M! .^.^.' I'V !^iv;.'i!;J.£;s Vi-^'.iM 8'? 'I' S^ Lwa #1 CORNELL UNIVERSITY LIBRARY MATHEMATICS ,,,f OJlJEj-L UNWERSITV LIBRARY 3 1924 063 723 377 The original of tiiis book is in tine Cornell University Library. There are no known copyright restrictions in the United States on the use of the text. http://www.archive.org/details/cu31924063723377 TREATISE A L G E B E A Peofs. OLIVER,. WAIT and JONES CORNELL UNIVERSITY. SECOND EDITION. ITHACA, N.Y.: DUDLEY F. FINCH. 1887. Entered according to Act of Congress, in the year 1S82, by JAMES E. OLIVEB, LUCIEN A. WAIT AND GEOEGE W. JONES, in the office of the Librarian of Congress, at Washington. J. S. CcsHiNQ & Co., Peintbbs, Boston. PEEFAOE. In writing this treatise on algebra, the authors have had two rules for their guidance.' As to matter : "Assume no previous knowledge of algebra, but lay down the primary definitions and axioms, and, building on these, develop the elementary principles in logical order; add such simple illustrations as shall make familiar these prin- ciples and their uses." As to form : " Make clear and precise definition of every word and symbol used in a technical sense ; make formal statement of every generarprinciple, and, if not an axiom, prove it rigor- ously ; make formal statement of every general problem, and give a rule for its solution, with reasons, examples, and checks ; add such notes as shall indicate motives, point out best arrange- ments, make clear special cases, and suggest extensions and new uses." In working out the plan here outlined, wide departures have been made from the standard text-books. Many new things have been introduced, not, indeed, because they were new, but necessarily, 'either as definitions in giving larger meanings to old words, or as axioms and theorems in stating and proving the elementarj' principles, or as problems and notes in showing new uses of principles already proved : e.g., many fundamental prin- ciples were found to be omitted by elementary writers because too difficult for a beginner, and by subsequent writers as already known. A typical case is that of logarithms : that " the product of two powers of any same base is a power of that base whose exponent is the sum of the exponents of the factors" is gener- ally proved for commensurable powers, but assumed, without proof, for incommensurable powers ; and the whole theory of logarithms, so important,, and their use, so common, are thus left to rest on faith. In a few cases new words and new symbols iv PEEFACE. have been introduced : notably the signs "'",",='=, and the copulas >,<,=:. It is beheved that the need will justify the innovation. Moreover, the tendency of modern work is to change the tra- ditional boundaries of algebra so as to utilize graphic represen- tation, the elements of infinitesimal analysis, and the calculus of operations, and only thus can the subject be presented most naturally and philosophically. A good example is that of the so-called " imaginaries," which, rightly presented, are as real as any other numbers. The authors set out to write a text-book for the use of their own classes in the University, i.e., for young men who had already studied the elements of algebra and geometry, and who had had some scholarly training ; and, though an elementary book, at no time have they thought of it as a book for beginners. The wants of their classes have ever been before them ; but the work has grown upon their hands until it embraces many topics that, from their nature or- their treatment, are quite beyond the range of ordinary college instruction. As a text-book, therefore, for use in ordinary classes it must be abridged ; yet its wide range makes it all the more valuable to teachers, as a book of refer- ence, and to those bright scholars who wish either to place their knowledge of algebra on a sure foundation, or to make that knowledge a stepping-stone to the higher analysis. Many thanks are due to Mr. James McMahon and Mr. A. S. Hathaway, instructors in mathematics in the University, for their very valuable assistance in the preparation of the^ text, and to Mr. Albert Jonas and Mr. E. C. Murphy, for useful suggestions, and for help in verification of the text and in proof-reading. Another edition will contain chapters on : theory of equations, integer analysis, symbolic methods, determinants and groups, probabihties, and insurance ; with a full alphabetical index. COK"TEE"TS. I.— PRIMARY DEFINITIONS AND SIGNS. SECTION PAGE 1. Number 2 2. Eepresentation of Numbers 2 3. Positive and Negative Numbers 3 4. Special Signs 6 5. Copulas and Statements 7 6. Addition 9 7. Subtraction 10 8. Multiplication 11 9. Division 13 10. Involution , . . 14 11. , Evolution 15 12. Expressions 16 18. Functions 19 14. Goefficients. — Like and Unlike Terms 21 15.. Degree 21 16. Examples 23 II.— PRIMARY OPERATIONS. 1. Logical Terms 28 2. Combinatory Properties of Operations 27 3. Axioms 32 • 4. Addition Commutative and Associative 35 5. Sign of Product 36 6. Multiplication Commutative and AssQciative ' 37 7. Multiplication Distributive as to Addition 45 8. Proportion 48 9. Process of Addition '. 52 10. Process of Subtraction . . ; 52 11. Process of Multiplication 53 12. Process of Division 60 13. Operations on Fractions 65 14. Examples , 67 VI CONTENTS. ni. — MEASURES, MULTIPLES, AND FACTORS. SECTION PAGB 1. Definitions 79 2. Asioma 81 3. Measures and Multiples 81 4. Prime and Composite Numbers. — Factors 83 5. Process of Finding- the Highest Common Mfeasure ... 90 6. Process of Finding the Lowest Common Multiple .... 95 7. Process of Factoring 96 8. Examples 102 rv. — PERMUTATIONS AND COMBHSrATIONS. 1. Definitions 106 2. Permutations 106 3. Combinations , 112 4. Examples 118 v. — POWERS AND ROOTS OF POLYNOMIALS. 1. Product of Binomial Factors 121 2. The Binomial Theorem 122 3. The Polynomial Theorem 125 4. Roots of Polynomials 127 5. Absolute and Relative Error 129 6. Roots of Numerals 132 7. Examples 134 VL — CONTINUED FRACTIONS. 1. Form of Continued Fractions. — Convergents 137 2. Conversion of Common Fractions 138 3. Conversion of Surds 141 / 4. Computation of Convergents 144 5. General Properties I47 6. Secondary Convergents I53 7. Examples .' ' igg Vn. — INCOMMENSURABLES, LIMITS, INFINITESIMALS, AND DERIVATIVES. 1. Variables and Constants. — Continuity I57 2. Incommensurables '_ 150 CONTENTS. VU SECTION PAGE 3. Limits 161 4. Infinitesimals and Infinites 163 5. Derivatives 165 6. First Principles 167 7. Primary Operations on Incommensurables 168 8. General Properties of Limits .' 169 9. General Properties of Derivatives 173 . 10. Indeterminate Forms . . ' 178 11. Graphic Tiepresentation of Functions 181 12. Integration 184 13. Examples 189 VIII. — POWEES AND BOOTS. 1. Fractional Powers 194 2. Combinations of Commensurable Powers 195 3. Continuity of Commensurable Powers 199 4. Incommensurable Powers 205 5. Combinations of Powers in General 207 6. Continuity of Powers in General 209 7. Derivatives of Powers 213 8. Kadicals 215 9. Operatioijs on Radicals 223 10. Examples 227 IX. — LOGARITHMS. 1. General Properties 233 2. Special Properties, Base 10 236 3. Computation of Logarithms 237 4. Tables of Logarithms 239 5. Operations with Common Logarithms 240 6. Examples 250 X. — IMAGHSTAEIES. 1. Definitions and Graphic Representation 251 2. Addition and Subtraction 259 3. Multiplication and Division 264 4. Powers and Roots 268 5. Abridged Representation 278 6. Examples 279 Viii CONTENTS. XL — EQUATIONS. SEOTIOH ^^°^ 1. Statements 281 2. Solution of Equations. — Unknowns 282 3. Degree of Equation 283 4. General Properties 284 5. Simple Equations involving One Unknown 291 6. Elimination 293 7. Simple Equations, Two or More Unknowns 298 8. Graphic Representation of Simple Equations involving Two Unknowns 307 9. Bezout's Method, Unknown Multipliers 308 10. Special Problems of the First Degree 310 11. Quadratic Equations involving One Unknown 818 12. Graphic Representation of Quadratic Equations .... 318 13. Solution of Quadratic Equations by Aid of Continued Fractions 323 14. Maxima and Minima 324 15. Simultaneous Equations 330 16. Special Problems involving Quadratics 340 17. Binomial Equations 341 18. Logarithmic and Exponential Equations 342 19. Examples 343 XII. — SERIES. 1. Arithmetic Progression 361 2. Geometric Progression 364 3. Harmonic Progression 367 4. Convergence and Divergence 369 5. Indeterminate Series 375 6. Imaginary Series 378 7. Expansion of Functions in Infinite Series 381 8. Method of Unknown Coefficients 383 9. Binomial Theorem 390 10. Finite Differences 393 11. ^Interpolation 396 12. Taylor's Theorem 400 13. Computation of Logarithms 403 14. Examples 405 ALG-EBRA. I. PRIMARY DEFINITIONS AND SIGNS. Algebea is that branch of Mathematics which treats of the relations of numbers. It is distinguished from Arithmetic, as haying wider generalizations, as using signs and letters more freely, and as recognizing negatives and imaginaries. The ap- plications of many words common in Arithmetic are greatly extended in Algebra, and their definitions are correspondingly enlarged. The sj'mbols explained below constitute a symbolic language, a species of short-hand writing, wherein numbers and their relations are more conveniently expressed than in the ordinary language of words. In this language the signs stand for words and phrases, and generally have the same grammatical relations as the words and phrases themselves. The words may be restored at any time. The reader should constantly practice translating from one form to the other till both are familiar. This symbolic language is one of the characteristic features of Algebra ; and among its many advantages are these : clear- ness, brevitj', and generality of statement ; the ability to mass directly under the eye, and thus to bring before the mind as a whole, all the steps in a long and intricate investigation ; and the facility of tracing a number through all the changes it may undergo. Spme other sciences, for example Chemistry -and Logic, have a symbolic language of their own. 2 PKIMAEY DEFINITIONS AlfD SIGNS. [I- § 1. KUMBER. In measuring anything, some unit of the same kind is first assumed, and the relation the thing measured bears to this unit, i.e., the operation that if performed upon the unit will produce the given thing, is expressed by a number. The unit, being acted upon, is the operand, the number is the operator, and the thing produced is the result. Such numbers are also called abstract numbers, because all their properties and relations are independent of the particular units used ; and the units and the measured things are concrete numbers. Abstract and concrete numbers are also called quan- tities. Abstract numbers likewise arise fi"om the combination of other abstract numbers : and in this way, their relations form the chief subject-matter of Algebra. Two abstract numbers are equal if, operating upon the same unit in the same way, they produce the same result. An abstract number is an integer if the thing measured be made up of entire units ; a simple fraxition, if the thing be one or more of the equal parts that the unit may be divided into. Integers, simple fractions, and such other numbers as can be reduced to integers or simple fractions, are commensurable num- bers : those which cannot be so reduced are either incommensu- rables or imaginaries. § 2. REPRESENTATION OF NUMBERS. NuMBEES are represented by Arabic numerals, or by letters. Among the more common forms are these : 0, 1, 2, 3, . . . , 10, read : naught, one, two, three, ...,ten; a, 13, y, 8, e, 6, tt, . A, 2, read : alpha, beta, gamma, delta, epsilon, theta, pi, phi, large delta, large sigma; -, -, |, read : four sevenths, x over y, half pi. a', b", c", d<"", read : a prime, b second, c fourth, d m'"; Pat Pii Pii read : p sub zero, p sub one, p sub x. i 3.] POSITIVE AKD NEGATIVE NUMBEES. 3 The accents, numerals, and letters, attached to other numerals and letters, are indices. An index attached below its letter is a suffix or subscript, and is read szib. The index of a power [§ 10] is an exponent. The letter or numeral to which the index is attached is the stem. Sometimes the indices are written without the stem ; this form of writing is called the umbral notation. E.g., instead of ai_ a ; ag, 4 ; . . . «;, j, write 1,2; 3, 4 ;.../, 7l. The accent and subscript notation has two chief advantages : It gives a very great number of distinct symbols. It permits numbers analogously relnted to the problem in hand to be represented by the same letters. E.g., p', p",. p'", or pi, 2hi JP31 maj- stand for the princi- pals of three promissory- notes ; then t', t", t'", or t^, t^, t^, will natnrall}^ stand for the three times for which these three notes are respec- tivelj' given, and r', ?■", r'", or r^, rg, rg, for the three rates. The value of a letter or other sj-mbol is the number for which it stands. Ordinarily the same letter stands for but one number during any one investigation, but for different numbers in differ- ent investigations ; and different letters, or the same letter with different indices, for different numbers in the same investigation. § 3. POSITIVE AND NEGATIVE NUMBERS. When the measuring unit is taken in the same sense as the quantity measured, the number is positive; when in the opposite sense, the number is negative. In which sense the unit shall be taken, is a matter of custom or convenience. Manifestly, if two quantities, opposite in sense, are measured by the same unit, one number is positive and the other negative : E.g., if distances to the north or east from a given point are positive, distances to the south or west are negative: i.e., if the measuring unit is a northerly or easterly unit, then southerly or westerly distances are expressed by negative numbers, and vice versa. 4 PKIMAEY DEFINITIOXS AXD SIGNS. [I. So, if the revolutions of a wheel forward are positive, revo- lutions backward are negative ; if assets are positive, liabilities are negative ; if dates a.d. are positive, dates B.C. are negative ; if the readings of a thermometer above zero are positive, the readings below zero are negative. The primary notion of a negative number is that of one which, when taken with a positive number of the same kind, goes to diminish it, to cancel it altogether, or to reverse it. E.g., liabilities neutralize (negative) so much of assets, thereb}' diminishing the net assets or leaving a net liabilitj-. If two numbers of the same kind, when taken together, exactl}' cancel each other, they are opposites, one of the other. Manifestl}-, of two opposites, one is positive and the other is negative. So, if numbers are used as indices of two algebraic operations which when performed successively tend to neutralize each other, a positive number is commonly used for one index and a nega- ti^-e number for the other ; and sometimes, as with exponents of powers [§ 10], custom has permanently determined which index shall be positive and which negative. When denoted by Arabic numerals, positive numbers are writ- ten with the sign -f- or with no sign, and negative numbers with the sign — , and it is evident at sight whether the numbei* is positive or negative. E.g., if the measuring unit be -SI of assets, then +100, or simply 100 without the sign, expresses the net value of an estate whose assets exceed its liabilities by $ 100 ; and —100, that of an estate whose liabilities exceed its assets by $ 100. But, if a number be denoted by a letter, it is not evident upon its face, and often it is not necessary to know, whether that let- ter denotes a positive or a negative number. E.g.. in the above example, n may stand either for + 100 or for - 100, at the pleasure of the writer. If, however, n stands for+100, then -n stands for -100 ; and if n stands for -100, then - N stands for + 100. In either case + n and - n are opposites. § 3.] POSITIVE AND NEGATIVE NUMBEES. 5 In this use of the signs + and — , they are called signs of quality, since thej' indicate the qualitj-, in an important particu- lar, of the quantities measured, and of the numbers before which they stand. These signs are also used to indicate the operations of addition and subtraction [§§ 6, 7], and are then called signs of operation ; but, as the reader -will see wlien he comes to the studj- of these operations, the two uses are alwaj-s in accord, and the signs may often be understood in either waj' at pleasure. Sometimes signs performing both oflSces occur before the same number [§§ 6,7]. The sign + before a number denotes either the number itself, or its opposite, whichever of them is positive ; the sign " denotes whichever of them is negative ; i.e., a number preceded by "*■ is essentially positive, and a number preceded by~ is essentiallj^ negative. JE.g., if N stands either for 100 or for —100, +n, read n taken positive, stands for +100 ; and ~n, read n taken negative, stands for - 100. So, "^100 may alwaj'S be written for +100, and ~100 for —100 ; but not ■'■N for n or + sr, nor "n for — n, unless the value of n be positive. Manifestly, +N and ~k are opposites ; and so are +100 and "100. Note. — The reader should observe that some things admit of negatives and some do not. JE.g., time may be counted backwards as well as forwards from any given date ; so may distance from any given point ; so may he^t and cold from an arbitrary zero ; so may money of account, as above ; but with real dollars, say five of them, he will find, when he tries to count past none, — five, four, three, two, one, none, — that he is attempting to do what is impossible. So, when he comes to the study of the so-called imaginaries, he will find that for some things they have a real existence, but for other things they have not. So, for some things, fractions have no existence. E.g., I of a man, or | of an atom, or 1^ events or facts, would be unmeaning. 6 PKIMAEY DEFINITIONS AND SIGNS. [I- § 4. SPECIAL SIGNS. The sign of continuation is ... , read and so on. E.g., 1, -2, +3, ..., +9 means 1, -2, +3, -4, +6, -6, +7, -8, +9. The signs of inference are •.-, read since or because, and .•., read therefore. E.g., -.■ 80 cts.<$l, .-. 400 ets.<$5; or 400cts.<$5, •.• 80cts.<$l. The signs of grouping axe {),[], \ ], ~', | . They show that all within the brackets, under the liorizontal bar, or before the vertical bar, is taken together as one number, and subject to the same operation; viz., that which is indicated by the sign preceding or following it, or by the index attached to it. When two or more numbers joined bj- the signs + and — [§§ 6, 7] are grouped together by a bar or brackets, they form an aggregate. E.g., (l+2+3)Xo— 2 is the product of two aggregates [§ 8]. Wlien two statements are identical, except only for a few characteristic words or signs, then, as a matter of convenience, the two statements may be written together as a double state- ment, bj' placing the pairs of corresponding words or signs one above the other. E.g., ■.• the battle of Salamis was fought 480 B.C. and that of Waterloo 1815 a.d.,, - fvSo - ^-^1^* 2295 yrs. ^ ^Slltf °- So, it^ ^^ 30° ^ ^^l'^'^'" yf te'-day . to-day. ' ' IS 1 warmer to-day """'M yesterday. In such double statements, all the words and signs in the apper line, together with the common parts, go to make up the first statement ; and all the words and signs in the lower line, together with the common parts, the second statement. In the same way, three or more statements may be written together. I When, of a double statement, only one part can be true, but which that is, is unknown, such statement is ambiguous. § 5.] COPTJLAS AND STATEMENTS. I § 5. COPULAS AND STATEMENTS. Two numbers are equal when, in every combination which contains either of them, the other maj- talce its place without changing the result. When one number is equal to another, the two are joined by the sign = , read equals, or is equal to, and the whole is an equation; or by the sign s, read is identical loitli, and the whole is an identity. E.g., 100 cents = 1 dollar ; 100 cents = 100 cents ; k = a;. An identity' is an equation wherein the two numbers I'emain equal, however the values of smy of the letters may change. Ever3' identity is an equation, but not everj- equation is an identitj'. Hence = maj' always take the place of s , but s not always of = . The sign = is also used for " stands for" and "represents." E.g., p s principal, t s time, r s rate, i = interest. When one number is not equal to another, they are joined by the signs ^t,,^, <, >, <, >•, read : not equal to, not identical with, less than, greater, thani, smaller than, larger than. E.g., 8<)cts.=7t$l,- 100cts.^$l, 80cts.<$l, 120cts.>$l, 80cts.<$l, 120 cts. >$1. So, <, >, iS, 9^) mean not less than, not greater than, etc. The words "greater" and " less" are here used in a technical sense, and may be expressed by higher and lower in speaking of temperatures and elevations, by north of or east of and south of or west of in Surveying and Geography, by later and earlier in comparing two dates, and so on ; but " larger" and " smaller" take account of the size of the two numbers only. E.g., 30 ft. up > 50 ft. down, and 30 ft. down > 50 ft. down ; i.e., +30>-50, and -30 > -50. But +30 < -50, and "30 < -50. If two numbers be equally large, the sign is :=: ; if not equally large, ^. k(7., +1600 =-1600; +1600 5fe-1700. 8 PErMAEY DEFINITIONS AND SIGNS. U- In general, anj- positive number, however small, is greater than any negative number, however large ; and, of two negative numbers, the smaller is greater than the larger. Manifestly, the greater a number the less is its opposite ; but a number and its opposite are equally large. The signs =, =, =, =#=, #, ^, <, >> <> >> <> ^' ^> ^ are signs of assertion or copulas. Equations, identities, and inequalities are statements, and when of general truths, they are formulae. The first member is all that precedes the copula, and the second member, all that follows it. A continued statement is one having more than two members ; it is equivalent to as many simple statements as there are copu- las, and each copula, unless preceded by a comma, connects the two members immediately adjacent to it. E.g., l<3<5<-7 9<-9 is equivalent to the group of independent statements 1<3, 3<5, 5<-7, -79^9. So,---a<6 .•.2a<26<3& is equivalent to the chain of connected statements •.• a<& .-. 2a<2&, and 2 6<3&, and .-. 2a<3&. But-.-a<& .-. a, <2a, <26 is equivalent to the chain of connected statements •.•a<6 .-. a (which <2a) <2 6, and is read : Since a is smaller than 6, therefore a, which is smaller than 2a, is smaller than 2b. This is, in effect, a brief form for a logical chain of statements. The office of the commas is to parenthesize what is between tliem, and compare directlj' what precedes the first comma and what follows the last comma ; the basis of comparison being found in what the commas enclose. The first comma is read which. So, a^ — a, =b, is equivalent to the two independent statements a^ — a and a = b ; and here, too, the office of the comma is to carry forward the first member, a, and compare it with b which follows the comma. The comma is read and. § 6-] ADDITION. 9 § 6. ADDITIOIT. The sum of two or more concrete numbers of the same kind is a new concrete number got by joining together the several things measured, and then measuring the, aggregate by the same unit that measured the original numbers. The sum of two or more abstract numbers is a new abstract number which, if used as an operator upon any unit, will give the same result as if the original numbers were first used as operators upon the unit and their results were then added. Addition is the process of finding the sum of two or more numbers. If the numbers added be commensurable, then, at bottom, addition is but counting either by entire units or by the aliquot parts of a unit : on (forward) if positive numbers be added ; off (backward) if negative numbers be added. The sign of addition is + ; re&d plus, or the sum of ... and ... E.g., 50cts. + 60cts. + 90cts. =$2; 50 + 60 + 90 = 2'00. In Algebra the word " addition '' is used in a broader sense than in Arithmetic, and covei's negative as well as positive numbers. U.g., he who has $10,000 cash and $4,000 debts is worth but $6,000; i.e., $10,000 cash + $4,000 debts = $6,000 net assets; +10,000+-4,000 =+6,000. So, a train which has run east 10 miles, then west 20 miles over the same track, is lO'miles west of the start- ing-point ; i.e., 10 east-miles + 20 west-miles = 10 west-miles ; +10 +-20 =-10. But a train which has run west 10 miles, then west 20 miles more, is 30 miles west of the starting-point; i.e., 10 west-miles + 20 west-miles = 30 west-miles ; -10 +-20 =-30. Though the numbers to be added must always be of the same kind, they are often expressed by letters whose values are not known, or in units whose values are diflTerent, and which there- fore cannot be reduced to one sum. E.g., 5''33'"30' + 12'^47-°30» = 18''21'°. 10 PRIMAEY DEFESTITIOKS AND SIGNS. [I. Manifestly, the sum of two opposites is 0. E.g., 90 ft. up is the opposite of 90 ft. down ; i.e., + 90 is the opposite of — 90 ; and the sum of the two is 0. So, +a and —a, ~a and +a, 26 — 3c and 3c— 2b. §7. SUBTEACTION. Stjbtkaction is the inverse of addition, and consists in find- ing what number must be added to one number, the subtra- hend, to give another number, the minuend. The result is the remainder, and the sign is — , i-ead minus or the excess of ... over ... One or both of the numbers may be negative, and the minuend may be less than the subtrahend. E.g., $50— $40= $10, $40- $50 ="$10. -$50 --$40 =-$10, -$40 --$50 =+$10. ■ So, if of two men A has $ 10,000 cash and no debts, and B has $5,000 debts but no assets, then A is $ 15,000 better off than B, i.e., +10,000 --5,000 =+15,000; and B is $ 15,000 worse off than A, i.e., -5,000 -+10,000 =-15,000. So, ■.• the battle of Salamis was fought 480 B.C., and that of Waterloo 1815 a.d., .■. "Waterloo was fought 2295 ;years after Salamis, i.e., +1815 --480 =+2295; and Salamis was fought 2295 years before "Waterlop, i.e., -480 -+1815 =-2295. So, if to-day a thermometer read 10° below zero, and yes- terday it read 20° below zero, then it is 10° warmer to-day than yesterday, i.e., -10 --20 =+10; and it was 10° colder yesterday than to-day, i.e., -20 --10 =-10. The difference between two numbers is the remainder found by subtracting the less from the greater ; the sign is ~. E.g., 16~12 = 12~16 = 4; -16~+12=+16~-12 = 28. MULTIPLICATIOK. 11 § 8. MULTIPLICATION. The product of a concrete number, the multiplicand, by an abstract number, the multiplier, is a concrete number of the same kind as the multiplicand, and bearing to the multiplicand the same relation as the multiplier bears to unity. . The product of two or more abstract numbers is a new abstract number such that, if a unit be multiplied by it, the product is the same as the final product obtained by multiplying the unit by the first of the numbers, the product so found by the second of them, and so on. Multiplication is the process of finding the product of two or more numbers ; the numbers are the factors of the product. Multiplication by a ^ P°^^ ^^ integer is but a repeated ' negative . addition ^^ ^^^ multiplicand { *° 0, and multiplication ' subtraction ' from by a i P°"^*^^^ fraction is the repeated V ^'^<^^*^°^ *° ' negative subtraction from of the equal parts into which the multiplicand's divided. In the last analysis, multiplication is but a counting, on or off, according as the multiplier is positive or negative ; but it is a counting by groups, each equal to the multiplicand, if the multiplier be an integer, and by aliquot parts of such groups if it be a fraction, instead of by single units as in addition. - E.g., five, ten, fifteen, twenty, twenty-five, thirtj', gives the product of five by six, or of "five by "sis. So, "five, "ten, "fifteen, gives the product of "five by three, or of five by "three. . So, one half of five, two halves of five, three halves of five, giyes the product of five by f , or of "five by "f. So, I of f , f of f , 4 of f , gives the product of -f by |, or of "f by "f . So, f , I, I gives the product of | by 3 or of "| by "3. 12 PEIMAEY DEFINITIONS AND SIGNS. [I- The signs of multiplication are X , read by, and • , read into. E.g., 50 cts. X 8 = $4; 8-50 cts. = $4. So, placing the factors one after the other, with no sign be- tween them, means multiplication of the first into the second, or of the second by the first. E.g., ah is the product of a into h, or of 6 bj' a, and ah^a-b=.h X a. "When the product of two numbers is multiplied bj' a third number, such multiplication is the continued muUijMoation of the three numbers ; so for four numbers, for five numbers, and so on ; and the product of such multiplication is the continued product of the several factors. E.g., 5x6x7 = 210, and 5 ■ 6 • 7 = 210. The continued product of the natural numbers 1 • 2 • 3 ... is indicated bj- the sign ! placed after the last factor, or bj' the sign |_ placed before and under the last factor. E.g. , 5 ! or 1 5 , read factorial 5, =l'2-3-4-5, = 120 ; nl or Im,' read factorial m, =l"2-3-...n. Some peculiar properties of negatives appear in multiplication. E.g., if a train, now at a, is running east 20 miles an hour, then five hours hence it will be 100 miles east of a, i.e., +20x+5=+100; but five hours ago it was 100 miles west of a, i.e., +20 X -5 =-100. So, if the train is backing, i.e. running west, then five hours hence it wiU be 100 miles west of a, i.e., -20 X +5 =^100; but five hours ago it was 100 miles east of a, i.e., -20 X -5 =+100. Two numbers whose product is 1 are reciprocals of each other. E.g., 4 is the reciprocal of J ; — 3 of — ^ ; ^ of f . Manifestly, the larger a number, the smaller its reciprocal. The product of a number by an integer is a multiple of that number: the double, triple, quadruple, ••-, when the multiplier is 2, 3, 4, .... § 9-] DIVISION. 13 §9. DIVISION. Division is the inverse of multiplication, and consists in find- ing either factor, when the product and the other factor are given. The product is now called the dividend, the given factor is the divisor, and the result is the quotient. E.g., -.■ the product of 6 by 10 is 50, .-.the quotient of 50 by^ lols^s' So, ••• the product of a into 6 is ab, . • . the quotient of a6 by -{ ? , ' The { ^iyjgQj. ' being the multiplier, is an abstract number [§ 8] ; and the ■{ Vp f ^^^ dividend are alike in kind. When both factors are abstract, the two definitions of division agree, as will appear later. E.g., -.■ the product of $5 by 4 is $20, .-.the quotient of $20 by^ l\s\f The signs of division are : , read the ratio of ... to ..., and -i-, read divided by, or the quotient of ... by E.g., $20:15 = 4; $20-^4 = $5; 20:5 = 4; 20 -=-4 = 5. So, writing the dividend over the divisor with a horizontal line between them means division. The dividend is then called the numerator, the divisor.the denominator, and the whole expression a fraction. Hence a fraction is the expression for the quotient in a division as yet unperformed. Note. This definition of a fraction differs from that hereto- fore given [§ 1], but later it wiU appear that the two definitions are in fuU accord. If the dividend be a multiple of the divisor, then the quotient is an integer and the division is complete ; but if the dividend be not a multiple of the divisor, its excess over the greatest multiple that is contained in it is the remainder. E.g., 27:5 = 5, quotient with 2 remainder. 14 PRIMARY DEPIKIXIONS AND SIGNS. U- §10. INVOLUTION. A J P°**'*^^ inteciral power of a number is the continued ) negative '' ^ \ product ^^ jjjj.^y ^^ ^jjg gj^g^ number. The number whose power is sought is the base. The symbol that shows how many times the base is used as ^ multiplier ^^ ^j^^ exponent) it is written at the right and above the Z; and is ^P^^- for a ^P°^ power. E.g., lxaxaxa = aS read tJiird power of a, a third power, or a cvhe. lxaxa=d', read secondpower of a, a second power, or a square. 1 X.a=a^, read first power of a, a. first power, or simply a. 1 = a", read zerotJi power of a, a zeToth power. l-T-a= a~^, read minus first power of a, or a minus first power . 1 -f- a -j- a = a"^, read minus second power of a, or a minus second power. l-j-a-=-o-=-a= a~', read minus third power of a, or a minus third power. A roof of a number is one of the equal 'factors into which it may be resolved. The number whose root is sought is the base ; the symbol that shows into how many equal factors the base is resolved is the root-index. The radical sign, -^, is writ- ten before the base, and the root-index is ql the left and above it ; or else the reciprocal of the root-index is attached to the base as an exponent. The root-index 2 need not be written. E.g. , .^4, or simply V* = 4* = ±2 ; -^243 = 243* = 7. A fractional power of a number is either a root of the number or some integral power of such root. The exponent is then a simple fraction whose denominator shows into how many equal § 11] EVOLUTION. 15 factors tlie base is resolved, and whose numerator shows how many times one of these factors is used as i ^^. ^^ '^^ ' ■' ' divisor. E.g., 64^ = 1 X 4 X 4 = 16, read 64, ^ power, equals 16. 64 ^= 1 ^4-=-4 = 3ig-, read 64, —^ power, equals ^^. So, a^ = Va, c' s {-{/cy, x^ = ( V^;)", ^-i = {i/k)-K The words " integral," " fractional," " positive," and " nega- tive " apply to the exponents only, and not at all to the results of the operations indicated ; i.e. , a pos^itive integral power is one whose exponent is a positive integer, and so on. Integral and fractional powers are commensurahle powers. Those powers whose exponents are incommensurable are called incommensurable powers ; they are defined in [VIII. § 4]. Involution is the process of finding the powers of numbers ; its sign is the position of the exponent. Note. — The reader may compare what is here said of posi- tive and negative exponents, as indices of repeated multiplica- tion and division of a unit by the base, or by one of the equal factors of the base, with what is said in § 3 of operations which tend to neutralize each other. He will then see the peculiar propriety of expressing repeated multiplication by a positive exponent, and repeated division, the inverse of multiplication, by a negative exponent. §11. EVOLUTION. ' EvoLtrrioN is the inverse of involution, and consists in finding a base that, when raised to the power denoted by the index, produces the given number. ' The result is the root. The logarithm of a number is the exponent of that power to which a base must be raised to give the number. The finding of logarithms is another inverse of involution. E.g., •.• \Qf.= 100, .•. 2 is the logarithm of 100 taken to the base 10 ; it is written logio 100 = 2, and read log, base 10, o/lOO equals 2. So, logi„1000 = 3, logiolO = l, logi„l=0, logio.l = -l. .16 PEIMABY DEFINITIONS AND SIGNS. [I. § 12. EXPRESSIONS. An Algebeaic Expression is a number or combination of numbers written in algebraic form. It is called an "expression" or a " number," according as the thought is of the symbol or of the A'alue which the symbol represents. Unless a single letter or numeral, an expression is made up of simpler expressions affected or combined by signs of operation ; and the order of these operations is as follows : 1. Every letter or numeral, with its indices, if any, denotes a number by itself; and so does everj' expression united by a bar or parenthesis. These numbers, in turn, may be affected by exponents, etc. ; but each exponent affects onl}' the single numeral, letter, or parenthesis it is written to ; and if a power of a power is to be denoted, the new base must be parenthesized. E.g., 2? 2,^ a?b\x-y){x + yY is the product of 2», 3^ a^ h\ (x-y), and {x + yY; but [(2''3)^a]' is the cube of the product of a b}' the square of 2^3. So, (a')" is the cth power of a'; but a''" is the b'th power of a. So, a' is the 6' th power of a. 2. When a product is denoted by writing the factors together without the sign x or ■ , or when a quotient is denoted by a fraction, the product or quotient is affected, as a single number, by the adjacent sign ^, log, x , •, ., s-, +,—,+, or -. E.g. , ^^ab-x'y^-.S* denotes that positive the square root of 2ab is multiplied into x'y', and the product di- vided by 3* ; but -\/2ab-x'y^:S* is any square root of 2 a& • x'y' : 3*. So, log^y- is the logarithm of f^/^ ; but log I ■ 2^ is the product of log f into y'. 3. In this book, when successive numbers are separated, some by the sign X , •, :, or -=-, and some by + or — , the multiplica- tions and divisions are first performed, and then the products and quotients are added or subtracted ; and if several of these sio-ns of § 12.] EXPEESSIONS. 17 multiplication and division occur in succession, or several signs of addition and subtraction, the left-hand operation is first per- formed. But the usage among algebraists is not uniform. E.g., 3:2-6 — 6-^3x2 + 1 denotes that from f • 6 is sub- tracted f X 2, and to the remainder, 5, is added 1. Those parts of an expression which are joined bj- the signs -)- or — are terms, and terms are ■{ . K which ■{ -, , , . ' ' simple ' do not contain the sign + or — except in an index. An expression of one term onlj- is a monomial, of two terms a binomial, of three terms a trinomial, of four terms a quadri- nomial; of two or more terms a polynomial. An expression is ^ 7 -^ i when the numbers are expressed . wholly by numerals ; , finUe ^^^^ ^^^ ^^^^^^ ^^ ' wholly or m part by letters ; ' infinite ,. . T J . j limited, operations implied is^ ,,^^^^1^^!. A finite expression is^J £*™;f^g„f„^ when there is implied { °° other operation than addition, subtraction, multiplication, division, and involution to commensurable powers. , , . ... rational -, ., , can -u j. j An algebraic expression is -I when it ■{ be ireea ' surd cannot .entire or intearal , i free from divisors and roots, from roots ; if^^^^i^^^i when ^ ^^^ ^^^^ ^^^^ divisors. E.g.,Sbc, A, a'V+iJB''"'"^ [»" being a positive integer] are entire simple rational monomials. So, 3_+2^^ Sy ^ a+1^ (a + x-^) are complex fractional a a+1 a— 1 monomials. a + x~^ is a fractional binomial with simple terms. (a ^ &) _[- ^^ is a binomial with complex terms. ^ m + n 2x 3 V 3bc + 5 XV— 7 mn and 1 ^ — na"-^x are rational ■^ a a+1 trinomials ; the first is entire, the second fractional. 18 PEIMAEY DEFLtriTIONS ANI> SIGNS. [I. 2 2 a. 3 So, y/a±l:fl—^a and 5pq^ — 3pq^ + pq —^pq are quadrinomials, but reducible to monomials ; viz. , to 2 and 2^pq^. The above examples are literal ; the following are numerical : ^5+-y/7, 2 +3 , 1+V— 1 are binomial surds. 1 + -^2 — -y/— 3 is a trinomial surd. 1.1 X 1.01 X 1.001 X ••• is an inflnitelj'' continued product. is an infinite continued fraction, but one Oil "•" whose value is -i/2 — 1 , an irrational 2-1-1 V ' ~ finite number, as will appear later. An expression may be entire, rational, etc., as to some of its letters only. E.g. , "' ' V " is rational as to a, m and n, m + n and it is entu'e as to a ; but it is irrational as to 6 and c, and it is fractional as to m and n. When the terms of an expression are so related to each other that each successive term is derivable by some fixed law from the previous terms, the expression is a series. E.g., l + x-\-a? + 3?-\ \-3f is a finite series if r is any given integer ; but \ + x + oi? + !i?-\---- + af-\ is an infinite series. In this series af is called the general term, because by giving to r in turn the values 0, 1, 2, 3, . . . , or any of them, all the terms of the series, or any of them, are found. When the values of the several letters in a literal expression fire known, then the value of the expression may be found by substituting these values in place of the letters, and performing the operations indicated. E.g., if a = 2, 6 = 3, c = 4, then a6c=24:, a-f-6 — c=l, a : (& -|- c) = f . So, if a; = a -f; & and y=a — b, then x+y = 2a, x — y = 2b, xy = a^ — b'. § 13-] FUNCTIONS. 19 A literal expression may be entire, fractional, rational, etc., but its numerical value not so ; or the reverse. U.g., a; is entire, a;-i fractional, V* irrational, j/^ratignal; but, if a; = -^ and 2/'= -^2, then the value of x is fractional, that of x'^ is- entire, that of -y/x is rational, and that of y^ is irrational. Manifestly, if all the letters stand for integers and the expres- sion is entire, its value is an integer. H.g. , if a and 6 are integers, (a — 6) (2 a^ + 3 6^) is integi-al. "^ As to any of its letters, an expression is symmetric when its value remains unchanged however those letters exchange places. E.g. , xyz and x + y + e we symmetric a.stox,y, and », or as to any two of them. So, w + x — y — ziB symmetric as to w and x, and as to y and z ; but not as to w and y, to w and z, to x and, y, nor to x and z. An expression is co-fiverted or transformed, when changed, in form but not in value ; developed or expanded, when transformed into a series. § 13. FUNCTIONS. If a number is so related to other numbers that its value depends upon ''their values, it is a function of those numbers : ^°^ impS ^'^°<'"°°' ^^^° ^ nTexprlsed ^"^ *^™« ""^ *^«^« numbers. The numbers are the arguments of the function. E.g. , in M = 3 xy, u is an explicit function of the arguments X and y ; but X is an implicit function of the arguments u and y, and y is an implicit function of the arguments u and x. So, in y^ = u:3x, y is an implicit function of u and x ; but in y =-y/{u: 3x), y is an explicit function. An explicit function of one or more numbers is known (given or determined) in terms of tiiose numbers. It is symmetric, algebraic, transcendental, rational, etc., according as the expres- sion which gives its value is symmetric, algebraic, etc. 20 PEIMAEY DEFLNITIONS AKD SIGNS. [I. If one number (function) depends upon its arguments ill the same way as another number depends upon its own arguments, i.e., if the expressions involved are of the same form, then the first number is the same function of its arguments, as the second number is of its arguments. E.g., if a^ + x=a and y' + y = b, then ^ " is the same^ f'^P}!*^!* function of <" '^ as ^ ^ is of ^ ^Z- ' X ' imphcit 'a ' y ' b. So, the expression x+2y is the same function of x and y as a + 2 & is of a and b, and the same as ?/ + 2 a; is of y and x. A function may be denoted by the letters/, f, <^, ... , with or without indices, and followed by the arguments, either enclosed in a parenthesis or not. rr^ A^ „4. ) the same „ j. j the same letter To denoted •>.«. , function S ^.a, ^.^ i.^. • i is ' a different ' a different letter or index used. E.g., if /(a;) = a?—ax, then f{y) =y^ — ay during the same investigation ; but f{y) cannot stand for a? — ay, nor for ay — y^. So, if F(a3, a) = a^ — aa;, then ■p{y,b)=y^ — by, ¥{a,x) = a^—xa, .... But if F'(a!, a)=.a?— a^x, or any other form, then f'(6, y) = b^ — y^b, the same form. If p(a!,2/) = F(2/,a!), then either is^ ^^^^^^^ric f-°«- ^'- --^ V- E.g., if F (a;, y) =f{x) .f(y) , or sf{xy) , then F denotes a symmetric function ; but not if F(a;, y) =f{x) -/'(y) , or =/(a; : y) . So, ^ {x, y) + {y, x) , but not ^ («, 2/) - <^ (j/, a;) , is sym- metric. So, if F(a!,2/,2), T{x,z,y), y{y,z,x), v{y,x,z), v{z,x,y), and f(z, y, x) be^ ^|J^ ^^ identical, then either is J a sjtd metric friction of a; v z ^ an unsymmetric ^""^''^"n o^ '*') 2/> »• §§ 14, 15.] ' COEFFICIENTS. — DEGEEE. 21 § 14. COEFFICIENTS.— LIKE AND UNLIKE TEEMS. "When a number is the product of several factors, they are its co-factors; and anj^ one of them, or the product of an}' two or more of them, is a coefficient of the product of the remaining co-factors. A coefficient is numerical, literal, or mixed, accord- ing as it is a numeral, a letter or letters,' or a numeral and let- ters combined. E.g., in labc, 7 is the coefficient of a6c, 7a of he, lab of c, Ib'ofac^ 7c of ab, Ibcota, Usuallj- the numeral alone, together with the sign of the num- ber, -f or — , is counted as the coefficient. Terms which differ only in their coefficients are like (similar) terms ; other terms are unlike. JE.g., 5 ax and 7aa; are lUie, but 5 ax and 7 by are unlike. So, 5 ax and 7 6a! are like if 5 a and 7 6 are counted as the coefficients of a; ; but unlike if 5 and 7 be coefficients of ax and bx. So, 3V(a'+6')) 5aV(«'+&'). (76 + 9c) V(a'+60 are like surds. But 3 V(«'+ 6') ,5a.^{a^ + b-), (7 6 + 9 c) V(a' + c') are air unlike surds. § 15. DEGEEE. I'HB sum of the exponents in a simple term is its degree. The degree of a polynomial is that of the term whose degree is high- est of all. A polynomial made up of simple terms all of the same degree is homogeneous. Expressions having the same de- gree are homogeneous with each other. E.g., a^ + 3 a^ 6 + 3 a6^ -f 6' is homogeneous, of the 3d degree. So, a", a"-i6, a"-2&^ ..., a^-'h', ..., a6"-S 6" are homo-, geneous with each other and of the nth degree. and aa?, l^xy, (?y^ are of the 2d degree and homogeneous with each other as to x and y ; but of the 3d, 4th, and 5th degrees respectively, and not homogeneous, as to all the letters. 22 PKIMAEY DEFINITIONS AilD SIGNS. [I. So, the trinomial a'^ + b~^ + c"' is of the —1st degree, and not homogeneous ; and a?- + b' + c' is of the 3d degree and not homogeneous ; and ma'b~''x'-{-n'a^b~^xy +p^a*b~''y^ is homogeneous and of the 2d degree as to x and y, and homogeneous and of the — 3d degree as to a, b, x, and y ; but it is not homogeneous if m, n, and p be also counted, for then the first term is of the —2d degree, the second term is of the —1st degree, and the last term is of the 0th degree. So, the binomials ^ - 11' and ^a' - fs-^/b*- ? |^ &'* are b bar \ yb'J homogeneous and respectively of the 1st and the | degree. The degree of a product is the sum of the degrees of the fac- tors. The degree of any power of an expression is the product of the degree of the expression by the exponent of the power. The degree of a quotient is the excess of the degree of the dividend above the degree of the divisor. A power or product or quotient of homogeneous expressions is homogeneous, and a sum of homogeneous expressions of any same degree is homo- geneous and of that degree. E.g., (d'+b')i-(x'+f)i:{cib + xy)i is of the 1st degree and homogeneous as to all the letters, but it is of the 0th degree and not homogeneous a,s to a and b only, or as to x and y only. § 16.] EXAMPLES. 23 §16. EXAMPLES. 1. In the sentence {x + ay—(x^ay=:iax, point out the verb, nouns, conjunctions, and phrases, and state their grammatical relations. , §2. 2. Translate and read in words the following symbols : 3^ a'„, 6"i, cW, d(„,,„ n(»'', d.,, p'V-, xl,„,. 3. Write in symbols ': p sub naught, q second, x prime sub r prime, large x fourth sub a prime, large /sub i third and sub k. §3. 4. If a = 2 and 6 = — 3, which of the following numbers are positive, and which negative ? a, b, —a, —b, 2a, 56, —8a, —116. §5. 5. Connect each of the following pairs of numbers by the appropriate sign > or < ; also by the sign > or -^ : 0,1; 0,-1; -1,0; 2,1; 1,-2; -2,-1; -1,-2; ^x,~2x; +x,^2x; ~a + ~b,^a — ~b. 6. Eead in words the statements ; If a < 6 ^0, then a ^ 6 If a > 6, ^ 0, then a ^ 6 and explain the meaning of the copulas used therein. 7. Correct the following continued statements by introducing or suppressing commas : 3<— 4,<1>-1; -.-x^a, .-. 3a;>3a, > 2a. §§ 6, 7. 8. Read in words the following formulse : -ci+-&=:-(+a + +6) ; +a--6>0; -a-+6<0. 9. Correct the following statements by introducing the proper brackets : 5-3 + 1 = 1; 5-3-1 = 3; -5+4-l=-8. 24 PRIMARY DEFINITIONS AND SIGNS. P. 10. Read ia words the statements : (a + b)+(a — b) = 2a; (a + b) — (a — b) = 2b; and, considering a and b to stand for any two numbers whatever, read these two statements as general truths. , §§ 8, 9. 11. Separate the portions of the following continued statements where necessary to avoid false equations or inequalities : 2x3 = 6+4=10-^5 = 2; 2ay + 3xr/=5xy — osy = 4:xy. 12. Read and verify the statements : 1 ! . 2 ! • 3 ! • 4 ! = 1*. 2'. 32. 41 ; 3 ! !,= (3 !) !,= (3 [)"■ 4 • 5. 13. Correct the following statements by introducing the proper brackets : 30-^3x5=2; 30-r-10^&=15 ; 5x-'ixxl+2=p3x. § 10.. 14. Translate into words : (a + by+{x — yy; a^+ax' + ab x + abc; +6 +bc +c +ca [(a + 6).c-(a;-2/)n».|[(a-6)-c]'-(a;-2/)''|-3. 15. Interpret the following expressions and statements : 2*; a*; (2a;)-S; (6t/)~^; 8* = 4; 2°= 3'' = 4»= .... 16. Introduce brackets so that 2^ shall equal 64 ; 256. 17. Whatpower of a;is [(0)8)2]^? (a^)^''? {afyi aiCsV? a;'^? § 11. 18. Find the value of : logsS, logal, log22, legal, log22-^2, logjS, logjl, log82, logs^, log J 2, logjl, logje. 19. Of what number is 4 the logarithm to base 2 ? to base 4 ? to base \'i to base |-? 20. To the base 10, of what number is 3 the loo-arithm? 2' 1 '' 0? -1? -2?f?|? -I? -I? 21. To what base is 2 the logarithm of 9 ? of 27? of 4? 22. To what base is \ the logarithm of 5 ? of y5 ? of i ? §16.] EXAMPLES. 25 § 12. 23. If a=l, 6 = -3, c = 5, find the value of : a'V + 1 l-a?(? , 26^-4ac ffl' + 2a& + 62 a^ + d^ a^-c^ fe'^-c^ 6ii_26c + c2 24. If a = 25, 6 = 9, c = -4, d = -l, find the values of : Va' - 2 ^&8 + 3 -^c* - 4 ^d' ; V- &c + 3 Vocd - 4 V- ^^f* + V-c^*^- 25. If a = 0, 6 = — 2, c = 4, d = — 6, find the value of : 3^(262-a)+2{2a), i-y), cf,(x + y), cj,{x-y), and find the values of ^(0), (l), <^(— 2). 29. If ^(a;, y, z,t) = i>P + 3yz + 1^, write the expressions for : (x) + <^ (y) -I- ^' («) ; F {x, yz) + f {y, zx) + f (z, xy) . § 14. 81. Show what factors must be taken as coefficients in order that the following sets of terms shall be like : 3ab,3bc; 5aa?, 2axy; 7m?i^, 4m-y'W; 2abc, 3bcd, icdx; ^xy", ^yz^ {a? + y^) . § 15. 32. In Ex. 23-25 state the degree of eaoh one of the expres- sions, and show which of them are homogeneous. 26 PEIMAEY OPEEATIONS. [II- n. PRIMARY OPERATIONS. § 1. LOGICAL TERMS. A DEFINITION is a statement of the sense in which a word or symbol is used. A theorem is a general truth : if self-evident, it is an axiom; if auxiliary to a following theorem, it is a lemma; if an ohvious consequence of a previous theorem, it is a corollary. A theorem consists of two parts, the hypothesis or data, and the conclusion which, if not self-evident, is to be established by a demonstration. A converse of a theorem is another theorem that has for data the conclusion, or the conclusion and any of the data, of the first theorem, and for conclusion some datum of the first theorem. E.g., the theorem "If from equal numbers equals be subtracted, the remainders are equal," is an axiom, wherein the clause before the comma is the hypothesis, and the clause after the comma is the conclusion. It needs no demonstration. Its converses are : ' ' If the remainders be equal, the numbers from which equals are subtracted are equal," and "If the remainders be equal, the numbers subtracted from equals are equal." Of demonstrations three kinds are found in Algebra : (a) Direct proof, wherein the conclusion follows as a direct and necessary consequence of certain axioms and definitions, and of other theorems already proved. (6) Proof by exclusion, also called reductio ad absurdum, or indirect proof, wherein are first enumerated all possible conclu- sions from the given data, and then the truth of one of them is established by the exclusion as absurd of all the rest. (c) Proof by induction, which consists of three steps :' 1. Proof, either direct or indirect, that the theorem is true when applied to one or more cases at the beginning of a series of particular cases of the general theorem. § 2.] COMBINATORY PEOPEKTIES OP OPERATIONS. 27 2. Proof that, if the theorem be true up to any one case in- clusive, then it must also be true for the next higher case in the series. 3 . Proof by progressive steps that, since , beginning with the cases actually proved (1), it is true for the next, and the next, and the next, indefinitely (2) , therefore it is universally true. A problem is anything to be done ; usually, in Algebra, it is to find numbers or expressions that will satisfy given conditions. These numbers or expressions, together with the process of find- ing them, constitute the solution of the problem. A solution is -l 9'®'^^™ when it gives ■{ ^ „ of the numbers, ' particular ° ' some ■ ' or expressions, or sets of numbers or expressions, that satisfy the given conditions. Usually the general solution is sought, with a demonstration showing, by previous theorems and problems, that the solution satisfies the given conditions and is general. A check, or test, is a comparison of results designed to detect any accidental errors in the work. A postulate assumes as self-evident that the solution of a problem is possible. The letters^ «--; at the end of a-! Strn"*"" ^^and for J . , demonstrandum „t. , „„„ , , „ < proved. quod erat \ f^^^^^^^^ - which was to be \ i^^^^_ § 2. COMBIBTATOEY PEOPEETIES OF OPERATIONS. Ak Algebraic Operation is an act \>j which two or more numbers, the elements, are combined together to produce one number, the result. Manifestly, the result is a function of the elements. An operation is^ ^^^^ when^ "^^^ ^^^^ two elements are combined. If a complex operation consist of two or more simple operations, and if they be all of the same kind, it is a continued operation. E.g., the continued addition of three numbers consists of first adding two of them, and then adding the third number to this sum'. 28 PEIMAJBY OPEEATIOKS. [II. Of the two elements of a simple operation, one, the operand, is conceived of as acted upon by the other, the operator, in a way shown by the sign of operation. E.g., in 6 + 2 = 8, 6 — 2 = 4, 6 X 2 = 12, 6:2 = 3, the operand is 6 ; the operator is 2 ; the results are 8, 4, 12, 3 ; the operations are addition, subtraction, multiplication, division ; and the signs of operation are + , — , X , : . So, in 16^=256, -^16 or V16 = ± 4, log2l6 = 4, the operand is 16 ; the operator is 2 ; the results are 256, ±4,4; the operations are involution, evolution, the finding of a loga- rithm ; and the signs of operation are, the position of the expo- nent, -y/, the word "log." I um-deferndnate, An operation is < multi-determinate, when, from giveni elements, ' indeterminate, I only one result, it gives < several different results, but none intermediate. ' an infinite number of results in a continuous series. The rational operations (addition, subtraction, multiplication, division, and involution to integral powers) and the finding of logarithms are generally unideterminate ; but evolution is gener- ally multideterminate ; and operations with special elements are often indeterminate. E.g., 6+2, =8; 6-2, =4; 6x2, = 12; 6:2, = 3; 3^,= 9 ; log g 9, = 2; ai-e unideterminate ; but -^9, = either + 3 or — 3, is multideterminate ; and 0:0, 0", logoO, logjl, are indeterminate. When the result and the^ operator . . operand ' operand °^ , uuc^ operator may be found by an operation called the ^ •''' , inverse of the origiilal or direct operation, wherein the operand, operator, and result nro the J result, operator, and operand, ,. , „ ^, ^■^^ *^^ ^ result, operand, and operator, respectively, of the direct operation. Hence an inverse operation is the undoing of what was done by the direct operation, and it ends where the direct operation began. § 2.] COMBINATOBY PKOPEETIES OP OPERATIONS. 29 An inverse operation may be defined as an operation ' ' the eflfect of which the direct operation simply annuls," It consists not in any new procedure, " but in a series of guesses suggested by prior general knowledge of the results of the direct operation, and tested by the direct operation itself." — Boole. E.g., 6-2= 4 •.• 4 + 2 = 6; 6:2= 3 ■.- 3x2 = 6; V9 =±3 ■■• (+3)2 = 9 and (-3/ = 9; log39= 2 •.• 3^ =9; An inverse operation is, therefore, described by the two words ' ' guess " and ' ' test." The error of one guess helps the next one. E.g., To divide 756 by 27 : Guess 30 ; that is too large, for the product, 27 X 30, is 810, which is larger than 756. Guess 20 ; that is too small, for the product, 27 X 20, is 540, and the remainder, 216^ is larger than 27. Guess 8 as the quotient of the remainder 216 : 27; this guess is right, for the product, 27 x 8, is 216 ; a,nd the whole quotient is 28, the sum of 20 and 8. An inverse operation may or may not be multideterminate when the direct operation is unideterminate ; and the two in- verses may or may not be of the same kind. E.g., Direct Operations. First Inverses. Second Inverses. 6 + 2= 8, 8-2= 6, 8-6 = 2; 6X2 = 12, 12 : 2= 6, 12 : 6 = 2 ; 6^ =36, V36 ^±6, log636 = 2; wherein the two inverses of addition are both subtraction and unideterminate ; and of multiplication, they are both division and unideterminate ; but of involution, the first is evolution and multideterminate, and the second is the finding of a logarithm and unideterminate. A direct simple opera,tion is sometimes the repetition of more elementary operations. E.g., addition of a { negative ^°^*®Ser, ± m, is counting a unit ■{ °J^ wi times ; and addition of a-[ ^gg^^^g fraction, ± ^i 30 PEIMAEY OPERATIONS. [U. is counting m times ■{ ^ such a number as, if counted on n times, would add a unit. ■ Bositive . i , -I adding So, multiplication by a^ P^J^^j^^ mteger, ± m, is ^ subtmcting the multiplicand m times ■{ ^^^ ; and multiplication by a . positive fj.aj,tio, ±t!*, is ^ ^*^f °S ^ times { *° ^ such ' negative ' w' ' subtractmg ' from a number as, if added n times to 0, would give the multiplicand. So, involution by a ^ P°gatiVe integral exponent, ± m, is , multiplymg ^ ^ ^j^^ ^^^^ ^ ^^^^ . ^^g^ involution by a ' dividmg •' < "^e ^-*-^l -P--*' ± ™' ^^ ^ rSSr ^' '^ *^"^^' by such a number as, if multiplied n times into 1, would give the base. Thus the operations of addition, multiplication, and involution aU come from the more elementary operation of counting. So, often, when the operator is a-{ ^g^t^e ^"^^S^rj as ± m, then the more elementary <{ •,„_,„„ operation is performed m times in succession upon the operand ; and when the operator is I positive fJ.a^gyQ^ ^g ±— , then some operation is performed ' negative ' n' ^ *^ m times which, if performed n times, would be equivalent to 11 1 TP r*f" the more elementary ■{ . operation. The modulus of a simple operation is that operator, if any, which always makes the result equal to the operand. E.g.,-.- x + = x and x — = x, [« any number a; X 1 = a; and x : l = x, and a^ =x and y/x = x ; .-. the modulus of addition and subtraction is 0, the modulus of multiplication and division is 1, and the modulus of involution and evolution is 1. §2.] COMBINATORY PROPERTIES OE OPERATIONS. 31 Aa operation is ^ <'ommutative elements ^ °^° ^, > non-commutatwe eieuitjubs ^ cannot excliange places without changing the result ; i.e., when the result ^' -i an uns™Smetric ^^'^^^^o^ of the elements. JE.g., Commutative Operations. Non-commutative Operations. 2 + 3 = 3 + 2, 2-3^=3-2, 2x3 = 3x2. 2:3:?fc3:2, 2^ ^ 3S ^2 ^^3, I0g32:#=l0g23. A continued operation is { "**"'"''' *'"f when, as long as the ^ ' non-associative ' ° elements do not exchange places, they-j ^^^ . be grouped at will without changing the result. E.g., Associative Operations. Non-associative Operations. (12+4) + 2 = 12 + (4+2), (12-4) - 2 ^t 12 - (4-2), (12x4) X 2 = 12 X (4x2), (12 : 4) : 2^^=12 : (4:2), (12^)2 :^12(^'). . , i- • 1 distributive . ~ . , , . A second operation is-( ^^^_aistribvAwe "' *" ^ ^'^^^ operation when the final result ■{ ■ , the same, whether the second opera- tor act upon the result of the first operation, or upon the separate elements of the first operation, and then these results are com- bined by the first operation. An operation distributive as to addition .is also called simply distributive, or linear. E.g., Distributive Operations. Non-distributive Operations. Multiplication as to addition. Addition as to multiplication. 12 + 6x3 = 12x3+6x3; 12x6+3 =jfc 12+3 x6"+3. InTOlution as to multiplication. Involution as to addition. 12x6' =12^X6^ 12+6' :#=122+6^ Evolution as to mqltiplication. Evolution as to addition. ■v/27x8 =^27X^8; ^27±8 :#-^27±-^8; Finding of logarithms as to addition. log6(216 + 36) ^ log6216 + loge36 ; Finding of logarithms as to multiplication. log8(216 X 36)#lbg6216 X log^Se. 32 PBIMAKY OPBEATIONS. [H. § 3. AXIOMS. 1 . Numbers equal to the same mimber are equal to each other. 2. If to equal numbers equals be added, the sums are equal. 3. If from equal numbers equals be subtracted, the remain- dees are equal. 4. If equal numbers be multiplied by equals, the products are equal. 5. If equal numbers be divided by equals, the quotients are equal. 6. If equal numbers be raised to like integral powers, the POWERS are equal. 7. If of two equal numbers like roots be taken, every root of the first number is equal to some root of the other. 8. If of three numbers the first be-{ ^^^ ®^ than the second, and the second be equal to or ■{ ^^^^^ than the third, then is the first { ^^^^^ than the third. 9. If one number be \ P^^ ®^ than another, and if to each of them be added the same number or equal numbers, then is the first sum^ fe's*s^*^^ ^^^^ ^'^^ °^^^^- 10. If one number be ■{ F^®^ ^^ than another, and if from each of them be subtracted the same number or equal numbers, then is the first remainder ■{ P than the other. 11. If one number be-{ ?^ than another, and if each of them be subtracted from the same number or from equal num- bers, then is the first remainder { , than the other. ' ' greater 12. If one set of numbers be-^ P^^ ^^ than another set of as many more, each than each, then is the sum of the first set -{ P^^ ^ than the sum of the others. § 3-] AXIOMS. 33 13. If one number be^ ^^s^^^ ^^^^ another, and if each of them be multiplied or divided by the same or equal positive num- bers, then is the first product or quotient -{ S^®^^^^ than the other. 14. If one number be-j f^^^^^ than another, and if each of them be multiplied or divided by the same or equal negative numbers, then is the first product or quotient -j , than the other. ^^* ^^ 15. If of three numbers the first be -i ^''Sf than the second, ' smaller ' and the second be equal to or-{ ^^S®'' than the third, then is the first -{ g^^ngp than the third. 1 P "PP" AT* 16. If one number be ■{ g_^|ii than another, and if each of them be multiplied -by or into the same number or equal num- bers, then is the first product <{ „!f iip than the other. 17. If one number be -{ gj-ffip than another, and if each of them be divided by the same number or by equal numbers, then is the first quotient ■{ J,°ii than the other. 18. If one number be ■{ =,, than another, and if the same number or equal numbers be divided bj' each of them, then is the first quotient.) i„ ' „ than the other. 19. If one set of numbers be -{ g„fn„j. than another set of as many more, each than each, then is the product of the first set ■ arger ^j^^^ ^j^^ product of the others. ' smaller ^ 20. If one number be -j „j_fii„„ than another, and if like posi- tive powers or roots of them be taken, then is the first power or root^ ^^^^^ than the other. 34 PEIMAEY OPERATIONS. [H- th. 21. If one number be ■{ ^^^ than another, and if like nega- tive powers or roots of them be taken, then is the first power or root-{ l^^^^^ than the other. 22. If two numbers be opposites, one of them is positive and the other is negative ; they are equally large ; and their sum is naught. 23. If all the letters of an entire expression stand for integers, the value of the expression is an integer. Note 1. For convenience, and because quite evident, aU the propositions above given are called axioms, although, in strict- ness, some of them are deducible from others. E.g., Ax. 1 is deducible from Ax. 8. For let A, B, c be three numbers such that a = c and b = c ; then either A = B, orA>B, orA B ; then •.•A>B and b = c, [liyP' .-. A>c, [ax. 8 a consequence from the supposition a > b, which is contrary to the hypothesis of the theorem, and therefore absurd. .'. the supposition a> b, which led to this absurd conse- quence, is itself absurd ; and a > b. So it may be proved that a < b ; and -.-A^B and a^9"'t^ve, °-f ^^'^ ^ opposite '^""'^ < to ^''^ d'^>^^d. Note. Th. 2 and Cor. 2 are summarized in the familiar rule for sign of product or quotient : " Like signs give + ; unlike, — ." 2, 3. §6.] MULTIPLICATION COMMUT. AOTJ ASSOC. 37 § 6. MULTIPLIOATIOK COMMUTATIVE ANB ASSOCIATIVE. Thbok. 3. The product of two or more numbers is the same, in whatever order the factors are m,ultiplied, and however they are grouped. (a) Two positive integers, a, b For let » * * • # » then will a X b = b X a. * « * be a collection of stars, trees, or any other units, con- sisting of a horizontal lines, and 6 vertical columns ; then ••• if a, the number of stars in one column, be multiplied by b, the number of columns, the product, a X 6, gives the whole number in the collection ; and '.• if b, the number of stars in one line, be multiplied by a, the number of lines, the product, b X a, gives the whole number in the collection ; .-. ax& = 6xa. Q.E.D. [ax. 1 (6) Three positive integers, a, b, c ; then will axb xc=bxaxc=cxaxb=cxbxa =a XGXb=cxaxb=bxaxc—bxcxa = bxcxa=cxbxa = axbxc = axcxb. For let a a a ... a a a a ... a a a a ... a be a collection of groups of a units each, in 6 hori- zontal lines and c vertical columns ; and •.• axb is the number of units in one column, -•. cTxb X c is the number of units in all the c columns, i.e., in the whole collection ; then •.•axe is the number of units in one line, .*. axe Xb is the number of units in all the b lines, i.e., in the whole collection ; -•. ax6xc = axcx6. q. e. d. [ax. 1 38 PELMAHY OPERATIONS. [H. th. So let 6 & 6 ... b b b b ... b b b b ... b be a collection of groups of 6 units each, in a lines and c columns ; then •.• each of the c columns has bxa units, and •.• each of the a lines contains b X c units, .'. & X a X c = 6 X c X a. Q.B.n. But*.* a X & = & X a, axc = cxa, bxc = cxb, and axb xc = cx axb, axcxb = bxaxc, b xcx a=ax bxc; [(a) .-. ax 6 X c = bxax c = c X ax & = cx 6 X a =axcx6=cxax&=6xaxc=&xcxa = 6xcxa = cX&Xa=ax&Xc = axcx6. q.e.d. Note. In this proof the reader wiU observe that the bar can be placed over a group of factors, or removed, at pleasure, when that group stands at the left end of the series, but not otherwise. E.g., axbxc = axbxc; for in either case, the product of a by & is first found and then that product is multiplied by c. But a X 6 X c is a very different matter ; for in this case, the product of b by c is first found, and a is then multiplied by this product. (c) Any number of positive integers. 1. The theorem is true for three factors. [(a, 6) 2. If it be true up to n factors inclusive, it is true also for ji + 1 factors. For let the n + 1 factors, a, 6, c, . . . i, j, 7c, be grouped and multiplied together in any desired way, and let the product be p ; then •-• p is got by multiplying the product, say Q, of some of these factors, by the remaining factor, or by the product, say e, of the remaining factors, .-. P = Q XE. 3. §6.] MULTII'LiOATION COMMUT. AND ASSOC. 39 Let E be that one of these products which has the factor k, and let s be the product of the other factors of e ; then •.■ neither q, k, s nor the product Q X s has more than n factors, .•.in each of them the several factors may take any desired order, [hyp. above -•. P, = Q X R, = Q X s X A; = QX sx fe [(&) = axbx ■■■X j XTc = axbx---Xj XJc. 3. But the theorem is true for three factors, [(a, 6) .-. it is true for four factors ; [2, above So for five factors, for six factors, q. e. d. Note. This proof is an example of proof by induction. [§ 1 (d) Any number of integers, "'"a, ~b, +c, . . ., +1, ~m, "n, whereof k factors are negative. For in whatever order the factors are multiplied, p, their product, = ± +a X "*"& X • • • X """m x "*■«, [(c) a.-{^ ,. number when kis-{ , , ' rlu'a' •. ' negative ' odd. [th. 2 or. 1 (e) Any number of factors, a, b, c, ... -,-,.. ., whereof some, or all, are simple fractions. '' For-.- in the product axb X cx ■■• X^X^X ■•■, = ±+aX+6x+cX •••xY^xY-) X---> each unit of the product a x & X c x • • • is divided <* into X equal parts and r of them are taken, and then each of these ax6xcX"-Xr parts is divided into y equal parts, and s of them are taken, and so on, i.e., the unit is divided into xxyX ••• equal parts, and ax6xcX-"XrXsX--'0f them are taken ; r s ax6xcx---'xrxs--- .-. ax &XCX ••• X-X-X--- = X y xxy X < - X 6 X - xcx---eachof t y X is divided into y parts, and s of them are taken, S 7* So in the product a X - X 6 X - xcx---eachof theaunits y X 40 PKIMABT OPERATIONS. [H- th. and then each of these aX s parts is taken 6 times, and then each of these a X s X 6 parts is divided into x equal parts, and r of them are taken, and then each of these a X s X & X r parts is taken c times, and so on; i.e., the unit is divided into y x a; X ■ • • equal parts, and ax sxb xrx ex ■••of them are taken. .-.ax^XfiX^XCX -«X«x6xrXCX- y X yxxx--- But-.- yxxx ••• = xxyx---, [_{d) and axsx6xrxcx^-^ = ax6xcx--^xrxs; . . in each of these two products, the unit is divided into the same number of equal parts, and the same number of these parts are taken. So for all other possible products of these factors ; .-. the products are all equal. q.e.d. (/) Any number of factors, whereof some or all are neither integers nor simple fractions, but which are all commensurable. For, let A, B , c, ... 5 ,_,... be the values of such factors when X Y reduced, wherein the letters all stand for integers ; [I. § 1 df. then '.• the value of the product axbxcx^^^X — X-X ••• X Y is the same, whatever the order or grouping of the factors, [(d, e) . • . the product of the given factors is the same, etc. Q. e. d. THEORY OF TRACTIONS. CoE. 1 . Tlie product of two or more fractions is a fraction , I numerator . .■• i ^ j. ^i i numerators j. .7 "^^"''^ denominator '' '^' P'"^^'^ °f ^^< denominators ''f ^^'\ given fractions. ijn in' "W, Let -, —, -— , ... be any fractions, d d' d" thenwiU^X^'x-^ ^X^'X- . d d' dxd' X ••• (a) n, d, ... all integers. This case was proved in the demonstration of Th. 3 (e) . 3. § 6.] MULTIPLICATION COMMUT. AND ASSOC. 41 (&) The fractions and their numerators and denominators any commensurables whatever. For, let /, /', ... be the values of -, — ,, ... respectively ; d d' then -.• n=fxd, n' =/' x d', ... , [I. § 9 df. .-. nxn'x---={fxd)x{f'xd')x--- [ax. 4 = {fXf'x---)x(dxd'x---), [th,3 n X n' X d Xd' X = ?X^X--- «•*=•»• [I-§9df. d d' Cor. 2. The reciprocal of any fraction is the same fraction in- . -, • J- J.- T. I numerator . ., , denominator verted; i.e., a fraction whose < ■, . . is the< . ' ' •> > denominator ' numerator of the given fraction. For, let n and d be any commensurable numbers ; ,, d ,,n dx n 1 then -.• -X- = ; = !> n d n X d .-. - is the reciprocal of -• q. e.d. [I. § 8 df. n d CoE. 3. The quotient of one number by another is the product of the first by the reciprocal of the other. (a) The divisor and dividend both simple fractions. For, let - and — be any two fractions, d d' ,, /n ^, d'\ ^n' n „ fd' ^n'\ w d'xn' n . then ••• - X — X — , = - X — X - = - X -; — -. = - , \d n'J d' d \n' d'J d n'xd' d .: !^:^ = r^x^!. Q.E.D. [L§9df. d d' d n' Note. Cors. 1, 2, 3 lead to the reduction of complex prod- ucts and quotients of commensurables to simple fractions, either directly or by progressive steps. By Cor. 2, the reciprocal of any commensurable is a commensurable ; hence, by Cor. 3, if both numerator and denominator be commensurables, so is the fraction; by Cors. 1, 3, any product or quotient of such frac- tions is got as a simple fraction, and so on. Compare Th. 5, Cor. 8. 42 PEEMAEY OPEEATIO^rS. [11. Ih. (&) Tlie dividend and divisor any commensurables whatever. For, let N, D be any two commensurables ; then •-• D is commensurable, .•. _ is likewise commensurable ; [(a) nt. D .-. ( N X - J X D = N X- ( - X D ) = N ; [th. 3 .-. Nxi = N:D. Q.E. D. [I. §9df. D CoR. 4. The product of the reciprocals of two or more numbers is the reciprocal of their product. T S For, let a, 6, ... -, -, ... be any numbers, X y then •.•[_x^X---X-X-X---)XlaX&X---X-X-X"-| \a r s J \ X y J = 1; 11 X y T , r s ■ • -X-rX---X-X-X--- and ax6x---X-X-X--- a r s X y are reciprocals . q . e . d . Ck>R. 5. If the numerator of a fraction be multiplied by any number, the fraction is multiplied by that number. For, let - be any fraction, and fc any multiplier ; then ^^4^ = (nxk)x\F=(nX-}iXJc = ''^^Xk. q.e.d. d d \ dj d Note. In this corollary and the two corollaries that follow, " multiplied by" includes " divided by," since to divide by k is but to multiply by its reciprocal. Cor. 6. If the denominator of a fraction be multiplied by any number, the fraction is divided by that number. For, let - be any fraction, and k any multiplier ; a then ^ =n X-i- = «X (^ X ^\ = (nx i^ xi = ^:fe. dxk dxk \d k) \ d) k d 3. § 6.] MXTLTIPLICATION COMMUT. AND ASSOC. 43 CoE. 7. If both terms of a fraction be multiplied by the same number, the value of the fraction is not changed. For, let - be any fraction, and k any multiplier ; +v,„„ nxlc n^k n ^ , '^''' a^u=d''l = d ^■^•"- t^^--^ CoE. 8. If there be a series of multiplications and divisions, the final result is the same, in whatever order they are performed, and however the elements are grouped; but whenever any group is made to follow the sign of division, the sign of operation of each element of the group is reversed. Note. The reader will observe the analogy between this cor- ollary and Th. 1, Cor. 3. He will see that, if three or more numbers are joined bj' the signs X and :, he may introduce or remove brackets just as if they were joined by the signs + and — . QUOTIENT OF A PRODUCT BY ITS FACTORS.' CoE. 9. If the product of several factors be divided by one of them, or by the product of two or more of them, the quotient is the product of the remaining factors. For •.• the product of the remaining factors by the divisor is the product of aU the factors, [th. 3 . • . the product of the remaining factors is the quotient of the product of all the factors bj- the divisor, q.e.d. PRODUCT OF INTEGRAL POWERS. CoE. 10. The product of two or more integral powers of any same number is a power of that number whose exponent is the sum of the exponents in the factors. ' For, let A be any number, and I, m,n ... any positive integers, then •.• a' = 1xaxaxax---? times, A"=lXAXAXAX ■■• m times, and A""=l : A : A : A : •••« times, and SO on. [I. §10df. .-. a' X A^X A'-"X ••• = (lXAXAX'--? times) x(lXAXAX-"m times) X (1 : A : A : • ■ • m times) X • •• = lXAXAX---(Z + mH ) times : a: a: •••(ti-J ) times = 1 XAXAX--'(Z + m + -"— n ) times [or. 9 _^I+m-7i..._ Q.E.D. 44 PEIMABY OPERATIONS. [II. th. Cob. 11. An integral power of any integral power of a base, is that power of the base whose expoTient is the product of the two given exponents. Let A be any number, m and n any integers , then will (a") "* = a"". (a) m. positive. For (a")"^", = 1 X a"x a"x ••• m times, (6) m negative. For (a") "", = 1 : a" : a" : • • • +m times, __ 1 . .*»+»+"• mtimes = 1 : a"*""" = a"'»", or = A™ if the sign of quality be erased. , Q. E. D. CoE. 12. TTie •{ ■?„„,,•„„/ of like integral powers of two or more numbers is the same power of the-{ P'''°^^ of those numbers. Let A, B, 0, ... be any numbers, n any integer, positive or [I. § 10 df. [cr. 10 Q.E.D. [I. § 10 df. [cr. 10 negative, then wiU a" x b" : o" • • • = a X b : c • • • . (a) n positive. For •, .• A"=l X AX AX •••ntimes. B"=lXBXBX---»i times. o"=lxcxox---m tunes, and so on ; .' •. A"XB":0''--- = (lXAXAX ••■ m times) x(lXBXBX ■•■ ntimes) : (1 XO XCX •■• m times) •• • = 1 XAXB :o •••X AXB : ••• X ••• n times = A X B : C ••• . Q.E.D. (6) n negative. For •, ,• a'"= 1 : A : A : ••• +n times,. b"" = 1 : B : B : • • • +W times. c "=l:o:o:--.+w times, and so on ^ .' '. A~"x b"" : c""--- = ( 1 : A : A : • • • +n times) x(l:B:B:..-+n times) : (1 : c : c : • • • +n times) • • • = 1:axb:c---:axb:C'--: •••+« times [or. 8 = (axb : c •••)"". n w T, i. § 7.] MULTIPLICATION DISTEIBTTTIVE AS TO ADDITION. 45 § 7. MULTIPLICATION DISTRIBUTIVE AS TO ADDITION. Theoe. 4. The sum of two or more like numbers is the product of the common factor by the sum of the coefficients. Let +m • o, +« • a, ~p- a, — a, ... be any lite numbers, ± whereof a is the common factor, and +m, +m, ~p, —, ... are the coefficients ; ±~. then will "•'m •a+'*'n-a + ~p-a-\ aH ±^ = (+m++n +15 +— + •■•)•«. For ••• ■''m ■a= a + a + a-\ counted on "♦'m times, +n-a= a + a + a-{ counted on "'"Ji times, -p.a = — a — a — a counted off +p times, ±0- ■ — o = the a!th part of a counted, on or off, r times, ..■^m-a++n-a+~p-a+ — -a + --- , = a + a + a-{ counted (+m ++n +"p •••) times ± the -th part of a H ; *»• .-. the whole sum is (+m ++w + p+~-i ) • a. Q. e. d. CoE. The sum of two or more fractions having a common denominator is a fraction whose numerator is the sum of their numerators, and whose denominator is the comrnon denominator. For,let^,±$, . d d nominator, ^, n ,n' , then ■%^-; + -' d d .. be any fractions having a common de- • \ dj \ dj = (n±n'+--)x^ [th. 3 cr. 3 [th. «*'*'+•••. Q.E.D. Note. In this corollary, and in general, subtraction is but a case of addition. E.Q., to subtract the fraction ^ is to add its opposite, — — • " ' d a 46 PEIMAEY OPERATIONS. [II. th. Theoe. 5. The product of two or more polynomials is the sum of the several products of each term of the first factor by each term, of the second factor by .each term of the third factor, and so on. r r' (a) Two factors, a + bH 1 1 — anda'+h'-\ 1 1 X x' wherein a, b, ... , a', b', ... are any integers, positive or negative, and ~, — , ... are any simple fractions. r X x' For (a + & + ... + - + ---) X (a' +b' + ■■■+% + ■■■) = (a + & + •••-! 1 ) counted +a' times, on or off, + (a + &H 1 1 ) counted +6' times, on or off, X + + the —7 tb part of (a + & H 1- - H ), on or off, «' X + = a counted a' times + b counted a' times ■} +- counted a' times + ■ s counted V times ^ counted V times + ■ + a counted V times + 6 counted V times H h - counted V times A + + the _th part of a + the Lth part of 6 + r^ r + the — ,th part of -A x' ^ X + = axo' + &xa'+ ■ ** + ax6'+6x6' + + r' r' + S X a' + +^X0' + + - X -, + X x' Q.E.p. Note. Manifestly, if a term in either factor is negative, the corresponding partial product is negative or positive according as the co-factor of this term is positive or negative. B. § 7.] MULTIPLICATION DISTEIB. AS TO ADDITION. 47 (&) Three or more factors. For ••• the product of two factors is the sum of the partial products of each term of one factor by each term of the other, [(a) and •.• the product of this product by a third factor is the sum of the partial products of each term of this product by each term of the third factor ; [(a) .*. the product of three factors is, etc. q.e.d. So, for any number of factors. q.e.d. FORM OF PKODUCT. Cob. 1. The form, of a product is independent of the values of the letters that enter into it; i.e., the same numerals, letters, exponents, coefficients, and signs, occur and combine in the same order, whatever the numbers for which the letters stand. CoK. 2. If each factor be symmetric as to two or more letters, the product is also symmetric as to the same letters. CoR. 3. If any values be given to the letters, or if any definite relations be assumed between their values, the value of the prod- uct equals the product of the values of the factors. Cor. 4. The sum of the coefficients of a product is the con- tinued product of the sum of the coefficients of the first factor, by the sum of the coefficients of the second factor, and so on. Cor. 5. The degree of ilie-{ ■, ^ . term of a product, as to any letter or letters, is the sum of the degrees of the \ jJ[jp„f terms of the factors, as to the same letter or letters. In particular, the degree of the product is the sum of the degrees of the several factors. Cor. 6. If each factor be homogeneous as to any letter or let- ters, then the product is homogeneous as to the same letter or letters. Cob. 7. TJie whole number of terms in any product, before reduction, is the continued product of the number of terms in the several factors ; and the product of two or more polynomials can never be.reduced to less than two terms; viz. : the term of highest degree and the term of lowest degree as to any letter or letters. 48 PEIMAEY OPEKATIONS. [II. th. CoE. 8. The value of every rational expression whose elements are commensurable numbers is, a commensurable number. For •••in such an expression the elements enter onlj- as ele- ments of sums, differences, products, quotients, and integral powers, and •.• these results enter only as elements of new sums, etc., and so on, and •.• the sums, etc., of commensurables are commensur- ables ; [th. 3 cr. 3 nt. , th. 4 cr. .•. the sums, etc., of the elements are commensurables, .-.the sums, etc., of these results and the original elements are commensurables, and so on ; .•. the final result is commensurable. q. e. d. § 8. PKOPOETION. FouE numbers are proportional (in proportion) when the first is such multiple, part, or parts, of the second, as the third is of the fourth ; i.e., when the quotient of the first by the second equals the quotient of the third by the fourth. E.g., if a : 6 = c : d, then a, b, c, d are proportionals, taken in the order given. A proportion is also written in the forms a:b::c:d and r = ;;i and it is read : a is to b as c is to d, or the ratio of&toh equals the ratio ofcto d, or, more briefly, a to b equals c to d. These quotients are now called ratios ; the dividends, ante- cedents; the divisors, consequents; the first and fourth terms, extremes; the second and third terms, means; the fourth term, a fourth proportional to the other three. Three numbers are proportional when the quotient of the first by the second equals the quotient of the second by the third. It is a case of four proportionals wherein the two means are the same number. The second number is a mean proportional between the first and third, and the third is a third proportional to the first and second. E.g., a : b = b : c, wherein 6 is a mean proportional between a and c, and c is a third proportional to a and 6. 6. § 8.] PEOPORTION. 49 Six or more numbers are in continued proportion when the first is to the second as the third is to the fourth, as the fifth is to the sixth, and so on. E.g., a:b= c: d = e:f= •••. • By aid of Th. 7 (6) this proportion maj^ be written in the form a: c: e: •■• = b:d:f: ••■, wherein a, c, e, ... are the antecedents, and 6, d, f, ... the consequents. This notation must not be con- founded with that used on p. 43 and elsewhere. Theok. 6. If four numbers be proportional, the product of the extremes equals the product of the means; and, conversely, if the product of two numbers equal the product of two others, the four numbers form, a proportion, wherein the factors of one product are the extremes and the factors of the other product are the means. (a) Let a:b = c:d, then will ad = be. For •.• {a:b)xbd=-{c:d)xbd, [ax. 4 -•. ad=bc. Q.E.D. [th. 3cr. 9 (&) Let ad = be, then will a:b = c:d. For •.■ ad:bd = bc:bd, [ax. 5 .•. a:b = c:d. Q.E.D. [th. 3 or. 7, cr. 5 nt. Cor. 1. If four numbers be proportional, either ■{ ^g^^ is ,. ., 1 means , ., .r. i eoatreme. the quotient of the product of the \ extremes^^ *^^ ''*^''' ^ mean. For, let a:b = c:d, then •.• ad=bc, ' [th. .-. a =bc:d, b = ad:c, c = ad:b, d = bc:a. [ax. 5 Cor. 2. If three numbers be proportional, either extreme equals the quotient of the square of the mean by the other extreme, and the mean equals the square root of the product of the extremes. For, let a : 6 = & : c, then •.■ ac = 6^ [th. .-. a =l^:c, b=-y/ac, c = b^:a. q.e.d. [ax. 5,7 Note. The equation ad = be may be resolved into eight dif- ferent proportions, four of them with a and d for extremes and b and c for means, and four of them with b and c for extremes and. a and d for means. The reader may write them out; he will find two of them given in Th. 7 (a, 6) . 50 PBIMABT OPERATIONS. [II. th. Thkor. 7. If four numbers be proportional, they are propor- tional : (a) Inversely: the second to the first as the fourth to the third. Let a:b = c:d, then will b:a=d:c. For •.• ad =bc, [th. 6 .-. b:a = d:c. q.e. d. [th. 6 ev. (6) Alternately: the first to the third as the second to the fourth. Let a:b = c:d, then will a:c=b:d. For ••• ad =bc, ' [th. 6 .-. a:c = 6:d. q.e.d. [th. 6 cv. (c) By addition or subtraction (composition or division) : the-{ *"™ . , of the first ■{ f the second, to the first or second, ^ *^^ < 7Zainder °^ ^^ *^^^ ^ Zt *^^ f"'"'^^' *° ^^^ ^^''^ °' fourth. Let a : 6 = c : d, then will a±b:a = c±d:c, and a±b:b = c±d:d. For •.• ad=bc, [th. 6 • •. ac±bc =:ac±ad and ad±bd = bc±bd, [ax. 2,3 i.e., (^a±b)c =a(c±d) and{a±b)d =b{c±d), .-. a±b:a = c±d:c and a ±b:b=c ± d: d. q.e.d. (d) By addition cmd subtraction (composition and division): the sum of the first and second to their remainder as the sum of the third and fourth to their remainder. Let a:b = c:d, then will a + b: a — b = c + d: c — d. For •■• a+6:a = c + d:c and a — b: .-. a + b:c + d = a:c and a — b: .•. a + b:c + d = a — b:c — d, .'. a + b:a—b = c+d:c — d. CoE. Conversely, if four numbers be proportional, (a) in- versely, (b) alternately, (c) by addition or subtraction, or (d) by addition and subtraction; then is the first to the second as the third to the fourth. The reader may prove, by retracing the steps, from conclusion to data, in each of the above demonstrations. a-- = c- -d: ;c. [(c) c- -d-- = a: ;c, m [ax. 1 Q. E. D • m w -,v.»' -«•'-, •-, = !:;•-' [ax.5,th.3cr.l 7-9- §8.] PROPORTION. 51 Thsoe. 8. If there be two or more sets of proportionals, the products of their corresponding terms are proportional. Leta:6 = c:d, a' : &' = c' : d', a" : 6"= c" : d", ••-, then will aa'a" ■■■ : bb'b" ■■■ = cc'c" ■■■ : dd'd" ■■: For •.• ad = bc, a'd' = b'c', a"d" =b"c",---, [th. 6 .-. ad-a'd'-a"d"---- =bc-b'c'-b"c"--; [ax. 4 .-. aa'a"----dd'd"--- =bb'b'---- cc'c"--; [th.3 .-. aa'a"- ■■ -.bb'b" ■•■ =cc'c"--- : dd'd"---. q.e.d. [th.Gcv. Cor. 1 . If there be two sets of proportionals, the quotients of their corresponding terms are proportional. For, let a : 6 = c : d and a' -.b' = c' : d', then ■.•ad =bc and a'd' =b'c', [th. 6 a4 _ be • p 2l — be a'd' ~6'c'' *■■'£«'■* a b c d Til /. < CoE. 2. jy four numbers be proportional, their Mice integral powers are proportional. The reader may write in formula, and prove. Theoe. 9. If six or more numbers be in continued proportion, the sum of the antecedents is to the sum of the consequents as any antecedent is to its consequent. Let a: 6 = c: d = e:/= •••, then will a + c + e-] : h + d+f-i = a:b = c:d="'. For •.' ad = bc, af=be, •••, [th. 6 -•. ab + ad + af-\ =ba + bc + be + "-, [ax. 2 .-. a(& + d+/+-) = 6(a + c + e+-), [th.4 .-. a-{-c + e-\ :b+d+f-\ — =a:b. q.b.d. [th.Gcv. CoE. 1. Jf a:b = c: d = e: f= •••, then ha + kc + le + -":hb+kd+lf+." = a:b, wherein h, k, 1, ••• are any numbers. The reader may state in words, and prove. CoE. 2. i/"a:b = c:d = e:f=—, then ha" + kc" + le" + ••• : hb" + kd" + lf"+ ••• = a" : b", wherein h, k, 1, ••• are any numbers and n any integer. The reader may state in words, and prove. 62 PEIKAEY OPERATIONS. [II. pr. § 9. PROCESS OF ADDITION. PrOB. 1. To ADD TWO OB MORE NDMBEKS. (a) The numbers like : To the common factor prefix the sum of the coefficients, [th. 4 E.g., 10 ft. down + 20 ft. up + 60 ft. up = 70 ft. up, 10 ft. up +20 ft. down + 60 ft. down = 70 ft. down. So, lOa;— 15a!+20a!— 25a!+30a!=60a;— 40a;=20a!, [th. 2 lOay + 20by- 30 cy = (10a + 20 6 - BOc)y. (&) The numbers unlike : Write the numbers together, with their proper signs, in any convenient order. [th. 1 E.g., 19a;2/2 — 29 mn + 39 a — 49 is irreducible.- So, 10 ay +20by — 30 cy is usually not reduced, but may be written (10ffl + 206 — 30c)y. (c) Some numbers like and some unlike : Unite into one sum each set of like numbers, and write these partial sums, together with the remaining terms, in any order. E.g., (a» + 3 a^ft + 3 ab" + b^) + (a^-3a'c + S ac? - c«) = 2a' + 3aX6-c) + 3a(6'' + c2) + (&'-c'). So, Bxy+lxy+^^xy-lbx^+lo?+%''+^x'-^f n b b b b ^ a-15b + 2c ^ ^ llm + lOw ^ a-Sc ^ b n b § 10. PROCESS OP SUBTRACTION. PrOB. 2. To StTBTRACT ONE NDMBEK FROM ANOTHER. To the minuend add the opposite of the subtrahend, [th. 1 cr. 2 E.g., 90 ft. up -60 ft. up = 90 ft. up +60 ft. down = 30 ft. up, 60ft.up-90ft.up= 60ft.up+90ft.down= 30ft.down, 90 ft. up -60 ft. down = 90 ft. up +60 ft. up = 150 ft. up ; i.e., +90 -+60 = +30, +60 -+90 = "30, +90--60 = +150. So, [2a' + 3a' {b - c) + 3a{b' + (?) + (6' - c^)] -[a«-3a2c + 3a +3 by +5, +8> -3 by +11, -8< +3by+ll,le.,-8> +3 by "11, -8< -3 by +&,i.e.,-8> "3 by "5. +8 < +10 by +2,i.e.,+8>+10by "2, +8 > -10 by +18, ^ -8 <+10 by +18, i.e.,-8 >+10 by-18, -8 > -10 by +2. E.g., +8- +3= +5 +8- -3 = +ll -8- +3 = -ll -8- -3= -5 +8-+10= -2 +8--10=+18 -8-+10 = -18 -8--10= +2 § 11. PKOCESS OF MULTIPLICATION. PeOB. 3. To MULTIPLY ONE NUMBER BT ANOTHER. (a) A monomial by a monomial : To the product of the numerical coefficients annex the several literal factors, each taken as many times as it is found in both multiplicand and multiplier together. [th. 3 -1- Mark the product { _if the factors are taken in -j sense.. E.g., +9a6-3 X+Ta^c^ =+63 aW, -5 xy^ z-^X^la? yz'^ = "35 x*fz-^, +9a-^V x-la^c-^ ^-QSa-^b^c- -bxyH-' X -lx-^f^=^2>f)X-^^. the same a contrary [th. 2 or. 1 (&) A polynomial by a monomial : Multiply each term of the multiplicand by the multiplier; add the partial products. [th. 4 E.g., (3 V + 7y~^!^— f a'"*^) X "|a;2/-V (c) A polynomial by a polynomial : Multiply each term of the multiplicand by each term of the multiplier; add the partial products. [th. 5 E.g., (d'-ab + b^) X (a + b) = a^-a^b+ ab!' + a^b-ab' + b^ = a» + 6». 54 PKIMAKY OPERATIONS. [II. pr. Note 1. Checks: The work is tested by division, [pr. 4 and sometimes by the principles laid down in [th. 5 cr. 2-7]. Note 2. Arrangement : The work is shortened by arrang- ing the terms of both factors, and of the product, according to the powers of some one letter (called the letter of arrangement) , and by grouping together like partial products. E.g., (a» + 3a2& + 3a&2 + 6S)x(a24.2a6 + &2) [a,let.ofar. is written a^ + ^a^h + Zah^ + W a^ + 2ab +&2 a» + 3 a<6 + 3 a'6^ + 1 d'W + 2 + 6 + 6 + 2 + 1 + 3 + 3 ah* + W = a' + 5 a*b +10 a^W + 10 a^h^+ 5 ab* + b". Note 3. Cross-multiplication: The work is shortened by grouping and adding mentally lUte partial products, and writing their sum only. E.g., in the example of Note 2, the computer says : and writes : Sa^b X a^ is 3a*b, a' x 2a6 is 2a*b, whose sum is 5a*b, 5a*b 3ab^ X a^ is 3a^b\ Sa^ft X 2ab is 6a'6S a' X b' is a't"*, whose sum is 10 a' 6^, " 10a'&' 6» X a^ is a'b^ 3ab^ x 2ab is 6a^b\ 3a'b X 6' is 3a^b^, whose sum is 10a^6^ ' lOa^^ 6' X 2a6 is 2ab*, 3ab'x 6^ is 3ab*, whose sum is 5 ab*, 5 db* W X 6' is 6», 6" and the whole product, as above, is a" + ba*b+ lOa^b^ + lOd'b' + 5ab* + &». So, to multiply 384 by 287, product 110208, the computer says : and writes : 4x7 = 28; 8 2; 8x7 = 56, 58; 4x8 = 32, 90; 9; 3x7 = 21, 30; 8x8 = 64, 94; 4x2 = 8,102; 2 10; 3X8 = 24, 34; 8x2 = 16,50; 5; 3x2 = 6, 11; 11 3. §11.] PBOCBSS OF MXTLTrPLICATION. 65 Note 4. Detached Coefficients : When both multiplicand and multiplier are arranged dy some one letter, i.e., are such that, after their coefllcients are detached, the remaining factors of successive terms will stand in one constant ratio, the work is shortened by the use of these detached coefficients, thus : Take the terms of both multiplicand and multiplier in such order that, when the coefficients are detached, the remaining factors (let- ters of arrangement) of successive terms shall have a constant ratio. Write the coefficients, suppressing the letters of arrangement, with for the coefficient of any term wanting in either series. Multiply the coefficients, and add those partial products that pertain to like terms of the final product. In the final product restore the suppressed factors : in the first term by actual multiplication, and in the other terms by means of the constant ratio. E.g., (a' + 3a^6-t-3a62 + 60x,(a' + 2a& + &2), , wherein the constant ratio of the literal parts is & : a in both fac- tors, gives 1 -f 3 + 3 + 1 l-i-2H- 1 1+3+ 3+ 1 + 2+ 6+ 6+2 + 1+ 3+3 + 1 1 + 5+10 + 10+5 + 1; and the product, when the letters of arrangement are restored, is ' a' + 5a*b + 10a'>V + 10a'b' + 5ab' + b^. Check: 1+3+3 + 1=8, 1 + 2 + 1 = 4, 8X4 = 32 and 1+5 + 10 + 10 + 5 + 1 = 32. So, 16(a^ + 29;^ + 4)x(a!-l) + 4(a^-2a5 + 3)x(«^-3) 12 4 1-2 3 gives 1 -1 2 4 -r -2 -4 1 16 1 -3 1 -2 3 -3 6 -9 1 -2 6 -9 4 16 16 -32 64 -64 4-8 24 "36 4-8 24 -36 20 8 -32 88-100, = 20a!*+ 8a^- 32a^+88a; -100. 66 PKIMAKY OPERATIONS. [II. pr. So, {ax - 3a?x^ + a^a^) X (b + Sa" bx^ + a* bx), [ratio, a^x\ gives 1 "3 1 , 13 1 Checks: Let a^a;* = ±l; then, Y~^ i 1t3 + 1-1±3+1=-5 3-9 3 =1-7+1. 1 -3 1 10-701, =abx—7d'bx'+a^ba?. q.e.d. This method is a familiar one ia Aiithmetic. E.g., 1089x237 = 258,093, or lth + 0h + 8t + 9u 237 2h + 3t + 7u 7623 7th + 6h + 2t + 3u 3267 3tth + 2th + 6h+7t 2178 2hth+ltth + 7th + 8h 258093 2hth + 5tth + 8th + 0h + 9t + 3u The first form is simply a case of detached coefficients, wherein the denominations and the relations of the several numerals are shown bj' their positions with reference to ea'ch other ; as, in the last form, they are shown by words and signs. Note 5. Type-forms : The work is often shortened by the use of certain simple t3'pe-forms, which the reader may prove by actual multiplication and then memorize. He may translate them into words and read them as theorems. They are : 1] (x + a) ■ {x + b) = af+{a + b)x + ab, 2J («+&)• (a -b) = a'- b\ 3] (a + &)= = a^ + 2a& + &^ 4] {a-by = d'-'iab + b\ 6] {a + b + c + -Y = a? + W-\- a contrary E.g., eSa-^b^d' — 35a!*2/~*z* —a-'b-'d-" la c=-y/5, gives (a' -|- b' -f 0^ — 3 abc) : (a -)- b + o) = A^ -f B^ + C^ — AB — AC — BC z=Aa^a?+^h'^y''+b-&an^xy-2a^x^5-3Wy-yJb. Note 7. Symmetry : If the dividend and divisor be both sym- metric as to two or more letters, and if there be no remainder, then the quotient will be a symmetric function of the same let- ters. It is then often sufficient to get a few characteristic terms, and to write the rest therefrom by sj^mmetry. E.g., to divide {a^ +1^ +(?+ 2 ah + 2hc + It ca) by (a+6-fc), wherein both elements arc symmetric functions of a, b, c. Manifestly a is the first terai of the quotient, -•. b and c are also probable terms of the quotient ; and •.• the product (a4-6-|-c)-(a+&4-c) is the given dividend, .-. the division is complete, and a -|- 6 + c is the quotient. 6. § 13.] OPEEATIONS ON FEACTIONS. 65 So {oi?y + asy^+ifz + yz^+ z^x + za?+ 3 xyz) : (x + y + z), a quotient of symmetric functions of x, y, and z, gives xy for one term of the quotient ; . • . yz and zx are also probable terms of the quotient ; and •.■ theproduct(a;+2/+z)(a;2/+2/»+2a;) is thegiven dividend, .•. the division is complete, and xy + yz + zx is the quo- tient sought. Had the last term of the dividend been 4 xyz, or any other number except 3 xyz, there would have been a remainder. The reader must therefore use great caution if he employs " sym- metrj"" in division. He may safply use it as suggestive of the true answer, but hardly ever as conclusive. Note 8. Contraction : When only the first few terms of a quotient are wanted, the work is shortened by omitting all par- tial prodi;icts that do not affect the required terms. E.g., (l+x + a^ + a? + ---) : (1 - 2a;-(-3ar^-4a!=-l-—) to four terms 1111 gives 2 6 8 -3 -9 4 13 4 4, [nt. 4 and the quotient, as far as wanted, is l+3a; + 4a;^ + 4a;=. § 13. OPEEATIOKS ON FEAGTIONS. PeOB. 5. To KEDTTCE A FRACTION TO LOWER TERMS. Divide both terms by any same number that divides them without a remainder; the quotients are the terms of the reduced fraction. ^ 36aW^36^, [th.3cr.7,cr.5nt. 24a°6a; 2 ax wherein the divisor is 12a*& ; and this common operator may be written under the sign = , so that the whole stands in the form Z&a*b^(? ^ 36c" 24c a" bx (12 oil) 2 ax Note. For reduction to lowest terms, see 66 PEIMABY OPERATIONS. [II. pr. PhOB. 6. To REDUCE A FRACTION TO A GIVEN NEW DENOMINA- TOR OR NUMERATOR. Divide the new denominator or numerator by the old, and multiply both terms of the fraction by the quotient. [th. 3 cr. 7, cr. 5 nt. 3a^u E.g., to reduce — -^ to an equivalent fraction whose denom- " 2a^b inator is a" be: •.• a'be : 2a^6 = ^ac, 3a^y _ ^aca^y 2a^b (!«) a'bc 2a^z So to reduce —j- to an equivalent fraction whose numer- o a c ator is Qa^yz: ••• Q3?yz : 2a?z = 3y, 2afz _ Qa?yz 3a^c (3rt 9a^cy Note. By this rule any entire or mixed numjber is reduced to a simple fraction. '^ 1 d So x + 2a-\ — = — I ! X X PrOB. 7. To REDUCE TWO OR MORE FRACTIONS TO A COMMON DENOMINATOR. Over the continued product of the denominators, write the prod- uct of each numerator into all the denominators except its own. [th. 3 cr. 7, cr. 5 nt. E.g. ^, ^, 3(« — ^) _ 35a^i!/ 42a5c Qax{a — h) ■ ■' 2a' a; ' 7 liaa;' liaa;' Uax Note. The fractions may be reduced by Pr. 6 to any common denominator whatever ; but this usually leads to complex frac- tions, which the rule of Pr. 7 avoids when the given fractions are simple. For reduction to lowest common denominator, see III. § 6. 6-9. § 14.] EXAMPLES. '67 PbOB. 8. To ADD FRACTIONS. Beduce the several fractions to a common denominator, and write the sum of the new numerators over the common denomi- nator. Eg. 35e° 3 (a -5) _ 216c° + 6aa;(ffl - 6) 2ax 7 Uax Note. Subtraction is but a case of addition ; add the opposite of the subtrahend. ■^■' 2aaj 7 Uax PkOB. 9. To MULTIPLY FKACTIONS. Write the product of the numerators over the product of the denominators. [th. 3 or. 1,3 ■^■' 2ax 7 Uax Note. Division is but a case of multiplication ; multiply by the reciprocal of the divisor. _g, 3&^' . 3 (g - &) ^8b^y^ 7 ^ 76o° ■^■' 2ax ' 7 2ax 3(a — b) 2ax(ci — b)' § 14. EXAMPLES. §§ 9, 10. PROBS. 1, 2. 1-8. Free from brackets and reduce to simplest form : (a) removing first the inner brackets, and proceeding outwards ; (6) removing first the outer brackets, and proceeding inwards ; (c) freeing together aU terms of a kind, from all the brackets. 1. a-[6-(c-d)]. 2. a — {a + & — [a + & — c — (a — 6 + «)]^ 3. -{(l + 2a! + 9a;2)+[(3 + 2a3-a;2) - (2 + 5a; + 7^2) + (- 3 + 3a; - 2a!^)] ^. 4. i[f(a;-a) + (3/-6)] + i[2(a!-a)+i(a-a;) + 2(2/-6)]. 5. -■J^[(5a-4&4-3c)-(-8ffl + 46-c)] -[(6a-8c)-(a-6 + 9c)]^. 6. i(ax' + bx + c)-l l(ax' - 6a; + c) + J (^aa? + bx- |c) + i{-ax' + 2bx + c)^. 68 PKIMAEY OPERATIONS. [II. 8. 1.25 [1.12a; -.24 (a; -.5)] -^[.21(a; + 1) - .15 (1 - .IBa;)- .12(a;-l- .25a;)J. 9. Add {a-2p)3^ + {q-b)x' + {3c-2r)x + {3p-a)3? -(2(e'-x)-{c-l)x-(b + q)3? -(p- a)a^ -oi^ + 3bx'-lc-2r)x, and arrange the sum to ascending powers of x. 10. Arrange a'+6'+c'+3a26 + 36=c + 3c2a + 3a62 + 36c= + 3ca^+6a6c (a) to ascending powers of a, using vertical bars, (6) to ascending powers of 6, using horizontal bars, (c) to ascending powers of c, using brackets. 11. Addie'+Sxf-x^+a^y + a^z + Sa^y^ + Sic^z^+Sxy^z — 3 xyz^— &x^yz — a?y + y* — y:^—3x^if—3 xy^z — 3xy'— 3xyz^— 3fz+3y^z^—6xyH — a?z + 3fz + :i^ + 3a?yz-3oiPz^+3xy'z + x-^-3y^z^ + y^ + 3xyz', and arrange the sum to descending powers of x, and the coefficients to descending powers of y. 12. Froma'— 4a'62— 8a^6'— 17a6*-126', subtract successively a^-2a*b-3a^lf, 2a*b-4.a?h'^-&a?V, 3a^l^-&d?W-9ab\ and 4.0^^ -8 ab* - 12V>, 13. K s = {a+h+c)x+{a+b+c)y, v ={b+c)x+{2b—c)y, \={c + a)x + {2c — a)y, and vr=(a+b)x-\-(2a—b)y; find the values of (s — ii) + (v— w), (s— v) + (w— u), and (s — w) + (u— v) , and the sum of these three sums. 14. Express by brackets, each preceded hy + ; each, by — ', each beginning with a + term ; taking the terms (a) two together, in their order, (&) three together, with an inner bracket embracing the last two of each triplet : -3c + 4d-2e+3/+2a-5&; -2e + 3/ + 2a-56-3c-4d; 2a-5&-3c-4cZ-2e+3/; a + 6 + c — a — & + c + a — 6 — c — a + 6 — c; abc — abd + abe — acd + ace — bed + bce — bde + ade. §14.] EXAMPLES. 69 15-79. Multiply and divide as shown by the signs ; use the methods given in the problems and notes specified : § 11. PBOB. 3, and note 3. (a-s- 2a^- 3a! + 1). (2a!2- 3a; + 4) . (a?+y^+ z^+ xy + yz—zx)-{x — y + z). (^a^-j-ax — b^) ■ (a^+ bx — a^) ■ {x — a + b) . l{a-l)o^+{a-iy3?+{a-lfx']-[{a+l)x + (a+iy + (a + l)'a;-i]. ^ Use vertical bars to join coefflcients of like powers of x in the product. (0*"+ 3 6"— 2 c") • («-" — 3 &-"+ 2 c-») . 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33. 34. 35. + b ■ ab x'+c +d x+cd ; x-—a -b x+ab ay'+c +d x+cd. + b x + ab • so'— a -b x+ab; of— a -b x+ab af—c -d x+cd. (03 + a) • (a; + &) • (a; + c) ■ (a; + cZ) , at one operation. § 11. PKOB. 3, NOTE 4. (a^ - Say'y'^ + Sxy* - y^)- {x'^ - ia^y^ + GaFy* - 4: xf + f) . {a^-2'a^+l)-(2a?-3x + 4:)-(x + l). (a^ — mx+ 7n?) • {a? + mx + im?) ■ (x^ + m?3? + m*) . Show that a; • (a; + 1) • (a; + 2) ■ (x + 3) + 1 = (a;2 + 3 a; + 1)^ Show that (2/ - 1) • 2/ • (2/ + 1) • (2/ + 2) + 1 = (s/= + 2/ - 1)'- What function must xheofy so that Exs. 26 and 27 shall be precisely the same equation ? (af + by*^- ci^) ■ {afz'^ - b'f^ + cy^) . (2a; + 3) .(33! -4); {Zy -b)-{2y + 1). (a;2 + 3a; + 2).(a;=-3a; + 2); {2 -lf)-{l+2f). {a? + Zx'y + Zxy'' + f)-{a?+2xy + y^\ {20^ -Zx^y + 2i^)-{2a? + Zxf + "1^). (2a;- 5)^ {y + 2f + 3fy; {2 -Zz-Z%' + 27?Y. 13 X 15 ; 35 X 79 ; 234 X 432 ; 135.7 X 12.34. 18^; 37^ 109^; 163^ 725^ 1881^; 70.23^ 70 PBIMAEY OPERATIONS. [H. § 11. PROB. 3, NOTE 6. 36. (a!4-2).(a!+3); (a;+2)-(a;-3); (a;-2)-(a!+3); (a!-2).(a!-3). 37. (y+a)-(y+b); {y-ayjy-h); {y-a)-(y+by,(y+a)iy-b). 38. (x + a + b)-{x + c-\-d); {x — a + b)-{x — c + d). 39. (3a''a^ + 5 6V)-(3aW + 56='c»). 40. {ax' + f)-(ax'-f}; (2a^ + 3y'z)-{2o^-3fz). 41. Xm'zi-n''yi)-(m'zi + n'yi); {2 + ^3)-{2- ^B). 42. (x-a)-(x + a); {x'-a')-{a^+a^); - ; (a!»-a'')-(a)"+a''), 43. (l-a;).(H-a;)-(l+a^)-(l+a^)-(l+a^)-(l+a!2"). 44. ix + 3yy; jx-SyY; (a; ±3)^; {2a^±3fy. 45. (a + 6'=^)=; (a-F+c)'; (oTS ±c^=^)'. 46. (x+y + zY; {2x + 3y~4:zy; (xy + yz + zxy. 47. (a + 6 + c) • ( — a + 6 + c) • (a — 6 + c) • (a + & — c) . 48. {a + 2b-3c-d)-(a-2b + 3c-d). 49. (a + ma; — war*) • (a — ma; + ma^) . 51. (a'' + aa; + a;^) • (a — a:) X (a" — aa; + w^) • (a + a;) . 52. (x"-^ + a;"-22/ + x'-'y^ -{ (- a;?/"-^ + y"-^) -(x — y). 53. (x"-' — x"-^y + af-^y^ ± xy"-' ^ 2/""^) ■ (a; + 2/) . 54. (p+pr+pr^ +pr^ -\ + pf~^) ■ {r — 1) . § 11. PROB. 3, NOTE 6. 55. (m + '0 + «"-F^^ + (w + i) — a; + 2/)". 56. (m — ■y + a; — ?/)^ — (m — -y — a; — 2/)^ 57. (a + 6 +3a + 6-m+M + m+n) • (a + 6 — 3a + 6-m+m + m + n) 2.™ 1 »,i j_»„ J^^^./-^ Lj, a«_i_AZ.™_L»i2 58. (a + & + 3a + 6='.m + w'^ + m + «)-(a+&— 3a+6'^-m+n' + m + w) . 59. ( V2 + V3 + V5) • (- V2 + V3 + V^) • ( V2 - V^ + V^) .(V2+V3-V5)- § 11-] EXAMPLES. 71 § 11. PROB. 3, NOTE 7. 60. (a& +cdy + (bo + ady+ {ca + My = %a?W+ 6 ahcd. . 61. {ax—'byy + {hx-cyy+{cx-ayy = {ay — hxy + {by — cxf + {cy — axy. 62. {-x + y + z){x-y + z){x-\-y-z) = ^a?y - ^a?. 63. (a + 6).(c + d)+(a + &).(c-d) + (a-6).(c + c?) + (a-&).(c-d). 64. {ax + 6?/) . (6a; + ay) + (aa; — by) ■ {bx — ay) . 65. {x + y + zy-{af + f + ^)=.S{x + y).{y+z).{z + x). 66. (a + 6 + c)3 = 2a3 + 3Sa26 + 6a6c. 67. If 2a = a + 6 + c + ..., then (2a)8 = 2a' + Sa^ft + 6 Sa6c, 2a-2a' = 2a' + 2a^&, Sa • %ab= Sa^ft + 3 2a&c. 68. (— a + 6 + c + c«)-(a — & + c + d)-(a + 6 — c + d)-(a + 6 + c-d) = -2a* + 2 2a^6' + 8 abed. 69. (aa; + by -\- cz) • {bx + cy + az) • {ex + ay + bz) = abc{a? + Tf + ^) + (a' + 6' + c') xyz + 3 a6c • xyz + (aft^ + be" + ca^) {xy^ + yz" + za?) + {a'b + b^c + (?a){x'y+y'-z+z^x). ^ Test the result severally by [th. 5 crs. 2-7]. § 11. PKOB. 3, NOTE 8. 70. (l-|a; + fa;=-fa;^ + "-)^ to four terms. 71. (l + .167a!+.014a;2 + .001a;5)2 X (1 - .333a;+ .056a;2 - .006a;'+ ■••)• Carry nothing beyond the third decimal place, and retain nothing beyond the term in a?. § 12. PKOB. 4. 72. 3a=6:a6; -Zax:-a?; mn-^:-m?n\ -r^sri; 2r-Vr^ 73. (a;^ + 2aa; + &):a;; (Ja^^-f ast/'^ + fy-*): -Sa;'^/-'- 74. (2/' + 52/+ 6): (2/ + 2); (15a;'+a;2r'+42/"') = (3a;H 2?/-^) 75. (a™*" — a" 6" + a" 6"* — ft^^") : (a" — 6") . 72 PEIMAKY OPERATIONS. [U. 76. a^ + a + & + c a^ + «6 + ac + ad + bc + bcl + cd X + abed -^ i a^ + abc + abd + acd + bcd § 12. PKOB. 4, NOTE 2. ■■ + a + b x + ab. 77.x:(x + a), (a + x): {b + x), a':{a + xy, a:{l+x), (1 + 2a;):(l — 3a;), 1 : (1 — 2a; + a;^), each to four places of the quotient, then write the complete quotients by annexing the remainders written over the divisors. 78. (l + 3?—8y^+6xy):{l + x—2y), first to ascending powers of X, second to descending powers of y. Test by multiplication. 79. {18xyz + 27^-a? + 8f):(x-Sz~2y). Test severally by all the principles in [th. 5. crs. 2-7]. 80. Show that if aP +px + g be divided by a; — o, the remainder is the value the dividend has when a is substituted for x. 81. So, if ar''+pa;^ + g'a; +r be dividedby « — a. 82. So, if x^ -^-po? + q'3?+ ro; + s be divided by a; — a. § 12. PEOB. 4, NOTES 3, 4. 83-86. Divide by detached coefficients, and by synthetic division : 83. ar'-9a;2 + 26a;-24 : a;-4; 2a?-4:a?-Qx+\l : x-2. 84. {a^ — Zoi?y'^ + xf):{x — Zy):{x + y). 85. (2ar'+10a;^-7a;3-14a;2^11a;-2) : {x'+bx-^) : {a?-x+l). 86. (1 + 2 a?) : (1 + a; + K^) ; (x2°+ y^") : (x^+ f) , each to four terms of the quotient ; write the complete quotients. Make two divisions, the first to ascending powers of x, the other to descending powers of x. 87. {af—2x*+3a^—ia?+5x—6):(x—2), the quotient : (a;- 2), • •• ; write the last quotient and the numerical remainders in a series ; use sj'nthetic division. §11-] EXAMPLES. 73 88-101. Divide as shown by the signs; follow the processes given in the notes exemplified: PKOB. i, NOTE 5. (4a!'- 92/8) : {2(i^ + 3if); §12. 88. (a^-y'):(x'~f); 89. (a'" - b'") : (a + 6) ; {a'"*^ + B^-'+i) : (a + b). 90. (a« + a«&2 + a<&^ + a'b^ + 6«) : (a* + a^b + a^fe^ + ab^ + 6*) . 91. [{a? + a') ■{sc?-a^)^:l{a^ + ax + a') .(ay'-ax + a^)] . 92. {^[+t-c^): (a + b-c); (d'-b^^) : (a ~b +c). 93. (J+^' + s?): (a, + 2/+«); (a!=_^^ri«):(a,_2/+2,). 94. (a;"™-l) : (a;"*-!); (aj*""- 1) : (a;"- 1). § 12. PEOB. 4, NOTE 6. 55. 9fg:{3f^gi±2uK^y; a+b' : (a+b ±3-x+^y; bymak- iijg suitable substitutions in Ex. 77. Get the quotients as far as the cubes of u^v^ and x + y, then write the complete quotients. 96. a a^ + 2a^ a!2 + 2a^ a; + a* -T-x' + a aj + a^ + b + 4a6 + 6a2& + 4a=& + b + 2ab + 26^ + 6a62 + 6a2 6= + b' + 26» + 4a6' + 6* § 12. PEOB. 4, NOTE 7. 97. (a!=+2a;-13 + 2a;-i + a;-2) : (aj + S+aj-^). 98. 99. (a« + 65 + c'-3a&c): (a + b+c); (a^-b^ + (?+3abc): (a-b + c). la?(y-z) + f{z-x) + ^(x-y)] :[a=^(2/-< + 2/'(«- a;) + 22(a;- -y)]. )• § 12. PEOB. 4, NOTE 8. 100.(l-.2a;+.04a^-.008a!3+"-):(l + .la!+.01a;'+.001»»+ Carry nothing bej'ond the third decimal place, nor beyond a;^. 101. (»s + lla^— 102a;+ 181): (a; — 3.213), with same limita- tion as in Ex. 100. 74 PEIMAEY OPERATIONS. [II. 102. If N be any dividend; Di, Dj, ••• any divisors; Qi, % the quotient and remainder got by dividing n by Di ; Qj, Rj the quotient and remainder got bj- dividing Qi b}- Dj, •••, show that N, = Qi • Di + Rj, = (Q2D2 + Ks) • Dx + Rl = Q2 • D2D1 + 1=2 • Dl + Rl, = (QsI>8 + R3) • D2D1 + R2 • Di + Rl = Q8D3D2D1 + R3 • D2D1 + R2 • Di + Rl, and so on, = Q>.I>nD„_i.--D3D2Di+R„-D„_i.--D2DlH + Rj • Di + Rl ; and if Di = D2 = D3 = ••• , then that N = 0,0," + R„Di"-l + R„.iDi-2 + - + R3D12 + R2D1+K1. 103. By the method of Ex. 102 develop «*+8a^+24a;2+32a!+16 to powers of a!+l ; of «— 1 ; of a; + 3 ; of a^ + a!+l ; also in the form aw -|- bk (a; + 1) + cx{x+ 1) {x + 2) + Bx(x+ 1) (x + 2) (x + 8), wherein a, b, c, d are free from x. 104. Express Bse^-lGaf + 2ix-X ^^ ^ ^^^ ^^ fractions '^ ix — 2y whose numerators are free from x. First solution : Develop the numerator to powers of a; — 2 [Ex.102], viz., 3(a;-2)3 + 2(a!-2)2-4(a;-2)+7; then, 3a!°-16a^+24a;-l _ 3(a!-2)° 2(a;-2)^ 4(!g-2) 7 {x-2y ~ {X-2Y + {X-2Y (x-2y "^ (x-2y = -§- + _? i_+_^.Q.E.F. a!-2 (a!-2)2 {x-2y {x-2f Second solution : Divide both numerator and denominator by a; — 2 three times in succession ; then, 3a!^-16a^+24a!-1 ^ 3a!^-10a! + 4 7 (a;- 2)* (x~2y (x-2y 3a!-4 4 , 7 (a; -2)2 {x-2y (a; -2)* = -A_ + _2 ^_ 7 . a;_2^(a;-2)2 (a; - 2)« (a; - 2^ § 14-] EXAMPLES. 75 105. Express -~^ — „,^"^ as a sum of fractions whose numerators are free from x. 106. Express , „T, as a sum of entire terms, (a; — 3)^ and of fractions whose numerators are free from x. 107. Express ^ , — - as a sum of entire terms a;(a;+l)(a; + 2) and of fractions whosemumerators are free from x, and whose denominators are a;, x{x+l), x(x+l)(x+2); either by first developing the numerator as to 05+ 2, (a; + 1) (as + 2) , a; (a; + 1) (a; + 2) , or by dividing both numerator and denominator successively by x + 2, a! + l, X. 108. Express — — — - as a sum of entire terms ^ a;(a!-l)(a!-2) and of fractions whose numerators are free from x, and whose denominators are x, x{x—l), x(x—l)(x—2). § 13. PROB. 5. 109-112. Eeduce to lower terms the fractions : a^+3a;+2. a^— 3a;+2. a^— 2a;— 15. acx'+(ad-bc)x—bd 109. 110. a^+ix+S' x^-ix+3' a^+2a;-35' a'aF-b'' a*-b*' a'±2ab + b^' 4«2±12a; + 9' 111 ic^-(3y-izy (4a;^+3a; + 2)'-(2g^ + 3a; + 4)^ {2x + 3yy-16z^' (Sse' + x-iy-ix'-x-sy 112, m? — n^ p* — q^ i^ — ^ , x'" — y"" . x'"—i m*-n'^' p^-^' r»-s=' a^^-f"' ¥^^ § 13. PROB. 6. 113. Eeduce to equivalent fractions, with the common numerator a* — b*, the fractions : a-b a±b a?-b\ a^+b\ g^+a^b+a^+W , a^-g'b+ab'-b' a+b' a-b' a^+ft^' a'-&'' ffl^-a^B+aft^-ft'' a^+al'b+ab'+W 76 PEIMAEY OPERATIONS. [H. 114. Reduce to equivalent fractions with the common denomi- nator a^ + a a? + ab X + dbc, the fractions : + b + ac + c + bc a?-h x + bc a? — a x-\-ac a^ — a x + ab — c — c -b x+a ' x+b ' x + c x—c x—b x — a x^ + a x + ab' a? + a a; + ac ' x' + h x-\-bc + b + c + c § 13. PROB. 7. 115. Reduce to a common denominator the sets of fractions : 12 3. X X a? a^ ax xy yz zx^ 1— a^ (1— a;)^' a' + ax x' — ax a^ — x^^ a 3a 2ax 2 3 2a; — 3 . a — x a + x a? — a?^ x 2a; — 1 4a;^ — l' a? — be W — ca rX 1+a; 1— a;' 1 +a; 1— a; 116. 1+— i— + ^ 1+a! 1 —a; 1-. 117 c^+& a—b a-\- a — b a + x a — x'a — x a + x' a + b o? — \? a^ + i 118. 119. 120. + + (a — 6) (a — c) (6 — c) (6 — a) (c — a){c — 6) ^T (a—b) -{a- c) (6 — c) • (6 — a) {c — a)-(c — b) 3?" a?" af aP 3^—1 a?' + l a;"— 1 a;'' + l 121 y'«' I (y'-y)-(g^-5^) , (y'-c^).(^''-c') § 14.] EXAMPLES. 77 § 13. PEOB. 9. 122-124. Multiply and divide, as shown by the signs, the fol- lowing fractions. Talie care to keep every fraction in its lowest terms, and to cancel where possible : 122. -±z^:t^. _^, fi+n , L_^\ . fi_lV x~y a — o x — y \ xj \ xj \ xl x*-b* . x' + lx oi?-Wa? a!*-26a^+6V 123 124 a?-'2hx + l^ x-b a^ + b^ ' oi?-hx + W a? — a? a?~a? a — x a? — ax + a? a?+ iax + a? a* + 33' o? + a? a + x a? + ax + x^ a^—2ax + x'' 125. Show that a:6:c = a:6:c = a:&XC. [th. 3 cr. 8 126. Show that a:b:c:d = a:h:c : d ■.a:b:cxd=:a:bxcxd. 127. Show that ^-^ d d' d" d-d':d" d-d'-n" = n : TiPTdP : d : WTrJ=n : d : n< : d' : dJ^TriP. 128. Eemove the bars and reduce 2 a : 3 a^ • 4 a" : 6 a* • a" : a^ ■ a. 129. Eemove the brackets and reduce to lowest terms : 130. Reduce the following complex fractions to simple fractions, (a) by performing the operations indicated, (6) by multiplying both numerator and denominator by a suitable multiplier : x-y a'-V m'+mn + n" p^ + q" '" 1+xy a'+b' m° + w° p' + q^ _ x(x-y) ' a+b ' m'-n^ ' p'-(f ' 1+xy a — b m?—mn + r? p^ — q^ 1 1 _ 1 _ m—n m+n fUT' i_l ' l+ l' m — n m+n T+x l—x 1+1 m + n m — n X 78 PKIMARY OPERATIONS. [II. 131-136. As an exercise on fractions, prove the theorems of proportion [ths. 6-9] and their corollaries. 131. If - = -, thenwiUad = 6c; if f = -, then. wiU ac = 6^ b d be 132. If ad = 6c, then will - = - ; if ac = b\ then will ? = - ■ b d be ^o+''— , [II. th.3 .-. N • n' = h. c. msr. X 1. c. mlt. Q. e. d. [II. ax. 1 CoE. 10. Every common { jf jp ^f ^'^° "'" w^ore numbers , measure ^,7 • i Ji.c.msr. ''"< multiple ''■^^^''''-U.c.mU. APPLICATION TO FRACTIONS. CoE. 11. If the terms of a simple fraction be prime to each other, the fraction cannot be reduced to an equivalent simple frac- tion in lower terms. For, let — be a fraction, wherein a, b, ■•■, g, h, ••■ are G'-H*-" p all different primes, and let - be any equivalent simple fraction ; then .- — , "" =-) [hyp- GO-n" ■■• Q .-. a'-b'"- X Q = G'-H* ••• XP, [II. ax.4 whose two members, being the same number, can be factored in only one waj' ; [th. .-. among the factors of ■{ ^ are \ ■*,' ^j' '"' .-. ^^ is a multiple Of ^^;^I;;;'^ I* a" ' B^ • • • and - is not in lower terms than q.e.d. Q G^-H*-" CoE. 12. If a fraction be in its lowest terms, so is every integral power of it) and conversely. CoE. 13. A fraction can be resolved into but one set offadors and divisors, a", b', •••, wherein a, b, ••• are primes, all different, and a, b, ••• are integers, some of them negative. Note. By aid of Cor. 13, Cors. 5-10 are extended and applied to fractions as well as entire numbers. 6. §4.] PimiE AND COMPOSITE NUMBEES.— FACTOES. 89 CoE. 14. The -j 7 ' " 7x ■ of two fractions is a fraction whose numerator is the ■{ , " " , " of their numerators, and wlioSe de- nominator is the ■{ y ' 'of their denominators. ' ft. c. msr. •' For, let -, = a'-b'"-, — ,, = a»'-b'' •••, be any two fractions, D D wherein a, b, ••• are primes, and the exponents a, 6, •••, a', 6', •••, are integers, some of them negative or zero ; ^""^ ^^* ^ Si & "• ^^ *^^ ^ greatest exponents in the pairs of exponents a,a\ b,b', ■•■; .T_ .,. I h. c. msr. , ^i ^ • ■ i both then •-■ the ■{ ■, ,. has every measure that is m ^ .,, of the fractions, and has no others, [§ 1 dfs. ., I h. c. msr. » n n' . ,, ■, , , a''i-b*i ••■, •■• ^^^ ^ 1. c. mlt. °^ b' d" '^ *^ l""""^""* ^ A"..B^ ...; wherein those factors, which have negative exponents make up the denominator of the ■{ ■.' ' ,, ' sought, q.b.d. So for three or more fractions. PKDIE AND COMPOSITE MEASTJKES. Cob. 15. The entire number a" • b' • • • has (a + 1) • (b + 1) • • • different entire measures, prime and composite (and their oppo- sites), whose sum is [(a"+^-1) : (a-1)] • [(b'+^-I) : (b-1)] •••. For •.• A" has (a+l) measures, a", a'-^ a''-^ •••, a^ a', I, and •.■ b' has (6+1) measures, and so on, and •.- the several products got by multiplying the (a + l) measures in turn by the (6+1) measures, and so on, are all different one from another, and •.■ there are (a+l)-(6+l) ••• of these products, all told, -•. there are (a+l)-(6+l) ••• different measures, q.e.d. And •.• the sum of the measures is the sum of all the different products of the measures a°, a°"S •••, a}, 1, by the measures b', b'"S .••, bS 1, by •••, .-, thesum = (A» + - + l)-(B»+"- + l)- [H. th.5 = [(A»+i-l): (A-l)]-C(B'+^-l):(B-l)]-. [II. 6 90 JIEASUKES, MULTIPLES, AND FACTORS. [III. pr. § 5. PEOCESS OF FINBING THE HIGHEST COMMON MEASURE. PkOB. 1. To FIND THE HIGHEST COMMON MEASURE OP TWO OR MOKE NUMBERS, v (a) Tlie prime factors of all the numbers known : Multiply together all the different prime factors, each with the least exponent it has in any one of the numbers, [th. 5 cr. 5, cr. 13 nt. H.g., of 9a'&'c, Sabred, and 1 Ci aV c"— 12 ab*, the common prime factors are 3, a, b'; the h.c.msr. is S-a-b". So ofl^a;?/"', ^a^y, 2x^y~^{x + y) the h.c.msr. is Jaiy"*. (6) The prime factors not known; two entire numbers: Divide the higher number (the larger if a numeral, and that of higher degree if literal) by the lower; the divisor by the remainder, if any ; that divisor by the second remainder, and so on, till notli- ing remains. J. , , suppress from any divisor, .. „ , ., . At pleasure, { i^jf^auce into anUMdend,''''y ^^^ire factor that . ., I dividend . ,. IS pnme to the ■{ j- ■ corresponding. At pleasure, suppress from any divisor and the corresponding dividend, any common measure of them; but reserve it as a factor of the final result. The last divisor, as above, multiplied by the rese>'ved factors^ if any, is the h. c. msr. sought. Let A and b be any two numbers, a the higher, q the quotient of A by B ; Ri, E2, E3, •", R„_i, R„, the successive remainders, whereof r„ is a measure of e„_i ; then is r„ the h. c. msr. sought. 1. If no factors be introduced or suppressed. For but Rl = A-QB, [I. §9df. whatever common measures a and b have, the same measures has Rj ; [th. 2 or. 4 A = Ri + QB, whatever common measures b and Rj have, the same measures has a, [th. 2 cr. 3 whatever common measures b and Ri have, the same common measures, and no others, have A and b ; 1- §5.] FINDING THE HIGHEST COMMON MBASUEE. 91 SO whatever common measures Kj and K2 have, the same and no others have b and Kj, .". the same and no others have a and b, and so on ; so whatever common measures e„_i and r„ have, the same and no others have r„_2 and e„_i, the same and no others have e„_3 and k„_2, and so on, .•. the same and no others have a and b ; but ••• E„ is the h. c. msr. of h„_i and e„, [byp. .'. E„ is the h. c. msr. of A and B. q.e.d. 2. If factors not common be introduced or suppressed. For • . ■ the h. c. msr. of the given polynomials is that of any two successive remainders of the series, [1 and -.• the h. c. msr. of these remainders is not changed when either of them is modified by the introduction or suppression of a factor prime to the other ; [th. 5 cr. 8 .-. the h. e.msr. of these two modified remainders is the h. c.msr. sought. So for any two remainders subsequent thereto. So for the modified e„_i and e„. q.e.d. 3. If a common factor he suppressed and reserved. For • . • the h. c. msr. of the given polynomials is that of any two successive remainders of the series, [1 and • . • when both of these remainders are modified bj' the sup- pression of a factor common to them, their h. c. msr. is divided by the same factor ; [th. 5 cr. 7 . • . the product of the h. c. msr. of these two modified remain- ders by the, suppressed factor is the h. c.msr. sought. So for anj' two remainders subsequent thereto. So for the modified e„_i and e„. q.e.d. E.g., tofind theh.c.msr. of a^+a;— 12 and a?—\Qx + 2\. x'+x-U a?-10x + '2\ x^+ gj— 12 or [-lla;-f33 (-ll a^— 3x I X— 3|a!-|-4 4a;-12 I and a; — 3 is the h. c. msr. sought. 1 -12 1 -10 21 1 1 -12 -] 1)-11 33 3 1 -3 4 -12 Q.E.r. 92 MEASUEES, MUXTIPLES, AND FACTORS. [III. pr. So to find the h. c. msr. of 4 aa;^ + 4 ao; — 48 a and 4ax^ — 40 aa; + 84a: ••• 4a is a common factor, and •.• of the remaining factors, a^+ w — 12 and oe'—lOx + 21, the h. c. msr. is a; — 3, [above •.' 4a(a;— 3), =4aa;— 12a, is the h. c. msr. sought, q.b.p. (c) The prime factors not known; three or more entire numbers : Find the h. c. msr. of any two of them (preferably the two lowest), then the h. c. msr. of this measure and the next number, and so on till all are used; the h. c. msr. last found is tJie h. c. msr. sought. [th. 5 cr. 6 E.g., to find the h.c. msr. ofa!2+a!-12, a^- 10a; + 21, and a?-6a?-19x + 84:: ••• ofK^+a;— 12 and ar*— 10a! + 21 the h. c.msr. is a;— 3, and •.• a; — 3 measures a;' — 6 a^ — 19 a; + 84, .-. a; — 3 is the h. c. msr. sought. q.e.f. (d) Some or all of the numbers fractions : Divide the h. c. msr. of the entire numbers and the numerators by the I. c. mlt. of the denominators. [th. 5 cr. 14, pr. 2 rr * AA^i. 1. „»^ + a!— 12 ,a;2— 10a; + 21 E.g., to find the h. c. msr. of — ■ and ■ : ^ x—b x+b •.• the h.c. msr. of the numerators is a; — 3, [above and •.• the 1. emit, of the denominators is a?— 2b, [inspection J. 3 is the h. c. msr. sought. q.e.p. 3? -lb Note 1 . In the process of case (6) each of the remainders Ki, Rj, ••• is the sum of a multiple of the first number and a multiple of the second number. Note 2. The arrangement of terms may be as to the ascending powers of some letter, or as to the descending powers, at pleasure. E.g., 2»'+lla;2+20a;+21 and a?—x-&, or 21 +20a!+llar'+2a;' and & +x — ci?. That arrangement is commonly best which makes the trial divisor smallest ; and at any step of the work the highest or lowest term of the divisor may be used as trial divisor at pleasure. 1. § 5.] FINDING THE HIGHEST COMMON MEASUEE. 93 The work is often shortened bj' using detached coefficients, and sometimes by synthetic division. It is also shortened by arrangement in columns and by not writing down quotients and products, but only remainders. E.g., toflndtheh.c.msr.of2a;Ha;2— 4a;— 3and2a^— 5a;+3: or and 2 2 -5 4) 1 -4 -3 2 -5 5 3 6 -7 6- -15 9 8- -12 2 -3 2 -3 3 or 2 1 - -4 -3 6- -7 4) 8- -12 2 -3 2-5 3 -2 and So -2 3 2a; — 3 is the h.c. msr. sought. q. e. f. theh.c.msr. ofa!'+3a;2+5a3 + 3 and a^+ea^+dx+i is a;+l : 1 a?+ 30^+ 5a;-|-3 9a!°+12a; + 3 - 21a?- 280^- 7x 22^-\-22oi {22a? x+1 3-7a; 13 5 3 1 -6 -7 22 22 3a;+l 16 9 4 a;*+6a;^+9a! + 4 a;^+ 3a:'+5a; + 3 3a;2+4a; + l 3a;^+3a; x+l Q.E. F. So by synthetic division, to find the h. c. msr. of a^+3a»6+5a=6''+5a6'+26*and2a8+5a'6+4a6«+&=; 13 5 5 2 -4 15 -10 12 7) 7_ 14 -2 -2 -1 a^ + 2 a6 + 6^ is the h. c. msr. sought. q. b. f. 94 MEASXTBES, MTJLTIPLBS, AND FACT0B8. [III. pr. So to find the h. cmsr. of the numerals 679, 301 : 679 or 679 602 2 301 301 2 77 3 231 77 3 70 1 70 70 1 7 10 70 7 10 and Q.E.r. then the h. c. msr. sought is 7. Note 3. K either of the two numbers be a product of known factors, or if both of them be such products, the work in (6) is shortened as foUows : Let A and b be any two numbers, and let a = Ai • Aj ••■ a„ and b = Bi ■ B2 ••• b„, wherein Ai, Aj,* ••• are prime to each other, and so are Bj, B2, •••, but Ai, A2, ■•• Bj, Bg, ••• are not necessarily primes or powers of primes ; every factor of the h. c. msr. of a and b which is a prime or a power of a prime, can be a factor of but one term of the series Aj, A2, •••, and so of the series Bj, Bj, •••, the h. c. msr. of a and b is the product of the m • n h. c. msrs. of the pairs of numbers Ai, Bi .••, formed by combining each of the m numbers in the series Ai, Aj, ... A„ with each of the n numbers in the series Bi, B2, ••• b„, these measures for the most part are detected by simple inspection, if A = Ai • A2 • A3 = (a^" - 6^) • (a!^ - 2/2) . (a^ft^- a^ (/") , B = Bi.B2.B3 = (a' + 6').(a^ + 2/').(a& + a!2/), Ai and Bi contain only a and &, A2 and B2 only a? and 2/, A3 and Bj onlj' factors with all four letters a, b, x, and y, Ai, A2, and A3 are prime to each other, and so are Bx, B2, and B3. [ax. 5 Ai is also, for the same reason, prime to B2 and Bs, a^ to Bi and B3, and A3 to Bj and Bj, h. 0. msr. A, B = h. c. msr. (aj, Bi) x h. c. msr. (A2, B2) X h. c. msr. (A3, B3) = {a + b)x{x+y)x(ab + a!y). q.e.f. and E.g and then and And 2. §6.] FINDING THE LOWEST COMMON MULTIPLE. 95 § 6. PEOOESS OF FINDING THE LOWEST COMMON MULTIPLE. PkOB. 2. To FIND THE LOWEST COMMON MULTIPLE OF TWO OR MOKE NUMBERS. (a) The prime factors of all tJie numbers known : Multiply together all the different prime factors, each with the great- est exponent it has in any one of the numbers, [th. 5 cr. 5, cr. 13 nt. E.g., of 9ab^o-\ 12a^b^d*, and 15a* + 21d'bd, the different prime factors, to their highest powers, are 2^ 3^ a^ &', c°, d\ 5a^+7bd, .-. the 1. emit, is 36-a^-b^-d*-{5a^ + 7bd), = 180 a* lfd^ + 252 a^b*d'. q.e.f. So to find the 1. c. mlt. of a^ - b\ a^ - 6', and a* - &* : •.• a^ -b^ = {a - b)- {a + b) , a^-ly^ = la-b)-(a' + ab + b^), and a*-b* = (a-b)-(a + b)-{a' + b^); -■. thej.c. mlt. sought is (o-6)-(a + 6).(a2 + a6 + 60-(a' + 6')- Q-e-f. (6) The prime factors not known; two entire numbers: Divide the product of the two numbers by their h.c.msr.; the quotient is the I. c. mlt. sought. [th. 5 cr. 9 Or, divide either number by their h.c.msr. and multiply the quotient by the other number. E.g., to find the 1. c. mlt. of a^ + £» -12 and a^ - 10a; + 21 : •.• their h. c. msr. is a; — 3, [pr. 1 (6) ex. .-. (a;2 + a!-12)-(a=-10a! + 21): (a!-3), = (»= + a;-12)-(a;-7), =a^ -6x^ -ldx + 8i, is the 1. 0. mlt. sought. q. e. f. (c) The prime factors not known ; three or more entire numbers : Find the I. emit, of any two of the numbers (preferably the two highest) ; then the I. c. mlt. of this multiple and the next number, and so on, till all the numbers are used; the I. emit, last found is the I. c. mlt. sought. [th. 5 cr. 6 E.g., to find the 1. c. mlt. of 289, 823, 361 : The 1. c. mlt. of 289 and 323 is 5491, [(a) and the l.c.mlt. of 5491 and 361 is 104329. q.e.f. [(a) 96 MEAStTEES, MULTIPLES, AND FACTORS. [IH. pr. Note. The solutions of Pr. 1 (a) and Pr. 2 (a) extend to the h. c. msr. and 1. c. mlt. of anj' numbers that are resolved into factors prime to one another, whether into prime factors or not. j;.5r., •.• 3a2-10a + 3, 6b^ + 7b — 20, m' + w' are all prime to one another, [inspection of (3a2-10a + 3)'. , (662 + 76_20)2. (m' + «») and(3a^-10a + 3)2. .(662+76-20)=. (m' + «=)-' the h. c. msr. is (3a^-10a + 3)= ■ (66^ + 76-20)2 .{m'> + nY\ and the 1. c. mlt. is {Sa' -10a + By • (662 + 76-20)" • (m' + »i«). § 7. PROCESS OF FACTOEINQ. PrOB. 3. To FACTOR AN ENTIRE NUMBER. IN GENERAL. Take out all monomial factors by inspection; by inspection also, or by trial, find an entire measure of the remaining factor; then of this measure, and of its co-factor; and so on, till no composite factor remains. Write the prime factors in order, and mark each one of them with that exponent which shows how many times it has been used. IN PARTICULAR. (a) TJie number an integer : Divide the number, and the successive quotients in order, by the primes 2, 3, 5, •••, usi7ig each divisor as many times as it measures the successive dividends. TJie successful divisors, and the last undivided dividend, are the prime factors sought. Note. No divisor larger than the square root of the dividend need be tried. For •.■ dividend = divisor x quotient, [I. § 9 df . .•.if divisor > -^dividend, then quotient •< -^dividend ; [II. ax. 18 i.e., if there be a factor larger than -^dividend, there is also a factor smaller than -^dividend, which is impossible, since all factors smaller than -^dividend have already been tried, and have failed. 3- § 7.] PROCESS OF FACTOEING. 97 Hence every composite number has some factor not larger than its own square root ; and if a number have no such factor then it is known to be prime. E.g., of 11908 710, 2 is a successful divisor once, 3 twice, 5 once, 11 once, 23 once, and the square root of the quotient, 523, is smaller than 23 ; .-. the prime factors are 2, 3, 3, 6, 11, 23, 523, and 11908710=2-32-5-11.23-523. (6) T7ie number a polynomial thai can be reduced to some type-form whose factors are known : Seduce the number to the type-form, and write its factors di- rectly, in the form of the factors of the type. E.g., a^+ 2aa; + a^ — 25m^w^ =(a; + a)^ — (5mw)^ = {x-^a-\-5mn).(x + a — 5mn). [11.3,2 (c) The number a polynomial with one letter of arra^igement : Find the h. c. msr. of the coefficients, and divide by it. By trial find a polynomial factor of degree not higher than half the degree of the polynomial. Try no factor unless its first and last coefficients measure the first and last coefficients of the number, respectively. Try no factor unless its value measures that of the polynomial when the letters have convenient integral values given to them. If all the coefficients in the polynomial be positive, try no factor whose first and last coefficients are not both positive. For no integer or simple literal monomial can measure a poly- nomial unless it measures every term of it. q. e. d. And if there be a factor whose degree is higher than half the degree of the polynomial, then its co-factor is of degree lower than half the degree of the polynomial, [11. th. 5 cr. 5 i.e., lower than the degree of the factor tried, and the lower factor, not the higher, is best sought, q.e.d. And •.• the -{ p®* term of the dividend is the \ ^ term of the divisor multiplied by the \ j^^^ term of the quotient, 98 MEASURES, MULTIPLES, AND FACTORS. [131. pr. .*. every entire measure of the dividend has its ■{ , , term first a measure of the ■{ , . term of the dividend, q.e.d. And'.' if there be an entire measure of the polynomial, the co-factor is then entire, [ax. 8 .'. whenever the letters have integral values then the value of the co-factor is an integer, [II. ax. 23 I.e., the value of the factor then measures the value of the polynomial. Q. e. d. The last clause of the rule is based on principles stated later. E.g., tofactor40aa^ + 130aa!2/H-75a2/*: -.• a is a common factor, and 5 the h. c. msr. of 40, 130, and 75, .'. the expression is resolved into the three factors 5, a, 8a^ + 26a^-^-152/^ wherein 1, 2, 4, 8 are the measures of 8, and 1, 3, 5, 15, of 15 ; and '.' all the coefflcients are positive, .-. the possible measures of 8a;^+26fl52/+152/*, on its face, are: x + y, 2x + y, ix+y, 8x + y, x + 3y, 2x + 3y, 4.x + 3y, 8x + 3y, x + 5y, 2x + 5y, 4:X + 5y, 8x + 5y, x + lby, 2x + 'i5y, Ax + loy, 8x + 15y. In 8a^+26 xy+16y^ and in these sixteen possible measures put x=l and y =1 ; then 8a?+26xy+15y^ = 4:9, whose measures are 1 , 7, and 49, and only 4a;-f-32/, = 7, and 2a; + 5?/, = 7, pass this test ; and 4:X + 3y and 2x + 5y are found by actual multipli- cation or division to be the factors sought. So tofactorp, = 7k=— 30a^-f 62a; — 45: The only possible linear factors, on its face, are a;±l, a; ±3, a; ±5, x±9, a; ±15, a; ±45, 7a! ±1, 7a! ±3, 7a! ±5, 7a! ± 9, 7a; ±15, 7a! ±45. In Ta:"— 30a!^+62a!— 45, and in these twenty-four possible factors', put a! = 1 ; 3- §7-] PEOCESS OP FACTORING. 99 then p = — 6, and the only possible factors of it are a;+l,=2; a;-3,=-2; a; + 5,=6; 7a;— 1, =6; 7a! — 5, =2; 7a; — 9, =-2. So put a; = 2 ; then p= 15, and out of the six possible factors above the only ones stUl possible are a; + l,=3; a;-3,=-l; 7a;-9,=5; then ••• of these three possible factors 7a; — 9 succeeds, and gives a;^— 3 a; + 5 for quotient, and the others fail, -•. a;^— 8 a; +5 is prime; and 7 as — 9, a;^ — 3a; '+ 5 are the factors sought. Note. For further discussion of this case see XI. th. 4. (d) The number a polynomial ; severalletters of arrangement : Arrange the number as to the powers of any one of the letters (preferably that one whose powers are most numerous), and unite all terms having any same power of this letter into a com- plex term. Find the h.c. msr. of the coefficients of the different powers of the letter of arrangement, and take it out as a factor of the polynomial; then the co-factor has no prime measures free from this letter. Arrange the polynomial, or the co-factor just found, as to any other letter, and proceed as before, and so on for all the letters; of the co-factor left, the prime factors, if any, will each contain all the letters, and can only be found by trial; but: Try no factor of more than half the degree of this co-factor as to any letter or letters; Try no factor that will not measure this co-factor if any one or more of its letters be made zero. If the polynomial be symmetric as to any of its letters, try no . factor that is not either symmetric as to those letters, or one of a set of possible factors that together are symmetric. So, if the polynomial be partially symmetric as to any letters, (i.e. , if for some interchanges among those letters its value would be unchanged,) try only those factors which, singly or in groups, are likewise either symmetric or partially symmetric. 100 MEASUEES, MULTIPLES, AND FACTORS. [IH. pr. E.g., to factor '2a?+ Q3?y + 'ixf—da?z -{■ xyz ^^y^z — %xz^ -yz--Z^: •.• 2, the coefficient of a;', is prime to 6y— 3«, the coefficient of 0^, [inspection .". there is no entire measure free from x. So •■• the coefficient of a' is prime to that of «^ [inspection .-. there is no entire measure free from z. But-.- the coefficients of j/^, ?/, j/" have a h. c. msr. 2x + z, . • . 2 a; + « is a factor of the polynomial, and the co-factor is x'+Sxy + 2y^—2xz — yz — 3z^, whereof every factor has all the letters, but reduces to a factor of 2y^— yz—3 z^, = y+z ■ 2y—'Sz, when x = 0, afa,eix>r ot3^—2xz—3z-, =x+z-x — Sz, •vvheny = 0, a factor ofx'+3xy+2y', = x+y-x + 2y, when» = 0; and ••• the trinomials a; + y + 2, x + 2y — 3z, and no others, fulfil these conditions, and are found by trial to succeed, .'. the factors of the given polynomial are 2x+z, x+y+z, x + 2y — 3z. Q.E. F. So, to factor a?—2xy + y^—2xz — 2yz + z^: \ •.• —2y — 2z and y^ — 2yz + z^, the coefficients of x and of a^, are prune to each other, .*. there is no entire measure free from x ; so there is no entire measure free from y, and none free from z ; .'. every factor has all the letters, but reduces to a factor of y'^— 2yz + ^, i.e. to ± {y—z), when a!= 0, a factor of a;^— 2 xz + «-, i.e. to ± {x—z) , when y = 0, a factor of a^— 2 xy-\- y^, i.e. to ± {x—y) , when z = 0; and ••• no trinomial fulfils all three conditions, .•. the given polynomial is a prime. Or, •.• the given polynomial is symmetric as to a;, «/, z, .•. the factors, if any, must be symmetric as to x, y, z, either as a set, or singly ; but ••• such a set would consist of at least three factors, 3- § 7.] PEOCESS OF FACTORING. 101 and •.• this polynomial, being of the second degree, can have but two factors, .". the factors are not sj-mmetric as a set. And •.• there can be no single symmetric factor except a; +2/ +«, and -.- a?-'2,xy-^y''-2xz~2yz + z^=^{x + y + zy, .*. there ai'e no factors symmetric singly, . • . the polynomial is a prime. Note 1. The proofs for the rule in case (d) are substantially the same as those given in case (c) . Note 2. The work is often aided by introducing new letters of arrangement, as to which the polynomial is more simple, or is homogeneous. E.g., to factor p, ~&x^if— 2Qv?y*%^ + 2bx^y^z — ix^y^z^ -Sx^yz^+Qz^. Let u = x'y, v = z^, and seek the factors of P, = Qu^ - 2Qu*v + 26u^'>^ — Sw'if — 8uv* + &v'. Try no factors except of the form au+bv, or ctt^+ duv+ sv^, wherein a, b, c, e are measures of 6, and the value of the pro- posed factor is a measure of p when for u and v are put any convenient integers. When u,v = l,l, thenp = l, and a + b and c +d + e, measures of p, each = 1. When u, 'U = l, 2, then p = 223, a prime, and a + 2b and c + 2d + 4e, measures of p, each of them = 223, which is manifestly larger than the other con- ditions permit, or else = 1. When M,'U = 2,1, thenp = 30, and 2a+b and 4c-f-2D-l-E, measures of p, are measures of 30. But •.• no integers a, b, measures of 6, satisfy all these con- ditions, .-. there is no measure of the form am -f B-y. And •.• the onlj' integers c, d, e that satisfy them are 2, —4, 3, and •.• 2u' — 4:UV + 3v' is found on trial to measure p, and the quotient is 3 w' — 4 m^v + 2 «^ ; .-. F = (2u^-4:uv + 3v^)-{3u^-iw'v + 2v^) = {2a^y^-4:a^yz^ + 3z)-{3a?f-4:ay'yz^+2z^). 102 MEASTJBES, MULTIPLES, AND FACTORS. [IIL So to factor r,=x — BX + cx' — Btx? -| , wherein a, b, ••• are positive : Let —y = x, then p becomes x + By + cy^ + d^ -\ , whose factors are often more easilj' found. So tofactorp, =36a^ — 25a;^ + 4: Let M = a^, ■« = 1, then p becomes SGu"— 25uv + 4^", whose factors are 4m — v, du — iv; i.e., 4x^-1, 9x^-4, =(2a;+l)-(2a!-l),(3a!+2)-(3a!-2). .-. p = (2a; + l)-(2a!-l).(3a; + 2).(3a;-2). Note 3. A polynomial may often be resolved into surd or imaginary factors. E.g., x-y ={x^+y^-)-{x^'-y^-). So 2a!-32/=(V2a;+V32/)-(V2a'-V32/) = (V2a;+V32/)-(^2a!+^32/).«/2a!-^32/) So !i? + \ =(a;+V-l)-(a;-V-l)- § 8. EXAMPLES. §§ 5, 6. PKOBS. 1, 2. • •• 12. Find the h. c. msr. and 1. c. mlt. of: 1. x-\,a?-l\x-2,3?-l; Z{o?-a?x),4,{x'+ax),b{a^-a*). 2. l-a?,{\ + xy; l-2a;, l-4a^, ]-8a!', 1-160^, 1-32!b'. 3. a?-ir2,x—^,a?—la?+Qx\ af+ai^+a?+x'+x+l,a?-x+l. 4. A-\-bx + a?, 9 — 2X — 3?, li + lx + a?, 20+x — a?. 5. 529(a^ + x-6), '!82(2sje'+7x + 3), 935(2a!2-3a;-2), 6. na^ + 3na^y — 27ix7f—2ny', ima? + ma^y—2mxy^—Smi^. 7. ai^—pa?+{q—l)x'+px — q, ai*—q3^+(p—l)x'+qx—p. 8. a?+(4ca + b)x'+{3a''+4:ab)x + 3a'b, a?+{2a-h)ot?-{3a?+2ab)x + Zd'b. 9. a?^' + ^'-a?-\, {a-2 + a-^)-{e-2+e-'). 10. x' + y'^+z'+2{xy + yz + zx), {^l^x + ^y + zy-(Jsx + ^yy. 11. ar^ + ix-^ + lx-^ + l, ia,-=-i; -^^f^. ^±^. ar—2xy + y' x — y §8.] EXAMPLES. 103 12. oF+alx- -b\ ■ab — c x^+2a -3& +4c or— %ab + 8ac -12 6c 9^+ 6a6 — 8ac -12&C a5+24a6c 13. 14, 15, 16. 17. 18. 19. 20. a;— 24a6c a;^— 2a -36 +4c 15. Eeduee to lowest terms by means of the h. c.msrs. of their numerators and denominators : _^--6k+5 . 1+3 a; -4 K^- 12 a?*. l+a!2 + 25a;* 7a;^-12a; + 5' 8a!»- a^» -I- ajSa -f- a- 4- 1 . 1 - a;- + aj^" - a^" ' a« + a=a; - aa^ - a^ ' x'°+ a'° ■4a;2-2a; + l' l+3a;-15a^-25a!* a;* — a* . a^+a;' a;-^+lla;-' + 3Q ■9a;-i- a!y~-^ + 2 +a;~-'y . k° + ; 9a;-3 + 53a;-2-9a;-i-18 xy-^ + x-^y x^ +y-^ 20. Reduce to lowest common denominator, by means of the 1. emits, of the given denominators, and add : 1 8 5 7 9 2(a + a;)' 4(a-a;)' &{a?+x')' S,{a?-3?)' 10(a+aa;+a!2)' a^+j/' tj(?—y^ x' + xy+y^ 3? — xy + y^ x — y x + y a? — 2/^' a^+ 2/*' x' +y^ ' a? — y^ ^ x + y x — y 1111 4a^{x + y)' 2x\x' + y')' 4a^(a a + b a — b a? — c + d a + b x — cd a? + a -b x — ab a' + V sP + b + d aP-a + d x — ad a?-b x — bc x' + a + c of — a — c a^ + W x + bd ■y) 2^(x^-f) a' 4- &' a^ — b^ a^ + 6 — c ar'-ft -d x + ac x + a^ x' + a -d y? — a + 6 a'- + x—bc x + bd x^ + G -d x — ad x — ab x — cd 21. 22. § 7. PROS. 3. Factor, or prove to be prime : 30;, 37; 72; 120; 323; 367; 1331; 1683; 8279; 15625. Make a table of the. prime numbers fi-om to 400. 104 MEASURES, MULTIPLES, AND FACTORS. [III. Note. The work is aided by arranging the odd numbers 1, 3, 6, 7, 9, •••, 399 upon paper ruled in squares, and marking off as composite every 3d, beginning with 3^, every 5th, beginning with 6^, etc. The multiples of any prime, p, thus marked off, have a common difference, 2 p, and often lie in convenient diagonal lines. All the multiples of p thus got from one another may be tested by merely testing the highest of them by division. Why are the small primes most frequent ? 23. Use the above table to factor 9991, or to prove it prime. 24. Tabulate the prime factors of the numerals 1, 2, 3, •••, 100. • •• 46. Factor, or prove to be prime : 25. x + V + a + S-x + l+3a; a^+ixy + y' + 5x + 5y + 6. 26. x' + y^ + z^~2xy±2xz:f2yz; ar' + tf — z^ ±2xy. 27. 2an^ + 2a'c' + 2Vc''-a''-b*- I'n'*B-lalike==»*» n ! ^b'* 2 alike, n-2 alike 1,(m 2V' Pn™ r alike, n-r alike j. I . (u _ j-) f 112 PEEMOTATIONS. AND COJIBIKATIONS. [IV. th. § 3. COMBINATIONS. PrOB. 2. To FIND THE SEVERAL COMBINATIONS OF « THINGS, TAKEN 1, 2, 3, •■• AT A TIME. To each of the n things, in turn, annex each of the things that follow it; the results are the couplets. To each of the couplets, in turn, annex each of the things that follow all its elements; the results are the triplets; and so on. E.g., of the four things a, b, c, d the combinations are : single things : a, b, c, d; couplets : ab, ac, ad, be, bd, cd ; triplets : abc, abd, acd, bed. So of 210, the prime factors are : 2, 3, 5, 7 ; the 2d degree factors : 2-3, 2.5, 2-7, 3-5, 3-7, 5-7; the 3d degree factors: 2.3-5, 2.3.7, 2.5-7, 3.5-7. By this process are formed all the possible sets in -which the several things are arranged in their normal order ; viz. : every such couplet possible, and from these .couplets, everv such triplet possible, and so on ; and the sets so formed embrace every pos- sible combination. For if any sets were formed with the order of the letters changed, such sets, though diflferent permutations, would be but the same combinations repeated. [§ 1 Theoe. 3. The number of combinations of n things, all differ- ent, taken i at a time, is ' r ! . (n - r) ! For take any r of the n things, and let them change places; then ••• of these r things, there are r ! permutations, but only one combiuatiou, and so for every set of r things ; -•. there, are r ! times more permutations of n things, taken r at a time, than ■'.here are combinations. I.e., p,n = r!-c,w, 3. §3.] COMBINATIONS. 113 whereia c,n = the number of combinations of n things taken r at a time. But •.• p,m = — — — , (n — r) ! n ! •■• C,n = — — -. Q.E.D. r\-(n — r)\ Note 1. A useful way of writing the formulae is : „ n „ n(n — l) w(n— l)(w^2) w(m — l)-..(n— r+1) 3 ! r ! or this: Cin = -, 0,71 = - • , =Oin. , 1 ' 1 2 '2 nm — In — 2 w — 2 C3W = -. . = Com , ... ' 1 2 3 ' 3 n n — 1 ,n — 2 n — r + 1 n — r + 1 12 3 r r wherein the successive terms of the series are got by multiplying the preceding terms by fractions of the form -, /c -f- 1 whose numerators decrease, and denominators in- crease, by one, at every step. n ^ For •.• each of them = — — — '- — — » rl{n — r)l .-. they are equal to each other. [11. ax. 1 In particular: ,CoW=C„m =1, =P„«naltte- CiM = c„_i»i = n, = p„n „_i rtike- w! 02^ = C„_2n = —77 -T—, = Pn» 2aUke, n-2alik«' 2 ! (n — 2) ! n\ Co n = C„_3 m = —-— -r-r, — Pm '^ S alike, n-3 alike- I {n — o)'. Note. Another and independent proof of Cor. 1 comes from its interpretation, and is as follows : •.• for every set of r things taken out of n things there is left one set of n — r things, and but one. 114 PERMUTATIONS AND COMBINATIONS. [IV. ths. .•. the number of combinations of any n things, when taken r at a time, and when taken n — r at a time, is the same. q.e.d. So if any same n things, whereof r are alike and n — r alike, be permuted in any same n places, (\ iff(*rfiiit then •.• when any two of the permutations are ■{ vi^ ' the r things occupy ■{ . , combination of places, .•. the number of permutations, Pn^^roiike, n-rauke equals the number of combinations, c,?i. q.b. d. Cor. 2. Cr(n + 1) = c^n + Cr_in. For and ,_(?i + l)-w.(w — l)-.(w-r + 2) "rv- 1 ' r\ c,n + c,_iw n .(n- 1) ...{n- '•+1) , 71 .(n- -1).. ■{n—r +2) r\ (r- ■1)! n ...(n- -r- ■f2).(« -r+1) «... (n- r+2) • r T\ r ! n- .(n- ■1) ...(n- r + 2).(w + l) .'. C,(w + l) = C,n + C(,_i)«. Q.E.D. Note. Another and independent proof of Cor. 2 is as follows : Let a, 6, c, •■• fc be any n things all different, and I another ; then ' . • c,m = the number of combinations of the n things a • • • A;, taken r at a time, and •.• c,_ims the number of combinations of the n things a ••■ fc, taken r — 1 at a time, and •.• no combinations of the n +1 things a ••• Z, taken r at a time, can be formed except those of the n things a ■■• A;, taken r at a time, and those of the n things a . • ■ ft, taken r — 1 at a time and followed by the new thing I, .-. Cr(n + 1) = c^n + c,.i7p. This note embodies a new rule for forming the combinations of n things taken r at a time. The reader may state it. It also serves to interpret the formula, and show what property of the combinations the formula expresses. 3. 4. §3.] COMBINATIONS. 115 Theor. 4. If there he n things, all different, p, q, r, ••• he any numbers such that p+q+r+--. = n, then there are — p! • q ! ■ r ! ••• ways in which these n things can be made up into sets, whereof the first set contains p things, the second set q things, the third set r things, and so on. E.g., ten soldiers may be formed into three guards, of 2, 3, and 5 men respectively, in '- — , = 2520, different ways. For let the first p things constitute the first set, the next q things the second set, and so on, and let the n things change places in every way possible, forming, in all, n ! permutations ; then •.' within each set of p things there are p ! permutations, within each set of q things q ! permutations, and so on, and ■ . • each of the p ! permutations combines with each of the q ! permutations, so that each of the double sets gives pl-ql permutations, and so on ; .-. for every way in which the sets are made up there are pl-q[-rl--- permutations, i.e., p„n=i9!-g!-r!"-Cj,,,,r,...w; n ! p'.-ql-rl--- Note. Expressed in the notation of this theorem, CoK. 1. If the number of sets be given, the greatest possible value o/ Cp, q, r, ... n *s when no two of the numbers p, q, r, ••• differ by more than a unit, one from the otMr. For, itp>q + l, then ■.■ pl-q\=p-(p~l)l-ql and (p-l)].(q+l)\ = (q + l)-{p-l)l-ql, .•.p!-g!>(p-l)!-(g+l)! m! . n] ''pl.q\-rl— {p-l)\-{q+l)\-rl- i.e. , c^, J, r, ... « < c,-!, J + 1, ,... «) 116 PEBMUTATIONS AKD COMBINATIONS. [IV. ths. and Cp_ J, ,_...« is not the greatest possible if p exceeds q by more than a unit. So of any other pair of them. ■'• Cp, ,,,_...«. is greatest when etc. q.e. d. In particular : If n be an -{ , -, number, then c,n is greatest when r=-{ J/^ + in CoK. 2. Tliere are '■ r — ways ofmakinq up a !-b!-(p!)»-(q !)"••• '^ "^ ^ ^ n things, all different, into a collection of a sets of p things each, b sets of q things each , and so on; wherein a:p + hq-i = n . E.g., a boat-club of 10 men can be divided into three pair- oars and a four in '■ , =3150, dif- 3!.1!.(2!)''.(4!)' ferent ways. n ' For ••• there are : ways of pl.p\...a timeS'g!-g!"-6 times-" making up n things into sets, whereof the first a sets contain p things each, the next 6 sets contain g things each, ••-, [th. and •.•of these ways, by reason of the permutation of the a sets among themselves, the 6 sets among them- selves, • ", there are al.bl--- for every way in which the collection of a -|- 6 -| sets is made up, ■ = a ! • 6 ! ••• times the number of ways n ipiy-iqiy- in which the collection can be made up ; n I .•. that number is '■ q.e.d. al.bl— (p\)'.{qiy... Theoe. 5. If there be n sets of things, containing p, q, r, ••• things respectively, and if combinations of n things be made up by taking one thing from each set, then the number of such com- binations IS p • q • r • • • . For, let then sets be 01,02,03, "-a,, bx,b2,bi,.:b„ Ci,Ci,Cg,.:C„ • ••, and write the first combination OiftiCi"- ; then while the 61 Cj-" stand fast, substitute Oj, 03, ••• o, in turn for Oi, thus forming p combinations. 6,6. §3.] COMBINATIONS. 117 So in each of these,' in turn, substitute &2, bg, •■■ h, for &i, thus foriping q combinations from one of them, and p • q combinations from all of them. So in each of these, in turn, substitute Cg, c^, ■■■ c„ for Cj, thus forming r combinations from one of them, and p-q-r combinations from all of them. So •", thus forming p. g.r... combinations, q.e.d. CoE. 1. If there be a set of p things, a set of q things, a set of r things, •••, there can be made wp Oip.Cjq-Ctr .•• combina- tions by taking i things from the first set, j things from the second set, k tilings from the third set, and so on. CoE. 2. With the data of Cor. 1 the number of permutations «s (i+j+b + —)! -CiP-Cjq-Ckr"-. Theok. 6. If there be n numbers, all different, and if all pos- sible homogeneous products of the rth degree (combinations with repetition) be made of them, including their rth powers and the products of their 1st, 2d, 3d, ••■ (r — l)th powers combined in all possible luays, so that there shall be r factors in each product, and no more, then the number of such products is c n -c fn+r-n _ P(D+l)-(n+r-l) r ! For, let a, b, c, ••• he n numbers, all different, and in each of these c,_ ^th repeiwonB **■ products let the letters be put in alphabeti- cal order, e.g., aaa---, bdde--- ; and then, while the first letter in each product stands fast, let the second letter be replaced by the letter next after it in the alphabet ; the third letter, by the letter next but one after it in the alphabet ; ... the rth letter, by the letter that is r — 1 steps beyond it in the alphabet, e.g., aaaa--- by abed---, bdde--- by befh--- ; then •■• each of the c,,^th repetitions** products is thus changed into a combination wherein no two elements are alike, and no element is beyond the (n + r — l)th letter of the alphabet, . ■. each product is changed into some one of the c,(w+r^ 1) combinations of r letters, without repetitions, of (n + r— 1) letters; 118 PERMUTATIONS AND COMBINATIONS. [IV. th. and ••• all the combinations so formed are unlike, either in theii first letters or in their second letters or •••, in the same way as are the products from which they were got, .'. to each of the products there corresponds a different one of the combinations of n + r — 1 things taken r at a time without repetitions ; •"• C,,^ii,repeUUons»l > C,(n + r— 1). Again, let the elements of each of the c,(n + r — 1) combina- tions be put in alphabetical order, and then, while the first ele- ment in each combination stands fast, let the second element be replaced by the letter next before it in the alphabet ; the third element, by the letter two places before it in the alphabet, and so on; then ■.• each of the Cr(w-|-r— 1) combinations thus gives a product wherein no element is beyond the nth letter of the alphabet, and no two letters stand in inverse alphabetical order, though some may be repeated, .*. each combination gives one of the c,, ^ja repetiaonj ^^ prod- ucts ; and ".• all the combinations so formed are unlike, .-. Cr{n+r-\) > C,,^threpeHtion.«- -■• C,,^threpeaHoii.« = C,(w + r— 1). Q. E. D. § 4. EXAMPLES. §2. 1. Find the number of permutations of 10 things, all different, taken 3 at a time ; 5 at a time ; 7 at a time ; all together. 2. Find the number of permutations of 10 things, taken all together, when 3 are alike and 7 alike ; when 2 are aUke, 3 alike, and 5 alike. 3. In how many different ways can the letters of the continued product a'6' be written? of a^ftV? of cJf)---(a; + Z) = (a; + a)-(a;-|-a)". n factors ^{x + aY; and •■• 2i(a"-Z) =a +a + a ••• n terms =Cyn-a, ' 22(a."Z) =a^ + a^ + a^--. CaW terms ^c^n-/^, 2r(a ••• = a' + a' + a' ••• c,n terms = c,n • oT, 2n-i(«'" 0=*""^+a''~^H c„_iWterms = c„_iTO.a"-*, 2„(a---Z) = a" once -^ =c„n.a"; .•. (a;+a)'*=Con • a" • a;"+Cim • a ■ of^^+Cjn • a^. a;""?. H l-c^n-a'-as'-'H |-c„_in-a»-i-a!+c„w-a"-a!°, i.e., (x + ft)" = a" + w«af-i ^ ^(^— 1) ^^^^-^ ^ ... ^n(n -l)...(n-r + l) ^ _ , „_ r! Q.E.D. 1. §2.] THE BINOMIAL THEOEEM. 123 Note 1. The theorem is also proved by aid of [IV. th. 2]. For (a; + a)" = {x + a)-{x + a).(x + a) ■■■ n factors = x-x-x- • x + x-x-x---a + + 05 • a; • a • ■ • a; + a; ■ a • a; • • • a; + a'a;-a!-"a; • a-a •x-a • x-a -) \-a-a-a---a + X-X-X- + + a;.a;-a- + x-a-x- + + a ■ a • a; • • • a; • a; = Pn^nalike • «",+ P„W n-lalike ' « • a!""^ + P« « „_2 ^i^e, 2 alike ' 0^ • ^""^ -\ h P„W „_, alike, r alike ' a' X""'' -\ 1- P„W„alike ' «" = a!" +waa!"-^ _^ w (w - 1) ^2^„_g ^ ^ ^„^ ^^ ^^ ^^ z 1 Note 2. The theorem is also proved by induction. 1 . Tlie law is true for the second power. For •.• {x + ay = iif + 2ax + a^, [multiplication .-. (a; + a)"=a;"+naa;"-^H [-«", when,n = 2. 2. If the law be true for the kth power", it is also true for tfiie (k + l)th. For, write (a; + a)»= a;* + Tcaa^-'- + ^ (^ ~ ^) a^x''-''+ ■■■ • Z 1 _^ ]c(]c-i)...(k-h+l) ^»^-»^ . . . ^ „*_ [-^yp_ Multiply both members by a;-f-a ; then (a; + «)*+'= 3!*+^+ A; + 1, ^_^_Mk-l) 2^-1^. 2! + fc + a' *+i Jc{Jc-l)...(7c-h + l) hi k(k-l)-(Jc-h + 2) (h-l)l = .t;*+i + (fc +I)aa;«+i^+ll^a2a!*-i + - 2! + (fe+l)fe(/c-l)-(fc-fe+2) ^,^_,+ , 7i! + ..■+»' 3. T/je tow; is true, whatever the exponent k. For ••• it is true for 7c =2, [1 .•. it is true for k = 3. [2 So for A; = 4, for /c = 5, •■• for A; = w. q.b.d. 124 POWERS AND ROOTS OF POLYNOMIALS. [V. th. Cor. 1 . 7/" X and a be any numbers and n any positive integer, 2] (s-a)°=s''-nax°-^+ '^^'^~-'^^ a^x'-' Tnax^-^ia". Cor. 2. TJie series is finite. For •.• the series is a continued product of finite polynomials, .'. it is itself finite. q.e.d. Note. Anotlier and independent proof of Cor. 2 is as follows : For • . • the several coeflicients form a series ^ n n n — 1 n n—1 n—2 ^'1' i"^~' r 2 ■ 3 '■■■' wherein each term is formed by multiplying the preceding one by a fraction of the form ; [IV. th.3 nt. k + 1 '- and ••• the numerator of this fraction grows less by a unit at each step, and the denominator greater, .'. some term of the series, and all after it, is 0, and the series terminates. CoE. 3. The coefficients of any two terms eqvMly distant from the extremities of the development are identical. CoR. 4. The sum of the coefficients of -(x + a)" is 2'. For, let a; = 1 , a = 1 ; then •.• (a! + a)" = (l + l)" = 2», and •.• (1 + 1)" = 1"+ w • 1 • 1"-!+ ^(^ -^ ) . 1^ . 1-2+ — + 1" s ^- [th- 1 , , n(n—l) , ,1 . . 2" =l + w+ "^"~^^ +- + l- Q-E.D. 2 ! CoE. 5. The sum of the coefficients of (x — a)° is 0. CoE. 6. In the development of (x + a)° the sum of 1st, 3d, 5th, ••• coefficients, and the sum of the 2d, Ath, 6th, ••• coefficients, are equal; and each sum is 2°"^. For •.• the sum of all of them is 2", [or. 4 and •.• sum(lst+3d+-..)-sum(2d+4th+.")=0, [cr.5 .-. sum(lst+3d+"-) = sum(2d+4th+-")=2":2=2"-^ 2. §3.] THE POLYNOMIAL THBOEBM. 125 Note. Cors. 4, 5, 6 may be written in formula, thus : 3] c„n + c„_i% + c„_2n-\ h Can + CiW = 2", 4] c„n — c„_iJi + c„_2n ip CjW ± Ci?i = 0, 6] c„w + c„_2W + c„_4n +... = c„_in+c„_3w+c,„_sw=2'-i. § 3. THE POLYNOMIAL THEOEEM. Theok. 2. i/" a, b, c, ••• 1 6e any m numbers; n a positive integer; p, q, r, ••• z any positive integers (including 0), smcA f^ai p + q + rH +z = n, {Aew; 6] _ n\ n!-0!-0!"- ' n] + w-l)!-l!-0! n ! w - 2) ! • 2 ! • ! ?i-2)!.l!.l!.0! n\ ^a'-'^-b^-(P---P • Sa"-2.&2.c''"-Z'' M - 3) ! • 3 ! • ! • n-3)!.2!-l!.0! n! w-3)!.l!-l!.l !-0! n\ ^a^-'-V-c"--!" — ■%a''-^-V-c'--^ — f Sa"-s.6i.ci-d^e»."?' p\-q\-r\---z\ %a'-¥-c'---V [the general term + • This theorem is but the generalization of the binomial theorem, and is proved in the same way. The reader may review here what is said of symmetry in multiplication [II. pr. 3, nt. 7] . He may also compare [TV. th. 5]. He will observe that he is actually forming the homogeneous products there spoken of. They are, however, of the nth degree here, instead of the fcth degree as there, and there are m numbers instead of n. 126 POWERS AND BOOTS OP POLYNOMIALS. [V. pr. CoK. 1 . Xet a + bx + cx^H- dx=+ • ascending powers ofs.; then will 7] (a + bx + cx^ + dx= +•••)" &e a series arranged to n!-a" 'b n! (n-l)!l! n ! • a" 2^,2 (n-2)!2! n!-a"-^c (n-l)!l! x^+ ^b' (n-3)! 3! n ! • a"-2bc a'+na-^bx+ ^^^'^^ a-^b^ 2 ! + na"-'c (n-2)!l!l! n!-a°-M (n-1)! 1! 2 n{n--lXn-2) 3y 3! x'+- x«+. + n(n-l)a''-^bc + na"-^d and, if p be any positive integer, and r, s, t ••• be any other positive integers, such that • r + 1 • s + 2 • t + • • • = p, the co- wl efficient of sP in' the development is S- ■ a'-b'-c'. r!-s!.t!. Cor. 2. If all the m numbers a, b, c, ••• 1 be positive, the sum of the coefficients of the development o/ (a + b + c + •■•1)" is m" ; if one of them be negative, the sum is (m — 2)" ; if two of them be negative, the sum is (m — 4)°, ••. ; and so on. CoE. 3. The development has as many sums of symmetric terms of the form given above as there are way sin which m positive in- tegers p, q, r, ••• z can be chosen, so that their sum shall be n. ■E.g., if m = 4, and n = 6, the four integers p, q, r, s may be either of the following : 6,0,0,0; 5,1,0,0; 4,2,0,0; 4,1,1,0; 3,3,0,0; 3,2,1,0; ' 3,1,1,1; 2,2,2,0; 2,2,1,1; and there are nine terms in the development. CoE, 4. The development has '■'^~*"™ "~ — i-^ separate terms. n!-(m — 1)! For this is the greatest number of terms possible in any integral polynomial of the mth degree homogeneous and having m letters. E.g., (a -\-bx+cay'y has 6! 41.2! , = 15, separate terms. [IV. th. 6 1- § *•] BOOTS OF POLYNOMIALS. 127 § 4. EOOTS OF POLYNOMIALS. PbOB. 1. To FIND THE Wth ROOT OF A POLYNOMIAL. Arrange the terms of the polynonitial in the order of the powers of some one letter, a perfect power first. If the first term be not a perfect^ power, divide the polynomial by such a monomial as will make it a perfect power, and reserve the root of this monomial as a factor of the result. Take the nth root of the first term. Raise this root to the (n — l)th power and multiply by n. Divide the second term of the polynomial by this product (the trial divisor) and add the quotient to the root first found. liaise the whole root to the nth power and subtract it from the polynomial. Divide the first term of the remainder by tlie trial divisor; add the quotient to the root found;' raise the whole root to the nth power ; subtract from the polynomial ; and so on. Let p = the given polynomial, and a H d + e -] = its nth, root, both arranged hy { ascendh™^ powers of some letter x ; and let A H d = the terms already found ; then •.• p — aH d" = (aH d + eH )" — aH d" = mA"-i • E + terms with ^ \^7^f^ powers , of a;, ^^^S^e^ .-. E = first term of quotient, (p — aH d") : tia""^, and p — A + ••• D +e", = TOA"-^ • P + terms with { ^^^ powers of x. has not the { ,^^ ®f power of ; lowest t'"-.. w. a;mp-A + ...D. So the successive terms of p are exhausted, as new terms of the root are found. q. e. d. Note 1. The work is an effort to retrace the steps taken in getting the power whose root is now sought. It is a process of trial, by progressive steps, lilie division and other inverse opera- tions, and its success is established by raising the root to the required power and comparing it with the given potynomial. [II. § 2, p. 29 128 POWERS AND BOOTS OF POLYNOjnALS. [V. pr. Note 2. Complete Divisor: In square root and cube root certain modifications maj' be introduced into the rule which shorten the work : In square root the trial divisor is double the first term of the root ; and a complete divisor is got by doubling the root already found and adding the new term of the root. When the complete divisor is multiplied by this new term of the root, and the prod- uct is subtracted from the last remainder, the whole root found is thereby squared and subtracted from the polynomial. E.g., a? + 2ah + b^ + '2,ac + 2hc + c^\a + h + c 2a + b \ 2ab + W 2a4-2&+c I 2ac + 26c + c^ In cube root the trial divisor is three times the square of the first term of the root, and the complete divisor is the sum of three times the square of the root already found, three times the product of this root by the new term of the root, and the square of the new term ; and when the complete divisor is multiplied by the new term and subtracted from the last remainder, the whole root found is thereby cubed and subtracted from the polynomial. E.g., \a + b + c 3g' + 3a6 + 6 M Zd'b-\-Zali'+V 3 a' + 6 a6 + 3 6' + 3 ae + 3 6e + c' M 3 a' c + 6 gftc + 3 ac' + 3 ft'-'c + 860"+ c' The reader may deduce like rules for getting the 4th, 5th, ••• roots, by means of the complete divisor, from the formula Note 3. Koots or Roots : For a root whose index is composite, it is generally better to factor the index and take in succession the roots indicated hy such factors. [II. th. 3 cr. 9, nt. E.g. , the 4th root is the square root of the square root ; the Gth root is the cube root of the square root ; the 8th root is the square root of the square root of the square root ; and so on. Note 4. Roots op Fractions : To find the root of a fraction, write the root of the numerator over that of the denominator. 1. § 6.] ABSOLUTE AND RELATIVE EEEOE. 129 §5. ABSOLUTE AND RELATIVE EEEOE. I When a number is given approximately only, the absolute error is the excess of the assumed value above the true value ; and it is ^ P ,. if the assumed value be -{ ? than the true ' negative ' less value. The relative error is the ratio, absolute error : true value. The possible error, whether absolute or relative, is the smallest number than which the actual error Is known not to be larger. JE.g., if of a long decimal a few figures only be given, the last figure written is usually increased by 1 when the first figure dropped is 5 or more ; and the possible error is then only half a unit of the last place written. A number is \ a^fyproximate *° ^fid'^'''^^ ^^^^^ its absolute error is not larger than -j * unit in its nth place towards the right. E.g., if a; ~ .2037 > .0005, then .204 is. approximate to three figures, and .20 is correct to two figures. So, for 100 K, 20.4 is approximate to three figures. The copula =, read approaches, joins numbers which difler by a number very small as to either of them. It is, therefore, used to join an assumed value to the true value of a number when the relative error becomes verj' small. E.g., if a be the true value of a number, x the assumed value, and a the absolute error, then x = a + a, and x= a when a becomes verjr small. 3 3 So, 3a + a^ = 3a, — 1-1=-, when a becomes very small. a a "^ In numerical work the degree qf approximation depends on the relative error. E.g., an inch in the earth's diameter, and a million miles in a star's distance, are alike inappreciable ; but a thousandth of an inch in a microscopic measurement is enormous. In pure mathematics the degree of approximation depends solely upon the time and patience of the computer ; but of num- bers based on measurement the positive relative error is seldom sraaUer than a millionth. 130 POWEES .AND ROOTS OP POLYNOMIALS. [V. th. Theob. Z. If a number he approximate to n significant figures and no more, thepossible relative error > 1 : 10° and ^ 1 : 10""^. For •.• any number •< 10" units of its own nth place, and •«£ 10""^ such units, and •.• poss. abs. err. h: 1 such unit, [hyp. and ■■• poss. rel. err. = poss. abs. err. : true number, [df. .-. poss. rel. err. > 1 : 10" and 9^. 1 : 10""^ q.e.d. [II. ax. 18 CoK. A number whose relative error is not larger than 1 : 10" is approximate to at least n significant figures. For •.• the number < 10" units of the nth place, and •.' its rel. err. ^ 1 : 10", [hyp. .-. its abs. err., = number X rel. err., ■< 1 unit of the nth place ; i.e., the number is approximate to m figures. q.e.d. in J mi. I absolute »., I sum j,. Theoe. 4. The \ ^^^^^.^^ errcr^ofthe \ ^^^^^^ of two or more , 1 equals .-, .<• ^i, • i absolute numbers \ ^J^^^ac^ies ^^' '"""^ °^ ^^'''' ^ relative ^'""^'- For, let a,b, ••• = the true values of two or more numbers, X, 2/, • ■ • = their assumed values, a, /S,--- = their absolute errors ; then: (a) •.• a! + 2/+...=(a + a) + (6+(8)+- = (a + 6 + ...) + (a + ^+...), .-. the abs. err. of sum, x + y -\ , = a + |S -) , i.e., = sum of abs. errs. q.e.d. (6) •.• x-y-={a + a)-(b+l3)- = a-b--- + terms which contain either a and not a, or j3 and not 6, or •••, as a factor; + terms with two or more of the abs. errs, u, yS, . . . as factors, .". abs. err. prod, a; -t/..., =x-y--- — a-b---, = the sum of terms aU having one or more of the abs. errs, a, /8, ••• for factors ; , , a . i8 a- B .•. rel. err. prod, x-y •■• =- + '^ -{ 1 \ -I ; ^ a b a-b i.e., rel. err. prod, x-y ■■■ = - +^+ ••• = sum of rel. errs. Q.E.D. 3-5. §5.] ABSOLUTE AND RELATIVE EEKOE. 131 CoE. 1. If the abs. errs, u, /8, ••• 6e each not larger than i, and if m.,n, ■■■ be any finite numbers, then abs. err. (ms + ny^ ) = ma + n/3H 3^ (+m + +nH )-e. In particular : abs. err. {x — y) = a — fi, 3* 2 e. Cor. 2. Mel. err. mx =: rel. err. x, if la have no error. CoE. 3. The relative error of a quotient approaches the differ- ence of the relative errors of the elements. For •.• divd. = divr. X quot., .•. rel. eipr. divd. = rel. err. divr. + rel. err. quot. ; [th. .-. rel. err. quot. = rel. err. divd. — rel. err. divr. q.e.d. In particular : the relative error of the reciprocal of a number • approaches the opposite of the relative error of the number itself. Theor. 5. The relative error of a positive integral power of a numiber approaches the product of the relative error of the number by the exponent of the power; and that of a root approaches the quotient of the relative error of the number by the root-index. Let X be any approximate number, and n any positive integer ; then will rel. err. x" = n- rel. err. x, and rel. err. -^/x = - ■ rel. err. x. (a) ■.■ af, ={a+ay, =a'^+n ■ a"-^ • a+ ^ ' ^"'~-^^ - a°-^ • a^+--, , „ a;"— a"- a , n-(n—l) /aV , .-. rel. err. a;", = -— , =«• H V; — ^ ■ {-] +•■■, a" «■ 2 ! \aj = n •- = m • rel. err. «. q.e.d. a (6) •.• x^^i^x)", [I. §11, elf. and ■.■ rel. err. a;, = rel. err. (-y/a!)", = w-rel. err. -Y/a;; [(a) .-. rel. err. -T/a; = - ■ rel. err. a;. q.e.d. ^ n Note. Ths. 3-5 enable the computer: (a) to see how far his results can be depended upon as approximate ; (6) to carry each part of his work so far that the final result shall be as approximate as he desires, or as the data, if themselves only approximate, permit, wasting no labor upon needless or unreli- able figures. Eesults correct to the last figure, e.g. for standard tables, are only got by computing with several extra decimals. 132 POWERS AND KOOTS OP POLYNOMIALS. [V. pr. §6. ROOTS OF NUMEEALS. PeOB. 2. To FIND THE JITH ROOT OF A NUMERAL. Separate the numeral into periods of t\ figures ea,ch, both to the left and to the right of the decimal point. Take the nth root of the largest perfect nth power contained in the left-hand period. Subtract this power from the period, and to the remainder annex the next period to form the first dividend. Raise the root-figure to the {n—\)th potver, and multiply byn. Divide the first dividend by this product (the trial divisor), and annex the quotient-figure to the root first found. Raise the whole root to the nth power, subtract from the first two periods, and to the remainder annex the next period for the second dividend. Raise the root found to the (n—l)th power, and multiply by n for a new trial divisor; and so vn. Note 1. Numerals are polynomials, but polynomials in which the terms overlie and hide each other ; and virtually the rule is the same for finding the roots of both. The separation into periods is a matter of convenience only. It comes from this : that the figures of the root of different orders are best got separatelj^, and that> since the nth power of even tens has n O's, therefore the first n figures, counting from the decimal point to the left, are of no avail in getting the tens of the root, and are set aside and reserved tiU wanted in getting the units' figure. So the nth power of even hundreds has 2 n O's, and the first 2n figures, two periods, are set aside and reserved till wanted in getting the tens ; and so on. So, in getting roots of decimal fractions, the nth power of tenths has n decimal figures, and the first n figures, one period, are used in getting the tenths' figure of the root ; the nth power of hundredths has 2n decimal figures, and so on. The same thing appears from this : that the root is easiest got if the denom- inator be a perfect nth power ; and this it is only when it consists of 1 with n O's, or 2n O's, or 3 » O's, and so on ; that is, when the number of decimal figures used is n, or 2n, or 3 n, and so on. 2. § 6.] BOOTS OP NUMERALS. 133 Note 2. Approximation : The root of a numeral may be got to any degree of approximation by reducing it to a fraction whose denominator equals or exceeds the nth power of the denominator sought, and then extracting the root. 6912 , 8/6912 19 + 771 „ S/A A- 1 i-U A 6912 , 8 16912 JE.g., -J216' \m' 25. Find the values, each to five decimal places, of [contraction V185, V912, -^625, -^587, -^729, •) ; then •.• a? = a?±b, whence a^—a^= ±6 and x — a = — a ± b X h a + a ± b_ a + X — . ~~ = a± 2a 2a±--. } a+x 2a±*' a + x and its convergents, if the entire number a be included, are : 2a 2a 2a± — 4a^±6 2a E.g., V2=V(1+1) = 1 + |3.1_ and the convergents are : 2 -\ , 1, li, If, 1^, ..., =1, |, |, g, ^.., whose squares are : 1 £ i? 289 __ =2-1, 2+i, 2-—, 2+—, ' 4' 25' 144' ' 4 25' 144' So V3 = V(4-1) = 2-L-1_^ and the convergents are : 4 _ ..., „ 7 26 97 ' 4' 15' 56' ■"' whose squares are : . 49 676 9409 ' 16' 225' 3136' '"' = 3 + 1, 3 + —, 3+—, 3-|--i-, — . ' 16 225' 3136' So V7=V(4 + 3) = 2 + ^ 8 ^+4^§- and the convergents are : 4 + • • • , 2, 2i, 211, 2H, ..., =2, ^, |, ^,..., 142 CONTLNITED TEACTIONS. [VI. pr. whose squares are : , 121 2500 54289 „„„.9„ 27„,81 ' 16' 361' 7744' ' ' 16' 361' 7744' Or V7==V(9-2)=3-|-2 „ =3-^1 and the convergents are : 6— •••, 3 — ••• „ 16 90 508 _Q 8 45 127 ' ¥' 34' 192''"' -'*' 3' 17' W"' whose squares are : 9,!^^?^,l^^..,=7+2,7+A,7+J„7+ ' ' 9' 289' 2304' ' '9' 289' 2304' ' Note 1. The rule is given in formula ; the reader may trans- late it into words. In general, he will find any such formula translatable both as a theorem and as a rule. The first is a statement of facts and is put in the indicative mood ; the other is a direction, an order, and is put in the imperative mood. Note 2. If _ be small, the errors of the squares of the suc- cessive convergents, and therefore of the convergents themselves, diminish rapidly-. For (a)V=aS~(a2±&) = 6. &o — ,= ■ ^ =a'±b + -—, \ 2a J ia^ 4a^ iar 4,0? So /'i^±M'Y = 16a°±24a<& + 9a^6' ^ 4a2±& / l&a*±Sa?b + b^ =:a=±6q: ^! , Ua*±d,a^b + W ~(d?±h) = ^ =b.(^\\(l±±\ 2. §3.] CONVERSION OP SURDS. 143 So in the above numerical examples, the smaller — is. the more rapidly does the series converge. 6 2 E.g. , the series got by taking -^7 = -,/(9 — 2) , wherein —= -, converges much more rapidly than that got by tak- ing V7 = V(4 + 3) , wherein ^ = |- Zj -1 So v'3 = V(4— 1) gives — =-, and the square of the a" 4: .. fourth convergent differs from 3 by only o-ioa i whereas ^2 = ^(1+1) gives -=-, and the square of the fourth CI X "I convergent differs from 2 by — — • Note 3. Another conversion may be made thus : =F& •.• aff'—a'=±b, whence x-\-a = - — -> 21 .-. a;=:-aq:-^ = -aqF— - — - b •^ a — x a + a± a—x a — x 2a± f, wherein the convergents are the opposites, each of each, of those found \>y the first conversion ; i.e., by the first process the convergents of the positive root were found, and by the second, those of the nega- tive root, equally large but of contrary sense. E.g., V3 = V(4-l) = -2+j^l_ and the convergents are 4 , „ 7 26 97 ~^' ~4' ~15' ~56'""" I^ For other uses of continued fractions see the computation of logarithms [IX. § 3] , and the solution of quadratic equations [XI.''§ 13]. 144 CONTINUED PBACTIONS. [VI. pr. § 4. COMPUTATION OF CONVBEGENTS. / PkOB. 3. To FIND THE CONVERGENTS OF A CONTINUED FRACTION. First Method. For the first convergent reject all after the first partial fraction, for the second convergent reject all after the second partial fraction, and so on; reduce to simple fractions the complex fractions that remain. The examples given above have all been solved by this method. Second Method. Form two series, { " ^"^ ^'' " 1 Di, Dj, Da, ■• wherein { ^^ = ^i' ""'-^'^^ =""'■ ^^' ' Di = di, Ds = dida + nj = Di- da + nj, , Ns = nidjds + niDg = Nj . dj + Nj • nj, • ' D3 = did2d3 + n2ds + ding = D2-d3 + Di-n8, . Q-| J Nt = Nij_i • dit+Ni_2 . nj. . -< ' Dt = Kk-i ■ dk+D^.j • nj, ' Dk, then are Ni Na Nj Nk _Nk_i-dk + Nk_2-nk Di Dj Ds Dk Dk_i-dk+Dk_2-nk the 1st, 2d, 3d, ••• lith, convergents. The reader may translate this formula into words. 1. The law is true for the third convergent. For the first three convergents, by the first method, are % nidj ^^^ nidj-d^ + n^^-n^ di ^1^2 + ^2 ■ di cZa- dj + na" ^3+ C?i' Jlj' _ 2' 3"i" — LU, as above. q. e. d. D2-d3 + Di.W3 2. Jf the law he true for the kf/i convergent, it is true also for the Qi+l)th convergent. For -.-iN-^^a , ^ J=i:|.±2W%, [hyp. an identity, whatever expression or value d^ may stand for, and therefore an identity when d^ is replaced hjd,+^. 8. § 4.] COMPUTATION OP CONVEEGENTS. 145 "« + ! ■di + di + . . I- "■!• _!!-fc^)- (d*-1 • «^s+ Dfe-2 • %) • «^*+l + Dj.! • nj + i _ I?f'^H-l+lTt-l-%+l - „ -- -J j . li.H^.D. 3. The law is trite whatever the index k. For -.• it is true for Jc = 8, [1 .-. it is true for & = 4. [2 So for fc = 5 ; for & = 6 ; for & = 7 ; and so on. Q. e. d. E.g., of the fraction — 5 a;-J c the convergents are : x-\ , ax ax-x + a-c aic^ + ac x' x' + b' {x' + b)-x + x-c' ofi + b asc^ + ac {aa? + ac) -x + ax-d + ad /'a^ + S a;'\ ■ x + {i^ + h)-d x* + b + c + d aj^+6d' So of the fraction — x o■^ — the convergents are : c+ > hx x' + bcx (b + d)x'+bcdx a' x+ab' {a+c)x+abc x'+ {ab+ad+cd)x+abcd^ 146 CONTINXrED FRACTIONS. [VI. pr. In particular, when nj, nj, • • • nj^, each = 1 , then the formula becomes : »* Dj-i • <^* + Di-a' The reader may translate this formiila into words. He may also demonstrate it anew, putting 1 for n-i, n^, ••• n^, n^+\, in the general demonstration. E.g., of the fraction — 1 a-\ 1 6 + — the convergents are : c-\ , 1. &. 6-c + l 6c + 1. a' a6 + l' (a6 + l)-c + a' ahc + a + c^ Compare the result of the previous example, when x = l. Note. Formulae [3, 4] may be made to include convergents of the mixed number n^ + -J— n^ d2-\ , as follows: Let ^N_i = l, No^no, wherein «o = any number, perhaps ; then will I ^i =No-di + N_i.mi, Kg = Nj . dj + No • Mj, ^ Di = Do • di + D_i • ni, D2 = Di • (^2 + Do • n^, and Ni _ Nq . di + K_i • Wi Nj _ Ni • ^2 + Nq • Wg Di Do-di+D-i-nj D2 Di'd2+Do'n3 as the reader may verify ; but it is convenient to speak of — , = Wo+ -^j-^ = ° \ ^, as Dj «1 cti the 1st convergent, even when Wq has a value not ; Ld of -? and -^ as the 0th and "Ist convergents. E.g., of IT, =3- 1 , the convergents after — , =-, are l — 3-7+1 _22. 22.15+3 _333. 333-1+22 355. 1-7+0' 7' 7-15+1' 106' 106-1+7' 113' — ; as in §2. 1- § 5.] GBNEKAL PEOPERTIES. 147 § 5. GENEEAL PEOPEBTIES. In this section, let Vj, Vj, Vg, ••• be the true values to which _ _ Nj, _ Wi Cj) = C2 J, = -— , = -= — - TC2 °* <^i + ;j— L '^s ^ _ N2,s _ »l2 ds + -..% d,' are convergents ; whence Theoe. 1 . If the partial numerators and denominators be all positive, the convergents Ci,i, Ci_2, ■•• are alternately greater and less than the true value Vi. Let «!, Ms, ••• di, c?2, •■■ be all positive ; then will Ci > Ti, C2 < Vi, C3 > Vi, - Cj -i ^ Vi, ^ ^ g^g^_ For '^>:r^^ [II. ax. 18 di di + Y2 i.e., Cj > Vj. Q.B.D. So-.-l^:^-^, .-. ^«,<^^»2_, [n.ax.i8 i.e., C2 < Vj. Q.B.D. ds d, + y, d,+ -^ '^'+A + ^: ... -^ Wo >-T— «2 i.e., C3 > Vi. Q.B.D. ITT I > J ^ odd. r. ^ T. 5] ■•. C, -{^Vi,^^^^g^_ Q.E.D. 148 CONTINUED FEACTIONS. [VI. th. Theoe. 2. If any convergent, Cjj, he subtracted from the next following convergent, Cij^.i, then will the remainder, -* I>k+1 Dk' ^ ' ' Dk-JPk+l „. Nj + i Ni DfNj + i — Nj-Dj + i Since r- = r ' it IS to prove that L'S + I Di D. • D: D«^-N4 + i-Nj-D4+i=(-l)'.ni-n2."Wj4^1. 1 . T/ie tow is true for k = 1 . For •■• Ni = mi, Di = di, N2=nicZ2, D2 = (iid2+m2) .•. Dj-Nj — Ni'D3= di-Wida — ni-('"iid2 + ^2) = — ni-n2 = ( — l)^-Wi-n2. q.e.d. 2. i/" D,j_i . N^ — Nk-i • Dk = ( — 1)''"' •^ni • 112 • • • nt, i/iew wiM Dk-Nk+i — Nk-Dk+i = ( — !)''■ ni- 112 •••rii+i. For Dj • Ni + i-Nj • Dj + i= Dj . (Nj • di + i +Ni_i • n^ + i) = (Nj.i ■ Di — Dj.i • Nj) • Wj + i = — (Dj_i • N4 — Ni_i • Dj.) . Wj + i = -(-l)'"^-%-i2 — %-Ws+i [hyp. = ( — l)'-ni-m2"-n*+i- Q.e.d. 3. r/ie Zaw is true whatever the index k. For •-• it is true for A; = 1, [1, above .•. it is true for A; = 2. [2, above So forA; = 3, forft = 4, forA; = 5, •••. q.e.d. In particular: If nj, n^, ••• each = 1, then will Cor. 1. TAe error of any convergent, —, of -y—, 1 Dk QiH- ;5 — ; wJien di, dj, ••• are ail positive, 'h+ •") is less than , and much more is it less than — — . Dk-DkH-l Dk For ••• the true value lies between — and -^, fth. 1 and ••• these differ by only <-— > i>*-D*+i d/ .*. etc. Q.E.D. 2- § 5.] GENEEAX, PEOPEETIES. 149 Cor. 2. Cv=- — -^^ — -+~ — - — ~ 1-(— 1?"' ' For Wl • tla Til Cj - Ci = - -^ — -, [th. C, — C, = - and Cj-Cj_i = (-1) S-l%-W2- Dj_lDs -' ■ ■ ' Di Di-D2 Da-Da ■" *- ^ ■ Di_i-Dj /«. particular: If nj, Wa, ■•• each = 1, then will 111 (-1)*-^ -" ' Dj Di-D2 Da-Da Dj_i-Ds Note. This formula gives a rule by which any continued frac- tion may be reduced to a series. E.g., V28 = V(25 + 3)=5+A_^^ whose 1st, 2d, ■•• convergents have the denominators 10, 103, 1060, 10909, 112270, •-. ; 3Q2 Q3 Q4 /20 = 0H I - I ■■^ 10 10-103 103-1060 1060-10909 ' a series whose successive terms grow smaller very rapidly. Cor. 3. jy nj, na, --- di, da, --• be all positive, the successive differences grow smaller and smaller. For •.• Ci_i~Ci = — -— andc.~Cj+i= — — — 1 Lcr-J- and Dj-l Dj DjDj+i Dt-t^t + l + Dj-l-Wt+i — Dj-l-nj + l ^ D.-<^.+i^ >1; Dj-l-Wj + i Ci-1 ~ Cj > C' ~ Cj + 1. Q. B. D. [3 150 CONTINUED FBACTIONS. [VI. th. CoK. 4. Tlie successive convergents approach to each other, und therefore to tlie true value which lies between them, rapidly when the ratios di • da : nj, dj • dj : iij, • • • are large. For *^*-'~°* =i + ^il^^±L, [above £'.gr., tie convergents to ^{a^ + h), =a+- — j approach the true value rapidly when — is small. [§ 2 Cor. 5. If ni, nj, ••• each = 1, and di, d2, ••• be all entire, aU the convergents are simple fractions in their lowest terms, and tJieir consecutive ■{ '^^nom^nators "'"^ ^''*™^ *° *"'^'' °*'*®''' For •.• every common measure of Nj, Dj is a measure of NfDj+i~Dj.Ni+i, =1. [ni.th.2cr.4 .•. the h. c. msr. of Nj, Dj is 1. q.e. d. So ofNj, Nt+i, and of Dj, Dj+i. Cor. 6. if nj, Dj, ••• each = 1, and dj, dj, ■•• &e positive, then between two consecutive convergents there lies no fraction whose denominator is smaller than the largest of their denominators. For, let - , i^fc — , = any simple fraction wherein d ■< Dj+i ; then N ^ Nj _ NjDj~DjN* whose numerator, being entire and not 0, [II. ax. 23, hyp. either := or > 1, and whose denominator ^Dj-Dj+i, - [J^JP- ^-^"^ — I — ; [n. ax. 17,18 D Di ' Di • Di * + l i.e., - differs more from -!t than — i±i does, D Dj Dj + i and cannot lie between them. q. e. d. 3> 4- § 5.] GENERAL PEOPEKTIES. 151 Theoe. 3. The difference ^tti _ ^ = (-l)""!- Pk+i-d^+2 Dk + 2 Dk D^-Dk + a For • • ^^^--'^* — °'''^''+^~^*'°*+^ Dj + 2 Dj, r>i-Di+2 whose numerator = d, • (n^+i • (^,+2 + n^ • n^^^) = (Dj-Nj + i — Nj-Dj, + i).d!j + 2 = (-l)'-%-"«4+i-<^»f 2, [th.2 •■• 6tC. Q.E.D. CoK. 1. i/" ni, Hj, ••• awd di, d2, ••• he all positive, then Ci > Cg > C{ > •■•, and C2 < C4 < Ce < ••■. C0E.2. y,^no + ^ + "'"^°'''^^ + °^"'"^^° +- 1 ■ D2 D^ • D4 D4 • Ds Ni miW2d3 mi"-W4d5 Theor. 4. Jn anj/ continued fraction ^ c = (-l)''-"i-"k+i For ••• c.H.i=|+.._%^„ and Vi = ^.. ^,^^ -•. Vi is what Cj^.ibecomes when for (^j+iis put dt+i+v^+s, and ■■•o.,,-o.^ (-^>;-;-----^- Dt'Dj + l ^ (-1)''-Wl---W;i + 1 (-1)'^-'»1 •••'% + ! Vl - Cj = (-1)'^-Wl---W;i + 1 "»»-(Dj + l+D*-Vi + 2) Q.E.D. 152 CONTrNTJBD FKACTIOKS. [VI. pr. 4. Cor. 1. If Di, na, ••• eac/t = 1, and d^+j, di+j, ••• eacA < 1, (ften Vi ~ Cit Zi'es between and »k-Dk + l Dk-(Dk^.i+Dk) n, For •■■ v-.o, =^i±i_ then lies between and 1, .-. Df(Dt+i+Ds-Vi+2)) the denominator of Vi~Cj, lies between Dj-Dj+i and i>s-(Dj+i + Di) ; .-. etc. Q.E.D. Note. So, if of ni, ria, ••• any are negative but all s: 1, and if di, ^25 ••• < 2, then Vj+a lies between — 1 and + 1 inclusive, and Vi ~ Cj between ^ and Di-CDj + l-Wj) D*- (!>* + !+ Dj) Cor. 2. i/" nj, nj, ••• each = 1, and dj, dj, ••• eac/i < 1, every convergent differs less than the previous one from the true value. For -.• D4 + 2 = Ds + l-di + 2 + Dj Dj, ■•■ »*+i-i>4+2>Di-(Dj+i + Ds) ; and •.• c,+i~vi>- — rEr-'°'~'''^r75^ — 3-BT' tcr-i ^k + 1' •Vit+i >i, Ni 1 . 1 .•. X ■^ ;, I.e., < :• Q.E.D. Di Dj • J D • t th.5. §6.] SECONDARY CONVEEGBNTS. 153 § 6. SECONDARY CONVEBGEITTS. If rii, Wj, ••• each = 1, and dj, di, •••he all positive integers, then the series Co, Ci, C2, C3, ••■ may be resolved into two series : CoiC2)C4,"-, all too small, and €1,03, Cj, •••, all too large, '^' c,-c, = -^, -, [th.3 wherein C2 — Co ^ and C3-C1 Put 1,2,3, and 1,2,3, then a series Do • D2 D2 • D4 — ds —ds = — ^, C5-C3 = -, •". Dl-Dg Ds-Dj ■• (Z2-I , in turn, for d^ in ^i-<^2 + ^o^ = Cj, Di • ^2 + Bo •• di—1, in turn, for d^ in C4, and so on ; a series of secondary convergents is formed, lying be- tween the primary convergents Co, C2, C4, ••• ; and the whole series, whose terms are all too small, is : Ni + Kq 2-yi+ Nq 3 • Xi + No (dg— 1)-iTi+No ^ "' Di + Do' 2.Di+Do' 3-Di + Do'""(d2-l)-Di+Do' " >r3 + N2 2 • yr3 + Kg 3 • N3 + N2 ___ (di—iys^+^2 ^ _ D3 + D2' 2-D3+D2' 3-03 + 02'"' (^4— 1)-D3+D2* *' or Co, Co|i, Co|2, ••• Co|a^_i, C2, C211, C212, ••• C2|i^_i, C4, ■••. So, put 1, 2, 3, ••• ^3—1, in turn, for dg in Cj, and 1, 2, 3, ••• d^—l, in turn, for ds in Cj, and so on ; then a second series is formed, whose terms are all too large : N2 + N1 2-N2 + N1 K4+N3 .2-N4+y3 D2+D1 2-D2 + D1 B4 + D3 2-D4+D3 or Ci, Ci|i, Ci|2, ••• Ci|^^_i, C3, C311, C312, ••■ C3|a^_i, C5, •". Theok. 5. The terms of the first series Cq, Co|i, Co 12, ••• grow greater and greater; of the second series Ci, Ci|i, Ci|2, ••• less and less ; and the differences of successive terms, smaller and smaller. Ni + No No 1 r^.% o ^°^ ^»l^-*^» =5H^-Bo = ro.(l., + Do)' ^'"'-^ ^°'--^°'-- (B. + Bo)-'(2B.+Boy ^•^'^''"•^- N2 + N1 Ni -1 So Ciii-Ci = Co — Ci 1 1 = — 1 and so on. 154 CONTINUED ABACTIONS. [VI. tli. Theok. 6. Ck I , ~ Ck+i = , wherein d^ i r = r • Dk+i+Ok- For, reduce C4|r~ c^+i to the common denominator d^i^-Dj; then the numerator, (r-N4+i+Nt)-Dt+i~(r-Dj+i+Dj)-Nij.i, = Nj-Dj + i~Dj-Ni+i, =1. Q.E.D. CoR. 1. Ck|r is in its lowest terms. For •.• every common measure of Nj|,, Dj|, is a measure of Ns|,-D4+i~Dj|,-Nt+i, =1, [III. th.2cr.4,th. .-. the h. c.msr. of Nj|,, Dji^is 1. q.e.d. CoK. 2. Between c^ | , o.nd Vi there lies no simple fraction with terms .so small as those o/ Cj( | ,. For •.• c,,,~c,+i = - -— , .•. between Oji, and c^+i there lies no simple fraction with denominator so small as i)j | , ; [compare th. 2 cr. 5 and ••• Vi lies between Cj|, and Cj+i, .•. between c,,,, and Vi there lies no simple fraction with terms so small as those of Cj I ,. q.e.d. Theok. 7. Ci,|,~Ci,|.= ~ — Dk I r • Ui I s For, reduce c^ | , ~ c^ | , to the common denominator Dj | , • d^ | , ; then the numerator, (Ni+i • r+Nj) ■ (Dj+, ■ S+Dj) ~ (Dj+i • r + Dj) • (Nj+1 • S+Nj)» = (Nj • Dj+i~ D4 • Nj+i) • (r ~ s)=r~s. Q. E. D. [th. 2 Theor. 8. When r > ^djj+g) *^6n 0^1, differs less from Vj i/ian (Zoes any simple fraction with terms as small as those of c^ | ,. For •.• y^^ ^* + (^^H-i+-)-^^+' , ^c,,., [sis 1 ^(a'-^+l): 6. Express -y/c, = ■\/(in^ 4- c — m') , as a continued fraction, and show that whatever the numbers m, m'. m-f ^ , — m^ =±m'± , , c — m 2m-^ 2m'+ 2m-j 2m'+"-. 7. Develop into sei;ies the sixth convergents of [th. 2 cr. 2 3'.14159265, |i|, V145, ^| ; ^(a'+b), ^(a'-b). 156 CONTINUED FRACTIONS. [VI- 8. Show that: 1 Ui Wi «2 "^ fh + i^. di-m + -= % di + ^ -'^1+ ^-^n" ,„ d + ^L. n" d'——r- n'll d"-'±- ''^d" + n d'" -d'"+-". ^ ' 355 9. Write all the primary and secondary convergents of — — in 113 order ; and find the fraction that differs least from it of all those whose denominators 3*- 100. [§ 6, th. 6 cr. 2 N N 355 1 10. Find all the fractions - such that d3«- 50, < D D 113 50.D 11. Prove that { ''' = (% + d,_i • d,) ^ l"-' + n,.,- d,i ^*-»- 12. The continued fraction - 1 occurs in totany, zoology, and astronomy. ^T... How does this fraction differ from ^1-? Find twelve convergents, and prove that : N*= 2 -Nj.s +1 ■Nj.g= — = N,-Nj_,^.2— Nr_2'Ns_, ^ = i(Nr • Ns_, + 3+ N,_3 . Nj_,) , wherein i = — _ . "8 — ... ... ... ... J that .-. K, measures N2,, Na„ ••• ; andthatN, • n,+, — n,+, ■ n, = (— l)'ii,_,- n, if r > i. 13. Convert the series ao+a(,aia!+aoai«2*^+"'± «o""<'fn'B" into the continued fraction, and find its first four convergents, 1 -^r- o,«x Oi^+l — §1.] VABIABLBS AKD CONSTANTS. - CONTINUITY. 167 VII. INCOMMENSURABLES, LIMITS, INFINITESI- MALS, AND DERIVATIVES. §1. VAEIABLBS AND CONSTANTS. — CONTINUITY. When the conditions of an investigation are such that one number takes a series of different values, that number is a vari- able; but a number that keeps its one value unchanged, is a constant; and the particular values that maj^ be given to vari- ables are constants. The remainder got bj- subtracting one value from the next is the increment of the value first named. When one variable is a function of another, the first is a dependent variable, and the other is an independent variable. From the fixed values of the constants and the values that may be given to the independent variables, the corresponding values of the dependent variables, or functions, may be computed. ^.gf., while a sum of money remains at interest, the principal and rate are constants, but the time and accrued interest are variables, of which either may be taken as the independent variable, and the other is dependent upon it ; for when the prin- cipal, rate, and^ i^erest ^^^ S^^®'^' ^^^^ W^^ ^® thereby de- termined ; and to different values given to { j^(.gj,gg^ different , „ I interest -, values of <^ . . correspond. So, the radius, circumference, and area of a circle are all func- tions of each other, and all grow together if the circle increases ; but the ratio of the circumference to the radius is constant, and so is the ratio of the area to the square of eitlier of them. When the variable, in passing from one value to another, passes through every intermediate value in order, then the vari- able is continuous; otherwise it is discontinuous. E.g., time is a continuous quantity, ever increasing by a steady growth, and the time of day, expressed in hours and parts of an hour, is a continuous number ; but if entire hours only be counted and the fractions rejected, the number is discontinuous. 158 INC0JDIENST7EABLES, LIMITS, ETC. [VII. If a function of a continuous variable remain real and finite as long as the variable is real and finite, if it can take but one value, or a limited number of values, for any one value of the variable, and if, in passing from one value to another, it passes through every intermediate value — such a function is a con- tinuous function; otherwise it is discontinuous. It is implied that for any small increment of the variable the increment of the functiou is also small, and that to the variable an increment can always be given so small that the increment of the function shall be smaller than any assigned number. E.g., interest is earned continuously, and may be computed for a 3'ear, a day, a minute, a second, a millionth part of a second, or any other fraction of a second, however small ; interest is, there- fore, a continuous function of the time. But in ordinarj' busi- ness fractions of a day are neglected, and interest, having definite sensible increments, is a discontinuous function of the time. So, with a falling body, the force of gravity is constant, but the time, velocity acquired, and distance fallen are A'ariables ; and the velocity and distance are continuous functions of the' time. So, the area of a regular polygon inscribed in a given circle is a function of the number of sides, and varies with the number of sides ; but neither the number of sides nor the area is a con- tinuous number. For while there may be regular polj'gons with 3, 4, 5, ••• or any integral number of sides, it is absurd to speak of such a pol3'gon of 3^ sides, 4f sides, and so on. So, the approximate value of the fraction ^, expressed by the decimals .3, .33, .333, ••• is a function of the number of 3's employed, but that number is discontinous and so is the value. So, the convergents of a continued fraction are functions of the partial numerators and denominators, but not continuous. So, in the equation 4 £0^ — 9 2/2 = 36, y = ±i-^{iaP — 36), and for all values of «< "3 and >"'"3, y ia a continuous function, but for all values of x from "3 to +3, y is discontinuous. So, if y = 1 : X, y is a continuous function for all values of x except a; = 0, where y leaps from a very large negative to a veiy large positive value. § 2.] INC05IMENSUEABL15S. 159 § 2. IKCOMMENSUEABLES. If, in any operation upon numbers, the result cannot be expressed as a commensurable number, either an integer or a fraction, but commensurable numbers can be found both greater and less than the true result that approach indefinitely near to it and to each other, such result is an incommensurable number. E.g., the square root of 2 is an incommensurable. (a) It is not an integer. For (0)2 = 0, (±1)2=1, (±2)2 = 4, and (±3)2, (± 4)2, ... each > 2. , - (&) It is not a simple fraction. For if possible, let -^2 = — , a simple fraction in its lowest terms ; then 2 = ^asimplefraetioninitslowestterms,[III.th.o,cr.3 i.e., an integer is equal to an irreducible fraction, which is absurd ; .-. n^ 2 is not a simple fraction. q.e.u. (c) Commensurables, both greater and less than ^2, can be found, that shall differ from it by less than any assigned number, however small. For •.• (±1)2=1, and (±2)2 = 4, .-. ±1< V2. and ± 2 > ■y/2, and each of them ~ -^2 < 1 . So -.• (±1.4)3=1.96, and (±1.5)2 = 2.25, .-. ± 1.4 < ■^/2, and ± 1.5 > ■y/2, and each of them ~ V2 < •!■ So •.• (±1.41)2=1.9881, and (± 1.42)2 = 2.0164, .-. ± 1.41 < V2> and ± 1.42 > ^2, and eachof them ~ -y/2 < .01. So indefinitely, however small the difference assigned. 160 INCOMMENStJEABLES, LIMITS, ETC. [VII. So the square roots of 4, 9, IG, 25, 36, 49, 64, and 81 are commensurables ; but of all other integers, and of most fractions, lying between 1 and 100 they are incommensurables ; and so of other roots. So the logarithms, to the base 10, of 10, 100, and 1000 are commensurable ; but of all other integers be- tween 1 and 10000 they are incommensurables. Incommensurable numbers often represent the attempt to ex- press the numerical measure of a quantity in terms of a unit that has no common measure with it. If expressed iu terms of some other unit, the number might be commensurable. E.g., the diagonal of a square is incommensurable with its side ; but in terms of the half diagonal, or any other exact part of itself, say ^^jths, |-|ds, -ffths, •••, it is commensurable. So, time may be expressed in days, in lunar months, or in years ; but it is very unlikely that a given length of time, exactly expressed in any one of these units, would be commensurable in either of the others. So, if two distances, a b and c d, be taken at random, the chances are few that ab is a measure of CD, or that they even have a common measure. If thej- seem to have one, it is probably because most measurements are inexact, and only rough approximations are used instead of the true numbers, which are commonlj' incommensurable. The words addition, subtraction, multiplication, division, and involution to commensurable powers, were defined in I. §§ 6-11 ; and those definitions were made so broad as to cover all kinds of numbers. The axioms laid down in II. § 3 also apply to all numbers. i^ncommensurable powers and logarithms are defined in VIII. §4, IX.§1. The combinatory properties of commensurable numbers were proved in II. §§ 4, 6, 7 ; the same properties are proved for in- commensurables in Vn. § 7. § 3.] LIMITS. 161 § 3. LIMITS. Whek a variable takes successive values that approach nearer and nearer to a given constant, so that the difference between the variable and the constant is very small, and may become and remain smaller than an}- number named or conceived of, then the constant is the limit of the variable ; and this definition applies ■whether the variable be always greater, or alwaj's less, or some- times greater and sometimes less, than the constant. E.g., 1, 1.4, 1.41, 1.414, 1.4142, ... are successive approximations to the true value of -v/2, and if the series be extended, a succession of terms may be found whose differences from -y^2 are smaller than any assigned number, and steadily grow smaller and smaller as the series goes on, but which terms are each less than ■y/2. So 2, 1.5, 1.42, 1.415, 1.4143, ••• are each greater than y'2, but approach it nearer and nearer without end ; . . while -y/2 can never be exactly expressed in decimals, it is the limit to which both the series approach. So as shown under continued fractions, .^72 = 1 +— i and the successive convergents, 2H — 3 7 17 41 99 239 2' 5' 12' 29' 70' 169' '" are alternately greater and less than -^2, the true value, but approach it nearer and nearer as a limit. So 1 is the limit !LE_ when n increases without bounds. n For •.•!Lil=i±-, and -, =1±-~1, =0 when w=oo, n n n n ^wherein " = " = grows smaller and smaller and approaches as its limit, and " = 00" = grows larger and larger without bounds ; .•. lim = li when n = 00. q.e.d. 162 IXCOJniEXSTTEABLES, LIMITS, ETC. [VII. So i is the limit of the series .3, -.33, .333, .3333, ■••. So if from tlie series i, (J)^ {\y, (|)^ ■•■ (-J-)" a new series of sums be formed hy taking then the limit of this series, when w = oo, is 1. For si~l=i, S2~l=i = (*)', S3~l=i = (i)^•••, and s„~l = (^)"=0, when 7i = oo. q.e.d. So, if a regular polygon be inscribed in a circle, and another be circumscribed about it, and if the number of their sides be doubled again and again, the area of the circle is the limit of the areas of both the polygons, and the circumference of the circle is the limit of their perimeters. The two areas approach nearer and nearer to the area of the circle and to each other ; but one is alwaj-s a little greater and the other a little less than the circle ; and so of the perimeters. So, the surface and volume of a cone are the limits respectively of the surface and volume of an inscribed, and of a circumscribed, P3-ramid ; the surface and volume of a cylinder are the limits of the surface and volume of an inscribed, and a circumscribed", prism, and so on. In these examples, as in all others, the constants -y/2, 1, |-, ••■ are limits, not simplj' because the successive values of the vari- ables approach nearer and nearer to them, for they approach nearer and nearer to many other numbers not their limits. JE.g., tlie series 1, 1.4, 1.41, 1.414, 1.4142, ••■ approaches hearer and nearer to 10000, which is not its limit at all. So, the area of the inscribed polygon approaches nearer and nearer to the area of the circumscribed square, not its limit. The constants are limits because, as the series is extended, some one of its terms, and aU the terms that follow it, will differ from the constant by a number smaller than any assigned num- ber, be that number never so small ; and further, because, how- ever far the series is extended, there is no point beyond which its successive terms are each of them equal to the limit. §4.] rNPINITESIMALS AND INFINITES. ,163 § 4. INFINITESIMALS ANB INFINITES. A VARIABLE is ivfinitesimcd if it can take values smaller than any assignable magnitude, infinite if it can take values larger than any assignable magnitude, finite if neither infinite nor infinitesimal. All constants except are finite. Strictly, the word "infinitesimal" applies only to that part of the series of values of the variable which are smaller than any number that can be named or conceived of, and ' ' infinite " to that part of the series of values which are larger than can be named or conceived of. Manifestly, the difference between a variable and its limit is an infinitesimal. The reader must carefully note the distinction between an infinitesimal and absolute nothing. The latter comes from sub- tracting any Dumber from itself; the former from dividing any number into small parts and then continually subdividing one of these parts. An infinitesimal always has some magnitude, but absolute nothing means the total absence of anything to measure. So, between the infinites of mathematics and the absolute infinity of space and duration, there is the same impassable gulf. Absolute infinity means that boundlessness to which nothing can be added, and from which nothing can be taken away, and there are no means by which it can be measured ; but a mathematical infinite is simply " a number larger than can be named or con- ceived of," and one such infinite may be larger than another, or any number of times another. The essential properties of infinitesimals aud infinites, upon which the mathematician rests, are : that, while following the law by which successive values are determined, the one may be made smaller and smaller, and the other larger and larger, at pleasure. -r . , 1 infinitesimals » ,, In comparing two or more -J • /. ■, any one of them may be assumed at pleasure as the base; and if the limiting ratio (limit of ratio) of any other of them to the base be finite, then that other number is of the same order as the base. If this limiting ratio be^ ^ then that other number is-{ j-S^fe^*™'" 164 mCOJIMENSTJEABLES, LIMITS, ETC. [VII. as to the assumed base. If the limiting ratio of an -l )"fi"|'^6simal ° ' infinite to the nth power of the base (whatever n) be a finite number, then the ■{ j^g^j^g is of the nth order as to the base, and . ■ infinitesimals » ., j •i- xi i ^ -i. two ■{■/.■. are of the same order ii thej^ have finite limiting ratios to the same power of the base. E.g., if upon any straight line ab a semicircle be described, and from c, any point of the circumference, CD be drawn perpendicular to ab, and AC and cb be joined, then ABC and cbd are similar right triangles, and AB : bc = bc : db. Let c move towards b, then AB is constant and bc and db are variables Let c approach indefinitely near to b, then BC is an infinitesimal of the first order, and DB of the second order, as to bc. For •.• AB • DB = bc^, .-. lim (bc^ : db) = ab a finite length So, if in the triangle abc, right-angled at c, perpendicu- lars be let fall from c on ab at D, from D on bc at e, from e on DB at F, from f on be at g, and so on ; then the triangles abc, cbd, DBE, EBF, FBG, ■■• are all similar, and BC : AB = DB : BC = BE : DB = FB : BE = BG : FB = •••. Conceive the point c to move towards b, and to approach indefinite!}' near to it, then the ratios grow smaller and smaller, and finall}' become infinitesimals, and the lengths db, be, fb, BG, ••• are infinitesimals of the 1st, 2d, 3d, 4th, ■•• orders as to the constant length ab. [above Q.E. D. [df. § 5.] DERIVATIVES. 165 § 5. DEEIVATIVES. If to a variable a small increment be given, and if the corres- ponding increment of a function of the variable be determined, then the limit of the ratio of the increment of the function to the increment of the variable, when the increment of the vari- able is taken indefinitely small, is the derivative of the function as to the variable. E.g., let a square pjTamid be cut bj- planes parallel to the base ; the sections are squares, and they grow larger as the cutting planes recede from the vertex. Take the sides of two squares 6 inches and 7 inches ; then (7^ - 6^) : (7-6) ='13:1 = 13. Take the sides of two square 6 inches and 6.1 inches then (6.P-6^) : (6.1 - 6) = 1.21 : .1 = 12.1. Take the sides of two squares 6 inches and 6.01 inches, then (6.01^—6^) : (6.01 - 6) = .1201 : .01 = 12.01. Take the sides of two squares 6 indhes and 6.001 inches, then (6.001^ - 6^) : (6.001 - 6) = .012001 : .001 = 12.001 ; It thus appears that as the difference of sides grows smaller, 1, .1, .01, vOOl, ■•■towards 0, so also the difference of areas grows smaller indefinitely, 13, 1.21, .1201, .012001, •••towards 0, but that the ratio of these differences, though growing smaller, has 12 and not for its limit. 13, 12.1, 12.01, 12.001, •■• towards 12 ; ^ i.e., just as the side of the square reaches and passes 6 inches in its growth, at that instant the area is growing 12 times as fast as the side ; as it reaches and passes 7 inches, 14 times as fast ; as it reaches and passes 8 inches, 16 times as fast, and so on ; and, in general, as it reaches and passes x inches, 2x times as fast. When two variables grow smaller and smaller together, their ratio does not necessarily, nor generally, become infinitesimal. JE.g., if a be anj^ number, however small, and mb, nb be smaller than a, then mb : nb = m : n, whatever m and n may be. 166 INCOMMENSUEABLES, LIMITS, ETC. [VII. So leta! = l, — 1 — ) ) ,"• towards 0, '20 400 8000 160000' and 2/=l, -, 7, r, — , ••• towards 0; 2 4: o lb then a;:« = l, — , > ) -^ , ••• towards 0, ^ ' 10 100 1000 10000 and y:x=l, 10, 100, 1000, 10000, ••• towards 00. 2222222 . ,„ So let X = — , — -, -—, 7— , — -, — -, ■;—-, ••• towards 0, 1-2 2-3 3-4 4-5 5-6 6-7 7- i i 1 1 i 1 22' 32' 42' 52' 62' 7^ and 2/= 1, -,, -2, -, -, -5, -, -"towards 0; 1 4 16 25 36 49 . , „ then x:y=l, -, -, _,_, ^, _,... towards 2. If 2/ he a funct^ion of x, then the phrase " derivative of y as to X," is written T>^y, wherein d stands for " derivative of," and the subscript x for " as to x." This phrase is read more briefly, " the X derivative of y." So,.Dj,a; = the derivative of x as to y, or tlie y derivative of x. Manifestly, DyX is the reciprocal of D^y, i.e., T>^y-T>yX= 1, for ■-• — '— ■ . — -^ = 1, however nearly the fractions have mc. y inc. x come to their limits v^x, D^y, Under the general heading of this chapter the reader will find three classes of problems, and the theorems that follow lay the foundation for r.ules for their solution : 1 . Those which involve the limits of variables. 2. Those which involve the ratios of two infinitesimals. 3. Those which involve the sums of an infinite number of infinitesimals. To the first class belong the various examples given under the head of incommensurables and limits ; to the second class belong those under the head of derivatives ; and to the third, the computation of areas and volumes, with other like problems. The process last named is called integration. §^] rmsT PEiNciPLES. 167 §6. FIRST PRINCIPLES. Theoe. I. If two variables, the one increasing and the other decreasing, approach each other so that their difference becomes infinitesimal, they have a common limit that lies between them. 1. Each of them has some limit. Fqi-, if either had no limit, they would pass each other. 2. They have the same limit. For, if they had different limits they could come no nearer together than their limits. [§ 3 df. lim. 3. This common limit lies between the two variables. For it is greater than the less, and less than the greater of them. CoR. If two constants always lie between two such variables, they are equal to the common limit and to each other. For, if possible, let one of them be greater than the limit ; then the greater variable can get no nearer the limit than this constant, which is absurd. [§ 3 df . lim. So, neither of the constants can be less than the limit. .•. they are equal to the limit, and to each other. q.e.d. Theok. 2 . The product of a finite number into a?i iifinitesimal is an infinitesimal, and of the same order. Let n be any finite number, and a an infinitesimal ; then will m ■ a be an infinitesinaal, and of the same order as a. 1 . n • a is an infinitesimal. For, take (3 any finite number however small, and a ■< /? : m ; then w ■ a < /8, and is inf'l. q.e.d. [§ 4 df. infl. II. ax. 16 2. w • a is an infinitesimal of the same order as a. For m • a : a, == n, is finite. q.e.d. [df . infl. of same ord. CoK. 1 . The sum of a finite number, n, of infinitesimals is infl. For their sum < n times the largest of them, [II. ax. 12 and ■.• that product is infinitesimal, [thj .-. the sum is infinitesimal. q.e.d. CoK. 2. If there be any finite number of commensurable vari- ables, x', y', z', •", a7id as many more, x", y", z", •••, such that s'~x" = 0, y'~y" = 0, z'~z" = 0,---; then will x' + y' + z' + --- ~ x" + y" + z"+--- ^ 0, and x' ■ y' • z' ■ x" ■ y" • z"+---=0. 168 INCOMMENSUEABLES, LIMITS, ETC. [VII. § 7. PRIMARY OPERATIONS ON INCOMMENSUEABLES. Theor. '^>. The addition of incommensurables is commutative and associative. For, let a, &, c, ■•• be any incommensurables, I x', x" I a;' < « < x", J 1 i. M/'i y" be commensurable \ y' n, then is (x : y) an infinitesimal of the (m — n) th order as to the base. For • . • lim (x : ^"') = h and lim {y : /S") = fc, wherein h and 7c are finite numbers, [iiJP- \im(x:y) ^\im^^ ^h ^ g^it^ j^^^^er ; lim|8"-" hm {y : /3^y ¥ .• , (a : 2/) is an infinitesimal of the (m— ?i)th order, q.e.d. 172 INCOMMENSUEABLES, LIMITS, ETC. [VII. Cor. 1. The product x-j is an infinitesimal of the (m + n)i/i order. Cor. 2. y : x is an infinite of the (m — n)th order. Cor. 3. If x, y be infinites of the mth and nth orders, and if m > n, then : X : J is an infinite of the (m — n)th order, X- J is an infinite of the (m + n)th order, y : X is an infinitesimal of the (m — n)th order. If there be two or more numbers not all equal, then any num- ber which is greater than the least of them and less the greatest is a mean. The average of two or more numbers is the quotient of their sum by their number. Theor. 12. If x', x", x'", ••• be a set of variables, and y') y") j"'i •••as many more, all positive or all negative, and suchthat lim(x':y') = l, lim(x":y") = l, lim(x"' : y"') = l, ••• ; and if the number of variables in each set increase without bounds, then the Emits of the sums of the two sets, unless infinite, are equal. For •.• lim(x':y') = l, \im{x" : y")=l, l\m(x"' : y"') = l, •••, , x'+x''+x"'+- fx' x" x'" \ C'lyP- ana • -r- — n-^ — ttt- — = mean — , — ,, -— , ••• , y'+y"+y"'+- W y" y'" J .•.lim«'' + '^" + '^"'+-^l, y' + y" + y"'+- . \im(x'+x"+x'"+—) ^^ lim (y'+y'i+y"'+-) .-. \im(x'+x"+x"'-] ) = lim(y'+y"+y"' +■••). Q.-E.D. CoR. If lira (x' : y') = m, lim(x": y") = m, •••,andifx\x",— y') y") ••■ &8 all positive or all negative, then lim[(x'-f-x"+-):(y'+y"+-)3 = m- Note. This theorem is of great service in geometry in com- puting areas, volumes, etc., bounded by curved lines or surfaces. Divide into narrow bands whose limits are rectangles, or thin plates whose limits are prisms, and then get the limit of the sum of such rectangles or prisms ; these limits are the areas or volumes sought. This operation is called integration. 1. § 9.] GEKEEAL PEOPEKTEES OP DEEIVATIVES. 173 § 9. GENEEAL PEOPEETIES OF DEEIVATIVES. PrOB. 1. To FIND THE DEEIVATITE, AS TO ANT VARIABLE, OF A FUNCTION OF THAT^ VARIABLE . In the function give the variable an increment; from the re- sulting eaypression subtract the function; divide the remainder by the increment of the variable, and get the limit of the quotient as that increment approaches zero. E.g., to find the derivative of a^ : Let X = a;^ ; substitute x + h for x, and let x' = (a; + hy ; then -.- x' — X ={x + hy—x' = 2xh + h\ h ... lim ^~^ = 2a! when A = 0, h i.e., Di(a^) =2a!. q.e.f. So to find the derivative of a? : Let X = a;' ; substitute a; + A for x, and let x'= (x + hY; then-.-'x'-x ={x + hy-a? = ^x'h + Zxh^+W. .-. 5^.=5 =3a!'+3xh + h'', h ... iim5^^^:^ = 3a!2 whenA = 0, h ^ i.e., i>x(3^) =3a^. So to find the derivative of x'^ : Let x = x-^; substitute a; + A for x, and let x'= (a; + h) "^ ; then •.• x'— X =(a! + 7j)-*— a;-^ ^Jl 1 ^ -h x + h x x{x + hy x'-x -1 h x{x + h) ... iijn?—^ = ~ whena=0, h or i.e., D,i =-i. Q-E.F. 174 LNCOMMENSUEABLES, LIMITS, ETC. [VII. th. Theor. 13. The derivative, as to any variable, of the sum of two or more functions of that variable, is the sum of their derivatives. Let u, V ••• be any functions of a variable », and x tbeir sum ; then will d^x = d.u + d^v H For let X take any infinitesimal increment h, and let x' stand for the new value of x, x' for the corresponding value of X, d' for that of u, v' for that of v, •••, so that x'~ X + h, x'= X + incx, v'= u + incu, v'=v + inc v, then ••• X =n+vH always, V^JIP- .-. X' =tj'+T'+— , -•. x'— X =n'— u+v'— vH , [II.ax.3 i.e., incx =incu +incv + •••, incx incu , incv h h h + •", [n. ax.5 .-. limS£l = lim^-H£H + limiH£I+...when7» = 0, [th. 7 h h h i.e., 0,x, =D,(u + v + -"), = D,U + I>iV-| . I Q.E.D. Theok. 14. The derivative, as to any variable, of the product of two or more functions of that variable, is the sum of the prod- ucts of the derivatives of the several factors each multiplied by all the other factors. Let u, V, w, ••• be any functions of a variable x, and x their product ; then willDjX = V'W"'D,u + U'W-"D^v+u-V".D^-w+'". For let X take any infinitesimal increment h, and a' be the new value of x, so that a;' = a; + /i, x' = x + incx, ■••, then •.• X =U'V'W ••• always, [hyP- .•. x' =tj'.v'.w'"-, i.e., x+incx = (u + incu) •(v + incv) -(w + incw) ••• = tj.V'W — |-V'W'"incu+u-W'"incv+>" + terms with two or more infin'l factors, 15. § 9.] GENEKAl PHOPEBTIBS OF DBEIVATIVES. 175 .*. incx = V'W"-incu + u-W'"incv + u.v-"incw+ ••• + terms with two or more infln'l factors, . incx ,, „ incu , ^.^ incv , • „ incw , h h h h + terms with one or more infln'l factors, .-. D,X, =D^(U-V.W-"), = VW"'DjU + U'W'"D,V + U-V"'DjWH + terms that vanish. q.e.d. [th. 7 Cor. In particular, the derivative, as to any variable, of the product of two functions of that variable, is the sum of the prod- ucts of the derivatives of the two functions each multiplied by the other function. i.e., D,(U'V) = U-D3,T+V'DjU. Note. Theorem 14 may be written in the form : D,(U-V-W-") D,U D,V D^W = h' " + •••. tj.VW-" U V i w Theoe. 15. Tlie derivative, as to any variable, of a fraction whose terms are functions of that variable, is a fraction whose numerator is the product of the denominator into the derivative of the numerator less the product of the numerator into the deriv- ative of the denominator, and whose denominator is the square of the given denominator. Let u, V be any functions of a variable x, and x their quotient ; ., .„ V-D,U — U-D,V then will D.x = — v^ For let X take any infinitesimal increment h, and x' be the new value of x, so that x'=x-^h, x' = x + incx, D' = u + incu, v' = v + incv, then ••• X = - always. , . u + incr I.e., x+incx = — ■ , V + incv 176 rNCOMMENSUBABLES, LIMITS, ETC. [VII. th incx _TJ + incu V + incv _u V _; V • incu — U'incv T^+V. incv V incu incv inc-x h h h v^ + v • incv T V • D.TT — 1 [J-D,V V^ Q.E.D. [th,7, th. 8,cr.l CoE. 1. If TJ be constant and v a function ofx, D«u = 0, and p,- = ~^ ' "'^ = p ■ d^— V v' V CoE. 2. If\be constant and u a function ofx, D^v = 0, am,d d»- = 2s? = _ . d^u. V V V Note. Theorem 15 may be written in the form : Theoe. 16. The derivative, as to any variable, of a function of a function of that variable, is the product of the derivatives of the immediate functions which compose it, each taken as to the variable on which the immediate function depends. Let u be any immediate function of a variable x, and x any immediate function of u ; then will Da,x = d„x • d^tj. For let X take an infinitesimal increment h, then v and x will take corresponding increments ; J incx incx incu ttt xi- o -in and • ■ — ; — =: ; — , [II. th. 3, crs. 1, 7 ft mou ft . • . lim ^ = lim l-H^ . lim '^5^, when h = 0, [th.6, cr.,th. 8 ft mcu h i.e., D^X =D„X.DjU. Q.E.D. 17. § 9.] GENERAL PEOPEETIES OF DEEIVATIVES. 177 Theok. 17. The derivative, as to any variable, of a com- mensurable power of that variable, is the product of the given exponent into a power of the variable whose exponent is a unit less than the given exponent. Let X be any variable, and n any commensurable number ; then will d^o;" = nx"~K (a) n a positive integer : For •.• «" = the product a; -a; •»•••, w-times repeated, .-. D^x" = x''~^ •■D^x+af~^ •i>xX-\ — ,w-times repeated [th.l4 = 7ia;"~^-Dj,a;. But •.• D^x = 1, ,f .-. n^x" = nar-\ ',| (6) n a positive fraction, E ; p, q both positive integers: For let X = X', then ••■ ' x« = a!^ • ■■ , gx'-^ •D. ,s.=px^-', . [(a),th.l6 r>«x p aS^^ p af-^ p (a!f)«^-« gV -' q q [II. th. 3, crs. 10, 11 i.e.. D.a^ = naf-^ Q-E.D. (c) n any negative number, — m : For let x s a;""', then i>.x=B.A. ==^ [th.l5,cr.l ^=n^ = -mx—\ [(«),(&) i.e., D^a^ = nx''-\ Q-e.d. CoK. Ifvbe any function of x, then D.u" = nu"-i • D.u. Wth. 16 178 ESrCOMMENSTJEABLES, LIMITS, ETC. [VII §10. INDETBEMINATB FOEMS. If there be au expression that, by the definitions of the sym- bols used, may take au infinite number of difierent values lying in a continuous series, such an expression is indeterminate. [See 11. §2, p. 28. E.g., the expressions 0:0, oo : oo, oo — oo, are indeterminate. For the quotient : may be any quotient that, multiplied by or into the divisor 0, gives the dividend as product ; and any finite quotient may do this. L • « • aiv. And the quotient oo : oo may be an j' quotient that, multiplied by or into the divisor oo, gives the dividend oo as product ; and any quotient, not 0, may do this. And the remainder oo— oo may be any remainder that, added to the subtrahend oo, gives the minuend « as sum ; and any remainder may do this. So the quotient x:yis indeterminate if of x, y it be known only that both = 0, or that both = oo. And the remainder x — y\s indeterminate if of », y it be known only that both = oo. For any number may be such a quotient or remainder. If for a particular value of any variable of which its terms are functions a fraction take the form 0:0, it maj' be regarded as approaching this form bj' gradual change of the variable, and its true value is strictly the limit of the ratio of two infinitesimals. This value is finite when the terms of the fraction are infini- tesimals of the same order [§§ 4, 5], and it is indeterminate only so long as the law is unknown subject to which they = 0. E.g., when a; = 1, a^—1 : a!^—l becomes : 0, but when a; = 1 + ft, this fraction becomes (l + fe)"-! ^ fe3 + 3/i^+3ft ^ 7t^+3ft + 3 (1 + A)2_l' A2+27i h+2 ' = 3:2 when Ti = 0, i.e., when a; = 1, and its true value, when a; =1, is 3 : 2. §10.] INDETEEMINATB FOEMS. 179 The reader will see that this process is equivalent to reducing the given fractio;ti to its lowest terms, then substituting 1 for x. In general, fractions take the form : because of some com- mon factor of their terms that vanishes for a particular value of the variable. If this factor can be found and divided out, and the particular value be substituted for the variable, the result is the true value of the given fraction ; and this method is particu- larly useful for fractions whose terms are entire. In the above example the vanishing factor is a; — 1 , and the fraction, when this factor is divided out, becomes — "'"'""^ == — a;+l 2 Theorem 18 will show another method of evaluation. Expressions that approach the forms oo : oo, oo — oo, maybe reduced to equivalent expressions that approach the form : 0. E.g. , let X, x' be functions of any same variable k, such that when a; = a, then also x, x' both = oo. Put X, x' under the forms tr : v, u' : v', wherein u, u', v, v' are all functions of x such that, when x = a, V is an infinitesimal of any order, and xi is finite or an infinitesimal of a lower order than v, [th.ll, cr. 2 and v' is an infinitesimal of any order, and u' is finite or an infinitesimal of a lower order than v', ,, I . u Tj' u-v'— Tj'-v . then X— X, =oo — oo, = = ; — , = — V v' v-v' B.g., if u, V, u', v' = a; + 2, k-I, as^^l, a?— 2a^+a;, then •.• u is finite, v, u' are infinitesimals of the first order, and v' is an infinitesimal of the second order, when a; = 1, .■. X — x', =00 — 00, = (« + 2 ■'^^^^2¥+ie—x—l ■x^—l):x{x—lf, :^0:0, = (a;^— a; — l):a!— 1, =00, whena!=l. ' [div. outvan.fac.(a! — 1)^- It has been shown above that the forms called indeterminate belong to that class of limiting expressions wherein the variables cease to have finite values. They difi"er from other limiting ex- 180 INCOMMENSUKABLES, LIMITS, ETC. [VII. th. pressions of the same class in this, that their limits cannot be determined without more knowledge of the relations of the vari- ables than appears upon the face of the expressions themselves. E.g., when x, y both = 0, the quotient (8 — a;) : (4 — y) , not an indeterminate form, = 2, no matter how x, y may be related ; but the quotient x:y, =0:0, may have any limit whatever, depending on the relations of the variables a;, y. From this point of view the form oo • may be added to the list of indeterminate forms ; for although it does not, like the other three forms, take au infinite number of different values by the mere definition of the symbols taken absolutely, vet, like them, it may take any value whatever, considered as a limiting expression, i.e. as the limit of the product of any two variables, one of which = oo and the other = 0. An expression that approaches the form oo • may be reduced to an equivalent expression that approaches the form : 0. Theoe. 18. If for a particular value of a variable two func- tions of that variable both vanish, the true value of their quotient is the quotient of the values of their derivatives for that value of the variable. Let X, x' be two functions of a variable x such that x^, x'^, their values when a is put for x, both vanish ; [x„ = [x'„=0 then will x„ :x' a=I>, :X. : D^x',. For, in X, ,x'^ , put a + h for x; then • •• ^a + n = Xo+» ~ -X., and x'„ ; + » = x'.+»- -X'., • •. x„ + » • X o + » = ^o + » ~ -x„: x' a + » , — x„ ■a + >" lim(x,+j:x'„+,) = lim ( ^'+»-^° : x,+t-x„ s ^^^^^ j^ ^ Q^ h h = lim ^°+''~^' : lim ^''■+»~^'' , [th. 8, cr. 1 h h = D^Xa : D^X'„. Q.E.D. 18. §11.] GEAPHICAIi EEPEESBNTATION OP FUNCTIONS. 181 § 11. GRAPHICAL EEPEESENTATION OF FUNCTIONS. Foe convenience in treating of integration and other subjects discussed later, the geometric words, origin, axis, abscissa, and ordinate, are here defined, and the reader is introduced to the method of representing by a geometric locus an algebraic equation between two variables, or a function of a single variable. Let p be any point, x'x any straight line lying in the plane of the paper, and o a fixed point on x'x ; from p draw PA perpendicular to x'x and x'- meeting it at A ; then x'x is the ref- erence line, or axis, the fixed point o is the reference point, or origin, the line OA is the abscissa of the point p, AP is its ordinate, and oa, ap together are the coordinates of p. If the figure lie before the reader so that x'x is a horizontal line with x to the right of x', then the direction x'x is ordinarily taken as the positive direction and xx' as the negative direction [I. § 3] ; and abscissas measured to the right from o are posi- tive, while those measured to the left are negative. So, ordi- nates measured upward from the axis are positive, and those measured downward are negative. An abscissa is generally represented by the letter x, and an ordinate by y. So, the line x'x is called the axis of abscissas, or the axis of x ; and the line t't, drawn perpendicular to x'x through o, is called the axis ofordinates, or the axis of y. The position of a point is determined, and the point may be constructed, when its coordinates are given. When the coordinates are not given, but are connected by a given relation {e.g., that their sum is constant), an infinite num- ber of points may be found that satisfy the conditions, for if any value be assumed for the abscissa, the given relation between the coordinates serves to determine the corresponding value or values of the ordinate. 182 INCOIVIMEKSTTEABLES, LUnTS, ETC. [VII. In general, these points all lie in some line, straight or curved, called their locus; and the relation between the variable coordinates may be expressed bj' a single equation between two variables, called the equation of tJie locus. In this equation either variable is a function of the other. The equation is satis- fied by the coordinates of every point of the locus, and by those of no other point. Such equations are generally written in the form y=fx, wherein x, the abscissa,, may be regarded as an independent variable, and y, the ordinate, as a function of x ; and the shape of the locus of the extremities of the ordinates shows the manner in which fx varies with x. E.g., the locus of points whose coordinates satisfy the rela- tion expressed by the equation y = mx is a straight line through the origin. Let ox be the axis, o the origin, ^!-^''^ p, p' any two points whose co- ,,^^1 ordinates oa, ap, oa', a'p' are ^< ' L so related that ap = m • oa, and a'p' = m. oa', i.e., so that y = mx for each of them ; then is pp' a straight line through o. For •.• AP : OA = a'p' : oa', [hyP- and ■.• AP is parallel to a'p', [constr. .*. the straight line op passes through p', and is the locua sought. [geom. So, the locus of the equation y = mx + & is a straight line that cuts the axis of y at a distance 6 above the origin. As above, construct the straight line that represents the equation y = mx ; draw any two ordinates ap, a'p', — and extend them to Q, q', ' , so that AQ = AP -f- 6, a'q' = a'p'-|- 6, wherein b is any consti).nt ; then is qq' a straight line parallel to opp', and the locus sought. § 11.] GRAPHICAL KEPEESBNTATION OV FUNCTIONS. 183 So, the locus of the equation a^+2/^=»^, wherein r is constant', is a circle whose centre is ,the origin and whose radius is r. The reader may see this from the prin- ciple of geometry that " in a right triangle the square of the hypotenuse equals the sum of the squares of the other two sides." So, the locus of the equation y' =px is a parabola whose axis is the axis of x, whose vertex is at the origin, and whose parameter is p ; and the locus of the equation a?=py is a parabola whose axi3 is the axis of y. The reader will '^ recognize these equations as the algebraic expres- sion of the geo- metric property of the parabola, that "the square of a perpendicu- lar from any point of the curve to its axis equals the product of its parauieter into that part of the axis intercepted between the vertex and the foot of the perpendicular." So, the locus of the equation xy = (? is the rectangular hyperbola, taken with ref- erence to its asymptotes as axes of coordinates. These figures also represent graph- ically the functions mx, mx + b, y'(r^— a;^), -^/pn!.'—, — , and sh6w how they vary with «. 184 rNCOJEMENSURABLES, LIMITS, ETC. [vn. t § 12. INTEGRATION. Theor. 19. If there be a variable x,and iftx be a function of X whose derivative as to x is f x and is continuous; and if the variable begin ivith the value Xq, = a, and take n more successive values Xi, Xj, • • • x^, = b ; and if, while a and b stand fast, n = CO and Xi — x,,, Xj — xj, ••• each = ; then the sum of the series of products (xj— Xo)f'xo, (x2— Xi)f'xi, •••(x„— x„_i)f' x^.j, approaches fb — fa as its limit. ILLUSTEATIVE EXAMPLES. That the reader may clearly understand the meaning of the theorem and its proof, and that he may see how this method of summation was first suggested, and follow the historical order of investigation, special applications of it to the finding of areas and volumes are given before the formal proof : To find the area of the figure included between two given ordinates, the axis of abscissas, and the parabolic curve whose equation is a?=py: Let the two given ordinates corre- spond to the abscissas OQ, = a, and OR, = 6 ; divide qr into n parts ; let the abscissas of the m + 1 successive points (in- cluding Q and r) be Xq, »„ x^, • • • a;„ : and the corresponding ordmates y^, y^, y^, — y„, and let n rectangles be formed as in the figure ; s, the area sought, is the limit of the sum of Sj, Sj, *8i ■■■ *n> tlie areas of the n rectangles, when n = ca and »! — Xq, x^ Si = 2/o-(aa— a^o) 1 then and ■Xi,"- each =0, p -^ •(Xi—Xo), and OXq , — Dxq Xd , Xi — • Xn ^ ^™ x^-Xa ^^^^ a!i- iCo = 0, [th.l2, nt. [geom. [hyp. [§ 5, df. deriv. 19. §12.] TNTEGKATION. 185 wherein ei is some variable that = when a^— Xf, == 0, and si = — [ki"— a;o'+(a!i— a!|,)ei]. So sa = — [«/— «!»+ (Kj— a!i) £2] , «3 = g- [a'a^- ^2'+ (!»3- a'2)«3] , and s„ = — [«„'- iB^i+ («„- a!„_i)ej, wherein ej, cji ■•• each = when n = and a^— a!o, ••• = ; •■• 2s = — [a!„5-a!o=+2(a!i-a;„)ei]. op But ■■• 2(a!i— a!o)£i3*(a!„— a!„)e„ wherein e, is the largest of the e's, .•. S(!Bi— a;o)ei= when Ss = s, • •• s = J- («„»- ajoO 3p = A(63_„a). Q.^.,. So to find the volume of the solid generated by the same figure revolving about ox, the tangent at the vertex : then ■.• V, the volume sought, i^ the limit of the sum of Vi, v^, Vg, •••v„, the volumes of the n cylinders of revolu- tion generated by the n rectangles when «. = 00 and (Bi — aJo, ajg— a!i, •••each = 0, [th. 12 and •.■ Vi =Tryo'{xi—Xo) [geom. = T^ • 5 a;„* («!- Xa) [hyp. lOp = ^ {"^^-^^ + .1) ((Bi- Xo) [§ 5, df . deriv. 5p^ \bi— a^o = g^2 L«!i- «'o'+ (»!- a^o) «i] , 186 ESrCOJIMENSUEABLES, LIMITS, ETC. [VII. th. and •.• V2 =7^C(a;2°— a'/)+(a'2— »i)£2], ^'3 = r^ C i'^s- ^2) + (a^s- Ka) fa], op ■■■ Si) = ^[(95«"-a5o") + S(a!i-a\))€i]; op"' and V =-^f6'— a^). q.e.f. [as above 5p^ PEOOl? OP THE THBOEEM. For ... J^~J'^° =/'a;o, wheiia!i— a!o = 0, [§5,df.deriv. iCj — Xq ~Fzr^ —J '"o+^ii wherein ej is some variable that = when Ki — Kq = 0, • • • A -fi>o = (aJi - iCo) • (/' <«o + ei) ■ So fx2— /»! = (032 - a^i) •(/'«! + ea) , fxs-fx^ ={Xi-X2)-{fx2 + £a), ... . . . J and fx„ —fx„_i = (a;„- a;„.i) (/'a;„_i + e„) ; wherein ej, €2; •■• each = 0, when n = co and ajj— Xg, ••■ =0, •■• /iCn— /^o = (a;i- a;o)/'a;o-l h (««— ^n-i) /'»„-! + (aa- a;o) ci-1 h (««- ^n-i) f»- £11. ax. 2 But •.• 2(fl!i— a!o)ei9^(a!„— a!o)c„ wherein e, is the largest of the e's, and •.• e, = 0, when »!.= 00 and Ki — ajo, ••• each = 0, [above • •• fb—fa=\iml{xj,—Xo)f'xo-j h(a;„— a!„_i)/'a;„_i]. q.e.d. Note. The theorem may be written in the form : lim S ' /' a; • inc x—fh —fa, when ihc a; = , wherein Sa/'^-incais/'aro-incaJoH t-/'a:n-i-inca;„_i, and Xa = a, a!„ = 6, inea!, = a!,+i— «,. IB. § 12.] INTEGKATION. 187 EXAMPLES OF THE DIRECT APPLICATION OP THE THEOEEM. To flnd V, the volume of a segment of a sphere of radius r, whose bounding planes are ' y distant a, b from the centre. Let CDT be a semicircle of radius r ; take CD for the axis of x ; let AF, BE be two ordinates distant a, b from the centre o ; and let c ^ o a b d the whole revolve about cd ; then the area of abep is the limit of the sum of the areas of a large number of rectangles ; and V, the volume of the solid generated by abep, is the limit of the sum of the volumes of the corresponding cylinders of revolution. Take ■n-y/{a;j,+i—Xp), =7r(r^— a;/) (ajp+j— ojp), as the type- term of this series ; then ■.■ fx = 7r{r'—x^), .-. fx =7r{l^X-ia?), V =7r[_{i^b-i¥)-{r^a- i.e \j=fb-fa i«0] = 7r(& - a) [r^- i(a^+a6 + 6')] = i-,r(6-a)[(r^-aO + ('^-&')+(4'^-^T6')] wherein c is the thickness of the segment, r^, r^, r^ the radii of its bases and middle section. Q.e.f. So the volume of the hemisphere generated by the quadrant odt = ,r(/r-/0) =,r(r«- i9-») = f ,rrs. And the whole volume of the sphere = |irr'' Q.E.r. NoxE. If in \Trc(ri+r2+'^r^)i the general expressioij for the volume of a spherical segment, r be put for rj, for r^, ^rVS for rj, r for c, the result is the volume of the hemisphere ; and if be put for i\, for rj, r for rj, 2r fore, the result is the volume of the sphere. The results thus found are identical with those given above. 188 ikcojMmenstjeables, limits, etc. [VII. So to And H, the height fallen through in a given time bj' a body starting from rest, on the assumption that, within anj' distance required in practice, the velocity of a fall- ing body increases uniformly, and hence that the velocity acquired at any instant is proportional to the time of falling from rest ; [laws of motion then v = gt, wherein g is some constant ; v is the velocity at the end of t seconds from starting, i.e., the number of feet the body would fall through in the next second if its rate did not change during that second. Let the entire tune, t, be divided into n intervals, ending respectively at ii, *2)"- =/(a!) , show by the properties of similar triangles that the equation of the chord PqPi is {y - 2/o) : (« - ico) = {yi - y with a rectilinear base the product of the average ordinate by the base is the area. 194 , POWERS AND ROOTS. [VIII. VIII. POWERS AND ROOTS. §1, FRACTIONAL POWERS. The words power, root, base, exponent, and root-index are defined in I. § 10. A root-index is always assumed to be a positive integer ; but an exponent may be any number whatever. The value of a fractional power is commonly ambiguous. E.g., 100^ = ±10; 9"* = ±^V- So, as appears later, everj' base except has three distinct cube roots, four distinct fourth roots, and so on. Some of these roots, however, are neither purely positive nor purely negative ; they are called imaginaries, or, better, complexes, and discussed in chap. X. Different powers of a base are in the same series when they are integral powers of the same root. An integral power of a base belongs to all series aUke. E.g., 9-1, 9-^ 9°, 9^ 9^, 9^, 9^ ••• are the— 2d, —1st, 0th, 1st, 2d, 3d, 4th, •.• powers of -^9, i.e., of -3 and of +3 : they form the two series I, -^, 1, -3, 9, -27, 81, •", powers of --^9, and i, J, 1, 3, 9, 27,81, ■••, powers of +V9 ! but the integral powers ^, 1, 9, 81, belong to both series. When several powers of the same base occur together, they are assumed to be all taken in the same series. E.g., the value of 4"^ — 3 • 4^ + 4^ is either ^_3.+2+(+2)^ = 2i, or A_3.-2+(-2y, =-2^, according as 4~*, 4*, 4^ are taken as powers of "'■2 or of "2 ; butnot JL_3.-2+(+2)3, =14i,nor^-3-+2 + (+2)^ = 1^. So, ^9a'--y/Aa'=±3aT2a = ±a, but not 3 a + 2 ci, nor — 3 a — 2 a. Powers of different bases are like powers if they have the same exponent. E.g., y/a, -y/b, s/ab; a\ V, ab' ; 2», 3", 6". § 2.] COMBINATIONS OP COMMENSURABLE POWERS. 195 §2. COMBINATIONS OF COMMENSURABLE POWERS. ly^ That every commensurable power of a real positive base has at least one reaL value is shown from independent consider- ations in th. 5, which may therefore be read here if preferred. Theoe. 1. Any commensurable poiuer of a base has the same value or values whether the exponent be in its lowest terms or not. Let fc, p', q be any positive integers ; p, = ±p', any integer, positive or negative ; a, any base ; then will every value of a*' be a value of a', and conversely. For • . • '-^A = one of any kq equal factors into which a can be resolved, [I. § 10 df. root and ■ . • the product of any k of these factors is equal to that of any other k of them, .-. all the q partial products so formed are equal, and each is a value of -^a, and every single value of ^a is a value of -^(-^a). So •.• [^(Va)]''=[^(Va)] •[•\/(,Va)] ■■■fcg factors = [-v/(Va)]*-[v'(Va)]'- g factors = (a/a) • i-V^) ■ (-^a) - q factors = a, -•. every value of -^(-^a) is a value of ^a. [df. root .-. Va=a/(-^a), ' i.e., every value of either member is a value of the other. So -.• a^ = [^(-Va)]';^ [I. §10df.fract.pwr. = '^^L 11 .-. the product a', =\-iJ -a.^ ■■• p times, has but one positive value. _p p So with A ?, = 1 : A«. CoR. If the base and exponent be both finite, so is the positive value of the power. ' (a) Jlie exponent an integer, either positive or negative. For •.• the power is the continuous^ P'^°^."''* of 1 by a finite number of finite { ^Slf'' ^df • int. pwr. .-. the power is finite. q.e.d. [VII. th.ll cr.3 (b) The exponent a fraction. Let the base a-{ 1, and let the exponent n lie between the integers i and i + 1 ; then ■.• 71 — i and i + 1 — n are positive and commensurable, .-. A»-', a'+'-", both^ Jl; [th.5 § 3.] CONTINTJITY OF COJVCVDENSUfiABLB POWEES. 201 and •.• A" = A"-'-A*, a'+^ = a'+^-^-a", [th.3 .'. A"^ J a', a' + i^ J a", i.e., +A" lies between a* and a'+^, which are )3oth finite and positive. • [(a) .•. +A" is finite. q.e.d. Theor. 6 . Of a commensurable -j -^ * * . ^ power of a variable positive base with qiven exponent, the positive value is j""* *'^'^'"^'*^*'*9' "^ -^ 'a decreasing continuous function of the base. Let A be the variable base, and n, = ± -, the given ^P^^^^l^® g' ° 'negative exponent ; then : (a) The larger the base a, the ■{ '^'''d^J the power. 1 1. For •■■ A= 1 -Af-A^-" g times, 1 .-. the larger a« is, the larger is a ; [II. § 3 ax. 19 i.e., the larger a is, the larger is a«; p 11 and -.• A« = 1 -A' -A'-" p times, 1 P .-. the larger a« is, the larger is A' ; p .•. the larger a is, the larger is A«; _p ' p and the smaller is a «, = 1 : +a» ; i.e., when A increases, A 'increases, but A ^decreases, q.b.d. (6) When a passes through every value from to +qo in order, +A" passes through every value from ■{ ^ . n in order. For, let B be any positive number ; ^ 1 1 then • . • the power B" has a positive value +6", fth. 5 11 . ■ . if A = +B», +A", = + (+B») ", takes the value b, [th. 4 .-. every number b from to +oo becomes in turn a value of +A" ; and •■• the larger the base, the ^ °,, the power, [(a) .•. when A passes through every value from to +oo in order, '''A" passes through every value from \j^ f° n' in order, i.e., it is continuous. q.e.d. 202 POWERS AND EOOTS. [VIII. Cor. 1. If the base and exponent be both finite, every infini- tesimal change in the base gives an infinitesimal change in the positive value of the power, and conversely. Cor. 2. If the base appivach a limit Aq, the positive value' of the power approaches a limit "'"Ao", and conversely. Theor. 7. Of a variable commensurable power of a constant positive base larger than unity : 1 . The positive value is an increasing function of the exponent; 2. The exponent can be so taken that the power shall lie be- tween any two positive numbers, however close together. Let A, >1, be the base ; and let n', n" be any two values of the exponent n, such that n'< n" ; then : 1 . +a" is an increasing function of n. For •.■ +A"" = +A"'-+A""-"' [th.3 = +a"' • a positive power of A = +a"' • a number greater than 1 , [th. 5(1) .-. +A''">+A"'; i.e., +a" increases with n. q.e.d. 2. n can be so taken that +a" shall lie between any positive numbers b, c. whereof b < o. 1 Take a', q any positive integers so great that a'>a, -< — 1; 1 Q B and let h = — — ; a'q 1 ^'^ 1 then •.■ (1+-) =1+a'q 1- other positive terms [bin.th. Q Q >H-a' >A, .-. A* b ; for, if mh be any integer > A-1' then A"", =(l + A-l)'»^ >l+m^(A-l)>B. [bin. th. And the series has terms < b ; for, if A"'* be any term >-5 •' B then A"*"'*, = 1 : A"'*, is a term 1 ; and let 6, = 1 : b, and c, = 1 : c, be any two positive numbers ; then -.• +A'* increases with the exponent, and takes values be- tween the positive numbers b, c, [th. 7 and •.• +a" is the reciprocal of +a", [th.4cr.3 -•- +a" decreases as the exponent increases, [II. ax. 18 and takes values between b and c. q.e.d. CoK. 2. When the base differs sensibly from 0, 1, and oo, and the exponent is not oo, then every infinitesimal change in the ex- ponent gives an infinitesimal change in the positive value of the power, and conversely. Cor. 3. When the base differs sensibly from 0, 1, and oo, and the exponent approaches a limit no, the positive value of the power approaches a limit +a°o ; and conversely. 204 POWERS AND ROOTS. [VIII. Note. ' The principles established in theors. 5,6,7 are sum- marized as follows : Every commensurable power of a positive base has one and but one real positive value [th. 5] ; ■{ decreasina <'°'^**'*^°''* function of the base if the base vary and the exponent be constant if the exponent vary and the base be constant and ■{ ^ -,-, than unity [th. 7, th. 7cr.l]. Wliether the base or the exponent varies, the commensurable power takes values [indeed, an infinite number of them] between any two positive numbers however close together [above, th. 7, cr.l] . When both base and exponent are finite, and the base =?fc 1, any infinitesimal change in either gives an infinitesimal change in the positive value of the power, and conversely [th. 6 cr.l, th.7 cr. 2] ; and if either the base or the exponent approach a limit while the other is constant, so does the positive value of the power, and conversely [th. 6 cr. 2, th. 7 cr. 3] . The positive value of the power is finite when the base and the exponent are finite [th. 5 cr.] . This value is ■{ ",, than unity if the exponent be positive, and ■{ , than unity if the exponent be negative, when the base is ■( ",, than unity [th. 5] . It appears later [th. 12] that the powers of a constant base take a continuous series of values when the exponent takes a continuous series. But when the varying exponent or base is restricted to commensurable values, then between any two values taken by the power there lie an infinite number of values not so taken. §4.] INCOMMENSUEABLE POWERS. 205 §4. INCOMMENSURABLE POWEES. Hitherto no meaning has been given to the symbol a" when n is incommensurable ; and any meaning that may now be given to it should fulfil, if possible, the following conditions : 1. It should give a single definite positive value to the symbol A" when A has a given positive value and n is incommensurable. 2. It should not conflict with any use that the symbol a" has when w is commensurable. 3. It should preserve all the fundamental properties that the symbol a" has when m is commensurable : in particular, theorems 2,3,4 should be true for all real exponents whatever. The following theorem lays a foundation for the definition : Theor. 8. If there he a constant positive base not nor 1 nor 00, and two variable commensurable exponents, one increas- ing and the other decreasing toward a common incommensurable limit not oo, then : 1. The positive values of the two variable commensurable powers have a common limit., which lies between them and is not nor oo . 2. This common limit depends upon the value of the base, and of the common limit of the exponents, but not upon the law by which either exponent approaches its limit. 3. This common limit is not a commensurable power of the base. Let A be any constant positive base not nor 1 nor oo ; let X, y be any variable commensurable exponents, x increasing and y decreasing toward a common incommensurable limit n not infinite ; and let x', y' stand respectively for a particular series of values of x, y that approach n as their common limit, and so with x", y", with x'", y'", •••, then : 1. +A^', +A^' have a common limit that lies between them and is not nor oo. For • . • the exponents x', y' each = n, their common limit, [hyp. .-. x', y' come to differ from each other by less than any assigned number, .•• +A''',+A''' come to differ from each other by less than any assigned number ; [th. 7 or. 2 206 POWERS AND BOOTS. [VIII. and •.• x'<.y' always, and x' increases while y' decreases, .-. +A'' 1, be any positive base, and n any-{ ^ .. 1 fl T'O'PT' exponent ; then is the positive value of a" \ °,, than 1 . (a) n commensurable. [th. 5 (&) n incommensurable. Let w',m"be any two commensurable variables, both ^P ,. ' ' •^ . ' 'negative, approaching n as their common limit in such wise that always w'< m < m" : then ■.• A >1 and w', n" are both ^ Positive, p, ' ' negative, l jf .•. of a"", A"", the positive values both ^ ^1; [(a) and •.• the positive value of A" lies between them, [th.8,df. .• the positive value of a"^ ^ 1. q.b.d. 2. Let A < 1, and let n be ^ Po^itjye = ^ ' ' negative: then •■• ->1, A .-. ofA", =(i)"", the positive value ^ ^ 1 for -w<| i^egative, r-^ ^ ' > ' positive, >- ~ I positive. I.e., for?i-<^ .. Q.E.D. ' ' negative. ^ CoE. i/" A" = 1 , , if7ie« etifter a = 1 or n = 0. For if neither a=1 nor w = 0, then is a" larger or smaller than unity, which is contrary to the hypothesis ; .-. either a=1 or n = 0. q.e.d. 210 POWERS AND ROOTS. [VIII. Theor. 12. If there be a variable positive base a, and a constant -l^ ^'' l^ exponent n, then: ' negative ^ ' 1 . To each value of the base there corresponds one and but one , .^7 1 an increasing „' ,. ..■, , positive value of the power; -j ^ decreasinq -^ -^ ""'^^' 2. To each positive value of the power there corresponds one and but one positive value of the base i \ „ ^ • function of the power. 3. Tlie positive values of the power and of the base are con- tinuous functions of each other. 1. (a) n commensurable. [th.5,6 (6) n incommensurable. For let A" be the limit of a series of commensurable powers of a ; then ■.• each of these powers has one and but one positive value, .-. A" has one and but one positive value. q.b.d. So, let a', a" be any two values of a, whereof a'-< a" : then ■.• a": a'>1, .•.(a":a')"^ ^1; [lem. .-. a"», =(a":a')"-a'% { ^a'", i.e. , the larger the base, the i ^^^(^^ the ^ 1°^^^^^^ power. Q.E.D. 2. Conversely : •.• A = (A»)», .". to each positive value of a" there corresponds one and but one positive value of a ; <» an increasing ^ "^ ' ' a decreasmg tion of A". Q.B.D. [1 3. Let A, always increasing, pass in order through every positive value from to +oo : ,, „ I an increasing » .■ « ri then •.• A", i ^ decreasing ^^'^^^'^^ ^^ ^' C^ takes in order every value fromO", =^ 0,tooo», = ^ * [2 .•. A" is a continuous function of a. q.e.d. [df. So A, =(a")s, is a continuous function of a", q.e.d. §e.] CONTINUITY OF POWERS IN GENERAL. 211 Theok. 13. If there he a constant positive base a-{ ^„ ■' ^ ' smaller than unity, and a variable exponent n ; then : 1. To each value of the exponent there corresponds one and , . ... 7 ^ j7 I an increasinq „ .. „ but one positive value of the power; \ , . ^function of the exponent. 2. To each positive value of the power there corresponds one and but one value of the exponent; { j ■ function of the power. 3. The exponent and the positive value of the power are con- tinuous functions of each other. 1 . For, when the base aud exponent are given, there is one and but one positive value of the power, q.e.d. [th.l2. 1 AT iu- 1 • I an increasing «,•-,, , And this value is ^ ^^ decreasing ^"'^''*i°'^ °^ ^^^ exponent ; for let ni, nj be any values of n whereof nj > % ; then •.• A-2 = A"!-^ • A"i, [th. 10 and ••• of a"2""i the positive value ^ ^1, [lemma; W2—Mi>0 .•. of a"! the positive value ■{ ^ that of a"i. q.e.d. [II. ax.l6 2. (a) To each positive value b of a" there corresponds one value of n. For let b', b" be any positive variables such that always b'-J ^B") ^b") ^^^ approaching b as their common limit ; i and let variable commensurable exponents n', n" be so taken that always the positive value of A"' shall lie between b' and b, and the positive value of A""shall lie between b and b" : [th. 7(2) , cr. 1 then •.• of the variable commensurable powers a"', a"" one in- creases and the other decreases toward b as their common limit, [byp. and the exponents n', n" have a common limit n that lies between them, [th.7cr. 3cnv. .*. the value of this common limit is a value of the expo- nent n corresponding to the value b of the power a". Q.E.D. [df. incom. pwr. 212 POAVERS AND KOOTS. [VIII, (6) To each positive value b of a" there corresponds hut one value of n. For if A", A" each = B, then •.• a»-»=a":a" = b:b = 1, [th.lOer. -•. m— n = 0. [cr.tolem. th. 12 .-. m =n. Q.E.D., (c) Tlie exponent is \ l^^J^/^^J^^ function of the positive value of the power. For this is equivalent to the statement, already proved, that the positive value of the power is^ Tdecrlasing^ function of the exponent. [1 3. For to every value of the exponent there corresponds one and Tiut one positive value of the power, and conversely ; [1,2 and ••■ as the exponent increases the positive value of the p-r^ :m: trsTs', -<^ — -ly ■' ci' 2 .-. as the exponent passes in order through all values from ~oo to +00, the positive value of the power passes in order through all values from ■{ _^. i„ n ' and conversely, as the power passes from -j ^ to ' *^® . exponent passes through all values from ~oo to +00 ; and -.• the power is the limit of a corresponding commensura- ble power that changes infinitesimally when the ex- ponent changes infinitesimally, and conversely, ■ -•. every infinitesimal change in either the exponent or the power gives an infinitesimal change in the other, .-. both exponent and power are continuous functions of each other. q.e.d. [df. contin. func. § 7.] DERIVATIVE OF A POWER. 213 § 7. ■ DERIVATIVES OF POWERS. DERIVATIVE OF A POWER OF A VARIABLE BASE. Theok. 14. The derivative as to any variable base of a power of that base is the product of the given exponent into a power of the base whose exponent is a unit less than the given exponent. Let X be any variable and n any number ; then will d^£c" = w ■ a!"~^ (a, b, c) n commensurable. [VII. th. 17 (d) n incommensurable. For let n' be a commensurable variable independent of x and such that n' = n, and let x take any increment h ; then •.• a;"', K" take the increments (a;+7i)"' — a;"', (a;+/i)"— »% and •.• x"' = x'^, {x+h^' = {x+hY, as n'=n, [df. incom.pwr. .*. (x + h)"-' — x"' = (x + h)" — x" as n' = n, whatever h may be, [VII. th.7 i.e., inca;"' = inca!" , inca;"' . inca;" f . i. n • and = , as m' = n, however small inc a;, inca; ' inca; T inca;"' . ,. inca;" , . -, . . „ .-. lim — - — = lim-^ — - as n' = n and inca; = 0, inc X ^,c^ inc x i.e., D^x"' ^^a;". ' [df. deriv. But ■.' D^af' =w'.a;"'-\ [VII. th. 17 and ■•• n'-a!"'~^ = w-a!"~^ as n' = n, ' [VII. th.8 -•. D^a;" =n-a;""^. q.e.d. [VII. th.6cr. DERIVATIVE OF A VARIABLE POWER. Theor. 15. The derivative as to any variable of that power of a base whose exponent is the variable, is the quotient of the given power by a constant whose value depends upon the base alone. Let A be any base, m^ a certain function of that base, x any variable ; then will d^a" = a"' : m^. For let x take any increment h ; then •.• a'^+»-a» = a''.(a*-1), [th.lO 214 POWERS AND BOOTS. [VIII. A'+'^ — A.' , A*— I = A • , But 7t D,A* = A" -lira when A = 0. h lim ^—^ = M^"-'^) when A = 0, [VII. th. 18 h D.ft = DjA* when ft = 0, an expression free from x and a function of a only ; and •.•A* has a single value for any one value of h, [th.l3(l) .•. has a single value, h and lim , when h=0, has a single value, dependent h on A alone. Put — = lim^— ^, when7i=0, Ma ft then DjA°'=a':m;a. q.b.d. Cor. Ifebe such a number that m^ = 1, then D^e^ = e'^. Note. The function m^ is called the modulus of that system of logarithms whose base is a ; its value is found by methods in [XII. pr. 11 ] . The base e is the base of the Napierian system of logarithms. ^,, DEKITATITE OP A LOGAKITHM. J\i Theob. 16. Tlie derivative as to any variable of a logarithm of that variable is the quotient of the modulus of the system by the variable. Let X be any variable, a the base, and m^ the modulus of the system ; then will o^logAa; = m^ : a;. For put y = logiO; ; then ■.■ X =A«; [I. §11 df. log .•. Dj,a; = DjA'' = A'': Ma. [th. 15 But ••• D,2/= 1 : D^a; = Ma : A.", .•. Djlogos = Ma : a;. q.e.d. Cor. i>Jog^:s. = 1 : x. § 8.] EADIOALS. 215 §8, RADICALS. A radical is an indicated root of a number. There may be a coeflBcient ; and then the whole expression is called a radical, and the indicated root is the radical factor. A radical is ^ . , . , if the root ^ , be found and ' irrational ' cannot exactly expressed in commensurable real numbers [I. §1], or in rational literal expressions [I. § 12]. Its value is-{ . . .. ., I do not involve ., . - » .. if it-{ . , the even root of a negative. E.g., -^256, ^8, -^-8, -^a', ■^/ (a^ + 2 ab + b^) are radi- cals that have the rational values -2, 2, -2, a, ±{a + b), besides certain irrational values discussed later. But ^x, V*^) a/***) -fa -a"", f(a^ + &^)* are irrational, and V-1' V-«'' -v/-2«', |a.(-a)S 5(„ + 6y_i)f are iraagiuaries ; the first two of them commensu- rable, and the others not. An expression that contains a radical is a radical expression. A radical expression that cannot be freed from roots is an irrational expression, or surd [I. § 12]. An equation that contains surds is rationalized when it is re- placed by an equivalent' equation free from surds. E.g., the equation x = y/2, i.e., x = ^2 or ^2, when rationalized, becomes a? = 2. Eoots of raitional bases, and integral powers of such roots, with rational coefficients, if any, are simple radicals ; and a radi- cal is in its simplest form when its coefficient is real and entire, its exponent positive and less than unity, its root-index the smallest possible, and its base a real and entire expression con- taining no factor to a power whose degree equals or exceeds the root-index. If a simple radical be surd, it is a simple surd. The degree of a simple radical is the value of its root-index. 216 POWERS AND BOOTS. [VIH A simple radical is quadratic, cubic, quartic or biquadratic, ••• when the root-index is 2, 3, 4, •■•. E.g., f(a2 + 6^)*, 3a6^ -^(a^-ftc"), aP-a^, ^-3, are simple quadratic, cubic, and quartic surds in their simplest forms. But *Va'' -v/«S V8> V-S. Ua' ■" ^^^ conformable ; but t/(^ ■ 3) ' V^ ' V(2 •■ 3) , -^5 ; •••are non-conformable. ,218 POAVERS AND BOOTS. [VIII. th. Theor. 18. Tlie sum of a finite number of simple non-con- formable surds cannot be rational. Let Oj-^Ai, a2-^A2, ■■■ a^V-*^' — C/'^u i/Bi, — -{/s„, be any simple non-conformable surds ; and let c, as well as Oi, Ai, Bx, ••• a„, A„, B„, be rational : then the relation -^Bj + -{/b^ -\ (- •>/b„ = c is impossible. (a) One surd, c ^ ; or two surds, c = ; i.e., -(/Bi = c is impossible, q.e.d. [df. surd and -^Bi + -^Bj = is impossible. q.e.d. [th.l7cr.l- (6) Ttvo quadratic surds. If possible, let V^i + V^s = Cj then Bj + 2 •y'Bi 82 + 82 = 0^, .-. 2-^/8132 = 0^ — 81 — 82, i.e., a sm-d equals a rational number, [th.l7cr.2 which is impossible ; [df. surd .-. -^Bi4--^B2:?^ C. Q.E.D. (c) Three quadratic surds. If possible, let Vbi + V^2 + V^s = c ; then •.• -v/b2 + Vb3 = c— -v/bi, [hyp. .-. B2 + 2VB2B3+B3 = C^ — 2 c VBi+Bi, .-. 2c-v/Bi + 2Vb2B3 = c^ + Bi — B2 — B3. So, 2 c-^B2 + 2 VBaBi = c^ -t- B2 — B3 — Bi ; 2cVb3 + 2-V/B1B2 = c^ + B3 — Bi — B2 ; i.e., the sum of two non-conformable surds is rational, which is impossible ; [(6) or else -^Bi is conformable to -y/BaBg, ysato VB3B1, -^/Bsto VB1B2, and c^-l-Bi-Ba- Bg = 0^+ B2-B3- Bj = 0^+83— Bi- 82 = 0, whence Bi ^ 82 = Bg, and -v'bi ± -y'Bi ± -,/Bi = c , which is also impossible ; [(a) ••• VBi+VB2+ V^ST^C. Q.E.D. 18, § 8.] BADICALS. 219 (d) Any number n of quadratic surds, c = 0. 1. The assumed surd equations ' ViH- Vb24- Vb3= 0, Vbi+Vb2+VJ*3+ V^4= 0, •••, VBi+VBa+V^sH \-^/Sn= may 6e redlMced to the equivalent surd equations E3 = Sg -y/Bj B3, K4 = S4 -y/^s Bi) • ■ ° ) Kn = Si, -Y/Bn_i B,,, and to the rational equations T3=0, T4=0, ■•■, T„ = 0, wherein Eg, Sg, Tg are rational functions of Bj, Bj, Bg ; ^4) S4, T4, 0/ B], • • • , B4 ; • • • ; Kn, Sn, T^, 0/ Bi, • • • B^. For if Vbi + Vb2 + V^s = 0, then •.- Vbi = — •\/b2 — Vbs) .-. Bi — B2 — Bg = 2 VBsBg, I.e., ' E3=S3VB2B3, q.e.d. and (bi — B2 — Bg)^ — 4b2B3 = 0, i.e., T3=0. Q.E.D. So, in the lp,st two equations replace -y/Bg by -^Bg + -^/b^ ; then •.• [bi — B2 — (VBs+VBi)^^ — 4b2(Vb3 + Vb4)^ = 0, .-. SBi^ — 2SB1B2 + 8BgB4 = 4(Bi + Bs — Bg — B4) -y/Bg B4, i.e., K4 = S4VB3B4, Q.E.D. and E4^ — S4^B3B4 = 0, i.e., T4=0. Q.E.D. So, if the law holds true for k surds, it holds true for fc+ 1 surds. For in the equation Tj,= replace t/Bj by -y/s^ + Vbs+h i.e. , replace Bj by Bj + Bj+i + 2 VBjBj+i ; then •.• Ts = becomes k^+i = Sj+i Vb*Bj+i, q.e.d. •■• ^ k+l — S j^.iBjBj+1^ 0, i.e., Tj^.i = 0. Q.E.D. But ■.• the law holds true for 3 surds and for 4 surds, .•. it holds true for 5 surds, for 6 surds, ... for w surds. Q.E.D. 220 POWERS AND BOOTS. [VUI. th. 2. The assumed surd equations Ej = S3 VB2 B3, Ri = S4 V^S B4, • ■ • R„ = Sn V^n-l Bn are all impossible. For •.■ V^i) Vb2) V^s) ••• VB» are non-eonformable surds, . • . Vb2 Ba; Vbs B4, • • • VBn-iB„ are surds ; [th. 1 7 cr. 2 .-. in each of these assumed equations a rational number stands equal to a surd, which is impossible, [df . surd or else k„, e„', e„",' •-, s„, s„', s„", •••, all = 0, wherein K„',---are whatE„,s„beeome when Bi,"-B„are permuted; e.g., S4, 84', S4" are Bi + B2 — B3 — B4, Bj + B3 — Bj — B4, Bi + Bi — B2 — Bg ; and if e„, e„', r„", •••, s„, s„', s„", ■•• all = 0, then Bi = B2 = • • • = b„, and Vbi) ■ • ■ VBn are conformable, which is impossible, [hyp. ••• VBi+ VB2+VB3^0, VBi+VB2+VB3+VB4=?^0,—, Vbi+\/B2H I-Vb»=?^0. q.e.d. (e) Any number of quadratic surds, c =?t 0. Take Vd,.+i a simple surd, and Dj = Bi-d^i : c^ • • • d„ = b„- r>„+i : c^; then •.• -y/oi-i 1- Vo« =i^ Vd»+i) [(d) • •• VbiH b-y/'Un^G- Q.E.D. [mult, by (c : Vi>n+i) (/) Any number n, of surds not all quadratic, c = 0. 1. The assumed surd equations -^Bi + -^Ba + ^B3 = 0, VBi+ ^B2+ ^B3+ ^B4 = 0, Vb, + -^Ba + ^B3 + ... + ^B„ = 0, wherein each simple surd is in its lowest terms, may be reduced respectively to the equivalent surd equations E3=S3.V3, R4=S4.V4, •.•, K„ = S„ • V„, and to the equivalent rational equatiorls T3=0, T4=0, •••, T„=0, wherein Kj, 83,13 are rational functions of Bi, B2, B3 ; ••• ; R4, S4,T4, 0/ Bi, •••,64; •", R„, 8n, T„, o/Bx, •.•,B„. and T3 = the surd Bg ' • B3', = ^'Bf^'^^' , 1 1 V4 = the surd B3"' • B4", = -^83-" B4'", . . . , h =l.c. mlt. 0/ q, r ; q', r' = the integers h : r, h : q ; k =l.c. mlt. 0/ r, s ; r", s"= the integers k : s, k : r; •■-. and 18, § 8.] RADICALS. 221 For in the assumed surd equation -^Bj + -^Ba = 0, and in the equivalent rational equation b/=±B2'', replace Bj by i-VB,+ ^B,y, =B2(1+V3)«; then b/ = ± Bj'' ( 1 + Ta)^', .-. B,^=±B/[l + MV3+ ^^'^^^~-^^ Y3^ + -]. [bin.th. But this equation can contain not more than h—1 surds ; for if Vs', V3*+\ Va'^^ -"be present they are conformable toVs°, VsS V3^ •••; .*. the equation, reversed, reduces to the form Y,"-' + Ai Va'-^ + A2 Va'-^ + . • • + Aj_2 • Va + A,_i = 0, wherein Aj, A2, A3, ••• Aj_i are rational. Let X = Vs^-^H h Aj+i ; w = Vj* - Bj-'" B3'' , = 0. [df . V3 Divide "w by x : the remainder y has no power of V3 above Va*"^. So, divide X by T : therem'derzhasnopwr.ofvgabovevg*"^; •••; and •.• -y/'b2~'"b3'' is a simple surd in its simplest form, .-. V3*— Ba^^'Ba'', or w, has no rational factor, [df. sim. form -■. w, X have no rational common factor ; ,*. the divisions go on till a remainder is reached having only the first power of Vj ; and then, one free from Vg. Let Eg— S3V3, Ta=these remainders, wherein R3, S3, Tgare rational; then ••• ^^=0, x = 0, .-. each successive remainder is 0, i.e., Eg— S3V3=0, T3 = 0. Q.E.D. So, in the assumed surd equation -^Bi + -^Bg + -y/B3 = , and the equivalent rational equation T3 = 0, replace Bg by (Vb3+a/b4)% =B3(1+v,)- then the surd equation V*i,+ "v/^2 + -V^a + V^4 = is equiva- lent to an equation x'== Owith no surds but V4, ■ ■ • v/"^ Let w' = V4* — B3-*"-b/" ; and divide w' by x', x' by t', ••• ; then the final remainders give K4— S4Vi= 0, 14= 0. q.e.d. ' So for any number of surds. q.e.d. 2. The equations Es= S3V3, k^ = S4V4, ••• are all impossible. ' For Kg, K4, ••• are rational, and S3V3, S4V4, ••• are surds. ^ ^-X-J.B " 222 POWERS AND ROOTS. [Vni. prs. {g) Any number of surds not all quadratic, c ^ 0. For, if possible, let -^Bi + -^Bj H + Vn = c ; and multiply by -v/b„+i, any surd non-conformable to the others ; then ^B„+i • Vi H H =V(« — V^); ^j a+^(a^-b) l a-V(a^-6) ^^_^_^_ 226 POWBBS AND ROOTS. [Vni.pr.9 Note. Sometimes a square root of a surd of the form « + V^ + V^ + V*^ ™^y ^^ found. Write V» + V2/ + V'«=V(«+V6 + Vc + Vf^). then x+y+z+2^xyJt2y/ixz-\-2y/yz=a+-yJb + ^c+^d. "Write x+y+z=a, 2V«2/=V^) '^■\/xz=-s/c, 2y/yz = ^d, and find values for x, y, z that satisfy these equations. E.g., to find a square root of 9 + 2-^/3 + 2^b + 2^15. Write a; + 2/ + « = 9, 2^xy = 2^3, 2-^xz = 2-^h, 2y/yz = 2Vlo. then a;=l, 2/ = 3, z = 5, and the root sought is 1+V3+V5- PrOB. 9. To FIND A CUBE ROOT OF A BINOMIAL SURD. Let a+ -v/6 be a binomial surd, and x +-y/y= Va + -yjb, wherein x, ^y are to be found. Cube both members of this equation ; then •.• a^ + 3a;2/ + (3a^ + 2/)V2/ = a + V^ -•. a?' + 3a;2/=a, (Saj^ + j/) V2/ = V* ! [th.l8,cr.l and a!= + 3a!2/-(3a^ + 2/)V2/ = a— V&) i.e., {x—^yf= a-^h , .-. x—^y =-^a — y/b. But a; + V?/ = -Va + ^h. [ hyp. Multiply these last two equations together ; then •.• a? — y= ■y/a? — h = m, say, .-. y =x' — m. Replace y by a^ — m in the equation x' + Zxy = a; then v? + 3x{x^ — m)=a, i.e., ia^ — Smx = a. From this point on there is no general solution, but particular examples may be solved by finding a value of x by inspection from the equation 4 a^ — 3 mx = a. E.g., to find the cube root of 10 + 6 V3 ; then a=10, 6=108, m= -^(100-108) = -2; .-. 4a!2+6a;=10, -•• a;= 1, 2/ = 3, and 1 + V3 is the root sought. §10.] EXAMPLES. 227 § 10. EXAMPLES. 1 . Replace the radical signs by fractional exponents in : Va'; ^9^; ^^; -^{Vx'f); ^{h' + ^+f). 2. Replace the fractional exponents by radical signs in : § 2, THEOES. 2, 3, 4. • ••11. Multiply or divide as indicated : 3. (21a)*: (26)*; {x+Z)^ ■ {x - 2,)^ ; (20 a&) *• (5 ac)*. 4. (2a&c)*^(3acd)*^(66d)*; 2(a6c)*.-3(a2c)*:-4(&2c)i. ' 1 1 ,1 1 11 5. (a + 6)" • (a + &)" • (a — 6)" • (a - 6)" • (a^ + W)v+~': 6. a;* : »"* ; a* • a~* • a"^ : a"* ; c^-a"^: (a" • a^ • a~**) . 7. (5 a"* + 6 6*) • (5 a"* - 6 6*) ; {x^y + y^z^) ■ (a;* - y~^z^) . 8. {S- a* 6* + 6^) • (a* + &*) ; {x-"- - y") : («"* - y~^) ;, (_3a-5_,_2a-*6-i).(-2a-S-3a-*6). 9. (a* - a* + 1 - a~* + a~^) . (a* + 1+ «."*) . (a* - a"*) . a^. 10. (« - 2/) : {x-i - 2/"*) ; (cc" - rO : (a:* - 2/"*) • 11. (2a!«2/-»-5a^2/-"+7a^2/-i-5a!2+2iB2/):(a!22/-3-a!22/-2+a!2/-i). 12. Simplify the fraction : o^ — ax'^ + a^x''^ — x'^ a^ — a'x * + a^x-'^ ■ '-x-^ ■ ax '+a^x~ 13. Get the square of : (as_j,j)j. sJb-^xi; x-(ay)^; i^x'^ -2ar^xK 14. Get the cube of : 2(3 a)*; 3a*6~*a;2/-i; (a* -6*)*; 3i+-y/y; a~*-a;i 15. Express in simplest form : » 1 (a;' 2/°™)" (,86^j ' LU"'V J ' lA^' y) (a^if'y 228 POWERS AND ROOTS. [VIII. 16. Get the product of: [(a-*)* + |(a^&)^^*]-[(a-^)*-^(a*6)*^^].(a*+a*6*). 17. Get the square of : 2aix + 3axi — 2a'x~^ — 3a^x-' ; a.''h+a-'^b''ar^bK 18. Get the cube of : |a;*2/-i_|a;-*2/i; ^oT^ — ix'^a^ ; ^a'^b'^ + 5a^bK 19. Find the square root of : a;-2- ex-'y-^ + ^y~'; a-'^-ia'^ - 2a-*+ 12a-^+9a-^ ; 4a-^-12a"*6* + 96*+16a"*ci-246*c*+16c*; -1 ' ^ -L '-'' v^ x^ 2v^ a ^+ (asb)' —2b2'-x!'rxy ■ ^-^ h -^ ^-2 X ^y (xv')^ 20. Find the cube root of : ^ ^'^ ^a"*- 27a!-='a^ - (a^a)"* + (3a;-ia*)2- 21. Find the fourth root of : ■^a? - ^x^y~^+i^x^y~^ - 2bQix^y~^ + &2by-^. § 5, THEORS. 9, 10, 11. • ••25. Multiply or divide as indicated : 22. a!^'./':«^'; (24a) ^^ . (66)^/3 . {x + 3)\{x-3)\ 23. a;^^^a,^^^a,^^^ a^2.a^:aJ; (/^ - 6^«) . (/^ O • 24. loi^"'"* • • 102'»«^-, = 20 • 500 ; lO^-^^ - : iQiaiios...^ ^ggg : 20. 25. (/'-/') :(a:^^-2/^*); [a^%2(a&)^*+6^'] :(a^^+6^i). • ••28. Get the powers and roots as indicated : 26. (10l•»"«•■■)^ = 20^ (10»'W3)i, ^y2; (10"'^)^^% (lO")". 27. {2d/ix+3ax'^^y; (4a*^Vl2a^ia;^^+9/'') *; (a!^V^)^i. § 10.] EXAMPLES. 229 § 7, THEOKS. 14, 15, 16. In the following examples e is assumed to be such a number that M, = 1, and a is any constant. • ••32. Find the derivatives as too; of the variable powers : 111 L _i _i 29. ^•, a"; of; e'" ; a'"' ; a;"''; e*; a'; af; e'; a'; x'". 30. e''"; e« ; a^" ; ««"'; e"^ ; a"'"; »«' ; x'^' ; x^". 31. e''(l— a'); eCa+a^)' ; e-<^'«; a^(a'-a^) ; a-v'C'J^-^^). • ••35. Find the derivatives as to x of the logarithms : 33. log.e"; log„a"'; log^ af; log,e-»; log^a""'; logia!-^ 34. log. (a + 6 a;") ; log, [log, (a + &»;")]; log, (e"' - e""') . § 9, PKOB. 1. .••39. Reduce to simplest form : 36. 125*; 567*; 392*; 1008*; 216*; 72*; 162*; 48^; 160^. 37. (llif)* (6|i)*; (10^)^; (6f|)-*; 2500*; ^296352. 38. -^X^lx-^yz^; ^56a^6=c^ ^112a-=6-V; -^64tt'6-«cS 39. V(72a^&-726+18a-=6); -^[a;*y-i-a;?/2-3a!2(a;-2/)]. § 9, PKOB, 2. • ••43. Free from coeflBcients : 40.-6V5; 2V«; 2a;V2; 4aV5&; 4^6; ha^^y; fV^i- 41. W2&; 5 Vic; 27a^; tVCI)*! 1(1)^ ! |a;r'-(l2/«'-')*^ 42. 3a=V'2a''^'; k^^^V, Sa^iVa?/; ia&cV3a'6. .„ a-& j ffl+6 . a;V / g" \^ . /a^z.^aa. f a^-y . a + &\a-6' 2^ Wy ' ^ ^'^ \a!^+2a;y+2/2 § 9, PKOB. 3. • ••45. Reduce to the same degree : 44. a*, a*; a*, &*; 3*, 4* ; ^ab, -^ac, -^1^ -^Qi^-G). 45. a*, 6*; a^, 6*, c^; a;*, a;^, a;^, a;^, x^ \ (Sa;)*, 2^*, 42*. 230 POWERS AND ROOTS. [VIII. 46. Which is the greater : (i) * or (|) * ? -(/2 or -^3 ? ^9 or -^18? m^or (m+1)* whenm>3? § 9, PKOB. 4. • ■•51. Add or subtract as indicated : 47. V18-V8; V128-2V50 + 7V72; 6Vf-3Vf 48. 9V80-2V125-5V245+V320; 3V| + 4ViV- 49. 625*- ?• 136* + 8- 320*; 8.(|)* + ^-12* - |.27*, 50. 6(8a'6)*+4a(a'6*)*-125(a«60*; a^ft* + 2a6^ + 6^'. 51. 2^i + 37^; |(|fO)4_^ (108)1. |a&*-|(6:a-')*. § 9, PKOB. 5. • ••59. Multiply or divide as indicated : ' 52. 3V2-2V3; 8V6:2V2; ?>y/l-2,^l; 3*^2*. 53. 3V6-2V3-4V5:12V10; 4 V3-3 V5-5-^2; 2*.3*^4*. 54. TV-(f)*=a)-(H)*; 5*. 4*. 3^60^; (|)*:(i)*. 55^ (A)-(l)* = ^(f)~*; VK- &0:V(« -&):» • (a^ + W)^ ; V-a-V-6-\/-a-\/-&-\/-a-^-6-\/-a-\/-6^ § 9, PROB. 6. • ••64. Find the required powers or roots : 60. (3V3)^ (2^6)"; (V2-V3)^ (V10-V5)'; (3*-3-*)l «!• (Vi-VI)'; (2*-2-?)^ (3*-3-*)^ (4* + 4-*y. 62. (V|a!-|2/)'; [a»6(a»6c)*]* ; (2a!*2/*2i)<; [(6a!*2/-*)*]'* 63. i;y-2»a"'&'»c2'»; -^(27a'a;)*; (a*a!-i+ a-*a!)'- 64. [(a + &)*-(ci-&)^]''; (a*+&*-c*)'; (a*6-*+a-*6*)'. §10.] EXAMPLES. 231 § 9, PEOB. 7. ■ ••69. Eedace to equivalent fractions with rational denominators : 65 J-- -L.- _§_. 3V8 . 2x _i.. i^. /a\f. MW • V3' V2' 2V3' 2V2' 3^^ gl ' g^^f W ' UJ '' 2 . V2-1 . V3-V2 . 21 1+V5 V3 + 1' V2 + 1' V3 + V2' V10-V3' 3V5-2V3 15 . (a + b)i+(a-b)i V10 + V20+V40-V5-V80' (a + b)i-(a-b)i (3 + V3)-(3 + V5)-(V5-2) . 1 (5-V5)-(l+V3) ' «+V[^+V(c + V<^)] V2^(V3+l)-(2-V3) . V2-(V2-3) (V2-l)-(3V3-5)^(2+V2)' (V2+8)-(V3-V5) 71. Reduce to simplest form : 1 1 x+(a^-l)i , g!-(g^-l)^ a-(a^-af)i a+(a^-oiF)i' a;-(a^-l)*' x + Caj^-l)* (a)^4.1)^+(a^_l)i (a;^ + i)^_(a^_i)^ _ x-(iK'+l)i 72. In the equation (a;^— 2/*) : {x — y) = 3f+ a^y + a^2/^ + a;^2/' + xy*+y^ put a; = a^ and y = b^; thence find {a' — 6^) : (a* — 6^) , and apply this and similar results to reduce to equivalent fractions with rational denominators : 1 10 3^- 2^ . ^'5-^4 . 1 . 2a+&^ ai-bi' 2-^6' 3^+2^' ^5 + ^4' „s_j,j' ga-fei" 73. Show that (V^+V^+V")' (V«+V&-Vc)' {a+b-c-2^ab) form a complementary group ; and thence reduce to equivalent fractions with rational denominators : 1 1+V2 . i-yct . ■Va+V&+V«' 1 + V2 + V3' 1-V«-V&' ■ V2 - V3 . a 1 + V2-V3' V(&+Vc)+V('^ + Ve) 66. 67. 68. 69. 70. 71. 232 POWERS AND KOOTS. [VIII. 74. Find the value of: fa ^i /6 1 X '^^^ H 2_j^^ — ^ when x = l-s/3. 75. Show that y = i{e'-e-') if e'' = y + ^{1 + y'). § 9, PEOB. 8. • ••78. Find the square root of: 76. 7 + 2V10; 7 + 4V3; 2-^3; 16-6V7; V18-V16^ 77. 8V3-6V5; 75-12V21; V27 + V15; -9+CV3. 78. a6 + c2+V(a'-c^)(6'-c^); 2[1 +(l-c^)^] ; a!!/ -2 a; (0:2/ -a;^)*; l-2aV(l-a^)- 79. Find the fourth root of : 28-16V3; 49+20V6; a'+b'+Gab-iiah^+ah^). 80. If ^x+yy + ^z = ^(a+2^b + 2-y/o + 2^d) , show that X, y, z must satisfy the four conditions x + y +z = a, xy = b, xz = c, yz = d, and hence show that the square root of 6 +2 V2 + 2 V3 + 2 V6 may be found. 81. Find the square root of : 10 + 2 V6 + 2 VIO + 2 V15 ; 8 + 2 V2+2 V5 + 2V10 ! 15-2 V15-2 V21 + 2 V35 ; 11 + 2 V6+4V3+ 6 V2. 82. Show that the square root of 10 + 2 V6 + 2 ^U + 2 ^21 cannot be expressed in the form -^a + -y/b + y'c. 83. Find the square root of : 15 - 2 V3 - 2 V15 + 6 V2 - 2 V6 + 2 V5 - 2 V30. § 9, PEOB. 9. 84. Find the cube root of : 7 + 5V2; 16 + 8V5; 22 + 10V7; 38+17V5; 21 V6 - 23 V.5 ; 3a- 2a' + (1 + 2a^) V(l - a")- ths. 1-3, § 1.] GENERAL PROPEETrES. 233 IX. LOGARITHMS. § 1. GENERAL PROPERTIES. The logarithm of a number is the exponent of that power to which another number, the base, must be raised to give the number first named. [I. § II E.g., in the equation a' = n, a is the base, n the number ; and X the exponent of the power of a and the logarithm to base a of the number n. The equation a; = logi]sr expresses the relation last named. The equation N = logA~^a; means that n is the number, a', whose logarithm to base a is a; ; it is read, n is the anti-loga- HtJim of X to base a. E.g., 0=logAl and A = log~^0, whatever a may be. So, l = log22, 2 = log39, 3=log464, 4=log5625,..., and 2=log2-^l, 9=log3-'2, 64=log4-^3, 625=log5-^4, ••-. So, -l=log2i, -2=log3|, -3=log4^\, -4=logs^,-, and -l=logj2, -2=log,9, -3 = logj64, -4=log, 625 •••. If the base be well known it may be suppressed, and these two equations may then be written x = logN, n = log~^a;. If while A is constant N take in succession all possible values from to 00, the corresponding values of x when taken together constitute a system of logarithms to base a. Operations upon or with logarithms are therefore operations upon or with the exponents of the powers of any same base ; and the principles established for such powers apply directly to logarithms, with but the change of name noted above. Theor. 1. The logarithm of unity to any base is zero. [df. pwr. Theob. 2. The logarithm of any number to itself as base is unity. [df . pwr. ^ Theob. 3. To any positive base ■{ ^jj than unity, every positive number has one and but one real logarithm : J aninoreasing f^^^^^^^ ^f ^/^g number. [VIII. th. ] 3 ' a decreasing •> •> l V 234 LOGAEITHMS. [IX.ths, Note. If either the base or the number be negative, there may or may not be one real logaritlim. E.g., +100 has the logarithm 2 to base +10 or ~10, and both +10 and "10 have the logarithm ^ to base +100 ; but ~100 has no real logarithms to base +10 or ~10, " nor has +10 or ~10 a real logarithm to base "100. So, """lOOO has the logarithm 3 to base *10, and "10 has the logarithm -J to base '''1000 ; but """lOOO has no real logarithm to base *10, and """lO has no real logarithm to base """lOOO. In this chapter, and in general where logarithms to the base 10 nre used as aids in numerical computations, the number as well as the base is assumed to be positive unless the contrary be stated. / Theor. 4. If the base he positive and-{ ^^tiip^ H^f^n unity, the logarithms of all numbers greater than unity are { „„„„*.•.,„'. (if all numbers positive and less than unity., {„■*■„' [VlH.lem.th. 12 Theok. 5. If the base be positive andJ^ %, than unity, and if the number be a positive variable that approaches zero, then \^its logarithm approaches { jSre'/r,>%^' t^^^- ^^- ^^ Theor. 6. TJie logarithm of a ■{ ^ ,. . of two numbers is the 1 sum of the logarithms of the factors. ' rTryr-r ii on 1 excess of log. div'd over log. div'r. - • s. , ^■9-, logA(B-c:D) = logiB + logAC-logiT). Theor. 7. The logarithm ofa-{^°^^^ of a number is the < ?«£. ^f *'- '<'^-^'- ^/ ^'- — - 'y ^^ < IZtZL [VIII. ths. 4, 10 E.g., log, (B^ ^0) = 2 log^B + ilog,c. CoR. The logarithm of the square root of the product of two numbers is iJie half sum of their logarithms to the same base. [th. 6 lS-9; log^V(B-c) =l(l0g,B+l0g,c). 4-8, § 1.] GENERAL PROPERTIES. 235 Theok. 8. If the logarithm of any same number he taken to two different bases, the first logarithm equals the product of the second logarithm into the logarithm of the second base taken to the first base, and vice versa. Let N be any number, a, b two bases ; then will logAN = logBN.logiB, and logBii = logiN-logBA. For let y = logoN ; [df . log then •.• N =3", and logj^N = y ■ log^B, [th. 7 .-. 10giN = logBN-logAB. c Cor. 4. The modulus of any system of logarithms is the loga- rithm, in that system, of the Napierian base e. [VIIT. th. 15 nt. Let A be the base of any system of logarithms, and m^ the modulus ; then •.• logAa5 = logAe-log,a!, [th.8 wherein log^e is a constant, independent of a;, .-. D^logia; = logie-D„logea;, i.e.; ^ =log,e.-, ■ [VILth.16 .-. Mi' =logAe. Q.E.D. E.g., M„ = logioe = logi„2.71828- [ .- =.4342944—. 236 LOGARITHMS. [IX.th.9 § 2. SPECIAL PEOPERTIES, BASE 10. I The logarithm of an exact power of 10 is an integer, [df.log E.g., of-, 1000, 100, 10, 1, .1, .01, .001, - the logarithms to base 10 are ..., +3, +2, +1, 0, --1, -2, -3, .... But of any other number the logarithm is fractional or incom- mensurable, and consists of a whole number, the characteristic, and an endless decimal, the mantissa. [VIII. § 4 df. incom. pwr. As a matter of convenience the mantissa is always taken posi- tive ; and the characteristic is the exponent, positive or negative, of the integral power of 10 next below the given number. A negative characteristic is indicated by the sign — above it. E.g., of the numbers 2000, 20, .2, .002, the logarithms to base 10 are 3.30103.--, 1.30103. .., 1.30103..., 3.30103..., whose characteristics are 3, 1, 1, 3, and whose common mantissa is +.30103.... Theor. 9. If a number be multiplied {or divided) by any integral power o/lO, the logarithm of the product (or quotient) and the logarithm of the number have the same mantissa. For •.• the logarithm of a product is the sum of the logarithms of its factors. [th. 6 and •.• the logarithm of the multiplier is integral, [hyp. .-. the mantissa of the sum is identical with the mantissa of the logarithm of the multiplicand. q.b.d. So, if a number be divided by an integral power of 10. CoR. For all numbers that consist of the same significant fig- ures in the same order, the m,antissa of the logarithm is constant, but the characteristic changes with the position of the decimal point in the number. E.g., of the numbers 79500, 795, 7.95, .0795, .000795, the logarithms to base 10 are 4.9004, 2.9004, 0.9004, 2.9004, 4.9004. pr.l,'§3.] COMPUTATIDSf OP LOGAKITHMS. 237 § 3. COMPUTATION OF LOGARITHMS. PrOB. 1. To COMPUTE THE LOGAEITHM OP A NUMBER TO A GIVEN BASE. FIKST METHOD, BY CONTINUED FRACTIONS. Form, the. eosponential equation, a'' = n, wherein n is the number, a the base, and x the logarithm sought. [df . log Sy trial find two consecutive integers, x' and x' + l, between which X lies, and write x = x' + y"^; wherein x' is known and y"' is some positive number less than unity. In the equation a^ = n, replace x by x'+y~^, giving a^'+^"'=n, 1 and divide both members by a^', giving Ay= n : a'^', = n', say. Raise both members of the equation a? = n' to the jth power, giving a = n'^. By trial find two consecutive integers, j' and y' + 1, between, which y lies, write y = y' -|- z~^, and so on, as above. Then x = x'+- = x'+V l = x'+V 1 and the convergents, which approach x as their limit, are : I x'y'+l x'y'z' + z' + x' y' y'z' + l E.g., given 10' = 5, to find x, i.e. to find logioS. Put X =0+2/-!, then •.• 10» =5, 5''=10,y=l+- z 6'"^^ =10, i' = 2, 2'=b, z = 2 + - \4:J ' Vv 125' \125J 4' t and SO on. _1 _1_ _1_ _1_ 2+- 2+- 1 2 + - 1 and the convergents are ^ ^ + ' 2 7 65 ^' 3' 10' 93' ■"• 238 LOGARITHMS. [IX. pr. These convergents are alternately too large and too small ; but their errors are respectively less than 1. _1_^J_. 1 _ 1 . 1 3' 3-10 30' 10-93 930' 93 • next denominator' which denominator is not less than 93 + 10, =103 ; [VI.ths.1,2 .-. —, =. 69892 •", is too small, and differs from the 93' 1 true value by less than 9579 The true logarithm of five to seven decimal places, as shown 65 by the tables, is .6989700, so that — actually differs from it by less than half of one ten-thousandth. So, logio2 =logi„10 - logi„5 = 1 - .69897 = .30103. So, log4 =2.1og2= .60206; log8=3-log2 =.90309; log625=4-log5 = 2.79588 ; Iog|=log4-log5 = 1.90309. SECOND METHOD, BY SUCCBSSIVE SQUAEE HOOTS ^OF PEOBUCTS. Take two numbers whose logarithms are known, the one greater and the other less than the given number. Find the square root of their product and the logarithm of this root, the half sum of their logarithms. [th. 10 Multiply this root by whichever of the two numbers lies at the other side of the given number, and find the square root of the prod- uct, and the half sum of the logarithms of the factors ; and so on. E.g., to find the logarithm of 5 to the base 10 : Take 10 whose logarithm is 1, and 1 whose logarithm is ; Kumber. then V(lOXl) =3.16227766; V(10X 3.16227766) =5.62341325; V(3.16228 X 5.62341) = 4.21696535 ; ^(5.62341 X 4.21697) = 4.86967671 ; V(5.62341 X 4.86968) = 5.23299218 ; ^(4.86968 X 5.23299) = 5.04806762 ; V(4.86968 X 5.04807) = 4.95807276 ; ^(.6875 + .703125) = .69531 V'(4.95807 X 5.04807) = 5.00028680 ; ^(.69531 + .70312) = .69921 V(4.95807 X 5.00029) = 4.97709632 ; J(.69921 + .69531) = .69726 J(.69921 + .69726) = .69823 Logarithm. ^(1 + 0) = .5 *{l+.5) = .75 J(.5 + .75) = .625 ^(.75 + .625) = .6875 i(.75 + .6875) = .71875 ^(.6875 + .71875) = .70312 1, §4.] TABLES OF LOGAEITHMS. 239 § 4. TABLES OF LOGARITHMS. If for successive equidistant values of a variable the corre- sponding values of a function of this variable be arranged in order, the function is tabulated; the variable is the argument of the table [I. § 13] and the successive values of the function are the tabular numbers. The values of the argument are commonly placed in the margin of the table. If the logarithms, to any one base, of the successive integers from 1 to a given number, say 1000, or 10000, be arranged for ready reference, they form a table of logarithms. Such tables are in use to three places of decimals, to four, five, six, seven, and even ten, twenty, or more places. In general, the greater the number, of decimal places, the greater the accuracy, and the greater the labor of using the tables. For the ordinary use of the engineer, navigator, chem- ist, or actuary, four- or flve-place tables are sufHcient ; but most refined computations in Astronomy or Geodesy require at least seven-plq,ce tables,. Most logarithmic tables are arranged on the same general plan as the four-place table given on pp. 248, 249. Tliis table gives the mantissa only ; the computer can readil^^ supply the characteristic. To save space, the first two figures of each argument are printed at the left of the page, and the third figure at the top of the page over the corresponding logarithm. To save time, labor, and injury to the eyes, the computer should use a well-arranged table, and then train himself to cer- tain, habits. The best tables have, the numbers grouped by spaces, or by light and heavy lines, into blocks of three or five lines, and three or rive columns, corresponding to the right-hand, figures of the arguments of the table. The usual patterns are |0|12 3|4 5 6|7 8 9|0|12 3| ••■ for three-line blocks, and |0 1 2 3 4|5 6 7 8 9|0 1 2 3 4|-.- for five-line blocks', as in the table on pp. 248, 249. Instead of tracing single lines of figures across the page and down the column, the computer should learn to guide himself by correspondences of position in the blocks. 240 LOGARITHMS. [IX. pr. § 5. OPERATIONS "WITH COMMON' LOGARITHMS. PrOB. 2. To TAKE OUT THE LOGARITHM OF A GIVEN NUMBER. (a) One, two, or three significant figures. If the number have one significant figure, annex two zeros; if two significant figures, annex one zero; for the mantissa icrite the four figures that lie opposite the first two figures and under the third figure, and for the characteristic write the exponent of the power o/ 10 next below the given number. E.g., log 567 = 2.7536; log 5.6 = 0.7482 ; log .05 = 2.6990; If a number have more than three significant figures, the mantissa of its logarithm is not found in the tablo, but lies between two tabular mantissas whose arguments are two three- figure numbers next larger and next smaller than the given number. - [th. 3 E.g., mantissa log 500.6 lies between .6990, .6998, i.e., between mantissa logs 500, 501. (6) Four or more significant figures. Find the mantissa of the logarithm of the first three figures as above; subtract this mantissa from the next larger tabular man- tissa, and take such part of the difference as the remaining figures are of a unit having the rank of the third figure; add this prod- uct, as a correction, to the mantissa of the first three figures. E.g., to find log 500.6; then •.• log 500 = 2.6990, log 501 = 2.6998, [tables and log 501 -log 500 =.0008, 500.6 -500 = .6, .-. log 500.6 = 2.6990 + .6 of .0008=2.6995. Note 1 . If the given number lie nearer the larger of the two arguments, its mantissa is easiest found by subtracting from the larger of the two tabular mantissas such part of their dif- ference as the excess of the larger argument over the given number is of a unit having the rank of the third figure. E.g., to find mantissa log 500.6 ; then •.• mantissa logs 500, 501 = .6990, .6998, [tables and •.• log 501— log 600 = .0008, 501 —500.6 = .4, .-. mantissa log 500.6 = .6998 — .4 of .0008 = .6995. 2, § 5.] OPERATIONS WITH COMMON LOGARITHMS. 241 or Note 2. The rule for interpolating or applying the correction rests upon a property which logarithms have in common with most other functions, and which the reader may observe for himself if he will examine the table carefully, viz. : that the differences of logarithms are very nearly proportional to the dif- ferences of their ^umbers when these differences are small. They are not exactly proportional, but the error is so small as to be inappreciable when using a four-place table. The seven-place tables give the logarithms of all five-figure numbers, and the errors for the sixth, seventh, and eighth figures, as far as due to this cause, are inappreciable. So the rule above given " for ap- plying the correction" is universal. Note 3. The computer should train himself to find the correc- tion and add it to the tabular mantissa (or subtract it) mentally, and to write down only the final result. To aid in this mental computation, small tables of proportional parts are often printed at the side of the principal table. Two forms of such tablets are here shown : the first most accurate, and the other of easiest use. E.g., to find mantissa log 22674 ; then •.• log 227 -log 226 = .3560 -.354.1 = .0019, .•. the correction to be added to .3541 is .7 of .0019-1- .04 of .0019 ; and is found thus : opposite 7 find 13.3 or 13 .3541 opposite 4 find ^ _J- 4-14 Add ; the correction is 14 14 giving .3555 Or •.• 22700-22674 = 26, .-. the correction to be subtracted from .3560 is .2 of .0019 -f- .06 of .0019 ; and is found thus : opposite 2 find 3.8 or 4 .3560 opposite 6 find J^ 1 —5 Add ; the correction is 5 5 .3555 19 18 1 1.9 1.8 2 3.8 3.6 3 5.7 5.4 4 7.6 7.2 5 9.5 9.0 6 11.4 10.8 7 13-.3 12.6 8 15.2 14.4 9 17.1 16.2 19 18 1 2 2 2 4 4 3 6 5 4 8 7 5 10 9 6 11 11 7 13 13 8 15 14 9 17 16 242 LOGARITHMS. [IX. prs. PrOB. 3. To FIND A NUMBER FROM ITS LOGARITHM. (a) Tlie mantissa found in the table. Write down the two figures opposite to the given mantissa in the left-hand column, and following them the figure at the top of the column in which the mantissa is found. Place the decimalpoint so that the number shall he next abova. that power of 10 whose exponent is the given characteristic. ^.^.,log-'2. 7536=567 ; log-^ 0.7482 =5.6 ; log "^ 2. 6990 =.05. (6) Tlie mantissa not found in the table. Take out the first three figures for the tabular mantissa next less, as above; from the given mantissa subtract this tabular mantissa, and divide tlie difference by the difference between the tabular mantissa next less and that next greater. Annex the quotient to the three figures first found. Place the decimal point as above. E.g., to find log-^ 2.6995. then •.• log-^ 2.6990 = 500, log-' 2.6998 = 501, [tables and •.• 2. 6995 -2.6990 = .0005, 2.6998-2.6990 = .0008 ; .-. the number sought is 500 +(.0005 : .0008), = 500.6. Note 1. The process is but the inverse of that for taking out logarithms, and the reason of the rule is the same for both. This four-place table allows only one-figure corrections, and so gives only four-figure numbers. In general, an m-place table gives n-figure numbers ; but sometimes, when the mantissa is large, the mth figure may be two or three units in error, and then the number is approximate only for n — 1 figures [V. § 5]. Note 2. If the given mantissa lie nearer the larger of the two tabular mantissas, the correction may be applied to the larger argument by subtraction. E.g., to find log"^ .3555 ; then ••• the next tabular mantissas .3541, .3560 differ by .0019, and correspond to 226, 227, as arguments, and •.• .3555 -.3541 = .0014, .3560- .3555 = .0005, .-. the number sought is 226 + 14, or 227 — fVi = 22674. ■3-5/§5.] OPERATIONS WITH COMMON LOGARITHMS. 243 V If the tablets of proportional parts be used, the, work, written out, appears as follows : 14 226 ' or 6 227 13.S +.7 3^ -.2 .7 + 4 1.2 - 6 .8 226.74 1.1 226.74 PrOB. 4. To FIND, BY ONE OPEKATION, THE ALGEBRAIC SUM OF SEVERAL LOGARITHMS. Arrange the logarithms vertically, and take the algebraic sum of each column of digits, beginning at the right and carrying as in ordinary addition; if this sum for any column be negative, make it positive by adding one or more tens to it and subtract as many units from the next column. E.g., to find the algebraic sum ia the margin, 3.1037 adding upward, the computer says : — 0.6986 9, 7,16, 10, 17, +2.2409 1, 3,-6,-14,-11, 9, 2 off, -2.5892 -2, 3,-5, -1,-10, 0, 1 off, +1.2529 -1, 1,-4, -2, -8,-7, 3, loff, =1.3097 -1,-2, 0, -2, 1, and, adding downward, for a check, he says : 7, 1, 10, 8, 17, 1, 4,-4,-13,-11, ,9, 2 off, and so on. PeOB. 5. To DIVIDE A LOGARITHM WHOSE CHAEAOTERISTIC IS NEGATIVE. Write down the number of times the divisor goes into that mul- tiple of itself which is equal to, or next less than, the negative characteristic ; carry on the positive remainder to the mantissa, and divide. E.g., 4.1234 : 3 = ("6 + 2.1234) : 3 = 2.7078. So, 3.4770. I = 8.4310:2 =4.2155. 244 LOGARITHMS. [IX.prs. PeOB. 6. To AVOID NEGATIVE CHAEACTBRISTICS. Modify the logarithms by adding 10 to their characteristics when negative; use the sums, differences, or exact multiples of the modified logarithms where the subject-matter is such that the com- puter cannot mistake the general magnitude of the results. To divide a modified logarithm, add such a multiple of 10 as will make the modified logarithm exceed the true logarithm by 10 times the divisor; then divide. E.g., if loga = 2.3010, log6=1.4771, to find log (aH~*) , = i(21oga-31og6). BT TBUB L06ABITHMS. 2.3010-2 = 4.6020 1.4771 -3 = 2.4313 BY HODIFIED LOSABITHMS. 8.3010-2 = 6.6020 9.4771-3 = 8.4313 , 5 )8.1707 9.6341 5 )2.1707 T.6341 At each step of the work with modified logarithms, any tens in the characteristics are rejected, or tens, if necessary, are added, so as to keep the characteristics between and 9 inclu- sive. Before dividing by 5, in the example just above, 4 tens were added, making the dividend 48.1707. Note. The arithmetical complement of the logarithm of a number is the modified logarithm of the reciprocal of the num- ber. It is got by subtracting the given logarithm, modified, if necessary, from 10 ; it may be read from the table by subtract- ing each figure from 9, beginning with the characteristic and ending with the last significant figure but one, subtracting the last significant figure from 10, and annexing as many zeros as the given logarithm ends with. The arithmetical complement of the arithmetical complement is the original logarithm. E.g., ar-com 3.4908000 = 6.5092000, and conversely. In any algebraic sum, a subtractive logarithm can be replaced by its arithmetical complement taken additively. In most cases, however, the method of prob. 4 appears preferable. E.g., in the. example under prob. 4, the terms —0.6986. — 2.5892 might be replaced by 9.3014, 1.4108. •\6-8, § 5.] OPERATIONS WITH COMMON LOGARITHMS. 245 PeOB. 7. To COMPOTE BY LOGAKITHMS THE PKODUCTS, QUO- TIENTS, POWERS, AND ROOTS OF NUMBERS. 1 . For a product : add the logarithms of the factors, and take out the antilogurithm of the sum. 2. For a quotient: from the logarithm of the dividend subtract that of the divisor, and take out the antilogarithm. 3. For a power : multiply the logarithm of the base by the ex- ponent of the power sought, and take out the antilogarithm. 4. For a root : divide the logarithm of the base by the root- index, and take out the antilogarithm. E.g., to find the value of (.01519 • 6.318 : 7.254)* : HUMEEKS. LOGABITHMS. .01519 2.1815 X 6.318 +0.8006 ^7.254 -0.8605 2.1216x1 .001522. 3.1824 and the number sought is 0.001522 Note. Not only simple operations, as in the above example, but complex operations, can be performed by logarithms. Some- times the expression whose value is sought must first be prepared by factoring. E.g., to find the value of ■^/{h^—b'), wherein h,b are any given numbers and may represent the lengths of the hypothenuse arid base of a right triangle : then .^{h''-b') = log-^{]ogh-\-b + \ogh-b). PeOB. 8. To SOLVE THE EXPONENTIAL EQUATION A' = B. Divide the logarithm ofs by the logarithm of the base a of the exponential : the quotient is x, the exponent sought. For •.• Af = B, , ' .-. a; log A = log B, .-. a;'=logB wlOgA. Q.E.D. 246 LOGAEITHMS. [IX. pr. PkOB. 9. To ESTIMATE THE AMOUNT OP POSSIBLE ERROR IN A LOGARITHM OR ANTILOGARITHM GOT FROM THE TABLE, AND IN THE SOLUTIONS OF PROBS. 7, 8 : Let p he the number of decimal places in the table used; a', b', ••• x', (a"b''-")'j *'ie number of units of their last decimal places contained m a, b, • ■ • x, a" b" • • • ; a,fi,---,the possible rela- tive errors, all taken positive, of a., b, ••• : then (a) Poss. err. log x = 10"" + .43 pass. rel. err. x. (6) Poss. rel. err. x = 1 : 2x'+ 2.3 -poss. err. log x. (c) Poss. re?, err. A"" b"-" [inpr. 7] = 1:2 (a'-b"—)' + 2.3 (+m ++n + •••) ■ IQ-p + (+ma++n;8 +••■)• (d) Poss. rel. err. x [m pr. 8] ^ 1 10-P+.43a 10-" + .43/3 . 2x' log A logB For •.• DxlogioX = Mio -, [VIII. th. 15, a = 10 X .•. Mio = x-DjjlogioX =— 1-: i_=.43, [tablelogs inclogx . .43 incx xncx X .incx i.e., mclogx=.43- X , ine X . inc log x and = j^— = 2.3inc. logx: (a) -.• logx, as got from x by p-place logarithm-tables, has a , possible error composed of : two possible half -units in jpth decimal place, from the omitted decimals of the printed logarithm and of the correction for interpolation, inc. or err. of x and an increment or error, = .43 '- '- ; [above .-. poss. err. logx = (} + J)10-'' -1- .43 • ?^^ t: 10"* -|-. 43 -poss. rel. err. X. q.e.d. (6) •.• X, as got from logx by the same table, has a possible error composed of : 9, § 5.] OPERATIONS WITH COMMON LOGARITHMS. 247 a possible half-unit in last decimal place, for the omitted decimals, and an increment or error, = 2.3 • x • inc. log x, . • . poss. err. x = ^ in last decimal place of x-j- 2.3 • x • poss. err. log x, .-. poss. rel. err. X = 1 : 2x' + 2.3-poss. err. logx. q.e.d. (c) •.• poss. err. log A = 10-" + .43 a, [(a) .•. poss. err. log A" = m (10-* + .43 a So, poss. err. log B" —m(10-" + . 43/3), •••, .'. poss. err. log (a^b"-") = (+m++w + -") 10-" +.43(+ma++W;8 + •••), .-. poss. rel. eri'. (a"b"---) = 1 : 2x' + 2.3[(+m++w+"-)10-"+.43(+ma+-")], wherein x s a™ b" • • • ; .-. poss. rel. err. (a^b""-) = l:2x'+2.3(+TO++m+-")10-"+(+ma + ---)- Q.E.D. (d) ■.• X = logB : logA, . • . poss. rel. err. x= poss. rel. err. from omitted decimals of x +poss. rel. err. log A+poss. rel. err. logs [V. th. 5 cr. 3 = 1+10:^+^^10^+^. Q.,.,.C(a) 2x log A logB. Note. If in (d) the divisions logs: log a be performed by logarithms, then • . • log X = log log b — log log a, .-. poss. err. logx=poss. err. log-logA+poss. err. log-log B = 10^" + .43 poss. rel. err. log a + 10"" + .43 poss: rel. err. log b [ (a) = 2.10-" + .43f l^" + -^^- + ^°"+-^^^' \ log A log B .-. poss. rel. err. x, = — + 2. 3 poss. err. logx [(&) = _L + 4.6 . 10- + 1°:^+^ + 1°:^+^^, 2x' log A logB which differs from the former result only by the term 4.6 • lO"" arising from the omitted decimals of the table used in performing the division, and obtainable also from (c) by making m = n = l, a = /8 = 0. 248 LOGABITHMS. N 1 2 3 4 5 6 7 8 9 lO 0000 0043 0086 0128 0170 0212 0253 0294 0334 0374 n 0414 0453 0492 0531 0569 0607 0645 0682 0719 0755 12 0792 0828 0864 0899 0934 0969 1004 1038 1072 1106 13 1139 1173 1206 1239 1271 1303 1335 1367 1399 1430 14 1461 1492 1523 1553 1584 1614 1644 1673 1703 1732 15 1761 1790 1818 1847 1875 1903 1931 1969 1987 2014 16 2041 2068 2095 2122 2148 2175 2201 2227 2253 2279 17 2304 2330 2355 2380 2405 2430 2465 2480 2504 2529 18 2553 2577 2601 2625 2648 2G72 2696 2718 2742 2765 19 2788 2810 2833 2856 2878 2900 2923 2945 2967 2989 20 3010 3032 3054 3075 3096 3118 3139 3160 3181 3201 21 3222 3243 3263 3284 3304 3324 3345 3366 3385 3404 22 3424 3444 3464 3483 3502 3523 3641 3560 3579 3598 23 3617 3636 3655 3674 ETeos 3711 3729 3747 3766- 3784 24 3803 3820 3838 3856 3874 3892 3909 3927 3945 3962 25 3979 3997 4014 4031 4048 4065 4082 4099 4116 4133 26 4150 4166 4183 4200 4216 4232 4249 4265 4281 4298 27 4314 4330 4346 4362 4378 4393 4409 4425 4440 4456 28 4472 4487 4502 4518 4533 4548 4564 4579 4594 4609 29 4624 4639 4654 4669 4683 4698 4713 4728 4742 4757 30 4771 4786 4800 4814 4829 4843 4857 4871 4Ef86 4900 31 4914 4928 4942 4956 4969 4983 4997 5011 5024 5038 32 5051 5065 6079 5092 5105 5119 5132 6145 5159 6172 33 6185 5198 5211 5224 6237 5250 6263 5276 5289 6302 34 5315 5328 5340 6353 6366 5378 5391 5403 5416 5428 35 5441 5453 5465 6478 5490 5602 5514 5527 5539 6651 36 5563 5575 5587 5599 5611 6623 5635 5647 6658 5670 37 5682 5694 6706 5717 5729 5740 5752 6763 5775 5786 38 5798 5809 6821 5832 5843 5855 5866 6877 5888 5899 39 5911 5923 5933 5944 6955 5966 6977 5988 5999 6010 40 ■6021 6031 6042 0053 6064 6075 6085 6096 6107 6117 41 6128 6138 6149 6160 6170 6180 6191 6201 6212 6222 42 6232 6243 6253 6263 6274 6284 6294 6304 6314 6326 43 G3o5 C345 G'3~)5 6365 6375 6385 6395 6405 6415 6426 44 6435 C444 6454 6464 6474 6484 6493 6603 6513 0522 45 6532 6542 6561 6561 6571 6580 6590 6599 6609 6618 46 6628 6637 6646 6656 6665 6676 6684 6693 0702 0712 47 6721 6730 6739 6749 6758 6767 6776 6785 6794 6803 48 6812 6821 6830 6839 6848 6857 6866 6875 6884 6893 49 6902 6911 6920 6928 6937 6946 6955 6964 6972 6981 50 6990 6998 7007 7016 7024 7033 7042 7050 7059 7067 51 7076 7084 7093 7101 7110 7118 7126 7136 7143 7162 52 7160 7168 7177 7186 7193 7202 7210 7218 7226 7235 53 7243 7251 7259 7267 7276 7284 7292 7300 7308 7316 54 7324 7332 7340 7348 7366 7364 7372 7380 7388 7396 LOGARITHMS. 249 N 1 2 3 4 5 6 7 8 55 7404 7412 7419 7427 7436 7443 7451 7459 7466 7474 56 7482 7490 7497 7505 7513 7520 7528 7636 7543 7551 57 7559 7666 7574 7582 7589 7597 7604 7612 7619 7627 58 7634 7642 7649 7657 7664 7672 7679 7686 7694 7701 59 7709 7716 7723 7731 7738 7745 7752 7760 7767 7774 60 7782 7789 7796 7803 7810 7818 7825 7832 7839 7846 61 7853 7860 7868 7875 7882 7889 7896 7903 7910 .7917 62 7924 7931 7938 7945 7952 7959 7966 7973 7980 7987 63 7993 8000 8007 8014 8021 8028 8035 8041 8048 8055 64 8062 8069 8075 8082 8089 8096 8102 8109 8116 8122 65 8129 8136 8142 8149 8156 8162 8169 8176 8182 8189 66 8195 8202 8209 8215 8222 8228 8235 8241 8248 8264 67 8261 8267 8274 8280 8287 8293 8299 8306 8312 8319 68 8325 8331 8338 8344 8361 8367 8363 8370 8376 8382 69 8388 8395 8401 8407 8414 8420 8426 8432 8439 8446 70 8451 8457 8463 8470 8476 8482 8488 8494 8600 8506 71 8513 8519 8525 8531 8537 8643 8549 8655 8561 8567 72 8573 8579 8585 8591 8597 8603 8609 8615 8621 8627 73 8633 8639 8645 8661 8657 8663 8669 8675 8681 8686 74 8692 8698 8704 8710 8716 8722 8727 8733 8739 8745 75 8751 8756 8762 8768 8774 8779 8786 8791 8797 8802 76 8808 8814 8820 8825 8831 8837 8842 8848 8864 8869 '77 8865 8871 8876 8882 8887 8893 8899 8904 8910 8915 78 8921 8927 8932 8938 8943 8949 8954 8960 8965 8971 79 8976 8982 8987 8993 8998 9004 9009 9015 9020 9025 80 9031 9036 9042 9047 9053 9058 9063 9069 9074 9079 81 9085 9090 9096 9101 9106 9112 9117 9122 9128 9133 82 9138 9143 9149 9154 9159 9165 9170 9175 9180 9186 83 9191 9196 9201 9206 9212 9217 9222 9227 9232 9238 84 9243 9248 9253 9258 9263 9269 9274 9279 9284 9289 85 9294 9299 9304 9309 9315 9320 9325 9330 9335 9340 86 9345 9350 9355 9360 9365 9370 9375 9380 9386 9390 87 9395 9400 9405 9410 9415 9420 9425 9430 9435 9440 88 9445 9450 9455 9460 9465 9469 9474 9479 9484 9489 89 9494 9499 9504 9509 9513 9518 9623 9528 9533 9538 90 9542 9547 9552 9557 9562 9566 9571 9576 9581 9586 91 9590 9595 9600 9605 9609 9614 9619 9624- 9628 9G33 92 9638 9643 9647 9652 9657 9661 9666 9671 9675 9680 93 9685 9689 9694 9699 9703 9708 9713 9717 9722 9727 94 9731 9736 9741 9745 9750 9754 9759 9763 9768 9773 95 9777 9782 9786 9791 9795 9800 9805 9809 9814 9818 96 9823 9827 9832 9836 9841 9845 9850 9854 9859 9863 97 9868 9872 9877 9881 9886 9890 9894 9899 9903 9908 •98 9912 9917 9921 9926 9930 9934 9939 9943 9948 9952 99 995G 9961 9965 9969 9974 9978 9983 9987 9991 9996 / / 250 I^GAEITHMS. [IX. §0. § 6. EXAMPLES. 1. What is the logarithm of 144 : to base 2 V3? to base 2^12? to base (2^12)-i? 2. What is the characteristic of : log27? log;2? log321? log2i3? logj21? logj21-i? 3. Findlog5312o ; log7343-i; logjSl; log^343; log^343-i. § 3, PKOB. 1. 4. By continued fractions derive the logarithms, to base 10, of 3 and 7 to four decimal places. Thence find the logarithms of : 9, 2.7, .81, 70, 4.9, 343, 21, 63, .441, .7-', 18.9-i. § 5, PEOB. 2. 5. From the table take out the logarithms of : 12, 120, 123, 124, 123.4, 1.234, 12350, .001235. § 5, PEOB. 3. 6. From the table find the antilogarithms of : 1.0792, 2.0792, 2.0899, 2.0984, 2.0913, 0.0913, 4.0917. § 5, PROBS. 4-8. 7. By logarithms find the values of : 2^5^85^ V(97^-9^) V12--^65 ~^83.64 x 39.56^ S'T ' 81-^572 ' y5.-^.i8' .08145^^^1.968' 8. From the logarithm of 2 find the number of digits in : 2", 2^°", 5™, 20^ 160^ 25^, 6.25«-^, 25 ^ 50"™- 9. By logarithms multiply 575.25 by 1 .06 ^; by 1 .03^»; by LOIS*". 10. By logarithms find ^1000, -^.00010098, -^.0000000037591. 11. Whatpower is2of 1.05? 3 of 1.04? 4 of 1.03? 5 of 1.02? 12. If the number of births per year be 1 in 45, and of deaths 1 in 60, in how many years wiU the population double, taking no account of other sources of increase or decrease ? § 5, PKOB. 9. 13. Find the possible error in each of the examples inNos. 7-12. X. § 1.] DEFINITIONS AND GRAPHIC EBPRESENTATION. 251 X. IMAGINARIES. Before taking up this chapter, the reader may refer to what is said of numbers in I. § 1 and of negatives in I. § 3 ; and particularly the note at the end of I. § 3. He will observe that, for some kinds of quantity, negatives as well as fractions are impossible. He may not be surprised, therefore, to learn that, even if the operation denoted by imaginary numbers can be conveniently performed upon only one kind of magnitude, they have most of the properties of real numbers and play an important part in algebra. These operations can, however, be performed, though less simply, upon ^11 kinds of magnitude, as appears in chapter XV. § 1. DEFINITIONS AND GEAPHIO EEFEESENTATION. In measuring any thing some unit of like kind is first assumed, and the relation the thing measured bears to this unit, both as to magnitude and as to sense or quality, is expressed by a num- ber [I. § 1] . Conversely, this number expresses that operation which must be performed upon the unit to produce the thing : the unit being then the operand, the number the operator, ancj the thing the result of the operation. POSITIVE AND NEGATIVE NUMBERS. In the method of graphic representation of numbers here de- scribed, a finite straight line pointing in an assumed direction is chosen as the concrete unit ; and the relation that any straight line pointing in the { .. direction has to this unit is ex- pressed by a ■{ P , . number. f J 1 negative If the reader so place himself before the unit that to him it becomes horizontal and points to the right, > , then any horizontal line pointing to the right, >, has its length and direction in terms of the unit line expressed by a positive number. If the line be taken up and reversed, so that it is still horizontal but points to the left, -^ , then 252 IMAGIXAEIES. [X. the relation of its length and direction to the unit is expressed by a negative number. The length remains as before ; but the qualitj', or direction, is reversed. VECTORS. A vector, or directed right line, is any line whose length and direction are considered, but not its location. Its two extremi- ties are distinguished from each other as its initial point and its terminal point. Its direction is the direction of the terminal point from the initial point, and would be reversed if these points were interchanged. As the name implies, a vector may be regarded as the repre- sentative of the operation of carrying a particle from its initial point to its terminal point. The direction of a vector may be designated by the order in which its two extremities are named, or by an arrow-head. ^^ .. vectors are those having the same length and , the same direction. ' opposite directions. E.g., the vectors ab, cd, e are equal to each other, but are opposite to the vectors ba, dc,/. A vector quantity is any concrete quantity whose magnitude and direction only are con- sidered, and which is naturally represented by a measured and directed right line or vector. E.g. , the direction and velocity or force of the wind, or of an electric current, is a vector quantity, and may be represented by an arrow. COMPLETE EEVEESALS. When the operand is a vector, the operation of multiplying it by ~1 consists in reversing its quality or direction, and is ex- hibited thus : multiplier (operator) A B C D e / ^ product multiplicaud A (reanlt) (operand) § 1.] DEFINITIONS AND GRAPHIC EEPEBSBNTATION. 253 So, even if the operand be not a vector, yet if it have a re- versible quality, the vector oa may still be taken as the repre- sentative of the operand ;~ and, since to multiply the operand by ~1 is simply to change its quality into the opposite quality, this multiplication is grapliically represented by the reversal" aob, while the result is represented by the vector ob. Hence, iii what follows, the vectors used may be either the aotual operands and results, or merely their representatives. If 'now there be a continuous rotary, motion, as with a spoke of a wheel, the direction or quality of the vector oa is alter- nately reversed and restored : E.g., a half revolution, one reversal, is multiplication by ~i. So, a whole revolution, two reversals, is multiplication by ~1 twice, i.e., multiplication by (~1)^, =+1. So, a revolution and a half, three reversals, is multiplication by ~1 three times, i.e., multiplication by (~1)^, =~1. > < — H — j '' > > < » <- no rey'l. > 1 rev*!. 2 rev'l. 3 rev'l. So, multiplying a vector by "2 doubles the vector and reverses it ; multiplying by (-2)^ doubles it twice and reverses it twice ■, and so on ; and the like is true whether the operand be a vector or not. — > < — > < By such multiplication two distinct effects are produced : the one quantitative, the ordinary multiplication of arithmetic, which consists in stretching the line multiplied ; the other qualitative, which consists in reversing the direction of the line. Every such multiplier or number may be regarded as itself the product of two factors : its tensor, the quantitative or stretch- ing factor ; and its versor, the qualitative or turning factor. If the tensor -{ ^1, its effect is to-{ u'Jf j-p the multiplicand. E.g., the number ~3 is the product of tensor 3 and versor ~1. So, the number *f is the product of tensor | and versor *1. 254 IMAGINAEIES. [X. PAr.TIAL KETEKSALS. — UrAGIXAKIES. But during its rotation the line has filled various intermediate positions wherein the numbers expressing its relation to the unit were neither purely positive nor purely negative numbers : E.g. , in the positions are : intermediate, +2, •"■^ its relations or ratios to a unit — "•"2, intermediate, ~2, and are represented thus : These intermediate numbers are imagi- naries, or imaginary numbers, and may be defined as numbers, not 0, that are neither purely positive nor purely negative. By way of distinction, positive and neg- ative numbers, the ordinary numbers of arithmetic and algebra, are real numbers. It appears later [XIII.] that every imaginary number of ordinary algebra involves an even root of a negative, and arises from an attempt to violate a condition of maximum or minimum : as in seeking the base of a right triangle whose height shall exceed the hypothenuse. The square root of a negative real number is a pure imagi- nary ; all other imaginaries are complexes. E.g., the value of ^~i: is not +2, whose square is +4, nor ~2, whose square is also ''"4 ; it is something different from either, and intermediate between them in character. So, most roots, whether odd or even, and whether of positive or negative bases, have imaginary values, as appears later. THE SYMBOL -y/~l. The symbol y'"! denotes a number whose square is~l : i.e., it is a number such that unit X V'l X V~l=°'^i*><"l' [I-§ lOdf.root Hence, whatever meaning is given to multiplication by "1, a § 1.] DEFINITIONS AND GRAPHIC EEPEESENTATION. 255 consequent meaning must be given to multiplication by y'"! such that two successive multiplications by ^"1 shall produce the same result as one multiplication by ~1. If the unit be a horizontal line pointing to the right, then the product, unit X V-"-' i® ^ vertical line of unit length poiutihg either upward or downward ; for if the horizontal unit-line be first revolved to a perpendicular either way, then the same amount of further rotation will bring it to the opposite hori- zontal position. Here multiplication by y'"! consists in revolv- ing the multiplicand-line through a right angle, either anti- docJcwise or clockwise. So, when the unit or operand is any vector whatever, ~1 has two distinct square roots, say i and *', whose effects as multi- pliers are to revolve the line through a right angle anti-clock- wise and clockwise respectively. Hence the effect of i' as a multiplier is the same as if the multiplicand-line were first multiplied by i and then reversed, i.e., were multiplied by —i ; hence i' = —i, since both numbers give the same result when multiplying any same unit [I. § 1]. Since division is the inverse of multiplication, and consists in finding one factor when the product and the .other factor are given [I. § 9], to divide a given vector by i is to find another vector that, if multiplied by i, would produce the given vector. The quotient is the vector got by revolving the dividend-vector through a right angle clockwise ; for manifestly, when this quotient-vector is revolved through a right angle anti-clockwise, i.e., is multiplied by i, the original direction is restored. Hence, to divide ainy vector by i is the same thing as to multiply it by — i ; and, in like manner, to divide any vector by — t is to revolve it through a right angle anti-clockwise, i.e., to multiply it by i. E.g. , the unit > gives the products and quotients : unitxl unitx i unitX'l unitxl unitx— J unitx^l unit : 1 unit : —i unit : "1 unit ; 1 unit ; i unit : 1 A -> .< > < 256 IMAGINAKIES. [X- So, the unit \ gives the products and quotients : unitxl unitx '' unitX"! unit : 1 unit ; — i unit "1 unitxl unitx — i unitX 1 unit : 1 unit : i unit : '1 THE SYMBOLS -y/ 1) -\/ 1) BTO. The Qperation of multiplying by ~1 consists in reversing the quality- of the multiplicand, and is represented by one reversal of tlie line that represents the multiplicand ; and the operation of multiplying by y/'l is one which if twice performed reverses the quality of the multiplicand, and is represented by a half re- versal of the line that represents the multiplicand. So, multiplying by -^~1 is an operation which three times per- formed reverses the multiplicand, and it is represented by one- third of one reversal of the line. So, multiplying by -^"1 is an operation which four times per- formed reverses the multiplicand, and it is represented by one- fourth of one reversal of the line ; and so on. The representatives of -^"1, -v'""!) ■y/~'^i ••• are the rotatioife shown in the following figures, wherein lines of the same length as the unit make with that unit angles of ^ir, ^tt, -^tt, ••,•■ / MULTIFLE ROOTS. But-.- (-1)^=-1, (-1)==-1, (-!)»=-!, (-!)'=-!, ..., .e., •.• 1, 3, 5, 7, •■• (any odd number) reijersals has the same effect as one reversal, .-. -,/"! may be represented by one-Jialf of 1, 3, 5, 7, ••• reversals ; § 1.] DEFINITIONS AND GRAPHIC REPRESENTATION. 257 and ••• one-half of 5, 9, 13, ••• reversals are 2^, 4|-, 6|-, ••• re- versals, and have the same effect as a half reversal, and one-half of 7, 11, 15, ••• reversals are 3^, 5J, 7^, •■• reversals, and have the same effect as 1^ reversals, .•. ~1 has only two distinct square roots in this system. So, -^"1 may be represented by ^, f , -f, |-, ••• reversals : and -.' -I, J^,Jj4, ••• reversals = 2^,4^, 6-J, ••• rev'ls = -J rev'l, and |, i^,-^'-, ••• reversals = 3, 5, 7, ••• rev'ls = f rev'l, and J^, -y-, • • • reversals = 3|, 5f , 7|, • • • rev'ls = | rev'ls ; .-. ~1 has three cube roots represented b}' the curved arrow-lines of the figures, and but three. -ilT So, -^ 1 may be represented by \, f , f , f , reversals, and ~1 has four fourth roots I'epresented by the arrow-lines of the figures, and but four ; and so en. POSITIVE AND NEGATIVE EOTATION. Anti-clockwise rotation indicated by the figures is positive rotation, or rotation through a positive angle; and clockwise rotation is negative rotation, or rotation through a negative angle. E.g., in the third figure above the two arrows indicate posi- tive and negative rotation respectively : rotation through the positive angle fir and through the negative angle —\tt. The roots of ~1 represented by negative rotation are therefore identical with those represented by positive rotation when taken in reverse order. E.g., (-l)-i=(-l)f, (-l)-l = (-l)f, ei)-f = ei)i The reader may draw diagrams to illustrate. 5^58 IMAGINAMES. [X. MODULUS, ARGUMENT, VERSI-TENSOR. Every number considered in algebra, whether real or imagi- nary, may be expressed in the form r • ("1)", wherein r is the tensor or quantitative factor of the number, and (~1)" is the versor or qualitative factor. When the number r("l)" operates upon any vector, the result is a vector of like kind, such that r is the ratio of their lengths or magnitudes and n is tiic ratio 6 : TT, which their difference of direction, 6, has to two right angles. If m be an even number, r("'l)" is positive ; if odd, negative ; if fractional, some or all of the values are imaginary. The tensor r is also called the modulus of the number ; 0, =n7r, is its argument or versorial angle; and the number r- (■"!)" is a versi-tensor. Every abstract number, whether real or imaginary, may be regarded as a versi-tensor. E.g., +4, ~3, 2i, —i are versi-tensors whose tensors are +4, + 3, +2, +1, and versorial angles 0, tt, -J-tt, f tt. The reader should clearly distinguish between a vector and a tensor or versi-tensor. Vectors are lines, i.e., quantities or concrete numbers, and may represent any concrete numbers, operands, or results, that admit of the same progressive change of quality as vectors undergo ; but tensors and versi-tensors are abstract numbers, i.e., ratios or operators, and are here represented by the relations of lines as to length and direction. The product of any vector by a versi-tensor is a vector of like kind ; that of two versi-tensors is a versi-tensor [§ 3] . The properties of versi-tensors are here explained and de- monstrated by aid of the appropriate lines ; but they would be as true, though perhaps not as evident, if standing alone in their symbolic form. It appears presentlj' that versi-tensors are susceptible of all the ordinary operations of numbers when those operations are properly defined, and that the ordinary numbers of arithmetic and algebra are but special cases of these more general numbers. The same rules govern all sorts of numbers, and under these rules all sorts of numbers may be associated, and operated upon together without confusion or error. §2.] ADDITION AND SUBTEACTION. 259 § 2. ADDITION AND SUBTRAGTIOH-. In adding two or more numbers, two different results may be souglit: (1) the aritbmetic sum, or sum total, wherein no re- gard is paid to signs of quality ; (2) the algebraic or net sum, wherein the quality and relations of the numbers are considered. M.g., if a railway-train has run sixty miles east, and then forty miles west over the same track, the total mileage is one hundred miles ; but the distance it now stands east of the starting point is but twenty miles. So, if a sportsman walk ten miles east, then ten north and ten west, he walks thii-ty miles, but is only ten miles distant, and due north, from camp. So, if several forces not all parallel to each other be applied to a body at the same point, the effective thrust, their resultant, is a single force acting along a line that may be parallel to none of tliem and is less than their arithmetic sum. Two or more vectors are added by placing the initial point of the second upon the terminal point of the first, the initial point of the third upon the terminal point of the second, and so on, without changing. their lengths or directions'; and the •yecio?- smto is that line which joins the first initial to the last terminal point. E.g., of the three lines ob, bc, cd, below, the vector sum is the line od, whatever their length and direction ; and this group of three lines, so far as the effect is concerned, in carrying the point from o to d, maj' be replaced by the single straight line od. 260 IMAGINARIES. [X. In particular, the vector sum or difference of the two perpen- diculars of a right triangle is the hypotenuse ; and the vector sum of two adjacent sides of a rectangle is a diagonal. H.g. , in the figures below, ox -f- xp = op and ox + or = op. X X A Conversely, a line may be replaced bj' any group of two or more lines that form a broken line and have the same initial and terminal points as the given line ; and the diagonal of a parallelogram may be replaced by two adjacent sides. E.g., in the figures above od may be replaced by OB+BO-f cd, and OP by ox + xp, or ox + or. The lines added are vectors (carriers) , and their sum is a vec- tor that reaches from the first initial to the last terminal point. So, when abstract numbers, operators, are added together, viz., tensors, versors, and versi-tensors, their sum is a single operator that, acting upon a unit operand, produces the same result as if the several operators had acted separately upon the unit, and the results had then been added together. The sum of the several numbers is the same whatever vector be used as operand : for the vector sums got by using different operands, being obtained by like constructions, and so being homologous lines of similar figures, as also are the operands, bear like rela- tions to the respective operands. The components of a vector are any two perpendicular vectors of which it is the sum. A vertical vector has no horizontal component, and a horizontal vector has no vertical component. An operator that produces a vector perpendicular to the operand, or, more generally, that half reverses the quality of anything, is a pure imaginary; and an operator that produces an oblique vector is a complex imaginary. E.g., in the right triangle oxp, let oa be the unit of length, and let ox, xp be respectively parallel and perpen- dicular to OA, and contain oa, in length, a, b times ; §2.] ADDITION AND SUBTRACTION. 261 then the symbols a, M, a + bi, stand for numbers that act- ing as operators on the unit give the lines ox, xp, op. If the unit be horizontal, if r be the length of any vector, and 6 be its inclination to the unit, then rcos^ is the length of its horizontal component, and rsin^ of its vertical component. The horizontal component is produced by an operator whose tensor is rcos^ and whose versor is 1 ; the vertical coufponent is produced by an operator whose tensor is rsin^ and whose versor is i. Hence the oblique vector is produced by the opera- tor r (cosfl + isin^) ; and the operator r ( — 1)" [n — $:ir'] is equivalent to the operator r(cos^ -f-isin^). The first gives the number in its versi-tensorial form, as the product of a tensor and a versor ; the second in its complex form, as the sum of its two elements ; i.e., of a real number and a pure imaginary. If X, yi be the elements of any number, and r, 6 the modulus and argument, then a;=rcos^, y=r8'mO, r=y{9?+y'''), 6=ta.vr^{y:x). Any number r{cos6 + isi.n6) \s-{ %, than another num- ber, when its modulus or tensor, r, is-J =,, , than the modu- . lus of the other : it is ■{ ? than the other nujnber when its real element, rcoaO, is-{ P than the real element of the other. The relations expressed by the signs > , < are inde- pendent of quality or direction, and they depend only upon the lengths of the vectors produced, while the relations expressed by the signs >, < depend only upon the horizontal projections. A number is infinitesimal, finite, or infinite when its modulus is infinitesimal, finite, or infinite ; and the arguments of and 00 are generally indeterminate. H.g., -l±3i^l±i, -l±3iH-n'+n"+-_ For let OA be the vector unit, and let x, y, z, •■■, operating on the unit, produce the vectors ox, ot, oz, •••, and let oa, ox, •■■ sthe lengths of the vectors oa, ox, ••• ; then but and :r{-iy ■ OA, OP : ^r'(-l)"'-ox, p = r'{-iy-r{-iy-oI; o\iff ox OP ox = r • OA, and op = r' • ox Zaox = 6, and Zxop = 6', op = rr' ■ OA, and Z. aop = + 0', OP may be produced by acting on oa with the single operator whose ten- sor is rr', and whose versorial angle is 5 + 0', or whose versor is ("!)"+"'. r'(-l)"' • r(-l)" ■ OA = ?-7-'(-l)"+"' ■ OA, /(-!)»'. r(-l)" =?-?-'(-l)''+»'. Q.E.D. [df. product if OQ be produced when the operator whose tensor is r", = oz : OA, and whose versorial angle is 6", =aoz, acts upon op ; OQ is also got when a single operator, with a tensor rr'r", = OQ : oa, and an angle -f- 0'+ 6", = aoq, acts on OA, . . r" (-!)»" • r'(-l)"' ■ r(-l)" ■ oa = rr'r"(-'iy +"'+''" ■ ol, .-. r"{-iy"-r'(-iy-r{-lY = rr'r"(-lY+'''+"". q.e.d. Cor. If one number be divided by another, the modulus of the quotient is the quotient of the moduli, and the argument of the quotient is the argument of the dividend less that of the divisor. In particular, of a nvmber and its reciprocal, the moduli are reciprocals and the arguments are opposites. So, then 266 IMAGINAEIES. [X. ths. then and and and and Theor. 3. Multiplication is commutative and associative. Let X, y, z ••• he any numbers, severally equal to r.(-l)", r'^iy, r".(-l)"",-; the product 03 • 2/ ■ z ■ ■■■=r-r'-r' the product x-y-z--^-=r-r'-r" r-r'-r" — =r-r'-r" — , : n+n'+n"+- .(-■[\n+ni+n"+... ■(-l)"+*^'+^'+-, so for any other order or grouping, [II. ths. 1,3 . • . the product x-y-z — is the same whatever the order and grouping of the factors. q.e.d. ' Theor. 4. Multiplication is distributive as to addition, (a) The product of the sum of two numbers by a third : Let X, y, z be any three numbers ; then will z-x + y = z-x + z-y. ;^P Let OAbe any vector unit, and let X, y, z, operating on the unit, produce ox, OT, oz, and let r, r', r" ; 6, (!', 6" be the moduli and argu- ments of x, y, z. Complete the parallelogram xoY-p ; then OP = ox + OT. Turn ox, ox by the angle 6", and stretch them in the ratio r", making OQ = « • ox, or = « • oy. Complete the parallelogram qor-s ; then ••• OQ : ox = OR : OT, and Zxoq = Zyor, [constr. .-. OxOYP is similar to Oqors, .•. Zfos = 6" and os = z-o¥; OS = OQ + OR, ~' ■z-ox. + z-oi, Q^' and 2 • OP = a • ox + z: ■ OY ; i.e., z • ox + OT : .-. z-x + y-oA = z-x-oA + z-y-OA = z-x + z-y-oA, .-. the product z-x + y = the product z-x+z-y. q.e.d. 3, 4, § 3.] MULTIPLICATION AND DIVISION. ' 267 (6) The product of the sum of;three or more numbers by another : Let X, y, z, ••■ he three or more numbers, and v another ; then m\\v-x + y-{-z-\ =v-x + v-y + v-«+ ■ For v-Xrhy -{-«-\ = V ■ X -\- V . y'-\- z -\ £(a) = v-x + v-y + v-z-{ = v-x + v-y + v-z + v — = V-X + V-y +V,-Z^ . Q.E.D. (c) TTie product of two or more polynomials : Jjetx + y + z + ■■•■, x'+y'+z'-] , be two polynomials ; then x + y + z-\- ■■■ • x'+y'+z'-\ = x-x'+y'-{-z'-\ + y -x' + y' + z'-\ + z.x'+y'+z'+-. + ..' = x-x'-\-x-y'+x-z'-\ -i-y.x'+y-y'+y-z'+-- + z-x'+z-y'+z-z'-\ + •". Q.E.D. So, if this product be multiplied by a third polynomial ' x" + y"+z''-\ , a fourth, and so on. CoK. The product of two or more complexes is the product of the sums of their elements used as polynomials. Let X, y be any complexes such that x =p + qi, y =p'-\- q'i ; then will x-y=p + qi-p'+q'i =pp'— qq'+i (pQ'+p'q) . So for three or more factors. Note. If x, y be put in the trigonometric form r- (cos^ + isin^), r'- (cos0' + isin6'), then x-y = rr'- [(cosflcos^'— sin^sin^') + i (sin cos 6' ■+■ cos sin ^') ] = rr'- [cos {e + $') + isin ((9 + «')]• [trig. So for three or more factors. 268 IMAGnSTARIES. [X. th. §4. POWERS AND BOOTS. A ^ P°^^ IT^ integral power, [?-(~l)'']=''°', of any versi-tensor r(-l)", is the continued i P™ J^J^ 1 X r{-iy >f. r(-l)" ••• m times, = !>*(~1)"]», is the pth power of any versi- tensor whose gth power is r(~l)": i.e., its effect when multi- plying any vector quantity is to stretch the multiplicand in the ratio r', and to turn it as would the versor [(~1)"]', or - times as far as would the versor ("!)". 5, §4.] POWBES AND BOOTS. 269 An incommensurable power [r(~l)'']"' [m incommensurable] , is the limit of [r(~l)"]'"', wherein m' is a commensurable vari- able whose limit is m: i.e., [{-("l)"]" denotes the versi-tensor whose ■ g and prime to h, this difference of arguments is not a multiple of 2 tt, all Jc of the ~th powers are distinct in value ; Jc and ••• their Jc arguments all differ by multiples of — — > iC when taken in order, after rejecting all entire multiples of 27r, each differs from the next by — ; iC i.e., the arguments of the powers are equidifferent. q.e.d. Cor. 2. (a) If a commensurable exponent m' approacJi some limit m, whether commensurable or incommensurable, tJien every value of tJie potoer [r ("1)°]""' approaches some value of [r("l)'']"' as a limit. 272 IMAGINABIES. [X- tl^s- (6) If m be incommensurable, the argument of the power [r(~l)°]'" may be indeterminate. For •.• (a) the common modulus +7'"'' of all values of [?•("!)"]"' approaches as a limit the modulus +r'" of [?•("!)"]"', and •.• the argument m' {n-\-2k)7r of any particular value of [r(~l )"]'"' approaches as a limit the argument wi(n + 2]c)ir of that corresponding value of [)-(~l)"]"' which is in the same series, .-. every value of the power [r("l)"]'"' approaches some value of [r("l)'']'" as a limit, q.e.d. And ••• (6) as the commensurable m' approaches the incom- mensurable limit m, the successive convergents have larger and larger denominators, [continued fractions .*. the number of distinct values of the m'th power in- creases without limit as m'^m; and ••• for any value of m' these numerous values of the power have their arguments equidifferent, , . asTO'=m the arguments of consecutive values of the power approach one another more and more closel}-, and in the limiting case, when the exponent is the incom- mensurable m, the argument of the power may be regarded as quite indeterminate, i.e., as continuous. Q.E.D. Note. By convention, however, the values of an incom- mensurable real power of a real positive base are often re- stricted to the single real positive value. So, by convention, every power of the Napierian base e [XII. th. 28, ap. 4, cr.J is restricted to its real positive value, though the powers of the equivalent number 2.71828 •■• are nol so restricted ; i.e., Ve=l-64872... only; but V2'71828-"= ±1.64872"-. 6, 7, § 4.] POWEES AND ROOTS. 273 Theok. 7. The product of like powers of two or more bases is the same power of the product of the bases. e e' Let the bases be r(~l)"-, r'(~l)^ •••, whose moduli are r,r',---, and whose arguments are $, 0', ■•• ; and let m be any real exponent : then •.• of the powers [r(-l)S]'», [r'(-l)i?]'", -r, the moduli are r", r'"*, •■•, and the arguments are m6, mO', ••• ; e_ e' .-. of the product of powers [r(-l)'^]'"- [r'(-l)'^]'"." the modulus is ?•"'•?•''"•••,= (r »•'•••) ", and the argument is m$ + m6'+ •••, =m{6 + 6'+ •••) ; and ••• the product of the given bases has modulus rr'--- and argument + 6'+---, -•. its mth power has modulus (rr' •••)'" and argument m,{6 + 6'-\- ■••) ; i.e., the product of mth powers of the bases, and the mth power of the product of the bases, have the same modulus and argument, and are equal, q.e.d. Cor. The quotient of like powers of two bases is the same power of the quotient of the bases. Note. When the exponent m is commensurable, and the arguments 0, 6', •■• of the given bases are so related that the values of their sum 6 + 6' + ■•■ cannot differ from one another except by certain of the multiples of 2 tt, it may happen that the power of the product or quotient has more distinct values than the product or quotient of the powers. [comp.VIII. th. 2 nt. 2 JE.g., let two given bases, and their products, be 1 + i, 2i, — 2 + 2 1, whose moduli and arguments are : V2, i7r + 27l7r; 2, ^7r + 27cir'; V8)f'r + 2Z7r, wherein h, k, 1 = any integers, positive, negative, or zero ; then, in general, (1 + 0°* ■ (2i)" = ("2 + 2i)'" ; i.e., every value of either member is a value of the other member, for the modulus of either member is +82, and the argument of either member is m • (fir + h27r), wherein I, —h + k, is any integer whatever. 274 IMAGINAKIES. [X. ths. 7-9, But if m = |, and if it happen that in the investigation from which the bases 1 +i, 2i, -2 + 2i arise, the 2i is got as the square of the 1 + i, while the ~2+2j presents itself independentlj", then •.• A; = 2 /i, while Z remains unrestricted, .-. the argument of (1 +i)"' (2i)'" is i[fir + (/i + 2/i)27r], =i7r + /i-27r, while the argument of (~2 + 2i)'" is |(i,r+Z-2^), =i,r+Z.|7r; i.e., the product of the powers has only one value, while the power of the product has three distinct values. Theoe. 8. The product of two powers of any same base, in any same series, is that power of the base whose exponent is the sum of their exponents, and is in the same series. Note. Different powers of a base are in the same series, when they arise from attributing to the base the same argument and not arguments differing by one or more entire revolutions ; i.e., when their bases are identical and not merely equivalent. [comp.VIII.§l, Vlll.th.lO Let the base be a, =r(~l)s, whose modulus and argument are r and 6 ; and letp, q, •■■ be any real exponents^; then •.• A", A', ••• A''+'+'" have the moduli r^, r«, ••■ rJ'+«+- and the arguments p6, qQ, ••• {p + q-\ )6, [th. 5 and • . • r" • r« • • • = r''+»+-, [VIII. th. 10 and pe + qe+-={p + q + -)e, .•. the product of the moduli of a*, a', ••• is the modulus of A*'+«+-, and the sum of the arguments of iP, a', ... is the argument of A*+»+- ; i.e., the product A^-A'..- =A^'+'" ; q.e.d. [th.2 and •■• the argument of this product is {p + q-\ )0, and not {p + q-\ \-2kTr)$, .'. the product is in the same series as the factors, q.e.d. CoK. The quotient of two powers of any same base, in any same series, is a power of the base whose exponent is the differ- ence of the given exponents; and it is in the same series. pr.l, §4.] PO WEES AND BOOTS. 275 Theoe. 9. A 'power of a power of any base is that power of the base whose exponent is the product of the given exponents. Let the base be a, = r("l)"- ; and let m, n be any exponents ; then • . • A™ has the modulus r" and the argument mO^ .-. (a™)" has the modulus (r"")" and the argument n{mff) ; i.e., (a"")" has the modulus r"*" and the argument mnO ; but A"" has the sanie modulus r"" and argument mnO ; .-. (a'")" = a""'. q.e.d. Note. If a base b be not identical with a" butbhly equivalent (th. 8 nt.), and if n have a denominator q ; then b" may have values not included among those of a"" ; and Theor. 9 may be stated as follows : Of any 'number known merely to be equivalent to a given power of a given base, any given power includes among its values all values of that power of the given base whose exponent is the product of the given exponents. [comp. VIII. th. 4 nt. E.g. , if 6, the argument of a, be a + 2 hir, and if B, = a" but ^ a"*, have argument m$ + 2JcTr, wherein h, k may take in succession all integral values, then ••• A"? has the argument — ■ a + -!-±- -2 it, and ■.• B» has the argument — • a + — • 2ir, .-. B« takes every value of a"?,, but it may be that b« takes other values besides. Pkob. 1. To FIND THE mth boot of ant real number, ±a" : Put X for the roots sought; then: To find the nth root of a", write s =a (~1)", a(-l)s, a(-l)i - a(-l/"^; [th. 5 I.e., I.e. f 2ir 2ir\ write x = a(cosO + ismO), a( cos^ + ism — J, To find the nth root of — a", write x = a ("1)°, a(-l)5, 5 2n— 1 , a(-l)i, ••• a(-l) n ' / ■T,..ir"\ / Stt... 3ir\ write x = a cos- + ism- , a cos \-\sin — , 276 IMAGINAEIES. [X. pr. 1. To find the square root of a? : then ••• a;= a(cosO + isinO), a(cosfir + isin-|7r), and •■• cosO = l, sinO = 0; cos7r = — 1, sinTr = 0, [trig. .-. x = a, —a. 2. To find the square root of —a": then ••• a! = a(cos^7r + isin^7r), a(cosf 7r+ Jsinf tt), and ••• cos^ir = 0, sin-|-ir= 1 ; cosf 7r= 0, sin |-7r= — l, [trig. .•. x= ai, —ai. 3. To find the cube root of a' : then •.• a; = a(cos0 4-isinO), a(cosf 7r + f sin|7r), a(cos|^7r + i sin|-7r) , and •.■ cosO = l, sinO = 0; cos|7r = — ^, sia^7r = ^^3; cos|7r=-|, sin|,r = -|V35 [trig. 4. To find the cube root of — a' : then -•. a; = a(cos|7r + isin^7r), a(co8f7r + «sinf x), a (cos f TT + 1 sin|ir) , and •.• cosj7r=J, sini7r= J-y/^ ; cos7r = — 1, sin7r = 0; cos fir = I, sin|7r = — -^--y/S, [trig. .-. x=ia{l+i^8),—a, ia{l—i-y/3). 5. To find the fourth root of a* : then -.• a!=a(cosO + isinO), a(cos|7r + isin|7r), a(c03f7r + J8in|7r), a(cos|7r + JsinfTr), and ■.• cosO = l, sinO=0; cosi7r=0, sin^5r=l; cos7r=— l,8inir = 0; cosf?r=0, sin|w = — 1 ;[trig, .-. x = a, ai, —a, —ai. 6. To find the fourth root of —a*: then •.• a;=a(cosi7r + isinj7r), a(cos|ir + isin|7r), a (cos Itt + isinf tt) , a (cos|t + «sin|ff) , and •.• cosiir = Vi5 Binj7r = V^; cosf7r= — Vii sinfrr = V|; cos|7r = — Vi, sin|7r = — Vi; C0s|7r = VJ, 8in|7r = — t/^; [trig. .-. x=ia(^2 + i^2), ia(--y/2+i^2), ^a(-V2-iV2), ia{^2-i^2). 1. 2, § 4.] POWERS AND ROOTS. 277 7. To find the fifth root of a^ : then,'.' a; = a(cosO + ismO), a(cosf ^r + isin|5r), •••, and '.' oosO=l, sinO = 0; eos|7r= i(-y/5 — 1), sin|,r=iV(10 + 2V5); -, [trig. -•. x = a, ia[(V5-l)+iV(10 + 2V5)], ia[-(V5 + l)+iV(10-2V5)], i« [— (V5 + 1) — «■ V(10 - 2V5)]> ia[(V5 - 1) -» V(10 + 2V5)]- 8. To find the fifth root of -a=: then '.■ x=a(cos^Tr + ism\ir), a(cos|Tr + «sin|7r), •••, and .-. cos|^ = i(V5 + l), sin|ir=^V(10-2V5); - [trig. .'. a; = ia[(V5+l)+JV(10-2V5)], ia[-(V3-l)+*V(10 + 2V5)], -«, ia[-(V5-l-iV10+2V5)], ia[(V5 + l)-iV(10-2V5)]. And so for other roots. PbOB. 2. To FIND THE nth ROOT OF AN IMAGINARY a + 6t : write r = ^ (a^ + b^) , 0= tan~^ (bra) ; e then a + bi = r ("I)'', =r(cos^ + isin6) ; anc? (a + bi)n = rn(~l)uT, r5(~l) n^r , rE(~l) n"- , •••, if 6 , . . 0\ i/ ^ + 27r , . . e + 27r\ = ra cos - + 1 si«- , ra cos 1- i sm- — ■ , ••• \ n ny' \ n i / E.g., to find the fourth root of 1 — -y/"3 : then ■.' a=l, b = --y/3, r = 2, ^ = f7r, |n-, J^tt, ^O-tt; .'. l-V-3 = 2(-l)f, 2(-l)i, 2(-l)¥, 2(n)¥, = 2 (cos -|7r + i sin -fir) , 2 (cos fir + * sin fn-) , 2(cos^+ isin-i/-7r), 2(cos^/ir+isin^7r), .-. (l-V-3)*=2i(-l)*, 2i(-l)t, 2i(-l)*, 2i(-l)^ = 2i(cos|Tr + 1 sin^Tr) , 2i(cosf tt +i sinf tt) , 2i(cos^+»sin^7r), 2J(cos-i/ir +»sinJgair), = 1.0299 + .59451, - .5945 + 1.0299*, -1.0299 -.5945 1, .5945 - 1.0299 1. 278 IMAGINABIES. [X. § 5. ABRIDGED EEPEBSENTA.TION. In many important applications of the theory of imaginaries their representation is abridged as follows : A fixed point or origin o is chosen, and a unit-line oa point- ing to the right is taken as the common operand of all the Imaginaries to be represented ; then, of anyimaginarj- op: oa, op is the representative vector^ and p is the representative point; for, since the operand oa is the same for all the imaginaries, the resulting vectors or even their terminal points are sufficient to distinguish one operator from another. In this abridged representation, the thing chiefly present to the mind is the point p ; and every number, real or imaginary, is conceived to be written at its representative point, in the plane oap. E.g., if p, Q be the representative points of any numbers p, q, then p is further than q^ from ^the^origin ^^^" !> ^ ^?; the middle point of pq is the representative point of i{p + q) ; and PQ is the representative vector of (g —p). So, if A, B, c, D be the representative point? of a, 6, c, d respectively, and if a + c = 6 + d, then abcd is a parallelogram whose equal sides ab, dc are representative vectors of the equal numbers (6 — a) , (c — d) , and whose centre is the representa- tive point of i (a -)- 6 -J- c -f- d) . If a variable pass from one value to another by continuous change, then its representative point moves along some locus, the pa true. Two statements are equivalent if one be both a necessary and a sufficient condition of the. other, i.e., if they be false together and true together. Two or more statements are { ^'^^°°'''^ ^ when, if some of ' incompatible them be true, the others must be { }^^ \ independent when, whichever of them be true or false, the rest may, just as well. be true, or be false. rp, „ I necessary conditions , associated Inere are n < . ,.".. among m < . ,., , ' contradictions ° > incompatible statements when some m— » of these statements are independent, and if these be true, the remaining statements are ■{ ,, }^ ' E.g., the equation a = 3 is equivalent to the equation 2a; = 6, and it is a sufficient, but not a necessary condition, of the in- equality a; < 4. The three statements are associated and have two necessary conditions among them, since, if the first be true, so are both the others. The last two are necessarj^ conditions of the first ; the third is not, but the second is, a sufficient condition of the first. So, the inequalities x=a\ [^a(-l+iv3)]'' = a=, [_ia{-l-i-y/S)Y=a\ Equations that involve the same unknown elements, and are satisfied by the same values of them, are simultaneous equations ; and those values are simultaneous values. E.g.,ii the equations 2a; + 52/= 19, 6a; — Sj/^S [a;, ?/ un- known] be simultaneous, 2, 3 is a pair of roots, •.• 2.2 + 5-3 = 19, and 6.2-3-3 = 3. So, of the simultaneous equations x — y=:5, x' + y'=lB 2,~3 ; 3, "2 are pairs of roots ; but not 2,^2 ; 3,~3. So, if two plane curves be expressed by two equations involv- ing two variables, for. the points of meeting both curves have the same coordinates, and for these points, but for no others, the two equations are sirnultaneous. The roots of an equation are sometimes called its solution. §§2,3.] DEGREE OP EQUATIOlSr. 283 §3, DEGEEB OF EQUATION. If, without extracting roots, an equation involving one un- known element be so transformed that both members are entire as to that element, the degree of the equation is the degree of that term wherein the degree of the unknown element is highest. If the equation contain every power of the unlinown element, from the highest to the zero power inclusive, it is a complete equation ; if not, it is incomplete. E.g., the equation abx = cd + ef [a; unknown] is of the first degree, a simple equation. So, the equation y^+3y = 49 [y unknown] is of the second degree; it is a complete quadratic equation; but the equation y^ = 49 is an incomplete quadratic ;, and the equation y^ + 0y = 49 is a complete quadratic. So, the equation r^ + 5r^ + or = 426 [r unknown] is of the third degree ; it is a complete cubic equation. So, the equation Jc*+12¥ + eQJc^ + 9oJc=12BG [S; un- known] is of the fourth degree ; it is a complete biquadratic equation. So, the equation s = a(r'' — 1) : (r — 1) is of the «th de- gree if r be the unknown element ; of the first degree, if s or a be the unknown element ; an exponential equation if m be the unknown element.' An equation may contain one unknown element or more. E.g., the equations above have each one unknown element; but the equation aaf-\-2 hxy + by^ + 2gx-{- 2fy + d = is a complete quadratic with two unknown elements ; and as? -f hy"^ + cz^ -|- 2fyz -\- 2gzx + 2 hxy + 2lx+2my -y^nz+d = [a;, y, z unknown] is a complete quad- ratic involving three unknown elements. If an entire equation involve two or more unknown elements, the degree of the equation is the sum of the exponents of the elements in that term in which their sum is greatest. E.g., the equation 3xy^+2x?+y^+x+y+27 = is a cubic, but not complete. 284 EQUATIONS. [XI. tlis. §4. GENEEAL PEOPEETIES. Theor. 1 . If to both members of an equation the same num- ber be added, the roots of the equation are not changed thereby. Let p = Q be any equation, and n any number ; then are the roots of equations p = Q, p + n = Q + n, identical. For ••• p + N = Q + N when p = q, and then only, .-. every root or set of roots of the equation p = Q, sat- isfies the equation p + n= q +n, and conversely ; i.e., every root of either equation is a root of the other. Q.E.D. CoK. 1 . If any term be transposed from one side of an equa- tion to the other and its sign reversed the roots of the equation are not changed thereby. E.g., the roots of the equations as? + bx-\-c = Q, av? -\-bx= — c, are identical. Cor. 2. If the signs of all the terms of an equation be changed the roots of the equation are not changed thereby. Theor. 2. If both members of an equation be multiplied by any same number, not a function of the unknown elements and not nor as, the roots of the equation are not changed thereby. Let the equation p = q be any equation, and n any number not a function of the unknown elements, and not nor oo ; then are the roots of the equations p = q, N'P = n-q identical. For, write the equations in the form p — q = 0, n(p — q) = 0; then •.• N is not a function of the unknown elements, and not 0, nor 00, .'. n(p — q) vanishes when p — Q vanishes, and then only, and conversely ; . • . every root of either equation is a root of the other. Q.E.D. Note 1. If n be 0, n(p — q) vanishes for any values of the Unknown elements that make p — Q finite. If N be CO, n(p — q) may not vanish when p — Q vanishes. 1, 2, § 4.] GENERAL PEOPEETIBS. 285 If N be a function of the unknown elements, sr may vanish for other values of those elements than those values that make p — Q vanish, and may thus cause n(p — q) to vanish. If N be a function of the unknown elements, n may become infinite for some of the values that make f — q vanish, and n(p — q) may not vanish. In each of these four cases the equations p— Q=0, n(p— q) =0 may not have all their roots identical. E.g., let X be any entire function of x, and u any constant ; then •.• whatever factors x has, the product (x — a)-x. has another factor, x — a. .:• whatever roots the equation x = has, the equation (a; — a) ■ X = has also the root a. So, of the equation 0? — 5x -{-6 =0, the roots are 2, 3 ; but of the equation a:^ — 5a^ + 6x = 0, the roots are 0, 2, 3, i.e., by multiplying the equation by a; a new root, . 0, is introduced which does not satisfy the original equa- tion a:? — 5x + 6 — 0, and is not a root of it. So, of the equation a^ — 5af + 6x = the roots are 0, 2, 3 ; but of the equation a^— 5a;-(-6 = the roots are 2, 3, only, i.e., b^' dividing the equation by x, one root, 0, is lost. So, of the equation 3— 33=15— 2a!, the single root is 12 ; but if this equation be multiplied bj' a; — 1, the resulting equation a;^ — 13a! + 12 = 0, has two roots, 12, 1. So, of the equation x^—1—ax — a the roots are 1, a — 1 ; and if both members be multiplied by x: (a; — 1), the re- sulting equation, a;(a; -f-1) =ax, still has the root a! = a— 1, for which the multiplier becomes neither 00 nor 0, but it has-{ . j q for which the multiplier becomes -{ ?" Note 2. If the function x be not entire, but contain a term of the form a : (x—a) ; then x may become infinite, when x=a, and (x— a)-K may take the form • qo, which may or maj- not vanish; and multiplying the equation x = by a! — a mayor may not introduce a new root into the equation. 286 EQUATIONS. [XL tlis. E.g. , multiplication by tlie factors x, x + 1 generally intro- duces the roots 0,~1 ; 1 3 but if the equation 2 = - + a; — IH ■ be multiplied X x+1 by x-(x+l), 0,~1 are not introduced as new roots. 3 For •-• X contains a term which is infinite when a; = — 1 , x + 1 , and •.• this infinite term, when multiplied by the zero, w + l, is the finite number 3, .*. (a! + l)-x does not vanish when x=~l; and ~1 is not a root of the new equation. So, X • X does not vanish when a; = ; and is not a root. But if the equation 1 = 6 be multiplied by x — 1, the resulting equation is x'— 7aj + 6 = 0, whose roots are 6,1, whereof 6 satisfies the original equation and is a root of it ; but •■• 1 does not satisfy it, and is not a root of it, .-. by the use of the factor a; — 1 a new root (a stranger) has been introduced into the equation. The reason is manifest : the factor x—1 is not needed to clear the equation of fractions ; for if the terms of the origi- nal equation be all transposed to one side and reduced to lowest terms, the equation becomes 7 — = 0, 1— a! i.e., 7— (l-|-a;)=0, whence a;=6 ; and there is no other root; i.e., the numerator and 'denominator vanish together when x — 1, and the value of the fraction : is 2. 1 ™3 1 So, the equation 1 -| x + — = may be cleared of 6 ar — 1 fractions by multiplying by 6(a5— 1) (x+1), and be- comes 7a^-|-6a;— 13 = 0, whose roots are 1, —\^; but •.• —^ satisfies the original equation, and 1 does not, .-. the factor a;— 1 iutroduces a new root 1, but a!+l does not introduce a new root. 2, 3, § 4.] GENREAL PEOPEETIES. 287 The reader may search out the reason for this difference. So, if the equation -^— 1 ; = be multi- X — a x + a x^ — cr plied by all its denominators, the resulting equation is (2a; — 1) (a;^ — a") = 0, whose roots are ^, +a, —a; but if it be multiplied by the least common denominator, the resulting equation, 2 a;— 1 = 0, has a single root, -J-. Of these three roots only ^ satisfies the original equation. Theor^ 3. If the two members of mi equation be raised to the same integral power, the results are equal; but it is possible that the new equation may have some roots not found in the old one. For if p = Q, wherein p or q or both of them are functions of some unknown element, say x, then p- = Q^ p3 = Q^ . . . , p» = Q», [II. as. 6 p2_Q2=0, p^-q3=0, •••, P»-Q»=0, i.e., (p-q)(p + q) = 0, (P-Q)(p2 + PQ + Q2) = 0, ••■, (p - q) (p-1 + p»-2q • • • + Q"-^) = 0. But these equations are satisfied either if such values be given the unknown that p — q=0, or that P+Q = 0, p^+pq+q==0, ■•■p»-^+p"-^q--- + q''-i= 0; and in general the roots of the equation p — Q = are not the same as the roots of the equations p + q = 0, p^ + pQ + Q' = 0, •", p"^i + p''"^QH hQ""^=0. E.g., it x = 5, then a;^=25, and x = +b,~o; but only +5 satisfies the original equation and is its root. So, if y{9—x) = x—9, then a;^— 17a;+72 = 0, anda;=8,9; but 9, not 8, satisfies the original equation, and is its root. Were that equation -^(9— a;) = a; — 9, the root were 8, not 9. Note. Unless the reader be sure that every step he has taken . . valid, ^. g ^-^^^ gg^ijjj successive transformed equation ' reversible, ' is true^ whenever ^^^ previous ones are true, his results can serve merely to suggest values of the roots for trial. If any step has been||^;^^^^'^3ij,ig the problem may have^ other solu- 288 EQUATIONS. [XI. ths. tions than he has found. In particular, he must have multiplied by no more factors containing the unknown than were necessary to clear of fractions, and must have -j f ^ no solutions in taking like "i i of both members ; or else he must test his results by substituting in the original equation or equations, and say : if they satisfy the equation they are among its true roots ; if not, they are strangers introduced in course of the work. The results arfe to be trusted only after they are tested. Theor. 4. If all the terms of a rational integral equation involving one unknown element be transposed to one side, then : 1. The polynomial so formed has for a factor the excess of the unknown element over any root of the equation; 2. Conversely, if this polynomial have for a factor the excess of the unknown element over any given number, that number is a root of the equation. 1 . Let the equation x = be any equation wherein x stands for some rational integral function of an unknown ele- ment X, say Aa;"+Ba;""^-fca!""^-| f-Ea;^+sa; + T, and let a be a root of the equation x = 0, then is x measured by a; — a. For, divide x by x—a, and put Q, b for quotient and remainder ; then •.• X = Q • (a; — a) + E, for every value of x, wherein k is independent of x and constant ; and ••• x=0 and x — a=0 whena;=o, [liyP- .-. K = 0; and • . ■ the division of x by a; — a is effected without remainder, .•. x — a is a measure of x. q.e.d. 2. Let X be any rational integral function of x, and let a; — a be a measure of x ; then is a a root of the equation x = 0. For • . • X = Q • (a; — a) , for every value of x, and there is no remainder, [hyp. and ■■■ x — a = when a; = a, .•. x=Q-0 = 0, when a; is replaced by a, i.e., a satisfies the equation x= 0, and is a root of it. q.e.d. 3, 4, § 4.] GENERAL PROPERTIES. 289 Cor. 1. Every factor of x that is itself a function of x may be put equal to' 0, and the roots of the equations so formed are roots of the equation x = 0. Cor. 2. No rational integral equation x = has more roots than the function x has linear factors [factors of tlie form x—a'] ; one? if the equation he of the nth degree^ it has not more than n roots. CoE. 3 . If there he two rational functions of the same variable, neither of which is higher than the nth degree, and if these two functions he equal for more than n finite values of the variable; then are the two functions identical. Let AK" + BK"-^ H h sa3 + T = a's)" + b'sj^-i H f-s'a; + T' be a true equation for more than n finite values of x ; then will A = a', b = b', ••■ s = s', t = t', and xx'' ■\-Bx^^-\ h sa; + t = A'a;''+ B'aj^-^H [-s'x + t. For, if not, the equation (a— A')a;" + (b - b') a!"-iH h (s — s')a; + (t— t') = has not more than n roots ; [cr. 2 which is contrary to hypothesis ; .-. A = a', B = b', •". Q.E.D. Note. The roots may not be all different. For if the function x have the same factor used two or more times, then the equation x = is said to have two or more equal roots. In general, if x = (a! — a)''- (sc — 6)'"-, wherein p, q, ■■■ are positive integers such that p+q-\ = n; then a is a p-fold root, b a qfold root, and so on. E.g., ii? — Zaa? + 'ia^x — a^ = {x — ay, and the three roots of thp equation a?—Zav? + 2>a^x — a^=Q are a, a, a. So, the equation (a; + a)^(a; — 6)^= has —a, —a, b, b for its four roots. It appears later that a set of equal roots are the limits of a set of unequal roots, and that if the equation x = be of the nth degree, it has n roots, equal or unequal, real or imaginary. 290 EQUATIONS. [XL th.e. Theoe. 5. If,of the rational integral equation x=0, tlie absolute term = 0, some root x' of the equation = 0. Let equatiou x = be written t = — sa; — es^ ax", and let s, e, ••• a stand fast, while x, t vary, and t = ; then will some root x' = 0. For •.• D,,T, =-s-2Ea;' , [VII. ths. 13,17 = — s, a finite number ; [a' = . • . the ratio inc t : inc x' is finite ; and the two infinitesi- mals are of the same order. • But •.• T = when x'=0, . • . if T ~ be small, so is as' ~ ; ' i.e., x' = when t = 0. q.b.d. CoE. 1. If the absolute term be 0, then is a root of the equation. For, if be put for x, the equation x = is satisfied, q.e.d. CoE. 2. If A., the coefficient of the highest power of the unknown element in x, = 0, then a root of the equation = oo. For, if X be replaced by y''- in the equation x = 0, that equation takes the form A I B I C I 1 I'' pS i__n whence T2/" + S2/"~'-1-e2/""^-| + C2/^-|-b?/ + a=0 ; [mult.by/* and if the absolute term a = 0, then some root y'= 0, and some root a;', = 1 : y'= oo. q.b.d. Note. If the last two, three, ••• of the coeflScients •••, e, s, t be zero, or approach zero, so do as many of the roots ; and if the first two, three, •••of the coefficients a, B, c, ••• approach zero, as ma:ny of the roots approach infinity. E.g., if A, B, c, E, s be infinitesimals of the first order, t be zero, and d, q be finite, then three of the roots are infinites, each of the order -J, two are infinitesimals, each of the order ^, and one is zero. pr.l, § 5.] SIMPLE EQUATIONS INVOLVING ONE UNKNOWN. 291 § 5. SIMPLE EQUATIONS INVOLVING ONE UNKNOWN. PkOB. 1. To SOLVE A SIMPLE EQUATION INVOLVING ONE UNKNOWN ELEMENT. Multiply both members of the equation by the I. c. mlt. of the denominators, if any. [th. 2 Transpose to one member all terms that involve the unknown element, and to the other member all other terms. [th. 1 Reduce both members to their simplest form, exhibiting or can- celling any com/mon factors. Divide both members by the coefficient of the unknown ele- ment. [II. ax. 5 To test the work, replace the unknown element by the result so found, in the original equation. E.g., i£|(a;+12) = i(6 + 3a:)-ia;, [a; unk. then •.• 7a; + 84=36 + 18a3 — 7a3, [mult.by42 and' (7 -18 + 7) a; = 36 -84, [trans. 84, 18a;, -7a; i.e., —4a; =—48, and a; =12; [div. by — 4 and •.• |(12 + 12) = -f(6 + 36)-2, [repl.a;byl2 .-. 12 is the root sought. o .„ (2a + &)62 , , a^b' „_ , 6 3abc So, if -^^ ■ — '—x-{ - = 3ca! + -a; , a{a-hby (a + &)^ a a + 6' then •.• (2a+&)&'(a + &)» + a^&^ = 3aG(a + byx + b{a + byx-3a''bG{a + by, .-. a^b^ + 3a^bc(a-^by ==(Sac + b){a + byx- (2a + &) &'(« + b)x, i.e. , a^6 [a& + 3 c(a + &) '] = a (ffl + &) [a6 + 3 c(a + 6)^ x, _ a^&[a& + 3c (» + &)'] ^ ab ■'■ '"~ a {a + b)[ab-^ Sc {a -{-by] a + &' This value satisfies the given equation, and is the root sought. Note 1. A simple equation can have but one root. For any such equation may take the form ax — b = 0, one linear factor. [th. 4 cr. 2 292 EQUATIONS. [XI. thB. Note 2. Equations not simple sometimes reduce to simple equations, and may be solved like them. E.g., it ^x-^lx-^{l-x)']=l; then •-• VC»'-V(1-«')] = V^-1' [trans. Va;, change signs a; — V(l — «) = a; —2^x + 1, [sqr. both mem. ^(^l — x) = 2^x—l, [cancel x , change signs l—x = 4:X — 4:^x+l, [sqr. both mem. iy'x =5£c, [trans. 16a; =25a;^, [sqr. both mem. a;(16— 25a;)=0, [trans. , factor a; = or = ^f . Both of these results satisfy the given equation, and are roots. For Vif-V[if-V(l-Tl)]=l if ^^^ second radical be negative, and the other two positive ; and i/O — V[0 — V(l — 0)] — 1 i^ ^^^ ^^^^ ^^'^ radicals take their negative values. But if the signs of the radicals be restricted, the equation may have no solution. E.g., Va'-V[a'-V(l -«')]= 1. ^a,_+/[a;_^Xl -«')]= 1- Note 3. General Discussion : Every simple equation in- volving one unknown may be reduced to the form ax + b = a'x + 6', whose general solution gives a; = (6' — 6) : (a — a') ; and there are three cases : (a) a4=a' ; then x has a single value, positive, negative, or zero, that satisfies the equation. (6) a = a', bi=h'; then a; = qo. This result may be interpreted in the language of limits by saying that if a, a' be variables, or either of them, and if a^al but a = a', then x grows larger and larger without bounds. E.g., if A, a' travel along the same road in the same direc- tion at a, a' miles an hour, and if a' be (b'—h) miles ahead of a, 5,6, §6.] ELIMIXATION. 293 then the quotient (&' — b) : (a — a') is the time before they will be together. If the hourlj- gain, a — a', be Small, that, time is long ; if there be no gain, i.e., if a = a', they will never be together, and there is no value of x that satisfies the equation. (c) a = a', b = b' ; then cb = : 0, and the equation is satisfied by any number whatever. In the example above (6) , a, a' are now together and they will always be together. § 6. ELIMINATION. Thboe. 6. If there be two or more unknown elements and a system of two or more independent simultaneous equations that involve them, then the roots are not changed thereby if any one of these equations be replaced by the sum of this equation and any other or others of them. Let the equations p = q, p'= q', p" = q", ••. be any system of simultaneous equations, and for the equation p = q put the equation p + p' = Q + q', or p + p' + p" = q + q' + q", or ••• ; [II. ax. 2 then will the roots of the system of equations, P + P' + ..-. =q + q'+..., p' = q'', f" = q", ..., be identical with the roots of the system first given. For •.• when p=q, p' = q', p" = q", ••■, then also p + p'+p"+"- = q + q'+q"+-; .•. whatever set of values satisfy the equations F = Q, p'=q', p"=q", ..., the same values satisfy the equation p + p' + p"H =Q + q' + q"H ; and conversely. q.e.d. Cob. In such a system of equations the roots are not changed if before the addition one or more of the equations be multiplied by any factor not a function of the unknown elements, and not 0. 294: EQUATIONS. [XI. th. Theoe. 7. If one equation of a system he solved for any one un- known element, in terms of the other unknown elements that enter into it, then the roots of the system are not changed thereby, if in the other equations this element be replaced by the value so found. For, letp = Q, p' = q', p" = q", ••• be a system of equations involving x,y,z, ••• in any way ; solve the equation p = Q for X, giving x=f{y, z, •••)) ^-i^^ substitute this expression for x in the other equations, giving them the new forms r' = s', r" = s", ••• ; then •.• X a.ndf(y, z, •■■) have the same values, .-. whatever values of x,y,z, ■•• make identities of the equations p= q, p'= q', p"= q", •••, the same values make identities of the equations e'=s', r"=s", •••, and, conversely, whatever values oi y, z, ••■ make identities of the equations r'=s', e"=s", •••, the same values of y, z, ••• make identities of the equations p = Q, p' = q', p" = q", •■•, i.e., both systems have the same roots. q.e.d. ' Note. By aid of Theors. 1, 2,3,6 a system of n independent simultaneous equations containing n unknown elements may be reduced to a new system of n — 1 equations, containing ra — 1 unknown elements, whose roots are identical with the roots of the original system, and these n—1 equations to n — 2 equa- tions, •", to two equations, to one equation. The process by which, one after another, the several un- knowns are removed from the system of equations is a case of elimination. In general when from two or more (say n) given equations a new equation is got that is, free from at least n — 1 of their elements, those elements are eliminated between the given equations; and the new equation, or its first member when the second member is zero, is the resultant of the given equations. The elimination is reversible when, whichever « — 1 of the given equations, together with the resultant, were known to be true, the remaining equation would necessarily be likewise true ; otherwise, the elimination is irreversible. 7, § 6.] ELIMINATION. , 295 Pl^OB. 2. To ELIMINATE AN UNKNOWN ELEMENT FROM A PAIR OF EQGATIONS INVOLVING THE SAME TWO UNKNOWN ELEMENTS. (a) Simple equations. 3?IRST METHOD, ADDITION AND StTBTEACTION. Find the least common multiple of the coefficients of tJiat element which is to be eliminated; divide it, in turn, by these coefficients, and multiply the two equations through by their quotients. Subtract one equation from the other, member from member. E.g., to eliminate x from the pair of equations 6a; + 72/=i85, 2 a; + 3 2/ = 33: then ■.• the I.e. mlt. of the coefficients 2, 6 is 6, . . 6a; + 72/ = 85, 6a! + 92/ = 99, [mult. by 1,3 2 ?/ = 1 4 . [subtract So, to eliminate x from the pair of equations Ex + M = 0, L'a; + M'=0, wherein l, m, l', m', are any expressions that do not contain x, but which may contain other unknown elements : then ••• LL'a; + L'M = 0, LL'a; + LM' = 0, [mult.byL',L .-. lm'— l'm =0. [subtract Note. The work is often best arranged as follows : Write the given equations under each other, and at the right, their respective multipliers with such signs that the new equation may be the algebraic sum of the products of the given equations by their multipliers. If there be two columns of multipliefs, one to eliminate each unknown, write first the cohimn to be first used. When small, the partial products can be obtained and added mentally, and only the sums written down. Detached coeffi- cients can be used in part of the work. E.g., the first of the above examples becomes : 6a; + 72/ = 85 2x + 3y = S3 3 -7 -1 3 or 6a;+72/ = 85 2 3 33 3-1 -7 3 Ax =255 = 24 4 24 -231 2 14 2y = U ■ x=6, y=7 x=6, y=7. 296 EQUATIONS. [XI. pr. SECOND METHOD, COMPARISON. Solve both equations for that element which is to be eliminated. Put the two values thus found equal to each other. E.g., to eliminate x from the pair of equations Gx + 7y = 85, 2x + 3y = S3: then a; = ^(85 -72/) = 1(33-31/), [sol. both eq. for a: THIRD METHOD, SUBSTITUTION. Solve either equation for that element which is to be eliminated. In the other equation replace this element by the value .so found. E.g. , to eliminate x from the pair of equations 6x + 7y = 85, 2x + 3y = S3: then •.• x = i{3B — 3y), [sol. 2d eq. for a; .-. 99 — 92/ + 72/ = 85. [repl. a;in Isteq. (6) Equations of degree higher than the first. Of the three methods of elimination shown above (a) some- times one, and sometimes another is most available. In the pair of equations I = ar""^ s = [to elim. n r — 1 the method of substitution is best : multiply the first equation by r, and replace at" bj' Ir in the other ; , , Ir — a then s = r — 1 In the same pair of equations [to elim. a the method of comparison is best: solve both equa- tions for a, and put the values equal ; then J- = ^(^-') . In the pair of equations x'+ y = ll, y^+x = l [to elim. x the first method is less easy ; but the other two are available. 3? = n-y, 3^ = {7-y'y=:i9-Uy^ + yS 11 — 2/ = 49 — 142/^ + 2/^- [comparison x={7-f), x' = i9-Uy^ + y*; af-\-y = 11 gives 49 — 142/^-|-2/*+2/=ll. [substitution 2, § 6.] PLIMINATION. 297 FOURTH METHOD, DIVISION. Reduce the equations to the form p = 0, Q = 0, ivherein p, q are functions of x, j. Divide F by q, Q by the remainder, and so on, as in finding the h. c. msr. of two entire numbers, until some remainder is found that is free from the element to be eliminated, or that has a com- mon measure with p, q, and the successive remaindets. If this remainder do not contain such a common measure, equate it to for the resultant sought. If a common measure ofp, q be found, divide each of them, or any two successive remainders, by this measure, and with the quo- tients proceed as before to find a resultant. If the solution of the given equations be sodght, then : Solve the resultant for the unknown element involved in it; re- place this element in the next previous remainder by the values thus found; equate to 0, and solve for the other unknown element. Equate to the common measure, if any, o/ p, q ; if the new equation thus found involve but one unknown element, solve it therefor; but if it involve both unknown elements, give to either of them any value whatever, and solve for the other. E.g., to eliminate y from equations p = 0, Q = 0, wherein visx-y*—2x+l ■'jf—'3(?—a?—x—\-y^+a?—x'—x-\-l-y+2, and Q is a; • 2/^+ ar* — 2 a;— 1 -y^ — x'-\-x—l -y — i-af—x—l : Divide p by q; the remainder, Ti, = x-y^ — 2x-\-l-y-[-2; so, divide q by r ; the remainder, s, = x'—l-y — 2; then •.• B, s have the h. c. msr. y — 2, and the quotients are xy—1, se'—l, .-. the resultant is a;^— 1 = 0, whose roots are ~1, and equations p=0, Q=0 are satisfied when, and only when, either a; = +], +1-2/— 1=0; or x=-l,-l-y—l=Q ; or a; = any value, y = 2. So,, to apply the fourth method to the pair of equations, Sx^-Axy+by^-ex-^-ly^UO, 2a^-^3xy-{-5y'=103: "Write p, Q in the form La;^+ wra; + n, L'a!^+ u'x + n', wherein l, m, n, l', m', n', may contain y but not x. 298 EQUATIONS. [XI. prs. §7. SIMPLE EQUATIONS, TWO OE MORE UNKNOWNS. PkOB. 3. To SOLVE A PAIR OF SIMPLE EQUATIONS, WHEREOF ONE HAS TWO UNKNOWN ELEMENTS, AND THE OTHER BUT ONE. Solve that equation which has but one unknown element, [pi'.l Replace this element by its value in the second equation, and solve for the other unknown element. [th. 4 cr. 2 E.g. , to find a;, y from the pair of equations 6a; + 72/ = 85, 4» = 24: . then 3!= 6, 36 + 7?/ = 85, y='l. PrOB. 4. To SOLVE A PAIR OF SIMPLE EQUATIONS, WHEREOF BOTH HAVE THE SAME TWO UNKNOWN ELEMENTS. Combine the two equations so as to eliminate one unknown ele- ment, and form an equation involving the other unknown element. Solve this equation for its unknown element, replace this element by its value in either of the given equations, and solve the equation so found for the other unknown element. For a check, replace the two unknown elements by their values in either of the original equations. E.g., to find x, y from the pair of equations 6x + 7y = 85, 2x + 3y = B3; then •.• 1(85 — 6a;) = 1(33 — 2a!), [eWm.y .-. 265 -18a; =231 -14a;, .-. —4a; =—24, a;=6; .-. 36 + 72/ = 85, y=7. So, to find X, y from the pair of equations ax + by = c, a'x + b'y = c' : then • . • ab'x + bb'y = cV, ba'x -\- bb'y = be', [II. ax. 4 .-. (ab'-a'b)x=(cb'-c'b), x= ''^'~'^'^ , ab' — a'b cb' — c'b , , ac' — a'c •'■ ""^i K+by = c, y=-T, IV ab' — a'b ab' — a'b 3,4, § 7.] SIMPLE EQUATIONS, TWO OR MOEE UNKNOWNS. 299 Note 1. The values of the two unknown elements maj- be got independently of each other, by separate eliminations ; or else, having found one of them, the other may be written bj- symmetry. E.g., if ax + by = c, a'x + h'y = c', [above then •.• these equations are not altered by interchanging a with 6, a' with 6', and x with y, .-. a; is the same function of a, &, a', 6' asisyof b,a,b',a'; .*. the value of either x or y is found from that of the other by interchanging a with b and a' with b'. So, if x + y = a, x — y=b: then { ^ is the half ^ ^Xrence °* «' *• Note 2. Incompatible equations : If two given equations be incompatible, no solution is possible. ^.gr., the equations 2x + 3y=l3, 2a! + 3?/=15 are incompatible ; ' for their resultant, = 2, is absurd. Note 3. Dependent equations : If one equation be depen- dent on the other, and derivable from it, there is no single solu- tion, but an infinite number of solutions. E.g., the equations 2x + 3y = 18, 6a; + 92/ = 39 are but one equation in two forms, and any value maj' be given to either of the unknown elements, and the corresponding value of the other computed. Note 4. General foemul^: The two equations ax+by=c, a'x + b'y = c' are the type-fOrms of every pair of two-unknown first-degree equations ; their solution giyes : x={cb'-c'b) : (ab'-a'b), y = {ac'-a'c):{ab'-a'b). The solution of this pair of equations embraces the solution of all such pairs of equations : the reader may translate the formulae into a, practical rule for such solutions without the intermediate steps. 300 EQUATIONS. [XI. pr. Note 5. General discussion : There are three general cases. (a) ab'^a'b; then X, y have single values, positive, negative, or zero, that satisfy both the equations. (&) a6' = a'6, cV^c'b; then a; = 00, y = . For •.• db' =a'b, cb' =f=c'b, [hyp. a:a' = b:b', c:c'=^b:b', [II.ax.13 a:a' =^c:c', [II. ax. 8 and ac' =^ a'c. q.e.d. [II. ax. 13 And •.• a6'— a'& = 0, cb'-c'b^O, cui' — a'c=^Q, X, = (cb'—c'b) : {ab'— a'b) , = oo ; and y, ={ac'— a'c) : {ab'— a'b), = CO. q.e.d. This result may be interpreted, in the language of limits, by saying that if a, a', b, 6', be variables, or either of them, and if a&' ^ a'b but ab' = a'b, then x, y grow larger without bounds. E.g., if ax + by=c, a'x + b'y = c' be the equations of two straight lines, then the values of x, y that satisfy both equations are the co-ordinates of the meeting-point of the two lines. If aV = a'b, then a : a' = 6 : 6', the two lines approach paral- lelism, the point of intersection recedes to a great distance, and the values of x, y become very great. If ab' = a'b, then a : a' = 6 : 6', the two lines are parallel, they have no meeting-point, and there are no values of X, y that satisfy both equations, (c) ab'=a'b, cb'=c'b; then a; = : 0, y = 0:0, and the equations are equivalent, and satisfied by giving any value to one unknown element and computing the corresponding value for the other. For ■•■ ab' =a'b, cb' =c'b, [byp. .•. a:a' = b:b', c:c' = b:b', .-. a:a' = c:c', Q.E.D. 4, § 7.] SIMPLE EQUATIONS, TWO OE MOEE trNKNOWTSTS. 301 In the above example (b) the two lines, under the special conditions given in (c) , are coincident, and every point is a common point. In general any value may be assumed at ran- dom for one co-ordinate, and the other may then be computed. If ab' = a'b, there also appear the following special cases : (d) a' = 0, &' = 0, c'^0; then x= —be' -.0, y = ac' : 0, which values are infinite. (6) a' = 0, 6' = 0, c' = 0; then a; = : 0, y = Q:Q, which values are indeterminate. (/) 6 = 0, 6' = 0, ac'4=a'c; then a; = : 0, y = (ac' — a'c) : 0, of which values ope is in- determinate and the other infinite. ig) 6 = 0, 6' = 0, ac'^a'c; then the equations are equivalent, and a; = : = c : a = c' : a', and ^ = 0:0, and is indeterminate. (7i) a=0, 6 = 0, 6'=0,'a'TfcO, c#=0; then a; = : 0, y = — a'c : 0, of which values one is indeter- minate and the other infinite. («■) a=0, 6 = 0, 6' = 0, c = 0; then a; = c' : a', y = : 0, of which values one is determinate and the other indeterminate. {j) a = 0, a' = 0, 6-0, 6' = 0; then a;=0:0, y = Q:0, which values are indeterminate; but not both are finite unless c = 0, c' = 0. The reader may interpret these results, and illustrate them by the meeting, when possible, of two straight lines. Two important special cases appear when c, c' both vaniah. (fc) ab'^a'b, c = 0, c' = 0; then a; = 0, y = 0. (0 ab' = a'b, c=0, c' = 0; then a; = : 0, y = : 0, which values are indeterminate. The reader may interpret these results, and illustrate them by the meeting of two straight lines : he will observe that in both cases the two lines pass through the origin ; in the first they meet there ; in the second they coincide throughout. 302 EQUATIONS. [XI. pr. PrOB. 5. To SOLVE A SYSTEM OF ?l INDEPENDENT SIMPLE EQUATIONS THAT INVOLVE THE SAME U UNKNOWN ELEMENTS. Combine the n equations, two and two, in n — 1 ways, so that each equation is used at least once, and so as to eliminate the same unknown element at each operation; thereby form n— 1 equations involving the same n — 1 unknown elements. So, combine these n — 1 equations, and thereby form n — 2 equations involving the same n — 2 unknown elements; and so on till there results one equation involving hut one unknown element. Solve this equation, and replace the unknown element by its value in one of the two equations involving two unknown elements. Solve this equation for the second unknown element, and replace these two elements by their values in one of the three equations involving three unknown elements; and so on. E.g., to find x, y, z from the system of equations a!+2y+3»=14, 3a;+22/+2=10, 6a;+9?/+132=63 : then 2a;— 2«= — 4, 15a!— 17«= -36,[elim.2/fr.eqs.l,2;2,3 .-. 4a; =4, [elim. z .-. a;=l, y=2, z = 3. [repl. a;, ?/ So, to fin.d X, y, z, from the system of equations ' ax+by+cz=d, a'x+b'y+c'z=d', a"x+b"y+c"z=d": d — by — czd' — b'y — c'z d" — b"v — c"z then •.• x= ^ = - 2 = 2 , a a' a" . • . a'd — a'by — a'cz — ad' — ab'y — ac'z and a"d' — a"b'y — a"c'z = a'd" — a'b"y — a'c"z, [elim. a; ^ ad'-a'd+ {a'c-ac')z _^ a'd"-a"d'+ (a"c'-a'c") z ■ ■ ^ ab'-a'b ~ a'b"-a"b' .-. {a'b'i - a"b') {ad' - a'd) + (a'b" - a"b') (a'c - ac')z = (ab'-a'b) {a'd"-a"d') + (ab'-a'b) (a"c'-a'c")z, „ ^ (ab'-a'b) (a'd"-a"d') - (a'b"-a"b')(ad'-a'd) (ab'-a'b) (a'c" - a"c') - (a'b"-a"b') (ac' - a'c) ^ ab'd" + a'b"d + a"bd' - a"b'd - d'bd" - ab"d' ab'c" + a'b"c + a"bc' - a"b'o- a'bc" - ab"c>' The reader may write the values of x, y by symmetry. 5, § 7.] SIMPLE EQUATIONS, TWO OE MOEE UNKNOWNS. 303_ Note 1. The equations must be so combined that no m of the n — 1 equations got by the elimination of one unlinown elc'^ ment shall represent less than m +1 of the original m equations ; and that no m of the n — 2 equations got by the elimination of two unknown elements shall represent less than m + 1 of those n — 1 equations ; and so on. Otherwise the m — 1 equations, or the m — 2 equations, •••, will not be independent, and no determinate solution will finally be got. Note 2. All the unknown elements not involved in evert EQUATION. An unknown element that does not appear in any equation may be considered as already eliminated from it, and the work is shortened by so much. Those unknown elements that appear in the feswest equations may be eliminated first. E.g., to find X, y, z, t, u from the system of equations 9x—2z+ w = 41, (1) 72/- 52 - t =12, (2) 4:y — 3x + 2u= 5, (3) 3y-iu + 3t = 7, (4) 7»-5w=ll: (5) Of these equations, x appears in two, y in three, z in three, u in four, t in two. Equations 1, 3 may be combined to eliminate x, and equa- tions 2, 4 to eliminate t, and there result two new equations involving y, z, u. These two equations may be combined to eliminate y, and there resalts one equation involving z, u. This last equation may be combined with equation 5 to elimi- nate either a; or m at pleasure. Note 3. Particular artifices : The equations may have a symmetry, as to the unknown elements or functions of them, that permits shorter processes than those of the general rule. Sometimes the sum of the unknown elements, or of the func- tions of them, may be got first. 304 EQUATIONS. [XI. pr. E.g., to find x, y, z, t, v from the system of equations y + z+ t + v = a, (1) z +t +v+x = b, (2) t +v + x + y = c, (3) v+x-\^y + z=d, (4) x + y+z + t =e; (5) then ••• 4:X + iy + 4z + 4:t + iv = a + b + c+d + e, [add . . x + y + z + t + v = i{a + b + c + d + e), and x = i{—3a-\-b + c + d + e), [sub. eq.l y = ^{a — 8b + c+d + e), and so on. So, to find X, y, z from the system of equations then 1 1_4 1 1_11 1 1_1. X y 15' y z 60' z x 4' 2,2,2 7 1,1,1 7 — 1 1 — — TTC' — 1 1 — — ;r^i X y z 10 X y z 20 [add, div.by 2 1_ 7 11_1 1_ 7 1_ 1 1 ' X 20 60 6' 2/ 20 4 lO' te _ 7 4 _ 1 . 20 15 .12' . a;=6, 2/=10, «^ = 12. Note 4. The ndmber of equations greater than that op UNKNOWN elements. So many equations as there are unknown elements may be taken at random, and solved. If the roots so found satisfy the remaining equations, the S3'stem is possible ; but, if not, the system is impossible. E.g., to find X, y from the system of three two-unknown equations 8x+7y = n, 5x—2y=l, 8x + y = 10: Take the first two equations and solve ; then -.• x=l, 2/ = 2 in these two equations, and ■.• these roots satisfy the third equation, . • . this system of equations is possible, and the roots are 1, 2. But not possible is the system of equations 3a; + 7^=17, 5x—2y = l, 8x + y=12. 5, § 7.] SIMPLE EQUATIONS, TWO OE MORE UNKNOWNS. 305 In general, if there be m+n compatible equations, and only m unknown elements, there are n equations of condition ; and the constants must have such relations that these equations of con- dition are all satisfied. ^■9 -J given the system of three two-unknown equations ax + by = c, a'x + b'y = c', a"x + b"y = c" ; +, . „ cb'—c'b ac' — a'c re c j. i. then •■• x = —-~ — -, 2/ = -— -, [from first two eq. ab'—a'b ab'—a'b . •. that a"x + b"y = c' be a true equation, „,, cb' — c'b , J,,, ac' — a'c ,, j. i, u i„ a" ; -f b" • — ; = c" must hold true ; ab'—a'b ab'—a'b i.e., ab'c"—ab"c'+ a'b"c — a'bc"+ a"bc' — a"b'c = is the required equation of condition, and establishes the necessary relation between the given constants. By this process all the unknown elements ar^ eliminated from the given equations. So, in general, from n equations n — 1 unknown elements may be eliminated. Note 5. The number of unknown elements greater than THAT OF EQUATIONS. If there he m+n unknown elements and onl}' m equations, all compatible, to some n of these elements arbi- trary values may be given, and the roots of. the m equations will contain these arbitrary values, or some of them, and be them- selves arbitrary, and the equations are indeterminate. E.g., to find x, y from the single two-unknown simple equation 2a; -f- 32/ =12: Put y=- -5, -4, -3, -2, "1, 0, +1, +2, +3, +4, +5,..., then x = ...13i, 12, 10^, 9, 7i, 6, 4|, 3, 1^, 0,-1^,...; or put » = ••■ -5, -4, -3, -2, -1, 0, +1, +2, +3, +4, +5,.-. ; then y = ...+7i,+6|, +6,+5.^,+4|,+4,+3i,+2|, +2,+^, +|,... ; So, a series of values are given to x increasing by 1, and there result a series of values for y decreasing by |, This may be illustrated geometrically, by taking x, y as the running co- ordinates of a point on a straight line whose equation is 2x+3y—12. Such series are called arithmetic progressions. [XII. § 1 £06 EQUATIONS. [XI. pr. If the results be limited by the condition that they shall all be integers, or all positive integers, it may happen that there are very few such roots, and certain modifications may be made in the method of solution. E.g., to find all possible pairs of positive integral roots that satisfy the single equation 2x + 3y=12: then x=6 — y — ^y. Put i-y = z; then y=2z, x=6 — 3z. Put z = •••, -3, -2, -1, 0, +1, +2, +3, ••• ; then 2/= •••,-6, -4, -2, 0, +2, +4, +6, ■••, and »= ■••,15, 12, 9,6, 3, 0, -3, •••, wherein 6,0; 3,2; 0, 4 are the only pairs of roots admissible. The progressions are here noticeable again ; that for y in- creases by 2, and that for x decreases by 3, and they both go on either way forever. So, to find sets of positive integral values for x, y, s that satisfy the pair of equations x + 2y + 3z=22, 3x — 6y + 2z = -2: then •.• lly + 7z = &8, [elim. a .-. 2 = 9 — 22/ +1(5 +32/). [solvefora Put t = i(5 + 3y); then 2/ = 2«— 2+|(i+l). Put u = ^{t + l); then t = Bu — l. .-. 2/=2(3m-1) — 2 + M =7m — 4, ' »=9-2(7m-4) +3M-1 =16-11m, and fl;=22-2(7M-4) -3(16-11m) =19m-18. Put u=-, -3, -2, -1, 0, 1, 2, 3, •••; [dif. +1 then 95= •••, -75, "56, "37, -18, 1, 20, 39, •••, [dif. +19 2/=..., -25, -18, -11, -4, 3,10, 15, •■•, [dif. +7 » = •••, 49, 38, 27, 16, 5, -6, "17, •••, [dif. -11 and the only set of positive integral roots is 1, 3, 5. But of sets, whereof two are positive and one negative, or vice versa, there are an infinite number ; and of sets whereof all three are negative there are none. 6, § 8.] GEAJPHIC KEPEESENTATION. 307 § 81 GRAPHIC EEPRESENTATION OF SIMPLE EQUATIONS INVOLVING TWO UNKNOWNS. EvEET simple equation involving two unknown elements may be reduced to the type-form, y = mx + b. [VII. § 11 E.g., the equation Ax + By = c becomes y = — -x+-, wherem s m, - = 6. B ' B Every such equation may therefore be represented by a straight line ; and conversely, every straight line has its equation. E.g., the equation 6x + 7y = 8b reduces to y= — faj + 12-f, wherein —^ = m, 12^ = 6, of the type-form. This equation is represented by the line cd below, This figure serves also to illustrate the solution of indeterminate equations [§7,nt. 5], wherein a; is a variable and y a function of x. \ If there be two simple equations in- ^V^ volving the same two variables, and if it j \\^ be required to find roots that satisfy both | \ \v^^ of them, then the two loci, platted with =; — ^ — a\t> ^^^ reference to the same origin, reference- ^ line, and scale, will meet in a point whose co-ordinates are the roots sought. E.g.,\t 6a;-J-7y = 12, 2x + Zy = 4l be a pair of simultaneous equations whose loci are cd, ef, and if these loci meet at p ; then the lengths of the co-ordinates ob, bp are the common roots of the two equations. But if the coefHcients of the variables in one equation be nearly equal to those in the other, then the loci are nearly parallel, and the point of intersection may recede to a great distance ; if they be identical with those in the other, or equimultiples of them, then the two loci are \ „„_„i]pr if the absolute term of the 308 EQUATIONS. [XL first -{ P® . the like multiple of that of the other ; and there , an infinite number of „ „„„+„ are •< common roots. ' no If one of a pair of equations involving the same two variables be 2/ = 0, the locus of this equation is the line ox, and the solu- tion of the pair of equations y = 0, y = 'mx + b reduces to the solution of the single simple equation involving one unknown element, ma;+&=0, wherein the locus of x, y is the point where the pair of lines Cross, and whose co-ordinates are , 0. m §9. BEZOUT'S METHOD, UNKNOWN MULTIPLIEES. Let aiX+hiy-\-CyZ+---=hi, aiX + h^y+c^z^ = /i. ■2) "n* + Ky + c„2 -I — = K, be a system of n simple equations involving any same n un- known elements. Multiply the first equation by Zcj, the second by fcj, •.-, the nth by fc„, wherein k^ = 0, and hi, ••• A;„ are unknown ; then •.■ aiX + h^y + CiZ+ ••• =\, h2a2X + 'kibiy + 'kiCiZ-\ =^2^21 \a„x + Wy + \c^z+ — =KK, + (bi + hh-^ )-KK)y + {Ci+hc.,-\ [-k„c„)z+ — = Ai -t-fcj/ijH \-KK- Put all the coeflBcients except that of x equal to 0, i.e., put bi+ksb^A \-k„b„=0, Ci+k^c^i f-fc„Cn=0, .•-, and thus form a system of n — 1 equations involving the same n — l unknown elements, k^, ■•• A;„. "Whichever of k^, k^, ••■ A;„ be taken as 1, and the others as un- known, the ratios fcj : fcj : •.• &„ come out the same ; but if kc, or fcj or • ■ • be 00 when ^i = 1 , then ^2 or Aij or . . • should be taken as 1, and ki, ■.• as unknown, whence ^1= 0. § 9.] bezout's method, unknown multipliers. S09 So, by aid of the multipliers Zj ••• Z„_i, reduce this system to a system of « — 2 equations involving the same n — 2 unknown elements, say Zj"- Z„_i, and so on ; and finally to two equations involving two unknown elements, say rj, rj, and to one equa- tion involving one unknown element, say t. Solve this equation for t, then solve for r-j, r^, then for •••, then for Zj •■• Z„_i, then for Ajg ■•• fc„, then for x,y,z, •••. E.g., to find X, y from the pair of equations ax + by = c, a'x + b'y = c' : then (a + A;a')a! + (6 + A;6')t/ = c + fco'. Put b + Tcb' = 0; ., , b , c + 7cc' cb' — c'b then Jc = — -- and x = — !—— = —- -■ 0' a + ka' ab' — a'b So, put (a + A:a') = ; ,, , a ■, c ~ kc' ac' — a'c then ft = T and y = a' 6-ft&' a6'-a'6 So, to find X, y, z from the system of equations ax-\-by+cz= d, a'x+b'y+c'z = d', a"x+b"y+c"z = d": then (a+]c'a'+Tc"a")x+ (b+k'b'+]c"b")y + (c+k'c'+k"c")z = d+k'd'+k"d". Put b + k'b' + k"b" = 0, c + k'c' +k"c" = 0; [&',fc"unk. then b+hc + {b' + hc')h' + (&" + 7ic")ft" = 0. Put 6"-f-Ac" = 0; 6" , ,, 6 + ftc b"c-bo" then 7i =--77, and k' = - —j-—- = —- — -j— c" b' + hc' b'c" — b"c' be' - b'c So, Jc" = But ••• X = 6'c" _ b"c' d + k'd' + k"d" a + k'a'+k"a"' b"c-bc" ,, 6c'-6'c ,„ , &"c-6c" , , be' -b'c „' a- b'c"-b"c' b'c"-b"c' The reader may reduce this fraction to a simple fraction, and write the values of y, z by symmetry. 310 EQUATIONS. [XI. pr § 10. SPECIAL PROBLEMS OF THE FIRST DEGREE. In a special problem certain elements are given and certain other elements have given relations to those first named, and are to be found. These relations are the same whether ex- pressed in ordinary language or in symbolic language; If in symbolic language, their expression gives an equation or a system of equations ; and the elements whose values are to be found are the unknown elements of these equations. The solution of a problem embraces three distinct parts : (1) putting it into equation ; (2) solving the equation or system of equations ; (3) discussing the results under special conditions. A problem is of the first degree if its solution depend on the solution of an equation or system of equations of the first degree only. PrOB. 6. To PUT A SPECIAL PROBLEM INTO EQUATION. By careful study of the enunciation of the problem, ascertain which of the elements named in it are known, and which are unknown; represent both the known and the unknown elements by symbols; and express in symbolic language all the relations that subsist between them. Tliese symbolic expressions are the eqitatioris sought. Note 1. It may be convenient to express all the unknown elements by aid of a single symbol. E.g., to divide $6341 among a,b, c, so that b shall have $420 more than a, and c $560 more than b : Put X for a's share, x + 420 for b's, a; + 420 + 560 for c's ; then a; + aj^ 420+ a; + 420 + 560 = 6321, a single one-un- known simple equation. So, to divide the number 144 into four parts, such that the first part increased by 5, the second decreased by 5, the third multiplied by 5, and the fourth divided by 6, shall all equal the same number : Put X for the number to which the several results are equal ; then X — 5 + X + 5 + x: 5+a!.5 = 144. 6, § 10.] SPECIAL PEOBLEMS OF THE FIRST DEGREE. 311 Note 2. It may be convenient to express different unknown elements by different symbols ; and to form a system of simul- taneous equations involving two or more unknown elements. II. g., a vintner at one time sells 20 dozen of port wine, and 30 dozen of sherry, and for the whole receives $600 ; and at another time he sells 30 dozen of port and 25 dozen of sherry, at the same price as before, and for the whole receives $700. Put X for the price of a dozen of port, and y for that of a dozen of sherry ; then 20a;+302/ = 600, 30a! + 25?/= 700, a pair of two- unkuown simple equations. So, if a certain rectangular bowling-green were 5 yards longer and 4 yards broader, it would contain 113 yards more ; but if it were 4 yards longer and 5 yards broader, it would contain 116 j-ards more. Put X, y for the length and breadth ; then (a; + 5) -(2/4-4) = a;?/ -1-113, (a;-f-4).(2/-|-5)=a;2/ + 116. So, if A, B, c, D engage to do a certain piece of work ; if A, B together can do it in 12 days ; a, d in 15 days; c, D in 18 days ; and if b, c begin the work, after 3 days A joins them, after 4 days more d joins them, and all working together they finish it in 2 days, in what time can each man do it working alone ? Put X, y, z, u for the number of days needed by a, b, c, d ; ., 1,1 1 1,1 1 1,1 1 9,9,6,2 , then -4--=—, —+--=-:, -+-=TS' -H 1 f-- = l> X y 12 X u 10 z u lo y z x u a system of four simple four-unknown equations. Note 3. Discussion op the Soldtion: To discuss the solu- tion of a problem whose answer is numerical, is to trj^ whether all the conditions of the problem are satisfied by all or any of the numbers that are found to satisfy the equations into which the problem was translated ; and, if not, to observe what other conditions the unknown elements must satisfy besides those taken account of in putting the problem into equation. ' 312 EQUATIONS. [XI. prs. To discuss the solution of a problem whose answer is literal is to observe between what limiting numerical values of the known elements the problem is possible ; and whether any singulari- ties or remarkable circumstances occur within these limits. E.g., in a certain two-digit number the first digit is half the other, and if 27 be added to the number, the order of the digits is reversed ; what is the number ? Put X for first digit, y for second digit ; then -.■ 2a; = 2/, IQx + y + 21 =^10y+ x, .-. x = Z, 2/ = 6, the number is 36 ; and 36 -f- 27 = 63. Were this the statement : in a certain two-digit number, the first digit is half the other, and if 24 be added to the number, the order of the digits is reversed ; then ■.• 2x = 2/, lOx-l-y-f 24 = 102/-f-a;, .•. a; = 2f, y= 5^, and the number is impossible. The statement of the problem puts a limitation upon the values of X, y not expressed by the equation : they must be integers. Were this the statement : of two numbers the first is half the second, and if to ten times the first the second and 24 be added, the sum is the sum of ten times the second added to the first ; then the same equations as before would express the rela- tions, and the values 2|-, 5-J- would satisfy all the conditions. For 2-21 = 5^, 10-2f -|-5^-|-24= 10-5i-f 2|. And were this the statement : in a certain two-digit number the first digit is half the other, and if a be added to the num- ber, the order of the digits is reversed ; then 2a; = 2/, \Qx + y + a=\Oy + x, x = ^a, y = ^a; the special condition is imposed that a shall be a multiple of 9 not greater than 36 nor less than — 36 ; i.e., a is 36, 27, 18, 9, 0, "9, "18, "27, -36, and the number is 48, 36, 24, 12, 0, "12, "24, -36, -48. 6-8, §11.] QUADRATIC EQUATION'S, ONE UNKNOWN. 313 § 11. QUADEATIC EQUATIONS INVOLVING ONE UNKNOWN. Pros. 7. To solve an incomplete qdadratio equation. Reduce the equation to the type-form x^ = q, and take the square root of both members; then x = ± •y'q. E.g., to find x from the equation K*'- 10) + t't! (6 a;'- 100) = 3a;=- 65 : then •.- 10a;^-100 + 18a^-300 = 90a;2-1950, [mult.bySO .-. -62a^=-1550, .-. a?=2b and a;=±5. Note. There are two square roots, opposites of each other ; they are both real if the q of the type-form be positive, and both imaginary if the q be negative. PrOB. 8. To SOLVE A COMPLETE QUADRATIC EQUATION. Reduce the equation to the typeform x^ + px = q. Add \^^ to both m,embers of the equation; take the square root; and solve the equations thus found. The result is of the form x = — Jp±-^-^(p^+4q). E.g., to find a; from the equation 3a3^ + 9a;= 120 : then •.• af + 3x = i0, ' [div.'byS .-. of+3x + 2i = A2l, [add(f)^=2^ -•. X:\-l^= ±6^, [extrl sqr. rts. of bothmem. .-. x = —l^ ± 6^= 5 or — 8 ; and 5, 8 are both roots. So, to find X from the equation aii? + bx + c = 0: then a? + ~x + —- = — ;— „— , a 4 a^ 4 a'' and X = -b±^/(b^-'^a■o) . ^nd both values are roots. 2a Note 1. Double Signs : Since either x+p or —{x+p) is a square root of iK^+px + ^p^, the given quadratic is satisfied as well when -{x+p) = ii/{p^+4.q) as when x+p=^^{p^+ iq); but this gives only the two values for x written above. 314 ' EQUATIONS. [XI. pr. Note 2. Discussion of the equation 3? -\-px = q, four SPECIAL CASES. The roots are : (a) p positive, q negative. Two real roots, both negative, if p^ + 4g be positive. Two real roots, both negative, equal to —^p,[fp^-\-4:q = 0. Two imaginary roots, conjugates, if p^ + Aq be negative. (&) p, q both negative. Two real roots, both positive, ifp^ + iq be positive. Two real roots, both positive, equal to —ip, if p^ + 4g = 0. Two imaginary roots, conjugates, it p^ + iqhe negative. (c) p, q both positive. Two real roots, the smaller positive, the larger negative. (d) p negative, q positive. Two real roots, the smaller negative, the larger positive. Note 3. Sums and products of roots. The sum of the two roots is —p, and their product is —q. The reader may prove. Note 4. The absolute term, 0. If g = 0, then of the equation a^ +px = the two roots are and —p, both real. Note 6. Solution bt factoring. Write the equation a? +px —q=0 in the form 3!'+px + ip'-i{p'^+Aq) = 0, i.e., in the form (a! + ip)^ — ^(^^+4^) = ; then •.• [x + ip-i^(p'+4:q)-].[x + ip+i^(p''+4q);\ = 0, and •.• this product vanishes when, and only when, one of its factors vanishes, .•. the roots of the equations and x + ip + i^(p'' + 4:q) = [th.4cr.l are the roots of the given equation. ••• a'=-*p + iV(p' + 4g), x=-ip-i-^{p^ + 4:q). 8, § 11.] QU ADEATIC EQUATIONS, ONE UNKNO^yN. 315 la practice the factoring is often made, at sight. E.g., to find x from the equation aP — 5x+6 = 0: then •.• a^— 5a; + 6 =(» — 2) (a; — 3), [factoring ,*. the roots are 2 and 3. Note 6. General etjle. The rule for solving incomplete quadratic equations may be stated in a more general form : Reduce the equation to the type-form ax' + bx -f- c = 0. Multiply {or divide) both members of the equation by a, or by any factor or multiple of a that shall make the coefficient of the first term a perfect square. Add to both Wtevibers of the equation whatever is necessary to make the first member a perfect square, and take the square root. Solve the simple equations thus found. The rule in this form often avoids fractions. Both rules rest on that for finding the square root, and are the same in principle. The solution of the equation gives a;= ^^ — ^• ^ ® 2a The reader may translate this formula into a working rule for finding the value of x without writing the intermediate steps. E.g. , to find x from the equation S.x' -\-2x=120 : then •.• 9a!2 + 27a;=360, [mult, by 3 and 9x' + 27x \Sx + 4l^ £sq. rt. of 1st mem. 9af 6a; + 4-i|27a; 27a; + 20^ and x=5 or — 8 ; or, by direct substitution in the formula, ^^-9±V(9--4.3--120)^g or -8. 2-3 316 EQUATIONS. [XI. ths. NoTK 7. Discussion of the equation ao/*^ + 6a! + c = ; THREE SPECIAL CASES : C = 0, 6 = 0, a = 0. (a) If c, the absolute term, he ; then the equation a3?+'bxz=() gives a;=0 and x='—b:a, two real roots, whereof one is 0. (6) Jfb, the coefficient of the first power ofx, be ; then the equation ax'+c=0 gives aj= ±-y/(— c : a), two real roots, opposites, if a, c be of contrary signs ; two imaginary roots, conjugates, if a, cbe of the same sign, (c) If a, the coefficient of the second power ofx, be ; then •.- ^^-& + V(&--4ac)^ -b -^/{b^ -4ac) ^ 2a 2a .-. a; = (-6+V6') :0, {- b - ^b-) : ; [a = i.e., a; = — 0:0, 26:0 if V*^ T^e + 6 ; and a; = -26:0, 0:0 if -y/fi^ be - 6. In either case there is an infinite and an indeterminate root. But this indeterminate root may be determined. For •.• a = 0, .•. when x^Ki, the equation a3f + bx + c=0 becomes 6a; + c = 0, whose single root is — c : 6. It may also be determined by multiplying both terms of the fraction -^± V(&'-4ac) ^^ -b^ -^{V" -4ac) ; tten X- V-(b'-4ac) ^ 2c 2a[-6TV(*^-4«c)] -6TV(^^-4ac) = — c:b or — c:0 when a = 0. This case is especially important as showing the value of the limits of the roots of the equation when a == ; and it is to be noted that as a = one of the roots = qo, and the other = — c : 6. This is also evident if the equation be written in the form a!~^(6 +ca!~') = — a. [div. eq. bx + c = —asy' hya^ For, if a = 0, then either a;"' = 0, and x = 4ac. ' -| real and equal when ■{&^ = 4ac. I imaginary | &^<4ac. Of the real and unequal roots he may show which is the larger ; and of all real roots he may show the conditions that make them positive or negative. He may also show that in every case the sum of the two roots is —b:a, and their product c : a, and that if x', x" stand for the two roots, then fflar' + 6a; + c = a (cc — a;') (« — x"). Note 8. Equations solved as quadratics : Every equation of either of the following forms, or reducible thereto, is solved by aid of quadratics : (a) a»=" + 6a;''+c=0, (6) (aa^" + baf + cy +p{ax'''' + 6a;" + c)" + g = 0, (c) {aa?" + bx'^ + cy'^±(ex^+fy'" = 0, (d) ' (a«^'' + 6a;» + c)="'±(dx2" + ea!")2"=0, wherein a, &, c, d, e, /, jp, q are independent of x, and may be real or imaginary. Whether a given equation p = 0, whose degree is even, be of form (a), appears at once. If it be not, then to see whether it reduces either to form (&) or to form (c) , find e, the entire part of the square root of p : if the remainder p — k^ be of the form j3K + g, the equation reduces to k = |-[— p+VCP^ — 4?)]i and if also R or some root of r be of the form aa^" + 6af + c, the equation reduces further to (5) ; or if p — r^ be ± a perfect 318 EQUATIONS. [XI. square, s^, the equation reduces to r= sV''^l^ ^^^ perhaps to (c). Or, arrange p to ascending powers of x, and find r', so much of -y/p that r'^ has the degree of p as to a; ; then if p — r'^ be ± a perfect square, s'^, the given equation reduces to r=s'v''''1, and perhaps to (d) . E.g., if 9a5*-52a^ + 64 = 0; then •.• 81a!* — 468a;2 + 676 = 100, [mult.by 9, add 100 .-. 9a;2^26±10, .•.ay' =4 or ^-, .-. X = ± 2 or ±^: four real roots. So, if (9c(;*-52a;= + 80)2 + 9(9a^-52a;2 + 80) -400 = 0; then •.• 4(9a;*-52a;2 + 80)=+36(9a!*-52a;2+80)+81 = 1681, .-. 2(9a!*-52x=+80) = -9±41 = 32 or -50, .-. 9a;*-52a!= + 80 =16 or -25, . . a;=±2, ±f, ±|V(26±V— 209) : eight roots. § 12. GRAPHIC EEPRESENTATION OF QUADRATIC FUNCTIONS. Let aaf + &a; + c be any quadratic function of x, and put y equal to it ; then different values may be given to x, the cor- responding values of y computed, and the function platted. The plat is a parabola whose axis is vertical. E.g. , in the equation y = a? + 2x—3. Put «=••■, -5,-4, -3,-2, -1, 0, +1, +2, +3, +4, +5,...; then y=—,12, 5, 0,-3,-4,-3, 0, 5,12,21,32,..., and the plat of the function is as shown in the figure, p. 319. If there be a pair of equations involving x,y y = tip + 2x—3, y = 0, their solution is reduced to the solution of a single quad- §12.] QUADRATIC FUNCTIONS. 319 ratio equation in-\^olving one unknown element, a?+ 2a!— 3 = ; and the roots of this equation are the abscissas of the points where the curve whose equation is y=a:F-{-2x—3 cuts the axis of abscissas whose equation is y = 0. The ordinates of the points of inter- section are manifestly 0. If the curve that represents the equa- tion y = x--\-2x — 3 remain fixed on the paper while the horizontal line that represents the equation 2/ = moves downwards, taking in succession the ^ positions o'x', o"x", •••, each ordinate of the curve is increased by the same length, and the value of y in the given equation' is increased by the same number ; and, by the simple change of the absolute term, the two roots of a quadratic equation maj- ap- proach each other, then become equal, then imaginary. JE.g., of !x?+2x = 3 the two roots are —3, 1, of x^+ 2x=Q the two roots are —2, 0, of a^-j-2x = —l the two roots are —1, —1, of a^+ 2 a; = —2 the two roots are -i-t-V-i, -1-V-i- In all such cases it is said that a straight line cuts the curve in two points, real and separate, real and coincident, or imaginary, just as it is said that every quadratic equation has two roots, real and unequal, real and equal, or imaginary' ; and thougli it may seem strange to the beginner to say that one line cuts another in two points when it only touches it, or to say that it cuts it in two points when it does not cut it at all, yet the language and the demonstrations of Algebra gain greatlj- by this generality ; and the pairs of roots so described have most of the algebraic properties of other pairs of roots : in particular, they each satisfy the given equation, and their sum is the —p and their product the — g of the type-form. [ 320 EQUATIONS. [XI. pr. Prob. 9. To PLAT THE EQUATION ax' -{-bxy + cy^= d, using NO IRRATIONAL FUNCTIONS OF X, y. a, b, C, d, X, ?/, ^LL REAL. (a) When 6^ > 4 ac, and ccZ-J^ 0. ±d Compute M, = +1-*^ ; and n, = +|^ \b^— 4ac \ c To V, an auxiliary variable, give any convenient series of val- ues; and for each value of \ find a pair of simultaneous values of X, y to satisfy the given equation : vi.., . = ^1.2M, ,- = Z^.n-I^.^. 2v 2v 2v c Plat each of the points x, y ; and join them by a curve. (&) When 4ac>6^: then always cd>0. [x,yveal Compute m', = *[ ^^-- ; and n', = ^l-- \4ae — b^ \c To the auxiliary variable v give any convenient series of values; and for each of them find values of x, y to satisfy the equation: Plat each of the points x, y ; and join them by a curve. (c) When 6^ = 4ac: then always cd-jtiO. [w, 2/ real Compute n', = ^1- ; the plat is two parallel straight lines ■^ 2c ' -^ 2c (a)-.' {2cy + bxy—(b^ — 4ac)-x'^4:cd, [giv. eq. .-. [2c2/+6a; + a;V(&^-4ac)][2c2/+6a;-a;V(6^-4ac)] = 4ccZ; . • . whatever value be given to v, when 2cy + bx + x^(b^—iac)= 2^{±cd)-v, then 2cy + bx — x ^ (b^— 4:ac) = ±2 -y/ {±cd) :v, v^ T 1 I i cd v^ m 1 .•. x = — ^^^— -A r; : — = — -^-— -m; Q.E.D.relim.w, solve V \6^ — 4ac V v' ± 1 and -.■ 4:cy + 2bx = 2 ■y/{±cd), [add eqs. above 9, § 12.] QUADBATIC PU]S"CT10NS. (6) then and And then y — —z — ^-N ~ Q.E.D. rrepl. CK, sol.for« 2 c 2v c {2cy + bxY= icd — (4ac — &^<) • s? [giv. eq. = [2 Vcd + K V(4«« - &')] [2 Vc«^ - a;V(4a«-62)]; whatever value be given to v, when 2Vcd + a;v(4ac-62) = i±-^(2c2/ + 6a;), 1 — V -.- 2Vcd-»V'(4ac-&') = l-i; 2Vcd+a;V(4ae-6^) ^ /l+jwV _4«_ I ccZ iv A^cd=l±^{2cy + hx), 1-v' \-i^ 2v bm.' [divide 2m'; Q.E.D. [sol.fora; [add eqs. above Q.E.D. [repl. X, sol. for y Q.E.D. 1+v^ 1 + v' c Acd is the sum of the positive quantities {2cy + bxy, (4ac- 62).a!2, cd<0. when (2cy + bxy = 4:cd _ (2cy + bx — 2 -y/cd) (2cy+bx + 2 ^cd) = 0, .^, Id bx Id bx ...either 2/ = ^--- or y = -^--^^, and conversely. q.e.d. E.g., to plat the equation Sx' + btey + Ty^— 4:25. [ig.,i>. 322 Here a, b, c, d = 3, 5,7,425 ; 4ac — bf =+59, and the case is (&) ; m'=V(7 -425: 59) = 7.101, n'= V(425 : 7) = 7.792, ^=5.072; and bm' the coeflflcients of 2 m', n', — in the values of x, y are : 322 EQUATIONS. [XI. prs. When V =0 ^ ^ ^ ^ *1 ^ *2 *3 ±5 00 ••., 5 3 2 3 2 ' ,, '2v ^ ="5 *3 *4 *12 ±. *12 H ^3 =^5 „ then - = — — — — 1 — — — — •••, H-y2 13 5 5 13 13 5 5 13 , l-«2 , 12 4 3 5 ^ -5 -3 -4 "12 _, and - = 1 — - - — — — 1 •", 1+v^ 13 5 5 13 13 6 6 13 and 2/ = (l,l|,...,^,-l)x7.792-(0,^,...,^,0)x5.072, i.e., x= *5.46 *8.52 *11.36, *13.11 *14.20.-., , ^ „a , 5.24 , 3.19 , 0.62 ,-1.68 =Fr ^7 and y= /.79 ^ g^^^ { g^g ^ 8.73 ^ 7.68 ^'O' -' *1 *1 *1 ^2 for V— — — — — 1 ■", 5 3 2 3 , _ ^„ ,-9-14 .-9.28 ,-8.73 ,-7.68 and y= 7./9 {_^_^^ ^_^g ^^^gg { ^_gg -.., for v= OS *5 *3 =^2 ^ .... 2 Give the same coefficients (except in order and sign) any values 'yi,i;2 of « such that ^2= —Vi, or = ± 1 : 'Uj, or = ± (1 — Vi) : (1 +Vi), or = ± (l + 'Wi) : (1 —Vi). Such values are 0, oo, =*=! ; and ±i, ±K ±2, ±3; and ±i, ± ?, ± -, ±5. 3 2 5 3 2 So, to plat the equation 3 a!^ + 5 a;?/ + 72/^ — 14a! — 51 2/ = 330 : here 3 (a; + 1) ^ + 5 (a; + 1) (j/ - 4) + 7 (2/ - 4)^ = 425, and the plat is as in the above example, except that the origin or datum-point to which the curve is referred will now be a unit to the right of, and 4 units below, the origin of the former plat. Note. When- d = 0, no auxiliary v is needed : plat (a) re- duces to a pair of lines y = ^\^—b + y/{b^—4ac)^G~^x and 2/= i-[— & — -y'(&'— 4ac)]c-'x; plat (6) to a point a!=0, y==0; plat (c) to two coincident lines y = — ^bc~^x. 0,10, §13.] SOLUTION BY CONTINUED PEACTIONS. 323 §13. SOLUTION OF QUADRATIC EQUATIONS BY AID OF CONTINUED FBACTIONS. PeOB. 10. To SOLVE A QUADRATIC EQUATION BT AID OF CON- TINUED FRACTIONS. First root : Write the equation in the form x(p+x) = g; then x = ^ =^9' =^ 1 „ = •••, p+x pA p-i-±— q ' and the convergents are : i>+ * q pq p^q + q^ p'q + 2pg^ _ p' p^ + q' f + 2pq pi+ip^q + q' ' Second root: Write the equation in the form o? = —px-\-q. q q q then « = — » + - = — » q — — p -q „=..., ^ ' X ^ p ^ pA q ' ■^ X ^ ' p and the convergents are : ^ — P^ + 1 _P^ + '2pq p^ + Sp^q + q" _ p ' p'^ + q ' p^ + 2pq ' Of these two sets of convergents, when taken two and two in order, the products are — g, and the sums approach —p. E.g., to find x from the equation a? -\-bx = i; 2 2 then the two roots are — 9 and — 5 2 5 + ^ . ^5-..., and the convergents are : o H 2 10 _M 290 and-5 -— -— ^79 5' 27' 145' 779' '"' ^ ' 5' 27' 145' whose products taken two and two are all — 2, and whose sums so taken are — 4|, —^t^, — 4 || °|, ■••, that differ from -5 by |, ^, ^, •.•. The reader may find the approximate values of x from the equation aa3^+ &a! + c = 0, and translate the formulee so found into words. In particular, he may find the approximate values of X when a = ; and show how the convergence of the con- tinued fraction depends upon the reality of the roots. This form of solution by continued fractions applies only to quadratics ; another form is given in XIII. 324 EQUATIONS. [XI. th. 8, §14. MAXIMA AND MINIMA. If a; be a variable, and y be a function of x, and if as x in- creases, y increase for a time and then decrease, the greatest value that y thus attains is a maximum ; but if as a; increases, y decrease for a time and then increase, the least value that y thus attains is a minimum. So for any two variables. The normal { .'. of any function u of one or more in- ' mmima •' dependent variables x,y, ... are such values of w that, if u were a little { ? ' some of a;, y, ... would become imaginary: they depend upon x, y, ... being restricted to real values. Ab- normal maxima and minima arise from other restrictions : as in the example below, where a certain rectangle is restricted to have its corners at or between the vertices of a certain triangle. If, by solving a quadratic or otherwise, the relation of u to a;, 2/, ... be expressed in the form p + q •y'u = 0, wherein P, Q are rational functions of a;, j/, ... and u a function of u ; then is m a normal ■{ ■ ■ whenever its value is such that v vanishes ' mmimum , J . A. -^ ic ■ • • v J. ■ 1 a decreasmg and IS not itself a maximum or minimum, but \s < . . =■ ' ' an increasing IHCI'G&SG function of u ; for a slight further \ ■, in the value of u makes -y/v imaginary, while its equal — p : q, a rational function of x,y, ..., remains real. Theor. 8. Maodmum and minimum values of a continuous function occur alternately. For •.• just after passing through the first maximum value the function is decreasing, and just before passing through the second maximum value the function is increasing ; [df. max., min. and • . • in [jassing from a decreasing state to an increasing state the function miist pass through a minimum value ;[df. .-. between two maximum values lies at least one minimum value. Q.E.D. So between two minimum values lies at least one maximum value. Q.E.D. pr.ll,§14.] MAXIMA AND MINIMA. 325 PrOB. 11. To DETERMINE MAXIMA AND MINIMA BY SOLVING QUADRATIC EQUATIONS. Sy an equation express the relation between x, a variable, and u, a function of x to he maximized or minimized. Solve for x ; and if the value of x thus found involve an even root of a function o/u, equate that function to and solve for u. See whether the values of u so found be maxima or minima. E.g., in a triangle to inscribe a rectangle of maximum area : 1. There is such a rectangle. For let ABC be any triangle, an its altitude, bc its base, defg any rectangle inscribed in it. then •.• DEFG approaches a minimum value, zero, as gp approaches A, and another minimum value, zero, as gf approaches bc, for some intermediate position of gf there is a maxi- mum value of DEFG. [th. 8 2. Let M s area DEFG, a; = DE, 2/ = dg, & = bo, 7i = an; then -.• xy = u, x:h — y = b:h, .-. u =by{h-y):h, y = ^h±^il{bh'-4hu):b'], .-. the maximum value of u is ^bh, and gf lies half-way between the vertex and base. So, about a sphere to circumscribe a cone of minimum volume : 1. There is such a cone. For let DEF be an}^ circle and abc an isosceles triangle cir- cumscribed about it and tangent to it at d, e, f ; let AD be the axis of the tri- angle, and let the whole figure revolve about ad ; then ■.• as the point a recedes from the circle, the lines ab, AC approach parallelism, and the triangle abc grows larger and larger without bounds, .■. the cone abc grOws larger and larger without bounds. 32G EQ^JATIO^"S. [XI. pr. And ••• as the point a approaches the circle the lines ab, ac approach parallelism with bc, and the triangle abo again grows larger and larger without bounds, .-. the cone abc grows larger and larger without bounds ; . • . for some intermediate position of a there is a minimum value of the volume of the cone. [th. 8 2. Let V = volume of cone abc, y = ad, its altitude, X = BD, radius of base, r = radius of sphere ; then V = ^Troiy'y ; and ••• AB-AF = AD-A0, ab^ = ad^ + bd^, af^ = ao^ — of", AO = AD — CD, [geom. (/ + af) ■ {y - r -r') = y^.{y- ry, x'y — i^f: {y — 2r), Y =\irr^y^: {y-2r), y =[3v±V(9v"-245rr-3v)]:27rr^ and -.• that y be real, 9v"-<247rr'v, the minimum value of v is f vrr', and the corresponding values of y, x are ir, r-,/2, i.e., the minimum circumscribed cone has its altitude double the diameter of the sphere, the area of its base two great circles, its volume double the volume of the sphere, and its whole surface double that of the sphere : as the reader may prove. gj! 2a; 4- 21 So, to ascertain if the fraction — — have any limi- ox — 14 tations on its value, for real values of x : T ^ af-'2x + 21 Let y = ■ ; ^ 6a; -14 then •.• a;=l + 32/±3V[(2/-2)(2/ + V-)]> [soL f or a; .-. that X be real, the factors y — 2 and y+^ must have the same sign, i.e., y may not lie between 2 and — -i^, but may have any other value, .-. 2 is a minimum and — ^ a maximum value of x. 11, § 14.] liJAXIMA AND MINIMA. 327 The reader may plat this function [a hyperbola], and the meaning of these statements will then be clearer to him. So, to find the limitations on the value of the quadratic func- tion aa? + bx + c for real values of x : Let y = as? + bx + c; then -.• « = [— & ±^(6^ — 4ac + 4a2/)] : 2a, .•. for real values of cc, 6^— 4ac+4a?/ cannot be negative, ', , 6^— 4ac , , , negative , . , positive. I.e., y -\ ^ cannot be ^- ^.,. when a is -< ^ ,. ' ' 4 a positive ' negative, 1 < 4ac— 6^ J 4ac — 6^ . ., , minimum , ■*• Vi -J , and is its < . value. ■^ ' > 4a 4a ' maximum The plat of the function is a parabola whose axis is vertical and vertex ■{ -, when a is ^ ^ . . ; and this parabola ' upward ' negative ' ^ { , cut the axis of a; when 6^—4 ac is -J =... ' does ' ' positive. The four cases are represented by the four cuts below. So, if 2/ = , , ' , , ; a'x^+Vx + c' then -jh-Vy) 2 (a — a'y) Vr(&'^-4ffl'c')y^-2 (55'-2ac'-2a'c)y+5^-4ac] 2 (a — a'y) .•. that a; be real, (6'2 _ 4a'c')2/^ - 2 (&6' -i 2 ac' - 2 a'c)y + 6^ _ 4ac < 0. Write this quadratic function of y in the form (6'2_4a'c')(2/-a)(2/-;8), wherein u, /3 are the roots of the equation got by putting this function equal to ; then three special cases are to be noted : 328 EQUATIONS. [XI. pr. (a) b' ^ — 4 a'c' positive. If a, /3 be real and a< /?, then, that x may be real, y must not lie between o, /3 ; i.e., y has a for a maximum and /? for a minimum value. If a, p be real and equal, or imaginary, the product {y — fi) (y — a) is always positive, and there is no limitation on the value of y. (b) h" — i a'c' negative. If a, ^ be real and a<.p, then, that x may be real, y must lie only between a, /8 ; i.e., y has a for a minimum and /3 for a maximum value. If o, (8 be real and equal, or imaginary, then the product (6'^ — 4 a'c') (2/ —a) (y — /3) is negative, and no real value of X is possible except for the particular values (c) b'^— 4a'c' zero. Then the quadratic function in y reduces to the form py + q; and that x be real this function may not be negative ; and if p be ^ ^"''^^^ 'y+^- cannot be { '^^gf "'^ ^ -^ ' negative, " ' p ' positive ; 1 < ? J -i I maximum , . 9 •■• V^Zr > and its -! . . value is " ' y- p' ' minimum p Note. It is sometimes better not to solve for the independent variable, but to express in terms of it the function to be maximized or minimized : noting that if a be a positive constant, m, n odd posi- tive integers, (u); ' minimum, ' ' ' ^v /> 1 . , -- , / \ p minima, but ±a — u, a:u, u ", Kl/fu) are-i ' ' ' T \ / 1 maxima. H.g., to divide a real number 2 a into two real parts whose product is a maximum : Let a — z and a + 2; be the two parts ; then • . • {q, — z) • {a +z) = a^ — z'', and ••• the product oF — z'^ is greatest when z = 0, .-. the parts are a and a. q.e.d. 11. § 14.] MAXIMA AND MINIMA. 329* There is no minimum ; for, as z grows larger, a^ — g' grows less without bounds. Let a — 2 = a; ; then o + z = 2a — a;, and the product o^— »2 = a; (2 a — a;) . To plat the locus of the equation y^ = x(2a—x): Take ox=2a, and on ox as a diameter describe a semi- y^ circle ; take b any point on / ox, and draw bp perpen- / dicular to ox ; / then ••• BP^=OB-Bx, [geom. o A c and .y = BP, a; = OB, 2 a — ksbx, .•. the semicircle is the locus sought, and y^ is greatest when b is at the centre, i.e., when ob, bx, bp, each = a. So, to resolve a real number a^ into two real positive factors whose sum is a minimum : Let X, y be the two parts ; then •.• {x+yy = {x-yy + ixy = ix-yy + 4:a?, .-. (a + 2/) is least when a; — 2/ = 0, i.e., when x = y — a. There is no maximum ; for, as a; ~ ?/ grows larger without bounds, so does x + y. From these two examples it appears that of all rectangles with the same perimeter the square has the greatest area, and that of all rectangles with the same area the square has the least perimeter. So, often the same conditions that make a variable u a maximum or minimum when some other variable v is constant, also make v a maximum or a minimum when u is constant. Other maxima and minima may be found by aid of the above. E.g. , to make — -^ — - a maximum : [a, 6 positive Make the reciprocal, ax-yh-.x, a minimum ; then •.• the product of ax and & : a; is the constant a&, .'. their sum is a minimum at 2-^ab ; i.e., the given function is a maximum at ^a~^h~^. 330 equati6ns. [XI. prs. then or and § 15. SIMULTANEOUS EQUATIONS. PrOB. 12. To SOLVE TWO EQUATIONS INVOLVING THE SAME TWO UNKNOWN ELEMENTS WHEN ONE OF THE EQUATIONS IS SIMPLE. Eliminate one of the unknown elements. [pr. 2 Solve the resultant for the other unknown element and replace this element by its value in the simple equation. Solve this equation for the first unknown element. E.g., to find the values of x, y from the pair of equations 3 a; + 2 2/ =20, d of + 5xy + 7y^ = A25 : x = i{20-~2y), [sol. first eq. for a; i(20— 22/)2+.|2/(20-2y) + 7/=425,[repl.a;insec.eq. 15y^ + 20y =875, and 2/= 7 or —8^, [sol. quad. for j/ 3a; +2-7 =20, and a;=2, [repl. j/ in first eq. 3a; -2-8^=20, and a;=12|-, the two pairs of roots are : 2, 7 ; 12|-, — 8-J. That both pairs of roots satisfy the two equations appears by direct substitution, and that there ought to be two pairs of roots is evident from the plat. x= 0, 2, 4, 6, 8, 10; in the equation 3 a; + 2 ?/ = 20 y= 10, 7, 4, 1, -2, -6, in the eq'n 3x'' + 5xy+7y^ — 4:25 2/=+7.8, +7,+6.2,+4.9,+3.5,+1.5, y = -7.8,-8.2, -8.9, -9.2,-9.3,-8.6, Note 1. The two pairs of roots maj- coincide. E.g., to find x, y from the pair of equations 7a;2+6a!2/+82/2+12a;+lG2/-88 = 0, 23a;+222/ = 68 : then the resultant is y^—2y-\-\ = (i, and a;, ?/=2, 1; 2,1. The geometric interpretation of the equality of the roots is that the loci represented intersect in coincident points; i.e., that they are tangent. A slight change in either equation so changes the locus that the points separate or disappear. Then the two roots are real and separate, or imaginary. 12, 13, §15.] SIMULTANEOUS EQUATIONS. 331 Note 2. Special expedients may be useful. H.g. , to find x, y from the pair of equations x + yz=l, xy=12: then •.• (x — yy=(x + yy — ixy=l, ••■ x — y=±l; and ■.• 2x={x + y) + {x-y), 2y={x + y)-{x-y), .-. a;=4or3, 2/=3or4; and the two pairs of roots are : 4, 3 ; 3,4. So, to find X, 'If from the pair of equations a;^-|-2/^= 125, x — y = 5: then -.- {^i, + yy+{x-yy = i{x' + f), .-. (a; + 2/)2 = 225 and x + y=±\6, .-. a;= 10 or — 5, y=b or — 10, and the two pairs of roots are : 10, 5 ; — 5, — 10. PrOB. 13. To SOLVE TWO EQUATIONS INVOLVING THE SAME TWO UNKNOWN ELEMENTS WHEN BOTH EQUATIONS AEE QUADRATIC. Eliminate one of the unknown elements by division; solve the resultant biquadratic equation for the other. Replace this element by each of the four roots so found, in the equation formed by equating to zero any remainder or divisor that contains the other unknown element in the first degree, and solve for that element. E.g., to find the four values of x, y from the pair of equations 2«2/ + 02/2— 195 = 0, Za? — 4:Xy — 'i=Q: then 2y-a;+(5y^-195)| ?,a?-4.y-x-l |3a;- (23/- 582) 2l &y.ii?-Sf--x-\4:y 6j/-a!2 + 8(5y^-195)a; -(232/='-585)a;-14y 2y • -22/(232/2-585)a;-282/2 -2y(23y^-585)a;-(5y^-195)(23y2-58.5) (5/-195) (232/2-585) -28^^ 332 EQUATIONS. [XI. pr. Equate this last remainder to zero ; then •.• 115^-74382/^ + 114075 = 0, 4563 If = 25 or = ±5 or ± 115' 117 V345' and a; = ± 7 or T J^- [repl. y This process consists in ifeplacing the two given equations by two new equations got from the last two remainders, the one free from x, and the other having x only in the first degree. So, to find X, y from the pair of quadratic equations x' + 2y'^ = Zxy, 16a;— 12^= 5a^ ; then •.• the resultant of these two equations is by^— li'f + 8?/^= 0, whose linear factors are y-,y, (52/ -4) (2/ -2), .-. 2/ = 0, 0,1, 2, and a;=0, 0, |, 4. The locus of the equation a^ + 22/^ — 3a^ = 0, i.e., of the equation {x — '2y){x — y) = 1, 2, -•• m, 1, 2, ••• n. This method is similar in principle to that of Note 1. E.g., In the example of Note 1, Po + P2 = 0, Qq — Q2 = 0» wherein Po, Qo, Pj, Q2 =/» /'> ax^ + hxy + (nf, a'a?+b'xy -\-c'y^.; Vi,Y2'=a+bv + cv^, a' + Vv + c'v^ ; and -y is found from the quadratic (cf'—c'/)v^ + (6/' — b'f) v + af'—a'f= 0. Note 3. Sometimes the solution of a pair of equations may be simplified by changing the unknown elements: notably by making use of the following relations, connecting the sum of two numbers, their difference, tljeir product, the difference of their squares, the sum of their squares : half sum + half difference , = greater number. half sum — half difference = less number. product sum X difference = difference of squares. sum of squares + twice product = square of sum. sum of squares — twice product = square of difference, (half sum)^ + (half difference)^ = half sum of squares, (half sum)^ — (half difference)^ = product. 334 EQUATIONS. [XL pi'. When each equation is symmetric as to x,y, it is commouly best to take symmetric functions of x, y for the new unknown elements. When one equation is symmetric as to x, y, and the other as to X, —y, it is often best to take x + y, x — y for the new unknown elements. Sometimes equations not originally thus symmetric may be made so by transformation. E.g., the resultant of the pair of equations 3xy — ix — iy — 0, x^ + y^ + x + y — 26 = is 92^ _ 15 y3_ 242/ +6242/ -416 = 0, which is not easily reduced to a quadratic. But put (x + y)^—2xy for x' + y^, and write the equations : 3xy-4:{x + y)=0, (x+yy+(x+y)-2xy-2e=0, then a; + 2/ = 6, xy = 8 or x + y = — ^, xy = — ^\ and the four values of x, y are found from these two pairs of equations, each consisting of a simple, equation and a quadratic. So, to find X, y from the pair of equations x — y = lxy, x' + y^ = ^xy; square the first equation and subtract from the second to find values of xy ; join each of these equations with the first equation to find values of x, y. So, in the equations a; + 2/ = 4, a;* + 2/* = 82, put M + 'y = a;, u — V'=y; then {u + v)+{u — v) = A, u=2; and (u+vy + {u-vy = 82, u'^ + 6uV+v* = 4:l, ... ■u< + 24i;2-25 = 0, .•. •y^=l or —25, v=±l or ±5i. .-. X =3, 1, 2 + 5i, 2—bi; 2/ =1, 3, 2 — bi, 2 + bi. So, to find the five values of x, y from the pair of equations a; + 2/ = 4, a^ + 2/^ = 244; 13, § 15.] SIMULTANEOUS EQUATIONS. 335 then •.• {u + v) + (u — v) = ^, (u + vy + {u — vy = 2U, .-. M=2; u' + 10u^v^ + 5uv*=zl22, .-. the five values of v are oo,+l,~l, +i^3, — i-y/3, the five values of a; are +oo, +3,+l, 2+1^3, 2— i-y/3, and the five values of ?/ are -aD,+l,+S, 2—i^3, 2+i-y/3. Note 4. The meaning of these infinite solutions may be interpreted as follows : Consider the equation a; + j/ = 4 as the limiting form of an equation a; + &2/=4, whose coefficient 6 gradually approaches unity as a limit : one of the pairs of values of x, y grow larger and larger without bounds, and the solution is either «=+co, y=~ao, or a; = ~oo, y=+co, according as 5 is a little less than unity or a little greater. Note 5. Sometimes the roots of higher equations may be found by the method of division. E.g., of the pair of equations y{!Jif +y^) = 4:(x + yy, xy = 4:(x + y), the resultant is y^ — Sy'^O; and •■• this function of y is divisible by y^, .•. the equation has three roots ; it has also one root 8. But •.- the general resultant of a cubic and a quadratic equa- tion is of the sixth degree, .-. this resultant has lost its two highest terms, and the equation has two roots oo ; .-. the values of y are co, oc, 0, 0, 0, 8, and the six corresponding values of x, found from the equation a; = 4 2/ : (y — 4) are 4,4,0,0,0,8. The geometrical interpretation of these roots is, that of the six points of intersection of the loci that represent the two equa- tions two are at an infinite distance and lie on the line a; = 4, three are at the origin, and one is at the point whose co-ordi- nates are 8, 8 ; or, in the language of limits, if one of the curves 336 EQUATIONS. [XI. pr. change its form slightly, by the gradual change (say) of a sin- gle coefficient, and thus approach its present form, then two of the points of intersection recede to an infinite distance, three of them not coincident approach the origin, and one approaches the point 8, 8. So, of the pair of equations x'+y(xy—l) = 0, f—!c(xy+l)=0, the resultant, found from the last remainder, is 42/''-4/-2/=0, and the second last remainder gives {2f + l)x-2f=Q, .-. y = or 2/* = ^(l±V2); i.e., y may have the value or any one of the eight values of ^/i(l±V2), =^(*±Vi). and X, =2y^ : (22/^+1) , may have the value or any one of the eight values of ^1 : (2 ± 2 V2), = ^(-i± Vi)- Note 6. Special methods of solution : Many sets of simul- taneous equations may be solved by special devices. The ex- amples given below are meant merely as suggestions to the reader. He is advised to try his own ingenuity upon each example before studj'ing the solution here shown ; and afterward, to see how far the principle of each solution applies to other examples. 1. To find the values of x, y from the pair of equations a^ + ajy + 2/* = 133, (1) a?- xy + f=l: (2) (3) [div.(l)by(2) (4) [add (2), (3) (5) [sub. (4) from (3) (6) [add (3), (5) (7) [sub. (5) from (2) = ±1; .-. a!,2/=+3,+2; "3,-2; +2, +3 ; "2,-3. 2. If a^-a;2 + 2/^ -2/2=84, o^ ■\-^y^ ■\-f=^^ -. then (a^+2/')'-2a^2/'-(-'B' + 2/')=84, (a;2 + 2/2) +3^2/2 = 49. then ^-V^y+f =19, and a!^ +2/2 = 13; •. xy =6, a;2 + 2a;2/+2/2 = 25. and ^-1xy^f = \; . ■. x + y=±5, x — y 13, § 15.] SIMULTAiJ^EOUS EQUATIONS. 337 Put u-=3? + y'',v=,x'^y'-; ' (1) then •.• M^ — w — 2'y=84, u + v—A.^; (2) .-. w' + u^^^l&i; [add (1) and twice (2) .-. M = 13 or —14. (1) Put 0)2 + 2/2 =13; then •.• a?y^ = Z%, a;= + 36a;-2= 13 ; .-. aj^= 9 or 4, y^ = 4 or 9 ; , .-. a; = ±3 or ±2, ?/ =±2 or ±3 ; i.e., x,y=+3+2 ; +3,-2 r3,+2 ; "3,-2 ;+2,+3 ; +2,-3 ; "2 +3 ;-2,-3; eight pairs of roots. (2) Put 3!^ + 2/2 = -14; then a? 2/^ = 63 , a;^+ 63a;-='= — 14 ; .-. af = -7±iy/U, 2/' = -7t*V14; .-. X =±v(-7±*vi4), y =±v(-7TiVi4); eight pairs of roots. The plats of these two equations intersect in only eight real points ; the other eight points of intersection are imaginary. 3. It^{x+y) + ^{x~y) = ^a, ^(!^+f)+^(a-'-f)=b: then 2a;+2V(a^-2/^)=a, 2af+2-y/{ai^-y^)=b',[sqT. and x' — y^ = ^a^ — ax + a^, x* — y^ = ^¥ — h^x' + a^, [div. by 2, transp., sqr. I.e., y'^=ax — ^a', 'jf = b^x' — lh'^; .'. b^oi?-lb* = {ax-lay; i.e., {a?-W)x'-^a^-x + {-^a^ + ib'') = 0; .-. x = la!'±b{a'-2W)']:i{a?-W), f= (ax - ia'y= ab ■ [ab ± {a' ^2b'')]:4.{a'-i^) ; y =±^[ab ■ (ab ±a'if2b'): 4(0^- S^)]. 4. It xix+y+z)=18, y(x+y+z)= 12, z{x+y+z) = 6: then {x + y + z){x + y + z) = 36, [add .-. x + y + z = ±6, .-. x=±S, y = ±2, z=±l. [div. 338 EQUATIONS. [XI. pr. 5. If xyz = a?{y + z) = W{z + x) = (?{x + y) : then 1 =a^fl+l)=.,.fl + l) = o^('i+lY \zx xyj \xy yzj \yz ' zxj T>.. 111. Put M, 1), w s — 1 — , — ; yz zx xy then ••• l = a'{v + w)=:b''(w + u) = (^(u + v), 1 1,1,1 a' b^ (? , , 1/1 , 1 , lA 2\a^ b' These roots may be grouped in six different ways : x = 2, 2, 3, 3, 4, 4; 2/ =3, 4, 4, 2, 2, 3; «=4, 3, 2, 4, 3, 2. So, for the sets of values from the other values of u, v. 7. If y'+ yz+z'=7, (1) z^+zx + x'=13, (2) af + xy-\-y' = 3: (3) then {x—y)(x-i-y + z) = 6, (z -y)(x + y +z) = 10, [sub.(l, 3)fr. (2) .-. x — y: z — y= 3 : 5, i.e., 5x — oy = 3z — 3y and 2y + 3z = 5x; (4) but •.• «^ + a!(z + a;) =13, [(2) .-. z^+^(2y + 3z)-l{2y + 8z)=13, [sab. for a; fr. (4) i.e., 4y2 + 222/2 + 4922 = 325, (5) .-. y = ±2, ±-^; 2 = + 3, ±^77^; 33 = ^1, ±- '^ V19 V19 V19 8. If af-yz = a (1), y'-zx=b (2), »2-a;y = c (3) : From the square of one subtract the product of the other two ; then x{a^ + y^ + ^ — 3 xyz) = a^ — 6c = a, y{ii? + 'if + z^ — 3 xyz) = 6^ — ca = b, and z{a? + f + ^ — 3xyz)=c^ — ab = c. . • . Ay = Bs;, Az = ca; ; .•. (a^ — Bc)a;2=aA^; [sub. for y, 2 in (1) ^/g • A g^ — 6c ■'■ '^ ~ V(a' - BC) ~ V(a' + 6" + 0=* - 3 a6c) ' 6^ — eg ia(l±iV3)- [3,4 10. To solve the equation a^ = — a" : then •.'- the equation a;° = — a' gives x^ = a^ if la; replace a;, .-. x = ai, ^a{ — i±^B), ~ai, ^a{i±^S). And so on for other roots. Note. Another method of solution is shown in X. Prob. 1, in finding the wth roots of a" and of — a". §18. LOGARITHMIC AND EXPONENTIAL EQUATIONS. The methods of solving such equations are set forth in IX. Probs. 3, 8. E.g., to find a; from the equation 15='' + 6 •15"' = 51975 : then .' IS'' =225 or —231, [sol.quad. .-. a; = log225 : log 15 = 2.3522 : 1.1761 = 2 ; but of the equation 15"'=— 231 no solution is possible. § 19.] EXAMPLES. 343 § 19. EXAMPLES. §1. 1. From the following statements pick out the suflBeient condi- tions, the necessary conditions, the equivalent statements, the associated statements, the incompatible statementb, the independent statements : a;<6, x=8, a;<5, a;>4, 2a;=6, aP=9, a;'=S, w^gS. 2. Give examples in which one statement is a ^ necessary, ^^ ^ ' sufficient, not a ■{ ' condition of another. ' necessary, 3. Show that if one statement ■{ ^® . a -J iiecessary ^j^ion ' be not ' sufficient of another, then the latter ^ !® , a ^ sufficient ^jy ' ' IS. not I necessary of the former ; give examples of these four cases. § 5, PKOB. 1. ■■• 24. Solve the equations : 4. 12 — 5a; = 13— a;; 1 — 5a;=7a!+3; 6a!— 5(3a;— 7) — 21 = 0. 5. a — 2a; = a; — 6; m — nx=px + q; ax—b(x — l) — c = 0. 6. (a;+l)(a;-l) = a;(a;-2) ; (a! + 4) (a;-2) = (a;-9) (a!-3). 7. (x+a)(x—b) = {x—c)(x+d); {x~m){x+n)=x(x—q). 8. aa; — m — 2 \bx — n — 3 [ca; — p — 4(da; — g)] | = 0. 9 2a; — i^^=— • 7g! + 4 5-a; ^22 a!_ 8-7a! ^3~3'9 3 32 6" 10. i(a!-l)-i(a!-2)=| ; -(5x-6)--(x-l) = x-2. 11. l(a; + 10)-|(3a;-4) + |(3a;-2)(2a;-3) = «^-A. 12. ±-^+^ = ^ ' 13. 14. a;— 3 a;— 4 a;_3 a;— 4 x — b a — 6' a;— 2 a;— 3 a;— 4 a;— 5 344 EQUATIONS. [XI. ''■ iHyitViH 7 =0; b _ a+b x—a x—b x—c 16. (x-ay+(x-by+{x-cy=3{x-a}{x-b)(^x-c). 17. (a;2-3x + 4)J = a;-3; [2(1 -k) (3 - 2a;)]* = 2a; + 1. 18. (8-4a;)* + (13-4a;)*=5; 2x+^[ia^+-y/(l-4x)'} = l. 19. 18: V(2a; + 3) = V(2iB-3)+V(2a! + 3). 20. 3V(a;-|) + 7V(a' + A)=10V(a'+T!ir)- 21. V(3a' + l)-V[2-a' + 2V(l-a')] = l- 22. -^(V3 + 9;v7)+-^(V3-a!V7)=y + a^y^ + xf + y* = l, x' + y^=2. § 7, PEOB. 5. • •• 48. Solve the systems of equations : 40. 3x — 4:y + 5z=4:, 8x — y — z = 6, 7x — 5y—3z = —l. 41. x + y + z = &, x — y + z=2, x + y — z — Q. 42. x—2y-5z=2Q, Sx-5y-3z=22, -8x+lly+9z=-b7. 43. 2{x+l)-3{y-l) + z-2 = 2, 2{x-l) + 4:(y+l)-5{z-l) = 3, 3(2a! + 2)-2(2/-l) + 3(»+l)=29. 44 1-1 + 1 = 11 ?_? + A = Zl §4.1_i. = 20 ' X y z 6 ' a; 2/ 3z 18' ce i/ 7« 2l' 45. 3^x+5i-y-l^z=51, y^ + g^ = 2a2+|a;^ c2/ + 6« = a, 2^a;+3|2/-liz =23|, ' z' +x'=2b'+ly', az+cx = b, lix+2^y+ |z =31|f^; af+y' = 2(f +^z'; bx+ay = c. ^46. a; + 2y+3» + 4M =20, x + 2y + 3z — 4.u=12, x+2y — 3z + 4:U =8, a; — 22/ + 3« + 4m = 8. 47. 3a; — 42/ + 3» + 3'y — 6m=11, 3a; — 52/ + 22 — 4m = 11, lQy — 3z+3u — 2v=^2, 5»+4M + 2'y — 2a;=3, 6u — 3v + 4:X — 2y=z6. 48. 5a;— 2(^ + 2 + ^) = — 1, — 12y + 3 (« + v + a;) = 3, 4z — 3(i) + a; + 2/)=2, 8v — (a; +2/ + «)= — 2. Denote x-{-y + z + vhjs; from these equations respectively express x, y, z, v in terms of s ; substitute these values in any one of the equations ; solve for s ; and thence find x, y, z, v. §8. • •• 52. Plat the lines that represent the equations : 49. a; = 0, y=0, a; = 4, a;= — 4, 2/=4, y=—4:, x=±a, y=±b, [«) &j lines of any known length 50. y = x, y=—x, y = 2x, 2/=— 3 as, 2t/ = 3a;, 3y=—2x, ly = mx, ly=— mx, [Z, m any two given numbers 346 EQUATIONS. [XT. 51. 2/=a; + 2, y=—x + 2, y = x — 2, y=—x — 2, y = mx + c, [m anj- given number, c any given line 52. 2a; + 32/ + 5 = 0, Sa; — 2?/ — 5 = 0, Za; + m2/ + c = 0, [^ m any given numbers, c any given line 53. Find the lengths of the intercepts upon the axes of the lines whose equations are given in Exs. 62-55. 54. Find the co-ordinates of the points of intersection of the lines whose equations are y = x, y=—x; y = x + 2, y = x — 2; 2x-\-3y + 5 = Q, 3x — 2y — 5 = 0; y = mx+c, y=— mx -f- c, [m any given number, c any given line 55. Find the co-ordinates of the vertices of the triangles bounded by the lines that represent the equations : 2x+3y + 5 = 0, Sx — 2y — 5 = 0, x=b; ax + by + c=^0, a'x + b'y+c' = 0, a^'x + b"y + c" = 0. 56. Find the co-ordinates of the vertices of the parallelograms bounded by the lines that represent the equations : ■2x + Sy + 5 = 0, 2a; + 3.!/ — 5 = 0, 3a;— 2y-f 5 = 0, Sx—2y — 5 = 0, §9. 57. By aid of Bezout's method solve examples 42-51. § 10, PKOE. 6. 58. Find two numbers, such that their sum is 27 ; and that, if four times the first be added to three times the second, the sum is 93. 59. Find two numbers, such that twice the first and three times the second together make 18 ; and if double the second be taken from five times the first, 7 remains. 60. A flagstaff is sunk in the ground one-sixth part of its height, the flag occupies 6 feet, and the remainder of the staff is three-quarters of its whole length ; what is the height of the flagstaff? 61. The diameter of a five-franc piece is 37 millimeters, and of a two-franc piece is 27 millimeters ; thirty pieces laid in contact in a straight line measure one meter ; how many of each kind are there ? § 19-] EXAMPLES. 347 62. Find three numbers such that the sum of the trst and second is 15 ; of the first and third, 16 ; and of the second and third, 17. 63. The sum of the three digits of which a number consists is 9 ; the first digit is one-eighth of the number consist- ing of the last two, and the last digit is likewise one- eighth of the number consisting of the first two. 64. At an examination there were 17 candidates, of whom some were passed, some conditioned, and the rest re- jected; if one less had been rejected, and one less conditioned, the number of those passed would have been twice those rejected, and five times those condi- tioned ; how many of each class were there ? 65. There are three candidates at an election, at which it is necessary that at least one more than half the entire number of electors should vote for the successful can- didate ; A fails to obtain an absolute majority, although he has 20 votes more than b ; but supposing that c, whose votes are only three-tenths of b's, had withdrawn, and that one-fourth of his supporters voted for a, then a would have been barely successful ; how manj' voted for each candidate ? 66. A gentleman left a sum of money to be diyided among four servants ; the first was to have half as much as the other three together, the second one-third as much as the other three, and the third one-fourth as much as the other three ; the first, moreover, was to have $ 70 more than the last ; how much should each get ? 67. A father divides his estate among his children as follows : to the first a dollars and ' the nth part of the remainder ; to the second, 2 a dollars and the nth part of the remain- der ; to the third, 3 a dollars and the nth part of the remainder ; and so on. It results that in the entire division of the estate each child receives the same amount. Find the value of the estate, the number of children, and the amount each one receives. 348 EQUATIONS. [XI. 68. In a company of a persons each man gave m dollai'S to the poor, and eacli woman n dollars ; the whole amount collected was ka dollars ; how many men were there, and how many women? Show that, if m>n, then m > A; > ?i. Show that the example is possible only when (m—A;)a,(ifc—n)a are multiples of m — m and have the same sign as m—n. 69. Upon a horizontal straight line let o be a fixed point, let a lie a units to the left of o, and b, h units to the right of o ; find on this line a third point x such that if m be the middle of bx, then ao is one-third of am. Show that if 4 a > 6, x lies to the left of o ; if .4 a = 6, x coincides with o ; if 4 a < 6, x lies to the right of o. 70. A reservoir holding v gallons is filled in h hours by p pipes, aU of the same size, and by the rain falling uniformly on a roof of s Square yards. Another reservoir holding v' gallons is filled in h' hours by p' pipes of the same size as the others, and the rain falling uniformly and with the same intensity as befo^-e upon a roof of s' square yards. Find X, the inflow per hour of each pipe, and y, the rain- fall per hour on each square yard of roof. Explain the meaning of the problem if for particular values of the constants either x ov y on both of them be negative. 71. Two circles of radii r, r' lie in the same plane and have their centres d units apart ; find the point where the exte- rior common tangents cut the line that joins the centers. Show by the formula that if the smaller circle grows while the larger stands fast, the point recedes farther and far- ther away ; that when the growing circle is of the same size as the other, that point has gone to infinity (does not exist) ; and that when the growing circle passes the other, the point reappears upon the other side at infinity, and creeps back toward the circles. 72. Find the four terms of a proportion that exceed by the same number the four numbers a, &, c, d. Discuss the solution when (1) ad = bc, (2) a + d = b + c. § 19-] EXAMPLES. 349 73. Given the series a + b, ap + bq, ap^ + bq^, api+b(f, ap* + bq^, • • ., to find two numbers x, y, such that each term of this series after the second can be got by multiplying the one before it by x, and the one before that by y, and adding the products. 74. Given the series a + & + c, ap + bq + cr, ap^ + b(f + c?-^, • • ■ , to find three numbers a;, j/, 2, such that each term of this series after the third may be found by multiplying the one before it by a;, the one before that by y, and the one before that by », and adding the products. 75. A laborer receives a dollars a day when he works, and forfeits h dollars a day when idle. At the end of m days he receives Ti dollars ; how many days does he work, and how many is he idle ? What relation exists between the given elements if his forfeits just cancel his earnings? if his forfeits exceed his earnings? , Give numerical illustrations. 76. A father is now a times as old as his son ; Te, years hence he will be b times as old ; what are their ages now ? Give numerical values to a, &, A;, and interpret the results. Show that : A; > if a > 6 ; A; = if a = 6 ; /c < if ct < 6. 77. The sum of two numbers is a, and the difference of their squares is W \ what are the numbers ? Interprettheresultsif (1) W>u?\ (2) W=a^; (3) }^a?\ (2) 1i'' = a^\ (3) ¥ f-ia^)-K'^-W) + {{<^-T'-,ci') = 0- 354 EQUATIONS. [XI. 100. ^{3a?- 7) + 1(25 - 4a;=) = Kbx' - 14) . 101. 2(2a!2-6)-^ + (a;2-3)-i=6(3a^-l)-i. § 11. PKOB. 8. •■■ 117. Solve both by completing the square and by factoring : 102. a;2-8a; + 15 = 0, a;2+ 10a; = - 24, a^-5a; + 4 = 0. 103. 6a^-19a;+10 = 0, 7a;2-3a; = 160, 110a2-21x+l = 0. 104. (5a!-3)'-7=44a;+5, (3a!-5) (2a;-5) = (a;+3)(a;-l). 105. 6a^ + |a;+,^% = 0, (a- 2)-^ - 2(a! + 2)-i = f . i„„ 3a;— 2 2a; — 5 8 a; + 3 , a; — 3 2a; — 3 106. =-, — ■ = • 2a; — 5 3a; — 2 3 a; + 2 a; — 2 a;— 1 i„„ a; + a,a; + & a , b a; + a,a! + &,a; + c „ lu/. 1 - = --j--, 1 --I = 6. I X — a x — o a x — a x — o x — c 108. a?-{f,+Zi)x+\{ll+\'ii)=(i, a;=-(4+3i)a;+(7+5i)=0. 109. 3a; + 2Va;— 1 = 0, a?i— 13a;2-„= 14. , llOi a;* - 14 af= + 40 = 0, a;* + f a;"* = 3i, ^2x—lx = -b2. 111. a; + 5-V(« + 5) = 6, ^x + yx-^ = 2\. 112. V(2a! + 7)+V(3a'-18) = V(7a'+l)- 113. a! + Va; + V(a'+2) + V('«^ + 2a;) = a. 114. a;2+3=2V(a;^-2a;+2)+2a;, ■yJ{!(?-2x+2)-^=Z-x. 115. 3ai'+16a!-2V(a^ + 5a;+l)=2. 116. a;2 — 2V(3a;^ — 2aa; + 4) = |ffl(a; + |-a + l). 117. na:;= + a; + n + l = 0, a!5 + a;= - 4a; — 4 = 0. 118. Form the quadratic equations whose pairs of roots are : 2,3; 1,-4; 3±2i; — l±i; ±3+2i; 4+5i, l + 2». 119. Form equations by putting equal the quadratic functions : 2a;^ + a! — 6; 63;^— 19a;+15 ; 3?—2mx + m?—'r?; a?— (m + m) a; + (m +p) (n —p) ; (x — aY—lf; a;2_5(l + t)a; + 13i, a;^ + (7 + 5i)a; + 6 +17j. Solve, and by aid of the roots factor the functions. 120. If a and 13 be the roots of the equation af—px + g = 0, find the value of aj3-^ + /SaT^ and of a? + /3'. 121. What value of c gives the equation 5a;^ + 3a! + c = equal roots ? § 19.] , EXAMPLES. 355 122. Eliminate a; from the two equations ad^+l)x + c = 0, o'a^+6'a! + c'=0. 123. Sliow that the quadratic function aa!^+&a;+c may be writ- ten in the forms — {(2ax + bY— {b'— 4ac) } -I 4 0!i a.iid — \2ax+b+^(b^-4:ac)\{2ax + b-^{b^—4:ac)\; hence derive the condition for real and unequal, for equal, and for imaginary factors. By this method factor the function 3 ay'+ Sa; + 2, and find for what values of x the function vanishes. § 12, PKOE. 9. 124. Plat the quadratic functions ; hence find the real values of X, if any, that make these functions vanish : a^ — Ax + 3, a.^-4a; + 3^, a;^ — 4a! + 4, a^-4a5 + 4f, x^ + x + 6, -x^ — x — Q, — 3a;2-10a;+13. § 13, PEOE. 10. 125. Find Ave convergents to the roots of the equations : a;2 + a;-6=0, a;2_3a3 + 2 = 0, a;- — 6a; + 9 = 0, 3x^ + 4:x=7, 4a^-3a;=10, So;'- 10a;= 20. § 14, PBOB. 11. 126. Find the maximum or minimum values of the functions a^_4a; + 3, 10 + 4:X-af, x'-Gx + d, -a!^ + 6a!-9; and the corresponding values of x. 127. From the plat of the functions in the examples of § 12, state which of them have maximum values and which minimum, and find these values. Show that each of these functions has a ■{ ^^^^^^ value if the vertex of the corresponding parabola be ■{ ^o^Q^ards, i.e., if the coefficient of a;^ be^ negative. 128. Show that '^ ~ "^ has no value between 1 and 5. a; — 3 129. Find the maximum value of (x + a) (x — b) : aP. 130. Show that a(a + x) : (a — x) can have any value. 356 EQUATIONS. [XI 131. Find the maximum or minimum values of a!^ + 2a; + ll . x' — x+l . a + x a — a; a!^+4a;+10' x' + x — l' a — x a+x To the last apply the principle that if the product of two variables be constant, tiieir sum is a minimum when they are equal. 132. Prove that the quotient (x + a): {x' + bx + c^) always lies between two fixed fiuite bounds if a? + ab and 6^<4c^ ; that there are two bounds between which it cannot lie if a^+ c^ > db and &^ > 4c^ ; and that it may take all values if a? + , 3y-x=y\ 143. x'+y''==^xy, x — y = \xy; a?+xy = &, x'+y = 5. , NOTE 1. 144. a^ + xy+2y^=7i, 2a? + 2xy + y^ = 73. 145. a? + y^ = a?,' xy = b''; x' — y^ = a', xy = h^. 146. a? + 3xy = bl, xy + 4.y'^ = 115 \ a?+f = Q, xy = 2. 147. !r2+a;2/+4^2=6, Zv? + 8y^=U;. 8^+2/^=1, i^-irf=l. NOTE 2. 148. a!2-3a^+22/2=0, a?+?/==a^-2/'; oi?-f=8,a?-f=2Q. 149. x' + xy — &y'^=Q, 2x + 8y + a?+3xy + 2y'=19. 150. 8a;= + 3a;22/ + a;2/==18, 2oi?+5x'y+Sxy^=:24:. § 19] EXAMPLES. . 357 NOTE 3. 151. x + y = 5, a5* + 2/*=97; x-if = 3, af-tf = 3093. 162. a^ + y^^Uaf'y^, x + y = 9; x^ + yi^l, a? + y==n. 153. a;2 + 2/2=7 + a^, oi? + y'= 6xy-l. ' NOTE 5. 154. a;* -0:2 + 2/* -2/2 = 84, a^ + a^2/' + 2/' = 49. 155. an/(a; + 2/)=30, a^ + ?/3 = 35. NOTE 6. 156. 4 (a; + 2/) = 3x2/, « + 2/ + a^ + 2/^ = 26. 157. a!=(a; + 2/) = 80, a^(2a;- 32/) = 80. 158. a;* + a;'2/' + 2/'=133, »2- a;?/ + 2/' = 7. 159. 9?-^f — {x-Vy)=a, a;* + 2l^ + a; + 2/- 2(a^ + 2/8) = 6. 160. a; + 2/+Va;2/=14, a;^ + 2/^ + a;?/ = 84. 161. x2 + 2/ = 4a;, 2/2 + 2! = 42/; a.-3 + a;2/2= 10, 2/3 + a!22/ = 5. 162. r^-^fJ^3x-{-3y = 31%, a;=+2/«-3a!- 3y = 324 ; a^= aa! + ly, 2/^= a2/ + 6a; ; bx + ay = ah, hx + ay = ixy. 163. 10a2_|_i5a.^^g^5_2a2^ 102/^ + 15a;2/= 3a6-262; 164. 6x^-3x — 4y=25, x" + 2x -3y= 18; 165. a;2/+6a; + 72/ = 50, 3xy + 2x+5y=72. 166. a; + 2/ =10, V^^r' + VZ/a'"' = 1 5 167. V(a^ + 2/')+V(a''-2/')=22/,, a;* -2/* = a*. 168. 83!* -2/* = 14, a!«2/i = 22/2. • ••172. Find the values of a;, y, z from the sets of equations : 169. yz=liG, bx+ay = ab, cx + az=ac. 170. x + y + z = x-'^ + y-'^ + z-^ = i, xyz = l; 171. xy=a(x + y), xz=:b{x+z), yz = c{y + z). 172. x + y + z=Q, 4a; + 2/ = 2z, x^ + y^ + z^=U. §16. 173. A boat-crew rows 3^ miles down a river and back again in an hour and 40 minutes ; if the river have a current of two miles an hour, at what rate does the crew row ? 174. A number is composed of two digits ; the first exceeds the second by unity, but the number itself falls short of the sum of the squares of its digits by 26 ; what is the number ? 175. A number is composed of two digits ; the first exceeds the second by 2 ; the sum of the squares of the given number and of the number got by reversing the digits is 4034 ; what is the number ? 358 EQUATIONS. [XI. 17G. Find the lengths of the segments of a line a, if m times the square of one be equal to n times the rectangle under the whole line and the other. 177. The driving-wheels of a locomotive are 2 feet longer in diameter than the running-wheels ; the running-wheels make 140 turns more than the driving-wheels in a mile ; what are the diameters? [ratio circum. : diam.= 22 : 7 178. A set off from London to York, and b at the same time from York to London, and they traveled uniformly ; a reached York 4 hours, and b reached London 9 hours, after they met ; in what time did each make the journey ? 179. A broker bought a number of hundred-dollar railway shares at a certain rate discount for $ 7500, and afterwards, at the same rate premium, he sold them all but 60 for $ 5000 ; • how many did he buy, and what did he give a share ? 180. Divide a line 3 feet long into two parts such that the circle standing on one segment as diameter shall be equal to the square standing on the other. 181. The number 563 in the decimal scale is less than the same number in a higher scale by 232 ; what is the radix of the higher scale ? 182. What is the price of eggs when two more in a shilling's worth lowers the price one penny a dozen ? 183. There are two numbers whose product is the difference of their squares, and the sum of whose squares is the differ- ence of their cubes ; what are the numbers ? 184. The sum of the squares of the numerator and denominator of a fraction is 389, and the difference of the fraction and its reciprocal is \^ ; find the fraction. 185. Find two numbers such that their sum, their product, and the sum of their squares shall be equal to each other. 186. Find two numbers whose product is p, and the difference' of whose cubes is m times the cube of their difference. 187. Find a fraction the product of whose numerator and denominator is 180, and such that if its numerator and denominator be each increased by 10, its value is doubled. § 19-] EXAMPLES. 359 188. A rectangular space, whose length and breadth are 42 and 78 feet, is surrounded by a ditch 5 feet deep, and capable of holding 220 tons of water ; what is the breadth of the ditch, counting 6 tons of water for a cubic fathom? 189. There is a fraction such that if the numerator be increased and the denominator diminished by 2, the reciprocal of the fraction is the result ; but if the numerator be dimin- ished and the denominator increased by 2, the result is less than the reciprocal by lj\ ; what is the fraction? Solve the same problem in general terms, replacing 2 and 190. Two boys set off from the right angle of a right-triangular field, running in opposite directions, with speeds in the ratio of 13:11; they first meet at the middle point of the hypothenuse, and again at a point 30 yards distant from the starting-point ; find the lengths of the three sides. 191. Two cubical vessels together hold 407 cubic inches ; when one vessel is placed on the other, the total height is 11 inches ; find the contents of each. 192. A number consists of two digits, the difference of whose squares is 40, and if it be multiplied by the number con- sisting of the same digits taken in reverse order, the product is 2701 ; find the number. 193. A vessel can be filled with water by two pipes ; by one of these pipes alone the vessel would be filled 2 hours sooner than by the other ; and the vessel can be filled by both pipes together in 1-|- hours ; find the time that each pipe alone would take to fill the vessel. 194. A vessel is to be filled with water by two pipes ; the first pipe is kept open during three-fifths of the time which the second would take to fill the vessel ; then the first pipe is closed and the second is opened ; had both pipes been kept open, the vessel would have been filled 6 hours sooner, and the first pipe would have brought in two-thirds of the water which the second pipe did bring ; how long would each pipe alone take to fill the vessel ? 360 KQUATIONS. [XI. 195. A number consists of three digits ; the first is to the second as the second is to the third ; the number itself is to the sum of its digits as 124 to 7; and if 594 be added to it, the digits are reversed ; what is the number ? 196. The diagonal of a box is 125 inches, the area of the lid is 4500 square inches, and the sum of three conterminous edges is 215 inches ; find the lengths of these edges. . 197. One side of a room is 5 feet longer than the other side, and 1000 square feet of paper is needed to cover its walls ; if it were 3 feet higher, the same paper would be needed for 3 only of its walls, the bare wall being one of its longer sides ; what are the dimensions of the room? § 17. PROB. 14. 198. Solve the binominal equations : a;'— 1 = 0, a^+l = 0, a;= = -8, a!« = 16, a;"'+l = 0, a!'2 + l = 0, a;^-l=0. 199. Find the square root to three decimal places of : 5 + 12!;, 12+5i, 161-2401, 13 + 7*, 7 + 13i. 200. Prove that the n roots of the equation x" = a + hi, are all given by the expression V • I COS — ■ f- » sm n n wherein r is the tensor and the versorial angle of the number a+ hi, and h has any n consecutive values in the series of natural numbers between ~oo and +oo . § 18. 201. Find the value of x from the exponential equations : x+2 u+l 2'= 8, 2'+« = 8"^', 3S=2=27i=i, 9'^=3, 83=^1 = 2. 202. By aid of the table of logarithms find x from the equations : 10' = 3, 4' = 10, .3' =.8, 3=''^+^ = 100''-', lb^^+*^'21*'-^. 203. Solve the equations : 3^=^- 7.3=== 18, 2"'^- 5.2'' + 6 = 0, 2'+i + 4' = 80, 4.32*+' — 5. 3*+^ =12. 204. If nA;="+" + 26fc"+" + c=0, prove that X = [log^ - 6^" ± y/{h^Jc"' — aclC") I — log(a/r)] : r logic. XII. th. !,§!.] ARITHMETIC PEOGEESSION. 361 XII. SERIES. FoK definition of series, see I. § 12. The first and last terms of a series are its extremes ; the other terms are its means. § l.~ ARITHMETIC PEOGEESSIOlsr. An Aeithmetic Peogkession is a series such that each term after the first is formed by adding a constant to the next pre- ceding term. The constant added is the common difference. The abbreviations are : a for first term, I for last term, d for common difference, n for number of terms, s for sum of all the terms. ^^^en d is ^ ^;^ the series is i ^^^S!^ progression. E.g., 1, 3, 5, 7, 9, is an ascending series, wherein d = +2, a=l, Z = 9, n = 5, s = 25. So, 9, 7, 5, 3, 1, ~1, ~3, is a descending progression, wherein d = -2, a=9, ?=-3, m=7, s=21. Theoe. 1. In an aritJimetic progression 1] l = a-[-(n-l)d. For ■.■ a + d, a + 2d, a + 3d, —, a + (k — l)d are the 2d, 3d, 4th, •■• fcth terms, [df. .•. a-f-(w — l)d = ^ the last of a series of w terms, q.e.d. CoK. In an arithmetic progression 2] a=:l — (n~l)d, 3] d = ^, ■' n—1 A-, ^ — *_L1 4] n = —■ 1- 1 • -' d The reader may prove, solving formula 1 in turn for a, d, n. 362 SEEIES. [XII. th. 2, Theoe. 2. In an arithmetic progression 5] s = \n{a + • For ••• s = a+(a+d) + (a+2d)-| \-Q — d) + l, n terms, and s = l + {l — d) + (l — 2,d)-\ |-(a+d)+a, nterms, •. 2s = {a + l) + {a + l) + (a + l)-\ H(a+0) « times, = M ■ (a + Z) . s = ^n(a + V). Q.E.D. Cor. 1 . In an arithmetic progression -■ n 7] Z =2s_ -I n ' 8] 2s a + l The reader may prove, solving formula 5 in turn for a, I. n. Cor. 2. In an arithmetic progression -' ~ 2« ' -' n(n-l) 111 ^_ d+2i±^\(n+dy-%ds-] -* 2d ' 12] s=^n[2l~{n-l)d'], -■ 2n ' 14] d = 21s-an) 15-1 d-2a±vr(2a-c;)^ + 8ds1 -■ 2d ' 16] s=|m[2a + (w-l)d], 17] a = ^|d±V[(2Z + d)2-8ds]^, 18] I =\\-d±^i{2a-dy + Sds-]\, 19] (Z + a)(;-a) -^ 2s-(Z-fa) ' 201 ' (? + a)a-a + d) -■ , 2d pr. 1,§1.] ARITHMETIC PEOGKESSION. SfiS The reader may prove formulse 9-12, combining 1, 5 so as to eliminate a, then solving in turn for I, d, n, s ; formulae 13-16, by eliminating I, then solving for a, d, n, s; formula: 17-20, by eliminating m, then solving for a, I, d, s. Note, 1. The formulae involving a may be got from those involving I, and vice versa, by symmetry, writing a in place of I, I in place of a, and — d in place of + d ; and thus seven of the fourteen formulae 1, 2, 6, 7, 9-18 may be written directly from the other seven ; for if any arithmetic progression be reversed, then a becomes I, I becomes a, and d becomes —d. Note 2. Formulae 11, 15 give two values for n. If either of these values be negative or fractional, it may be rejected as inconsistent with the conditions of the problem. [XI. pr. 6 nt. 3 PrOB. 1. To INSERT m ARITHMETIC MEANS BETWEEN a, I. Divide the remainder, 1 — a, by m.+ l for the common differ- ence; and to a add one, two, three, ••■ times this difference. E.g., to insert 5 means between 12 and 48 : then •.• (48 — 12) : (5 + 1) = 6, the common difference, , .-. the series sought is 12, 18, 24, 30, 36, 42, 48. Note. By aid of this problem, from every arithmetic pro- gression a new arithmetic progression may be formed by insert- ing the same number of arithmetic means between every two consecutive terms ; and the common difference of this new pro- gression is the quotient of the common difference of the other divided by one more than the number of terms so inserted. So from any arithmetic progression a new progression may be formed by taking equidistant terms ; E.g., if two means be inserted between two consecutive terms : then 6, 12, 18, 24, 30,... becomes 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, .•• and if of this new progression the first, fifth, ninth, ... terms be taken, a third progression is formed, 6, 14, 22, 30, ... whose common difference is 4 • 2, = 8. 364 SEKIES. [XII. ths. §2. GEOMETRIC PROGRESSION. A Geometric Progression is a series such that each term after the first is formed by multiplying the next preceding term by a constant multiplier. The multiplier is the common ratio. The abbreviations are : a for first term, I for last term, rfor com- mon ratio, n for number of terms, s for sum of all the terms. When r ^ 5 1, the series is { ^'^^"^^^ pf ogres^on. E.g., 1, 2, 4, 8, 16, is an ascending series, wherein r="'"2, «=!, 1 = 16, n = 5, s=31. So 1, ~2, 4, ~8, 16, is an ascending series, wherein r—~2, a=l, Z=16, n = 5, s=ll. But 16, 8, 4, 2, 1, i, i, is a descending series, wherein r = i, a =16, l = i, n=7, s = 31f. Theor. 3. In a geometric progression 21] Z = aj^ri. . For ■.• «!-, ar^, ar*," -ar*"^ are the 2d, 3d, 4th, "-ftth terms, [df. .•. ar^'^ = I, the last of a series of n terms. q.e.d. Cor. In a geometric progression 22] a = l: r"-S 23] r = "--i/(l:a), 24] ^^i+ log/-log« . -^ logr The reader may prove, solving formula 21 in turn for a, r, n. Theoe. 4. In a geometric progression „,, or" — a 1— r" > 25] s = 7-=«r; -" r— 1 1 —r For ■.• s =a +ar+a'r'-\ 1- o}^-* + ar""*, [df, .'. rs — s = a'r^ — a, ... s =55n^=akii:. Q.E.D. r— 1 l—r 3}/r§2.] GEOMETRIC PROGRESSION. 365 Cor. 1 . In a geometric progression 26] a= (^-^) !, 27] or" — sr = a ~ s, 28] ^_ los(rs-s + a)-loga . logr The reader may prove, solving formula 25 in turn for a, r, n. He will observe that formula 27 [r unknown] is of the jith degree, for which there is no general solution. In numerical equations the solution is always possible. / Cor. 2. In an infinite decreasing geometric progression, the limit of r" is ; and the value of s is the quotient a : (1 — r) . CoK. 3. In a geometric progression 29] I ^Jr-Ds.r^-^ 30] r l-r»-i + -L- = 0, s—l s—l o 1] ^^ \o^l-\o^\lr-{r-l)s1 ^ ^^ logr 321 c= '^~' 33] a{s-a) "-^ = l{s- 1)'-^ 34] l(s — l)''-^=a{s — a) ""S 35] ,^ Iog?-loga • ^^ log(s-a)-log(s-0 n—lnn n—l/gn 37] a = lr — s{r — l), I ^ s(r-l) + a^ r s — a- 38] 39] 40] s-r _ lr — a ' r-i' 366 SERIES. [XII. th. 5, The reader may prove formulse 29-32, combining formula 21, 25 so as to eliminate a, then solving in turn for l,r,n,s; formula 33-36 by eliminating r, then solving for a, I, n, s; formulse 37-40 by eliminating n, then solving for a, I, r, s. He will observe that formulse 30, 33, 34 have no general solu- tions. In numerical equations their solution is always possible. Note. The formulae involving a may be got from those involving I, and vice versa, by symmetry, writing a in place of I, I in place of a, and r~^ in place of /"^^ ; and thus seven of the fourteen formulse 21 , 22, 25-34, 37, 38 may be written directly from the other seven ; for if any geometric progression be re- versed, then a becomes I, I becomes a, and r"*"^ becomes r~^. Prob. 2. To INSERT m geometric means between a, I. Take the (m -f- l)th root of the quotient 1 : a for the common ratio; and multiply a by the first, sepond ■■■ powers of this ratio. E.g., To insert three means between 3 and 48 : then •.• -v/(48 : 3) = 2, the common ratio, .•. the series sought is 3, 6, 12, 24, 48. Note. By aid of this problem, from every geometric progres- sion a new geometric progression may be formed by inserting the same number of geometric means between every two con- secutive terms ; and the common ratio of this new progression is that root of the common ratio of the other whose index is one more than the number of means so inserted. So, from any geometric progression a new progression may be formed by taking equidistant terms. E.g., if two means be inserted between two consecutiveterms, then 8, 6, 12, 24,..- becomes 3, 3^2, 3^4,6,6^2,6^4,12,12^2,12^4, 24,..., and if of this new progression the first, fifth, ninth, •■• terms be taken a third progression is formed 3, 6^2, 12^4, - whose common ratio is the fourth power of either of the three values of -^2. )rs.2,3, §3.] HARMONIC PEOGRBSSION. 367 § 3. HAEMONIO PEOGEESSION. A Haemonio Progression is a series such that any three jonsecutive terms being taken, the ratio of the first to the third jquals the ratio of the excess of the first over the second to the jxcess of the second over the third. E.g., if p, q, r, be any three consecutive terms of a harmonic progression, then p:r=p — q:q — r. Theoe. 5. If a series of numbers be in harmonic progression 'heir reciprocals are in arithmetic progression; and conversely. Let p, g, r be. any, three consecutive terms of a harmonic progression ; then will r~^ — q~^ = q~^ —p~^. Por- •.• p:r=p — q:q—r, [df. .-. pq—pr=pr — qr, [II.th.6 .-. r~^ — q~^ = q~^ — p~^ ', Q.e.d. [div. bypgr So for the converse. PeOB. 3. To INSERT m HAEMONIC MEANS BETWEEN TWO EX- ;remes, a, I. Mnd m arithmetic rneans between a"-' and 1"^, and take their •eciprocals. E.g., to insert two harmonic means between 12 and 48 ; hen •.• ^-it = ^'s, and ^ = 3=^, .•. the arithmetic progression is -j^, -^j ■^-^, ■^, [pr-1 md the harmonic progression is 12, 16, 24, 48, [th. 5 vherein 12:24 = 12-16:16-24, 16:48=16-24:24-48. Note. The analogies and relations of the three progressions ippear below : If p, q, r be three numbers I arithmetic \p:p; n < geometric progression, then p — q:q — r— \p:q; I harmonic , I i' : '' i I arithmetic I k{p + r). md the < geometric mean of p,r\a< -y/pr. \ harmonic | 2pr : (p + r) . So, the geometric mean of p, r is the geometric mean of the irithmetic and harmonic means of p, r. 368 SERIES. [XII. ths. Theor. 6. If four numbers, p, q, r, s, be so related that p — q, p — r, p — s form a harmonic progression, then : (a) q — r, q — s, q— p likewise form a harmonic progression ; and so do r — s, r — p, r — q ; and s — p, s — q, s — r. (6) T7ie relations between p, q, r, s shown in (a) hold true also : 1. Among any four numbers, n+p, n+q, n+r, n+s, whose differences equal the differences o/p, q, r, s ; 2. Among any equimultiples o/p, q, r, s ; or of their reciprocals; 3. Among ?£±^, M±b, ?£±b^ aMib^ • cp + d cq + d cr + d cs + d wherein a, b, c, d are any numbers. 112 (a) •.• the condition that [-■ — = [tb. 5 p—q p—s p—r is that (p + r) -{q + s) =2pr + 2qs; [free fr. fracts., red. 112 and • . • the condition that 1- - q — r q—p Q' — s is that {q + s)-{r+p) = 2qs -{-2rp, [^ch. p,q,r,s to q,r,s,p i.e., that {p + r)-{q + s)=2pr + 2qs, as above, .■. when p— q, p— r, p— s form a harmonic progression, so do q — r, q — s, q—p. q.e.d. So do r — s, r—p, r—q; and s—p, s — q, s — r. (b) ••• relation (a) involves p, q, r, s only by their differences, .-. it holds for any numbers n+p,n+q, n+r, n+s. q.e.d. 2. ■ . • equation {p + r) ■{q + s) = 2pr + 2qs is not changed when for p, ••• s are put np, •■• ns; or n:p--- n: s; .-. the equation is true for these, if torp, ••■ s. q.e.d. [above „ ap + ba,bc — adaq + b rj- ■ • 3. •.• -J^-:i—=- + - — -_, ^7 =••-, •••, [division cp + d c crp + cd cq + d and •■• whenrelation(a)holdsforp,"-sitholdsforc^p,"'c'^s,[2 i.e., for (?p+cd, ■•• (?s + cd, [1 i.e., for bc-ad ^ _ _ bc-ad ^ j-^ c^p + cd^ 0^8 + cd^ J, a , be — ad a , be — ad ri I.e., ior- + — , ..._-| ; [1 c ,e same in whatever way the terms are arranged or grouped. Let s =Ti+T2+T3H , any convergent series ; let the same series be arranged or grouped in any other way, say (T2 + Ti) + (t, + T3)+-; id let s„' be the sum of the first n groups ; len will s„'= s when ?i = co . or •.• in s are found all the terms of s„' and more ; . • . s„' < s, and lim s„' > s. [df • limit s„ < lim s„', and s, = lim s„, > lim s„' ; .-. Ums ' = s. Q-E-D- 370 SEEIES. [XII. ths. Theok. 8. If the terms of a convergent series he multiplied by any same finite number, the new series thus formed is convergent. Let Ti + T2 + Tj -I +T„ H be any convergent series, and Jc a constant ; then is the series Jeii +fcr2 + Tcr^ H + fcr„ -\ convergent. For .• K„, =T„+i +T„+2 +-..,=0 wlienn = oo, [hyp. ■ •. kR^,=ki„+i + kT„+2-i )=0 whenn = oo. q.e.d. Cor. If the terms of a convergent series be multiplied by any finite numbers not larger than a given finite number, the new series thus formed is convergent, Theor. 9. If, after a given term, the terms of a series form a decreasing geometric progression, the series is convergent. Let Xi + Ta + TaH \-i„ + r-T^-yr^-y„-\ be a series such that the terms after a given term Tj form a geometric progression with r smaller than 1 ; then is this series convergent. For •.' Ti + ••• +T4 is a finite constant number, Sj, and ■•• Tj+i+ — =Tj+i.(l + r+r'+---) — Ti+i : 1 — r, when w = c» , [th. 4 cr. 2 -•. s = SjH — ^±i-, a finite number. q.e.d. 1 — r ^ Theor. 10. if one series be convergent, and if the terms of another series be not larger than the corresponding terms of the first series, the second series is convergent. Let +T1++T2++T3-J \-'^n-\ 136 a convergent series, and let Ti' + T2' + T3' -| f- t„' + • • • be another series such that Ti'^^Ti, T2'3S.T2, Ta'SHs, •", T„'3*-T„, ••■; then is the second series convergent. For •.• T„'+i3«-T„+i, T„'+2 9^T„+2, f, •'• K„ ^ E„, and ••• R„ = 0, [hyp- .•. E„'= 0. ^ Q.E.D. Cor. 1 . If one series he divergent, and if the terms of another series he not smaller than the corresponding terms of the first series, the second series is divergent. 8-11, §4.] CONVEEGENCE AND DiyEEGENCE. 371 CoE. 2. If one series be convergent, and if in a second series the ratio of each term to the term before it be not larger than the cor- responding ratio in the first series, the second series is convergent. Theok. 11. If, after a given term, the ratio of each term of a series to the term before it be smaller than some fixed number that is itself smaller than unity, the series is convergent. Let Ti + Ta + TjH 1-TjH be a series such that after a given term Tj the ratios t^^ : Tj, ■•• each < A < 1 ; then is the series convergent. For form a new series ii + i2+Ts+ - +T^ + il+i + — +rj ... identical with first series for the first Jc terms, and thereafter a geometric progression whose ratio is h ; then •■• this second series is convergent, [th. 9 and •-■ the terms of the first series are not larger than the cor- responding terms of the second series, [typ- .-. the first series is convergent. q.e.d. [th. 10 Note 1 . It is not suflBcient that the ratios Tj^i : Tj, • • • be simply less than a unit. JE.g. , the harmonic series 1-| 1 1 ( . [above ^ o 4 Note 2. Application or the theorem: : To apply this theorem, find the law of the ratio t„_,_i : t„, which in general is some function of n ; then determine whether this ratio r, as n increases, finally becomes and remains smaller than some fixed number h, that is itself smaller than a unit. ' , If smaller than h, the series is convergent. If smaller than a unit simply, there is doilbt. If a unit or larger than a unit, the series is divergent. E.g.,g^ren\ + ± + l^+...+^+...: then • . • the ratio t„ : t„_i = r- =0 when m == oo, the series is convergent. 2' 3' n" So, given 1 +_+_ + ...+_ + . 372 SERIES. [XII. th. then •.• the ratio t„ : t„_i= ^^~ ' = (1 )^==1 when »i=(», .-. there is doubt. But, if the series be grouped thus : 15^) ■^"■' i.e., in groups wherein the denominator of the first term of each group is an integral power of 2 ; then •.• the several groups are less than -j -, -, ■•-, and •.• the series 1 -\ 1 1 1 is convergent, [th. 9 2 4 8 .•. the first series is convergent. q.e.d. [th. 10 So, given l + —-\ !-•••: ' & 2^ 3» then ... s = l+fl + lVf- + - + - + - ^2" 37 V4^ 5^ 6* 7^", l. [th. 10 So, the series 1 -\ ■. — H 1 , ^ 2 (log 2y^ 3 (log 3)'^ 1 1 1+771 TTTT—, ^ + 2 log 2 (log log 2y 3 log 3 (log log 3)" + ..■, and so on, 1 convergent u > 1 r v. are -{ ,. ^° . when p CI- [group as above Note 3. Genekal test op convergence. The series l+2; + i; + -' ^+2(log2)'' + 3(log3)-+'"'^*°-' are each of them ■{ ,i:°Z.„p„t when p ^ 1. [ex. above These series, when compared with most other series, furnish a test of their convergence. [th.lO, th. 8 11, §4.] CONVERGENCE AND DIVERGENCE. 373 It is to be noted that in the divergent series 1 + - 4- i H . -; 1 2 3' the nth term, -, is an infinitesimal when n=oo. Let this infinitesimal be counted the base [VII. § 4, dt. order of infls.J ; then — , the nth term of the convergent series 14---4--i H ) [P>1] is an infinitesimal of an order higher than the first order bj- a finite number p — 1. And, conversely, a series whose terms are infinitesimals of an order, p, finitely higher than the first order, is convergent. But if _p>l, and p = l when n = cx>, there is doubt, and the series may then be tested by the series 1 + ^^^ f ^^^ + ..., and so on, i.e., from the series -, — , ■ — ^ —— , ..., a base n n ■ log n n ■ log n • log log n may generally be chosen for which, when w = oo, the order of the term t„ of the series to be tested is .j °i ^ 7 Jijgher than the first order, and the series is -j -,. ^^^S^^ _ ' divergent Note 4. BotnsiDs of eerok : In the summation of most series; only a finite number of terms is used, and only ap- proxima,tions to the true value are found ; and it is then important to know between what bounds the error lies. That approximation is s„ and the error is — e^. [V. § 5 df.,?i finite. E.g., in the first example of Note 2, _ , , _ 1 , 1 I and ••• the terms of this series are not greater than those of 1,1 1,1 n (w+1)! (n+1)! n + 1 (n+1)! (« + l) ,+• ... E >_ — I :fi L_\ ==^_; rth.4cr.2 ""■^(n + l)! V n + 1/ n-nl' '" i.e., the error lies between and n-nl In particulax, Se ~ s < — -— = — — ; Sjo ~ s < ; ■ 6 • 6 ! 4320 10 • 10 ! 374 SERIES. [XII. ths. Note 5. Series akeanged to powers of a variable : If a series be arranged to the powers of some variable x, thus • Ao + AiX + AiOC?-\ l--4„a;" + -", then the ratio t„+i : t„ =x{a„ : a„_i) = x : (a„_i : a„) , convergent I < and the series is ■{ in doubt it x-i ^ a„_j : a„, when n = oo. divergent , > ^' i E.g., the series l+a; + 2!ar' + 3! »;'+••• is divergent, however small x may be ; for the ratio a„_i : a„ = 1 : n = 0, when m = oo. So, the series 1 -\ 1 — --\ — ■-) is convergent, however large x may be. rj ,, . X , oF , x^ , .1 convergent . „ i ■< 1 ; So, the series - H — is i ^i- ^ it x<2--< ' 12 3 diverge*; '^Sl; for the ratio a„_i : a„ = 1 when n= cc ; and this series is divergent if x=l. So, the series «.+ 2^ + 3a^+ ... is i ^"."J^'-f^^* if x ^ <}; That value of x which leaves the series in doubt, viz., +lim (a„_i : A„) when n = oo, is the radius of convergence of the series. E.g., if r = radius of convergence, then in the first of the ex- amples above r==0; in the second, j-=oo; in the third and fourth, 7-= 1. If some of the powers of x be wanting, the general method of Note 2 must be applied. ^.^.,the series 1 + ^ + ?!+ ... +^!!!:2 is^ convergent ^ ' 3 7 2"— 1 divergent for the ratio t„+i : t„ = a^(2"- 1) : (2"+^- 1) = a^ : 2 when n = oo ; and the radius of convergence is t/2. 11, 12, § 5.] INDETEEMINATB SERIES. 375 / §5. INDETEEMINATE SERIES. An infinite series, that has different sums when its terms are arranged or grpuped in different ways is indeterminate. E.g., the sura of the series +1, "1, +1, "1, +1, „. may be either (1-1) + (i_i) + (i_i) + ..., =^0, or l + (-l + i) + (-i + i) + ..., =1. Indeterminate series, although not always divergent, are here classed with non-converggnt series. Theor. 12. An infinite series that has positive and negative terms that separately form divergent series is indeterminate. For take any positive term or group of positive terms for +Ti, leaving positive terms whose sum +Ei is infinite, and from the negative terms, whose sum is infinite, taKe enough terms so that their sum "Tj is larger than +Ti, leaving negative terms whose sum "Rj is infinite ; and from +Ri form +T3 larger than "Tg, leaving +R3 infinite ; and from -Rs form ~'X^ larger than +T3, leaving "e^ infinite ; and so on ; then •.• the new series +Ti, "Tj, +T3, "T4, ... gives ("''Ti+"T2) + (+T3+~T4)H — , = some negative number, and "^Ti+(""T2++T3)+(-T4++T5)4 — , = some positive number, .-. the series is indeterminate. q.e.d. Note 1. This result appears also from this, that the sum of the given series reduces to the difference of the sums of two divergent series and is of the form 00 — 00, an indeterminate expression. Cor. 1. A series s is non-convergent if the series got by making all the terms of s positive be divergent. For, if the series be divergent when all the terms are made positive, it is either of the form 00— qo, a— 00, 00— a, when part of the terms are made negative ; i.e., it is either indeterminate or divergent. q.e.d. Note 2. Manifestly a given series may be reduced to the form +Ti, "T2, +13, ~T4, ••• in an infinite number of ways, giving an infinite number of such double values. 376 SERIES. [XII. ths. Note 3. An indeterminate series may sometimes be arranged so as to Iiave terms alternately positive and negative and growing smaller and smaller ; and if the terms approach 0, the sum for such arrangement has a single finite value, but for different arrangements different values. If, for a particular arrangement, a series have a single finite value, however grouped, the series is convergent for that arrangement. E.g., if 8 = ---+---+---+ -towards*!; ^ 1 2 3 4 5 6 then the two values of s both lie between ,0 and 2 ; and s„ ~ s„+i = 1 when n = oo. So, if s = 1 1 1 1 towards 0, 2 3 4 5 6 ' then s 1 , • . • ' E2 is positive, \, <|, •". But, if this series be arranged thus : + Un-S ' 4w — ly '2w then s1 + |-|, <1 + |-| + 1 + 1,...; i.e., s|, <■••. The reader may group the positive terms by threes, or by fours, or ..., and the negative terms singly, by twos, or by threes, or ..., at his pleasure, taking care that the terms of his new series be always in descending order of magnitude. 12,13, §5.] INDETERMINATE SERIES. 377 Theor. 13. A series s is convergent if the series got by making all the terms of s positive be convergent. For let +s' = the series of positive terms in s, and ~s"= the series of negative terms in s ; then •.■ +s' + +s" is finite, [hyP- -•. """s', +s" are both finite, .'. +s', +s" is the same however its terms are arranged or grouped. [th.7 Let the terms of s be arranged and grouped in any way, and let s„ = sum of the first n groups of that arrangement, and +s^( = sum of the to' positive terms of s contained in s„, and ~s"«= sum of the n" negative terms of s contained in s„ ; then •.• s„ = s4, — +s;'") .•. lim s„ = lim s^/ — lim +s"„ = s'— +s". But ••• +s', +s" are finite constant numbers, [above .•. s, =+s'— "''s", is a finite constant number, q.e.d. Cor. 1 . If a series s be ■{ " ' . the series got by ■' ' non-convergent, " " making all the terms of s positive is ■{ , . ^ . Cor. 2. If an indeterminate series be convergent for a particu- lar arrangement +Ti, "Ts, +T3, ... ^t^, "^1^+1, ..., the ratio t^^i : t^ becomes and remains smaller than unity, but approaches unity as its limit. [th.ll nt. 2 For if the ratio Tj^i : Tj approach a limit h smaller than unity, the series is convergent and not indeterminate, [th. 13, th.ll nt.2 Note. If indeterminate series be classed with divergent series as above, then, in the light of theors. 12, 13, it appears that theors. 7-11, with their notes and corollaries, apply to series with negative terms, and that those theorems are general for all series of real terms. Indeterminate series are unsafe ; and, by reason of their slow^ convergence, they are worthless. 378 SERIES. [XII. ths. § 6. IMAGINARY SERIES. A SERIES whose terms are part or all imaginary is an imagi- nary series. If each term of the series Tj, Tj, ... be resolved into its two components Pi, Qii; Pj, Q2*; •••, the two series s', =Pi+P24 — , ands"i, = V (^ + Q^) ' ' " ' .•. s', s" are convergent, [ths. 10,13 .-. s, =s'+s"i, is convergent. q.e.d. Theoe. 15. If for any imaginary series the series of moduli be divergent, the imaginary series is non-convergent. For •.• +s'++s", the sum of the component series s', s"t, with all their terms made real and positive, is not less than the divergent series of moduli, ["'"p++Q<^(p^+Q^) .-. one or both of the series +s', +s" are divergent, .-. one or both of the series s', s" are non-convergent, [th. 12 cr. 1 .•. s, =s'+s"i, is non-convergent. q.e.d. CoE. If sbe-l °^ ^ '^ ' . so is its series of moduli. > •' < non-convergent, •' Note. Theors. 14, 15, when applied to series of real numbers, become theor. 13 and its converse, since the modulus of a real number is that number taken positive. In the Ught of theors. 14, 15, it appears that theors. 7-11, with their notes and corollaries, apply to series with imaginary terms, and that those theorems are general for all series. Theor. 16 shows that every series to rising powers of a variable has a radius of couvero-ence. 14-16, §6.] IMAGINARY SERIES. 379 Thuor. 16 (Abel's theorem). If a series, Aq+AiZ+AjZ^H — , arranged to rising powers of a variable z,be-i <^°™'"®''9'^'** jo^g„ " ■' ' ' non-convergent mod z = a constant r, it is | "owerg'en.J «,ftenewr mod z^ ^'^■ ' ' non-convergt ' »"• ^divergent whenever mod «{ ^ ^ .-. s, = Ao + A,z -H A,«^ + ..., is ^ convergent ' u 1 1 1 ^ 1 I 1 non-convergent whenever mod » ^ ^ ' q.e.d. [ths. 14,15 CoK. If, in a series arranged to rising powers of z, mod z I convergent \ < , increase from to co, the series is -i in doubt wlien modz-{ =r. j divergent \ > \r a constant, called the radius of convergence of the series. In most series r is lim ratio mod a„ : mod a„+i. [th. 1 1 nt. 5 Note. Graphic representation : Denote by z the represen- tative point of any number z ; i.e., the extremity of that vector from the origin whose ratio to the unit-line is z ; and so for other numbers. Let Ao, Ai«, AjS^, ... be any series arranged to rising powers of z ; and from o as centre, with radius equal to the radius of convergence of the series, draw a circle ; this circle, called the circle of convergence, embraces the region I within I convergent. i. upon which z lies when the series is \ in doubt, j without I divergent. If a series be arranged to rising powers of {z — a), then the circle of convergence has a for centre and r for radius, and the I convergent 1 within series is \ in doubt when z lies I upon this circle ; I divergent j without for mod {z — a)\ = r. 1 >r. 380 SERIES. [XTI. th. Theok. 17. In a series arranged to rising powers of a variable z, if modz be less than the radius of convergence of the series, an increment can be given to z so small that the increment of the series shall be less than any assigned number. For let s = Ao+Ai» + A2Z^H , take inod« less than r, the radius of convergence, and to z give an increment h so small that mod (2 + 7i) < r ; then •.• sands + incs, =Ao+Ai(« + 7i)+A2(z + A)"H , are both convergent series, [tiyP- (z 4- h)^— z^ .•. inc s, = h(Ai + A2 ^ / i ) , IS convergent ; h .'. inc s : A is a convergent series when h is finite ; [th. 8 and ■.• h may approach so that [(« + 7j)"— »"] : h is larger than but approaches n«""\ [bin. th. . ■. inc s : 7i = a finite limit when 7i = ; [th. 10 .-. incs, = ft.a finite number, = when ^=0. q.e.d. CoE. 1. D^s, =Ai + 2a2Z+3asZ^H 5 d/s, d/s, •■-, are all series whose common radius of convergence is r. Cob. 2. For all values of mod z{ 7 .„ than r, the . ■ senes- a finite continuous one-value an infinite or indeterminate function Ao + AjZ + AjZ^ + ••• is { function of z. If s be a series to rising powers of a variable 2, and z be a finite function of z that is equal to s for continuous values of z from to r, but unequal for a value of z larger than r, then s and z are discontinuous when » :=: r [theory of functions] , and r is the radius of convergence of s and the smallest value of z for which z is discontinuous. In the graphic representation of imaginaries, if the points of discontinuity, a,b,c, •■• of the function {a — zY{b—zy{c — zy ••• (Pj Qi i'i •■• any fractions or negative integers) be platted, and the function be equal to a series to rising powers of z — 7e, then, with k as the centre of convergence, the radius of con- vergence of the series is the distance from 71; to the nearest point of discontinuity. 17, §7.] EXPAKSION OP FUNCTIONS. 881 §7. EXPANSION OF FUNCTIONS IN INFINITE SERIES. If z = Ti + T2 + Ts H [z, Ti, Tj, T3 ••• functions of z] for all values of 2 that make the second member a convergent series, the series is an expansion of z in functions of z. -An ordinary function of a variable cannot, in general, be equal to any one infinite series for all values of that variable. E.g., if 2 be a variable that increases from to 00, then the ^ I finite and positive I '^ ^ i fraction is -j infinite when z\ =1 : '-—^ I finite and negative 1 > 1 : but ••• the series is infinite when z > 1, . • . the series 1 +2+2^ H , which equals the fraction for all values of « from to 1 , ceases to equal it When z > 1 . So, the series — z — z^— z' ■ ■ • , wherein z = 1 : », is an expansion of (l-a;)-!, = -z(l-z)-i; and the two are equal when z< 1, i.e, when a; > 1 ; but the series is divergent, and the twO' are unequal when z > 1 . I real ] < 1 ; So, the radical -^(1— z) is ■{ zero when z-{ =1; I imaginary I > 1 ; and it is shown later that an expansion of t/(1— z) is l-iz-iz'-^z'--; [bin.th. but this equality is impossible when z > 1 ; for the series-function remains real for all real values of z, and the radical becomes imaginar}' when z > 1. „ .„ , , negative integer, ,, „ • fraction , „s„ So, If n be any { ^^^^^.^^^ ' the a. So, if p, q, r be any fractions or negative integers, and if 2 = (a — z)''(6— z)'(c — z)'-", then z cannot equal a series to rising powers of z when z is larger than the smallest of the numbers a, b, c, •■•. 38-2 SERIES. [XII. ths. 18, 19 Theoe. 18. The sum of an infinite series Ao+AiX+AjX^H — , whose radius of convergence is greater than 0, approaches the limit Aq when x = 0. For •.■ Ai+A2a; +A8a!^-| — has the same radius of convergence as the given series, .•. it is convergent for small values of a;, .•. the product x{ax + li-^x + a^^ -\ )== 0, when a! = 0, -•• Ao + Aia; + A2a;^+ ••• = Aq when a;= 0. q.e.d. CoE. In the infinite series Aq + Aj x + Aj x^ H 1- a, x' H , X may he made so -j , that AfX' shall he any number of times larger than the sum of all the terms of-{ , ^ degree. Theoe. 19. If two series, arranged to rising powers of any same variable, be equal for all values of the variable tJiat make them both convergent, the coefficients of like powers of the vari- able are equal. Let Ao + Ai^ + AjOj^H = Ao'+Ai'a; +A2'a^H , when x^^r, wherein if the series have different radii of convergence, 'r is the least of the two ; then will Aq = Aq', Ai = Ai', Aj = Aj', • • ■ . For •.■ the two series are equal when a; .< r, [typ. . % they approach equal limits when x = ; i.e., Ao = Ao'. [th.l8 .-. Aia; + A2a^H — = Ai'a!+A2'a!^+"' whena! to the rising pow- ers of some letter in the denominator of the fraction, and with unknown coefficients. Free the equation from fractions. Equate the coefficients of the like powers of the letter of arrangement in the two members, each to each, and solve the equations thus found for the unknown coefficients. [th. 19 ^.g.,put ^ I'^^'l „ sA+Ba; + oa^ + Dai''+-; then •.• l + 2a; = A+ b -3a -3c + 5b a!» + . x-\- c -3b + 6a A=l, B — 3a=2, c — 3b + 5a = 0, •••; ., .-. A = l, b = 2 + 3a = 5, c = 3b — 5a= 10, ••■ ; and the series is 1 +5a;+ 10a;^ + 5a!^ — 35a;^"-, wherein every coefficient after the second equals three timeS the coefficient next before less five times the one before that. ' KEOUEEING SERIES. A series like that in the example above is a recurring series; it is a compound geometric progression, each of whose terms is the sum of the products of the two or more next preceding terms by constant multipliers. The group of multipliers is the scale. E.g., c=3b— 5a, d = 3c— 5b, e=3d— 5c, •", [px. pr,4 and (3, — 5) is the scale for the series of coefficients ; and of the series 1, 5a;, lOx^, ba?, -35a;^, •■•, the scale is l + 2a; 3 a;, ~ha?, and the sum is the fraction l_3a- + 5a;2 384 SBEIES. [Xn. pre. PeOB. 5. To FIND THE SCALE AKD SUM OF A KECURRING SERIES. (a) Scale of two terms, m, n. Write T3 = mT2 + nTi, 14 = 0115+11x2, 13 = 11114 + 1113. Solve the first two equations for m, n, and test the values thus found by the third equation. Write s = ^i(l-'") + ^» . 1 — m — n For •-- S = Ti + T2 + T3 + T4H = Ti + T2 + (mT2 + WTi) + (wiTj + WT2) + ••• = Ti + T2 + m(T2 + T3 + T4+-")+w(Ti+T2 + T3+-") = Ti + T2 + m(s — Ti) + ns, ... g=Ti(lziw)±T2_ ^^^ [sol. for s 1 — m — n E.g., to find the scale and sum of the recurring series Write 10a;^ = m-5a! + w.l and 6a? = m-10oe' + n-5x; solve for m, n ; and test by equation —2>ba^ = m-6a? + n-\0Q?; then m = Zx, n = — ox'; and 3^1-3r. + 5a;^ l + 2a> . l-3x + 6a^ l-Sa+Sa^ (6) Scale of three terms, m, n, p. Write T4 = mT3 + nT2 + pTi, Tj = mT4 + nT3 + pTj, T6=mTs + nT4+pT3, T7 = mT6 + n'E5+pT4. Solve the first three equations for m, n, p, and test by the fourth. Write s^T,(l-m-n) + T,(l-m)+T3. 1 — m— n— p For •-• S =Ti + T2 + T8 + T4H = Ti + T2 + T3 + (mT3 + riTa +pi^) + (mT4 + UTs +PI2) + ••• = Ti + T2 + T3 + m(T3 + T4 + T5 H ) + w(T2 + T3+T4+"-)+i5(Ti + T2 + T3+-") = Ti + T2 + T3 + m(s — Ti — Tj) + w(s — Ti) +pa, . g _ Ti(l — m — w) + T2(l — m) +T3 \—m — n—p Q.E.D. (c) So, for scale of four or more terms. 5, 6, §8.] TTNENOWIV COEFFICIENTS. 385 EXPANSION OF SUKD3. PeOB. 6. To EXPAND A SUED INTO A SERIES : ^ Put the surd equal to a series arranged to the rising powers of some letter in the surd, and with unknown coefficients. Free the equation from radicals. Equate the coefficients of the liTce powers of the letter of arrangement, each to each, and solve the equations thus found for the unknown coefficients. [th.l9 E.g., to expand -y/^a^+bx) : Put -y/(a^ + &a;) = A + Ba; + ca;'' + Da;' + Ea!* + Ffl5°H ; then a2 + 6a; = A^ + 2 ABo; + 2 AC A^=a^ 2ab = &, 2ac + b^ a!^+2AD 05^4- 2ae »*+■ 2 BC 2 BD 0, 2ad + 2bc=0, ■ ^. 2a' ° 8a^ bx b''a? D = - ft^a? and V(«' + &^) = « + ^-.F75+T7:75-- 1 5 16 a' + • So,V5=V(4+l)=2+|-^ + ,^, So, put ^{a^+bx): 16384 A + BX + C^+ DK^+ECC*+ + • then ■•• a'+&a;=A'+3A^B aj+SAB^Ias^+B^' +3a^c| +3a2d + 6 ABC A=«, B = -, C=- bx 9a»' 6V ai^+SA^E +3ac2 +3b^g +6abd a!*+. ^ 3a^ 9 a* So,^9 = ^(8 + l) = 2 + ^- ^ ^ 105* '~ 81a*' "" 243 a^f 56W IQb'^x* : + 81 a« 243 a" 5 + 288 20786 Note. This method of expanding (1+a;)' shows that a series a. + -bx + co?^ exists whc/se gth power is identically (1 +xy ; and so this series, when convergent, is a gth root of (1 +xy. There are q such series corresponding to the q, gth roots of unity. [X. th. ?? 386 SEEiEs. [Xn. pr. KESOL0TION OP FRACTIONS. PrOB. 7. To RESOLVE INTO A SUM OF PARTIAL FRACTIONS A FRACTION WHOSE TERMS AEE ENTIRE EONCTIONS OF ANT ELEMENT : If the degree of the numerator he not lower than that of the denominator, reduce the fraction to a mixed number. Resolve the denominator of the fraction into its prime factors. Equate the fraction to a set of fractions forined as follows: For every prime factor not repeated write a fraction tohose denominator is that prime factor; and for any prime factor repeated k times write k fractions whose denominators are the first, second, third, ••• kth powers of the factor. For the numerator of any fraction iorite an entire function of the given element with unknown coefficients, and of degree lower by unity than the prime factor that enters into its denominator. Free this equation from fractions. Equate the coefficients of the several powers of the letter of arrangement, each to each, and solve the equations thus found for the unknown coefficients of the numerators. E.g., to resolve — i— : [a;'— l = (a;-l)(a^+a!+l) Write -J-^J^+^^+-^; ar — 1 ar + w + l x — 1 then •.■ l = (A+c)a^ + ( — A + B + o)a! — B + c [free fr. frac?] .•. A + C = 0, — A + B + C = 0, — B + C=l .-. A = — ^, B = — I, C = J, , 1 x+2 , 1 and = — ■ • as^-l SCar' + ic + l) 3(a;-l) So, write — -i- = 1 1- - (a; - 1) (a; - 2) (a; - 3) ~ a; - 1 a; - 2 a; - 3 ' then •.• 2a^-10a; + 14 = A(a^— 5a; + 6) + B(a^— 4a!+ 3) + c{a?-3x + 2), -•. A+B+o = 2, 5a+4b+3c=10, Ga+3b+2c=14, -•. A = 3, b=-2, 0= 1, an^ 2a!^-10a; + 14 ^ 3 2_ _|_ 1 («— l)(a; — 2)(a! — 3) x — 1 x — 2x — 3 7, §8.] imBaS-OWN COEFFICIENTS. 387 Since, as appears from Note 1 below, the identity 2a;2-10a; + 14 ■ =A(a!-2) (a;-3) + B(a;-1) (95-3) +c(a;-l) (a;-2) holds true for every value of x, it is more readily solved as follows : Put x=l\ then 2— 10 + 14 = a--1 --2, and a = 3. Put x=2; then b=-2. Put a; = 3; then 0=1. So, write ^^ + ^ + ^ s ^ + ° + '^ ' ; then A =2, b=~3, c = 4, and 2a^ + a!+3 _ 2 3 ^ 4 {x + iy £»+! (ic+l)^ {x + lf This fraction may also be resolved as follows : • r I i\3 -7 — TTTi + T — TT^' [div.bya;+l (aj + l)" (a; + 1)2 (a; + l)^ A 2a!— 1 2 3 rj- 1, , 1 and •.' -■= -, rdiv.bvK+l 2a;^ + a; + 3_ 2 3 as before. (a;+l)= sB + l {x+iy {x+\y o ., 4a;^+3ai^— g^— 4a;— 1 _ a bk+c . Da; + E °' ^" ® (a! + l)(a;2_|_a, + i)2 - ^:j:i+a;2+a,+i "^(3^+0;+ 1/ ' then 4a;*+3a;8-a?— 4a;— 1 = A(a;^+a;+ 1)^+ (bx+c • »H^«+1 +Da;+E) (a;+ 1) . Put a; = — 1; then A = 3. And -.• (Ba! + c)(9;^ + a! + l) +Da! + B [repl.A,div. bya;+ 1 = [4a;* + 3a;^-*2-4a;-l— 3(fl;^+a! + l)^] : (a;+l) = a;' — 4 a;^ — 6 a; — 4, .•. Bai+c=a;— 5, Da; + E = — 2a!+l, [div.bya;^+a;+l , Ax^ + S!x^-a?-ix-l _ 3 x-5 2a!-l ^^ (a! + l)(a;2+a;+l)2 a!+l a;2^a!H-l (a;2+a;+l/ The division without remainder by (x+ 1) is a useful ch^ck. 388 SERIES. [XII. prs, Note 1. When unknown coefficients are got by giving special values to a variable x, the work does not of itself show whether any development of the proposed form be possible, but only shows what the coefficients must be if the development be possible. That every fraction is resolvable into partial fractions as here proposed appears, however, as follows. Let the given fraction be — , wherein tr, r, w are entire vw and prime to one another ; let Xi be any value of x, for which v = ; let Ui, Wi be the constants that u, w become when the variable x is replaced by the par- ticular value Xi ; then _L,^_H^+TIli£zlHl:Z, = ^ + J2L, vw Vl^i ■ v 'Wi • WT v v'w wherem a = — ^, u'=— = : (x—Xi), v'= v: (a;— a;,) ; Wi Wi \ 1/' \ i/ for •.• the entire expressions WjU — tJi"w, v, =0 when x = Xi, .-. each of them is divisible by a; — %. [XI. th. 4. tl' B . U So, w w v'w V V" wherein Kjisanyvalueof ajjfor which v'=0, v'=V2, w=W2;"' ■ ■ vw~l^v'^v'"^v"'^ 'V w = - + -; [Q = A + B-a!— aji + c-o!— iBi-a;— a!2+ — and the given fraction is resolved as proposed. If the denominator v,w have three or more factors, then one of them, say v, can be factored again, and so on. If v be a power «", then - is resolvable by division into 4+^^ + ... + L ^ E.g., above, where v= (a; + 1)'. Note 2. One of the uses of Prob. 7 is in the integration of rational fractions : ^ J (x-l){x-2){x-8) J \x-l x-2 x-Sj = 3 log (x—1) —2 log (x—2) -|-log(a;— 3) + a constant. 7, 8, § 8.] tmSKOWN COEPPICIBNTS. 389 REVEBSION OF SEEIBS. PeOB. 8. If A VARIABLE BE EQUAL TO A SERIES OP POWERS OF ANOTHER VARIABLE, TO FIND THE VALUE OF THE SECOND VARIABLE IN TERMS OF THE FIRST : Put the letter of arrangement of the given series equal to a new series arranged to powers of the required letter of arrange- ment with unknown coefficients, and in the neio series replace the new letter of an-rangement by the given series. Equate the coefficients of the Wee powers of the old letter of ■ arrangement^ each to each, and solve the equations thus found for the unknown coefficients. E.g., to revert the series y = ax + h3? -^ cv? -\ : Put x=.A.y + B?/^+ cy^-\ , and replace y, y^, y^, • ■ (aa; + 6a^+ca^H ), (ax + ba^+Ga?-\ y. then X = A.ax + a6 + Ba^ ai' + AC a^-{-- + 2 Bab + ca^ Aa=l, A6 + Ba^=0, AC+2Ba6 + ca^=0, 2b' -ac ) ■■■) ■1 A = -, a "=-^ and 1 62, a a^ = 2b''- ac a' a" So, to revert the series y = m + ax + ba? + ca? -\ : then •■• y — m = ax + ba? + ca?+---, .: x = l{y-m)-\{y-my+^-^^{y-my+. a a^ a^ So, to revert the series y = ao? -\-boi^ + ca? -\ : XT, ^ 1 b « , 2V — ac , . then a? = -y ^f -\ 5 — r+—- a a^ a" So, to revert the series y = ax + ba^ + caf + ••• : Put x = Ay + Bf + cf + —; then ••• x = Aax + Ab la^ + Ac a^ + ---, + Ba«| +3Ba26 + ca^ .•. Aa=l, A& + Ba'=0, Ac + 3Ba^& + ca'' = 0, •• and x = -y -f-^ — f+—. a a* a\ by 390 SEEEES. [XII. ths, §9. BINOMIAL THEOREM. The DR. 20. Tjf a + b be any binomial, and n any real number, then (a + b)° =a° + na-^b + " (° ~ ^) a^-'b^H- ••• ^ D(n-l)(n-2)-(n-r+l) ^„-,^,^...rv. th.l r ! For, put x = b:a; then (a+6)'' = a»(l + 6Ta)"=a»(l + a!)». (a) n commensurable. Put (l+a;)"sl+Ba!+ca!^+Dar'+".; [B,c,D,;.-unkn.,pr.6nt. n(l+a;)"-i=B+2ca;+3Da;2+.--, [VII.th.l7,cr.l; a^_) . [mult.byl + o! then and n{l + x)'' =B + 2c + B k + Sd + 2c But and and n(l + »)" = n + wBO! + nca^ + nna^ + • [above B + 2c + B x + So + 2c x'^-j = n + wna; + nca^ + WDa!'+ ■ n(n — l) B = n, C = — ^^ r-i-, D : 2! B = n, 2c + B = WB, 3D+2c = nc, •••, n(-w— l)(w— 2) 3! ' "■■ (l+a;)" 2! 3! (a + 6)"= a" + na"-' & + r^iltzll an-252 ^ . . . _ q_j,_i,_ (6) n incommensurable, a case of limits. Note. Although the form of the series does not depend on the ratio b : a, yet the series is worthless unless convergent. E.g., V5 = V(*+l) = 4* + i-4-5 -1-4-^ +31^.4-5-..., and the convergents are 2, 2\, 2|-|, 2^^, ••• ; but V5=V(l+4)=l+i-4-^4? + 3V-i'-> -. and the convergents are 1, 3, 1, 5, •••, which are useless. So, V3 = V(4 - 1) = 2, If, 111, im, - ; but y'— 3=-y/(l— 4)= 1, -1, -3, -7, •", which is absurd. 20,21, §9.] BINOMIAL THEOEEM. 391 Theor. 21. The series 1 + n + "('^"-^^ -[ 2! D(D-l)...(n-r + l) D(n-l)-(n-r+l)(n-r) r! "^ (r + 1)! "' is convergent if n be positive. 1. r may be made so large that T^+g • (r + 1)''+^< t^+i • r"+^. For [T,,, . (r + 1)"«] : (T.^, . ,-+1 ) = (^)"''' . (^) =-(-9"-('-^) and - may be taken so small that A<1. q.e.d. [th. 18, or. 2. The series is convergent. For ••• after r becomes larger than some fixed finite value r', each product t^i • r"+^ is smaller than the product before it, / [1 and ••• T,,+i-r'"+^ is some finite number, say Jc, .-. the series is convergent. q.e.d. [th. 11, nt. 3 Cor. The expansion of (a + a)" is-{ " .ifnbe ■^ •> \ ' ■> 1 non-convergent •' , po^ti^e; ^^^ ^ ^j. ( j^. .^ , convergent if^>>^^ ' negative; •' ^ ' ' non-convergent •' ' < Note. The expansion of (a + o)" is indeterminate if n lie between and~l; for then the successive terms of the series do not grow larger, and are alternately positive and negative. But if the negative terms be made positive, the series is the expansion of (a — ct)", whose value, a negative power of 0, is infinite. The expansion of {a-\-a)'^ is divergent if m < — 1 ; for then the successive terms of the series grow larger and larger. 392 SEKIES. [XII. pr. 9, PeOB. 9. To EXPAKD A POWER OF A BINOMIAL: Reduce the given expression to the type-form (a + b)" and apply the binomial formula. [th. 20 2 <-yy+{-yy, 3! = oii^ — 4:a^y + 6a?y^—4:xy^-{'y*. So, (2a-36)-»= (2a)-*+-4.(2a)-«-(-36) (2a)-«.(-36)2 -4 .-5 .„_..e 2! + -+ "^'"^'~^'"~^^+^) -2a-"-+^'-(-36)'-+ ..., r! ^ 1 4-(36) 4-5-(36)' 4-5-6-(3&)° (,2ay {2ay 2!.(2a)= 3!.(2a)' , 4-5-6--(r+3)(36)'- . "^■■■■^ j-!.(2d)'-+* ■^"■' So, (x + y)^ = xi+hx-iy + L~l.l.a!-iy^ .1 -1 -3 1 _§ . , 2 2 2 3! ^ ^ ^1 -1 -3 ^5 3-2r 1 -^> , 2 2 2 2 2 J-! ^ ^ = x^ + -x~^y - -x'h" + — a!"M- —x'^v* 2 " 8 ^16 ^ 128 * , , l-3.5-7...(2r-3) -5:zl ^ + '"± ^ ^X 3 W'T.... 2'-r\ Note. If »i be a positive integer, the series ends with the (7i + l)th term, since the coeflacients of the following terms become ; but if n be a negative integer, or a fraction, positive or negative, the series does not end, and is infinite. th.22, §10.] FINITE DIFFEEENCES. 393 § 10. FINITE DIFFERENCES. If there be any series of numbers, and if a second series be formed by subtracting each term of the first series from that which follows it, in order ; a third series, by subtracting each term of the second series from that which follows it, and so on ; then the terms of the second series are called the differences of the first order, or first differences; the terms of the "third series are the differences of the second order, or second differences; and so on. i:.g.,ii 1, 4, 9, 16, 25, 36, then ' 3, 5, 7, 9, 11, 13, 0, 0, 0, 0, 0, 0, So, if a, b, c, d, e, then 62, be a series, are the first differences, are the second differences, are the third differences. be any series, are the first differences, are the second differences, are the third differences. &i= c —b, &2= Ci— &l. Ci= d —c, 02= «! — Ci, as, and so on : wherein ai= 6 — a, and so on. The series a, ai, aj, 0,31 ••• is the auxiliary series; and the object of the theorems that follow is to show how to find one term and the sum of any number of terms of the principal series, by aid of the auxiliary series. Theok. 22. If a, b, c, d, e, ••• he any series, and a, aj, as, 83, ••• its auxiliary series, and if t^ be the (n + l)i/i term of the principal series; then t^ = a + nai + °^°~ ^ "^ ^ ^^-1+ ^n- 1.' The law is true when n = 1 , 2, and 3. For •.• ai=b —a, .•. b =a + ai. [df. Q.B.D. [w = l 394 SERIES. [XII. ths. So, Z*i=ai-ra2, c — b-\-bi, d = c + Ci, .-. c,— 6 + 61, = a+2ai + a2. q.e.d. [n=2 So, Ci = Oj + 2 «2 + as, .-. d,= c + Ci, = a + 3ai + 3a2 + as. q.e.d. [n=3 2. if the law he true when n = k, it is true when n = k + 1 . For, let q, r be the (A;+l)tli and (A; + 2)th terms of the principal series, then •.• 5 = a + CiA;-ai + C2ifc-a2+ \-cJi-ar-\ +0^, [hyp. and gi=ai + Cifc-a2H + c,_ik-ar-\ 1-Cifc.aj + aj+i, .-. r,= (? + gi, = a + Ci(fc+l).% + C2(fc+l)-a2H f-c,(A;+l)-a, -\ hCi(7(; + l).aj + aj+i. q.e.d. [IV. th.3,cr.2 3. Tlie lam is true universally. For -.• it is true when m = 3, [1 .'. it is true when ?i = 3 + 1 = 4, [2 .'. it is true when w = 4 + 1 = 5, .•. it is true when m = 6, 7, 8, •.-. q.e.d. Note 1 . The reader may compare this proof with the third proof of the binomial theorem. [V. th. 1, nt. 2 Note 2. This theorem is of special value when the auxiliary ' series is short, ending in zeros. E.g., of the series 1, 8, 27, 64, 125, •••, the auxiliary series is 1, 7, 12, 6, 0, 0, •••; and Tio=l+9-7 + — •12 + ?^^^-6 = 1000. 2 6 So, of the series 7, 16, 27, 40, 55, •••, the auxiliary series is 7, 9, 2, 0, 0, ••• ; and Tm=7 + 9-9 + ^-2 = 160, T„ =7 + (M-l)9+^(n-l) (?2-2)2 = n(n + 6). 22-24, §10.] FINITE DIFPEEENCBS. 395 Theok. 23. If the terms of a series he like, entire functions of their number in the series, the auxiliary senes end^uaith that term whose number is one greater than the degree of the function. Let the general term of the series be T„, = A + Bn+cn^4 +KM"': [m apos. integer then the general term t„' of the series of first differences %j Oi, Ci, •••, IS T„+i — T„, = B + (n + f — n^) H f- K (n + V- w") = B+c(2n + l)H |-K(m-w"'-i+..-), which contains no higher power of n than n™"'. So, . the general term t„" of the series of second differences flfj) 6ai C2, •••, is T^i' — T„', and contains no higher power of n than m*""^; •••. So, the general term of the series a„_i, 6„_i, c„_i, •••, contains only the first power of m. So, the general term of the, series a„, &„, c«, ••■, is free from m, i.e., is constant, and all the subsequent series, a„+i, 6,^i, ■••, •••, consist of zeros. q.e.d. Theor. 24. If the terms of a series be like entire functions of their number in the series, the form of these functions is identical with that found by aid of the auxiliary series. For •.• A + Bn + cm^H-.- +£«"* , . ,. , (n—l)(n — 2) = a + (w-l)ai + i ^i ^CS2+." (n-l)(n-2)...(n-m) m ! for all integral values of n, [ths. 22, 23 -•. these two functions, each an entire function of n of the mth degree, are equal for more than m values of the variable n, .•. they are identical. q.e.d. [XI. th. 4, er. 8 396 SERIES [XII. th. 25. Theor. 25. If a, h, •■■ \ be u terms of any series, and a, ai, aa, aj ••■ its auxiliary series, then r u I 1 1 1 1 n(n— 1) , n(n— 1) (n— 2) , a + b + cH 1- 1 = na + % '' ai + -^ ff ^a2 + "-. For, from the given series form a new series, 0, a, a + b, a + b + c, •••, a + b + c-\ \-l, wherein the (n + 1 ) th term is the sum of the first n terms of the given series ; then ••• 'the first differences of the new series are the terms of the given series, the second differences of the new series are the first differences of the given series ; and so on ; .•. the auxiliary series of the new series is 0, a, Oj, a2, a^,-", and its («+l)th term is A I O I Q.E.D. [th.22 § 11. INTERPOLATION. If for a series of values of a variable (the arguments) there be a corresponding series of values of some function of that variable, the insertion of intermediate values of the function cor- responding to intermediate values of the variable is interpolation. PrOB. 10. To INTERPOLATE VALUES BETWEEN THE TERMS OF A GIVEN SERIES. (a) The form of the function Tcnoum: Apply the law offormu- tion, as shown by the form of the function of n. E.g., of the series 1, 4, 9, 16, •■•, the (2^)th term is (2^)^=6i. So, of the series 1, 4, 7, 10, 13, •■•, the (3|-)th term is 7+i.3, = 8. So, of the series 1, 4, 16, 64, •■■, the (3|)th term is 1 ■ 4', = 32, pr. 10, § 11.] INTERPOLATION. 397 (&) The arguments equidiffefent, and the form of the function unknown : From the given series form the auxiliary series, and find the nth term of the given series hy aid of the formula of theor. 22. Assume the law of formation to fee that shown in the form of the nth term, and get intermediate terms by the application of this law, as in case (a) . E.g., of the series 1, 3, 6, 10, 15, 21, •••, the auxiliary series is 1, 2, 1, ; the nth term is l+2(n-l) + i(n-l),(«-2),=in(n + l); and the (2|-)th term is ^ • 2^ • 3|-, = 4f . So, of the series 1, 1.414, 1.732, 2, 2.236, 2.450, the values of the square roots of 1, 2, 3, 4, 5, 6, correct to three decimal places, the auxiliary series is 1, .414, -.096, .046, -.028, .02,'".; and the approximate value of V^i i^ 1 + f .414 - f .096 - xV -046 - y|^ .028 = 1.581. Note 1. This rule assumes that the law of formation of the series is that found by aid of the auxiliary series and the formula of theor. 22. The right to make this assumption appears as follows : if the auxiliary series terminates, the for- mula gives a law of formation by which the integral terms may be found, viz., that the function be a rational integral function of the argument; and, since the function so found is a con- tinuous function, by its aid intermediate terms may be got. Whether the original series was got by this law does not appear ; but as this is the simplest law made known by the data, and as this law does give the integral terms, it is as- sumed as the law of formation of intermediate terms. If the given series consist of two terms o, b, then the auxiliary series is a, ai, and the formula of interpolation for T„+i is a -|- ««!, the ordinary formula of proportional parts in common use with arithmetical tables. E.g., log 600 =2.6990, log 501 = 2.6998, and log 500.6 = 2.6990 -|- .6 X .0008 = 2.6995. 398 SERIES. [XII. pr. If the given series consist of 'three terms, a, 6, c, then the auxiliary series is a, %, 02, and the formula of interpolation foi' Tn+i is , , n(n — 1) a + nai-\ — ^ — -' a^. E.g., log 150 = 2.1761, log 160= 2.2041, logl70 = 2.2304 ; then the auxiliary series is 2.1761, .0280, — .0Q17, y and log 163 = 2.1761 + 1.3 x .0280 - hl2LA x .0017, = 2.2122. So, if the given series have but four terms, five terms, and so on. If the given series be infinite, the formula of interpolation is also infinite, and it is available when convergent, i.e., when no term of the auxiliary series a, a^, a^, ••• exceeds a given finite limit ; for since the series Of coefficients 1 + n + " v ~ -) is convergent when n is positive [th. 21], so is the formula of interpolation a + na^ + \n{n — l)a2 + ••• convergent, [th. 8, cr. When available, this formula is better adapted to computation than is the more general formula of case (c) . (c) The arguments not equidifferent : Let Xi, X2, Xj, ••• x,^i he any arguments not equidistant, and yu Jzi ys) ••• ym '^e corresponding values of the function, to in- terpolate a value of the function y, corresponding to a given, argument x; compute y by the formula (a^-a;2)(a;i-a!3)...(a;i-a!„+i) * ^ (a; - aQ {x-Xs)-..{x- x^-,) {x^-x^) (aJa-ajg) ■.. {x^-x^{) * _(_ (x-xy){x-x;)-(x-x^) (a'm+i-aa)(a'»H-i-a'2)-(a'«+i-a'») °^'' 10, § 11.] INTBEPOLATION. 399 For, assume y to be an entire function of x of the with degree, and write y = A. + '&x + ca?-\ t-Mx", then is this function identical with that written above. For ■.• they are equal when a!=a;i, when x^x^, when x=x,^i; i.e., for m + 1 values of x, .•. they are identical. < [XI. th.4,cr. E.g., if Xi, ajj, 0:3= 150, 160, 180, and 2/1,2/2,2/3 = 2.1761, 2.2041, 2.2553, to interpolate a value of y corresponding to »= 163 : then y= ^~^'^ 2.1761 + ^^'"-^ '^ 2.2041 + ^^^2.2553 -10.-30 10.-20 30-20 = 2.2122. Note 2. When x^, X2, •••, have a constant difference 1, the forjliula of case (c) is equivalent to that of ease (b) . For •-• each of these formulae makes y an entire function of X of the mth degree, and ■.• both formulae give the same value to y for more than m values of x, .•. the two functions are identical. [XI. th.4, cr. Note 8. The principle of interpolation is illustrated graphi- cally in the platting of curves by means of points. The abscissas of the points represent arguments ; the corresponding ordinates represent the known values of the function ; and any intermediate ordinate represents an intermediate value of the function. Graphically the interpolation is effected by joining the given points by the simplest smooth curve that can be drawn through them, and measuring the ordinate that corre- spdnds to any given argument. The most reliable part of this curve is commonly that which is not too near either end. 400 SERIES. [XII. the. § 12. Taylor's theoeem. Lemma. If t(x. + y) be any finite continuous function of the sum, X + y for all values of that sum between a and b, then for all such values r*,, f (x + y) = Dj,f (x + y ) . For if X be increased by h while y stands fast, then vj{x + y) = lim- ^^'" + y + /O -f(x + y) ^ [df . deriv. and if y be increased by h while x stands fast, then j>Jix + y) = lim^ ^ + ^ + ^^ "A"' + ^^ • h .-. i>J{x + y)= dJ{x + y) . [II. ax. 1 Theor. 26. If t(x + y) be a continuous function of the sum (x + y) that does not become infinite when y = 0, its expansion in powers ofy can contain no negative powers ofj. For if possible let the expansion contain a term ciy~", wherein c is independent of y ; then ••• cy~'"=co when y = 0, ••• /(a; + 2/) = w, which is contrary to the hypothesis, . • . this expansion can contain no negative powers of y. Theor. 27. i/"f (x +y) and its successive derivatives be finite and continuous functions of the sum (x + y) , the expansion of f(x + y) can contain no fractional power ofy. For if possible let the expansion contain a term cy"+f, wherein c is free from y, n is a. positive integer, and - is a proper fraction ; " then the (w + l)th derivative of this term as to 2/ is c y^~\ wherein c' is free from y, and - — 1 is negative, and ••• c'yi~^ = 00 when y = 0, .-./(»+« (a; +2/) = «>, which is contrary to the hypothesis ; . • . this expansion can contain no fractional powers of y. 26-28, § 12.] TAYLOE's THBOEEM. 401 Note. It is shown in the theory of functions that if a func- tion of y and its ^/-derivatives be finite, continuous, and one- valued for all values of y smaller than a constant r, the function may be expanded to a series of rising integral powers of y that is convergent when y is smaller than r. This is equivalent to saying that f{x + y) may be expanded to rising integral powers of y when f{x + y), f'{x + y), •••'are finite, continuous, and one-valued functions ot x + y from y=0 to y = r. Theor. 28. (Taylor's Theokem.) i/" f (x -f- y) 6e continuous, and if it be possible to expand this function in a series to positive integral powers of y, then f(x + y) = fx + 2f'x + |^f"x + |!f"'x-|---|-^f<°'x+-, wherein fx, f'x, f "x, f '"x ... f'^'x ... are what f(x + y) and its successive derivatives become when y = 0. 'For, T[)atf(x + y) = A + By + cy^ + i>y^-\ [■Ky''+—, wherein a, b, c, d, ... k, ... are finite and continuous functions of X, but free from y," and whose first derivatives as to X are all finite ; V^JV- then A =fx ; and •.• T,J(x + y) = i>^A + iD^B.y + i,^c.y'-' + — +i>^Ky^+—, and j>J(x + y) = + B + 2cy+3Dy^ + -+nKy^-'+ — , and •.• J)J(x + y) = -Dj{x + y), [leva. . . v^A + T,,B-y + i>^c-y^+ — =B + 2c-y + 3Dy^,+ — for all finite values of y. [th.l8, cr.l .-. B = D,A=/'a5, 2c = i>^B=f"x, .-. c=^if"x; S-D=-D^c=if"'x, .-. D = — /'"a;...; and so on. o ! ■■.f(.x + y)=fx + ^f Theoe. 29. J/n 6e any, number greater than 1, a any positive base, and m^ tJie modulus of the system; then log^N = log^(N-l) For take y = , whence '^V — then 2n-1 1-2/ N-1 log.(l+2/) = M,(2/-^ + ^-^ + -)- and log^(i_2/) = M^(-2/-|-^-^-). Cth.28ap.2 ••• log.i±^=2M,(2/+^'+^ + f +•••)• [IX.th.6 1 — y 6 o i and •.• log^N - log^(N - 1) = logA^j-^ = ^^^^Y^y ' .-. l0g^N = l0g,(N-l) + 2M,(2/ + ^ + ^ + ^ + -) = log.(.-l) + 2M.(^+^^^3 + -). This series is convergent if n >1. [th. 11 nt.4 CoK. i. jy N — 1 &e any positive fraction, however small, then log^{^-l) = log.^ - ^^^{j^-^ + 3(2n^-1)^ + "■)■ CoE. 2. If A. = e, the Napierian base, then m^ = I, and ^o^eN = %e(N-l) + 2(^^^ + 3^^^3 + -} 404 SERIES. [XII. prs. 10, 11. PeOB. 11. To COMPUTE A TABLE OP NAPIERIAN LOGARITHMS : Beginning with 2, compute the logarithm of every prime num- ber in order. [th. 29 cr. 2 For the logarithms of composite numbers, add together the log- arithms of their factors. E.g., log.2= log. 1 +2 A + — , + — +— , - '' ^ ° l3 3-3^ 0-3^ 7-3' [IX. th. 6 logl =0 .66666667: 1 = .66666667 7407407 :'3 .= 24(!!)136 823045:5 = 164609 91449 : 7 = 13064 10161 :9 = 1129 1129:11= 103 125:13= 10 14: 15= 1 .693147 So, log,3 = log,2 + 2 A + -A-^ + -i^+ . \5 3-0'' 5-0'' So, log.4 = 2 . log.2 = 1 .386294. So,log,5 = log.4 + 2g + ^ + ^4 = 1.098612. = 1.609438. So, logJO = log.2 + log. 5 = 2.302585. PeOB. 12. To COMPUTE A TABLE OF COMMON LOGAEITHMS ! For prime numbers, multiply the Napierian logarithms, found as above, by .43429448. / For composite numbers, add the logarithms of their factors, j For, logio N = log.N : log, 10, [IX. th. 8 / =Iog.N: 2.302585 = log,NX .43429448, Wherein Mio, = .43429448, is the reciprocal of 2.302585. E.g., Iogio2 = .693147 X .43429448 = .301030. [pr. 9 So, Iog,„3 = 1.098612 x .43429448 = .477121. Note. The work is further shortened bj- interpolation. § 14.] EXAMPLES. 405 ' • § 14. EXAMPLES. §1- • ••4. Find the last term and the sum of 5 terms, 20 terms, 35 terms, 50 terms, 2n terms, 2n + l terms, of the series: 1. The natural numbers ; the odd numbers ; the even numbers. 2. The numbers of the form r+ Jcx wherein r, k are constant integers and x a variable integer. ''3. The distances passed over in successive seconds by a falling body, starting from rest (16.1, 48.3, 80.5, ••• feet, or '4.9, 14.7, 24.5, •■■ meters). 4. 1, -2, +3, -4, ... ; 1, -3, +5, "7, +9, .•• ; 3, 2|, 2^, .... 5. One hundred stones are placed in a line on the ground a meter apart, and a basket is placed a meter from the first stone ; how many kilometers must a man run, who, start- ing from the basket, picks up all the stones, one by one, and returns to the basket each time he picks up a stone ? ... 8. Find the five elements of the arithmetic progressions : fQ. 1, 3, 5, ... 99; 1, 3, 5, ... 2A;-1; 4 + 5 + 6 + •.■ = 5350. 7. 5...7means...75 ; 3... 11 means 11 ; 2|-...3 means ■•• 20. ' 8. ... 5 terms .•■ 19 ... 7 means ... 67 ; 1, x, ■■■ 4tx, 19 ; 1^ 1-50 = 204; 9. Fill out the arithmetic progressions : 0^ h3..--|-4=10, =18, =2(4fc+l). [;% any integer 10. Find the distances passed over by a body falling from rest in successive quarter seconds ; and in successive periods of 5 seconds. [ex. 3 11. A stone thrown into the air took 5 seconds to rise and fall to the same level ; how high was it thrown ? [ex. 3 12. Find the condition that a, 6, c may be the pth, qth, rih terms of an arithmetic progression ; if this condition be satisfied, and if a, b, c be positive integers, show that p, q, r may be the ath, 6th, cth terms of an arithmetic progression, and that the product of the common differ- ences of the two progressions is unity. 406 SERIES. [xn. 13. Divide unity into 4 parts in arithmetic progression, such that the sum of their cubes shall be -j-V- 14. The interior angles of a rectilinear figure are in arithmetic progression ; the least angle is 120° and the common difference 5° ; find the number of sides. 15. A three-digit number is 26 times the sum of' its digits; the digits are in arithmetic progression ; if 396 be added to the number, the digits are reversed : find the number. 16. At 4 P.M., A, riding 4 miles an hour, is 11 miles ahead of b ; B increases his speed regularly ^ ot a mile every hour, and has ridden since starting at 11 p.m. the day before, 72i miles ; when did a pass b, and when will b pass a? §2. ... 19. Find the last term, and the sum, of 10 terms, n terms, 00 terms, of the series : 17. The integral powers of ±2; ±3; ±k; ±\; ±\; ±]-- [A;>1 A o fC ''■ ^-2+4-' ^ + 2l^+i + -' 1+0+^+-. 19. l+() + ()-|+...; .672672...; | + | + f + | + -. 20. A man invests $100 half-yearly in stocks that pay 3 per cent half-yearly dividends, and invests the dividends as they are received ; how much will he have invested at the end of 10, 20, 30 years? 21. A man at 20 insures his life for $2000, paying therefor a premium of $20 half-yearly ; what is the gain or loss to the insurance company if he die at 30, 40, 50, 60, 70, estimating that it costs the companj' 10 per cent of its premiums to collect and care for them, and that money is worth 5 per cent per annum ? 22. Show, that ^.Ui--- - .666"- ; ■^2.370370--. = 1.333—. 23. Find four geometric means between 1 and 32 ; two between .1 and 100 ; three between ^ and 9 ; three between 2 and \. § 14.] EXAMPLES. 407 24. The sum of three numbers in geometric progression is 13, and the product of the mean and the sum of the extremes is 30 ; what are the numbers ? 25. Show that, if n geometric means lie between a and c, their n product is {acy. 26. If the common ratio of a geometric progression be less than ^, prove that every term is greater than the sum of all the terms that follow it. 27. What is the condition that a, 6, c may be the pth, 9th, rth terms of a geometric progression ? If this condition be satisfied, and log^a, log^d, logj^c be positive whole num- bers a', b\ c', show that a^, a*^, a'' are the a'th, 6'th, c'th terms of a geometric progression. 28. If there be an infinite number of infinite decreasing geo- inetric progressions, wherein the ratio is common, and the first term of each is the wth term of that just before it, show that their sum is a : (1 — r) (1 — r""^) . 29. There are two infinite decreasing geometric progressions, each beginning with 1, whose sums are s, s' : prove that the sum of the series formed by multiplying their corre- sponding terms is ss' : (s -(- s'— 1). §3. 30. Continue in both directions the harmonic progressions : 2, 3, 6 ; 3, 4, 6 ; 1, 1^, If ; to five terms, to n terms. 31. The difference of two numbers is 8 and their harmonic mean is If ; what are the numbers? y r / V :: ^. 32. What is the condition that a, b, c be the pth, gth, rth terms of a harmonic progression ? 33. If a, b, c, ... be in geometric progression, and a? = b'' = c'' = ..., then^, q, r. ... are in harmonic progression. 34. Prove that the arithmetic, geometric, and harmonic means of two numbers greater than unity are in descending order of magnitude. ■^ r \ 408 SERIES. [XII. §4. "" • ••53. Determine which of the series are convergent : . ' q^ 2.3,4/^ . 1,3,3^*^ . 1,2,3,'*^ ^^- l + 2 + 3 + "V' +2 + ¥+^:"' 100+ 100 + Too "^^ ■ 36. 1 + ^— -+^ _ + ___._+ . c.^^^ 07 1 , 2 1 , 3 _1_ 4'J_ . a + /t 1 CT + 2/t 1 . . J_ 1_ J_ 1 \ ^ ^^M -^ ^' ^^" 1.2 + 2. 3+3 •4''"(r; V(l-2)^V(2-3)^V(3-4) + "j^- ^^- r:3+2:4+3~:3+ ' ^+2+2r3 + 2:3T4+ • 1 1 1 40. , ,, +- TTT ;rTT + - u a(a + 6) (a + 6)(a + 26) (a + 26)(a + 36) C -%^ 41 5 6 7 . 1.23 1.2^3"'"2^3^4 3^4^5 ' 2^3 3.4'^4.5 "^ / 42. 1+^ + ^ + -; i+^+^ + iT + "- 43 i + i + ill , 1:3^ + .... _2^ 2^^ 2^4^^6'' ^^- ^+2 + 2^4 + 2^4^6^ '4!+ 6!+ 8! ^/ 44. Find Ss, and its bounds of error, in each of the above series. 45. Write the above series to powers of x, so that a;" shall have for coefficient the nth term of the series, and determine the radius of convergence in each case. §5. 46. 1-2+3-4 + -; 1-1+1-1+-. 47. a — & + c + a + & — c — a-f6 + c + a — 6-l-c-l — . ..Q o 3_l4 5, . 1 1,1 1 , . ^^- 2-1 + 3-4 + -' ^-2 + 5-4+ ' AQ 1 1^1 1. .2 + 1 2^+1 ,2'+l ... 50. 3 5 7 2 2^ 2^ g-l 2(2a-l) 3(3a-l) j2 2^ 3^ SI J L + J • J L + J__, 1-2 2.3 3.4 ' 1-3 2-4^3.5 § 14.] EXAMPLES. 409- 52 _2 §_j.J 5_ . ,_1 1-3 1-3-5 3-6, 5.7 7-9 9-11 ' 2 2-4 2-4.6 53. 1-1- ' l'-3 l^-3^-5 . 2^.42 2^.4^6^ §6. 54. Write the series in §§ 4, 5 to rising powers of a;, «— 1, a; + 1, £0—2, x+2, x—l+i, a;— 2 + 1, so that x", (a;—!)", ..-, shall have for coefflcient the nth term of the series, and construct the circles of convergence of the resulting series. - » 55. Determine in advance from the character of the functions in Exs. 66-8, 66, 67, 69-71, what will be the radius of convergence of their expansions to rising powers of x. 56 57 58, 58. Expand into series to rising powers of x, the fractions : 1 3 a; - 2 ^ . 5-10a;\^ 1 ^» 3-2a;' (a;- l)(a!-2)(a;-3) ' 2 -aj-Sa!^' 1 -aj + a;^' VJ X ?) ' ^{a^-a?) 68. Find the values correct to four decimal places of : V3 ; V5 ; ^9 ; ^31 ; ^17 ; ^80 ; ^33 ; V^, a/^i VIO- 103. Given the squares of 1, 3, 5, 7, 9, 11; interpolate the squares of 2, 4, 6, 8, 10. 104. Given the amount of one dollar at compound interest : f or 1 year, 1.06; for 2 years, 1.1236; for 3 years, 1.19102; for 4 years, 1.26248; for 5 ^^ears, 1.33823; interpolate the amounts for ^, 1^, 2^, 3|-, 4^ years. §12. 105. Prove the equation : log.a; = |[loge(a;+l) + log,(a;-l)] + (2a^-l)-i + i(2a^-l)-' + i(2a!^-l)-« + .... 106. Assuming the expansion of log,(l +x) and of e", show that (1 +»i~^a!)"=e*(l — ^n~'a!^), when n = oo. 107. Find the coefficient of a;" in the expansion of -21 — ~ . 108. Expand to rising powers of x ; also, to falling powers : log (a + hx + ex') , log \^{af +px + q) : {x^ +p'x + q')2- 109. Expand to five terms by Maclaurin's theorem : i(e' + e-»), iie'-e-^), ^(e^ + e-"), ^ {e<- - e-^) i-\ a;(e«_l)-i, {a + bx + cx^y. §13. 110. Compute a table of Napierian and of common logarithms, each correct to four places, of the numbers from 1 to 20. •'!' s$. I^^i fjl' "./''^