7772 7818 7864 7909 7935 8000 8046 8091 8137 8182 45 96 8227 8272 83i8 8363 8408 8453 8498 8543 8588 8632 45 97 8677 8722 8767 881 1 8856 8900 8945 8989 9034 9078 45 98 9123 9167 921 1 9255 93oo 9344 9 388 9432 9476 9520 44 99 9364 9607 9631 2 9695 3 9739 9782 9826 9870 99'3 99 5 7 43 N. 1 4 5 6 7 8 9 D. 200 LOGARITHMS. COMPUTATION OF LOGARITHMS. Theorem I. 462. In any two systems of logarithms, the logarithms of like numbers have a constant ratio. Demonstration. — Let the bases of the systems be a and a', and let x be any number whose logarithms in the two systems are s and z '. That is, a z = x and a'~ = x. (i) a z = a!* Let a m — a'. Substituting in (i), a z — a wz' .•. z = mz or z - = m, 2 But since a and a' are constant, in is also constant. Cor. i. — The logarithm of a number consists of two factors, one of which is a function of the base and the other a function of the number. This is evident, since changing the base introduces or removes a constant factor and makes no other change in the logarithm. 463. The Modulus of a System is this constant factor, depending on the base, and is usually represented by M. The logarithm of x will therefore be Mf{x), in which M is a function of the base of the system and is therefore constant. Cor. 2. — TJie base may be so chosen as to malce the modulus i . For in the equation a m = a', it is evident that a' may be so taken as to give m any value whatever; m may therefore be made such as to cancel the factor M, or modulus ; so that, in z — mz', the constant factor of 2' shall be 1. 464. Baron Napier, who first suggested this use of expo- nents, took such a base for his system of logarithms, and they are called from the inventor the Napierian system. LOGARITHMS. 201 The Napierian Logarithm of z will therefore be simply f(x), the modulus beiug i. Note. — The Napierian system of logarithms is sometimes called the natural system, on account of its relation to other systems. Napierian logarithms are also called hyperbolic, logarithms, by reason of their rela- tions to certain areas connected with the hyperbola. The base of this system, commonly represented by e, is 2.718281 + . (Art. 468.) 465. Logarithms of different systems may be expressed by writing the base of the system subscript to the abbreviation log. Thus, log e indicates a Napierian, and logio a common logarithm. When no subscript figure or letter is used, the abbreviation log. must be understood to mean the common logarithm. We may also write M\o, M e , etc., for the moduli of the different systems. Coe. 3. — Log a x = M a log 6 x ; M a being the modulus of the system whose base is a. For log (l x = M„f(x) and log e x=f(x). (Arts. 463-464.) It follows therefore that 466. The modulus of any system is the ratio of any logarithm in that system to the Napierian logarithm of the same number. Hence, 467. A table of logarithms with any base may be con- structed by multiplying the logarithms of the Napierian system by the modulus of the required system. Cor. 4. — The logarithms of the same number in different systems are to each other as their moduli. For log a X = M a fix) and log a ' X = M a 'f(x). • lo g« ■*' _ M± m log a ' X ~ M a '' 202 COMPUTATION OF Theorem II. 468. The differential of the logarithm of a variable is equal to the modulus of the system multiplied by the differential of the variable divided by the variable. Let u = log a x. (i) Adding- dx to x, u + du = log,, (x + dx). Subtracting (i), du = log„ (i + — \ = M a f(x + — Y (2) (Arts. 457, 2°, and 463.) To find / H — -J, let x t and x<> be any two values of x. We may then write x 1 = x.,", (3) And log a x t = n log, x.,. (4) Differentiating (3), dx 1 = ?ix.,"- l dx 2 . (5) Differentiating (4), d(log a x^ = nd(\og* ./-,), (6) , . . dx t dx , . (5) "5- (3), -^ (7) (6) -*- (7), dx x x x dx s = n — -. x 3 d{\og a x x d(\0g a X 2 ) dx x dx s ' x 1 x 2 , dx du oc — . dx du = m — . X dx m — x = ,/,/(, + f). m = M a , dx X =/(-?> du dx = Ma . X du = d(\og.x) = *j. That is, (Arts. 377-380), du oc — . .-. du = m — . (8) From (2) and (8), Hence (Art. 419, Cor.), And Hence du = M a —. (9) If a = e, du = dQog e x) = —. (10) 468'/. To find the value of e, we have from (2), substituting e for a, , . dx _, . , / dx\ 1 e d " = 1 H . For convenience put du\= — = -. x \ x I n i 1 Then c = 1 + . (11) 11 Raising (11) to the ?;th power, 1 n(n — 1) 1 n (n — 1) (» — 2) 1 e = 1 + n - + — ■ . -5 + - • —. + etc. n 1-2 « ? 1-2-3 n Since = — , an infinitesimal, n = cc , and w a; a = 1 + - + — + - -+- ■- + etc. (Art. 412.) (12) I 1-2 1-2-3 I-2-34 .•. e = 2.71828182S459045 +. LOGARITHMS. 203 469. To find a formula for computing the Napierian logarithms of numbers, assume log* (i + x) = A + Bx + Ox~ 4- Bx s + Ex i + etc. (i) Differentiating successively, and removing factors common to both members. = B + 2 Ox + $Bx* + 4 Ex 3 + etc. (2) 1 -f x 1 - = 2O + 2- 3 Dx + 3.4^2 4- etc. (3) + ( T + a .\3 = 3^ + 3 • 4^» + etc. (4) = AE + etc. (5) + «) 4 - 4- z -' 1 Hence, making a = 0, From (0. etc. 477. The Law of a Series must therefore express two things. 1st. The rate of increase of the quantity. 2d. The intervals of time at which its values are taken for the terms of the series. The law of a series is usually expressed in the form of a general rule for the formation of any term from the preceding term or terms ; or we may have several terms of a series given from which to determine the law. 208 SERIES. 478. Since there is no limit to the number of different laws which may govern the formation of series, there will be an unlimited variety of series. Our space will only allow the discussion of a few of the most important. 479. For convenience we number the terms of a series from left to right, beginning with some term which we call the first term ; but as it is evident that any series may be extended both ways, we shall not only have terms numbered i, 2, 3, 4, 5, etc., to the right, but also those numbered o, — i, — 2, —3, — 4, etc., to the left, 480. The problems to be discussed relating to series are, 1st. Finding any required term of a series. 2d. Interpolation of terms. 3d. Summation of series. 4th. Reversion of series. 481. To find any term of a series requires a formula which will give the value of the variable quantity at any given time ; as in the series of the squares, to find the 10th term is the same thing as finding the area of the square at the end of 10 seconds. 482. Interpolation of terms is the process of finding one or more terms intermediate between any two terms of a series. Thus, if we find the area of the square (Art, 476) at the end of 5^ seconds, we shall have a term of the series between the 5th and 6th, and equidistant from each ; that is, equidistant in time, but not in value. 483. The practical value of this problem may be illus- trated by supposing the altitude of the sun to be known for noon, and for each hour after noon till sunset. These alti- tudes will form a series, and if the law can be found we can find the sun's altitude for any intermediate time, say for 2\, 2\, and 2| o'clock. These will form three terms between the third and fourth terms of the series. SERIES. 200 484. It is evident that the formula for finding any term of a series will apply to interpolation, for if we can find the value of a variable at the end of 10 seconds, we can by the same process find it for q\ or of seconds. 485. The Summation of series is the process of finding the sum of any number of terms of a series. The method will of course depend on the law of the series. 486. A Converging Series is one in which the sum of an infinite number of terms infinite. 487. A Diverging Sei'ies is one in which the sum of an infinite number of terms is infinite. 488. A series is increasing or decreasing according as its successive terms increase or decrease. DIFFERENCE SERIES. 489. Take the series i . 5 . 15 . 35 . 70 . 126 . 210. (1) By subtracting the first term from the second, the second from the third, and so on, we have a series of differences called the first order of differences. From these differences another set of differences may be formed in the same way, called the second order of differ- ences, and from these the third order, and so on till the differences become zero. The series (1) and its several orders of differences will be as follows : Series, 1 5 15 35 70 126 210 1st order of diff. 4 10 20 35 56 84 2d order of diff. 6 10 15 21 28 3d order of diff. 4567 4th order of diff. 1 1 1 5 th order of diff. o p 210 SERIES. 490. A series which, like the above, gives an order of differences equal to zero, is called a Difference Series. Not only is series (i) a difference series, but each set of differences is also a difference series. The series 4, 5, 6, 7, etc., having its first differences con- stant (1 . 1 . 1, etc.), is called a difference series of the first order ; 6 . 10 . 15, etc., having the second differences constant, is of the second order. So also 4 . 10 . 20, etc., is of the third order, and 1 . 5 . 15 . 35, etc., of the fourth order. It is also evident that this series may be made the differ- ences of a series of the fifth order, and so on indefinitely. 491. A difference series of the first order, is called an JEquidifferent Series, because each term is formed by adding a constant difference to the term preceding. Note. — This series is commonly called an Arithmetical Series or Progression. 492. To find formulas fo*r the n th term and the sum of n terms of a difference series of any order, take the series, « x a 2 a i a i a 5 0« 1st order of differences, a. i —a i a 3 —a, a i —a 3 a 3 — a 4 a 6 —a B 2d order of differences, a 3 — 2a 2 +n 1 a 4 -2« 3 +a 3 a- a —2a i + a i a 6 —2a h +a i 3d order of differences, "4 — 303 + 30-3 — «i 06— 304 + 30J— «j "« — 3«„ + 3«4—0 3 4th order of differences, 05— 4'*4 +6a 3 — 402 +0i 6 —40s + 6 '4 — 4« a + a s 5 th order of differences, 0g — 505 + IO 04 _ I0a x + 502— 0r If we put the first terms of these successive orders of differences = d lt d.,, d 3 , etc., we shall have, (7 t = a»— a x . d.> — a 3 —2a 2 +«,. d 3 = « 4 — 30 3 + 30 3 — "l- d 4 = a 5 — 4a i + 6a 3 — 4a i +a 1 . d. = a*—sa s \-ioa i —ioa i + $a 9 —a l . SERIES. 211 In these equations we find the coefficients are the same as for the n^ power of a binomial. From this we can write , n(n— i) n(n—i)(n—2) d n = a„+i— na n + — a n -\ — a n -2 + i-2 1.2.3 n(n— 1) {n— 2)(n— 3) 1.2.3.4 Reversing the order of terms, a n -z — etc. , , n (n — 1) n(n— i)(n— 2) d n = ± a 1 =f na 2 ± — ^ a 3 =f — ! f -± ' a, ± etc., 1-2 1-2.3 in which the upper signs will be used when n is even, and the tower when n is odtf. From the values of d lt d 2 , etc., we get, a 2 = a 1 +d x . a 3 = a Y + 2d t +d.,. a i = #1 + 3^1 + 3^2 + ^3- a 5 = a 1 +4d 1 +6d.,+4d 3 +d i . And from the law of the coefficients, which is evident, / nj (w— 1) («— 2) , (n — i)(n— 2){n— 3), a„ = a, + (« — 1)^ + v 'd s + - -d s + etc. (A) 1-2 1 • 2-3 493. From this formula any term of a difference series of any order may be found, when enough of its terms are known to give the first terms of the several orders of differences. The number of terms of the formula used in any case will depend on the order of the series. Thus for series of the first order (Equidifferent series), all the differences after d x will be zero. Hence the formula will become a n = a t ■+ (n — 1) d (A)' corresponding to the common formula for the n th term of an Arithmetical Series. Note. — The subscript 1 is omitted from d as unnecessary. 494. To find the Sum of n terms of the Difference Series, form another series of which the given series shall be the first order of differences ; thus, 0, a v a t -f a 9 , a t + r/ 3 4- n %i etc. 212 SERIES. It is evident that the {n + i)^term of this series is the sum of n terms of the given series ; hence, if we apply formula (A) and find the (« + i) (A term of this last series, \vu shall have the sum of n terms of the given series, as required. To make this application we must make in (A), n = n + i, a l = o, d 1 = a lt d., = d u etc. Making these substitutions, we have, putting a„ +1 = 8 n , a , n(n — i) . n (n — i) (n — 2) , ,„, S n = na 1 + -* } -d x + — K - LS > d + etc. (B) 1-2 1-2-3 Note.— S is used for the sum of a series, with a subscript letter or figure to indicate the number of terms included. 495. If the series be Equidifferent, this becomes ~ n (n — i) , /T>X( S n = na 1 + — v — — ' d, (B)' 1.2 which, by substituting the value of d from (A)', becomes S n = a ^-^n. (B)" This is the common formula for the sum of an Arith- metical Series. 496. The formula for the n"' term of a series is also used for interpolation. (Art. 484.) In that formula n, which represents the number of any term, may be more properly regarded as representing the time at which the value of the variable is taken for any term. Hence the formula for the n th term applies equally well when n is fractional as when it is integral. If therefore it be required to interpolate 3 terms between the 9th and 10th terms of a series, we have only to make 11 = 9I, 9I, and 9f, successively in the formula for the n th term of the series. When the terms are known between which other terms are to be interpolated, the preceding terms of the series may be disregarded, and these two terms may be called the first and second terms of the series. One term to be interpolated will be the i| term, two will be i\ and if, three will be i\, 1 1, \\, and so on. SERIES. 213 Note. — Terms thus interpolated are called Means, and the terms between which they are inserted are called Extremes. 497. When several equidifferent means are to be inserted between two extremes, it may be convenient to use a formula for (I, obtained from Eq. (A)' ; thus, On = «i + (n — i) d, gives d = ^£-/ ; (C) in which a„ and a t represent the two extremes, and n the number of terms in the completed series, or tioo more than the number to be interpolated. Thus to insert 4 means between 5 and 20, we have 20 — 5 * = 6^T = 3 " Hence the series complete is 5 • 8 • 11 . 14 . 17 . 20. By formula (A), a series may also be carried backward, calling the numbers of the terms from the first term, o, —1, —2, —3, —4, etc. This may be illustrated by the series, 1 • 4 • 10 • 20 • 35. Thus, — 5 —4—3—2-10 123 4 5 6 — 10 — 4 — 1 o o o (1 4 10 20 35) 56 6 3 1 o o 1 (3 6 10 15) 21 — 3 —2 —1 o 1 2 (3 4 5) 6 1 1 1 1 1 1 (1 1) 1 The first line shows the number of each term of the series. These terms are formed by extending the equidifferent series, and, from that, forming terms of the series of the second order, and then those of the third. We may also find any one of these terms by formula (A). (Art. 492.) 214 SERIES. i. Given the series r . 8 . 27 . 64. 125 . etc., to find: 1st. the 15th term ; 2d, the sum of 15 terms ; 3d, the first of three terms interpolated between the 5th and 6th: 4th, the — 4th term; 5th, the n ,h term ; 6th, the sum of n terms. SOLUTIONS. 1 8 27 64 125 216 7 19 37 61 91 12 18 24 30 6 6 6 o o 1st. From formula (A) we have, 14 -13 14 • 13 • 12 , « 15 = 1 + 14- 7 + ^ ?I2 + -^ — * 6; 2 2-3 «! B = 1 + 98 + IO92 + 2184 = 3375. 2d. By formula (B), *5 • 14 15 • 14- 13 Si 1 15 + 7 + 12 + 15 • 14 - 13 • 12 2 2.3 2-3-4 8 15 = 15 + 735 + 5460 + 8190 = 14400. 3d. The first of three terms inserted between the fifth and sixth will be the 5} term. Using formula (A), we have, ai= I+4 i. 7+ l|j L 3i I2+ 4i 1 3i 1 2| 6 ^ 4 2 2-3 a 5 x = 1 + 29! + 82^ + 31& = i44||- 4th. By the same formula the — 4 th term is a—i „ r+ (- 5)7+ <-5)(- I s + t-3(-' i >(-7> ,U 2 2-3 a-i — 1 — 35 + 180— 210 = —64. 5th. The same formula gives the 11 th term, («-i)(h-2) (n-i)(7i-2)(n-3) a n - i + (»— 1)7 + - fi2+i -— o, 1-2 I-2-3 a„ = i + 7(n—i) + 6(?i—i)(n — 2) + (n—i)(n — 2)(7i-3) - n 3 . SERIES. 215 6th. The sum of n terms from formula (B) is a _ n + n JP-j) ? + n{n-i)(n- 2) ^ + n {n-i) (n-z) (n- 3) 6 . 2 2-3 2.3-4 .-. Sn = n + ?,n(n— i) + 2n(n— i)(n— 2) + \n()i— i)(n — 2) (»— 3) ~n (n + i)~ _ nnn + i)y EXAMPLES. Find the n th term and the sum of n terms of the following series, and apply the formulas thus obtained by making n = different numbers. Also interpolate terms until the formulas are familiar. I h 4, 7> 10, 13, etc. 2 3> 61, 10 , 1 3 J, 17, etc. 3 2, 7, 12, i7< etc. 4. 2, 6, i3 3 2 3 , etc. 5. i ? 2, 3. 4, 5, etc. 6. i, 3> 5- 7, 9, etc. 7- 2, 4, 6, 8, etc. 8. i, 3> 6, 10, etc. 9- A 2 2 3 2 , 4 2 etc. 10. 1 3 , 2 3 , 3 3 , 4 3 etc. 1 1. 1, 4- 10, 2 °. 35, etc. 12. i, 5> 15, 35, 70, 126, etc. 13- 14. 1, What 6, is t 21, ie 56. ten 126, 252. n, the — xst 462, term etc. term of Ex. 11 above? 15. Which of the above series gives the number of balls that can be piled in a pyramid whose base is an equilateral triangle ? Which the number that can be piled in a pyramid whose base is a square ? •210 SERIES. 1 6. How many balls can be piled in a triangular pyram having 10 balls on eacb side of the lowest tier? 17. How many in a quadrangular pyramid, having tl same number on each side in the lowest tier ? 18. How many in an oblong rectangular pile 20 balls Ion and 5 balls wide ? 19. Find the — 10th term of Ex. 12 above. 20. If a body fall 16 feet in one second, 3 times as far th next second, 5 times as far the third, and so on, how far wil if fall the tenth second? How far in 10 seconds? How fa in i\ seconds? How far in 5^ seconds? How far in 1 seconds ? 2 1 . Find from a„ = a y + {11 — i)d, Q n (a l + a n ) 6 B = -, the following Formulas for Equidifferent Series : «! = (In — (n — i)d. (1) (2) (3) 2S„ CL\ = ti — a„. ti (ti — i)d 2 «i = \ ± V(a n + W ~ ^S n . (4) a n = «i + {>i — *)d. (5) a a = — " - «,. (6) n «„=*. + <^M. (7) tl 2 a n = - - ± V2dS n + (a v -Jdf- ( 8 ) 2 10 SERIES. __ n (a, + a n )^ 2 (9) & = - [?a x + (» — i) (h, a b , a fy etc. If each term depends on one preceding term only, we have in which m is the constant multiplier. This gives On m = — • d, etc. -4ws. — I, +2. «s + s«i + l °d, etc - ^ W5 - r > — 3) + 3- 3. a 3 , a 3 + a 2 , a 3 + 2^ + a, , a 3 + 3a 2 4- 3a, + d, ai + 4a-2 + ba x + 4d, a z -\-sa. 2 +\oa } , + iod, a 3 -\-6a,+ 15^ + 20^ fl 3 + 7« 2 + 21a, + 35^ etc. ^l«5. — 1, +4, — 6, +4. 4. a 4 , a 4 + «3 , a 4 + 2« 3 + a, , a i 4- 3«s + 3«a + «i , a 4 4- 6a 3 + i5«s + 2oa i + 15^ f/ 4 + 7«a + 21a., 4-35«i + 35^ «4 + 8«3+28a. 2 4-56rt, -\-70d, a i + ga i -\-T>^ a i J <-^A a \ + 12 ^d, etc. -4res. 1, — 5, 4- 10, — 10, 4- 5. 505. The student will observe that Examples 1 to 4 are general expressions for difference series of the 1st, 2d, 3d, and 4th orders respectively, of which d, d, d, d, etc., are the constant differences. These constant differences form a series whose scale is 1, and we may call it a difference series of the zero order. 506. From these solutions we infer the following principles : i°. Every difference series is also a recurring series. 2 . TJie order of a series as a difference series is one less than the order of the same series as a recurring series. 3 . All difference series of the same order have the same scale when regarded as recurring series. 4°. These scales are the same as the coefficients of the n a poioer of a — x, omitting the first, changing their signs and reversing the terms ; n being the order of the series considered as a recurring series. SERIES, 221 507. Any recurring series not having such a scale cannot be a true difference series ; but when a series has an order of differences very nearly constant, the application of the formulas for difference series will give approximate results. For illustration take the following example: i. Given log. 200 " 210 2.30103, 2.32222, 220 = 2.34242, 230 = 2.36173, 240 = 2.38021, 2 5° = 2-39794, to find log. 205. Solution. — The given logarithms form a series, of which log. 205 is the i\ term. Finding the differences, we have, Series, 2.30103 2.32222 2.34242 2.36173 2.38021 2.39794 1st diff., .02119 .02020 .01931 .01848 .01773 2d " —.00099 —.00089 —.00083 —.00075 3d " .00010 .00006 00008 4th " —.00004 .00002 By Formula (A), 1.1 1,1.3 ai% = 2.30103 + ^(.02119) + *-* (.00099) + - — (.00010) 1-2 1-2-3 + — (.00004). 1.2-3.4 /. am = log. 205 = 2. 30103 + .010595 + .000124+ .000006 + . 000002 = 2.31175+, which agrees with the log. 205 from the table, although the series is not a perfect difference series. 2. Find in like manner log. 215. 3. Find in like manner log. 225. 4. Find in like manner log. 232. 222 SEKIES. 508. The Equimultiple Series, which is a recurring series of the first order, and whose terms are each formed by multiplying the preceding term by a constant multiplier, may be written, «i , ajn, a x m 2 , a A m z , etc., in which «, is the first term and m the constant multiplier. This obviously gives, a n — a x m n ~ x . (D) Also, S n = eh +- a x m + a,m 2 -f a x m n ~\ Multiplying by m, mS n = a-on + ajW 2 + a^n 5 . . . . + a 1 m n . Subtracting the first from the second, mS n — S„ = a x m n — a x , alld £ = ^jrJh = •>&_■=£. (E) m — i m — i ' 509. If m be a positive integer, greater than unity, the series will be increasing; and if negative, the terms will increase numerically, but will be alternately + and — . If m be a proper fraction, the series will decrease ; and if the number of terms be infinite, a n will be by Formula (D), a M = (hm™ = o, a, (m™ — i) and S~ = m — i or 8 n = -^— • (E)' i — m 510. The method of interpolation already given applies to all series when a formula for a„ can be found. (Art. 496.) It is therefore applicable to equimultiple series, and we may use for this purpose Formula (D). SERIES. 223 511. To find a term between the 4th and 5 th, make n = 4f Then a i}i = a x m ZYi or a x wfl. To interpolate two terms between the 4th and 5th, n must be made 4^ and 4! successively. 512. Interpolation may also be performed by finding the multiplier of the series formed by the interpolated and adja- cent terms. This will abbreviate the work when many terms are to be interpolated. Putting m' for this multiplier, and n' for the number of terms to be interpolated between any two terms, we have 1 m! = ?w" 7 + I . (F) Thus, to interpolate 3 terms between any two terms of the series, 1, 16, 256, etc., i_ i in which m — 16, m' = i6 3+1 = i6 T = 2. If these terms be inserted between 16 and 256, the series will be, 16, 32, 64, 128, 256. EXAMPLES. 513. Perform the examples, p. 215, by the principles of recurring series, so far as they are applicable; also the follow- ing : 1. Find «2o and S. i(t , also a n and S n in 1, 2, 4, 8, 16, etc. 2. Find a w and #10, also a n and S„ in 3, 9, 27, 81, etc. 3. Find am and S 10 , also a n and S„ in 5, 10, 20, 40, etc. 224 S E K I E s . 4. Find a a and 8 m in 8, 4, 2, 1, etc. 5. Find «„, and #„ in 4, 1, J, etc. 6. Find m! for interpolating a single term between any two of the last series, and find what the term between the 5th and 6th will be. 7. Interpolate two terms between each two of the series 1, 8, 64, etc., by finding m'. 8. Find a w and S i0 , also a B and S 5 of the series h — 3> +9. ~ 2 7, + etc. 9. Find the scale of 1, 2X, 3Z 2 , 5.T 3 , io.r 4 , 21.T 5 , 43Z 6 , and carry the series to the 10th term. 10. Find the scale and a n of the series, 2, 5.T, 8a; 2 , 1 1. r 3 , 142c*, 17a; 5 . 11. Find from q^m* — 1) * = m-i ' the following Formulas for Equimultiple Series: 4 = ~ (1) a, = \ years, allow- ing compound interest at 4%. 9. Find the sum that will amount to $1000 in 5 years, on interest at $%. 10. Find the present worth of an annuity to commence in 2 years and to continue 10 years. 11. Find the present worth of an annuity in arrears 3 years and to continue 7 years longer. Note. — Use formula from Problem 6. 12. Find the present worth of a perpetual annuity of 8100 to commence in 5 years, at 6% compound interest. 13. The same as above, to commence now. 14. The same as above, to commence 5 years ago. 15. Find the time when an annual payment of $100 should have begun to cancel at maturity a note for $500 given Jan. 1, 1875, and due Jan. 1, 18S0, with compound interest at $%. 16. A man travels from a certain point northward 10 miles the first day, 9 miles the second, 8 miles the third, and so on, continuing the series by the same law. How far north will he be at the end of 5 days? How far at the end of 10 days ? 11 days? 20 days? 22 days? 30 days? How far north will he travel the 15th day? 17. A man travels as above 10 miles the first day. and each succeeding day -^ as far as the day before. How far north will he be in 5 days? In 10 days? How long would it require for him to travel 25 miles? How far would he travel if he should continue forever? SERIES. 229 18. A ball falling from the height of ioo feet rebounds 50 feet. If it continue to rebound one half the distance it falls, how many times will it rebound ? How far will it move before coming to rest ? 522. The following identical equation i m(m+p)(m + 2p) . . {m + rp) rp \ m{m+p) . . [m + (r—i)p~\ « I (m+p) (m + 2p) . . (m + rp) \ (E) furnishes a method for the summation of series whose terms are of the form of the first member. The following examples will illustrate : 1. Find 8 n and S m of — + -*- + — + etc. 1.2 2-3 3-4 Solution. — Here a = 1 ; m = 1, 2, 3, etc. ; p = 1 ; and r = I. Substituting in 2d member of (E), S, ' n = \(\ + l + i . . . + - -i-i ... - — ) 1 V n - n 11 + 1/ — 1 — = . Ans. n+i n + 1 If n = co , 1 = 1 = I : n+l co 2. Find S„ of -i-g + ^— + — --7 + etc. 3-8 6-i2 9-16 Solution. — Multiplying the series by 12 gives 111 - + H + etc., 1-2 2-3 3-4 which is the same as in Ex. 1. £L = A. 230 SERIES. 3. Find S„ of ^— + --- - 5 — 4- — -_ + etc 1-2-3 2-3-4 3-4-5 Solution, a — 4, 5. 6, etc. ; to = 1, 2, 3, etc. ; p = 1 ; r = 2. Substituting in (E), 5. = I (-*- + -5- + -^ + -*- + etc. - -4- - -A. - -5_ _ etc.) \i -2 2.3 3-4 45 23 3.4 4.5 / = A (— - + — 4- — + — + etc. ). "Vi -2 2-3 3-4 4-5 / Applying (E) again to this series, beginning with the term , we have, a — 1 ; to = 2, 3, 4, etc. ; p = 1 ; r = 1. $» = i + I +t + i + et c- - s - 1 - 5 - etc = f 5 = h (— + -) = f- -4«»- - \i • 2 2/ 4 Find #,„ of the following: 1 4 7 4. H 1 £ — + etc. i-3-5 3-5-7 5-7-9 5. -i ^-+^ 5_- + ete. 3-5 5-7 7-9 9 ' IJ 1 1 1 6. 1 h -- - 4- etc. i-3 -'-4 3-5 1 1 1 7. Y etc. i-3 2-4 3-5 8. -1- 4- -±- + — i-- 4- etc. i-5 5-9 9-^3 1 1 1 + - + - -2 + etc. 1 • 2 • 3 • 4 2-3.4.5 3-4-5" 6 1 1 1 io. l. _ 1 4- etc. 1 • 3 • 5 * 7 3-5-7-9 5 • 7 • 9 " T T I 2 2 2 3 2 11. — + - H 2 -, 4- etc. 1.2.3.4 2-3.4.5 3 -4. 5 • 6 I 2 2 2 3 2 12. y- - H — + etc. 1-3-5-7 3-5-7-9 5 -7 -9 • " SERIES. 231 REVERSION OF SERIES. 523. When we have y = a series which is a function of x, finding x as a function of y is called Reverting the Series. Given, y = x + x 2 + x 3 + a; 4 + x 5 -f etc., to revert the series. Solution. — Assume x = Ay + By 2 + Cy 3 + Dy* + etc. Substituting this value of a; in (i), y = Ay + B + A 2 if + C + 2AB + ^l 3 y 3 -\- D \y i + etc. + B 2 + 3^ 2 5 + J 4 Hence, Art. 419, A = 1 , 5 + .42 = o; .-. B = -1. C + 2.-l£ + ^l 3 = o; .-. C = 1. D + 2 AC+B 2 + sA 2 B + A i = o; .-. # = — 1. The law of the series is evident and we may write x = y — i f + y 3 — y* + f — etc - (0 (*) EXAMPLES. Revert the following series : 1. y = x + 2X 2 + 3.r 3 -f 4.T 4 + etc. 2. y = 1 — x + x 2 — x 3 + .r 4 — eti 3. ?/ = x — \x 2 + Ice 8 — \x^ + etc. 4. y ■= 1 + x + 2x 2 + $x 3 + etc. CHAPTER XX. LOCI OF EQUATIONS. 525. We have seen that an equation with two or more unknown quantities is indeterminate ; that is, there are no definite values that can be assigned as the only values of these quantities. (Art. 241.) For example, in the equation x + y = 5, we may have x = 1 and y = 4, x = 2 and y = 3, x = 3 and y = 2, and so on indefinitely. And x not only may be any integral number between + 00 and — co, but it may have any value, integral, frac- tional, or incommensurable between those limits. Hence, x and y are variables, and pass from one value to another by infinitesimal increments ; that is, they pass through all inter- mediate values, as a point moving from one position to another along a line passes through all intermediate points. As the number of these points is infinite, so the number of different values of a variable is infinite. (Art. 390.) 526. The relation of an equation containing two variables, to a line considered as the path of a / point moving in a plane surface, is more fully illustrated by the following method. e/P d/ Fig. I. 1$ Q 527. Assume the two lines XX' and YY' (Fig. 1) at right angles to each other, intersecting at A. Take any equation with two variables, as y = 2X + 3. LOCI OF EQUATIONS. 233 In this eqn< ition, If x = o, y = 3- If X = — I, y = : I. If X = I, y = 5- If X = 2, ?/ =■ —I If X = 2, y = 7- If £C = —3, y = — 3 If * = 3, y = 9- If as = — 4j 2/ = -5 If x = 4, y = ii- If x = — 5, y = -7 If x = 5, !/ = J 3- If x = — 6, y = —9 If, now, we adopt some convenient unit, and measure the positive values of x from A towards X (the negative values being of course measured in the opposite direction), we shall find on XX' the several points i, 2, 3, 4, etc., — 1, —2, —3, —4, etc. If from each of these points we erect a perpendicular equal to the corresponding value of y, we shall have the lines bi, C2, d3, and b' ( — 1), c' ( — 2), d' ( — 3), etc. If we have carefully drawn the lines to the proper measure, we may now place a ruler upon them, and connect all the points a, b, c, d, etc., by one straight line PQ. We may also take fractional values for x and find correspond- ing values for y from the given equation, and all the lines representing the values of y will terminate upon this line PQ. In like manner, any point on the line PQ represents a set of values for x and y by its distances from the lines YY' and XX'. Hence, the equation y = 2x + 3 is said to be the equation of the line PQ. 528. It is important for the student to become familiar with the definitions of the following terms used in discussions of this kind. Def. 1. The Axes, or Axes of Reference are the assumed lines XX' and YY'. 2. Tile Or iff in is their point of intersection A. 3. The Axis of Abscissas is the line XX'. 234 LOCI OF EQUATION'S. 4- TJie Axis of Ordinates is the line YY'. 5. The Ordinate of a point is its distance from the axis of abscissas. Thus the ordinates of b, c, d, etc., Fig. I, are bi, C2, d3, etc. 6. The Foot of the Ordinate is the point where it meets the axis of Abscissas. 7. Tlte Abscissa of a point is the distance from the origin to the foot of its ordinate, or the distance of the point from the axis of ordinates. 8. The Co-ordinates of a point are its abscissa and ordinate. 9. TJie Locus of an Equation is the line which the equation represents. 10. Constructing a Locus is drawing the line (as shown above) represented by an equation. 11. Abscissas are positive when measured to the right, and negative when measured to the left of the axis of ordinates. 12. Ordinates are positive above and negative below the axis of abscissas. 529. The four parts into which the plane is divided by the axes are called the first, second, third, and fourth angle respectively, beginning with the angle on the right of the axis of ordinates and above the axis of abscissas, and going round to the left. Thus, YAX is the first, YAX' the second, VAX' the third, and VAX the fourth angle. 530. Abscissas are usually represented by the letter x, ordinates by g. In the first angle x and g are both positive. In the second angle x is negative and g positive. In the third angle x and g are both negative. In the fourth angle x is positive and g is negative. LOCI OF EQUATIONS. 235 531. Construct the loci of the following equations : 2. y — 2>x + 2. 5. y = — 3 x + 2. 3. y = 2X — I. • 6. y = — x. 4. V = 3»- 7- 2/ = 2. Note. — The last gives a line every point of which has its ordinate 2. Equations of the first degree always represent straight lines. They need not be in the form given above, but may for convenience be put in that form before constructing the loci. Construct the following loci : 8# x-y _x + y_ 2 6 9- 2X — 3?/ = 3 10. X 5 =0. 2 y 532. When we make x = o, we find the point where the line crosses the axis of ordinates, aud when y .= o, the point where it crosses the axis of abscissas. y = 2x + 3, Thus in the equation if y = o, we have, 2X + 3 = O, and x = — f. This is the distance A (— f) (Fig. I), and it is the root of the equation 2.v + 3 — 0. Thus we see how we may construct the roots of equations having one unknown quantity. 11. Construct the root of 3 SOLUTION. Put the equation in the form, x — 4 x — 2 x 236 X — 4 x — 2 X — 8 Make y = ; — , 2 3 6 * and construct the locus. The abscissa of the point where this locus cuts the axis of the abscissas will be the root required. 236 LOCI OF EQUATIONS 533. It appears from this, that equations of the first degree should give straight lines; for if the line could cut the axis of abscissas a second time, a second root would be found, which is not possible for such an equation. (Art. 232.) 12. Construct the locus y = x z — 2X 2 and find the real roots of x + 2, x z — 2a; 2 — x + 2 = o. Note — It is to be observed that the imaginary roots cannot be thus constructed. Fig. II. (\ I 2 Fig. Ill Fig. IV Assuming the axes XX' and YY' (Fig. II), and making X = 0, y = 2 ; X = I 2 J y = 9 . X = I, y = ; X = —I, y = ; X = lh y = 5 • — s > X = -I* y = — 4|; X — 2, y = °; X = — 2, y = —12; X = 3' y = 8; X = 4, y = 3°- LOCI OF EQUATIONS, 237 The roots required are 2, 1, and — 1, represented by the distances (A2), (Ai), and [A (— 1)]. This example illustrates the three roots of the equation of the third degree, showing how the line is cut by the axes of abscissas three times. 13. Construct the locus x s + 3X z + 6x + 1 = y, and find the value of x when y = o. (See Fig. III.) Fig. VI Fig. VII. UJ lY Making X = 0, y = 1; X = 1, y = n; X = — I, 2/ = -3; X = 2, ^ = 33 5 X = — 2, # = —7; x = —3, 2/ = ~ l 7 By constructing these points and others intermediate, by making x = \, — \, etc., and sketching in the curve to join the points, we may measure the distance AB, which will give us approximately the real root of the equation. The construction also shows the other roots to be imaginary. 238 LOCI OF EQUATIONS. In like manner construct and find real roots of y = x 14- IS- 16. 17- 18. 19. r 2 - 5. (See Fig. IV.) y = a? — 2x* + 1. (See Fig. V.) y — & _ q& _ 3 x + 18. (See Fig. VI.) y — tf — 2x* + i. (See Fig. VII.) y = x* - 2X* — 1. (See Fig. VIII.) y — X 4 _ 2X 2 + 2. (See Fig. IX.) 20. y = x 5 — 3a 4 — 5» 3 + i5z 2 + 4£— 12. (See Fig. X.) Fig. VIII. -x- UJ Fig. IX Fig. X Note. — The object of this chapter is to furnish illustrations of the principles to be developed relating to the general Theory of Equations, and not to give a practical method of finding the roots of equations. CHAPTER XXI. THEORY OF EQUATIONS. 534. When a problem involving but one unknown quantity produces an equation of the first or second degree, we can easily solve it in a general manner, by using general symbols, as letters, for the known quantities, thus obtaining a formula by which the unknown quantity may be found for all special cases of the problem, by mere arithmetical computation. Such formulas were obtained in Arts. 240, 340. 535. But when a problem gives rise to an equation of the third or fourth degree, the same may be done, but with much more difficulty, except in special cases. Indeed, so complicated is the reduction of the general equation of the fourth degree, that it is seldom employed in practice. 536. For equations above the fourth degree, no general method of reduction has yet been found. But when such equations arise, by putting for the known quantities numerical values, instead of general symbols, thus forming equations with numerical coefficients, called numerical equations, the real roots may be readily found. By this method, the arithmetical part of the solution of the problem is performed before the algebraic, and the equation must be reduced for each special case of the problem. 537. The object of the present chapter is the discussion of the methods employed to find the real roots of numerical equations of a higher degree than can be readily reduced by the methods already given. Observe we do not say methods of reducing higher equations, for the reduction of an equation 240 NUMERICAL HIGHER EQUATIONS. implies the use of Axiom i, Art. 38, in such manner as to bring out the unknown quantity as an explicit function of the known quantities. This is not done with higher equations; but by various devices depending on the General Theory of Equations;, the real roots are discovered without a process of reduction. 538. To facilitate the discovery of the real roots of Higher Numerical Equations, they are reduced to the form x n _j_ A&n- 1 + AaJK~ % i n _,x + A m = o. (1) in which Ai, A i} etc., and n are integral and the exponents are all posit ire. The reductions necessary to give this form to an equation may be any or all of the following: 1. To make the exponents positive. 2. To make the exponents integral. 3. To make the coefficient of x n unity. 4. To make the coefficients A x , A i} etc., integral. These transformations may be made as follows : 539. To Make the Exponents Positive. Eule. — Multiply the equation byxvrith a positive exponent equal numerically to the largest negative exponent in the equation. Note. — This will evidently accomplish the desired transformation, and will not affect the roots of the equation, since both members are equally affected. It is in fact only clearing the equation of fractious. 540. To Make the Exponents Integral. Rule. — Multiply each exponent by the least amnion multi- ple of the denominators of the fractional exponents. This will obviously make the exponents integral, but it will at the same time change the roots of the equation. It will NUMERICAL HIGHER EQUATION'S. 241 therefore be of no advantage to find, the roots of the trans- formed equation, unless we can ascertain what function its roots will be of the primitive roots. This we may easily do, for if m be the common multiple used, the transformation will be made by substituting y m for x. Thus, Let the original equation be ill x a + A x x h -f A,x c + A z = o. Substituting y m = x, III {y m f + A l {y™) h + A i (tf»y+ A 3 = o. Eeducing, m m m if + A^xf + A*y~ e + A, = o. If m be made a multiple of a, b, and c, that is, abc, the equation becomes y bc + A^y™ + A,y ab + A 3 = o, in which each exponent has been multiplied by m = abc. But since y™ = x, the values of y must be raised to the m th power to give the values of x. Hence, When the exponents of x in an equation are multiplied by m, the roots of the transformed equation raised to the m th power will be the roots of the primitive equation. 541. To Make the Coefficient of ,x n Unity. Rule. — Divide the equation by that coefficient. This evidently does not affect the roots. 11 242 NUMERICAL 11 I < MI E R E Q U A T I O N S . 542. To Make the Coefficients integral without Changing the Coefficient of ae 71 . Utile. — Multiply the coefficient of x n ~ x by kj that of x n ~ 2 by J&; that of aP~ 3 by k\ and so on, to the coefficient of .'". making k such a number as will render each coefficient integral. v This rule is obtained by substituting | for x, and clearing of fractions ; thus, V Substituting t = « in (0> yn yn-1 yn-2 f- + AA : + AA ; J . . . . 4- A n = O. Clearing of fractions, if + A,ky n ^ x + AJcttf 1 -* . . . . + AJ n — o. If, now, any of the coefficients, A x , A,, etc., are fractional, a value for k maybe taken such as to remove the denominators. Since V 7 I = x, y = kx. That is, the roots have been multiplied by k. and when found must be divided by k. to give the primitive roots. EXAMPLES. Reduce the following equations to the form (i), B 4 l I. 2X* — $x* + 7£ 3 -| = o. Solution. — Multiplying the exponents of x bv 3, we have, 2y 5 - $if + Hi - \ = o, in which y z = X. (Art. 540.) Dividing by 2 gives, y- - f*/ 4 + ly ~ I = o. (Art, 541.) NUMERICAL HIGHER EQUATIONS. 243 Applying Rule, Art. 542, ,5 5* 4 , 7**. 3* 5 s- 1 g* + — 2 — — — = O, 228 in which z = ley. Making k = 2, to cancel denominators, z 5 — 5s 4 + 56s — 12 = 0. in which z = 2y, and since y 3 = x, z = 2xK 2. 32^ — 5$* -j- 8a; — 2$ — 1 =0. 3- 5* r * — 3«* + 9^* + 5 = o. 4. |aj* — iJ + |o£ — \ = o. 543. We have thus seen how any equation with a single unknown quantity having rational coefficients and exponents, may be reduced to the form (1) (in which the coefficient of x n is unity, the other coefficients are integral, and the exponents integral and positive), without making any unknown change in the roots. We now proceed to discuss this equation, for the purpose of discovering its real roots. For the sake of brevity we shall use for it the symbol f(x) = o. I. DIVISIBILITY. Theorem I. 544. If f(x) be divided by x — a, the remainder will be f{a), that is, it will be what f{x) becomes when a is substituted for x. Demonstration.— Let q be the quotient and r the remainder obtained by the division. Then f(x) = (x — a)q + r, an equation true independently of the value of x. It is therefore true when x — a. Substituting a for x, we have, /(«) = r. 244 NUMERICAL HIGHER EQUATIONS. Cor. i. — When a is a root of f(x) = o, fix) is divisible by x — a, and not otherwise. For if a be a root, f(a) = o. (Art. 217.) .*. r = o, and the division is complete. But if a be not a root, f(a) is not o,and r is not o. and the division is not perfect. Cor. 2. — If a be a root of f(x) = o, a is a factor of the absolute term. For, it is evident that if f(x) is divisible by x — a, the absolute term is divisible by a. 545. This theorem gives an easy method of determining whether any number be a root of an equation. Try what factors of the absolute terms are roots of the following equations : 1. x 3 — &x 2 + n.c + 20 = o. 2. x 5 — x 4 — 252^ + 85a; 2 — 962; + 36 = o. 3. .t 5 — 1 = o. 4. a 4 — 2X 5 + 8x — 16 = 0. 5. x 3 — iiz 2 + 433 — 65 = o. 6. x 5 — ix 3 -\- 2X -\- 15 = o. In making these trials, use the method of synthetic division. (Art. 151.) Thus, to try —3 in Ex. 6, we write, 1+0 — 7 + 0+ 2 + 15 I — 3 — 3 + 9 — 6+18 — 60 — 3 + 2 — 6 + 20 — 45. remainder. The remainder is —45. and —3 is not a root. 546. "We have also from this theorem a method of deter- mining the value of a function when any number is substituted for x. Thus, when — 3 is substituted for x in Ex. 6, the first member reduces to — 45. Note. — It should be observed that this theorem is applicable to all equations and not merely to those reduced to the form of (1). Find the value of the first member of each of the preceding equations, when 1, 2, 3, — 1, — 2, and — 3 are substituted for x. NUMERICAL HIGHER EQUATIONS. 245 II. NUMBER OF ROOTS. Theorem II. 547. Every equation of an integral degree has the same number of roots as there are units in its degree. Demonstration. — To prove this theorem, it is necessary to assume that every such equation has at least one root, since the proof of this is not within the scope of elementary Algebra. But with this assumption we may easily prove the theorem, for, representing this root by ctu and dividing the first member, when put in the form f(x) = o, by x — a x , we have (x—afiq =f(x) = o, which is satisfied when x — a\ = o, or when q = o. But q is a function of x one unit lower in degree than the original function, and q — o has by the assumption one root, which may be divided out as before, and this process may be continued so long as q is a function of x ; hence the theorem is true, and f(x) — o has n roots and no more. Cor. i. — The first member off(x) = o is the product of the n binomial factors, (x — «j) (x — a. 2 ) (x — a 3 ) . . . . (x — a n ), in which a 1} « 2 , etc., are its roots. Note. — It is not to be understood that n\ , a?, etc., are all different numbers, for there is nothing in the proof of the theorem to forbid their being all equal. III. FORMATION OF EQUATIONS. Cor. 2. — An equation mag be formed, having any roots whatever, by subtracting each root from x, and putting the product of the binomials thus formed equal to zero. (Cor. i.) Cor. 3. — The coefficients of the several powers of x in the first member of f(x) = o, will be as follows : Of x n ~ x , the sum of the roots with their signs changed. Of x n ~ 2 , the sum of the products of the roots taken two and two. (Art. z&T,.) 246 NUMERICAL HIGHER EQUATIONS. Of x n ~ 3 , the sum of the products of the roots, with their sign* changed, taken three and three ; and in general, The coefficient of x" i> will be the sum of the products of the roots, with /heir signs changed, taken p at a time. This follows from the binomial formula, in which it is evident that the coefficients are formed in the same manner as in / ■')• Observe that in forming f(x) by multiplying n binomial factors, the second terms of these factors are the roots with their signs cfiangi d ; hence the products which form the several coefficients are all formed by the mots with their signs changed, but those products which have an < ven number of factors, as those taken, 2, 4, 6, etc , at a time, will have the same sign whether the signs of the roots be changed or not. By this corollary an equation having any given roots may be formed by changing the signs of the roots, and forming the several coefficients in accordance with this law. Cor. 4. The absolute term or coefficient of x° is the product of all the roots ivith their signs changed. Note— Let the student notice how these principles applied to equations of the second degree give the same results as those already obtained in Chapter XI, Art. 329. 548. The truth of Theorem II is illustrated by the loci of equations, where the number of roots is represented by the number of times a straight line can cut the locus of the equa- tion. This will be seen, by an examination of the loci already constructed, to be equal to the number of units in the degree. Observe that it is not the number of times the axis of abscissas cuts the locus, for these are the real roots only, but the number of times that any straight line can cut the locus includes the whole number of roots, real and imaginary. EXAMPLES. Form the equations having the following roots, both by Cor. 2 and Cor. 3. 1. 5, 3, 3> — 1, — 3> — 5- 2. a, b, c, d. 3. 2, 1 + V— 3> 1 — \ /Z= ~3- 4. 2, — 2. — 3 + V— 2 > — 3 — V— 2. 5. 2 ± a/^5> 3 ± V- 7.- — 1 ± V— i- NUMERICAL HIGHER EQUATIONS. 247 IV. FORMS OF ROOTS. Theorem III. 549. The imaginary roots of f(x) = o will be found in conjugate pairs. That is, if a+ -\/ —b be a root, a — -\/—b will also be a root. Demonstration. — Since the sum and the product of the roots are both real, there cannot be an odd number of imaginary roots, nor an even number, except when found in conjugate pairs whose sum and product are both real. Cor. i. f(x) will have a real quadratic factor for each pair of imaginary roots. For [x - (a + \/~b)~] x [x -(a- \/~b)] = (x - a? + b. Cor. 2. f {x) may be separated into real factors ; of the first degree, corresponding to each real root ; and of the second degree, corresponding to each p air of imaginary roots. Cor. 3. — The coefficients off(x) maybe so changed as to give two equal or two unequal real roots in place of each pair of imaginary roots. For, the factors which produce the quadratic factor (x— af + b are imaginary when b is positive, real and equal when b is zero, and real and unequal when b is negative. Bat a change of b from positive to negative through zero will produce no change in f(x), except in its coefficients. Hence, the coefficients may be so changed as to require b — o, or b < o. Cor. 4. — The product of the imaginary roots of f(x) = o is always positive. For, (a + \/— b) (a — \/— b) = a i + b, which is always positive. Hence, Cor. 5. — In an equation whose roots are all imaginary, the coefficient of x° will be positive. 248 NUMEEICAL HIGHER EQUATIONS. Cor. 6. — Every equation of an odd degree has at least one real roof, opposite in sign to the absolute term, but an equation of an even degree mag have all its roots imaginary. Cor. 7. — An equation of an even degree toliose absolute term is negative, tuill have at least two real roots, one positive and one negative. This theorem and its corollaries are illustrated by the loci of equa- tions. For example, Figs. VII, VIII, and IX represent a locus of the 4th order in different positions with reference to the axis of abscissas. By reference to equations (16), (17), and (18) (Art. 5341, which give those different positions, it will be observed that they differ only in the abso- lute term, an increase in this term carrying the locus upward and a decrease downward. This is evidently as it should be, for a change in the absolute term produces a corresponding change in each of the ordi- nates. It appears therefore that a curve of an even degree, going as it does to infinity in only one direction and that upward, may be so placed, by giving a proper value to the absolute term of its equation, that the axis of abscissas will not cut it at all, as in Fig. IX, making all the values of x imaginary when y = o. Or, as in Fig. VIII, the axis may cut the curve twice only, giving two real and two imaginary roots. In passing from the position Fig. IX to Fig. VIII, it would pass the position Fig. VII, in which the roots are all real, two positive and two negative, the positive roots being equal, and also the negative. By making the absolute term of the equation i, a position for the locus would be found in which the roots would all be real and unequal. It appears also from this illustration that when the equation is so changed as to drop out a real root, it must drop out two such roots ; hence the number of imaginary roots will be even. Fig. X shows a curve of the oth order, or one whose equation is of the 5th degree. This curve goes to infinity both upward and downward, as do all curves whose equations are of an odd degree; see Figs. II, III, V, and VI. These curves cannot be moved up or down so far that they will not be cut at least once by the axis of abscissas. The roots of their equations, therefore, cannot all be imaginary. In equation 19 (Fig. X), making the absolute term —25 will make all the roots but one imaginary, and that will be positive; making the absolute term 1 or any greater number will leave but one real root, and that will be negative. NUMERICAL HIGHER EQUATIONS. 249 Theorem IV. 550. If f (x) has all its roots imaginary, it will have a positive value for every real value of x. Demonstration. — By the supposition, f(x) is the product of factors of the second degree, each formed from a pair of imaginary roots ; as, (a; — a + -y/— b)(x — a — -y/— b) = (x — a)- + b. Each of these factors is + for all values of ,r ; hence their product is +. Notice how this is illustrated hy Fig. IX. The curve being all above the axis of abscissas, when the roots are imaginary the ordinates are positive for all values of x. Cor. — The sign of f (x) for any real value of x will depend on the real roots. Theorem V. 551. The equation f(x) = o can have no fractional root. Demonstration. — Let - represent a fraction reduced to its lowest terms, and suppose this fraction to be a root of f(x) = o ; then substi- tuting, we have, a" . n n ~ l . a n ~ 2 . j- + Ax -— r + A* =— = + + An = O. b" b"~ l o n ~ z Multiplying by b"-\ y- + A iffl"- 1 + Aia n ~ 2 b +.... + Anb"- 1 = o. b All the terms after the first in this equation are integral, and the first term is an irreducible fraction. But the sum of integers and an irre- ducible fraction cannot be zero. Hence the last equation is absurd, and - is not a root of f(x) = o. Cor. — The real roots of f(x) = o will be either integral or incommensurable, since these are the only real quantities, except fractions. Note. — Incommensurable Hoots are such as cannot be meas- ured with the unit of measure employed, and therefore require an infinite number of terms to express them. Such a quantity is y/2, which gives rise to an endless decimal. A Fraction is one or more of the finite parts of a unit, and can always be expressed by a finite numerator and denominator. 250 NUMERICAL HIGHEi; E Q U A T J X S . V. SIGNS OF THE ROOTS. Theorem VI. 552. Changing the signs of the alternate terms of f{x) = o changes the signs of its roots. Demonstration. — By Theorem II, Cor. 3, changing the signs of the roots changes the signs of the alternate terms, beginning with the term containing x n ~ *. But changing the signs of the alternate terms, begin- ning with x", gives the same equation by afterwards changing all the signs, which does not affect the roots. Or the theorem may be proved as follows : If +a and —a be substi- tuted for x in /{.' , tin' only difference in the result will be opposite signs for the terms containing the odd powers of x. Hence if a lie a root of f(x) = 0, —a will be a rout of the same equation with the signs of the odd powers of x changed, or with the signs of the even powers changed, since this will give the same result. Theorem VII. 553. In f{x) = o, the number of positive roots cannot be greater than the number of variations if sign, nor the number of negative roots greater than the permanences. Note. — When two consecutive signs in an equation are alike, it is called a Permanence ; when unlike, a Variation. Demonstration. — Let the signs of f(x), taken in their order, be + — ++ — +. If now a new positive root (a) be introduced, f(x) will be multiplied by x— a. This multiplication will give, using only the signs, + — H- + — + + — + — ++ — + — + + — + — + ± — + — from which we see that using either sign, where there is an ambiguity we have added a variation. In like manner, multiplying by + + will add a permanence. NUMERICAL HIGHER EQUATIONS. 251 Cor. i. — If the roots of an equation be all real, the number of positive roots ivill equal the number of variations, and the number of negative roots the number of permanences. For, in that case tlie whole number of roots will equal the whole number of permanences and variations together ; hence by the theorem the corollary must be true. Cor. 2. — If an equation be incomplete, and the signs before and after the missing term or terms be alike, the equation must have imaginary roots. For the intervening terms whose coefficients are zero may be either + or — , and we shall have, when there is one missing term, + ± + or — ± — , and in either case we may count either two permanences or two variations. But the positive roots cannot be greater than the least number of variations, nor the negative roots greater than the least number of permanences ; hence the two cannot be equal to the degree of the equation. For example, in the equation X s + 33J 3 — x + 7 = o, we shall have the signs + ± + ± — +. From which, counting as few variations as possible, we have 2 ; hence there cannot be more than 2 positive roots. Counting as few perma- nences as possible we have i, and there cannot be more than i negative root. But there are 5 roots in all, hence there must be at least 2 imaginary roots. Cor. 3. — If two or more consecutive terms are wanting, the equation will have imaginary roots whatever may be the signs of the preceding and following terms. Eow many positive and how many negative roots can the following equations have, and how many of their roots must be imaginary ? 1. .r 7 — 51 4 + 6x 2 - 5/ I 1 = o, 2. x 6 — 2a* 2 + 5=0. 3. X 5 + X +1=0. 4. x 5 — x 2 — 5 =0, 252 NUMERICAL HIGHER EQUATIONS. VI. LIMITS OF ROOTS. 554. A Superior Limit to the roots of an equation, is a number known to be larger than the largest root. 555. An Inferior Limit is a number known to be less than the least root. It is sometimes convenient to find such limits which shall confine the search for roots within as narrow bounds as possible. This may be done by the following theorems : Theorem VIII. 556. If, in f(x) = o, two different numbers (not roots) be substituted for x, the signs of the result will be alike when there is an even, and unlike when there is an odd number of real roots situated between the numbers substituted. Demonstration. — Let (as— a^ {.r—a,) (x-a 3 ) . . . (x—a n -j,) (i) (p being the number of imaginary roots) be the product of the factors of f\.r), which are formed from the real roots of f(x) = o. The sign of this product tor any value of x will be the same as of /(.n for the same value of x ; for the product of the imaginary factors is always +. (Art. 550). Let it be supposed that the roots a lt a.,, a A , etc., are in the order of their magnitude, a x being greatest, a % the next, and so on. If, now, a number {a') greater than a, be substituted for x, the factors of (1) will all be positive, and the result will be positive ; but if a number {a") less than ", and greater than a. 2 be substituted for .r, there will be one negative factor, and the rest will be positive. The result will therefore be negaticc. In like manner if the number substituted be diminished till it is lrss than a s , the result will change its sign again and become positive. Thus we see that for every root which is passed in diminishing the number substituted, the sign of the result changes, an even number of changes giving like signs and an odd number unlike. If there be two or more equal roots they will be passed at the same time, but this does not affect the truth of the theorem. NUMERICAL HIGHER EQUATIONS. 253 This theorem is also illustrated by the loci of equations. Referring to the preceding chapter it will be observed that the ordinate of any point of a locus is the value obtained by substituting the value of the abscissa for x in /(«). By taking any two abscissas, it will be seen that the corresponding ordinates have opposite signs when the number of intermediate roots is odd, and the same sign when that number is even. Note. — It must not be forgotten that zero is an even number. Cor. — If a number less than the least root of f(x) = o be substituted for x, the result will be positive when the equation has an even number of real roots, and negative ivhen an odd number. Theorem IX. 557. If A m be the largest, and A h the first negative coeffi- dent of f(x) = o, then {A m ) h + i is a superior limit to its roots. Demonstration. — Any value of x which renders f(x) positive, and of which it can be shown that all greater values will render it positive, is a superior limit to the roots. Suppose all the terms after the first negative term to be negative, and the coefficients of the negative terms each to be equal to A m - As this will be the most unfavorable case pos- sible, if we can show that when i x = or > (A,,,)* + i, (i) then x" > A m (.i:"- ft + as*-*- 1 . . . x + i), (2) the theorem will be proved, forf(x) will then be positive. Subtracting 1 from each member of (1) and raising it to the h t!l power, (> = or > A m , or (,r — i)*- 1 (x — 1) = or > A m ; X h ~ l (X — I) > Am, x -(h-\) and 1 > Am . x — 1 Multiplying by x n , x n > A or x n > A - 1 X>'~h+\ — 1 254 -\ r m e r i c a l ii i <; he:; e y u a t i o x b . Making the division, X» > A m (X'~ ! > + X"-^- 1 . . . X + i). 1 Hence, (A m ) h + i is a superior limit to the roots off(x) = o, A,„ hi inn the greatest negative coefficient and h the number of terms preceding t/w first negative coefficient. Cor. — //" the signs of the alternate terms of an equation he changed, a superior limit to the roots of the resulting equation will, by changing its sign, become an inferior Until to the roots of the original equation. (Theorem VI.) VII. LIMITING EQUATION. 558. In the following discussions,, /'(•'') represents the first differential coefficient of fix), and (.< ) the greatest common divisor of f{x) and f'(x). Theorem X. 559. The real roots of f (x) = o are situated between those of f{x) = o; that is, if a t , a z , a 3 , etc., are roots of _/'(./•) = o, then f'(x) — o has a root between a t and a», also between a 2 and a z , and so on. Demonstration. — Suppose tlie roots of f(.r) to be real. Then Vve have, /(.?■) = (x—a x ) (x—a z ) (jj— a 8 ) .... (x-a,,). f(x) = (x—a 2 ) (x-o.,) ( since one of its roots is between the two equal roots of f(x) = o. But the change in the absolute term of the equation of the locus, which caused the axis of abscissas to pass through m, makes no change in f'(x). (Art. 413, Rule II.) Hence, The roots of f'(x) = o are the abscissas of the points m, m', etc. VIII. EQUAL ROOTS. Theorem XL 561. If f(x) = o has m roots each equal to a, f (x) = o will have m — i such roots ; and (x — a) m ~ l will be a common divisor of f(x) and f (x). Demonstration.— This is evident from Theorem X, for if any two roots of /(.r) = o become equal, the root of f'(x) = o between them must be the same, and if m roots of f(x) = o become equal, m — 1 roots of f'(.r) — o must also be the same. This will obviously give (x—a)"'~ l as a common divisor of f(x) and f'(x). This common divisor will have other factors if f(x) has other equal roots, and not otherwise, 256 NUMERICAL HIGHER EQUATIONS. This theorem may aJso be proved by reference to the form of f{x) and f'(x), under Theorem X. A comparison of these functions shows: That if a lt a.,, a 3 , etc., are all unequal, tbere is no common divisor of the two, for each factor of f(.r) is wanting in some one term of / ./ :). But if any two of the roots a l , a.,, a a , etc., are equal, making two of the factors of j\x) equal, then that factor will be found in every term of f'(x), an( l "iU therefore be a common divisor of the two functions. So if three of the factors of f(.r) are equal, that factor would be found twice in f\x), and therefore twice in the common divisor. Also if there be two sets of equal factors, the same would be true of each. Cor. i. — If f(x) = o has no equal roots, there is no com- mon divisor of f(x) and f'(x). Cor. 2. — If (p (x) = o has m roots each equal to a, f(x) = o will have w, and f{x) = o, m + i such roots. Cor. 3. — If 4>(x) = o has no equal roots, f(x) is divisible by (p(x) twice, and each of the roots of (x), and '(x) = 3X' 2 — 2.i' — 8, we find that divisor to be x — 2. Putting x — 2 = o, we have x = 2, one of the roots of (x) = o by x — 2, we have x- + x — 6 = o. Fi'om this we get x = — i ± y^6 + i = — £ ±f = 2 or — 3. We have now the three roots of

(x) = o, viz. : 2, 2, and —3, we know that 2, which is twice a root of

(x) = o, is twice a root of f(x) = o. ( Theorem XI, Cor. 2.) Hence the roots of f(x) = o are known to be 2, 2, 2, —3, and —3, as soon as we have the roots of ■) = o are all integral (Art. 551, Theorem V), and may be found by the following- theorem : Theorem XII. 565. If a r be a root of f{x) = o, and A„ be divided by a, and A n _ x added to the quotient, and if this sum be divided by a,, and A n - 2 added to the quotient, and /I/is process be continued till all tlie coefficients of /(.r) have been used, the result will be zero. Demonstration. — A careful consideration of the manner in which the coefficients are formed from the roots, will make the truth of the theorem plain. A n is the product of the roots with their signs changed, .-. — the product of all the roots, except a r , trittt their signs changed. A„-\ is the sum of the products of the roots, with their signs changed, taken n — 1 at a time. Each of the terms of A„-\ will therefore be divisible by a r except one, and that term will equal — ' ". Therefore adding — - to A„-i, cancels the onlv term which does not contain the a r factor a r , and makes A„-\ divisible by a r . In like manner, An 2 is the sum of the products of the roots, with their signs changed, taken n— 2 at a time, and the terms not containing a r will equal the last quotient obtained, with its sign changed. Hence, adding that quotient to A n -2 cancels the terms not containing the factor a r , and renders A„ -2 divisible by a r . In the same way each coefficient is made divisible by a ... But .1, is the sum of the roots, and when the terms not containing the factor a r are cancelled, there will remain —a r , which divided by a r gives — r, and this added to 1, the coefficient <»!' .<■■, i;'ives zero. 566. To find the integral roots of an equation by the application of this theorem, use the following Rule. — I. Write in a line the integral factors of the absolute term (A„). NUMERICAL HIGHER EQUATIONS. 259 II. Divide A„ by each of these factors, and write each quotient below its divisor. III. Add An-\ to each quotient and write the sum below. IV. Divide each sum by that factor of A n which stands above it, and continue in like manner to add the successive coefficients and to divide, until the coefficients are all used. V. If, in the course of the operation, the division by any one of the factors is imperfect, that factor is not a root and the ivorh with it will cease. To illustrate the rule take Ex. 2. (Art. 562.) The opera- tion will be as follows : Divisors, 1, 2, 4, 8, 16, — 1, —2, —4, —8, — 16, Fac.of A„. 16, 8, 4, 2, 1, —16, —8, —4, —2, —1, Quotients. Add —48 = A n — 1. Add -32, -40, -44, -46, — 32, —20, —11, 32 = A n -2. -47. -64. 64, -56, 2S, -52, 13 — 50, —49, Sums. Quotients. Add O, 12, +21, 0, 6, 36 = A n -3. 96, -96, 60, -3D, 45, Sums. Quotients. Add 36, 42, 36, 21, — 57 = An-i. -6o, 60, 6, -3. Sums. Quotients. Add — 21, —36, -21, -l8, 18 = An-5- 3, -3, -60, 30, Sums. Quotients. Add -3, 0, -3, 0, 8 = A n -6. 15, -15, 48, -24, Sums. Quotients. Add 5, 8, 5. 4, -6 = A,,--. -7, + 7, -16, 8, Sums. Quotients. — 1. —2, — 1, —1, + 1, — 1, 2, — 1, Sums. Quotients. 260 NUMERICAL HIGHER EQUATIONS. The work need be carried no farther to show that the integral roots are i, 2, — 1 and — 2. This does not determine whether these roots are once or more than once roots of the equation, but the first member may now be divided by the factors x— 1, .1—2, x+i and x+2, and the quotient put equal to zero will be an equation whose roots will be the other four roots of the primitive equation. Making this division by the synthetic method as follows : 1 — 6 + 8 + 18 - 57 + 36 + 32 - 48 + 16 I 2 2 — 8 + o + 36 — 42 — 12 + 40 — 16 o ] —2 I - 4 + + 18 — 21 — 6 + 20 — 8 + c — 2 + 12 — 24 + 12 + 18 — 24+8 i - 6 + 12 — 6 — g + 12 — 4 + j_£ + 1 — 5 + 7 + 1 — 8 + 4 1 -5 + 7 4- 1 — 8 + 4 + , -1 — 1 + 6 — 13 + 12 — 4 1— 6+13 — 12+ 4+ o We have for the equation having the other four roots, X 4 — 6x 3 + 13a? — 123! + 4 = 0. Applying the rule for integral roots, 4, —1, -2, -4, x » —4. —2, —1, I, 2, 4. 2, — 12, -8, — 10, — 11, -16, —14, -13, -8, -5, 16, 7, 13, 5. 8, 29, 20, 5, 4, -29, -10, -6, — 1, — 2, -35, -16, —1, — 1, 35- 8. We find 1 and 2 are roots of the equation. Dividing out these roots gives x 2 — 3.?' + 2 = 0, whose roots are r and 2. Hence the roots of the primitive equation are 2, 2, 2, 1, 1, 1, —2, and —1. NUMERICAL HIGHER EQUATIONS. 261 567. Let the student now apply the principles already developed to the following EXAMPLES. Each of the following questions should be answered in reference to the equations below : i. How many roots has the equation ? (Theorem II.) 2. What is their product and what their sum ? (Theorem II, Cor. 3.) 3. What numbers may be its integral roots ? (Theorem I, Cor. 2.) 4. What limits to its roots can be fixed ? (Theorem IX.) 5. Has it an even or an odd number of real roots? (Theorems II and III.) 6. Has it an even or an odd number of negative roots? (Art. 547, Cor. 4.) 7. Has it an even or an odd number of positive roots ? (Art. 547, Cor. 4.) 8. How many positive and how many negative roots can the equation have ? (Theorem VII.) 9. What are its commensurable roots ? (Theorem XII.) 10. What is the value of the first member when 5 is substi- tuted for x ? (Theorem I.) 1 1. What equation has the same roots with opposite signs ? (Theorem VI. ) 12. What equation has its roots twice as large ? (Art. 542.) 13. What equation has intermediate roots ? (Theorem X.) 1. x 6 + 45a: 4 — 246a; 2 + 200 = o. 2. .t 4 + $x 3 — 2X — 156 = o. 3. z 6 — sx* 4- 3.r 3 + 6x* — 297 = o. 4. x 5 + 9Z 4 + 7-r 3 — 3a; 2 — 4X + 10 = o. -MI-.' N I U i: K I < A L II I ( I II E It E < > D A T I ij S 5- X 5 — 22T 4 + 3-r 3 — 3^ 2 4- 2X — 1 =0. 6. x~' — 5a-' 6 + 3X 4 + 9a; 2 + 12 = o. Transform the following to the form (1) and give the changes produced in the roots by the transformation. (Arts. 538, 542.) 7- 8. 9- 10. 1 1. x % _ g^t _|_ >j£s -f. *x* _ 1 — o. 3^5 -[_ ya;o — 6.rs + 2./ — 2=0. 111 2X- -{- 3X* + 4£T — 2.r +2 = 0. 9^ + 32 4 — 2£ a 4- 2 = O. a; 2 — a; x 5 — a: 2 a^ — x Remove the equal roots from the following : (Theorem XL) 12. x 5 + a 4 — 4a; 3 — 4a: 2 4- 42; + 4 = o. 13. x G — Sx 5 + 26a; 4 — 44a: 3 + 41a; 2 — 2o.r + 4 = 0. 14. a; 6 — 3a. 4 — 45a; 2 — 81 = o. Produce equations having the following roots: h —2, 3- 3> 4, V 2 , —V*- —5> V— 2, — V— 2. — 1 + V—3, — 1 — V— 3, i 2. 1 ± V—3, 3 ± V— 2. I >• 2 7 "S> 2, I "25 rZr 2, I, 2, 2, 2. V- 2, \/2, — A/2. NUMERICAL HIGHER EQUATIONS. 263 X. INCOMMENSURABLE ROOTS. 568. The process of finding the Incommensurable Hoots of an equation depends on a theorem called, from its discoverer, "Sturm's Theorem." * This theorem assumes that the equation has no equal roots ; but as we have already seen how the equal roots may be removed, we may prepare any equation for the application of the theorem. 569. Assuming, then, that f (x) has no equal roots, and forming f'(x), divide f(x) by f'(x), and represent the remainder with its signs changed by f )l . 2 .( x )- In like manner divide f'(x) by /„_ 2 (x), and repre- sent the remainder with its signs changed by /„_ 3 (.r), and proceed in the same manner until a remainder, f (x), is found. Notes. — i. The subscripts n — 2, n — 3, . . . o, indicate the degree of the function. f (.r) is therefore independent of X, and such a remain- der will be found ; for, each division may be continued until the remain- der is a unit lower in degree than the divisor, and the division will at no time be complete, since f(x) has no equal roots. (Theorem XI.) 2. The use of these functions is such as not to forbid multiplying or dividing them by any positive numerical factor. They may therefore be simplified by rejecting all such factors, and to avoid fractions in dividing, any dividend or divisor may be multiplied or divided by any positive a umber. 570. The functions / ra _ 2 (x), / w _ 3 (x), etc., are called the Sturmian Functions, and with f(x) and f'(x) con- stitute the functions to which Sturm's theorem relates. The relations of these functions to each other is expressed in the following equations, in which Q x , Q 2 , etc., are the successive quotients. * See page 307, Note 5. 264 STURM'S THE OB EM. f{x) = /'(*)x ft -/_(*). (i) /' <*) = /_ (x) Xft- /„_a (*). (2) /_ (z) = /_ (*) x Q 3 -/„_ 4 (.«■). (3) /,(*) = /i(*)X &-,-/.(*). ( 4 ) 571. Consecutive Functions are those adjacent, in the order /(z), /' (as), /_ (.r), /_(a:) /o(»). Theorem XIII. 572. JVo /wo consecutive functions can become zero for ttie same value of x. Demonstration. — Suppose that /„_._. (.n and f n -s{x) could became zero for the same value of x. Then by Kq nation (2), /' (;r) = o, and by- Equation (i), f(x) = o. But / (,r) and f(x) cannot become zero for the same value of x . (Art. 559.) .\ f n -i (%) and/„_ 3 (.r) cannot be zero at the same time. The same may be proved of any two consecutive functions. Cor. — If f (x) or any one of the Sturmian functions reduces to zero for any value of x, the adjacent functions have opposite signs for the same value of x. If /„_ 2 (.r) = o, in Equation (2), we have f'(X) = ~fn-B(X), and in like manner for any other functions. XI. STURM'S THEOREM. 573. If in f{x). f (x), /„_, (x), /„_ 3 (x) /, (x), two different numbers l» substituted for x, and the signs of the resulting values of the functions for each substitution be set separately in the order of the functions, the difference in the number of variations in I In 1 two cases will be equal to the number of real roots of f(x) = o situated between th numbers substituted. STURM'S THEOREM. 2G5 Demonstration. — ist. If the number substituted for x be supposed to change from one value to another, so as to pass through all inter- mediate values, the several functions will change their values in a simi- lar manner, and whenever the value of any function passes from + to — or from — to + , it will pass through zero. 2d. When any intermediate function becomes zero for any value of x, the adjacent functions have opposite signs for the same value of x. j Hence a change of sign in any intermediate function (that is, in any function except the first and last) can have no effect on the number of : variations. This is obvious, for the variations of + + — and + — — ; are the same. 3d. As none of the intermediate functions can affect the number of ! variations by any change that may occur in their signs, and as the last function, being independent of x, never changes its sign, any change in j the number of variations must be produced by a change of sign in f(x). But f(x) will change its sign whenever the number substituted passes a root of f(x) = 0. If a number greater than «i be substituted, f(x) and /' (.c) will both be positive and will form a permanence. If now this number be supposed to decrease when it passes «i , f(x) will change its sign, while f \x) will remain positive, since it has no root so great as ct\ . This change of sign will make the first two signs — + , giving in place of a permanence a variation, and the number of variations will be increased by one. As the value of x continues to decrease, it will pass a root of /' (a;) = o before it comes to «a (Th. X), and therefore f'x will change its sign before f{x) changes again. This will make the first two signs , but without affecting the variations. When therefore the value of x passes ai , fix) will become positive and the first two signs will become + — , adding another variation. In the same way a varia- tion will be added whenever x passes a root of f(x) — o. Hence, the theorem is proved. 574. Sturm's TJicorcm is applied to finding the situation of the incommensurable roots of an equation. It may also be used to find the commensurable roots, but the methods already given are sufficient to determine these. A single example will illustrate its application. 1. Find the situation of the real roots of x 3 — 3a; 2 + 6x — 5=0. SOLUTION. f(x) = X 3 - 3.^ + 635—5. /' (x) -f- 3 = a 2 — 2a! + 2. (Art. 569, Note 2.) /, (X) = — 2X + 3. f (x) = - I. 2GG STURM'S THEOREM. Substituting in these functions different values for x, we have the signs as follows : fix). /'(*). fi(x). fo(x). + oo gives + + — — one variation, o " — + + — two variations. — 00 " — + + — two There is therefore one real root between + co and o, and no root between — co and o. That is, tbere is one positive and no negative root. Hence, two of the roots are imaginary. To find the situation of the real root, we substitute as follows : 1 gives — + + — two variations. 2 " + + — — one variation. The root is therefore between i and 2, or i + a decimal. We may find more exactly the situation of this root by substituting i.i, 1.2, 1.3, etc., as follows- I.I gives — + + — two variations 1.2 " — + + — two " 1-3 " — + + — two 1.4 " + + + — one variation. Hence the root is between 1.3 and 1.4 ; that is, it is 1.3 + . In the same way other figures of the root could be found, but the substitutions would be tedious, and we shall hereafter show an easier ' method of carrying out the work after the first one or two figures have been found. 575. Find by Sturm's TJieorem the first two figures of each incommensurable root of tbe following equations : 2. x z 4- 3.r 2 — 32 4- 1 = o. 3. x 5 — 5-r 4 — 10a; 3 + iox — 1=0. 4. x 4 4- 3-T 3 — 6x +2 = 0. 5. x s — 5-r 2 +7=0. 6 . x 5 — x -f- 5 =0. 7. x 7 — 7X 6 + jx 5 — 14X* — 7.T 3 + yx 2 + 142; +1=0. 8. x 3 — 6x 2 + 3.r + 5 = o. 9. X s 4- 6a: 2 — t,x .+ 9 = o. 10. .r 3 + 5a; 2 — ix 4- 2 = o. HORNER'S METHOD. 267 HORNER'S METHOD OF APPROXIMATION.* 576. Horner's 31ethod of finding the successive figures of an incommensurable root is based on the following Problem. — To find an equation whose roots shall he less than the roots of a given equation by a given number. Solution. — Let x be the number by which the roots of f(x) = o are to be diminished, and put y for the unknown quantity in the trans- formed equation. Then y = x — x' or x = y + x'. Substituting y + x' for x, we have, (y + x') n + Aiiy + x')"- 1 + Ai(y + x') n ~ 2 .... An-ziy + x'f + An-\ (y+x 1 ) + A n = f(y+x') = o. (i) Developing the different powers of (y + x') and collecting the terms containing like powers of y, and writing B\, -Z? 2 , B3, etc., for the coefficients, we have, yn + By- 1 + Bay"- 2 + B,,-iy + B H = f(y + %') = o. (2) Substituting for y its value, (x—x'), this equation becomes, (x—x) n + Bi (x—x') n ~ v + B 2 (x—x') 11 - 2 .... .&_, {x-x') + B n = f(x) = o. (3) If equation (3) be divided by (x—x'), the remainder will be B n ; and if the quotient be divided by (x—x'), the remainder will be B n —\, and so on, by successive divisions by (x—x'), the remainders will be the succes- sive coefficients of (2), beginning with the last. But the first member of equation (3) is equal to f(x) ; hence if these successive divisions by (x—x') be performed upon f(x), the remainders will be the coefficients required. In the following example the transformation is effected both by substitution and by the method of division : 1. Transform the equation a 8 — 5a* + $x— 7 = o, to another whose roots shall be 2 less than those of the given equation. 1st. By Substitution, (y + 2) 3 - 5 (y + 2 f + 3 {y + 2 ) - 7 = o. * Sed pasje 307, Note 6. 268 Horner's method Developing the equation, Or, y 3 + by 2 + 12 + 8 = 0. — 5I — 20 — 20 + 3 + 6 - 7 y 3 + f - sy - j 3 = 0, ,4ns. 2d. By Division. 1 - 5 + 3 - 7 | 2 + 2 — 5 — 5 3 = B; — 3 - 3 «- 1 2 — 2 - 1 «- i 5 = 52 2 J,3 + yi _ 5^ _ I3 — Qi ^4^ Notes. — 1. The coefficients of the first terms of the successive quo- tients are omitted, as not essential to the work. 2. The coefficients of the transformed equation are printed in full- face type, and preceded by an inverted comma. 577. To apply this by Horner's method of approximating the incommensurable roots of an equation, take Ex. 1, Art. 574. x 3 — 3Z 2 + 6x — 5 =0. (0 "We found that this equation has but one real root, and that this root is between 1.3 and 1.4. To find other figures of the root, diminish the roots first by 1, as follows : — 3 + 6 - 5 + 1 — 2 + 4 — 2 a. 4 ,- 1 + 1 — 1 — 1 * + 3 + 1 This gives « + y* + oy ! + yy — 1 = o, (2) an equation whose roots are 1 less than those of the primitive equation. Its real root therefore lies between .3 and .4. HORNER S METHOD. 269 Diminishing its roots by .3, as before, 1 + + 3 — 1 + .3 + .09 + .927 3-09 .18 .073 : + 3.27 and we have + .9s 2 + 3.27s — .073 = o, (3) an equation whose roots are 1.3 less than those of the primitive equation. Its real root will therefore be the remaining figures of the real root of the primitive equation. Hence it is less than .1, its cube less than .001, and its square less than .01, and the omission of the first two terms will make but little difference in its root. We may therefore find the root approximately by using only the last two terms. Thus, 3.27s — .073 = 0, or 3.27s = .073, and ■073 z = — — = .02 + 3-27 This gives, probably, the first figure of the root of (3), and the third figure of the root of (1). Diminishing the roots of (3) by .02, we have, « + 3-27 .01* - -073 + .065768 .92 + 3.2884 ( — .007232 ,02 + 0.0188 •94 < + 02 3.3072 96 The new equation is w 3 + .9610° + 3.3072W — .007232 = o. (4) We know by this result that .02 is not greater than the first figure of the root of (3), for if it were, it would give a positive value on being substituted for x (Th. VIII), but the result of division shows that it gives a negative value. (Th. I.) In the same manner that .02 was found from (3) we now find the next figure from (4) to be .002, and proceed to diminish the roots of (4) by .002, a process which may be continued indefinitely. 270 HORNER S METHOD. 578. The operation of diminishing the roots by these successive figures may be written more compactly by omitting to re-write the coefficients of each new equation, and using them as they stand when first found. We give Ex. i written in that manner, and designate the coefficients of each successive transformed equation by half- parentheses and full-face type, marking them by a subscript figure to indicate whether they belong to the first, second, or third transformed equation. Thus, i( + 3, indicates a coefficient of the first transformed equation, and 2 (3.27 a coefficient of the second transformed equation. By a careful examination the student will see that the process is equivalent to the transformations above. Observe that when a decimal is to be added to an integer, it is annexed for greater economy of space. i — 3 + i + 6 — 2 — 5 | 1.3221 + 4 — 2 + I + 4 — i .(- 1 .927 — I + I i(+ 3.09 + .18 2 (- .073 + .065768 .( + + 0-3 •3 + 3.27 .0184 3(~ + .007232 .006618248 + + .6 •3 + + 3-2884 .0188 4(~ + .000613752 000331 115 + .92 .02 3( + + 3.3072 .001924 b(- .000282637 + + •94 .02 + + 3309*24 .001928 3( + + .962 .002 4( + + 3.311052 .00009661 + + .964 .002 + 3.31114861 *( + .966i Let the student extend the above, and find two or three more places of the root. He will observe (hat it is unnecessary to use more than two places of decimals in the first column, four in the second, and six in the third. HORNER'S METHOD. 271 Let him also apply this method to equations in Art. 575, of which two figures of the roots have already been found. In finding negative roots, change the signs of the alternate terms beginning with x 71 ' 1 , and thus change the signs of the roots. (Art. 552.) The positive roots of the equation thus changed will, when their signs are changed, be the negative roots of the primitive equation. 2. Find the fifth root of 5. The equation is x 5 — 5 = o, which by Art. 296, 2 , has but one real root, and, without the application of Sturm's Theorem, we know the first figure to be 1. We may therefore proceed at once to apply Horner's Method, as follows : + + + 1 + 1 + + + 1 + — 1 + 5 1 + + I + 1 + 1 2 + + 1 + 3 + 4 4 + + 2 + I + 3 3 + + 4i(+ 6 5 + + 3 + 1 + 6 4 ( + 10 + + 4i( + 1 10 l(+ 5 If we now divide 4 by 5 to find the next figure, we get .8, but this is evidently too large ; for by Art. 556, the first remainder obtained in the next division must be negative. If, therefore, we use a figure so large as to give a positive result, we must reduce it. Let the student complete the work in this example. 3. Find the cube root of 1953 125 by Horner's method. 4. Find the real roots of x i — x 2 -f 2X — 1 =0. 5. Find one root of x 5 — 2.1 4 — yc 2 — $x — 38756 = o. 6. Find a root of x 3 — 2 =0. 7. Find the roots of 5.V 3 — %z — 1 =0. 373 hoexee's method. 579. To apply Sturm's Theorem and Horner's Method of Approximation, it is not necessary to reduce the equation to the form, Art. 538. To use synthetic division for the applica- tion of Horner's Method, we must however make the first coefficient unity. This will give for Example 7, the coef- ficients 1 +0 — .6 — .2 8. Find the roots of x z — jx + 7 = o. 9. Find the roots of x s — 5.?; — 5=0. 10. Find the roots of x 3 + 3a; 2 + 3^ + 5 =0. 580. The foregoing processes of finding the real roots of numerical equations may be summed np as follows: 1st. Reduce the equation to the form x n 4- A,x n ~ l + A< 2 x n ~2 . . . . 4- A n ^x 4- A„ = o, by Arts. 539-542. 2d. Try the factors of the last term by Art. 566 for com- mensurable roots. 3d. Divide out the roots thus found. (Art. 544, Cor. 1.) 4th. Apply Sturm's Theorem to the depressed equation, and if in the course of the process a common divisor between /(•'') and f (x) be found, form two equations by Theorem XI, and apply Sturm's Tlieorem to these equations to find the situation of the incommensurable roots. 5th. Find an approximation to each of the incommensurable roots by Horner's Method. 581. The imaginary roots may be found, when there are only two, by removing the other roots and reducing as a quadratic. hoknee's method. 273 582. When fin equation has two nearly equal roots, it will be necessary to find, by Sturm's Theorem, a sufficient number of figures to separate the roots. For example, in the following equation, x 3 + ll % 2 — 102a; 4- 181 =0, we find, by Sturm's Theorem, that the roots are all real, two being positive and one negative. We also find the positive roots situated between 3 and 4. Diminishing the roots by 3, gives the coefficients of the transformed equation, 1 4- 20 — 9 -f i, two of whose roots are between o and 1. Applying Horner's Method, trying successively .1, .2, etc., we have, 1 + 20 — 9 +1 | .1 + .1 + 2 - 01 — '699 + 20.1 — 6. 994 + . 301 The remainder being -f , both positive roots are either greater or less than .1. (Theorem VIII.) 1 4- 20 — 9 4-1 I .2 4- .24- 4.04 — .992 4- 20.2 — 4.96,+ .008 For the same reason these roots are both either greater or less than .2, and if we were to try .3, .4, etc., we should still find a positive remainder. Horner's Method, therefore, does not dis- tinguish between these roots. But by further application of Sturm's Theorem, we find the positive roots of the primitive equation are situated between 3.2 and ^.^, and by Horner's Method we may then find the remaining figures. Diminishing the roots by 3.2 gives the coefficients, 1 4- 20.6 — .88 4- .008 J .01 4- .01 4- .2061 — .006739 4- 20.61 — .6739,+ -001261 and w r e see that both roots of the primitive equation are either greater or less than 3.21. 1 4- 20.6 — .88 4- .008 I .02 4- .02 4- -4124 — -O Q935 2 4- 20.62 — .46764— .001352 We now know that one root is greater and one less than 3.22. Let the student find each of these roots to 5 decimal places. 274 KECUERIKG EQUATIONS. RECURRING EQUATIONS. 583. A Recurring liquation is one in which the coefficients of x n ~ r and x r are numerically equal, the cor- responding coefficients all having either like ox unlike signs. 584. A Reciprocal Equation is one whose roots are reciprocals of each other; that is, if a be a root, - is also a root. Theorem XIV. 585. A recurring equation is also Reciprocal. Demonstration. — The general recurring equation is x" + J-jCC'-i + A 2 x n ~ 2 . . . ± A 2 x 2 ± A^x ± i = o. (i) Substituting - for x, x — + A, — , + A- ——i. . . . ± A* — ± A, - ± i = o. (2) Clearing of fractions, i + A x x + A 2 x- . . . ± A.,x n ~ 2 ± A-lX' 1 ~ 1 ± x" = o. (3) Which is the same as (i) ; hence the equation is satisfied when - is put for x. x In equation (i), the double signs in the second half indicate that those terms are either all of the same sign as the corresponding terms of the first half, or all of contrary sign. Theorem XV. 586. A Recurring Equation of an odd degree has — i or + i as a root, according as the corresponding terms have like or tinlike signs. Demonstration. — Whenever the corresponding coefficients have unlike signs, the substitution of + 1 for x will cause them to cancel each RECURRING EQUATIONS. 275 other; and when they have like signs, — i substituted for x will do the same ; since one of these terms will be an even and the other an odd power of x. Cor. — Such an equation may have its degree reduced one unit by dividing by x — i or x + i. Theorem XVI. 587. A Recurring Equation of an even {the 2n th ) degree, in which the coefficient of x n is zero and the like coefficients have unlike signs, has both -\-i and — i as roots. Demonstration. — Represent the equation by x 2n + J-!* 2 "- 1 + A 2 x-><--- ... — A 2 x- — A^x — i = o. It is evident that both +i and — i will cause corresponding terms to cancel. Cor. — Such an equation may be divided by x 2 — i, and its degree thus reduced two units. Theorem XVII. 588. Every Recurring Equation of an even degree, whose corresponding terms have like signs, may be reduced to an equation of one-half that degree. Demonstration. — Let the equation be x 2 ' 1 + ^4 1 a; 2n — i + A s x 2n ~ 2 . . . + A„x n . . . + A»x 2 + A x x + i = o. (i) Dividing by x" and uniting terms, (* + L) + Al ^-, + _L) + As ( ;C n- 2 + _!__) . . . An = o. ( 2 ) Make then, 2 — X + X ' 8 9 — 2 = & + I x-' 3 _ -33 - it' 3 + I ^3 5 276 RECUERIXG EQUATIONS. and in general a"» h may be expressed in terms of 2, the highest power being z m . Substituting these values in (2), we have tm equation of the n th degree. 589. Eecurring equations may often be reduced as quad- ratics, by reducing the degree in accordance with the preceding theorems. The following examples will make the student familiar with the principles. EXAMPLES. 1. x 5 — 7> xi + 2xZ — 2x2 + S x ~~ l — °- ' Solution. — By Theorem XV, + 1 is a root. Dividing out this root we have the equation, x i — 2X 3 — 2X + 1 = o. Dividing by a; 2 , a: 2 + - : — 2 (x + -) = o. xl \ ay Substituting s = x + - , x z 2 — 2 — 2S = o ; z = 1 ± \/3 ; x + - = 1 ± -v/3 ; X <& + 1 = (1 ± y3) ■** ; = £AJ5 ± 1 a/±7^. (a) a; 2 The roots written separately are : — 1 ; I+ ^ + J V7^ ; 2 LtJ5_i VT^a; 2 2 1 — BINOMIAL EQUATIONS, 277 Note. — Observe, that in equation (a) the double sign under the radical comes from the double sign in the numerator of the preceding term ; hence, the upper signs of these must be taken together, and also the lower. But the double sign between the two terms has no depend- ence on the others, and may therefore be taken either way. X 3 — 2X~ + 2X — I = O. x 4 — 3a; 3 -+- 4X 2 — $X + I = O. x* + 7X 3 — jx — 1 = o. 3.r 5 — zx 4 + 5a; 3 — 5.r 2 + 2X — 3 = 0. x 6 — 3« 5 + 5a 4 — 5a; 2 + 32; — 1=0. x 4 - — x 3 + x 2 — x + 1 = o. X 4 — X 5 + X — I =: O. ax 4 — 2X 3 -\- 2x — a = o. 5X 4 + Sx 3 + 9a; 2 + Sx + 5 = o. BINOMIAL EQUATIONS. 590. A Binomial Equation is one having but two terms, and is of the form yn ± a n — o. (1) By substituting ax for y, it takes the form a"^ ±a n = o, or x n ± 1 = o. (2) The roots of (2) multiplied by a will give the roots of (1). 591. The real roots of a binomial equation may be found by the usual method of evolution or by Horner's method of approximation. 592. The Imaginary Roots can be found by the method (Art. 303) which involves Trigonometry, or, if the degree of the equation be not too high, by solving the equation in form (2) as a recurring equation. 278 BINOMIAL EQUATION'S, EXAMPLES. i. Find the roots of x 5 — i = o. One root i3 i, which being removed by division gives x 4 + x 3 + x 2 + x+i = o. Dividing bvx 2 , x 2 + — + x A 1-1=0. x 2 x Putting x + S 2 + 2 — I = o ; z = — | ± yi+I = i Vs = x + ~ ' & + (i^iVl)ai = — ij = -i±iV5 ±V '(W ^Y - i ; !B=i(-I± A /5± V-IOT2 fJ0. The roots of £ 5 — i = o are, therefore, i, i(-i + V5 + V-10-2 V5) (-1 + \/5 - V- 10-2 fs) (— I — y/s + V— IO+2 V$) H-I-V5- V-IO + 2 ^). The fifth roots of any other number will be these roots multiplied by the real 5th root of that number. For example, the roots of x b — 32 = o are the above, each multiplied by 2. Find the roots of the following : 2. 7? ±1 = 0. 6. a? + 7 = o. 3. .r 1 ±1=0. 7. .i -5 — 4 = 0. 4. .r 6 ±1=0. 8. a 7 -f- 5 r= o. 5. a* + 5 = o. 9. z-° — 3 = o. EXPONENTIAL EQUATIONS. 279 EXPONENTIAL EQUATIONS. 593. An Exponential Equation is an equation in which one or more of the exponents contains an unknown quantity. i. Given a x = b to find x. Solution. — Taking the logarithms of both members, x log a = log b. (Art. 460.) log b x = r - e -. log a 2. Given x® = a. Solution. — Taking the logarithms ■x log x = log a. Find by inspection from the table of logarithms and by trial, the value of x. For example, let log a = 8. Then, x log x = 8, and by inspection, x = 8.6, nearly. 3. Given 10^ = 57 to find x. 4. Given 27^ = 84 to find x. 5. Given x 2 ® = 13 to find x. 6. Given x x+% = 5 to find x. 7. In how many years will a sum of money double at compound interest, at 6% ? 8. In how many years will a dollars amount to A at compound interest, at v , ? 9. If a young man spends 25 cents a day for cigars, in how many years might he buy a farm worth $5000 with the same money, by depositing it once a quarter, at 5% compound interest ? <\ r w APPENDIX. PROBABILITIES. 594. The Probability that any event will happen is the ratio of the favorable to the whole number of chances. This is on the supposition that the chances are all equally good. 595. Let it be known that a bag contains 10 balls, num- bered from i to 10. If one ball be drawn, and there be no reason why it should be one number rather than another, we say there is one chance in ten that it will be a particular number, as No. 2 or No. 5. The chance then that No. 5 will be drawn is ^, there being 10 events each equally likely to happen and only one of them being favorable. If, of the 10 balls, 3 are white, 2 black and 5 red, the chance that a red ball will first be drawn is ■£$ or \ ; that a black ball will be drawn is ^ or \ ; and that a white ball will be drawn is ^. The chance that the first ball drawn will be either black or red, is ^ + \ = -^ ; and the chance that it will be either white, black or red, is \ + \\--r$ = 1. In this case the chance becomes a certainty, and is represented by 1. 596. If another bag contain 4 white, 3 black and 3 red balls, the chance of drawing the first time a red ball from this is T %, while the chance of drawing a red from the first bag is t%. The chance that both balls will be red is the product of the chances for each, or T 5 „ x ^ = -^. This is evident, for any one of the balls in the first bag may be drawn wit li any ope in the other, making the whole number of chances PROBABILITIES. 281 the product of the whole number in one bag by the whole number in the other. Also, since each of the favorable chances in one case may happen with any one of the chances in the other, the product of the favorable chances in one case, multiplied by the favorable chances in the other case, will give the number of favorable chances when the two events are to occur together. EXAMPLES. i. What is the chance of throwing sixes, with two dice, at the first throw ? Solution. — The chance that either of the dice will turn up a six is evidently J. Hence the chances of double sixes is a x i = ^\, Am. 2. What is the chance of throwing sixes twice in succession ? Solution. — The chance that two throws will give the same is Sf X 36 - T5S6> A.IIS. 3. What is the chance that a throw of two dice will be greater than 6 ? 4. What is the chance that in drawing 100 numbers from a box, any three numbers will be drawn consecutively ? 5. What is the chance that three points taken at random on a circle will be on the same semicircle ? 6. What is the chance that two coppers tossed at random will both turn up heads? What is the chance that three coppers will do the same ? 7. What is the chance that six coppers will turn up half heads and half tails ? 8. What is the chance of drawing three white balls suc- cessively from a bag containing 10 white, 7 black and 5 red balls, the ball drawn being replaced before the next draw ? 9. What would the chance be in Prob. 8, if the ball be not replaced ? 10. What is the chance that one of each color will be drawn in the first three draws, not replacing balls drawn ? 282 cardan's formula. CARDAN'S FORMULA. 597. Cardan's formula for the reduction of cubic equa- tions is obtained as follows : The general form of a cubic equation is a? + ax 2 + bx -\- c = o: (i) From this equation the second term may be made to disap- pear by the substitution of y for x, giving (y--) + a \y - 1) + h \y - -) + c = °- This diminishes each root of the equation by the quantity -^ (Art. S7 6.) Developing and reducing, = o. if + of — \a % + I y + -fcc? -lab + c Substituting^ for b — ^a 2 , and q for faa 3 — \ab + c gives f + py + q = o. (2) Making y = z — ^ p 3 the equation becomes z? + qz s — — - = o, 27 which by Art. 329, (A), gives = - 9 ' ± \/ r 2 -n3 4 27 cardan's formula. 283 and zt=(-£±V q - + ^J, therefore, V 2 4 27/ or, rationalizing the denominator and omitting the double signs, because they give no more values than single signs, 598. In equation (2), the coefficient of y 2 being the sum of the roots is o (Art. 547, Cor. 3); hence, representing two of the roots by m ± VJi, the third root will be — 2m. These roots give the equation V z — (3 m? + n ) V + 2 ( m5 — mn ) — °- (Art. 547, Cor. 2.) Equating with (2) gives the identical equation, y s + py + q = y 3 — (3 m2 + n ) y + 2 ( wi3 — w*w) > p = — (3™ 2 + w) ; q = 2 (m 3 — ?ww). Hence, |/ ^ + ^ = (^ 2 - fw) V- 3». (3) The roots — 2m and m ± Vw admit of three cases : 1. When n is positive and the roots are all real. 2. When n is o and tivo of the roots are equal. 3. When n is negative and tivo of the roots are imaginary. The first case makes the second member of (3) imaginary, and therefore the terms of formula (A) become imaginary. The second and third cases make (3) real, and formula (A) real. Hence, 284 cardan's formula. When the roots of a cubic equation are all real and unequal, Cardan's formula has its terms imaginary. This will occur when p [Eq. (2)] is negative and - < — — • 4 27 599. The following examples illustrate the application of the formula to each of these cases: 1. Find the roots of z 3 + 6x — 20 = o. By the formula, < X = (10 + •\/ioo + 8) 4 + (10 — /y/100 + 8) s = (10 + 6 ^3)* + (10 - 6 -\/3)* = 1 f ^3 + 1 - \/3 = 2. Removing this root from the equation, 1 + + 6 — 20 \ 2 2 + 4 + 20 2 + 10 The depressed equation is x* + 2X + 10*= o ; and x — — 1 ± y/— 9 = — 1 ± 3 \/^. The roots are 2, — 1 — i 3, and —1—53. 2. Find the roots of a- 3 — 3% — 2 = o. By the formula, x = (1 + \/i — i) 1 + (1 — V 1 — J )* = 1 + 1 = 2. Dividing by x — 2 and reducing the depressed equation, we find X = — I ± 0. CONTINUED FEACTIONS. 289 fraction, |, is too great ; and, consequently, 2^ being greater than the true denominator, the fraction, 1 x 3 r, _l_ I 7 n ' 2 + 3" ^ 7 will be /ess than the true value of the continued fraction. 604. Similar reasoning will, evidently, hold in respect to any number of terms, and will apply equally to the general form (2), as to the particular example we have considered. Hence, If we include in the reduction an odd number of partial fractions, the result ivill be too great; if an even number, the result will be too small. 605. The fractions, 1 1 a{ x' 1' , and -= be the convergents cor- responding to cf„_o, «„_] , a„, and a n+l . Then, since the n th convergent is formed according to the above law, we shall have N_ Ma n + L . JST ~ M'on + L 1 ' {3) If now we substitute in -==. , a„ A for a„ , we shall -ft' ' « n+ i P obviously find -=-,• Thus, P_ M \ a " + T~) + L {Ma n + L )a n+ , + M . P' ~ \rL , ' \ , t. ~ ( M ' a » + L ') a ^ + M ' ' M '^ + i-) +L ' or, from (3), -^ = ^ + j/'" (4) Consequently, if the law holds good for // convergents, it will for n + 1. Hence, CONTINUED FRACTIONS. 291 608. To Find the Numerator and Denominator of any Con- vergent after the Second, as the (n + i) th , we have the following Eule. — Multiply the numerator and denominator of the n th convergent by the (n + i) th partial denominator, and add to the products, respectively, the numerator and denominator of the (n — i) th convergent. 609. The numerator and denominator of any convergent must be respectively greater than those of the preceding ; each numerator and each denominator being at least equal to the sum of the two next preceding. 610. Moreover, each convergent is found by substituting in the preceding, for the last partial denominator, an expres- sion known to approach more nearly to the true denominator. Hence, evidently, each convergent approximates more closely than the preceding to the true value of the continued fraction. i. Find the successive convergents of the continued fraction, i 2 + I I I 2 +- Am* I I 3 4 an(\ 3SI ■a-llb. •$, ■$, -g, yj, dim g^y. The first four convergents are approximate values of the continued fraction; the last, |f|, is the true value. 611. A continued fraction is sometimes mixed, or made up of a whole number and a fraction. Thus, i 3 + i 2 + (5) 3 + sTetc. 292 CONTINUED FRACTIONS. Iii such cases, the integral part may be reserved and added to the convergents; or it may be taken, with i as a denomina- tor, for the first convergent. 612. Thus, in the above example, we shall have the con- vergents, ,1 ,3 ,16 nr 3 7 2 4 127 This form, " "*" i (6) is sometimes assumed as the general form of a continued frac- tion; the place of the integral part, when it is wanting, beiug filled with o. In that case, the first convergent is evidently too small, the second too great, and so on, those of an odd order being too small, and those of an even order too great. (Art. 604.) Note. — If the integral part be zero, the first convergent will of course be £. 613. If the second convergent of Art. 606 be subtracted from the first, the remainder is unity divided by the product of the denominators. If the third be subtracted from the second, the remainder is minus unity divided by the product of the denominators. Suppose it has been proved that this law extends to n — 1 convergents ; that is, r t nr> T' nr 1 , (7) L M LM' - L'M L' M' ~ L'M' ± 1 ~ L'M' ' M M' N M Ma n + L N' ~ M' M'a n + L' L'M— LM' LM' - L'M Then -^7 — -™ = -m — M'N' M'N' (8) the numerator of which is the same as that of (7), with a contrary sign. Hence, the principle proved in regard to the first three convergents, applies equally to the whole series. For, CONTINUED FRACTIONS. 293 If each convergent he subtracted from that which next pre- cedes, the numerator of the difference will he ± i, and the denominator will he the product of the denominators of the two convergent^. 614. Again, the true value of the continued fraction lies between any two successive convergents, and differs from either of them less than they differ from each other. (Art. 605.) M That is, the convergent -^7, differs from the true value of the continued fraction by less than M'N' But (Art. 608), M' < N, and .-. M't < M'N'. •'• mW < W* Hence ' Cor. 1. — The error in taking any convergent whatever for the true value of the continued fraction is numerically less than unity divided by the square of the denominator of that con- vergent. 615. The denominator of each convergent is greater than the next preceding by some whole number. (Art. 609.) Hence, if the fraction be infinite, we may find a convergent whose denominator shall be greater than any given quantity; and, consequently, Cor. 2. — We may find a convergent which shall differ from the true value of the continued fraction by less than any given quantity. 616. Suppose that M and M' have a common divisor, D. Then D will of course divide I'M and LM', multiples of M and M' , and consequently the difference of those multiples, LM' - LM = ± 1. 294 CONTINUED FRACTIONS. Therefore D must divide ± \, which has no integral divisor but unity. .*. D = i. Hence, Cor. 3. — Every convergent is in its lowest terms. 617. One of the most obvious uses of continued fractions is to express approximately, in small numbers, fractious whose terms are large. Thus, 17 11 1 1 3 + tt 3 + rr 3 + 17 "17 17 „ , 1 -5- 2 + o' Here we first divide both numerator and denominator of 4,£ by 17. We then reduce -ff to a mixed number, 3-^, and again divide both terms of -fa by 8 and reduce to a mixed number, and so on. These operations evidently produce no change in the value of the given fraction. Kow the several convergents of the continued fraction found are ^, f, and ££. I I Q— We find - = — - , too great : 3 59 J and - = — -, too small, 7 59 but differing from the true value by ouly 7 fj. 2. If the fraction proposed had been -ff, we should have found i. x x x x x — a II. Find the greatest integer contained in x x , and denote it by a x , and denote the fractional excess of x x above a x by — • Then x, = a, + -• Xi III. Apply the same process to x 2 , and so on. Thus, 1 l . l x = a -\ — = a -\ = a -\ x x i , • i «i + - «i + x* i , a* -\ — > etc. x s If x < i, we shall have a = o. We shall always have x x , x s , etc., > i. For if x x = or < i, we have - = or > i, and a is not x x the greatest integer contained in x. 620. Whenever we find a denominator. x n , equal to a whole number, we shall have x n = a n) and the continued fraction will terminate. This will happen if the quantity x can be exactly expressed by a common fraction. If the quantity is not equal to a common fraction (i. e., if it is incommensurable), the continued fraction will extend to infinity. 296 CONTINUED FRACTIONS. 621. i. Given n = 3. 141 59, employing only five decimal places. (Art. 42, 4th.) Eeduce n to a continued fraction, and find approximate values. Ans. n = 3 _j 7 + - 1 i5 + H , etc. 25 Convergents, 3, 3. 2 -, £§£, fff, etc. Note. — The second approximate value, " 7 3 , was found by Ar- chimedes ; the fourth, fff, by Adrian Metius. 2. The common or tropical year consists of 365.242241 mean solar days. Find approximate values for this time. Ans. 365 1, 3652V, 365/3, 365T6T' etc Note. — The third approximation shows an excess of the solar year above 365 days of ^ of a day. To preserve the coincidence between the solar and civil year, therefore, eight years in thirty-three must contain 366 days each. That is, a day must be added to every fourth year seven times in succession, and the eighth time to the fifth year. 3. The sidereal month (i. e., the time of the moon's sidereal revolution) consists of 27.321661 days, or the moon revolves 1000000 times in 2732 1661 days. Find approximate values of this ratio. Ans. 27, - 8 /, ?££-, t^V-j ctc . Note. — These ratios show that the moon revolves about 3 times in 82 days ; 28 times in 765 days ; or, more exactly, 143 times in 3907 days. 622. Continued fractions are also employed in finding the roots of equations, and in extracting the roots of numbers. 1. Extract the square root of 3; i.e., find a root of the equation, & — 3 = °- 0) Here a;=H Diminishing the roots of (1) by 1, we have, f. 4. 2lJ _ 2 = o, (2) an equation whose roots are equal to -• CONTINUED FRACTIONS. 297 Transforming (2), we find, 2.r, 2 — 2.r, — 1 = 0. (3) This gives, x t = 1 H Transforming (3) in the same manner as (1), we have, X? — 20i\ — 2=0, (4) and x 2 = 2 -) x 3 "We find, in like manner, 2X |f, |i f£. 623. A continued fraction of this kind, in which any number of the partial denominators are continually repeated in the same order, is called Periodic. 624. It will be found that every incommensurable root of an equation of the second degree may be expressed by a periodic continued fraction. Of course, when the first period is found, such a fraction may be developed to any extent by simply repeating the period. 2. Extract the square root of 2. Convergents, 1, f, |, \\, f£, f$, etc. MISCELLANEOUS EXAMPLES AND PROBLEMS. i. Eeduce the following fraction to its lowest terms, X s — 5a; 2 — 4X + 20 mmmmm x 3 + 5a; 2 — 42; — 20 2. Add -2 to — -L_^_. 5« + 3* 7 ft -h 9 X , . a — a; 2«+z 3. bubtract — 5 = from — — • 2 a 2 + 3aa; + a: 2 « 2 — a; 2 _ , 4X + 1 5a; — 1 4. Keduce = x — 2. '5 3 5. A travels 5 miles an hour, and B starts on the same road 3 hours later than A and travels 5 \ miles an hour. When will B overtake A ? 6. Find the time between 5 and 6 o'clock when the hour and minute hand of a watch are together. 7. Find the square root of 4a; 2 — i2xy + 9?/ 2 + 42:2: — 6yz + z 2 . 8. Find the greatest common divisor of :\ miles per hour, respectively. Find the distance travelled on level road. 13. A man bought a house which cost him 4% on the purchase money to put it in repair. At the end of one year, having received no rent, he sold it for $1192, by which he gained 10% on the original cost, besides paying him 5^ on his investment as interest for the year. What did he pay for the house ? 14. In a town meeting a resolution was adopted by a majority equal to ^ of the number voting with the minority; but if 100 of those voting with the majority had voted with the minority, the majority in favor of the resolution would have been only 1. Find the number of voters on each side. 15. Reduce Vx — Va + yx -fa — b = Vb. 16. Reduce [(x — a) 2 + tab + b 2 ]* = x — a + b. „ ., x — Vx 2 — a 2 (x 2 -f ax)? — (x % — a % Y 17. Reduce (x + Vx 2 — a 2 )* (x 2 — a 2 )~i 18. Reduce 2x Vi — x 4 = a (1 + xf). 19. Vx — or 1 — V* — x l = (x — 1) or 1 . 20. A and B start together to walk around a circular course. In half an hour A has walked 3 complete circuits and B 4^. Assuming that each walks at uniform speed, tind when B next overtakes A. 21. The distance from A to B is 15 miles. The road is up-hill for the first 5 miles, then level for 4 miles, and then down-hill the rest of the distance. A man walks from A to B in 3 hours 52 minutes, and back in 4 hours; he then walks half way to B and back in 3 hours 55 minutes. Find his rate of walking up-hill. down-hill 7 and on level ground. 300 MISCELLANEOUS EXAMPLES. X 2 b 2 22. If 11 varies as , and if when x = - , y = J a + x a * 3 a 6 j — Y % , find the equations between x and y. a 1 + 23. Find the sura of 9 terms of an equidifferent series whose middle term is 18. 24. Find the sum of n terms of the series, 1 1 1 1 + V2 3 + 2 V2 7 + 5 V2 etc. 25. A number consists of 3 digits. The whole number is equal to the square of the number formed by the first two digits; also the first digit exceeds twice the second by unity. "What is the number ? 26. Prove that the number of ways in which m positive signs and n negative signs may be placed in a row so that no two negative signs shall be together, is equal to C „ . 27. A and B start at the same time and travel towards each other. In 7 days A is 5 miles more than his own day's jour- ney nearer the half-way house than B. In 10 days both have passed the half-way house and they are 100 miles apart, and B is 3 days longer than A upon the whole journey. Required their distance apart at starting and rate of walking. 28. A farmer sowed one bushel of wheat, and the second year sowed all the first year's crop, and thus continued sowing each year the whole crop of the preceding year. The 10th year the product was 1048576 bushels. What was the yearly rate of increase, on the supposition that it was the same, each year ? 29. Two men start from different points, at the same time, to walk towards each other; when they meet, one of them turns back, and on reaching his starting-point, again turns and walks .in the same direction as at first. Each arrives at the other's starting-point at the same time. Where did they first meet? What is the ratio of their rates of walking? Where did they meet the second time? MISCELLANEOUS EXAMPLES. 301 30. A man wishes to surround a given area by hurdles. Placing them one foot apart, he lacks 80 ; and putting them a yard apart, he has 50 hurdles too many. How many hurdles has he, and at what distance apart must they be, so as just to enclose the space ? 31. In the bottom of a cistern containing 192 gallons of water, two outlets are opened. After 3 hours, one of them is stopped, and the cistern is emptied by the other in 1 1 hours. Had 6 hours elapsed before the stoppage, it would have re- quired only 6 hours more to empty it. How many gallons did each outlet discharge in an hour, supposing the discharge uniform ? 32. Seduce (4 + $x — x~)? = 2*2$ + (x 2 + 32; — 4)^. 33. Find the relation between the coefficients of ax 2 -f bx -f- c = o, that one root may be one-half the other. 34. Divide 1 1 1 into three parts, such that the products of the parts taken two and two may be in the ratio of 4, 5, and 6. 35. Show that the number of ways in which mn things can be divided among m persons so that each shall have n of \mn them, is > \Z^ - 36. The m th term of an equidifferent series is - , and the 71 th term is — ■ Show that the sum of mn terms is - — ■ m 2 37. Find the sum of n terms of the reciprocals of an equi- multiple series whose first term is a and the multiplier m. 38. Find #„ of 2$, 4^, 81V 1 6 A etc. 39. If a is an equidifferent mean between b and c. and c an harmonic mean between a and b, show that b is an equi- multiple mean between a and c. 40. If n is a positive integer and x a positive fraction less. j x nJr ^- 1 x n than 1, show that — < n 4- 1 n 41. If a and b are positive, and m a positive fraction less than 1, show that (a -f b) m a>~ m < a + mb. 302 MISCELLANEOUS EXAMPLES. „. \ x 2 4- y = 7 ) to find all the values of x 42. Given { , - , { x + y l = n \ and y. 43. Reduce ax = by — y 2 ; a; 2 = y 2 + (/; — ?/) 2 . 44. Prove that w 5 — n is always divisible by 30 ; and, if n be odd, by 240. 45. The income of a certain railroad company would pay a dividend of d c / if there were no preferred stock ; but 8400000 is such stock, and is guaranteed l\%, the ordinary stockholders receiving only 5$. Find the amount of ordinary stock. 46. The population of a certain town in 1820 was 2375; in 1830, 2948; in 1840, 3800; in 1850, 5005; in 1S60, 6636; and in 1870, 8768. By the same law of increase, find the population in 1845, I ^54, 1862, and 1880. 47. A man has a plank whose ends are of unequal width. Find the distance from the narrow end that it must be cut, to make the parts equal. 48. Find the scales of the following series: 1, 4X, i8x 2 , So.c 3 , 35 6.? 4 , etc. 1, 2x, 3.T 2 , 8X 3 , 13X 4 , 30.T 5 , 5 52*, etc. 49. The population of a country increases 25 % every 10 years. In what time will it double? 50. If the student who is attempting to solve this problem belongs to a class of 50, of whom -^ cannot solve it and T 4 „ can solve it, and of the remainder f stand 2 chances to 1 to solve it. and f stand an even chance to fail, what is the chance that he will be successful ? FORMULAS. 9- 10. ii. 12. 14. 16. 17- 18. 19. 20. (a + x) (a — x) = a 2 — x 2 . (a 4- x) 2 = a 2 + 2 ax 4- ^ 2 . (a - z) 2 2 ax + aA fl ( 7) J \ (a + z) re = a n + Ha*- 1 2: 4 ^ * a^z 2 2 r2 , w (n — 1 ) O — 2 ) 4 ~ ■ a n ~ s x 3 + eta If x 2 4- 2a£ + 5 = o, x = — «± V« 2 — b. P m = n (n — 1) (n — 2) .... (u — m + 1). P„ = \n . P, C m = -J w (« — 1) (:?, + 77?) — 1 / a a \ rp\m (m+p)... [m+(r—i)p] (m+p)(m+2p) . . . (in + rp)J 33- if y z + py + q = ©, 34. - = 00 ; — = o ; - = a, or indeterminate. O 00 O 35. -0 = - 2 = -* = -6 = _*» = +• ,5_ 1 — 3 _ _5 ___7—- !«+! __ < NOTES The following brief sketches of eminent mathematicians who have made valuable contributions to our knowledge of the subjects treated in this volume and to whom reference has been made, are drawn from the most reliable sources. Note I. (P. 104.) Neirton, Sir Isaac, an illustrious English philosopher and mathematician, born at Woolsthorpe, in Lincolnshire, on the 25 th of December, 1642 (old style). He entered Trinity College, Cambridge, as a sub-sizar, in June, 166 1, before which date it does not appear that he had been a profound student of mathematics. It has been said that he commenced the study of Euclid's Elements, but he found the first propositions so self-evident that he threw the book aside as too trifling. In 1664 he discovered the Binomial Theorem, in 1665 took the degree of B. A., and probably in the same year discovered the Differential Calculus, or Method of Fluxions, as he called it. It was in the autumn of the same year that Newton con- ceived the idea of universal gravitation, the suggestion coming from the fall of an apple. It would exceed the limits of this notice even to mention his many remarkable works in Philosophy, Astronomy and Mathematics. Near the end of his life he said, " I know not what I may appear to the world, but to myself I seem to have been only like a boy playing on the sea-shore and diverting myself in now and then finding a smoother pebble or a prettier shell than ordinary, whilst the great ocean of truth lay all undiscovered before me." He died at Kensington on the 20th of March, 1727, and was buried in Westminster Abbey. 306 NOTES. Note II. (P. 186.) M'Lanrin , Colin, a Scottish mathematician, was born in Kilmodan, Argyllshire, in Feb. 1698, and died in Edinburgh, June 14th, 1746. He was a graduate of the University of Glasgow, and in 17 17 was appointed Professor of Mathematics in Marischal College, Aberdeen, which posi- tion he held till 1725, when at the recommendation of Sir Isaac Newton he was called to the mathematical chair of Edinburgh. He held this professorship for over twenty years. His principal works are, " Geometrica Organica" "A Treatise on the Percussion of Bodies," " On Fluxions," said to be the most complete treatise on the subject and the author's most profound work ; " A Treatise on Algebra," and "An Account of Sir Isaac Newton's Philosophical Discoveries." Note III. (P. 191.) Napier, John, Baron of Merchiston, was bom at Merchiston Castle, near Edinburgh, Scotland, in 1550. He was educated at the University of St. Andrew's, and is celebrated as the inventor of Logarithms. His loga- rithmic tables were first published in 1614 under the title " Mirifici Logarithmorum Canonis Description Napier also enriched the science of Trigonometry by the general theorem for the resolution of all the cases of right-angled spherical triangles. He died in 16 17. Note IV. (P. 191.) Brir/f/s, Henry, an eminent English mathematician, born at Warleywood, near Halifax, about 1556. He was educated at St. John's College, Cambridge. In 1596 he was chosen Professor of Geometry in Gresham College, London. He became first Savilian Professor of Geometry at Oxford in 16 1 9. He is chiefly distinguished for the improvement and con- struction of logarith ms. No sooner was Napier's system of Loga- rithms published, than Prof. Briggs began the application of the rules in his " Imitatio Napierea." He greatly improved upon NOTES. 307 Napier's plan, by adopting 10 as the base of his system, and he has the honor of being the author of the system now in general use. He published in 1624 a work entitled " Logarithmica Arithmetica" containing the logarithms of all integral numbers to 20000, and also from 90000 to 100000, calculated to fourteen places. He died in 1630. Note V. (P. 263.) Sturm, Jacques Charles Francois, an eminent Swiss mathematician, was born at Geneva, in September, 1803. He was tutor to the son of Madame de Stael, with whom he visited Paris in 1823. In 1827 Sturm and his friend Colladon obtained the grand prize of Mathematics, proposed by the Academy of Sciences in Paris, for the best memoir on the compression of liquids. He discovered in 1829 the celebrated theorem which bears his name. He became Professor of Mathematics at the College Eollin in 1830, a member of the Institute in 1836, and Professor of Analysis at the Polytechnic School in 1840. He died in 1855. Note VI. (P. 267.) Homer f TV. 6?., was an eminent English mathemati- cian, born near the close of the last century. He was a teacher of mathematics in Bath, and died in 1837. About fifty years ago he discovered the Method of Synthetic Division, otherwise known as the "Method of Dividing by Detached Coefficients." In 1819 he com- municated to the Royal Society his method of solving algebraic equations of all degrees, entitled, " A New Method of solving Numerical Equations of all orders, by continuous Approximations." Previous to this time there was no direct and reliable method of finding the roots of equations beyond the fourth degree. By his method the process is comparatively brief and simple. This method is regarded as among the most valuable contributions to the science of Mathematics in modern times. The first elementary writer that saw the value of it, says De Morgan, was Prof. J. R. Young, who 308 NOTES. introduced it into his Treatise on Algebra, published in 1826. Prof. Young says it is the shortest method of extracting roots of higher equations that he has seen. Note VII. (P. 282.) Cavdan, Jerome, an Italian physician and mathema- tician, born at Pavia in 150 1. He graduated as doctor of medicine at Padna in 1525, and was successively professor of mathematics and medicine at Milan and Bologna. He dealt much in Astrology and was a professed adept in magical arts. Among his numerous writings are, "Ars Magna," "De Rem in SuMilitate," " De Rerum Varietate," " De Vita Propria? and several medical works. In 1545 he published in his "Ars Magna" a method of solving cubic equations, now known as " Cardan's Formula." He was the first that noticed negative roots. He died at Eome in 1576. Note VIII. (P. 286.) Desert rtes 9 Rene, (Lat. Renatus Cartesius) an illus- trious French philosopher and mathematician, born at La Haye, in Touraine, March 31, 1596. He was educated at the College of La Fleche. On leaving college, at the age of nineteen, he resolved to reject all scholastic dogmas and to free himself of prejudices, and then to receive nothing that was not supported by reason and experiment. To perfect his education he determined to travel, and to this end entered the Dutch army in 1616 and came into the service of the Duke of Bavaria in 1619. I' 1 1620 he was in the battle of Prague, but soon renounced the military profession and gave himself to more congenial pursuits. In 1637 he produced his celebrated "Discourse on the Method of Reasoning Well and of investigating Scientific Truth? in which were included treatises on Metaphysics, Dioptrics and Geometry. This last treatise included the method now known as the Cartesian Geometry. The formula given in our Appendix for the reduction of biquadratic equations is due to him. He died at Stockholm in February, 1650. ANSWERS Page 11, Art. 16. i . The square of the side. 2. Twice the radius. 3. The circumference is 3.1 416 times the diameter. The area is .7854 times the square of its diameter. 4. Interest is the product of principal, rate and time. Amount is the product of principal, 1 + rate and time. 5. The product. 6. The product. Page 26, Art. 92. I. — . 7- + • r 1 . Has no sign. 2. — . 8. + . 12. 4-. 3. a is — and b is +. 9- — . 13- +• 4-6. Given. 10. + • 14. Has no sign. or Page 27, Art. 96. 2. (a + by = a 2 + 2ab + b 2 . 3. (a — by = a 2 — 2ab -f- b 2 . 4. (a + If + (« - by = 2 {n 2 + b 2 ). 5. a 2 — b 2 ~- a — b = a + b, a 2 — b 2 = a + b. a — b x$ — y* 1 1 6. — -^ = 0* _ y\. X* + 2/T 7- (* + */) 2 - (x - yf = 4xy. 5. 10. 6. 4. Prtr/e 28, Art. 98. 7. ± 2. 8. + 2. 9. 144. 10. 16. 310 ANSWERS, i. Given. 2. i ia + Sx — 2a 2 + ax. 3. a-- + 44* -\-2c1b- — ab. 4. 6a 2 b + ioai 2 — Sax 2 — a 2 x. 5. «Z< 2 — zu l b + a 3 — 6ac — 4ttC 2 — c 3 . 6. 2> x y % + 9 r2 i/ + 7 x2 y 2 - 7. #z — 2to 4- ex. Page 31, Art. 100. 8. 3«*#2 — «5s — a s b. 9. aa^y — 2ax\f. 10. Vei + V«c. 1 1. 3a 2 Z> — aJ + 2b, or J (3ft 2 — a + 2). 12. r/ 2 (3.r + 2« — 35). 13. n (m — 2d 4- b). 14. Given. 15. (a + £)* — (a — J)* P« x + 7« 3. 5«J. 4. 2« 2 + ax. 17. Va — #• Page 33, Art. 114. 5. {a — 1) (3 + y) 9. o. - 2 %- 6. — 2 Vz 2 + a 2 . 7. 2# — 2C. 8. 6z 8 + to 2 . Page 35, Art. 110. 10. o. 1 1 . 40b. 12. 2 (b — c 13. — 6 2 . m). 1. 2rt — 2J — 2c — 42+21/ — 4. 2. 3 fl _ 3 & _ 3 c + 3 to — 9. 3. 2a — 2C. 4. 2£ (a + J — c). i 7 1 1 11,11 5. a a toca _|_ 2%?o*c«, or (2* + i)a^jM, 6. 2a 2 (c — 5). 7. ZCESb* — Xlf. „ 1,1 S , J , 8. 2a-0- — 2rt-"'//S — rt0. 9. — 2C V# + #• 10. 4a- ^. 11. 2« + // + & = 2rt, or 2ff + 2b. 12. 4to? — 2am. Page 38, Art. 130. 1, 2. Given. 3. \a 3 x 2 yH. 4. ^tocz 3 . 5. 2ia*bx*y 8 . 6. 5r< w+7 ' +1 to + * +1 a;z. + wi + 2 — +ra+S 7. DO" m 8. 3a 2 to. 9. 4. 10. a^J 2 *. 11. a^b 2 ™. ANSWERS. 311 12. a im b 2m . 13. zMtf 1 . 14. x 2 >^y. '2n-—Sn ,i2?n-—m 15. x*"—*' l y 16. 6a i bx'-. 17. =F« 4 Z/ 4 .r>. 18. — 2a l b 2 X 2 . , J 10 19. oa 3 x'»'. is. 18 20. a? a; * . J'rtf/e 39, Art. 131. 21. Given. 22. « 4 — 2a 2 £ 3 + z 4 . 23. 2a z b — 2a 2 bx 2 — 20b 3 . 24. a 5 + ft 4 .? — ax* — x 5 . 25. 2r< 2 a; + 2abx 2 + 2«ca; 3 — ax A bx 3 ex*. 26. 4« 2 — b 2 x 2 + 2&ra 3 — c'x*. 2 7 . «3ot _ a mffln _(_ fl 2»i£n _ £3ra 28. I — 2A' 2 4- 2X 3 — 2a; 5 + X 6 . Page 40, Art. 133. 30. 3a 7 J — $a % b 2 + 2a 5 b s + 7 r^ 4 /> 4 — ja 3 b 5 — 5« 2 J 6 + 5^ 31. a 5 — 7,a 3 x 2 4- a 2 x 3 + 2f/.r 4 — a; 5 . 32. .r 6 4- x*y 2 — x 2 xf //" 33- (t s — atx — 5« 4 £ 4 + 8a 3 x 5 — 4« 2 £ 6 + ax 7 . Page 44, Art. 144. 5. ^ab 3 c hi 2 , or 4ad 2 W 6. \a ^b~hd~ 2 , or ytbd 2 7- 8. $a 2 c 5 d 2 ~~b 3 9. Given. 10. — 4f< w - 1 Z<"- 1 . 11. — 2*7 4 Z> 2 . S aP ~b n b 2 z« Page 45, Art. 145. 4-z.i 15. — yt*o*. 16. ab?. a 12. 57"'//-", or 13. — 4tfZ< 2 . 14. — \a~ x b~ 2 , or — 4ab 2 17. — j = — ab~\ 18. — a m "-b nl 19. — a m;i - m b n ' 1 - n . 20. tfi+lymr-\ 21. a n+2 b n + i x n . Page 4(>, Art. 140. 1. J.r — 2f?.r 2 4- ytV/x 3 . 2. 2 4- rt"Z/2m _ 3ft 3»J2»n # 3. -2f/ w— " 24- Aa 2m ~ n b 2n ~ m . 4. 2a m b s ~ 2n — 3<7 w + 2M fr ! . 312 A NSWERS, 5. 2 ;t m - 2n b s — 3a m b s+2a . 6. 2a~ m -» ] u* '-») — 3/. 5. a 4 — e&c 2 -j- " 4 - ,3 11 11 3 6. a* + ft-a' + (fix* + a* 7. ft 4- ft-a;2 — x. 8. a w + a 11 . 9. ft" 6" — a n -W\ _a -I7-1 7—2 10. ft a — ft 36 3 + * 11. ft 3 — 5« 2 a; — «£. I'ftf/e 48, Art. 148. 12. a*. 13. I — 2£ + X 2 . 14. & 2 + 26a 4- x*. 15. a 3 4- 30% + 3-^ 2 + 9/ 32i/ 4 + W 2ft— 13 16. 2ft 2 — 2ft — 2 4- $.~>, Arts. 1Z5-15U. To give the answers to problems in Factoring Avould destroy their value to the student. They are therefore omit- ted. The same reason may be inferred when other answers are omitted. 1'aae fil, Art. ItiS. i, 2. Given. 3. a (x + a) 4. x + 5. 5. b (a + b) 6. x {a 4- b) 7. a — b. 8. x — 1. 9. a — x. 10. (x — i) 2 (x — 2). 11. X + 2. 12. 3 13- 3 + 3- 14. X + 2. 15. £ 2 — 4. IVff/e 62, Art. 173. 1. a 4 + «.t 3 — ff 3 .r — 2X + 15. a 4 4- 5^' 3 + 5 X ~ — 5^ — 6 - a 4 4- flfte — ff-c 3 — a 4 . JV/f/e 6*6, .4r/. 193 a 2 — xy 4- y~ Given. 6a*d?e[ 2 yibc 2 yz. x — y r i _ x 2 + 1 A' 2 — I I X* _ ^2 4. yi « + 3 2 3 "(^+7)' a — a x 4- « 1 — a 3. r' 2 4- CO + &*• 4. „ 8 — a 6 &2 4- a 4 * 4 — fc 8 & 6 + W 5. r ? 12 — rt 6 £ 6 + J 12 . rr> 2 -f- a. x 4- 1 6. 7. x + : 8. £2 + x 3 + a- — 1 2 J - 1 a + T ANSWERS. 315 a x ab a 4 + a 2 — 2 Page 67, Art. 195. 2.V 2 -)- 2 .r + i « — a; 6. 2rt5 7. o. a 2 c — c a 6 — 2fl 4 + 2 a 2 — I a 6 + 1 a 4 + 2 a 3 + 2 ft 2 + 2 a + a* — 1 Page 68, Art. 196. s 6 + 5^ 5 + 5 y + 25 a; 2 — 2.T — 35 ff 3 + a 2 + « « + 1 6. „- x i — 1 Page 69, Art. 19 S. x* — x s y + x 2 y 2 — xy s ^ x 4 + x 2 y 2 _ X* — y 4 ' re 4 — y 4 ' a 4 — y 4 a 4 + 1 ^ 2a 4 — 2 _ 3a 6 — 3a 4 + 3. ,.2 _ 2 a — 1 4 (a 7 + a 6 — a — 1 ) 8 (« 8 + a 6 1)' 2 (a 7 — a 6 4- 2a 5 — 2a 4 4- 2« 3 — 2« 2 4- a — 1) m 8 (a 8 + a« _ a 2 _ !) _ 5 8 (a 8 4- a 6 — a 2 — 1) 4 (x 2 — 4) x 3 — 3a 2 + 4 # x 5 4- 3a 2 — 4 4 - 34 _ 5X 2 + 4 5 34 _ 5 ^qT" 4 ' ^ — 5^4-4" 5. C. D. is (x 4 - 1) (x 3 4- 3^ _ 3 x 4- 3) (^ 2 + 3). Numerators are a (a 3 + t,x 2 — 33 4- 3) (z 2 + 3) ; (* 2 + i) 2 (^-i)(^ 2 + 3); And (a 4 — 1) (a; 3 4- 3a 2 — 3a +-3). s + y . (x — y ) O 2 — ?/ 3 ) _ (.r 2 + ?/ 2 ) (a 2 + y 3 ) a 4 -?/ t6' f //< 316 I, 2. Given. 3« (« + x) 3- ¥ 4- 2 (a -\- b) a — b 5- o. 6. — i. ANSWERS. Page 70, Art. 109. x + 4 a; — 4 8 -*_ ' a; + rt 2 9- " — 3 x d + 2x 2 — 9.r — il -a- 2 a; 4 -f x 3 — x — i x a + x i x + r i a; a X X a + 3 W- X 3 i i >' ( a 8 _ £3)2 3^ 2XI/Z 2 9«< 2 d a: 2 — 3a; + 9 a; 2 — 4 Page 71, 4- 5- ^«. 20i 1 a; + 4 * — 3* 6. -5-. .2 + 4 '" a; 2 — rt 5 s IO x % — ax 4 + d' Page 72, Art. 202. 5- 6. a; 6 — cfi.r 3 + a 4 a^ — a 3 x 2 + a c z 5 + a 3 x 3 + a 5 aa: + 7a; + 4a + 28 «.r + 5a; + 6a + 30 a — 1 (x* — x + 1) (a 2 — a + 1) 9. — (a; 5 + x* + a; 3 + a; 2 + a; + 1 ). IVj/e 72, Art. 203. x 4 -c' 4 + a J (1 - O (3 - « 2 ) a 2 + ay + if x 2 — f ANSWERS, 317 Page 73, Arts. 205, 206. 25a 2 — $ab _ 25a 2 acex c 10 + C 5 X 6 _|_ x i% a 1 —l$x % 4- x* ' 5a. 2. 3^ (1 — x). b* (c 3 + d 3 ) 5 ' a («6 2 + c 2 «") ' , Q/ 8 -y 4 + i)(a+ 1) h a* + ^ + 1 # 4 (a; 12 ?/ 12 — z 10 ?/ 10 + z?y s — a- 6 ?/ 8 + a- 4 ?/ 4 — xhp + 1 ) y 6 (// 6 — rt # 5 + a ^y i — a3 ?f + fl4 # 2 — a5 + « 6 c — .S c 14 c 2 — 169 rr — 4 8a 4- 16 a 4 — y 5 Page 75, Art. 210. a 3 - a *~b + sab 2 4 - 5 3 a (aaTTjs) 2a 2 — 2a -f- 1 a (a — 1) az a 4- # a 4 4- 6a 2 + 1 a — 1 x 1 + ax 4- a 2 2; — a 2a# a + ic a — x ab y- x — 1 10. a 2 — x\ 11. a + x. 12. aW — a 2 - -i 8 4- 1 « 2 - & 2 13- 1. 14. 1 x X J 5- a. 16. I+ ^ 2 ' 17- a — 1. 18. 1 -5. Given. x = 12. * = 7. Pagre 82, Art. 233. abc 9. x ab + ac J -c be 318 ANSWERS. mn — ab io. x = — ■ — i a + o — m — n _ tfc + al? + be* — (a + b + c) ab + ac 4- be — i 1 6. x = 4. 17. # = 6. 18. x = 4. 19. a; = 7b. 20. a; = ^6 (1 — a 2 ). 12. 2 = 5. 13. x = m — n 14. x = 5. 15. 2; = 7. C + 2(tf — J) 21. K= S -. zb 22. 2; = 2. PROBLEMS. 1. 2 dols., 6 halves, 30 qrs., 90 dimes, and 450 half dimes. 2. 36 years. 3. 210 acres. 4. - - = the greater, and = the less. 2 2 c. — = one part, and — = the other nart. 2 x 2 aw — c» + J , era + « — J 6. The parts are and 11 + 1 « 4- 1 7. $80000. 8. — $600. 9. 1st, 6 miles ; 2d, 15 miles ; 3d, 00 miles. Page 89. i7- 10. a years 11. $93. 52 8o??^c SZOU7W , . „ 12. — ^ ; c being feet. 13 14 5280m — nc s 1 + a abc , — =- days. r/Z> + ac + fo J 15. 40 gallons. 16. c-b a a n — m a m 4- 11 19. Brandy 30. wine 40, and water 70 gallons. 20. A's, ^8si; B's, Igif ; C's, $83^-. 21. 30 and 7. Prob. 4 gives the formula. ANSWERS. 319 Page 'JO. 22. 60 apples and 20 oranges. 23. x = — — ; in which x = original number of oranges, J m — n ° ° mx = the number of apples, a = number of apples, and b = number of oranges sold, and n = ratio of oranges to apples after the sale. 2a (1 — mn) „ . ,, 24. — = No. sold m all. mn + m — 2 — — - = No. of apples at first. mn + 7U — 2 — — = No. of oranges at first. mn + m — 2 25. 1st, 140 qts. ; 2d, 60 qts. ; 3d, 45 qts. ; 4th, 80 qts. 26. 70; 25 ; 36 ; 15 ; 20. 27. 1^ of a mile per hour. 28. 1 mile per hour. 29. 1000. Page 95, Art. 251. 1. Given. 2. x — 2, y = 3. 3. x = 12, ?/ = 18. 5. a = 11, y = 9. // — bd ad — h 6. x = r , y = r- a — b J a — b 7. x = 7i? (a + b), y = h (a + V)- 8. x — 7L y = 5. 9. a: = 244, y = — 172. 10. 2; = 8, y — 12. 4a + b b — 2Ci 11. x = ^-z — , y = 6 J 12 b'r — be _ «'c — ac' l2 - x = ah > _ a > b > V — a'b^-~ab'' 13. The equations are not independent and represent but one condition; viz., that y = \y. 320 ANSWERS. b' — b ab' — n'b 14. x — ;, V — —j— a — a' J a — a 15. x = oq, y == 00. The two conditions are incompatible except for infinite quantities. 16. x = 3, # = 4. 17., = —-, y = -j- 18. x = 6, y = 4. 19. x — 4, y = 15. Page 96. PROBLEMS. i. The number is 462. 2. 6 oranges and 10 apples. 3. The fraction is -ff. 4. The 1st in 12, the 2d in 20, and the 3d in 30 hours. 5. The problem is indeterminate, only one condition being given from which to determine two unknown quantities, viz., that the interest on the sum for 2 months shall be §10. 6. A in 25 days and B in i6| days. 7. A, I ; B, -j% ; and C, ^. C worked 6 days. _ ac — ab , ,, , . ab — be 8. —r 01 the 1st, and j- of the 2d. a — a — b Page 07. 9. Loaves 6 cts. each ; Fruit 20 cts. 10. The fraction is f. (;» 2 + mn) c — (« 2 + ab) p 11. 1st 2d, (a -f £) (w + n) (nib — an) ' (M 8 + WMl) C — (6 2 4- tfJ)p (a + b) (m 4- ») («« — mb) 12. The problem is indeterminate. / 13. $400 and $100, at 2% and 4%. 14. The fraction is ^. 15. A's, 500 dollars ; B's, — 500 dollars. 16. A, $900 ; B, $600. The mortgage $500. ANSWERS. 321 Note. — The student will observe that the preceding problem contains really two independent problems. Page 09, Art. 252. i. Given. 2. x = 3, y = 2, and z = i. (a 2 — #c) w + (/> 3 — rtc) » + (c 2 — a b) r 3. x — y = a 3 + b 3 4- c 3 — 3«#c (£ 2 — «^) m 4- (c 3 — i—m') — iniH'\Hii , -^{jn^i^n')' p , _ mn'ri m'n"- m" [i — m')—n>i')i"(m' — m")(i—mm') m(m'n—m"J{ n' — m'n)— nn"(m'— m"n T )(i— mm' ) ' 8 - A > --^-r-L7, 17/ 5 B> ub + &rf — ad ' ' ad -+■ bd — ab' p 2«fo/ ~ 2 abed ^> 7J7J I 77 77J? ' ^' ad -{- ab — bd ' ' abd — acd + bed 9. A's, $1000 ; B's, — $500 ; C's, $0. IO - 3 ? 5, and 6 miles. JV/f/r 106, Art. 274. 5. ft 2 + 2ax 4- a: 2 and « 2 — 202; + a; 2 . 6. 4a 2 4- 4«& + Z/ 2 and « 2 — 4^ 4- 4b 2 . I b b* . ± b & 7. a~ -\ j - + etc. and a~ r s — etc. 2tt a 8«' 3 20 s 8«? 8. .r 6 + 6.r 5 y 4- 15./' 4 // 2 + 2o.r 3 // 3 + 15/*!/ + 6xif + if; X s — 7x e y + 2ix 5 f — 35^y + 35-'' 3 i/ 4 — 2ix 2 y 5 +jxy 6 — //<. 9. 32a 5 -|- i6oa i b + i,20(fib~ 4- 320« 2 Z/ 3 4- i6o«£ 4 4- 32^; 27a 3 — 8ia 2 6 4- 8i^ 2 — 27J 3 . I X X •2 io 1 . 4- .4- etc. a (fi a 3 a i I 2X -zx* AX S a 3 a 3 a* « 5 I 2.T X % 4.T 3 11. a 3 4 l - 4 4- — ^ — etc. ; 3a 3 9« :! 8i«» I 22; x 2 AX 3 a* - -. — etc. 3«3 9a 5 8if/J s s 1 erf* erf 12. a? + %a% + ^¥ 4-^-^4- etc. ; i6d* 64a 2 B 7 T 9 .111 CLU * a- 2a- 8«? i6«*~ ANSWERS. 10 5'" 323 1 s^ W 2 1015c 8 i3- T + ^r + fr- + — tt + etc. ; a* 2a s 8a- ioa~s~ s — fare + J £-a 2 t 2 — 5^ i6a* 5c 4 — etc. 14. aV 4- + etc. ; by — + -f- + Hh + etc. «2^ 2 a- £3 Srr-u - - 15. r/ 3 + 4^5 — Sac + 4^ — \6bc + 16c 2 . 16. 4a 2 — \2ffix + 4rt& 2 — $axy 4- 4«2 4- 9« 2 .e 2 — 6ab 2 x -f- 6ea -3 ?/ — 6«.rz + Z/ 4 — 2#\n/ + 2# 2 z 4- .r 2 */ 2 — 2.^2 + z 2 . 17. a 2 aj 2 4- 2abxy — 6axz 4- ioax 4- b y 2 — 6byz 4- 10 by 4- 92 2 — 302 + 25. 18. a 3 4- 3« 2 6 — 3« 2 c + yf~d — 6bcd 4- b 3 + 3«i 2 — 3//^ 4- $b 2 d — 6acd — c 3 4- yic 2 4- 3^c 2 4- ^c 2 d 4- 6^&r/ 4- d 3 4- 3«r/ 2 4- 3W 2 — 3-Y/ 3 — 6«#c. 19. 8^' 3 — 36x' 2 y 4- i2.« 3 z — 2jy 3 4- 54a;?/ 3 4- 2yy 2 z 4- z 3 + 6xz 2 — gyz 2 — 3^xyz. 20. a 3 x 3 4- yi 2 bx 2 y 4- 3a 3 .r 3 .i' 4- 3a* mx* — ia 2 nx 2 4- 6bmyz — 6/w»2 — Gbnnry — 6bnyz 4- # 3 ^ 3 4- ytb 2 xy 2 4- $b 2 y 2 z 4- $b 2 my 2 — 3^ 2 »// 2 4- damxz — 6amnx — Gnnxz 4- z z 4- 3«#2 3 + 3^' 2 + 3^?2 2 — 3^2 3 + dabmxy — 6abnxy 4- w 3 + ynn 2 x 4- 2>bm 2 y 4- 3?« 2 « — 3m 2 w 4- dabxyz — n 3 4- ^an 2 x 4- 3^ 3 3- a(*± W-3)- 4 .—5 and 5 (£ ± J V— 3) ANSWERS. .-lrf*. 305-310. 6. a. 7. 5 Vo.r — 2 \/— «&• 9- 10. II. 12. i3- 14. i5- 16. i7- 18. 19. 20. 21. 22. 2 3- 24. JPaflre 22?. a 3 b + «& 3 — « 2 £ + ab % 4- rf&fti + tffai. a* 4- fo t z. 1 a- -t- o-\ a — Z>. a - b. (a+»)(tf*+8*)(«t*+«*). ■V — «&cd. 8a 3 . a 2 + ». a + V — x. V — x — V — a. a (a* — X s ). a* + **. m* + «= *w 5 + w 2 1 + re l+»i a~mn~b mn + a m b n l + m 1 + n m + n m + n -f a~n b^ 1 + a mn b mn . 8. 2 V« 2 — tf 2 . 25. a n — A". 26. (P + aJs 4- Ja. 27. a 3 — ofib 4- 62. 28. cfi + a 2 5* 4- ff^Jt 4- a b 1,4 ,5 + a*b* 4- A : . X'l 12,2 8,4 29. fl » - ft 5 0> + rt^5 * t ■ -78 — tf*ft* 4- //\ 2-2 20 7 1 - , 1 30. « » — a" 3 0* + a 6 P 16,8 1_4, .,5 — cf*0* 4- a a — afyi 4- cf^ft* — a^fa 4- a 2 ^ 2 — a*o* + a 3 6- — err . 31. 1024. 32. 1024. 33- i° 2 4- 34. 1024. 2. 2 V3. 3 ii- 3- 3 / V / 2- 4- 1^3- Prtflre 125, 4r*. 3i2. 4. — 12. 5. — 6 y/2. Art. 313. 5. vV 6. 1 4- -v/^a. ANSWERS. 325 i. a. 2. I. Page 12S, Art. 31S. 4CIX 4 2 a/ i + x 3- " 2 - 7. -V2+| V— I. 8. Zero. 2 a 2 9- a 2 — a; 2 5- J 3- 6. 2 + 3 V3 — a/5- ,P« y * —x *~y* + , IS 10 11 , 8 . „ 2 2 , 4 2 4 2 2 6 8 8 ° y "' — x a y 5 4-a^y 4 — ^// : ' +-' :i // 5 — ■ i " j y z +y s z 10 + # 6 1 o A 34 „ £ £. „ 1 a^ — ay 3 4- a^t/a _ a z y + a"-?/ 3 — a 3 //j y* — a 3 1'age 131, Art. 319. i. Given. 2. x — I. 3- a = 25- (a - 6)* 4. x- = 2(7 — b 1. .?- — 5-T — 14 = o. 2. 2 % — (a 4- £) X 4- tf£ = o. 3. x % — (a — b) x — ab = o. / ab Y 6. x = 3. 7. X =■ VIC. 8. x = cc, or ± 5. n r 2 5 9. X TS . Page 135, Art. 330. 4. x 2 4- (a + b)x + ab = o. 5. .<- — 6x + 12 = o, 6. 2* + 4>r + 5 = o. 1. * = ± f. 2. x — ± 3. 3 . a J =$[a-J±(« + &)V^]. 4. x = 3, or — f.^ 5. ./• = 2 ± V— 23. 6. .r = 5, or f. 7. a; = ±| V2. 8. 'x '*= I (a 4- b 4- c ± y ( fi + p + ( .-i _ r /£ _ ac—ic). 9. .r = ± 5. 10. x = 1, or — 2. 1 1. .? = ± V— be = —?Vbc, or —*Vbc. rage 137, Art. 335. 12. .r = * (1 + a/i7)- 13- x = \(a ± v« 2 . + b). 14. x = 3, or — 4. is- x = \, or — f. 16. x = 1 (9 ± V^45)- i7- * ±a 18. ;z 2 — mr ± Va? {m* — n*) 4- A s >n 2 ]. ,9. s = V»(i±^)- ANSWERS. 327 i. 15 and 6, or 35 and — 14. 2. 10 and 4. 3. 17 and 9. Par/c 138. PROBLEMS. 7. 36 and 64. 4. \ 'a 2 ± b 2 5- 3 6 - 6. 58 and 37. 4 miles per hour. 10. 2 2 25 miles from 0. n. 13. 5000 in 1840, 4000 in 1850, 5200 in i860, 5300 in 1870. 14. 12 and 18. 15. 144. 16. 3 inches. 17. The man 36 yrs., son 16. Page 139 18. [n— a±V{ct— n) 2 — 4b]. ) r hV3 \and m 2 's/n — 1 y/411 — 1 — V 3 Vw — I 19. 7. 20. 75 at $4. or— 120 at — $2.50. 21. 256 sq. yds. JY/rye i^/, Art. 339. 1. » = a ± \ Vi — 4") w - 2. a?= (-liivr+^r. 3. « = ± ^« 2 - [- \ ± JVi + 8 {m 4- wTJ 2 . 2 4 5. x = 27, or — 64. 6. a; = 64, or 729. • \ ± Vn(n — 1) + i 7. x — : ^ -— — » I ± Vn(n— 1) + i Prtgre i^2. 8. a: = 4, or — 1, 9. .? = ± 20 V3 328 8 ANSWERS. IO. x = o, or — \ (b ± V b 2 — A°)- 1 1. x = 5, or 26|. 12. x= ±i V3. 13- a; = 3, or — £. 14. a; = — (1 ± 2 V— 2) 6 , or o. 729 15. X = 0, i or 2. 16. X = ± 1. 17. X = 18. a; = — 1, or \ ( — I ± I ± V- 3), or|(i± V— 3) 19. x = ± 3«- 20. a: = 2. 21. a; = 22. £ = a (1 ± ??) 2 1 ± 2W 2 aft 23. S = ±i- Pr/flre i4S, Art. 350. 5. a- 6. x 7. a; 8. 3; 9. x 10. .r 11. x V 12. .T 13. X 14. # 15. a; 5; V = i- 7, or 2 ; y = 2, or 7, y = S ;. T /*==). 3, or 2 ; y = 2,, or 3. ± 2, or =F ^ 6 - V5 ; y = ± 3, or ± f V5. 2, or i ; 7/ = 3< or — 24. 4, 16, or 14 ± V58 ; 5> — 7, or — 1 ± V^8. \ (1 ± V— 7) ; y = i(iT a/ 1 ^). 4, or — 2 ; ?/ = 2, or — 4. ±3; y = ±i- ±2; y = ±i. ANSWERS 329 1 6. x y 17. X 18. X 19. X 20. X 21. .r 22. a; 23. a; 24. x 25. iC 26. a; 27. z 28. iB 29. a; 30. a; I [a ± Vcfi + b ± *Jb - 2 (a* ± a V« 2 + &)]; i [a ± Va* + b T^i-2 (« 2 ± a y^"+T)]. 1, or — 2 ; y =r — 2, or 1. 16, or 4 ; y = 4, or 16. 9, or 1 ; y = 1, or 9. 2 ; y = 1. 4; 2/ = 2. 2, or 4 ; y = 4, or 2. 5 ; 2/ = 3- ± 6, or + 3 ; y = ± 3, or ± 6. ± 5 ; y = ± !• 2- ; y = 1. 4, or 32 ; y = — 3, or f 3, 2, or — z ± -j Vio ; 2, 3, or — 2 + i -v/xo. 5, or I ; y = 2, or ± 2 5 # = 3- 2. 7, it, and 23. 3. 6, 13, and 25. Page 149. PROBLEMS. 4. 4 and 5 yards 5- 36. Page 150. 6. 15, 12, and 9. 7. A's, $80 at 5%; B's, $120 at 6%. 8. A's, $100 ; B's, $150. 9. ,-u = ( — an* + b \/b 2 m 2 — a 2 )n 2 + n*), and (r — a 4 Ifi — a* -s (— £« 2 ± a Vb*m* — a 2 m? + w 4 ). 330 ANSWERS. 10. The parts of a are — \ab — m + n ± \/(ab — m— n) 2 — 4m >i\, and 20 —. [ab + m — n ^f V{ab — m — nf — 4m a]. 20 The parts of b are 211 [ab + m — n ± V(ub — m — nf — 41/in], and — [ab — m + n ^ V(ab — m — n) 2 — 4inn~\. 211 L 11. yTi. 12. i (3 + V^~3) and ± (3 - V^3). 13- i(3 ± V5) and i(i ± Vs). 6 ±2 V5 6 ± 2 V5 '5- ± 3 and ± 1. 16. A had 200 acres, at $1.50 per acre ; B had 400 acres, at $.75 per acre. Page 153, Art. 357. 1. 2. Given. 3. x > 4, the limit of #. 4. 2; > 2£, the limit of a;. 5. x > «, a: < #, the limits of x. 6. x < 5, a; > 3 ; .*. 4 is the only integral value of x. 7. x < 6 and » > 4 ; .•. 3, 4, and 5, are the integral values of x. 8. The number is 19 or 20. 9. He sold 60 apples. 10. 20 and 5. Page 100. P R B L E M S . i. The third proportional is 100. 2. The first term is 54. 3. The mean proportional is 15. ANSWERS. 331 T abc , 4. In -7T-, clays. * ab J 5. 16 and 20 tons. 6. A, 54! hours at 9^ miles per hour = 494-ff miles ; B, 59! hours at n T \ miles per hour = 57o T 6 ¥ miles ; A, 3 hours at o miles per hour = o miles ; B, o hours at 2 miles per hour = o miles. 7. The numbers are 30, 48, 50. 8. 6, or 3 (— 1 + V— 3) ; and 9, or § (— 1 ± V^~i). 9. The number is 863. 10. The numbers are a, 2a, and 3a. 11. The conditions are not independent. Page 162, Art. 381. 1. Given. in 2. x = - 2 ; xif = m ; » r : x 2 = y 2 2 : 3^2. 3. a; = m (a + y); x I : :r 2 = a + y T : a + y 2 - 4. x = m (tf + if) ; x x : x 2 = y Y 2 + y z 3 : y 2 2 + y 2 3. 5- « = rr+ya ; a?i : x * = y* + y* : 2/1 + vi • 6. The values of y are 25, nf ; of x, 12, 4f. 7. The values of x are ff , 2?> xs> °- 8. The value of m is 1. 9. The value of y is f \ 2X % . 10. y = 2(s + i). 11. 19200 inches. 5 seconds. Paflre iOo, ^?*f. 35*. 1. 362880. 3. 126. 2. 15120. 4. 126. 332 ANSWERS. _ n (n — i) (n — m + i) 5* °»» — n \l l _ n (n — i) . . . . (n — m + i) \u — tit ^n — m — \n — m \m n (n — i) . . . . [n — (n — m) + i] to — m \n — m \m ' n (n — i) . . . (m + i) \m ii \n — m \m 6. a = i 5 . 7. Cs = 84. 8. No more. 9. [m. 10. Zero. \n — m \m 11. 8, or — 1. The 2d Ans. not applicable. 12. m = n. 13. m = n, or w + 1. l S- 454053 6 °o- 16. 3991680. Page 175, Art. 417. -6. Given. -£ = (x-b) (x-c) + (x-a) {x-c) + {x-a) (x-b). (It/ X 2 + 2X — I dx (X + l) 2 9- iz = 1 — x A dy dx (x 2 + i) 2 10. dy dx to" - 2 > r/ 2 // _ 3 a;^ ^ t/a; 2 + -! + —• 4x? x 3 535 (/// 1 11. — — = • dx (1 — a;) 2 12. ^ = 5 + 2Cx + z Dx\ i3- 14. dx r?>/ t/.c ; = — 4 (1 — z) 3 - ANSWERS. 333 J da; 16. ^ = - TO (i - xT~\ dx 17. -^ = 7a; 6 — 20a; 3 + 6a;. dx 18. g = (*-i) {x+2) (x-5) + (*-i) (.,+ 2) (*+ 3 ) + (*-i) (*-S) (*+3) + (3+2) (a-5) (*+3). 19. Given. rfy \/« (y — %) 20. «# 2 V# Vxy 3 dx _ 1 "' ^ ~" ~ 2 (1 +~^i* 22. 4 = £— . f&e (1 — a; 2 ) -3 efy 2a: 23 ' ^' "~ 3 (« + « 2 ) f ' 24. du = 2xyhlx 4- yxh^dy + $x 2 y 2 dx 4- 2X 3 ydy. du dx 2 5- 3= = * 3 - » 2. Given. 3. a; — a; 2 4- a; 3 — as* + etc. 4. 1 + x — a; 3 + etc. x 5. a? 1 - —= - etc. 2a;* 2 • 4a; 3 2 • 8a; 2 . a a — 1 . , . 6. — H (- (a — 1) (1 + x + aH 4- etc.) x l x v 1 a; xx % 7.-7 i + — - 7 - etc a 3 2«s 2 • 4«s 8. l—- x + ^x*— 1 a; 3 4- etc. a a^ a 3 a 4 9. — 1 — 2a; — 2a -2 — a; 3 4- 2a; 4 4- 7.T 5 4- etc. 334 ANSWERS IO. 1 1. 13- 14. IS- 16. 17- 18. 19. 20. 1 + x + x~ + .r 5 + jc* + et^. 1 — 3* + 3^ — 33* + 3- i;5 — etc. « ~~ a 2 + a 3 "~ ^ + etc * X X 2 x z , a a 1 a 3 i I l * i i - — etc. Sizs 3* 1 1 1 + z* + x + »* — xi + a 4 — etc. — a* + 32* — 5.r + 73$ — 92$ + etc. 1 + x* + a$ 4- a* + x 4- ajf + etc. #3 -j- #* + s£* + cc» + etc. 1, 2. Given. 3 3 4 + Page 183, Art. 430. 3i 1 r 4 (* — 2) 35 (•'• + 5) 10* J_ 5 1 3 (X + 1)2 + 30JT+1) ~~ X £ . X + 2 2 (X — i) 2 (z 2 + 3 + 4)' + 672 — I05.T 48.T — TOO 49./- 49 (a- 2 + 2x + 7) 2 49 (/ 2 + 2X + 7) "3 J 25 40 [x — 5) s x 4o (» + 3) 4- 1. + a — a* a 4 x — 1 ' (x — i) 2 x 1 1 4 (3 — 1) 4(* + l) 2 (a 2 4- 1) 122 f + 9 (X — 2) 2 27 (.? — 2) 27 (.1- +l) 9 fa + l) ANSWERS, 335 II. (a + b) {x — b) (a + b) (x +- a ) + 7 r. + 23 9 (a + 2) x — 2 4 (a; — 2) 2 12 (a; + 2) 2 5 3 + 18 (a; — 1) 2 (a- 4- 1) Page ISO, Art, 433. a 5 -f 5« 4 & 4- io« 3 6 2 4- iort 2 & 3 4- 5«J 4 + b 5 . 2. «■ 1 2 a 3 i 2 . 8ai J" i6«' 3 -. — etc. 1 3 5 6b 2 to6 3 156 4 i_5 & „2 "T" „3 rt 2/ £ 3 r 4 + etc. a 1 1 + 8a; 3 5?/ 3 i6a; ; + etc. ^+ 2 I + f 2 +f 3 +etc. + x X* ' 4n 6 % zn m 3 4 : — 3?« 3 gnfl iSivi* I 271 5W 2 4 0» ? s « - i -1 5 H 1 + ma 3m 3 9/n 3 1 2)i 5 M 2 w* 3m 3 9m 3 I 4 4 3 2 fo. z a - -2j - — 4 - — * 3a;s 93;? 81a; 3 a 3 + 6a- 2 4- 12a; + 8. 7W 4 n> + etc. 243m 3 " ti + etc. tirm 3 4cm 3 -^— - + etc. 81m" 3 " — etc. 11. 12. i3- 14. IS- 11 1 1 q ** + j 4- — - -2, + etc. a 3 2a; 3 2a: 2 8a;* « 5 a; 5 4- 5 « 4 ia 4 ?/ + 1 oa z b\r % if + 1 oa 2 b 3 x 2 y s 4- 5 a£ 4 a;?/ 4 4- J 5 // 5 . 11 J?/ S 2 ?/ 2 Z» 3 ?/ 3 t r 5 3 1 5 E etc. 2« 3 a;^ 8a 3 a; 3 " i6f/ 3 a- 3 arx* 4 11 S 8 "I „ 5 6 , 7 7 ~t~ elt " ^a; 3 " 8 8 ' _ 5 6 - 1 7 2^-a; 3 8a*a;* ioflW 5 3:36 A N S W EKS. 16. 25a 2 — loax + 49.T- 1 1 2X 17. 3 « + — T 3-« J 2.T 2 4^ -7-8 + ~ 7 - et °- 18. 19. 5 . . . . (2?? — s) , , 7 * 2«— 1 . 1 . 2 • 3 ■(?£ — i) ■J -A- S 2n ~ 3) fl i-na*-i. >«— 1 • I • 2 • 3 -(M — i) Plriflre 10?, 4r«. 11 fl. = 5i 6. m' = 1, ^5? = TTS' 7- W' = 2, a,j = 2, « lS = 4, a 2l = 16, ff2l = 32- S. "10 = — 19683, # I0 = — 14762; a s = 81 S s = 61. 9. The scale is 22 s , — 3a; 2 , 3a;. Series continued is 86a; 7 , 171.T 8 , 341a; 9 , 672a; 10 , etc. 10. The scale is —a; 2 , 2a;. « I2 = 35a;". Page 227, Art. 520. 1. The fifth term is ^. „ 1 i t 1 2 - 3> To? T*> TF- 3- 5. 6|, 8|, i 2 i 25. 4. See Key. Page 227, Art. 521. 1. a = p (1 + r)*. rs ( 1 + r)* 2. p = 3- * = (1 + r)< - 1 (1 + r)' ANSWE RS. Page 228. 339 4- 5- 6. 7- 8. 9- io. 12. 13- 14. 15- 16. 17- w = 7- (1 + r)< a "r(i + ?•)' g[(i + r) y — 1] w - r (1 + r )^' $595.58. $871.73. £783.53- _ a[ (i + r) m — 1] w ~" r(i+r)u a [(1 + r) 7 — 1] W ~ r(i+r)« $1245.43. $1666.66. $2230.38. April 27th, 1875. Answers in order in miles 160, 55, 55, 10, — n, — 270, — 4. 40.951 miles; 61.25+ miles; 2.73+ days; 1 00 miles. 18. 00 , 300 ft. 5 "2T> 1 T2- Page 229. Page 230, Ari. 522. 7- ^. 8. 1. 9- tV 10. 11. 7 ?0- -P« a: 2 — «Jc x + abed = 0. — b + ac — aid — c + ad — acd -d + £c + hd 4- ctf — hed 3. x 3 — 8 = 0. 4. x i + 6.r 3 + 7a 2 - - 24a; — 44 = 0. 5. .'E 6 — Sx 5 + 31a 4 — 402 s + 6x z + 288 = 0. Page 251, Art. 553. 1. 4 positive, 1 negative, 2 imaginary. 2. 2 positive, 2 negative, 2 imaginary. 3. o positive, 1 negative, 4 imaginary. 4. 1 positive, o negative, 4 imaginary. raye 262, Art. 567. 1-6. Not desirable to give. 7. z 4 — 6z 3 + 28Z 2 + 2 — 16 = o. z - 2X' % . ANSWERS. 341 8. z % — 6z 3 + 28z 2 — 482 — 32=0. z = 2xK 9. z n — 642 s — 3842* — 5122 s — 4096 = o. z = 22T&. 10. 2 5 + 22T 4 + 243^ — I3122 =0. 2 = 9^8. 11. Z 2 -\- Z — 6=0. Z = 2X. 12. X -J" I = O. 13. All the roots are included in the equal roots, and the result of removing them is 1 = 1, an equation of the zero degree and having no roots. 14. The same as the last. The equations are as follows : 1. x 3 — 2X % — 5a; + 6 = o. 2. x 4 — 7a: 3 4- 10a; 2 4- 142' — 24 = o. 3. x 3 4- 5a; 2 4- 2a; + 10 = o. 4. 2X* — a; 3 — i6x 4-8 = 0. 5. x* — 8a; 3 4- 27a; 2 — 46a; 4- 44 = o. 6. a; 6 — 12a: 5 4- 58a; 4 — 144a; 3 4- 193a' 2 — 132a; 4- 36 = o. 1 -r4 _l_ o ? ^3 _1- I 1 7 .2 4 9, r 1 35 — ,-, 7. x •+- 2-g-a 4- -35 -x — Ttt x -r T4 4" — °- 8. 6x 5 — 41^ + 97a- 3 — 97a; 2 4- 41a - — 6 = 0. 9. a 4 — 4 = o. 10. x 5 4- ioa^ 4- 40a; 3 4- 80a; 2 4- 80a; 4-32 =0. Page 266, Art. 575. 2. x = — 3.84-. 3. x = 6.5+. 4. x = 0.3 4-. 5. x=— i.o+, 1.3 + , and 4.6+. 6. a? = — 1.3 + . 7. a; = ±1.4+ and ± J-7+- 8. x = 1.4 + , 5.24-, and - — 0.64-. 9. x = — 6.64-. 10. x = 0.4 4-, 0.74-, and —6.2+. 342 ANSWERS. rage 271, Arts. 578, 579. 2. X — I.3797+. 3. x = 125. 4. x = 0.601+ and — 1.6 -4-. 5. x = 8.8 + . 6. x = 1. 259921 +. 7. x = 0.90 + . 8. x = 1.3 + , 1.6 + , and — 3.04 + . 9. # = 2.6 + . 10. a; = — 2.5+. Page 277, Art. 589. 1. Given. 2. sb = i, or \ (1 ± V— 3)- 3. a; = ± (3 ± V- 7 ± V - 62 + 6 V- 7). 4. x = ± 1, or i (— 7 ± 3 Vs)- 5. a; = 1, or ^ (— 1 ± V'71 ± v — 72 + V'71). 6. x = ± 1, or i (3 ± V^5 ± ^8 + V^f). 7. — 2.805 +, and —3-77+. 43. x = — a ± V« 2 + £ 2 ; V 2 «2 + 2^ + V« 2 + ^ 2/ = * ± 2 44. See Key. 45. $600000. 46. In 1845, Population 4354 ; " 1854, " 5602; " 1862, " 6943; " 1880, " 1 1478. I 47. - y^—r, \V ± W\ (6 2 + d'2)]; I being the length and b and £' the width of the ends. 48. The first is 2x 2 + 4X ; the second is 2X Z + $x 2 -f o. 49. In 41.06+ years. 50. The chance is |^. > RETURN CIRCULATION DEPARTMENT TOi^ 202 Main Library LOAN PERIOD 1 HOME USE 2 3 4 5 6 ALL BOOKS MAY BE RECALLED AFTER 7 DAYS 1-month loans may be renewed by calling 842-3405 1-year loans may be recharged try bringing the books to the Circulation Desk Renewals and recharges may be mad9 4 days prior to due date DUE AS STAMPED BELOW rECC |rMAR 10 IS 85 UNIVERSITY OF CALIFORNIA, BERKELEY FORM NO. DD6, 60m, 1/83 BERKELEY, CA 94720 ©s KNEBAi UC - BERKELEY UNIVERSITY OF CALIFORNIA LIBRARY